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1310.0374
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aainstitutetext: Key Laboratory of Micro-nano Measurement-Manipulation and
Physics (Ministry of Education) and School of Physics, Beihang University,
Beijing 100191, Chinabbinstitutetext: Department of Physics, Liaoning
University, Shenyang 110036 , Chinaccinstitutetext: International Research
Center for Nuclei and Particles in the Cosmos, Beihang University, Beijing
100191, China
# Search for $C=+$ charmonium and $XYZ$ states in $e^{+}e^{-}\to\gamma+~{}H$
at BESIII
Yi-Jie Li a , Guang-Zhi Xu b , Kui-Yong Liu a,c , Yu-Jie Zhang
[email protected] [email protected] [email protected] [email protected]
###### Abstract
Within the framework of nonrelativistic quantum chromodynamics, we study the
production of $C=+$ charmonium states $H$ in $e^{+}e^{-}\to\gamma~{}+~{}H$ at
BESIII with $H=\eta_{c}(nS)$ (n=1, 2, 3, and 4), $\chi_{cJ}(nP)$ (n=1, 2, and
3), and ${}^{1}D_{2}(nD)$ (n=1 and 2). The radiative and relativistic
corrections are calculated to next-to-leading order for $S$ and $P$ wave
states. We then argue that the search for $C=+$ $XYZ$ states such as
$X(3872)$, $X(3940)$, $X(4160)$, and $X(4350)$ in
$e^{+}e^{-}\to\gamma~{}+~{}H$ at BESIII may help clarify the nature of these
states. BESIII can search $XYZ$ states through two body process
$e^{+}e^{-}\to\gamma H$, where $H$ decay to $J/\psi\pi^{+}\pi^{-}$,
$J/\psi\phi$, or $D\bar{D}$. This result may be useful in identifying the
nature of $C=+$ $XYZ$ states. For completeness, the production of $C=+$
charmonium in $e^{+}e^{-}\to\gamma+~{}H$ at B factories is also discussed.
## 1 Introduction
During the last 10 years, many heavy quarkonium or heavy quarkonium-like $XYZ$
states had been discovered (more details can be found in Ref.Brambilla:2010cs
and related papers). The $X(3872)$ state is the first and the most famous
state among them. It was discovered by the Belle collaborationChoi:2003ue ,
and confirmed by the CDF Acosta:2003zx , D0Abazov:2004kp , BaBarAubert:2004ns
, LHCbAaij:2011sn , and CMSChatrchyan:2013cld collaborations. One of the most
conspicuous properties of $X(3872)$ is its mass, which is close to the
$D^{0}\bar{D}^{\star 0}$ threshold within $1$ MeV; hence, $X(3872)$ is
suggested to be a $D^{0}\bar{D}^{\star 0}$ molecule Braaten:2003he ;
Close:2003sg ; Wong:2003xk ; Voloshin:2003nt . The contribution of the charged
component $D^{+}D^{\star-}$ is also considered in Ref.Gamermann:2009uq ;
Aceti:2012cb . The molecule model may be puzzled to explain the production
cross-sections of $X(3872)$ in hadron colliders ( which may be large in some
phenomenological modelsArtoisenet:2010uu ) Suzuki:2005ha . The quantum
numbers of $X(3872)$ have been determined to be $J^{PC}=1^{++}$ by LHCb
collaboration Aaij:2013zoa . The $J^{PC}$ of $X(3872)$ is the same as
$\chi_{c1}(nP)$. On the contrary, the mass $3.872$ GeV seems too low for a
$\chi_{c1}(2P)$ state. The coupled-channel and screening effects may draw its
mass down to $3.87$ GeV Li:2009zu . However, next-to-leading order (NLO)
prediction of $X(3872)$ production in hadron colliders within nonrelativistic
quantum chromodynamics (NRQCD) disfavors the interpretation of $X(3872)$ as
pure $\chi_{c1}(2P)$ Butenschoen:2013pxa . The possibility that $X(3872)$
might be a mixture state with the $\chi_{c1}(2P)$ and the $D^{0}\bar{D}^{\star
0}$ components was proposed in Ref.Meng:2005er . The prompt $X(3872)$
hadroproduction is studied at NLO in $\alpha_{s}$Meng:2013gga and the result
is consistent with the CMS Chatrchyan:2013cld and the CDF dataAcosta:2003zx .
This idea is also favored the data of some other measurements and predictions
Suzuki:2005ha ; Meng:2007cx ; Li:2009ad ; Li:2009zu .
Besides $X(3872)$, other $C=+$ $XYZ$ states are listed in Table 1. These
states are particularly interesting and the interpretations for their nature
are still inconclusiveMolina:2009ct . $X(3915)$ ($X(3945)$ or $Y(3940)$) and
$Z(3930)$ are assigned as the $\chi_{c0}(2P)$ and $\chi_{c2}(2P)$ states by
the Particle Data GroupBeringer:1900zz . However this identification may be
called into questionGuo:2012tv . The experimental results for these $C=+$
states have induced renewed theoretical interest in understanding the nature
of charmonium-like states. The double charmonium production in $e^{+}e^{-}$
annihilation at B factoriesAbe:2002rb ; Aubert:2005tj turned out to be a
possible way to identify the $C=+$ charmonium or charmonium-like states,
recoiling against the easily reconstructed $1^{--}$ charmonium $J/\psi$ and
$\psi(2S)$. In addition to $\eta_{c},\eta_{c}(2S)$, $\chi_{c0}$, $X(3940)$
(decaying into $D\bar{D^{*}}$), and $X(4160)$ (decaying into
$D^{*}\bar{D^{*}}$) have also been observed in double charmonium production at
B factories. However, $\chi_{c1}$ and $\chi_{c2}$ states are missing in
production associated with $J/\psi$ at B factories. Identifying the $C=+$
charmonium states $H$ in the $e^{+}e^{-}\to\gamma^{*}\to\gamma+H$ process at B
factories is also proposedChung:2008km ; Braguta:2010mf . The quantum
chromodynamics (QCD) corrections of $e^{+}e^{-}\to\gamma^{*}\to\gamma+H$ at B
factories are calculated in Ref.Li:2009ki ; Sang:2009jc . The relativistic
correction of $e^{+}e^{-}\to\gamma^{*}\to\gamma+\eta_{c}$ is also included in
Ref.Sang:2009jc . Indirect measurement of quarkonium in the two-photon process
is also proposedSang:2012cp .
Table 1: $C=+$ $XYZ$ states. $X(3915)$, $X(3945)$, and $Y(3940)$ is considered as $\chi_{c0}(2P)$ for compatible properties. $Z(3930)$ is considreed as $\chi_{c2}(2P)$Eidelman:2012vu ; Brambilla:2010cs . State | $m(\Gamma)$ in MeV | $J^{PC}$ | Production (Decay) | Ref
---|---|---|---|---
$X(3872)$ | 3871.68$\pm$0.17 ( $<1.2$) | $1^{++}$ | $B\to K\,(\pi^{+}\pi^{-}J/\psi)$ | Choi:2003ue
| | | $p\bar{p}\to(\pi^{+}\pi^{-}J/\psi)+...$ | Acosta:2003zx ; Abulencia:2006ma
| | | $B\to K\,(\omega J/\psi)$ | Abe:2005ix ; delAmoSanchez:2010jr
| | | $B\to K\,(D^{0}\bar{D}^{*})$ | Gokhroo:2006bt ; Aubert:2007rva
| | | $B\to K\,(\gamma J/\psi,\gamma\psi(2S))$ | Aubert:2006aj
| | | $pp\to(\pi^{+}\pi^{-}J/\psi)+...$ | Aaij:2011sn ; Chatrchyan:2013cld ; Aaij:2013zoa
$X(3915)$ | $3917.5\pm 2.7$ ($27\pm 10$ ) | $0^{++}$ | $B\to K\,(\omega J/\psi)$ | Abe:2004zs ; Aubert:2007vj
| | | $e^{+}e^{-}\to e^{+}e^{-}\,(\omega J/\psi)$ | delAmoSanchez:2010jr ; Lees:2012xs
$X(3940)$ | $3942^{+9}_{-8}$ ( $37^{+27}_{-17}$ ) | $J^{P+}$ | $e^{+}e^{-}\to J/\psi\,(D\bar{D}^{*})$ | Abe:2007sya
$Y(4140)$ | $4143.0\pm 3.1$ ( $12^{+9}_{-\ 6}$) | $J^{P+}$ | $B\to K\,(\phi J/\psi)$ | Aaltonen:2011at
$X(4160)$ | $4156^{+29}_{-25}$ ( $139^{+110}_{-60}$) | $J^{P+}$ | $e^{+}e^{-}\to J/\psi\,(D^{*+}\bar{D}^{*-})$ | Abe:2007sya
$Y(4274)$ | $4274.4^{+8.4}_{-6.7}$ ( $32^{+22}_{-15}$) | $J^{P+}$ | $B\to K\,(\phi J/\psi)$ | Aaltonen:2011at
$X(4350)$ | $4350.6^{+4.6}_{-5.1}$ ( $13.3^{+18.4}_{-10.0}$ ) | 0/2++ | $e^{+}e^{-}\to e^{+}e^{-}\,(\phi J/\psi)$ | Shen:2009vs
Recently, BesIII reports the cross-sections of $e^{+}e^{-}\to\gamma
X(3872)$Yuan:2013lma ; Ablikim:2013dyn
$\displaystyle\sigma[e^{+}e^{-}\to\gamma X(3872)]\times{\rm
Br}[J/\psi\pi\pi]<0.13{\rm pb\ \ at\ 90\%\ CL.\ }$ $\displaystyle\hskip
8.5359pt\sqrt{s}=4.009{\rm GeV}$ $\displaystyle\sigma[e^{+}e^{-}\to\gamma
X(3872)]\times{\rm Br}[J/\psi\pi\pi]=0.32\pm 0.15\pm 0.02{\rm pb}$
$\displaystyle\hskip 8.5359pt\sqrt{s}=4.230{\rm GeV}$
$\displaystyle\sigma[e^{+}e^{-}\to\gamma X(3872)]\times{\rm
Br}[J/\psi\pi\pi]=0.35\pm 0.12\pm 0.02{\rm pb}$ $\displaystyle\hskip
8.5359pt\sqrt{s}=4.260{\rm GeV}$ $\displaystyle\sigma[e^{+}e^{-}\to\gamma
X(3872)]\times{\rm Br}[J/\psi\pi\pi]<0.39{\rm pb\ \ at\ 90\%\ CL.\ }$
$\displaystyle\hskip 8.5359pt\sqrt{s}=4.360{\rm GeV}$ (1)
Where ${\rm Br}[J/\psi\pi\pi]$ means ${\rm Br}[X(3872)\to J/\psi\pi\pi]$. And
the studies of $\psi(4160)\to X(3872)\gamma$ Margaryan:2013tta and
$\psi(4260)\to X(3872)\gamma$ Guo:2013zbw are proposed to probe the molecular
content of the $X(3872)$.
Many NLO relativistic and radiative corrections for heavy quarkonium
production are considered within nonrelativistic QCD (NRQCD)Bodwin:1994jh . By
introducing the color octet mechanism, one can obtain the infrared-safe
calculations for the decay rates of P wave Brambilla:2008zg ; Lansberg:2009xh
; Hwang:2010iq and D waveHe:2008xb ; He:2009bf ; Fan:2009cj quarkonium
states. The color octet contributions of the diphoton decay of P wave
quarkonium states are calculated in Ref.Ma:2002ev . $O(\alpha_{s}v^{2})$
corrections to the decays of $h_{c},h_{b}$ and $\eta_{b}$ are studied in
Ref.Guo:2011tz ; Li:2012rn . The NLO QCD correctionsZhang:2006ay ; Wang:2011qg
; Zhang:2008gp ; Zhang:2009ym ; Gong:2007db ; Gong:2008ce ; Gong:2009ng ;
Gong:2009kp ; Ma:2008gq ; Dong:2011fb ; Bodwin:2013ys , relativistic
correctionsHe:2007te ; Bodwin:2006ke ; Bodwin:2007ga ; Elekina:2009wt ;
Jia:2009np ; He:2009uf ; Fan:2012dy ; Fan:2012vw , and
${\mathcal{O}}(\alpha_{s}v^{2})$ corrections Dong:2012xx ; Li:2013qp largely
compensate for the discrepancies between theoretical values and experimental
measurements at B factories. The contributions of higher-order QCD corrections
for charmonium production Campbell:2007ws ; Gong:2008hk ; Gong:2008sn ;
Gang:2012js ; Ma:2010yw ; Ma:2010vd ; Shao:2012iz ; Butenschoen:2010rq ;
Butenschoen:2013pxa ; Meng:2013gga and polarization Chao:2012iv ;
Butenschoen:2012px ; Gong:2012ug ; Shao:2012fs in hadron colliders are also
significant. The relativistic corrections to $J/\psi$ hadroproduction are
significantFan:2009zq ; Xu:2012am ; Li:2013csa .
We calculate the production of $C=+$ charmonium at $e^{+}e^{-}$ annihilation
at BESIII to test the nature of $C=+$ $XYZ$ states. Our paper is organized as
follows. The calculation framework is given in Sec. 2. The numerical results
of the cross-sections of $C=+$ charmonium are discussed in Sec. 3. A
discussion of $X(3872)$ and other $C=+$ $XYZ$ states is given in Sec. 4. The
summary is given in Sec. 5.
## 2 The frame of the calculation
In the NRQCD factorization framework, we can express the amplitude in the rest
frame of $H$ asChung:2008km ; Li:2009ki ; Sang:2009jc
$\displaystyle{\cal A}(e^{-}(k_{1})e^{+}(k_{2})\rightarrow
H_{c\bar{c}}({}^{2S+1}L_{J})(2p_{1})+\gamma)$ (2) $\displaystyle=$
$\displaystyle\sum\limits_{L_{z}S_{z}}\sum\limits_{s_{1}s_{2}}\sum\limits_{jk}\int{\rm
d}^{3}\vec{q}\Phi_{c\bar{c}}(\vec{q})\langle s_{1};s_{2}\mid
SS_{z}\rangle\langle 3j;\bar{3}k\mid 1\rangle$ $\displaystyle\times{\cal
A}\left[e^{-}(k_{1})e^{+}(k_{2})\rightarrow
c_{j}^{s_{1}}(p_{1}+q)+\bar{c}^{s_{2}}_{k}(p_{1}-q)+\gamma(k)\right],$
where $\langle 3j;\bar{3}k\mid 1\rangle=\delta_{jk}/\sqrt{N_{c}}$, $\langle
s_{1};s_{2}\mid SS_{z}\rangle$ is the color Clebsch-Gordan coefficient for
$c\bar{c}$ pairs projecting out appropriate bound states, and $\langle
s_{1};s_{2}\mid SS_{z}\rangle$ is the spin Clebsch-Gordan coefficient. ${\cal
A}\left[e^{-}(k_{1})e^{+}(k_{2})\rightarrow
c_{j}^{s_{1}}(p_{1}+q)+\bar{c}^{s_{2}}_{k}(p_{1}-q)+\gamma(k)\right]$ is the
quark level scattering amplitude. In the rest frame of $H$, $q=(0,\vec{q})$,
and $p_{1}=(\sqrt{m_{c}^{2}+\vec{q}^{2}},0,0,0)$.
$\Phi^{H}_{c\bar{c}}(\vec{q})$ is the $c\bar{c}$ component wave function of
hadron $H$ in momentum space. For $v^{2}=\vec{q}^{2}/m_{c}^{2}\ll
1$Bodwin:1994jh , we can expand Eq.(2) with $v^{2}$:
$\displaystyle{\cal A}(q)$ $\displaystyle=$ $\displaystyle{\cal
A}(0)+\left.\frac{\partial{\cal
A}(\vec{q})}{\partial\vec{q}^{\alpha}}\right|_{q=0}\vec{q}^{\alpha}+\left.\frac{\partial^{2}{\cal
A}(\vec{q})}{\partial\vec{q}^{\alpha}\partial\vec{q}^{\beta}}\right|_{q=0}\frac{\vec{q}^{\alpha}\vec{q}^{\beta}}{2}$
(3) $\displaystyle+\left.\frac{\partial^{3}{\cal
A}(\vec{q})}{\partial\vec{q}^{\alpha}\partial\vec{q}^{\beta}\partial\vec{q}^{\delta}}\right|_{q=0}\frac{\vec{q}^{\alpha}\vec{q}^{\beta}\vec{q}^{\delta}}{3!}+....$
Here ${\cal A}(q)={\cal A}\left[e^{-}(k_{1})e^{+}(k_{2})\rightarrow
c_{j}^{s_{1}}(p_{1}+q)+\bar{c}^{s_{2}}_{k}(p_{1}-q)+\gamma(k)\right]$. We
consider the Fourier transform between the momentum space and position space
as: Bodwin:1994jh ; Xu:2012am ,
$\displaystyle\int{\rm d}^{3}\vec{q}\ \ \Phi_{c\bar{c}}(\vec{q})$
$\displaystyle\propto$ $\displaystyle\sqrt{Z_{c\bar{c}}^{H}}R_{c\bar{c}}(0)$
$\displaystyle\int{\rm d}^{3}\vec{q}\ \
\vec{q}^{\alpha}\Phi_{c\bar{c}}(\vec{q})$ $\displaystyle\propto$
$\displaystyle\sqrt{Z_{c\bar{c}}^{H}}R^{\prime}_{c\bar{c}}(0)$
$\displaystyle\int{\rm d}^{3}\vec{q}\ \
\vec{q}^{\alpha}\vec{q}^{\beta}\Phi_{c\bar{c}}(\vec{q})$
$\displaystyle\propto$
$\displaystyle\sqrt{Z_{c\bar{c}}^{H}}R^{\prime\prime}_{c\bar{c}}(0)$
$\displaystyle\int{\rm d}^{3}\vec{q}\ \
\vec{q}^{\alpha}\vec{q}^{\beta}\vec{q}^{\delta}\Phi_{c\bar{c}}(\vec{q})$
$\displaystyle\propto$
$\displaystyle\sqrt{Z_{c\bar{c}}^{H}}R^{\prime\prime\prime}_{c\bar{c}}(0).$
(4)
Here $Z_{c\bar{c}}^{H}$ is the possibility of $c\bar{c}$ component in hadron
$H$. $R_{c\bar{c}}(0)$ is the radial Schrodinger wave function at the origin.
$R^{l}_{c\bar{c}}(0)$ is the derivative of the radial Schrodinger wave
function at the origin
$\displaystyle R^{l}_{c\bar{c}}(0)=\left.\frac{{\rm
d}^{l}R_{c\bar{c}}(r)}{{\rm d}^{l}r}\right|_{r=0}$ (5)
$R_{c\bar{c}}(0)$ corresponds to the ${\cal O}(v^{0})$ S wave matrix element,
$R^{\prime}_{c\bar{c}}(0)$ corresponds to the ${\cal O}(v^{0})$ P wave matrix
element, $R^{\prime\prime}_{c\bar{c}}(0)$ corresponds to the ${\cal O}(v^{2})$
S wave matrix element or ${\cal O}(v^{0})$ D wave matrix element, and
$R^{\prime\prime\prime}_{c\bar{c}}(0)$ corresponds to the ${\cal O}(v^{2})$ P
wave matrix element.
$R_{c\bar{c}}(0)$ is also written as long-distance matrix elements (LDMEs) as
discussed in Ref.Xu:2012am . For example,
$\displaystyle\langle
0|\mathcal{O}^{\chi_{c1}}(^{3}P_{1}^{[1]})|0\rangle=\frac{27}{2\pi}|R^{\prime}_{1P}(0)|^{2},$
(6)
We calculated the relativistic corrections for the S wave and P wave states
and obtain two LDMEs for $\eta_{c}$, four LDMEs for $\chi_{cJ}$, and one LDMEs
for ${}^{1}D_{2}$ states. To simplify the discussion of the numerical result,
we assumed that
$\displaystyle<0|\mathcal{O}^{\chi_{cJ}}({}^{3}P_{J}^{[1]})|0>$
$\displaystyle=$
$\displaystyle(2J+1)<0|\mathcal{O}^{\chi_{cJ}}({}^{3}P_{0}^{[1]})|0>.$ (7)
$v^{2}=\frac{\langle
0|\mathcal{P}^{H}(^{2s+1}L_{J}^{[c]})|0\rangle}{m_{c}^{2}\langle
0|\mathcal{O}^{H}(^{2s+1}L_{J}^{[c]})|0\rangle}.$ (8)
Then there is only one LDME for $S$ wave, $P$ wave, and $D$ wave respectively.
More details can be found in Ref.Xu:2012am .
The relativistic correction $K$ factor is
$\displaystyle K_{v^{2}}[\eta_{c}]$ $\displaystyle=$
$\displaystyle-\frac{5v^{2}}{6}-\frac{rv^{2}}{1-r},$ $\displaystyle
K_{v^{2}}[\chi_{c0}]$ $\displaystyle=$
$\displaystyle-\frac{\left(55r^{2}-28r+13\right)v^{2}}{10\left(3r^{2}-4r+1\right)}-\frac{rv^{2}}{1-r},$
$\displaystyle K_{v^{2}}[\chi_{c1}]$ $\displaystyle=$
$\displaystyle-\frac{\left(21r^{2}+30r-11\right)v^{2}}{10\left(r^{2}-1\right)}-\frac{rv^{2}}{1-r},$
$\displaystyle K_{v^{2}}[\chi_{c2}]$ $\displaystyle=$
$\displaystyle-\frac{\left(90r^{3}+113r^{2}+4r-7\right)v^{2}}{10(r-1)\left(6r^{2}+3r+1\right)}-\frac{rv^{2}}{1-r},$
(9)
where $r=4m_{c}^{2}/s$. $-\frac{rv^{2}}{1-r}$ is the relativistic correction
of the phase space. If we select $r\to 0$, the $K_{v^{2}}$ factor is
consistent with the $K$ factor at large $p_{T}$ in Ref.Xu:2012am .
Our leading order (LO) cross-sections of $e^{+}e^{-}\to\gamma^{*}\to\gamma+H$
is consistent with Ref.Chung:2008km ; Li:2009ki ; Sang:2009jc . The QCD
corrections of $e^{+}e^{-}\to\gamma^{*}\to\gamma+H$ is consistent with
Ref.Li:2009ki ; Sang:2009jc . And the relativistic corrections of
$e^{+}e^{-}\to\gamma^{*}\to\gamma+\eta_{c}$ is consistent with Ref.Sang:2009jc
; Fan:2012dy ; Fan:2012vw .
We can obtain a similar amplitude for the $D\bar{D}$ component in the molecule
model. We can estimate the off-resonance amplitude of $e^{+}e^{-}\to H+\gamma$
from the $D\bar{D}$ component. The parton-level amplitudes may be compared
with the hadron-level amplitudes:
$\displaystyle{\cal A}\left[e^{-}(k_{1})e^{+}(k_{2})\rightarrow
c\bar{c}(2p_{1})+\gamma\right]\sim{\cal
A}\left[e^{-}(k_{1})e^{+}(k_{2})\rightarrow D\bar{D}(2p_{1})+\gamma\right]$
(10)
By contrast, the $R^{l}_{c\bar{c}}(0)\sim v^{2l}R^{S}_{c\bar{c}}(0)\gg
R_{D\bar{D}}(0)$ with the $S$ wave $l=0$ and $P$ wave $l=1$ for the binding
energies of $c\bar{c}$ and $D\bar{D}$ are several hundreds of MeV and several
MeV, respectively. If $Z_{c\bar{c}}^{H}\sim Z_{D\bar{D}}^{H}$, we can consider
the $c\bar{c}$ contributions only.
In the numerical calculation, we consider the charm quark mass as half of the
hadron mass consistent with the physics phase space. With a large charm quark
mass, the wave functions at the origin are identified as the Cornell potential
result in Ref.Eichten:1995ch . The sellected parameters are as follows:
$\displaystyle m_{c}=m_{H}/2,\hskip 56.9055pt\alpha_{s}=0.23,\hskip
68.28644pt\alpha=1/133,$ $\displaystyle v^{2}=0.23,\hskip
69.70915ptR_{1S}=1.454{\rm GeV}^{3},\hskip 25.6073ptR_{2S}=0.927{\rm
GeV}^{3},$ $\displaystyle R_{3S}=0.791{\rm GeV}^{3},\hskip
28.45274ptR^{\prime}_{1P}=0.131{\rm GeV}^{5},\hskip
22.76228ptR^{\prime}_{2P}=0.186{\rm GeV}^{5},$ $\displaystyle
R^{\prime\prime}_{1D}=0.031{\rm GeV}^{7}.$ (11)
The wave functions at origin for higher states are estimated as
$\displaystyle R_{4S}$ $\displaystyle=$ $\displaystyle 2\times
R_{3S}-R_{2S}=0.655{\rm GeV}^{3},$ $\displaystyle R^{\prime}_{3P}$
$\displaystyle=$ $\displaystyle(R^{\prime}_{1P}+R^{\prime}_{2P})/2=0.159{\rm
GeV}^{5},$ $\displaystyle R^{\prime\prime}_{2D}$ $\displaystyle=$
$\displaystyle R^{\prime\prime}_{1D}=0.031{\rm GeV}^{7}.$ (12)
In the numerical result, ”$\sigma_{LO}$” is the LO cross-section,
”$\sigma_{v^{2}}$” is the cross-section including the LO and the relativistic
correction, ”$\sigma_{\alpha_{s}}$” is the cross-section including the LO and
the radiative correction, and ”$\sigma_{\alpha_{s},v^{2}}$” is the cross-
section including the LO, the relativistic correction, and the radiative
correction. In addition, ”LO” is the LO cross-section, ”RC” is the
relativistic correction, ”QCD” is the radiative correction, and ”Total” is the
cross-section including the LO, the relativistic correction, and the radiative
correction.
For the LO, the cross-section is ${\cal O}(\alpha_{s}^{0}v^{0})$. As
$\alpha_{s}=0.23\pm 0.03$ and $v^{2}=0.23\pm 0.03$ are reasonable estimates,
we can estimate that the uncertainty of the numerical result from $\alpha_{s}$
and $v^{2}$ is $<10\%$.
## 3 Pure $C=+$ charmonium states
We can estimate the cross-sections for pure $C=+$ charmonium states $H$ in
$e^{+}e^{-}\to\gamma~{}+~{}H$ at BESIII with $H=\eta_{c}(nS)$ (n=1, 2, 3, and
4), $\chi_{cJ}(nP)$ (n=1, 2, and 3), and ${}^{1}D_{2}(nD)$ (n=1 and 2). The
mass of the lower states can be found in Ref.Beringer:1900zz , and the mass of
the higher states is selected from Ref.Li:2009zu .
Figure 1: The cross-sections of $e^{+}e^{-}\to\eta_{c}+\gamma$ as a function of $\sqrt{s}$ in fb. The cross-section ”$\sigma_{LO}$”, ”$\sigma_{v^{2}}$”, ”$\sigma_{\alpha_{s}}$”, and ”$\sigma_{\alpha_{s},v^{2}}$” are defined near the end of Section 2. Figure 2: The cross-sections of $e^{+}e^{-}\to\eta_{c2}(1D,2D)+\gamma$ as a function of $\sqrt{s}$ in fb. Table 2: The cross-sections of $e^{+}e^{-}\to H+\gamma$ for $\eta_{c}(nS)$ with $n=1,2,3,4$ and $\eta_{c2}{(nD)}$ for $n=1,2$ charmonium states in fb. The labels LO, RC, QCD and Total are defined near the end of Section 2. The mass of $\eta_{c}(3S)$, $\eta_{c}(4S)$, $\eta_{c2}(1D)$, and $\eta_{c2}(2D)$ are selected from Ref.Li:2009zu . The other mass can be found in Ref.Beringer:1900zz . $\sqrt{s}$(GeV) | 4.00 | 4.25 | 4.50 | 4.75 | 5.00 | 10.6 | 11.2
---|---|---|---|---|---|---|---
$\eta_{c}$(2981) | LO | 2781 | 2494 | 2192 | 1906 | 1652 | 117 | 95
| RC | -1332 | -1033 | -814 | -650 | -526 | -25 | -20
| QCD | -909 | -807 | -700 | -598 | -508 | -22 | -16
| Total | 540 | 653 | 678 | 658 | 617 | 70 | 58
$\eta_{c}(2S)$(3639) | LO | 563 | 684 | 706 | 679 | 629 | 58 | 48
| RC | -730 | -563 | -442 | -352 | -284 | -13 | -10
| QCD | -177 | -221 | -231 | -222 | -205 | -13 | -10
| Total | -344 | -100 | 33 | 105 | 141 | 32 | 27
$\eta_{c}(3S)$(3994) | LO | | 233 | 337 | 374 | 377 | 44 | 36
| RC | | -450 | -352 | -279 | -225 | -10 | -8
| QCD | | -72 | -107 | -121 | -123 | -10 | -8
| Total | | -228 | -122 | -27 | 29 | 24 | 20
$\eta_{c}(4S)$(4250) | LO | | | 133 | 198 | 225 | 34 | 28
| RC | | | -279 | -221 | -178 | -8 | -6
| QCD | | | -41 | -63 | -73 | -8 | -7
| Total | | | -186 | -86 | -26 | 17 | 15
$\eta_{c2}(1D)$(3796) | LO | 4.0 | 6.4 | 7.3 | 7.3 | 7.0 | 0.71 | 0.58
$\eta_{c2}(2D)$(4099) | LO | | 1.5 | 2.9 | 3.5 | 3.7 | 0.47 | 0.38
The cross-section of $e^{+}e^{-}\to\eta_{c}+\gamma$ as a function of
$\sqrt{s}$ is shown in Fig.1. The cross-sections of
$e^{+}e^{-}\to\eta_{c2}(1D,2D)+\gamma$ as a function of $\sqrt{s}$ are shown
in Fig.2. The numerical results for $nS$ with $n=1,2,3,4$ and $nD$ with
$n=1,2$ are listed in Table 2. We determined that the radiative and
relativistic corrections are negative and large for $\eta_{c}(nS)$,
respectively. The LO cross-sections for $\eta_{c2}(1D,2D)$ is very small at
BESIII; hence, the high order corrections are ignored.
The cross-sections of $e^{+}e^{-}\to\chi_{cJ}+\gamma$ as a function of
$\sqrt{s}$ are shown in Fig.3, Fig.4, and Fig.5 for $J=0,1,2$, respectively.
The numerical results for $\chi_{cJ}(nP)$ with $n=1,2,3$ are listed in Table
3, Table 4, and Table 5 for $J=0,1,2$, respectively. We determined that the
QCD corrections are large but negative and the relativistic corrections are
large and positive. Hence, many $P$ wave states can be searched at BESIII.
Figure 3: The cross-sections of $e^{+}e^{-}\to\chi_{c0}+\gamma$ as a function of $\sqrt{s}$ in fb. The cross-section ”$\sigma_{LO}$”, ”$\sigma_{v^{2}}$”, ”$\sigma_{\alpha_{s}}$”, and ”$\sigma_{\alpha_{s},v^{2}}$” are defined near the end of Section 2. Table 3: The cross-sections of $e^{+}e^{-}\to\chi_{c0}(nP)+\gamma$ with $n=1,2,3$ in fb. The labels LO, RC, QCD and Total are defined near the end of Section 2. The $\chi_{c0}(2P)$ is considreed as $X(3915)$($X(3945)$/$Y(3940)$) Eidelman:2012vu ; Brambilla:2010cs . The mass of $\chi_{c0}(3P)$ are selected from Ref.Li:2009zu . The other mass can be found in Ref.Beringer:1900zz . $\sqrt{s}$(GeV) | 4.00 | 4.25 | 4.50 | 4.75 | 5.00 | 10.6 | 11.2
---|---|---|---|---|---|---|---
$\chi_{c0}$(3415) | LO | 877 | 328 | 132 | 53 | 21 | 1.81 | 1.6
| RC | 825 | 268 | 107 | 48 | 22 | -0.77 | -0.63
| QCD | -528 | -228 | -107 | -52 | -26 | -0.38 | -0.29
| Total | 1173 | 368 | 131 | 49 | 17 | 1.42 | 1.22
$\chi_{c0}(2P)$(3918) | LO | | 1991 | 665 | 271 | 119 | 1.30 | 1.18
| RC | | 3102 | 680 | 230 | 96 | -0.64 | -0.54
| QCD | | -1013 | -384 | -177 | -89 | 0.39 | 0.30
| Total | | 4080 | 962 | 324 | 127 | 1.04 | 0.94
$\chi_{c0}(3P)$(4131) | LO | | | 1073 | 384 | 164 | 0.82 | 0.75
| RC | | | 1600 | 391 | 140 | -0.44 | -0.38
| QCD | | | -551 | -223 | -107 | 0.29 | 0.23
| Total | | | 2121 | 554 | 198 | 0.67 | 0.61
Figure 4: The cross-sections of $e^{+}e^{-}\to\chi_{c1}+\gamma$ as a function of $\sqrt{s}$ in fb. The cross-section ”$\sigma_{LO}$”, ”$\sigma_{v^{2}}$”, ”$\sigma_{\alpha_{s}}$”, and ”$\sigma_{\alpha_{s},v^{2}}$” are defined near the end of Section 2. Table 4: The cross-sections of $e^{+}e^{-}\to\chi_{c1}(nP)+\gamma$ with $n=1,2,3$ in fb. The labels LO, RC, QCD and Total are defined near the end of Section 2. The mass of $\chi_{c1}(2P,3P)$ are selected from Ref.Li:2009zu . And the mass of $\chi_{c1}(1P)$ can be found in Ref.Beringer:1900zz . $\sqrt{s}$(GeV) | 4.00 | 4.25 | 4.50 | 4.75 | 5.00 | 10.6 | 11.2
---|---|---|---|---|---|---|---
$\chi_{c1}$(3511) | LO | 7186 | 3874 | 2392 | 1597 | 1124 | 23.5 | 18.5
| RC | 4448 | 1296 | 459 | 168 | 52 | -4.8 | -3.8
| QCD | -3327 | -1791 | -1091 | -715 | -492 | -6.5 | -4.9
| Total | 8307 | 3379 | 1760 | 1051 | 685 | 12.3 | 9.7
$\chi_{c1}(2P)$(3901) | LO | | 8854 | 4244 | 2495 | 1624 | 25.7 | 20.0
| RC | | 9585 | 2297 | 789 | 312 | -4.9 | -3.9
| QCD | | -4041 | -1967 | -1152 | -741 | -7.7 | -5.70
| Total | | 14397 | 4573 | 2131 | 1195 | 13.2 | 10.3
$\chi_{c1}(3P)$(4178) | LO | | | 1073 | 384 | 164 | 0.82 | 0.75
| RC | | | 1600 | 391 | 140 | -0.44 | -0.38
| QCD | | | -551 | -223 | -107 | 0.29 | 0.23
| Total | | | 2121 | 554 | 198 | 0.67 | 0.61
Figure 5: The cross-sections of $e^{+}e^{-}\to\chi_{c2}+\gamma$ as a function of $\sqrt{s}$ in fb. The cross-section ”$\sigma_{LO}$”, ”$\sigma_{v^{2}}$”, ”$\sigma_{\alpha_{s}}$”, and ”$\sigma_{\alpha_{s},v^{2}}$” are defined near the end of Section 2. Table 5: The cross-sections of $e^{+}e^{-}\to\chi_{c2}(nP)+\gamma$ with $n=1,2,3$ in fb. The labels LO, RC, QCD and Total are defined near the end of Section 2. $\chi_{c2}(2P)$ is considreed as $Z(3930)$, Eidelman:2012vu ; Brambilla:2010cs . The mass of $\chi_{c2}(3P)$ are selected from Ref.Li:2009zu . And the mass of $\chi_{c2}(1P)$ can be found in Ref.Beringer:1900zz . $\sqrt{s}$(GeV) | 4.00 | 4.25 | 4.50 | 4.75 | 5.00 | 10.6 | 11.2
---|---|---|---|---|---|---|---
$\chi_{c2}$(3556) | LO | 10149 | 4724 | 2590 | 1562 | 1004 | 9.66 | 7.37
| RC | 8587 | 2385 | 880 | 376 | 173 | -1.16 | -0.93
| QCD | -5056 | -2455 | -1384 | -851 | -557 | -6.27 | -4.82
| Total | 13679 | 4655 | 2087 | 1086 | 621 | 2.22 | 1.63
$\chi_{c2}(2P)$(3927) | LO | | 13419 | 5581 | 2931 | 1927 | 11.29 | 8.53
| RC | | 17835 | 3965 | 1355 | 565 | -1.22 | -0.99
| QCD | | -6423 | -2822 | -1533 | -926 | -7.25 | -5.52
| Total | | 24862 | 6723 | 2754 | 1368 | 2.82 | 2.03
$\chi_{c2}(3P)$(4208) | LO | | | 8938 | 3607 | 1886 | 8.55 | 6.40
| RC | | | 14212 | 2949 | 995 | -0.83 | -0.68
| QCD | | | -4210 | -1803 | -977 | -5.43 | -4.10
| Total | | | 18941 | 4753 | 1904 | 2.28 | 1.62
The NRQCD requires that the energy of photon at the center of the mass frame
of $e^{+}e^{-}$
$\displaystyle
E_{\gamma}=\frac{s-M_{H}^{2}}{2\sqrt{s}}\sim\sqrt{s}-M_{H}+{\cal
O}\left[(1-M_{H}/\sqrt{s})^{2}\right]$ (13)
be larger than $\Lambda_{QCD}\sim 300\ {\rm MeV}\sim m_{c}v^{2}$. Although
this process is a QED process, the prediction is not reliable and only a
reference value if this requirement is not satisfied. If we replace photon
with gluon, the soft photon contributions correspond to the long-distance
color octet contributionsBodwin:1994jh ; Sang:2009jc .
## 4 $C=+$ $XYZ$ states
$X(4160)$ and $Y(4274)$ are found in the B decay $B\to K+H\to K+\phi J/\psi$
by CDF collaborationAaltonen:2011at . No signal of $X(4160)$ or $Y(4274)$ is
reported by B factories. Hence, the cross-sections for $X(4160)$ or $Y(4274)$
at BESIII may be too small. The cross-sections of $e^{+}e^{-}\to\gamma H$ for
$X(3872)$, $X(3940)$, $X(4160)$, and $X(4350)$ are discussed here. The
$1^{--}$ resonance contributions are ignored here.
### 4.1 $X(3872)$
In the light of the mixture state of the $\chi_{c1}(2P)$ and
$D^{0}\bar{D}^{\star 0}$ molecule, the cross-sections of $X(3872)$ at hadron
collides can be expressed asMeng:2013gga :
$d\sigma[X(3872)\to J/\psi\pi^{+}\pi^{-}]=d\sigma[\chi_{c1}(2P)]{\times}k,$
(14)
where $k=Z^{X(3875)}_{c\bar{c}}{\times}Br[X(3872)\to J/\psi\pi^{+}\pi^{-}]$.
$Br[X(3872)\to J/\psi\pi^{+}\pi^{-}]$ is the branching fraction for $X(3872)$
decay to $J/\psi\pi^{+}\pi^{-}$. $Z^{X(3875)}_{c\bar{c}}$ is the possibility
of the $\chi_{c1}(2P)$ component in $X(3872)$. And $k=0.018\pm 0.04$
Meng:2005er ; Meng:2013gga .
Figure 6: The cross-sections of $e^{+}e^{-}\to\chi_{c2}+\gamma$ as a function
of $\sqrt{s}$ in fb. The cross-section ”$\sigma_{LO}$”, ”$\sigma_{v^{2}}$”,
”$\sigma_{\alpha_{s}}$”, and ”$\sigma_{\alpha_{s},v^{2}}$” are defined near
the end of Section 2. The uncertainty bind of $\sigma_{\alpha_{s},v^{2}}$ is
from the uncertainty of $k=0.018\pm 0.04$. Table 6: The cross-sections of
$e^{+}e^{-}\to X(3872)+\gamma\to J/\psi\pi\pi+\gamma$ in fb. The labels LO,
RC, QCD and Total are defined near the end of Section 2.
$\sqrt{s}$(GeV) | 4.15 | 4.2 | 4.25 | 4.3 | 4.35 | 4.45 | 4.55
---|---|---|---|---|---|---|---
LO | 221$\pm$49 | 180$\pm$40 | 150$\pm$33 | 127$\pm$28 | 110$\pm$24 | 84$\pm$19 | 66$\pm$15
RC | 310$\pm$69 | 208$\pm$46 | 146$\pm$32 | 106$\pm$24 | 80$\pm$18 | 47$\pm$10 | 30$\pm$7
QCD | -100$\pm$22 | -82$\pm$18 | -69$\pm$15 | -59$\pm$13 | -51$\pm$11 | -39$\pm$9 | -31$\pm$7
Total | 431$\pm$96 | 306$\pm$68 | 227$\pm$51 | 175$\pm$39 | 138$\pm$31 | 92$\pm$20 | 65$\pm$14
$\sqrt{s}$(GeV) | NRQCD prediction for continue | BESIII Yuan:2013lma ; Ablikim:2013dyn
---|---|---
4.009 | | $<$130 at 90% CL.
4.160 | $401\pm 89$ |
4.230 | $255\pm 57$ | $320\pm 150\pm 20$
4.260 | $215\pm 48$ | $350\pm 120\pm 20$
4.360 | $133\pm 29$ | $<$130 at 90% CL.
4.415 | $105\pm 23$ |
4.660 | $47\pm 10$ |
To clarify the nature of $X(3872)$, we also give the numerical calculation of
$e^{+}e^{-}\to\gamma X(3872)\to J/\psi\pi^{+}\pi^{-}\gamma$ in this picture
$\displaystyle\sigma[e^{+}e^{-}\to\gamma X(3872)]\times{\rm Br}[X\to
J/\psi\pi\pi]$ (15) $\displaystyle=$
$\displaystyle\sigma[e^{+}e^{-}\to\gamma\chi_{c1}(2P)(3872)]{\times}(0.018\pm
0.004)$
The cross-sections as a function of $\sqrt{s}$ is shown in Fig.6. Many
$1^{--}$ states with $M_{H}<5~{}$ GeV are also observed. We can predict the
cross-sections from continuous contributions at this point, and the result is
listed in Table 6. We ignore the $1^{--}$ resonances contributions here. We
emphasize that if we select $\sqrt{s}=4.009{\rm GeV}$, the energy of photon
$E_{\gamma}=134~{}$ MeV and smaller than $\Lambda_{QCD}\sim m_{c}v^{2}\sim
300\ {\rm MeV}$. Hence, NRQCD cannot accurately predict the cross-sections
with a soft photon with $\sqrt{s}=4.009{\rm GeV}$Bodwin:1994jh . If
$\sqrt{s}=4.160{\rm GeV}$, the energy of photon is $E_{\gamma}=270{\rm MeV}$.
Although this process is a QED process, the prediction is not reliable and
only a reference valueSang:2009jc . We determined that the NRQCD prediction of
the continuous contributions can be compared with the BESIII data of the
cross-sections of $e^{+}e^{-}\to\gamma X(3872)$ Yuan:2013lma ; Ablikim:2013dyn
in Eq.(1).
When we only considered the continuum production, the resonance contributions
can be estimated as that:
$\displaystyle\sigma_{Res}[s]=\frac{12\pi\Gamma[Res\to
e^{+}e^{-}]\Gamma[Res\to\gamma X]}{(s-M^{2})^{2}+(M\Gamma_{tot}[Res])^{2}}.$
(16)
We take into account only one resonance here and ignore continuum and other
resonances here. If we ignore the interference between one resonance and
continuum and other resonances, the $gamma$ energy dependence of the
$\Gamma[Res\to\gamma X]$, and $D\bar{D}$ contributions of decay of
$Res\to\gamma X$, we can estimate the resonance contributions. With $X(3872)$
considered as $2P$ states, the largest decay widths are $\psi(4040)$ and
$\psi(4160)$, which are considered as the mixing of $\psi(3S)$ and $\psi(2D)$
Li:2012vc ; Barnes:2005pb . The $\Gamma[Res\to\gamma X]$ for other states will
be less than $1$ keV Barnes:2005pb , and $\Gamma_{tot}\sim 100~{}$MeV,
$\Gamma[Res\to e^{+}e^{-}]\sim 1~{}$keV. Hence, we ignore the contributions
from other resonances. With the parameters for $\psi(4040)$ and
$\psi(4160)$Beringer:1900zz ; Barnes:2005pb :
$\displaystyle\Gamma[\psi(4040)\to e^{+}e^{-}]=0.87~{}{\rm keV}\hskip
12.80365pt\Gamma[\psi(4040)\to\gamma X]=40~{}{\rm keV}\hskip
15.36429pt\Gamma_{tot}[\psi(4040)]=80~{}{\rm MeV}$
$\displaystyle\Gamma[\psi(4160)\to e^{+}e^{-}]=0.83~{}{\rm keV}\hskip
12.80365pt\Gamma[\psi(4160)\to\gamma X]=140~{}{\rm keV}\hskip
7.11317pt\Gamma_{tot}[\psi(4160)]=103~{}{\rm MeV}$
Hence, we can determine the contributions of these parameters to
$X(3872)\gamma\to J/\psi\pi^{+}\pi^{-}\gamma$
$\displaystyle(\sigma_{\psi(4040)}[4.23]+\sigma_{\psi(4160)}[4.23])\times
k=(62\pm 14)fb$
$\displaystyle(\sigma_{\psi(4040)}[4.26]+\sigma_{\psi(4160)}[4.26])\times
k=(37\pm 8)fb$ (17)
If we considered the interference, the result would be more complex. On the
other hand, we have calculated the quark-level intermediate states, which do
not clearly deal with the hadron-level intermediate states.
### 4.2 $X(3940)$ and $X(4160)$
$X(3940)$ and $X(4160)$ are observed in $e^{+}e^{-}\to J/\psi\,(D\bar{D})$ at
B factories Abe:2007sya . $\eta_{c}$ and $\chi_{c0}$ are recoiled with
$J/\psi$, but $\chi_{c1}$ and $\chi_{c2}$ are missedAbe:2007sya . The
theoretical predictions are consistent with the experimental dataLiu:2002wq ;
Liu:2004ga ; Wang:2011qg ; Dong:2011fb . So there should be large
$\eta_{c}(nS)$ and $\chi_{c0}(nP)$ component in $X(3940)$ and $X(4160)$,
respectively. The mass of $\eta_{c}(3S)$ and $\chi_{c0}(3P)$ are predicted as
$3994$ MeV and $4130$ MeV respectivelyLi:2009zu . Compared with Table 2 and
Table 3, we can found that the cross-sections of $\eta_{c}(3S)$ is small even
negative at $\sqrt{s}<$ 5 GeV. But $\chi_{c0}(3P)$ is large. The cross-
sections as a function of $\sqrt{s}$ is shown in Fig 7. Here
$Z_{c\bar{c}}^{X}\leq 1$ is the possibility of $\eta_{c}(3S)$ and
$\chi_{c0}(3P)$ component in $X(3940)$ and $X(4160)$ respectively. The BESIII
collaboration can search $X(3940)$ and $X(4160)$ in the process
$e^{+}e^{-}\to\gamma\ +X(D\bar{D})$. The result may be useful in identifying
the nature of $X(3940)$ and $X(4160)$.
Figure 7: The cross-sections of $e^{+}e^{-}\to X(3940)(X(4160))+\gamma$ as a
function of $\sqrt{s}$ in fb.
### 4.3 $X(4350)$
$X(4350)$ are found in $\gamma\gamma\to H\to\phi J/\psi$ at B factories
Shen:2009vs . And $J^{PC}$ is $0^{++}$ or $2^{++}$. So there should be large
$\chi_{c0}(nP)$ or $\chi_{c2}(nP)$ component in $X(4350)$. In Ref.Li:2009zu ,
The mass of $\chi_{c2}(3P)$ is 4208 MeV. Ignore more detail of the mass, we
considered it as $\chi_{c0}(nS)$ or $\chi_{c2}(nP)$, the wave function at
origin are estimated as
$\displaystyle R^{\prime}=R^{\prime}_{3P}$ $\displaystyle=$
$\displaystyle(R^{\prime}_{1P}+R^{\prime}_{2P})/2=0.159{\rm GeV}^{5},$ (18)
The cross-sections of $e^{+}e^{-}\to X(4350)+\gamma$ as a function of
$\sqrt{s}$ is show in Fig.8. Here $Z_{c\bar{c}}^{X}$ is the possibility of
$\chi_{c0}(nP)$ or $\chi_{c2}(nP)$ component in $X(4350)$. The cross-section
for $\chi_{c2}(nP)$ is larger than $\chi_{c0}(nP)$ by a factor of $6$. The
result may be useful in identifying the nature of $X(4350)$.
Figure 8: The cross-sections of $e^{+}e^{-}\to X(4350)+\gamma$ as a function
of $\sqrt{s}$ in fb. The cross-section ”$\sigma_{LO}$”, ”$\sigma_{v^{2}}$”,
”$\sigma_{\alpha_{s}}$”, and ”$\sigma_{\alpha_{s},v^{2}}$” are defined near
the end of Section 2. And $Z_{c\bar{c}}^{X}$ is the possibility of
$\chi_{c0}(nP)$ or $\chi_{c2}(nP)$ component in $X(4350)$.
## 5 Summary and discussion
While BESIII and Belle have collected a large amount of data, some final
states may be searched by the experimentalists. We can estimate the possible
event number at BESIII and Belle. The possible event number is
$\displaystyle N=\sigma[e^{+}e^{-}\to\gamma+c\bar{c}[n]]\times
Z_{c\bar{c}}^{H}\times Br\times{\cal L}\times\epsilon,$ (19)
where $\epsilon$ is the efficiency of detectors selected as $20\%$, $Br$ is
the branch ratio of $H$ to the decay mode, and ${\cal L}$ is the luminosity.
The result is listed in Table 7.
Table 7: The possible event number of $C=+$ charmonium and $XYZ$ states through $e^{+}e^{-}\to\gamma+H$ at BESIII and Belle. The efficiency of detectors are selected as $20\%$. The integrated luminosity is $1.0fb^{-1}@4.23$ GeV, $1.0fb^{-1}@4.26$ GeV, $0.5fb^{-1}@4.66$ GeV, and $1ab^{-1}@10.6$ GeV. The decay mode of $nKm\pi$ corresponds to $D\bar{D}$ decay, and the branch ratio is estimated as $1\%$. H | Decay | $Br$ | $Z_{c\bar{c}}^{H}$ | 4.23 | 4.26 | 4.66 | 10.6
---|---|---|---|---|---|---|---
$\eta_{c}$ | $K\bar{K}\pi$ | $7.2\%$ | 1 | 9 | 9 | 5 | 1012
$\chi_{c0}$ | $2\pi^{+}2\pi^{-}$ | $2.2\%$ | 1 | 2 | 2 | | 6
$\chi_{c1}$ | $\gamma l^{+}l^{-}(\gamma J/\psi)$ | $4.1\%$ | 1 | 29 | 27 | 5 | 101
$\chi_{c2}$ | $\gamma l^{+}l^{-}(\gamma J/\psi)$ | $2.3\%$ | 1 | 23 | 20 | 3 | 10
$\eta_{c2}(1D)$ | $\gamma\gamma K\bar{K}\pi$ | $1.5\%$ | 1 | | | | 2
$\eta_{c}(2S)$ | $K\bar{K}\pi$ | $1.9\%$ | 1 | | | | 123
$X(3872)(\chi_{c1}(2P))$ | $\pi^{+}\pi^{-}l^{+}l^{-}(\pi^{+}\pi^{-}J/\psi)$ | $0.6\%$ | 0.36 | 6 | 5 | 1 | 6
$X(3915)(\chi_{c0}(2P))$ | $\pi^{+}\pi^{-}\pi^{0}l^{+}l^{-}(\omega J/\psi)$ | $1\%$ | 1 | 9 | 8 | | 2
$Z(3930)(\chi_{c2}(2P))$ | $nKm\pi(D\bar{D})$ | $1\%$ | 1 | 57 | 46 | 4 | 6
$X(3940)(\eta_{c}(3S))$ | $nKm\pi(D\bar{D})$ | $1\%$ | 1 | | | | 48
As a summary, we study the production of $C=+$ charmonium states $H$ in
$e^{+}e^{-}\to\gamma~{}+~{}H$ at BESIII with $H=\eta_{c}(nS)$ (n=1, 2, 3, and
4), $\chi_{cJ}(nP)$ (n=1, 2, and 3), and ${}^{1}D_{2}(nD)$ (n=1 and 2) within
the framework of NRQCD. The radiative and relativistic corrections are
calculated to next-to-leading order for $S$ and $P$ wave states. We then argue
that the search for $C=+$ $XYZ$ states such as $X(3872)$, $X(3940)$,
$X(4160)$, and $X(4350)$ in $e^{+}e^{-}\to\gamma~{}+~{}H$ at BESIII may help
clarify the nature of these states. BESIII can search $XYZ$ states through two
body process $e^{+}e^{-}\to\gamma H$, where $H$ decay to
$J/\psi\pi^{+}\pi^{-}$, $J/\psi\phi$, or $D\bar{D}$. This result may be useful
in identifying the nature of $C=+$ $XYZ$ states. For completeness, the
production of $C=+$ charmonium in $e^{+}e^{-}\to\gamma+~{}H$ at B factories is
also discussed.
###### Acknowledgements.
The authors would like to thank Professor C.P. Shen for useful discussion.
This work was supported by the National Natural Science Foundation of China
(Grants No.11075011 and No. 11375021), the Foundation for the Author of
National Excellent Doctoral Dissertation of China (Grants No. 2007B18 and No.
201020), the Fundamental Research Funds for the Central Universities, and the
Education Ministry of LiaoNing Province.
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|
arxiv-papers
| 2013-10-01T16:25:38 |
2024-09-04T02:49:51.859817
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Yi-Jie Li, Guang-Zhi Xu, Kui-Yong Liu, Yu-Jie Zhang",
"submitter": "Yu-Jie Zhang Dr.",
"url": "https://arxiv.org/abs/1310.0374"
}
|
1310.0406
|
# Hilbert’s 6th Problem: Exact and Approximate Hydrodynamic Manifolds for
Kinetic Equations
Alexander N. Gorban Department of Mathematics, University of Leicester, LE1
7RH, Leicester, UK [email protected] Ilya Karlin Department of Mechanical and
Process Engineering, ETH Zürich 8092 Zürich, Switzerland
[email protected]
###### Abstract.
The problem of the derivation of hydrodynamics from the Boltzmann equation and
related dissipative systems is formulated as the problem of slow invariant
manifold in the space of distributions. We review a few instances where such
hydrodynamic manifolds were found analytically both as the result of summation
of the Chapman–Enskog asymptotic expansion and by the direct solution of the
invariance equation. These model cases, comprising Grad’s moment systems, both
linear and nonlinear, are studied in depth in order to gain understanding of
what can be expected for the Boltzmann equation. Particularly, the dispersive
dominance and saturation of dissipation rate of the exact hydrodynamics in the
short-wave limit and the viscosity modification at high divergence of the flow
velocity are indicated as severe obstacles to the resolution of Hilbert’s 6th
Problem. Furthermore, we review the derivation of the approximate hydrodynamic
manifold for the Boltzmann equation using Newton’s iteration and avoiding
smallness parameters, and compare this to the exact solutions. Additionally,
we discuss the problem of projection of the Boltzmann equation onto the
approximate hydrodynamic invariant manifold using entropy concepts. Finally, a
set of hypotheses is put forward where we describe open questions and set a
horizon for what can be derived exactly or proven about the hydrodynamic
manifolds for the Boltzmann equation in the future.
###### 2010 Mathematics Subject Classification:
76P05, 82B40, 35Q35
###### Contents
1. 1 Introduction
1. 1.1 Hilbert’s 6th Problem
2. 1.2 The main equations
3. 1.3 Singular perturbation and separation of times in kinetics
4. 1.4 The structure of the paper
2. 2 Invariance equation and Chapman–Enskog expansion
1. 2.1 The idea of invariant manifold in kinetics
2. 2.2 The Chapman–Enskog expansion
3. 2.3 Euler, Navier–Stokes, Burnett, and super–Burnett terms for a simple kinetic equation
3. 3 Algebraic hydrodynamic invariant manifolds and exact summation of the Chapman–Enskog series for the simplest kinetic model
1. 3.1 Grin of the vanishing cat: $\epsilon$=1
2. 3.2 The pseudodifferential form of the stress tensor
3. 3.3 The energy formula and ‘capillarity’ of ideal gas
4. 3.4 Algebraic invariant manifold in Fourier representation
5. 3.5 Stability of the exact hydrodynamic system and saturation of dissipation for short waves
6. 3.6 Expansion at $k^{2}=\infty$ and matched asymptotics
4. 4 Algebraic invariant manifold for general linear kinetics in 1D
1. 4.1 General form of the invariance equation for 1D linear kinetics
2. 4.2 Hyperbolicity of exact hydrodynamics
3. 4.3 Destruction of hydrodynamic invariant manifold for short waves in the moment equations
4. 4.4 Invariant manifolds, entanglement of hydrodynamic and non-hydrodynamic modes and saturation of dissipation for the 3D 13 moments Grad system
5. 4.5 Algebraic hydrodynamic invariant manifold for the linearized Boltzmann and BGK equations: separation of hydrodynamic and non-hydrodynamic modes
5. 5 Hydrodynamic invariant manifolds for nonlinear kinetics
1. 5.1 1D nonlinear Grad equation and nonlinear viscosity
2. 5.2 Approximate invariant manifold for the Boltzmann equation
1. 5.2.1 Invariance equation
2. 5.2.2 Invariance correction to the local Maxwellian
3. 5.2.3 Micro-local techniques for the invariance equation
6. 6 The projection problem and the entropy equation
7. 7 Conclusion
## 1\. Introduction
### 1.1. Hilbert’s 6th Problem
The 6th Problem differs significantly from the other 22 Hilbert’s problems
[77]. The title of the problem itself is mysterious: “Mathematical treatment
of the axioms of physics”. Physics, in its essence, is a special activity for
the creation, validation and destruction of theories for real-world phenomena,
where “We are trying to prove ourselves wrong as quickly as possible, because
only in that way can we find progress” [38]. There exist no mathematical tools
to formalize relations between Theory and Reality in live Physics. Therefore
the 6th Problem may be viewed as a tremendous challenge in deep study of ideas
of physical reality in order to replace vague philosophy by a new logical and
mathematical discipline. Some research in quantum observation theory and
related topics can be viewed as steps in that direction, but it seems that at
present we are far from an understanding of the most logical and mathematical
problems here.
The first explanation of the 6th Problem given by Hilbert reduced the level of
challenge and made the problem more tractable: “The investigations on the
foundations of geometry suggest the problem: To treat in the same manner, by
means of axioms, those physical sciences in which mathematics plays an
important part; in the first rank are the theory of probabilities and
mechanics”. This is definitely “a programmatic call” [23] for the
axiomatization of the formal parts of existent physical theories and no new
universal logical framework for the representation of reality is necessary. In
this context, the axiomatic approach is a tool for the retrospective analysis
of well-established and elaborated physical theories [23] and not for live
physics.
For the general statements of the 6th Problem it seems unclear now how to
formulate criteria of solutions. In a further explanation Hilbert proposed two
specific problems: (i) axiomatic treatment of probability with limit theorems
for foundation of statistical physics and (ii) the rigorous theory of limiting
processes “which lead from the atomistic view to the laws of motion of
continua”. For complete resolution of these problems Hilbert has set no
criteria either but some important parts of them have been already claimed as
solved. Several axiomatic approaches to probability have been developed and
the equivalence of some of them has been proven [45]. Kolmogorov’s axiomatics
(1933) [97] is now accepted as standard. Thirty years later, the complexity
approach to randomness was invented by Solomonoff and Kolmogorov (see the
review [149] and the textbook [108]). The rigorous foundation of equilibrium
statistical physics of many particles based on the central limit theorems was
proposed [96, 30]. The modern development of the limit theorems in high
dimensions is based on the geometrical ideas of the measure concentration
effects [73, 138] and gives new insights into the foundation of statistical
physics (see, for example, [47, 139]). Despite many open questions, this part
of the Hilbert programme is essentially fulfilled – the probability theory and
the foundations of equilibrium statistical physics are now well-established
chapters of mathematics.
The way from the “atomistic view to the laws of motion of continua” is not so
well formalized. It includes at least two steps: (i) from mechanics to
kinetics (from Newton to Boltzmann) and (ii) from kinetics to mechanics and
non-equilibrium thermodynamics of continua (from Boltzmann to Euler and
Navier–Stokes–Fourier).
The first part of the problem, the transition from the reversible–in–time
equations of mechanics to irreversible kinetic equations, is still too far
from a complete rigorous theory. The highest achievement here is the proof
that rarefied gas of hard spheres will follow the Boltzmann equation during a
fraction of the collision time, starting from a non-correlated initial state
[105, 43]. The BBGKY hierarchy [13] provides the general framework for this
problem. For the systems close to global thermodynamic equilibrium the global
in time estimates are available and the validity of the linearized Boltzmann
equation is proven recently in this limit for rarefied gas of hard spheres
[12].
The second part, model reduction in dissipative systems, from kinetics to
macroscopic dynamics, is ready for a mathematical treatment. Some limit
theorems about this model reduction are already proven (see the review book
[127] and the companion paper by L. Saint-Raymond [128] in this volume), and
open questions can be presented in a rigorous mathematical form. Our review is
focused on this model reduction problem, which is important in many areas of
kinetics, from the Boltzmann equation to chemical kinetics. There exist many
similar heuristic approaches for different applications [60, 113, 125, 130].
It seems that Hilbert presumed the kinetic level of description (the
“Boltzmann level”) as an intermediate step between the microscopic mechanical
description and the continuum mechanics. Nevertheless, this intermediate
description may be omitted. The transition from the microscopic to the
macroscopic description without an intermediate kinetic equation is used in
many physical theories like the Green–Kubo formalism [101], the Zubarev method
of a nonequilibrium statistical operator [148], and the projection operator
techniques [68]. This possibility is demonstrated rigorously for a rarefied
gas near global equilibrium [12].
The reduction from the Boltzmann kinetics to hydrodynamics may be split into
three problems: existence of hydrodynamics, the form of the hydrodynamic
equations and the relaxation of the Boltzmann kinetics to hydrodynamics.
Formalization of these problems is a crucial step in the analysis.
Three questions arise:
1. (1)
Is there hydrodynamics in the kinetic equation, i.e., is it possible to lift
the hydrodynamic fields to the relevant one-particle distribution functions in
such a way that the projection of the kinetics of the relevant distributions
satisfies some hydrodynamic equations?
2. (2)
Do these hydrodynamics have the conventional Euler and Navier–Stokes–Fourier
form?
3. (3)
Do the solutions of the kinetic equation degenerate to the hydrodynamic regime
(after some transient period)?
The first question is the problem of existence of a hydrodynamic invariant
manifold for kinetics (this manifold should be parameterized by the
hydrodynamic fields). The second one is about the form of the hydrodynamic
equations obtained by the natural projection of kinetic equations from the
invariant manifold. The third question is about the intermediate asymptotics
of the relaxation of kinetics to equilibrium: do the solutions go fast to the
hydrodynamic invariant manifold and then follow this manifold on the path to
equilibrium?
The answer to all three questions is essentially positive in the asymptotic
regime when the Mach number $M\\!a$ and the Knudsen number $K\\!n$ tend to
zero [6, 46] (see [127, 128]). This is a limit of very slow flows with very
small gradients of all fields, i.e. almost no flow at all. Such a flow changes
in time very slowly and a rescaling of time $t_{\rm old}=t_{\rm
new}/\varepsilon$ is needed to return it to non-trivial dynamics (the so-
called diffusive rescaling). After the rescaling, we approach in this limit
the Euler and Navier–Stokes–Fourier hydrodynamics of incompressible liquids.
Thus in the limit $M\\!a,K\\!n\to 0$ and after rescaling the 6th Hilbert
Problem is essentially resolved and the result meets Hilbert’s expectations:
the continuum equations are rigorously derived from the Boltzmann equation.
Besides the limit the answers are known partially. To the best of our
knowledge, now the answers to these three questions are: (1) sometimes; (2)
not always; (3) possibly.
Some hints about the problems with hydrodynamic asymptotics can be found in
the series of works about the small dispersion limit of the Korteweg–de Vries
equation [106]. Recently, analysis of the exact solution of the model
reduction problem for a simple kinetic model [57, 136] has demonstrated that a
hydrodynamic invariant manifold may exist and produce non-local hydrodynamics.
Analysis of more complicated kinetics [91, 87, 88, 19, 20] supports and
extends these observations: the hydrodynamic invariant manifold may exist but
sometimes does not exist, and the hydrodynamic equations when
$M\\!a\nrightarrow 0$ may differ essentially from the Euler and
Navier–Stokes–Fourier equations.
At least two effects prevent us from giving positive answers to the first two
questions outside of the limit $M\\!a,K\\!n\to 0$:
* •
Entanglement between the hydrodynamic and non-hydrodynamic modes may destroy
the hydrodynamic invariant manifold.
* •
Saturation of dissipation at high frequencies is a universal effect that is
impossible in the classical hydrodynamic equations.
These effects appear already in simple linear kinetic models and are studied
in detail for the exactly solvable reduction problems. The entanglement
between the hydrodynamic and non-hydrodynamic modes manifests itself in many
popular moment approximations for the Boltzmann equation. In particular, it
exists for the three-dimensional 10-moment and 13-moment Grad systems [87, 91,
60, 19, 20] but the numerical study of the hydrodynamic invariant manifolds
for the BGK model equation [88] demonstrates the absence of such an
entanglement. Therefore, our conjecture is that for the Boltzmann equation the
exact hydrodynamic modes are separated from the non-hydrodynamic ones if the
linearized collision operator has a spectral gap between the five times
degenerated zero and the rest of the spectrum.
The saturation of dissipation seems to be a universal phenomenon [124, 52, 53,
102, 133, 91, 60]. It appears in all exactly solved reduction problems for
kinetic equations [91] and in the Bhatnagar–Gross–Krook [7] (BGK) kinetics
[88] and is also proven for various regularizations of the Chapman–Enskog
expansion [124, 52, 133, 60].
The answer to Hilbert’s 6th Problem concerning transition from the Boltzmann
equation to the classical equations of motion of compressible continua
($M\\!a\nrightarrow 0$) may turn out to be negative. Even if we can overcome
the first difficulty, separate the hydrodynamic modes from the non-
hydrodynamic ones (as in the exact solution [57] or for the BGK equation [88])
and produce the hydrodynamic equations from the Boltzmann equation, the result
will be manifestly different from the conventional equations of hydrodynamics.
### 1.2. The main equations
We discuss here two groups of examples. The first of them consists of kinetic
equations which describe the evolution of a one-particle gas distribution
function $f(t,\mbox{\boldmath$x$};\mbox{\boldmath$v$})$
$\partial_{t}f+\mbox{\boldmath$v$}\cdot\nabla_{x}f=\frac{1}{\epsilon}Q(f),$
(1.1)
where $Q(f)$ is the collision operator. For the Boltzmann equation, $Q$ is a
quadratic operator and, therefore, the notation $Q(f,f)$ is often used.
The second group of examples are the systems of Grad moment equations [69, 9,
85, 60]. The system of 13-moment Grad equations linearized near equilibrium is
$\begin{split}\partial_{t}\rho&=-\nabla\cdot{\mbox{\boldmath$u$}},\\\
\partial_{t}{\mbox{\boldmath$u$}}&=-\nabla\rho-\nabla
T-\nabla\cdot\mbox{\boldmath$\sigma$},\\\
\partial_{t}T&=-\frac{2}{3}(\nabla\cdot{\mbox{\boldmath$u$}}+\nabla\cdot{\mbox{\boldmath$q$}}),\end{split}$
(1.2)
$\begin{split}\partial_{t}\mbox{\boldmath$\sigma$}&=-2\overline{\nabla{\mbox{\boldmath$u$}}}-\frac{4}{5}\overline{\nabla{\mbox{\boldmath$q$}}}-\frac{1}{\epsilon}\mbox{\boldmath$\sigma$},\\\
\partial_{t}{\mbox{\boldmath$q$}}&=-\frac{5}{2}\nabla
T-\nabla\cdot\mbox{\boldmath$\sigma$}-\frac{2}{3\epsilon}{\mbox{\boldmath$q$}}.\end{split}$
(1.3)
In these equations, $\mbox{\boldmath$\sigma$}({\mbox{\boldmath$x$}},t)$ is the
dimensionless stress tensor, $\mbox{\boldmath$\sigma$}=(\sigma_{ij})$, and
${\mbox{\boldmath$q$}({\mbox{\boldmath$x$}},t)}$ is the dimensionless vector
of heat flux, $\mbox{\boldmath$q$}=(q_{i})$. We use the system of units in
which Boltzmann’s constant $k_{\rm B}$ and the particle mass $m$ are equal to
one, and the system of dimensionless variables:
${\mbox{\boldmath$u$}}=\frac{\delta{\mbox{\boldmath$u$}}}{\sqrt{T_{0}}},\
\rho=\frac{\delta\rho}{\rho_{0}},\ T=\frac{\delta
T}{T_{0}},\mbox{\boldmath$x$}=\frac{\rho_{0}}{\eta(T_{0})\sqrt{T_{0}}}\mbox{\boldmath$x$}^{\prime},\
t=\frac{\rho_{0}}{\eta(T_{0})}t^{\prime},$ (1.4)
where ${\mbox{\boldmath$x$}}^{\prime}$ are spatial coordinates, and
$t^{\prime}$ is time.
The dot denotes the standard scalar product, while the overline indicates the
symmetric traceless part of a tensor. For a tensor
$\mbox{\boldmath$a$}=(a_{ij})$ this part is
$\overline{\mbox{\boldmath$a$}}=\frac{1}{2}(\mbox{\boldmath$a$}+\mbox{\boldmath$a$}^{T})-\frac{1}{3}{I}\mbox{tr}(\mbox{\boldmath$a$}),$
where ${I}$ is unit matrix. In particular,
$\overline{{\nabla}{\mbox{\boldmath$u$}}}=\frac{1}{2}({\nabla}{\mbox{\boldmath$u$}}+({\nabla}{\mbox{\boldmath$u$}})^{T}-\frac{2}{3}{I}{\nabla}\cdot{\mbox{\boldmath$u$}}),$
where $I=(\delta_{ij}$ is the identity matrix.
We also study a simple model of a coupling of the hydrodynamic variables, $u$
and $p$
($p(\mbox{\boldmath$x$},t)=\rho(\mbox{\boldmath$x$},t)+T(\mbox{\boldmath$x$},t)$),
to the non-hydrodynamic variable $\sigma$, the 3D linearized Grad equations
for 10 moments $p$, $u$, and $\sigma$:
$\begin{split}\partial_{t}p&=-\frac{5}{3}\nabla\cdot{\mbox{\boldmath$u$}},\\\
\partial_{t}{\mbox{\boldmath$u$}}&=-\nabla
p-\nabla\cdot\mbox{\boldmath$\sigma$},\\\
\partial_{t}\mbox{\boldmath$\sigma$}&=-2\overline{\nabla{\mbox{\boldmath$u$}}}-\frac{1}{\epsilon}\mbox{\boldmath$\sigma$}.\end{split}$
(1.5)
Here, the coefficient $\frac{5}{3}$ is the adiabatic exponent of the 3D ideal
gas.
The simplest model and the starting point in our analysis is the reduction of
the system (1.5) to the functions that depend on one space coordinate $x$ with
the velocity $u$ oriented along the $x$ axis:
$\begin{split}\partial_{t}p&=-\frac{5}{3}\partial_{x}u,\\\
\partial_{t}u&=-\partial_{x}p-\partial_{x}\sigma,\\\
\partial_{t}\sigma&=-\frac{4}{3}\partial_{x}u-\frac{1}{\epsilon}\sigma,\end{split}$
(1.6)
where $\sigma$ is the dimensionless $xx$-component of the stress tensor and
the equation describes the unidirectional solutions of the previous system
(1.5).
These equations are elements of the staircase of simplifications, from the
Boltzmann equation to moment equations of various complexity, which was
introduced by Grad [69] and elaborated further by many authors. In particular,
Levermore proved hyperbolicity of the properly constructed moment equations
[107]. This staircase forms the basis of the Extended Irreversible
Thermodynamics (EIT [85]).
### 1.3. Singular perturbation and separation of times in kinetics
The kinetic equations are singularly perturbed with a small parameter
$\epsilon$ (the “Knudsen number”) and we are interested in the asymptotic
properties of solutions when $\epsilon$ is small. The physical interpretation
of the Knudsen number is the ratio of the “microscopic lengths” (for example,
the mean free path) to the “macroscopic scale”, where the solution changes
significantly. Therefore, its definition depends on the properties of
solutions. If the space derivatives are uniformly bounded, then we can study
the asymptotic behavior $\epsilon\to 0$. But for some singular solutions this
problem statement may be senseless. The simple illustration of rescaling with
the erasing of $\epsilon$ gives the set of travelling automodel solutions for
(1.1). If we look for them in a form
$f=\varphi(\boldsymbol{\xi},\mbox{\boldmath$v$})$ where
$\boldsymbol{\xi}=(\mbox{\boldmath$x$}-\mbox{\boldmath$c$}t)/\epsilon$ then
the equation for $\varphi(\boldsymbol{\xi},\mbox{\boldmath$v$})$ does not
depend on $\epsilon$:
$(\mbox{\boldmath$v$}-\mbox{\boldmath$c$})\cdot\nabla_{\xi}\phi=Q(\phi).$
In general, $\epsilon$ may be considered as a variable that is neither small
nor large and the problem is to analyze the dependence of solutions on
$\epsilon$.
For the Boltzmann equation (1.1) the collision term $Q(f)$ does not enter
directly into the time derivatives of the hydrodynamic variables, $\rho=\int
f{\mathrm{d}}\mbox{\boldmath$v$}$,
$\mbox{\boldmath$u$}=\int\mbox{\boldmath$v$}f{\mathrm{d}}\mbox{\boldmath$v$}$
and
$T=\int(\mbox{\boldmath$v$}-\mbox{\boldmath$u$})^{2}f{\mathrm{d}}\mbox{\boldmath$v$}$
due to the mass, momentum and energy conservation laws
$\int\\{1;\mbox{\boldmath$v$};(\mbox{\boldmath$v$}-\mbox{\boldmath$u$})^{2}\\}Q(f){\mathrm{d}}\mbox{\boldmath$v$}=0.$
The following dynamical system point of view is valid for smooth solutions in
a bounded region with no-flux and equilibrium boundary conditions, but it is
used with some success much more widely. The collision term is “fast”
(includes the large parameter $1/\varepsilon$) and does not affect the
macroscopic hydrodynamic variables directly. Therefore, the following
qualitative picture is expected for the solutions: (i) the collision term goes
quickly almost to its equilibrium (the system almost approaches a local
equilibrium) and during this fast initial motion the changes of hydrodynamic
variables are small, (ii) after that the distribution function is defined with
high accuracy by the hydrodynamic variables (if they have bounded space
derivatives). The relaxation of the collision term almost to its equilibrium
is supported by monotonic entropy growth (Boltzmann’s $H$-theorem). This
qualitative picture is illustrated in Fig. 1.
Such a “nonrigorous picture of the Boltzmann dynamics” [29] which operates by
the manifolds in the space of probability distributions is a seminal tool for
production of qualitative hypotheses. The points (‘states’) in Fig. 1.
correspond to the distributions $f(\mbox{\boldmath$x$},\mbox{\boldmath$v$})$,
and the points in the projection correspond to the hydrodynamic fields in
space.
Figure 1. Fast–slow decomposition. Bold dashed lines outline the vicinity of
the slow manifold where the solutions stay after initial layer. The projection
of the distributions onto the hydrodynamic fields and the parametrization of
this manifold by the hydrodynamic fields are represented.
For the Grad equations (1.2)-(1.3), (1.5) and (1.6) the hydrodynamic variables
$\rho,\mbox{\boldmath$u$},T$ are explicitly separated from the fluxes and the
projection onto the hydrodynamic fields is just the selection of the
hydrodynamic part of the set of all fields. For example, for (1.6) this is
just the selection of
$p(\mbox{\boldmath$x$}),\mbox{\boldmath$u$}(\mbox{\boldmath$x$})$ from the
whole set of fields
$p(\mbox{\boldmath$x$}),\mbox{\boldmath$u$}(\mbox{\boldmath$x$}),\boldsymbol{\sigma}(\mbox{\boldmath$x$})$.
The expected qualitative picture for smooth solutions is the same as in Fig.
1.
For finite-dimensional ODEs, Fig. 1 represents the systems which satisfy the
Tikhonov singular perturbation theorem [141]. In some formal sense, this
picture for the Boltzmann equation is also rigorous when $\epsilon\to 0$ and
is proven in [6]. Assume that
$f^{\epsilon}(t,\mbox{\boldmath$x$},\mbox{\boldmath$v$})$ is a sequence of
nonnegative solutions of the Boltzmann equation (1.1) when $\epsilon\to 0$ and
there exists a limit
$f^{\epsilon}(t,\mbox{\boldmath$x$},\mbox{\boldmath$v$})\to
f^{0}(t,\mbox{\boldmath$x$},\mbox{\boldmath$v$})$. Then (under some additional
regularity conditions), this limit
$f^{0}(t,\mbox{\boldmath$x$},\mbox{\boldmath$v$})$ is a local Maxwellian and
the corresponding moments satisfy the compressible Euler equation. According
to [127], this is “the easiest of all hydrodynamic limits of the Boltzmann
equation at the formal level”.
The theory of singular perturbations was developed starting from complex
systems, from the Boltzmann equation (Hilbert [78], Enskog [35], Chapman [24],
Grad [69, 70]) to ODEs. The recently developed geometric theory of singular
perturbation [36, 37, 84] can be considered as a formalization of the
Chapman–Enskog approach for the area where complete rigorous theory is
achievable.
A program of the derivation of (weak) solutions of the Navier–Stokes equations
from the (weak) solutions of the Boltzmann equation was formulated in 1991 [6]
and finalized in 2004 [46] with the answer: the incompressible Navier–Stokes
(Navier–Stokes–Fourier) equations appear in a limit of appropriately scaled
solutions of the Boltzmann equation.
We use the geometry of time-separation (Fig. 1) as a guide for formal
constructions and present further development of this scheme using some ideas
from thermodynamics and dynamics.
### 1.4. The structure of the paper
In Sec. 2 we introduce the invariance equation for invariant manifolds. It has
been studied by Lyapunov (Lyapunov’s auxiliary theorem [112], Theorem 2.1
below). We describe the structure of the invariance equations for the
Boltzmann and Grad equations and in Sec. 2.2 construct the Chapman–Enskog
expansion for the solution of the invariance equation.
It may be worth stressing that the invariance equation is a nonlinear equation
and there is no known general method to solve them even for linear
differential equations. The main construction is illustrated on the simplest
kinetic equation (1.6): in Sec. 2.3 the Euler, Navier–Stokes, Burnett, and
super–Burnett terms are calculated for this equation and the “ultraviolet
catastrophe” of the Chapman–Enskog series is demonstrated (Fig. 3).
The first example of the exact summation of the Chapman–Enskog series is
presented in detail for the simplest system (1.6) in Sec. 3. We analyze the
structure of the Chapman–Enskog series and find the pseudodifferential
representation of the stress tensor on the hydrodynamic invariant manifold.
Using this representation, in Sec. 3.3 we represent the energy balance
equation in the “capillarity–viscosity” form proposed by Slemrod [136]. This
form explains the macroscopic sense of the dissipation saturation effect: the
attenuation rate does not depend on the wave vector $k$ for short waves (it
tends to a constant value when $k^{2}\to\infty$). In the highly non-
equilibrium gas the capillarity energy becomes significant and it tends to
infinity for high velocity gradients.
In the Fourier representation, the invariance equation for (1.6) is a system
of two coupled quadratic equations with linear in $k^{2}$ coefficients (Sec.
3.4). It can be solved in radicals and the corresponding hydrodynamics has the
acoustic waves decay with saturation (Sec. 3.5). The hydrodynamic invariant
manifold for (1.6) is analytic at the infinitely-distant point $k^{2}=\infty$.
Matching of the first terms of the Taylor series in powers of $1/k^{2}$ with
the first terms of the Chapman–Enskog series gives simpler hydrodynamic
equations with qualitatively the same effects and even quantitatively the same
saturation level of attenuation of acoustic waves (Sec. 3.6). We may guess
that the matched asymptotics of this type include all the essential
information about hydrodynamics both at low and high frequencies.
The construction of the invariance equations in the Fourier representation
remains the same for a general linear kinetic equation (Sec. 4.1). The exact
hydrodynamics on the invariant manifolds always inherits many important
properties of the original kinetics, such as dissipation and conservation
laws. In particular, if the original kinetic system is hyperbolic then for
bounded hydrodynamic invariant manifolds the hydrodynamic equations are also
hyperbolic (Sec. 4.2).
In Sec. 4, we study the invariance equations for three systems: 1D solutions
of the 13 moment Grad system (Sec. 4.3), the full 3D 13 moment Grad system
(Sec. 4.4), and the linearized BGK kinetic equation (Sec. 4.5). The 13 moment
Grad system demonstrates an important effect: the invariance equation may lose
the physically meaningful solution for short waves. Therefore, existence of
the exact hydrodynamic manifold is not compulsory for all the usual kinetic
equations. Nevertheless, for the BGK equation with the complete advection
operator $\mbox{\boldmath$v$}\cdot\nabla$ the invariance equation exists for
short waves too (as is demonstrated numerically in [88]).
For nonlinear kinetics, the exact solutions to the invariance equations are
not known. In Sec. 5 we demonstrate two approaches to approximate invariant
manifolds. First, for the nonlinear Grad equation we find the leading terms of
the Chapman–Enskog series in the order of the Mach number and exactly sum
them. For this purpose, we construct the approximate invariant manifold and
find the solution for the nonlinear viscosity in the form of an ODE (Sec.
5.1). For the 1D solutions of the Boltzmann equation we construct the
invariance equation and demonstrate the result of the first Newton–Kantorovich
iteration for the solution of this equation (Sec. 5.2 and [53, 60]). Use of
the approximate invariant manifolds causes a problem of dissipativity
preservation in the hydrodynamics on these manifolds. There exists a unique
modification of the projection operator that guarantees the preservation of
entropy production for hydrodynamics produced by projection of kinetics onto
an approximate invariant manifold even for rough approximations [59]. This
construction is presented in Sec. 6. In Conclusion, we discuss solved and
unsolved problems and formulate several hypotheses.
## 2\. Invariance equation and Chapman–Enskog expansion
### 2.1. The idea of invariant manifold in kinetics
Very often, the Chapman–Enskog expansion for the Boltzmann equation is
introduced as an asymptotic expansion in powers of $\epsilon$ of the solutions
of equation (1.1), which should depend on time only through time dependence of
the macroscopic hydrodynamic fields. Historically, the definition of the
method is “procedure oriented”: an expansion is created step by step with the
leading idea that solutions should depend on time only through the macroscopic
variables and their derivatives. In this approach what we are looking for
often remains hidden.
The result of the Chapman–Enskog method is not a solution of the kinetic
equation but rather the proper parametrization of microscopic variables
(distribution functions) by the macroscopic (hydrodynamic) fields. It is a
lifting procedure: we take the hydrodynamic fields and find for them the
corresponding distribution function. This lifting should be consistent with
the kinetics, i.e. the set of the corresponding distributions (collected for
all possible hydrodynamic fields) should be invariant with respect to a shift
in time. Therefore, the Chapman–Enskog procedure looks for an invariant
manifold for the kinetic equation which is close to the local equilibrium for
a small Knudsen number and smooth hydrodynamic fields with bounded
derivatives. This is the “object oriented” description of the Chapman–Enskog
procedure.
The puzzle in the statement of the problem of transition from kinetics to
hydrodynamics has been so deep that Uhlenbeck called it the “Hilbert paradox”
[143]. In the reduced hydrodynamic description, the state of a gas is
completely determined if one knows initially the space dependence of the five
macroscopic variables $p$, $u$, and $T$. Uhlenbeck has found this impossible:
“On the one hand it couldn’t be true, because the initial-value problem for
the Boltzmann equation (which supposedly gives a better description of the
state of the gas) requires the knowledge of the initial value of the
distribution function $f(\mbox{\boldmath$r$},\mbox{\boldmath$v$},t)$ of which
$p$, $u$, and $T$ are only the first five moments in v. But on the other hand
the hydrodynamical equations surely give a causal description of the motion of
a fluid. Otherwise how could fluid mechanics be used?”
Perhaps, McKean gave the first clear explanation of the problem as a
construction of a ‘nice submanifold’ where ‘the hydrodynamical equations
define the same flow as the (more complicated) Boltzmann equation does’ [115].
He presented the problem by a partially commutative diagram and we use this
idea in slightly revised form in Fig. 2.
Figure 2. McKean diagram. The Chapman–Enskog procedure aims to create a
lifting operation, from the hydrodynamic variables to the corresponding
distributions on the invariant manifold. IM stands for Invariant Manifold. The
part of the diagram in the dashed polygon is commutative.
The invariance equation just expresses the fact that the vector field is
tangent to the manifold. The invariance equation has the simplest form for
manifolds parameterized by moments, i.e. by the values of the given linear
functionals. Let us consider an equation in a domain $U$ of a normed space $E$
with analytical (at least, Gateaux-analytical) right hand sides
$\partial_{t}f=J(f).$ (2.1)
A space of macroscopic variables (moment fields) is defined with a surjective
linear map to them $m:f\mapsto M$ ($M$ are macroscopic variables). Below when
referring to a manifold parameterized with the macroscopic fields $M$ we use
the notation $\mbox{\boldmath$f$}_{M}$. We are looking for an invariant
manifold $\mbox{\boldmath$f$}_{M}$ parameterized by the value of $M$, with the
self-consistency condition $m(\mbox{\boldmath$f$}_{M})=M$.
The invariance equation is
$\boxed{J(\mbox{\boldmath$f$}_{M})=(D_{M}\mbox{\boldmath$f$}_{M})m(J(\mbox{\boldmath$f$}_{M})).}$
(2.2)
Here, the differential $D_{M}$ of $\mbox{\boldmath$f$}_{M}$ is calculated at
the point $M=m(\mbox{\boldmath$f$}_{M})$.
Equation (2.2) means that the time derivative of $f$ on the manifold
$\mbox{\boldmath$f$}_{M}$ can be calculated by a simple chain rule: calculate
the derivative of $M$ using the map $m$,
$\dot{M}=m(J(\mbox{\boldmath$f$}_{M}))$, and then write that the time
dependence of $f$ can be expressed through the time dependence of $M$. If we
find the approximate solution to eq. (2.2) then the approximate reduced model
(hydrodynamics) is
$\partial_{t}M=m(J(\mbox{\boldmath$f$}_{M})).$ (2.3)
The invariance equation can be represented in the form
$\partial^{\rm micro}_{t}\mbox{\boldmath$f$}_{M}=\partial^{\rm
macro}_{t}\mbox{\boldmath$f$}_{M},$
where the microscopic time derivative, $\partial^{\rm
micro}_{t}\mbox{\boldmath$f$}_{M}$ is just a value of the vector field
$J(\mbox{\boldmath$f$}_{M})$ and the macroscopic time derivative is calculated
by the chain rule,
$\partial^{\rm
macro}_{t}\mbox{\boldmath$f$}_{M}=(D_{M}\mbox{\boldmath$f$}_{M})\partial_{t}M$
under the assumption that dynamics of $M$ follows the projected equation
(2.3).
We use the natural (and naive) moment-based projection (2.3) till Sec. 6 where
we demonstrate that in many situations the modified projectors are more
suitable from thermodynamic point of view. In addition, the flexible choice of
projectors allows us to treat various nonlinear functionals (like scattering
rates) as macroscopic variables [56, 65].
If $\mbox{\boldmath$f$}_{M}$ is a solution to the invariance equation (2.2)
then the reduced model (2.3) has two important properties:
* •
Preservation of conservation laws. If a differentiable functional $U(f)$ is
conserved due to the initial kinetic equation (2.1) then the functional
$U_{M}=U(\mbox{\boldmath$f$}_{M})$ conserves due to reduced system (2.3), i.e.
it has zero time derivative due to this system.
* •
Preservation of dissipation. If the time derivative of a differentiable
functional $H(f)$ is non-positive due to the initial kinetic equation, then
the time derivative of the functional $H_{M}=H(\mbox{\boldmath$f$}_{M})$ is
also non-positive due to reduced system.
These elementary properties are the obvious consequences of the invariance
equation (2.2) and the chain rule for differentiation. Indeed, for every
differentiable functional $S(f)$ we introduce a functional
$S_{M}=S(\mbox{\boldmath$f$}_{M})$. Then for the time derivative of $S_{M}$
due to projected equation (2.3) coincides with the time derivative of $S(f)$
at point $f=\mbox{\boldmath$f$}_{M}$ due to (2.1). (Preservation of time
derivatives.) Despite the very elementary character of these properties, they
may be extremely important in the construction of the energy and entropy
formulas for the projected equations (2.3) and in the proof of the $H$-theorem
and hyperbolicity.
The difficulties with preservation of conservation laws and dissipation
inequalities may occur when one uses the approximate solutions of the
invariance equation. For these situations, two techniques are invented:
modification of the projection operation (see [51, 59] and Sec. 6 below) and
modification of the entropy functional [72, 71]. They allow to retain the
dissipation inequality for the approximate equations.
It is obvious that the invariance equation (2.2) for dynamical systems usually
has too many solutions, at least locally, in a vicinity of any non-singular
point. For example, every trajectory of (2.1) is a 1D invariant manifold and
if a manifold $\mathcal{L}$ is transversal to a vector field $J$ then the
trajectory of $\mathcal{L}$ is invariant.
Lyapunov used the analyticity of the invariant manifold for finite-dimensional
analytic vector fields $J$ to prove its existence and uniqueness near a fixed
point $\mbox{\boldmath$f$}_{0}$ if $\ker m$ is a invariant subspace of the
Jacobian $(DJ)_{0}$ of $J$ at this point and under some “no resonance”
conditions (the Lyapunov auxiliary theorem [112]). Under these conditions,
there exist many smooth non-analytical manifolds, but the analytical one is
unique.
###### Theorem 2.1 (Lyapunov auxiliary theorem)
Let $\ker m$ have a $(DJ)_{0}$-invariant supplement $(\ker m)^{\prime}$,
$E=\ker m\oplus(\ker m)^{\prime}$. Assume that the restriction $(DJ)_{0}$ onto
$\ker m$ has the spectrum $\kappa_{1},\ldots,\kappa_{j}$ and the restriction
of this operator on the supplement $(\ker m)^{\prime}$ has the spectrum
$\lambda_{1},\ldots,\lambda_{l}$. Let the two following conditions hold:
1. (1)
$0\notin{\rm conv}\\{\kappa_{1},\ldots,\kappa_{j}\\}$;
2. (2)
The spectra $\\{\kappa_{1},\ldots,\kappa_{j}\\}$ and
$\\{\lambda_{1},\ldots,\lambda_{l}\\}$ are not related by any equation of the
form
$\sum_{i}n_{i}\kappa_{i}=\lambda_{k}$
with integer $n_{i}$.
Then there exists a unique analytic solution $\mbox{\boldmath$f$}_{M}$ of the
invariance equation (2.2) with condition
$\mbox{\boldmath$f$}_{M}=\mbox{\boldmath$f$}_{0}$ for
$M=m(\mbox{\boldmath$f$}_{0})$, and in a sufficiently small vicinity of
$m(\mbox{\boldmath$f$}_{0})$.
This solution is tangent to $(\ker m)^{\prime}$ at point
$\mbox{\boldmath$f$}_{0}$.
Recently, the approach to invariant manifolds based on the invariance equation
in combination with the Lyapunov auxiliary theorem were used for the reduction
of kinetic systems [93, 94, 95].
### 2.2. The Chapman–Enskog expansion
The Chapman–Enskog and geometric singular perturbation approach assume the
special singularly perturbed structure of the equations and look for the
invariant manifold in a form of the series in the powers of a small parameter
$\epsilon$. A one-parametric system of equations is considered:
$\partial_{t}f+A(f)=\frac{1}{\epsilon}Q(f).$ (2.4)
The following assumptions connect the macroscopic variables to the singular
perturbation:
* •
$m(Q(f))=0$;
* •
for each $M\in m(U)$ the system of equations
$Q(f)=0,\;\;m(f)=M$
has a unique solution $\mbox{\boldmath$f$}^{\rm eq}_{M}$ (in Boltzmann
kinetics it is the local Maxwellian);
* •
$\mbox{\boldmath$f$}^{\rm eq}_{M}$ is asymptotically stable and globally
attracting for the fast system
$\partial_{t}f=\frac{1}{\epsilon}Q(f)$
in $(\mbox{\boldmath$f$}^{\rm eq}_{M}+\ker m)\cap U$.
Let the differential of the fast vector field $Q(f)$ at equilibrium
$\mbox{\boldmath$f$}^{\rm eq}_{M}$ be $\mathcal{Q}_{M}$. For the
Chapman–Enskog method it is important that $\mathcal{Q}_{M}$ is invertible in
$\ker m$. For the classical kinetic equations this assumption can be checked
using the symmetry of $\mathcal{Q}_{M}$ with respect to the entropic inner
product (Onsager’s reciprocal relations).
The invariance equation for the singularly perturbed system (2.4) with the
moment parametrization $m$ is:
$\boxed{\frac{1}{\epsilon}Q(\mbox{\boldmath$f$}_{M})=A(\mbox{\boldmath$f$}_{M})-(D_{M}\mbox{\boldmath$f$}_{M})(m(A(\mbox{\boldmath$f$}_{M}))).}$
(2.5)
The fast vector field vanishes on the right hand side of this equation because
$m(Q(\mbox{\boldmath$f$}))=0$. The self-consistency condition
$m(\mbox{\boldmath$f$}_{M})=M$ gives
$m(D_{M}\mbox{\boldmath$f$}_{M})m(J)=m(J)$
for all $J$, hence,
$m[A(\mbox{\boldmath$f$}_{M})-(D_{M}\mbox{\boldmath$f$}_{M})m(A(\mbox{\boldmath$f$}_{M}))]=0.$
(2.6)
If we find an approximate solution of (2.5) then the corresponding macroscopic
(hydrodynamic) equation (2.3) is
$\partial_{t}M+m(A(\mbox{\boldmath$f$}_{M}))=0.$ (2.7)
Let us represent all the operators in (2.5) by the Taylor series (for the
Boltzmann equation $A$ is the linear free flight operator, $A=v\cdot\nabla$,
and $Q$ is the quadratic collision operator). We look for the invariant
manifold in the form of the power series:
$\mbox{\boldmath$f$}_{M}=\mbox{\boldmath$f$}^{\rm
eq}_{M}+\sum_{i=1}^{\infty}\epsilon^{i}\mbox{\boldmath$f$}^{(i)}_{M}$ (2.8)
with the self-consistency condition $m(\mbox{\boldmath$f$}_{M})=M$, which
implies $m(\mbox{\boldmath$f$}^{\rm eq}_{M})=M$,
$m(\mbox{\boldmath$f$}^{(i)}_{M})=0$ for $i\geq 1$. After matching the
coefficients of the series in (2.5), we obtain for every
$\mbox{\boldmath$f$}^{(i)}_{M}$ a linear equation
$\mathcal{Q}_{M}\mbox{\boldmath$f$}^{(i)}_{M}=P^{(i)}(\mbox{\boldmath$f$}^{\rm
eq}_{M},\mbox{\boldmath$f$}^{(1)}_{M},\ldots,\mbox{\boldmath$f$}^{(i-1)}_{M}),$
(2.9)
where the polynomial operator $P^{(i)}$ at each order $i$ can be obtained by
straightforward calculations from (2.5). Due to the self-consistency,
$m(P^{(i)})=0$ for all $i$ and the equation (2.9) is solvable. The first term
of the Chapman–Enskog expansion has a simple form
$\boxed{\mbox{\boldmath$f$}^{(1)}_{M}=\mathcal{Q}_{M}^{-1}(1-(D_{M}\mbox{\boldmath$f$}^{\rm
eq}_{M})m)(A(\mbox{\boldmath$f$}^{\rm eq}_{M})).}$ (2.10)
A detailed analysis of explicit versions of this formula for the Boltzmann
equation and other kinetic equations is presented in many books and papers
[24, 79]. Most of the physical applications of kinetic theory, from the
transport processes in gases to modern numerical methods (lattice Boltzmann
models [137]) give examples of the practical applications and deciphering of
this formula. For the Boltzmann kinetics, the zero-order approximation,
$\mbox{\boldmath$f$}^{(0)}_{M}\approx\mbox{\boldmath$f$}^{\rm eq}_{M}$
produces in projection on the hydrodynamic fields (2.7) the compressible Euler
equation. The first-order approximate invariant manifold,
$\mbox{\boldmath$f$}^{(1)}_{M}\approx\mbox{\boldmath$f$}^{\rm
eq}_{M}+\epsilon\mbox{\boldmath$f$}^{(1)}_{M}$, gives the compressible Navier-
Stokes equation and provides the explicit dependence of the transport
coefficients from the collision model. This bridge from the “atomistic view to
the laws of motion of continua” is, in some sense, the main result of the
Boltzmann kinetics and follows precisely Hilbert’s request but not as
rigorously as it is desired.
The calculation of higher order terms needs nothing but differentiation and
calculation of the inverse operator $\mathcal{Q}_{M}^{-1}$. (Nevertheless
these calculations may be very bulky and one of the creators of the method, S.
Chapman, compared reading his book [24] to “chewing glass”, cited by [15]).
Differentiability is needed also because the transport operator $A$ should be
bounded to provide strong sense to the manipulations (see the discussion in
[128]). The second order in $\epsilon$ hydrodynamic equations (2.3) are called
Burnett equations (with $\epsilon^{2}$ terms) and super-Burnett equations for
higher orders.
### 2.3. Euler, Navier–Stokes, Burnett, and super–Burnett terms for a simple
kinetic equation
Let us illustrate the basic construction on the simplest example (1.6).
$\displaystyle\mbox{\boldmath$f$}=\left(\begin{array}[]{c}p(x)\\\ u(x)\\\
\sigma(x)\end{array}\right),\;m=\left(\begin{array}[]{ccc}1&0&0\\\
0&1&0\end{array}\right),\;M=\left(\begin{array}[]{c}p(x)\\\
u(x)\end{array}\right),\;\ker m=\left\\{\left(\begin{array}[]{c}0\\\ 0\\\
y\end{array}\right)\right\\},$ $\displaystyle
A(\mbox{\boldmath$f$})=\left(\begin{array}[]{c}\frac{5}{3}\partial_{x}u\\\
\partial_{x}p+\partial_{x}\sigma\\\
\frac{4}{3}\partial_{x}u\end{array}\right),\;Q(\mbox{\boldmath$f$})=\left(\begin{array}[]{c}0\\\
0\\\ -\sigma\end{array}\right),\;\mathcal{Q}_{M}^{-1}=\mathcal{Q}_{M}=-1\mbox{
on }\ker m,$ $\displaystyle\mbox{\boldmath$f$}^{\rm
eq}_{M}=\left(\begin{array}[]{c}p(x)\\\ u(x)\\\
0\end{array}\right),\;D_{M}\mbox{\boldmath$f$}^{\rm
eq}_{M}=\left(\begin{array}[]{cc}1&0\\\ 0&1\\\
0&0\end{array}\right),\;\mbox{\boldmath$f$}^{(1)}_{M}=\left(\begin{array}[]{c}0\\\
0\\\ -\frac{4}{3}\partial_{x}u\end{array}\right).\;$
We hasten to remark that (1.6) is a simple linear system and can be integrated
immediately in explicit form. However, that solution contains both the fast
and slow components and it does not readily reveal the slow hydrodynamic
manifold of the system. Instead, we are interested in extracting this slow
manifold by a direct method. The Chapman-Enskog expansion is thus the tool for
this extracting which we shall address first.
The projected equations in the zeroth (Euler) and the first (Navier–Stokes)
order of $\epsilon$ are
$\mbox{ (Euler)
}\begin{array}[]{ll}\partial_{t}p=-\frac{5}{3}\partial_{x}u,\\\
\partial_{t}u=-\partial_{x}p;\end{array}\;\;\;\mbox{ (Navier-Stokes)
}\begin{array}[]{ll}\partial_{t}p=-\frac{5}{3}\partial_{x}u,\\\
\partial_{t}u=-\partial_{x}p+\epsilon\frac{4}{3}\partial_{x}^{2}u.\end{array}$
It is straightforward to calculate the two next terms (Burnett and super-
Burnett ones) but let us introduce convenient notations to represent the whole
Chapman-Enskog series for (1.6). Only the third component of the invariance
equation (2.5) for (1.6) is non-trivial because of self-consistency condition
(2.6). and we can write
$-\frac{1}{\epsilon}\sigma_{(p,u)}=\frac{4}{3}\partial_{x}u-\frac{5}{3}(D_{p}\sigma_{(p,u)})(\partial_{x}u)-(D_{u}\sigma_{(p,u)})(\partial_{x}p+\partial_{x}\sigma_{(p,u)}).$
(2.14)
Here, $M={(p,u)}$ and the differentials are calculated by the elementary rule:
if a function $\Phi$ depends on values of $p(x)$ and its derivatives,
$\Phi=\Phi(p,\partial_{x}p,\partial^{2}_{x}p,\ldots)$ then $D_{p}\Phi$ is a
differential operator,
$D_{p}\Phi=\frac{\partial\Phi}{\partial
p}+\frac{\partial\Phi}{\partial(\partial_{x}p)}\partial_{x}+\frac{\partial\Phi}{\partial(\partial_{x}^{2}p)}\partial^{2}_{x}+\ldots$
The equilibrium of the fast system (the Euler approximation) is known,
$\sigma_{(p,u)}^{(0)}=0$. We have already found
$\sigma_{(p,u)}^{(1)}=-\frac{4}{3}\partial_{x}u$ (the Navier–Stokes
approximation). In each order of the Chapman–Enskog expansion $i\geq 1$ we get
from (2.14):
$\sigma_{(p,u)}^{(i+1)}=\frac{5}{3}(D_{p}\sigma_{(p,u)}^{(i)})(\partial_{x}u)+(D_{u}\sigma_{(p,u)}^{(i)})(\partial_{x}p)+\sum_{j+l=i}(D_{u}\sigma_{(p,u)}^{(j)})(\partial_{x}\sigma_{(p,u)}^{(l)})$
(2.15)
This chain of equations is nonlinear but every $\sigma_{(p,u)}^{(i+1)}$ is a
linear function of derivatives of $u$ and $p$ with constant coefficients
because this sequence starts from $-\frac{4}{3}\partial_{x}u$ and the
induction step in $i$ is obvious. Let $\sigma_{(p,u)}^{(i)}$ be a linear
function of derivatives of $u$ and $p$ with constant coefficients. Then its
differentials $D_{p}\sigma_{(p,u)}^{(i)}$ and $D_{u}\sigma_{(p,u)}^{(i)}$ are
linear differential operators with constant coefficients and all terms in
(2.15) are again linear functions of derivatives of $u$ and $p$ with constant
coefficients.
For $\sigma_{(p,u)}^{(2)}$ ($i+1=2$) the operators in the right hand part of
(2.15) are: $(D_{p}\sigma_{(p,u)}^{(1)})\allowbreak=0$,
$(D_{u}\sigma_{(p,u)}^{(1)})=-\frac{4}{3}\partial_{x}$, and in the third term
in each summand either $l=0$, $j=1$ or $l=1$, $j=0$. Therefore, for the
Burnett term,
$\sigma_{(p,u)}^{(2)}=-\frac{4}{3}\partial^{2}_{x}p.$
For the super Burnett term in $\sigma_{(p,u)}^{(3)}$ ($i+1=3$) the operators
in the right hand part of (2.15) are
$(D_{p}\sigma_{(p,u)}^{(2)})=-\frac{4}{3}\partial^{2}_{x}$,
$(D_{u}\sigma_{(p,u)}^{(2)})=0$ and in the third term, only summand with
$l=j=1$ may take non-zero value:
$(D_{u}\sigma_{(p,u)}^{(1)})(\partial_{x}\sigma_{(p,u)}^{(1)})=(-\frac{4}{3}\partial^{2}_{x})(-\frac{4}{3}\partial_{x}u)=\frac{16}{9}\partial^{3}_{x}u.$
Finally, $\sigma_{(p,u)}^{(3)}=-\frac{4}{9}\partial^{3}_{x}u$ and the
projected equations have the form
$\displaystyle\begin{array}[]{ll}\partial_{t}p=-\frac{5}{3}\partial_{x}u,\\\
\partial_{t}u=-\partial_{x}p+\epsilon\frac{4}{3}\partial_{x}^{2}u+\epsilon^{2}\frac{4}{3}\partial_{x}^{3}p\end{array}\mbox{
(Burnett). }$ (2.18)
$\displaystyle\begin{array}[]{ll}\partial_{t}p=-\frac{5}{3}\partial_{x}u,\\\
\partial_{t}u=-\partial_{x}p+\epsilon\frac{4}{3}\partial_{x}^{2}u+\epsilon^{2}\frac{4}{3}\partial_{x}^{3}p+\epsilon^{3}\frac{4}{9}\partial_{x}^{4}u\end{array}\mbox{
(super Burnett). }$ (2.21)
To see the properties of the resulting equations, we compute the dispersion
relation for the hydrodynamic modes. Using a new space-time scale,
$x^{\prime}=\epsilon^{-1}x$, and $t^{\prime}=\epsilon^{-1}t$, and representing
$u=u_{k}\varphi(x^{\prime},t^{\prime})$, and
$p=p_{k}\varphi(x^{\prime},t^{\prime})$, where
$\varphi(x^{\prime},t^{\prime})=\exp(\omega t^{\prime}+ikx^{\prime})$, and $k$
is a real-valued wave vector, we obtain the following dispersion relations
$\omega(k)$ from the condition of a non-trivial solvability of the
corresponding linear system with respect to $u_{k}$ and $p_{k}$:
$\omega_{\pm}=-\frac{2}{3}k^{2}\pm\frac{1}{3}i|k|\sqrt{15-4k^{2}},$ (2.22)
for the Navier–Stokes approximation,
$\omega_{\pm}=-\frac{2}{3}k^{2}\pm\frac{1}{3}i|k|\sqrt{15+16k^{2}},$ (2.23)
for the Burnett approximation (2.18), and
$\omega_{\pm}=\frac{2}{9}k^{2}(k^{2}-3)\pm\frac{1}{9}i|k|\sqrt{135+144k^{2}+24k^{4}-4k^{6}},$
(2.24)
for the super-Burnett approximation (2.21).
Figure 3. Attenuation rates [91]. Solid: exact summation; diamonds:
hydrodynamic modes of the kinetic equations with $\epsilon=1$ (1.6) (they
match the solid line per construction); circles: the non-hydrodynamic mode of
(1.6), $\epsilon=1$; dash dot line: the Navier–Stokes approximation; dash: the
super–Burnett approximation; dash double dot line: the first Newton’s
iteration (3.19). The result for the second iteration (3.20) is
indistinguishable from the exact solution at this scale.
These examples demonstrate that the real part is non-positive, ${\rm
Re}(\omega_{\pm}(k))\leq 0$ (Fig. 3), for the Navier–Stokes (2.22) and for the
Burnett (2.23) approximations, for all wave vectors. Thus, these
approximations describe attenuating acoustic waves. However, for the super-
Burnett approximation, the function ${\rm Re}(\omega_{\pm}(k))$ (2.24) becomes
positive as soon as $|k|>\sqrt{3}$. The equilibrium is stable within the
Navier–Stokes and the Burnett approximation, and it becomes unstable within
the super-Burnett approximation for sufficiently short waves. Similar to the
case of the Bobylev instability of the Burnett hydrodynamics for the Boltzmann
equation, the latter result contradicts the dissipative properties of the Grad
system (1.6): the spectrum of the kinetic system (1.6) is stable for arbitrary
$k$ (see Fig. 3). For the 13-moment system (1.2)-(1.3) the instability of
short waves appears already in the Burnett approximation [60, 91] (see section
3 below). For the Boltzman equation this effect was discovered by Bobylev [9].
In Fig. 3, we also represent the attenuation rates of the hydrodynamic and
non-hydrodynamic mode of the kinetic equations (1.6). The characteristic
equation of these kinetic equations reads:
$3\omega^{3}+3\omega^{2}+9k^{2}\omega+5k^{2}=0.$ (2.25)
The two complex-conjugate roots of this equation correspond to the
hydrodynamic modes, while for the non-hydrodynamic real mode,
$\omega_{nh}(k)$, $\omega_{nh}(0)=-1$, and $\omega_{nh}\rightarrow-0.5$ as
$|k|\rightarrow\infty$. The non-hydrodynamic modes of the Grad equations are
characterized by the common property that for them $\omega(0)\neq 0$. These
modes are irrelevant to the Chapman–Enskog branch of the invariant manifold.
Thus, the Chapman–Enskog expansion:
* •
Gives excellent, but already known on phenomenological grounds, zero and first
order approximations – the Euler and Navier–Stokes equations;
* •
Provides a bridge from microscopic models of collisions to macroscopic
transport coefficients in the known continuum equations;
* •
Already the next correction, not known phenomenologically and hence of
interest, does not exist because of non-physical behavior.
The first term of the Chapman–Enskog expansion gives the possibility to
evaluate the coefficients in the phenomenological equations (like viscosity,
thermal conductivity and diffusion coefficient) from the microscopic models of
collisions. The success of the first order approximation (2.10) is compatible
with the failure of the higher order terms. The Burnett and Super-Burnett
equations have non-physical properties, negative viscosity for high gradients
and instability for short waves. The Chapman–Enskog expansion has to be
truncated after the first order term or not truncated at all.
Such a situation when the first approximations are useful but the higher terms
become senseless is not very novel. There are at least three famous examples:
* •
The “ultraviolet catastrophe” in higher order terms because of physical
phenomena at very short distances [33] and the perturbation series
divergencies [147] are well known in quantum field theory, and many approaches
have been developed to deal with these singularities [146];
* •
Singularities and divergence in the semiclassical Wentzel-Kramers-Brillouin
(WKB) approach [42, 145, 82];
* •
The small denominators affect the convergence of the Poincaré series in the
classical many body problem and the theory of nearly integrable systems. They
may even make the perturbation series approach senseless [3].
Many ideas have been proposed and implemented to deal with these
singularities: use of the direct iteration method instead of power series in
KAM [98, 2, 3], renormalization [39, 21, 117], summation and partial summation
and rational approximation of the perturbation series [114, 34] and string
theories [144, 28] in quantum field theory [146]. Various ad hoc analytical
and numerical regularization tricks have been proposed too. Exactly solvable
models give the possibility of exhaustive analysis of the solutions. Even in
the situation when they are not applicable directly to reality we can use them
as benchmarks for all perturbation and approximation methods and for
regularization tricks.
We follow this stream of ideas with the modifications required for kinetic
theory. In the next section we describe algebraic invariant manifolds for the
kinetic equations (1.2)-(1.3), (1.5), (1.6) and demonstrate the exact
summation approach for the Chapman–Enskog series for these models. We use
these models to demonstrate the application of the Newton method to the
invariance equation (2.5).
## 3\. Algebraic hydrodynamic invariant manifolds and exact summation of the
Chapman–Enskog series for the simplest kinetic model
### 3.1. Grin of the vanishing cat: $\epsilon$=1
At the end of the previous section we introduced a new space-time scale,
$x^{\prime}=\epsilon^{-1}x$, and $t^{\prime}=\epsilon^{-1}t$. The rescaled
equations do not depend on $\epsilon$ at all and are, at the same time,
equivalent to the original systems. Therefore, the presence of the small
parameter in the equations is virtual. “Putting $\epsilon$ back $=1$, you hope
that everything will converge and single out a nice submanifold” [115].
In this section, we find the invariant manifold for the equations with
$\epsilon$=1. Now, there is no fast–slow decomposition of motion. It is
natural to ask: what is the remainder of the qualitative picture of slow
invariant manifold presented in Fig. 1? Or an even sharper question: what we
are looking for?
The rest of the fast-slow decomposition is the zeroth term in the
Chapman–Enskog expansion (2.8). It starts from the equilibrium of the fast
motion, $\mbox{\boldmath$f$}^{\rm eq}_{M}$. This (locally) equilibrium
manifold corresponds to the limit $\epsilon=0$. The first terms of the series
for $\sigma$ for (1.6),
$\sigma=-\epsilon\frac{4}{3}\partial_{x}u-\epsilon^{2}\frac{4}{3}\partial_{x}^{2}p-\epsilon^{3}\frac{4}{9}\partial_{x}^{3}u+\ldots,$
(3.1)
also bear the offprint of the zeroth approximation, $\sigma^{(0)}=0$, even
when we take $\epsilon=1$. The Chapman–Enskog procedure derives recurrently
terms of the series from the starting term, $\mbox{\boldmath$f$}^{\rm
eq}_{M}$.
The problem of the invariant manifold includes two difficulties: (i) it is
difficult to find any global solution or even prove its existence and (ii)
there often exists too many different local solutions. The auxiliary Lyapunov
theorem gives the first solution of the problem near an equilibrium and
several seminal hints for the further attempts. One of them is: use the
analyticity as a selection criterion. The Chapman–Enskog method demonstrates
that the inclusion of the system in the one-parametric family (parameterized
by $\epsilon$) and the requirement of analyticity up to the limit $\epsilon=0$
allows us to select a sensible solution to the invariance equation. Even if we
return to a single system with $\epsilon=1$, the structure of the constructed
invariant manifold remembers the limit case $\epsilon=0$… This can be
considered as a manifestation of the effect of “the grin of the vanishing
cat”: ‘I’ve often seen a cat without a grin,’ thought Alice: ‘but a grin
without a cat! It’s the most curious thing I ever saw in my life!’ (Lewis
Carroll, Alice’s Adventures in Wonderland.) The small parameter disappears (we
take $\epsilon=1$) but the effect of its presence persists in the analytic
invariant manifold. There are some other effects of such a grin in kinetics
[67].
The use of the term “slow manifold” for the case $\epsilon=1$ seems to be an
abuse of language. Nevertheless, this manifold has some offprints of slowness,
at least for smooth solutions bounded by small number. The definition of slow
manifolds for a single system may be a non-trivial task [27, 60]. There is a
problem with a local definition because for a given vector field the
“slowness” of a submanifold cannot be invariant with respect to
diffeomorphisms in a vicinity of a regular point. Therefore we use the term
“hydrodynamic invariant manifold”.
### 3.2. The pseudodifferential form of the stress tensor
Let us return to the simplest kinetic equation (1.6). In order to construct
the exact solution, we first analyze the global structure of the
Chapman–Enskog series given by the recurrence formula (2.15). The first three
terms (3.1) give us a hint: the terms in the series alternate. For odd
$i=1,3,\ldots$ they are proportional to $\partial_{x}^{i}u$ and for even
$i=2,4,\ldots$ they are proportional to $\partial_{x}^{i}p$. Indeed, this
structure can be proved by induction in $i$ starting in (2.15) from the first
term $-\frac{4}{3}\partial_{x}u$. It is sufficient to notice that
$(D_{p}\partial^{(i)}_{x}p)=\partial_{x}^{(i)}$, $(D_{p}\partial^{i}_{x}u)=0$,
$(D_{u}\partial^{i}_{x}p)=0$, $(D_{u}\partial^{i}_{x}u)=\partial_{x}^{(i)}$
and to use the induction assumption in (2.15).
The global structure of the Chapman–Enskog series gives the following
representation of the stress $\sigma$ on the hydrodynamic invariant manifold
$\sigma(x)=A(-\partial_{x}^{2})\partial_{x}u(x)+B(-\partial_{x}^{2})\partial_{x}^{2}p(x),$
(3.2)
where $A(y)$, $B(y)$ are yet unknown functions and the sign ‘$-$’ in the
arguments is adopted for simplicity of formulas in the Fourier transform.
It is easy to prove the structure (3.2) without any calculation or induction.
Let us use the symmetry property of the kinetic equation (1.6): it is
invariant with respect to the transformation $x\mapsto-x$, $u\mapsto-u$,
$p\mapsto p$ and $\sigma\mapsto\sigma$ which transforms solutions into
solutions. The invariance equation inherits this property, the initial
equilibrium ($\sigma=0$) is also symmetric and, therefore, the expression for
$\sigma(x)$ should be even. This is exactly (3.2) where $A(y)$ and $B(y)$ are
arbitrary even functions. (If they are, say, twice differentiable at the
origin then we can represent them as functions of $y^{2}$).
### 3.3. The energy formula and ‘capillarity’ of ideal gas
Traditionally, $\sigma$ is considered as a viscous stress tensor but the
second term, $B(-\partial_{x}^{2})\partial_{x}^{2}p(x)$, is proportional to
second derivative of $p(x)$ and it does not meet usual expectations
($\sigma\sim\nabla u$). Slemrod [135, 136] noticed that the proper
interpretation of this term is the capillarity tension rather than viscosity.
This is made clear by inspection of the energy balance formula. Let us derive
the Slemrod energy formula for the simple model (1.6). The time derivative of
the kinetic energy due to the first two equations (1.6) is
$\begin{split}\frac{1}{2}\partial_{t}\int_{-\infty}^{\infty}u^{2}\,{\mathrm{d}}x&=\int_{-\infty}^{\infty}u\partial_{t}u\,{\mathrm{d}}x=-\int_{-\infty}^{\infty}u\partial_{x}p\,{\mathrm{d}}x-\int_{-\infty}^{\infty}u\partial_{x}\sigma\,{\mathrm{d}}x\\\
&=-\frac{1}{2}\partial_{t}\frac{3}{5}\int_{-\infty}^{\infty}p^{2}\,{\mathrm{d}}x+\int_{-\infty}^{\infty}\sigma\partial_{x}u\,{\mathrm{d}}x\end{split}$
(3.3)
Here we used integration by parts and assumed that all the fields with their
derivatives tend to $0$ when $x\to\pm\infty$.
In $x$-space the energy formula is
$\frac{1}{2}\partial_{t}\left(\frac{3}{5}\int_{-\infty}^{\infty}p^{2}\,{\mathrm{d}}x+\int_{-\infty}^{\infty}u^{2}\,{\mathrm{d}}x\right)=\int_{-\infty}^{\infty}\sigma\partial_{x}u\,{\mathrm{d}}x$
(3.4)
This form of the energy equation is standard. Note that the usual factor
$\rho$ in front of $u^{2}$ is absent because we work with the linearized
equations.
Let us use in (3.4) the representation (3.2) for $\sigma$ and notice that
$\partial_{x}u=-\frac{3}{5}\partial_{t}p$:
$\int_{-\infty}^{\infty}\sigma\partial_{x}u\,{\mathrm{d}}x=\int_{-\infty}^{\infty}A(-\partial_{x}^{2})\partial^{2}_{x}u\,{\mathrm{d}}x-\frac{3}{5}\int_{-\infty}^{\infty}(\partial_{t}p)[B(-\partial_{x}^{2})\partial_{x}^{2}p]\,{\mathrm{d}}x$
The operator $B(-\partial_{x}^{2})\partial_{x}^{2}$ is symmetric, therefore,
$\int_{-\infty}^{\infty}(\partial_{t}p)[B(-\partial_{x}^{2})\partial_{x}^{2}p]\,{\mathrm{d}}x=\frac{1}{2}\partial_{t}\left(\int_{-\infty}^{\infty}p[B(-\partial_{x}^{2})\partial_{x}^{2}p]\,{\mathrm{d}}x\right)$
The quadratic form,
$U_{c}=\frac{3}{5}\int_{-\infty}^{\infty}p(B(-\partial_{x}^{2})\partial_{x}^{2}p)\,{\mathrm{d}}x=-\frac{3}{5}\int_{-\infty}^{\infty}(\partial_{x}p)(B(-\partial_{x}^{2})\partial_{x}p)\,{\mathrm{d}}x$
(3.5)
may be considered as a part of the energy. Moreover, if the function $B(y)$ is
negative then this form is positive. Due to Parseval’s identity we have
$U_{c}=-\frac{3}{5}\int_{-\infty}^{\infty}k^{2}B(k^{2})|p_{k}|^{2}\,{\mathrm{d}}k.$
(3.6)
Finally, the energy formula in $x$-space is
$\boxed{\begin{aligned}
\frac{1}{2}\partial_{t}\int_{-\infty}^{\infty}\left(\frac{3}{5}p^{2}+u^{2}-\frac{3}{5}(\partial_{x}p)(B(-\partial_{x}^{2})\partial_{x}p)\right)\,{\mathrm{d}}x\\\
=\int_{-\infty}^{\infty}(\partial_{x}u)(A(-\partial_{x}^{2})\partial_{x}u)\,{\mathrm{d}}x\end{aligned}}$
(3.7)
In $k$-space it has the form
$\begin{split}\frac{1}{2}\partial_{t}\int_{-\infty}^{\infty}\left(\frac{3}{5}|p_{k}|^{2}+|u_{k}|^{2}-\frac{3}{5}k^{2}B(k^{2})|p_{k}|^{2}\right)\,{\mathrm{d}}k=\int_{-\infty}^{\infty}k^{2}A(k^{2})|u_{k}|^{2}\,{\mathrm{d}}k\end{split}$
(3.8)
It is worth mentioning that the functions $A(k^{2})$ and $B(k^{2})$ are
negative (see Sec. 3.4). If we keep only the first non-trivial terms,
$A=B=-\frac{4}{3}$, then the energy formula becomes
$\displaystyle\frac{1}{2}\partial_{t}\int_{-\infty}^{\infty}\left(\frac{3}{5}p^{2}+u^{2}+\frac{4}{5}(\partial_{x}p)^{2}\right)\,{\mathrm{d}}x=-\frac{4}{3}\int_{-\infty}^{\infty}(\partial_{x}u)^{2}\,{\mathrm{d}}x;$
(3.9)
$\displaystyle\frac{1}{2}\partial_{t}\int_{-\infty}^{\infty}\left(\frac{3}{5}|p_{k}|^{2}\,{\mathrm{d}}k+|u_{k}|^{2}+\frac{4}{5}k^{2}|p_{k}|^{2}\right)\,{\mathrm{d}}k=-\frac{4}{3}\int_{-\infty}^{\infty}k^{2}|u_{k}|^{2}\,{\mathrm{d}}k.$
(3.10)
Slemrod represents the structure of the obtained energy formula as
$\begin{split}\partial_{t}({\rm MECHANICAL\ ENERGY})+\partial_{t}({\rm
CAPILLARITY\ ENERGY})\\\ ={\rm VISCOUS\ DISSIPATION}.\end{split}$ (3.11)
The capillarity terms $(\partial_{x}p)^{2}$ in the energy density are standard
in the thermodynamics of phase transitions.
The bulk capillarity terms in fluid mechanics were introduced into the
Navier–Stokes equations by Korteweg [100] (for a review of some further
results see [32]). Such terms appear naturally in theories of the phase
transitions such as van der Waals liquids [132], Ginzburg–Landau [1] and
Cahn–Hilliard equations [17, 16], and phase fields models [25]. Surprisingly,
such terms are also found in the ideal gas dynamics as a consequence of the
Chapman–Enskog expansion [134, 133]. In higher-order approximations, the
viscosity is reduced by the terms which are similar to Korteweg’s capillarity.
Finally, in the energy formula for the exact sum of the Chapman–Enskog
expansion we see terms of the same form: the viscous dissipation is decreased
and the additional term appears in the energy (3.7), (3.8).
### 3.4. Algebraic invariant manifold in Fourier representation
It is convenient to work with the pseudodifferential operators like (3.2) in
Fourier space. Let us denote $p_{k}$, $u_{k}$ and $\sigma_{k}$, where $k$ is
the ‘wave vector’ (space frequency).
The Fourier-transformed kinetic equation (1.6) takes the form ($\epsilon=1$):
$\begin{split}\partial_{t}p_{k}&=-\frac{5}{3}iku_{k},\\\
\partial_{t}u_{k}&=-ikp_{k}-ik\sigma_{k},\\\
\partial_{t}\sigma_{k}&=-\frac{4}{3}iku_{k}-\sigma_{k}.\end{split}$ (3.12)
We know already that the result of the reduction should be a function
$\sigma_{k}(u_{k},p_{k},k)$ of the following form:
$\sigma_{k}(u_{k},p_{k},k)=ikA(k^{2})u_{k}-k^{2}B(k^{2})p_{k},$ (3.13)
where $A$ and $B$ are unknown real-valued functions of $k^{2}$.
The question of the summation of the Chapman–Enskog series amounts to finding
the two functions, $A(k^{2})$ and $B(k^{2})$. Let us write the invariance
equation for unknown functions $A$ and $B$. We can compute the time derivative
of $\sigma_{k}(u_{k},p_{k},k)$ in two different ways. First, we use the right
hand side of the third equation in (3.12). We find the microscopic time
derivative:
$\partial_{t}^{\rm
micro}\sigma_{k}=-ik\left(\frac{4}{3}+A\right)u_{k}+k^{2}Bp_{k}.$ (3.14)
Secondly, let us use chain rule and the first two equations in (3.12). We find
the macroscopic time derivative:
$\begin{split}\partial_{t}^{\rm
macro}\sigma_{k}&=\frac{\partial\sigma_{k}}{\partial
u_{k}}\partial_{t}u_{k}+\frac{\partial\sigma_{k}}{\partial
p_{k}}\partial_{t}p_{k}\\\
&=ikA\left(-ikp_{k}-ik\sigma_{k}\right)-k^{2}B\left(-\frac{5}{3}iku_{k}\right)\\\
&=ik\left(\frac{5}{3}k^{2}B+k^{2}A\right)u_{k}+k^{2}\left(A-k^{2}B\right)p_{k}.\end{split}$
(3.15)
The microscopic time derivative should coincide with the macroscopic time
derivative for all values of $u_{k}$ and $p_{k}$. This is the invariance
equation:
$\partial_{t}^{\rm macro}\sigma_{k}=\partial_{t}^{\rm micro}\sigma_{k}.$
(3.16)
For the kinetic system (3.12), it reduces to a system of two quadratic
equations for functions $A(k^{2})$ and $B(k^{2})$:
$\begin{split}F(A,B,k)&=-A-\frac{4}{3}-k^{2}\left(\frac{5}{3}B+A^{2}\right)=0,\\\
G(A,B,k)&=-B+A\left(1-k^{2}B\right)=0.\end{split}$ (3.17)
The Taylor series for $A(k^{2})$, $B(k^{2})$ correspond exactly to the
Chapman–Enskog series: if we look for these functions in the form
$A(y)=\sum_{l\geq 0}a_{l}y^{l}$ and $B(y)=\sum_{l\geq 0}b_{l}y^{l}$ then from
(3.17) we find immediately $a_{0}=b_{0}=-\frac{4}{3}$ (these are exactly the
Navier–Stokes and Burnett terms) and the recurrence equation for $a_{i+1}$,
$b_{i+1}$:
$\begin{split}a_{n+1}&=\frac{5}{3}b_{n}+\sum_{m=0}^{n}a_{n-m}a_{m},\\\
b_{n+1}&=a_{n+1}+\sum_{m=0}^{n}a_{n-m}b_{m}.\end{split}$ (3.18)
The initial condition for this set of equations are the Navier–Stokes and the
Burnett terms $a_{0}=b_{0}=-\frac{4}{3}$.
The Newton method for the invariance equation (3.17) generates the sequence
$A_{i}(k^{2})$, $B_{i}(k^{2})$, where the differences, $\delta
A_{i+1}=A_{i+1}-A_{i}$ and $\delta B_{i+1}=B_{i+1}-B_{i}$ satisfy the system
of linear equations
$\left(\begin{array}[]{cc}\frac{\partial F(A,B,k^{2})}{\partial
A}\left.\right|_{(A_{i},B_{i})}&\frac{\partial F(A,B,k^{2})}{\partial
B}\left.\right|_{(A_{i},B_{i})}\\\ \frac{\partial G(A,B,k^{2})}{\partial
A}\left.\right|_{(A_{i},B_{i})}&\frac{\partial G(A,B,k^{2})}{\partial
B}\left.\right|_{(A_{i},B_{i})}\\\
\end{array}\right)\left(\begin{array}[]{c}\delta A_{i+1}\\\ \delta B_{i+1}\\\
\end{array}\right)+\left(\begin{array}[]{c}F(A_{i},B_{i},k^{2})\\\
G(A_{i},B_{i},k^{2})\\\ \end{array}\right)=0.$
Rewrite this system in the explicit form:
$\left(\begin{array}[]{cc}-(1+2k^{2}A_{i})&-\frac{5}{3}k^{2}\\\
1-k^{2}B_{i}&-(1+k^{2}A_{i})\\\
\end{array}\right)\left(\begin{array}[]{c}\delta A_{i+1}\\\ \delta B_{i+1}\\\
\end{array}\right)+\left(\begin{array}[]{c}F(A_{i},B_{i},k^{2})\\\
G(A_{i},B_{i},k^{2})\\\ \end{array}\right)=0.$
Let us start from the zeroth-order term of the Chapman–Enskog expansion
(Euler’s approximation), $A_{0}=B_{0}=0$. Then, the first Newton’s iteration
gives
$A_{1}=B_{1}=-\frac{4}{3+5k^{2}}.$ (3.19)
The second Newton’s iteration also gives the negative rational functions
$\begin{split}A_{2}&=-\frac{4(27+63k^{2}+153k^{2}k^{2}+125k^{2}k^{2}k^{2})}{3(3+5k^{2})(9+9k^{2}+67k^{2}k^{2}+75k^{2}k^{2}k^{2})},\\\
B_{2}&=-\frac{4(9+33k^{2}+115k^{2}k^{2}+75k^{2}k^{2}k^{2})}{(3+5k^{2})(9+9k^{2}+67k^{2}k^{2}+75k^{2}k^{2}k^{2})}.\end{split}$
(3.20)
The corresponding attenuation rates are shown in Fig. 3. They are stable and
converge fast to the exact solutions. At the infinity, $k^{2}\to\infty$, the
second iteration has the same limit, as the exact solution:
$k^{2}A_{2}\to-\frac{4}{9}$ and $k^{2}B_{2}\to-\frac{4}{5}$ (compare to Sec.
3.6).
Thus, we made three steps:
1. (1)
We used the invariance equation, Chapman–Enskog procedure and the symmetry
properties to find a linear space where the hydrodynamic invariant manifold is
located. This space is parameterized by two functions of one variable (3.13);
2. (2)
We used the invariance equation and defined an algebraic manifold in this
space. For the simple kinetic system (1.6), (3.12) this manifold is given by
the system of two quadratic equations which depends linearly on $k^{2}$
(3.17).
3. (3)
We found that Newton’s iterations for the invariant manifold demonstrate much
better approximation properties than the truncated Chapman–Enskog.
### 3.5. Stability of the exact hydrodynamic system and saturation of
dissipation for short waves
Stability is one of the first questions to analyze. There exists a series of
simple general statements about the preservation of stability, well-posedness
and hyperbolicity in the exact hydrodynamics. Indeed, any solution of the
exact hydrodynamics is the projection of a solution of the initial equation
from the invariant manifold onto the hydrodynamic moments (Figs. 1, 2) and the
projection of a bounded solution is bounded. (In infinite dimension we have to
assume that the projection is continuous with respect to the chosen norms.)
Several statements of this type are discussed in Sec. 4. Nevertheless, a
direct analysis of dispersion relations and attenuation rates is instructive.
Knowing $A(k^{2})$ and $B(k^{2})$, the dispersion relation for the
hydrodynamic modes can be derived:
$\omega_{\pm}=\frac{k^{2}A}{2}\pm
i\frac{|k|}{2}\sqrt{\frac{20}{3}(1-k^{2}B)-k^{2}A^{2}}.$ (3.21)
It is convenient to reduce the consideration to a single function. Solving the
system (3.17) for $B$, and introducing a new function,
$X(k^{2})=k^{2}B(k^{2})$, we obtain an equivalent cubic equation:
$-\frac{5}{3}(X-1)^{2}\left(X+\frac{4}{5}\right)=\frac{X}{k^{2}}.$ (3.22)
Since the hydrodynamic manifold should be represented by the real-valued
functions $A(k^{2})$ and $B(k^{2})$ (3.13), we are only interested in the
real-valued roots of (3.22).
An elementary analysis gives: the real-valued root $X(k^{2})$ of (3.22) is
unique and negative for all finite values $k^{2}$. Moreover, the function
$X(k^{2})$ is a monotonic function of $k^{2}$. The limiting values are:
$\lim_{|k|\rightarrow
0}X(k^{2})=0,\quad\lim_{|k|\rightarrow\infty}X(k^{2})=-0.8.$ (3.23)
Under the conditions just mentioned, the function under the root in (3.21) is
negative for all values of the wave vector $k$, including the limits, and we
come to the following dispersion law:
$\omega_{\pm}=\frac{X}{2(1-X)}\pm
i\frac{|k|}{2}\sqrt{\frac{5X^{2}-16X+20}{3}},$ (3.24)
where $X=X(k^{2})$ is the real-valued root of equation (3.22). Since
$0>X(k^{2})>-1$ for all $|k|>0$, the attenuation rate, ${\rm
Re}(\omega_{\pm})$, is negative for all $|k|>0$, and the exact acoustic
spectrum of the Chapman–Enskog procedure is stable for arbitrary wave lengths
(Fig. 3, solid line). In the short-wave limit, from (3.24) we obtain:
$\lim_{|k|\rightarrow\infty}{\rm Re}\omega_{\pm}=-\frac{2}{9},;\
\;\;\lim_{|k|\rightarrow\infty}\frac{{\rm Im}\omega_{\pm}}{|k|}=\pm\sqrt{3}.$
(3.25)
### 3.6. Expansion at $k^{2}=\infty$ and matched asymptotics
For large values of $k^{2}$, a version of the Chapman–Enskog expansion at an
infinitely-distant point is useful. Let us rewrite the algebraic equation for
the invariant manifold (3.17) in the form
$\begin{split}\frac{5}{3}B+A^{2}&=-\varsigma(\frac{4}{3}+A),\\\
AB&=\varsigma(A-B),\end{split}$ (3.26)
where $\varsigma=1/k^{2}$. For the analytic solutions near the point
$\varsigma=0$ the Taylor series is:
$A=\sum_{l=1}^{\infty}\alpha_{l}\varsigma^{l}$,
$B=\sum_{l=1}^{\infty}\beta_{l}\varsigma^{l}$, where
$\alpha_{1}=-\frac{4}{9}$, $\beta_{1}=-\frac{4}{5}$,
$\alpha_{2}=\frac{80}{2187}$, $\beta_{2}=\frac{4}{27}$, … . The first term
gives for the frequency (3.21) the same limit:
$\omega_{\pm}=-\frac{2}{9}\pm i{|k|}{\sqrt{3}},$ (3.27)
and the higher terms give some corrections.
Let us match the Navier–Stokes term and the first term in the $1/k^{2}$
expansion. We get:
$A\approx-\frac{4}{3+9k^{2}},\;\;B\approx-\frac{4}{3+5k^{2}}$ (3.28)
and
$\sigma_{k}=ikA(k^{2})u_{k}-k^{2}B(k^{2})p_{k}\approx-\frac{4ik}{3+9k^{2}}u_{k}+\frac{4k^{2}}{3+5k^{2}}p_{k}.$
(3.29)
This simplest non-locality captures the main effects: the asymptotic for short
waves (large $k^{2}$) and the Navier–Stokes approximation for hydrodynamics
for smooth solutions with bounded derivatives and small Knudsen and Mach
numbers (small $k^{2}$).
The saturation of dissipation at large $k^{2}$ is a universal effect and
hydrodynamics that do not take this effect into account cannot pretend to be a
universal asymptotic equation.
This section demonstrates that for the simple kinetic model (1.6):
* •
The Chapman–Enskog series amounts to an algebraic invariant manifold, and the
“smallness” of the Knudsen number $\epsilon$ used to develop the
Chapman–Enskog procedure is no longer necessary.
* •
The exact dispersion relation (3.24) on the algebraic invariant manifold is
stable for all wave lengths.
* •
The exact result of the Chapman–Enskog procedure has a clear non-polynomial
character. The resulting exact hydrodynamics are essentially nonlocal in
space. For this reason, even if the hydrodynamic equations of a certain level
of the approximation are stable, they cannot reproduce the non-polynomial
behavior for sufficiently short waves.
* •
The Newton iterations for the invariance equations provide much better results
than the Chapman–Enskog expansion. The first iteration gives the Navier–Stokes
asymptotic for long waves and the qualitatively correct behavior with
saturation for short waves. The second iteration gives the proper higher order
approximation in the long wave limit and the quantitatively proper asymptotic
for short waves.
In the next section we extend these results to a general linear kinetic
equation.
## 4\. Algebraic invariant manifold for general linear kinetics in 1D
### 4.1. General form of the invariance equation for 1D linear kinetics
For linearized kinetic equations, it is convenient to start directly with the
Fourier transformed system.
Let us consider two sets of variables: macroscopic variables $M$ and
microscopic variables $\mu$. The corresponding vector spaces are $E_{M}$
($M\in E_{M}$) and $E_{\mu}$ ($\mu\in E_{\mu}$), $k$ is the wave vector and
the initial kinetic system in the Fourier space for functions $M_{k}(t)$ and
$\mu_{k}(t)$ has the following form:
$\begin{split}\partial_{t}M_{k}&=ikL_{MM}M_{k}+ikL_{M\mu}\mu_{k};\\\
\partial_{t}\mu_{k}&=ikL_{\mu
M}M_{k}+ikL_{\mu\mu}\mu_{k}+C\mu_{k},\end{split}$ (4.1)
where $L_{MM}:E_{M}\to E_{M}$, $L_{M\mu}E_{\mu}\to E_{M}$, $L_{\mu M}:E_{M}\to
E_{\mu}$, $L_{\mu\mu}:E_{\mu}\to E_{\mu}$, and $C:E_{\mu}\to E_{\mu}$ are
constant linear operators (matrices).
The only requirement for the following algebra is: the operator $C:E_{\mu}\to
E_{\mu}$ is invertible. (Of course, for further properties like stability of
reduced equations we need more assumptions like stability of the whole system
(4.1) and negative definiteness of $C$.)
We look for a hydrodynamic invariant manifold in the form
$\mu_{k}=\mathcal{X}(k)M_{k},$ (4.2)
where $\mathcal{X}(k):E_{M}\to E_{\mu}$ is a linear map for all $k$.
The corresponding exact hydrodynamic equation is
$\boxed{\partial_{t}M_{k}=ik[L_{MM}+L_{M\mu}\mathcal{X}(k)]M_{k}.}$ (4.3)
Calculate the micro- and macroscopic derivatives of $\mu_{k}$ (4.2) exactly as
in (3.14), (3.15):
$\begin{split}\partial_{t}^{\rm micro}\mu_{k}&=[ikL_{\mu
M}+ikL_{\mu\mu}\mathcal{X}(k)+C\mathcal{X}(k)]M_{k};\\\ \partial_{t}^{\rm
macro}\mu_{k}&=[ik\mathcal{X}(k)L_{MM}+ik\mathcal{X}(k)L_{M\mu}\mathcal{X}(k)]M_{k}.\end{split}$
(4.4)
The invariance equation for $\mathcal{X}(k)$ is again a system of algebraic
equations (a quadratic matrix equation):
$\boxed{\mathcal{X}(k)=ikC^{-1}[-L_{\mu
M}+(\mathcal{X}(k)L_{MM}-L_{\mu\mu}\mathcal{X}(k))+\mathcal{X}(k)L_{M\mu}\mathcal{X}(k)].}$
(4.5)
This is a general invariance equation for linear kinetic systems (4.1). The
Chapman–Enskog series is a Taylor expansion for the solution of this equation
at $k=0$. Thus, immediately we get the first terms:
$\mathcal{X}(0)=0,\;\mathcal{X}^{\prime}(0)=-iC^{-1}L_{\mu
M},\;\mathcal{X}^{\prime\prime}(0)=2C^{-1}(C^{-1}L_{\mu
M}L_{MM}-L_{\mu\mu}C^{-1}L_{\mu M}).$
The sequence of the Euler, Navier–Stokes and Burnett approximations is:
$\begin{split}\partial_{t}M_{k}=&ikL_{MM}M_{k}\;\;\mbox{(Euler)};\\\
\partial_{t}M_{k}=&ikL_{MM}M_{k}+k^{2}L_{M\mu}C^{-1}L_{\mu
M}M_{k}\;\;\mbox{(Navier--Stokes)};\\\
\partial_{t}M_{k}=&ikL_{MM}M_{k}+k^{2}L_{M\mu}C^{-1}L_{\mu M}M_{k}\\\
&+ik^{3}L_{M\mu}C^{-1}(C^{-1}L_{\mu M}L_{MM}-L_{\mu\mu}C^{-1}L_{\mu
M})M_{k}\;\;\mbox{(Burnett)}.\end{split}$ (4.6)
Let us use the identity $\mathcal{X}(0)=0$ and the fact that the functions in
the $x$-space are real-valued. We can separate odd and even parts of
$\mathcal{X}(k)$ and write
$\mathcal{X}(k)=ik\mathcal{A}(k^{2})+k^{2}\mathcal{B}(k^{2}),$ (4.7)
where $\mathcal{A}(y)$ and $\mathcal{B}(y)$ are real-valued matrices. For
these unknowns, the invariance equation is even closer to the simple example
(3.17):
$\begin{split}\mathcal{A}(k^{2})=&C^{-1}[-L_{\mu
M}+k^{2}(\mathcal{B}(k^{2})L_{MM}-L_{\mu\mu}\mathcal{B}(k^{2}))\\\
&-k^{2}\mathcal{A}(k^{2})L_{M\mu}\mathcal{A}(k^{2})+k^{4}\mathcal{B}(k^{2})L_{M\mu}\mathcal{B}(k^{2})],\\\
\mathcal{B}(k^{2})=&-C^{-1}[(\mathcal{A}(k^{2})L_{MM}-L_{\mu\mu}\mathcal{A}(k^{2}))\\\
&+k^{2}\mathcal{A}(k^{2})L_{M\mu}\mathcal{B}(k^{2})+k^{2}\mathcal{B}(k^{2})L_{M\mu}\mathcal{A}(k^{2})].\end{split}$
(4.8)
### 4.2. Hyperbolicity of exact hydrodynamics
Hyperbolicity is an important property of the exact hydrodynamics. Let us
recall that the linear system represented in Fourier space by the equation
$\partial_{t}u_{k}=-iA(k)u_{k}$
is hyperbolic if for every $t\geq 0$ the operator $\exp(-itA(k))$ is uniformly
bounded as a function of $k$ (it is sufficient to take $t=1$). This means that
the Cauchy problem for this system is well-posed forward in time.
This system is strongly hyperbolic if for every $t\in\mathbb{R}$ the operator
$\exp(-itA(k))$ is uniformly bounded as a function of $k$ (it is sufficient to
take $t=\pm 1$). This means that the Cauchy problem for this system is well-
posed both forward and backward in time.
###### Proposition 4.1 (Preservation of hyperbolicity)
Let the original system (4.1) be (strongly) hyperbolic. Then the reduced
system (4.3) is also (strongly) hyperbolic if the lifting operator
$\mathcal{X}(k)$ (4.2) is a bounded function of $k$.
###### Proof.
Hyperbolicity (strong hyperbolicity) is just a requirement of the uniform
boundedness in $k$ of the solutions of (4.1) for each $t>0$ (or for all $t$)
with uniformly bounded in $k$ initial conditions. For the exact hydrodynamics,
solutions of the projected equations are projections of the solutions of the
original system. Let the original system (4.1) be (strongly) hyperbolic. If
the lifting operator $\mathcal{X}(k)$ is a bounded function of $k$ then for
the uniformly bounded initial condition $M_{k}$ the corresponding initial
value $\mu_{k}=\mathcal{X}(k)M_{k}$ is also bounded and, due to the
hyperbolicity of (4.1), the projection of the solution is uniformly bounded in
$k$ for all $t\geq 0$. In the following commutative diagram, the upper
horizontal arrow and the vertical arrows are the bounded operators, hence the
lower horizontal arrow is also a bounded operator.
$\begin{CD}(M_{k}(0),\mu_{k}(0))@>{{\mbox{Time shift (initial
eq.)}}}>{}>(M_{k}(t),\mu_{k}(t))\\\
@A{\mbox{Lifting}}A{}A@V{}V{\mbox{Projection}}V\\\ M_{k}(0)@>{{\mbox{Exact
hydrodynamics}}}>{}>M_{k}(t)\end{CD}$ (4.9)
∎
To analyze the boundedness of the lifting operator we have to study the
asymptotics of the solution of the invariance equation at the infinitely-
distant point $k^{2}=\infty$. If this is a regular point then we can find the
Taylor expansion in powers of $\varsigma=\frac{1}{k^{2}}$,
$A=\sum_{l}\alpha_{l}\varsigma^{l}$ and $B=\sum_{l}\beta_{l}\varsigma^{l}$.
For the boundedness of $\mathcal{X}(k)$ (4.7) we should take in these series
$\alpha_{0}=\beta_{0}=0$. If the solution of the invariance equation is a real
analytic function for $0\geq k^{2}\geq\infty$ then the condition is sufficient
for the hyperbolicity of the projected equation (4.3). If $\mathcal{X}(k)$ is
an exact solution of the algebraic invariance equation (4.5) then the
hydrodynamic equation (4.3) gives the exact reduction of (4.1). Various
approximations give the approximate reduction like the Chapman–Enskog
approximations (4.6).
The expansion near an infinitely-distant point is useful but may be not so
straightforward. Nevertheless, if such an expansion exists then we can
immediately produce the matched asymptotics.
Thus, as we can see, the summation of the Chapman–Enskog series to an
algebraic manifold is not just a coincidence but a typical effect for kinetic
equations. For a specific kinetic system we have to make use of all the
existing symmetries like parity and rotation symmetry in order to reduce the
dimension of the invariance equation and to select the proper physical
solution. Another simple but important condition is that all the kinetic and
hydrodynamic variables should be real-valued. The third selection rule is the
behavior of the spectrum near $k=0$: the attenuation rate should go to zero
when $k\to 0$.
The Chapman–Enskog expansion is a Taylor series (in $k$) for the solution of
the invariance equation. In general, there is no reason to believe that the
first few terms of the Taylor series at $k=0$ properly describe the asymptotic
behavior of the solutions of the invariance equation (4.5) for all $k$.
Already the simple examples such as (3.12) reveal that the exact hydrodynamic
is essentially nonlocal and the behavior of the attenuation rate at
$k\to\infty$ does not correspond to any truncation of the Chapman–Enskog
series.
Of course, for a numerical solution of (4.5), the Taylor series expansion is
not the best approach. The Newton method gives much better results and even
the first approximation may be very close to the solution [20].
In the next section we show that for more complex kinetic equations the
situation may be even more involved and both the truncation and the summation
of the whole series may become meaningless for sufficiently large $k$. In
these cases, the hydrodynamic solution of the invariance equations does not
exist for large $k$ and the whole problem of hydrodynamic reduction has no
solution. We will see how the hydrodynamic description is destroyed and the
coupling between hydrodynamic and non-hydrodynamic modes becomes permanent and
indestructible. Perhaps, the only advice in this situation may be to change
the set of variables or to modify the projector onto these variables: if
hydrodynamics exist, then the set of hydrodynamic variables or the projection
onto these variables should be different.
### 4.3. Destruction of hydrodynamic invariant manifold for short waves in
the moment equations
In this section we study the one-dimensional version of the Grad equations
(1.2) and (1.3) in the $k$-representation:
$\begin{split}\partial_{t}\rho_{k}&=-iku_{k},\\\
\partial_{t}u_{k}&=-ik\rho_{k}-ikT_{k}-ik\sigma_{k},\\\
\partial_{t}T_{k}&=-\frac{2}{3}iku_{k}-\frac{2}{3}ikq_{k},\\\
\partial_{t}\sigma_{k}&=-\frac{4}{3}iku_{k}-\frac{8}{15}ikq_{k}-\sigma_{k},\\\
\partial_{t}q_{k}&=-\frac{5}{2}ikT_{k}-ik\sigma_{k}-\frac{2}{3}q_{k}.\end{split}$
(4.10)
The Grad system (4.10) provides the simplest coupling of the hydrodynamic
variables $\rho_{k}$, $u_{k}$, and $T_{k}$ to the non-hydrodynamic variables,
$\sigma_{k}$ and $q_{k}$, the latter is the heat flux. We need to reduce the
Grad system (4.10) to the three hydrodynamic equations with respect to the
variables $\rho_{k}$, $u_{k}$, and $T_{k}$. That is, in the general notations
of the previous section, $M=\rho_{k},u_{k},T_{k}$ $\mu=\sigma_{k},q_{k}$ and
we have to express the functions $\sigma_{k}$ and $q_{k}$ in terms of
$\rho_{k}$, $u_{k}$, and $T_{k}$:
$\sigma_{k}=\sigma_{k}(\rho_{k},u_{k},T_{k},k),\;\;\;q_{k}=q_{k}(\rho_{k},u_{k},T_{k},k).$
The derivation of the invariance equation for the system (4.10) goes along the
same lines as in the previous sections. The quantities $\rho$ and $T$ are
scalars, $u$ and $q$ are (1D) vectors, and the (1D) stress ‘tensor’ $\sigma$
is again a scalar. The vectors and scalars transform differently under the
parity transformation $x\mapsto-x$, $k\mapsto-k$. We use this symmetry
property and find the representation (4.2) of $\sigma,q$ similar to (3.13):
$\begin{split}\sigma_{k}&=ikA(k^{2})u_{k}-k^{2}B(k^{2})\rho_{k}-k^{2}C(k^{2})T_{k},\\\
q_{k}&=ikX(k^{2})\rho_{k}+ikY(k^{2})T_{k}-k^{2}Z(k^{2})u_{k},\end{split}$
(4.11)
where the functions $A,\dots,Z$ are the unknowns in the invariance equation.
By the nature of the CE recurrence procedure for the real-valued in $x$-space
kinetic equations, $A,\dots,Z$ are real-valued functions.
Let us find the microscopic and macroscopic time derivatives (4.4). Computing
the microscopic time derivative of the functions (4.11), due to the two last
equations of the Grad system (4.10) we derive:
$\begin{split}\partial_{t}^{\rm
micro}\sigma_{k}&=-ik\left(\frac{4}{3}-\frac{8}{15}k^{2}Z+A\right)u_{k}\\\
&+k^{2}\left(\frac{8}{15}X+B\right)\rho_{k}+k^{2}\left(\frac{8}{15}Y+C\right)T_{k},\\\
\partial_{t}^{\rm
micro}q_{k}&=k^{2}\left(A+\frac{2}{3}Z\right)u_{k}+ik\left(k^{2}B-\frac{2}{3}X\right)\rho_{k}\\\
&-ik\left(\frac{5}{2}-k^{2}C-\frac{2}{3}Y\right)T_{k}.\end{split}$
On the other hand, computing the macroscopic time derivative of the functions
(4.11) due to the first three equations of the system (4.10), we obtain:
$\begin{split}\partial_{t}^{\rm
macro}\sigma_{k}&=\frac{\partial\sigma_{k}}{\partial
u_{k}}\partial_{t}u_{k}+\frac{\partial\sigma_{k}}{\partial\rho_{k}}\partial_{t}\rho+\frac{\partial\sigma_{k}}{\partial
T_{k}}\partial_{t}T_{k}\\\
&=ik\left(k^{2}A^{2}+k^{2}B+\frac{2}{3}k^{2}C-\frac{2}{3}k^{2}k^{2}CZ\right)u_{k}\\\
&+\left(k^{2}A-k^{2}k^{2}AB-\frac{2}{3}k^{2}k^{2}CX\right)\rho_{k}\\\
&+\left(k^{2}A-k^{2}k^{2}AC-\frac{2}{3}k^{2}k^{2}CY\right)T_{k};\\\
\end{split}$ $\begin{split}\partial_{t}^{\rm macro}q_{k}&=\frac{\partial
q_{k}}{\partial u_{k}}\partial_{t}u_{k}+\frac{\partial
q_{k}}{\partial\rho_{k}}\partial_{t}\rho u_{k}+\frac{\partial q_{k}}{\partial
T_{k}}\partial_{t}T_{k}\\\
&=\left(-k^{2}k^{2}ZA+k^{2}X+\frac{2}{3}k^{2}Y-\frac{2}{3}k^{2}k^{2}YZ\right)u_{k}\\\
&+ik\left(k^{2}Z-k^{2}k^{2}ZB+\frac{2}{3}k^{2}YX\right)\rho_{k}\\\
&+ik\left(k^{2}Z-k^{2}k^{2}ZC+\frac{2}{3}k^{2}Y^{2}\right)T_{k}.\\\
\end{split}$
The invariance equation (4.5) for this case is a system of six coupled
quadratic equations with quadratic in $k^{2}$ coefficients:
$\begin{split}&F_{1}=-\frac{4}{3}-A-k^{2}(A^{2}+B-\frac{8Z}{15}+\frac{2C}{3})+\frac{2}{3}k^{4}CZ=0,\\\
&F_{2}=\frac{8}{15}X+B-A+k^{2}AB+\frac{2}{3}k^{2}CX=0,\\\
&F_{3}=\frac{8}{15}Y+C-A+k^{2}AC+\frac{2}{3}k^{2}CY=0,\\\
&F_{4}=A+\frac{2}{3}Z+k^{2}ZA-X-\frac{2}{3}Y+\frac{2}{3}k^{2}YZ=0,\\\
&F_{5}=k^{2}B-\frac{2}{3}X-k^{2}Z+k^{4}ZB-\frac{2}{3}YX=0,\\\
&F_{6}=-\frac{5}{2}-\frac{2}{3}Y+k^{2}(C-Z)+k^{4}ZC-\frac{2}{3}k^{2}Y^{2}=0.\end{split}$
(4.12)
There are several approaches to to deal with this system. One can easily
calculate the Taylor series for $A,B,C,X,Y,Z$ in powers of $k^{2}$ at the
point $k=0$. In application to (4.11) this is exactly the Chapman–Enskog
series (the Taylor series for $\sigma$ and $q$). To find the linear and
quadratic in $k$ terms in (4.11) we need just a zeroth approximation for
$A,B,C,X,Y,Z$ from (LABEL:invariance131):
$A=B=-\frac{4}{3},\;C=\frac{2}{3},\;X=0,\;Y=-\frac{15}{4},\;Z=\frac{7}{4}.$
(4.13)
This is the Burnett approximation:
$\displaystyle\sigma_{k}$ $\displaystyle=$
$\displaystyle-\frac{4}{3}iku_{k}+\frac{4}{3}k^{2}\rho_{k}-\frac{2}{3}k^{2}T_{k},$
$\displaystyle q_{k}$ $\displaystyle=$
$\displaystyle-\frac{15}{4}ikT_{k}-\frac{7}{4}k^{2}u_{k}$
The dispersion relation for this Burnett approximation coincides with the one
obtained by Bobylev [9] from the Boltzmann equation for Maxwell molecules, and
the short waves are unstable in this approximation.
Direct Newton’s iterations produce more sensible results. Thus, starting from
$A=B=C=X=Y=Z=0$ we get the first iteration
$\displaystyle A_{1}$ $\displaystyle=$
$\displaystyle-20\frac{141k^{2}+20}{867k^{2}k^{2}+2105k^{2}+300},$
$\displaystyle B_{1}$ $\displaystyle=$
$\displaystyle-20\frac{459k^{2}k^{2}+810k^{2}+100}{3468k^{2}k^{2}k^{2}+12755k^{2}k^{2}+11725k^{2}+1500},$
$\displaystyle C_{1}$ $\displaystyle=$
$\displaystyle-10\frac{51k^{2}k^{2}-485k^{2}-100}{3468k^{2}k^{2}k^{2}+12755k^{2}k^{2}+11725k^{2}+1500},$
$\displaystyle X_{1}$ $\displaystyle=$
$\displaystyle-\frac{375k^{2}(21k^{2}-5)}{2(3468k^{2}k^{2}k^{2}+12755k^{2}k^{2}+11725k^{2}+1500)},$
$\displaystyle Y_{1}$ $\displaystyle=$
$\displaystyle-\frac{225(394k^{2}k^{2}+685k^{2}+100)}{4(3468k^{2}k^{2}k^{2}+12755k^{2}k^{2}+11725k^{2}+1500)},$
$\displaystyle Z_{1}$ $\displaystyle=$
$\displaystyle-15\frac{153k^{2}+35}{867k^{2}k^{2}+2105k^{2}+300}.$
The corresponding hydrodynamics are non-local but stable and were first
obtained by a partial summation (regularization) of the Chapman–Enskog series
[49].
A numerical solution of the invariance equation (LABEL:invariance131) is also
straightforward and does not produce any serious problem. Selection of the
proper (Chapman–Enskog) branch of the solution, is set by the asymptotics:
$\omega\to 0$ when $k\to 0$.
The dispersion equation for frequency $\omega$ is
$\begin{split}\omega^{3}&-k^{2}\left(\frac{2}{3}Y+A\right)\omega^{2}\\\
&+k^{2}\left(\frac{5}{3}-\frac{2}{3}k^{2}Z-\frac{2}{3}k^{2}C-k^{2}B+\frac{2}{3}k^{2}AY+\frac{2}{3}k^{2}k^{2}CZ\right)\omega\\\
&+\frac{2}{3}k^{2}(k^{2}X-k^{2}Y+k^{2}k^{2}BY-k^{2}k^{2}XC)=0.\end{split}$
(4.14)
Figure 4. The dispersion relation for the linearized 1D Grad system (4.10).
The solution for the whole kinetic system (4.10) features five $\omega$’s,
while the motions on the hydrodynamic invariant manifold has three of them for
each $k<k_{c}$ and destroys for $k\geq k_{c}$. The bold solid line shows the
hydrodynamic acoustic mode (two complex conjugated roots). The bold dashed
line for $k<k_{c}$ is the hydrodynamic diffusion mode (a real root). At
$k=k_{c}$ this line meets a real root of nonhydrodynamic mode (thin dash-dot
line) and for $k>k_{c}$ they turn into a couple of complex conjugated roots
(bold double-dashed line at $k>k_{c}$). The four-point stars correspond to the
third Newton iteration for the diffusion mode. A dash-and-dot line at the
bottom of the plot shows the isolated non-hydrodynamic mode (single real root
of (2.25).
The real-valued solution to the invariance equation (LABEL:invariance131) does
not exist for sufficiently large $k$. (A telling simple example of such a
behavior of real algebraic sets gives the equation $k^{2}(1-k^{2})+A^{2}=0$.)
Above a critical value $k_{c}\approx 0.3023$, the Chapman–Enskog branch in
(LABEL:invariance131) disappears, and two complex conjugated solutions emerge.
This situation becomes clear if we look at the dispersion curves (Fig. 4). For
$k<k_{c}$ the Chapman–Enskog branch of the dispersion relation consists of
three hydrodynamic modes starting from 0 at $k=0$. Two non-hydrodynamic modes
start from strictly negative values at $k=0$ and are real-valued. They
describe the relaxation to the hydrodynamic invariant manifold from the
initial conditions outside this manifold. (This is, in other words, relaxation
of the non-hydrodynamic variables, $\sigma_{k}$ and $q_{k}$ to their values
$\sigma_{k}(\rho_{k},u_{k},T_{k},k)$ and $q_{k}(\rho_{k},u_{k},T_{k},k)$.) For
$k<k_{c}$ the non-hydrodynamic modes are real-valued, the relaxation goes
exponentially, without damped oscillations. At $k=k_{c}$, one root from the
non-hydrodynamic branch crosses a real-valued root of the hydrodynamic branch
and they together transform into a couple of complex conjugated roots when
$k>k_{c}$. It is impossible to capture two pairs of complex modes by an
equation for three macroscopic variables and, at the same time, it is
impossible to separate two complex conjugated modes between two systems of
real-valued equations.
For small $k$, when the separation of time between the “fast” collision term
and the “not-so-fast” advection is significant, there is an essential
difference between the relaxation of hydrodynamic and non-hydrodynamic
variables: $\rho$ and $u$ do not change in collision and their relaxation is
relatively slow, but $\sigma$ and $q$ are directly affected by collisions and
their relaxation to $\sigma_{k}(\rho_{k},u_{k},T_{k},k)$ and
$q_{k}(\rho_{k},u_{k},T_{k},k)$ is fast. Nevertheless, when $k$ grows and
achieves $k_{c}$ the difference between the hydrodynamic and non-hydrodynamic
variables becomes less pronounced. In such a case, the 4-dimensional invariant
manifold may describe the relaxation better. For this purpose, we can create
the invariance equation for an extended list of four ‘hydrodynamic variables’
and repeat the construction. Instead of the selection of the Chapman–Enskog
branch only, we have to select a continuous branch which includes the roots
with $\omega\to 0$ when $k\to 0$.
The 2D algebraic manifold given by the dispersion equation (4.14) and the
invariance equation (LABEL:invariance131) represents the important properties
of the hydrodynamic invariant manifold (see Fig 4). In particular, the crucial
question is the existence of the Chapman–Enskog branch and the description of
the connected component of this curve which includes the germ of the
Chapman–Enskog branch near $k=0$.
Iterations of the Newton method for the invariance equation converge fast to
the solution with singularity. For $k<k_{c}$ the corresponding attenuation
rates converge to the exact solution and for $k>k_{c}$ the real part of the
diffusion mode ${\rm Re}\omega\to-\infty$ with Newton’s iterations (Fig. 4).
The corresponding limit system has the infinitely fast decay of the diffusion
mode when $k>k_{c}$. This regularization of singularities by the infinite
dissipation is quite typical for the application of the Newton method to
solution of the invariance equation. The ‘solid jet’ limit for the extremely
fast compressions gives us another example [55] (see also Sec. 5.2).
### 4.4. Invariant manifolds, entanglement of hydrodynamic and non-
hydrodynamic modes and saturation of dissipation for the 3D 13 moments Grad
system
The thirteen moments linear Grad system consists of 13 linearized PDE’s (1.2),
(1.3) giving the time evolution of the hydrodynamic fields (density $\rho$,
velocity vector field $u$, and temperature $T$) and of higher-order
distinguished moments: five components of the symmetric traceless stress
tensor $\sigma$ and three components of the heat flux $q$ [69]. With this
example, we conclude the presentation of exact hydrodynamic manifolds for
linearized Grad models.
A point of departure is the Fourier transform of the linearized three-
dimensional Grad’s thirteen-moment system:
$\displaystyle\partial_{t}\rho_{k}$ $\displaystyle=$
$\displaystyle-i\mbox{\boldmath$k$}\cdot\mbox{\boldmath$u$}_{k},$
$\displaystyle\partial_{t}\mbox{\boldmath$u$}_{k}$ $\displaystyle=$
$\displaystyle-i\mbox{\boldmath$k$}\rho_{k}-i\mbox{\boldmath$k$}T_{k}-i\mbox{\boldmath$k$}\cdot\mbox{\boldmath$\sigma$}_{k},$
$\displaystyle\partial_{t}T_{k}$ $\displaystyle=$
$\displaystyle-\frac{2}{3}i\mbox{\boldmath$k$}\cdot(\mbox{\boldmath$u$}_{k}+\mbox{\boldmath$q$}_{k}),$
$\displaystyle\partial_{t}\mbox{\boldmath$\sigma$}_{k}$ $\displaystyle=$
$\displaystyle-2i\overline{\mbox{\boldmath$k$}\mbox{\boldmath$u$}_{k}}-\frac{4}{5}i\overline{\mbox{\boldmath$k$}\mbox{\boldmath$q$}_{k}}-\mbox{\boldmath$\sigma$}_{k},$
$\displaystyle\partial_{t}\mbox{\boldmath$q$}_{k}$ $\displaystyle=$
$\displaystyle-\frac{5}{2}i\mbox{\boldmath$k$}T_{k}-i\mbox{\boldmath$k$}\cdot\mbox{\boldmath$\sigma$}_{k}-\frac{2}{3}\mbox{\boldmath$q$},$
where $k$ is the wave vector, $\rho_{k}$, $\mbox{\boldmath$u$}_{k}$ and
$T_{k}$ are the Fourier images for density, velocity and temperature,
respectively, and $\mbox{\boldmath$\sigma$}_{k}$ and $\mbox{\boldmath$q$}_{k}$
are the nonequilibrium traceless symmetric stress tensor
($\overline{\mbox{\boldmath$\sigma$}}=\mbox{\boldmath$\sigma$}$) and heat flux
vector components, respectively.
Decompose the vectors and tensors into the parallel (longitudinal) and
orthogonal (lateral) parts with respect to the wave vector $k$, because the
fields are rotationally symmetric around any chosen direction $k$. A unit
vector in the direction of the wave vector is
$\mbox{\boldmath$e$}=\mbox{\boldmath$k$}/k$, $k=|\mbox{\boldmath$k$}|$, and
the corresponding decomposition is
$\mbox{\boldmath$u$}_{k}=u_{k}^{\|}\,\mbox{\boldmath$e$}+\mbox{\boldmath$u$}_{k}^{\perp}$,
$\mbox{\boldmath$q$}_{k}=q_{k}^{\|}\,\mbox{\boldmath$e$}+\mbox{\boldmath$q$}_{k}^{\perp}$,
and
$\mbox{\boldmath$\sigma$}_{k}=\frac{3}{2}\sigma_{k}^{\|}\overline{\mbox{\boldmath$e$}\mbox{\boldmath$e$}}+2\mbox{\boldmath$\sigma$}_{k}^{\perp}$,
where $\mbox{\boldmath$e$}\cdot\mbox{\boldmath$u$}_{k}^{\perp}=0$,
$\mbox{\boldmath$e$}\cdot\mbox{\boldmath$q$}_{k}^{\perp}=0$, and
$\mbox{\boldmath$e$}\mbox{\boldmath$e$}:\mbox{\boldmath$\sigma$}_{k}^{\perp}=0$.
In these variables, the linearized 3D 13-moment Grad system decomposes into
two closed sets of equations, one for the longitudinal and another for the
lateral modes. The equations for $\rho_{k}$, $u_{k}^{\|}$, $T_{k}$,
$\sigma_{k}^{\|}$, and $q_{k}^{\|}$ coincide with the 1D Grad system (4.10)
from the previous section (the difference is just in the superscript ∥). For
the lateral modes we get
$\begin{split}\partial_{t}\mbox{\boldmath$u$}_{k}^{\perp}&=-ik\,\mbox{\boldmath$e$}\cdot\mbox{\boldmath$\sigma$}_{k}^{\perp},\\\
\partial_{t}\mbox{\boldmath$\sigma$}_{k}^{\perp}&=-ik\overline{\mbox{\boldmath$e$}\mbox{\boldmath$u$}_{k}^{\perp}}-\frac{2}{5}ik\overline{\mbox{\boldmath$e$}\mbox{\boldmath$q$}_{k}^{\perp}}-\mbox{\boldmath$\sigma$}_{k}^{\perp},\\\
\partial_{t}\mbox{\boldmath$q$}_{k}^{\perp}&=-ik\,\mbox{\boldmath$e$}\cdot\mbox{\boldmath$\sigma$}_{k}^{\perp}-\frac{2}{3}\mbox{\boldmath$q$}_{k}^{\perp}.\end{split}$
(4.15)
The hydrodynamic invariant manifold for these decoupled systems is a direct
product of the invariant manifolds for (4.10) and for (4.15). The
parametrization (4.11), the invariance equation (LABEL:invariance131), the
dispersion equation for exact hydrodynamics (4.14) and the plots of the
attenuation rates (Fig. 4) for (4.10) are presented in the previous section.
For the lateral modes the hydrodynamic variables consist of the 2D vector
$\mbox{\boldmath$u$}_{k}^{\perp}$. We use the general expression (4.7) and
take into account the rotational symmetry for the parametrization of the non-
hydrodynamic variables $\mbox{\boldmath$\sigma$}_{k}^{\perp}$ and
$\mbox{\boldmath$q$}_{k}^{\bot}$ by the hydrodynamic ones:
$\mbox{\boldmath$\sigma$}_{k}^{\perp}=ikD(k^{2})\overline{\mbox{\boldmath$e$}\mbox{\boldmath$u$}_{k}^{\perp}},\;\;\mbox{\boldmath$q$}_{k}^{\bot}=-k^{2}U(k^{2})\mbox{\boldmath$u$}_{k}^{\perp}.$
(4.16)
Figure 5. The dispersion relation for the linearized 3D 13 moment Grad system
(1.2), (1.3). The bold solid line shows the hydrodynamic acoustic mode (two
complex conjugated roots). The bold dotted line represents the shear mode
(double degenerated real-valued root). The bold dashed line for $k<k_{c}$ is
the hydrodynamic diffusion mode (a real-valued root). At $k=k_{c}$ this line
meets a real-valued root of non-hydrodynamic mode (thin dash-and-dot line) and
for $k>k_{c}$ they turn into a couple of complex conjugated roots (bold
double-dashed line at $k>k_{c}$). Dash-and-dot lines at the bottom of the plot
show the separated non-hydrodynamic modes. All the modes demonstrate the
saturation of dissipation.
There are two unknown scalar real-valued functions here: $D(k^{2})$ and
$U(k^{2})$. We equate the microscopic and macroscopic time derivatives of the
non-hydrodynamic variables and get the invariance conditions:
$\begin{split}&\frac{\partial\mbox{\boldmath$\sigma$}_{k}^{\perp}}{\partial\mbox{\boldmath$u$}_{k}^{\perp}}\cdot(-ik\mbox{\boldmath$e$}\cdot\mbox{\boldmath$\sigma$}_{k}^{\perp})=-ik\overline{\mbox{\boldmath$e$}\mbox{\boldmath$u$}_{k}^{\perp}}-\frac{2}{5}ik\overline{\mbox{\boldmath$e$}\mbox{\boldmath$q$}_{k}^{\perp}}-\mbox{\boldmath$\sigma$}_{k}^{\perp},\\\
&\frac{\partial\mbox{\boldmath$q$}_{k}^{\perp}}{\partial\mbox{\boldmath$u$}_{k}^{\perp}}\cdot(-ik\mbox{\boldmath$e$}\cdot\mbox{\boldmath$\sigma$}_{k}^{\perp})=-ik\,\mbox{\boldmath$e$}\cdot\mbox{\boldmath$\sigma$}_{k}^{\perp}-\frac{2}{3}\mbox{\boldmath$q$}_{k}^{\perp},\end{split}$
(4.17)
We substitute here $\mbox{\boldmath$\sigma$}_{k}^{\perp}$ and
$\mbox{\boldmath$q$}_{k}^{\perp}$ by the expressions (4.16) and derive the
algebraic invariance equation for $D$ and $U$, which can be transformed into
the form:
$\begin{split}&15k^{4}D^{3}+25k^{2}D^{2}+(10+21k^{2})D+10=0,\\\
&U=-\frac{3D}{2+3k^{2}D}.\end{split}$ (4.18)
The solution of the cubic equation (4.18) with the additional condition
$D(0)=-1$ matches the Navier-Stokes asymptotics and is real-valued for all
$k^{2}$ [20]. The dispersion equation gives a twice-degenerated real-valued
shear mode. All 13 modes for the three-dimensional, 13 moment linearized Grad
system are presented in Fig. 5 with 5 hydrodynamic and 8 non-hydrodynamic
modes. This plot includes also 8 modes (3 hydrodynamic and 5 non-hydrodynamic
ones) for the one-dimensional system (4.10). Entanglement between hydrodynamic
and non-hydrodynamics modes appears at the same critical value of $k\approx
0.3023$ and the exact hydrodynamics does not exist for larger $k$.
### 4.5. Algebraic hydrodynamic invariant manifold for the linearized
Boltzmann and BGK equations: separation of hydrodynamic and non-hydrodynamic
modes
The entanglement of the hydrodynamic and non-hydrodynamic modes at large wave
vectors $k$ destroys the exact hydrodynamic for the Grad moment equations. We
conjecture that this is the catastrophe of the applicability of the moment
equations and the hydrodynamic manifolds are destroyed together with the Grad
approximation. It is plausible that if the linearized collision operator has a
spectral gap (see a review in [119]) between the five time degenerated zero
and other eigenvalues then the algebraic hydrodynamic invariant manifold
exists for all $k$. This remains an open question but the numerical
calculations of the hydrodynamic invariant manifold available for the
linearized kinetic equation (1.1) with the BGK collision operator [7, 54]
support this conjecture [88].
The incompressible hydrodynamic limit for the scaled solutions of the BGK
equation was proven in 2003 [126].
The linearized kinetic equation (1.1) has the form
$\partial_{t}f+\mbox{\boldmath$v$}\cdot\nabla_{x}f=Lf,$ (4.19)
where $f(t,\mbox{\boldmath$v$},\mbox{\boldmath$x$})$ is the deviation of the
distribution function from its equilibrium value $f^{*}(\mbox{\boldmath$v$})$,
$L$ is the linearized kinetic operator. Operator $L$ is symmetric with respect
to the entropic inner product
$\langle\varphi,\psi\rangle_{f^{*}}=\int\frac{\varphi(\mbox{\boldmath$v$})\psi(\mbox{\boldmath$v$})}{f^{*}(\mbox{\boldmath$v$})}\,{\mathrm{d}}^{3}\mbox{\boldmath$v$}.$
(4.20)
In the $L_{2}$ space with this inner product, $\ker L=({\rm im}L)^{\bot}$ is a
finite dimensional subspace. It is spanned by five functions
$f^{*}(\mbox{\boldmath$v$}),\mbox{\boldmath$v$}f^{*}(\mbox{\boldmath$v$}),v^{2}f^{*}(\mbox{\boldmath$v$}).$
The hydrodynamic variables (for the given $t$ and $x$) are the inner products
of these functions on $f(t,\mbox{\boldmath$x$},\mbox{\boldmath$v$})$, but it
is more convenient to use the orthonormal basis with respect to the product
$\langle\cdot,\cdot\rangle_{f*}$,
$\varphi_{1}(\mbox{\boldmath$v$}),\ldots,\varphi_{5}(\mbox{\boldmath$v$})$.
The macroscopic variables are $M_{i}=\langle\varphi_{i},f\rangle_{f^{*}}$
($i=1,2,\ldots,5$).
It is convenient to represent $f$ in the form of the direct sum of the
macroscopic and microscopic components
$f=P_{\rm macro}f+P_{\rm micro}f,$
where
$P_{\rm
macro}f=\sum_{i}\varphi_{i}\langle\varphi_{i},f\rangle_{f^{*}},\;\;P_{\rm
micro}f=(f-\sum_{i}\varphi_{i}\langle\varphi_{i},f\rangle_{f^{*}})$
After the Fourier transformation the linearized kinetic equation is
$\partial_{t}f_{k}=-i(\mbox{\boldmath$k$},\mbox{\boldmath$v$})f_{k}+Lf_{k}.$
(4.21)
The lifting operation $\mathcal{X}(\mbox{\boldmath$k$}):M_{k}\mapsto f_{k}$
(4.2) should have the form
$\mathcal{X}(\mbox{\boldmath$k$})(M)=\sum_{i}M_{ik}\varphi_{i}(\mbox{\boldmath$v$})+\sum_{i}M_{ik}\psi_{i}(\mbox{\boldmath$k$},\mbox{\boldmath$v$}),$
where $\langle\varphi_{i},\psi_{j}\rangle_{f^{*}}=0$ for all
$i,j=1,2,\ldots,5$. We equate the microscopic and macroscopic time derivatives
(4.4) of $f$ and get the invariance equation (4.5):
$\begin{split}L\psi_{j}=&i\mbox{\boldmath$k$}\cdot[P_{\rm
micro}(\mbox{\boldmath$v$}\varphi_{j})+P_{\rm
micro}(\mbox{\boldmath$v$}\psi_{j})\\\
&-\sum_{l}\psi_{l}\langle\varphi_{l},\mbox{\boldmath$v$}\varphi_{j}\rangle_{f^{*}}-\sum_{l}\psi_{l}\langle\varphi_{l},\mbox{\boldmath$v$}\psi_{j}\rangle_{f^{*}}].\end{split}$
(4.22)
For the solution of this equation, it is important that ${\rm im}L={\rm
im}P_{\rm micro}$ and the both operators $L$ and $L^{-1}$ are defined and
bounded on this microscopic subspace. The linearized BGK collision integral is
simply $L=-P_{\rm micro}$ (the relaxation parameter $\epsilon=1$) and the
invariance equation has in this case an especially simple form.
In [88] the form of this equation has been analyzed further and it has been
solved numerically by several methods: the Newton iterations and continuation
in parameter $k$. The attenuation rates for the Chapman–Enskog branch have
been analyzed. All the methods have produced the same results: (i) the real-
valued hydrodynamic invariant manifold exists for all range of $k$, from zero
to large values, (ii) hydrodynamic modes are always separated from the non-
hydrodynamic modes (no entanglement effects), and (iii) the saturation of
dissipation exists for large $k$.
## 5\. Hydrodynamic invariant manifolds for nonlinear kinetics
### 5.1. 1D nonlinear Grad equation and nonlinear viscosity
In the preceding sections we represented the hydrodynamic invariant manifolds
for linear kinetic equations. The algebraic equations for these manifolds in
$k$-space have a relatively simple closed form and can be studied both
analytically and numerically. For nonlinear kinetics, the situation is more
difficult for a simple reason: it is impossible to cast the problem of the
invariant manifold in the form of a system of decoupled finite–dimensional
problems by the Fourier transform. The equations for the invariant manifolds
for the finite–dimensional nonlinear dynamics have been published by Lyapunov
in 1892 [112] but even for ODEs this is a nonlinear and rather non-standard
system of PDEs.
There are several ways to study the hydrodynamic invariant manifolds for the
nonlinear kinetics. In addition to the classical Chapman–Enskog series
expansion, we can solve the invariance equation numerically or
semi–analytically, for example, by the iterations instead of the power series.
In the next section, we demonstrate this method for the Boltzmann equation. In
this section, we follow the strategy that seems to be promising: to evaluate
the asymptotics of the hydrodynamic invariant manifolds at large gradients and
frequencies and to match these asymptotics with the first Chapman–Enskog
terms. For this purpose, we use exact summation of the “leading terms” in the
Chapman–Enskog series.
The starting point is the set of the 1D nonlinear Grad equations for the
hydrodynamic variables $\rho$, $u$ and $T$, coupled with the non-hydrodynamic
variable $\sigma$, where $\sigma$ is the $xx$-component of the stress tensor:
$\displaystyle\partial_{t}\rho$ $\displaystyle=$
$\displaystyle-\partial_{x}(\rho u);$ (5.1) $\displaystyle\partial_{t}u$
$\displaystyle=$
$\displaystyle-u\partial_{x}u-\rho^{-1}\partial_{x}p-\rho^{-1}\partial_{x}\sigma;$
(5.2) $\displaystyle\partial_{t}T$ $\displaystyle=$
$\displaystyle-u\partial_{x}T-(2/3)T\partial_{x}u-(2/3)\rho^{-1}\sigma\partial_{x}u;$
(5.3) $\displaystyle\partial_{t}\sigma$ $\displaystyle=$
$\displaystyle-u\partial_{x}\sigma-(4/3)p\partial_{x}u-(7/3)\sigma\partial_{x}u-\frac{p}{\mu(T)}\sigma.$
(5.4)
Here $p=\rho T$ and $\mu(T)$ is the temperature-dependent viscosity
coefficient. We adopt the form $\mu(T)=\alpha T^{\gamma}$, where $\gamma$
varies from $\gamma=1$ (Maxwell’s molecules) to $\gamma=1/2$ (hard spheres)
[24].
Our goal is to compute the correction to the Navier–Stokes approximation of
the hydrodynamic invariant manifold, $\sigma_{\rm NS}=-(4/3)\mu\partial_{x}u$,
for high values of the velocity. Let us consider first the Burnett correction
from (5.1)-(5.4):
$\sigma_{\rm
B}=-\frac{4}{3}\mu\partial_{x}u+\frac{8(2-\gamma)}{9}\mu^{2}p^{-1}(\partial_{x}u)^{2}-\frac{4}{3}\mu^{2}p^{-1}\partial_{x}(\rho^{-1}\partial_{x}p).$
(5.5)
Each further $n$th term of the Chapman–Enskog expansion contributes, among
others, a nonlinear term proportional to $(\partial_{x}u)^{n+1}$. Such terms
can be named the high-speed terms since they dominate the rest of the
contributions in each order of the Chapman–Enskog expansion when the
characteristic average velocity is comparable to the thermal speed. Indeed,
let $U$ be the characteristic velocity (the Mach number). Consider the scaling
$u=U\widetilde{u}$, where $\widetilde{u}=O(1)$. This velocity scaling is
instrumental to the selection of the leading large gradient terms and the
result below is manifestly Galilean–invariant.
The term $(\partial_{x}u)^{n+1}$ includes the factor $U^{n+1}$ which is the
highest possible order of $U$ among the terms available in the $n$th order of
the Chapman–Enskog expansion. Simple dimensional analysis leads to the
conclusion that such terms are of the form
$\mu(p^{-1}\mu\partial_{x}u)^{n}\partial_{x}u=\mu g^{n}\partial_{x}u,$
where $g=p^{-1}\mu\partial_{x}u$ is dimensionless. Therefore, the
Chapman–Enskog expansion for the function $\sigma$ may be formally rewritten
as:
$\sigma\\!=\\!-\\!\mu\\!\left\\{\\!\frac{4}{3}\\!-\\!\frac{8(2\\!-\\!\gamma)}{9}g\\!+\\!r_{2}g^{2}\\!+\\!\dots\\!+\\!r_{n}g^{n}\\!+\\!\dots\\!\right\\}\\!\partial_{x}u\\!+\\!\dots\\!$
(5.6)
The series in the brackets is the collection of the high-speed contributions
of interest, coming from all orders of the Chapman–Enskog expansion, while the
dots outside the brackets stand for the terms of other natures. Thus after
summation the series of the high-speed corrections to the Navier–Stokes
approximation for the Grad equations (5.1) takes the form:
$\boxed{\sigma_{\rm nl}=-\mu R(g)\partial_{x}u,}$ (5.7)
where $R(g)$ is a yet unknown function represented by a formal subsequence of
Chapman–Enskog terms in the expansion (5.6). The function $R$ can be
considered as a dynamic modification of the viscosity $\mu$ due to the
gradient of the average velocity.
Let us write the invariance equation for the representation (5.7). We first
compute the microscopic derivative of the function $\sigma_{\rm nl}$ by
substituting (5.7) into the right hand side of (5.4):
$\begin{split}\partial_{t}^{\rm micro}\sigma_{\rm
nl}&=-u\partial_{x}\sigma_{\rm
nl}-\frac{4}{3}p\partial_{x}u-\frac{7}{3}\sigma_{\rm
nl}\partial_{x}u-\frac{p}{\mu(T)}\sigma_{\rm nl}\\\
&=\left\\{-\frac{4}{3}+\frac{7}{3}gR+R\right\\}p\partial_{x}u+\dots,\end{split}$
(5.8)
where dots denote the terms irrelevant to the high speed approximation (5.7).
Second, computing the macroscopic derivative of $\sigma_{\rm nl}$ due to
(5.1), (5.2), and (5.3), we obtain:
$\partial_{t}^{\rm macro}\sigma_{\rm
nl}=-[\partial_{t}\mu(T)]R\partial_{x}u-\mu(T)\frac{{\mathrm{d}}R}{{\mathrm{d}}g}[\partial_{t}g]\partial_{x}u-\mu(T)R\partial_{x}[\partial_{t}u].$
(5.9)
In the latter expression, the time derivatives of the hydrodynamic variables
should be replaced with the right hand sides of (5.1), (5.2), and (5.3),
where, in turn, $\sigma$ should be replaced by $\sigma_{\rm nl}$ (5.7). We
find:
$\partial_{t}^{\rm macro}\sigma_{\rm
nl}=\left\\{gR+\frac{2}{3}(1-gR)\times\left(\gamma
gR+(\gamma-1)g^{2}\frac{{\mathrm{d}}R}{{\mathrm{d}}g}\right)\right\\}p\partial_{x}u+\dots$
(5.10)
Again we omit the terms irrelevant to the analysis of the leading terms.
Equating the relevant terms in (5.8) and (5.10), we obtain the approximate
invariance equation for the function $R$:
$\boxed{(1-\gamma)g^{2}\left(1-gR\right)\frac{{\mathrm{d}}R}{{\mathrm{d}}g}+\gamma
g^{2}R^{2}+\left[\frac{3}{2}+g(2-\gamma)\right]R-2=0.}$ (5.11)
It is approximate because in the microscopic derivative many terms are
omitted, and it becomes more accurate when the velocities are multiplied by a
large factor. When $g\to\pm\infty$ then the viscosity factor (5.11) $R\to 0$.
For Maxwell’s molecules ($\gamma=1$), (5.11) simplifies considerably, and
becomes the algebraic equation:
$g^{2}R^{2}+\left(\frac{3}{2}+g\right)R-2=0.$ (5.12)
The solution recovers the Navier–Stokes relation in the limit of small $g$ and
for an arbitrary $g$ it reads:
$R_{\rm MM}=\frac{-3-2g+3\sqrt{1+(4/3)g+4g^{2}}}{4g^{2}}.$ (5.13)
The function $R_{\rm MM}$ (5.13) is plotted in Fig. 6. Note that $R_{\rm MM}$
is positive for all values of its argument $g$, as is appropriate for the
viscosity factor, while the Burnett approximation to the function $R_{\rm MM}$
violates positivity.
Figure 6. Viscosity factor $R(g)$ (5.11): solid - $R(g)$ for Maxwell
molecules; dash - the Burnett approximation of $R(g)$ for Maxwell molecules;
dots - the Navier–Stokes approximation; dash-dots - Viscosity factor $R(g)$
for hard spheres, the first approximation (5.14).
For other models ($\gamma\neq 1$), the invariance equation (5.11) is a
nonlinear ODE with the initial condition $R(0)=4/3$ (the Navier–Stokes
condition). Several ways to derive analytic results are possible. One
possibility is to expand the function $R$ into powers of $g$, around the point
$g=0$. This brings us back to the original sub-series of the Chapman–Enskog
expansion (5.6)). Instead, we take advantage of the opportunity offered by the
parameter $\gamma$. Introduce another parameter $\beta=1-\gamma$, and consider
the expansion:
$R(\beta,g)=R_{0}(g)+\beta R_{1}(g)+\beta^{2}R_{2}(g)+\dots.$
Substituting this expansion into the invariance equation (5.11), we derive
$R_{0}(g)=R_{\rm MM}(g)$,
$R_{1}(g)=-g(1-gR_{0})\frac{R_{0}+g({\mathrm{d}}R_{0}/{\mathrm{d}}g)}{2g^{2}R_{0}+g+(3/2)},$
(5.14)
etc. That is, the solution for models different from Maxwell’s molecules is
constructed in the form of a series with the exact solution for the Maxwell
molecules as the leading term. For hard spheres ($\beta=1/2$), the result to
the first-order term reads: $R_{\rm HS}\approx R_{\rm MM}+(1/2)R_{1}$. The
resulting approximate viscosity factor is shown in Fig. 6 (dash-dots line).
The features of the approximation obtained are qualitatively the same as in
the case of Maxwell molecules.
Precisely the same result for the nonlinear elongational viscosity obtained
first from the Grad equations [89] was derived in [129, 44] from the solution
to the BGK kinetic equation in the regime of so-called homoenergetic extension
flow. This remarkable fact gives more credit to the derivation of hydrodynamic
manifolds from nonlinear Grad equations.
The approximate invariance equation (5.11) defines the relevant physical
solution to the viscosity factor for all values of $g$. The hydrodynamic
equations are now given by (5.1), (5.2), and (5.3), where $\sigma$ is replaced
by $\sigma_{\rm nl}$ (5.7). First, the correction concerns the nonlinear
regime, and, thus, the linearized form of the new equations coincides with the
linearized Navier–Stokes equations. Second, the solution (5.13) for Maxwell
molecules and the result of the approximation (5.14) for other models and also
the numerical solution [91] suggest that the modified viscosity $\mu R$
vanishes in the limit of very high values of the velocity gradients. However,
a cautious remark is in order since the original “kinetic” description is
Grad’s equations (5.1)-(5.4) and not the Boltzmann equation. The first Newton
iteration for the Boltzmann equation gives a singularity of viscosity at a
large negative value of divergency (see below, Sec. 5.2).
### 5.2. Approximate invariant manifold for the Boltzmann equation
#### 5.2.1. Invariance equation
We begin with writing down the invariance condition for the hydrodynamic
manifold of the Boltzmann equation. A convenient point of departure is the
Boltzmann equation (1.1) in a co-moving reference frame,
$D_{t}f=-(\mbox{\boldmath$v$}-\mbox{\boldmath$u$})\cdot\nabla_{x}f+Q(f),$
(5.15)
where $D_{t}$ is the material time derivative,
$D_{t}=\partial_{t}+\mbox{\boldmath$u$}\cdot\nabla_{x}.$ The macroscopic
(hydrodynamic) variables are:
$M=\left\\{n;n\mbox{\boldmath$u$};\frac{3nk_{\rm
B}T}{\mu}+nu^{2}\right\\}=m[f]=\int\\{1;\mbox{\boldmath$v$};v^{2}\\}f\,{\mathrm{d}}\mbox{\boldmath$v$},$
where $n$ is number density, $u$ is the flow velocity, and $T$ is the
temperature; $\mu$ is particle’s mass and $k_{\rm B}$ is Boltzmann’s constant.
These fields do not change in collisions, hence, the projection of the
Boltzmann equation on the hydrodynamic variables is
$D_{t}M=-m[(\mbox{\boldmath$v$}-\mbox{\boldmath$u$})\cdot\nabla_{x}f].$ (5.16)
For the given hydrodynamic fields $M$ the local Maxwellian $f^{\rm LM}_{M}$
(or just $f^{\rm LM}$) is the only zero of the collision integral $Q(f)$.
$f^{\rm LM}=n\left(\frac{2\pi k_{\rm
B}T}{\mu}\right)^{-3/2}\exp\left(-\frac{\mu(\mbox{\boldmath$v$}-\mbox{\boldmath$u$})^{2}}{2k_{\rm
B}T}\right).$ (5.17)
The local Maxwellian depends on space through the hydrodynamic fields.
We are looking for an invariant manifold $\mbox{\boldmath$f$}_{M}$ in the
space of distribution functions parameterized by the hydrodynamic fields. Such
a manifold is represented by a lifting map $M\mapsto f_{M}$ that maps the
hydrodynamic fields in 3D space, three functions of the space variables,
$M=\\{n(\mbox{\boldmath$x$}),\mbox{\boldmath$u$}(\mbox{\boldmath$x$}),T(\mbox{\boldmath$x$})\\}$,
into a function of six variables
$\mbox{\boldmath$f$}_{M}(\mbox{\boldmath$x$},\mbox{\boldmath$v$})$. The
consistency condition should hold:
$m[f_{M}]=M.$ (5.18)
The differential of the lifting operator at the point $M$ is a linear map
$(D_{M}\mbox{\boldmath$f$}_{M}):\delta M\to\delta f$.
It is straightforward to write down the invariance condition for the
hydrodynamic manifold: The microscopic time derivative of $f_{M}$ is given by
the right hand side of the Boltzmann equation on the manifold,
$D^{\rm
micro}_{t}f_{M}=-(\mbox{\boldmath$v$}-\mbox{\boldmath$u$})\cdot\nabla_{x}f_{M}+Q(f_{M}),$
while the macroscopic time derivative is defined by the chain rule:
$D^{\rm
macro}_{t}f_{M}=-(D_{M}f_{M})m[(\mbox{\boldmath$v$}-\mbox{\boldmath$u$})\cdot\nabla_{x}f_{M}].$
The invariance equation requires that, for any $M$, the outcome of two ways of
taking the derivative should be the same:
$\boxed{-(D_{M}\mbox{\boldmath$f$}_{M})m[(\mbox{\boldmath$v$}-\mbox{\boldmath$u$})\cdot\nabla_{x}\mbox{\boldmath$f$}_{M}]=-(\mbox{\boldmath$v$}-\mbox{\boldmath$u$})\cdot\nabla_{x}\mbox{\boldmath$f$}_{M}+Q(\mbox{\boldmath$f$}_{M})}.$
(5.19)
One more field plays a central role in the study of invariant manifolds, the
defect of invariance:
$\begin{split}\Delta_{M}&=D^{\rm macro}_{t}\mbox{\boldmath$f$}_{M}-D^{\rm
micro}_{t}\mbox{\boldmath$f$}_{M}\\\
&=-(D_{M}\mbox{\boldmath$f$}_{M})m[(\mbox{\boldmath$v$}-\mbox{\boldmath$u$})\cdot\nabla_{x}\mbox{\boldmath$f$}_{M}]+(\mbox{\boldmath$v$}-\mbox{\boldmath$u$})\cdot\nabla_{x}\mbox{\boldmath$f$}_{M}.\end{split}$
(5.20)
It measures the “non-invariance” of a manifold $\mbox{\boldmath$f$}_{M}$.
Let an approximation of the lifting operation $M\to\mbox{\boldmath$f$}_{M}$ be
given. The equation of the first iteration for the unknown correction
$\delta\mbox{\boldmath$f$}_{M}$ of $\mbox{\boldmath$f$}_{M}$ is obtained by
the linearization (We assume that the initial approximation,
$\mbox{\boldmath$f$}_{M}$, satisfies the consistency condition and $m[\delta
f]=0$.):
$(D_{M}\mbox{\boldmath$f$}_{M})m[(\mbox{\boldmath$v$}-\mbox{\boldmath$u$})\cdot\nabla_{x}\delta\mbox{\boldmath$f$}_{M}]-(\mbox{\boldmath$v$}-\mbox{\boldmath$u$})\cdot\nabla_{x}\delta\mbox{\boldmath$f$}_{M}+L\delta\mbox{\boldmath$f$}_{M}=\Delta_{M}.$
(5.21)
Here, $L_{M}$ is a linearization of $Q$ at $f_{M}^{\rm LM}$. If
$\mbox{\boldmath$f$}_{M}$ is a local equilibrium then, the integral operator
$L_{M}$ at each point $x$ is symmetric with respect to the entropic inner
product (4.20). The equation of iteration (5.21) is linear but with non-
constant in space coefficients because both $(D_{M}\mbox{\boldmath$f$}_{M})$
and $L_{M}$ depend on $x$.
It is necessary to stress that the standard Newton method does not work in
these settings. If $\mbox{\boldmath$f$}_{M}$ is not a local equilibria then
$L_{M}$ may be not symmetric and we may lose such instruments as the Fredholm
alternative. Therefore, we use in the iterations the linearized operators
$L_{M}$ at the local equilibrium and not at the current approximate
distribution $\mbox{\boldmath$f$}_{M}$ (the Newton–Kantorovich method). We
also do not include the differential of the term
$(D_{M}\mbox{\boldmath$f$}_{M})m$ in (5.21). The reason for this incomplete
linearization of the invariance equation (5.19) is that it provides
convergence to the slowest invariant manifold (at least, for linear vector
fields), and other invariant manifolds are unstable in iteration dynamics. The
complete linearization does not have this property [53, 60].
#### 5.2.2. Invariance correction to the local Maxwellian
Let us choose the local Maxwellian $\mbox{\boldmath$f$}_{M}=f^{\rm LM}$ (5.17)
as the initial approximation to the invariant manifold in (5.21). In order to
find the right hand side of this equation, we evaluate the defect of
invariance (5.20) $\Delta_{M}=\Delta^{\rm LM}$:
$\Delta^{\rm LM}=f^{\rm LM}D,$ (5.22)
where
$\boxed{\begin{aligned}
D=&\left(\frac{\mu(\mbox{\boldmath$v$}-\mbox{\boldmath$u$})^{2}}{2k_{\rm
B}T}-\frac{5}{2}\right)(\mbox{\boldmath$v$}-\mbox{\boldmath$u$})\cdot\frac{\nabla_{x}T}{T}\\\
&+\frac{\mu}{k_{\rm
B}T}\left[(\mbox{\boldmath$v$}-\mbox{\boldmath$u$})\otimes(\mbox{\boldmath$v$}-\mbox{\boldmath$u$})-\frac{1}{3}\mbox{\bf
1}(\mbox{\boldmath$v$}-\mbox{\boldmath$u$})^{2}\right]:\nabla_{x}\mbox{\boldmath$u$}.\end{aligned}}$
(5.23)
Note that there is no “smallness” parameter involved in the present
consideration, the defect of invariance of the local Maxwellian is neither
“small” or “large” by itself. We now proceed with finding a correction $\delta
f$ to the local Maxwellian on the basis of the linearized equation (5.21)
supplemented with the consistency condition,
$m[\delta f]=0.$ (5.24)
Note that, if we introduce the formal large parameter,
$L\leftarrow\epsilon^{-1}L$ and look at the leading-order correction $\delta
f\leftarrow\epsilon\delta f$, disregarding all the rest in equation (5.21), we
get a linear non-homogeneous integral equation,
$\begin{split}\Lambda(\delta f/f^{\rm
LM})=&\left(\frac{\mu(\mbox{\boldmath$v$}-\mbox{\boldmath$u$})^{2}}{2k_{\rm
B}T}-\frac{5}{2}\right)(\mbox{\boldmath$v$}-\mbox{\boldmath$u$})\cdot\frac{\nabla_{x}T}{T}\\\
&+\frac{\mu}{k_{\rm
B}T}\left[(\mbox{\boldmath$v$}-\mbox{\boldmath$u$})\otimes(\mbox{\boldmath$v$}-\mbox{\boldmath$u$})-\frac{1}{3}\mbox{\bf
1}(\mbox{\boldmath$v$}-\mbox{\boldmath$u$})^{2}\right]:\nabla_{x}\mbox{\boldmath$u$},\end{split}$
(5.25)
where
$\Lambda\varphi=\int
w(\mbox{\boldmath$v$}^{\prime},\mbox{\boldmath$v$}_{1}^{\prime}|\mbox{\boldmath$v$},\mbox{\boldmath$v$}_{1})f^{\rm
LM}(\mbox{\boldmath$v$}_{1})[\varphi(\mbox{\boldmath$v$}_{1}^{\prime})+\varphi(\mbox{\boldmath$v$}^{\prime})-\varphi(\mbox{\boldmath$v$})-\varphi(\mbox{\boldmath$v$}_{1})]{\mathrm{d}}\mbox{\boldmath$v$}_{1}^{\prime}{\mathrm{d}}\mbox{\boldmath$v$}^{\prime}{\mathrm{d}}\mbox{\boldmath$v$}_{1}$
is the linearized Boltzmann collision operator ($w$ is the scattering kernel;
standard notation for the velocities before and after the binary encounter is
used). It is readily seen that (5.25) is nothing but the standard equation of
the first Chapman-Enskog approximation, whereas the consistency condition
(5.24) results in the unique solution (Fredholm alternative) to (5.25). This
leads to the classical Navier-Stokes-Fourier equations of the Chapman-Enskog
method.
Thus, the first iteration (5.21) for the solution of the invariance equation
(5.19) with the local Maxwellian as the initial approximation is matched to
the first Chapman-Enskog correction to the local Maxwellian. However, equation
(5.21) is much more complicated than its Chapman-Enskog limit: equation (5.21)
is linear but integro-differential (rather than just the linear integral
equation (5.25)), with coefficients varying in space through both
$(D_{M}f^{\rm LM})$ and $L$. We shall now describe a micro-local approach for
solving (5.21).
#### 5.2.3. Micro-local techniques for the invariance equation
Introducing $\delta f=f^{\rm LM}\varphi$, equation (5.21) for the local
Maxwellian initial approximation can be cast in the following form,
$\Lambda^{*}\varphi-(\mbox{\boldmath$V$}^{*}\cdot\nabla)\varphi=D,$ (5.26)
where the enhanced linearized collision integral $\Lambda^{*}$ and the
enhanced free flight operator $(\mbox{\boldmath$V$}^{*}\cdot\nabla)$ act as
follows: Let us denote $\Pi$ the projection operator ($\Pi^{2}=\Pi$),
$\Pi g=\left(f^{\rm LM}\right)^{-1}D_{M}f^{\rm LM}m[f^{\rm LM}g].$ (5.27)
Then in (5.26) we have:
$\displaystyle\Lambda^{*}\varphi$ $\displaystyle=$
$\displaystyle\Lambda\varphi+(\Pi-1)(r\varphi),$ $\displaystyle r$
$\displaystyle=$
$\displaystyle(\mbox{\boldmath$v$}-\mbox{\boldmath$u$})\cdot\frac{\nabla_{x}n}{n}+\frac{\mu}{k_{\rm
B}T}(\mbox{\boldmath$v$}-\mbox{\boldmath$u$})\otimes(\mbox{\boldmath$v$}-\mbox{\boldmath$u$}):\nabla_{x}\mbox{\boldmath$u$}$
$\displaystyle+\left(\frac{\mu(\mbox{\boldmath$v$}-\mbox{\boldmath$u$})^{2}}{2k_{\rm
B}T}-\frac{3}{2}\right)(\mbox{\boldmath$v$}-\mbox{\boldmath$u$})\cdot\frac{\nabla_{x}T}{T},$
$\displaystyle(\mbox{\boldmath$V$}^{*}\cdot\nabla)\varphi$ $\displaystyle=$
$\displaystyle(1-\Pi)((\mbox{\boldmath$v$}-\mbox{\boldmath$u$})\cdot\nabla_{x}\varphi).$
The structure of the invariance equation (5.26) suggests the way of inverting
the enhanced operator $\Lambda^{*}-(\mbox{\boldmath$V$}^{*}\cdot\nabla)$:
* •
Step 1: Discard the enhanced free flight operator. The resulting local in
space linear integral equation, $\Lambda^{*}[\varphi]=D$, is similar to the
Chapman-Enskog equation (5.25), and has unique solution by the Fredholm
alternative:
$\varphi_{\rm
loc}(\mbox{\boldmath$x$})=\left(\Lambda_{\textbf{\mbox{\boldmath$x$}}}^{*}\right)^{-1}[D(\mbox{\boldmath$x$})].$
(5.28)
Here we have explicitly indicated the space variables in order to stress the
fact of locality. (For a given $x$, both $D(\mbox{\boldmath$x$})$ and
$\varphi_{\rm loc}(\mbox{\boldmath$x$})$ are functions of $x$ and $v$ and
$\Lambda_{\textbf{\mbox{\boldmath$x$}}}$ is an integral in $v$ operator.)
* •
Step 2: Fourier-transform the local solution:
$\hat{\varphi}_{\rm loc}(\mbox{\boldmath$k$})=\int
e^{-i{\textbf{\mbox{\boldmath$k$}}\cdot\textbf{\mbox{\boldmath$x$}}}}\varphi_{\rm
loc}(\mbox{\boldmath$x$}){\mathrm{d}}\mbox{\boldmath$x$}.$ (5.29)
* •
Step 3: Replace the Fourier-transformed enhanced free flight operator with its
main symbol and solve the linear integral equation:
$[\Lambda_{\textbf{\mbox{\boldmath$x$}}}^{*}+i(\mbox{\boldmath$V$}_{\textbf{\mbox{\boldmath$x$}}}^{*}\cdot\mbox{\boldmath$k$})][\hat{\varphi}(\mbox{\boldmath$x$},\mbox{\boldmath$k$})]=\hat{D}(\mbox{\boldmath$x$},\mbox{\boldmath$k$}),$
(5.30)
where
$\hat{D}(\mbox{\boldmath$x$},\mbox{\boldmath$k$})=\Lambda_{\textbf{\mbox{\boldmath$x$}}}^{*}[\hat{\varphi}_{\rm
loc}(\mbox{\boldmath$k$})].$ (5.31)
* •
Step 4: Back-transform the result:
$\varphi=(2\pi)^{-3}\int
e^{i{\textbf{\mbox{\boldmath$k$}}\cdot\textbf{\mbox{\boldmath$x$}}}}\hat{\varphi}(\mbox{\boldmath$x$},\mbox{\boldmath$k$}){\mathrm{d}}\mbox{\boldmath$k$};$
(5.32)
the resulting $\varphi$ is a function of $x$ and $v$.
Several comments are in order here. The above approach to solving the
invariance equation is the realization of the Fourier integral operator and
parametrix expansion techniques [131, 142]. The equation appearing in Step 3
is in fact the first term of the parametrix expansion. At each step of the
algorithm, one needs to solve linear integral equations of the type familiar
from the standard literature on the Boltzmann equation. Solutions at each step
are unique by the Fredholm alternative. In practice, a good approximation for
such linear integral equations is achieved by a projection on a finite-
dimensional basis. Even with these approximations, evaluation of the
correction to the local Maxwellian remains rather involved. Nevertheless,
several results in limiting cases were obtained, and are reviewed below.
Figure 7. Acoustic dispersion curves for the frequency-response nonlocal
approximation (5.33) (solid line) and for the Burnett approximation of the
Chapman-Enskog expansion [9] (dashed line). Arrows indicate the direction of
increase of $k^{2}$.
For the unidirectional flow near the global equilibrium
($n=n_{0},\mbox{\boldmath$u$}=0,T=T_{0}$) for Maxwell’s molecules the
iteration gives the following expressions for the $xx$ component of the stress
tensor $\sigma$ and the $x$ component of the heat flux $q$ for 1D solutions
(in the corresponding dimensionless variables (1.4)):
$\boxed{\begin{aligned}
\sigma&=-\frac{2}{3}n_{0}T_{0}\left(1-\frac{2}{5}{\partial_{x}^{2}}\right)^{-1}\left(2{\partial_{x}u}-3{\partial_{x}^{2}T}\right);\\\
q&=-\frac{5}{4}n_{0}T^{3/2}_{0}\left(1-\frac{2}{5}{\partial_{x}^{2}}\right)^{-1}\left(3{\partial_{x}T}-\frac{8}{5}{\partial_{x}^{2}u}\right).\end{aligned}}$
(5.33)
The corresponding dispersion curves are presented in Fig. 7, where the
saturation effect is obvious.
Already at the first iteration the nonlinear terms are strongly coupled with
the nonlocality in expressions for $\sigma$ in $q$ (see [53, 60]). Viscosity
tends to positive infinity for high speed of compression (large negative ${\rm
div}\mbox{\boldmath$u$}$). In other words, the flow becomes “infinitely
viscous” when ${\partial_{x}u}$ approaches the critical negative value
$-u_{x}^{*}$. This infinite viscosity threshold prevents a transfer of the
flow into nonphysical region of negative viscosity if
${\partial_{x}u}<-u_{x}^{*}$ because of the “infinitely strong damping” at
$-u_{x}^{*}$.
The large positive values of $\partial_{x}u$ means that the gas diverges
rapidly, and the flow becomes nonviscid because the particles retard to
exchange their momentum. On the contrary, its negative values (near
$-u^{*}_{x}$ ) describe an extremely strong compression of the flow, which
results in a ‘solid jet’ limit with an infinite viscosity [55].
As an example, we present the result of the above micro-local analysis for the
part of the stress tensor $\sigma$ which does not vanish when $T$ and $n$ are
fixed:
$\boxed{\begin{aligned}
\sigma(x)&=-\frac{1}{6\pi}n(x)\int^{+\infty}_{-\infty}{\mathrm{d}}y\int^{+\infty}_{-\infty}{\mathrm{d}}k\exp(ik(x-y))\frac{2}{3}{\partial_{y}u(y)}\\\
&\times\left[\left(n(x)\lambda_{3}+\frac{11}{9}{\partial_{x}u(x)}\right)\left(n(x)\lambda_{4}+\frac{27}{4}{\partial_{x}u(x)}\right)+\frac{k^{2}v^{2}_{T}(x)}{9}\right]^{-1}\\\
&\times\left[\left(n(x)\lambda_{3}+\frac{11}{9}{\partial_{x}u(x)}\right)\left(n(x)\lambda_{4}+\frac{27}{4}{\partial_{x}u(x)}\right)\right.\\\
&\left.+\frac{4}{9}\left(n(y)\lambda_{4}+\frac{27}{4}{\partial_{y}u(y)}\right)v^{-2}_{T}(x)(u(x)-u(y))^{2}{\partial_{x}u(x)}\right.\\\
&\left.-\frac{2}{3}ik(u(x)-u(y)){\partial_{x}u(x)}\right]\left(n(y)\lambda_{3}+\frac{11}{9}{\partial_{y}u(y)}\right)^{-1}\\\
&+O\left({\partial_{x}\ln T(x)},{\partial_{x}\ln n(x)}\right).\end{aligned}}$
(5.34)
The answer in this form does not depend on the detailed collision model. Only
the general properties like conservation laws, $H$-theorem and Fredholm’s
alternative for the linearized collision integral are used. All the specific
information about the collision model is collected in the positive numbers
$\lambda_{3,4}$. They are represented by quadratures in [53, 60]. The
‘residual’ terms describe the part of the stress tensor governed by the
temperature and density gradients.
The simplest local approximation to this singularity in $\sigma$ has the form
$\boxed{\sigma=-\mu_{0}(T)n\left(1+\frac{\partial_{x}u}{u_{x}^{*}}\right)^{-1}{\partial_{x}u}.}$
(5.35)
For the viscosity factor $R$ (5.7) this approximation gives (compare to (5.13)
and Fig. 6).
$R=\frac{const}{1+{\partial_{x}u}/{u_{x}^{*}}}$ (5.36)
The approximations with singularities similar to (5.35) with $u_{x}^{*}=3/7$
have been also obtained by the partial summation of the Chapman–Enskog series
[49, 50].
As we can see, the invariance correction results in a strong coupling between
non-locality and non-linearity, and is far from the conventional Navier–Stokes
and Euler equation or other truncations of the Chapman–Enskog series. Results
of the micro-local correction to the local Maxwellian are quite similar to the
summation of the selected main terms of the Chapman–Enskog expansion. In
general, the question about the hydrodynamic invariant manifolds for the
Boltzmann equation remains less studied so far because the coupling between
the non-linearity and the non-locality brings about new challenges in
calculations and proofs. There is hardly a reason to expect that the invariant
manifolds for the genuine Boltzmann equation will have a nice analytic form
similar to the exactly solvable reduction problem for the linearized Grad
equations. Nevertheless, some effects persist: the saturation of dissipation
for high frequencies and the nonlocal character of the hydrodynamic equations.
## 6\. The projection problem and the entropy equation
The exact invariant manifolds inherit many properties of the original systems:
conservation laws, dissipation inequalities (entropy growth) and hyperbolicity
of the exactly reduced system follow from these properties of the original
system. The reason for this inheritance is simple: the vector field of the
original system is tangent to the invariant manifold and if $M(t)$ is a
solution to the exact hydrodynamic equations then, after the lifting
operation, $f_{M(t)}$ is a solution to the original kinetic equation.
In real-world applications, we very rarely meet the exact reduction from
kinetics to hydrodynamics and should work with the approximate invariant
manifolds. If $\mbox{\boldmath$f$}_{M}$ is not an exact invariant manifold
then a special projection problem arises [51, 59, 121]: how should we define
the projection of the vector field on the manifold $\mbox{\boldmath$f$}_{M}$
in order to preserve the most important properties, the conservation laws
(first law of thermodynamics) and the positivity of entropy production (second
law of thermodynamics). For hydrodynamics, the existence of the ‘natural’
moment projection $m$ (5.16) masks the problem.
The problem of dissipativity preservation attracts much attention in the
theory of shock waves. For strong shocks it is necessary to use the kinetic
representation, for rarefied gases the Boltzmann kinetic equation gives the
framework for studying the structure of strong shocks [26]. One of the common
heuristic ways to use the Boltzmann equation far from local equilibrium
consists of three steps:
1. (1)
Construction of a specific ansatz for the distribution function for a given
physical problem;
2. (2)
Projection of the Boltzmann equation on the ansatz;
3. (3)
Estimation and correction of the ansatz (optional).
The first and, at the same time, the most successful ansatz for the
distribution function in the shock layer was invented in the middle of the
twentieth century. It is the bimodal Tamm–Mott-Smith approximation (see, for
example, the book [26]):
$f(\mbox{\boldmath$v$},\mbox{\boldmath$x$})=f_{\rm
TMS}(\mbox{\boldmath$v$},z)=a_{-}(z)f_{-}(\mbox{\boldmath$v$})+a_{+}(z)f_{+}(\mbox{\boldmath$v$}),$
(6.1)
where $z$ is the space coordinate in the direction of the shock wave motion,
$f_{\pm}(\mbox{\boldmath$v$})$ are the downstream and the upstream Maxwellian
distributions, respectively. The macroscopic variables for the Tamm–Mott-Smith
approximation are the coefficients $a_{\pm}(z)$, the lifting operation is
given by (6.1) but is remains unclear how to project the Boltzmann equation
onto the linear manifold (6.1) and create the macroscopic equation.
To respect second law of thermodynamics and provide positivity of entropy
production, Lampis [104] used the entropy density $s$ as a new variable. The
entropy density is defined as a functional of $f(\mbox{\boldmath$v$})$,
$s(\mbox{\boldmath$x$})=-\int f(\mbox{\boldmath$x$},\mbox{\boldmath$v$})\ln
f(\mbox{\boldmath$x$},\mbox{\boldmath$v$})\,{\mathrm{d}}^{3}\mbox{\boldmath$v$}$.
For each distribution $f$ the time derivative of $s$ is defined by the
Boltzmann equation and the chain rule:
$\partial_{t}s=-\int\ln
f\partial_{t}f\,{\mathrm{d}}^{3}\mbox{\boldmath$v$}=\mbox{ entropy flux
}+\mbox{ entropy production }.$ (6.2)
The distribution $f$ in (6.2) is defined by the Tamm–Mott-Smith approximation:
1. (1)
Calculate the density $n$ and entropy density $s$ on the Tamm–Mott-Smith
approximation (6.1) as functions of $a_{\pm}$, $n=n(a_{+},a_{-})$,
$s=s(a_{+},a_{-})$;
2. (2)
Find the inverse transformation $a_{\pm}(n,s)$;
3. (3)
The lifting operation in the variables $n$ and $s$ is
$f_{(n,s)}(\mbox{\boldmath$v$})=a_{-}(n,s)f_{-}(\mbox{\boldmath$v$})+a_{+}(n,s)f_{+}(\mbox{\boldmath$v$}).$
This combinational of the natural projection (6.2) and the Tamm–Mott-Smith
lifting operation provides the approximate equations on the Tamm–Mott-Smith
manifold with positive entropy production. Several other projections have been
tested computationally [81]. All of them violate second law of thermodynamics
because for some initial conditions the entropy production for them becomes
negative at some points. Indeed, introduction of the entropy density as an
independent variable with the natural projection of the kinetic equation on
this variable seems to be an attractive and universal way to satisfy the
second law of thermodynamics on smooth solutions but near the equilibria this
change of variables becomes singular.
Another universal solution works near equilibria (and local equilibria). The
advection operator does not change entropy. Let us consider a linear
approximation to a space–uniform kinetic equation near equilibrium
$f^{*}(\mbox{\boldmath$v$})$: $\partial_{t}\delta f=Kf$. The second
differential of entropy generates a positive quadratic form
$\langle\varphi,\psi\
\rangle_{f^{*}}=-(D^{2}S)(\varphi,\psi)=\int\frac{\varphi\psi}{f^{*}}\,{\mathrm{d}}^{3}\mbox{\boldmath$v$}.$
(6.3)
The quadratic approximation to the entropy production is non-negative:
$-\langle\varphi,K\varphi\ \rangle_{f^{*}}\geq 0.$ (6.4)
Let $T$ be a closed linear subspace in the space of distributions. There is a
unique projector $P_{T}$ onto this subspace which does not violate the
positivity of entropy production for any bounded operator $K$ with property
(6.4): If $-\langle P_{T}\varphi,P_{T}KP_{T}\varphi\rangle_{f^{*}}\geq 0$ for
all $\varphi$, $\psi$ and all bounded $K$ with property (6.4), then $P_{T}$ is
an orthogonal projector with respect to the entropic inner product (6.3) [58,
59]. This projector acts on functions of $v$. For a local equilibrium
$f^{*}(\mbox{\boldmath$x$},\mbox{\boldmath$v$})$ the projector is constructed
for each $x$ and acts on functions
$\varphi(\mbox{\boldmath$x$},\mbox{\boldmath$v$})$ point-wise at each point
$x$. Liu and Yu [110] also used this projector in a vicinity of local
equilibria for the micro–macro decomposition in the analysis of the shock
profiles and for the study nonlinear stability of the global Maxwellian states
[111]. Robertson studied the projection onto manifolds constructed by the
conditional maximization of the entropy and the micro–macro decomposition in
the vicinity of such manifolds [123]. He obtained the orthogonal projectors
with respect to the entropic inner product and called this result “the
equation of motion for the generalized canonical density operator”.
The general case can be considered as a ‘coupling’ of the above two examples:
the introduction of the entropy density as a new variable, and the orthogonal
projector with respect to entropic inner product. Let us consider all smooth
vector fields with non-negative entropy production. The projector which
preserves the nonnegativity of the entropy production for all such fields
turns out to be unique. This is the so-called thermodynamic projector [51, 58,
59, 60]. Let us describe this projector $P$ for a given state $f$, closed
subspace $T_{f}={\rm imP_{T}}$, and the differential $(DS)_{f}$ of the entropy
$S$ at $f$. For each state $f$ we use the entropic inner product (6.3) at
$f^{*}=f$. There exists a unique vector $g(f)$ such that $\langle
g,\varphi\rangle_{f}=(DS)_{f}(\varphi)$ for all $\varphi$. This is nothing but
the Riesz representation of the linear functional $D_{x}S$ with respect to
entropic scalar product. If $g\neq 0$ then the thermodynamic projector of the
vector field $J$ is
$\boxed{P_{T}(J)=P^{\bot}(J)+\frac{g^{\|}}{\langle
g^{\|}|g^{\|}\rangle_{f}}\langle g^{\bot}|J\rangle_{f},}$ (6.5)
where $P_{T}^{\bot}$ is the orthogonal projector onto $T_{f}$ with respect to
the entropic scalar product, and the vector $g$ is split onto tangent and
orthogonal components:
$g=g^{\|}+g^{\bot};\;g^{\|}=P^{\bot}g;\>g^{\bot}=(1-P^{\bot})g.$
This projector is defined if $g^{\|}\neq 0$. If $g^{\|}=0$ (the equilibrium
point) then $J=0$ and $P(J)=P^{\bot}(J)=0$.
Figure 8. The main geometrical structures of model reduction with an
approximate invariant manifold (the ansatz manifold): $J(f)$ is the vector
field of the system under consideration, $\partial_{t}f=J(f)$, the lifting map
$M\mapsto\mbox{\boldmath$f$}_{M}$ maps a macroscopic field $M$ into the
corresponding point $\mbox{\boldmath$f$}_{M}$ on the ansatz manifold, $T_{M}$
is the tangent space to the ansatz manifold at point
$\mbox{\boldmath$f$}_{M}$, $P$ is the thermodynamic projector onto $T_{M}$ at
point $\mbox{\boldmath$f$}_{M}$, $PJ(\mbox{\boldmath$f$}_{M})$ is the
projection of the vector $J(\mbox{\boldmath$f$}_{M})$ onto tangent space
$T_{M}$, the vector field ${\mathrm{d}}M/{\mathrm{d}}t$ describes the induced
dynamics on the space of macroscopic variables,
$\Delta_{M}=(1-P)J(\mbox{\boldmath$f$}_{M})$ is the defect of invariance, the
affine subspace $\mbox{\boldmath$f$}_{M}+\ker P$ is the plane of fast motions,
and $\Delta_{M}\in\ker P$. The invariance equation is $\Delta_{M}=0$.
The selection of the projector in the form (6.5) guaranties preservation of
entropy production. The thermodynamic projector can be applied for the
projection of the kinetic equation onto the tangent space to the approximate
invariant manifold if the differential of the entropy does not annihilate the
tangent space to this manifold. (Compare to the relative entropy approach in
[128].)
Modification of the projector changes also the a simplistic picture of the
separation of motions (Fig. 1). The modified version is presented in Fig. 8.
The main differences are:
* •
The projection of the vector field $J$ on the macroscopic variables $M$ goes
in two steps,
$J(\mbox{\boldmath$f$}_{M})\mapsto PJ(\mbox{\boldmath$f$}_{M})\mapsto
m(PJ(\mbox{\boldmath$f$}_{M})),$
the first operation $J(\mbox{\boldmath$f$}_{M})\mapsto
PJ(\mbox{\boldmath$f$}_{M})$ projects $J$ onto the tangent plane $T_{M}$ to
the ansatz manifold at point $\mbox{\boldmath$f$}_{M}$ and the second is the
standard projection onto macroscopic variables $m$. Therefore, the macroscopic
equations are $\partial_{t}M=m(PJ(\mbox{\boldmath$f$}_{M}))$ instead of (2.3).
* •
The plane of fast motion is now $\mbox{\boldmath$f$}_{M}+\ker P$ instead of
$\mbox{\boldmath$f$}_{M}+\ker m$ from Fig. 1.
* •
The entropy maximizer on $\mbox{\boldmath$f$}_{M}+\ker P$ is
$\mbox{\boldmath$f$}_{M}$, exactly as the local Maxwellians
$\mbox{\boldmath$f$}_{M}^{\rm LM}$ are the entropy maximizers on
$\mbox{\boldmath$f$}_{M}^{\rm LM}+\ker m$. Thus, the entropic projector allows
us to represent an ansatz manifold as a collection of the conditional entropy
maximizers.
For details of the thermodynamic projector construction we refer to [59, 60].
Some examples with construction of the thermodynamic projector with
preservation of linear conservation laws are presented in [48].
Another possible modification is a modification of the entropy functional.
Recently, Grmela [71, 72] proposed to modify the entropy functional after each
step of the Chapman–Enskog expansion in order to transform the approximate
invariant manifold into the manifold of the conditional entropy maximizers.
This idea is very similar to the thermodynamic projector in the following
sense: any point $\varphi$ on the approximate invariant manifold is the
conditional entropy maximum on the linear manifold $\varphi+\ker P_{T}$, where
$T=T_{\varphi}$ is the tangent subspace to the manifold at point $\varphi$.
Both modifications represent the approximate invariant manifold as a set of
conditional maximizers of the entropy.
## 7\. Conclusion
It is useful to solve the invariance equation. This is a particular case of
the Newton’s famous sentence: “It is useful to solve differential equations”
(“Data æquatione quotcunque fluentes quantitæ involvente fluxiones invenire et
vice versa,” translation published by V.I. Arnold [5]). The importance of the
invariance equation has been recognized in mechanics by Lyapunov in his thesis
(1892) [112]. The problem of persistence and bifurcations of invariant
manifolds under perturbations is one of the most seminal problems in dynamics
[98, 2, 3, 4, 80, 140].
Several approaches to the computation of invariant manifolds have been
developed: Lyapunov series [112], methods of geometric singular perturbation
theory [36, 37, 84] and various power series expansions [35, 24, 8]. The graph
transformation approach was invented by Hadamard in 1901 [75] and developed
further by many authors [80, 41, 74, 99]. The Newton-type direct iteration
methods in various forms [125, 51, 52, 53, 103] proved their efficiency for
model reduction and calculation of slow manifolds in kinetics. There is also a
series of numerical methods based on the analysis of motion of an embedded
manifold along the trajectories with subtraction of the motion of the manifold
‘parallel to themselves’ [40, 62, 120, 60].
The Chapman–Enskog method [35, 24] was proposed in 1916. This method aims to
construct the invariant manifold for the Boltzmann equation in the form of a
series in powers of a small parameter, the Knudsen number $K\\!n$. This
invariant manifold is parameterized by the hydrodynamic fields (density,
velocity, temperature). The zeroth-order term of this series is the
corresponding local equilibrium. This form of the solution (the power series
and the local equilibrium zeroth term) is, at the same time, a selection rule
that is necessary to choose the hydrodynamic (or Chapman–Enskog) solution of
the invariance equation.
If we truncate the Chapman–Enskog series at the zeroth term then we get the
Euler hydrodynamic equations, the first term gives the Navier-Stokes
hydrodynamics but already the next term (Burnett) is singular and gives
negative viscosity for large divergence of the flow and instability of short
waves. Nevertheless, if we apply, for example, the Newton–Kantorovich method
[53, 62, 60] then all these singularities vanish (Sec. 5.2).
The Chapman–Enskog expansion appears as the Taylor series for the solution of
the invariance equation. Truncation of this series may approximate the
hydrodynamic invariant manifold in some limit cases such as the long wave
limit or a vicinity of the global equilibrium. Of course, the results of the
invariant manifold approach should coincide with the proven hydrodynamic
limits of the Boltzmann kinetics [6, 109, 46, 127, 128] ‘at the end of
relaxation’.
In general, there is no reason to hope that a few first terms of the Taylor
series give an appropriate global approximation of solutions of the invariance
equation (4.5). This is clearly demonstrated by the exact solutions (Sec. 3,
4).
The invariant manifold idea was present implicitly in the original Enskog and
Chapman works and in most subsequent publications and textbooks. An explicit
formulation of the invariant manifold programme for the derivation of fluid
mechanics and hydrodynamic limits from the Boltzmann equation was published by
McKean [115] (see Fig. 2 in Sec 2.1). At the same time, McKean noticed that
the problem of the invariant manifold for kinetic equations does not include
the small parameter because by the rescaling of the space dependence of the
initial conditions we can remove the coefficient in front of the collision
integral: there is no difference between the Boltzmann equations with
different $K\\!n$. Now we know that the formal ‘small’ parameter is necessary
for the selection of the hydrodynamic branch of the solutions of the
invariance equation because this equation can have many more solutions. (For
example, Lyapunov used for this purpose analyticity of the invariant manifold
and selected the zeroth approximation in the form of the invariant subspace of
the linear approximation.)
The simplest example of invariant manifold is a trajectory (invariant curve).
Therefore, the method of invariant manifold may be used for the construction
and analysis of the trajectories. This simple idea is useful and the method of
invariant manifold was applied for solution of the following problems:
* •
For analysis and correction of the Tamm–Mott-Smith approximation of strong
shock waves far from local equilibrium [51], with the Newton iterations for
corrections;
* •
For analysis of reaction kinetics [18] and reaction–diffusion equations [116];
* •
For lifting of shock waves from the piece-wise solutions of the Euler equation
to the solutions of the Boltzmann equation hear local equilibrium for small
$K\\!n$ [110];
* •
For analytical approximation of the relaxation trajectories [61]. (The method
is tested for the space-independent Boltzmann equation with various
collisional mechanisms.)
The invariant manifold approach to the kinetic part of the 6th Hilbert’s
Problem concerning the limit transition from the Boltzmann kinetics to
mechanics of continua was invented by Enskog almost a century ago, in 1916
[35]. From a physical perspective, it remains the main method for the
construction of macroscopic dynamics from dissipative kinetic equations.
Mathematicians, in general, pay less attention to this approach because
usually in its formulation the solution procedure (the algorithm for the
construction of the bulky and singular Chapman–Enskog series) is not separated
from the problem statement (the hydrodynamic invariant manifold).
Nevertheless, since the 1960s the invariant manifold statement of the problem
has been clear for some researchers [115, 53, 60].
Analysis of the simple kinetic models with algebraic hydrodynamic invariant
manifolds (Sec. 3) shows that the hydrodynamic invariant manifolds may exist
globally and the divergence of the Chapman–Enskog series does not mean the
non-existence or non-analyticity of this manifold.
The invariance equation for the more complex Grad kinetic equations
(linearized) is also obtained in an algebraic form (see Sec. 4.3 and [91, 60]
for 1D and Sec. 4.4 and [20] for 3D space). An analysis of these polynomial
equations shows that the real-valued solution of the invariance equation in
the $k$-space may break down for very short waves. This effect is caused by
the so-called entanglement of hydrodynamic and non-hydrodynamic modes.
The linearized equation with the BGK collision model [7] includes the genuine
free flight advection operator and is closer to the Boltzmann equation in the
hierarchy of simplifications. For this equation, there are numerical
indications that the hydrodynamic modes are separated from the non-
hydrodynamic ones and the calculations show that the hydrodynamic invariant
manifold may exist globally (for all values of the wave vector $k$) [88].
It seems more difficult to find a nonlinear Boltzmann equation with exactly
solvable invariance equation and summarize the Chapman–Enskog series for a
nonlinear kinetic equation exactly. Instead of this, we select in each term of
the series the terms of the main order in the power of the Mach number $M\\!a$
and exactly summarize the resulting series for the simple nonlinear 1D Grad
system (Sec. 5.1, [91, 60]). This expansion gives the dependence of the
viscosity on the velocity gradient (5.7), (5.11), (5.13).
The exact hydrodynamics projected from the invariant manifolds inherits many
useful properties of the initial kinetics: conservation laws, dissipation
inequalities, and (for the bounded lifting operators) hyperbolicity (Sec.
4.2). Also, the existence and uniqueness theorem may be valid in the
projections if it is valid for the original kinetics. In applications, for the
approximate hydrodynamic invariant manifolds, the projected equation may
violate many important properties. In this case, the change of the projector
operator solves some of these problems (Sec. 6). The construction of the
thermodynamic projector guarantees the positivity of entropy production even
in very rough approximations [51, 59].
At the present time, Hilbert’s 6th Problem is not completely solved in its
kinetic part. More precisely, there are several hypotheses we can prove or
refute. The Hilbert hypothesis has not been unambiguously formulated but
following his own works in the Boltzmann kinetics we can guess that he
expected to receive the Euler and Navier–Stokes equations as an ultimate
hydrodynamic limit of the Boltzmann equation.
Now, the Euler limit is proven for the limit $K\\!n,M\\!a\to 0$, $M\\!a\ll
K\\!n$, and the Navier–Stokes limit is proven for $K\\!n,M\\!a\to 0$,
$M\\!a\sim K\\!n$. In these limits, the flux is extremely slow and the
gradients are extremely small (the velocity, density and temperature do not
change significantly over a long distance). The system is close to the global
equilibrium. Of course, after rescaling these solutions restore some dynamics
but this rescaling erases some physically important effects. For example, it
is a simple exercise to transform an attenuation curve with saturation from
Fig. 3 into a parabola (Navier–Stokes) or even into a horizontal straight line
(no attenuation, the Euler limit) with arbitrary accuracy by the rescaling of
space and time.
We can state at present that beyond this limit, the Euler and Navier–Stokes
hydrodynamics do not provide the proper hydrodynamic limit of the Boltzmann
equation. A solution of the Boltzmann equation relaxes to the equilibrium [29]
and, on its way to equilibrium, the classical hydrodynamic limit will be
achieved as an intermediate asymptotic (after the proper rescaling). This
recently proven result fills an important gap in our knowledge about the
Boltzmann equation but from the physics perspective this is still the limit
$K\\!n,M\\!a\to 0$ (with the proof that this limit will be achieved on the
path to equilibrium).
The invariant manifold hypothesis was formulated clearly by McKean [115] (see
Sec. 2.1, Fig. 2 and Sec. 3.1): the kinetic equation admits an invariant
manifold parameterized by the hydrodynamic fields, and the Chapman–Enskog
series are the Taylor series for this manifold. Nothing is expected to be
small and no rescaling is needed. After the publication of the McKean work
(1965), this hypothesis was supported by exactly solved reduction problems,
explicitly calculated algebraic forms of the invariance equation and direct
numerical solutions of these equations for some cases like the linearized BGK
equation.
In addition to the existence of the hydrodynamic invariant manifold some
stability conditions of this manifold are needed in practice. Roughly
speaking, the relaxation to this manifold should be faster than the motion
along it. An example of such a condition gives the separation of the
hydrodynamic and non-hydrodynamic modes for linear kinetic equations (see the
examples in Sec. 4 and 3). It should be stressed that the strong separation of
the relaxation times (Fig. 1) is impossible without a small parameter. For the
“$\varepsilon=1$ approach” we can expect only some dominance of the relaxation
towards the hydrodynamic manifold over the relaxation along it.
The capillarity hypothesis was proposed very recently by Slemrod [135, 136].
He advocated the $\varepsilon=1$ approach and studied the exact sum of the
Chapman–Enskog series obtained in [57, 91]. Slemrod demonstrated that in the
balance of the kinetic energy (3.7) a viscosity term appears (3.11) and the
saturation of dissipation can be represented as the interplay between
viscosity and capillarity (Sec. 3.3).
On the basis of this idea and some heuristics about the relation between the
moment (Grad) equations and the genuine Boltzmann equation, Slemrod suggested
that the proper exact hydrodynamic equation should have the form of the
Korteweg hydrodynamics [100, 32, 134] rather than of Euler or Navier–Stokes
ones.
The capillarity–like terms appear, indeed, in the energy balance for all
hydrodynamic equations found as a projection of the kinetic equations onto the
exact or approximate invariant hydrodynamic manifolds. In that (‘wide’) sense,
the capillarity hypothesis is plausible. In the more narrow sense, as does the
validity of the Korteweg hydrodynamics, the capillarity hypothesis requires
some efforts for reformulation. The interplay between nonlinearity and
nonlocality on the hydrodynamic manifolds seems to be much more complex than
in the Korteweg equations (see, for example, Sec. 5.2, eq. (5.34), or [53,
60]). For a serious consideration of this hypothesis we have to find out for
which asymptotic assumption we expect it to be valid (if $\varepsilon=1$ then
this question is non-trivial).
In the context of the exact solution of the invariance equations, three
problems become visible:
1. (1)
To prove the existence of the hydrodynamic invariant manifold for the
linearized Boltzmann equation.
2. (2)
To prove the existence of the analytic hydrodynamic invariant manifold for the
Boltzmann equation.
3. (3)
To match the low-frequency, small gradient asymptotics of the invariant
manifold with the high-frequency, large gradient asymptotics and prove the
universality of the matched asymptotics in some limits.
The first problem seems to be not extremely difficult. For its positive
solution, the linearized collision operator should be bounded and satisfy the
spectral gap condition.
For the nonlinear Boltzmann equation, the existence of the analytic invariant
manifold seems to be plausible but the singularities in the first
Newton–Kantorovich approximation (Sec. 5.2) may give a hint about the possible
difficulties in the highly non-linear regions. In this first approximation,
flows with very high negative divergence cannot appear in the evolution of
flows with lower divergence because the viscosity tends to infinity. This
‘solid jet’ [55] effect can be considered as a sort of phase transition.
The idea of an exact hydrodynamic invariant manifold is attractive and the
approximate solutions of the invariance equation can be useful but the
possibility of elegant asymptotic solution is very attractive too. Now we know
that we do not know how to state the proper problem. Can the observable
hydrodynamic regimes be considered as solutions of a simplified hydrodynamic
equation? Here a new, yet non-mathematical notion appears, “the observable
hydrodynamic regimes”. We can speculate now, that when the analytic invariant
manifold exists, then together with the low-frequency, low-gradient
Chapman–Enskog asymptotics the high–frequency and high–gradient asymptotics of
the hydrodynamic equations are also achievable in a constructive simple form
(see examples in Sec. 3.6 and Sec. 5.1). The bold hypothesis #3 means that in
some asymptotic sense only the extreme cases are important and the behavior of
the invariant manifold between them may be substituted by matching
asymptotics. We still do not know an exact formulation of this hypothesis and
can only guess how the behaviour of the hydrodynamic solutions becomes
dependent only on the extreme cases. Some hints may be found in recent works
about the universal asymptotics of solutions of PDEs with small dissipation
[31] (which develop the ideas of Il’in proposed in the analysis of boundary
layers [83]).
We hope that problem #1 about the existence of hydrodynamic invariant
manifolds for the linearized Boltzmann equation will be solved soon, problem
#2 about the full nonlinear Boltzmann equation may be approached and solved
after the first one. We expect that the answer will be positive: hydrodynamic
invariant manifolds do exist under the spectral gap condition.
Once the first two problems will be solved then the entire object, the
hydrodynamic invariant manifold will be outlined. For this manifold, the
various asymptotic expansions could be produced, for low frequencies and
gradients, for high frequencies, and for large gradients. Matching of these
expansions and analysis of the resulting equations may give material for the
exploration of hypothesis #3. Some guesses about the resulting equations may
be formulated now, on the basis of the known results. For example we can
expect that non-locality may be reduced to the substitution of the time
derivative $\partial_{t}$ in the system of fluid dynamic equations by
$(1-W\Delta)\partial_{t}$, where $\Delta$ is the Laplace operator and $W$ is a
positive definite matrix (compare to Sec. 3.6). It seems interesting and
attractive that the resulting equations may be new and, at the same time,
simple and beautiful hydrodynamic equations.
From the mathematical perspective, the approach based on the invariance
equation now creates more questions than answers. It changes the problem
statement and the exact solutions give us some hints about the possible
answers.
## Acknowledgements
We are grateful to M. Gromov for stimulating discussion, to M. Slemrod for
inspiring comments and ideas and to L. Saint-Raymond for useful comments. IK
gratefully acknowledges support by the European Research Council (ERC)
Advanced Grant 291094-ELBM.
## About the authors
Professor Alexander N. Gorban holds a personal chair in Applied Mathematics at
the University of Leicester since 2004. He worked for Russian Academy of
Sciences, Siberian Branch (Krasnoyarsk, Russia) and ETH Zürich (Switzerland),
was a visiting professor and research scholar at Clay Mathematics Institute
(Cambridge, US), IHES (Bures–sur-Yvette, Île de France), Courant Institute of
Mathematical Sciences (NY, US) and Isaac Newton Institute for Mathematical
Sciences (Cambridge, UK). Main research interests: Dynamics of systems of
physical, Chemical and biological kinetics; Biomathematics; Data mining and
model reduction problems.
Professor Ilya Karlin is Faculty Member at the Department of Mechanical and
Process Engineering, ETH Zurich, Switzerland. He was Alexander von Humboldt
Fellow at the University of Ulm (Germany), CNR Fellow at the Institute of
Applied Mathematics CNR “M. Picone” (Rome, Italy), and Senior Lecturer in
Multiscale Modeling at the University of Southampton (England). Main research
interests: Exact and non-perturbative results in kinetic theory; Fluid
dynamics; Entropic lattice Boltzmann method; Model reduction for combustion
systems.
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|
arxiv-papers
| 2013-10-01T17:39:06 |
2024-09-04T02:49:51.871002
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "A. N. Gorban, I. Karlin",
"submitter": "Alexander Gorban",
"url": "https://arxiv.org/abs/1310.0406"
}
|
1310.0407
|
# Orthogonally additive, orthogonality preserving, holomorphic mappings
between C∗-algebras
Jorge J. Garcés [email protected] Departamento de Análisis Matemático, Facultad
de Ciencias, Universidad de Granada, 18071 Granada, Spain. , Antonio M.
Peralta [email protected] Departamento de Análisis Matemático, Facultad de
Ciencias, Universidad de Granada, 18071 Granada, Spain. , Daniele Puglisi
[email protected] Department of Mathematics and Computer Sciences,
University of Catania, Catania, 95125, Italy and María Isabel Ramírez
Departamento de Algebra y Análisis Matemático, Universidad de Almería, 04120
Almería, Spain [email protected]
###### Abstract.
We study holomorphic maps between C∗-algebras $A$ and $B$. When
$f:B_{A}(0,\varrho)\longrightarrow B$ is a holomorphic mapping whose Taylor
series at zero is uniformly converging in some open unit ball
$U=B_{A}(0,\delta)$ and we assume that $f$ is orthogonality preserving on
$A_{sa}\cap U$, orthogonally additive on $U$ and $f(U)$ contains an invertible
element in $B$, then there exist a sequence $(h_{n})$ in $B^{**}$ and Jordan
∗-homomorphisms $\Theta,\widetilde{\Theta}:M(A)\to B^{**}$ such that
$f(x)=\sum_{n=1}^{\infty}h_{n}\widetilde{\Theta}(a^{n})=\sum_{n=1}^{\infty}{\Theta}(a^{n})h_{n},$
uniformly in $a\in U$. When $B$ is abelian the hypothesis of $B$ being unital
and $f(U)\cap\hbox{inv}(B)\neq\emptyset$ can be relaxed to get the same
statement.
Authors partially partially supported by the Spanish Ministry of Economy and
Competitiveness, D.G.I. project no. MTM2011-23843, and Junta de Andalucía
grant FQM3737.
2010 MSC: Primary 46G20, 46L05; Secondary 46L51, 46E15, 46E50.
Keywords and phrases: C∗-algebra, von Neumann algebra, orthogonally additive
holomorphic functions, orthogonality preservers, orthomorphism, non-
commutative $L_{1}$-spaces.
## 1\. Introduction
The description of orthogonally additive $n$-homogeneous polynomial on
$C(K)$-spaces and on general C∗-algebras, developed by Y. Benyamini, S.
Lassalle, J.L.G. Llavona [1] and D. Pérez, and I. Villanueva [14] and C.
Palazuelos, A.M. Peralta and I. Villanueva [12], respectively (see also [5]
and [4, §3]), led Functional Analysts to study and explore orthogonally
additive holomorphic functions on $C(K)$-spaces (see [6, 10]) and subsequently
on general C∗-algebras (cf. [13]).
We recall that a mapping $f$ from a C∗-algebra $A$ into a Banach space $B$ is
said to be _orthogonally additive_ on a subset $U\subseteq A$ if for every
$a,b$ in $U$ with $a\perp b$, and $a+b\in U$ we have $f(a+b)=f(a)+f(b)$, where
elements $a,$ $b$ in $A$ are said to be _orthogonal_ (denoted by $a\perp b$)
whenever $ab^{*}=b^{*}a=0$. We shall say that $f$ is _additive on elements
having zero-product_ if for every $a,b$ in $A$ with $ab=0$ we have
$f(a+b)=f(a)+f(b)$. Having this terminology in mind, the description of all
$n$-homogeneous polynomials on a general C∗-algebra, $A,$ which are
orthogonally additive on the self adjoint part, $A_{sa}$, of $A$ reads as
follows (see section §2 for concrete definitions not explained here).
###### Theorem 1.
[12] Let $A$ be a C∗-algebra, $B$ a Banach space, $n\in\mathbb{N},$ and let
$P:A\to B$ be an $n$-homogeneous polynomial. The following statements are
equivalent:
1. $(a)$
There exists a bounded linear operator $T:A\to X$ satisfying
$P(a)=T(a^{n}),$
for every $a\in A,$ and $\|P\|\leq\|T\|\leq 2\|P\|$.
2. $(b)$
$P$ is additive on elements having zero-products.
3. $(c)$
$P$ is orthogonally additive on $A_{sa}$.$\hfill\Box$
The task of replacing $n$-homogeneous polynomials by polynomials or by
holomorphic functions involves a higher difficulty. For example, as noticed by
D. Carando, S. Lassalle and I. Zalduendo [6, Example 2.2.], when $K$ denotes
the closed unit disc in $\mathbb{C}$, there is no entire function
$\Phi:{\mathbb{C}}\to{\mathbb{C}}$ such that the mapping $h:C(K)\to C(K)$,
$h(f)=\Phi\circ f$ factors all degree-2 orthogonally additive scalar
polynomials over $C(K)$. Furthermore, similar arguments show that, defining
$P:C([0,1])\to\mathbb{C}$, $P(f)=f(0)+f(1)^{2}$, we cannot find a triplet
$(\Phi,\alpha_{1},\alpha_{2})$, where $\Phi:C[0,1]\to\mathbb{C}$ is a
∗-homomorphism and $\alpha_{1},\alpha_{2}\in\mathbb{C}$, satisfying that
$P(f)=\alpha_{1}\Phi(f)+\alpha_{2}\Phi(f^{2})$ for every $f\in C([0,1])$.
To avoid the difficulties commented above, Carando, Lassalle and Zalduendo
introduce a factorization through an $L_{1}(\mu)$ space. More concretely, for
each compact Hausdorff space $K$, a holomorphic mapping of bounded type
$f:C(K)\to\mathbb{C}$ is orthogonally additive if and only if there exist a
Borel regular measure $\mu$ on $K$, a sequence $(g_{k})_{k}\subseteq
L_{1}(\mu)$ and a holomorphic function of bounded type $h:C(K)\to L_{1}(\mu)$
such that $\displaystyle h(a)=\sum_{k=0}^{\infty}g_{k}~{}a^{k},$ and
$f(a)=\int_{K}h(a)~{}d\mu,$
for every $a\in C(K)$ (cf. [6, Theorem 3.3]).
When $C(K)$ is replaced with a general C∗-algebra $A$, a holomorphic function
of bounded type $f:A\to\mathbb{C}$ is orthogonally additive on $A_{sa}$ if and
only if there exist a positive functional $\varphi$ in $A^{*}$, a sequence
$(\psi_{n})$ in $L_{1}(A^{**},\varphi)$ and a power series holomorphic
function $h$ in $\mathcal{H}_{b}(A,A^{*})$ such that
$h(a)=\sum_{k=1}^{\infty}\psi_{k}\cdot a^{k}\hbox{ and }f(a)=\langle
1_{{}_{A^{**}}},h(a)\rangle=\int h(a)\ d\varphi,$
for every $a$ in $A$, where $1_{{}_{A^{**}}}$ denotes the unit element in
$A^{**}$ and $L_{1}(A^{**},\varphi)$ is a non-commutative $L_{1}$-space (cf.
[13]).
A very recent contribution due to Q. Bu, M.-H. Hsu, and N.-Ch. Wong [2], shows
that, for holomorphic mappings between $C(K)$, we can avoid the factorization
through an $L_{1}(\mu)$-space by imposing additional hypothesis. Before
stating the detailed result, we shall set down some definitions.
Let $A$ and $B$ be C∗-algebras. When $f:U\subseteq A\to B$ is a map and the
condition
(1) $a\perp b\Rightarrow f(a)\perp f(b)$
(respectively,
(2) $ab=0\Rightarrow f(a)f(b)=0\ )$
holds for every $a,b\in U$, we shall say that $f$ _preserves orthogonality_ or
is _orthogonality preserving_ (respectively, $f$ _preserves zero products_) on
$U$. In the case $A=U$ we shall simply say that $f$ is _orthogonality
preserving_ (respectively, $f$ _preserves zero products_). Orthogonality
preserving bounded linear maps between C∗-algebras were completely described
in [3, Theorem 17] (see [4] for completeness).
The following Banach-Stone type theorem for zero product preserving or
orthogonality preserving holomorphic functions between $C_{0}(L)$ spaces is
established by Bu, Hsu and Wong in [2, Theorem 3.4].
###### Theorem 2.
[2] Let $L_{1}$ and $L_{2}$ be locally compact Hausdorff spaces and let
$H:B_{C_{0}(L_{1})}(0,r)\to C_{0}(L_{2})$ be a bounded orthogonally additive
holomorphic function. If $H$ is zero product preserving or orthogonality
preserving, then there exist a sequence $(\mathcal{O}_{n})$ of open subsets of
$L_{2}$, a sequence $(h_{n})$ of bounded functions from
$L_{2}\cup\\{\infty\\}$ into $\mathbb{C}$ and a mapping $\varphi:L_{2}\to
L_{1}$ such that for each natural $n$ the function $h_{n}$ is continuous and
nonvanishing on $\mathcal{O}_{n}$ and
$f(a)(t)=\sum_{n=1}^{\infty}h_{n}(t)\left(a(\varphi(t))\right)^{n},(t\in
L_{2}),$
uniformly in $a\in B_{C_{0}(L_{1})}(0,r)$.$\hfill\Box$
The study developed by Bu, Hsu and Wong restricts to commutative C∗-algebras
or to orthogonality preserving and orthogonally additive, $n$-homogeneous
polynomials between general C∗-algebras. The aim of this paper is to extend
their study to holomorphic maps between general C∗-algebras. In Section 4, we
determine the form of every orthogonality preserving, orthogonally additive
holomorphic function from a general C∗-algebra into a commutative C∗-algebra
(see Theorem 16).
In the wider setting of holomorphic mappings between general C∗-algebras, we
prove the following: Let $A$ and $B$ be C∗-algebras with $B$ unital and let
$f:B_{A}(0,\varrho)\longrightarrow B$ be a holomorphic mapping whose Taylor
series at zero is uniformly converging in some open unit ball
$U=B_{A}(0,\delta)$. Suppose $f$ is orthogonality preserving on $A_{sa}\cap
U$, orthogonally additive on $U$ and $f(U)$ contains an invertible element.
Then there exist a sequence $(h_{n})$ in $B^{**}$ and Jordan ∗-homomorphisms
$\Theta,\widetilde{\Theta}:M(A)\to B^{**}$ such that
$f(x)=\sum_{n=1}^{\infty}h_{n}\widetilde{\Theta}(a^{n})=\sum_{n=1}^{\infty}{\Theta}(a^{n})h_{n},$
uniformly in $a\in U$ (see Theorem 18).
The main tool to establish our main results is a newfangled investigation on
orthogonality preserving pairs of operators between C∗-algebras developed in
Section 3. Among the novelties presented in Section 3, we find an innovating
alternative characterization of orthogonality preserving operators between
C∗-algebras which complements the original one established in [3] (see
Proposition 14). Orthogonality preserving pairs of operators are also valid to
determine orthogonality preserving operators and orthomorphisms or local
operators on C∗-algebras in the sense employed by A.C. Zaanen [19] and B.E.
Johnson [11], respectively.
## 2\. Orthogonally additive, orthogonality preserving, holomorphic mappings
on C∗-algebras
Let $X$ and $Y$ be Banach spaces. Given a natural $n$, a (continuous)
$n$-homogeneous polynomial $P$ from $X$ to $Y$ is a mapping
$P:X\longrightarrow Y$ for which there is a (continuous) multilinear symmetric
operator $A:X\times\ldots\times X\to Y$ such that $P(x)=A(x,\ldots,x),\
\text{for every}\ x\in X.$ All the polynomials considered in this paper are
assumed to be continuous. By a $0$-homogeneous polynomial we mean a constant
function. The symbol $\mathcal{P}(^{n}X,Y)$ will denote the Banach space of
all continuous $n$-homogeneous polynomials from $X$ to $Y$, with norm given by
$\displaystyle\|P\|=\sup_{\|x\|\leq 1}\|P(x)\|.$
Throughout the paper, the word operator will always stand for a bounded linear
mapping.
We recall that, given a domain $U$ in a complex Banach space $X$ (i.e. an
open, connected subset), a function $f$ from $U$ to another complex Banach
space $Y$ is said to be _holomorphic_ if the Frchet derivative of $f$ at
$z_{0}$ exists for every point $z_{0}$ in $U$. It is known that $f$ is
holomorphic in $U$ if and only if for each $z_{0}\in X$ there exists a
sequence $\left(P_{k}(z_{0})\right)_{k}$ of polynomials from $X$ into $Y$,
where each $P_{k}(z_{0})$ is $k$-homogeneous, and a neighborhood $V_{z_{0}}$
of $z_{0}$ such that the series
$\sum_{k=0}^{\infty}P_{k}(z_{0})(y-z_{0})$
converges uniformly to $f(y)$ for every $y\in V_{z_{0}}$. Homogeneous
polynomials on a C∗-algebra $A$ constitute the most basic examples of
holomorphic functions on $A$. A holomorphic function $f:X\longrightarrow Y$ is
said to be of bounded type if it is bounded on all bounded subsets of $X$, in
this case its Taylor series at zero, $f=\sum_{k=0}^{\infty}P_{k},$ has
infinite radius of uniform convergence, i.e.
$\limsup_{k\rightarrow\infty}\|P_{k}\|^{\frac{1}{k}}=0$ (compare [7, §6.2],
see also [8]).
Suppose $f:B_{X}(0,\delta)\to Y$ is a holomorphic function and let
$\displaystyle f=\sum_{k=0}^{\infty}P_{k}$ be its Taylor series at zero which
is assumed to be uniformly convergent in $U=B_{X}(0,\delta)$. Given
$\varphi\in Y^{*}$, it follows from Cauchy’s integral formula that, for each
$a\in U$, we have:
$\varphi P_{n}(a)=\frac{1}{2\pi i}\int_{\gamma}\frac{\varphi f(\lambda
a)}{\lambda^{n+1}}d\lambda,$
where $\gamma$ is the circle forming the boundary of a disc in the complex
plane $D_{\mathbb{C}}(0,r_{1}),$ taken counter-clockwise, such that
$a+D_{\mathbb{C}}(0,r_{1})a\subseteq U$. We refer to [7] for the basic facts
and definitions used in this paper.
In this section we shall study orthogonally additive, orthogonality
preserving, holomorphic mappings between C∗-algebras. We begin with an
observation which can be directly derived from Cauchy’s integral formula. The
statement in the next lemma was originally stated by D. Carando, S. Lassalle
and I. Zalduendo in [6, Lemma 1.1] (see also [13, Lemma 3]).
###### Lemma 3.
Let $f:B_{A}(0,\varrho)\longrightarrow B$ be a holomorphic mapping, where $A$
is a C∗-algebra and $B$ is a complex Banach space, and let $\displaystyle
f=\sum_{k=0}^{\infty}P_{k}$ be its Taylor series at zero, which is uniformly
converging in $U=B_{A}(0,\delta)$. Then the mapping $f$ is orthogonally
additive on $U$ (respectively, orthogonally additive on $A_{sa}\cap U$ or
additive on elements having zero-product in $U$) if, and only if, all the
$P_{k}$’s satisfy the same property. In such a case, $P_{0}=0$.$\hfill\Box$
We recall that a functional $\varphi$ in the dual of a C∗-algebra $A$ is
_symmetric_ when $\varphi(a)\in\mathbb{R}$, for every $a\in A_{sa}$.
Reciprocally, if $\varphi(b)\in\mathbb{R}$ for every symmetric functional
$\varphi\in A^{*}$, the element $b$ lies in $A_{sa}$. Having this in mind, our
next lemma also is a direct consequence of the Cauchy’s integral formula. A
mapping $f:A\to B$ between C∗-algebras is called _symmetric_ whenever
$f(A_{sa})\subseteq B_{sa}$, or equivalently, $f(a)=f(a)^{*}$, whenever $a\in
A_{sa}$.
###### Lemma 4.
Let $f:B_{A}(0,\varrho)\longrightarrow B$ be a holomorphic mapping, where $A$
and $B$ are C∗-algebras, and let $\displaystyle f=\sum_{k=0}^{\infty}P_{k}$ be
its Taylor series at zero, which is uniformly converging in
$U=B_{A}(0,\delta)$. Then the mapping $f$ is symmetric on $U$ (i.e.
$f(A_{sa}\cap U)\subseteq B_{sa}$) if, and only if, $P_{k}$ is symmetric (i.e.
$P_{k}(A_{sa})\subseteq B_{sa}$) for every
$k\in\mathbb{N}\cup\\{0\\}$.$\hfill\Box$
###### Definition 5.
Let $S,T:A\to B$ be a couple of mappings between two C∗-algebras. We shall say
that the pair $(S,T)$ is orthogonality preserving on a subset $U\subseteq A$
if $S(a)\perp T(b)$ whenever $a\perp b$ in $U$. When $ab=0$ in $U$ implies
$S(a)T(b)=0$ in $B$, we shall say that $(S,T)$ preserves zero products on $U$.
We observe that a mapping $T:A\to B$ is orthogonality preserving in the usual
sense if and only if the pair $(T,T)$ is orthogonality preserving. We also
notice that $(S,T)$ is orthogonality preserving (on $A_{sa}$) if and only if
$(T,S)$ is orthogonality preserving (on $A_{sa}$).
Our next result assures that the $n$-homogeneous polynomials appearing in the
Taylor series of an orthogonality preserving holomorphic mapping between
C∗-algebras are pairwise orthogonality preserving.
###### Proposition 6.
Let $f:B_{A}(0,\varrho)\longrightarrow B$ be a holomorphic mapping, where $A$
and $B$ are C∗-algebras, and let $\displaystyle f=\sum_{k=0}^{\infty}P_{k}$ be
its Taylor series at zero, which is uniformly converging in
$U=B_{A}(0,\delta)$. The following statements hold:
1. $(a)$
The mapping $f$ is orthogonally preserving on $U$ (respectively, orthogonally
preserving on $A_{sa}\cap U$) if, and only if, $P_{0}=0$ and the pair
$(P_{n},P_{m})$ is orthogonality preserving (respectively, orthogonally
preserving on $A_{sa}$) for every $n,m\in\mathbb{N}$.
2. $(b)$
The mapping $f$ preserves zero products on $U$ if, and only if, $P_{0}=0$ and
for every $n,m\in\mathbb{N},$ the pair $(P_{n},P_{m})$ preserves zero
products.
###### Proof.
$(a)$ The “if” implication is clear. To prove the ”only if” implication, let
us fix $a,b\in U$ with $a\perp b$. Let us find two positive scalars $r,C$ such
that $a,b\in B(0,r)$, and $\|f(x)\|\leq C$ for every $x\in
B(0,r)\subset\overline{B}(0,r)\subseteq U$. From the Cauchy estimates we have
$\|P_{m}\|\leq\frac{C}{r^{m}},$ for every $m\in\mathbb{N}\cup\\{0\\}.$ By
hypothesis $f(ta)\perp f(tb)$, for every $r>t>0$, and hence
$P_{0}(ta)P_{0}(tb)^{*}+P_{0}(ta)\left(\sum_{k=1}^{\infty}P_{k}(tb)\right)^{*}+\left(\sum_{k=1}^{\infty}P_{k}(ta)\right)\left(\sum_{k=0}^{\infty}P_{k}(tb)\right)^{*}=0,$
and by homogeneity
$P_{0}(a)P_{0}(b)^{*}=-P_{0}(a)\left(\sum_{k=1}^{\infty}t^{k}P_{k}(b)\right)^{*}+\left(\sum_{k=1}^{\infty}t^{k}P_{k}(a)\right)\left(\sum_{k=0}^{\infty}t^{k}P_{k}(b)\right)^{*}.$
Letting $t\to 0$, we have $P_{0}(a)P_{0}(b)^{*}=0$. In particular, $P_{0}=0$.
We shall prove by induction on $n$ that the pair $(P_{j},P_{k})$ is
orthogonality preserving on $U$ for every $1\leq j,k\leq n$. Since
$f(ta)f(tb)^{*}=0$, we also deduce that
$P_{1}(ta)P_{1}(tb)^{*}+P_{1}(ta)\left(\sum_{k=2}^{\infty}P_{k}(tb)\right)^{*}+\left(\sum_{k=2}^{\infty}P_{k}(ta)\right)\left(\sum_{k=1}^{\infty}P_{k}(tb)\right)^{*}=0,$
for every $\frac{\min\\{\|a\|,\|b\|\\}}{r}>t>0,$ which implies that
$t^{2}P_{1}(a)P_{1}(b)^{*}=-tP_{1}(a)\left(\sum_{k=2}^{\infty}t^{k}P_{k}(b)\right)^{*}-\left(\sum_{k=2}^{\infty}t^{k}P_{k}(a)\right)\left(\sum_{k=1}^{\infty}t^{k}P_{k}(b)\right)^{*},$
for every $\frac{\min\\{\|a\|,\|b\|\\}}{r}>t>0$, and hence
$\left\|P_{1}(a)P_{1}(b)^{*}\right\|\leq
tC\|P_{1}(a)\|\sum_{k=2}^{\infty}\frac{\|b\|^{k}}{r^{k}}t^{k-2}$
$+tC^{2}\left(\sum_{k=2}^{\infty}\frac{\|a\|^{k}}{r^{k}}t^{k-2}\right)\left(\sum_{k=1}^{\infty}\frac{\|b\|^{k}}{r^{k}}t^{k-1}\right).$
Taking limit in $t\to 0$, we get $P_{1}(a)P_{1}(b)^{*}=0$. Let us assume that
$(P_{j},P_{k})$ is orthogonality preserving on $U$ for every $1\leq j,k\leq
n$. Following the argument above we deduce that
$P_{1}(a)P_{n+1}(b)^{*}+P_{n+1}(a)P_{1}(b)^{*}=-tP_{1}(a)\left(\sum_{j=n+2}^{\infty}t^{j-n-2}P_{j}(b)\right)^{*}$
$-t\sum_{k=2}^{n}t^{k-2}P_{k}(a)\left(\sum_{j=n+1}^{\infty}t^{j-n-1}P_{j}(b)\right)^{*}-tP_{n+1}(a)\left(\sum_{j=2}^{\infty}t^{j-2}P_{j}(b)\right)^{*}$
$-t\left(\sum_{k=n+2}^{\infty}t^{k-n-2}P_{k}(a)\right)\left(\sum_{j=1}^{\infty}t^{j-1}P_{j}(b)\right)^{*},$
for every $\frac{\min\\{\|a\|,\|b\|\\}}{r}>|t|>0$. Taking limit in $t\to 0$,
we have
$P_{1}(a)P_{n+1}(b)^{*}+P_{n+1}(a)P_{1}(b)^{*}=0.$
Replacing $a$ with $sa$ ($s>0$) we get
$sP_{1}(a)P_{n+1}(b)^{*}+s^{n+1}P_{n+1}(a)P_{1}(b)^{*}=0$
for every $s>0$, which implies that
$P_{1}(a)P_{n+1}(b)^{*}=0.$
In a similar manner we prove that $P_{k}(a)P_{n+1}(b)^{*}=0$, for every $1\leq
k\leq n+1$. The equalities $P_{k}(b)^{*}P_{j}(a)=0$ ($1\leq j,k\leq n+1$)
follow similarly.
We have shown that for each $n,m\in\mathbb{N}$, $P_{n}(a)\perp P_{m}(b)$
whenever $a,b\in U$ with $a\perp b$. Finally, taking $a,b\in A$ with $a\perp
b$, we can find a positive $\rho$ such that $\rho a,\rho b\in U$ and $\rho
a\perp\rho b$, which implies that $P_{n}(\rho a)\perp P_{m}(\rho b)$ for every
$n,m\in\mathbb{N}$, witnessing that $(P_{n},P_{m})$ is orthogonality
preserving for every $n,m\in\mathbb{N}$.
The proof of $(b)$ follows in a similar manner. ∎
We can obtain now a corollary which is a first step toward the description of
orthogonality preserving, orthogonally additive, holomorphic mappings between
C∗-algebras.
###### Corollary 7.
Let $f:B_{A}(0,\varrho)\longrightarrow B$ be a holomorphic mapping, where $A$
and $B$ are C∗-algebras, and let $\displaystyle f=\sum_{k=0}^{\infty}P_{k}$ be
its Taylor series at zero, which is uniformly converging in
$U=B_{A}(0,\delta)$. Suppose $f$ is orthogonality preserving on $A_{sa}\cap U$
and orthogonally additive (respectively, orthogonally additive and zero
products preserving). Then there exists a sequence $(T_{n})$ of operators from
$A$ into $B$ satisfying that the pair $(T_{n},T_{m})$ is orthogonality
preserving on $A_{sa}$ (respectively, zero products preserving on $A_{sa}$)
for every $n,m\in\mathbb{N}$ and
(3) $f(x)=\sum_{n=1}^{\infty}T_{n}(x^{n}),$
uniformly in $x\in U$. In particular every $T_{n}$ is orthogonality preserving
(respectively, zero products preserving) on $A_{sa}$. Furthermore, $f$ is
symmetric if and only if every $T_{n}$ is symmetric.
###### Proof.
Combining Lemma 3 and Proposition 6, we deduce that $P_{0}=0$, $P_{n}$ is
orthogonally additive and $(P_{n},P_{m})$ is orthogonality preserving on
$A_{sa}$ for every $n,m$ in $\mathbb{N}.$ By Theorem 1, for each natural $n$
there exists an operator $T_{n}:A\to B$ such that $\|P_{n}\|\leq\|T_{n}\|\leq
2\|P_{n}\|$ and
$P_{n}(a)=T_{n}(a^{n}),$
for every $a\in A$.
Consider now two positive elements $a,b\in A$ with $a\perp b$ and fix
$n,m\in\mathbb{N}$. In this case there exist positive elements $c,d$ in $A$
with $c^{n}=a$ and $d^{m}=b$ and $c\perp d$. Since the pair $(P_{n},P_{m})$ is
orthogonality preserving on $A_{sa}$, we have
$T_{n}(a)=T_{n}(c^{n})=P_{n}(c)\perp P_{m}(d)=T_{m}(d^{m})=T_{m}(b).$ Now,
noticing that given $a,b$ in $A_{sa}$ with $a\perp b$, we can write
$a=a^{+}-b^{-}$ and $b=b^{+}-b^{-}$, where $a^{\sigma},b^{\tau}$ are positive,
$a^{+}\perp a^{-},$ $b^{+}\perp b^{-}$ and $a^{\sigma}\perp b^{\tau},$ for
every $\sigma,\tau\in\\{+,-\\},$ we deduce that $T_{n}(a)\perp T_{m}(b)$. This
shows that the pair $(T_{n},T_{m})$ is orthogonality preserving on $A_{sa}$.
When $f$ orthogonally additive and zero products preserving the pair
$(T_{n},T_{m})$ is zero products preserving on $A_{sa}$ for every
$n,m\in\mathbb{N}$. The final statement is clear from Lemma 4. ∎
It should be remarked here that if a mapping $f:B_{A}(0,\delta)\longrightarrow
B$ is given by an expression of the form in (3) which uniformly converging in
$U=B_{A}(0,\delta)$ where $(T_{n})$ is a sequence of operators from $A$ into
$B$ such that the pair $(T_{n},T_{m})$ is orthogonality preserving on $A_{sa}$
(respectively, zero products preserving on $A_{sa}$) for every
$n,m\in\mathbb{N}$, then $f$ is orthogonally additive and orthogonality
preserving on $A_{sa}\cap U$ (respectively, orthogonally additive and zero
products preserving).
## 3\. Orthogonality preserving pairs of operators
Let $A$ and $B$ be two C∗-algebras. In this section we shall study those pairs
of operators $S,T:A\to B$ satisfying that $S,T$ and the pair $(S,T)$ preserve
orthogonality on $A_{sa}$. Our description generalizes some of the results
obtained by M. Wolff in [17] because a (symmetric) mapping $T:A\to B$ is
orthogonality preserving on $A_{sa}$ if and only if the pair $(T,T)$ enjoys
the same property. In particular, for every ∗-homomorphism $\Phi:A\to B$, the
pair $(\Phi,\Phi)$ preservers orthogonality. The same statement is true
whenever $\Phi$ is a ∗-anti-homomorphism, or a Jordan ∗-homomorphism, or a
triple homomorphism for the triple product
$\left\\{a,b,c\right\\}=\frac{1}{2}(ab^{*}c+cb^{*}a)$.
We observe that $S,T$ being symmetric implies that $(S,T)$ is orthogonality
preserving on $A_{sa}$ if and only if $(S,T)$ is zero products preserving on
$A_{sa}$. We shall offer here a newfangled and simplified proof which is also
valid for pairs of operators.
Let $a$ be an element in a von Neumann algebra $M$. We recall that the _left_
and _right_ _support projections_ of $a$ (denoted by $l(a)$ and $d(a)$) are
defined as follows: $l(a)$ (respectively, $d(a)$) is the smallest projection
$p\in M$ (respectively, $q\in M$) with the property that $pa=a$ (respectively,
$aq=a$). It is known that when $a$ is hermitian $d(a)=l(a)$ is called the
_support_ or _range projection_ of $a$ and is denoted by $s(a)$. It is also
known that, for each $a=a^{*}$, the sequence $(a^{\frac{1}{3^{n}}})$ converges
in the strong∗-topology of $M$ to $s(a)$ (cf. [15, §1.10 and 1.11]).
An element $e$ in a C∗-algebra $A$ is said to be a _partial isometry_ whenever
$ee^{*}e=e$ (equivalently, $ee^{*}$ or $e^{*}e$ is a projection in $A$). For
each partial isometry $e$, the projections $ee^{*}$ and $e^{*}e$ are called
the left and right support projections associated to $e$, respectively. Every
partial isometry $e$ in $A$ defines a Jordan product and an involution on
$A_{e}(e):=ee^{*}Ae^{*}e$ given by
$a\bullet_{{}_{e}}b=\frac{1}{2}(ae^{*}b+be^{*}a)$ and
$a^{\sharp_{{}_{e}}}=ea^{*}e$ ($a,b\in A_{2}(e)$). It is known that
$(A_{2}(e),\bullet_{{}_{e}},{\sharp_{{}_{e}}})$ is a unital JB∗-algebra with
respect to its natural norm and $e$ is the unit element for the Jordan product
$\bullet_{{}_{e}}$.
Every element $a$ in a C∗-algebra $A$ admits a _polar decomposition_ in
$A^{**}$, that is, $a$ decomposes uniquely as follows: $a=u|a|$, where
$|a|=(a^{*}a)^{\frac{1}{2}}$ and $u$ is a partial isometry in $A^{**}$ such
that $u^{*}u=s(|a|)$ and $uu^{*}=s(|a^{*}|)$ (compare [15, Theorem 1.12.1]).
Observe that $uu^{*}a=au^{*}u=u$. The unique partial isometry $u$ appearing in
the polar decomposition of $a$ is called the range partial isometry of $a$ and
is denoted by $r(a)$. Let us observe that taking $c=r(a)|a|^{\frac{1}{3}}$, we
have $cc^{*}c=a$. It is also easy to check that for each $b\in A$ with
$b=r(a)r(a)^{*}b$ (respectively, $b=br(a)^{*}r(a)$) the condition $a^{*}b=0$
(respectively, $ba^{*}=0$) implies $b=0$. Furthermore, $a\perp b$ in $A$ if
and only if $r(a)\perp r(b)$ in $A^{**}$.
We begin with a basic argument in the study of orthogonality preserving
operators between C∗-algebras whose proof is inserted here for completeness
reasons. Let us recall that for every C∗-algebra $A$, the _multiplier algebra_
of $A$, $M(A)$, is the set of all elements $x\in A^{**}$ such that for each
$Ax,xA\subseteq A$. We notice that $M(A)$ is a C∗-algebra and contains the
unit element of $A^{**}$.
###### Lemma 8.
Let $A$ and $B$ be C∗-algebras and let $S,T:A\to B$ be a pair of operators.
1. $(a)$
The pair $(S,T)$ preserves orthogonality (on $A_{sa}$) if and only if the pair
$(S^{**}|_{M(A)},T^{**}|_{M(A)})$ preserves orthogonality (on $M(A)_{sa}$);
2. $(b)$
The pair $(S,T)$ preserves zero products (on $A_{sa}$) if and only if the pair
$(S^{**}|_{M(A)},T^{**}|_{M(A)})$ preserves zero products (on $M(A)_{sa}$).
###### Proof.
$(a)$ The “if” implication is clear. Let $a,b$ be two elements in $M(A)$ with
$a\perp b$. We can find two elements $c$ and $d$ in $M(A)$ satisfying
$cc^{*}c=a$, $dd^{*}d=b$ and $c\perp d$. Since $cxc\perp dyd$, for every $x,y$
in $A$, we have $T(cxc)\perp T(dyd)$ for every $x,y\in A$. By Goldstine’s
theorem we find two bounded nets $(x_{\lambda})$ and $(y_{\mu})$ in $A$,
converging in the weak∗ topology of $A^{**}$ to $c^{*}$ and $d^{*}$,
respectively. Since
$T(cx_{\lambda}c)T(dy_{\mu}d)^{*}=T(dy_{\mu}d)^{*}T(cx_{\lambda}c)=0$, for
every $\lambda,\mu$, $T^{**}$ is weak∗-continuous, the product of $A^{**}$ is
separately weak∗-continuous and the involution of $A^{**}$ also is
weak∗-continuous, we get
$T^{**}(cc^{*}c)T^{**}(dd^{*}d)=T^{**}(a)T^{**}(b)^{*}=0=T^{**}(b)^{*}T^{**}(a),$
and hence $T^{**}(a)\perp T^{**}(b)$, as desired.
The proof of $(b)$ follows by a similar argument. ∎
###### Proposition 9.
Let $S,T:A\to B$ be operators between C∗-algebras such that $(S,T)$ is
orthogonality preserving on $A_{sa}$. Let us denote $h:=S^{**}(1)$ and
$k:=T^{**}(1)$. Then the identities
$S(a)T(a^{*})^{*}=S(a^{2})k^{*}=hT((a^{2})^{*})^{*},$
$T(a^{*})^{*}S(a)=k^{*}S(a^{2})=hT((a^{2})^{*})^{*}h,$
$S(a)k^{*}=hT(a^{*})^{*},\hbox{ and, }k^{*}S(a)=T(a^{*})^{*}h$
hold for every $a\in A$.
###### Proof.
By Lemma 8, we may assume, without loss of generality, that $A$ is unital.
$(a)$ For each $\varphi\in B^{*}$, the continuous bilinear form
$V_{\varphi}:A\times A\to\mathbb{C}$,
$V_{\varphi}(a,b)=\varphi(S(a)T(b^{*})^{*})$ is orthogonal, that is,
$V_{\varphi}(a,b)=0$, whenever $ab=0$ in $A_{sa}$. By Goldstein’s theorem [9,
Theorem 1.10] there exist functionals $\omega_{1},\omega_{2}\in A^{*}$
satisfying that
$V_{\varphi}(a,b)=\omega_{1}(ab)+\omega_{2}(ba),$
for all $a,b\in A$. Taking $b=1$ and $a=b$ we have
$\varphi(S(a)k^{*})=V_{\varphi}(a,1)=V_{\varphi}(1,a)=\varphi(hT(a)^{*})$
and
$\varphi(S(a)T(a)^{*})=\varphi(S(a^{2})k^{*})=\varphi(hT(a^{2})^{*}),$
for every $a\in A_{sa}$, respectively. Since $\varphi$ was arbitrarily chosen,
we get, by linearity, $S(a)k^{*}=hT(a^{*})^{*}$ and
$S(a)T(a^{*})^{*}=S(a^{2})k^{*}=hT((a^{2})^{*})^{*}$, for every $a\in A$. The
other identities follow in a similar way, but replacing
$V_{\varphi}(a,b)=\varphi(S(a)T(b^{*})^{*})$ with
$V_{\varphi}(a,b)=\varphi(T(b^{*})^{*}S(a))$. ∎
###### Lemma 10.
Let $J_{1},J_{2}:A\to B$ be Jordan ∗-homomorphism between C∗-algebras. The
following statements are equivalent:
1. $(a)$
The pair $(J_{1},J_{2})$ is orthogonality preserving on $A_{sa}$;
2. $(b)$
The identity
$J_{1}(a)J_{2}(a)=J_{1}(a^{2})J_{2}^{**}(1)=J_{1}^{**}(1)J_{2}(a^{2}),$
holds for every $a\in A_{sa}$;
3. $(c)$
The identity
$J_{1}^{**}(1)J_{2}(a)=J_{1}(a)J_{2}^{**}(1),$
holds for every $a\in A_{sa}$.
Furthermore, when $J_{1}^{**}$ is unital,
$J_{2}(a)=J_{1}(a)J_{2}^{**}(1)=J_{2}^{**}(1)J_{1}(a),$ for every $a$ in $A.$
###### Proof.
The implications $(a)\Rightarrow(b)\Rightarrow(c)$ have been established in
Proposition 9. To see $(c)\Rightarrow(a)$, we observe that
$J_{i}(x)=J_{i}^{**}(1)J_{i}(x)J_{i}^{**}(1)=J_{i}(x)J_{i}^{**}(1)=J_{i}^{**}(1)J_{i}(x)$,
for every $x\in A$. Therefore, given $a,b\in A_{sa}$ with $a\perp b$, we have
$J_{1}(a)J_{2}(b)=J_{1}(a)J_{1}^{**}(1)J_{2}(b)=J_{1}(a)J_{1}(b)J_{2}^{**}(1)=0$.
∎
In [17, Proposition 2.5], M. Wolff establishes a uniqueness result for
∗-homomorphisms between C∗-algebras showing that for each pair $(U,V)$ of
unital ∗-homomorphisms from a unital C∗-algebra $A$ into a unital C∗-algebra
$B$, the condition $(U,V)$ orthogonality preserving on $A_{sa}$ implies $U=V$.
This uniqueness result is a direct consequence of our previous lemma.
Orthogonality preserving pairs of operators can be also used to rediscover the
notion of orthomorphism in the sense introduced by Zaanen in [19]. We recall
that an operator $T$ on a C∗-algebra $A$ is said to be an _orthomorphism_ or a
_band preserving_ operator when the implication $a\perp b\Rightarrow T(a)\perp
b$ holds for every $a,b\in A$. We notice that when $A$ is regarded as an
$A$-bimodule, an operator $T:A\to A$ is an orthomorphism if and only if it is
a _local operator_ in the sense used by B.E. Johnson in [11, §3]. Clearly, an
operator $T:A\to A$ is an orthomorphism if and only if $(T,Id_{A})$ is
orthogonality preserving. The following non-commutative extension of [19,
THEOREM 5] follows from Proposition 9.
###### Corollary 11.
Let $T$ be an operator on a C∗-algebra $A$. Then $T$ is an orthomorphism if
and only if $T(a)=T^{**}(1)a=aT^{**}(1)$, for every $a$ in $A$, that is, $T$
is a multiple of the identity on $A$ by an element in its center.$\hfill\Box$
We recall that two elements $a,$ $b$ in a JB∗-algebra $A$ are said to
_operator commute_ in $A$ if the multiplication operators $M_{a}$ and $M_{b}$
commute, where $M_{a}$ is defined by $M_{a}(x):=a\circ x$. That is, $a$ and
$b$ operator commute if and only if $(a\circ x)\circ b=a\circ(x\circ b)$ for
all $x$ in $A$. An useful result in Jordan theory assures that self-adjoint
elements $a$ and $b$ in $A$ generate a JB∗-subalgebra that can be realized as
a JC∗-subalgebra of some $B(H)$ (compare [18]), and, under this
identification, $a$ and $b$ commute as elements in $L(H)$ whenever they
operator commute in $A$, equivalently $a^{2}\circ b=2(a\circ b)\circ
a-a^{2}\circ b$ (cf. Proposition 1 in [16]).
The next lemma contains a property which is probably known in C∗-algebra, we
include an sketch of the proof because we were unable to find an explicit
reference.
###### Lemma 12.
Let $e$ be a partial isometry in a C∗-algebra $A$ and let $a,b$ be two
elements in $A_{2}(e)=ee^{*}Ae^{*}e$. Then $a$, $b$ operator commute in the
JB∗-algebra $(A_{2}(e),\bullet_{{}_{e}},{\sharp_{{}_{e}}})$ if and only if
$ae^{*}$ and $be^{*}$ operator commute in the JB∗-algebra
$(A_{2}(ee^{*}),\bullet_{{}_{ee^{*}}},{\sharp_{{}_{ee^{*}}}})$, where
$x\bullet_{{}_{ee^{*}}}y=x\circ y=\frac{1}{2}(xy+yx)$, for every $x,y\in
A_{2}(ee^{*})$. Furthermore, when $a$ and $b$ are hermitian elements in
$(A_{2}(e),\bullet_{{}_{e}},{\sharp_{{}_{e}}})$, $a$, $b$ operator commute if
and only if $ae^{*}$ and $be^{*}$ commute in the usual sense (i.e.
$ae^{*}be^{*}=be^{*}ae^{*}$).
###### Proof.
We observe that the mapping
$R_{e^{*}}:(A_{2}(e),\bullet_{{}_{e}})\to(A_{2}(ee^{*}),\bullet_{{}_{ee^{*}}})$,
$x\mapsto xe^{*}$ is a Jordan ∗-isomorphism between the above JB∗-algebras.
So, the first equivalence is clear. The second one has been commented before.
∎
Our next corollary relies on the following description of orthogonality
preserving operators between C∗-algebras obtained in [3] (see also [4]).
###### Theorem 13.
[3, Theorem 17], [4, Theorem 4.1 and Corollary 4.2] Let $T$ be an operator
from a C∗-algebra $A$ into another C∗-algebra $B$ the following are
equivalent:
1. $a)$
$T$ is orthogonality preserving (on $A_{sa}$).
2. $b)$
There exits a unital Jordan ∗-homomorphism $J:M(A)\to B_{2}^{**}(r(h))$ such
that $J(x)$ and $h=T^{**}(1)$ operator commute and
$T(x)=h\bullet_{{}_{r(h)}}J(x),\hbox{ for every $x\in A$},$
where $M(A)$ is the multiplier algebra of $A$, $r(h)$ is the range partial
isometry of $h$ in $B^{**}$, $B_{2}^{**}(r(h))=r(h)r(h)^{*}B^{**}r(h)^{*}r(h)$
and $\bullet_{{}_{r(h)}}$ is the natural product making $B_{2}^{**}(r(h))$ a
JB∗-algebra.
Furthermore, when $T$ is symmetric, $h$ is hermitian and hence $r(h)$
decomposes as orthogonal sum of two projections in $B^{**}$.$\hfill\Box$
Our next result gives a new perspective for the study of orthogonality
preserving (pairs of) operators between C∗-algebras.
###### Proposition 14.
Let $A$ and $B$ be C∗-algebras. Let $S,T:A\to B$ be operators and let
$h=S^{**}(1)$ and $k=T^{**}(1)$. Then the following statements hold:
1. $(a)$
The operator $S$ is orthogonality preserving if and only if there exit two
Jordan ∗-homomorphisms $\Phi,\widetilde{\Phi}:M(A)\to B^{**}$ satisfying
$\Phi(1)=r(h)r(h)^{*}$, $\widetilde{\Phi}(1)=r(h)^{*}r(h),$ and
$S(a)=\Phi(a)h=h\widetilde{\Phi}(a),$ for every $a\in A$.
2. $(b)$
$S,T$ and $(S,T)$ are orthogonality preserving on $A_{sa}$ if and only if the
following statements hold:
1. $(b1)$
There exit Jordan ∗-homomorphisms
$\Phi_{1},\widetilde{\Phi}_{1},\Phi_{2},\widetilde{\Phi}_{2}:M(A)\to B^{**}$
satisfying $\Phi_{1}(1)=r(h)r(h)^{*}$, $\widetilde{\Phi}_{1}(1)=r(h)^{*}r(h),$
$\Phi_{2}(1)=r(k)r(k)^{*}$, $\widetilde{\Phi}_{2}(1)=r(k)^{*}r(k),$
$S(a)=\Phi_{1}(a)h=h\widetilde{\Phi}_{1}(a),$ and
$T(a)=\Phi_{2}(a)k=k\widetilde{\Phi}_{2}(a),$ for every $a\in A$;
2. $(b2)$
The pairs $({\Phi}_{1},{\Phi}_{2})$ and
$(\widetilde{\Phi}_{1},\widetilde{\Phi}_{2})$ are orthogonality preserving on
$A_{sa}$.
###### Proof.
The “if” implications are clear in both statements. We shall only prove the
“only if” implication.
$(a)$. By Theorem 13, there exits a unital Jordan ∗-homomorphism
$J_{1}:M(A)\to B_{2}^{**}(r(h))$ such that $J_{1}(x)$ and $h$ operator commute
in the JB∗-algebra $(B_{2}^{**}(r(h)),\bullet_{{}_{r(h)}})$ and
$S(x)=h\bullet_{{}_{r(a)}}J_{1}(a)\hbox{ for every $a\in A$}.$
Fix $a\in A_{sa}$. Since $h$ and $J_{1}(a)$ are hermitian elements in
$(B_{2}^{**}(r(h)),\bullet_{{}_{r(h)}})$ which operator commute, Lemma 12
assures that $hr(h)^{*}$ and $J_{1}(a)r(h)^{*}$ commute in the usual sense of
$B^{**}$, that is,
$hr(h)^{*}J_{1}(a)r(h)^{*}=J_{1}(a)r(h)^{*}hr(h)^{*},$
or equivalently,
$hr(h)^{*}J_{1}(a)=J_{1}(a)r(h)^{*}h.$
Consequently, we have
$S(a)=h\bullet_{{}_{r(h)}}J_{1}(a)=hr(h)^{*}J_{1}(a)=J_{1}(a)r(h)^{*}h,$
for every $a\in A$. The desired statement follows by considering
$\Phi_{1}(a)=J_{1}(a)r(h)^{*}$ and $\widetilde{\Phi}_{1}(a)=r(h)^{*}J_{1}(a).$
$(b)$ The statement in $(b1)$ follows from $(a)$. We shall prove $(b2)$.
By hypothesis, given $a,b$ in $A_{sa}$ with $a\perp b$, we have
$0=S(a)T(b)^{*}=\left(h\widetilde{\Phi}_{1}(a)\right)\left(k\widetilde{\Phi}_{2}(b)\right)^{*}$
$=h\widetilde{\Phi}_{1}(a)\widetilde{\Phi}_{2}(b)^{*}k^{*}$
Having in mind that $\widetilde{\Phi}_{1}(A)\subseteq r(h)^{*}r(h)B^{**}$ and
$\widetilde{\Phi}_{2}(A)\subseteq B^{**}r(k)^{*}r(k)$, we deduce that
$\widetilde{\Phi}_{1}(a)\widetilde{\Phi}_{2}(b)^{*}=0$ (compare the comments
before Lemma 8), as we desired. In a similar fashion we prove
$\widetilde{\Phi}_{2}(b)^{*}\widetilde{\Phi}_{1}(a)=0$,
${\Phi}_{2}(b)^{*}{\Phi}_{1}(a)=0={\Phi}_{1}(a){\Phi}_{2}(b)^{*}.$ ∎
## 4\. Holomorphic mappings valued in a commutative C∗-algebra
The particular setting in which a holomorphic function is valued in a
commutative C∗-algebra provides enough advantages to establish a full
description of the orthogonally additive, orthogonality preserving,
holomorphic mappings which are valued in a commutatively C∗-algebra.
###### Proposition 15.
Let $S,T:A\to B$ be operators between C∗-algebras with $B$ commutative.
Suppose that $S$, $T$ and $(S,T)$ are orthogonality preserving, and let us
denote $h=S^{**}(1)$ and $k=T^{**}(1)$. Then there exits a Jordan
∗-homomorphism $\Phi:M(A)\to B^{**}$ satisfying $\Phi(1)=r(|h|+|k|)$,
$S(a)=\Phi(a)h,$ and $T(a)=\Phi(a)k,$ for every $a\in A$.
###### Proof.
Let $\Phi_{1},\widetilde{\Phi}_{1},\Phi_{2},\widetilde{\Phi}_{2}:M(A)\to
B^{**}$ be the Jordan ∗-homomorphisms satisfying $(b1)$ and $(b2)$ in
Proposition 14. By hypothesis, $B$ is commutative, and hence
$\Phi_{i}=\widetilde{\Phi}_{i}$ for every $i=1,2$ (compare the proof of
Proposition 14). Since the pair $({\Phi}_{1},{\Phi}_{2})$ is orthogonality
preserving on $A_{sa}$, Lemma 10 assures that
${\Phi}_{1}^{**}(1){\Phi}_{2}(a)={\Phi}_{1}(a){\Phi}_{2}^{**}(1),$
for every $a\in A_{sa}$. In order to simplify notation, let us denote
$p={\Phi}_{1}^{**}(1)$ and $q={\Phi}_{2}^{**}(1)$.
We define an operator ${\Phi}:M(A)\to B^{**}$, defined by
$\Phi(a)=pq\Phi_{1}(a)+p(1-q)\Phi_{1}(a)+q(1-p)\Phi_{2}(a).$
Since $p{\Phi}_{2}(a)={\Phi}_{1}(a)q$, it can be easily checked that $\Phi$ is
a Jordan ∗-homomorphism such that $S(a)=\Phi(a)h,$ and $T(a)=\Phi(a)k,$ for
every $a\in A$. ∎
###### Theorem 16.
Let $f:B_{A}(0,\varrho)\longrightarrow B$ be a holomorphic mapping, where $A$
and $B$ are C∗-algebras with $B$ commutative, and let $\displaystyle
f=\sum_{k=0}^{\infty}P_{k}$ be its Taylor series at zero, which is uniformly
converging in $U=B_{A}(0,\delta)$. Suppose $f$ is orthogonality preserving on
$A_{sa}\cap U$ and orthogonally additive (equivalently, orthogonally additive
and zero products preserving). Then there exist a sequence $(h_{n})$ in
$B^{**}$ and a Jordan ∗-homomorphism $\Phi:M(A)\to B^{**}$ such that
$f(x)=\sum_{n=1}^{\infty}h_{n}\Phi(a^{n})=\sum_{n=1}^{\infty}h_{n}\Phi(a^{n}),$
uniformly in $a\in U$.
###### Proof.
By Corollary 7, there exists a sequence $(T_{n})$ of operators from $A$ into
$B$ satisfying that the pair $(T_{n},T_{m})$ is orthogonality preserving on
$A_{sa}$ (equivalently, zero products preserving on $A_{sa}$) for every
$n,m\in\mathbb{N}$ and
$f(x)=\sum_{n=1}^{\infty}T_{n}(x^{n}),$
uniformly in $x\in U$. Denote $h_{n}=T_{n}^{**}(1)$.
We shall prove now the existence of the Jordan ∗-homomorphism $\Phi$. We
prove, by induction, that for each natural $n$, there exists a Jordan
∗-homomorphism $\Psi_{n}:M(A)\to B^{**}$ such that
$r(\Psi_{n}(1))=r(|h_{1}|+\ldots+|h_{n}|)$ and $T_{k}(a)=h_{k}\Psi_{n}(a)$ for
every $k\leq n$, $a\in A$. The statement for $n=1$ follows from Corollary 7
and Proposition 14. Let us assume that our statement is true for $n$. Since
for every $k,m$ in $\mathbb{N}$, $T_{k}$, $T_{m}$ and the pair $(T_{k},T_{m})$
are orthogonality preserving, we can easily check that $T_{n+1}$,
$T_{1}+\ldots+T_{n}$ and
$(T_{n+1},T_{1}+\ldots+T_{n})=(T_{n+1},(h_{1}+\ldots+h_{n})\Psi_{n})$ are
orthogonality preserving. By Proposition 15, there exists a Jordan
∗-homomorphism $\Psi_{n+1}:M(A)\to B^{**}$ satisfying
$r(\Psi_{n+1}(1))=r(|h_{1}|+\ldots+|h_{n}|+|h_{n+1}|)$,
$T_{n+1}(a)=h_{n+1}\Psi_{n+1}(a^{n+1})$ and
$(T_{1}+\ldots+T_{n})(a)=(h_{1}+\ldots+h_{n})\Psi_{n+1}(a)$ for every $k\leq
n$, $a\in A$. Since for each, $1\leq k\leq n$,
$h_{k}\Psi_{n+1}(a)=h_{k}r(|h_{1}|+\ldots+|h_{n}|+|h_{n+1}|)\Psi_{n+1}(a)$
$=h_{k}(|h_{1}|+\ldots+|h_{n}|)\Psi_{n+1}(a)$
$=h_{k}(|h_{1}|+\ldots+|h_{n}|)\Psi_{n}(a)=h_{k}\Psi_{n}=T_{k}(a),$
for every $a\in A$, as desired.
Let us consider a free ultrafilter $\mathcal{U}$ on ${\mathbb{N}}$. By the
Banach-Alaoglu theorem, any bounded set in $B^{**}$ is relatively
weak∗-compact, and thus the assignment
$a\mapsto\Phi(a):=w^{*}-\lim_{\mathcal{U}}\Psi_{n}(a)$ defines a Jordan
∗-homomorphism from $M(A)$ into $B^{**}$. If we fix a natural $k$, we know
that $T_{k}(a)=h_{k}\Psi_{n}(a)$, for every $n\geq k$ and $a\in A$. Then it
can be easily checked that $T_{k}(a)=h_{k}\Phi(a),$ for every $a\in A$, which
concludes the proof. ∎
The Banach-Stone type theorem for orthogonally additive, orthogonality
preserving, holomorphic mappings between commutative C∗-algebras, established
in Theorem 2 (see [2, Theorem 3.4]) is a direct consequence of our previous
result.
## 5\. Banach-Stone type theorems for holomorphic mappings between general
C∗-algebras
In this section we deal with holomorphic functions between general
C∗-algebras. In this more general setting we shall require additional
hypothesis to establish a result in the line of the above Theorem 16.
Given a unital C∗-algebra $A$, the symbol inv$(A)$ will denote the set of
invertible elements in $A$. The next lemma is a technical tool which is needed
later. The proof is left to the reader and follows easily from the fact that
inv$(A)$ is an open subset of $A$.
###### Lemma 17.
Let $f:B_{A}(0,\varrho)\longrightarrow B$ be a holomorphic mapping, where $A$
and $B$ are C∗-algebras with $B$ unital, and let $\displaystyle
f=\sum_{k=0}^{\infty}P_{k}$ be its Taylor series at zero, which is uniformly
converging in $U=B_{A}(0,\delta)$. Let us assume that there exists $a_{0}\in
U$ with $f(a_{0})\in\hbox{inv}(B)$. Then there exists $m_{0}\in\mathbb{N}$
such that
$\displaystyle\sum_{k=0}^{m_{0}}P_{k}(a_{0})\in\hbox{inv}(B)$.$\hfill\Box$
We can now state a description of those orthogonally additive, orthogonality
preserving, holomorphic mappings between C∗-algebras whose image contains an
invertible element.
###### Theorem 18.
Let $f:B_{A}(0,\varrho)\longrightarrow B$ be a holomorphic mapping, where $A$
and $B$ are C∗-algebras with $B$ unital, and let $\displaystyle
f=\sum_{k=0}^{\infty}P_{k}$ be its Taylor series at zero, which is uniformly
converging in $U=B_{A}(0,\delta)$. Suppose $f$ is orthogonality preserving on
$A_{sa}\cap U$, orthogonally additive on $U$ and
$f(U)\cap\hbox{inv}(B)\neq\emptyset$. Then there exist a sequence $(h_{n})$ in
$B^{**}$ and Jordan ∗-homomorphisms $\Theta,\widetilde{\Theta}:M(A)\to B^{**}$
such that
$f(a)=\sum_{n=1}^{\infty}h_{n}\widetilde{\Theta}(a^{n})=\sum_{n=1}^{\infty}{\Theta}(a^{n})h_{n},$
uniformly in $a\in U$.
###### Proof.
By Corollary 7 there exists a sequence $(T_{n})$ of operators from $A$ into
$B$ satisfying that the pair $(T_{n},T_{m})$ is orthogonality preserving on
$A_{sa}$ for every $n,m\in\mathbb{N}$ and
$f(x)=\sum_{n=1}^{\infty}T_{n}(x^{n}),$
uniformly in $x\in U$.
Now, Proposition 14 $(a)$, applied to $T_{n}$ ($n\in\mathbb{N}$), implies the
existence of sequences $(\Phi_{n})$ and $(\widetilde{\Phi}_{n})$ of Jordan
∗-homomorphisms from $M(A)$ into $B^{**}$ satisfying
$\Phi_{n}(1)=r(h_{n})r(h_{n})^{*}$,
$\widetilde{\Phi}_{n}(1)=r(h_{n})^{*}r(h_{n}),$ where $h_{n}=T_{n}^{**}(1)$,
and
$T_{n}(a)=\Phi_{n}(a)h_{n}=h_{n}\widetilde{\Phi}_{n}(a),$
for every $a\in A$, $n\in\mathbb{N}$. Moreover, from Proposition 14 $(b)$, the
pairs $({\Phi}_{n},{\Phi}_{m})$ and
$(\widetilde{\Phi}_{n},\widetilde{\Phi}_{m})$ are orthogonality preserving on
$A_{sa}$, for every $n,m\in\mathbb{N}$.
Since $f(U)\cap\hbox{inv}(B)\neq\emptyset$, it follows from Lemma 17 that
there exists a natural $m_{0}$ and $a_{0}\in A$ such that
$\displaystyle\sum_{k=1}^{m_{0}}P_{k}(a_{0})=\sum_{k=1}^{m_{0}}\Phi_{k}(a_{0}^{k})h_{k}=\sum_{k=1}^{m_{0}}h_{k}\widetilde{\Phi}_{k}(a_{0}^{k})\in\hbox{inv}(B).$
We claim that $r(h_{1})^{*}r(h_{1})+\ldots+r(h_{m_{0}})^{*}r(h_{m_{0}})$ is
invertible in $B^{+}$ (and in $B^{**}$). Otherwise, we could find a projection
$q\in B^{**}$ satisfying
$(r(h_{1})^{*}r(h_{1})+\ldots+r(h_{m_{0}})^{*}r(h_{m_{0}}))q=0$. This would
imply that
$\left(\sum_{k=1}^{m_{0}}P_{k}(a_{0})\right)q=\left(\sum_{k=1}^{m_{0}}\Phi_{k}(a_{0}^{k})h_{k}\right)q=0,$
contradicting that
$\displaystyle\sum_{k=1}^{m_{0}}P_{k}(a_{0})=\sum_{k=1}^{m_{0}}\Phi_{k}(a_{0}^{k})h_{k}$
is invertible in $B$.
Consider now the mapping $\Psi=\sum_{k=1}^{m_{0}}\widetilde{\Phi}_{k}$. It is
clear that, for each natural $n$, $\Psi$, $\widetilde{\Phi}_{n}$ and the pair
$(\Psi,\widetilde{\Phi}_{n})$ are orthogonality preserving. Applying
Proposition 14 $(b)$, we deduce the existence of Jordan ∗-homomorphisms
$\Theta,\widetilde{\Theta},\Theta_{n},\widetilde{\Theta}_{n}:M(A)\to B^{**}$
such that $(\Theta,\Theta_{n})$ and
$(\widetilde{\Theta},\widetilde{\Theta}_{n})$ are orthogonality preserving,
$\Theta(1)=r(k)r(k)^{*}$, $\widetilde{\Theta}(1)=r(k)^{*}r(k)$,
$\Theta_{n}(1)=r(h_{n})r(h_{n})^{*}$,
$\widetilde{\Theta}_{n}(1)=r(h_{n})^{*}r(h_{n})$,
$\Psi(a)=\Theta(a)k=k\widetilde{\Theta}(a)$
and
$\widetilde{\Phi}_{n}(a)=\Theta_{n}(a)r(h_{n})^{*}r(h_{n})=r(h_{n})^{*}r(h_{n})\widetilde{\Theta}_{n}(a),$
for every $a\in A$, where
$k=\Psi(1)=r(h_{1})^{*}r(h_{1})+\ldots+r(h_{m_{0}})^{*}r(h_{m_{0}})$. The
invertibility of $k$, proved in the previous paragraph, shows that
$\Theta(1)=1$. Thus, since $(\widetilde{\Theta},\widetilde{\Theta}_{n})$ is
orthogonality preserving, the last statement in Lemma 10 proves that
$\widetilde{\Theta}_{n}(a)=\widetilde{\Theta}_{n}(1)\widetilde{\Theta}(a)=\widetilde{\Theta}(a)\widetilde{\Theta}_{n}(1),$
for every $a\in A$, $n\in\mathbb{N}$. The above identities guarantee that
$\widetilde{\Phi}_{n}(a)=\Theta(a)r(h_{n})^{*}r(h_{n})=r(h_{n})^{*}r(h_{n})\widetilde{\Theta}(a),$
for every $a\in A$, $n\in\mathbb{N}$.
A similar argument to the one given above, but replacing
$\widetilde{\Phi}_{k}$ with ${\Phi}_{k}$, shows the existence of a Jordan
∗-homomorphism $\Theta:M(A)\to B^{**}$ such that
${\Phi}_{n}(a)=\Theta(a)r(h_{n})r(h_{n})^{*}=r(h_{n})r(h_{n})^{*}{\Theta}(a),$
for every $a\in A$, $n\in\mathbb{N}$, which concludes the proof. ∎
## References
* [1] Y. Benyamini, S. Lassalle, J.G. Llavona; Homogeneous orthogonally additive polynomials on Banach lattices, _Bull. London Math. Soc._ 38, no. 3, 459-469 (2006).
* [2] Q. Bu, M.-H. Hsu, N.-Ch. Wong, Zero Products and norm preserving orthogonally additive homogeneous polynomials on C∗-algebras, preprint 2013.
* [3] M. Burgos, F.J. Fernández-Polo, J.J. Garcés, J. Martínez Moreno, A.M. Peralta, Orthogonality preservers in C*-algebras, JB*-algebras and JB*-triples, _J. Math. Anal. Appl._ , 348, 220-233 (2008).
* [4] M. Burgos, F.J. Fernández-Polo, J. J. Garcés, A.M. Peralta, Orthogonality preservers Revisited, _Asian-European Journal of Mathematics_ 2, No. 3, 387-405 (2009).
* [5] D. Carando, S. Lassalle, I. Zalduendo, Orthogonally additive polynomials over $C(K)$ are measures – a short proof, _Integr. equ. oper. theory_ 56, 597-602 (2006).
* [6] D. Carando, S. Lassalle, I. Zalduendo, Orthogonally Additive Holomorphic functions of Bounded Type over $C(K)$, _Proc. of the Edinburgh Math. Soc._ 53, 609-618 (2010).
* [7] S. Dineen, _Complex Analysis on infinite dimensional Spaces_ , Springer 1999.
* [8] T.W. Gamelin, Analytic functions on Banach spaces. In Complex potential theory (Montreal, PQ, 1993), 187233, NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., 439, Kluwer Acad. Publ., Dordrecht, 1994.
* [9] S. Goldstein, Stationarity of operator algebras, _J. Funct. Anal._ 118, no. 2, 275-308 (1993).
* [10] J.A. Jaramillo, A. Prieto, I. Zalduendo, Orthogonally additive holomorphic functions on open subsets of $C(K)$, _Rev. Mat. Complut._ 25, no. 1, 31-41 (2012).
* [11] B.E. Johnson, Local derivations on C∗-algebras are derivations, _Trans. Amer. Math. Soc._ 353, 313-325 (2001).
* [12] C. Palazuelos, A.M. Peralta, I. Villanueva; Orthogonally Additive Polynomials on C∗-Algebras, _Quart. J. Math._ 59, 363-374 (2008).
* [13] A.M. Peralta, D. Puglisi, Orthogonally Additive Holomorphic functions on C∗-algebras, _Operators and Matrices_ 6, Number 3, 621-629 (2012).
* [14] D. Pérez, I. Villanueva, Orthogonally additive polynomials on spaces of continuous functions, _J. Math. Anal. Appl._ 306, 97-105, (2005).
* [15] S. Sakai; $C^{*}$-algebras and $W^{*}$-algebras, Springer-Verlag, Berlin, 1971.
* [16] D. Topping, Jordan algebras of self-adjoint operators, Mem. Amer. Math. Soc. 53, 1965.
* [17] M. Wolff, Disjointness preserving operators in C∗-algebras, _Arch. Math._ 62, 248-253 (1994).
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|
arxiv-papers
| 2013-10-01T17:42:01 |
2024-09-04T02:49:51.885806
|
{
"license": "Public Domain",
"authors": "Jorge J. Garc\\'es, Antonio M. Peralta, Daniele Puglisi, Mar\\'ia I.\n Ram\\'irez",
"submitter": "Antonio M. Peralta",
"url": "https://arxiv.org/abs/1310.0407"
}
|
1310.0511
|
# Does the problem of global warming exist at all?
Insight from the temperature drift induced by inevitable colored noise
V.D. Rusov1111Corresponding author: Vitaliy D. Rusov, E-mail: [email protected],
V.P. Smolyar1, M.V. Eingorn1,2,
T.N. Zelentsova1, E.P. Linnik1, M.E. Beglaryan1, B. Vachev3
###### Abstract
In the present paper we state a problem of the colored noise nonremovability
on the climatic 30-year time scale, which essentially changes the angle of
view on the known problem of global warming.
1Department of Theoretical and Experimental Nuclear Physics,
Odessa National Polytechnic University, 1 Shevchenko ave., Odessa 65044,
Ukraine
2CREST and NASA Research Centers, North Carolina Central University,
Fayetteville st. 1801, Durham, North Carolina 27707, U.S.A.
3Institute for Nuclear Research and Nuclear Energy, Tsarigradsko Chaussee
Blvd. 72, Sofia 1784, Bulgaria
## 1 Introduction
It is known that the historical surface temperature data set HadCRUT provides
a record of surface temperature trends and variability since 1850. The most
well established version of this data set, HadCRUT3 [2], has been produced
benefiting from recent improvements to the sea surface temperature data set
which forms its marine component and from improvements to the station records
which provide the land data. A new version of this data set, HadCRUT4 [3],
improves and updates the gridded land-based Climatic Research Unit temperature
database and virtually does not differ from its predecessor – HadCRUT3 [2].
In the framework of both versions, basing on a careful analysis, a
comprehensive set of uncertainty estimates has been derived to accompany the
data and the following perfectly clear conclusion has been made: ”…Since the
mid twentieth century the uncertainties in global and hemispheric mean
temperatures are small, and the temperature increase greatly exceeds its
uncertainty. In earlier periods the uncertainties are larger, _but the
temperature increase over the twentieth century is still significantly larger
than its uncertainty_” (Fig. 1a).
Figure 1: HadCRUT3 global temperature time series at annual resolution (blue)
and smoothed (orange) annual resolution, obtained via averaging of the blue
data with (a) 21-year and (b) 30-year moving intervals.
At the same time, analyzing the methodology of the temperature data
reconstruction given in these rigorous researches [2, 3], it is necessary to
note one important, in our opinion, key point about these papers. It consists
in the complete absence of the commentaries on the physical reasons of
ignoring the nature and type of the noise accompanying the procedure of
successive temperature averaging on different time scales. This is a major
point, since every time scale the temperature of the global climatic system is
being averaged on, is characterized, in general case, by its own type of the
observed noise, which is known to be determined by the spectral density of the
temperature fluctuations in the studied record. In other words, switching from
one time scale (monthly) to another (annual or thirty-year) requires the
knowledge of the corresponding spectral density of the temperature
fluctuations in order to estimate the most important quantity – the
temperature variance. Meanwhile, the averaged temperature variance on the
given time scale may be characterized _in certain cases_ by a _nonremovable_
variance of the colored noise.
The effect of colored noise variance nonremovability on a given time scale
means just that the colored noise is generated by the temperature fluctuations
in the climatic system. Therefore, its variance is actually a temperature
variance equivalent on this time scale. Alternatively speaking, a nonremovable
noise is not an ”interference” masking the real temperature, but rather a
characteristic measure of the temperature fluctuations on a given time scale,
which cannot be disregarded or ”cleaned out” by some kind of time-averaging
procedure.
The analysis of conditions and consequences of such effect of the colored
noise nonremovability typical to the power spectrum of the earth climatic
system (ECS) temperature fluctuations around tricennial time scale (the
climatic state 222Climate is commonly defined as a statistical ensemble of ECS
states characterized by a corresponding set of thermodynamical parameters
(temperature, pressure, etc.) averaged over a long period of time. The
classical period is 30 years, as defined by the World Meteorological
Organization. Hence, it follows that the weather deviation from the climatic
norm cannot be considered as a climate change.) is the primary goal of this
Letter.
## 2 Allan variance and a temperature drift induced by colored noise
In order to retrieve the ”true” variance magnitude for the colored noise of
$1/f^{\alpha}$ type (where $\alpha\geqslant 1$) which dominates in ECS in the
climate state, it is necessary to find a good equivalent of the ”original”
virtual temperature record in some way. Such equivalent record must
characterize the mean global trend while not being ”cleaned” by any special
time-averaging procedures such as the ones used for HadCRUT3 database [2, 3].
For this purpose we calculated the Allan variance [1, 4, 6, 7] for every
historical temperature record observed at 5113 meteorological stations over
the world as used in the HadCRUT3 calculations
$\sigma_{A}^{2}(\tau)=\frac{1}{2}\left\langle\left(y_{i+1}(\tau)-y_{i}(\tau)\right)^{2}\right\rangle,$
(1)
which is a variance of the first differences averaged over the time interval
$\tau$ of the signal $y$. The Allan variance behavior depends on the form of
the noise power spectral density (PSD). For a noise with PSD of the
$1/f^{\alpha}$ form the Allan variance (1) is proportional to
$\tau^{\alpha-1}$, where $\alpha=0$ stands for white (uncorrelated) noise,
$\alpha=1$ for ”$1/f$” (flicker) noise, and $\alpha\geqslant 2$ for correlated
low frequency (drift) noise [1, 4, 6, 7].
Figure 2: Allan variance plot for a set of temperature time series, obtained
at the 5113 meteorological stations over the world. The green line corresponds
to a temperature record at Geneva-Cointrin (Switzerland) weather station. The
color lines reproduce the isoclines, i.e. the lines of equal density of points
that form the general ”relief” of the processed weather stations data.
Analysis of the Fig. 2 demonstrates that the experimental temperature record
obtained at Geneva-Cointrin weather station over the years 1753 – 2011 is a
sufficient equivalent of the ”original” virtual temperature record for our
purposes, since it matches the mean global trend well and is not ”cleaned” by
any kind of time-averaging procedures used in HadCRUT3 [2, 3].
Basing on the comparative analysis of Geneva-Cointrin and HadCRUT3 global
temperature time series, performed by means of Allan variance (Fig. 3) and PSD
(Fig. 4), one may conclude the following. It is obvious, for example, that the
time-averaging procedures used in HadCRUT3 analysis suppressed the Allan
variance and PSD of the temperature fluctuations completely (relative to the
original Geneva-Cointrin data). In addition to that they changed the noise
type, i.e. the white noise to the flicker noise (see the yellow area on Fig. 3
and Fig. 4a), while the original flicker ($1/f$) and drift ($1/f^{2}$) noises
survived, but were highly depressed (see the green and ping areas on Fig. 3
and Fig. 4b). On the other hand, the original drift noise variance apparently
reaches the value of $\sigma_{A}^{2}$ (see the star at Fig. 3) around the
tricennial time scale (the climatic state).
Figure 3: Allan variance plot for temperature records, obtained at Geneva-
Cointrin weather station (the green line) and within the framework of HadCRUT3
(the blue line). Figure 4: PSD of temperature fluctuations for (a) HadCRUT3
and (b) Geneva-Cointrin global temperature time series. The insets are laid
over the PSD of temperature fluctuations adopted from [5].
The latter means that if the drift noise around the tricennial time scale (see
the star at Fig. 3) is nonremovable when ECS is in a state of _climate_ , the
standard deviation $\sigma_{A}$ reaches the value of $\sim 0.45$ and
definitely should be taken into account when constructing the HadCRUT3 global
temperature time series at smoothed annual resolution (the yellow line on Fig.
1a). Its adoption obviously may change the interpretation of HadCRUT3 analysis
drastically (see Fig. 1b, the gray color).
Considering the extreme importance of such conclusion, let us discuss the
possible physical reasons of the colored noise nonremovability in climatic
states description below. For this purpose let us examine a point system,
e.g., ”water – vapor”, with two phase transitions taking place in it, having
the interacting order parameters $X$ and $Y$. At a point of phase transition
lines intersection a potential of such system may be written down in the form
of expansion [11, 13, 12]:
$\Phi=\Phi_{0}-\alpha_{1}X^{2}-\alpha_{2}Y^{2}-\alpha_{12}XY+\beta_{1}X^{4}+\beta_{2}Y^{4}+\beta_{12}X^{2}Y^{2}.$
(2)
Introducing the fluctuating forces in the form of additive terms
$\Gamma_{1}(t)$ and $\Gamma_{2}(t)$, where $\Gamma_{1}(t)$ and $\Gamma_{2}(t)$
are the Gaussian delta-correlated noises, it is possible to pass to a system
of coupled Langevin equations, following [11, 13, 12]:
$\partial X/\partial
t=-2\beta_{12}XY^{2}-4\beta_{1}X^{3}+2\alpha_{1}X+\alpha_{12}Y+\Gamma_{1}(t),$
(3)
$\partial Y/\partial
t=-2\beta_{12}YX^{2}-4\beta_{2}Y^{3}+2\alpha_{2}Y+\alpha_{12}X+\Gamma_{2}(t).$
(4)
A plan for solution of such system is rather simple. The system (3)-(4) is
solved numerically using, for example, the Euler method with different
parameters. The obtained numerical solutions $X(t)$ and $Y(t)$ then undergo a
Fast Fourier Transform and result in a spectral density of fluctuations. In a
simplest case, when the solutions have divergent spectral characteristics, the
parameters of the system, according to [11, 13, 12], are $\beta_{12}=1/2$,
$\beta_{1}=\alpha_{12}=1$, $\alpha_{1}=\alpha_{2}=\beta_{2}=0$, and the system
(3)-(4) takes on the following form:
$\partial X/\partial t=-XY^{2}-4X^{3}+Y+\Gamma_{1}(t),$ (5)
$\partial Y/\partial t=-YX^{2}+X+\Gamma_{2}(t).$ (6)
In the absence of the external noise the solution has asymptotics $X(t)\to
t^{-1/2}$ and $Y(t)\to t^{1/2}$, when $t\to\infty$. The results of numerical
modeling presented in [11, 13, 12] show that the frequency dependence of the
spectral density $S_{X}(f)$ of fluctuations $X(t)$ has the form of
$1/f^{\alpha}$, where $\alpha\approx 1$. In other words, the system (3)-(4)
generates the $1/f$-noise, i.e. the flicker noise. Meanwhile, the spectral
density of the order parameter $Y(t)$ fluctuations depends on the frequency
like $S_{Y}(f)\propto 1/f^{\mu}$, where $1.5\leqslant\mu\leqslant 2$, i.e.
generates the drift noise.
The potential in which the system performs a random walk described by the
system (5)-(6) has the form
$\Phi=\Phi_{0}-XY+X^{4}+(1/2)X^{2}Y^{2}.$ (7)
As follows from (7), this is a double-well potential or, more precisely
speaking, a two-valley potential, i.e. a potential surface has two valleys. In
the absence of the external noise the phase trajectory of a system, depending
on the initial conditions, is enclosed entirely in one of the valleys. In the
presence of an external small-amplitude noise the system performs a random
walk inside the valley. Increasing the noise intensity leads to the system
jumps from one valley to the other, and $X(t)$ displays the $1/f$-noise, while
the order parameter $Y(t)$ displays a $1/f^{2}$-noise.
Thereby this example indicates that the origin of the heat pulsations with a
spectral density inversely proportional to a frequency may be related to the
interaction between the nonequilibrium phase transitions in the system of two
coupled Langevin equations which transforms the white noise into two
oscillation modes with spectral densities proportional to $1/f$ and $1/f^{2}$
[11, 13, 12].
The intersection and interaction of two phase transitions is a rather
widespread phenomenon. Therefore, the model (3)-(4) [6] may be universal
enough to serve as a basis for the explanation of the nonremovable colored
noise formation mechanism in a wide range of processes involving the
nonequilibrium phase transitions. Let us show this for a point nonlinear
system ”water – vapor” in the particular climatic model below.
## 3 Climatic two-well potential and bifurcation model of the Earth global
climate
In our earlier papers we proposed a bifurcation model for the energy balance
of the Earth global climate. Its theoretical solution was in a good agreement
with the known experimental data on Earth surface paleotemperature evolution
for the last 420 thousand years and 740 thousand years, obtained within the
antarctic projects Vostok and EPICA Dome C respectively. In the framework of
the proposed model a concept of climatic sensitivity was also introduced,
which manifests a property of temperature instability in the form of the so-
called hysteresis loop, as shown in [9, 10]. Basing on this concept and using
the bifurcation equation of the model, we reconstructed a time series for the
global ice volume for the last 1 million years which fits well the
experimental time series of $\delta^{18}O$ concentrations in marine sediments.
A base for this result lies in a fact that an effective mechanism of climatic
”cold-warm” oscillations formation had been built into the model and was
provided by the interaction of the nonequilibrium phase transitions in a
”fresh water – water vapor” system in the Earth boundary layer [9, 10].
If we add to the stated above that our bifurcation model of the Earth global
climate, developed in [11, 13, 12], was built upon a climatic double-well
potential
$U(T,t)=\frac{1}{4}T^{4}+\frac{1}{2}a(t)T^{2}+b(t)T,$ (8)
where
$a(t)=-\frac{1}{4\delta\sigma}a_{\mu}H_{\oplus}(t),$ (9)
$b(t)=-\frac{1}{4\delta\sigma}\left[\frac{\eta_{\alpha}S_{0}}{4}+\frac{1}{2}\beta+\frac{1}{2}b_{\mu}H_{\oplus}(t)\right],$
(10)
which is qualitatively identical (given $Y(t)\sim\mathrm{const}$) to a two-
valley potential (7), it becomes clear that ECS is really capable of
generating the colored noise which is not actually a masking ”interference”,
but rather an internal property of the ECS temperature pulsations. Here
$H_{\oplus}$ is the Y-component 333The physical sense of the Y-component of
the Earth magnetic field adoption is discussed in [8] in detail. of the Earth
magnetic field intensity; $T$ is the mean global temperature of the Earth
surface at the time $t$; $S_{0}=1366.2~{}W/m^{2}$ is the solar constant;
$\delta=0.95$ is the emissivity of the Earth surface; $\sigma=5.67\cdot
10^{-8}~{}W/(m^{2}K^{4})$ is the Stefan-Boltzmann constant;
$\eta_{\alpha}=0.01513~{}K^{-1}$, $a_{\mu}=0.5398~{}W/(m^{2}K^{2})$,
$b_{\mu}=-310~{}W/(m^{2}K)$; finally, $\beta=0.006~{}W/(m^{2}K)$ is the carbon
dioxide accumulation rate in the atmosphere normalized by a unit temperature.
Following this conclusion, let us consider a basic equation of the bifurcation
model of the Earth global climate for the average Earth surface temperature on
the 30-year time scale [8, 9, 10]:
$\frac{m^{*}}{4\delta\sigma}\frac{dT}{dt}=T^{3}+a(t)T+b(t)+\xi(t),$ (11)
where the numerical value $0.129$ was used for $m^{*}$, and
$\langle\xi(t)\rangle=0,~{}~{}~{}\langle\xi(t)\xi(t^{\prime})\rangle=2D\delta(t-t^{\prime}).$
(12)
Obviously, the solution of the stochastic differential equation (11) fits into
the total uncertainty limits of HadCRUT3 global temperature time series at
smoothed annual resolution completely (Fig. 1b, the red line), where a
nonremovable noise variance predominates. At the same time, as the numerical
integration of Eq. (1) using the Runge-Kutta fourth order method with
different initial conditions reveals, the obtained solution (Fig. 1b, the red
line) is strongly stable. It means that in the presence of a nonremovable
colored noise on the climatic 30-year time scale the interpretation of the
HadCRUT3 analysis data changes drastically, replacing the known ”global
warming” paradigm with an alternative theory of climate as a highly stable ECS
state which is characterized by a temperature $\sim 287~{}K$ within the
colored noise variance.
## 4 Conclusion
One of the major results of the present paper is a statement of the problem of
the possible nonremovability of the colored noise on the climatic 30-year time
scale, which changes the angle of view on the known problem of global warming
radically.
One of the basic climate-generating nonlinear ”fresh water – water vapor”
subsystem was involved to explain the mechanism of the nonremovable colored
noise formation. The appearance of the nonremovable thermal pulsations with
the spectral density inversely proportional to the frequency is explained by
the interaction of the nonequilibrium phase transitions in such system. In
other words, is has been shown that ECS is really capable of generating the
nonremovable colored noise which is an internal property of the temperature
pulsations and not just a masking ”interference”.
However it must be admitted that all the arguments adduced here apply not to a
direct, but to an indirect proof of the colored noise nonremovability on the
climatic 30-year time scale. This is also true for our model representations,
which, although have passed a reliable verification by fitting the known
experimental paleotemperature data on a large time scale, have the same
arguments of indirect action. At the same time it should be noted that the
problem of the possible noise nonremovability formulated in the present paper
makes any claims about global warming inappropriate until the problem of the
colored noise origin on the climatic 30-year time scale is definitely solved.
## Acknowledgements
This work is partially supported by EU FP7 Marie Curie Actions, SP3-People,
IRSES project BlackSeaHazNet (PIRSES-GA-2009-246874).
The work of M. Eingorn was supported by NSF CREST award HRD-0833184 and NASA
grant NNX09AV07A.
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|
arxiv-papers
| 2013-10-01T22:40:40 |
2024-09-04T02:49:51.893687
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "V.D. Rusov, V.P. Smolyar, M.V. Eingorn, T.N. Zelentsova, E.P. Linnik,\n M.E. Beglaryan, B. Vachev",
"submitter": "Vladimir Smolyar",
"url": "https://arxiv.org/abs/1310.0511"
}
|
1310.0843
|
# Polygonal $\mathcal{VH}$ complexes
Jason K.C. Polák Math. & Stats.
McGill Univ.
Montreal, QC, Canada H3A 2K6 [email protected] and Daniel T. Wise Math. &
Stats.
McGill Univ.
Montreal, QC, Canada H3A 2K6 [email protected]
###### Abstract.
Ian Leary inquires whether a class of hyperbolic finitely presented groups are
residually finite. We answer in the affirmative by giving a systematic version
of a construction in his paper, which shows that the standard $2$-complexes of
these presentations have a $\mathcal{VH}$-structure. This structure induces a
splitting of these groups, which together with hyperbolicity, implies that
these groups are residually finite.
Supported by NSERC
## 1\. Introduction
Recall that a group is _residually finite_ if every nontrivial element maps to
a nontrivial element in a finite quotient. For instance, $GL_{n}(\mathbb{Z})$
is residually finite, and hence so too are its subgroups. Ian Leary describes
the following group $G$ in [Lea]:
$\left\langle
a,b,c,d,e,f\bigg{|}\begin{array}[]{ccc}abcdef,&ab^{-1}c^{2}f^{-1}e^{2}d^{-1},&a^{2}fc^{2}bed,\\\
ad^{-2}cb^{-2}ef^{-1},&af^{-2}cd^{-1}eb^{-2},&ad^{2}cf^{2}eb^{2}\end{array}\right\rangle$
(1)
Leary shows that $G\cong\pi_{1}X$ where $X$ is a nonpositively curved square
complex. He constructs $X$ by subdividing the standard 2-complex of
Presentation (1). Leary asks whether $G$ and some similar groups are
residually finite, and he reports that his investigations with the GAP
software indicated that $G$ has few low-index subgroups. We note that $G$ is
perfect – which is part of the reason Leary was drawn to investigate groups
like $G$ in conjunction with his Kan-Thurston generalization.
The main goal of this note is to explicitly describe conditions under which
$2$-complexes such as $X$ can be subdivided into nonpositively curved
$\mathcal{VH}$-complexes. This develops the ad-hoc method described by Leary.
Moreover, it seems that Leary’s method might have been guided by the theory of
$\mathcal{VH}$ complexes, and revealing the lurking $\mathcal{VH}$-structure
allows us to answer Leary’s question. Indeed, since $X$ is a compact
nonpositively curved $\mathcal{VH}$-complex, $\pi_{1}X=G$ has a so-called
quasiconvex hierarchy and since $G$ is word-hyperbolic, it is virtually
special and hence residually finite via results from [Wis].
We now describe the parts of this paper. In Section 2, we review material
about nonpositively curved $\mathcal{VH}$-complexes. In Section 3, we describe
a criterion for subdividing certain $2$-complex into a $\mathcal{VH}$-complex.
Our _squaring construction_ is a systematic version of the construction
suggested by Leary. Finally, we illustrate this technique in Section 4 where
we show that Leary’s groups are residually finite.
## 2\. Nonpositively curved $\mathcal{VH}$-complexes
A _square complex_ $X$ is a combinatorial $2$-complex whose $2$-cells are
squares in the sense that their attaching maps are closed length $4$ paths in
$X^{1}$. We say $X$ is _nonpositively curved_ if $\operatorname{link}(x)$ has
girth $\geq 4$ for each $x\in X^{0}$. Recall that the _link_ of a 0-cell $x$
is the graph whose vertices correspond to corners of $1$-cells incident with
$x$, and whose edges correspond to corners of $2$-cells incident with $x$.
Roughly speaking, $\operatorname{link}(x)$ is isomorphic to the
$\epsilon$-sphere about $x$ in $X$.
A _$\mathcal{VH}$ -complex_ $X$ is a square complex such that the $1$-cells of
$X$ are partitioned into two disjoint sets $H$ and $V$ called _horizontal_ and
_vertical_ respectively, and the attaching map of every $2$-cell is a length
$4$ path that alternates between vertical and horizontal $1$-cells.
A $\mathcal{VH}$-complex is nonpositively curved if and only if there are no
length $2$ cycles in each $\operatorname{link}(x)$ – indeed, each link is
bipartite because of the $\mathcal{VH}$ structure. The main result that we
will need about $\mathcal{VH}$-complexes is the following result which is a
specific case of the main theorem in [Wis]:
###### Theorem 2.1.
If $X$ is a compact nonpositively curved $\mathcal{VH}$-complex such that
$\pi_{1}X$ is word-hyperbolic then $X$ is virtually special. Consequently,
$\pi_{1}X$ embeds in $GL_{n}(\mathbb{Z})$, and so $\pi_{1}X$ is residually
finite.
The feature of compact nonpositively curved $\mathcal{VH}$-complexes that
enables us to apply Theorem 2.1 is that $\pi_{1}X$ has a so-called quasiconvex
hierarchy, which means that it can be built from trivial groups by a finite
sequence of HNN extensions and amalgamated free products along quasiconvex
subgroups. This type of hierarchy occurs for fundamental groups of
$\mathcal{VH}$-complexes because they split geometrically as graphs of graphs
as has been explored for instance in [Wis06].
## 3\. Squaring polygonal complexes
A _bicomplex_ is a combinatorial $2$-complex $X$ such that the $1$-cells of
$X^{1}$ are partitioned into two sets called _vertical_ and _horizontal_ , and
the attaching map of each $2$-cell traverses both vertical and horizontal
$1$-cells. We note that each such attaching map is thus a concatenation
$V_{1}H_{1}\cdots V_{r}H_{r}$ of nontrivial paths where each $V_{i}$ is the
concatenation of vertical $1$-cells and likewise each $H_{i}$ is a horizontal
path. We refer to the $V_{i}$ and $H_{i}$ as _sides_ of the 2-cell, so if the
attaching map is $V_{1}H_{1}\cdots V_{r}H_{r}$ then the 2-cell has $2r$ sides.
$X$ has a _repeated $\mathcal{VH}$-corner_ if there is a concatenation $vh$ of
a single horizontal and single vertical $1$-cell that occurs as a “piece” in
$X$, in the sense that it occurs in two ways as a subpath of attaching maps of
$2$-cells. (Specifically, $vh$ or $h^{-1}v^{-1}$ could occur in two distinct
attaching maps, or in two ways within the same attaching map.)
A $2$-cell of $X$ with attaching map $V_{1}H_{1}\cdots V_{r}H_{r}$ satisfies
the _triangle inequality_ if for each $i$ we have:
$|V_{i}|\leq\sum_{j\neq i}|V_{j}|\ \ \text{and}\ \ |H_{i}|\leq\sum_{j\neq
i}|H_{j}|$
The goal of this section is to prove the following:
###### Theorem 3.1 ($\mathcal{VH}$ subdivision criterion).
Let $X$ be a bicomplex. If each $2$-cell of $X$ satisfies the triangle
inequality and if $X$ has no repeated $\mathcal{VH}$-corners, then $X$ can be
subdivided into a nonpositively curved $\mathcal{VH}$-complex.
_(Hyperbolicity Criterion)_ Moreover, if each $2$-cell has at least $6$ sides,
and $X$ is compact, then $\pi_{1}X$ is hyperbolic.
###### Example 3.2.
It is instructive to indicate some simple examples with no repeated
$\mathcal{VH}$-corners but where Theorem 3.1 fails without the triangle
inequality:
1. (1)
$\langle v,h\mid v^{m}h^{n}\rangle$
2. (2)
$\langle v,h\mid v^{m}h^{n},v^{-m}h^{n}\rangle$
3. (3)
$\langle v,h\mid hv^{m}h^{-1}v^{n}\rangle$
4. (4)
$\langle u,v,h\mid h^{-1}uhv^{-1}u^{-1},h^{-1}vhu^{-1}v^{-1}\rangle$
In each case, $X$ is the standard 2-complex of the presentation given above.
The first example is homeomorphic to a nonpositively curved
$\mathcal{VH}$-complex, but it does not have a $\mathcal{VH}$-complex
subdivision that is consistent with the original $\mathcal{VH}$-decomposition
of the 1-skeleton. The second example has no nonpositively curved
$\mathcal{VH}$-subdivision – indeed, the group has torsion when $n,m\neq 0$.
The third example has a nonpositively curved $\mathcal{VH}$-subdivision
exactly when $m=\pm n$. The fourth example is word-hyperbolic. However, it
does not have any $\mathcal{VH}$-subdivision. Let us sketch this last claim.
Without loss of generality, we may assume that $h$ is horizontal, and then an
application of the Combinatorial Gauss-Bonnet Theorem [GS91] shows that $u$
and $v$ are vertical. Each two cell in a subdivision has at most four points
of positive curvature that must occur at the $\mathcal{VH}$-corners. If there
were a $\mathcal{VH}$-subdivision, then each would have curvature exactly
$\tfrac{\pi}{2}$ and so the divided two-cell $C^{\prime}$ would be a
rectangular grid. This however is impossible as the length of the path $u$
must be strictly less than the length of the path $v^{-1}u^{-1}$.
The main tool used to prove Theorem 3.1 is a procedure that subdivides a
single polygon $C$ into a $\mathcal{VH}$-complex, as we can then subdivide
$X^{1}$ and then apply this procedure to subdivide each $2$-cell of $X$. We
will therefore focus on a bicomplex that consists of a single $2$-cell $C$
that we refer to as a polygon. The observation is that when $C$ satisfies the
triangle inequality, there is a pairing between the vertical edges on
$\partial C$, and also a pairing between the horizontal edges on $\partial C$
(the exact type of “pairing” we need is described below). We add line segments
to $C$ that connect the midpoints of paired edges, and our
$\mathcal{VH}$-subdivision of $C$ is simply the dual (see Figure 2).
###### Definition 3.3 (Admissible Pairing).
Let $P$ be a polygon and let $E_{1},\dots,E_{k}$ be distinct sides of $P$.
Suppose each edge $E_{i}$ is subdivided into $|E_{i}|$ edges. An _admissible
pairing_ with respect to these sides is an equivalence relation $\sim$ on the
subdivided edges satisfying the following:
1. (1)
If $u\sim v$, $u\in E_{i}$ and $v\in E_{j}$ then $i\not=j$.
2. (2)
If $u\sim v$ and $w\sim x$, then $w$ and $x$ lie in the same connected
component of $P\setminus u\cup v$.
3. (3)
Each equivalence class has two members.
We now demonstrate that the triangle inequality implies, and is in fact
equivalent to the existence of an admissible pairing.
###### Lemma 3.4.
Suppose that $E_{1},\dots,E_{r}$ are sides of a polygon, that each $E_{i}$ is
subdivided into $n_{i}$ edges, and that $\sum n_{i}$ is even. Then there
exists an admissible pairing of these vertices if and only if for each $i$,
the inequality $n_{i}\leq\sum_{j\not=i}n_{j}$ holds.
Figure 1. The sequence of paired edges is indicated by the numbered line
segments above.
###### Proof.
Suppose that $E_{1},\dots,E_{r}$ are ordered clockwise around the polygon. We
construct the admissible pairing via steps, each step pairing two edges. Each
step will consist of choosing $E_{i}$ with $|E_{i}|$ maximal, taking an
unpaired edge $v\in E_{i}$ closest to a vertex of the edge, and pairing it
with an edge $u$ in some $E_{j}$ closet to $v$ with $i\not=j$.
By convexity considerations, if we manage a pairing that satisfies (1) and (3)
using such a procedure, it will also be admissible. Suppose we represent the
number of unpaired edges in each step of this pairing by an ordered vector so
that before any edges are paired, our vector is $(n_{1},\dots,n_{r})$. Using
our pairing strategy, each step will consist of subtracting $1$ from two
components, one component being maximal.
It thus suffices to prove that any vector $(n_{1},\dots,n_{r})$ can be
completely reduced to the zero vector if and only if the triangle inequality
$n_{i}\leq\sum_{j\not=i}n_{j}$ holds for each $i$. Suppose that there is an
admissible pairing. If the triangle inequality does not hold, then there
exists an $i$ such that $n_{i}>\sum_{j\not=i}n_{j}$. But each vertex in
$V_{i}$ must be paired with a vertex outside of $V_{i}$, and this is obviously
impossible.
Now suppose the triangle inequality holds. If $n_{i}=1$ for all $i$, then
there is an admissible pairing since $\sum n_{i}$ is even. Otherwise, choose
an $i$ such that $n_{i}\geq n_{j}$ for each $j$. Subtract $1$ from $n_{i}$ and
an adjacent component. It is easy to verify that the triangle inequality holds
in the new vector. Eventually we will get a vector with all entries being $1$
whose sum is even, and so there is an admissible pairing. See Figure 1 for an
example of this algorithm applied to a polygon. ∎
We now prove the main result.
###### Proof of Theorem 3.1.
_( $\mathcal{VH}$ Subdivision Criterion)_ By subdividing $X^{1}$, we can
ensure that for each $2$-cell of $X$ with attaching map $V_{1}H_{1}\cdots
V_{r}H_{r}$, both $\sum|V_{i}|$ and $\sum|H_{i}|$ are even. Note that the
triangle inequalities are preserved by subdivision.
By Lemma 3.4, for each $2$-cell $C$, there is an admissible pairing for the
vertical $1$-cells of $C$, and an admissible pairing for the horizontal
$1$-cells of $C$. These pairings form a collection of line segments within $C$
that join barycenters of paired edges, and we let $\Gamma$ be the graph
consisting of the union of these lines segments. As in Figure 2, the dual of
$\Gamma$ within $C$ forms the 1-skeleton of our desired $\mathcal{VH}$-complex
$C^{\prime}$. We note that $\partial C$ embeds as a subgraph of this dual, and
$\partial C\cong\partial C^{\prime}$.
In this way, we subdivide each $2$-cell of $X$ to obtain a 2-complex
$X^{\prime}$ whose $2$-cells are the subdivisions $C^{\prime}$ of the various
$2$-cells $C$. Observe that $X^{\prime}$ is a $\mathcal{VH}$-complex since
each $C^{\prime}$ is a $\mathcal{VH}$-complex and the $\mathcal{VH}$-structure
on $\partial C^{\prime}$ agrees with the $\mathcal{VH}$-structure on $X^{1}$.
Finally, $X^{\prime}$ is nonpositively curved precisely if each
$\operatorname{link}(x)$ has no 2-cycles. This is clear when $x$ lies in the
interior of some $C^{\prime}$. When $x\in(X^{\prime})^{1}=X^{1}$, a 2-cycle in
$\operatorname{link}(x)$ corresponds precisely to a repeated
$\mathcal{VH}$-corner.
_(Hyperbolicity Criterion)_. The compact nonpositively curved square complex
$X^{\prime}$ has universal cover $\widetilde{X}^{\prime}$ which is
$\mathrm{CAT}(0)$, and by Gromov’s flat plane theorem [Bri95],
$\pi_{1}X^{\prime}$ is hyperbolic if and only if $\widetilde{X}^{\prime}$ does
not contain an isometrically embedded copy of $\mathbb{E}^{2}$ – whose cell
structure would be a finite grid in our case.
Observe that each square in a flat plane lies in some $C^{\prime}$ in the flat
plane. The interior $0$-cells of $C^{\prime}$ must have valence exactly four,
and the $0$-cells in the interior of a horizontal or a vertical side must have
valence exactly three.
Each $0$-cell on a $\mathcal{VH}$-corner must have even valence either $2$ or
$4$, but again the latter is impossible for the embedding in the flat plane
shows that there is a repeated $\mathcal{VH}$-corner. Furthermore, since there
are at least six sides, there are at least six $\mathcal{VH}$-corners.
Combinatorial Gauss-Bonnet applied to $C^{\prime}$ shows that
$\displaystyle 2\pi$
$\displaystyle=\sum_{v\in\mathrm{Int}(C)}\kappa(v)+\sum_{v\in\partial(C))}\kappa(v)\geq
6\cdot\frac{\pi}{2}=3\pi.$
Thus $C^{\prime}$ cannot lie in an infinite grid. ∎
Figure 2. The two systems of dual curves in $C$ are indicated on the left, the
dual of $\Gamma$ is indicated in the middle, and the
$\mathcal{VH}$-subdivision $C^{\prime}$ is indicated on the right.
## 4\. Application to Leary’s examples
There are two examples from Leary’s paper [Lea] that we shall consider. The
first example is the group:
$\left\langle
a,b,c,d,e,f\bigg{|}\begin{array}[]{ccc}abcdef,&ab^{-1}c^{2}f^{-1}e^{2}d^{-1},&a^{2}fc^{2}bed,\\\
ad^{-2}cb^{-2}ef^{-1},&af^{-2}cd^{-1}eb^{-2},&ad^{2}cf^{2}eb^{2}\end{array}\right\rangle$
(2)
Leary proved that this group is nontrivial, torsion-free, and acyclic, and
asked whether it is also residually finite.
The standard 2-complex $X$ of this presentation is a bicomplex whose vertical
$1$-cells correspond to $\\{a,c,e\\}$ and whose horizontal $1$-cells
correspond to $\\{b,d,f\\}$. It is easily verified that the $2$-cells satisfy
the triangle inequality.
Finally, this group is hyperbolic because each polygon has at least six sides,
so we can apply the hyperbolicity criterion in Theorem 3.1. The group is
therefore residually finite by Theorem 2.1.
A second family of examples with which Leary is concerned is defined as
follows. We let $n\in\mathbb{N}$ with $n\geq 4$, and for each
$i\in\mathbb{Z}/n\mathbb{Z}$ we define the two words
$A_{i}=a_{i}a_{i+2}a_{i}^{-2}a_{i+2}^{-1}a_{i}$ and
$B_{i}=b_{i}b_{i+2}b_{i}^{-2}b_{i+2}^{-1}b_{i}$ and the eight words given by:
$a_{i}A_{i}B_{i}A_{i+1}B_{i}A_{i+2}B_{i}A_{i+3}B_{i}\ \ \text{ and }\ \
b_{i}B_{i}A_{i}^{-1}B_{i}A_{i+1}^{-1}B_{i}A_{i+2}^{-1}B_{i}A_{i+3}^{-1}$
This group is again hyperbolic by our hyperbolicity criterion (this also
follows since the presentation is $C^{\prime}(\frac{1}{6})$). The triangle
inequalities are readily verified, and there are no repeated
$\mathcal{VH}$-corners by design.
Acknowledgement: We are grateful to the referee for improving the exposition
of this paper.
## References
* [Bri95] Martin R. Bridson. On the existence of flat planes in spaces of nonpositive curvature. Proc. Amer. Math. Soc., 123(1):223–235, 1995.
* [GS91] S. M. Gersten and H. Short. Small cancellation theory and automatic groups. II. Invent. Math., 105(3):641–662, 1991.
* [Lea] I.J. Leary. A metric Kan–Thurston theorem. Preprint. arXiv:1009.1540.
* [Wis] Daniel T. Wise. The structure of groups with a quasiconvex hierarchy. pages 1–189.
* [Wis06] Daniel T. Wise. Subgroup separability of the figure 8 knot group. Topology, 45(3):421–463, 2006.
|
arxiv-papers
| 2013-10-02T21:04:11 |
2024-09-04T02:49:51.903013
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Jason K.C. Pol\\'ak and Daniel T. Wise",
"submitter": "Jason Polak",
"url": "https://arxiv.org/abs/1310.0843"
}
|
1310.0855
|
DPF2013-148
Charmless three-body decays of $b$-hadrons
Thomas Latham111On behalf of the LHCb collaboration.
Department of Physics, University of Warwick,
Coventry, CV4 7AL, United Kingdom
> A review of recent results from LHCb and the $B$-factories on the charmless
> decays of $b$-hadrons into three-body final states is presented.
> PRESENTED AT
>
>
>
>
> DPF 2013
> The Meeting of the American Physical Society
> Division of Particles and Fields
> Santa Cruz, California, August 13–17, 2013
>
## 1 Introduction
Charmless decays of $b$-hadrons can proceed through both $b\\!\rightarrow u$
tree and $b\\!\rightarrow s,d$ loop (penguin) diagrams, which can interfere.
Since they have a relative weak phase of $\gamma$ and the diagrams appear at
similar orders, this can give rise to large direct $C\\!P$ violation. In the
decays of neutral $B$ mesons, time-dependent analyses allow measurements of
mixing-induced $C\\!P$ asymmetries. Comparing the values of these asymmetries
with those measured in tree-dominated decays such as
$B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{0}_{\rm\scriptscriptstyle S}$ or
$B^{0}_{s}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\phi$ can
be a sensitive test of the Standard Model (SM), with significant deviations
being a sign that new physics particles could be appearing in the loops.
Recent results from LHCb for the direct $C\\!P$ asymmetries, defined as
${\cal A}^{C\\!P}(B\\!\rightarrow f)=\frac{\Gamma(\kern
1.79993pt\overline{\kern-1.79993ptB}{}\\!\rightarrow\kern
1.79993pt\overline{\kern-1.79993ptf}{})-\Gamma(B\\!\rightarrow
f)}{\Gamma(\kern 1.79993pt\overline{\kern-1.79993ptB}{}\\!\rightarrow\kern
1.79993pt\overline{\kern-1.79993ptf}{})+\Gamma(B\\!\rightarrow f)}\,,$ (1)
of the decays $B^{0}\\!\rightarrow K^{+}\pi^{-}$ and $B^{0}_{s}\\!\rightarrow
K^{-}\pi^{+}$ [1] exhibit large central values***The inclusion of charge
conjugate processes is implied throughout, except in ${\cal A}^{C\\!P}$
definitions.
$\displaystyle{\cal A}^{C\\!P}(B^{0}_{s}\\!\rightarrow\pi^{+}K^{-})$
$\displaystyle=$ $\displaystyle\phantom{-}0.27\phantom{0}\pm
0.04\phantom{0}\mathrm{\,(stat)}\pm 0.01\phantom{0}\mathrm{\,(syst)}\,,$
$\displaystyle{\cal A}^{C\\!P}(B^{0}\\!\rightarrow K^{+}\pi^{-})$
$\displaystyle=$ $\displaystyle-0.080\pm 0.007\mathrm{\,(stat)}\pm
0.003\mathrm{\,(syst)}\,.$
The first of these constitutes the first observation of $C\\!P$ violation in
the $B^{0}_{s}$ system with a significance of $6.5\,\sigma$, while the latter
is the world’s most precise single measurement of that quantity. Combining
these results with related quantities in the expression
$\displaystyle\Delta\equiv\frac{{\cal A}^{C\\!P}(B^{0}\\!\rightarrow
K^{+}\pi^{-})}{{\cal
A}^{C\\!P}(B^{0}_{s}\\!\rightarrow\pi^{+}K^{-})}+\frac{{\cal
B}(B^{0}_{s}\\!\rightarrow\pi^{+}K^{-})}{{\cal B}(B^{0}\\!\rightarrow
K^{+}\pi^{-})}\frac{\tau_{B^{0}}}{\tau_{B^{0}_{s}}}=-0.02\pm 0.05\pm 0.04\,,$
it is found that everything is consistent with the SM expectation ($\Delta=0$)
[2].
It is necessary to form such a combination of quantities in order to test for
compatibility, or otherwise, with the SM because the source of the strong
phase difference is not well understood in two-body decays. Three-body decays,
on the other hand, allow direct measurements of the relative strong phases
through an amplitude analysis of the Dalitz plot. Determining both the
magnitudes and the phases of the intermediate states provides greater
information for constraining theoretical models. In addition, modelling the
interferences can help to resolve trigonometric ambiguities in the measurement
of weak phases, see for example Ref. [3].
## 2 Direct $C\\!P$ violation in $B^{+}\\!\rightarrow h^{+}h^{+}h^{-}$ decays
Searches for direct $C\\!P$ violation in $B^{+}\\!\rightarrow h^{+}h^{+}h^{-}$
decays, where $h=\pi,K$ are motivated both by the large asymmetries seen in
$B\\!\rightarrow K\pi$ decays and $B$-factory results that have shown evidence
for direct $C\\!P$ asymmetries in $B^{+}\\!\rightarrow\rho^{0}(770)K^{+}$ [4,
5] and $B^{+}\\!\rightarrow\phi(1020)K^{+}$ [6]. The recent LHCb analysis of
$B^{+}\\!\rightarrow K^{+}h^{+}h^{-}$ decays makes measurements of the global
$C\\!P$ asymmetry as well as the local asymmetries in regions of the Dalitz-
plot.
The analysis, full details of which can be found in Ref. [7], uses the
1.0$\mbox{\,fb}^{-1}$ of $pp$ collision data collected during 2011 by the LHCb
detector [8]. The raw asymmetry of measured yields
${\cal A}^{C\\!P}_{\rm RAW}=\frac{N_{B^{-}}-N_{B^{+}}}{N_{B^{-}}+N_{B^{+}}}$
(2)
is determined from a simultaneous fit to the sample of $B^{+}$ and $B^{-}$
candidates. The raw asymmetry must be corrected for both production and
detection asymmetries
${\cal A}^{C\\!P}={\cal A}^{C\\!P}_{\rm RAW}-{\cal A}_{P}(B^{\pm})-{\cal
A}_{D}(K^{\pm})\,,$ (3)
which are determined from the control channel
$B^{+}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+}$, where
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ decays to $\mu^{+}\mu^{-}$,
according to the relation
${\cal A}_{D}(K^{\pm})+{\cal A}_{P}(B^{\pm})={\cal A}^{C\\!P}_{\rm
RAW}({J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+})-{\cal
A}^{C\\!P}({J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+})\,.$ (4)
This channel is well suited for this role due to its similar topology to the
signal channel and since its $C\\!P$ asymmetry is consistent with zero and
precisely determined, ${\cal A}^{C\\!P}({J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{+})=(0.1\pm 0.7)\%$ [9].
The results of the fit to the data sample are shown in Figure 1 and the values
of the $C\\!P$ asymmetries are found to be
$\displaystyle{\cal A}^{C\\!P}(B^{+}\\!\rightarrow K^{+}\pi^{+}\pi^{-})$
$\displaystyle=$ $\displaystyle\phantom{-}0.032\pm 0.008\mathrm{\,(stat)}\pm
0.004\mathrm{\,(syst)}\pm 0.007({J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{+})\,,$ $\displaystyle{\cal A}^{C\\!P}(B^{+}\\!\rightarrow
K^{+}K^{+}K^{-})$ $\displaystyle=$ $\displaystyle-0.043\pm
0.009\mathrm{\,(stat)}\pm 0.003\mathrm{\,(syst)}\pm
0.007({J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+})\,.$
The significance of $C\\!P$ violation in each decay mode is $2.8\,\sigma$ and
$3.7\,\sigma$, respectively.
Figure 1: Distributions of the $B$-candidate invariant mass for (a)
$B^{+}\\!\rightarrow K^{+}\pi^{+}\pi^{-}$ decays and (b) $B^{+}\\!\rightarrow
K^{+}K^{+}K^{-}$ decays. The left (right) plots show the $B^{-}$ ($B^{+}$)
decays.
The variation of the raw asymmetry over the Dalitz plot is also studied and
the results are shown in Figure 2. In some regions there are extremely large
asymmetries present, in particular around the $\rho^{0}$ resonance in
$B^{+}\\!\rightarrow K^{+}\pi^{+}\pi^{-}$ but also in regions that are not
clearly associated with a resonance. The local $C\\!P$ asymmetries in the
region where
$m^{2}_{K^{+}\pi^{-}}<15\,(\\!{\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}})^{2}$
and
$0.08<m^{2}_{\pi^{+}\pi^{-}}<0.66\,(\\!{\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}})^{2}$
in $B^{+}\\!\rightarrow K^{+}\pi^{+}\pi^{-}$ and in the region
$m^{2}_{K^{+}\kern-1.12003ptK^{-}{\rm
high}}<15\,(\\!{\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}})^{2}$ and
$1.2<m^{2}_{K^{+}\kern-1.12003ptK^{-}{\rm
low}}<2.0\,(\\!{\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}})^{2}$ are determined
to be
$\displaystyle{\cal A}^{C\\!P}_{\rm local}(B^{+}\\!\rightarrow
K^{+}\pi^{+}\pi^{-})$ $\displaystyle=$ $\displaystyle\phantom{-}0.678\pm
0.078\mathrm{\,(stat)}\pm 0.032\mathrm{\,(syst)}\pm
0.007({J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+})\,,$
$\displaystyle{\cal A}^{C\\!P}_{\rm local}(B^{+}\\!\rightarrow
K^{+}K^{+}K^{-})$ $\displaystyle=$ $\displaystyle-0.226\pm
0.020\mathrm{\,(stat)}\pm 0.004\mathrm{\,(syst)}\pm
0.007({J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+})\,,$
respectively.
Figure 2: Variation of the raw asymmetry over the Dalitz plot in (a)
$B^{+}\\!\rightarrow K^{+}\pi^{+}\pi^{-}$ and (b) $B^{+}\\!\rightarrow
K^{+}K^{+}K^{-}$ decays.
The BaBar experiment has recently made an update of their analysis of
$B^{+}\\!\rightarrow K^{+}K^{+}K^{-}$, in order to provide a direct comparison
with the LHCb results for the asymmetry as a function of
$m_{K^{+}\kern-1.12003ptK^{-}}$ [10]. This comparison is shown in Figure 3.
The shapes of the distributions are extremely similar, albeit with a small
offset, which is determined to be $0.045\pm 0.021$ ($0.053\pm 0.021$) for the
$m_{K^{+}\kern-1.12003ptK^{-}{\rm low}}^{2}$
($m_{K^{+}\kern-1.12003ptK^{-}{\rm high}}^{2}$) spectrum. However, it must be
remembered that the LHCb distribution is that of the raw asymmetry and hence
has not been corrected for production and detection effects. These are of the
order of 1% and act in the direction to decrease the mild discrepancy.
Figure 3: Asymmetry as a function of (left) $m_{K^{+}\kern-1.12003ptK^{-}{\rm
low}}^{2}$ (right) $m_{K^{+}\kern-1.12003ptK^{-}{\rm high}}^{2}$ for
$B^{+}\\!\rightarrow K^{+}K^{+}K^{-}$ decays. The BaBar (LHCb) data are the
open (filled) circles.
Very similar findings to those from $B^{+}\\!\rightarrow K^{+}h^{+}h^{-}$
decays are made in a preliminary analysis of
$B^{+}\\!\rightarrow\pi^{+}\pi^{+}\pi^{-}$ and
$B^{+}\\!\rightarrow\pi^{+}K^{+}K^{-}$ [11], both in terms of the large local
asymmetries and the opposite sign of the asymmetries between the two modes. In
addition, the local asymmetries are observed mainly in regions not clearly
associated with a well established resonance. This could indicate that
$\pi^{+}\pi^{-}\\!\rightarrow K^{+}\kern-1.60004ptK^{-}$ rescattering is
playing a role in the generation of the strong phase difference. Amplitude
analyses of these modes using the larger dataset now available at LHCb
(3$\mbox{\,fb}^{-1}$) will provide more information to resolve this puzzle.
## 3 Dynamics of $B^{+}\\!\rightarrow p\overline{}ph^{+}$ decays
The large asymmetries seen in $B^{+}\\!\rightarrow h^{+}h^{+}h^{-}$ decays
raise the question about the role of $\pi^{+}\pi^{-}\leftrightarrow
K^{+}\kern-1.60004ptK^{-}$ rescattering in these modes. The closely related
decays $B^{+}\\!\rightarrow p\overline{}ph^{+}$ can shed some light on this
issue since it is expected that $h^{+}h^{-}\leftrightarrow p\overline{}p$
rescattering should be much smaller. The threshold enhancements seen in many
$B\\!\rightarrow p\overline{}pX$ decays provide further motivation for
studying these decays. The analysis, which uses the LHCb 1.0$\mbox{\,fb}^{-1}$
data sample collected during 2011, studies the dynamics of the decays as well
as the $C\\!P$ asymmetries. Full details can be found in Ref. [12].
Fits to the $B$-candidate invariant mass distribution, shown in Figure 4,
yield $7029\pm 139$ ($656\pm 70$) signal events for the mode
$B^{+}\\!\rightarrow p\overline{}pK^{+}$ ($B^{+}\\!\rightarrow
p\overline{}p\pi^{+}$), where the uncertainties are statistical only. The fit
model contains contributions from signal, cross-feed (where the kaon in the
signal mode is mis-identified as a pion or vice versa), combinatorial and
partially-reconstructed backgrounds. The $C\\!P$ asymmetries for
$B^{+}\\!\rightarrow p\overline{}pK^{+}$ are determined by repeating the fits
to the $B$-candidate invariant mass in bins of both the $p\overline{}p$ and
$K^{+}\overline{}p$ invariant masses and separating by the charge of the $B$
candidate. The results are shown in Figure 5 and are consistent with zero in
all bins, albeit with large uncertainties.
Figure 4: Distributions of the $B$-candidate invariant mass for (left)
$B^{+}\\!\rightarrow p\overline{}pK^{+}$ decays and (right)
$B^{+}\\!\rightarrow p\overline{}p\pi^{+}$ decays.
Figure 5: $C\\!P$ asymmetry as a function of (left) $m_{p\overline{}p}$
(right) $m_{K^{+}\overline{}p}$ for $B^{+}\\!\rightarrow p\overline{}pK^{+}$
decays.
The decay dynamics are studied by constructing differential production spectra
as a function of the invariant masses and the cosine of the angle,
$\theta_{p}$, between the daughter meson and the opposite-sign baryon in the
$p\overline{}p$ rest frame. The distributions as a function of $p\overline{}p$
invariant mass are shown in Figure 6 and show very clear threshold enhancement
behaviour, similar to other $B\\!\rightarrow p\overline{}pX$ decays. The
distributions as a function of $\cos\theta_{p}$ are shown in Figure 7 and
exhibit strikingly opposite behaviour between the two decay modes, the
forward/backward asymmetries being
$\displaystyle A_{\mathrm{FB}}(B^{+}\\!\rightarrow p\overline{}pK^{+})$
$\displaystyle=$ $\displaystyle\phantom{-}0.370\pm 0.018\mathrm{\,(stat)}\pm
0.016\mathrm{\,(syst)}\,,$ $\displaystyle A_{\mathrm{FB}}(B^{+}\\!\rightarrow
p\overline{}p\pi^{+})$ $\displaystyle=$ $\displaystyle-0.392\pm
0.117\mathrm{\,(stat)}\pm 0.015\mathrm{\,(syst)}\,.$
This behaviour can also clearly be seen when examining the
$B^{+}\\!\rightarrow p\overline{}pK^{+}$ Dalitz plot shown in Figure 8, which
has been background-subtracted using the sPlot technique [13]. The other clear
features are the vertical bands at high $p\overline{}p$ invariamt mass, which
are contributions from charmonium intermediate states. These have been studied
separately in an analysis reported in Ref. [14]. There is also some structure
at low $m_{K^{+}\overline{}p}$, which is shown more clearly in the signal
sPlot invariant mass projection in Figure 9. A two-dimensional fit to the
$B$-candidate invariant mass and $m_{K^{+}\overline{}p}$ is performed in this
region in order to extract the yield of the $\kern
1.00006pt\overline{\kern-1.00006pt\mathchar 28931\relax}(1520)$ resonance. The
signal is found to have a significance of $5.1\,\sigma$, which constitutes
first observation of the decay $B^{+}\\!\rightarrow\kern
1.00006pt\overline{\kern-1.00006pt\mathchar 28931\relax}(1520)p$ with a
branching fraction of
$\displaystyle{\cal B}(B^{+}\\!\rightarrow\kern
1.00006pt\overline{\kern-1.00006pt\mathchar
28931\relax}(1520)p)=(3.9\,^{+1.0}_{-0.9}\mathrm{\,(stat)}\pm
0.1\mathrm{\,(syst)}\pm 0.3({\rm BF}))\times 10^{-7}\,,$
where the third uncertainty is from the branching fraction of
$B^{+}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+}$,
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\\!\rightarrow p\overline{}p$.
Figure 6: Differential production spectra as a function of $m_{p\overline{}p}$
for (left) $B^{+}\\!\rightarrow p\overline{}pK^{+}$ decays (right)
$B^{+}\\!\rightarrow p\overline{}p\pi^{+}$ decays.
Figure 7: Differential production spectra as a function of $\cos\theta_{p}$
for (left) $B^{+}\\!\rightarrow p\overline{}pK^{+}$ decays (right)
$B^{+}\\!\rightarrow p\overline{}p\pi^{+}$ decays. Figure 8: Dalitz plot
distribution for $B^{+}\\!\rightarrow p\overline{}pK^{+}$ signal events. The
black solid curves are lines of constant $\cos\theta_{p}$. Figure 9:
Distribution of $K^{+}\overline{}p$ invariant mass for $B^{+}\\!\rightarrow
p\overline{}pK^{+}$ signal events in the region
$1.440<m_{K^{+}\overline{}p}<1.585{\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$.
## 4 Results from $B^{0}\\!\rightarrow h^{+}h^{-}\pi^{0}$ decays
The Belle collaboration have recently reported the results of a search for the
decay $B^{0}\\!\rightarrow K^{+}K^{-}\pi^{0}$, using a data sample of 772
million $B\kern 1.79993pt\overline{\kern-1.79993ptB}{}$ pairs. Full details of
the analysis are given in Ref. [15]. A fit is performed to $\Delta E$, the
difference between the energy of the $B$ candidate and the beam energy, and
the output of a neural network of event-shape variables. The latter variable
is a powerful discriminant against the dominant background from continuum
light-quark production. The fit yields $299\pm 83$ signal events, where the
uncertainty is statistical only. The projections of the fit are shown in
Figure 10. The signal has a significance of $3.5\,\sigma$, which constitutes
the first evidence of this decay, with a branching fraction of
$\displaystyle{\cal B}(B^{0}\\!\rightarrow K^{+}K^{-}\pi^{0})=(2.17\pm
0.60\mathrm{\,(stat)}\pm 0.24\mathrm{\,(syst)})\times 10^{-6}\,.$ (5)
Figure 10: Projections of the maximum likelihood fit to $B^{0}\\!\rightarrow
K^{+}K^{-}\pi^{0}$ candidate events for the variables (left) $\Delta E$ and
(right) the neural network of event-shape variables.
The BaBar collaboration have recently updated their time-dependent Dalitz-plot
analysis of the decay $B^{0}\\!\rightarrow\pi^{+}\pi^{-}\pi^{0}$ to use their
full $\mathchar 28935\relax{(4S)}$ dataset of 471 million $B\kern
1.79993pt\overline{\kern-1.79993ptB}{}$ pairs. The primary goal of this
analysis is to measure the CKM angle $\alpha$ using the Snyder-Quinn method
[16]. Full details of the analysis can be found in Ref. [17]. A thorough
robustness study was conducted, which showed that while the extraction of the
fit parameters and most of the derived quasi-two-body parameters was robust,
unfortunately the extraction of $\alpha$ itself was not. However, hints of
direct $C\\!P$ violation were seen in the two parameters
$\displaystyle A_{\rho\pi}^{+-}$ $\displaystyle=$
$\displaystyle\frac{\Gamma(\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\\!\rightarrow\rho^{-}\pi^{+})-\Gamma(B^{0}\\!\rightarrow\rho^{+}\pi^{-})}{\Gamma(\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\\!\rightarrow\rho^{-}\pi^{+})+\Gamma(B^{0}\\!\rightarrow\rho^{+}\pi^{-})}\,,$
(6) $\displaystyle A_{\rho\pi}^{-+}$ $\displaystyle=$
$\displaystyle\frac{\Gamma(\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\\!\rightarrow\rho^{+}\pi^{-})-\Gamma(B^{0}\\!\rightarrow\rho^{-}\pi^{+})}{\Gamma(\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\\!\rightarrow\rho^{+}\pi^{-})+\Gamma(B^{0}\\!\rightarrow\rho^{-}\pi^{+})}\,.$
(7)
The result of the 2D scan for these parameters is shown in Figure 11. The
consistentcy with the no direct $C\\!P$ violation point is quantified as
$\Delta\chi^{2}=6.42$.
Figure 11: Likelihood scan in the $A_{\rho\pi}^{+-}$ vs. $A_{\rho\pi}^{-+}$
plane.
## 5 Studies of $B^{0}_{(s)}\\!\rightarrow K^{0}_{\rm\scriptscriptstyle
S}h^{+}h^{-}$ decays
Time-dependent flavour-tagged Dalitz-plot analyses of $B$ decays to
$K^{0}_{\rm\scriptscriptstyle S}h^{+}h^{-}$ final states are sensitive to
mixing-induced $C\\!P$-violating phases. For example, the recent BaBar
measurement $\beta_{\rm eff}(\phi K^{0}_{\rm\scriptscriptstyle S})=(21\pm 6\pm
2)^{{}^{\circ}}$ in the decay $B^{0}\\!\rightarrow
K^{0}_{\rm\scriptscriptstyle S}K^{+}K^{-}$ [6]. Such an analysis is not
possible with the current LHCb statistics, however it is possible to search
for the previously unobserved $B^{0}_{s}$ decays to these final states.
The analysis, which uses the LHCb 1.0$\mbox{\,fb}^{-1}$ data sample collected
during 2011, has separate optimisations of the selection for the suppressed
and favoured decays in each final state. In addition, most of the
reconstructed $K^{0}_{\rm\scriptscriptstyle S}$ mesons decay downstream of the
LHCb Vertex Locator and so do not have information from that subdetector,
while the remaining $\sim\frac{1}{3}$ do have such information. This leads to
rather different efficiencies for the two types of
$K^{0}_{\rm\scriptscriptstyle S}$ candidates (referred to as Downstream and
Long, respectively) and hence the need to treat each category separately in
the analysis. Full details of the analysis can be found in Ref. [18].
Figures 12 and 13 show the results of the fits to the
$B^{0}_{(s)}\\!\rightarrow K^{0}_{\rm\scriptscriptstyle S}h^{+}h^{-}$
candidate events when the selection is applied for the favoured modes and
suppressed modes, respectively. The decay $B^{0}_{s}\\!\rightarrow
K^{0}_{\rm\scriptscriptstyle S}K^{\pm}\pi^{\mp}$ is unambiguously observed and
the BaBar observation of $B^{0}\\!\rightarrow K^{0}_{\rm\scriptscriptstyle
S}K^{\pm}\pi^{\mp}$ [19] is confirmed. The decay $B^{0}_{s}\\!\rightarrow
K^{0}_{\rm\scriptscriptstyle S}\pi^{+}\pi^{-}$ is observed for the first time
with a significance of $5.9\,\sigma$, while no significant evidence is
obtained for the decay $B^{0}_{s}\\!\rightarrow K^{0}_{\rm\scriptscriptstyle
S}K^{+}K^{-}$.
Figure 12: Invariant mass distributions of (top)
$K^{0}_{\rm\scriptscriptstyle S}K^{+}K^{-}$, (middle)
$K^{0}_{\rm\scriptscriptstyle S}K^{\pm}\pi^{\mp}$, and (bottom)
$K^{0}_{\rm\scriptscriptstyle S}\pi^{+}\pi^{-}$ candidate events, with the
loose selection for (left) Downstream and (right) Long
$K^{0}_{\rm\scriptscriptstyle S}$ reconstruction categories. In each plot, the
total fit model is overlaid (solid black line) on the data points. The signal
components are the black short-dashed or dotted lines, while cross-feed decays
are the black dashed lines close to the signal peaks. The combinatorial
background contribution is the green long-dash dotted line. Partially
reconstructed contributions from various sources are also shown.
Figure 13: Invariant mass distributions of (top)
$K^{0}_{\rm\scriptscriptstyle S}K^{+}K^{-}$, (middle)
$K^{0}_{\rm\scriptscriptstyle S}K^{\pm}\pi^{\mp}$, and (bottom)
$K^{0}_{\rm\scriptscriptstyle S}\pi^{+}\pi^{-}$ candidate events, with the
tight selection for (left) Downstream and (right) Long
$K^{0}_{\rm\scriptscriptstyle S}$ reconstruction categories. In each plot, the
total fit model is overlaid (solid black line) on the data points. The signal
components are the black short-dashed or dotted lines, while cross-feed decays
are the black dashed lines close to the signal peaks. The combinatorial
background contribution is the green long-dash dotted line. Partially
reconstructed contributions from various sources are also shown.
The branching fractions of all the modes are measured with respect to
$B^{0}\\!\rightarrow K^{0}_{\rm\scriptscriptstyle S}\pi^{+}\pi^{-}$, for which
the world average branching fraction is $(2.48\pm 0.10)\times 10^{-5}$ [9].
The ratios of branching fractions are determined to be
$\displaystyle\frac{{\cal B}\left(B^{0}\\!\rightarrow
K^{0}_{\rm\scriptscriptstyle S}K^{\pm}\pi^{\mp}\right)}{{\cal
B}\left(B^{0}\\!\rightarrow K^{0}_{\rm\scriptscriptstyle
S}\pi^{+}\pi^{-}\right)}$ $\displaystyle=$ $\displaystyle 0.128\pm
0.017\;{\rm(stat.)}\;\pm 0.009\;({\rm syst.})\,,$ $\displaystyle\frac{{\cal
B}\left(B^{0}\\!\rightarrow K^{0}_{\rm\scriptscriptstyle
S}K^{+}K^{-}\right)}{{\cal B}\left(B^{0}\\!\rightarrow
K^{0}_{\rm\scriptscriptstyle S}\pi^{+}\pi^{-}\right)}$ $\displaystyle=$
$\displaystyle 0.385\pm 0.031\;{\rm(stat.)}\;\pm 0.023\;({\rm syst.})\,,$
$\displaystyle\frac{{\cal B}\left(B^{0}_{s}\\!\rightarrow
K^{0}_{\rm\scriptscriptstyle S}\pi^{+}\pi^{-}\right)}{{\cal
B}\left(B^{0}\\!\rightarrow K^{0}_{\rm\scriptscriptstyle
S}\pi^{+}\pi^{-}\right)}$ $\displaystyle=$ $\displaystyle 0.29\phantom{0}\pm
0.06\phantom{0}\;{\rm(stat.)}\;\pm 0.03\phantom{0}\;({\rm syst.})\;\pm
0.02\phantom{0}\;(f_{s}/f_{d})\,,$ $\displaystyle\frac{{\cal
B}\left(B^{0}_{s}\\!\rightarrow K^{0}_{\rm\scriptscriptstyle
S}K^{\pm}\pi^{\mp}\right)}{{\cal B}\left(B^{0}\\!\rightarrow
K^{0}_{\rm\scriptscriptstyle S}\pi^{+}\pi^{-}\right)}$ $\displaystyle=$
$\displaystyle 1.48\phantom{0}\pm 0.12\phantom{0}\;{\rm(stat.)}\;\pm
0.08\phantom{0}\;({\rm syst.})\;\pm 0.12\phantom{0}\;(f_{s}/f_{d})\,,$
$\displaystyle\frac{{\cal B}\left(B^{0}_{s}\\!\rightarrow
K^{0}_{\rm\scriptscriptstyle S}K^{+}K^{-}\right)}{{\cal
B}\left(B^{0}\\!\rightarrow K^{0}_{\rm\scriptscriptstyle
S}\pi^{+}\pi^{-}\right)}$ $\displaystyle\in$ $\displaystyle[0.004;0.068]\;{\rm
at\;\;90\%\;CL}\,,$
where $f_{s}/f_{d}$ refers to the uncertainty on the ratio of hadronisation
fractions of the $b$ quark to $B^{0}_{s}$ and $B^{0}$ mesons [20].
## 6 Summary
A review of recent results of the analyses of charmless three-body decays of
$b$-hadrons has been presented. With the $B$-factories exploiting their final
datasets and LHCb starting to analyse the 2$\mbox{\,fb}^{-1}$ 2012 data sample
there should be many more interesting results to come in the near future, both
in $B$ meson decays and in the almost completely unexplored territory of the
decays of the $\mathchar 28931\relax^{0}_{b}$ and other $b$-baryons.
ACKNOWLEDGMENTS
Work supported by the European Research Council under FP7 and by the United
Kingdom’s Science and Technology Facilities Council.
## References
* [1] R. Aaij et al. (LHCb Collaboration), Phys. Rev. Lett. 110, 221601 (2013), arXiv:1304.6173 [hep-ex].
* [2] M. Gronau, Phys. Lett. B 492, 297 (2000), hep-ph/0008292.
* [3] T. Latham and T. Gershon, J. Phys. G 36, 025006 (2009), arXiv:0809.0872 [hep-ph].
* [4] A. Garmash et al. (Belle Collaboration), Phys. Rev. Lett. 96, 251803 (2006), hep-ex/0512066.
* [5] B. Aubert et al. (BaBar Collaboration), Phys. Rev. D 78, 012004 (2008), arXiv:0803.4451 [hep-ex].
* [6] J. P. Lees et al. (BaBar Collaboration), Phys. Rev. D 85, 112010 (2012), arXiv:1201.5897 [hep-ex].
* [7] R. Aaij et al. (LHCb Collaboration), Phys. Rev. Lett. 111, 101801 (2013), arXiv:1306.1246 [hep-ex].
* [8] A. A. Alves, Jr. et al. (LHCb Collaboration), JINST 3, S08005 (2008).
* [9] J. Beringer et al. (Particle Data Group), Phys. Rev. D 86, 010001 (2012).
* [10] J. P. Lees et al. (BaBar Collaboration), arXiv:1305.4218 [hep-ex].
* [11] R. Aaij et al. (LHCb Collaboration), LHCb-CONF-2012-028.
* [12] R. Aaij et al. (LHCb Collaboration), Phys. Rev. D 88, 052015 (2013), arXiv:1307.6165 [hep-ex].
* [13] M. Pivk and F. R. Le Diberder, Nucl. Instrum. Meth. A 555, 356 (2005), physics/0402083 [physics.data-an].
* [14] R. Aaij et al. (LHCb Collaboration), Eur. Phys. J. C 73, 2462 (2013), arXiv:1303.7133 [hep-ex].
* [15] V. Gaur et al. (Belle Collaboration), Phys. Rev. D 87, 091101 (2013), arXiv:1304.5312 [hep-ex].
* [16] A. E. Snyder and H. R. Quinn, Phys. Rev. D 48, 2139 (1993).
* [17] J. P. Lees et al. (BaBar Collaboration), Phys. Rev. D 88, 012003 (2013), arXiv:1304.3503 [hep-ex].
* [18] R. Aaij et al. [LHCb Collaboration], to appear in JHEP, arXiv:1307.7648 [hep-ex].
* [19] P. del Amo Sanchez et al. (BaBar Collaboration), Phys. Rev. D 82, 031101 (2010), arXiv:1003.0640 [hep-ex].
* [20] R. Aaij et al. (LHCb Collaboration), JHEP 1304, 001 (2013), arXiv:1301.5286 [hep-ex].
|
arxiv-papers
| 2013-10-02T21:58:05 |
2024-09-04T02:49:51.909186
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Thomas Latham",
"submitter": "Thomas Latham",
"url": "https://arxiv.org/abs/1310.0855"
}
|
1310.1052
|
# Diagonal changes for surfaces in hyperelliptic components
A geometric natural extension of Ferenczi-Zamboni moves
Vincent Delecroix and Corinna Ulcigrai
(2013)
###### Abstract
We describe geometric algorithms that generalize the classical continued
fraction algorithm for the torus to all translation surfaces in hyperelliptic
components of translation surfaces. We show that these algorithms produce all
saddle connections which are best approximations in a geometric sense, which
generalizes the notion of best approximation for the classical continued
fraction. In addition, they allow to list all systoles along a Teichmueller
geodesic and all bispecial words which appear in the symbolic coding of linear
flows. The elementary moves of the described algorithms provide a geometric
invertible extension of the renormalization moves introduced by S. Ferenczi
and L. Zamboni for the corresponding interval exchange transformations.
###### Contents
1. 1 Introduction
1. 1.1 Geometric continued fraction algorithm for the torus
2. 1.2 Diagonal changes algorithms for translation surfaces
1. 1.2.1 Translation surfaces, wedges and quadrangulations.
2. 1.2.2 Diagonal changes via staircase moves
3. 1.3 Applications of diagonal changes algorithms
1. 1.3.1 Geometric best approximations
2. 1.3.2 Bispecial words in the language of cutting sequences
3. 1.3.3 Applications to Teichmüller dynamics
4. 1.4 Comparison with other algorithms in the literature
5. 1.5 Structure of the paper
6. 1.6 Acknowledgements
2. 2 Diagonal changes on the space of quadrangulations
1. 2.1 Parameters on quadrangulations
2. 2.2 Bi-partite interval exchanges and quadrangulations
3. 2.3 Staircase moves and Ferenczi-Zamboni moves
4. 2.4 Diagonal changes algorithms given by staircase moves.
5. 2.5 Invertibility, self-duality and Markov structure on parameter space
3. 3 Existence of quadrangulations and staircase moves
1. 3.1 Quadrangulations in hyperelliptic components
1. 3.1.1 Hyperelliptic components of strata of translation surfaces
2. 3.1.2 Two geometric results in hyperelliptic components
3. 3.1.3 Triangulations on the sphere and Ferenczi-Zamboni trees of relations
2. 3.2 Existence of quadrangulations, proof of Theorem 1.8
3. 3.3 Existence of staircase move, proof of Theorem 1.9
4. 3.4 Non hyperelliptic components
4. 4 Best-approximations and bispecial words via staircase moves
1. 4.1 Best approximations via staircase moves and applications
1. 4.1.1 Staircase moves produce the same geometric objects
2. 4.1.2 Systoles and Lagrange values along Teichmueller geodesics
2. 4.2 Description of the language via staircase moves
1. 4.2.1 Bispecial words as cutting sequences of best approximations
2. 4.2.2 Cutting sequences by staircase moves
## 1 Introduction
We begin this introduction by describing in §1.1 a geometric version of the
standard (additive) continued fraction algorithm, in terms of changes of bases
for lattices. One of the key properties of the continued fraction algorithm is
that it generates all rational best approximations of an irrational number.
This property has a geometric interpretation: the continued fraction algorithm
produces all saddle connections which are geometric best approximations (see
Definition 1.1).
In this paper we define diagonal changes algorithms which provide geometric
generalizations of the continued fraction algorithm for linear flows on
translation surfaces of higher genera (tori are translation surfaces of genus
1). Basic definitions appear in §1.2.1 and the algorithm is described in
§1.2.2.
The diagonal changes algorithms have several nice properties which are
described in §1.3 of this introduction: they produce all geometric best
approximations (see §1.3.1), allow to construct all bispecial words in the
symbolic coding of linear flows (see §1.3.2) and detect all systoles along a
Teichmüller geodesic (see §1.3.3).
### 1.1 Geometric continued fraction algorithm for the torus
Let $\Lambda\subset\mathbb{C}$ be a lattice. The standard continued fraction
algorithm provides a way to construct a sequence of vectors in $\Lambda$ that
are good approximation of the vertical direction. Let us present a geometric
version of this algorithm. We choose a basis $(w_{\ell},w_{r})$ of $\Lambda$
such that:
* •
$\operatorname{Re}(w_{\ell})<0$ and $\operatorname{Re}(w_{r})>0$,
* •
$\operatorname{Im}(w_{\ell})>0$ and $\operatorname{Im}(w_{r})>0$.
It is clear that such a basis exists if $\Lambda$ does not contain vertical or
horizontal non-zero vectors. The basis $(w_{\ell},w_{r})$ forms a _wedge_ that
contains the vertical direction; in other words, the vertical is contained in
the positive cone generated by this basis. The parallelogram
$Q=Q(w_{\ell},w_{r})$ formed from these two vectors is a fundamental domain
for the action of $\Lambda$ on $\mathbb{C}$. We say that the parallelogram $Q$
is _left-slanted_ (respectively _right-slanted_) if the vertical half-axis
$\\{z;\,\operatorname{Re}(z)=0\ \text{and}\ \operatorname{Im}(z)>0\\}$ crosses
the left (resp. right) top side, that is the side parallel to $w_{r}$ (resp.
$w_{\ell}$). An example is shown in figure 1.
(a) left slanted (b) right slanted
Figure 1: examples of left and right slanted parallelograms
One step of the algorithm is as follows. If the parallelogram $Q$ defined by
the basis $(w_{\ell},w_{r})$ is left slanted, consider the new basis
$w^{\prime}_{\ell}=w_{\ell}$ and $w^{\prime}_{r}=w_{d}=w_{r}+w_{\ell}$.
Geometrically, the new parallelogram $Q^{\prime}$ with sides
$(w^{\prime}_{\ell},w^{\prime}_{r})$ is obtained by cutting the old one along
a diagonal and pasting the lower triangle as in Figure 2(a). Remark that,
after this operation, the vertical axis is contained in the parallelogram
$Q^{\prime}$. We call such move a _left move_. If the parallelogram is right
slanted, then we made a _right move_ in a symmetric way (see Figure Figure
2(b)).
(a) a left move (b) a right move
Figure 2: a left move and a right move for the two examples of Figure 1
Let us set $w^{(0)}_{\ell}=w_{\ell}$ and $w^{(0)}_{r}=w_{r}$. Applying
successively the above step we get a sequence of bases
$(w^{(n)}_{\ell},w^{(n)}_{r})$ of $\Lambda$ for which the imaginary parts of
both vectors in the base tend to infinity. Notice that the algorithm may stop
after a finite number of steps, but this is the case if and only if the
lattice $\Lambda$ contains a vertical vector. Let us also remark that one can
also define a cut and paste operation which is the inverse operation to the
diagonal change defined above. Thus, the algorithm can also be defined in
backward time. The backward orbit is infinite if and only if $\Lambda$ does
not contain horizontal vectors. In the sequel, we assume that $\Lambda$ does
neither contain vertical nor horizontal vectors.
Let us recall some well known Diophantine approximation properties of this
sequence of bases. Let $\Gamma$ be the set of primitive vectors of $\Lambda$
with positive imaginary part. One can decompose $\Gamma$ as union of
$\Gamma_{\ell}$ and $\Gamma_{r}$ which denote respectively the primitive
vectors with positive and negative real part. Remark that, for any
$n\in\mathbb{N}$, $w^{(n)}_{\ell}$ belongs to $\Gamma_{\ell}$ and
$w^{(n)}_{r}$ belongs to $\Gamma_{r}$.
###### Definition 1.1.
A vector $v\in\Gamma_{r}$ is a _(right) geometric best approximation_ if
$\forall
u\in\Gamma_{r},\quad\operatorname{Im}(u)<\operatorname{Im}(v)\Rightarrow|\operatorname{Re}(u)|>|\operatorname{Re}(v)|.$
The definition of _left geometric best approximations_ is obtained by
replacing $\Gamma_{r}$ by $\Gamma_{\ell}$.
###### Remark.
In geometric terms, $v$ is a right best approximation if and only if the
rectangle
$R(v):=\left[0,\operatorname{Re}(v)\right]\times\left[0,\operatorname{Im}(v)\right]$
does not contains any vector of $\Lambda$ in its interior.
The geometric continued fraction algorithm _constructs_ all geometric best
approximations in the following sense:
###### Theorem 1.2.
Let $\Lambda$ be a lattice in $\mathbb{C}$ that does not contain neither
horizontal nor vertical vectors. Then the sequence of bases
$(w^{(n)}_{\ell},w^{(n)}_{r})$ built from the algorithm is uniquely defined up
to a shift in the numbering. Moreover, the vectors $w^{(n)}_{\ell}$ and
$w^{(n)}_{r}$ are exactly the geometric best approximations.
The above theorem can be interpreted and proved in terms of Diophantine
approximation: intermediate convergents of a real number $\alpha$ are exactly
the approximation of the first kind (see [28, thm 15 p. 22]). We will prove
this statement in much more generality in Theorem 1.12.
The quotient $\mathbb{T}_{\Lambda}=\mathbb{C}/\Lambda$ is a flat torus on
which the origin is marked. On $\mathbb{T}_{\Lambda}$ there is a family of
linear flows, which are the quotients of the straight line flows
$\varphi^{\theta}_{t}:z\mapsto z+te^{\sqrt{-1}\,\theta}$ where $\theta$ is a
fixed element in the circle $S^{1}=\mathbb{R}/(2\pi)\mathbb{Z}$. A _saddle
connection_ is a trajectory of a linear flow from the marked point to itself.
There is a one to one correspondence between saddle connections and primitive
vectors of $\Lambda$. The algorithm hence produces saddle connections which
give better and better approximation of the vertical linear flow.
In §1.3.3 we recall the well-known connection of the continued fraction
algorithm with the geodesic flow on the modular surface and explain that the
geometric continued fraction algorithm also detects _systoles_ for the
geodesic flow.
### 1.2 Diagonal changes algorithms for translation surfaces
We start this section by defining translation surfaces, which are
generalizations of flat tori. We then introduce the notion of wedges and their
associated quadrangulations. Using them, we define algorithms which consist of
diagonal changes and provide a generalization for translation surfaces of the
continued fraction.
#### 1.2.1 Translation surfaces, wedges and quadrangulations.
A translation surface can be defined by gluing polygons in the following way.
Let $(P_{i})_{i}$ be a finite collection of polygons in the plane
$\mathbb{C}$, with a pairing of edges such that for each edge $e$ of a polygon
$P_{i}$ there is an edge $\sigma(e)$ of a polygon $P_{j}$ such that $e$ and
$\sigma(e)$ are parallel, of the same length and have opposite outgoing normal
vector (with respect to their polygon). Let us identify each edge $e$ with the
corresponding edge $\sigma(e)$ by the unique translation that sends $e$ to
$\sigma(e)$. The quotient $X$ of $\sqcup P_{i}$ under those identification is
called a _translation surface_. We will always assume that a translation
surface is connected. Flat tori (see §1.1) are examples of translation
surfaces built from one parallelogram and see Figure 5 for a translation
surface built from 3 quadrilaterals.
Let $\Sigma=\Sigma(X)$ be the finite subset of points of $X$ which are images
of vertices of the polygons $P_{i}$ in $X$. Such points are called
_singularities_ of $X$. The surface $X$ carries a flat (Euclidean) metric on
$X\backslash\Sigma$ induced by the Euclidean metric on the plane, with conical
singularities at the points in $\Sigma$ with cone angles of the form $2\pi k$
with $k\in\mathbb{N}$. A _cone-point_ with cone angle $2\pi k$ has a
neighborhood which is isometric to a finite $k$-sheeted cover of the plane
branched at the origin, which can be parametrized by polar coordinates
$(\rho,\theta)$ where $\rho\in\mathbb{R}^{+}$ and $\theta\in\mathbb{R}/(2\pi
k)\mathbb{Z}$.
The surface $X$ also inherits a _translation structure_ from $\mathbb{C}$,
which is an atlas on $X\backslash\Sigma$ whose transition maps are
translations. On $X\backslash\Sigma$ there is a well defined notion of
(oriented) directions and hence one can define _linear flows_ which correspond
to moving along lines in a given direction in $S^{1}$. The flow
$\varphi^{\theta}_{t}$ in direction $\theta\in S^{1}$ is explicitly given in
local charts by $\varphi^{\theta}_{t}:z\mapsto z+te^{\sqrt{-1}\,\theta}$. Note
that the flow is not well defined at $x\in(X\backslash\Sigma)$ if its orbit
$\varphi^{\theta}_{t}(x)$ goes into a singularity.
A translation surface $X$ is in particular a Riemann surface endowed with a
non-zero Abelian differential. The complex structure is obtained from the
translation charts and the differential form, generally denoted $\omega$, is
obtained by lifting $dz$. Conversly, a compact Riemann surface with a non-zero
Abelian differential $\omega$ determines a translation surface (by finding
local coordinates $z$ such that $\omega$ is locally $dz$). If $x\in X$ is a
conical singularity of angle $2\pi k$ then we can write locally $\omega$
around $x$ as $z^{k-1}dz$. For more details on the various definitions of
translation surfaces, we refer to [35] and [48].
We consider the following notion of isomorphism between translation surfaces.
If the surface $S$ is defined from some polygons and identifications of their
edges then we allow the two following operations. The _cut operation_ consists
in cutting a polygon along a segment that joins two of its vertices and, in
the new set of polygons, identify the two newly created sides. The _paste
operation_ consists in gluing two polygons that were identified. Two surfaces
$X$ and $X^{\prime}$ defined respectively from $((P_{i})_{i},\sigma)$ and
$((P^{\prime}_{j})_{j},\sigma^{\prime})$ are _isomorphic_ if there exists a
sequence of cut and paste operations that goes from $((P_{i})_{i},\sigma)$ to
$((P^{\prime}_{j})_{k},\sigma^{\prime})$ (where we consider that two polygonal
representation are equal if we can pass from one to the other by translating
the polygons). The _stratum_ $\mathcal{H}(k_{1}-1,\ldots,k_{n}-1)$ of
translation surfaces is the set of isomorphisms classes of translation
surfaces with conical singularities with angles $2\pi k_{1}$, …, $2\pi k_{n}$,
or, equivalently, of non-zero Abelian differentials with zeros of order
$k_{1}-1$, …, $k_{n}-1$. If there are $m_{i}$ singularities with total angle
$\pi k_{i}$ we use the notation
$\mathcal{H}((k_{1}-1)^{m_{1}},\ldots,(k_{n}-1)^{m_{n}})$.
An _affine diffeomorphism_ $\Psi:X\to X^{\prime}$ between two translation
surfaces, is an homeomorphism which maps $\Sigma(X)$ to $\Sigma(X^{\prime})$
and is affine in the coordinate charts. Because of connectedness of
$X\backslash\Sigma(X)$, the linear part of the affine diffeomorphism is
constant and may be identified to a matrix in
$\operatorname{GL}(2,\mathbb{R})$. We call this matrix the _derivative_ of
$\Psi$. Two translation surfaces $X$ and $Y$ are _translation equivalent_ if
there exists an affine diffeomorphism $\Psi:X\to Y$ whose derivative is the
identity matrix. It is easy to see that two translation surfaces $X$ and $Y$
are translation equivalent if and only if $Y$ is obtained from $X$ by a
sequence of cut and paste operations.
##### Bundles of saddle connections.
Let $X$ be a translation surface with singularities $\Sigma$ and let
$\varphi_{t}^{\theta}$, $\theta\in S^{1}$, be the family of linear flows on
$X$. A _saddle connection_ in $X$ is the orbit of some linear flow that joins
two singularities of $X$. Note that if $X$ is built from a union of polygons,
any side $v$ of a polygon gives a saddle connection on $X$.
If in direction $\theta$ there is no saddle connection, then the flow
$\varphi_{t}^{\theta}$ is minimal (meaning that any infinite trajectory is
dense in $X$). This result was first proven by M. Keane in the context of
interval exchange transformations [27] and the corresponding condition for
interval exchange transformations (orbits of discontinuity points are infinite
and distinct) is often called _Keane’s condition_. On the flat torus
$\mathbb{C}/(\mathbb{Z}\oplus\mathbb{Z}\sqrt{-1})$ the directions of saddle
connections are exactly the rational ones (ie the angles $\theta\in S^{1}$ for
which the slope $\tan(\theta)$ is a rational number). For a general
translation surface the set of directions for which there exists a saddle
connection is countable but has no particular algebraic structure.
The _displacement vector_ (sometimes called _holonomy vector_) associated to
an oriented saddle connection is the vector in $\mathbb{C}$ which gives the
displacement between the initial and final point seen as an element of
$\mathbb{C}$. More precisely, a saddle connection is a set of points
$(\varphi^{\theta}_{t}(x))_{t\in I}$ for some point $x\in X\backslash\Sigma$
and some interval $I=[a,b]\subset\mathbb{R}$, its displacement vector is
$(b-a)e^{\sqrt{-1}\theta}$. Given a side of a polygon $P_{i}$ that defines the
surface, its sides are saddle connections and their displacement are simply
the sides seen as complex vectors. The displacement can also be seen as the
integral of the Abelian form $\omega$ along the saddle connection. It is well
known that for any translation surface $X$ the set of displacement vectors of
saddle connections on $X$ is a discrete subset of $\mathbb{C}$, see for
example [46] or [35].
We call _natural orientation_ of a saddle connection $\gamma$ the unique
orientation of $\gamma$ such that its displacement vector has non-negative
imaginary part. We say that a saddle connection _starts_ (respectively _ends_)
at a singularity if that singularity is the first endpoint (respectively last
endpoint) of the saddle connection according to its natural orientation. A
saddle connection is _left slanted_ (respectively _right slanted_) if with its
natural orientation its real part is negative (resp. positive), as shown in
Figure 3(a) (resp. Figure 3(b)).
(a) left slanted (b) right slanted
(c) a wedge
Figure 3: left and right slanted saddle connections and a wedge
Let $\Gamma=\Gamma(X)$ denote the set of all saddle connections on a given
translation surface $X$ and let $\Gamma^{\ell}$ (respectively $\Gamma^{r}$)
the subset of all left-slanted (respectively right-slanted) saddle
connections. Saddle connections in $\Gamma$ can be subdivided as follows into
subsets, which (following the notation introduced by L. Marchese in [32]) we
will call _bundles of saddle connections_. Assume that the singularity set
$\Sigma$ consist of $n$ singularities of cone-angles $2\pi k_{1},\dots 2\pi
k_{n}$. Remark that, if the conical angle at $p_{i}\in\Sigma$ is $2\pi k_{i}$,
from $p_{i}$ there are $k_{i}$ outgoing trajectories of the vertical linear
flow and $k_{i}$ outgoing trajectories of the horizontal linear flow (since
$p_{i}$ has a neighborhood isomorphic to $k_{i}$ planes). For each
$p_{i}\in\Sigma$, choose a reference horizontal ray $v_{i}$ starting from
$p_{i}$. For any two linear trajectories $\gamma,\gamma^{\prime}$ starting at
$p_{i}$ we denote by $\angle(\gamma,\gamma^{\prime})\in[0,2\pi k_{i})$ the
angle between them. Each saddle connection $\gamma$ starting at $p_{i}$
belongs to one of the $k_{i}$ _outgoing half planes_ , that is the angle
$\angle(\gamma,v_{i})$ with respect to the chosen horizontal $v_{i}$ from
$p_{i}$ satisfies
$2\pi j\leq\angle(\gamma,v_{i})<2\pi j+\pi,\quad\text{for a unique}\ 0\leq
j<k_{i}.$
Two saddle connections belong to the same _bundle_ if and only if they start
from the same singularity $p_{i}$ and _belong to the same half-plane_. Remark
that there are $k$ bundles of saddle connections on $X$, where
$k=k_{1}+\dots+k_{n}$ is the total angle. We will label them with the integers
$1$, …, $k$ and denote them by $\Gamma_{1}$, …, $\Gamma_{k}$.
##### Wedges.
In the case of the torus, the diagonal changes algorithm produces a sequence
of bases of saddle connections which form a wedge and provide better and
better approximations of the vertical. On a translation surface, the
algorithms we consider will produce a sequence of collections of $k$ _wedges_
(defined below), one for each of the $k$ vertical rays in $X$ emanating from
the singularities.
###### Definition 1.3 (wedge).
A _wedge_ $w$ on a translation surface $X$ is a pair of saddle connections
$w=(w_{\ell},w_{r})$ such that:
* (i)
$w_{\ell}$ and $w_{r}$ start from the same conical singularity of $X$,
* (ii)
$w_{\ell}$ is left-slanted and $w_{r}$ is right-slanted,
* (iii)
$(w_{\ell},w_{r})$ consist of two edges of an embedded triangle in $S$.
A picture of a wedge is shown in Figure 3(c). Remark that $(i)$ and $(iii)$
are equivalent to asking that the saddle connections $w_{\ell}$ and $w_{r}$
forming the wedge belong to the same bundle. Remark also that a wedge has the
property that it contains a unique vertical trajectory, that is there is
exactly one trajectory of the vertical flow which starts from the conical
singularity shared by $w_{\ell}$ and $w_{r}$ and intersects the interior of
the triangle with edges $w_{\ell}$ and $w_{r}$.
##### Quadrangulations.
Let us now define special decompositions of $X$ into polygons that are
quadrilaterals. A _quadrilateral_ $q$ in a flat surface $X$ is the image of an
isometrically embedded quadrilateral in $\mathbb{C}$ so that the vertices of
$q$ are singularities of $X$ and there is no other singularities of $X$ in
$q$.
###### Definition 1.4 (admissible quadrilateral).
A quadrilateral $q$ in $X$ is _admissible_ if there is exactly one trajectory
of the vertical linear flow of $X$ starting from one of its vertices and
exactly one ending in a vertex. Equivalently, it is admissible if left-slanted
and right-slanted saddle connections alternate while we turn around the
quadrilaterals.
Examples of admissible and non-admissible quadrilaterals are given in Figure
4.
(a) admissible (b) non admissible
(c) non admissible
Figure 4: examples of admissible and non-admissible quadrilaterals
Let $q$ be an adimssible quadrilateral. We will refer to the saddle
connections starting from the same singularity as the _bottom sides_ of the
quadrilateral $q$ and to the ones ending in the same singularity as the _top
sides_ of $q$. Furthermore, we will call _bottom right side_ (resp. _bottom
left side_) the right-slanted (resp. left-slanted) bottom side of $q$ and
_top right side_ (resp. _top left side_) the left-slanted (resp. right-
slanted) top side of $q$. Remark that from the definition it follows that the
_bottom_ sides of an admissible quadrilateral $q$ form a wedge. We will refer
to it as _the wedge of the quadrilateral_ $q$.
###### Definition 1.5 (quadrangulation).
A _quadrangulation_ $Q$ of $X$ is a decomposition of $X$ into a union of
admissible quadrilaterals.
Given a quadrangulation $Q$, we write $q\in Q$ if $q$ is a quadrilateral in
the decomposition and we call _wedges of the quadrangulation_ $Q$ the
collection of wedges of all quadrilaterals in $Q$. An example of a
quadrangulation is given in Figure 5: the quadrilaterals $q_{1},q_{2},q_{3}$
give a quadrangulation of a surface in genus $2$ with one $6\pi$ conical
singularity.
Figure 5: a quadrangulation of a surface in
$\mathcal{H}(2)=\mathcal{C}^{hyp}(3)$
Let us stress that quadrilaterals in a quadrangulation are by definition
admissible. As each quadrilateral is glued to some other, each top side of a
quadrilateral is also the bottom side of another quadrilateral, thus it
belongs to a wedge. Hence, the wedges of $Q$ on the surface $X$ completely
determine the quadrangulation. In §2.1 we will introduce a combinatorial datum
given by a pair of permutations that describes how quadrilaterals are glued to
each other.
#### 1.2.2 Diagonal changes via staircase moves
Let $Q$ be a quadrangulation of a translation surface. A _diagonal change_
consists in replacing the left or right part of the wedge of a quadrilateral
$q\in Q$ by the diagonal of the quadrilateral $q$. We consider elementary
moves on the set of wedges (the _staircase moves_) which, by performing
simultaneous diagonal changes, produce a new set of wedges which correspond to
a new quadrangulation $Q^{\prime}$ of $X$. The moves of the geometric
continued fraction algorithm in §1.1 are a special case of staircase moves.
##### Staircases and staircase moves.
Let $Q$ be a quadrangulation of a translation surface $X$ and let
$w=(w_{\ell},w_{r})$ be the wedge of a quadrilateral $q\in Q$. We denote by
$w_{d}$ the _diagonal_ saddle connection of $q$ which starts at the
singularity of $w$ and ends at the top singularity of $q$.
As in the case of the torus, we say that a quadrilateral $q$ is _left-slanted_
if the vertical issued from the bottom singularity crosses the top left side
of $q$ and _right-slanted_ if it crosses the top right side (see Figure 1 for
an illustration). Remark that the diagonal $w_{d}$ of $q$ form a wedge with
$w_{\ell}$ (respectively with $w_{r}$) if and only if $q$ is left-slanted
(respectively right-slanted) (see Figure 1). Therefore, for each quadrilateral
we have the following alternatives:
* •
if the quadrilateral $q$ is _left-slanted_ , either we keep the wedge
$(w_{\ell},w_{r})$ or we do a _left-diagonal change_ , that is we replace it
by $(w_{\ell},w_{d})$ (which in this case is again a wedge);
* •
if the quadrilateral $q$ is _right-slanted_ , either we keep the wedge
$(w_{\ell},w_{r})$ or we do a _right-diagonal change_ , that is we replace it
by the $(w_{d},w_{r})$ (which in this case also is a wedge);
The key geometrical object which allow to perform diagonal changes
consistently and hence define elementary moves are _staircases_ :
###### Definition 1.6 (staircase).
Given a quadrangulation $Q$ of $X$, a _left staircase_ $S$ _for_ $Q$
(respectively a _right staircase_ $S$ _for_ $Q$) is a subset $S\subset X$
which is the union of quadrilaterals $q_{1},\dots,q_{n}$ of $Q$ that are
cyclically glued so that the top left (resp. top right) side of $q_{i}$ is
identified with the bottom right (resp. bottom left) side of $q_{i+1}$ for
$1\leq i<k$ and of $q_{1}$ for $i=n$.
A left (respectively right) staircase $S$ is _well slanted_ if all its
quadrilaterals are left (resp. right) slanted.
An example of a right-staircase (which explain the choice of the name
_staircase_) is given in Figure 6(a): remark that the two sides labeled by
$w_{1,\ell}$ are identified, so that the staircase is the union of $3$
quadrilaterals. An example of a well slanted staircase is the right staircase
in Figure 6(a) (all three quadrilaterals all right slanted), while the
staircase in Figure 6(b) is not well slanted (it is a right-staircase in which
$q_{1}$ and $q_{3}$ are right slanted but $q_{2}$ is left slanted).
We remark that a left staircase (respectively right staircase) $S$ in $X$ is a
topological cylinder whose boundary consists of a union of saddle connections
which are all left slanted (resp. all right slanted). Remark also that a
staircase $S$ for $Q$ has a natural decomposition as union of admissible
quadrilaterals induced by the quadrangulation $Q$ of $S$.
(a) a well slanted right staircase (b) diagonal changes in the same staircase
Figure 6: diagonal changes in a right staircase
###### Definition 1.7 (staircase move).
Given a quadrangulation $Q$ and a well slanted left-staircase $S$
(respectively a well slanted right staircase $S$), the _staircase move_ in $S$
is the operation which consists in doing simultaneously left (resp. right)
diagonal changes in all the quadrilaterals of $S$.
Remark that given a quadrangulation there may be none or several well slanted
staircases. In the first case no staircase move is possible while in the
latter there is a choice of staircase moves.
The importance of staircases lies in the following elementary result (see
Lemma 2.6): if $Q$ is a quadrangulation of a surface $X$ and $S$ be a well
slanted staircase in $Q$, the staircase move in $S$ produces a new
quadrangulation $Q^{\prime}$ of $X$. Furthermore, one can show that staircase
moves are the minimal possible ways to combine individual diagonal changes
consistently in order to keep a quadrangulation (see Lemma 2.7).
##### Diagonal changes algorithms for surfaces in hyperelliptic strata.
We prove the existence of quadrangulations and diagonal changes given by
staircase moves for a class of translation surfaces which belong to the so
called _hyperelliptic components of strata_. Here below we provide an
introduction to hyperelliptic components, but we refer to §3.1.1 for more
details.
An affine automorphism $s:X\to X$ of a translation surface $X$ is an
_hyperelliptic involution_ if it is an involution, that is $s^{2}$ is the
identity, and the quotient of $X\backslash\Sigma(X)$ by $s$ is a (punctured)
sphere. An example of a surface which admits an hyperelliptic involution is
given in Figure 5. The surface is obtained from three quadrilaterals, one
which is fixed by the involution (the quadrilateral $q_{2}$) and the other two
which are exchanged ($q_{1}$ and $q_{3}$). On the picture, the hyperelliptic
involution can be seen as a rotation by $180$ degrees. One can show that if a
translation surface admits an hyperelliptic involution, then this involution
is unique.
Strata of translation surfaces are generally not connected and their connected
components were classified by M. Kontsevich and A. Zorich [29]. Hyperelliptic
components are the connected components of strata with the property that each
surface in them admits an hyperelliptic involution. From the Kontsevich-Zorich
classification, it follows that in each stratum
$\mathcal{H}(k_{1},\ldots,k_{n})$ there are either one, two or three connected
components, some of which are hyperelliptic. For each integer $k\geq 1$ there
is exactly one hyperelliptic component which contains surfaces with total
conical angle $2\pi k$. We denote this component by $\mathcal{C}^{hyp}(k)$. If
$k$ is odd, then $\mathcal{C}^{hyp}(k)\subset\mathcal{H}(k-1)$ while if $k$ is
even $\mathcal{C}^{hyp}(k)\subset\mathcal{H}(k/2-1,k/2-1)$ (see also Theorem
3.1) . For $1\leq k\leq 4$ (that correspond to genus $1$ or $2$), the strata
are connected and we have the following equalities:
$\mathcal{H}(0)=\mathcal{C}^{hyp}(1)$ (this is the torus case),
$\mathcal{H}(0,0)=\mathcal{C}^{hyp}(2)$, $\mathcal{H}(2)=\mathcal{C}^{hyp}(3)$
and $\mathcal{H}(1,1)=\mathcal{C}^{hyp}(4)$.
As in the genus $2$ example in Figure 5 above, if $X$ belongs to a
hyperelliptic component $\mathcal{C}^{hyp}(k)$ it turns out that all
quadrilaterals in the quadrangulation are either parallelograms $q$, in which
case $s(q)=q$, or come into pairs $q_{i},q_{j}$ such that $q_{i}\neq q_{j}$
and $s(q_{i})=q_{j}$ in which case $q_{i}$ and $q_{j}$ have parallel
diagonals. This will be proved in Lemma 3.2.
Our main results for translation surfaces in hyperelliptic components are the
following two theorems.
###### Theorem 1.8.
Let $X$ be a surface in a hyperelliptic component $\mathcal{C}^{hyp}(k)$ that
admits no horizontal and no vertical saddle connections. Then $X$ admits a
quadrangulation.
###### Theorem 1.9.
Let $Q$ be a quadrangulation of a surface $X$ in $\mathcal{C}^{hyp}(k)$ and
assume that no quadrilateral in $Q$ has a vertical diagonal. Then, there
exists at least one well slanted staircase in $Q$.
These two results allow us to define diagonal changes algorithms given by
staircase moves in hyperelliptic components. Start from a quadrangulation $Q$
of $X\in\mathcal{C}^{hyp}(k)$, which exists by Theorem 1.8. Theorem 1.9
implies that there exists a staircase move for $Q$. Remark that there can be
more than one well slanted staircase and hence several possible moves.
Diagonal changes algorithms correspond to a systematic way of choosing which
staircase moves to perform. In the torus case, where quadrangulations consist
of only one quadrilateral (a parallelogram), there is no choice. In §2.4 we
give some examples of various diagonal changes algorithms. Nevertheless, we
will show that the actual choice of an algorithm in some sense does not
matter, since the sequence of wedges and well slanted staircases produced by
_any_ sequence of staircase moves is the same (see Theorem 1.12 below).
In various works S. Ferenczi and L. Zamboni (see for example [18, 19]) defined
and studied an induction algorithm for interval exchange transformations with
symmetric permutations, namely the permutations in $S_{n}$ defined by
$i\mapsto n-i+1$ for $1\leq i\leq n$. These interval exchange transformations
may be obtained as first return maps of linear flows on sufaces in
$\mathcal{C}^{hyp}(n-1)$. We call their induction the _Ferenczi-Zamboni
induction_ (see also §1.4). Staircase moves provide a geometric invertible
extension of the elementary moves in the Ferenczi-Zamboni induction, in a
sense that is made precise in Section 2. We note that Theorem 1.9 is
originally proved in [18] in the context of interval exchange transformations.
In view of these two results, a natural question would be to investigate other
components of strata of translation surfaces. We do not know if in general any
translation surface admit a quadrangulation. Nevertheless, in §3.4 we provide
examples of quadrangulations of translation surface in which no staircase move
is possible.
### 1.3 Applications of diagonal changes algorithms
In this section we summarize properties of diagonal changes algorithms and
highlight some of its applications: they detect geometric best approximations
(see §1.3.1), allow to produce bispecial factors for symbolic codings of
linear flows (see §1.3.2) and may be used to construct the sequence of
systoles along a Teichmüller geodesic (see §1.3.3).
#### 1.3.1 Geometric best approximations
The notion of _geometric best approximation_ is a generalization for saddle
connections on translation surfaces of the one for the torus (see Definition
1.1). To define geometric best approximations for higher genera surfaces it is
natural to compare only saddle connections which belong to the same bundle.
Recall that if $X$ has conical singularities with cone angles $2\pi
k_{1},\dots,2\pi k_{n}$, there are $k=k_{1}+\dots+k_{n}$ bundles of saddle
connections (see the beginning of §1.2.1 for the definition). Let us label
them and denote them by $\Gamma_{1},\ldots,\Gamma_{k}$. Each $\Gamma_{i}$ can
be decomposed as $\Gamma_{i}^{\ell}\cup\Gamma_{i}^{r}$ where
$\Gamma_{i}^{\ell}$ (respectively $\Gamma_{i}^{r}$) consists of left-slanted
(respectively right-slanted) saddle connections in $\Gamma_{i}$.
We will adopt the following convention. Remark that given a saddle connection
on $X$ we can associate to it a pair $(i,v)$ where $v\in\mathbb{C}$ is its
displacement (or holonomy) vector and $0\leq i<k$ is such that the saddle
connection belongs to the bundle $\Gamma_{i}$. Conversely, knowing the bundle
to which the saddle connection belong and its displacement vector
$v\in\mathbb{C}$ completely determines the saddle connection. Thus, we can
abuse the notation by identifying saddle connections with their displacement
vector as long as the bundle is clear from the context.
###### Notation.
For a saddle connection $v$ on a translation surface, let
$\operatorname{Re}(v)$, $\operatorname{Im}(v)$ and $|v|$ denote respectively
the real part, the imaginary part and the absolute value of the displacement
vector of $v$.
Given a bundle $\Gamma_{i}$ of saddle connections, we will denote by the
complex number $v\in\mathbb{C}$ the saddle connection in $\Gamma_{i}$ that has
$v$ as its displacement vector and we will hence write $v\in\Gamma_{i}$.
###### Definition 1.10.
A saddle connection $v\in\Gamma_{i}^{r}$ is a _right (geometric) best
approximation_ if
$\forall
u\in\Gamma_{i}^{r},\quad\operatorname{Im}u<\operatorname{Im}v\Rightarrow|\operatorname{Re}u|>|\operatorname{Re}v|.$
A similar definition for _left (geometric) best approximation_ is obtained by
replacing $\Gamma_{i}^{r}$ by $\Gamma_{i}^{\ell}$.
As for the torus, we can rephrase the definition in terms of singularity-free
rectangles. Let us call an _immersed rectangle_ $R\subset X$ a subset without
singularities in its interior which is obtained by isometrically immersing in
$X$ an Euclidean rectangle with horizontal and vertical sides in $\mathbb{C}$
(recall that immersed means _locally_ injective opposed to embedded which
means _globally_ injective). We remark that an immersed rectangle does not
have to be embedded in $X$ and can have self-intersections. The following
equivalent geometric characterization is proved at the beginning of section
4.1.
###### Lemma 1.11.
A saddle connection $v$ on $X$ is a geometric best approximation if and only
if there exists an immersed rectangle $R(v)$ in $X$ which has $v$ as its
diagonal.
One of the important properties of diagonal changes is that any sequence of
staircase moves produces all geometric best approximations (see Theorem 1.2
for the torus case). Let us recall that if $X$ is a surface in hyperelliptic
component with neither horizontal nor vertical saddle connections, then by
Theorem 1.8 it admits quadrangulations and for each of them, in virtue of
Theorem 1.9, there is at least one staircase move. Furthermore, we will see
that, starting from any quadrangulation $Q^{(0)}$, by self-duality of the
algorithm one can define _backwards moves_ (see §2.5) and hence produce a bi-
infinite sequence $(Q^{(n)})_{n\in\mathbb{Z}}$ of quadrangulations of $X$
obtained by a sequence of staircase moves. In Theorem 4.1 we state and prove a
more precise version of the following result.
###### Theorem 1.12.
Let $X$ be a surface in $\mathcal{C}^{hyp}(k)$ that has neither horizontal nor
vertical saddle connections. Let $(Q^{(n)})_{n\in\mathbb{Z}}$ be _any_
sequence of quadrangulations of $X$ where $Q^{(n+1)}$ is obtained from
$Q^{(n)}$ by a staircase move. Then the saddle connections belonging to the
wedges of the quadrangulations $Q^{(n)},n\in\mathbb{Z}$, are exactly all
geometric best approximations of $X$.
#### 1.3.2 Bispecial words in the language of cutting sequences
Let $X$ be a translation surface such that the vertical flow on $X$ is minimal
(for example without vertical saddle connections) and let $Q$ be be a
quadrangulation of $X$. Let us denote by $q_{1},\dots,q_{k}$ its
quadrilaterals and let us label the saddle connections in $Q$ as follows. To
the saddle connections $w_{i,\ell}$ and $w_{i,r}$ which form the wedge $w_{i}$
of the quadrilateral $q_{i}\in Q$ let us associate respectively the labels
$(i,\ell)$ and $(i,r)$. Given an infinite orbit of the vertical flow in $X$,
its _cutting sequence_ with respect to $Q$ is the infinite word on the
alphabet $\mathcal{A}=\\{1,\ldots,d\\}\times\\{\ell,r\\}$ that corresponds to
the sequence of names of saddle connections of $Q$ crossed by that orbit.
It follows from minimality of the vertical flow on $X$ that each cutting
sequence of an infinite orbit is made of the same pieces, in the sense that
the set of finite words in $\mathcal{A}^{*}$ that appear in a cutting sequence
does not depend on the cutting sequence but only on $X$. The set of finite
words that appear in a cutting sequence (or all cutting sequences) is the
_language of $Q$_ and is denoted $\mathcal{L}_{Q}$. Note that
$\mathcal{L}_{Q}$ can also be defined in terms of symbolic coding of bipartite
interval exchanges (see §2). In the torus case, or equivalently interval
exchanges of two intervals which are rotations of the circle, the coding is on
a two letter alphabet $\\{\ell,r\\}$. The sequences that are obtain for the
torus are called _Sturmian words_ and have several characterization (for
example in terms of balance or complexity, see [37]). For higher genera cases,
there is a characterization of such sequences in [5] and [17] based on
bifurcations.
A word in $\mathcal{L}_{Q}$ is called _left special_ (resp. _right special_)
if it may be extended in two ways on the left (resp. on the right). It is
_bispecial_ if it is left and right special. An important questions in
symbolic dynamics is to describe the set of bispecial words in a language. The
diagonal changes algorithm provides a full answer to this question.
###### Theorem 1.13.
Let $X$ be a surface in $\mathcal{C}^{hyp}(k)$ without vertical saddle
connections and let $Q$ be a quadrangulation of $X$. Let
$(Q^{(n)})_{n\in\mathbb{N}}$ be _any_ sequence of quadrangulations obtained by
a sequence of staircase moves starting from $Q$. Then, the set of bispecial
words of the language $\mathcal{L}_{Q}$ is exactly the set of cutting
sequences of diagonals of all quadrangulations in
$(Q^{(n)})_{n\in\mathbb{N}}$.
Furthermore, cutting sequences of diagonals can be constructed recursively
from moves of the algorithm in terms of substitutions, as explained in §4.2
(see Theorem 4.10). Thus, diagonal changes algorithms can be used to construct
a list of bispecial words. We derive Theorem 1.13 from Theorem 1.12, since we
show in §3 that in our context bispecial words correspond to geometric best
approximations. A combinatorial proof of Theorem 1.13 was first given in [18]
in the context of interval exchange transformations.
#### 1.3.3 Applications to Teichmüller dynamics
In this section we mention other applications of diagonal changes algorithms
in Teichmüller dynamics. Let us first recall the well-known connection between
classical continued fractions and the geodesic flow on the modular surface
(see for example [39] and also [1] for a more geometric approach in the same
spirit as §1.1).
##### Tori and the modular surface.
The _modular surface_ is the quotient
$\mathcal{M}_{1}=\mathbb{H}/\operatorname{SL}(2,\mathbb{Z})$ of the upper half
plane $\mathbb{H}=\\{z;\ \operatorname{Im}z>0\\}$ by the action of
$\operatorname{SL}(2,\mathbb{Z})$ by Moebius transformations. Its unit tangent
bundle $T^{1}\mathcal{M}_{1}$ is isomorphic to
$\operatorname{SL}(2,\mathbb{R})/\operatorname{SL}(2,\mathbb{Z})$ (see for
example [4]). It is well-known that the _space of unimodular lattices_ is
isomorphic to $\mathcal{M}_{1}$ and the space $\mathcal{H}^{1}(0)$ of tori of
unit area is isomorphic to
$\operatorname{SL}(2,\mathbb{R})/\operatorname{SL}(2,\mathbb{Z})$. The
correspondence is obtained by mapping the lattice $\Lambda\subset\mathbb{C}$,
or equivalently the flat torus $\mathbb{T}^{2}_{\Lambda}$, to the point
$w_{2}/w_{1}\in\mathbb{H}$, where $w_{1}$ and $w_{r}$ form a direct base of
the lattice $\Lambda$ and are respectively the shortest and the second
shortest saddle connections on $\mathbb{T}^{2}_{\Lambda}$.
The _geodesic flow_ $g_{t}$ on the unit tangent bundle of the modular surface
$T^{1}\mathcal{M}_{1}\cong\operatorname{SL}(2,\mathbb{R})/\operatorname{SL}(2,\mathbb{Z})$
is given by the action of the $1$-parameter group of diagonal matrices
$\left\\{g_{t}=\left(\begin{array}[]{cc}e^{t}&0\\\
0&e^{-t}\end{array}\right);\ t\in\mathbb{R}\right\\}$ (1)
by left multiplication on $\operatorname{SL}(2,\mathbb{R})$. Orbits of $g_{t}$
project to geodesics on $\mathcal{M}_{1}$ with respect to the hyperbolic
metric. The continued fraction algorithm can be used to describe the _Poincaré
first return map_ of the geodesic flow on a suitably chosen section of
$T^{1}\mathcal{M}_{1}$ (this classical connection, known since Hedlund and
Morse, was nicely pinpointed by Series in [39]).
Furthermore, the geometric continued fraction algorithm can be used to
describe the set of vectors in $\Lambda$, or equivalently the set of saddle
connections on $\mathbb{T}^{2}_{\Lambda}$, which become short along a geodesic
in the following sense. The _systole function_ is
$\operatorname{sys}(\Lambda)=\\{\min|v|;\ v\in\Lambda\backslash\\{0\\}\\}$.
Recall that compact sets in $\mathcal{M}_{1}$ can be characterized as sets on
which the systole function is bounded (by Mahler’s compactness criterion).
Given a flat torus $\mathbb{T}^{2}_{\Lambda}\in T^{1}\mathcal{M}_{1}$,
consider the systole function evaluated along the _geodesic_ passing though
it, that is the map $t\mapsto\operatorname{sys}(g_{t}\Lambda)$. We say that a
vector $v\in{\Lambda}$ (or equivalently the corresponding saddle connection on
$\mathbb{T}^{2}_{\Lambda}$) _realizes the systole at time $t$_ if
$|g_{t}v|=\operatorname{sys}(g_{t}\Lambda)$. Then the vectors in
$v\in{\Lambda}$ that realizes systoles for some $t\in\mathbb{R}$ are exactly
the vectors $w^{(n)}_{\ell}$ and $w^{(n)}_{r}$ in the sequence of bases
$\left((w^{(n)}_{\ell},w^{(n)}_{r})\right)_{n\in\mathbb{Z}}$ built from the
geometric continued fraction algorithm.
##### Systoles along Teichmüller geodesics.
Diagonal changes algorithms play a role in describing short saddle connections
along _Teichmüller geodesics_ analogous to the role played by the standard
continued fraction for the torus.
Let $\mathcal{H}(k_{1}-1,\ldots,k_{n}-1)$ be a stratum of translation surfaces
(as defined in §1.2.1) and let
$\mathcal{H}^{1}(k_{1}-1,\ldots,k_{n}-1)\subset\mathcal{H}(k_{1}-1,\ldots,k_{n}-1)$
consist of translation surfaces of area one. Seen as a topological space,
$\mathcal{H}^{1}(k_{1}-1,\ldots,k_{n}-1)$ is never compact. Nevertheless, as
in the case of tori, compact sets can be defined using the systole function
$\operatorname{sys}(X)=\min\\{|v|;\ v\in\Gamma(X)\\}$
where $X$ is a translation surface and as before $\Gamma(X)$ is the set of
saddle connections on $X$.
The linear action of $\operatorname{SL}(2,\mathbb{R})$ on $\mathbb{C}$
identified to $\mathbb{R}^{2}$ induces an action of
$\operatorname{SL}(2,\mathbb{R})$ on translation surfaces: given a translation
surface $X$ obtained gluing polygons $P_{i}\subset\mathbb{C}$ and
$A\in\operatorname{SL}(2,\mathbb{R})$, the surface $A\cdot X$ is obtained
gluing the polygons $AP_{i}$ using the same identifications. This is well
defined since the linear action preserves pairs of parallel congruent sides.
The restriction of the $\operatorname{SL}(2,\mathbb{R})$-action on
$\mathcal{H}^{1}(k_{1}-1,\ldots,k_{n}-1)$ to the diagonal subgroup $g_{t}$ in
(1) is known as the _Teichmüller geodesic flow_.
Let $X$ be a translation surface and, as in the case of the torus, consider
the systole function $t\mapsto\operatorname{sys}(g_{t}X)$. We say that a
saddle connection $v$ on $X$ _realizes the systole at time $t$_ if
$\operatorname{sys}(g_{t}X)=|g_{t}v|$.
###### Theorem 1.14.
Let $X$ be a surface in a hyperelliptic component of a stratum
$\mathcal{C}^{hyp}(k)$ with no horizontal nor vertical saddle connections. Let
$(Q^{(n)})_{n\in\mathbb{Z}}$ be a sequence of quadrangulations of the surface
$X$ where $Q^{(n+1)}$ is obtained from $Q^{(n)}$ by a staircase move. Then,
the set of saddle connections on $X$ which realize the systoles along the
Teichmüller geodesic passing through $X$ is a subset of the sides the
quadrangulations $Q^{(n)}_{n}$, $n\in\mathbb{Z}$.
Theorem 1.14 is proved as a consequence of Theorem 1.12 in §4.1.2.
##### Pseudo-Anosov diffeomorphisms.
We mention another application of diagonal changes algorithms which we prove
in [12]. An important problem in dynamics is to study the set of closed orbits
of a flow. We show in [12] that diagonal changes algorithms can be used to
effectively produce a list, ordered by length, of all closed orbits of the
Teichmüller flow in each hyperelliptic component $\mathcal{C}^{hyp}(k)$. Since
closed Teichmüller geodesics are in one to one correspondence with conjugacy
classes of _pseudo-Anosov diffeomorphisms_ , one can equivalently list pseudo-
Anosov conjugacy classes, ordered by dilation. Furthermore, diagonal changes
are much better suited for this problem than other algorithms such as Rauzy-
Veech induction or train-track splittings, as explained in §1.4.
##### Lagrange spectra.
Recently, Lagrange spectra for translation surfaces, which are a
generalization of the classical Lagrange spectrum in Diophantine
approximation, were defined and studied by P. Hubert, L. Marchese and C.
Ulcigrai in [24]. If $\mathcal{C}$ is a connected component of a stratum of
translation surfaces, its Lagrange spectrum $\mathcal{L}(\mathcal{C})$ is the
set of values $\mathcal{L}(\mathcal{C}):=\\{1/{a(X)};\
X\in\mathcal{C}\\}\subset\mathbb{R}\cup\\{+\infty\\}$, where
$a(X):=\liminf_{|\operatorname{Im}(v)|\to\infty}\frac{\\{|\operatorname{Im}(v)||\operatorname{Re}(v)|;\
\text{v saddle connection on $X$}\\}}{\operatorname{Area}(X)},$ (2)
where $\operatorname{Area}(X)$ is the area of the surface $X$ with respect to
its flat metric. Equivalently, one has that $a(X)=s^{2}(X)/2$, where
$s(X):=\liminf_{t\to\infty}\operatorname{sys}(g_{t}X)/\operatorname{Area}(X)$,
see [46] and [24].
If $X$ belongs to a hyperelliptic component $\mathcal{C}^{hyp}(k)$, we show in
Theorem 4.6 that $a(X)$ can be computed using diagonal changes algorithms.
Furthermore, in [24] it is shown that $\mathcal{L}(\mathcal{C})$ is the
closure of the values $1/a(X)$, $X\in\mathcal{C}$ for which the Teichmüller
geodesic through $X$ is closed. Thus, diagonal changes algorithms can be used
to get finer and finer approximations of the Lagrange spectrum
$\mathcal{L}(\mathcal{C}^{hyp}(k))$, by first listing closed Teichmüller
geodesics in $\mathcal{C}^{hyp}(k)$ and then computing the corresponding
Lagrange values.
### 1.4 Comparison with other algorithms in the literature
In this section we compare diagonal changes algorithms with other induction
algorithms in the literature: Ferenczi-Zamboni induction, Rauzy-Veech
induction, da Rocha induction and train-track splittings. From a Diophantine
point of view, we mention the analogy between Y. Cheung’s $Z$-convergents and
our best approximations. From a combinatorial point of view we mention a link
between the combinatorics of diagonal changes and cluster algebras
combinatorics.
The Ferenczi-Zamboni induction (FZ induction for short), which is called by
the authors _self-dual induction_ , is an induction algorithm for interval
exchange transformations (IETs), first introduced for IETs of $3$ intervals in
a series of papers jointly with Holton [13, 14, 15], then in [18] for all
symmetric IETs (namely those with combinatorics given by the permutation
$\pi(k)=n-k+1$, $1\leq k\leq n$). Very recently Ferenczi in [21] developed a
new induction for any IETs. The algorithm was defined and developed with the
main aim of giving a combinatorial description of the IETs language and in
particular to produce the list of bispecial words, see [14, 17]. The FZ
algorithm was also used by Ferenczi and Zamboni to produce examples of IETs
with special ergodic and spectral properties (see [15, 16, 19]).
The diagonal changes that we describe are a geometric version for translation
surfaces in hyperelliptic components of FZ induction for symmetric IETs. While
the proofs of the existence of FZ-moves and that FZ induction sees all
bispecial words given in [18] are purely combinatorial, the proofs of the
analogous results for staircase moves in this paper are very geometric. Some
of the definitions and proofs for FZ induction are combinatorially quite heavy
and we believe that one of the advantages of our geometric approach is to make
the induction easier to understand and proofs simpler and more transparent.
Recently Ferenczi extended the FZ induction for any IET [21]. Following our
paper, he also gave in [22] a geometric counterpart in the language of
diagonal changes. Many of the geometric properties of staircase moves seem to
extend also for these general algorithms, which we stress are not given by
quadrangulations and staircase moves.
Another very well known induction algorithm for translation surfaces and IETs
is _Rauzy-Veech induction_. The Rauzy-Veech induction for translation surfaces
is a geometric invertible extension of the Rauzy induction for interval
exchanges in the same way the staircase moves for translation surfaces are an
extension of FZ-moves for IETs. Rauzy-Veech induction has been a key tool to
prove conjectures on the ergodic properties of IETs and linear flows on
translation surfaces. The dynamics of the induction itself has been studied in
detail (see for example [44] or most recently [2] and [3]).
Despite the many applications of the Rauzy-Veech induction to ergodic
problems, diagonal changes algorithms are a much better suited tool to attack
some dynamical questions, in particular to list geometric best approximations
in each bundle and to enumerate conjugacy classes of pseudo-Anosovs or
equivalently Teichmüller closed geodesics. The heuristic explanation for this
is that the Rauzy-Veech algorithm involves a choice of a conical singularity
and of a separatrix which gives a transveral for the IET. Therefore, the
domain on which the induction is defined, namely the space of zippered
rectangles introduced by Veech in [44], is a _finite-to-one_ cover of
connected components of strata of translation surfaces. In [12], on the other
hand, we explain that the space of quadrangulations, which is the analogous
for staircase moves of the space of zippered rectangles, yield a faithful
representation of hyperelliptic components.
The idea of an induction algorithm which, contrary to Rauzy-Veech induction,
did not require the choice of a separatrix was long advocated, in particular
by P. Arnoux. In the setting of IETs a similar idea is also at the base of the
induction invented by L. da Rocha (see [31]) and of the induction described by
Cruz and da Rocha in [9] for rotational IETs. We remark that the latter,
similarly to FZ induction for symmetric IETs, also uses a bipartite IET
structure. Recently Inoue and Nakada in [26] defined a geometric extension of
the Cruz-da Rocha induction by using zippered rectangles of a bipartite form
and showed that this extension is dual to Rauzy-Veech induction on zippered
rectangles.
Rauzy-Veech induction and diagonal changes algorithms may be seen as train-
tracks algorithms. Train-tracks are combinatorial objects embedded in
surfaces, that allow to describe measured foliations (such as the vertical
foliation in a translation surface). Train-tracks splittings have been used in
particular to provide a way to describe and enumerate conjugacy classes of
pseudo-Anosov diffeomorphisms, see for examples [36], [43] and [30]. In the
context of translation surfaces and Teichmüller dynamics, several results
which exploit train-tracks splittings and a related symbolic coding of the
Teichmüller flow were obtained by U. Hamenstädt, see for example [23]. Our
diagonal changes algorithms use train-tracks of a very special form (which
correspond to the bipartite nature of the IETs arising from quadrangulations,
see §2.2). The train-tracks splittings allowed in our induction are the one
which preserve this bipartite structure. Train-tracks algorithms often have
the drawback that there is a large choice of possible moves and the graphs
which describe combinatorial data are very large. In the case of our
algorithms, the combinatorial graph associated to the moves (defined in §2.5)
has a much more manageable size (see the comparison table in [12]).
Furthermore, as explained in [12], one can produce pseudo-Anosov
diffeomorphisms from certain paths in the graph without having to check a
rather subtle irreducibility condition which is needed when considering loops
in the graph of train-tracks (see for example [43, Proposition 3.7]).
In [40], J. Smillie and C. Ulcigrai characterized the language of cutting
sequences for linear trajectories on translation surfaces obtained from
regular $2n$-gons (this characterization could also be proven for double
regular $n$-gons with $n$ odd, see D. Davis [10]). The characterization is
based on an induction algorithm which uses affine diffeomorphisms in the Veech
group, see also [41]. One can show that this algorithm turns out to be a
diagonal changes algorithm. Ferenczi in [20] considered the interval exchanges
which arise as Poincaré maps of linear flows in regular $2n$-gons and
described the FZ-moves which arise when performing FZ-induction starting from
them. One can also see that the diagonal changes algorithm by Smillie and
Ulcigrai is an acceleration of the geometric extension of the moves in [20].
Let us now mention the connections with $Z$-convergents and then cluster
algebras. The notion of geometric best approximation for translation surfaces
that we define in this paper is very close to the notion of $Z$-convergents
for translation surfaces introduced by Y. Cheung (see his joint paper [8] with
P. Hubert and H. Masur for the definition). The definition is parallel to the
notion of best approximation in the space of higher dimensional lattices that
was used by Y. Cheung in [7]. The $Z$-convergents were further used by P.
Hubert and T. Schmidt [25] to provide transcendence criterion in the context
of translation surfaces. In all these works on translation surfaces, the
sequence of $Z$-convergents are considered from a theoretical point of view:
no actual description of these sets were given. Diagonal changes algorithms
provide an explicit construction of best approximations.
Finally, we remark that it turns out that the combinatorics which appear in
diagonal changes (in particular the graph $\mathcal{G}$) is related to cluster
algebras. Recently R. Marsh and S. Schroll in [34] explained this connection.
In the case of FZ induction, they explain how one can put in one-to-one
correspondence the trees of relations introduced in [18] with triangulations
on the sphere and diagonal changes for these triangulations with the FZ-moves
on the trees of relations defined by [6]. The combinatorics of these moves are
exactly our staircase moves seen on the sphere (recall that in hyperelliptic
components, each surface is a double cover of the sphere).
### 1.5 Structure of the paper
In §2 we give a formal definition of staircase moves on the space of
parameters which describe quadrangulations. We also explain the link between
quadrangulations and bipartite interval exchanges and hence between staircase
moves and FZ moves. Finally, we prove that staircase moves display a form of
self-duality and Markov structure.
In §3 we first give the definition of hyperelliptic components. We then prove
that translation surfaces in hyperelliptic components always admit a
quadrangulation (Theorem 1.8) and that each of these quadrangulations has a
well slanted staircase (Theorem 1.9).
The applications of diagonal changes algorithms given by staircase moves
mentioned above are considered in §4. We first prove that staircase moves
produce exactly all geometric best approximations (Theorem 1.12). We then show
how this result can be used to study the systole function along Teichmüller
geodesics. Finally, we prove that bispecial words are exactly cutting
sequences of best-approximations (Theorem 4.10).
### 1.6 Acknowledgements
We would like to thank S. Ferenczi, E. Lanneau and S. Schroll for useful
discussions. We are grateful to P. Hubert, who invited the second author to
Marseille for a scientific visit during which diagonal changes were initially
conceived and who also immediately pointed out the connection with FZ
induction. V. Delecroix is supported by the ERC Starting Grant
“Quasiperiodics” and C. Ulcigrai is partially supported by the EPSRC Grant
EP/I019030/1, which we thankfully acknowledge for making the authors
collaboration possible.
## 2 Diagonal changes on the space of quadrangulations
We begin this section by describing in §2.1 the combinatorial and length data
which define a quadrangulation. We then describe the link between
quadrangulations and bipartite interval exchange maps (see §2.2). The
induction developed by Ferenczi and Zamboni operates on bipartite interval
exchanges. In §2.3 and §2.4 we give a more formal definition of staircase
moves and the associated diagonal changes algorithms and explain the relation
with FZ induction. Finally, in §2.5 we show that the staircase moves are
invertible and provide a Markov structure to the parameter space of
quadrangulations. In particular, we show that our staircase moves provide a
geometric realization of the natural extension of elementary FZ moves. We also
show that the inverse of a staircase move is again a staircase move. In this
sense these types of inductions are sometimes described as self-dual
inductions.
### 2.1 Parameters on quadrangulations
Let $Q$ be a quadrangulation of a translation surface $X$. We saw in the
introduction that $Q$ is determined by the collection of wedges of
quadrilaterals in $Q$. In addition to wedges, the quadrangulation $Q$ also
determines a pair of permutations which describe how the quadrilaterals of the
quadrangulation $Q$ are glued to each other as follows (refer to Figure 7).
Figure 7: a quadrilateral $q_{i}$ glued with $q_{\pi_{r}(i)}$ and
$q_{\pi_{\ell}(i)}$
###### Definition 2.1.
Let $Q$ be a quadrangulation with $k$ quadrilaterals and let us label the
quadrilaterals by the integers $\\{1,\ldots,k\\}$. Let $q_{i}$ denote the
quadrilateral labelled $i$. The _combinatorial datum_
${\underline{\pi}}={\underline{\pi}}_{Q}$ of the labelled quadrangulation $Q$
is a pair $(\pi_{\ell},\pi_{r})$ of permutations of $\\{1,\ldots,k\\}$ such
that
* (i)
for each $1\leq i\leq k$, the top left side of $q_{i}$ is glued to the bottom
right side of $q_{\pi_{\ell}(i)}$.
* (ii)
For each $1\leq i\leq k$, the top right side of $q_{i}$ is glued to the bottom
left side of $q_{\pi_{r}(i)}$.
Thus $\pi_{\ell}(i),\pi_{r}(i)$ describe to which wedges the top sides of the
quadrilateral $q_{i}$ belong, as illustrated by Figure 7.
(a) a quadrangulation $Q$ with $\pi_{\ell}=(1,2,3)$ and $\pi_{r}=(1)(2,3)$ (b)
the graph $G_{Q}$ associated to $Q$
Figure 8: the graph $G_{Q}$ associated to a quadrangulation $Q$ of a surface
in $\mathcal{H}(2)=\mathcal{C}^{hyp}(3)$
We mention that the combinatorial datum
${\underline{\pi}}_{Q}=(\pi_{\ell},\pi_{r})$ of a labelled quadrangulation $Q$
can also be described by a graph $G_{Q}$, whose vertices are in one-to-one
correspondence with the quadrilaterals $q_{1},\dots,q_{k}$ and will be denoted
by the corresponding index $1\leq i\leq k$. The edges of $G_{Q}$ are labelled
by $r$ or $l$ and are such that for each $1\leq i\leq k$ there is an
$\ell$-edge from $i$ to $\pi_{\ell}(i)$ and $r$-edge from $i$ to $\pi_{r}(i)$.
An example is given in Figure 8. These graphs are used by Ferenczi and Zamboni
in [19].
Let $Q$ be a labelled quadrangulation and let $w_{1},\dots,w_{k}$ be the
wedges corresponding to $q_{1},\dots,q_{k}$. Remark that quadrilaterals in a
quadrangulation (or equivalently, wedges) are in one to one correspondence
with bundles of saddle connections. Thus, labelling the quadrilaterals in $Q$
by $q_{1},\dots,q_{k}$ automatically induces also a labelling of bundles by
$\Gamma_{1},\dots,\Gamma_{k}$ so that each $w_{i,\ell}$ (resp. $w_{i,r}$)
belong to the bundle $\Gamma_{i,\ell}$ (resp. $\Gamma_{i,r}$).
Since for each $w_{i,\ell}$ and $w_{i,r}$ the bundle to which they belong
(resp. $\Gamma_{i,\ell}$ or $\Gamma_{i,r}$) is clear from the context, we will
without confusion identify the saddle connections in the wedges with the
complex numbers which give their displacement vectors. Using this notation and
remarking that by construction $w_{i,\ell}$ and $w_{\pi_{\ell}(i),r}$ are the
left sides of the quadrilateral $q_{i}$ while $w_{i,r}$ and
$w_{\pi_{r}(i),\ell}$ are its right sides, we have
$w_{i,\ell}+w_{\pi_{\ell}(i),r}=w_{i,r}+w_{\pi_{r}(i),\ell},\qquad 1\leq i\leq
k.$ (3)
The equations in (3) are called _train-track relations_.
Conversely, we can construct a surface with a quadrangulation by starting from
a combinatorial datum $\pi=(\pi_{\ell},\pi_{r})$ in $S_{k}\times S_{k}$ and a
length datum
$\underline{w}=((w_{1,\ell},w_{1,r}),\ldots,(w_{k,\ell},w_{k,r}))\in\left((\mathbb{R}_{-}\times\mathbb{R}_{+})\times(\mathbb{R}_{+}\times\mathbb{R}_{+})\right)^{k},$
where $\mathbb{R}_{+}=\\{t\in\mathbb{R};\ t>0\\}$ and
$\mathbb{R}_{-}=\\{t\in\mathbb{R};\ t<0\\}$. If $\underline{w}$ satisfies the
train-track relations (3) we can build a labelled quadrangulation $Q$ that we
denote $({\underline{\pi}},\underline{w})$. When we write
$Q=({\underline{\pi}},\underline{w})$ we assume implicitely that
$\underline{w}$ satisfies the train-track relations.
### 2.2 Bi-partite interval exchanges and quadrangulations
Let us define bipartite interval exchange transformations and show that they
arise as Poincaré first return maps of the vertical linear flow in a
quadrangulation. Given $Q=({\underline{\pi}},\underline{w})$, the union of the
wedges of $Q$ provide a convenient section for the vertical flow on the
associated surface. The first return map on this section has a bipartite
structure: for each $1\leq i\leq k$ the points on the wedge $w_{i}$ are
divided in two sets depending on their future (the left part go to
$q_{\pi_{\ell}(i)}$ and the right part to $q_{\pi_{r}(i)}$) and there is
another partition with respect to their past (the left part comes from
$q_{\pi_{r}^{-1}(i)}$ and the right part comes from $q_{\pi_{\ell}^{-1}(i)}$).
(a) A bipartite IET (b) The suspension of a the bipartite IET in Figure 9(a)
Figure 9: a bipartite interval exchange transformations with 3 intervals and
one of its suspension. The resulting translation surface belongs to
$\mathcal{H}(2)=\mathcal{C}^{hyp}(3)$
A _bipartite interval exchange map_ is a piecewise isometry $T:I\to I$ defined
on the _disjoint union_ $I=\bigsqcup_{i=1}^{k}I_{i}$ of $k$ open bounded
intervals $I_{1},\ldots,I_{k}$. Each interval $I_{i}$ is partitioned in two
different ways as union of two intervals and $T$ maps isometrically the
intervals in the first partition to the intervals in the second partition, so
that the image of a right interval (resp. a left interval) is a left (resp.
right) interval (see Figure 9(a)).
More formally, let ${\underline{\pi}}=(\pi_{\ell},\pi_{r})$ where $\pi_{\ell}$
and $\pi_{r}$ are two permutations of $\\{1,\ldots,k\\}$. Let
${\underline{\lambda}}=((\lambda_{1,\ell},\lambda_{1,r}),\dots,(\lambda_{k,\ell},\lambda_{k,r}))\in(\mathbb{R}_{-}\times\mathbb{R}_{+})^{k}$
be such that
$\lambda_{i,\ell}+\lambda_{\pi_{\ell}(i),r}=\lambda_{i,r}+\lambda_{\pi_{r}(i),\ell},\qquad\forall
1\leq i\leq k.$ (4)
The relations given by the second formula in (4) are the _train-track
relations_ for the lengths, analogous to the ones for the wedges (3).
For $i\leq i\leq k$, set
$I_{i}=\left(\lambda_{i,\ell},\lambda_{i,r}\right)\subset\mathbb{R}$ and let
$\begin{array}[]{cc}I_{i,\ell}=\left(\lambda_{i,\ell},0\right),&I_{i,r}=\left(0,\lambda_{i,r}\right),\\\
J_{i,\ell}=\left(\lambda_{i,\ell},\lambda_{i,\ell}+\lambda_{\pi_{\ell}(i),r}\right),&J_{i,r}=\left(\lambda_{i,r}+\lambda_{\pi_{r}(i),\ell},\lambda_{i,r}\right).\end{array}$
Remark that $\\{I_{i,\ell},I_{i,r}\\}$ is obviously a partition of
$I_{i}\backslash\\{0\\}$ and the train track relations (4) imply that
$\\{J_{i,\ell},J_{i,r}\\}$ is a partition of
$I_{i}\backslash\\{\lambda_{i,d}\\}$ where
$\lambda_{i,d}=\lambda_{i,\ell}+\lambda_{\pi_{\ell}(i),r}=\lambda_{i,r}+\lambda_{\pi_{r}(i),\ell}$.
###### Definition 2.2.
The _bipartite interval exchange map_ with data
$({\underline{\pi}},{\underline{\lambda}})$ is the map from
$I=I_{1}\sqcup\ldots\sqcup I_{k}$ that maps by translation $J_{i,l}$ to
$I_{\pi_{\ell}(i),r}$ and $J_{i,r}$ to $I_{\pi_{r}(i),\ell}$. The map is not
defined at the points $\lambda_{i,d}\in I_{i}$, $1\leq i\leq k$.
We introduced bipartite IETs so that the following holds. Let us call
_interior_ of a wedge $w=(w_{\ell},w_{r})$ the union of the interiors of the
saddle connections $w_{\ell}$ and $w_{r}$ together with their common
singularity point.
###### Lemma 2.3 (cross sections of quadrangulations).
Given a quadrangulation $Q=({\underline{\pi}},\underline{w})$, the Poincaré
first return map $F$ of the vertical flow on the union of the interiors of the
wedges of $Q$ is conjugate to the bipartite IET
$T=({\underline{\pi}},{\underline{\lambda}})$, where the vector
${\underline{\lambda}}$ is given by the real parts of the wedges. More
precisely, if $p$ is the projection $p$ that maps a point $z$ of the wedge
$w_{i}$ to the point $\operatorname{Re}(z)\in I_{i}$, we have $pF=Tp$.
Remark that for each $1\leq i\leq k$ the IET $T$ is defined at all points of
$I_{i}$ except at the point $\lambda_{i,d}\in I_{i}$, which corresponds to the
unique point of the wedge $w_{i}$ whose trajectory hits an endpoint of a wedge
(and hence $F$ is not defined there). Clearly the Lebesgue measure on $I$ is
invariant under $T$. The pull back of the Lebesgue measure $p$ is the
absolutely continuous _transverse measure_ invariant under the Poincaré map.
Conversely, starting from a given bipartite IET $T$ we can construct as
follows a family of quadrangulations on a surface $X$ for which $T$ is the
Poincaré first return map on the union of the wedges, see Figure 9(b).
###### Definition 2.4.
A _suspension data_ ${\underline{\tau}}$ for the bipartite IET
$({\underline{\pi}},{\underline{\lambda}})$ is a vector
${\underline{\tau}}=\left((\tau_{1,\ell},\tau_{1,r}),\dots,(\tau_{k,\ell},\tau_{k,r})\right)$
in $(\mathbb{R}_{+}\times\mathbb{R}_{+})^{k}$ that satisfies the train-track
relations
$\tau_{i,\ell}+\tau_{i,r}=\tau_{\pi_{r}(i),\ell}+\tau_{\pi_{\ell}(i),r},\quad\text{for
$i=1,\ldots,k$.}$
To the interval exchange data $({\underline{\pi}},{\underline{\lambda}})$ and
the suspension datum ${\underline{\tau}}$ we associate a quadrangulation
$Q=({\underline{\pi}},{\underline{\lambda}},{\underline{\tau}})=({\underline{\pi}},\underline{w})$
where the wedges of $Q$ are
$w_{i,\ell}=\lambda_{i,\ell}+\sqrt{-1}\,\tau_{i,\ell}$ and
$w_{i,r}=\lambda_{i,r}+\sqrt{-1}\,\tau_{i,r}$. The following result can be
seen as a converse of Lemma 2.3.
###### Lemma 2.5 (suspensions of bipartite IETs).
Given a bipartite IET $T=({\underline{\pi}},{\underline{\lambda}})$ and a
suspension datum ${\underline{\tau}}$ for $T$, let
$Q=({\underline{\pi}},{\underline{\lambda}},{\underline{\tau}})$ be the
associated quadrangulation. Then the Poincaré map of the vertical flow on the
associated surface on the union of the interior of the wedges of $Q$ is
conjugated to $T$.
We remark that the vertical flow on the translation surface given by
$Q=({\underline{\pi}},\underline{w})$ can also be represented as a special
flow over the corresponding bipartite IET
$T=({\underline{\pi}},{\underline{\lambda}})$. The components of the vector
${\underline{\tau}}$ give the heights of the corresponding Rohlin towers. One
can embedd geometrically these towers into the surface as shown in Figure
9(b). Note that with this representation by Rohlin towers, the section is
naturally given by horizontal intervals in the surface.
### 2.3 Staircase moves and Ferenczi-Zamboni moves
In the introduction we already gave a geometric definition of a staircase move
(Definition 1.7). Let us now describe the corresponding operation on
quadrangulation data.
Given a quadrangulation $Q=({\underline{\pi}},\underline{w})$, let us recall
that the top right side of the quadrilateral $q_{i}$ is glued to the
quadrilateral $q_{\pi_{r(i)}}$. Thus, if
$\\{i,\pi_{r}(i),\dots\pi_{r}^{n}(i)\\}$ is a _cycle_ of $\pi_{r}$, that is
$\pi_{r}^{j}(i)\neq i$ for $1\leq j\leq n$ but $\pi_{r}^{n+1}(i)=i$, the
corresponding quadrilaterals $\\{q_{i},q_{\pi_{r}(i)},\dots
q_{\pi_{r}^{n}(i)}\\}$ are glued to each other through top right/bottom left
sides. Similarly, since the top left side of $q_{i}$ is glued to
$q_{\pi_{\ell}(i)}$, the quadrilaters $\\{q_{i},q_{\pi_{\ell}(i)},\dots
q_{\pi_{\ell}^{n}(i)}\\}$ indexed by a cycle of $\pi_{\ell}$ are glued to each
other through top left/bottom right sides.
###### Notation.
Given a cycle $c\in\pi_{\ell}$ (respectively a cycle $c\in\pi_{r}$) we denote
by $S_{c}$ the left (respectively right) staircase for $Q$ which is obtained
as union of the quadrilaterals in $Q$ indexed by the cycle $c$. Abusing the
notation, we will denote by $S=S_{c}$ both the collection of quadrilaterals
and their union as a subset of $X$, so we will both write $S\subset X$ and
$q\in S$ where $q$ is one of the quadrilaterals of $Q$ contained in $S$.
Let $Q=({\underline{\pi}},\underline{w})$ be a quadrangulation. For each wedge
$w_{i}=(w_{i,\ell},w_{i,r})$ of a quadrilateral $q_{i}\in Q$, we denote by
$w_{i,d}$ (or by $w_{i,d^{+}}$) the (forward) diagonal of the quadrilateral,
which is given by
$w_{i,d}=w_{i,d^{+}}:=w_{i,\ell}+w_{\pi_{\ell}(i),r}=w_{i,r}+w_{\pi_{r}(i),\ell},$
(5)
where the above equality holds by the train-track relations (3) for
$\underline{w}$. Remark that a right (resp. left) staircase $S_{c}$ associated
to a cycle $c$ of $\pi_{r}$ (resp. $\pi_{\ell}$) is well slanted (see
Definition 1.6) if and only if $\operatorname{Re}(w_{i,d})<0$
($\operatorname{Re}(w_{i,d})>0$) for all $i\in c$.
Let $c$ be a cycle of $\pi_{r}$ and assume that the corresponding staircase
$S_{c}$ is well slanted. Let us show that the staircase move in $S_{c}$
produces a new quadrangulation and describe its data (refer to Figure 10 and
see also Lemma 2.6 below). Since in a diagonal change, we replace a side of a
wedge with its diagonal it is clear that after the staircase move we obtained
a new length data $\underline{w}^{\prime}$ given by
$w_{i}^{\prime}=\left\\{\begin{array}[]{ll}(w_{i,d},w_{i,r})&\text{if $i\in
c$,}\\\ w_{i}&\text{otherwise.}\end{array}\right.$ (6)
From the well slantedness of the staircase $S_{c}$, it follows that also
$w_{i}^{\prime}$, $1\leq i\leq k$ are wedges, that is
$w^{\prime}_{i,\ell}\in\mathbb{R}_{-}\times\mathbb{R}_{+}$ and
$w^{\prime}_{i,r}\in\mathbb{R}_{+}\times\mathbb{R}_{+}$. Furthermore, the
wedges $\underline{w}^{\prime}$ determine a new quadrangulation $Q^{\prime}$
since, as shown in Figure 10, $w_{i}^{\prime}$ for $i\in c$ is the wedge of
the quadrilateral $q_{i}^{\prime}$ which has $w_{\pi_{r}(i),d}$ as right top
edge and $w_{\pi_{l}\pi_{r}(i),\ell}$ as left top edge . This also shows that
the quadrilateral glued to the top right side of $q_{i}^{\prime}$ is
$q_{\pi_{r}(i)}^{\prime}$ while the quadrilateral glued to the top left side
of $q_{i}^{\prime}$ is $q_{\pi_{\ell}(\pi_{r}(i))}$, as shown in Figure 10.
Thus, the combinatorics
${\underline{\pi}}^{\prime}=(\pi_{\ell}^{\prime},\pi_{r}^{\prime})$ of the new
quadrangulation $Q^{\prime}$ is given by
$\pi^{\prime}_{\ell}(i)=\left\\{\begin{array}[]{ll}\pi_{\ell}\circ\pi_{r}(i)&\text{if
$i\in c$,}\\\
\pi_{\ell}(i)&\text{otherwise.}\end{array}\right.\qquad\text{and}\qquad\pi_{r}^{\prime}=\pi_{r}.$
(7)
We will denote by $c\cdot{\underline{\pi}}$ the new combinatorial datum
${\underline{\pi}}^{\prime}$ given by the above formulas. It follows from the
formula for ${\underline{\pi}}^{\prime}$ that the train-track relations for
${\underline{\pi}}^{\prime}=c\cdot{\underline{\pi}}$ are satisfied by
$\underline{w}^{\prime}$.
Figure 10: right staircase move on the parameters
$({\underline{\pi}},\underline{w})$ of a quadrangulation
Similary, if $c$ is a cycle of $\pi_{\ell}$ and $S_{c}$ is well slanted, the
staircase move in $S_{c}$ produces a new quadrangulation
$Q^{\prime}=(c\cdot{\underline{\pi}},\underline{w}^{\prime})$ where
$\underline{w}^{\prime}$ and
$c\cdot{\underline{\pi}}=(\pi_{\ell}^{\prime},\pi^{\prime}_{r})$ is given by
$w^{\prime}_{i}=\left\\{\begin{array}[]{ll}(w_{i,\ell},w_{i,d})&\text{if $i\in
c$,}\\\ w_{i}&\text{otherwise,}\end{array}\right.$ (8)
$\pi_{\ell}^{\prime}=\pi_{\ell}\qquad\text{and}\qquad\pi^{\prime}_{r}(i)=\left\\{\begin{array}[]{ll}\pi_{r}\circ\pi_{\ell}(i)&\text{if
$i\in c$,}\\\ \pi_{r}(i)&\text{otherwise.}\end{array}\right.$ (9)
We remark that the operation on the permutation ${\underline{\pi}}$ does not
depend on the length datum and the operation on the wedges $\underline{w}$ is
linear. Thus, to describe the new length datum $\underline{w}^{\prime}$, we
introduce the $2k\times 2k$ matrix $A_{{\underline{\pi}},c}$ as follows. We
index the rows and columns of $A_{{\underline{\pi}},c}$ by the $2k$ indices
$(1,\ell),(1,r)$, $(2,\ell),(2,r)$, …, $(k,\ell),(k,r)$. Let $I_{2k}$ the be
$2k\times 2k$ identity matrix and for $1\leq i,j\leq k$ and
$\varepsilon,\nu\in\\{l,r\\}$ let $E_{(i,\varepsilon),(j,\nu)}$ be the
$2k\times 2k$ matrix whose entry in row $(i,\varepsilon)$ and column $(j,\nu)$
is $1$ and all the other entries are zero. We set
$A_{{\underline{\pi}},c}=\left\\{\begin{array}[]{ll}I_{2k}+\sum_{i\in
c}E_{(i,\ell),(\pi_{\ell}(i),r)}&\quad\text{if $c$ is a cycle of
$\pi_{r}$},\\\ I_{2k}+\sum_{i\in c}E_{(i,r),(\pi_{r}(i),\ell)}&\quad\text{if
$c$ is a cycle of $\pi_{\ell}$}.\end{array}\right.$ (10)
Thus, with the convention that $\underline{w}$ and $\underline{w}^{\prime}$
denote column vectors, one can verify from equations (5), (6) and (8) that we
can write $\underline{w}^{\prime}=A_{{\underline{\pi}},c}\ \underline{w}$.
Thus, we proved the following:
###### Lemma 2.6 (staircase move on data).
Given a labelled quadrangulation $Q=({\underline{\pi}},\underline{w})$ and a
cycle $c$ of ${\underline{\pi}}$, if the staircase $S_{c}$ is well slanted,
when performing on $Q$ the staircase move in $S_{c}$ one obtains a new
labelled quadrangulation
$Q^{\prime}=({\underline{\pi}}^{\prime},\underline{w}^{\prime})$ with
${\underline{\pi}}^{\prime}=c\cdot{\underline{\pi}},\qquad\underline{w}^{\prime}=A_{{\underline{\pi}},c}\
\underline{w},$
where $c\cdot{\underline{\pi}}$ and $A_{{\underline{\pi}},c}$ are given by
formulas (7), (9) and (10) above.
One can moreover show that staircases are the smallest unions of
quadrilaterals in which one can simultaneously perform diagonal changes to
obtain a new quadrangulation, in the following sense.
###### Lemma 2.7.
Let $Q=({\underline{\pi}},\underline{w})$ be a quadrangulation and let
$\mathcal{I}_{\ell},\mathcal{I}_{r}\subset\\{1,\ldots,d\\}$ be such that the
quadrilaterals $q_{i}$ with $i\in\mathcal{I}_{\ell}$ are left slanted and the
quadrilaterals $q_{i}$ with $i\in\mathcal{I}_{r}$ are right slanted.
The new set of wedges obtain after individual diagonal changes in the
quadrilaterals $Q_{i}$ for
$i\in\mathcal{I}=\mathcal{I}_{\ell}\cup\mathcal{I}_{r}$ is associated to a
quadrangulation if and only if the set of indices $\mathcal{I}_{\ell}$
(respectively $\mathcal{I}_{r}$) is a union of cycles of $\pi_{\ell}$ (resp.
$\pi_{r}$).
We leave the proof to the reader.
One can verify that staircase moves provide a geometric extension of the
elementary moves on bipartite IETs which appear in the FZ induction [18], in
the following sense.
###### Remark.
Let $Q=({\underline{\pi}},{\underline{\lambda}},{\underline{\tau}})$ be a
quadrangulation of a surface in $\mathcal{C}^{hyp}(k)$ and
$T=({\underline{\pi}},{\underline{\lambda}})$ be the corresponding bipartite
IET. Let
$Q^{\prime}=({\underline{\pi}}^{\prime},{\underline{\lambda}}^{\prime},{\underline{\tau}}^{\prime})$
be the quadrangulation obtained from $Q$ by performing a staircase move in $c$
and let $T^{\prime}$ be corresponding bipartite IET. Then $T^{\prime}$ is the
bipartite IET obtained from $T$ by one elementary step of a FZ move.
An alternative description of the geometric extension can be given in terms of
Rohlin towers. The action of a staircase move at the level of Rohlin towers
associated to a quadrangulation is the stacking operation shown in Figure 11.
Figure 11: diagonal changes seen on suspension
### 2.4 Diagonal changes algorithms given by staircase moves.
Let $Q=Q^{(0)}$ be a given starting quadrangulation. An _algorithm_ produces a
sequence of quadrangulations $Q^{(1)}$, $Q^{(2)}$, …in such way that
$Q^{(n+1)}$ is obtained from $Q^{(n)}$ by a sequence of staircase moves. As we
already mentioned there might be several possible staircase moves. Remark that
if $S_{1}$ and $S_{2}$ are (disjoint) well slanted staircases in $Q$, the
staircase moves in $S_{1}$ and $S_{2}$ commute, so that the order in which
they are performed does not matter and the two moves can be performed
simultaneously. If $Q^{\prime}$ is obtained from $Q$ by performing staircase
moves in a subset of the well slanted staircases of $Q$, we will say that
$Q^{\prime}$ is obtained from $Q$ by _simultaneous staircase moves_.
Let us first define the greedy algorithm, which corresponds to the algorithm
introduced in [18] for bipartite IETs.
###### Definition 2.8 (greedy algorithm).
The _greedy diagonal changes algorithm_ starting from $Q=Q^{(0)}$ produces the
sequence $(Q^{(n)})_{n\in\mathbb{N}}$ of quadrangulations where $Q^{(n+1)}$ is
obtained from $Q^{(n)}$ by performing simultaneous staircase moves in all well
slanted staircases for $Q^{(n)}$.
Let us remark that a left (resp. right) staircase move does not modify
$\pi_{\ell}$ (resp. $\pi_{r}$). Thus, even if the quadrilaterals in a
staircase change, the left (resp. right) staircases (each seen as union of the
corresponding quadrilaterals) do not change during a left (resp. right)
staircase move. Thus, it makes sense to define the _multiplicity_ of a left
(resp. right) staircase $S_{c}$ as the maximum $n$ such that we can perform
$n$ consecutive left (right) staircase moves in $S_{c}$. The following
algorithm may be thought as a generalization of the multiplicative continued
fraction algorithm (associated to the Gauss map) that is an acceleration of
the additive one (associated to the Farey map).
###### Definition 2.9 (left/right algorithm).
The _left/right algorithm_ starting at $Q^{(0)}=Q$ produces a sequence
$(Q^{(n)})_{n\in\mathbb{N}}$ where, if $n$ is even, $Q^{(n+1)}$ is obtained
from $Q^{(n)}$ by performing in each left slanted staircase as many left
staricase moves as the multiplicity of the staircase, while if $n$ is odd
$Q^{(n+1)}$ is obtained by doing the same for all right slanted staicases for
$Q^{(n)}$.
Remark that in the left/right algorithm $Q^{(n+1)}$ is in general obtained
from $Q^{(n)}$ by several staircase moves that are not simultaneous. A version
of this algorithm was already used in [11] for the stratum
$\mathcal{H}(2)=\mathcal{C}^{hyp}(3)$.
In [12] we describe a third diagonal change algorithm, which we call _geodesic
algorithm_ , which is determined by the return map to a Poincaré section of
the Teichmüller flow. Another different version of a diagonal change algorithm
at the level of interval exchanges was used in [20] to describe interval
exchanges that comes from flat surfaces built from $2n$-gons. One can check
that their algorithm is actually the “additive” version at the level of IETs
of the algorithm described by Smillie and Ulcigrai in [40, 41].
Let us say that a diagonal changes algorithm given by staircase moves is a
_slow algorithm_ if each of its moves is given by simultaneous staircase
moves. The greedy algorithm is an example of a slow algorithm, while the
left/right algorithm, the geodesic algorithm and the one described by Smillie
and Ulcigrai in [40, 41] are not. Theorem 4.3 in §4.1.1 shows that _any_ slow
algorithm actually produce the same geometric objects and therefore the choice
of an actual algorithm is not so important.
Further information on the relation between different (not necessarily slow)
algorithms can be deduced from [12], where we give a detailed description of
the structure of the set of quadrangulations on a given surface $X$. In
particular, we show that given a surface $X$ in a hyperelliptic component, for
any two quadrangulations $Q_{1}$ and $Q_{2}$ of $X$ there exists a sequence of
backward and forward staircase moves from $Q_{1}$ to $Q_{2}$.
### 2.5 Invertibility, self-duality and Markov structure on parameter space
In this section we introduce the space of (labelled) quadrangulations of
surfaces in a component $\mathcal{C}^{hyp}(k)$. We prove that staircase moves
are invertible and self-dual on this set of quadrangulations (see Theorem
2.14).
Let us fix $k$ and build the space of all labelled quadrangulations of
surfaces in $\mathcal{C}^{hyp}(k)$. Start from a fixed combinatorial datum
${\underline{\pi}}$ of such a surface and consider the oriented graph
$\mathcal{G}=\mathcal{G}({\underline{\pi}})$ defined as follows. The vertices
are the set of combinatorial data that may be obtained from $\pi$ by a
sequence of staircase moves. There is an edge from $\pi$ to $\pi^{\prime}$
labelled by $c$ if and only if $c\cdot\pi=\pi^{\prime}$. In Figure 12 we show
the graph associated to $\pi_{\ell}=(1,3)$ and $\pi_{r}=(1,2)$. The notation
for cycles used in the figure, which makes clear whether a cycle belong to
$\pi_{\ell}$ or $\pi_{r}$, is the following: if $c$ is a cycle of $\pi_{\ell}$
then we write it as a word of length $k$ on the alphabet $\\{\cdot,\ell\\}$,
where the $i^{th}$ letter of the word is $\ell$ if and only if $i\in c$. For
example, the cycle $c=\\{1,3\\}$ is denoted $\ell\cdot\ell$. Cycles of
$\pi_{r}$ are denoted in the same way using words on the alphabet
$\\{\cdot,r\\}$.
Figure 12: the graph $\mathcal{G}$ of combinatorial data for quadrangulations
in $\mathcal{C}^{hyp}(3)\simeq\mathcal{H}(2)$
As we will see in §3.1, if ${\underline{\pi}}=(\pi_{\ell},\pi_{r})$ is a
combinatorial datum of a quadrangulation of a surface in
$\mathcal{C}^{hyp}(k)$, there exists an involution $\iota$ of
$\\{1,\ldots,k\\}$, that corresponds to the action of the hyperelliptic
involution on the quadrilaterals. Moreover $\pi_{\ell}\,\pi_{r}\,\iota$ is a
$k$-cycle and is invariant under the operation $c\cdot\pi$ associated to a
staircase move, i.e. the $k$-cycles associated to the vertices of
$\mathcal{G}$ are the same. It is proven in [6] that this invariant is
complete, i.e. that two pairs ${\underline{\pi}}$ and
${\underline{\pi}}^{\prime}$ belongs to the same graph if and only if
$\pi_{\ell}\,\pi_{r}\,\iota=\pi^{\prime}_{\ell}\,\pi^{\prime}_{r}\,\iota^{\prime}$.
The same result is proved in [12] using the ergodicity of the Teichmueller
flow on $\mathcal{C}^{hyp}(k)$. In particular, starting from different
combinatorial data ${\underline{\pi}}$ and ${\underline{\pi}}^{\prime}$ that
correspond to two quadrangulations of surfaces in the same component
$\mathcal{C}^{hyp}(k)$, then the graph $\mathcal{G}({\underline{\pi}})$ and
$\mathcal{G}({\underline{\pi}}^{\prime})$ are isomorphic. More precisely,
there exists a permutation $\sigma$ in $S_{k}$ such that the isomorphism is
given by
$(\pi_{\ell},\pi_{r})\mapsto(\sigma\pi_{\ell}\sigma^{-1},\sigma\pi_{r}\sigma^{-1})$.
For each combinatorial datum ${\underline{\pi}}=(\pi_{\ell},\pi_{r})$ in
$\mathcal{G}$, let us introduce the cones
$\Delta_{{\underline{\pi}}}\subset(\mathbb{R}^{2})^{k}$ and
$\Theta_{\underline{\pi}}\subset(\mathbb{R}^{2})^{k}$ that parametrize all
possible lengths and heights of wedges with combinatorial datum
${\underline{\pi}}$, that is the lengths and heights which satisfy the train-
track relations given by ${\underline{\pi}}$. Formally
$\displaystyle\Delta_{{\underline{\pi}}}=\\{$
$\displaystyle\left((\lambda_{1,\ell},\lambda_{1,r}),\dots,(\lambda_{k,\ell},\lambda_{k,r})\right)\in(\mathbb{R}_{-}\times\mathbb{R}_{+})^{k};$
$\displaystyle\lambda_{i,\ell}+\lambda_{\pi_{\ell}(i),r}=\lambda_{i,r}+\lambda_{\pi_{r}(i),\ell}\quad\text{for
$1\leq i\leq k$}\\}$ $\displaystyle\Theta_{{\underline{\pi}}}=\\{$
$\displaystyle\left((\tau_{1,\ell},\tau_{1,r}),\dots,(\tau_{k,\ell},\tau_{k,r})\right)\in(\mathbb{R}_{+}\times\mathbb{R}_{+})^{k};$
$\displaystyle\tau_{i,\ell}+\tau_{\pi_{\ell}(i),r}=\tau_{i,r}+\tau_{\pi_{r}(i),\ell}\quad\text{for
$1\leq i\leq k$}\\}.$
Then the _space of labelled quadrangulations_ of surfaces in
$\mathcal{C}^{hyp}(k)$ is
$\mathcal{Q}_{k}=\\{({\underline{\pi}},{\underline{\lambda}},{\underline{\tau}});\
{\underline{\pi}}\in\mathcal{G},\
{\underline{\lambda}}\in\Delta_{\underline{\pi}},\
{\underline{\tau}}\in\Theta_{\underline{\pi}}\\}.$
In [12] we show that each hyperelliptic component $\mathcal{C}^{hyp}(k)$ is
essentially the same as $\mathcal{Q}_{k}/\sim$ where $\sim$ is the equivalence
relation generated by staircase moves.
Given
$({\underline{\pi}},{\underline{\lambda}},{\underline{\tau}})\in\mathcal{Q}_{k}$
and a cycle $c$ of $\pi_{r}$ or $\pi_{\ell}$, remark that the heights
${\underline{\tau}}\in\Theta_{\underline{\pi}}$ play no role in determining
whether $S_{c}$ is well-slanted. Thus, let
$\Delta_{{\underline{\pi}},c}\subset\Delta_{\underline{\pi}}$ be the subset of
lengths data for which the staircase $S_{c}$ is well slanted. Recall that the
(forward) diagonal
$w_{i,d}=w_{i,\ell}+w_{\pi_{\ell}(i),r}=w_{i,r}+w_{\pi_{r}(i),\ell}$ of
$q_{i}$ is left (resp. right) slanted if and only if its real part
$\lambda_{i,d}=\operatorname{Re}w_{i,d}$ is greater than $0$ (resp. less than
$0$). Thus, formally, we have
$\Delta_{{\underline{\pi}},c}:=\left\\{\begin{array}[]{ll}\\{{\underline{\lambda}}\in\Delta_{{\underline{\pi}}}\
|\quad\lambda_{i,d}<0\ \forall i\in c\\},&\text{if $c$ is a cycle of
$\pi_{r}$,}\\\ \\{{\underline{\lambda}}\in\Delta_{{\underline{\pi}}}\
|\quad\lambda_{i,d}>0\ \forall i\in c\\},&\text{if $c$ is a cycle of
$\pi_{\ell}$.}\\\ \end{array}\right.$ (11)
Then one can perform a staircase move in $S_{c}$ if and only if
${\underline{\lambda}}\in\Delta_{c}$. Using the Definitions (7) and (9) of
$c\cdot{\underline{\pi}}$ and the definition (10) of $A_{{\underline{\pi}},c}$
and remarking that $A_{{\underline{\pi}},c}$ acts linearly both on the real
and imaginary part of each saddle connection in $\underline{w}$, we can
formally define a staircase move on the parameter space as follows:
###### Definition 2.10.
Let ${\underline{\pi}}=(\pi_{\ell},\pi_{r})\in\mathcal{G}$ and let $c$ be a
cycle of $\pi_{r}$ or $\pi_{\ell}$. The staircase move
$\widehat{m}_{{\underline{\pi}},c}$ on $\mathcal{Q}_{k}$ is map defined on
$\\{{\underline{\pi}}\\}\times\Delta_{{\underline{\pi}},c}\times\Theta_{\underline{\pi}}\subset\mathcal{Q}_{k}$
which sends $({\underline{\pi}},{\underline{\lambda}},{\underline{\tau}})$ to
$\widehat{m}_{{\underline{\pi}},c}({\underline{\pi}},{\underline{\lambda}},{\underline{\tau}})=(c\cdot{\underline{\pi}},\,A_{{\underline{\pi}},c}\
{\underline{\lambda}},A_{{\underline{\pi}},c}\ {\underline{\tau}})$.
Geometrically, the inverse of a staircase move in $X$ is simply a staircase
move in the surface obtained from $X$ by counterclockwise rotation by 90
degrees. To formalize the action by rotation, we introduce the operator $R$ on
the parameter space of quadrangulations $\mathcal{Q}_{k}$. Remark that if
$q\subset\mathbb{C}$ is an admissible quadrilateral, multiplying by the
imaginary unit $\sqrt{-1}$ we get the rotated quadrilateral $\sqrt{-1}q$ which
is still admissible. Thus, if $Q$ is a labelled quadrangulation for $X$, then
the collection of quadrilaterals $q^{\prime}=\sqrt{-1}q$ also determine a
quadrangulation of $X$, which we denote by $\sqrt{-1}Q$. We denote by
$Q^{\prime}$ the quadrangulation $\sqrt{-1}Q$ labelled so that the wedge
$v_{i}^{\prime}$ of the quadrilateral $q_{i}^{\prime}$ contains the same
vertical ray which was contained in the wedge $v_{i}$ of $q_{i}$, as shown in
Figure 13). As we prove below, this convention for the labelling (but not for
example the more naive convention of calling $q_{i}^{\prime}$ the
quadrilateral $\sqrt{-1}q$) guarantees that the operator $R$ that sends $Q$ to
$Q^{\prime}$ is a well defined operation on the space $\mathcal{Q}_{k}$ of
labelled quadrangulations. The explicit formulas for the wedges and
combinatorial datum of $q^{\prime}\in Q^{\prime}$ can be easily obtained from
$Q=({\underline{\pi}},\underline{w})$ by looking at Figure 13 and lead to the
following formal definition:
(a) $Q$ (b) $\sqrt{-1}\,Q$
(c) $Q^{\prime}$
Figure 13: a quadrangulation seen from the vertical labelled $i$, its rotation
by $\pi/2$ and its new labels
###### Definition 2.11.
The _rotation operator_ $R$ sends
$Q=({\underline{\pi}},\underline{w})\in\mathcal{Q}_{k}$ to
$RQ=({\underline{\pi}}^{\prime},\underline{w}^{\prime})$ given by the
following formulas:
$\pi^{\prime}_{\ell}=\pi_{\ell}\,\pi_{r}\,\pi_{\ell}^{-1},\quad\pi^{\prime}_{r}=\pi_{\ell}^{-1}$
and
$q^{\prime}_{i}=\sqrt{-1}\,q_{\pi_{\ell}^{-1}(i)}\quad
w^{\prime}_{i,\ell}=\sqrt{-1}\,w_{i,r}\quad
w^{\prime}_{i,r}=-\sqrt{-1}\,w_{\pi_{\ell}^{-1}(i),\ell}.$
Let us show that is a well defined operator from $\mathcal{Q}_{k}$ to
$\mathcal{Q}_{k}$. It is clear from the geometric description and
admissibility of quadrilaterals that $\underline{w}^{\prime}$ is also a vector
of wedges and that they satisfy the train-track relations for
${\underline{\pi}}^{\prime}$. Hence, if
$\underline{w}^{\prime}={\underline{\lambda}}^{\prime}+\sqrt{-1}{\underline{\tau}}$,
we have that
${\underline{\lambda}}^{\prime}\in\Delta_{{\underline{\pi}}^{\prime}}$ and
${\underline{\tau}}^{\prime}\in\Theta_{{\underline{\pi}}^{\prime}}$. Thus,
since
$\mathcal{Q}_{k}=\mathcal{G}\times\Delta_{{\underline{\pi}}^{\prime}}\times\Theta_{{\underline{\pi}}^{\prime}}$,
one only needs to verify that
${\underline{\pi}}^{\prime}=(\pi_{\ell}\,\pi_{r}\,\pi_{\ell}^{-1},\pi_{\ell}^{-1})$
belong to the same graph $\mathcal{G}=\mathcal{G}(\pi)$. This is proved in §
3.1.3 (see Corollary 3.7) and can be shown either from the combinatorial
description in [6] or from the connectedness of $\mathcal{C}^{hyp}(k)$ proved
in [12].
The operator $R$ is invertible and one can check that the inverse rotation
$R^{-1}:\mathcal{Q}_{k}\to\mathcal{Q}_{k}$ is given by
$({\underline{\pi}}^{\prime},\underline{w}^{\prime})=R^{-1}({\underline{\pi}},\underline{w})$
where
$\pi^{\prime}_{\ell}=\pi_{r}^{-1},\quad\pi^{\prime}_{r}=\pi_{r}\,\pi_{\ell}\,\pi_{r}^{-1}\quad\text{and}\quad
w^{\prime}_{i,l}=\sqrt{-1}\,w_{\pi_{r}^{-1}(i),r},\quad
w^{\prime}_{i,r}=-\sqrt{-1}\,w_{i,\ell}.$ (12)
Let us remark that $R$ exchanges the role of ${\underline{\lambda}}$ and
${\underline{\tau}}$, more precisely if
$({\underline{\pi}}^{\prime},w^{\prime})=R({\underline{\pi}},w)$ then
$w^{\prime}_{i,\ell}=-\tau_{i,r}+\sqrt{-1}\,\lambda_{i,r}\quad\text{and}\quad
w^{\prime}_{i,r}=\tau_{\pi_{\ell}^{-1}(i),\ell}-\sqrt{-1}\,\lambda_{\pi_{\ell}^{-1}(i),\ell}.$
(13)
So far, for a given admissible quadrilateral $q_{i}$ in a quadrangulation
$Q=(\pi,\underline{w})$ we only considered the forward diagonal
$w_{i,d}=w_{i,d^{+}}=w_{i,l}+w_{\pi_{\ell}(i),r}$ connecting the bottom vertex
to the top one.
###### Definition 2.12.
Let $q_{i}$ be a quadrilateral in a quadrangulation
$Q=({\underline{\pi}},\underline{w})$. The _backward diagonal_ $w_{i,d^{-}}$
of $q$ is the diagonal joining the left vertex to the right vertex of $q_{i}$.
The definition is given so that the forward diagonal $w_{i,d^{+}}^{\prime}$ of
the quadrilateral $q^{\prime}_{i}$ in $Q^{\prime}=RQ$ is obtained by rotating
the backward diagonal of $q_{\pi_{\ell}^{-1}(i)}$, that is
$w^{\prime}_{i,d^{+}}=\sqrt{-1}\,w_{\pi_{\ell}^{-1}(i),d^{-}}=\sqrt{-1}\,(w_{\pi_{\ell}^{-1}(i),r}-w_{\pi_{\ell}^{-1}(i),\ell}).$
(14)
It is clear geometrically that left (right) staircases becomes right (left)
staircases after rotation. More precisely, if $c$ is a left cycle of
${\underline{\pi}}=(\pi_{\ell},\pi_{r})$, then it is also a right cycle of
${\underline{\pi}}^{\prime}=(\pi_{\ell}\,\pi_{r}\,\pi_{\ell}^{-1},\pi_{\ell}^{-1})$.
On the other hand, if $c=\\{i_{1},\dots,i_{n}\\}$ is a right cycle of
${\underline{\pi}}=(\pi_{\ell},\pi_{r})$, then
$\pi_{\ell}\,c:=\\{\pi_{\ell}(i_{1}),\dots,\pi_{\ell}(i_{n})\\}$ is a left
cycle of
${\underline{\pi}}^{\prime}=(\pi_{\ell}\,\pi_{r}\,\pi_{\ell}^{-1},\pi_{\ell}^{-1})$.
Thus, let us define
$c^{\prime}:=\left\\{\begin{array}[]{ll}c&\text{if $c$ is a cycle of
$\pi_{\ell}$,}\\\ \pi_{\ell}\,c&\text{if $c$ is a cycle of
$\pi_{r}$.}\end{array}\right.$ (15)
Then, if $S_{c}$ is a right (resp. left) staircase for $Q$, it corresponds to
the left (resp. right) staircase $S_{c}^{\prime}$ for $Q^{\prime}=RQ$ under
the action of $R$, that is, $S_{c^{\prime}}$ is the union of the rotated
quadrilaterals $\sqrt{-1}q$, $q\in Q$.
Recall that $\Delta_{{\underline{\pi}},c}$ is defined so that we can perform a
staircase move in $S_{c}$ exactly when
${\underline{\lambda}}\in\Delta_{{\underline{\pi}},c}$, i.e. $S_{c}$ is well
slanted (see (11)). Similarly, we define the set of parameters such that the
rotated staircase $S_{c^{\prime}}$ for $Q^{\prime}=RQ$ (where $c^{\prime}$ is
given by (15)) is well slanted so that we can perform a move in $Q^{\prime}$.
if $c^{\prime}$ is a cycle in ${\underline{\pi}}^{\prime}$, It is clear that
this set depends only on ${\underline{\tau}}$ since the forward diagonal
$w^{\prime}_{i,d^{+}}$ of the quadrilateral $q^{\prime}_{i}$ is obtained by
rotating the backward diagonal of the quadrilateral $q_{\pi_{\ell}^{-1}(i)}$
of $Q$ and this exchanges the role of lengths and suspension datas (see
Equations (14) and (13)). Thus this set of parameters is
$\\{{\underline{\pi}}\\}\times\Delta_{\underline{\pi}}\times\Theta_{{\underline{\pi}},c}$
where
$\Theta_{{\underline{\pi}},c}=\left\\{\begin{array}[]{ll}\\{{\underline{\tau}}\in\Theta_{\underline{\pi}};\
\tau_{i,d^{-}}=\tau_{i,r}-\tau_{i,\ell}<0,\quad i\in c\\}&\text{if $c$ is a
left cycle,}\\\ \\{{\underline{\tau}}\in\Theta_{\underline{\pi}};\
\tau_{i,d^{-}}=\tau_{i,r}-\tau_{i,\ell}>0,\quad i\in c\\}&\text{if $c$ is
right cycle.}\\\ \end{array}\right.$
From the definitions and the exchange in the role of lengths and suspension
datas (see Equation (13)), we also get the following result.
###### Lemma 2.13.
Let ${\underline{\pi}}=(\pi_{\ell},\pi_{r})$ be a combinatorial datum of a
quadrangulation $Q\in\mathcal{Q}_{k}$ and let
${\underline{\pi}}^{\prime}=(\pi_{\ell}\,\pi_{r}\,\pi_{\ell}^{-1},\pi_{\ell}^{-1})$
be the combinatorial datum of $Q^{\prime}=RQ$. Let $c$ be a cycle of
${\underline{\pi}}$ and let $c^{\prime}$ be the corresponding cycle in
${\underline{\pi}}^{\prime}$ given by (15). Then
* (i)
$R$ maps
$\\{{\underline{\pi}}\\}\times\Delta_{\underline{\pi}}\times\Theta_{{\underline{\pi}},c}$
bijectively onto
$\\{{\underline{\pi}}^{\prime}\\}\times\Delta_{{\underline{\pi}}^{\prime},c^{\prime}}\times\Theta_{{\underline{\pi}}^{\prime}}$,
* (ii)
$R$ maps
$\\{{\underline{\pi}}\\}\times\Delta_{{\underline{\pi}},c}\times\Theta_{\underline{\pi}}$
bijectively onto
$\\{{\underline{\pi}}^{\prime}\\}\times\Delta_{{\underline{\pi}}^{\prime}}\times\Theta_{{\underline{\pi}}^{\prime},c^{\prime}}$.
###### Theorem 2.14 (self-duality).
Let $\pi$ be a permutation, let $c$ be a cycle of $\pi$. Then
$\widehat{m}_{{\underline{\pi}},c}:\\{{\underline{\pi}}\\}\times\Delta_{{\underline{\pi}},c}\times\Theta_{\underline{\pi}}\rightarrow\\{c\cdot{\underline{\pi}}\\}\times\Delta_{c\cdot{\underline{\pi}}}\times\Theta_{c\cdot{\underline{\pi}},c}$
(16)
is a bijection. Moreover, if $c$ is a cycle of $\pi_{\ell}$ the inverse is
given by
$\widehat{m}_{{\underline{\pi}},c}^{-1}=R^{-1}\circ\widehat{m}_{{\underline{\pi}}^{\prime},c^{\prime}}\circ
R,$ (17)
where ${\underline{\pi}}^{\prime}=R\cdot{\underline{\pi}}$ and $c^{\prime}$ is
given by (15).
The proof of the Theorem, which follows from the definitions and the Lemma, is
given here below. Equation (17) is a formulation of the _self-duality
property_ of staircase moves (we refer for example to Schwheiger [38] for the
definition of duality). Geometrically it simply means that the inverse of a
left (respectively right) staircase move is given by a right (respectively
left) staircase move in the rotated staircase.
Let us explain in which sense the bijection in (16) shows that there is a
_loss of memory_ phenomenon (or Markov property). The space of
quadrangulations $\mathcal{Q}_{k}$ projects on the corresponding space of
bipartite IETs, which is given by
$\\{({\underline{\pi}},{\underline{\lambda}});\,{\underline{\pi}}\in\mathcal{G},\
{\underline{\lambda}}\in\Delta_{\underline{\pi}}\\}$. Let
$m_{{\underline{\pi}},c}$ be the projection of
$\widehat{m}_{{\underline{\pi}},c}$ on the bipartite IETs space. In other
words, $m_{{\underline{\pi}},c}$ is the map defined on
$\\{{\underline{\pi}}\\}\times\Delta_{{\underline{\pi}},c}$ which sends
$({\underline{\pi}},{\underline{\lambda}})$ to
$m_{{\underline{\pi}},c}({\underline{\pi}},{\underline{\lambda}})=(c\cdot{\underline{\pi}},A_{{\underline{\pi}},c}\,{\underline{\lambda}})$.
###### Corollary 2.15 (Markov property).
The map
$m_{{\underline{\pi}},c}:\\{{\underline{\pi}}\\}\times\Delta_{{\underline{\pi}},c}\to\\{c\cdot{\underline{\pi}}^{\prime}\\}\times\Delta_{c\cdot{\underline{\pi}}^{\prime}}$
is a bijection.
The corollary shows that given any (oriented) path in the graph $\mathcal{G}$,
which corresponds to a sequence of staircase moves, there exists a
quadrangulation $Q=({\underline{\pi}},\underline{w})$ from which we can apply
this sequence of moves. In this sense, staircase moves have a Markov
structure. For the greedy algorithm, one can use the sets
$\Delta_{{\underline{\pi}},c}$ to define a natural Markov partition on
$\mathcal{Q}_{k}$ that is a finite partition $\mathcal{P}$ of
$\mathcal{Q}_{k}$ so that the image of each atom of $\mathcal{P}$ is union of
atoms. As shown in [11], this is not the case for the left/right algorithm for
which we should keep in memory one step of the history.
###### Proof of Theorem 2.14.
By (13) and by definition of $\Delta_{\pi,c}$ and $\Theta_{c\cdot\pi,c}$ it is
clear that the image of the map $\widehat{m}_{{\underline{\pi}},c}$ is
$\\{c\cdot{\underline{\pi}}\\}\times\Delta_{c\cdot{\underline{\pi}}}\times\Theta_{c\cdot{\underline{\pi}},c}$.
Using also Lemma 2.13, it follows that all compositions in the statement make
sense.
Let $c$ be a cycle of $\pi_{r}$ and $c^{\prime}$ be the cycle associated to
$c$ by (15). Let us denote by
$({\underline{\pi}}^{\prime},\underline{w}^{\prime})=R({\underline{\pi}},\underline{w})$,
$(\pi^{\prime\prime},\underline{w}^{\prime\prime})=\widehat{m}_{{\underline{\pi}}^{\prime},c^{\prime}}\,({\underline{\pi}}^{\prime},\underline{w}^{\prime})$
and
$({\underline{\pi}}^{\prime\prime\prime},\underline{w}^{\prime\prime\prime})=R^{-1}({\underline{\pi}}^{\prime\prime},\underline{w}^{\prime\prime})$.
We first compute $\pi^{\prime},\pi^{\prime\prime}$ and
$\pi^{\prime\prime\prime}$ to get the action of the composition
$R^{-1}\widehat{m}_{\pi^{\prime},c^{\prime}}R$ on combinatorial data. By
formulas (12) for $R$, we have that
$\pi^{\prime}_{\ell}=\pi_{\ell}\pi_{r}\pi_{\ell}^{-1}\quad\text{and}\quad\pi^{\prime}_{r}=\pi_{\ell}^{-1}.$
Now recall that $c^{\prime}$ is associated to a left slanted staircase in the
rotated quadrangulation $Q^{\prime}=RQ$, so
$\widehat{m}_{{\underline{\pi}}^{\prime},c^{\prime}}$ is a left staircase
move. Thus, by definition of a left staircase move we get that
$\pi^{\prime\prime}=c^{\prime}\cdot\pi^{\prime}$ is given by
$\pi^{\prime\prime}_{\ell}=\pi^{\prime}_{\ell}=\pi_{\ell}\,\pi_{r}\,\pi_{\ell}^{-1}\quad\text{and}\quad\pi^{\prime\prime}_{r}(i)=\left\\{\begin{array}[]{ll}\pi^{\prime}_{r}\,\pi^{\prime}_{\ell}(i)&\text{if
$i\in c^{\prime}$,}\\\
\pi^{\prime}_{r}(i)&\text{otherwise}\end{array}\right.=\left\\{\begin{array}[]{ll}\pi_{r}\,\pi_{\ell}^{-1}&\text{if
$i\in c^{\prime}$,}\\\ \pi_{\ell}^{-1}&\text{otherwise.}\end{array}\right.$
Finally, by formulas (12) for $R^{-1}$, we have that
$\pi^{\prime\prime\prime}_{\ell}=(\pi^{\prime\prime}_{r})^{-1}=\left\\{\begin{array}[]{ll}\pi_{\ell}\pi_{r}^{-1}(i)&\text{if
$\pi_{\ell}\,\pi_{r}^{-1}(i)\in c^{\prime}$,}\\\
\pi_{\ell}(i)&\text{otherwise}\end{array}\right.\quad\text{and}\quad\pi^{\prime\prime\prime}_{r}=\pi^{\prime\prime}_{r}\pi^{\prime\prime}_{\ell}(\pi^{\prime\prime}_{r})^{-1}.$
By the definition of $c^{\prime}$, te condition
$\pi_{\ell}\,\pi_{r}^{-1}(i)\in c^{\prime}$ is equivalent to
$\pi_{r}^{-1}(i)\in c$ and since $c$ is a right cycle, it is also equivalent
to $i\in c$. Now, to compute the expression of $\pi^{\prime\prime\prime}_{r}$,
let us consider separately the cases $i\in c$ and $i\notin c$. As shown above,
if $i\in c$ we also have $\pi_{\ell}\,\pi_{r}^{-1}(i)\in c^{\prime}$ and thus
$(\pi^{\prime\prime}_{r})^{-1}(i)=\pi_{\ell}\pi_{r}^{-1}(i)$. Hence
$\pi^{\prime\prime}_{\ell}(\pi^{\prime\prime}_{r})^{-1}(i)=\pi_{\ell}(i)$.
Since, when $c$ is a right cycle, $c^{\prime}=\pi_{\ell}\,c$ we then have that
$\pi_{\ell}(i)\in c^{\prime}$ and hence, by the above expression for
$\pi_{r}^{\prime\prime}$ we get
$\pi^{\prime\prime}_{r}\,\pi^{\prime\prime}_{\ell}\,(\pi^{\prime\prime}_{r})^{-1}(i)=\pi_{r}(i).$
Now consider the case $i\notin c$. We get
$\pi^{\prime\prime}_{\ell}(\pi^{\prime\prime}_{r})^{-1}(i)=\pi_{\ell}\pi_{r}(i)$.
Now $c$ and its complement are stable under $\pi_{r}$ and hence,
$\pi_{\ell}\pi_{r}(i)\not\in c^{\prime}$. Hence, we obtain
$\pi^{\prime\prime}_{r}\pi^{\prime\prime}_{\ell}(\pi^{\prime\prime}_{r})^{-1}(i)=\pi_{r}(i).$
Thus, in both cases $\pi^{\prime\prime\prime}_{r}=\pi_{r}$. One can verify
from the formulas for the combinatorial datum of a right staricase move that
$c\cdot\pi^{\prime\prime\prime}=\pi$. This show that
$\pi^{\prime\prime\prime}$ is the combinatorial datum of the inverse staircase
move in $S_{c}$.
Let us now compute the wedges $w^{\prime}$, $w^{\prime\prime}$ and
$w^{\prime\prime\prime}$. From the formulas for $R$ and a left staircase move
in $S_{c^{\prime}}$ we get
$\begin{array}[]{ll}w^{\prime}_{i,\ell}=\sqrt{-1}\,w_{i,r}&w^{\prime}_{i,r}=-\sqrt{-1}\,w_{\pi_{\ell}^{-1}(i),\ell}\\\
w^{\prime\prime}_{i,\ell}=w^{\prime}_{i,\ell}=\sqrt{-1}\,w_{i,r}&w^{\prime\prime}_{i,r}=\left\\{\begin{array}[]{ll}w^{\prime}_{i,\ell}+w^{\prime}_{\pi_{\ell}^{\prime}(i),r}&\text{if
$i\in c^{\prime}$,}\\\
w^{\prime}_{i,r}&\text{otherwise.}\end{array}\right.\end{array}$
Thus, since $\pi_{\ell}^{\prime}=\pi_{\ell}\,\pi_{r}\,\pi_{\ell}^{-1}$,
combining the above expressions we get that
$w^{\prime\prime}_{i,r}=\left\\{\begin{array}[]{ll}\sqrt{-1}\,w_{i,r}-\sqrt{-1}\,w_{\pi_{r}\,\pi_{\ell}^{-1}(i),\ell}&\text{if
$i\in c^{\prime}$,}\\\
-\sqrt{-1}\,w_{\pi_{\ell}^{-1}(i),\ell}&\text{otherwise.}\end{array}\right.$
From the formula for $R^{-1}$ we then get
$w^{\prime\prime\prime}_{i,\ell}=\sqrt{-1}w^{\prime\prime}_{(\pi_{r}^{\prime\prime})^{-1}(i),r},\qquad
w^{\prime\prime\prime}_{i,r}=-\sqrt{-1}w^{\prime\prime}_{i,\ell}=-\sqrt{-1}\,(\sqrt{-1}\,w_{i,r})=w_{i,r}.$
To compute $w^{\prime\prime\prime}_{i,\ell}$, let us use the expression
computed above for $(\pi_{r}^{\prime\prime})^{-1}$ and consider separately two
cases. If $i\in c$, then
$(\pi_{r}^{\prime\prime})^{-1}(i)=\pi_{\ell}\,\pi_{r}^{-1}(i)$ which belongs
to $c^{\prime}$ (since $c$ is invariant under $\pi_{r}$ and by definition of
$c^{\prime}$). Thus, for $i\in c$ we get that
$w^{\prime\prime\prime}_{i,\ell}=\sqrt{-1}\,w^{\prime\prime}_{\pi_{\ell}\,\pi_{r}^{-1}(i),r}=-w_{\pi_{\ell}\,\pi_{r}^{-1}(i),r}+w_{\pi_{r}\,\pi_{\ell}^{-1}\pi_{\ell}\,\pi_{r}^{-1}(i),\ell}=w_{i,\ell}-w_{\pi_{\ell}\,\pi_{r}^{-1}(i),r}.$
On the other hand, if $i\notin c$,
$(\pi_{r}^{\prime\prime})^{-1}(i)=\pi_{\ell}(i)$, which is not in
$c^{\prime}$, thus
$w^{\prime\prime\prime}_{i,\ell}=\sqrt{-1}w^{\prime\prime}_{\pi_{\ell}(i),r}=w_{i,r}.$
One can check that this is indeed the expression for the wedges of the inverse
of the staircase move in $S_{c}$. The case when $c$ is a cycle of $\pi_{\ell}$
is analogous.
∎
## 3 Existence of quadrangulations and staircase moves
In this section we prove the existence of quadrangulations for any surface
that belongs to an hyperelliptic component of a stratum (Theorem 1.8) and the
existence of well slanted staircases for any of these quadrangulations
(Theorem 1.9). We first start with a precise definition of hyperelliptic
components of strata in terms of double cover of quadratic differentials.
### 3.1 Quadrangulations in hyperelliptic components
We have already seen in §1.2.1 that translation surfaces can be constructed by
gluing polygons or equivalently by assigning a non-zero Abelian differential
on a Riemann surface. We first describe a more general construction which
produces Riemann surfaces with quadratic differentials. We then define
orientation covers of quadratic differentials, that are a particular case of
translation surfaces. Then we define hyperelliptic components as the set of
orientation covers of quadratic differentials that belong to some fixed
stratum.
#### 3.1.1 Hyperelliptic components of strata of translation surfaces
While a translation surface is obtained by gluing polygons by translations, a
quadratic differential can be obtained by gluing polygons by translations and
rotation by $180$ degrees.
Let $P_{i}\subset\mathbb{C}$ be a collection of polygons whose edges are
identified into pairs such that:
1. 1.
either the two edges in the pair are parallel with opposite normal vector
(with respect to their polygons) and we identify the two edges by the unique
translation that sends one to the other,
2. 2.
or the two edges are parallel but have the same normal vector and we identify
them under the unique rotation by 180 degrees (ie a map of the form
$z\mapsto-z+c$) that maps one edge to the other.
The quotient of $\cup P_{i}$ by the identifications of the edges is a surface
$X$ which carries the structure of a Riemman surface with a quadratic
differential $q$ (which is induced from the form $dz^{2}$ on the polygons). If
in this construction all pairs are of the first form then the construction
reduces to the one described in §1.2.1 and $X$ is a translation surface, or,
equivalently, a Riemann surface $X$ which carries an Abelian differential
$\omega$. Let $\Sigma\subset X$ denote as before the singularity set
corresponding to the images of the vertices of the polygons. A quadratic
differentials has conical singularities with angles of the form $k\pi$ with
$k$ integer (instead of $2\pi k$ as in the case of Abelian differentials).
Moreover, while an Abelian differential determines on $X\backslash\Sigma$ a
well defined notion of lines in direction $\theta\in S^{1}$, a quadratic
differential only determines a notion of (non-oriented) lines in direction
$\theta\in\mathbb{P}^{1}\mathbb{R}$.
We define two quadratic differentials $(X,q)$ and $(X^{\prime},q^{\prime})$ to
be isomorphic if there exists an homeomorphism $X\rightarrow X^{\prime}$ such
that $q=f^{*}q^{\prime}$. We can also define this notion of isomorphism as cut
and paste operations on polygons, similarly to the definition given in §1.2.1
for translation surfaces. We denote by $\mathcal{Q}(k_{1}-2,\ldots,k_{n}-2)$
the equivalence class of quadratic differentials with conical singularities of
angles $\pi k_{1},\ldots,\pi k_{n}$. The number $k_{i}-2$ correspond to the
degree of the quadratic differential as $q$ can be written locally as
$z^{k_{i}-2}dz^{2}$ around a singularity with conical angle $\pi k_{i}$. Note
that we have the topological restriction that $\sum_{i=1}^{n}k_{i}=4g-4+2n$
where $g$ is the genus of the surface. If there are $m_{i}$ singularities with
total angle $\pi k_{i}$ we use the notation
$\mathcal{Q}((k_{1}-2)^{m_{1}},\ldots,(k_{n}-2)^{m_{n}})$.
Let $(X,q)$ be a quadratic differential. We associate to $q$ its canonical
_orientation cover_ : it is the Abelian differential $(\tilde{X},\omega)$,
unique up to isomorphism, such that there exists a degree $2$ map
$\pi:\tilde{X}\rightarrow X$ and such that $\pi^{*}q=\omega^{2}$. The stratum
in which $\tilde{X}$ belongs is easily computed as follows: each singularity
of angle $\pi k_{i}$ with $k_{i}$ even is not ramified and gives two
singularities on $\tilde{X}$ of angle $\pi k_{i}$; each singularity of angle
$k_{i}$ with $k_{i}$ odd is ramified and gives a singularity on $\tilde{X}$ of
angle $2\pi k_{i}$. As an example, the orientation covers of surfaces in
$\mathcal{Q}(2,3^{2})$ belong to $\mathcal{H}(1^{2},4)$. Because a degree two
cover is always normal, an orientation cover always comes with an involution
whose quotient is the corresponding quadratic differential.
When a quadratic differential varies in its stratum, its orientation cover
varies in a connected component of the corresponding stratum of Abelian
differentials. When the stratum of quadratic differentials is a sphere (i.e. a
stratum of the form $\mathcal{Q}(k_{1}-2,\ldots,k_{n}-2)$ with
$k_{1}+\ldots+k_{n}=2n-4$) such locus is called a _hyperelliptic locus_. In
this case the involution is an _hyperelliptic involution_. The points of an
hyperelliptic surface which are fixed by the hyperelliptic involution are
called _Weierstrass points_. They might be conical singularities or regular
points. In the latter case, they projects down to conical singularities of
angle $\pi$ on the sphere that are called _poles_ (because they correspond to
singularities of the form $z^{-1}dz^{2}$ for the quadratic differential).
Because of Hurwitz formula, a hyperelliptic surface of genus $g$ has $2g+2$
Weirstrass points.
In most cases, _hyperelliptic loci_ have positive codimension in the
corresponding stratum of Abelian differentials, but an infinite family of
hyperelliptic loci have full dimension and form connected components.
###### Theorem 3.1 ([29], section 2.1 p.5–7).
In each stratum $\mathcal{H}(2g-2)$ (respectively $\mathcal{H}(g-1,g-1)$) the
hyperelliptic locus built as the orientation cover of quadratic differentials
in $\mathcal{Q}(k-2,-1^{k+2})$ for $k=2g-1$ (resp. $k=2g$) forms a connected
component. These are the only hypelliptic loci that form connected components
of stratum.
Recall from the Introduction that we denote by $\mathcal{C}^{hyp}(k)$ the
hyperelliptic component of $\mathcal{H}(k-1)$ if $k$ is odd or of
$\mathcal{H}(k/2-1,k/2-1)$ if $k$ is even. Surfaces in $\mathcal{C}^{hyp}(k)$
have total conical angle $2k\pi$ and hence any quadrangulation on them is made
by $k$ quadrilaterals.
#### 3.1.2 Two geometric results in hyperelliptic components
Using the description of surfaces in an hyperelliptic component
$\mathcal{C}^{hyp}(k)$ as double covers of quadratic differentials in the
stratum $\mathcal{Q}(k-2,-1^{k+2})$ of quadratic differentials, we prove two
important results. The first one shows that a quadrangulation of a surface in
a hyperelliptic component of a stratum is always preserved by the
hyperelliptic involution. The second one is a cut and paste construction that
will be used in some of the following proofs.
###### Lemma 3.2.
Let $Q$ be a quadrangulation of a surface $X$ in a hyperelliptic component
$\mathcal{C}^{hyp}(k)$.
1. 1.
Each staircase for $Q$ is fixed (as a set) by the hyperelliptic involution of
$X$.
2. 2.
If $q\in Q$ is a quadrilateral then its image under the hyperelliptic
involution is another quadrilateral that belongs to the same left and right
staircases for $Q$ to which $q$ belongs.
###### Proof.
Let us first show that a quadrangulation can be continuously deformed in such
way that staircases become metric cylinders. We then prove the result when all
staircases are metric cylinders. Finally, we show that the property for the
latter is preserved under deformation in the component $\mathcal{C}^{hyp}(k)$
and hence holds for all surfaces in that component.
Let $X$ be a surface in $\mathcal{C}^{hyp}(k)$ and $Q=Q^{(0)}$ be a
quadrangulation of $X$. Let us label its quadrilaterals and denote them by
$q_{1},\dots,q_{k}$. Let ${\underline{\pi}}=(\pi_{\ell},\pi_{r})$ be its
combinatorial datum and $\underline{w}=\underline{w}^{(0)}$ its length datum,
so that $Q=({\underline{\pi}},\underline{w})$. Let us introduce the length
datum $w^{(1)}_{i,\ell}=-1+\sqrt{-1}$ and $w^{(1)}_{i,r}=1+\sqrt{-1}$ for all
$i=1,\ldots,k$. Remark that the quadrangulation
$Q^{(1)}=({\underline{\pi}},\underline{w}^{(1)})$ is a quadrangulation made by
squares whose sides have length $\sqrt{2}$ and hence staircases are metric
cylinders. Consider the straight line in the parameter space of length data
that goes from $\underline{w}^{(0)}$ to $\underline{w}^{(1)}$ given by
$\underline{w}^{(t)}=(1-t)\underline{w}^{(0)}+t\underline{w}^{(1)}$. Since
both the train-track relations and the positivity conditions
($\lambda_{i,\ell}<0<\lambda_{i,r}$ and $\tau_{i,\ell},\tau_{i,r}>0$) are
convex, $\underline{w}^{(t)}$ is a valid length datum for ${\underline{\pi}}$
for all $0\leq t\leq 1$. We hence get a path of quadrangulations
$Q^{(t)}=({\underline{\pi}},\underline{w}^{(t)})$ and a continuous path
$X^{(t)}$ of translation surfaces.
We first claim that the hyperelliptic involution of $X^{(1)}$ maps each
quadrilateral $q^{(1)}_{i}$ in $Q^{(1)}$ to another quadrilateral of $Q^{(1)}$
reversing the orientation. Indeed, since $X^{(1)}$ is made by squares with
side length $\sqrt{2}$, the saddle connections of length $\sqrt{2}$ on
$X^{(1)}$ are exactly the sides of $Q^{(1)}$. Since the hyperelliptic
involution preserves the flat metric of $X^{(1)}$, it must preserve this set.
Thus, each quadrilateral $q^{(1)}_{i}$ of $Q^{(1)}$ is sent, reversing the
orientation, to another quadrilateral $q^{(1)}_{\iota(i)}$ where $\iota$ is an
involution of $\\{1,\ldots,k\\}$.
Now we claim that the map $\iota$ actually preserves staircases, that is $i$
and $\iota(i)$ belongs to the same cycles of both $\pi_{\ell}$ and $\pi_{r}$.
Let consider a surface $X$ in $\mathcal{Q}(k-2,-1^{k+2})$ and a maximal
cylinder $C$ in it. Because it is a sphere, each closed curve separates the
surface into two connected components. Now the circumference of a cylinder is
a closed curve so the zero of degree $k$ in $Y$ belongs to only one side of
the cylinder. The other side contains only poles and hence it has to contain
two poles. If we lift such cylinder to the corresponding hyperelliptic
component it consists of one cylinder which contains two Weirstrass points in
its middle. This proves that any cylinder in any surface that belongs to the
hyperelliptic component is fixed (as a set) by the hyperelliptic involution.
Hence the conclusion of the Lemma holds for the quadrangulation $Q^{(1)}$ of
$X^{(1)}$.
Now it remains to deduce that the hyperelliptic involution on $X^{(t)}$ for
$t\in[0,1]$ also sends the quadrilateral $i$ to the quadrilateral
$q_{\iota(i)}$ reversing the orientation. Heuristically, this is because the
quadrangulations $Q^{(t)}$ of $X^{(t)}$, $t\in[0,1]$, are obtained by a
continuous deformation and the hyperelliptic involution is continuous on
$\mathcal{C}^{hyp}(k)$. We warn the reader that it makes no sense to speak of
continuity of the hyperelliptic involution on $\mathcal{C}^{hyp}(k)$. We need
to consider the so called universal curve on $\mathcal{C}^{hyp}(k)$, that is
the set of equivalence class $(X,x)$ where $X\in\mathcal{C}^{hyp}(k)$ and
$x\in X$. This universal curve is also a connected component of a stratum
(with a point with conical angle $2\pi$). The hyperelliptic involution acts on
the universal curve by action on the second coordinate and is continuous on
it. As it is an isometry, the hyperelliptic involution sends saddle
connections to saddle connections. We would like to argue that the
hyperelliptic involution is continuous on the set of saddle connection, but
the problem is that the map $X\mapsto\Gamma(X)$ which to a surface associate
its set of saddle connections (seen as a discrete subset of
$(\mathbb{R}\times\mathbb{R}_{+})^{k}$) is not continuous, as saddle
connections may appear or disappear. Nevertheless, if $X^{(t)}$ is a
continuous path of surfaces and $\gamma^{(t)}:[0,1]\rightarrow X^{(t)}$ and
$\eta^{(t)}:[0,1]\rightarrow X^{(t)}$ are such that
* •
the maps $(s,t)\mapsto\gamma^{(t)}(s)$ and $(s,t)\mapsto\eta^{(t)}(s)$ are
continuous from $[0,1]\times[0,1]$ to $X$,
* •
for each $t$, $\gamma^{(t)}$ and $\eta^{(t)}$ are saddle connections
parametrized with constant speed,
* •
at time $t=0$, the saddle connections coincide, i.e. we have
$\gamma^{(0)}(s)=\iota\circ\eta^{(0)}(s)$,
then for all time $t\in[0,1]$, we have
$\gamma^{(t)}(s)=\iota\circ\eta^{(t)}(s)$. This simply follows from a
continuity argument. We may apply this to our saddle connections that form the
sides of our quadrangulations, namely $\gamma^{(t)}(s)=sv$ where $v$ is
thought as an element of $\mathbb{C}$. ∎
###### Lemma 3.3.
Let $X$ be a surface in a hyperelliptic component $\mathcal{C}^{hyp}(k)$ and
let $s:X\rightarrow X$ be the hyperelliptic involution. Let $\gamma$ be a
saddle connection in $X$ that is not fixed by the hyperelliptic involution.
Then $X\backslash(\Sigma\cup\gamma\cup s\gamma)$ has two connected components
both of them having $\gamma$ and $s\gamma$ on their boundary. Let $X_{1}$ and
$X_{2}$ be obtained from these two connected components by identifying
$\gamma$ and $s\gamma$ by translation. Then $X_{1}$ and $X_{2}$ are (non
empty) translation surfaces in hyperelliptic components. Furthermore, if
$k_{1}\geq 1$ and $k_{2}\geq 1$ are such that
$X_{1}\in\mathcal{C}^{hyp}(k_{1})$ and $X_{2}\in\mathcal{C}^{hyp}(k_{2})$, we
have $k=k_{1}+k_{2}$.
###### Proof.
Let $Y$ be the quotient of $X$ under the hyperelliptic involution. The image
of $\gamma$ (which is also the image of $s\gamma$) in $Y$ is a segment that
does not contain a pole in its interior (this is because a saddle connection
in $X$ is preserved under the hyperelliptic involution if and only if it
contains a Weirstrass point in its interior). We obtain a closed curve on the
sphere which is a loop (both ends are the zero of the quadratic differential)
and hence separates the sphere into two components whose boundaries each
consists of a copy of the segment image of $\gamma$. Let us now add a pole in
the middle point of each segment, hence defining a new quadratic differential
on each surface. Taking the double covers of these new quadratic differentials
we obtain two surfaces $X_{1}$ and $X_{2}$ as in the statement. The relation
$k=k_{1}+k_{2}$ follows from computing total conical angles. ∎
#### 3.1.3 Triangulations on the sphere and Ferenczi-Zamboni trees of
relations
From Lemma 3.2, we know that a quadrangulation of a surface that belongs to a
hyperelliptic component of a stratum is necessarily fixed by the hyperelliptic
involution of the surface. In particular, it makes sense to consider the
quotient of the quadrangulation on the sphere. We see in this section that
this quotient is naturally a triangulation that it is intimately related to
the so called _trees of relations_ that appear in work by Ferenczi and
Zamboni, see [18].
Let $q$ be a quadratic differential on the sphere $\mathbb{C}\mathbb{P}^{1}$
which belongs to $\mathcal{Q}(k-2,-1^{k+2})$ and let $z_{0}$ denotes the point
of $\mathbb{C}\mathbb{P}^{1}$ at which $q$ has the zero of degree $k-2$. We
call a _triangle_ on $(\mathbb{C}\mathbb{P}^{1},q)$ an open embedded triangle
in $(\mathbb{C}\mathbb{P}^{1},q)$ whose boundary consists of saddle
connections between $z_{0}$ and itself that may pass through one pole. Notice
that, since the conical angle at a pole is $\pi$, an edge which passes through
a pole actually consists of two copies of the same segment. A _triangulation_
of $(\mathbb{C}\mathbb{P}^{1},q)$ is a set of triangles on $q$ such that their
interiors have empty intersection and their union is the whole
$\mathbb{C}\mathbb{P}^{1}$. An example of a triangulation is shown in Figure
14(b).
(a) a quadrangulation in $\mathcal{C}^{hyp}(5)$ (b) its quotient in
$\mathcal{Q}(3,-1^{7})$
(c) its tree of relations
Figure 14: from a quadrangulation of a surface in $\mathcal{C}^{hyp}(5)$ to
the tree of relations
Given a triangulation $T$ on the sphere, we canonically associate its _dual
graph_ $G_{T}$. The vertices $v_{t}$ are the triangles $t\in T$ and we join
two vertices $v_{t}$ and $v_{t^{\prime}}$ by an edge if the corresponding
triangles $t$ and $t^{\prime}$ share an edge which has no pole on it. An
example of such graph is given in Figure 14(c).
###### Lemma 3.4.
Let $G_{T}$ be the dual graph associated to the triangulation $T$ of a
quadratic differential $(\mathbb{C}\mathbb{P}^{1},q)$ in a stratum
$\mathcal{Q}(k-2,-1^{k+2})$. Then $G_{T}$ is a tree.
###### Proof.
The connectedness of $G_{T}$ comes from the connectedness of
$\mathbb{C}\mathbb{P}^{1}$. Hence, to prove that it is a tree it is enough to
show that the number of edges of $G_{T}$ is its number of vertices minus one.
By definition, the vertices are the triangles of $T$ so there are $k$ of them.
Now, because it is a triangulation and there are $k+2$ poles the number of
edges is $(3k-(k+2))/2=k-1$. ∎
Recall that quadrangulations of surfaces in $\mathcal{C}^{hyp}(k)$ have by
definition the additional property that the quadrilaterals are admissible
(recall Definition 1.4). In the quotient, we can see this property as a
compatibility condition on the triangles. More precisely, in any triangle
there is exactly one vertex such that the vertical segment emanating from that
vertex is contained in the triangle (as illustrated in Figure 15(a), see also
Figure 14(b) for an example). We can then assign labels to each side of a
triangle as follows. Let us consider the unique vertical from a vertex of $t$
which is contained in $t$ and orient it so that it starts from the vertex. We
label $d$ the side opposite to the vertex from which the vertical starts,
which is the unique side crossed by the considered vertical. We then label
$\ell$ and $r$ the other two sides of $t$ (which form a wedge which contains
the considered vertical), so that rotating counterclockwise around the vertex
one sees first the side labelled $r$, then the vertical, then the side
labelled $\ell$, as shown in Figure 15(a).
Let $Q$ be a quadrangulation of a surface in $\mathcal{C}^{hyp}(k)$ and $T$
the triangulation obtained taking its quotient by the hyperelliptic
involution. Then the admissibility of the quadrilaterals in $Q$ implies that
that pairs of sides of triangles of $T$ which are identified carry the same
label, see Figure 15(b) and 15(c). Thus, we can assign labels in
$\\{\ell,r,d\\}$ to the edges of the dual tree $G_{T}$ associated to the
triangulation $T$, by assigning to each edge of $G_{T}$ the common label of
the dual pair of identified triangle edges (see the example in Figure 14(c)).
This labelled tree is called the _tree of relations_ in [18] (we warn the
reader that $\ell$, $r$ and $d$ are respectively replaced in [18] by
$\hat{+}$, $\hat{-}$ and $\hat{=}$). As shown in [18], the tree encodes indeed
the train-track relations for the the length datum $\underline{w}$ of $Q$ as
follows. If the edge of the tree connecting the vertices $i$ to $j$ carries
the label $r$ (resp. $l$), the wedges of the quadrilaterals $q_{i}$ and
$q_{j}$ obtained by double covers of the triangles dual to the vertices $i$
and $j$ are such that $w_{i,r}=w_{j,r}$ (resp. $w_{i,\ell}=w_{j,\ell}$). If
the edge connecting $i$ to $j$ carries the label $d$, the quadrilaterals
$q_{i}$ and $q_{j}$ have parellel isometric diagonals, that is
$w_{i,d}=w_{j,d}$. One can show that this set of equations is equivalent to
the set of train-track relations
$w_{i,\ell}+w_{\pi_{\ell}(i),r}=w_{i,r}+w_{\pi_{r}(i),\ell}$ for $1\leq i\leq
k$ (a sketch is given in [18]).
(a) Labels on triangles (b) admissible gluing
(c) non admissible gluing
Figure 15: labels on triangles and admissibility of configurations
If $Q$ is a labelled quadrangulation, also the triangles of the induced
triangulation $T$ inherit labels $1\leq i\leq k$. More precisely, each
quadrilateral is cut in two triangles by its backward diagonal. Bottom
triangles and top triangles are exchanged by the hyperelliptic involution. Let
us consider a triangle $t$ on the sphere. Its preimage in $Q$ is a union of a
top and a bottom triangle. The bottom one belongs to some $q_{i}$ and we set
$i$ as the label for $t$.
We can then equivalently describe the tree of relations with three involutions
$\sigma_{\ell}$, $\sigma_{r}$ and $\sigma_{d}$ of $\\{1,\ldots,k\\}$. Define
$\sigma_{\ell}$ so that if the triangles $i$ and $j$ share an edge labelled
$\ell$ then $\sigma_{\ell}(i)=j$ and $\sigma_{\ell}(j)=i$. We define similarly
$\sigma_{r}$ and $\sigma_{d}$. We use the notation ${\underline{\sigma}}$ for
the triple $(\sigma_{\ell},\sigma_{r},\sigma_{d})$ and call it the
_combinatorial datum_ of the triangulation $T$. The following Lemma relates
the combinatorial datum ${\underline{\pi}}$ of a quadrangulation $Q$ to the
combinatorial datum ${\underline{\sigma}}$ of the quotient triangulation $T$.
Equivalently, it links the tree of relations $G_{T}$ and the graph $G_{Q}$.
###### Lemma 3.5.
If $T$ is a labelled triangulation with combinatorial datum
${\underline{\sigma}}=(\sigma_{\ell},\sigma_{r},\sigma_{d})$ induced by a
labelled quadrangulation $Q$ of a surface in $\mathcal{C}^{hyp}(k)$ with
combinatorial datum ${\underline{\pi}}=(\pi_{\ell},\pi_{r})$ then
$\sigma_{d}=\iota$ is the action of the hyperelliptic involution on the
quadrilaterals of $Q$ and
$\pi_{\ell}=\sigma_{r}\circ\sigma_{d}\qquad\text{and}\qquad\pi_{r}=\sigma_{\ell}\circ\sigma_{d}.$
In particular $\pi_{\ell}^{-1}=\iota\pi_{\ell}\iota$ and
$\pi_{r}^{-1}=\iota\pi_{r}\iota$.
###### Proof.
By construction, the labels on the sphere are built in such way that
$\sigma_{d}$ corresponds to the action of the hyperelliptic involution. Recall
that quadrilaterals in $Q$ are cut in triangles by the backward diagonals. Now
$\pi_{\ell}$ can be seen on the bottom triangles as first crossing the
diagonal (hence applying $\sigma_{d}$) and then crossing the top left side
which is right slanted (hence applying $\sigma_{r}$). So
$\pi_{\ell}=\sigma_{r}\sigma_{d}$. Reasoning in the same way for $\pi_{r}$ we
get the other formula. ∎
Let us remark that one can show that $\sigma_{\ell}\sigma_{r}\sigma_{d}$ is a
$k$-cycle, since it corresponds geometrically to turning around the
singularity of angle $\pi k$ on the sphere. In [6], it is shown that this
$k$-cycle is a complete invariant that classifies pair of permutations in the
same graph $\mathcal{G}$. Their main result can be rephrased as follows:
###### Theorem 3.6 ([6]).
Let $Q$ be a quadrangulation of a surface in $\mathcal{C}^{hyp}(k)$ and
$T_{Q}$ be the quotient triangulation. Let
${\underline{\pi}}=(\pi_{\ell},\pi_{r})$ and
${\underline{\sigma}}=(\sigma_{\ell},\sigma_{r},\sigma_{d})$ be respectively
the combinatorial datum of $Q$ and $T_{Q}$. Then, the permutation
$\sigma_{\ell}\sigma_{r}\sigma_{d}=\pi_{r}\sigma_{d}\pi_{\ell}$ is a $k$-cycle
which is invariant under the operation of staircase moves
${\underline{\pi}}\mapsto c\cdot{\underline{\pi}}$. Moreover, two
combinatorial data ${\underline{\pi}}$ and ${\underline{\pi}}^{\prime}$ that
correspond to quadrangulations of surfaces in $\mathcal{C}^{hyp}(k)$ can be
joined by a sequence of staircase moves and hence belong to the same graph
$\mathcal{G}=\mathcal{G}({\underline{\pi}})$ if and only if
$\pi_{r}\pi_{\ell}\sigma_{d}=\pi^{\prime}_{r}\pi^{\prime}_{\ell}\sigma_{d}^{\prime}$.
The following corollary of this result is used to show that the rotation
operator $R:\mathcal{Q}_{k}\to\mathcal{Q}_{k}$ defined in § 2.5 is well
defined.
###### Corollary 3.7.
Let ${\underline{\pi}}=(\pi_{\ell},\pi_{r})$ be a combinatorial datum of a
labelled quadrangulation $Q$ in $\mathcal{Q}_{k}$ and let
${\underline{\pi}}^{\prime}=(\pi_{\ell}\,\pi_{r}\,\pi_{\ell}^{-1},\pi_{\ell}^{-1})$.
Then ${\underline{\pi}}^{\prime}$ belongs to $\mathcal{G}(\pi)$.
###### Proof.
Consider the quadrangulation
$Q^{\prime}=RQ=(\underline{w}^{\prime},{\underline{\pi}}^{\prime})$ (recall
Definition 2.11). It follows from the definition of $R$ that if $\iota$
denotes the action of the hyperelliptic involution on the labels of $Q$, the
action $\iota^{\prime}$ of of the hyperelliptic involution on the labels of
$Q^{\prime}$ is given by $\iota^{\prime}=\pi_{\ell}\,\iota\,\pi_{\ell}^{-1}$.
Then, from the definition of $R$ and the equality
$\iota\pi_{\ell}^{-1}=\pi_{\ell}\iota$ which follows from from Lemma 3.5, one
has
$\pi^{\prime}_{\ell}\,\pi^{\prime}_{r}\,\iota^{\prime}=(\pi_{\ell}\,\pi_{r}\,\pi_{\ell}^{-1})\,\pi_{\ell}^{-1}\,(\pi_{\ell}\iota\,\pi_{\ell}^{-1})=\pi_{\ell}\pi_{r}\pi_{\ell}^{-1}(\iota\pi_{\ell}^{-1})=\pi_{\ell}\pi_{r}\pi_{\ell}^{-1}\pi_{\ell}\iota=\pi_{\ell}\pi_{r}\iota.$
This shows that ${\underline{\pi}}^{\prime}\in\mathcal{G}({\underline{\pi}})$
by Theorem 3.6, remarking that, with the notation in the Theorem, we have
$\iota=\sigma_{d}$ and $\iota^{\prime}=\sigma_{d}^{\prime}$ by Lemma 3.5. ∎
We remark finally that it is possible to define an operation on trees of
relations (see [6] or [34]) that corresponds to a combinatorial staircase
move, that is to the map which sends ${\underline{\pi}}\mapsto
c\cdot{\underline{\pi}}$ (see the definitions in (7) and (9)). R. Marsh and S.
Schroll in [34] generalize these operations to trees with $k$ labels on edges
(here we have $k=3$ labels, namely $\ell$, $r$ and $d$) and show a link with
cluster algebra combinatorics. They intepret moves on trees as changes of
diagonals in $k$-angulations of polygons. For $k=3$, their triangulations are
a combinatorial version of the metric triangulations of the sphere that we
described above.
### 3.2 Existence of quadrangulations, proof of Theorem 1.8
We now prove that for any surface $X\in\mathcal{C}^{hyp}(k)$ there exists
quadrangulations (Theorem 1.8). Before proceeding to the proof, we state and
prove two lemmas that are valid for any translation surface, not necessarily
in an hyperelliptic component. The first one is about existence of wedges and
the second one about existence of admissible quadrilaterals.
###### Lemma 3.8.
Let $X$ be a translation surface which has no horizontal and no vertical
saddle connections. Then in any bundle of $X$ there are infinitely many left
and right best approximations.
Moreover, for any bundle $\Gamma_{i}$ of $X$ we have
$\min\ \\{\operatorname{Im}(v);\ \text{$v\in\Gamma_{i}$ is a best
approximation and
$|\operatorname{Re}(v)|<r$}\\}<\frac{\operatorname{Area}(X)}{r}.$
Given a best approximation $v$, the quantity
$|\operatorname{Re}(v)|\operatorname{Im}(v)$, also called _area of the best
approximation_ $v$, corresponds to the area of the immersed rectangle $R(v)$
given by Lemma 1.11. The above statement shows that this quantity is
uniformely bounded from above. The optimal constant on a given surface is
related to the Minkowski constant in the context of Cheung’s Z-convergents,
see [25]. The lower bound of areas of best approximations is related to the
Lagrange spectrum, see [24] and §1.3.3. Finally, let us mention that there is
a better bound for the systole (the length of the shortest saddle connection)
following from J. Smillie and B. Weiss’ argument in [42] (see the Appendix A
in [24]), namely
$\operatorname{sys}(X)\leq 2\
\sqrt{\frac{\operatorname{Area}(X)}{\pi(2g-2+n)}},$
where $g$ is the genus and $n$ the number of singularities of $X$. We remark
though that the proof of the above bound cannot be adapted to get bounds on
the length of shortest saddle connection in a given bundle.
The first part of the proof of Lemma 3.8 is very similar to arguments used to
prove minimality of the vertical flow under Keane’s condition. The second
statement in the Lemma is an adaptation of the proof of an upper bound on the
systole by Vorobets [46] to each bundle.
###### Proof of Lemma 3.8.
Let $X$ be a translation surface with no horizontal and no vertical saddle
connections. Let $I$ be an horizontal segment in $X$ and assume that one of
its endpoints, say $p$, is a singularity of $X$ and that $I$ does not contain
any other singularity in its interior. We claim that there exists $t_{1}>0$
and $-\operatorname{Area}(X)/|I|\leq t_{2}<\operatorname{Area}(X)/|I|$ such
that for $t=t_{1}$ and $t=t_{2}$, $\varphi_{t}(I)$ contains a point of
$\Sigma$ in its interior (that is there exists $x$ in the interior of $I$ such
that $\varphi_{t}(x)\in\Sigma$).
Since the area of $X$ is finite, the set $\cup_{t\geq 0}\varphi_{t}(I)$ has to
self-intersect. Let $s$ be the minimum first return time, that is the minimum
$t>0$ such that there exists $x\in I$ for which $\varphi_{t}(x)\in I$. Clearly
$s\leq\operatorname{Area}(X)/|I|$. If there exist a singularity inside
$\cup_{0<t<s}\varphi_{t}(I)$, that is there exists $0<t_{0}<s$ and $x_{0}$
such that $\varphi_{t_{0}}(x_{0})\in\Sigma$, we are done as we can take
$t_{1}=t_{2}=t_{0}$. If there is none, it follows that $\varphi_{s}$ is
continuous on $I$. If $p\in\varphi_{s}(I)$, we are done. We cannot have
$\varphi_{s}(I)=I$, otherwise there would be a vertical saddle connection.
Thus, we can assume that the other endpoint of $I$, that we will denote by
$y$, belongs to the interior of $\varphi_{s}(I)$. In this case, there is a
point $z\in I$ such that $\varphi_{-s}(z)=p$ and we can take $t_{2}=-s$. Let
$x\in I$ be such that $\varphi_{s}(x)=y$ (note that the distance between $p$
and $z$ is the same as the distance between $x$ and $y$). Consider now the
interval $I^{\prime}\subset I$ which has $x$ and $y$ as endpoints. Reasoning
as before, $\bigcup_{t>0}\varphi_{t}(I^{\prime})$ has to self intersect. Let
$s^{\prime}>0$ be the minimum first return time of $I^{\prime}$ in $I$. If
there exist a singularity inside
$\cup_{0<t<s^{\prime}}\varphi_{t}(I^{\prime})$, then we are done. Otherwise,
$\varphi_{s^{\prime}}(I^{\prime})$ is an interval that intersects $I$ and
which is disjoint from $\varphi_{s}(I)\cap I$ by definition of first return
time. Hence it has to contain $p$ in its interior and we can set
$t_{1}=s^{\prime}$.
We now apply the claim to bundles of saddle connections. Let us fix a positive
real number $r>0$ and let $\psi_{t}$ be the horizontal flow in $X$. For any
given bundle $\Gamma_{i}$ starting at a singularity $p\in\Sigma$, pick the
vertical segment $I_{r}$ issued from $p$ that belongs to the bundle and whose
length is ${\operatorname{Area}(X)}/r$. Applying the claim to $\psi_{t}$ from
$I_{r}$ (remark the the property of having no horizontal and no vertical
saddle connections is preserved by rotation of $\pi/2$), we get the existence
of a minimum $t_{1}>0$ such that $\psi_{t_{1}}(I_{r})$ contains a singularity.
By construction, this gives a right geometric best approximation. Similarly we
obtain the existence of a minimum $t^{\prime}_{1}<0$ such that
$\psi_{t^{\prime}_{1}}(I_{r})$ contains a singularity. This gives us a left
geometric best approximation. We know from the claim that either
$\min(|t_{1}|,|t^{\prime}_{1}|)<r$. We hence obtain a left and a right
geometric best approximation whose imaginary part is less than
$\operatorname{Area}(X)/r$ and for one of them, the real part is less than
$r$. This proves the quantitative estimate of the statement. By considering
decreasing values of $r$, this construction provides saddle connections whose
real part tends to $0$ (and imaginary part tends to $\infty$). ∎
We remark that the conclusion of Lemma 3.8 can still be proved under a weaker
assumption, that is that the surface $X$ has no horizontal _or_ no vertical
saddle connections. More precisely, if in a bundle $\Gamma_{i}$ there is a
vertical saddle connection $w$ but no horizontal saddle connection then there
is no best approximation $w^{\prime}$ with
$\operatorname{Im}(w^{\prime})>\operatorname{Im}(w)$ but there are still
infinitely many with arbitrarily small imaginary part.
###### Lemma 3.9 (diagonal determine quadrilateral).
Let $X$ be a translation surface without vertical saddle connections and let
$v$ be a saddle connection which is a geometric best approximation. Then there
exists a unique admissible (in particular embedded) quadrilateral $q$ whose
sides are all geometric best approximations and that has $v$ as foward
diagonal.
If moreover $v$ is left slanted (respectively right slanted), then there
exists a unique right (resp. left) slanted admissible quadrilateral in $X$
whose sides are best approximations and so that $v$ is its bottom left side
(resp. right side).
We remark also that the second part of the Lemma does not give any information
on left slanted (resp. right slanted) admissible quadrilaterals in $X$ whose
sides are geometric best approximations and have $v$ as bottom left (resp.
right) side. There might indeed be either none or several such quadrilaterals,
as it is clear from the last part of proof below.
In the proof of Lemma 3.9, we will use the following Lemma.
###### Lemma 3.10.
Let $X$ be a translation surface and let $P\subset X$ be an isometrically
immersed convex polygon that contains no singularities in its interior or in
the interior of its sides and whose vertices belong to $\Sigma$. Then the
interior of $P$ is embedded in $X$.
###### Proof.
Let $P_{0}\subset\mathbb{C}$ be convex polygon and let $f:P_{0}\to X$ be an
isometric immersion so that the image $P=f(P_{0})$ is the given immersed
polygon. We assume that $P$ contains no singularities in its interior or in
the interior of its sides and that its vertices belong to $\Sigma$. We need to
prove that $f$ is globally injective. Assume by contradiction that there
exists two distinct points $p_{1},p_{2}$ in the interior of $P_{0}$ such that
$f(p_{1})=f(p_{2})$ and consider the segment $\gamma$ connecting $p_{1}$ to
$p_{2}$. Then $f(\gamma)$ is an isometrically immersed closed curve on $X$ and
hence a closed geodesic with respect to the flat metric. Thus, there exists a
cylinder $C$ foliated by closed flat geodesics which contain $f(\gamma)$.
Since $P=f(P_{0})$ does not contain singularities, if $\gamma^{\prime}$ is
another segment inside $P$ which is obtained from $\gamma$ by parallel
transport (that is $\gamma^{\prime}=\gamma+c$ for some $c\in\mathbb{C}$),
$f(\gamma^{\prime})$ is also obtained by parallel transport of $f(\gamma)$
inside $X$ and hence is still a closed flat geodesic. Now consider the longest
segments inside $P_{0}$ which are parallel to $\gamma$. Because of convexity,
one of them necessarily starts at at a vertex of $P_{0}$. Now, we can find
$c\in\mathbb{C}$ such that $\gamma^{\prime}=\gamma+c$ is contained in $P_{0}$
and starts from that singularity. By construction, the other endpoint of
$\gamma^{\prime}$ is either inside $P_{0}$ or in the interior of its sides.
Because the two endpoints of $\gamma^{\prime}$ are identified by $f$ this
contradicts the fact that the interior of $P$ and the interior of its sides
are free of singularities. ∎
###### Proof of Lemma 3.9.
By definition of best approximation, $v$ is the diagonal of an immersed
rectangle $R(v)\subset X$. Let $v_{\ell}$ and $v_{r}$ respectively be the left
and right vertical sides of $R$, see Figure 16(a). Flow $v_{\ell}$
(respectively $v_{r}$) horizontally to the right (respectively to the left)
until the first time it hits a singularity, that we call $p_{\ell}$
(respectively $p_{r}$) as shown in Figure 16(a). Both singularities hit are
unique since otherwise $X$ would have a vertical saddle connection. Consider
the immersed convex quadrilateral which has as vertices $v_{r},v_{\ell}$ and
the endpoints of $v$ (see Figure 16(b)). Since by construction it does not
contain conical singularities in its interior, by Lemma 3.10 it is embedded.
Thus, we constructed an admissible quadrilateral which has $v$ as forward
diagonal. Furthermore, each of the sides of $q$ is a geometric best
approximation since by construction each is the diagonal of an immersed
rectangle without singularities in its interior (see Figure 16(b)).
(a) (b)
(c)
Figure 16: building a quadrilateral from a diagonal or a side (proof of Lemma
3.9)
The uniqueness comes from the construction: given an admissible quadrilateral
$q$ whose sides are best approximations, its forward diagonal $v$ is a best
approximation and we can build $q$ by flowing horizontally as above the
vertical sides of the immersed rectangle $R(v)$ associated to $v$.
For the second part of the statement, we consider the same construction.
Consider a fixed left slanted best approximation $v$ in some bundle
$\Gamma_{i}^{\ell}$. We want to determine which diagonals $u$ may have
produced $v$ by horizontal flowing the vertical left side of the associated
rectangle $R(u)$. Let $v^{\prime}$ be the slanted saddle connection in
$\Gamma_{i}^{\ell}$ which in next to $v$ in the natural order given by
increasing imaginary part. One can see, looking at Figure 16(c), that all the
possible such diagonals $u$ are exactly the left slanted saddle connection
$v^{\prime}$ and all the right slanted saddle connections $v_{r}$ which
satisfy
$\operatorname{Im}(v)\leq\operatorname{Im}(v_{r})\leq\operatorname{Im}(v^{\prime})$
(possibly none). In particular, only $v^{\prime}$ is the diagonal of a right
slanted quadrilateral as in the second part of the lemma. The right slanted
saddle connections are all possible diagonals of the set (possibly empty) of
left slanted admissible quadrilaterals with sides which are best
approximations and $v$ as bottom left side. ∎
We are now ready to prove Theorem 1.8. Let us first remark that the statement
is trivial for the torus case, since for any given lattice with neither
horizontal nor vertical vector there always exists a basis which form the
wedge of an admissible quadrilateral.
###### Proof of Theorem 1.8.
Let $q$ be an admissible quadrilateral whose sides are all best
approximations, whose existence is guaranteed by Lemma 3.8 and Lemma 3.9. We
denote its bottom sides by $v_{\ell}$, $v_{r}$ and its top sides by
$v^{\prime}_{r}$ and $v^{\prime}_{\ell}$. Now, consider its image $s(q)$ under
the hyperelliptic involution $s$. It is easy to see that, since all sides of
$q$ are best approximations, either $q=s(q)$ or $q$ and $s(q)$ have disjoint
interiors. In both cases, for each side $v$ of $q$ if $v=s(v)$ we do nothing,
while if $v\not=s(v)$ we cut and paste as in Lemma 3.3. After this operation,
we a obtain a surface made by one or two quadrilateral (if respectively or
$q\not=s(q)$) and at most four surfaces $X_{\ell}$, $X_{r}$, $X^{\prime}_{r}$
and $X^{\prime}_{\ell}$ that contain respectively $v_{\ell}$, $v_{r}$,
$v^{\prime}_{r}$ and $v^{\prime}_{\ell}$ (with the convention that we assume
that $X_{z}$ is empty if $v_{z}=s(v_{z})$). Moreover, each of these surfaces
belongs to a hyperelliptic component with strictly smaller total angle by
Lemma 3.3. On each non empty surface among $X_{\ell}$, $X_{r}$,
$X^{\prime}_{r}$ and $X^{\prime}_{\ell}$ let us consider a saddle connection
given by Lemma 3.8, let us complete it to an admissible quadrilateral by Lemma
3.9 and then iterate the above construction. In finitely many steps, the
construction thus produces $k$ admissible quadrilateral which provide a
quadrangulation of the original surface $X$. ∎
Let us remark that the proof does not extend to other components of strata. We
can still use a cut and paste construction but the resulting surfaces
$X_{\ell}$, $X_{r}$, $X^{\prime}_{\ell}$ and $X^{\prime}_{r}$ might be
connected to each other. In particular, if two of them are connected we obtain
a surface in which we want to complete a set of two saddle connections into a
quadrangulation.
### 3.3 Existence of staircase move, proof of Theorem 1.9
In this section, we give the proof of Theorem 1.9.
###### Proof of Theorem 1.9.
The proof proceeds by induction on the number of quadrilaterals, or in an
equivalent way on the integer $k$ such that the surface belongs to
$\mathcal{C}^{hyp}(k)$. The case of the torus ($k=1$) is trivial, since a
staircase made of one quadrilateral is always well slanted.
Let $Q=({\underline{\pi}},\underline{w})$ be an admissible quadrangulation of
a surface in $\mathcal{C}^{hyp}(k)$ and denote by $\iota$ the action of the
hyperelliptic involution $s$ on the quadrilaterals (i.e. $\iota(i)=j$ if and
only if $q_{j}=s(q_{i})$). Let us prove by contradiction that there exists at
least one well slanted staircase in which it is possible to make a diagonal
change. If no staircase move for $Q$ is possible, we claim that there exists a
right staircase $S$ which contains both left and right slanted quadrilaterals.
Indeed, no right staircase consists of only right slanted quadrilaterals,
otherwise it would be well slanted and a right move would be possible. If all
right staircases consist of only left slanted quadrilaterals, all left
staircase moves are possible. Thus, there exists $S$ with both left and right
slanted quadrilaterals. In particular, in $S$ there exist two consecutive
quadrilaterals $q_{i}$ and $q_{\pi_{r}(i)}$ which are respectively left
slanted and right slanted. We remark that it follows that
$\iota(i)\not=\pi_{r}(i)$, since otherwise the diagonals of $q_{i}$ and
$q_{\pi_{r}(i)}$ would be parallel and hence $q_{i}$ and $q_{\pi_{r}(i)}$
would have the same slantedness. In particular, the common edge
$w_{\pi_{r}(i),\ell}$ of $q_{i}$ and $q_{\pi_{r}(i)}$ does not contain a
Weierstrass point and hence $w_{\pi_{r}(i),\ell}\neq s(w_{\pi_{r}(i),\ell})$.
Let us cut the quadrangulation $Q$ along the edge $w_{\pi_{r}(i),\ell}$
(between $q_{i}$ and $q_{\pi_{r}(i)}$) and along its image under hyperelliptic
involution, which is the edge $w_{\iota(i),\ell}$ (between
$s(q_{\pi_{r}(i)})=q_{\iota(\pi_{r}(i))}$ and $s(q_{i})=q_{\iota(i)}$). From
Lemma 3.3, we know that after cutting along these edges we obtain two
connected components and that, after identifying on each of them the
corresponding copies of $w_{\pi_{r}(i),\ell}$ and $w_{\iota(i),\ell}$ by
parallel translations, we obtain two quadrangulations of surfaces with
strictly less quadrilaterals. We denote by $X^{\prime}$ the surface containing
$q_{i}$. By inductive assumption, there exists a staircase move in
$X^{\prime}$. Since $q_{i}$ is left-slanted, the saddle connection
$w_{i,\ell}$ does not change during the move. Hence, the move lifts to $X$ and
by glueing back the two components we can globally define a staircase move on
$X$. ∎
From Lemma 1.9, it is easy to see that the Keane’s condition (no vertical
saddle connections) is exactly the condition needed for any diagonal changes
algorithm not to stop (for the analogous of this Lemma in the case of Rauzy-
Veech induction see [47]).
###### Lemma 3.11.
Let $Q$ be a quadrangulation of a surface $X$ in $\mathcal{C}^{hyp}(k)$. There
exists an infinite sequence of staircase moves starting from $Q$ such that the
real part of each saddle connection in the wedges of $Q$ tends to zero if and
only if $X$ has no vertical saddle connection.
Moreover, if $X$ has no vertical saddle connection then for any infinite
sequence of staircase moves starting from $Q$ there are infinitely many left
and right diagonal changes and the width of each wedge goes to zero.
###### Proof.
Let us first prove the second part of the Lemma. Assume that $X$ has no
vertical saddle connection and let $Q^{(n)}$ be a sequence of quadrangulations
obtained by staircase moves starting from $Q=Q^{(0)}$. Assume by contradiction
that for some $1\leq i\leq k$ the quadrilateral $q_{i}^{(n)}$ undergos only
finitely many left changes, i.e. there exists $n_{1}$ so that for $n\geq
n_{1}$ we have $w^{(n)}_{i,r}=w^{(n_{1})}_{i,r}$. Then, because
$\operatorname{Re}(w^{(n)}_{i,r})\neq 0$ and the area of the surface is
finite, the sequence $(\operatorname{Im}(w^{(n)}_{i,\ell}))_{n\in\mathbb{N}}$
has to be bounded. Because of the discreteness of the set of saddle
connections, this implies that there exists $n_{2}\geq n_{1}$ such that for
$n\geq n_{2}$, also $w^{(n)}_{i,\ell}=w^{(n_{1})}_{i,\ell}$ and hence
$q^{(n)}_{i}=q^{(n+1)}_{i}$ for any $n\geq n_{2}$. Since the top sides of
$q^{(n)}_{i}$ are bottom sides for $q^{(n)}_{\pi_{\ell}(i)}$ and
$q^{(n)}_{\pi_{r}(i)}$ respectively, this implies also that for $n\geq n_{2}$
the quadrilateral $q_{\pi_{\ell}(i)}$ undergos only right diagonal changes and
the quadrilateral $q_{\pi_{\ell}(i)}$ undergos only left diagonal changes. In
particular, repeating the same argument $w^{(n)}_{\pi_{\ell}(i),l}$ and
$w^{(n)}_{\pi_{r}(i),r}$ are ultimately constant. Because of the connectedness
of the surface, or equivalently because the group generated by $\pi_{\ell}$
and $\pi_{r}$ acts transitively on $\\{1,\ldots,k\\}$ we can repeat the
argument and show that the quadrangulations $Q^{(n)}$ are ultimately constant,
contradicting the assumption that the sequence is obtained by staircase moves
(which are by definition not identity).
Let us now prove the first part. Let $X\in\mathcal{C}^{hyp}(k)$ and let us
first assume that there is an infinite sequence of staircase moves from the
quadrangulation $Q=Q^{(0)}$ of $X$ such that the associated sequence of
quadrangulations $Q^{(n)}=({\underline{\pi}}^{(n)},\underline{w}^{(n)})$ is
such that both $\operatorname{Re}(w^{(n)}_{i,\ell})$ and
$\operatorname{Re}(w^{(n)}_{i,r})$ tend to zero for any $1\leq i\leq k$. Then,
necessarily, since the set of saddle connections is discrete,
$\operatorname{Im}(w^{(n)}_{i,\ell})$ and $\operatorname{Im}(w^{(n)}_{i,r})$
tend to infinity. Assume by contradiction that there is a vertical saddle
connection $v$ on $X$ and let $\Gamma_{j}$, $1\leq j\leq k$, be the bundle
which contains it. Since by definition of wedges the sides of a wedge form a
triangle embedded in the surface and the imaginary parts of the wedge
$w^{(n)}_{j}$ both go to infinity, for $n$ sufficiently large $v$ is contained
in the triangle with sides $w_{i,\ell}$ and $w_{i,r}$. This contradicts the
fact that the interior of the triangle is free of singularities. Thus $X$ has
no vertical saddle connections.
Conversely, let $X\in\mathcal{C}^{hyp}(k)$ be without vertical saddle
connections and $Q^{(0)}$ be a quadrangulation of $X$. Because $X$ has no
vertical saddle connection then no quadrangulation on $X$ is vertical.
Applying inductively Theorem 1.9 from $Q^{(0)}$ we obtain an infinite sequence
of quadrangulations. Using the first part of the proof, the width of each
wedge necessarily tends to 0. ∎
### 3.4 Non hyperelliptic components
In this section we provide examples of translation surfaces which do not
belong to a hyperelliptic component of a stratum and admit quadrangulations
for which there are no possible staircase moves. Our strategy consists in
finding quadrangulation with $k$ quadrilaterals for which both $\pi_{\ell}$
and $\pi_{r}$ are $k$-cycles. This construction is possible in many component
of stratum but not in $\mathcal{C}^{hyp}(k)$ if $k\geq 3$. Then, once we found
this combinatorial datum we find a length datum in order that there is at
least one left-slanted and one right-slanted quadrilaterals.
We first consider the stratum $\mathcal{H}(0,0,0)$, which is the smallest
stratum which does not contain a hyperelliptic component. Let
$\pi_{\ell}=(1,2,3)=\pi_{r}=(1,2,3)$ and consider the wedges
$w_{1,\ell}=(-1.3,2),\quad w_{1,r}=(1,1),\quad
w_{2,\ell}=w_{3,\ell}=(-1.3,2)\quad\text{and}\quad w_{2,r}=w_{3,r}=(1.7,1)$
One can check that these length data satisfy the train-track relations for
${\underline{\pi}}=(\pi_{\ell},\pi_{r})$ and hence correspond to a
quadrangulation $Q$ (see Figure 17). Moreover we have
$w_{1,d}=(-0.3,3),\quad w_{2,d}=(0.4,3)\quad\text{and}\quad w_{3,d}=(-0.3,3).$
Hence, there is no well slanted staircase in $Q$.
Figure 17: a quadrangulation of a surface in $\mathcal{H}(0,0,0)$ with no well
slanted staircase
One can notice that the surface $X$ associated to $Q$ admits a hyperelliptic
symmetry that exchanges $q_{1}$ and $q_{3}$ while fixes $q_{2}$. In other
words, the quadrangulation is fixed by the hyperelliptic involution of $X$.
The quotient of $X$ by the hyperelliptic involution belongs to
$\mathcal{Q}(0,0,-1^{4})$. One can also check that in that case, the graph
associated to the triangulation described in §3.1.3 is no more a tree.
Now we construct another example which belongs to $\mathcal{H}(4)$. This
stratum is the smallest one which contains more than one component one of
which is hyperelliptic. Let $\pi_{r}=(1,2,3,4,5)$ and $\pi_{\ell}=(2,1,3,5,4)$
and consider the wedges
$\begin{array}[]{lllll}w_{1,r}=(2,1)&w_{2,r}=(1.5,1)&w_{3,r}=(2.5,1)&w_{4,r}=(3.5,1)&w_{5,r}=(1,1)\\\
w_{1,\ell}=(-1.5,2)&w_{2,\ell}=(-2.5,2)&w_{3,\ell}=(-0.5,2)&w_{4,\ell}=(-1.5,2)&w_{5,\ell}=(-3,2).\end{array}$
Then one can check that the train track relations are satisfied and hence
$Q=({\underline{\pi}},\underline{w})$ is a quadrangulation (see Figure 18).
Moreover we have
$w_{1,d}=(-0.5,3),\quad w_{2,d}=(1,3),\quad w_{3,d}=(1,3),\quad
w_{4,d}=(0.5,3),\quad w_{5,d}=(-0.5,3).$
This shows that there is no well slanted staircase in $Q$.
Figure 18: a quadrangulation of a surface in a non hyperelliptic component of
$\mathcal{H}(4)$ with no well slanted staircase
## 4 Best-approximations and bispecial words via staircase moves
In this section we prove Theorem 1.12 and show more generally that all best
approximations in each bundle are produced by any slow diagonal changes
algorithm (see Theorem 4.1). We then deduce several results. We first show, by
proving Theorem 4.3, that the geometric objects, namely wedges and well
slanted staircases, produced by any sequence of staircase moves are the same.
We then prove that the saddle connections which realize the systoles along a
Teichmueller geodesics are contained in the set of best approximations
(Theorem 1.14). Finally we prove that cutting sequences of bispecial words
coincide with best approximations (Theorem 1.13) and explain how they can be
generated recursively using diagonal changes (see Theorems 4.10).
### 4.1 Best approximations via staircase moves and applications
In this section we prove Theorem 1.12. Let us first prove the equivalent
geometric characterization of best approximations as diagonals of immersed
rectangles (Lemma 1.11).
###### Proof of Lemma 1.11.
In this proof, we will explicitly avoid the identification of saddle
connections in a bundle $\Gamma_{i}$ with their displacement vectors in
$\mathbb{C}$ and we will denote by $\operatorname{hol}(\gamma)\in\mathbb{C}$
the displacement vector of a saddle connection $\gamma$ on $X$ and by
$\operatorname{hol}(\Gamma_{i})$ the set of displacement vectors of saddle
connections in $\Gamma_{i}$. For each saddle connection $\gamma$ in
$\Gamma_{i}^{r}$ (respectively in $\Gamma_{i}^{\ell}$) let
$\widetilde{R}(\gamma)$ be the rectangle given by
$\widetilde{R}(\gamma)=[0,\operatorname{Re}\left(\operatorname{hol}(\gamma)\right)]\times[0,\operatorname{Im}\left(\operatorname{hol}(\gamma)\right)]\quad\mathrm{(resp.}\
\widetilde{R}(\gamma)=[\operatorname{Re}\left(\operatorname{hol}(\gamma)\right),0]\times[0,\operatorname{Im}\left(\operatorname{hol}(\gamma)\right)]\mathrm{\,)}.$
(18)
Using this notation, we first remark that Definition 1.10 can be rephrased as
follows: a saddle connection $\gamma\in\Gamma_{i}^{r}$ (respectiveley
$\gamma\in\Gamma_{i}^{\ell}$) is a _(geometric) best approximation_ if and
only if the rectangle $\widetilde{R}(v)$ does not contain any element of
$\operatorname{hol}(\Gamma_{i}^{r})$ (resp.
$\operatorname{hol}(\Gamma_{i}^{\ell})$) in its interior.
Let $v$ be a saddle connection starting at a point $p_{0}\in\Sigma$ and,
assuming that $v$ is right slanted, let $\Gamma_{i}^{r}$ be the bundle to
which $v$ belongs (the case of $v\in\Gamma_{i}^{\ell}$ is analogous). Suppose
first that there exists an immersed rectangle $R(v)\subset X$ which has $v$ as
a diagonal and does not contain singularities in its interior. The image of
$R(v)$ by the developing map
$\operatorname{dev}_{p_{0}}:R(v)\rightarrow\mathbb{C}$ given by
$p\mapsto\int_{p_{0}}^{p}\omega$ is exactly the rectangle $\widetilde{R}(v)$
in (18). If by contradiction $v$ is not a best approximation, by the remark at
the beginning of the proof $\operatorname{hol}(\Gamma_{i}^{r})$ intersects the
interior of $\widetilde{R}(v)$. Thus there is a saddle connection
$\gamma\in\Gamma_{i}^{r}$ whose holonomy $\operatorname{hol}(\gamma)$ belongs
to the interior of $\widetilde{R}(v)$. Since $\gamma$ belongs to the same
bundle than $v$, this means that $\gamma$ is contained in $R(v)$ and hence the
endpoint of $\gamma$ is a singularity in the interior of $R(v)$, which
contradicts the initial assumption.
Conversely, assume that $\gamma\in\Gamma_{i}^{r}$ is a best approximation (the
proof for $\gamma\in\Gamma_{i}^{\ell}$ is analogous). Then we claim that we
can immerse the rectangle $\widetilde{R}(v)$ given by (18) in $X$ so that its
image $R(v)$ is an immersed rectangle which has $\gamma$ as diagonal and does
not contain singularities in its interior. Define the immersion $\iota$ by
sending $z=\rho e^{i\theta}\in\widetilde{R}(v)$ to the point
$\iota(z)=\gamma_{\rho}^{\theta}(p_{0})$ which has distance $\rho$ from
$p_{0}$ and belongs to the unique linear trajectory
$(\gamma_{t}^{\theta}(p_{0}))_{t\geq 0}$ in direction $\theta$ which starts at
$p_{0}$ and such that such that
$|\angle(\gamma_{t}^{\theta}(p_{0}),\gamma)|<\pi/2$. To see that $\iota$ is
well defined, it is enough to check that these trajectories do not hit
singularities. This will show at the same time that the image $R(\gamma)$ of
$\iota$ does not intersect $\Sigma$. If by contradiction there is a
singularity $p_{1}\in\Sigma$ in the interior of $R(\gamma)$, since the saddle
connection $\gamma^{\prime}$ connecting $p_{0}$ to $p_{1}$ is inside
$R(\gamma)$, it belongs to the same bundle than $\gamma$ and has holonomy in
$\widetilde{R}(\gamma)$, thus the interior of $\widetilde{R}(\gamma)$
intersects $\operatorname{hol}(\Gamma_{i}^{r})$, which contradicts the
equivalent definition of best approximation given by the remark at the
beginning of this proof. ∎
#### 4.1.1 Staircase moves produce the same geometric objects
Let us now prove the following theorem, which is a more precise formulation of
Theorem 1.12 in the introduction.
###### Theorem 4.1.
Let $X$ be a surface in $\mathcal{C}^{hyp}(k)$ with neither horizontal nor
vertical saddle connections. Let $(Q^{(n)})_{n\in\mathbb{Z}}$ be any sequence
of labeled quadrangulations
$Q^{(n)}=({\underline{\pi}}^{(n)},\underline{w}^{(n)})$ of $X$ where
$Q^{(n+1)}$ is obtained from $Q^{(n)}$ by simultaneous staircase moves. Then,
for each $1\leq i\leq k$ the saddle connections in the sequence
$(w_{i,\ell}^{(n)})_{n\in\mathbb{Z}}$ (resp.
$(w_{i,r}^{(n)})_{n\in\mathbb{Z}}$) are exactly all best approximations in
$\Gamma_{i}^{\ell}$ (resp. $\Gamma_{i}^{r}$) ordered by increasing imaginary
part.
###### Proof.
We first prove that any saddle connections belonging to the wedges of one of
the quadrangulations in $(Q^{(n)})_{n\in\mathbb{Z}}$ is a best approximation.
Let $Q=({\underline{\pi}},\underline{w})=Q^{(n)}$ be a quadrangulation in the
sequence and let $w_{\ell}$ be a left slanted saddle connection belonging to
the wedge of some $q\in Q$. The proof for right slanted saddle connections in
the wedges is analogous. Let $S$ be the right staircase that contains $q$, so
that $w_{\ell}$ belongs to the interior of the staircase $S$. Let
$R(w_{\ell})\subset X$ be the image of the rectangle $\widetilde{R}(w_{\ell})$
in $\mathbb{C}$ which has $w_{\ell}$ as its diagonal, shown in Figure 19(a).
Since each saddle connection belonging to the boundary of $S$ is left slanted,
$\widetilde{R}(w_{\ell})$ is contained in the universal cover $\widetilde{S}$
of $S$ shown in Figure 19(b) and thus $R(w_{\ell})$ is contained inside $S$.
Since the staircase $S$ does not contain any singularity in its interior, it
follows that $R(w_{\ell})$ is an immersed rectangle which contains no point of
$\Gamma_{i}^{r}$ in its interior. This shows that $w_{\ell}$ is a geometric
best approximation by Lemma 1.11.
(a) in a staircase (b) in the universal cover
Figure 19: immersed rectangle around a side in a staircase which becomes
embedded in its universal cover
Let us now prove that all geometric best approximations in $\Gamma_{i}^{r}$
for any fixed $1\leq i\leq k$ appear in the sequence
$(w_{i,r}^{(n)})_{n\in\mathbb{Z}}$ in their natural order. Since we just
proved the saddle connections in $\Gamma_{i}^{r}$ given by the sequence
$(w_{i,r}^{(n)})_{n\in\mathbb{Z}}$ are geometric best approximations and by
construction they are naturally ordered by increasing imaginary part, it is
enough to show that if $w_{i,r}^{(n)}$ and $w_{i,r}^{(n+1)}$ are two
successive saddle connections in $\Gamma_{i}^{r}$ according to this natural
order, there is no geometric best approximation with imaginary part strictly
in between $\operatorname{Im}w_{i,r}^{(n)}$ and
$\operatorname{Im}w_{i,r}^{(n+1)}$. For this, we claim that it is enough to
show that if $R\subset\mathbb{C}$ is the rectangle
$R=[0,\operatorname{Re}(w_{i,r}^{(n)})]\times[0,\operatorname{Im}(w_{i,r}^{(n+1)})]$
shown in Figure 20), then
$\Gamma_{i}^{\ell}\ \cap R=\left\\{w_{i,r}^{(n)},w_{i,r}^{(n+1)}\right\\}.$
Indeed, this implies that there are no saddle connection $v\in\Gamma_{i}^{r}$
with
$\operatorname{Im}w_{i,r}^{(n)}<\operatorname{Im}v<\operatorname{Im}w_{i,r}^{(n+1)}$
and $0<\operatorname{Re}v\leq\operatorname{Re}(w_{i,r}^{(n)})$. And if
$v\in\Gamma_{i}^{r}$ satisfies
$\operatorname{Im}w_{i,r}^{(n)}<\operatorname{Im}v<\operatorname{Im}w_{i,r}^{(n+1)}$
and $\operatorname{Re}v>\operatorname{Re}(w_{i,r}^{(n)})$ then it is not a
best approximation.
Figure 20: diagonal change seen on the displacement vectors
By construction, since $w_{i,r}^{(n)}$ and $w_{i,r}^{(n+1)}$ are consecutive
saddle connections, $w_{i,r}^{(n+1)}$ is the diagonal $d_{i}^{(n)}$ of the
quadrilateral $q_{i}$ in $Q^{(n)}$. Thus the top right saddle connection of
$q^{(n)}_{i}$, that we will denote by $w_{j,\ell}^{(n)}$, joins the endpoint
of $w_{i,r}^{(n)}$ and $w_{i,d}^{(n)}$. Notice that $R$ is the union of the
three smaller rectangles $R_{1},R_{2},R_{3}$ which have as diagonals
respectively the saddle connections $w_{i,r}^{(n)}$, $w_{i,d}^{(n)}$ and
$w_{j,\ell}^{(n)}$ (see Figure 20). By the previous part of the proof and by
definition of geometric best approximation, each of the rectangles $R_{1}$ and
$R_{2}$ do not contain elements of $\Gamma_{i}^{r}$ in their interior. Thus,
if by contradiction there exist an element $v\in\Gamma_{i}^{r}$ in the
interior of $R$, there is an element of $u\in\Gamma_{i}^{r}$ inside $R_{3}$.
Thus, there is also a saddle connection $u\in\Gamma_{j}^{\ell}$ inside the
image of $R_{3}$ on the surface contradicting that, by the first part of the
theorem, also $w_{j,\ell}^{(n)}$ is a best approximation. This concludes the
proof. ∎
Theorem 4.1 has the following corollary for sequences obtained by forward
moves only:
###### Corollary 4.2.
Let $X$ be a surface in $\mathcal{C}^{hyp}(k)$ with no vertical saddle
connections. Let
$(Q^{(n)})_{n\in\mathbb{N}}=\left(({\underline{\pi}}^{(n)},\underline{w}^{(n)})\right)_{n\in\mathbb{N}}$
be any sequence of labeled quadrangulations of $X$ where $Q^{(n+1)}$ is
obtained from $Q^{(n)}$ by simultaneous staircase moves. Then:
* (i)
For each $1\leq i\leq k$ the saddle connections in the sequence
$(w_{i,r}^{(n)})_{n\in\mathbb{N}}$ (resp.
$(w_{i,\ell}^{(n)})_{n\in\mathbb{N}}$) are exactly all best approximations $v$
in $\Gamma_{i}^{r}$ (resp. in $\Gamma_{i}^{\ell}$) which have
$\operatorname{Im}v\geq\operatorname{Im}w_{i,r}$ (resp.
$\operatorname{Im}v\geq\operatorname{Im}w_{i,\ell}$), or, equivalently,
$|\operatorname{Re}v|\geq|\operatorname{Re}w_{i,r}|$ (resp.
$|\operatorname{Re}v|\geq|\operatorname{Re}w_{i,\ell}|$).
* (i)
For each $i$, $1\leq i\leq k$, the set of diagonals $(w^{(n)}_{i,d})_{n}$
coincide with the set of best approximations $v$ in $\Gamma_{i}$ such that
$\operatorname{Im}(v)>\max(\operatorname{Im}(w_{i,\ell}),\operatorname{Im}(w_{i,r}))$;
or equivalently to the set of bottom sides of the quadrilaterals
$(q_{i}^{(n)})_{n}$ different from the one of $q_{i}^{(0)}$.
###### Proof.
Part $(i)$ follows from Theorem 4.1 since geometric best approximations are
produced ordered by increasing imaginary part. Remark that if $v$ and $u$ are
left best approximations then $\operatorname{Im}v<\operatorname{Im}u$ (resp.
$|\operatorname{Re}v|<|\operatorname{Re}u|$ if and only if
$|\operatorname{Re}v|<|\operatorname{Re}u|$ (resp.
$|\operatorname{Re}v|<|\operatorname{Re}u$).
Recall from Lemma 3.11, that if $X$ has no vertical saddle connection then
each diagonal of $Q^{(n)}$ eventually becomes a side of a wedge. Hence Part
(ii) follows from Part (i). ∎
Combining Theorem 1.12 with Lemma 3.9 (diagonals uniquely determine their
quadrilaterals) we can now prove that any sequence of staircase moves produce
not only the same sequence of saddle connections, but also the same sequence
of wedges and well slanted staircases:
###### Theorem 4.3.
Let $X$ be a surface in $\mathcal{C}^{hyp}(k)$ without vertical saddle
connections and let $Q$ be a quadrangulation of $X$. Let
$(Q_{1}^{(n)})_{n\in\mathbb{N}}$, $(Q_{2}^{(n)})_{n\in\mathbb{N}}$ be any two
sequences of quadrangulations of the surface $X$ such that
$Q^{(0)}_{1}=Q^{(0)}_{2}=Q$ and, for $i=1,2$, $Q_{i}^{(n+1)}$ is a new
quadrangulation obtained from $Q_{i}^{(n)}$ by simultaneous staircase moves.
* (i)
The collection of the wedges of the quadrangulations in the sequence
$(Q_{1}^{(n)})_{n\in\mathbb{N}}$ is the same as a set than the collection of
the wedges of the quadrangulations in the sequence
$(Q_{2}^{(n)})_{n\in\mathbb{N}}$.
* (ii)
The set of well slanted staircases associated to the quadrangulations in
$(Q_{1}^{(n)})_{n\in\mathbb{N}}$ is the the same than the set of well slanted
staircases associated to the quadrangulations in
$(Q_{2}^{(n)})_{n\in\mathbb{N}}$.
###### Proof.
Let $(Q_{1}^{(n)})_{n\in\mathbb{N}}$, $(Q_{2}^{(n)})_{n\in\mathbb{N}}$ be as
in the statement. Because of Theorem 3.11, each diagonal in $Q_{1}^{(n)}$ will
eventually become a side. This is also true for $Q_{2}^{(n)}$. By Corollary
4.2, the set of diagonals in $(Q_{1}^{(n)})_{n}$ and $(Q_{2}^{(n)})_{n}$
coincide. Now, by Lemma 3.9, each diagonal uniquely determines its
quadrilateral. It follows that the set of quadrilaterals and the set of wedges
in $(Q_{1}^{(n)})_{n}$ and $(Q_{2}^{(n)})_{n}$ are the same, thus concluding
the proof of $(i)$.
Let us now prove $(ii)$. Since we just showed that quadrilaterals for
$(Q_{1}^{(n)})_{n}$ and $(Q_{2}^{(n)})_{n}$ are the same, it is enough to show
that each such quadrilateral uniquely determines the well slanted staicase to
which it belongs. Let $q=q_{i}$ be a right slanted quadrilateral in
$Q_{1}^{(n)}$ for some $n\in\mathbb{N}$ (the case when $q$ is left slanted is
similar) and let $v$ its right top side. We only need to prove that there is a
unique quadrilateral $q^{\prime}$ which is right slanted and has $v$ as it
bottom left side, since such quadrilateral is necessarily a neighbour of $q$
in a well slanted right staircase. From Theorem 1.12, we know that $v$ is a
geometric best approximation and from Lemma 3.9 existence and uniqueness is
guaranteed. Repeating this argument, we see that the right well slanted
staircase which contains $q$ is uniquely determined. ∎
#### 4.1.2 Systoles and Lagrange values along Teichmueller geodesics
Recall from §1.3.3 that the systole on a translation surface is the length of
the shortest saddle connection. In this section we prove Theorem 1.14 on
systoles along Teichmueller geodesics and then state and prove Theorem 4.6
which shows that diagonal changes can be used to compute the quantity $a(X)$
along closed Teichmüller geodesics.
The following general Lemma holds for any translation surface (not necessarily
in a hyperelliptic component).
###### Lemma 4.4.
Let $X$ be a translation surface and let $v$ be a saddle connection on $X$
which realizes the systole for some time $t$ along the Teichmueller geodesics
$(g_{t}X)_{t\in\mathbb{R}}$. Then $v$ is a geometric best approximation.
###### Proof.
Let $v\in\Gamma_{i}$. Let us prove the first part. Assume that $v$ on $X$
realize the systole for some time $t>0$. Since the property of being a best
approximations is invariant under the geodesic flow $(g_{t})_{t\in\mathbb{R}}$
(since immersed rectangles with horizontal and vertical sides are mapped to
immersed rectangles of the same form), we can replace $X$ by $g_{-t}X$ and
assume that $t=0$. Thus, for any saddle connection $u$ in $X$, the flat lenght
$|u|$ of $u$ is greater or equal than $|v|$. In particular, the semicircle in
$\mathbb{C}$ centered in the origin and of radius $|v|$ does not contain any
point of $\Gamma_{i}$ in its interior. This implies in particular that the
rectangle in $\mathbb{C}$ which has $v$ as diagonal and vertical and
horizontal sides does not contain any point of $\Gamma_{i}$ in its interior
and hence that $v$ is a geometric best approximation. ∎
###### Proof of Theorem 1.14.
The Theorem now follows immediately as a corollary of Theorem 4.1 and Lemma
4.4 above: let $X$ and $Q$ be as in the assumptions. Assume that $v$ realizes
the systole for some time $t_{0}$. Then by the first part of Lemma 4.4, $v$ is
a best approximation. Thus, by Theorem 1.12 it appears as one of the wedges. ∎
If one is interested only in saddle connections which realize the systoles
along a _Teichmueller geodesic ray_ $(g_{t}X)_{t\geq 0}$ starting from $X$,
one needs an extra assumption to avoid missing saddle connections which might
realize minima for small values of $t$. The following result can be deduced
from Theorem 1.14.
###### Corollary 4.5.
Let $X$ be a surface in $\mathcal{C}^{hyp}(k)$ with neither horizontal nor
vertical saddle connections. Let $Q=({\underline{\pi}},\underline{w})$ be a
quadrangulation of $X$ for which each side $v$ satisfies
$|\operatorname{Re}(v)|>\operatorname{sys}(X)$. Let
$\\{Q^{(n)},n\in\mathbb{N}\\}$ be any sequence of quadrangulations obtained
from $Q^{(0)}=Q$ by simultaneous staircase moves. Then the saddle connections
on $X$ which realize the systole along the Teichmueller geodesic ray
$(g_{t}X)_{t\geq 0}$ are a subset of the sides of the quadrangulations in
$\\{Q^{(n)},n\in\mathbb{N}\\}$.
###### Proof.
Since by assumption each side $v$ of $Q$ satisfies
$|\operatorname{Re}(v)|>\operatorname{sys}(X)$, From Part $(i)$ of Corollary
4.2 we know that the set of sides of $(Q^{(n)})_{n}$ contains all best
approximations $v$ that satisfy
$|\operatorname{Re}(v)|\leq\operatorname{sys}(X)$. Hence, because of Lemma
4.4, it is enough to show that saddle connections $v$ that realize the systole
at a positive time satisfies
$|\operatorname{Re}(v)|\leq\operatorname{sys}(X)$. Now, by definition of the
systole we have $\operatorname{sys}(g_{t}X)\leq e^{t}\operatorname{sys}(X)$.
Thus, if $v$ is a saddle connection which realizes a systole at time $t>0$ we
have $|\operatorname{Re}(g_{t}v)|\leq|g_{t}v|=\operatorname{sys}(g_{t}X)\leq
e^{t}\operatorname{sys}(X)$. As
$|\operatorname{Re}(g_{t}v)|=e^{t}|\operatorname{Re}(v)|$, we obtain that
$|\operatorname{Re}(v)|\leq\operatorname{sys}(X)$. ∎
We now deduce from Theorem 4.1 that the values of the Lagrange spectrum
$\mathcal{L}(\mathcal{C}^{hyp}(k))$ of a hyperelliptic component (defined in
§1.3.3 of the Introduction) can be computed using staircase moves. Let us
recall that the definition of $a(X)$ is given in (2).
###### Theorem 4.6.
Let $X$ be a surface in $\mathcal{C}^{hyp}(k)$ with neither horizontal nor
vertical saddle connections and let $(Q^{(n)})_{n\in\mathbb{N}}$ be any
sequence of labeled quadrangulations
$Q^{(n)}=({\underline{\pi}}^{(n)},\underline{w}^{(n)})$ of $X$ where
$Q^{(n+1)}$ is obtained from $Q^{(n)}$ by simultaneous staircase moves. Then
$a(X)=\liminf_{n\to+\infty}a(\underline{w}^{(n)}),\quad\text{where}\quad
a(\underline{w}^{(n)}):=\min_{v\,\text{in}\,\underline{w}^{(n)}}|\operatorname{Re}v||\operatorname{Im}v|,$
(19)
where the minimum in the definition of $a(\underline{w}^{(n)})$ is taken over
all areas of saddle connections belonging to the wedges in
$w^{(n)}_{1},\dots,w^{(n)}_{k}$.
###### Proof.
Let us assume for simplicity that $\operatorname{Area}(X)=1$. Let us recall
that it is shown in [24] that the quantity $a(X)$ (which is defined in (2) in
the introduction) is also equal to $s^{2}(X)/2$ where
$s(X)=\liminf_{t\to\infty}\operatorname{sys}(g_{t}X)$. Set $Q=Q^{(0)}$ and let
$\lambda_{min}(Q)=\min_{v\,\text{in}\,w^{(0)}}|\operatorname{Re}(v)|$ where as
in the definition of $a(\underline{w}^{(n)})$ the minimum is taken over all
saddle connections that belongs to the wedges of $Q$. Consider a time
$t_{0}>0$ such that
$\operatorname{sys}(g_{t_{0}}X)<e^{t_{0}}\lambda_{min}(Q)=\lambda_{min}(g_{t_{0}}Q)$
(such time exists since the systole function is bounded from above on each
stratum of translation surfaces of unit area). Because of Corollary 4.5, the
saddle connections in the wedges $\\{\underline{w}^{(n)};n\in\mathbb{N}\\}$
contain all saddle connections that realize the systoles at time larger than
$t_{0}$.
Let $(t_{k})_{k\in\mathbb{N}}$ be the sequence of times when the systole
function has a local minimum for $t\geq t_{0}$ and let $v_{k}$ be a saddle
connection in the wedges $\underline{w}^{(n_{k})}$ that realizes the systole,
that is such that $|g_{t_{k}}v_{k}|=\operatorname{sys}(g_{t_{k}}X)$. Since
$t_{k}$ is a local minimum of $t\mapsto|g_{t}v_{k}|$, it follows that
$g_{t_{k}}v_{k}$ is the diagonal of a square, so
$\operatorname{sys}(g_{t_{k}}X)=\sqrt{2}\operatorname{Im}v_{k}=\sqrt{2}|\operatorname{Re}v_{k}|$
and
$a(\underline{w}^{(n_{k})})=\operatorname{Im}v_{k}|\operatorname{Re}v_{k}|=(\operatorname{sys}(g_{t_{k}}X)/\sqrt{2})^{2}$.
Thus, since the liminf of a sequence is invariant under reordering (more
precisely if $\pi:\mathbb{N}\rightarrow\mathbb{N}$ is a bijection and
$(u_{n})_{n\in\mathbb{N}}$ is a sequence of real numbers then $\liminf
u_{n}=\liminf u_{\pi(n)}$),
$a(X)=\frac{(\liminf_{t\to\infty}\operatorname{sys}(g_{t}X))^{2}}{2}=\frac{(\liminf_{k\to\infty}\operatorname{sys}(g_{t_{k}}X))^{2}}{2}=\liminf_{k\to\infty}a(\underline{w}^{(n_{k})})\geq\liminf_{n\to\infty}a(\underline{w}^{(n)}).$
The opposite inequality, that is
$a(X)\leq\liminf_{n\to\infty}a(\underline{w}^{(n)})$, is obvious from the
definition (2) of $a(X)$ and the invariance of liminf under reordering, since
saddle connections belonging ot the wedges $\underline{w}^{(n)}$ are a subset
of all saddle connections with positive imaginary parts. ∎
### 4.2 Description of the language via staircase moves
In this section we prove that diagonal changes allow to effectively construct
the list of bispecial words in the language of cutting sequences. We first
show in §4.2.1 that there is a correspondence between bispecial words and
geometric best approximations (see Lemma 4.7). Theorem 1.13 about bispecial
words then follows from Theorem 4.3 of the preceding section. In §4.2.2 we
show that cutting sequences of best approximations can be constructed by
recursive formulas determined by a sequence of staircase moves (see Theorem
4.10 for the precise statement).
#### 4.2.1 Bispecial words as cutting sequences of best approximations
Except in the proof of Theorem 1.13, we consider in this section general
translation surfaces, i.e. we do not assume that they belong to an
hyperelliptic component $\mathcal{C}^{hyp}(k)$.
Given a labeled quadrangulation $Q=({\underline{\pi}},\underline{w})$ of a
translation surface $X$, recall that $\mathcal{L}_{Q}$ denotes the language of
cutting sequences of trajectories of the vertical flow on $X$ (see §1.3.2).
The alphabet of $\mathcal{L}_{Q}$ is
$\mathcal{A}=\\{1,\ldots,k\\}\times\\{\ell,r\\}$ where $(i,\ell)$ and $(i,r)$
are respectively the labels of the saddle connections $w_{i,\ell}$ and
$w_{i,r}$ of the wedge $w_{i}$ in $Q$.
###### Lemma 4.7.
Let $X$ be a translation surface with total angle $k$ and without vertical nor
horizontal saddle connections. Let $Q=({\underline{\pi}},\underline{w})$ be a
labeled quadrangulation of $X$. A word $W=A_{1}\dots A_{n}$ in
$\mathcal{L}_{Q}$ is bispecial if and only if it is the cutting sequence of a
geometric best approximation $v$ in a bundle $\Gamma_{i}$ with
$\operatorname{Im}v\geq\operatorname{Im}w_{i,d}$. Furthermore, if $W$ is a not
empty word in $\mathcal{L}_{q}$:
* (i)
If $W$ is a left special word, then its left extensions are $(i,\ell)$ and
$(i,r)$ for some $i\in\\{1,\ldots,k\\}$.
* (ii)
If $W$ is right special, then its right extensions are $(\pi_{r}(j),\ell)$ and
$(\pi_{\ell}(j),r)$ for some $j\in\\{1,\ldots,k\\}$.
* (iii)
If $W$ is bispecial and its left and right extensions are respectively
$(i,\ell)$, $(i,r)$ and $(\pi_{r}(j),\ell)$, $(\pi_{\ell}(j),r)$ then the
words $(i,\ell)\,W\,(\pi_{\ell}(j),r)$ and $(i,r)\,W\,(\pi_{r}(j),\ell)$ are
in $\mathcal{L}_{Q}$ and exactly one of $(i,\ell)\,W\,(\pi_{r}(j),\ell)$ or
$(i,r)\,W\,(\pi_{\ell}(j),r)$ is in $\mathcal{L}_{Q}$.
We remark that properties (i), (ii)and (iii) of the above lemma constitutes
the characterization of the language that comes from interval exchange
transformations (see [5] and [17]).
In the proof of Theorem 1.13, given word in the language we want to associate
to it a set of orbits of the vertical flow that have that word as a cutting
sequence. The following definition is convenient to pass from combinatorics to
geometry.
###### Definition 4.8.
Let $Q$ be a quadrangulation of a translation surface $X$ with no vertical
saddle connections. Let $w\in\mathcal{L}_{Q}$ be a non-empty word. We define
the _beam_ or _cylinder_ ${[}W{]}$ associated to $W$ as the set of finite
orbits of the vertical flow whose coding is exactly $W$ and are maximal with
respect to that property.
The following Lemma describes the geometric shape of a beam. Examples of beams
are shown in Figure 21.
(a) beam of a letter (b) beam of a word
Figure 21: examples of beams of trajectories illustrating Definition 4.8
###### Lemma 4.9.
Let $Q$ be a quadrangulation of a translation surface $X$ with no vertical
saddle connections. Let $W$ be a non-empty word in $\mathcal{L}_{Q}$. Then the
beam ${[}W{]}$ is an immersed polygon delimited on the left and the right by
vertical separatrices. The bottom side is delimited either by one side of $Q$
or by a pair of sides $w_{i,\ell}$ and $w_{i,r}$ for some $1\leq i\leq k$. The
top side is delimited by one side of $Q$ or by a pair of sides
$w_{\pi_{\ell}(j),r}$ and $w_{\pi_{r}(j),\ell}$ for some $1\leq j\leq k$.
###### Proof of Lemma 4.9.
We will denote by $w({A_{k}})$ the saddle connection in a wedge corresponding
to the label $A_{k}\in\mathcal{A}$, that is $w(A_{k})=w_{i,\ell}$ if
$A_{k}=(i,\ell)$ or $w(A_{k})=w_{i,r}$ if $A_{k}=(i,r)$. Let $W=A_{1}\ldots
A_{n}$ be a non empty word in $\mathcal{L}_{Q}$ and let $i$ and $j$ be
respectively such that $w(A_{1})$ is a top side of $q_{i}$ and $w(A_{n})$ is a
bottom side of $q_{j}$. Let us first remark that, by definition of a
quadrilateral, if $x$ is a point on $w(A_{n})$ then the first saddle
connection crossed by the forward orbit $(\varphi_{t}(x))_{t>0}$ is either
$w_{\pi_{\ell}(j),r}$ or $w_{\pi_{r}(j),\ell}$. Similarly, for $x$ on
$w(A_{1})$ the first saddle connection crossed by the backward orbit
$(\varphi_{t}(x))_{t<0}$ is either $w_{i,\ell}$ or $w_{i,r}$. Furthermore, the
sets of points on $w(A_{n})$ that first hit backward or forward a given side
is a connected subsegment of $w(A_{n})$.
We now proceed by induction on the length $n$ of the word $W$. For a word
$W=A$ of length $1$, one can see from the previous remark that the beam is a
polygon such that two of its vertices are the endpoints of the associated
saddle connection, as shown in Figure 21(a).
(a) (b)
(c)
Figure 22: possible splitting of the beam of trajectories ${[}w{]}$ for
$w=A_{1}\dots A_{n}\in\mathcal{L}_{Q}$
For the inductive step, refer to Figure 22. Assume that the result holds for
all words of length $n$ and consider a word $A_{1}\ldots A_{n+1}$ of length
$n+1$. As before, let $j$ be such that $A_{n}$ is a bottom side of $q_{j}$ and
$j^{\prime}$ be such that $w(A_{n+1})$ is a bottom side of $q_{j^{\prime}}$.
For each orbit in $[W]$ consider the intersection with the wedge $w(A_{n})$
that corresponds to the $n$-th crossing of the sides of $Q$. By induction
hypothesis, this set of points is a connected segment $J$ in $w(A_{n})$. By
the initial remark, we know that the vertical trajectories emanating from $J$
either
1. 1.
all cross the wedge $w(A_{n+1})$ for $A_{n+1}=(\pi_{r}(j),\ell)$ as in Figure
22(a),
2. 2.
or all cross $A_{n+1}=(\pi_{\ell}(j),r)$) as in Figure 22(b),
3. 3.
or one of them hit the conical singuarity which is the top vertex of the
quadrilateral $q_{j}$ as in Figure 22(c).
In the two first cases, the beam ${[}A_{1}\cdots A_{n}A_{n+1}{]}$ is obtained
simply prolonging the trajectories of the beam ${[}A_{1}\cdots A_{n}{]}$
until, after crossing $w(A_{n+1})$, they hit the top side of $q_{j^{\prime}}$.
In the third case, the trajectories are split into two connected subsets of
trajectories, accordingly to whether after $w(A_{n})$ trajectories cross
$w_{\pi_{r}(j),\ell}$ or $w_{\pi_{\ell}(j),r}$. In all cases, it is clear from
the construction that the beam ${[}A_{1}\cdots A_{n+1}{]}$ is again an
immersed polygon with the same properties. ∎
###### Proof of Lemma 4.7.
From Lemma 4.9 the possible right extensions of a non-empty word
$W=A_{1}\ldots A_{n}$ in $\mathcal{L}_{Q}$ are of the form $(\pi_{\ell}(j),r)$
and $(\pi_{r}(j),\ell)$ for the integer $j$ such that
$A_{n}\in\\{j\\}\times\\{\ell,r\\}$ (see Figure 22). Similarly, its left
extensions are of the form $(i,\ell)$ and $(i,r)$ for $i$ such that
$A_{1}\in\\{i\\}\times\\{\ell,r\\}$. This proves items (i) and (ii).
(a) (b)
Figure 23: cutting sequences of geometric best approximations are bispecial
We now prove that cutting sequences of best approximations with imaginary
parts as in the statement of the Lemma are exactly all bispecial words. Let
$v$ be a geometric best approximation in $\Gamma_{i}$ with
$\operatorname{Im}v\geq\operatorname{Im}w_{i,d}$. If
$\operatorname{Im}v=\operatorname{Im}w_{i,d}$ then $v=w_{i,d}$ and the cutting
sequence of $v$ is the empty word, which is bispecial. Let us hence assume
that $\operatorname{Im}v>\operatorname{Im}w_{i,d}$ and let $W=A_{1}\ldots
A_{n}$ be the cutting sequence of $v$ where $n\geq 1$. Let $i$ and $j$ be so
that $A_{1}\in\\{i\\}\times\\{\ell,r\\}$ and
$A_{n}\in\\{j\\}\times\\{\ell,r\\}$. Since $v$ is a best approximation, by
Lemma 1.11 there exists an immersed rectangle $R(v)\subset X$ with no
singularity in its interior and no singularity on its sides other than the
endpoints of $v$. Without loss of generality, we may assume that $v$ is right
slanted. Let $v_{\ell}$ and $v_{r}$ be respectively the left and right
vertical side of $R(v)$, as shown in Figure 23(a). Since
$\operatorname{Im}v>\operatorname{Im}w_{i,d}>\operatorname{Im}w_{i,r}$, the
beginning of $v$ belongs to the sector determined by the wedge
$(w_{i,\ell},w_{i,r})$ and $w_{i,r}$ crosses the vertical side $v_{r}$, as
shown in Figure 23(a). We now claim that $w_{\pi_{r}(j),\ell}$ crosses the
other vertical side, that is $v_{\ell}$. Indeed, since $w_{\pi_{r}(j),\ell}$
is left slanted and $R(v)$ cannot its starting point in its interior, either
$w_{\pi_{r}(j),\ell}$ crosses $v_{\ell}$ or it crosses the bottom side of
$R(v)$. The latter possibility cannot happen since otherwise
$w_{\pi_{r}(j),\ell}$ would have to intersect $w_{i,r}$, which is excluded
from the definition of quadrangulations.
Let us call vertical trajectory in $R(v)$ any finite trajectory which is
obtained intersecting a vertical trajectory with $R(v)$. It follows from what
we proved that the first side of $Q$ hit by any vertical trajectory in $R(v)$
is $w_{i,r}$, while $w_{\pi_{r}(j),\ell}$ is the last side of $Q$ hit, see
Figure 23(b). We claim that in between these two hitting times the cutting
sequence of the vertical trajectory in $R(v)$ is the same than the cutting
sequence $W$ of $v$. Indeed, since $R(v)$ does not contain singularities and
sides of $Q$ cannot cross neither $w_{i,r}$ nor $w_{\pi_{r}(j),\ell}$, they
have to cross both $v_{\ell}$ and $v_{r}$. Thus, any vertical segment in
$R(v)$ has cutting sequence $(i,r)\,W\,(\pi_{\ell}(j),r)$. Now flow
horizontally $v_{\ell}$ to the left and to $v_{r}$ to the right. If $\psi_{t}$
denotes the horizontal flow, for any $t>0$ such $\psi_{s}(v_{r})$ does not
contain any singularity for $0<s\leq t$, the vertical trajectory
$\psi_{t}(v_{r})$ has coding $(i,r)\,W\,(\pi_{r}(j),\ell)$ (see Figure 23(b)).
Similarly, for any $t<0$ such $\psi_{s}(v_{\ell})$ does not contain any
singularity for $-t\leq s<0$, the vertical trajectory $\psi_{t}(v_{\ell})$ has
coding $(i,\ell)\,W\,(\pi_{\ell}(j),r)$. This shows that $W$ is bispecial.
(a) (b)
Figure 24: bispecial words are cutting sequences of geometric best
approximations
Let us now assume that $W=A_{1}\ldots A_{n}$ is a non-empty bispecial word. We
know from Lemma 4.9 the that letters that may be append to its left are
$(i,\ell)$ and $(i,r)$ where $i$ is such that
$A_{1}\in\\{i\\}\times\\{\ell,r\\}$. Similarly the letters that may be append
to its right are $(\pi_{\ell}(j),r)$ and $(\pi_{r}(j),\ell)$ where
$A_{n}\in\\{j\\}\times\\{\ell,r\\}$. From the Lemma 4.9 the beam $[W]$ is an
immersed polygon whose sides are either vertical or part of the sides of $Q$.
Because $W$ is bispecial, both the top and bottom of $Q$ consists of two sides
and in particular they contain the top singularity of $q_{j}$ and the bottom
singularity of $q_{i}$ respectively. Consider the saddle connection $v$ which
connects these two singularities and let us assume without loss of generality
that it is left slanted (as in Figure 24(a)). Let us show that it is a best
approximation by constructing an immersed rectangle that has $v$ as its
diagonal. Consider the vertical trajectory $v_{\ell}$ in the beam that hits
the top singularity of $q_{j}$ and the vertical trajectory $v_{r}$ in the beam
emanating from the bottom singularity of $q_{i}$. Let us consider the
quadrilateral $P$ built from the beam $[W]$ by cutting its left and right
parts up $v_{\ell}$ and $v_{r}$, see the dark quadrilateral in Figure 24(a)).
Then flow vertically forward each point on the top sides and backward each
point on the bottom sides until they first hit an horizontal trajectory.
Extending $P$ by these trajectories, we obtain a rectangle $R$ which contains
$P$, as shown in Figure 24(b). By construction $R$ is a rectangle which has
$v$ as a diagonal and it does not contain singularities in its interior (since
$P$ is contained in the interior of the beam and when extending top and bottom
sides one hits a horizontal before hitting a singularity by definition of
quadrangulation). This shows that $v$ is a best approximation and, arguing as
in the previous part of the proof, it also follows that $v$ has cutting
sequence $W$.
∎
Exploting Lemma 4.7, we can now deduce Theorem 1.13 from Theorem 1.12.
###### Proof of Theorem 1.13.
Let $X$ be a surface in $\mathcal{C}^{hyp}(k)$ with no vertical saddle
connections. Let $Q$ be a quadrangulation of $X$ and let $\mathcal{L}_{Q}$ be
the associated language. Let $(Q^{(n)})_{n\in\mathbb{N}}$ be a sequence of
quadrangulations $Q^{(n)}=({\underline{\pi}}^{(n)},\underline{w}^{(n)})$
obtained starting from $Q$ by simultaneous staircase moves. Then, by Lemma
4.7, the set of bispecial words coincide with the set of geometric best
approximations $v$ in some $\Gamma_{i}$ such that $\operatorname{Im}v\geq
w_{i,d}$. By Corollary 4.2, these are exactly the diagonals in
$(Q^{(n)})_{n}$. ∎
#### 4.2.2 Cutting sequences by staircase moves
In this section we show how to produce all cutting sequences of best
approximations from the sequence of staircase moves, see Theorem 4.10. The key
step is Lemma 4.11 which describe the combinatorial operation that allows to
deduce the cutting sequence of a diagonal of an admissible quadrilateral
obtained by staircase moves from the cutting sequences of its sides.
###### Theorem 4.10.
Let $X\in\mathcal{C}^{hyp}(k)$ be a translation surface with no vertical
saddle connections. Let $Q$ be a quadrangulation of $X$ and let
$\mathcal{L}_{Q}$ be the associated language of cutting sequences. Let
$\\{Q^{(n)}\\}_{n\in\mathbb{N}}$ be any sequence of labeled quadrangulations
$Q^{(n)}=Q({\underline{\pi}}^{(n)},\underline{w}^{(n)})$ starting from
$Q^{(0)}=Q$ and such that $Q^{(n+1)}$ is obtained from $Q^{(n+1)}$ by
performing a staircase move in the staircase $S_{c_{n}}$ for $Q^{(n)}$ given
by a cycle $c_{n}$ of ${\underline{\pi}}^{(n)}$. Set
$D_{i}^{(0)}=\emptyset,\qquad{L}_{i}^{(0)}=(\pi_{\ell}^{-1}(i),\ell),\qquad{R}_{i}^{(0)}=(\pi_{r}^{-1}(i),r),\qquad\textrm{for}\
1\leq i\leq k.$ (20)
Let ${L}_{i}^{(n)},{R}_{i}^{(n)}$ and $D_{i}^{(n)}$ for $n\geq 1$ be given by
the following recursive formulas:
$\displaystyle{L}^{(n+1)}_{i}$ $\displaystyle=$
$\displaystyle\left\\{\begin{array}[]{ll}{L}^{(n)}_{i}{R}^{(n)}_{\pi_{\ell}^{(n)}(i)}&\text{if
$i\in c_{n}$ and $c_{n}$ is a cycle of $\pi^{(n)}_{r}$},\\\
{L}^{(n)}_{i}&\text{otherwise,}\end{array}\right.$ (23)
$\displaystyle{R}^{(n+1)}_{i}$ $\displaystyle=$
$\displaystyle\left\\{\begin{array}[]{ll}{R}^{(n)}_{i}{L}^{(n)}_{\pi_{r}^{(n)}(i)}&\text{if
$i\in c_{n}$ and $c_{n}$ is a cycle of $\pi^{(n)}_{\ell}$},\\\
{R}^{(n)}_{i}&\text{otherwise,}\end{array}\right.$ (26)
$\displaystyle{D}^{(n+1)}_{i}$ $\displaystyle=$
$\displaystyle\left\\{\begin{array}[]{ll}{D}^{(n)}_{i}{R}^{(n)}_{\pi^{(n)}_{l}\pi_{r}^{(n)}(i)}&\text{if
$i\in c_{n}$ and $c_{n}$ is a cycle of $\pi_{r}$},\\\
{D}^{(n)}_{i}{L}^{(n)}_{\pi^{(n)}_{r}\pi_{\ell}^{(n)}(i)}&\text{if $i\in
c_{n}$ and $c_{n}$ is a cycle of $\pi_{\ell}$},\\\ {D}^{(n)}_{i}&\text{if
$i\notin c_{n}$.}\end{array}\right.$ (30)
Then the bispecial words of $\mathcal{L}_{Q}$ are exactly all words which
appear in the sequences $(D_{i}^{(n)})_{n\in\mathbb{N}}$ for $1\leq i\leq k$.
We will prove Theorem 4.10 from Theorem 1.13 by showing that for any
$n\in\mathbb{N}$ the word $D_{i}^{(n)}$ given by the recursive formulas in the
statement is the cutting sequence of the diagonal $w_{i,d}^{(n)}$ for any
$1\leq i\leq k$. We remark that an analogous Theorem in the setup of interval
exchange transformations is proved by Ferenczi and Zamboni in [18]. In their
context, the analogous of the words $L_{i}^{(n)}$ and $R_{i}^{(n)}$ that are
needed to build the bispecial words $D_{i}^{(n)}$ can be interpreted as
cutting sequences of Rohlin towers for the bipartite IETs
$({\underline{\pi}}^{(n)},{\underline{\lambda}}^{(n)})$ (see §2.2).
Let us first prove a preliminary Lemma that shows how the cutting sequence of
a diagonal of quadrilateral in a quadrangulation can be deduced from the
cutting sequences of the sides and the combinatorial datum (see also Figure
25).
###### Lemma 4.11.
Let $Q=({\underline{\pi}},\underline{w})$ be obtained from
$Q^{\prime}=({\underline{\pi}}^{\prime},\underline{w}^{\prime})$ by a sequence
of staircase moves. Let $W_{i,\ell},W_{i,r}$ be respectively the cutting
sequences of the saddle connections $w_{i,\ell}$ and $w_{i,r}$ with respect to
the labelling of $Q^{\prime}$. Then the cutting sequence $D_{i}$ of a diagonal
$w_{i,d}$ in $Q$ is given by
$D_{i}=\left\\{\begin{array}[]{lll}W_{i,r}\,(j,r)\,(\pi_{r}(i),\ell)\,W_{\pi_{r}(i),\ell}&\text{if
$w_{i,r}\not=w^{\prime}_{i,r}$ and
$w_{\pi_{r}(i),\ell}\not=w^{\prime}_{\pi_{r}(i),\ell}$,}&\ref{subfig:case_nn}\\\
(\pi_{r}(i),\ell)\,W_{\pi_{r}(i),\ell}&\text{if $w_{i,r}=w^{\prime}_{i,r}$ and
$w_{\pi_{r}(i),\ell}\not=w^{\prime}_{\pi_{r}(i),\ell}$,}&\ref{subfig:case_en}\\\
W_{i,r}\,(j,r)&\text{if $w_{i,r}\not=w^{\prime}_{i,r}$ and
$w_{\pi_{r}(i),\ell}=w^{\prime}_{\pi_{r}(i),\ell}$,}&\ref{subfig:case_ne}\\\
\emptyset&\text{if $w_{i,r}=w^{\prime}_{i,r}$ and
$w_{\pi_{r}(i),\ell}=w^{\prime}_{\pi_{r}(i),\ell}$,}&\ref{subfig:case_ee}\end{array}\right.$
(31)
where $j=(\pi_{r}^{\prime})^{-1}\pi_{r}(i)$. Similarly, we have
$D_{i}=\left\\{\begin{array}[]{ll}\emptyset&\text{if
$w_{i,\ell}=w^{\prime}_{i,\ell}$ and
$w_{\pi_{\ell}(i),r}=w^{\prime}_{\pi_{\ell}(i),r}$,}\\\
(\pi_{\ell}(i),r)\,W_{\pi_{\ell}(i),r}&\text{if
$w_{i,\ell}=w^{\prime}_{i,\ell}$ and
$w_{\pi_{\ell}(i),r}\not=w^{\prime}_{\pi_{\ell}(i),r}$,}\\\
W_{i,\ell}\,(j,\ell)&\text{if $w_{i,\ell}\not=w^{\prime}_{i,\ell}$ and
$w_{\pi_{\ell}(i),r}=w^{\prime}_{\pi_{\ell}(i),r}$,}\\\
W_{i,\ell}\,(j,r)\,(\pi_{\ell}(i),r)\,W_{\pi_{\ell}(i),r}&\text{if
$w_{i,\ell}\not=w^{\prime}_{i,\ell}$ and
$w_{\pi_{\ell}(i),r}\not=w^{\prime}_{\pi_{r}(i),\ell}$,}\end{array}\right.$
(32)
where $j=(\pi^{\prime}_{\ell})^{-1}\,\pi_{\ell}(i)$.
(a) (b)
(c) (d)
Figure 25: the four cases in the proof of Lemma 4.11
###### Proof.
We prove only (31) as the case of (32) is the same after vertical reflection.
Let $p_{i}$ for $1\leq i\leq k$ be the vertex of the wedge $w_{i}$ of $Q$.
Consider the quadrilateral $q_{i}\in Q$. The diagonal $w_{i,d}$ divides
$q_{i}$ in two triangles. Let us consider the right triangle $T_{r}$ which has
sides $w_{i,r}$, $w_{i,d}$ and $w_{\pi_{r}(i),\ell}$. Remark that right most
vertex of $T$, that is the endpoint of $w_{i,r}$, is $p_{\pi_{r}(i)}$.
Since $q_{i}$ and hence also $T_{r}$ does not contain any singularity in its
interior, any saddle connection of $Q^{\prime}$ which crosses the diagonal
$w_{i,d}$ has to cross either the union of the interior of the two other sides
$w_{i,r}$ and $w_{\pi_{r}(i),\ell}$ of the triangle, or has $p_{\pi_{r}(i)}$
as an endpoint. Remark that there at most two saddle connections of
$Q^{\prime}$ which intersect $w_{i,d}$ and ends in $p_{\pi_{r}(i)}$ before
leaving $T_{r}$, namely
$w^{\prime}_{\pi_{r}(i),\ell}=w^{\prime}_{\pi^{\prime}_{r}(j),\ell}$ and
$w^{\prime}_{j,r}$ where $j=(\pi^{\prime}_{r})^{-1}\pi_{r}(i)$. The saddle
connection $w^{\prime}_{\pi_{r}(i),\ell}$ crosses $w_{i,d}$ if and only if
$w_{\pi_{r}(i),\ell}\neq w^{\prime}_{\pi_{r}(i),\ell}$ (case (b) and (d) in
(31) and Figure 25). On the other hand, the saddle connection
$w^{\prime}_{j,r}$ crosses $w_{i,d}$ if and only if $w_{i,r}\neq
w^{\prime}_{i,r}$ (case (c) and (d)).
In the case $w_{\pi_{r}(i),\ell}\neq w^{\prime}_{\pi_{r}(i),\ell}$ and
$w^{\prime}_{i,r}\not=w_{i,r}$ (see Figure 25(a)) the cutting sequence of
$w_{i,d}$ is obtained by concatenation of the one of $w_{i,r}$, the two
letters $(j,r)$ and $(\pi_{r}(j),\ell)$ and then the cutting sequence of
$w_{\pi_{r}(i),\ell}$. The other three cases, which are somewhat degenerate,
are obtained similarly, referring to Figures 25(b), 25(c) and 25(d). ∎
Recall that diagonal change consists in replacing one side of a wedge by the
diagonal. Hence Lemma 4.11 already provide a way to obtain recursively the
cutting sequences of sides and diagonals. In order to simplify notations and
gather all four cases Theorem 4.10, we defined words $L_{i}$ and $R_{i}$.
These words are _extended_ cutting sequences of sides, that is cutting
sequences preceded by the labels of some of the incoming edges in the starting
vertex. Keeping the same notation as in the Lemma, let us define $L_{i}$ and
$R_{i}$ from the cutting sequence of the sides by
$\displaystyle L_{i}$ $\displaystyle=$
$\displaystyle\left\\{\begin{array}[]{ll}((\pi^{\prime}_{r})^{-1}(i),r)&\text{if
$w_{i,\ell}=w^{\prime}_{i,\ell}$},\\\
((\pi^{\prime}_{r})^{-1}(i),r)\,(i,\ell)\,W_{i,\ell}&\text{if $w_{i,\ell}\neq
w^{\prime}_{i,\ell}$}.\\\ \end{array}\right.$ (35) $\displaystyle R_{i}$
$\displaystyle=$
$\displaystyle\left\\{\begin{array}[]{ll}((\pi^{\prime}_{\ell})^{-1}(i),\ell)&\text{if
$w_{i,r}=w^{\prime}_{i,r}$},\\\
((\pi^{\prime}_{\ell})^{-1}(i),\ell)\,(i,r)\,W_{i,r}&\text{if $w_{i,r}\neq
w^{\prime}_{i,r}$},\\\ \end{array}\right.$ (38)
###### Proof of Theorem 4.10.
Let us first show by induction on $n$ that the words $L_{i}^{(n)}$,
$R_{i}^{(n)}$ given by the recursive formulas (23) and (26) in the statement
are respectively the words defined from cutting sequence of sides by (35) and
(38).
For $n=0$ the definitions in (20) also coincide with the definitions given by
(35) and (38). Let us fix $n\in\mathbb{N}$ and assume that for all $1\leq
i\leq k$ the words $L_{i}^{(n)},R_{i}^{(n)}$ given by (35) and (38) satisfy
the recursive formulas in the statement and let us show that then the same is
also true for $n+1$. Let us assume that $Q^{(n+1)}$ is obtained from $Q^{(n)}$
by a left staircase move in $S_{c_{n}}$ (i.e. $c_{n}$ is a cycle of
$\pi^{(n)}_{\ell}$).
By definition of a left move, $w_{i,\ell}^{(n+1)}=w_{i,\ell}^{(n)}$ (and hence
$W_{i,r}^{(n+1)}=W_{i,r}^{(n)}$) for every $1\leq i\leq k$ and
$w_{i,r}^{(n+1)}=w_{i,r}^{(n)}$ (and hence $W_{i,r}^{(n+1)}=W_{i,r}^{(n)}$)
unless $i\in c_{n}$. Thus, from (35) and (38) we obtain that
$L_{i}^{(n+1)}=L_{i}^{(n)}$ for all $1\leq i\leq k$ and
$R_{i}^{(n+1)}=R_{i}^{(n)}$ for all $i\notin c_{n}$.
Consider now $i\in c_{n}$. In that case $w_{i,r}^{(n+1)}=w_{i,d}^{(n)}$. We
will consider four possible cases that correspond to the four cases in Lemma
4.11 and Figure 25. Case (a) is the only case that happens for any $n$
sufficiently large. Cases (b), (c) and (d) only happen for initial steps of
the induction and should be treated separately. In Lemma 4.11 we set
$Q^{\prime}=Q^{(0)}$ and $Q=Q^{(n)}$ and with this notation in mind one can
refer to Lemma 4.11 and Figure 25. Using the same notation introduced in Lemma
4.11, we denote $j:=(\pi_{r}^{(0)})^{-1}\pi_{\ell}(i)$.
Case (a): $w^{(n)}_{i,r}\neq w^{(0)}_{i,r}$ and
$w^{(n)}_{\pi_{r}^{(n)}(i),\ell}\not=w^{(0)}_{\pi_{r}^{(n)}(i),\ell}$.
We first apply Lemma 4.11 to $w_{i,r}^{(n+1)}=w_{i,d}^{(n)}$ and get
$W^{(n+1)}_{i,r}=D^{(n)}_{i}=W_{i,r}^{(n)}(j,r)(\pi_{r}^{(0)}(i),\ell)W_{\pi_{r}(i)^{(n)},\ell}=W_{i,r}^{(n)}L^{(n)}_{\pi^{(n)}_{r}(i)}$
Now, using (38) for $R_{i}^{(n)}$ and $R_{i}^{(n+1)}$ we get
$R^{(n+1)}_{i}=((\pi_{r}^{(0)})^{-1}(i),\ell)\,(i,r)\,W^{(n+1)}_{i,r}=((\pi_{r}^{(0)})^{-1}i,\ell)\,(i,r)\,W_{i,r}^{(n)}\,L_{\pi^{(n)}_{r}(i)}=R^{(n)}_{i}\,L^{(n)}_{\pi_{r}^{(n)}(i)}.$
Case (b): $w_{i,r}^{(n)}=w^{(0)}_{i,r}$ and
$w_{\pi_{r}^{(n)}(i),\ell}^{(n)}\neq w^{(0)}_{\pi_{r}^{(n)}(i),\ell}$.
From Lemma 4.11 we get that
$W^{(n+1)}_{i,r}=D^{(n)}_{i}=(\pi_{r}^{(n)}(i),r)W_{\pi_{r}^{(n)}(i),\ell}^{(n)}$
and from (38) we obtain
$R^{(n+1)}_{i}=((\pi^{(0)}_{r})^{-1}(i),\ell)\,(i,r)\,W^{(n+1)}_{i,r}=((\pi^{(0)}_{r})^{-1}(i),\ell)\,(i,r)\,(\pi_{r}^{(n)}(i),r)W_{\pi_{r}^{(n)}(i),\ell}^{(n)}=R^{(n)}_{i}\,L^{(n)}_{\pi^{(n)}_{r}(i)}.$
Case (c): $w_{i,r}^{(n)}\neq w^{(0)}_{i,r}$ and
$w_{\pi_{r}^{(n)}(i),\ell}^{(n)}=w^{(0)}_{\pi_{r}^{(n)}(i),\ell}$.
From Lemma 4.11 we get that $W^{(n+1)}_{i,r}=D^{(n)}_{i}=W_{i,r}^{(n)}(j,r)$
and from (38) we obtain
$R^{(n+1)}_{i}=((\pi^{(0)}_{r})^{-1}(i),\ell)\,(i,r)\,W^{(n+1)}_{i,r}=((\pi^{(0)}_{r})^{-1}(i),\ell)\,(i,r)\,W_{i,r}^{(n)}(j,r)=R^{(n)}_{i}L^{(n)}_{\pi_{r}^{(n)}(i)}.$
Case (d): $w_{i,r}^{(n)}=w^{(0)}_{i,r}$ and
$w_{\pi_{r}^{(n)}(i),\ell}^{(n)}=w_{\pi_{r}^{(n)}(i),\ell}^{(0)}$
In that case, $q_{i}$ is a quadrilateral in both $Q^{(0)}$ and $Q^{(n)}$.
Hence $L_{\pi^{(n)}_{r}(i)}^{(n)}=(i,r)$,
$R_{i}^{(n)}=((\pi_{\ell}^{(0)})^{-1}(i),\ell)$ and
$W^{(n+1)}_{i,r}=D^{(n)}_{i}=\emptyset$. We get
$R_{i}^{(n+1)}=((\pi^{(0)}_{r})^{-1}(i),\ell)\,(i,r)\,W^{(n+1)}_{i,r}=((\pi^{(0)}_{r})^{-1}(i),\ell)\,(i,r)=R_{i}^{(n)}\,L^{(n)}_{\pi_{r}^{(n)}(i)}.$
Hence, in all cases the recursive relation (26) holds for $n+1$. The case of
right staircase move is symmetric in which only the $L_{i}$ change and proves
that (23) holds in that case. This conclude the proof that $L_{i}^{(n)}$ and
$R_{i}^{(n)}$ given recursively in (23) and (26) coincide with the definition
(35) and (38).
Let us now verify the relations (30) about cutting sequence of diagonals. For
$n=0$, The cutting sequences $D^{(0)}_{i}$ of the diagonals $w_{i,d}^{(0)}$
are clearly the empty word for all $1\leq i\leq k$. Now assume that the
relation holds for $n$. We consider as before the case where $i$ has a left
diagonal change at step $n$. We apply Lemma 4.11 to the quadrilaterals
$q_{i}^{\prime}$ obtained after the move. Since its diagonal is
$w_{i,d}^{(n+1)}\neq w_{i,d}^{(n)}$, we get
$D^{(n+1)}_{i}=W_{i,r}^{(n+1)}R^{(n+1)}_{\pi_{r}^{(n+1)}(i)}.$
Since by definition of a move $W_{i,r}^{(n+1)}=D^{(n)}_{i}$ and
$\pi_{r}^{(n+1)}=\pi_{r}^{(n)}\pi_{\ell}^{(n)}$, this shows that the inductive
assumption (30) also holds for $n+1$ and conclude the proof. ∎
## References
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* [6] J. Cassaigne, S. Ferenczi, L. Zamboni, Combinatorial trees arising in the study of interval exchange transformations, European J. of Combin., 32 no. 8, p. 1428–1444 (2011).
* [7] Y. Cheung, Hausdorff dimension of the set of Singular Pairs, Ann. of Math. (2), 173 no. 1, p. 127–167 (2011).
* [8] Y. Cheung, P. Hubert, H. Masur, Dichotomy for the Hausdorff dimension of the set of nonergodic directions, Invent. math., 183 no. 2, p. 337–383 (2011).
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* [10] D. Davis, Cutting sequences, regular polygons and the Veech group, Geom. Dedicata, 162, p. 231–261 (2013).
* [11] V. Delecroix, Divergent trajectories in the periodic windtree model, J. Mod. Dyn., 7 no. 1, p. 1–29 (2013).
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|
arxiv-papers
| 2013-10-03T18:29:26 |
2024-09-04T02:49:51.921247
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/",
"authors": "Vincent Delecroix and Corinna Ulcigrai",
"submitter": "Vincent Delecroix",
"url": "https://arxiv.org/abs/1310.1052"
}
|
1310.1118
|
# Evolution of choices over time: The U.S. Presidential election 2012 and the
NY City Mayoral Election, 2013
Mukkai Krishnamoorthy, Wesley Miller
Rensselaer Polytechnic Institute, Troy, NY Raju Krishnamoorthy
Columbia University, New York, NY
###### Abstract
We conducted surveys before and after the 2012 U.S. Presidential election and
prior to the NY City Mayoral election in 2013. The surveys were done using
Amazon Turk. This poster describes the results of our analysis of the surveys
and predicts the winner of the NY City Mayoral Election.
## 1 Introduction
Several authors have published analyses of poll and survey data of the U.S.
Presidential election 2012 [1][2], with vastly different predictions. Both
sampling differences in voter populations and differences in methods of
analysis may have accounted for the difference in the prediction. In the
current experiment, we perform a survey and an indirect analysis, using Amazon
Turk, as opposed to analysis of direct voter preferences using Amazon Turk
[4][3]. We have focused on voters’ two most pressing concerns, one national
and the other international, from a choice of 5 in each category.
A lot of data analysis of the presidential election in 2012 has been done. In
particular, please see [1]. Other data analysis [2] did not fare that well.
One of the reasons for such a difference could be the sampling of voter
population. Another reason could be due to data analysis. Please see these two
other papers for using Amazon Turk [4][3] in Election Prediction. As opposed
tp these two papers, we do an indirect analysis - not measuring direct voter
preferences.The perspective voters were asked to give on a pressing national
and international concern. Our approach is to do a seconday analysis,
concentrating more on the most important issue (one of a national concern and
the other of international concern among choices of 5 in each category) that
the surveyed people had in their minds.
These are the limitations on our data:
* •
Did all those surveyed for the presidential election 2012, vote?
* •
Were the same people surveyed before and after the election?
* •
Will those surveyed for the NY City Mayoral Primary vote?
* •
Sample size is extremely small, compared to the voting population (200 for the
US Presidential election and 100 for the NY City Mayoral)
Even with these limitations, we feel that it exhibits interesting patterns.
## 2 Data Collection
The data was collected through Amazon Mechanical Turk. Our pool of
participants was people of voting age in the United States, the voting
population. We selected 200 people before the election and asked each one of
them which domestic issue would affect his/her voting decision most when
considering the economy, healthcare, tax reform, education, and national
security. We then asked each one of them which international issue would have
the greatest effect on his/her voting decision when considering withdrawal
from wars, international security, global trade issues, more active
engagement, and international partners. Once the election was over and
President Obama had been re-elected, we posted the same survey for 200 more
people, asking this time which of the issues affected their decisions the most
when they voted. The third set of data was collected from a pool of 100
participants and asked the same set of questions (that we asked during the
presidential election).
## 3 Amazon Turk Experimental Results
Presidential Before
---
| National | International
---|---
| The Economy | 145
---|---
Healthcare | 30
Education | 11
Tax Reform | 12
National Security | 2
| Withdrawal from Wars | 109
---|---
International Security | 39
Global Trade Issues | 40
More Active Engagement | 6
International Partners | 6
Presidential After
| National | International
---|---
| The Economy | 147
---|---
Healthcare | 36
Education | 9
Tax Reform | 7
National Security | 1
| Withdrawal from Wars | 105
---|---
International Security | 43
Global Trade Issues | 42
More Active Engagement | 3
International Partners | 7
Mayoral Before
| City | National
---|---
| The Economy | 71
---|---
Healthcare | 15
Education | 7
Tax Reform | 7
City Security | 0
| Withdrawal from Wars | 39
---|---
National Security | 29
Global Trade Issues | 15
More Active Engagement | 11
State Partners | 6
## 4 Observations
The results of ordering all the surveys (shown in the previous section) remain
the same. This tells us about the general anxiety level of the people. Even
aftera gap of one year, rankings (of concerns) did not change. Even though our
sample size is small, we believe in the authenticity of data because the
survey after the election did not vary even slightly from those taken before
the election. What is more striking is the approximate percentage of people
prefering the choices (in almost all three cases). There is a small change of
percentages of healthcare and International Security (before and after
election) as both are hot button issues among the voting population.
Based on presidential outcome, we conjecture that Mr. de Blasio, the
democratic party’schoice, will win the November Mayoral election in New York
City (Nate Silver predicted Ms. Quinn, who lost in the democratic primaries
[5]).
## 5 Conclusion
There was evident variation in the sets of data; however, it is so small that
it can be attributed to the small sample size relative to the voting
population. There is no evidence of a change in voter opinion between the two
surveys and the mayoral election.
## 6 Acknowledgements
The Authors wish to thank Dr Janaki Krishnamoorthy and Prof Gurpur Prabhu with
their constructive suggestions and editing the manuscript. First author wishes
to thank Mr. Sean O’Sullivan for establishing Rensselaer Center for Open
Source Software where part of this work is carried out.
## References
* [1] Nate Silver, _Five-Thirty-Eight Blogs from New York Times_. http://fivethirtyeight.blogs.nytimes.com/2012/11/05/in-ohio-polls-show-benefit-of-auto-rescue-to-obama/ November 5, 2012. 2012\.
* [2] Dick Morris, _Prediction Romney 325, Obama 213_. http://www.dickmorris.com/prediction-romney-325-obama-213/ November 5, 2012.
* [3] John Sides, _How Representative Are Amazon Mechanical Turk Workers?_ http://themonkeycage.org/2012/12/19/how-representative-are-amazon-mechanical-turk-workers/
* [4] Paul Cuff, Sanjeev Kulkarni, Mark Wang and John Strum, _Voting Research- Voting Theory_ http://www.princeton.edu/~cuff/voting/theory.html
* [5] Jacob Kornbluh, _Nate Silver: Quinn the most likely democrat, Jewish Press, Sept 6, 2013._ http://www.jewishpress.com/news/breaking-news/nate-silver-quinn-the-most-likely-democrat/2013/06/09/
|
arxiv-papers
| 2013-10-03T21:09:48 |
2024-09-04T02:49:51.937194
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Mukkai Krishnamoorthy, Wesley Miller and Raju Krishnamoorthy",
"submitter": "Mukkai Krishnamoorthy",
"url": "https://arxiv.org/abs/1310.1118"
}
|
1310.1184
|
arxiv-papers
| 2013-10-04T07:02:42 |
2024-09-04T02:49:51.942360
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Gurpreet Singh Saini, Manoj Kumar",
"submitter": "Gurpreet Saini",
"url": "https://arxiv.org/abs/1310.1184"
}
|
|
1310.1229
|
# Grain Destruction in a Supernova Remnant Shock Wave
John C. Raymond11affiliation: Harvard-Smithsonian Center for Astrophysics, 60
Garden St., Cambridge, MA 02138, USA; [email protected] Parviz
Ghavamian22affiliation: Dept. of Physics, Astronomy & Geosciences, Towson
University, Towson, MD 21252 Brian J. Williams33affiliation: NASA Goddard
Space Flight Center, Greenbelt, MD 20771 William P. Blair,44affiliation:
Department of Physics and Astronomy, Johns Hopkins University, 3400 N. Charles
St., Baltimore, MD 21218, USA Kazimierz J. Borkowski55affiliation: Department
of Physics, North Carolina State University, Raleigh, NC 27695 Terrance J.
Gaetz11affiliation: Harvard-Smithsonian Center for Astrophysics, 60 Garden
St., Cambridge, MA 02138, USA; [email protected] Ravi
Sankrit66affiliation: SOFIA Science Center, NASA Ames Research Center, M/S
232-12, Moffett Field, CA 94035
###### Abstract
Dust grains are sputtered away in the hot gas behind shock fronts in supernova
remnants, gradually enriching the gas phase with refractory elements. We have
measured emission in C IV $\lambda$1550 from C atoms sputtered from dust in
the gas behind a non-radiative shock wave in the northern Cygnus Loop.
Overall, the intensity observed behind the shock agrees approximately with
predictions from model calculations that match the Spitzer 24 $\mu$m and the
X-ray intensity profiles. Thus these observations confirm the overall picture
of dust destruction in SNR shocks and the sputtering rates used in models.
However, there is a discrepancy in that the CIV intensity 10′′ behind the
shock is too high compared to the intensities at the shock and 25′′ behind it.
Variations in the density, hydrogen neutral fraction and the dust properties
over parsec scales in the pre-shock medium limit our ability to test dust
destruction models in detail.
dust; ISM: individual (Cygnus Loop); ISM: supernova remnants; shock waves;
ultraviolet: ISM
## 1 Introduction
Destruction of dust by supernova remnant (SNR) shock waves controls the
dust/gas ratio in the ISM (Draine, 2009; Dwek & Scalo, 1980; Dwek, 1998). In
SNRs, it also controls the gas phase abundances of refractory elements such as
C, Si and Fe, which are highly depleted in the pre-shock gas but contribute
strongly to SNR X-ray spectra. It also determines the infrared cooling rate,
which dominates over the X-ray cooling rate in shocks faster than about 400
$\rm km~{}s^{-1}$ (Arendt et al., 2010a). However, the destruction rate is
poorly known (Nozawa et al., 2006), and different types of grains are
destroyed at different rates (Serra Diaz-Cano & Jones, 2008).
In radiative shock waves, in which the density increases as the gas cools,
grains collide with each other at high speed due to betatron acceleration. The
colliding grains shatter, altering the size distribution (Shull, 1978;
Borkowski & Dwek, 1995; Jones et al., 1996; Slavin et al., 2004). In non-
radiative shocks, for which the cooling time is large compared to dynamical
time scales, sputtering dominates over grain shattering (Borkowski et al.,
2006). SNR shocks faster than 300 $\rm km~{}s^{-1}$ are typically non-
radiative (Cox, 1972). In the shocked plasma, proton and He++ collisions
sputter atoms off the grains (Draine & Salpeter, 1979). The grains initially
move at 3/4 of the shock speed relative to the plasma, and they gradually slow
down due to gas drag and Coulomb drag. For a shock moving perpendicular to the
magnetic field (quasi-perpendicular shock), the motion is gyrotropic, while
for a quasi-parallel shock the motion is along the flow direction. Until a
grain slows down due to gas drag, its sputtering rate is enhanced by the
increased collision speed. This nonthermal sputtering is especially important
at moderate shock speeds up to about 400 $\rm km~{}s^{-1}$ (Dwek et al.,
1996), typical of middle-aged SNRs such as the Cygnus Loop. Laboratory studies
and computer simulations give sputtering rates (Bianchi & Ferrara, 2005), but
the simulated surfaces may differ from actual interstellar grains, and
therefore the rates are uncertain.
Spitzer observations have provided important new constraints on the mass, size
distribution, temperature distribution and destruction rates of ISM grains in
SNRs. Borkowski et al. (2006) and Williams et al. (2006, 2011) constructed
models of grain heating and destruction in non-radiative shocks to match
Spitzer observations of young LMC SNRs. The spectra and intensities could be
matched by models with fairly standard parameters, but the inferred pre-shock
dust-to-gas ratio in the ambient gas near the LMC remnants was typically 1/5
the average LMC value obtained from extinction studies (Weingartner & Draine,
2001). Arendt et al. (2010a) studied the dust destruction in Puppis A, and the
change in grain size distribution did not match that expected from parameters
derived from X-ray spectra. Winkler et al. (2013) find evidence for a higher
grain destruction rate in SN1006 than expected.
Most dust destruction in the ISM occurs in shocks at modest speeds in middle-
aged SNRs simply because they account for most of the volume swept out during
SNR evolution. A detailed Spitzer study of grain destruction in the Cygnus
Loop was carried out by Sankrit et al. (2010). They obtained 24 $\mu$m and 70
$\mu$m images of a non-radiative shock in the northern Cygnus Loop. Relatively
faint optical and UV emission is produced in a narrow ionization zone just
behind the shock, and it provides several diagnostics for plasma temperatures
(Chevalier & Raymond, 1978; Ghavamian et al., 2001). Because the shock is non-
radiative, there is no significant contribution of emission lines to the IR
spectrum (Williams et al., 2011). The Cygnus Loop was chosen for this
investigation because it is bright, and because the small foreground E(B-V)
means that it can be observed in the UV. It is also nearby, $<$ 640 pc (Blair
et al., 2009; Salvesen et al., 2009), so that the dust destruction zone is
spatially resolved by X-ray and IR instruments. Sankrit et al. (2010) selected
a region where the shock parameters had been determined from H$\alpha$, UV and
X-ray observations (Ghavamian et al., 2001; Raymond et al., 2003).
The Spitzer 24 $\mu$m image is shown along with H$\alpha$ and X-ray images in
Figure 1. Models similar to those of Borkowski et al. (2006) were able to
match the 24 $\mu$m intensity falloff with distance, the variation in the 24
$\mu$m/70 $\mu$m ratio and the IR to X-ray flux ratio. The declines in the 24
$\mu$m/70 $\mu$m ratio and the IR to X-ray ratio clearly demonstrate
destruction of dust, but there remain ambiguities involving the density and
depth of the emitting region, as well as the porosity of the grains. Sankrit
et al. (2010) concluded that non-thermal sputtering due to the motion of the
grains through the plasma is required to match the variation in the 24
$\mu$m/70 $\mu$m ratio. That process is more important in the 400 $\rm
km~{}s^{-1}$ shocks of the Cygnus Loop than at the higher temperatures of the
young SNRs investigated by Borkowski et al. (2006), Williams et al. (2006) and
Winkler et al. (2013) because of the lower thermal speeds of protons and
$\alpha$ particles behind the slower shock. The best fit model of Sankrit et
al. (2010) has only half the dust-to-gas ratio expected for the diffuse ISM.
In spite of the quality of the recent Spitzer observations, crucial questions
about the heating and destruction of interstellar dust in SNR shocks remain.
The inference that the dust-to-gas ratio is only half the expected value could
mean that either the derived dust mass is too small due to incorrect heating
rates or emissivities, or else the destruction rate is underestimated. In this
paper we report the detection of emission in the C IV $\lambda$1550 doublet
from carbon atoms liberated from grains behind the shock in the region
observed by Sankrit et al. (2010). Each neutral carbon atom liberated from a
grain is quickly ionized to C VI or C VII in the hot post-shock gas, but
during the time it spends in each ionization stage it can be excited. Thus
each sputtered carbon atom emits about 30 photons in the C IV doublet before
it is ionized to C V. We derive the rate at which carbon is liberated from
grains and compare the observed intensities with model predictions.
## 2 Observations and Data Reduction
We observed three positions with the Cosmic Origins Spectrograph (COS) (Green
et al., 2012) on HST on 2012 April 25-26. Figure 1 shows the 3 observed
positions overlaid on H$\alpha$, Chandra X-ray and 24 $\mu$m images. The 1.5”
proper motion since the H$\alpha$ image was obtained was taken into account
based on the value of 4.1” in 39 years measured by Salvesen et al. (2009).
Table 1 shows the positions and exposure times. We used the G160M grating
centered at 1577 Å with the PSA aperture covering the spectral ranges 1386 to
1559 Å and 1577 to 1751 Å. The positions were chosen to be 0.4′′, 10′′, and
25′′ behind the shock. The first position was intended to be slightly behind
the shock position delineated by H$\alpha$ because it takes a finite time, and
therefore distance, to ionize carbon up to and through the C IV state. For a
distance to the Cygnus Loop of 640 pc (Salvesen et al., 2009), a post-shock
density somewhat above 1 $\rm cm^{-3}$ (Raymond et al., 2003) and a post-shock
temperature of about $2\times 10^{6}$ K, that distance corresponds to about
0.4′′. Unfortunately, the H$\alpha$ filament bifurcates at that position, and
the COS aperture lies between the two segments. From Figure 1 it can be seen
that the Position 1 aperture was centered on the H$\alpha$ filament about
1.5′′ behind the brightness peak.
It should be noted, however, that appearances can be deceiving. The SNR blast
wave is rippled as a result of velocity variations caused by density
inhomogeneities in the ambient ISM, and the H$\alpha$ filaments are actually
tangencies between the line of sight and the thin (unresolved) emitting region
behind the shock (Hester, 1987). A schematic diagram of the rippled shock
surface and several lines of sight is shown in Figure 2. The uppermost line of
sight would correspond to a bright filament, while the lowermost is close to a
different tangent point, so that it would appear bright. Thus the apparent
change in brightness as a function of apparent distance behind the shock would
contain a secondary brightness peak unrelated to the outermost filament.
The data were processed with the standard COS pipeline except that we fit a
background plus emission lines to the spectrum rather than using the
background-subtracted spectrum in order to get more a reliable estimate of the
uncertainties. The apparent continuum contains both the real hydrogen 2-photon
continuum and the detector background, but we have not attempted to separate
those contributions. The data were binned by 32 pixels for the fits.
Figures 3 to 5 show the C IV and He II emission lines at the three positions.
The errors are based on the RMS deviations from broad wavelength regions
around the lines with an additional contribution from the photon statistics of
the lines. For each position we show the best Gaussian fits to the profiles,
where the wavelength separation of the C IV doublet is fixed at the laboratory
value and the intensity ratio is fixed at 2:1. The Position 3 He II fit is an
exception, because the line is barely detected and the formal best fit has an
unreasonably large width, so the fit shown assumes the Position 2 He II width.
The C IV $\lambda$1550 doublet and the He II $\lambda$1640 line were the only
features detected in the wavelength range sampled. The next brightest features
expected are the Si IV $\lambda$$\lambda$ 1393,1402 doublet, the O III]
$\lambda$$\lambda$ 1664,1666 lines and the O IV] multiplet at $\lambda$1400\.
Those lines were not expected to be detectable because of the low abundance of
Si, the high ionization rate of Si IV, and the small excitation cross sections
of the O III and O IV intercombination lines. Thus, while these lines are
easily detected in radiative shock waves (Raymond et al., 1988), they are not
seen in non-radiative shocks (Raymond et al., 1995, 2003).
The instrument profile of the COS aperture for an extended source is not
Gaussian. Moreover, the Gaussian fits leave correlated residuals, which casts
doubt on the accuracy of the fit. Therefore, we measured intensities by simply
integrating the fluxes above the background over the line profiles, and we use
the integrated errors as estimates of the 1 $\sigma$ uncertainties. We do,
however, use the Gaussian fits as the only reasonable measure of the line
widths. The intensity ratio of the C IV doublet was fixed at its intrinsic
value of 2:1 for the fits to determine line widths. The best fit line widths
for the He II line were wider than those for the C IV lines by about the
amount expected if both helium and carbon are thermally broadened with a
temperature of about $1.5\times 10^{6}$ K, but the uncertainties are larger
than the difference.
Table 2 shows the measured parameters for the three positions. The fluxes were
corrected for a reddening E(B-V) = 0.08 using the Cardelli et al. (1989)
galactic extinction function, as adopted by Raymond et al. (2003). It is
apparent that the C IV flux at Position 1 is smaller than that at Position 2.
This unexpected result is discussed below.
In a thin sheet of emitting gas seen edge-on, resonance scattering can affect
the observed intensities [e.g., Long et al. (1992)], reducing the intensity
ratio of the doublet from its intrinsic 2:1 value and reducing the total C IV
intensity. Raymond et al. (2003) used Far Ultraviolet Spectroscopic Explorer
(FUSE) observations of the O VI doublet to exploit this effect to constrain
the shock parameters and geometry, and they showed that the optical depth in O
VI $\lambda$1032 is $\sim$ 1\. Considering the lower abundance of carbon and
the shorter ionization time of C IV, the optical depth in the C IV
$\lambda$1548 line should be $\sim$0.1. However, the FUSE spectra were
acquired somewhat to the NW of our positions, and they averaged over 20′′x4′′
regions, so a larger optical depth at Position 1 is possible. We performed
Gaussian fits with the I(1548)/I(1550) ratio unconstrained and found the ratio
to be consistent with the optically thin ratio of 2:1. The best fit ratio is
actually slightly above 2, but lower ratios corresponding to optical depths as
large as $\tau_{1548}=0.82$ are within the uncertainties, and that would
reduce the total C IV flux at position 1 by at most 25%. Thus a small optical
depth is indicated by the I(1548)/I(1550) ratio.
We must also consider whether background emission from the Galaxy makes a
significant contribution to the observed fluxes. Martin & Bowyer (1990)
measured C IV fluxes of 2700 to 5700 $\rm ph~{}cm^{-2}~{}s^{-1}~{}sr^{-1}$ at
high galactic latitudes, and they obtained only upper limits somewhat below
those values at low galactic latitudes where the Cygnus Loop lies. Even the
highest Galactic background values are 35 times smaller than what we observe
at Position 3, and we conclude that the measured fluxes originate in the
Cygnus Loop.
## 3 Analysis
We assume shock parameters based on optical, UV and X-ray studies of a portion
of the same H$\alpha$ filament located about 5.7′ farther to the NW. That
region has a more complex H$\alpha$ morphology due to several ripples of the
shock surface [analogous to Blair et al. (2005)], so we chose a set of
positions along strip 1 of Sankrit et al. (2010). Ghavamian et al. (2001)
measured a shock speed of 300-365 $\rm km~{}s^{-1}$ and a ratio of electron to
proton temperatures $T_{e}/T_{p}$ at the shock of 0.7-1.0 from the width of
the H$\alpha$ narrow component and the intensity ratio of the broad and narrow
components. Subsequently, van Adelsberg et al. (2008) were unable to match
both the broad component width and the narrow-to-broad intensity ratio,
perhaps because of a contribution of a shock precursor to the narrow
component. However, the electron temperature determined from X-rays (Raymond
et al., 2003; Salvesen et al., 2009) supports the conclusion that $T_{e}$ is
nearly equal to $T_{p}$, and in that case the broad component width indicates
a shock speed at the upper end of the range given by Ghavamian et al. (2001).
UV observations of a section of the Balmer filament a few arcminutes NW of our
Position 1 by FUSE showed that the proton and oxygen kinetic temperatures were
close to equilibration (Raymond et al., 2003). The relative intensities of the
C IV and He II lines in a UV spectrum from the Hopkins Ultraviolet Telescope
(HUT) indicated about half the solar carbon abundance, meaning that half the
carbon entered the shock in the gas phase or in very small grains that were
vaporized within the HUT aperture, that is, within 5′′ of the shock (Raymond
et al., 2003). That paper also used the optical depths in the O VI lines to
estimate a depth along the line-of-sight of 0.7-1.5 pc, pre-shock density of
0.3-0.5 $\rm cm^{-3}$ and a pre-shock neutral fraction of about 0.5. Salvesen
et al. (2009) measured proper motions of filaments along the northern Cygnus
Loop. Their filament 6 coincides with our Position 1, and the proper motion is
0.105”/yr or 333 $\rm km~{}s^{-1}$ at 640 pc. Katsuda et al. (2008) analyzed
Chandra observations of the NE region of the Cygnus Loop, and our Position 1
is located near the SE end of their Area 1. The emission measure they derive
is compatible with a pre-shock density of 0.5 $\rm cm^{-3}$ and a line-of-
sight depth of 1.5 to 2 pc. However, their derived electron temperature of
about 0.27 keV is about twice what one would expect from the 333 $\rm
km~{}s^{-1}$ shock speed given by the proper motion and the 640 pc upper limit
to the distance of Blair et al. (2009).
To interpret the line fluxes, we need to know how many photons each atom emits
before it is ionized. Since the post-shock temperature is far above the
temperature where C IV and He II are found in ionization equilibrium (log T =
5.0 and 4.7, respectively), each atom survives for a time
$\tau_{ion}=1/(n_{e}q_{i})$ and it is excited at a rate $n_{e}q_{ex}$.
Therefore, it emits on average $q_{ex}/q_{i}$ photons, where $q_{ex}$ is the
excitation rate and $q_{i}$ is the ionization rate, before it is ionized.
Using Version 6 of CHIANTI (Dere et al., 2009), specifically the He II
excitation computed by Connor Ballance for that database and the C IV
excitation rate from Griffin et al. (2000), with the ionization rates of Dere
(2007), we find that each C atom emits 31 $\lambda$1550 photons, while each He
atom emits 0.078 $\lambda$1640 photons. Thus for solar abundances (Asplund et
al., 2009), the C: He ratio of 0.0032 implies an intensity ratio I(C IV)/I(He
II) = 1.33 (in $\rm erg~{}cm^{-2}~{}s^{-1}$). We will also use the similar
number for hydrogen; each neutral H atom passing through the shock produces
0.25 H$\alpha$ photons (Chevalier et al., 1980).
The H$\alpha$ image shown in Figure 1 was obtained in 1999 at the 1.2 m
telescope at the Fred Lawrence Whipple Observatory. It was calibrated based on
the optical spectrum at a nearby position, and H$\alpha$ fluxes in several
apertures are given in Raymond et al. (2003). We determined the H$\alpha$
fluxes within the COS apertures at the three positions, and they are shown in
Table 2. The intensity ratios can be combined with the numbers of photons per
atom to infer the neutral fraction of hydrogen entering the shock. We asssume
that helium is entirely neutral or singly ionized. Substantial numbers of He I
$\lambda$584 and He II$\lambda$ 304 photons can ionize H and He I, but the
shock produces relatively few photons above 54.4 eV, and the photoionization
cross section at those energies is relatively small. We use the ratio of
$\lambda$1640 intensity to H$\alpha$ at Position 1 to derive a neutral
fraction of 0.11 with an uncertainty of a factor of 1.8, including a 32%
measurement uncertainty in the $\lambda$1640 intensity (2-$\sigma$) and
uncertainty estimates in reddening correction and H$\alpha$ calibration. We
therefore estimate a hydrogen neutral fraction of 0.06 to 0.20. That is
smaller than the more model-dependent estimate of Raymond et al. (2003), but
in keeping with the upper limit of 0.2 from the limit on the ratio of He II
$\lambda$ to H$\alpha$ (Ghavamian et al., 2001).
### 3.1 Gas phase and sputtered carbon contributions to C IV
The observed intensities are the sum of emission from C atoms liberated from
dust grains downstream of the shock and emission from near the shock as it
curves around the Cygnus Loop, projected onto the line of sight. On the other
hand, He is very quickly ionized, and since it is not depleted onto grains,
the $\lambda$1640 emission is produced only at the shock front. Based on the
temperatures and densities above, it originates within 1′′ of the shock. The
aperture at Position 1 includes both “gas phase” C IV emission and emission
from carbon sputtered from dust grains that passed through parts of the shock
that appear ahead of Position 1 in projection. Therefore, we take the Position
1 ratio of I(C IV)/I(He II) = 1.1 to be an upper limit to the ratio due to
emission at the shock from carbon that passes through the shock in the gas
phase. This includes emission from PAHs and very small grains that are
vaporized near the shock (Micelotta et al., 2010). Comparison of the value of
1.1 with the theoretical value of 1.33 indicates that at most 80% of the
carbon is in the gas phase or PAHs at the shock. HUT measured a C IV: He II
ratio of 0.73 for a section of this filament farther to the NW. The 10′′ HUT
wide aperture was placed along the filament, so it includes C IV emission from
carbon that is vaporized from grains within 5′′ of the shock. This gives a
more stringent limit of 0.45 for the fraction of carbon entering the shock in
the gas phase or PAHs and it suggests a significant amount of sputtering
within 5′′ of the shock. The HUT upper limit is comparable to the estimated
dust-to-gas ratio of about one half the Galactic value from Sankrit et al.
(2010).
We assume that the ratio of C IV to He II produced at the shock is constant
and use it to subtract off the contribution to the C IV emission from the
shocks projected onto the line of sight at Positions 2 and 3. Taking the C
IV:He II ratio at the shock to be $<1.1$, we find that the C IV emission from
carbon liberated from dust is 16 to 27 and 3.3 to 5.5$\times 10^{-16}~{}\rm
erg~{}cm^{-2}~{}s^{-1}$ at positions 2 and 3, respectively. Those fluxes imply
that carbon is being sputtered from grains at rates of 4.6 to $7.7\times
10^{5}$ atoms $\rm cm^{-2}~{}s^{-1}$ at Position 2 and 0.9 to $1.5\times
10^{5}$ atoms $\rm cm^{-2}~{}s^{-1}$ at Position 3.
### 3.2 Line widths
The measured widths of the $\lambda$1640 profiles are larger than those of the
C IV lines as expected if their kinetic temperatures are equal, but the widths
are equal within the uncertainties. The thermal width of the helium line is
141 $\rm km~{}s^{-1}$ (FWHM) at $1.7\times 10^{6}$ K expected for the 350 $\rm
km~{}s^{-1}$ shock speed obtained by Ghavamian et al. (2001) from the
H$\alpha$ profile, compared with 82 $\rm km~{}s^{-1}$ for carbon. The COS line
profile for diffuse emission that fills the aperture is not exactly known
beyond the statement that the width is about 200 $\rm km~{}s^{-1}$ (France et
al., 2009), so the observed line widths are consistent with thermal broadening
and equal carbon and helium and hydrogen kinetic temperatures. This is in
contrast with shocks in the solar wind (Korreck et al., 2007; Zimbardo, 2011)
and faster SNR shocks (Raymond et al., 1995; Korreck et al., 2004), where more
massive ions have much higher temperatures, but the uncertainties permit a
wide range of kinetic temperatures. Moreover, the larger grains slow down very
gradually in the shocked plasma due to Coulomb collisions, though they may
gyrate about the magnetic field and move with the bulk flow (Dwek et al.,
1996; Sankrit et al., 2010). Carbon atoms sputtered from these grains will
initially move at a high speed, potentially giving a line width comparable to
the H$\alpha$ line width, as we discuss below.
## 4 Physical Models
Sankrit et al. (2010) found a dust-to-gas ratio about half the typical
galactic value, and we used their model to predict the C IV emission, using
the fraction of dust remaining as a function of distance behind the shock to
estimate the sputtering rate and assuming 31 photons per C atom. We measure a
higher C IV intensity than expected from a simple plane-parallel model,
suggesting that emission from gas phase carbon makes a significant
contribution or that the simple plane parallel model is not adequate. We
therefore make more detailed models of the destruction of grains and the
emission from sputtered carbon, and we use those models with the shock
geometry inferred from the H$\alpha$ image to predict the C IV brightness for
comparison with the observed values.
### 4.1 Grain Destruction Models
Sankrit et al. (2010) computed models of grain destruction and IR emission to
match Spitzer observations of this part of the Cygnus Loop, and we have used a
model close to their lowest temperature model, which matches the shock speed
determined from the proper motion, to predict the C IV emission from carbon
sputtered from dust. The model, which is described more fully in Williams
(2010), includes the enhanced sputtering due to the motion of grains through
the hot plasma. Sputtering of a dust grain in a hot plasma is a function of
both the temperature (energy per collision) and density (frequency of
collisions) of the gas. We include sputtering by both protons and alpha
particles, assuming cosmic abundances, such that nα = 0.1np. We use sputtering
rates from Nozawa et al. (2006), augmented by calculations of an enhancement
in sputtering yields for small grains by Jurac et al. (1998). We use the pre-
shock grain size distributions of Weingartner & Draine (2001) and calculate
the sputtering for grain sizes from 1 nm to 1 $\mu$m as a function of the
sputtering timescale, $\tau$, defined as the integral of the proton density
over the time since the gas was shocked. As a comparison, we also computed
sputtering rates for model BARE-GR-S of Zubko et al. (2004), and found rates
that were over twice as large near the shock and about 50% higher for $\tau$
corresponding to our Position 2 and 30% higher for Position 3.
The total mass in grains is calculated by integrating over the grain-size
distribution, which changes as a function of $\tau$ due to sputtering.
Relative motions between the dust grains and the hot gas, which result in
“non-thermal” sputtering, are included by solving the coupled differential
equations for grain radius and velocity given in Draine & Salpeter (1979).
Dust grains enter the shock with a velocity of 3vs/4 relative to the
downstream plasma, and slow down due to collisions with the ambient gas.
Initially, sputtering is a combination of both thermal and non-thermal
effects, with the non-thermal effects going to zero as gas drag and to a
lesser extent Coulomb drag slow the grains with respect to the gas. Just
behind the shock, the motion of the grains through the plasma approximately
doubles the sputtering rate. By the time the gas reaches our Positions 2 and
3, the grains smaller than about 0.01 $\mu$m have slowed enough that
sputtering rates approach the thermal value, while grains about 0.1 $\mu$m
still experience the enhanced rate. Based on the results of Ghavamian et al.
(2001) and Raymond et al. (2003) we assume equal electron and ion temperatures
of 160 eV, which corresponds to a shock speed of 366 $\rm km~{}s^{-1}$ .
Katsuda et al. (2008) find a higher electron temperature of about 270 eV for
regions to the NW of our positions, but we adopt the shock speed based on
measured proper motions (Salvesen et al., 2009).
Figure 6 shows the predicted dust destruction rates and fractions of dust
remaining as a function of $\tau$ behind the shock. Silicates behave somewhat
differently than carbonaceous grains, and we show the silicate curves for
comparison. For a post-shock density of 2 $\rm cm^{-3}$, the shock proper
motion of 0.105′′ per year and an assumed compression of a factor of 4 by the
shock (so that the post-shock gas moves at 0.0265′′ per year relative to the
shock), Positions 2 and 3 correspond to $2.4\times 10^{10}$ and $5.9\times
10^{10}~{}\rm cm^{-3}~{}s$, respectively. The model assumes that 25% of the
carbon is initially in atomic or molecular form or in PAHs that are destroyed
very rapidly close to the shock.
Figure 7 shows how grains are decelerated behind the shock by the gas drag
(Baines et al. 1965; Draine & Salpeter 1979), for grains with initial
(preshock) radii of 0.005, 0.01, 0.02, 0.04, and 0.1 $\mu$m. (We did not
include the Coulomb drag in these calculations, as it is generally less
important than the gas drag in hot X-ray emitting plasmas due to the small
values of charge on the grains). For large grains, deceleration is modest on
temporal scales of interest in this paper because the slowing-down time (drag
time) is long. As the drag time scales linearly with the grain density,
carbonaceous grains are decelerated more rapidly than silicate grains. For
small grains, both sputtering and gas drag become important, and their
combined effects lead to rapid deceleration. This is particularly pronounced
for silicate grains because their sputtering yields are larger than for
carbonaceous grains (Nozawa et al. 2006). For example, silicate grains with
the initial grain size of 0.005 $\mu$m quickly slow down and are completely
sputtered away at $\tau=3.5\times 10^{10}$ cm-3 s.
Figure 8 shows the mass fraction in grains as a function of velocity relative
to the post-shock plasma. Again, solid lines depict carbonaceous grains with
initial radii of 0.005, 0.01, 0.02, 0.04, and 0.1 $\mu$m from bottom to top.
Much of the carbon liberated from grains comes from the smaller ones, so much
of the carbon is produced when grains have slowed by about a factor of 2
relative to the gas. This will affect the width of the C IV lines from
sputtered carbon. If the shock is a parallel shock, the dust is not initially
compressed by a factor of 4 along with the gas, but it is compressed as the
velocity decreases. That could affect the spatial distribution of C IV
emission and IR intensity behind the shock. Figure 8 also indicates the
velocities of the grains from which the carbon is sputtered. A preliminary
calculation of the line profile at Position 2 assuming a perpendicular shock
and a magnetic field in the plane of the sky shows an emission plateau 540
$\rm km~{}s^{-1}$ wide (FWZI), with a central component of about 200 $\rm
km~{}s^{-1}$ (FWHM). For other magnetic geometries, those widths would be
multiplied by the cosine of the angle between the shock normal and the
magnetic field and by the sine of the angle between the magnetic field and the
line of sight. However, the motion of particles along the field relative to
the plasma will affect the velocity distribution at a given location. The
measured FWHM at Position 2 is just within the uncertainties of the predicted
central component width. The predicted profile at Position 3 consists mostly
of the broad plateau, which is compatible with the 460 $\rm km~{}s^{-1}$ upper
limit to the measured velocity width at that position.
### 4.2 Comparison to observations
As can be seen in Figure 1, the shock is not a simple planar sheet
conveniently oriented along our line of sight. Rather, it is a rippled sheet
that appears bright where it is tangent to our line of sight (Hester, 1987).
Therefore, we cannot simply compare the model prediction in Figure 6 with the
observed fluxes without considering the geometrical structure of the shock
front. Indeed, the sputtering rate $dF/d\tau$ from Figure 6 drops steeply, and
for $\tau$ about $5\times 10^{10}~{}\rm cm^{-3}~{}s$ it is very low, so that
multiplying that value by the column density of carbon and the photon yield
per atom yields a C IV flux below that observed.
To describe the shape of the shock front, we use the H$\alpha$ image shown in
Figure 1. The intensities from a 3 pixel wide average along the line
connecting Positions 1, 2 and 3 are shown in Figure 9. It was shown above that
the neutral fraction in the pre-shock gas is 0.06 to 0.2, and each H atom
produces 0.25 H$\alpha$ photons just after it passes through the shock, so the
H$\alpha$ brightness is directly related to the flux of particles through the
shock at each pixel. We next assume that the ratio of carbon to hydrogen is
$3\times 10^{-4}$ by number (Asplund et al., 2009) with 75% in dust and use
the model shown in Figure 6 to compute the C IV intensity at each pixel along
the cut through Positions 1, 2 and 3 assuming a post-shock density of 2 $\rm
cm^{-3}$. Note that the intensity has two components. First, there is C IV
produced immediately behind the shock from carbon that was in the gas phase or
PAHs, which is proportional to the local H$\alpha$ brightness. Second, there
is the C IV from carbon sputtered from grains that passed through the shock in
pixels farther toward the outside of the remnant. We assume that the H$\alpha$
is formed at the shock front, and its brightness indicates the mass flux
through the shock at each position as shown in Figure 9. We then use the
emission as a function of spatial offset from the shock derived from Figure 6
to compute the C IV emission from gas at all downstream pixels and sum the
contributions from the shocks the positions along Figure 9.
Figure 10 shows the predicted C IV intensities along with the observed values,
where we have assumed a pre-shock neutral fraction of 0.2, at the upper end of
the range determined above. The predictions lie above the observations at
Positions 1 and 3, but below the observations at Position 2. Note that the
predicted rate of liberation of carbon from grains, $dF/d\tau$ in Figure 6,
drops steeply with $\tau$, and much of the emission at Position 3 arises from
gas that was shocked relatively close to Position 3 (in projection) rather
than gas that passed through the shock seen as the H$\alpha$ filament.
Overall, the approximate agreement between the observed and predicted fluxes
is encouraging considering the uncertainties in the gas-phase abundances at
the shock, the sputtering rate, the grain size distribution, the post-shock
temperature and the pre-shock neutral fraction. If some of the H$\alpha$
arises in a shock precursor (Hester et al., 1994; Raymond et al., 2011), the
mass flux through the shock and therefore the predicted C IV emission would be
overestimated. On the other hand, if the pre-shock neutral fraction is
overestimated, the mass flux is underestimated and the predicted C IV is
underestimated. If the post-shock temperature we have assumed is
underestimated, the sputtering rate is also underestimated. Another
uncertainty is that some of the H$\alpha$ emission can arise from the
photoionization precursor. Though this is absent behind the shock in the plane
parallel case, the curvature of the SNR blastwave means that some of the
precursor emission will be seen in projection behind the shock, leading to an
overprediction of the C IV intensity. We have no way to resolve these
ambiguities, but conclude that a model of grain destruction with current
sputtering rates, combined with plausible parameters for the shock, predicts a
level of C IV emission from C atoms liberated from grains in rough agreement
with observations.
However, the discrepancy between Position 2 and the other 2 positions remains.
There are several possible explanations.
Neutral fraction variations: We have assumed that the hydrogen neutral
fraction is constant throughout the relevant part of the ISM. In a region
where the neutral fraction is larger, a given H$\alpha$ brightness would
translate into a lower mass flux than the value used in the model, and the C
IV brightness would be smaller. We derived the neutral fraction from the
Position 1 observation, so this explanation would require that the neutral
fraction changes between Positions 1 and 2.
Gas phase carbon variations: The model assumes that the fraction of carbon in
the gas phase at the shock is 0.25 everywhere. The value quite likely varies
by a factor of 2 in the ISM (Jenkins, 2009; Sofia & Parvathi, 2009). Figure 11
shows that reducing the gas phase fraction to about 10% would bring the
Position 1 C IV flux into agreement but then the model underpredicts the
intensity at Position 2.
Sputtering rate: The sputtering rate is poorly known (Nozawa et al., 2006),
and it scales with the post-shock density, which our models assume to be 2
$\rm cm^{-3}$. Increasing the sputtering rate would increase the C IV
intensity from sputtered carbon, especially in pixels just behind the shock. A
combination of smaller fraction of carbon in the gas phase and higher
sputtering rate might in principle decrease the level of disagreement, but it
would not give a higher C IV intensity at Position 2 than at Position 1.
Sputtering rates computed with the Zubko et al. (2004) size distribution would
be higher everywhere, and they would not help resolve the discrepancy.
Optical depth: If resonant scattering in the edge-on sheet of gas just behind
the shock reduces the C IV intensity at Position 1 by a factor of 2, but does
not affect Positions 2 or 3, that would solve the problem. However, that would
require an optical depth of 3.5 in the $\lambda$1548 line, which would imply a
C IV doublet ratio of only 1.2, which is not compatible with the observed
spectrum.
ISM inhomogeneity: The $H\alpha$ intensity may not be an adequate proxy for
the flux of material through the shock. The IR surface brightness from Spitzer
peaks perhaps 30” behind the bright H$\alpha$ filament, which is not
consistent with either a planar shock picture or with the convolution of the
H$\alpha$ brightness with the sputtering model. The bright IR emission could
indicate that the shock seen in projection about 20” behind the H$\alpha$
filament is now passing through a region with very low hydrogen neutral
fraction, or it could be that the shock passed through a dense or dusty clump
about 1000 years ago. A similar variation in density along the shock path was
noted by Winkler et al. (2013) in SN1006.
Overall, while we have been able to detect the C IV emission from carbon
sputtered from grains, it is likely that projection effects (caused by
variations in density, changes in neutral fraction, variable dust properties
or some combination of these) limit our ability to provide a definitive test
of grain destruction models with these data.
## 5 Summary
A simple, model-independent estimate of the C IV emission from carbon atoms
sputtered from dust grains behind the shock was made by assuming that the
ratio of C IV to He II emission from carbon in the gas phase at the shock is
constant over the region observed. An attempt to model the emission in more
detail by using the H$\alpha$ intensity to map out the geometrical structure
of the shock produced general agreement to about a factor of 1.5, but a
significant discrepancy among the positions remained. In particular, the
models cannot explain why the C IV is brighter at Position 2 than at Position
1. We considered several explanations for this discrepancy, but the most
likely is that the properties of the ISM, in particular density, neutral
fraction or dust properties, vary over parsec scales. Overall, we find that
the C IV emission from carbon sputtered from grains is compatible with the
sputtering rate and grain size distribution assumed by Sankrit et al. (2010),
but preferably with a smaller fraction of carbon in the gas phase and PAHs
than assumed by those models.
The complexity of the structure along the line of sight prevents us from
deriving stronger constraints at present. Further progress could be achieved
by 1) observing a simpler shock structure in the Cygnus Loop or another SNR,
2) resolving the discrepancy between shock speed derived from proper motions
and post-shock temperature from the X-ray spectra, 3) sorting out the
geometrical complexity of this part of the Cygnus Loop, for instance by
obtaining more observations of the He II line and a higher resolution
H$\alpha$ image, or 5) obtaining deeper X-ray spectra to better determine the
temperature, density and elemental abundances of the shocked plasma.
This work was performed under grant HST-GO-12885 to the Smithsonian
Astrophysical Observatory. P.G was supported under grant HST-GO-12545.08, and
T.J.G. acknowledges support under NASA contract NAS8-03060.
Facilities: HST (COS) FLWO:1.2m
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Table 1
COS Observations
Position | RA(2000) | Dec(2000) | Dist. from Shock | Exp. Time
---|---|---|---|---
1 | 20 54 43.611 | 32 16 03.53 | 0.43′′ | 2500
2 | 20 54 43.055 | 32 15 56.46 | 10′′ | 7801
3 | 20 54 42.221 | 32 15 45.85 | 25′′ | 14501
Table 2
C IV $\lambda$1550 and He II $\lambda$1640 Fluxes and Widths
| Observed | Dereddened
---|---|---
Position | F${}_{1550}^{a}$ | w${}_{1550}^{b}$ | F${}_{1640}^{a}$ | w${}_{1640}^{b}$ | I${}_{1550}^{a}$ | I${}_{1640}^{a}$ | I${}_{H\alpha}^{c}$
1 | 11.2$\pm$0.96 | $247_{-68}^{+110}$ | 10.1$\pm$1.6 | $280_{-70}^{+104}$ | 20.7 | 18.3 | 17.4
2 | 14.8$\pm$0.59 | $237_{-34}^{+38}$ | 5.5$\pm$0.94 | $384_{-134}^{+195}$ | 27.3 | 9.95 | 7.36
3 | 3.0$\pm$0.44 | $324_{-92}^{+135}$ | 1.1$\pm$0.69 | $1340_{-1020}^{+240}$ | 5.54 | 1.99 | 2.79
a $10^{-16}~{}erg~{}cm^{-2}~{}s^{-1}$
b FWHM uncorrected for instrument profile: km s-1
c from H$\alpha$ image from Mt. Hopkins 1.2-m telescope
Figure 1: COS aperture positions overlaid on H$\alpha$, Spitzer 24 $\mu$m
images and Chandra X-ray images. The right hand panel shows Positions 1, 2 and
3 (left to right) overlaid on a 3 color superposition of H$\alpha$ (red), 24
$\mu$m (green) and X-rays (blue). The image scale is indicated by the 10′′ and
15′′ spacings between the COS aperture positions. Figure 2: Schematic diagram
of the rippled shock surface and three lines of sight at different distances
behind the shock tangency. The outermost line of sight is tangent to the
shock, giving a bright filament, while the innermost line of sight would give
a secondary brightness peak due to near tangency with the second ripple. The
observer is located far to the left in this schematic. Figure 3: C IV
doublet and He II $\lambda$1640 line at position 1. The solid curve is the
best Gaussian fit, the solid histogram is the data, and the dashed histogram
shows the uncertainties. Figure 4: C IV doublet and He II $\lambda$1640 line
at position 2. Figure 5: C IV doublet and He II $\lambda$1640 line at
position 3. Figure 6: Left panel; predictions for the fractions of carbon and
silicates remaining in grains as a function of $\tau=n_{e}t$ according to the
model described in section 4.1. Right panel; rates at which carbon and
silicates are sputtered from grains as a function of $\tau$. Figure 7: Grain
velocities relative to the shocked plasma as a function of $\tau$ for grain
sizes 0.1, 0.04, 0.02, 0.01, and 0.005$\mu$m (top to bottom). Solid curves are
for carbonaceous grains, and dashed curves for silicates. Figure 8: Mass
fraction in grains as a function of velocity relative to the shocked plasma
for grain sizes 0.1, 0.04, 0.02, 0.01, and 0.005$\mu$m (top to bottom). Solid
lines pertain to carbonaceous grains, and dashed lines show silicate grains
for comparison. Figure 9: H$\alpha$ surface brightness along a line through
Positions 1, 2 and 3 from a 2.8′′ (3 pixel) wide average from the image in
Figure 1. Positions 1, 2 and 3 are located at x-values 0, 10 and 25. There is
a star at -50. Figure 10: Predicted C IV surface brightness due to carbon
that is in the gas phase at the shock (short dashed), carbon sputtered from
grains (long dashed) and the total (solid). COS observations at positions 1, 2
and 3 are shown with 2$\sigma$ error bars. Figure 11: Predicted C IV surface
brightness due to carbon for gas phase fractions of carbon at the shock of
0.25 (solid), 0.40 (dashed) and 0.15 (dotted). COS observations at positions
1, 2 and 3 are shown with 2$\sigma$ error bars.
|
arxiv-papers
| 2013-10-04T11:52:24 |
2024-09-04T02:49:51.950217
|
{
"license": "Public Domain",
"authors": "John C. Raymond, Parviz Ghavamian, Brian J. Williams, William P.\n Blair, Kazimierz J. Borkowski, Terrance J. Gaetz and Ravi Sankrit",
"submitter": "John C. Raymond",
"url": "https://arxiv.org/abs/1310.1229"
}
|
1310.1273
|
# On the symmetric doubly stochastic matrices that are determined by their
spectra
Bassam Mourad111 Fax:+961 7 768174. [email protected] Hassan Abbas Department
of Mathematics, Faculty of Science V, Lebanese University, Nabatieh, Lebanon
Department of Mathematics, Faculty of Science I, Lebanese University, Beirut,
Lebanon
###### Abstract
A symmetric doubly stochastic matrix $A$ is said to be determined by its
spectra if the only symmetric doubly stochastic matrices that are similar to
$A$ are of the form $P^{T}AP$ for some permutation matrix $P.$ The problem of
characterizing such matrices is considered here. An “almost” the same but a
more difficult problem was proposed by [ M. Fang, A note on the inverse
eigenvalue problem for symmetric doubly stochastic matrices, Lin. Alg. Appl.,
432 (2010) 2925-2927] as follows: “Characterize all the $n$-tuples
$\lambda=(1,\lambda_{2},...,\lambda_{n})$ such that up to a permutation
similarity, there exists a unique symmetric doubly stochastic matrix with
spectrum $\lambda.$” In this short note, some general results concerning our
two problems are first obtained. Then, we completely solve these two problems
for the case $n=3.$ Some connections with spectral graph theory are then
studied. Finally, concerning the general case, two open questions are posed
and a conjecture is introduced.
###### keywords:
Doubly stochastic matrices , Inverse eigenvalue problem, Spectral
characterization
###### MSC:
15A12, 15A18, 15A51, 05C50
## 1 Introduction
An $n\times n$ real matrix $A$ having each row and column sum equal to 1 is
called doubly quasi-stochastic. If, in addition, $A$ is nonnegative then $A$
is said to be doubly stochastic. The set of all $n\times n$ doubly-stochastic
matrices is denoted by $\Delta_{n}$ and the set of all the symmetric elements
in $\Delta_{n}$ will be denoted by $\Delta^{s}_{n}.$ For $0\leq a\leq n,$
denote by $\Delta^{s}_{n}(a)$ to be the set of all elements of
$\Delta^{s}_{n}$ with trace $a.$
Three particular elements of $\Delta^{s}_{n}$ are of interest to us. The first
is $I_{n}$ which is the $n\times n$ identity matrix and the second $J_{n}$
which is the $n\times n$ matrix whose all entries are $\frac{1}{n}.$ The third
is $C_{n}$ which denotes the $n\times n$ matrix whose diagonal entries are all
zeroes and whose off-diagonal entries are equal to $\frac{1}{n-1}$ (here for
$n\geq 2$). In addition, let
$e_{n}=\frac{1}{\sqrt{n}}(1,1,...,1)^{T}\in\mathbb{R}^{n}$ where $\mathbb{R}$
denotes the real line. For two matrices (and in particular for row vectors)
$A$ and $B,$ the line-segment joining them is denoted by $[A,B],$ and for any
two sets $E$ and $F$ we write $E-F$ to denote the set of elements in $E$ which
are not in $F.$
Note that it is clear from the definition that an $n\times n$ real matrix $A$
is doubly quasi-stochastic if and only if $Ae_{n}=e_{n}$ and
$e_{n}^{T}A=e_{n}^{T}$ if and only if $AJ_{n}=J_{n}A=J_{n}.$
The symmetric doubly stochastic inverse eigenvalue problem asks which sets of
$n$ real numbers occur as the spectrum of an $n\times n$ symmetric doubly
stochastic matrix. For more information on this problem see, e.g., [7, 8, 9,
10, 13, 14, 15, 16, 17, 18].
Regarding the inverse eigenvalue problem for symmetric doubly stochastic
matrices, the following “inaccurate” proposition was presented in [8].
###### Proposition 1.1
Let $\lambda=(1,\lambda_{2},...,\lambda_{n})$ be in $\mathbb{R}^{n}$ with
$1>\lambda_{2}\geq...\geq\lambda_{n}\geq-1.$ If
$\frac{1}{n}+\frac{1}{n(n-1)}\lambda_{2}+\frac{1}{(n-1)(n-2)}\lambda_{3}+...+\frac{1}{(2)(1)}\lambda_{n}\geq
0.$
then there is a positive (i.e. all of its entries are positive) doubly
stochastic matrix $D$ in $\Delta_{n}^{s}$ such that $D$ has spectrum
$\lambda$.
In [7] the author presented a counterexample of the preceding proposition as
follows.
###### Theorem 1.2
Let $\lambda=(1,0,-2/3).$ Then there does not exist a $3\times 3$ symmetric
positive doubly stochastic matrix with spectrum $\lambda.$
In [4] it was pointed out that the 2-tuple $(1,-1)$ which is the spectrum of
$C_{2}$ is also another counterexample. Moreover, it should be mentioned here
that the proof of the preceding theorem (which is the main result of [7]) is
done by showing that the matrix $A=\left(\begin{array}[]{ccc}0&2/3&1/3\\\
2/3&0&1/3\\\ 1/3&1/3&1/3\\\ \end{array}\right)$ has spectrum $\lambda$ and the
only matrices in $\Delta_{3}^{s}$ that are similar to $A$ are of the form
$P^{T}AP$ for some permutation matrix $P.$ Based on this, the author suggested
the following problem.
###### Problem 1.3
Characterize all the $n$-tuples $\lambda=(1,\lambda_{2},...,\lambda_{n})$ with
$1\geq\lambda_{2}\geq...\geq\lambda_{n}\geq-1$ such that up to a permutation
similarity, there exists a unique symmetric doubly stochastic matrix $A$ with
spectrum $\lambda$ $($such $\lambda$ is said to characterize $A$
permutationally or $A$ is said to be permutationally characterized by
$\lambda$$).$
Note now that in the language of the preceding problem, the $3\times 3$ matrix
$A$ presented above is permutationally characterized by $(1,0,-2/3).$
Two matrices $A$ and $B$ are said to be permutationally similar if $B=P^{T}AP$
for some permutation matrix $P.$ Next, we say that a doubly stochastic matrix
$A$ is determined by its spectra (DS for short) in $\Delta_{n}$ if for every
element $B$ of $\Delta_{n}$ which is similar to $A,$ then $B$ is
permutationally similar to $A.$ If in addition $A$ is symmetric then $A$ is
said to be DS in $\Delta_{n}^{s}$ if $B\in\Delta_{n}^{s}$ is similar to $A$
implies that $B$ is permutationally similar to $A.$ For a symmetric doubly
stochastic matrix $A,$ obviously $A$ is DS in $\Delta_{n}$ implies that $A$ is
DS in $\Delta_{n}^{s}.$ However, it is not known whether the converse is true
or false and though it is an interesting open problem, it will not be dealt
with here. Also, though the problem of characterizing all doubly stochastic
matrices that are DS in $\Delta_{n}$ is very interesting and we will touch on
some aspects of this problem, however here we are particularly more interested
in the following problem which is very related to Problem 1.3.
###### Problem 1.4
Characterize all symmetric doubly stochastic matrices that are DS in
$\Delta_{n}^{s}.$
All above problems appear to be very difficult and it seems that there is no
systematic way under which these problems can be approached (see Section 3).
Practically nothing is known about them except perhaps what is mentioned
earlier. In addition, we note that if a symmetric doubly stochastic matrix $A$
is permutationally characterized by $(1,\lambda_{2},...,\lambda_{n}),$ then
obviously $A$ is DS in $\Delta_{n}^{s}.$ So that in order to solve Problem
1.3, we need to solve Problem 1.4 first and then for every solution $X$ of
this last problem, we have to find the spectrum of $X.$
The rest of the paper is organized as follows. Section 2 is mainly concerned
with obtaining some general results for our two problems. In Section 3, we
completely solve Problem 1.3 and Problem 1.4 for the case $n=3$ which is one
of the main results of this paper. In Section 4, we study the close connection
of Problem 1.4 with spectral graph theory; more precisely with “regular graphs
that are DS.” We conclude in Section 5 by posing two open questions and by
introducing a conjecture related to the general case.
## 2 Some general results
We start our study with the following two lemmas that explore some aspects of
the spectral properties of doubly stochastic matrices and are consequences of
the Perron-Frobenius theorem (see, e.g. [12]). But first recall that a square
nonnegative matrix $A$ is irreducible if $A$ is not permutationally similar to
a matrix of the form $\left(\begin{array}[]{ccc}A_{1}&0\\\ A_{2}&A_{3}\\\
\end{array}\right)$ where $A_{1}$ and $A_{2}$ are square. Otherwise, $A$ is
said to be reducible.
###### Lemma 2.1
Every doubly stochastic matrix is permutationally similar to a direct sum of
irreducible doubly stochastic matrices.
###### Lemma 2.2
Let $A$ be an $n\times n$ irreducible doubly stochastic matrix. If $A$ has
exactly $k$ eigenvalues of unit modulus, then these are the $k$th roots of
unity. In addition, if $k>1,$ then $k$ is a divisor of $n$ and $A$ is
permutationally similar to a matrix of the form
$\left(\begin{array}[]{ccccc}0&A_{1}&0&\ldots&0\\\ 0&0&A_{2}&\ldots&0\\\
\vdots&\vdots&\vdots&\ddots&\vdots\\\ 0&0&0&\ldots&A_{k-1}\\\
A_{k}&0&0&\ldots&0\\\ \end{array}\right)$
where $A_{i}$ is doubly stochastic of order $\frac{n}{k}\times\frac{n}{k}$ for
$i=1,...k.$
As a result, we have the following.
###### Theorem 2.3
Every permutation matrix is DS in $\Delta_{n}.$
Proof. Suppose first that $A$ is an irreducible permutation matrix and let
$X^{-1}AX$ be a doubly stochastic matrix, then clearly $X^{-1}AX$ is
irreducible and all of its eigenvalues are of unit modulus. Therefore by the
preceding lemma $X^{-1}AX$ is a permutation matrix. Now if $A$ is reducible
then the proof can be completed by using Lemma 2.1.
An immediate consequence is the following corollary.
###### Corollary 2.4
Let $\lambda=(1,\lambda_{2},...,\lambda_{n})$ be in $\mathbb{R}^{n}$ where
$\lambda_{i}\in\\{-1,1\\}$ for $i=2,...,n$ and such that
$1+\lambda_{2}+...+\lambda_{n}\geq 0.$ Then $\lambda$ characterizes
permutationally a vertex (i.e. symmetric permutation matrix) of
$\Delta_{n}^{s}.$
###### Lemma 2.5
Let $X$ be an invertible matrix such that $X^{-1}J_{n}X$ is symmetric doubly
stochastic. Then $X^{-1}J_{n}X=J_{n}.$
Proof. Since $X^{-1}J_{n}X$ is symmetric doubly stochastic, then by the
spectral theorem for symmetric matrices, there exists an orthogonal matrix $U$
whose first column is $e_{n}$ and the remaining columns are orthogonal to
$e_{n}$ (i.e. the sum of all components in each of the remaining columns is
zero) such that $U^{T}X^{-1}J_{n}XU=(1\oplus 0_{n-1})$ where $0_{n-1}$ is the
$n-1\times n-1$ zero matrix. Hence $X^{-1}J_{n}X=U(1\oplus 0_{n-1})U^{T}.$ But
then a simple check shows that $U(1\oplus 0_{n-1})U^{T}=J_{n}$ and the proof
is complete.
###### Corollary 2.6
The matrices $I_{n},$ $J_{n}$ and $C_{n}$ are DS in $\Delta_{n}^{s}.$
Proof. The first part is obvious, and the second part follows from the
preceding lemma. For the third part, we note that
$C_{n}=\frac{n}{n-1}J_{n}-\frac{1}{n-1}I_{n}$ and then for any invertible
matrix $X$ such that $X^{-1}C_{n}X$ is symmetric doubly stochastic we obtain
$X^{-1}C_{n}X=\frac{n}{n-1}X^{-1}J_{n}X-\frac{1}{n-1}I_{n}.$ Therefore
$X^{-1}C_{n}X+\frac{1}{n-1}I_{n}=\frac{n}{n-1}X^{-1}J_{n}X$ and where the
left-hand side is a nonnegative matrix with row and column sum equals to
$1+\frac{1}{n-1}.$ Thus
$X^{-1}J_{n}X=\frac{n-1}{n}(X^{-1}C_{n}X+\frac{1}{n-1}I_{n})$ is symmetric
doubly stochastic and then by the preceding lemma, the proof is complete.
Next we need the following auxiliary materials.
###### Lemma 2.7
The inverse of an invertible doubly quasi-stochastic matrix is doubly quasi-
stochastic.
Proof. Multiplying to the left of $AA^{-1}=I_{n}$ by $J_{n}$ we obtain
$J_{n}AA^{-1}=J_{n}.$ Since $A$ is doubly quasi-stochastic, then
$J_{n}A^{-1}=J_{n}.$ Similarly, multiplying to the right of $A^{-1}A=I_{n}$ by
$J_{n},$ we obtain $A^{-1}J_{n}=J_{n}.$ Thus $A^{-1}$ is doubly quasi-
stochastic.
###### Lemma 2.8
If $A$ is an $n\times n$ irreducible doubly stochastic matrix such that
$B=X^{-1}AX$ is doubly stochastic for some invertible matrix $X$, then there
exists a doubly stochastic matrix $Y$ such that $B=Y^{-1}AY.$
Proof. See [12, Theorem 4.1, p. 123].
###### Corollary 2.9
The matrices $I_{n},$ $J_{n}$ and $C_{n}$ are DS in $\Delta_{n}.$
Proof. If $B=X^{-1}J_{n}X$ is doubly stochastic, then by the preceding lemma,
there exists $Y\in\Delta_{n}$ such that $B=Y^{-1}J_{n}Y.$ Hence $B=J_{n}.$ The
rest of proof can be completed by using a similar argument as that of
Corollary 2.6.
It is easy to see that each symmetric doubly stochastic matrix $D_{a}$ of
trace $a$ which lies on the line-segment joining $I_{n}$ to $C_{n}$ has the
property that $0\leq a\leq n.$ Also recall that
$J_{n}=\frac{n-1}{n}C_{n}+\frac{1}{n}I_{n}$ so that $J_{n}$ is on this line-
segment $[I_{n},C_{n}].$ With this in mind, we have the following theorem.
###### Theorem 2.10
Any point $D_{a}$ that lies on the line-segment $[I_{n},C_{n}]$ is DS in
$\Delta_{n}$ and hence it is also DS in $\Delta_{n}^{s}.$
Proof. We split the proof into two cases.
* 1.
For $0\leq a\leq 1,$ then $D_{a}$ is a convex combination of $J_{n}$ and
$C_{n}.$ From the trace of $D_{a}$, we easily obtain $D_{a}=aJ_{n}+(1-a)C_{n}$
so that $D_{a}=aJ_{n}+(1-a)\frac{n}{n-1}J_{n}-\frac{1}{n-1}I_{n}$ or
$D_{a}=\frac{n-a}{n-1}J_{n}-\frac{1-a}{n-1}I_{n}.$ Note that $D_{a}$ is a
positive matrix and so it is irreducible. Now if $X$ is an invertible matrix
such that $B=X^{-1}D_{a}X$ is doubly stochastic, then by the preceding lemma,
there exists a doubly stochastic matrix $Y$ such that $B=Y^{-1}D_{a}Y.$ Hence
$B=X^{-1}D_{a}X=Y^{-1}D_{a}Y=\frac{n-a}{n-1}Y^{-1}J_{n}Y-\frac{1-a}{n-1}Y^{-1}I_{n}Y.$
Thus $X^{-1}D_{a}X=D_{a}$ and this shows that $D_{a}$ is DS in $\Delta_{n}.$
* 2.
For $1\leq a\leq n,$ then $D_{a}$ is a convex combination of $I_{n}$ and
$J_{n}.$ From the trace of $D_{a}$, it is easy to see that in this case
$D_{a}=\frac{a-1}{n-1}I_{n}+(1-\frac{a-1}{n-1})J_{n},$ and that the proof can
be completed in a similar way to that of the previous case.
Knowing that the eigenvalues of $C_{n}$ are given by
$(1,-\frac{1}{n-1},...,-\frac{1}{n-1}),$ then we have the following
conclusion.
###### Corollary 2.11
Let $\lambda$ be any point that lies on the line-segment
$[(1,...,1),(1,-\frac{1}{n-1},...,-\frac{1}{n-1})]$ of $\mathbb{R}^{n}.$ Then
$\lambda$ characterizes a unique element of $\Delta_{n}^{s}.$
It should be noted that if two doubly stochastic matrices $A$ and $B$ are DS
in $\Delta_{n}^{s}$ (or $\Delta_{n}$) then their direct sum $A\oplus B$ may
not be DS in $\Delta_{n}^{s}.$ To see this, it suffices to check that in
$\Delta_{2k}^{s},$ the matrices $J_{2}\oplus J_{2k-2}$ and $J_{k}\oplus J_{k}$
have the same spectrum so that they are similar (as they are symmetric).
Moreover, for $k\geq 3,$ $\frac{1}{k}$ is an entry of the latter and is not an
entry of the first so that they not permutationally similar. However, we have
the following.
###### Theorem 2.12
The matrix $C_{n}\oplus C_{n}$ is DS in $\Delta_{2n}.$
Proof. If $Z\in\Delta_{2n}$ is similar to $C_{n}\oplus C_{n}$ then obviously
the spectrum of $Z$ is $(1,1,-1/(n-1),...,-1/(n-1)).$ Since $Z$ is reducible
and has the eigenvalue 1 repeated twice, then $Z$ is permutationally similar
to a direct sum of two doubly stochastic matrices $A$ and $B.$ But the traces
of $A$ and $B$ are zeroes so that necessarily the spectrum of $A$ and $B$ is
the same and is equal to $(1,-1/(n-1),...,-1/(n-1)).$ By corollary 2.9,
$A=B=C_{n}.$
Using a virtually identical proof to that of the preceding theorem, we
conclude by mathematical induction the following.
###### Corollary 2.13
For any positive integers $n_{1},...,n_{k},$ the matrix
$C_{n_{1}}\oplus...\oplus C_{n_{k}}$ is DS in $\Delta_{n_{1}+...+n_{k}}.$
Our next result is concerned with some other doubly stochastic matrices that
are DS in $\Delta_{2n}.$ For this purpose, we introduce the following
notations. In $\Delta_{2n}^{s},$ define $I=\left(\begin{array}[]{cc}0&I_{n}\\\
I_{n}&0\\\ \end{array}\right),$ $J=\left(\begin{array}[]{cc}0&J_{n}\\\
J_{n}&0\\\ \end{array}\right)$ and $C=\left(\begin{array}[]{cc}0&C_{n}\\\
C_{n}&0\\\ \end{array}\right)$ then it can be easily checked that
$C=\frac{n}{n-1}J-\frac{1}{n-1}I$ so that $J$ belongs to the line-segment
$[I,C].$ With this in mind, we conclude with the following result.
###### Theorem 2.14
Any point on the line-segment $[I,C]$ is DS in $\Delta_{2n}$ and hence in
$\Delta_{2n}^{s}.$
Proof. We first prove that $J$ and $I$ are DS in $\Delta_{2n}.$ For, if there
exists $Z\in\Delta_{2n}$ which is similar to $J,$ then obviously the spectrum
of $Z$ is $(1,0,...,0,-1)\in\mathbb{R}^{2n}.$ By Lemma 2.2, $Z$ is
permutationally similar to a matrix of the form
$\left(\begin{array}[]{cc}0&D\\\ D^{T}&0\\\ \end{array}\right)$ where
$D\in\Delta_{n}.$ So that $DD^{T}\in\Delta_{n}^{s}$ and its eigenvalues are
$(1,0,...,0)$ and therefore $DD^{T}$ is similar to $J_{n}.$ By Lemma 2.5,
$DD^{T}=J_{n}$ and since rank($D$)=rank($DD^{T}$)=rank($J_{n}$)=1, then
$D=J_{n}$ and therefore $Z=J.$
Now suppose that $S\in\Delta_{2n}$ is similar to $I,$ then $S$ has spectrum
$(\underbrace{1,...,1}_{\text{n times}},\underbrace{-1,...,-1}_{\text{n
times}})$ and therefore $S$ is permutationally similar to
$\underbrace{C_{2}\oplus...\oplus C_{2}}_{\text{n times}}$ but this in turn
means that $I$ and $S$ are permutationally similar. Since
$C=\frac{n}{n-1}J-\frac{1}{n-1}I$ then a similar proof to that of Corollary
2.6 shows that $C$ is also DS in $\Delta_{2n}.$ Finally, using a similar
argument to that of Theorem 2.10, the proof can be easily completed.
Recall that two matrices are cospectral if they have the same spectra. We
conclude this section by proving that a symmetric doubly stochastic matrix
that is DS in $\Delta_{n}$ may be cospectral to another element of
$\Delta_{n}.$ But first, we need the following result for which the proof can
be found in [22].
###### Lemma 2.15
Let $M=\left(\begin{array}[]{cc}A&B\\\ C&D\\\ \end{array}\right)$ where $A$
and $D$ are square. If $AC=CA,$ then
$\det(M)=\det(AD-CB).$
###### Theorem 2.16
Let $A$ be in $\Delta_{n}$ and define $M=\left(\begin{array}[]{cc}0&J_{n}\\\
A&0\\\ \end{array}\right)$ and let $J=\left(\begin{array}[]{cc}0&J_{n}\\\
J_{n}&0\\\ \end{array}\right)$ be as defined earlier. Then $M$ and $J$ are
cospectral.
Proof. The characteristic polynomial of $M$ is given by
$p_{M}(\lambda)=\det\left(\begin{array}[]{cc}-\lambda I_{n}&J_{n}\\\
A&-\lambda I_{n}\\\ \end{array}\right).$ By the preceding lemma,
$p_{M}(\lambda)=\det(\lambda^{2}I_{n}-AJ_{n})=\det(\lambda^{2}I_{n}-J_{n})$
i.e. $\lambda^{2}$ is an eigenvalue of $J_{n}.$ On the other hand,
$p_{J}(\lambda)=\det(\lambda^{2}I_{n}-J_{n}J_{n})=\det(\lambda^{2}I_{n}-J_{n}).$
Thus $M$ and $J$ are cospectral.
## 3 Particular cases
Recall that Birkhoff’s theorem states that $\Delta_{n}$ is a convex polytope
of dimension $(n-1)^{2}$ where its vertices are the $n\times n$ permutation
matrices. On the other hand, $\Delta_{n}^{s}$ is a convex polytope of
dimension $\frac{1}{2}n(n-1)$, and its vertices were determined in [11, 5]
where it is proved that if $A$ is a vertex of $\Delta_{n}^{s}$, then
$A=\frac{1}{2}(P+P^{T})$ for some permutation matrix $P$, although not every
$\frac{1}{2}(P+P^{T})$ is a vertex.
### 3.1 The case $n=2$
It is easy to see $\Delta_{2}=\Delta_{2}^{s}$ i.e. every $2\times 2$ doubly
stochastic matrix is necessarily symmetric. Moreover, $\Delta_{2}$ is the
line-segment joining $I_{2}$ to $C_{2}.$ So that every $2\times 2$ doubly
stochastic matrix is determined by its spectra and every point of the line-
segment joining $[(1,1),(1,-1)]$ characterizes a unique $2\times 2$ doubly
stochastic matrix.
### 3.2 The case $n=3$
Here we solve completely Problem 1.3 and hence Problem 1.4 for the case $n=3.$
The convex polytope $\Delta^{s}_{3}$ sits in the 6-dimensional vector space of
all $3\times 3$ real symmetric matrices, and following [11, 5]
$\Delta^{s}_{3}$ is the convex hull of the following matrices:
$I_{3},\mbox{ }X=\left(\begin{array}[]{ccc}1&0&0\\\ 0&0&1\\\ 0&1&0\\\
\end{array}\right),\mbox{ }Y=\left(\begin{array}[]{ccc}0&0&1\\\ 0&1&0\\\
1&0&0\\\ \end{array}\right),\mbox{ }Z=\left(\begin{array}[]{ccc}0&1&0\\\
1&0&0\\\ 0&0&1\\\ \end{array}\right),\mbox{
}C_{3}=\left(\begin{array}[]{ccc}0&1/2&1/2\\\ 1/2&0&1/2\\\ 1/2&1/2&0\\\
\end{array}\right).$
Our main observation is the following:
###### Lemma 3.1
$J_{3}=\frac{1}{3}(X+Y+Z)$, $C_{3}=\frac{3}{2}J_{3}-\frac{1}{2}I_{3}$ and the
triangle $XYZ$ is equilateral with respect to the Frobenius norm. In addition,
$I_{3}C_{3}$ is an axis of symmetry for $\Delta_{3}^{s}$, and every point on
$I_{3}C_{3}$ commutes with all other points in $\Delta_{3}^{s}$.
Thus it is clear from the preceding lemma that $\Delta_{3}^{s}$ is
3-dimensional and has the shape seen in Figure 1.
Figure 1: The shape of $\Delta_{3}^{s}$.
Our next goal is to prove that the only symmetric doubly stochastic matrices
of $\Delta_{3}^{s}(1)$ (which is the closed triangle $XYZ$) that are DS in
$\Delta_{3}^{s}$ are $J_{3}$ and its vertices $X,$ $Y$ and $Z.$
Using Maple for example, it is easy to check the following lemma.
###### Lemma 3.2
For $0\leq x\leq 1$ and $0\leq y\leq 1$ with $0\leq x+y\leq 1,$ the symmetric
doubly stochastic matrix $A=xX+yY+(1-x-y)Z$ has eigenvalues
$\left(1,\sqrt{3x^{2}+3y^{2}+3xy+1-3x-3y},-\sqrt{3x^{2}+3y^{2}+3xy+1-3x-3y}\right).$
Now if we let the domain $D$ be defined by $0\leq x\leq 1,$ $0\leq y\leq 1$
and $0\leq x+y\leq 1,$ then it is easy to see that $D$ is the closed triangle
whose vertices are $O=(0,0),A=(1,0)$ and $B=(0,1).$ Define the function $f$
over $D$ by:
$f(x,y)=3x^{2}+3y^{2}+3xy+1-3x-3y.$
Concerning the function $f,$ we have the following.
###### Lemma 3.3
Over the domain $D,$ the function $f$ has zero as absolute minimum and 1 as an
absolute maximum. Thus over the domain $D,$ we have $0\leq
3x^{2}+3y^{2}+3xy+1-3x-3y\leq 1.$
Proof. Since $f$ is differentiable then the only places where $f$ can assume
these values are points inside $D$ where the first partial derivatives satisfy
$f_{x}=f_{y}=0,$ and points on the boundary.
* 1.
Potential points inside $D$: Solving the system
$\left\\{\begin{array}[c]{l}6x+3y-3=0\\\ 3x+6y-3=0\\\ \end{array}\right.$
yields the unique solution $x=y=\frac{1}{3}$ with
$f(\frac{1}{3},\frac{1}{3})=0.$
* 2.
Potential points on the boundary of $D$: We have to check the 3 sides of the
triangle $OAB$ one side at a time.
1\. On the segment $[O,A],$ $fx,y)=f(x,0)=3x^{2}-3x+1$ which can be regarded
as a function of $x$ where $0\leq x\leq 1,$ and such that its derivative
$f^{\prime}(x,0)=6x-3=0$ for $x=1/2.$ Therefore we have 3 potential points
where their images by $f$ are given by $f(0,0)=1,$ $f(1,0)=1,$ and
$f(1/2,0)=1/4.$
2\. On the segment $[O,B],$ clearly (as $x$ and $y$ play a symmetric role in
the function $f(x,y)$) we obtain the following potential points: $f(0,0)=1,$
$f(0,1)=1,$ and $f(0,1/2)=1/4.$
3\. On the segment $[A,B],$ we have already accounted for the values of $f$ at
the endpoints of $[A,B],$ so that we only need to look at the interior points
of $[A,B].$ Clearly $f(x,1-x)=3x^{2}-3x+1$ and $f^{\prime}(x,1-x)=6x-3=0$ for
$x=1/2.$ Hence, $(1/2,1/2)$ is the final potential point with
$f(1/2,1/2)=1/4.$
Thus our claim is valid.
###### Remark 3.4
The surface $z=f(x,y)$ where $(x,y)\in D$ and any horizontal plane $z=d$ where
$0\leq d\leq 1$ intersect at exactly one point which is $(1/3,1/3)$ for $d=0$
and intersect at the three points $(0,0),$ $(1,0)$ and $(0,1)$ for $d=1.$
Moreover, for $0<d<1$ they intersect in an infinite number of points (see
Figure 2).
Figure 2: The surface $z=f(x,y)$ over $D.$
As a consequence, we have the following corollary.
###### Lemma 3.5
The only elements of $\Delta_{3}^{s}(1)$ that are DS in $\Delta_{3}^{s}$ are
$J_{3}$ and $X,$ $Y$ and $Z.$
Proof. First note that the line segments $[J_{3},X],$ $[J_{3},Y]$ and
$[J_{3},Z]$ are permutationally similar as $X,$ $Y$ and $Z$ are and any point
outside these line segments can not be permutationally similar to a point on
them (from the geometry of the triangle $XYZ$). Let $M$ be any point in
$\Delta_{3}^{s}(1)$ and consider the following two cases:
* 1.
If $M\in\Delta_{3}^{s}(1)-[J_{3},X]\cup[J_{3},Y]\cup[J_{3},Z]$ then
$M=xX+yY+(1-x-y)Z$ for some $(x,y)\in D.$ Define $\alpha=\sqrt{f(x,y)}$ where
$(x,y)$ varies over the domain $D.$ Then by the preceding lemma,
$0\leq\alpha\leq 1.$ Moreover, it is easy to check that the matrix $N$ given
by $N=\alpha X+(1-\alpha)J_{n}$ has eigenvalues $(1,\alpha,-\alpha).$ So that
by Lemma 3.2 the two symmetric doubly stochastic matrices matrices $M$ and $N$
have the same spectrum. Since we are dealing with symmetric matrices, then
they are similar. Thus in this case $M$ is not DS in $\Delta_{3}^{s}.$
* 2.
For the case where $M$ is in
$[J_{3},X]\cup[J_{3},Y]\cup[J_{3},Z]-\\{J_{3},X,Y,Z\\},$ without loss of
generality let $M=dX+(1-d)J_{3}$ for some $0<d<1.$ We want to show that there
there exists $K\in\Delta_{3}^{s}(1)$ which is similar to $M$ but not
permutationally similar. For, let $K=xX+yY+(1-x-y)Z$ where $(x,y)$ varies over
$D.$ First the condition on $(x,y)\in D$ in terms of $d$ for which $M$ and $K$
are similar is given by $d=\sqrt{f(x,y)}.$ Such $(x,y)$ always exists due to
the continuity of $f(x,y)$ in $D.$ Also we want to impose the other constraint
that at least one entry of $M$ is not an entry of $K$ or vice versa so that
they are not permutationally similar. Clearly
$M=\left(\begin{array}[]{ccc}1/3+2/3d&1/3-1/3d&1/3-1/3d\\\
1/3-1/3d&1/3-1/3d&1/3+2/3d\\\ 1/3-1/3d&1/3+2/3d&1/3-1/3d\\\
\end{array}\right)$ and $K=\left(\begin{array}[]{ccc}x&1-x-y&y\\\ 1-x-y&y&x\\\
y&x&1-x-y\\\ \end{array}\right),$ and since $M$ has at most two distinct
entries which are $1/3+2/3d$ and $1/3-1/3d$ so that we need to impose the
constraint that $x\neq 1/3+2/3d$ and $x\neq 1/3-1/3d.$ An inspection shows
that $x=1/3+2/3d$ or $x=1/3-1/3d$ if and only if $(3x-1)^{2}=4f(x,y)$ or
$(3x-1)^{2}=f(x,y)$ if and only if $(x-(2y-1))^{2}=0$ or $(x-y)(2x+y-1)=0.$ So
that our second constraint amounts to $x$ not being an element of
$\\{y,2y-1,(1-y)/2\\}.$ Thus we only need to exclude these 3 particular values
of $x$ and since $0<d<1$ then by Remark 3.4, an infinite number of such $x$
exists (since each of the 3 planes $x=y,$ $x=2y-1$ and $x=(1-y)/2$ intersects
the curve $d=\sqrt{f(x,y)}$ in a finite number of points) so that we conclude
that $M$ is not DS in $\Delta_{3}^{s}.$
Finally, $X,$ $Y$ and $Z$ are DS by Theorem 2.3 and $J_{3}$ is DS by Corollary
2.9.
For $1\leq a\leq 3,$ let $\Delta_{3}^{s}(a)$ intersect $[I_{3},J_{3}],$
$[I_{3},X],$ $[I_{3},Y]$ and $[I_{3},Z]$ in $D_{a},$ $X_{a},$ $Y_{a}$ and
$Z_{a}$ respectively (for $0\leq a\leq 1,$ we only need to replace $I_{3}$ by
$C_{3},$ in this statement). Then clearly $\Delta_{3}^{s}(a)$ is the closed
triangle $X_{a}Y_{a}Z_{a}$ and the 3 vertices $X_{a},$ $Y_{a},$ and $Z_{a}$
are permutationally similar since $X,$ $Y,$ and $Z$ are. With these notations,
we have the following.
###### Lemma 3.6
The only points of $\Delta_{3}^{s}(a)$ that are DS in $\Delta_{3}^{s}$ are
$\\{D_{a},X_{a},Y_{a},Z_{a}\\}.$
Proof. For $1\leq a\leq 3,$ let $M_{a}$ be any point in
$\Delta_{3}^{s}(a)-\\{D_{a},X_{a},Y_{a},Z_{a}\\},$ and let the line through
$I_{3}$ (resp. $C_{3}$ for $0\leq a\leq 1$) and $M_{a}$ intersect
$\Delta_{3}^{s}(1)$ in $M.$ Clearly $M$ is in
$\Delta_{3}^{s}(1)-\\{J_{3},X,Y,Z\\}$ and $M$ is not DS by the preceding
lemma. Then there exists $N$ in $\Delta_{3}^{s}(1)-\\{J_{3},X,Y,Z\\}$ such
that $N$ and $M$ are similar but not permutationally similar. Let $N_{a}$ be
the intersection of $[I_{3},N]$ (resp. $[C_{3},N]$ for $0\leq a\leq 1$) with
$\Delta_{3}^{s}(a),$ then clearly $M_{a}$ and $N_{a}$ are similar but not
permutationally similar.
If $M_{a}=D_{a},$ then $D_{a}$ is DS in $\Delta_{3}^{s}.$ Now if
$M_{a}\in\\{X_{a},Y_{a},Z_{a}\\},$ then it is enough to study the case where
$M_{a}=X_{a}.$ If there exists
$N_{a}\in\Delta_{3}^{s}(a)-\\{D_{a},X_{a},Y_{a},Z_{a}\\},$ such that $X_{a}$
and $N_{a}$ are similar but not permutationally similar, then there exists
$N\in\Delta_{3}^{s}(1)-\\{J_{3},X,Y,Z\\}$ such that $X$ and $N$ are similar
but not permutationally similar which is a contradiction to $X$ being DS in
$\Delta_{3}^{s}.$
From the preceding 3 lemmas, we conclude one of our main results which
completely solves Problem 1.4 in the case $n=3.$
###### Theorem 3.7
The only symmetric doubly stochastic matrices that are DS in $\Delta_{3}^{s}$
are those lying on one of the following line-segments $[I_{3},X],$
$[I_{3},Y],$ $[I_{3},Z],$ $[C_{3},X],$ $[C_{3},Y],$ $[C_{3},Z],$ or
$[I_{3},C_{3}].$
As a conclusion, we solve Problem 1.3 for the case $n=3.$
###### Corollary 3.8
The only points of $\mathbb{R}^{3}$ that characterize permutationally elements
of $\Delta_{3}^{s}$ are those belonging to
$[(1,1,1),(1,1,-1)]\cup[(1,-1/2,-1/2),(1,1,-1)]\cup[(1,1,1),(1,-1/2,-1/2)].$
Proof. It is enough to check that the spectrum of any of the 3 line-segments
$[I_{3},X],$ $[I_{3},Y],$ $[I_{3},Z]$ is $[(1,1,1),(1,1,-1)],$ and the
spectrum of any of $[C_{3},X],$ $[C_{3},Y],$ $[C_{3},Z]$ is
$[(1,-1/2,-1/2),(1,1,-1)].$ The last part is true by Corollary 2.11.
## 4 Connections with spectral graph theory
In this section, we present some close connections between Problem 1.4 and the
topic known “regular graphs that are DS” (see, e.g., [2, 6, 19, 20]). First
let us introduce some related notations (see, e.g. [1]). The adjacency matrix
of a simple graph $G$ will be denoted by $A(G)$ which is a symmetric
nonnegative (0,1)-matrix and its eigenvalues $\lambda_{1},...\lambda_{n}$ form
the spectrum of $G$ which is a multiset and will be denoted by $\sigma(G).$
The graph $G$ is called integral if all of its eigenvalues are integers, and
it is called circulant if $A(G)$ is circulant. In addition, $G$ is said to be
$k$-regular if the degree of each of its vertices is $k.$ A strongly regular
graph $G$ with parameters $(v,k,\lambda,\mu)$ is a $k$-regular graph which is
not complete nor edgeless and satisfying the following two conditions:
(i) For each pair of adjacent vertices there exist $\lambda$ vertices adjacent
to both.
(ii) For each pair of non-adjacent vertices there exist $\mu$ vertices
adjacent to both.
We use the usual notation $K_{n}$ to denote the complete graph on $n$ vertices
where each vertex is connected to all other vertices. Moreover, the complete
bipartite graph $K_{n_{1},n_{2}}$ has vertices partitioned into two subsets
$V_{1}$ and $V_{2}$ of $n_{1},n_{2}$ elements each, and two vertices are
adjacent if and if only if one is in $V_{1}$ and the other is in $V_{2}.$
Two graphs are said to be isomorphic if and only if their adjacency matrices
are permutationally similar. Two graphs are said to be cospectral or
isospectral if they have the same spectrum. A graph $G$ is said to be DS if
any graph $H$ which is cospectral to $G$ is isomorphic to $G.$ In general, the
problem of determining whether a graph $G$ is DS or not is still open though
many partial results are known (see [20] for the latest developments on this
problem).
We are particularly interested in the subproblem of finding which regular
graphs are DS due to its link with Problem 1.4. To explain this, we need some
more notations. But first recall that if $G$ is a $k$-regular graph with $n$
vertices, then the spectral radius of $A(G)$ equals $k$ and it is an
eigenvalue of $G$ with corresponding unit eigenvector equals to $e_{n}.$ Now
let $\Omega_{n}^{s}(k)$ be the set of all $n\times n$ nonnegative symmetric
matrices with each row and each column equals to $k,$ and $\Lambda_{n}(k)$
denote the subset of $\Omega_{n}^{s}(k)$ formed by of all (0,1)-matrices with
$k$ 1’s in each row and each column. In addition, let $\Lambda_{n}^{0}(k)$ be
the set of those elements of $\Lambda_{n}(k)$ that have zero trace. Then
clearly $A\in\Lambda_{n}^{0}(k)$ if and only if $A$ is the adjacency matrix of
some $k$-regular graph with $n$ vertices and $k$ edges. Also note that if $A$
is in $\Omega_{n}^{s}(k)$ if and only if $\frac{1}{k}A$ is an element of
$\Delta_{n}^{s}.$ So that if we extend the notion of DS to all elements of
$\Omega_{n}^{s}(k),$ then obviously $A\in\Lambda_{n}(k)$ is DS in
$\Omega_{n}^{s}(k)$ if and only if $\frac{1}{k}A$ is DS in $\Delta_{n}^{s}.$
Also, note that a $k$-regular graph $G$ is DS if and only if $A(G)$ is DS in
$\Lambda_{n}^{0}(k)$
It is well-known that 1-regular graphs are DS (see [6]); a fact that can be
easily derived from Theorem 2.3. In addition, the fact that the complete graph
$K_{n}$ is DS can be seen from Corollary 2.9 since
$\frac{1}{k}A(K_{n})=C_{n}.$ On the one hand, a disjoint union of complete
graphs is DS; a fact that can be deduced from Corollary 2.13, and on the other
hand, $K_{n,n}$ is DS by Theorem 2.14.
Although proving that graphs are DS is a much more harder task than just
showing they are not DS and the same is true for Problem 1.4, one can benefit
from the fact that cospectral regular graphs that are not isomorphic (i.e.
cospectral mates) give rise to symmetric doubly stochastic matrices that are
not DS in $\Delta_{n}^{s}.$ So that all known results concerning finding
cospectral mates for regular graphs can lead to exclude elements from
$\Delta_{n}^{s}(0)$ as solutions to Problem 1.4. In what follows, we mention
among the many such situations, 3 particular examples (see [2] for other
situations). The first is concerned with strongly regular graphs where it is
well known that connected strongly regular graphs with parameters
$(v,k,\lambda,\mu)$ have eigenvalue k appearing once and two other eigenvalues
with prescribed multiplicity. In general there are many non-isomorphic graphs
for a fixed parameter and the number of non-isomorphic graphs can grow
dramatically (see e.g. [2]). The second deals with cospectral integral regular
graphs where for example in [21](see also the references within) the authors
prove the existence of infinitely many pairs of cospectral integral graphs
which results in the existence of infinitely many pairs of symmetric doubly
stochastic matrices that are not DS in $\Delta_{n}^{s}.$ The final case is
concerned with circulant graphs (which are regular) where in [3] it is proved
that there are infinitely many cospectral non-isomorphic circulant graphs.
## 5 Two related open questions and a conjecture
We conclude this paper with the following two open questions for which the
answer to any of them can help shed some light on Problem 1.4 for general $n.$
(1) If $G$ is a $k$-regular graph that is DS. Does this imply that
$\frac{1}{k}A(G)$ is DS in $\Delta_{n}^{s}$?
Note that as mentioned earlier this is true for 1-regular graphs, disjoint
union of complete graphs, the graphs $K_{n}$ and $K_{n,n}.$
(2) What are the elements of $\Delta_{n}^{s}(0)$ that are DS in
$\Delta_{n}^{s}$?
we know that $C_{n}$ and the zero trace $n\times n$ permutation matrices are
among these ones.
Finally, based on the solution for the case $n=3,$ we propose the following
conjecture.
###### conjecture 5.1
For $0<a\leq n,$ the only elements of $\Delta_{n}^{s}(a)$ that are DS in
$\Delta_{n}^{s}$ are points on the line segments $[I_{n},C_{n}],$ $[I_{n},P]$
and $[C_{n},P]$ where $P$ is a vertex of $\Delta_{n}^{s}.$
## Acknowledgments
This work is supported by the Lebanese University research grants program for
the Discrete Mathematics and Algebra research group.
## References
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* [2] A. E. Brouwer, W. H. Haemers, Spectra of graphs, Springer, 2011.
* [3] J. Brown, Isomorphic and nonisomorphic, isospectral circulant graphs, available from arXiv:0904.1968v1, 2009.
* [4] R. Brualdi, From the Editor-in-chief, Lin. Alg. Appl., 434, (2011) pp. 449-853.
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|
arxiv-papers
| 2013-10-04T14:01:19 |
2024-09-04T02:49:51.958358
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Bassam Mourad and Hassan Abbas",
"submitter": "Bassam Mourad",
"url": "https://arxiv.org/abs/1310.1273"
}
|
1310.1343
|
# Vibronic phenomena and exciton–vibrational interference in two-dimensional
spectra of molecular aggregates
Vytautas Butkus Department of Theoretical Physics, Faculty of Physics,
Vilnius University, Sauletekio 9-III, 10222 Vilnius, Lithuania Center for
Physical Sciences and Technology, Gostauto 9, 01108 Vilnius, Lithuania Leonas
Valkunas Department of Theoretical Physics, Faculty of Physics, Vilnius
University, Sauletekio 9-III, 10222 Vilnius, Lithuania Center for Physical
Sciences and Technology, Gostauto 9, 01108 Vilnius, Lithuania Darius
Abramavicius [email protected] Department of Theoretical Physics,
Faculty of Physics, Vilnius University, Sauletekio 9-III, 10222 Vilnius,
Lithuania State Key Laboratory of Supramolecular Complexes, Jilin University,
2699 Qianjin Street, Changchun 130012, PR China
###### Abstract
A general theory of electronic excitations in aggregates of molecules coupled
to intramolecular vibrations and the harmonic environment is developed for
simulation of the third-order nonlinear spectroscopy signals. The model is
applied in studies of the time-resolved two-dimensional coherent spectra of
four characteristic model systems: weakly / strongly vibronically coupled
molecular dimers coupled to high / low frequency intramolecular vibrations.
The results allow us to classify the typical spectroscopic features as well as
to define the limiting cases, when the long-lived quantum coherences are
present due to vibrational lifetime borrowing, when the complete exciton-
vibronic mixing occurs and when separation of excitonic and vibrational
coherences is proper.
## I Introduction
Excitonic energy spectrum of molecular aggregates experiences essential
transformation due to the presence of high-frequency intramolecular
vibrations. As a result, coupling between electronic excitations and
intramolecular vibrations known as vibronic coupling turn to be responsible
for a host of spectroscopically-observed phenomena. The vibronic effects have
been investigated intensively by different theoretical methods since the
foundation of molecular (Frenkel) exciton theory Davydov-book ; V.I.Broude1985
. Along with the advance of nonlinear spectroscopic techniques, some new
insights related to coupling between electronic degrees of freedom of
molecular aggregates and intramolecular vibrations were observed in third-
order spectroscopic signals, for example, in two-dimensional (2D) coherent
spectra demonstrating vibrational wave-packet motion, long-lived coherences,
vibrational anisotropy beats, polaron formation, etc.nemeth-sperling-JCP2010 ;
Egorova2007 ; Smith2011 ; Dahlbom2002 ; Gelzinis2011 ;
ZhaoYang_molecular_ring_JCP2013 . Probably the most extensively discussed
issue lately is the impact of discrete vibrational resonances on the
electronic coherences and _vice versa._ These coherences are observed in the
2D electronic spectroscopy, but its possible role in energy transfer is
currently under discussion Chin2013 ; Christensson_JPCB2012 ; Kreisbeck2012 .
Range and diversity of molecular systems, where vibronic coupling is very
significant, appears to be extremely wide. Historically, molecular crystals
were the first systems where the vibronic coupling was considered and the
theoretical basis of the spectral characterization was developed by analyzing
their stationary spectra Fulton1964 ; Philpott1971 ; Davydov1970 ;
Davydov1971a . Further development of the theoretical approach was addressed
to studies of vibronic excitations in H and J aggregates and in molecular
films Friesner1981 ; Scherer1984 . Strong coupling to discrete intramolecular
high-frequency modes of the ${\rm C=C}$ stretch vibration at around
$1400\,\mathrm{\mathrm{cm}^{-1}}$ together with the strong electrostatic
interaction between the molecules are the most evident properties of the
J-aggregates. Coupling to discrete low-frequency intramolecular modes (160
$\mathrm{\mathrm{cm}^{-1}}$, for exampleMilota2013_JPCA_VibrJaggr ;
KobayashiBook1996 ) has also been considered. Significant vibronic features
are prevalent in spectra of aggregated and strongly-coupled molecular dimeric
dyes, the formation of which is usually the first step towards the large-scale
molecular aggregation West1965 ; Kopainsky1981 ; Baraldi2002 ; Moran2006 ;
Seibt2008 ; Bixner2012 .
Photosynthetic pigment–protein (P–P) complexes could be considered as yet
another class of molecular systems, where weak vibronic coupling (however,
only recently observed) was found to be important Kolli2012 ;
Christensson_JPCB2012 ; Adolphs2006 ; Lee-Fleming2007 ; Womick2011 ;
Richards2012 . Since pigment molecules within P–P formations are weakly-
coupled and the surrounding protein framework is ready to dissipate any
vibrational motion of the pigments, the domination of electronic coupling over
vibronic effects is commonly assumed. Therefore, long-lasting oscillations in
coherent 2D spectra were initially explained by purely excitonic coherences
engel-nat2007 ; ColliniScholes2010 ; Panitchayangkoon2011 . Recently, vibronic
components and mixing of both, electronic and vibronic, ingredients have been
reported Christensson2011 ; Jonas_PNAS2012 .
If we were to represent the above-mentioned systems as points on a schematic
two-dimensional phase space, where the axes indicate vibrational frequency and
electronic resonance interaction, the most of it would be covered as presented
in Fig. 1. We can make a classification of the points scattered over the plot
by considering the possible time-resolved experiment with ultra-short laser
pulses of typical bandwidth of $\sim 1000-2000$ $\mathrm{\mathrm{cm}^{-1}}$.
In the top–left corner of the figure we then have the weakly-coupled systems
with high-frequency vibrations. In this case the experiment would resolve a
few peaks of the vibronic progression at most and the splittings due to
electronic coupling would be overlapping. The mixed case where electronic
resonance interactions and vibronic progression would be resolvable along with
strong quantum-mechanical mixing of both types of transitions, is present in
the top-right part of the scheme. The laser spectrum would cover only a few
peaks in this case. On the bottom–left corner we have the mixed systems again,
but the laser pulse spectrum could cover all peaks. And, finally, on the
bottom–right corner we have the case where the full vibrational progression
could be observed in the experiment and the electronic splitting would be
well-resolved.
Fig. 1: Experimentally and theoretically investigated molecular systems
(dimeric dyes, weakly-coupled P-P complexes, J-aggregates and films),
characterized by different electronic resonance interactions $J$ and
vibrational frequencies $\omega_{0}$. Dashed line indicates the region of
exciton-vibronic resonance ($\omega_{0}=2J$), the numbers next to symbols are
the references to the corresponding studies. Stars indicate the model dimer
systems considered in this paper.
To cover all these cases in an unified model, we present the molecular
exciton–vibronic theory developed for the purpose of its application in
describing the two-dimensional electronic spectroscopy signals. It is based on
the Holstein-type exciton–vibronic Hamiltonian with assumption of multi-
particle vibronic state basis Holstein1959 ; Fulton1964 . Dissipation is
included by coupling the vibrational coordinate to the harmonic bath. Special
attention is paid to exciton–vibronic resonances, at which the most pronounced
mixing of states is present Polyutov2012 ; Chenu2013 ; Jonas_PNAS2012 ;
Butkus2013 . Four different models of dimer systems are chosen for
consideration as indicated by stars in Fig. 1: two being considerably away
from the exciton–vibronic resonance (D1 and D2) and another two corresponding
to mixed conditions (D3 and D4). As one can observe, these models represent
four typical molecular systems: weakly-coupled P–P complexes with high and
low-frequency vibrations (D1 and D3), the J-aggregate (D2) and a molecular dye
(D4). Therefore, the conclusions drawn from the results of model systems are
general in terms of its application to different molecular aggregates.
## II Vibrational aggregate model
Various models of a molecule coupled to continuum of bath vibrations were
developed within the framework of the perturbative system–bath interaction
expansionmukbook . The bath is then described by the spectral density
function, which represents auto-correlations of the electronic _site_ energy
fluctuations due to the environment. The most popular model assumes the
Brownian particle-like vibrational motion of the molecule in a solvent May2011
; Valkunas2013 . This model is usually enough to obtain proper spectral
lineshapes in simulations of systems with no expressed high-frequency
vibrations at fixed temperature. For systems with well-resolved high-frequency
modes of vibrations the spectral density approach is applied by including a
$\delta$-shaped or finite-bandwidth peak into the bath spectral density
function. The $\delta$-peak does give ever-lasting coherent beats in the
coherent 2D spectramancal-sperling-jcp2010 ; Egorova2008 , while in case of
finite-width peak decay of oscillations is obtained due to pure dephasing
Butkus-Abramavicius-Valkunas-JCP2012 ; Seibt2013 . However, this method has
two deficiencies. Firstly, it neglects the effects caused by quantum-
mechanical mixing of the vibronic levels of different molecules when the
vibronic splitting is comparable to the intermolecular excitonic coupling.
Secondly, it does not include vibrational relaxation as the vibrations are
assumed to be in thermal equilibrium at fixed temperature. These could be
important effects when the coupling to vibrations is strong.
There have been several studies of nonlinear coherent spectra of molecular
dimers with exciton–vibronic mixing included Chenu2013 ; Jonas_PNAS2012 ;
Krcmar2013_CP . However, the realistic molecular aggregates contain several
dozens or hundreds of molecules. We develop a general description applicable
for molecular aggregates with an arbitrary number of chromophores.
Let us start with the displaced oscillator model of a molecule. It dictates
that the Hamiltonian of a single (say $m$-th) molecule in an aggregate can be
given by
$\displaystyle\hat{H}_{m}$
$\displaystyle=\left[\frac{\hat{p}_{m}^{2}}{2}+\frac{\omega_{m}^{2}}{2}\hat{q}_{m}^{2}\right]|{\rm
g}^{m}\rangle\langle{\rm g}^{m}|$
$\displaystyle+\left[\epsilon_{m}+\frac{\hat{p}_{m}^{2}}{2}+\frac{\omega_{m}^{2}}{2}(\hat{q}_{m}-d_{m})^{2}\right]|{\rm
e}^{m}\rangle\langle{\rm e}^{m}|.$ (1)
Here $\hat{p}_{m}$ and $\hat{q}_{m}$ are the momentum and coordinate operators
of the intramolecular vibrational motion, $\omega_{m}$ is the vibrational
frequency and $d_{m}$ is the displacement in the excited state. The effective
mass of the oscillator is taken as unity. Ground and excited state wavevectors
for the $m$-th molecule, $|{\rm g}^{m}\rangle$ and $|{\rm e}^{m}\rangle$ (we
use superscript indices for later convenience) respectively, in the space of
electronic states of the single molecule comprise the complete basis set, thus
$|{\rm g}^{m}\rangle\langle{\rm g}^{m}|+|{\rm e}^{m}\rangle\langle{\rm
e}^{m}|=1$. After introducing operators for electronic excitations
$\hat{B}_{m}^{\dagger}$ , so that $|{\rm
e}^{m}\rangle=\hat{B}_{m}^{\dagger}|{\rm g}^{m}\rangle$, and its Hermitian
conjugate $\hat{B}_{m}$, we can write
$\displaystyle\hat{H}_{m}$
$\displaystyle=\frac{\hat{p}_{m}^{2}}{2}+\frac{\omega_{m}^{2}}{2}\hat{q}_{m}^{2}$
$\displaystyle+\left(\epsilon_{m}+\lambda_{m}-\omega_{m}^{2}d_{m}\hat{q}_{m}\right)\hat{B}_{m}^{\dagger}\hat{B}_{m}.$
(2)
Here we defined the reorganization energy
$\lambda_{m}=\omega_{m}^{2}d_{m}^{2}/2$. As the molecule can be electronically
excited just once, we must have $\hat{B}_{m}^{\dagger}|{\rm e}^{m}\rangle=0$
or $\hat{B}_{m}\hat{B}_{m}^{\dagger}+\hat{B}_{m}^{\dagger}\hat{B}_{m}=1,$
which reflects the fermionic property.
By inserting the bosonic creation and annihilation operators for the
vibrational degrees of freedom
$\displaystyle\hat{p}_{m}$ $\displaystyle={\rm
i}\sqrt{\frac{\omega_{m}}{2}}\left(\hat{b}_{m}^{\dagger}-\hat{b}_{m}\right)\leavevmode\nobreak\
{\rm and}\leavevmode\nobreak\
\hat{q}_{m}=\sqrt{\frac{1}{2\omega_{m}}}\left(\hat{b}_{m}^{\dagger}+\hat{b}_{m}\right)$
into Eq. (2), one gets the fully quantized Hamiltonian of the $m$-th molecule,
$\displaystyle\hat{H}_{m}$
$\displaystyle=\omega_{m}\left(\hat{b}_{m}^{\dagger}\hat{b}_{m}+\frac{1}{2}\right)$
$\displaystyle+[\epsilon_{m}+\lambda_{m}-\omega_{m}\sqrt{s_{m}}\left(\hat{b}_{m}^{\dagger}+\hat{b}_{m}\right)]\hat{B}_{m}^{\dagger}\hat{B}_{m}.$
(3)
Here the Huang–Rhys factor is defined as $s_{m}\equiv\lambda_{m}/\omega_{m}$.
This brings the vibrational ladder of states in the electronic ground state
$|\mathrm{g}_{i}^{m}\rangle=\frac{\left(\hat{b}_{m}^{\dagger}\right)^{i}}{\sqrt{i!}}|0\rangle$
(4)
and in the electronic excited state
$|\mathrm{e}_{i}^{m}\rangle\equiv\hat{B}_{m}^{\dagger}|\mathrm{g}_{i}^{m}\rangle=\hat{B}_{m}^{\dagger}\frac{\left(\hat{b}_{m}^{\dagger}\right)^{i}}{\sqrt{i!}}|0\rangle.$
(5)
$|0\rangle$ is the vacuum state in terms of electronic and vibrational
excitations.
### II.1 Hamiltonian of the vibrational aggregate
The Hamiltonian of an aggregate of realistic molecules involves three
components: electronic states, vibrational structure for each electronic state
and the Coulomb coupling between all electronic and vibronic levels. The first
two are described by extending the Hamiltonian of a single molecule into the
space of a set of molecules within the Heitler–London approximation, which
assumes that the aggregate states are constructed from the direct products of
the molecular single excitations Davydov-book ; Amerongen2000 ; May2011 ;
Valkunas2013 . We consider only single and double excitations. The Coulomb
coupling between the $m$-th and $n$-th molecule is denoted by the resonant
electronic coupling constant $J_{mn}$ and the corresponding term is as
follows:
$\hat{H}_{{\rm Coulomb}}=\sum_{m\neq
n}J_{mn}\hat{B}_{m}^{\dagger}\hat{B}_{n}.$ (6)
We neglect electrostatic interactions between vibrations in the ground state.
Within this model the Hamiltonian for the vibrational aggregate is given by
$\displaystyle\hat{H}$
$\displaystyle=\sum_{m}\left[\epsilon_{m}+\lambda_{m}-\omega_{m}\sqrt{s_{m}}\left(\hat{b}_{m}^{\dagger}+\hat{b}_{m}\right)\right]\hat{B}_{m}^{\dagger}\hat{B}_{m}$
$\displaystyle+\sum_{m}\omega_{m}\left(\hat{b}_{m}^{\dagger}\hat{b}_{m}+\frac{1}{2}\right)+\sum_{m\neq
n}J_{mn}\hat{B}_{m}^{\dagger}\hat{B}_{n}.$ (7)
Similarly as to the electronic aggregate we get bands corresponding to
electronic states, but now the ground state $|{\rm g}\rangle$ of the aggregate
is not a single quantum level, but a band of vibrational states. Thus, there
are states with all chromophores in their electronic ground states, while
vibrational excitations are arbitrary:
$|\mathrm{g}_{(i_{1}i_{2}...i_{N})}\rangle\equiv|\prod_{m}{\rm
g}_{i_{m}}^{m}\rangle=\left[\prod_{m}\frac{\left(\hat{b}_{m}^{\dagger}\right)^{i_{m}}}{\sqrt{i_{m}!}}\right]|0\rangle.$
(8)
Here $i_{m}$ is a quantum number of vibrational excitation of the $m$-th
molecule. Thus $|{\rm g}_{i_{m}}^{m}\rangle$ now denotes the electronic ground
state of the $m$-th molecule being in the $i_{m}$-th vibrational level.
The singly-excited states are obtained by assuming that one of the molecules
is in its electronic excited state, while the others are in their arbitrary
vibrational ground states. We thus get the set of states
$|\mathrm{e}_{n,(i_{1}i_{2}...i_{N})}\rangle\equiv|{\rm
e}_{i_{n}}^{n}\negmedspace\prod_{{m\atop m\neq n}}\negmedspace{\rm
g}_{i_{m}}^{m}\rangle=\hat{B}_{n}^{\dagger}\negthickspace\left[\prod_{m}\frac{\left(\hat{b}_{m}^{\dagger}\right)^{i_{m}}}{\sqrt{i_{m}!}}\right]|0\rangle.$
(9)
The doubly-excited states are obtained similarly,
$|\mathrm{f}_{kl,(i_{1}i_{2}...i_{N})}\rangle\equiv|{\rm e}_{i_{k}}^{k}{\rm
e}_{i_{l}}^{l}\negthickspace\negthickspace\prod_{{m\atop m\neq
k,l}}\negthickspace\negthickspace{\rm
g}_{i_{m}}^{m}\rangle=\hat{B}_{k}^{\dagger}\hat{B}_{l}^{\dagger}\negthickspace\left[\prod_{m}\frac{\left(\hat{b}_{m}^{\dagger}\right)^{i_{m}}}{\sqrt{i_{m}!}}\right]|0\rangle.$
(10)
State ordering $k<l$ is satisfied here. A complete basis set is included into
the model since all possible combinations (multi-particle states) of vibronic
and vibrational excitations are considered, cf. single-particle approximation,
where only states $|{\rm e}_{i_{n}}^{n}\prod_{{m\atop m\neq n}}{\rm
g}_{0}^{m}\rangle$ are includedPhilpott1971 ; Spano2009a . The index notation
is further simplified by introducing the $N$-component vector
$\bm{i}=(i_{1}i_{3}...i_{N})$. Then the basis states can be written as
$|\mathrm{g}_{\bm{i}}\rangle$, $|\mathrm{e}_{n,\bm{i}}\rangle$ and
$|\mathrm{f}_{kl,\bm{i}}\rangle$.
In this setup electronic and vibrational subsystems are coupled only through
term
$\left(\hat{b}_{m}^{\dagger}+\hat{b}_{m}\right)\hat{B}_{m}^{\dagger}\hat{B}_{m}$
in Hamiltonian (Eq. (7)). It thus induces the shifts of electronic energies by
creation or annihilation of vibrational quantum. Otherwise, electronic and
vibrational subsystems are independent. The basis set is chosen accordingly.
The other basis set is possible by using shifted vibrational excitations in
the electronic excited states Polyutov2012 ; eisfeld:134103 . However, our
approach gives convenient form for various matrix elements and allows us to
easily incorporate the environment as shown below. Hamiltonian of the ground
state manifold in this basis is diagonal,
$\displaystyle H_{\bm{i},\bm{j}}^{({\rm gg})}$
$\displaystyle=\left[\sum_{m}\omega_{m}\left(i_{m}+\frac{1}{2}\right)\right]\bm{\delta}_{\bm{ij}},$
(11)
where $\delta_{\bm{ij}}\equiv\prod_{m}\delta_{i_{m}j_{m}}$. Similarly, the
Hamiltonian of singly-excited states is given by
$\displaystyle H_{\bm{i},\bm{j}}^{({\rm e}_{n}{\rm e}_{k})}$
$\displaystyle=\delta_{nk}\left[\epsilon_{n}+\lambda_{n}+\sum_{m}\omega_{m}\left(i_{m}+\frac{1}{2}\right)\right]\delta_{\bm{ij}}$
$\displaystyle-\delta_{nk}\omega_{n}\sqrt{s_{n}}\langle
i_{n},j_{n}\rangle\prod_{{m\atop m\neq n}}\delta_{i_{m}j_{m}}$
$\displaystyle+(1-\delta_{nk})J_{nk}\delta_{\bm{ij}},$ (12)
where we have defined the vibrational wavefunction overlap $\langle
i_{n},j_{n}\rangle=\sqrt{i_{n}}\delta_{i_{n},j_{n}+1}+\sqrt{j_{n}}\delta_{i_{n},j_{n}-1}$.
For the double-exciton states we have
$\displaystyle H_{\bm{i},\bm{j}}^{({\rm f}_{kl}{\rm
f}_{k^{\prime}l^{\prime}})}=$
$\displaystyle\quad\delta_{kk^{\prime}}\delta_{ll^{\prime}}\left[\epsilon_{k}+\epsilon_{l}+\lambda_{k}+\lambda_{l}+\sum_{m}\omega_{m}\left(i_{m}+\frac{1}{2}\right)\right]\bm{\delta}_{\bm{ij}}$
$\displaystyle\quad-\delta_{kk^{\prime}}\delta_{ll^{\prime}}\omega_{k}\sqrt{s_{k}}\langle
i_{k},j_{k}\rangle\prod_{{m\atop m\neq k}}\delta_{i_{m}j_{m}}$
$\displaystyle\quad+\delta_{kk^{\prime}}\delta_{ll^{\prime}}\omega_{l}\sqrt{s_{l}}\langle
i_{l},j_{l}\rangle\prod_{{m\atop m\neq l}}\delta{}_{i_{m}j_{m}}$
$\displaystyle\quad+\left[\delta_{kk^{\prime}}(1-\delta_{ll^{\prime}})J_{ll^{\prime}}+\delta_{ll^{\prime}}(1-\delta_{kk^{\prime}})J_{kk^{\prime}}\right]\bm{\delta}_{\bm{ij}}.$
(13)
The exciton energies (eigenstate basis) $\varepsilon_{{\rm e}}$ and
$\varepsilon_{{\rm f}}$ are obtained by numerically diagonalizing matrices
defined above. The bands of singly- and doubly-excited states are however much
more complicated than those of the electronic aggregate due to coupling
between the singly-excited vibronic subbands. In the eigenstate basis all
these substates become mixed. The unitary transformation to the eigenstate
basis is thus as follows:
$\displaystyle|\mathrm{e}_{p}\rangle$
$\displaystyle=\sum_{n}\sum_{\bm{i}}\psi_{p,\bm{i}}^{n}|\mathrm{e}_{n,\bm{i}}\rangle,$
(14) $\displaystyle|\mathrm{f}_{r}\rangle$ $\displaystyle=\sum_{{kl\atop
k<l}}\sum_{\bm{i}}\Psi_{r,\bm{i}}^{kl}|\mathrm{f}_{kl,\bm{i}}\rangle.$ (15)
Note that for high vibronic numbers $i,j$ the Franck–Condon parameter becomes
small and these states do not contribute to the spectra. In general, if one
includes $\nu$ vibrational levels in description of each of $N$ molecules,
this results in $N\nu^{N}$ singly-excited states, and $N(N-1)\nu^{N}/2$
doubly-excited states, enumerated by indices $p$ and $r$ in the previous
expressions, respectively.
For electronic excitations we consider the dipole operator defined as
$\hat{\bm{P}}=\sum_{m}^{N}\bm{d}_{m}(\hat{B}_{m}^{\dagger}+\hat{B}_{m}),$ (16)
where $\bm{d}_{m}$ is the electronic transition dipole vector of the $m$-th
molecule. This form essentially reflects the Frank–Condon approximation where
the electronic transition is not coupled to vibrational system. The dipole
moments representing transitions from the ground state to singly-excited
states and from singly-excited state to the doubly-excited states are given by
$\displaystyle\bm{\mu}_{{\rm g}_{\bm{i}}}^{{\rm e}_{p}}$
$\displaystyle=\langle{\rm g}_{\bm{i}}|\hat{\bm{P}}|{\rm
e}_{p}\rangle=\sum_{m}^{N}\bm{d}_{m}\psi_{p,\bm{i}}^{m}$
and
$\displaystyle\bm{\mu}_{{\rm e}_{p}}^{{\rm f}_{r}}$ $\displaystyle=\langle{\rm
e}_{p}|\hat{\bm{P}}|{\rm
f}_{r}\rangle=\sum_{m,n}^{N}\sum_{\bm{i}}\bm{d}_{k}\psi_{p,\bm{i}}^{n}\Psi_{r,\bm{i}}^{(mn)}.$
The transition amplitudes thus have the mixed electronic–vibronic nature
encoded in eigenvectors $\psi_{p,\bm{i}}^{n}$ and $\Psi_{r,\bm{i}}^{(mn)}$.
### II.2 Coupling to the bath
We next include the relaxation using a microscopic dephasing theory, based on
the linear coupling of the vibronic coordinate to the harmonic overdamped
bathAbramavicius2009 . Hence, we assume that the vibronic coordinate is
damped. The bath is described as a set $\left\\{\alpha\right\\}$ of harmonic
oscillators, whose Hamiltonian is:
$\displaystyle\hat{H}_{{\rm B}}$ $\displaystyle=$
$\displaystyle\sum_{\alpha}\frac{1}{2}\hat{p}_{\alpha}^{2}+\frac{1}{2}w_{\alpha}^{2}\hat{x}_{\alpha}^{2}.$
(17)
Here $\hat{p}_{\alpha}$ is the momentum and $\hat{x}_{\alpha}$ is the
coordinate operators and $w_{\alpha}$ is the frequency of the $\alpha$-th bath
oscillator. The system–bath interaction is then given in the bilinear form
$\hat{H}_{{\rm
SB}}=\sum_{m\alpha}z_{m\alpha}\hat{x}_{\alpha}\hat{q}_{m}=\sum_{m\alpha}\sqrt{\frac{z_{m\alpha}^{2}}{2\omega_{m}}}\hat{x}_{\alpha}\left(\hat{b}_{m}^{\dagger}+\hat{b}_{m}\right).$
(18)
We add these two operators to complete the Hamiltonian in Eq. (7). Thus, the
off-diagonal fluctuations of vibronic levels translate into diagonal
fluctuations of the electronic-only aggregates due to electronic excitation
creation/annihilation operators in Eq. (18). More explicitly, coupling $z$
induces the vibronic off-diagonal couplings and causes vibrational
intramolecular relaxation. Resonance intermolecular interaction $J$ will
extend into the electronic energy relaxation between different molecules. The
non-zero fluctuating matrix elements in the site basis (Eqs. (8)–(10)) are
very simple:
$\displaystyle\left(\hat{H}_{{\rm SB}}\right)_{\bm{i},\bm{j}}^{({\rm gg})}$
$\displaystyle=\langle{\rm g}_{\bm{i}}|\hat{H}_{{\rm SB}}|{\rm
g}_{\bm{j}}\rangle=\mathcal{H}(\bm{i},\bm{j}),$ (19)
$\displaystyle\left(\hat{H}_{{\rm SB}}\right)_{\bm{i},\bm{j}}^{({\rm
e}_{n}{\rm e}_{k})}$ $\displaystyle=\langle{\rm e}_{n,\bm{i}}|\hat{H}_{{\rm
SB}}|{\rm e}_{k,\bm{j}}\rangle=\delta_{nk}\mathcal{H}(\bm{i},\bm{j}),$ (20)
$\displaystyle\left(\hat{H}_{{\rm SB}}\right)_{\bm{i},\bm{j}}^{({\rm
f}_{kl}{\rm f}_{k^{\prime}l^{\prime}})}$ $\displaystyle=\langle{\rm
f}_{kl,\bm{i}}|\hat{H}_{{\rm SB}}|{\rm
f}_{k^{\prime}l^{\prime},\bm{j}}\rangle=\delta_{kk^{\prime}}\delta_{ll^{\prime}}\mathcal{H}(\bm{i},\bm{j}).$
(21)
Here we defined an auxiliary function of bath–space fluctuations
$\mathcal{H}(\bm{i},\bm{j})=\sum_{m\alpha}\sqrt{\frac{z_{m\alpha}^{2}}{2\omega_{m}}}\langle
i_{m},j_{m}\rangle\hat{x}_{\alpha}\prod_{{s\atop s\neq
m}}\delta_{i_{s}j_{s}}.$ (22)
Notice, that interband fluctuations are absent, so the interband relaxation
(electronic relaxation to the ground state) is not included. Transformation to
the eigenstate basis yields the fluctuations of the eigenstate
characteristics. In the ground state manifold we have eigenstates equivalent
to the site basis since the corresponding Hamiltonian is diagonal (Eq. (11)).
For the manifold of singly-excited states we get
$\left(\hat{H}_{{\rm SB}}\right)_{p_{1}p_{2}}^{({\rm
ee})}=\sum_{m}^{N}\sum_{\bm{i},\bm{j}}\psi_{p_{1},\bm{i}}^{m\ast}\psi_{p_{2},\bm{j}}^{m}\mathcal{H}(\bm{i},\bm{j}),$
(23)
and for the manifold of doubly-excited states
$\left(\hat{H}_{{\rm SB}}\right)_{r_{1}r_{2}}^{({\rm ff})}=\sum_{{m,n\atop
m>n}}^{N}\sum_{\bm{i},\bm{j}}\Psi_{r_{1},\bm{i}}^{(mn)\ast}\Psi_{r_{2},\bm{j}}^{(m_{1}n_{1})}\mathcal{H}(\bm{i},\bm{j}).$
(24)
The quantities of interest, which describe the relaxation properties, are the
correlation functions of fluctuating Hamiltonian elements. Firstly, we assume
that fluctuations of different chromophores are independent. Therefore, we can
sort out and associate the bath coordinates to specific molecules. Since the
bath oscillators are independent, correlation functions of the operator in the
Heisenberg representation with respect to the thermal equilibrium are
uncorrelated,
$\langle\hat{x}_{\alpha}(t)\hat{x}_{\beta}(0)\rangle=\delta_{\alpha\beta}\langle\hat{x}_{\alpha}(t)\hat{x}_{\alpha}(0)\rangle$,
and we can obtain separated baths of different molecules. Secondly, we assume
that the different molecules have statistically the same surroundings, so the
system–bath coupling is fully characterized by the following single
fluctuation correlation function:
$C_{0}(t)=\sum_{m\alpha}\frac{z_{m\alpha}^{2}}{2\omega_{m}}\langle\hat{x}_{\alpha}(t)\hat{x}_{\alpha}(0)\rangle.$
(25)
For the infinite number of bath oscillators they can be conveniently expressed
using the spectral density $\mathcal{C}^{\prime\prime}\left(\omega\right)$
mukbook :__
$C_{0}\left(t\right)=\frac{1}{\pi}\intop_{-\infty}^{+\infty}\frac{1}{1-\mathrm{e}^{-\beta\omega}}\mathrm{e}^{-\mathrm{i}\omega
t}\mathcal{C}^{\prime\prime}(\omega)\mathrm{d}\omega,$ (26)
where $\beta=\left(k_{{\rm B}}T\right)^{-1}$ is the inverse thermal energy.
Using these functions we get the eigenstate fluctuation correlation functions
$C_{ab,cd}(t)=\langle\left(\hat{H}_{{\rm
SB}}(t)\right)_{ab}\left(\hat{H}_{{\rm
SB}}\right)_{cd}\rangle=C_{0}(t)h_{ab,cd}$ for different manifolds. For the
electronic ground state manifold where a single eigenstate is equivalent to
the original basis state $|\mathrm{g}_{\bm{i}}\rangle$ it yields
$C_{\bm{i}\bm{j},\bm{k}\bm{l}}^{({\rm gg})}(t)=C_{0}(t)\sum_{m}\langle
i_{m},j_{m}\rangle\langle k_{m},l_{m}\rangle\prod_{{s\atop s\neq
m}}\delta_{i_{s}j_{s}}\delta_{k_{s}l_{s}}.$ (27)
We use the shorthand vector notations
$\bm{i}_{s}^{-}=(i_{1},i_{2},...,i_{s-1},i_{s}-1,i_{s+1},...,i_{N})$ and
$\bm{i}_{s}^{+}=(i_{1},i_{2},...,i_{s-1},i_{s}+1,i_{s+1},...,i_{N})$, which
allow us to explicitly write:
$\displaystyle C_{\bm{i}\bm{j},\bm{k}\bm{l}}^{({\rm gg})}(t)=$ (28)
$\displaystyle\quad
C_{0}(t)\sum_{s}^{N}\left\\{\sqrt{i_{s}j_{s}}\delta_{\bm{i}\bm{j}_{s}^{+}}\delta_{\bm{k}\bm{l}_{s}^{+}}+\sqrt{(i_{s}+1)j_{s}}\delta_{\bm{i}_{s}^{+}\bm{j}}\delta_{\bm{k}\bm{l}_{s}^{+}}\right.$
$\displaystyle\quad\left.+\sqrt{i_{s}(k_{s}+1)}\delta_{\bm{i}\bm{j}_{s}^{+}}\delta_{\bm{k}_{s}^{+}\bm{l}}+\sqrt{(i_{s}+1)(k_{s}+1)}\delta_{\bm{i}_{s}^{+}\bm{j}}\delta_{\bm{k}_{s}^{+}\bm{l}}\right\\}.$
Similarly, one can obtain the correlation functions involving the singly- and
doubly-excited states $C_{p_{1}p_{2},p_{3}p_{4}}^{({\rm ee})}(t)$,
$C_{p_{1}p_{2},r_{1}r_{2}}^{({\rm ef})}(t)$ and
$C_{r_{1}r_{2},r_{3}r_{4}}^{({\rm ff})}(t)$ (see Appendix A for the
corresponding expressions).
### II.3 Population transfer
As the bath induces off-diagonal fluctuations in all three bands of states one
has to consider the population transfer inside the excited and ground
manifolds (the populations of the doubly-excited states are never created so
the transport is not relevant there). The propagator $G_{{\rm e}_{p_{2}}{\rm
e}_{p_{1}}}(t_{2})$ denotes the conditional probability of the excitation to
be transferred to state $|{\rm e}_{p_{2}}\rangle\langle{\rm e}_{p_{2}}|$ from
$|{\rm e}_{p_{1}}\rangle\langle{\rm e}_{p_{1}}|$ in time $t_{2}$. Similarly,
$G_{{\rm g}_{\bm{j}}{\rm g}_{\bm{i}}}(t_{2})$ is the propagator in the
electronic ground manifold. In this model the bath is considered as the
intermolecular modes which should be Markovian while intramolecular
vibrational coordinates are considered explicitly. Hence, the Redfield theory
applies for the Markovian bath. Within the secular Redfield theory May2011 ,
both types of propagators satisfy the Pauli master equation,
$\frac{\partial}{\partial t}G_{ab}(t)=\sum_{{c\atop c\neq a}}k_{a\leftarrow
c}G_{cb}(t)-\sum_{{c\atop c\neq a}}k_{c\leftarrow a}G_{ab}(t).$ (29)
Here indices $a$, $b$ and $c$ can be either excited state numbers, or vectors,
indicating vibrational ground states. $k_{a\leftarrow c}$ are the transfer
rates in the excited (further on denoted by $k_{{\rm e}_{p_{2}}\leftarrow{\rm
e}_{p_{1}}}$) or ground state ($k_{{\rm g}_{\bm{j}}\leftarrow{\rm
g}_{\bm{i}}}$). Using the Redfield relaxation theory, one can obtain simple
expressions for the rates
$\displaystyle k_{{\rm e}_{p_{2}}\leftarrow{\rm e}_{p_{1}}}$
$\displaystyle=h_{{\rm e}_{p_{2}}{\rm e}_{p_{1}},{\rm e}_{p_{1}}{\rm
e}_{p_{2}}}C^{\prime\prime}\left(\omega_{{\rm e}_{p_{1}}{\rm
e}_{p_{2}}}\right)\left[\coth\left(\frac{\beta\omega_{{\rm e}_{p_{1}}{\rm
e}_{p_{2}}}}{2}\right)+1\right]$ (30)
for the excited state population transfer. For the ground state vibrational
relaxation, one has only two subsets of nonzero terms,
$k_{{\rm g}_{\bm{i}_{s}^{-}}\leftarrow{\rm
g}_{\bm{i}}}=i_{s}C^{\prime\prime}(\omega_{s})\left[\coth\left(\frac{\beta\omega_{s}}{2}\right)+1\right]$
(31)
and
$k_{{\rm g}_{\bm{i}_{s}^{+}}\leftarrow{\rm
g}_{\bm{i}}}=\left(i_{s}+1\right)C^{\prime\prime}(-\omega_{s})\left[\coth\left(-\frac{\beta\omega_{s}}{2}\right)+1\right].$
(32)
With these transformation expressions now it is possible to develop the
general theory describing the spectroscopic properties of vibronic aggregates.
## III Results
In the theory of the vibronic aggregate described above we derived all
identities necessary to simulate the third-order signals in the frame of the
second-order cumulant expansion of the system response function mukbook . The
system response function of an electronic-only aggregate is defined as a sum
of contributions (or so-called Liouville space pathways) responsible for
_bleaching_ of the ground state (B), photon-induced (stimulated) _emission_
from the excited state (E) and _induced_ _absorption_ of a photon in the
excited state (A). They are conveniently represented by the double-sided
Feynman diagrams, which show the system evolution during the delay times
$t_{1}$, $t_{2}$ and $t_{3}$ between the interactions. In diagrams, ground- or
excited-state populations or coherences evolve during delay time $t_{2}$ (Fig.
2a). Additionally, since population state can be transferred during $t_{2}$ in
the excited state, the so-called population transfer pathways $\tilde{S}_{{\rm
E}}$ and $\tilde{S}_{{\rm A}}$ are added up (Fig. 2b). In the case of vibronic
aggregate, this formalism has to be extended to take into account multi-level
ground state. Therefore, additional diagrams with coherences and population
transfer in the ground-state manifold have to be included. This ingredient and
the resulting final expressions for the two-dimensional coherent spectra is
described in Appendix B.
Fig. 2: System response function of the rephasing signal of the vibronic
aggregate is represented by 6 double-sided Feynman diagrams without population
transfer (a) and with population transfer (b). The diagrams are denoted as
responsible for stimulated emission (E), excited state absorption (A) and
ground-state bleaching (B) processes.
To discuss the outcomes of the developed system response function theory for
the molecular aggregate, we consider a molecular dimer (MD) as the simplest
molecular complex exhibiting vibronic phenomena, as well as the
exciton–vibrational interference. The vibrational frequencies, site energies
and Huang–Rhys factors of the constituent molecules are taken to be the same
and are denoted by $\omega_{0}\equiv\omega_{1}=\omega_{2}$,
$\epsilon\equiv\epsilon_{1}=\epsilon_{2}=1200\,\mathrm{\mathrm{cm}^{-1}}$ and
$s=s_{1}=s_{2}$, respectively. Also, we analyze the models in the case of weak
system–bath coupling (with Huang–Rhys factor equal to $s=0.05$) and strong
coupling ($s=0.5$). Four distinct parameter sets are used and we denote the
corresponding models as D1-D4, indicated by stars in Fig. (1).
In the D1 model the resonant coupling constant is taken to be $J=100$
$\mathrm{\mathrm{cm}^{-1}}$ and the vibrational frequency is chosen to be
$\omega_{0}=1400$ $\mathrm{\mathrm{cm}^{-1}}$. Such parameters are typical for
the photosynthetic pigment–protein complexes, for example, the photosynthetic
antenna of cryptophyte protein phycoerythrin 545 (the Huang–Rhys factor is
0.1) Kolli2012 . We denote this model as the weakly-coupled P-P complex with
high-frequency vibration.
In the D2 model resonant coupling of $J=600\,\mathrm{\mathrm{cm}^{-1}}$ and
vibrational frequency $\omega_{0}=250\,\mathrm{\mathrm{cm}^{-1}}$ is used.
These numbers are typical parameters of J-aggregates, coupled to low-frequency
intramolecular vibrations. For example, in 2D electronic spectra of PVA/C8O3
tubular J-aggregates, oscillations associated to the $160$
$\mathrm{\mathrm{cm}^{-1}}$ vibration is observed and the strongest coupling
between the molecules is in a range of $640$–$1110$
$\mathrm{\mathrm{cm}^{-1}}$ as it was shown by Milota _et al._
Milota2013_JPCA_VibrJaggr . In the same study, the experimental Fourier maps
were obtained. In J-aggregates the coupling to vibrations for individual
chromophores is known to decrease due to exciton delocalization Spano2009a .
It means that, if the aggregate is approximated as a dimer, the Huang–Rhys
factor of the monomer should be multiplied by factor of $N/2$ where $N$ is a
number of chromophores in the aggregate in the case of complete state
delocalization. Therefore, a very strong coupling to vibrations should be
considered. In our case, D2 model with $s=0.5$ represents the typical
J-aggregate better.
Parameters of the D3 model ($J=100\,\mathrm{\mathrm{cm}^{-1}}$,
$\omega_{0}=250\,\mathrm{\mathrm{cm}^{-1}}$) are, as in the D1 model, typical
for the P-P complexes. In such molecular systems strong coupling to discrete
low-frequency vibrations are present. For example, in the measurements of two-
color photon echo of the light-harvesting complex phycocyanin-645 from
cryptophyte marine algae, long-lived oscillations possibly associated to the
$194\,\mathrm{\mathrm{cm}^{-1}}$ vibrational mode were observed Richards2012 .
Similar parameters were also considered to be relevant for the
Fenna–Mathews–Olsen (FMO) photosynthetic light-harvesting complex Chenu2013 .
Therefore, we assume that the D3 model effectively represents the weakly-
coupled P-P complex coupled to a low-frequency vibrational mode.
Presence of strong resonance electronic interaction between molecules and
strong coupling with high-frequency vibrations is typical for many dimeric
dyes. Hence, in the D4 model, the main parameters are set to
$J=600\,\mathrm{\mathrm{cm}^{-1}}$ and
$\omega_{0}=1400\,\mathrm{\mathrm{cm}^{-1}}$ to be similar to ones of perylene
bisimide dye with the Huang–Rhys factor of $0.6$ Seibt2006 .
The bath, whose degrees of freedom are not treated explicitly, is represented
by the Debye spectral density
$C^{\prime\prime}(\omega)=2\lambda\omega/(\omega^{2}+\gamma^{2})$ which
represents the low-frequency fluctuations. In order to get the similar
homogeneous broadening in all cases of Huang–Rhys factors, the value of
$\lambda s=25\,\mathrm{\mathrm{cm}^{-1}}$ is kept constant throughout all
simulations. The solvent damping energy is set to $\gamma=50$
$\mathrm{\mathrm{cm}^{-1}}$. The molecular transition dipole vectors are taken
to have unitary lengths and their orientations are spread by an angle
$\alpha=2\pi/5$. Temperature is set to $150$ K ($\beta^{-1}\approx
104\,\mathrm{\mathrm{cm}^{-1}}$).
Fig. 3: Dependencies of the singly-excited state energies on the electronic
resonance interaction ($J$ coupling) in the case of the vibrational frequency
$\omega_{0}=250\,\mathrm{\mathrm{cm}^{-1}}$ (a) and
$1400\,\mathrm{\mathrm{cm}^{-1}}$ (b). The Huang–Rhys factor is $s=0.05$
(black lines) and $s=0.5$ (gray lines). Resonant coupling constants
corresponding to models D1-D4 are indicated by the red vertical lines.
Let us consider the manifold of singly-excited states of all D1–D4 models. It
consists of superpositions of electronic singly-excited states and vibrational
excitations of the constituent molecules. The energy dependence on the
resonant coupling constant reveals a complex composition of the states within
the singly-excited state manifold (Fig. 3). For uncoupled molecules ($J=0$)
the ladder-type pattern of vibrational energy states is present as the
energies are equally separated by $\omega_{0}$. Increasing coupling produces
the excitonic splitting which can be seen as the red shift of the lowest
energy state and appearance of two ladder-type progressions. However, the
interaction of vibronic and electronic states induces repulsion of the energy
levels, which is mostly evident where the ladders experience crossing, i.e. in
the vicinity of the so-called avoided crossing regions Polyutov2012 . We
denote the corresponding parameters for which the crossings occur as the
exciton–vibronic resonances. The complete mixing of the electronic and
vibronic substates is obtained for these resonances. The energy level
repulsion effect is more pronounced in the case of $s=0.5$ (see the gray lines
in Fig. 3).
In models D1 and D2 the vibrational frequency $\omega_{0}$ and resonant
coupling constant $J$ differs significantly and we are reasonably away from
the resonance as can be seen in Figures 1 and 3 (the corresponding resonant
coupling values are indicated by vertical lines in the later one). Therefore,
these models can be considered as rather pure systems of vibrational and
electronic aggregates, respectively. On the contrary, parameters of the D3 and
D4 models assure that the system is very close to the exciton-vibronic
resonances and the spectroscopic signals will be more complex due to mixing.
Properties of the model dimers are reflected in linear absorption spectra
(Fig. 4). The D1 system has intermolecular coupling of the same order as the
absorption linewidth. Hence, both electronic transitions (and excitonic
splitting) become hidden inside the single peak 12000 cm-1 when the Huang–Rhys
factor is $s=0.05$. Another peak at $\sim 13500\,\mathrm{\mathrm{cm}^{-1}}$
comes from the one-quantum level of the vibrational progression and becomes
stronger for $s=0.5$ (red dashed line).
Fig. 4: Absorption spectra of dimers D1-D4 in case of Huang–Rhys factors
$s=0.05$ (black solid line) and $s=0.5$ (red dashed line).
The D2 model is completely opposite to the D1. The excitonic splitting is
large and two absorption peaks approximately at
$11500\,\mathrm{\mathrm{cm}^{-1}}$ and $12700\,\mathrm{\mathrm{cm}^{-1}}$ show
the excitonic system character. As the vibrational frequency is small, we find
the vibrational progression on both excitonic lines dependent on the
Huang–Rhys factor. The D1 and D2 systems, more or less, behave "additively"
where the excitonic contributions and the vibrational progressions add up in
absorption.
Models D3 and D4 are very different. In the D3, both parameters, the excitonic
resonance interaction and the vibrational frequency, are small and the
absorption spectrum shows a single broad line at $\sim
12000\,\mathrm{\mathrm{cm}^{-1}}$. While excitonic and vibrational
contributions are mixed, as shown in Fig. 3a, surprisingly, the absorption
spectrum is relatively simple with a single electronic peak shaped by the
vibrational progression. However the shape is strongly dependent on the
Huang–Rhys factor: for $s=0.05$, one can guess two excitonic bands (black
solid line), while for $s=0.5$, the excitonic spectrum disappears and the
vibrational progression is observed.
The fine features of mixed system is better seen in the model D4, which has
large energy splittings between levels compared to the D3. The D4 model shows
non-trivial spectrum even for small value of the HR factor. There is a single
lower-excitonic peak at 11500 cm-1, but the higher-excitonic peak is split
into two ($\sim 12500\,\mathrm{\mathrm{cm}^{-1}}$ and $\sim
13000\,\mathrm{\mathrm{cm}^{-1}}$). The large HR factor makes the spectrum
even more complicated where we find four peaks and they all are due to
superpositions of vibrational and electronic nature. Hence both D3 and D4
systems reflect the mixed _vibronic_ features of the molecular dimer.
The two-dimensional electronic spectroscopy has been suggested as being able
to distinguish between the origin of transitions of such systems. So we
analyze transition types,which could be resolved by means of this spectroscopy
for our models D1–D4. The 2D spectra reveal as a set of peaks – all of them
contain oscillatory contributions in the population delay time. The so-called
Fourier maps are useful for the analysis of the origin of the oscillations in
2D spectra Butkus-Zigmantas-Abramavicius-Valkunas-CPL2012 ;
Milota2013_JPCA_VibrJaggr ; Christensson2013 ; Seibt2013 ; Calhoun2009 ;
Panitchayangkoon2011 ; Turner2012 . Thus, the maps are calculated by fitting
the evolution of each point of the 2D spectrum by the exponentially decaying
function and performing the Fourier transform of the residuals over the delay
interval $t_{2}$,
$A(\left|\omega_{1}\right|,\omega_{2},\omega_{3})=\intop_{0}^{\infty}\mathrm{e}^{-\mathrm{i}\omega_{2}t_{2}}S_{{\rm
residuals}}(\left|\omega_{1}\right|,t_{2},\omega_{3})\mathrm{d}t_{2}.$ (33)
The amplitude and phase which completely describe the oscillations of every
point of the 2D spectrum are then extracted from the complex function
$A(\left|\omega_{1}\right|,\omega_{2},\omega_{3})$. As the dependence of the
amplitude on frequency $\omega_{2}$ oscillation is available for every point
of $\omega_{1}$ and $\omega_{3}$, we suggest first to introduce a
representative variable that would characterize which oscillation frequencies
are important, in general. The maximum of the Fourier amplitude as a function
of $\omega_{2}$ can be used for that:
$\displaystyle\mathcal{A}(\omega_{2})$ $\displaystyle=\mbox{$\max$}\left[{\rm
Abs}\,A(\left|\omega_{1}\right|,\omega_{2},\omega_{3})\right]_{\omega_{2}={\rm
const.}}.$
The $\mathcal{A}(\omega_{2})$ dependencies on the oscillation frequency for
the D1-D4 models are depicted in Fig. 5 and the Fourier maps of several
dominant frequencies are presented in Figures 6 and 7. We next discuss the
models separately.
Fig. 5: Maximum of the Fourier amplitudes characterizing oscillations in the
2D spectra of the model dimers D1–D4 (a–d panels, respectively) in case of
Huang–Rhys factors $s=0.05$ (black solid lines) and $s=0.5$ (red dashed
lines). The frequency values of the peaks are indicated in the graph.
### III.1 D1 model. Weakly-coupled P-P complex with high-frequency vibration
Fig. 6: Oscillations in 2D spectra of weakly-coupled P-P complex with high-
frequency vibration (D1 model) and J-aggregate (D2 model) in case of weak and
strong coupling to vibrations ($s=0.05$ and $s=0.5$, respectively). 2D
rephasing spectra at $t_{2}=0$ and two most significant Fourier maps are
represented in rows of every model with. Schemes of the oscillations-providing
contributions ($\circ$ – excited state absorption, $\square$ – stimulated
emission and $\diamond$ – ground state bleaching) are presented next to the
maps. The size of the symbols are proportional to the amplitude of the
corresponding contribution.
Two dominant frequencies of $190\,\mathrm{\mathrm{cm}^{-1}}$ and
$1400\,\mathrm{\mathrm{cm}^{-1}}$ representing oscillations in spectra of the
D1 system are resolved when $s=0.05$ (Fig. 5a). Hence we consider Fourier maps
at these two frequencies. The former corresponds to the excitonic energy
splitting, but the frequency is smaller than $2J$
($190\,\mathrm{\mathrm{cm}^{-1}}\approx 1.8J$) due to slight energy level
repulsion, present even away from the exciton–vibronic resonance. The map for
$\omega_{2}=190\,\mathrm{\mathrm{cm}^{-1}}$ is typical for electronic
coherence, as the oscillating behavior corresponding to the excited state
absorption and ground state bleaching contributions are positioned
symmetrically with respect to the diagonal line and the oscillations are in-
phase (Fig. 6a). Since the distance between the positions is smaller than the
homogeneous linewidth, the most intensive oscillations are present on the
diagonal due to constructive interference. The Fourier map at
$\omega_{2}=1400\,\mathrm{\mathrm{cm}^{-1}}$ is a typical reflection of the
vibrational/vibronic coherence as the oscillations are present both on the
diagonal line and on the cross-peaks, characterized by complex phase
dependence Butkus2013 . The phase of oscillations is shifted by $\pi$ at the
center of the lower diagonal peak compared to the centers of the other peaks,
which is also typical for beatings of vibrational/vibronic coherences Butkus-
Zigmantas-Abramavicius-Valkunas-CPL2012 . Two more off-diagonal oscillating
features at around $\omega_{3}=10500\,\mathrm{\mathrm{cm}^{-1}}$ are out of
bounds in presented Fourier maps, hence they would be off-resonant in a
typical experiment.
Increasing the Huang–Rhys factor to $s=0.5$ causes stronger mixing in the
system. The shape of the Fourier map at $\omega_{2}=\omega_{0}$ does not
change notably, however, its amplitude increases by factor of
$\leavevmode\nobreak\ 3$. The Fourier map at
$\omega_{2}=120\,\mathrm{\mathrm{cm}^{-1}}\approx 1.2J$ closely resembles the
map at $\omega_{2}=190\,\mathrm{\mathrm{cm}^{-1}}$ when $s=0.05$. Additional
contributions of the excited state absorption appear above the diagonal.
Features in this map are not very smooth since the lifetime of oscillations is
much shorter (note that symbol sizes in Fig. 5a, representing the amplitudes
of contributions in the schemes for $s=0.05$ and $s=0.5$ are, however,
similar).
### III.2 D2 model. J-aggregate.
The strongest frequencies for model D2 are 250 $\mathrm{\mathrm{cm}^{-1}}$ and
1250 $\mathrm{\mathrm{cm}^{-1}}$ (Fig. 5b). The Fourier map of the D2 system
at $\omega_{2}=\omega_{0}=250\,\mathrm{\mathrm{cm}^{-1}}$ clearly shows the
large contribution from the ground state and excited state vibrations on the
diagonal and less significant excited state absorption features in the cross-
peaks (Fig. 6b). The oscillations are more intensive below the diagonal, which
is consistent with the maps of the above-mentioned PVA/C8O3
J-aggregateMilota2013_JPCA_VibrJaggr . If compared, the relative intensities
of oscillations associated with electronic ($\omega_{2}\approx 2J$) and
vibrational ($\omega_{2}\approx\omega_{0}$) transition, one would find that
the relative intensity of electronic coherences has a tendency to decrease
when increasing the Huang–Rhys factor. Thus, for $s\gg 1$, maps would be
completely dominated by the vibrational coherences. The maps at
$\omega_{2}=1200\,\mathrm{\mathrm{cm}^{-1}}$ and
$\omega_{2}=1230\,\mathrm{\mathrm{cm}^{-1}}$ in Fig. 6b are typical for
electronic coherences as the oscillations are diagonal-symmetric. Note that
the energy splitting is much larger than the homogeneous linewidth and the two
peaks in the maps are distinguished, cf. to the corresponding Fourier maps in
the D1 model.
### III.3 D3 model. Weakly-coupled P-P complex with low-frequency vibration
Fig. 7: Oscillations of a weakly-coupled P-P complex with low-frequency
vibration (D3 model) and strongly-coupled dimeric dye (D4 model). Presentation
is analogous to Fig. 6.
Assignment of oscillations in the D3 with strongest peaks shown in Fig. 5c is
complicated since the parameters are close to the exciton-vibronic resonance
(Fig. 3a). It might appear that there is only a continuum of low-frequency
oscillations in the spectra for $s=0.05$ since the maximum amplitude
dependence on the frequency does not contain any peaks (Fig. 5c). However,
there are short-lived oscillations at
$\omega_{2}=180\,\mathrm{\mathrm{cm}^{-1}}$ and
$\omega_{2}=250\,\mathrm{\mathrm{cm}^{-1}}$, but their Fourier maps are not
distinguishable (Fig. 7a). Increasing the Huang–Rhys factor to $s=0.5$
enhances the $\omega_{2}=\omega_{0}$ oscillation which, as it can be seen in
the scheme next to the map in Fig. 7a, is a mixture of many different
contributions.
### III.4 D4 model. Strongly-coupled dimeric dye
The absorption spectrum of the D4 model changes drastically when increasing
the Huang–Rhys factor (Fig. 4). Both transition frequencies and intensities
are redistributed due to sensitivity of the energy spectrum at the avoided
crossing region. In the 2D spectra for $s=0.05$, there are 3 clearly separable
long-lived oscillations of frequencies $\omega_{2}=0.8J\approx
470\,\mathrm{\mathrm{cm}^{-1}}$, $\omega_{2}=1.1J\approx
1060\,\mathrm{\mathrm{cm}^{-1}}$ and $\omega_{2}=\omega_{0}$ (Fig. 5d). The
later two correspond to the excitonic energy splitting and vibrational
coherence, respectively, while the $470\,\mathrm{\mathrm{cm}^{-1}}$
oscillation signifies beatings between the lower and upper states in the
avoided crossing region (the corresponding energy level gaps are indicated in
Fig. 3b). For $s=0.5$ the level repulsion effect is even more pronounced, as
the gap between the lowest energy states decreases from $1.8J$ to $1.1J$ and
the gap of the avoided crossing region increases from $0.8J$ to $1.3J$.
The Fourier maps allow us to separate electronic and vibrational coherences in
this particular mixed case. When $s=0.05$ (Fig. 7b) the Fourier map for
$\omega_{2}=1060\,\mathrm{\mathrm{cm}^{-1}}$ is typical for electronic
coherence. The only signature of coupling to vibrations is the oscillatory
contribution of the excited state absorption appearing above the stimulated
emission. It indicates that doubly-excited state manifold is effectively
shifted up due to vibronic coupling and, therefore, the peaks are elongated
along $\omega_{3}$ axis in the Fourier maps. The
$\omega_{2}=1400\,\mathrm{\mathrm{cm}^{-1}}$ map is exceptionally created by
the ground state vibrations, however, the energy level structure in the
excited state manifold is reflected as the distance between some oscillating
features in the Fourier maps are found to be equal to
$1060\,\mathrm{\mathrm{cm}^{-1}}$ (see the labels with arrows in Fig. 7b).
The $\omega_{2}=1400\,\mathrm{\mathrm{cm}^{-1}}$ oscillation becomes mixed if
$s=0.5$. As it can be seen in the corresponding scheme of oscillations,
contributions from all types of diagrams appear and heavily congest the
Fourier map. The lowest diagonal peak becomes oscillating due to stimulated
emission and ground state bleaching contributions. The map for the
$\omega_{2}=640\,\mathrm{\mathrm{cm}^{-1}}$ oscillation is similar to one for
$\omega_{2}=1060\,\mathrm{\mathrm{cm}^{-1}}$ presented above. Stronger
coupling to vibrations induce appearance of additional oscillations in the
excited state manifold, seen as two peaks above the diagonal.
## IV Discussion
2D electronic spectroscopy is the ultimate tool capable to directly reflect
coherent system dynamics, manifested by spectral oscillations and beatings.
Frequency and decay rate of oscillations indicate the energy difference of
states involved in the coherent superposition and the coherent state lifetime,
respectively. Positions of emerging oscillations in spectra are conveniently
depicted by the use of Fourier maps, thus providing us one more additional
dimension. For example, peaks in the maps, symmetric with respect to the
diagonal, reflect the electronic coherence evolution in the excited state (D1
and D2 models in Fig. 6). The information about the phase of oscillations
provides additional information about the ongoing processes. Therefore,
fitting the experimental data by means of the simulated Fourier maps would
lead to unambiguous conclusions.
### IV.1 Nature of coherences
Let us discuss about the quantum coherences in molecular aggregates. Quantum
mechanical description of _electronic_ excitation treats the rigid constituent
molecule’s skeleton as the potential energy surface for electrons. The
resulting electron density dynamics after photo-excitation can be approximated
by the oscillatory electric dipole moment. Thus, the coupling between the
molecules produces the discrete energy levels in the single excited manifold
of the _electronic_ aggregate. In a similar way, if intramolecular vibrations
are considered in an isolated molecule, the harmonic/anharmonic oscillator
model for the electronic ground and excited states can be applied. This also
results in discrete spectrum of _vibrational_ levels. These two pictures merge
due to the intermolecular coupling and the electronic and vibrational
subsystems mix up. It is then natural to try to quantify, how much of
electronic or vibrational character is inherited in the composite system.
However, this often leads to many ambiguities, for example, in linear spectra
of J-aggregatesFidderCP2007 .
There have been many attempts to unambiguously distinguish between
vibrational, vibronic and excitonic coherences visible as oscillations in 2D
spectra. However, the question of how to do that is proper only if mixing in
the system is low. As it was shown here, two conditions for low mixing can be
distinguished: (i) the coupling between vibrational and electronic subsystem
has to be weak (small Huang–Rhys factor); (ii) the system must not be in a
vicinity of exciton–vibronic resonance, represented by the avoided crossing
region in the energy spectrum. These conditions are best met for high-
frequency intramolecular vibrations in weakly-coupled P–P complexes and low-
frequency vibrations in strongly-coupled aggregates, the D1 and D2 models,
respectively.
In the case of substantial mixing of the coherences of electronic and
vibrational character, the information about the transition composition can be
evaluated from coherent oscillations in some cases. It is most obvious in the
D2 model representing the case of the J-aggregate, when the electronic system
is strongly coupled to low-frequency vibration (second row in Fig. 6b).
Despite the strong mixing the Fourier map for the electronic frequency
($\omega_{2}=1230\,\mathrm{\mathrm{cm}^{-1}}$) contains diagonal-symmetric
features, similar to those present in the weak mixing case. In the D4 model,
which stands as an equivalent of the strongly-coupled dimeric dye, the mixture
of coherences for $\omega_{2}=1060\,\mathrm{\mathrm{cm}^{-1}}$ and
$\omega_{2}=1400\,\mathrm{\mathrm{cm}^{-1}}$ can also be disentangled
($s=0.05$, Fig. 7b). Firstly, the Fourier map at
$\omega_{2}=1060\,\mathrm{\mathrm{cm}^{-1}}$ contains diagonal-symmetric
peaks, which would suggest, that this particular coherence is rather
electronic. Secondly, there are features in the map at
$\omega_{2}=1400\,\mathrm{\mathrm{cm}^{-1}}$ separated by
$1400\,\mathrm{\mathrm{cm}^{-1}}$ and $1060\,\mathrm{\mathrm{cm}^{-1}}$ as
well as the peak on the diagonal exhibiting high-frequency oscillations. The
later fact as well as the obviously stronger oscillations below the diagonal
shows that the origin of the $1400\,\mathrm{\mathrm{cm}^{-1}}$ oscillation is
rather vibrational. The similar analysis can be applied to the D4 model with
$s=0.5$, where the mere evidence of vibrational content is the diagonal
oscillating peak in the $\omega_{2}=1400\,\mathrm{\mathrm{cm}^{-1}}$ map.
One cannot discriminate between coherences of dominating electronic or
vibrational character in weakly-interacting photosynthetic complexes, coupled
to low-frequency vibrations. This is clearly demonstrated by the D3 model in
both cases of weak and strong electron–phonon coupling (first and second rows
in Fig. 7, respectively). The coherences in the system are highly mixed and no
typical patterns, which were present in the Fourier maps of the other systems,
are found here. The Fourier map in case of strong coupling to vibrations is
composed of many contributions, evolving in the ground and excited states (the
second row in Fig. 7), thus, indicating complete state character mixing.
Hence, the _electronic_ or _vibrational_ transitions are properly qualified,
while _vibronic_ and _how-much-vibronic_ is a vague concept and should be
avoided. Instead one should treat such coherences as simply mixed, which is
completely a proper concept in quantum mechanics.
### IV.2 Lifetime of coherences in aggregates
The fact that some coherences are less visible in the Fourier maps is to high
degree related to their lifetimes. Obviously, oscillations which decay fast
will be vaguely captured by the Fourier transform or even will not be present
in the maps at all. Let us now concentrate on the maximum of the Fourier
amplitude dependence on frequency, $\mathcal{A}(\omega_{2})$, in case of
$s=0.05$, presented by the solid lines in Fig. 5. The widths of the peaks are
given by the lifetime of the corresponding oscillations.
The lifetime of the vibrational ground state coherence depends only on the
overlap of vibrational frequency and bath spectral density. It can be deduced
from Eqs. (31) and (32). For example, the lifetime of $|{\rm g}_{0}{\rm
g}_{0}\rangle\langle{\rm g}_{0}{\rm g}_{1}|$ coherence
$\tau_{01}=2(\gamma_{{\rm g}_{0}{\rm g}_{0}}+\gamma_{{\rm g}_{0}{\rm
g}_{1}})^{-1}$ is $\sim 3$ ps for $\omega_{0}=1400\,\mathrm{\mathrm{cm}^{-1}}$
and the width of the corresponding peak in the $\mathcal{A}(\omega_{2})$
dependence is $\sim 40\,\mathrm{\mathrm{cm}^{-1}}$ (Fig. 5a and d). The
lifetime of $\omega_{0}=250\,\mathrm{\mathrm{cm}^{-1}}$ coherence is $\sim
500$ fs, thus, the corresponding peak in $\mathcal{A}(\omega_{2})$ is very
broad and, therefore, hardly distinguishable (Fig. 5b and c). This effect
essentially depends on the spectral density function (including the shape and
the amplitude) and its value at the corresponding vibrational frequency
Jankowiak_JPCB2013_SpectralDensities .
The lifetimes of coherences in the excited state manifold are not that
trivial. On one hand, transfer rates relating electronic states of purely
electronic aggregates depend on the absolute value of the bath spectral
density at the corresponding frequency. Additionally, they depend on extend of
delocalization of a particular state. On the other hand, transfer rates
between vibronic states of a single molecule are the same as of the ground
state vibrational states. In our case, these two pictures are merged and the
lifetimes of mixed coherences cannot be expressed in simple terms.
It has been shown, that the lifetimes of vibronic coherences increase
significantly, if the electronic transitions are close to vibrational
frequencies even if the Huang–Rhys factor is small ($s<0.1$) Chenu2013 ;
Christensson_JPCB2012 . This is evident in the $\mathcal{A}(\omega_{2})$
dependencies, as well: the lifetime of the $\omega_{2}=1.8J$ oscillation in
the D1 model is smaller than the corresponding lifetime of the frequency
oscillation in the D4 model by factor of $\sim 1.8$ while the lifetimes of the
$\omega_{2}=\omega_{0}$ coherences are identical. If compared, $\omega_{2}=2J$
oscillations in the D2 model decay at least $3$ times faster than the
$\omega_{2}=1.8J$ oscillations in the D4 model.
If the Huang–Rhys factors are large ($s\gtrsim 0.5$), low-frequency
vibrational coherences in the ground state decay faster than in the case of
weak coupling to vibrations discussed above (see dashed lines in Fig. 5). This
is due to the lower value of the reorganization energy, which is
$\lambda=50\,\mathrm{\mathrm{cm}^{-1}}$ for $s=0.5$ (cf.
$\lambda=500\,\mathrm{\mathrm{cm}^{-1}}$ for $s=0.05$). Stronger interaction
with vibrations induces more mixing in the system. Therefore, we can see long-
lived coherences of $\omega_{2}=2.05J$ in the D2 model. We can thus conclude
that the electronic coherences effectively borrow some lifetime from the
vibrational coherences due to the quantum mechanical mixing. The mixing and
borrowing of the dipole strength in excitonic systems is a well-known
phenomenon, however the lifetime borrowing is poorly discussed.
### IV.3 Energetic disorder
Energetic disorder is yet another important parameter for the coherent state
evolution. It was shown for molecular dimer with vibrations, that in the case
of substantial energy disorder, coherences of prevailing vibrational/vibronic
character will dominate over those of rather electronic character Butkus2013 .
From the discussion above it follows that this effect will be more significant
for rather pure systems: weakly-coupled P–P complexes with high-frequency
vibrations and J-aggregates (D1 and D2 models, respectively). In both cases
vibrational coherences will dominate in the Fourier maps, while the electronic
coherences will dephase fast because of combined influence of the energetic
disorder and the lifetime of the state. In the mixed systems (D3 and D4) the
result will be more complicated. The reason behind is that the mixing occurs
when resonances match, i.e. at the exciton-vibronic resonance. Disorder will
make the matching less significant for most of ensemble members show less
mixing. Hence, the electronic and vibronic character for a disordered ensemble
is better defined and should be better identified in experiments.
## V Conclusions
In this paper, theory of molecular aggregate with intramolecular vibrations
for coherent spectroscopy was presented. It accounts for incoherent and
coherent effects caused by excitonic coupling and exciton–vibronic
interaction. The molecular dimer model is used for simulation purposes of
typical systems in a wide range of parameters to reflect pigment–protein
complexes, J-aggregates or dimeric molecular dyes.
Regarding the question of distinguishing the electronic, vibronic or
vibrational coherences, we conclude that the question itself is fully defined
and proper only if the character of states is pure (electronic or
vibrational). We have shown that such a separation is indeed possible for
systems, where the resonant coupling and vibrational frequency is off-
resonant, i.e. the system is away from the so-called exciton–vibronic
resonance. The analysis of oscillations in mixed systems is of qualitative
significance and allows us to to tell if the specified oscillation is of the
mixed origin. The frequencies of transitions, involved in the mixture can be
resolved.
The positions of oscillating features must be taken into careful consideration
when analyzing experimental data. There are spectral regions of mixed systems,
where oscillations exceptionally due to coherences in the ground or excited
states can be found, thus providing more information about system composition.
For example, the property that features in the Fourier maps are asymmetric
with respect to the diagonal is the signature that the corresponding state is
mixed.
Lifetime of excitonic coherences is mostly determined by the coupling to
discrete modes of intramolecular vibrations. These modes, on their own, are
coupled to the continuum of low-frequency bath fluctuations, represented by
the spectral density. Thus, the overlap of the spectral density function and
frequencies of intramolecular vibrations as well as the form of spectral
density function directly influences the lifetime of electronic coherences.
## Appendix A Correlation functions involving singly- and doubly-excited
states
Singly-excited eigenstates are obtained by unitary transformation, and we get
the same symmetry properties as for
$\displaystyle C_{p_{1}p_{2},p_{3}p_{4}}^{({\rm ee})}(t)$
$\displaystyle=C_{0}(t)\sum_{m,n}^{N}\sum_{\bm{i},\bm{j}}\sum_{\bm{k},\bm{l}}\sum_{a}^{N}\psi_{p_{1},\bm{i}}^{m\ast}\psi_{p_{2},\bm{j}}^{m}\psi_{p_{3},\bm{k}}^{n\ast}\psi_{p_{4},\bm{l}}^{n}$
$\displaystyle\times\langle i_{a},j_{a}\rangle\langle
k_{a},l_{a}\rangle\prod_{{s\atop s\neq
a}}\delta_{i_{s}j_{s}}\delta_{k_{s}l_{s}}.$ (34)
Here the first sum is over different chromophores, the second and third sum is
over the vibrational levels of the whole aggregate and finally the sum over
$a$ is over the different vibrational modes (which is identical to the number
of sites since each site has one vibrational coordinate). We then get the
following result:
$\displaystyle C_{p_{1}p_{2},p_{3}p_{4}}^{({\rm ee})}(t)$
$\displaystyle=C_{0}(t)\sum_{\bm{i},\bm{k}}\sum_{s}^{N}\left\\{\sqrt{i_{s}k_{s}}\xi_{\bm{i}_{s}^{-}}^{(p_{1}p_{2})}\xi_{\bm{k}_{s}^{-}}^{(p_{3}p_{4})}+\sqrt{(i_{s}+1)k_{s}}\xi_{\bm{i}_{s}^{+}}^{(p_{1}p_{2})}\xi_{\bm{k}_{s}^{-}}^{(p_{3}p_{4})}\right.$
$\displaystyle\left.+\sqrt{i_{s}(k_{s}+1)}\xi_{\bm{i}_{s}^{-}}^{(p_{1}p_{2})}\xi_{\bm{k}_{s}^{+}}^{(p_{3}p_{4})}+\sqrt{(i_{s}+1)(k_{s}+1)}\xi_{\bm{i}_{s}^{+}}^{(p_{1}p_{2})}\xi_{\bm{k}_{s}^{+}}^{(p_{3}p_{4})}\right\\}.$
Here
$\xi_{\bm{i}_{s}^{\pm}}^{(p_{a}p_{b})}=\sum_{n}^{N}\psi_{p_{a},\bm{i}}^{n\ast}\psi_{p_{b},\bm{i}_{s}^{\pm}}^{n}.$
(35)
For the functions involving the double excitations we can write similarly
$\displaystyle C_{p_{1}p_{2},r_{1}r_{2}}^{({\rm ef})}(t)$
$\displaystyle=C_{0}(t)\sum_{\bm{i},\bm{k}}\sum_{s}^{N}\left\\{\sqrt{i_{s}k_{s}}\xi_{\bm{i}_{s}^{-}}^{(p_{1}p_{2})}\Xi_{\bm{k}_{s}^{-}}^{(r_{1}r_{2})}+\sqrt{(i_{s}+1)k_{s}}\xi_{\bm{i}_{s}^{+}}^{(p_{1}p_{2})}\Xi_{\bm{k}_{s}^{-}}^{(r_{1}r_{2})}\right.$
$\displaystyle\left.+\sqrt{i_{s}(k_{s}+1)}\xi_{\bm{i}_{s}^{-}}^{(p_{1}p_{2})}\Xi_{\bm{k}_{s}^{+}}^{(r_{1}r_{2})}+\sqrt{(i_{s}+1)(k_{s}+1)}\xi_{\bm{i}_{s}^{+}}^{(p_{1}p_{2})}\Xi_{\bm{k}_{s}^{+}}^{(r_{1}r_{2})}\right\\}$
and
$\displaystyle C_{r_{1}r_{2},r_{3}r_{4}}^{({\rm ff})}(t)$
$\displaystyle=C_{0}(t)\sum_{\bm{i},\bm{k}}\sum_{s}^{N}\left\\{\sqrt{i_{s}k_{s}}\Xi_{\bm{i}_{s}^{-}}^{(r_{1}r_{2})}\Xi_{\bm{k}_{s}^{-}}^{(r_{3}r_{4})}+\sqrt{(i_{s}+1)k_{s}}\Xi_{\bm{i}_{s}^{+}}^{(r_{1}r_{2})}\Xi_{\bm{k}_{s}^{-}}^{(r_{3}r_{4})}\right.$
$\displaystyle\left.+\sqrt{i_{s}(k_{s}+1)}\Xi_{\bm{i}_{s}^{-}}^{(r_{1}r_{2})}\Xi_{\bm{k}_{s}^{+}}^{(r_{3}r_{4})}+\sqrt{(i_{s}+1)(k_{s}+1)}\Xi_{\bm{i}_{s}^{+}}^{(r_{1}r_{2})}\Xi_{\bm{k}_{s}^{+}}^{(r_{3}r_{4})}\right\\},$
where
$\Xi_{\bm{i}_{s}^{\pm}}^{(r_{1}r_{2})}=\sum_{{m,n\atop
m>n}}^{N}\Psi_{r_{1},\bm{i}}^{(mn)\ast}\Psi_{r_{2},\bm{i}_{s}^{\pm}}^{(mn)}.$
(36)
## Appendix B Response functions for 2D photon echo spectra
We consider the photon-echo signals of the 2D electronic spectroscopy in the
impulsive limit. In this limit, the laser pulses are assumed as infinitely
short and, therefore, the measured intensity of electric field is proportional
to the system response function, the expressions of which are presented in
this appendix following the notation used in Fig. 2 Abramavicius2009 . The
expressions for the response involve the spectral lineshape functions
$g_{ab,cd}(t)\equiv h_{ab,cd}g_{0}(t)$. These are given by the linear integral
transformation of the bath correlation functions,
$g_{0}(t)=\int_{0}^{t}\mathrm{d}t^{\prime}\int_{0}^{t^{\prime}}\mathrm{d}t^{\prime\prime}\left\langle
C_{0}(t^{\prime\prime})C_{0}(0)\right\rangle$ mukbook . The response functions
of the photon-echo (rephasing) signal when transport is ignored are then
($\bm{t}\equiv\left\\{t_{3},t_{2},t_{1}\right\\}$)
$\displaystyle S_{{\rm
B}}(\bm{t})=\mathrm{i}^{3}\theta(\bm{t})\sum_{\bm{i}\bm{j}}\sum_{p_{1}p_{2}}p_{{\rm
g}_{\bm{i}}}(\delta_{\bm{i},\bm{j}}G_{{\rm g}_{\bm{i}}{\rm
g}_{\bm{i}}}(t_{2})+\zeta_{\bm{i},\bm{j}})$ (37)
$\displaystyle\times\left\langle\bm{\mu}_{{\rm g}_{\bm{i}}}^{{\rm
e}_{p_{1}}}\bm{\mu}_{{\rm e}_{p_{1}}}^{{\rm g}_{\bm{j}}}\bm{\mu}_{{\rm
g}_{\bm{j}}}^{{\rm e}_{p_{2}}}\bm{\mu}_{{\rm e}_{p_{2}}}^{{\rm
g}_{\bm{i}}}\right\rangle\mathrm{e}^{{\rm i}\xi_{{\rm e}_{p_{1}}{\rm
g}_{\bm{i}}}t_{1}-{\rm i}\xi_{{\rm g}_{\bm{i}}{\rm g}_{\bm{j}}}t_{2}-{\rm
i}\xi_{{\rm e}_{p_{2}}{\rm g}_{\bm{j}}}t_{3}}$
$\displaystyle\times\mathrm{e}^{\phi_{{\rm e}_{p_{1}}{\rm g}_{\bm{j}}{\rm
e}_{p_{2}}{\rm g}_{\bm{i}}}(0,t_{1},t_{1}+t_{2}+t_{3},t_{1}+t_{2})},$
$\displaystyle S_{{\rm
E}}(\bm{t})=\mathrm{i}^{3}\theta(\bm{t})\sum_{\bm{i}\bm{j}}\sum_{p_{1}p_{2}}p_{{\rm
g}_{\bm{i}}}(\delta_{p_{1}p_{2}}G_{{\rm e}_{p_{1}}{\rm
e}_{p_{2}}}(t_{2})+\zeta_{p_{1}p_{2}})$ (38)
$\displaystyle\times\left\langle\bm{\mu}_{{\rm g}_{\bm{i}}}^{{\rm
e}_{p_{1}}}\bm{\mu}_{{\rm g}_{\bm{i}}}^{{\rm e}_{p_{2}}}\bm{\mu}_{{\rm
e}_{p_{2}}}^{{\rm g}_{\bm{j}}}\bm{\mu}_{{\rm e}_{p_{2}}}^{{\rm
g}_{\bm{j}}}\right\rangle\mathrm{e}^{{\rm i}\xi_{{\rm e}_{p_{1}}{\rm
g}_{\bm{i}}}t_{1}-\mathrm{i}\xi_{{\rm e}_{p_{2}}{\rm e}_{p_{1}}}t_{2}-{\rm
i}\xi_{{\rm e}_{p_{1}}{\rm g}_{\bm{j}}}t_{3}}$
$\displaystyle\times\mathrm{e}^{\phi_{{\rm e}_{p_{1}}{\rm g}_{\bm{j}}{\rm
e}_{p_{2}}{\rm g}_{\bm{i}}}(0,t_{1}+t_{2},t_{1}+t_{2}+t_{3},t_{1})}$
and
$\displaystyle S_{{\rm
A}}(\bm{t})=-\mathrm{i}^{3}\theta(\bm{t})\sum_{\bm{i}}\sum_{p_{1}p_{2}}\sum_{r}p_{{\rm
g}_{\bm{i}}}(\delta_{p_{1}p_{2}}G_{{\rm e}_{p_{1}}{\rm
e}_{p_{1}}}(t_{2})+\zeta_{p_{1}p_{2}})$ (39)
$\displaystyle\times\left\langle\bm{\mu}_{{\rm g}_{\bm{i}}}^{{\rm
e}_{p_{1}}}\bm{\mu}_{{\rm g}_{\bm{i}}}^{{\rm e}_{p_{2}}}\bm{\mu}_{{\rm
e}_{p_{2}}}^{{\rm f}_{r}}\bm{\mu}_{{\rm f}_{r}}^{{\rm
e}_{p_{1}}}\right\rangle\mathrm{e}^{\mathrm{i}\xi_{{\rm e}_{p_{1}}{\rm
g}_{\bm{i}}}t_{1}-\mathrm{i}\xi_{{\rm e}_{p_{2}}{\rm
e}_{p_{1}}}t_{2}-\mathrm{i}\xi_{{\rm f}_{r}{\rm e}_{p_{1}}}t_{3}}$
$\displaystyle\times\mathrm{e}^{\phi_{{\rm e}_{p_{1}}{\rm f}_{r}{\rm
e}_{p_{2}}{\rm g}_{\bm{i}}}(0,t_{1}+t_{2}+t_{3},t_{1}+t_{2},t_{1})}.$
Here, the complex variable
$\xi_{ab}=\omega_{ab}-\mathrm{i}\frac{1}{2}(\gamma_{a}+\gamma_{b})$ is used to
take into account the state dephasing due to finite lifetime,
$\gamma_{a}=\frac{1}{2}\sum_{a^{\prime}\neq a}k_{a\leftarrow a^{\prime}}$.
$p_{{\rm g}_{\bm{i}}}$ is the Boltzmann probability for the system to be in
the $\bm{i}$-th vibrational state prior the excitation and $\theta(\bm{t})$ is
the product of Heaviside functions, $\theta(t_{1})\theta(t_{2})\theta(t_{3})$.
The auxiliary function is
$\displaystyle\phi_{{\rm e}_{p_{1}}c{\rm e}_{p_{2}}{\rm
g}_{\bm{i}}}(\tau_{4},\tau_{3},\tau_{2},\tau_{1})=$ $\displaystyle\quad-
g_{{\rm e}_{p_{1}}{\rm e}_{p_{1}}}(\tau_{43})-g_{cc}(\tau_{32})-g_{{\rm
e}_{p2}{\rm e}_{p2}}(\tau_{21})$ (40) $\displaystyle\quad+g_{{\rm
e}_{p_{1}}c}(\tau_{32})+g_{{\rm e}_{p_{1}}c}(\tau_{43})-g_{{\rm
e}_{p_{1}}c}(\tau_{42})$ $\displaystyle\quad-g_{{\rm e}_{p_{1}}{\rm
e}_{p_{2}}}(\tau_{32})+g_{{\rm e}_{p_{1}}{\rm e}_{p_{2}}}(\tau_{31})+g_{{\rm
e}_{p_{1}}{\rm e}_{p_{2}}}(\tau_{42})$ $\displaystyle\quad-g_{{\rm
e}_{p_{1}}{\rm e}_{p_{2}}}(\tau_{41})+g_{c{\rm e}_{p_{2}}}(\tau_{21})+g_{c{\rm
e}_{p_{2}}}(\tau_{32})-g_{c{\rm e}_{p_{2}}}(\tau_{31})$ $\displaystyle\quad-
g_{c{\rm g}_{\bm{i}}}(\tau_{21})+g_{c{\rm g}_{\bm{i}}}(\tau_{24})+g_{c{\rm
g}_{\bm{i}}}(\tau_{31})-g_{c{\rm g}_{\bm{i}}}(\tau_{34}),$
where $c$ stands for either doubly-excited state ${\rm f}_{r}$, either ground
state ${\rm g}_{\bm{j}}$. Response function components with transport are
$\displaystyle\tilde{S}_{{\rm
B}}(\bm{t})=\mathrm{i}^{3}\bm{\theta}(\bm{t})\sum_{\bm{i}\bm{j}}\sum_{p_{1}p_{2}}p_{{\rm
g}_{\bm{i}}}\zeta_{\bm{i}\bm{j}}G_{{\rm g}_{\bm{j}}{\rm g}_{\bm{i}}}(t_{2})$
(41) $\displaystyle\times\left\langle\bm{\mu}_{{\rm g}_{\bm{i}}}^{{\rm
e}_{p_{1}}}\bm{\mu}_{{\rm e}_{p_{1}}}^{{\rm g}_{\bm{i}}}\bm{\mu}_{{\rm
g}_{\bm{j}}}^{{\rm e}_{p_{2}}}\bm{\mu}_{{\rm e}_{p_{2}}}^{{\rm
g}_{j}}\right\rangle\mathrm{e}^{\mathrm{i}\xi_{{\rm e}_{p_{1}}{\rm
g}_{\bm{i}}}t_{1}-\mathrm{i}\xi_{{\rm e}_{p_{2}}{\rm
g}_{\bm{j}}}t_{3}+\varphi_{{\rm e}_{p_{2}}{\rm g}_{\bm{j}}{\rm e}_{p_{2}}{\rm
e}_{p_{1}}}^{\ast}(\bm{t})},$
$\displaystyle\tilde{S}_{{\rm
E}}(\bm{t})=\mathrm{i}^{3}\bm{\theta}(\bm{t})\sum_{\bm{i}\bm{j}}\sum_{p_{1}p_{2}}p_{{\rm
g}_{\bm{i}}}\zeta_{p_{1}p_{2}}G_{{\rm e}_{p_{2}}{\rm e}_{p_{1}}}(t_{2})$ (42)
$\displaystyle\times\left\langle\bm{\mu}_{{\rm g}_{\bm{i}}}^{{\rm
e}_{p_{1}}}\bm{\mu}_{{\rm g}_{\bm{i}}}^{{\rm e}_{p_{1}}}\bm{\mu}_{{\rm
e}_{p_{2}}}^{{\rm g}_{\bm{j}}}\bm{\mu}_{{\rm e}_{p_{2}}}^{{\rm
g}_{\bm{j}}}\right\rangle\mathrm{e}^{\mathrm{i}\xi_{{\rm e}_{p_{1}}{\rm
g}_{\bm{i}}}t_{1}-\mathrm{i}\xi_{{\rm e}_{p_{2}}{\rm
g}_{\bm{j}}}t_{3}+\varphi_{{\rm e}_{p_{2}}{\rm g}_{\bm{j}}{\rm e}_{p_{2}}{\rm
e}_{p_{1}}}^{\ast}(\bm{t})},$
and
$\displaystyle\tilde{S}_{{\rm
A}}(\bm{t})=-i^{3}\bm{\theta}(\bm{t})\sum_{\bm{i}}\sum_{p_{1}p_{2}}\sum_{r}p_{{\rm
g}_{\bm{i}}}\zeta_{p_{1}p_{2}}G_{{\rm e}_{p_{2}}{\rm e}_{p_{1}}}(t_{2})$ (43)
$\displaystyle\times\left\langle\bm{\mu}_{{\rm g}_{\bm{i}}}^{{\rm
e}_{p_{1}}}\bm{\mu}_{{\rm g}_{\bm{i}}}^{{\rm e}_{p_{1}}}\bm{\mu}_{{\rm
e}_{p_{2}}}^{{\rm f}_{r}}\bm{\mu}_{{\rm f}_{r}}^{{\rm
e}_{p_{2}}}\right\rangle\mathrm{e}^{{\rm i}\xi_{{\rm e}_{p_{1}}{\rm
g}_{\bm{i}}}t_{1}-{\rm i}\xi_{{\rm f}_{r}{\rm e}_{p_{2}}}t_{3}+\varphi_{{\rm
f}_{r}{\rm e}_{p_{2}}{\rm e}_{p_{2}}{\rm e}_{p_{1}}}^{\ast}(\bm{t})}.$
Here
$\displaystyle\varphi_{cb{\rm e}_{p_{2}}{\rm e}_{p_{1}}}(\bm{t})=-g_{{\rm
e}_{p_{1}}{\rm e}_{p_{1}}}(t_{1})-g_{bb}(t_{3})-g_{cc}^{\ast}(t_{3})$
$\displaystyle-g_{b{\rm e}_{p_{1}}}(t_{1}+t_{2}+t_{3})+g_{b{\rm
e}_{p_{1}}}(t_{1}+t_{2})+g_{b{\rm e}_{p_{1}}}(t_{2}+t_{3})$ $\displaystyle-
g_{b{\rm e}_{p_{1}}}(t_{2})+g_{c{\rm e}_{p_{1}}}(t_{1}+t_{2}+t_{3})-g_{c{\rm
e}_{p_{1}}}(t_{1}+t_{2})$ $\displaystyle-g_{c{\rm
e}_{p_{1}}}(t_{2}+t_{3})+g_{c{\rm
e}_{p_{1}}}(t_{2})+g_{cb}(t_{3})+g_{bc}^{\ast}(t_{3})$ $\displaystyle+2{\rm
i}\Im[g_{c{\rm e}_{p_{2}}}(t_{2}+t_{3})-g_{c{\rm e}_{p_{2}}}(t_{2})-g_{c{\rm
e}_{p_{2}}}(t_{3})$ $\displaystyle+g_{b{\rm e}_{p_{2}}}(t_{2})-g_{b{\rm
e}_{p_{2}}}(t_{2}+t_{3})+g_{b{\rm e}_{p_{2}}}(t_{3}).$ (44)
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|
arxiv-papers
| 2013-10-04T17:29:46 |
2024-09-04T02:49:51.969236
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Vytautas Butkus, Leonas Valkunas, Darius Abramavicius",
"submitter": "Vytautas Butkus",
"url": "https://arxiv.org/abs/1310.1343"
}
|
1310.1371
|
# Robust and highly performant ring detection algorithm for 3d particle
tracking using 2d microscope imaging
Eldad Afik111To whom correspondence should be addressed; email:
[email protected]
Department of Physics of Complex Systems, Weizmann Institute of Science,
Rehovot 76100, Israel
###### Abstract
Three-dimensional particle tracking is an essential tool in studying dynamics
under the microscope, namely, fluid dynamics in microfluidic devices, bacteria
taxis, cellular trafficking. The 3d position can be determined using 2d
imaging alone by measuring the diffraction rings generated by an out-of-focus
fluorescent particle, imaged on a single camera. Here I present a ring
detection algorithm exhibiting a high detection rate, which is robust to the
challenges arising from ring occlusion, inclusions and overlaps, and allows
resolving particles even when near to each other. It is capable of real time
analysis thanks to its high performance and low memory footprint. The proposed
algorithm, an offspring of the circle Hough transform, addresses the need to
efficiently trace the trajectories of many particles concurrently, when their
number in not necessarily fixed, by solving a classification problem, and
overcomes the challenges of finding local maxima in the complex parameter
space which results from ring clusters and noise. Several algorithmic concepts
introduced here can be advantageous in other cases, particularly when dealing
with noisy and sparse data. The implementation is based on open-source and
cross-platform software packages only, making it easy to distribute and
modify. It is implemented in a microfluidic experiment allowing real-time
multi-particle tracking at $70\text{\,}\mathrm{Hz}$, achieving a detection
rate which exceeds 94% and only 1% false-detection.
To keep the main presentation succinct many details of the proposed algorithm
and the application for particle tracking under the microscope were left out
of the main text. These can be found below. The Supplementary Information
opens with a more detailed and technical presentation of the algorithm,
followed by the discussion of the application of the proposed algorithm for
particle tracking, in particular in the light of the available alternatives;
the closing section contains the supporting figures: (i) an empirical
calibration curve of the ring radius to the out-of-focus distance is shown in
Supplementary Fig. S1; the error bars provide an estimate of the precision of
the proposed method resulting from the combination of the algorithm with the
optical system together; (ii) Supplementary Fig. S2 shows a sample of 3d
particle trajectories reconstructed based on the proposed method; and finally
(iii) examples from the comparative assessment of the algorithm robustness
referred to in the manuscript and described in the Methods section can be
found in Supplementary Fig. S3.
The study of dynamics often relies on tracking objects under the microscope.
Indeed, precise and robust particle tracking is essential in many fields,
including studies of micro-Rheology [1, 2], chaotic dissipative flows [3, 4],
feedback for micro-manipulation [5], and other soft condensed matter physics
and engineering problems. Moreover, microfluidic systems play a growing role
as part of lab-on-a-chip apparatus in micro-chemistry [6, 7], bioanalytics [8,
9], and other bio-medical research and engineering applications [10]. Yet,
detailed characterisation of the flow and transport phenomena at the micro-
scale is still a non-trivial task. In general, the motion is three dimensional
and automated tracking is cumbersome from the perspective of both
instrumentation and software. Setting up several viewing angles as done for
large systems [11] becomes even more complicated in microscopic systems, while
scanning through the third axis, e.g. confocal microscopy, clearly compromises
temporal resolution and concurrency. In this work the three-dimensional
positions of fluorescent particles are inferred from the information encoded
in the diffraction rings which result from out-of-focus imaging, converting
the 3d localisation problem to an image analysis problem of ring detection.
The development of the method presented here was motivated by the study of
pair dispersion in a chaotic flow [12], taking place in a microfluidic tube of
$140\text{\,}\mathrm{\SIUnitSymbolMicro m}$ – the observation volume is larger
by more than three orders of magnitude with respect to those reported in Refs.
[5] and [13]. The experiments consist of tracking tracers advected by the
flow, at seeding levels of several tens to hundreds in the observation window,
where it is necessary that the particles are resolved even when nearby to each
other. The typical flow rates dictate sampling rates of
$70\text{\,}\mathrm{Hz}$ whereas the statistical nature of the problem
requires data acquisition over weeks. Using a standard epi-fluorescence
microscope and the fact that the parameters of the most visible ring can be
mapped to the 3d position of the tracer, the particle localisation problem is
converted to a circle detection problem. The constraints set by the nature of
the experiment require an image analysis algorithm that is robust not only to
the noise of the image acquisition process, but to rings overlaps, inclusions
and occlusions as well. The typical complexity of the images is exemplified in
a sub-frame from our experiment presented in Fig. 1a. In addition, the data
flow is of about $180\text{\,}\mathrm{GB}\text{/}\mathrm{h}$, an overwhelming
rate which demands the optimisation of the algorithm for real-time analysis.
In this presentation I will focus on the development of an algorithm for this
purpose. The key steps for achieving high-performance are introduced following
the presentation of the main concepts which contribute to the robustness of
the algorithm. The application for particle tracking is presented and
discussed in the Supplementary Information; further technical details of the
optical apparatus can be found in the Methods section.
Imagine for the moment that you have successfully identified which of the
pixels in the image reside on a ring. The issue of doing so will be addressed
later on. Given this set of coordinates, it may seem straightforward to find
the parameters of the circles which best fit them. However there is a missing
piece of information here, that is, which sub-sets of pixels belong together
to form a ring. Moreover, we do not know a priori how many rings there are in
the image and there may be false detected coordinates which we would like to
disregard. Therefore, we need some method to classify/cluster the coordinates
into sub-sets, each sub-set matching a single ring.
A circle in a two-dimensional image is uniquely specified by three parameters.
In this work two designate the centre of the circle and the third specifies
its radius. The parameter space of all possible circles is therefore three-
dimensional. One can detect circles in an image by mapping the image intensity
field to the circle parameter space. Peaks in this parameter space imply a
circle well represented in the image. One approach to achieve this mapping is
via a discrete Radon transform, which for the purposes of this presentation
translates to convolving the image with a mask of a ring [14]. Since each
candidate radius calls for a separate convolution, this results in visiting
all the pixels in the image over and over. Recall that the outer-most ring is
sufficient for 3d localisation of the imaged particles. Hence it is worth
noting that even at moderate rings densities, the pixels lying on the outer-
most ones consist a small fraction of the pixel population, less than 2% in my
case. When there are more than a couple of potential radii this procedure
would perform a plethora of useless computations [14]. This fact directs to
another approach, which may seem equivalent yet explicitly exploits this
information sparsity – the circle Hough transform [15]: each pixel votes for
all the candidate points in the parameter space of which it may be part. In
this way, every pixel is visited once, potentially reducing the computation
time by orders of magnitude. The discrete version of the parameter space is
commonly referred to as the array of accumulators. During the voting procedure
each vote increments an accumulator by one. Alas, in the literature of
Computer Vision and Pattern Recognition it is well known that the standard
circle Hough transform is rather demanding both for large memory requirements,
which grow with the radii range, as well as for its 3d nature which renders
peak finding in the parameter space a difficult task to tackle [16, 17].
One way to address these computational challenges is to resort to lower
dimensionality circle Hough transforms, but these usually miss circles having
nearby centres and are less robust, resulting in higher false positive and
false negative errors rates [16]. Another path is to randomly sub-sample the
information content in the image, giving way to non-deterministic methods; see
Ref. [17] and references therein. However, due to their random nature these
methods suffer from inferior detection rates and accuracy when compared with
the deterministic ones [17].
For these reasons I developed an algorithm which is an offspring of the full
3d-circle Hough transform, yet the local maxima detection issues are addressed
and it shows high performance and a small memory footprint.
## Results
####
a b
c
d
Figure 1: Snapshots from the experiment and a demonstration of the algorithm
robustness. (a) typical image complexity is exemplified in an unprocessed sub-
frame consisting of 1/9 part of the full frame, corresponding to lateral
dimension of $215\text{\,}\mathrm{\SIUnitSymbolMicro
m}$$\times$$315\text{\,}\mathrm{\SIUnitSymbolMicro m}$. The axial range
available for the particles is $140\text{\,}\mathrm{\SIUnitSymbolMicro m}$.
(b) the corresponding analysis result; in red are the radii in pixels units.
(c) & (d) time sequences of sub-frames ($400\text{\,}\mathrm{ms}$ each). Red
coloured particles in (c) demonstrate pair dispersion, in which the algorithm
is required to resolve rings with similar parameters. The yellow particle in
(d) shows radius change corresponding to a downwards translation. Each sub-
frame in (c) & (d) images a box which lateral dimensions is
$190\text{\,}\mathrm{\SIUnitSymbolMicro
m}$$\times$$270\text{\,}\mathrm{\SIUnitSymbolMicro m}$.
The key steps of the algorithm are conceptually outlined as follows: (i)
detect directed ridges; (ii) map the directed ridges to the parameter space
of circles; (iii) detect local maxima via radius dependent smoothing and
normalisation; (iv) classify the coordinates of the ridge pixels according to
the peaks in the circle parameter space, and fit each sub-set to a circle,
achieving sub-pixel accuracy. This outline is presented graphically in Fig. 2
for a small sub-frame containing two fluorescent particles.
a
b
c
f
e
d
Figure 2: Algorithm outline. (a) raw sub-image containing two fluorescent
particles; note that the inner rings of each particle are thinner than the
outer most one. This scale separation admits suppression of all but the outer
most ring via Gaussian smoothing (to ease visualisation the contrast was
enhanced in the images on the expense of the central peak of the diffraction
pattern); (b) ridge detection: the ridges are defined using a differential
geometric descriptor and shown here as arrows representing $X_{-}$, the
principal direction, corresponding to $k_{-}$, the least principal curvature,
which is plotted in the background. The arrows originate from the ridge pixel.
Note that the inner rings are successfully suppressed based on the scale
separation. To ease visualisation every second detected ridge is omitted; (c)
circle Hough transform: directed ridges $\to$ circle parameter space; (d)
local maxima detection: radius dependent smoothing of the parameter space as
well as normalisation by 1/r and thresholding greatly emphasise the local
maxima representing the rings in the image; (e) sub-pixel accuracy: based on
the detected rings, annulus masks (blue and green annuli in the figure) allow
classification of ridge pixels (red points) and sub-pixel accuracy is achieved
via circle fitting. Note the discarded directed ridges of the central peak (in
(b)) as they do not belong to any local maxima in the processed circle
parameter space (d); (f) the output: best fit circle for the ridge pixels of
the outer-most ring of each particle.
#### Directed ridge detection & votes collection
The first step is locating the pixels of interest. Like many other feature
detection algorithms, the standard circle Hough transform relies on an edge
detection step, where edges are the borders of dark and bright regions. As the
images contain rings rather then filled circles, I chose to implement an
algorithm that detects ridges, thin curves which are brighter than their
neighbourhood, rather than edges. This exhibits better consistency. First note
that the ring of interest is thicker than the inner ones. This is advantageous
as the image admits scale selection [18] – the inner rings can be suppressed
using a Gaussian smoothing having the appropriate scale, approximately that of
the most visible ring. Ridges are then found using a differential geometric
descriptor [18], which defines ridge pixels using the following two
properties: (i) Negative least principal curvature, $k_{-}<0$; (ii)
$k_{-}$is a local minimum along the direction of the associated principal
direction $X_{-}$. The principal curvatures are the eigenvalues of the
Hessian, the matrix of the image second spatial derivatives. Here $k_{-}$ and
$X_{-}$ denote the smaller principal curvature and the corresponding
eigenvector. This is demonstrated for the two fluorescent particles in Fig.
2a, where this sub-image is analysed for directed ridges, represented by
arrows overlaid on the $k_{-}$ field as background of Fig. 2b. Note that
$X_{-}$ is collinear with the direction to the centre of the ring. I use this
to significantly reduce the complexity of the voting procedure, in a similar
way to the gradient directed circle Hough transform [19] – each directed ridge
pixel votes for all candidate points in the 3d parameter space of circles,
provided that the circle centre is within the $r_{min}$ to $r_{max}$ range,
directed along $X_{-}$.
#### Local maxima detection in a noisy parameter space: radius dependent
smoothing & normalisation
As votes from the ridge pixels accumulate, each ring in the image transforms
into two mirroring coaxial cones, aligned along the radius axis, having a
joint apex. This procedure results in a discrete scalar function over a 3d
box. This is demonstrated in Fig. 2c. The coordinates of the apexes, which are
the local maxima of this function, are the candidate circle parameters. In
practice, there are many sources which render the resulting circle parameter
space very noisy: The raw image is a discrete representation of the intensity
field, the image acquisition process itself is not noiseless, and the image
complexity mentioned above, all may result in errors in the detected ridge
position and direction, as well as false ridge detection and false negatives.
Note that the deviation from ideal voting due to the error in determining the
ridge direction grows linearly with the radius. Therefore, each equi-radius
level of the ring parameter space is smoothed using Gaussian weights, whose
width is proportional to the radius. Next, note that finding local maxima in
the 3d parameter space requires the comparison of accumulators in different
equi-radius levels, which asks for some normalisation as larger rings are
expected to receive more votes. This leads to a natural normalisation by
$1/r$, following which an ideal ring is expected to receive $2\pi$ votes. Fig.
2d shows how this procedure simplifies the parameter space. Local maxima can
then be located by nearest-neighbours comparison.
#### Ridge points classification & sub-pixel accuracy
The local maxima identified in the parameter space induce a classification on
the ridge coordinates: for each peak an annulus mask is formed and a best
fitting circle is found for all ridge coordinates within the annulus. Ridge
pixels which are not covered by any annulus mask are not fitted for. This is
desired as these usually result from non-circular features in the image or
noise. See the example in Fig. 2e. In this way sub-pixel precision is
achieved.
#### Empirical detection and error rates
Examining the robustness of the algorithm on the experimental data reveals a
detection rate that exceeds 94% with only 1% false-detection; for further
details see Robustness assessment in the Methods section.
To demonstrate the excellence of these results let us compare the robustness
with a recently published algorithm — the EDCircles algorithm introduced in
Ref. [20]. Founded on the mathematical theory of perception [21] it detects
contiguous edge segments and employs the Helmholtz principle for controlling
false detections. It was chosen as the competitor for this examination for two
main reasons: (i) it is parameter-free and so the comparison is insensitive
to the choice of input parameters; and (ii) the results presented in the
manuscript [20] are very promising – the EDCircles was demonstrated to exhibit
a much better detection rate when compared with the state-of-the-art lower
dimensionality Circle Hough Transform implemented in OpenCV [22], referred to
as 21HT in Ref. [16].
In practice, the EDCircles showed a detection rate lower than 61% and nearly
2% false-detection. While the EDCircles detected 21% of the rings missed by
the algorithm proposed here, the latter detected successfully more than 88% of
those missed by its competitor for this comparison; examples can be found in
Supplementary Fig. S3; further details of the test and results are summarised
in Comparative assessment of the algorithm robustness in the Methods section .
For precision and accuracy estimation, in particular in the light of particle
localisation, see Experimental details and Precision assessment in the Methods
section as well as Supplementary Fig. S1 and the accompanying caption.
#### Key algorithmic optimisations for memory requirement & temporal
performance
As was mentioned above, it is desired for our purposes to have the images
processed in real-time. Here I briefly outline the key ideas behind the
optimisation of the algorithm, the full details are available in the open-
source code itself (see the Methods section). The first key point for the
algorithm optimisation is the splitting of the voting procedure – the votes
are recorded for each ridge pixel as it is detected, such that ridge detection
and votes collection are done in _one-pass_.
The population of the parameter space is performed at a separate stage, which
leads to the second key point. Instead of holding an array for the full
parameter space, only two sub-spaces are maintained, consisting of three
consecutive equi-radius levels; the first for the raw parameter sub-space, the
second for the smoothed and normalised one, where local maxima are searched
for. The equi-radius levels are populated and processed one by one. Only the
accumulators exceeding an integer vote threshold are regarded as hotspots and
are mapped to the smoothed and normalised sub-space. Each time a radius-level
is completed, regarded here as the “top” one, the hotspots in the level
beneath, the “middle” one, are verified to exceed a pre-set floating point
threshold, a fraction of $2\pi$. Those which do are searched for local maxima
by a nearest neighbour comparison within a $3\times 3\times 3$ voxels box.
Once this search is completed, the “bottom” level is no longer needed and a
cyclic permutation takes place where the “bottom” level becomes the new “top”.
This allows _memory recycling_ and avoids the need to initialise big arrays of
zeros to represent the full 3d parameter space.
_Registering modified array elements and undoing_ is the last key point. The
radius-levels are required to be blank prior to their population. Recalling
the sparsity of the parameter space (see Figs. 2c and 2d for example), going
over all its elements is a waste of processing time. Instead, each time an
array element is modified for the first time its indices are registered. Once
the search for peaks is done, all modifications to the “bottom” radius-level
are undone, preparing it for reuse. This lifts the need to clean the whole
array.
The combination of _memory recycling_ with _registering modified array
elements and undoing_ reduces the computation time and in my case results in a
nearly ten times less memory consumption. In fact, the size of the arrays
representing the parameter space kept in memory is now fixed and no longer
grows with the radii range. For preliminary testing purposes, I first
implemented the algorithm outlined in the beginning of the Results section (as
well as in Fig. 2) using convolutions and other array based operations. The
final implementation, inspired by the circle Hough transform and including the
above optimisations, is more than 50 times faster. This is attributable to the
reduction in the number of operations required once the sparsity of the data
is taken advantage of. Further details and explanations can be found in the
Methods section and the Detailed algorithm section of the Supplementary
Information.
## Discussion
In this work I have presented a new algorithm to analyse images of complex
annular patterns. Image complexity and noise often result in a challenging
parameter space where local maxima are difficult to find, a problem not
addressed within the classical Hough transform algorithm. The main novelty
introduced here to overcome this difficult task and to gain robustness are the
radius dependent smoothing and normalisation. The resulting detection and
error rates are very promising, even more so in the light of alternative
methods. As it was already mentioned in the introduction the non-deterministic
or randomised methods typically provide a gain in the temporal performance but
suffer in reliability when it comes to detection and error rates [17]. The
EDCircles algorithm [20] was chosen as a competitor for the comparative
assessment of robustness mainly as it was reported to outperform the state-of-
the-art implementation of the natural competitor — OpenCV’s deterministic
Circle Hough Transform [22]. The algorithm proposed here demonstrated a
detection rate higher by more than 50% and a nearly three times smaller false
reports rate.
Several algorithmic concepts have been introduced to improve memory
requirements and temporal performance, of highest importance are those
referred above as _registering modified array elements and undoing_ as well as
_memory recycling_. These have been empirically shown to reduce memory
consumption by nearly ten times and result in an over fifty times faster
analysis rate. These can be advantageous for other algorithms as well,
particularly when the data is sparse.
Though the development of this algorithm was motivated by the analysis of
fluorescence microscopy images, it is more general and can be applied to other
cases as well. The interpretation of the Hough transform as a
classification/clustering algorithm has a wider potential than merely image
analysis. To name a physical example is the case of particle jets emerging
from several sources of unknown loci. A dataset consisting of the positions
and momenta of the particles at a certain time is analogous to that of the
directed ridges and the jet sources can be identified with the local maxima
over the parameter space.
The method I introduced above is currently implemented in an experiment which
requires long unsupervised measurements lasting for days at high temporal
resolution, sampling a volume of interest which contains tens to hundreds of
particles. Thanks to the fact that the whole volume of interest is sampled at
once by a single 2d image, concurrency is achieved. The use of LED and the
relatively short exposure times can be potentially exploited to avoid photo-
damage and bleaching. It is demonstrated to be robust to the overlap,
inclusion and occlusion of the ring pattern of the imaged particles. It
features high performance admitting real-time applications. A discussion of
this approach for particle tracking in light of other methods [34, 35, 36, 37,
38] can be found in the second section of the Supplementary Information.
This method paves the way for studies of 3d flows in microfluidic devices. Its
robustness to the vicinity of particles to each other allows to study the
dynamics of particle pairs [12], triples, etc. As such it also has a potential
for biomedical research. A possible immediate application is the detailed
characterisation of the transport induced by the presence of cells in confined
flows, a phenomenon presented in Ref. [23]. Together with the development of
high signal tracers, labelling techniques and sensitive cameras, this method
may be useful in other life sciences studies such as cellular trafficking
[24], cell migration [25] and bacterial taxis [26].
## Methods
#### Algorithm implementation
I implemented this algorithm relying on freely available open-source and
cross-platform software packages only. The source is available online
(https://github.com/eldad-a/ridge-directed-ring-detector). Most of the heavy
lifting is achieved using the Cython language [27]. It has a Python-like
syntax from which a C code is automatically generated and compiled. This
allows the code to be short and easy to read while enjoying the performance of
C. For example, this implementation exploits the Numpy/Cython strided direct
data access [27, 28] by fully sorting the votes. In the image pre-processing
step the image is smoothed using a Gaussian convolution and the smoothed image
spatial derivatives are calculated using a 5$\times$5 2nd order Sobel
operator; these operations are done using OpenCV’s Python bindings [22].
The equi-radius levels of the circle parameter space are smoothed using
Gaussian weights
$\exp\left\\{-\frac{1}{2}\left[\frac{x-x_{hotspot}}{\sigma}\right]^{2}\right\\}$,
whose width is proportional to the radius $\sigma(r)\propto r$. The explicit
form, $\sigma(r)=0.05r+0.25$, was found empirically. The slope coefficient is
interpreted as accounting for $\sim 0.1$rad uncertainty in the direction of
$X_{-}$.
#### Experimental details
The imaging system consists of an inverted fluorescence microscope (IMT-2,
Olympus), mounted with a Plan-Apochromat 20$\times$/0.8NA objective (Carl
Zeiss) and a fluorescence filter cube; a Royal-Blue LED (Luxeonstar) served
for the fluorophore excitation. A CCD (GX1920, Allied Vision Technologies) was
mounted via zoom and 0.1$\times$ c-mount adapters (Vario-Orthomate 543513 and
543431, Leitz), sampling at $70\text{\,}\mathrm{Hz}$,
$968\text{\,}\mathrm{px}$$\times$$728\text{\,}\mathrm{px}$, covering
$810\text{\,}\mathrm{\SIUnitSymbolMicro
m}$$\times$$610\text{\,}\mathrm{\SIUnitSymbolMicro m}$ laterally.
The experiments were conducted in a microfluidic device, implemented in
polydimethylsiloxane elastomer by soft lithography, consisting of a
curvilinear tube (see grey broken line in Supplementary Fig. S2a). The
rectangular cross-section of the tube was measured to be of
$140\text{\,}\mathrm{\SIUnitSymbolMicro m}$ depth and
$185\text{\,}\mathrm{\SIUnitSymbolMicro m}$ width. The working fluid consisted
of polyacrylamide in aqueous sugar (sucrose and sorbitol) syrup, seeded with
fluorescent particles (1 micron 15702 Fluoresbrite® YG Carboxylate particles,
PolySciences Inc.). The flow was driven by gravity.
For the empirical calibration, the same working fluid was sandwiched between
two microscope glass slides. The separation distance between the slides was
set to $161\text{\,}\mathrm{\SIUnitSymbolMicro m}$ by micro-spheres (4316A PS
NIST certified calibration and traceability, Duke Standards) serving as
spacers. The microscope objective was translated in steps of
$2\text{\,}\mathrm{\SIUnitSymbolMicro m}$ to acquire images of the tracers at
different off-focus distances $\Delta z$. The microscope focus knob was
manipulated by a computer controlling a stepper motor. During the rest stages
of the objective, the ring radii of every detected tracer were averaged over
210 frames spanning $3\text{\,}\mathrm{s}$. Due to the high viscosity of the
fluid, 1100 times larger than water viscosity, tracer motion due to diffusion
is negligible during this time interval. The median of the estimated standard-
deviations of the data presented in Supplementary Fig. S1a is
$0.03\text{\,}\mathrm{px}$ and the maximal is $0.27\text{\,}\mathrm{px}$. In
practice, to account for the uncertainties in finding the focal position and
due to optical aberrations, 25 tracers dispersed throughout the observation
volume were accounted for. Their curves were aligned via shifting $\Delta z$
by the larger root of a quadratic polynomial fit. Then, the conversion
function $r^{-1}(r)$ was obtained by inversion of the quadratic polynomial fit
accounting for all the data together; see Supplementary Fig. S1b. The
resulting root-mean-squared-error, $\sqrt{\langle\left(\Delta
z-r^{-1}(r)\right)^{2}\rangle}=$ $1.97\text{\,}\mathrm{\SIUnitSymbolMicro m}$,
and the maximal measured absolute error is
$5.35\text{\,}\mathrm{\SIUnitSymbolMicro m}$; these reflect the uncertainty
due to the empirical calibration procedure taken here. Finally, the out-of-
focus distance of the objective $\Delta z$ has to be converted to the physical
distance via multiplication by the ratio of the refractive indices, 1.58 in
this case. The observed axial range exceeds
$180\text{\,}\mathrm{\SIUnitSymbolMicro m}$.
#### Robustness assessment
In order to estimate the robustness of the algorithm, images from the chaotic
flow experiment were analysed and cropped to a sub-region of
$340\text{\,}\mathrm{px}$$\times$$370\text{\,}\mathrm{px}$. The analysis
results of 600 such sub-frames were examined. These sub-frames contained 14.3
rings on average, out of which 67.7% were in ring clusters (overlap and
inclusion configurations). This examination shows an average of 6.8% False-
Negative errors. In some sub-frames a ghost ring would appear accompanied by a
strong distortion of the tracer image in its real position. This is attributed
the microfluidic walls and observed only when a particle is very close to the
wall. Excluding from the statistics two such tracers, the False-Negative error
rate is reduced, corresponding to a detection rate of 94.7%. This examination
does not show any significant sensitivity to rings overlap and inclusion. On
the contrary, 95.5% of the rings in clusters were identified correctly while
only 0.8% of the reported rings in clusters were non-existing particles.
#### Comparative assessment of the algorithm robustness
To demonstrate the high-quality of the above results, a comparison was made
against the on-line demo of the EDCircles, provided by the authors of Ref.
[20]. The tests were conducted on a subset of 151 images taken from the same
experiment as in the Robustness assessment above, cropped to the same sub-
region of $340\text{\,}\mathrm{px}$$\times$$370\text{\,}\mathrm{px}$. In this
case the images were first cropped and exported to the PNG format prior to the
analysis, the format for which the EDCircles on-line demo exhibited the best
detection rate. In this comparison rings whose centre lay outside the cropped
image were not considered, as well as those whose visible perimeter was less
than a half of the complete one; the average ring number was found to be 13.4.
Few examples are presented in Supplementary Fig. S3.
The EDCircles demo detected 60.9% of the rings (False-Negative error rate of
39.2%); in contrast, the method proposed here showed a detection rate of
94.3%. While 1.7% of the reported detections by the EDCircles demo were False-
Positive, only 0.6% of the rings reported by the new algorithm presented here
were non-existing or erroneous ones. Out of those particles missed by the
proposed method 21.1% were detected by the EDCircles demo; in contrast, 88.6%
of those missed by the competitor algorithm were detected by the one proposed
here.
#### Performance assessment
The performance assessment is based the analysis of 1500 full frames
containing 50 particles on average; for a typical example see Supplementary
Fig. S2a. The test was run on an i7-3820 CPU desktop, running Ubuntu linux
operating system. A single process analysed at an average rate of
$6.28\text{\,}\mathrm{Hz}$. To achieve higher performance as required for our
experiments, I use the multiprocessing package of Python, exploiting the
multi-core processors available on modern computers. The analysis rate scales
linearly with the number of processes. No deterioration of the processing rate
per core was noted (tested up to twice the core number with hyper-threading).
Based on a producer-consumer model, one can even transparently distribute the
workload among several computers if needed. This is partially attributable to
the small memory footprint of the algorithm.
#### Precision assessment
To estimate the precision of the presented localisation method, smoothing
splines were applied to the reconstructed trajectories providing an estimator
for the error variance $\sigma^{2}$; see Ref. [29]. The mean values are as
follows: $\sigma_{x}=$ $0.134\text{\,}\mathrm{\SIUnitSymbolMicro m}$,
$\sigma_{y}=$ $0.135\text{\,}\mathrm{\SIUnitSymbolMicro m}$ and $\sigma_{z}=$
$0.434\text{\,}\mathrm{\SIUnitSymbolMicro m}$, for $x$ and $y$ denoting the
lateral coordinates, and $z$ the axial one. This axial uncertainty corresponds
to 0.3% of the axial range covered by the particles. According to the data
provided in Ref. [5] a value of 0.1% was achieved and a similar one in Ref.
[13]. The uncertainty estimates reported here account for noise which rise not
only due to the image analysis and the multi-particle scenario, but also due
to other sources, namely the motion of the particles and the linking
procedure. The details are as follows.
From a 3m30s measurement, 2014 trajectories which span more than 1s were
analysed (discarding shorter ones). Particle positions, converted to microns,
were linked to reconstruct their trajectories; for a sub-sample see
Supplementary Fig. S2b. The linking algorithm was adapted from the code
accompanying Ref. [30], generalised to n-dimensions, the kinematic model was
modified to account for accelerations, and a memory feature was introduced to
account for occasional misses. Finally, a natural cubic smoothing spline was
applied to smooth-out noise and obtain an estimate of particle velocity and
acceleration [31, 32]. The smoothing parameter was automatically set using
Vapnik’s measure, using a code adapted from the Octave splines package [33].
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## Acknowledgements
I thank P. Reisman for introducing me to the Hough transfrom, O. Schwatz for
helpful discussions regarding the optical setup, A. Frishman and S. van der
Walt for their useful comments on the manuscript. Special thanks go to Y.
Kaplan and T. Afik for their help in robustness assessment, and to V.
Steinberg and M. Feldman for helpful discussions of this work and its
presentation. This work is supported by grants from the German-Israel
Foundation (GIF) and the Lower Saxony Ministry of Science and Culture
Cooperation (Germany).
## Additional Information
#### Competing financial interests:
The author declares no competing financial interests.
#### How to cite this article:
Afik, E. Robust and highly performant ring detection algorithm for 3d particle
tracking using 2d microscope imaging. Sci. Rep. 5, 13584; doi:
10.1038/srep13584 (2015)
This work is licensed under a Creative Commons Attribution 4.0 International
License. The images or other third party material in this article are included
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license, users will need to obtain permission from the license holder to
reproduce the material. To view a copy of this license, visit
http://creativecommons.org/licenses/by/4.0/
## Supplementary Information for:
## Robust and highly performant ring detection algorithm for 3d particle
tracking using 2d microscope imaging
Eldad Afik
Department of Physics of Complex Systems,
Weizmann Institute of Science,
Rehovot 76100, Israel
email: [email protected]
## Detailed algorithm
As mentioned in the main text, the standard circle Hough transform is often
avoided not only for its challenging local maximum detection in noisy 3d space
but for its heavy memory requirements as well. The standard circle Hough
transform requires a 3-dimensional array of accumulators. The coordinates of
each array element are the parameters of a candidate circle. The value of the
accumulator at these coordinates indicates how well this circle is represented
in the image. Code optimisation for high-performance and small memory
footprint is achieved following this scheme:
1. 1.
_Image pre-processing step:_ the image is smoothed using a Gaussian
convolution and the smoothed image spatial derivatives are calculated using a
5$\times$5 2nd order Sobel operator [22]. Using these derivatives, the local
least principal curvature $k_{-}$ is estimated as the smaller eigen-value of
the Hessian matrix.
2. 2.
_One-pass ridge detection and votes collection:_ for each pixel in $k_{-}$
which is smaller than a pre-defined curvature threshold (the latter is no
greater than zero), the corresponding $X_{-}$ is calculated. If this pixel is
found to be a local minimum along the direction of $X_{-}$, its coordinates
are recorded in the ridge container. At this stage, its votes are collected as
well, that is, the potential circles parameters to which it may belong.
3. 3.
_Sort the votes stack according to the radii:_ this allows performing the
parameter space incrementing procedure equi-radius level by level. In order to
achieve higher performance, the votes are further sorted by the row index and
then by the column index exploiting the numpy/cython strided direct data
access [27, 28]. For this reason each circle parameter triple is represented
as an integer using a bijection.
4. 4.
_Circle parameter space population and local maximum detection via radius-
dependent smoothing and normalisation:_ This is done using two arrays
representing a sub-space of the full circle parameter space. Each consists of
3 consecutive equi-radius levels; the first for the raw accumulators sub-
space, the second for the smoothed and normalised one, where local maxima are
to be searched for. There are two votes thresholding steps: an integer
threshold for the raw accumulators and a floating point threshold, a fraction
of $2\pi$, for the smoothed and normalised array elements. In describing the
procedure it is assumed that the $r-2$ and $r-1$ levels have already been
populated in both sub-space triples and the $r$ levels are blank, i.e. all
zeros. As long as the votes drawn from the votes stack point to the same
radius, the corresponding radius-level is populated by incrementing the
indicated accumulator. Recall that the votes are fully sorted hence all votes
pointing to a certain voxel will come out from the stack in a row. Every time
a new circle parameter triple is encountered, its coordinates are recorded as
modified. In case the previously incremented voxel has surpassed the 1st votes
threshold its coordinates are recorded as a hotspot – a circle candidate. Once
there are no more votes for this $r$-level, it is mapped to the second
subspace: for each hotspot voxel a spatial average is calculated, weighted by
a Gaussian function, which width is linearly dependent on the radius; the
value of the average is then normalised by $1/r$. After mapping all the
hostpots of the current $r$-level, a local maximum is searched for among the
hotspots of the $(r-1)$-level which pass the 2nd votes threshold. This is done
using a nearest neighbours comparison within a $3\times 3\times 3$ voxels box.
Array elements which are local maximum and exceed the threshold are registered
as rings. Once all hotspots have been processed, all modifications to the
$(r-2)$-level are undone as its data are no longer needed. By this it is made
ready to be regarded as the next $r$-level and a cyclic permutation among the
levels takes place. In practice, this is performed by accessing the equi-
radius levels using the modulo operation – the radius indices are calculated
using $r\pmod{3}$.
5. 5.
_Sub-pixeling via circle fit:_ the detected ridge coordinates are subjected to
a circle fit via the non-exclusive classification induced by the results of
the directed circle Hough transform. The coordinates in the ridge container
are clustered based on annuli masks dictated by the detected rings and sub-
pixel accuracy of the rings parameters is achieved.
### Additional notes
* •
The ridge detection can be used to achieve a compressed representation of the
features in the image. This can be done by storing a hash table associating
ridge coordinates as keys with their corresponding $X_{-}$ as values.
* •
The algorithm is not restricted to directed ridges as it can be replaced by
directed edges in case these are better descriptors of the features in the
image. This is achieved by replacing the Hessian by the Gradient. In this
case, the gradient magnitude replacing $k_{-}$ has to be a local maximum along
the gradient direction.
* •
To reduce false detection, the radii range is extended such that the Hough
transform is over the range $\left[r_{min}-1,r_{max}+1\right]$, but local
maximum detection are searched for within the original range.
* •
In case additional performance per processing unit is required, one could use
a lower resolution in discretising the circle parameter space. Measuring the
effect of this on the accuracy is left for future work.
* •
Using several colours, the method should be, in principle, extendible to even
higher particle densities.
## Application of the proposed algorithm for particle tracking and discussion
of alternative methods
When tracking small light emitting objects, such as fluorescent particles
under the microscope, the appearance of rings is often a sign of the object
going out of focus. Normally this results in the loss of the tracked object,
which is thereafter considered as a hindering background source. However,
these rings carry information of the 3-dimensional position of the particle.
This has been used for localising a single light scattering magnetic bead
based on matching the radial intensity profile to an empirical set of
reference images [5]. An axial range of $10\text{\,}\mathrm{\SIUnitSymbolMicro
m}$ was demonstrated and a temporal resolution of $25\text{\,}\mathrm{Hz}$ was
achieved using the knowledge of the particle’s previous position. In fact, for
fluorescent particles the radius of the most visible ring of each particle
precisely indicates its axial position – the radius follows a simple scaling
with the particle distance from the focal plane (see Supplementary Fig. S1). A
similar approach was recently described in [13], where the measurements were,
once again, limited to a single particle in the observation volume, with an
axial range of $3\text{\,}\mathrm{\SIUnitSymbolMicro m}$ and temporal
resolution of $10\text{\,}\mathrm{Hz}$.
In comparison with other existing methods for 3d particle tracking, the method
presented here is advantageous when it comes to long measurements, temporal
resolution and concurrency, as well as real-time applications. The confocal
scanning microscope requires scanning the volume of interest. Therefore it is
slower and cannot yet provide instantaneous information of the whole volume.
Unlike Holographic microscopy [34, 35], the proposed method does not pose long
and heavy computational demands which is restrictive for real-time
applications or when large datasets are required for statistics.
One could expect the optical method discussed here to produce patterns which
are symmetric about the focal plane. When this applies, it may result in an
ambiguity with respect to whether the particle is above or below focus. Our
optical arrangement (see the Methods section) shows clear diffraction rings
only on one side. Furthermore, as particles approach focus, the radius of the
outer-most ring becomes too small to resolve. For these reasons the focal
plane is placed outside the volume of interest (as reflected in the
Supplementary Fig. S1). Optical astigmatism offers a mean for discriminating
between the two sides of the optical axis [36, 37, 38]. The introduction of a
cylindrical lens results in the deformation of a circular spot into an
ellipsoidal one as a fluorescent particle goes further away from focus, with
the ellipse major axis of a particle above focus aligned perpendicular to a
one below. In Ref. [36] the axial range was limited to a couple of microns
above and below focus; in Refs. [37, 38] it was restricted to less than a
micron. Within these ranges the tracer image can be approximated by an
elliptical gaussian pattern. However, extending the range generates elliptical
rings as well; see Figure 1 in Ref. [36]. This requires dealing with two
species of patterns, spots and rings. Moreover, deforming circular rings into
elliptical ones, the dimensionality of the parameter space increases, and so
does the technical complexity of the image analysis. Therefore the advantage
of the stronger signal, by working closer to focus on both its sides, is
expected to have a heavy computational cost once the range is extended such
that diffraction rings appear as well. The method presented here requires
working away from focus. Rings visibility decreases as the fluorescence signal
spreads over a larger area, thus setting the lower bound for the exposure
time. Nevertheless, I have found that the fluorescence signal-to-noise ratio
allowed tracking particles moving chaotically at speeds exceeding
$400\text{\,}\mathrm{\SIUnitSymbolMicro m}\text{/}\mathrm{s}$.
## Supplementary figures
a
b
Supplementary Figure S1: Calibration curve. (a) The empirical relation between
the outer-most ring radius and the out-of-focus distance $\Delta z$; the
latter measures the translation of the objective from the position at which a
particle would be in focus. The plot shows the data for two different
fluorescent particles (denoted by blue and green in the plot). The quadratic
polynomial fit provides an approximation for the $r(\Delta z)$ relation. The
data was acquired by scanning through the vertical axis of the observation
volume by objective translation steps of $2\text{\,}\mathrm{\SIUnitSymbolMicro
m}$ (see Experimental details in the Methods section). Each data point is an
average of the measured radius over 210 frames spanning
$3\text{\,}\mathrm{s}$, taken while the objective is stationary. Error bars
reflect the standard-deviation; the median standard-deviation of the presented
datasets is $0.03\text{\,}\mathrm{px}$ and the maximal is
$0.27\text{\,}\mathrm{px}$.
(b) The conversion function $r^{-1}(r)$ was obtained by the inversion of the
quadratic polynomial fit, based on 25 tracers dispersed in the observation
volume (see Experimental details in the Methods section). The resulting root-
mean-squared-error $\sqrt{\langle\left(\Delta z-r^{-1}(r)\right)^{2}\rangle}=$
$1.97\text{\,}\mathrm{\SIUnitSymbolMicro m}$, and the maximal measured
absolute error is $5.35\text{\,}\mathrm{\SIUnitSymbolMicro m}$; these estimate
the uncertainty due to the calibration procedure followed here. Finally, the
out-of-focus distance of the objective $\Delta z$ needs to be converted to a
physical distance via multiplication by the refractive indices ratio, 1.58 in
this case. Thus the observed axial range exceeds
$180\text{\,}\mathrm{\SIUnitSymbolMicro m}$.
ab
Supplementary Figure S2: 2d out-of-focus images $\to$ 3d trajectories. (a) A
single raw full 2d frame imaging a volume of
$810\text{\,}\mathrm{\SIUnitSymbolMicro
m}$$\times$$610\text{\,}\mathrm{\SIUnitSymbolMicro
m}$$\times$$140\text{\,}\mathrm{\SIUnitSymbolMicro m}$, including 60 tracers
in one period of the curvilinear tube (which boundaries are denoted by the
broken grey line). Smaller rings result from tracers being closer to the focal
plane, which is positioned above the tube. (b) A sub-sample of 40 trajectories
reconstructed from a time-lapse sequence of such frames, which spans 12
seconds of data acquisition. The colour coding indicates the speed ranging
from $80\text{\,}\mathrm{\SIUnitSymbolMicro m}\text{/}\mathrm{s}$ (blue) to
$400\text{\,}\mathrm{\SIUnitSymbolMicro m}\text{/}\mathrm{s}$ (red); the mean
flow is rightwards. Corresponding axes are denoted by colours. The isometric
view of the bottom panel can be obtained by three rotations, starting from the
orientation of the upper panel: $-90\text{\,}\mathrm{\SIUnitSymbolDegree}$
about the green axis, $-45\text{\,}\mathrm{\SIUnitSymbolDegree}$ about the
blue axis, and approximately $35\text{\,}\mathrm{\SIUnitSymbolDegree}$ about
the new horizontal axis.
a b c d
Supplementary Figure S3: Examples from the comparative assessment of the
algorithm robustness. Output examples of the proposed algorithm can be found
on the left column (reported rings are shown in dashed red) to be compared
with those of EDCircles [20] on the right column (circles in purple, ellipses
in blue); further details of the test and results can be found in the Methods
section. In 3.3% of the 151 tested images the alternative algorithm showed
comparable results to those of the new one, that is both exhibited the same
detection and error rates – an example is shown in (a); in all other examined
images the proposed method outperformed its competitor – examples are shown in
(b), (c) & (d). The new algorithm resolves rings even when the signal-to-noise
ratio is limiting for the opponent – this is typical for tracers which are
farther out-of-focus, hence their rings are larger and fainter. Overlapping
rings of similar radii are another challenge resolved by the new algorithm; in
contrast these are often missed or merged into ellipses by the opponent – see
(c) & (d); similarly, optical artefacts often result in errors for the
alternative method, in contrast to the new proposed one – e.g. see small ring
reported by the opponent to be an ellipse in (d).
|
arxiv-papers
| 2013-10-02T08:36:32 |
2024-09-04T02:49:51.978354
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/",
"authors": "Eldad Afik",
"submitter": "Eldad Afik",
"url": "https://arxiv.org/abs/1310.1371"
}
|
1310.1522
|
# Unified Model of Temperature Dependence of Core Losses in Soft Magnetic
Materials Exposed to Nonsinusoidal Flux Waveforms and DC Bias Condition
Adam Ruszczyk [email protected] ABB Corporate Research, Starowiślna
13a, 31-038 Kraków, Poland, Krzysztof Sokalski [email protected]
Institute of Computer Science, Czȩstochowa University of Technology, Al. Armii
Krajowej 17, 42-200 Czȩstochowa, Poland
###### Abstract
Assuming that Soft Magnetic Material is a Complex System and expressing this
feature by scaling invariance of the power loss characteristic, the unified
model of the temperature dependence of Core Losses in Soft Magnetic Materials
Exposed to Nonsinusoidal Flux Waveforms and DC Bias Condition has been
constructed. In order to verify this achievement the appropriate measurement
data concerning power losses and the all independent variables have been
collected. The model parameters have been estimated and the power losses
modeling has been performed. Comparison of the experimental values of power
losses with their calculated values has showed good agreement.
###### pacs:
75.50.-y, 61.85.+p
The following article has been submitted to Applied Physics Letters. If it is
published, it will be found online at http://apl.aip.org
## Introduction
Application of soft magnetic materials in electronic devices requires
knowledge about losses under different conditions of exposition: sinusoidal
and nonsinusoidal flux waveforms of different shapes, with and without DC bias
condition. During the two last decades the two classes of core loss’ models
have been elaborated. The first class consists of models which are based on
the Steinmetz Equation (bib:Steinmetz, ), (bib:Albach, ), (bib:Reinert, ),
(bib:Li, ),(bib:Venka, ), (bib:Boss1, ), (bib:Ecklebe, ), (bib:Ecklebe1,
),(bib:FFio, ). However the second class is based on the assumption that the
shape of the waveform does not matter and as a result only look at peaks
(bib:Sokal1, ),(bib:Sokal2, ),(bib:Sokal3, ),(bib:Ruszcz, ) and (bib:Cale, ).
Non of them presents satisfactory algoritm enabling us to calculate of core
losses v.s. temperature of sample with and without presence of conditions for
exposition mentioned above. Therefore, this paper is devoted to solution of
this problem.
## I Scaling and Unified Core Loss Model
On the base of our recent papers (bib:Sokal1, ),(bib:Ruszcz, ) we derive the
unified model of the total core loss versus the four independent variables:
$f$-frequency, $\triangle B$-pik to pik magnetic induction, $H_{DC}$-DC bias
and $T$-temperature:
$P_{tot}=F(f,\triangle B,H_{DC},T).$ (1)
In order to apply scaling to (1) the right hand side has to be homogeneous
function in general sense. This assumption has to be satisfied both, by the
experimental data and by the mathematical model. However, according to results
of researches presented in (bib:ABB1, ), (1) and measurement data formed by
the action of DC-bias are not uniform in the required sense. This problem we
have solved in the previous paper (bib:Ruszcz, ) by using the method invented
by Van den Bossche et al. (bib:Boss1, ). They have mapped the DC-bias into
primary magnetization curve. Using their idea we have used the following
mapping:
$H_{DC}\rightarrow[M_{0},M_{1},M_{2},M_{3}],$ (2)
where $M_{i}=tanh{(H_{DC}\cdot c_{i})}$ and $c_{i}$ are free parameters to be
determined from the experimental data. The number of $M_{i}$ components is
optional. The introduced mapping (2) enables us to write down the following
condition for $P_{tot}(f,\triangle B,[M_{0},M_{1},M_{2},M_{3}],T)$ to be a
homogenous function in general sense:
$\displaystyle\exists\\{a,b,c,d,g\\}\in{\bf
R^{5}}:\forall\lambda\in\bf{R_{+}}$ $\displaystyle
P_{tot}(\lambda^{a}f,\lambda^{b}(\triangle
B),\lambda^{c}[M_{0},M_{1},M_{2},M_{3}],\lambda^{d}T)=$
$\displaystyle\lambda^{g}P_{tot}(f,\triangle B,[M_{0},M_{1},M_{2},M_{3}],T).$
(3)
Substituting for $\lambda$ the following expression:
$\lambda=(\triangle B)^{-1/b}$ we derive the most general form for $P_{tot}$
which satisfies (3):
$P_{tot}=(\triangle B)^{\beta}\,F\left(\frac{f}{(\triangle
B)^{\alpha}},\frac{[M_{0},M_{1},M_{2},M_{3}]}{(\triangle
B)^{\gamma}},\frac{T}{(\triangle B)^{\delta}}\right),$ (4)
where,
$\alpha=\frac{a}{b},\beta=\frac{g}{b},\gamma=\frac{c}{b},\delta=\frac{d}{b}$
and $F(\cdot,\cdot,\cdot)$ is an arbitrary function to be determined.
## II The modeling of $F(\cdot,\cdot,\cdot)$
In order to determine $F(\cdot,\cdot,\cdot)$ we assume its form to be
factorable:
$\displaystyle F\left(\frac{f}{(\triangle
B)^{\alpha}},\frac{[M_{0},M_{1},M_{2},M_{3}]}{(\triangle
B)^{\gamma}},\frac{T}{(\triangle B)^{\delta}}\right)=$
$\displaystyle\Phi\left(\frac{f}{(\triangle
B)^{\alpha}},\frac{[M_{0},M_{1},M_{2},M_{3}]}{(\triangle
B)^{\gamma}}\right)\,\Theta\left(\frac{T}{(\triangle B)^{\delta}}\right).$ (5)
$\Phi(\cdot,\cdot)$ is a version of very well working model function derived
in (bib:Ruszcz, ):
$\displaystyle\Phi(\frac{f}{(\triangle
B)^{\alpha}},H_{DC})=\Sigma_{i=1}^{4}\Gamma_{i}\left(\frac{f}{(\triangle
B)^{\alpha}}\right)^{i\,(1-x)}+$
$\displaystyle\Sigma_{i=0}^{3}\Gamma_{i+5}\left(\frac{f}{(\triangle
B)^{\alpha}}\right)^{(i+y)(1-x)}\frac{tanh(H_{DC}\cdot c_{i})}{(\triangle
B)^{\delta}}.$ (6)
Basing on some computer experiments we have selected for $\Theta(\cdot)$ the
following Padé approximant (bib:pade, ):
$\Theta=\left(\frac{\psi_{0}+\theta\,(\psi_{1}+\theta\,\psi_{2})}{1+\theta\,(\psi_{3}+\theta\,\psi_{4})}\right)^{1-z},$
(7)
where $\theta=\frac{T+\tau}{\Delta B^{\gamma}}$, $T^{\circ}C$ is measured
temperature, $\tau$ and $z$ are tuning parameters, $\psi_{i}$ are Padé
expansion coefficients.
## III Experimental Data, Estimations of Parametr’s and Modeling
Table 1: Selected 60 records of the measurement data of SIFERRIT N87 $T[^{o}C]$ | $\triangle B[T]$ | $f[kHz]$ | $H_{DC}[\frac{A}{m}]$ | $P_{tot}[\frac{W}{m^{3}}]$ | $T[^{o}C]$ | $\triangle B[T]$ | $f[kHz]$ | $H_{DC}[\frac{A}{m}]$ | $P_{tot}[\frac{W}{m^{3}}]$
---|---|---|---|---|---|---|---|---|---
28,1 | 0,395 | 1 | 8,634 | 4064,3 | 28,1 | 0,391 | 1 | 20,146 | 4469,0
28,1 | 0,374 | 1 | 60,634 | 6332,4 | 28,3 | 0,351 | 1 | 86,651 | 6463,6
17,7 | 0,398 | 2 | 7,8014 | 9452,1 | 17,8 | 0,398 | 2 | 20,555 | 10663,8
18,9 | 0,396 | 2 | 35,583 | 12745,8 | 18,5 | 0,377 | 2 | 89,240 | 16015,6
26,2 | 0,400 | 5 | 6,570 | 21131,3 | 26,4 | 0,400 | 5 | 17,820 | 23110,0
26,5 | 0,398 | 5 | 33,230 | 28057,3 | 27,1 | 0,386 | 5 | 89,400 | 35209,8
28,4 | 0,401 | 10 | 5,892 | 41549,0 | 28,6 | 0,401 | 10 | 17,477 | 45257,9
28,8 | 0,400 | 10 | 31,820 | 54650,9 | 29,7 | 0,393 | 10 | 73,960 | 63821,6
30,8 | 0,386 | 10 | 105,00 | 64632,1 | 28,4 | 0,490 | 1 | 11,694 | 6611,0
28,4 | 0,488 | 1 | 24,299 | 7196,0 | 28,4 | 0,451 | 1 | 78,390 | 8771,6
19,1 | 0,497 | 2 | 10,120 | 15234,1 | 19,2 | 0,496 | 2 | 23,718 | 16781,0
19,3 | 0,485 | 2 | 54,63 | 19235,9 | 19,8 | 0,475 | 2 | 76,86 | 20100,2
27,7 | 0,502 | 5 | 8,92 | 34634,8 | 27,4 | 0,503 | 5 | 15,02 | 36195,2
27,7 | 0,501 | 5 | 21,5 | 37496,6 | 28,6 | 0,496 | 5 | 47,5 | 41259,7
31,7 | 0,499 | 10 | 20,52 | 71226,8 | 32,15 | 0,494 | 10 | 45,04 | 76876,5
32,6 | 0,487 | 10 | 67,14 | 80858,2 | 28,5 | 0,588 | 1 | 14,42 | 10042,9
28,7 | 0,561 | 1 | 57,97 | 11239,6 | 28,7 | 0,541 | 1 | 78,08 | 11255,7
29,1 | 0,58 | 2 | 12,82 | 19689,9 | 28,7 | 0,576 | 2 | 54,36 | 22043,0
30,1 | 0,592 | 5 | 42,4 | 52126,7 | 31,1 | 0,599 | 10 | 10,29 | 92648,6
31,3 | 0,595 | 10 | 31,23 | 96446,4 | 28,9 | 0,684 | 1 | 22,05 | 14150,5
28,1 | 0,389 | 1 | 33,507 | 5358,8 | 28,4 | 0,346 | 1 | 91,066 | 6376,4
18,2 | 0,386 | 2 | 68,034 | 15049,1 | 18,7 | 0,367 | 2 | 110,59 | 16027,7
29 | 0,669 | 1 | 41,33 | 14417,5 | 34,7 | 0,586 | 10 | 61,25 | 96583,3
30,2 | 0,616 | 5 | 36,05 | 54344,9 | 28,7 | 0,586 | 2 | 33,49 | 21002,2
28,5 | 0,580 | 1 | 36,01 | 10790,0 | 42,1 | 0,496 | 50 | 47,53 | 289491,2
31,5 | 0,499 | 10 | 7,57 | 65879,7 | 28,1 | 0,500 | 5 | 31,42 | 39530,2
20,2 | 0,469 | 2 | 87,44 | 20547,5 | 19,7 | 0,480 | 2 | 68,36 | 20073,3
28,5 | 0,443 | 1 | 85,100 | 8702,4 | 28,3 | 0,473 | 1 | 54,300 | 8296,5
30,2 | 0,387 | 10 | 99,190 | 64410,1 | 29,2 | 0,396 | 10 | 61,172 | 62814,4
27,5 | 0,386 | 5 | 97,779 | 35945,6 | 26,8 | 0,394 | 5 | 58,800 | 32614,3
Figure 1: Projection of the measurement points and the scaling theory points
(5)-(7) in $[f/(\triangle B^{\alpha})^{(1-x)},P_{tot}/(\triangle B^{\beta})]$
plane. Figure 2: Projection of the measurement points and the scaling theory
points (5)-(7) in $[tanh(H\,c_{1},P_{tot}/(\triangle B)^{\beta}]$ plane.
The B-H Loop measurements have been performed for SIFERRIT N87. The Core
Losses per unit volume have been calculated as the enclosed area of the B-H
loop, multiplied by the frequency f. The following factors influence the
accuracy of measurements: 1) Phase Shift Error of Voltage and Current $<4\%$,
2) Equipment Accuracy $<5,6\%$, 3) Capacitive Couplings negligible (capacitive
currents are relatively lower compared to inductive currents), and 4)
Temperature $<4\%$. For details of the applied measurement method and the
errors of the relevant factors we refer to (bib:Ecklebe, ), (bib:Ecklebe1, ).
The parameter values of (4)-(7) have been estimated by minimization of
$\chi^{2}$ using the Simplex method of Nelder and Mead (bib:pade, ) and the
our experimental data. The measurement series consists of $60$ points, see
TABLE 1. Standard deviation per point is equal to
$15[\frac{W}{m^{3}T^{\beta}}]$ Applying the formulae (4)-(7) and the estimated
parameter values TABLE 2 we have drawn the three scatter plots Fig. 1, Fig. 2
and Fig. 3, which compare estimated points with the experimental ones in the
three projections, respectively. Note that, in order to prevent generation of
large numbers in the estimation process the unit of frequency was kHz while
other magnitudes were expressed in SI unit system.
Table 2: The set of estimated model’s parameters of (4)-(7) for $\delta=0$ $\alpha$ | $\beta$ | $x$ | $\Gamma_{1}$ | $\Gamma_{2}$ | $\Gamma_{3}$ | $\Gamma_{4}$ | $\Gamma_{5}$
---|---|---|---|---|---|---|---
${\small-11,628}$ | -8,6382 | 0,52629 | -1,4083 | 739,55 | 1253,4 | 4238,5 | 0,12264
$\Gamma_{6}$ | $\Gamma_{7}$ | $\Gamma_{8}$ | y | $\psi_{3}$ | $c_{3}$ | $\psi_{4}$ | $\psi_{5}$
-30,972 | -51,869 | -4201,45 | 0,28877 | 14,4558 | 0,1648 | -1,27E-01 | 0,28302
$c_{2}$ | $\tau$ | $\gamma$ | $\psi_{2}$ | $\psi_{1}$ | $c_{1}$ | $c_{0}$ | z
-0,1808 | 7,77E-02 | -0,17954 | 2,3966 | -0,8993 | -2,44E-02 | -0,4877 | 4,84E-02
Figure 3: Projection of the measurement points and the scaling theory points
(5)-(7) in $[\frac{T+\tau}{(\triangle B)^{\gamma}},P_{tot}/(\triangle
B)^{\beta}]$ plane.
## IV Conclusions
Efficiency of the scaling in solving problems concerning of power losses in
Soft Magnetic Materials has been confirmed all ready in the recent papers
(bib:Sokal1, )-(bib:Ruszcz, ). However, this paper is the first one which
presents application of scaling in modeling of temperature dependence of the
core loss. The presented method is universal, which means that it works for
wide spectrum of expositions and different soft magnetic materials. Moreover
the presented model formulae (4)-(7) are not closed and can be adapted for a
current problem by fitting the forms of both factors $\Phi$ and $\Theta$. At
the end one must say that success in applying the scaling depends on property
of data. The data must obey the scaling.
##
## References
* (1) C.P. Steinmetz, On the law of hysteresis, Trans. Amer. Inst. Elect. Eng., 9, 3-64 (1892).
* (2) M. Albach, T. Durbaum, and A. Brockmeyer, IEEE Power Electronics Specialists Conference, pp. 1463 1468 (1996).
* (3) J. Reinert, A. Brockmeyer J. Reinert, A. Brockmeyer, and R.W. De Doncker, Calculation of losses in ferro- and ferrimagnetic materials based on the modified Steinmetz equation, Annual Meeting of the IEEE Industry Applications Society, 1999.
* (4) Jieli Li, T. Abdallah, and C. R. Sullivan, Improved calculation of core loss with nonsinusoidal waveforms, in Annual Meeting of the IEEE Industry Applications Society, 2001, pp. 2203-2210.
* (5) K. Venkatachalam, C. R. Sullivan, T. Abdallah, and H. Tacca, Accurate prediction of ferrite core loss with nonsinusoidal waveforms using only Steinmetz parameters IEEE Workshop on Computers in Power Electronics (COMPEL), 2002.
* (6) Alex Van den Bossche, Vencislav Valchev, Georgi Georgiev, Measurement and loss model of ferrites in nonsinusoidal waves, IEEE Power Electronics Specialists Conference, 2004.
* (7) Jonas Mühlethaler, Jürgen Biela, Johann Walter Kolar and Andreas Ecklebe, Core-Loss Calculation for Magnetic Components Employed in Power Electronic Systems, IEEE TRANSACTIONS ON POWER ELECTRONICS, 27, pp.964-973 (2012).
* (8) J. Mühlethaler, J. Biela, J.W. Kolar, A. Ecklebe, Core Losses Under the DC Bias Condition Based on Steinmetz Parameters, IEEE Transactions on Power Electronics, 27, pp.953-963 (2012).
* (9) F. Fiorillo and A. Novikov, An improved approach to power lossess in magnetic laminations under nonsinusoidal induction waveform, IEEE Trans. Magnet., 26, pp.2559-2561 (1990).
* (10) K. Sokalski, J. Szczyg owski, M. Najgebauer and W. Wilczy nski, Thermodynamical Scaling of Eddy Current Losses in Magnetic Materials, Proc. 12th IGTE Symposium, 2006.
* (11) K. Sokalski, J. Szczygłowski, M. Najgebauer and W. Wilczyński, [12] K. Sokalski, J. Szczyg owski, M. Najgebauer and W. Wilczy nski, Losses scaling in soft magnetic materials, COMPEL: Int. J. Comput. Math. Electr. Electron. Eng.,26, 640-649 ( 2007), COMPEL: Int. J. Comput. Math. Electr. Electron. Eng.,26, 640-649 ( 2007).
* (12) K. Sokalski, J. Szczyg owski, and W. Wilczy nski, Scaling conception of power loss separation in soft magnetic materials, http://arxiv.org/abs/1111.0939v1.
* (13) A. Ruszczyk, K. Sokalski, J. Szczygłowski, Scaling in Modeling of Core Losses in Soft Magnetic Materials Exposed to Nonsinusoidal Flux Waveforms and DC Bias, SMM21 Conference, Budapest 2013.
* (14) J. Cale, S.D. Sudhoff, S. D. and R.R. Chan, A Field-Extrema Hysteresis Loss Model for High-Frequency Ferrimagnetic Materials, IEEE Transactions on Magnetics, vol. 44, issue 7, pp. 1728-1736 (2008).
* (15) K. Sokalski, J. Szczygłowski, ABB REPORT, PLCRC/50002437/02/1725/2011.
* (16) William H. Press, Saul A. Teukolsky, William T. Vetterling, Brian P. Flannery, Numerical Recipes in Fortran 77, The Art of Scientific Computing, Second Edition, Volume 1 of Fortran Numerical Recipes, Published by the Press Syndicate of the University of Cambridge 1997, p. 194.
|
arxiv-papers
| 2013-10-05T23:05:14 |
2024-09-04T02:49:51.991005
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Adam Ruszczyk and Krzysztof Sokalski",
"submitter": "Krzysztof Sokalski prof",
"url": "https://arxiv.org/abs/1310.1522"
}
|
1310.1535
|
Legendrian torus knots in S^1× S^2
Feifei Chen
School of Mathematical Sciences
Peking University
100871, China
Fan Ding
LMAM, School of Mathematical Sciences
Peking University
100871, China
Youlin Li
Department of Mathematics
Shanghai Jiao Tong University
Shanghai 200240, China
We classify the Legendrian torus knots in $S^1\times S^2$ with its standard tight contact structure up to Legendrian isotopy.
§ INTRODUCTION
Considerable progress has been made towards the classification of
Legendrian knots and links in contact 3-manifolds. The
classification of Legendrian unknots (in any tight contact
3-manifolds) is due to Eliashberg and Fraser, cf. [5] and
[6]. The classification of Legendrian torus knots and the
figure eight knot in $S^3$ with its standard tight contact
structure is due to Etnyre and Honda [8]. For a general
introduction to this topic see [7]. Recently, Legendrian
twist knots are classified in [10] and Legendrian cables of
positive torus knots are classified in [9]. Legendrian
knots in contact 3-manifolds other than $S^3$ (with its standard
contact structure) are also studied. For example, Legendrian
linear curves in all tight contact structures on $T^3$ are
classified in [13] and Legendrian torus knots in the $1$-jet
space $J^1(S^1)$ with its standard tight contact structure are
classified in [2]. Legendrian rational unknots in lens
spaces are studied in [1] and [12]. Legendrian torus
knots in lens spaces are studied in [17]. The purpose of the
present paper is to give a complete classification of Legendrian
torus knots in $S^1\times S^2$ with its standard tight contact
The standard tight contact structure $\xi_{\st}$ on $S^{1}\times S^{2}\subset S^{1}\times \R^3$
is given by $\ker (x_3\mathrm{d}\theta+x_1\mathrm{d}x_2-x_2\mathrm{d}x_1)$,
where $\theta$ denotes the $S^1$-coordinate and $(x_1,x_2,x_3)$ cartesian coordinates on $\R^3$.
Here we think of $S^1$ as $\R /2\pi \Z$. This is the unique positive tight contact structure on $S^1\times S^2$
up to isotopy; see <cit.>. Moreover, $\xi_{\st}$ is trivial as an abstract real $2$-plane bundle.
Suppose $K$ is an oriented Legendrian knot in $(S^1\times S^2,\xi_{\st})$.
For any preassigned choice of nowhere zero vector field $v$ in $\xi_{\st}$
(up to homotopies through such vector fields) we can define the rotation number
$\rot_v(K)$ to be the signed number of times that the tangent vector field $\tau$ to $K$ rotates in $\xi_{\st}$
relative to $v$ as we travel once around $K$ in the direction specified by its orientation. Usually we omit $v$ in $\rot_v(K)$.
Let $T_0=\{(\theta,x_1,x_2,x_3)\in S^1\times S^2: x_3= 0\}$, then
$T_0$ is a Heegaard torus, i.e., the closures of the components of $(S^1\times S^2)\setminus T_0$
are two solid tori. A knot in $S^1\times S^2$ is called a torus knot if it is (smoothly) isotopic to a knot on $T_0$.
Consider the solid torus $V_0=\{(\theta,x_1,x_2,x_3)\in S^1\times S^2: x_3\ge 0\}$.
The curve on $T_0=\partial V_0$ given by $\theta=0$, oriented positively in the $x_1x_2$-plane,
is a meridian of $V_0$, and let $m_0\in H_1(T_0)$ denote the class of this meridian;
the curve given by $(x_1,x_2,x_3)=(1,0,0)$, oriented by the parameter $\theta$,
is a longitude, and let $l_0\in H_1(T_0)$ denote the class of this longitude.
Then $(m_0,l_0)$ is a basis for $H_1(T_0)$. An oriented knot in $S^1\times S^2$
is called a $(p,q)$-torus knot if it is isotopic to an oriented knot on $T_0$
homologically equivalent to $pm_0+ql_0$ (with $p$ and $q$ coprime). A $(\pm 1,0)$-torus knot is trivial,
i.e., bounds a disk in $S^1\times S^2$. A $(p,1)$-torus knot is isotopic to $S^1\times \{(0,0,1)\}$,
oriented by the parameter $\theta$. For this knot type, we have
Two oriented Legendrian knots
in $(S^1\times S^2,\xi_{\st})$ of the same oriented knot type as $S^1\times \{ (0,0,1)\}$
are Legendrian isotopic if and only if their rotation numbers agree.
Let $K$ be a $(p,q)$-torus knot with $q\ge 2$. There is a Heegaard torus $T$ on which $K$ sits.
In Section 2, we shall prove that the framing of $K$ induced by $T$
(i.e., the framing corresponding to two simple closed curves which are the intersection of $T$
and the boundary of a tubular neighborhood of $K$), is independent of the Heegaard torus $T$ we choose.
For a Legendrian $(p,q)$-torus knot $K$ with $q\ge 2$, we define the twisting number $\tw (K)$ to
be the number of counterclockwise (right) $2\pi$ twists of $\xi_{\st}$ along $K$, relative to the framing of $K$ induced by $T$. We have
Let $K$ be a Legendrian $(p,q)$-torus knot and $K'$ a Legendrian $(p',q)$-torus knot
in $(S^1\times S^2,\xi_{\st})$ with $q\ge 2$. Then $K$ and $K'$ are Legendrian isotopic if and only if
their oriented knot types and their classical invariants $\tw$ and $\rot$ agree.
In Section 2, we give the topological classification of torus knots in $S^1\times S^2$. In Section 3,
we recall the classification of Legendrian torus knots in the 1-jet space $J^1(S^1)$ of the circle with its
standard contact structure, and give an analogous result for Legendrian torus knots in a solid torus with
a suitable tight contact structure. In Section 4, we prove Theorems <ref> and <ref>.
Acknowledgements: Authors would like to thank John Etnyre for helpful conversations, careful reading
of the paper and invaluable comments. Part of this work was carried out while the third author was
visiting Georgia Institute of Technology and he would like to thank them for their hospitality.
The third author was partially supported by NSFC grant 11001171 and the China Scholarship Council grant 201208310626.
§ TOPOLOGICAL TORUS KNOTS IN $S^1\TIMES S^2$
For $r_1,r_2\in \Q\setminus \Z$, we describe the Seifert manifold
$M(D^2;r_1,r_2)$ as follows. Let $\Sigma$ be an oriented pair of
pants. For each connected component $T_i$ of
$-(\partial\Sigma\times S^1)=T_1\cup T_2\cup T_3$, denote the
homology class in $H_1(T_i)$ of the connected component of
$-\partial(\Sigma\times\{ 1\})$ in $T_i$ by $\mu_i$ and the
homology class of the $S^1$ factor in $H_1(T_i)$ by $\lambda_i$.
For $i=1,2$, let $V_{i}=D^{2}\times S^{1}$. Then $M(D^2;r_1,r_2)$
is obtained from $\Sigma\times S^1$ by gluing $V_i$ to $T_i$,
$i=1,2$, using a diffeomorphism $\varphi_i:\partial V_i\to T_i$
sending the meridian $\partial (D^2\times\{ 1\})$ to a circle in
$T_i$ homologically equivalent to $p_i\mu_i-q_i\lambda_i$, where
$p_i,q_i$ are coprime and $\frac{q_i}{p_i}=r_i$. Note that
$M(D^2;r_1,r_2)$ corresponds to $M(0,1;r_1,r_2)$ in <cit.>.
Let $K$ be an oriented knot in $S^1\times S^2$. Denote a tubular
neighborhood of $K$ (diffeomorphic to a solid torus) by $\nu K$.
Let $T$ be a Heegaard torus in $S^1\times S^2$ on which $K$ sits.
Then $\partial(\nu K)\cap T$ are the two longitudes of $\nu K$
determined by the framing of $K$ induced by $T$.
If $K$ is a $(p,q)$-torus knot in $S^1\times S^2 $ with $q\ge 2$,
then the framing of $K$ induced by a Heegaard torus $T$ on which
$K$ sits is independent of the Heegaard torus $T$ we choose.
The compact manifold $S^1\times S^2 \setminus \Int(\nu K)$ is a
Seifert fibred space having $\partial(\nu K)\cap T$ as two regular
fibers. Fix such a Seifert fibration of $S^1\times S^2 \setminus
\Int(\nu K)$, then the essential surfaces are either vertical or
horizontal (cf. <cit.>). If $q>2$, then
$S^1\times S^2 \setminus \Int(\nu K)$ has a unique (up to isotopy)
essential surface which is a vertical annulus and separating. If
$q=2$, then $S^1\times S^2 \setminus \Int(\nu K)$ has a unique
vertical essential annulus and a horizontal essential annulus. The
vertical annulus is separating, and the horizontal one is
non-separating. So the Seifert fibred space $S^1\times S^2
\setminus \Int(\nu K)$ has a unique essential separating annulus
whose boundary is isotopic to $\partial(\nu K)\cap T$ in
$\partial(\nu K)$. Hence the framing of $K$ induced by a Heegaard
torus $T$ on which $K$ sits is determined by the the compact
manifold $S^1\times S^2 \setminus \Int(\nu K)$, and thus
independent of the Heegaard torus $T$ we choose.
We classify the torus knots in $S^1\times S^2$ as follows.
For $q\ge 2$, a $(p,q)$-torus knot and a $(p',q)$-torus knot are isotopic if and only if $p'\equiv p\mod 2q$ or $p'\equiv -p\mod 2q$.
For $p,q$ coprime and $q\ge 2$, let $K(p,q)$ be an oriented knot on
$T_0\subset S^1\times S^2$ homologically equivalent to $pm_0+ql_0$
(for the definitions of $T_0,m_0,l_0$, see Section 1). We prove
that $K(p,q)$ and $K(p',q)$ are isotopic in $S^1\times S^2$ if and
only if $p'\equiv p\mod 2q$ or $p'\equiv -p\mod 2q$. We divide the
proof into 4 steps.
Step 1. We prove that if $p'\equiv p\mod 2q$ or $p'\equiv -p\mod
2q$, then $K(p',q)$ and $K(p,q)$ are isotopic.
Write $r_{\theta}$ for the rotation of $S^2\subset\R^3$ about the
$x_3$-axis through an angle $\theta$. Define a diffeomorphism $r$
of $S^1\times S^2$ by
$r(\theta,\mathbf{x})=(\theta,r_{\theta}(\mathbf{x}))$. Since
$r^2(\theta,\mathbf{x})=(\theta,r_{2\theta}(\mathbf{x}))$ and
$\pi_1(SO(3))\cong \Z_2$, $r^2$ is isotopic to the identity (cf.
<cit.>). The diffeomorphism $r^2$ sends a knot on $T_0$ homologically
equivalent to $pm_0+ql_0$ to a knot on $T_0$ homologically
equivalent to $(p+2q)m_0+ql_0$. Thus $K(p,q)$ is isotopic to
$K(p+2q,q)$ in $S^1\times S^2$. Hence if $p'\equiv p\mod 2q$, then
$K(p',q)$ is isotopic to $K(p,q)$ in $S^1\times S^2$.
Define a diffeomorphism $b$ of $S^1\times S^2$ by
$b(\theta,x_1,x_2,x_3)=(\theta,x_1,-x_2,-x_3)$. Then $b$ is isotopic to
the identity and sends a knot on $T_0$ homologically equivalent to
$pm_0+ql_0$ to a knot on $T_0$ homologically equivalent to
$-pm_0+ql_0$. Thus $K(p,q)$ is isotopic to $K(-p,q)$ in $S^1\times
S^2$. Combining this with the preceding paragraph, we conclude
that if $p'\equiv -p\mod 2q$, then $K(p',q)$ is isotopic to
$K(p,q)$ in $S^1\times S^2$.
Step 2. We prove that if $K(p,q)$ and $K(p',q)$ are isotopic in
$S^1\times S^2$, then $p'\equiv p\mod q$ or $p'\equiv -p\mod q$.
Since $p$ and $q$ are coprime, we may choose $s,t\in\Z$ such that
$ps-tq=1$. The closure of the complement of a tubular neighborhood
of $K(p,q)$ in $S^1\times S^2$ is the Seifert manifold
$M(D^2;\frac{s}{q},-\frac{s}{q})$. Similarly, we choose
$s',t'\in\Z$ such that $p's'-t'q=1$. Then the closure of the
complement of a tubular neighborhood of $K(p',q)$ is
$M(D^2;\frac{s'}{q},-\frac{s'}{q})$. Now suppose $K(p,q)$ and
$K(p',q)$ are isotopic in $S^1\times S^2$. Then
$M(D^2;\frac{s}{q},-\frac{s}{q})$ and
$M(D^2;\frac{s'}{q},-\frac{s'}{q})$ are orientation-preserving
diffeomorphic. By <cit.>, we
have $s'\equiv s\mod q$ or $s'\equiv -s\mod q$. If $s'\equiv s\mod
q$, then there exists an integer $k$ such that $s'=s+kq$. Combined
with $ps-tq=1$ and $p's'-t'q=1$, this gives $(p'-p)s=(t'-t-kp')q$.
Since $q$ and $s$ are coprime, $q$ divides $p'-p$. Thus $p'\equiv
p\mod q$. Similarly, if $s'\equiv -s\mod q$, then $p'\equiv -p\mod
Step 3. We prove that for $q>2$, $K(p,q)$ is not isotopic to
$K(p+q,q)$ in $S^1\times S^2$.
First note that $r(K(p,q))$ is isotopic to $K(p+q,q)$. Thus if
$K(p,q)$ is isotopic to $K(p+q,q)$, then $r(K(p,q))$ is isotopic
to $K(p,q)$ and there is an orientation-preserving diffeomorphism
$g$, isotopic to $r$, such that the restriction of $g$ on $K(p,q)$ is the
identity. Note that $g(T_0)$ is also a Heegaard torus. Thus the framing of
$K(p,q)$ induced by $T_0$ is the same as the framing of $K(p,q)$
induced by $g(T_0)$. Hence after an isotopy, we may assume that
$g$ is the identity on a tubular neighborhood $N$ (diffeomorphic
to a solid torus) of $K(p,q)$.
The compact manifold $(S^1\times S^2)\setminus\Int(N)$ is diffeomorphic to the
Seifert manifold $M(D^2;\frac{s}{q},-\frac{s}{q})$. Recall that
$M(D^2;\frac{s}{q},-\frac{s}{q})$ is obtained from $\Sigma\times
S^1$ by gluing $V_i$ to $T_i$ ($i=1,2$), using a diffeomorphism
$\varphi_i:\partial V_i\to T_i$ sending the meridian $\partial
(D^2\times\{ 1\})$ to a circle in $T_i$ homologically equivalent
to $q\mu_1-s\lambda_1$ for $i=1$, or $q\mu_2+s\lambda_2$ for $i=2$
(for the notation $\Sigma
,T_i(i=1,2,3),\mu_i,\lambda_i,V_i(i=1,2)$, see the first paragraph
of this section). The simple closed curve $S^1\times \{ (0,0,-1)\}$ corresponds to a core
of $V_1$ and the simple closed curve $S^1\times \{(0,0,1)\}$ corresponds to a core of
$V_2$. Consider the restriction of $g$ to $(S^1\times
S^2)\setminus\Int(N)$, still denoted by $g$, as a
self-diffeomorphism of $M(D^2;\frac{s}{q},-\frac{s}{q})$ which is
the identity on the boundary. The compact surface $A=T_0\setminus\Int (N)$ is an
essential vertical annulus in $M(D^2;\frac{s}{q},-\frac{s}{q})$.
By <cit.>, $g(A)$ is isotopic (relative to
the boundary) to a vertical annulus. Thus we may assume that
$g(A)$ is a vertical annulus (disjoint from $V_1$ and $V_2$).
(30, 10)$\alpha$
(80, 80)$\beta$
(50, 50)$C$
(100, 0)$\Sigma$
(172, 10)$\alpha$
(220, 80)$\beta$
(200, 67)$C'$
(244, 0)$\Sigma$
(314, 10)$\alpha$
(365, 80)$\beta$
(335, 50)$C'$
(385, 0)$\Sigma$
The oriented pair of pants $\Sigma$, the oriented arc $C$ shown in the left, and two possible oriented arcs $C'$ shown in the middle and right.
Let $C$ denote the arc in $\Sigma$ such that $C\times S^1$ is the
annulus $A$ (see the left of Figure <ref>). Let $C'$ denote
the arc in $\Sigma$ such that $C'\times S^1$ is the annulus $g(A)$
(see the middle and right of Figure <ref>). Let $B$ denote
the component of $\partial\Sigma$ such that $B\times S^1$ is
$T_3$. The two points $C\cap B$ divides $B$ into $2$ arcs $\alpha$
and $\beta$ (see the left of Figure <ref>). In
$M(D^2;\frac{s}{q},-\frac{s}{q})$, the torus $(\alpha \cup
C)\times S^1$ bounds a solid torus $N_0$ containing one of $V_1$
and $V_2$, say $V_1$. Orient $\alpha$ as a part of $\partial
\Sigma$. Orient $C$ such that the orientation on $\alpha$ and the
orientation on $C$ give an orientation on $\alpha\cup C$ (see the
left of Figure <ref>). Note that $g$ is the identity on
$T_3$. Orient $C'$ such that the orientation on $\alpha$ and the
orientation on $C'$ give an orientation on $\alpha\cup C'$ (see
the middle and right of Figure <ref>). Denote the class in
$H_1((\alpha\cup C)\times S^1)$ of $(\alpha\cup C)\times\{ 1\}$ by
$\mu$ and the class in $H_1((\alpha\cup C)\times S^1)$ of a fiber
by $\lambda$. Then $q\mu-s\lambda$ is the class of a meridian of
$N_0$. Denote the class in $H_1((\alpha\cup C')\times S^1)$ of
$(\alpha\cup C')\times\{ 1\}$ by $\mu'$ and the class in
$H_1((\alpha\cup C')\times S^1)$ of a fiber by $\lambda'$. In
$M(D^2;\frac{s}{q},-\frac{s}{q})$, the torus $(\alpha \cup
C')\times S^1$ bounds a solid torus $N_0^{\prime}$ containing one
of $V_1$ and $V_2$, and $g$ sends $N_0$ onto $N_0^{\prime}$.
Suppose that $g(C\times\{ 1\})\subset C'\times S^1$ wraps around
the $S^1$ factor $k$ times as we travel once around $C$. If
$N_0^{\prime}$ contains $V_1$ (see the middle of
Figure <ref>), then $q\mu'-s\lambda'$ is the class of a
meridian of $N_0^{\prime}$. The diffeomorphism $g$ sends the class
$q\mu-s\lambda\in H_1((\alpha\cup C)\times S^1)$ to the class
$q\mu'+(kq-s)\lambda'\in H_1((\alpha\cup C')\times S^1)$. Since
$q\mu'+(kq-s)\lambda'$ needs to be the class of a meridian of
$N_0^{\prime}$, we have $k=0$. Thus we may assume that $g$ is the
identity on $V_1$ (cf. <cit.>). Then $g$, as a
self-diffeomorphism of $S^1\times S^2$, is the identity near
$S^1\times\{ (0,0,-1)\}$. Hence by <cit.>, $g$ is
isotopic to the identity. But $g$ is isotopic to $r$ and $r$ is
not isotopic to the identity (cf. <cit.>), and we get
a contradiction. Hence $N_0^{\prime}$ contains $V_2$ (see the
right of Figure <ref>). Then $q\mu'+s\lambda'$ is the class
of a meridian of $N_0^{\prime}$ and $kq-s=s$. Thus $2s$ is divided
by $q$. Since $q,s$ are coprime, we conclude that $2$ is divided
by $q$, contrary to the assumption that $q>2$.
Step 4. If $K(p,q)$ and $K(p',q)$ are isotopic in $S^1\times S^2$,
then $p'\equiv p\mod 2q$ or $p'\equiv -p\mod 2q$.
First assume that $q>2$. If $K(p,q)$ and $K(p',q)$ are isotopic,
then by Step 2, there exists an integer $k$ such that $p'=p+kq$ or
$p'=-p+kq$. If $k$ is odd, then by Step 1, $K(p',q)$ is isotopic
to $K(p+q,q)$. Hence $K(p+q,q)$ is isotopic to $K(p,q)$ in
$S^1\times S^2$, contrary to the conclusion in Step 3. Thus $k$ is
even and $p'\equiv p\mod 2q$ or $p'\equiv -p\mod 2q$.
Assume now that $q=2$. If $K(p,2)$ and $K(p',2)$ are isotopic in
$S^1\times S^2$, then by Step 2, there exists an integer $k$ such
that $p'=p+2k$. If $k$ is even, then $p'\equiv p\mod 4$. If $k$ is
odd, then $p+k$ is even since $p$ is odd ($p$ and $2$ are
coprime). Hence by $p'=-p+2(p+k)$, we have $p'\equiv -p\mod 4$.
§ LEGENDRIAN TORUS KNOTS IN $J^1(S^1)$ AND IN A SOLID TORUS
For fixing notation, we give definitions and properties of
Legendrian $(p,q)$-torus knots in $J^1(S^1)$ and in a solid torus.
§.§ Legendrian torus knots in $J^1(S^1)$
Let $J^1(S^1)=T^*S^1\times\R =S^1\times \R^2=\{ (\theta,y,z):\theta\in S^1=\R/2\pi\Z,y,z\in\R\}$
be the $1$-jet space of $S^1$ with its standard contact structure $\xi_0=\ker(\mathrm{d}z-y\mathrm{d}\theta)$.
One can visualize a Legendrian knot $K\subset J^1(S^1)$ in its front projection to
a strip $[0,2\pi]\times \R$ in the $\theta z$-plane. The Thurston-Bennequin invariant
of $K$ is $\tb(K)=\mathrm{writhe}(K)-\frac{1}{2}\# (\mathrm{cusps}(K))$, where the quantities
on the right are computed from the front projection of $K$. This invariant has a definition
that does not rely on the front projection, and which shows that $\tb$ is a Legendrian isotopy invariant, cf. [2].
For an oriented Legendrian knot $K$ in $J^1(S^1)$, we may define its rotation number
in terms of its front projection as $\rot(K)=\frac{1}{2}(c_--c_+)$, with $c_{\pm}$ the number
of cusps oriented upwards or downwards, respectively; cf. [3]. This is the same as
the rotation number defined by the nowhere zero vector field $\partial_y$ in $\xi_0$, cf. <cit.>.
By a torus knot in $J^1(S^1)$, we mean a knot that sits on a torus isotopic to
the torus $T_1=\{(\theta,y,z)\in J^1(S^1):y^2+z^2=1\}$. Consider the solid torus $M_1=\{(\theta,y,z)\in J^1(S^1):y^2+z^2\leq1\}$.
The curve on $T_1=\partial M_1$ given by $\theta=0$, oriented positively in the $yz$-plane,
is a meridian of $M_1$, and let $m_1\in H_1(T_1)$ denote the class of this meridian; the curve
given by $(y,z)=(1,0)$, oriented by the parameter $\theta$, is a longitude, and
let $l_1\in H_1(T_1)$ denote the class of this longitude. Then $(m_1,l_1)$ is a positive basis for $H_1(\partial M_1)$.
For $p,q$ coprime, a $(p,q)$-torus knot in $J^1(S^1)$ is an oriented knot
isotopic to an oriented knot on $T_1$ homologically equivalent to $pm_1+ql_1$.
A $(\pm 1,0)$-torus knot in $J^1(S^1)$ is trivial. A $(p,1)$-torus knot in $J^1(S^1)$
is isotopic to $S^1\times \{(0,0)\}$, oriented by the variable $\theta$. For $q\ge 2$,
if a $(p,q)$-torus knot is isotopic to a $(p',q)$-torus knot in $J^1(S^1)$, then $p=p'$.
This can be seen as follows. Let $K$ and $K'$ be a $(p,q)$-torus knot and a $(p',q)$-torus knot
in $J^1(S^1)$, respectively. Embed $J^1(S^1)$ into $S^3$ as an open tubular
neighborhood of an unknot in $S^3$. Then $K$ and $K'$ become a $(p+cq,q)$-torus knot and a $(p'+cq,q)$-torus
knot in $S^3$ for some $c\in\Z$. Using different framings of the unknot to define the embedding,
$c$ can be any integer. Thus if $K$ and $K'$ are isotopic in $J^1(S^1)$, then the corresponding
$(p+cq,q)$-torus knot and $(p'+cq,q)$-torus knot are isotopic in $S^3$ for each $c\in \Z$.
Thus by the classification of torus knots in $S^3$, we have $p=p'$.
For an oriented Legendrian knot $K$ in a contact $3$-manifold,
we have a positive stabilization $S_+(K)$ and a negative stabilization $S_-(K)$ (cf. [8]).
Stabilizations are well defined and commute with each other. For an oriented Legendrian knot $K$ in $J^1(S^1)$,
we have $\tb(S_{\pm}(K))=\tb(K)-1$, $\rot(S_{\pm}(K))=\rot (K)\pm 1$.
Notice that the stabilization affects the classical invariants in this way for any oriented Legendrian knot $K$
in any contact $3$-manifold, as long as the classical invariants can be defined (cf. [8]).
The results in the following three paragraphs can be deduced from <cit.>.
The maximum Thurston-Bennequin invariant of a Legendrian knot in $J^1(S^1)$ isotopic to $S^1\times\{(0,0)\}$ is $0$.
Any Legendrian knot isotopic to $S^1\times\{(0,0)\}$ with $\tb=0$ is Legendrian isotopic to
$S^1\times\{(0,0)\}$. A Legendrian knot in $J^1(S^1)$ isotopic to $S^1\times\{(0,0)\}$ with non-maximum $\tb$ can be destabilized in $J^1(S^1)$.
For $p\ge 1$ and $q\ge 2$, the maximum Thurston-Bennequin invariant of a Legendrian $(p,q)$-torus
knot in $J^1(S^1)$ is $p(q-1)$. Any two Legendrian $(p,q)$-torus knots with maximum Thurston-Bennequin
invariant are Legendrian isotopic. A Legendrian $(p,q)$-torus knot in $J^1(S^1)$ with non-maximal $\tb$ can be destabilized in $J^1(S^1)$.
For $p<0$ and $q\ge 2$, the maximum Thurston-Bennequin invariant of a Legendrian $(p,q)$-torus knot
in $J^1(S^1)$ is $pq$. The possible values of $\rot$ (for $\tb=pq$ being maximum) are shown to lie
in $\{ \pm(p+2lq):l\in\Z,0\le l<-\frac{p}{q}\}$. A Legendrian $(p,q)$-torus knot in $J^1(S^1)$
with non-maximal $\tb$ can be destabilized in $J^1(S^1)$.
By <cit.>, two oriented Legendrian torus knots in $J^1(S^1)$
are Legendrian isotopic if and only if their oriented knot types and their classical invariants $\tb$ and $\rot$ agree.
§.§ Legendrian torus knots in a solid torus
Let $V$ be an oriented solid torus. Let $m\in H_1(\partial V)$ be the class of an oriented meridian
of $V$ and $l\in H_1(\partial V)$ the class of an oriented longitude of $V$. The meridian and the
longitude are oriented in such a way that $m,l$ form a positive basis for $H_1(\partial V)$.
By a torus knot in $V$, we mean a knot in $\Int(V)$ that sits on a torus parallel to $\partial V$
(i.e. this torus and $\partial V$ bound a thickened torus in $V$). For $p,q$ coprime, a $(p,q)$-torus knot in $V$
is an oriented knot in $\Int (V)$ that sits on a torus $T$ parallel to $\partial V$
such that this oriented knot is homologically equivalent to $pm+ql$ in the thickened torus bounded by $T$ and $\partial V$.
Similar to the cases in $J^1(S^1)$, we have: a $(\pm 1,0)$-torus
knot in $V$ is trivial; a $(p,1)$-torus knot in $V$ is isotopic to
a core of $V$; for $q\ge 2$, if a $(p,q)$-torus knot in $V$ is
isotopic to a $(p',q)$-torus knot in $V$, then $p=p'$. For a
$(p,q)$-torus knot $K$ with $q\ge 2$, the framing of $K$ induced
by a torus $T$ parallel to $\partial V$ on which $K$ sits is
independent of the torus $T$ we choose. This can be seen by
embedding $V$ in $S^1\times S^2$ and using
Proposition <ref>.
Now let $\xi$ be a positive tight contact structure on $V$ with
convex boundary having two dividing curves each in the homology
class $l$. For a Legendrian $(p,q)$-torus knot $K$ in $(V,\xi)$
with $q\ge 2$, we define the twisting number $\tw(K)$ to be the
number of counterclockwise (right) $2\pi$ twists of $\xi$ along
$K$, relative to the framing of $K$ induced by a torus $T$
parallel to $\partial V$ on which $K$ sits. Since $\xi$ is trivial
as an abstract real $2$-plane bundle, using a nowhere zero vector
field in $\xi$, we can define the rotation number of an oriented
Legendrian knot in $(V,\xi)$.
Let $L$ be an oriented Legendrian core of $(V,\xi)$ such that $l$
is the class of a parallel copy of $L$ determined by the contact
framing. Such a Legendrian core exists by <cit.>.
Furthermore, by <cit.>, we have a contact
embedding $\phi$ from $(V,\xi)$ to $(J_1(S^1),\xi_0)$ whose image
is $M_1$ (possibly perturbing $\partial V$) and sending $l$ to
For a Legendrian $(p,q)$-torus knot $K$ in $(V,\xi)$ with $q\ge
2$, $\phi(K)$ is a Legendrian $(p,q)$-torus knot in
$(J_1(S^1),\xi_0)$, and $\tw(K)=\tb(\phi(K))-pq$. Using the
nowhere vanishing vector field in $(V,\xi)$, which is sent to
$\partial_{y}$ by $\mathrm{d}\phi$, to define the rotation number
of $K$, we have $\rot(K)=\rot(\phi(K))$.
The following proposition is essentially contained in [8], and can be easily derived from Subsection 3.1 by a contact flow.
With $V,\xi,L$ as above,
* for $p,q$ coprime and $q\ge 2$, two Legendrian $(p,q)$-torus knots in $(V,\xi)$ are Legendrian isotopic
if and only if their classical invariants (twisting number and rotation number) agree;
* an oriented Legendrian knot in $(\Int(V),\xi)$ isotopic to $L$ is Legendrian isotopic to a stabilization of $L$.
§ LEGENDRIAN TORUS KNOTS IN $S^1\TIMES S^2$
First, we prove the main results.
Let $K$ and $K'$ be two oriented
Legendrian knots in $(S^1\times S^2,\xi_{\st})$ which have the
same oriented knot type as $S^1\times \{ (0,0,1)\}$ and the same
rotation number. The knot $K$ has a tubular neighborhood $N$ with
convex boundary having two dividing curves each in the homology
class $\lambda$, where $\lambda$ is the class in $H_1(\partial N)$
of a parallel copy of $K$ determined by the contact framing.
Similarly, the knot $K'$ has a tubular neighborhood $N'$ with
convex boundary having two dividing curves each in the homology
class $\lambda'$, where $\lambda'$ is the class in $H_1(\partial
N')$ of a parallel copy of $K'$ determined by the contact framing.
This allows one to find a contactomorphism $\phi:N\to N'$ sending
$K$ to $K'$ and $\lambda$ to $\lambda'$ (and sending a meridian of
$N$ to a meridian of $N'$). Since a meridian of $N$ (respectively,
$N'$) is also a meridian of $(S^1\times S^2)\backslash \Int(N)$
(respectively, $(S^1\times S^2)\backslash \Int(N')$), $\phi$ can
be extended to a diffeomorphism of $S^1\times S^2$. Furthermore,
by Theorem 3.14 of [8], $\phi$ can be extended to a
contactomorphism, still denoted by $\phi$, of $(S^1\times
In [4], we have a contactomorphism $r_c$ of $(S^1\times S^2,\xi_{\st})$
isotopic to the diffeomorphism $r$ (for the definition of $r$, see the proof of Proposition <ref>)
such that for the oriented Legendrian knot $K_0$, which is shown in Figure <ref>,
in $(S^1\times S^2,\xi_{\st})$, $r_c(K_0)$ is Legendrian isotopic to the positive stabilization $S_+(K_0)$ of $K_0$.
In particular, $\rot (r_c(K_0))=\rot (K_0)+1$. Thus for an oriented Legendrian
knot $K_1$ in $(S^1\times S^2,\xi_{\st})$ homotopic to $q$ times the standard generator
of the fundamental group $\pi_1(S^1\times S^2)\cong\Z$, we have $\rot(r_c(K_1))=\rot(K_1)+q$.
According to [4], any contactomorphism of $(S^1\times S^2,\xi_{\st})$ acting trivially on
homology is contact isotopic to a uniquely determined integer power of $r_c$. So $\phi$ is contact
isotopic to $r_c^m$ for some integer $m$. Hence $\rot (K')=\rot (K)+m$. Since $\rot (K)=\rot(K')$,
we conclude that $m=0$ and thus $\phi$ is contact isotopic to the identity. Thus $K$ and $K'$ are Legendrian isotopic.
The Legendrian knot $K_0$ in $(S^1\times S^2,\xi_{\st})$.
Perturb $T_0$ to be a convex torus $T_0'$
with two dividing curves each in the homology class $l_0'$, where
$l_0'\in H_1(T_0')$ corresponds to $l_0\in H_1(T_0)$ under the
perturbation. Let $m_0'\in H_1(T_0')$ denote the class
corresponding to $m_0\in H_1(T_0)$ under the perturbation. For the
notation $T_0,l_0,m_0$, see Section 1. Let $N_0$ and $N_1$ denote
the closures of the components of $(S^1\times S^2)\backslash T_0'$
containing $S^1\times \{ (0,0,1)\}$ and $S^1\times\{ (0,0,-1)\}$,
respectively. Let $K_0$ (respectively, $K_1$) be an oriented
Legendrian core of $N_0$ (respectively, $N_1$) such that $l_0'$ is
the class of a parallel copy of $K_0$ (respectively, $K_1$)
determined by the contact framing. One may consider $K_0$ as the
Legendrian knot $K_0$ shown in Figure <ref> and $K_1$ as a
Legendrian push-off of $K_0$ along the vertical direction. In
particular, $\rot (K_0)=\rot (K_1)$.
Let $K_2$ be a Legendrian knot in $(S^1\times S^2,\xi_{\st})$ which
sits on a Heegaard torus $T$. Denote the closures of the components of
$(S^1\times S^2)\backslash T$ by $N,N'$. Let $K_2'$ be an oriented Legendrian
core of $N'$ such that $K_2'$ is isotopic to $S^1\times \{ (0,0,1)\}$ in $S^1\times S^2$
as oriented knots. Stabilize $K_2'$ if necessary to make $\rot(K_2')=\rot(K_1)$.
By Theorem <ref>, $K_2'$ is Legendrian isotopic to $K_1$. Thus we may assume
that $K_2$ is in $(S^1\times S^2)\backslash K_1$. Since $(S^1\times S^2)\backslash K_1$
is contactomorphic to $(J^{1}(S^{1}), \xi_0)$, cf. <cit.>,
using a contact flow, we may push $K_2$ into $\Int (N_0)$.
Now let $K$ and $K'$ be two oriented Legendrian torus knots in $(S^1\times S^2,\xi_{\st})$
which have the same oriented knot type and classical invariants $\tw$ and $\rot$.
By the preceding paragraph, we may assume that $K$ and $K'$ are Legendrian torus knots
in $N_0$ with the same invariants $\tw$ and $\rot$. Use $m_0',l_0'$ to define $(p,q)$-torus
knots in $N_0$. Then a $(p,q)$-torus knot in $N_0$ is also a $(p,q)$-torus knot
in $S^1\times S^2$. Without loss of generality, we may assume that $K$ is a
Legendrian $(p,q)$-torus knot in $N_0$ and $K'$ is a Legendrian $(p',q)$-torus
knot in $N_0$ with $q\ge 2$. By Proposition <ref>, $p'\equiv p\mod 2q$ or $p'\equiv -p\mod 2q$.
If $p'\equiv p\mod 2q$, then by interchanging the roles of $K$ and $K'$ if necessary,
we may assume that $p'=p+2kq$, where $k$ is a non-negative integer. There is a contactomorphism $g$
of $(S^1\times S^2,\xi_{\st})$ which sends $K_0$ to $S_+S_-(K_0)$ and is contact isotopic to the
identity (cf. the proof of Theorem <ref>). We may assume that $g$ sends $N_0$
into $\Int (N_0)$. The class of a parallel copy of $S_{+}S_{-}(K_{0})$ determined by the contact framing is
$l_0'-2m_0'$. Then $g$ sends a $(p_0,q)$-torus knot (corresponding to $p_{0}m_0'+ql_0'$)
in $N_0$ to a $(p_{0}-2q,q)$-torus knot (corresponding to $p_0m_0'+q(l_0'-2m_0')=(p_0-2q)m_0'+ql_0'$)
in $N_0$. Hence $g^k(K')$ is a $(p,q)$-torus knot in $N_0$.
By Proposition <ref> (1), $g^k(K')$ and $K$
are Legendrian isotopic in $ (N_0,\xi_{\st})$. Thus $K$ and $K'$ are Legendrian isotopic in $(S^1\times S^2,\xi_{\st})$.
If $p'\equiv -p\mod 2q$, let $T'$ be a torus in $\Int (N_0)$ parallel to $\partial N_0$
on which $K'$ sits. Let $N_0'$ be the solid torus in $\Int (N_0)$ which has boundary $T'$.
Let $K_0'$ be an oriented Legendrian core of $N_0'$ such that $K_0'$ is isotopic to
$S^1\times \{ (0,0,1)\}$ in $S^1\times S^2$ as oriented knots. Stabilize $K_0'$ if necessary
to make $\rot (K_0')=\rot (K_0)=\rot (K_1)$. By Proposition <ref> (2),
$K_0'$ is Legendrian isotopic to $S_ +^kS_-^k(K_0)$ in $(N_0,\xi_{\st})$, where $k$ is a
non-negative integer. There is a contactomorphism $h$ of $(S^1\times S^2,\xi_{\st})$ which
sends $K_0'$ to $K_1$ and is contact isotopic to the identity (cf. the proof of
Theorem <ref>). Using a contact flow in $(S^1\times S^2)\backslash K_1$,
we may assume that $h(K')$ is in $\Int (N_0)$. Since $K_0'$ is Legendrian isotopic
to $S_{+}^{k} S_{-}^{k}(K_{0})$ in $(N_0,\xi_{\st})$, the class of a parallel copy
of $K_0'$ determined by the contact framing is $l_0'-2km_0'$. Note that a meridian of $K_1$
corresponds to $-m_0'$. Since $h$ sends $K_0'$ to $K_1$ and is a contactomorphism, $h$
sends $m_0'$ to $-m_0'$ and sends $l_0'-2km_0'$ to $l_0'$, thus sends $l_0'$ to $l_0'-2km_0'$,
and hence sends the $(p',q)$-torus knot $K'$ (corresponding to $p'm_0'+ql_0'$) to a $(-p'-2kq,q)$-torus
knot (corresponding to $p'(-m_0')+q(l_0'-2km_0')=(-p'-2kq)m_0'+ql_0'$) in $N_0$.
By the preceding paragraph, $h(K')$ and $K$ are Legendrian isotopic in $(S^1\times S^2,\xi_{\st})$.
Thus $K$ and $K'$ are Legendrian isotopic in $(S^1\times S^2,\xi_{\st})$.
(60, -5)$L_{0}$
(370, -5)$L_{1}$
(220, -5)$L_{-1}$
Three Legendrian torus knots in $(S^1\times S^2,\xi_{\st})$.
A Legendrian isotopy.
As described in <cit.>, one can represent
$(S^1\times S^2,\xi_{\st})$ by the Kirby diagram with one
$1$-handle in the standard contact structure on $S^3$. We define
the rotation number of an oriented Legendrian knot in $(S^1\times
S^2,\xi_{\st})$ as described in <cit.>. In
Figure <ref>, $L_0$ is a Legendrian $(2,1)$-torus knot with
$\rot(L_0)=0$ and $\tw (L_0)=-1$, $L_{-1}$ is a Legendrian
$(2,-1)$-torus knot with $\rot (L_{-1})=-1$ and $\tw (L_{-1})=0$,
and $L_1$ is a Legendrian $(2,-1)$-torus knot with $\rot (L_1)=1$
and $\tw (L_1)=0$. By Proposition <ref>, $L_0,L_{-1},L_1$ are isotopic in $S^1\times S^2$.
Furthermore, by Theorem <ref>, $L_0$ is Legendrian
isotopic to $S_+(L_{-1})$ and $S_-(L_1)$. An explicit Legendrian
isotopy between $L_0$ and $S_-(L_1)$ is shown in
Figure <ref>. In the second step of this Legendrian isotopy,
we perform a move of type $6$ from <cit.>.
We give some propositions on the invariants of Legendrian
$(p,q)$-torus knots, $q\geq 2$, in $(S^1\times S^2, \xi_{\st})$.
For a Legendrian $(p,q)$-torus knot, $q\geq2$, in $(S^1\times S^2,
\xi_{\st})$, the maximal $\tw$ invariant is $0$.
According to the proof of Theorem <ref>, we can push a
Legndrian $(p,q)$-torus knot in $(S^1\times S^2, \xi_{\st})$ into
a Legendrian $(p',q)$-torus knot in $(N_0, \xi_{\st})$. According
to Section 3, if $p'>0$, then it has $\tw\leq -p'<0$, and if
$p'<0$, then it has $\tw\leq 0$. Note that the $\tw$ invariant of
a Legndrian $(p,q)$-torus knot in $(S^1\times S^2, \xi_{\st})$
coincides with that of its push-off in $(N_0, \xi_{\st})$. So the
maximal $\tw$ invariant of a Legendrian $(p,q)$-torus knot in
$(S^1\times S^2, \xi_{\st})$ is nonpositive. On the other hand,
there exists a Legendrian $(p',q)$-torus knot in $(N_0,
\xi_{\st})$, with $p'\equiv p\mod 2q$ and $p'<0$, which has $\tw$
invariant $0$. So the proposition holds.
For a Legendrian $(p,q)$-torus knot, $q\geq2$, in $(S^1\times S^2,
\xi_{\st})$, with the maximal $\tw$ invariant, it has $\rot\in\{
\pm p+2dq: d\in \Z\}$.
There exists an integer $d'$ such that $\pm p-2d'q<0$. Choose a
Legendrian $(p-2d'q,q)$-torus knot in $(N_0, \xi_{\st})$ which has
maximal $\tw$ invariant. Then, according to Section 3, its
rotation number belongs to $\{ \pm( p-2d'q+2d''q): 0\leq
d''<\frac{-p+2d'q}{q}, d''\in \Z\}$. Choose a Legendrian
$(-p-2d'q,q)$-torus knot in $(N_0, \xi_0)$ which has maximal $\tw$
invariant. Then its rotation number belongs to $\{ \pm(
-p-2d'q+2d''q): 0\leq d''<\frac{p+2d'q}{q}, d''\in \Z\}$. Both of
these two Legendrian knots are Legendrian $(p,q)$-torus knots in
$(S^1\times S^2, \xi_{\st})$. Note that the rotation number of a
Legendrian $(p,q)$-torus knot in $(S^1\times S^2, \xi_{\st})$
coincides with that of its push-off in $(N_0, \xi_{\st})$. Since
$d'$ can be arbitrarily large, the rotation number of a Legendrian
$(p,q)$-torus knot in $(S^1\times S^2, \xi_{\st})$ with maximal
$\tw$ invariant can be, and can only be, any number of $\{\pm
p+2dq: d\in \Z\}$.
A Legendrian $(p,q)$-torus knot, $q\geq2$, in $(S^1\times S^2, \xi_{\st})$
with non-maximal $\tw$ can be destabilized in $(S^1\times S^2, \xi_{\st})$.
We can push a Legendrian $(p,q)$-torus knot in $(S^1\times S^2,
\xi_{\st})$ to be a Legendrian $(p',q)$-torus knot in $(N_0,
\xi_{\st})$ such that $p'\equiv p\mod 2q$ and $p'<0$. Then the
$\tw$ invariant of the Legendrian $(p',q)$-torus knot is
non-maximal. According to Section 3, we can destabilize it in
$(N_0, \xi_{\st})$. So we can destabilize the Legendrian
$(p,q)$-torus knot in $(S^1\times S^2, \xi_{\st})$.
[1] K. L. Baker and J. B. Etnyre, Rational linking
and contact geometry, Perspectives in Analysis, Geometry, and
Topology, Progr. Math. 296 (Birkhäuser, Basel, 2012), 19-37.
[2] F. Ding and H. Geiges, Legendrian knots and links classified by classical
invariants, Commun. Contemp. Math. 9 (2007), 135-162.
[3] F. Ding and H. Geiges, Legendrian helix and
cable links, Commun. Contemp. Math. 12 (2010), 487-500.
[4] F. Ding and H. Geiges, The diffeotopy group of
$S^1\times S^2$ via contact topology, Compositio Math. 146 (2010),
[5] Y. Eliashberg and M. Fraser, Classification of
topologically trivial Legendrian knots, Geometry, Topology, and
Dynamics (Montréal, 1995), CRM Proc. Lecture Notes Vol. 15
(Amer. Math. Soc., Providence, 1998), 17-51.
[6] Y. Eliashberg and M. Fraser, Topologically
trivial Legendrian knots, J. Symplectic Geom. 7 (2009), 77-127.
[7] J. B. Etnyre, Legendrian and transversal knots,
Handbook of Knot Theory (Elsevier, Amsterdam, 2005), 105-185.
[8] J. B. Etnyre and K. Honda, Knots and contact
geometry I: Torus knots and the figure eight knot, J. Symplectic
Geom. 1 (2001), 63-120.
[9] J. B. Etnyre, D. J. LaFountain and B. Tosun,
Legendrian and transverse cables of positive torus knots, Geom.
Topol. 16 (2012), no. 3, 1639-1689.
[10] J. B. Etnyre, L. L. Ng and V. Vértesi,
Legendrian and transverse twist knots, J. Eur. Math. Soc. (JEMS), 15 (2013), no. 3, 969-995.
[11] H. Geiges, An introduction to contact topology,
Cambridge Studies in Advanced Mathematics, vol. 109 (Cambridge
University Press, Cambridge, 2008).
[12] H. Geiges and S. Onaran, Legendrian rational
unknots in lens spaces, arXiv:1302.3792.
[13] P. Ghiggini, Linear Legendrian curves in $T^3$,
Math. Proc. Camb. Philos. Soc. 140 (2006), 451-473.
[14] R. E. Gompf, Handlebody construction of Stein
surfaces, Ann. of Math. (2) 148 (1998), 619-693.
[15] A. Hatcher, Notes on Basic 3-Manifold
Topology, available at
[16] C. Hodgson and J. H. Rubinstein, Involutions and
isotopies of lens spaces, Knot theory and manifolds (Vancouver,
1983), Lecture Notes in Mathematics, vol. 1144, ed. D. Rolfsen
(Springer, Berlin, 1985), 60-96.
[17] S. C. Onaran, Legendrian knots in lens spaces,
|
arxiv-papers
| 2013-10-06T03:45:27 |
2024-09-04T02:49:51.997266
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Feifei Chen, Fan Ding, Youlin Li",
"submitter": "Youlin Li",
"url": "https://arxiv.org/abs/1310.1535"
}
|
1310.1536
|
# An Information-Spectrum Approach to the Capacity Region of the Interference
Channel
Authors Xiao Ma, Lei Lin, Chulong Liang, Xiujie Huang, and Baoming Bai This
paper was presented in part at the 2012 IEEE International Symposium on
Information Theory. This work was supported by the 973 Program
(No.2012CB316100) and the NSFC (No.61172082).X. Ma, and C. Liang are with the
Department of Electronic and Communication Engineering, Sun Yat-sen
University, Guangzhou 510006, Guangdong, China (email:
[email protected]).L. Lin is with the Department of Applied Mathematics,
Sun Yat-sen University, Guangzhou 510006, Guangdong, China (email:
[email protected]).X. Huang is with the Department of Electrical
Engineering, University of Hawaii, Honolulu 96822, HI, USA.B. Bai is with the
State Lab. of ISN, Xidian University, Xi’an 710071, Shaanxi, China.
###### Abstract
In this paper, we present a general formula for the capacity region of a
general interference channel with two pairs of users. The formula shows that
the capacity region is the union of a family of rectangles, where each
rectangle is determined by a pair of spectral inf-mutual information rates.
Although the presented formula is usually difficult to compute, it provides us
useful insights into the interference channels. In particular, when the inputs
are discrete ergodic Markov processes and the channel is stationary
memoryless, the formula can be evaluated by BCJR algorithm. Also the formula
suggests us that the simplest inner bounds (obtained by treating the
interference as noise) could be improved by taking into account the structure
of the interference processes. This is verified numerically by computing the
mutual information rates for Gaussian interference channels with embedded
convolutional codes. Moreover, we present a coding scheme to approach the
theoretical achievable rate pairs. Numerical results show that decoding gain
can be achieved by considering the structure of the interference.
###### Index Terms:
Capacity region, interference channel, information spectrum, limit
superior/inferior in probability, spectral inf-mutual information rate.
## I Introduction
The interference channel (IC) is a communication model with multiple pairs of
senders and receivers, in which each sender has an independent message
intended only for the corresponding receiver. This model was first mentioned
by Shannon [1] in 1961 and further studied by Ahlswede [2] in 1974. A basic
problem for the IC is to determine the capacity region, which is currently one
of long-standing open problems in information theory. Only in some special
cases, the capacity regions are known, such as strong interference channels,
very strong interference channels and deterministic interference channels [3,
4, 5, 6]. For a general IC, various inner and outer bounds of the capacity
region have been obtained. In 2004, Kramer derived two outer bounds on the
capacity region of the general Gaussian interference channel (GIFC) [7]. The
first bound for a general GIFC unifies and improves the outer bounds of Sato
[8] and Carleial [9]. The second bound follows directly from the outer bounds
of Sato [10] and Costa [11], which is derived by considering a degraded GIFC
and is even better than the first one for certain weak GIFCs. The best inner
bound (the so-called HK region) is that proposed by Han and Kobayashi [4],
which has been simplified by Chong et al. and Kramer in their independent
works [12] and [13]. In recent years, Etkin, Tse and Wang [14] showed by
introducing the idea of approximation that HK region [4] is within one bit of
the capacity region for the GIFC.
In [15], the authors proposed a new computational model for the two-user GIFC,
in which one pair of users (called primary users) are constrained to use a
fixed encoder and the other pair of users (called secondary users) are allowed
to optimize their code. The maximum rate at which the secondary users can
communicate reliably without degrading the performance of the primary users is
called the accessible capacity of the secondary users. Since the structure of
the interference from the primary link has been taken into account in the
computation, the accessible capacity is usually higher than the maximum rate
when treating the interference as noise, as is consistent with the spirit of
[16][17]. However, to compute the accessible capacity [15], the primary link
is allowed to have a non-neglected error probability. This makes the model
unattractive when the capacity region is considered. For this reason, we will
relax the fixed-code constraints on the primary users in this paper. In other
words, we will compute a pair of transmission rates at which both links can be
asymptotically error-free.
In this paper, we consider a more general interference channel which is
characterized by a sequence of transition probabilities. By the use of the
information spectrum approach [18][19], we present a general formula for the
capacity region of the general interference channel with two pairs of users.
The formula shows that the capacity region is the union of a family of
rectangles, in which each rectangle is determined by a pair of spectral inf-
mutual information rates. The information spectrum approach, which is based on
the _limit superior/inferior in probability_ of a sequence of random
variables, has been proved to be powerful in characterizing the limit behavior
of a general source/channel. For instance, in [18] and [20], Han and Verdú
proved that the minimum compression rate for a general source equals its
_spectral sup-entropy rate_ and the maximum transmission rate for a general
point-to-point channel equals its _spectral inf-mutual information rate_ with
an optimized input process. Also the information spectrum approach can be used
to derive the capacity region of a general multiple access channel [21]. For
more applications of the information spectrum approach, see [19] and the
references therein.
The rest of the paper is structured as follows. Sec. II introduces the
definition of a general IC and the concept of the spectral inf-mutual
information rate. In Sec. III-A, a general formula for the capacity region of
the general IC is proposed; while, in Sec. III-B, a trellis-based algorithm is
presented to compute the pair of rates for a stationary memoryless IC with
discrete ergodic Markov sources. In Sec. III-C, numerical results are
presented for a GIFC with binary-phase shift-keying (BPSK) modulation. Sec. IV
provides the detection and decoding algorithms for channels with structured
interference. Sec. V concludes this paper.
In this paper, a random variable is denoted by an upper-case letter, say $X$,
while its realization and sample space are denoted by $x$ and $\mathcal{X}$,
respectively. The sequence of random variables with length $n$ are denoted by
$X^{n}$, while its realization is denoted by ${\bf x}\in\mathcal{X}^{n}$ or
$x^{n}\in\mathcal{X}^{n}$. We use $P_{X}(x)$ to denote the probability mass
function (pmf) of $X$ if it is discrete or the probability density function
(pdf) of $X$ if it is continuous.
Figure 1: General interference channel ${\bf W}$.
## II Basic Definitions And Problem Statement
### II-A General IC
Let $\mathcal{X}_{1}$, $\mathcal{X}_{2}$ be two finite input alphabets and
$\mathcal{Y}_{1}$, $\mathcal{Y}_{2}$ be two finite output alphabets. A general
interference channel ${\bf W}$ (see Fig. 1) is characterized by a sequence
${\bf W}=\\{W^{n}(\cdot,\cdot|\cdot,\cdot)\\}_{n=1}^{\infty}$, where
$W^{n}:\mathcal{X}_{1}^{n}\times\mathcal{X}_{2}^{n}\rightarrow\mathcal{Y}_{1}^{n}\times\mathcal{Y}_{2}^{n}$
is a probability transition matrix. That is, for all $n$,
$\displaystyle W^{n}({\bf y}_{1},{\bf y}_{2}|{\bf x}_{1},{\bf x}_{2})$
$\displaystyle\geq$ $\displaystyle 0$ $\displaystyle\sum\limits_{{\bf
y}_{1}\in\mathcal{Y}_{1}^{n},{\bf y}_{2}\in\mathcal{Y}_{2}^{n}}W^{n}({\bf
y}_{1},{\bf y}_{2}|{\bf x}_{1},{\bf x}_{2})$ $\displaystyle=$ $\displaystyle
1.$
The marginal distributions $W_{1}^{n},W_{2}^{n}$ of the $W^{n}$ are given by
$\displaystyle W_{1}^{n}({\bf y}_{1}|{\bf x}_{1},{\bf x}_{2})$
$\displaystyle=$ $\displaystyle\sum_{{\bf
y}_{2}\in\mathcal{Y}_{2}^{n}}W^{n}({\bf y}_{1},{\bf y}_{2}|{\bf x}_{1},{\bf
x}_{2}),$ (1) $\displaystyle W_{2}^{n}({\bf y}_{2}|{\bf x}_{1},{\bf x}_{2})$
$\displaystyle=$ $\displaystyle\sum_{{\bf
y}_{1}\in\mathcal{Y}_{1}^{n}}W^{n}({\bf y}_{1},{\bf y}_{2}|{\bf x}_{1},{\bf
x}_{2}).$ (2)
###### Definition 1
An $(n,M_{n}^{(1)},M_{n}^{(2)},\varepsilon_{n}^{(1)},\varepsilon_{n}^{(2)})$
code for the interference channel ${\bf W}$ consists of the following
essentials:
* a)
message sets:
$\displaystyle\mathcal{M}_{n}^{(1)}=\\{1,2,\ldots,M_{n}^{(1)}\\},\,\,\,$
$\displaystyle{\rm for\,\,\,Sender~{}1}$
$\displaystyle\mathcal{M}_{n}^{(2)}=\\{1,2,\ldots,M_{n}^{(2)}\\},\,\,\,$
$\displaystyle{\rm for\,\,\,Sender~{}2}$
* b)
sets of codewords:
$\begin{array}[]{ll}\\{{\bf x}_{1}(1),{\bf x}_{1}(2),\ldots,{\bf
x}_{1}(M_{n}^{(1)})\\}\subseteq\mathcal{X}_{1}^{n},&{\rm
for\,\,\,Encoder~{}1}\\\ \\{{\bf x}_{2}(1),{\bf x}_{2}(2),\ldots,{\bf
x}_{2}(M_{n}^{(2)})\\}\subseteq\mathcal{X}_{2}^{n},&{\rm
for\,\,\,Encoder~{}2}\end{array}$
For Sender 1 to transmit message $i$, Encoder 1 outputs the codeword ${\bf
x}_{1}(i)$. Similarly, for Sender 2 to transmit message $j$, Encoder 2 outputs
the codeword ${\bf x}_{2}(j)$.
* c)
collections of decoding sets:
$\begin{array}[]{ll}\mathcal{B}_{1}=\\{\mathcal{B}_{1i}\subseteq\mathcal{Y}_{1}^{n}\\}_{i=1,...,M_{n}^{(1)}},&{\rm
for\,\,\,Decoder~{}1}\\\
\mathcal{B}_{2}=\\{\mathcal{B}_{2j}\subseteq\mathcal{Y}_{2}^{n}\\}_{j=1,...,M_{n}^{(2)}},&{\rm
for\,\,\,Decoder~{}2}\end{array}$
where
$\mathcal{Y}_{1}^{n}=\bigcup\limits_{i=1}^{M_{n}^{(1)}}\mathcal{B}_{1i},\,\,\mathcal{B}_{1i}\bigcap\mathcal{B}_{1i^{\prime}}=\emptyset$
for $i\neq i^{\prime}$ and
$\mathcal{Y}_{2}^{n}=\bigcup\limits_{j=1}^{M_{n}^{(2)}}\mathcal{B}_{2j},\,\,\mathcal{B}_{2j}\bigcap\mathcal{B}_{2j^{\prime}}=\emptyset$
for $j\neq j^{\prime}$. That is, $\mathcal{B}_{1}$ and $\mathcal{B}_{2}$ are
the disjoint partitions of $\mathcal{Y}_{1}^{n}$ and $\mathcal{Y}_{2}^{n}$
determined in advance, respectively. After receiving ${\bf y}_{1}$, Decoder 1
outputs $\hat{i}$ whenever ${\bf y}_{1}\in\mathcal{B}_{1\hat{i}}$. Similarly,
after receiving ${\bf y}_{2}$, Decoder 2 outputs $\hat{j}$ whenever ${\bf
y}_{2}\in\mathcal{B}_{2\hat{j}}$.
* d)
probabilities of decoding errors:
$\begin{array}[]{ll}&\varepsilon_{n}^{(1)}=\frac{1}{M_{n}^{(1)}M_{n}^{(2)}}\sum\limits_{i=1}^{M_{n}^{(1)}}\sum\limits_{j=1}^{M_{n}^{(2)}}W_{1}^{n}(\mathcal{B}^{c}_{1i}|{\bf
x}_{1}(i),{\bf x}_{2}(j)),\\\
&\varepsilon_{n}^{(2)}=\frac{1}{M_{n}^{(1)}M_{n}^{(2)}}\sum\limits_{i=1}^{M_{n}^{(1)}}\sum\limits_{j=1}^{M_{n}^{(2)}}W_{2}^{n}(\mathcal{B}^{c}_{2j}|{\bf
x}_{1}(i),{\bf x}_{2}(j)),\end{array}$
where $``c"$ denotes the complement of a set. Here we have assumed that each
message of $i\in\mathcal{M}_{n}^{(1)}$ and $j\in\mathcal{M}_{n}^{(2)}$ is
produced independently with uniform distribution.
Remark: The optimal decoding to minimize the probability of errors is defining
the decoding sets $\mathcal{B}_{1i}$ and $\mathcal{B}_{2j}$ according to the
the maximum likelihood decoding [22]. That is, the two receivers choose,
respectively,
$\hat{i}=\arg\max_{i}{\rm Pr}\\{{\bf y}_{1}|{\bf x}_{1}(i)\\}$
and
$\hat{j}=\arg\max_{j}{\rm Pr}\\{{\bf y}_{2}|{\bf x}_{2}(j)\\}$
as the estimates of the transmitted messages.
###### Definition 2
A rate pair $(R_{1},R_{2})$ is achievable if there exists a sequence of
$(n,M_{n}^{(1)},M_{n}^{(2)},\varepsilon_{n}^{(1)},\varepsilon_{n}^{(2)})$
codes such that
$\displaystyle\lim_{n\rightarrow\infty}\varepsilon_{n}^{(1)}=0$
$\displaystyle{\rm and}$
$\displaystyle\lim_{n\rightarrow\infty}\varepsilon_{n}^{(2)}=0,$
$\displaystyle\liminf_{n\rightarrow\infty}\frac{\log M_{n}^{(1)}}{n}\geq
R_{1}$ $\displaystyle{\rm and}$
$\displaystyle\liminf_{n\rightarrow\infty}\frac{\log M_{n}^{(2)}}{n}\geq
R_{2}.$
###### Definition 3
The set of all achievable rates is called the capacity region of the
interference channel ${\bf W}$, which is denoted by $\mathcal{C}({\bf W})$.
### II-B Preliminaries of Information-Spectrum Approach
The following notions can be found in [19].
###### Definition 4 (liminf in probability)
For a sequence of random variables $\\{Z^{n}\\}_{n=1}^{\infty}$,
$p\textrm{-}\liminf_{n\rightarrow\infty}Z^{n}\overset{\triangle}{=}\sup\\{\beta|\lim_{n\rightarrow\infty}{\rm
Pr}\\{Z^{n}<\beta\\}=0\\}.$
###### Definition 5
If two random variables sequences ${\bf
X}_{1}=\\{{X}_{1}^{n}\\}_{n=1}^{\infty}$ and ${\bf
X}_{2}=\\{{X}_{2}^{n}\\}_{n=1}^{\infty}$ satisfy that
$P_{{X}_{1}^{n}{X}_{2}^{n}}({\bf x}_{1},{\bf x}_{2})=P_{{X}_{1}^{n}}({\bf
x}_{1})P_{{X}_{2}^{n}}({\bf x}_{2})$ (3)
for all ${\bf x}_{1}\in\mathcal{X}_{1}^{n}$, ${\bf
x}_{2}\in\mathcal{X}_{2}^{n}$ and $n$, they are called independent and denoted
by ${\bf X}_{1}\bot{\bf X}_{2}$.
Similar to [18], we have
###### Definition 6
Let $S_{I}\stackrel{{\scriptstyle\triangle}}{{=}}\\{({\bf X}_{1},{\bf
X}_{2})|{\bf X}_{1}\bot{\bf X}_{2}\\}$. Given an $({\bf X}_{1},{\bf X}_{2})\in
S_{I}$, for the interference channel ${\bf W}$, we define the spectral inf-
mutual information rate by
$\displaystyle\underline{I}({\bf X}_{1};{\bf Y}_{1})$ $\displaystyle\equiv$
$\displaystyle
p\textrm{-}\liminf_{n\rightarrow\infty}\frac{1}{n}\log\frac{P_{Y_{1}^{n}|X_{1}^{n}}({Y}_{1}^{n}|{X}_{1}^{n})}{P_{{Y}_{1}^{n}}({Y}_{1}^{n})},$
(4) $\displaystyle\underline{I}({\bf X}_{2};{\bf Y}_{2})$
$\displaystyle\equiv$ $\displaystyle
p\textrm{-}\liminf_{n\rightarrow\infty}\frac{1}{n}\log\frac{P_{Y_{2}^{n}|X_{2}^{n}}({Y}_{2}^{n}|{X}_{2}^{n})}{P_{{Y}_{2}^{n}}({Y}_{2}^{n})},$
(5)
where
$\displaystyle P_{Y_{1}^{n}|X_{1}^{n}}({\bf y}_{1}|{\bf x}_{1})$
$\displaystyle=$ $\displaystyle\sum_{{\bf x}_{2},{\bf
y}_{2}}P_{X_{2}^{n}}({\bf x}_{2})W^{n}({\bf y}_{1},{\bf y}_{2}|{\bf
x}_{1},{\bf x}_{2}),$ (6) $\displaystyle P_{Y_{2}^{n}|X_{2}^{n}}({\bf
y}_{2}|{\bf x}_{2})$ $\displaystyle=$ $\displaystyle\sum_{{\bf x}_{1},{\bf
y}_{1}}P_{X_{1}^{n}}({\bf x}_{1})W^{n}({\bf y}_{1},{\bf y}_{2}|{\bf
x}_{1},{\bf x}_{2}).$ (7)
## III The Capacity Region of General IC
In this section, we derive a formula for the capacity region $\mathcal{C}({\bf
W})$ of the general IC.
### III-A The Main Theorem
###### Theorem 1
The capacity region $\mathcal{C}({\bf W})$ of the interference channel ${\bf
W}$ is given by
$\mathcal{C}({\bf W})=\bigcup_{({\bf X}_{1},{\bf X}_{2})\in
S_{I}}\mathcal{R}_{\bf W}({\bf X}_{1},{\bf X}_{2}),$ (8)
where $\mathcal{R}_{\bf W}({\bf X}_{1},{\bf X}_{2})$ is defined as the
collection of all $(R_{1},R_{2})$ satisfying that
$\displaystyle 0\leq R_{1}$ $\displaystyle\leq$
$\displaystyle\underline{I}({\bf X}_{1};{\bf Y}_{1}),$ (9) $\displaystyle
0\leq R_{2}$ $\displaystyle\leq$ $\displaystyle\underline{I}({\bf X}_{2};{\bf
Y}_{2}).$ (10)
To prove Theorem 1, we need the following lemmas.
###### Lemma 1
Let
$({\bf X}_{1}=\\{{X}_{1}^{n}\\}_{n=1}^{\infty},{\bf
X}_{2}=\\{{X}_{2}^{n}\\}_{n=1}^{\infty})$
be any channel input such that $({\bf X}_{1},{\bf X}_{2})\in S_{I}$. The
corresponding output via an interference channel ${\bf W}=\\{W^{n}\\}$ is
denoted by $({\bf Y}_{1}=\\{{Y}_{1}^{n}\\}_{n=1}^{\infty},{\bf
Y}_{2}=\\{{Y}_{2}^{n}\\}_{n=1}^{\infty})$. Then, for any fixed $M_{n}^{(1)}$
and $M_{n}^{(2)}$, there exists an
$(n,M_{n}^{(1)},M_{n}^{(2)},\varepsilon_{n}^{(1)},\varepsilon_{n}^{(2)})$ code
satisfying that
$\varepsilon_{n}^{(1)}+\varepsilon_{n}^{(2)}\leq{\rm
Pr}\\{T^{c}_{n}(1)\\}+{\rm Pr}\\{T^{c}_{n}(2)\\}+2e^{-n\gamma},$ (11)
where
$\begin{array}[]{l}T_{n}(1)=\\{({\bf x}_{1},{\bf
y}_{1})|\frac{1}{n}\log\frac{P_{Y_{1}^{n}|X_{1}^{n}}({\bf y}_{1}|{\bf
x}_{1})}{P_{{Y}_{1}^{n}}({\bf y}_{1})}>\frac{1}{n}\log
M_{n}^{(1)}+\gamma\\},\\\ T_{n}(2)=\\{({\bf x}_{2},{\bf
y}_{2})|\frac{1}{n}\log\frac{P_{Y_{2}^{n}|X_{2}^{n}}({\bf y}_{2}|{\bf
x}_{2})}{P_{{Y}_{2}^{n}}({\bf y}_{2})}>\frac{1}{n}\log
M_{n}^{(2)}+\gamma\\}\end{array}$
and $\gamma>0$ is an arbitrarily small number.
###### Proof:
The proof is similar to that of [18, Lemma 3].
Codebook generation. Generate $M_{n}^{(1)}$ independent codewords ${\bf
x}_{1}(1),...,{\bf x}_{1}(M_{n}^{(1)})\in\mathcal{X}_{1}^{n}$ subject to the
probability distribution $P_{X_{1}^{n}}$. Similarly, generate $M_{n}^{(2)}$
independent codewords ${\bf x}_{2}(1),...,{\bf
x}_{2}(M_{n}^{(2)})\in\mathcal{X}_{2}^{n}$ subject to the probability
distribution $P_{X_{2}^{n}}$.
Encoding. To send message $i$, Sender 1 sends the codeword ${\bf x}_{1}(i)$.
Similarly, to send message $j$, Sender 2 sends ${\bf x}_{2}(j)$.
Decoding. Receiver 1 chooses the $i$ such that $({\bf x}_{1}(i),{\bf
y}_{1})\in T_{n}(1)$ if such $i$ exists and is unique. Similarly, Receiver 2
chooses the $j$ such that $({\bf x}_{2}(j),{\bf y}_{2})\in T_{n}(2)$ if such
$j$ exists and is unique. Otherwise, an error is declared.
Analysis of the error probability. By the symmetry of the random code
construction, we can assume that $(1,1)$ was sent. Define
$E_{1i}=\\{({\bf x}_{1}(i),{\bf y}_{1})\in T_{n}(1)\\},\,\,E_{2j}=\\{({\bf
x}_{2}(j),{\bf y}_{2})\in T_{n}(2)\\}.$
For Receiver 1, an error occurs if $({\bf x}_{1}(1),{\bf y}_{1})\notin
T_{n}(1)$ or $({\bf x}_{1}(i),{\bf y}_{1})\in T_{n}(1)$ for some $i\neq 1$.
Similarly, for Receiver 2, an error occurs if $({\bf x}_{2}(1),{\bf
y}_{2})\notin T_{n}(2)$ or $({\bf x}_{2}(j),{\bf y}_{2})\in T_{n}(2)$ for some
$j\neq 1$. So the ensemble average of the error probabilities of Decoder 1 and
Decoder 2 can be upper-bounded as follows:
$\begin{array}[]{l}\overline{{\varepsilon}_{n}^{(1)}+{\varepsilon}_{n}^{(2)}}=\overline{\varepsilon_{n}^{(1)}}+\overline{\varepsilon_{n}^{(2)}}\\\
\leq{\rm Pr}\\{E_{11}^{c}\\}+{\rm Pr}\\{\bigcup\limits_{i\neq 1}E_{1i}\\}+{\rm
Pr}\\{E_{21}^{c}\\}+{\rm Pr}\\{\bigcup\limits_{j\neq 1}E_{2j}\\}.\end{array}$
It can be seen that
$\begin{array}[]{ll}&{\rm Pr}\\{\bigcup\limits_{i\neq
1}E_{1i}\\}\leq\sum\limits_{i\neq 1}{\rm Pr}\\{E_{1i}\\}\\\
&=\sum\limits_{i\neq 1}{\rm Pr}\\{({\bf x}_{1}(i),{\bf y}_{1})\in
T_{n}(1)\\}\\\ &\stackrel{{\scriptstyle(a)}}{{=}}\sum\limits_{i\neq
1}\sum\limits_{({\bf x}_{1},{\bf y}_{1})\in T_{n}(1)}P_{X_{1}^{n}}({\bf
x}_{1})P_{Y_{1}^{n}}({\bf y}_{1})\\\
&\stackrel{{\scriptstyle(b)}}{{\leq}}\sum\limits_{i\neq 1}\sum\limits_{({\bf
x}_{1},{\bf y}_{1})\in T_{n}(1)}P_{X_{1}^{n}}({\bf
x}_{1})P_{Y_{1}^{n}|X_{1}^{n}}({\bf y}_{1}|{\bf
x}_{1})\frac{e^{-n\gamma}}{M_{n}^{(1)}}\\\ &\leq\sum\limits_{i\neq
1}\frac{e^{-n\gamma}}{M_{n}^{(1)}}=(M_{n}^{(1)}-1)\frac{e^{-n\gamma}}{M_{n}^{(1)}}\leq
e^{-n\gamma},\end{array}$
where $(a)$ follows from the independence of ${\bf x}_{1}(i)~{}(i\neq 1)$ and
${\bf y}_{1}$ and $(b)$ follows from the definition of $T_{n}(1)$. Similarly,
we obtain
${\rm Pr}\\{\bigcup\limits_{j\neq 1}E_{2j}\\}\leq e^{-n\gamma}.$ (12)
Combining all inequalities above, we can see that there must exist at least
one $(n,M_{n}^{(1)},M_{n}^{(2)},\varepsilon_{n}^{(1)},\varepsilon_{n}^{(2)})$
code satisfying (11). ∎
###### Lemma 2
For all $n$, any
$(n,M_{n}^{(1)},M_{n}^{(2)},\varepsilon_{n}^{(1)},\varepsilon_{n}^{(2)})$ code
satisfies that
$\begin{array}[]{l}\varepsilon_{n}^{(1)}\geq{\rm
Pr}\\{\frac{1}{n}\log\frac{P_{Y_{1}^{n}|X_{1}^{n}}({Y}_{1}^{n}|{X}_{1}^{n})}{P_{{Y}_{1}^{n}}({Y}_{1}^{n})}\leq\frac{1}{n}\log
M_{n}^{(1)}-\gamma\\}-e^{-n\gamma},\\\ \varepsilon_{n}^{(2)}\geq{\rm
Pr}\\{\frac{1}{n}\log\frac{P_{Y_{2}^{n}|X_{2}^{n}}({Y}_{2}^{n}|{X}_{2}^{n})}{P_{{Y}_{2}^{n}}({Y}_{2}^{n})}\leq\frac{1}{n}\log
M_{n}^{(2)}-\gamma\\}-e^{-n\gamma},\end{array}$ (13)
for every $\gamma>0$, where $X_{1}^{n}~{}({\rm resp.,}\,X_{2}^{n})$ places
probability mass $1/M_{n}^{(1)}~{}({\rm resp.,}\,1/M_{n}^{(2)})$ on each
codeword for Encoder 1 (resp., Encoder 2) and (3), (6), (7) hold.
###### Proof:
The proof is similar to that of [18, Lemma 4]. By using the relation
$\frac{P_{Y_{1}^{n}|X_{1}^{n}}({\bf y}_{1}|{\bf x}_{1})}{P_{{Y}_{1}^{n}}({\bf
y}_{1})}=\frac{P_{X_{1}^{n}|Y_{1}^{n}}({\bf x}_{1}|{\bf
y}_{1})}{P_{{X}_{1}^{n}}({\bf x}_{1})}$
and noticing that $P_{{X}_{1}^{n}}({\bf x}_{1})=\frac{1}{M_{n}^{(1)}}$, we can
rewrite the first term on the right-hand side of the first inequality of (13)
as
${\rm Pr}\\{P_{X_{1}^{n}|Y_{1}^{n}}(X_{1}^{n}|Y_{1}^{n})\leq e^{-n\gamma}\\}.$
By setting
$L_{n}=\\{({\bf x}_{1},{\bf y}_{1})|P_{X_{1}^{n}|Y_{1}^{n}}({\bf x}_{1}|{\bf
y}_{1})\leq e^{-n\gamma}\\},$
the first inequality of (13) can be expressed as
${\rm Pr}\\{L_{n}\\}\leq\varepsilon_{n}^{(1)}+e^{-n\gamma}.$ (14)
In order to prove this inequality, we set
$\mathcal{A}_{i}=\\{{\bf
y}_{1}\in\mathcal{Y}_{1}^{n}|P_{X_{1}^{n}|Y_{1}^{n}}({\bf x}_{1}(i)|{\bf
y}_{1})\leq e^{-n\gamma}\\}.$
It can be seen that
$\begin{array}[]{l}{\rm
Pr}\\{L_{n}\\}=\sum\limits_{i=1}^{M_{n}^{(1)}}P_{X_{1}^{n}Y_{1}^{n}}({\bf
x}_{1}(i),\mathcal{A}_{i})\\\
=\sum\limits_{i=1}^{M_{n}^{(1)}}P_{X_{1}^{n}Y_{1}^{n}}({\bf
x}_{1}(i),\mathcal{A}_{i}\bigcap\mathcal{B}_{1i})+\sum\limits_{i=1}^{M_{n}^{(1)}}P_{X_{1}^{n}Y_{1}^{n}}({\bf
x}_{1}(i),\mathcal{A}_{i}\bigcap\mathcal{B}_{1i}^{c})\\\
\leq\sum\limits_{i=1}^{M_{n}^{(1)}}P_{X_{1}^{n}Y_{1}^{n}}({\bf
x}_{1}(i),\mathcal{A}_{i}\bigcap\mathcal{B}_{1i})+\sum\limits_{i=1}^{M_{n}^{(1)}}P_{X_{1}^{n}Y_{1}^{n}}({\bf
x}_{1}(i),\mathcal{B}_{1i}^{c})\\\
=\sum\limits_{i=1}^{M_{n}^{(1)}}\sum\limits_{{\bf
y}_{1}\in\mathcal{A}_{i}\bigcap\mathcal{B}_{1i}}P_{X_{1}^{n}Y_{1}^{n}}({\bf
x}_{1}(i),{\bf y}_{1})+\varepsilon_{n}^{(1)}\\\
=\sum\limits_{i=1}^{M_{n}^{(1)}}\sum\limits_{{\bf
y}_{1}\in\mathcal{A}_{i}\bigcap\mathcal{B}_{1i}}P_{X_{1}^{n}|Y_{1}^{n}}({\bf
x}_{1}(i)|{\bf y}_{1})P_{Y_{1}^{n}}({\bf y}_{1})+\varepsilon_{n}^{(1)}\\\
\stackrel{{\scriptstyle(a)}}{{\leq}}e^{-n\gamma}\sum\limits_{i=1}^{M_{n}^{(1)}}\sum\limits_{{\bf
y}_{1}\in\mathcal{B}_{1i}}P_{Y_{1}^{n}}({\bf y}_{1})+\varepsilon_{n}^{(1)}\\\
=e^{-n\gamma}P_{Y_{1}^{n}}(\bigcup\limits_{i=1}^{M_{n}^{(1)}}\mathcal{B}_{1i})+\varepsilon_{n}^{(1)}\leq
e^{-n\gamma}+\varepsilon_{n}^{(1)},\end{array}$
where $\mathcal{B}_{1i}$ is the decoding region corresponding to codeword
${\bf x}_{1}(i)$ and $(a)$ follows from the definition of $\mathcal{A}_{i}$.
Therefore, the first inequality of (13) is proved. Similarly, we can obtain
the second inequality of (13). ∎
Now we prove Theorem 1.
###### Proof:
1) To prove that an arbitrary rate pair $(R_{1},R_{2})$ satisfying (9) and
(10) is achievable, we define
$M_{n}^{(1)}=e^{n(R_{1}-2\gamma)}\,\,\,{\rm
and}\,\,\,M_{n}^{(2)}=e^{n(R_{2}-2\gamma)}$
for an arbitrarily small constant $\gamma>0$. Lemma 1 guarantees the existence
of an
$(n,M_{n}^{(1)},M_{n}^{(2)},\varepsilon_{n}^{(1)},\varepsilon_{n}^{(2)})$ code
satisfying
$\begin{array}[]{ll}\varepsilon_{n}^{(1)}+\varepsilon_{n}^{(2)}&\leq{\rm
Pr}\\{\frac{1}{n}\log\frac{P_{Y_{1}^{n}|X_{1}^{n}}({Y}_{1}^{n}|{X}_{1}^{n})}{P_{{Y}_{1}^{n}}({Y}_{1}^{n})}\leq
R_{1}-\gamma\\}\\\ &\,\,\,\,\,\,+{\rm
Pr}\\{\frac{1}{n}\log\frac{P_{Y_{2}^{n}|X_{2}^{n}}({Y}_{2}^{n}|{X}_{2}^{n})}{P_{{Y}_{2}^{n}}({Y}_{2}^{n})}\leq
R_{2}-\gamma\\}+2e^{-n\gamma}\\\ &\leq{\rm
Pr}\\{\frac{1}{n}\log\frac{P_{Y_{1}^{n}|X_{1}^{n}}({Y}_{1}^{n}|{X}_{1}^{n})}{P_{{Y}_{1}^{n}}({Y}_{1}^{n})}\leq\underline{I}({\bf
X}_{1};{\bf Y}_{1})-\gamma\\}\\\ &\,\,\,\,\,\,+{\rm
Pr}\\{\frac{1}{n}\log\frac{P_{Y_{2}^{n}|X_{2}^{n}}({Y}_{2}^{n}|{X}_{2}^{n})}{P_{{Y}_{2}^{n}}({Y}_{2}^{n})}\leq\underline{I}({\bf
X}_{2};{\bf Y}_{2})-\gamma\\}+2e^{-n\gamma}.\end{array}$
From the definition of the spectral inf-mutual information rate, we have
$\lim_{n\rightarrow\infty}\varepsilon_{n}^{(1)}=0\,\,\,{\rm
and}\,\,\,\lim_{n\rightarrow\infty}\varepsilon_{n}^{(2)}=0.$
2) Suppose that a rate pair $(R_{1},R_{2})$ is achievable. Then, for any
constant $\gamma>0$, there exists an
$(n,M_{n}^{(1)},M_{n}^{(2)},\varepsilon_{n}^{(1)},\varepsilon_{n}^{(2)})$ code
satisfying
$\frac{\log M_{n}^{(1)}}{n}\geq R_{1}-\gamma\,\,\,{\rm and}\,\,\,\frac{\log
M_{n}^{(2)}}{n}\geq R_{2}-\gamma$ (15)
for all sufficiently large $n$ and
$\lim_{n\rightarrow\infty}\varepsilon_{n}^{(1)}=0\,\,\,{\rm
and}\,\,\,\lim_{n\rightarrow\infty}\varepsilon_{n}^{(2)}=0.$
From Lemma 2, we get
$\begin{array}[]{l}\varepsilon_{n}^{(1)}\geq{\rm
Pr}\\{\frac{1}{n}\log\frac{P_{Y_{1}^{n}|X_{1}^{n}}({Y}_{1}^{n}|{X}_{1}^{n})}{P_{{Y}_{1}^{n}}({Y}_{1}^{n})}\leq
R_{1}-2\gamma\\}-e^{-n\gamma}\\\ \varepsilon_{n}^{(2)}\geq{\rm
Pr}\\{\frac{1}{n}\log\frac{P_{Y_{2}^{n}|X_{2}^{n}}({Y}_{2}^{n}|{X}_{2}^{n})}{P_{{Y}_{2}^{n}}({Y}_{2}^{n})}\leq
R_{2}-2\gamma\\}-e^{-n\gamma}\end{array}.$ (16)
Taking the limits as $n\rightarrow\infty$ on both sides, we have
$\begin{array}[]{l}\lim\limits_{n\rightarrow\infty}{\rm
Pr}\\{\frac{1}{n}\log\frac{P_{Y_{1}^{n}|X_{1}^{n}}({Y}_{1}^{n}|{X}_{1}^{n})}{P_{{Y}_{1}^{n}}({Y}_{1}^{n})}\leq
R_{1}-2\gamma\\}=0\\\ \lim\limits_{n\rightarrow\infty}{\rm
Pr}\\{\frac{1}{n}\log\frac{P_{Y_{2}^{n}|X_{2}^{n}}({Y}_{2}^{n}|{X}_{2}^{n})}{P_{{Y}_{2}^{n}}({Y}_{2}^{n})}\leq
R_{2}-2\gamma\\}=0\end{array}.$ (17)
From the definitions of $\underline{I}({\bf X}_{1};{\bf Y}_{1})$ and
$\underline{I}({\bf X}_{2};{\bf Y}_{2})$, we can see that
$R_{1}-2\gamma\leq\underline{I}({\bf X}_{1};{\bf Y}_{1})$ and
$R_{2}-2\gamma\leq\underline{I}({\bf X}_{2};{\bf Y}_{2})$, which completes the
proof since $\gamma$ is arbitrary.
∎
### III-B The Algorithm to Compute Achievable Rate Pairs
Theorem 1 provides a general formula for the capacity region of a general IC.
However, it is usually difficult to compute the spectral inf-mutual
information rates given in (9) and (10). In order to get insights into the
interference channels, we make the following assumptions:
* 1)
the channel is stationary and memoryless, that is, the transition probability
of the channel can be written as
$W^{n}({\bf y}_{1},{\bf y}_{2}|{\bf x}_{1},{\bf
x}_{2})=\prod_{i=1}^{n}W(y_{1,i},y_{2,i}|x_{1,i},x_{2,i});$
* 2)
sources are restricted to be stationary and ergodic discrete Markov processes.
With the above assumptions, the spectral inf-mutual information rates are
reduced as
$\displaystyle\underline{I}({\bf X}_{1};{\bf Y}_{1})$ $\displaystyle=$
$\displaystyle\lim_{n\rightarrow\infty}\frac{1}{n}I({X}_{1}^{n};{Y}_{1}^{n}),$
(18) $\displaystyle\underline{I}({\bf X}_{2};{\bf Y}_{2})$ $\displaystyle=$
$\displaystyle\lim_{n\rightarrow\infty}\frac{1}{n}I({X}_{2}^{n};{Y}_{2}^{n}),$
(19)
which can be evaluated by the Monte Carlo method [23][24][25] using BCJR
algorithm [26] over a trellis. Actually, any stationary and ergodic discrete
Markov source can be represented by a time-invariant trellis and (hence) is
uniquely specified by a trellis section. A trellis section is composed of left
(or starting) states and right (or ending) states, which are connected by
branches in between. For example, Source ${\bf x}_{1}$ can be specified by a
trellis $\mathcal{T}_{1}$ as follows.
* •
Both the left and right states are selected from the set
$\mathcal{S}_{1}=\\{0,1,...,|\mathcal{S}_{1}|-1\\}$;
* •
Each branch is represented by a three-tuple
$b=(s_{1}^{-}(b),x_{1}(b),s_{1}^{+}(b))$, where $s_{1}^{-}(b)$ is the left
state, $s_{1}^{+}(b)$ is the right state, and the symbol
$x_{1}(b)\in\mathcal{X}_{1}$ is the associated label. We also assume that a
branch $b$ is uniquely determined by $s_{1}^{-}(b)$ and $x_{1}(b)$;
* •
At time $t=0$, the source starts from state $s_{1,0}\in\mathcal{S}_{1}$. If at
time $t-1~{}(t>0)$, the source is in the state $s_{1,t-1}\in\mathcal{S}_{1}$,
then at time $t~{}(t>0)$, the source generates a symbol
$x_{1,t}\in\mathcal{X}_{1}$ according to the conditional probability
$P(x_{1,t}|s_{1,t-1})$ and goes into a state $s_{1,t}\in\mathcal{S}_{1}$ such
that $(s_{1,t-1},x_{1,t},s_{1,t})$ is a branch. Obviously, when the source
runs from time $t=0$ to $t=n$, a sequence $x_{1,1},x_{1,2},...,x_{1,n}$ is
generated. The Markov property says that
$P(x_{1,t}|x_{1,1},...,x_{1,{t-1}},s_{1,0})=P(x_{1,t}|s_{1,t-1}).$
So the probability of a given sequence $x_{1,1},x_{1,2},...,x_{1,n}$ with the
initial state $s_{1,0}$ can be factored as
$P(x_{1,1},x_{1,2},...,x_{1,n}|s_{1,0})=\prod\limits_{t=1}^{n}P(x_{1,t}|s_{1,t-1}).$
Similarly, we can represent ${\bf x}_{2}$ by a trellis $\mathcal{T}_{2}$ with
the state set $\mathcal{S}_{2}=\\{0,1,...,|\mathcal{S}_{2}|-1\\}$. Each branch
is denoted by $b=(s_{2}^{-}(b),x_{2}(b),s_{2}^{+}(b))$, where $s_{2}^{-}(b)$
is the left state, $s_{2}^{+}(b)$ is the right state and the symbol
$x_{2}(b)\in\mathcal{X}_{2}$ is the associated label. Assume that source ${\bf
x}_{2}$ starts from the state $s_{2,0}\in\mathcal{S}_{2}$. If at time
$t-1~{}(t>0)$, the source is in the state $s_{2,t-1}\in\mathcal{S}_{2}$, then
at time $t~{}(t>0)$, the source generates a symbol $x_{2,t}\in\mathcal{X}_{2}$
according to the conditional probability $P(x_{2,t}|s_{2,t-1})$ and goes into
a state $s_{2,t}\in\mathcal{S}_{2}$ such that $(s_{2,t-1},x_{2,t},s_{2,t})$ is
a branch. The probability of a given sequence $x_{2,1},x_{2,2},...,x_{2,n}$
can be factored as
$P(x_{2,1},x_{2,2},...,x_{2,n}|s_{2,0})=\prod\limits_{t=1}^{n}P(x_{2,t}|s_{2,t-1}).$
In what follows, we have fixed the initial states as $s_{1,0}=0$ and
$s_{2,0}=0$, and removed them from the equations for simplicity.
Next we focus on the evaluation of
$\lim\limits_{n\rightarrow\infty}\frac{1}{n}I({X}_{1}^{n};{Y}_{1}^{n})$, while
$\lim\limits_{n\rightarrow\infty}\frac{1}{n}I({X}_{2}^{n};{Y}_{2}^{n})$ can be
estimated similarly. Specifically, we can express the limit as
$\lim_{n\rightarrow\infty}\frac{1}{n}I({X}_{1}^{n};{Y}_{1}^{n})=\lim_{n\rightarrow\infty}\frac{1}{n}H(Y_{1}^{n})-\lim_{n\rightarrow\infty}\frac{1}{n}H(Y_{1}^{n}|X_{1}^{n}),$
(20)
where $\lim\limits_{n\rightarrow\infty}\frac{1}{n}H(Y_{1}^{n})$ and
$\lim\limits_{n\rightarrow\infty}\frac{1}{n}H(Y_{1}^{n}|X_{1}^{n})$ can be
estimated by similar methods111For continuous ${\bf y}_{1}$, the computations
of $\lim\limits_{n\rightarrow\infty}\frac{1}{n}h(Y_{1}^{n})$ and
$\lim\limits_{n\rightarrow\infty}\frac{1}{n}h(Y_{1}^{n}|X_{1}^{n})$ can be
implemented by substituting pdf for pmf.. As an example, we show how to
compute $\lim\limits_{n\rightarrow\infty}\frac{1}{n}H(Y_{1}^{n})$. According
to the Shannon-McMillan-Breiman theorem [27], it can be seen that, with
probability 1,
$\lim_{n\rightarrow\infty}-\frac{1}{n}\log
P(y_{1}^{n})=\lim_{n\rightarrow\infty}\frac{1}{n}H(Y_{1}^{n}),$
where $y_{1}^{n}$ stands for $(y_{1,1},y_{1,2},...,y_{1,n})$. Then evaluating
$\lim\limits_{n\rightarrow\infty}\frac{1}{n}H(Y_{1}^{n})$ is converted to
computing
$\lim_{n\rightarrow\infty}-\frac{1}{n}\log
P(y_{1}^{n})\approx-\frac{1}{n}\log\left(\prod_{t=1}^{n}P(y_{1,t}|y_{1}^{t-1})\right)=-\frac{1}{n}\sum_{t=1}^{n}\log
P(y_{1,t}|y_{1}^{t-1})$
for a sufficiently long typical sequence $y_{1}^{n}$. Here, the key is to
compute the conditional probabilities $P(y_{1,t}|y_{1}^{t-1})$ for all $t$.
Since both ${\bf y}_{1}$ and ${\bf y}_{2}$ are hidden Markov sequences, this
can be done by performing the BCJR algorithm over the following product
trellis.
* •
The product trellis has the state set
$\mathcal{S}=\mathcal{S}_{1}\times\mathcal{S}_{2}$, where “$\times$” denotes
Cartesian product.
* •
Each branch is represented by a four-tuple
$b=(s^{-}(b),x_{1}(b),x_{2}(b),s^{+}(b))$, where
$s^{-}(b)=(s_{1}^{-}(b),s_{2}^{-}(b))$ is the left state,
$s^{+}(b)=(s_{1}^{+}(b),s_{2}^{+}(b))$ is the right state. Then
$x_{1}(b)\in\mathcal{X}_{1}$ and $x_{2}(b)\in\mathcal{X}_{2}$ are the
associated labels in branch $b$ such that
$(s_{1}^{-}(b),x_{1}(b),s_{1}^{+}(b))$ and
$(s_{2}^{-}(b),x_{2}(b),s_{2}^{+}(b))$ are branches in $\mathcal{T}_{1}$ and
$\mathcal{T}_{2}$, respectively.
* •
At time $t=0$, the sources start from state
$s_{0}=(s_{1,0},s_{2,0})\in\mathcal{S}$. If at time $t-1~{}(t>0)$, the sources
are in the state $s_{t-1}=(s_{1,t-1},s_{2,t-1})\in\mathcal{S}$, then at time
$t~{}(t>0)$, the sources generate symbols
$(x_{1,t}\in\mathcal{X}_{1},x_{2,t}\in\mathcal{X}_{2})$ according to the
conditional probability $P(x_{1,t}|s_{1,t-1})P(x_{2,t}|s_{2,t-1})$ and go into
a state $s_{t}=(s_{1,t},s_{2,t})\in\mathcal{S}_{2}$ such that
$(s_{t-1},x_{1,t},x_{2,t},s_{t})$ is a branch.
Given the received sequence ${\bf y}_{1}$, we define
* •
Branch metrics: To each branch $b_{t}=\\{s_{t-1},x_{1,t},x_{2,t},s_{t}\\}$, we
assign a metric
$\displaystyle\rho(b_{t})$ $\displaystyle\overset{\triangle}{=}$
$\displaystyle P(b_{t}|s_{t-1})P(y_{1,t}|x_{1,t}x_{2,t})$ (21)
$\displaystyle=$ $\displaystyle
P(x_{1,t}|s_{1,t-1})P(x_{2,t}|s_{2,t-1})P(y_{1,t}|x_{1,t}x_{2,t}),$ (22)
In the computation of
$\lim\limits_{n\rightarrow\infty}\frac{1}{n}H(Y_{1}^{n}|X_{1}^{n})$, the
metric is replaced by $P(b_{t}|s_{t-1},x_{1,t})P(y_{1,t}|x_{1,t}x_{2,t})$.
* •
State transition probabilities: The transition probability from $s_{t-1}$ to
$s_{t}$ is defined as
$\displaystyle\gamma_{t}(s_{t-1},s_{t})$ $\displaystyle\overset{\triangle}{=}$
$\displaystyle P(s_{t},y_{1,t}|s_{t-1})$ (23) $\displaystyle=$
$\displaystyle\sum_{b_{t}:s^{-}(b_{t})=s_{t-1},s^{+}(b_{t})=s_{t}}\rho(b_{t}).$
(24)
* •
Forward recursion variables: We define the _a posteriori_ probabilities
$\alpha_{t}(s_{t})\overset{\triangle}{=}P(s_{t}|y_{1}^{t}),\,\,\,t=0,1,...n.$
(25)
Then
$P(y_{1,t}|y_{1}^{t-1})=\sum_{s_{t-1},s_{t}}\alpha(s_{t-1})\gamma_{t}(s_{t-1},s_{t}),$
(26)
where the values of $\alpha_{t}(s_{t})$ can be computed recursively by
$\alpha_{t}(s_{t})=\frac{\sum_{s_{t-1}}\alpha_{t-1}(s_{t-1})\gamma_{t}(s_{t-1},s_{t})}{\sum_{s_{t-1},s_{t}}\alpha_{t-1}(s_{t-1})\gamma_{t}(s_{t-1},s_{t})}.$
(27)
In summary, the algorithm to estimate the entropy rate
$\lim\limits_{n\rightarrow\infty}\frac{1}{n}H(Y_{1}^{n})$ is described as
follows.
###### Algorithm 1
1. 1.
Initializations: Choose a sufficiently large number $n$. Set the initial state
of the trellis to be $s_{0}=0$. The forward recursion variables are
initialized as $\alpha_{0}(s)=1$ if $s=0$ and otherwise $\alpha_{0}(s)=0$.
2. 2.
Simulations for Sender 1: Generate a Markov sequence ${\bf
x}_{1}=(x_{1,1},x_{1,2},...,x_{1,n})$ according to the trellis
$\mathcal{T}_{1}$ of source ${\bf x}_{1}$.
3. 3.
Simulations for Sender 2: Generate a Markov sequence ${\bf
x}_{2}=(x_{2,1},x_{2,2},...,x_{2,n})$ according to the trellis
$\mathcal{T}_{2}$ of source ${\bf x}_{2}$.
4. 4.
Simulations for Receiver 1: Generate the received sequence ${\bf y}_{1}$
according to the transition probability $W^{n}({\bf y}_{1},{\bf y}_{2}|{\bf
x}_{1},{\bf x}_{2})$.
5. 5.
Computations:
* a)
For $t=1,2,...,n$ , compute the values of $P(y_{1,t}|y_{1}^{t-1})$ and
$\alpha_{t}(s_{t})$ recursively according to equations (26) and (27).
* b)
Evaluate the entropy rate
$\lim_{n\rightarrow\infty}\frac{1}{n}H(Y_{1}^{n})=-\frac{1}{n}\sum_{t=1}^{n}\log
P(y_{1,t}|y_{1}^{t-1}).$
Similarly, we can evaluate the entropy rate
$\lim\limits_{n\rightarrow\infty}\frac{1}{n}H(Y_{1}^{n}|X_{1}^{n})$.
Therefore, we obtain the achievable rate $\underline{I}({\bf X}_{1};{\bf
Y}_{1})=\lim\limits_{n\rightarrow\infty}\frac{1}{n}I({X}_{1}^{n};{Y}_{1}^{n})$.
### III-C Numerical Results
Figure 2: Symmetric Gaussian interference channel.
We consider the model as shown in Fig. 2, where the channel inputs ${\bf
x}_{1}(i)$ and ${\bf x}_{2}(j)$ are BPSK sequences with power constraints
$P_{1}$ and $P_{2}$, respectively; the additive noise ${\bf n}_{1}$ and ${\bf
n}_{2}$ are sequences of independent and identically distributed (i.i.d.)
standard Gaussian random variables; the channel outputs ${\bf y}_{1}$ and
${\bf y}_{2}$ are
$\displaystyle{\bf y}_{1}$ $\displaystyle=$ $\displaystyle{\bf
x}_{1}(i)+\sqrt{a}{\bf x}_{2}(j)+{\bf n}_{1},$ (28) $\displaystyle{\bf y}_{2}$
$\displaystyle=$ $\displaystyle{\bf x}_{2}(j)+\sqrt{a}{\bf x}_{1}(i)+{\bf
n}_{2}.$ (29)
We assume that ${\bf x}_{1}$ and ${\bf x}_{2}$ are the outputs from two
(possibly different) generalized trellis encoders driven by independent and
uniformly distributed (i.u.d.) input sequences, as proposed in [15]. As
examples, we consider two input processes. One is referred to as “UnBPSK”,
standing for an i.u.d. BPSK sequence; the other is referred to as “CcBPSK”,
standing for an output sequence from the convolutional encoder with the
generator matrix $G(D)=[1+D+D^{2}\,\,\,1+D^{2}]$ driven by an i.u.d. input
sequence.
Fig. 3 shows the trellis representation of the signal model when Sender 1 uses
CcBPSK and Sender 2 uses UnBPSK. Fig. 4 shows the numerical results. There are
three rectangles, OECH, ODBG and OFAI, each of which is determined by a pair
of spectral inf-mutual information rates. Specifically, the rectangle OECH
corresponds to the case when both senders use UnBPSK as inputs; the rectangle
ODBG corresponds to the case when Sender 1 uses UnBPSK as input and Sender 2
uses CcBPSK as input; and the rectangle OFAI corresponds to the case when
Sender 1 uses CcBPSK as input and Sender 2 uses UnBPSK as input. The point “A”
can be achieved by a coding scheme, in which Sender 1 uses a binary linear
(coset) code concatenated with the convolutional code and Sender 2 uses a
binary linear code, and the point “B” can be achieved similarly; while the
points on the line “AB” can be achieved by time-sharing scheme. The point “C”
represents the limits when the two senders use binary linear codes but regard
the interference as an i.u.d. additive (BPSK) noise. It can be seen that the
area of the pentagonal region ODBAI is greater than that of the rectangle
OECH, which implies that knowing the structure of the interference can be used
to improve potentially the bandwidth-efficiency.
Figure 3: The trellis section of (CcBPSK, UnBPSK) with 32 branches. For each
branch $b$, $s^{-}(b)$ and $s^{+}(b)$ are the left state and the right state,
respectively; while the associated symbols $x_{1}(b)$ and $x_{2}(b)$ are the
transmitted signals at Sender 1 and Sender 2, respectively.
Figure 4: The evaluated achievable rate pairs of a specific GIFC, where
$P_{1}=P_{2}=7.0~{}{\rm dB}$ and $a=0.5$. The rectangle OECH with legend
“(UnBPSK, UnBPSK)” corresponds to the case when both senders use UnBPSK as
inputs; the rectangle ODBG with legend “(UnBPSK, CcBPSK)” corresponds to the
case when Sender 1 uses UnBPSK as input and Sender 2 uses CcBPSK as input; and
the rectangle OFAI with legend “(CcBPSK, UnBPSK)” corresponds to the case when
Sender 1 uses CcBPSK as input and Sender 2 uses UnBPSK as input.
## IV Decoding Algorithms for Channels with Structured Interference
The purpose of this section has two-folds. The first is to present a coding
scheme to approach the point “B” in Fig. 4. The second is to show the decoding
gain achieved by taking into account the structure of the interference.
### IV-A A Coding Scheme
We design a coding scheme using Kite codes222The main reason that we choose
Kite codes is that it is convenient to set up the code rates. Actually, given
data length, the code rates of Kite codes can be “continuously” varying from
$0.1$ to $0.9$ with satisfactory performance, as shown in [28] [29].. Kite
codes are a class of low-density parity-check (LDPC) codes, which can be
decoded using the sum-product algorithm (SPA) [30, 31]. As shown in Fig. 5,
Sender 1 uses a Kite code (with a parity-check matrix ${\bf H}_{1}$) and
Sender 2 uses a Kite code (with a parity-check matrix ${\bf H}_{2}$)
concatenated with the convolutional code with the generator matrix
$G(D)=[1+D+D^{2}\,\,\,1+D^{2}].$
Figure 5: A coding scheme for the two-user GIFC.
_Encoding_ : For Sender 1, a binary sequence ${\bf
u}_{1}=(u_{1,1},u_{1,2},...,u_{1,L_{1}})$ of length $L_{1}$ is encoded by a
Kite code into a coded sequence ${\bf c}_{1}=(c_{1,1},c_{1,2},...,c_{1,N})$ of
length $N$. For Sender 2, a binary sequence ${\bf
u}_{2}=(u_{2,1},u_{2,2},...,u_{2,L_{2}})$ of length $L_{2}$ is firstly encoded
by a Kite code into a sequence ${\bf
v}_{2}=(v_{2,1},v_{2,2},...,v_{2,N^{\prime}})$ of length $N^{\prime}$ and then
the sequence ${\bf v}_{2}$ is encoded by the convolutional code with the
generator matrix $G(D)$ into a coded sequence ${\bf
c}_{2}=(c_{2,1},c_{2,2},...,c_{2,N})$ of length $N$.
_Modulation_ : The codewords ${\bf c}_{k}$ are mapped into the bipolar
sequences ${\bf x}_{k}=(x_{k,1},x_{k,2},...,x_{k,N})$ with
$x_{k,i}=\sqrt{P_{k}}(1-2c_{k,i})$ ($k=1,2$), where $P_{k}$ is the power. Then
we transmit ${\bf x}_{k}$ for $k=1,2$ over the interference channel.
_Decoding_ : After receiving ${\bf y}_{1}$, Receiver 1 attempts to recover the
transmitted message ${\bf u}_{1}$. Similarly, after receiving ${\bf y}_{2}$,
Receiver 2 attempts to recover the transmitted message ${\bf u}_{2}$. We will
consider several decoding algorithms in the next subsection to recover the
transmitted messages.
### IV-B Decoding Algorithms
In this subsection, depending on the knowledge about the interference, we
present four decoding schemes, including “knowing only the power of the
interference”, “knowing the signaling of the interference”, “knowing the CC”
and “knowing the whole structure”. We focus on the decoding of Receiver 1,
while the decoding of Receiver 2 can be implemented similarly.333There is no
decoding scheme “Knowing the CC” for User 2 because User 1 has no
convolutional structure. All these decoding algorithms will be described as
_message processing/passing_ algorithms over normal graphs [32].
#### IV-B1 Message processing/passing algorithms over normal graphs
Figure 6: A normal graph of a general (sub)system.
As shown in Fig. 6, a normal graph consists of edges and vertices, which
represent variables and subsystem constraints, respectively. Let ${\bf
Z}=\left\\{Z_{1},Z_{2},\cdots,Z_{n}\right\\}$ be $n$ distinct random variables
that constitute a subsystem $S^{(0)}$. This subsystem can be represented by a
normal subgraph with edges representing ${\bf Z}$ and a vertex $S^{(0)}$
representing the subsystem constraints. Each half-edge (ending with a dongle)
may potentially be coupled to some half-edge in other subsystems. For example,
$Z_{1}$ and $Z_{m}$ are shown to be connected to subsystems $S^{(1)}$ and
$S^{(m)}$, respectively. In this case, the corresponding edge is called a
full-edge. Associated with each edge is a _message_ that is defined in this
paper as the pmf/pdf of the corresponding variable. As in [33], we use the
notation $P_{Z_{i}}^{(S^{(i)}\rightarrow S^{(0)})}(z)$ to denote the message
from $S^{(i)}$ to $S^{(0)}$. In particular, we use the notation
$P_{Z_{i}}^{(|\rightarrow S^{(0)})}(z)$ to represent the initial messages
“driving” the subsystem $S^{(0)}$. For example, such initial messages can be
the _a priori_ probabilities from the source or the _a posteriori_
probabilities computed from the channel observations. Assume that all messages
to $S^{(0)}$ are available. The vertex $S^{(0)}$, as a message processor,
delivers the outgoing message with respect to any given $Z_{i}$ by computing
the likelihood function
$P_{Z_{i}}^{(S^{(0)}\rightarrow
S^{(i)})}(z)\propto\mbox{Pr}\left\\{S^{(0)}\mbox{ is satisfied
}|Z_{i}=z\right\\},z\in\mathcal{Z}$ (30)
by considering all the available messages as well as the system constraints.
We claim that $P_{Z_{i}}^{(S^{(0)}\rightarrow S^{(i)})}(z)$ is exactly the so-
called extrinsic message because the computation of the likelihood function is
irrelevant to the incoming message $P_{Z_{i}}^{(S^{(i)}\rightarrow
S^{(0)})}(z)$.
#### IV-B2 Knowing only the power of the interference
Figure 7: The normal graphs: (a) stands for the normal graph of “knowing only
the power of the interference” and “knowing the signaling of the interference”
for Decoder 1; (b) stands for the normal graph of “knowing the CC” and
“knowing the whole structure” for Decoder 1.
The decoding scheme for “knowing only the power of the interference” is the
simplest one, which can be described as a message processing/passing algorithm
over the normal graph as shown in Fig. 7. In this scheme, the interference
from Sender 2 is treated as a Gaussian distribution with mean zero and
variance $aP_{2}$, where “$P_{2}$” is the power and “$a$” is the square of
interference coefficient. That is, Receiver 1 assumes that
$X_{2,j}\sim\mathcal{N}(0,aP_{2})$ for $j=1,2,\cdots,N$. Since
$N_{1,j}\sim\mathcal{N}(0,1)$ for $j=1,2,\cdots,N$, the decoding algorithm is
initialized by the initial messages as follows
$\begin{split}P_{C_{1,j}}^{(\Sigma_{1}\rightarrow
K_{1})}(c)=\mbox{Pr}\left\\{C_{1,j}=c|\mathbf{y_{1}},X_{2,j}\sim\mathcal{N}(0,aP_{2}),j=1,2,\cdots,N\right\\}\\\
\propto\frac{1}{\sqrt{2\pi(1+aP_{2})}}\exp\left\\{-\frac{[y_{1,j}-\sqrt{P_{1}}(1-2c)]^{2}}{2(1+aP_{2})}\right\\},c\in\mathbb{F}_{2}\end{split}$
(31)
for $j=1,2,\cdots,N$. Then the decoding algorithm uses SPA to compute
iteratively the extrinsic messages $P_{U_{1,i}}^{(K_{1}\rightarrow|)}$ and
$P_{C_{1,j}}^{(K_{1}\rightarrow\Sigma_{1})}$. Once these are done, we make the
following decisions:
$\hat{u}_{1,i}=\left\\{\begin{array}[]{ll}0,~{}\mbox{if}~{}P_{U_{1,i}}^{(|\rightarrow
K_{1})}(0)P_{U_{1,i}}^{(K_{1}\rightarrow|)}(0)>P_{U_{1,i}}^{(|\rightarrow
K_{1})}(1)P_{U_{1,i}}^{(K_{1}\rightarrow|)}(1),\\\
1,~{}\mbox{otherwise}.\end{array}\right.$ (32)
$\hat{c}_{1,j}=\left\\{\begin{array}[]{ll}0,~{}\mbox{if}~{}P_{C_{1,j}}^{(\Sigma_{1}\rightarrow
K_{1})}(0)P_{C_{1,j}}^{(K_{1}\rightarrow\Sigma_{1})}(0)>P_{C_{1,j}}^{(\Sigma_{1}\rightarrow
K_{1})}(1)P_{C_{1,j}}^{(K_{1}\rightarrow\Sigma_{1})}(1),\\\
1,~{}\mbox{otherwise}.\end{array}\right.$ (33)
for $i=1,2,\cdots,L_{1}$ and $j=1,2,\cdots,N$. The details about the decoding
algorithm are shown as below.
###### Algorithm 2 (“knowing only the power of the interference”)
* •
Initialization:
1. 1.
Initialize $P_{U_{1,i}}^{(|\rightarrow K_{1})}(u)=\frac{1}{2}$ for
$i=1,2,\cdots,L_{1}$ and $u\in\mathbb{F}_{2}$.
2. 2.
Compute $P_{C_{1,j}}^{(\Sigma_{1}\rightarrow K_{1})}(c)$ for $j=1,2,\cdots,N$
and $c\in\mathbb{F}_{2}$ according to (31).
3. 3.
Set a maximum iteration number $J$ and iteration variable $j=1$.
* •
Repeat while $j\leq J$:
1. 1.
Compute extrinsic messages $P_{U_{1,i}}^{(K_{1}\rightarrow|)}$ and
$P_{C_{1,j}}^{(K_{1}\rightarrow\Sigma_{1})}$ for $i=1,2,\cdots,L_{1}$ and
$j=1,2,\cdots,N$ using SPA.
2. 2.
Make decisions according to (32) and (33). Denote
${\bf\hat{u}}_{1}=\left(\hat{u}_{1,1},\hat{u}_{1,2},\cdots,\hat{u}_{1,L_{1}}\right)$
and
${\bf\hat{c}}_{1}=\left(\hat{c}_{1,1},\hat{c}_{1,2},\cdots,\hat{c}_{1,N}\right)$.
3. 3.
Compute the syndrome ${\bf S}_{1}={\bf\hat{c}}_{1}\cdot{\bf H}_{1}^{T}$. If
${\bf S}_{1}=\mathbf{0}$, output ${\bf\hat{u}}_{1}$ and ${\bf\hat{c}}_{1}$ and
exit the iteration.
4. 4.
Set $j=j+1$. If ${\bf S}_{1}\neq\mathbf{0}$ and $j>J$, report a decoding
failure.
* •
End decoding.
#### IV-B3 Knowing the signaling of the interference
The decoding algorithm for this scheme is almost the same as Algorithm 2, see
Fig. 7. The difference is that $X_{2,j}\sim\mathcal{B}(1/2)$ (Bernoulli-1/2
distribution444Strictly speaking, $X_{2,j}$ is a shift/scaling version of
$\mathcal{B}(1/2)$.) for $j=1,2,\cdots,N$. So the computation of
$P_{C_{1,j}}^{(\Sigma_{1}\rightarrow K_{1})}(c)$ is changed into
$\begin{split}P_{C_{1,j}}^{(\Sigma_{1}\rightarrow
K_{1})}(c)=\mbox{Pr}\left\\{C_{1,j}=c|\mathbf{y_{1}},X_{2,j}\sim\mathcal{B}(1/2),j=1,2,\cdots,N\right\\}\\\
\propto\frac{1}{2}\frac{1}{\sqrt{2\pi}}\exp\left\\{-\frac{\left[y_{1,j}-\sqrt{P_{1}}(1-2c)-\sqrt{aP_{2}}\right]^{2}}{2}\right\\}\\\
+\frac{1}{2}\frac{1}{\sqrt{2\pi}}\exp\left\\{-\frac{\left[y_{1,j}-\sqrt{P_{1}}(1-2c)+\sqrt{aP_{2}}\right]^{2}}{2}\right\\},\\\
c\in\mathbb{F}_{2}\end{split}$ (34)
Then the decoding algorithm of “knowing the signaling of the interference” can
be shown as below.
###### Algorithm 3 (“knowing the signaling of the interference”)
* •
Initialization:
1. 1.
Initialize $P_{U_{1,i}}^{(|\rightarrow K_{1})}(u)=\frac{1}{2}$ for
$i=1,2,\cdots,L_{1}$ and $u\in\mathbb{F}_{2}$.
2. 2.
Compute $P_{C_{1,j}}^{(\Sigma_{1}\rightarrow K_{1})}(c)$ for $j=1,2,\cdots,N$
and $c\in\mathbb{F}_{2}$ according to (34).
3. 3.
Set a maximum iteration number $J$ and iteration variable $j=1$.
* •
Repeat while $j\leq J$:
1. 1.
Compute extrinsic messages $P_{U_{1,i}}^{(K_{1}\rightarrow|)}$ and
$P_{C_{1,j}}^{(K_{1}\rightarrow\Sigma_{1})}$ for $i=1,2,\cdots,L_{1}$ and
$j=1,2,\cdots,N$ using SPA.
2. 2.
Make decisions according to (32) and (33), respectively.
3. 3.
Compute the syndrome ${\bf S}_{1}={\bf\hat{c}}_{1}\cdot{\bf H}_{1}^{T}$. If
${\bf S}_{1}=\mathbf{0}$, output ${\bf\hat{u}}_{1}$ and ${\bf\hat{c}}_{1}$ and
exit the iteration.
4. 4.
Set $j=j+1$. If ${\bf S}_{1}\neq\mathbf{0}$ and $j>J$, report a decoding
failure.
* •
End decoding.
#### IV-B4 Knowing the CC
“Knowing the CC” means that Decoder 1 knows the structure of the convolutional
code. This scheme can be described as a message processing/passing algorithm
over the normal graph as shown in Fig. 7. Actually, the vertex $T_{1}$ is a
combination of three subsystems, convolutional encoder, modulation and GIFC
constraint, which can be specified by a trellis $\mathcal{T}$ with parallel
branches [15]. Therefore, the BCJR algorithm can be used to compute the
extrinsic messages $P_{C_{1,j}}^{(T_{1}\rightarrow K_{1})}(c)$ for
$j=1,2,\cdots,N$ over the trellis $\mathcal{T}$. Since the structure of Kite
code for Sender 2 is unknown, the constraint of vertex $K_{2}$ is inactive. In
this case, the pmf of variable $V_{2,k}$ ($k=1,2,\cdots,N^{\prime}$) is
assumed to be Bernoulli-1/2 distribution. There are two strategies to
implement the BCJR algorithm. One is called “BCJR-once”, in which the BCJR
algorithm is performed only once. The other strategy is called “BCJR-repeat”,
in which the BCJR algorithm is performed more than once. In this scheme, the
decoding decisions on $C_{1,j}$ are modified into
$\hat{c}_{1,j}=\left\\{\begin{array}[]{ll}0,~{}\mbox{if}~{}P_{C_{1,j}}^{(T_{1}\rightarrow
K_{1})}(0)P_{C_{1,j}}^{(K_{1}\rightarrow
T_{1})}(0)>P_{C_{1,j}}^{(T_{1}\rightarrow
K_{1})}(1)P_{C_{1,j}}^{(K_{1}\rightarrow T_{1})}(1),\\\
1,~{}\mbox{otherwise},\end{array}\right.$ (35)
for $j=1,2,\cdots,N$. These two decoding procedures are described in Algorithm
4 and Algorithm 5, respectively.
###### Algorithm 4 (BCJR-once)
* •
Initialization:
1. 1.
Initialize pmf $P_{C_{1,j}}^{(K_{1}\rightarrow
T_{1})}\left(c\right)=\frac{1}{2}$ and $P_{C_{2,j}}^{(|\rightarrow
T_{1})}\left(c\right)=\frac{1}{2}$ for $j=1,2,\cdots,N,c\in\mathbb{F}_{2}$ and
$P_{V_{2,k}}^{(|\rightarrow T_{1})}\left(v\right)=\frac{1}{2}$ for
$k=1,2,\cdots,N^{\prime},v\in\mathbb{F}_{2}$.
2. 2.
Compute extrinsic messages $P_{C_{1,j}}^{(T_{1}\rightarrow
K_{1})}\left(c\right)$ for $j=1,2,\cdots,N$, $c\in\mathbb{F}_{2}$ using BCJR
algorithm over the parallel branch trellis $\mathcal{T}$.
3. 3.
Set a maximum iteration number $J$ and iteration variable $j=1$.
* •
Repeat while $j\leq J$:
1. 1.
Compute extrinsic messages $P_{U_{1,i}}^{(K_{1}\rightarrow|)}$ and
$P_{C_{1,j}}^{(K_{1}\rightarrow T_{1})}$ for $i=1,2,\cdots,L_{1}$ and
$j=1,2,\cdots,N$ using SPA.
2. 2.
Make decisions according to (32) and (35).
3. 3.
Compute the syndrome ${\bf S}_{1}={\bf\hat{c}}_{1}\cdot{\bf H}_{1}^{T}$. If
${\bf S}_{1}=\mathbf{0}$, output ${\bf\hat{u}}_{1}$ and ${\bf\hat{c}}_{1}$ and
exit the iteration.
4. 4.
Set $j=j+1$. If ${\bf S}_{1}\neq\mathbf{0}$ and $j>J$, report a decoding
failure.
* •
End Decoding
###### Algorithm 5 (BCJR-repeat)
* •
Initialization:
1. 1.
Initialize pmf $P_{C_{1,j}}^{(K_{1}\rightarrow
T_{1})}\left(c\right)=\frac{1}{2}$ and $P_{C_{2,j}}^{(|\rightarrow
T_{1})}\left(c\right)=\frac{1}{2}$ for $j=1,2,\cdots,N,c\in\mathbb{F}_{2}$ and
$P_{V_{2,k}}^{(|\rightarrow T_{1})}\left(v\right)=\frac{1}{2}$ for
$k=1,2,\cdots,N^{\prime},v\in\mathbb{F}_{2}$.
2. 2.
Set a maximum iteration number $J$ and iteration variable $j=1$.
* •
Repeat while $j\leq J$:
1. 1.
Compute extrinsic messages $P_{C_{1,j}}^{(T_{1}\rightarrow
K_{1})}\left(c\right)$ for $j=1,2,\cdots,N$, $c\in\mathbb{F}_{2}$ using BCJR
algorithm over the parallel branch trellis $\mathcal{T}$.
2. 2.
Compute extrinsic messages $P_{U_{1,i}}^{(K_{1}\rightarrow|)}$ and
$P_{C_{1,j}}^{(K_{1}\rightarrow T_{1})}$ for $i=1,2,\cdots,L_{1}$ and
$j=1,2,\cdots,N$ using SPA.
3. 3.
Make decisions according to (32) and (35).
4. 4.
Compute the syndrome ${\bf S}_{1}={\bf\hat{c}}_{1}\cdot{\bf H}_{1}^{T}$. If
${\bf S}_{1}=\mathbf{0}$, output ${\bf\hat{u}}_{1}$ and ${\bf\hat{c}}_{1}$ and
exit the iteration.
5. 5.
Set $j=j+1$. If ${\bf S}_{1}\neq\mathbf{0}$ and $j>J$, report a decoding
failure.
* •
End Decoding
#### IV-B5 Knowing the whole structure
The scheme “knowing the whole structure” for Receiver 1 can also be described
as a message processing/passing algorithm over the normal graph shown in Fig.
7. Since knowing the whole structure of the interference, Receiver 1 can
decode iteratively utilizing the structure of both users. Using the BCJR
algorithm, $P_{C_{1,j}}^{(T_{1}\rightarrow K_{1})}\left(c\right)$ and
$P_{V_{2,k}}^{(T_{1}\rightarrow K_{2})}\left(v\right)$ are computed
simultaneously over the parallel branch trellis $\mathcal{T}$. The iterative
decoding algorithm is presented in Algorithm 6.
###### Algorithm 6 (“knowing the whole structure”)
* •
Initialization:
1. 1.
Initialize pmf $P_{C_{1,j}}^{(K_{1}\rightarrow
T_{1})}\left(c\right)=\frac{1}{2}$ and $P_{C_{2,j}}^{(|\rightarrow
T_{1})}\left(c\right)=\frac{1}{2}$ for $j=1,2,\cdots,N,c\in\mathbb{F}_{2}$ and
$P_{V_{2,k}}^{(K_{2}\rightarrow T_{1})}\left(v\right)=\frac{1}{2}$ for
$k=1,2,\cdots,N^{\prime},v\in\mathbb{F}_{2}$.
2. 2.
Set a maximum iteration number $J$ and iteration variable $j=1$.
* •
Repeat while $j\leq J$:
1. 1.
Compute extrinsic messages $P_{C_{1,j}}^{(T_{1}\rightarrow
K_{1})}\left(c\right)$ for $j=1,2,\cdots,N$, $c\in\mathbb{F}_{2}$ and
$P_{V_{2,k}}^{(T_{1}\rightarrow K_{2})}\left(v\right)$ for
$k=1,2,\cdots,N^{\prime}$, $v\in\mathbb{F}_{2}$ using BCJR algorithm over the
parallel branch trellis $\mathcal{T}$.
2. 2.
Compute extrinsic messages $P_{U_{1,i}}^{(K_{1}\rightarrow|)}$ and
$P_{C_{1,j}}^{(K_{1}\rightarrow T_{1})}$ for $i=1,2,\cdots,L_{1}$ and
$j=1,2,\cdots,N$ using SPA.
3. 3.
Compute extrinsic messages $P_{V_{2,k}}^{(K_{2}\rightarrow
T_{1})}\left(v\right)$ for $k=1,2,\cdots,N^{\prime},v\in\mathbb{F}_{2}$ using
SPA.
4. 4.
Make decisions according to (32) and (35).
5. 5.
Compute the syndrome ${\bf S}_{1}={\bf\hat{c}}_{1}\cdot{\bf H}_{1}^{T}$. If
${\bf S}_{1}=\mathbf{0}$, output ${\bf\hat{u}}_{1}$ and ${\bf\hat{c}}_{1}$ and
exit the iteration.
6. 6.
Set $j=j+1$. If ${\bf S}_{1}\neq\mathbf{0}$ and $j>J$, report a decoding
failure.
* •
End Decoding
### IV-C Numerical Results
In this subsection, simulation results of the decoding algorithms are shown
and analyzed. Simulation parameters of Fig. 8 and Fig. 9 are presented in
TABLE IV-C. In these two figures, we let the power constraints of two senders
be same, that is, $P_{1}\equiv P_{2}=P$. Here, “Gaussian” stands for the
scheme “knowing only the power of the interference”, “BPSK” stands for the
scheme “knowing the signaling of the interference”, “BCJR1” stands for the
scheme “BCJR-once”, “CONV” stands for the scheme “BCJR-repeat” and “Know All
Structure” stands for the scheme “knowing the whole structure”. From Fig. 8
and Fig. 9, we can easily see that the decoding gains get larger as more
details of the structure of the interference are known.
[!t] Parameters of the BER performance simulations Parameters Values Square of
interference coefficient $a$ 0.5 Maximum iteration number $J$ 200 Kite Code of
Sender 1 $N=10000,L_{1}=8782$ Kite Code of Sender 2
$N^{\prime}=5000,L_{2}=4862$ Generator matrix $G(D)$
$[1+D+D^{2}\,\,\,1+D^{2}]$ Code rate pair $\left(R_{1},R_{2}\right)$
$\left(0.8782,0.4862\right)$
Figure 8: The error performance of Receiver 1. “Gaussian” stands for the
scheme “knowing only the power of the interference”, “BPSK” stands for the
scheme “knowing the signaling of the interference”, “BCJR1” stands for the
scheme “BCJR-once”, “CONV” stands for the scheme “BCJR-repeat” and “Know All
Structure” stands for the scheme “knowing the whole structure”.
Figure 9: The error performance of Receiver 2. “Gaussian” stands for the
scheme “knowing only the power of the interference”, “BPSK” stands for the
scheme “knowing the signaling of the interference” and “Know All Structure”
stands for the scheme “knowing the whole structure”.
Another objective is to find out a code rate pair nearest to the point “B” in
Fig. 4 with bit error rate (BER) performance of $10^{-4}$. So we do the
simulations with different code rate pairs. In the simulations, we adopt the
scheme “knowing the whole structure” and gradually decrease the code rates
from the point “B” with a step length $0.01$. Simulation parameters for
different code rate pairs are listed in TABLE IV-C, while the simulation
results are presented using a 3D graph in Fig. 10. From the figure, it is
obvious that as the code rates of two users are decreasing, the BER also
decreases. Finally, we find out the “best” code rate pair is $(0.71,0.48)$ for
User 1 and User 2. The theoretical value of the point “B” is about
$(0.878,0.486)$. So we can see that the gap between the result using our
decoding scheme and the theoretical value is small.
[!t] Parameters of the simulations for different code rate pairs. Parameters
Values Square of interference coefficient $a$ $0.5$ Maximum iteration number
$J$ $200$ Code length $N$ of Kite Code of Sender 1 $10000$ Code length
$N^{\prime}$ of Kite Code of Sender 2 $5000$ Generator matrix $G(D)$
$[1+D+D^{2}\,\,\,1+D^{2}]$ Step length $100$ Range of message length $L_{1}$
$7100\sim 8800$ Range of message length $L_{2}$ $4000\sim 4900$
Figure 10: Error performance of two users with different code rate pairs
$(R_{1},R_{2})$. Blue plane represents BER level, green surface stands for the
error performance of Receiver 1 and red surface stands for the error
performance of Receiver 2.
## V Conclusions
In this paper, we have proved that the capacity region of the two-user
interference channel is the union of a family of rectangles, each of which is
defined by a pair of spectral inf-mutual information rates associated with two
independent input processes. For the stationary memoryless channel with
discrete Markov inputs, the defined pair of rates can be computed, which show
us that the simplest inner bounds (obtained by treating the interference as
noise) could be improved by taking into account the structure of the
interference processes. Also a concrete coding scheme to approach the
theoretical achievable rate pairs was presented, which showed that the
decoding gain can be achieved by considering the structure of the
interference.
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|
arxiv-papers
| 2013-10-06T03:53:46 |
2024-09-04T02:49:52.004088
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Xiao Ma and Lei Lin and Chulong Liang and Xiujie Huang and Baoming Bai",
"submitter": "Lei Lin",
"url": "https://arxiv.org/abs/1310.1536"
}
|
1310.1599
|
[40mm,40mm]1525
# How can one observe gravitational angular momentum radiation from a
dynamical source near null infinity?
Chih-Hung Wang [email protected] Leichung Waldorf high school, Taichung,
406, Taiwan Yu-Huei Wu [email protected] Center for Mathematics and
Theoretical Physics, National Central University, Chungli, 320, Taiwan.
###### Abstract
To answer a question of how can one observed angular momentum radiation near
null infinity, one can first transform the dynamical twisting vacuum solution
and make it satisfy Bondi coordinate conditions in the asymptotical region of
the null infinity. We then obtain the Bondi-Sachs news function and also find
the relationship of how does the angular momentum contribute to the news
functions from the exact solution sense. By using the Komar’s integral of
angular momentum, the gravitational angular momentum flux of the dynamical
twisting space-time can be obtained. All of our results can be compared with
the Kerr solution, Robbinson-Trautmann or Schwarzschild solution. This study
can provide a theoretical basis to understand the correlations of
gravitational radiations near a rotating dynamical horizon and null infinity.
###### pacs:
04.20.Ha, 04.20.Jb, 04.20.Gz,97.60.Lf
Introduction and motivation.– A theoretical framework of studying
gravitational outgoing radiations and mass loss at null infinity was
originally established by Bondi et al Bondi-Sachs and further developed by
Newman and Unit NU . In asymptotically flat empty space-times with outgoing
radiation condition and certain coordinate conditions satisfying, the
framework provides a systematical analysis to study gravitational energy flux
and also non-linear effects of gravitational radiations for general
asymptotically flat solutions of Einstein field equations. Besides the study
of asymptotical behaviors near the null infinity, the authors apply
asymptotical expansions to investigate gravitational radiations on the
neighborhood of another space-time boundary, horizon Wu2007 Wu-Wang ,
however, the authors do not satisfy with the setting of the slow rotation
approximation. The gravitational energy flux and angular-momentum flux across
a slowly rotating dynamical horizons (DHs) were obtained in Wu2007 Wu-Wang .
The merger of two black holes (BHs) is an important source to generate
gravitational waves, and it is normally accompanied by the recoil velocity and
spin flip Merritt-Ekers-02 Baker-06 . In the study of recoil velocity and
spin precession during the binary BHs merger, we need to understand not only
gravitational radiations near DHs but also gravitational waves propagating to
null infinity. It is physically important to established the correlations of
geometric quantities between the DHs and null infinity. In Macedo-Saa-08 ,
they used Robinson-Trautman (RT) space-time RT62 , which is an exact solution
of Einstein field equations containing spherical GWs, to study gravitational
wave recoils. Moreover, Rezzolla, Macedo and Jaramillo Rezzolla-et-al-10
study the antikick of head-on collision of two nonspinning BHs, which has been
observed from numerical-relativity calculations, in RT space-times. Although
RT space-time is an algebraic special solution, i.e. shear-free, it is a
dynamical solution and has GWs propagating to null infinity. Bondi coordinate
is a physical coordinate which allows us to study the gravitational radiation.
Therefore, we must transform it to the Bondi coordinate and the news function
appears. The asymptotical behavior of RT space-time in the Bondi coordinate
has been first investigated by Foster and Newman Foster-Newman67 , and they
only considered small deviation from spherical symmetry. In Gonna-Kramer-98 ,
Gonna and Krammer study the pure and gravitational radiations of RT space-time
and obtain Bondi-Sachs news function, i.e. the gravitational free data.
It is interesting to generalize their works to include the spins in the merger
of binary BHs. So, how can one define angular momentum of a dynamical source?
Since it is not proper to use RT space-time to study binary BHs merger with
spins, the dynamical twisting vacuum solution SKMHH-03 , i.e. hypersurface
non-orthogonal, may provide a good candidate to study spinning BHs since the
Kerr black hole is a stationary vacuum solution of the twisting space-times.
In this paper, we perform a coordinate transformation on twisting vacuum
solutions to the Bondi coordinates and apply Newman-Unti (NU) asymptotical
expansion NU to obtain news function of twisting space-times. One can clearly
see how does the angular momentum contribute to the gravitational radiation
(the news function). We use the twisting solution of P. 437 Sec. 29 SKMHH-03
in the complex coordinate $(u,r,\zeta,\overline{\zeta})$ and later transform
it to the Bondi complex coordinates
$(U,R,\zeta^{\prime},\overline{\zeta}^{\prime})$. By writing a twisting vacuum
solution in the Bondi coordinate, we then obtain the NU mass and also angular
momentum. After obtaining the NU mass, one could combine the news function to
obtain the Bondi mass formula. From this work, we may know how the angular
parameter contributes to the Bondi mass. We also observe a formula that
related with angular momentum and the news function. It is easy to show that
when angular parameter vanishes, the solution will return to Robinson-Trautman
(RT) spacetime in the Bondi coordinates. The results will be used to study the
merger of two spinning BHs and also the influences of the gravitational
angular-momentum flux on the spin flip Wang-Wu 2013b in the future.
Unfortunately, no explicit expression for the angular momentum in terms of the
Kerr parameters $m$ and $a$ is given from the spinor construction of angular
momentum, e.g., Bergqvist and Ludvigsen, Bramson’s superpotential, Ludvigsen-
Vickers angular momentum (See Szabados-04 for the detail). Thus our angular
momentum is calculated by using Komar integral Komar59 . We show that our
calculation will return to Kerr solution and yield the $ma$. Our convention in
this paper is $(+---)$. Note that we do not require axial symmetric.
Twisting vacuum solutions in the Bondi coordinates.– Here we use the Newman-
Unti affine parameter rather than Bondi luminosity distance. So that our
results can be compared with the results of Newman-UntiNU or Foster-Newman
Foster-Newman67 . The covariant tetrad of twisting spacetime in
$(u,r,\zeta,\overline{\zeta})$ coordinate SKMHH-03 is
$\displaystyle\ell_{a}$ $\displaystyle=$
$\displaystyle(1,0,L,\overline{L}),\;n_{a}=(H,1,A,\overline{A}),$ (1)
$\displaystyle m_{a}$ $\displaystyle=$
$\displaystyle(0,0,0,-B),\;\overline{m}_{a}=(0,0,-\overline{B},0),$ (2)
where $A:=W+LH$ and $B:=\frac{1}{\eta P}$ and $A,B,\eta$ are complex and are
functions of $(u,r,\zeta,\overline{\zeta})$ and $L,W$ are functions of
$(u,\zeta,\overline{\zeta})$. We then have the NP coefficients and since the
spacetime is algebraic special for $\ell$, we have $\kappa=\sigma=\lambda=0$
and
$\displaystyle H$ $\displaystyle:=$ $\displaystyle\frac{K}{2}-r(\ln
P)_{,u}-\frac{mr}{r^{2}+\varpi^{2}},$ $\displaystyle K$ $\displaystyle:=$
$\displaystyle 2P^{2}{\rm Re}[\partial(\overline{\partial}\ln
P-\overline{L}_{,u})],\;2i\varpi:=P^{2}(\overline{\partial}L-\partial\overline{L}),$
$\displaystyle W$ $\displaystyle=$
$\displaystyle\frac{L_{,u}}{\eta}+i\partial\varpi,\;\partial\equiv\partial_{\zeta}-L\partial_{u},$
$\displaystyle\eta^{-1}$ $\displaystyle:=$
$\displaystyle-(r+i\varpi),\;\;(B\overline{B})^{-1}=\eta\overline{\eta}P^{2}=\frac{P^{2}}{r^{2}}+O(\frac{1}{r^{3}}),$
where we use $r_{0}=0,M=0$ and $\varpi,P$ is real in SKMHH-03 .
The contravariant tetrad in $(u,r,\zeta,\overline{\zeta})$ coordinate is
$\displaystyle\ell^{a}$ $\displaystyle=$
$\displaystyle(0,1,0,0),\;n^{a}=(1,-H,0,0),$ (3) $\displaystyle m^{a}$
$\displaystyle=$
$\displaystyle(-\frac{L}{\overline{B}},\frac{W}{\overline{B}},\frac{1}{\overline{B}},0),\;\overline{m}^{a}=(-\frac{\overline{L}}{B},\frac{\overline{W}}{B},0,\frac{1}{B}).$
(4)
We need to perform a coordinate transformation $\tilde{g}^{ab}=\frac{\partial
X^{a}}{\partial x^{i}}\frac{\partial X^{b}}{\partial x^{j}}g^{ij}$ to
transform coordinate $(u,r,\zeta,\overline{\zeta})$ to Bondi coordinate
$(U,R,\zeta^{\prime},\overline{\zeta}^{\prime})$.
$\displaystyle U$ $\displaystyle=$ $\displaystyle
U_{0}+\frac{U_{1}}{r}+O(\frac{1}{r^{2}}),$ (5) $\displaystyle R$
$\displaystyle=$ $\displaystyle
R_{-1}r+R_{0}+\frac{R_{1}}{r}+O(\frac{1}{r^{2}}),$ (6)
$\displaystyle\zeta^{\prime}$ $\displaystyle=$
$\displaystyle\zeta_{0}+\frac{\zeta_{1}}{r}+O(\frac{1}{r^{2}}),\;\overline{\zeta}^{\prime}=\overline{\zeta}_{0}+\frac{\overline{\zeta}_{1}}{r}+O(\frac{1}{r^{2}}).$
(7)
The metric of the twisting spacetime is
$\displaystyle g^{00}$ $\displaystyle=$ $\displaystyle
g^{uu}=-2\frac{\overline{L}L}{\overline{B}B}=\frac{g^{00}_{0}}{r^{2}}+O(\frac{1}{r^{3}}),$
$\displaystyle g^{01}$ $\displaystyle=$ $\displaystyle
1+\frac{2HL\overline{L}-L\overline{A}-\overline{L}A}{B\overline{B}}=1+\frac{g^{01}_{0}}{r}+O(\frac{1}{r^{2}}),$
$\displaystyle g^{02}$ $\displaystyle=$
$\displaystyle\frac{\overline{L}}{\overline{B}B}=\frac{g^{02}_{0}}{r^{2}}+O(\frac{1}{r^{3}}),\;g^{03}=\frac{L}{\overline{B}B}=\frac{g^{03}_{0}}{r^{2}}+O(\frac{1}{r^{3}}),$
$\displaystyle g^{11}$ $\displaystyle=$
$\displaystyle-2H-2\frac{A\overline{A}-AH\overline{L}-\overline{A}HL+H^{2}L\overline{L}}{B\overline{B}}$
$\displaystyle=$
$\displaystyle-\tilde{K}+\frac{2\tilde{m}}{r}+r\partial_{u}\ln
P+O(\frac{1}{r^{2}}),$ $\displaystyle g^{12}$ $\displaystyle=$
$\displaystyle\frac{\overline{W}}{B\overline{B}}=\frac{g^{12}_{0}}{r}+O(\frac{1}{r^{2}}),$
$\displaystyle g^{13}$ $\displaystyle=$
$\displaystyle\frac{W}{B\overline{B}}=\frac{g^{13}_{0}}{r}+O(\frac{1}{r^{2}}),$
$\displaystyle g^{23}$ $\displaystyle=$ $\displaystyle
g^{\zeta\overline{\zeta}}=-\frac{1}{B\overline{B}}=\frac{g^{23}_{0}}{r^{2}}+O(\frac{1}{r^{3}}).$
where
$\displaystyle g^{01}_{0}$ $\displaystyle=$
$\displaystyle-P^{2}(L\overline{L})_{,0},\;\;g^{02}_{0}=P^{2}\overline{L},\;\;g^{03}_{0}=P^{2}L,$
$\displaystyle g^{12}_{0}$ $\displaystyle=$
$\displaystyle-P^{2}\overline{L}_{,0},\;\;g^{13}_{0}=-P^{2}L_{,0},$
$\displaystyle g^{00}_{0}$ $\displaystyle=$
$\displaystyle-2\overline{L}LP^{2},\;\;g^{23}_{0}=-P^{2},$
$\displaystyle\tilde{K}$ $\displaystyle=$ $\displaystyle
K-2P^{2}|L_{,0}|^{2},$ $\displaystyle\tilde{m}$ $\displaystyle=$
$\displaystyle m+P^{2}[-2\Sigma
L_{,0}\overline{L}_{,0}+L_{,0}\overline{\partial}\Sigma+\overline{L}_{,0}\partial\Sigma].$
and the asymptotic values of $H,B,W$ are
$\displaystyle H$ $\displaystyle=$
$\displaystyle\frac{1}{2}K-\frac{m}{r}-r(\ln P)_{,u}+O(\frac{1}{r^{2}}),$ (8)
$\displaystyle B$ $\displaystyle=$ $\displaystyle-\frac{r}{P}+O(1),$ (9)
$\displaystyle W$ $\displaystyle=$ $\displaystyle-L_{,0}r+i(-\Sigma
L_{,0}+\partial\Sigma),$ (10)
where $L=L(u,\zeta,\overline{\zeta})$, $P=P(u,\zeta,\overline{\zeta})$,and
$\Sigma=\Sigma(u,\zeta,\overline{\zeta})$.
Bondi coordinate conditions are
$\displaystyle\tilde{g}^{00}=\tilde{g}^{02}=\tilde{g}^{03}=0,\;\;\tilde{g}^{01}=-1,$
(11)
which we choose affine parameter here.
Transform Newman-Unti real coordinate $(U,R,x^{2},x^{3})$ to Bondi complex
coordinate $(U,R,\zeta^{\prime},\overline{\zeta}^{\prime})$.– Here we use a
complex coordinate $(\zeta,\overline{\zeta})$ to make our whole calculation
simpler. We need to use a coordinate transformation to transfer Newman-Unti
real coordinate $(x^{2},x^{3})$ 111 Newman-Unti originally use
$(x^{3},x^{4})$, here our coordinate runs from $(x^{0},x^{1},x^{2},x^{3})$.
into the complex coordinate
$(y^{2},y^{3})=(\zeta^{\prime},\overline{\zeta}^{\prime})$ in order to compare
with Newman-Unti NU . We have
$\displaystyle
y^{2}=\zeta^{\prime}=\frac{1}{2}(x^{2}+ix^{3}),\;\;\;y^{3}=\overline{\zeta}^{\prime}=\frac{1}{2}(x^{2}-ix^{3}).$
(12)
From $\tilde{g^{\alpha\beta}}=\frac{\partial y^{\alpha}}{\partial
x^{m}}\frac{\partial y^{\beta}}{\partial x^{n}}g^{mn}_{NU}$, we get
$\displaystyle\tilde{g^{11}}$ $\displaystyle=$ $\displaystyle\frac{\partial
y^{1}}{\partial x^{m}}\frac{\partial y^{1}}{\partial
x^{n}}g^{mn}_{NU}=g^{11}_{NU},$ (13) $\displaystyle\tilde{g^{01}}$
$\displaystyle=$ $\displaystyle
1,\;\;\tilde{g^{12}}=\frac{1}{2}g^{12}_{NU}+i\frac{1}{2}g^{13}_{NU},$ (14)
$\displaystyle\tilde{g^{13}}$ $\displaystyle=$
$\displaystyle-i\frac{1}{2}g^{13}_{NU}+\frac{1}{2}g^{12}_{NU},$ (15)
$\displaystyle\tilde{g^{22}}$ $\displaystyle=$
$\displaystyle\frac{1}{4}g^{22}_{NU}+\frac{1}{4}g^{33}_{NU}+i\frac{1}{2}g^{23}_{NU}$
(16) $\displaystyle=$ $\displaystyle 2P^{2}_{NU}\sigma^{0}R^{-3}+O(R^{-4}),$
(17) $\displaystyle\tilde{g^{23}}$ $\displaystyle=$
$\displaystyle\frac{1}{4}g^{22}_{NU}-\frac{1}{4}g^{33}_{NU}$ (18)
$\displaystyle=$
$\displaystyle-P^{2}_{NU}R^{-2}-3\sigma^{0}\overline{\sigma}^{0}P^{2}_{NU}R^{-4}+O(R^{-5}),$
(19) $\displaystyle\tilde{g^{33}}$ $\displaystyle=$
$\displaystyle\frac{1}{4}g^{22}_{NU}-\frac{1}{4}g^{33}_{NU}-i\frac{1}{2}g^{23}_{NU}$
(20) $\displaystyle=$ $\displaystyle
2P^{2}_{NU}\overline{\sigma}^{0}R^{-3}+O(R^{-4}),$ (21)
and we use the results of Newman-Unti NU
$\displaystyle g^{11}_{NU}$ $\displaystyle=$
$\displaystyle-2P_{NU}^{2}\nabla\overline{\nabla}\ln
P_{NU}-(\Psi^{0}_{2}+\overline{\Psi}^{0}_{2})R^{-1}+O(R^{-2}),$ $\displaystyle
g^{22}_{NU}$ $\displaystyle=$
$\displaystyle-2P^{2}_{NU}R^{-2}+2P^{2}_{NU}(\sigma^{0}+\overline{\sigma}^{0})R^{-3}+O(R^{-4}),$
$\displaystyle g^{23}_{NU}$ $\displaystyle=$
$\displaystyle-2iP^{2}_{NU}(\sigma^{0}-\overline{\sigma}^{0})R^{-3}+O(R^{-4}),$
$\displaystyle g^{33}_{NU}$ $\displaystyle=$
$\displaystyle-2P^{2}_{NU}R^{-2}-2P^{2}_{NU}(\sigma^{0}+\overline{\sigma}^{0})R^{-3}+O(R^{-4}),$
where $P_{NU}=P_{NU}(x^{2},x^{3})$ and later we will prove $f=P_{NU}$. Note
that we first introduce $f$ from $O(r)$ of $\tilde{g}^{11}$. The definition of
Newman-Unti mass integral is $M_{NU}:=\oint m_{NU}dS$ where the Newman-Unti
mass is defined as $m_{NU}:={\rm Re}\Psi_{2}^{0}$.
Results from coordinate transformation.– From $\tilde{g}^{00}$ the $O(r^{-2})$
term vanishes, we get
$\displaystyle
U_{1}=-fPU_{0,2}U_{0,3}-\frac{P^{3}L\overline{L}}{f}+P^{2}\overline{L}U_{0,2}+P^{2}LU_{0,3}.$
From $\tilde{g}^{01}=1$ and $O(1)$ vanishes, we get
$\displaystyle R_{-1}=U_{0,0}^{-1}$ (22)
From $O(r^{-1})$ term vanishes, we obtain an identity:
From $\tilde{g}^{02}=O(r^{-3})$ and $\tilde{g}^{03}=O(r^{-3})$ , we get
$\displaystyle\zeta_{1}$ $\displaystyle=$ $\displaystyle
P(P\overline{L}_{0}-fU_{0,3}),\;\;\overline{\zeta}_{1}=P(PL_{0}-fU_{0,2}).$
(23)
From $\tilde{g}^{R\zeta^{\prime}}=\tilde{g}^{12}=\tilde{g}^{13}=O(r^{-1})$, we
get
$\displaystyle\zeta_{0,0}=0,\;\overline{\zeta}_{0,0}=0,\;\zeta_{0,2}=1,\;\overline{\zeta}_{0,3}=1,$
(24)
and $\zeta_{0}=\zeta$, $\overline{\zeta}_{0}=\overline{\zeta}$.
From $O(r)$ of $\tilde{g}^{RR}=\tilde{g}^{11}$, we obtain a differential
equation $(\ln R_{-1}+\ln P+K)_{,0}=0$ and thus get
$\displaystyle R_{-1}=fP^{-1}$ (25)
where $f$ is a function that depends on $(\zeta,\overline{\zeta})$. Thus,
$U_{0,0}=\frac{P}{f}$.
From $O(1)$, we obtain $R_{0,0}$. After integration, we can further obtain
$\displaystyle
R_{0}=f^{2}U_{0,23}+\overline{L}(f_{,2}P-fP_{,2})+L(f_{,3}P-fP_{,3})$
$\displaystyle-\frac{1}{2}fP(\partial_{2}\overline{L}+\partial_{3}L)+fP_{,0}L\overline{L}-fP(L\overline{L})_{,0}.$
(26)
From $O(r^{-1})$, we obtain the Newman-Unti mass
$m_{NU}=\frac{1}{2}(\Psi^{0}_{2}+\overline{\Psi}^{0}_{2})={\rm
Re}\Psi^{0}_{2}$ after integration,
$\displaystyle m_{NU}$ $\displaystyle=$
$\displaystyle-2PL\overline{L}R_{-1,0}R_{0,0}+\frac{R_{1,0}R_{-1}-R_{-1,0}R_{1}}{P}$
(27) $\displaystyle-$ $\displaystyle P(L\overline{L})_{,0}R_{0,0}R_{-1}$
$\displaystyle+$
$\displaystyle[P\overline{L}(R_{-1,0}R_{0,2}+R_{0,0}R_{-1,2})+C.C]$
$\displaystyle-$ $\displaystyle
2\frac{P_{,0}R_{-1}R_{1}}{P^{2}}+\frac{R_{-1}^{2}}{P}\tilde{m}-[R_{-1}R_{0,2}\overline{L}_{,0}P+C.C.]$
$\displaystyle-$ $\displaystyle(R_{-1,2}R_{0,3}+R_{0,2}R_{-1,3})P.$
which we need $R_{0},R_{-1},R_{1}$. From
$\tilde{g}^{23}=\tilde{g}^{\zeta^{\prime}\overline{\zeta}^{\prime}}$, we get
$\displaystyle f=P_{NU}$ (28)
from $O(r^{-2})$. From $O(r^{-3})$, we obtain $R_{0}$ which yield the same
result with the one from $\tilde{g}^{RR}$. From
$\tilde{g}^{22}=\tilde{g}^{\zeta^{\prime}\zeta^{\prime}}$ and $O(r^{-3})$, we
obtain the shear term.
$\displaystyle\sigma^{0}$ $\displaystyle=$
$\displaystyle\frac{f}{P}(P^{2}\overline{L})_{,3}+\frac{Pf\overline{L}_{,0}^{2}}{2}+2\frac{f^{2}}{P}P_{,0}U_{0,3}\overline{L}$
(29) $\displaystyle+$
$\displaystyle(f^{2}U_{0,3})_{,3}-\frac{f^{3}}{P^{2}}(U_{0,3})^{2}.$ (30)
One can calculate the Bondi news function by applying $\partial_{u}$
$\displaystyle\dot{\overline{\sigma^{0}}}$ $\displaystyle=$
$\displaystyle\frac{\partial}{\partial
u}"U_{0,2}\;\;\textrm{terms}"+\frac{\partial}{\partial
u}"L\;\;\textrm{terms}"$ (31) $\displaystyle=$ $\displaystyle
f(\frac{(P^{2}L)_{,2}}{P})_{,0}+\frac{f}{2}(PL_{,0}^{2})_{,0}+2f^{2}(\frac{P_{,0}U_{0,2}L}{P})_{,0}$
$\displaystyle+$
$\displaystyle[(f^{2})_{,2}U_{0,2}+f^{2}U_{0,22}]_{,0}-f^{3}[\frac{P_{,0}(U_{0,2})^{2}}{P^{2}}]_{,0},$
and the Bondi News function is defined as $\frac{\partial}{\partial
U}{\overline{\sigma^{0}}}=\frac{P}{f}\frac{\partial}{\partial
u}{\overline{\sigma^{0}}}$, thus one can work out the Bondi mass
$\Psi_{2}^{0}+\sigma^{0}\frac{\partial}{\partial U}\overline{\sigma}^{0}$ from
$m_{NU}$ by using (27) and (31). The mass loss comes from the news function
$\frac{\partial}{\partial U}{\overline{\sigma^{0}}}$. When $L=0$, the results
should return to the Gonna-Kramer-98 written by using spherical coordinate.
Also, $\Psi^{0}_{4}=-\frac{\partial^{2}}{\partial U^{2}}\overline{\sigma}^{0}$
can be calculated.
Komar’s angular momentum.– From $x^{2}=\theta,x^{3}=\phi$ and
$\zeta=\sqrt{2}e^{i\phi}\cot\frac{\theta}{2}$ Penrose , we have
$\frac{\partial}{\partial\zeta}=Q^{2}\frac{\partial}{\partial\theta}+Q^{3}\frac{\partial}{\partial\phi}$
where $Q^{2}=-\frac{1}{\sqrt{2}}e^{-i\phi}\sin^{2}\frac{\theta}{2}$ and
$Q^{3}=-i\frac{1}{2\sqrt{2}}e^{-i\phi}\tan\frac{\theta}{2}$. The rotation
vector (asymptotically Killing vector) $\partial/\partial\phi$ is
$\displaystyle\frac{\partial}{\partial\phi}=\phi^{a}=-i\frac{3}{4}(\zeta\frac{\partial}{\partial\zeta}-\overline{\zeta}\frac{\partial}{\partial\overline{\zeta}})$
(32)
where
$\frac{\partial}{\partial\zeta}=\overline{B}\tilde{m}^{a}+L\tilde{n}^{a}+A\tilde{\ell}^{a}$.
Note that "$\tilde{}$" represents the null rotation that make $m,\overline{m}$
tangent for two sphere on null infinity. We obtain the Komar angular momentum
for the twisting space time Komar59 is
$\displaystyle J_{K}$ $\displaystyle=$
$\displaystyle\frac{1}{2\pi}\oint\nabla^{a}\tilde{\phi}^{b}d\tilde{S_{ab}}$
(33) $\displaystyle=$ $\displaystyle\frac{1}{2\pi}\oint i\frac{-3}{2}(\zeta
L-\overline{\zeta}\overline{L}){\rm Re}\tilde{\gamma}dS$ (34)
$\displaystyle\approx$ $\displaystyle\frac{1}{2\pi}\int i\frac{3}{4}(\zeta
L-\overline{\zeta}\overline{L})m_{NU}d\theta d\phi$ (35)
where $dS_{ab}=l_{[a}n_{b]}dS$, "$\approx$" represents approximate on null
infinity, $dS\approx r^{2}\sin\theta d\theta\phi$, $m_{NU}:={\rm
Re}\tilde{\Psi^{0}_{2}}$ and we use the results of Newman-Unti NU
$\gamma=-\frac{\Psi^{0}_{2}}{2r^{2}}+O(r^{-3})$. This angular momentum will
yield $ma$ for Kerr solution.
Conclusions.– We have build up the relationship of how the angular momentum
contribute to the gravitational radiation (the news function) from the exact
solution (the twisting space-time). Dynamical twisting space-time is a
solution that allows the freedom of gravitational radiation from the exact
solution sense and it characterizes spin. Therefore, we must transform it to
the Bondi coordinate and the news function appears. Though there is no
satisfactory way to have an explicit expression for the angular momentum in
terms of the Kerr parameters from spinor construction. However, we use Komar
integral for the twisting space-time and get a general expression that is
related with NP ${\rm Re}\gamma$. ${\rm Re}\gamma$ is something like the
surface gravity of horizon ${\rm Re}\epsilon$ for $\ell$. Further from the
results of Newman-Unti, it can be rewritten as ${\rm Re}\Psi_{2}$, i.e., the
Newman-Unti mass term. Thus it would be very easy to check that our results
should go back to the Kerr solution. We hope that these results will help us
to understand the dynamics of the merger of two spinning BHs and see the
influences of the gravitational angular momentum flux on the spin flip problem
which we have write down the initial state and the final state of the merge
Wang-Wu 2013b . Also, we have worked out Komar angular momentum for Kerr black
hole in Wu2007 and for null infinity in this paper. It would be interesting
to build up a further correlation between horizon and null infinity.
Acknowledgment.– The key ideas of this paper were carried out by YH Wu during
the visit of Albert Einstein Institute (AEI), Golm, July 2013. YH Wu and CH
Wang would like to express deep gratitude to Dr. Jose-Luis Jaramillo for
academic discussion and AEI for hospitality and travel support. CH Wang would
like to thank GR 20 conference center for the travel support. YH Wu would like
to thank Prof. Yun-Kau Lau to first draw her attention to this problem.
Appendix: Kerr solution and its angular momentum.– To calculate Komar’s
angular momentum, one needs to make $m,\overline{m}$ tangent on two sphere at
null infinity for Kerr solution. Firstly, we can write down the Kerr solution
in Bondi coordinate. From the coordinate transformation
$k^{a^{\prime}}=\frac{\partial x^{a^{\prime}}}{\partial x^{a}}k^{a}$, we have
$U_{0,0}=\frac{P_{Kerr}}{f}=1$, then $U_{0}=u$,
$U_{1}=-\frac{a\zeta\overline{\zeta}}{P_{Kerr}^{2}}$,
$R_{-1}=\frac{f}{P_{Kerr}}=1$, $R_{0}=0$, $\zeta_{0}=\zeta$,
$\zeta_{1}=ia\zeta$. The leading order of the contravariant tetrad in Bondi
coordinate $(U,R,\zeta^{\prime},\overline{\zeta}^{\prime})$ is
$\displaystyle\ell^{a}$ $\displaystyle=$
$\displaystyle(0,1,0,0),\;n^{a}=(1,-H_{Kerr},0,0)$ $\displaystyle m^{a}$
$\displaystyle=$
$\displaystyle(\frac{P_{Kerr}L_{Kerr}}{r},\frac{P_{Kerr}L_{Kerr}}{r},-\frac{P_{Kerr}}{r},0),$
where we ignore the higher order terms $O(r^{-2})$. For the Kerr solution, we
use
$\displaystyle P_{Kerr}$ $\displaystyle=$ $\displaystyle
1+\frac{1}{2}\zeta\overline{\zeta}=1+\cot(\frac{\theta}{2})^{2},\;K=1,$
$\displaystyle L_{Kerr}$ $\displaystyle=$
$\displaystyle-i\frac{a\overline{\zeta}}{P^{2}_{Kerr}}=-i\frac{a\sin^{2}\theta}{2}=-W_{Kerr},$
$\displaystyle\varpi_{Kerr}$ $\displaystyle=$
$\displaystyle-a\frac{2-\zeta\overline{\zeta}}{2+\zeta\overline{\zeta}},\;B_{Kerr}=-\frac{1}{\eta
P_{Kerr}}$ $\displaystyle H_{Kerr}$ $\displaystyle=$
$\displaystyle\frac{1}{2}-\frac{mr}{r^{2}+\varpi_{Kerr}^{2}}.$
Secondly, we null rotate it to make $m,\overline{m}$ tangent on the two sphere
at null infinity. We perform type II null rotation
$m^{\prime}=m+bn$,$\ell^{\prime}=\ell+\overline{b}m+b\overline{m}+b\overline{b}n$,
and $n^{\prime}=n$ where $b=-\frac{P_{Kerr}L_{Kerr}}{r},$ and it satisfy the
gauge conditions $\pi=-\overline{\tau},{\rm Im}\rho=0,{\rm Re}\epsilon=0.$
Thus, after null rotation, the Komar angular momentum for Kerr solution is
$ma$ and we also check this point with GRtensor Maple.
## References
* (1) Bondi H van der Burg M G J and Metzner A W K, Proc Roy Soc (London) A269 21 (1962). Sachs, R. K, Proc. Roy. Soc (London), A264, 309-338 (1961).
* (2) J. Foster and E. T. Newman, J. Math. Phys. 8,189 (1967).
* (3) Komar, A., Phys. Rev., 113, 934-936, (1959).
* (4) Newman , E. T. and Unit, T. W. J., J. Math. Phys. 3, 891-901 (1962).
* (5) Baker, J. et al. , Phys. Rev. Lett. 96 111102 2006
* (6) D. Merritt and R. Ekers, Science 297, 1310 (2002);
* (7) Szabados L B 2004,Living Rev. Rel. 7. 4.
* (8) Penrose, R., and Rindler, W., Spinors and space-time, Vol. 1 and Vol.2 (Cambridge University Press, Cambridge; New York, 1984 and 1986).
* (9) R. P. Macedo and A. Saa, Phys. Rev. D 78, 104025 (2008).
* (10) U von der Gönna and D Kramer 1998 Class. Quantum Grav. 15 215
* (11) I. Robinson and A. Trautman, Proc. R. Soc. A 265, 463 (1962).
* (12) L. Rezzolla, R. P. Macedo and J. L. Jaramillo, Phys. Rev. Lett. 104,221101 (2010).
* (13) H. Stephani, D. Kramer, M. A. H. MacCallum, C. Hoenselaers, and E. Herlt,Cambridge University Press (2003).
* (14) Yu-Huei Wu, PhD thesis, University of Southampton (2007).
* (15) Yu-Huei Wu and Chih-Hung Wang,Phys. Rev. D 83, 084044 (2011). Yu-Huei Wu and Chih-Hung Wang, Phys. Rev. D 80, 063002 (2009).
* (16) Wang and Wu, paper in preparation (2013). Yu-Huei Wu, "Understand law of precession during the merger of binary black hole", GR20, Poland.
|
arxiv-papers
| 2013-10-06T16:42:06 |
2024-09-04T02:49:52.017624
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Chih-Hung Wang and Yu-Huei Wu",
"submitter": "Yu-Huei Wu",
"url": "https://arxiv.org/abs/1310.1599"
}
|
1310.1604
|
# The Science Cases for Building
a Band 1 Receiver Suite for ALMA
J. Di Francesco11affiliation: National Research Council Canada, 5071 West
Saanich Rd, Victoria, BC, V9E 2E7, Canada 22affiliation: Dept. of Physics &
Astronomy, University of Victoria, Victoria, BC, V8P 1A1, Canada , D.
Johnstone11affiliation: National Research Council Canada, 5071 West Saanich
Rd, Victoria, BC, V9E 2E7, Canada 22affiliation: Dept. of Physics & Astronomy,
University of Victoria, Victoria, BC, V8P 1A1, Canada , B.
Matthews11affiliation: National Research Council Canada, 5071 West Saanich Rd,
Victoria, BC, V9E 2E7, Canada 22affiliation: Dept. of Physics & Astronomy,
University of Victoria, Victoria, BC, V8P 1A1, Canada , N.
Bartel33affiliation: Dept. of Physics and Astronomy, York University, Toronto,
M3J 1P3, ON, Canada , L. Bronfman44affiliation: Dept. de Astronomía,
Universidad de Chile, Casilla 36-D, Santiago, Chile , S.
Casassus44affiliation: Dept. de Astronomía, Universidad de Chile, Casilla
36-D, Santiago, Chile , S. Chitsazzadeh22affiliation: Dept. of Physics &
Astronomy, University of Victoria, Victoria, BC, V8P 1A1, Canada
55affiliation: Dept. of Physics and Astronomy, The University of Western
Ontario, London, ON, N6A 3K7, Canada , H. Chou66affiliation: Academia Sinica,
Institute of Astronomy and Astrophysics, P.O. Box 23-141, Taipei 10617, Taiwan
, M. Cunningham77affiliation: School of Physics, University of New South
Wales, Sydney, NSW 20152, Australia , G. Duchêne88affiliation: Astronomy
Dept., University of California, Berkeley, CA 94720-3411, USA 99affiliation:
Université Joseph Fourier - Grenoble 1/CNRS, LAOG UMR 5571, BP 53, 38041
Grenoble, France , J. Geisbuesch1010affiliation: National Research Council
Canada, P.O. Box 248, Penticton, BC, V2A 6J9, Canada , A.
Hales1111affiliation: National Radio Astronomy Observatory, 520 Edgemont Road,
Charlottesville, Virginia 22903, USA , P.T.P. Ho66affiliation: Academia
Sinica, Institute of Astronomy and Astrophysics, P.O. Box 23-141, Taipei
10617, Taiwan M. Houde55affiliation: Dept. of Physics and Astronomy, The
University of Western Ontario, London, ON, N6A 3K7, Canada , D.
Iono1212affiliation: National Astronomical Observatory of Japan, 2-21-1 Osawa,
Mitaka, Tokyo, 181-8588, Japan , F. Kemper66affiliation: Academia Sinica,
Institute of Astronomy and Astrophysics, P.O. Box 23-141, Taipei 10617, Taiwan
, A. Kepley1111affiliation: National Radio Astronomy Observatory, 520 Edgemont
Road, Charlottesville, Virginia 22903, USA , P.M. Koch66affiliation: Academia
Sinica, Institute of Astronomy and Astrophysics, P.O. Box 23-141, Taipei
10617, Taiwan , K. Kohno1313affiliation: Institute of Astronomy, The
University of Tokyo, 2-21-1 Osawa, Mitaka,Tokyo 181-0015, Japan , R.
Kothes1010affiliation: National Research Council Canada, P.O. Box 248,
Penticton, BC, V2A 6J9, Canada , S-P. Lai1414affiliation: Institute of
Astronomy and Dept. of Physics, National Tsing Hua University, Taiwan , K.Y.
Lin66affiliation: Academia Sinica, Institute of Astronomy and Astrophysics,
P.O. Box 23-141, Taipei 10617, Taiwan , S.-Y. Liu66affiliation: Academia
Sinica, Institute of Astronomy and Astrophysics, P.O. Box 23-141, Taipei
10617, Taiwan , B. Mason1111affiliation: National Radio Astronomy Observatory,
520 Edgemont Road, Charlottesville, Virginia 22903, USA , T.J.
Maccarone1515affiliation: Department of Physics, Texas Tech University,
Lubbock, TX, 79409-1051, USA , N. Mizuno1212affiliation: National Astronomical
Observatory of Japan, 2-21-1 Osawa, Mitaka, Tokyo, 181-8588, Japan , O.
Morata66affiliation: Academia Sinica, Institute of Astronomy and Astrophysics,
P.O. Box 23-141, Taipei 10617, Taiwan , G. Schieven11affiliation: National
Research Council Canada, 5071 West Saanich Rd, Victoria, BC, V9E 2E7, Canada ,
A.M.M. Scaife1616affiliation: School of Physics and Astronomy, University of
Southampton, Southampton, Hampshire, S017 1BJ, UK , D. Scott1717affiliation:
Dept. of Physics and Astronomy, University of British Columbia, Vancouver, BC,
V6T 1Z1, Canada , H. Shang66affiliation: Academia Sinica, Institute of
Astronomy and Astrophysics, P.O. Box 23-141, Taipei 10617, Taiwan , M.
Shimojo1212affiliation: National Astronomical Observatory of Japan, 2-21-1
Osawa, Mitaka, Tokyo, 181-8588, Japan , Y.-N. Su66affiliation: Academia
Sinica, Institute of Astronomy and Astrophysics, P.O. Box 23-141, Taipei
10617, Taiwan , S. Takakuwa66affiliation: Academia Sinica, Institute of
Astronomy and Astrophysics, P.O. Box 23-141, Taipei 10617, Taiwan , J.
Wagg1818affiliation: European Southern Observatory, Alonso de Cordova 3107,
Vitacura, Casilla 19001, Santiago 19, Chile 1919affiliation: Astrophysics
Group, Cavendish Laboratory, JJ Thomson Avenue, Cambridge, CB30HE, UK , A.
Wootten1111affiliation: National Radio Astronomy Observatory, 520 Edgemont
Road, Charlottesville, Virginia 22903, USA , F. Yusef-Zadeh2020affiliation:
Dept. of Physics and Astronomy and Center for Interdisciplinary Research in
Astronomy, Northwestern University, Evanston, IL 60208, USA
## 1 Executive Summary
The ALMA Band 1 project aims to provide a low-cost solution to one of the
original design goals of the Atacama Large Millimeter/submillimeter Array
(ALMA), access to frequencies of $\sim$40 GHz at high resolution and
sensitivity from the southern hemisphere. In this document, we present a set
of compelling science cases for construction of the ALMA Band 1 receiver
suite. For these, we assume in tandem the updated nominal Band 1 frequency
range of 35-50 GHz and its likely extension up to 52 GHz that together
optimize the Band 1 science return.
A comprehensive comparison of ALMA and the Jansky VLA (JVLA) over 40-50 GHz
finds ALMA having similar sensitivity at lower frequencies but the edge in
sensitivity (e.g., up to a factor of $\sim$2) at higher frequencies. In
addition, ALMA’s larger primary beams allow this sensitivity to be obtained
over wider fields. Furthermore, ALMA Band 1 images will have significantly
greater fidelity than those from the JVLA since ALMA has a larger number of
instantaneous baselines. ALMA’s smaller dishes (and the ACA, if needed) in
principle can allow the recovery of more extended emission. Finally, ALMA Band
1 will likely include frequencies of 50-52 GHz that the JVLA simply cannot
observe.
The scope of the science cases ranges from nearby stars to the re-ionization
edge of the Universe. Two cases provide additional leverage on the present
ALMA Level One Science Goals and are seen as particularly powerful motivations
for building the Band 1 receiver suite: (1) detailing the evolution of grains
in protoplanetary disks, as a complement to the gas kinematics, requires
continuum observations out to 35 GHz ($\sim$9 mm); and (2) detecting CO 3–2
line emission from galaxies like the Milky Way during the epoch of re-
ionization, i.e., 6 $<z<$ 10, also requires Band 1 receiver coverage. Indeed,
Band 1 will increase the volume of the observable Universe in CO lines by a
factor of 8. The range of Band 1 science is very broad, however, and also
includes studies of galaxy clusters (i.e., via the Sunyaev-Zel’dovich Effect),
very small dust grains in the ISM, the Galactic Center, solar studies, pulsar
wind nebulae, radio supernovae, X-ray binaries, dense cloud cores, complex
carbon-chain molecules, ionized gas (e.g., in HII regions), masers, magnetic
fields in the dense ISM, jets and outflows from young stars, the co-evolution
of star formation with active galactic nuclei, and the molecular mass in
moderate redshift galaxies.
## 2 Introduction
The Atacama Large Millimeter/submillimeter Array (ALMA) will be a single
research instrument composed of at least fifty 12-m antennas in its 12-m Array
and twelve 7-m high-precision antennas plus four 12-m antennas in its compact
array (the Atacama Compact Array; ACA), located at a very high altitude of
5000 m on the Chajnantor plateau of the Chilean Andes. The weather conditions
at the ALMA site will allow transformational research into the physics of the
cold Universe across a wide range of wavelengths, from radio to submillimeter.
Thus, ALMA will be capable of probing the first stars and galaxies and
directly imaging the disks in which planets are formed. ALMA will be the pre-
eminent astronomical imaging and spectroscopic instrument at
millimetre/submillimetre wavelengths for decades to come. It will provide
scientists with capabilities and wavelength coverage that complement those of
other key research facilities of its era, such as the James Webb Space
Telescope (JWST), 30-m class Giant Segmented Mirror Telescopes (GSMTs), and
the Square Kilometer Array (SKA).
ALMA will revolutionize many areas of astronomy and an amazing breadth of
science has already been proposed (see, for example, the ALMA Design Reference
Science Plan). The technical requirements of the ALMA Project are, however,
driven by three specific Level One Science Goals:
(1) The ability to detect spectral line emission from CO or CII in a normal
galaxy like the Milky Way at a redshift of $z=3$, in less than 24 hours of
observation.
(2) The ability to image the gas kinematics in a solar-mass
protostellar/protoplanetary disk at a distance of 150 pc (roughly the distance
of the star-forming clouds in Ophiuchus or Corona Australis), enabling one to
study the physical, chemical, and magnetic field structure of the disk and to
detect the tidal gaps created by planets undergoing formation.
(3) The ability to provide precise images at an angular resolution of 0.1′′.
Here the term “precise image” means an accurate representation of the sky
brightness at all points where the brightness is greater than 0.1% of the peak
image brightness. This requirement applies to all sources visible to ALMA that
transit at an elevation greater than 20∘.
ALMA was originally envisioned to provide access to all frequencies between 31
GHz and 950 GHz accessible from the ground. During a re-baselining exercise
undertaken in 2001, the entire project was scrutinized to find necessary cost
savings. The two lowest receiver frequencies, Bands 1 and 2, covering 31–45
GHz and 67–90 GHz respectively, were among those items delayed beyond the
start of science operations. Nevertheless, Band 1 was re-affirmed as a high
priority future item for ALMA.
In May 2001, John Richer and Geoff Blake prepared the document Science with
Band 1 (31–45 GHz) on ALMA as part of the re-baselining exercise. Key
arguments for Band 1 receivers included their abilities to: (1) enable
exciting science opportunities, bringing in a wider community of users; (2) be
a significantly faster imaging and survey instrument than the upgraded VLA
(now known as the Jansky VLA or JVLA), especially due to the larger primary
beam; (3) provide access to the southern sky at these wavelengths; (4) allow
excellent science possible even in “poor” weather; (5) be a relatively cheap
and reliable receiver to build and maintain; and (6) be a very useful
engineering/debugging tool for the entire array given the lower contribution
of the atmosphere at many of its frequencies relative to other bands.
The Richer/Blake document was followed by an ASAC Committee Report in October
2001, after the addition of Japan into the ALMA project re-opened the question
of observing frequency priorities for those Bands which had been put on hold
during re-baselining. The unanimous recommendation of the ASAC was to put Band
10 as top priority, followed by a very high priority Band 1. At that time, the
key science cases for Band 1 receivers were seen to be (1) high-resolution
Sunyaev-Zel’dovich effect (SZE) imaging of cluster gas at all redshifts; and
(2) mapping the cold ISM in Galaxies at intermediate and high redshift.
The scientific landscape has changed significantly since 2001 and thus it is
time to re-examine the main science drivers for ALMA Band 1 receivers, even
reconsidering the nominal frequency range of Band 1 itself to optimize the
science return. In addition, the ALMA Development process has begun, and now
is the time to put forth the best case for longer wavelength observing with
ALMA. In October 2008, two dozen astronomers from around the globe met in
Victoria, Canada to discuss Band 1 science. This paper summarizes the
outstanding cases made possible with Band 1 that were highlighted at that
meeting and since. In Section 3, we describe the new nominal Band 1 frequency
range of 35-50 GHz, and its likely extension to 52 GHz. In Section 4, we
present two science cases that reaffirm and enhance the already established
ALMA Project Level One Science Goals. Section 5 discusses both weather
considerations at the ALMA site and compares the observing capabilities of
ALMA and the JVLA over Band 1 frequencies. In Section 6, we provide a
selection of continuum and line science cases that reinforce the breadth and
versatility of the Band 1 receiver suite. Finally, Section 7 briefly
summarizes the report.
## 3 The Band 1 Frequency Range
Band 1 was originally defined as 31.3–45 GHz, with the lower end set to the
lower edge of a frequency range assigned to radio astronomy and the upper set
to include SiO $J$=1–0 emission at 43 GHz. Receiver technology advances,
however, have made it possible to widen and shift the Band 1 range and
optimize the science return of Band 1. For example, a wider range and shift to
higher frequencies will allow molecular emission from galaxies at a wider
range of (slightly lower) redshifts to be explored. Also, it will allow
molecular emission from several new species in our Galaxy to be probed. (Of
course, this shift does in turn remove the ability to detect molecular
emission from some higher redshift galaxies and some other Galactic
transitions.) Furthermore, a shift to higher frequencies for Band 1 will
improve (slightly) the angular resolution of continuum observations and better
exploit the advantages of the dry ALMA site.
A review of the nominal frequency range by the Band 1 Science Team (i.e.,
several authors of this document) in June 2012 resulted in a proposed new Band
1 frequency range definition, nominally 35–50 GHz with a likely extension up
to 52 GHz. The shift up to 50 GHz will allow the important line CS $J$=1–0 at
48.99 GHz to be observable with ALMA. In addition, the nominal range of 35-50
GHz alone is itself $\sim$10% wider than before. As it will provide the
highest sensitivities, the nominal range will be preferred for high-redshift
science. The extension to 50-52 GHz, which the JVLA cannot observe, may be
somewhat adversely affected by atmospheric O2, resulting in lower relative
sensitivity. Since numerous transitions of other interesting molecules have
rest frequencies at 50-52 GHz, however, this extension will allow such
emission to be probed toward sources in our Galaxy. This document has been
updated in September 2012 to reflect the new nominal frequency range and the
extension. See Section 5 for a comparison of the sensitivities and imaging
characteristics of ALMA and the JVLA over Band 1 frequencies.
## 4 Level One Science Cases for Band 1
In this section, we present two science cases that reaffirm and enhance the
already established ALMA Project Level One Science Goals: Evolution of Grains
in Disks Around Stars (§4.1) and The First Generation of Galaxies (§4.2).
Further science cases are presented in §6.
### 4.1 Evolution of Grains in Disks Around Stars
#### 4.1.1 Protoplanetary Disks
Planet formation takes place in disks of dust and gas surrounding young stars.
It is within these gas-rich protoplanetary disks that dust grains must
agglomerate from the sub-micron sizes associated with the interstellar medium
to larger pebbles, rocks and planetesimals, if planets are ultimately to be
formed. The timescale of this agglomeration process is thought to be a few
tens of Myr for terrestrial planets, while the process leading to the
formation of giant planet cores remains uncertain. Core accretion models
require at least a few Myr to form Jovian planets (Pollack et al. 1996), while
dynamical instability models could form giant planets on orbital timescales
($t\ll 1$ Myr; Boss 2005).
Gravitational instability models require high disk masses to form planets. So
far, most accurate disk mass estimates come from submillimeter and millimeter
observations, where the dust is optically thin. Andrews & Williams (2007a,
2007b) show that submillimeter observations of dozens of protoplanetary disks
reveal that only one system could be gravitationally unstable, conflicting
with the high frequency of Jovian planets seen around low mass stars. Have
these relatively young (1–6 Myr) systems already formed planets, or is most of
the dust mass locked into larger grains and therefore not accounted for in
submillimeter and millimeter observations? If grain growth to centimeter sizes
has occurred, most of a disk’s dust mass would reside in the large particle
population, which would emit at longer millimeter and centimeter wavelengths.
Figure 1 from Greaves et al. (in prep.) compares disk masses for objects in
Taurus and Ophiuchus derived from 9 mm and 1.3 mm dust fluxes. The longer
wavelength masses are found to be generally higher than the shorter wavelength
values, indicating that a significant fraction of the disks’ total dust masses
are indeed locked up in larger grains.
Figure 1: Disk masses measured from 9 mm continuum emission compared to those
measured from 1.3 mm continuum emission in the regions of Taurus and
Ophiuchus. Many disks show higher mass measurements at the longer wavelength,
indicating the presence of larger grains than those detected at 1.3 mm
measurements. (Greaves et al., in prep.)
Identifying where and when dust coagulation occurs is critical to constrain
current models of planetary formation. Growth from sub-micron to micron-sized
particles can be traced with infrared spectroscopy and imaging polarimetry.
The next step, growth beyond micron sizes, is readily studied by determining
the slope of the spectral energy distribution (SED) of the dust thermal
emission at submillimeter and millimeter wavelengths. The dust mass opacity
index at wavelengths longer than 0.1 millimeter is approximately a power-law
whose normalization depends on the dust properties, such as composition, size
distribution, and geometry (Draine 2006). The index of the power law is
commonly given by $\beta$. The presence of large grains is detectable through
a decrease in $\beta$, which can be derived directly from the slope of the
Rayleigh-Jeans tail of the SED, $\alpha$, where $\beta=\alpha-2$ when the
emission is optically thin. Numerous studies have revealed that the $\beta$
values of disks are substantially lower than the typical ISM value of $\sim 2$
(e.g., Testi et al. 2003; Weintraub et al. 1989; Adams et al. 1990; Beckwith
et al. 1990; Beckwith & Sargent 1991; Mannings & Emerson 1994).
The key stumbling block to the interpretation of $\beta$ occurs when the disk
is not resolved spatially. The amount of flux detected at a given wavelength
is a function of both $\beta$ and the size of the disk (Testi et al. 2001).
Resolving the ambiguity therefore is truly a matter of resolution, and
sufficient resolution is only offered at these wavelengths by interferometers.
Among the three high level science goals of ALMA is the ability to detect and
image gas kinematics in protoplanetary disks undergoing planetary formation at
150 pc. At ALMA’s observing wavelengths, its capability for imaging the
continuum dust emission in these disks is also second-to-none. At present,
however, the longest wavelength that ALMA can reach is 3.6 mm. Given that dust
particles emit very inefficiently at wavelengths longer than their sizes, the
present ALMA design will not be sensitive to particles larger than $\sim 3$
mm. This situation negates ALMA’s potential ability to follow the dust grain
growth from mm-sized to cm-sized pebbles in protoplanetary disks.
Figure 2 shows the SEDs for three different circumstellar disk models,
computed using the full dust radiative transfer MCFOST code (Pinte et al.
2006; Pinte et al. 2009). The model parameters are representative of
protoplanetary disks (although there is substantial object-to-object
variation). The circumstellar disk is passively heated by a 4000 K, 2 L⊙
central star and the system is located 160 pc away. The dust component of the
disk is assumed to be fully mixed with the gas and the latter is assumed to be
in vertical hydrostatic equilibrium. The disk extends radially from 1 AU to
100 AU. The total dust mass in the model is $10^{-3}$ M⊙ (the gas component is
irrelevant for continuum emission calculations, so its mass is not set in the
model, though a typical 100:1 gas:dust ratio is generally assumed). The dust
population is described by a single power-law size distribution $N(a)\propto
a^{-3.5}$ with a minimum grain size of 0.03 $\mu$m and extending to 10 $\mu$m,
1 mm or 1 cm depending on the model. The dust composition is taken to be the
“astronomical silicates” model from Draine (2003).
Figure 2: Spectral energy distribution plot showing the differences between
three disk models having different maximum grain sizes. The solid curve is the
model with a${}_{\rm max}=1$ cm, which keeps declining with roughly constant
slope all the way to 1 cm. The two dashed curves are for a${}_{\rm max}=10\
\mu$m and 1 mm. The top one, which breaks around 5 mm is the model with
a${}_{\rm max}=1$ mm. It’s interesting to note how the fluxes are very much
the same for a${}_{\rm max}=1$ mm or 1 cm, except precisely towards ALMA’s
Band 1. There is at least an order of magnitude difference in power at 1 cm
between the max${}_{\rm size}=1$ mm versus the max${}_{\rm size}=1$ cm disks.
These models indicate that observations at the ALMA Band 1 regime are crucial
for determining whether grain-growth to cm-sizes is indeed occurring.
Figure 2 reveals that observations in the ALMA Band 1 spectral region are
crucial for determining whether grain-growth to cm-sizes is indeed occurring.
The 1 cm flux density of the max${}_{size}=$1 cm disk model is $\sim
50\,\mu$Jy, comparable to the 1 $\sigma$ sensitivities provided by ALMA’s Band
1 with 1 minute integration. Besides ALMA, there are no existing or planned
southern astronomical facilities capable of observing to such depths at these
frequencies. Therefore, if ALMA Band 1 receivers are not built there will be
no way of putting ALMA observations of protoplanetary disks in the context of
coagulation of dust grains to centimeter sizes. Though such information could
be acquired in part with the JVLA (for sufficiently northern sources), ALMA
Band 1 receivers would yield superior data for comparison with those of other
Bands, given greater similarities in spatial frequency coverage. (Spatial
frequency coverage depends on the latitude of the observatory and the
declination of the source.)
By complementing observations in other ALMA Bands, Band 1 will provide a
crucial longer wavelength lever to minimize the uncertainty in $\alpha$.
Evidence for small pebbles has been detected in several disks (Rodmann et al.
2006). The prime example is TW Hya, a protoplanetary disk 50 pc from the Sun
(Wilner et al. 2000). Its SED is well matched by an irradiated accretion disk
model fit from 10s of AU to an outer radius of 200 AU and requires the
presence of particle sizes up to 1 cm in the disk (see Figure 3). The measured
$\beta$ is $0.7\pm 0.1$ (Calvet et al. 2002). To date, no trend in $\beta$ has
been detected with stellar luminosity, mass or age (Ricci et al. 2010). Lower
$\alpha$ values are associated with less 60 $\mu$m excess, however, suggesting
that settling or agglomeration processes could be removing the smallest
grains, decreasing the shorter wavelength emission (Acke et al. 2004). (See
§6.1.1 for further discussion of probes of small grains with the ALMA Band 1
receivers.)
At the resolution provided by its longest baselines at $\sim$40 GHz
($\sim$0.14′′), ALMA will easily resolve protoplanetary disks at the distance
of the closest star-forming regions (50–150 pc). These resolved images will
provide the most accurate determination of the disk’s dust mass. The dust
distribution at centimeter wavelengths can then be compared to millimeter and
submillimeter images, revealing where in the disk dust coagulation is
occurring. For example, previous investigations of the radial dependency of
dust properties in disks by Guilloteau et al. (2009) and Isella et al. (2010)
were conducted at 1 mm and 3 mm, and as such they were sensitive to only
millimeter-sized grains. Note, however, that Melis et al. (2011) used the
Jansky VLA to map the 7 mm emission from the protoplanetary disk around the
young source L1527 IRS at $\sim$1.5′′ and tentatively detected a dearth of
“pebble-sized” grains. ALMA Band 1 receivers will help clarify this situation.
As described above, Band 1 data will be sensitive to larger grains. Moreover,
through detection of concentrations of such large grains, protoplanets in
formation can be identified. These condensations are predicted by simulations
of gravitational instability models (see Figure 4a; Greaves et al. 2008) and
have been detected in the nearby star HL Tau (Figure 4b; Greaves et al. 2008).
Figure 3: Spectral energy distribution of TW Hya, showing the fit to the SED
for an irradiated accretion disk model with a maximum particle size of 1 cm
(Calvet et al. 2002).
Detecting dust emission at centimeter wavelengths also requires high
sensitivity, because its brightness is several orders of magnitude lower than
in the submillimeter. In addition, at wavelengths longer than 7 mm (i.e.,
$\nu$ $<$ 45 GHz), the contribution from other radiative processes, such as
ionized winds, can contribute significantly to the total flux and complicate
the interpretation of detected emission. Rodmann et al. (2006) found that the
contribution of free-free emission to the total flux is typically 25% at a
wavelength of 7 mm. Observations of continuum emission at the 35-50 GHz (6–9
mm) spectral range enabled by Band 1 would increase substantially the sampling
rate in the region where emission is detected from both the free-free and
thermal dust emission components. The synergy with the JVLA will provide a
longer wavelength lever for sources observed in common, providing an estimate
of the free-free contribution to the Band 1 flux. Such data would not be
essential, however, given wide frequency coverage within Band 1 alone. For
example, multiple continuum observations could be used to quantify accurately
the relative amounts of free-free and dust emission through changes in
spectral slope, and thereby determine precisely the contribution from large
dust grains (i.e., protoplanetary material).
Figure 4: (Left) Image from an SPH simulation showing the surface density
structure of a 0.3 M⊙ disk around a 0.5 M⊙ star. A single dense clump has
formed in the disk, at a radius of 75 AU and with a mass of $\sim 8$ MJup.
(Right) VLA 1.3 cm images toward HL Tau. The main image shows natural
weighting with a beam of 0.11′′ FWHM. The arrow indicates the jet axis. Upper
inset: compact central peak subtracted. Lower inset: uniform weighting, with a
beam of 0.08′′ FWHM. The compact object lies to the upper right hand side.
This condensation was also detected at 1.4 mm with the BIMA array (Welch et
al. 2004).
In summary, ALMA Band 1 receivers would provide the sensitivity to long
wavelength emission needed to probe dust coagulation and growth in
protoplanetary disks observed at higher-frequency bands. Of course, ALMA Band
1 will allow such investigations of sources too far south to observe with the
JVLA. (For the highest resolutions, the improved phase stability available at
ALMA will also be very important.) Furthermore, as comparisons with higher
frequencies are better when there is similar spatial frequency coverage,
however, sources are best observed at different wavelengths from the same
latitude, favouring ALMA data over JVLA data even for northern sources.
#### 4.1.2 Debris Disks
Around main sequence stars, pebble-sized bodies are produced differently than
in disks around pre-main-sequence stars. Here, destructive collisional
cascades from even larger planetesimals through to centimeter, millimeter, and
then micron-sized particles provides ongoing replenishment of the debris
population (Wyatt 2009; Dullemond & Dominik 2005). The methods for detecting
large (i.e., centimeter-sized) grains is the same as in protoplanetary disks,
despite their origin in destructive rather than agglomerative processes. In
each case, the longer the wavelength at which continuum emission is detected,
the larger the grains that must be present in the system.
Using ALMA Band 7, Boley et al. (2012) detected the debris disk of Fomalhaut,
and noted its sharp inner and outer boundary. Band 1 images, however, could
show higher contrast features in debris disks compared to other ALMA Bands,
due to the longer resonant lifetimes of the larger particles that dominate the
emission. This sensitivity in turn will help detect any edges and gaps in the
disks. Dramatic changes in the morphology of debris disks as a function of
wavelength have already been observed (e.g., Maness et al. 2008), but not yet
at the long wavelengths Band 1 will probe. When observed, such structures are
often considered signposts to the existence of planets.
Detections of debris disks in Band 1 will be challenging compared to detecting
forming condensations in protoplanetary disks. Debris disks typically have
relatively low surface brightnesses and large spatial distributions 100s of AU
in radii. They also can be found much closer to the Sun than the nearest
protoplanetary disks. Indeed, the closest disks could subtend as much as
$\sim$150′′ on the sky (assuming a 300 AU diameter disk at 2 pc). Therefore,
ALMA’s large field-of-view relative to other long wavelength instruments, such
as the JVLA, will be very advantageous for imaging these objects. (Mosaicking
will still be required to image the largest ones on the sky.) In addition, the
ALMA 12-m Array’s smaller minimum baselines and the ACA will provide higher
sensitivity to the low surface brightness emission from these objects.
### 4.2 The First Generation of Galaxies:
Molecular gas in galaxies during the era of re-ionization
The first generation of luminous objects in the Universe began the process of
re-ionizing the intergalactic medium (IGM). The detection of large-scale
polarization in the cosmic microwave background (CMB), caused by Thomson
scattering of the CMB by the IGM during re-ionization, suggests that the
Universe was significantly ionized as far back as $z~{}\approx$ 11.0 $\pm$ 1.4
(Dunkley et al. 2009). The “near” edge of the era of re-ionization has been
inferred from the detection of the Gunn-Peterson effect (Gunn & Peterson 1965)
toward galaxies with $z\gtrsim 6$ (Fan et al. 2006a,b). The nearly complete
absorption of all continuum shortward of the Ly$\alpha$ break is due to
moderate amounts of neutral hydrogen in the IGM, suggesting re-ionization was
complete by $z$ $\approx$ 6\. The Gunn-Peterson effect also insures that at
these redshifts the Universe is opaque at wavelengths shorter than $\sim$
1$\,\mu$m.
Figure 5: VLA redshifted CO $J$=3–2 map of the quasar J1148+5251 using the
combined B- and C-array data sets (covering the total bandwidth, 37.5 MHz or
240 km s-1), from Walter et al. (2004). Contours are shown at –2, –1.4, 1.4,
2, 2.8, and 4 $\times\sigma$ (1 $\sigma$ = 43 $\mu$Jy beam-1). The beam size
(0.35″$\times$0.30″) is shown in the bottom left corner; the plus sign
indicates the SDSS position (and positional accuracy) of J1148+5251.
To study the first generations of galaxies, and to understand the origins of
the black hole-bulge mass relation, it will be necessary to study the star-
formation properties of galaxies in the $6\lesssim z\lesssim 11$ range. Quasar
hosts and other sources are rapidly being discovered at the near end of this
range (e.g., Cool et al. 2006; Mortlock et al. 2008; Glikman et al. 2008;
Willott et al. 2009), and searches are underway for even more distant objects
(e.g., Ota et al. 2008; Bouwens et al. 2009).
Recently, CO has been detected111Note that interferometers in general have an
advantage over single-dish telescopes when detecting molecular emission at
high redshift since their high-resolution imaging capabilities provide the
spatial information needed to associate a detection with a specific object. in
galaxies at redshifts $>$6\. These and other observations in the cm/mm of
$z>6$ galaxies are summarized by Carilli et al. (2008; see also the large
surveys of CO at $z$ $>$ 6 by Wang et al. 2010, 2011a and references therein).
Current instrumentation sensitivities are such that detections are limited to
hyperluminous infrared galaxies, i.e. L${}_{\rm FIR}>10^{13}$ L⊙. Only a small
fraction of galaxies are this luminous. The best-studied such object is
J1148+5251 with a redshift of $z=6.419$ (see Carilli et al. 2008). For
example, Walter et al. (2004) imaged the CO $J$=3–2 emission (Figure 5) using
the VLA, from which they were able to infer the dynamical mass. Walter et al.
(2009) were not able to detect the [NII] line at 205 $\mu$m, but did detect
the CO $J$=6–5 transition. More recently, Wang et al. (2011b) detected the
lower-energy CO $J$=2–1 transition and Reichers et al. (2009) imaged CO
$J$=7–6 and CI (${}^{3}P_{2}$–${}^{3}P_{1}$) emission towards this source.
These and other (dust continuum) observations show that there was already a
significant abundance of metals and dust by this epoch.
Figure 6 shows the observable frequency of rotational transitions of 12CO,
from $J$=1–0 through $J$=10–9, as a function of redshift. Also shown are the
frequency ranges of the ALMA Bands (excluding Band 2 for clarity). Note that
this Figure shows the new nominal range of Band 1 of 35-50 GHz, as this range
will yield the highest sensitivities. As the Figure shows, Band 1 receivers
will be able to detect galaxies in $J$=3–2 at $6\lesssim z\lesssim 9$, i.e.,
in the redshifts of the era of re-ionization ($z\mathrel{\raise
1.50696pt\hbox{$\scriptstyle>$}\kern-6.00006pt\lower
1.72218pt\hbox{{$\scriptstyle\sim$}}}6$), while higher Bands can only observe
higher-$J$ lines that may be less excited. (For example, Band 3 receivers
would be able to detect $J$=6–5 emission in the range $4.8\lesssim z\lesssim
7.2$.) Moreover, Band 1 receivers will enable coverage for $J$=2–1 and $J$=1–0
emission at $3.6\lesssim z\lesssim 5.6$ and $1.3\lesssim z\lesssim 2.3$,
respectively. Assuming a 150 $\mu$Jy CO $J$=2–1 line of width $\sim$600 km s-1
at $z=5.7$, a 5 $\sigma$ detection would take less than 4 hours with the
50-antenna ALMA 12-m Array.
Figure 6: Observable frequencies of 12CO rotational transitions and [CII]
2P3/2–2P1/2 as a function of redshift. The frequency ranges of the ALMA Bands
are also shown. Note that the range for Band 1 reflects the new nominal range
of 35-50 GHz.
Band 1 will also allow multiline observations toward certain subsets of
redshifts. For example, galaxies at $1.3\lesssim z\lesssim 2.3$ can be
observed in Band 1 but also at $J$=4–3 and $J$=3–2 in Band 4 (NB: a small gap
exists at $z$ $\approx$ 1.8). Figure 6 also shows that in addition the [CII]
2P3/2–2P1/2 line can be observed toward a subset of these galaxies at
$1.6\lesssim z\lesssim 2.2$ using Band 9. The [CII] line can also be observed
toward galaxies at $2.8\lesssim z\lesssim 5.9$ using Bands 7 and 8 (NB: a
small gap in redshift coverage exists at $z$ $\approx$ 4).
As with other ALMA Bands, high-redshift science will be done with Band 1 in a
targeted mode, i.e., towards known high-$z$ sources. An instantaneous $\sim$8
GHz range of frequency coverage, however, will allow significant sensitivity
to other sources proximate on the sky to the known target source but at quite
different redshifts. (If the target sources are within clustered environments,
other sources may even be found at similar redshifts.) Indeed, “blank-sky”
surveys, made by pointing ALMA towards one location but stepping through the
entire Band 1 frequency range, are an enticing possibility (see, e.g., Aravena
et al. 2012). In particular, the ALMA 12-m Array’s antennas provide a much
larger instantaneous field-of-view than the JVLA’s antennas, allowing wider
searches of blank sky.
In summary, ALMA Band 1 will allow for wide-band observations of molecular
emission from many interesting galaxies in the era of re-ionization. Band 1
allows for observations of lower-$J$ lines that are complementary to lines
detected with higher frequency bands. In particular, ALMA’s southern location
will allow observations of objects not observable (well or at all) with the
JVLA. Also, its larger field-of-view gives it an edge in areal “blank sky”
coverage for detecting at similar or different redshifts sources proximate to
known targets.
#### 4.2.1 Quasar Host Galaxies
The discovery of molecular gas in quasar host galaxies at $z\sim 6$, when the
Universe was less than 1 Gyr old (Walter et al. 2003; Bertoldi et al. 2003;
Carilli et al. 2007), has opened a new window on the study of gas in systems
that contributed to the re-ionization of the Universe. Studies of how the
molecular gas properties should evolve, and how they can be used to reveal the
dynamics of these massive systems, have recently prompted a new generation of
semi-analytic models with the further aim of understanding how high-redshift
quasars fit within the context of large-scale structure formation. Li et al.
(2007, 2008) have used state of the art N-body simulations to show that the
observed optical properties of high-redshift quasars can be explained if these
objects formed early on in the most massive dark matter halos ($\sim 8\times
10^{12}$ M⊙). These models predict that the most luminous quasars should
evolve due to an increase of major mergers, which one would expect to find
evidence for in the CO line profiles and the spatial distribution of the
molecular gas (Narayanan et al. 2008). Detailed radiative transfer models of
the FIR spectral energy distribution of these systems have been driven by the
observations of one $z=6.42$ quasar (namely J1148+5251; Walter et al. 2003,
2004). Larger samples of CO-detected quasars are needed to provide better
constraints on the models and constrain dynamical masses to compare with
infrared measurements of black-hole masses (e.g., from MgII lines) and explore
the (possible) evolution of the relation between the masses of central black
holes and bulges. Current 3 mm surveys of high-$J$ CO line emission in $z\sim
6$ FIR-luminous quasars are being conducted with the PdBI, having successfully
detected CO line emission in eight objects (Wang et al. 2010, 2011a).
Lower-$J$ lines, like those accessible with ALMA Band 1, will trace the more
abundant lower density gas in these systems. Here again, ALMA’s southern
location will prove to be an advantage for targets too far south to be well
observed with the JVLA.
#### 4.2.2 Lyman-$\alpha$ Emitters
The rarity of the luminous quasars at early times suggests that their UV
emission was unlikely to have contributed significantly to the re-ionization
of the Universe (e.g., Fan et al. 2001). A more important type of galaxy in
the context of cosmic re-ionization are the Lyman-$\alpha$ emitters (hereafter
LAEs). These galaxies were discovered through their excess emission in narrow-
band images centered on the redshifted Lyman-$\alpha$ line (e.g. Hu et al.
1998; Rhoads et al. 2000; Taniguchi et al. 2005), and constitute a significant
fraction of the star-forming galaxy population at $z\sim 6$. While the star-
formation rates in LAEs inferred from their UV continuum emission are a few
tens of solar masses per year (e.g., Taniguchi et al. 2005), their number
density and the shape of the Lyman-$\alpha$ emission line provide important
probes of physical conditions in the Universe around the epoch of re-
ionization. As such, it is very important that we understand the properties
related to their star-formation activity. In particular, we need to quantify
the amount of molecular gas available for fuel. Wagg, Kanekar & Carilli (2009)
used the Green Bank Telescope to search for CO $J$=1-0 line emission in two
$z>6.5$ LAEs, including the highest spectroscopically confirmed redshift LAE
at $z=6.96$ (Iye et al. 2006). The limits to the CO line luminosity implied by
the non-detections of CO J=1–0 in these two objects suggest modest molecular
gas masses ($\mathrel{\raise
1.50696pt\hbox{$\scriptstyle<$}\kern-6.00006pt\lower
1.72218pt\hbox{{$\scriptstyle\sim$}}}$ 1010 M⊙). This conclusion, however, is
based on observations of only two objects, and future studies would benefit
from the sensitivity gained by observing higher-$J$ CO transitions, whose flux
density may scale as $\nu^{2}$ due to a contribution to the molecular gas
excitation by the cosmic microwave background radiation (19 K at $z=6$). With
other facilities, it has been proven challenging to detect even the higher
energy CO $J$=2–1 line from Lyman-$\alpha$-emitting galaxies at these
redshifts, using existing facilities (Wagg & Kanekar 2011). At these
redshifts, such studies would require ALMA, including the Band 1 receivers.
Again, ALMA’s southern location is advantageous for the detection of more
southern LAEs.
## 5 Suitability of Band 1 for ALMA vs. JVLA
Here we compare the relative capabilities of ALMA and the Jansky Very Large
Array (JVLA) over Band 1 frequencies in common. The JVLA currently has
observing capability over the nominal Band 1 frequency range of 35–50 GHz,
through its receivers in the Ka-band (26.5–40 GHz) and Q-band (40–50 GHz).
ALMA Band 1, however, will likely be extended to 50-52 GHz, frequencies the
JVLA cannot observe. In the following, we compare the differences in site
conditions and array characteristics that show that Band 1 observing is
superior with ALMA than with the JVLA.
### 5.1 Site Conditions
ALMA is located on the Llano de Chajnantor at a higher altitude (5040 m) than
the JVLA on the Plains of San Agustin (2124 m). Opacity in Band 1 consists of
a wet component of atmospheric water vapor and a dry component of non-H2O
gases, like O2. The quantity of the wet component, as measured by precipitable
water vapor (PWV) affects more the lower end of the Band 1 frequency range.
The dry component, however, dominates at the upper end. Nevertheless, the ALMA
site is very well-suited for Band 1 observing. Even during the worst octile of
weather, however, the typical optical depth through the Band 1 Receiver range
is less than 0.1. Though other frequency ranges like Band 3 can still use such
weather, the addition of cloud cover and water droplets in the air make even
lower frequency observations more attractive.
The PWV over the JVLA during the years 1990-1998 was measured to range between
4.5 mm in winter and 14 mm in summer with a $\pm$2 mm scatter throughout the
year (Butler 1998; VLA Memo 237). In comparison, the PWV over ALMA during the
years 1995-2003 was measured to range between 1.2 mm in winter to 3.5-7.0 mm
in summer (median $\approx$ 1.4 mm), using opacity data obtained by Otórola et
al. (2005; ALMA Memo 512) and conversions provided by D’Addario & Holdaway
(2003; ALMA Memo 521). For frequencies $<$45 GHz, Butler (2010; VLA Test Memo
232) found empirically a linear relation between opacity and PWV, where
opacities varied from 6% to 10% from 1 mm to 14 mm. Assuming this relation is
applicable to both observatories, we find the atmospheric opacities at $<$45
GHz over ALMA to be generally half those over JVLA.
Phase stability over the JVLA was measured with a 300-m baseline test
interferometer at 11.3 GHz, and median characteristics from one year of data
were reported by Butler & Desai (1999; VLA Test Memo 222). They found median
phase variation rms values ranging from 2-2.5∘ in winter nighttime to $>$10∘
in summer daytime. Scaling these values to the zenith and converting to path
delay rms fluctuations, these phases convert to 430-540 fsec to 2100 fsec,
respectively. For ALMA, D’Addario & Holdaway, using six years of data from a
similar 300-m baseline test interferometer, determined a median path delay
fluctuation of 500 fsec. (A seasonal breakdown was not provided.) Though the
data are somewhat scant, the overall median path delay at ALMA is about equal
that of the best median path delays at the JVLA in winter nighttime. Note,
however, that phase stability can be mitigated by water vapor radiometer data
available at both sites.
### 5.2 Array Characteristics
The most important difference between the characteristics of ALMA and the JVLA
is that they are located at very different latitudes, the former at
$-23^{\circ}$ and the latter at $+34^{\circ}$. Some sources too far south to
be observed at the JVLA (or at least observed well) will be observable with
ALMA. (The Australia Telescope Compact Array (ATCA) can also observe some Band
1 frequencies from the southern hemisphere but at much lower relative
sensitivity than ALMA or the JVLA. Hence, we do not consider it further.)
Important targets in the southern hemisphere that are better observed at ALMA
than the JVLA (if at all) include Sgr A*, the center of our Galaxy, the
Magellanic Clouds, the closest neighboring galaxies, and TW Hya, the closest
protoplanetary disk. Indeed, any source observed with ALMA in higher frequency
bands can be more effectively observed at 35-50 GHz with Band 1 receivers.
Also, with numerous satellite observatories providing full sky coverage (e.g.,
JWST, Spitzer, Herschel), having full-sky coverage from ground-based
facilities at important frequencies is optimal.
Table 1: Summary of general properties of the ALMA Band 1 and JVLA | ALMA |
---|---|---
| Band 1 | JVLA
Latitude | $-23\arcdeg$ | $+34\arcdeg$
Altitude (m) | 5040 | 2124
No. of antennas | 50 | 25
Antenna diameter | 12 | 25
Pointing accuracy (arcsec) | 0.6 | 2–3
Frequencies (GHz) | 35–52 | 26.5–40 (band Ka)
| | 40–50 (band Q)
Aperture efficiency, $A_{\rm e}$ | 0.78 | 0.34–0.39
$\Delta\nu_{\rm max}$ (Hz) | 3820 | 1
Single-field sensitivity ($\propto ND^{2}$) | 7200 | 17000
effective | 5600 | 5800-6600
Mosaic image sensitivity ($\propto ND$) | 600 | 680
effective | 530 | 420–390
Image fidelity ($\propto N^{3}$) | 130000 | 20000
Table 1 summarizes the differences between ALMA and the JVLA. Comparing their
attributes, we note that ALMA’s 12-m Array has antennas of smaller surface
area than those of the JVLA (12-m diameter vs. 25-m) but these are larger in
number (50 in the 12-m Array vs. 27) and have higher pointing accuracies (0.6″
vs. 2-3″) and aperture efficiencies at Band 1 frequencies (0.78 vs.
0.34–0.39). Combining these numbers (except pointing accuracy), the effective
surface area of the ALMA 12-m Array is a factor of 0.85–0.98 that of the JVLA.
Adding Band 1 to the ACA antennas would minimize even this small difference.
ALMA has the same 8 GHz maximum bandwidth as the JVLA with its new WIDAR
correlator. ALMA’s present correlator has a lower maximum spectral resolution
than the JVLA’s, however, i.e., a maximum of 3.82 kHz vs. 1 Hz, respectively.
(ALMA’s correlator of course could be similarly upgraded in the future.)
Figure 7: Images from JVLA and ALMA observations simulated with CASA. The observations were set toward a “blank” sky at 45 GHz with 8 GHz (continuum) bandwidth, with JVLA in its D-configuration while ALMA in its “12” configuration provided in CASA. Both array configurations give rise to a similar angular resolution of $\sim$1$\farcs$6 FWHM. The white dotted circles denote the corresponding primary beam sizes. There resulting 1 $\sigma$ rms noise levels after 2 hours of on-source integration are 9.6 $\mu$Jy and 4.5 $\mu$Jy, respectively, for JVLA and ALMA, which are in general agreement with the estimated noise level shown in Table 2. Table 2: Comparison of Point-Source Sensitivity between JVLA and ALMA | JVLA | ALMA
---|---|---
no. of antennas | 25 | 50
polarization | dual | dual
weather | winter | auto (5.2 mm) PWV
source position | zenith | zenith
on-source time | 60 s | 1 hr | 60 s | 1 hr
bandwidth | 1 MHz | 1 MHz
freq. | 35 GHz | 3.2 mJy | 0.41 mJy | 3.0 mJy | 0.38 mJy
| 40 GHz | 3.6 mJy | 0.47 mJy | 3.1 mJy | 0.40 mJy
| 45 GHz | 5.1 mJy | 0.68 mJy | 3.6 mJy | 0.47 mJy
| 50 GHz | 25.5 mJy | 3.29 mJy | (not available)
bandwidth | 8 GHz | 8 GHz
freq. | 40 GHz | 50 $\mu$Jy | 5.4 $\mu$Jy | 35 $\mu$Jy | 4.5 $\mu$Jy
| 45 GHz | 78 $\mu$Jy | 10 $\mu$Jy | 41 $\mu$Jy | 5.3 $\mu$Jy
Given differing antenna numbers, sizes, and baselines, the two observatories
differ in various imaging metrics222These metrics were defined and used to
compare ALMA to other existing interferometers in the 2005 NRC document The
Atacama Large Millimetre Array: Implications of a Potential Descope.:
Figure 8: Images from CASA simulations of JVLA and ALMA mosaic observations
of 45 GHz continuum. The left-hand panels show the model image convolved with
the synthesized beams. The pointing patterns for the mosaicked observations
are shown with white dots. The right-hand panels show the resulting observed
images. Both simulated observations are executed with eight hours of on-source
time in total toward the zenith. The ALMA and JVLA were assumed to be in their
“12” and “D” configurations, (both provided in CASA), respectively, which
resulted in similar synthesized beam sizes of 1.7′′ $\times$ 1.7′′. The
achieved noise level by ALMA is around three times better than that by JVLA.
(i.e,. 10 $\mu$Jy beam-1 for ALMA vs. 30 $\mu$Jy beam-1 for JVLA). Observation
overheads (e.g., calibration scans) and phase decoherence due to site location
were not included in the simulations, both of which will lead to greater
degradation in the JVLA images. Figure 9: Images from CASA simulations of
observations of extended 45 GHz emission with the JVLA and ALMA. The left-hand
panels show the model image (a superposition of the G41.1-0.3.b template
provided by the CASA guide with three extended Gaussian sources (two 18′′ in
size and one 48′′ in size) convolved with the synthesized beams. The middle
panels show the resulting images from the simulations. The right-hand panels
show the difference between the model and observation images. Both simulated
observations were executed with one hour of on-source time in total toward the
zenith. The ALMA and JVLA are assumed to be in their “12” and “D”
configurations (both provided in CASA), respectively, which resulted in
similar synthesized beam sizes of 1.7′′ $\times$ 1.7′′. The achieved noise
level by ALMA is around five times better than that by JVLA (i.e., 10 $\mu$Jy
beam-1 for ALMA vs. 50 $\mu$Jy beam-1 for JVLA). Observation overheads (e.g.,
calibration scans) and phase decoherence due to site location were not
included in the simulations, both of which will lead to greater degradation in
the JVLA images.
* •
Comparing the face-value “single-field sensitivity” metric ($ND^{2}$; where
$D$ is the antenna diameter and $N$ is the number of antennas), ALMA appears
about half as “sensitive” as the JVLA (7200 vs. 17000). Factoring in aperture
efficiencies to give effective values of $D$, however, the metrics are
actually much more similar (5600 vs. 5800-6600). Table 2 provides more
realistic comparisons of JVLA and ALMA sensitivities for point sources across
the proposed Band 1 frequency range, estimated using their respective
sensitivity calculators333For JVLA and ALMA, see
https://science.nrao.edu/facilities/evla/calibration-and-tools/exposure and
http://almascience.eso.org/call-for-proposals/sensitivity-calculator,
respectively. For these calculations, we assume the original ALMA
specifications for Band 1 receiver performance, i.e., the same 40–80 K as for
the JVLA’s Ka/Q-band receivers.. Note that the JVLA sensitivities require the
JVLA’s best weather (“winter”) while a relatively high PWV level (5.2 mm) was
actually chosen for ALMA here. From these calculations, we see continuum
sensitivities of ALMA for Band 1 are actually similar to better than those of
the JVLA. For example, a 1 $\sigma$ rms of $\sim$5 $\mu$Jy beam-1 is expected
at 40 GHz after 1 hour of integration at both observatories. At higher
frequencies (e.g., $>$45 GHz), however, the point source sensitivity of ALMA
is better than that of JVLA by factors of 1.4–1.9, depending on bandwidth.
(Simulations of JVLA and ALMA observations suggest even larger improvements;
see below.) In addition, Figure 7 shows simulated “blank-sky” observations at
45 GHz carried out with CASA, giving another perspective on this comparison.
(ALMA’s improved pointing accuracy and better phase stability were not fully
incorporated into these calculations.) Note also that ALMA’s 12-m diameter
antennas provide a field-of-view for single-pointing observations that is more
than twice as wide as what the JVLA’s 25-m diameter antennas provide (see
Table 3), so ALMA’s similar or better sensitivity is obtained over a wider
area in a single pointing.
* •
Comparing the “mosaic image sensitivity” metric ($ND$), again on face value,
ALMA’s 12-m Array and the JVLA appear already quite similar (600 vs. 680,
respectively). Factoring in only the improved aperture efficiencies of ALMA at
its lowest frequencies vs. those of the JVLA at its highest frequencies, the
comparison is in ALMA’s favour by a factor of $\sim$1.3 (530 vs. 420-390). As
with the single-pointing comparison above, the superior weather at the ALMA
site will increase this factor further. For example, Figure 8 shows mosaic
simulations for JVLA and ALMA of a galaxy, in relatively similar compact
configurations over the same 8 hours of integration. The ALMA observations are
performed with fewer pointings than those of the JVLA. The resulting 1
$\sigma$ rms noise level of the ALMA image is a factor of three better than
that of the JVLA image.
* •
ALMA’s larger number of baselines yield a higher “image fidelity” metric
($N^{3}$) by a factor of $>$6 (130000 for ALMA vs. 20000 for the JVLA) over
similar observation durations. Basically, ALMA’s larger number of baselines
allow more spatial frequencies to be sampled per unit time, yielding more
accurate images. Figure 9 shows an example of ALMA’s higher intrinsic fidelity
relative to that of the JVLA, especially for extended emission, based on
simulations of a high-mass star-forming region. The difference between the
model and observation images (right panel) is noticeably smaller for the ALMA
case than for the JVLA one.
Table 3: Comparison of angular scale coverage between JVLA and ALMA at 45 GHz | JVLA | ALMA
---|---|---
Configuration | A | D | most extended | most compact
Bmin (km) | 0.68 | 0.035 | 0.04 | 0.015
Bmax (km) | 36.4 | 1.03 | 16 | 0.15
$\theta_{PRIMARY}$ | 60 | 60 | 135 | 135
$\theta_{HFBW}$ | 0.043 | 1.5 | 0.08 | 9
$\theta_{LAS}$ | 1.2 | 32 | 35 | 93
* •
At present, ALMA has maximum baselines that are a factor of $\sim$2 smaller
than the JVLA’s (15-18 km vs. 36.4 km), meaning that the JVLA can in principle
produce images of resolution up to a factor of 2 higher than ALMA can at the
same frequency. ALMA will be in turn more sensitive to extended emission,
however. First, ALMA’s smaller dishes mean that its minimum baselines are
shorter than those of the JVLA (16-m vs. 35-m; see Table 3 for a comparison),
allowing higher sensitivity to extended, low-surface-brightness emission.
Second, ALMA can include the ACA antennas, each of 7-m diameter but together
in a close-packed configuration, in principle allowing even further
sensitivity to extended emission.
In summary, ALMA Band 1 can be superior to the JVLA at its highest frequencies
in many ways, including:
* •
Access to southern sources, given ALMA’s southern hemisphere location;
* •
Wide-field sensitive imaging, due to ALMA’s larger number of smaller, high
precision antennas located at an excellent site;
* •
High image fidelity, given ALMA’s larger number of antennas;
* •
Sensitivity to extended emission, if appropriate, due to ALMA’s shorter
minimum baselines and the ACA;
* •
Likely coverage of 50-52 GHz, frequencies not possible with the current JVLA
receivers;
* •
Recovery of short spacing visibilities, by using the Atacama Compact Array,
and the total power single-dish observations;
* •
Combination with other ALMA bands, for many multi-band projects; and
* •
Lower overheads, by applying for and using a single observatory.
As shown in §4, the top science cases for Band 1 can stand shoulder-to-
shoulder with the primary Level 0 goals of ALMA. Thus, the primary motivation
for the enhancement is not as a “poor weather” back-up receiver but rather the
excellent science that can be achieved. In the following sections, we explore
the large and broad variety of science cases beyond the top cases identified
in §4 that the ALMA Band 1receiver suite will be able to address.
## 6 A Broad Range of Science Cases
Along with the two science cases presented above in §4, there is a wealth of
scientific opportunity available to the wide ALMA community when the Band 1
receiver suite is built. Here we highlight a selection of science cases which
would significantly benefit from Band 1 receivers on ALMA.
### 6.1 Continuum Observations with ALMA Band 1
The astrophysical continuum radiation at wavelengths of $\sim$1 cm is
relatively unexplored. Yet, this radiation is key to understanding radio
emission mechanisms and probing regions that are optically thick at shorter
wavelengths. The sensitivity and resolution of ALMA Band 1 will allow: (1)
improved understanding of galaxy clusters through the Sunyaev-Zel’dovich
Effect; (2) a diagnostic of the smallest interstellar dust grains; (3) studies
of jets from young stars; (4) an understanding of the nature of pulsar wind
nebulae; (5) the detection of radio SNe, with constraints on stellar
precursors and remnants; (6) a diagnostic of X-ray binaries; and (7) improved
probes of Sgr A*, the supermassive black hole at the center of the Galaxy.
#### 6.1.1 The Sunyaev-Zel’dovich Effect
Much of what we know about galaxy clusters has come from X-ray observations of
thermal bremsstrahlung emission of the intra-cluster medium (ICM). For
example, the angular resolution of Chandra has been crucial to advancing our
understanding in this area and has resulted in a renaissance in astrophysical
studies of galaxy clusters. In recent years, the Sunyaev-Zel’dovich Effect
(SZE) has provided an increasingly important view of these cosmic structures
(Birkinshaw 1999). Since the SZE signal is proportional to the product of the
electron density and its temperature ($\sim n_{e}\,T_{e}$, compared to
$n_{e}^{2}\sqrt{T_{e}}$ for the X-rays), it gives a complementary view of the
physical state of the ICM, one more sensitive to hot phases that also directly
measures local departures from thermal pressure equilibrium. To date, the
majority of SZE observations have been carried out at comparatively low
angular resolution (beams $>1^{\prime}$ in size), yielding information about
the overall bulk cluster properties. Advances in instrumentation have begun
making higher angular resolution measurements of the SZE possible, revealing
previously unsuspected shock-heated gas in the ICM of clusters previously
thought to be dynamically relaxed (Komatsu et al. 2001, Kitayama et al. 2004,
Mason et al. 2010, Korngut et al. 2011, Plagge et al. 2012). These
$10^{\prime\prime}$ to $20^{\prime\prime}$ SZE images are the current state of
the art. A Band 1 receiver suite on ALMA will surpass this benchmark, making
possible detailed studies of the ICM using the SZE on larger samples and with
greater sensitivity than before.
ALMA Band 1 receivers will be capable of addressing a wide range of basic
questions about the observed structure and evolution of clusters. For example,
what are the structures of ICM shocks and the mechanisms responsible for
converting gravitational potential energy into thermal energy in the ICM
(Markevitch et al. 2007, Sarazin et al. 1988)? What is the influence of Helium
ion sedimentation within the cluster atmosphere (Ettori et al. 2006)? What is
the nature of the AGN-inflated “bubbles” seen in the cores of some clusters
(Pfrommer et al. 2005), and what is the role of cosmic rays in the ICM? What
is the nature of the underlying ICM turbulence (e.g., Kolmogorov versus
Kraichnan)? A particularly rich area will be the detailed study of ICM shocks,
which are common since infalling sub-clusters are typically transsonic.
Several galaxy cluster mergers have been observed recently with Chandra and
XMM in X-rays with resolutions at the arcsecond level where substructures
become visible (Markevitch, et al. 2000, 2002). The features of interest for
these studies will typically fit within one or a few ALMA Band 1 fields-of-
view and require longer integrations (several to $\sim$10 hours per pointing).
Note that Band 1 receivers also may have the sensitivity to detect the SZE
from the halos of massive individual ellipticals or massive groups.
Another important area where high-resolution SZE imaging will have an impact
is the interpretation of SZE survey data. ACT (Dunkley et al. 2011), SPT
(Williamson et al. 2011), and Planck (Planck Collaboration, 2011) have all
conducted 1000$+\,{\rm deg^{2}}$ surveys to detect and catalog galaxy clusters
via the SZE. These surveys provide unique and valuable information about
cosmology but their interpretation depends upon assumptions about the
relationship between the SZE signal and the total virial mass of the halos
observed. It is known that both gravitational (cluster merger) and non-
gravitational processes (AGN and supernova feedback, bulk flows444By bulk
flow, we refer to the motion of a cluster itself through its surrounding
medium, producing a kinematic contribution to the observed SZE signal; in
theory, this contribution has a different spectral dependence than the thermal
SZE and may be distinguishable with good spatial coverage., cosmic ray
pressure) give rise to considerable scatter and potential biases (e.g.,
Morandi et al. 2007) in this relationship. Cluster mergers have a particularly
dramatic effect on the SZE, typically generating transsonic (Mach $\sim
2$-$4$) shock fronts which can enhance the peak SZE in the cluster by an order
of magnitude (Poole et al. 2007, Wik et al. 2008).
The systematic astrophysical uncertainties just described are the limiting
factor in making cosmological inferences from the small published samples of a
few dozen SZE-selected clusters (e.g., Sehgal et al. 2011). ALMA Band 1
receivers are the only foreseen prospect for efficient high-resolution
observations of the large southern hemisphere samples of SZE-selected clusters
that will directly improve inferences from these surveys. They will be used to
image (at $5^{\prime\prime}-10^{\prime\prime}$ resolution) galaxy clusters
discovered in the low-resolution ($\sim$1′) surveys, detecting shocks and
mergers and identifying ICM substructure, and providing a direct,
phenomenological handle on important survey systematics. Indeed, the
sensitivity and resolution of an ALMA Band 1 receiver suite allows for
efficient follow-up observations of cluster detections made by blind southern
hemisphere SZE surveys. Thus, a study of a selection of clusters from these
survey experiments in a statistical manner becomes feasible and new important
insights into the mass-observable relation and its scatter and dependence on
cluster physics can potentially be obtained. The ability to understand cluster
selection in detail is essential to derive reliable constraints on
cosmological models from SZE cluster surveys (see e.g., Geisbuesch et al.
2005; Geisbuesch & Hobson 2007).
The coming decade will also see an explosion of optical and X-ray cluster
data. The German/Russian satellite eRosita, due to launch in 2014, will carry
out the first all-sky X-ray survey since ROSAT (Merloni et al. 2012). Among
other things, it is expected to catalog $\sim$100000 clusters out to $z=1.3$
(Cappellutti et al. 2011). Also, the Dark Energy Survey (DES; Dark Energy
Survey Collaboration 2005) is a $5000\,{\rm deg^{2}}$, mostly southern sky
survey also expected to find $\sim$100000 galaxy clusters. Targeted SZE
observations with ALMA Band 1 receivers will be invaluable to determine the
properties of clusters at redshifts where X-ray spectrscopy and gravitational
lensing begin to fail. These high-$z$ clusters, such as the ACT-discovered SZE
cluster “El Gordo” at $z=0.89$, weighing in at $M=(2.16\pm 0.32)\times
10^{15}$ M⊙ (Menanteau et al. 2011), offer leverage on so-called “pink
elephant” tests capable of constraining cosmological or gravitational theories
based on the existence of individual extreme objects, i.e., provided their
properties are accurately determined. Importantly, note that an ACA equipped
with Band 1 receivers will be comparable in capability to the OVRO/BIMA arrays
used in the current decade to measure the bulk SZE properties of large
northern hemisphere cluster samples (Bonamente et al. 2008). Extending this
capability to the southern hemisphere over the next decade is important to
realize the full potential of these rich cluster samples.
Given the large number of ALMA baselines, the resulting high image fidelity
and dynamic range of the data will be advantageous to SZE studies, in
particular the detailed ones. In addition, long baseline data from ALMA can be
used to remove accurately the intrinsic and background (i.e., gravitationally
lensed) discrete source populations. These latter objects are a signal of
substantial interest from another point of view, but they also set a
significant “confusion noise” floor to millimeter single-dish observations,
especially considering the factor of $2-3$ boost in source confusion in
clusters due to gravitational lensing (Blain et al. 2002).
Figure 10: Simulated $1.5$ hour ALMA Band 1 (left) and Band 3 (right)
observations of a galaxy cluster covering $5^{\prime}\times 5^{\prime}$. The
shock is represented as a Gaussian component $5^{\prime\prime}\times
25^{\prime\prime}$ in extent with a peak SZE of $y=10^{-4}$, considerably
weaker than the amplitude observed in RXJ1347-1145 by Mason et al. (2010). The
Band 3 data were tapered to the innate resolution of the Band 1 map, $\sim
10^{\prime\prime}$ (FWHM). ACA baselines were not included in this simulation
but the overplotted contours show the ACA Band 1 image (using a
$45^{\prime\prime}$ taper) of the bulk ICM in this system in a simulated 12 hr
integration after subtraction of the shock signal. The bulk ICM is modeled as
an elliptical isothermal $\beta$ model with $R_{core}=(150,250)\,{\rm kpc}$,
$\beta=0.7$, and $y_{o}=3\times 10^{-5}$ at $z=0.7$, characteristic of
disturbed, merging systems.
ALMA will have a considerably higher sensitivity for these observations than
the JVLA, owing to an order of magnitude higher surface brightness
sensitivity, or ALMA Band 3, owing to lower system temperatures and larger
primary beam. In Figure 10, we show simulated Band 1 and Band 3 observations
(using the ALMA 12-m Array and the ACA) that cover the virial region ($D\sim
5^{\prime}$) of a moderately massive SZE cluster with a merger shock. For
these simulations, we considered a hypothetical project to detect a feature
with a Compton $y=10^{-4}$, characteristic of strong shocks in major mergers,
with a characteristic feature size of $5^{\prime\prime}-20^{\prime\prime}$.
The required flux density sensitivity is similar in both cases after allowing
for resolution effects, about 1 $\sigma$ = $8-9\,{\rm\mu Jy}$ rms in both
instances. We find that a clear detection is achieved in only $1.5$ hours of
Band 1 observing, but nearly $40$ hours are required at Band 3. The ACA Band 1
measurement of the bulk ICM signature (a 12 hr observation is needed for good
SNR) is also shown, tapered to a $45^{\prime\prime}$ FWHM beam. Yamada et al.
(2012) find similar results in their detailed study of SZE imaging with ALMA
and the ACA at $\lambda\approx 1\,{\rm cm}$.
In summary, ALMA’s southern location matching large galaxy cluster surveys,
intrinsically high image fidelity, and sensitivity to extended low-brightness
features (e.g., relative to the JVLA) will make Band 1 observations very
compelling probes of physics of galaxy clusters using the Sunyaev-Zel’dovich
Effect.
#### 6.1.2 Very Small Grains and Spinning Dust
The last decade has seen the discovery of surprisingly bright cm-wavelength
radio emission from a number of distinct galactic objects but most notably
dark clouds (e.g., Finkbeiner et al. 2002; Casassus et al. 2008 (see Figure
11); Scaife et al. 2009). The spectrum of this new component of continuum
radiation can be explained by electric dipole radiation from rapidly rotating
(“spinning”) very small dust grains (VSGs), as calculated by Draine & Lazarian
(1998; DL98). This emission has been also seen as a large-scale foreground in
CMB maps, spatially correlated with thermal dust emission and having a
spectrum peaking at $\sim$40 GHz.
All of the existing work aimed at diagnosing this continuum emission is
derived from CMB experiments on large angular scales, where the bulk of the
radio signal occurs, e.g., recently by the Planck satellite. Details on small
angular scales are crucial, however, for probing star formation and
circumstellar environments. Simply, progress in the understanding of the solid
and gaseous states of the ISM requires sufficient resolution to separate the
distinct environments. Directly measuring the VSG abundance and solid state
physics is very exciting because VSGs play a central role in the chemical and
thermal balance of the ISM. For example, the smallest grains account for most
of the surface area available for catalysis of molecular formation.
Figure 11: Three-colour image of the $\rho$ Oph W photo-dissociation region
(Casassus et al. 2008). Red: MIPS 24 $\mu$m continuum Green: IRAC 8 $\mu$m
continuum, dominated by the 7.7 $\mu$m PAH Band Blue: 2MASS Ks-band image. The
$x-$ and $y-$axes show offset in RA and Dec from $\rho$ Oph W, in degrees. The
contours follow the 31 GHz emission, with levels at 0.067, 0.107, 0.140,
0.170, and 0.197 MJy sr-1.
DL98 proposed that the grain size distribution in their spinning dust model
would be dominated by VSGs, thought to be mostly PAH nanoparticles. The size
distribution of VSGs is poorly known, however, since studies of interstellar
extinction are relatively insensitive to its details. The existence of VSGs
has been supported by several assertions. First, a significant amount of
carbonaceous nanoparticles in the ISM could explain observations of
unidentified IR emission features. Second, the strong mid-infrared emission
component seen by IRAS must result from the reprocessing of starlight by
ultrasmall grains. Indeed, the fraction of the ISM carbon content proposed to
exist in VSGs considerably exceeds that implied to exist in the MRN dust size
distribution. (The MRN dust distribution is known to underestimate this
fraction.)
Observationally determining PAH content in dust clouds is not straightforward.
Where there is a strong source of UV flux present, it is possible to identify
PAHs by their spectral emission features. In the case of pre-stellar and Class
0 cloud cores, however, these features are absent. With observations from ALMA
Band 1 receivers constraining the spinning dust SED at similar resolution to,
e.g., Spitzer or the forthcoming MIRI instrument on the JWST, it will be
possible to measure the VSG size distribution directly from the data.
This work will also be important in the context of circumstellar and
protoplanetary disks, where the proposed population of VSGs may have important
implications for disk evolution. Certainly, spinning dust emission will
provide a better measure of the small grain population within circumstellar
disks than PAH emission since favorable excitation conditions for PAHs exist
only in the outermost layers of the disk. Since all the VSGs in the disk
should contribute spinning dust emission, such emission will provide a much
better probe of the mass in VSGs. Combining this information with the PAH
emission features would then also give us a useful measure of sedimentation in
disks.
Spinning dust emission from a VSG population will in theory dominate the
thermal emission from disks (around Herbig Ae/Be stars) at frequencies $\leq$
50 GHz by significant factors (Rafikov 2006). The existence of these VSGs has
been confirmed observationally from PAH spectral features seen in the disks of
Herbig Ae/Be stars (Acke & van den Ancker 2004) but it has not been detected
in protoplanetary disks due to a lack of strong UV flux. Since spinning dust
emission has been observed to be spatially correlated with PAH emission
(Scaife et al. 2010), spinning dust may provide a unique window on the small
grain population of these disks. In the context of disk evolution, these
recent measurements conflict with the established view that dust grains are
expected to grow as disks age. It may be the case that dust fragmentation is
important in disks (Dullemond & Dominik 2005), or there exists a separate
population of very small carbonaceous grains distinct from the MRN
distribution (Leger & Puget 1984; Draine & Anderson 1985). This second
proposition has not only important implications for the study of circumstellar
disks but also more generally for the complete characterization of dust and
the ISM.
The arcsecond resolution necessary for these measurements will be achievable
with several ALMA configurations and Band 1. From the models of Rafikov
(2006), the difference between a thermal dust spectrum with $\beta$ $\approx$
1 and the predicted spinning dust contribution for a brown dwarf disk would be
observable at 5 $\sigma$ in a matter of minutes with ALMA Band 1 receivers.
With longer observation times and consequently higher sensitivity, it will be
also possible to distinguish between different grain size distributions and
physical conditions within the disk (such as grain electric dipole moments,
rotational kinematics, optical properties and catalysis of molecule
formation).
In summary, spinning dust emission provides a unique insight into the VSG
population under conditions where it is not possible to observe using mid-IR
emission. The high resolution and excellent sensitivity of ALMA are ideal for
differentiating the distinct environments where the VSG population resides and
will be crucial for probing star formation and circumstellar regions.
Specifically, Band 1 receivers will allow routine surveys of the new continuum
component at its spectral maximum. The smaller minimum baselines of ALMA will
make it more ideal for probing (especially at southern declinations) the more
extended instances of spinning dust emission, e.g., cores, than the JVLA.
Also, ALMA Band 1 observations of more compact objects like disks (see §5.1)
will be better suited for comparison with those at higher frequency bands than
those from the JVLA, given the more similar spatial frequency coverage
afforded by observing from the same latitude.
#### 6.1.3 Jets from Young Stars
Radio continuum emission is observed from the jets and winds of young stellar
objects and is due to the interaction of free electrons, i.e., “free-free
emission.” The radio images appear elongated and jet-like and are usually
located near the base of large optical Herbig-Haro flows (Reipurth & Bally
2000). These regions usually have only sub-arcsecond sizes, indicating the
youth of the emitting material and the short dynamical times involved. The
emitted flux is usually weak, with a flat to positive spectral index with
increasing frequency, and it can be obscured by the stronger thermal emission
from dust grains at higher frequencies (e.g., Anglada 1995). Multi-wavelength
studies of the brightest radio jets at centimeter wavelengths trace either
earlier and stronger sources or more massive systems. The triple system
L1551-IRS 5, one of the most studied low-mass systems (Rodriguez et al. 1998,
2003; Lim & Takakuwa 2006), is illustrative of the sub-arcsecond scales
required (Figure 12).
Figure 12: Background: The VLA+Pie Town continuum image of L1551 IRS 5 at 3.5
cm obtained by Rodriguez et al. (2003) in their Figure 1. The size of the beam
(0.18 X 0.12″; P.A. = 35∘) is shown in the bottom left-hand corner. Black
rectangles mark the positions and deconvolved dimensions of the 7 mm compact
protoplanetary disks. The dashed lines indicate the position angles of the jet
cores. Inset: map of the south jet from the X-Wind model convolved with the
beam and plotted with the same contour levels from Figure 4 of Shang et al.
(2004).
Ground-based, interferometric studies of radio jets provide the best
opportunity to resolve the finest scales of the underlying source, comparable
or better than optical studies of jets by HST. Such finely detailed images can
provide the ability to differentiate between theoretical ideas about the
nature of these jets, i.e., the launch region, the collimation process, and
the structure of the inner disks. Modeling efforts with the radio continuum
emission presented in Shang et al. (2004) demonstrated one such possibility in
constraining theoretical parameters using earlier millimeter and centimeter
interferometers (Figure 12). Band 1 observations will discriminate between
competing jet launch theories tied to the disk location of the launch point by
achieving better than 0.1′′ angular resolution.
The high sensitivity of ALMA Band 1 observations will also allow detection of
radio emission from less luminous sources. ALMA will thus have the potential
to discover a significant number of new radio jets, providing a catalog from
which evolutionary changes in the physical properties can be deduced. As well,
multi-epoch surveys will be able to follow the evolution of the freshly
ejected material down to a few AU from the driving sources through movies. The
35-52 GHz frequency range of Band 1 will show contributions to the observed
emission from both the ionized component of the jet and the thermal emission
from the dust. These data, together with detailed theoretical modelling will
uncover a complete understanding of properties of the spectral energy
distribution (SED) from the ionized inner regions of young stellar jets.
Relative to the JVLA, Band 1 observations with ALMA may have modest
improvements in sensitivity at frequencies in common. Of course, southern
sources will be much better observed with ALMA. Moreover, the wider field-of-
view of ALMA will more easily allow for observations of multiple jets across
crowded regions such as within young protoclusters.
#### 6.1.4 Spatial and Flaring Studies of Sgr A*
Figure 13: (a) Left A 22 GHz image of the Sgr A* region at
$0.36^{\prime\prime}\times 0.18^{\prime\prime}$ resolution (PA=2∘) constructed
by combining JVLA A- and B- array data.
Near-IR and radio observations provide compelling evidence that the compact
nonthermal radio source Sgr A* is identified with a 4 $\times$ $10^{6}$ M⊙
black hole at the center of the Galaxy (Reid and Brunthaler 2004; Ghez et al.
2008; Gillessen et al. 2009). It is puzzling, however, that the bolometric
luminosity of Sgr A* due to synchrotron thermal emission from hot electrons in
the magnetized accretion flow is several orders of magnitude lower than
expected from the accretion of stellar winds. There have been two different
approaches to address this puzzling issue. One is to search for the base of a
jet from Sgr A* and identify interaction sites of a jet with the ionized and
molecular material surrounding Sgr A*. The other is to study the correlations
of the variable emission from Sgr A* at centimeter and millimeter bands.
Studies of images and variability are well suited using ALMA’s Band 1 and will
be complementary to each other in addressing the key question as to why Sgr A*
is so underluminous. Note that Sgr A* is located at a declination of -29∘,
making it a more attractive target for ALMA than the JVLA.
Regarding jets, recent JVLA observations at radio wavelengths presented a
tantalizing detection of a jet-like linear feature appearing to emanate from
Sgr A* (Yusef-Zadeh et al. 2012). Figure 13 shows a 23 GHz image of the inner
30′′ of Sgr A*. A new linear feature is noted running diagonally crossing the
bright N and W arms of the mini-spiral, along which several blobs (b, c, d, h1
and h2) are detected. What is interesting about the direction in which the
linear feature is detected is that several radio blobs have X-ray and FeII/III
counterparts also along the axis of the linear structure. In addition, the
extension of the linear feature appears to be polarized at 8 GHz, suggesting
that this feature is a synchrotron source. The radio-polarized linear jet-like
structure is best characterized by a mildly relativistic jet-driven outflow
from Sgr A*, and an outflow rate $\gamma\dot{M}\sim 10^{-6}$
$\hbox{M}_{\odot}$ yr-1.
The linear arrangements of antennas in the JVLA configurations can lead to
linear structures in the residual beam pattern due to deconvolution errors.
ALMA’s configurations, however, should lead to data with better, more-uniform
uv coverage and will establish the reality of the linear structure. In
particular, Band 1 will be most effective in studying the faint jet-like
feature from Sgr A*. Dust emission from the immediate environment of Sgr A*
dominates fluxes at shorter wavelengths relative to optically thin non-thermal
emission from the jet with a steep energy spectrum. Thus, observations with
Band 1 are critical for measuring properly the morphology, spectral index and
polarization characteristics of the jet emanating from Sgr A*. Although Sgr A*
is a unique object in the Galaxy, similar motivations also apply to other non-
thermal radio continuum sources such as microquasars, e.g., 1E1740.7-2942,
that have faint radio jets and are located in the inner Galaxy.
Regarding the correlations of variable emission from Sgr A*, recent radio
measurements have detected a time delay of $\sim$30 $\pm$ 10 minutes between
the peaks of 7 mm and 13 mm radio continuum emission toward Sgr A* (Yusef-
Zadeh et al. 2006). This behaviour is consistent with a picture of a flare in
which the synchrotron emission is initially optically thick. Flaring at a
given frequency is produced through the adiabatic expansion of an initially
optically thick blob of synchrotron-emitting relativistic electrons. The
intensity grows as the blob expands, then peaks and declines at each frequency
that the blob becomes optically thin. This peak first occurs at 43 GHz and
then at 22 GHz about 30 minutes later. Theoretical light curves of flare
emission, as shown in Figure 14, show that it occurs at high near-infrared
frequencies first and is increasingly delayed at successively lower ALMA
frequencies that are initially optically thick.
Figure 14: Theoretical light curves of Stokes I for optically thick
synchrotron emission at five different bands corresponding ALMA Bands 3, 6, 7
and 9 as a function of expanding blob radius. These light curves assume an
energy power law index p=1 where n(E)$\propto$ E-p.
The limited time coverage of JVLA observations at radio wavelengths (due to
the low maximum elevation of Sgr A* at the JVLA) means that there can be a
large uncertainty in determining the underlying background flux level of a
particular flare, as well as difficulty identifying flares in different bands.
Observations of Sgr A* with a long time coverage using ALMA’s Band 1 can fit
the corresponding light curves simultaneously to place much tighter
constraints on the derived physical parameters of the flare emission region.
Two parameters of high interest are the expansion speed of the hot plasma and
the initial magnetic field. These quantities characterize the nature of
outflow and cooling processes relevant to millimeter emission. The fitting of
a light curve at one frequency will automatically generate models for any
other frequency. We should be able to test the time delay between the peaks of
flare emission within Band 1.
What has emerged from past observing campaigns to study Sgr A* is that radio,
submillimeter, near-infrared, and X-ray emission can be powerful probes of the
evolution of the emitting region since they are all variable. We now know that
flare emission at infrared wavelengths is due to optically thin synchrotron
emission that is detected when a flare is launched (Eckart et al. 2006). The
relationship between radio and near-infrared/X-ray flare emission has remained
unexplored due the very limited simultaneous time coverage between radio and
infrared telescopes. The continuous variations of the radio flux on hourly
time scales also make the identification of radio counterparts to infrared
flares difficult. In spite of the limited time coverage, the strong flaring in
near-infrared/X-ray wavelengths has given us an opportunity to examine if
there is a correlation with variability at radio frequencies. A key motivation
for observing Sgr A* is to compare its flaring activity with the adiabatic
expansion picture. One of the prediction of this model is a time delay between
the peaks of optically thin near-infrared emission and optically thick radio
emission, as discussed above. From this model, a near-infrared flare of short
duration of 0.5-1 hr is expected to have a radio counterpart of duration of
$\sim 2$ hr shifted in time by 3-5 hr.
Figure 15: The light curves of Sgr A* on 2007 April 4 obtained with XMM in
X-rays (top), VLT and HST in NIR (middle), and IRAM-30m and VLA at 240 GHz and
43 GHz, respectively (bottom). The NIR light curves in the middle panel are
represented as H (1.66 $\mu$m) in red, Ks and Ks-polarization mode (2.12
$\mu$m) in green and light blue, respectively, L’ (3.8$\mu$m) in black (Dodds-
Eden et al. 2009), and NICMOS of HST in blue at 1.70 $\mu$m. In the bottom
panel, red and black colors represent the 240 GHz and 43 GHz light curves,
respectively.
Figure 15 shows composite light curves of Sgr A* obtained with XMM, VLT, HST,
the IRAM 30-m Telescope, and the VLA on 2007 April 4. These curves reveal that
there was no significant variation at 240 GHz during the period when the
strong near-infrared/X-ray flare took place. The IRAM observation shows an
average flux of 3.42 Jy $\pm$ 0.26 Jy between 5 hr and 6h UT when the powerful
near-infrared flare took place. The millimeter flux is mainly arising from the
quiescent component of Sgr A*. Comparing the light curves of the 43 GHz and
240 GHz data, there is no evidence for a simultaneous radio counterpart to the
near-infrared/X-ray flare with no time delays. Given the limited coverage in
time with the VLA, it is clear that we can not be confident about the time
delay between radio and near-infrared/X-ray peaks. There is also no overlap in
time between the VLA and Subaru data to test the adiabatic picture of flare
emission by making simultaneous NIR and radio observations. In future, ALMA
and VLT will have the best time overlap to test this important aspect of flare
emission from Sgr A*. Although Sgr A* is a unique object in the Galaxy,
similar arguments could be made for numerous transient sources found in the
inner Galaxy.
Figure 16: Radio emission as a function of frequency expected from G2 cloud
(red) when compared to quiescent emission from Sgr A*, as shown in blue
(Narayan, Ozel, & Sironi 2012). Left and right panels show predictions based
on different assumptions on the energy spectrum of nonthermal particles (p).
Finally, we note the utility of ALMA Band 1 receivers to trace close
encounters of gas clouds with Sgr A*. For example, a 3 MEarth cloud of ionized
gas and dust named G2 has been recently determined to be on a collision course
with Sgr A*. VLT observations indicate that the G2 cloud approaches pericenter
in mid-2013 and it will be disrupted and portions will likely be accreted by
the massive black hole residing there (Gillessen et al. 2012). At the
pericenter distance, the velocity of the gas cloud will be 5400 km s-1.
Accordingly, the cloud is expected to produce a bow shock that can easily
accelerate electrons into a power-law distribution of index $p=2.5-3.5$,
assuming standard shock conditions (Narayan et al. 2012). Depending on $p$,
the expected additional emission from Sgr A* ranges from 0.6 Jy to 4 Jy, over
a dynamical timescale of $\sim$6 months. The model behind the additional radio
emission from the disruption of G2 by the black hole could have been tested
directly with ALMA Band 1 observations. Though Band 1 receivers will not be
ready for the interaction of G2 with Sgr A* by 2013, this close encounter is
likely not an isolated event, and future disruptions of other, similar clouds
in the Sgr A* region by the black hole could be monitored with Band 1.
In summary, ALMA Band 1 receivers will provide important constraints to models
of Sgr A*, the supermassive black hole in the center of the Galaxy. ALMA’s
southern location will allow for improved observations of Sgr A* than possible
at the JVLA site, due to the southern declination of the object. For example,
the longer time Sgr A* is present over the horizon improves studies of
variability, and also improves sensitivity and spatial frequency coverage for
observations of associated phenomena at Band 1 frequencies.
#### 6.1.5 Acceleration Sites in Solar Flares
When a solar flare occurs, some of the particles in the corona are accelerated
from a few hundred eV up to a few MeV within less than one second. The non-
thermal electrons accelerated by a flare flow along the magnetic field lines
of the flare, emitting microwaves while propagating through the corona.
Finally, they collide with the dense and cool plasma in the chromosphere and
lose the energy by radiation and thermalization. In most flares, two hard
X-Ray (HXR) sources are observed at the footpoints of the flare loop, and one
microwave source is observed around the top of the loop (see Figure 17).
Previous observations of these sources had been done by HXR and microwave
solar telescopes with low spatial resolution (e.g., $\sim 10$ arcsec) and low
dynamic range (10–100). Hence, it has been hard to investigate the structures
and time evolution of the sources behind particle acceleration, especially
since we do not yet know where the acceleration site is in a flare. Some
indirect evidence suggests that the acceleration site is located above the
flare loop, in a location filled with $\sim 10$ MK thermal plasma (Masuda et
al. 1994, Aschwanden et al. 1996, Sui and Holman 2003), but there is no direct
evidence yet. Currently, it is also impossible to investigate the the
relationship of the acceleration site with the thermal structures, like the
in-flow of magnetic reconnection detected by the EUV observations (Yokoyama et
al. 2001). Therefore, there has been no significant progress in the study of
the particle acceleration in the last decade.
Figure 17: Images of a solar flare at X-ray, EUV, and radio wavelengths. The
top row panels show radio and EUV images in the pre-flare phase. On the left
are 17 GHz contours overlaid on a greyscale 34 GHz image (both averaged over
the period 23:00-00:15 UT), while the right panel shows a 195 Å image from
00:18:19 UT together with two 17 GHz contours for context. The remaining rows
of panels show the 96′′ $\times$ 96′′ region outlined in the pre-flare images.
The left panels show the RHESSI greyscale image of 12-20 keV HXR overlaid with
17 GHz total intensity radio contours (solid curves) and RHESSI 100-150 keV
HXR contours (dashed curves). The right panels show a 195 Å image of the same
region overlaid with solid grey contours for the RHESSI 12-20 keV HXR and
dashed black contours for the RHESSI 100-150 keV HXR. The panel labels refer
to the times of the 17 GHz images (left) and the TRACE images (right). Figure
from White et al. (2003).
Breakthroughs in the study of the particle acceleration in a solar flare may
be possible by solar ALMA observations even with ALMA’s current specs, because
its spatial resolutions and dynamic ranges are one order magnitude higher than
the current solar HXR and microwave telescopes. Nevertheless, the possibility
is very tiny for two important reasons: 1) the field-of-view of ALMA Band 3,
the presently lowest observing frequency receiver of ALMA, is about 60′′. That
field-of-view is not large enough for most flare observations and also it
would be very hard to observe simultaneously the region above the flare loop
predicted to be the acceleration site and the flare loop itself. Moreover, the
size of the field-of-view is directly related to the possibility of observing
flares, since the duration of solar observations by ALMA is limited. 2) If the
acceleration site is above the flare loop, as suggested by indirect evidence,
we can easily infer that the magnetic field strengths at the site is a few
tens of Gauss. The emissivity of the microwaves emitted by the gyro-
synchrotoron mechanism, however, strongly depends on the magnetic field
strength. Therefore, emission at frequencies of 230 GHz and higher from the
acceleration site is very weak. Such high frequency emission has been detected
only from the main sources of large flares by submillimeter single-dish
observations (e.g., Kaufmann, et al. 2004). Therefore, a lower frequency band
with the high spatial resolution and dynamic range of ALMA is needed to
observe the non-thermal emission from the acceleration site. Flare
observations with ALMA Band 1, with a single-pointing field-of-view of about
100′′ in the 35–50 GHz frequency range, will obtain significantly better
results for the particle acceleration studies of a solar flare. If the Band 1
receiver has also the capability to observe circular polarization, even higher
scientific returns will be achieved, because the circular polarization of the
gyro-synchrotoron emission will reveal the magnetic field strength of the
emitting region.
The JVLA can also observe the Sun at similar frequencies as those of Band 1,
but JVLA solar observations have several disadvantages. First, the JVLA has a
more reduced $u$–$v$ coverage. To synthesize a solar image, snapshot data are
needed because the non-thermal emission from a solar flare changes within less
than one second. Hence, ALMA’s larger number of baselines means that a larger
number of data points will be instantaneously sampled on the u-v plane.
Second, since the JVLA antennas are larger than the ALMA antennas and the JVLA
cannot sample as many short spacings, the maximum angular scale observable
with the JVLA is $\sim 32^{\prime\prime}$, making it harder to reconstruct
flare loops than with ALMA. Finally, the field-of-view of the JVLA, $\sim
60^{\prime\prime}$, is relatively small.
The total flux of gyro-syncrhotron emission emitted from a solar flare follows
a power-law distribution with frequency in the optically-thin frequency range,
so lower frequency observations are more sensitive in detecting flares. The
typical turnover frequency of flares is about 10 GHz. Therefore, the total
flux of emission in the Band 1 frequency range is one to two orders of
magnitude larger than that in Band 3. Nobeyama polarimeter data have shown
that the total flux average from 700 solar flares at 35 GHz is 46.3 SFU
($4.63\times 10^{5}$ Jy). Special care has to be taken to deal with such a
large input flux.
#### 6.1.6 Pulsar Wind Nebulae
Pulsars generate magnetized particle winds that inflate an expanding bubble
called a pulsar wind nebula (PWN) whose outer edge is confined by the slowly
expanding supernova ejecta. Electrons and positrons are accelerated at the
termination shock some $0.1\,$pc distant from the pulsar. Those relativistic
particles interact with the magnetic field inside the wind-blown bubble to
produce synchrotron emission across the entire electromagnetic spectrum.
Particles accelerated at the shock form toroidal structures, known as wisps,
and some of them are collimated along the rotation axis of the pulsar,
contributing to the formation of jet-like features. The synchrotron emission
structure in the post-shock and jet regions provide direct insight on the
particle acceleration process, magnetic collimation, and the magnetization
properties of the winds in PWNe. These observations have so far (except for
the Crab Nebula) been limited to X-ray wavelengths with the Chandra satellite
(e.g., Helfand et al. 2001).
Figure 18: Two-colour VLBI image of SN 1986J highlighting the emergence of a
central component. The red colour and the contours represent the 5.0 GHz radio
brightness. The contours are drawn at 11.3, 16. 22.6É90.5% of the peak
brightness of 0.55 mJy/bm. The blue to white colours show the 15 GHz
brightness of the compact, central component. The scale is given by the width
of the picture of 9 mas. North is up and east to the left. For more
information on the emergence of the compact source, see Bietenholz et al.
(2004).
ALMA has the sensitivity and resolution necessary to detect PWNe features at
high radio frequencies, where we can detect the emission from relativistic
particles that have much longer lifetimes than in X-rays. At cm/mm-
wavelengths, flat-spectrum synchrotron PWNe stand out over steep-spectrum SNRs
(e.g., as seen in the Vela PWN (Hales et al. 2004), discussed in § 6.1.6
below, and illustrated in Figure 18 (Bietenholz et al. 2004)) with minimal
confusion from the Rayleigh-Jeans tail of submm dust. ALMA Band 1 receivers
will allow observations in the frequency regime where PWNe dominate, and
bridge an important gap in frequency coverage, where spectral features such as
power-law breaks occur and linear polarization observations do not suffer from
significant Faraday rotation. Here, even the modest improvements in
sensitivity of ALMA in Band 1 over the JVLA at similar frequencies will be
important. Also, of course, southern PWNe will be much better probed with
ALMA.
#### 6.1.7 Radio Supernovae
Radio supernovae occur when the blast wave of a core-collapse supernova (SN)
sweeps through the slowly expanding wind left over from the progenitor red
supergiant. Particle acceleration and magnetic field amplification lead to
synchrotron radiation in a shell bounded by the forward and reverse shocks
(Chevalier 1982). In general, free-free absorption of the radiation in the
ionized foreground medium coupled with the expansion of the SN causes the
radio light curve first to rise at high frequencies and subsequently at
progressively lower frequencies while the optical depth decreases. When the
optical depth has reached approximately unity, the radio light curve peaks and
decreases thereafter (e.g., Weiler et al. 2002). These characteristics allow
estimates to be made of the density profiles of the expanding ejecta and the
circumstellar medium and also of the mass loss of the progenitor. Resolved
images of SNe provide information, e.g., on the structure of the shell, size,
expansion velocity, age, deceleration, and magnetic field, in addition to
refined estimates of the density profiles and the mass loss (Bartel et al.
2002). Radio observations of SNe can be regarded as a time machine, where the
history of the mass loss of the progenitor is recorded tens of thousands of
years before the star died. Finally, the SN images can be used to make a movie
of the expanding shell of radio emission and to obtain a geometric estimate of
the distance to the host galaxy (Bartel et al. 2007).
ALMA Band 1 receivers will allow exciting science to be done in the areas of
radio light curve measurements, imaging of a nearby SN and, in conjunction
with VLBI, imaging of more distant SNe. Depending on the medium, the delay
between the peak of the radio light curve at 20 cm and 1 cm can be as long as
10 years, as for instance was the case of SN 1996cr (Bauer et al. 2008).
Absorption can also occur in the source itself. In case of SN 1986J, a new
component appeared in the radio spectrum and in the VLBI images about 20 years
after the explosion and then only at or around 1 cm wavelength. The component
is located in the projected center of the shell-like structure of the SN and
may be emission from a very dense clump fortuitously close to that center, or
possibly from a pulsar wind nebula in the physical center of the shell (Figure
18, Bietenholz et al. 2004, 2010). Observations in Band 1 minimize the
absorption effect relative to observations at longer wavelengths and thus
allow investigations of SNe at the earliest times without compromising too
much on the signal to noise ratio of a source with a steep spectrum. ALMA with
Band 1 receivers has the sensitivity to measure the radio light curves of 10s
to 100 SNe. In addition, ALMA may be then also particularly sensitive in
finding “SN factories” in starburst galaxies (e.g., Lonsdale et al. 2006)
where relatively large opacities would otherwise hinder or prevent discovery.
ALMA with Band 1 receivers will allow high-dynamic range images of SN 1987A in
the Large Magellanic Cloud with a resolution of about 300 FWHM beams across
the area of the shell in 2014. Such data would be a significant improvement
over presently obtainable images (Gaensler et al. 2007; Lakićević et al.
2012). Also, since the size of the SN increases by one Band 1 FWHM beam width
per 3 years, the expansion of the shell can be monitored accurately and in
detail, making this SN an important target for ALMA.
In summary, ALMA Band 1 receivers could make strides in observing high-
frequency synchrotron from supernovae, allowing important measurements of
their properties. ALMA’s location in the southern hemisphere makes
investigations of southern SNe (expecially SN 1987A) especially compelling.
Note that ALMA’s southern berth also would make it an important element of
VLBI arrays operating in Band 1, providing southern baselines and high
sensitivity. Previous SN VLBI observations at 1 cm wavelength have provided
clues about physical conditions at the earliest times after the transition
from opaqueness to transparency, and SN VLBI with Band 1 will surely focus on
this area of research.
#### 6.1.8 X-ray Binaries
X-ray binaries (i.e., binary star systems with either a neutron star or a
black hole accreting from a close companion) frequently show jet emission.
Most of these systems are transients. Typically, 1-2 black hole X-ray binaries
undergo a transient outburst per year, while neutron stars outburst at a
slightly higher rate. Outbursts typically last several months (although there
are some which are both considerably longer or shorter), and during outbursts,
X-ray luminosities can change by as much as 7 orders of magnitude. The radio
luminosities of systems seen to date correlate well with the hard X-ray
luminosities (i.e., those above $\sim$20 keV), albeit with considerable, yet
poorly understood scatter.
When the X-ray spectra become dominated by thermal X-ray emission, the radio
emission often turns off (e.g., Tananbaum et al. 1972; Fender et al. 1999),
but the extent to which the flux turns down is still poorly constrained. This
turndown is not seen in neutron star X-ray binaries (Migliari et al. 2004).
The reduced radio emission in black hole X-ray binaries when they have soft
X-ray spectra can be explained by models of jet production in which the jet
power scales with the polodial component of the magnetic field of the
accretion flow (e.g., Livio, Ogilvie & Pringle 1999), and may have
implications for the radio loud/quiet quasar dichotomy (e.g., Meier 1999;
Maccarone, Gallo & Fender 2003). The still-present radio emission from neutron
stars in their soft state may be indicating that the neutron star boundary
layers play an important role in powering jets (Maccarone 2008). The soft
states of X-ray transients are short-lived. During them, there may be decaying
emission from transient radio flares launched during the state transitions.
Therefore, to place better upper limits on the radio jets produced during the
soft state, a high sensitivity, high frequency system with a very high duty
cycle is needed.
The radio properties of X-ray binaries with neutron star primaries are much
more poorly understood than those of black hole X-ray binaries. This situation
is partially because the neutron star X-ray binaries are fainter in X-rays
than are the black hole X-ray binaries. There is, however, additionally some
evidence that neutron star X-ray binaries show a steeper relation between
X-ray luminosity and radio luminosity than do the black hole X-ray binaries,
with $L_{R}\propto L_{X}^{0.7}$ for the black holes and $L_{R}\propto
L_{X}^{1.4}$ for the neutron stars. This difference may be explained if the
neutron stars are radiatively efficient (i.e., with the X-ray luminosity
scaling with the accretion rate) while the black holes are not (i.e., with the
X-ray luminosity scaling with the square of the accretion rate, as has been
proposed by Narayan & Yi 1994) – see Koerding et al. (2006). Radio/X-ray
correlations for neutron star X-ray binaries are, to date, based on small
numbers of data points from few sources, and the most recent work (Tudose et
al. 2009) indicates that the situation may be far more complex than the
picture presented above.
In summary, Band 1 frequencies are important for resolving the relationship
between radio and X-ray flares in transient events from neutron star and black
hole binaries. ALMA with Band 1 receivers would provide the ability to catch
such events at southern declinations. ALMA’s high sensitivity is especially
important to constrain the downturns at radio wavelengths seen in many events.
### 6.2 Line Observations with ALMA Band 1
As with the continuum science cases, numerous examples of scientific
opportunity will be available to ALMA users interested in the numerous lines
located in the Band 1 frequency range from molecular rotational transitions
and radio recombination lines. Here we discuss some science cases that involve
high sensitivity observations of lines, including studies of (1) chemical
differentiation in cloud cores; (2) the chemistry of complex carbon-chain
molecules; (3) ionized gas in the dusty nuclei of starburst galaxies; (4) the
photoevaporation of protoplanetary disks; (5) inflows and outflows from HII
regions; (6) masers; (7) magnetic field strengths in dense gas; (8) molecular
outflows from young stars; (9) the co-evolution of star formation and active
galactic nuclei; and (10) the molecular gas content of star-forming galaxies
at $z$ $\sim$ 2.
#### 6.2.1 Fine Structure of Chemical Differentiation in Cloud Cores
Previous single-dish millimeter molecular line observations have found that
molecular abundance distributions differ significantly between individual dark
cloud cores. A widely accepted interpretation of this chemical differentiation
is that there exists non-equilibrium gas-phase chemical evolution through ion-
molecule reactions within dark cloud cores. Younger cores are rich in “early-
type” carbon-chain molecules such as CCS and HC3N, while more evolved cores,
closer to protostellar formation via gravitational collapse, are rich in
“late-type” molecules such as NH3 and SO (Suzuki et al. 1992). Recent high-
resolution millimeter-line observations, however, have revealed that there are
even finer variations of molecular distributions within cores down to
$\sim$3000 AU scales, and that these fine-scale chemical fluctuations cannot
be explained by the simple scenario of chemical evolution of cores (Takakuwa
et al. 2003, Buckle et al. 2006). The explanation suggested for this behaviour
is that there is first molecular depletion onto grain surfaces in these
regions and then subsequent reaction and desorption of molecules back to the
gas phase through clump-to-clump collisions or energy injection from newly
formed protostars (e.g., Buckle et al. 2006). The molecules that can
differentiate between regions with “early–type” chemistry, before any collapse
of a protostellar object, and the “late–type” chemistry, apparent after the
formation of a protostellar core, have their ground-state (strongest)
transitions in ALMA Band 1. These heavy saturated organic molecules can only
be formed on the surfaces of dust grains, and so their appearance in the
interstellar medium signals the presence of a central heating source, likely a
protostar. ALMA Band 1 receivers will provide the most sensitive test of when
a central heating source turns on, since ALMA will then have the resolution
and sensitivity to detect the presence of these complex molecules within a
dense core of more diffuse, unprocessed gas.
Other recent work (see Garrod, Weaver & Herbst 2008 and references therein)
has shown some surprising detections of saturated complex organic molecules
around apparently quiescent dust cores, consistent with model predictions for
the “warm-up” chemistry expected when a core is undergoing gravitational
collapse and forming an internal heating source. According to models, a later
stage in this sequence occurs when complex saturated molecules produced on
grain surfaces react as the gas warms up, producing “hot core” chemistry, with
even more complex products.
In summary, ALMA Band 1 receivers will allow probes of the smallest length
scales of chemical variation in cloud cores to clarify the relationship among
different molecular abundance distributions (in conjunction with chemical
models). These projects will require both ALMA’s excellent spatial resolution
and in particular its ability to recover the larger-scale structure of cores
through observations with the ACA. Indeed, ALMA’s higher sensitivity to
extended, surface brightness emission and high fidelity make observations of
such lines preferable to observations of them with the JVLA. Also, ALMA Band 1
will likely include 50-52 GHz, a frequency range unavailable with the JVLA
that contains many interesting lines, including C3H2 11,1–00,0 at 51.8 GHz.
Table 4 lists some molecular transitions needed for the chemical studies
within these clouds that are observable over 35-52 GHz.
Table 4: Molecular Transitions between 35 GHz and 52 GHz SO | 23–22 | 36.202040 GHz
---|---|---
HC3N | 4–3 | 36.392332 GHz
HCS+ | 1–0 | 42.674205 GHz
SiO | 1–0 | 43.42376 GHz
HC5N | 17–16 | 45.264721 GHz
CCS | 43–32 | 45.379033 GHz
HC3N | 5–4 | 45.490316 GHz
CCCS | 8–7 | 46.245621 GHz
C3H2 | 21,1–20,2 | 46.755621 GHz
C34S | 1–0 | 48.206956 GHz
CH3OH | 10–00 | 48.372467 GHz
CS | 1–0 | 48.99096 GHz
HDO | 32,1–32,2 | 50.23630 GHz
HC5N | 19–18 | 50.58982 GHz
DC3N | 6–5 | 50.65860 GHz
O2 | N=35-35, J=35-34 | 50.98773 GHz
CH3CHO | 1(1,1)-0(0,0) | 51.37391 GHz
NH2D | 1(1,0)–1(1,1) | 51.47845 GHz
CH2CHCHO | 111–000 | 51.59607 GHz
C3H2 | 11,1–00,0 | 51.841418 GHz
#### 6.2.2 Complex Carbon Chain Molecules
Band 1 receivers will provide the opportunity to search with ALMA for new
complex organic molecules, including the amino acids and sugars from which
life on Earth may have originally evolved. In addition, these complex
molecules provide a powerful tool for understanding star formation and the
processes surrounding it.
There are several reasons why Band 1 is the best place to search for complex
molecules. First, the heavier a molecule, the lower will be its rotational
transition frequencies. The many abundant lighter molecules (e.g., CO, HCN,
CN) have their lowest transitions in Band 3, and so do not appear at all in
Band 1. Therefore, Band 1 does not suffer from contamination from these common
molecules, and so line confusion is much less of a problem. Second, system
temperatures at Band 1 frequencies will be significantly lower than in higher
bands, giving extra sensitivity to detect weak transitions from less abundant
complex molecules, such as glycolaldehyde, the simple sugar known to exist in
the interstellar medium. Table 5 lists some complex carbon-chain molecules
whose transitions have been already detected in the ISM. Note that searches
for complex molecules can be made with Band 1 also using lines in absorption
against bright background objects like, e.g., young stars or quasars.
There is now a significant body of evidence to suggest that complex biological
molecules, such as amino acids and sugars needed for evolution of life on
Earth, evolved in the interstellar medium (e.g., see Holtom et al. 2005; Hunt-
Cunningham & Jones 2004; Bailey et al. 1998). Band 1 receivers will be one of
the best instruments in the world to test this hypothesis observationally.
As with the molecular transitions described in §6.2.1, ALMA’s sensitivity to
low surface brightness line emission through the smaller minimum baselines of
the 12-m Array and the ACA itself makes exploring complex carbon-chain
molecular chemistry preferable with ALMA than the JVLA over 35-50 GHz. In
addition, the likely addition of 50-52 GHz to the Band 1 frequency range is
not available at the JVLA.
Table 5: Some detected ISM complex carbon chain molecules CH2CHCN | propenitrile
---|---
CH2CNH | ketenimine
CH3C4H | methyldiacetylene
CH3CCCN | methyl cyanoacetylene
CH3CH2CN | ethyl cyanide
CH3CHO | acetaldehyde
CH3CONH2 | acetamide
CH3OCH3 | ethyl butyl ether
CH3OCHO | methyl formate
C6H- | hexatriyne anion
C8H | octatetraynyl
H2CCCC | cumulene carbene
HCCCNH+ | $\cdots$
#### 6.2.3 Radio Recombination Lines
In the radio and submillimeter, we have access to an extinction-free ionized
gas tracer: radio recombination lines (RRLs). These lines can measure the
density, filling factor, temperature, and kinematics of the ionized gas in
young star-forming regions that are still heavily obscured by dust. Measuring
the properties of the ionized gas in these regions allows us to probe the
properties of the interstellar medium and the stars in a very early stage of
star formation. RRLs in the ALMA Band 1 frequency range (e.g., H53$\alpha$ at
43.309 GHz) trace ionized gas with densities of $10^{4}\ {\rm cm}^{-3}$, which
is similiar to the densities of young HII regions (Churchwell 2002).
Using RRLs detected in ALMA Band 1, we can:
* •
measure the properties of the ionized gas and young massive stars in the dusty
nuclei of starburst galaxies (see Figure 19; Kepley et al. 2011),
* •
detect the photoevaporation of protoplanetary disks (Pascucci, Gorti &
Hollenbach 2012), and
* •
quantify the properties of inflows and outflows from HII regions (Peters et
al. 2012) and possibly gas ionized by jets from young stars (Shepherd et al.
2013).
In the past, RRLs were difficult to observe – particularly in external
galaxies – because they are faint and broad lines. Today, the high sensitivity
and wide bandwidths of facilities like ALMA make RRLs more accessible. The
wide band widths also allow us to stack RRLs. RRL properties change slowly
with frequency, so stacking all RRLs observed within a band improves the
sensitivity of the observations without increasing the observing time or
affecting the properties of the line.
RRLs are brighter at higher frequencies, but they also are further apart in
frequency space. ALMA Band 1 frequencies are ideal for RRL detection because
the lines are bright and we can detect 3-4 lines in the 8 GHz of bandwidth
provided by the ALMA correlator. At lower frequencies, the lines will be
fainter; at higher frequencies, we cannot stack as many lines.
Figure 19: RRLs can measure the ionized gas properties in the dusty nuclei of
starburst galaxies. The left panel shows JVLA observations of the 1cm
continuum emission, which is mostly free-free emission, from the nuclear
starburst of the edge-on galaxy NGC 253. The right panel shows JVLA
observations of the H58$\alpha$ emission from the same galaxy. The background
image shows optical HST images. Paschen $\alpha$ is red, I band is green, and
B band is blue. Figure from Kepley et al. (2011).
Modeling RRL emission requires a sensitive measurement of the free-free
continuum. At the ALMA Band 1 frequencies, the free-free continuum begins to
dominate over the synchrotron and dust continua, making measuring the free-
free component straightforward. Modeling RRLs at frequencies higher than
$\sim$100 GHz requires disentangling free-free and dust emission.
In summary, ALMA Band 1 receivers will allow the RRLs in its frequency range
to be observed towards many possible targets, including the dusty nuclei of
starburst galaxies, photoevaporating disks, and HII regions. The southern
location of ALMA will allow southern examples of these sources to be easily
observed to high sensitivity.
#### 6.2.4 Maser Science
Masers (Microwave Amplifications by Stimulated Emission of Radiation)
frequently occur in regions of active star formation, from molecular
transitions whose populations are either radiatively or collisionally
inverted. A photon emitted from this material will interact with other excited
molecules along its path, stimulating further emission of identical photons.
This process leads to the creation of a highly directional beam that has
sufficient intensity to be detected at very large distances.
Masers are observed from a variety of molecular and atomic species and each
serves as a signpost for a specific phenomenon, a property which renders
masers powerful astrophysical tools (Menten 2007). More precisely, masers are
formed under specific conditions, and the detection of maser emission
therefore suggests that physical conditions (e.g., temperature, density, and
molecular abundance) in the region where the maser forms lie within a defined
range (c.f., Cohen 1995, Ellingsen 2004, and references therein). Therefore,
interferometric blind and targeted surveys of maser species can lead to the
detection of objects at interesting evolutionary phases (Ellingsen 2007).
Table 6: ALMA bands with known maser lines (Menten 2007) Species | ALMA Bands
---|---
H2O | $-$B3, B5, B6, B7, B8, B9
CH3OH | $-$B1, B3, B4, B6
SiO | $-$B1, B2, B3, B4, B5, B6, B7
HCN | $-$B3, B4, B6, B7, B9
Theoretical models of masers strongly depend on physical conditions as well as
the geometry of the maser source. A successful model should be able to
reproduce observational characteristics of observed maser lines but also to
predict new maser transitions (e.g., the models of Sobolev 1997 for Class II
methanol masers and Neufeld 1991 for water masers). In that respect,
interferometry is essential for the successful search of candidate lines and
confirmation of their maser nature. ALMA, in particular, will resolve closely
spaced maser spots and help further establish precise models of masing sources
by determining if the detected maser signals are associated with thermal
emission (Sobolev 1999), which is essential for improving theoretical models.
With Band 1, ALMA will cover an even wider frequency range, making it ideal
for multi-transition observations of various maser species across the
millimeter and submillimeter windows. Examples of species with observed maser
radiation in the different ALMA bands are given in Table 6, while Tables 7 & 8
list SiO and methanol maser transitions that have been observed or predicted
to be within Band 1.
Maser radiation can be linearly or circularly polarized depending on the
magnetic properties of the molecule. Polarimetric studies of maser radiation
with interferometers can therefore yield information on the morphology of the
magnetic field threading the region on small scales, with the plane-of-sky and
line-of-sight components of the field being probed using linear and circular
polarization measurements, respectively (e.g., see Harvey-Smith 2008,
Vlemmings 2006). Polarization data are essential for improving on the theory
of maser polarization first introduced by Goldreich (1973a), which applies to
a linear maser region, a constant magnetic field, the simplest energy states
for a masing transition, and asymptotic limits. Observations at higher spatial
resolution are needed to verify and improve on more realistic and extensive
models (Watson 2008).
In summary, the ALMA Band 1 frequency range contains numerous CH3OH and SiO
maser lines that can be observed to trace very distinct conditions in the ISM
and probe maser production mechanisms. With ALMA’s high resolutions and
sensitivities in the south, the Band 1 receivers will be able to trace easily
masers from southern sources, and provide highly complementary data to masers
observed in the higher frequency ALMA Bands.
Table 7: Observed SiO maser lines in the Band 1 of ALMA (Menten 2007). Transitions | Frequency (GHz)
---|---
v=0 (J= 1 $\to$ 0) | $-$42.373359
v=3 (J= 1 $\to$ 0) | $-$42.519373
v=2 (J= 1 $\to$ 0) | $-$42.820582
v=0 (J= 1 $\to$ 0) | $-$42.879916
v=1 (J= 1 $\to$ 0) | $-$43.122079
v=0 (J= 1 $\to$ 0) | $-$43.423585
Table 8: Observed (Menten 2007) and predicted (designated with a star, Cragg et al. 2005) methanol maser lines in Band 1 Transitions | Frequency (GHz)
---|---
4(-1) $\to$ 3(0)E | $-$36.1693
7(-2) $\to$ 8(-1)E | $-$37.7037
6(2) $\to$ 5(3)A+ | $-$38.2933
6(2) $\to$ 5(3)A- | $-$38.4527
7(0) $\to$ 6(1)A+ | $-$44.0694
2(0) $\to$ 3(1)E ∗ | $-$44.9558
9(3) $\to$ 10(2)E ∗ | $-$45.8436
#### 6.2.5 Magnetic Field Strengths from Zeeman Measurements
Magnetic fields are believed to play a crucial role in the star formation
process. Various theoretical and numerical studies explain how magnetic fields
can account for the support of clouds against self-gravity, the formation of
cloud cores, the persistence of supersonic line widths, and the low specific
angular momentum of cloud cores and stars (McKee & Ostriker 2007). The
Òstandard modelÓ suggests that the initial mass-to-(magnetic) flux ratio,
M/$\Phi_{init}$, is the key parameter governing the fate of molecular cores.
Namely, if the M/$\Phi_{init}$ of a core is greater than the critical value,
the core will collapse and form stars on short time scales, but for cores with
M/$\Phi_{init}$ smaller than the critical value the process of ambipolar
diffusion will take a long time to reduce the magnetic pressure (Mouschovias &
Spitzer 1976; Shu et al. 1987). On the other hand, recent MHD simulations
suggest that turbulence can control the formation of clouds and cores. In such
cases, the mass-to-flux ratio in the center of a collapsing core will be
larger than that in its envelope, the opposite of the ambipolar diffusion
results (Dib et al. 2007). Therefore, measuring the magnetic field strengths
and the mass-to-flux ratios in the core and envelope provide a critical test
for star formation theories.
Despite its central importance, the magnetic field is the most poorly measured
parameter in the star formation process. The main problem is that magnetic
fields can be measured only via polarized radiation, which requires extremely
high sensitivity for detections. As a result, the observed data on magnetic
fields is sparse compared with those related to the densities, temperatures,
and kinematics in star-forming cores. The large collecting area of ALMA
provides the best opportunity to resolve the sensitivity problem for magnetic
field measurements.
The key to determining mass-to-flux ratios is the measurement of the strength
of magnetic fields. This measurement can be made directly through detection of
the Zeeman effect in spectral lines. Observations of Zeeman splitting involve
detecting the small difference between left and right circular polarizations,
which is generally very small in interstellar conditions (with the exception
of masers). Successful non-maser detections of the Zeeman effect in molecular
clouds have only been carried out with HI, OH, and CN lines because these
species have the largest Zeeman splitting factors ($\sim$2 – 3.3 Hz/$\mu$G)
among all molecular lines (Crutcher et al. 1996, 1999; Falgarone et al. 2008).
Thermal HI and OH lines, however, probe relatively low-density gas ($n$(H)
$<10^{4}$ cm-3). Also, CN detections are difficult; Crutcher (2012) described
only 8 CN Zeeman detections towards 14 positions observed with significant
sensitivity.
ALMA Band 1 receivers provide the opportunity to detect the Zeeman effect from
the CCS 43–32 line at 45.37903 GHz and hence greatly advance our understanding
in star formation. CCS has been widely recognized as being present only very
early in the star-forming process through chemical models (Aikawa et al. 2001,
2005) and observations (Suzuki et al. 1992; Lai & Crutcher 2000). Therefore
the mass-to-flux ratio derived from the CCS Zeeman measurements will be very
close to the initial values before the onset of gravitational collapse. CCS
43–32 also has a relatively large Zeeman splitting factor ($\sim$ 0.6
Hz/$\mu$G; Shinnaga & Yamamoto 2000) compared to most molecules. ALMA’s
antennas and site will be excellent at these “long” wavelengths, providing the
stability and accuracy needed for such sensitive polarization work. The
linearly polarized detectors on ALMA’s antennas will also be ideally suited to
measurement of Stokes V signatures from CCS.
Figure 20: The expected detection limits (3 $\sigma$) with integration time of
1 hr and 10 hr for a range of magnetic field strengths and CCS line intensity.
Using the BIMA survey results from Lai & Crutcher (2000), Figure 20
demonstrates that detections of CCS Zeeman effects can be achieved if the ALMA
specifications for Band 1 receivers are met. Zeeman effect detection depends
on two factors: the magnetic field strength and the line intensity. The two
lines in Fig. 20 show the 3 $\sigma$ detection limits for Stokes V spectrum
with channel width of 0.024 km s-1 and 1 hr or 10 hr integration time for a
range of magnetic field strengths and line intensities. The channel width is
chosen to have at least 6 channels across the FWHM of the total intensity
spectrum (Stokes I). If we scale the line intensity from Lai & Crutcher (2000)
assuming the intensity distribution is uniform within the 30$\arcsec$ BIMA
beam, the expected line intensity would be around 0.1-0.4 Jy for ALMA
observations with 10$\arcsec$ beam. Therefore, Fig. 20 shows that for the
magnetic fields of 0.2-1 mG (typical values estimated from the application of
the Chandrasehkar-Fermi method to dust polarimetry in dense cores), we can
detect the CCS Zeeman effect with reasonable on-source integration time (less
than 10 hr).
Note that the SiO v=1, J=1–0 transition at 43.12 GHz could be also used to
probe magnetic fields using the Zeeman effect, under certain circumstances.
Though its Zeeman splitting factor is lower than that of the CCS 43–32 line,
the Zeeman effect may be detectible in situations where the SiO line is
extraordinarily bright, e.g., as a maser (see McIntosh, Predmore & Patel
1994). (Note, however, that non-Zeeman interpretations of circularly polarized
SiO emission have also been advanced; see Weibe & Watson 1998).
In summary, ALMA Band 1 receivers will provide the opportunity to measure the
initial mass-to-flux ratio of molecular cores through the detection of the
Zeeman effect. ALMA’s linear feeds are ideally suited to measuring Stokes V
and ALMA’s ability to recover extended, low surface brightness emission
through the shorter baselines of the 12-m Array and the inclusion of the ACA
will be critical. E.g., Roy et al. 2011 noted that the JVLA only recovered
1-13% of the integrated emission of CCS 21–10 observed in single-dish
observations using the JVLA’s most compact (D) configuration.) The results
from Zeeman splitting from ALMA will allow us to test realistically the
expectations from theoretical and numerical models for the first time.
#### 6.2.6 Molecular Outflows from Young Stars
The Submillimeter Array (SMA) has proven to be a successful instrument for the
study of the youngest molecular outflows and jets from the most deeply
embedded sources (e.g., Hirano et al. 2006; Palau et al. 2006; Lee et al.
2007a,b, 2008, 2009). The detection of excitation from rotational transitions
of SiO up to levels $J$=8–7 and CO up to $J$=3–2 have uniquely identified a
molecular high-velocity jet-like component located within outflow shells. This
component displays similarities to the optical forbidden line jets observed in
T-Tauri stars (Hirano et al. 2006; Palau et al. 2006; Codella et al. 2007;
Cabrit et al. 2007). These observations have provided a new probe of how jets
are launched and collimated during the earliest protostellar phase.
One unique opportunity offered by the Band 1 frequency range is observation of
the $J$=1–0 transition of the SiO molecule at 43.424 GHz. This transition has
not yet been detected nor surveyed around even the brightest molecular
outflows, except using single-dish telescopes (Haschick & Ho 1990). One
feature of this line that may be potentially distinct from the higher-$J$
transitions of SiO is that it may be tracing the outer and more diffuse gas
located on the outskirts of outflow shells that can be easily excited by
shocks. Potential morphological and kinematic studies of the regions where the
outflows interact with their own pre-natal clouds could be contrasted with
other transitions using knowledge of their excitation conditions. In
particular, the improved sensitivity to extended emission and higher image
fidelity of ALMA make observations of SiO $J$=1–0 toward outflows more
attractive with ALMA than with the JVLA.
#### 6.2.7 Co-Evolution of Star Formation and Active Galactic Nuclei
Roughly half of the high-redshift objects detected in CO line emission are
believed to host an active galactic nucleus (AGN). Although they are selected
based on their AGN properties, optically luminous high-redshift quasars
exhibit many characteristics indicative of ongoing star formation, e.g.,
thermal emission from warm dust (Wang et al. 2008) or extended UV continuum
emission. Indeed, galaxies with AGNs in the local Universe reveal a strong
correlation between the mass ($m$) in their supermassive black hole (SMBH) and
that of their stellar bulge (measured from the stellar velocity dispersion
($\sigma$); e.g., Kormendy & Richstone 1995; Magorrian et al. 1998; Gebhardt
et al. 2000). Such a correlation can be explained if the SMBH formed coevally
with the stellar bulge, implying that the luminous quasar activity signaling
the formation of a sub-arcsecond SMBH at high-redshift should be accompanied
by starburst activity. High spatial resolution observations of CO line
emission in high-redshift quasars can be used to infer the dynamical masses,
which are found to be comparable to the derived molecular gas + black hole
masses, meaning that their stellar component cannot contribute a large
fraction of the total mass.
There is mounting evidence that quasar host galaxies at redshifts $z$ = 4–6
have SMBH masses up to an order of magnitude larger than those expected from
their bulge masses and the local relation (Walter et al. 2004; Riechers et
al., in prep.), suggesting that the SMBH may have formed first. The possible
time evolution of the $m-\sigma$ relation is of fundamental importance in
studies of galaxy evolution, and this new finding needs to be made more
statistically robust. Future observations of high-redshift AGN with the Band 1
receivers on ALMA would allow us to address this question through the study of
low-$J$ CO line emission in galaxies beyond redshifts $z\approx 1.3$ (see
§4.2). ALMA especially allows studies of examples of such objects in the south
that are not well observable (if at all) with the JVLA.
#### 6.2.8 The Molecular Gas Content of Star-Forming Galaxies at $z\sim 2$
While low-$J$ CO line emission has only been detected in a few high-redshift
objects, high-$J$ CO line emission has been detected in more than sixty
sources, most of which are classified as either submillimeter galaxies (SMGs)
or far-infrared (FIR) luminous QSOs (see Carilli et al. 2011 for a review).
Most of these studies have been conducted with sensitive interferometers and
single-dish facilities operating in the 3 mm band (e.g., ALMA Band 3), which
is sensitive to higher-$J$ CO line transitions at high redshift, as is
illustrated in Figure 6. These lines generally trace warmer and denser gas,
and so previous data may have led to a bias in our understanding of the
molecular gas properties of high-redshift galaxies (e.g., Papadopoulos &
Ivison 2002). The addition of Band 1 receivers on ALMA will allow comparisons
of the cold gas traced by the low-$J$ transitions ($J$=2–1/1–0) in galaxies
from moderate redshifts ($z\approx 1.3$) to those which existed when the
Universe was re-ionized sometime before $z\mathrel{\raise
1.50696pt\hbox{$\scriptstyle>$}\kern-6.00006pt\lower
1.72218pt\hbox{{$\scriptstyle\sim$}}}6$.
Although many previous studies of CO line emission in high-redshift galaxies
have focused on those starburst galaxies and AGN undergoing episodes of
extreme star formation (e.g., $\gg$100 M⊙ yr-1), significant masses of
molecular gas ($>10^{10}$ M⊙) have been discovered in more modest star-forming
galaxies at $z=1.5-2.0$ (Daddi et al. 2008). These “BzK” galaxies are selected
for their location in a B-$z$-K colour diagram (Daddi et al. 2004) and have
star-formation rates of $\sim$100 M⊙ yr-1 (Daddi et al. 2007), while their
number density is roughly a factor of 30 larger than that of the more extreme
SMGs at similar redshifts. Observations of CO $J$=2–1 line emission in these
BzK galaxies reveal comparable masses of molecular gas to that of the SMGs, so
their star-formation efficiencies appear lower. The excitation conditions of
their molecular gas (temperature and density) are similar to those of the
Milky Way (Dannerbauer et al. 2008), as indicated by the “turnover” in the CO
line spectral energy distribution occurring at the J=3–2 transition, i.e.,
lower than that of the SMGs which typically occurs at the J=6–5 or J=5–4
transition (Weiss et al. 2005). To develop a full spectral energy distribution
for the CO line excitation, observations of these galaxies in the $J$=1–0
transition are needed with Band 1 receivers on ALMA. Such data will also
provide a more robust estimate of the total molecular gas mass, along with the
spatial resolution needed to constrain the gas kinematics, as has been done
for the SMGs (Tacconi et al. 2006). Indeed, recent high-resolution studies of
CO $J$=1–0 from lensed Lyman Break galaxies (Riechers et al. 2010) and
unlensed BzK galaxies (Aravena et al., in prep.) have been made with the JVLA.
Also, CO $J$=1–0 emission has been detected with the JVLA or GBT towards SMGs
Ivison et al. 2010, 2011; Frayer et al. 2011; Riechers et al. 2011a,b). ALMA
observations will allow similar important investigations to occur towards
southern objects, especially those traced by ALMA itself in its higher-
frequency Bands.
## 7 Summary
The Band 1 receiver suite has been considered an essential part of ALMA from
the earliest planning days. Even through the re-baselining exercise in 2001,
the importance of Band 1 was emphasized. With the ALMA Development Plan
underway, we have undertaken an updated review of the scientific opportunity
at these longer wavelengths. This document presents a set of compelling
science cases over this frequency range. The science cases reflect the new
proposed range of Band 1, 35-50 GHz (nominal) with an extension up to 52 GHz,
which was in fact chosen to optimize the science return from Band 1. The
science cases range from nearby stars and galaxies to the re-ionization edge
of the Universe. Two provide additional leverage on the present ALMA Level One
Science Goals and are seen as particularly powerful motivations for building
the Band 1 receiver suite: (1) detailing the evolution of grains in
protoplanetary disks, as a complement to the gas kinematics, requires
continuum observations out to $\sim 35\,$GHz ($\sim 9\,$mm); and (2) detecting
CO 3 – 2 spectral line emission from Galaxies like the Milky Way during the
era of re-ionization, $6<z<10$ also requires Band 1 receiver coverage. Band 1
receivers will also allow the pursuit of a diverse range of science cases that
take advantage of the ALMA’s particular strengths over other facilities (e.g.,
the JVLA).
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|
arxiv-papers
| 2013-10-06T17:07:32 |
2024-09-04T02:49:52.026687
|
{
"license": "Public Domain",
"authors": "J. Di Francesco, D. Johnstone, B. C. Matthews, N. Bartel, L. Bronfman,\n S. Casassus, S. Chitsazzadeh, H. Chou, M. Cunningham, G. Duchene, J.\n Geisbuesch, A. Hales, P. T. P. Ho, M. Houde, D. Iono, F. Kemper, A. Kepley,\n P. M. Koch, K. Kohno, R. Kothes, S.-P. Lai, K.Y. Lin, S.-Y. Liu, B. Mason, T.\n J. Maccarone, N. Mizuno, O. Morata, G. Schieven, A. M. M. Scaife, D. Scott,\n H. Shang, M. Shimojo, Y.-N. Su, S. Takakuwa, J. Wagg, A. Wootten, and F.\n Yusef-Zadeh",
"submitter": "James Di Francesco",
"url": "https://arxiv.org/abs/1310.1604"
}
|
1310.1606
|
# A note on a relationship between the inverse eigenvalue problems for
nonnegative and doubly stochastic matrices and some applications111Preprint
submitted to Elsevier
Bassam Mourad Bassam mourad, Department of Mathematics, Faculty of Science V,
Lebanese University, Nabatieh, Lebanon ([email protected]).
(November, 2012)
###### Abstract
In this note, we establish some connection between the nonnegative inverse
eigenvalue problem and that of doubly stochastic one. More precisely, we prove
that if $(r;\lambda_{2},...,\lambda_{n})$ is the spectrum of an $n\times n$
nonnegative matrix $A$ with Perron eigenvalue $r$, then there exists a least
real number $k_{A}\geq-r$ such that $(r+\epsilon;\lambda_{2},...,\lambda_{n})$
is the spectrum of an $n\times n$ nonnegative generalized doubly stochastic
matrix for all $\epsilon\geq k_{A}.$ As a consequence, any solutions for the
nonnegative inverse eigenvalue problem will yield solutions to the doubly
stochastic inverse eigenvalue problem. In addition, we give a new sufficient
condition for a stochastic matrix $A$ to be cospectral to a doubly stochastic
matrix $B$ and in this case $B$ is shown to be the unique closest doubly
stochastic matrix to $A$ with respect to the Frobenius norm. Some related
results are also discussed.
### keywords.
nonnegative matrices, stochastic matrices, doubly stochastic matrices, inverse
eigenvalue problem
### AMS.
15A12, 15A18, 15A51
## 1 Introduction
An $m\times n$ matrix $A$ with real entries is said to be nonnegative if all
of its entries are nonnegative. If in addition, each row sum of $A$ is equal
to $1,$ then $A$ is called stochastic. Generally, an $n\times n$ matrix $A$
over the field of the real numbers $\mathbb{R}$ having each row sum equals to
a nonnegative number $r\in\mathbb{R}^{+}$, is said to be an $r$-generalized
stochastic matrix (note that $A$ is not necessarily nonnegative). If $A$ and
its transpose $A^{T}$ are $r$-generalized stochastic matrices then $A$ is said
to be an $r$-generalized doubly stochastic matrix. The set of all
$r$-generalized $n\times n$ doubly stochastic matrices with entries in
$\mathbb{R}$ is denoted by $\Omega^{r}(n)$. A generalized doubly stochastic
matrix is an element of $\Omega(n)$ where
$\Omega(n)=\bigcup_{r\in\mathbb{R}^{+}}\Omega^{r}(n).$
Of special importance are the _nonnegative_ elements in $\Omega(n)$ and in
particular the nonnegative elements in $\Omega^{1}(n)$ which are called the
_doubly stochastic_ matrices and have been the object of study for a long time
see [4, 8, 11, 15, 16, 17, 18, 19, 20, 22], and earlier work can be found in
[2, 9, 24, 27].
The Perron-Frobenius theorem states that if $A$ is a nonnegative matrix, then
it has a nonnegative eigenvalue $r$ (that is the Perron root) which is greater
than or equal to the modulus of each of the other eigenvalues (see e.g. [25,
14]). To this eigenvalue $r$ of $A$ corresponds a nonnegative eigenvector $x$
which is also referred to as the Perron-Frobenius eigenvector of $A.$ In
particular, it is well-known that if $A$ is an $n\times n$ stochastic matrix
then its Perron eigenvalue $r=1$ and its corresponding unit eigenvector is the
column vector $x=e_{n}=\frac{1}{\sqrt{n}}(1,1,...,1)^{T}\in\mathbb{R}^{n}.$
Obviously, this is also true for $A$ and $A^{T}$ when $A$ is doubly
stochastic. More generally, for any $X\in\Omega^{r}(n)$, $e_{n}$ is also an
eigenvector for both $X$ and $X^{T}$ corresponding to the eigenvalue $r$.
Therefore $X\in\Omega^{r}(n)$ if and only if $Xe_{n}=re_{n}$ and
$e_{n}^{T}X=re_{n}^{T}$ if and only if $XJ_{n}=J_{n}X=rJ_{n},$ where $J_{n}$
is the $n\times n$ matrix with each of its entries is equal to $\frac{1}{n}$.
Two matrices $A$ and $B$ are said to be _cospectral_ if they have the same set
of eigenvalues. Throughout this paper, if $A=(a_{ij})$ is any square matrix,
then the spectrum of $A$ is denoted by $\sigma(A).$ In addition, $x_{j}(A)$
and $a_{j}$ denote the _sum_ and the _smallest_ entry of the $j$th column of
$A$ respectively. For any real number $a,$ the absolute value of $a$ will be
denoted by $|a|,$ and the $n\times n$ identity matrix will be denoted by
$I_{n}.$ Next, we introduce the following notation which first appeared in
[5]. To indicate that the $n$-list $\\{\lambda_{1},\ldots,\lambda_{n}\\}$ is
the spectrum of an $n\times n$ nonnegative matrix $A$ with Perron eigenvalue
$\lambda_{1},$ we will write the first component with a semi-column as
$(\lambda_{1};\lambda_{2},\ldots,\lambda_{n})$ and say that it is realized by
$A.$
Recall that the inverse eigenvalue problem for special kind of matrices is
concerned with constructing a matrix that maintains the required structure
from its set of eigenvalues. For Jacobi matrices, this has been studied
recently in [6], for symmetric quasi anti-bidiagonal matrices, this has been
done in [21] and more recently for block Toeplitz matrices, a study was
presented in [29], see [3] for more on these topics.
In this paper, we are concerned with the following inverse eigenvalue
problems. The nonnegative inverse eigenvalue problem (NIEP) can be stated as
the problem of finding necessary and sufficient conditions for an $n$-tuples
$(\lambda_{1};\lambda_{2},\ldots,\lambda_{n})$ (where
$\lambda_{2},...,\lambda_{n}$ might be complex) to be the spectrum of an
$n\times n$ nonnegative matrix $A$ see [1, 10, 13, 14, 28] and the references
therein. Similarly, the stochastic inverse eigenvalue problem (SIEP) asks
which sets of $n$ complex numbers can occur as the spectrum of an $n\times n$
stochastic matrix $A$. In addition, the doubly stochastic inverse eigenvalue
problem denoted by (DIEP), is the problem of determining the necessary and
sufficient conditions for a complex $n$-tuples to be the spectrum of an
$n\times n$ doubly stochastic matrix. Now the nonnegative $r$-generalized
stochastic (resp. doubly stochastic) inverse eigenvalue problem can be defined
analogously. However, for $r>0$ it is obvious that this last problem is
equivalent to that of (SIEP) (resp. DIEP) since
$(r;\lambda_{2},...,\lambda_{n})$ is realized by an $n\times n$ nonnegative
$r$-generalized stochastic (resp. doubly stochastic) matrix if and only if
$\frac{1}{r}(r;\lambda_{2},...,\lambda_{n})$ is realized by an $n\times n$
stochastic (resp. doubly stochastic) matrix.
It is well-known (see [9]) that (NIEP) is equivalent to (SIEP). More
precisely, if the $n$-tuples $(\lambda_{1};\lambda_{2},...,\lambda_{n})$ is
the spectrum of an $n\times n$ nonnegative matrix $A,$ then
$(\lambda_{1};\lambda_{2},...,\lambda_{n})$ is also the spectrum of a
nonnegative $\lambda_{1}$-generalized stochastic matrix. In addition, (SIEP)
and (DIEP) are known not to be equivalent (see [9]) and hence the obvious
question to consider here is how these two problems are connected. Our
intention in this paper, is to establish some relation between these two
problems. More generally, if $\lambda=(\lambda_{1},...,\lambda_{n})$ is the
spectrum of an $n\times n$ nonnegative matrix $A,$ what can be done to
$\lambda$ in order to have it realized by an $n\times n$ doubly stochastic
matrix. In addition, the problem of finding sufficient conditions on a
stochastic matrix to be cospectral to a doubly stochastic matrix is also
considered here.
Finally, we conclude this section by some results from [12] but first we need
to introduce some more relevant notation. Let
$V_{n-1}=I_{n-1}-(1+\frac{1}{\sqrt{n}})J_{n-1}$ and define the block matrix
$U_{n}=\left(\begin{array}[]{cc}\frac{1}{\sqrt{n}}&\frac{\sqrt{n-1}}{\sqrt{n}}e_{n-1}^{T}\\\
\frac{\sqrt{n-1}}{\sqrt{n}}e_{n-1}&V_{n-1}\\\ \end{array}\right).$ Then the
first result can be stated as follows.
###### Lemma 1.1
For any matrix $A\in\Omega^{1}(n),$ there exists an $(n-1)\times(n-1)$ matrix
$X$ such that $A=U_{n}(1\oplus X)U_{n}$ and conversely, for any
$(n-1)\times(n-1)$ real matrix $X,$ $U_{n}(1\oplus X)U_{n}\in\Omega^{1}(n).$
###### Remark 1.2
The preceding lemma is also valid if $U_{n}$ is replaced by any real
orthogonal matrix $V$ whose first column is $e_{n}$ and in this case
$A=V(1\oplus X)V^{T}$ (see [26]).
The second one is the following.
###### Theorem 1.3
[12] Let $A$ be an $n\times n$ real matrix. Then
$B^{*}=(I_{n}-J_{n})A(I_{n}-J_{n})+J_{n}$ is the unique closest matrix to $A$
in $\Omega^{1}(n)$ with respect to the Frobenius norm
## 2 Main observations
We start this section by introducing the following auxiliary result which is
presented in Perfect [23] and is due to R. Rado.
###### Theorem 2.1
([23]) Let $A$ be any $n\times n$ matrix with eigenvalues
$\lambda_{1},...,\lambda_{n}.$ Let $X_{1},X_{2},...,X_{r}$ be $r$ eigenvectors
of $A$ corresponding respectively to the eigenvalues
$\lambda_{1},...,\lambda_{r}$ with $r\leq n$ and let
$X=[X_{1}|X_{2}|...|X_{r}|]$ be the $n\times r$ matrix whose columns are
$X_{1},X_{2},...,X_{r}.$ Then for any $r\times n$ matrix $C,$ the matrix
$A+XC$ has eigenvalues
$\gamma_{1},...,\gamma_{r},\lambda_{r+1},...,\lambda_{n}$ where
$\gamma_{1},...,\gamma_{r}$ are the eigenvalues of the matrix $\Lambda+CX$
where $\Lambda=diagonal(\lambda_{1},...,\lambda_{r}).$
As a conclusion, we have the following elementary lemma.
###### Lemma 2.2
Let $(r,\lambda_{2},...,\lambda_{n})$ be the spectrum of an $r$-generalized
$n\times n$ stochastic (resp. doubly stochastic) matrix $A=(a_{ij}).$ Then for
any real $\epsilon,$ the $n$-list $(r+\epsilon,\lambda_{2},...,\lambda_{n})$
is the spectrum of an $(r+\epsilon)$-generalized stochastic (resp. doubly
stochastic) $n\times n$ matrix $S.$ In particular, there exists $k\geq 0$ such
that the $n$-list $(r+\epsilon,\lambda_{2},...,\lambda_{n})$ is the spectrum
of a nonnegative $(r+\epsilon)$-generalized stochastic (resp. doubly
stochastic) $n\times n$ matrix $S$ for all $\epsilon\geq k.$
Proof. Since $e_{n}$ is an eigenvector for $A$ corresponding to the eigenvalue
$r,$ then taking the matrices $X=\epsilon e_{n}$ and $C=e_{n}^{T}$ in the
preceding theorem, we obtain the $(r+\epsilon)$-generalized stochastic (resp.
doubly stochastic) $n\times n$ matrix $S=A+\epsilon e_{n}e_{n}^{T}$ with
$\sigma(S)=(r+\epsilon,\lambda_{2},...,\lambda_{n}).$ For the second part, we
let $k=|\min(a_{ij})|$ and then for any $\epsilon\geq k,$ the matrix $S$ is
nonnegative. This completes the proof.
Now for nonnegative matrices, we have the following.
###### Theorem 2.3
Let $\sigma(A)=(r;\lambda_{2},...,\lambda_{n})$ be the spectrum of an $n\times
n$ nonnegative matrix $A=(a_{ij}).$ Then there exists a real $k_{A}\geq-r$
such that $(r+\epsilon;\lambda_{2},...,\lambda_{n})$ is the spectrum of a
nonnegative $(r+\epsilon)$-generalized $n\times n$ doubly stochastic matrix
$D,$ for all $\epsilon\geq k_{A}.$
Proof. First let $x_{j}(A)$ be denoted by $x_{j}$ for all $j=1,2,...,n$ and
without loss of generality, we can assume that $A$ is nonnegative
$r$-generalized stochastic matrix. Then for any vector
$y=\sqrt{n}(y_{1},y_{2},...,y_{n})^{T}$ in $\mathbb{R}^{n},$ Theorem 2.1 tells
us that the spectrum $\sigma(B)$ of the matrix $B=A+e_{n}y^{T}$ is clearly
equal to
$\sigma(B)=\left(r+\sum\limits_{j=1}^{n}y_{j},\lambda_{2},...,\lambda_{n}\right).$
Now the key idea is to study the conditions on all the $y_{j}$s for which $B$
is a nonnegative generalized doubly stochastic matrix. Clearly the matrix $B$
is given by:
$B=\left(\begin{array}[]{cccccc}a_{11}&a_{12}&.&.&a_{1n-1}&r-\sum\limits_{j=1}^{n-1}a_{1j}\\\
a_{21}&a_{22}&.&.&a_{2n-1}&r-\sum\limits_{j=1}^{n-1}a_{2j}\\\ .&.&.&.&.&.\\\
.&.&.&.&.&.\\\
a_{n1}&a_{n2}&.&.&a_{nn-1}&r-\sum\limits_{j=1}^{n-1}a_{nj}\end{array}\right)+\left(\begin{array}[]{cccccc}y_{1}&y_{2}&.&.&y_{n-1}&y_{n}\\\
y_{1}&y_{2}&.&.&y_{n-1}&y_{n}\\\ .&.&.&.&.&.\\\ .&.&.&.&.&.\\\
y_{1}&y_{2}&.&.&y_{n-1}&y_{n}\\\ \end{array}\right)$
Now observe that each row sum of $B$ is equal to $r+\sum_{j=1}^{n}y_{j},$ and
for all $j=1,2,...n,$ the sum of the $j$th column of $B$ is $x_{j}+ny_{j}.$ By
equating each $j$th column sum of $B$ to the $j$th row sum which is
$r+\sum_{j=1}^{n}y_{j},$ we obtain the system of $n$ linear equations in the
$n$ unknowns $y_{1},...,y_{n}$ given by:
$\displaystyle\left\\{\begin{array}[]{c}ny_{1}+x_{1}=r+\sum\limits_{j=1}^{n}y_{j}\\\
ny_{2}+x_{2}=r+\sum\limits_{j=1}^{n}y_{j}\\\ \vdots\\\
ny_{n-1}+x_{n-1}=r+\sum\limits_{j=1}^{n}y_{j}\\\
ny_{n}+x_{n}=r+\sum\limits_{j=1}^{n}y_{j}\end{array}\right.$ (6)
Since $\sum_{j=1}^{n}x_{j}=nr,$ then the sum of any $n-1$ equations of (1) is
equal to the remaining equation and the system is in fact of $n-1$ equations
in $n$ unknowns and has an infinite number of solutions (which is obviously a
line solution). Now if we let $x_{m}=max(x_{j}),$ then we take $y_{m}$ as the
only parameter for this system, and then its solution is easily given by:
$\displaystyle\left\\{\begin{array}[]{c}y_{1}=r+y_{m}-(1/n)(2x_{1}+x_{2}+...+x_{m-1}+x_{m+1}+...+x_{n})\\\
y_{2}=r+y_{m}-(1/n)(x_{1}+2x_{2}+x_{3}+...+x_{m-1}+x_{m+1}+...+x_{n})\\\
\vdots\\\
y_{m-1}=r+y_{m}-(1/n)(x_{1}+...+x_{m-2}+2x_{m-1}+x_{m+1}+...+x_{n})\\\
y_{m+1}=r+y_{m}-(1/n)(x_{1}+...+x_{m-1}+2x_{m+1}+x_{m+2}+...+x_{n})\\\
\vdots\\\
y_{j}=r+y_{m}-(\frac{x_{1}+x_{2}+...+x_{m-1}+x_{m+1}+...+x_{j-1}+2x_{j}+x_{j+1}+...+x_{n}}{n})\\\
\vdots\\\
y_{n}=r+y_{m}-(1/n)(x_{1}+x_{2}+...+x_{m-1}+x_{m+1}+...+x_{n-1}+2x_{n})\\\
\end{array}\right.$ (16)
So that for $j\neq m$ the $ij$-entry of $B=(b_{ij})$ is clearly given by:
$b_{ij}=a_{ij}+r+y_{m}-(1/n)(x_{1}+...+x_{m-1}+x_{m+1}+...+x_{j-1}+2x_{j}+x_{j+1}+...+x_{n}),$
and
$b_{im}=a_{im}+y_{m}.$
As $a_{j}$ be the smallest entry of the $j$th column of $A,$ then obviously
the conditions for which $B$ is nonnegative are given by:
$\displaystyle\left\\{\begin{array}[]{c}a_{1}+r+y_{m}-(1/n)(2x_{1}+x_{2}+...+x_{m-1}+x_{m+1}+...+x_{n-1})\geq
0\\\ a_{2}+r+y_{m}-(1/n)(x_{1}+2x_{2}+...++x_{m-1}+x_{m+1}+...+x_{n-1})\geq
0\\\ \vdots\\\
a_{m-1}+r+y_{m}-(1/n)(x_{1}+...+x_{m-2}+2x_{m-1}+x_{m+1}+...+x_{n})\geq 0\\\
a_{m}+y_{m}\geq 0\\\
a_{m+1}+r+y_{m}-(1/n)(x_{1}+...+x_{m-1}+2x_{m+1}+x_{m+2}+...+x_{n})\geq 0\\\
\vdots\\\
a_{n}+r+y_{m}-(1/n)(x_{1}+x_{2}+...+x_{m-1}+x_{m+1}+...+x_{n-1}+2x_{n})\geq
0.\\\ \end{array}\right.$ (25)
It is easy to see that system (3) has an infinite number of solutions. Let
$k_{A}$ be the smallest value of $y_{m}$ for which (3) is satisfied. Since
$na_{j}\leq x_{j}$ for all $j\neq m$ then we have
$a_{j}-(1/n)(x_{1}+...+x_{m-1}+x_{m+1}+...+x_{j-1}+2x_{j}+x_{j+1}+...+x_{n})\leq
0,$ so that $r+k_{A}\geq
r+k_{A}+a_{j}-(1/n)(x_{1}+...+x_{m-1}+x_{m+1}+...+x_{j-1}+2x_{j}+x_{j+1}+...+x_{n})\geq
0,$ for all $j\neq m.$ Moreover $a_{m}\leq r$ and then $r+k_{A}\geq
a_{m}+k_{A}\geq 0.$ Thus $k_{A}\geq-r.$ Now for any $\epsilon\geq k_{A}$ there
exists $\alpha\geq 0$ such that such that $\epsilon=k_{A}+\alpha$ and then
$(r+\epsilon,\lambda_{2},...,\lambda_{n})$ is obviously realized by $B+\alpha
J_{n}$ and the proof is complete.
###### Example 2.4
Let $A$ be the stochastic matrix defined by
$A=\left(\begin{array}[]{ccc}1/3&1/3&1/3\\\ 1/4&1/4&1/2\\\ 1/6&1/6&2/3\\\
\end{array}\right)$ with $\sigma(A)=(1,0,1/4).$ Then $x_{1}=x_{2}=3/4,$
$x_{3}=3/2$ and therefore the parameter for the system is $y_{3}$ with
$B=A+\left(\begin{array}[]{ccc}1+y_{3}-(2x_{1}+x_{2})/3&1+y_{3}-(x_{1}+2x_{2})/3&y_{3}\\\
1+y_{3}-(2x_{1}+x_{2})/3&1+y_{3}-(x_{1}+2x_{2})/3&y_{3}\\\
1+y_{3}-(2x_{1}+x_{2})/3&1+y_{3}-(x_{1}+2x_{2})/3&y_{3}\\\
\end{array}\right)=\left(\begin{array}[]{ccc}7/12+y_{3}&7/12+y_{3}&1/3+y_{3}\\\
1/2+y_{3}&1/2+y_{3}&1/2+y_{3}\\\ 5/12+y_{3}&5/12+y_{3}&2/3+y_{3}\\\
\end{array}\right).$
Clearly the smallest entry of $B$ is $y_{3}+1/3$ and therefore $k_{A}=-1/3,$
and then we obtain the nonnegative $\frac{1}{2}$-generalized doubly stochastic
matrix $X=\left(\begin{array}[]{ccc}1/4&1/4&0\\\ 1/6&1/6&1/6\\\
1/12&1/12&1/3\\\ \end{array}\right)$ whose spectrum is
$\sigma(X)=(1/2,0,1/4).$ Finally, note that $X+1/2J_{n}$ is doubly stochastic
with spectrum $(1,0,1/4).$
It should be stressed that the boundary $k_{A}=-r$ can be achieved by the
matrix $A$ whose each entry in the first column is $r$ and all the remaining
entries are zeroes and in this case, $A$ is cospectral to the nonnegative
$r$-generalized doubly stochastic matrix $rJ_{n}.$
## 3 Applications
One of the obvious applications of the preceding theorem lies in the fact that
any known sufficient conditions for the resolution of (NIEP) will eventually
lead to some sufficient conditions for the resolution of (DIEP).
Recall from [7] that if $(\lambda_{2},...,\lambda_{n})$ is any list of complex
numbers which is closed under complex conjugation, then there exists a least
real number $\lambda_{1}\geq 0$ such that $(h;\lambda_{2},...,\lambda_{n})$ is
realized by some $n\times n$ nonnegative matrix $A_{h}$ for all
$h\geq\lambda_{1}.$ Thus by Theorem 2.3,
$\frac{1}{h+k_{A_{h}}}(h+k_{A_{h}};\lambda_{2},...,\lambda_{n})$ is realized
by an $n\times n$ doubly stochastic matrix.
To present another main application, recall that in [9], the author obtained
some sufficient conditions for a stochastic matrix to be similar to a doubly
stochastic matrix. In connection with this, we prove that the preceding
theorem can be simply used to obtain sufficient conditions for a stochastic
matrix to be cospectral to a doubly stochastic matrix which is one of the main
results of this paper.
###### Theorem 3.1
Let $A=(a_{ij})$ be an $n\times n$ stochastic matrix with spectrum
$(1,\lambda_{2},...,\lambda_{n}).$ Let $a_{j}$ and $x_{j}$ be the smallest
entry and the sum of the $j$th column of $A$ respectively. If
$\displaystyle\left\\{\begin{array}[]{c}x_{1}\leq\quad 1+na_{1}\\\
x_{2}\leq\quad 1+na_{2}\\\ \vdots\\\ x_{n}\leq 1+na_{n}\\\ \end{array}\right.$
(30)
then there exists an $n\times n$ doubly stochastic matrix $B$ with spectrum
$(1,\lambda_{2},...,\lambda_{n}).$ Moreover, $B$ is the unique closest doubly
stochastic matrix to $A$ with respect to the Frobenius norm and with the
property that $B$ is cospectral to $A.$
Proof. In system (3), substituting $r=1$ and using the fact that
$x_{1}+x_{2}+...+x_{n}=n,$ the system becomes
$\displaystyle\left\\{\begin{array}[]{c}a_{1}+y_{m}+\frac{x_{m}}{n}-\frac{x_{1}}{n}\geq
0\\\ a_{2}+y_{m}+\frac{x_{m}}{n}-\frac{x_{2}}{n}\geq 0\\\ \vdots\\\
a_{m-1}+y_{m}+\frac{x_{m}}{n}-\frac{x_{m-1}}{n}\geq 0\\\ a_{m}+y_{m}\geq 0\\\
a_{m+1}+y_{m}+\frac{x_{m}}{n}-\frac{x_{m+1}}{n}\geq 0\\\ \vdots\\\
a_{n}+y_{m}+\frac{x_{m}}{n}-\frac{x_{n}}{n}\geq 0.\\\ \end{array}\right..$
(39)
Choosing $y_{m}=\frac{1}{n}-\frac{x_{m}}{n}$ in (5), we obtain (4). Next, it
is not hard to check that the system in (4) implies that the matrix $B$ in the
proof of the preceding theorem is doubly stochastic and has the required
spectrum. For the second part, Theorem 1.3 tells us that it suffices to prove
that $B=(I_{n}-J_{n})A(I_{n}-J_{n})+J_{n}.$ Since for a stochastic matrix $A,$
we have $AJ_{n}=J_{n}$ then
$(I_{n}-J_{n})A(I_{n}-J_{n})+J_{n}=A-J_{n}A+J_{n}.$ Moreover, each $ij$-entry
of the matrix $J_{n}A$ is clearly equals to $\frac{x_{j}(A)}{n}.$ With this in
mind, a simple check now shows that all entries of the two matrices $B$ and
$A-J_{n}A+J_{n}$ are the same and the proof is complete.
###### Example 3.2
Consider the following stochastic matrix
$A=\left(\begin{array}[]{ccc}2/3&1/3&0\\\ 1/3&2/3&0\\\ 1/2&1/2&0\\\
\end{array}\right)$ which has spectrum $\sigma(A)=(1,1/3,0).$ Clearly $A$
satisfies the conditions of the preceding theorem and from the above argument,
$A$ is cospectral to the matrix $B=\left(\begin{array}[]{ccc}1/2&1/6&1/3\\\
1/6&1/2&1/3\\\ 1/3&1/3&1/3\\\ \end{array}\right)$ which is the unique doubly
stochastic matrix that is closest to $A$ with respect to the Frobenius norm.
It is worth mentioning that by choosing a particular $y_{m}$ in system $(3)$
results in a nonnegative generalized doubly stochastic matrix $B$ whose row
and column sum does not depend on the entries of $A.$ As illustration, we have
the following theorem.
###### Theorem 3.3
Let $A=(a_{ij})$ be an $n\times n$ nonnegative $r$-generalized stochastic
matrix with spectrum $(r;\lambda_{2},...,\lambda_{n}).$ Then there exists an
$n\times n$ nonnegative generalized doubly stochastic matrix $B$ with spectrum
$(nr;\lambda_{2},...,\lambda_{n}).$
Proof. In system $(3),$ let
$y_{m}=(1/n)(x_{1}+...+x_{m-1}+x_{m+1}+...+x_{n}).$ Since each entry of $A$ is
less than or equal $r$ then $x_{j}\leq nr$ for all $j=1,2,...,n.$ Therefore
with this choice of $y_{m},$ the system (3) is satisfied and in this case we
obtain the nonnegative generalized doubly stochastic matrix $B$ whose each row
and column sum equals to $nr.$
## 4 Some related results
In [9], the author also studied the problem of finding sufficient conditions
for a real matrix to be similar to an element of $\Omega^{1}(n)$ and obtained
the following result.
###### Theorem 4.1
[9] Let $A$ be a real $n\times n$ matrix with spectrum
$\sigma(A)=\\{1,\lambda_{2},...,\lambda_{n}\\}.$ Then $A$ is similar to a
matrix with row and column sum 1 if and only if the space of left eigenvectors
of $A$ corresponding to 1 is not orthogonal to the space of right eigenvectors
of $A$ corresponding to 1.
Replacing the word ”similar” by ”cospectral”, we have the following result.
###### Theorem 4.2
Let $A$ be a real $n\times n$ matrix with spectrum
$\sigma(A)=\\{1,\lambda_{2},...,\lambda_{n}\\}.$ Then $A$ is always cospectral
to a matrix with row and column sum 1.
Proof. Let $C_{n-1}$ be the companion matrix of the $(n-1)$-list
$\\{\lambda_{2},...,\lambda_{n}\\}.$ Then $C_{n-1}$ is an $(n-1)\times(n-1)$
real matrix with spectrum $\\{\lambda_{2},...,\lambda_{n}\\}.$ Therefore by
Lemma 1.1, the matrix $B=U_{n}(1\oplus C_{n-1})U_{n}$ is in $\Omega^{1}(n)$
and whose spectrum obviously satisfies $\sigma(B)=\sigma(A).$
###### Corollary 4.3
Let $A$ be a real $n\times n$ matrix with spectrum
$\sigma(A)=\\{1,\lambda_{2},...,\lambda_{n}\\}.$ Then there exists $k_{A}\geq
0$ such that $(1+k_{A};\lambda_{2},...,\lambda_{n})$ is the spectrum of an
$n\times n$ nonnegative $(1+k_{A})$-generalized doubly stochastic matrix.
Proof. Consider the matrix $B=(b_{ij})=U_{n}(1\oplus C_{n-1})U_{n}$ given in
the proof of the preceding theorem and let $k_{A}=|\min(b_{ij})|.$ Then
$(1+k_{A};\lambda_{2},...,\lambda_{n})$ is the spectrum of the nonnegative
$(1+k_{A})$-generalized doubly stochastic matrix $B+k_{A}J_{n}.$
###### Remark 4.4
It should be noted that in the proof of Theorem 4.2, if we replace $U_{n}$ by
any orthogonal matrix $V$ whose first column is $e_{n},$ then the matrix
$B^{\prime}=V(1\oplus C_{n-1})V^{T}$ is also an element of $\Omega^{1}(n)$ and
whose spectrum $\sigma(B^{\prime})=\sigma(A).$ Of course for each choice of
$V,$ there corresponds a different nonnegative $k_{A}$ such that the preceding
corollary is valid.
## Acknowledgments
This work is supported by the Lebanese University research grants program for
the Discrete Mathematics and Algebra research group.
## References
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* [2] R. Brualdi, Some applications of doubly-stochastic matrices, Lin. Alg. Appl. 107, (1988) pp. 77-89.
* [3] M. T. Chu and G. H. Golub, Inverse eigenvalue problems: Theory, algorithms and applications, Oxford University Press, 2005.
* [4] M. Fang, A note on the inverse eigenvalue problem for symmetric doubly stochastic matrices, Lin. Alg. Appl., 432, issue 11, (2010) pp. 2925-2927.
* [5] M. Fielder, Eigenvalues of non-negative symmetric matrices, lin. Alg. Appl. 9, (1974) pp. 119-142.
* [6] K. Ghanbari, A survey on inverse and generalized inverse eigenvalue problems for Jacobi matrices, Appl. Math. Comput., 195, (2008) pp. 355-363.
* [7] W. Guo, Eigenvalues of nonnegative matrices. In: Lin Alg. Appl. 266, (1997) pp. 261-270.
* [8] S. G. Hwang and S. S. Pyo, The inverse eigenvalue problem for symmetric doubly stochastic matrices, Lin. Alg. Appl., 379, (2004) pp. 77-83.
* [9] C. R. Johnson, Row stochastic matrices similar to doubly-stochastic matrices, Lin. Multilin. Alg., 10, (1981) pp. 113-130.
* [10] C. R. Johnson, T. Laffey and R. Loewy, The real and the symmetric nonnegative inverse eigenvalue problems are different, Proc. Amer. Math. soc. Vol. 124 (Number 12), (1996) pp. 3647-3651.
* [11] I. Kaddoura and B. Mourad, On a conjecture concerning the inverse eigenvalue problem for $4\times 4$ symmetric doubly stochastic matrices, Int. Math. Forum 3, 31,(2008) pp. 1513-1519.
* [12] R. N. Khoury, Closest matrices in the space of generalized doubly stochastic matrices, J. Math. Anal. and Apppl., 222,(1998) pp.562-568.
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* [14] H. Minc, Non-negative matrices, Berlin Press, New York, 1988.
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* [17] B. Mourad, A note on the boundary of the set where the decreasingly ordered spectra of symmetric doubly stochastic matrices lie, Lin. Alg. Appl., 416, (2006) pp. 546-558.
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* [19] B. Mourad, H. Abbas, A. Mourad, A. Ghaddar and I. Kaddoura, An algorithm for constructing doubly stochastic matrices for the inverse eigenvalue problem, Lin. Alg. Appl., 439, (2013) pp. 1382-1400.
* [20] B. Mourad, Generalization of some results concerning eigenvalues of a certain class of matrices and some applications, Linear and Multilinear Algebra, (2013) vol. 61, issue 9, pp. 1234-1243.
* [21] A. Nazari, Z. Beiranvand, The inverse eigenvalue problem for symmetric quasi anti-bidiagonal matrices, Appl. Math. Comput., 217, (2011) pp. 9526-9531.
* [22] A. Nazari, F. Sherafat, On the inverse eigenvalue problem for nonnegative matrices of order two to five Lin. Alg. Appl., 436, (2012) pp. 1771-1790.
* [23] H. Perfect, Methods for constructing certain stochastic matrices II, Duke. Math. J. 22, (1955) pp. 305-311.
* [24] H. Perfect and L. Mirsky, Spectral properties of doubly-stochastic matrices, Monatsh. Math., 69, (1965) pp. 35-57.
* [25] E. Seneta, Nonnegative matrices, Springer, 2nd edition, 2006.
* [26] R. Sinkhorn, Concerning the magnitude of the entries in a doubly stochastic matrix Lin. Multilin. Alg., 10, (1981) pp.107-112.
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* [28] H. R. Suleimanova , Stochastic matrices with real characteristic numbers, Doklady Akad. Nauk SSSR (N.S.) 66 (1949), pp. 343-345.
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|
arxiv-papers
| 2013-10-06T17:18:47 |
2024-09-04T02:49:52.039376
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/",
"authors": "Kassem Rammal, Bassam Mourad, Hassane Abbas, Hassan Issa",
"submitter": "Bassam Mourad",
"url": "https://arxiv.org/abs/1310.1606"
}
|
1310.1666
|
# The discrete mKdV equation revisited: a Riemann-Hilbert approach
00footnotetext:
Junyi Zhu, Xianguo Geng and Yonghui Kuang
School of Mathematics and Statistics, Zhengzhou University,
Zhengzhou, Henan 450001, People’s Republic of China Email: [email protected]
###### Abstract
We study the plus and minus type discrete mKdV equation. Some different
symmetry conditions associated with two Lax pairs are introduced to derive the
matrix Riemann-Hilbert problem with zero. By virtue of regularization of the
Riemann-Hilbert problem, we obtain the complex and real solution to the plus
type discrete mKdV equation respectively. Under the gauge transformation
between the plus and minus type, the solutions of minus type can be obtained
in terms of the given plus ones.
## 1 Introduction
The discrete mKdV (dmKdV) equation [3]
$u_{t}(n,t)=\left(1\pm u^{2}(n,t)\right)[u(n+1,t)-u(n-1,t)],$ (1.1)
is an integrable equation in mathematical physics, and it is an important
member of the discrete Ablowitz-Ladik equations [1, 2, 3, 4]. For specific
purpose, We call equation (1.1) the plus and minus type dmKdV. In this paper,
we study the plus type dmKdV equation with the help of the Riemann-Hilbert
(RH) method following [5], then the solutions of the minus type can be
obtained by virtue of a gauge transformation.
The plus type dmKdV equation (1.1) admits the following Lax pair formulation
[6]:
$\psi(n+1,t)=\gamma_{n}(I+Q_{n})Z\psi(n,t),\quad\psi_{t}(n,t)=(k\sigma_{3}+\tilde{Q}_{n})\psi(n,t),$
(1.2)
where $I$ is the identity matrix, $\gamma_{n}=\sqrt{\det(I+Q_{n})}^{-1}$, and
the matrices $Q_{n},Z,\tilde{Q}_{n}$ take the form
$\displaystyle Q_{n}=\left(\begin{matrix}0&u(n,t)\\\
-u(n,t)&0\end{matrix}\right),\quad Z=\left(\begin{matrix}z&0\\\
0&1\end{matrix}\right),$ (1.3)
$\displaystyle\tilde{Q}_{n}=Q_{n}+Z^{-1}Q_{n-1}Z,\quad
k=\frac{1}{2}(z-z^{-1}),$
with $z$ is a spectral parameter. We note that the Lax pair formulation (1.2)
can be rewritten as
$\psi(n+1,t)=(I+Q_{n})Z\psi(n,t),\quad\psi_{t}(n,t)=(k\sigma_{3}+\tilde{Q}_{n}-Q_{n}Q_{n-1})\psi(n,t),$
(1.4)
which are the ones in [6].
The plus type dmKdV equation (1.1) admits another Lax pair formulation [3]
$\varphi(n+1,t)=\gamma_{n}(E+Q_{n})\varphi(n,t),\quad\varphi_{t}(n,t)=(\omega\sigma_{3}+\hat{Q}_{n})\varphi(n,t),$
(1.5)
where $Q_{n}$ is defined as (1.3) and
$\displaystyle E=\left(\begin{matrix}z&0\\\
0&z^{-1}\end{matrix}\right),\quad\gamma_{n}=\frac{1}{\sqrt{\det(E+Q_{n})}},$
(1.6)
$\displaystyle\hat{Q}_{n}=EQ_{n}+Q_{n-1}E,\quad\omega=\frac{1}{2}(z^{2}-z^{-2}).$
We note that the Lax pair (1.5) can be rewritten in a similar form as (1.4)
which are the ones as in [3].
It is known that self-dual network can also be reduced to the discrete
analogue of the mKdV equation [7]. we note that there are many other
differential-difference equations which can be transformed into the dmKdV
equation [8, 9, 10, 11, 12, 13, 14, 15]. The dmKdV equation has widely
applications in the fields as plasma physics, electromagnetic waves in
ferromagnetic, antiferromagnetic or dielectric systems, and can be solved by
the method of inverse scattering transform, Hirota bilinear, Algebro-geometric
approach and others [3, 16, 17, 18, 19, 20, 6, 21, 22, 23, 24, 25, 26, 27, 28,
29, 30, 31, 32].
In this paper, we firstly consider the Lax pair (1.2) and introduce a special
symmetry condition, which imply that $u(n,t)$ in (1.1) can be extended to
complex value. For simplicity, we suppose that $u(n,t)\neq\pm i$ and
$\gamma_{n}$ is chosen as one of branches. It is noted that the complex
solution in this paper is different from the complexiton solution introduced
by W.X. Ma [33, 34, 35, 36], in which the complexiton solutions are obtained
in the sense of complex eigenvalues, and are still real. Next we consider the
linear system (1.5) under the usual symmetry condition which confine the
potential $u(n,t)$ to be real. We note that to obtain the real solution of the
plus type dmKdV equation, one needs to introduce some constraint condition.
For one soliton solution as an example, we assume that $(2n+1)\eta+2\cosh
2\xi\sin 2\eta t=m\pi$, where $m=0,\pm 1,\pm 2,\cdots$ and the discrete
spectrum for $N=1$ is defined as $z_{1}=e^{\xi+i\eta}$.
The organization of this paper is as follows. In section 2, we derive the
complex solution of the plus type dmKdV equation by virtue of RH problem
associated with linear system (1.2). In section 3, we derive the real solution
of the plus type dmKdV equation by virtue of RH problem associated with linear
system (1.5). In section 4, we study the gauge transformation between the plus
and minus type dmKdV equation, from which the solution of minus type can be
obtained in terms of the given plus one.
## 2 Complex solution of the dmKdV equation
### 2.1 The spectral analysis
For the sake of convenience, we write the spectral equation (1.2) in terms of
the matrix
$J(n)=\psi(n)Z^{-n}e^{-k\sigma_{3}t}.$
Hence, the dmKdV equation allows the Lax representation:
$J(n+1)=\gamma_{n}(I+Q_{n})ZJ(n)Z^{-1},$ (2.7)
and
$J_{t}(n)=k[\sigma_{3},J(n)]+\tilde{Q}_{n}J(n).$ (2.8)
Here and after we suppress the variables dependence for simplicity of
notation.
Now we introduce matrix Jost functions $J_{\pm}(n,z)$ of the spectral equation
(2.7) obeying the asymptotic conditions
$J_{\pm}(n,z)\rightarrow I,\quad n\rightarrow\pm\infty.$ (2.9)
Then there exists the scattering matrix $S(z)$ admitting
$J_{-}(n,z)=J_{+}(n,z)Z^{n}S(z)Z^{-n},\quad
S(z)=\left(\begin{matrix}a_{+}(z)&-b_{-}(z)\\\
b_{+}(z)&a_{-}(z)\end{matrix}\right).$ (2.10)
Here we assume that the Jost functions and the scattering matrix satisfy the
symmetry condition
$J_{\pm}^{T}(n,z^{-})=J_{\pm}^{-1}(n,z),\quad\det J_{\pm}(n,z)=1,$ (2.11)
and
$S^{T}(n,z^{-})=S^{-1}(n,z),\quad\det S(n,z)=1,\quad z^{-}=z^{-1}.$ (2.12)
In the following, we consider the asymptotic behavior of the solution
$J(n,z)$. To this end, we first let
$J(n,z)=J^{(0)}(n)+z^{-1}J^{(1)}(n)+O(z^{-2}),\quad z\rightarrow\infty,$
(2.13)
and substitute it into the spectral equation (2.7). This yields
$J_{11}^{(0)}(n+1)=\gamma_{n}J_{11}^{(0)}(n),\quad
J_{22}^{(0)}(n+1)=\gamma_{n}^{-1}J_{22}^{(0)}(n),$ (2.14)
and $J_{12}^{(0)}(n)=0$,
$J_{12}^{(1)}(n)=-u(n)J_{22}^{(0)}(n),\quad
J_{21}^{(0)}(n+1)=-u(n)J_{11}^{(0)}(n+1).$ (2.15)
Next, according to the symmetry (2.11), we let
$J^{-1}(n,z)=\tilde{J}_{(0)}+z\tilde{J}_{(1)}+O(z^{2}),\quad z\rightarrow 0,$
and find similarly $\tilde{J}_{(0)21}(n)=0$,
$\displaystyle\tilde{J}_{(0)11}(n+1)=\gamma_{n}\tilde{J}_{(0)11}(n),\quad\tilde{J}_{(0)22}(n+1)=\gamma_{n}^{-1}\tilde{J}_{(0)22}(n),$
(2.16)
$\displaystyle\tilde{J}_{(0)12}(n+1)=-u(n)\tilde{J}_{(0)11}(n+1),\quad\tilde{J}_{(1)21}(n)=-u(n)\tilde{J}_{(0)22}(n).$
We will now discuss the analytic of the Jost solutions. The spectral equation
(2.7), as a iterative relation, can be written as
$J(n,z)=\nu_{+}(n)\lim\limits_{N\rightarrow\infty}\prod\limits_{l=n}^{N}(Z^{-1}(I-Q_{l}))J(N+1,z)Z^{N-n+1},$
(2.17)
here and after we introduce two new functions $\nu_{\pm}(n)$ as following
$\nu_{+}(n)=\prod\limits_{l=n}^{\infty}\gamma_{l},\quad\nu_{-}(n)=\prod\limits_{l=-\infty}^{n-1}\gamma_{l}.$
(2.18)
We note that the first column of the matrix equation (2.17) involves two
positive power series in $z$, while the second column involves two negative
power series in $z$. Thus the first column $J_{+}^{[1]}(n,z)$ of the Jost
function $J_{+}$ is analytical for $|z|<1$, denoted by ${\mathbb{C}}_{I}$, and
the second column $J_{+}^{[2]}(n,z)$ is analytical for $|z|>1$ or
(${\mathbb{C}}_{O}$). By the same way one can show that the column
$J_{-}^{[1]}(n,z)$ is analytical for $|z|>1$ or (${\mathbb{C}}_{O}$). We
introduce a matrix function
$\Phi_{+}(n,z)=\left(J_{-}^{[1]},J_{+}^{[2]}\right)$
which is analytical in ${\mathbb{C}}_{O}$ and solves the spectral equation
(2.7).
It follows from the symmetry condition (2.11) that the rows
$(J_{-})_{[1]}^{-1}$ and $(J_{+})_{[2]}^{-1}$ are analytical in
${\mathbb{C}}_{I}$. Thus the matrix function
$\Phi_{-}^{-1}(n,z)=\left(\begin{array}[]{c}(J_{-})_{[1]}^{-1}\\\
(J_{+})_{[2]}^{-1}\end{array}\right)$
is analytical in ${\mathbb{C}}_{I}$ and solves the adjoint spectral problem of
(2.7).
By virtue of the definition (2.10) of the scattering matrix , we find
$\Phi_{+}(n,z)=J_{\pm}Z^{n}S_{\pm}Z^{-n},$ (2.19)
where
$S_{+}=\left(\begin{matrix}a_{+}&0\\\ b_{+}&1\end{matrix}\right),\quad
S_{-}=\left(\begin{matrix}1&b_{-}\\\ 0&a_{+}\end{matrix}\right).$
Hence, on use of (2.11), we obtain
$\det\Phi_{+}=\det J_{\pm}\det S_{\pm}=a_{+}(z).$ (2.20)
Following the same procedure as the one used for $\Phi_{+}$, one obtains
$\displaystyle\Phi_{-}^{-1}(n,z)=Z^{n}T_{\pm}Z^{-n}J_{\pm}^{-1},\quad\det\Phi_{-}^{-1}(n,z)=a_{-}(z),$
(2.21) $\displaystyle T_{+}=\left(\begin{matrix}a_{-}&b_{-}\\\
0&1\end{matrix}\right),\qquad T_{-}=\left(\begin{matrix}1&0\\\
b_{+}&a_{-}\end{matrix}\right).$
Asymptotic formulae for these sectionally analytic functions can be derived
from equations (2.13) to (2.16),
$\Phi_{+}(n,z)\rightarrow\Phi_{+}^{(0)}(n)=\left(\begin{matrix}\nu_{-}(n)&0\\\
-u(n-1)\nu_{-}(n)&\nu_{+}(n)\end{matrix}\right),\quad z\rightarrow\infty,$
(2.22)
and
$\Phi_{-}^{-1}(n,z)\rightarrow\Phi_{-(0)}^{-1}(n)=\left(\begin{matrix}\nu_{-}(n)&-u(n-1)\nu_{-}(n)\\\
0&\nu_{+}(n)\end{matrix}\right),\quad z\rightarrow 0.$ (2.23)
where $\nu_{\pm}(n)$ defined in (2.18). Indeed, to obtain equation (2.22), we
know, from the definition of $\Phi_{+}$ and the asymptotic expansion (2.13),
that
$\Phi_{+}^{(0)}(n)=\left(\begin{matrix}J_{-11}^{(0)}(n)&0\\\
J_{-21}^{(0)}(n)&J_{+22}^{(0)}(n)\end{matrix}\right),$
Iterating the relations (2.14) and (2.15), we find
$\displaystyle
J_{-11}^{(0)}(n)=\gamma_{n-1}J_{-11}^{(0)}(n-1)=\cdots=\nu_{-}(n),$
$\displaystyle J_{-21}^{(0)}(n)=-u(n-1)J_{-11}^{(0)}(n)=-u(n-1)\nu_{-}(n),$
$\displaystyle
J_{+22}^{(0)}(n)=\gamma_{n}J_{+22}^{(0)}(n+1)=\cdots=\nu_{+}(n),$
which give (2.22) in terms of the boundary condition (2.9). Equation (2.23)
can be obtained from (2.16) in a same way.
We note that the symmetry condition about these sectionally analytic functions
can be obtained from that of the Jost solutions as
$\Phi_{+}^{T}(n,z^{-})=\Phi_{-}^{-1}(n,z).$ (2.24)
In addition, equations (2.19) and (2.20) imply that $a_{+}(z)$ and $a_{-}(z)$
are analytical in the domain of ${\mathbb{C}}_{O}$ and ${\mathbb{C}}_{I}$
respectively. Furthermore they admit the following asymptotic behavior
$a_{+}(z)\rightarrow\nu,~{}z\rightarrow\infty;\quad
a_{-}(z)\rightarrow\nu,~{}z\rightarrow 0,$ (2.25)
where $\nu=\nu_{+}(n)\nu_{-}(n)=\prod_{l=-\infty}^{\infty}\gamma_{l}$.
It is noted that the potential $u(n)$ can be reconstructed by the analytic
functions. Indeed, from the first equation of (2.15), we find
$u=-\frac{J_{12}^{(1)}(n)}{J_{22}^{(0)}(n)}=-\lim\limits_{z\rightarrow\infty}\frac{(z\Phi_{+})_{12}}{(\Phi_{+})_{22}}=-\frac{\Phi_{+12}^{(1)}}{\Phi_{+22}^{(0)}},$
(2.26)
while the second equation of (2.15) is an identity.
### 2.2 RH problem and its regularization
Now we can introduce the RH problem
$\displaystyle\Phi_{-}^{-1}(n,z)\Phi_{+}(n,z)=Z^{n}G(z)Z^{-n},\quad|z|=1,$
(2.27) $\displaystyle
G(z)=T_{+}S_{+}=T_{-}S_{-}=\left(\begin{matrix}1&b_{-}(z)\\\
b_{+}(z)&1\end{matrix}\right).$
The normalization of the RH problem is given by (2.22) which is noncanonical.
Hence the dmKdV potential can be retrieved by virtue of the solution of RH
problem.
In order to obtain the soliton solutions of the dmKdV equation, we take
$G(z)=I$ and suppose $a_{+}(z)$ has simple zeros at
$z_{j}\in{\mathbb{C}_{O}},~{}j=1,\cdots,N$. From the symmetry (2.24), we know
that
$\det\Phi_{+}(z_{j})=0,~{}\det\Phi_{-}^{-1}(z^{-}_{l})=0,~{}j,l=1,\cdots,N$.
In this case, problem (2.27) is called the RH one with zeros which can be
solved by virtue of its regularization.
To obtain the relevant regular problem, we introduce a rational matrix
function
$\chi_{j}^{-1}=I+\frac{z_{j}-z^{-}_{j}}{z-z_{j}}P_{j},\quad
P_{j}=\frac{|y_{j}\rangle\langle\tilde{y}_{j}|}{\langle\tilde{y}_{j}|y_{j}\rangle},$
where the eigenvector $|y_{j}\rangle$ solves
$\Phi_{+}(n,z_{j})|y_{j}\rangle=0$. Since $\Phi_{+}(n,z_{j})$ admits the
linear system (2.7) and (2.8), then we have
$\displaystyle\Phi_{+}(n,t,z_{j})Z^{-1}(z_{j})|y_{j}\rangle(n+1,t)=0,$
$\displaystyle\Phi_{+}(n,t,z_{j})(|y_{j}\rangle_{t}-k_{j}\sigma_{3}|y_{j}\rangle)(n,t)=0,$
which imply that
$|y_{j}\rangle(n,t)=Z^{n}(z_{j})e^{k_{j}\sigma_{3}t}|y_{j}\rangle_{0},\quad
k_{j}=(z_{j}-z_{j}^{-})/2,$ (2.28)
where $|y_{j}\rangle_{0}$ is an arbitrary constant vector. In addition, one
finds that $\langle\tilde{y}_{j}|=|y_{j}\rangle^{T}$ satisfies
$\langle\tilde{y}_{j}|\Phi_{-}^{-1}(n,z_{l}^{-})=0$.
Therefore the product $\Phi_{+}(z)\chi_{j}^{-1}(z)$ is regular at the point
$z_{j}$ and $\chi_{l}(z)\Phi_{-}^{-1}(z)$ is regular at $z^{-}_{l}$, where
$\chi_{l}=I-\frac{z_{l}-z^{-}_{l}}{z-z^{-}_{l}}P_{l}.$ (2.29)
The regularization of all the other zeros is performed similarly and
eventually we obtain the following representation for the analytic solution
$\Phi_{\pm}=\phi_{\pm}\Gamma,\quad\Gamma=\chi_{N}\chi_{N-1}\cdots\chi_{1},$
(2.30)
where the holomorphic matrix functions $\phi_{\pm}$ solve the regular RH
problem
$\phi_{-}^{-1}(n,z)\phi_{+}(n,z)=I.$ (2.31)
We note that the soliton matrix $\Gamma$ can be decomposed into simple
fractions
$\Gamma=I-\sum\limits_{j,l=1}^{N}\frac{1}{z-z^{-}_{l}}|y_{j}\rangle(D^{-1})_{jl}\langle\tilde{y}_{l}|,\quad
D_{lj}=\frac{\langle\tilde{y}_{l}|y_{j}\rangle}{z_{j}-z_{l}^{-}}.$ (2.32)
In the following, we will establish the relationship between the solution of
dmKdV equation and the soliton matrix. Taking into account the asymptotic
formula (2.22) and the expression of $\Gamma$ (2.32), we choose
$\Phi_{+}^{(0)}=\phi_{+}$. Then, in view of (2.30), we find
$\Gamma(n,z)=I+z^{-1}\Gamma^{(1)}(n)+O(z^{-2}),\quad z\rightarrow\infty.$
(2.33)
and $\Phi_{+12}^{(1)}(n)=\nu_{-}(n)\Gamma_{12}^{(1)}(n)$. In addition, the
assumption $G(z)=I$ implies that $b_{\pm}(z)=0$ and then $a_{+}(z)a_{-}(z)=1$
in view of (2.12). From (2.25), we know that $\nu=\nu_{+}(n)\nu_{-}(n)=1$.
Hence the potential $u(n)$ can be rewritten as
$u(n)=-\frac{\nu_{-}(n)\Gamma_{12}^{(1)}(n)}{\nu_{+}(n)}=-\nu_{-}^{2}(n)\Gamma_{12}^{(1)}(n).$
(2.34)
Next we will establish the relationship between $\nu_{-}$ and $\Gamma$. Since
$G(z)=I$, the RH problem (2.27) reduces to $\Phi_{+}=\Phi_{-}$, from which we
can consider the asymptotic behavior of $\Phi_{+}$ near $z=0$. Indeed, the
asymptotic formulae (2.21) and (2.22) imply that
$\Phi_{+}\rightarrow\Phi_{-(0)}(n)=\left(\begin{matrix}\nu_{+}(n)&u(n-1)\nu_{-}(n)\\\
0&\nu_{-}(n)\end{matrix}\right),\quad z\rightarrow 0.$
Thus from (2.30) we obtain
$\displaystyle\Gamma(n,z)|_{z=0}$
$\displaystyle=(\Phi_{+}^{(0)})^{-1}(n)\Phi_{+}|_{z=0}$ (2.35)
$\displaystyle=\left(\begin{matrix}\nu_{+}^{2}(n)&u(n-1)\\\
u(n-1)&\nu_{-}^{2}(n-1)\end{matrix}\right),$
which implies that $\nu_{-}^{2}(n)=\Gamma_{22}(n+1,z=0)$. As a result, the
potential $u(n)$ takes the form
$u(n)=-\Gamma_{12}^{(1)}(n)\Gamma_{22}(n+1,z=0).$ (2.36)
### 2.3 Complex soliton solutions
In this section, we will derive the soliton solutions of the dmKdV equation
(1.1). To this end, we let
$z_{j}=e^{\xi_{j}+i\eta_{j}},~{}\xi_{j}>0,\quad|y_{j}\rangle_{0}=\left(\begin{array}[]{c}e^{a_{j}+i\alpha_{j}}\\\
1\end{array}\right).$
Hence the vector $|y_{j}\rangle,~{}(j=1,2,\cdots,N)$ take the form
$|y_{j}\rangle=e^{\frac{1}{2}(\theta_{j}(n)+i\phi_{j}(n))}\left(\begin{array}[]{c}e^{\frac{1}{2}(X_{j}(n,t)+i\varphi_{j}(n,t))}\\\
e^{-\frac{1}{2}(X_{j}(n,t)+i\varphi_{j}(n,t))}\end{array}\right),$ (2.37)
where
$\displaystyle X_{j}(n,t)=n\xi_{j}+2\sinh\xi_{j}\cos\eta_{j}t+a_{j},$ (2.38)
$\displaystyle\varphi_{j}(n,t)=n\eta_{j}+2\cosh\xi_{j}\sin\eta_{j}t+\alpha_{j},$
with $\theta_{j}(n)=X_{j}(n,0),\phi_{j}(n)=\varphi_{j}(n,0)$.
In particularly, for $N=1$, equation (2.32) reduces to
$\Gamma(n,z)=I-\frac{z_{1}-z_{1}^{-}}{z-z_{1}^{-}}\frac{|y_{1}\rangle\langle\tilde{y}_{1}|}{\langle\tilde{y}_{1}|y_{1}\rangle},$
(2.39)
from which we have the complex form of one-soliton solution to dmKdV equation
$u(n,t)=\frac{z_{1}^{2}-1}{2}{\rm
sech}\\{X_{1}(n+1,t)+i\varphi_{1}(n+1,t)\\}.$ (2.40)
For $N=2$, we find
$\displaystyle\Gamma^{(1)}(n,z)=$ $\displaystyle-\frac{1}{\det
D}\left\\{D_{22}|y_{1}\rangle\langle\tilde{y}_{1}|-D_{21}|y_{2}\rangle\langle\tilde{y}_{1}|\right.$
(2.41)
$\displaystyle\qquad\left.+D_{11}|y_{2}\rangle\langle\tilde{y}_{2}|-D_{12}|y_{1}\rangle\langle\tilde{y}_{2}|\right\\},$
and
$\displaystyle\Gamma(n,0)=I+$ $\displaystyle\frac{1}{\det
D}\left\\{z_{1}\left[D_{22}|1\rangle\langle\tilde{1}|B-D_{21}|2\rangle\langle\tilde{1}|B\right]\right.$
(2.42)
$\displaystyle\qquad\left.+z_{2}\left[D_{11}|2\rangle\langle\tilde{2}|B-D_{12}|1\rangle\langle\tilde{2}|B\right]\right\\},$
where $\det D$ is obtained according to the definition of (2.32) and (2.37) as
$\det
D=\Xi\Omega_{2}(n),\quad\Xi=\left(\prod\limits_{j,l=1}^{2}(z_{j}-z_{l}^{-})\right)^{-1}\frac{2e^{\theta_{1}+\theta_{2}+i(\phi_{1}+\phi_{2})}}{z_{1}z_{2}},$
(2.43)
and
$\displaystyle\Omega_{2}(n)=$
$\displaystyle(z_{1}-z_{2})^{2}\cosh\\{\vartheta_{1}(n,t)+\vartheta_{2}(n,t)\\}$
(2.44)
$\displaystyle+(z_{1}z_{2}-1)^{2}\cosh\\{\vartheta_{1}(n,t)-\vartheta_{2}(n,t)\\}-(z_{1}^{2}-1)(z_{2}^{2}-1),$
with
$\vartheta_{j}(n,t)=X_{j}(n,t)+i\varphi_{j}(n,t).$
In addition, from (2.37), (2.41) and (2.42), we know that
$\Gamma_{12}^{(1)}(n)=-\frac{V_{2}(n+1)}{\Omega_{2}(n)},\quad\Gamma_{22}(n,0)=\frac{\Omega_{2}(n)}{\Omega_{2}(n+1)},$
(2.45)
where
$\displaystyle V_{2}(n)=$
$\displaystyle(z_{2}-z_{1})(z_{1}z_{2}-1)\left[z_{1}(z_{2}^{2}-1)\cosh\\{\vartheta_{1}(n,t)\\}\right.$
(2.46)
$\displaystyle\quad\left.-z_{2}(z_{1}^{2}-1)\cosh\\{\vartheta_{2}(n,t)\\}\right].$
Hence the solution of the plus type dmKdV equation for $N=2$ can be given by
$u(n)=\frac{V_{2}(n+1)}{\Omega_{2}(n+1)}.$ (2.47)
It is noted that the solution $u(n)$ can also be derived through
$\Gamma_{12}(n+1,z=0)$ by (2.35). It is verified that the representations of
solution by $\Gamma_{12}(n+1,z=0)$ are same as the ones in (2.40) and (2.47).
## 3 Real solutions of the dmKdV equation
### 3.1 The spectral analysis
In the section, we consider the inear system (1.5) and assume that the
solution $u(n,t)$ is a real function. After the transformation
$J(n)=\varphi(n)E^{-n}e^{-\omega\sigma_{3}t},$
the dmKdV equation allows the Lax representation:
$J(n+1)=\gamma_{n}(E+Q_{n})J(n)E^{-1},$ (3.1)
and
$J_{t}(n)=\omega[\sigma_{3},J(n)]+\hat{Q}_{n}J(n).$ (3.2)
We assume that the function $J(n,z)$ admits the following symmetry conditions
$J^{\dagger}(n,\bar{z})=J^{-1}(n,z),\quad\sigma_{3}J(n,-z)\sigma_{3}=J(n,z),$
(3.3)
where $\bar{z}=(z^{*})^{-1}$ with $z^{*}$ denotes the complex conjugate of
$z$.
The Jost functions $J_{\pm}(n,z)$ and the scattering matrix $S(z)$ can be
introduced in the same way as in (2.9) and (2.10). It is readily verified that
the matrices $J_{\pm}(n,z)$ and $S(z)$ are unimodular, and satisfy the
symmetry conditions (3.3).
We note that similar considerations apply to the asymptotic behavior of the
Jost functions $J_{\pm}(n,z)$, one find
$\displaystyle J(n,z)=J^{(0)}(n)+z^{-1}J^{(1)}(n)+O(z^{-2}),\quad
z\rightarrow\infty,$ (3.4) $\displaystyle
J(n,z)=J_{(0)}(n)+zJ_{(1)}(n)+O(z^{2}),\quad z\rightarrow 0,$
where the diagonal matrices $J^{(0)}(n)$ and $J_{(0)}(n)$ admit the following
iterative relations
$J^{(0)}(n+1)=\left(\begin{matrix}\gamma_{n}&0\\\
0&\gamma_{n}^{-1}\end{matrix}\right)J^{(0)}(n),\quad
J_{(0)}(n+1)=\left(\begin{matrix}\gamma_{n}^{-1}&0\\\
0&\gamma_{n}\end{matrix}\right)J_{(0)}(n).$ (3.5)
In addition, the solution can be constructed by
$u(n)=-\frac{J_{12}^{(1)}(n)}{J_{22}^{(0)}(n)}.$ (3.6)
The analytical properties of the Jost functions $J_{\pm}(n,z)$ are the same as
the ones in complex section above, and can be used to define the same
sectionally holomorphic $\Phi_{+}(n,z)$ and $\Phi_{-}^{-1}(n,z)$ as (2.19) and
(2.21) with $Z$ replaced by $E$. Furthermore, we have the symmetry condition
about $\Phi_{\pm}(n,z)$
$\Phi_{+}^{\dagger}(n,\bar{z})=\Phi^{-1}(n,z),$ (3.7)
and the asymptotic behavior
$\Phi_{+}(n,z)\rightarrow\Phi_{+}^{(0)}(n)=\left(\begin{matrix}\nu_{-}(n)&0\\\
0&\nu_{+}(n)\end{matrix}\right),\quad z\rightarrow\infty,\\\ $ (3.8)
and
$\Phi_{-}^{-1}(n,z)\rightarrow\tilde{\Phi}_{-}^{(0)}(n)=\left(\begin{matrix}\nu_{-}(n)&0\\\
0&\nu_{+}(n)\end{matrix}\right),\quad z\rightarrow 0,$ (3.9)
where the real functions $\nu_{\pm}(n)$ are defined as in (2.18).
### 3.2 The regularization of the RH problem and the soliton solutions
The RH problem associated with $\Phi_{\pm}(n,z)$ can be constructed as in
(2.27), while the normalization of the RH problem is given by (3.8).
In order to obtain the real soliton solutions of the dmKdV equation, we take
$G(z)=I$ and suppose $a_{+}(z)=\det\Phi_{+}(n,z)$ has simple zeros at $\pm
z_{j}\in{\mathbb{C}_{O}},~{}j=1,\cdots,N$. From the symmetries (3.7), we know
that $\det\Phi_{+}(\pm
z_{j})=0,~{}\det\Phi_{-}^{-1}(\pm\bar{z}_{l})=0,~{}j,l=1,\cdots,N$.
For convenience, we introduce the notations
$k_{2j}=z_{j},\quad k_{2j-1}=-z_{j}.$
Then the soliton matrix can be written in the form
$\Gamma(n,z)=\chi_{2N}\chi_{2N-1}\cdots\chi_{2}\chi_{1},$ (3.10)
where
$\chi_{l}=I-\frac{k_{l}-\bar{k}_{l}}{z-\bar{k}_{l}}P_{l},\quad\chi_{j}^{-l}=I+\frac{k_{j}-\bar{k}_{j}}{z+k_{j}}P_{j},\quad
P_{j}=\frac{|j\rangle\langle j|}{\langle j|j\rangle},$ (3.11)
with the eigenvector $\langle j|=|j\rangle^{\dagger}$ and $|j\rangle$ solves
$\Phi_{+}(n,k_{j})|j\rangle=0$. In this case, one may find that
$|2j\rangle=\sigma_{3}|2j-1\rangle$ and $P_{2j}=\sigma_{3}P_{2j-1}\sigma_{3}$.
Hence the product $\Phi_{+}(z)\chi_{j}^{-1}(z)$ is regular at the point
$k_{j}$ and $\chi_{l}(z)\Phi_{-}^{-1}(z)$ is regular at $\bar{k}_{l}$.
Since $\Phi_{+}(n,z)$ solves the linear system (3.1) and (3.2), we know that
the eigenvector $|j\rangle$ takes the form
$|j\rangle(n,t)=E^{n}(k_{j})e^{\omega_{j}\sigma_{3}t}|j_{0}\rangle,\quad\omega_{j}=\omega(k_{j}).$
(3.12)
The regular RH problem can be derived similarly as (2.31) and (2.30), where
the soliton matrix $\Gamma$ has the following decomposition
$\Gamma=I-\sum\limits_{j,l=1}^{2N}\frac{1}{z-\bar{k}_{l}}|j\rangle(D^{-1})_{jl}\langle
l|,\quad D_{lj}=\frac{\langle l|j\rangle}{k_{j}-\bar{k}_{l}}.$ (3.13)
For $N=1$, we take $z_{1}=e^{\xi+i\eta}$, that is $k_{2}=z_{1},k_{1}=-z_{1}$,
$|2\rangle=\left(\begin{array}[]{c}e^{\theta+i\phi}\\\
e^{-(\theta+i\phi)}\end{array}\right),\quad|1\rangle=\sigma_{3}|2\rangle,$
(3.14)
where
$\theta=n\xi+\sinh 2\xi\cos 2\eta t+\alpha,\quad\phi=n\eta+\cosh 2\xi\sin
2\eta t+\beta.$ (3.15)
In this case, the soliton matrix takes the form
$\Gamma(n,t,z)=I-\frac{D_{-}}{z-\bar{z}_{1}}-\frac{D_{+}}{z+\bar{z}_{1}},$
(3.16)
where
$D_{-}=\frac{z_{1}-\bar{z}_{1}}{2}\left(\begin{matrix}\frac{e^{2\theta}}{z_{1}e^{-2\theta}+\bar{z}_{1}e^{2\theta}}&\frac{e^{2i\phi}}{z_{1}e^{-2\theta}+\bar{z}_{1}e^{2\theta}}\\\
\frac{e^{-2i\phi}}{z_{1}e^{2\theta}+\bar{z}_{1}e^{-2\theta}}&\frac{e^{-2\theta}}{z_{1}e^{2\theta}+\bar{z}_{1}e^{-2\theta}}\end{matrix}\right),\quad
D_{+}=-\sigma_{3}D_{-}\sigma_{3}.$ (3.17)
From (3.16) and (3.17), we have
$\Gamma(n,z=0)=\frac{z_{1}}{\bar{z}_{1}}\left(\begin{matrix}\frac{z_{1}e^{2\theta}+\bar{z}_{1}e^{-2\theta}}{z_{1}e^{-2\theta}+\bar{z}_{1}e^{2\theta}}&0\\\
0&\frac{z_{1}e^{-2\theta}+\bar{z}_{1}e^{2\theta}}{z_{1}e^{2\theta}+\bar{z}_{1}e^{-2\theta}}\end{matrix}\right),$
(3.18)
and
$\Gamma^{(1)}(n)=-(z_{1}^{2}-\bar{z}_{1}^{2})\left(\begin{matrix}0&\frac{e^{2i\phi}}{z_{1}e^{-2\theta}+\bar{z}_{1}e^{2\theta}}\\\
\frac{e^{-2i\phi}}{z_{1}e^{2\theta}+\bar{z}_{1}e^{-2\theta}}&0\end{matrix}\right),$
(3.19)
where $\Gamma^{(1)}(n)$ is defined by the asymptotic behavior
$\Gamma(n,z)=I+z^{-1}\Gamma^{(1)}(n)+O(z^{-2}),\quad z\rightarrow\infty.$
(3.20)
Next we will give the solution of dmKdV equation (1.1) for $N=1$. To this end,
we take the asymptotic behavior of the sectionally holomorphic $\Phi_{+}(n,z)$
as
$\Phi_{+}(n,z)=\Phi_{+}^{(0)}(n)+z^{-1}\Phi_{+}^{(1)}(n)+O(z^{-2}),\quad
z\rightarrow\infty,$
which together with $\Phi_{+}(n,z)=\Phi_{+}^{(0)}(n)\Gamma(n,z)$ imply that
$\Phi_{+}^{(1)}(n)=\Phi_{+}^{(0)}(n)\Gamma^{(1)}(n)$. Note that the solution
$u(n,t)$ can be rewritten as
$u(n,t)=-\frac{\Phi_{+12}^{(1)}(n)}{\Phi_{+22}^{(0)}(n)}=-\frac{\nu_{-}(n)}{\nu_{+}(n)}\Gamma_{12}^{(1)}(n),$
(3.21)
in view of (3.6), (3.8) and the definition of $\Phi_{+}(n,z)$. On the other
hand, since $S^{\dagger}(\bar{z})=S^{-1}(z),\det S(z)=1$, and
$a_{+}(z)=\det\Phi_{+}(n,z)\rightarrow\nu_{+}(n)\nu_{-}(n),~{}z\rightarrow\infty$,
as well as $G(z)=I$ in the RH problem as in (2.27), then
$\nu=\nu_{+}(n)\nu_{-}(n)=1.$ (3.22)
Taking notice of the RH problem reduces to $\Phi_{+}(n,z)=\Phi_{-}(n,z)$,
which allows us to discuss the asymptotic behavior of $\Phi_{+}(n,z)$ near
$z=0$,
$\Phi_{+}(n,z)\rightarrow\tilde{\Phi}_{-}^{(0)}(n),\quad z\rightarrow 0,$
(3.23)
where $\tilde{\Phi}_{-}^{(0)}(n)$ defined in (3.9). Now using again
$\Phi_{+}(n,z)=\Phi_{+}^{(0)}(n)\Gamma(n,z)$, we know that
$\Gamma(n,z=0)=\left(\begin{matrix}\nu_{-}^{-2}(n)&0\\\
0&\nu_{+}^{-2}(n)\end{matrix}\right).$ (3.24)
Hence the solution $u(n,t)$ can be reconstructed from (3.24), (3.22) and
(3.26)
$u(n,t)=-\Gamma_{22}(n,z=0)\Gamma_{12}^{(1)}(n),$ (3.25)
For $N=1$, we have the one soliton solution of dmKdV equation (1.1) from
(3.18) and (3.19) as
$u(n,t)=e^{2\xi}\sinh 2\xi{\rm sech}(2\theta+\xi),$ (3.26)
in terms of the assumption $\eta+2\phi=0$, where $\theta$ and $\phi$ are
defined in (3.15).
## 4 Gauge transformation
In this section, we discuss the Gauge transformation between the plus type
dmKdV equation and the minus one. Here we confine ourselves to the system
(1.4). In [6], the minus type dmKdV equation is the compatibility condition of
the Lax pair
$\chi(n+1)=\tilde{L}_{n}\chi(n),\quad\chi_{t}(n)=\tilde{M}_{n}\chi(n),$ (4.1)
where $\tilde{L}_{n}=(I+U_{n})Z,\tilde{M}_{n}=k\sigma_{3}+\tilde{U}_{n}$ and
$\displaystyle U_{n}=\left(\begin{matrix}0&\tilde{u}(n)\\\
\tilde{u}(n)&0\end{matrix}\right),$ (4.2)
$\displaystyle\tilde{U}_{n}=U_{n}+Z^{-1}U_{n-1}Z-U_{n}U_{n-1}.$
We let $\chi(n)=G_{n}\psi(n)$. If $\psi(n)$ and $Q_{n}$ solve the linear
equations (1.4), then
$\displaystyle\tilde{L}_{n}=G_{n+1}L_{n}G_{n}^{-1},$ (4.3)
$\displaystyle\tilde{M}_{n}=G_{n,t}G_{n}^{-1}+G_{n}M_{n}G_{n}^{-1},$
and
$\tilde{u}(n)=-iu(n-2),$ (4.4)
where $L_{n}=(I+Q_{n})Z,M_{n}=k\sigma_{3}+\tilde{Q}_{n}-Q_{n}Q_{n-1}$ and
$G_{n}=\rho_{n}^{\pm}\left(\begin{matrix}1-iu(n-1)\tilde{u}(n)\lambda&-u(n-1)-i\tilde{u}(n)\lambda\\\
-\tilde{u}(n)\lambda+iu(n-1)\lambda^{2}&u(n-1)\tilde{u}(n)\lambda+i\lambda^{2}\end{matrix}\right),$
(4.5)
with
$\rho_{n}^{+}=\prod\limits_{k=n}^{+\infty}\frac{1+u^{2}(k)}{1-\tilde{u}^{2}(k)},\quad\rho_{n}^{-}=\prod\limits_{k=-\infty}^{n-1}\frac{1-\tilde{u}^{2}(k)}{1+u^{2}(k)}.$
Indeed, $\tilde{L}_{n}=G_{n+1}L_{n}G_{n}^{-1}$ implies that the matrix $G_{n}$
can be represented in the following form
$G_{n}=\left(\begin{matrix}a_{0}(n)+a_{1}(n)\lambda&b_{0}(n)+b_{1}(n)\lambda\\\
c_{1}(n)\lambda+c_{2}(n)\lambda^{2}&d_{1}(n)\lambda+d_{2}(n)\lambda^{2}\end{matrix}\right).$
Then we have a set of equations
$a_{0}(n+1)(1+u^{2}(n))=a_{0}(n)(1-\tilde{u}^{2}(n)),$ (4.6a)
$b_{0}(n+1)=-u(n)a_{0}(n+1),\quad c_{1}(n)=-\tilde{u}(n)a_{0}(n),$ (4.6b)
$\tilde{u}(n)b_{0}(n)=u(n)c_{1}(n+1)+d_{1}(n+1)-d_{1}(n),$ (4.6c)
$d_{2}(n+1)(1+u^{2}(n))=d_{2}(n)(1-\tilde{u}^{2}(n)),$ (4.6d)
$b_{1}(n)=-\tilde{u}(n)d_{2}(n),\quad c_{2}(n+1)=u(n)d_{2}(n+1),$ (4.6e)
$a_{1}(n+1)-a_{1}(n)-u(n)b_{1}(n+1)=\tilde{u}(n)c_{2}(n),$ (4.6f)
$b_{0}(n)=u(n)a_{1}(n+1)+b_{1}(n+1)-\tilde{u}(n)d_{1}(n),$ (4.6g)
$c_{2}(n)=-\tilde{u}(n)a_{1}(n)+c_{1}(n+1)-u(n)d_{1}(n+1).$ (4.6h)
Hence, equation (4.6a) implies $a_{0}(n)=\rho_{n}^{\pm}$, then $b_{0},c_{1}$
and $d_{1}$ can be obtained from (4.6b) and (4.6c). We take
$d_{2}(n)=\alpha\rho_{n}^{\pm}$ by (4.6d), then by (4.6e) and (4.6f),
$b_{1},c_{2}$ and $a_{1}$ is at hand, where $\alpha$ is some constant. Thus
$G_{n}$ in (4.5) is obtained. In addition, the last two equation (4.6g) and
(4.6h) product
$\alpha\tilde{u}(n+1)=u(n-1),\quad\tilde{u}(n+1)=-\alpha u(n-1),$ (4.7)
in view of the identity
$\rho_{n+1}^{\pm}(1+u^{2}(n))=\rho_{n}^{\pm}(1-\tilde{u}^{2}(n))$. Thus (4.4)
is proven. It is remarked that the second equation of (4.3) is valid for
$G_{n}$ given by (4.5), since the gauge transformation between (1.4) and (4.1)
implies
$\tilde{L}_{n,t}-\tilde{M}_{n+1}\tilde{L}_{n}+\tilde{L}_{n}\tilde{M}_{n}=G_{n+1}(L_{n,t}-M_{n+1}L_{n}+L_{n}M_{n})G_{n}^{-1}.$
In this equation, $L_{n,t}-M_{n+1}L_{n}+L_{n}M_{n}=0$ implies the plus type
equation of (1.1), while
$\tilde{L}_{n,t}-\tilde{M}_{n+1}\tilde{L}_{n}+\tilde{L}_{n}\tilde{M}_{n}=0$
gives the minus one.
We note that the gauge transformation about the similar problem of (1.5) gives
rise to
$\tilde{u}(n)=-iu(n-1).$ (4.8)
Hence the solutions of minus type dmKdV equation can be obtained by (4.4) or
(4.8) from the given plus ones. It is interesting to remark that the soliton
solutions to minus type dmKdV equation can be also obtained without needing to
consider the nonvanishing boundary conditions [31], and the solutions are
complex value by equation (4.4) or (4.8) for above two cases.
## Acknowledgments
Projects 11001250 and 11171312 are supported by the National Natural Science
Foundation of China. The work of JY Zhu is partially supported by the
Foundation for Young Teachers in Colleges and Universities of Henan Province.
## References
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|
arxiv-papers
| 2013-10-07T03:34:53 |
2024-09-04T02:49:52.046355
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Junyi Zhu, Xianguo Geng and Yonghui Kuang",
"submitter": "Junyi Zhu",
"url": "https://arxiv.org/abs/1310.1666"
}
|
1310.1701
|
# Strong Gravitational Lensing with Gauss-Bonnet correction
J. Sadeghi a, and H. Vaez a
a _Physics Department, Mazandaran University_ ,
_P.O.Box 47416-95447, Babolsar, Iran_
Email: [email protected]: [email protected]
###### Abstract
In this paper we investigate the strong gravitational lensing in a five
dimensional background with Gauss-Bonnet gravity, so that in 4-dimensions the
Gauss-Bonnet correction disappears. By considering the logarithmic term for
deflection angle, we obtain the deflection angle $\hat{\alpha}$ and
corresponding parameters $\bar{a}$ and $\bar{b}$. Finally, we estimate some
properties of relativistic images such as $\theta_{\infty}$, $s$ and $r_{m}$.
Keywords: Gravitational lensing; Gauss-Bonnet correction
## 1 Introduction
The deviation of the light rays in the gravitational fields is referred to
gravitational lensing. The gravitational lensing (GL) in the weak limit has
been used to test the General Relativity since its beginning [1, 2]. But, this
theory in the weak limit was not able to describe the high bending and looping
of the light rays. Hence, scientist community stated this phenomenon in the
strong filed regime. In the strong field limit, the light rays pass very close
to black hole and one set of infinitive relativistic ”ghost” images would be
produce on each side of black hole. These images are produced due to the light
rays wind one or several times around the black hole before reaching to
observer. At first, this phenomenon was proposed by Darwin [3]. Several
studies of null geodesics in the strong gravitational field have been done in
the past years [4]-[7]. In 2000, Virbhadra and Ellis showed that a source of
light behind a schwarzschild black hole would product an infinitive series of
images on each side of the massive object [8]. Theses relativistic images are
formed when the light rays travel very close to the black hole horizon, wind
several times around the black hole before appearing at observer. By an
alternative method, Frittelli et al. obtained an exact lens equation, integral
expression for deflection and compared their results with Virbhadra et al [9].
A new technic was proposed by Bozza et al. to find the position of the
relativistic images and their magnification [10]. They used the first two
terms of approximation to study schwarzschild black hole lensing. This method
was applied to other works such as Eiroa, Romero and Torres studied a
Reissner-Nordstrom black hole lensing[11]; Petters calculated relativistic
effects on microlensing events [13]. Afterward, the generalization of Bozza’s
method for spherically symetric metric was developed in [14]. Bozza compared
the image patterns for several interesting backgrounds and showed that by the
separation of the first two relativistic images, we can distinguish two
different collapsed objects. Further studies were developed for other black
holes and metrics [15]-[31].
The gravitational lenses are important tools for probing the universe. In
Refs. [32, 33] Narasimha and Chitre predicted that the gravitational lensing
of dark matter can give the useful data about the position of the dark matter
in the universe . Also, the gravitational lens are used to detect the exotic
objects in the universe, such as cosmic strings [34]-[36].
On the other hand, the gravitational theories in higher dimensions have
attracted considerable attention. One of these higher dimension gravities is
the supersymmetric string theory. Einstein-Gauss-Bonnet ($EGB$) theory, which
emerges as the low-energy limit of this theory, can be considered as an
effective model of gravity in higher dimensions. This theory yields a
correction to Einstein-Hilbert action. The Gauss-Bonnet term involves up to
second- order derivatives of the metric with the same degrees of freedom as
the Einstein theory [37, 38]. The variation of EGB action has different
solutions and the spherically symmetric solution in the presence of Gauss-
Bonnet gravity was obtained by Boulwar and Deser [39] and charged black hole
one is found by Wiltshire [40]. The properties of the Gauss-Bonnet black holes
have been studied in Refs. [41]-[53].
In this paper we study the strong gravitational lensing and obtain the
logarithmic deflection angle and corresponding coefficients. In the final we
investigated some properties of relativistic images.
The paper is organized as follows: In Section 2, we briefly present the
Einstein-Gauss-Bonnet gravity. Section 3 is devoted to investigate the strong
gravitational lensing in the presence of Gauss-Bonnet term. We consider the
logarithmic term which was proposed by Bozza, and obtain its parameters
$\bar{a}$ and $\bar{b}$. In section 4, some properties of relativistic images
will be studied. Finally, in the last section we present summery.
## 2 Einstein-Gauss-Bonnet gravity
The action of Einstein-Gauss-Bonnet gravity in five dimensional is given by
[39],
$I=-\frac{1}{16\pi
G_{5}}\int{d^{5}x\sqrt{-g}\,\left(R+\frac{\alpha}{2}L_{GB}\right)},$ (1)
where $R$ and $\alpha$ are Ricci scalar and Gauss-Bonnet constant
respectively. $G_{5}$ is five-dimensional Newton’s constant and $L_{GB}$ is
the Gauss-Bonnet term as follows,
$L_{GB}=R^{2}-4R_{ab}R^{ab}+R_{abcd}R^{abcd},$ (2)
here $R_{ab}$ and $R_{abcd}$ are Ricci tensor and Riemann tensor respectively.
Note that the indexes run over the components of five dimensional space. The
exact and spherically metric solution of the above action have been founded by
Boulware and Deser [39],
$ds^{2}=-f(r)dt^{2}+f(r)^{-1}dr^{2}+c(r)d\Omega_{3}^{2}\,\,,$ (3)
where
$\displaystyle f(r)=1+\frac{r^{2}}{2\alpha}\left(1-\sqrt{1+\frac{8\alpha
M}{r^{4}}}\right),\quad\quad c(r)=r^{2}.$ (4)
Figure 1: The figure shows the variation of horizon, photon sphere radius and
minimum impact parameter with respect to $\alpha$.
Here $M$ is related to $ADM$ mass and note that we set $G=c=1$. For
simplicity, we introduce the dimensionless quantities as $a=\frac{\alpha}{M}$
and $x=\frac{r}{\sqrt{2M}}$. So, we have,
$\displaystyle
f(x)=1+\frac{x^{2}}{a}\left(1-\sqrt{1+\frac{2a}{x^{4}}}\right).$ (5)
When $a$ tends to zero the warp factor of the Myers-Perry metric is obtained
[12]. The solution of $f(x)=0$, $x_{h}=\frac{1}{2}\sqrt{4-2a}$ is the horizon
radius of the black hole. The variation of the horizon is plotted with respect
to $a/M$ in figure 1.
## 3 Lens equation, Deflection angle with Gauss-Bonnet correction
The lens equation for a source of light and an observer situated at large
distances from a lens(deflector) is given by [17],
$D_{os}\tan\beta=\frac{D_{ol}\sin\theta-
D_{ls}\sin(\hat{\alpha}-\theta)}{\cos(\hat{\alpha}-\theta)}.$ (6)
Where, $D_{ls}$ and $D_{os}$ stand for the lens-source and observer-source
diameter distance, respectively. The angular positions of source and images
with respect to the optical axis (the line joining the observer and center of
the lens) are represented by $\beta$ and $\theta$. The deflection of the light
rays denotes by $\hat{\alpha}$ which can be positive, $\hat{\alpha}>0$
(bending toward the lens) or be negative, $\hat{\alpha}<0$ (bending away from
the lens). In the next section, we will obtain the deflection angle. The
particular distance from the center of the lens to the null geodesic at the
source position is called impact parameter, which is given by following
expression,
$\displaystyle u=D_{ol}\,\sin\theta.$ (7)
We can find the angular positions of images by the intersection of two
functions $\tan\theta-\tan\beta$ and
$\frac{D_{ls}}{D_{os}}(\tan\theta+\tan(\hat{\alpha}-\theta))$ vs $\theta$ for
the same side and vs $-\theta$ for opposite side. In addition to the primary
and secondary image positions (due to the weak limit), there is a sequence of
intersections that show the angular positions of the relativistic images.
These points are very close to each other, so they are not distinguishable.
For this reason, we call them relativistic images. These images are due to the
bending of light rays more than $3\pi/2$.
Now, we are going to investigate the deflection angle in the presence of
Gauss-Bonnet correction gravity. By using the null geodesic equation for the
following standard background metric,
$ds^{2}=-\mathcal{A}(r)dt^{2}+\mathcal{A}^{-1}(r)dr^{2}+\mathcal{C}(r)\,d\phi^{2}+\mathcal{D}(r)d\psi^{2},$
(8)
one can find the following equations,
$\displaystyle\dot{t}=\frac{E}{\mathcal{A}(r)},$
$\displaystyle\dot{\phi}=\frac{L_{\phi}}{\mathcal{C}(r)},$
$\displaystyle\dot{\psi}=\frac{L_{\psi}}{\mathcal{D}(r)},$ (9)
$(\dot{r})^{2}=\frac{1}{\mathcal{B}(r)}\left[\frac{\mathcal{D}(r)E-\mathcal{A}(r)L^{2}_{\psi}}{\mathcal{A}(r)\mathcal{D}(r)}-\frac{L^{2}_{\phi}}{\mathcal{C}(r)}\right].$
(10)
where $E$ is the energy of photon and $L_{\phi}$ and $L_{\psi}$ are angular
momentums in $\phi$ and $\psi$ directions. Here a dot denotes derivation with
respect to affine parameter. If we consider the $\theta$ component of geodesic
equations in the equatorial plane $(\theta=\pi/2$), we have
$\displaystyle\dot{\phi}\left[\mathcal{D}(r)\dot{\psi}\right]=\dot{\phi}L_{\psi}=0.$
(11)
Here, if we consider $\dot{\phi}=0$, the deflection angle of light ray becomes
zero and this is illegal, therefor we set $L_{\psi}=0$. For a light ray coming
from infinity the deflection angle in the directions $\phi$ is given by [26],
$\displaystyle\hat{\alpha_{\phi}}=I_{\phi}(x_{0})-\pi,$
where
$\displaystyle
I(x_{0})=2\int^{\infty}_{x_{0}}\left[\frac{{\mathcal{C}}(x)}{{\mathcal{C}}(x_{0})}{\mathcal{A}}(x_{0})-\mathcal{A}(x)\right]^{-\frac{1}{2}}\frac{dx}{x},$
(13)
where $x_{0}$ is the closet approach distance for the light ray when it passes
near the lens. The impact parameter for the closet approach is expressed by,
$\displaystyle
u(x_{0})=\sqrt{\frac{\mathcal{C}(x_{0})}{\mathcal{A}(x_{0})}}=x\sqrt{\frac{1}{1+\frac{x^{2}(1-\sqrt{1+\frac{2a}{x^{4}}})}{a}}}.$
(14)
The above relation is obtained from the null geodesic equation (10) with
setting $dr/d\phi=0.$ By using (7) and (14) one can relate the image position
to the closet approach and this relation allows us to write the deflection
angle as a function of image position. The image position is plotted as a
function of the closest approach in figure 2. We see that the image positions
and distances between relativistic images reduce by increasing the Gauss-
Bonnet parameter. For the large values of $x_{0}$ the curves coincide for any
value of Gauss-Bonnet parameter and this means that primary and secondary
image position remain unchanged.
Figure 2: The angular position of images with respect to $x_{0}$ at
$\alpha/M=0$, $\alpha/M=.5$ and $\alpha/M=1$. (Mass 4,31$\times
10^{6}M_{\odot}$, the distance $D_{ol}=8.5Kpc$, and $\mu
as\equiv$microarcseconds)
There is a minimum value for the closest approach that is called the photon
sphere radius and is a $r=$const null geodesic. The photon sphere is the root
of derivative of the impact parameter with respect to $x_{0}$ which is given
by,
$\displaystyle x_{ps}=\,(4-2a)^{\frac{1}{4}}.$ (15)
The dashed curve in the figure 1 shows the variation of photon sphere radius.
It decreases with increasing the Gauss-Bonnet parameter and tends to zero at
$\alpha=2$. When $x_{0}$ asymptotically approaches the photon sphere radius,
the photon reveals around the lens more times and the deflection angle
diverges as $x_{0}$ tends to photon sphere. We can rewrite the equation (13)
as,
$I(x_{0})=2\int^{1}_{0}F(z,x_{0})\,dz,$ (16)
and
$F(z,x_{0})=\frac{1}{\sqrt{\mathcal{A}(x_{0})-\mathcal{A}(x)\frac{\mathcal{C}(x_{0})}{\mathcal{C}(x)}}},$
(17)
where $z=1-\frac{x_{0}}{x}$. The function $F(z,x_{0})$ diverges as $z$
approaches to zero. Therefore, we can split the integral (16) in two parts,
the divergent part $I_{D}(x_{0})$ and the regular one $I_{R}(x_{0})$, as
follows [14]
$I_{D}(x_{0})=2\int^{1}_{0}F_{0}(z,x_{0})\,dz,$ (18)
$I_{R}(x_{0})=2\int^{1}_{0}\left[F(z,x_{0})-F_{0}(z,x_{0})\right]\,dz.$ (19)
Here we expand the argument of the square root in $F(z,x_{0})$ up to the
second order in $z$
$F_{0}(z,x_{0})=\frac{1}{\sqrt{p(x_{0})z+q(x_{0})z^{2}}},$ (20)
where
$\displaystyle
p(x_{0})=\frac{x_{0}}{c(x_{0})}\left[c^{\prime}(x_{0})f(x_{0})-c(x_{0})f^{\prime}(x_{0})\right]$
$\displaystyle=-\frac{2\left(-x_{0}^{4}-2a+2x_{0}^{2}\sqrt{\frac{x_{0}^{4}+2a}{x_{0}^{4}}}\right)}{x_{0}^{4}+2a}$
(21) $\displaystyle
q(x_{0})=\frac{x_{0}^{2}}{2c(x_{0})}\left[2c^{\prime}(x_{0})c(x_{0})f^{\prime}(x_{0})-2c^{\prime}(x_{0})^{2}f(x_{0})+f(x_{0})c(x_{0})c^{\prime\prime}(x_{0})-c^{2}(x_{0})f^{\prime\prime}(x_{0})\right]$
$\displaystyle=\frac{-x_{0}^{8}-4x_{0}^{4}a-4a^{2}+6x_{0}^{6}\sqrt{\frac{x_{0}^{4}+2a}{x_{0}^{4}}}+4x0^{2}\sqrt{\frac{x_{0}^{4}+2a}{x_{0}^{4}}}a}{(x_{0}^{4}+2a)^{2}}.$
(22)
As $a$ goes to zero, $p$ and $q$ tend to five dimensional schwarzschild ones,
$p=-\frac{4}{x0^{2}}+2$ and $q=\frac{6}{x0^{2}}-1$. For $x_{0}>x_{ps}$,
$p(x_{0})$ is nonzero and the leading order of the divergence in $F_{0}$ is
$z^{-1/2}$, which have a finite result. As $x_{0}\longrightarrow x_{ps}$,
$p(x_{0})$ approaches zero and the divergence is of order $z^{-1}$, that makes
the integral divergent logarithmically . Therefor, the deflection angle can be
approximated in the following form [14]
$\hat{\alpha}=-{\bar{a}}\,\log\left(\frac{u}{u_{ps}}-1\right)+{\bar{b}}+O(u-u_{ps}),$
(23)
Figure 3: The coefficients $\bar{a}$ and $\bar{b}$ as functions of the Gauss-
Bonnet parameter.
where
$\displaystyle\bar{a}=\frac{1}{\sqrt{q(x_{ps})}}\,\approx\frac{\sqrt{2}}{2}+0.128\,a+0.161\,a^{2}$
$\displaystyle\bar{b}=-\pi+b_{R}+\bar{a}\,\log\frac{x_{ps}^{2}\left[\mathcal{C}^{\prime\prime}(x_{ps})\mathcal{F}(x_{ps})-\mathcal{C}(x_{ps})\mathcal{F}^{\prime\prime}(x_{ps})\right]}{u_{ps}\sqrt{\mathcal{F}^{3}(x_{ps})\mathcal{C}(x_{ps})}}\approx
0.6902+0.154\,a+0.373\,a^{2}\,,$ $\displaystyle
b_{R}=I_{R}(x_{ps}),\,\,\,\,\,\,u_{ps}=\sqrt{\frac{\mathcal{C}(x_{ps})}{\mathcal{F}(x_{ps})}}\,.$
(24)
When $a$ tends to zero, we have $a=\frac{\sqrt{2}}{2}$ and $b=0.6902$, that
these values belong to Myers-Perry metric [12]. Using (23) and (3), we can
investigate the properties of strong gravitational lensing in the presence of
Gauss- Bonnet correction. The variations of the $u_{ps}$ is shown in figure 1.
Also, coefficients $\bar{a}$, $\bar{b}$, and the deflection angle
$\hat{\alpha}$ have been plotted with respect to the Gauss- Bonnet correction
in figures 3-4. We see that by increasing $\alpha$, the deflection angle
$\hat{\alpha}$ and $\bar{a}$ increase and $\bar{b}$ decreases. The deflection
angle becomes diverge as $\alpha\longrightarrow 2$.
Figure 4: Deflection angle in presence of Gauss-Bonnet term at
$x_{0}=1.01x_{ps}$.
Figure 5: The variation of compacted images position as a function of Gauss-
Bonnet parameter.
Figure 6: The variation of angular separation $s$ with respect to $\alpha$.
Figure 7: The relative magnification $r_{m}$ versus $\alpha$.
## 4 Relativistic images properties
In the previous section, we investigated the strong gravitational lensing by
using a simple and reliable logarithmic formula for deflection angle that was
obtained by Bozza et al. and obtained corresponding parameters $\bar{a}$ and
$\bar{b}$. Now we study some properties of relativistic images in the presence
of Gauss-Bonnet gravity. When a source, lens, and observer are highly aligned,
we can write the lens equation in strong gravitational lensing, as following
[14]
$\beta=\theta-\frac{D_{ls}}{D_{os}}\Delta\alpha_{n},$ (25)
where $\Delta\alpha_{n}=\alpha-2n\pi$ is the offset of deflection angle in
which all the loops are subtracted, and the integer $n$ indicates the $n$-th
image. The image position $\theta_{n}$ and the image magnification $\mu_{n}$
can be approximated as obtained in Ref [10],
$\theta_{n}=\theta^{0}_{n}+\frac{u_{ps}(\beta-\theta_{n}^{0})e^{\frac{\bar{b}-2n\pi}{\bar{a}}}D_{os}}{\bar{a}D_{ls}D_{ol}},$
(26)
$\mu_{n}=\frac{u_{ps}^{2}(1+e^{\frac{\bar{b}-2n\pi}{\bar{a}}})e^{\frac{\bar{b}-2n\pi}{\bar{a}}}D_{os}}{\bar{a}\beta
D_{ls}D_{ol}^{2}},$ (27)
where
$\theta_{m}^{0}=\theta_{ps}(1+e^{(\bar{b}-m\pi)/\bar{a}}),$ (28)
$\theta_{n}^{0}$ is the angular position of $\alpha=2n\pi$. They separate the
outer most image $\theta_{1}$ from the others images which are packed together
at $\theta_{\infty}$. Therefore, the separation between $\theta_{1}$ and
$\theta_{\infty}$ and ratio of their magnification can be considered by,
$\displaystyle s=\theta_{1}-\theta_{\infty}$
$\displaystyle\mathcal{R}=\frac{\mu_{1}}{\sum^{\infty}_{n=2}\mu_{n}}.$ (29)
The asymptotic position of the set of images $\theta_{\infty}$ can be obtained
from the minimum of the impact parameter as,
$\displaystyle\theta_{\infty}=\frac{u_{ps}}{D_{ol}}.$ (30)
By considering equation (30), we can approximate equations (4) as,
$\displaystyle
s=\theta_{\infty}e^{\frac{\bar{b}}{\bar{a}}-\frac{2\pi}{\bar{a}}},$
$\displaystyle\mathcal{R}=e^{\frac{2\pi}{\bar{a}}}.$ (31)
Another property that can be defined for relativistic images is the relative
magnification of the outermost relativistic image with the other ones. This is
shown by $r_{m}$ which is related to $\mathcal{R}$ as,
$\displaystyle r_{m}=2.5\,\log\mathcal{R}\,.$ (32)
If we suppose a five dimensional black hole with mass $4.31\times
10^{6}M_{\odot}$ ( Galaxy center mass) and the distance between the observer
and black hole is $D_{OL}=8.5\,kpc$ (The distance between the sun and galaxy
center) [54], we can study the effect of the Gauss-Bonnet parameter on these
quantities. Our results are presented in figure 4-6 and Table 1.
| $\alpha/M$ | | | ${\theta_{\infty}}$ | | $s$ | | $r_{m}$ | | | $a$ | | $b$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
| $0$ | | | 20.02002002 | | 0.001043418 | | 9.647597725 | | | 0.707106781 | | -0.690292235 |
| $0.3$ | | | 19.6255146 | | 0.001970453 | | 8.894645615 | | | 0.766964988 | | -0.777741402 |
| $0.6$ | | | 19.18509246 | | 0.00377615 | | 8.071759368 | | | 0.845154255 | | -0.928671426 |
| $0.9$ | | | 18.68212151 | | 0.008402205 | | 7.15484996 | | | 0.953462589 | | -1.06498697 |
| $1.2$ | | | 18.08715205 | | 0.020603386 | | 6.10167655 | | | 1.118033989 | | -1.294291956 |
| $1.5$ | | | 17.33784593 | | 0.058672906 | | 4.823798862 | | | 1.414213562 | | -1.761807496 |
| $1.8$ | | | 16.1916094 | | 0.210208553 | | 2.973589325 | | | 2.294157338 | | -3.6829744 |
| $1.95$ | | | 15.12420331 | | 0.480609627 | | 1.364376354 | | | 5.00123 | | -10.96179596 |
Table 1: Numerical estimations for the coefficients and observables of strong
gravitational lensing with Gauss-Bonnet correction . (Not that the numerical
values for $\theta_{\infty}$ and $s$ are of order microarcsec).
## 5 Summary
The light rays can be deviated from a straight way in the gravitational field
as predicted by General Relativity in which this deflection of light rays is
known as gravitational lensing. In the strong field limit, the deflection
angle of the light rays passing very close to the black hole, becomes so large
that, the light rays wind several times around the black hole before appearing
at the observer. Therefore the observer would detect two infinite set of faint
relativistic images produced on each side of the black hole. On the other
hand, the gravitational theories in higher dimensions have been attracting
considerable attention in recent decades. Einstein-Gauss-Bonnet theory that
emerges as the low-energy limit of supersymmetric string theory, is one of the
candidates for higher dimension theory. We considered five dimensional metric
with Gauss-Bonnet correction and studied the strong gravitational lensing and
obtained the deflection angle and corresponding parameters $\bar{a}$ and
$\bar{b}$. We saw that by increasing $\alpha$, the deflection angle
$\hat{\alpha}$ and $\bar{a}$ increase and $\bar{b}$ decreases. The deflection
angle became diverge as $\alpha\longrightarrow 2$.
Finally, we estimated some properties of relativistic images which can be
detected by astronomical instruments. Our results have been presented in
Figures 5-7. In figures 5 and 7, the variations of $\theta_{\infty}$ and
$r_{m}$, shown that the position of compacted images and relative
magnification reduce with increasing $\alpha$. Also the angular separation is
an increasing function and diverges, as $\alpha$ tends to two (figure 6).
Furthermore, we saw that the position of images reduces with $\alpha$ (see
figure 2). But you should note the decreasing rate of $\theta_{\infty}$ is
more than $\theta_{1}$, therefore $s$ will be increasing function.
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|
arxiv-papers
| 2013-10-07T08:46:53 |
2024-09-04T02:49:52.052952
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "J. Sadeghi, H. Vaez",
"submitter": "Hassan Vaez",
"url": "https://arxiv.org/abs/1310.1701"
}
|
1310.1706
|
# ELECTRON CLOUD EFFECTS IN ACCELERATORS††thanks: Work supported by the US
DOE under contract DE-AC02-05CH11231.
M. A. Furman Center for Beam Physics [email protected] LBNL Berkeley CA
94720
and CLASSE Cornell University Ithaca NY 14853
###### Abstract
We present a brief summary of various aspects of the electron-cloud effect
(ECE) in accelerators.
For further details, the reader is encouraged to refer to the proceedings of
many prior workshops, either dedicated to EC or with significant EC contents,
including the entire “ECLOUD” series [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12,
13, 14, 15, 16, 18, 17, 19, 20, 21, 22]. In addition, the proceedings of the
various flavors of Particle Accelerator Conferences [23] contain a large
number of EC-related publications. The ICFA Beam Dynamics Newsletter series
[24] contains one dedicated issue, and several occasional articles, on EC. An
extensive reference database is the LHC website on EC [25].
## 1 Introduction
The qualitative picture of the development of an electron cloud for a bunched
beam is as follows:
1. 1.
Upon being injected into an empty chamber, a beam generates electrons by one
or more mechanisms; these electrons are usually referred to as primary, or
seed, electrons.
2. 2.
These primary electrons get rattled around the chamber from the passage of
successive bunches.
3. 3.
As these electrons hit the chamber surface they yield secondary electrons,
which are, in turn, added to the existing electron population.
This process repeats with the passage of successive bunches. An essential
ingredient of the build-up and dissipation of the EC is the secondary electron
yield (SEY) of the chamber surface, characterized by the function $\delta(E)$,
where $E$ is the electron-wall impact energy. The function $\delta(E)$ has a
peak $\delta_{\rm max}$ typically ranging in $1-4$ at an energy $E=E_{\rm
max}$ typically ranging in $200-400$ eV. A convenient phenomenological
parameter is the effective SEY, $\delta_{\rm eff}$, defined to be the average
of $\delta(E)$ over all electron-wall collisions during a relevant time
window. Unfortunately, there is no simple a-priori way to determine
$\delta_{\rm eff}$, because it depends in a complicated way on a combination
of many of the beam and chamber parameters.
If $\delta_{\rm eff}<1$, the chamber wall acts as a net absorber of electrons
and the EC density $n_{e}$ grows linearly in time following beam injection
into an empty chamber. The growth saturates when the net number of electrons
generated by primary mechanisms balances the net number of electrons absorbed
by the walls.
If $\delta_{\rm eff}>1$, the EC initially grows exponentially. This
exponential growth slows down as the space-charge fields from the electrons
effectively neutralize the beam field, reducing the electron acceleration.
Ultimately, the process stops when the EC space-charge fields are strong
enough to repel the electrons back to the walls of the chamber upon being
born, at which point $\delta_{\rm eff}$ becomes $=1$. At this point, the EC
distribution reaches a dynamical equilibrium characterized by rapid temporal
and spatial fluctuations, determined by the bunch size and other variables.
For typical present-day storage rings, whether using positron or proton beams,
the average $n_{e}$ reaches a level $\sim 10^{10-12}$ m-3, the energy spectrum
of the electrons typically peaks at an energy below $\sim 100$ eV, and has a
high-energy tail reaching out to keV’s. In more detail, however, the EC
distribution reaches a dynamical equilibrium characterized by temporal and
spatial fluctuations. The temporal fluctuations span a typical range
$10^{-12}-10^{-6}$ s, depending on the bunch length and intensity, and on the
bunch train length and fill pattern. Spatial fluctuations typically span the
range $10^{-9}-10^{-2}$ m, depending on the transverse bunch size and
transverse dimensions of the vacuum chamber, and external magnetic field if
any. The density $n_{e}$ gradually decays following beam extraction, or during
the passage of a gap in the beam. The decay rate is controlled by the low-$E$
value (typically $E\>\hbox{\lower 2.58334pt\hbox{$\sim$}\hbox
to0.0pt{\hss\raise 2.58334pt\hbox{$<$}}}\>20$ eV) of $\delta(E)$. In general,
there is no simple, direct correlation between the rise time and the fall time
of the buildup of $n_{e}$ [26]. Figure 1 illustrates the build-up of the
electron cloud in the LHC.
Figure 1: Cartoon illustrating the build-up of the electron cloud in the LHC
for the case of 25-ns bunch spacing. The process starts with photoelectrons
and is amplified by the secondary emission process. This cartoon was generated
by F. Ruggiero.
The ECE combines many parameters of a storage ring such as bunch intensity,
size and spacing, beam energy [27], vacuum chamber geometry, vacuum pressure,
and electronic properties of the chamber surface material such as photon
reflectivity $R_{\gamma}$, effective photoelectric yield (or quantum
efficiency) $Y_{\rm eff}$, the SEY, the secondary emission spectrum [28, 29],
etc.
In regions of the storage ring with an external magnetic field, such as dipole
bending magnets, quadrupoles, etc., the EC distribution develops
characteristic geometrical patterns. For typical magnetic fields in the range
$B=0.01-5$ T and typical EC energies $<100$ eV, the electrons move in tightly-
wound spiral trajectories about the field lines. Thus in practice, in a
bending dipole, the electrons are free to move in the vertical ($y$)
direction, but are essentially frozen in the horizontal ($x$). As a result,
the $y$-kick imparted by the beam on a given electron has an $x$ dependence
that is remembered by the electron for many bunch passages. It often happens
that the electron-wall impact energy equals $E_{\rm max}$ at an $x$-location
less than the horizontal chamber radius. At this location
$\delta(E)=\delta_{\rm max}$, hence $n_{e}$ is maximum, leading to
characteristic high-density vertical stripes symmetrically located about $x=0$
[30]. For quadrupole magnets, the EC distribution develops a characteristic
four-fold pattern, with characteristic four-fold stripes [31].
In summary, the electron-cloud formation and dissipation:
* •
Is characterized by rich physics, involving many ingredients pertaining to the
beam and its environment.
* •
Involves a broad range of energy and time scales.
* •
Is always undesirable in particle accelerators.
* •
Is often a performance-limiting problem, especially in present and future
high-intensity storage rings.
* •
Is challenging to accurately quantify, predict and extrapolate.
The electron cloud has been shown to be detrimental to the performance of many
storage rings, and is a concern for future such machines, which typically call
for high beam intensity and compact vacuum chambers. At any given storage
ring, adverse effects may include one or more of the following: sudden, large,
vacuum pressure rise; beam instabilities; emittance growth; interference with
diagnostic instrumentation; excessive heat deposition on the chamber walls;
etc. Mitigation mechanisms have been required in most cases in order to reach,
or exceed, the design performance of the machine.
A more extensive summary of the ECE and its history is presented in Ref. [32].
## 2 Primary and secondary electrons
The main sources of primary electrons are: photoemission from synchrotron-
radiated photons striking the chamber walls; ionization of residual gas; and
electron generation from stray beam particles striking the walls of the
chamber. Depending on the type of machine, one of these three processes is
typically dominant. For example, in positron or electron storage rings, upon
traversing the bending magnets, the beam usually emits copious synchrotron
radiation with a $\sim$keV critical energy, yielding photoelectrons upon
striking the vacuum chamber. In proton rings, the process is typically
initiated by ionization of residual gas, or from electron generation when
stray beam particles strike the chamber. A notable exception is the LHC, which
is the first proton storage ring ever built in which the beam emits
significant synchrotron radiation, $\sim 0.4$ photons per proton per bending
magnet traversal, with a photon critical energy $\sim 44$ eV [33]. In this
case, photoemission is the dominant primary mechanism.
Primary emission mechanisms are usually insufficient to lead to a significant
EC density. However, the average electron-wall impact energy is typically
$\sim$100–200 eV, at which the SEY function $\delta(E)$ is significant. If the
effective SEY is $>1$, secondary emission readily exponentiates in time, which
can lead to a large amplification factor, typically a few orders of magnitude,
over the primary electron density, and to strong temporal and spatial
fluctuations in the electron distribution [34]. This compounding effect of
secondary emission is usually the main determinant of the strength of the
ECEs, and is particularly strong in positively-charged bunched beams (in
negatively-charged beams, the electrons born at the walls are pushed back
towards the walls with relatively low energy, typically resulting in
relatively inefficient secondary emission).
Photoemission and secondary electron emission depend differently on the beam
properties: photoelectron emission behaves linearly in beam intensity, is very
sensitive to beam energy, and is independent of the sign of the beam particle
charge, while secondary emission behaves nonlinearly in beam intensity, is not
very sensitive to beam energy, and is sensitive to the sign of the beam
particle charge. These features allow, in principle, to disentangle the
effects of primary from secondary electrons, given sufficient flexibility in
the machine operation as in CESRTA (see below).
## 3 Conditioning and Mitigation
Storage ring vacuum chambers are fabricated of ”technical metals.” Such
materials have rough surfaces and contain impurities, typically concentrated
at the surface. For such surfaces, the SEY gradually decreases in time with
machine operation owing to the bombardment of the very electrons in the cloud.
Such “conditioning effect” has been consistently observed in storage rings,
and is of course beneficial to the performance of the machine. Typically, it
is observed that $\delta_{\rm max}$ decreases rapidly (typically hours to
days) upon machine operation startup, and then effectively reaches a limit.
Indeed, as $\delta_{\rm max}$ decreases, the EC intensity decreases, leading
to a diminished electron-wall bombardment, hence to a slower conditioning
rate. This exponential slowing down, in effect, sets a practical limit on the
lowest value of $\delta_{\rm max}$ that is achievable via this phenomenon.
Recent experience at the SPS and LHC [35] is consistent with prior experience
at many other machines, namely that $\delta_{\rm max}$ decreases rapidly but
does not go far enough to avoid all EC detrimental effects.
Even if $\delta_{\rm max}$ were to decrease via the conditioning effect to its
natural limit [36, 37], it is not guaranteed to be low enough to avoid
undesirable ECE’s. For this reason, deliberate mitigation mechanisms are
typically implemented in present-day and future storage rings. Mitigation
mechanisms can be classified into passive and active. Passive mechanisms that
have been employed at various machines include:
* •
Coating the chamber with low-emission substances such as TiN [38, 39], TiZrV
[40, 41, 42, 43, 44, 45, 19, 46, 19, 46] and amorphous carbon (a-C) [47, 48].
* •
Etching grooves on the chamber surface in order to make it effectively
rougher, thereby decreasing the effective quantum efficiency via transverse
grooves [49] or the effective SEY via longitudinal grooves [50, 51].
* •
Implementing weak solenoidal fields ($\sim$10–20 G) to trap the electrons
close to the chamber walls, thus minimizing their detrimental effects on the
beam [52, 53]
In terms of active mechanisms, clearing electrodes [54, 55] show significant
promise in controlling the electron cloud development. If an electron cloud is
unavoidable and problematic, active mechanisms that have been employed to
control the stability of the beam include tailoring the bunch fill pattern
[56] and increasing the storage ring chromaticity [34]. Fast, single-bunch,
feedback systems are under active investigation as an effective mechanism to
stabilize electron-cloud induced coherent instabilities [57, 58].
## 4 Simulation of the ECE
Broadly speaking, depending on the approximations implemented, EC simulation
codes in use today are of three kinds:
* •
Build-up codes.
* •
Instability codes.
* •
Self-consistent codes.
Build-up codes make the approximation that the beam is a prescribed function
of space and time, and therefore is nondynamical. The electrons, on the other
hand, are fully dynamical. With this kind of code one can study the build-up
and decay of the EC, its density distribution, and its time and energy scales,
but not the effects of the EC on the beam111Actually, these codes do allow the
computation of the dipole wake induced by the EC on the beam, which in turn
allows a first-order computation of the coherent tune shift of successive
bunches of the beam.. These codes may include a detailed model of the
electron-wall interaction, and come in 2D and 3D versions. 2D codes are well
suited to study the EC in certain isolated regions of a storage ring, such as
in the body of magnets, and field-free regions. 3D codes are used to study the
EC in magnetic regions that are essentially 3D in nature, such as fringe
fields and wigglers.
Instability codes aim at studying the effects on the beam by an initially
prescribed EC. In these codes the beam particles are fully dynamical, while
the dynamics of the cloud electrons is limited. For example, the electron-wall
interaction may be simplified or non-existent, and/or the electron
distribution may be refreshed to its initial state with the passage of
successive bunches.
Self-consistent codes aim to study the dynamics of the beam and the electrons
under their simultaneous, mutual, interaction. Such codes are far more
computationally expensive than either of the above-mentioned “first-order”
codes, and represent the ultimate logical stage of the above-mentioned
simulation code efforts.
In many cases of interest, the net electron motion in the longitudinal
direction, i.e. along the beam direction, is not significant, hence the
electron cloud is sensibly localized. For this reason, in first approximation,
it makes sense to study it at various locations around the ring independently
of the others. In addition, given that the essential dynamics of the electrons
is in the transverse plane, i.e. perpendicular to the beam direction, two-
dimensional simulations are also a good first approximation to describe the
build-up and decay. In some cases, such as the PSR, electron generation,
trapping and ejection from the edges of quadrupole magnets is now known to be
significant, and these electrons act as seeds for the EC buildup in nearby
drift regions [59].
A comprehensive online repository containing code descriptions and contact
persons has been developed by the CARE program [60].
Self-consistent codes are beginning to yield useful results. We present here
one such example obtained with the code WARP/POSINST, pertaining to the SPS
[61]. In this case, a train of three beam batches, each consisting of 72
bunches, was simulated using a massively parallel computer at NERSC. The goal
of the simulation was primarily to assess the impact of the evolution of the
proton distribution in the beam on the EC density, as compared to the EC
density evolution produced by a build-up code, in which the proton
distribution is frozen in time. Fig. 2 shows some of the results of this
exercise. The conclusion is that, after 1000 turns, the actual proton
distribution leads to a 50–100% increase in the estimate of $n_{e}$ relative
to the case in which the proton distribution is kept frozen at its initial
state. While this result is suggestive, it must still be considered
preliminary because of the approximations employed, notably that of a constant
focusing lattice and the fact that the EC distribution was reinitialized at
avery turn (a fully self-consistent simulation, in which both the EC and the
proton distributions evolve in time in response to each other has also been
carried out [61]).
Figure 2: The EC density $n_{e}$ as a function of time over a 6-$\mu$s time
window, showing the passage of a 3-batch beam (the revolution period is
$\sim$23 $\mu$s). Each trace represents the evolution after the number of
revolutions indicated. For example, the blue trace (“turn 400”) shows the
window after 399 turns have elapsed. The red trace (“turn 0”) shows the
evolution of $n_{e}$ at beam injection; this trace is in excellent agreement
with the result of a build-up code, as it should, in which the proton
distribution is kept frozen at its initial state (a 3D gaussian distribution).
In this exercise, the electron distribution was reinitialized at every turn at
the beginning of the train. Thus the fact that the “turn 1000” trace is a
factor $\sim 2$ times larger than the “turn 0” trace is attributable only to
the evolution of the proton distribution in the beam after 1000 turns.
## 5 The CESRTA program
A significant, dedicated systematic R&D program to understand the EC and low-
emittance tuning has been ongoing at Cornell University for $\sim 5$ years
based on the CESR storage ring. The e+e- collider CESR was decommissioned and
the CLEO detector removed. Wigglers were added to the storage ring, along with
an extensive array of diagnostic instrumentation intended to analyze the EC.
This revamped storage ring (the CESR Test Accelerator, or CESRTA) is intended
as a prototype for the damping rings of a possible future e+e- linear collider
[62]. A major report will describe the R&D effort in detail [63].
As a test accelerator, CESRTA has unprecedented operational flexibility,
specifically:
* •
Essentially all beam time is devoted to machine studies.
* •
The injector allows for an almost arbitrary fill pattern.
* •
The beam species is selectable (e+ or e-), although the two species move in
opposite directions in the beam pipe.
* •
The beam energy is tunable within the range $\sim 2-5$ GeV
* •
The bunch intensity is selectable.
The new diagnostic devices include: retarding-field analyzers (RFA’s) at many
locations, magnetized or not; shielded pick-ups (SPU’s); a microwave
transmission setup; filtered and gated beam position monitors (BPM’s); etc. In
addition, an array of special-purpose devices have been installed including:
an in-situ SEY measuring device; a low-magnetic-field chicane, transplanted
from PEP-II at SLAC; various sections of beam pipe with low-emission coatings
or grooved surfaces; and clearing electrodes. RFA’s allow the measurement of
the spatially-resolved, time-averaged, electron flux at the walls of the
chamber. The SPU’s allow the measurement of the electron flux at the walls of
the chamber with a time resolution of $\sim 1$ ns. The BPM’s, by themselves or
in combination with a beam pinger and a feedback damping system, allow the
measurement of bunch-by-bunch frequency spectra and coherent tunes. x-ray
beam-size monitors allow the measurement of beam size bunch-by-bunch and turn-
by-turn.
As part of the CESRTA R&D, a broad-based program of developing, comparing and
benchmarking electron cloud buildup simulation codes, and to a much lesser
extent beam dynamics codes, was initiated in 2008 and continues today.
Specifically CESRTA input parameters have been used as input to the simulation
codes ECLOUD [64, 65], CLOUDLAND [66, 67], POSINST [68, 69], WARP/POSINST [70]
and PEHTS [71], and the results compared against measurements. By iterating
this process, EC-related parameters that are not well known were pinned down,
allowing more reliable extrapolations to the future ILC damping rings. The
main parameters that are not well known are those pertaining to the electronic
surface properties, i.e. photon reflectivity; photoemission yield or quantum
efficiency (QE); photoemission spectrum; and secondary electron yield and
spectrum [72].
In addition, a new photon-tracking code, SYNRAD3D [73], has been developed and
implemented, which allows the tracking of synchrotron radiation emitted by the
beam as it traverses magnetic elements. The code allows for the description of
the actual beam size at the emission point, as well as the actual description
of the vacuum chamber geometry and external magnetic fields for the entire
ring. Models for the photon reflectivity and quantum efficiency have been
incorporated. The outcome of this code is the photoelectron emission
distribution along the perimeter of the chamber cross section at any desired
point in the ring. This photoelectron distribution is fed as an input to the
above-mentioned build-up codes. A simpler code of this nature was developed
earlier in the context of the LHC EC effort [74].
By adjusting the bunch train length and adding a “witness bunch” at various
distances after the end of the train, one is able to disentangle the effects
of the photoelectrons from the secondary electrons. A comparison of a
simulation vs. measurements at CESRTA is shown in Fig. 3 [75].
Figure 3: Measured tune shifts (black points) vs. bunch number, for a train of
ten 0.75-mA/bunch, 5.3 GeV, positron bunches with 14 ns spacing, followed by
witness bunches [75]. Red points are computed (using POSINST) based on a
simplified assumption for the incident photon distribution consisting of a
direct component plus a uniform background (free parameter) of scattered
photons. Blue points are computed using results for the photoelectron emission
distribution obtained from SYNRAD3D (with no free parameters for the
radiation) as input to POSINST. The good agreement between measurements and
simulations gives confidence in the EC model implemented in the code POSINST.
The computation employing the SYNRAD3D results is clearly in better agreement
with the measurements than that using a simplified photoemission distribution
model.
## 6 Conclusions
* •
The ECE is an ubiquitous phenomenon for intense beams. The phenomenon spans a
broad range of charged-particle storage rings.
* •
The ECE is important inasmuch as it limits machine performance, especially for
high-intensity future machines.
* •
The ECE is interesting, as it involves in an essential way various areas of
physics, such as: surface geometry and surface electronics; beam intensity and
particle distribution; beam energy; residual vacuum pressure in the chamber;
certain magnetic features of the storage ring; and other areas.
* •
Simulation codes are getting better and better in their detailed modeling
capabilities and predictive ability.
* •
Enormous progress has been made since 1995, with a disproportionate credit due
to CESRTA and CERN over the past few years. Better and more refined electron
detection mechanisms are now deployed. Simulation codes are getting better and
better calibrated against measurements.
* •
Phenomelogical rules of thumb are appearing that tell us the conditions under
which the ECE is serious, but not (yet) the conditions under which it s
guaranteed to be safe.
## 7 Epilogue
This workshop is dedicated to the memory of Francesco Ruggiero (1957-2007). I
met Francesco on many occasions during my career. I feel honored to have met
him and grateful for what I learned from him. I am especially grateful to
Francesco for his strong support of electron-cloud R&D effort at CERN and
elsewhere. The knowledge that has come out of this program, plus the recent
experience at the LHC and SPS, have already greatly benefitted the field as a
whole, and will continue to benefit the design and reliability of accelerators
worldwide for a long time to come. This workshop is rightfully dedicated to
Francesco’s memory.
## 8 acknowledgments
Over the years I have greatly benefitted from discussions and/or collaboration
with many colleagues at ANL, BNL, CERN, Cornell, FNAL, Frascati, KEK, LANL,
LBNL, SLAC and TechX—I am grateful to all of them, too numerous to list here.
I want to express my special thanks to Roberto Cimino and Frank Zimmermann for
organizing this productive and enlightening workshop. We are grateful to NERSC
for supercomputer support.
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|
arxiv-papers
| 2013-10-07T09:03:06 |
2024-09-04T02:49:52.058901
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "M.A. Furman (LBNL, Berkeley and Cornell U., CLASSE)",
"submitter": "Scientific Information Service CERN",
"url": "https://arxiv.org/abs/1310.1706"
}
|
1310.1716
|
# Time delay in the recoiling valence-photoemission of Ar endohedrally
confined in C60
Gopal Dixit [email protected] Center for Free-Electron Laser Science, DESY,
Notkestrasse 85, 22607 Hamburg, Germany Max Born Institute, Max-Born-Strasse
2A, 12489 Berlin, Germany Himadri S. Chakraborty [email protected]
Department of Natural Sciences, Center for Innovation and Entrepreneurship,
Northwest Missouri State University, Maryville, Missouri 64468, USA Mohamed
El-Amine Madjet [email protected] Center for Free-Electron Laser Science,
DESY, Notkestrasse 85, 22607 Hamburg, Germany Qatar Energy and Environment
Research Institute (QEERI), Qatar Foundation, Doha, Qatar
###### Abstract
The effects of confinement and electron correlations on the relative time
delay between the 3s and 3p photoemissions of Ar confined endohedrally in C60
are investigated using the time dependent local density approximation - a
method that is also found to mostly agree with recent time delay measurements
between the 3s and 3p subshells in atomic Ar. At energies in the neighborhood
of 3p Cooper minimum, correlations with C60 electrons are found to induce
opposite temporal effects in the emission of Ar 3p hybridized symmetrically
versus that of Ar 3p hybridized antisymmetrically with C60. A recoil-type
interaction model mediated by the confinement is found to best describe the
phenomenon.
###### pacs:
32.80.Fb, 61.48.-c, 31.15.E-
With the tremendous advancement in technology for generating attosecond (as)
isolated pulses as well as attosecond pulse trains, it becomes possible to
study fundamental phenomena of light-matter interaction with unprecedented
precision on an as timescale hentschel ; goulielmakis1 ; krausz . In
particular, the relative time delay between the photoelectrons from different
subshells on as timescale, a subject of intense recent activities, is expected
to probe important aspects of electron correlations that predominantly
influence the photoelectron. Pump-probe experiments have been performed to
measure the relative delay in the photoemission processes, where extreme
ultra-violet (XUV) pulses are used to remove an electron from a particular
subshell and subsequently a weak infrared (IR) pulse accesses the temporal
information of the emission event pazourek .
Streaking measurements were carried out to probe photoemission from the
valence and the conduction band in single-crystalline magnesium
neppl2012attosecond and tungsten cavalieri2007attosecond . A streaking
technique was also employed to measure the relative delay of approximately
21$\pm$5 as between the 2s and 2p subshells of atomic Ne at 106 eV photon
energy schultze2010delay . Despite several theoretical attempts
mauritsson2005accessing ; kheifets2010delay ; moore2011time ;
ivanov2011accurate ; nagele2012time ; dahlstrom2012diagrammatic ;
kheifets2013time to explain this measured delay in Ne, only about a half of
the delay could be reproduced, keeping the time delay in Ne photoemissions
still an open problem. Recently, the relative delay between the 3s and 3p
subshells in Ar is measured at three photon energies by interferometric
technique using attosecond pulses klunder2011probing ; guenot2012photoemission
. Theoretical methods (e.g. time-dependent nonperturbative method
mauritsson2005accessing , diagrammatic many-body perturbation theory
dahlstrom2012diagrammatic , Random phase approximation with exchange (RPAE)
guenot2012photoemission ; kheifets2013time , and multi-configurational
Hartree-Fock (MCHF) carette2013multiconfigurational ) have been employed to
investigate this relative delay in Ar, although agreements between theory and
experiment is rather inconclusive. A ubiquitous understanding in all these
studies is the dominant influence of electron correlations to determine the
time behavior of outgoing electrons. Thus, it is fair to expect that the
process near a Cooper minimum or a resonance will be particularly nuanced.
It is therefore of spontaneous interest to extend the study to test the effect
of correlations on the temporal photoresponse of atoms in material
confinements. A brilliant natural laboratory for such is an atom endohedrally
captured in a fullerene shell; see Fig. 1 which envisions the process. There
are two compelling reasons for this choice: (i) such materials are highly
stable, have low-cost sustenance at room temperature and are enjoying a rapid
improvement in their synthesis techniques popov2013 and (ii) effects of
correlations of the central atom with the cage electrons have been predicted
to spectacularly influence the atomic valence photoionization madjet2007giant
. In this Letter, by considering Ar@C60, we show that a confinement-induced
correlation effect of C60 at energies surrounding the Ar 3p Cooper minimum
produces a faster and a slower emission of the Ar 3p electrons hybridized,
respectively, in a symmetric and an antisymmetric mode with a near-degenerate
C60 orbital.
Figure 1: (Color online). Schematic for probing the effects of correlations
from the confinement on the relative time delay in the emission of an atom
encaged endohedrally inside C60.
Time dependent local density approximation (TDLDA), with Leeuwen and Baerends
(LB) exchange-correlation functional to produce accurate asymptotic behavior
van1994exchange of ground and continuum electrons, is employed to calculate
the dynamical response of the system to the external electromagnetic field. To
demonstrate the accuracy of the method for an isolated atom, the total
photoionization cross section and the partial 3s and 3p cross sections of Ar
are presented in Fig. 2a and compared with available experiments
mobus1993measurements ; samson2002precision . As seen, our TDLDA total and 3s
cross sections are in excellent agreement with experimental results and the
positions of the 3s and 3p Cooper minima at, respectively, 42 and 48 eV are
well reproduced. The dominance of 3p contribution over 3s in this energy range
(Fig. 2a) also automatically implies the accuracy of our TDLDA 3p result.
Figure 2: (Color online). Top: TDLDA 3p, 3s and total photoionization cross
sections for atomic Ar are compared with experiments for 3s
mobus1993measurements and total samson2002precision . For 3s the computed
cross section is scaled to reproduce the measurement at the Cooper minimum.
Bottom: The relative TDLDA time delay between 3s and 3p of Ar and its
comparison with measurements (solid black circles, Ref.
guenot2012photoemission ; open red squares, Ref. klunder2011probing ). RPAE
results kheifets2013time at three experimental energies are also cited.
The absolute time delay in Ar pump-probe photoemission contains two
contributions: one due to the absorption of XUV photon and the other due to
the probe pulse. Owing to the weak probe pulse, the probe-assisted delay
contributions can be estimated dahlstrom2012diagrammatic as a function of the
kinetic energy of electrons from different Ar subshells. This allowed
evaluation of the relative delay in recent measurements klunder2011probing ;
guenot2012photoemission . This delay therefore connects to the energy
derivative of the quantum phase of complex photoionization amplitude
yakovlev2010attosecond \- the Wigner-Smith time delay wigner1955lower ;
smith1960lifetime ; de2002time . Several methods ivanov2011accurate ;
nagele2012time ; dahlstrom2012theory ; ivanov2013extraction have been
utilized to extract the Wigner-Smith time delay directly from the
measurements.
The photoionization amplitude from an initial bound state ($n_{i}l_{i}$) to a
final continuum state ($kl$) can be expressed as
$\displaystyle f(\hat{\bf{k}})$ $\displaystyle=$
$\displaystyle(8\pi)^{3/2}\sum_{\begin{subarray}{c}l=l_{i}\pm 1\\\
m=m_{i}\end{subarray}}(-i)^{l}e^{i\eta_{l}(\hat{\bf{k}})}Y_{lm}^{*}(\hat{\bf{k}})\langle\phi_{kl}||r+\delta
V||\phi_{n_{i}l_{i}}\rangle$ (5)
$\displaystyle\times\sqrt{(2l+1)(2l_{i}+1)}\left(\begin{array}[]{ccc}l&1&l_{i}\\\
0&0&0\end{array}\right)\left(\begin{array}[]{ccc}l&1&l_{i}\\\
-m&0&m_{i}\end{array}\right).$
Here, $\delta V$ is the complex induced potential which embodies TDLDA many-
body correlations. The phase $\eta_{l}$ includes contributions from both the
short range and Coulomb potentials, whereas the phase of the complex matrix
element in Eq. (5) is the correlation phase. For Ar, the correlation near
Cooper minima primarily arises from the coupling of 3p with 3s channels. The
total phase is the sum of these three contributions. The time delay profile is
computed by differentiating the TDLDA total phase in energy.
Our TDLDA relative Wigner-Smith delay between Ar 3s and 3p,
$\tau_{\textrm{3s}}-\tau_{\textrm{3p}}$, is compared with the experimental
data of Guénot et al. guenot2012photoemission and of Klünder et al.
klunder2011probing in Fig. 2b. As seen, the relative delay is strongly energy
dependent. Note that the TDLDA results are in excellent agreement with both
sets of experimental results at 34.1 and 37.2 eV. The third measurement at
40.3 eV, which is in the vicinity of the 3s Cooper minimum, is negative in
Ref. klunder2011probing in contrast to its positive value in Ref.
guenot2012photoemission . Note that our result captures the correct sign as in
Klünder et al. at 40.3 eV. In general, 3p$\rightarrow$kd photochannel is
dominant over 3p$\rightarrow$ks at most energies. Close to the 3p Cooper
minimum, however, 3p$\rightarrow$kd begins to rapidly decrease to its minimum
value, enabling 3p$\rightarrow$ks to significantly contribute to the net 3p
delay. The s- and d-wave emissions have different angular distributions but
their Wigner delays are independent of emission directions. Thus, assuming
that all 3p photoelectrons are detected (integration over solid angle), the
net 3p delay must be a statistical combination, that is, the sum of the delays
weighted by the channel’s individual cross section branching ratios. As
illustrated in Fig. 2b, upon including 3p$\rightarrow$ks along with
3p$\rightarrow$kd (purple curve) this way, the shape of the TDLDA delay
strikingly alters near 3p Cooper minimum. We stress that the delay near a
Cooper minimum needs to be addressed with great care which can reveal new
physics, as shown below for an endohedrally confined Ar atom.
We also include recent RPAE results kheifets2013time for three experimental
energies in Fig. 2b. As seen, RPAE and experiments match only at 34.1 eV. The
superior performance of TDLDA in explaining the measurements is thus evident.
While both TDLDA and RPA are many-body linear response theories, they have
significant differences in the details, particularly, in treating electron
correlations onida2002electronic . Variants of the Kohn-Sham LDA+LB scheme
were successfully utilized to describe attosecond strong-field phenomena
petretti2010alignment ; heslar2011high ; farrell2011strong ; toffoil2012 ;
hellgren2013 , underscoring the reliability of many-body correlations that
TDLDA characteristically offers.
This success of TDLDA method for free Ar encouraged us to use the approach to
investigate the delay in an Ar atom endohedrally sequestered in C60. The
jellium model is employed for computing the relative delay madjet2010 . This
model enjoyed earlier successes in codiscovering with experimentalists a high-
energy plasmon resonance scully2005 , interpreting the energy-dependent
oscillations in C60 valence photo-intensities rudel2002 , and predicting giant
enhancements in the confined atom’s photoresponse from the coupling with C60
plasmons madjet2007giant . Significant ground state hybridization of Ar 3p is
found to occur with the C60 3p orbital, resulting in 3p[Ar+C60] and 3p[Ar-C60]
from, respectively, the symmetric and antisymmetric wavefunction mixing. These
are spherical analogs of bonding and antibonding states in molecules or
dimers. Such atom-fullerene hybridization was predicted before chakraborty2009
and detected in the photoemission experiment on multilayers of Ar@C60
morscher2010strong . In fact, the hybridization gap of 1.5 eV between
3p[Ar+C60] and 3p[Ar-C60] in our calculation is in good agreement with the
measured value of 1.6$\pm$0.2 eV morscher2010strong .
Figure 3: (Color online). TDLDA quantum phases for ionization via d-waves from
bonding 3p[Ar+C60] and antibonding 3p[Ar-C60] levels and via p-wave from Ar
3s@ are compared with their counterparts in free Ar.
The TDLDA Wigner-Smith phases for relevant ionization channels for confined
and free Ar are presented in Fig. 3. We use the symbol “@” to denote states
belonging to the confined Ar. The narrow resonance spikes below 40 eV are due
to single electron Rydberg-type excitations in C60. This energy zone also
includes the C60 plasmon resonances, although their effects are suppressed by
the Coulomb phase that dominates the extended region above ionization
thresholds. We note that the Ar 3s Cooper minimum shifts slightly lower in
energy to 36.5 eV from the confinement, but the confinement moves the two 3p
minima, each in the bonding and antibonding channels, somewhat higher in
energy. What is rather dramatic in Fig. 3 is that the quantum phase
corresponding to 3p[Ar+C60]$\rightarrow$kd@ (thick solid black) makes a
downward $\pi$ phase shift, whereas the phase associated with
3p[Ar-C60]$\rightarrow$kd@ (thick solid red) suffers a upward 2$\pi$ phase
shift at their respective Cooper minimum. Further note that both these
contributions together yield a net phase that shifts up by $\pi$ as in the
case of free-Ar 3p$\rightarrow$kd channel (dashed black curve in Fig. 3) at
its Cooper minimum.
This contrasting phase behavior between hybrid 3p emissions is likely the
effect of symmetric and antisymmetric wavefunction shapes on the matrix
elements through dynamical correlations. Using the well-known Fano scheme of
perturbative interchannel coupling fano1961 the lead contribution to the
matrix element $\langle\delta V\rangle$ (Eq. (5)) is javani2012
$\langle\delta V\rangle_{\alpha}(E)=\displaystyle\sum_{\beta}\int
dE^{\prime}\frac{\langle\Psi_{\beta}(E^{\prime})|\frac{1}{|{\bf
r}_{\alpha}-{\bf r}_{\beta}|}|\Psi_{\alpha}(E)\rangle}{E-E^{\prime}}\langle
z\rangle_{\beta}(E^{\prime}),$ (6)
where $\alpha$ denotes each of the 3p[Ar$\pm$ C60]$\rightarrow$kd@ channels.
$\Psi$ are channel-wavefunctions that involve both bound (hole) and continuum
(photoelectron) states, and $\langle z\rangle_{\beta}$ is the single channel
matrix element of each perturbing channel $\beta$. Thus, the summation over
channels incorporates bound states as the hole states. Two points can be
noted: First, $\langle\delta V\rangle$ dominates near the Cooper minimum of a
channel $\alpha$, since the “unperturbed” $\langle z\rangle_{\alpha}$ is
already small at these energies; second, $\langle\delta V\rangle$ depends on
the coupling matrix element in the numerator of Eq. (6) that involves overlaps
between the bound state of a $\alpha$ channel with that in a perturbing
$\beta$ channel. These overlaps are critical, since 3p[Ar+C60] wavefunction
has a structure completely opposite to that of 3p[Ar-C60] over the C60 shell
region where each of them strongly overlaps with a host of C60 wavefunctions
to build correlations. These opposing modes of overlap from one hybrid to
another flip the phase modification direction between two hybrid 3p emissions
around a respective Cooper minimum, as seen in Fig. 3.
Figure 4: (Color online). Top: Absolute time delay for ionizations in
3p[Ar$\pm$C60]$\rightarrow$kd@ and 3s@$\rightarrow$kp@ channels. For the two
hybrid channels, results modified by incorporating s-wave delays are also
shown. Bottom: Relative delays $\tau_{\textrm{3s@}}-\tau_{\textrm{3p
[Ar}\pm\textrm{C}_{60}]}$, including the s-wave contributions;
$\tau_{\textrm{3s}}-\tau_{\textrm{3p}}$ of free Ar is also shown for
comparison.
Depending on the upward (downward) shift in the quantum phase, the resulting
photoelectron exhibits positive (negative) time delay and hence emerges
slower(faster) from the ionization region. This is evident in Fig. 4a, which
features various absolute delays: Channels 3p[Ar+C60]$\rightarrow$ kd@ and
3p[Ar-C60]$\rightarrow$ kd@ exhibit, respectively, a fast and a slow emission
over relatively narrow ranges about their Cooper minima. Note that the peak
delay of the antibonding electron is approximately double to the peak
advancement (negative delay) of the bonding electron. The delay profile
becomes softer and broader in energy by including the contribution from
s-wave, but the general trend of a rapid and a slow ejection, respectively, in
the bonding and antibonding channels survives.
The conservation of the quantum phase, i.e., the net phase shift of $\pi$ in
the upward direction (as in the free Ar) for 3p in Ar@C60, can be understood
in the language of a collision type interaction between two hybrid 3p
electrons. The phase behaves like the linear momentum in a two-body collision
which is a conserved quantity. Its energy derivative, i.e., the time delay,
can be thought to be commensurate with the collision force, the time
derivative of the momentum, since time and energy are conjugate variables.
This implies, that if one hybrid electron goes through an advanced emission,
the other hybrid must delay or time-recoil appropriately to keep the net delay
roughly close to the delay of free Ar. Of course, here the process is
underpinned by the orbital mixing. Therefore, the phenomenon can be pictured
as the photo-liberation of two recoiling electrons in the temporal domain from
the atom-fullerene hybridization. Hence, it is also likely to exist in the
ionization of molecules, nanodimers, and fullerene onions that support hybrid
electrons.
The time delays in the photoionization of 3p hybrids (with s-wave contribution
included) relative to 3s@, $\tau_{\textrm{3s@}}-\tau_{\textrm{3p
[Ar}\pm\textrm{C}_{60}]}$, are presented in Fig. 4b. One notes in Fig. 4a that
3s@$\rightarrow$kp@ produces an absolute delay profile, which is negative for
most energies and, on an average, comparable to the absolute delay in
3p[Ar-C60]$\rightarrow$kd@+ks@. Consequently, their (fast) emergence at about
similar speeds keeps their relative delay close to $\tau=0$, but with a bias
toward negative values. On the other hand, for the
3p[Ar+C60]$\rightarrow$kd@+ks@ channel the relative delay remains mostly
strongly negative. However, the rich structures in the delay profiles
emphasize that the Cooper minimum regions are particularly attractive for time
delay studies.
The 3p bonding-antibonding gap of 1.5 eV requires the energy of the probe
pulse to be smaller than this gap. Otherwise, the sideband of one level will
begin to overlap with the harmonics of the other. Also, by varying the
polarization angle between XUV and IR pulses one can potentially probe both
independent contributions, i.e., the relative delay between 3s orbital and 3p
bonding/anti-bonding orbital, i.e., by extending the standard RABBIT method
veniard1995 , where the polarization of XUV pulse is the same as the IR pulse.
Therefore, techniques based on interferometry, such as RABBIT and PROOF
chini2010 , have potentials to probe the relative delay between 3p
bonding/antibonding and 3s electrons. One may also perform the streaking
experiments using IR as well as THz pulses for accessing the delay. We suggest
that future experiments be performed on the time delay in Ar and Ar@C60 over
broader photon energy ranges including the 3p Cooper minimum to unravel new
physics from confinement and correlations.
In conclusion, our TDLDA relative Wigner-Smith time delay between 3s and 3p
subshells in free Ar are in excellent agreement with the measured delay except
near the 3s Cooper minimum, where, however, the TDLDA is consistent with the
sign of one set of measurements. In the case of confined Ar, due to the
electron correlation, the delays of the 3p bonding and 3p antibonding
emissions are governed by a recoil-type emission in the time-domain mediated
by the host C60. It is found that the emission from the 3s@ level is slightly
faster than the emission from the 3p bonding level but is substantially
faster, by 100 as and above, than the emission from the 3p antibonding level.
We further demonstrate that the delay of Ar 3p electron, free or confined,
leads to significant modifications in the vicinity of the Cooper minimum by
including the s-wave photochannel.
###### Acknowledgements.
G. D. acknowledges Misha Ivanov, Tim Laarmann and Oliver Mücke for useful
discussions. The research is supported by the NSF, USA.
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|
arxiv-papers
| 2013-10-07T09:46:43 |
2024-09-04T02:49:52.065767
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Gopal Dixit, Himadri S. Chakraborty, and Mohamed El-Amine Madjet",
"submitter": "Gopal Dixit dr.",
"url": "https://arxiv.org/abs/1310.1716"
}
|
1310.1718
|
# Segregated Vector Solutions for linearly coupled Nonlinear Schrödinger
Systems
Chang-Shou Lin and Shuangjie Peng Taida Institute for Mathematical Sciences
and Department of Mathematics, National Taiwan University, Taipei, 10617,
Taiwan [email protected] School of Mathematics and Statistics, Central
China Normal University, Wuhan, 430079, P. R. China [email protected]
###### Abstract.
We consider the following system linearly coupled by nonlinear Schrödinger
equations in $\mathbb{R}^{3}$
$\left\\{\begin{array}[]{ll}-\Delta
u_{j}+u_{j}=u^{3}_{j}-\varepsilon\sum\limits_{i\neq
j}^{N}u_{i},&x\in\mathbb{R}^{3},\vspace{0.2cm}\\\ u_{j}\in
H^{1}(\mathbb{R}^{3}),\quad j=1,\cdots,N,\end{array}\right.$
where $\varepsilon\in\mathbb{R}$ is a coupling constant. This type of system
arises in particular in models in nonlinear $N$-core fiber.
We examine the effect of the linear coupling to the solution structure. When
$N=2,3$, for any prescribed integer $\ell\geq 2$, we construct a non-radial
vector solutions of segregated type, with two components having exactly $\ell$
positive bumps for $\varepsilon>0$ sufficiently small. We also give an
explicit description on the characteristic features of the vector solutions.
## 1\. Introduction
We consider the following nonlinear Schrödinger systems which are linearly
coupled by $N$ equations
$\left\\{\begin{array}[]{ll}-\Delta
u_{j}+u_{j}=u^{3}_{j}-\varepsilon\sum\limits_{i\neq
j}^{N}u_{i},&x\in\mathbb{R}^{3},\vspace{0.2cm}\\\ u_{j}\in
H^{1}(\mathbb{R}^{3}),\quad j=1,\cdots,N,\end{array}\right.$ (1.1)
where $\varepsilon\in\mathbb{R}$. These systems arise when one considers
stationary pulselike (standing wave) solutions of the time-dependent
$N$-coupled Schrödinger systems of the form
$\left\\{\begin{array}[]{l}-i\frac{\partial}{\partial
t}\Phi_{j}=\Delta\Phi_{j}-\Phi_{j}+|\Phi_{j}|^{2}\Phi_{j}-\varepsilon\sum^{N}\limits_{i\neq
j}\Phi_{i},\ \ \hbox{in}\ \mathbb{R}^{3}\times\mathbb{R}^{+},\\\
\Phi_{j}=\Phi_{j}(x,t)\in\mathbb{C},t>0,\ j=1,\cdots,N.\\\ \end{array}\right.$
(1.2)
This type of system arises in nonlinear optics. For example, the propagation
of optical pulses in nonlinear $N$-core directional coupler can be described
by $N$ linearly coupled nonlinear Schrödinger equations. Here $\Phi_{j}$
($j=1,\cdots,N$) are envelope functions and $\varepsilon$, which is the
normalized coupling coefficient between the cores, is equal to the linear
coupling coefficient times the dispersion length. The sign of $\varepsilon$
determines whether the interactions of fiber couplers are repulsive or
attractive. In the attractive case the components of a vector solution tend to
go along with each other leading to synchronization, and in the repulsive case
the components tend to segregate with each other leading to phase separations.
These phenomena have been documented in numeric simulations (e.g., [1] and
references therein).
Nonlinear Schrödinger equations have been broadly investigated in many
aspects, such as existence of solitary waves, concentration and multi-bump
phenomena for semiclassical states (see e.g. [10], [24] and the references
therein). The study on system of Schrödinger equations began quite recently.
Mathematical work on systems with the nonlinearly coupling terms (e.g. the
term $\sum_{i\neq j}^{N}u_{i}$ in (1.1) being replaced by $u_{j}\sum_{i\neq
j}^{N}u_{i}^{2}$ ) has been studied extensively in recent years, for example,
[6, 8, 9, 11, 12, 14, 16, 17, 20, 21, 22, 23] and references therein, where
phase separation or synchronization has been proved in several cases.
However, for the linearly coupled system (1.1), as far as the authors know, it
seems that there are few results. In [5], when $N=2$, solitons of linearly
coupled systems of semilinear non-autonomous equations were studied by using
concentration compactness principle, and existence of both positive ground and
bound states was proved under some decay assumptions on the potentials at
infinity. In [2], this type of non-autonomous systems was also considered by
using a perturbation argument. Concerning on autonomous systems, we also
mention some results. If $N=2$ and the dimension is one, for $\varepsilon<0$,
(1.1) has in addition to the semi-trivial solutions $(\pm U,\,0),\,(0,\,\pm
U)$, two types of soliton like solutions, given by
$\displaystyle(U_{1+\varepsilon},\,U_{1+\varepsilon}),\,\,(-U_{1+\varepsilon},\,-U_{1+\varepsilon}),\quad\hbox{for}\,\,\,-1\leq\varepsilon\leq
0,\,\,(\hbox{symmetric}\,\,\,\hbox{states}),$
$\displaystyle(U_{1-\varepsilon},\,-U_{1-\varepsilon}),\,\,(-U_{1-\varepsilon},\,U_{1-\varepsilon}),\quad\hbox{for}\,\,\,\varepsilon\leq
0,\,\,(\hbox{anti-symmetric}\,\,\,\hbox{states}),$
where, for $\lambda>0$, $U_{\lambda}$ is the unique solution of
$\begin{cases}-u^{\prime\prime}+\lambda u=u^{3},\quad
u>0,\quad\text{in}\;\mathbb{R},\\\
u(0)=\max\limits_{x\in\mathbb{R}}u(x),\,\,u(x)\in
H^{1}(\mathbb{R}).\end{cases}$
By using numerical methods, a bifurcation diagram is reported in [1] where it
is indicated that for $\varepsilon\in(-1,0)$, there exists a family of new
solutions for (1.1), bifurcating from the branch of the anti-symmetric state
at $\varepsilon=-1$. This kind of results was rigorously verified in [3] for
small value of the parameter $\varepsilon<0$. More precisely, in [3], it was
proved that a solution with one $2$-bump component having bumps located near
$\pm|\ln(-\varepsilon)|$, while the other component having one negative peaks
exists. This type of results was generalized recently in an interesting paper
[4] to two and three dimensional cases. In [4], it was proved that if
$\mathcal{P}$ denotes a regular polytope centered at the origin of
$\mathbb{R}^{d}\,(d=2,3)$ such that its side is larger than the radius of the
circumscribed circle or sphere, then there exists a solution with one multi-
bump component having bumps located near the vertices of
$\ln(-\varepsilon)\mathcal{P}$, while the other component has one negative
peak as $\varepsilon\to 0^{-}$. So in [4], the first component of the
solutions has more than one bump, while the second component is negative and
has only one bump. We emphasize here that the solutions obtained in [4]
bifurcate also from the branch of anti-symmetric state at $\varepsilon=0$.
Furthermore, as pointed out in [3], for $\varepsilon<0$, vector solutions with
one component being multi-bump do not exist near symmetric states, but only
near the anti-symmetric ones. Hence, an interesting problem is: can we find
solutions bifurcating from the symmetric state if $\varepsilon>0$? In this
paper, our main purpose is to prove that, for any prescribed integer $\ell\geq
2$, (1.1) has new solutions, different from the previous ones, with the
feature that two components have exactly $\ell$ positive bumps when
$\varepsilon>0$ is sufficiently small.
To state our main results, we introduce some notations.
The Sobolev space $H^{1}(\mathbb{R}^{3})$ is endowed with the standard norm
$\|u\|_{\mathbb{R}^{3}}=\Bigl{(}\int_{\mathbb{R}^{3}}(|\nabla
u|^{2}+u^{2})\Bigl{)}^{\frac{1}{2}}.$
Denote by $U$ the unique solution of the following problem
$\begin{cases}-\Delta u+u=u^{3},\quad u>0,\quad\text{in}\;\mathbb{R}^{3},\\\
u(0)=\max\limits_{x\in\mathbb{R}^{3}}u(x),\,\,u(x)\in
H^{1}(\mathbb{R}^{3}).\end{cases}$ (1.3)
It is well known that $U(x)=U(|x|)$ satisfies
$\lim\limits_{|x|\to+\infty}|x|e^{|x|}U=A>0,\,\,\hbox{and}\,\,\lim\limits_{|x|\to+\infty}\frac{U^{\prime}(|x|)}{U(x)}=-1.$
Moreover, $U(x)$ is non-degenerate, that is,
$Kernel(\mathbb{L})=span\Bigl{\\{}\frac{\partial U(x)}{\partial
x_{i}}:\,\,i=1,2,3\Bigl{\\}},$
where $\mathbb{L}$ is the linearized operator
$\mathbb{L}:\,\,H^{1}(\mathbb{R}^{3})\to
L^{2}(\mathbb{R}^{3}),\,\,\mathbb{L}(u)=:\Delta u-u+3U^{2}u.$
Let
$x^{j}=\displaystyle\Bigl{(}r\cos\frac{2(j-1)\pi}{\ell},r\sin\frac{2(j-1)\pi}{\ell},0\Bigr{)}:=\bigl{(}{x^{\prime}}^{j},\,\,0\bigr{)},\,\,j=1,\cdots,\ell,$
(1.4)
and
$\begin{array}[]{l}y^{j}=\displaystyle\Bigl{(}\rho\cos\frac{(2j-1)\pi}{\ell},\rho\sin\frac{(2j-1)\pi}{\ell},0\Bigr{)}:=\bigl{(}{y^{\prime}}^{j},\,\,0\bigr{)},\,\,\,j=1,\cdots,\ell,\end{array}$
(1.5)
where $r,\,\rho\in[r_{0}|\ln\varepsilon|,\,\,r_{1}|\ln\varepsilon|]$ for some
$r_{1}>r_{0}>0$.
In this paper, for any function $W:\mathbb{R}^{3}\to\mathbb{R}$ and
$\xi\in\mathbb{R}^{3}$, we define $W_{\xi}=W(x-\xi)$.
We first consider the following problem linearly coupled by two nonlinear
Schrödinger equations
$\left\\{\begin{array}[]{ll}-\Delta u+u=u^{3}-\varepsilon
v,&x\in\mathbb{R}^{3},\vspace{0.2cm}\\\ -\Delta v+v=v^{3}-\varepsilon
u,&x\in\mathbb{R}^{3}.\end{array}\right.$ (1.6)
The main result can be stated as follows
###### Theorem 1.1.
For any integer $\ell\geq 2$, there exists $\varepsilon_{0}$ such that for
$\varepsilon\in(0,\varepsilon_{0})$, problem (1.6) has a solution $(u,v)\in
H^{1}(\mathbb{R}^{3})\times H^{1}(\mathbb{R}^{3})$ satisfying
$u^{\varepsilon}\sim\sum\limits_{j=1}^{\ell}U_{x_{\varepsilon}^{j}},\,\,v^{\varepsilon}\sim\sum\limits_{j=1}^{\ell}U_{y_{\varepsilon}^{j}},$
where $x_{\varepsilon}^{j}$ and $y_{\varepsilon}^{j}$ are respectively defined
by (1.4) and (1.5) with
$r_{\varepsilon}=\frac{2\sin\frac{\pi}{\ell}}{2\sin\frac{\pi}{\ell}-\sqrt{2(1-\cos\frac{\pi}{\ell})}}|\ln\varepsilon|+o(|\ln\varepsilon|),\quad\rho_{\varepsilon}=\frac{2\sin\frac{\pi}{\ell}}{2\sin\frac{\pi}{\ell}-\sqrt{2(1-\cos\frac{\pi}{\ell})}}|\ln\varepsilon|+o(|\ln\varepsilon|).$
Moreover, as $\varepsilon\to 0^{+}$,
$\|u^{\varepsilon}(\cdot)-v^{\varepsilon}(T_{\ell}\cdot)\|_{H^{1}}+\|u^{\varepsilon}(\cdot)-v^{\varepsilon}(T_{\ell}\cdot)\|_{L^{\infty}}\to
0.$
Here $T_{\ell}\in SO(3)$ is the rotation on the $(x_{1},x_{2})$ plane of
$\frac{\pi}{\ell}$.
Theorem 1.1 says that
$|x^{i}_{\varepsilon}-y^{j}_{\varepsilon}|/|\ln\varepsilon|\to a_{i,j}>0$
($i,j=1,\cdots,\ell$) as $\varepsilon\to 0$. Hence Theorem 1.1 gives
segregated types of solutions for system (1.6) with the essential support of
the two components being segregated for $\varepsilon$ sufficiently small.
We also construct segregated vector solutions for the following three coupled
systems, which arise when one considers the propagation of pulses in a
$3$-core couplers with circular symmetry:
$\left\\{\begin{array}[]{ll}-\Delta
u+u=u^{3}-\varepsilon(v+\omega),&x\in\mathbb{R}^{3},\vspace{0.2cm}\\\ -\Delta
v+v=v^{3}-\varepsilon(u+\omega),&x\in\mathbb{R}^{3},\vspace{0.2cm}\\\
-\Delta\omega+\omega=\omega^{3}-\varepsilon(u+v),&x\in\mathbb{R}^{3}.\end{array}\right.$
(1.7)
###### Theorem 1.2.
For any integer $\ell\geq 2$, there exists $\varepsilon_{0}$ such that for
$\varepsilon\in(0,\varepsilon_{0})$, problem (1.7) has a solution
$(u^{\varepsilon},v^{\varepsilon},\omega^{\varepsilon})\in(H^{1}(\mathbb{R}^{3}))^{3}$
satisfying
$u^{\varepsilon}\sim\sum\limits_{j=1}^{\ell}U_{x_{\varepsilon}^{j}},\quad
v^{\varepsilon}\sim\sum\limits_{j=1}^{\ell}U_{y_{\varepsilon}^{j}},\quad\omega^{\varepsilon}\sim
U,$
where $x_{\varepsilon}^{j}$ and $y_{\varepsilon}^{j}$ are the same as those of
Theorem 1.1 if $\ell>2$, but for $\ell=2$
$r_{\varepsilon}=|\ln\varepsilon|+o(|\ln\varepsilon|),\quad\rho_{\varepsilon}=|\ln\varepsilon|+o(|\ln\varepsilon|).$
Moreover, as $\varepsilon\to 0^{+}$,
$\|u^{\varepsilon}(\cdot)-v^{\varepsilon}(T_{\ell}\cdot)\|_{H^{1}}+\|u^{\varepsilon}(\cdot)-v^{\varepsilon}(T_{\ell}\cdot)\|_{L^{\infty}}\to
0.$
###### Remark 1.3.
The segregation nature of these solutions are demonstrated from the
$L^{\infty}$ estimates in the theorems and will be more clear in Propositions
3.1 and 4.1 stated later after we find a good approximate solution and fix the
notations. Roughly speaking, as $\varepsilon\to 0$, the segregated solutions
may have a large number of bumps near infinity while the locations of the
bumps for $u$ and $v$ have an angular shift.
###### Remark 1.4.
In [4], to guarantee the existence of the solutions, the side of the polytope
should be greater than the radius, which implies that the number of the
solutions cannot be very large (at least in two dimensional case). In our
results, the number of the bumps can be very large, and the energy of the
solutions can become so large as we expected. Moreover, all the bumps are
positive, which implies that these solutions bifurcate from the symmetric
state at $\varepsilon=0$. Hence our results are in striking contrast with
those of [4].
###### Remark 1.5.
Our argument also works well for the following more general problems in
various dimensional case
$\left\\{\begin{array}[]{ll}-\Delta
u_{j}+u_{j}=|u_{j}|^{p-2}u_{j}-\varepsilon\sum\limits_{i\neq
j}^{N}u_{i},&x\in\mathbb{R}^{d},\vspace{0.2cm}\\\ u_{j}\in
H^{1}(\mathbb{R}^{d}),\quad j=1,\cdots,N.\end{array}\right.$
Here $N=2,3$, $d>1$, and $2<p<2^{*}$, where $2^{*}=2d/(d-2)$ if $d\geq 3$ and
$2^{*}=+\infty$ if $d=2$. We point out that our results are most likely wrong
for $d=1$, which is verified by the numerical computation in [1].
To prove the main results, we will employ the well-known Lyapunov-Schmidt
reduction (see, e.g., [19]) to glue the functions $U_{x^{j}}$ (or $U_{y^{j}}$)
($j=1,\cdots,\ell$). In performing this technique, to find critical points of
the reduced functionals, a basic requirement is that the error terms of the
functionals, which come from the finite dimensional reduction, should be of
higher order small data of the main terms in the reduced functionals. However,
in our linearly coupled systems, different from the nonlinearly coupled ones
(see, e.g., [13] and [18]), if we choose $(U,U)$ as an approximate solution,
the error terms from the linear coupling dominate the main terms (which are
generated from the interaction between the neighbor bumps) of the reduced
functionals. To overcome this difficulty, we should modify another approximate
solution $(U,0)$. This idea is essentially from [4], where an approximate
solution $(U_{\varepsilon},V_{\varepsilon})$ bifurcates from $(U,0)$. However,
comparing with [4], we encounter two more problems. Firstly, we need a new
approximate solution and a precise estimate on it. To this end, we will make a
modification on $(U,0)$ carefully by using the reduction technique (see
section 2). This procedure provides us a more accurate approximate solution
with required estimate. Secondly, after performing a second reduction, we need
to solve a two-dimensional critical point problem, which requires us to choose
a very delicate domain and make a precise analysis on the reduced functionals.
So, we need a very accurate estimate on the energy of the reduced functional,
which also needs the help of the approximate solution. Hence, here we will
perform the reduction twice and deal with more complicated reduced
functionals.
To find vector solutions with two components having the prescribed number of
bumps, we will employ the idea proposed by Wei and Yan in [24], where
infinitely many positive solutions were constructed for single Schrödinger
equations. This idea is also effective in finding infinitely many non-radial
positive solutions for semilinear elliptic problems with critical or super-
critical Sobolev growth (see, for example, [25, 26, 27]) and Schrödinger
systems with nonlinear coupling (see, for example, [18]).
This paper is organized as follows. In section 2, we will perform a reduction
argument for the first time and modify the vector function $(U,0)$ so that we
can get an accurate approximate solution and a precise estimate on it. In
section 3, using the approximate solution, we will formulate a more precise
version of the main results which give more precise descriptions about the
segregated character of the solutions. We will also carry out the reduction
for the second time to a finite two-dimensional setting and prove Theorem 1.1.
The study of existence of segregated solutions for a system coupled by three
nonlinear Schrödinger equations will be briefly discussed in section 4 by
using our framework of methods. We conclude with the energy expansion in the
appendix.
## 2\. An approximate solution
In this section, to look for a proper approximate vector solution, we need to
modify $(U,0)$. Let $H^{1}_{r}(\mathbb{R}^{3})$ and
$L^{2}_{r}(\mathbb{R}^{3})$ denote the corresponding spaces of radial
functions. For $(u,v)\in H^{1}_{r}(\mathbb{R}^{3})\times
H^{1}_{r}(\mathbb{R}^{3})$, we define
$\|(u,v)\|=\|u\|_{H^{1}(\mathbb{R}^{3})}+\|v\|_{H^{1}(\mathbb{R}^{3})}$.
Solving in $H^{1}_{r}(\mathbb{R}^{3})\times H^{1}_{r}(\mathbb{R}^{3})$ the
equations
$\left\\{\begin{array}[]{ll}-\Delta
u_{1}+u_{1}-3U^{2}u_{1}=0,&x\in\mathbb{R}^{3},\vspace{0.2cm}\\\
-\Delta\tilde{v}_{1}+\tilde{v}_{1}=-U,&x\in\mathbb{R}^{3},\end{array}\right.$
we get $u_{1}=0$ and $\tilde{v}_{1}<0$. Let $c(x)\in
H^{1}_{r}(\mathbb{R}^{3})$ satisfy
$-\Delta c(x)+c(x)=\tilde{v}_{1}^{3},$
then $v_{1}=\tilde{v}_{1}+\varepsilon^{2}c(x)$ solves
$-\Delta v_{1}+v_{1}=-U+\varepsilon^{2}\tilde{v}_{1}^{3}.$
Now for $k\geq 2$, by the Fredholm Alternative Theorem we can define
$(u_{k},v_{k})\in H^{1}_{r}(\mathbb{R}^{3})\times H^{1}_{r}(\mathbb{R}^{3})$
by solving
$\left\\{\begin{array}[]{ll}-\Delta
u_{k}+u_{k}-3U^{2}u_{k}=-kv_{k-1},&x\in\mathbb{R}^{3},\vspace{0.2cm}\\\
-\Delta v_{k}+v_{k}=-ku_{k-1},&x\in\mathbb{R}^{3}.\end{array}\right.$ (2.1)
We can also see that $v_{2}=0$.
###### Remark 2.1.
Here we execute the second modification by defining $v_{1}$ so that the norm
of the error terms in $H^{1}(\mathbb{R}^{3})\times H^{1}(\mathbb{R}^{3})$ can
be dominated by $C\varepsilon^{4}$ (see Proposition 2.2 later).
We want to find suitable $(w(x),h(x))\in H^{1}_{r}(\mathbb{R}^{3})\times
H^{1}_{r}(\mathbb{R}^{3})$ such that
$(U_{\varepsilon},\,v_{\varepsilon})=:\Bigl{(}U+\sum\limits_{i=1}^{4}\frac{\varepsilon^{i}}{i!}u_{i}+\varepsilon^{4}w,\,\sum\limits_{i=1}^{4}\frac{\varepsilon^{i}}{i!}v_{i}+\varepsilon^{4}h\Bigr{)}$
(2.2)
solves problem (1.6).
Inserting (2.2) into (1.6) and employing (2.1), we find
$\left\\{\begin{array}[]{ll}-\Delta
w+w-3U^{2}w=\displaystyle\frac{H_{\varepsilon}(u_{2},u_{3},u_{4},v_{4},U)}{\varepsilon^{4}}+l_{\varepsilon}(h,w)+\frac{R_{\varepsilon}(\varepsilon^{4}w)}{\varepsilon^{4}},\vspace{0.2cm}\\\
-\Delta
h+h=\displaystyle\frac{\bar{H}_{\varepsilon}(v_{1},v_{3},v_{4},u_{4})}{\varepsilon^{4}}+\bar{l}_{\varepsilon}(h,w)+\displaystyle\frac{\bar{R}_{\varepsilon}(\varepsilon^{4}h)}{\varepsilon^{4}},\end{array}\right.$
(2.3)
where
$\displaystyle
H_{\varepsilon}(u_{2},u_{3},u_{4},v_{4},U)=\Bigl{(}U+\displaystyle\sum\limits_{i=2}^{4}\frac{\varepsilon^{i}}{i!}u_{i}\Bigr{)}^{3}-U^{3}-3U^{2}\sum\limits_{i=2}^{4}\frac{\varepsilon^{i}}{i!}u_{i}-\frac{\varepsilon^{5}}{4!}v_{4},$
$\displaystyle
l_{\varepsilon}(w,h)=3\Bigl{(}\Bigl{(}U+\displaystyle\sum\limits_{i=2}^{4}\frac{\varepsilon^{i}}{i!}u_{i}\Bigr{)}^{2}-U^{2}\Bigr{)}w-\varepsilon
h,$ $\displaystyle
R_{\varepsilon}(\varepsilon^{4}w)=3\Bigl{(}U+\sum\limits_{i=2}^{4}\frac{\varepsilon^{i}}{i!}u_{i}\Bigr{)}(\varepsilon^{4}w)^{2}+(\varepsilon^{4}w)^{3},$
$\displaystyle\frac{\bar{H}_{\varepsilon}(v_{1},v_{3},v_{4},u_{4})}{\varepsilon^{4}}=\Bigl{(}\sum\limits_{i=1}^{4}\displaystyle\frac{\varepsilon^{i}}{i!}v_{i}\Bigr{)}^{3}-\varepsilon^{3}\tilde{v}_{1}^{3}-\frac{\varepsilon^{5}}{4!}u_{4},$
$\displaystyle\bar{l}_{\varepsilon}=-\varepsilon
w+3\Bigl{(}\displaystyle\sum\limits_{i=1}^{4}\frac{\varepsilon^{i}}{i!}v_{i}\Bigr{)}^{2}h,$
$\displaystyle\bar{R}_{\varepsilon}(\varepsilon^{4}h)=3\Bigl{(}\displaystyle\sum\limits_{i=1}^{4}\frac{\varepsilon^{i}}{i!}v_{i}\Bigr{)}(\varepsilon^{4}h)^{2}+(\varepsilon^{4}h)^{3}.$
Direct calculation yields that
$\displaystyle\Bigl{|}\frac{H_{\varepsilon}(u_{2},u_{3},u_{4},v_{4},U)}{\varepsilon^{4}}\Bigl{|}$
$\displaystyle\leq$ $\displaystyle C,$
$\displaystyle\frac{\bar{H}_{\varepsilon}(v_{1},v_{3},v_{4},u_{4})}{\varepsilon^{4}}$
$\displaystyle=$
$\displaystyle\Bigl{(}\Bigl{(}\sum\limits_{i=1}^{4}\displaystyle\frac{\varepsilon^{i}}{i!}v_{i}\Bigr{)}^{3}-\varepsilon^{3}\tilde{v}_{1}^{3}-\frac{\varepsilon^{5}}{4!}u_{4}\Bigr{)}\Bigl{/}\varepsilon^{4},$
$\displaystyle=$
$\displaystyle\Bigl{(}\Bigl{(}\sum\limits_{i=1}^{4}\displaystyle\frac{\varepsilon^{i}}{i!}v_{i}\Bigr{)}^{3}-\varepsilon^{3}v_{1}^{3}+\varepsilon^{3}(v_{1}^{3}-\tilde{v}_{1}^{3})-\frac{\varepsilon^{5}}{4!}u_{4}\Bigr{)}\Bigl{/}\varepsilon^{4}$
$\displaystyle=$ $\displaystyle O(\varepsilon),$
where we have used the fact $v_{2}=0$ and
$|v_{1}-\tilde{v}_{1}|=O(\varepsilon^{2})$.
Since the kernel of operator
$\mathbb{L}\left(\begin{array}[]{ll}w\vspace{0.2cm}\\\
h\end{array}\right)=\left(\begin{array}[]{ll}-\Delta
w+w-3U^{2}w\vspace{0.2cm}\\\ -\Delta
h+h\end{array}\right):\,\,H^{1}_{r}(\mathbb{R}^{3})\times
H^{1}_{r}(\mathbb{R}^{3})\to L^{2}_{r}(\mathbb{R}^{3})\times
L^{2}_{r}(\mathbb{R}^{3})$
is $\\{(0,0)\\}$ in $H^{1}_{r}(\mathbb{R}^{3})\times
H^{1}_{r}(\mathbb{R}^{3})$, we know that the operator $\mathbb{L}$ has bounded
inverse in $H^{1}_{r}(\mathbb{R}^{3})\times H^{1}_{r}(\mathbb{R}^{3})$.
Define
$\left(\begin{array}[]{ll}\bar{w}\vspace{0.2cm}\\\
\bar{h}\end{array}\right)=\mathbb{L}^{-1}\left(\begin{array}[]{ll}\displaystyle\frac{H_{\varepsilon}(u_{2},u_{3},u_{4},v_{4},U)}{\varepsilon^{4}}+l_{\varepsilon}(h,w)+\frac{R_{\varepsilon}(\varepsilon^{4}w)}{\varepsilon^{4}}\vspace{0.2cm}\\\
\displaystyle\frac{\bar{H}_{\varepsilon}(v_{1},v_{3},v_{4},u_{4})}{\varepsilon^{4}}+\bar{l}_{\varepsilon}(h,w)+\frac{\bar{R}_{\varepsilon}(\varepsilon^{4}h)}{\varepsilon^{4}}\end{array}\right)=:\mathbb{A}\left(\begin{array}[]{ll}w\vspace{0.2cm}\\\
h\end{array}\right)$
and the set
$\mathbb{S}=\\{(w,h)\in H^{1}_{r}(\mathbb{R}^{3})\times
H^{1}_{r}(\mathbb{R}^{3}):\,\,\|(w,\,h))\|\leq|\varepsilon|^{-\sigma}\\},$
where $\sigma>0$ is sufficiently small.
Then by direct calculation, we find for
$(w,h),\,(w_{1},h_{1}),\,(w_{2},h_{2})\in\mathbb{S}$,
$\displaystyle\|(\bar{w},\,\bar{h})\|\leq
C(1+\varepsilon)\leq|\varepsilon|^{-\sigma},$
$\displaystyle\|(\bar{w}_{1}-\bar{w}_{2},\bar{h}_{1}-\bar{h}_{2})\|=\|\mathbb{A}(w_{1}-w_{2},h_{1}-h_{2})\|$
$\displaystyle\hskip
110.96556pt\leq|\varepsilon|\|(w_{1}-w_{2},h_{1}-h_{2})\|<\frac{1}{2}\|(w_{1}-w_{2},h_{1}-h_{2})\|.$
Therefore, the operator $\mathbb{A}$ maps $\mathbb{S}$ into $\mathbb{S}$ and
is a contraction map. So, by the contraction mapping theorem, there exists
$(w,h)\in\mathbb{S}$, such that $(w,h)=\mathbb{A}(w,h)$. Direct computation
yields
$\displaystyle\Bigl{|}\int_{\mathbb{R}^{3}}\frac{H_{\varepsilon}(u_{2},u_{3},u_{4},v_{4},U)}{\varepsilon^{4}}\varphi+\displaystyle\frac{\bar{H}_{\varepsilon}(v_{1},v_{3},v_{4},u_{4})}{\varepsilon^{4}}\psi\Bigr{|}$
$\displaystyle\leq$ $\displaystyle
C\|(\varphi,\psi)\|,\,\,\forall\,\,(\varphi,\psi)\in
H^{1}_{r}(\mathbb{R}^{3})\times H^{1}_{r}(\mathbb{R}^{3}).$
As a result, we see
$\|(w,h)\|\leq C\varepsilon^{4}.$ (2.4)
Now we consider the asymptotic behavior of $u_{i},v_{i},(i=1,\cdots,4)$ at
infinity. We claim that for any fixed small $\tau>0$, there exists a positive
constant $C$ depending on $\tau,u_{i},v_{i},(i=1,\cdots,4)$ such that
$|u_{i}(r)|+|v_{i}(r)|\leq
Ce^{-(1-\tau)r},\,\,(i=1,\cdots,4),\,\,\,\forall\,\,r>1.$ (2.5)
Indeed, by induction, we suppose $|v_{i-1}|\leq C_{i-1}e^{-(1-\tau)r}$. Since
$\displaystyle-\Delta e^{-(1-\tau)r}+e^{-(1-\tau)r}-3U^{2}e^{-(1-\tau)r}$
$\displaystyle=$
$\displaystyle\Bigl{(}1-(1-\tau)^{2}+\frac{N-1}{r}-3U^{2}\Bigr{)}e^{-(1-\tau)r},$
we can choose $\bar{C}_{i},R_{i}$ depending on $u_{i},\tau,i$ and $C_{i-1}$
such that $\bar{C}_{i}e^{-(1-\tau)r}$ is a super-solution of the first
equation of (2.1) on $\mathbb{R}^{3}\setminus B_{R_{i}}(0)$. By comparison
theory of elliptic equations, we conclude
$u_{i}\leq\bar{C}_{i}e^{-(1-\tau)r},\,\,\forall\,\,r\geq R_{i}.$
With the same argument, we can also prove that
$u_{i}\geq-\bar{C}_{i}e^{-(1-\tau)r},\,\,\forall\,\,r\geq R_{i}.$
Hence, we can choose $C_{i}$ depending on $u_{i},\tau,i,C_{i-1}$ such that
$|u_{i}(r)|\leq C_{i}e^{-(1-\tau)r},\,\,\forall\,\,r>1.$
Similarly, we can prove that $|\tilde{v}_{1}|\leq Ce^{-(1-\tau)r}$,
$|c(x)|\leq Ce^{-(1-\tau)r}$ and also $|v_{i}|\leq C_{i}e^{-(1-\tau)r}$ for
$r>1$.
The above results can be summarized as
###### Proposition 2.2.
There exists $\varepsilon_{0}>0$ such that for
$\varepsilon\in(-\varepsilon_{0},\,\varepsilon_{0})$, problem (1.6) has a
solution $(U_{\varepsilon},\,v_{\varepsilon})\in
H^{1}_{r}(\mathbb{R}^{3})\times H^{1}_{r}(\mathbb{R}^{3})$ satisfying
$U_{\varepsilon}\to U$, $v_{\varepsilon}\to 0$ in $H^{1}_{r}$ as
$\varepsilon\to 0$. Moreover,
$\begin{array}[]{ll}U_{\varepsilon}=U+\sum\limits_{i=2}^{4}\displaystyle\frac{\varepsilon^{i}}{i!}u_{i}+w,\,\,\,v_{\varepsilon}=\sum\limits_{i=1}^{4}\displaystyle\frac{\varepsilon^{i}}{i!}v_{i}+h.\end{array}$
(2.6)
Here
$\|(w,h)\|\leq\tilde{C}\varepsilon^{4},$ (2.7)
$\tilde{C}>0$ is independent of $\varepsilon$. $u_{i}$ and $v_{i}$ satisfy
$|u_{i}(r)|+|v_{i}(r)|\leq Ce^{-(1-\tau)r},\,\,\,\forall\,\,r>1,$ (2.8)
where $\tau>0$ is any fixed small constant, $C$ depends on
$\tau,u_{i},v_{i},\,(i=1,\cdots,4)$.
With the same argument we can also construct a solution for problem (1.7)
which is linearly coupled by three equations.
The main result is
###### Proposition 2.3.
There exists $\varepsilon_{0}>0$ such that for
$\varepsilon\in(-\varepsilon_{0},\,\varepsilon_{0})$, problem (1.7) has a
solution
$(U_{\varepsilon},\,v_{\varepsilon},\omega_{\varepsilon})\in(H^{1}_{r}(\mathbb{R}^{3}))^{3}$
satisfying $U_{\varepsilon}\to U$, $v_{\varepsilon}\to 0$ and
$\omega_{\varepsilon}\to 0$ in $H^{1}_{r}$ as $\varepsilon\to 0$. Moreover,
$\begin{array}[]{ll}U_{\varepsilon}=U+\sum\limits_{i=2}^{4}\displaystyle\frac{\varepsilon^{i}}{i!}u_{i}+w,\,\,\,v_{\varepsilon}=\sum\limits_{i=1}^{4}\displaystyle\frac{\varepsilon^{i}}{i!}v_{i}+h,\,\,\,\omega_{\varepsilon}=\sum\limits_{i=1}^{4}\displaystyle\frac{\varepsilon^{i}}{i!}\omega_{i}+g.\end{array}$
(2.9)
Here
$\|(w,h,g)\|=:\|w\|_{H^{1}(\mathbb{R}^{3})}+\|h\|_{H^{1}(\mathbb{R}^{3})}+\|g\|_{H^{1}(\mathbb{R}^{3})}\leq\bar{C}\varepsilon^{4},$
(2.10)
$\bar{C}>0$ is independent of $\varepsilon$. $u_{i},\,v_{i}$ and $\omega_{i}$
satisfy
$|u_{i}(r)|+|v_{i}(r)|+|\omega_{i}(r)|\leq
Ce^{-(1-\tau)r},\,\,\,\forall\,\,r>1,$ (2.11)
where $\tau>0$ is any fixed small constant, $C$ depends on
$\tau,u_{i},v_{i},\omega_{i}\,(i=1,\cdots,4)$.
###### Proof.
Solve
$\left\\{\begin{array}[]{ll}-\Delta
u_{1}+u_{1}-3U^{2}u_{1}=0,&x\in\mathbb{R}^{3},\vspace{0.2cm}\\\
-\Delta\tilde{v}_{1}+\tilde{v}_{1}=-U,&x\in\mathbb{R}^{3},\vspace{0.2cm}\\\
-\Delta\tilde{\omega}_{1}+\tilde{\omega}_{1}=-U,&x\in\mathbb{R}^{3},\end{array}\right.$
then $u_{1}=0$, $\tilde{v}_{1}\in H^{1}_{r}(\mathbb{R}^{3})$,
$\tilde{\omega}_{1}\in H^{1}_{r}(\mathbb{R}^{3})$. Let $c(x),\,d(x)\in
H^{1}_{r}(\mathbb{R}^{3})$ satisfy
$-\Delta c(x)+c(x)=\tilde{v}_{1}^{3},\,\,-\Delta
d(x)+d(x)=\tilde{\omega}_{1}^{3},$
we see that $v_{1}=\tilde{v}_{1}+\varepsilon^{2}c(x)$ and
$\omega_{1}=\tilde{\omega}_{1}+\varepsilon^{2}d(x)$ solve
$-\Delta
v_{1}+v_{1}=-U+\varepsilon^{2}\tilde{v}_{1}^{3},\,\,\,-\Delta\omega_{1}+\omega_{1}=-U+\varepsilon^{2}\tilde{\omega}_{1}^{3}.$
For $k\geq 2$, we can define
$(u_{k},v_{k},\omega_{k})\in(H^{1}_{r}(\mathbb{R}^{3}))^{3}$ by solving
$\left\\{\begin{array}[]{ll}-\Delta
u_{k}+u_{k}-3U^{2}u_{k}=-k(v_{k-1}+\omega_{k-1}),&x\in\mathbb{R}^{3},\vspace{0.2cm}\\\
-\Delta
v_{k}+v_{k}=-k(u_{k-1}+\omega_{k-1}),&x\in\mathbb{R}^{3},\vspace{0.2cm}\\\
-\Delta\omega_{k}+\omega_{k}=-k(u_{k-1}+v_{k-1}),&x\in\mathbb{R}^{3}.\end{array}\right.$
(2.12)
Proceeding as we prove Proposition 2.2, we can find
$(w,h,g)\in(H^{1}_{r}(\mathbb{R}^{N}))^{3}$ such that (2.10) and (2.11) hold
true and $(U_{\varepsilon},v_{\varepsilon},\omega_{\varepsilon})$ defined by
(2.9) satisfies problem (1.7). ∎
## 3\. Segregated vector solutions for 2 coupled Schrödinger system
We will use $(U_{\varepsilon},v_{\varepsilon})$ to construct multi-bump
solutions for (1.6). It follows from Proposition 2.2 that
$(U_{\varepsilon},v_{\varepsilon})$ has the form
$\begin{array}[]{ll}U_{\varepsilon}=U+\varepsilon^{2}p_{\varepsilon}(r)+w,\,\,\,v_{\varepsilon}=\varepsilon
q_{\varepsilon}(r)+h,\end{array}$ (3.1)
where
$p_{\varepsilon}(r)\leq Ce^{-(1-\tau)r},\,\,q_{\varepsilon}(r)\leq
Ce^{-(1-\tau)r},\,\,\,\|(w,h)\|\leq C\varepsilon^{4}.$
Here $C$ is independent of $\varepsilon$, and $\tau>0$ is defined in (2.8).
For any integer $\ell\geq 2$, set
$m=2\sin\frac{\pi}{\ell},\,\,n=\sqrt{2\Bigl{(}1-\cos\frac{\pi}{\ell}\Bigr{)}}.$
Then it can be easily check that
$m>n>0,\quad 2<\frac{m}{m-n}<4.$
Let $x^{j}$ and $y^{j}$ be defined by (1.4) and (1.5) respectively. In this
section, we assume
$(r,\rho)\in\mathcal{D}_{\varepsilon}\times\mathcal{D}_{\varepsilon}=:\Bigl{[}\frac{|\ln\varepsilon|}{m-n+\frac{\mu\ln|\ln\varepsilon|}{|\ln\varepsilon|}},\,\frac{|\ln\varepsilon|}{m-n}\Bigr{]}\times\Bigl{[}\frac{|\ln\varepsilon|}{m-n+\frac{\mu\ln|\ln\varepsilon|}{|\ln\varepsilon|}},\,\frac{|\ln\varepsilon|}{m-n}\Bigr{]},$
(3.2)
where the constant $\mu>m-n$. For any function $W:\mathbb{R}^{3}\to\mathbb{R}$
and $\xi\in\mathbb{R}^{3}$, we define $W_{\xi}=W(x-\xi)$. Set
$U_{\varepsilon,r}=\sum\limits_{i=1}^{\ell}U_{\varepsilon,x^{i}},\,\,v_{\varepsilon,r}=\sum\limits_{i=1}^{\ell}v_{\varepsilon,x^{i}},\,\,U_{\varepsilon,\rho}=\sum\limits_{i=1}^{\ell}U_{\varepsilon,y^{i}},\,\,v_{\varepsilon,\rho}=\sum\limits_{i=1}^{\ell}v_{\varepsilon,y^{i}},$
and
$Y_{\varepsilon,j}=\frac{\partial U_{\varepsilon,x^{j}}}{\partial
r},\,\,Z_{\varepsilon,j}=\frac{\partial
U_{\varepsilon,y^{j}}}{\partial\rho},\quad j=1,\cdots,\ell.$
Define
$\begin{split}H_{s}=\bigl{\\{}u:\,&u\in H^{1}(\mathbb{R}^{3}),u\;\text{is even
in}\;x_{h},h=2,3,\\\
&u(r\cos\theta,r\sin\theta,x^{\prime})=u(r\cos(\theta+\frac{2\pi
j}{\ell}),r\sin(\theta+\frac{2\pi j}{\ell}),x^{\prime})\bigr{\\}},\end{split}$
and
$\mathbb{E}=\Bigl{\\{}(u,v)\in H_{s}\times
H_{s},\;\sum\limits_{j=1}^{\ell}\int_{\mathbb{R}^{3}}U_{\varepsilon,x^{j}}^{2}Y_{\varepsilon,j}u=0,\,\,\sum\limits_{j=1}^{\ell}\int_{\mathbb{R}^{3}}U_{\varepsilon,y^{j}}^{2}Z_{\varepsilon,j}v=0\Bigr{\\}}.$
(3.3)
To prove Theorem 1.1, it suffices to prove
###### Proposition 3.1.
For any integer $\ell\geq 2$, there exists $\varepsilon_{0}>0$ such that for
$\varepsilon\in(0,\,\varepsilon_{0})$, problem (1.6) has a solution $(u,v)$
with the form
$u=U_{\varepsilon,r}+v_{\varepsilon,\rho}+\varphi_{\varepsilon},\,\,v=U_{\varepsilon,\rho}+v_{\varepsilon,r}+\psi_{\varepsilon},$
where $(\varphi_{\varepsilon},\psi_{\varepsilon})\in\mathbb{E}$ satisfies
$\|(\varphi_{\varepsilon},\,\psi_{\varepsilon})\|=o(\varepsilon^{\frac{m}{m-n}})$.
Let
$\begin{array}[]{ll}I(u,v)=&\displaystyle\frac{1}{2}\int_{\mathbb{R}^{3}}\bigl{(}|\nabla
u|^{2}+u^{2}+|\nabla v|^{2}+v^{2}\bigl{)}\vspace{0.2cm}\\\
&-\displaystyle\frac{1}{4}\int_{\mathbb{R}^{3}}\bigl{(}u^{4}+v^{4}\bigl{)}+\varepsilon\int_{\mathbb{R}^{3}}uv,\quad(u,v)\in
H_{s}\times H_{s},\end{array}$
and
$J(\varphi,\psi)=I(U_{\varepsilon,r}+v_{\varepsilon,\rho}+\varphi,U_{\varepsilon,\rho}+v_{\varepsilon,r}+\psi).$
Expand $J(\varphi,\psi)$ as follows:
$J(\varphi,\psi)=J(0,0)-l(\varphi,\psi)+\frac{1}{2}\bar{L}(\varphi,\psi)-R(\varphi,\psi),\quad(\varphi,\psi)\in\mathbb{E},$
(3.4)
where
$\begin{array}[]{ll}l(\varphi,\psi)=&\displaystyle\int_{\mathbb{R}^{3}}\bigl{(}(U_{\varepsilon,r}+v_{\varepsilon,\rho})^{3}-\sum\limits_{j=1}^{\ell}U_{\varepsilon,x^{j}}^{3}-\sum\limits_{j=1}^{\ell}v_{\varepsilon,y^{j}}^{3}\bigr{)}\varphi\vspace{0.2cm}\\\
&+\displaystyle\int_{\mathbb{R}^{3}}\bigl{(}(U_{\varepsilon,\rho}+v_{\varepsilon,r})^{3}-\sum\limits_{j=1}^{\ell}U_{\varepsilon,y^{j}}^{3}-\sum\limits_{j=1}^{\ell}v_{\varepsilon,x^{j}}^{3}\bigr{)}\psi\end{array}$
$\begin{split}\bar{L}(\varphi,\psi)=&\int_{\mathbb{R}^{3}}\bigl{(}|\nabla\varphi|^{2}+\varphi^{2}-3(U_{\varepsilon,r}+v_{\varepsilon,\rho})^{2}\varphi^{2}\bigr{)}\vspace{0.2cm}\\\
&+\int_{\mathbb{R}^{3}}\bigl{(}|\nabla\psi|^{2}+\psi^{2}-3(U_{\varepsilon,\rho}+v_{\varepsilon,r})^{2}\psi^{2}\bigr{)}+2\varepsilon\int_{\mathbb{R}^{3}}\varphi\psi,\end{split}$
and
$R(\varphi,\psi)=\int_{\mathbb{R}^{3}}\bigl{(}(U_{\varepsilon,\rho}+v_{\varepsilon,r})\varphi^{3}+(U_{\varepsilon,\rho}+v_{\varepsilon,r})\psi^{3}\bigr{)}+\frac{1}{4}\int_{\mathbb{R}^{N}}(\varphi^{4}+\psi^{4}).$
###### Remark 3.2.
Here, in the expression of the linear part $l(\varphi,\psi)$, there are no
terms from the coupled term $\varepsilon\int_{\mathbb{R}^{3}}uv$ since we use
$(U_{\varepsilon},\,v_{\varepsilon})$ to construct the vector solutions. We
will see later in the proof of Proposition 3.1 that this choice of the
approximate solution guarantees that the error terms of the reduced functional
are dominated by $\varepsilon^{\frac{m+\sigma}{m-n}}$, which is of higher
order small datum of the main terms. However, if we use $(U,U)$ as an
approximate solution, then in the expression of $l(\varphi,\psi)$, the terms
from the coupling like
$\varepsilon\int_{\mathbb{R}^{N}}(\sum_{j=1}^{\ell}U_{x^{j}})\psi+\varepsilon\int_{\mathbb{R}^{3}}(\sum_{j=1}^{\ell}U_{y^{j}})\varphi$
will appear, which implies $\|l(\varphi,\psi)\|=O(\varepsilon)$. Hence the
error terms of the reduced functional are of order $O(\varepsilon^{2})$, which
will dominate the main terms and we have no way to solve the reduced
functional.
It is easy to check that $\bar{L}(\varphi,\psi)$ can be generated by a bounded
linear operator $L$ from $\mathbb{E}$ to $\mathbb{E}$, which is defined as
$\displaystyle\bigl{\langle}L(u,v),(\varphi,\psi)\bigr{\rangle}$
$\displaystyle=$ $\displaystyle\int_{\mathbb{R}^{3}}\bigl{(}\nabla
u\nabla\varphi+u\varphi-3(U_{\varepsilon,r}+v_{\varepsilon,\rho})^{2}u\varphi\bigr{)}$
$\displaystyle+\int_{\mathbb{R}^{3}}\bigl{(}\nabla
v\nabla\psi+v\psi-3(U_{\varepsilon,\rho}+v_{\varepsilon,r})^{2}v\psi\bigr{)}+\varepsilon\int_{\mathbb{R}^{3}}(u\psi+v\varphi).$
Now, we discuss the invertibility of $L$.
###### Lemma 3.3.
There exists $\varepsilon_{0}>0$, such that for
$\varepsilon\in(0,\,\varepsilon_{0})$, there is a constant $\varrho>0$,
independent of $\varepsilon$, satisfying that for any
$(r,\rho)\in\mathcal{D}_{\varepsilon}\times\mathcal{D}_{\varepsilon}$,
$\|L(u,v)\|\geq\varrho\|(u,v)\|,\quad(u,v)\in\mathbb{E}.$
###### Proof.
Suppose to the contrary that there are $\varepsilon_{n}\to 0^{+}$ (as
$n\to+\infty$),
$(r_{n},\rho_{n})\in\mathcal{D}_{\varepsilon_{n}}\times\mathcal{D}_{\varepsilon_{n}}$,
and $(u_{n},v_{n})\in\mathbb{E}$, with
$\bigl{\langle}L(u_{n},v_{n}),(\varphi,\psi)\bigr{\rangle}=o_{n}(1)\|(u_{n},v_{n})\|\|(\varphi,\psi)\|,\quad\forall\;(\varphi,\psi)\in\mathbb{E}.$
(3.5)
We may assume that $\|(u_{n},v_{n})\|=1$. We see from (3.5),
$\begin{array}[]{ll}&\displaystyle\int_{\mathbb{R}^{3}}\bigl{(}\nabla
u_{n}\nabla\varphi+u_{n}\varphi-3(U_{\varepsilon_{n},r_{n}}+v_{\varepsilon_{n},\rho_{n}})^{2}u_{n}\varphi\bigr{)}\vspace{0.2cm}\\\
&+\displaystyle\int_{\mathbb{R}^{3}}\bigl{(}\nabla
v_{n}\nabla\psi+v_{n}\psi-3(U_{\varepsilon_{n},\rho_{n}}+v_{\varepsilon_{n},r_{n}})^{2}v_{n}\psi\bigr{)}\vspace{0.2cm}\\\
&+\varepsilon_{n}\displaystyle\int_{\mathbb{R}^{3}}(u_{n}\psi+v_{n}\varphi)=o_{n}(1)\|(\varphi,\psi)\|,\,\,\forall\,\,(\varphi,\psi)\in\mathbb{E}.\end{array}$
(3.6)
In particular,
$\begin{array}[]{ll}&\displaystyle\int_{\mathbb{R}^{3}}\bigl{(}|\nabla
u_{n}|^{2}+u_{n}^{2}-3(U_{\varepsilon_{n},r_{n}}+v_{\varepsilon_{n},\rho_{n}})^{2}u_{n}^{2}\bigr{)}\vspace{0.2cm}\\\
&+\displaystyle\int_{\mathbb{R}^{3}}\bigl{(}|\nabla
v_{n}|^{2}+v_{n}^{2}-3(U_{\varepsilon_{n},\rho_{n}}+v_{\varepsilon_{n},r_{n}})^{2}v_{n}^{2}\bigr{)}+2\varepsilon_{n}\displaystyle\int_{\mathbb{R}^{3}}u_{n}v_{n}=o_{n}(1),\end{array}$
(3.7)
and
$\displaystyle\int_{\mathbb{R}^{3}}\bigl{(}|\nabla
u_{n}|^{2}+u_{n}^{2}+|\nabla v_{n}|^{2}+v_{n}^{2}\bigr{)}=1.$ (3.8)
Let
$\bar{u}_{n}(x)=u_{n}(x-x^{1}),\,\,\,\bar{v}_{n}(x)=v_{n}(x-y^{1}).$
We may assume the existence of $u$, such that as $n\to+\infty$,
$\bar{u}_{n}\to u,\quad\text{weakly in}\;H^{1}_{loc}(\mathbb{R}^{3}),\hskip
28.45274pt\bar{u}_{n}\to u,\quad\text{strongly
in}\;L^{2}_{loc}(\mathbb{R}^{3}).$
Moreover, $u$ is even in $x_{h}$, $h=2,3.$
By symmetry, we see
$\int_{\mathbb{R}^{3}}U_{\varepsilon_{n},x_{n}^{1}}^{2}Y_{\varepsilon_{n},1}u_{n}=0.$
It follows from $(U_{\varepsilon_{n}},v_{\varepsilon_{n}})\to(U,0)$ in
$H^{1}(\mathbb{R}^{3})\times H^{1}(\mathbb{R}^{3})$ that
$\displaystyle\Bigl{|}\int_{\mathbb{R}^{3}}U_{\varepsilon_{n},x_{n}^{1}}^{2}Y_{\varepsilon_{n},1}u_{n}-\int_{\mathbb{R}^{3}}U_{x_{n}^{1}}^{2}\frac{\partial
U_{x_{n}^{1}}}{\partial r_{n}}u_{n}\Bigl{|}$ $\displaystyle\leq$
$\displaystyle\Bigl{|}\int_{\mathbb{R}^{3}}(U_{\varepsilon_{n},x_{n}^{1}}^{2}-U_{x_{n}^{1}}^{2})Y_{\varepsilon_{n},1}u_{n}\Bigl{|}+\Bigl{|}\int_{\mathbb{R}^{3}}U_{x_{n}^{1}}^{2}\Bigl{(}Y_{\varepsilon_{n},1}-\frac{\partial
U_{x_{n}^{1}}}{\partial r_{n}}\Bigr{)}u_{n}\Bigl{|}\to 0,\,\,\,(n\to+\infty).$
Hence
$\int_{\mathbb{R}^{3}}U_{x_{n}^{1}}^{2}\frac{\partial U_{x_{n}^{1}}}{\partial
r_{n}}u_{n}\to 0,$
which implies
$\int_{\mathbb{R}^{3}}U^{2}\frac{\partial U}{\partial x_{1}}u=0.$ (3.9)
Let $\varphi\in C_{0}^{\infty}(B_{R}(0))$ be even in $x_{h}$, $h=2,3$. Define
$\varphi_{n}(x)=:\varphi(x-x^{1})\in C_{0}^{\infty}(B_{R}(x^{1}))$. We may
identify $\varphi_{n}(x)$ as elements in $H_{s}$ by redefining the values
outside $B_{R}(x^{1})$ with the symmetry.
From the fact that $U_{\varepsilon_{n}}\to U$ and $v_{\varepsilon_{n}}\to 0$
in $H^{1}(\mathbb{R}^{3})$, we deduce
$\begin{array}[]{ll}&\displaystyle\int_{\mathbb{R}^{3}}(U_{\varepsilon_{n},r_{n}}+v_{\varepsilon_{n},\rho_{n}})^{2}u_{n}\varphi_{n}\vspace{0.2cm}\\\
=&\displaystyle\int_{\mathbb{R}^{3}}(U^{2}_{\varepsilon_{n},r_{n}}+2U_{\varepsilon_{n},r_{n}}v_{\varepsilon_{n},\rho_{n}}+v_{\varepsilon_{n},\rho_{n}}^{2})u_{n}\varphi_{n}=\displaystyle\int_{\mathbb{R}^{3}}U^{2}u\varphi+o_{n}(1).\end{array}$
(3.10)
Then choosing $(\varphi,\psi)=(\varphi_{n},0)$ in (3.6) and considering
(3.10), we can use the argument in [24], to prove that $u$ solves
$-\Delta u+u-3U^{2}u=0,\hskip 28.45274ptx\in\mathbb{R}^{3}.$ (3.11)
Since we work in the space of functions which are even in $x_{2}$ and $x_{3}$,
we see $u=c\frac{\partial U}{\partial x_{1}}$ for some $c$, which implies that
$u=0$ since $u$ satisfies (3.9).
To deal with $v_{n}$, we first claim that for any $v(x)\in H_{s}$, $v(x)$ is
even with respect to the ray with an angle of $\pi/\ell$.
Indeed, suppose that $|(x_{1},x_{2})|=a$, then
$\displaystyle v(x)$ $\displaystyle=:$ $\displaystyle
v(a\cos(\frac{\pi}{\ell}+\theta),a\sin(\frac{\pi}{\ell}+\theta),x_{3})$
$\displaystyle=$ $\displaystyle
v(a\cos(\frac{\pi}{\ell}+\theta),-a\sin(\frac{\pi}{\ell}+\theta),x_{3})$
$\displaystyle=$ $\displaystyle
v(a\cos(-\frac{\pi}{\ell}-\theta),a\sin(-\frac{\pi}{\ell}-\theta),x_{3})$
$\displaystyle=$ $\displaystyle
v(a\cos(\frac{\pi}{\ell}-\theta),a\sin(\frac{\pi}{\ell}-\theta),x_{3}).$
Now as we deal with $u_{n}$, we can check
$\bar{v}_{n}\to 0,\quad\text{weakly in}\;H^{1}_{loc}(\mathbb{R}^{3}),\hskip
28.45274pt\bar{v}_{n}\to 0,\quad\text{strongly
in}\;L^{2}_{loc}(\mathbb{R}^{3}).$
Similar to (3.10), using the fact that $U_{\varepsilon_{n}}\to U$ and
$v_{\varepsilon_{n}}\to 0$ in $H^{1}(\mathbb{R}^{3})$ as $n\to+\infty$, we
deduce that
$\begin{array}[]{ll}&\displaystyle\int_{\mathbb{R}^{3}}(U_{\varepsilon_{n},r_{n}}+v_{\varepsilon_{n},\rho_{n}})^{2}u^{2}_{n}+\displaystyle\int_{\mathbb{R}^{3}}(U_{\varepsilon_{n},\rho_{n}}+v_{\varepsilon_{n},r_{n}})^{2}v^{2}_{n}\vspace{0.2cm}\\\
=&\displaystyle\int_{\mathbb{R}^{3}}\Bigl{(}\sum\limits_{j=1}^{\ell}U_{x^{j}}^{2}\Bigr{)}u_{n}^{2}+\displaystyle\int_{\mathbb{R}^{3}}\Bigl{(}\sum\limits_{j=1}^{\ell}U_{y^{j}}^{2}\Bigr{)}v_{n}^{2}+o_{n}(1).\end{array}$
(3.12)
Hence we find
$\begin{array}[]{ll}o_{n}(1)&=\displaystyle\int_{\mathbb{R}^{3}}\bigl{(}|\nabla
u_{n}|^{2}+u_{n}^{2}-3(U_{\varepsilon_{n},r_{n}}+v_{\varepsilon_{n},\rho_{n}})^{2}u_{n}^{2}\bigr{)}\vspace{0.2cm}\\\
&\hskip 14.22636pt+\displaystyle\int_{\mathbb{R}^{3}}\bigl{(}|\nabla
v_{n}|^{2}+v_{n}^{2}-3(U_{\varepsilon_{n},\rho_{n}}+v_{\varepsilon_{n},r_{n}})^{2}v_{n}^{2}\bigr{)}+2\varepsilon_{n}\displaystyle\int_{\mathbb{R}^{N}}u_{n}v_{n}\vspace{0.2cm}\\\
&=1+Ce^{-R},\end{array}$ (3.13)
which is impossible for large $n$ and large $R$.
As a result, we complete the proof. ∎
###### Lemma 3.4.
There is a constant $C>0$, independent of $\varepsilon$, such that
$\|R(\varphi,\psi)\|\leq
C\|(\varphi,\psi)\|^{3},\quad\|R^{\prime}(\varphi,\psi)\|\leq
C\|(\varphi,\psi)\|^{2},\quad\|R^{\prime\prime}(\varphi,\psi)\|\leq
C\|(\varphi,\psi)\|.$
###### Proof.
The proof can be completed by direct calculation and we omit it. ∎
Now we perform the finite-dimensional reduction procedure.
###### Proposition 3.5.
There exists $\varepsilon_{0}>0$ such that for
$\varepsilon\in(0,\,\varepsilon_{0})$, there is a $C^{1}$ map from
$\mathcal{D}_{\varepsilon}\times\mathcal{D}_{\varepsilon}$ to $H_{s}\times
H_{s}$: $(\varphi,\psi)=(\varphi(r,\rho),\psi(r,\rho))$, satisfying
$(\varphi,\psi)\in\mathbb{E}$, and
$J^{\prime}_{(\varphi,\psi)}(\varphi,\psi)=0,\quad\hbox{on}\,\,\,\mathbb{E}.$
Moreover, there is a constant $C>0$ independent of $\varepsilon$, such that
$\|(\varphi,\psi)\|\leq
C\Bigl{(}\frac{e^{-|x^{1}-x^{2}|}}{|x^{1}-x^{2}|}+\frac{e^{-|y^{1}-y^{2}|}}{|y^{1}-y^{2}|}+\varepsilon
e^{-(1-\tau)|x^{1}-y^{1}|}+\varepsilon^{4}\Bigr{)}.$ (3.14)
###### Proof.
It follows from the proof of Lemma 3.6 below, that $l(\varphi,\psi)$ is a
bounded linear functional in $\mathbb{E}$. Thus, there is an
$f_{\varepsilon}\in\mathbb{E}$, such that
$l(\varphi,\psi)=\bigl{\langle}f_{\varepsilon},(\varphi,\psi)\bigr{\rangle}.$
Thus, finding a critical point for $J(\varphi,\psi)$ in $\mathbb{E}$ is
equivalent to solving
$f_{\varepsilon}-L(\varphi,\psi)+R^{\prime}(\varphi,\psi)=0.$ (3.15)
By Lemma 3.3, $L$ is invertible. Thus, (3.15) can be rewritten as
$(\varphi,\psi)=A(\varphi,\psi)=:L^{-1}(f_{\varepsilon}+R^{\prime}(\varphi,\psi)).$
(3.16)
Set
$\begin{array}[]{ll}D=\Bigl{\\{}(\varphi,\psi):(\varphi,\psi)\in\mathbb{E},\|(\varphi,\psi)\|\leq\displaystyle\frac{e^{-(1-\sigma)|x^{1}-x^{2}|}}{|x^{1}-x^{2}|}&+\displaystyle\frac{e^{-(1-\sigma)|y^{1}-y^{2}|}}{|y^{1}-y^{2}|}\vspace{0.2cm}\\\
&+\varepsilon^{1-\sigma}e^{-(1-\tau)|x^{1}-y^{1}|}+\varepsilon^{4-\sigma}\Bigr{\\}},\end{array}$
where $\sigma>0$ is small.
From Lemma 3.4 and Lemma 3.6 below, for $\varepsilon$ small,
$\begin{array}[]{ll}\|A(\varphi,\psi)\|&\leq
C\|f_{\varepsilon}\|+C\|(\varphi,\psi)\|^{2}\vspace{0.2cm}\\\
&\leq\displaystyle\frac{e^{-(1-\sigma)|x^{1}-x^{2}|}}{|x^{1}-x^{2}|}+\displaystyle\frac{e^{-(1-\sigma)|y^{1}-y^{2}|}}{|y^{1}-y^{2}|}+\varepsilon^{1-\sigma}e^{-(1-\tau)|x^{1}-y^{1}|}+\varepsilon^{4-\sigma},\end{array}$
(3.17)
and
$\begin{split}\|A(\varphi_{1},\psi_{1})-A(\varphi_{2},\psi_{2})\|&=\|L^{-1}R^{\prime}(\varphi_{1},\psi_{1})-L^{-1}R^{\prime}(\varphi_{2},\psi_{2})\|\\\
&\leq
C\bigl{(}\|(\varphi_{1},\psi_{1})\|+\|(\varphi_{2},\psi_{2})\|\bigr{)}\|(\varphi_{1},\psi_{1})-(\varphi_{2},\psi_{2})\|\\\
&\leq\frac{1}{2}\|(\varphi_{1},\psi_{1})-(\varphi_{2},\psi_{2})\|.\end{split}$
Therefore, $A$ maps $D$ into $D$ and is a contraction map. So, there exists
$(\varphi,\psi)\in\mathbb{E}$, such that $(\varphi,\psi)=A(\varphi,\psi)$.
Moreover by (3.16), we have
$\|(\varphi,\psi)\|\leq
C\Bigl{(}\frac{e^{-|x^{1}-x^{2}|}}{|x^{1}-x^{2}|}+\frac{e^{-|y^{1}-y^{2}|}}{|y^{1}-y^{2}|}+\varepsilon
e^{-(1-\tau)|x^{1}-y^{1}|}+\varepsilon^{4}\Bigl{)}.$
∎
###### Lemma 3.6.
There is a constant $C>0$ independent of $\varepsilon$, such that
$\|f_{\varepsilon}\|\leq
C\Bigl{(}\frac{e^{-|x^{1}-x^{2}|}}{|x^{1}-x^{2}|}+\frac{e^{-|y^{1}-y^{2}|}}{|y^{1}-y^{2}|}+\varepsilon
e^{-(1-\tau)|x^{1}-y^{1}|}+\varepsilon^{4}\Bigl{)}.$
###### Proof.
We see
$\displaystyle\displaystyle\int_{\mathbb{R}^{3}}\bigl{(}(U_{\varepsilon,r}+v_{\varepsilon,\rho})^{3}-\sum\limits_{j=1}^{\ell}U_{\varepsilon,x^{j}}^{3}-\sum\limits_{j=1}^{\ell}v_{\varepsilon,y^{j}}^{3}\bigr{)}\varphi$
$\displaystyle=$
$\displaystyle\int_{\mathbb{R}^{3}}\Bigl{(}(\sum\limits_{j=1}^{\ell}U_{\varepsilon,x^{j}})^{3}-\sum\limits_{j=1}^{\ell}U_{\varepsilon,x^{j}}^{3}+(\sum\limits_{j=1}^{\ell}v_{\varepsilon,y^{j}})^{3}-\sum\limits_{j=1}^{\ell}v_{\varepsilon,y^{j}}^{3}+3U_{\varepsilon,r}^{2}v_{\varepsilon,\rho}+3U_{\varepsilon,r}v^{2}_{\varepsilon,\rho}\Bigr{)}\varphi$
$\displaystyle=$
$\displaystyle\int_{\mathbb{R}^{3}}\Bigl{(}3\sum\limits_{j\neq
i}^{\ell}U^{2}_{\varepsilon,x^{i}}U_{\varepsilon,x^{j}}+3\sum\limits_{j\neq
i}^{\ell}v_{\varepsilon,y^{i}}^{2}v_{\varepsilon,y^{j}}+3U_{\varepsilon,r}^{2}v_{\varepsilon,\rho}+3U_{\varepsilon,r}v^{2}_{\varepsilon,\rho}\Bigr{)}\varphi.$
It follows from Proposition 2.2, Proposition A.1 and Hölder inequality that
for $i\neq j$
$\displaystyle\Bigl{|}\int_{\mathbb{R}^{3}}U^{2}_{\varepsilon,x^{i}}U_{\varepsilon,x^{j}}\varphi\Bigl{|}$
$\displaystyle=$
$\displaystyle\Bigl{|}\int_{\mathbb{R}^{3}}(U_{x^{i}}+\varepsilon^{2}p_{\varepsilon}(|x-x^{i}|)+w(x-x^{i}))^{2}(U_{x^{j}}+\varepsilon^{2}p_{\varepsilon}(|x-x^{j}|)+w(x-x^{j}))\varphi\Bigl{|}$
$\displaystyle\leq$ $\displaystyle
C\Bigl{(}\frac{e^{-|x^{i}-x^{j}|}}{|x^{i}-x^{j}|}+\varepsilon
e^{-(1-\tau)|x^{i}-x^{j}|}+\varepsilon^{4}\Bigl{)}\|\varphi\|_{H^{1}(\mathbb{R}^{3})},$
$\displaystyle\Bigl{|}\int_{\mathbb{R}^{3}}v^{2}_{\varepsilon,y^{i}}v_{\varepsilon,y^{j}}\varphi\Bigl{|}$
$\displaystyle=$ $\displaystyle
C\Bigl{|}\int_{\mathbb{R}^{3}}(\varepsilon^{2}q_{\varepsilon}^{2}|x-y^{i}|+h^{2}(|x-y^{i}|))(\varepsilon
q_{\varepsilon}|x-y^{j}|+h(|x-y^{j}|))\varphi\Bigl{|}$ $\displaystyle\leq$
$\displaystyle
C(\varepsilon^{3}e^{-(1-3\tau)|y^{i}-y^{j}|}+\varepsilon^{4})\|\varphi\|_{H^{1}(\mathbb{R}^{3})},$
and
$\Bigl{|}\int_{\mathbb{R}^{3}}(3U_{\varepsilon,r}^{2}v_{\varepsilon,\rho}+3U_{\varepsilon,r}v^{2}_{\varepsilon,\rho})\varphi\Bigl{|}\leq
C\sum\limits_{i,j=1}^{\ell}(\varepsilon
e^{-(1-\tau)|x^{i}-y^{j}|}+\varepsilon^{4})\|\varphi\|_{H^{1}(\mathbb{R}^{3})}.$
Therefore,
$\displaystyle\Bigl{|}\displaystyle\int_{\mathbb{R}^{3}}\bigl{(}(U_{\varepsilon,r}+v_{\varepsilon,\rho})^{3}-\sum\limits_{j=1}^{\ell}U_{\varepsilon,x^{j}}^{3}-\sum\limits_{j=1}^{\ell}v_{\varepsilon,y^{j}}^{3}\bigr{)}\varphi\Bigl{|}$
$\displaystyle\leq$ $\displaystyle C\Bigl{(}\sum\limits_{i\neq
j}^{\ell}\frac{e^{-|x^{i}-x^{j}|}}{|x^{i}-x^{j}|}+\varepsilon\sum\limits_{i,j=1}^{\ell}e^{-(1-\tau)|x^{i}-y^{j}|}+\varepsilon^{4}\Bigl{)}\|\varphi\|_{H^{1}(\mathbb{R}^{3})}.$
Similarly,
$\displaystyle\Bigl{|}\displaystyle\int_{\mathbb{R}^{3}}\bigl{(}(U_{\varepsilon,\rho}+v_{\varepsilon,r})^{3}-\sum\limits_{j=1}^{\ell}U_{\varepsilon,y^{j}}^{3}-\sum\limits_{j=1}^{\ell}v_{\varepsilon,x^{j}}^{3}\bigr{)}\psi\Bigl{|}$
$\displaystyle\leq$ $\displaystyle C\Bigl{(}\sum\limits_{i\neq
j}^{\ell}\frac{e^{-|y^{i}-y^{j}|}}{|y^{i}-y^{j}|}+\varepsilon\sum\limits_{i,j=1}^{\ell}e^{-(1-\tau)|x^{i}-y^{j}|}+\varepsilon^{4}\Bigl{)}\|\psi\|_{H^{1}(\mathbb{R}^{3})}.$
As a result, we complete the proof. ∎
Now we are ready to prove Proposition 3.1. Let
$(\varphi_{r,\rho},\psi_{r,\rho})=(\varphi(r,\rho),\psi(r,\rho))$ be the map
obtained in Proposition 3.5. Define
$F(r,\rho)=I(U_{\varepsilon,r}+v_{\varepsilon,\rho}+\varphi_{r,\rho},U_{\varepsilon,\rho}+v_{\varepsilon,r}+\psi_{r,\rho}),\quad\forall\;(r,\rho)\in\mathcal{D}_{\varepsilon}\times\mathcal{D}_{\varepsilon}.$
With the same argument in [10, 19], we can easily check that for $\varepsilon$
sufficiently small, if $(r,\rho)$ is a critical point of $F(r,\rho)$, then
$(U_{\varepsilon,r}+v_{\varepsilon,\rho}+\varphi_{r,\rho},U_{\varepsilon,\rho}+v_{\varepsilon,r}+\psi_{r,\rho})$
is a critical point of $I$.
###### Proof of Proposition 3.1.
The boundedness of $L$ in $H_{s}\times H_{s}$ and Lemma 3.4 imply that
$\|L(\varphi_{r,\rho},\psi_{r,\rho})\|\leq
C\|(\varphi_{r,\rho},\psi_{r,\rho})\|,\quad|R(\varphi_{r,\rho},\psi_{r,\rho})|\leq
C\|(\varphi_{r,\rho},\psi_{r,\rho})\|^{3}.$
So, Proposition 3.5 and Lemma 3.6 combined by Proposition A.2 give
$\begin{split}F(r,\rho)=&I(U_{\varepsilon,r}+v_{\varepsilon,\rho},U_{\varepsilon,\rho}+v_{\varepsilon,r})-l(\varphi_{r,\rho},\psi_{r,\rho})+\frac{1}{2}\bigl{\langle}L(\varphi_{r,\rho},\psi_{r,\rho}),(\varphi_{r,\rho},\psi_{r,\rho})\bigr{\rangle}-R(\varphi_{r,\rho},\psi_{r,\rho})\\\
=&I(U_{\varepsilon,r}+v_{\varepsilon,\rho},U_{\varepsilon,\rho}+v_{\varepsilon,r})+O\bigl{(}\|f_{\varepsilon}\|\|(\varphi_{r,\rho},\psi_{r,\rho})\|+\|(\varphi_{r,\rho},\psi_{r,\rho})\|^{2}\bigr{)}\\\
=&\displaystyle\sum\limits_{j=1}^{\ell}I(U_{\varepsilon,x^{j}},v_{\varepsilon,x^{j}})+\displaystyle\sum\limits_{j=1}^{\ell}I(U_{\varepsilon,y^{j}},v_{\varepsilon,y^{j}})\\\
&-\displaystyle\sum\limits_{i<j}^{\ell}C_{ij}\frac{e^{-|x^{i}-x^{j}|}}{|x^{i}-x^{j}|}-\displaystyle\sum\limits_{i<j}^{\ell}C_{ij}\frac{e^{-|y^{i}-y^{j}|}}{|y^{i}-y^{j}|}+\varepsilon\displaystyle\sum\limits_{i,j=1}^{\ell}\bar{C}_{ij}e^{-|x^{i}-y^{j}|}\\\
&+O(\varepsilon
e^{-(1-\tau)|y^{1}-y^{2}|}+\varepsilon^{2}e^{-(1-\tau)|x^{1}-y^{1}|}+\varepsilon
e^{-(1-\tau)|x^{1}-x^{2}|}+\varepsilon^{4})\\\
&+O\Bigl{(}\frac{e^{-|x^{1}-x^{2}|}}{|x^{1}-x^{2}|}+\frac{e^{-|y^{1}-y^{2}|}}{|y^{1}-y^{2}|}+\varepsilon
e^{-(1-\tau)|x^{1}-y^{1}|}\Bigl{)}^{2},\end{split}$
where $\bar{C}_{ij}$ and $C_{ij}$ are those in Proposition A.2.
Recalling
$r,\rho\in\mathcal{D}_{\varepsilon}=:\Bigl{[}\frac{|\ln\varepsilon|}{m-n+\frac{\mu\ln|\ln\varepsilon|}{|\ln\varepsilon|}},\,\frac{|\ln\varepsilon|}{m-n}\Bigr{]},$
where $m=2\sin\frac{\pi}{\ell},\,n=\sqrt{2(1-\cos\frac{\pi}{\ell})}$,
$\mu>m-n>0$, and noting
$\frac{1}{m-n+\frac{\mu\ln|\ln\varepsilon|}{|\ln\varepsilon|}}=\frac{1}{m-n}-\frac{\mu}{(m-n)^{2}}\frac{\ln|\ln\varepsilon|}{|\ln\varepsilon|}+O\Bigl{(}\frac{\ln|\ln\varepsilon|}{|\ln\varepsilon|}\Bigl{)}^{2},$
(3.18)
we can check
$\displaystyle O(\varepsilon
e^{-(1-\tau)|y^{1}-y^{2}|}+\varepsilon^{2}e^{-(1-\tau)|x^{1}-y^{1}|}+\varepsilon
e^{-(1-\tau)|x^{1}-x^{2}|}+\varepsilon^{4})$
$\displaystyle+O\Bigl{(}\frac{e^{-|x^{1}-x^{2}|}}{|x^{1}-x^{2}|}+\frac{e^{-|y^{1}-y^{2}|}}{|y^{1}-y^{2}|}+\varepsilon
e^{-(1-\tau)|x^{1}-y^{1}|}\Bigl{)}^{2}$ $\displaystyle=$ $\displaystyle
O(\varepsilon^{\frac{m}{m-n}+\sigma}),$
where $\sigma>0$ is a small number such that $2<\frac{m}{m-n}+\sigma<4$ for
$\ell\geq 2$.
Hence, considering the symmetry again, we find
$\begin{split}F(r,\rho)=&\displaystyle\sum\limits_{j=1}^{\ell}I(U_{\varepsilon,x^{j}},v_{\varepsilon,x^{j}})+\displaystyle\sum\limits_{j=1}^{\ell}I(U_{\varepsilon,y^{j}},v_{\varepsilon,y^{j}})\\\
&-\displaystyle\sum\limits_{i<j}^{\ell}C_{ij}\frac{e^{-|x^{i}-x^{j}|}}{|x^{i}-x^{j}|}-\displaystyle\sum\limits_{i<j}^{\ell}C_{ij}\frac{e^{-|y^{i}-y^{j}|}}{|y^{i}-y^{j}|}+\varepsilon\displaystyle\sum\limits_{i,j=1}^{\ell}\bar{C}_{ij}e^{-|x^{i}-y^{j}|}+O(\varepsilon^{\frac{m}{m-n}+\sigma})\\\
=&C_{\varepsilon}+\ell\Bigl{(}\bar{C}\varepsilon
e^{-\sqrt{r^{2}+\rho^{2}-2r\rho\cos\frac{\pi}{\ell}}}-\frac{C}{mr}e^{-mr}-\frac{C}{m\rho}e^{-m\rho}\Bigl{)}+O(\varepsilon^{\frac{m}{m-n}+\sigma}),\end{split}$
(3.19)
where $C_{\varepsilon}$ depends on $\varepsilon$ but is independent of $r$ and
$\rho$, $C=C_{12},\,\bar{C}=\bar{C}_{11}$.
Now we prove that the maximizer of $F(r,\rho)$ in
$\mathcal{D}_{\varepsilon}\times\mathcal{D}_{\varepsilon}$ is an interior
point of $\mathcal{D}_{\varepsilon}\times\mathcal{D}_{\varepsilon}$. To this
end, we consider the function
$G(r,\rho)=\bar{C}\varepsilon
e^{-\sqrt{r^{2}+\rho^{2}-2r\rho\cos\frac{\pi}{\ell}}}-\frac{C}{mr}e^{-mr}-\frac{C}{m\rho}e^{-m\rho},\,\,\,r,\rho\in\mathcal{D}_{\varepsilon}.$
In order to check that $G(r,\rho)$ achieves maximum at some point
$(r_{0},\rho_{0})$ in the interior of
$\mathcal{D}_{\varepsilon}\times\mathcal{D}_{\varepsilon}$, we need to
estimate both the value of $G(r,\rho)$ on the boundary of
$\mathcal{D}_{\varepsilon}\times\mathcal{D}_{\varepsilon}$ and the value of
$G(r_{0},\rho_{0})$.
Set
$\check{r}=\frac{|\ln\varepsilon|}{m-n+\frac{\mu\ln|\ln\varepsilon|}{|\ln\varepsilon|}},\,\hat{r}=\frac{|\ln\varepsilon|}{m-n},$
and define
$\rho_{\theta}=\frac{|\ln\varepsilon|}{m-n+\frac{\mu\ln|\ln\varepsilon|}{|\ln\varepsilon|}\theta},\quad\,\theta\in[0,1].$
Then $\check{r}\leq\rho_{\theta}\leq\hat{r}$ for $\theta\in[0,1]$, and
$\rho_{\theta}=\frac{|\ln\varepsilon|}{m-n}-\frac{\mu\theta}{(m-n)^{2}}\ln|\ln\varepsilon|+O\Bigl{(}\frac{\ln^{2}|\ln\varepsilon|}{|\ln\varepsilon|}\Bigl{)},$
(3.20)
$\sqrt{\hat{r}^{2}+\rho_{\theta}^{2}-2\hat{r}\rho_{\theta}\cos\frac{\pi}{\ell}}=\frac{n}{m-n}|\ln\varepsilon|-\frac{\mu\theta
n}{2(m-n)^{2}}\ln|\ln\varepsilon|+O\Bigl{(}\frac{\ln^{2}|\ln\varepsilon|}{|\ln\varepsilon|}\Bigr{)}.$
(3.21)
Hence
$\begin{array}[]{ll}G(\hat{r},\rho_{\theta})=&-C\varepsilon^{\frac{m}{m-n}}|\ln\varepsilon|^{-1}-c\varepsilon^{\frac{m}{m-n}}|\ln\varepsilon|^{\frac{\mu\theta
m}{(m-n)^{2}}-1}+\tilde{c}\varepsilon^{\frac{m}{m-n}}|\ln\varepsilon|^{\frac{\mu\theta
n}{2(m-n)^{2}}}\vspace{0.2cm}\\\
&+o\Bigl{(}\varepsilon^{\frac{m}{m-n}}|\ln\varepsilon|^{\frac{\mu\theta
n}{2(m-n)^{2}}}+\varepsilon^{\frac{m}{m-n}}|\ln\varepsilon|^{\frac{\mu\theta
m}{(m-n)^{2}}-1}\Bigr{)},\end{array}$ (3.22)
where $C,\,c$ and $\tilde{c}$ are positive constants independent of
$\varepsilon$.
Set
$f(\theta)=\frac{\mu\theta m}{(m-n)^{2}}-1-\frac{\mu\theta n}{2(m-n)^{2}}.$
Since $\mu>m-n>0$, we see
$f(1)=\frac{\mu m}{(m-n)^{2}}-1-\frac{\mu n}{2(m-n)^{2}}>\frac{\mu
m}{(m-n)^{2}}-1-\frac{\mu n}{(m-n)^{2}}=\frac{\mu}{m-n}-1>0.$
Considering $f(0)=-1<0$, there exists a unique $\bar{\theta}\in(0,1)$ such
that
$\frac{\mu\bar{\theta}m}{(m-n)^{2}}-1=\frac{\mu\bar{\theta}n}{2(m-n)^{2}}.$
(3.23)
Moreover, if $\theta\in(\bar{\theta},\,1]$, then
$\frac{\mu\theta m}{(m-n)^{2}}-1>\frac{\mu\theta n}{2(m-n)^{2}},$
which implies $G(\hat{r},\rho_{\theta})<0$. But, if
$\theta\in[0,\,\bar{\theta})$, then
$\frac{\mu\theta m}{(m-n)^{2}}-1<\frac{\mu\theta n}{2(m-n)^{2}},$ (3.24)
which means $G(\hat{r},\rho_{\theta})>0$ and
$G(\hat{r},\rho_{\theta})<c_{1}\varepsilon^{\frac{m}{m-n}}|\ln\varepsilon|^{\frac{\mu\bar{\theta}n}{2(m-n)^{2}}}$
for some constant $c_{1}>0$ independent of $\varepsilon$.
Therefore, we get that for $\varepsilon$ sufficiently small
$G(\hat{r},\rho_{\theta})\leq
2c_{1}\varepsilon^{\frac{m}{m-n}}|\ln\varepsilon|^{\frac{\mu\bar{\theta}n}{2(m-n)^{2}}},\quad\,\,\forall\,\,\theta\in[0,1],$
which says that
$\max\limits_{\rho\in\mathcal{D}_{\varepsilon}}G(\hat{r},\rho)\leq
2c_{1}\varepsilon^{\frac{m}{m-n}}|\ln\varepsilon|^{\frac{\mu\bar{\theta}n}{2(m-n)^{2}}}.$
(3.25)
Similarly,
$\max\limits_{r\in\mathcal{D}_{\varepsilon}}G(r,\hat{r})\leq
2c_{1}\varepsilon^{\frac{m}{m-n}}|\ln\varepsilon|^{\frac{\mu\bar{\theta}n}{2(m-n)^{2}}}.$
(3.26)
###### Remark 3.7.
It can be verified from (3.22) and (3.24) that for
$\theta\in(0,\,\bar{\theta})$, there exists a constant $c_{2}>0$ independent
of $\varepsilon$ such that for $\varepsilon$ sufficiently small,
$\max\limits_{\rho\in\mathcal{D}_{\varepsilon}}G(\hat{r},\rho)\geq
c_{2}\varepsilon^{\frac{m}{m-n}}|\ln\varepsilon|^{\frac{\mu\theta
n}{2(m-n)^{2}}}.$
Now we estimate
$\max\limits_{\rho\in\mathcal{D}_{\varepsilon}}G(\check{r},\rho)$.
Since for $\varepsilon>0$ sufficiently small,
$\sqrt{\check{r}^{2}+\rho^{2}-2\check{r}\rho\cos\frac{\pi}{\ell}}\geq
n\check{r},$
it follows from (3.18) and the fact $\mu>m-n>0$ that for $\varepsilon$
sufficiently small,
$\displaystyle G(\check{r},\rho)$ $\displaystyle\leq$
$\displaystyle-\frac{C}{m\check{r}}e^{-m\check{r}}+\bar{C}\varepsilon
e^{-\sqrt{\check{r}^{2}+\rho^{2}-2\check{r}\rho\cos\frac{\pi}{\ell}}}$
$\displaystyle\leq$
$\displaystyle-\frac{C}{m\check{r}}e^{-m\check{r}}+\bar{C}\varepsilon
e^{-n\check{r}}$ $\displaystyle\leq$ $\displaystyle-
C_{1}\varepsilon^{\frac{m}{m-n}}|\ln\varepsilon|^{\frac{\mu
m}{(m-n)^{2}}-1}+\bar{C}_{1}\varepsilon^{\frac{m}{m-n}}|\ln\varepsilon|^{\frac{\mu
n}{(m-n)^{2}}}$ $\displaystyle<$ $\displaystyle 0,$
where $C_{1}$ and $\bar{C}_{1}$ are positive constants independent of
$\varepsilon$. Hence,
$\max\limits_{\rho\in\mathcal{D}_{\varepsilon}}G(\check{r},\rho)\leq 0.$
(3.27)
The same argument yields
$\max\limits_{r\in\mathcal{D}_{\varepsilon}}G(r,\check{r})\leq 0.$ (3.28)
At last, we estimate $G(r_{0},\rho_{0}))$. Taking $\theta=\bar{\theta}$ in
(3.20), we find for $\varepsilon$ sufficiently small
$\displaystyle G(r_{0},\rho_{0})$ $\displaystyle\geq$ $\displaystyle
G(\rho_{\bar{\theta}},\rho_{\bar{\theta}})$ $\displaystyle=$
$\displaystyle\bar{C}\varepsilon
e^{-n\rho_{\bar{\theta}}}-\frac{2C}{m\rho_{\bar{\theta}}}e^{-m\rho_{\bar{\theta}}}$
$\displaystyle=$
$\displaystyle\bar{C}\varepsilon^{\frac{m}{m-n}}|\ln\varepsilon|^{\frac{\mu\bar{\theta}n}{(m-n)^{2}}}-\frac{2(m-n)C}{m}\varepsilon^{\frac{m}{m-n}}|\ln\varepsilon|^{\frac{\mu\bar{\theta}m}{(m-n)^{2}}-1}+o\Bigl{(}\varepsilon^{\frac{m}{m-n}}|\ln\varepsilon|^{\frac{\mu\bar{\theta}n}{(m-n)^{2}}}\Bigl{)}$
$\displaystyle\geq$
$\displaystyle\frac{\bar{C}}{2}\varepsilon^{\frac{m}{m-n}}|\ln\varepsilon|^{\frac{\mu\bar{\theta}n}{(m-n)^{2}}},$
since by (3.23),
$\frac{\mu\bar{\theta}n}{(m-n)^{2}}>\frac{\mu\bar{\theta}n}{2(m-n)^{2}}=\frac{\mu\bar{\theta}m}{(m-n)^{2}}-1.$
The above estimate and (3.25)-(3.28) show that for $\varepsilon>0$
sufficiently small, $(r_{0},\rho_{0})$ is in the interior of
$\mathcal{D}_{\varepsilon}\times\mathcal{D}_{\varepsilon}$. Comparing the
above estimate on $G(r_{0},\rho_{0})$ and (3.25)-(3.28) with (3.19), we
conclude that $F(r,\rho)$ achieves (local) maximum also in the interior of
$\mathcal{D}_{\varepsilon}\times\mathcal{D}_{\varepsilon}$.
As a consequence, we complete the proof. ∎
## 4\. Segregated solutions for system coupled by three equations
In this section, we consider the following system linearly coupled by three
equations
$\left\\{\begin{array}[]{ll}-\Delta
u+u=u^{3}-\varepsilon(v+\omega),&x\in\mathbb{R}^{3},\vspace{0.2cm}\\\ -\Delta
v+v=v^{3}-\varepsilon(u+\omega),&x\in\mathbb{R}^{3},\vspace{0.2cm}\\\
-\Delta\omega+\omega=\omega^{3}-\varepsilon(u+v),&x\in\mathbb{R}^{3}.\end{array}\right.$
(4.1)
Let
$(U_{\varepsilon},v_{\varepsilon},\omega_{\varepsilon})\in(H^{1}_{r}(\mathbb{R}^{3}))^{3}$
be the solution of (4.1) obtained in Proposition 2.3. In this part, we will
use the same notations as those in previous sections. Define
$\omega_{\varepsilon,r}=\sum\limits_{j=1}^{\ell}\omega_{\varepsilon,x^{j}},\,\,\,\omega_{\varepsilon,\rho}=\sum\limits_{j=1}^{\ell}\omega_{\varepsilon,y^{j}}$
and
$\mathbf{E}=\\{(\varphi,\psi,\phi)\in(H^{1}_{r}(\mathbb{R}^{3}))^{3}:\,\,(\varphi,\psi)\in\mathbb{E},\,\phi\in
H_{s}\\},$
where $\mathbb{E}$ is defined as (3.3), $r,\,\rho\in\mathcal{D}_{\varepsilon}$
and $\mathcal{\mathcal{D}_{\varepsilon}}$ is defined by (3.2) for $\ell>2$ but
by
$\mathcal{D}_{\varepsilon}=\Bigl{[}\frac{|\ln\varepsilon|}{1+\frac{\mu\ln|\ln\varepsilon|}{|\ln\varepsilon|}},\,|\ln\varepsilon|\Bigr{]},\,\,(\mu>1)$
for $\ell=2$.
To prove Theorem 1.2, we only need to verify
###### Proposition 4.1.
For any integer $\ell\geq 2$, there exists $\varepsilon_{0}>0$ such that for
$\varepsilon\in(0,\varepsilon_{0})$, problem 4.1 has a solution $(u,v,\omega)$
with the form
$\left\\{\begin{array}[]{ll}u=U_{\varepsilon,r}+v_{\varepsilon,\rho}+\omega_{\varepsilon}+\varphi,\vspace{0.2cm}\\\
v=\omega_{\varepsilon,r}+U_{\varepsilon,\rho}+v_{\varepsilon}+\psi,\vspace{0.2cm}\\\
\omega=v_{\varepsilon,r}+\omega_{\varepsilon,\rho}+U_{\varepsilon}+\phi,\end{array}\right.$
where $(\varphi,\psi,\phi)\in\mathbf{E}$ satisfies
$\|(\varphi,\psi,\phi)\|=\left\\{\begin{array}[]{ll}o(\varepsilon^{\frac{m}{m-n}}),&\hbox{if}\,\,\,\ell>2,\vspace{0.2cm}\\\
o(\varepsilon^{2}),&\hbox{if}\,\,\,\ell=2.\end{array}\right.$
###### Proof.
The proof is similar to that of Proposition 3.1, we only give a sketch here.
Define
$\begin{array}[]{ll}\bar{I}(u,v,\omega)=&\displaystyle\frac{1}{2}\int_{\mathbb{R}^{3}}\bigl{(}|\nabla
u|^{2}+u^{2}+|\nabla
v|^{2}+v^{2}+|\nabla\omega|^{2}+\omega^{2}\bigl{)}\vspace{0.2cm}\\\
&-\displaystyle\frac{1}{4}\int_{\mathbb{R}^{3}}\bigl{(}u^{4}+v^{4}+\omega^{4}\bigl{)}+\varepsilon\int_{\mathbb{R}^{3}}(uv+u\omega+v\omega),\quad\forall\,\,\,(u,v,\omega)\in(H_{s})^{3},\end{array}$
and
$\begin{array}[]{ll}\bar{J}(\varphi,\psi,\phi)=&\bar{I}(U_{\varepsilon,r}+v_{\varepsilon,\rho}+\omega_{\varepsilon}+\varphi,\,\omega_{\varepsilon,r}+U_{\varepsilon,\rho}+v_{\varepsilon}+\psi,\,v_{\varepsilon,r}+\omega_{\varepsilon,\rho}+U_{\varepsilon}+\phi),\vspace{0.2cm}\\\
&\quad\quad\forall\,\,\,(\varphi,\psi,\phi)\in\mathbf{E}.\end{array}$
Proceeding as we prove Proposition 3.5, we find that for $\varepsilon$
sufficiently small, there is a $C^{1}$ map from
$(\mathcal{D}_{\varepsilon})^{2}$ to $\mathbf{E}$:
$(\varphi,\psi,\phi)=(\varphi(r,\rho),\psi(r,\rho),\phi(r,\rho))$, satisfying
$\bar{J}^{\prime}_{(\varphi,\psi,\phi)}(\varphi,\psi,\phi)=0,\quad\hbox{on}\,\,\,\mathbf{E},$
and
$\|(\varphi,\psi,\phi)\|=O\Bigl{(}\frac{e^{-|x^{1}-x^{2}|}}{|x^{1}-x^{2}|}+\frac{e^{-|y^{1}-y^{2}|}}{|y^{1}-y^{2}|}+\varepsilon
e^{-(1-\tau)|x^{1}-y^{1}|}+\varepsilon e^{-(1-\tau)|x^{1}|}+\varepsilon
e^{-(1-\tau)|y^{1}|}+\varepsilon^{4}\Bigr{)}.$ (4.2)
We should point out here that when we carry out the finite dimensional
reduction, we do not impose an orthogonal decomposition on $\phi$ (see the
definition of $\mathbf{E}$), since the kernel of the operator
$\Delta-(1-3U^{2})I$ in $H_{s}$ is $\\{0\\}$.
It follows from Proposition A.3 and (4.2) that
$\begin{split}&\bar{F}(r,\rho)=:\bar{J}(\varphi(r,\rho),\psi(r,\rho),\phi(r,\rho))\\\
=&\displaystyle\sum\limits_{j=1}^{\ell}\bar{I}(U_{\varepsilon,x^{j}},v_{\varepsilon,x^{j}},\omega_{\varepsilon,x^{j}})+\displaystyle\sum\limits_{j=1}^{\ell}\bar{I}(U_{\varepsilon,y^{j}},v_{\varepsilon,y^{j}},\omega_{\varepsilon,y^{j}})+\bar{I}(U_{\varepsilon},v_{\varepsilon},\omega_{\varepsilon})\\\
&-\displaystyle\sum\limits_{i<j}^{\ell}C_{ij}\frac{e^{-|x^{i}-x^{j}|}}{|x^{i}-x^{j}|}-\displaystyle\sum\limits_{i<j}^{\ell}C_{ij}\frac{e^{-|y^{i}-y^{j}|}}{|y^{i}-y^{j}|}+\varepsilon\displaystyle\sum\limits_{i,j=1}^{\ell}\bar{C}_{ij}e^{-|x^{i}-y^{j}|}+\varepsilon\displaystyle\sum\limits_{j=1}^{\ell}\tilde{C}_{j}(e^{-|x^{j}|}+e^{-|y^{j}|})\\\
&+O(\varepsilon e^{-(1-\tau)|y^{1}-y^{2}|}+\varepsilon
e^{-(1-\tau)|x^{1}-x^{2}|}+\varepsilon^{2}(e^{-(1-\tau)|x^{1}|}+e^{-(1-\tau)|y^{1}|}+e^{-(1-\tau)|x^{1}-y^{1}|})+\varepsilon^{4})\\\
&+O\Bigl{(}\frac{e^{-|x^{1}-x^{2}|}}{|x^{1}-x^{2}|}+\frac{e^{-|y^{1}-y^{2}|}}{|y^{1}-y^{2}|}+\varepsilon
e^{-(1-\tau)|x^{1}-y^{1}|}+\varepsilon e^{-(1-\tau)|x^{1}|}+\varepsilon
e^{-(1-\tau)|y^{1}|}\Bigl{)}^{2}\\\
=&C_{\varepsilon}+\ell\Bigl{(}\bar{C}\varepsilon
e^{-\sqrt{r^{2}+\rho^{2}-2r\rho\cos\frac{\pi}{\ell}}}+\tilde{C}\varepsilon(e^{-r}+e^{-\rho})-\frac{C}{mr}e^{-mr}-\frac{C}{m\rho}e^{-m\rho}\Bigl{)}+O(\varepsilon^{\frac{m}{m-n}+\sigma}),\end{split}$
where $C_{\varepsilon}>0$ depends on $\varepsilon$ but is independent of $r$
and $\rho$. $\bar{C},\,\tilde{C}$ and $C$ are positive constants independent
of $\varepsilon,\,r$ and $\rho$.
Define function
$\bar{G}(r,\rho)=\bar{C}\varepsilon
e^{-\sqrt{r^{2}+\rho^{2}-2r\rho\cos\frac{\pi}{\ell}}}+\tilde{C}\varepsilon(e^{-r}+e^{-\rho})-\frac{C}{mr}e^{-mr}-\frac{C}{m\rho}e^{-m\rho},\,\,\,r,\rho\in\mathcal{D}_{\varepsilon}.$
We want to verify that $\bar{G}(r,\rho)$ achieves maximum at some point
$(r_{0},\rho_{0})$ which is in the interior of
$\mathcal{D}_{\varepsilon}\times\mathcal{D}_{\varepsilon}$. We have three
cases: (1) $\ell=2$; (2) $\ell=3$; (3) $\ell>3$.
Case (1): $\ell=2$.
In this situation, $m=2,\,n=\sqrt{2}$,
$|x^{1}-y^{1}|=\sqrt{r^{2}+\rho^{2}-2r\rho\cos\frac{\pi}{\ell}}>(1+\sigma)\max\\{r,\,\rho\\}$
for some $\sigma>0$. Hence, without loss of generality, we suppose
$\displaystyle\bar{G}(r,\rho)$ $\displaystyle=$
$\displaystyle(\tilde{C}\varepsilon
e^{-r}-\frac{C}{2r}e^{-2r})+(\tilde{C}\varepsilon
e^{-\rho}-\frac{C}{2\rho}e^{-2\rho})$ $\displaystyle=:$ $\displaystyle
G(r)+G(\rho),\,\,\,r,\rho\in\mathcal{D}_{\varepsilon}.$
Therefore we need to modify $\mathcal{D}_{\varepsilon}$ as
$\mathcal{D}_{\varepsilon}=\Bigl{[}\frac{|\ln\varepsilon|}{1+\frac{\mu\ln|\ln\varepsilon|}{|\ln\varepsilon|}},\,|\ln\varepsilon|\Bigr{]},\quad\mu>1.$
Using the argument to prove Proposition 3.1 (see Remark 3.7), we can find
$\bar{r}_{0}$ which is interior points of $\mathcal{D}_{\varepsilon}$ such
that
$\displaystyle
G(\bar{r}_{0})=\max\limits_{r\in\mathcal{D}_{\varepsilon}}{G(r)}\geq
C_{1}\varepsilon^{2}|\ln\varepsilon|^{\tilde{\theta}}\geq
C_{1}\varepsilon^{2}\geq C_{1}\varepsilon^{\frac{2}{2-\sqrt{2}}}$
for some $\tilde{\theta}>0$ and $C_{1}>0$.
Suppose that $\bar{G}(r,\rho)$ achieves maximum at
$(r_{0},\rho_{0})\in\mathcal{D}_{\varepsilon}\times\mathcal{D}_{\varepsilon}$,
then
$\bar{G}(r_{0},\rho_{0})\geq 2G(\bar{r}_{0})\geq
2C_{1}\varepsilon^{2}|\ln\varepsilon|^{\tilde{\theta}}.$ (4.3)
On the other hand, there exist $\sigma>0$ and $C_{2}>0$ such that
$\begin{array}[]{ll}&\bar{G}(\check{r},\rho)\leq-
C_{2}\varepsilon^{2}|\ln\varepsilon|^{2\mu-1}+G(\bar{r}_{0})<(1-\sigma)\bar{G}(r_{0},\rho_{0}),\quad\,\,\forall\,\,\rho\in\mathcal{D}_{\varepsilon},\vspace{0.2cm}\\\
&\bar{G}(\hat{r},\rho)\leq
C_{2}\varepsilon^{2}+G(\bar{r}_{0})<(1-\sigma)\bar{G}(r_{0},\rho_{0}),\quad\,\,\forall\,\,\rho\in\mathcal{D}_{\varepsilon},\vspace{0.2cm}\\\
&\bar{G}(r,\check{r})\leq
G(\bar{r}_{0})-C_{2}\varepsilon^{2}|\ln\varepsilon|^{2\mu-1}<(1-\sigma)\bar{G}(r_{0},\rho_{0}),\quad\,\,\forall\,\,r\in\mathcal{D}_{\varepsilon},\vspace{0.2cm}\\\
&\bar{G}(r,\hat{r})\leq
G(\bar{r}_{0})+C_{2}\varepsilon^{2}<(1-\sigma)\bar{G}(r_{0},\rho_{0}),\quad\,\,\forall\,\,r\in\mathcal{D}_{\varepsilon},\end{array}$
(4.4)
where $\check{r}$ and $\hat{r}$ are modified respectively as
$\check{r}=\frac{|\ln\varepsilon|}{1+\frac{\mu\ln|\ln\varepsilon|}{|\ln\varepsilon|}},\quad\hat{r}=|\ln\varepsilon|.$
Therefore, $(r_{0},\rho_{0})$ is an interior point of
$\mathcal{D}_{\varepsilon}\times\mathcal{D}_{\varepsilon}$. Comparing (4.4)
with $\bar{F}(r,\rho)$ and (4.3), we conclude that $\bar{F}(r,\rho)$ has
(local) maximizer in the interior of
$\mathcal{D}_{\varepsilon}\times\mathcal{D}_{\varepsilon}$.
Case (2): $\ell=3$.
In this case, $m=\sqrt{3},\,n=1$, and it is possible that
$r\sim\rho\sim|x^{1}-y^{1}|=\sqrt{r^{2}+\rho^{2}-2r\rho\cos\frac{\pi}{\ell}}$.
So we consider
$\bar{G}(r,\rho)=\bar{C}\varepsilon
e^{-\sqrt{r^{2}+\rho^{2}-r\rho}}+\tilde{C}\varepsilon(e^{-r}+e^{-\rho})-\frac{C}{\sqrt{3}r}e^{-\sqrt{3}r}-\frac{C}{\sqrt{3}\rho}e^{-\sqrt{3}\rho},\,\,\,r,\rho\in\mathcal{D}_{\varepsilon}.$
Now we analyze $\bar{G}(r,\rho)$ on
$\partial(\mathcal{D}_{\varepsilon}\times\mathcal{D}_{\varepsilon})$.
Firstly, again using (3.18), we see
$\begin{array}[]{ll}&\bar{G}(\check{r},\rho)\leq\hat{C}\varepsilon
e^{-\check{r}}-\displaystyle\frac{C}{\sqrt{3}\check{r}}e^{-\sqrt{3}\check{r}}<0,\forall\,\,\rho\in\mathcal{D}_{\varepsilon},\vspace{0.2cm}\\\
&\bar{G}(r,\check{r})\leq\hat{C}\varepsilon
e^{-\check{r}}-\displaystyle\frac{C}{\sqrt{3}\check{r}}e^{-\sqrt{3}\check{r}}<0,\forall\,\,r\in\mathcal{D}_{\varepsilon},\end{array}$
(4.5)
where $\hat{C}$ and $C$ are positive constants independent of $\varepsilon$.
On the other hand, suppose that, in $\mathcal{D}_{\varepsilon}$,
$\bar{G}(\hat{r},\rho)$ achieves maximizer
$\bar{\rho}\in(\check{r},\,\hat{r})$. Arguing as we prove Proposition 3.1 (see
Remark 3.7), we find
$\bar{G}(\hat{r},\bar{\rho})\geq
C_{3}\varepsilon^{\frac{\sqrt{3}}{\sqrt{3}-1}}|\ln\varepsilon|^{\theta_{0}}$
(4.6)
for some $\theta_{0}>0$ and $C_{3}>0$.
Since
$\tilde{C}\varepsilon
e^{-\hat{r}}-\frac{C}{\sqrt{3}\hat{r}}e^{-\sqrt{3}\hat{r}}=O(\varepsilon^{\frac{\sqrt{3}}{\sqrt{3}-1}})$
and
$e^{-\sqrt{\hat{r}^{2}+\bar{\rho}^{2}-\hat{r}\bar{\rho}}}<e^{-\bar{\rho}},$
we see
$\displaystyle
C_{3}\varepsilon^{\frac{\sqrt{3}}{\sqrt{3}-1}}|\ln\varepsilon|^{\theta_{0}}$
$\displaystyle\leq$
$\displaystyle\bar{G}(\hat{r},\bar{\rho})<\bar{C}\varepsilon
e^{-\bar{\rho}}+O(\varepsilon^{\frac{\sqrt{3}}{\sqrt{3}-1}})+\tilde{C}\varepsilon
e^{-\bar{\rho}}-\frac{C}{\sqrt{3}\bar{\rho}}e^{-\sqrt{3}\bar{\rho}}.$
Hence, there exists $\sigma>0$ such that
$\displaystyle\bar{G}(\bar{\rho},\bar{\rho})$ $\displaystyle=$
$\displaystyle\bar{C}\varepsilon e^{-\bar{\rho}}+2\bigl{(}\tilde{C}\varepsilon
e^{-\bar{\rho}}-\frac{C}{\sqrt{3}\bar{\rho}}e^{-\sqrt{3}\bar{\rho}}\bigr{)}$
$\displaystyle>$
$\displaystyle\bar{G}(\hat{r},\bar{\rho})+\tilde{C}\varepsilon
e^{-\bar{\rho}}-\frac{C}{\sqrt{3}\bar{\rho}}e^{-\sqrt{3}\bar{\rho}}$
$\displaystyle>$ $\displaystyle(1+\sigma)\bar{G}(\hat{r},\bar{\rho}),$
which implies
$\max\limits_{r,\rho\in\mathcal{D}_{\varepsilon}}\bar{G}(r,\rho)\geq\bar{G}(\bar{\rho},\bar{\rho})>(1+\sigma)\bar{G}(\hat{r},\bar{\rho})=(1+\sigma)\max\limits_{\rho\in\mathcal{D}_{\varepsilon}}\bar{G}(\hat{r},\rho).$
Similarly,
$\max\limits_{r,\rho\in\mathcal{D}_{\varepsilon}}\bar{G}(r,\rho)\geq(1+\sigma)\max\limits_{r\in\mathcal{D}_{\varepsilon}}\bar{G}(r,\hat{r}).$
These two estimates and (4.5) imply that $\bar{F}(r,\rho)$ has (local)
maximizer in the interior of
$\mathcal{D}_{\varepsilon}\times\mathcal{D}_{\varepsilon}$.
Case (3): $\ell>3$.
In this situation,
$\sqrt{r^{2}+\rho^{2}-2r\rho\cos\frac{\pi}{\ell}}=|x^{1}-y^{1}|<(1-\sigma)\min\\{r,\,\rho\\}$
for some $\sigma>0$. Then
$\bar{G}(r,\rho)=\bar{C}\varepsilon
e^{-\sqrt{r^{2}+\rho^{2}-2r\rho\cos\frac{\pi}{\ell}}}-\frac{C}{mr}e^{-mr}-\frac{C}{m\rho}e^{-m\rho},\,\,\,r,\rho\in\mathcal{D}_{\varepsilon}.$
This is exactly $G(r,\rho)$ in the proof of Proposition 3.1 and we omit the
rest of the proof.
As a result, we complete the proof. ∎
## Appendix A Energy expansions
In this section, we will expand the energy
$I(U_{\varepsilon,r}+v_{\varepsilon,\rho},\,v_{\varepsilon,r}+U_{\varepsilon,\rho})$,
which is defined as
$\begin{array}[]{ll}I(u,v)=&\displaystyle\frac{1}{2}\int_{\mathbb{R}^{3}}\bigl{(}|\nabla
u|^{2}+u^{2}+|\nabla v|^{2}+v^{2}\bigl{)}\vspace{0.2cm}\\\
&-\displaystyle\frac{1}{4}\int_{\mathbb{R}^{3}}\bigl{(}u^{4}+v^{4}\bigl{)}+\varepsilon\int_{\mathbb{R}^{3}}uv,\quad(u,v)\in
H_{s}\times H_{s}.\end{array}$
Recall that $(U_{\varepsilon},v_{\varepsilon})$ has the form
$\begin{array}[]{ll}U_{\varepsilon}=U+\varepsilon^{2}p_{\varepsilon}(r)+w,\,\,\,v_{\varepsilon}=\varepsilon
q_{\varepsilon}(r)+h,\end{array}$ (A.1)
where
$p_{\varepsilon}(r)\leq Ce^{-(1-\tau)r},\,\,q_{\varepsilon}(r)\leq
Ce^{-(1-\tau)r},\,\,\,\|(w,h)\|\leq C\varepsilon^{4}.$ (A.2)
The following estimates can be found in [4].
###### Proposition A.1.
Suppose that $u(x),v(x)\in H_{r}^{1}(\mathbb{R}^{N})\,(N\geq 1)$ satisfy
$u(r)\sim r^{\alpha}e^{-\beta r},\,\,\,v(r)\sim r^{\gamma}e^{-\eta
r},\,\,\,(r\to+\infty),$
where $\alpha,\,\gamma\in\mathbb{R},\,\beta>0,\,\eta>0$. Let
$y\in\mathbb{R}^{N}$ with $|y|\to+\infty$. We have
(i) if $\beta<\eta$, then
$\int_{\mathbb{R}^{N}}u_{y}v\sim|y|^{\alpha}e^{-\beta|y|}.$
(ii) if $\beta=\eta$, suppose for simplicity, that $\alpha\geq\gamma$. Then
$\int_{\mathbb{R}^{N}}u_{y}v\sim\left\\{\begin{array}[]{ll}e^{-\beta|y|}|y|^{\alpha+\gamma+\frac{1+N}{2}}&\hbox{if}\,\,\gamma>-\frac{1+N}{2},\vspace{0.2cm}\\\
e^{-\beta|y|}|y|^{\alpha}\ln|y|&\hbox{if}\,\,\gamma=-\frac{1+N}{2},\vspace{0.2cm}\\\
e^{-\beta|y|}|y|^{\alpha}&\hbox{if}\,\,\gamma<-\frac{1+N}{2}.\end{array}\right.$
###### Proposition A.2.
We have
$\displaystyle
I(U_{\varepsilon,r}+v_{\varepsilon,\rho},\,v_{\varepsilon,r}+U_{\varepsilon,\rho})$
$\displaystyle=$
$\displaystyle\displaystyle\sum\limits_{j=1}^{\ell}I(U_{\varepsilon,x^{j}},v_{\varepsilon,x^{j}})+\displaystyle\sum\limits_{j=1}^{\ell}I(U_{\varepsilon,y^{j}},v_{\varepsilon,y^{j}})$
$\displaystyle-\displaystyle\sum\limits_{i<j}^{\ell}C_{ij}\frac{e^{-|x^{i}-x^{j}|}}{|x^{i}-x^{j}|}-\displaystyle\sum\limits_{i<j}^{\ell}C_{ij}\frac{e^{-|y^{i}-y^{j}|}}{|y^{i}-y^{j}|}+\varepsilon\displaystyle\sum\limits_{i,j=1}^{\ell}\bar{C}_{ij}e^{-|x^{i}-y^{j}|}$
$\displaystyle+O(\varepsilon
e^{-(1-\tau)|y^{1}-y^{2}|}+\varepsilon^{2}e^{-(1-\tau)|x^{1}-y^{1}|}+\varepsilon
e^{-(1-\tau)|x^{1}-x^{2}|}+\varepsilon^{4}),$
where $C_{ij},\bar{C}_{ij}\,(i,j=1,\cdots,\ell)$ are positive constants
independent of $\varepsilon$, $r$ and $\rho$.
###### Proof.
Write
$\begin{array}[]{ll}&I(U_{\varepsilon,r}+v_{\varepsilon,\rho},\,v_{\varepsilon,r}+U_{\varepsilon,\rho})\vspace{0.2cm}\\\
=&I(U_{\varepsilon,r},v_{\varepsilon,r})+I(U_{\varepsilon,\rho},v_{\varepsilon,\rho})\vspace{0.2cm}\\\
&-\displaystyle\frac{1}{4}\int_{\mathbb{R}^{3}}\Bigl{(}\bigl{(}U_{\varepsilon,r}+v_{\varepsilon,\rho}\bigr{)}^{4}-U^{4}_{\varepsilon,r}-v^{4}_{\varepsilon,\rho}-4\displaystyle\sum\limits_{i,j=1}^{\ell}U^{3}_{\varepsilon,x^{i}}v_{\varepsilon,y^{j}}\Bigr{)}\vspace{0.2cm}\\\
&-\displaystyle\frac{1}{4}\int_{\mathbb{R}^{3}}\Bigl{(}\bigl{(}U_{\varepsilon,\rho}+v_{\varepsilon,r}\bigr{)}^{4}-U^{4}_{\varepsilon,\rho}-v^{4}_{\varepsilon,r}-4\displaystyle\sum\limits_{i,j=1}^{\ell}U^{3}_{\varepsilon,y^{i}}v_{\varepsilon,x^{j}}\Bigr{)}\vspace{0.2cm}\\\
&+\varepsilon\displaystyle\int_{\mathbb{R}^{3}}\Bigl{(}(U_{\varepsilon,r}+v_{\varepsilon,\rho})(v_{\varepsilon,r}+U_{\varepsilon,\rho})-U_{\varepsilon,r}v_{\varepsilon,r}-U_{\varepsilon,\rho}v_{\varepsilon,\rho}-2\displaystyle\sum\limits_{i,j=1}^{\ell}v_{\varepsilon,x^{i}}v_{\varepsilon,y^{j}}\Bigr{)}\vspace{0.2cm}\\\
=:&I(U_{\varepsilon,r},v_{\varepsilon,r})+I(U_{\varepsilon,\rho},v_{\varepsilon,\rho})-I_{1}-I_{2}+\varepsilon
I_{3}.\end{array}$ (A.3)
Now we estimate each term in (A.3).
For $I_{1}$, from (A.2) we see
$\begin{array}[]{ll}I_{1}&=\displaystyle\int_{\mathbb{R}^{3}}\Bigl{[}4\Bigl{(}\sum\limits_{i=1}^{\ell}U_{\varepsilon,x^{i}}\Bigr{)}^{3}\sum\limits_{i=1}^{\ell}v_{\varepsilon,y^{i}}-4\displaystyle\sum\limits_{i,j=1}^{\ell}U^{3}_{\varepsilon,x^{i}}v_{\varepsilon,y^{j}}+4\sum\limits_{i=1}^{\ell}U_{\varepsilon,x^{i}}\Bigl{(}\sum\limits_{i=1}^{\ell}v_{\varepsilon,y^{i}}\Bigr{)}^{3}\Bigr{]}\vspace{0.2cm}\\\
&\hskip
11.38092pt+O\Bigl{(}\displaystyle\int_{\mathbb{R}^{N}}\Bigl{(}\sum\limits_{i=1}^{\ell}U_{\varepsilon,x^{i}}\Bigl{)}^{2}\Bigl{(}\sum\limits_{i=1}^{\ell}v_{\varepsilon,y^{i}}\Bigr{)}^{2}\Bigl{)}\vspace{0.2cm}\\\
&=O\Bigl{(}\displaystyle\int_{\mathbb{R}^{3}}\sum\limits_{i\neq
j}^{\ell}U^{2}_{\varepsilon,x^{i}}U_{\varepsilon,x^{j}}\sum\limits_{i=1}^{\ell}v_{\varepsilon,y^{i}}\Bigl{)}+O(\varepsilon^{3}e^{-(1-\tau)|x^{1}-y^{1}|}+\varepsilon^{2}e^{-(1-\tau)|x^{1}-y^{1}|}+\varepsilon^{4})\vspace{0.2cm}\\\
&=O(\varepsilon
e^{-(1-\tau)|x^{1}-x^{2}|}+\varepsilon^{4})+O(\varepsilon^{2}e^{-(1-\tau)|x^{1}-y^{1}|}).\end{array}$
(A.4)
Similarly,
$\begin{array}[]{ll}I_{2}=O(\varepsilon
e^{-(1-\tau)|y^{1}-y^{2}|}+\varepsilon^{4})+O(\varepsilon^{2}e^{-(1-\tau)|x^{1}-y^{1}|}).\end{array}$
(A.5)
Calculating $I_{3}$, we obtain
$I_{3}=\int_{\mathbb{R}^{3}}\Bigl{(}\displaystyle\sum\limits_{i,j=1}^{\ell}U_{\varepsilon,x^{i}}U_{\varepsilon,y^{j}}-\displaystyle\sum\limits_{i,j=1}^{\ell}v_{\varepsilon,x^{i}}v_{\varepsilon,y^{j}}\Bigr{)}.$
On the other hand, by (A.2) and Proposition A.1, we see
$\begin{array}[]{ll}\displaystyle\int_{\mathbb{R}^{3}}U_{\varepsilon,x^{i}}U_{\varepsilon,y^{j}}&=\displaystyle\int_{\mathbb{R}^{3}}\bigl{(}U_{x^{i}}+\varepsilon^{2}p_{\varepsilon}(|x-x^{i}|)+w(|x-x^{i}|)\bigr{)}\vspace{0.2cm}\\\
&\hskip
28.45274pt\times\bigl{(}U_{y^{j}}+\varepsilon^{2}p_{\varepsilon}(|x-y^{j}|)+w(|x-y^{j}|)\bigr{)}\vspace{0.2cm}\\\
&=\displaystyle\int_{\mathbb{R}^{3}}U_{x^{i}}U_{y^{j}}+O(\varepsilon
e^{-(1-\tau)|x^{1}-y^{1}|}+\varepsilon^{4})\vspace{0.2cm}\\\
&=\bar{C}_{ij}e^{-|x^{i}-y^{j}|}+O(\varepsilon
e^{-(1-\tau)|x^{1}-y^{1}|}+\varepsilon^{4}),\end{array}$ (A.6)
and similarly,
$\int_{\mathbb{R}^{3}}\displaystyle\sum\limits_{i,j=1}^{\ell}v_{\varepsilon,x^{i}}v_{\varepsilon,y^{j}}=O(\varepsilon^{2}e^{-(1-2\tau)|x^{1}-y^{1}|}+\varepsilon^{5}).$
Hence,
$I_{3}=\displaystyle\sum\limits_{i,j=1}^{\ell}\bar{C}_{ij}e^{-|x^{i}-y^{j}|}+O(\varepsilon
e^{-(1-\tau)|x^{1}-y^{1}|}+\varepsilon^{4}).$ (A.7)
At last, we estimate
$I(U_{\varepsilon,r},v_{\varepsilon,r})+I(U_{\varepsilon,\rho},v_{\varepsilon,\rho})$.
We find
$\displaystyle I(U_{\varepsilon,r},v_{\varepsilon,r})$ $\displaystyle=$
$\displaystyle\displaystyle\sum\limits_{j=1}^{\ell}I(U_{\varepsilon,x^{j}},v_{\varepsilon,x^{j}})-\frac{1}{4}\displaystyle\int_{\mathbb{R}^{3}}\Bigl{[}\Bigl{(}\displaystyle\sum\limits_{j=1}^{\ell}U_{\varepsilon,x^{j}}\Bigr{)}^{4}-\displaystyle\sum\limits_{j=1}^{\ell}U^{4}_{\varepsilon,x^{j}}-4\displaystyle\sum\limits_{i<j}^{\ell}U^{3}_{\varepsilon,x^{j}}U_{\varepsilon,x^{i}}\Bigr{]}$
$\displaystyle-\frac{1}{4}\displaystyle\int_{\mathbb{R}^{3}}\Bigl{[}\Bigl{(}\displaystyle\sum\limits_{j=1}^{\ell}v_{\varepsilon,x^{j}}\Bigr{)}^{4}-\displaystyle\sum\limits_{j=1}^{\ell}v^{4}_{\varepsilon,x^{j}}-4\displaystyle\sum\limits_{i<j}^{\ell}v^{3}_{\varepsilon,x^{j}}v_{\varepsilon,x^{i}}\Bigr{]}$
$\displaystyle=$
$\displaystyle\displaystyle\sum\limits_{j=1}^{\ell}I(U_{\varepsilon,x^{j}},v_{\varepsilon,x^{j}})$
$\displaystyle-\displaystyle\int_{\mathbb{R}^{3}}\Bigl{(}\displaystyle\sum\limits_{i<j}^{\ell}U^{3}_{\varepsilon,x^{j}}U_{\varepsilon,x^{i}}+3\displaystyle\sum\limits_{i,j=1}^{\ell}U^{2}_{\varepsilon,x^{j}}U^{2}_{\varepsilon,x^{i}}+\displaystyle\sum\limits_{i<j}^{\ell}v^{3}_{\varepsilon,x^{j}}v_{\varepsilon,x^{i}}+3\displaystyle\sum\limits_{i,j=1}^{\ell}v^{2}_{\varepsilon,x^{j}}v^{2}_{\varepsilon,x^{i}}\Bigr{)}.$
Similar to (A.6), we have for $i\neq j$
$\displaystyle\displaystyle\int_{\mathbb{R}^{3}}\displaystyle\sum\limits_{i<j}^{\ell}U^{3}_{\varepsilon,x^{j}}U_{\varepsilon,x^{i}}$
$\displaystyle=$
$\displaystyle\displaystyle\sum\limits_{i<j}^{\ell}C_{ij}\frac{e^{-|x^{i}-x^{j}|}}{|x^{i}-x^{j}|}+O(\varepsilon
e^{-(1-\tau)|x^{1}-x^{2}|}+\varepsilon^{4}),$
$\displaystyle\displaystyle\int_{\mathbb{R}^{3}}\displaystyle\sum\limits_{i,j=1}^{\ell}U^{2}_{\varepsilon,x^{j}}U^{2}_{\varepsilon,x^{i}}$
$\displaystyle=$
$\displaystyle\displaystyle\sum\limits_{i<j}^{\ell}C^{\prime}_{ij}\frac{e^{-2|x^{i}-x^{j}|}}{|x^{i}-x^{j}|^{2}}+O(\varepsilon
e^{-(1-\tau)|x^{1}-x^{2}|}+\varepsilon^{4}),$
and
$\displaystyle\int_{\mathbb{R}^{3}}\Bigl{(}\displaystyle\sum\limits_{i<j}^{\ell}v^{3}_{\varepsilon,x^{j}}v_{\varepsilon,x^{i}}+3\displaystyle\sum\limits_{i,j=1}^{\ell}v^{2}_{\varepsilon,x^{j}}v^{2}_{\varepsilon,x^{i}}\Bigr{)}=O(\varepsilon^{4}).$
Therefore,
$I(U_{\varepsilon,r},v_{\varepsilon,r})=\displaystyle\sum\limits_{j=1}^{\ell}I(U_{\varepsilon,x^{j}},v_{\varepsilon,x^{j}})+\displaystyle\sum\limits_{i<j}^{\ell}C_{ij}\frac{e^{-|x^{i}-x^{j}|}}{|x^{i}-x^{j}|}+O(\varepsilon
e^{-(1-\tau)|x^{1}-x^{2}|}+\varepsilon^{4}).$ (A.8)
With the same argument, we check
$I(U_{\varepsilon,\rho},v_{\varepsilon,\rho})=\displaystyle\sum\limits_{j=1}^{\ell}I(U_{\varepsilon,y^{j}},v_{\varepsilon,y^{j}})+\displaystyle\sum\limits_{i<j}^{\ell}C_{ij}\frac{e^{-|y^{i}-y^{j}|}}{|y^{i}-y^{j}|}+O(\varepsilon
e^{-(1-\tau)|x^{1}-x^{2}|}+\varepsilon^{4}).$ (A.9)
Now, inserting (A.4), (A.5), (A.7), (A.8) and (A.9) into (A.3), we complete
the proof.
∎
###### Proposition A.3.
We have
$\begin{split}&\bar{I}(U_{\varepsilon,r}+v_{\varepsilon,\rho}+\omega_{\varepsilon},\omega_{\varepsilon,r}+U_{\varepsilon,\rho}+v_{\varepsilon},v_{\varepsilon,r}+\omega_{\varepsilon,\rho}+U_{\varepsilon})\\\
=&\displaystyle\sum\limits_{j=1}^{\ell}\bar{I}(U_{\varepsilon,x^{j}},v_{\varepsilon,x^{j}},\omega_{\varepsilon,x^{j}})+\displaystyle\sum\limits_{j=1}^{\ell}\bar{I}(U_{\varepsilon,y^{j}},v_{\varepsilon,y^{j}}+\omega_{\varepsilon,y^{j}})+\bar{I}(U_{\varepsilon},v_{\varepsilon},\omega_{\varepsilon})\\\
&-\displaystyle\sum\limits_{i<j}^{\ell}C_{ij}\frac{e^{-|x^{i}-x^{j}|}}{|x^{i}-x^{j}|}-\displaystyle\sum\limits_{i<j}^{\ell}C_{ij}\frac{e^{-|y^{i}-y^{j}|}}{|y^{i}-y^{j}|}+\varepsilon\displaystyle\sum\limits_{i,j=1}^{\ell}\bar{C}_{ij}e^{-|x^{i}-y^{j}|}+\varepsilon\displaystyle\sum\limits_{j=1}^{\ell}\tilde{C}_{j}(e^{-|x^{j}|}+e^{-|y^{j}|})\\\
&+O\bigl{(}\varepsilon e^{-(1-\tau)|y^{1}-y^{2}|}+\varepsilon
e^{-(1-\tau)|x^{1}-x^{2}|}+\varepsilon^{2}(e^{-(1-\tau)|x^{1}|}+e^{-(1-\tau)|y^{1}|}+e^{-(1-\tau)|x^{1}-y^{1}|})+\varepsilon^{4}\bigl{)},\end{split}$
where $\bar{C}_{ij},\,\tilde{C}_{j}$ and $C_{ij}\,(i,j=1,\cdots,\ell)$ are
positive constants independent of $\varepsilon$, $r$ and $\rho$.
###### Proof.
The proof is similar to that of Proposition A.2 and we omit it here. ∎
Acknowledgment. The authors are grateful to Shusen Yan for helpful discussion.
S. Peng thanks Taida Institute for Mathematical Sciences for the warm
hospitality during his visit.
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|
arxiv-papers
| 2013-10-07T09:56:55 |
2024-09-04T02:49:52.072417
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Chang-Shou Lin and Shuangjie Peng",
"submitter": "Shuangjie Peng",
"url": "https://arxiv.org/abs/1310.1718"
}
|
1310.1738
|
# Fast creation of conditional quantum gate and entanglement using a common
bath only
Nan Qiu State Key Laboratory of Low Dimensional Quantum Physics, Department
of Physics, Tsinghua University, Beijing 100084, People s Republic of China
Xiang-Bin Wang [email protected] State Key Laboratory of Low
Dimensional Quantum Physics, Department of Physics, Tsinghua University,
Beijing 100084, People s Republic of China Jinan Institute of Quantum
Technology, Shandong Academy of Information and Communication Technology,
Jinan 250101, People s Republic of China
###### Abstract
We propose a scheme to for fast conditional phase shift and creation
entanglement of two qubits that interact with a common heat bath. Dynamical
decoupling is applied in the scheme so that it works even in the regime of
strong interaction between qubits and environment. Our scheme does not request
any direct interaction between the two qubits.
###### pacs:
03.67.-a, 03.67.Mn, 03.67.Pp
_Introduction._ Quantum entanglement and quantum conditional phase shift play
the central role in quantum information processing and quantum commutation QI
, for example, quantum cryptography with the Bell theorem Bell , quantum dense
coding dc , quantum teleportation qt . It is therefore an important task to
generate entangled states. Any gate that entangles two qubits, e.g., the
conditional phase-shift gate is universal for quantum computation, when
assisted by single-qubit gates QCEG . It means that entangling two-qubit gates
provide the ability to perform universal quantum computation. However, it is
often very fragile due to environmental perturbations. The method of dynamical
decoupling (DD) PDD ; CDD ; UDD ; PUDD ; CUDD ; QDD ; TNDD ; TNDD1 ; MNDD ;
IMNDD ; IMNDD1 can be used to protect the coherence of qubits in noisy
environment, e.g. it can remove the interaction between the system and
environment. DD can also be applied for engineered quantum interaction between
qubits and the interaction between qubits and baths DDEBI1 ; DDEBI2 .
A smart scheme EnDB shows that entanglement between two qubits can be
generated if the two qubits interact with a common bath in thermal
equilibrium, but not interact directly with each other. This model enhance the
usefulness of environment. However, it requests the interaction between qubits
and the common bath be weak. Hence it will cost a long time to prepare
entangled state by such a model.
Here we present an efficient method to generate quantum entanglement and make
the conditional phase shift gate using only a common heat bath through
dynamical decoupling PDD . Our method can work in the regime of strong
interaction between qubits and environment thus quantum entanglement between
qubits can be generated rather fast. Compared with the existing method EnDB ,
our method can work much more efficiently. In the strong interaction regime,
the coherence of the two qubits would be destroyed rapidly by the environment
if there is no designed quantum engineering, e.g., the dynamical decoupling.
Nested UDD MNDD ; IMNDD ; IMNDD1 could protect a multi-qubit state, but the
nonlocal correlation of qubits is locked by them. Therefore, it cannot
generate an entangling two-qubit gate. Since entangling two-qubit gates result
in changing nonlocal correlation of qubits, the control field should reduce
the effect on nonlocal correlation of qubits. The control field should reduce
the decoherence on the one hand keep the effective two-body Hamiltonian of the
two qubits which generates entangle.
The common bath could induce effectively nonlinear couplings in a quantum
many-body (multispin) system LCBGm . The advantage of the system-bath coupling
is taken by dynamical control so as to realize cooling or heating on a single
qubit system Toc . Entangled qubits in the common both can be protected with
un-simultaneously DD DDincom . Here we simultaneously use UDD on the two
qubits UDD ; PUDD to minimize decoherence, but make the conditional phase be
non-negligible value. Thus entangling two-qubit gates could be performed fast
in strong coupling regime.
_Model._ Two two-level atoms interacting with a common bosonic bath may be
described by an extended spin-boson Hamiltonian CL
$H_{total}=H_{S}+H_{B}+H_{int}$ (setting $\hbar=1$ ), where
$\displaystyle
H_{S}=\frac{\Omega_{1}}{2}\sigma_{1}^{z}+\frac{\Omega_{2}}{2}\sigma_{2}^{z}$
(1) $\displaystyle H_{B}=\sum\limits_{j}{\omega_{j}a_{j}^{{\dagger}}a_{j}}$
(2) $\displaystyle
H_{int}=(\sigma_{1}^{z}+\sigma_{2}^{z})\left[{\sum\limits_{j}{\lambda_{j}\left({a_{j}^{{\dagger}}+a_{j}}\right)}}\right].$
(3)
Here $\Omega_{i}$ is the transition frequency of the $i$th qubit,
$\sigma_{i}^{z}$ is the Pauli spin operator of the $i$th qubit, and the
environment is represented by a collective bosonic bath with annihilation
(creation) operators $a_{j}^{({\dagger})}$.
_Control pulses._ Consider now $N_{d}$ instantaneous $\pi$-pulses of
$\sigma_{1}^{x}$ ($\sigma_{2}^{x}$) applied to our system at time $t_{n_{1}}$
($t_{n_{2}}$), with $1\leq n_{1}\leq N_{d}$ ($1\leq n_{2}\leq N_{d}$). Upon
application of one such pulse, one has, in the frame of applied pulses,
$\sigma_{1}^{z}\rightarrow-\sigma_{1}^{z}$
($\sigma_{2}^{z}\rightarrow-\sigma_{2}^{z}$). It hence convenient to introduce
the so-called switching function $f_{1(2)}(t)$ , where
$\displaystyle
f_{1(2)}(t)=\sum\limits_{n_{1(2)}}^{N_{d}}{(-1)^{n_{1(2)}+1}\theta(t-t_{n_{1(2)}})\theta(t-t_{n_{1(2)}+1})},$
(4)
with $\theta(t)$ is the Heaviside function.
In the interaction picture this yields
$\displaystyle H_{I}$ $\displaystyle=$
$\displaystyle(\sigma_{1}^{z}f_{1}(t)+\sigma_{2}^{z}f_{2}(t))\times$ (5)
$\displaystyle\left[{\sum\limits_{j}{\lambda_{j}\left({a_{j}^{{\dagger}}\exp(i\omega_{j}t)+a_{j}\exp(-i\omega_{j}t)}\right)}}\right].$
The closed-form equation for the time-evolution operator (see Appendix A)
takes the simple form
$\displaystyle U\left(t\right)$ $\displaystyle=$
$\displaystyle\exp\left[{-i\int\limits_{0}^{t}{H_{I}(t_{1})dt_{1}}-i\Theta(t)\sigma_{1}^{z}\sigma_{2}^{z}}\right]$
(6) $\displaystyle=$
$\displaystyle\exp\left(-iH_{d}t\right)\exp\left(-iH_{p}t\right),$
where
$\displaystyle\Theta(t)=-\int\limits_{0}^{t}\int\limits_{0}^{t_{1}}$
$\displaystyle\left({f_{1}\left({t_{1}}\right)f_{2}\left({t_{2}}\right)+f_{2}\left({t_{1}}\right)f_{1}\left({t_{2}}\right)}\right)\times$
(7)
$\displaystyle\sum\limits_{j}{\left|{\lambda_{j}}\right|^{2}\sin\left[{\omega_{j}\left({t_{1}-t_{2}}\right)}\right]}{dt_{1}dt_{2}},$
$\displaystyle
H_{p}\equiv\frac{{\Theta(t)\sigma_{1}^{z}\sigma_{2}^{z}}}{{t}},$ (8)
$\displaystyle
H_{d}\equiv\frac{{\sum\limits_{j}{(\xi_{j}(t)a_{j}^{\dagger}+\xi_{j}^{*}(t)a_{j})}}}{t},$
(9)
$\displaystyle\xi_{j}(t)=\int\limits_{0}^{t}{ds\lambda_{j}e^{i\omega_{j}s}\left({\sigma_{1}^{z}f_{1}(s)+\sigma_{2}^{z}f_{2}(s)}\right)}.$
(10)
The evolution of the system, given by Eq.(6), describes a reservoir-modified
i-swap transformation, and also expresses the decoherence induced by the
reservoir. The Hamiltonian $H_{p}$ generates an entangled gate. However, the
Hamiltonian $H_{d}$ would destroy the coherence. The control field changes
both of them. The uncorrelated initial state is given by,
$\displaystyle\rho_{tot}(0)=\left|\Psi\right\rangle\left\langle\Psi\right|\otimes\frac{{e^{-\frac{{H_{B}}}{{k_{B}T}}}}}{{{\rm{Tr_{B}}}\left({e^{-\frac{{H_{B}}}{{k_{B}T}}}}\right)}},$
(11)
with $k_{B}$ denotes Boltzmann s constant. We then focuses on the evolution of
the reduced state
$\displaystyle\rho_{S}(t)={\rm{Tr_{B}}}(U(t)\rho_{tot}(0)U^{\dagger}(t)).$
(12)
By taking the trace over the field variables of Eq. (12) we get
$\displaystyle\rho_{S}\left(t\right)=\sum\limits_{n,m}^{4}{\rho_{n,m}(0)e^{\left({i\Theta(t)a_{n,m}-\Upsilon\left(t\right)b_{n,m}}\right)}\left|{\phi_{n}}\right\rangle\left\langle{\phi_{m}}\right|},$
(13)
where
$\left|{\phi_{1}}\right\rangle\equiv\left|{g_{1}g_{2}}\right\rangle,\left|{\phi_{2}}\right\rangle\equiv\left|{g_{1}e_{2}}\right\rangle,\left|{\phi_{3}}\right\rangle\equiv\left|{e_{1}g_{2}}\right\rangle,\left|{\phi_{4}}\right\rangle\equiv\left|{e_{1}e_{2}}\right\rangle,\left({a_{mn}}\right)=\left({\begin{array}[]{*{20}c}0&2&2&0\\\
{-2}&0&0&{-2}\\\ {-2}&0&0&{-2}\\\ 0&2&2&0\\\
\end{array}}\right),\left({b_{mn}}\right)=\left({\begin{array}[]{*{20}c}0&2&2&8\\\
2&0&0&2\\\ 2&0&0&2\\\ 8&2&2&0\\\ \end{array}}\right)$,
$\left|{g_{i}}\right\rangle$ is the ground state of the $i$ qubit,
$\left|{e_{i}}\right\rangle$ is the excited state of the $i$ qubit.
Figure 1: (Color online) The functions
$G_{P}\equiv\frac{{10}}{{\omega^{2}}}[-2M\times\Im_{N_{t}}(\omega,\Delta)+\aleph]$,
$G_{D}\equiv\frac{{10\coth(\frac{\omega}{{2T}})}}{{\omega^{2}}}\Re_{N_{t}}(\omega,\Delta)$,$G_{S}\equiv
0.1\times J(\omega)$,
$G_{PS}\equiv\frac{{J(\omega)}}{{\omega^{2}}}[-2M\times\Im_{N_{t}}(\omega,\Delta)+\aleph]$,
$G_{DS}\equiv\frac{{J(\omega)\coth(\frac{\omega}{{2T}})}}{{\omega^{2}}}\Re_{N_{t}}(\omega,\Delta)$.
Parameters are: $\omega_{c}=34$MHz; $\eta=135$; $\Omega_{1}=10$MHz;
$\Omega_{2}=10$MHz; $T=1$K; $\Delta=16$ns; $M=3$; $N_{d}=8$.
Created by the common bath, the phase $\Theta(t)$ establishes the nonlocal
correlation (entanglement) between the two qubits.
Now we consider to simultaneously use UDD ($f_{1}(t)=f_{2}(t)=f(t)$) on the
two qubits during the evolution of qubits-bath system form $(l-1)\Delta$ to
$l\Delta$, with a number $l$, a time period $\Delta$. This simultaneous
control can on one hand eliminate decoherence on the other hand keep the the
nonlocal correlation (entanglement) between the two qubits. UDD UDD ; PUDD
was originally proposed for suppressing the pure dephasing of a single qubit.
If the pure-dephasing is described by $\sigma_{z}$-type error (we use standard
notation for Pauli matrices), then a UDD sequence of instantaneous $\pi$
pulses of the $\sigma_{x}$ form is applied at
$\displaystyle t_{j}=t\sin^{2}(\frac{{j\pi}}{{2N_{d}+2}}),j=1,2,...,N_{d},$
(14)
with $N_{d}+1$ pulse intervals during the time period $(0,t]$. For convenience
we also define $t_{N_{d}+1}=t$. For odd $N_{d}$, an additional control pulse
is applied at time $t_{N_{d}+1}$. ReferencePUDD proved that such a control
sequence can protect the expectation value of $\sigma_{x}$ to the $N_{d}$th
order in a universal fashion, irrespective of qubit-environment coupling. This
can be shown by an effective Hamiltonian that only contains even powers of
$\sigma_{z}$.
_Fast generation of quantum entanglement._ After we perform UDD operation
above, as the most important quantity, the phase $\Theta$ is given by
$\displaystyle\Theta(t){\text{ =
}}\int\limits_{\text{0}}^{\infty}{d\omega\frac{{J\left(\omega\right)}}{{\omega^{2}}}}[-2M\times\Im_{N_{d}}(\omega,\Delta)+\aleph].$
(15)
Here $J(\omega)$ is the spectrum of standard Ohmic bath
$\displaystyle J(\omega)$ $\displaystyle=$
$\displaystyle\sum\limits_{j}{\left|{\lambda_{j}}\right|^{2}\delta\left({\omega-\omega_{j}}\right)}$
(16) $\displaystyle=$ $\displaystyle\eta\omega e^{-\omega/\omega_{c}},$
where $\eta$ is the dimensionless parameter determining the coupling strength
between qubit and bath, $\omega_{c}$ is the high-energy cutoff value. The
functional $\Im_{N_{d}}$ is
$\displaystyle\Im_{N_{d}}(\omega,\Delta)$ $\displaystyle=$
$\displaystyle\omega^{2}\int\limits_{0}^{\Delta}{\int\limits_{0}^{t_{1}}{dt_{1}dt_{2}f\left({t_{1}}\right)f\left({t_{2}}\right)}}\sin\left[{\omega_{j}\left({t_{1}-t_{2}}\right)}\right],$
and
$\displaystyle\aleph=-\frac{1}{{{\rm{2}}i}}\int\limits_{0}^{\infty}\frac{{J\left(\omega\right)}}{{\left({1-\cos\omega\Delta}\right)}}$
$\displaystyle\times\left\\{{\left[{1-e^{i\omega\Delta
M}-M\left({1-{\mathop{\rm
e}\nolimits}^{i\omega\Delta}}\right)}\right]\left|{f(\omega,\Delta)}\right|^{2}-h.c.}\right\\}d\omega,$
with $M=\frac{t}{\Delta}$. In the Eq.(Fast creation of conditional quantum
gate and entanglement using a common bath only),
$\displaystyle f(\omega,\Delta)=1+(-1)^{N_{d}+1}{\mathop{\rm
e}\nolimits}^{i\omega\Delta}+2\sum\limits_{p=1}^{N_{d}}{(-1)^{p}e^{i\omega\Delta\delta_{p}}}$
(19)
with $\delta_{p}=\sin^{2}\frac{\pi\times p}{2\times(N_{d}+1)}$, is determined
by $f(t)$ via the following relation
$\displaystyle
f\left({\omega,\Delta}\right)=-i\omega\int\limits_{0}^{\Delta}{dte^{i\omega
t}f\left(t\right)}.$ (20)
The entangling gate is performed by the effective Hamiltonian $H_{p}$. The
concurrence of the two qubits oscillates between zero and one, when the value
of $\Theta(t)$ rises.
The decoherence function is given by
$\displaystyle\Upsilon\left(t\right)=\int_{0}^{\infty}{d\omega\frac{{J\left(\omega\right)\coth(\frac{\omega}{{2k_{B}T}})}}{{\omega^{2}}}\Re_{N_{t}}(\omega,\Delta)},$
(21)
with
$\displaystyle\Re_{N_{t}}(\omega,T)=\left|{\frac{{1-e^{i\omega\Delta
M}}}{{1-e^{i\omega\Delta}}}f\left({\omega,\Delta}\right)}\right|^{2}.$ (22)
Figure 2: (Color online) Comparison of entanglement concurrence for UDD and
free evolution. Curve 1: $\eta=1$; $\eta_{1}=\eta_{2}=0$; $\omega_{c}=30$MHz;
$T=0.08$mK; $\Delta$=60ns; $M=16$; $\Omega_{1}=10$MHz; $\Omega_{2}=10$MHz;
$N_{d}=9$. Carve 2: $\eta=\eta_{1}=\eta_{2}=10$;
$\omega_{c}=\omega_{c_{1}}=\omega_{c_{2}}=30$MHz; $T=1$mK; $\Delta$=29ns;
$M=9$; $\Omega_{1}=10$MHz; $\Omega_{2}=10$MHz; $N_{d}=7$. Carve 3:
$\eta=\eta_{1}=\eta_{2}=100$;
$\omega_{c}=\omega_{c_{1}}=\omega_{c_{2}}=30$MHz; $T=1$K; $\Delta$=16ns;
$M=8$;$\Omega_{1}=10$MHz; $\Omega_{2}=10$MHz; $N_{d}=8$. $\eta_{i}$ is the
dimensionless parameter controlling the coupling between the $i$th qubit and
its individual bath. $\omega_{c_{i}}$ is the high-energy cutoff frequency of
the $i$th qubit of individual bath.
The UDD modulation function $f(t)$ changes both the phase (associated with
$G_{PS}$) and decoherence (associated with $G_{DS}$). The function
$|f(\omega,t)|$ is minimized to its $N_{d}$-th order in time as shown in Refs.
UDD ; PUDD . As one can see in the Fig.1, after simultaneous UDD, the peak
position of $\Re$ (associated with $G_{D}$) is moved to a position much larger
than the cutoff frequency $\omega_{c}$ in spectrum density functional
$J(\omega)$ (If the spectrum is not soft LBUDD1 ; LBUDD2 ; LBUDD3 ). So the
overlap between functional $\Re$ and the spectrum density $J(\omega)$ is
negligible, which means the decoherence is almost suppressed. For the same
reason, $\aleph$ is also almost eliminated. However, as shown with Ref. TUDD ,
the phase $\Theta$ is quadratic in functional $f(t)$, so that the UDD sequence
does not reduce this term. In this case, the functional $\Im$ (associated with
$G_{P}$) takes non-negligible value in low frequency region, while the phase
evolution $\Theta(t)$ (associated with $G_{DS}$) is still in action.
The progress for entanglement creation also generates a quantum gate. The
entangling gate $U=\exp(i\Theta\sigma_{1}^{z}\sigma_{2}^{z})$ refers to
conditional phase gate, with $\Theta\propto M$. If we have the information of
the spectrum density which can be detected with DD MSDD , we can design the
periodic time $\Delta$ and UDD sequence to achieve the entangling gate more
effectively. The general case (common bath and individual baths) is considered
in Appendix B.
_Conclusion._ The associated-environment is used to create the entanglement,
when simultaneous UDD is applied. The strong coupling between the qubits and
the environment is considered. Without UDD, the decoherence would destroy the
correlation between the qubits before the entanglement of two qubits grows up,
as shown in Fig.2. When simultaneous UDD is used, the decoherence is
significantly reduced. The common bath with strong coupling can generate the
entanglement in a short time. Within such a short time, the decoherence is
negligible. When the common bath is a single-mode harmonic oscillator, our
scheme also work. We look forward to developing this scheme in spin squeezing.
###### Acknowledgements.
We thank Jiangbin Gong for motivating discussion. This work is supported in
part by the 10000-Plan of Shandong province and the National High-Tech Program
of China grant No. 2011AA010800 and 2011AA010803, NSFC grant No. 11174177 and
60725416.
## APPENDIX
### .1
To obtain a closed-form expression for the time-ordered unitary operator
$\displaystyle
U(t)=T_{\leftarrow}\exp\left({-i\int\limits_{0}^{t}{H_{I}(t_{1})dt_{1}}}\right),$
(23)
we resort to the Magnus expansion of the exponent of $U(t)=\exp(\Omega(t))$.
The first few terms of the expansion are
$\displaystyle\Omega(t)$ $\displaystyle=$
$\displaystyle-i\int\limits_{0}^{t}{H_{I}(t_{1})dt_{1}}+\frac{1}{2}\int\limits_{0}^{t}{dt_{1}\int\limits_{0}^{t_{1}}{dt_{2}\left[{H_{I}(t_{1}),H_{I}(t_{2})}\right]}}$
$\displaystyle+\cdots.$
We now take advantage of the remarkable property of bosonic bath operators,
namely, that the commutator of the interaction Hamiltonian at two different
times is a C-number function in the bath operators:
$\displaystyle\left[{H_{I}(t_{1}),H_{I}(t_{2})}\right]=$
$\displaystyle-2i\left({f_{1}\left({t_{1}}\right)f_{2}\left({t_{2}}\right)+f_{2}\left({t_{2}}\right)f_{1}\left({t_{1}}\right)}\right)$
$\displaystyle\times\sum\limits_{j}{\left|{\lambda_{j}}\right|^{2}\sin\left[{\omega_{j}\left({t_{1}-t_{2}}\right)}\right]\sigma_{1}^{z}\sigma_{2}^{z}}.$
Since $\sigma_{1}^{z}\sigma_{2}^{z}$ commutes with all its powers, the fact
that this commutator is a C-number implies that only the first two terms of
the expansion are non-zero. Now, the closed-form equation for the time-
evolution operator takes the simple form
$\displaystyle
U\left(t\right)=\exp\left[{-i\int\limits_{0}^{t}{H_{I}(t_{1})dt_{1}}-i\Theta(t)\sigma_{1}^{z}\sigma_{2}^{z}}\right],$
(26)
where
$\displaystyle\Theta(t)=\int\limits_{0}^{t}{\int\limits_{0}^{t_{1}}}$
$\displaystyle\left({f_{1}\left({t_{1}}\right)f_{2}\left({t_{2}}\right)+f_{2}\left({t_{1}}\right)f_{1}\left({t_{2}}\right)}\right)$
(27)
$\displaystyle\times\sum\limits_{j}{\left|{\lambda_{j}}\right|^{2}\sin\left[{\omega_{j}\left({t_{1}-t_{2}}\right)}\right]}{dt_{1}dt_{2}}.$
### .2
In the case of two qubits are unsymmetrically coupled with their common bath
and individual baths, the Hamiltonian in the interaction picture takes the
form as
$\displaystyle H_{I}$ $\displaystyle=$
$\displaystyle\sigma_{1}^{z}f_{1}(t)\left[{\sum\limits_{j}{\lambda_{j}\left({a_{j}^{{\dagger}}e^{(i\omega_{j}t)}+a_{j}e^{(-i\omega_{j}t)}}\right)}}\right]$
$\displaystyle+\sigma_{1}^{z}f_{1}(t)\left[{\sum\limits_{j}{\lambda_{1,j}\left({a_{1,j}^{{\dagger}}e^{(i\omega_{1,j}t)}+a_{1,j}e^{(-i\omega_{1,j}t)}}\right)}}\right]$
$\displaystyle+\sigma_{2}^{z}f_{2}(t)\left[{\sum\limits_{j}{\lambda_{j}^{{}^{\prime}}\left({a_{j}^{{\dagger}}e^{(i\omega_{j}t)}+a_{j}e^{(}-i\omega_{j}t)}\right)}}\right]$
$\displaystyle+\sigma_{1}^{z}f_{2}(t)\left[{\sum\limits_{j}{\lambda_{2,j}\left({a_{2,j}^{{\dagger}}e^{(i\omega_{2,j}t)}+a_{2,j}e^{(-i\omega_{2,j}t)}}\right)}}\right].$
The phase is given by
$\displaystyle\Theta(t)$ $\displaystyle=$
$\displaystyle-2\int\limits_{0}^{t}{dt_{1}\int\limits_{0}^{t_{1}}{dt_{2}\sum\limits_{j}{\lambda_{j}\lambda^{\prime}_{j}{\rm
X}(t_{1},t_{2},\omega_{j})}}}$ (29) $\displaystyle=$
$\displaystyle-2\int\limits_{0}^{t}{dt_{1}\int\limits_{0}^{t_{1}}{dt_{2}\int\limits_{0}^{\infty}{d\omega}\bar{J}(\omega){\rm
X}(t_{1},t_{2},\omega)}},$
where ${\rm X}(t_{1},t_{2},\omega)=f(t_{1})f(t_{2})\sin[\omega(t_{1}-t_{2})]$,
$\bar{J}\left(\omega\right)=\sqrt{J\left(\omega\right)J^{{}^{\prime}}\left(\omega\right)}$.
The decoherence function is written as
$\displaystyle\Upsilon\left(t\right)=\int_{0}^{\infty}{d\omega\frac{{\tilde{J}\left(\omega\right)\coth(\frac{\omega}{{2T}})}}{{\omega^{2}}}\Re_{N_{t}}(\omega,T)},$
(30)
where $\Theta_{1,2}=\Theta_{1,3}=2\Theta$,
$\Theta_{2,4}=\Theta_{3,4}=-2\Theta$, $\Theta_{1,4}=\Theta_{2,3}=0$,
$\Upsilon_{1,2}=\Upsilon_{1,3}=2\Upsilon$,
$\Upsilon_{2,4}=\Upsilon_{3,4}=2\Upsilon$, $\Upsilon_{1,4}=8\Upsilon$,
$\Upsilon_{2,3}=2\Upsilon$,
$\tilde{J}_{1,2}\left(\omega\right)=J^{{}^{\prime}}\left(\omega\right)+J_{2}\left(\omega\right)$,
$\tilde{J}_{1,3}\left(\omega\right)=J\left(\omega\right)+J_{1}\left(\omega\right)$,
$\tilde{J}_{1,4}\left(\omega\right)=\frac{J\left(\omega\right)+J^{{}^{\prime}}\left(\omega\right)+2\bar{J}\left(\omega\right)+J_{1}\left(\omega\right)+J_{2}\left(\omega\right)}{4}$,
$\tilde{J}_{2,3}\left(\omega\right)=J\left(\omega\right)+J^{{}^{\prime}}\left(\omega\right)-2\bar{J}\left(\omega\right)+J_{1}\left(\omega\right)+J_{2}\left(\omega\right)$,
$\tilde{J}_{2,4}\left(\omega\right)=J\left(\omega\right)+J_{1}\left(\omega\right)$,
$\tilde{J}_{3,4}\left(\omega\right)=J^{{}^{\prime}}\left(\omega\right)+J_{2}\left(\omega\right)$.
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|
arxiv-papers
| 2013-10-07T11:44:17 |
2024-09-04T02:49:52.080854
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Nan Qiu and Xiang-Bin Wang",
"submitter": "Xiang-Bin Wang",
"url": "https://arxiv.org/abs/1310.1738"
}
|
1310.1872
|
# Complex absorbing potential method
for the perturbed Dirac operator 111First version 12 february 2012; second
version 23 May 2012 (ArXiv version)
J. Kungsman
Department of Mathematics
Uppsala University
SE-751 06 Uppsala, Sweden M. Melgaard
Department of Mathematics
School of Mathematical and Physical Sciences
University of Sussex
Brighton BN1 9QH
Great Britain The second author acknowledges support by Science Foundation
Ireland
(October 7, 2013)
###### Abstract
The Complex Absorbing Potential (CAP) method is widely used to compute
resonances in Quantum Chemistry, both for nonrelativistic and relativistic
Hamiltonians. In the semiclassical limit $\hbar\to 0$ we consider resonances
near the real axis and we establish the CAP method rigorously for the
perturbed Dirac operator by proving that individual resonances are perturbed
eigenvalues of the nonselfadjoint CAP Hamiltonian, and vice versa. The proofs
are based on pseudodifferential operator theory and microlocal analysis.
###### Contents
1. 1 Introduction
2. 2 Preliminaries
3. 3 Dirac and CAP Hamiltonians
4. 4 Complex distortion and resonances
1. 4.1 Complex distortion
2. 4.2 Resonances
5. 5 Main results
6. 6 Properties of CAP Hamiltonians
7. 7 Quasimodes and resonances
8. 8 Proof of main results
1. 8.1 Approximating a single eigenvalue when $R_{0}^{\prime}<R_{1}$
2. 8.2 Approximating a single eigenvalue when $R_{1}\leq R_{0}^{\prime}$
9. A Semiclassical maximum principle
## 1 Introduction
One of the most successful methods for computing resonances in Quantum
Chemistry is the Complex Absorbing Potential (CAP) method, partly because it
yields good approximations to the true resonances and, partly, because it is
easy to implement numerically (see, e.g., Muga et al. [Mu’04]).
Within the semiclassical limit, i.e., as Planck’s “constant” $\hbar$ tends to
zero, we study the CAP method rigorously when the governing Hamiltonian is a
semiclassical Dirac operator
$\mathbb{D}=-ic\hbar\sum_{j=1}^{3}\alpha_{j}\partial_{x_{j}}+\beta
mc^{2}+\mathbb{V}(x),$
acting on
${\mathbf{L}}^{2}({\mathbb{R}}^{3};{\mathbb{C}}^{4})=\bigoplus_{j=1}^{4}{\mathbf{L}}^{2}({\mathbb{R}}^{3})=:({\mathbf{L}}^{2}({\mathbb{R}}^{3}))^{4}$.
Here the $\\{\alpha_{j}\\}_{j=1}^{3}$ and $\beta=\alpha_{4}$ are $4\times 4$
Dirac matrices obeying the anti-commutation relations
$\alpha_{j}\alpha_{k}+\alpha_{k}\alpha_{j}=2\delta_{jk}\mbox{\boldmath$I$\unboldmath}_{4},\quad
1\leq j,k\leq 4,$
where $\mbox{\boldmath$I$\unboldmath}_{n}$ is the $n\times n$ identity matrix.
The potential $\mathbb{V}$ is assumed to have compact support.
We define resonances through the method of complex distortion which has been
widely applied in the context of Schrödinger operators but which subsequently
was carried over to Dirac resonances in [Se’88]. Thus the resonances
$z(\hbar)=E(\hbar)\pm\Gamma(\hbar)/2$ appear as eigenvalues of a non-
selfadjoint operator $\mathbb{D}_{\theta}$ associated with $\mathbb{D}$. In
applications one is interested in computing the resonance energy $E$ and the
width $\Gamma$, which is the inverse of the life-time of the corresponding
resonant state. One way to do so is the CAP method, i.e., to augment the
Hamiltonian by an imaginary potential and consider eigenvalues of the
perturbed Hamiltonian as good approximations of the true resonances. In this
paper we justify this method in the semiclassical approximation for resonances
with $\Gamma(\hbar)={\mathcal{O}}(\hbar^{N})$, $N\gg 1$, and show that such
resonances give rise to eigenvalues of the CAP Hamiltonian
$\mathbb{J}:=\mathbb{D}-i\mathbb{W}$ within distance at most
$\hbar^{-5}\log(\hbar^{-1})\Gamma(\hbar)+{\mathcal{O}}(\hbar^{\infty})$. Also
the converse implication is proved. Both of these results hold under the
assumption that the CAP is zero in the interaction region, i.e. the support of
the potential $\mathbb{V}$, and “switched on” outside this region. In
numerical implementations, however, the “switch-on” point is moved inward
towards the interaction region as much as possible to minimize the number of
grid points used. If the classical Hamiltonian vector fields generated by the
eigenvalues of the principal symbol of $\mathbb{D}$ are nontrapping (see
Definition 3.2), one can allow the supports to intersect which at worst
increases the error by a factor $\hbar^{-1}$. This requires the use of an
Egorov type theorem for matrix valued Hamiltonians, which enables one to
express the time evolution of quantum observables (self-adjoint operators) in
the semiclassical limit in terms of a classical dynamics of principal (matrix)
symbols. The mentioned results deal with single resonances/eigenvalues and
give no information regarding multiplicities; clusters of resonances will be
treated in a future work.
Despite its success in Physics and Chemistry, only few rigorous justifications
of the method exist. For (nonrelativistic, scalar valued) Schrödinger
operators with compactly supported electric potentials, Stefanov [St’05] was
the first to establish results similar to the above-mentioned ones. In the
“non-intersecting” case, he starts from a resonance and then, by considering a
cutoff resonant state (see Section 8.1), he constructs a quasimode (see
Section 7) which generates a perturbed resonance. In the “intersecting” case,
the previous scheme of proof only applies after a refined microlocal analysis,
involving a propagation-of-singularities argument. Recently Kungsman and
Melgaard carried over Stefanov’s results to matrix valued Schrödinger
operators [KuMe’10]. The matrix valued setting is more complicated, in
particular, in the “intersecting” case, where one has to begin by solving
Heisenberg’s equations of motion semiclassically. Then, by applying a
localization result away from the semiclassical wavefront set, it is possible
to investigate how singularities propagate in this situation. The Egorov type
statement, which is part of the proof by Kungsman and Melgaard [KuMe’10]
differs from the scalar case because one also needs to propagate the matrix
degrees of freedom. To push through this scheme of proof for matrix valued
Schrödinger operators, it was necessary to impose an additional technical (and
restrictive) assumption in [KuMe’10]. An interesting feature of the present
work, for the perturbed Dirac operator (also a matrix structure), is that one
can avoid such technicalities, thus obtaining more natural and better results,
and the afore-mentioned scheme of proof (using cutoff resonant states, Egorov
type result, propagation of singularity argument and quasimodes), developed in
[KuMe’10], can be carried through, using a “full” version of the matrix valued
Egorov type theorem, see Lemma 8.1. We interpret this as yet another evidence
of the fact that Dirac’s description of the electron is a better physical
model.
Other rigorous results on resonances for Dirac operators are found in [Pa’91,
Pa’92, BaHe’92, AmBrNo’01, Kh’07].
## 2 Preliminaries
Notation. Throughout the paper we denote by $C$ (with or without indices)
various positive constants whose precise value is of no importance and their
values may change from line to line; the “constants” usually depend on various
parameters but not on $\hbar$. For $x_{0}\in{\mathbb{R}}^{3}$ and $R>0$ the
notation
$B(x_{0},R)=\\{x\,:\,|x-x_{0}|<R\\}$
means an open ball centered at $x_{0}$ having radius $R$. For
$x\in{\mathbb{R}}^{3}$ we denote $\langle x\rangle:=(1+|x|^{2})^{1/2}$. For a
complex number $\zeta\in{\mathbb{C}}\setminus[-\infty,0)$, we denote by
$\zeta^{\frac{1}{2}}$ its branch of the square root with positive real part.
The set
$D(\zeta,r)=\\{z\in{\mathbb{C}}:|z-\zeta|<r,\>\zeta\in{\mathbb{C}},\>r>0\\}$
defines an open disk in ${\mathbb{C}}$ with center in $\zeta$ and radius $r$.
Complex rectangles $\\{z\in{\mathbb{C}}\,:\,l\leq\operatorname{{\rm
Re}\,}z\leq r,\,b\leq\operatorname{{\rm Im}\,}z\leq t\\}$ are written
$\displaystyle[l,r]+i[b,t].$ (2.1)
We shall denote by $\mathrm{M}_{4}({\mathbb{C}})$ the set of all $4\times 4$
matrices over ${\mathbb{C}}$, equipped with the operator norm denoted by
$\|\cdot\|_{4\times 4}$. We let
${\mathcal{H}}:={\mathbf{L}}^{2}({\mathbb{R}}^{3},{\mathbb{C}}^{4})$ be the
space of (equivalence classes of) ${\mathbb{C}}^{4}$-valued functions
$\mbox{\boldmath$u$\unboldmath}=(u_{1},u_{2},u_{3},u_{4})^{t}$ on
${\mathbb{R}}^{3}$ endowed with the inner product
$\langle\mbox{\boldmath$u$\unboldmath},\mbox{\boldmath$v$\unboldmath}\rangle=\sum_{j=1}^{4}\int_{{\mathbb{R}}^{3}}u_{i}\overline{v_{i}}\,dx$
such that
$\langle\mbox{\boldmath$u$\unboldmath},\mbox{\boldmath$u$\unboldmath}\rangle=:\|\mbox{\boldmath$u$\unboldmath}\|^{2}$
is finite. The space $C_{0}^{\infty}({\mathbb{R}}^{3})$ consists of all
infinitely differentiable functions on ${\mathbb{R}}^{3}$ with compact
support. We let $D_{x_{j}}=-i\partial/\partial x_{j}$ and
$D^{\gamma}=D_{x_{1}}^{\gamma_{1}}D_{x_{2}}^{\gamma_{1}}D_{x_{3}}^{\gamma_{3}}$
with standard multi-index notation
$\gamma=(\gamma_{1},\gamma_{2},\gamma_{3})\in{\mathbb{N}}_{0}^{3}$. The
semiclassical Sobolev space of order one is denoted by
${\mathbf{H}}^{1}({\mathbb{R}}^{3},{\mathbb{C}}^{4})$ and is equipped with the
norm
$\|\mbox{\boldmath$u$\unboldmath}\|_{{\mathbf{H}}^{1}}^{2}=\sum_{j=1}^{4}\int_{{\mathbb{R}}^{3}}(|\hbar\nabla
u_{j}|^{2}+|u_{j}|^{2})\,dx.$
Moreover, the Schwartz space of rapidly decreasing functions and its dual
space of tempered distributions are denoted by $\sc\mbox{S}\hskip
1.0pt({\mathbb{R}}^{3},{\mathbb{C}}^{4})$ and $\sc\mbox{S}\hskip
1.0pt^{\prime}({\mathbb{R}}^{3},{\mathbb{C}}^{4})$, respectively. For
$\chi_{1},\chi_{2}\in C^{\infty}_{0}({\mathbb{R}}^{n},[0,1])$ we use
$\chi_{1}\prec\chi_{2}$ to indicate that $\chi_{2}=1$ in a neighborhood of
$\operatorname{supp\,}\chi_{1}$ (i.e., the support of $\chi_{1}$). We always
assume cut-off functions take their values in $[0,1]$.
Operators. If $A$ is an operator on
${\mathbf{L}}^{2}({\mathbb{R}}^{3},{\mathbb{C}}^{4})$ its domain is denoted
$\operatorname{Dom\,}(\mbox{\boldmath$A$\unboldmath})$. The spectrum of $A$ is
the disjoint union of the discrete and essential spectra of $A$ and is
designated by
$\operatorname{spec\,}(\mbox{\boldmath$A$\unboldmath})=\operatorname{spec}_{\operatorname{d}}(\mbox{\boldmath$A$\unboldmath})\cup\operatorname{spec}_{\operatorname{ess}}(\mbox{\boldmath$A$\unboldmath})$.
Moreover, its resolvent set is denoted by
$\rho(\mbox{\boldmath$A$\unboldmath})$ and its resolvent is
$\mbox{\boldmath$R$\unboldmath}(\zeta)=(\mbox{\boldmath$A$\unboldmath}-\zeta)^{-1}$.
The spaces of bounded and compact operators between Hilbert spaces
$\mathcal{H}_{1}$ and $\mathcal{H}_{2}$ are denoted by
$\mathcal{B}({\mathcal{H}_{1}},{\mathcal{H}_{2}})$ and
$\mathcal{B}_{\infty}({\mathcal{H}_{1}},{\mathcal{H}_{2}})$, respectively. If
$\mathcal{H}:=\mathcal{H}_{1}=\mathcal{H}_{2}$ we use the notation
$\mathcal{B}(\mathcal{H})$ and $\mathcal{B}_{\infty}(\mathcal{H})$,
respectively. The commutator of two operators $A$ and $B$, when defined, is
denoted
$[\mbox{\boldmath$A$\unboldmath},\mbox{\boldmath$B$\unboldmath}]=\mbox{\boldmath$AB$\unboldmath}-\mbox{\boldmath$BA$\unboldmath}$.
The number of eigenvalues or resonances (counting multiplicities) of $A$ on a
set $\Omega\subset{\mathbb{C}}$ will be denoted
$\operatorname{Count\,}(\mbox{\boldmath$A$\unboldmath},\Omega)$. Scalar-
valued, respectively matrix-valued, operators are denoted by capitals,
respectively boldface capitals, e.g.
$\mbox{\boldmath$\chi$\unboldmath}=\chi\mbox{\boldmath$I$\unboldmath}_{4}$. If
$\mbox{\boldmath$A$\unboldmath}\in{\mathcal{B}}_{\infty}({\mathcal{H}})$ and
if for some orthonormal basis $\\{\mbox{\boldmath$f$\unboldmath}_{j}\\}$ of
${\mathcal{H}}$ the sum
$\displaystyle\sum_{j}\langle(\mbox{\boldmath$A$\unboldmath}^{\ast}\mbox{\boldmath$A$\unboldmath})^{1/2}\mbox{\boldmath$f$\unboldmath}_{j},\mbox{\boldmath$f$\unboldmath}_{j}\rangle$
(2.2)
is finite, then this property is independent of the choice of orthonormal
basis and we say that $A$ is of trace class, in symbols
$\mbox{\boldmath$A$\unboldmath}\in{\mathcal{B}}_{1}({\mathcal{H}})$, and the
trace norm $\|\mbox{\boldmath$A$\unboldmath}\|_{{\mathcal{B}}_{1}}$ is given
by (2.2). Equivalently $\mbox{\boldmath$A$\unboldmath}\in{\mathcal{B}}_{1}$ if
and only if the sequence
$\mu_{1}(\mbox{\boldmath$A$\unboldmath})\geq\mu_{2}(\mbox{\boldmath$A$\unboldmath})\geq\cdots$
of eigenvalues of
$(\mbox{\boldmath$A$\unboldmath}^{\ast}\mbox{\boldmath$A$\unboldmath})^{1/2}$,
called singular values of $A$, is summable. The singular values satisfy Ky
Fan’s inequalities
$\displaystyle\mu_{i+j-1}(\mbox{\boldmath$A$\unboldmath}+\mbox{\boldmath$B$\unboldmath})$
$\displaystyle\leq\mu_{i}(\mbox{\boldmath$A$\unboldmath})+\mu_{j}(\mbox{\boldmath$B$\unboldmath})$
(2.3)
$\displaystyle\mu_{i+j-1}(\mbox{\boldmath$A$\unboldmath}\mbox{\boldmath$B$\unboldmath})$
$\displaystyle\leq\mu_{i}(\mbox{\boldmath$A$\unboldmath})\mu_{j}(\mbox{\boldmath$B$\unboldmath})$
(2.4)
for $i,j\geq 0$ and
$\mbox{\boldmath$A$\unboldmath},\mbox{\boldmath$B$\unboldmath}\in{\mathcal{B}}_{\infty}$
and also
$\displaystyle\mu_{j}(\mbox{\boldmath$A$\unboldmath}\mbox{\boldmath$B$\unboldmath})$
$\displaystyle\leq\|\mbox{\boldmath$B$\unboldmath}\|\mu_{j}(\mbox{\boldmath$A$\unboldmath})$
(2.5)
$\displaystyle\mu_{j}(\mbox{\boldmath$B$\unboldmath}\mbox{\boldmath$A$\unboldmath})$
$\displaystyle\leq\|\mbox{\boldmath$B$\unboldmath}\|\mu_{j}(\mbox{\boldmath$A$\unboldmath})$
(2.6)
whenever $\mbox{\boldmath$A$\unboldmath}\in{\mathcal{B}}_{\infty}$ and
$\mbox{\boldmath$B$\unboldmath}\in{\mathcal{B}}$. When $A$ is of trace class
it is possible to extend the relation
$\det(\mbox{\boldmath$1$\unboldmath}-\mbox{\boldmath$A$\unboldmath})=\prod_{j}(1-\lambda_{j})$
for $A$ of finite rank, where $\lambda_{j}$ are the eigenvalues of $A$,
repeated according to multiplicity, so that
$\det(\mbox{\boldmath$1$\unboldmath}-\mbox{\boldmath$A$\unboldmath})\neq 0$ if
and only if $\mbox{\boldmath$1$\unboldmath}-\mbox{\boldmath$A$\unboldmath}$ is
invertible and
$\det(\mbox{\boldmath$1$\unboldmath}-\mbox{\boldmath$A$\unboldmath})\leq
e^{\|\mbox{{\scriptsize\boldmath$A$\unboldmath}}\|_{{\mathcal{B}}_{1}}}=e^{\sum\mu_{j}(\mbox{{\scriptsize\boldmath$A$\unboldmath}})}.$
(2.7)
holds for any $A$ of trace class; see, e.g., [Sj’02] for details.
Pseudodifferential operators. For the (trivial) cotangent bundle of
${\mathbb{R}}^{3}$ we write ${\mathsf{T}}^{\ast}{\mathbb{R}}^{3}$ and it is
sometimes convenient to think of it as the product of space and frequency,
i.e.
${\mathsf{T}}^{\ast}{\mathbb{R}}^{3}={\mathbb{R}}_{x}^{3}\times{\mathbb{R}}_{\xi}^{3}$.
Let $m:{\mathsf{T}}^{\ast}{\mathbb{R}}^{3}\to{\mathbb{R}}_{+}$ be a so called
order function, i.e. a smooth function such that there are $C,N>0$ so that
$m(x,\xi)\leq C\big{(}1+(x-y)^{2}+(\xi-\eta)^{2}\big{)}^{N/2}m(y,\eta)$
for all $(x,\xi),(y,\eta)\in{\mathsf{T}}^{\ast}{\mathbb{R}}^{3}$. Then we
define ${\mathsf{S}}(m)\subset
C^{\infty}{({\mathsf{T}}^{\ast}{\mathbb{R}}^{3})}\otimes\mathrm{M}_{4}({\mathbb{C}})$
to consist of all $\mbox{\boldmath$a$\unboldmath}\in
C^{\infty}{({\mathsf{T}}^{\ast}{\mathbb{R}}^{3})}\otimes\mathrm{M}_{4}({\mathbb{C}})$
such that for all multi-indices $\alpha,\beta\in{\mathbb{N}}_{0}^{3}$ there
are constants $C_{\alpha,\beta}>0$ with
$\|\partial_{\xi}^{\alpha}\partial_{x}^{\beta}\mbox{\boldmath$a$\unboldmath}(x,\xi)\|_{4\times
4}\leq C_{\alpha,\beta}m(x,\xi)\quad\text{for all
}(x,\xi)\in{\mathsf{T}}^{\ast}{\mathbb{R}}^{3}.$
For $\mbox{\boldmath$a$\unboldmath}\in{\mathsf{S}}(m)$ we can define a
corresponding Weyl quantization $\mbox{\boldmath$A$\unboldmath}={\rm
op}^{W}[{\mbox{\boldmath$a$\unboldmath}}]$ on
${\mathbf{L}}^{2}({\mathbb{R}}^{3},{\mathbb{C}}^{4})$ by
$(\mbox{\boldmath$A$\unboldmath}\mbox{\boldmath$u$\unboldmath})(x)=\frac{1}{(2\pi\hbar)^{3}}\iint\limits_{{\mathsf{T}}^{\ast}{\mathbb{R}}^{3}}e^{i(x-y)\cdot\xi/\hbar}\mbox{\boldmath$a$\unboldmath}\Big{(}\frac{x+y}{2},\xi\Big{)}\mbox{\boldmath$u$\unboldmath}(y)\,dy\,d\xi.$
For symbols that are bounded with all their derivatives we have the celebrated
result by Calderon-Vaillancourt [DiSj’99, Theorem 7.11].
###### Proposition 2.1.
Let $\mbox{\boldmath$a$\unboldmath}\in{\mathsf{S}}(1)$. Then ${\rm
op}^{W}[{\mbox{\boldmath$a$\unboldmath}}]$ defines a continuous operator on
${\mathbf{L}}^{2}({\mathbb{R}}^{3},{\mathbb{C}}^{4})$.
We recall that if $m_{1},\,m_{2}$ are order functions and
$\mbox{\boldmath$a$\unboldmath}\in{\mathsf{S}}(m_{1})$,
$\mbox{\boldmath$b$\unboldmath}\in{\mathsf{S}}(m_{2})$ then $m_{1}m_{2}$ is an
order function and there exists
$\mbox{\boldmath$a$\unboldmath}\\#\mbox{\boldmath$b$\unboldmath}\in{\mathsf{S}}(m_{1}m_{2})$
so that
$\displaystyle{\rm op}^{W}[{\mbox{\boldmath$a$\unboldmath}}]{\rm
op}^{W}[{\mbox{\boldmath$b$\unboldmath}}]={\rm
op}^{W}[{\mbox{\boldmath$a$\unboldmath}\\#\mbox{\boldmath$b$\unboldmath}}].$
(2.8)
If for $\mbox{\boldmath$a$\unboldmath}\in{\mathsf{S}}(m)$ there are
$\mbox{\boldmath$a$\unboldmath}_{j}\in{\mathsf{S}}(m)$ so that for any
$N\in{\mathbb{N}}$ and $\alpha,\beta\in{\mathbb{N}}_{0}^{3}$ there exists
$C_{N,\alpha}>0$ such that
$\|\partial_{\xi}^{\alpha}\partial_{x}^{\beta}(\mbox{\boldmath$a$\unboldmath}-\sum_{j=0}^{N-1}\hbar^{j}\mbox{\boldmath$a$\unboldmath}_{j})\|\leq
C_{N,\alpha}\hbar^{N}m$
then we write $\mbox{\boldmath$a$\unboldmath}\sim\sum_{j\geq
0}\hbar^{j}\mbox{\boldmath$a$\unboldmath}_{j}$ and we call
$\mbox{\boldmath$a$\unboldmath}_{0}$ and $\mbox{\boldmath$a$\unboldmath}_{1}$
the principal, and subprincipal symbol of ${\rm
op}^{W}[{\mbox{\boldmath$a$\unboldmath}}]$, respectively. The principal symbol
of $\mbox{\boldmath$a$\unboldmath}\\#\mbox{\boldmath$b$\unboldmath}$ in (2.8)
is given by the product of the principal symbols of $a$ and $b$.
If
$\mbox{\boldmath$a$\unboldmath}\sim\sum\hbar^{j}\mbox{\boldmath$a$\unboldmath}_{j}$
and
$\mbox{\boldmath$b$\unboldmath}\sim\sum\hbar^{j}\mbox{\boldmath$b$\unboldmath}_{j}$
the symbol $\mbox{\boldmath$a$\unboldmath}\\#\mbox{\boldmath$b$\unboldmath}$
has the asymptotic expansion
$\displaystyle\mbox{\boldmath$a$\unboldmath}\\#\mbox{\boldmath$b$\unboldmath}(x,\xi)=\sum_{j,k,l\in{\mathbb{N}}_{0}}\frac{\hbar^{j+k+l}}{j!}\Big{(}\frac{i}{2}(\partial_{x}\partial_{\eta}-\partial_{\xi}\partial_{y})\Big{)}^{j}\mbox{\boldmath$a$\unboldmath}_{k}(x,\xi)\mbox{\boldmath$b$\unboldmath}_{l}(y,\eta)\Big{|}_{\begin{subarray}{c}y=x\\\
\eta=\xi\end{subarray}}.$ (2.9)
An operator $\mbox{\boldmath$A$\unboldmath}={\rm
op}^{W}[{\mbox{\boldmath$a$\unboldmath}}]$ is called elliptic at
$(x_{0},\xi_{0})$ if $\mbox{\boldmath$a$\unboldmath}^{-1}(x_{0},\xi_{0})$
exists and belongs to ${\mathsf{S}}(m^{-1})$. We say that $A$ is elliptic if
it is elliptic at every point. In the affirmative case, one can construct a
parametrix $\mbox{\boldmath$q$\unboldmath}\in{\mathsf{S}}(m^{-1})$ which is an
asymptotic inverse of $a$ in the sense of symbol products:
###### Lemma 2.2.
Suppose $\mbox{\boldmath$a$\unboldmath}\in{\mathsf{S}}(m)$ is elliptic in the
sense that $\mbox{\boldmath$a$\unboldmath}^{-1}(x,\xi)$ exists for all
$(x,\xi)\in{\mathsf{T}}^{\ast}{\mathbb{R}}^{d}$ and belongs to the class
${\mathsf{S}}(m^{-1})$. Then there exists a parametrix
$\mbox{\boldmath$q$\unboldmath}\in{\mathsf{S}}(m^{-1})$ with an asymptotic
expansion of the form
$\mbox{\boldmath$q$\unboldmath}\sim\mbox{\boldmath$a$\unboldmath}^{-1}+\hbar(\mbox{\boldmath$a$\unboldmath}^{-1}\\#\mbox{\boldmath$r$\unboldmath})+\hbar^{2}(\mbox{\boldmath$a$\unboldmath}^{-1}\\#\mbox{\boldmath$r$\unboldmath}\\#\mbox{\boldmath$r$\unboldmath})+\cdots$
(2.10)
such that $\mbox{\boldmath$r$\unboldmath}\in{\mathsf{S}}(1)$ and
$\mbox{\boldmath$a$\unboldmath}\\#\mbox{\boldmath$q$\unboldmath}\sim\mbox{\boldmath$q$\unboldmath}\\#\mbox{\boldmath$a$\unboldmath}\sim\mbox{\boldmath$I$\unboldmath}_{4}.$
For the proof of Theorem 5.2 it is convenient to have the following notion of
microlocality:
###### Definition 2.3.
We say that
$\mbox{\boldmath$u$\unboldmath}\in{\mathbf{L}}^{2}({\mathbb{R}}^{3},{\mathbb{C}}^{4})$
is microlocally ${\mathcal{O}}(\varepsilon(\hbar))$ at $(x_{0},\xi_{0})$ if
there is $\mbox{\boldmath$a$\unboldmath}\in{\mathsf{S}}(1)$, invertible at
$(x_{0},\xi_{0})$, such that
$\|{\rm
op}^{W}[{\mbox{\boldmath$a$\unboldmath}}]\mbox{\boldmath$u$\unboldmath}\|={\mathcal{O}}(\varepsilon(\hbar)),$
uniformly as $\hbar\to 0$.
###### Lemma 2.4.
For $a$ as in Definition 2.3 we can find
$\mbox{\boldmath$\chi$\unboldmath}_{0}\in{\mathsf{S}}(1)$ with support away
from $(x_{0},\xi_{0})$, such that
$\mbox{\boldmath$a$\unboldmath}+\mbox{\boldmath$\chi$\unboldmath}_{0}$ is
everywhere invertible.
###### Proof.
First assume
$\mbox{\boldmath$a$\unboldmath}(x_{0},\xi_{0})=\mbox{\boldmath$I$\unboldmath}_{4}$.
Let $\lambda_{\textrm{min}}(x,\xi)$ be equal to the smallest of the
eigenvalues of $a$ at $(x,\xi)$. Then there is $\varepsilon>0$ so that
$\lambda_{\textrm{min}}(x,\xi)>1/2$ for all $(x,\xi)\in
B((x_{0},\xi_{0}),\varepsilon)$. Pick
$M>\sup_{{\mathbb{R}}^{2n}}\|\mbox{\boldmath$a$\unboldmath}(x,\xi)\|$ and
choose $\chi_{0}$ to be a non-negative smooth function such that
$\chi_{0}(x,\xi)=\begin{cases}0\quad&\text{in
}B((x_{0},\xi_{0}),\frac{\varepsilon}{2}),\\\ M\quad&\text{in
}\mathbb{R}^{2n}\setminus B((x_{0},\xi_{0}),\varepsilon)\end{cases}.$
Letting
$\mbox{\boldmath$\chi_{0}$\unboldmath}=\chi_{0}\mbox{\boldmath$I$\unboldmath}_{4}$
it is clear that
$\mbox{\boldmath$a$\unboldmath}+\mbox{\boldmath$\chi$\unboldmath}_{0}$ is
everywhere positive definite.
In the general case we consider
$\widetilde{\mbox{\boldmath$a$\unboldmath}}(x,\xi):=\mbox{\boldmath$a$\unboldmath}(x,\xi)\mbox{\boldmath$a$\unboldmath}^{-1}(x_{0},\xi_{0})$.
Then $\widetilde{\mbox{\boldmath$a$\unboldmath}}$ satisfies
$\widetilde{\mbox{\boldmath$a$\unboldmath}}(x_{0},\xi_{0})=\mbox{\boldmath$I$\unboldmath}_{4}$
so by the first part of the proof there is
$\widetilde{\mbox{\boldmath$\chi$\unboldmath}}_{0}$, supported away from
$(x_{0},\xi_{0})$, such that
$\widetilde{\mbox{\boldmath$a$\unboldmath}}+\widetilde{\mbox{\boldmath$\chi$\unboldmath}}_{0}$
is elliptic. Thus
$\mbox{\boldmath$a$\unboldmath}(x,\xi)+\widetilde{\mbox{\boldmath$\chi$\unboldmath}}_{0}(x,\xi)\mbox{\boldmath$a$\unboldmath}(x_{0},\xi_{0})$
is everywhere invertible and
$\mbox{\boldmath$\chi$\unboldmath}_{0}(x,\xi):=\widetilde{\mbox{\boldmath$\chi$\unboldmath}}_{0}(x,\xi)\mbox{\boldmath$a$\unboldmath}(x_{0},\xi_{0})$
belong to ${\mathsf{S}}(1)$ and has support away from $(x_{0},\xi_{0})$. ∎
The following lemma shows the strength of Definition 2.3.
###### Lemma 2.5.
Assume that
$\mbox{\boldmath$u$\unboldmath}\in{\mathbf{L}}^{2}({\mathbb{R}}^{3},{\mathbb{C}}^{4})$
is microlocally ${\mathcal{O}}(\varepsilon(\hbar))$ at $(x_{0},\xi_{0})$.
Then, for any $\mbox{\boldmath$b$\unboldmath}\in{\mathsf{S}}(1)$ with
sufficiently small support near $(x_{0},\xi_{0})$, it holds that
$\|{\rm
op}^{W}[{\mbox{\boldmath$b$\unboldmath}}]\mbox{\boldmath$u$\unboldmath}\|={\mathcal{O}}(\varepsilon(\hbar)+\hbar^{\infty}),$
uniformly as $\hbar\to 0$.
###### Proof.
Let $\mbox{\boldmath$\chi$\unboldmath}_{0}$ be as in Lemma 2.4 and $a$ as in
Definition 2.3. Then we can find a parametrix
$\mbox{\boldmath$q$\unboldmath}\in{\mathsf{S}}(1)$ with
${\rm op}^{W}[{\mbox{\boldmath$q$\unboldmath}}]{\rm
op}^{W}[{\mbox{\boldmath$a$\unboldmath}+\mbox{\boldmath$\chi$\unboldmath}_{0}}]=\mbox{\boldmath$1$\unboldmath}+\mbox{\boldmath$R$\unboldmath},$
where $\|\mbox{\boldmath$R$\unboldmath}\|={\mathcal{O}}(\hbar^{\infty})$.
Therefore, for any $\mbox{\boldmath$b$\unboldmath}\in{\mathsf{S}}(1)$,
$\displaystyle{\rm
op}^{W}[{\mbox{\boldmath$b$\unboldmath}}]\mbox{\boldmath$u$\unboldmath}={\rm
op}^{W}[{\mbox{\boldmath$b$\unboldmath}}]{\rm
op}^{W}[{\mbox{\boldmath$q$\unboldmath}}]{\rm
op}^{W}[{\mbox{\boldmath$a$\unboldmath}+\mbox{\boldmath$\chi$\unboldmath}_{0}}]\mbox{\boldmath$u$\unboldmath}-{\rm
op}^{W}[{\mbox{\boldmath$b$\unboldmath}}]\mbox{\boldmath$R$\unboldmath}\mbox{\boldmath$u$\unboldmath}.$
Using $\|{\rm
op}^{W}[{\mbox{\boldmath$a$\unboldmath}}]\mbox{\boldmath$u$\unboldmath}\|={\mathcal{O}}(\varepsilon(\hbar))$
we obtain
${\rm
op}^{W}[{\mbox{\boldmath$b$\unboldmath}}]\mbox{\boldmath$u$\unboldmath}={\rm
op}^{W}[{\mbox{\boldmath$b$\unboldmath}\\#\mbox{\boldmath$q$\unboldmath}\\#\mbox{\boldmath$\chi$\unboldmath}_{0}}]\mbox{\boldmath$u$\unboldmath}+{\mathcal{O}}(\varepsilon(\hbar)).$
Since $\mbox{\boldmath$\chi$\unboldmath}_{0}$ has no support in a neighborhood
of $(x_{0},\xi_{0})$ (see Lemma 2.4) we can choose $b$ with sufficiently small
support around $(x_{0},\xi_{0})$ so that
$\operatorname{supp\,}(\mbox{\boldmath$b$\unboldmath})\cap\operatorname{supp\,}(\mbox{\boldmath$\chi$\unboldmath}_{0})=\emptyset$.
The lemma now follows from (2.9) and Proposition 2.1. ∎
## 3 Dirac and CAP Hamiltonians
We introduce various assumptions and we define perturbed Dirac operators.
Moreover, we introduce the CAP Hamiltonians.
The free Dirac operator. The free semiclassical Dirac operator, describing the
motion of a relativistic electron or positron without external forces, is the
unique self-adjoint extension of the symmetric operator defined on
$C_{0}^{\infty}({\mathbb{R}}^{3},{\mathbb{C}}^{4})$ in the Hilbert space
${\mathcal{H}}={\mathbf{L}}^{2}({\mathbb{R}}^{3},{\mathbb{C}}^{4})$ by
$\mathbb{D}_{0}:=c\mbox{\boldmath$\alpha$\unboldmath}\cdot\frac{\hbar}{i}\nabla+\beta
mc^{2}=-ic\hbar\sum_{j=1}^{3}\alpha_{j}\frac{\partial{}}{\partial{x_{j}}}+\beta
mc^{2},$
where $\nabla=(\partial_{x_{1}},\partial_{x_{2}},\partial_{x_{3}})$ is the
gradient, $c$ the speed of light, $m$ the electron mass, $\hbar$ the
semiclassical parameter, and
$\mbox{\boldmath$\alpha$\unboldmath}:=(\alpha_{1},\alpha_{2},\alpha_{3})$ with
$\alpha_{1}$, $\alpha_{2}$, $\alpha_{3}$, $\beta$ being Hermitian $4\times 4$
matrices, which satisfy the anti-commutation relations
$\displaystyle\begin{cases}\alpha_{i}\alpha_{j}+\alpha_{j}\alpha_{i}=2\delta_{ij}\mbox{\boldmath$I$\unboldmath}_{4},\quad&\text{for
}i,j=1,2,3,\\\ \alpha_{i}\beta+\beta\alpha_{i}=0,\quad&\text{for
}i=1,2,3,\end{cases}$
and $\beta^{2}=\mbox{\boldmath$I$\unboldmath}_{4}$. For instance, one can use
the “standard representation”
$\alpha_{i}=\begin{pmatrix}0&\sigma_{i}\\\
\sigma_{i}&0\end{pmatrix},\quad\beta=\begin{pmatrix}\mbox{\boldmath$I$\unboldmath}_{2}&0\\\
0&-\mbox{\boldmath$I$\unboldmath}_{2},\end{pmatrix}$
where
$\sigma_{1}=\begin{pmatrix}0&1\\\
1&0\end{pmatrix},\quad\sigma_{2}=\begin{pmatrix}0&-i\\\
i&0\end{pmatrix},\quad\sigma_{3}=\begin{pmatrix}1&0\\\ 0&-1\end{pmatrix}$
(3.1)
are $2\times 2$ Pauli matrices. It is well-known that the resulting self-
adjoint operator $\mathbb{D}_{0}$ has domain
$\operatorname{Dom\,}(\mathbb{D}_{0})={\mathbf{H}}^{1}({\mathbb{R}}^{3},{\mathbb{C}}^{4})$
and
$\operatorname{spec\,}(\mathbb{D}_{0})=\operatorname{spec}_{\operatorname{ess}}(\mathbb{D}_{0})=(-\infty,-mc^{2}]\cup[mc^{2},\infty)$;
see, e.g., [Th’92].
Perturbed Dirac operator. To describe the interaction of a particle with
external fields we perturb $\mathbb{D}_{0}$ by a potential $\mathbb{V}\in
C^{\infty}({\mathbb{R}}^{3})\otimes\mathrm{M}_{4}({\mathbb{C}})$, viewed as a
multiplication operator on ${\mathcal{H}}$.
###### Assumption 3.1.
Let the potential
$\mathbb{V}\,:\,{\mathbb{R}}^{3}\to\mathrm{M}_{4}({\mathbb{C}})$ be Hermitian,
smooth for all $x\in{\mathbb{R}}^{3}$, and compactly supported; the number
$R_{0}^{\prime}>0$ is chosen such that $\operatorname{supp\,}\mathbb{V}\subset
B(0,R_{0}^{\prime})$.
Under Assumption 3.1 it is well-known that
$\mathbb{D}:=\mathbb{D}_{0}+\mathbb{V}$ is self-adjoint on
$\operatorname{Dom\,}(\mathbb{D}_{0})={\mathbf{H}}^{1}({\mathbb{R}}^{3},{\mathbb{C}}^{4})$.
Moreover it follows from Weyl’s theorem that
$\operatorname{spec}_{\operatorname{ess}}(\mathbb{D})=\operatorname{spec}_{\operatorname{ess}}(\mathbb{D}_{0})=\operatorname{spec\,}(\mathbb{D}_{0})$;
see, e.g., [Th’92, Section 4.3]. Henceforth we shall emphasize the dependence
of $\hbar$ in $\mathbb{D}$ by writing $\mathbb{D}(\hbar)$.
Hamiltonian flow. Let $\mbox{\boldmath$d$\unboldmath}_{0}$ be the principal
symbol of $\mathbb{D}(\hbar)$ and let its eigenvalues be denoted by
$\lambda_{j}$, $j=1,\ldots,4$.
The Hamiltonian trajectories (or bicharacteristics), denoted by
$(x_{j}(t),\xi_{j}(t))=:\Phi_{j}^{t}(x_{0},\xi_{0})$, $j=1,\ldots,4$, are
defined as the solutions of Hamilton’s equations
$\displaystyle\begin{cases}x_{j}^{\prime}(t)&=\nabla_{\xi}\lambda_{j}(x_{j}(t),\xi_{j}(t))\\\
\xi_{j}^{\prime}(t)&=-\nabla_{x}\lambda_{j}(x_{j}(t),\xi_{j}(t))\end{cases},\qquad(x_{j}(0),\xi_{j}(0))=(x_{0},\xi_{0}).$
Nontrapping condition. We introduce the following nontrapping condition for
the Hamiltonian flow generated by the eigenvalues $\lambda_{j}(x,\xi)$,
$j=1,\ldots,4$.
###### Definition 3.2.
We say that an energy band $J\subset{\mathbb{R}}$ is nontrapping for
$\mathbb{D}(\hbar)$ if for any $R>0$ there exists $T_{R}>0$ such that
$|x_{j}(t)|>R\text{ for }\lambda_{j}(x_{0},\xi_{0})\in J\text{ provided
}|t|>T_{R}\>\mbox{ and }\>j=1,\ldots,4.$
Hyperbolicity condition. To avoid the difficulty of energy level crossings in
certain situations, we shall introduce the following assumption.
###### Assumption 3.3.
Distinct eigenvalues are said to satisfy the hyperbolicity condition if
$|\lambda_{j}(x,\xi)-\lambda_{k}(x,\xi)|\geq C\langle\xi\rangle\quad\text{for
all }(x,\xi)\in{\mathsf{T}}^{\ast}{\mathbb{R}}^{3}$
for some constant $C>0$.
###### Example 3.4.
To illustrate Assumption 3.3 we consider the Dirac operator describing a
particle of mass $m$ and charge $e$ subject to external time-independent
electromagnetic fields $\mbox{\boldmath$E$\unboldmath}(x)=-\nabla\phi(x)$ and
$\mbox{\boldmath$B$\unboldmath}(x)=\nabla\times\mbox{\boldmath$A$\unboldmath}(x)$:
$\mathbb{D}_{\mbox{{\scriptsize\boldmath$A$\unboldmath}},\phi}(\hbar)=c\mbox{\boldmath$\alpha$\unboldmath}\cdot\left(\frac{\hbar}{i}\nabla-\frac{e}{c}\mbox{\boldmath$A$\unboldmath}(x)\right)+\beta
mc^{2}+e\phi(x).$
The principal symbol of
$\mathbb{D}_{\mbox{{\scriptsize\boldmath$A$\unboldmath}},\phi}(\hbar)$ is
$\mbox{\boldmath$d$\unboldmath}_{0,\mbox{{\scriptsize\boldmath$A$\unboldmath}},\phi}(x,\xi)=c\mbox{\boldmath$\alpha$\unboldmath}\cdot\left(\xi-\frac{e}{c}\mbox{\boldmath$A$\unboldmath}(x)\right)+\beta
mc^{2}+e\phi(x).$ (3.2)
For any $(x,\xi)$ the symbol
$\mbox{\boldmath$d$\unboldmath}_{0,\mbox{{\scriptsize\boldmath$A$\unboldmath}},\phi}(x,\xi)$
is a Hermitian $4\times 4$ matrix with two doubly degenerated eigenvalues
$\lambda^{\pm}(x,\xi)=e\phi(x)\pm\sqrt{\left(c\xi-e\mbox{\boldmath$A$\unboldmath}(x)\right)^{2}+m^{2}c^{4}}$
associated with projection matrices
$\mbox{\boldmath$\lambda$\unboldmath}_{0}^{\pm}(x,\xi)=\frac{1}{2}\left(\mbox{\boldmath$I$\unboldmath}_{4}\pm\frac{\mbox{\boldmath$\alpha$\unboldmath}\cdot(c\xi-e\mbox{\boldmath$A$\unboldmath}(x))+\beta
mc^{2}}{\sqrt{(c\xi-e\mbox{\boldmath$A$\unboldmath}(x))^{2}+m^{2}c^{4}}}\right)$
onto the respective eigenspaces in ${\mathbb{C}}^{4}$. Since $A$ and $\phi$
satisfy Asusmption 3.1 we may choose
$m(x,\xi):=\sqrt{\left(c\xi-e\mbox{\boldmath$A$\unboldmath}(x)\right)^{2}+m^{2}c^{4}},$
as an order function for the symbol
$\mbox{\boldmath$d$\unboldmath}_{0,\mbox{{\scriptsize\boldmath$A$\unboldmath}},\phi}$.
In particular,
$|\lambda^{+}(x,\xi)-\lambda^{-}(x,\xi)|=2m(x,\xi),$
which shows that Assumption 3.3 holds true.
Cordes [Co’82] imposes a similar condition on the eigenvalues of the symbol of
an operator in a strictly hyperbolic system, and Bolte-Glaser [BoGl’04a,
Theorem 3.2] prove a semiclassical version of Egorov’s theorem under
Assumption 3.3.
Complex absorbing potential Hamiltonian.
###### Assumption 3.5.
Suppose $W\in{\mathbf{L}}^{\infty}({\mathbb{R}}^{3},{\mathbb{C}})$ is smooth
and let $\mathbb{W}=W\mbox{\boldmath$I$\unboldmath}_{4}$ be the operator on
${\mathbf{L}}^{2}({\mathbb{R}}^{3},{\mathbb{C}}^{4})$ induced by
multiplication. Suppose, moreover, that $W$ satisfy the following properties:
1. (i)
$\operatorname{{\rm Re}\,}W\geq 0$;
2. (ii)
There is an $R_{1}>0$ such that $\operatorname{supp\,}W\subset\\{|x|\geq
R_{1}\\}$;
3. (iii)
For some $\delta_{0}>0$ and $R_{2}>R_{1}$ we have $\operatorname{{\rm
Re}\,}W\geq\delta_{0}$ for $|x|>R_{2}$;
4. (iv)
$|\operatorname{{\rm Im}\,}W|\leq C\sqrt{\operatorname{{\rm Re}\,}W}$ for some
constant $C$.
By property (i) $-i\mathbb{W}$ contributes a negative imaginary term which is
necessary in order for the CAP to be absorbing. Property (ii) means absorption
of the wave packet takes place away from the interaction region. If, on the
other hand, $\mathbb{D}$ is assumed to be nontrapping on
$\operatorname{supp\,}W$ in the sense of Definition 3.2 and satisfy the
hyperbolicity condition in Assumption 3.3 this condition can be relaxed, see
Theorem 5.2. Property (iii) is a strengthening of property (i) required to
prove that eigenvalues of the CAP Hamiltonian defined below implies the
existence of resonances nearby. We also allow $W$ to have a non-zero imaginary
part as long as it is dominated by the real part in the sense of property
(iv).
In particular, we see that $i\mathbb{W}$ is not Hermitian. We now define two
CAP operators. First,
$\mathbb{J}_{\infty}(\hbar):=\mathbb{D}(\hbar)-i\mathbb{W}(x)\quad\mbox{ on
}\quad{\mathcal{H}}.$
Second, given $R>R_{2}$ let ${\mathcal{H}}_{R}(\hbar)$ be the restriction of
${\mathcal{H}}$ to the ball $B(0,R)$ and let $\mathbb{D}_{R}(\hbar)$ be the
Dirichlet realization of $\mathbb{D}(\hbar)$ there. Define
$\mathbb{J}_{R}(\hbar):=\mathbb{D}_{R}(\hbar)-i\mathbb{W}(x),$
We see that both $\mathbb{J}_{\infty}(\hbar)$ and $\mathbb{J}_{R}(\hbar)$ are
closed unbounded operators with
$\operatorname{Dom\,}(\mathbb{J}_{\infty}(\hbar))=\operatorname{Dom\,}(\mathbb{D}(\hbar))\quad\mbox{
and
}\quad\operatorname{Dom\,}(\mathbb{J}_{R}(\hbar))=\operatorname{Dom\,}(\mathbb{D}_{R}(\hbar))$
Furthermore, since $\operatorname{{\rm Re}\,}W\geq 0$, we see that
${\mathbb{C}}_{+}$ is contained in their resolvent sets.
###### Remark 3.6.
In Physics and Chemistry the function $W$ is usually chosen to be real-valued,
but, as in [St’05], we have defined $W$ as a complex-valued function.
## 4 Complex distortion and resonances
In the spirit of Aguilar-Balslev-Combes theory of resonances we summarize the
spectral deformation theory for the Dirac operator, following Hunziker’s
approach, and we define resonances. Basic facts are stated without proofs; we
refer to [Hu’86, HiSi’96, Kh’07] for details.
### 4.1 Complex distortion
We perform complex distortion outside of $B(0,R_{2})\cup B(0,R_{0}^{\prime})$
and for this purpose we introduce a smooth vector field $g$ with the following
properties.
###### Assumption 4.1.
Let $g:{\mathbb{R}}^{3}\to{\mathbb{R}}^{3}$ be a smooth function which
satisfies:
(i) $g(x)=0\text{ for }|x|\leq R_{0}\text{ where
}R_{0}>\max(R_{0}^{\prime},R_{2});$
(ii) $g(x)=x\text{ for }|x|>R_{0}+\eta\text{ for some }\eta>0;$
(iii) $\sup_{x\in{\mathbb{R}}^{3}}\|(Dg)(x)\|<\sqrt{2}$ with $(Dg)(x)$ being
the Jacobian matrix of $g$.
The parameter $R_{0}$ will be chosen suitably in different circumstances. This
will not affect the set of resonances we study.
Henceforth we impose Assumption 4.1. For fixed $\varepsilon\in(0,1)$ and
$\theta\in
D_{\varepsilon}:=\Big{\\{}\theta\in{\mathbb{C}}\,:\,|\theta|<r_{\varepsilon}:=\frac{\varepsilon}{\sqrt{1+\varepsilon^{2}}}\Big{\\}},$
we let $\phi_{\theta}:{\mathbb{R}}^{3}\to{\mathbb{R}}^{3}$ be defined by
$\phi_{\theta}(x)=x+\theta g(x)$ and we denote the Jacobian determinant of
$\phi_{\theta}$ by $J_{\theta}$. We then define
$\mbox{\boldmath$U$\unboldmath}_{\theta}\,:\,\mbox{\boldmath$\sc\mbox{S}\hskip
1.0pt$\unboldmath}({\mathbb{R}}^{3},{\mathbb{C}}^{4})\to\mbox{\boldmath$\sc\mbox{S}\hskip
1.0pt$\unboldmath}({\mathbb{R}}^{3},{\mathbb{C}}^{4})$ for
$\theta\in(-r_{\varepsilon},r_{\varepsilon})$ by
$\mbox{\boldmath$U$\unboldmath}_{\theta}\mbox{\boldmath$f$\unboldmath}(x)=J_{\theta}^{1/2}(x)\mbox{\boldmath$f$\unboldmath}(\phi_{\theta}(x)).$
One has:
(P1): The map $\mbox{\boldmath$U$\unboldmath}_{\theta}$ extends, for
$\theta\in(-r_{\varepsilon},r_{\varepsilon})$, to a unitary operator on
${\mathbf{L}}^{2}({\mathbb{R}}^{3},{\mathbb{C}}^{4})$.
###### Definition 4.2.
Let ${\mathcal{A}}$ be the linear space of all entire functions
$\mbox{\boldmath$f$\unboldmath}=(f_{i})_{1\leq i\leq 4}$ such that for any
$0<\varepsilon<1$ and $k\in{\mathbb{N}}$ we have
$\lim_{\begin{subarray}{c}|z|\to\infty\\\ z\in
C_{\varepsilon}\end{subarray}}|z|^{k}|f_{i}(z)|=0\quad\text{for }1\leq i\leq
4,$
where
$C_{\varepsilon}=\\{z\in{\mathbb{C}}^{3}\,:\,|\operatorname{{\rm
Im}\,}z|\leq\varepsilon|\operatorname{{\rm Re}\,}z|,\,|\operatorname{{\rm
Re}\,}z|>\max(R_{0}^{\prime},R_{2})\\}.$ (4.1)
We now define the class of analytic vectors:
###### Definition 4.3.
Let ${\mathcal{B}}\subset{\mathbf{L}}^{2}({\mathbb{R}}^{3},{\mathbb{C}}^{4})$
be the set of
$\mbox{\boldmath$\psi$\unboldmath}\in{\mathbf{L}}^{2}({\mathbb{R}}^{3},{\mathbb{C}}^{4})$
such that there exists $\mbox{\boldmath$f$\unboldmath}\in{\mathcal{A}}$ with
$\mbox{\boldmath$f$\unboldmath}(x)=\mbox{\boldmath$\psi$\unboldmath}(x)$ for
$x\in{\mathbb{R}}^{3}$.
Then:
(P2): The set ${\mathcal{B}}$ is dense in
${\mathbf{L}}^{2}({\mathbb{R}}^{3},{\mathbb{C}}^{4})$.
This statement follows from the fact that ${\mathcal{B}}$ is a linear space
which contain the set of Hermite functions which has a dense span.
Moreover, for ${\mathcal{B}}$ to be a set of analytic vectors for
$\mbox{\boldmath$U$\unboldmath}_{\theta}$ (see e.g. [HiSi’96]), we need the
following fact; wherein we allow $\theta$ to become non-real.
(P3): For all $\theta\in D_{\varepsilon}$ we have
* (i)
For all $\mbox{\boldmath$f$\unboldmath}\in{\mathcal{B}}$ the map
$\theta\mapsto\mbox{\boldmath$U$\unboldmath}_{\theta}\mbox{\boldmath$f$\unboldmath}$
is analytic.
* (ii)
$\mbox{\boldmath$U$\unboldmath}_{\theta}{\mathcal{B}}$ is dense in
${\mathbf{L}}^{2}({\mathbb{R}}^{3},{\mathbb{C}}^{4})$.
We are now ready to define the family of spectrally deformed Dirac operators.
###### Definition 4.4.
For $\theta\in D_{\varepsilon}^{+}:=D_{\varepsilon}\cap\\{{\rm Im}\,z\geq
0\\}$ we let
$\mathbb{D}_{\theta}(\hbar):=\mbox{\boldmath$U$\unboldmath}_{\theta}\mathbb{D}(\hbar)\mbox{\boldmath$U$\unboldmath}_{\theta}^{-1}=\mbox{\boldmath$U$\unboldmath}_{\theta}\mathbb{D}_{0}(\hbar)\mbox{\boldmath$U$\unboldmath}_{\theta}^{-1}+\mbox{\boldmath$U$\unboldmath}_{\theta}\mathbb{V}\mbox{\boldmath$U$\unboldmath}_{\theta}^{-1}=:\mathbb{D}_{0,\theta}(\hbar)+\mathbb{V}(\phi_{\theta}(x)).$
We have:
(P4): For $\theta_{0}\in D_{\varepsilon}^{+}$ the eigenvalues of
$\mathbb{D}_{\theta_{0}}(\hbar)$ are independent of the spectral deformation
family $\\{\mbox{\boldmath$U$\unboldmath}_{\theta_{0}}\\}$.
Khochman [Kh’07, Lemma 3] proves the following representation of the free
deformed Hamiltonian
$\mathbb{D}_{0,\theta}(\hbar)=\mbox{\boldmath$U$\unboldmath}_{\theta}\mathbb{D}_{0}(\hbar)\mbox{\boldmath$U$\unboldmath}_{\theta}^{-1}$.
###### Lemma 4.5.
For $\theta\in D_{\varepsilon}$
$\mathbb{D}_{0,\theta}(\hbar)=-\frac{1}{1+\theta}ic\hbar\mbox{\boldmath$\alpha$\unboldmath}\cdot\nabla+\beta
mc^{2}+\mbox{\boldmath$Q$\unboldmath}_{\theta}(x,\hbar\partial_{x_{j}}),$
where
$\mbox{\boldmath$Q$\unboldmath}_{\theta}(x,\hbar\partial_{x_{j}})=\sum_{|\gamma|\leq
1}\mbox{\boldmath$a$\unboldmath}_{\gamma}(x,\theta)(\hbar\partial_{x_{j}})^{\gamma}$
with $\mbox{\boldmath$a$\unboldmath}_{\gamma}(x,\cdot)$ analytic and
${\mathcal{O}}(\theta)$, and
$\mbox{\boldmath$a$\unboldmath}_{\gamma}(\cdot,\theta)\in
C_{0}^{\infty}(B(0,R_{0}+2\eta),{\mathbb{C}}^{4})$.
###### Proof.
Since
$\mathbb{D}_{\theta,0}=\mbox{\boldmath$U$\unboldmath}_{\theta}\mathbb{D}_{0}\mbox{\boldmath$U$\unboldmath}_{\theta}^{-1}=-ic\hbar\sum_{j=1}^{3}\alpha_{j}\mbox{\boldmath$U$\unboldmath}_{\theta}\partial_{j}\mbox{\boldmath$U$\unboldmath}_{\theta}^{-1}+\beta
mc^{2}$
we need only compute, for any
$\mbox{\boldmath$f$\unboldmath}\in\mbox{\boldmath$\sc\mbox{S}\hskip
1.0pt$\unboldmath}({\mathbb{R}}^{3},{\mathbb{C}}^{4})$,
$\displaystyle\mbox{\boldmath$U$\unboldmath}_{\theta}\partial_{j}\mbox{\boldmath$U$\unboldmath}_{\theta}^{-1}\mbox{\boldmath$f$\unboldmath}(x)$
$\displaystyle=J_{\theta}^{1/2}(x)\Big{(}\partial_{j}J_{\theta}^{-1/2}(\phi_{\theta}^{-1}(x))\mbox{\boldmath$f$\unboldmath}(\phi_{\theta}^{-1}(x))\Big{)}(\phi_{\theta}(x))$
$\displaystyle=-\frac{1}{2}J_{\theta}^{-1}(x)\partial_{j}J_{\theta}(x)\mbox{\boldmath$f$\unboldmath}(x)+\sum_{k=1}^{3}\partial_{j}\phi_{\theta,k}^{-1}\big{(}\phi_{\theta}(x)\big{)}\partial_{k}\mbox{\boldmath$f$\unboldmath}(x)$
where we use the notation
$\phi_{\theta}^{-1}=(\phi_{\theta,1}^{-1},\phi_{\theta,2}^{-1},\phi_{\theta,3}^{-1})$.
By Assumption 4.1 we have
$\partial_{j}\phi_{\theta,k}^{-1}(\phi_{\theta}(x))=(1+\theta)^{-1}\delta_{jk}$,
$k=1,2,3$, provided $|x|>R_{0}+\eta$. Thus, if $\chi\in
C_{0}^{\infty}(B(0,R_{0}+2\eta))$ is taken to equal $1$ near $B(0,R_{0}+\eta)$
we have
$\displaystyle\mbox{\boldmath$U$\unboldmath}_{\theta}\partial_{j}\mbox{\boldmath$U$\unboldmath}_{\theta}^{-1}\mbox{\boldmath$f$\unboldmath}(x)=-\frac{1}{2}J_{\theta}^{-1}(x)\partial_{j}J_{\theta}(x)\mbox{\boldmath$f$\unboldmath}(x)+\frac{1}{1+\theta}\partial_{j}\mbox{\boldmath$f$\unboldmath}(x)(1-\chi)$
$\displaystyle\phantom{ooooooooooooooooooooooooooooooooooo}+\sum_{k=1}^{3}\partial_{j}\phi_{\theta,k}^{-1}\big{(}\phi_{\theta}(x)\big{)}\partial_{k}\mbox{\boldmath$f$\unboldmath}(x)\chi$
$\displaystyle=\frac{1}{1+\theta}\partial_{j}\mbox{\boldmath$f$\unboldmath}(x)$
$\displaystyle\phantom{oo}+\Big{\\{}-\frac{1}{2}J_{\theta}^{-1}(x)\partial_{j}J_{\theta}(x)\mbox{\boldmath$f$\unboldmath}(x)-\frac{1}{1+\theta}\partial_{j}\mbox{\boldmath$f$\unboldmath}(x)\chi+\sum_{k=1}^{3}\partial_{j}\phi_{\theta,k}^{-1}\big{(}\phi_{\theta}(x)\big{)}\partial_{k}\mbox{\boldmath$f$\unboldmath}(x)\chi\Big{\\}}$
where the terms in brackets are what makes $Q_{\theta}$ after multiplication
by $-ic\hbar\alpha_{j}$ and summation over $j=1,2,3$. ∎
###### Remark 4.6.
In particular we see that, for $\theta\in D_{\varepsilon}$,
$\theta\mapsto\mathbb{D}_{0,\theta}$ is a holomorphic family of type (A) in
the sense of Kato (see [Ka’95, p. 375]).
The above representation can be modified to the following more variable one:
###### Lemma 4.7.
For $\theta\in D_{\varepsilon}$ we have, using the principal branch of the
cube root,
$\displaystyle\mathbb{D}_{\theta}=-J_{\theta}^{-1/3}ic\hbar\mbox{\boldmath$\alpha$\unboldmath}\cdot\nabla+\beta
mc^{2}+\widetilde{\mbox{\boldmath$Q$\unboldmath}}_{\theta}(x,\hbar\partial_{x_{j}}),$
where
$\widetilde{\mbox{\boldmath$Q$\unboldmath}}_{\theta}(x,\hbar\partial_{x_{j}})=\sum_{|\gamma|\leq
1}\widetilde{\mbox{\boldmath$a$\unboldmath}}_{\gamma}(x,\theta)(\hbar\partial_{x_{j}})^{\gamma}$
with the $\widetilde{\mbox{\boldmath$a$\unboldmath}}_{\gamma}(\cdot,\theta)$
supported in $\\{R_{0}<|x|<R_{0}+2\eta\\}$.
Using Lemma 4.7 we are now ready to show the following:
###### Proposition 4.8.
For $\theta\in D_{\varepsilon}^{+}$, $\operatorname{{\rm Re}\,}z>mc^{2}$ and
any $K\in{\mathbb{Z}}_{+}$ there is $C_{K}>0$ such that
$\|(\mathbb{D}_{\theta}-z)^{-1}\|\leq\frac{C_{K}}{\operatorname{{\rm
Im}\,}z}\quad\text{for }\operatorname{{\rm Im}\,}z>\hbar^{K},$
provided $\hbar$ is small enough.
###### Proof.
We prove the result by studying the quantity
$\operatorname{{\rm Im}\,}\langle
J_{\theta}^{1/3}(\mathbb{D}_{\theta}-\operatorname{{\rm
Re}\,}z)\mbox{\boldmath$u$\unboldmath},\mbox{\boldmath$u$\unboldmath}\rangle.$
(4.2)
Take $\chi\in C_{0}^{\infty}(B(0,R_{0}))$ which equals $1$ near
$B(0,R_{0}^{\prime})$. Then, using the fact that $J_{\theta}(x)=1$ and
$\mathbb{D}_{\theta}=\mathbb{D}$ for $|x|<R_{0}$,
$\displaystyle\operatorname{{\rm Im}\,}\langle
J_{\theta}^{1/3}(\mathbb{D}_{\theta}-\operatorname{{\rm
Re}\,}z)\mbox{\boldmath$u$\unboldmath},\mbox{\boldmath$u$\unboldmath}\rangle=\operatorname{{\rm
Im}\,}\langle(\mathbb{D}-\operatorname{{\rm
Re}\,}z)\mbox{\boldmath$\chi$\unboldmath}\mbox{\boldmath$u$\unboldmath},\mbox{\boldmath$\chi$\unboldmath}\mbox{\boldmath$u$\unboldmath}\rangle$
$\displaystyle+\operatorname{{\rm Im}\,}\langle(\mathbb{D}-\operatorname{{\rm
Re}\,}z)\mbox{\boldmath$\chi$\unboldmath}\mbox{\boldmath$u$\unboldmath},(\mbox{\boldmath$1$\unboldmath}-\mbox{\boldmath$\chi$\unboldmath})\mbox{\boldmath$u$\unboldmath}\rangle+\operatorname{{\rm
Im}\,}\langle(\mathbb{D}-\operatorname{{\rm
Re}\,}z)(\mbox{\boldmath$1$\unboldmath}-\mbox{\boldmath$\chi$\unboldmath})\mbox{\boldmath$u$\unboldmath},\mbox{\boldmath$\chi$\unboldmath}\mbox{\boldmath$u$\unboldmath}\rangle$
$\displaystyle+\operatorname{{\rm Im}\,}\langle
J_{\theta}^{1/3}(\mathbb{D}_{\theta}-\operatorname{{\rm
Re}\,}z)(\mbox{\boldmath$1$\unboldmath}-\mbox{\boldmath$\chi$\unboldmath})\mbox{\boldmath$u$\unboldmath},(\mbox{\boldmath$1$\unboldmath}-\mbox{\boldmath$\chi$\unboldmath})\mbox{\boldmath$u$\unboldmath}\rangle$
$\displaystyle=\operatorname{{\rm Im}\,}\langle
J_{\theta}^{1/3}(\mathbb{D}_{\theta}-\operatorname{{\rm
Re}\,}z)(\mbox{\boldmath$1$\unboldmath}-\mbox{\boldmath$\chi$\unboldmath})\mbox{\boldmath$u$\unboldmath},(\mbox{\boldmath$1$\unboldmath}-\mbox{\boldmath$\chi$\unboldmath})\mbox{\boldmath$u$\unboldmath}\rangle,$
because $\mathbb{D}$ is symmetric. Consequently, in order to estimate
$\operatorname{{\rm Im}\,}\langle
J_{\theta}^{1/3}(\mathbb{D}_{\theta}-\operatorname{{\rm
Re}\,}z)\mbox{\boldmath$u$\unboldmath},\mbox{\boldmath$u$\unboldmath}\rangle$,
it suffices to consider $u$ with support in ${\mathbb{R}}^{3}\setminus
B(0,R_{0}^{\prime})$, meaning we can replace $\mathbb{D}_{\theta}$ by
$\mathbb{D}_{0,\theta}$.
By Lemma 4.7 we may write
$\mathbb{D}_{0,\theta}=-J_{\theta}^{-1/3}ic\hbar\mbox{\boldmath$\alpha$\unboldmath}\cdot\nabla+\beta
mc^{2}+\widetilde{\mbox{\boldmath$Q$\unboldmath}}_{\theta}$
where $\widetilde{\mbox{\boldmath$Q$\unboldmath}}_{\theta}$ is a first order
differential operator having smooth coefficients supported in some subset $U$
of $B(0,R_{0}+2\eta)\setminus B(0,R_{0})$. Thus (4.2) becomes
$\displaystyle\operatorname{{\rm Im}\,}\langle
J_{\theta}^{1/3}(\mathbb{D}_{\theta}-\operatorname{{\rm
Re}\,}z)\mbox{\boldmath$u$\unboldmath},\mbox{\boldmath$u$\unboldmath}\rangle$
$\displaystyle=mc^{2}\langle\operatorname{{\rm
Im}\,}(J_{\theta}^{1/3})\beta\mbox{\boldmath$u$\unboldmath},\mbox{\boldmath$u$\unboldmath}\rangle+\operatorname{{\rm
Im}\,}\langle
J_{\theta}^{1/3}\widetilde{\mbox{\boldmath$Q$\unboldmath}}_{\theta}\mbox{\boldmath$u$\unboldmath},\mbox{\boldmath$u$\unboldmath}\rangle$
$\displaystyle\phantom{ooooooooooooooooooooo}-\operatorname{{\rm
Re}\,}z\langle\operatorname{{\rm
Im}\,}(J_{\theta}^{1/3})\mbox{\boldmath$u$\unboldmath},\mbox{\boldmath$u$\unboldmath}\rangle$
$\displaystyle\leq(mc^{2}-\operatorname{{\rm Re}\,}z)\langle\operatorname{{\rm
Im}\,}(J_{\theta}^{1/3})\mbox{\boldmath$u$\unboldmath},\mbox{\boldmath$u$\unboldmath}\rangle+C\|\mbox{\boldmath$u$\unboldmath}\|_{{\mathbf{H}}^{1}(U)}\|\mbox{\boldmath$u$\unboldmath}\|.$
Now, the first term on the right is non-positive. Moreover,
$\displaystyle\|\mbox{\boldmath$u$\unboldmath}\|_{{\mathbf{H}}^{1}(U)}$
$\displaystyle=\|(\mbox{\boldmath$1$\unboldmath}-\mbox{\boldmath$\chi$\unboldmath})\mbox{\boldmath$u$\unboldmath}\|_{{\mathbf{H}}^{1}(U)}\leq
C_{1}\|(\mathbb{D}_{0,\theta}-z)(\mbox{\boldmath$1$\unboldmath}-\mbox{\boldmath$\chi$\unboldmath})\mbox{\boldmath$u$\unboldmath}\|=C_{1}\|(\mathbb{D}_{\theta}-z)(\mbox{\boldmath$1$\unboldmath}-\mbox{\boldmath$\chi$\unboldmath})\mbox{\boldmath$u$\unboldmath}\|$
$\displaystyle\leq
C_{1}(\|(\mathbb{D}_{\theta}-z)\mbox{\boldmath$u$\unboldmath}\|+\|[\mathbb{D}_{0,\theta},\mbox{\boldmath$\chi$\unboldmath}]\mbox{\boldmath$u$\unboldmath}\|)$
$\displaystyle\leq
C_{1}(\|(\mathbb{D}_{\theta}-z)\mbox{\boldmath$u$\unboldmath}\|+\hbar\|\mbox{\boldmath$u$\unboldmath}\|_{\operatorname{supp\,}(\nabla\chi)})$
since $z\not\in\operatorname{spec\,}(\mathbb{D}_{0,\theta})$ (see Sec. 4.2).
Next take $\chi_{2}$ having the same properties as $\chi$ but also
$\chi_{2}\prec\chi$. Then, in the same way,
$\displaystyle\|\mbox{\boldmath$u$\unboldmath}\|_{\operatorname{supp\,}(\nabla\chi)}=\|(\mbox{\boldmath$1$\unboldmath}-\mbox{\boldmath$\chi$\unboldmath}_{2})\mbox{\boldmath$u$\unboldmath}\|_{\operatorname{supp\,}(\nabla\chi)}\leq
C_{2}(\|(\mathbb{D}_{\theta}-z)\mbox{\boldmath$u$\unboldmath}\|+\hbar\|\mbox{\boldmath$u$\unboldmath}\|_{\operatorname{supp\,}(\nabla\chi_{2})}).$
Continuing in this way, with $\chi_{K}\prec\cdots\prec\chi_{2}\prec\chi$, we
obtain, for any $K\in{\mathbb{Z}}_{+}$,
$\|\mbox{\boldmath$u$\unboldmath}\|_{{\mathbf{H}}^{1}(U)}\leq
C_{K}(\|(\mathbb{D}_{\theta}-z)\mbox{\boldmath$u$\unboldmath}\|+\hbar^{K}\|\mbox{\boldmath$u$\unboldmath}\|_{\operatorname{supp\,}(\nabla\chi_{K})}),\quad\text{for
}z\not\in\operatorname{spec\,}(\mathbb{D}_{0,\theta}).$ (4.3)
Therefore
$\displaystyle\operatorname{{\rm Im}\,}\langle
J_{\theta}^{1/3}(\mathbb{D}_{\theta}-\operatorname{{\rm
Re}\,}z)\mbox{\boldmath$u$\unboldmath},\mbox{\boldmath$u$\unboldmath}\rangle\leq
C_{K}(\|(\mathbb{D}_{\theta}-z)\mbox{\boldmath$u$\unboldmath}\|+\hbar^{K}\|\mbox{\boldmath$u$\unboldmath}\|)\|\|\mbox{\boldmath$u$\unboldmath}\|$
which together with
$\displaystyle\operatorname{{\rm Im}\,}\langle
J_{\theta}^{1/3}(\mathbb{D}_{\theta}-z)\mbox{\boldmath$u$\unboldmath},\mbox{\boldmath$u$\unboldmath}\rangle=\operatorname{{\rm
Im}\,}\langle J_{\theta}^{1/3}(\mathbb{D}_{\theta}-\operatorname{{\rm
Re}\,}z)\mbox{\boldmath$u$\unboldmath},\mbox{\boldmath$u$\unboldmath}\rangle-(\operatorname{{\rm
Im}\,}z)\langle\operatorname{{\rm
Re}\,}(J_{\theta}^{1/3})\mbox{\boldmath$u$\unboldmath},\mbox{\boldmath$u$\unboldmath}\rangle$
gives
$C\|(\mathbb{D}_{\theta}-z)\mbox{\boldmath$u$\unboldmath}\|\geq(C_{0}\operatorname{{\rm
Im}\,}z-C_{K+1}\hbar^{K+1})\|\mbox{\boldmath$u$\unboldmath}\|\geq\frac{C_{0}}{2}\operatorname{{\rm
Im}\,}z\|u\|,\quad\operatorname{{\rm Im}\,}z>\hbar^{K},$
provided $\hbar$ is sufficiently small. This proves the lemma. ∎
### 4.2 Resonances
Let
$\Sigma_{\theta}:=\left\\{\,z\in{\mathbb{C}}\,:\,z=\pm
c\Big{(}\frac{\lambda}{(1+\theta)^{2}}+m^{2}c^{2}\Big{)}^{1/2},\,\lambda\in{\rm[}0,\infty{\rm)}\,\right\\},$
where we have taken the principal branch of the square root function, and put
(see Figure 1)
$S_{\theta_{0}}:=\bigcup_{\theta\in
D_{\varepsilon,\theta_{0}}^{+}}\Sigma_{\theta}$
where
$D_{\varepsilon,\theta_{0}}^{+}:=\left\\{\,\theta\in
D_{\varepsilon}^{+}\,:\,\arg(1+\theta)<\arg(1+\theta_{0}),\;\frac{1}{|1+\theta|}<\frac{1}{|1+\theta_{0}|}\,\right\\}.$
We have the following results, where the second asserts that the essential
spectrum of $\mathbb{D}_{0,\theta}(\hbar)$ is invariant under the influence of
a potential satisfying Assumption 3.1.
(P5):
$\operatorname{spec}_{\operatorname{ess}}(\mathbb{D}_{0,\theta}(\hbar))=\Sigma_{\theta}$.
(P6):
$\operatorname{spec}_{\operatorname{ess}}(\mathbb{D}_{\theta}(\hbar))=\Sigma_{\theta}$
.
In view of Property (P4) the following definition makes sense.
###### Definition 4.9.
The set of resonances of $\mathbb{D}(\hbar)$ in
$S_{\theta_{0}}\cup{\mathbb{R}}$, designated
$\operatorname{Res\,}(\mathbb{D}(\hbar))$ (with $\theta_{0}$ suppressed), is
the set of eigenvalues of $\mathbb{D}_{\theta_{0}}(\hbar)$. If $z_{0}$ is a
resonance, then the spectral (or Riesz) projection
$\mbox{\boldmath$\Pi$\unboldmath}_{z_{0}}=\frac{1}{2\pi
i}\oint\limits_{|z-z_{0}|\ll 1}(\mathbb{D}_{\theta}(\hbar)-z)^{-1}\,dz$
makes sense and has finite rank. We define the multiplicity of $z_{0}$ to be
the rank of $\mbox{\boldmath$\Pi$\unboldmath}_{z_{0}}$.
We will restrict ourselves to the study of resonances having positive
energies. Namely, we assume that the resonances are located in a rectangle
${\mathcal{R}}$ satisfying the following:
###### Assumption 4.10.
We say that a complex rectangle ${\mathcal{R}}$ as in (2.1) satisfies the
assumption $(\textbf{A}_{{\mathcal{R}}}^{+})$ if $l>mc^{2}$, $b<0<t$ and there
exists $\theta_{0}\in D_{\varepsilon}^{+}$ such that
${\mathcal{R}}\cap\Sigma_{\theta_{0}}=\emptyset$. (cf. Figure 1).
In Figure 1 we show a typical scenario when we fix a $\theta_{0}\in
D_{\varepsilon}^{+}$ to uncover the resonances in $S_{\theta_{0}}$.
$mc^{2}$$-mc^{2}$$\mathcal{R}$$\operatorname{Res}(\mathbb{D})$$\Gamma_{\theta_{0}}$$S_{\theta_{0}}$
Figure 1: The set $S_{\theta_{0}}$ and a rectangle ${\mathcal{R}}$ satisfying
$(\textbf{A}_{{\mathcal{R}}}^{+})$.
The following upper bound on the number of resonances, not necessarily close
to the real axis, will be used repeatedly throughout the paper. A proof can be
found in Khochman [Kh’07], who follows Nedelec’s work on matrix valued
Schrödinger operators [Ne’01] (in turn inspired by Sjöstrand [Sj’97]).
###### Theorem 4.11.
Let $\mathbb{V}$ satisfy Assumption 3.1 and let ${\mathcal{R}}$ be a complex
rectangle satisfying Assumption $(\textbf{A}_{{\mathcal{R}}}^{+})$. Then
$\operatorname{Count\,}\left(\mathbb{D}(\hbar),\operatorname{Res\,}(\mathbb{D}(\hbar))\cap{\mathcal{R}}\right)\leq
C({\mathcal{R}})\hbar^{-3}.$
We will need the following important a priori resolvent estimate for
$\mathbb{D}_{\theta}(\hbar)$ away from the critical set, which is useful for
applying the semiclassical maximum principle (see, e.g., [TaZw’98] or [St’05,
Corollary 1]). Due to lack of space, we omit its lengthy proof (which is based
on ideas from Sjöstrand and Zworski [SjZw’91] and Sjöstrand [Sj’97]).
###### Proposition 4.12.
Let Assumption 3.1 hold. Let ${\mathcal{R}}$ be a complex rectangle satisfying
Assumption $(\textbf{A}_{{\mathcal{R}}}^{+})$ and assume
$g:(0,\hbar_{0}]\to{\mathbb{R}}_{+}$ is $o(1)$. Then there are constants
$A=A({\mathcal{R}})>0$ and $\hbar_{1}\in(0,\hbar_{0})$ such that
$\displaystyle\|(\mathbb{D}_{\theta}(\hbar)-z)^{-1}\|\leq
Ae^{A\hbar^{-3}\log\frac{1}{g(\hbar)}}\quad\text{for all
}z\in{\mathcal{R}}\setminus\bigcup_{z_{j}\in\operatorname{Res\,}(\mathbb{D}(\hbar))\cap{\mathcal{R}}}D(z_{j},g(\hbar)),$
(4.4)
for all $0<\hbar\leq\hbar_{1}$.
## 5 Main results
Henceforth we always impose Assumption 3.1 and Assumption 3.5. Moreover,
$\mathbb{J}(\hbar)$ represents either $\mathbb{J}_{\infty}(\hbar)$ or
$\mathbb{J}_{R}(\hbar)$. Throughout we shall assume that
$mc^{2}<l_{0}<r_{0}<\infty$ (here $l_{0}$ and $r_{0}$ are independent of
$\hbar$).
#### The case $R_{0}^{\prime}<R_{1}$
Bear in mind that
$\operatorname{supp\,}\mathbb{W}\subset{\mathbb{R}}^{3}\setminus B(0,R_{1})$.
We obtain the following result, which shows how a single resonance of
$\mathbb{D}(\hbar)$ generates a single eigenvalue of $\mathbb{J}(\hbar)$
nearby, and vice versa.
###### Theorem 5.1.
1\. Let $R_{0}^{\prime}<R_{1}$. Suppose $z_{0}(\hbar)$ is a resonance of
$\mathbb{D}(\hbar)$ in
$[l_{0},r_{0}]+i\Big{[}-\frac{\hbar^{5}}{C\log\frac{1}{\hbar}},0\Big{]},\quad
C\gg 1.$
Then there is an $\hbar_{0}\in{\rm(}0,1{\rm]}$ such that, for
$0<\hbar\leq\hbar_{0}$, $\mathbb{J}(\hbar)$ has an eigenvalue in
$\big{[}\operatorname{{\rm
Re}\,}z_{0}(\hbar)-\varepsilon(\hbar)\log\frac{1}{\hbar},\operatorname{{\rm
Re}\,}z_{0}(\hbar)+\varepsilon(\hbar)\log\frac{1}{\hbar}\big{]}+i[-\varepsilon(\hbar),0]$
(5.1)
where $\varepsilon(\hbar)=-\hbar^{-5}\operatorname{{\rm
Im}\,}z_{0}(\hbar)+{\mathcal{O}}(\hbar^{\infty})$.
2\. Let $R_{0}^{\prime}<R_{1}$. Suppose $w_{0}(\hbar)$ is an eigenvalue of
$\mathbb{J}(\hbar)$ in
$[l_{0},r_{0}]+i\Big{[}-\Big{(}\frac{\hbar^{4}}{C\log\frac{1}{\hbar}}\Big{)}^{2},0\Big{]},\quad
C\gg 1.$
Then there is an $\hbar_{0}\in{\rm(}0,1{\rm]}$ such that, for
$0<\hbar\leq\hbar_{0}$, $\mathbb{D}(\hbar)$ has a resonance in (5.1) with
$\varepsilon(\hbar)=\hbar^{-4}\sqrt{-\operatorname{{\rm
Im}\,}w_{0}(\hbar)}+{\mathcal{O}}(\hbar^{\infty})$.
#### The case $R_{1}<R_{0}^{\prime}$
As the following theorem shows we only worsen the error by at most a factor
$\hbar^{-1}$ if we allow the supports of $\mathbb{V}$ and $\mathbb{W}$ to
intersect. To establish it we need to impose both the nontrapping assumption
and the hyperbolicity condition.
###### Theorem 5.2.
Let $R_{1}<R_{0}^{\prime}$. Suppose that $\mathbb{D}(\hbar)$ is nontrapping
for $|x|>R_{1}$ on the interval $J=[l_{0},r_{0}]$; in the sense of Definition
3.2. Moreover, let Assumption 3.3 be satisfied and suppose $z_{0}(\hbar)$ is a
resonance of $\mathbb{D}(\hbar)$ in
$[l_{0},r_{0}]+i\Big{[}-\frac{\hbar^{6}}{C\log\frac{1}{\hbar}},0\Big{]},\quad
C\gg 1.$
Then there is an $\hbar_{0}\in{\rm(}0,1{\rm]}$ such that, for
$0<\hbar\leq\hbar_{0}$, $\mathbb{J}(\hbar)$ has an eigenvalue in (5.1) with
$\varepsilon(\hbar)=-\hbar^{-6}\operatorname{{\rm
Im}\,}z_{0}(\hbar)+{\mathcal{O}}(\hbar^{\infty}).$
## 6 Properties of CAP Hamiltonians
Herein we study the spectral properties of the CAP Hamiltonians. We give an
estimate of the number of eigenvalues of $\mathbb{J}(\hbar)$ on a rectangle.
The result is an analogue of the estimate in Theorem 4.11 for
$\mathbb{D}(\hbar)$, however this time for the number of eigenvalues of
$\mathbb{J}(\hbar)$ rather than the resonances of $\mathbb{D}(\hbar)$. Our
approach is inspired by Stefanov [St’05].
Since the following result is independent of $\hbar$ we need not indicate that
we have a family of $\hbar$-dependent operators.
###### Lemma 6.1.
The resolvent $(\mathbb{J}-z)^{-1}$ exists as a meromorphic operator in
$\operatorname{{\rm Im}\,}z>-\delta_{0}$ with the poles being the eigenvalues
of finite multiplicity.
###### Proof.
Let $\chi_{1}+\chi_{2}+\chi_{3}=1$ be a smooth partition of unity with
$\chi_{1}=1$ near $B(0,R_{0}^{\prime})$ and supported in
$B(0,\tfrac{R_{0}^{\prime}+R_{1}}{2})$, $\chi_{2}$ compactly supported and
$\chi_{3}$ supported in $|x|>R_{2}$. Let $\widetilde{\chi}_{j}\succ\chi_{j}$
have the same support properties. The fact that $\\{\widetilde{\chi}_{j}\\}$
is not a partition of unity does not matter. Define $W_{1}$ to equal
$\delta_{0}$ for $|x|<R_{2}$ and $W$ otherwise. For $\operatorname{{\rm
Im}\,}z_{0}>0$ fixed (see below) the operator
$\mbox{\boldmath$E$\unboldmath}(z,z_{0})=\widetilde{\mbox{\boldmath$\chi$\unboldmath}}_{1}(\mathbb{D}-z_{0})^{-1}\mbox{\boldmath$\chi$\unboldmath}_{1}+\widetilde{\mbox{\boldmath$\chi$\unboldmath}}_{2}(\mathbb{D}_{0}-i\mathbb{W}-z_{0})^{-1}\mbox{\boldmath$\chi$\unboldmath}_{2}+\widetilde{\mbox{\boldmath$\chi$\unboldmath}}_{3}(\mathbb{D}_{0}-i\mathbb{W}_{1}-z)^{-1}\mbox{\boldmath$\chi$\unboldmath}_{3}$
depends analytically on $z$ in $\operatorname{{\rm Im}\,}z>-\delta_{0}$.
Moreover
$(\mathbb{J}-z)\mbox{\boldmath$E$\unboldmath}(z,z_{0})=\mbox{\boldmath$1$\unboldmath}+\mbox{\boldmath$K$\unboldmath}(z,z_{0})$
where
$\displaystyle\mbox{\boldmath$K$\unboldmath}(z,z_{0})$
$\displaystyle=[\mathbb{D}_{0},\widetilde{\mbox{\boldmath$\chi$\unboldmath}}_{1}](\mathbb{D}-z_{0})^{-1}\mbox{\boldmath$\chi$\unboldmath}_{1}+(z_{0}-z)\widetilde{\mbox{\boldmath$\chi$\unboldmath}}_{1}(\mathbb{D}-z_{0})^{-1}\mbox{\boldmath$\chi$\unboldmath}_{1}$
$\displaystyle+[\mathbb{D}_{0},\widetilde{\mbox{\boldmath$\chi$\unboldmath}}_{2}](\mathbb{D}_{0}-i\mathbb{W}-z_{0})^{-1}\mbox{\boldmath$\chi$\unboldmath}_{2}+(z_{0}-z)\widetilde{\mbox{\boldmath$\chi$\unboldmath}}_{2}(\mathbb{D}_{0}-i\mathbb{W}-z_{0})^{-1}\mbox{\boldmath$\chi$\unboldmath}_{2}$
$\displaystyle+[\mathbb{D}_{0},\widetilde{\mbox{\boldmath$\chi$\unboldmath}}_{3}](\mathbb{D}_{0}-i\mathbb{W}_{1}-z)^{-1}\mbox{\boldmath$\chi$\unboldmath}_{3}$
$\displaystyle=:\mbox{\boldmath$K$\unboldmath}_{1}(z_{0})+\mbox{\boldmath$K$\unboldmath}_{2}(z,z_{0})+\mbox{\boldmath$K$\unboldmath}_{3}(z_{0})+\mbox{\boldmath$K$\unboldmath}_{4}(z,z_{0})+\mbox{\boldmath$K$\unboldmath}_{5}(z).$
By construction $\mbox{\boldmath$K$\unboldmath}(z,z_{0})$ depends analytically
on $z$ in $\operatorname{{\rm Im}\,}z>-\delta_{0}$. Furthermore it follows by
the Rellich-Kondrachov embedding theorem that
$\mbox{\boldmath$K$\unboldmath}(z,z_{0})$ is a compact operator on
${\mathbf{L}}^{2}({\mathbb{R}}^{3},{\mathbb{C}}^{4})$. Since for
$\operatorname{{\rm Im}\,}z_{0}>0$ sufficiently large we have
$\|\mbox{\boldmath$K$\unboldmath}(z,z_{0})\|\leq C\max\\{|\operatorname{{\rm
Im}\,}z|^{-1},(\operatorname{{\rm Im}\,}z_{0})^{-1}\\}$ we see that for
$z=z_{0}$ and $\operatorname{{\rm Im}\,}z_{0}$ large enough
$\|\mbox{\boldmath$K$\unboldmath}(z,z_{0})\|\leq 1/2$. By the analytic
Fredholm theorem, for fixed $z_{0}$ as above,
$(\mbox{\boldmath$1$\unboldmath}+\mbox{\boldmath$K$\unboldmath}(z,z_{0}))^{-1}$
exists as a meromorphic operator in $\operatorname{{\rm Im}\,}z>-\delta_{0}$.
Similarly a left parametrix is constructed by interchanging
$\mbox{\boldmath$\chi$\unboldmath}_{j}$ and
$\widetilde{\mbox{\boldmath$\chi$\unboldmath}}_{j}$ for $j=1,2,3$. The left
and right inverses will share the same poles and agree elsewhere and thus
$\displaystyle(\mathbb{J}-z)^{-1}=\mbox{\boldmath$E$\unboldmath}(z,z_{0})(\mbox{\boldmath$1$\unboldmath}+\mbox{\boldmath$K$\unboldmath}(z,z_{0}))^{-1}$
(6.1)
so that $(\mathbb{J}-z)^{-1}$ is meromorphic in $\operatorname{{\rm
Im}\,}z>-\delta_{0}$ with finite rank residues at the poles which are the
eigenvalues. ∎
The following result and its proof is similar to [St’05, Proposition 2].
###### Proposition 6.2.
Let Assumption 3.1 and Assumption 3.5 hold. If ${\mathcal{R}}$ satisfies
Assumption $(\textbf{A}_{{\mathcal{R}}}^{+})$ then the number of eigenvalues
in ${\mathcal{R}}$ satisfies
$\displaystyle\operatorname{Count\,}(\mathbb{J}(\hbar),{\mathcal{R}})={\mathcal{O}}(\hbar^{-4}).$
(6.2)
###### Proof.
In addition to the requirements imposed in the proof of Lemma 6.1, assume
$z_{0}$ is such that we can find $r_{0},\varepsilon_{0}>0$ so that
${\mathcal{R}}\subset D(z_{0},r_{0})\subset
D(z_{0},r_{0}+\varepsilon_{0})\subset\\{\operatorname{{\rm
Im}\,}z>-\delta_{0}\\}$. By (6.1) it suffices to estimate the number of points
$z$ in $D(z_{0},r_{0})$ where
$\mbox{\boldmath$1$\unboldmath}+\mbox{\boldmath$K$\unboldmath}(z,z_{0})$ is
not invertible. Since
$\|\mbox{\boldmath$K$\unboldmath}_{5}(z)\|={\mathcal{O}}(\hbar)$ we may write
$\displaystyle\mbox{\boldmath$1$\unboldmath}+\mbox{\boldmath$K$\unboldmath}(z,z_{0})$
$\displaystyle=\big{(}\mbox{\boldmath$1$\unboldmath}+\widetilde{\mbox{\boldmath$K$\unboldmath}}(z)\big{)}(\mbox{\boldmath$1$\unboldmath}+\mbox{\boldmath$K$\unboldmath}_{1}(z_{0})+\mbox{\boldmath$K$\unboldmath}_{3}(z_{0})+\mbox{\boldmath$K$\unboldmath}_{5}(z))$
$\displaystyle\widetilde{\mbox{\boldmath$K$\unboldmath}}(z,z_{0})$
$\displaystyle:=(\mbox{\boldmath$K$\unboldmath}_{2}(z,z_{0})+\mbox{\boldmath$K$\unboldmath}_{4}(z,z_{0}))\big{(}\mbox{\boldmath$1$\unboldmath}+\mbox{\boldmath$K$\unboldmath}_{1}(z_{0})+\mbox{\boldmath$K$\unboldmath}_{3}(z_{0})+\mbox{\boldmath$K$\unboldmath}_{5}(z)\big{)}^{-1}$
provided $\hbar$ is small enough. Thus
$\mbox{\boldmath$1$\unboldmath}+\mbox{\boldmath$K$\unboldmath}(z,z_{0})$ is
not invertible if and only if
$\mbox{\boldmath$1$\unboldmath}+\widetilde{\mbox{\boldmath$K$\unboldmath}}(z,z_{0})$
is not invertible. Now, since
$\mbox{\boldmath$1$\unboldmath}+\widetilde{\mbox{\boldmath$K$\unboldmath}}(z,z_{0})$
need not belong to ${\mathcal{B}}_{1}$ (see below) and since the singular
points of
$\mbox{\boldmath$1$\unboldmath}+\widetilde{\mbox{\boldmath$K$\unboldmath}}(z,z_{0})$
are included among those of
$\mbox{\boldmath$1$\unboldmath}-\widetilde{\mbox{\boldmath$K$\unboldmath}}^{4}(z,z_{0})$,
we are going to estimate the number of zeros of
$f(z):=\det(\mbox{\boldmath$1$\unboldmath}-\widetilde{\mbox{\boldmath$K$\unboldmath}}^{4}(z,z_{0})).$
By (2.7), (2.4), (2.5) and (2.3) it suffices to obtain upper bounds of
$\mu_{j}(\mbox{\boldmath$K$\unboldmath}_{2})$ and
$\mu_{j}(\mbox{\boldmath$K$\unboldmath}_{4})$. To this end, let
$\widetilde{R}>R_{0}^{\prime}+R_{1}$ and consider the flat torus
$\mathbb{T}:=({\mathbb{R}}/\widetilde{R}{\mathbb{Z}})^{3}$ obtained by
identifying opposite faces of the cube
$\\{x\in{\mathbb{R}}^{3}:|x_{j}|<\widetilde{R},\,j=1,2,3\\}$. We assume
$\mathbb{T}$ carries the metric induced by the Euclidean metric on
${\mathbb{R}}^{3}$ and trivial spin structure. Denote by
$\mathbb{D}_{0,\mathbb{T}}$ the corresponding free semiclassical Dirac
operator on $\mathbb{T}$. Then, viewing $B(0,\tfrac{R_{0}^{\prime}+R_{1}}{2})$
as a subset of $\mathbb{T}$,
$\mathbb{D}_{\mathbb{T}}:=\mathbb{D}_{0,\mathbb{T}}+\mathbb{V}(x)$ coincides
with $\mathbb{D}$ near $B(0,\tfrac{R_{0}^{\prime}+R_{1}}{2})$. It is well-
known that $\mathbb{D}_{0,\mathbb{T}}$ satisfies the Weyl law (in fact this
follows from the Weyl law for $\Delta_{\mathbb{T}}$ in view of the
Schrödinger-Lichnerowicz formula)
$\displaystyle\operatorname{Count\,}(\mathbb{D}_{0,\mathbb{T}},[-\lambda,\lambda])={\mathcal{O}}\Big{(}\frac{\lambda^{3}}{\hbar^{3}}\Big{)},$
(6.3)
and since $\mathbb{V}$ is a bounded multiplication operator the Weyl
asymptotics remain true also for $\mathbb{D}_{\mathbb{T}}$. Denote by
$\lambda_{1}\leq\lambda_{2}\leq\cdots$ the eigenvalues of
$\mathbb{D}_{0,\mathbb{T}}$. Then (6.3) implies
$\mu_{j}\big{(}(\mathbb{D}_{\mathbb{T}}-i)^{-1}\big{)}=|i-\lambda_{j}|^{-1}\leq\frac{C}{1+\hbar
j^{1/3}},$
and by the resolvent equation the same estimate holds for
$\mu_{j}\big{(}(\mathbb{D}_{\mathbb{T}}-z_{0})^{-1}\big{)}$. From the identity
$(\mathbb{D}-z_{0})^{-1}\mbox{\boldmath$\chi$\unboldmath}_{1}=\widetilde{\mbox{\boldmath$\chi$\unboldmath}}_{1}(\mathbb{D}_{\mathbb{T}}-z_{0})^{-1}\mbox{\boldmath$\chi$\unboldmath}_{1}-(\mathbb{D}-z_{0})^{-1}[\mathbb{D},\widetilde{\mbox{\boldmath$\chi$\unboldmath}}_{1}](\mathbb{D}_{\mathbb{T}}-z_{0})^{-1}\mbox{\boldmath$\chi$\unboldmath}_{1}$
we now obtain
$\mu_{j}(\mbox{\boldmath$K$\unboldmath}_{2})\leq\frac{C}{1+\hbar j^{1/3}}.$
By taking a possibly larger torus we see that
$\widetilde{\mbox{\boldmath$\chi$\unboldmath}}_{2}(\mathbb{D}_{0}-i\mathbb{W}-z_{0})^{-1}\mbox{\boldmath$\chi$\unboldmath}_{2}=(\mathbb{D}_{0,\mathbb{T}}-i)^{-1}(\mathbb{D}_{0,\mathbb{T}}-i)\widetilde{\mbox{\boldmath$\chi$\unboldmath}}_{2}(\mathbb{D}_{0}-i\mathbb{W}-z_{0})^{-1}\mbox{\boldmath$\chi$\unboldmath}_{2}$
where
$(\mathbb{D}_{0,\mathbb{T}}-i)\widetilde{\mbox{\boldmath$\chi$\unboldmath}}_{2}(\mathbb{D}_{0}-i\mathbb{W}-z_{0})^{-1}\mbox{\boldmath$\chi$\unboldmath}_{2}$
is bounded. Thus, by (6.3), also
$\mu_{j}(\mbox{\boldmath$K$\unboldmath}_{4})\leq\frac{C}{1+\hbar j^{1/3}}.$
It follows that
$\sum_{j}\mu_{j}(\widetilde{\mbox{\boldmath$K$\unboldmath}}^{4})\leq\sum_{j}\frac{C}{(1+\hbar
j^{1/3})^{4}}\leq\sum_{j}\frac{C}{1+\hbar^{4}j^{4/3}}\leq C\hbar^{-4},$
and from (2.7) we obtain $|f(z)|\leq e^{C\hbar^{-4}}$ for $z\in
D(z_{0},r_{0}+\varepsilon_{0})$. Thus, since $f(z_{0})=1$, an application of
Jensen’s formula relative to $D(z_{0},r_{0}+\varepsilon_{0})$ and
$D(z_{0},r_{0})$ gives (6.2). ∎
Finally we establish an a priori resolvent estimate for the complex scaled CAP
Hamiltonian $\mathbb{J}_{\theta}$, which takes into account the distance to
its eigenvalues $w_{j}$; this is the analogue of Proposition 4.12 above.
###### Proposition 6.3.
Let Assumption 3.1 and Assumption 3.5 hold. Let ${\mathcal{R}}$ be a complex
rectangle satisfying Assumption $(\textbf{A}_{{\mathcal{R}}}^{+})$ and assume
$g:(0,\hbar_{0}]\to{\mathbb{R}}_{+}$ is $o(1)$. Then there are constants
$A=A({\mathcal{R}})>0$ and $\hbar_{1}\in(0,\hbar_{0})$ such that
$\|(\mathbb{J}(\hbar)-z)^{-1}\|\leq
Ae^{A\hbar^{-4}\log\frac{1}{g(\hbar)}},\quad
z\in{\mathcal{R}}\setminus\bigcup_{w_{j}(\hbar)\in\operatorname{spec\,}(\mathbb{J}_{\theta}(\hbar))\cap{\mathcal{R}}^{\prime}}D(w_{j}(\hbar),g(\hbar)),$
where ${\mathcal{R}}\subsetneq{\mathcal{R}}^{\prime}$.
The following proof is partially sketchy to avoid repeating arguments.
###### Proof.
In this proof, once again, we suppress the subscript in
$\mathbb{J}_{\infty}(\hbar)$ and its dependence on $\hbar$. Using the notation
from Lemma 6.1 and Proposition 6.2 we have
$(\mathbb{J}-z)^{-1}=\mbox{\boldmath$E$\unboldmath}(\mbox{\boldmath$1$\unboldmath}+\mbox{\boldmath$K$\unboldmath}_{1}+\mbox{\boldmath$K$\unboldmath}_{3}+\mbox{\boldmath$K$\unboldmath}_{5})^{-1}(\mbox{\boldmath$1$\unboldmath}-\widetilde{\mbox{\boldmath$K$\unboldmath}}+\widetilde{\mbox{\boldmath$K$\unboldmath}}^{2}-\widetilde{\mbox{\boldmath$K$\unboldmath}}^{3})(\mbox{\boldmath$1$\unboldmath}-\widetilde{\mbox{\boldmath$K$\unboldmath}}^{4})^{-1}$
so it suffices to estimate
$(\mbox{\boldmath$1$\unboldmath}-\widetilde{\mbox{\boldmath$K$\unboldmath}}^{4})^{-1}$
away from the set of eigenvalues of $\mathbb{J}$. To this end we have (see
[GoKr’69, Ch. V, Theorem 5.1])
$\|(\mbox{\boldmath$1$\unboldmath}+\widetilde{\mbox{\boldmath$K$\unboldmath}}^{4}(z))^{-1}\|\leq\frac{\det(\mbox{\boldmath$1$\unboldmath}+|\widetilde{\mbox{\boldmath$K$\unboldmath}}^{4}(z)|)}{|\det(\mbox{\boldmath$1$\unboldmath}+\widetilde{\mbox{\boldmath$K$\unboldmath}}^{4}(z))|}.$
For the numerator we have as before
$\det(\mbox{\boldmath$1$\unboldmath}+|\widetilde{\mbox{\boldmath$K$\unboldmath}}^{4}(z)|)\leq
e^{\|\widetilde{\mbox{{\scriptsize\boldmath$K$\unboldmath}}}(z)\|_{{\mathcal{B}}_{1}}}\leq
e^{C\hbar^{-4}}$. The denominator can be treated as in [Sj’97, Section 8],
i.e. by first factoring out its zeros and then use the upper bound for the
eigenvalue counting function to obtain
$|\det(\mbox{\boldmath$1$\unboldmath}+\widetilde{\mbox{\boldmath$K$\unboldmath}}_{0}(z))|\geq
Ce^{C\hbar^{-4}\log\frac{1}{g}}\quad\text{for
}\operatorname{dist\,}(z,\operatorname{spec\,}(\mathbb{J}_{\theta})\cap{\mathcal{R}})\geq
g(\hbar).$
Putting these facts together gives the assertion. ∎
###### Remark 6.4.
The results above, established for $\mathbb{J}_{\infty}(\hbar)$ and its
resolvent, can easily be carried over to the CAP Hamiltonian
$\mathbb{J}_{R}(\hbar)$ and its resolvent.
## 7 Quasimodes and resonances
In this section we present the main result that enables us to relate so-called
quasimodes of $\mathbb{D}(\hbar)$ with resonances of $\mathbb{D}(\hbar)$. It
informs us that if we have a set of linearly independent quasimodes, which can
be thought of as square integrable approximate resonant states, for energies
in a real interval $I$ and if this set remains linearly independent under
small perturbations (in the semiclassical sense), then there are as many
resonances as there are quasimodes and these are located with real parts near
$I$ and having small imaginary parts. Such a result was first established by
Tang and Zworski [TaZw’98] for Schrödinger operators. We give a version which
is valid for the perturbed Dirac operator. Our proof is adopted from Stefanov
[St’99] who even managed to treat higher multiplicities and clusters of
resonances in the case when quasimodes are very close to each other. He showed
that such clusters of quasimodes generate (asymptotically) at least the same
number of resonances. In [St’05] he improved the latter result in several ways
by modifying the reasoning in [St’99, Theorem 1]. The underlying ideas,
however, are the same as in Tang and Zworski [TaZw’98] (see also [Sj’02,
Theorem 11.2]).
Let $\chi,\widetilde{\chi}\in C_{0}^{\infty}({\mathbb{R}}^{3})$ with
$\mathbf{1}_{B(0,R)}\prec\chi\prec\widetilde{\chi}$ and let
$z_{0}\in\operatorname{Res\,}(\mathbb{D}(\hbar))$. Then, for $z$ in a
neighborhood of $z_{0}$ we have, with $N$ finite,
$\mbox{\boldmath$\chi$\unboldmath}(\mathbb{D}_{\theta}-z)^{-1}\widetilde{\mbox{\boldmath$\chi$\unboldmath}}=\mbox{\boldmath$A$\unboldmath}_{0}(z,\hbar)+\sum_{j=1}^{N}(z-z_{0}(\hbar))^{-j}\mbox{\boldmath$A$\unboldmath}_{j}(\hbar)$
(7.1)
for some operator $\mbox{\boldmath$A$\unboldmath}_{0}(z,\hbar)$, holomorphic
in $z$ near $z_{0}(\hbar)$, and finite rank operators
$\mbox{\boldmath$A$\unboldmath}_{j}$, $1\leq j\leq N$, independent of $z$.
###### Lemma 7.1.
Let $\chi\in C_{0}^{\infty}({\mathbb{R}}^{3})$ with $\chi=1$ on $B(0,R)$ for
some $R>0$. Then, for any
$z_{0}(\hbar)\in\operatorname{Res\,}(\mathbb{D}(\hbar))$, we have
$\displaystyle\mbox{\boldmath$\chi$\unboldmath}(\mathbb{D}_{\theta}(\hbar)-z)^{-1}\mbox{\boldmath$\chi$\unboldmath}=\mbox{\boldmath$A$\unboldmath}_{0}(z,\hbar)\mbox{\boldmath$\chi$\unboldmath}+\sum_{j=1}^{N}(z-z_{0}(\hbar))^{-j}\mbox{\boldmath$A$\unboldmath}_{1}(\hbar)\mbox{\boldmath$Q$\unboldmath}_{j}(\hbar)$
for some operators $\mbox{\boldmath$Q$\unboldmath}_{j}$, holomorphic at
$z_{0}(\hbar)$.
###### Proof.
For notational reasons we denote $\chi=\chi_{1}$ and
$\widetilde{\chi}=\chi_{N}$ and introduce the sequence of intermediate cut-off
functions
$\chi_{1}\prec\chi_{2}\prec\cdots\prec\chi_{N}.$
Multiply (7.1) by $\mathbb{D}_{\theta}-z$ from the right to get
$\displaystyle\mbox{\boldmath$\chi$\unboldmath}_{1}+\mbox{\boldmath$\chi$\unboldmath}_{1}$
$\displaystyle(\mathbb{D}_{\theta}-z)^{-1}[\widetilde{\mbox{\boldmath$\chi$\unboldmath}},\mathbb{D}_{\theta}]=\mbox{\boldmath$A$\unboldmath}_{0}(z)(\mathbb{D}_{\theta}-z)+\sum_{j=1}^{N}(z-z_{0})^{-j}\mbox{\boldmath$A$\unboldmath}_{j}(\mathbb{D}_{\theta}-z)$
$\displaystyle=\mbox{\boldmath$A$\unboldmath}_{0}(z)(\mathbb{D}_{\theta}-z)-\mbox{\boldmath$A$\unboldmath}_{1}+\sum_{j=1}^{N}(z-z_{0})^{-j}\big{(}\mbox{\boldmath$A$\unboldmath}_{j}(\mathbb{D}_{\theta}-z)-\mbox{\boldmath$A$\unboldmath}_{j+1}\big{)}$
with the convention that $\mbox{\boldmath$A$\unboldmath}_{N+1}=0$. Upon
multiplying by $\mbox{\boldmath$\chi$\unboldmath}_{l}$ from the right and
using
$[\widetilde{\mbox{\boldmath$\chi$\unboldmath}},\mathbb{D}_{\theta}]\mbox{\boldmath$\chi$\unboldmath}_{l}=0$
we realize that all singular terms on the right must vanish, that is
$\mbox{\boldmath$A$\unboldmath}_{j}(\mathbb{D}_{\theta}-z_{0})\mbox{\boldmath$\chi$\unboldmath}_{l}=\mbox{\boldmath$A$\unboldmath}_{j+1}\mbox{\boldmath$\chi$\unboldmath}_{l},\quad
j,l=1,\ldots,N-1.$
Using the latter identity repeatedly results in
$\displaystyle\mbox{\boldmath$A$\unboldmath}_{j}\chi_{1}$
$\displaystyle=\mbox{\boldmath$A$\unboldmath}_{j-1}(\mathbb{D}_{\theta}-z_{0})\mbox{\boldmath$\chi$\unboldmath}_{1}=\mbox{\boldmath$A$\unboldmath}_{j-1}\mbox{\boldmath$\chi$\unboldmath}_{2}(\mathbb{D}_{\theta}-z_{0})\mbox{\boldmath$\chi$\unboldmath}_{1}$
$\displaystyle\phantom{a}\vdots$
$\displaystyle=\mbox{\boldmath$A$\unboldmath}_{1}(\mathbb{D}_{\theta}-z_{0})\mbox{\boldmath$\chi$\unboldmath}_{j-1}(\mathbb{D}_{\theta}-z_{0})\mbox{\boldmath$\chi$\unboldmath}_{j-2}\cdots\mbox{\boldmath$\chi$\unboldmath}_{2}(\mathbb{D}_{\theta}-z_{0})\chi_{1}.$
By multiplying (7.1) from the right by $\chi$ and using the previous relation
we obtain the lemma with
$\mbox{\boldmath$Q$\unboldmath}_{j}=(\mathbb{D}_{\theta}-z_{0})\mbox{\boldmath$\chi$\unboldmath}_{j-1}(\mathbb{D}_{\theta}-z_{0})\mbox{\boldmath$\chi$\unboldmath}_{j-2}\cdots\mbox{\boldmath$\chi$\unboldmath}_{2}(\mathbb{D}_{\theta}-z_{0})\chi_{1}.\qed$
We state and prove the main result of this section for positive energies.
###### Theorem 7.2.
Assume $mc^{2}<l_{0}\leq l(\hbar)\leq r(\hbar)\leq r_{0}<\infty$. Assume that
for any $\hbar\in(0,\hbar_{0}]$ there is $m(\hbar)\in{\mathbb{Z}}_{+}$,
$E_{j}(\hbar)\in[l(\hbar),r(\hbar)]$ and normalized
$\mbox{\boldmath$u$\unboldmath}_{j}(\hbar)\in\operatorname{Dom\,}(\mathbb{D})$
(quasimodes) for $1\leq j\leq m(\hbar)$, having support in a ball $B(0,R)$
where $R<R_{0}$ does not depend on $\hbar$. Assume, moreover, that
$\displaystyle\|(\mathbb{D}(\hbar)-E_{j}(\hbar))\mbox{\boldmath$u$\unboldmath}_{j}(\hbar)\|\leq\rho(\hbar)$
(7.2) and $\displaystyle\text{all
}\tilde{\mbox{\boldmath$u$\unboldmath}}_{j}(\hbar)\in{\mathcal{H}}\text{ such
that
}\|\tilde{\mbox{\boldmath$u$\unboldmath}}_{j}(\hbar)-\mbox{\boldmath$u$\unboldmath}_{j}(\hbar)\|\leq\frac{\hbar^{N}}{M},\quad
1\leq j\leq m(\hbar),$ $\displaystyle\text{are linearly independent},$ (7.3)
where $\rho(\hbar)\leq\hbar^{4+N}/(C\log\hbar^{-1})$, $C\gg 1$, $N\geq 0$ and
$M>0$. Then there exists $C_{0}=C_{0}(l_{0},r_{0})>0$ such that for any $B>0$
and $K\in{\mathbb{Z}}_{+}$ there is an
$\hbar_{1}=\hbar_{1}(A,B,M,N)\leq\hbar_{0}$ such that for any
$\hbar\in(0,\hbar_{1}]$ there will be at least $m(\hbar)$ resonances of
$\mathbb{D}(\hbar)$ in
$\displaystyle[l(\hbar)-b(\hbar)\log\frac{1}{\hbar},r(\hbar)+b(\hbar)\log\frac{1}{\hbar}]+i[-b(\hbar),0],$
(7.4)
where
$b(\hbar)=\max{(C_{0}BM\rho(\hbar)\hbar^{-4-N},e^{-B/\hbar},\hbar^{K})}.$
We remark that Theorem 7.2 is stronger than what is needed for the present
work where we only work with one quasimode at a time.
###### Proof.
Denote by $z_{1},\ldots,z_{l}$ all _distinct_ resonances in
$\displaystyle{\mathcal{R}}_{2}:=[l-2w,r+2w]+i\Big{[}-2A\hbar^{-3}(\log\frac{1}{S})S,S\Big{]}.$
(7.5)
where $S=\max(e^{3}M\hbar^{-N}\rho(\hbar),e^{-2B/\hbar},\hbar^{K+4})$ for some
$B>0$ (cf. Appendix A) and
$w=12A\hbar^{-3}(\log\frac{1}{\hbar})(\log\frac{1}{S})S.$
Clearly $S$ and $w$ satisfies (A.2). It is easy to see that
${\mathcal{R}}_{2}\cap{\mathbb{C}}_{-}$ is contained in the box (7.4) for
$\hbar$ small enough so it suffices to show that there are at least $m$
resonances in ${\mathcal{R}}_{2}$. Fix $\chi\in
C_{0}^{\infty}({\mathbb{R}}^{3})$ with $\chi\succ\mathbf{1}_{B(0,R)}$. Let
$\Pi$ be the orthogonal projection onto
$\cup_{j}\mbox{\boldmath$A$\unboldmath}_{1}^{(j)}({\mathcal{H}})$ with
$\mbox{\boldmath$A$\unboldmath}_{1}^{(j)}$ being the residue at $z_{j}$, cf.
(7.1), and let
$\mbox{\boldmath$\Pi$\unboldmath}^{\prime}=\mbox{\boldmath$1$\unboldmath}-\mbox{\boldmath$\Pi$\unboldmath}$
be the complementary projection. In view of Lemma 7.1
$\mbox{\boldmath$F$\unboldmath}(z):=\mbox{\boldmath$\Pi$\unboldmath}^{\prime}\mbox{\boldmath$\chi$\unboldmath}(\mathbb{D}_{\theta}(\hbar)-z)^{-1}\mbox{\boldmath$\chi$\unboldmath}$
is holomorphic in a neighborhood of ${\mathcal{R}}_{2}$. We are going to use
this fact to show that the estimate in (4.4) holds in the whole of the smaller
box
${\mathcal{R}}_{1}:=[l-w,r+w]+i\Big{[}-A\hbar^{-3}(\log\frac{1}{S})S,S\Big{]}\subset{\mathcal{R}}_{2}$
The bound $\|\mbox{\boldmath$F$\unboldmath}(z)\|\leq C/S$ (cf. Proposition
A.1) for $\operatorname{{\rm Im}\,}z=S$ follows from Proposition 4.8. From
Proposition 4.12 with $g=S$ it follows that (4.4) is fulfilled for
$z\in{\mathcal{R}}_{2}\cap\\{z:\operatorname{dist\,}(z,\operatorname{Res\,}(\mathbb{D}))\geq
S\\}$. Consider now the set obtained by adjoining to ${\mathcal{R}}_{1}$ the
set of unions of disks $D(z_{j},S)$ that have a point in common with
${\mathcal{R}}_{1}$ (see Figure 2).
$w$${\mathcal{R}}_{1}$${\mathcal{R}}_{2}$$A\hbar^{-3}\log\frac{1}{S}$$S$
Figure 2: Connected unions of disks centered at resonances with radius $S$
that intersect with ${\mathcal{R}}_{1}$ never intersect the complement of
${\mathcal{R}}_{2}$.
If we can show that the set so obtained is contained in ${\mathcal{R}}_{2}$,
provided $\hbar$ is small enough, where $F$ is holomorphic, then it would
follow from the (classical) maximum principle that (4.4) holds in all of
${\mathcal{R}}_{1}$ since we know it holds on the boundary of the extended
set. To this end, notice how it follows from Theorem 4.11 that the diameter of
any connected chain of disks centered at resonances having radii $S$ is
${\mathcal{O}}(\hbar^{-3}S)$ while the shortest distance from
${\mathcal{R}}_{1}\cap{\mathbb{C}}_{-}$ (to where the resonances are confined)
to the complement of ${\mathcal{R}}_{2}$ is $A\hbar^{-3}(\log S^{-1})S$. Since
the latter is greater than the former, provided $\hbar$ is sufficiently small,
it follows that any such union of disks that intersect with
${\mathcal{R}}_{1}$ cannot intersect the complement of ${\mathcal{R}}_{2}$.
Thus
$\|\mbox{\boldmath$F$\unboldmath}(z)\|\leq
Ae^{A\hbar^{-3}\log\tfrac{1}{S}}\quad\text{for all }z\in{\mathcal{R}}_{1}.$
We are now in a position to apply Proposition A.1 so that, by letting $z\to
E_{j}$,
$\displaystyle\|\mbox{\boldmath$\Pi$\unboldmath}\mbox{\boldmath$u$\unboldmath}_{j}-\mbox{\boldmath$u$\unboldmath}_{j}\|=\|\mbox{\boldmath$\Pi$\unboldmath}^{\prime}\mbox{\boldmath$\chi$\unboldmath}\mbox{\boldmath$u$\unboldmath}_{j}\|=\|\mbox{\boldmath$F$\unboldmath}(z)(\mathbb{D}-z)\mbox{\boldmath$u$\unboldmath}_{j}\|\leq\frac{e^{3}\rho(\hbar)}{S},$
where we have also used the quasimode property (7.2). It follows from our
choice of $S$ and the assumption (7.3) that
$\\{\mbox{\boldmath$\Pi$\unboldmath}\mbox{\boldmath$u$\unboldmath}_{j}\\}_{j=1}^{m}$
is linearly independent. Consequently,
$\operatorname{Count\,}(\mathbb{D},{\mathcal{R}}_{2})=\sum_{j=1}^{l}\operatorname{rank}\mbox{\boldmath$A$\unboldmath}_{1}^{(j)}\geq\operatorname{rank}\mbox{\boldmath$\Pi$\unboldmath}\geq
m,$
which concludes the proof. ∎
With minor modifications Theorem 7.2 holds also with
$\mathbb{D}_{\theta}(\hbar)$ replaced by $\mathbb{J}(\hbar)$ (and resonances
by eigenvalues). We re-phrase it for the precise statement.
###### Corollary 7.3.
Assume $mc^{2}<l_{0}\leq l(\hbar)\leq r(\hbar)\leq r_{0}<\infty$. Assume that
for any $\hbar\in(0,\hbar_{0}]$ there is $m(\hbar)\in{\mathbb{Z}}_{+}$,
$E_{j}(\hbar)\in[l(\hbar),r(\hbar)]$ and normalized
$\mbox{\boldmath$u$\unboldmath}_{j}(\hbar)$ (quasimodes) for $1\leq j\leq
m(\hbar)$, having support in a ball $B(0,R)$ where $R<R_{0}$ does not depend
on $\hbar$. Assume, moreover, that
$\displaystyle\|(\mathbb{J}(\hbar)-E_{j}(\hbar))\mbox{\boldmath$u$\unboldmath}_{j}(\hbar)\|\leq\rho(\hbar)$
(7.6) and $\displaystyle\text{all
}\tilde{\mbox{\boldmath$u$\unboldmath}}_{j}(\hbar)\in{\mathcal{H}}\text{ such
that
}\|\tilde{\mbox{\boldmath$u$\unboldmath}}_{j}(\hbar)-\mbox{\boldmath$u$\unboldmath}_{j}(\hbar)\|\leq\frac{\hbar^{N}}{M},\quad
1\leq j\leq m(\hbar),$ $\displaystyle\text{are linearly independent},$ (7.7)
where $\rho(\hbar)\leq\hbar^{5+N}/(C\log\hbar^{-1})$, $C\gg 1$, $N\geq 0$ and
$M>0$. Then there exists $C_{0}=C_{0}(l_{0},r_{0})>0$ such that for any $B>0$
and $K\in{\mathbb{Z}}_{+}$ there is an
$\hbar_{1}=\hbar_{1}(A,B,M,N)\leq\hbar_{0}$ such that for any
$\hbar\in(0,\hbar_{1}]$ there will be at least $m(\hbar)$ eigenvalues of
$\mathbb{J}(\hbar)$ in
$\displaystyle[l(\hbar)-b(\hbar)\log\frac{1}{\hbar},r(\hbar)+b(\hbar)\log\frac{1}{\hbar}]+i[-b(\hbar),0],$
(7.8)
where
$b(\hbar)=\max{(C_{0}BM\rho(\hbar)\hbar^{-5-N},e^{-B/\hbar})}.$
###### Proof.
This proof works just as above if we use
$\operatorname{Count\,}(\mathbb{J}(\hbar),{\mathcal{R}})={\mathcal{O}}(\hbar^{-4})$
(see Proposition 6.2) and define $w$ accordingly. From
$\displaystyle-\operatorname{{\rm
Im}\,}\langle(\mathbb{J}-z)\mbox{\boldmath$u$\unboldmath},\mbox{\boldmath$u$\unboldmath}\rangle$
$\displaystyle=\|\sqrt{\operatorname{{\rm
Re}\,}(W)}\mbox{\boldmath$u$\unboldmath}\|^{2}+\operatorname{{\rm
Im}\,}z\|\mbox{\boldmath$u$\unboldmath}\|^{2}\geq\operatorname{{\rm
Im}\,}z\|\mbox{\boldmath$u$\unboldmath}\|^{2}$ (7.9)
it follows that
$\displaystyle\|(\mathbb{J}-z)^{-1}\|\leq\frac{1}{\operatorname{{\rm
Im}\,}z}\quad\text{for }\operatorname{{\rm Im}\,}z>0.$ (7.10)
Therefore, by Proposition 6.3, we can apply the semiclassical maximum
principle to the holomorphic function
$\mbox{\boldmath$F$\unboldmath}(z)=(\mbox{\boldmath$1$\unboldmath}-\mbox{\boldmath$\Pi$\unboldmath})(\mathbb{J}-z)^{-1}\mbox{\boldmath$\chi$\unboldmath}$,
where $\Pi$ is defined similarly as before. ∎
###### Remark 7.4.
Notice that due to (7.10) we do not need to take $c(\hbar)$ as large as in
Theorem 7.2 which possibly gives us an improved error estimate.
## 8 Proof of main results
### 8.1 Approximating a single eigenvalue when $R_{0}^{\prime}<R_{1}$
We are now in position to prove that a single resonance of $\mathbb{D}(\hbar)$
generates a single eigenvalue of $\mathbb{J}(\hbar)$ nearby in the sense
stated in Theorem 5.1, and vice versa. To show these results we use Theorem
7.2 and Corollary 7.3, respectively, with $m=1$.
###### Proof of Theorem 5.1.
We suppress the dependence of $\hbar$ for all operators below.
1\. Take $\chi\in C_{0}^{\infty}(B(0,R_{1}))$ with $\chi=1$ in a neighborhood
of $B(0,R_{0}^{\prime})$. Let $u$ be an eigenfunction of $\mathbb{D}_{\theta}$
associated with the eigenvalue $z_{0}$. Since
$\mbox{\boldmath$\chi$\unboldmath}\mathbb{W}=\mbox{\boldmath$0$\unboldmath}$
we have
$(\mathbb{J}-\operatorname{{\rm
Re}\,}z_{0})\mbox{\boldmath$\chi$\unboldmath}\mbox{\boldmath$u$\unboldmath}=[\mathbb{D}_{\theta},\mbox{\boldmath$\chi$\unboldmath}]\mbox{\boldmath$u$\unboldmath}+i\operatorname{{\rm
Im}\,}z_{0}\mbox{\boldmath$\chi$\unboldmath}\mbox{\boldmath$u$\unboldmath}.$
Furthermore, it follows from (4.3) that
$\|[\mathbb{D}_{\theta},\mbox{\boldmath$\chi$\unboldmath}]\mbox{\boldmath$u$\unboldmath}\|={\mathcal{O}}(\hbar^{\infty})\|\mbox{\boldmath$u$\unboldmath}\|$.
Consequently,
$\|(\mathbb{J}-\operatorname{{\rm
Re}\,}z_{0})\mbox{\boldmath$\chi$\unboldmath}\mbox{\boldmath$u$\unboldmath}\|\leq(-\operatorname{{\rm
Im}\,}z_{0}+{\mathcal{O}}(\hbar^{\infty}))\|\mbox{\boldmath$u$\unboldmath}\|.$
(8.1)
Another application of (4.3) gives
$\|\mbox{\boldmath$u$\unboldmath}\|\leq\|\mbox{\boldmath$\chi$\unboldmath}\mbox{\boldmath$u$\unboldmath}\|+\|(\mbox{\boldmath$1$\unboldmath}-\mbox{\boldmath$\chi$\unboldmath})\mbox{\boldmath$u$\unboldmath}\|\leq\|\mbox{\boldmath$\chi$\unboldmath}\mbox{\boldmath$u$\unboldmath}\|+{\mathcal{O}}(\hbar^{\infty})\|\mbox{\boldmath$u$\unboldmath}\|$
and, therefore, we obtain, for $\hbar$ sufficiently small, that
$(1-o(1))\|\mbox{\boldmath$u$\unboldmath}\|\leq\|\mbox{\boldmath$\chi$\unboldmath}\mbox{\boldmath$u$\unboldmath}\|$,
and thus $\|\mbox{\boldmath$u$\unboldmath}\|\leq
C\|\mbox{\boldmath$\chi$\unboldmath}\mbox{\boldmath$u$\unboldmath}\|$. Then
(8.1) implies that
$\|(\mathbb{J}-\operatorname{{\rm
Re}\,}z_{0})\mbox{\boldmath$\chi$\unboldmath}\mbox{\boldmath$u$\unboldmath}\|\leq(-\operatorname{{\rm
Im}\,}z_{0}+{\mathcal{O}}(\hbar^{\infty}))\|\mbox{\boldmath$\chi$\unboldmath}\mbox{\boldmath$u$\unboldmath}\|$
and, by interpreting
$\mbox{\boldmath$\chi$\unboldmath}\mbox{\boldmath$u$\unboldmath}/\|\mbox{\boldmath$\chi$\unboldmath}\mbox{\boldmath$u$\unboldmath}\|$
as a quasimode for $\mathbb{J}(\hbar)$, an application of Corollary 7.3 yields
$\rho(\hbar)=-\operatorname{{\rm
Im}\,}z_{0}+{\mathcal{O}}(\hbar^{\infty})\leq\frac{\hbar^{5}}{C\log\frac{1}{\hbar}}.$
2\. Let
$\mbox{\boldmath$f$\unboldmath}\in{\mathbf{H}}^{1}({\mathbb{R}}^{3},{\mathbb{C}}^{4})$
be an eigenvector of $\mathbb{J}$ corresponding to $w_{0}$, i.e.
$\mathbb{J}\mbox{\boldmath$f$\unboldmath}=w_{0}\mbox{\boldmath$f$\unboldmath}$.
Let $\chi\in C_{0}^{\infty}(B(0,R_{0}))$, $0\leq\chi\leq 1$, be 1 near
$B(0,R_{2})$. We will show that
$\mbox{\boldmath$\chi$\unboldmath}\mbox{\boldmath$f$\unboldmath}/\|\mbox{\boldmath$\chi$\unboldmath}\mbox{\boldmath$f$\unboldmath}\|$
is a quasimode. Therefore we consider
$\displaystyle(\mathbb{D}-\operatorname{{\rm
Re}\,}w_{0})\mbox{\boldmath$\chi$\unboldmath}\mbox{\boldmath$f$\unboldmath}=[\mathbb{D},\mbox{\boldmath$\chi$\unboldmath}]\mbox{\boldmath$f$\unboldmath}+i\mbox{\boldmath$\chi$\unboldmath}\mathbb{W}\mbox{\boldmath$f$\unboldmath}+i\operatorname{{\rm
Im}\,}w_{0}\mbox{\boldmath$\chi$\unboldmath}\mbox{\boldmath$f$\unboldmath}.$
(8.2)
From (7.9) we have $\|\sqrt{\operatorname{{\rm
Re}\,}W}\mbox{\boldmath$f$\unboldmath}\|=\sqrt{-\operatorname{{\rm
Im}\,}w_{0}}\|\mbox{\boldmath$f$\unboldmath}\|$, which because of Assumption
3.5 (iv) and the fact that
$[\mathbb{D},\mbox{\boldmath$\chi$\unboldmath}]\mbox{\boldmath$f$\unboldmath}$
is supported in $|x|>R_{2}$, makes the norms of the first two terms on the
right hand side bounded by $C\sqrt{-\operatorname{{\rm
Im}\,}w_{0}}\|\mbox{\boldmath$f$\unboldmath}\|$. For the same reason $\chi$$f$
is uniformly bounded away from zero and
$\|\mbox{\boldmath$f$\unboldmath}\|\leq
C\|\mbox{\boldmath$\chi$\unboldmath}\mbox{\boldmath$f$\unboldmath}\|$. Thus
$\|(\mathbb{D}-\operatorname{{\rm
Re}\,}w_{0})\mbox{\boldmath$\chi$\unboldmath}\mbox{\boldmath$f$\unboldmath}\|\leq
C\sqrt{-\operatorname{{\rm
Im}\,}w_{0}}\leq\frac{\hbar^{4}}{C\log\frac{1}{\hbar}}.$
An application of Theorem 7.2 finishes the proof. ∎
### 8.2 Approximating a single eigenvalue when $R_{1}\leq R_{0}^{\prime}$
#### Semiclassical projections
Denote by
$\mbox{\boldmath$\lambda$\unboldmath}_{j}:{\mathsf{T}}^{\ast}{\mathbb{R}}^{3}\to\mathrm{M}_{4}({\mathbb{C}})$,
$j=1,\ldots,4$, the projection matrices onto the eigenspaces corresponding to
the eigenvalues $\lambda_{j}$ of the principal symbol
$\mbox{\boldmath$d$\unboldmath}_{0}$ of $\mathbb{D}$. Since the symbols
$\mbox{\boldmath$\lambda$\unboldmath}_{j}$ depend on $x$, their quantizations
$\mbox{\boldmath$\Lambda$\unboldmath}_{j}:={\rm
op}^{W}[{\mbox{\boldmath$\lambda$\unboldmath}_{j}}]$ are not projection
operators. Rather they satisfy [EmWe’96]
$\mbox{\boldmath$\Lambda$\unboldmath}_{j}^{2}-\mbox{\boldmath$\Lambda$\unboldmath}_{j}={\mathcal{O}}(\hbar),\quad
j=1,\ldots,4.$ (8.3)
In addition, $\mbox{\boldmath$\Lambda$\unboldmath}_{j}$ do not commute with
$\mathbb{D}(\hbar)$. One can improve the error on the right-hand side of (8.3)
by adding a suitable term of order $\hbar$ to the symbol $\mathbb{D}_{0}$ and,
subsequently, quantization results in an operator which is a projector up to
an error of order $\hbar^{2}$. Iteration of this process leads to an error of
arbitrary order $\hbar^{N}$ [EmWe’96].
Under Assumption 3.3 it is shown in [BoGl’04a, Proposition 2.1] that one can
construct
$\mbox{\boldmath$\lambda$\unboldmath}_{j}\sim\sum_{n\geq
0}\hbar^{n}\mbox{\boldmath$\lambda$\unboldmath}_{j,n}$
such that
$\mbox{\boldmath$\lambda$\unboldmath}_{j}\\#\mbox{\boldmath$\lambda$\unboldmath}_{j}\sim\mbox{\boldmath$\lambda$\unboldmath}_{j}\sim\mbox{\boldmath$\lambda$\unboldmath}_{j}^{\ast}\quad\text{and}\quad[\mbox{\boldmath$\lambda$\unboldmath}_{j},\mbox{\boldmath$d$\unboldmath}]_{\\#}\sim
0,$
and, moreover, in agreement with the discussion above, the corresponding
quantizations $\mbox{\boldmath$\Lambda$\unboldmath}_{j}$ satisfy the relations
($j=1,\ldots,4$):
$\displaystyle\mbox{\boldmath$\Lambda$\unboldmath}_{1}+\mbox{\boldmath$\Lambda$\unboldmath}_{2}\equiv\mbox{\boldmath$1$\unboldmath},$
$\displaystyle\mbox{\boldmath$\Lambda$\unboldmath}_{j}^{2}\equiv\mbox{\boldmath$\Lambda$\unboldmath}_{j}\equiv\mbox{\boldmath$\Lambda$\unboldmath}_{j}^{\ast},$
$\displaystyle\|[\mathbb{D},\mbox{\boldmath$\Lambda$\unboldmath}_{j}]\|\equiv
0,$
where $\equiv$ means modulo terms of norm ${\mathcal{O}}(\hbar^{\infty})$. The
operators $\mbox{\boldmath$\Lambda$\unboldmath}_{j}$, $j=1,\ldots,4$, are
called almost orthogonal projections.
#### Matrix valued Egorov theorem
We now indicate how to solve Heisenberg’s equation of motion semiclassically
in the sense that given $\mbox{\boldmath$A$\unboldmath}={\rm
op}^{W}[{\mbox{\boldmath$a$\unboldmath}}]$ with
$\mbox{\boldmath$a$\unboldmath}\in{\mathsf{S}}(1)$ we can, for all $t$, find
$\mbox{\boldmath$a$\unboldmath}(t)\in{\mathsf{S}}(1)$ such that
$\frac{\partial{}}{\partial{t}}{\rm
op}^{W}[{\mbox{\boldmath$a$\unboldmath}(t)}]=[\mathbb{D},{\rm
op}^{W}[{\mbox{\boldmath$a$\unboldmath}(t)}]]+\mbox{\boldmath$R$\unboldmath}(t),\quad\mbox{\boldmath$A$\unboldmath}(0)=\mbox{\boldmath$A$\unboldmath},$
with $\|\mbox{\boldmath$R$\unboldmath}(t)\|={\mathcal{O}}(\hbar^{N})$ for any
$N\in{\mathbb{N}}$. This means we can approximate the time evolution
$\mbox{\boldmath$A$\unboldmath}(t):=\exp(i\mathbb{D}t/\hbar)\mbox{\boldmath$A$\unboldmath}\exp(-i\mathbb{D}t/\hbar)$
of $A$ to any order.
We extract the following lemma from [BoGl’04a, Theorem 3.2 and the discussions
preceeding and proceeding it].
###### Lemma 8.1 (Matrix Egorov theorem).
Let Assumption 3.1 and Assumption 3.3 be satisfied. Suppose that
$\mbox{\boldmath$a$\unboldmath}\in{\mathsf{S}}(1)$ is block-diagonal with
respect to the $\mbox{\boldmath$\lambda$\unboldmath}_{j}$ in the sense that
$\mbox{\boldmath$a$\unboldmath}\sim\sum_{j=1}^{4}\mbox{\boldmath$\lambda$\unboldmath}_{j}\\#\mbox{\boldmath$a$\unboldmath}\\#\mbox{\boldmath$\lambda$\unboldmath}_{j}.$
Then, for any $T>0$, we can find
$\mbox{\boldmath$a$\unboldmath}(t)\in{\mathsf{S}}(1)$ for all $0\leq t\leq T$
such that
$\|\mbox{\boldmath$A$\unboldmath}(t)-{\rm
op}^{W}[{\mbox{\boldmath$a$\unboldmath}(t)}]\|={\mathcal{O}}(\hbar^{\infty})\quad\text{for
all }t\in[0,T].$
Moreover, the principal symbol is given by
$\mbox{\boldmath$a$\unboldmath}_{0}(x,\xi,t)=\sum_{j=1}^{4}\mbox{\boldmath$\mathfrak{t}$\unboldmath}_{jj}^{\ast}(x,\xi,t)\mbox{\boldmath$\lambda$\unboldmath}_{j,0}(\Phi_{j}^{t}(x,\xi))\mbox{\boldmath$a$\unboldmath}_{0}(\Phi_{j}^{t}(x,\xi))\mbox{\boldmath$\lambda$\unboldmath}_{j,0}(\Phi_{j}^{t}(x,\xi))\mbox{\boldmath$\mathfrak{t}$\unboldmath}_{jj}(x,\xi,t)$
where the $4\times 4$ unitary transport matrices
$\mbox{\boldmath$\mathfrak{t}$\unboldmath}_{jj}$ are given by
$\frac{d}{dt}\mbox{\boldmath$\mathfrak{t}$\unboldmath}_{jj}(x,\xi,t)+i\widetilde{\mathfrak{T}}_{jj,1}(\Phi_{j}^{t}(x,\xi))\mbox{\boldmath$\mathfrak{t}$\unboldmath}_{jj}(x,\xi,t)=0,\quad\mbox{\boldmath$\mathfrak{t}$\unboldmath}(x,\xi,0)=\mbox{\boldmath$I$\unboldmath}_{4}.$
Here
$\widetilde{\mathfrak{T}}_{jj,1}=-i\frac{\lambda_{j}}{2}\mbox{\boldmath$\lambda$\unboldmath}_{j,0}\\{\mbox{\boldmath$\lambda$\unboldmath}_{j,0},\mbox{\boldmath$\lambda$\unboldmath}_{j,0}\\}\mbox{\boldmath$\lambda$\unboldmath}_{j,0}-i[\mbox{\boldmath$\lambda$\unboldmath}_{j,0},\\{\lambda_{j},\mbox{\boldmath$\lambda$\unboldmath}_{j,0}\\}]+\mbox{\boldmath$\lambda$\unboldmath}_{j,0}\mathfrak{T}_{j,1}\mbox{\boldmath$\lambda$\unboldmath}_{j,0}$
where $\mathfrak{T}_{j,1}$ is the subprincipal symbol of ${\rm
op}^{W}[{\mbox{\boldmath$\lambda$\unboldmath}_{j}\\#\mbox{\boldmath$d$\unboldmath}\\#\mbox{\boldmath$\lambda$\unboldmath}_{j}}]$.
###### Remark 8.2.
Notice that Lemma 8.1 requires that both [BoGl’04a, Property (3.9)] and the
assumptions of [BoGl’04a, Lemma 3.3] are satisfied; but these conditions are
clearly fulfilled in our case.
By imposing additional assumptions we can say even more about ${\rm
op}^{W}[{\mbox{\boldmath$a$\unboldmath}(T)}]$.
###### Lemma 8.3.
Let Assumption 3.1, Definition 3.2 , and Assumption 3.3 hold. Then, provided
$T$ and $\hbar^{-1}$ are sufficiently large we can construct
$\mbox{\boldmath$a$\unboldmath}(x,\xi,t)$ as in Lemma 8.1 so that
$\mbox{\boldmath$a$\unboldmath}(x,\xi,T)=\mbox{\boldmath$I$\unboldmath}_{4}+{\mathcal{O}}(\hbar)$
in a neighborhood of $(x_{0},\xi_{0})$.
###### Proof.
By defining
$\mbox{\boldmath$a$\unboldmath}_{0}(x,\xi)=\sum_{j=1}^{4}\mbox{\boldmath$\chi$\unboldmath}_{j}(x,\xi)\mbox{\boldmath$\lambda$\unboldmath}_{j,0}(x,\xi)\mbox{\boldmath$\mathfrak{t}$\unboldmath}_{jj}(\Phi_{j}^{-T}(x_{0},\xi_{0}),T)\mbox{\boldmath$\lambda$\unboldmath}_{j,0}(\Phi_{j}^{-T}(x_{0},\xi_{0}))\\\
\times\mbox{\boldmath$\mathfrak{t}$\unboldmath}_{jj}^{\ast}(\Phi_{j}^{-T}(x_{0},\xi_{0}),T)\mbox{\boldmath$\lambda$\unboldmath}_{j,0}(x,\xi)$
it follows from Lemma 8.1, the identity (see [BoGl’04a, Equation (4.1)]),
$\mbox{\boldmath$\lambda$\unboldmath}_{j,0}(\Phi_{j}^{t}(x,\xi))\mbox{\boldmath$\mathfrak{t}$\unboldmath}_{jj}(x,\xi,t)\mbox{\boldmath$\lambda$\unboldmath}_{j,0}(x,\xi)=\mbox{\boldmath$\mathfrak{t}$\unboldmath}_{jj}(x,\xi,t)\mbox{\boldmath$\lambda$\unboldmath}_{j,0}(x,\xi)$
and its adjoint equation and the fact that
$\mbox{\boldmath$\lambda$\unboldmath}_{j,0}(x,\xi)\mbox{\boldmath$\lambda$\unboldmath}_{k,0}(x,\xi)=0$
whenever $\mbox{\boldmath$\lambda$\unboldmath}_{j,0}$ and
$\mbox{\boldmath$\lambda$\unboldmath}_{k,0}$ are projections corresponding to
distinct eigenvalues, that
$\mbox{\boldmath$a$\unboldmath}_{0}(x,\xi,T)=\sum_{j=1}^{4}\mbox{\boldmath$\chi$\unboldmath}_{j}(\Phi_{j}^{T}(x,\xi))\mbox{\boldmath$\lambda$\unboldmath}_{j,0}(x,\xi)$
and, in particular,
$\mbox{\boldmath$a$\unboldmath}_{0}(x_{0},\xi_{0},T)=\mbox{\boldmath$I$\unboldmath}_{4}$.
∎
#### Propagation of singularities
###### Lemma 8.4 (Propagation of singularities).
Let $R_{0}^{\prime}<R_{1}^{\prime}$ and suppose that for some
$z_{0}(\hbar)\in[l_{0},r_{0}]$ and
$\mbox{\boldmath$v$\unboldmath}(\hbar)\in\operatorname{Dom\,}(\mathbb{D})$
with $\operatorname{supp\,}\mbox{\boldmath$v$\unboldmath}(\hbar)\subset
B(0,R_{1}^{\prime})$ and $\|\mbox{\boldmath$v$\unboldmath}(\hbar)\|\leq C$ for
some $C>0$ we have
$(\mathbb{D}(\hbar)-z_{0}(\hbar))\mbox{\boldmath$v$\unboldmath}(\hbar)=\mbox{\boldmath$g$\unboldmath}(\hbar)$
with
$\|\mbox{\boldmath$g$\unboldmath}(\hbar)\|={\mathcal{O}}(\varepsilon(\hbar))$,
$\varepsilon(\hbar)={\mathcal{O}}(\hbar^{N})$ for some $N>0$. If
$(x_{0},\xi_{0})\in{\mathsf{T}}^{\ast}{\mathbb{R}}^{3}$ is such that the norms
of the $x$-projections of $\Phi_{j}^{T}(x_{0},\xi_{0})$, $j=1,\ldots,4$,
exceed $R_{1}^{\prime}$ for some $0<T<\infty$ then
$\mbox{\boldmath$v$\unboldmath}(\hbar)$ is microlocally
${\mathcal{O}}(\hbar^{-1}\varepsilon(\hbar)+\hbar^{\infty})$ at
$(x_{0},\xi_{0})$.
###### Proof.
Let $\mbox{\boldmath$a$\unboldmath}(t)\in S(1)$, $0\leq t\leq T$, be as in
Lemma 8.1 with $\mbox{\boldmath$a$\unboldmath}(0)$ invertible and supported
near the points $\Phi_{j}^{T}(x_{0},\xi_{0})$ for $j=1,\ldots,4$. Denote by
$\mbox{\boldmath$A$\unboldmath}(t)={\rm
op}^{W}[{\mbox{\boldmath$a$\unboldmath}(t)}]$ its quantization. Put
$l(t)=\|\mbox{\boldmath$A$\unboldmath}(t)\mbox{\boldmath$v$\unboldmath}\|$
so that $l(0)={\mathcal{O}}(\hbar^{\infty})$ since
$\mbox{\boldmath$a$\unboldmath}(0)$ has support where
$\mbox{\boldmath$v$\unboldmath}=0$. Then, using the fact that ${\rm
op}^{W}[{\mbox{\boldmath$a$\unboldmath}(t)}]$ approximately solves
Heisenberg’s equation of motion,
$\displaystyle l(t)\frac{d}{dt}l(t)$
$\displaystyle=\frac{d}{dt}\frac{l^{2}(t)}{2}=\operatorname{{\rm
Re}\,}\Big{\langle}\frac{d}{dt}\mbox{\boldmath$A$\unboldmath}(t)\mbox{\boldmath$v$\unboldmath},\mbox{\boldmath$A$\unboldmath}(t)\mbox{\boldmath$v$\unboldmath}\Big{\rangle}$
$\displaystyle=-\hbar^{-1}\operatorname{{\rm
Im}\,}\Big{\langle}\big{(}[\mathbb{D}-z_{0},\mbox{\boldmath$A$\unboldmath}(t)]+\mbox{\boldmath$R$\unboldmath}(t)\big{)}\mbox{\boldmath$v$\unboldmath},\mbox{\boldmath$A$\unboldmath}(t)\mbox{\boldmath$v$\unboldmath}\Big{\rangle}$
$\displaystyle=\hbar^{-1}\operatorname{{\rm
Im}\,}\Big{\langle}\mbox{\boldmath$A$\unboldmath}(t)(\mathbb{D}-z_{0})\mbox{\boldmath$v$\unboldmath},\mbox{\boldmath$A$\unboldmath}(t)\mbox{\boldmath$v$\unboldmath}\Big{\rangle}-\hbar^{-1}\operatorname{{\rm
Im}\,}\Big{\langle}\mbox{\boldmath$R$\unboldmath}(t)\mbox{\boldmath$v$\unboldmath},\mbox{\boldmath$A$\unboldmath}(t)\mbox{\boldmath$v$\unboldmath}\Big{\rangle}$
$\displaystyle=\hbar^{-1}\operatorname{{\rm
Im}\,}\Big{\langle}\mbox{\boldmath$A$\unboldmath}(t)\mbox{\boldmath$g$\unboldmath},\mbox{\boldmath$A$\unboldmath}(t)\mbox{\boldmath$v$\unboldmath}\Big{\rangle}-\hbar^{-1}\operatorname{{\rm
Im}\,}\Big{\langle}\mbox{\boldmath$R$\unboldmath}(t)\mbox{\boldmath$v$\unboldmath},\mbox{\boldmath$A$\unboldmath}(t)\mbox{\boldmath$v$\unboldmath}\Big{\rangle}.$
Since $\mbox{\boldmath$A$\unboldmath}(t)$ is bounded for all $t\in[0,T]$ and
$\mbox{\boldmath$R$\unboldmath}(t)$ can be made to satisfy
$\|\mbox{\boldmath$R$\unboldmath}(t)\mbox{\boldmath$v$\unboldmath}\|={\mathcal{O}}(\hbar^{\infty})$
we obtain
$\frac{d}{dt}l(t)={\mathcal{O}}(\hbar^{-1}\varepsilon(\hbar)+\hbar^{\infty})$
and since $l(0)={\mathcal{O}}(\hbar^{\infty})$ we see that
$l(t)={\mathcal{O}}(\hbar^{-1}\varepsilon(\hbar)+\hbar^{\infty})$ for all
$t\in[0,T]$. In particular, by Lemma 8.3,
$\mbox{\boldmath$a$\unboldmath}_{0}(x,\xi,T)$ equals the identity near
$(x_{0},\xi_{0})$ so that $\mbox{\boldmath$a$\unboldmath}(x,\xi,T)$ is
invertible near $(x_{0},\xi_{0})$ (see Section 2) provided $\hbar$ is small
enough. ∎
#### Proof of Theorem 5.2
We are now in position to prove Theorem 5.2.
###### Proof of Theorem 5.2.
Let $R_{0}^{\prime}<R_{1}^{\prime}<R_{0}$ and pick $\chi\in
C_{0}^{\infty}{B(0,R_{1}^{\prime})}$ with $\chi=1$ near $B(0,R_{0}^{\prime})$.
Then, with
$\mbox{\boldmath$v$\unboldmath}=\mbox{\boldmath$\chi$\unboldmath}\mbox{\boldmath$u$\unboldmath}$,
we obtain as in (8.2),
$(\mathbb{J}(\hbar)-\operatorname{{\rm
Re}\,}z_{0})\mbox{\boldmath$v$\unboldmath}=[\mathbb{D}(\hbar),\mbox{\boldmath$\chi$\unboldmath}]\mbox{\boldmath$u$\unboldmath}+i(\operatorname{{\rm
Im}\,}z_{0})\mbox{\boldmath$v$\unboldmath}-i\mathbb{W}\mbox{\boldmath$v$\unboldmath},$
(8.4)
where, as in (8.1), $\|(\mathbb{D}(\hbar)-\operatorname{{\rm
Re}\,}z_{0})\mbox{\boldmath$v$\unboldmath}\|={\mathcal{O}}(\varepsilon(\hbar))$
with $\varepsilon(\hbar)=-\operatorname{{\rm
Im}\,}z_{0}(\hbar)+{\mathcal{O}}(\hbar^{\infty})$.
It remains to estimate $\|\mathbb{W}\mbox{\boldmath$v$\unboldmath}\|$. To this
end, notice that under the nontrapping assumption (see Definition 3.2) Lemma
8.4 can be applied to any point $(x_{0},\xi_{0})\in E_{[l_{0},r_{0}]}$ with
$x_{0}\in\operatorname{supp\,}\mathbb{W}$ where
$E_{[l_{0},r_{0}]}=\cup_{j=1}^{4}\lambda_{j}^{-1}([l_{0},r_{0}])$. Near any
point $(x,\xi)$ in the complement of $E_{[l_{0},r_{0}]}$ the operator
$\mathbb{D}-z_{0}$ is elliptic and therefore
$\mbox{\boldmath$a$\unboldmath}(x,\xi)=\langle\xi\rangle^{-1}\\#(\mbox{\boldmath$d$\unboldmath}(x,\xi)-z_{0})$
will satisfy $\|{\rm
op}^{W}[{\mbox{\boldmath$a$\unboldmath}}]\mbox{\boldmath$v$\unboldmath}\|={\mathcal{O}}(\varepsilon(\hbar))$
with $\mbox{\boldmath$a$\unboldmath}\in{\mathsf{S}}(1)$ and ${\rm
op}^{W}[{\mbox{\boldmath$a$\unboldmath}}]$ elliptic at $(x,\xi)$. Take
$\chi\in C^{\infty}({\mathsf{T}}^{\ast}{\mathbb{R}}^{3})$ which equals 1 near
$E_{[l_{0},r_{0}]}$ and has support in $\\{|\xi|\leq C\\}$ for some $C>0$.
Consider any $(x_{0},\xi_{0})$ with $R_{1}<|x_{0}|<R_{1}^{\prime}$. For any
$(x_{0},\xi_{\alpha})\in\\{x_{0}\\}\times\\{|\xi|\leq C\\}$ and any
$\mbox{\boldmath$b$\unboldmath}_{\alpha}\in{\mathsf{S}}(1)$ supported in a
sufficiently small open neighborhood $U_{\alpha}$ of $(x_{0},\xi_{\alpha})$ it
holds, by Lemma 8.4 and Lemma 2.5, that $\|{\rm
op}^{W}[{\mbox{\boldmath$b$\unboldmath}_{\alpha}}]\mbox{\boldmath$v$\unboldmath}\|={\mathcal{O}}(\hbar^{-1}\varepsilon(\hbar)+\hbar^{\infty})$.
Now consider any $\mbox{\boldmath$b$\unboldmath}\in C_{0}^{\infty}(\cup
U_{\alpha})$ and extract a finite subcover $\\{U_{j}\\}_{j=1}^{N}$ of
$\operatorname{supp\,}\mbox{\boldmath$b$\unboldmath}$ and let
$\\{\mbox{\boldmath$\chi$\unboldmath}_{j}\\}_{j=1}^{N}$ be a smooth partition
of unity subordinate to it. It follows that
$\|{\rm
op}^{W}[{\mbox{\boldmath$b$\unboldmath}}]\mbox{\boldmath$v$\unboldmath}\|\leq\sum_{j=1}^{N}\|{\rm
op}^{W}[{\mbox{\boldmath$\chi$\unboldmath}_{j}\mbox{\boldmath$b$\unboldmath}}]\mbox{\boldmath$v$\unboldmath}\|={\mathcal{O}}(\hbar^{-1}\varepsilon(\hbar)+\hbar^{\infty}).$
Consider
$(\mbox{\boldmath$I$\unboldmath}_{4}-\mbox{\boldmath$\chi$\unboldmath}(x,\xi))\\#(\mbox{\boldmath$d$\unboldmath}(x,\xi)-z_{0})^{-1}\\#(\mbox{\boldmath$d$\unboldmath}(x,\xi)-z_{0})+\mbox{\boldmath$\chi$\unboldmath}(x,\xi)=\mbox{\boldmath$I$\unboldmath}_{4}+{\mathcal{O}}(\hbar),$
which is well-defined since
$\mbox{\boldmath$\chi$\unboldmath}(x,\xi)=\mbox{\boldmath$I$\unboldmath}_{4}$
near $E_{[l_{0},r_{0}]}$ where
$(\mbox{\boldmath$d$\unboldmath}(x,\xi)-z_{0})^{-1}$ does not exist and is
everywhere invertible provided $\hbar$ is small enough [ONE CAN ALSO TAKE
$(\mbox{\boldmath$d$\unboldmath}^{\dagger}-z_{0})\\#(\mbox{\boldmath$d$\unboldmath}-z_{0})+\mbox{\boldmath$\chi$\unboldmath}$].
It follows (see Lemma 2.2) that there is
$\mbox{\boldmath$q$\unboldmath}\in{\mathsf{S}}(1)$ such that
$\displaystyle\mbox{\boldmath$v$\unboldmath}={\rm
op}^{W}[{\mbox{\boldmath$q$\unboldmath}}]{\rm
op}^{W}[{(\mbox{\boldmath$I$\unboldmath}_{4}-\mbox{\boldmath$\chi$\unboldmath})\\#(\mbox{\boldmath$d$\unboldmath}-z_{0})^{-1}\\#(\mbox{\boldmath$d$\unboldmath}-z_{0})+\mbox{\boldmath$\chi$\unboldmath}}]\mbox{\boldmath$v$\unboldmath}-{\rm
op}^{W}[{\mbox{\boldmath$r$\unboldmath}}]\mbox{\boldmath$v$\unboldmath},$
with $\|{\rm
op}^{W}[{\mbox{\boldmath$r$\unboldmath}}]\|={\mathcal{O}}(\hbar^{\infty})$.
Pick $\chi_{0}=\chi_{0}(x)\in C_{0}^{\infty}(\pi_{x}(\cap_{j=1}^{N}U_{j}))$
which equals 1 in a neighborhood $V$ of $x_{0}$. Here
$\pi_{x}(x_{0},\xi_{0})=x_{0}$ denotes the projection of
${\mathsf{T}}^{\ast}{\mathbb{R}}^{3}$ onto its base manifold. Then
$\mbox{\boldmath$\chi$\unboldmath}_{0}\mbox{\boldmath$v$\unboldmath}={\rm
op}^{W}[{\mbox{\boldmath$\chi$\unboldmath}_{0}\\#\mbox{\boldmath$q$\unboldmath}\\#\mbox{\boldmath$\chi$\unboldmath}}]\mbox{\boldmath$v$\unboldmath}+{\mathcal{O}}(\varepsilon(\hbar)+\hbar^{\infty}),$
where
$\mbox{\boldmath$\chi$\unboldmath}_{0}\\#\mbox{\boldmath$q$\unboldmath}\\#\mbox{\boldmath$\chi$\unboldmath}$
is asymptotically equivalent to a symbol in ${\mathsf{S}}(1)$ supported in
$\cup U_{\alpha}$. We conclude that
$\|\mbox{\boldmath$v$\unboldmath}\|_{{\mathbf{L}}^{2}(V)}={\mathcal{O}}(\hbar^{-1}\varepsilon(\hbar)+\hbar^{\infty}).$
By compactness of $\\{R_{1}<|x|<R_{1}^{\prime}\\}$ we conclude that
$\|\mbox{\boldmath$v$\unboldmath}\|_{{\mathbf{L}}^{2}(\operatorname{supp\,}\mathbb{W})}={\mathcal{O}}(\hbar^{-1}\varepsilon(\hbar)+\hbar^{\infty})$.
It now follows from (8.4) that
$\|(\mathbb{J}(\hbar)-\operatorname{{\rm
Re}\,}z_{0})\mbox{\boldmath$\chi$\unboldmath}\mbox{\boldmath$u$\unboldmath}\|\leq(\hbar^{-1}\tilde{\varepsilon}(\hbar)+\hbar^{\infty})\|\mbox{\boldmath$\chi$\unboldmath}\mbox{\boldmath$u$\unboldmath}\|,$
so the theorem follows by an application of Corollary 7.3. ∎
## Appendix A Semiclassical maximum principle
The following result is sometimes referred to as the “semiclassical maximum
principle” [tang_zworski]. The version we state can be found in Stefanov
[St’05].
###### Proposition A.1.
Let $l(\hbar)\leq r(\hbar)$ and assume that
$\mbox{\boldmath$F$\unboldmath}(z,\hbar)$ is a holomorphic operator-valued
function in a neighborhood of
${\mathcal{R}}(\hbar)=[l(\hbar)-w(\hbar),r(\hbar)+w(\hbar)]+i\Big{[}-A\hbar^{-3}S(\hbar)\log\frac{1}{S(\hbar)},S(\hbar)\Big{]},$
(A.1)
where $e^{-B/\hbar}<S(\hbar)<1$ for some $B>0$ and
$3A\hbar^{-3}S(\hbar)(\log\frac{1}{\hbar})(\log\frac{1}{S(\hbar)})\leq
w(\hbar).$ (A.2)
If moreover $\mbox{\boldmath$F$\unboldmath}(z,\hbar)$ satisfies
$\displaystyle\|\mbox{\boldmath$F$\unboldmath}(z,\hbar)\|$ $\displaystyle\leq
e^{A\hbar^{-3}\log(1/S(\hbar))}$ (A.3) and
$\displaystyle\|\mbox{\boldmath$F$\unboldmath}(z,\hbar)\|$
$\displaystyle\leq\frac{C}{\operatorname{{\rm Im}\,}z}\qquad\text{on
}{\mathcal{R}}(\hbar)\cap\\{\operatorname{{\rm Im}\,}z=S(\hbar)\\},$ (A.4)
there there exists $\hbar_{1}=\hbar_{1}(S)>0$ such that
$\|\mbox{\boldmath$F$\unboldmath}(z,\hbar)\|\leq\frac{e^{3}}{S(\hbar)},\quad\text{for
all }z\in[l(\hbar),r(\hbar)]+i[-S(\hbar),S(\hbar)],$ (A.5)
for all $0<\hbar\leq\hbar_{1}$.
## References
* [AmBrNo’01] L. Amour, R. Brummelhuis, J. Nourrigat, Resonances of the Dirac Hamiltonian in the non relativistic limit, Ann. Henri Poincaré 2 (2001), 583–603.
* [BaHe’92] E. Balslev, B. Helffer, Limiting absorption principle and resonances for the Dirac operator. Adv. in Appl. Math. 13 (1992), no. 2, 186–215.
* [BoGl’04a] J. Bolte, R. Glaser, A semiclassical Egorov theorem and quantum ergodicity for matrix valued operators, Comm. Math. Phys. 247 (2004), no. 2, 391–419.
* [Co’82] H. O. Cordes, A version of Egorov’s theorem for systems of hyperbolic pseudo-differential equations, J. Funct. Anal. 48 (1982), 285–300.
* [DiSj’99] M. Dimassi, J. Sjöstrand, Spectral asymptotics in the semi-classical limit, London Math. Soc. Lecture Notes Series 268, Cambridge University Press, 1999.
* [EmWe’96] C. Emmrich, A. Weinstein, Geometry of transport equation in multicomponent WKB approximations, Commun. Math. Phys. 176 (1996), 701–711.
* [GoKr’69] I. C. Gohberg, M. G. Krein, Introduction to the theory of linear nonselfadjoint operators. Translated from the Russian by A. Feinstein. Translations of Mathematical Monographs, Vol. 18 AMS, Providence, R.I. 1969.
* [HiSi’96] P. D. Hislop, I. M. Sigal, Introduction to Spectral Theory with Applications to Schrödinger Operators, Applied Mathematics Series vol. 113, Springer, New York, 1996.
* [Hu’86] W. Hunziker, Distortion analyticity and molecular resonance curves, Ann. Inst. Henri Poincaré 45 (1986), 339–358.
* [Ka’95] T. Kato, Perturbation theory for linear operators. Second edition. Grundlehren der Mathematischen Wissenschaften, Band 132. Springer–Verlag, Berlin–New York, 1976.
* [Kh’07] A. Khochman, Resonances and spectral shift function for the semiclassical Dirac operator, Rev. Math. Phys. 19 (2007), no. 10, 1071–1115.
* [KuMe’10] J. Kungsman, M. Melgaard, Complex absorbing potential method for systems, Dissertationes Mathematicae 469 (2010), 58 pp.
* [Mu’04] J. G. Muga, J. P. Palao, B. Navarro, I. L. Egusquiza, Complex absorbing potentials, Physics Reports 395 (2004), 357-426.
* [Ne’01] L. Nedelec, Resonances for matrix Schrödinger operators, Duke Math. J. 106 (2001), 209–236.
* [Pa’91] B. Parisse, Résonances pour l’opérateur de Dirac, Helv. Phys. Acta 64 (1991), no. 5, 557–591.
* [Pa’92] B. Parisse, Résonances pour l’opérateur de Dirac II, Helv. Phys. Acta 65 (1992), no. 8, 1077–1118.
* [Se’88] P. Seba, The complex scaling method for Dirac resonances, Lett. Math. Phys. 16 (1988), no. 1, 51–59.
* [Sj’97] J. Sjöstrand, A trace formula and review of some estimates for resonances, In: Microlocal analysis and Spectral Theory (Lucca, 1996), p 377-437, NATO Adv. Sci. Inst. Ser. C. Math. Phys. Sci. 490, Kluwer, Dordrecht, 1997.
* [Sj’01] J. Sjöstrand, Resonances for bottles and trace formulae, Math. Nachr. 221 (2001), 95–149.
* [Sj’02] J. Sjöstrand, Lectures on resonances, unpublished, 2002.
* [SjZw’91] J. Sjöstrand, M. Zworski, Complex scaling and the distribution of scattering poles, J. AMS, 4 (4) (1991), 729–769.
* [St’99] P. Stefanov, Quasimodes and resonances: sharp lower bounds, Duke Math. J. 99 (1999), 75–92.
* [St’01] P. Stefanov, Lower bound of the number of Rayleigh resonances for arbitrary body, Indiana Univ. Math. J. 49 (2000), 405–426.
* [St’03] P. Stefanov, Sharp upper bounds on the number of resonances near the real axis for trapping systems. Amer. J. Math. 125 (2003), no. 1, 183–224.
* [St’05] P. Stefanov, Approximating resonances with the complex absorbing potential method, Comm. Partial Differential Equations 30 (2005), no. 10-12, 1843–1862.
* [TaZw’98] S.-H. Tang, M. Zworski, From quasimodes to resonances, Math. Res. Lett. 5 (1998), 261–272.
* [Th’92] B. Thaller, The Dirac Equation, Texts and Monographs in Physics, Springer Verlag, 1992.
|
arxiv-papers
| 2013-10-07T17:51:59 |
2024-09-04T02:49:52.090893
|
{
"license": "Public Domain",
"authors": "J. Kungsman, M. Melgaard",
"submitter": "Michael Melgaard",
"url": "https://arxiv.org/abs/1310.1872"
}
|
1310.1938
|
# QUANTUM SPINDOWN OF HIGHLY MAGNETIZED NEUTRON STARS
B. LAMINE, C. BERTHIERE, A. DUPAYS
Pulsars are highly magnetized and rapidly rotating neutron stars. The magnetic
field can reach the critical magnetic field from which quantum effects of the
vacuum becomes relevant, giving rise to magnetooptic properties of vacuum
characterized as an effective non linear medium. One spectacular consequence
of this prediction is a macroscopic friction that leads to an additionnal
contribution in the spindown of pulsars. In this paper, we highlight some
observational consequences and in particular derive new constraints on the
parameters of the Crab pulsar and J0540-6919.
## 1 Introduction
It is known from long time that quantum effects give to vacuum its own
electromagnetic properties, in analogy with a usual ponderable media (for a
recent review, see for example $\\!{}^{{\bf?}}$). Among experimentally
accessible consequences are photon/photon scattering $\\!{}^{{\bf?}}$ or
vacuum birefringence $\\!{}^{{\bf?}}$, both being targeted by laboratory
experiments. Astrophyscial objects such as Neutron Stars (NS) are also a
laboratory to test those quantum corrections, simply because they sustain a
very high magnetic field that enhances the quantum features of vacuum. In
particular, when the magnetic energy around a NS is comparable to the electron
rest energy, $B\sim B_{c}=\frac{m_{e}^{2}c^{2}}{e\hbar}$, a significant
magnetization arise in the vacuum that give rise to spectacular macroscopic
consequences on the spindown of this NS $\\!{}^{{\bf?},{\bf?}}$.
The physical principle of the quantum contribution to the spindown is rather
simple. The extremely high magnetic field, generated by the rotating dipole
$\bm{m}$, creates a time-dependent magnetization in the vacuum around the NS.
In return, this magnetization creates a magnetic field $\bm{B}_{\mathrm{qu}}$
which feeds back on the NS magnetic dipole. Due to retardation effect (finite
speed of light), this back-action leads to a torque
$\bm{m}\times\bm{B}_{\mathrm{qu}}$ that slows down the spinning of the NS. It
has been shown that the energy loss rate of an isolated pulsar via the
previous quantum channel, in the limit $B_{0}\ll B_{c}$, is given by
$\\!{}^{{\bf?},{\bf?}}$
$\dot{E}_{\mathrm{qu}}=-\alpha\left(\frac{3\pi^{2}}{4}\right)\frac{\sin^{2}{\theta}}{B_{c}^{2}\mu_{0}c}\frac{B_{0}^{4}R^{4}}{P^{2}}\;,$
(1)
where $\alpha$ is the fine structure constant, $\theta$ the inclination angle,
$B_{0}$ the surface magnetic field, $c$ the speed of light, $R$ the radius of
the NS and $P=2\pi/\Omega$ the rotation period. This contribution is an
additional one compared to the classical spindown, which scales differently
with respect to the physical parameters of the NS. Within the vacuum model
(oblique rotator in vacuum), the energy loss rate is given by the usual dipole
formula
$\dot{E}_{\mathrm{cl}}=-\frac{128\pi^{5}}{3}\frac{B_{0}^{2}\sin^{2}\theta
R^{6}}{P^{4}\mu_{0}c^{3}}\;,$ (2)
where we can see that the classical contribution scales as $P^{-4}$ instead of
the $P^{-2}$ for the quantum one. Both contributions are roughly of the same
order when
$\frac{B_{0}}{B_{c}}\sim\frac{R\Omega}{c}\frac{1}{\sqrt{\alpha}}\frac{1}{\sin\theta}$.
Therefore, even if the magnetic field is much smaller than the critical
magnetic field, the quantum contribution will dominate the classical one for
large period P (or low $\Omega$). Hence, the late-time evolution of the
spindown of a pulsar should generically be quantum-dominated, because the
period gradually increases as the NS loses energy. Of course, this conclusion
holds only in the simple model considered here, which is certainly incomplete.
Among the physical phenomena that could change the previous statement are for
instance the inclusion of a real magnetosphere, or taking into account an
alignment of the NS (ie $\sin\theta\rightarrow 0$ as times passes
$\\!{}^{{\bf?}}$).
## 2 Quantum spindown
Using equations (1)-(2) and assuming the total energy is only rotational
kinetic energy $E=\frac{1}{2}J\Omega^{2}$ ($J$ being the inertia moment), one
gets a new evolution equation for the rotation period,
$\dot{E}=\dot{E}_{\mathrm{cl}}+\dot{E}_{\mathrm{qu}}\quad\Rightarrow\quad\dot{P}=\frac{\mathcal{T}_{\mathrm{cl}}}{P}+\frac{P}{\mathcal{T}_{\mathrm{qu}}}\>,$
(3)
where the constants $\mathcal{T}_{\mathrm{cl}}$ and
$\mathcal{T}_{\mathrm{qu}}$ are easily obtained,
$\displaystyle\mathcal{T}_{\mathrm{cl}}$ $\displaystyle\simeq$ $\displaystyle
8.8\times
10^{-16}\,\text{s}\left(\frac{B_{0}}{10^{8}\,\text{T}}\right)^{2}\sin^{2}\theta\left(\frac{R}{10\,\text{km}}\right)^{4}\left(\frac{1.4\,\text{M}_{\odot}}{M}\right)\;;$
(4) $\displaystyle\mathcal{T}_{\mathrm{qu}}$ $\displaystyle\simeq$
$\displaystyle 2.1\times
10^{13}\,\text{s}\,\left(\frac{10^{8}\,\text{T}}{B_{0}}\right)^{4}\frac{1}{\sin^{2}\theta}\left(\frac{10\,\text{km}}{R}\right)^{2}\left(\frac{M}{1.4\,\text{M}_{\odot}}\right)\;.$
(5)
It is clear that the quantum contribution will be dominant once
$P>\mathcal{T}\equiv\sqrt{\mathcal{T}_{\mathrm{cl}}\mathcal{T}_{\mathrm{qu}}}$,
with $\mathcal{T}\simeq
140\,\text{ms}\left(\frac{10^{8}\,\text{T}}{B_{0}}\right)\left(\frac{R}{10\,\text{km}}\right)$
a characteristic time. From the measurement of $P_{0}$, $P_{1}$ and $n_{0}$,
respectively the present period, its first derivative and the braking index
($n\equiv 2-P\ddot{P}/\dot{P}^{2}$), it is possible to solve equation (3) and
determine $\mathcal{T}_{\mathrm{cl}}$ and $\mathcal{T}_{\mathrm{qu}}$,
$P(t)=\mathcal{T}\left[\left(1+\frac{P_{0}^{2}}{\mathcal{T}^{2}}\right)e^{\frac{t}{\mathcal{T}_{\mathrm{qu}}}}-1\right]^{1/2}\quad,\quad\mathcal{T}_{\mathrm{cl}}=P_{0}P_{1}\frac{n_{0}-1}{2}\quad,\quad\mathcal{T}_{\mathrm{qu}}=\frac{P_{0}}{P_{1}}\frac{2}{3-n_{0}}\;.$
(6)
As an explicit example, the evolution of the Crab pulsar from the previous
expressions is represented in the $(P,\dot{P})$ diagramm of figure 1. The
evolution starts from the birth of the Crab ($\leavevmode\nobreak\ 955$ years
ago) and last $50\,$kyr. Each dot in this plot is a pulsar from the ATNF
catalog $\\!{}^{{\bf?}}$. The interesting feature is that the evolution
naturally brings the Crab towards the so-called magnetar region in the upper
right corner, even if the magnetic field is smaller than the critical field
(see next section for an estimation of the Crab magnetic field, being a few
$10^{8}\,$T). This could support the idea already proposed in $\\!{}^{{\bf?}}$
that some of the so-called magnetars are in fact normal evolved pulsars. A
deeper analysis of this hypothesis is underway.
Figure 1: Evolution of the Crab pulsar. The grey line is the classical
evolution while the black line is the evolution taking into account the
quantum correction.
The quantum evolution given by (6) also has a consequence on the age of the
pulsar, obtained as
$t_{\text{age}}=\frac{P_{0}}{P_{1}}\frac{1}{n_{0}-3}\,\ln\frac{n_{0}-1+(3-n_{0})\frac{P_{i}^{2}}{P_{0}^{2}}}{2}\mathrel{\mathop{\sim}\limits_{P_{i}\ll
P_{0}}}\frac{P_{0}}{(3-n_{0})P_{1}}\ln\frac{n_{0}-1}{2}$ (7)
The last equality assumes that the present period $P_{0}$ is much greater than
the initial period $P_{i}$. This age is always greater than the characteristic
age $\frac{P_{0}}{2P_{1}}$ obtained classically with the same approximation
$P_{i}\ll P_{0}$. This consequence could be an observational signature of the
quantum evolution since it would show up as a systematic bias between the
kinematical age (or SNR age) and the characteristic age. Such discrepancies
are for example reported in $\\!{}^{{\bf?}}$.
## 3 Constraints on the mass and radius
From $\mathcal{T}_{\mathrm{cl}}$ and $\mathcal{T}_{\mathrm{qu}}$, it is
straightforward to determine $B_{0}$ and $\sin\theta$ as a function of the
mass $M$ and the radius $R$ of the NS, giving
$\displaystyle B_{0}$ $\displaystyle=$ $\displaystyle\frac{5\pi
B_{c}R}{2c\sqrt{\alpha}}\frac{1}{P_{0}}\sqrt{\frac{3-n_{0}}{n_{0}-1}}\;;$ (8)
$\displaystyle\sin^{2}\theta$ $\displaystyle=$
$\displaystyle\frac{3\alpha}{400\pi^{5}}\frac{\mu_{0}Jc^{5}}{R^{8}B_{c}^{2}}P_{0}^{3}P_{1}\frac{(n_{0}-1)^{2}}{3-n_{0}}\;.$
(9)
The condition $\sin^{2}\theta<1$ then provides constraints on the mass $M$ and
the radius $R$ of the pulsar. Such constraints are represented in the figure 2
as exclusion regions, for two similar pulsars, namely $J0534+2200$ (the Crab)
and $J0540-6919$; for those pulsars the braking index $n_{0}$ is confidently
measured and is given in table 1. For sake of simplicity we assumed
$J=\frac{2}{5}MR^{2}$. In the same figure are represented some families of
Equation Of State for the NS (extracted from $\\!{}^{{\bf?}}$). It is quite
remarkable that taking into account the quantum effect sets some constraints
on such EOS. In particular, the strange quark models (SQM) seem excluded. Of
course, it is not possible to draw any definite conclusion unless a more
realistic model is studied. For example, it is expected that the magnetosphere
could significantly change the previous conclusions.
Figure 2: Constraints in the $(M,R)$ diagramm. The colored regions are excluded, either by causality or by the condition $\sin\theta<1$. The solid lines corresponds to different models of NS equation of state. The dotted lines correspond to constant value of the inclination angle while vertical dashed lines correspond to constant magnetic field $B_{0}$. Table 1: Spin and breaking index. Name | $n_{0}$ | $P_{0}$ | $P_{1}$
---|---|---|---
J2000 | | (ms) | ($10^{-13}$)
J0534+2200 (Crab) | 2.51 | 33.1 | 4.23
J0540-6919 | 2.14 | 50.5 | 4.79
## 4 Conclusion
We showed that the predicted quantum-induced spindown in NS leads to
observational consequences that should be looked for carefully. In particular,
the evolution of a pulsar in the $(P,\dot{P})$ diagramm is qualitatively
changed for highly manetized pulsars, the true age of a pulsar significantly
differs from the characteristic age, and some constraints on the equation of
state can be obtained, through new relationships between the mass, the radius,
the inclination angle and the magnetic field of the NS.
## References
## References
* [1] R Battesti and C Rizzo. Magnetic and electric properties of a quantum vacuum. Reports on Progress in Physics, 76(1):6401, January 2013.
* [2] David d’Enterria and Gustavo G Silveira. Observing light-by-light scattering at the Large Hadron Collider. arXiv.org, page 7142, May 2013.
* [3] P Berceau, M Fouché, R Battesti, and C Rizzo. Magnetic linear birefringence measurements using pulsed fields. Physical Review A, 85(1):13837, January 2012.
* [4] A Dupays, C Rizzo, D Bakalov, and G F Bignami. Quantum Vacuum Friction in highly magnetized neutron stars. EPL (Europhysics Letters), 82(6):69002, June 2008.
* [5] Arnaud Dupays, Carlo Rizzo, and Giovanni Fabrizio Bignami. Quantum vacuum influence on pulsars spindown evolution. Europhysics Letters, 98(4):49001, May 2012.
* [6] M D T Young, L S Chan, R R Burman, and D G Blair. Pulsar magnetic alignment and the pulsewidth-age relation. Monthly Notices of the Royal Astronomical Society, 402(2):1317–1329, February 2010.
* [7] R N Manchester, G B Hobbs, A Teoh, and M Hobbs. The Australia Telescope National Facility Pulsar Catalogue. The Astronomical Journal, 129(4):1993–2006, April 2005.
* [8] M A McLaughlin, Z Arzoumanian, J M Cordes, D C Backer, A N Lommen, D R Lorimer, and A F Zepka. PSR J1740+ 1000: A young pulsar well out of the Galactic plane. Astrophysical Journal, 564(1):333, 2002.
* [9] James M Lattimer. Equation of state constraints from neutron stars. Astrophysics and Space Science, 308(1):371–379, April 2007.
|
arxiv-papers
| 2013-10-07T20:14:33 |
2024-09-04T02:49:52.101440
|
{
"license": "Public Domain",
"authors": "Brahim Lamine, Cl\\'ement Berthi\\`ere, Arnaud Dupays",
"submitter": "Brahim Lamine",
"url": "https://arxiv.org/abs/1310.1938"
}
|
1310.2110
|
# A road map for synthesizing the scaling patterns in ecology
Cang Hui Centre for Invasion Biology, Department of Botany and Zoology,
Stellenbosch University, Matieland 7602, South Africa; E-mail: [email protected]
###### Abstract
Ecology studies biodiversity in its variety and complexity. It describes how
species distribute and perform in response to environmental changes.
Ecological processes and structures are highly complex and adaptive. In order
to quantify emerging ecological patterns and investigate their hidden
mechanisms, we need to rely on the simplicity of mathematical language. This
becomes especially apparent when dealing with scaling patterns in ecology.
Indeed, nearly all of ecological patterns are scale dependent. Such scale
dependence hampers our predictive power and creates problems in our inference.
This challenge calls for a clear and fundamental understanding of how and why
ecological patterns change across scales. As Simon Levin stated in his
MacArthur Award lecture, the problem of relating phenomena across scales is
the central problem in ecology and other natural sciences. It has become clear
that there is currently a drive in ecology and complexity science to develop
new quantitative approaches that are suitable for analysing and forecasting
patterns of ecological systems. Here I provide a road map for future works on
synthesizing the scaling patterns in ecology, aiming (i) to collect and sort a
diverse array of ecological patterns, (ii) to present the dominant parametric
forms of how these patterns change across spatial and temporal scales, (iii)
to detect the processes and mechanisms using mathematical models, and finally
(iv) to probe the physical meaning of these scaling patterns. This road map is
divided into three parts and covers three main concepts of scale in ecology:
heterogeneity, hierarchy and size. Using scale as a thread, this road map and
its following works weave the kaleidoscope of ecological scaling patterns into
a cohesive whole.
PACS numbers
87.23
###### pacs:
Valid PACS appear here
††preprint: APS/123-QED
Ecological patterns are emerging structures observed in populations,
communities and ecosystems. Elucidating drivers behind ecological patterns can
greatly improve our knowledge on how ecosystems assemble, function and respond
to change and perturbation. Due to the non-random nature, most, if not all,
ecological patterns change with measurement, characteristic and organization
scales and exhibit distinct scaling properties. Such scaling properties can be
broadly grouped into patterns related to heterogeneity, hierarchy and size.
The road map introduces the three groups of scaling patterns. The emphasis
here is not to provide solutions to these outlined research questions; rather,
by grouping relevant scaling patterns under unique banners, I attempt to
highlight the challenges and connect the emerging clues for building a unified
theory for scaling patterns in ecology in the near future.
## I Heterogeneity
This section on the scaling pattern of heterogeneity aims to investigate how
aggregated structures of organisms, diversity and ecosystem service change
with measurement scales and which/why biological patterns resonate with
underlying processes at the same characteristic scales.
Aggregation. Species distributions are not uniform across space, reflecting
the interplay between habitat heterogeneity and the underlying nonlinear
biotic regulation ref1 . Such non-random, aggregated patterns not only can be
the indicative of non-equilibrium dynamics (e.g. during range expansion mihaja
; ecography ; cecile ) but also self-organized pattern emergence (e.g.
ploszhang ; ncer ; ncem ). When ecologists examine species distributions
across scales, the Modifiable Areal Unit Problem presents itself ref2 ; maup .
The problem can be described as the change in species distribution
characteristics as the unit of measurement changes, both in terms of size and
shape of the sample unit. Such scaling patterns of aggregation follow three
general parametric forms: logarithmic, power-law and lognormal shape ref1 .
Following on recent progress of using the Bayesian rule for cross-scale
extrapolation ref3 ; ref4 , further advancement in this field is to provide a
consistent description of aggregation when scaling up and down, and therefore
a universal basis of comparison for distributions in differing contexts. A
fully-functional model with predictive power for up- and down-scaling species
distribution is needed. Under certain conditions, this model should further
allow extrapolating fine-scale occupancy and population densities from coarse-
scale observations (e.g. ecoappl ; ecoscience ). Great potential exists to
apply such a predictive model in various cross-scale pattern analyses
espcially when detection is imperfect (e.g. springer ; jae ).
Space-for-time substitution. The directionality of community succession is an
important concept in conservation biology Odum ; pickett ; it is analogous to
the irreversibility of time in physics that has revolutionised the
understanding of complex adaptive systems jorg . By definition, succession is
an orderly process of community change after disturbance Odum . Knowing the
directionality of succession is necessary for (i) distinguishing new from
mature communities (i.e. defining the age of a community), (ii) understanding
how communities evolve and respond to disturbance (e.g. habitat loss and
climate change), and (iii) designing more efficient conservation and
restoration plans nuria . However, popularising the concept of directionality
in succession is challenging for two reasons. Firstly, acknowledging this
concept demands the acceptance of inherent bias in nature which contradicts
the null hypothesis of a random and isotropic world jorg . Secondly,
appropriate long-term data required for detecting the succession direction are
scarce, and indices and analytical methods for such computation are lacking.
To this end designing alternative tests (e.g. the space-for-time substitution;
pickett that can capture the essence of directionality and irreversibility in
community development but which can be applied to available data becomes
crucial.
The spatial and temporal scales of ecological processes are intertwined.
Processes that account for the spatial distribution of species also underpin
its temporal dynamics. This means that we can potentially forecast the future
or rebuild the history based solely on current spatial distribution, without
resorting to long-term time series ref5 ; ref6 . In other words, the need to
wait years and decades to measure changes in distribution can be averted
through the ability to make sufficiently accurate predictions based only on
the spatial distribution of species at the current time. As the ability to
forecast the temporal trend of a focal species provides crucial information on
its performance and viability ref5 ; ref7 , the methodology of space-for-time
substitution is extremely appealing, especially because our ability to obtain
spatial records has been drastically improved. This area of research calls for
a model that can relate the scaling pattern of species’ current spatial
distribution to the near-future population trend and performance.
Scale resonance. Just as two tuning forks of the same characteristic frequency
resonate, so do ecological patterns and processes working at the same scale.
Species distributions are regulated by a variety of abiotic and biotic
processes working in concert but at different scales mcgill . Those processes
identified as key biotic drivers using methods such as multivariate statistics
often resonate with the scale of the study. That is, information being picked
up represents a product of the measurement method, rather than the intrinsic
cross-scale mechanism. Such a pattern of scale resonance has been observed
when synthesizing a series of collaborative works on identifying the factors
of the distribution of Argentine ants at local ref8 , regional ref9 and
global scales ref10 . This finding brings into question many regional
management planning practices that are based on the upscale extrapolation of
local-scale studies. Future research needs to explore the mechanism behind
scale resonance in ecology and to present a statistical remedy for cross-scale
inference.
Co-distribution. To exploit resources while mitigating conflict, species often
partition available habitats, forming co-distribution patterns of association
or dissociation. Null models based on permutation have been widely applied for
detecting signals of association or dissociation from co-distribution
patterns, from which the type of biotic interactions can be inferred. Future
research needs to present a model that incorporates biotic interactions and
also captures the transition from fine-scale dissociation to coarse-scale
association (e.g. ref4 ), explaining why this co-distribution pattern changes
across scales and how this scale dependence affects the pairwise measure of
species turnover. It should be reconcile the debate between the Rich-Get-
Richer phenomenon in invasion biology and the opposing Competitive-Exclusion-
Principle.
Biodiversity. Species diversity patterns, such as the species-area curve
ref11a , endemics-area relationship, distance decay of similarity and
occupancy frequency distribution ref11 ; ref12 , are just a few interrelated
patterns of scale dependence emerging from complex ecological systems. The
integration of patterns of species diversity patterns is central to
understanding the processes that drive species assembly 125 . Changing the
measurement scale will lead to a coordinated change in all diversity patterns.
I envisage a new diversity pattern – delta diversity – that connects all
commonly known diversity scaling patterns, using delta diversity as building
blocks. This model should be able to further explain Raunkiaer’s bimodal law
of frequency ref13 and resolves the debate on the ceiling of species richness
in a community.
Ecosystem function and service Ecosystem services are by-products from the
function of ecosystem processes that sustain the basic needs of humans and
their socioeconomic activities. According to the Millennium Ecosystem
Assessment, ecosystem services are generated from interactions ranging from
specialist taxa to all biodiversity, and the functional units of the variety
of services range from local populations to global biogeochemical cycles. At
the local scale, we benefit from services of pollination, pest control, soil
fertility and seed dispersal that are related to biodiversity. At regional
scale, we benefit from services of air and water purification, flood and
drought mitigation, and waste decomposition that are delivered by plants and
micro-organisms. At the global level, we benefit from services of climate
stability and UV protection from plants and biogeochemical cycles.
Understanding how different ecosystem services change with spatial scales and
potentially conceptualizing into a model for extrapolating the level of
service across scales warrants great attention. Recent studies showed the
scale dependence of a crude ecosystem service indicator – biocapacity and the
resultant sustainability index ref14 ; ref15 , and the robustness and
invasibility of recipient ecosystems to biological invasions ref16 .
Exploiting ecosystem services within their maximum sustainable level can
ensure a reliable service provision without triggering a regime shift.
Achieving this balance is a challenge for conservation management and
sustainable development.
## II Hierarchy
The scaling pattern of hierarchy depicts how the structure and function of
asymmetrical ecological systems emerge and change with the system complexity.
Using ecological networks as the model system, this section aims to
investigate how cascade interactions affects the robustness and resilience of
networks, how network architectures, especially nestedness and
compartmentalization, emerge and function, and the role of network complexity
on the stability of ecological networks.
Cascade. Nodes and edges of a network are a good proxy of species and their
interactions in an ecosystem. Probing processes that can lead to the emergence
of large-scale network architectures, e.g. small-world networks and scale-free
networks, is a new wave in science. Scaling laws of food webs depict how a
biological network behaves as a function of its complexity. In reconciling
with May’s stability criterion of complex systems may , Cohen’s cascade model
cohen and its later development provide a phenomenological explanation of
some of the scaling laws and scaling invariant patterns. The principle of
Maximum Entropy that identifies the unbiased estimates under constraints has
been widely used in ecology. Ecological networks are efficient energy
transporting, non-equilibrium systems that are adaptive to changes and
disturbances. This calls for the development of models based on the recently
developed principle of Maximum Entropy Production in non-equilibrium
thermodynamics dewar , in an attempt to explain the pattern emergence in
complex adaptive networks, in particular, the cascade interactions in food
webs. Such models will provide a physical understanding on the network
emergence and shed light on solving the complexity-stability debate.
Nestedness. Nested structure has been observed in many networks, in particular
bipartite mutualistic networks (e.g. pollination networks and seed-dispersal
networks bas ). To have models with quantitative accuracy and predictive
power, one needs to rely on process-based models. A key feature in ecological
networks is the adaptive and innovative nature of the edges and nodes. Species
are constantly optimizing which partners they should interact with for maximum
fitness gain, as according to the optimal foraging theory. This can be
achieved at two time scales: 1) At the ecological time scale, interaction
switching reflects seeking optimal fitness gain ref17 ; anim ; 2) At the
evolutionary time scale, interaction switching reflects coevolutionary
dynamics between interacting traits.
Compartmentalization. One important hierarchical structure in ecological
networks is compartmentalization, i.e. the formation of functional modules,
where interactions are most likely to occur within the same module, that is to
say “like is connected to like” in a network ref18 . Energy flows
directionally in resource-exploitation networks (e.g. host-parasite networks),
forming compartmentalized structures. This research area calls for a process-
based model that optimizes energy transport via adaptive interaction switching
partners for local fitness gain. The strong predictive power requires further
demonstration by comparing the level of compartmentalization observed versus
predicted for real networks savannah . This, together with the previous two
research areas, provides a physical understanding of pattern emergence in
complex adaptive networks.
## III Size
This section on the scaling pattern of size investigates how form and function
change as organisms get larger or as their traits change. The section aims to
investigate how allometric laws of metabolism emerge and how phenotypes of
biological traits form through co-evolution with other traits, species and the
environment and how this affects the path of evolution and diversification.
Allometry. Allometric scaling is the most salient pattern of how biological
rates, especially metabolic rate, are regulated by organism size allo , known
as Kleiber’s law. A model, depicting how fractal-structured circulatory or
vascular systems transport energy and matter throughout the body of the
organism, has been proposed for explaining the emergence of allometric
scaling, which is further summarized succinctly as the Fourth Dimension of
Life west . Recently, a not fully developed concept suggests that natural
selection can increase the ontogenetic and population growth rates to a
ceiling, above which the population will crash due to intrinsic instability
lev , in principle similar to the origin of planetary rings that are only
distributed on a thin surface surrounding the planet, with all other
directions, although possible, have been eliminated through inter-particle
collisions. This progress calls for further development of a conceptual model
for allometric scaling based on the principle of Maximum Entropy Production
and self-organized criticality. This model should be able to simulate the
process of cell differentiation in multicellular organisms.
Trait. This research area focuses on addressing two questions. First, what are
the mechanisms determining phenotypic traits? Evolution via natural selection
relies on heritable phenotypic variation and has long been regarded as being
solely reliant on direct expression of gene variation. This assumption may be
an exaggeration, as factors other than genetic variation can also dictate the
outcomes of natural selection and thus evolution. At the forefront of these
alternative platforms is the rapidly expanding field of epigenetics – the
biochemical modification of DNA without changes in its sequence – that gives
rise to differential gene expression and thus different phenotypes epi . A
revised Price equation that incorporates epigenetic mechanism for explaining
phenotypic variation is needed. Second, how do traits affect biotic
interactions and thus the path of evolution? Biotic interactions, largely to
do with resource competition and exploitation, are realised by specific
phenotypic traits that are used for searching and handling resources, for
competing and defending resources, for memorizing and recognizing beneficial
resources, and for escaping from being exploited as resources by predators or
parasites. In this regard, Adaptive Dynamics is a power tool to explore how
trait-mediated interactions can lead to diverse evolutionary phenomena ulf ,
from the Red Queen dynamics to speciation ec ; ref19 .
## IV Epilogue
Nature never fails to amaze us. It coordinates the interplay of numerous
organisms in their environments, forming a complex functional system that
sustains us and many other species via ecosystem service. As the ultimate goal
of natural sciences, quantifying emerging patterns in nature and understanding
hidden mechanisms are the pinnacle of science. An important phenomenon in
quantifying ecological patterns is that nearly all of them are scale
dependent. This scale dependence creates problems in our inference, yet also
simultaneously provides opportunities for us to pry into the core of how
nature assembles, organizes and functions. Once again, we need to relate these
research areas back to the overarching research philosophy in studying natural
systems: (i) what patterns exist in nature (i.e. using statistical methods to
measure and quantify ecological patterns); (ii) how such patterns emerge (i.e.
proposing mathematical models to unveil mechanisms driving the ecological
pattern formation; (iii) why nature organizes itself in such a way (i.e. using
physical laws to reveal the adaptive and/or optimal nature of ecological
systems). This has been eloquently presented as the Ouroboros of scales in
Just Six Numbers by Martin Rees. William Blake also expressed the role of
scales in his Auguries of Innocence in 1803, ‘To see a world in a grain of
sand, and a heaven in a wild flower; Hold infinity in the palm of your hand,
and eternity in an hour’.
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|
arxiv-papers
| 2013-10-08T12:20:40 |
2024-09-04T02:49:52.113953
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Cang Hui",
"submitter": "Cang Hui",
"url": "https://arxiv.org/abs/1310.2110"
}
|
1310.2145
|
EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN)
CERN-PH-EP-2013-186 LHCb-PAPER-2013-055 8 October 2013
Observation of $\kern
3.73305pt\overline{\kern-3.73305ptB}{}^{0}_{(s)}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}f_{1}(1285)$ decays and measurement of the $f_{1}(1285)$ mixing angle
The LHCb collaboration†††Authors are listed on the following pages.
Decays of $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ and $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ mesons into
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\pi^{+}\pi^{-}\pi^{+}\pi^{-}$
final states, produced in $pp$ collisions at the LHC, are investigated using
data corresponding to an integrated luminosity of 3 fb-1 collected with the
LHCb detector. $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{(s)}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}f_{1}(1285)$ decays are seen for the first time, and the branching
fractions are measured. Using these rates, the $f_{1}(1285)$ mixing angle
between strange and non-strange components of its wave function in the
$q\overline{q}$ structure model is determined to be
$\pm(24.0^{\,+3.1\,+0.6}_{\,-2.6\,-0.8})^{\circ}$. Implications on the
possible tetraquark nature of the $f_{1}(1285)$ are discussed.
Submitted to Phys. Rev. Lett.
© CERN on behalf of the LHCb collaboration, license CC-BY-3.0.
LHCb collaboration
R. Aaij40, B. Adeva36, M. Adinolfi45, C. Adrover6, A. Affolder51, Z.
Ajaltouni5, J. Albrecht9, F. Alessio37, M. Alexander50, S. Ali40, G.
Alkhazov29, P. Alvarez Cartelle36, A.A. Alves Jr24, S. Amato2, S. Amerio21, Y.
Amhis7, L. Anderlini17,f, J. Anderson39, R. Andreassen56, M. Andreotti16,e,
J.E. Andrews57, R.B. Appleby53, O. Aquines Gutierrez10, F. Archilli18, A.
Artamonov34, M. Artuso58, E. Aslanides6, G. Auriemma24,m, M. Baalouch5, S.
Bachmann11, J.J. Back47, A. Badalov35, C. Baesso59, V. Balagura30, W.
Baldini16, R.J. Barlow53, C. Barschel37, S. Barsuk7, W. Barter46, V.
Batozskaya27, Th. Bauer40, A. Bay38, J. Beddow50, F. Bedeschi22, I. Bediaga1,
S. Belogurov30, K. Belous34, I. Belyaev30, E. Ben-Haim8, G. Bencivenni18, S.
Benson49, J. Benton45, A. Berezhnoy31, R. Bernet39, M.-O. Bettler46, M. van
Beuzekom40, A. Bien11, S. Bifani44, T. Bird53, A. Bizzeti17,h, P.M.
Bjørnstad53, T. Blake37, F. Blanc38, J. Blouw10, S. Blusk58, V. Bocci24, A.
Bondar33, N. Bondar29, W. Bonivento15, S. Borghi53, A. Borgia58, T.J.V.
Bowcock51, E. Bowen39, C. Bozzi16, T. Brambach9, J. van den Brand41, J.
Bressieux38, D. Brett53, M. Britsch10, T. Britton58, N.H. Brook45, H. Brown51,
A. Bursche39, G. Busetto21,q, J. Buytaert37, S. Cadeddu15, R. Calabrese16,e,
O. Callot7, M. Calvi20,j, M. Calvo Gomez35,n, A. Camboni35, P. Campana18,37,
D. Campora Perez37, A. Carbone14,c, G. Carboni23,k, R. Cardinale19,i, A.
Cardini15, H. Carranza-Mejia49, L. Carson52, K. Carvalho Akiba2, G. Casse51,
L. Castillo Garcia37, M. Cattaneo37, Ch. Cauet9, R. Cenci57, M. Charles54, Ph.
Charpentier37, S.-F. Cheung54, N. Chiapolini39, M. Chrzaszcz39,25, K. Ciba37,
X. Cid Vidal37, G. Ciezarek52, P.E.L. Clarke49, M. Clemencic37, H.V. Cliff46,
J. Closier37, C. Coca28, V. Coco40, J. Cogan6, E. Cogneras5, P. Collins37, A.
Comerma-Montells35, A. Contu15,37, A. Cook45, M. Coombes45, S. Coquereau8, G.
Corti37, B. Couturier37, G.A. Cowan49, D.C. Craik47, M. Cruz Torres59, S.
Cunliffe52, R. Currie49, C. D’Ambrosio37, P. David8, P.N.Y. David40, A.
Davis56, I. De Bonis4, K. De Bruyn40, S. De Capua53, M. De Cian11, J.M. De
Miranda1, L. De Paula2, W. De Silva56, P. De Simone18, D. Decamp4, M.
Deckenhoff9, L. Del Buono8, N. Déléage4, D. Derkach54, O. Deschamps5, F.
Dettori41, A. Di Canto11, H. Dijkstra37, M. Dogaru28, S. Donleavy51, F.
Dordei11, A. Dosil Suárez36, D. Dossett47, A. Dovbnya42, F. Dupertuis38, P.
Durante37, R. Dzhelyadin34, A. Dziurda25, A. Dzyuba29, S. Easo48, U. Egede52,
V. Egorychev30, S. Eidelman33, D. van Eijk40, S. Eisenhardt49, U.
Eitschberger9, R. Ekelhof9, L. Eklund50,37, I. El Rifai5, Ch. Elsasser39, A.
Falabella14,e, C. Färber11, C. Farinelli40, S. Farry51, D. Ferguson49, V.
Fernandez Albor36, F. Ferreira Rodrigues1, M. Ferro-Luzzi37, S. Filippov32, M.
Fiore16,e, M. Fiorini16,e, C. Fitzpatrick37, M. Fontana10, F. Fontanelli19,i,
R. Forty37, O. Francisco2, M. Frank37, C. Frei37, M. Frosini17,37,f, E.
Furfaro23,k, A. Gallas Torreira36, D. Galli14,c, M. Gandelman2, P. Gandini58,
Y. Gao3, J. Garofoli58, P. Garosi53, J. Garra Tico46, L. Garrido35, C.
Gaspar37, R. Gauld54, E. Gersabeck11, M. Gersabeck53, T. Gershon47, Ph. Ghez4,
V. Gibson46, L. Giubega28, V.V. Gligorov37, C. Göbel59, D. Golubkov30, A.
Golutvin52,30,37, A. Gomes2, P. Gorbounov30,37, H. Gordon37, M. Grabalosa
Gándara5, R. Graciani Diaz35, L.A. Granado Cardoso37, E. Graugés35, G.
Graziani17, A. Grecu28, E. Greening54, S. Gregson46, P. Griffith44, L.
Grillo11, O. Grünberg60, B. Gui58, E. Gushchin32, Yu. Guz34,37, T. Gys37, C.
Hadjivasiliou58, G. Haefeli38, C. Haen37, T.W. Hafkenscheid61, S.C. Haines46,
S. Hall52, B. Hamilton57, T. Hampson45, S. Hansmann-Menzemer11, N. Harnew54,
S.T. Harnew45, J. Harrison53, T. Hartmann60, J. He37, T. Head37, V. Heijne40,
K. Hennessy51, P. Henrard5, J.A. Hernando Morata36, E. van Herwijnen37, M.
Heß60, A. Hicheur1, E. Hicks51, D. Hill54, M. Hoballah5, C. Hombach53, W.
Hulsbergen40, P. Hunt54, T. Huse51, N. Hussain54, D. Hutchcroft51, D. Hynds50,
V. Iakovenko43, M. Idzik26, P. Ilten12, R. Jacobsson37, A. Jaeger11, E.
Jans40, P. Jaton38, A. Jawahery57, F. Jing3, M. John54, D. Johnson54, C.R.
Jones46, C. Joram37, B. Jost37, M. Kaballo9, S. Kandybei42, W. Kanso6, M.
Karacson37, T.M. Karbach37, I.R. Kenyon44, T. Ketel41, B. Khanji20, O.
Kochebina7, I. Komarov38, R.F. Koopman41, P. Koppenburg40, M. Korolev31, A.
Kozlinskiy40, L. Kravchuk32, K. Kreplin11, M. Kreps47, G. Krocker11, P.
Krokovny33, F. Kruse9, M. Kucharczyk20,25,37,j, V. Kudryavtsev33, K. Kurek27,
T. Kvaratskheliya30,37, V.N. La Thi38, D. Lacarrere37, G. Lafferty53, A.
Lai15, D. Lambert49, R.W. Lambert41, E. Lanciotti37, G. Lanfranchi18, C.
Langenbruch37, T. Latham47, C. Lazzeroni44, R. Le Gac6, J. van Leerdam40,
J.-P. Lees4, R. Lefèvre5, A. Leflat31, J. Lefrançois7, S. Leo22, O. Leroy6, T.
Lesiak25, B. Leverington11, Y. Li3, L. Li Gioi5, M. Liles51, R. Lindner37, C.
Linn11, B. Liu3, G. Liu37, S. Lohn37, I. Longstaff50, J.H. Lopes2, N. Lopez-
March38, H. Lu3, D. Lucchesi21,q, J. Luisier38, H. Luo49, E. Luppi16,e, O.
Lupton54, F. Machefert7, I.V. Machikhiliyan30, F. Maciuc28, O. Maev29,37, S.
Malde54, G. Manca15,d, G. Mancinelli6, J. Maratas5, U. Marconi14, P.
Marino22,s, R. Märki38, J. Marks11, G. Martellotti24, A. Martens8, A. Martín
Sánchez7, M. Martinelli40, D. Martinez Santos41,37, D. Martins Tostes2, A.
Martynov31, A. Massafferri1, R. Matev37, Z. Mathe37, C. Matteuzzi20, E.
Maurice6, A. Mazurov16,37,e, M. McCann52, J. McCarthy44, A. McNab53, R.
McNulty12, B. McSkelly51, B. Meadows56,54, F. Meier9, M. Meissner11, M.
Merk40, D.A. Milanes8, M.-N. Minard4, J. Molina Rodriguez59, S. Monteil5, D.
Moran53, P. Morawski25, A. Mordà6, M.J. Morello22,s, R. Mountain58, I. Mous40,
F. Muheim49, K. Müller39, R. Muresan28, B. Muryn26, B. Muster38, P. Naik45, T.
Nakada38, R. Nandakumar48, I. Nasteva1, M. Needham49, S. Neubert37, N.
Neufeld37, A.D. Nguyen38, T.D. Nguyen38, C. Nguyen-Mau38,o, M. Nicol7, V.
Niess5, R. Niet9, N. Nikitin31, T. Nikodem11, A. Nomerotski54, A. Novoselov34,
A. Oblakowska-Mucha26, V. Obraztsov34, S. Oggero40, S. Ogilvy50, O.
Okhrimenko43, R. Oldeman15,d, G. Onderwater61, M. Orlandea28, J.M. Otalora
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Palutan18, J. Panman37, A. Papanestis48, M. Pappagallo50, C. Parkes53, C.J.
Parkinson52, G. Passaleva17, G.D. Patel51, M. Patel52, G.N. Patrick48, C.
Patrignani19,i, C. Pavel-Nicorescu28, A. Pazos Alvarez36, A. Pearce53, A.
Pellegrino40, G. Penso24,l, M. Pepe Altarelli37, S. Perazzini14,c, E. Perez
Trigo36, A. Pérez-Calero Yzquierdo35, P. Perret5, M. Perrin-Terrin6, L.
Pescatore44, E. Pesen62, G. Pessina20, K. Petridis52, A. Petrolini19,i, A.
Phan58, E. Picatoste Olloqui35, B. Pietrzyk4, T. Pilař47, D. Pinci24, S.
Playfer49, M. Plo Casasus36, F. Polci8, G. Polok25, A. Poluektov47,33, E.
Polycarpo2, A. Popov34, D. Popov10, B. Popovici28, C. Potterat35, A. Powell54,
J. Prisciandaro38, A. Pritchard51, C. Prouve7, V. Pugatch43, A. Puig
Navarro38, G. Punzi22,r, W. Qian4, B. Rachwal25, J.H. Rademacker45, B.
Rakotomiaramanana38, M.S. Rangel2, I. Raniuk42, N. Rauschmayr37, G. Raven41,
S. Redford54, S. Reichert53, M.M. Reid47, A.C. dos Reis1, S. Ricciardi48, A.
Richards52, K. Rinnert51, V. Rives Molina35, D.A. Roa Romero5, P. Robbe7, D.A.
Roberts57, A.B. Rodrigues1, E. Rodrigues53, P. Rodriguez Perez36, S. Roiser37,
V. Romanovsky34, A. Romero Vidal36, M. Rotondo21, J. Rouvinet38, T. Ruf37, F.
Ruffini22, H. Ruiz35, P. Ruiz Valls35, G. Sabatino24,k, J.J. Saborido Silva36,
N. Sagidova29, P. Sail50, B. Saitta15,d, V. Salustino Guimaraes2, B. Sanmartin
Sedes36, R. Santacesaria24, C. Santamarina Rios36, E. Santovetti23,k, M.
Sapunov6, A. Sarti18, C. Satriano24,m, A. Satta23, M. Savrie16,e, D.
Savrina30,31, M. Schiller41, H. Schindler37, M. Schlupp9, M. Schmelling10, B.
Schmidt37, O. Schneider38, A. Schopper37, M.-H. Schune7, R. Schwemmer37, B.
Sciascia18, A. Sciubba24, M. Seco36, A. Semennikov30, K. Senderowska26, I.
Sepp52, N. Serra39, J. Serrano6, P. Seyfert11, M. Shapkin34, I.
Shapoval16,42,e, Y. Shcheglov29, T. Shears51, L. Shekhtman33, O. Shevchenko42,
V. Shevchenko30, A. Shires9, R. Silva Coutinho47, M. Sirendi46, N. Skidmore45,
T. Skwarnicki58, N.A. Smith51, E. Smith54,48, E. Smith52, J. Smith46, M.
Smith53, M.D. Sokoloff56, F.J.P. Soler50, F. Soomro38, D. Souza45, B. Souza De
Paula2, B. Spaan9, A. Sparkes49, P. Spradlin50, F. Stagni37, S. Stahl11, O.
Steinkamp39, S. Stevenson54, S. Stoica28, S. Stone58, B. Storaci39, M.
Straticiuc28, U. Straumann39, V.K. Subbiah37, L. Sun56, W. Sutcliffe52, S.
Swientek9, V. Syropoulos41, M. Szczekowski27, P. Szczypka38,37, D. Szilard2,
T. Szumlak26, S. T’Jampens4, M. Teklishyn7, G. Tellarini16,e, E. Teodorescu28,
F. Teubert37, C. Thomas54, E. Thomas37, J. van Tilburg11, V. Tisserand4, M.
Tobin38, S. Tolk41, L. Tomassetti16,e, D. Tonelli37, S. Topp-Joergensen54, N.
Torr54, E. Tournefier4,52, S. Tourneur38, M.T. Tran38, M. Tresch39, A.
Tsaregorodtsev6, P. Tsopelas40, N. Tuning40,37, M. Ubeda Garcia37, A.
Ukleja27, A. Ustyuzhanin52,p, U. Uwer11, V. Vagnoni14, G. Valenti14, A.
Vallier7, R. Vazquez Gomez18, P. Vazquez Regueiro36, C. Vázquez Sierra36, S.
Vecchi16, J.J. Velthuis45, M. Veltri17,g, G. Veneziano38, M. Vesterinen37, B.
Viaud7, D. Vieira2, X. Vilasis-Cardona35,n, A. Vollhardt39, D. Volyanskyy10,
D. Voong45, A. Vorobyev29, V. Vorobyev33, C. Voß60, H. Voss10, R. Waldi60, C.
Wallace47, R. Wallace12, S. Wandernoth11, J. Wang58, D.R. Ward46, N.K.
Watson44, A.D. Webber53, D. Websdale52, M. Whitehead47, J. Wicht37, J.
Wiechczynski25, D. Wiedner11, L. Wiggers40, G. Wilkinson54, M.P.
Williams47,48, M. Williams55, F.F. Wilson48, J. Wimberley57, J. Wishahi9, W.
Wislicki27, M. Witek25, G. Wormser7, S.A. Wotton46, S. Wright46, S. Wu3, K.
Wyllie37, Y. Xie49,37, Z. Xing58, Z. Yang3, X. Yuan3, O. Yushchenko34, M.
Zangoli14, M. Zavertyaev10,a, F. Zhang3, L. Zhang58, W.C. Zhang12, Y. Zhang3,
A. Zhelezov11, A. Zhokhov30, L. Zhong3, A. Zvyagin37.
1Centro Brasileiro de Pesquisas Físicas (CBPF), Rio de Janeiro, Brazil
2Universidade Federal do Rio de Janeiro (UFRJ), Rio de Janeiro, Brazil
3Center for High Energy Physics, Tsinghua University, Beijing, China
4LAPP, Université de Savoie, CNRS/IN2P3, Annecy-Le-Vieux, France
5Clermont Université, Université Blaise Pascal, CNRS/IN2P3, LPC, Clermont-
Ferrand, France
6CPPM, Aix-Marseille Université, CNRS/IN2P3, Marseille, France
7LAL, Université Paris-Sud, CNRS/IN2P3, Orsay, France
8LPNHE, Université Pierre et Marie Curie, Université Paris Diderot,
CNRS/IN2P3, Paris, France
9Fakultät Physik, Technische Universität Dortmund, Dortmund, Germany
10Max-Planck-Institut für Kernphysik (MPIK), Heidelberg, Germany
11Physikalisches Institut, Ruprecht-Karls-Universität Heidelberg, Heidelberg,
Germany
12School of Physics, University College Dublin, Dublin, Ireland
13Sezione INFN di Bari, Bari, Italy
14Sezione INFN di Bologna, Bologna, Italy
15Sezione INFN di Cagliari, Cagliari, Italy
16Sezione INFN di Ferrara, Ferrara, Italy
17Sezione INFN di Firenze, Firenze, Italy
18Laboratori Nazionali dell’INFN di Frascati, Frascati, Italy
19Sezione INFN di Genova, Genova, Italy
20Sezione INFN di Milano Bicocca, Milano, Italy
21Sezione INFN di Padova, Padova, Italy
22Sezione INFN di Pisa, Pisa, Italy
23Sezione INFN di Roma Tor Vergata, Roma, Italy
24Sezione INFN di Roma La Sapienza, Roma, Italy
25Henryk Niewodniczanski Institute of Nuclear Physics Polish Academy of
Sciences, Kraków, Poland
26AGH - University of Science and Technology, Faculty of Physics and Applied
Computer Science, Kraków, Poland
27National Center for Nuclear Research (NCBJ), Warsaw, Poland
28Horia Hulubei National Institute of Physics and Nuclear Engineering,
Bucharest-Magurele, Romania
29Petersburg Nuclear Physics Institute (PNPI), Gatchina, Russia
30Institute of Theoretical and Experimental Physics (ITEP), Moscow, Russia
31Institute of Nuclear Physics, Moscow State University (SINP MSU), Moscow,
Russia
32Institute for Nuclear Research of the Russian Academy of Sciences (INR RAN),
Moscow, Russia
33Budker Institute of Nuclear Physics (SB RAS) and Novosibirsk State
University, Novosibirsk, Russia
34Institute for High Energy Physics (IHEP), Protvino, Russia
35Universitat de Barcelona, Barcelona, Spain
36Universidad de Santiago de Compostela, Santiago de Compostela, Spain
37European Organization for Nuclear Research (CERN), Geneva, Switzerland
38Ecole Polytechnique Fédérale de Lausanne (EPFL), Lausanne, Switzerland
39Physik-Institut, Universität Zürich, Zürich, Switzerland
40Nikhef National Institute for Subatomic Physics, Amsterdam, The Netherlands
41Nikhef National Institute for Subatomic Physics and VU University Amsterdam,
Amsterdam, The Netherlands
42NSC Kharkiv Institute of Physics and Technology (NSC KIPT), Kharkiv, Ukraine
43Institute for Nuclear Research of the National Academy of Sciences (KINR),
Kyiv, Ukraine
44University of Birmingham, Birmingham, United Kingdom
45H.H. Wills Physics Laboratory, University of Bristol, Bristol, United
Kingdom
46Cavendish Laboratory, University of Cambridge, Cambridge, United Kingdom
47Department of Physics, University of Warwick, Coventry, United Kingdom
48STFC Rutherford Appleton Laboratory, Didcot, United Kingdom
49School of Physics and Astronomy, University of Edinburgh, Edinburgh, United
Kingdom
50School of Physics and Astronomy, University of Glasgow, Glasgow, United
Kingdom
51Oliver Lodge Laboratory, University of Liverpool, Liverpool, United Kingdom
52Imperial College London, London, United Kingdom
53School of Physics and Astronomy, University of Manchester, Manchester,
United Kingdom
54Department of Physics, University of Oxford, Oxford, United Kingdom
55Massachusetts Institute of Technology, Cambridge, MA, United States
56University of Cincinnati, Cincinnati, OH, United States
57University of Maryland, College Park, MD, United States
58Syracuse University, Syracuse, NY, United States
59Pontifícia Universidade Católica do Rio de Janeiro (PUC-Rio), Rio de
Janeiro, Brazil, associated to 2
60Institut für Physik, Universität Rostock, Rostock, Germany, associated to 11
61KVI-University of Groningen, Groningen, The Netherlands, associated to 40
62Celal Bayar University, Manisa, Turkey, associated to 37
aP.N. Lebedev Physical Institute, Russian Academy of Science (LPI RAS),
Moscow, Russia
bUniversità di Bari, Bari, Italy
cUniversità di Bologna, Bologna, Italy
dUniversità di Cagliari, Cagliari, Italy
eUniversità di Ferrara, Ferrara, Italy
fUniversità di Firenze, Firenze, Italy
gUniversità di Urbino, Urbino, Italy
hUniversità di Modena e Reggio Emilia, Modena, Italy
iUniversità di Genova, Genova, Italy
jUniversità di Milano Bicocca, Milano, Italy
kUniversità di Roma Tor Vergata, Roma, Italy
lUniversità di Roma La Sapienza, Roma, Italy
mUniversità della Basilicata, Potenza, Italy
nLIFAELS, La Salle, Universitat Ramon Llull, Barcelona, Spain
oHanoi University of Science, Hanoi, Viet Nam
pInstitute of Physics and Technology, Moscow, Russia
qUniversità di Padova, Padova, Italy
rUniversità di Pisa, Pisa, Italy
sScuola Normale Superiore, Pisa, Italy
Light flavorless hadrons, $f$, are not entirely understood as $q\overline{q}$
states. Some states with the same quantum numbers such as the $\eta$ and
$\eta^{\prime}$ exhibit mixing [1]. Others, such as the $f_{0}(500)$ and the
$f_{0}(980)$, could be mixed $q\overline{q}$ states, or they could be
comprised of tetraquarks [2, *Weinberg:2013cfa, *Hooft:2008we,
*Achasov:2012kk]. In addition some states, such as the $f_{0}(1500)$, are
discussed as being made solely of gluons [6, *Jaffe:1976ig]. Understanding if
the $f$ states are indeed explained by the quark model is crucial to
identifying other exotic structures. Previous investigations of $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ and $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ decays (called generically $\kern
1.79993pt\overline{\kern-1.79993ptB}{}$) into a
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ meson and a $\pi^{+}\pi^{-}$
[8, 9] or $K^{+}K^{-}$ [10, 11] pair have revealed the presence of several
light flavorless meson resonances including the $f_{0}(500)$ and the
$f_{0}(980)$. Use of $\kern
1.79993pt\overline{\kern-1.79993ptB}{}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}f$ decays has been suggested as an excellent way of both measuring
mixing angles and discerning if some of the $f$ states are tetraquarks [12,
13, *Fleischer:2011ib]. In this Letter the
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\pi^{+}\pi^{-}\pi^{+}\pi^{-}$
final state is investigated with the aim of seeking additional $f$ states.
(Mention of a particular process also implies the use of its charge conjugated
decay.)
Data are obtained from 3 fb-1 of integrated luminosity collected with the LHCb
detector [15] using $pp$ collisions. One third of the data was acquired at a
center-of-mass energy of 7 TeV, and the remainder at 8 TeV. The LHCb detector
is a single-arm forward spectrometer covering the pseudorapidity range
$2<\eta<5$, designed for the study of particles containing $b$ or $c$ quarks.
The detector includes a high precision tracking system consisting of a
silicon-strip vertex detector surrounding the $pp$ interaction region, a
large-area silicon-strip detector located upstream of a dipole magnet with a
bending power of about $4{\rm\,Tm}$, and three stations of silicon-strip
detectors and straw drift tubes placed downstream. The combined tracking
system provides a momentum measurement with relative uncertainty that varies
from 0.4% at 5 GeV to 0.6% at 100 GeV. (We work in units where $c$=1.) The
impact parameter (IP) is defined as the minimum track distance with respect to
the primary vertex. For tracks with large transverse momentum, $p_{\rm T}$,
with respect to the proton beam direction, the IP resolution is approximately
20$\,\upmu\rm m$. Charged hadrons are identified using two ring-imaging
Cherenkov (RICH) detectors. Photon, electron and hadron candidates are
identified by a calorimeter system consisting of scintillating-pad and pre-
shower detectors, an electromagnetic calorimeter and a hadronic calorimeter.
Muons are identified by a system composed of alternating layers of iron and
multiwire proportional chambers.
The LHCb trigger [16] consists of a hardware stage, based on information from
the calorimeter and muon systems, followed by a software stage that applies
event reconstruction. Events selected for this analysis are triggered by a
candidate ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\rightarrow\mu^{+}\mu^{-}$ decay, required to be consistent with coming
from the decay of a $b$-hadron by using either IP requirements or detachment
from the associated primary vertex. Simulations are performed using Pythia
[17] with the specific tuning given in Ref. [18], and the LHCb detector
description based on Geant4 [19, *Agostinelli:2002hh] described in Ref. [21].
Decays of $b$-hadrons are based on EvtGen [22].
Events are preselected and then are further filtered using a multivariate
analyzer based on the boosted decision tree (BDT) technique [23]. In the
preselection, all charged track candidates are required to have $p_{\rm T}$
$>$ 250 MeV, while for muon candidates the requirement is $p_{\rm T}$ $>$ 550
MeV. Events must have a $\mu^{+}\mu^{-}$ combination that forms a common
vertex with $\chi^{2}<20$, an invariant mass between $-48$ and +43 MeV of the
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ meson mass, and are constrained
to the ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ mass. The four pions
must have a vector summed $\mbox{$p_{\rm T}$}>1$ GeV, form a vertex with
$\chi^{2}<50$ for five degrees of freedom, and a common vertex with the
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ candidate with $\chi^{2}<90$
for nine degrees of freedom. The angle between the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}$ momentum and the vector from the
primary vertex to the $\kern 1.79993pt\overline{\kern-1.79993ptB}{}$ decay
vertex is required to be smaller than 2.56∘. Particle identification [24]
requirements are based on the difference in the logarithm of the likelihood,
DLL$(h_{1}-h_{2})$, to distinguish between the hypotheses $h_{1}$ and $h_{2}$.
We require DLL$(\pi-\mu)>-10$ and DLL$(\pi-K)>-10$. We also explicitly
eliminate candidate $\psi(2S)[$or
$X(3872)]\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\pi^{+}\pi^{-}$ events by rejecting any candidate where one
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\pi^{+}\pi^{-}$ combination is
within 23 MeV of the $\psi(2S)$ or 9 MeV of the $X(3872)$ meson masses. Other
resonant contributions such as $\kern
1.79993pt\overline{\kern-1.79993ptB}{}\rightarrow\psi(4160)\pi^{+}\pi^{-}$ are
searched for, but not found.
The BDT uses 12 variables that are chosen to separate signal and background:
the minimum DLL$(\pi-\mu)$ of the $\mu^{+}$ and $\mu^{-}$, the scalar $p_{\rm
T}$ sum of the four pions, and the vector $p_{\rm T}$ sum of the four pions;
relating to the $\kern 1.79993pt\overline{\kern-1.79993ptB}{}$ candidate: the
flight distance, the vertex $\chi^{2}$, the $p_{\rm T}$, and the
$\chi^{2}_{\rm IP}$, which is defined as the difference in $\chi^{2}$ of a
given primary vertex reconstructed with and without the considered particle.
In addition, considering the $\pi^{+}\pi^{+}$ and $\pi^{-}\pi^{-}$ as pairs of
particles, the minimum $p_{\rm T}$, and the minimum $\chi^{2}_{\rm IP}$ of
each pair are used. The signal sample used for BDT training is based on
simulation, while the background sample uses the sideband $200-250$ MeV above
the $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ mass peak from 1/3
of the available data. The BDT is then tested on independent samples from the
same sources. The BDT selection is optimized by taking the signal, $S$, and
background, $B$, events within $\pm$20 MeV of the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ peak from the preselection and
maximizing $S^{2}/(S+B)$ by using the signal and background efficiencies
provided as a function of BDT.
The ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\pi^{+}\pi^{-}\pi^{+}\pi^{-}$
invariant mass distribution is shown in Fig. 1. Multiple combinations are at
the 6% level and a single candidate is chosen based on vertex $\chi^{2}$ and
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ mass. We fit the mass
distribution using the same signal function shape for both $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ and $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ peaks. This shape is a double
Crystal Ball function [25] with common means and radiative tail parameters
obtained from simulation. The combinatorial background is parametrized with an
exponential function. There are 1193$\pm$46 $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ and 839$\pm$39 $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ decays. Possible backgrounds
caused by particle misidentification, for example $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\pi^{+}K^{-}\pi^{+}\pi^{-}$ decays, would appear as signal if the
particle identification incorrectly assigns the $K^{-}$ as a $\pi^{-}$. In
this case the invariant mass is always below the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ signal region. Evaluating all such
backgrounds shows negligible contributions in the signal regions. These and
other low-mass backgrounds are described by a Gaussian distribution.
Figure 1: Invariant mass distribution for
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\pi^{+}\pi^{-}\pi^{+}\pi^{-}$
combinations. The data are fit with Crystal Ball functions for $\kern
1.61993pt\overline{\kern-1.61993ptB}{}^{0}$ [(red) dashed curve] and $\kern
1.61993pt\overline{\kern-1.61993ptB}{}^{0}_{s}$ [(purple) dot-dashed curve]
signals, an exponential function for combinatoric background (black) dotted,
and a Gaussian shape for lower mass background (blue) long-dashed. The total
is shown with a (blue) solid curve.
In order to improve the four-pion mass resolution we kinematically fit each
candidate with the constraints that the $\mu^{+}\mu^{-}$ be at the
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ mass and that the
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\pi^{+}\pi^{-}\pi^{+}\pi^{-}$ be
at the $\kern 1.79993pt\overline{\kern-1.79993ptB}{}$ mass. The four-pion
invariant mass distributions for $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ and $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ decays within $\pm$20 MeV of the
$\kern 1.79993pt\overline{\kern-1.79993ptB}{}$ mass peaks are shown in Fig. 2.
The backgrounds, determined from fits to the number of events in the region
$40-80$ MeV above the $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$
mass, are subtracted.
Figure 2: Background subtracted invariant mass distributions of the four pions
in (a) $\kern 1.61993pt\overline{\kern-1.61993ptB}{}^{0}_{s}$ and (b) $\kern
1.61993pt\overline{\kern-1.61993ptB}{}^{0}$ decays are shown in the histogram
overlaid with the (black) filled points with the error bars indicating the
uncertainties. The open (red) circles show the helicity $\pm$1 components of
the signals.
There are clear signals around 1285 MeV in both $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ and $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ decays with structures at higher
masses. The ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ decay angular
distribution is used to probe the spin of the recoiling four-pion system. We
examine the distribution of the helicity angle $\theta$ of the $\mu^{+}$ with
respect to the $\kern 1.79993pt\overline{\kern-1.79993ptB}{}$ direction in the
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ rest frame, after correcting
for the angular acceptance using simulation. The resulting distribution is
then fit by the sum of shapes $(1-\alpha)\sin^{2}\theta$ and
$\alpha(1+\cos^{2}\theta)/2$, where $\alpha$ is the fraction of the helicity
$\pm$1 component. For scalar four-pion states the
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ helicity is 0, while for higher
spin states it is a mixture of helicity 0 and helicity $\pm$1 components. We
also show in Fig. 2 the helicity $\pm$1 yields. In the region near 1285 MeV
there is a significant helicity $\pm$1 component, as expected if the state we
are observing is the $f_{1}(1285)$.
There is also a large and wider peak near 1450 MeV in the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ channel. Previously we
observed a structure at a mass near 1475 MeV using $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\pi^{+}\pi^{-}$ decays that we attributed to $f_{0}(1370)$ decay.
However it could equally well be the $f_{0}(1500)$ meson, an interpretation
favored by Ochs [6]. While the $f_{0}(1500)$ is known to decay into four
pions, the structure observed in our data cannot be pure spin-0 because of the
significant helicity $\pm$1 component in this mass region. We do not pursue
further the composition of the higher mass regions in either $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ or $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ decays in this Letter.
We use the measured branching fractions of $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\pi^{+}\pi^{-}$ [8] and $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\pi^{+}\pi^{-}$ [9] for normalizations. The data selection is updated
from that used in previous publications to more closely follow the procedure
in this analysis. We find signal yields of 22 476$\pm$177 $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ events and 16 016$\pm$187
$\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ events within $\pm$20 MeV
of the signal peaks. The overall efficiencies determined by simulation are
(1.411$\pm$0.015)% and (1.317$\pm$0.015)%, respectively, for $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ and $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ decays, where the uncertainty is
statistical only. The relative efficiencies for the
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\pi^{+}\pi^{-}\pi^{+}\pi^{-}$
final states with respect to ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\pi^{+}\pi^{-}$ are 14.3% and 14.5% for $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ and $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ decays, with small statistical
uncertainties. We compute the overall branching fraction ratios
$\displaystyle{\cal{B}}(\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\pi^{+}\pi^{-}\pi^{+}\pi^{-})/{\cal{B}}(\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\pi^{+}\pi^{-})=0.371\pm 0.015\pm 0.022,$ $\displaystyle{\cal{B}}(\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\pi^{+}\pi^{-}\pi^{+}\pi^{-})/{\cal{B}}(\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\pi^{+}\pi^{-})=0.361\pm 0.017\pm 0.021.$
The systematic uncertainties arise from the decay model (5.0%), background
shape (0.8%), signal shape (0.8%), simulation statistics (1.9%), and tracking
efficiencies (2.0%), resulting in a total of 5.8%.
We proceed to determine the ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}f_{1}(1285)$ yields by fitting the individual four-pion mass spectra in
both $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ and $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ final states. The $f_{1}(1285)$
state is modeled by a relativistic Breit-Wigner function multiplied by phase
space and convoluted with our mass resolution of 3 MeV. We take the mass and
width of the $f_{1}(1285)$ as 1282.1$\pm$0.6 MeV and 24.2$\pm$1.1 MeV,
respectively [1]. The combinatorial background is constrained from sideband
data and is allowed to vary by its statistical uncertainty. Backgrounds from
higher mass resonances are parameterized by Gaussian shapes whose masses and
widths are allowed to vary. We restrict the fits to the interval 1.1$-$1.5
GeV, which contains 94.3% of the signal. The fits to the data are shown in
Fig. 3. The results of the fits are listed in Table 1 along with twice the
negative change in the logarithm of the likelihood ($-2\Delta\ln L$) if fit
without the signal, and the resulting signal significance. The systematic
uncertainties from the signal shape and higher mass resonances have been
included. Both final states are seen with significance above five standard
deviations. This constitutes the first observation of the $f_{1}(1285)$ in
$b$-hadron decays. As a consistency check, we also perform a simultaneous fit
to both $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ and $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ samples letting the mass and width
vary in the fit. We find the mass and width of the $f_{1}(1285)$ to be
1284.2$\pm$2.2 MeV and 32.4$\pm$5.8 MeV, respectively, where the uncertainties
are statistical only, consistent with the known values. To determine the
systematic uncertainty in the yields we redo the fits allowing $\pm 1\sigma$
variations of the mass and width values independently. We assign $\pm$2.7% and
$\pm$2.0% for the systematic uncertainties on the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ and $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ yields, respectively, from this
source.
We obtain the branching fraction ratios, using an efficiency of
0.1820$\pm$0.0036%, determined by simulation, for the
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}f_{1}(1285)$ final state as
$\displaystyle\frac{{\cal{B}}(\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}f_{1}(1285),~{}f_{1}(1285)\rightarrow\pi^{+}\pi^{-}\pi^{+}\pi^{-})}{{\cal{B}}(\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\pi^{+}\pi^{-})}=(3.82\pm 0.52^{\,+0.29}_{\,-0.32})\%,$
$\displaystyle\frac{{\cal{B}}(\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}f_{1}(1285),~{}f_{1}(1285)\rightarrow\pi^{+}\pi^{-}\pi^{+}\pi^{-})}{{\cal{B}}(\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\pi^{+}\pi^{-})}=(2.32\pm 0.54\pm 0.11)\%,$
$\displaystyle\frac{{\cal{B}}(\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}f_{1}(1285))}{{\cal{B}}(\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}f_{1}(1285))}=(11.6\pm 3.1^{\,+0.7}_{\,-0.8})\%.$
Figure 3: Fits to the four-pion invariant mass in (a) $\kern 1.61993pt\overline{\kern-1.61993ptB}{}^{0}_{s}$ and (b) $\kern 1.61993pt\overline{\kern-1.61993ptB}{}^{0}$ decays. The data are shown as points, the signals components as (black) dashed curves, the combinatorial background by (black) dotted curves, and the higher mass resonance tail by (red) dot-dashed curves. Table 1: Fit results for $\kern 1.61993pt\overline{\kern-1.61993ptB}{}^{0}_{s}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}f_{1}(1285)$ and $\kern 1.61993pt\overline{\kern-1.61993ptB}{}^{0}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}f_{1}(1285)$ decays. | Yield | $-2\Delta\ln L$ | Significance ($\sigma$)
---|---|---|---
$\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ | $110.2\pm 15.0$ | 58.1 | 7.2
$\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ | $\,49.2\pm 11.4$ | 29.5 | 5.2
For the latter ratio we use a $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}/\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ production ratio of
0.259$\pm$0.015 [26, *Aaij:2013qqa]; this uncertainty is taken as systematic.
The other systematic uncertainties are listed in Table 2. The shape of the
high-mass tail is changed in the case of $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ decays from a single Gaussian
to two relativistic Breit-Wigner shapes corresponding to the mass and width
values of the $f_{1}(1420)$ and the $f_{0}(1500)$ mesons. For the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ high mass shape we change from a
Gaussian shape to a second order polynomial. The decay model reflects the
allowed variation in the fraction of $\rho^{0}\rho^{0}$ and
$\rho^{0}\pi^{+}\pi^{-}$ decays. The total uncertainties are ascertained by
adding the individual components in quadrature separately for the positive and
negative values.
Table 2: Systematic uncertainties of the branching fractions ${\cal{B}}(\kern 1.61993pt\overline{\kern-1.61993ptB}{}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}f_{1}(1285),~{}f_{1}(1285)\rightarrow\pi^{+}\pi^{-}\pi^{+}\pi^{-})$ and the $\kern 1.61993pt\overline{\kern-1.61993ptB}{}^{0}/\kern 1.61993pt\overline{\kern-1.61993ptB}{}^{0}_{s}$ rate ratio. The “+” and “–” signs indicate the positive and negative uncertainties, respectively. All numbers are in (%). Source | $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ | $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ | Ratio
---|---|---|---
| + | – | + | – | + | –
Mass & width of $f_{1}$ | 2.0 | 2.0 | 2.7 | 2.7 | 1.5 | 1.5
Shape of high mass | 0.6 | 0 | 0 | 3.7 | 0 | 3.8
Efficiency | 2.0 | 2.0 | 2.0 | 2.0 | 0 | 0
Tracking | 2.0 | 2.0 | 2.0 | 2.0 | 0 | 0
Simulation statistics | 2.0 | 2.0 | 2.0 | 2.0 | 0 | 0
Total | 4.0 | 4.0 | 4.4 | 5.7 | 1.5 | 4.1
Considering the $f_{1}(1285)$ as a mixed $q\bar{q}$ state, we characterize the
mixing with a 2$\times$2 rotation matrix containing a single parameter, the
angle $\phi$, so that the wave functions of the $f_{1}(1285)$ and its partner,
indicated by $f_{1}^{*}$, are given by
$\displaystyle|f_{1}(1285)\rangle$ $\displaystyle=$
$\displaystyle\cos\phi|n\bar{n}\rangle-\sin\phi|s\bar{s}\rangle,$
$\displaystyle|f_{1}^{*}\rangle$ $\displaystyle=$
$\displaystyle\sin\phi|n\bar{n}\rangle+\cos\phi|s\bar{s}\rangle,$
$\displaystyle{\rm where~{}}|n\bar{n}\rangle$ $\displaystyle\equiv$
$\displaystyle\frac{1}{\sqrt{2}}\left(|u\bar{u}\rangle+|d\bar{d}\rangle\right).$
(1)
The decay widths can be written as [12]
$\displaystyle\Gamma(\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}f_{1}(1285))$ $\displaystyle=$ $\displaystyle
0.5|A_{0}|^{2}|V_{cd}|^{2}\Phi_{0}\cos^{2}\phi,$ $\displaystyle\Gamma(\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}f_{1}(1285))$ $\displaystyle=$
$\displaystyle|A_{s}|^{2}|V_{cs}|^{2}\Phi_{s}\sin^{2}\phi,$ (2)
where $A_{i}$ is the tree level amplitude, $V_{cd}$ and $V_{cs}$ are quark
mixing matrix elements, and $\Phi_{i}$ are phase space factors. The amplitude
ratio $|A_{0}|/|A_{s}|$ is taken as unity [12]. The width ratio is given by
$\frac{{\cal{B}}(\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}f_{1}(1285))}{{\cal{B}}(\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}f_{1}(1285))}=\frac{\tau_{0}}{2\tau_{s}}\frac{|V_{cd}|^{2}\Phi_{0}\cos^{2}\phi}{|V_{cs}|^{2}\Phi_{s}\sin^{2}\phi},$
(3)
where $\tau_{s}$ is the $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$
lifetime and $\tau_{0}$ is the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ lifetime. The angle $\phi$ is then
given by
$\tan^{2}\phi=\frac{1}{2}\frac{{\cal{B}}(\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}f_{1}(1285))}{{\cal{B}}(\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}f_{1}(1285))}\frac{\tau_{0}}{\tau_{s}}\frac{|V_{cd}|^{2}}{|V_{cs}|^{2}}\frac{\Phi_{0}}{\Phi_{s}}=0.1970\pm
0.053^{\,+0.014}_{\,-0.012}.$ (4)
The ratio of the phase space factors $\Phi_{0}/\Phi_{s}$ equals 0.855. The
other input values are $\tau_{s}=1.508$ ps [28], $\tau_{0}=1.519$ ps,
$|V_{cd}|=0.2245$, and $|V_{cs}|=0.97345$ [1]. We use the lifetime measured in
$\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\phi$ decays as the helicity components are in approximately the same
ratio as in ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}f_{1}(1285)$. No
uncertainties are assigned on these quantities as they are much smaller than
the other errors. The resulting mixing angle is
$\phi=\pm(24.0^{\,+3.1\,+0.6}_{\,-2.6\,-0.8})^{\circ}.$
The systematic uncertainty is computed from the systematic errors assigned to
the branching fractions.
The $f_{1}(1285)$ mixing angle has been estimated assuming that it is mixed
with the $f_{1}(1420)$ state. Yang finds
$\phi=\pm(15.8^{\,+4.5}_{\,-4.6})^{\circ}$ using radiative decays [29],
consistent with an earlier determination of
$\pm(15^{\,+\;\;5}_{\,-10})^{\circ}$ [30]. A lattice QCD analysis gives
$(31\pm 2)^{\circ}$, while an another phenomenological calculation gives a
range between $(20-30)^{\circ}$ [31, *Dudek:2013yja, *Close:1997nm]; see also
Ref. [33, *Cheng:2011pb] for other theoretical predictions. In this analysis
we do not specify the other mixed partner.
If the $f_{1}(1285)$ is a tetraquark state its wave function would be
$|f_{1}\rangle=\frac{1}{\sqrt{2}}\left([su][\bar{s}\bar{u}]+[sd][\bar{s}\bar{d}]\right)$
in order for it to be produced significantly in both $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ and $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ decays into
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}f_{1}(1285)$ decays. Using this
wave function, the tetraquark model described in Ref. [12] predicts
$\frac{{\cal{B}}(\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}f_{1}(1285))}{{\cal{B}}(\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}f_{1}(1285))}=\frac{1}{4}\frac{\tau_{0}}{\tau_{s}}\frac{|V_{cd}|^{2}\Phi_{0}}{|V_{cs}|^{2}\Phi_{s}}=1.14\%,$
(5)
with small uncertainties. Our measurement of this ratio of $(11.6\pm
3.1^{\,+0.7}_{\,-0.8})$% differs by 3.3 standard deviations from the
tetraquark interpretation including the systematic uncertainty.
Branching fraction ratios are converted into branching fractions using the
previously measured rates listed in Table 3. We correct the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ rates to reflect the updated
value of the $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ to $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ production fraction of
0.259$\pm$0.015 [26, *Aaij:2013qqa]. We determine
$\displaystyle{\cal{B}}(\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\pi^{+}\pi^{-}\pi^{+}\pi^{-})=(7.62\pm 0.36\pm 0.64\pm 0.42)\times
10^{-5},$ $\displaystyle{\cal{B}}(\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\pi^{+}\pi^{-}\pi^{+}\pi^{-})=(1.43\pm 0.08\pm 0.09\pm 0.06)\times
10^{-5}.$
where the first uncertainty is statistical, the second and third are
systematic, being due to the relative branching fraction measurements and the
errors in the absolute branching fraction normalization, respectively. For the
$\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ decay this
normalization error is due to the uncertainty on the production ratio of
$\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ versus $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ and is 5.8% [9]. For the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ mode the uncertainty is due to the
error of 4.1% on
${{\cal{B}}(B^{-}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{-})}$ [10].
Table 3: Branching fractions used for normalization. Rate | Value | Ref.
---|---|---
$\frac{{\cal{B}}(\kern 1.25995pt\overline{\kern-1.25995ptB}{}^{0}_{s}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\pi^{+}\pi^{-})}{{\cal{B}}(\kern 1.25995pt\overline{\kern-1.25995ptB}{}^{0}_{s}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\phi)}$ | $(19.79\pm 0.47\pm 0.52)$% | [8]
${\cal{B}}(\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\pi^{+}\pi^{-})$ | $(3.97\pm 0.09\pm 0.11\pm 0.16)\times 10^{-5}$ | [9]
${{\cal{B}}(\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\phi)}$ | $(10.50\pm 0.13\pm 0.64\pm 0.82)\times 10^{-4}$ | [10]
${{\cal{B}}(B^{-}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{-})}$ | $(10.18\pm 0.42)\times 10^{-4}$ | [10]
In conclusion, we report the first observations of $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ and $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}f_{1}(1285)$ decays. These are also the first observations of the
$f_{1}(1285)$ meson in heavy quark decays. We determine
$\displaystyle{\cal{B}}(\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}f_{1}(1285),~{}f_{1}(1285)\rightarrow\pi^{+}\pi^{-}\pi^{+}\pi^{-})=(7.85\pm
1.09^{\,+0.76}_{\,-0.90}\pm 0.46)\times 10^{-6},$
$\displaystyle{\cal{B}}(\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}f_{1}(1285),~{}f_{1}(1285)\rightarrow\pi^{+}\pi^{-}\pi^{+}\pi^{-})=(9.21\pm
2.14\pm 0.52\pm 0.38)\times 10^{-7},$ $\displaystyle{\cal{B}}(\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}f_{1}(1285))=(7.14\pm 0.99^{\,+0.83}_{\,-0.91}\pm 0.41)\times 10^{-5},$
$\displaystyle{\cal{B}}(\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}f_{1}(1285))=(8.37\pm 1.95^{\,+0.71}_{\,-0.66}\pm 0.35)\times 10^{-6},$
where we use the known branching fraction
${\cal{B}}(f_{1}(1285)\rightarrow\pi^{+}\pi^{-}\pi^{+}\pi^{-})=(11.0^{\,+0.7}_{\,-0.6})$%
[1]. Investigation of $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$
and $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ decays into
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\pi^{+}\pi^{-}\pi^{+}\pi^{-}$ has
revealed the presence of the ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}f_{1}(1285)$ state in both decay channels. This allows determination of
the $f_{1}(1285)$ mixing angle to be
$\pm(24.0^{\,+3.1+0.6}_{\,-2.6\,-0.8})^{\circ}$, even though the mixing
companion of this state is not detected. According to Ref. [12], our measured
value disfavors the interpretation of the $f_{1}(1285)$ as a tetraquark state.
We express our gratitude to our colleagues in the CERN accelerator departments
for the excellent performance of the LHC. We thank the technical and
administrative staff at the LHCb institutes. We acknowledge support from CERN
and from the national agencies: CAPES, CNPq, FAPERJ and FINEP (Brazil); NSFC
(China); CNRS/IN2P3 and Region Auvergne (France); BMBF, DFG, HGF and MPG
(Germany); SFI (Ireland); INFN (Italy); FOM and NWO (The Netherlands); SCSR
(Poland); MEN/IFA (Romania); MinES, Rosatom, RFBR and NRC “Kurchatov
Institute” (Russia); MinECo, XuntaGal and GENCAT (Spain); SNSF and SER
(Switzerland); NAS Ukraine (Ukraine); STFC (United Kingdom); NSF (USA). We
also acknowledge the support received from the ERC under FP7. The Tier1
computing centres are supported by IN2P3 (France), KIT and BMBF (Germany),
INFN (Italy), NWO and SURF (The Netherlands), PIC (Spain), GridPP (United
Kingdom). We are thankful for the computing resources put at our disposal by
Yandex LLC (Russia), as well as to the communities behind the multiple open
source software packages that we depend on.
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2024-09-04T02:49:52.121204
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "LHCb collaboration: R. Aaij, B. Adeva, M. Adinolfi, C. Adrover, A.\n Affolder, Z. Ajaltouni, J. Albrecht, F. Alessio, M. Alexander, S. Ali, G.\n Alkhazov, P. Alvarez Cartelle, A.A. Alves Jr, S. Amato, S. Amerio, Y. Amhis,\n L. Anderlini, J. Anderson, R. Andreassen, M. Andreotti, J.E. Andrews, R.B.\n Appleby, O. Aquines Gutierrez, F. Archilli, A. Artamonov, M. Artuso, E.\n Aslanides, G. Auriemma, M. Baalouch, S. Bachmann, J.J. Back, A. Badalov, C.\n Baesso, V. Balagura, W. Baldini, R.J. Barlow, C. Barschel, S. Barsuk, W.\n Barter, V. Batozskaya, Th. Bauer, A. Bay, J. Beddow, F. Bedeschi, I. Bediaga,\n S. Belogurov, K. Belous, I. Belyaev, E. Ben-Haim, G. Bencivenni, S. Benson,\n J. Benton, A. Berezhnoy, R. Bernet, M.-O. Bettler, M. van Beuzekom, A. Bien,\n S. Bifani, T. Bird, A. Bizzeti, P.M. Bj{\\o}rnstad, T. Blake, F. Blanc, J.\n Blouw, S. Blusk, V. Bocci, A. Bondar, N. Bondar, W. Bonivento, S. Borghi, A.\n Borgia, T.J.V. Bowcock, E. Bowen, C. Bozzi, T. Brambach, J. van den Brand, J.\n Bressieux, D. Brett, M. Britsch, T. Britton, N.H. Brook, H. Brown, A.\n Bursche, G. Busetto, J. Buytaert, S. Cadeddu, R. Calabrese, O. Callot, M.\n Calvi, M. Calvo Gomez, A. Camboni, P. Campana, D. Campora Perez, A. Carbone,\n G. Carboni, R. Cardinale, A. Cardini, H. Carranza-Mejia, L. Carson, K.\n Carvalho Akiba, G. Casse, L. Castillo Garcia, M. Cattaneo, Ch. Cauet, R.\n Cenci, M. Charles, Ph. Charpentier, S.-F. Cheung, N. Chiapolini, M.\n Chrzaszcz, K. Ciba, X. Cid Vidal, G. Ciezarek, P.E.L. Clarke, M. Clemencic,\n H.V. Cliff, J. Closier, C. Coca, V. Coco, J. Cogan, E. Cogneras, P. Collins,\n A. Comerma-Montells, A. Contu, A. Cook, M. Coombes, S. Coquereau, G. Corti,\n B. Couturier, G.A. Cowan, D.C. Craik, M. Cruz Torres, S. Cunliffe, R. Currie,\n C. D'Ambrosio, P. David, P.N.Y. David, A. Davis, I. De Bonis, K. De Bruyn, S.\n De Capua, M. De Cian, J.M. De Miranda, L. De Paula, W. De Silva, P. De\n Simone, D. Decamp, M. Deckenhoff, L. Del Buono, N. D\\'el\\'eage, D. Derkach,\n O. Deschamps, F. Dettori, A. Di Canto, H. Dijkstra, M. Dogaru, S. Donleavy,\n F. Dordei, A. Dosil Su\\'arez, D. Dossett, A. Dovbnya, F. Dupertuis, P.\n Durante, R. Dzhelyadin, A. Dziurda, A. Dzyuba, S. Easo, U. Egede, V.\n Egorychev, S. Eidelman, D. van Eijk, S. Eisenhardt, U. Eitschberger, R.\n Ekelhof, L. Eklund, I. El Rifai, Ch. Elsasser, A. Falabella, C. F\\\"arber, C.\n Farinelli, S. Farry, D. Ferguson, V. Fernandez Albor, F. Ferreira Rodrigues,\n M. Ferro-Luzzi, S. Filippov, M. Fiore, M. Fiorini, C. Fitzpatrick, M.\n Fontana, F. Fontanelli, R. Forty, O. Francisco, M. Frank, C. Frei, M.\n Frosini, E. Furfaro, A. Gallas Torreira, D. Galli, M. Gandelman, P. Gandini,\n Y. Gao, J. Garofoli, P. Garosi, J. Garra Tico, L. Garrido, C. Gaspar, R.\n Gauld, E. Gersabeck, M. Gersabeck, T. Gershon, Ph. Ghez, V. Gibson, L.\n Giubega, V.V. Gligorov, C. G\\\"obel, D. Golubkov, A. Golutvin, A. Gomes, P.\n Gorbounov, H. Gordon, M. Grabalosa G\\'andara, R. Graciani Diaz, L.A. Granado\n Cardoso, E. Graug\\'es, G. Graziani, A. Grecu, E. Greening, S. Gregson, P.\n Griffith, L. Grillo, O. Gr\\\"unberg, B. Gui, E. Gushchin, Yu. Guz, T. Gys, C.\n Hadjivasiliou, G. Haefeli, C. Haen, T.W. Hafkenscheid, S.C. Haines, S. Hall,\n B. Hamilton, T. Hampson, S. Hansmann-Menzemer, N. Harnew, S.T. Harnew, J.\n Harrison, T. Hartmann, J. He, T. Head, V. Heijne, K. Hennessy, P. Henrard,\n J.A. Hernando Morata, E. van Herwijnen, M. He\\ss, A. Hicheur, E. Hicks, D.\n Hill, M. Hoballah, C. Hombach, W. Hulsbergen, P. Hunt, T. Huse, N. Hussain,\n D. Hutchcroft, D. Hynds, V. Iakovenko, M. Idzik, P. Ilten, R. Jacobsson, A.\n Jaeger, E. Jans, P. Jaton, A. Jawahery, F. Jing, M. John, D. Johnson, C.R.\n Jones, C. Joram, B. Jost, M. Kaballo, S. Kandybei, W. Kanso, M. Karacson,\n T.M. Karbach, I.R. Kenyon, T. Ketel, B. Khanji, O. Kochebina, I. Komarov,\n R.F. Koopman, P. Koppenburg, M. Korolev, A. Kozlinskiy, L. Kravchuk, K.\n Kreplin, M. Kreps, G. Krocker, P. Krokovny, F. Kruse, M. Kucharczyk, V.\n Kudryavtsev, K. Kurek, T. Kvaratskheliya, V.N. La Thi, D. Lacarrere, G.\n Lafferty, A. Lai, D. Lambert, R.W. Lambert, E. Lanciotti, G. Lanfranchi, C.\n Langenbruch, T. Latham, C. Lazzeroni, R. Le Gac, J. van Leerdam, J.-P. Lees,\n R. Lef\\`evre, A. Leflat, J. Lefran\\c{c}ois, S. Leo, O. Leroy, T. Lesiak, B.\n Leverington, Y. Li, L. Li Gioi, M. Liles, R. Lindner, C. Linn, B. Liu, G.\n Liu, S. Lohn, I. Longstaff, J.H. Lopes, N. Lopez-March, H. Lu, D. Lucchesi,\n J. Luisier, H. Luo, E. Luppi, O. Lupton, F. Machefert, I.V. Machikhiliyan, F.\n Maciuc, O. Maev, S. Malde, G. Manca, G. Mancinelli, J. Maratas, U. Marconi,\n P. Marino, R. M\\\"arki, J. Marks, G. Martellotti, A. Martens, A. Mart\\'in\n S\\'anchez, M. Martinelli, D. Martinez Santos, D. Martins Tostes, A. Martynov,\n A. Massafferri, R. Matev, Z. Mathe, C. Matteuzzi, E. Maurice, A. Mazurov, M.\n McCann, J. McCarthy, A. McNab, R. McNulty, B. McSkelly, B. Meadows, F. Meier,\n M. Meissner, M. Merk, D.A. Milanes, M.-N. Minard, J. Molina Rodriguez, S.\n Monteil, D. Moran, P. Morawski, A. Mord\\`a, M.J. Morello, R. Mountain, I.\n Mous, F. Muheim, K. M\\\"uller, R. Muresan, B. Muryn, B. Muster, P. Naik, T.\n Nakada, R. Nandakumar, I. Nasteva, M. Needham, S. Neubert, N. Neufeld, A.D.\n Nguyen, T.D. Nguyen, C. Nguyen-Mau, M. Nicol, V. Niess, R. Niet, N. Nikitin,\n T. Nikodem, A. Nomerotski, A. Novoselov, A. Oblakowska-Mucha, V. Obraztsov,\n S. Oggero, S. Ogilvy, O. Okhrimenko, R. Oldeman, G. Onderwater, M. Orlandea,\n J.M. Otalora Goicochea, P. Owen, A. Oyanguren, B.K. Pal, A. Palano, M.\n Palutan, J. Panman, A. Papanestis, M. Pappagallo, C. Parkes, C.J. Parkinson,\n G. Passaleva, G.D. Patel, M. Patel, G.N. Patrick, C. Patrignani, C.\n Pavel-Nicorescu, A. Pazos Alvarez, A. Pearce, A. Pellegrino, G. Penso, M.\n Pepe Altarelli, S. Perazzini, E. Perez Trigo, A. P\\'erez-Calero Yzquierdo, P.\n Perret, M. Perrin-Terrin, L. Pescatore, E. Pesen, G. Pessina, K. Petridis, A.\n Petrolini, A. Phan, E. Picatoste Olloqui, B. Pietrzyk, T. Pila\\v{r}, D.\n Pinci, S. Playfer, M. Plo Casasus, F. Polci, G. Polok, A. Poluektov, E.\n Polycarpo, A. Popov, D. Popov, B. Popovici, C. Potterat, A. Powell, J.\n Prisciandaro, A. Pritchard, C. Prouve, V. Pugatch, A. Puig Navarro, G. Punzi,\n W. Qian, B. Rachwal, J.H. Rademacker, B. Rakotomiaramanana, M.S. Rangel, I.\n Raniuk, N. Rauschmayr, G. Raven, S. Redford, S. Reichert, M.M. Reid, A.C. dos\n Reis, S. Ricciardi, A. Richards, K. Rinnert, V. Rives Molina, D.A. Roa\n Romero, P. Robbe, D.A. Roberts, A.B. Rodrigues, E. Rodrigues, P. Rodriguez\n Perez, S. Roiser, V. Romanovsky, A. Romero Vidal, M. Rotondo, J. Rouvinet, T.\n Ruf, F. Ruffini, H. Ruiz, P. Ruiz Valls, G. Sabatino, J.J. Saborido Silva, N.\n Sagidova, P. Sail, B. Saitta, V. Salustino Guimaraes, B. Sanmartin Sedes, R.\n Santacesaria, C. Santamarina Rios, E. Santovetti, M. Sapunov, A. Sarti, C.\n Satriano, A. Satta, M. Savrie, D. Savrina, M. Schiller, H. Schindler, M.\n Schlupp, M. Schmelling, B. Schmidt, O. Schneider, A. Schopper, M.-H. Schune,\n R. Schwemmer, B. Sciascia, A. Sciubba, M. Seco, A. Semennikov, K.\n Senderowska, I. Sepp, N. Serra, J. Serrano, P. Seyfert, M. Shapkin, I.\n Shapoval, Y. Shcheglov, T. Shears, L. Shekhtman, O. Shevchenko, V.\n Shevchenko, A. Shires, R. Silva Coutinho, M. Sirendi, N. Skidmore, T.\n Skwarnicki, N.A. Smith, E. Smith, E. Smith, J. Smith, M. Smith, M.D.\n Sokoloff, F.J.P. Soler, F. Soomro, D. Souza, B. Souza De Paula, B. Spaan, A.\n Sparkes, P. Spradlin, F. Stagni, S. Stahl, O. Steinkamp, S. Stevenson, S.\n Stoica, S. Stone, B. Storaci, M. Straticiuc, U. Straumann, V.K. Subbiah, L.\n Sun, W. Sutcliffe, S. Swientek, V. Syropoulos, M. Szczekowski, P. Szczypka,\n D. Szilard, T. Szumlak, S. T'Jampens, M. Teklishyn, G. Tellarini, E.\n Teodorescu, F. Teubert, C. Thomas, E. Thomas, J. van Tilburg, V. Tisserand,\n M. Tobin, S. Tolk, L. Tomassetti, D. Tonelli, S. Topp-Joergensen, N. Torr, E.\n Tournefier, S. Tourneur, M.T. Tran, M. Tresch, A. Tsaregorodtsev, P.\n Tsopelas, N. Tuning, M. Ubeda Garcia, A. Ukleja, A. Ustyuzhanin, U. Uwer, V.\n Vagnoni, G. Valenti, A. Vallier, R. Vazquez Gomez, P. Vazquez Regueiro, C.\n V\\'azquez Sierra, S. Vecchi, J.J. Velthuis, M. Veltri, G. Veneziano, M.\n Vesterinen, B. Viaud, D. Vieira, X. Vilasis-Cardona, A. Vollhardt, D.\n Volyanskyy, D. Voong, A. Vorobyev, V. Vorobyev, C. Vo\\ss, H. Voss, R. Waldi,\n C. Wallace, R. Wallace, S. Wandernoth, J. Wang, D.R. Ward, N.K. Watson, A.D.\n Webber, D. Websdale, M. Whitehead, J. Wicht, J. Wiechczynski, D. Wiedner, L.\n Wiggers, G. Wilkinson, M.P. Williams, M. Williams, F.F. Wilson, J. Wimberley,\n J. Wishahi, W. Wislicki, M. Witek, G. Wormser, S.A. Wotton, S. Wright, S. Wu,\n K. Wyllie, Y. Xie, Z. Xing, Z. Yang, X. Yuan, O. Yushchenko, M. Zangoli, M.\n Zavertyaev, F. Zhang, L. Zhang, W.C. Zhang, Y. Zhang, A. Zhelezov, A.\n Zhokhov, L. Zhong, A. Zvyagin",
"submitter": "Sheldon Stone",
"url": "https://arxiv.org/abs/1310.2145"
}
|
1310.2353
|
# A note on the $3$-rainbow index of $K_{2,t}$
Tingting Liu, Yumei Hu 111supported by NSFC No. 11001196.
Department of Mathematics, Tianjin University, Tianjin 300072, P. R. China
E-mails: [email protected]; [email protected];
###### Abstract
A tree $T$, in an edge-colored graph $G$, is called a rainbow tree if no two
edges of $T$ are assigned the same color. For a vertex subset $S\in V(G)$, a
tree that connects $S$ in $G$ is called an $S$-tree. A $k$-rainbow coloring
of $G$ is an edge coloring of $G$ having the property that for every set $S$
of $k$ vertices of $G$, there exists a rainbow $S$-tree $T$ in $G$. The
minimum number of colors needed in a $k$-rainbow coloring of $G$ is the
$k$-rainbow index of $G$, denoted by $rx_{k}(G)$. In this paper, we obtain the
exact values of $rx_{3}(K_{2,t})$ for any $t\geq 1$.
Keywords
: edge-coloring, $k$-rainbow index, rainbow tree, complete bipartite graph.
## 1 Introduction
All graphs considered in this paper are simple, finite and undirected. We
follow the terminology and notation of Bondy and Murty [1]. Let $G$ be a
nontrivial connected graph of order $n$ on which is defined an edge coloring,
where adjacent edges may be the same color. A path $P$ is a rainbow path if no
two edges of $P$ are colored the same. The graph $G$ is rainbow connected if
$G$ contains a $u-v$ rainbow path for every pair $u,v$ of distinct vertices of
$G$. The minimum number of colors that results in a rainbow connected graph
$G$ is the rainbow connection number $rc(G)$ of $G$. These concepts were
introduced by Chartrand et al. in [2].
Another generalization of rainbow connection number was also introduced by
Chartrand et al. [3]. A tree $T$ is a rainbow tree if no two edges of $T$ are
colored the same. For a vertex subset $S\in V(G)$, a tree that connects $S$ in
$G$ is called an $S$-tree. Let $k$ be a fixed integer with $2\leq k\leq n$. An
edge coloring of $G$ is called a k-rainbow coloring if for every set $S$ of
$k$ vertices of $G$, there exists a rainbow $S$-tree. The k-rainbow index
$rx_{k}(G)$ of $G$ is the minimum number of colors needed in a $k$-rainbow
coloring of $G$. It is obvious that $rc(G)=rx_{2}(G)$. It follows, for every
nontrivial connected graph $G$ of order $n$, that
$rx_{2}(G)\leq rx_{3}(G)\leq\cdots\leq rx_{k}(G).$
Chakraborty et al. [4] showed that computing the rainbow connection number of
a graph is NP-hard. Thus, it is more difficult to compute $k$-rainbow index of
general graphs.
For complete bipartite graph, Chartrand et al. [2] obtained
$rc(K_{s,t})=min\\{\sqrt[s]{t},4\\}$, for integers s and t with $2\leq s\leq
t$. More results on the rainbow connection number can be found in the survey
[5]. For $3$-rainbow index, Li et al. [6] obtained the exact value of regular
complete bipartite $K_{r,r}$, $rx_{3}(K_{r,r})=3$, with $r\geq 3$.
In [7], we showed, for any integers s and t with $3\leq s\leq t$,
$rx_{3}(K_{s,t})\leq min\\{6,s+t-3\\}$, and the bound is tight. But this bound
can not be generalized to the graph $K_{2,t}$. So in the paper, we derive the
exact value of $rx_{3}(K_{2,t})$ for different $t(t\geq 1)$. We get the
following theorem.
###### Theorem 1.
For any integer $t\geq 1$,
$rx_{3}(K_{2,t})=\left\\{\begin{array}[]{lll}2,&\mbox{ if $t=1,2$;}\\\
3,&\mbox{ if $t=3,4$;}\\\ 4,&\mbox{ if $5\leq t\leq 8$;}\\\ 5,&\mbox{ if
$9\leq t\leq 20$;}\\\ k,&\mbox{ if $(k-1)(k-2)+1\leq t\leq k(k-1)$,~{}($k\geq
6$);}\\\ \end{array}\right.$
## 2 Proof of Theorem 1
In this section, we determine the $3$-rainbow index of complete bipartite
graphs $K_{2,t}$. First of all, we need some new techniques and notions.
Let $U$ and $W$ be the two partite sets of $K_{2,t}$, where
$U=\\{u_{1},~{}u_{2}\\},W=\\{w_{1},~{}w_{2},~{}\cdots,w_{t}\\}$. Suppose that
there exists a $3$-rainbow coloring $c$ : $E(K_{2,t})$ $\longrightarrow$
$\\{1,2,\cdots,k\\}$. Corresponding to the $3$-rainbow coloring, there is a
color code($w$) assigned to every vertex $w\in W$, consisting of an ordered
$2$-tuple ($a_{1},~{}a_{2}$), where $a_{i}=c(u_{i}w)\in\\{1,2,\cdots,k\\}$ for
$i=1,2$. In turn, for a subset $Y$ of $W$, given color codes of vertices in
$Y$ are acceptable if the corresponding coloring is $3$-rainbow coloring of
the graph induced by $Y\cup U$. Let $B$ be a set of colors. Color codes are
$B$-limited if both colors in every color code, but not necessarily distinct,
are from $B$. The maximum number of color codes which are not only $B$-limited
but also acceptable is denoted by $\beta_{B}$.
Note that we adopt the following thought in the proof: we first give a certain
$B$ with $k$ colors, then we consider the the maximum number of color codes
which are not only $B$-limited but also acceptable, the number is the tight
upper bound of $t$ with $rx_{3}(K_{2,t})=k$.
The following claims are easy to verify and will be used later.
###### Claim 1.
If $|B|$=1, then $\beta_{B}\leq 1$.
###### Claim 2.
If $|B|$=2, then $\beta_{B}\leq 2$.
###### Proof.
By contradiction. We may assume $\beta_{B}\geq 3$. For three vertices in $W$,
we can find a rainbow tree containing them. We know the rainbow tree
containing them uses at least an edge adjacent with every vertex of them, thus
the tree uses at least three edges whose coloring are from $B$. Since the
color codes are $B$-limited and $|B|$=2, the tree is not a rainbow tree, a
contradiction. ∎
###### Lemma 2.1.
For $t=1,2,$ $rx_{3}(K_{2,t})=2$, and $rx_{3}(K_{2,t})\geq 3$ for $t\geq 3$.
###### Proof.
Since $K_{2,1}$ is a tree, $rx_{3}(K_{2,1})=2$. For $t=2$,
$rx_{3}(K_{2,2})=rx_{3}(C_{4})=2$. From the Claim 2, we get if $t\geq 3$, then
$rx_{3}(K_{2,t})\geq 3$. ∎
The following lemma reminds us how to construct color codes to some extent and
is useful to show that an edge coloring is $3$-rainbow coloring by character
of color codes.
###### Lemma 2.2.
Let $c$ be an edge coloring of $K_{2,t}$ with $rx_{3}(K_{2,t})=k$ and
$S=\\{v_{1},v_{2},v_{3}\\}$ be any a set of three vertices in $K_{2,t}$. We
have the following.
$(1)$ $|S\cap W|=3$. When $k=3$, there is a rainbow $S$-tree if and only if
there exists $i\in\\{1,2\\}$ such that $c(u_{i}v_{j})$ are distinct
($j=1,2,3$); when $k\geq 4$, if there are at least $4$ colors used by the
color codes of three vertices, then there is a rainbow $S$-tree.
$(2)$ $|S\cap W|=2$. If both $i$-th ($i=1,2$) elements of two color codes are
distinct or at least three colors are used, then there is a rainbow $S$-tree.
$(3)$ $|S\cap W|=1$. If $a_{1}\neq a_{2}$ for any color code $(a_{1},a_{2})$,
then there is a rainbow $S$-tree.
###### Proof.
For $(1)$, firstly, when $k=3$, if there exists $i\in\\{1,2\\}$ such that
$c(u_{i}v_{j})$ are distinct ($j=1,2,3$), then we find a rainbow $S$-tree
$T=\\{u_{i}v_{1},~{}u_{i}v_{2},~{}u_{i}v_{3}\\}$. And if there exists no
$i\in\\{1,2\\}$ satisfying above condition, then we need to add at least two
other vertices to obtain the rainbow $S$-tree, which implies there are at
least four edges in rainbow $S$-tree. It contradicts that $k=3$. Secondly, for
$k\geq 4$, let code($v_{1}$)=$(c(v_{1}u_{1}),~{}c(v_{1}u_{2}))$,
code($v_{2}$)=$(c(v_{2}u_{1}),~{}c(v_{2}u_{2}))$,
code($v_{3}$)=$(c(v_{3}u_{1}),~{}c(v_{3}u_{2}))$. If there exists
$i\in\\{1,2\\}$ such that $c(u_{i}v_{j})$ are distinct ($j=1,2,3$), the
conclusion clearly holds. And if not, without loss of generality, assume
$c(v_{1}u_{1})=c(v_{2}u_{1})\neq c(v_{3}u_{1})$, then we can find a rainbow
$S$-tree $T=\\{v_{1}u_{2},v_{2}u_{1},v_{3}u_{1},v_{3}u_{2}\\}$ or
$T=\\{v_{1}u_{1},v_{2}u_{2},v_{3}u_{1},v_{3}u_{2}\\}$.
For $(2)$, suppose that $v_{1}=u_{1}\in U,~{}v_{2}=w_{1}\in
W,~{}v_{3}=w_{2}\in W$. we can easily find a rainbow $S$-tree
$T=\\{u_{1}w_{1},~{}u_{1}w_{2}\\}$ with length $2$ or
$T=\\{u_{1}w_{1},~{}w_{1}u_{2},~{}u_{2}w_{2}\\}$ with length $3$.
For $(3)$, suppose that $v_{1}=u_{1}\in U,~{}v_{2}=u_{2}\in
U,~{}v_{3}=w_{1}\in W$. Then the tree $T=\\{u_{1}w_{1},~{}w_{1}u_{2}\\}$ is a
rainbow tree containing $S$. ∎
###### Lemma 2.3.
For $t=3,4,$ $rx_{3}(K_{2,t})=3$, and $rx_{3}(K_{2,t})\geq 4$ for $t\geq 5$.
###### Proof.
First, we show the latter of conclusion that $rx_{3}(k_{2,t})\geq 4$ for
$t\geq 5$. By contradiction. We assume there exists $t\geq 5$ such that
$rx_{3}(k_{2,t})=3$ by Lemma 2.1.
From Lemma 2.2 (1) and (2), if $rx_{3}(K_{2,t})=3$, then for any three color
codes: code($w_{1}$), code($w_{2}$), code($w_{3}$), there exists
$i\in\\{1,2\\}$ such that $c(w_{1}u_{i}),~{}c(w_{2}u_{i}),~{}c(w_{3}u_{i})$
are different. Moreover, there is no same color code in this case.
Now we try to connect the problem to the game of chess. The only fact needed
about the game is that rooks are isolate if and only if any three of them lie
in the different rows or the different columns of the chessboard. We give each
square on the board a pair ($i,j$) of coordinates. The integer $i$ designates
the row number of the square and the integer $j$ designates the column number
of the square, where $i$ and $j$ are integers between $1$ and $3$. Our concern
is the maximum number of rooks which are isolate on the chess since it is the
upper bound of $t$ with $rx_{3}(K_{2,t})=3$. We consider the condition from
two factors:
(a) if the rooks lie in different rows and columns.
(b) if two of them lie in the same rows or columns.
It is easy to verify that at most $4$ rooks are isolate, such as
$(1,2),~{}(2,1),~{}(1,3),~{}(3,1)$ shown in Figure 1, a contradiction. Thus,
the conclusion holds.
Second, we give the vertices of $K_{2,3}$ any three color codes shown in
Figure 1 (b) and give the vertices of $K_{2,4}$ all color codes shown in
Figure 1 (b). It is easy to check that corresponding coloring is a $3$-rainbow
coloring. So $rx_{3}(K_{2,3})=rx_{3}(K_{2,4})=3$.
Figure 1: An example of (a) and (b) used in Lemma 2.3.
∎
From the proof of the Lemma 2.3, the following claim is easily obtained.
###### Claim 3.
If $|B|=3$, then $\beta_{B}=4$.
###### Lemma 2.4.
For $5\leq t\leq 8$, $rx_{3}(K_{2,t})=4$, and $rx_{3}(K_{2,t})\geq 5$ for
$t\geq 9$.
###### Proof.
Similarly, we first prove the latter of the lemma. By contradiction. we may
assume that there exists $t\geq 9$ such that $rx_{3}(k_{2,t})=4$. It follows
that $\beta_{B}\geq 9$.
Figure 2: The graph used in Lemma 2.4.
Let $B=\\{1,2,3,4\\}$ be a set of 4 colors. Let
$B_{1}=\\{1,2,3\\},~{}B_{2}=\\{1,2,4\\},~{}B_{3}=\\{1,3,4\\},~{}B_{4}=\\{2,3,4\\}$.
Then $|B_{i}|=3$, so $\beta_{B_{i}}=4~{}(i=1,2,3,4$). Since $B$ is the union
of four $B_{i}~{}(i=1,2,3,4)$, thus $\beta_{B}\leq 16$. And we find that a
color code is limited in at least two $B_{i}~{}(i=1,2,3,4)$. So we get
$\beta_{B}\leq 8$, a contradiction.
Then, we will get eight color codes such that the corresponding coloring is
$3$-rainbow coloring. We can seek eight rooks on the $4$-by-$4$ board, shown
in Figure $2$. By the Lemma 2.2, for any $t$ ($5\leq t\leq 8$) rooks in Figure
$2$, we can find a $3$-rainbow coloring of $K_{2,t}$. Thus
$rx_{3}(K_{2,t})=4$, ($5\leq t\leq 8$). ∎
###### Lemma 2.5.
For $9\leq t\leq 20$, $rx_{3}(K_{2,t})=5$, and $rx_{3}(K_{2,t})\geq 6$ for
$t\geq 21$.
###### Proof.
From the Claim $2$, we know $t\leq C_{5}^{2}\times 2=5\times 4=20$, if
$rx_{3}(k_{2,t})=5$. That is, $rx_{3}(k_{2,t})\geq 6$ for $t\geq 21$.
Next, we give $t$ vertices $t$ color codes $(9\leq t\leq 20)$ and the
corresponding coloring is $3$-rainbow coloring. When $9\leq t\leq 10$, we just
give $t$ vertices the first $t$ codes successively: (1,2), (2,3), (3,4),
(4,5), (3,1), (4,2), (5,3), (1,4), (2,5), (5,1) (see Figure 3). When $11\leq
t\leq 20$, we choose randomly $t-10$ color codes from the remaining color
codes in Figure 4(a) to give the $t-10$ vertices.
Then, it remains to show the coloring is $3$-rainbow coloring. Let $S$ be a
set of three vertices. By the Lemma 2.2, we can find a rainbow $S$-tree with
the exception of the case: $|S\cap W|=3$ and the only $3$ different colors
used by the color codes of $S$. Note that $3$ colors used by the color codes
of $S$ may be allowed in this case. But there must exist a color code
consisted of other two distinct colors. Thus we will find a rainbow $S$-tree
with length $5$, for example see Figure 3.
Figure 3.
When $t\geq 10$, if $3$ different colors used by the color codes of $S$, there
must be a color code consisted of other two distinct colors appearing in the
first $10$ color codes of $K_{2,t}$ by the strategy of coloring. When $t=9$,
we only to check the subcase that color codes of $S$ are limited in {2,3,4}.
It is easy to verify the fact there is a rainbow tree connecting $S$, which
correspond to the color codes $(2,3),~{}(3,4),~{}(4,2)$, respectively. So the
coloring is $3$-rainbow coloring. That is, for $9\leq t\leq 20$,
$rx_{3}(k_{2,t})\leq 5$. With the aid of Lemma 2.4, we get
$rx_{3}(k_{2,t})=5$, $9\leq t\leq 20$.
Figure 4: The graph (a) used in Lemma 2.5 and the graph (b)used in Lemma 2.6.
∎
###### Lemma 2.6.
For $(k-1)(k-2)+1\leq t\leq k(k-1)$, $rx_{3}(K_{2,t})=k$, $k\geq 6$.
###### Proof.
By the Claim $2$, if $rx_{3}(K_{2,t})=k$, then it has at most $C_{k}^{2}\times
2=k(k-1)$ acceptable color codes. Thus when $t\geq k(k-1)+1$,
$rx_{3}(K_{2,t})\geq k+1$. Similarly, when $t\geq(k-1)(k-2)+1$,
$rx_{3}(K_{2,t})\geq k$.
Now we give a $3$-rainbow coloring of $K_{2,t}$ ($(k-1)(k-2)+1\leq t\leq
k(k-1)$) with $k$ colors. When $k\geq 6$, $(k-1)(k-2)+1>\frac{1}{2}k(k-1)$,
thus $t>\frac{1}{2}k(k-1)$. We randomly give the first $\frac{1}{2}k(k-1)$
vertices $\frac{1}{2}k(k-1)$ color codes in upper triangle of the chessboard,
see Figure 4(b). Then the other $t-\frac{1}{2}k(k-1)$ vertices are received
any $t-\frac{1}{2}k(k-1)$ remaining color codes in Figure 4(b).
Next we show this kind of coloring is $3$-rainbow coloring. Let $S$ be the set
of three vertices. By Lemma 2.2, we only to check the case : $|S\cap W|=3$ and
only $3$ different colors used by the color codes of three vertices. Similar
to the proof of Lemma 2.5, we need to find a color code consisted of other two
distinct colors to construct a rainbow $S$-tree. Since the combinations of any
two colors have appeared in first $k(k-1)/2$ color codes, we can easily find
such a color code. Hence, in any case, there is a rainbow $S$-tree with length
at most $5$. That is, $(k-1)(k-2)+1\leq t\leq k(k-1)$, $rx_{3}(K_{2,t})\leq
k$. So $rx_{3}(K_{2,t})=k$ for $(k-1)(k-2)+1\leq t\leq k(k-1)$. ∎
Now we complete the proof of Theorem 1.
## References
* [1] J.A. Bondy, U.S.R. Murty, Graph Theory, GTM 244, $Springer$, New York, 2008.
* [2] G. Chartrand, G.L. Johns,K.A. MeKeon, P. Zhang, Rainbow connection in graphs , Math.Bohem 133(1)(2008)85-98.
* [3] G. Chartrand, F. Okamoto, P. Zhang, Rainbow trees in graphs and generalized connectivity, Networks DOI(2010).
* [4] S. Chakraborty, E. Fischer, A. Matsliah, R. Yuster, Hardness and algorithms for rainbow connection, J. Combin. Optim. 21(2010), 330-347.
* [5] X. Li, Y. Sun, Rainbow connections of graphs—A survey, Graphs and Combin 29(2013), 1-38.
* [6] L. Chen, X. Li, K. Yang, Y. Zhao, The 3-rainbow index of a graph. arXiv:1307.0079V3 [math.CO] (2013).
* [7] T. Liu, Y. Hu, some upper bounds for 3-rainbow index of graphs. submitted.
|
arxiv-papers
| 2013-10-09T05:24:51 |
2024-09-04T02:49:52.136197
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Tingting Liu and Yumei Hu",
"submitter": "Liu Tingting",
"url": "https://arxiv.org/abs/1310.2353"
}
|
1310.2355
|
# Some upper bounds for 3-rainbow index of graphs
Tingting Liu, Yumei Hu111supported by NSFC No. 11001196.
Department of Mathematics, Tianjin University, Tianjin 300072, P. R. China
E-mails: [email protected]; [email protected];
###### Abstract
A tree $T$, in an edge-colored graph $G$, is called a rainbow tree if no two
edges of $T$ are assigned the same color. A $k$-rainbow coloring of $G$ is an
edge coloring of $G$ having the property that for every set $S$ of $k$
vertices of $G$, there exists a rainbow tree $T$ in $G$ such that $S\subseteq
V(T)$. The minimum number of colors needed in a $k$-rainbow coloring of $G$ is
the $k$-rainbow index of $G$ , denoted by $rx_{k}(G)$. In this paper, we
consider $3$-rainbow index $rx_{3}(G)$ of $G$. We first show that for
connected graph $G$ with minimum degree $\delta(G)\geq 3$, the tight upper
bound of $rx_{3}(G)$ is $rx_{3}(G[D])+4$, where $D$ is the connected
$2$-dominating set of $G$. And then we determine a tight upper bound for
$K_{s,t}(3\leq s\leq t)$ and a better bound for $(P_{5},C_{5})$-free graphs.
Finally, we obtain a sharp bound for $3$-rainbow index of general graphs.
Keywords
: $3$-rainbow index; rainbow tree; connected $2$-dominating set.
## 1 Introduction
All graphs considered in this paper are simple, connected and undirected. We
follow the terminology and notation of Bondy and Murty [1]. An edge-colored
graph $G$ is rainbow connected if any two vertices are connected by a path
whose edges have distinct colors. The rainbow connection number $rc(G)$ of
$G$, introduced by Chartrand et al. [6], is the minimum number of colors that
results in a rainbow connected graph $G$.
Later, another generalization of rainbow connection number was introduced by
Chartrand et al.[5] in 2009. A tree $T$ is a rainbow tree if no two edges of
$T$ are colored the same. Let $k$ be a fixed integer with $2\leq k\leq n$. An
edge coloring of $G$ is called a $k$-rainbow coloring if for every set $S$ of
$k$ vertices of $G$, there exists a rainbow tree in $G$ containing the
vertices of $S$. The $k$-rainbow index $rx_{k}(G)$ of $G$ is the minimum
number of colors needed in a $k$-rainbow coloring of $G$. It is obvious that
$rc(G)=rx_{2}(G)$.
Let $k$ be a positive integer. A subset $D\subseteq V(G)$ is a $k$-dominating
set of the graph $G$ if $|N_{G}(v)\cap D|\geq k$ for every $v\in V\setminus
D$. The $k$-domination number $\gamma_{k}(G)$ is the minimum cardinality among
the $k$-dominating sets of $G$. Note that the $1$-domination number
$\gamma_{1}(G)$ is the usual domination number $\gamma(G)$. A subset $S$ is a
connected $k$-dominating set if it is a $k$-dominating set and the graph
induced by $S$ is connected. The connected $k$-domination number
$\gamma_{k}^{c}(G)$ represents the cardinalities of a minimum connected
$k$-dominating set. For $k=1$, we write $\gamma_{c}$ instead of
$\gamma_{1}^{c}(G)$.
Chakraborty et al. [3] showed that computing the rainbow connection number of
a graph is NP-hard. So it is also NP-hard to compute $k$-rainbow index of an
arbitrary graph.
Chandran et al. [4] use a strengthened connected dominating set (connected
$2$-way dominating set) to prove $rc(G)\leq rc(G[D])+3$. This led us to the
investigation of what is strengthening of a connected dominating set which can
apply to consider $3$-rainbow index of a graph.
Recently, for $3$-rainbow index, Li et al. did some basic results and they
obtained the following theorem.
###### Theorem 1.1.
[7] Let $G$ be a $2$-connected graph of order $n$ $(n\geq 4)$. Then
$rx_{3}(G)\leq n-2$, with equality if and only if $G=C_{n}$ or $G$ is a
spanning subgraph of $3$-sun or $G$ is a spanning subgraph of $K_{5}\setminus
e$ or $G$ is a spanning subgraph of $K_{4}$.
Here, a $3$-sun is a graph $G$ which is defined from $C_{6}$ =
$v_{1}v_{2}\cdots v_{6}v_{1}$ by adding three edges $v_{2}v_{4}$, $v_{2}v_{6}$
and $v_{4}v_{6}$.
Chartrand et al. [6] obtained that for integers s and t with $2\leq s\leq t$,
$rc(K_{s,t})=min\\{\sqrt[s]{t},4\\}$. Thus, $rx_{2}(G)=rc(G)\leq 4$. Li et al.
[7] consider the regular complete bipartite graphs $K_{r,r}$. They show
$rx_{3}(K_{r,r})=3$ for integer $r$ with $r\geq 3$.
In this paper, we focus on $3$-rainbow index. In section $2$, we adopt
connected $2$-dominating set to study $3$-rainbow index. A coloring strategy
is obtained which uses only a constant number of extra colors outside the
dominating set. We prove that $rx_{3}(G)\leq rx_{3}(G[D])+4$, where $D$ is the
connected $2$-dominating set of $G$. In section $3$, We determine a sharp
bound of $3$-rainbow index for $K_{s,t}$ ( $3\leq s\leq t$) and an upper bound
for $(P_{5},C_{5})$-free. In section $4$, we investigate a sharp upper for
$rx_{3}(G)$ of general graphs by block decomposition and an upper bound for
graphs with $\delta(G)\geq 3$ by connected $2$-dominating set.
## 2 A sharp upper of $3$-rainbow index in terms of connected $2$-dominating
set
###### Theorem 2.1.
Let $G$ be a connected graph with minimal degree $\delta\geq 3$. If $D$ is a
connected $2$-dominating set of $G$, then $rx_{3}(G)\leq rx_{3}(G[D])+4$ and
the bound is tight.
###### Proof.
We prove the theorem by demonstrating that $G$ has a 3-rainbow coloring with
$rx_{3}(G[D])+4$ colors. For $x\in V(G)\setminus D$, its neighbors in $D$ will
be called foots of $x$, and the corresponding edges will be called legs of
$x$.
We give $G[D]$ a 3-rainbow coloring using colors
$1,2,\cdots,k~{}(k=rx_{3}(G[D]))$. Let $H:=G\setminus D$. Partition $V(H)$
into sets $X,Y,Z$ as follows. $Z$ is the set of all isolated vertices of $H$.
In every nonsingleton connected component of $H$, choose a spanning tree. So
we construct a forest on $W:=V(H)\setminus Z$ and choose $X$ and $Y$ as any
one of the bipartitions defined by this forest. Color every $X-D$ edge with
$k+1$ or $k+2$ where each of $k+1,k+2$ appears at least once, every $Y-D$ edge
with $k+1$ or $k+3$ where each of $k+1,~{}k+3$ appears at least once, every
edge between $X$ and $Y$ with $k+4$. Since $G$ has a minimal degree
$\delta\geq 3$, every vertex in $Z$ will have at least three neighbors in $D$.
Color two of them with $k+1$ and $k+3$ and all the others with $k+4$. Next, we
show that under such an edge coloring for any three vertices in $D$ there
exists a rainbow tree containing them.
For three vertices $(x,y,z)\in D\times D\times D$, there is already a rainbow
tree containing them in $G[D]$. For three vertices $(x,y,z)\in$ $D\times
D\times V(H)$ (or $D\times V(H)\times V(H)$), join any one leg of $z$ (or
$k+1$, $k+3$ ($k+2$) legs of $y$ and $z$ ) with a rainbow tree containing the
corresponding foot (or two foots), $x$ and $y$ (or $x$) in $G[D]$. Now we
consider the case three vertices $(x,y,z)\in$ $V(H)\times V(H)\times V(H)$.
For three vertices $(x,y,z)\in Z\times Z\times Z$, join three edges which
color $k+1,k+4$ and $k+3$ with a rainbow tree containing the corresponding
foots $(x^{\prime},y^{\prime},z^{\prime})$ in $D$. For two vertices $(x,y)\in
Z\times Z$, $z\in W$, join a $k+1$ leg of $z$ and $k+3,~{}k+4$ legs of
$x,~{}y$ with a rainbow tree containing the corresponding foots in $G[D]$.
Consider one vertex $x\in Z$, two vertices $(y,z)\in W\times W$. If $(y,z)\in
X\times X$, join a $k+4$ leg of $x$ and $k+1,k+2$ legs of $y$ and $z$ with a
rainbow tree containing the corresponding foots in $G[D]$. If $(y,z)\in
X\times Y$ or $(y,z)\in Y\times Y$, join a $k+4$ leg of $x$ and $k+1,k+3$ legs
of $y$ and $z$ with a rainbow tree containing the corresponding foots in
$G[D]$. Then consider three vertices $(x,y,z)\in W\times W\times W$. If
$(x,y,z)\in X\times X\times X$, we know, for $x\in X$, $x$ has a neighbor
$y(x)\in Y$. $x-y(x)$ edge (colored $k+4$) and $k+3$ leg of $y(x)$, join $k+1$
leg of $y$ and $k+2$ leg of $z$ with a rainbow tree containing the
corresponding foots in $G[D]$. Similarly, in other cases, we can find a
rainbow tree containing them. Hence, $G$ has a 3-rainbow coloring with
$rx_{3}(G[D])+4$ colors.
The proof of tightness is given in the next section. ∎
## 3 Upper bounds for $3$-rainbow index of some special graphs
In this section, we consider two special graphs: complete bipartite graphs
$K_{s,t}$ and $(P_{5},C_{5})$-free graphs.
###### Theorem 3.1.
For any complete bipartite graphs $K_{s,t}$ with $3\leq s\leq t$,
$rx_{3}(K_{s,t})\leq min\\{6,s+t-3\\}$, and the bound is tight.
###### Proof.
Because $K_{s,t}$ with $3\leq s\leq t$ is a 2-connected graph, by Theorem 1.1,
we have, $rx_{3}(K_{s,t})\leq s+t-3$. The equality clearly holds for $s=t=3$
since $rx_{3}(K_{3,3})=3$. Thus, to complete the proof, it suffices to show
$rx_{3}(K_{s,t})\leq 6$, $3\leq s\leq t$. Let $U$ and $W$ be the two partite
sets of $K_{s,t}$, where $|U|=s$ and $|W|=t$. Suppose
$U=\\{u_{1},u_{2},\cdots,u_{s}\\},~{}W=\\{w_{1},w_{2},\cdots,w_{t}\\}$.
Clearly we can find a connected 2-dominating set
$D=\\{u_{1},u_{2},w_{1},w_{2}\\}$ of $K_{s,t}$. In addition, $K_{s,t}\setminus
D$ is connected, $Z=\emptyset$, by Theorem 2.1, $rx_{3}(K_{s,t})\leq
rx_{3}(G[D])+4=6$.
To prove the sharpness of the above upper bound, we derive the following
claim.
Claim. For any $s\geq 3$, $t\geq 2\times 6^{s}$, $rx_{3}(K_{s,t})=6$.
Firstly, we consider the graph $K_{3,t}$. We may assume that there exists a
3-rainbow coloring $c$ of $K_{3,t}$ with $k$ colors. Corresponding to this
3-rainbow coloring, for every vertex $w$ in $W$, there is a color code,
code($w$), assigned $a_{i}=c(u_{i}w)\in\\{1,2,\cdots,k\\}$, $1\leq i\leq 3$.
Observe that any three vertices have at least three distinct colors appeared
in their color codes. Thus, we know that at most two vertices have the common
code except possibly when $a_{1}\neq a_{2}\neq a_{3}$. Otherwise, there is no
rainbow tree containing these three vertices which have the same code and at
most two colors in color code. Therefore, when $t\geq 2k^{3}$, there must
exist three vertices $w^{\prime}$, $w^{\prime\prime}$,
$w^{\prime\prime\prime}$ such that code
($w^{\prime}$)=code($w^{\prime\prime}$)=code($w^{\prime\prime\prime}$)=$\\{a_{1},a_{2},a_{3}\\}$
and $a_{1}\neq a_{2}\neq a_{3}$. If a rainbow tree containing
$S=\\{w^{\prime},\ w^{\prime\prime},\ w^{\prime\prime\prime}\\}$, it must
contain $u_{1},u_{2},u_{3}$ and $w_{i}$ to guarantee its connectivity, where
$w_{i}$ belongs to $W$ and code($w_{i}$)=$\\{b_{1},b_{2},b_{3}\\}$, where
$a_{i}$, $b_{j}$ are different from each other, $i=1,2,3;\ j=1,2,3$. Thus
$k\geq 6$. So $rx_{3}(K_{3,t})=6$, when $t\geq 2\times 6^{3}$. Similarly, we
can prove $rx_{3}(K_{s,t})=6$, for $s\geq 4$, $t\geq 2\times 6^{s}$. Thus,
this claim also provides the tight proof of the Theorem 2.1. ∎
Here, we can simply check that the upper bound can not be generalized to the
graphs $K_{2,t}$. By the same method used in the above claim, We may assume
that there exists a 3-rainbow coloring $c$ of $K_{2,t}$ with $k$ colors.
Corresponding to this 3-rainbow coloring, there is a color code, code($w$),
assigned $a_{i}=c(u_{i}w)\in\\{1,2,\cdots,k\\}$ for $i=1,2$. Observe that at
most two vertices have the common code. It follows, $t\leq 2k^{2}$. Thus, $k$
is not less than a certain constant when $t$ is enough large.
To state next theorem, we need to make more definitions. A graph $G$ is called
a perfect connected dominant graph if $\gamma(X)=\gamma_{c}(X)$, for each
connected induced subgraph $X$ of $G$. If $G$ and $H$ are two graphs, we say
that $G$ is $H$-free if $H$ does not appear as an induced subgraph of $G$.
Furthermore, if $G$ is $H_{1}$-free and $H_{2}$-free, we say that $G$ is
$(H_{1},H_{2})$-free. Next, we determine the upper bound for $3$-rainbow index
of $(P_{5},C_{5})$-free graphs.
Zverovich has obtained the following result.
###### Theorem 3.2.
[14] A graph $G$ is a perfect connected-dominant graph if and only if $G$
contains no induced path $P_{5}$ and induced cycle $C_{5}$.
As shown in Theorem 2.1, in order to obtain a better bound of $3$-rainbow
index, we may turn to a smallest possible connected $2$-dominating set. For a
graph with minimal degree $\delta\geq 3$, B.Reed proved the following
conclusion in [13].
###### Theorem 3.3.
[13] If $G$ is connected graph with $\delta\geq 3$, then
$\gamma(G)\leq\frac{3n}{8}$.
For $(P_{5},C_{5})$-free graphs $\delta\geq 3$, we have
$\gamma_{c}(G)\leq\frac{3n}{8}$. Inspired by this result, the extension of the
idea of connected dominating set to connected $2$-dominating set is what gives
the following lemma.
###### Lemma 3.1.
Let $G$ be a connected graph of order $n$ with minimal degree $\delta\geq 2$.
If $D$ is a connected dominating set in a graph $G$, then there is a set of
vertices $D^{\prime}\supseteq D$ such that $D^{\prime}$ is a connected
2-dominating set and $|D^{\prime}|\leq\frac{1}{2}n+\frac{1}{2}|D|$.
###### Proof.
There are two types of the components of $G\setminus D$: singletons and
connected subgraphs. Let $P$ be the set of the singletons, and $Q$ be the set
of the connected components of $G\setminus D$. Note that $G\setminus D=P\cup
Q$. Since $\delta\geq 2$, for any vertex $v$ in $P$, it has at least two
neighbors in $D$. In every non-singleton connected component of $Q$, we choose
a spanning tree. This gives a spanning forest on $V(Q)$. Choose $X$ and $Y$ as
any one of the bipartitions defined by this forest. Without loss of
generality, we suppose that $|X|\leq|Y|$.
Stage $D^{\prime}=D$
while $\exists v\in V(Q)$ such that $|N(v)\bigcap D|=1$
$\\{$
If $v\in Y$
Pick a vertex $u\in N(v)\bigcap X$.
Let $D^{\prime}=D^{\prime}\bigcup\\{u\\}$
else
$D^{\prime}=D^{\prime}\bigcup\\{v\\}$
$\\}$
Clearly $D^{\prime}$ remains to be connected. Since stage ends only when any
vertex in $V(Q)$ has at least 2 neighbors in $D^{\prime}$. So the final
$D^{\prime}$ is a connected $2$-dominating set. Let $k$ be the number of
iterations executed. Since we add a vertex in $X$ to $D^{\prime}$, $|X|$
reduces by 1 in every iteration, $k\leq|X|\leq\frac{1}{2}(n-|D|)$, so
$|D^{\prime}|\leq|D|+k\leq|D|+\frac{1}{2}(n-|D|)=\frac{1}{2}n+\frac{1}{2}|D|$.
∎
For a connected $(P_{5},C_{5})$-free graph $G$ with $\delta\geq 3$, we can
derive the following result by Theorem 2.1, Theorem 3.2, Theorem 3.3 and Lemma
3.1.
###### Theorem 3.4.
For every connected $(P_{5},C_{5})$-free graphs $G$ with $\delta(G)\geq 3$,
$rx_{3}(G)\leq\frac{11}{16}n+3$.
###### Proof.
For every connected $(P_{5},C_{5})$-free graphs $G$ with $\delta(G)\geq 3$,
from the Theorem 3.2, $\gamma(G)=\gamma_{c}(G)$. And by the Theorem 3.3, we
have $\gamma(G)\leq\frac{3n}{8}$. Thus, $\gamma_{c}(G)\leq\frac{3n}{8}$.
Combining this with Lemma 3.1, the graph $G$ have a connected $2$-dominating
set $D$ with order less than $\frac{11}{16}n$. Observe that the connected
$2$-dominating set $D$ can get a 3-rainbow coloring using $|D|-1$ colors by
ensuring that every edge of some spanning tree gets distinct color. So the
upper bound follows immediately from Theorem 2.1. ∎
## 4 Upper bounds for $3$-rainbow index of general graphs
In this section, we derive a sharp bound for $3$-rainbow index of general
graphs by block decomposition. And we also show a better bound for $3$-rainbow
index of general graphs with $\delta(G)\geq 3$ by connected $2$-dominating
set.
Let $\mathcal{A}$ be the set of blocks of $G$, whose element is $K_{2}$; Let
$\mathcal{B}$ be the set of blocks of $G$, whose element is $K_{3}$; Let
$\mathcal{C}$ be the set of blocks of $G$, whose element $X$ is a cycle or a
block of order $4\leq|V(X)|\leq 6$; Let $\mathcal{D}$ be the set of blocks of
$G$, whose element $X$ is not a cycle and $|V(X)|\geq 7$.
###### Theorem 4.1.
Let $G$ be a connected graph of order $n~{}(n\geq 3)$. If $G$ has a block
decomposition $B_{1},B_{2},\cdots,B_{q}$, then $rx_{3}(G)\leq
n-|\mathcal{C}|-2|\mathcal{D}|-1$, and the upper bound is tight.
###### Proof.
Let $G$ be a connected graph of order $n$ with $q$ blocks in its block
decomposition. If $q=1$, then we have done by Theorem 1.1 and
$rx_{3}(K_{3})=2$, which satisfies the above bound. Thus, we suppose $q\geq
2$.
Note that $|\mathcal{A}\cup\mathcal{B}\cup\mathcal{C}\cup\mathcal{D}|=q$. From
the Theorem 1.1, we get $rx_{3}(X)\leq|X|-2$ for $X\in\mathcal{C}$ and
$rx_{3}(X)\leq|X|-3$ for $X\in\mathcal{D}$. Hence, it follows that
$\displaystyle rx_{3}(G)$ $\displaystyle\leq$
$\displaystyle\sum_{X\in\mathcal{A}}1+\sum_{X\in\mathcal{B}}2+\sum_{X\in\mathcal{C}}(|X|-2)+\sum_{X\in\mathcal{D}}(|X|-3)$
$\displaystyle=$ $\displaystyle n-|\mathcal{C}|-2|\mathcal{D}|-1.$
In order to prove that the upper bound is tight, we construct the graph $G$ of
order $n$, as shown in Figure 1, consisting of $(n-3r-7)$ $K_{2}$, $r$ cycles
of order 4 and one 7-length-cycle with a chord. It is clear that
$|\mathcal{C}|$=$r$, $|\mathcal{D}|=1$. We consider the size of a rainbow tree
$T$ contain the vertices $\\{u,v,w\\}$. $|E(T)|=n-4r-7+3r+4=n-r-3$ and
$rx_{3}(G)\leq n-|\mathcal{C}|-2|\mathcal{D}|-1=n-r-3$ by the above theorem.
we have $rx_{3}(G)=n-|\mathcal{C}|-2|\mathcal{D}|-1$.
Figure 1: Graph for Theorem 4.1
∎
We finish this section with general graphs with minimal degree at least $3$.
Here, we denote as $q_{max}(G)$ the maximum number of components of
$G\backslash u$ among all vertices $u\in V$. The following result is needed in
the sequel.
###### Theorem 4.2.
[9] Let G be a connected graph on $n$ vertices with minimum degree $\delta\geq
2$ and let $k$ be an integer with $1\leq k\leq\delta$. Then
$\gamma_{k}^{c}\leq n-q_{max}(G)(\delta-k+1)$
For general graphs with $\delta\geq 3$, we obtain an upper bound for
$3$-rainbow index from Theorem 2.1 and Theorem 4.2.
###### Theorem 4.3.
Let $G$ be a connected graph with minimal degree $\delta\geq 3$. Then
$rx_{3}(G)\leq n-q_{max}(\delta-1)+3$.
Note that the bound of $3$-rainbow index is better for the graphs with cut
vertices and larger minimal degree.
## References
* [1] J. A. Bondy, U. S. R. Murty, Graph Theory, Springer, 2008.
* [2] Y. Caro, A. Lev, Y. Roditty, Z. Tuza, R. Yuster, On rainbow connection, Electron. J. Combin 15(1), 2008, R57.
* [3] S. Chakraborty, E. Fischer, A. Matsliah, R. Yuster, Hardness and algorithms for rainbow connection, J. Combin. Optim. 21, 2010, pp. 330-347.
* [4] L. S. Chand, A. Das, D. Rajendraprasad, N. M. Varma, Rainbow connection number and connected dominating sets, Electronic Notes in Discrete Math. 38, 2011, pp. 239-244.
* [5] G. Chartrand, F. Okamoto, P. Zhang, Rainbow trees in graphs and generalized connectivity, Networks, DOI, 2010.
* [6] G. Chartrand, G. L. Johns, K. A. MeKeon, P. Zhang, Rainbow connection in graphs, Math. Bohem 133(1), 2008, pp. 85-98.
* [7] L. Chen, X. Li, K. Yang, Y. Zhao, The 3-rainbow index of a graph. arXiv:1307.0079V3 [math.CO] (2013).
* [8] X. Chen, X. Li, A solutuon to a conjecture on the rainbow connection number, Ars Combin., 104, 2012, pp. 193-196.
* [9] Adriana Hansberg, Bounds on the connected $k$-domination number in graphs, Discrete Applied Mathematics 158, 2010, pp. 1506-1510.
* [10] X. Li, S. Liu, Rainbow Connections number and the number of blocks, Graphs and Combin., in press.
* [11] X. Li, Y. Shi, Y. Sun, Rainbow connections of graphs—A survey, Graphs and Combin 29, 2013, pp. 1-38.
* [12] X. Li, Y. Sun, Rainbow connection numbers of line graphs, Ars Combin., 100, 2011, pp. 449-463.
* [13] B. Reed, Paths, stars, and the number three, Combinatorics, Probability Computing 5(3),1996, pp. 277-295.
* [14] I. E. Zverovich, Perfect connected-dominant graphs. Discuss. Math. Graph Theory 23, 2003, pp. 159-162.
|
arxiv-papers
| 2013-10-09T05:32:54 |
2024-09-04T02:49:52.141442
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Tingting Liu and Yumei Hu",
"submitter": "Liu Tingting",
"url": "https://arxiv.org/abs/1310.2355"
}
|
1310.2479
|
Spatio-temporal variation of conversational utterances on Twitter
Christian M. Alis, May T. Lim∗
National Institute of Physics, University of the Philippines, Diliman, 1101
Quezon City, Philippines
$\ast$ E-mail: [email protected]
## Abstract
Conversations reflect the existing norms of a language. Previously, we found
that utterance lengths in English fictional conversations in books and movies
have shortened over a period of 200 years. In this work, we show that this
shortening occurs even for a brief period of 3 years (September 2009-December
2012) using 229 million utterances from Twitter. Furthermore, the subset of
geographically-tagged tweets from the United States show an inverse proportion
between utterance lengths and the state-level percentage of the Black
population. We argue that shortening of utterances can be explained by the
increasing usage of jargon including coined words.
## Introduction
Utterances, the speaking turns in a conversation, relay short bits of
information. Though utterances adapt strongly to medium [1], utterance
shortening has been observed over a span of two centuries. Here we show that
utterances in the online social medium Twitter did not only significantly
shorten in a span of a few years but also varied geographically—providing
evidence of increasing usage of jargon brought about by formation of groups.
Our use of Twitter conversations provided us with a large, highly resolved and
current dataset.
Twitter (twitter.com) is an online social medium that allows its users to post
messages (tweets) of up to 140 characters in length, which are public by
default. Previous studies [2, 3] on Twitter conversations focused on modelling
the structure of conversations rather than the form of utterances. Recent
studies have ranged from characterizing the graph of the Twitter social
network [4, 5] to inferring the mood of the population [6, 7, 8, 9]. Owing to
the large number of Twitter users (about half-billion in June 2012 [10]) and
easy access via the provided application programming interfaces (API), Twitter
has become a platform for studying the usage of the English language. For
example, it has been found that longer Twitter messages (tweets) are more
likely to be credible [11] and, by determining where tweets were posted,
dialects [12] and geographical diffusion of new words [13] are observable.
Conversations in Twitter are typically performed in one of two ways:
privately, using direct messages; or publicly, using replies. Replies [14] are
tweets that begin with the username of the recipient prefixed with an at (@)
sign, for example, @bob Hello! How are you?. Since replies may be viewed by
other users aside from the recipient, replies are used for public
conversations [15] akin to having conversations while other people are
listening.
Conversation analysis usually investigates the structure of conversations [16]
by looking at the interaction of utterances instead of the individual
utterances themselves. Since we are more interested with the encoding of
information or idea into an utterance, this paper focused instead on the
construction of individual utterances and not in their interaction. More
specifically, the length of utterances are measured because the production
time and the amount of information of an utterance should be correlated with
its length.
Sentence lengths are not as widely studied as words, and conversational
utterances less so. The study of sentence lengths began with the work of Udny
Yule [17] in 1939 and eventually led to the discovery that sentence length
distributions may be approximated by a gamma distribution [18]. On the other
hand, the mean length of utterance is used to evaluate the level of language
development of children [19, 20]. The length of sentences and utterances are
usually measured in terms of words or morphemes but we used the number of
characters (orthographic length) as unit of length because Twitter imposes a
maximum tweet length in terms of characters. The use of orthographic length of
sentences has previously been shown to be a valid unit when comparing
utterance length distributions [1]. Furthermore, the orthographic length of
words is highly correlated with word length in terms of syllables [21].
In this paper, 229 million conversational utterances collected from 18
September 2009 to 14 December 2012 are first characterized. By comparing
expected (fitted) and empirical utterance length distributions, we show that
the character limit of tweets has little to no effect on the median utterance
length. We also identify some factors that significantly affect the utterance
length. The dataset is then disaggregated to reveal that utterances shortened
in a span of more than three years. Possible mechanisms of shortening are then
explored. Finally, the variation of utterance length across different US
states and its correlation with demographic and socioeconomic variables are
investigated.
## Results
### Aggregate utterance length distribution
The utterance length distribution (ULD) of the entire data set (Fig. 1A) is
bimodal and can be fitted with a gamma distribution after taking the
140-character limit into account [1]. It is bimodal due to the mixture of the
natural (unconstrained) ULD and shortened (constrained) ULD forced by the
140-character limit. To estimate the unconstrained ULD, a generalized gamma
distribution,
$\mathrm{Pr}(x)=\frac{\tilde{x}^{\alpha-1}e^{-\tilde{x}}}{{\Gamma(\alpha)}},$
(1)
where ̵̃$x$ is the utterance length, $\tilde{x}=(x-x_{0})/s$ is the scaled
utterance length, and $\alpha$, $x_{0}$ and $s$ are fitting parameters that
describe the shape, translation and ordinate scaling factor, respectively, was
fitted on the utterance length distribution from $x=1$ char. to a cut-off
length $x=x_{c}$ using least squares as was done in Ref. [1]. The estimated
natural ULD ($\alpha=1.46$, $x_{0}=1.01$ char., $s=30.0$ char.) fits the
empirical ULD with an $r^{2}=0.950$.
Both empirical and fitted (unconstrained) ULD are skewed to the right and the
quartiles (Q1=25th percentile; Q2=median=50th percentile; Q3=75th percentile)
are either the same (Q1=19 char., Q2=36 char.) or differs by 3 characters
(Q3${}_{\text{empirical}}=65$ char., Q3${}_{\text{fitted}}=62$ char.). From
here on, we used the quartiles of the empirical ULD to describe the
distributions.
Starting 10 October 2011, all URLs in tweets are automatically shortened by
Twitter [22] into a 20-character URL (http://t.co/xxxxxxxx) and this caused
the spike at $x=20$ char. in the ULD. The spike at $x=26$ char. is due to non-
English tweets while the spike at $x=3$ char. is due to the acronym LOL
(laughing out loud). Restricting utterances to English and removing URLs and
LOL result to a smoother ULD (Fig. 1B) but with the same quartiles as the
original distribution.
### Temporal dependence of utterance lengths
Utterance length distributions for tweets aggregated over a 24-hour period
that were sampled during Fridays follow the general characteristics of the
utterance length distribution for the entire dataset as shown by the
representative utterance length distributions in Fig. 2. The right peak of the
plots seems to get smaller and shifted to the left as the date becomes more
recent, suggesting shortening of utterances over time. This shortening is
clearly shown when the quartiles are plotted with respect to time (Fig. 3A).
The quartiles roughly follow their corresponding regression line except for 26
Nov 2010, which shows an unexpected spike due to spam.
As expected the linear regression line of the median (2nd quartile) is not at
the middle of Q1 and Q3 regression lines because the utterance length
distributions are skewed. The regression line of Q1 (Table 1 and Fig. 4, all)
is less steep than the regression line for the median, which, in turn, is less
steep than the regression line for Q3. The shortening of utterances is,
therefore, mostly due to the decreased occurrence of longer utterance lengths
rather than the shifting of the whole utterance length distributions to the
left.
Table 1 and Fig. 4 demonstrate the robustness of the decrease in utterance
length. The months included in the dataset differ for each year yet shortening
is still observed even if only utterances from the common included months of
September to December are considered (Table 1 and Fig. 4, Sep–Dec). Similarly,
the number of utterances per day and the percent of public data collected are
not constant throughout the entire dataset. To remove any size effects on the
results, $10^{5}$ utterances, an amount slightly smaller than the smallest
daily sample size, were sampled without replacement for each day (Fig. 3B) but
the same observations remained (Table 1 and Fig. 4, resampled). Another
possible reason for the shortening is the increased usage of link shorteners.
However, the shortening trend (Table 1 and Fig. 4, URLs removed) persisted
even if all links in the utterances were removed (Fig. 3C). Finally,
restricting the analysis to only English tweets (Fig. 3D) resulted to the same
observations (Table 1 and Fig. 4, English only).
### Possible mechanisms for shortening
A possible mechanism for utterance length shortening is the shortening of the
most frequent words either by a change in orthography (spelling) of the most
frequent words or their replacement by shorter words.
The median word length of all words is 4 characters (Fig. 5A) for all years
from 2009 to 2012. Although the median length of the 1000 most frequently used
words from 2009 to 2012 is constant at 4 characters (Fig. 5B), the peak (mode)
moved from 4 characters in 2009 to 3 characters (Fig. 5C) in the succeeding
years. Based on Kruskal-Wallis tests, the word length distributions of the
1000 most frequently used words for 2010–2012 are not significantly different
($H=1.112$, $p=0.5734$) with each other but are significantly different with
the distribution for 2009 ($H=10.31$, $p=0.0161$). However, the observed
shortening is not just due to a sudden shortening of the 1000 most frequently
occurring words from 2009 to 2010 because it was still observed in 2010–2012
(Table 1 and Fig. 4, 2010–2012)
From $60.32\pm 0.0185$% in 2009, the relative occurrence of the 1000 most
frequently used words (Fig. 5D) with respect to all words decreased to
$52.80\pm 0.0267$% in 2012. In that same timespan, the median utterance length
in words decreased from 8 words to 5 words (Fig. 5E) while the median tweet
length in words (Fig. 5F) decreased from 10 words to 8 words.
A topic is a word, usually in the form of #topic, or a phrase that is
contained in a tweet. Trending topics are the most prominent topics being
talked about in Twitter within a period of time. The shortening of trending
topics could potentially explain the observed shortening of utterances but
instead of decreasing, the median length of trending topics increased from 11
characters in 2009 to 13 characters in 2012 (Fig. 5G). Utterances about a
trending topic are shortening but the $r^{2}$-values (Table 1 and Fig. 4,
trending topics) are too small to cause the observed shortening of utterances.
The shortening of utterances is a global phenomenon and is not restricted to
the US since utterances that were geolocated outside the US also exhibited
shortening (Table 1 and Fig. 4, outside US). It was previously observed in
utterances from movies and books [1] albeit at a rate 1 and 3 orders of
magnitude smaller (-0.266 char./year in books; -0.001897 char./year in
movies), respectively. Although conversations do tend to get shorter in time,
our current findings show that it is occurring faster now on Twitter.
### Geographical variation of utterance lengths
Out of the 229 million utterances, only 795,048 utterances (0.347%) have
geographic information pointing to one of the US states (utterances-byloc.txt
in SI). The number of geolocated utterances per US state is strongly
correlated ($r^{2}=0.944$) with the 2010 census population of the US state and
ranges from 396 utterances in Wyoming to 96,120 utterances in California. The
medians are not correlated with the number of utterances ($r^{2}=0.104$)
although resampling to 300 utterances, a slightly smaller number of tweets
than the smallest sample size, resulted to changes in the quartiles, unlike in
the previous section where resampling did not change the quartiles for almost
all days after resampling.
To estimate how the quartiles change, the quartiles were bootstrapped using
$10^{4}$ repetitions but the bootstrapped values (Fig. 6A) turned out to be
the same as the empirical values. The spread in the bootstrapped medians is
very small that the interquartile range (IQR=Q3-Q1) of 40% of the bootstrapped
medians is zero. Any difference, therefore, in the median between two US
states is almost guaranteed to be significant. Both Kruskal-Wallis H-test
($H=8011$, $p<10^{-3}$) [23] and pairwise Mann-Whitney U-test [24] on the
empirical ULD of each US state conclude that not all ULD of the US states are
the same.
Plotting the medians over a US map (Fig. 6B) suggests southeastern and eastern
US states tend to have shorter utterance lengths. This clustering of
neighboring US states is very tenous, however, since pairwise Mann-Whitney
U-tests on the median utterance length of each US state yielded non-
neighboring US state pairings.
To check for possible correlates, the bootstrapped median utterance length was
regressed with demographic and socioeconomic information available in the
United States Census Bureau State and Country QuickFacts [25] (Table 2). Out
of the 51 variables (listed in SI Text S1), only the percent Black resident
population (latest data from 2011, $r^{2}=0.685$) and percent Black-owned
firms (latest data from 2007, $r^{2}=0.613$) have $r^{2}>0.5$. A detailed
description of both variables are in SI Text S1. The two variables are
strongly correlated though ($r^{2}=0.947$) so the correlation of the
bootstrapped median is really with the percentage distribution of Black
residents. The bootstrapped median is inversely proportional to the Black
resident population (Fig. 7A). Restricting utterances to English and removing
URLs improved the correlation to $r^{2}=0.707$.
For comparison, the median utterance length was also plotted against the
percent of persons 25 years and over who are high school graduates or higher
from 2007 to 2011 (Fig. 7B) and median household income from 2007 to 2011 in
thousands of dollars (Fig. 7C), which are both described in detail in SI Text
S1. Both variables are uncorrelated or only slightly correlated, at best, with
the median utterance length because the values of the coefficient of
determination are $r^{2}=0.397$ and $r^{2}=0.068$, respectively.
### Multivariate regression of median utterance length
We explored the possible dependence of the median utterance length on several
variables by considering linear combinations of the QuickFacts variables. To
ease the comparison of variable effect size and to avoid numerical problems,
the variables were standardized by subtracting the sample mean for the
variable then dividing by the sample standard deviation for the variable. That
is, the $z$-scores of the variables were considered in the multiple
regression. Aside from standardization, no other transformation e.g., power
transformation, was performed on the variables.
The parameter estimates of the linear model (Model 2) with percent Black
resident population $B$, percent high school graduates $H$, and median
household income $I$ as predictors are shown in Table
LABEL:tab:multiregression. Only the coefficient for $B$ is significantly
different from zero and its magnitude is 5 to 10 times larger than the
coefficients for $H$ and $I$. Further supported by an F-test ($F=2.87$, df =
2, $p=0.067$, $\alpha=0.05$), the three-variable model can be simplified into
the one-variable model.
Performing a stepwise regression
($\alpha_{\text{in}}=\alpha_{\text{out}}=0.05$) with race, educational
attainment and income QuickFacts variables as candidate predictors will yield
a two-variable model, Model 3. The variable $B$ is still included in the model
and the magnitude of its coefficient (-3.30) is about 4.5 times that of the
other predictor (0.73), percent of persons 25 years and over who are holders
of bachelor’s degree or higher from 2007 to 2011 (denoted as variable $C$).
The $r^{2}$ value using only $C$ as predictor is 0.093. The two-variable model
cannot be reduced to a single-variable model with either $B$ ($F=5.01$, df =
2, $p=0.030$, $\alpha=0.05$) or $C$ ($F=2.87$, df = 2, $p=0.067$,
$\alpha=0.05$) as the only predictor. The two-variable model improved $r^{2}$
by 0.03 or 4.3% from that of the single-variable model with $B$ as the only
predictor.
Expanding the set of candidate variables to all QuickFacts variables then
performing another stepwise regression
($\alpha_{\text{in}}=\alpha_{\text{out}}=0.05$) results to a five-variable
model, Model 4. Both variables $B$ and $C$ are included in the model with $B$
still having the largest coefficient magnitude. By adding three more
predictors to Model 3, $r^{2}$ increased by 0.121 or 16.9%. The adjusted
$r^{2}$ of Model 4 is larger by 0.114 or 16.2% than the two-variable model
and, since the latter is not equivalent to the former ($F=10.8$, df = 3,
$p<10^{-3}$, $\alpha=0.05$), the former can be considered as a better model
despite having more predictors. Model 4 suggests that shorter utterances are
correlated with US states having larger percentage of Blacks and lower
percentage of bachelor’s degree holders but has more owner-owned houses,
larger manufacturing output and less dense population.
The values of QuickFacts variables are regularly updated by the US Census
Bureau and only the most recent values are retained. By looking up each
variable in the source dataset, one can reconstruct the QuickFacts for
previous years (up to 2010). Repeating the regression analysis for the
different models using the data for previous years resulted to coefficient
estimates that are within the standard errors of the quoted variables above.
Thus, the coefficients remained essentially the same from 2010 to 2012.
## Discussion
The observation of geographic variability is not entirely unexpected because
of the existence of dialects. What is more surprising is that the utterance
length is (anti)correlated with the resident Black population. This factor
also dominates other predictors when combined with other demographic and
socioeconomic factors using multiple regression. A possible explanation is
that Blacks converse more distinctly and more characteristically than other
racial groups. Since utterances are only weakly correlated with median income
and educational attainment then perhaps the shorter utterance lengths is a
characteristic of their race—perhaps pointing towards the controversial
language of Ebonics [26]. The strong correlation does not imply causality, and
it is beyond the scope of this work to look for actual evidence of Ebonics in
the tweets.
Results show that people are communicating with fewer and shorter words. The
principles of least effort communications [27] provide us with two possible
implications. If the information content of each word remains the same then
the information content of each utterance is lesser and more utterances are
needed to deliver the same amount of information—a phenomenon that could be
verified by tracking the complete conversations between individuals, and not
just samples as we are doing now. On the other hand, if the amount of
information content of each utterance remains the same then encoding becomes
either more efficient (comprehension remains the same) or more ambiguous
through time. When ambiguity increases, speaker effort is minimized at the
expense of listener effort.
Based on anecdotal evidence, replies broken into several tweets are not more
frequent than before but shorter spelling and omission of words do seem to be
more prevalent. That is, encoding appears to become more efficient without
sacrificing as much precision.
The shortening, it seems, can be explained by increased usage of jargon, which
in turn provides evidence of segregation into groups. People who are engaged
in a conversation communicate using a shared context, which may utilize a more
specialized lexicon (jargon or even coined words). Although utterances are
expected to be less clear due to the use of fewer words, the use of context
prevents this from happening. The decrease in the frequency of words from
60.32% to 52.8% could mean that the use of jargon increased by about 60.32% -
52.8% = 7.52%. Furthermore, one of these groups might be composed of African
Americans hence the dependence on percent Black population can be readily
explained. Since no other demographic or socioeconomic variable is correlated
with utterance length then these groupings cannot be entirely demographic or
socioeconomic in nature.
There is no obvious remaining factor that could bias the temporal analysis of
utterance lengths after the shortening was shown to be robust. There are
several approaches in determining the proper location of users from tweets
[12, 28] but we used the simplest method of assigning the location of the user
to the location of the tweet. The geolocated tweets are relatively few and the
tweets (users) were then aggregated by US state. Statistical data from the US
Census Bureau were then used in the analysis. The inherent assumption,
therefore, is that the sample used by the Bureau can also be used to describe
the sample of Twitter users. A survey done by Smith and Brenner [29], however,
showed that among the different races, Blacks significantly use Twitter more
than other races. This could be the reason why only the dependence on Black
population was observed. More data are needed to verify if our assumption is
justified but our results are tantalizing enough to warrant a second look.
## Materials and Methods
Tweets were first retrieved using the Twitter streaming application
programming interface [30] and corresponds to 15% (before August 2010), 10%
(between August 2010 to mid-2012) or 1% (mid-2012 to present) of the total
public tweets. For ease of computation, we analyzed only tweets posted every
Friday from 18 September 2009 to 14 December 2012. Although issues in our
retrieval process prevented us from getting the entire sampled feed for the
entire data collection period, only Fridays with uninterrupted and complete
data were considered resulting to a total of 124 days analyzed. Conversational
utterances in the form of replies were selected by filtering for tweets that
begin with an at sign (@), which yielded 229 million utterances (utterances-
bydate.txt in SI).
The utterance length of a tweet was measured by first stripping off all
leading @usernames with the python regular expression ((^|\s)*@\w+?\b)+. The
utterance length is the number of remaining characters after leading and
trailing whitespace characters were removed. Utterances having lengths equal
to zero (0.483%) and greater than the maximum length of 140 char. (0.0935%)
were excluded from the dataset.
Tweet language identification was performed using langid.py [31], which claims
88.6% to 94.1% accuracy when identifying the language of a tweet over 97
languages. ldig [32] stands to be the most accurate automated language
identification system for tweets having a claim of 99% accuracy over 19
languages ($>98\%$ accuracy for English), however, it has not yet been
formally subjected to peer review. Nevertheless, we repeated the analysis
using ldig and found similar results and conclusions.
Tweets were geolocated using the geo and coordinates metadata of the tweets
and were categorized by US states using TIGER/Line shapefiles [33] prepared by
the US Census Bureau. A user must opt-in to have location information be
attached to their tweets. Previously, only exact coordinates (latitude and
longitude) are attached as location information and these become the values of
the geo and coordinates metadata of the tweet. More recently, users may opt to
select less granular location information e.g., neighborhood, city and
country, and these less precise place information are now the default. [34] A
user, though, may still choose exact location information or omit location
information for every tweet. Geolocation is possible with both mobile clients
and browsers but geolocation for the latter is not yet available for all
countries.
A parallel [35] version of the Space Saving [36] algorithm for selecting the
most frequent $k$ words was used instead of a naive histogram of word
occurrences because of the prohibitive amount of resources needed. The Space
Saving algorithm maintains a frequency count of up to $k$ words only. An
untracked word replaces the least frequent word if the maximum number of $k$
words are already being tracked. The parallel Space Saving algorithm involves
partitioning the data then running the Space Saving algorithm for each chunk.
The results of each chunk are merged using an algorithm similar to Space
Saving. The word frequency of both Space Saving and its parallel version are
approximate for near-$k$-ranked words. To have a guaranteed list of the 1000
most frequently occurring words, a much larger value of $k=10^{5}$ was used.
## Acknowledgments
Some computational resources were provided by an AWS in education grant and
the Advanced Science and Technology Institute, Department of Science and
Technology, Philippines.
## References
* 1. Alis CM, Lim MT (2012) Adaptation of fictional and online conversations to communication media. The European Physical Journal B 85: 1–7.
* 2. Ritter A, Cherry C, Dolan B (2010) Unsupervised modeling of Twitter conversations. In: Human Language Technologies: The 2010 Annual Conference of the North American Chapter of the Association for Computational Linguistics. Los Angeles, California: Association for Computational Linguistics. pp. 172–180.
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## Figure Legends
Figure 1: Utterance length distribution of the entire dataset. A. Unfiltered
utterance length distribution of the entire dataset B. Utterance length
distribution of English tweets with URLs and LOL removed. The solid line in
both plots is the best fit of Eq. (1). Figure 2: Representative utterance
length distributions per year. Utterance length distribution of every first
available Friday of December in the dataset. Figure 3: Utterance length
distribution over time. First quartile Q1 (square), median Q2 (circle) and
third quartile Q3 (triangle) of the A. original dataset, B. after resampling
into $10^{5}$ utterances per day, C. removing URLs and D. restricting to
English tweets. Figure 4: Slopes of utterance length quartiles temporal
regression lines. Visualization of Table 1. Figure 5: Exploring possible
mechanisms of shortening. Annual values of A. median word length of all words,
B. median word length of the 1000 most frequently occurring words, C. most
frequent word length of the 1000 most frequently occurring words, D. fraction
of 1000 most frequently occurring words relatively to all occurrences of
words, E. median utterance length in number of words F. median tweet length in
number of words, and G. median trending topic phrase length. Figure 6:
Utterance lengths across US states. A. Box plot of the utterance length
distribution of each US state sorted by increasing median utterance length.
The notches were estimated using 10,000 bootstrap repetitions but the
resulting bootstrapped median values are the same as the empirical median
values B. Contiguous US states colored with the bootstrapped median utterance
length. Figure 7: Median utterance length against demographic and
socioeconomic variables. The bootstrapped median utterance length plotted
against A. 2011 resident Black population in percent ($r^{2}=0.685$), B.
persons 25 years and over who are high school graduates or higher from 2007 to
2011, in percent ($r^{2}=0.397$) and C. Median household income from 2007 to
2011 in thousands of dollars ($r^{2}=0.068$). The linear regression line is
also shown in each plot.
## Supporting Information Legends
Text S1. Information on State and County QuickFacts variables.
Dataset S1. Utterance length frequencies by date. The rows of this comma-
separated file correspond to tweets posted on a certain UTC date. The first
column is the date in ISO format (yyyy-mm-dd) and the remaining columns list
the number of utterances with a length of 1 character, 2 characters, 3
characters and so on, until 139 characters. Only the frequencies of “valid”
utterance lengths (1-139 characters) are included.
Dataset S2. Utterance length frequencies by US state. The rows of this comma-
separated file correspond to tweets posted from a US state. The first column
is the abbreviated US state (e.g., AK) and the remaining columns list the
number of utterances with a length of 1 character, 2 characters, 3 characters
and so on, until 139 characters. Only the frequencies of “valid” utterance
lengths (1-139 characters) are included.
## Tables
Table 1: Slopes of utterance length quartiles temporal regression lines Subset | Q1 | Median (Q2) | Q3
---|---|---|---
| Slope | $r^{2}$ | Slope | $r^{2}$ | Slope | $r^{2}$
| (chars./year) | | (chars./year) | | (chars./year) |
All | -2.53 | 0.916 | -5.20 | 0.926 | -8.32 | 0.862
Sep–Dec | -5.29 | 0.812 | -7.85 | 0.894 | -9.97 | 0.889
Resampled | -2.54 | 0.918 | -5.25 | 0.927 | -8.23 | 0.860
URLs removed | -2.42 | 0.933 | -4.63 | 0.922 | -7.51 | 0.887
English only | -3.38 | 0.910 | -5.95 | 0.881 | -8.35 | 0.785
2010–2012 | -5.19 | 0.842 | -8.07 | 0.910 | -10.4 | 0.938
Trending topics | -3.41 | 0.153 | -6.83 | 0.294 | -7.61 | 0.440
Outside US | -5.57 | 0.838 | -8.15 | 0.904 | -10.4 | 0.909
Table 2: Single-variable linear regression of median utterance length with
selected US Census Bureau QuickFacts variables
Independent variable | Parameter estimate (standard error)
---|---
| 1a | 1b | 1c | 1d | 1e | 1f | 1g
2011 resident Black population in percent $B$ | -3.411*** | | | | | |
| (0.334) | | | | | |
Persons 25 years and over who are high school graduates or higher from 2007 to 2011 in percent $H$ | | 2.597*** | | | | |
| | (0.462) | | | | |
Median household income from 2007 to 2011 in thousands of dollars $I$ | | | 1.074 | | | |
| | | (0.574) | | | |
Persons 25 years and over who has bachelor’s degree or higher from 2007 to 2011 in percent $C$ | | | | 1.254** | | |
| | | | (0.567) | | |
2010 population per square mile $D$ | | | | | -1.247** | |
| | | | | (0.567) | |
Owner-occupied housing units in percent of total occupied housing units from 2007 to 2011 $O$ | | | | | | -0.846 |
| | | | | | (0.582) |
Total value of manufacturing shipments in 2007 $M$ | | | | | | | -1.310**
| | | | | | | (0.564)
Constant | 35.40*** | 35.40*** | 35.40*** | 35.40*** | 35.40*** | 35.40*** | 35.40***
| (0.334) | (0.462) | (0.574) | (0.567) | (0.567) | (0.582) | (0.564)
$r^{2}$ | 0.685 | 0.397 | 0.068 | 0.093 | 0.092 | 0.042 | 0.101
adjusted $r^{2}$ | 0.678 | 0.384 | 0.049 | 0.074 | 0.073 | 0.022 | 0.082
Standard errors are presented in parentheses below the corresponding parameter
estimates. Bold indicates significance at the 5% level, $n=50$
** $p<0.05$
*** $p<0.01$
|
arxiv-papers
| 2013-10-09T13:38:01 |
2024-09-04T02:49:52.162267
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Christian M. Alis, May T. Lim",
"submitter": "Christian Alis",
"url": "https://arxiv.org/abs/1310.2479"
}
|
1310.2494
|
Algorithmes auto-stabilisants pour la construction d’arbres couvrants et la
gestion d’entités autonomes
Self-stabilizing algorithms for spanning tree construction and for the
management of mobile entities
Lélia Blin
Rapport scientifique présenté en vue de l’obtention
de l’Habilitation à Diriger les Recherches
soutenue le 1 décembre 2011
à l’Université Pierre et Marie Curie - Paris 6
Devant le jury composé de :
Rapporteurs : Paola Flocchini, Professeur, Université d’Ottawa, Canada.
Toshimitsu Masuzawa, Professeur, Université d’Osaka, Japon. Rachid Guerraoui,
Professeur, École Polytechnique Fédérale de Lausanne, Suisse. Examinateurs :
Antonio Fernández Anta, Professeur, Université Rey Juan Carlos, Espagne.
Laurent Fribourg, DR CNRS, ENS Cachan, France. Colette Johnen, Professeur,
Université de Bordeaux, France. Franck Petit, Professeur, Université Pierre et
Marie Curie, France. Sébastien Tixeuil, Professeur, Université Pierre et Marie
Curie, France.
.
###### Contents
1. Summary of the document in English
2. Introduction
3. I Arbres couvrants sous contraintes
1. 1 Algorithmes auto-stabilisants et arbres couvrants
1. 1.1 Eléments de la théorie de l’auto-stabilisation
2. 1.2 Construction d’arbres couvrants
1. 1.2.1 Bref rappel de la théorie des graphes
2. 1.2.2 Bref état de l’art d’algorithmes auto-stabilisants pour la construction d’arbres couvrants
3. 1.3 Récapitulatif et problèmes ouverts
2. 2 Arbres couvrants de poids minimum
1. 2.1 Approches centralisées pour le MST
2. 2.2 Approches réparties pour le MST
3. 2.3 Approches auto-stabilisantes
1. 2.3.1 Algorithme de Gupta et Srimani
2. 2.3.2 Algorithme de Higham et Lyan
3. 2.3.3 Contributions à la construction auto-stabilisante de MST
4. 2.3.4 Algorithme de Korman, Kutten et Masuzawa
4. 2.4 Conclusion
3. 3 Autres constructions d’arbres couvrants sous contraintes
1. 3.1 Algorithmes auto-stabilisants sans-cycle
1. 3.1.1 Etat de l’art en auto-stabilisation
2. 3.1.2 Algorithme auto-stabilisant sans-cycle pour le MST
3. 3.1.3 Généralisation
2. 3.2 Arbre de Steiner
1. 3.2.1 Etat de l’art
2. 3.2.2 Contribution à la construction auto-stabilisante d’arbres de Steiner
3. 3.3 Arbre couvrant de degré minimum
1. 3.3.1 Etat de l’art
2. 3.3.2 Un premier algorithme auto-stabilisant
4. 3.4 Perspective: Arbre couvrant de poids et de degré minimum
4. II Entités autonomes
1. 4 Le nommage en présence de fautes internes
1. 4.1 Un modèle local pour un système de robots
2. 4.2 Les problèmes du nommage et de l’élection
3. 4.3 Algorithmes auto-stabilisants pour le nommage
1. 4.3.1 Algorithme déterministe
2. 4.3.2 Algorithme probabiliste
4. 4.4 Perspectives
2. 5 Auto-organisation dans un modèle à vision globale
1. 5.1 Etat de l’art des algorithmes dans le modèle CORDA discret
2. 5.2 Un modèle global minimaliste pour un système de robots
3. 5.3 Résultat d’impossibilités
4. 5.4 Algorithme d’exploration perpétuelle
1. 5.4.1 Algorithme utilisant un nombre minimum de robots
2. 5.4.2 Algorithme utilisant un nombre maximum de robots
5. 5.5 Perspectives
5. III Conclusions et perspectives
1. 6 Perspectives de recherche
1. 6.1 Compromis mémoire - temps de convergence
2. 6.2 Compromis mémoire - qualité de la solution
2. Research perspectives (in English)
1. Tradeoff between memory size and convergence time
2. Tradeoff between memory size and quality of solutions
### Summary of the document in English
In the context of large-scale networks, the consideration of _faults_ is an
evident necessity. This document is focussing on the _self-stabilizing_
approach which aims at conceiving algorithms “repairing themselves” in case of
transient faults, that is of faults implying an arbitrary modification of the
states of the processes. The document focuses on two different contexts,
covering the major part of my research work these last years. The first part
of the document (Part I) is dedicated to the design and analysis of self-
stabilizing algorithms for _networks of processes_. The second part of the
document (Part II) is dedicated to the design and analysis of self-stabilizing
algorithms for _autonomous entities_ (i.e., software agents, robots, etc.)
moving in a network.
###### Constrained Spanning Tree Construction.
The first part is characterized by two specific aspects. One is the nature of
the considered problems. The other is the permanent objective of optimizing
the performances of the algorithms. Indeed, within the framework of spanning
tree construction, self-stabilization mainly focused on the most classic
constructions, namely BFS trees, DFS trees, or shortest path trees. We are
interested in the construction of trees in a vaster framework, involving
constraints of a _global_ nature, in both static and dynamic networks. We
contributed in particular to the development of algorithms for the self-
stabilizing construction of minimum-degree spanning trees, minimum-weight
spanning tree (MST), and Steiner trees. Besides, our approach of self-
stabilization aims not only at the feasibility but also also includes the
search for _effective_ algorithms. The main measure of complexity that we are
considering is the memory used by every process. We however also considered
other measures, as the convergence time and the quantity of information
exchanged between the processes.
This study of effective construction of spanning trees brings to light two
facts. On one hand, self-stabilization seems to have a domain of applications
as wide as distributed computing. Our work demonstrate that it is definitively
case in the field of the spanning tree construction. On the other hand, and
especially, our work on memory complexity seems to indicate that self-
stabilization does not imply additional cost. As a typical example,
distributed MST construction requires a memory of $\Omega(\log n)$ bits per
process (if only to store its parent in the tree). We shall see in this
document that it is possible to conceive a self-stabilizing MST construction
algorithm of using $O(\log n)$ bits of memory per process.
###### Organization of Part I.
Chapter 1 summarizes the main lines of the theory of self-stabilization, and
describes the elementary notions of graph theory used in this document. It
also provides a brief state-of-the-art of the self-stabilizing algorithms for
the construction of spanning trees optimizing criteria not considered further
in the following chapters. Chapter 2 summarizes my contribution to the self-
stabilizing construction of MST. My related papers are [19, 17]. Finally,
Chapter 3 presents my works on the self-stabilizing construction of trees
optimizing criteria different from minimum weight, such as minimum-degree
spanning tree, and Steiner trees. My related papers are [19, 21, 23, 22].
###### Autonomous Entities.
The second part of the document is dedicated to the design and analysis of
self-stabilizing algorithms for _autonomous entities_. This latter term refers
to any computing entity susceptible to move in a space according to certain
constraints. We shall consider mostly physical robots moving in a discrete or
continuous space. We can make however sometimes reference to contexts
involving software agents in a network. For the sake of simplicity, we shall
use the terminology “ _robot_ ” in every case. Self-stabilization is a generic
technique to tolerate any transient failure in a distributed system that is
obviously interesting to generalize in the framework where the algorithm is
executed by robots (one often rather refers to _self-organization_ instead of
self-stabilization). It is worth noticing strong resemblances between the
self-stabilizing algorithmic for networks and the one for robots. For example,
the notion of _token_ circulation in the former framework seems very much
correlated with the circulation of _robots_ in the latter framework. In a
similar way, we cannot miss noticing a resemblance between the traversal of
graphs by messages, and graph _exploration_ by a robot. The equivalence (in
term of calculability) between the message passing model and the “agent model”
was already brought to light in the literature [12, 29]. This document seems
to indicate that this established equivalence could be extended to the
framework of auto-stabilization. The current knowledge in self-stabilizing
algorithms for robots is not elaborated enough to establish this
generalization yet. Also, the relative youth of robots self-stabilization
theory, and the lack of tools, prevent us to deal with efficiency (i.e.,
complexity) as it can be done in the context of network self-stabilization. In
this document, we thus focused on feasibility (i.e., calculability) of
elementary problems such as naming and graph exploration. To this end, we
studied various models with for objective either to determine the minimal
hypotheses of a model for the realization of a task, or to determine for a
given model, the maximum corruption the robots can possibly tolerate.
This study of robot self-stabilization underlines the fact that it is possible
to develop self-stabilizing solutions for robots within the framework of a
very constrained environment, including maximal hypotheses on the robots and
system corruption, and minimal hypotheses on the strength of the model.
###### Organization of Part II.
Chapter 4 presents my main results obtained in a model where the faults can be
generated by the robots _and_ by the network. These results include
impossibility results as well as determinist and probabilistic algorithms, for
various problems including _naming_ and _election_. My related paper is [24].
Chapter 5 summarizes then my work on the search for minimal hypotheses in the
discrete CORDA model enabling to achieve a task (in a self-stabilizing
manner). It is demonstrated that, in spite of the weakness of the model, it is
possible for robots to perform sophisticated tasks, among which is _perpetual
exploration_. My related paper is [18].
###### Perspectives.
The document opens a certain number of long-term research directions, detailed
in Chapter 6. My research perspectives get organized around the study of the
tradeoff between the memory space used by the nodes of a network, the
convergence time of the algorithm, and the quality of the retuned solution.
### Introduction
Dans le contexte des réseaux à grande échelle, la prise en compte des _pannes_
est une nécessité évidente. Ce document s’intéresse à l’approche _auto-
stabilisante_ qui vise à concevoir des algorithmes se réparant d’eux-même en
cas de fautes transitoires, c’est-à-dire de pannes impliquant la modification
arbitraire de l’état des processus. Il se focalise sur deux contextes
différents, couvrant la majeure partie de mes travaux de recherche ces
dernières années. La première partie du document (partie I) est consacrée à
l’algorithmique auto-stabilisante pour les _réseaux de processus_. La seconde
partie du document (partie II) est consacrée quant à elle à l’algorithmique
auto-stabilisante pour des _entités autonomes_ (agents logiciels, robots,
etc.) se déplaçant dans un réseau.
###### Arbres couvrants sous contraintes.
La première partie se caractérise par deux aspects spécifiques. Le premier est
lié à la nature des problèmes considérés. Le second est lié à un soucis
d’optimisation des performances des algorithmes. En effet, dans le cadre de la
construction d’arbres couvrants, l’auto-stabilisation s’est historiquement
principalement focalisée sur les constructions les plus classiques, à savoir
arbres BFS, arbres DFS, ou arbres de plus courts chemins. Nous nous sommes
intéressés à la construction d’arbres dans un cadre plus vaste, impliquant des
contraintes _globales_ , dans des réseaux statiques ou dynamiques. Nous avons
en particulier contribué au développement d’algorithmes pour la construction
auto-stabilisante d’arbres couvrants de degré minimum, d’arbres couvrants de
poids minimum (MST), ou d’arbres de Steiner. Par ailleurs, notre approche de
l’auto-stabilisation ne vise pas seulement la faisabilité mais inclut
également la recherche d’algorithmes _efficaces_. La principale mesure de
complexité visée est la mémoire utilisée par chaque processus. Nous avons
toutefois considéré également d’autres mesures, comme le temps de convergence
ou la quantité d’information échangée entre les processus.
De cette étude de la construction efficace d’arbres couvrants, nous mettons en
évidence deux enseignements. D’une part, l’auto-stabilisation semble avoir un
spectre d’applications aussi large que le réparti. Nos travaux démontrent que
c’est effectivement le cas dans le domaine de la construction d’arbres
couvrants. D’autre part, et surtout, nos travaux sur la complexité mémoire des
algorithmes semblent indiquer que l’auto-stabilisation n’implique pas de coût
supplémentaire. A titre d’exemple caractéristique, construire un MST en
réparti nécessite une mémoire de $\Omega(\log n)$ bits par processus (ne
serait-ce que pour stocker le parent dans son arbre). Nous verrons dans ce
document qu’il est possible de concevoir un algorithme auto-stabilisante de
construction de MST utilisant $O(\log n)$ bits de mémoire par processus.
###### Organisation de la partie I.
Le chapitre 1 rappelle les grandes lignes de la théorie de l’auto-
stabilisation, et décrit les notions élémentaires de théorie des graphes
utilisées dans ce document. Il dresse en particulier un bref état de l’art des
algorithmes auto-stabilisants pour la construction d’arbres couvrants
spécifiques ou optimisant des critères non considérés dans les chapitres
suivants. Le chapitre 2 présente mes travaux sur la construction d’arbres
couvrant de poids minimum (MST). Enfin le chapitre 3 a pour objet de présenter
mes travaux sur la construction d’arbres couvrants optimisés, différents du
MST, tel que l’arbre de degré minimum, l’arbre de Steiner, etc.
###### Entités autonomes.
La seconde partie du document est consacrée à l’algorithmique répartie auto-
stabilisante pour les _entités autonomes_. Ce terme désigne toute entité de
calcul susceptible de se déplacer dans un espace selon certaines contraintes.
Nous sous-entendrons le plus souvent des robots physiques se déplaçant dans un
espace discret ou continu. Nous pourrons toutefois parfois faire référence à
des contextes s’appliquant à des agents logiciels dans un réseau. Par abus de
langage, nous utiliserons la terminologie brève et imagée de _robot_ dans tous
les cas. L’auto-stabilisation est une technique générique pour tolérer toute
défaillance transitoire dans un système réparti qu’il est évidemment
envisageable de généraliser au cadre où les algorithmes sont exécutés par des
robots (on parle alors souvent plutôt d’ _auto-organisation_ que d’auto-
stabilisation). Il convient de noter de fortes similitudes entre
l’algorithmique auto-stabilisante pour les réseaux de processus et celle pour
les robots. Par exemple, la notion de circulation de _jetons_ dans le premier
cadre semble corrélée à la circulation de _robots_ dans le second cadre. De
manière similaire, on ne peut manquer de noter une similitude entre _parcours_
de graphes par des messages, et _exploration_ par des robots. L’équivalence
(en terme de calculabilité) entre le modèle par passage de messages et celui
par agents a déjà été mis en évidence dans la littérature [12, 29]. Ce
document semble indiquer une généralisation de cet état de fait à l’auto-
stabilisation. Les connaissances en algorithmique auto-stabilisante pour les
robots ne sont toutefois pas encore suffisamment élaborées pour établir cette
généralisation. De même, la relative jeunesse de l’auto-stabilisation pour les
robots ne permet de traiter de questions d’efficacité (i.e., complexité) que
difficilement. Dans ce document, nous nous sommes surtout focalisée sur la
faisabilité (i.e., calculabilité) de problèmes élémentaires tels que le
nommage ou l’exploration. A cette fin, nous avons étudié différents modèles
avec pour objectif soit de déterminer les hypothèses minimales d’un modèle
pour la réalisation d’une tâche, soit de déterminer, pour un modèle donné, la
corruption maximum qu’il est possible de toléré.
De cette étude de l’auto-stabilisation pour les robots, nous mettons en
évidence un enseignement principal, à savoir qu’il reste possible de
développer des solutions auto-stabilisantes dans le cadre d’environnements
très contraignants, incluant des hypothèses maximales sur la corruption des
robots et du système, et des hypothèses minimales sur la force du modèle.
###### Organisation de la partie II.
Le chapitre 4 présente mes principaux résultats obtenus dans un modèle où les
fautes peuvent être générées par le réseau et par les robots eux-mêmes. Ces
résultats se déclinent en résultats d’impossibilité, et en algorithmes
déterministes ou probabilistes, ce pour les problèmes du _nommage_ et de l’
_élection_. Le chapitre 5 résume ensuite mon travail sur la recherche
d’hypothèses minimales dans le modèle CORDA discret permettant de réaliser une
tâche. Il y est en particulier démontré que malgré la faiblesse du modèle, il
reste encore possible pour des robots d’effectuer des tâches sophistiquées,
dont en particulier l’ _exploration perpétuelle_.
###### Perspectives.
Le document ouvre un certain nombre de perspectives de recherche à long terme,
détaillées dans le chapitre 6. Ces perspectives s’organisent autour de l’étude
du compromis entre l’espace utilisé par les nœuds d’un réseau, le temps de
convergence de l’algorithme, et la qualité de la solution retournée.
## Part I Arbres couvrants sous contraintes
### Chapter 1 Algorithmes auto-stabilisants et arbres couvrants
Ce chapitre rappelle les grandes lignes de la théorie de l’auto-stabilisation,
et décrit les notions élémentaires de théorie des graphes utilisées dans ce
document. Il dresse en particulier un bref état de l’art des algorithmes auto-
stabilisants pour la construction d’arbres couvrants spécifiques (BFS, DFS,
etc.) ou optimisant des critères non considérés dans la suite du document
(diamètre minimal, etc.).
#### 1.1 Eléments de la théorie de l’auto-stabilisation
Une panne (appelée aussi _faute_) dans un système réparti désigne une
défaillance temporaire ou définitive d’un ou plusieurs composants du système.
Par composants, nous entendons essentiellement processeurs, ou liens de
communications. Il existe principalement deux catégories d’algorithmes
traitant des pannes : les algorithmes _robustes_ [126] et les algorithmes
_auto-stabilisants_. Les premiers utilisent typiquement des techniques de
redondance de l’information et des composants (communications ou processus).
Ce document s’intéresse uniquement à la seconde catégorie d’algorithmes, et
donc à l’approche auto-stabilisante. Cette approche vise à concevoir des
algorithmes se réparant eux-même en cas de fautes transitoires.
Dijkstra [48] est considéré comme le fondateur de la théorie de l’auto-
stabilisation. Il définit un système auto-stabilisant comme un système qui,
quelque soit son état initial, est capable de retrouver de lui même un état
_légitime_ en un nombre fini d’étapes. Un état légitime est un état qui
respecte la spécification du problème à résoudre. De nombreux ouvrages ont été
écrits dans ce domaine [51, 128, 46]. Je ne ferai donc pas une présentation
exhaustive de l’auto-stabilisation, mais rappellerai uniquement dans ce
chapitre les notions qui seront utiles à la compréhension de ce document.
Un _système réparti_ est un réseau composé de processeurs, ou _nœud_ , (chacun
exécutant un unique processus), et de mécanismes de communication entre ces
nœuds. Un tel système est modélisé par un graphe non orienté. Si les nœuds
sont indistingables, le réseau est dit _anonyme_. Dans un système non-anonyme,
les nœuds disposent d’identifiants distincts deux-à-deux. Si tous les nœuds
utilisent le même algorithme, le système est dit _uniforme_. Dans le cas
contraire, le système est dit _non-uniforme_. Lorsque quelques nœuds exécutent
un algorithme différent de l’algorithme exécuté par tous les autres, le
système est dit _semi-uniforme_. L’exemple le plus classique d’un algorithme
semi-uniforme est un algorithme utilisant un nœud distingué, par exemple comme
racine pour la construction d’un arbre couvrant.
L’hypothèse de base en algorithmique répartie est que chaque nœud peut
communiquer avec tous ses voisins dans le réseau. Dans le contexte de l’auto-
stabilisation, trois grands types de mécanismes de communication sont
considérés: (i) le modèle _à états_ , dit aussi modèle _à mémoires partagées_
[48], (ii) le modèle _à registres partagées_ [55], et (iii) le modèle _par
passage de messages_ [126, 107, 118] Dans le modèle à état, chaque nœud peut
lire l’état de tous ses voisins et mettre à jour son propre état en une étape
atomique. Dans le modèle à registres partagés, chaque nœud peut lire le
registre d’un de ses voisins, ou mettre à jour son propre état, en une étape
atomique, mais pas les deux à la fois. Dans le modèle par passage de messages
un nœud envoie un message à un de ses voisins ou reçoit un message d’un de ses
voisins (pas les deux à la fois), en une étape atomique. Les liens de
communication sont généralement considérés comme FIFO, et les messages sont
traités dans leur ordre d’arrivée. Dans son livre [107], Peleg propose une
classification des modèles par passage de messages. Le modèle ${\cal CONGEST}$
est le plus communément utilisé dans ce document. Il se focalise sur le volume
de communications communément admis comme raisonnable , à savoir $O(\log n)$
bits par message, où $n$ est le nombre de nœuds dans le réseau. Notons que si
les nœuds possèdent des identifiants deux-à-deux distincts entre 1 et $n$,
alors $\log n$ bits est la taille minimum requise pour le codage de ces
identifiants. Avec une taille de messages imposée, on peut alors comparer le
temps de convergence (mesuré en nombre d’étapes de communication) et le nombre
de messages échangés.
Notons qu’il existe des transformateurs pour passer d’un modèle à un autre,
dans le cas des graphes non orientés [51]. L’utilisation de l’un ou l’autre
des modèles ci-dessus n’est donc pas restrictive.
Si les temps pour transférer une information d’un nœud à un voisin (lire un
registre, échanger un message, etc.) sont identiques, alors le système est dit
_synchrone_. Sinon, le système est dit _asynchrone_. Si les temps pour
transférer une information d’un nœud à un voisin sont potentiellement
différent mais qu’une borne supérieure sur ces temps est connue, alors le
système est dit _semi-synchrone_. Dans les systèmes asynchrones, il est
important de modéliser le comportement individuel de chaque nœud. Un nœud est
dit _activable_ dès qu’il peut effectuer une action dans un algorithme donné.
Afin de modéliser le comportement des nœuds activables, on utilise un
ordonnanceur, appelé parfois _démon_ ou _adversaire_ , tel que décrit dans
[42, 89, 41]. Dans la suite du document, le terme d’adversaire est utilisé.
L’adversaire est un dispositif indépendant des nœuds, et possédant une vision
globale. A chaque pas de calcul, il choisit les nœuds susceptibles d’exécuter
une action parmi les nœuds activables. L’adversaire est de puissance variable
selon combien de processus activables peuvent être activés à chaque pas de
calcul:
* •
l’adversaire est dit _central_ (ou séquentiel) s’il n’active qu’un seul nœud
activable;
* •
l’adversaire est dit _distribué_ s’il peut activer plusieurs nœuds parmi ceux
qui sont activables;
* •
l’adversaire est dit _synchrone_ (ou parallèle) s’il doit activer tous les
nœuds activables.
L’adversaire est par ailleurs contraint par des hypothèse liées à l’équité de
ses choix. Les contraintes d’équité les plus courantes sont les suivantes:
* •
l’adversaire est dit _faiblement équitable_ s’il doit ultimement activer tout
nœud continument et infiniment activable;
* •
l’adversaire _fortement équitable_ s’il doit ultimement activer tout nœud
infiniment activable;
L’adversaire est dit _inéquitable_ s’il n’est pas équitable (ni fortement, ni
faiblement). Les modèles d’adversaires ci-dessus sont plus ou moins
contraignants pour le concepteur de l’algorithme. Il peut également découler
différents résultats d’impossibilité de ces différents adversaires .
On dit qu’un algorithme a _convergé_ (ou qu’il a _terminé_) lorsque son état
global est conforme à la spécification attendue, comme par exemple la présence
d’un unique leader dans le cas du problème de l’élection. Dans le modèle à
états ou celui à registres partagés, un algorithme est dit _silencieux_ [52]
si les valeurs des variables locales des nœuds ne changent plus après la
convergence. Dans un modèle à passage de messages, un algorithme est dit
silencieux s’il n’y a plus de circulation de messages, ou si le contenu des
messages échangés ne changent pas après convergence. Dolev, Gouda et Shneider
[52] ont prouvé que, dans un modèle à registres, la mémoire minimum requise
parun algorithme auto-stabilisant silencieux de construction d’arbre couvrant
est $\Omega(\log n)$ bits sur chaque nœud.
Les performances des algorithmes se mesurent à travers de leur complexité en
mémoire (spatiale) et de leur temps de convergence. On établit la performance
en mémoire en mesurant l’espace mémoire occupé en chaque nœud, et/ou en
mesurant la taille des messages échangés. Le temps de convergence d’un
algorithme est le temps qu’il met à atteindre la spécification demandée après
une défaillance. L’unité de mesure du temps la plus souvent utilisée en auto-
stabilisation est la _ronde_ [56, 37]. Durant une ronde, tous les nœuds
activables sont activés au moins une fois par l’adversaire. Notons que le
définition de ronde dépend fortement de l’équité de l’adversaire.
#### 1.2 Construction d’arbres couvrants
La construction d’une structure de communication efficace au sein de réseaux à
grande échelle (grilles de calculs, ou réseaux pair-à-pairs) ou au sein de
réseaux dynamiques (réseaux ad hoc, ou réseaux de capteurs) est souvent
utilisée comme brique de base permettant la réalisation de tâches élaborées.
La structure de communication la plus adaptée est souvent un arbre. Cet arbre
doit couvrir tout ou partie des nœuds, et posséder un certain nombre de
caractéristiques dépendant de l’application. Par ailleurs, la construction
d’arbres couvrants participe à la résolution de nombreux problèmes
fondamentaux de l’algorithmique répartie. Le problème de la construction
d’arbres couvrants a donc naturellement été très largement étudié aussi bien
en réparti qu’en auto-stabilisation. L’objectif général de la première partie
du document concerne les algorithmes auto-stabilisants permettant de maintenir
un sous-graphe couvrant particulier, tel qu’un arbre couvrant, un arbre de
Steiner, etc. Ce sous-graphe peut être potentiellement dynamiques: les nœuds
et les arêtes peuvent apparaitre ou disparaitre, les poids des arêtes peuvent
évoluer avec le temps, etc.
##### 1.2.1 Bref rappel de la théorie des graphes
Cette section présente quelques rappels élémentaires de théorie des graphes.
Dans un graphe $G=(V,E)$, un chemin est une suite de sommets
$u_{0},u_{1},\dots,u_{k}$ où $\\{u_{i},u_{i+1}\\}\in E$ pour tout
$i=0,\dots,k-1$. Un chemin est dit élémentaire si $u_{i}\neq u_{j}$ pour tout
$i\neq j$. Par défaut, les chemins considérés dans ce document sont, sauf
indication contraire, élémentaires. Les sommets $u_{0}$ et $u_{k}$ sont les
extrémités du chemin. Un cycle (élémentaire) est un chemin (élémentaire) dont
les deux extrémités sont identiques. En général, on notera $n$ le nombre de
sommets du graphe. Un graphe connexe est un graphe tel qu’il existe un chemin
entre toute paire de sommets. Un arbre est un graphe connexe et sans cycle.
###### Lemma 1.
Soit $T=(V,E)$ un graphe. Les propriétés suivantes sont équivalentes:
* •
$T$ est un arbre;
* •
$T$ est connexe et sans cycle;
* •
Il existe un _unique_ chemin entre toute paire de sommets de $T$;
* •
$T$ est connexe et la suppression d’une arête quelconque de $T$ suffit à
déconnecter $T$;
* •
$T$ est sans cycle et l’ajout d’une arête entre deux sommets non adjacents de
$T$ crée un cycle;
* •
$T$ est connexe et possède $n-1$ arêtes.
On appelle _arbre couvrant_ de $G=(V,E)$ tout arbre $T=(V,E^{\prime})$ avec
$E^{\prime}\subseteq E$. Dans un graphe $G=(V,E)$, un _cocycle_ est défini par
un ensemble $A\in V$; il contient toutes les arêtes $\\{u,v\\}$ de $G$ tel que
$u\in A$ et $v\notin A$. Les deux définitions ci-dessous sont à la base de la
plupart des algorithmes de construction d’arbres couvrants.
###### Définition 1 (Cycle élémentaire associé).
Soit $T=(V,E_{T})$ un arbre couvrant de $G=(V,E)$, et soit $e\in E\setminus
E_{T}$. Le sous-graphe $T^{\prime}=(V,E_{T}\cup\\{e\\})$, contient un unique
cycle appelé _cycle élémentaire_ associé à $e$, noté $C_{e}$.
###### Définition 2 (Echange).
Soit $T=(V,E_{T})$ un arbre couvrant de $G=(V,E)$, et soit $e\notin E_{T}$ et
$f\in C_{e}$, $f\neq e$. L’opération qui consiste à échanger $e$ et $f$ est
appelée _échange_. De cet échange résulte l’arbre couvrant $T^{\prime}$ où
$E_{T^{\prime}}=E_{T}\cup\\{e\\}\setminus\\{f\\}$.
(a) Arbre $T$ contenant l’arête $e$
(b) Arbre $T^{\prime}$ contenant l’arête $f$
Figure 1.1: Echange
##### 1.2.2 Bref état de l’art d’algorithmes auto-stabilisants pour la
construction d’arbres couvrants
Un grand nombre d’algorithmes auto-stabilisants pour la construction d’arbre
couvrants ont été proposés à ce jour. Gartner [72] et Rovedakis [117] ont
proposé un état de l’art approfondi de ce domaine. Cette section se contente
de décrire un état de l’art partiel, qui ne traite pas des problèmes abordés
plus en détail dans les chapitres suivants (c’est-à-dire la construction
d’arbres couvrants de poids minimum, d’arbres couvrants de degré minimum,
d’arbres de Steiner, etc.). Le tableau 1.1 résume les caractéristiques des
algorithmes présentés dans cette section.
###### 1.2.2.1 Arbre couvrant en largeur d’abord
Dolev, israeli et Moran [54, 55] sont parmi les premiers à avoir proposé un
algorithme auto-stabilisant de construction d’arbre. Leur algorithme construit
un _arbre couvrant en largeur d’abord_ (BFS pour Breadth First Search en
anglais). Cet algorithme est semi-uniforme, et fonctionne par propagation de
distance. C’est une brique de base pour la conception d’un algorithme auto-
stabilisant pour le problème de l’exclusion mutuelle dans un réseau
asynchrone, anonyme, et dynamique. Le modèle de communication considéré est
par registres, avec un adversaire centralisé. Les auteurs introduisent la
notion de _composition équitable_ d’algorithmes. Ils introduisent également
l’importante notion d’ _atomicité lecture/écriture_ décrite plus haut dans ce
document.
Afek, Kutten et Yung [2] ont proposé un algorithme construisant un BFS dans un
réseau non-anonyme. La racine de l’arbre couvrant est le nœud d’identifiant
maximum. Chaque nœud met à jour sa variable racine. Les configurations
erronées vont être éliminées grâce à cette variable. Dès qu’un nœud s’aperçoit
qu’il n’a pas la bonne racine, il commence par se déclarer racine lui même, et
effectue ensuite une demande de connexion en inondant le réseau. Cette
connexion sera effective uniquement après accusé de réception par la racine
(ou par un nœud qui se considère de façon erronée comme une racine). Afek et
Bremler-Barr [1] ont amélioré l’approche proposée dans [2]. Dans [2], la
racine élue pouvait ne pas se trouver dans le réseau car la variable racine
peut contenir un identifiant maximum erroné après une faute. Dans [1], la
racine est nécessairement présente dans le réseau. Datta, Larmore et Vemula
[44] ont proposé un algorithme auto-stabilisant reprenant l’approche de Afek
et Bremler-Barr [1]. Ils construisent de manière auto-stabilisante un BFS afin
d’effectuer une élection. Pour ce faire, ils utilisent des vagues de couleurs
différentes afin de contrôler la distance à la racine, ainsi qu’un mécanisme
d’accusé de réception afin d’arrêter les modifications de l’arbre. C’est donc
en particulier un algorithme silencieux [52].
Arora et Gouda [5, 6] ont présenté un système de réinitialisation après faute,
dans un réseau non anonyme. Ce système en couches utilise trois algorithmes:
un algorithme d’élection, un algorithme de construction d’arbre couvrant, et
un algorithme de diffusion. Les auteurs présentent une solution silencieuse et
auto-stabilisante pour chacun des trois problèmes. Comme un certain nombre
d’autres auteurs par la suite ([81, 26, 22]) ils utilisent la connaissance a
priori d’une borne supérieure sur le temps de communication entre deux nœuds
quelconques dans le réseau afin de pouvoir éliminer les cycles résultant d’un
configuration erronées après une faute.
Huang et Chen [82] ont proposé un algorithme semi-uniforme de construction
auto-stabilisante de BFS. Leur contribution la plus importante reste toutefois
les nouvelles techniques de preuves d’algorithmes auto-stabilisants qu’ils
proposent dans leur article.
Enfin, dans un cadre dynamique, Dolev [50] a proposé un algorithme auto-
stabilisant de routage, et un algorithme auto-stabilisant d’élection. Pour
l’élection, chaque nœud devient racine d’un BFS. La contribution principale
est le temps de convergence de chaque construction de BFS, qui est optimal en
$O(D)$ rondes où $D$ est le diamètre du graphe. De manière indépendante,
Aggarwal et Kutten [3] ont proposé un algorithme de construction d’un arbre
couvrant enraciné au nœud de plus grand identifiant, optimal en temps de
convergence, $O(D)$ rondes.
###### 1.2.2.2 Arbre couvrant en profondeur d’abord
La même approche que Dolev, israeli et Moran [54, 55] a été reprise par Collin
et Dolev [36] afin de concevoir un algorithme de construction d’arbres
couvrants en profondeur d’abord (DFS, pour Depth Fisrt Search en anglais).
Pour cela, ils ont utilisé un modèle faisant référence à des numéros de port
pour les arêtes. Chaque nœud $u$ connaît le numéro de port de chaque arête
$e=\\{u,v\\}$ incidente à $u$, ainsi que le numéro de port de $e$ en son autre
extrémité $v$. Avec cette connaissance, un ordre lexicographique est créé pour
construire un parcours DFS.
La construction auto-stabilisante d’arbres couvrants en profondeur d’abord va
souvent de paire dans la littérature avec le parcours de jeton. Huang et Chen
[83] ont proposé un algorithme auto-stabilisant pour la circulation d’un jeton
dans un réseau anonyme semi-uniforme. Le jeton suit un parcours en profondeur
aléatoire111Les auteurs précisent que l’algorithme peut être modifié pour
obtenir un parcour déterministe. L’algorithme nécessite toutefois la
connaissance a priori de la taille du réseau. Huang et Wuu [84] ont proposé un
autre algorithme auto-stabilisant pour la circulation d’un jeton, cette fois
dans un réseau anonyme uniforme. Ce second algorithme nécessite également la
connaissance a priori de la taille du réseau. L’algorithme de Datta, Johnen,
Petit et Villain [43], contrairement aux deux algorithmes précédents, ne fait
aucune supposition a priori sur le réseau. De plus, cet algorithme améliore la
taille mémoire de chaque nœud en passant de $O(\log n)$ bits à $O(\log\Delta)$
bits, où $\Delta$ est le degré maximum du réseau. Notons que tous ces
algorithmes de circulation de jeton sont non-silencieux car le jeton
transporte une information qui évolue le long du parcours.
###### 1.2.2.3 Arbre couvrant de plus court chemin
Le problème de l’arbre couvrant de plus court chemin (SPT pour Shortest Path
Tree en anglais) est la version pondéré du BFS: la distance à la racine dans
l’arbre doit être égale à la distance à la racine dans le graphe. Dans ce
cadre, Huang et Lin [85] ont proposé un algorithme semi-uniforme auto-
stabilisant pour ce problème. Chaque nœud calcule sa distance par rapport à
tous ses voisins à la Dijkstra. Un nœud $u$ choisit pour parent son voisin $v$
qui minimise $d_{v}+w(u,v)$, où $d_{v}$ est la distance supposée de $v$ à la
racine (initialisée à zéro), et $w(u,v)$ le poids de l’arête $u,v$. Dans un
cadre dynamique, le poids des arêtes peut changer au cours du temps. Johnen et
Tixeuil [88] ont proposé deux algorithmes auto-stabilisants de construction
d’arbres couvrants. Le principal apport de leur approche est de s’intéresser à
la propriété _sans-cycle_ introduite par [68]. Cette propriété stipule que
l’arbre couvrant doit s’adapter aux changements de poids des arêtes sans se
déconnecter ni créer de cycle. Cette approche est développée plus en détail
dans la section 3.1 en rapport avec mes propres contributions. Gupta et
Srimani [78] supposent le même dynamisme que Johnen et Tixeuil [88]. Ils ont
proposé plusieurs algorithmes auto-stabilisants, dont un algorithme semi-
uniforme construisant un arbre SPT. Le principal apport de cette dernière
contribution est de fournir un algorithme auto-stabilisant silencieux, optimal
en espace et en temps de convergence.
Burman et Kutten [26] se sont intéressés à un autre type de dynamisme:
l’arrivée et/ou le départ des nœuds, et/ou du arêtes du réseau. De plus, ces
auteurs ont proposé d’adapter l’atomicité lecture/écriture du modèle par
registre au modèle par passage de message. Ce nouveau concept est appelé
_atomicité envoi/réception_ (send/receive atomicity). Leur algorithme de
construction de SPT s’inspire de l’algorithme auto-stabilisant proposé par
Awerbuch et al. [10] conçu pour réinitialiser le réseau après un changement de
topologie (ce dernier algorithme utilise un algorithme de construction de SPT
comme sous-procédure).
###### 1.2.2.4 Arbre couvrant de diamètre minimum
Bui, Butelle et Lavault [27] se sont intéressés au problème de l’arbre
couvrant de diamètre minimum. De manière surprenante, ce problème a été peu
traité dans la littérature auto-stabilisante. L’algorithme dans [27] est conçu
dans un cadre pondéré, où le poids des arêtes sont positifs. Les auteurs
prouvent que ce problème est équivalent à trouver un _centre_ du réseau (un
nœud dont la distance maximum à tous les autres nœuds est minimum). Une fois
un centre identifié, l’algorithme calcule l’arbre couvrant de plus court
chemin enraciné à ce centre.
#### 1.3 Récapitulatif et problèmes ouverts
Le tableau 1.1 résume les caractéristiques des algorithmes évoqués dans ce
chapitre. Améliorer la complexité en espace ou en temps de certains
algorithmes de ce tableau sont autant de problèmes ouverts. Rappelons que la
seule borne inférieure non triviale en auto-stabilisation pour la construction
d’arbres couvrants n’est valide que dans le cadre silencieux (voir [53]).
Les deux chapitres suivants sont consacrés à la construction d’arbres
couvrants optimisant des métriques particulières, incluant en particulier
* •
les arbres couvrants de poids minimum,
* •
les arbres couvrants de degré minimum,
* •
les arbres de Steiner,
* •
les constructions bi-critères (poids et degré),
* •
etc.
| Articles | Semi-uniforme | Anonyme | Connaissance | Communications | Adversaire | Equité | Atomicité | Espace mémoire | Temps de convergence | Propriété
---|---|---|---|---|---|---|---|---|---|---|---
ST | [3] | | | | R | C | $f$ | $\oplus$ | $O(\log n)$ | $O(D)$ | dyn
| [83] | $\checkmark$ | $\checkmark$ | $n$ | R | D | | | $O(\log\Delta n)$ | |
DFS | [36] | $\checkmark$ | $\checkmark$ | | R | C | $f$ | $\oplus$ | $O(n\log\Delta)$ | $O(Dn\Delta)$ |
| [84] | | $\checkmark$ | $n$ | R | C | $f$ | | $O(\log n)$ | |
| [43] | $\checkmark$ | $\checkmark$ | | R | D | $f$ | | $O(\log\Delta)$ | $O(Dn\Delta)$ |
| [54] | $\checkmark$ | $\checkmark$ | | R | C | $f$ | $\oplus$ | $O(\Delta\log n)$ | $O(D)$ | dyn
| [5] | | | $n$ | R | C | | | $O(\log n)$ | $O(n^{2})$ | dyn
BFS | [2] | | | | R | D | $f$ | $\oplus$ | $O(\log n)$ | $O(n^{2})$ | dyn
| [82] | $\checkmark$ | $\checkmark$ | $n$ | R | D | $I$ | | | |
| [50] | | | | R | C | $f$ | $\oplus$ | $O(\Delta n\log n)$ | $\Theta(D)$ | dyn
| [1] | | | B222L’algorithme est semi-synchrone | M | D | | | $O(\log n)$ | $O(n)$ |
| [44] | | | | R | D | $f$ | $\oplus$ | $O(\log n)$ | $O(n)$ |
| [85] | $\checkmark$ | $\checkmark$ | | R | C | $I$ | | $O(\log n)$ | |
SPT | [88] | $\checkmark$ | $\checkmark$ | | R | D | $f$ | | $O(\log n)$ | | dyn
| [78] | $\checkmark$ | $\checkmark$ | | M/R | D | | | $O(\log n)$ | $O(D)$ |
| [26] | | | $D$ | M | D | | $\oplus$ | $O(\log^{2}n)$ | $O(D)$ | dyn
MDiam | [27] | | $\checkmark$ | | M | D | | | $O(n^{2}\log n$ | $O(n\Delta+d^{2}$ |
| | | | | | | | | $+n\log W)$ | $+n\log^{2}n)$ |
Table 1.1: Algorithmes auto-stabilisants asynchrones pour la construction
d’arbres couvrants. $D$ diamètre du graphe. $\Delta$ degré maximum du graphe.
ST: arbre couvrant; DFS: arbre en profondeur d’abord; BFS: arbre en largeur
d’abord; SPT: arbre de plus court chemin; MDiam: arbre de diamètre minimum; R:
registres partagés; M: passage de messages; Adversaire: Distribué (D), central
(C), inéquitable ($I$), faiblement équitable ($f$), fortement équitable ($F$).
Atomicité: $\oplus$ lecture ou écriture. Propriété: dynamique (dyn), sans-
cycle (SC).
### Chapter 2 Arbres couvrants de poids minimum
Ce chapitre a pour objet de présenter mes travaux sur la construction d’arbres
couvrants de poids minimum (Minimum Spanning Tree: MST). Ce problème est un de
ceux les plus étudiés en algorithmique séquentielle comme répartie. De façon
formelle, le problème est le suivant.
###### Définition 3 (Arbre couvrant de poids minimum).
Soit $G=(V,E,w)$ un graphe non orienté pondéré111Dans la partie répartie et
auto-stabilisante du document nous supposerons que les poids des arêtes sont
positifs, bornés et peuvent être codés en $O(\log n)$ bits, où $n$ est le
nombre de nœuds du réseau.. On appelle arbre couvrant de poids minimum de $G$
tout arbre couvrant dont la somme des poids des arêtes est minimum.
Ce chapitre est organisé de la façon suivante. La première section consiste en
un très bref état de l’art résumant les principales techniques utilisées en
séquentiel. La section suivante est consacrée à un état de l’art des
algorithmes répartis existant pour la construction de MST. La section 2.3 est
le cœur de ce chapitre. Elle présente un état de l’art exhaustif des
algorithmes auto-stabilisants pour le MST, ainsi que deux de mes contributions
dans ce domaine.
#### 2.1 Approches centralisées pour le MST
Cette section est consacrée à un état de l’art partiel des algorithmes
centralisés pour la construction de MST, et des techniques les plus couramment
utilisées pour ce problème.
Dans un contexte centralisé, trouver un arbre couvrant de poids minimum est
une tâche qui se résout en temps polynomial, notamment au moyen d’algorithmes
gloutons. Les premiers algorithmes traitant du problèmes sont nombreux. Leur
historique est même sujet à débat. Bor̊uvka [25] apparait maintenant comme le
premier auteur à avoir publié sur le sujet. Les travaux de [112, 121, 101]
restent cependant plus connus et enseignés.
La construction d’un MST se fait en général sur la base de propriétés
classiques des arbres, et utilise la plupart du temps au moins une des deux
propriétés suivantes, mis en évidence dans [120]:
###### Propriété 1 (Bleu).
Toute arête de poids minimum d’un cocycle de $G$ fait partie d’un MST de $G$.
###### Propriété 2 (Rouge).
Toute arête de poids maximum d’un cycle de $G$ ne fait partie d’aucun MST de
$G$.
(a) Cocycle
(b) Cycle
Figure 2.1: Les figures ci-dessus illustrent les propriétés bleu et rouge.
Dans la figure 2.1(a), le cocycle est constitué par les arêtes de poids
$\\{2,10,8\\}$; l’arête de poids $2$ fera partie de l’unique MST de ce graphe.
Dans la figure 2.1(b), le cycle est constitué par les arêtes de poids
$\\{10,11,8,6\\}$; l’arête de poids $11$ ne fera pas parti de l’unique MST.
La plupart des algorithmes traitant du MST peuvent être classés en deux
catégories: ceux qui utilisent la propriété, ou règle, bleue, et ceux qui
utilisent la règle rouge. L’algorithme de Prim [112] est l’exemple même de
l’utilisation de la propriété bleue. Au départ un nœud $u$ est choisi
arbitrairement, et le cocycle séparant ce nœud $u$ des autres nœuds est
calculé; l’arête minimum de ce cocycle, notée $\\{u,v\\}$, fait parti du MST
final. L’algorithme calcule ensuite le cocycle séparant le sous-arbre induit
par les nœuds $u$ et $v$ des autres nœuds du graphe. Ainsi de suite jusqu’à
l’obtention d’un arbre couvrant. Cet arbre est un MST(voir Figure 2.2).
L’algorithme de Bor̊uvka [25], redécouvert par Sollin [121], procède de la
même manière à la différence près qu’il choisit initialement de calculer les
cocycles induits par chaque nœud. Ainsi, il calcule à chaque étape des
cocycles de plusieurs sous-arbres. Il apparait donc comme une solution
permettant un certain degré de parallélisme. C’est par exemple cette technique
qui est à la base du fameux algorithme réparti de Gallager, Humblet, et Spira
[69].
(a) Cocycle induit par $A$
(b) Induit par $A,B$
(c) Induit par $A,B,C$
Figure 2.2: Algorithme de Prim
L’algorithme de Kruskal est quant à lui basé sur la propriété rouge. Il
choisit des arêtes dans l’ordre croissant des poids tant que ces arêtes ne
forment pas de cycle. Notons que lorsqu’une arête forme un cycle, celle-ci est
nécessairement de poids maximum dans ce cycle puisque les poids ont été
choisis dans l’orde croissant. Cette arête ne fait donc pas partie d’aucun MST
(voir Figure 2.3).
(a) Arête 2
(b) Arête 3
(c) Arête 4
(d) Arête 5
(e) Arête 8
Figure 2.3: Algorithme de Kruskal
A ce jour, l’algorithme de construction centralisée de plus faible complexité
est celui de Pettie et Ramachandran [110], qui s’exécute en temps
$O(|E|\alpha(|E|,n))$, où $\alpha$ est l’inverse de la fonction d’Ackermann.
Des solutions linéaires en nombre d’arêtes existent toutefois, mais utilisent
des approches probabilistes [65, 92].
#### 2.2 Approches réparties pour le MST
Cette section est consacrée à un état de l’art partiel des algorithmes
répartis pour la construction de MST. Dans un contexte réparti, le premier
algorithme publié dans la littérature est [39, 40]. Toutefois, aucune analyse
de complexité n’est fourni dans cet article. La référence dans le domaine est
de fait l’algorithme de Gallager, Humblet et Spira [69]. Cet algorithme
asynchrone est optimal en nombre de messages échangés, $O(|E|+n\log n)$, et a
pour complexité temporelle $O(n\log n)$ étapes en synchrone. L’optimalité de
la complexité en message est une conséquence d’un résultat de [64, 9] qui
démontre que le nombre minimum de bits échangés afin de construire un MST est
de $\Omega(|E|+n\log n)$. L’algorithme de Gallager, Humblet et Spira a valu à
ses auteurs le prix Dijkstra en 2004. Il a défini les fondements des notions
et du vocabulaire utilisés par la communauté du réparti pour la construction
du MST.
L’algorithme de Gallager, Humblet et Spira est basé sur la propriété rouge, à
la manière de l’algorithme de Bor̊uvka-Sollin [25, 121]. Les sous-arbres
construits sont appelés _fragments_. Initialement, chaque nœud du système est
un fragment. Par la suite, chaque fragment _fusionne_ grâce à l’arête sortante
du fragment de poids minimum. Autrement dit, grâce à l’arête de poids minimum
du cocycle. La difficulté en réparti est de permettre à chaque nœud de
connaître le fragment auquel il appartient, et d’identifier les arêtes à
l’extérieur du fragment et les arêtes à l’intérieur de celui-ci. A cette fin,
les nœuds ont besoin d’une vision globale , autrement dit d’effectuer un
échange d’information afin de maintenir à jour leur appartenance aux
différents fragments au fur et à mesure des fusions.
L’optimalité en message de la construction d’un MST étant obtenue par
l’algorithme [69], la suite des travaux dans ce domaine s’est attachée à
diminuer la complexité en temps. Les travaux dans ce domaine ont pour objet
principal de contrôler la taille des fragments afin d’améliorer la rapidité de
la mise à jour des nœuds, et donc la complexité en temps. La première
amélioration notable est celle d’Awerbuch [8] en temps $O(n)$ étapes, tout en
restant optimal en nombre de messages échangés. Dans la fin des années 90,
Garay, Kutten et Peleg relancent la recherche dans ce domaine. Dans [70], ils
obtiennent un temps $O(D+n^{0.614})$ étapes, où $D$ est le diamètre du graphe.
Cette complexité a été améliorée dans [102] en $O(D+\sqrt{n}\log^{*}n)$
étapes, au détriment du nombre de messages échangés, qui devient
$O(|E|+n^{3/2})$. Dans [108], il est prouvé une borne inférieure
$\widetilde{\Omega}(\sqrt{n})$ étapes dans le cas particulier des graphes de
diamètre $\Omega(\log n)$, où la notation $\widetilde{\Omega}$ signifie qu’il
n’est pas tenu compte des facteurs polylogarithmiques. Toujours dans le cas
des graphes de petit diamètre, Lotker, Patt-Shamir et Peleg [104] ont obtenu
une borne inférieure de $\widetilde{\Omega}(\sqrt[3]{n})$ étapes pour les
graphes de diamètre $3$, et $\widetilde{\Omega}(\sqrt[4]{n})$ pour graphe de
diamètre $4$. Dans les graphes de diamètre $2$ il existe un algorithme
s’exécutant en $O(\log n)$ étapes [104].
#### 2.3 Approches auto-stabilisantes
Cette section présente un état de l’art exhaustif de la construction auto-
stabilisante de MST. La présentation est faite de façon chronologique. Elle
débute donc par les travaux de Gupta et Srimani [77, 78], et de Higham et Lyan
[81]. Elle est suivie par deux contributions personnelles dans ce domaine [19,
17]. Elle est enfin conclue par les améliorations récentes apportées par
Korman, Kutten et Masuzawa [99]. Les caractéristiques de ces différents
algorithmes sont résumées dans la Table 2.1.
Articles | Système | Connaissance | Communications | Taille message | Espace mémoire | Temps de convergence | Non-silencieux | sans-cycle
---|---|---|---|---|---|---|---|---
[77, 78] | Semi | $n$ | M | $O(\log n)$ | $\Theta(n\log n)$ | $\Omega(n^{2})$ | |
[81] | Semi | $O(D)$ | M | $O(n\log n)$ | $O(\log n)$ | $O(n^{3})$ | $\checkmark$ |
[19] | A | | R | | $O(\log n)$ | $O(n^{3})$ | $\checkmark$ | $\checkmark$
[17] | A | | R | | $\Omega(\log^{2}n)$ | $O(n^{2})$ | |
[99] | A | | R | | $O(\log n)$ | $\mathbf{O(n)}$ | |
Table 2.1: Algorithmes auto-stabilisants pour le problème du MST. Dans cette
table, Semi signifie Semi-synchrone, et A signifie asynchrone. De même, M
signifie modèle par passage de messages, et R modèle à registres partagés.
##### 2.3.1 Algorithme de Gupta et Srimani
Le premier algorithme auto-stabilisant pour la construction d’un MST a été
publié par Gupta et Srimani [77, 78]. Cet article traite essentiellement d’une
approche auto-stabilisante pour le calcul des plus courts chemins entre toutes
paires de nœuds. Les auteurs utilisent ensuite ce résultat pour résoudre le
problème du MST. Ils considèrent des graphes dont les poids sont
uniques222Cette hypothèse n’est pas restrictive car on peut facilement
transformer un graphe pondéré à poids non distincts en un graphe pondéré à
poids deux-à-deux distincts. Il suffit par exemple d’ajouter au poids de
chaque arête l’identifiant le plus petit d’une de ses deux extrémités., et se
placent dans le cas de graphes dynamiques: le poids des arêtes peut évoluer
avec le temps, et les nœuds peuvent apparaître et disparaître. Les auteurs
utilisent un modèle par passage de messages similaire à un modèle par
registres partagés. Dans le modèle utilisé, chaque nœud $v$ envoie
périodiquement un message à ses voisins. Si $u$ reçoit un message d’un nœud
$v$ qu’il ne connaissait pas, il le rajoute à son ensemble de voisins. A
l’inverse, si un nœud $u$ n’a pas reçu de messages de son voisin $v$ au bout
d’un certain délai, alors $u$ considère que $v$ a quitté le réseau, et il
supprime donc ce nœud de la liste de ses voisins. L’algorithme a donc besoin
d’une borne sur le temps de communication entre deux nœuds. Il fonctionne donc
dans un système _semi-synchrone_. Si le réseau ne change pas pendant une
certaine durée, alors les messages échangés contiendront toujours la même
information. L’algorithme est donc un algorithme _silencieux_. Plus
spécifiquement, l’algorithme de Gupta et Srimani s’exécute de la façon
suivante. Fixons deux nœuds $u$ et $v$. Le nœud $u$ sélectionne l’arête $e$ de
poids $w(e)$ minimum parmi les arêtes de poids maximum de chaque chemin vers
$v$. Si $v$ est un nœud adjacent à $u$ et si $w(\\{u,v\\})=w(e)$ alors
$\\{u,v\\}$ est une arête du MST final. Cela revient à utiliser la propriété
bleu. En d’autres termes, chaque nœud calcule le cocycle de poids maximum, et
choisit l’arête de poids minimum de ce cocycle. Pour pouvoir calculer l’arête
de poids maximum de tous les chemins, les auteurs ont besoin de connaitre la
taille $n$ du réseau, et ils supposent que les nœuds ont des identifiants de 1
à $n$. Comme nous l’avons vu, chaque nœud conserve le poids de l’arête de
poids maximum allant à chaque autre nœud du réseau. Il a donc besoin d’une
mémoire de taille $\Theta(n\log n)$ bits. Le temps de convergence est
$\Omega(n^{2})$ rondes.
##### 2.3.2 Algorithme de Higham et Lyan
Higham et Lyan [81] ont proposé un algorithme _semi_ -synchrone dans le modèle
par passage de messages. Leur algorithme suppose que chaque nœud connaît une
borne supérieure $B$ sur le délai que met un message à traverser le réseau.
Cela revient à supposer un temps maximum de traversée d’une arête, donc à
considérer un réseau semi-synchrone.
L’algorithme utilise la propriété rouge, et fonctionne de la manière suivante.
Chaque arête doit déterminer si elle doit appartenir ou non au MST. Une arête
$e$ n’appartenant pas au MST inonde le graphe afin de trouver son cycle
élémentaire associé, noté $C_{e}$. Lorsque $e$ reçoit le message $m_{e}$ ayant
parcouru $C_{e}$, cette arête utilise les informations collectées par $m_{e}$,
c’est-à-dire les identifiants et les poids des arêtes se trouvant sur $C_{e}$.
Si $w_{e}$ n’est pas le poids le plus grand du cycle $C_{e}$, alors $e$ fait
parti du MST, sinon $e$ ne fait pas parti du MST. La borne supérieure $B$ est
utilisée comme un délai à ne pas dépasser. En effet si après un intervalle de
temps $B$, l’arête $e$ n’a reçu aucun message en retour de son inondation,
alors $e$ conclut que la structure existante n’est pas connexe. Elle décide
alors de devenir provisoirement une arête du MST, quitte à revoir sa décision
plus tard. Si une arête $e$ faisant partie provisoirement du MST ne reçoit pas
de message de recherche du cycle élémentaire $C_{e}$ au bout d’un temps $3B$,
alors elle peut considérer que toutes les arêtes font partie du MST, ce qui
une configuration erronée. Dans ce cas, $e$ déclenche un message de type
_trouver le cycle_. On remarque donc que, dans les deux cas, s’il existe au
moins une arête ne faisant pas parti de l’arbre couvrant, ou si toutes les
arêtes font partie de l’arbre, alors des messages sont générés. Cet algorithme
est donc non-silencieux.
En terme de complexité, chaque nœuds à besoin de $O(\log n)$ bits de mémoire
pour exécuter l’algorithme. La quantité d’information échangée (identifiants
et poids des nœuds des chemins parcourus) est de $O(n\log n)$ bits par
message.
##### 2.3.3 Contributions à la construction auto-stabilisante de MST
Cette section résume mes contributions à la conception d’algorithmes auto-
stabilisants de construction de MST. Elle présente en particulier deux
algorithmes. L’un est le premier algorithme auto-stabilisant pour le MST ne
nécessitant aucune connaissance a priori sur le réseau. De plus, cet
algorithme est entièrement asynchrone avec une taille mémoire et une taille de
message logarithmique. L’autre algorithme améliore ce premier algorithme en
optimisant le rapport entre le temps de convergence et la taille mémoire. Ces
deux algorithmes sont décrits dans les sous-sections suivantes.
###### 2.3.3.1 Transformation d’arbres de plus courts chemins en MST
Ma première contribution a été réalisée en collaboration avec Maria Potop-
Butucaru, Stéphane Rovedakis et Sébastien Tixeuil. Elle a été publié dans
[19]. L’apport de ce travail est multiple. D’une part, contrairement aux
travaux précédents (voir [77, 81, 78]) notre algorithme n’a besoin d’aucune
connaissance a priori sur le réseau. Il ne fait de plus aucune supposition sur
les délais de communication, et est donc asynchrone. Par ailleurs,
l’algorithme possède la propriété _sans-cycle_. Enfin il est le premier
algorithme à atteindre un espace mémoire de $O(\log n)$ bits avec des tailles
de message $O(\log n)$ bits.
La propriété sans-cycle est traités dans la section 3.1. Je ne vais donc
traiter ici que des autres aspects de notre algorithme. Sans perte de
généralité vis-à-vis des travaux précédents, nous considérons un réseaux
anonyme, sur lequel s’exécute un algorithme _semi-uniforme_. Autrement dit,
nous distinguons un nœud arbitraire parmi les nœuds du réseau. Ce dernier
jouera un rôle particulier. Nous appelons ce nœud la _racine_ de l’arbre.
Notons que dans un réseau avec des identifiants, on peut toujours élire une
racine; réciproquement, si le réseau dispose d’une racine alors les nœuds
peuvent s’attribuer des identifiants distincts [51]. Notons également, qu’il
est impossible de calculer de manière déterministe auto-stabilisante un MST
dans un réseaux anonyme (voir [78]). L’hypothèse de semi-uniformité offre une
forme de minimalité.
Nous travaillons dans un modèle de communications par registres, avec un
adversaire faiblement équitable, et une atomicité lecture/écriture. Plus
précisément, quand l’adversaire active un nœud, ce nœud peut de façon atomique
(1) lire sa mémoire et la mémoire de ses voisins, et (2) écrire dans sa
mémoire. (Nous avons besoin de cette atomicité afin de garantir la propriété
sans-cycle, mais nous aurions pu nous en passer pour les autres propriétés de
l’algorithme). La structure du réseau est statique, mais les poids des arêtes
peuvent changer au cours du temps, ce qui confère un certain dynamisme au
réseau. Afin de préserver le déterminisme de notre algorithme, on suppose que
les ports relatifs aux arêtes liant un nœud $u$ à ses voisins sont numéroté de
1 à $\deg(u)$.
Notre algorithme possède deux caractéristiques conceptuelles essentielles:
* •
d’une part, il maintient un arbre couvrant (cet arbre n’est pas nécessairement
minimum, mais il se sera au final);
* •
d’autre part, les nœuds sont munis d’ _étiquettes_ distinctes qui, à l’inverse
des identifiants, codent de l’information.
###### Brève description de l’algorithme.
Notre algorithme fonctionne en trois étapes:
1. i.
Construction d’un arbre couvrant.
2. ii.
Circulation sur l’arbre couvrant d’un jeton qui étiquette chaque nœud.
3. iii.
Amélioration d’un cycle élémentaire.
Nous décrivons chacune de ces trois étapes. En utilisant l’algorithme de
Johnen-Tixeuil [88], nous construisons un arbre de plus courts chemins
enraciné à la racine $r$, tout en maintenant la propriété sans-cycle (c’est
principalement le maintient de la propriété sans-cycle qui nous a motivé dans
le choix de cet algorithme). Une fois l’arbre couvrant construit, la racine
initie une circulation DFS d’un jeton. Pour cela, nous utilisons l’algorithme
de Petit-Villain [109]. Comme l’algorithme de Higham-Liang, notre algorithme
utilise la propriété rouge: il élimine du MST l’arête de poids maximum d’un
cycle. Toutefois, contrairement à [81] qui nécessite d’inonder le réseau pour
trouver les cycles élémentaires, notre algorithme utilise les cycles
élémentaires induits par la présence d’un arbre couvrant existant. Cela est
possible grâce aux deux caractéristiques essentielles de notre algorithme, à
savoir: (1) maintenance d’un arbre couvrant, et (2) utilisation d’
_étiquettes_ DFS. Grâce à l’arbre et aux étiquettes, les cycles élémentaires
sont facilement identifiés, ce qui permet d’éviter l’inondation.
Lorsque l’arbre couvrant de plus court chemin est construit, la racine $r$
déclenche une circulation de _jeton_. Ce jeton circule dans l’arbre en
profondeur d’abord (DFS), en utilisant les numéros de ports. Le jeton possède
un compteur. Ce compteur est initialisé à zéro lors du passage du jeton à la
racine. A chaque fois que le jeton découvre un nouveau nœud, il incrémente son
compteur. Quand le jeton arrive à un nœud dans le sens racine-feuilles, ce
nœud prend pour étiquette le compteur du jeton (voir Figure 2.4). Cette
étiquette, notés $\mbox{$\ell$}_{u}$ (pour label en anglais), a pour objet
d’identifier les cycles élémentaires associés aux arêtes ne faisant pas partie
de l’arbre couvrant. Pour ce faire, lorsque le jeton arrive sur un nœud $u$,
si ce nœud possède des arêtes qui ne font pas partie de l’arbre, alors le
jeton est _gelé_. Soit $v$ le nœud extrémité de l’arête $\\{u,v\\}$ ne faisant
pas partie de l’arbre. Si $\mbox{$\ell$}_{v}<\mbox{$\ell$}_{u}$, alors le nœud
$u$ déclenche une procédure d’amélioration de cycle. Grâce aux étiquettes,
l’algorithme construit ainsi l’unique chemin $P(u,v)$ entre $u$ et $v$ dans
l’arbre. Si les étiquettes sont cohérentes, chaque nœud $w$ calcule son
prédécesseur dans $P(u,v)$ de la façon suivante. Si
$\mbox{$\ell$}_{w}>\mbox{$\ell$}_{v}$ alors le parent de $w$ est son
prédécesseur dans $P(u,v)$, sinon son prédécesseur est le nœud $w^{\prime}$
défini comme l’enfant de $w$ d’étiquette maximum tel que
$\mbox{$\ell$}_{w^{\prime}}<\mbox{$\ell$}_{v}$. Le poids maximum d’une arête
de $P(u,v)$ est collecté. Si le poids de $\\{u,v\\}$ n’est pas ce maximum,
alors les arêtes du cycle sont échangées par échanges successifs. Le
déroulement de cet échange sera spécifié dans la section 3.1. La figure 2.4
illustre l’étiquetage des nœuds, ainsi que les échanges successifs d’arêtes.
Si au cours du parcours du cycle, l’étiquette courante n’est pas cohérente par
rapport à celles de ses voisins, le jeton est libéré, et il continue sa course
et l’étiquetage des nœuds. L’étiquetage sera en effet rectifié au passage
suivant du jeton puisque l’algorithme maintient en permanence une structure
d’arbre couvrant.
(a) Jeton au nœud 12
(b) Echange de 1 et 9
(c) Echange de 10 et 1
Figure 2.4: Illustration de l’étiquetage lors du déroulement de l’algorithme
de la section 2.3.3.1
###### Complexité.
Notons que les algorithmes de Johnen-Tixeuil et de Petit-Villain [88, 109]
utilisent un nombre constant de variables de taille $O(\log n)$ bits. Il en va
de même pour la partie que nous avons développée car nous manipulons trois
identifiants et un poids d’arête dans la partie amélioration de cycle.
L’algorithme a donc une complexité mémoire de $O(\log n)$ bits. Pour
fonctionner, l’algorithme a besoin d’une circulation permanente du jeton. Il
n’est donc pas silencieux et ne peut donc pas être considéré comme optimal en
mémoire, car, à ce jour, aucune borne inférieure n’a été donnée sur la mémoire
utilisée pour la construction non silencieuse d’arbres couvrants. Notre
algorithme traite toutefois le problème dans un cadre dynamique (les poids des
arêtes peuvent changer au cours du temps), et il est probablement beaucoup
plus difficile de rester silencieux dans un tel cadre.
Pour la complexité temporelle, le pire des cas arrive après un changement de
poids d’une arête. En effet une arête de l’arbre appartient à au plus $m-n+1$
cycles, obtenues en rajoutant à l’arbre une arête qui n’est pas dans l’arbre,
où $m$ est le nombre d’arêtes. Donc, avant de déterminer si elle est
effectivement dans le MST, l’algorithme déclenche une vérification pour
chacune des arêtes ne faisant pas partie de l’arbre dans chacun de ces $m-n+1$
cycles. Comme chaque amélioration nécessite $O(n)$ rondes, il découle une
complexité en temps de $O(n^{3})$ rondes.
###### 2.3.3.2 Utilisation d’étiquetages informatifs
L’algorithme de la section 2.3.3.1 utilise un étiquetage des nœuds par
profondeur d’abord. Quoique trivial, nous avons vu la capacité de cet
étiquetage à faciliter la conception d’algorithmes auto-stabilisants pour le
MST. Cela nous a conduit à concevoir un algorithme basé sur des schémas
d’étiquetage informatifs non triviaux. Ce travail a été réalisé en
collaboration avec Shlomi Dolev, Maria Potop-Butucaru, et Stéphane Rovedakis.
Il a été publié dans [17]. Nous sommes par ailleurs parti du constat qu’aucun
des algorithmes auto-stabilisants existant n’utilisait l’approche de
l’algorithme répartie le plus cité, à savoir celui de Gallager, Humblet et
Spira [69]. La composante la plus compliquée et la plus onéreuse dans [69] est
la gestion des fragments (i.e., permettre à un nœud de distinguer les nœuds de
son voisinage faisant partie du même fragment que lui de ceux d’un autre
fragment). Dans l’approche de Gallager, Humblet et Spira, pour chaque
fragment, une racine donne l’identifiant de ce fragment. Ainsi, tout nœud
appartenant à un même fragment possède un ancêtre commun. Nous donc avons eu
l’idée d’utiliser un étiquetage informatif donnant le plus proche ancêtre
commun de deux nœuds.
L’apport de cet algorithme est donc conceptuellement double:
* •
d’une part, il est le premier à utiliser une approche à la Gallager, Humblet
et Spira pour l’auto-stabilisation;
* •
d’autre part, il est le premier à utiliser un schéma d’étiquetage informatif
non trivial pour la construction auto-stabilisante de MST.
Ce double apport nous a permis d’améliorer le rapport entre le temps de
convergence et l’espace mémoire, plus précisément: mémoire $O(\log^{2}n)$
bits, et temps de convergence $O(n^{2})$ rondes. Par ailleurs, cet algorithme
est silencieux.
###### Brève description de l’algorithme.
Chaque nœud du réseau possède un identifiant unique. Pour l’étiquetage
informatif du plus proche ancêtre commun (LCA — pour least common ancestor)
dans un arbre enraciné, nous avons utilisé le travail de Harel et Tarjan [79]
où les auteurs définissent deux sortes d’arêtes: les _légères_ et les
_lourdes_. Une arête est dite lourde si elle conduit au sous-arbre contenant
le plus grand nombre de nœuds; elle est dite légère sinon. L’étiquetage est
constitué d’un ou plusieurs couples. Le premier paramètre d’un couple est
l’identifiant d’un nœud $u$. Le second est la distance au nœud $u$. Un tel
couple est noté $(\mbox{\sf Id}_{u},\mbox{\sf d}_{u})$. Le nombre de couples
est borné par $O(\log n)$, car, comme il est remarqué dans [79], il y a au
plus $\log n$ arêtes légères sur n’importe quel chemin entre une feuille et la
racine, d’où il résulte des étiquettes de $O(\log^{2}n)$ bits. Une racine $u$
sera étiquetée par $(\mbox{\sf Id}_{u},0)$. Un nœud $v$ séparé de son parent
$u$ par une arête lourde prendra l’étiquetage $(\mbox{\sf Id}_{r},\mbox{\sf
d}_{u}+1)$. Un nœud $v$ séparé de son parent $u$ par une arête légère prendra
l’étiquetage $(\mbox{\sf Id}_{r},\mbox{\sf d}_{u})(\mbox{\sf Id}_{v},0)$. Cet
étiquetage récursif est illustré dans la Figure 2.5.
Figure 2.5: Illustration du schéma d’étiquetage LCA
L’étiquette $\mbox{$\ell$}_{w}$ du plus petit ancêtre commun $w$ entre deux
nœuds $u$ et $v$, s’il existe, est calculée de la façon suivante. Soit $\ell$
une étiquette telle que
$\mbox{$\ell$}_{u}=\mbox{$\ell$}.(a_{0},a_{1}).\mbox{$\ell$}_{u}^{\prime}$ et
$\mbox{$\ell$}_{v}=\mbox{$\ell$}.(b_{0},b_{1}).\mbox{$\ell$}_{v}^{\prime}$.
Alors
$\mbox{$\ell$}_{w}=\left\\{\begin{array}[]{ll}\mbox{$\ell$}.(a_{0},a_{1})&\mbox{{si}
}(a_{0}=b_{0}\vee\mbox{$\ell$}\neq\emptyset)\wedge(\mbox{$\ell$}_{u}\prec\mbox{$\ell$}_{v})\\\
\mbox{$\ell$}.(b_{0},b_{1})&\mbox{{si}
}(a_{0}=b_{0}\vee\mbox{$\ell$}\neq\emptyset)\wedge(\mbox{$\ell$}_{v}\prec\mbox{$\ell$}_{u})\\\
\emptyset&\mbox{{sinon}}\end{array}\right.$
S’il n’existe pas d’ancêtre commun à $u$ et $v$, alors ces deux nœuds sont
dans deux fragments distincts. Par ailleurs, nous utilisons l’ordre
lexicographique sur les étiquettes, noté $\prec$, dans le but de détecter la
présence de cycles. Un nœud $u$ peut détecter la présence d’un cycle en
comparant son étiquette avec celle de son parent $v$. Si
$\mbox{$\ell$}_{u}\prec\mbox{$\ell$}_{v}$ alors le nœud $u$ supprime son
parent et devient racine de son propre fragment.
L’auto-stabilisation impose des contraintes supplémentaires non satisfaites
par l’algorithme de Gallager, Humblet et Spira: la configuration après panne
peut être un arbre couvrant qui n’est pas un MST, et il faut donc rectifier
cet arbre. Nous allons donc utiliser la propriété bleue pour fusionner les
fragments, et la propriété rouge pour supprimer les arêtes qui ne font pas
partie du MST final333Notons que ce n’est pas la première fois que les deux
propriétés bleue et rouge sont simultanément utilisées dans un même algorithme
distribué pour réseaux dynamiques (e.g., [105, 106]), mais jamais, à notre
connaissance, sous contrainte d’auto-stabilisation.. Chaque fragment
identifie, grâce à l’étiquetage, l’arête de poids minimum sortant de son
fragment, et l’arête de poids minimum interne au fragment ne faisant pas
partie de l’arbre. La première arête sert pour la fusion de fragments, et la
seconde sert pour la correction de l’arbre. Nous donnons priorité à la
correction sur la fusion. Pour la correction, soit $u$ le plus petit ancêtre
commun des extrémités de l’arête $e$ où $e$ est l’arête interne de plus petit
poids. S’il existe une arête $f$ de poids inférieur à $e$ sur le chemin entre
les deux extrémités de $e$ dans l’arbre courant, alors $f$ est effacé de
l’arbre couvrant. Pour la fusion, l’arête sortante de poids minimum est
utilisée.
###### Complexité.
La complexité en mémoire découle de la taille des étiquettes, à savoir
$O(\log^{2}n)$ bits. Pour la complexité temporelle on considère le nombre de
rondes nécessaires à casser un cycle ou à effectuer une fusion. Dans les deux
cas, une partie des nœuds ont besoin d’une nouvelle étiquette. Cette opération
s’effectue en $O(n)$ rondes. Comme il y a au plus $\frac{n}{2}$ cycles, il
faudra $O(n^{2})$ rondes pour converger. Pour ce qui concerne les fusions, le
pire des cas est lorsque chaque nœud est un unique fragment. Dans ce cas, il
faut effectuer $n$ fusions, d’où l’on déduit le même nombre de rondes pour
converger, à savoir $O(n^{2})$.
##### 2.3.4 Algorithme de Korman, Kutten et Masuzawa
Nous concluons cette section en mentionnant que nos contributions [19, 17] ont
été récemment améliorées par Korman, Kutten et Masuzawa [99]. Ces auteurs se
placent dans le même modèle que nos travaux. Leur article traite à la fois la
_vérification_ et la construction d’un MST. Le problème de la vérification a
été introduit par Tarjan [125] et est défini de la façon suivante. Etant
donnés un graphe pondéré et un sous-graphe de ce graphe, décider si le sous-
graphe forme un MST du graphe. La vérification de MST possède sa propre
littérature. Il est à noter que le problème est considéré comme plus facile
que la construction de MST, en tout cas de façon centralisée. Fort de leur
expérience en répartie aussi bien pour le MST que pour l’étiquetage informatif
([100, 98]), Korman, Kutten et Masuzawa reprennent l’idée de l’étiquetage
informatif pour le MST auto-stabilisant introduit dans [17]. Ils l’optimisent
afin d’être optimal en mémoire $O(\log n)$ bits, et afin d’obtenir une
convergence en temps $O(n)$ rondes. Pour cela ils reprennent l’idée de _niveau
de fragments_ introduite par Gallager, Humblet et Spira pour limiter la taille
des fragments, et pour que les fusions se fassent entre des fragments
contenant à peu près le même nombre de nœuds. Cette dernière démarche permet
d’obtenir au plus $O(\log n)$ fusions. L’étiquetage est composé de
l’identifiant du fragment, du niveau du fragment, et des deux identifiants des
extrémités de l’arête de poids minimum ayant servi à la fusion. Grâce à cette
étiquette, il est possible d’effectuer la vérification, ce qui leur permet de
corriger le MST courant, et donc d’être auto-stabilisant. Une des difficultés
de leur approche est de maintenir une mémoire de $O(\log n)$ bits. En effet,
lorsque le MST est construit, chaque nœud a participé à au plus $\log n$
fusions. Les étiquettes peuvent donc atteindre $O(\log^{2}n)$ bits. Pour
maintenir une mémoire logarithmique, les auteurs distribuent l’information et
la font circuler à l’aide de ce qu’ils appellent un _train_ , qui pipeline le
transfert d’information entre les nœuds.
#### 2.4 Conclusion
Ce chapitre a présenté mes contributions à l’auto-stabilisation visant à
construire des MST en visant une double optimalité, en temps et en mémoire. A
ce jour, en ce qui concerne les algorithmes silencieux, l’algorithme de Korman
et al. [99] est optimal en mémoire. Le manque de bornes inférieures sur le
temps d’exécution d’algorithmes auto-stabilisants ne permet pas de conclure
sur l’optimalité en temps de [99]. Nous pointons donc le problème ouvert
suivant :
###### Problème ouvert 1.
Obtenir une borne inférieure non triviale du temps d’exécution d’algorithmes
auto-stabilisants silencieux (ou non) de construction de MST.
Pour ce qui concerne la taille mémoire, il n’existe pas de borne dans le cas
d’algorithmes non-silencieux pour le MST, ni même pour la construction
d’arbres couvrants en général.
###### Problème ouvert 2.
Obtenir une borne inférieure non triviale de la taille mémoire d’algorithmes
auto-stabilisants non-silencieux de construction d’arbres couvrants.
Enfin, un des défis de l’algorithmique répartie est d’obtenir des algorithmes
optimaux à la fois en mémoire et en temps. Dans le cas du MST, nous soulignons
le problème suivant :
###### Problème ouvert 3.
Concevoir un algorithme réparti de construction de MST, optimal en temps et en
nombre de messages, dans le modèle ${\cal CONGEST}$.
### Chapter 3 Autres constructions d’arbres couvrants sous contraintes
Ce chapitre a pour objet de présenter mes travaux sur la construction d’arbres
couvrants optimisés, différents du MST. La première section est consacrée à
l’approche _sans-cycle_ évoquée dans le chapitre précédent. Cette même section
présente deux de mes contributions mettant en œuvre cette propriété. La
première est appliquée au MST dynamique auto-stabilisant, la seconde est
appliquée à la généralisation de la propriété sans-cycle à _toute_
construction d’arbres couvrants dans les réseaux dynamiques. La section 3.2
est consacrée au problème de l’arbre de Steiner, c’est-à-dire une
généralisation du problème MST à la couverture d’un sous-ensemble quelconque
de nœuds. Enfin les deux dernières sections du chapitre traitent du problème
de la minimisation du degré de l’arbre couvrant, l’une dans le cas des réseaux
non-pondérés, l’autre dans le cas des réseaux pondérés. Dans le second cas, on
vise la double minimisation du degré et du poids de l’arbre couvrant.
#### 3.1 Algorithmes auto-stabilisants sans-cycle
Dans cette section, nous nous intéressons à la propriété sans-cycle (loop-free
en anglais). Cette propriété est particulièrement intéressante dans les
réseaux qui supportent un certain degré de dynamisme. Elle garantit qu’une
structure d’arbre couvrant est préservée pendant tout le temps de
l’algorithme, jusqu’à convergence vers l’arbre couvrant optimisant la métrique
considérée. Les algorithmes sans-cycle ont été introduits par [68, 71].
Cette section, présente un état de l’art des algorithmes auto-stabilisant
sans-cycle, suivi d’un résumé de mes deux contributions à ce domaine : un
algorithme auto-stabilisant sans-cycle pour le MST, et une méthode de
transformation de tout algorithme auto-stabilisant de construction d’arbres
couvrants en un algorithme sans-cycle.
##### 3.1.1 Etat de l’art en auto-stabilisation
A notre connaissance, il n’existe que deux contributions à l’auto-
stabilisation faisant référence à la notion de sans-cycle : [35] et [88]. Ces
deux travaux s’intéressent à la construction d’arbres couvrants de plus court
chemin enracinés. L’article de Johnen et Tixeuil [88] améliore les résultats
de Cobb et Gouda [35]. En effet, contrairement à [35], [88] ne nécessite
aucune connaissance a priori du réseau. Dans ces deux articles, le dynamisme
considéré est l’évolution au cours du temps des valeurs des arêtes.
L’algorithme s’exécute de la façon suivante. Chaque nœud $u$ maintient sa
distance à la racine $r$, et pointe vers un voisin (son parent) qui le conduit
par un plus court à cette racine. Pour maintenir la propriété sans-cycle, un
nœud $u$ qui s’aperçoit d’un changement de distance dans son voisinage qui
implique un changement de parent, vérifie que le nœud voisin $v$ qui annonce
la plus courte distance vers la racine n’est pas un de ses descendants. Si $v$
n’est pas un descendant de $u$, alors $u$ change (de façon atomique et locale
— voir Figure 3.1) son pointeur vers $v$. Sinon, il reste en attente de la
mise à jour de son sous-arbre.
(a)
(b)
Figure 3.1: Changement de parent de manière atomique et locale dans un arbre
de plus courts chemins.
##### 3.1.2 Algorithme auto-stabilisant sans-cycle pour le MST
Les contributions ci-dessus soulèvent immédiatement la question de savoir s’il
est possible de traiter de façon auto-stabilisante sans-cycle des problèmes de
construction d’arbres dont le critère d’optimisation est _global_. Le problème
du plus court chemin peut être qualifié de local [76] car les changements
nécessaires à la maintenance d’un arbre de plus courts chemins sont locaux
(changement de pointeurs entre arêtes incidentes). En revanche, des problèmes
comme le MST ou l’arbre couvrant de degré minimum, implique des changement
entre arêtes arbitrairement distantes dans le réseau. Pour aborder des
problèmes globaux, nous nous sommes intéressés à la construction auto-
stabilisante sans-cycle d’un MST.
Ma première contribution dans le domaine a été effectuée en collaboration avec
Maria Potop-Butucaru, Stéphane Rovedakis et Sébastien Tixeuil [23]. Nous
traitons le problème du MST dans un réseau où les poids des arêtes sont
dynamiques. Un algorithme auto-stabilisant sans-cycle pour le MST a été décrit
dans la section 2.3.3.1. Cette description a cependant omis la vérification du
respect de la propriété sans-cycle, que nous traitons maintenant. L’idée
principale de notre algorithme consiste à travailler sur les cycles
fondamentaux engendrés par les arêtes ne faisant pas partie de l’arbre. Si une
arête $e$ ne faisant pas partie de l’arbre possède un poids plus petit qu’une
arête $f$ faisant partie de l’arbre et du cycle engendré par $e$, alors $e$
doit être échangée avec $f$. Ce changement ne peut être fait directement sans
violer la propriété sans-cycle. Nous avons donc mis en place un mécanisme de
changement atomique, arête par arête, le long d’un cycle engendré par $e$.
Notre algorithme possède donc deux caractéristiques conceptuelles essentielles
:
* •
Application de la propriété sans-cycle à des optimisations globales.
* •
Mécanisme de changement atomique arête par arête le long d’un cycle.
###### Brève description du mécanisme de changement atomique.
Rappelons que le modèle dans lequel l’algorithme s’exécute est un modèle à
registres partagés avec une atomicité lecture/écriture. Autrement dit,
lorsqu’un nœud est activé il peut en même temps lire sur les registres de ses
voisins et écrire dans son propre registre.
Comme nous l’avons vu dans la section 2.3.3.1, lorsque le jeton arrive sur un
nœud $u$ qui possède une arête $e=\\{u,v\\}$ ne faisant pas partie de l’arbre
couvrant courant, ce jeton est gelé et un message circule dans le cycle
fondamental $C_{e}$ à l’aide des étiquettes sur les nœuds. Ce message récolte
le poids de l’arête de poids maximum $f$, ainsi que les étiquettes de ses
extrémités. Supposons qu’au terme de cette récolte, l’arête $e=\\{u,v\\}$
doive être échangée avec $f$ (voir Figure 2.4). Soit $w$ le plus proche
ancêtre commun à $u$ et $v$. Supposons sans perte de généralité que l’arête
$f$ est comprise entre $u$ et $w$. On considère alors le chemin
$u,x_{1},x_{2},...,x_{k},u^{\prime}$ entre $u$ est $u^{\prime}$ où
$u^{\prime}$ est l’extrémité de $f$ la plus proche de $u$. Le nœud $u$ va
changer son pointeur parent en une étape atomique (ce qui maintient la
propriété sans-cycle) afin de pointer vers $v$, puis le nœud $x_{1}$ va
changer son pointeur vers $u$, puis le nœud $x_{2}$ va changer son pointeur
vers $x_{1}$, ainsi de suite jusqu’à ce que $u^{\prime}$ prenne pour parent
$x_{k}$. Tous ces changements sont effectués de manière atomique. La propriété
sans-cycle est donc préservée.
##### 3.1.3 Généralisation
Dans ce travail en collaboration avec Maria Potop-Butucaru, Stéphane Rovedakis
et Sébastien Tixeuil, publié dans [23], nous proposons un algorithme
généralisant la propriété sans-cycle à toute construction d’arbres couvrants
sous contrainte. Contrairement aux algorithmes auto-stabilisants précédents
[35, 88, 23], dont le dynamisme est dû uniquement aux changements de poids des
arêtes, le dynamisme considéré dans cette sous-section est l’arrivée et le
départ arbitraire de nœuds.
Soit $T$ un arbre couvrant un réseau $G$, optimisant une critère $\mu$ donné,
et construit par un algorithme $\cal{A}$. Supposons que le réseau $G$ subisse
des changements topologiques jusqu’à obtenir un réseau $G^{\prime}$. L’arbre
$T$ n’est pas forcément optimal pour le réseau $G^{\prime}$. Il faut donc
transformer l’arbre $T$ en un arbre optimisé pour le réseau $G^{\prime}$.
Cette transformation doit respecter la propriété sans-cycle pour passer de $T$
à $T^{\prime}$. Pour cela nous utilisons de la composition d’algorithmes,
dont, plus spécifiquement, la _composition équitable_ introduite par Dolev,
Israeli, et Moran [54, 55]. La composition équitable fonctionne de la manière
suivante. Soit deux algorithmes $\cal M$ et $\cal E$, le premier est dit
maître et le second est dit esclave .
* •
L’algorithme $\cal E$ est un algorithme statique qui calcule, étant donné un
graphe $G^{\prime}$, un arbre couvrant $T^{\prime}$ de $G^{\prime}$ optimisant
le critère $\mu$.
* •
L’algorithme $\cal M$ est un algorithme dynamique qui prend en entrée
$T^{\prime}$ et effectue le passage de $T$ à $T^{\prime}$ en respectant la
propriété sans-cycle.
Dolev, Israeli, et Moran [54, 55] ont prouvé que la composition équitable de
$\cal M$ avec $\cal E$ dans un contexte dynamique résulte en un algorithme qui
respecte la propriété sans-cycle et dont l’arbre couvrant satisfera le critère
d’optimisation $\mu$. Notre apport conceptuel dans ce cadre est le suivant :
* •
Conception d’un mécanisme générique pour la construction d’algorithmes auto-
stabilisants sans-cycle pour la construction d’arbres couvrants optimisant un
critère quelconque111Plus exactement, un algorithme réparti auto-stabilisant
dans un environnement statique doit exister pour ce critère., dans des réseaux
dynamiques.
Afin d’utiliser la composition équitable, il faut concevoir un algorithme
maître $\cal M$ qui nous permettra, par sa composition avec un algorithme de
construction d’arbre optimisé, de supporter le dynamisme en respectant la
propriété sans-cycle. L’algorithme $\cal M$ que nous avons proposé est un
algorithme (respectant la propriété sans-cycle) de construction d’arbres
couvrants en largeur. Cet algorithme est appelé BFSSC(BFS pour Breadth-first
search en anglais et SC pour sans-cycle). Son exécution repose sur l’existence
d’une racine $r$.
Soit $\cal E$ un algorithme de construction d’arbre optimisé pour le critère
$\mu$. Cet algorithme sert d’oracle pour notre algorithme BFSSC. Soit un
réseau $G$, et $T$ l’arbre couvrant de $G$ obtenu par $\cal E$. BFSSC effectue
un parcours en largeur d’abord de $T$ à partir de la racine $r$. Ce parcours
induit une orientation des arêtes de $T$, qui pointent vers la racine. Après
un changement topologique de $G$ en $G^{\prime}$, un appel à $\cal E$ permet
de calculer un nouvel arbre $T^{\prime}$. BFSSC utilise la combinaison de
l’orientation des arêtes de $T$ et de la connaissance de $T^{\prime}$ pour
passer de $T$ à $T^{\prime}$ par une succession d’opérations locales
atomiques, comme dans la section précédente.
Le coup additionnel temporel de cette composition est de $O(n^{2})$ rondes, à
ajouter à l’algorithme esclave $\cal E$. L’algorithme BFSSC utilise une espace
mémoire de $O(\log n)$ bits.
#### 3.2 Arbre de Steiner
Comme suite logique au MST, je me suis intéressée au problème de l’arbre de
Steiner. Ce problème est une généralisation du MST consistant à connecter un
ensemble $S$ quelconque de nœuds en minimisant le poids de l’arbre de
connexion. Les éléments de cet ensemble $S$ sont appelés les _membres_ à
connecter. Ce problème classique de la théorie des graphes est un des
problèmes NP-difficiles fondamentaux. Il est donc normal que les dernières
décennies aient été consacrées à trouver la meilleure approximation possible à
ce problème. De façon formelle le problème se présente de la façon suivante :
###### Définition 4 (Arbre de Steiner).
Soit $G=(V,E,w)$ un graphe non orienté pondéré, et soit $S\subset V$. On
appelle arbre de Steiner de $G$ tout arbre couvrant les nœuds de $S$ en
minimisant la somme des poids de ses arêtes.
Un algorithme est une $\rho$-approximation de l’arbre de Steiner s’il calcule
un arbre dont la somme des poids des arêtes est au plus $\rho$ fois le poids
d’un arbre de Steiner (i.e., optimal pour ce critère de poids). Cette section
présente un état de l’art pour le problème de Steiner ainsi que ma
contribution dans le domaine.
##### 3.2.1 Etat de l’art
D’un point de vu combinatoire, la première approximation est une
2-approximation dû à Takahashi et Matsuyama [124]. Dans leur article, les
auteurs fournissent trois algorithmes pour le problème de Steiner, qui
illustrent des techniques utilisées par la suite. Le premier algorithme est un
algorithme qui construit un MST. Cet MST est ensuite élagué , autrement dit
les branches de l’arbre qui ne conduisent pas à un membre sont supprimées pour
obtenir un arbre de Steiner approché. Cette approche permet d’obtenir une
$(n-|S|+1)$-approximation. Le deuxième algorithme construit un arbre de plus
courts chemins enraciné à un nœud membre, vers tous les autres nœuds membres.
Par cette approche, on obtient une $(|S|+1)$-approximation. Enfin le dernier
algorithme donne une 2-approximation. Il se base sur la même idée que
l’algorithme de Prim. Autrement dit, on choisit un nœud membre $u$ arbitraire,
que l’on connecte par un plus court chemin à un nœud membre $v$ le plus proche
de $u$. Ensuite, la composante connexe créé par $u,v$ est connectée par un
plus court chemin à un troisième nœud membre $w$ le plus proche de cette
composante connexe, et ainsi de suite jusqu’à obtenir une composante connexe
incluant tous les membres. L’approche combinatoire a une longue histoire [129,
133, 14, 113]. La meilleure approximation connue par des techniques
combinatoires est de $1,55$. Elle est dû à Robins et Zelikovsky [116].
La programmation linéaire est une autre approche pour trouver la meilleure
approximation à l’arbre de Steiner. En 1998, Jain [87] utilise une méthode
d’arrondis successifs pour obtenir une 2-approximation. En 1999, Rajagopalan
et Vazirani [114] posent la question suivante: est-il possible d’obtenir une
approximation significativement plus petite que 2 par une approche de
programmation linéaire. L’an dernier, [28] ont répondu positivement en
obtenant une approximation bornée supérieurement par $1,33$.
Dans le cadre de l’algorithmique répartie, Chen Houle et Kuo [33] ont produit
une version répartie de l’algorithme de [129], dont le rapport d’approximation
est 2. Gatani, Lo Re et Gaglio [73] ont ensuite proposé une version répartie
de l’algorithme d’Imase et Waxman [86]. Ce dernier est une version _on line_
de la construction de l’arbre de Steiner. Dans cette approche dynamique , les
membres arrivent un par un dans le réseau. L’arbre de Steiner est donc calculé
par rapport aux membres déjà en place. Dans l’algorithme d’Imase et Waxman, un
membre qui rejoint le réseau se connecte à l’arbre existant par le plus court
chemin. Par cette approche, les auteurs obtiennent une
$\log|S|$-approximation.
Lorsque je me suis intéressée au problème de l’arbre de Steiner, seuls deux
travaux auto-stabilisants existaient, [90] et [91], par les mêmes auteurs:
Kamei et Kakugawa. A ma connaissance, aucun autre publication n’a été produite
dans le domaine depuis, sauf ma propre contribution, que je détaille dans la
section suivante. Kamei et Kakugawa considèrent dans [90, 91] un environnement
dynamique. Le dynamisme étudié est toutefois assez contraint puisque le réseau
est statique, et seuls les nœuds peuvent changer de statut, en devenant
membres ou en cessant d’être membre. Dans les deux articles, les auteurs
utilisent le modèle à registres partagés avec un adversaire centralisé non
équitable. Dans leur premier article, ils proposent une version auto-
stabilisante de l’algorithme dans [124]. Dans cette approche, l’existence d’un
MST est supposée a priori, et seul le module d’élagage est fourni. Dans le
second article [91], les auteurs proposent une approche en quatre couches:
(i) Construction de fragments, i.e., chaque nœud non membre se connecte au
membre le plus proche par un plus court chemin, ce qui crée une forêt dont
chaque sous-ensemble est appelé fragment. (ii) Calcul du graphe réduit $H$,
i.e., concaténation des arêtes entre les différents fragments. (iii) Calcul
d’un MST réduit: le MST de $H$. (iv) Elagage de ce MST.
##### 3.2.2 Contribution à la construction auto-stabilisante d’arbres de
Steiner
J’ai contribué à l’approche auto-stabilisante pour le problème de l’arbre de
Steiner en collaboration de Maria Potop-Butucaru et Stéphane Rovedakis. Les
résultats de cette collaboration ont été publié dans [21]. Dans les deux
algorithmes que proposent Kamei et Kakugawa, il est supposé l’existence a
priori d’un algorithme auto-stabilisant de construction de MST. Nous avons
conçu une solution qui ne repose pas sur un module de MST. Cette solution est
basé sur l’algorithme _on-line_ d’Imase et Waxsman [86], dont il résulte une
solution capable de supporter un dynamisme plus important que celui de [90,
91], à savoir l’arrivée et le départ de nœud du réseau.
L’apport conceptuel de cet algorithme est donc double:
* •
d’une part, il est le premier algorithme à ne pas reposer sur l’existence a
priori d’un algorithme auto-stabilisant pour le MST;
* •
d’autre part, il est le premier à considérer un dynamisme important (départs
et arrivées de nœuds).
Ce double apport nous a de plus permis de fournir des garanties sur la
structure couvrante restante après une panne si l’algorithme avait eu le temps
de converger (super stabilisation).
###### Brève description de l’algorithme.
Notre travail se place dans le cadre d’un modèle par passage de messages, avec
un adversaire faiblement équitable, et une atomicité envoie/réception.
L’algorithme se décompose en quatre phases ordonnées et utilise un membre
comme racine. Il s’exécute comme suit: (i) chaque nœud met à jour sa distance
à l’arbre de Steiner courant, (ii) chaque nœud souhaitant devenir membre
envoie une requête de connexion, (iii) connexion des nouveaux membres après
accusé de réception des requêtes, (iv) mise à jour de l’arbre de Steiner
courant, dont mise à jour de la distance de chaque nœud de l’arbre à la racine
de l’arbre. L’algorithme utilise explicitement une racine et une variable
gérant la distance entre chaque nœud et la racine afin d’éliminer les cycles
émanant d’une configuration initiale potentiellement erronée. Un arbre de plus
courts chemins vers l’arbre de Steiner courant est maintenu pour tout nœud du
réseau. Les membres, ainsi que les nœuds du réseau impliqués dans l’arbre de
Steiner, doivent être déclarés connectés. Si un nœud connecté détecte une
incohérence (problème de distance, de pointeur, etc.) il se déconnecte, et
lance l’ordre de déconnexion dans son sous-arbre. Lorsque un membre est
déconnecté, il lance une requête de connexion via l’arbre de plus courts
chemins, et attend un accusé de réception pour réellement se connecter. Quand
il est enfin connecté, une mise à jour des distances par rapport au nouvel
arbre de Steiner est initiée.
En procédant de cette manière nous obtenons, comme Imase et Waxsman [86], une
$(\lceil\log|S|\rceil)$-approximation. La mémoire utilisée en chaque nœud est
de $O(\delta\log n)$ bits où $\delta$ est le degré de l’arbre couvrant
courant. La convergence se fait en $O(D|S|)$ rondes, où $D$ est le diamètre du
réseau.
#### 3.3 Arbre couvrant de degré minimum
La construction d’un arbre couvrant de degré minimum a très peu été étudiée
dans le domaine réparti. Pourtant, minimiser le degré d’un arbre est une
contrainte naturelle. Elle peut par ailleurs se révéler d’importance pratique
car les nœuds de fort degré favorisent la congestion des communications. Ils
sont également les premiers nœuds ciblés en cas d’attaque visant à déconnecter
un réseau. Par aileurs, dans le monde du pair-à-pair, il peut être intéressant
pour les utilisateurs eux même d’être de faible degré. En effet, si un
utilisateur possède une information très demandée, chaque lien (virtuel) de
communication sera potentiellement utilisé pour fournir cette information, ce
qui peut entrainer une diminution significative de sa propre bande passante.
En 2004 j’ai proposé avec Franck Butelle [15] le premier algorithme réparti
pour la construction d’arbre couvrant de degré minimum. Lorsque je me suis
intéressé à l’auto-stabilisation, c’est naturellement à la construction
d’arbre couvrant de degré minimum que je me suis consacrée en premier lieu. De
façon formelle, le problème est le suivant.
###### Définition 5 (Arbre couvrant de degré minimum).
Soit $G$ un graphe non orienté. On appelle arbre couvrant de degré minimum de
$G$ tout arbre couvrant dont le degré222Le degré de l’arbre est le plus grand
degré des nœuds est minimum parmi tous les arbres couvrants de $G$.
##### 3.3.1 Etat de l’art
Le problème de l’arbre couvrant de degré minimum est connu comme étant _NP-
difficile_ , par réduction triviale du problème du chemin Hamiltonien. Fürer
et Raghavachari [66, 67] sont les premiers à s’être intéressés à ce problème
en séquentiel. Ils ont montré que le problème est facilement approximable en
fournissant un algorithme calculant une solution de degré $\mbox{OPT}+1$ où
OPT est la valeur du degré minimum. L’algorithme est glouton, et fonctionne de
la manière suivante. Au départ un arbre couvrant quelconque est construit.
Puis, de façon itérative, on essaie de réduire le degré des nœuds de plus fort
degré, jusqu’à que ce ne soit plus possible. Pour diminuer le degré $k$ d’un
nœud $u$, on identifie ses enfants, notés $u_{1},...,u_{k}$, ainsi que les
sous-arbres enracinés en ces enfants, noté $T_{u_{1}},...,T_{u_{k}}$,
respectivement. Sans perte de généralité, considérons $v$ un descendant de
$u_{1}$ tel que $\deg(v)<\deg(u)-2$. Si $v$ possède une arête $e$ qui ne fait
pas parti de l’arbre, et dont l’extrémité $w$ est élément de $T_{u_{j}}$ avec
$j\neq 1$, alors l’algorithme échange $e$ avec l’arête $\\{u,v\\}$, ce qui a
pour conséquence de diminuer le degré de $u$ tout en maintenant un arbre
couvrant. La condition $\deg(v)<\deg(u)-2$ assure qu’après un échange, le
nombre de nœuds de degré maximum aura diminué (voir l’exemple dans la Figure
3.2). Ce procédé est en fait récursif, car s’il n’existe pas de descendant $v$
de $u$ tel que $\deg(v)<\deg(u)-2$, il peut se faire que le degré d’un des
descendants de $u$ puisse être diminué afin que, par la suite, on puisse
diminuer le degré de $u$. Cette opération est répétée jusqu’à ce qu’aucune
amélioration puisse être effectuée.
(a) Degré maximum $u$
(b) Degré maximum $u$ et $v$
(c) Arbre optimal pour le degré
Figure 3.2: Diminution des degrés de l’arbre couvrant.
L’algorithme présenté en 2004, en collaboration avec Franck Butelle [15], a
été conçu pour le modèle par passage de message ${\cal CONGEST}$. Le principe
est le suivant. L’algorithme construit tout d’abord un arbre couvrant
quelconque. Par la suite, les nœuds calculent (à partir des feuilles) le degré
maximum de l’arbre couvrant courant. Une racine $r$ est identifiée comme le
noeud ayant un degré maximum (en cas d’égalité, il choisit celui d’identifiant
maximum). Tous les enfants de $r$ effectuent un parcours en largeur d’abord du
graphe. Chaque parcours est marqué par l’identifiant de l’enfant ayant initié
le parcours. Une arête $e$ ne faisant pas parti de l’arbre est candidate à
l’échange avec une arête de l’arbre si et seulement si les deux conditions
suivantes sont réunies: (i) $e$est traversée par deux parcours en largeur
d’abord d’identifiants différents; (ii) les deux extrémités $u$ et $v$ de $e$
satisfont $\deg(u)<\deg(r)-2$ et $\deg(v)<\deg(r)-2$. L’algorithme suit
ensuite les grandes lignes de l’algorithme de Fürer et Raghavachari [66, 67]
pour les échanges, ainsi que pour la récursivité de la recherche des arêtes
échangeables. Pour identifier les arêtes candidates à l’échange notre
algorithme inonde le graphe de plusieurs parcours en largeur d’abord, le
nombre de messages échangés est donc très important. Malheureusement, à
l’heure actuelle, aucune étude n’a été faite sur la complexité distribuée de
ce problème.
###### Problème ouvert 4.
Dans le modèle ${\cal CONGEST}$, quel est le nombre minimum de messages à
échanger et quel est le nombre minimum d’étapes à effectuer pour la
construction d’une approximation à $+1$ du degré de l’arbre couvrant de degré
minimum ?
##### 3.3.2 Un premier algorithme auto-stabilisant
Je me suis consacré à la construction auto-stabilisante d’arbres couvrants de
degré minimum en collaboration avec Maria Potop-Butucaru et Stéphane Rovedakis
[20, 22]. Nous avons obtenu le premier (et pour l’instant le seul) algorithme
auto-stabilisant pour ce problème. L’algorithme est décrit dans le même modèle
que celui de la section 2.3.3.1, quoique dans un cadre semi-synchrone
puisqu’il utilise un chronomètre. Il est basé sur la détection de cycles
fondamentaux induits par des arêtes ne faisant pas partie de l’arbre couvrant
courant. En effet, si l’on considère un nœud $u$ de degré maximum dans
l’arbre, diminuer le degré de $u$ requiert d’échanger une de ses arêtes
incidentes, $f$, avec une arête $e$ ne faisant pas partie de l’arbre couvrant
courant, et créant un cycle fondamental dont $u$ est élément. Notons que cette
approche est plus efficace que celle de [15] car elle permet de diminuer
simultanément le degré des nœuds de plus grand degré, et de limiter l’échange
d’information (communication au sein de cycles plutôt que par inondation).
L’apport conceptuel de cet algorithme est donc double:
* •
d’une part, il est le premier algorithme auto-stabilisant pour la construction
de degré minimum à utiliser une approche de construction par cycle
élémentaire;
* •
d’autre part, il permet de diminuer le degré de tous les nœuds de degré
maximum en parallèle.
Ce double apport nous a permis d’obtenir un temps de convergence $O(mn^{2}\log
n)$ rondes pour un espace mémoire de $O(\log n)$ bits où $m$ est le nombre
d’arêtes du réseau.
###### Brève description de l’algorithme.
Notre algorithme fonctionne en quatre étapes:
1. i.
Construction et maintien d’un arbre couvrant.
2. ii.
Calcul du degré maximum de l’arbre couvrant courant.
3. iii.
Calcul des cycles élémentaires.
4. iv.
Reduction (si c’est possible) des degrés maximum.
La principale difficulté en auto-stabilisation est de faire exécuter ces
quatre étapes de façon indépendante, sans qu’une étape remette en cause
l’intégrité (auto-stabilisation) d’une autre. Ces étapes sont décrites ci-
après.
Construction et maintien d’un arbre couvrant. Pour construire et maintenir un
arbre couvrant, nous avons adapté à nos besoins l’algorithme auto-stabilisant
de construction d’arbres couvrants en largeur d’abord proposé par Afek, Kutten
et Yung [2]. L’arbre construit est enraciné au nœud ayant le plus petit
identifiant. Par la suite, diminuer le degré de l’arbre couvrant ne changera
pas l’identifiant de la racine, donc ne remettra pas en cause cette partie de
l’algorithme.
Calcul du degré maximum. Pour que chaque nœud sache si son degré est maximum
dans le graphe, nous utilisons la méthode classique de propagation
d’information avec retour (ou PIF pour Propagation of Information with
Feedback). De nombreuses solutions stabilisantes existent [16, 38]. Elles ont
pour avantage de stabiliser instantanément. Cependant, garantir la
stabilisation instantanée requiert des techniques complexes. Afin de faciliter
l’analyse de notre algorithme, nous avons proposé une solution auto-
stabilisante plus simple pour le PIF.
Identification des cycles fondamentaux333La technique présentée ici n’utilise
pas les étiquettes informatives. Elle a en effet été développée avant que nous
proposions les étiquettes informatives pour les cycles en auto-stabilisation..
Soit $e=\\{u,v\\}$ une arête ne faisant pas parti de l’arbre couvrant. Pour
chercher son cycle fondamental $C_{e}$, le nœud extrémité de $e$ d’identifiant
minimum, par exemple $u$, lance un parcours en profondeur d’abord de l’arbre
couvrant. Comme tout parcours en profondeur, il y a une phase de descente où
les nœuds sont visités pour la première fois, et une phase de remontée où le
message revient sur des nœuds déjà visités. Le message qui effectue le
parcours stocke l’identifiant des nœuds et leur degré pendant la phase de
descente, et efface ces données pendant la phase de remontée. Le parcours
s’arrête quand il rencontre l’autre extrémité de $e$, c’est-à-dire $v$.
Lorsque $v$ reçoit le message de parcours, celui-ci contient tous les
identifiants et les degré des nœuds de l’unique chemin dans l’arbre entre $u$
et $v$. La taille de ce message peut donc atteindre $O(n\log n)$ bits. Cette
idée est basé sur l’algorithme auto-stabilisant pour le MST de Higham et Lyan
[81]. L’algorithme résultant possède donc les mêmes défauts. En particulier,
il nécessite un chronomètre pour déclencher périodiquement une recherche de
cycles fondamentaux à partir des arêtes ne faisant pas partie de l’arbre
couvrant. Il est donc semi-synchrone et non-silencieux.
Réduction des degrés. Lorsque l’un des nœuds extrémités $v$ de l’arête $e$ ne
faisant pas parti de l’arbre récupère les informations relatives au cycle
fondamental $C_{e}$, l’algorithme détermine si $C_{e}$ contient un nœud de
degré maximum ou un nœud bloquant. Dans le cas d’un nœud de degré maximum, si
$u$ et $v$ ne sont pas des nœuds bloquants, alors $e$ est rajouté, et une des
arête incidentes au nœud de degré maximum est supprimée. Cette suppression est
effectuée à l’aide d’un message. L’important dans cette étape est de maintenir
un arbre couvrant et de maintenir une orientation vers la racine, ce qui peut
nécessiter une réorientation d’une partie de l’arbre couvrant avant la
suppression de l’arête. Si $u$ et $v$ sont des nœuds bloquants, alors ils
attendrons un message de recherche de cycle pour signaler leur état, et être
potentiellement débloqués par la suite.
###### Complexité
L’algorithme converge vers une configuration légitime en $O(|E|n^{2}\log n)$
rondes et utilise $O(\Delta\log n)$ bits de mémoire, ou $\Delta$ est le degré
maximum du réseau. Le nombre important de rondes est dû à la convergence de
chacune des étapes de notre algorithme. Il convient de noter que nous ne
sommes pas dans le modèle ${\cal CONGEST}$, et que pour récolter les
informations du cycle fondamental, nous utilisons des messages de taille
$O(n\log n)$ bits. En revanche, l’information stockée sur la mémoire des nœuds
est quant à elle de $O(\Delta\log n)$ bits. Enfin, l’utilisation d’un
chronomètre pour les arêtes ne faisant pas parti de l’arbre, rend cet
algorithme semi-synchrone et non-silencieux.
###### Remarque.
Fort de l’expérience acquise dans le domaine ces dernières années, un
algorithme auto-stabilisant plus performant (dont silencieux) pour ce problème
pourrait maintenant être proposé dans le modèle ${\cal CONGEST}$ avec une
taille mémoire optimale de $O(\log n)$ bits.
#### 3.4 Perspective: Arbre couvrant de poids et de degré minimum
Cette section a pour objet d’ouvrir quelques perspectives en liaison avec
l’optimisation d’arbres couvrants sous contrainte. Ayant traité séparément le
problème de l’arbre couvrant de poids minimum et celui de l’arbre couvrant de
degré minimum, il est naturel de s’intéresser maintenant à la combinaison des
deux problèmes. Ce chapitre est ainsi consacré à ce problème bi-critère. Un
bref état de l’art du problème de la construction d’arbres couvrants de poids
minimum et de degré borné est présenté ci-après. Cet état de l’art me
permettra de conclure par un certain nombre de pistes de recherche.
En 2006, Goemans [75] émet la conjecture que l’approximation obtenue par Fürer
et Raghavachari [66] peut se généraliser aux graphes pondérés. Autrement dit,
il conjecture que, parmi les MST, on peut trouver en temps polynomial un MST
de degré au plus $\Delta^{*}+1$, où $\Delta^{*}$ est le degré minimum de tout
MST. Ce problème d’optimisation bi-critère est référencé dans la littérature
par _arbre couvrant de poids minimum et de degré borné_ (en anglais Minimum
Bounded Degree Spanning Trees ou MBDST). Sa définition formelle est la
suivante. Soit $G$ un graphe. La solution cherchée est contrainte par un
entier $B_{v}$ donné pour chacun des nœuds $v$ du graphe. MBDST requiert de
trouver un MST $T$ tel que, pour tout $v$, on ait $\deg_{T}(v)\leq B_{v}$.
Soit OPT le poids d’un tel MST. Une $(\alpha,f(B_{v}))$-approximation de MBDST
est un arbre $T$ dont le poids est au plus $\alpha\;\mbox{OPT}$, et tel que
$\deg_{T}(v)\leq f(B_{v})$. Par exemple le résultat de Fürer et Raghavachari
[66] peut est reformulé par une $(1,k+1)$ approximation dans le cas des
graphes non pondérés (i.e., $B_{v}=k$ pour tout $v$).
Fischer propose, dans le rapport technique [58], d’étendre la technique
algorithmique de Fürer et Raghavachari [66] pour les graphes pondérés. Plus
précisément, cet auteur cherche un MST de degré minimum. Pour cela, il
introduit deux modifications à l’algorithme de [66]. D’une part, l’arbre
initial n’est pas quelconque, mais est un MST. D’autre part, les échanges
d’arêtes se font entre arêtes de poids identiques. Fischer annonce que ces
modifications permettent d’obtenir en temps polynomial un MST dont les nœuds
ont degré au plus $O(\Delta^{*}+\log n)$, où $\Delta^{*}$ est le degré minimum
de tout MST. (La même année, Ravi et al. [115] ont adapté leur travail sur
l’arbre Steiner au problème MBDST pour obtenir une $(O(\log n),O(B_{v}\log
n))$-approximation). En 2000, Konemann et al. [95] ont repris l’approche de
Fischer [58] et en ont effectué une analyse plus détaillée. Dans leur article,
la programmation linéaire est utilisée pour la première fois pour ce problème.
Les mêmes auteurs améliorent ensuite leurs techniques dans [96, 97] pour
obtenir une $(1,O(B_{v}+\log n))$-approximation. Chaudhuri et al. [31, 32]
utilisent quant à eux une méthode développée pour le problème du flot maximum,
pour obtenir une $(1,O(B_{v}))$-approximation. Enfin, Singh et Lau [119]
prouvent la conjecture de Goemans [75], en obtenant une
$(1,B_{v}+1)$-approximation, toujours sur la base de techniques de
programmation linéaire.
Dans un contexte réparti, seuls Lavault et Valencia-Pabon [103] traitent à ma
connaissance ce problème. Ils proposent une version répartie de l’algorithms
Fischer [58], garantissant ainsi la même approximation. Leur algorithme a une
complexité temporelle de $O(\Delta^{2+\epsilon})$ étapes, où $\Delta$ est le
degré du MST initial, et une complexité en nombre de messages échangés de
$O(n^{3+\epsilon})$ bits.
Cet ensemble de travaux sur le MBDST invitent à considérer les problèmes
suivants. D’une part, partant du constat qu’il est difficile de donner des
versions réparties d’algorithmes utilisant la programmation linéaire, nous
souhaiterions aborder le problème de façon purement combinatoire:
###### Problème ouvert 5.
Développer une approche combinatoire pour obtenir un algorithme polynomial
calculant une $(1,\Delta^{*}+1)$-approximation pour le problème de l’arbre
couvrant de poids minimum, et de degré borné, dans le cas des graphes
pondérés.
Bien sûr, tout algorithme combinatoire polynomial retournant une
$(1,\Delta^{*}+o(\log n))$-approximation serait déjà intéressante. En fait, il
serait déjà intéressant de proposer un algorithme réparti offrant la même
approximation que Fischer [58] dans le modèle $\cal CONGEST$.
## Part II Entités autonomes
### Chapter 4 Le nommage en présence de fautes internes
La seconde partie du document est consacrée à des _entités autonomes_ (agents
logiciels mobiles, robots, etc.) se déplaçant dans un réseau. Les algorithmes
sont exécutés non plus par les nœuds du réseau, mais par les entités
autonomes. L’ensemble du réseau et des entités autonomes forme un système dans
lequel le réseau joue le rôle d’environnement externe pour les entités
autonomes. Par abus de langage, nous utilisons dans ce document le terme
_robots_ pour désigner les entités autonomes. Ce chapitre est consacré à un
modèle dans lequel les robots n’ont qu’une vision locale du système (chacun
n’a accès qu’aux informations disponibles sur le nœud sur lequel il se
trouve). Le chapitre suivant sera consacré à un modèle dans lequel les robots
ont une vision globale du système.
La plupart des algorithmes auto-stabilisants pour les robots [74, 80, 13, 49]
cherchent à se protéger de pannes _externes_ , autrement dit de fautes
générées par l’environnement, mais pas par les robots eux-mêmes. Ce chapitre
présente une nouvelle approche de l’auto-stabilisation pour les robots, à
savoir la conception d’algorithmes auto-stabilisants pour des fautes internes
et externes, c’est-à-dire générées par les robots et par leur environnement.
Cette nouvelle approche a été étudiée en collaboration avec M. Potop-Butucaru
et S. Tixeuil [24].
La plupart des algorithmes conçus pour des robots utilisent les identifiants
de ces robots. Dans ce chapitre, les pannes internes induisent une corruption
de la mémoire des robots. Les identifiants des robots peuvent donc être
corrompus. Par conséquent, la tâche consistant à attribuer des identifiants
deux-à-deux distincts aux robots apparait comme une brique de base essentielle
à l’algorithmique pour les entités mobiles. Cette tâche est appelée le
_nommage_ (naming en anglais). Dans le cadre étudié dans la Partie I du
document, c’est-à-dire les algorithmes auto-stabilisants pour les réseaux,
l’existence d’identifiants deux-à-deux distincts attribués aux nœuds est
équivalente à l’existence d’un unique leader. Ce chapitre est consacré à cette
équivalence dans le cas des algorithmes auto-stabilisants pour les robots
susceptibles de subir des fautes internes et externes.
La première partie de ce chapitre est consacrée à la formalisation d’un modèle
pour l’étude de systèmes de robots sujets à des défaillances transitoires
internes et externes. Dans un deuxième temps, le chapitre est consacrée à des
résultats d’impossibilité pour le problème du nommage. Nous montrons que, dans
le cas général, le nommage est impossible à résoudre de façon déterministe,
mais qu’il l’est au moyen d’un algorithme probabiliste. Dans le cadre
déterministe, nous montrons que le nommage est possible dans un arbre avec des
liens de communication semi-bidirectionnels. Ces résultats complètent les
résultats d’impossibilité dans les réseaux répartis anonymes (voir [130,
131]). De plus, nos algorithmes peuvent servir de brique de base pour résoudre
d’autres problèmes, dont en particulier le regroupement — problème connu pour
avoir une solution uniquement si les robots ont un identifiant unique [45].
#### 4.1 Un modèle local pour un système de robots
Soit $G=(V,E)$ un réseau anonyme. Les robots sont des machines de Turing qui
se déplacent de nœuds en nœuds dans $G$ en traversant ses arêtes, et qui sont
capables d’interagir avec leur environnement. On suppose un ensemble de $k>0$
robots. Durant l’exécution d’un algorithme, le nombre de robots ne change pas
(i.e., les robots ne peuvent ni disparaitre ni apparaitre dans le réseau).
Chaque robot possède un espace mémoire suffisant pour stocker au moins un
identifiant, donc $\Omega(\log k)$ bits. Pour $u\in V$, les arêtes incidentes
à $u$ sont étiquetées par des numéros de port deux-à-deux distincts, entre 1
et $\deg(u)$. Chaque nœud du réseau possède un _tableau blanc_ qui peut
stocker un certain nombre d’information, sur lequel les robots peuvent lire et
écrire. Les arêtes sont _bidirectionnelles_ , c’est-à-dire utilisables dans
les deux sens. On considérera deux sous-cas selon qu’une arête peut être
traversée par deux robots dans les deux sens simultanément, ou uniquement dans
un sens à la fois. Dans le second cas, on dira que l’arête est _semi-
bidirectionnelle_. Une _configuration_ du système est définie par l’ensemble
des nœuds occupés par les robots, l’état des robots, et l’information contenue
dans tous les tableaux blancs. Les informations suivantes sont accessibles à
un robot r occupant un nœud $u$ du réseau :
* •
le numéro de port de l’arête par laquelle r est arrivé en $u$, et le degré de
$u$;
* •
l’état de chacun des robots présents sur le nœud $u$ en même temps que r;
* •
les données stockée sur le tableau blanc de $u$.
En fonction des informations ci-dessus, le robot change d’état, et décide
possiblement de se déplacer. Le système est _asynchrone_. L’adversaire qui
modélise l’asynchronisme est distribué, faiblement équitable (voir Chapitre
1). Les nœuds contenant au moins un robot sont dits activables. A chaque étape
atomique, l’adversaire doit choisir un sous-ensemble non vide $S\subseteq V$
de nœuds activables. (On dit que les nœuds de $S$ sont activés par
l’adversaire). L’algorithme a ensuite la liberté de choisir quel robot est
activé sur chacun des nœuds de $S$. Autrement dit, en une étape atomique, tous
les nœuds choisis par l’adversaire exécutent le code d’au moins un robot
localisé sur chacun d’entre eux. Dans ce cadre, une ronde est définie par le
temps minimum que mettent tous les nœuds activables à être activés par
l’adversaire.
Ce chapitre étudie la résistance d’un système de robots aux fautes
transitoires, internes et externes. Pour cela, nous supposons que chaque faute
dans le système peut modifier le système de façon arbitraire, c’est-à-dire,
plus précisément, (i) la mémoire (i.e., l’état) des robots (faute interne),
(ii) la localisation des robots (faute externe), et (iii) le contenu des
tableaux blancs (faute externe). Notons que la structure du réseau est
statique, et que, comme nous l’avons dit, il n’y a pas de modification du
nombre de robots. Le modèle de fautes ci-dessus généralise le modèle utilisé
dans [74, 80, 13, 49] qui ne considère que les fautes externes.
#### 4.2 Les problèmes du nommage et de l’élection
Comme il a été dit précédemment, une grande partie de la littérature sur les
robots supposent que ces derniers ont des identifiants non corruptibles. Dans
notre modèle, les robots peuvent avoir une mémoire erronée après une faute du
système, ce qui implique des valeurs d’identifiants potentiellement erronées.
La capacité de redonner aux robots des identifiants deux-à-deux distincts est
donc indispensable dans un système de robots avec fautes internes.
Le problème du nommage est formalisé de la manière suivante: Soit $S$ un
système composé de $k$ robots dans un graphe $G$. Le système $S$ satisfait la
spécification de nommage si les $k$ robots ont des identifiants entiers entre
1 et $k$ deux-à-deux distincts. Le problème de l’élection et équivalent au
problème du nommage, identifiant deux à deux distincts. Il se formalise de la
manière suivante.Soit $S$ un système composé de $k$ robots dans un graphe $G$.
Le système $S$ satisfait la spécification d’élection si un unique robot est
dans l’état leader et tous les autres robots sont dans l’état battu.
###### Théoreme 1.
Blin et al. [24]. Les problèmes du nommage et de l’élection auto-stabilisants
sont équivalents même en présence de fautes internes, c’est-à-dire que $k$
robots avec des identifiants deux-à-deux distincts peuvent élire un leader, et
$k$ robots disposant d’un leader peuvent s’attribuer des identifiants deux-à-
deux distincts.
La preuve du théorème ci-dessous est dans [24]. Intuitivement, pour résoudre
l’élection, on procède de la façon suivantes. Chaque robot effectue un
parcours Eulérien auto-stabilisant du graphe. A chaque arrivée sur un nœud, il
inscrit son identifiant sur le tableau blanc. Après la stabilisation des
parcours, tous les tableaux blancs ont la liste de tous les identifiants. Le
robot avec l’identifiant maximum peut se déclarer leader. Réciproquement, le
principe de l’algorithme est le suivant. Le leader $\mbox{\sc r}_{L}$ suit un
parcours Eulérien du graphe. Les autres robots procèdent de façon à rejoindre
$\mbox{\sc r}_{L}$ en suivant l’information que ce dernier laisse sur les
tableaux blancs. $\mbox{\sc r}_{L}$ prend 1 comme identifiant. Lorsqu’un robot
r rejoint $\mbox{\sc r}_{L}$, r reste avec $\mbox{\sc r}_{L}$ et prend comme
identifiant le nombre de robots actuellement avec $\mbox{\sc r}_{L}$, incluant
r et $\mbox{\sc r}_{L}$.
Tout comme la plupart des résultats d’impossibilité dans le modèle discret
(cf. Introduction), les résultats d’impossibilité relatifs au nommage et à
l’élection sont dus à l’existence de symétries entre les robots impossibles à
briser. Considérons par exemple le cas où $G$ est un cycle. Supposons qu’après
une défaillance du système, (i) chaque nœud de $G$ contient un seul robot
(i.e., $k=n$), (ii) les robots sont tous dans le même état (incluant le fait
qu’ils ont le même identifiant), et (iii) les tableaux blancs son vides.
Dans ce cas, l’adversaire pourra activer tous les robots indéfiniment. En
effet, à chaque activation, les robots effectuent la même action, et tous les
robots garderont le même état, de même que tous les nœuds garderont le même
tableau blanc. Par conséquent, résoudre le problème de nommage (ou de
l’élection) de façon déterministe dans un cycle est impossible, même dans un
environnement synchrone, avec une mémoire infini pour les robots et les
tableaux blancs.
Le présence d’arêtes bidirectionnelles est également un obstacle à la
résolution du nommage (et de l’élection). Supposons en effet, la présence de
deux robots avec la même information à l’extrémité d’une même arête, si les
robots peuvent traverser en même temps dans les deux sens cette arête, les
robots se croisent sans jamais briser la symétrie.
#### 4.3 Algorithmes auto-stabilisants pour le nommage
Dans cette section, nous décrivons tout d’abord un algorithme déterministe
dans un cadre contournant les deux obstacles mis en évidence dans la section
précédentes, c’est-à-dire la présence de cycles et d’arêtes bidirectionnelles.
Nous nous restreignons donc aux arbres dont les arêtes sont semi-
bidirectionnelles (i.e., non utilisables dans les deux sens en même temps).
Dans un second temps, nous décrivons un algorithme _probabiliste_ réalisant le
nommage dans tout réseau (connexe) avec arêtes bidirectionnelles.
##### 4.3.1 Algorithme déterministe
Soit $k$ robots placés de façon arbitraire dans un arbre. Nous supposons que
les tableaux blancs peuvent stocker $\Omega(k(\log k+\log\Delta))$ bits, où
$\Delta$ est le degré maximum du graphe. Chaque robot r a un identifiant
entier $\mbox{\sf Id}_{\mbox{\sc r}}\in[1,k]$. Cet entier peut être corrompu.
Sans perte de généralité, on suppose que la corruption d’un identifiant
préserve toutefois l’appartenance à $[1,k]$.
Succinctement, l’algorithme fonctionne de la manière suivante. Chaque nœud
stocke dans son tableau blanc jusqu’à $k$ paires (identifiant, numéro de
port). L’écriture sur chaque tableau blanc se fait en ordre FIFO afin
d’ordonner les écritures. Lorsqu’une paire est écrite dans un tableau
contenant déjà $k$ paires, la plus ancienne paire est détruite. Chaque robot
effectue un parcours Eulérien de l’arbre. Quand un robot r arrive sur un nœud
$u$, il vérifie si son identifiant $\mbox{\sf Id}_{\mbox{\sc r}}$ est présent
dans une des paires stockées sur le tableau blanc de $u$, noté
$\mbox{\sc{\small TB}}_{u}$. Si cet identifiant n’est pas présent, alors le
robot inscrit la paire $(\mbox{\sf Id}_{\mbox{\sc r}},p)$ sur
$\mbox{\sc{\small TB}}_{u}$ où $p$ est le numéro de port par lequel r partira
pour continuer son parcours Eulérien. Si l’identifiant de r est présent dans
une paire sur $\mbox{\sc{\small TB}}_{u}$, deux cas doivent être considérés.
S’il y a déjà un robot $\mbox{\sc r}^{\prime}$ localisé en $u$ avec le même
identifiant que r, alors r est activé et prend un nouvel identifiant, le plus
petit identifiant non présent sur le tableau. Le robot r continue ensuite son
parcours Eulérien, en notant la paire $(\mbox{\sf Id}_{\mbox{\sc r}},p)$
convenable sur $\mbox{\sc{\small TB}}_{u}$. Enfin, si r est le seul robot sur
$u$ avec identifiant $\mbox{\sf Id}_{\mbox{\sc r}}$, il teste si la dernière
arête $e$ associée à son identifiant est l’arête par laquelle il est entré sur
$u$. Si oui, alors cela est cohérent avec un parcours Eulérien et r continue
ce parcours. Si non, alors r sort de $u$ par l’arête $e$ afin de rencontrer le
robot portant le même identifiant que lui, s’il existe, et provoquer ainsi le
changement d’identifiant de l’un des deux.
Afin de prouver la correction de l’algorithme, il convient de noter deux
remarques importantes. D’une part, puisque le réseau est un arbre et que les
arêtes sont semi-bidirectionnelles, deux robots possédant le même identifiant
se retrouveront sur un même nœud en un nombre fini d’étapes. D’autre part,
l’espace mémoire de chaque tableau étant borné, est l’écriture étant FIFO, si
un tableau possède des informations erronées, celles-ci finiront par
disparaitre en étant recouvertes par des informations correctes.
Pour ce qui est de la complexité de l’algorithme, on peut montrer que
l’algorithme converge en $O(kn)$ rondes. Ce temps de convergence découle du
fait que deux robots portant le même identifiant mettrons dans le pire des cas
$O(n)$ rondes pour se rencontrer.
##### 4.3.2 Algorithme probabiliste
L’approche probabiliste permet de considérer le cadre général de graphes
arbitraires avec liens bidirectionnels. De fait, il est très simple d’obtenir
une solution, sans même utiliser de tableaux blancs sur les nœuds. Chaque
robot se déplace suivant une marche aléatoire uniforme. Lorsque plusieurs
robots se rencontrent, ils s’ignorent s’ils ont des identifiants différents.
Deux robots ayant le même identifiant se rencontrant sur un nœud choisissent
chacun un nouvel identifiant de manière aléatoire uniforme entre 1 et $k$.
Pour prouver la convergence, nous considérons le cas d’une configuration
initiale dans laquelle deux robots ont le même identifiant. Il est connu [127]
que la marche aléatoire non biaisée implique que les deux robots se
rencontreront en au plus $O(n^{3})$ rondes. Lorsque deux robots ayant le même
identifiant se rencontrent, les robots ont chacun une probabilité au moins
$\frac{1}{k}$ de choisir un identifiant utilisé par aucun autre robot.
L’algorithme probabiliste donc a un temps de stabilisation de $O(kn^{3})$.
#### 4.4 Perspectives
Les travaux présentés dans ce chapitre sont les premiers à considérer la
conception d’algorithmes auto-stabilisants pour des robots susceptibles de
subir des pannes internes, en plus des pannes externes usuellement traitées
dans la littérature. Nous avons montré qu’il était possible sous ces
hypothèses de réaliser le nommage et l’élection dans les arbres en
déterministe, et dans tous les graphes en probabiliste.
La restriction aux arbres, dans le cas déterministe, est motivée par les
symétries pouvant être induites par la présence de cycles. Une piste de
recherche évidente consiste donc à considérer le cas des graphes avec cycles
mais suffisamment asymétriques pour permettre le nommage.
Par ailleurs, le nommage et l’élection ne sont intéressants qu’en tant que
briques élémentaires pour la réalisation de tâches plus élaborées, comme le
rendez-vous ou la recherche d’intrus. Concevoir des algorithmes basés sur ces
briques élémentaires demande de composer avec soin des algorithmes auto-
stabilisants pour des robots. Si la composition d’algorithmes auto-
stabilisants pour les réseaux est maintenant bien comprise, il n’en va pas
nécessairement de même dans le cadre de l’algorithmique pour entités mobiles.
L’étude de la composition d’algorithmes auto-stabilisants pour entités mobiles
est à ma connaissance un problème ouvert.
### Chapter 5 Auto-organisation dans un modèle à vision globale
Le chapitre précédent était consacré à un modèle dans lequel les robots
n’avaient qu’une vision locale du système (chacun n’a accès qu’aux
informations disponibles sur le nœud sur lequel il se trouve). Ce chapitre est
consacré à un modèle dans lequel les robots ont une vision globale du système.
Le chapitre considère le modèle CORDA (asynchronisme) dans un modèle discret
(réseau). Il a pour objectif d’identifier les hypothèses minimales nécessaires
à la réalisation de tâches complexes de façon auto-stabilisante. Mes
contributions dans ce cadre ont été réalisées en collaboration de A. Milani,
M. Potop-Butucaru et de S. Tixeuil [18].
La première section de ce chapitre est consacrée à un bref état de l’art des
algorithmes conçus pour le modèle CORDA discret. Un modèle minimale est
ensuite formalisé, dans la section suivante. La tâche algorithmique utilisée
pour la compréhension de ce modèle est l’ _exploration perpétuelle_ , définie
comme suit. La position initiale des robots est arbitraire — elle peut
résulter par exemple d’une faute du système. Partant de cette position
initiale, les robots doivent agir de façon à ce que chaque nœud du réseau soit
visité infiniment souvent par chacun des robots. La section 5.3 est ainsi
consacrée à établir des bornes inférieures et supérieures sur le nombre de
robots pouvant réaliser l’exploration perpétuelle dans l’anneau. La section
5.4 résume quant à elle une de nos contributions principales, à savoir deux
algorithmes d’exploration utilisant respectivement un nombre minimal et
maximal de robots. La dernière section liste quelques perspectives sur le
modèle CORDA et sur le problème de l’exploration perpétuelle.
La contribution de ce chapitre à l’auto-organisation d’un système de robots
est donc double,
* •
d’une part, la mise en évidence d’un modèle minimaliste n’ajoutant aucune
hypothèse non inhérente à l’esprit d’un modèle observation-calcul-déplacement
tel que le modèle CORDA;
* •
d’autre part, la conception d’algorithmes pour les robots créant et maintenant
des asymétries entre les positions de ces robots permettant de donner une
direction à l’exploration en l’absence de références extérieurs (numéro de
ports, identifiant des nœuds, sens de direction, etc.).
#### 5.1 Etat de l’art des algorithmes dans le modèle CORDA discret
Une grande partie de la littérature sur l’algorithmique dédiée à la
coordination de robots distribués considère que les robots évoluent dans un
espace euclidien continu à deux dimensions. Les sujets essentiellement traités
dans ce domaine sont la formation de patterns (cercle, carré, etc.), le
regroupement, l’éparpillement, le rendez-vous, etc. Nous renvoyons à [123, 34,
7, 57, 63, 132] pour des exemples de résolutions de tels problèmes. Dans cadre
continu, le modèle suppose que les robots utilisent des capteurs visuels
possédant une parfaite précision qui permettent ainsi de localiser la position
des autres robots. Les robots sont également supposés capables de se déplacer.
Les déplacements sont souvent supposés idéaux, c’est-à-dire sans déviation par
rapport à une destination fixée.
Ce modèle est critiquable car la technologie actuelle permet difficilement de
le mettre en oeuvre. Les capteurs visuels et les déplacements des robots sont
en effet loin d’être parfaits. La tendance ces dernières années a donc été de
déplacer le cadre d’étude du modèle continu au modèle discret.
Dans le modèle discret, l’espace est divisé en un nombre fini d’
_emplacements_ modélisant une pièce, une zone de couverture d’une antenne, un
accès à un bâtiment, etc. La structure de l’espace est idéalement représentée
par un graphe où les nœuds du graphe représentent les emplacements qui peuvent
être détectés par les robots, et où les arêtes représentent la possibilité
pour un robot de se déplacer d’un emplacement à un autre. Ainsi, le modèle
discret facilite à la fois le problème de la détection et du déplacement. Au
lieu de devoir détecter la position exacte d’un autre robot, il est en effet
plus aisé pour un robot de détecter si un emplacement est vide ou s’il
contient un ou plusieurs autres robots. En outre, un robot peut plus
facilement rejoindre un emplacement que rejoindre des coordonnées
géographiques exactes. Le modèle discret permet également de simplifier les
algorithmes en raisonnant sur des structures finies (_i.e_ , des graphes)
plutôt que sur des structures infinies (le plan ou l’espace 3D). Il y a
cependant un prix à payer à cette simplification. Comme l’ont noté la plupart
des articles relatifs à l’algorithmique pour les entités mobiles dans le
modèle discret [94, 93, 59, 60, 47], le modèle discret est livrée avec le coût
de la _symétrie_ qui n’existe pas dans le modèle continu où les robots
possèdent leur propre système de localisation (e.g., GPS et boussole). Ce
chapitre est consacré à l’étude du modèle discret, et en particulier à la
conception d’algorithmes auto-stabilisant s’exécutant correctement en dépit
des symétries potentielles.
Un des premiers modèles pour l’étude de l’auto-organisation de robots
distribués, appellé _SYm_ , a été proposé par Suzuki et Yamashita [122]. Dans
le modèle SYm, les robots n’ont pas d’état (ils n’ont donc pas de mémoire du
passé), et fonctionne par cycle élémentaire synchrone. Un cycle élémentaire
est constitué des trois actions atomiques synchrones suivantes: observation,
calcul, déplacement (Look-Compute-Move en anglais). Autrement dit, à chaque
cycle, chaque robot observe tout d’abord les positions des autres robots (les
robots sont supposés capables de voir les positions de tous les autres
robots), puis il calcule le déplacement qu’il doit effectuer, et enfin il se
déplace selon ce déplacement. Le modèle _CORDA_ [62, 111] peut se définir
comme la variante asynchrone du modèle SYm. Il reprend toutes les hypothèses
du modèle SYm mais il suppose que les robots sont asynchrones. La littérature
traitant de l’auto-organisation de robots dans un modèle du type observation-
calcul-déplacement utilise presque systématiquement une variante de SYm ou de
CORDA. Le modèle peut en particulier supposer la présence ou non
d’identifiants distincts affectés aux robots, la capacité de stockage
d’information par les robots et/ou les nœuds du réseau dans lequel ils se
déplacent, la présence ou non d’un sens de la direction attribué au robots, la
possibilité ou non d’empiler des robots, etc. Ce sont autant d’hypothèses qui
influent sur la capacité d’observer, de calculer et de se déplacer.
A ma connaissance, tous les articles traitant du modèle CORDA discret
enrichissent ce modèle en supposant des hypothèses supplémentaires, comme par
exemple la présence d’identifiants distincts, ou d’un sens de direction.
Dans les modèles SYm et CORDA, l’ensemble des positions des robots détermine
une _configuration_ du système. Une _photographie instantanée_ (snapshot en
anglais), ou simplement _photo_ , du système à l’instant $t$ par un robot est
la configuration du système à cet instant $t$. La phase d’observation exécuté
par un robot consiste à prendre une _photo_ de la position des autres robots.
Durant sa phase de calcul, chaque robot calcule son action en fonction de la
dernière photo prise. Enfin, dans sa phase déplacement, chaque robot se
déplace en traversant _une seule_ arête incidente au nœud courant. Cette arête
est déterminée durant la phase de calcul. En présence de numéro de ports, ou
d’un sens de direction, établir la correspondance entre une arête identifiée
sur la photo et l’arête réelle à emprunter est direct. Nous verrons dans la
section suivante que l’absence de numéro de port et de sens de direction
n’empêche pas d’établir cette correspondance.
Dans le cadre du modèle CORDA discret, les deux problèmes les plus étudiés
sont sans doute le _regroupement_ [94, 93] et l’ _exploration_ , dans ses
versions avec arrêt [59, 60, 47, 30, 61] et perpétuelle [11]. La tâche de
regroupement demande aux robots de se réunir en un nœud du réseau ; la tâche
d’exploration demande aux robots de visiter chaque nœud du réseau. Ce chapitre
se focalise uniquement sur la tâche d’exploration.
Dans l’exploration avec arrêt, le fait que les robots doivent s’arrêter après
avoir exploré tous les nœuds du réseau requiert de leur part de se souvenir
quelle partie du réseau a été explorée. Les robots doivent donc être capable
de distinguer les différentes étapes de l’exploration (nœud exploré ou pas,
arête traversée ou pas, etc.) bien qu’ils n’aient pas de mémoire persistante.
La symétrie des configurations est le principal problème rencontré dans le
modèle discret. C’est pourquoi la plupart des articles du domaine se sont dans
un premier temps restreints à l’étude de réseaux particuliers tels que les
arbres [61] et les cycles [59, 94, 93, 47, 61]. Dans [30], les auteurs
considèrent les réseaux quelconques mais supposent que les positions initiales
des robots sont asymétriques.
Une technique classique pour éviter la présence de symétrie est d’utiliser un
grand nombre de robots pour créer des groupes de robots de taille différentes
et donc asymétriques. Une mesure de complexité souvent considérée est donc le
nombre minimum de robots requis pour explorer un réseau donné. Pour les arbres
à $n$ nœuds, $\Omega(n)$ robots sont nécessaires [60] pour l’exploration avec
arrêt, même si le degré maximum est $4$. En revanche, pour les arbres de degré
maximum 3, un nombre de robots exponentiellement plus faible, $O(\log
n/\log\log n)$, est suffisant. Dans un cycle de $n$ nœuds, l’exploration avec
arrêt par $k$ robots est infaisable si $k|n$, mais faisable si
$\mbox{pgcd}(n,k)=1$ avec $k\geq 17$ [59]. En conséquence, le nombre de robots
nécessaire et suffisant pour cette tâche est $\Theta(\log n)$ dans le cycle.
Enfin, dans [30], les auteurs proposent un algorithme d’exploration avec arrêt
dans un réseau quelconque avec des arêtes étiquetées par des numéro de ports,
pour un nombre impair de robots $k\geq 4$.
Dans [11], les auteurs résolvent le problème de l’exploration perpétuelle dans
une grille partielle anonyme (c’est-à-dire une grille anonyme à laquelle un
ensemble de nœuds et d’arêtes ont été supprimés). Cet article introduit la
contrainte d’ _excusivité_ mentionnée précédemment, qui stipule qu’au plus un
robot peut occuper le même nœud, ou traverser la même arête. Un certain nombre
de travaux utilisent des tours de robots pour casser la symétrie (voir [59]).
La contrainte d’exclusivité interdit ce type de stratégies. Nos contributions
personnelles considèrent également la contrainte d’exclusivité mais,
contrairement à [11], les robots n’ont pas de sens de la direction.
(a) $(\mbox{\sc r}_{2},\mbox{\sc f}_{2},\mbox{\sc r}_{1},\mbox{\sc f}_{z})$
(b) $(\mbox{\sc r}_{1},\mbox{\sc f}_{1},\mbox{\sc r}_{1},\mbox{\sc
f}_{2},\mbox{\sc r}_{1},\mbox{\sc f}_{z-1})$
Figure 5.1: Photos
#### 5.2 Un modèle global minimaliste pour un système de robots
Ce chapitre se focalise sur le modèle CORDA dans sa version élémentaire, sans
hypothèse supplémentaire. Les robots n’ont donc pas d’état, sont totalement
asynchrones, ont une vision globale du système (graphe et robots), et sont
capable de calculer et de déplacer. En revanche, il ne sont munis d’aucune
information supplémentaire. En particulier, ils sont anonymes, ne possèdent
pas de moyen de communication direct, et n’ont pas de sens de direction (ils
ne peuvent donc pas distinguer la droite de la gauche, ou le nord du sud).
Egalement, les nœuds sont anonymes, et les arêtes ne possèdent pas de numéro
de port. En terme de déplacement, il ne peut y avoir qu’au plus un robot par
nœud, et une arête ne peut être traversée que par un seul robot à la fois (pas
de croisement, i.e., arête semi-bidirectionnelle). Un algorithme ne respectant
ces dernières spécifications relatives au nombre de robots par nœuds et aux
croisements le long des arêtes entraine une _collision_ , et sera considéré
incorrecte. Dans de telles conditions minimalistes, les robots ne collaborent
que par l’intermédiaire de leurs positions qui dictent leurs actions à chacun.
Spécifions précisément le modèle CORDA discret pour un anneau et pour une
chaîne, selon les principes établis dans [18]. Les robots sont _asynchrones_ ,
_anonymes_ , _silencieux_ (ils ne possèdent pas de moyen de communication
direct), sont _sans état_ (pas de mémoire du passé, oblivious en anglais), et
n’ont _aucun sens de la direction_. L’asynchronisme est modélisé par un
adversaire qui décide quel robot, ou quel sous-ensemble de robots, est activé
parmi les robots activables par l’algorithme. Les robots sont soumis à la
contrainte d’exclusivité.
Afin de simplifier notre propos, nous avons besoin de formaliser la notion de
photo présentée précédemment. Une _photo_ S (pour snapshot) impliquant $k$
robots dans un anneau à $n$ nœuds, est une séquence non orientée (circulaire
dans le cas du cycle) de symboles r et f indexés par des entiers: $\mbox{\sc
r}_{i}$ signifie que $i$ nœuds consécutifs sont occupés par des robots, et
$\mbox{\sc f}_{j}$ signifie que $j$ nœuds consécutifs sont non occupés. Par
exemple, la photo $\mbox{\tt S}=(\mbox{\sc r}_{i_{1}},\mbox{\sc
f}_{j_{1}},\dots,\mbox{\sc r}_{i_{\ell}},\mbox{\sc f}_{j_{\ell}})$ décrit le
cas où $k$ robots sont divisés en $\ell$ groupes, et, pour $m=1,\dots,\ell$ le
$m$-ème groupe de robots occupe $i_{m}$ nœuds consécutifs dans l’anneau, et
les $m$-ème et $(m+1)$-ème groupes de robots sont séparés par $j_{m}\geq 1$
nœuds libres. Le résultat d’une observation par un robot r est une photo
$\mbox{\tt S}=(\mbox{\sc r}_{i_{1}},\mbox{\sc f}_{j_{1}},\dots,\mbox{\sc
r}_{i_{\ell}},\mbox{\sc f}_{j_{\ell}})$. Le calcul effectué par ce robot
résulte en une autre photo $\mbox{\tt S}^{\prime}$ à atteindre par un unique
déplacement effectué par r ou par un autre robot. Un algorithme sera donc
défini par un ensemble de transitions entre photos $\mbox{\tt
S}\rightarrow\mbox{\tt S}^{\prime}$ spécifiant la configuration $\mbox{\tt
S}^{\prime}$ image de S, pour chaque S. Par exemple, la transition
$(\mbox{\sc r}_{2},\mbox{\sc f}_{2},\mbox{\sc r}_{1},\mbox{\sc
f}_{n-5})\rightarrow(\mbox{\sc r}_{1},\mbox{\sc f}_{1},\mbox{\sc
r}_{1},\mbox{\sc f}_{2},\mbox{\sc r}_{1},\mbox{\sc f}_{n-6})$
stipule que le robot du groupe de deux robots à côté de $n-5$ nœuds libres
(voir Figure 5.1(a)) doit se déplacer d’un cran vers ces $n-5$ nœuds libres
(voir Figure 5.1(b)).
#### 5.3 Résultat d’impossibilités
Les résultats d’impossibilités que nous avons obtenus dans [18] sont résumés
dans cette section. Ils résultent de la mise en évidence de cas de symétrie
qu’il est impossible de briser dans le modèle CORDA discret minimaliste que
nous utilisons. Soit r un robot sur une chaîne, et soit $u$ un nœud à
l’extrémité de cette chaîne. Soit $v$ le nœud voisin de $u$. Tout algorithme
d’exploration doit spécifier à r localisé sur $v$ d’aller en $u$ car sinon $u$
ne serait jamais exploré. Il en résulte que si r est initialement placé en $v$
alors le fait que le robot soit sans état implique que l’adversaire pourra
systématiquement ordonner le déplacement de r de $v$ vers $u$ et de $u$ vers
$v$, indéfiniment, et le reste de la chaîne ne sera jamais explorée. (Notons
que cette impossibilité n’est pas vérifiée dans le cas où le robot aurait un
sens de direction, ou dans le cas de l’existence de numéros de port. En effet,
dans les deux cas, la fonction de transition est de la forme $(\mbox{\tt
S},d)\rightarrow(\mbox{\tt S}^{\prime},d^{\prime})$ où $d$ et $d^{\prime}$
sont soit des numéros de port, soit des directions.) L’exploration perpétuelle
d’une chaîne par $k>1$ robots est impossible, simplement parce que les robots
n’ont aucun moyen de se croiser du fait de la contrainte d’exclusivité.
Partant du fait que l’exploration perpétuelle est impossible dans une chaîne,
il est naturel des s’intéresser par la suite à l’exploration perpétuelle dans
un anneau et aux cas d’impossibilité dans cette topologie. L’impossibilité
d’explorer l’anneau avec un unique robot est illustrée sur les figures 5.2(a)
et 5.2(b). L’impossibilité d’explorer l’anneau avec un nombre pair de robots
est illustrée sur les figures 5.2-5.2. De même, il est simple de montrer
qu’explorer l’anneau de $n$ nœuds avec $k$ robots, $n-4\leq k\leq n$, est
impossible. Nous avons montré dans [18] qu’en revanche, $k=3$ et $k=n-5$ sont
des valeurs universelles, c’est-à-dire que pour $n$ assez grand, et non
multiple de $k$, l’exploration perpétuelle de l’anneau à $n$ nœuds est
possible avec $k$ robots, quelle que soit la configuration initiale.
(a) $(\mbox{\sc r}_{1},\mbox{\sc f}_{z})$
(b) $(\mbox{\sc r}_{1},\mbox{\sc f}_{z})$
Figure 5.2: Cas d’impossibilité avec un robot, ou avec un nombre pair de
robots
Déterminer la valeur minimale de $n$ pour laquelle 3 robots peuvent effectuer
l’exploration perpétuelle de l’anneau à $n$ nœuds requiert une étude de cas
spécifique. Nous montrons que cette valeur est 10. En utilisant le résultat de
Flocchini et al. [59] qui stipule en particulier qu’il est impossible
d’explorer un anneau de $n$ nœuds avec un nombre de robots $k$ divisant $n$
nous pouvons diminuer le nombre de cas à traiter à $n=4,5,7,8$. Nous avons
introduit la notion de _pellicule_. Une pellicule représente toutes les photos
possibles pour un nombre de nœuds et de robots fixés, ainsi que tous les
mouvements possibles entre ces photos (voir la figure 5.3(a) pour un anneau
avec 7 nœuds et 3 robots). Nous avons prouvé qu’une pellicule peut-être
_réduite_ à un sous-ensemble de photos particulières en supprimant les photos
qui sont trivialement un obstacle à l’exploration perpétuelle (voir la figure
5.3(b) pour un anneau avec 7 nœuds et 3 robots). Par exemple, les photos
permettant à l’adversaire de forcer le robot à effectuer un ping-pong
perpétuel entre deux nœuds peuvent être trivialement supprimées. Une fois
cette réduction faite, il reste à analyser la sous-pellicule restante. Cela
est effectué en simulant les déplacements induits par les photos de cette
sous-pellicule. Ainsi dans le cas d’un anneau à sept nœuds, on montre qu’il
n’existe pas d’algorithme déterministe permettant la visite perpétuelle par
trois robots (voir figure 5.4).
(a) Pellicule $G_{7,3}$ pour 7 nœuds et 3 robots
(b) Réduction de la pellicule $G_{7,3}$
Figure 5.3: Pellicules et réductions
On procède de même dans le cas des anneaux à 4, 5, ou 8 nœuds pour 3 robots.
Dans le cas de 4 ou 5 nœuds, la réduction résulte en une sous-pellicule vide.
Dans le cas de 8 nœuds la réduction résulte en la même sous-pellicule que
celle obtenue pour 7 nœuds. Il est donc impossible de faire l’exploration d’un
anneau de moins de dix nœuds avec exactement trois robots.
Figure 5.4: Dans la pellicule $G_{7,3}$, le nœud gris n’est jamais visité
#### 5.4 Algorithme d’exploration perpétuelle
Dans cette section, nous montrons tout d’abord qu’il existe un algorithme
d’exploration perpétuelle pour trois robots dans un anneau de $n$ nœuds, avec
$n\geq 10$ et $n$ différent d’un multiple de trois. Nous décrivons ensuite un
algorithme déterministe permettant de faire l’exploration perpétuelle avec
$n-5$ robots dans un anneau de $n$ nœuds, où $n>10$ et $k$ est impair.
(a) Photo $\mbox{\tt S}^{\circ}_{2}$
(b) Photo $\mbox{\tt S}^{\circ}_{3}$
(c) Photo $\overline{\mbox{\tt S}^{\circ}_{2}}$
(d) Photo $\mbox{\tt S}^{\circ}_{2}$
Figure 5.5: Exploration perpétuelle avec $3$ robots
##### 5.4.1 Algorithme utilisant un nombre minimum de robots
Notre algorithme traite différemment deux types de photos: les photos du
régime permanent, et les photos du régime transitoire. Les premières sont
appelées photos _permanentes_ , et les secondes photos _transitoires_.
Les photos permanentes présentent une asymétrie des positions des robots qui
permet de donner une direction à l’exploration (sens des aiguilles d’une
montre, ou sens inverse des aiguilles d’une montre). Cette asymétrie est créée
à la fois par les groupes de robots (placés sur des nœuds consécutifs) et par
les groupes de nœuds libres. Notre algorithme assure que la même asymétrie
sera maintenue tout au long de son exécution dans le régime permanent. Dans ce
régime, l’algorithme préserve une formation en deux groupes de robots, l’un
constitué par un robot et l’autre par deux robots. Le robot seul, noté
$\mbox{\sc r}_{C}$ (abusivement, car les robots n’ont pas d’identifiant), sera
toujours séparé par au moins deux nœuds libres des autres robots. Dans
l’algorithme, $\mbox{\sc r}_{C}$ avance dans la direction indiquée par un plus
grand nombre de nœuds libres. Les deux autres robots, $\mbox{\sc r}_{B}$ et
$\mbox{\sc r}_{A}$, avancent vers dans la direction indiquée par un plus petit
nombre de nœuds libres. Leur objectif est de rejoindre le robot $\mbox{\sc
r}_{C}$ (voir figure 5.5). Plus formellement, notre algorithme d’exploration
perpétuelle en régime permanent est décrit ci-dessous. Il a l’avantage de ne
rendre activable qu’un seul robot à chaque étape, ce qui force le choix de
l’adversaire. Dans cet algorithme paramètré par $z$, on suppose que
$z\neq\\{0,1,2,3\\}$.
Algorithme d’exploration perpétuelle avec un nombre minimum de robots
---
$\mbox{\tt S}^{\circ}_{2}=(\mbox{\sc r}_{2},\mbox{\sc f}_{2},\mbox{\sc r}_{1},\mbox{\sc f}_{z})$ | $\rightarrow\mbox{\tt S}^{\circ}_{3}=(\mbox{\sc r}_{1},\mbox{\sc f}_{1},\mbox{\sc r}_{1},\mbox{\sc f}_{2},\mbox{\sc r}_{1},\mbox{\sc f}_{z-1})$
$\mbox{\tt S}^{\circ}_{3}=(\mbox{\sc r}_{1},\mbox{\sc f}_{1},\mbox{\sc r}_{1},\mbox{\sc f}_{2},\mbox{\sc r}_{1},\mbox{\sc f}_{z})$ | $\rightarrow\overline{\mbox{\tt S}^{\circ}_{2}}=(\mbox{\sc r}_{2},\mbox{\sc f}_{3},\mbox{\sc r}_{1},\mbox{\sc f}_{z})$
$\overline{\mbox{\tt S}^{\circ}_{2}}=(\mbox{\sc r}_{2},\mbox{\sc f}_{3},\mbox{\sc r}_{1},\mbox{\sc f}_{z})$ | $\rightarrow\mbox{\tt S}^{\circ}_{2}=(\mbox{\sc r}_{2},\mbox{\sc f}_{2},\mbox{\sc r}_{1},\mbox{\sc f}_{z+1})$
Les photos transitoires nécessitent un traitement spécifique. Le nombre de
robots étant limité à $3$, le nombre de photos transitoires est limité à $5$.
Grâce à la pellicule $G_{n,3}$, nous avons pu construire un algorithme de
convergence pour passer du régime transitoire au régime permanent. Cet
algorithme est décrit ci-dessous (voir aussi la figure 5.6) . Notons que, dans
le cas $\mbox{\tt S}^{\bullet}_{1}$, l’adversaire a le choix d’activer un ou
deux robots, ce qui est délicat à traiter.
Algorithme de convergence avec un nombre minimum de robots .
---
$\mbox{\tt S}^{\bullet}_{2}=(\mbox{\sc r}_{2},\mbox{\sc f}_{y},\mbox{\sc r}_{1},\mbox{\sc f}_{z})$ | $\rightarrow\mbox{\tt S}^{\bullet}_{2}=(\mbox{\sc r}_{2},\mbox{\sc f}_{y-1},\mbox{\sc r}_{1},\mbox{\sc f}_{z+1})$ | avec $y<z,(y,z)\not\in\\{1,2,3\\}$
$\mbox{\tt S}^{\bullet}_{3}=(\mbox{\sc r}_{1},\mbox{\sc f}_{x},\mbox{\sc r}_{1},\mbox{\sc f}_{y},\mbox{\sc r}_{1},\mbox{\sc f}_{y})$ | $\rightarrow\overline{\mbox{\tt S}^{\bullet}_{3}}=(\mbox{\sc r}_{1},\mbox{\sc f}_{x},\mbox{\sc r}_{1},\mbox{\sc f}_{y-1},\mbox{\sc r}_{1},\mbox{\sc f}_{y+1})$ | avec $x\neq y\neq 0$
$\overline{\mbox{\tt S}^{\bullet}_{3}}=(\mbox{\sc r}_{1},\mbox{\sc f}_{x},\mbox{\sc r}_{1},\mbox{\sc f}_{y},\mbox{\sc r}_{1},\mbox{\sc f}_{z})$ | $\rightarrow\overline{\mbox{\tt S}^{\bullet}_{3}}=(\mbox{\sc r}_{1},\mbox{\sc f}_{x-1},\mbox{\sc r}_{1},\mbox{\sc f}_{y},\mbox{\sc r}_{1},\mbox{\sc f}_{z+1})$ | avec $x<y<z$
$\mbox{\tt S}^{\bullet}_{1}=(\mbox{\sc r}_{3},\mbox{\sc f}_{z})$ | $\rightarrow\overline{\mbox{\tt S}^{\bullet}_{2}}=(\mbox{\sc r}_{2},\mbox{\sc f}_{1},\mbox{\sc r}_{1},\mbox{\sc f}_{z-1})$ | quand $1$ robot activé
| $\rightarrow$ $\mbox{\tt S}^{\bullet}_{3}=(\mbox{\sc r}_{1},\mbox{\sc f}_{1},\mbox{\sc r}_{1},\mbox{\sc f}_{1},\mbox{\sc r}_{1},\mbox{\sc f}_{z-2})$ | quand $2$ robots sont activés
$\overline{\mbox{\tt S}^{\bullet}_{2}}=(\mbox{\sc r}_{2},\mbox{\sc f}_{1},\mbox{\sc r}_{1},\mbox{\sc f}_{z})$ | $\rightarrow\mbox{\tt S}^{\circ}_{2}=(\mbox{\sc r}_{2},\mbox{\sc f}_{2},\mbox{\sc r}_{1},\mbox{\sc f}_{z-1})$ |
Figure 5.6: Pellicule avec $3$ robots représentant les photos permanentes et
transitoires, ainsi que leur convergence
##### 5.4.2 Algorithme utilisant un nombre maximum de robots
Dans cette partie, nous décrivons succinctement l’algorithme d’exploration
perpétuelle par un nombre maximum de robots, à savoir $k=n-5$ robots pour $k$
impair, $k>3$ et $n\mod k\neq 0$. Comme pour le nombre minimum de robots,
l’algorithme présente un régime transitoire et un régime permanent.
Malheureusement, la phase transitoire implique un nombre de photos
(transitoires) très importants. De surcroît, le nombre de photos transitoires
pour lesquelles l’adversaire peut choisir d’activer plus d’un robot est
également très important. Le manque de place nous empêche de décrire ici
l’algorithme de passage du régime transitoire au régime permanent.
L’idée principale de l’algorithme en régime permanent est de créer la même
asymétrie que dans l’exploration perpétuelle avec 3 robots. Toutefois, dans le
cas présent, les rôles des nœuds libres et des robots sont inversés. Ainsi,
l’algorithme maintient deux ou trois groupes de nœuds libres, et deux ou trois
groupes de robots (voir la figure 5.7(d)). Par exemple, considérons une photo
$\mbox{\tt S}_{1}$ indiquant deux groupes de robots, l’un constitué de deux
robots, et l’autre constitué de $n-7$ robots. Ces deux groupes sont séparés
d’un côté par trois nœuds libres, et de l’autre par deux nœuds libres. Le
robot appartenant au plus grand groupe de robots rejoint alors le groupe de
deux robots à travers les trois nœuds libres. Ensuite, le robot appartenant
maintenant au groupe de trois robots rejoint le grand groupe de robots à
travers les deux nœuds libres. La configuration obtenue sera identique à celle
de la photo $\mbox{\tt S}_{1}$. Le processus peut donc être répété
indéfiniment.
(a) $\mbox{\tt S}^{\circ}_{2}$
(b) $\mbox{\tt S}^{\circ}_{3}$
(c) $\overline{\mbox{\tt S}^{\circ}_{3}}$
4
(d) $\overline{\mbox{\tt S}^{\circ}_{2}}$
(e) $\wideparen{\mbox{\tt S}^{\circ}_{3}}$
(f) $\mbox{\tt S}^{\circ}_{2}$
Figure 5.7: Exploration perpétuelle utilisant un nombre maximum de robots
#### 5.5 Perspectives
Les travaux présentés dans ce chapitre sont les premiers à considérer la
conception d’algorithmes auto-stabilisants pour les robots dans le modèle
CORDA sans hypothèse supplémentaire. Nous avons montré qu’il est possible,
sous certaines conditions élémentaires liant la taille de l’anneau au nombre
de robots, de réaliser l’exploration perpétuelle dans tout anneau, de façon
déterministe. Le choix de l’anneau est motivé par le fait que, malgré sa
simplicité, son étude permet de mettre en évidence des techniques déjà
sophistiquées. Il n’en reste pas moins qu’une piste de recherche évidente
consiste à considérer des familles de réseaux plus complexes, comme les
grilles et les tores, voire des réseaux quelconques. Par ailleurs, il serait
évidemment intéressant d’étudier le _regroupement_ dans le modèle CORDA
discret avec contrainte d’exclusivité. Notons néanmoins qu’il n’est même pas
clair comment définir ce problème dans ce cadre. On pourrait imaginer le
regroupement sur un sous-réseau connexe du réseau initial, mais d’autres
définitions sont possibles, potentiellement incluant la formation de formes
spécifiques (chemin, anneau, etc.).
## Part III Conclusions et perspectives
### Chapter 6 Perspectives de recherche
Ce document fournit un résumé de mes principales contributions récentes à
l’auto-stabilisation, aussi bien dans le cadre des réseaux que dans celui des
entités mobiles. Chacun des chapitres a listé un certain nombre de problèmes
ouverts et de directions de recherche spécifiques à chacune des thématiques
abordées dans le chapitre. Dans cette dernière partie du document, je
développe des perspectives de recherche générales, à longs termes, autour _des
compromis entre l’espace utilisé par les nœuds, le temps de convergence de
l’algorithme, et la qualité de la solution retournée par l’algorithme_.
Plusieurs paramètres peuvent en effet être pris en compte pour mesurer
l’efficacité d’un algorithme auto-stabilisant, dont en particulier le temps de
convergence et la complexité mémoire. L’importance du temps de convergence
vient de la nécessité évidente pour un système de retourner le plus rapidement
possible dans un état valide après une panne. La nécessité de minimiser la
mémoire vient, d’une part, de l’importance grandissante de réseaux tel que les
réseaux de capteurs qui ont des espaces mémoires restreints et, d’autre part,
de l’intérêt de minimiser l’échange d’information et le stockage d’information
afin de limiter la corruption.
La minimisation de la mémoire peut se concevoir au détriment d’autres
critères, dont en particulier le temps de convergence, et la qualité de la
solution espérée. Mes perspectives de recherche s’organisent autour de deux
axes :
* •
compromis mémoire - temps de convergence;
* •
compromis mémoire - qualité de la solution.
Ces deux axes sont bien évidemment complémentaires, et peuvent avoir à être
imbriqués. Dans un but de simplicité de la présentation, ils sont toutefois
décrits ci-dessous de façon indépendantes.
#### 6.1 Compromis mémoire - temps de convergence
Nous avons vu par exemple dans le chapitre 2 que, pour le cas de la
construction d’un arbre couvrant de poids minimum, un certain compromis
espace-temps peut être mis en évidence. Nous avons en effet conçu un
algorithme en temps de convergence $O(n^{2})$ utilisant une mémoire
$O(\log^{2}n)$ bits en chaque nœud, mais nous avons montré qu’au prix d’une
augmentation du temps en $O(n^{3})$, il est possible de se limiter à une
mémoire $O(\log n)$ bits par nœud. C’est précisément ce type de compromis que
nous cherchons à mettre en évidence111Notons qu’un meilleurs compromis a été
trouvé recemment [99]..
Nous comptons aborder le compromis mémoire - temps de convergence selon deux
approches. D’une part, nous allons considérer un grand nombre de problèmes
dans le cadre de l’optimisation de structures couvrantes, afin d’étudier si le
compromis mis en évidence pour le problème de l’arbre couvrant de poids
minimum peut s’observer dans d’autres cadres. Le problème sur lequel nous
comptons focaliser nos efforts est celui de la construction de _spanners_ ,
c’est-à-dire de graphes partiels couvrants. Les spanners sont principalement
caractérisés par leur nombre d’arêtes et leur facteur d’élongation. Ce dernier
paramètre est défini par le maximum, pris sur toutes les paires de nœuds
$(u,v)$, du rapport entre la distance $\mbox{dist}_{G}(u,v)$ entre ces deux
nœuds dans le réseau $G$ et la distance $\mbox{dist}_{S}(u,v)$ entre ces mêmes
nœuds dans le spanner $S$ :
$\mbox{\'{e}longation}=\max_{u,v}\frac{\mbox{dist}_{S}(u,v)}{\mbox{dist}_{G}(u,v)}\leavevmode\nobreak\
.$
La littérature sur les spanners a pour objectif le meilleur compromis entre
nombre d’arêtes et élongation, selon des approches centralisées ou réparties.
Dans [4], un algorithme réparti est ainsi proposé, construisant pour tout
$k\geq 1$, un spanner d’élongation $2k-1$ avec un nombre d’arêtes
$O(n^{1+1/k})$. Nous avons pour but de reprendre cette approche, mais dans un
cadre auto-stabilisant. Est-il possible de concevoir des algorithmes auto-
stabilisants offrant les mêmes performances que celles ci-dessus ? Quelle est
l’espace mémoire requis pour ce type d’algorithmes (s’ils existent) ? Peut-on
mettre en évidence des compromis espace - élongation - nombre d’arêtes ? Ce
sont autant de questions que nous comptons aborder dans l’avenir.
D’autre part, nous avons également pour souhait la mise en évidence de bornes
inférieures. L’établissement de bornes inférieures non-triviales est un des
défis de l’informatique (cf. P versus NP). Le cadre du réparti et de l’auto-
stabilisation ne simplifie pas forcément la difficulté de la tâche, mais
certaines restrictions, comme imposer aux algorithmes de satisfaire certaines
contraintes de terminaison (par exemple d’être silencieux), semble permettre
l’obtention de bornes non-triviales (voir [53]).
#### 6.2 Compromis mémoire - qualité de la solution
Nous avons également pour objectif l’étude du compromis entre l’espace mémoire
utilisé par un algorithme et le rapport d’approximation qu’il garantit pour un
problème d’optimisation donné. Ces dernières années, différents types de
compromis ont fait l’objet de recherches intensives. La théorie des
_algorithmes d’approximation_ est basée sur un tel compromis. Celle-ci a été
développée autour de l’idée que, pour certains problèmes d’optimisation NP-
difficiles, il est possible de produire de bonnes solutions approchées en
temps de calcul polynomial. Nous nous proposons d’étudier le compromis entre
l’espace mémoire utilisé et le rapport d’approximation dans le cas des
algorithmes auto-stabilisants, dont les nœuds sont restreints à utiliser un
espace limité.
Dans le cas de la construction d’arbres, la plupart des algorithmes répartis
de la littérature se focalisent sur des algorithmes dont les caractéristiques
sont à un facteur d’approximation $\rho$ de l’optimal, où $\rho$ est proche du
_meilleur_ facteur connu pour un algorithme séquentiel. C’est, par exemple, le
cas de la construction d’arbres de Steiner ($\rho=2$), ou d’arbres de degré
minimum ($\mbox{OPT}+1$). Cette approche, satisfaisante du point de vue des
performances en terme d’optimisation, peut se révéler très coûteuse en mémoire
dans un cadre auto-stabilisant. Pour optimiser la mémoire, il pourrait se
révéler plus efficace de relaxer quelque peu le facteur d’approximation. C’est
cette voie que je me propose d’étudier dans l’avenir.
### Research perspectives (in English)
This document has summarized my recent contributions in the field of self-
stabilization, within the networking framework as well as within the framework
of computing with mobile entities. Each chapter has listed open problems, and
some specific research directions related to the topics addressed in the
chapter. In this last chapter of the document, I am going to develop general
long term research perspectives organized around the study of _tradeoffs
between the memory space used by nodes, the convergence time of the algorithm,
and the quality of the solution returned by the algorithm_.
Several parameters can be taken into account for measuring the efficiency of a
self-stabilizing algorithm, among which the convergence time and the memory
space play an important role. The importance of the convergence time comes
from the evident necessity for a system to return to a valid state after a
fault, as quickly as possible. The importance of minimizing the memory space
comes from, on one hand, the growing importance of networks, such as sensor
networks, which involve computing facilities subject to space constraints,
and, on the other hand, the minimization of the amount of information exchange
and storage, in order to limit the probability of information corruption.
Minimizing the memory space can be achieved, though potentially to the
detriment of other criteria, among which the convergence time, and the quality
of the returned solution. My research perspectives thus get organized around
two main subjects:
* •
tradeoff between memory size and convergence time;
* •
tradeoff between memory size and quality of solutions.
These two subjects are obviously complementary, and thus must not be treated
independently from each other. Nevertheless, for the sake of simplifying the
presentation, they are described below as two independent topics.
#### Tradeoff between memory size and convergence time
As we have seen in Chapter 2, in the case of MST construction, some space-time
tradeoffs can be identified. We have indeed conceived an algorithm whose
convergence time is $O(n^{2})$, with a memory space of $O(\log^{2}n)$ bits at
every nodes, and we have shown that, to the prize of increasing the
convergence time to $O(n^{3})$, it is possible to reduce the memory space to
$O(\log n)$ bits per node. This is precisely this kind of tradeoffs that are
aiming at studying in the future222Note that a better tradeoff has recently
been identified in [99]..
We plan to tackle tradeoffs between memory space and convergence time
according to two approaches. First, we are going to consider a large number of
problems within the framework of optimizing spanning structures, and we will
analyze whether the kind of tradeoffs brought to light for MST construction
can be observed in different frameworks. One of the problems on which we plan
to focus our efforts is the construction of _spanners_ (i.e., spanning
subgraphs). Spanners are essentially characterized by their number of edges
and by their stretch factor. This latter parameter is defined by the maximum,
taken over all pairs $(u,v)$ of nodes, of the ratio between the distance
$\mbox{dist}_{G}(u,v)$ between these two nodes in the network $G$, and the
distance $\mbox{dist}_{S}(u,v)$ between the same two nodes in the spanner $S$:
$\mbox{stretch}=\max_{u,v}\frac{\mbox{dist}_{S}(u,v)}{\mbox{dist}_{G}(u,v)}\leavevmode\nobreak\
.$
The literature on spanners mostly focuses on the best tradeoff between the
number of edges and the stretch, according to centralized or distributed
approaches. In [4], a distributed algorithm is proposed which, for any
$k\geq$1, constructs spanners of stretch $2k-1$ with a number of edges
$O(n^{1+1/k})$. We aim at revisiting distributed spanner construction, in the
framework of self-stabilization. Is it possible to conceive self-stabilizing
algorithms offering the same performances as those described above? What would
be the memory space requirement of such algorithms (if they exist)? Can we
bring to light tradeoffs between memory space, stretch, and number of edges?
These questions are typical of the ones we plan to tackle in the future.
Our second approach aims at identifying lower bounds. Establishing lower
bounds is one of challenges of computer science (as exemplified by the P
versus NP question). The framework of distributed computing, and/or self-
stabilization does not necessarily simplify the difficulty of the task.
However, restrictions such as imposing the algorithms to satisfy certain
termination constraints (for example to be silent), have been proved to be
helpful for deriving lower bounds (see, e.g., [53]).
#### Tradeoff between memory size and quality of solutions
One of our objectives is also to study tradeoffs between, on the one hand, the
memory space used by the algorithm, and, on the other hand, the quality of the
solution provided by the algorithm, in the framework of optimization problems.
In this framework, various types of tradeoffs have been the object of
extensive researches these last years. The theory of _approximation
algorithms_ is precisely based on that sort of tradeoffs. It was developed
around the idea that, for many NP-hard optimization problems, it is possible
to compute “good” solutions (though not necessarily optimal) in polynomial
time. We are aiming at studying the tradeoff between memory space and
approximation ratio in the case of self-stabilizing algorithms (in a context
in which nodes are restricted to use a limited space).
In the case of spanning tree construction, most of the distributed algorithms
in the literature are based on sequential approximation algorithms with
approximation factor $\rho$ close to the best known approximation factor. That
is, for example, the case of Steiner tree construction ($\rho=2$), and
minimum-degree spanning tree ($\mbox{OPT}+1$). In the context of self-
stabilization, this approach, which is satisfying from the point of view of
optimization, can be very expensive as far as memory is concerned. For
optimizing memory, one may consider relaxing the quality of the approximation
factor. That is this approach that I suggest studying in the future.
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|
arxiv-papers
| 2013-10-09T14:27:43 |
2024-09-04T02:49:52.171604
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "L\\'elia Blin",
"submitter": "Lelia Blin",
"url": "https://arxiv.org/abs/1310.2494"
}
|
1310.2495
|
# Thermal neutron flux measurements in STAR experimental hall.
Y. Fisyak O. Tsai Z. Xu Brookhaven National Laboratory, Upton, New York
11973 University of California - Los Angeles, Physics Department, Los
Angeles, CA 90095-1547
###### Abstract
We report on measurements of thermal neutron fluxes at different locations in
the STAR experimental hall during pp $\sqrt{s}$ = 510 GeV Run 13 at RHIC. We
compared these measurements with calculations based on PYTHIA as minimum bias
events generator, the detailed GEANT3 simulation of the STAR detector and the
experimental hall, and using GCALOR as neutron transport code. A good (within
$\approx$ 30%) agreement was found at locations near ($\approx$1m) and very
far ($\approx$10m) from the beam pipe. For intermediate locations ($\approx$
5m) the simulation overestimates neutron flux by a factor of $\approx$3.
###### keywords:
thermal neutrons, measurements, simulation
## 1 Introduction
Since the time of $R\&D$ for the SSC detectors [1, 2] it has been understood
that the main source of background in a detector at modern colliders are
collisions at interaction point. The contribution from other sources (beam gas
interactions, beam halo particles, etc.) estimated to be below 10%[3].
Extensive simulations of background conditions were part of detectors
optimizations for SSC and LHC experiments. ATLAS[4] and CMS[5] have made
simulations for all types of backgrounds including neutrons. Estimations of
the neutron fluxes in experimental areas were based on simulations only,
without support from experimentally measured data. Only recently the ATLAS-MPX
collaboration[6] published results of absolute background measurements in the
ATLAS experimental hall including thermal neutrons and made a comparison with
results of simulations with GEANT3+GCALOR[7] and Fluka[8]. Their conclusion
was [6] : “Measured thermal neutron fluxes are found to be largely in
agreement with the original simulations, mostly within a factor of two.
Significant deviations are observed in the low radiation regions of ATLAS
cavern, where measured thermal neutron fluxes are found to be lower than
predicted by Monte Carlo simulations.”
The STAR detector at the Relativistic Heavy Ion Collider (RHIC)[9] is planning
series of upgrades in the near future with detectors using different types of
silicon sensors. Reliable estimations of neutron background at STAR are
required to evaluate different technologies for these upgrades. This necessity
and the lack of experimental results for neutron background estimates were our
motivations for this work. Same questions have been raised in context of
ongoing detector R&D for proposed Electron Ion Collider (EIC[10]):
* 1.
What are neutron background conditions currently at the STAR detector and will
be at EIC?
* 2.
How reliable can we estimate these conditions ?
To answer these questions we:
* 1.
made measurement of the absolute thermal neutron flux at different locations
in the STAR[9] Wide Angle Hall (WAH) during RHIC Run 13[11],
* 2.
compared experimental results with simulation in order to understand how
reliable this simulation is, and
* 3.
estimated fluxes of the intermediate energy neutrons using simulation results.
For the purpose of future discussions we will classify neutrons by kinetic
energy($E_{kin}$) as follows:
* 1.
intermediate energy neutrons with $E_{kin}$ in range 100 keV$\textendash$1
MeV, which are most damaging for electronics and silicon detectors, and
* 2.
thermal neutrons with $E_{kin}$ below 250 meV. This definition includes cold
($<25meV$), thermal as such ($25meV$), and part of epithermal (25
meV$<E_{kin}<$400 meV) neutrons. The thermal neutrons generate $\gamma-$quanta
producing noise in detector elements.
## 2 Measurements
### 2.1 $He^{3}$ counter
We used a $He^{3}$ filled proportional counter[12] ($He^{3}C$), loaned to us
by BNL Instrumentation Division, to measure fluxes of thermal neutrons in WAH.
* 1.
The thermal neutron were detected via reaction:
$n+He^{3}\rightarrow H^{1}+H^{3}+764keV,$
with cross section : $\sigma=5.4\sqrt{(}25.3~{}meV/E_{kin})$ [kbarn][13].
* 2.
The $He^{3}C$ specification[12] gave the neutron sensitivity $100\pm 10$
counts per $1Hz/cm^{2}$ of thermal neutron flux. This sensitivity was measured
with calibrated isotropic thermal neutron flux at a temperature of
$25^{0}$C[14].
* 3.
The signal was shaped with the threshold set to 20% of the maximum signal (764
keV), which corresponds to an unambiguous thermal neutron registration
(contamination of $\gamma$ and charged particles were due to only multiple
hits during signal collection time of the detector $\approx$ 5 $\mu s$ and
neglected herein).
* 4.
During the run $He^{3}C$ was positioned at 6 locations[15] of WAH (Fig.1): the
South and North (Fig.2) on the level of the second platform just outside of
MTD from the south and north sides of the detector, the Bottom on the floor
under MTD (Fig.3 and Fig.4), the West and East near the entrances to the
tunnel (Fig.5), and the Far Away (Fig.4) on the floor just after the entrance
to WAH.
Figure 1: STAR Wide Angle Hall GEANT3 geometry model (version y2013-1x)
including building elements (floor, roof, and walls), tunnel, shielding, RHIC
dipole magnet (DX), and the whole STAR detector. MTD stands for Muon Telescope
Detector, ZDC - Zero Degree Calorimeters, BBC - Beam Beam Counters, FMS -
Forward Meson Spectrometer. Figure 2: South (x = 428 cm, y = 183 cm, z = 0)
and North (x = -442 cm, y = 202 cm, z = 0 ) locations of $He^{3}C$. Figure 3:
Bottom (x = 15 cm, y = -390 cm, z = 53 cm) location of $He^{3}C$. Figure 4:
Bottom (x = 15 cm, y = -390 cm, z = 53 cm) and Far Away (x = -970 cm, y = -390
cm, z = -750 cm) locations of $He^{3}C$. Figure 5: West (x = 183 cm, y = 0, z
= 676 cm) and East (x = 135 cm, y = -20cm, z = -686 cm) locations of
$He^{3}C$.
The shaped $He^{3}C$ signal was fed to the so called STAR RICH scalers
(channel 16), and the rate of the scaler (Hz) was recorded in STAR online
database (each 15 s) and in STAR daq stream (with frequency 1 Hz) together
with others scalers (particularly, ZDC West, ZDC East, and ZDC West and East
coincidence). The $He^{3}C$ rate versus date of data taking for different
counter locations is shown in Fig.6.
Figure 6: Measured $He^{3}C$ rate (Hz) versus date at different counter
locations: South (during period: 03/13-04/03), West (04/03-04/17), East
(04/17-05/03), North (05/08-05/22), Bottom (05/23-06/05), and Far Away
(06/06-06/10). The location change is marked as red dots. Figure 7: The
$He^{3}C$ rate (C) versus event rate ($R$) for different counter locations.
The corrected counter rates ($C_{0}$, see text) per 1 MHz of inelastic events
at different locations are given in kHz.
### 2.2 Event rate
In this study we used the East and West ZDC scalers. In order to estimate
event rate [MHz] the following approach[16, 17] was used:
* 1.
$N_{BC}=9.383\times 111/120$: number of bunch crossings,
* 2.
$N_{WE}$: number of crossings that contain a coincidence of the West and East
counters with probability
$P_{WE}=N_{WE}/N_{BC}$,
* 3.
$N_{E}$: number of crossings that contain a hit in the East counter,
$P_{E}=N_{E}/N_{BC}$,
* 4.
$N_{W}$: number of crossings that contain a hit in the West counter,
$P_{W}=N_{W}/N_{BC}$,
* 5.
$P_{A}$: a probability to produce an East hit,
* 6.
$P_{B}$: a probability to produce a West hit,
* 7.
$P_{AB}$: a probability to produce at least one or more East and West
coincidences in the beam crossing.
Then we used 3 equations:
$\displaystyle P_{E}=P_{A}+P_{AB}\times(1-P_{A})$ $\displaystyle
P_{W}=P_{B}+P_{AB}\times(1-P_{B})$ $\displaystyle P_{WE}=P_{A}\times
P_{B}+P_{AB}\times(1-P_{A}\times P_{B})$
and solved them with respect to $P_{AB}$
$\displaystyle
P_{AB}=\frac{P_{WE}-P_{E}P_{W}}{1+P_{WE}-P_{E}-P_{W}}=1-e^{-\mu},$
where $\mu$ is the mean value of Poisson distribution.
Thus the coincidence rate (AB) corrected for random coincidence for A and B is
$\displaystyle N_{AB}=\mu\times N_{BC}=-ln(1-P_{AB})\times N_{BC}.$
The coincidence rate in ZDC corresponded to $\sigma$ = 2.81 mb[16] from 50 mb
of pp[18] inelastic cross section at $\sqrt{s}$ = 510 GeV. Thus the total
event rate: $R=50/2.81\times N_{AB}$.
### 2.3 Fluxes
The measured fluxes are obtained from the $He^{3}C$ rate (C) using the counter
sensitivity. Dependences of the measured C at the different locations on $R$
are shown in Fig.7. In order to normalize C to 1 MHz of pp interaction rate
($C_{0}$) and also account for saturation effects in $He^{3}C$ due to its dead
time, the dependences were approximated by $C=R\times(C_{0}+R\times C_{1}).$
The measurements of $C_{0}$ for different locations are presented in Table 1.
Table 1: The measured $He^{3}C$ rate ($C_{0}$), the estimated from the $He^{3}C$ rate neutron flux for $E_{kin}<250meV$ (RC) using the counter sensitivity (100$\pm$10 counts/($Hz/cm^{2}$)) and its efficiencies in the kinematical range ($87\%$), simulated (MC) thermal neutron flux $(Hz/cm^{2})$, and ratio RC to MC for the different $He^{3}C$ locations in WAH. All numbers are normalized per 1 MHz of pp inelastic collisions at $\sqrt{s}$ = 510 GeV. Location | $C_{0}$ (kHz) | RC $(Hz/cm^{2})$ | MC$(Hz/cm^{2})$ | ratio
---|---|---|---|---
South | 1.18 | 13.6 $\pm$ 1.4 | 34.7 $\pm$ 5.9 | 0.39 $\pm$ 0.08
West | 9.15 | 105.2 $\pm$ 10.5 | 124.1 $\pm$ 11.1 | 0.85 $\pm$ 0.11
East | 12.14 | 139.5 $\pm$ 13.9 | 105.3 $\pm$ 10.3 | 1.33 $\pm$ 0.18
North | 2.34 | 26.9 $\pm$ 2.6 | 39.9 $\pm$ 6.3 | 0.67 $\pm$ 0.13
Bottom | 0.66 | 7.6 $\pm$ 0.8 | 23.9 $\pm$ 4.9 | 0.32 $\pm$ 0.07
FarAway | 0.63 | 7.2 $\pm$ 0.7 | 7.0 $\pm$ 2.6 | 1.03 $\pm$ 0.40
## 3 Simulation
To estimate fluxes, PYTHIA version 6.4.26[19] as pp 510 GeV minimum biased
event generator and GEANT3+GCALOR[7] for propagation particles in WAH were
used. The STAR detector and WAH geometry description was taken as version
$y2013\\_1x$[20] used for RHIC Run 13. The only two essential changes from
default STAR simulation were:
(1) reducing $E_{kin}$ cut for neutral hadrons (CUTNEU) from 1 MeV to
$10^{-13}$ GeV, and
(2) increasing maximum particle time of flight cut (TOFMAX) from $5\times
10^{-4}$ to $1\times 10^{3}$ s.
The simulated $E_{kin}$ spectrum of neutrons in WAH and the spectrum
convoluted with the neutron cross section are presented in Fig.8. From this
spectrum we can conclude that $87\%$ of neutrons with $E_{kin}<250meV$ were
detected by $He^{3}C$.
The neutron’s time of flight distribution from the simulation is presented in
Fig.9. There are two distinct components in the distribution: the first one
with $\tau$ = 7.1 ms which corresponds to neutron dissipation from WAH and the
second component suppressed by a factor of $10^{7}$ with respect to the first
one with $\tau=891s$ which is due to neutron decays (in the simulation neutron
life time $\tau=887s$ was used). Unfortunately, with our maximum recording
rate of 1 Hz we could not detect the dissipation component.
Figure 8: The neutron kinetic energy spectrum in WAH and result of convolution
(red dashed line) of this spectrum with $He^{3}$ neutron cross section. The
integral of the convoluted spectrum corresponds to $87\%$ of the total
spectrum integral in region $<250meV$. Figure 9: The neutron’s time of flight.
The black and green histograms are for all and the thermal neutrons
($E_{kin}<250meV$), respectively. The red line corresponds to exponential fit
$e^{-t/\tau}$ with $\tau=7.1ms$ and the blue line to fit with $\tau=891s$.
Flux was defined as sum of track length of a particle collected in a given
volume in unit time divided by the volume size. The fluxes were normalized to
1 MHz rate of pp inelastic events at $\sqrt{s}$ = 510 GeV. Fluxes for all
neutrons and neutrons with $E_{kin}<250meV$ are show in Fig.10 and Fig.11,
respectively. The radial dependence of fluxes at Z $\approx$ 0 and Z $\approx$
675 cm for all neutrons, neutrons with $E_{kin}>100keV$ and neutrons with
$E_{kin}<250meV$ are shown in Fig.12.
Figure 10: Neutron flux in WAH. Figure 11: Thermal neutron ($E_{kin}<250meV$)
flux in WAH.
Figure 12: The radial dependence of fluxes at Z = 0 (a) and Z = 675 cm (b) for
all neutrons, neutrons with $E_{kin}>100keV$ and $E_{kin}<250meV$, and the
measured flux at the South, North and West locations.
## 4 Conclusions
From this study we conclude that we can estimate neutron background for STAR
detector with good precision. The results of the measurement and simulation
are presented with absolute values and their ratios in Table 1. The comparison
is good (within $30\%$) for the West, East and Far Away locations. However,
for the South, North and Bottom locations the simulation overestimated flux by
a factor of $\approx$ 3\. This conclusion is very close to one [6] which we
cited in the introduction. The mismatch between the measurement and the
simulation may be due to inaccurate description of geometry and material in
the WAH, which would affect the neutron dissipation from the interaction
region. The deviation could also be related to the neutron transport
parameters.
## 5 Acknowledgments
We thank Brookhaven National Laboratory Instrumentation Division and,
especially, G.Smith and N. Schaknowski, for the $He^{3}$ detector. We thank
the STAR Collaboration, the RHIC Operations Group and RCF at BNL. This work
was supported by the Offices of NP and HEP within the U.S. DOE Office of
Science.
## References
* [1] Donald E. Groom, “Radiation Levels in SSC Detectors,” Nucl. Instrum. Meth. A279 (1989) 1-6.
* [2] M.V.Diwan, et al., “Radiation environment and shielding for a high luminosity collider detector,” BNL-52492 Formal Report, SSCL-SR-1223.
* [3] A.I.Drozhdin, M.Huhtinen, and N.V.Mokhov, “Accelerator Related Background in the CMS Detector at LHC,” CERN/TIS-RP/96-08/PP.
* [4] Yu. Fisyak, “Study of neutron and gamma backgrounds in ATLAS,” CERN-ATL-CAL-94-039. S.Baranov, et al., “Estimation of Radiation Background, Impact on Detectors, Activation and Shielding Optimization in ATLAS,” ATL-GEN-2005-001.
* [5] M.Huhtinen, “Radiation Environment Simulations for the CMS Detector,” CERN CMS TN/95-198. Y.Fisyak, R.Breedon, “Comments on the simulation of background for the CMS muon system,” CMS TN/96-019.
* [6] M.Campbell et al., “Analysis of the Radiation Field in ATLAS Using 2008-2011 Data from the ATLAS-MPX Network,” ATL-GEN-PUB-2013-001 http://cds.cern.ch/record/1544435/files/ATL-GEN-PUB-2013-001.pdf
* [7] C.Zeitnitz and T.A.Gabriel, “The GEANT-GCALOR Interface and Benchmark Calculations for Zeus Calorimeters,” Nucl. Instrum. Meth. A349 (1994) 106-111
* [8] G. Collazuol, A. Ferrari, A. Guglielmi, and P.R. Sala, “Hadronic models and experimental data for the neutrino beam production,” Nucl. Instrum. Meth. A449, 609-623 (2000)
* [9] H.K.Ackerman et al., “STAR detector overview,” Nucl. Instrum. Meth. A499: 624,2003.
* [10] BNL-98815-2012-JA, JLAB-PHY-12-1652, arXiv:1212.1701
* [11] http://www.rhichome.bnl.gov/RHIC/Runs/index.html#Run-13
* [12] $He^{3}$ counter, RS_P4-1614-204 GE Power System Reuter-Stokes,
http://www.ge-mcs.com/download/reuter-stokes/GEA13545B_ThermalCount.pdf
* [13] http://www.nndc.bnl.gov/exfor/servlet/E4sGetTabSect?SectID=13235&req=61079&PenSectID=872
* [14] Nathan Johnson,GE Energy,Reuter-Stokes Measurements Solutions, [email protected], private communication.
* [15] STAR uses a right-handed coordinate system with its origin at the nominal interaction point and z-axis coinciding the the axis of the beam pipe. The x-axis points south, the y-axis points upward, and the z-axis points to the west. STAR Note 0229A.
* [16] James Dunlop, [email protected], private communication.
* [17] D.Cronin-Hennessy and P.F.Derwent, “The CDF Run I luminosity measurement,” Fermilab-Pub-99/162-E.
* [18] ATLAS Collaboration, “Measurement of the Inelastic Proton-Proton Cross-Section at $\sqrt{s}$=7 TeV with the ATLAS Detector,” arXiv:1104.0326
* [19] T. Sj$\ddot{o}$strand, S. Mrenna and P. Skands, JHEP05, 026 (2006)
* [20] https://drupal.star.bnl.gov/STAR/comp/simu/geometry-tags
|
arxiv-papers
| 2013-10-09T14:30:22 |
2024-09-04T02:49:52.190327
|
{
"license": "Public Domain",
"authors": "Yuri Fisyak, Oleg Tsai, Zhangbu Xu",
"submitter": "Yuri Fisyak",
"url": "https://arxiv.org/abs/1310.2495"
}
|
1310.2521
|
# Large momentum-dependence of the main dispersion “kink” in the high-$T_{c}$
superconductor Bi2Sr2CaCu2O8+δ
N. C. Plumb1111Present address: Swiss Light Source, Paul Scherrer Institut,
CH-5232 Villigen PSI, Switzerland, T. J. Reber1, H. Iwasawa2, Y. Cao1, M.
Arita2, K. Shimada2, H. Namatame2, M. Taniguchi2, Y. Yoshida3, H. Eisaki3, Y.
Aiura3 and D. S. Dessau1,4 1 Department of Physics, University of Colorado,
Boulder, CO 80309-0390, USA 2 Hiroshima Synchrotron Radiation Center,
Hiroshima University, Higashi-Hiroshima 739-0046, Japan 3 National Institute
of Advanced Industrial Science and Technology, Tsukuba, Ibaraki 305-8568,
Japan 4 JILA, University of Colorado and NIST, Boulder, CO 80309-0440, USA
[email protected], [email protected]
###### Abstract
Ultrahigh resolution angle-resolved photoemission spectroscopy (ARPES) with
low-energy photons is used to study the detailed momentum dependence of the
well-known nodal “kink” dispersion anomaly of Bi2Sr2CaCu2O8+δ. We find that
the kink’s location transitions smoothly from a maximum binding energy of
about 65 meV at the node of the $d$-wave superconducting gap to 55 meV roughly
one-third of the way to the antinode. Meanwhile, the self-energy spectrum
corresponding to the kink dramatically sharpens and intensifies beyond a
critical point in momentum space. We discuss the possible bosonic spectrum in
energy and momentum space that can couple to the $k$-space dispersion of the
electronic kinks.
###### pacs:
74.72.-h, 74.25.Jb, 74.25.Kc
## 1 Introduction
One of the defining characteristics of the electronic structure of the
high-$T_{c}$ cuprates is the presence of an especially prominent anomaly, or
“kink”, in the electronic dispersion, which corresponds to a strong feature in
the complex electronic self-energy spectrum
$\Sigma(\boldsymbol{k},\omega)=\Sigma^{\prime}(\boldsymbol{k},\omega)+i\Sigma^{\prime\prime}(\boldsymbol{k},\omega)$.
The origin of the kink — whether it is due to interactions of the electrons
with bosons (particularly phonons [1, 2] or magnetic excitations [3, 4, 5]) or
some other phenomenon [6] — is still heavily debated. Likewise the kink’s
connection to superconductivity, and whether the interactions it signifies may
either form or break Cooper pairs, or be altogether irrelevant, remains
unknown.
At the nodes of the $d$-wave superconducting gap, this kink appears at a
binding energy of roughly 60–70 meV [7, 8, 9, 10]. Meanwhile near the
antinode, a seemingly stronger kink is located at about 20–40 meV, depending
on doping [11, 12, 13]. While a possible connection between the nodal and
antinodal kinks remains a mystery, the new data here fills in details of the
evolving physics between these points. Such information is crucial for
obtaining a complete understanding of the behaviour and origin of the kink and
hence the electron-boson coupling in the high-$T_{c}$ superconductors.
## 2 Analysis and results
### 2.1 Experimental
The data presented here were obtained from Bi2Sr2CaCu2O8+δ (Bi2212) near
optimal doping with $T_{c}\approx 89$ K. Rotational alignment of the sample
better than $1^{\circ}$ was performed by Laue diffraction. The data were
collected in the superconducting state at 10 K using a photon energy of 7 eV.
Compared to conventional photon energies, the low photon energy greatly
improves the photoelectron escape depth, momentum resolution, and overall
spectral sharpness [14]. Total combined energy resolution of the light source
and analyser was about 7 meV. ARPES cuts were taken along the $(\pi,\pi)$
direction of the Fermi surface (FS).
### 2.2 Momentum-dependent self-energy
In the present work, we are especially concerned with the self-energy
contribution due to electrons coupling to a collective mode over a sharp
energy range, and we wish to isolate this from other interactions with smooth
energy dependencies (e.g., electron-electron scattering [15]). This is
accomplished by assuming a smooth (in this case linear) effective bare band
$\epsilon_{\text{eff}}(\boldsymbol{k})$ for each ARPES cut that connects
points on the dispersion far from the main kink. The real part of the
effective self-energy is then simply
$\Sigma^{\prime}_{\text{eff}}(\omega)=\omega-
v_{F}^{\text{eff}}[k_{m}(\omega)-k_{F}]$ (1)
where $k_{m}(\omega)$ is the measured dispersion, $v_{F}^{\text{eff}}$ is the
slope of $\epsilon_{\text{eff}}(k)$, and $k_{F}$ is the Fermi momentum.
$\Sigma^{\prime\prime}_{\text{eff}}(\omega)$ is then the Kramers-Kronig
transformation of $\Sigma^{\prime}_{\text{eff}}(\omega)$. Unlike
$\Sigma^{\prime}(\omega)$, $\Sigma^{\prime}_{\text{eff}}(\omega)$ is well-
behaved at its endpoints (by construction), and its Kramers-Kronig
transformation is easily computed. Our routine sets the in-gap points of
$\Sigma^{\prime}_{\text{eff}}(\omega)$ to zero and computes the transformation
by Fast Fourier transform assuming electron-hole symmetry. We have verified by
simulations that a possible violation of electron-hole symmetry [16] should
not significantly alter the findings here. We note that Eq. 1 is a
conventional definition of $\Sigma^{\prime}_{\text{eff}}$. Recently it was
shown that this definition undervalues the “true” bosonic part of the self-
energy by an overall scaling factor related to the coupling strength of
electron-electron interactions, $\lambda_{\text{el-el}}$ [17]. As this factor
influences the magnitude of the self-energy, not its distribution along
$\omega$ or $\boldsymbol{k}$ , neglecting it will not affect the conclusions
of the present study. A systematic assessment of $\lambda_{\text{el-el}}$ in
Bi2212, and hence the correct scaling factor to be applied to
$\Sigma^{\prime}_{\text{eff}}$, is currently underway.
Figure 1: (a) First quadrant of the Fermi surface of Bi2212. The colour scale
is the measured spectral intensity 10 meV below $E_{F}$. Two representative
cuts (i and ii) are indicated by red curves. The black curves are sketches of
the antibonding (AB) and bonding band (BB) sheets. The use of 7-eV photons
isolates the antibonding band. (b) Raw ARPES data from cuts i and ii. The
solid black curves are the peak positions of the fitted MDCs, while the dashed
red lines are the effective noninteracting bands for the dispersions (see
text). (c) MDC widths at cuts progressing away from the node. (d) Real (black)
and imaginary (red) components of the effective electronic self-energy for
cuts i and ii. (e) $\Sigma^{\prime}_{\text{eff}}$ and (f)
$-\partial\Sigma^{\prime\prime}_{\text{eff}}/\partial\omega$ as a function of
$\theta$ and $\omega$. The black dots are the peak locations of
$\Sigma^{\prime}_{\text{eff}}$ at each $\theta$, which we call
$\Omega_{\text{kink}}(\theta)$.
Figure 1(a) shows ARPES data collected along the FS in the first quadrant of
the Brillouin zone. The colour scale represents the measured intensity 10 meV
below $E_{F}$. The thick solid lines in Figure 1(a) are sketches of the
antibonding (AB) and bonding band (BB) Fermi surfaces based on a tight-binding
model [18]. For 7-eV photons, only the AB is detected [19], which greatly
simplifies the analysis. Two representative raw data cuts, corresponding to
Fermi surface angles $\theta=0.9^{\circ}$ and $\theta=16.3^{\circ}$, are
indicated by the red curves labelled i and ii, respectively. The spectra from
these cuts are shown in Figure 1(b). The dispersions from momentum
distribution curve (MDC) fits are overlaid on the spectra (solid black
curves). The dashed red lines are assumed effective bare bands used to
calculate corresponding effective self-energy spectra
$\Sigma_{\text{eff}}(\omega)$ at each $\theta$. These effective bare bands are
determined by linear fits from -230 meV to -200 meV that are constrained to
pass through the MDC peak location at $\omega=-\Delta(\theta)$ (i.e.,
$k_{F}$).
Figure 1(c) shows the Lorentzian MDC widths for each cut from i to ii, while
Figure 1(d) depicts $\Sigma^{\prime}_{\text{eff}}(\omega)$ (black, right axis)
and $\Sigma^{\prime\prime}_{\text{eff}}(\omega)$ (red, left axis) for cuts i
and ii. The full spectrum of $\Sigma^{\prime}_{\text{eff}}(\theta,\omega)$ is
plotted as a colour scale in Figure 1(e). We define the kink energy
$\Omega_{\text{kink}}$ as the location of the peak in
$\Sigma^{\prime}_{\text{eff}}(\omega)$ at each $\theta$. These values are
determined by quadratic fits over a range $\pm 20$ meV about the maximum of
each spectrum. The error bars show the standard deviations ($\pm\sigma$)
returned from the fits. Figure 1(f) depicts
$-\partial\Sigma^{\prime\prime}_{\text{eff}}/\partial\omega$ as a function of
$\omega$ and $\theta$. To reduce noise in the derivative, some light smoothing
was applied to the $\Sigma^{\prime}_{\text{eff}}$ spectrum. Together panels
(e)-(f) highlight the evolution of the self-energy over the nodal region,
which exhibits both dispersive behaviour and sharpening. The results are fully
consistent with the behaviour of the MDC widths in panel (c) and the
electronic dispersion anomalies in (b), providing an important verification of
the self-consistency of the data and analysis methods. It is worth noting that
the quantity $-\partial\Sigma^{\prime\prime}_{\text{eff}}/\partial\omega$ in
panel (f) is somewhat related to a useful parameter of strong coupling theory
— the Eliashberg boson coupling spectrum $\alpha^{2}F(\boldsymbol{k},\nu)$,
where $\nu$ is the energy axis for bosons. In an ungapped system at $T=0$,
$\Sigma^{\prime\prime}(\boldsymbol{k},\omega)=\pi\int_{0}^{|\omega|}d\nu\alpha^{2}F(\boldsymbol{k},\nu)$
[20], although the anisotropic gapping in cuprates can significantly alter
this relationship. Addressing this issue via suitable “gap referencing” is a
key objective of the present work.
Two key points are evident from Figure 1. First, $\Omega_{\text{kink}}$
evolves smoothly as a function of $\theta$ in the nodal region, shifting
toward $E_{F}$ by about 10 meV from $\theta=0$ to $\theta=15^{\circ}$. Second,
the nature of $\Sigma_{\text{eff}}$ appears to change abruptly past a critical
point in $\boldsymbol{k}$-space. While by eye the kink perhaps becomes more
dramatic going from node to antinode [21], this fact alone does not
necessarily mean that the self-energy strengthens, since $\Sigma$ is related
to the bare band velocity, which decreases away from the node. Indeed, the
results in Figure 1(c)-(f) show that over much of the near-nodal region
$\Sigma_{\text{eff}}(\omega)$ is relatively unchanged, despite the visual
appearance that the kink is “getting stronger”. However, for $\theta\gtrsim
10^{\circ}$ Figure 1(e) shows a rapid increase in the peak of
$\Sigma^{\prime}_{\text{eff}}(\omega)$. This corresponds with sharpening of
the step in $\Sigma^{\prime\prime}_{\text{eff}}(\omega)$ seen in Figure 1(f).
The findings in Figure 1 contrast with a previous study of overdoped Pb-Bi2212
where it was argued that the scattering rate near $E_{F}$ is independent of
$\boldsymbol{k}$ [22]. We also point out that these results contradict recent
claims that the energy $\Omega_{\text{kink}}$ is constant near the node and
then suddenly jumps at a “crossover” point on the FS $\sim 15^{\circ}$ away
from the node [23, 24], though there still may be a crossover parameterised
by, e.g., the intensity and sharpness of the features in
$\Sigma_{\text{eff}}(\omega,\theta)$, as seen in Figure 1(e)-(f).
### 2.3 Scattering $\mathbf{q}$-space analysis of the kink momentum
dependence
The large nodal ARPES kink seen in cuprates is generally explained as the
result of the coupling of the electrons to a bosonic mode of energy
$\Omega_{\text{boson}}$. In particular, $\Omega_{\text{kink}}$ may be able to
tell us which electrons interact with which bosons, and in principle this can
yield information about which (or even whether) bosons act as the “glue”
responsible for the formation of the Cooper pairs. The new finding of the
large, smooth dispersion of $\Omega_{\text{kink}}(\boldsymbol{k})$ is
therefore an important result that may connect directly to the coupling
mechanism of the electrons within a pair. Here we consider how to best connect
the $k$-dispersion of the kink to known data of the $q$-space dependence of
various bosonic modes.
In the simplest picture, the kink energies $\Omega_{\text{kink}}$ will be
exactly those of the coupling boson [8], though this ignores the “gap
referencing” which is simple for an $s$-wave superconductor
($|\Omega_{\text{kink}}|=\Omega_{\text{boson}}+\Delta$) but more complicated
for a $d$-wave superconductor in which $\Delta$ is strongly $k$-dependent. In
the presence of an anisotropic gap $\Delta(\boldsymbol{k})$, a bosonic mode
with energy $\Omega_{\text{boson}}(\boldsymbol{q})$ scattering an electron
purely from $\boldsymbol{k}$ to $\boldsymbol{k^{\prime}}$ is expected to
produce an ARPES dispersion kink below $E_{F}$ at [25]
$|\Omega_{\text{kink}}(\boldsymbol{k})|=\Omega_{\text{boson}}(\boldsymbol{q})+\Delta(\boldsymbol{k^{\prime}})$
(2)
which can be deduced by considering the set of photoholes at $\boldsymbol{k}$
that can be annihilated via electrons decaying from $\boldsymbol{k^{\prime}}$
and emitting bosons $\Omega_{\text{boson}}(\boldsymbol{q})$. An argument along
these lines (but for an isotropic gap) is presented in section 7.3 of [20]. We
will make use of this gap referencing relationship throughout the present work
in order to identify the boson dispersions appropriate to particular
scattering scenarios. Additional corrections for relating
$\Omega_{\text{kink}}$ to $\Omega_{\text{boson}}$ are believed to be too small
to account for the dispersive behaviour of
$\Omega_{\text{kink}}(\boldsymbol{k})$ [26] and therefore should not
qualitatively alter the present work.
Figure 2: Extracting the boson coupling mode dispersion assuming the
following scattering $\boldsymbol{q}$ directionalities in (a): horizontal and
vertical (H/V), intra-hole-pocket (Intra), and inter-hole-pocket (Inter). (b)
$|\Omega_{\text{kink}}|$ and $\Omega_{\text{boson}}^{*}$ as a function of FS
angle $\theta$. $\Omega_{\text{boson}}^{*}$ [Equation (3)] is the boson energy
for the special cases of scattering along symmetry directions such that
$\Delta(\boldsymbol{k^{\prime}})=\Delta(\boldsymbol{k})$, as depicted in (a).
(c), (d) Dispersions of $|\Omega_{\text{kink}}(\boldsymbol{q})|$ (red ■) and
$\Omega_{\text{boson}}^{*}(\boldsymbol{q})$ (blue ●), assuming that scattering
occurs horizontally/vertically in the Brillouin zone. Curves for
$|\Omega_{\text{kink}}(\boldsymbol{q})|$ and
$\Omega_{\text{boson}}^{*}(\boldsymbol{q})$ have been extracted considering
the short (c) and long (d) H/V scattering channels separately. The extracted
dispersions are compared to those of Cu-O phonons measured by INS in YBa2Cu3O7
(△) and YBa2Cu3O6 (▽) [27], as well as phonons observed by IXS in Bi2201 (◇)
[23]. The pink highlighted Cu-O bond-stretching (Cu-O BS) branch, in
particular, has been identified by some previous experiments as possibly
relevant to the nodal kink. (e), (f) Analogous plots assuming diagonal intra-
and inter-pocket scattering. The green shaded line in (f) is the approximate
dispersion of the high-energy branch of spin fluctuations (SF) observed in
optimally-doped Bi2212 [28]. The hatched area indicates that this region is
observed in the neutron data to be somewhat filled in by the width of the
dispersion peaks. For simplicity, error bars from Figures 1(e) and (f) are
shown only for the $\Omega_{\text{boson}}^{*}(\boldsymbol{q})$ curve in each
panel.
There is reason to believe that the predominant electron-boson scattering
relevant to the nodal kink falls along some symmetry direction, thus
simplifying the connection between $k$\- and $q$-space. For instance, spin
fluctuations observed by inelastic neutron scattering (INS) are peaked at
points on or near the $(\xi,\xi,0)$ line [29, 30, 31]. Likewise, both
experiment [32] and theory [1, 33] suggest the Cu-O “half-breathing” phonon
mode scatters electrons primarily along $(\xi,0,0)$ [equivalently
$(0,\xi,0)$], and there is evidence that this mode couples strongly to
electrons [34] and, in particular, may contribute to the nodal kink [19].
In the case of phonons, numerical calculations find that, on the whole, the
scattering matrix elements are fairly complicated [1, 33]. Nevertheless, they
are expected to evolve smoothly over the FS and exhibit some preference for
particular directionalities. Thus, despite the complexity of the full
scattering problem, one can reasonably expect to find qualitative agreement
between the actual phonon dispersion and the inference from ARPES — at least
over a limited portion of the FS.
To proceed with our analysis, assumed scattering $\boldsymbol{q}$’s along
symmetry directions are illustrated in Figure 2(a). We consider cases where
electrons may scatter horizontally, vertically, or diagonally via inter- or
intra-hole-pocket vectors. The kink energies obtained in Figure 1(e) are
plotted in Figure 2(b) as a function of FS angle $\theta$ (red ■). The blue
circles (●) are the corresponding gap-referenced boson energies
$\Omega_{\text{boson}}^{*}$. The asterisk (*) denotes that only the special
cases of scattering vectors shown in that panel apply. Under these
circumstances, $\Delta(\boldsymbol{k})=\Delta(\boldsymbol{k^{\prime}})$,
leading to
$\Omega_{\text{boson}}^{*}(\theta)=|\Omega_{\text{kink}}(\theta)|+\Delta(\theta)\text{.}$
(3)
In calculating $\Omega_{\text{boson}}^{*}(\theta)$, we used $\Delta(\theta)$
based on our ARPES-measured values, which were found to have excellent
agreement with the expected $d$-wave form, with maximum (antinodal) magnitude
$\Delta_{0}=30$ meV. The gap measurements shown here were performed using the
newly-developed tomographic density of states (TDoS) technique [35, 36], and
we have checked that the analysis and results that follow are essentially
unchanged if the symmetrised energy distribution curve (EDC) method is
employed [37].
The extracted $\boldsymbol{q}$-space dispersions of $\Omega_{\text{kink}}$ and
$\Omega_{\text{boson}}^{*}$ under these various scattering scenarios are shown
in Figure 2(c)–(f). The insets in each of these panels illustrate how the $q$
values on each horizontal axis were determined. For simplicity and
generalizability of the analysis, we consider each scattering channel
independently. For an assumed bosonic mode that would scatter electrons in the
$(\xi,0,0)/(0,\xi,0)$ directions, a given $\boldsymbol{k}$ point on the Fermi
surface could couple via two orthogonal vectors — one shorter than the node-
node distance in $\boldsymbol{q}$-space ($\xi\sim 0.35$) and the other longer
[labelled “H/V short” and “H/V long” in Figure 2(c) and (d), respectively].
However, in general these short and long $\boldsymbol{q}$ channels would not
be expected to contribute equally to the appearance of the kink, but rather
their relative scattering intensities would evolve around the Fermi surface
(only matching at the node-node distance, where they have the same length). To
disentangle these paired interactions, the
$\Omega_{\text{kink}}(\boldsymbol{q})$ and
$\Omega_{\text{boson}}^{*}(\boldsymbol{q})$ curves were extracted by treating
the short and long scattering vectors separately, as plotted in Figure 2(c),
(d). This is a logical choice, since one or the other (either the short or
long vector) would probably be more influential at any given point on the
Fermi surface. Meanwhile, the diagonal intra- and inter-hole-pocket vectors
[“Intra” and “Inter” in Figure 2(e) and (f) respectively] are also treated
independently, since they are distinct in how they couple the topology of the
Fermi surface.
In Figure 2(c)–(f), the extracted $\Omega_{\text{boson}}^{*}(\boldsymbol{q})$
curves (blue ●) are compared to various Cu-O phonon dispersions in YBa2Cu3O6+x
(YBCO) with $x=0$ (▽) and $x=1$ (△) [27], as well as two phonon branches
observed in Bi2Sr1.6La0.4Cu2O6+δ (Bi2201, ◇) [23]. Additionally, a sketch of
the dispersion of a high-energy branch of incommensurate spin fluctuations is
shown in Figure 2(f) (green shaded line).
Overall, the extracted $\Omega_{\text{boson}}^{*}(\boldsymbol{q})$ curves do
not provide clear support that the kink primarily originates from electron-
phonon interactions, although the data may not be wholly inconsistent with
this possibility. For instance, in Figure 2(d), there is a limited
$\boldsymbol{q}$-space region [$\boldsymbol{q}\approx(0.3\text{--}0.4,0,0)$]
where the extracted boson dispersion roughly overlaps with the Cu-O bond
stretching phonon branch, as pointed out in [23]. However, for larger values
of $\boldsymbol{q}$ extending toward $(0.5,0,0)$,
$\Omega_{\text{boson}}^{*}(\boldsymbol{q})$ diverges from the Cu-O bond
stretching phonon branch with a different slope. In this regard, our data show
greater overall similarity between $\Omega_{\text{boson}}^{*}(\boldsymbol{q})$
and the SF dispersion in Figure 2(f), which differ by merely a simple offset
in $\omega$ and/or $\boldsymbol{q}$, perhaps reflecting systematic differences
between the techniques and/or samples used in the studies. The rough
correspondence between the kink and the SF dispersion compares favorably with
spectral analysis of the spin response function extracted from ARPES data by
Chatterjee et al [38], though their formalism only considered spin
fluctuations and did not provide a comparison to phonon dispersions. Moreover,
that approach treated the spectral function holistically, rather than
isolating the kink feature and considering its explicit connection to the
bosonic spectral function.
The SF dispersion in Figure 2(f) is the high-energy branch of incommensurate
spin excitations so far observed in many cuprates [29, 30, 39, 31, 28]. It
converges with a low-energy branch near $\sim 40$ meV, where there is a well-
known $\boldsymbol{q}=(0.5,0.5)$-centred “resonance” peak in the spin
susceptibility at low $T$ [40, 41, 42, 43]. The effective self-energy obtained
from our analysis intensifies significantly at $\Omega_{\text{boson}}^{*}$
somewhat near the resonance energy. This is depicted in Figure 3, where the
peak height of $\Sigma^{\prime}_{\text{eff}}$ (red triangles) is plotted
versus $\Omega_{\text{boson}}^{*}$. INS data from optimally-doped Bi2212 (open
circles) show the difference in scattered neutron intensity from 100 K to 10
K, illustrating the location of the resonance [43]. Notably, within the
context of an orbital-overlap model, coupling to the Cu-O bond-stretching
phonon suggested by Figure 2(d) is not expected to intensify in this manner at
$\theta$ corresponding to $\Omega_{\text{boson}}^{*}$ close to the resonance
[44]. With that said, the data is again only in rough agreement with the SF
picture, and other studies imply that the strength of
$\Sigma_{\text{eff}}^{{}^{\prime}}$ is monotonic around the Fermi surface
[45], meaning that it would not obey the peak-shaped trend of the INS data
reproduced in Figure 3. However, this does not fully undermine the possibility
that the kink is SF-related, as it could instead signal a contribution to
$\Sigma_{\text{eff}}^{{}^{\prime}}$ at low energies (i.e., near the antinode)
from additional types of electron-boson interactions, as we will discuss.
Figure 3: Peak height of $\Sigma^{\prime}_{\text{eff}}$ as a function of
$\Omega_{\text{boson}}^{*}$. The results are compared to INS data [43]
highlighting the magnetic resonance at $\sim 40$ meV. The INS data points are
the neutron scattering intensity at 100 K subtracted from the signal at 10 K.
## 3 Discussion
The above analysis has relied on the key assumption of a dominant scattering
mode and directionality, which, while consistent with interpretations of some
INS and inelastic x-ray scattering (IXS) data and calculations, is not
exhaustive. A natural potential counter example is the case where all points
on the FS couple primarily to the van Hove singularities at the antinodes — a
situation in which the results would not be directly comparable with
conventional INS/IXS data, since the $\boldsymbol{q}$’s would no longer fall
along a straight line. Perhaps the best that can be said is that coupling
strictly to the antinodes would shift the
$\Omega_{\text{kink}}(\boldsymbol{q})$ dispersion down universally by
$\Delta_{0}\approx 30$ meV such that $\Omega_{\text{boson}}(\boldsymbol{q})$
would span an energy range of roughly 25–35 meV. These energies are home to
many phonons in the cuprates [46], which in principle could combine their
effects in some intricate way to produce the observed kink behaviour. A strong
antinodal coupling appears unlikely, however, because this scenario would
imply a huge shift of the nodal kink energy between the normal and
superconducting states and/or as a function of doping. Multiple ARPES studies
find no evidence of such a shift [10, 12, 21, 47, 25], although one case where
the kink was interpreted to be composed of multiple phonon mode couplings
arguably shows evidence of node-antinode scattering [48].
Reviewing the results, the analysis of
$\Omega_{\text{boson}}^{*}(\boldsymbol{q})$ for “long” $(\xi,0,0)$/$(0,\xi,0)$
scattering vectors found some region of agreement with the dispersion of a
Cu-O bond stretching phonon [Figure 2(d)], although the trends diverge as
$\xi$ extends out to 0.5. On the other hand,
$\Omega_{\text{boson}}^{*}(\boldsymbol{q})$ extracted for diagonal inter-hole-
pocket scattering is similar to the dispersion of spin fluctuations [Figure
2(f)], merely differing by a simple offset in energy and/or $\boldsymbol{q}$.
Additionally, Figure 3 illustrates that the strength of the self-energy
associated with the kink, plotted with respect to $\Omega_{\text{boson}}^{*}$,
has a qualitative resemblance to spin fluctuations, and this is contrary to
the behaviour predicted for electron-phonon coupling [44]. However, to the
extent the data might be viewed as favoring the spin fluctuation picture, it
poses an intriguing apparent paradox; A very detailed low-$h\nu$ ARPES study
found an isotope shift in the energy of the nodal kink [19], giving strong
merit to the phonon scenario. One possible explanation for this conflict is
that electron-phonon interactions might constitute a finite but relatively
small contribution to the total self-energy [49]. Alternatively, the results
may signal a role for coupling between spin and lattice degrees of freedom
[50, 51, 52].
The discovery of the large momentum dependence of the main nodal kink adds to
the richness of strong electron-boson coupling phenomena in cuprates. It was
recently shown that a newly-discovered ultra-low-energy kink $\sim 10$ meV
below $E_{F}$ [53, 25, 54, 55] has its own distinct momentum dependence that
runs counter to the behaviour of the deeper-energy kink studied here [56].
Specifically, unlike the larger main kink 65–55 meV below $E_{F}$, which
evolves toward lower binding energy while moving from node to antinode, the
ultra-low-energy kink closely follows the contour of the superconducting gap
in the nodal region, moving to higher binding energy approaching the antinode.
A natural, but spectroscopically demanding, next course of study will be to
investigate the possible convergence of these two energy scales near the
antinodal point and to see whether either or both this these connect with the
antinodal feature observed near 20–40 meV, which historically has been
regarded as a separate kink. Hence the data here, in concert with [56], open
the possibility that interactions with distinct physical origins could combine
within a narrow energy range in the antinodal region — perhaps with major
implications for high-$T_{c}$ superconductivity.
In conclusion, using low photon energy ARPES, we have mapped the detailed
momentum dependence of the primary kink in the nodal $k$-space region of near-
optimal Bi2212. From a simplifying treatment of the data that takes into
account effects of the $d$-wave superconducting gap, the kink’s dispersion
seems inconsistent with most phonons, though over a limited range of momentum
transfer [$\boldsymbol{q}\approx(0.3\text{--}0.4,0,0)$] it bears some
semblance to scattering due to a Cu-O bond-stretching mode. However, in terms
of the momentum dependence of the location and sharpness/intensity of the
self-energy feature, the greatest similarity is found with the dispersion of
the upper branch of incommensurate spin fluctuations.
_Note added_ — During review of this manuscript, a related article was
published [57].
Funding was provided by the DOE under project number DE-FG02-03ER46066.
Experiments were conducted at BL-9A of the Hiroshima Synchrotron Radiation
Center and BL5-4 of the Stanford Synchrotron Radiation Lightsource (SSRL).
SSRL is operated by the DOE, Office of Basic Energy Sciences. We thank D.
Reznik, T. P. Devereaux, and S. Johnston for valuable conversations.
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|
arxiv-papers
| 2013-10-09T15:16:21 |
2024-09-04T02:49:52.196073
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "N. C. Plumb, T. J. Reber, H. Iwasawa, Y. Cao, M. Arita, K. Shimada, H.\n Namatame, M. Taniguchi, Y. Yoshida, H. Eisaki, Y. Aiura, D. S. Dessau",
"submitter": "Nicholas Plumb",
"url": "https://arxiv.org/abs/1310.2521"
}
|
1310.2535
|
EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN)
CERN-PH-EP-2013-183 LHCb-PAPER-2013-050 9 October 2013
Search for the decay
$D^{0}\\!\rightarrow\mathup{{{\pi}}}^{+}\mathup{{{\pi}}}^{-}\mathup{{{\mu}}}^{+}\mathup{{{\mu}}}^{-}$
The LHCb collaboration†††Authors are listed on the following pages.
A search for the
$D^{0}\\!\rightarrow\mathup{{{\pi}}}^{+}\mathup{{{\pi}}}^{-}\mathup{{{\mu}}}^{+}\mathup{{{\mu}}}^{-}$
decay, where the muon pair does not originate from a resonance, is performed
using proton-proton collision data corresponding to an integrated luminosity
of $1.0\mbox{\,fb}^{-1}$ recorded by the LHCb experiment at a centre-of-mass
energy of $7\mathrm{\,Te\kern-1.00006ptV}$. No signal is observed and an upper
limit on the relative branching fraction with respect to the resonant decay
mode
$D^{0}\\!\rightarrow\mathup{{{\pi}}}^{+}\mathup{{{\pi}}}^{-}\phi(\rightarrow\mathup{{{\mu}}}^{+}\mathup{{{\mu}}}^{-})$,
under the assumption of a phase-space model, is found to be
$\mathcal{{\cal
B}}(D^{0}\\!\rightarrow\mathup{{{\pi}}}^{+}\mathup{{{\pi}}}^{-}\mathup{{{\mu}}}^{+}\mathup{{{\mu}}}^{-})/\mathcal{{\cal
B}}(D^{0}\rightarrow\mathup{{{\pi}}}^{+}\mathup{{{\pi}}}^{-}\phi(\rightarrow\mathup{{{\mu}}}^{+}\mu^{-}))<0.96\\\
$
at $90\%$ confidence level. The upper limit on the absolute branching fraction
is evaluated to be $\mathcal{{\cal
B}}(D^{0}\\!\rightarrow\mathup{{{\pi}}}^{+}\mathup{{{\pi}}}^{-}\mathup{{{\mu}}}^{+}\mathup{{{\mu}}}^{-})<5.5\,\times
10^{-7}$ at 90% confidence level. This is the most stringent to date.
Submitted to Phys. Lett. B
© CERN on behalf of the LHCb collaboration, license CC-BY-3.0.
LHCb collaboration
R. Aaij40, B. Adeva36, M. Adinolfi45, C. Adrover6, A. Affolder51, Z.
Ajaltouni5, J. Albrecht9, F. Alessio37, M. Alexander50, S. Ali40, G.
Alkhazov29, P. Alvarez Cartelle36, A.A. Alves Jr24, S. Amato2, S. Amerio21, Y.
Amhis7, L. Anderlini17,f, J. Anderson39, R. Andreassen56, J.E. Andrews57, R.B.
Appleby53, O. Aquines Gutierrez10, F. Archilli18, A. Artamonov34, M. Artuso58,
E. Aslanides6, G. Auriemma24,m, M. Baalouch5, S. Bachmann11, J.J. Back47, A.
Badalov35, C. Baesso59, V. Balagura30, W. Baldini16, R.J. Barlow53, C.
Barschel37, S. Barsuk7, W. Barter46, Th. Bauer40, A. Bay38, J. Beddow50, F.
Bedeschi22, I. Bediaga1, S. Belogurov30, K. Belous34, I. Belyaev30, E. Ben-
Haim8, G. Bencivenni18, S. Benson49, J. Benton45, A. Berezhnoy31, R. Bernet39,
M.-O. Bettler46, M. van Beuzekom40, A. Bien11, S. Bifani44, T. Bird53, A.
Bizzeti17,h, P.M. Bjørnstad53, T. Blake37, F. Blanc38, J. Blouw10, S. Blusk58,
V. Bocci24, A. Bondar33, N. Bondar29, W. Bonivento15, S. Borghi53, A.
Borgia58, T.J.V. Bowcock51, E. Bowen39, C. Bozzi16, T. Brambach9, J. van den
Brand41, J. Bressieux38, D. Brett53, M. Britsch10, T. Britton58, N.H. Brook45,
H. Brown51, A. Bursche39, G. Busetto21,q, J. Buytaert37, S. Cadeddu15, O.
Callot7, M. Calvi20,j, M. Calvo Gomez35,n, A. Camboni35, P. Campana18,37, D.
Campora Perez37, A. Carbone14,c, G. Carboni23,k, R. Cardinale19,i, A.
Cardini15, H. Carranza-Mejia49, L. Carson52, K. Carvalho Akiba2, G. Casse51,
L. Castillo Garcia37, M. Cattaneo37, Ch. Cauet9, R. Cenci57, M. Charles54, Ph.
Charpentier37, S.-F. Cheung54, N. Chiapolini39, M. Chrzaszcz39,25, K. Ciba37,
X. Cid Vidal37, G. Ciezarek52, P.E.L. Clarke49, M. Clemencic37, H.V. Cliff46,
J. Closier37, C. Coca28, V. Coco40, J. Cogan6, E. Cogneras5, P. Collins37, A.
Comerma-Montells35, A. Contu15,37, A. Cook45, M. Coombes45, S. Coquereau8, G.
Corti37, B. Couturier37, G.A. Cowan49, D.C. Craik47, M. Cruz Torres59, S.
Cunliffe52, R. Currie49, C. D’Ambrosio37, P. David8, P.N.Y. David40, A.
Davis56, I. De Bonis4, K. De Bruyn40, S. De Capua53, M. De Cian11, J.M. De
Miranda1, L. De Paula2, W. De Silva56, P. De Simone18, D. Decamp4, M.
Deckenhoff9, L. Del Buono8, N. Déléage4, D. Derkach54, O. Deschamps5, F.
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Dordei11, A. Dosil Suárez36, D. Dossett47, A. Dovbnya42, F. Dupertuis38, P.
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Eitschberger9, R. Ekelhof9, L. Eklund50,37, I. El Rifai5, Ch. Elsasser39, A.
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Gauld54, E. Gersabeck11, M. Gersabeck53, T. Gershon47, Ph. Ghez4, V. Gibson46,
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A. Gomes2, P. Gorbounov30,37, H. Gordon37, M. Grabalosa Gándara5, R. Graciani
Diaz35, L.A. Granado Cardoso37, E. Graugés35, G. Graziani17, A. Grecu28, E.
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van Herwijnen37, M. Heß60, A. Hicheur1, E. Hicks51, D. Hill54, M. Hoballah5,
C. Hombach53, W. Hulsbergen40, P. Hunt54, T. Huse51, N. Hussain54, D.
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Kandybei42, W. Kanso6, M. Karacson37, T.M. Karbach37, I.R. Kenyon44, T.
Ketel41, B. Khanji20, O. Kochebina7, I. Komarov38, R.F. Koopman41, P.
Koppenburg40, M. Korolev31, A. Kozlinskiy40, L. Kravchuk32, K. Kreplin11, M.
Kreps47, G. Krocker11, P. Krokovny33, F. Kruse9, M. Kucharczyk20,25,37,j, V.
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Lefrançois7, S. Leo22, O. Leroy6, T. Lesiak25, B. Leverington11, Y. Li3, L. Li
Gioi5, M. Liles51, R. Lindner37, C. Linn11, B. Liu3, G. Liu37, S. Lohn37, I.
Longstaff50, J.H. Lopes2, N. Lopez-March38, H. Lu3, D. Lucchesi21,q, J.
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U. Marconi14, P. Marino22,s, R. Märki38, J. Marks11, G. Martellotti24, A.
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1Centro Brasileiro de Pesquisas Físicas (CBPF), Rio de Janeiro, Brazil
2Universidade Federal do Rio de Janeiro (UFRJ), Rio de Janeiro, Brazil
3Center for High Energy Physics, Tsinghua University, Beijing, China
4LAPP, Université de Savoie, CNRS/IN2P3, Annecy-Le-Vieux, France
5Clermont Université, Université Blaise Pascal, CNRS/IN2P3, LPC, Clermont-
Ferrand, France
6CPPM, Aix-Marseille Université, CNRS/IN2P3, Marseille, France
7LAL, Université Paris-Sud, CNRS/IN2P3, Orsay, France
8LPNHE, Université Pierre et Marie Curie, Université Paris Diderot,
CNRS/IN2P3, Paris, France
9Fakultät Physik, Technische Universität Dortmund, Dortmund, Germany
10Max-Planck-Institut für Kernphysik (MPIK), Heidelberg, Germany
11Physikalisches Institut, Ruprecht-Karls-Universität Heidelberg, Heidelberg,
Germany
12School of Physics, University College Dublin, Dublin, Ireland
13Sezione INFN di Bari, Bari, Italy
14Sezione INFN di Bologna, Bologna, Italy
15Sezione INFN di Cagliari, Cagliari, Italy
16Sezione INFN di Ferrara, Ferrara, Italy
17Sezione INFN di Firenze, Firenze, Italy
18Laboratori Nazionali dell’INFN di Frascati, Frascati, Italy
19Sezione INFN di Genova, Genova, Italy
20Sezione INFN di Milano Bicocca, Milano, Italy
21Sezione INFN di Padova, Padova, Italy
22Sezione INFN di Pisa, Pisa, Italy
23Sezione INFN di Roma Tor Vergata, Roma, Italy
24Sezione INFN di Roma La Sapienza, Roma, Italy
25Henryk Niewodniczanski Institute of Nuclear Physics Polish Academy of
Sciences, Kraków, Poland
26AGH - University of Science and Technology, Faculty of Physics and Applied
Computer Science, Kraków, Poland
27National Center for Nuclear Research (NCBJ), Warsaw, Poland
28Horia Hulubei National Institute of Physics and Nuclear Engineering,
Bucharest-Magurele, Romania
29Petersburg Nuclear Physics Institute (PNPI), Gatchina, Russia
30Institute of Theoretical and Experimental Physics (ITEP), Moscow, Russia
31Institute of Nuclear Physics, Moscow State University (SINP MSU), Moscow,
Russia
32Institute for Nuclear Research of the Russian Academy of Sciences (INR RAN),
Moscow, Russia
33Budker Institute of Nuclear Physics (SB RAS) and Novosibirsk State
University, Novosibirsk, Russia
34Institute for High Energy Physics (IHEP), Protvino, Russia
35Universitat de Barcelona, Barcelona, Spain
36Universidad de Santiago de Compostela, Santiago de Compostela, Spain
37European Organization for Nuclear Research (CERN), Geneva, Switzerland
38Ecole Polytechnique Fédérale de Lausanne (EPFL), Lausanne, Switzerland
39Physik-Institut, Universität Zürich, Zürich, Switzerland
40Nikhef National Institute for Subatomic Physics, Amsterdam, The Netherlands
41Nikhef National Institute for Subatomic Physics and VU University Amsterdam,
Amsterdam, The Netherlands
42NSC Kharkiv Institute of Physics and Technology (NSC KIPT), Kharkiv, Ukraine
43Institute for Nuclear Research of the National Academy of Sciences (KINR),
Kyiv, Ukraine
44University of Birmingham, Birmingham, United Kingdom
45H.H. Wills Physics Laboratory, University of Bristol, Bristol, United
Kingdom
46Cavendish Laboratory, University of Cambridge, Cambridge, United Kingdom
47Department of Physics, University of Warwick, Coventry, United Kingdom
48STFC Rutherford Appleton Laboratory, Didcot, United Kingdom
49School of Physics and Astronomy, University of Edinburgh, Edinburgh, United
Kingdom
50School of Physics and Astronomy, University of Glasgow, Glasgow, United
Kingdom
51Oliver Lodge Laboratory, University of Liverpool, Liverpool, United Kingdom
52Imperial College London, London, United Kingdom
53School of Physics and Astronomy, University of Manchester, Manchester,
United Kingdom
54Department of Physics, University of Oxford, Oxford, United Kingdom
55Massachusetts Institute of Technology, Cambridge, MA, United States
56University of Cincinnati, Cincinnati, OH, United States
57University of Maryland, College Park, MD, United States
58Syracuse University, Syracuse, NY, United States
59Pontifícia Universidade Católica do Rio de Janeiro (PUC-Rio), Rio de
Janeiro, Brazil, associated to 2
60Institut für Physik, Universität Rostock, Rostock, Germany, associated to 11
61Celal Bayar University, Manisa, Turkey, associated to 37
aP.N. Lebedev Physical Institute, Russian Academy of Science (LPI RAS),
Moscow, Russia
bUniversità di Bari, Bari, Italy
cUniversità di Bologna, Bologna, Italy
dUniversità di Cagliari, Cagliari, Italy
eUniversità di Ferrara, Ferrara, Italy
fUniversità di Firenze, Firenze, Italy
gUniversità di Urbino, Urbino, Italy
hUniversità di Modena e Reggio Emilia, Modena, Italy
iUniversità di Genova, Genova, Italy
jUniversità di Milano Bicocca, Milano, Italy
kUniversità di Roma Tor Vergata, Roma, Italy
lUniversità di Roma La Sapienza, Roma, Italy
mUniversità della Basilicata, Potenza, Italy
nLIFAELS, La Salle, Universitat Ramon Llull, Barcelona, Spain
oHanoi University of Science, Hanoi, Viet Nam
pInstitute of Physics and Technology, Moscow, Russia
qUniversità di Padova, Padova, Italy
rUniversità di Pisa, Pisa, Italy
sScuola Normale Superiore, Pisa, Italy
## 1 Introduction
Flavour-changing neutral current (FCNC) processes are rare within the Standard
Model (SM) as they cannot occur at tree level and are suppressed by the
Glashow-Iliopoulos-Maiani (GIM) mechanism at loop level. In contrast to the
$B$ meson system, where the high mass of the top quark in the loop weakens the
suppression, the GIM cancellation is almost exact [1] in $D$ meson decays,
leading to expected branching fractions for $c\rightarrow
u\mathup{{{\mu}}}^{+}\mathup{{{\mu}}}^{-}$ processes in the range $(1-3)\times
10^{-9}$ [2, 3, 4]. This suppression allows for sub-leading processes with
potential for physics beyond the SM, such as FCNC decays of $D$ mesons, and
the coupling of up-type quarks in electroweak processes illustrated in Fig. 1,
to be probed more precisely.
The total branching fraction for these decays is expected to be dominated by
long-distance contributions involving resonances, such as
$D^{0}\rightarrow\mathup{{{\pi}}}^{+}\mathup{{{\pi}}}^{-}V(\rightarrow\mathup{{{\mu}}}^{+}\mu^{-})$,
where $V$ can be any of the light vector mesons $\phi$, $\rho^{0}$ or
$\omega$. The corresponding branching fractions can reach ${\cal{O}}(10^{-6})$
[2, 3, 4]. The angular structure of these four-body semileptonic $D^{0}$
decays provides access to a variety of differential distributions. Of
particular interest are angular asymmetries that allow for a theoretically
robust separation of long- and short-distance effects, the latter being more
sensitive to physics beyond the SM [4]. No such decays have been observed to
date and the most stringent limit reported is ${\cal
B}(D^{0}\\!\rightarrow\mathup{{{\pi}}}^{+}\mathup{{{\pi}}}^{-}\mathup{{{\mu}}}^{+}\mathup{{{\mu}}}^{-})<3.0\times
10^{-5}$ at 90% confidence level ($\mathrm{CL}$) by the E791 collaboration
[5]. The same processes can be probed using
$D^{+}_{(s)}\rightarrow\pi^{+}\mu^{+}\mu^{-}$ decays. Upper limits on their
branching fractions have been recently set to ${\cal
B}(D^{+}\rightarrow\pi^{+}\mu^{+}\mu^{-})<7.3\times 10^{-8}$ and ${\cal
B}(D^{+}_{s}\rightarrow\pi^{+}\mu^{+}\mu^{-})<4.1\times 10^{-7}$ at 90% CL by
the LHCb collaboration [6].
This Letter presents the result of a search for the
$D^{0}\\!\rightarrow\mathup{{{\pi}}}^{+}\mathup{{{\pi}}}^{-}\mathup{{{\mu}}}^{+}\mathup{{{\mu}}}^{-}$
decay, in which the muons do not originate from a resonance, performed using
$D^{*+}\rightarrow D^{0}\mathup{{{\pi}}}^{+}$ decays, with the $D^{*+}$ meson
produced directly at the $pp$ collision primary vertex. The reduction in
background yield associated with this selection vastly compensates for the
loss of signal yield. No attempt is made to distinguish contributions from
intermediate resonances in the dipion invariant mass such as the $\rho^{0}$.
Throughout this Letter, the inclusion of charge conjugate processes is
implied. The data samples used in this analysis correspond to an integrated
luminosity of 1.0$\mbox{\,fb}^{-1}$ at
$\sqrt{s}=7$$\mathrm{\,Te\kern-1.00006ptV}$ recorded by the LHCb experiment.
$\mathup{{{c}}}$$\mathup{{\overline{{u}}}}$$\mathup{{{u}}}$$\mathup{{\overline{{u}}}}$$\mathup{{{\gamma}}}/\mathup{{{Z}}^{\scriptstyle{0}}}$$\mathup{{{W}}^{\scriptstyle{+}}}$$\mathup{{{\mu}}^{\scriptstyle{-}}}$$\mathup{{{\mu}}^{\scriptstyle{+}}}$$\mathup{{\overline{{d}}}}$$\mathup{{{d}}}$$D^{0}$$\mathup{{{\pi}}}^{+}$$\mathup{{{\pi}}}^{-}$$\mathup{{{c}}}$$\mathup{{\overline{{u}}}}$$\mathup{{{u}}}$$\mathup{{\overline{{u}}}}$$\mathup{{{W}}^{\scriptstyle{+}}}$$\mathup{{{W}}^{\scriptstyle{-}}}$$\mathup{{{\mu}}^{\scriptstyle{+}}}$$\mathup{{{\mu}}^{\scriptstyle{-}}}$$\mathup{{\overline{{d}}}}$$\mathup{{{d}}}$$D^{0}$$\mathup{{{\pi}}}^{+}$$\mathup{{{\pi}}}^{-}$
Figure 1: Leading Feynman diagrams for the FCNC decay
$D^{0}\\!\rightarrow\mathup{{{\pi}}}^{+}\mathup{{{\pi}}}^{-}\mathup{{{\mu}}}^{+}\mathup{{{\mu}}}^{-}$
in the SM.
The analysis is performed in four dimuon mass ranges to exclude decays
dominated by the contributions of resonant dimuon final states. The regions at
low and high dimuon masses, away from the $\eta$, $\rho^{0}$ and $\phi$
resonant regions, are the most sensitive to non-SM physics and are defined as
the signal regions. The signal yield is normalised to the yield of resonant
$D^{0}\\!\rightarrow\mathup{{{\pi}}}^{+}\mathup{{{\pi}}}^{-}\phi(\rightarrow\mathup{{{\mu}}}^{+}\mathup{{{\mu}}}^{-})$
decays, isolated in an appropriate dimuon range centred around the $\phi$
pole.
## 2 The LHCb detector and trigger
The LHCb detector [7] is a single-arm forward spectrometer covering the
pseudorapidity range $2<\eta<5$, designed for the study of particles
containing $b$ or $c$ quarks. The detector includes a high-precision tracking
system consisting of a silicon-strip vertex detector surrounding the $pp$
interaction region, a large-area silicon-strip detector located upstream of a
dipole magnet with a bending power of about $4{\rm\,Tm}$, and three stations
of silicon-strip detectors and straw drift tubes placed downstream. The
combined tracking system provides a momentum measurement with relative
uncertainty that varies from 0.4% at 5${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$
to 0.6% at 100${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$, and impact parameter
resolution of 20${\,\upmu\rm m}$ for tracks with large transverse momentum.
Different types of charged hadrons are distinguished by information from two
ring-imaging Cherenkov detectors [8]. Photon, electron and hadron candidates
are identified by a calorimeter system consisting of scintillating-pad and
preshower detectors, an electromagnetic calorimeter and a hadronic
calorimeter. Muons are identified by a system composed of alternating layers
of iron and multiwire proportional chambers [9].
The trigger [10] consists of a hardware stage, based on information from the
calorimeter and muon systems, followed by a software stage, which applies a
full event reconstruction. The hardware trigger selects muons with transverse
momentum, $p_{\rm T}$, exceeding 1.48${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$,
and dimuons whose product of $p_{\rm T}$ values exceeds
$(1.3{\mathrm{\,Ge\kern-1.00006ptV\\!/}c})^{2}$. In the software trigger, at
least one of the final state muons is required to have momentum larger than
8${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$, and to have an impact parameter, IP,
defined as the minimum distance of the particle trajectory from the associated
primary vertex (PV) in three dimensions, greater than 100${\,\upmu\rm m}$.
Alternatively, a dimuon trigger accepts events with oppositely charged muon
candidates having good track quality, $p_{\rm T}$ exceeding
$0.5{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$, and momentum exceeding
$6{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$. In a second stage of the software
trigger, two algorithms select
$D^{0}\\!\rightarrow\mathup{{{\pi}}}^{+}\mathup{{{\pi}}}^{-}\mathup{{{\mu}}}^{+}\mathup{{{\mu}}}^{-}$
candidates. The first algorithm, used to increase the efficiency in the
highest dimuon mass region, requires oppositely charged muons with scalar sum
of $p_{\rm T}$ greater than $1.5{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ and
dimuon mass greater than $1{\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$. A second
algorithm selects events with two oppositely charged muons and two oppositely
charged hadrons with no invariant mass requirement on the dimuon.
Simulated events for the signal, using a phase-space model, and the
normalisation mode, are used to define selection criteria and to evaluate
efficiencies. The $pp$ collisions are generated using Pythia 6.4 [11] with a
specific LHCb configuration [12]. Decays of hadronic particles are described
by EvtGen [13]. The interaction of the generated particles with the detector
and its response are implemented using the Geant4 toolkit [14,
*Agostinelli:2002hh] as described in Ref. [16].
## 3 Candidate selection
Candidate
$D^{0}\\!\rightarrow\mathup{{{\pi}}}^{+}\mathup{{{\pi}}}^{-}\mathup{{{\mu}}}^{+}\mathup{{{\mu}}}^{-}$
decays are required to originate from $D^{*+}\rightarrow
D^{0}\mathup{{{\pi}}}^{+}$ decays. The $D^{0}$ candidate is formed by
combining two pion and two muon candidates where both pairs consist of
oppositely charged particles. An additional pion track is combined with the
$D^{0}$ candidate to build the $D^{*+}$ candidate. The $\chi^{2}$ per degree
of freedom of the vertex fit is required to be less than 5 for both the
$D^{*+}$ and the $D^{0}$ candidates. The angle between the $D^{0}$ momentum
vector and the direction from the associated PV to the decay vertex,
$\theta_{D^{0}}$, is required to be less than $0.8^{\circ}$. Each of the four
particles forming the $D^{0}$ meson must have momentum exceeding 3
${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ and $p_{\rm T}$ exceeding 0.4
${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$. The tracks must be displaced with
respect to any PV and have $\chi^{2}_{\rm IP}$ larger than 4. Here
$\chi^{2}_{\rm IP}$ is defined as the difference between the $\chi^{2}$ of the
PV fit done with and without the track under consideration.
Further discrimination is achieved using a boosted decision tree (BDT) [17,
*Roe, 19], which distinguishes between signal and combinatorial background
candidates. This multivariate analysis algorithm is trained using simulated
$D^{0}\\!\rightarrow\mathup{{{\pi}}}^{+}\mathup{{{\pi}}}^{-}\mathup{{{\mu}}}^{+}\mathup{{{\mu}}}^{-}$
signal events and a background sample taken from data mass sidebands around
the
$D^{0}\\!\rightarrow\mathup{{{\pi}}}^{+}\mathup{{{\pi}}}^{-}\mathup{{{\mu}}}^{+}\mathup{{{\mu}}}^{-}$
signal mass region. Only 1% of the candidates in the sidebands are used in the
training. The BDT uses the following variables: $\theta_{D^{0}}$, $\chi^{2}$
of the decay vertex and flight distance of the $D^{0}$ candidate, $p$ and
$p_{\rm T}$ of the $D^{0}$ candidate and of each of the four final state
tracks, $\chi^{2}$ of the vertex and $p_{\rm T}$ of the $D^{*+}$ candidate,
$\chi^{2}_{\rm IP}$ of the $D^{0}$ candidate and of the final state particles,
the maximum distance of closest approach between all pairs of tracks forming
the $D^{0}$ and $D^{*+}$ candidates, and the $p_{\rm T}$ and $\chi^{2}_{\rm
IP}$ of the bachelor pion from the $D^{*+}$ candidate.
The BDT discriminant is used to classify each candidate. Assuming a signal
branching fraction of $10^{-9}$, an optimisation study is performed to choose
the combined BDT and muon particle identification (PID) selection criteria
that maximise the expected statistical significance of the signal. This
significance is defined as $S/\sqrt{S+B}$, where $S$ and $B$ are the signal
and background yields respectively. The PID information is quantified as the
difference in the log-likelihood of the detector response under different
particle mass hypotheses (DLL) [8, 20]. The optimisation procedure yields an
optimal threshold for the BDT discriminant and a minimum value for
$\mathrm{DLL}_{\mathup{{{\mu}}}\pi}$ (the difference between the muon and pion
hypotheses) of 1.5 for both $\mathup{{{\mu}}}$ candidates. In addition, the
pion candidate is required to have $\mathrm{DLL}_{K\mathup{{{\pi}}}}$ less
than 3.0 and $\mathrm{DLL}_{p\mathup{{{\pi}}}}$ less than 2.0, and each muon
candidate must not share hits in the muon stations with any other muon
candidate. In the 2% of events in which multiple candidates are reconstructed,
the candidate with the smallest $D^{0}$ vertex $\chi^{2}$ is chosen.
The bachelor $\mathup{{{\pi}}}^{+}$ of the $D^{*+}\rightarrow
D^{0}\mathup{{{\pi}}}^{+}$ decay is constrained to the PV using a Kalman
filter [21]. This constraint improves the resolution for the mass difference
between the $D^{*+}$ and the $D^{0}$ candidates, $\Delta m\equiv
m(\pi^{+}\pi^{-}\mu^{+}\mu^{-}\pi^{+})-m(\pi^{+}\pi^{-}\mu^{+}\mu^{-})$, by a
factor of two, down to $0.3$${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$.
Candidates are selected with a $\Delta m$ value in the range
$140.0-151.4$${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$.
Candidates from the kinematically similar decay
$D^{0}\\!\rightarrow\mathup{{{\pi}}}^{+}\mathup{{{\pi}}}^{-}\mathup{{{\pi}}}^{+}\mathup{{{\pi}}}^{-}$
form an important peaking background due to the possible misidentification of
two oppositely charged pions as muons. A sample of this hadronic background is
retained with a selection that is identical to that applied to the signal
except that no muon identification is required. These candidates are then
reconstructed under the
$D^{0}\\!\rightarrow\mathup{{{\pi}}}^{+}\mathup{{{\pi}}}^{-}\mathup{{{\mu}}}^{+}\mathup{{{\mu}}}^{-}$
hypothesis and a subsample of the candidates, in which at least one such pion
satisfies the muon identification requirements, is used to determine the shape
of this peaking background in each region of dimuon mass,
$m(\mathup{{{\mu}}}^{+}\mathup{{{\mu}}}^{-})$. Under the correct mass
hypotheses the
$D^{0}\\!\rightarrow\mathup{{{\pi}}}^{+}\mathup{{{\pi}}}^{-}\mathup{{{\pi}}}^{+}\mathup{{{\pi}}}^{-}$
candidates are also used as a control sample to check differences between data
and simulation that may affect the event selection performance. Moreover, they
are used to determine the expected signal shape in each
$m(\mathup{{{\mu}}}^{+}\mathup{{{\mu}}}^{-})$ region by subdividing the
$D^{0}\\!\rightarrow\mathup{{{\pi}}}^{+}\mathup{{{\pi}}}^{-}\mathup{{{\pi}}}^{+}\mathup{{{\pi}}}^{-}$
sample in the same regions of $m(\pi^{+}\pi^{-})$.
Another potential source of peaking background is due to
$\mathit{\Lambda}_{c}(2595)^{+}\rightarrow\mathit{\Sigma}_{c}(2455)^{0}\mathup{{{\pi}}}^{+}$
decays, followed by the
$\Sigma_{c}(2455)^{0}\rightarrow\mathit{\Lambda}^{+}_{c}\mathup{{{\pi}}}^{-}$
and then $\mathit{\Lambda}^{+}_{c}\rightarrow pK^{-}\mathup{{{\pi}}}^{+}$
decays, with the two pions in the decay chain misidentified as muons and the
proton and the kaon misidentified as pions. Therefore, the
$\mathrm{DLL}_{K\mathup{{{\pi}}}}$ and $\mathrm{DLL}_{p\mathup{{{\pi}}}}$
requirements are tightened to be less than zero for the
low-$m(\mathup{{{\mu}}}^{+}\mathup{{{\mu}}}^{-})$ region, where the baryonic
background is concentrated, suppressing this background to a negligible level.
Another potentially large background from the
$D^{0}\rightarrow\mathup{{{\pi}}}^{+}\mathup{{{\pi}}}^{-}\eta$ decay, followed
by the decay $\eta\rightarrow\mathup{{{\mu}}}^{+}\mu^{-}\gamma$, does not peak
at the $D^{0}$ mass since candidates in which the
$m(\mathup{{{\mu}}}^{+}\mathup{{{\mu}}}^{-})$ is within $\pm
20$${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ of the nominal $\eta$ mass are
removed from the final fit. The remaining contribution to low values of the
$m(\mathup{{{\pi}}}^{+}\mathup{{{\pi}}}^{-}\mathup{{{\mu}}}^{+}\mu^{-})$
invariant mass is included in the combinatorial background.
## 4 Mass fit
The shapes and yields of the signal and background contributions are
determined using an unbinned maximum likelihood fit to the two-dimensional
$\left[m(\pi^{+}\pi^{-}\mu^{+}\mu^{-}\pi^{+}),\Delta m\right]$ distributions
in the ranges $1810-1920$ and $140-151.4$
${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$, respectively. This range is chosen
to contain all reconstructed
$D^{0}\\!\rightarrow\mathup{{{\pi}}}^{+}\mathup{{{\pi}}}^{-}\mathup{{{\mu}}}^{+}\mathup{{{\mu}}}^{-}$
candidates.
The
$D^{0}\\!\rightarrow\mathup{{{\pi}}}^{+}\mathup{{{\pi}}}^{-}\mathup{{{\mu}}}^{+}\mathup{{{\mu}}}^{-}$
data are split into four regions of
$m(\mathup{{{\mu}}}^{+}\mathup{{{\mu}}}^{-})$: two regions containing the
$\rho/\omega$ and $\phi$ resonances and two signal regions, referred to as
low-$m(\mathup{{{\mu}}}^{+}\mathup{{{\mu}}}^{-})$ and
high-$m(\mathup{{{\mu}}}^{+}\mathup{{{\mu}}}^{-})$, respectively. The
definitions of these regions are provided in Table 1.
The $D^{0}$ mass and $\Delta m$ shapes for
$D^{0}\\!\rightarrow\mathup{{{\pi}}}^{+}\mathup{{{\pi}}}^{-}\mathup{{{\mu}}}^{+}\mathup{{{\mu}}}^{-}$
candidates are described by a double Crystal Ball function [22, *CB2], which
consists of a Gaussian core and independent left and right power-law tails, on
either sides of the core. The parameters of these shapes are determined from
the
$D^{0}\\!\rightarrow\mathup{{{\pi}}}^{+}\mathup{{{\pi}}}^{-}\mathup{{{\pi}}}^{+}\mathup{{{\pi}}}^{-}$
control sample independently for each of the four
$m(\mathup{{{\mu}}}^{+}\mathup{{{\mu}}}^{-})$ regions.
The
$D^{0}\\!\rightarrow\mathup{{{\pi}}}^{+}\mathup{{{\pi}}}^{-}\mathup{{{\pi}}}^{+}\mathup{{{\pi}}}^{-}$
peaking background is also split into the predefined dimuon mass regions and
is fitted with a double Crystal Ball function. This provides a well-defined
shape for this prominent background, which is included in the fit to the
signal sample. The yield of the misidentified component is allowed to vary and
fitted in each region of the analysis. The combinatorial background is
described by an exponential function in the $D^{0}$ candidate mass, while the
shape in $\Delta m$ is described by the empirical function $f_{\Delta}(\Delta
m,a)=1-e^{-(\Delta m-\Delta{m_{0}})/a}$, where the parameter $\Delta{m_{0}}$
is fixed to $139.6\,{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$. The two-
dimensional shape used in the fit implicitly assumes that
$m(\pi^{+}\pi^{-}\mu^{+}\mu^{-}\pi^{+})$ and $\Delta m$ are not correlated.
All the floating coefficients are allowed to vary independently in each of the
$m(\mathup{{{\mu}}}^{+}\mathup{{{\mu}}}^{-})$ regions. Migration between the
regions is found to be negligible from simulation studies. The yield observed
in the $\mathup{{{\phi}}}$ region is used to normalise the yields in the
signal regions.
One-dimensional projections for the $D^{0}$ candidate invariant mass and
$\Delta m$ spectra, together with the result of the fits, are shown in Figs. 2
and 3, respectively. The signal yields, which include contributions from the
tails of the $m(\mathup{{{\mu}}}^{+}\mathup{{{\mu}}}^{-})$ resonances leaking
into the low- and high-$m(\mathup{{{\mu}}}^{+}\mathup{{{\mu}}}^{-})$ ranges,
are shown in Table 1. No significant excess of candidates is seen in either of
the two signal regions.
Table 1: $D^{0}\rightarrow\pi^{+}\pi^{-}\mu^{+}\mu^{-}$ fitted yields in the four $m(\mu^{+}\mu^{-})$ regions. The corresponding signal fractions under the assumption of a phase-space model, as described in Section 7, are listed in the last column. Range description 0 | $m(\mathup{{{\mu}}}^{+}\mathup{{{\mu}}}^{-})$ [${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$] 0 | $D^{0}\\!\rightarrow\mathup{{{\pi}}}^{+}\mathup{{{\pi}}}^{-}\mathup{{{\mu}}}^{+}\mathup{{{\mu}}}^{-}$ yield | 00Fraction
---|---|---|---
low-$m(\mathup{{{\mu}}}^{+}\mathup{{{\mu}}}^{-})$ | $\phantom{00000-}250-525$ | $\phantom{000000000}2\pm 2$ | 00 30.6%
$\mathup{{{\rho}}}$/$\mathup{{{\omega}}}$ | $\phantom{00000-}565-950$ | $\phantom{00000000}23\pm 6$ | 00 43.4%
$\mathup{{{\phi}}}$ | $\phantom{00000-}950-1100$ | $\phantom{00000000}63\pm 10$ | 00 10.1%
high-$m(\mathup{{{\mu}}}^{+}\mathup{{{\mu}}}^{-})$ | $\phantom{00000-}>1100$ | $\phantom{000000000}3\pm 2$ | 00 8.9%
Figure 2: Distributions of $m(\pi^{+}\pi^{-}\mu^{+}\mu^{-})$ for
$D^{0}\\!\rightarrow\mathup{{{\pi}}}^{+}\mathup{{{\pi}}}^{-}\mathup{{{\mu}}}^{+}\mathup{{{\mu}}}^{-}$
candidates in the (a) low-$m(\mathup{{{\mu}}}^{+}\mathup{{{\mu}}}^{-})$, (b)
$\mathup{{{\rho}}}$/$\mathup{{{\omega}}}$, (c) $\mathup{{{\phi}}}$, and (d)
high-$m(\mathup{{{\mu}}}^{+}\mathup{{{\mu}}}^{-})$ regions, with $\Delta m$ in
the range $144.4-146.6$ ${\mathrm{\,Me\kern-0.90005ptV\\!/}c^{2}}$. The data
are shown as points (black) and the fit result (dark blue line) is overlaid.
The components of the fit are also shown: the signal (filled area), the
$D^{0}\\!\rightarrow\mathup{{{\pi}}}^{+}\mathup{{{\pi}}}^{-}\mathup{{{\pi}}}^{+}\mathup{{{\pi}}}^{-}$
background (green dashed line) and the non-peaking background (red dashed-
dotted line).
The yields in the signal regions are compatible with the expectations from
leakage from the $m(\mathup{{{\mu}}}^{+}\mathup{{{\mu}}}^{-})$ resonant
regions. The number of expected events from leakage is calculated assuming the
$m(\mathup{{{\mu}}}^{+}\mathup{{{\mu}}}^{-})$ spectrum given by a sum of
relativistic Breit-Wigner functions, describing the $\eta$, $\rho/\omega$ and
$\phi$ resonances. The contribution from each resonance is scaled according to
the branching fractions as determined from resonant $D^{0}\rightarrow
K^{+}K^{-}\mathup{{{\pi}}}^{+}\mathup{{{\pi}}}^{-}$ and
$D^{0}\rightarrow\mathup{{{\pi}}}^{+}\mathup{{{\pi}}}^{-}\mathup{{{\pi}}}^{+}\mathup{{{\pi}}}^{-}$
decays [24]. The resulting shape is used to extrapolate the yields fitted in
the $\phi$ and $\rho$ regions into the
$m(\mathup{{{\mu}}}^{+}\mathup{{{\mu}}}^{-})$ signal regions. An additional
extrapolation is performed using the signal yield in the
$m(\mathup{{{\mu}}}^{+}\mathup{{{\mu}}}^{-})$ range $773-793$
${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$, where the contribution from the
$\omega$ resonance is enhanced. In this approach the interference among
different resonances is not accounted for and a systematic uncertainty to the
extrapolated yield is assigned according to the spread in their
extrapolations. The expected number of leakage events is estimated to be $1\pm
1$ in both the low- and high- $m(\mathup{{{\mu}}}^{+}\mathup{{{\mu}}}^{-})$
regions. This precision of this estimate is dominated by the systematic
uncertainty.
Figure 3: Distributions of $\Delta m$ for
$D^{0}\\!\rightarrow\mathup{{{\pi}}}^{+}\mathup{{{\pi}}}^{-}\mathup{{{\mu}}}^{+}\mathup{{{\mu}}}^{-}$
candidates in the (a) low-$m(\mathup{{{\mu}}}^{+}\mathup{{{\mu}}}^{-})$, (b)
$\mathup{{{\rho}}}$/$\mathup{{{\omega}}}$, (c) $\mathup{{{\phi}}}$, and (d)
high-$m(\mathup{{{\mu}}}^{+}\mathup{{{\mu}}}^{-})$ regions, with the $D^{0}$
invariant mass in the range $1840-1888$
${\mathrm{\,Me\kern-0.90005ptV\\!/}c^{2}}$. The data are shown as points
(black) and the fit result (dark blue line) is overlaid. The components of the
fit are also shown: the signal (filled area), the
$D^{0}\\!\rightarrow\mathup{{{\pi}}}^{+}\mathup{{{\pi}}}^{-}\mathup{{{\pi}}}^{+}\mathup{{{\pi}}}^{-}$
background (green dashed line) and the non-peaking background (red dashed-
dotted line).
## 5 Branching fraction determination
The
$D^{0}\\!\rightarrow\mathup{{{\pi}}}^{+}\mathup{{{\pi}}}^{-}\mathup{{{\mu}}}^{+}\mathup{{{\mu}}}^{-}$
branching fraction ratio for each
$m(\mathup{{{\mu}}}^{+}\mathup{{{\mu}}}^{-})$ signal region $i$ is calculated
using
$\frac{{{\cal
B}(D^{0}\\!\rightarrow\mathup{{{\pi}}}^{+}\mathup{{{\pi}}}^{-}\mathup{{{\mu}}}^{+}\mathup{{{\mu}}}^{-})^{i}}}{{{\cal
B}(D^{0}\\!\rightarrow\mathup{{{\pi}}}^{+}\mathup{{{\pi}}}^{-}\phi(\rightarrow\mathup{{{\mu}}}^{+}\mathup{{{\mu}}}^{-}))}}=\frac{N^{i}_{D^{0}\\!\rightarrow\mathup{{{\pi}}}^{+}\mathup{{{\pi}}}^{-}\mathup{{{\mu}}}^{+}\mathup{{{\mu}}}^{-}}}{N_{D^{0}\\!\rightarrow\mathup{{{\pi}}}^{+}\mathup{{{\pi}}}^{-}\phi(\rightarrow\mathup{{{\mu}}}^{+}\mathup{{{\mu}}}^{-})}}\times\frac{\epsilon_{D^{0}\\!\rightarrow\mathup{{{\pi}}}^{+}\mathup{{{\pi}}}^{-}\phi(\rightarrow\mathup{{{\mu}}}^{+}\mathup{{{\mu}}}^{-})}}{\epsilon^{i}_{D^{0}\\!\rightarrow\mathup{{{\pi}}}^{+}\mathup{{{\pi}}}^{-}\mathup{{{\mu}}}^{+}\mathup{{{\mu}}}^{-}}}.$
(1)
The yield and efficiency are given by
$N_{D^{0}\\!\rightarrow\mathup{{{\pi}}}^{+}\mathup{{{\pi}}}^{-}\mathup{{{\mu}}}^{+}\mathup{{{\mu}}}^{-}}$
and
$\epsilon_{D^{0}\\!\rightarrow\mathup{{{\pi}}}^{+}\mathup{{{\pi}}}^{-}\mathup{{{\mu}}}^{+}\mathup{{{\mu}}}^{-}}$,
respectively, for the signal channel, and by
$N_{D^{0}\\!\rightarrow\mathup{{{\pi}}}^{+}\mathup{{{\pi}}}^{-}\phi(\rightarrow\mathup{{{\mu}}}^{+}\mathup{{{\mu}}}^{-})}$
and
$\epsilon_{D^{0}\\!\rightarrow\mathup{{{\pi}}}^{+}\mathup{{{\pi}}}^{-}\phi(\rightarrow\mathup{{{\mu}}}^{+}\mathup{{{\mu}}}^{-})}$
for the reference channel. The values for the efficiency ratio
$\epsilon_{D^{0}\\!\rightarrow\mathup{{{\pi}}}^{+}\mathup{{{\pi}}}^{-}\mathup{{{\mu}}}^{+}\mathup{{{\mu}}}^{-}}/\epsilon_{D^{0}\\!\rightarrow\mathup{{{\pi}}}^{+}\mathup{{{\pi}}}^{-}\phi(\rightarrow\mathup{{{\mu}}}^{+}\mathup{{{\mu}}}^{-})}$
in the low-$m(\mathup{{{\mu}}}^{+}\mathup{{{\mu}}}^{-})$ and
high-$m(\mathup{{{\mu}}}^{+}\mathup{{{\mu}}}^{-})$ regions, as estimated from
simulations, are $0.24\pm 0.03$ and $0.69\pm 0.11$, respectively, where the
uncertainty reflects the limited statistics of the simulated samples. The
efficiencies for reconstructing the signal decay mode and the reference mode
include the geometric acceptance of the detector, the efficiencies for track
reconstruction, particle identification, selection and trigger. Both
efficiency ratios deviate from unity due to differences in the kinematic
distributions of the final state particles in the two decays. Moreover,
tighter particle identification requirements are responsible for a lower
efficiency ratio in the low-$m(\mathup{{{\mu}}}^{+}\mathup{{{\mu}}}^{-})$
region. The accuracy with which the simulation reproduces the track
reconstruction and particle identification is limited. Therefore, the
corresponding efficiencies are also studied in data and systematic
uncertainties are assigned.
An upper limit on the absolute branching fraction is given using an estimate
of the branching fraction of the normalisation mode. The
$D^{0}\\!\rightarrow\mathup{{{\pi}}}^{+}\mathup{{{\pi}}}^{-}\phi(\rightarrow\mathup{{{\mu}}}^{+}\mathup{{{\mu}}}^{-})$
branching fraction is estimated using the results of the amplitude analysis of
the $D^{0}\rightarrow K^{+}K^{-}\mathup{{{\pi}}}^{+}\mathup{{{\pi}}}^{-}$
decay performed at CLEO [25]. Only the fit fraction of the decay modes in
which the two kaons originate from an intermediate $\phi$ resonance are
considered and the
$D^{0}\\!\rightarrow\mathup{{{\pi}}}^{+}\mathup{{{\pi}}}^{-}\phi(\rightarrow\mathup{{{\mu}}}^{+}\mathup{{{\mu}}}^{-})$
branching fraction is calculated by multiplying this fraction by the total
$D^{0}\rightarrow K^{+}K^{-}\mathup{{{\pi}}}^{+}\mathup{{{\pi}}}^{-}$
branching fraction and using the known value of ${\cal
B}(\phi\rightarrow\mathup{{{\mu}}}^{+}\mu^{-})/{\cal B}(\phi\rightarrow
K^{+}K^{-})$ [24]. There are several interfering contributions to the
$D^{0}\\!\rightarrow\mathup{{{\pi}}}^{+}\mathup{{{\pi}}}^{-}\phi(\rightarrow
K^{+}K^{-})$ amplitude. Considering the interference fractions provided in
Ref.[25], the following estimate for the branching fraction is obtained,
${\cal
B}(D^{0}\\!\rightarrow\mathup{{{\pi}}}^{+}\mathup{{{\pi}}}^{-}\phi(\rightarrow\mathup{{{\mu}}}^{+}\mathup{{{\mu}}}^{-}))=(5.2\pm
0.6)\times 10^{-7}$. This estimate includes only the statistical uncertainty
and refers to the baseline fit model used for the CLEO measurement. Similar
estimates for ${\cal
B}(D^{0}\\!\rightarrow\mathup{{{\pi}}}^{+}\mathup{{{\pi}}}^{-}\phi(\rightarrow\mathup{{{\mu}}}^{+}\mathup{{{\mu}}}^{-}))$
are performed using all the alternative models considered in Ref.[25] assuming
the interference fractions to be the same as for the baseline model. The
spread among the estimates is used to assign a systematic uncertainty of
$17\%$ on ${\cal
B}(D^{0}\\!\rightarrow\mathup{{{\pi}}}^{+}\mathup{{{\pi}}}^{-}\phi(\rightarrow\mathup{{{\mu}}}^{+}\mathup{{{\mu}}}^{-}))$.
The above procedure to estimate ${\cal
B}(D^{0}\\!\rightarrow\mathup{{{\pi}}}^{+}\mathup{{{\pi}}}^{-}\phi(\rightarrow\mathup{{{\mu}}}^{+}\mathup{{{\mu}}}^{-}))$
is supported by the narrow width of the $\mathup{{{\phi}}}$ resonance
resulting in interference effects with other channels [25] that are negligible
compared to the statistical uncertainty. The estimate for ${\cal
B}(D^{0}\\!\rightarrow\mathup{{{\pi}}}^{+}\mathup{{{\pi}}}^{-}\phi(\rightarrow\mathup{{{\mu}}}^{+}\mathup{{{\mu}}}^{-}))$
is $(5.2\pm 1.1)\times 10^{-7}$, including both statistical and systematic
uncertainties, and is used to set an upper limit on the absolute
$D^{0}\\!\rightarrow\mathup{{{\pi}}}^{+}\mathup{{{\pi}}}^{-}\mathup{{{\mu}}}^{+}\mathup{{{\mu}}}^{-}$
branching fraction.
A possible alternative normalisation, with respect to the $\rho/\omega$ dimuon
mass region, would be heavily limited by the low statistics available and the
relatively high contamination from
$D^{0}\\!\rightarrow\mathup{{{\pi}}}^{+}\mathup{{{\pi}}}^{-}\mathup{{{\pi}}}^{+}\mathup{{{\pi}}}^{-}$,
as can be seen in Figure 2b.
## 6 Systematic uncertainties
Several systematic uncertainties affect the efficiency ratio. Differences in
the particle identification between the signal and the normalisation regions
are investigated in data. A tag-and-probe technique applied to $b\rightarrow
J/\psi X$ decays provides a large sample of muon candidates to determine the
muon identification efficiencies [20]. General agreement between simulation
and data is found to a level of 1%, which is assigned as a systematic
uncertainty.
The particle identification performance for hadrons is investigated by
comparing the efficiency in
$D^{0}\rightarrow\mathup{{{\pi}}}^{+}\mathup{{{\pi}}}^{-}\mathup{{{\pi}}}^{+}\mathup{{{\pi}}}^{-}$
candidates in data and simulation as a function of the
$\mathrm{DLL}_{K\mathup{{{\pi}}}}$ requirement. The largest discrepancy
between data and simulation on the efficiency ratio is found to be 4% and is
taken as a systematic uncertainty.
Several quantities, particularly the impact parameter, are known to be
imperfectly reproduced in the simulation. Since this may affect the
reconstruction and selection efficiency, a systematic uncertainty is estimated
by smearing track properties to reproduce the distributions observed in data.
The corresponding variation in the efficiency ratio yields an uncertainty of
5%. The BDT description in simulation is checked using background-subtracted
$D^{0}\\!\rightarrow\mathup{{{\pi}}}^{+}\mathup{{{\pi}}}^{-}\mathup{{{\pi}}}^{+}\mathup{{{\pi}}}^{-}$
candidates where no significant difference is seen. Therefore, no extra
systematic uncertainty is assigned.
The systematic uncertainty due to possible mismodelling of the trigger
efficiency in the simulation is assigned as follows. The trigger requirements
in simulations are varied reproducing the typical changes of trigger
configurations that occurred during data taking and an alternate efficiency
ratio is calculated in both the $m(\mathup{{{\mu}}}^{+}\mathup{{{\mu}}}^{-})$
signal regions. The largest difference between the alternate and the baseline
efficiency ratio, 5%, is found in the
low-$m(\mathup{{{\mu}}}^{+}\mathup{{{\mu}}}^{-})$ region. This difference is
assumed as the overall systematic uncertainty on the trigger efficiency.
The uncertainties on the efficiency ratio due to the finite size of the
simulated samples in the low- and high-
$m(\mathup{{{\mu}}}^{+}\mathup{{{\mu}}}^{-})$ regions are 12% and 16%
respectively. The production of significantly larger sample of simulated
events is impractical due to the low reconstruction and selection
efficiencies, particularly in the signal regions. In addition, the statistical
uncertainties of the fitted yields in data, listed in Table 1, dominate the
total uncertainty. The sources of uncertainty are summarised in Table 2.
Table 2: Relative systematic uncertainties averaged over all the $m(\mathup{{{\mu}}}^{+}\mathup{{{\mu}}}^{-})$ regions for the efficiency ratio. Source | Uncertainty (%)
---|---
Trigger efficiency | 5
Hadron identification | 4
Reconstruction and selection efficiency | 5
Muon identification | 1
Finite simulation sample size | 12–16
Total | 15–18
According to simulations, biases in the efficiency ratio introduced by varying
the relative contribution of
$D^{0}\rightarrow\rho^{0}(\rightarrow\pi\pi)\phi(\rightarrow\mu\mu)$ and
three-body
$D^{0}\\!\rightarrow\mathup{{{\pi}}}^{+}\mathup{{{\pi}}}^{-}\phi(\rightarrow\mathup{{{\mu}}}^{+}\mathup{{{\mu}}}^{-})$
decays are well within the assigned uncertainty. Varying the value of ${\cal
B}(D^{0}\\!\rightarrow\mathup{{{\pi}}}^{+}\mathup{{{\pi}}}^{-}\phi(\rightarrow\mathup{{{\mu}}}^{+}\mathup{{{\mu}}}^{-}))$
has a negligible effect on the number of leakage events, and no additional
systematic uncertainty is assigned.
The systematic uncertainties affecting the yield ratio are taken into account
when the branching fraction limits are calculated. The shapes of the signal
peaks are taken from the
$D^{0}\\!\rightarrow\mathup{{{\pi}}}^{+}\mathup{{{\pi}}}^{-}\mathup{{{\pi}}}^{+}\mathup{{{\pi}}}^{-}$
samples separately for each $m(\mathup{{{\mu}}}^{+}\mathup{{{\mu}}}^{-})$
region to account for variations of the shape as a function of
$m(\mathup{{{\mu}}}^{+}\mathup{{{\mu}}}^{-})$. The impact of alternative
shapes for the signal and misidentified
$D^{0}\\!\rightarrow\mathup{{{\pi}}}^{+}\mathup{{{\pi}}}^{-}\mathup{{{\pi}}}^{+}\mathup{{{\pi}}}^{-}$
decays on the fitted yields and the final limit are investigated. The signal
and misidentification background shapes in the signal regions are fitted using
the shapes obtained in the $\phi$ region, and from
$D^{0}\\!\rightarrow\mathup{{{\pi}}}^{+}\mathup{{{\pi}}}^{-}\mathup{{{\pi}}}^{+}\mathup{{{\pi}}}^{-}$
events reconstructed as
$D^{0}\\!\rightarrow\mathup{{{\pi}}}^{+}\mathup{{{\pi}}}^{-}\mathup{{{\mu}}}^{+}\mathup{{{\mu}}}^{-}$,
but without any muon identification requirements. The change in the result is
negligible.
The absolute branching fraction limit includes an extra uncertainty of 21%
from the estimate of the branching fraction of the normalisation mode.
## 7 Results
The compatibility of the observed distribution of candidates with a signal
plus background or background-only hypothesis is evaluated using the
$\mathrm{CL}_{s}$ method [26, 27], which includes the treatment of systematic
uncertainties. Upper limits on the non-resonant
$D^{0}\\!\rightarrow\mathup{{{\pi}}}^{+}\mathup{{{\pi}}}^{-}\mathup{{{\mu}}}^{+}\mathup{{{\mu}}}^{-}$
to
$D^{0}\\!\rightarrow\mathup{{{\pi}}}^{+}\mathup{{{\pi}}}^{-}\phi(\rightarrow\mathup{{{\mu}}}^{+}\mathup{{{\mu}}}^{-})$
branching fraction ratio and on the absolute
$D^{0}\\!\rightarrow\mathup{{{\pi}}}^{+}\mathup{{{\pi}}}^{-}\mathup{{{\mu}}}^{+}\mathup{{{\mu}}}^{-}$
branching fraction are determined using the observed distribution of
$\mathrm{CL}_{s}$ as a function of the branching fraction in each
$m(\mathup{{{\mu}}}^{+}\mathup{{{\mu}}}^{-})$ search region. The extrapolation
to the full $m(\mathup{{{\mu}}}^{+}\mathup{{{\mu}}}^{-})$ phase space is
performed assuming a four-body phase space model for
$D^{0}\\!\rightarrow\mathup{{{\pi}}}^{+}\mathup{{{\pi}}}^{-}\mathup{{{\mu}}}^{+}\mathup{{{\mu}}}^{-}$
for which fractions in each $m(\mathup{{{\mu}}}^{+}\mathup{{{\mu}}}^{-})$
region are quoted in Table 1.
Figure 4: Observed (solid curve) and expected (dashed curve) $\mathrm{CL}_{s}$
values as a function of ${\cal
B}(D^{0}\\!\rightarrow\mathup{{{\pi}}}^{+}\mathup{{{\pi}}}^{-}\mathup{{{\mu}}}^{+}\mathup{{{\mu}}}^{-})/{\cal
B}(D^{0}\\!\rightarrow\mathup{{{\pi}}}^{+}\mathup{{{\pi}}}^{-}\phi(\rightarrow\mathup{{{\mu}}}^{+}\mathup{{{\mu}}}^{-}))$.
The green (yellow) shaded area contains 68.3% and 95.5% of the results of the
analysis on experiments simulated with no signal. The upper limits at the
90(95)% $\mathrm{CL}$ are indicated by the dashed (solid) line. Figure 5:
Observed (solid curve) and expected (dashed curve) $\mathrm{CL}_{s}$ values as
a function of $\cal
B$($D^{0}\\!\rightarrow\mathup{{{\pi}}}^{+}\mathup{{{\pi}}}^{-}\mathup{{{\mu}}}^{+}\mathup{{{\mu}}}^{-}$).
The green (yellow) shaded area contains 68.3% and 95.5% of the results of the
analysis on experiments simulated with no signal. The upper limits at the
90(95)% $\mathrm{CL}$ are indicated by the dashed (solid) line.
The observed distribution of $\mathrm{CL}_{s}$ as a function of the total
branching fraction ratio for
$D^{0}\\!\rightarrow\mathup{{{\pi}}}^{+}\mathup{{{\pi}}}^{-}\mathup{{{\mu}}}^{+}\mathup{{{\mu}}}^{-}$
is shown in Fig. 4. A similar distribution for the absolute branching fraction
is shown in Fig. 5. The upper limits on the branching fraction ratio and
absolute branching fraction at 90% and 95% $\mathrm{CL}$ and the p-values
$(1-\mathrm{CL}_{b})$ for the background-only hypothesis are given in Table 3
and in Table 4. The p-values are computed for the branching fraction value at
which $\mathrm{CL_{s+b}}$ equals $0.5$. Despite the smaller event yield for
$D^{0}\\!\rightarrow\mathup{{{\pi}}}^{+}\mathup{{{\pi}}}^{-}\mathup{{{\mu}}}^{+}\mathup{{{\mu}}}^{-}$
relative to
$D^{0}\\!\rightarrow\mathup{{{\pi}}}^{+}\mathup{{{\pi}}}^{-}\phi(\rightarrow\mathup{{{\mu}}}^{+}\mathup{{{\mu}}}^{-})$,
the upper limit on the total relative branching fraction is of order unity due
to several factors. These are the low reconstruction and selection efficiency
ratio in the signal region, the systematic and statistical uncertainties, and
the extrapolation to the full $m(\mathup{{{\mu}}}^{+}\mathup{{{\mu}}}^{-})$
range according to a phase-space model.
Table 3: Upper limits on $\mathcal{{\cal B}}(D^{0}\\!\rightarrow\mathup{{{\pi}}}^{+}\mathup{{{\pi}}}^{-}\mathup{{{\mu}}}^{+}\mathup{{{\mu}}}^{-})/\mathcal{{\cal B}}(D^{0}\rightarrow\mathup{{{\pi}}}^{+}\mathup{{{\pi}}}^{-}\phi(\rightarrow\mathup{{{\mu}}}^{+}\mu^{-}))$ at 90 and 95% $\mathrm{CL}$, and p-values for the background-only hypothesis in each $m(\mathup{{{\mu}}}^{+}\mathup{{{\mu}}}^{-})$ region and in the full $m(\mathup{{{\mu}}}^{+}\mathup{{{\mu}}}^{-})$ range (assuming a phase-space model). Region | $90\%$ | $95\%$ | p-value
---|---|---|---
low-$m(\mathup{{{\mu}}}^{+}\mathup{{{\mu}}}^{-})$ | 0.41 | 0.51 | 0.32
high-$m(\mathup{{{\mu}}}^{+}\mathup{{{\mu}}}^{-})$ | 0.17 | 0.21 | 0.12
Total | 0.96 | 1.19 | 0.25
Table 4: Upper limits on $\mathcal{{\cal B}}(D^{0}\\!\rightarrow\mathup{{{\pi}}}^{+}\mathup{{{\pi}}}^{-}\mathup{{{\mu}}}^{+}\mathup{{{\mu}}}^{-})$ at 90 and 95% $\mathrm{CL}$ in each $m(\mathup{{{\mu}}}^{+}\mathup{{{\mu}}}^{-})$ region and in the full $m(\mathup{{{\mu}}}^{+}\mathup{{{\mu}}}^{-})$ range (assuming a phase-space model). Region | $90\%\,[\times 10^{-7}]$ | $95\%\,[\times 10^{-7}]$
---|---|---
low-$m(\mathup{{{\mu}}}^{+}\mathup{{{\mu}}}^{-})$ | 2.3 | 2.9
high-$m(\mathup{{{\mu}}}^{+}\mathup{{{\mu}}}^{-})$ | 1.0 | 1.2
Total | 5.5 | 6.7
It is noted that, while the results in individual
$m(\mathup{{{\mu}}}^{+}\mathup{{{\mu}}}^{-})$ regions naturally include
possible contributions from
$D^{0}\\!\rightarrow\rho(\rightarrow\mathup{{{\pi}}}^{+}\mathup{{{\pi}}}^{-})\mathup{{{\mu}}}^{+}\mathup{{{\mu}}}^{-}$
since differences in the reconstruction and selection efficiency with respect
to the four-body
$D^{0}\\!\rightarrow\mathup{{{\pi}}}^{+}\mathup{{{\pi}}}^{-}\mathup{{{\mu}}}^{+}\mathup{{{\mu}}}^{-}$
are negligible, the extrapolation to the full
$m(\mathup{{{\mu}}}^{+}\mathup{{{\mu}}}^{-})$ phase-space depends on the four-
body assumption. Distinguishing a $\rho$ component in the dipion mass spectrum
requires an amplitude analysis which would be hardly informative given the
small sample size and beyond the scope of this first search.
Contributions for non-resonant
$D^{0}\\!\rightarrow\mathup{{{\pi}}}^{+}\mathup{{{\pi}}}^{-}\mathup{{{\mu}}}^{+}\mathup{{{\mu}}}^{-}$
events in the normalisation mode $m(\mathup{{{\mu}}}^{+}\mathup{{{\mu}}}^{-})$
window are neglected in the upper limit calculations. Assuming a branching
fraction equal to the 90% $\mathrm{CL}$ upper limit set in the highest
$m(\mathup{{{\mu}}}^{+}\mathup{{{\mu}}}^{-})$ region, the relative
contribution of the non-resonant mode is estimated to be less than 3%, which
is small compared with other uncertainties.
## 8 Conclusions
A search for the
$D^{0}\\!\rightarrow\mathup{{{\pi}}}^{+}\mathup{{{\pi}}}^{-}\mathup{{{\mu}}}^{+}\mathup{{{\mu}}}^{-}$
decay is conducted using $pp$ collision data, corresponding to an integrated
luminosity of 1.0 $\mbox{\,fb}^{-1}$ at $\sqrt{s}=7$
$\mathrm{\,Te\kern-1.00006ptV}$ recorded by the LHCb experiment. The numbers
of events in the non-resonant $m(\mathup{{{\mu}}}^{+}\mathup{{{\mu}}}^{-})$
regions are compatible with the background-only hypothesis. The limits set on
branching fractions in two $m(\mathup{{{\mu}}}^{+}\mathup{{{\mu}}}^{-})$ bins
and on the total branching fraction, excluding the resonant contributions and
assuming a phase-space model, are
$\displaystyle\frac{\mathcal{{\cal
B}}(D^{0}\\!\rightarrow\mathup{{{\pi}}}^{+}\mathup{{{\pi}}}^{-}\mathup{{{\mu}}}^{+}\mathup{{{\mu}}}^{-})}{\mathcal{{\cal
B}}(D^{0}\rightarrow\mathup{{{\pi}}}^{+}\mathup{{{\pi}}}^{-}\phi(\rightarrow\mathup{{{\mu}}}^{+}\mu^{-}))}$
$\displaystyle<$ $\displaystyle
0.96\,(1.19),\mathrm{\,\,\,at\,\,the\,\,90\,(95)\%\,\,\mathrm{CL}},$
$\displaystyle\mathcal{{\cal
B}}(D^{0}\\!\rightarrow\mathup{{{\pi}}}^{+}\mathup{{{\pi}}}^{-}\mathup{{{\mu}}}^{+}\mathup{{{\mu}}}^{-})$
$\displaystyle<$ $\displaystyle 5.5\,(6.7)\times
10^{-7},\mathrm{\,\,\,at\,\,the\,\,90\,(95)\%\,\,\mathrm{CL}}.$
The upper limit on the absolute branching fraction is improved by a factor of
$50$ with respect to the previous search [5], yielding the most stringent
result to date.
## Acknowledgements
We express our gratitude to our colleagues in the CERN accelerator departments
for the excellent performance of the LHC. We thank the technical and
administrative staff at the LHCb institutes. We acknowledge support from CERN
and from the national agencies: CAPES, CNPq, FAPERJ and FINEP (Brazil); NSFC
(China); CNRS/IN2P3 and Region Auvergne (France); BMBF, DFG, HGF and MPG
(Germany); SFI (Ireland); INFN (Italy); FOM and NWO (The Netherlands); SCSR
(Poland); MEN/IFA (Romania); MinES, Rosatom, RFBR and NRC “Kurchatov
Institute” (Russia); MinECo, XuntaGal and GENCAT (Spain); SNSF and SER
(Switzerland); NAS Ukraine (Ukraine); STFC (United Kingdom); NSF (USA). We
also acknowledge the support received from the ERC under FP7. The Tier1
computing centres are supported by IN2P3 (France), KIT and BMBF (Germany),
INFN (Italy), NWO and SURF (The Netherlands), PIC (Spain), GridPP (United
Kingdom). We are thankful for the computing resources put at our disposal by
Yandex LLC (Russia), as well as to the communities behind the multiple open
source software packages that we depend on.
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|
arxiv-papers
| 2013-10-09T16:17:57 |
2024-09-04T02:49:52.203598
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "LHCb collaboration: R. Aaij, B. Adeva, M. Adinolfi, C. Adrover, A.\n Affolder, Z. Ajaltouni, J. Albrecht, F. Alessio, M. Alexander, S. Ali, G.\n Alkhazov, P. Alvarez Cartelle, A.A. Alves Jr, S. Amato, S. Amerio, Y. Amhis,\n L. Anderlini, J. Anderson, R. Andreassen, J.E. Andrews, R.B. Appleby, O.\n Aquines Gutierrez, F. Archilli, A. Artamonov, M. Artuso, E. Aslanides, G.\n Auriemma, M. Baalouch, S. Bachmann, J.J. Back, A. Badalov, C. Baesso, V.\n Balagura, W. Baldini, R.J. Barlow, C. Barschel, S. Barsuk, W. Barter, Th.\n Bauer, A. Bay, J. Beddow, F. Bedeschi, I. Bediaga, S. Belogurov, K. Belous,\n I. Belyaev, E. Ben-Haim, G. Bencivenni, S. Benson, J. Benton, A. Berezhnoy,\n R. Bernet, M.-O. Bettler, M. van Beuzekom, A. Bien, S. Bifani, T. Bird, A.\n Bizzeti, P.M. Bj{\\o}rnstad, T. Blake, F. Blanc, J. Blouw, S. Blusk, V. Bocci,\n A. Bondar, N. Bondar, W. Bonivento, S. Borghi, A. Borgia, T.J.V. Bowcock, E.\n Bowen, C. Bozzi, T. Brambach, J. van den Brand, J. Bressieux, D. Brett, M.\n Britsch, T. Britton, N.H. Brook, H. Brown, A. Bursche, G. Busetto, J.\n Buytaert, S. Cadeddu, O. Callot, M. Calvi, M. Calvo Gomez, A. Camboni, P.\n Campana, D. Campora Perez, A. Carbone, G. Carboni, R. Cardinale, A. Cardini,\n H. Carranza-Mejia, L. Carson, K. Carvalho Akiba, G. Casse, L. Castillo\n Garcia, M. Cattaneo, Ch. Cauet, R. Cenci, M. Charles, Ph. Charpentier, S.-F.\n Cheung, N. Chiapolini, M. Chrzaszcz, K. Ciba, X. Cid Vidal, G. Ciezarek,\n P.E.L. Clarke, M. Clemencic, H.V. Cliff, J. Closier, C. Coca, V. Coco, J.\n Cogan, E. Cogneras, P. Collins, A. Comerma-Montells, A. Contu, A. Cook, M.\n Coombes, S. Coquereau, G. Corti, B. Couturier, G.A. Cowan, D.C. Craik, M.\n Cruz Torres, S. Cunliffe, R. Currie, C. D'Ambrosio, P. David, P.N.Y. David,\n A. Davis, I. De Bonis, K. De Bruyn, S. De Capua, M. De Cian, J.M. De Miranda,\n L. De Paula, W. De Silva, P. De Simone, D. Decamp, M. Deckenhoff, L. Del\n Buono, N. D\\'el\\'eage, D. Derkach, O. Deschamps, F. Dettori, A. Di Canto, H.\n Dijkstra, M. Dogaru, S. Donleavy, F. Dordei, A. Dosil Su\\'arez, D. Dossett,\n A. Dovbnya, F. Dupertuis, P. Durante, R. Dzhelyadin, A. Dziurda, A. Dzyuba,\n S. Easo, U. Egede, V. Egorychev, S. Eidelman, D. van Eijk, S. Eisenhardt, U.\n Eitschberger, R. Ekelhof, L. Eklund, I. El Rifai, Ch. Elsasser, A. Falabella,\n C. F\\\"arber, C. Farinelli, S. Farry, D. Ferguson, V. Fernandez Albor, F.\n Ferreira Rodrigues, M. Ferro-Luzzi, S. Filippov, M. Fiore, C. Fitzpatrick, M.\n Fontana, F. Fontanelli, R. Forty, O. Francisco, M. Frank, C. Frei, M.\n Frosini, E. Furfaro, A. Gallas Torreira, D. Galli, M. Gandelman, P. Gandini,\n Y. Gao, J. Garofoli, P. Garosi, J. Garra Tico, L. Garrido, C. Gaspar, R.\n Gauld, E. Gersabeck, M. Gersabeck, T. Gershon, Ph. Ghez, V. Gibson, L.\n Giubega, V.V. Gligorov, C. G\\\"obel, D. Golubkov, A. Golutvin, A. Gomes, P.\n Gorbounov, H. Gordon, M. Grabalosa G\\'andara, R. Graciani Diaz, L.A. Granado\n Cardoso, E. Graug\\'es, G. Graziani, A. Grecu, E. Greening, S. Gregson, P.\n Griffith, L. Grillo, O. Gr\\\"unberg, B. Gui, E. Gushchin, Yu. Guz, T. Gys, C.\n Hadjivasiliou, G. Haefeli, C. Haen, S.C. Haines, S. Hall, B. Hamilton, T.\n Hampson, S. Hansmann-Menzemer, N. Harnew, S.T. Harnew, J. Harrison, T.\n Hartmann, J. He, T. Head, V. Heijne, K. Hennessy, P. Henrard, J.A. Hernando\n Morata, E. van Herwijnen, M. He\\ss, A. Hicheur, E. Hicks, D. Hill, M.\n Hoballah, C. Hombach, W. Hulsbergen, P. Hunt, T. Huse, N. Hussain, D.\n Hutchcroft, D. Hynds, V. Iakovenko, M. Idzik, P. Ilten, R. Jacobsson, A.\n Jaeger, E. Jans, P. Jaton, A. Jawahery, F. Jing, M. John, D. Johnson, C.R.\n Jones, C. Joram, B. Jost, M. Kaballo, S. Kandybei, W. Kanso, M. Karacson,\n T.M. Karbach, I.R. Kenyon, T. Ketel, B. Khanji, O. Kochebina, I. Komarov,\n R.F. Koopman, P. Koppenburg, M. Korolev, A. Kozlinskiy, L. Kravchuk, K.\n Kreplin, M. Kreps, G. Krocker, P. Krokovny, F. Kruse, M. Kucharczyk, V.\n Kudryavtsev, K. Kurek, T. Kvaratskheliya, V.N. La Thi, D. Lacarrere, G.\n Lafferty, A. Lai, D. Lambert, R.W. Lambert, E. Lanciotti, G. Lanfranchi, C.\n Langenbruch, T. Latham, C. Lazzeroni, R. Le Gac, J. van Leerdam, J.-P. Lees,\n R. Lef\\`evre, A. Leflat, J. Lefran\\c{c}ois, S. Leo, O. Leroy, T. Lesiak, B.\n Leverington, Y. Li, L. Li Gioi, M. Liles, R. Lindner, C. Linn, B. Liu, G.\n Liu, S. Lohn, I. Longstaff, J.H. Lopes, N. Lopez-March, H. Lu, D. Lucchesi,\n J. Luisier, H. Luo, O. Lupton, F. Machefert, I.V. Machikhiliyan, F. Maciuc,\n O. Maev, S. Malde, G. Manca, G. Mancinelli, J. Maratas, U. Marconi, P.\n Marino, R. M\\\"arki, J. Marks, G. Martellotti, A. Martens, A. Mart\\'in\n S\\'anchez, M. Martinelli, D. Martinez Santos, D. Martins Tostes, A. Martynov,\n A. Massafferri, R. Matev, Z. Mathe, C. Matteuzzi, E. Maurice, A. Mazurov, J.\n McCarthy, A. McNab, R. McNulty, B. McSkelly, B. Meadows, F. Meier, M.\n Meissner, M. Merk, D.A. Milanes, M.-N. Minard, J. Molina Rodriguez, S.\n Monteil, D. Moran, P. Morawski, A. Mord\\`a, M.J. Morello, R. Mountain, I.\n Mous, F. Muheim, K. M\\\"uller, R. Muresan, B. Muryn, B. Muster, P. Naik, T.\n Nakada, R. Nandakumar, I. Nasteva, M. Needham, S. Neubert, N. Neufeld, A.D.\n Nguyen, T.D. Nguyen, C. Nguyen-Mau, M. Nicol, V. Niess, R. Niet, N. Nikitin,\n T. Nikodem, A. Nomerotski, A. Novoselov, A. Oblakowska-Mucha, V. Obraztsov,\n S. Oggero, S. Ogilvy, O. Okhrimenko, R. Oldeman, M. Orlandea, J.M. Otalora\n Goicochea, P. Owen, A. Oyanguren, B.K. Pal, A. Palano, M. Palutan, J. Panman,\n A. Papanestis, M. Pappagallo, C. Parkes, C.J. Parkinson, G. Passaleva, G.D.\n Patel, M. Patel, G.N. Patrick, C. Patrignani, C. Pavel-Nicorescu, A. Pazos\n Alvarez, A. Pearce, A. Pellegrino, G. Penso, M. Pepe Altarelli, S. Perazzini,\n E. Perez Trigo, A. P\\'erez-Calero Yzquierdo, P. Perret, M. Perrin-Terrin, L.\n Pescatore, E. Pesen, G. Pessina, K. Petridis, A. Petrolini, A. Phan, E.\n Picatoste Olloqui, B. Pietrzyk, T. Pila\\v{r}, D. Pinci, S. Playfer, M. Plo\n Casasus, F. Polci, G. Polok, A. Poluektov, E. Polycarpo, A. Popov, D. Popov,\n B. Popovici, C. Potterat, A. Powell, J. Prisciandaro, A. Pritchard, C.\n Prouve, V. Pugatch, A. Puig Navarro, G. Punzi, W. Qian, B. Rachwal, J.H.\n Rademacker, B. Rakotomiaramanana, M.S. Rangel, I. Raniuk, N. Rauschmayr, G.\n Raven, S. Redford, S. Reichert, M.M. Reid, A.C. dos Reis, S. Ricciardi, A.\n Richards, K. Rinnert, V. Rives Molina, D.A. Roa Romero, P. Robbe, D.A.\n Roberts, A.B. Rodrigues, E. Rodrigues, P. Rodriguez Perez, S. Roiser, V.\n Romanovsky, A. Romero Vidal, M. Rotondo, J. Rouvinet, T. Ruf, F. Ruffini, H.\n Ruiz, P. Ruiz Valls, G. Sabatino, J.J. Saborido Silva, N. Sagidova, P. Sail,\n B. Saitta, V. Salustino Guimaraes, B. Sanmartin Sedes, R. Santacesaria, C.\n Santamarina Rios, E. Santovetti, M. Sapunov, A. Sarti, C. Satriano, A. Satta,\n M. Savrie, D. Savrina, M. Schiller, H. Schindler, M. Schlupp, M. Schmelling,\n B. Schmidt, O. Schneider, A. Schopper, M.-H. Schune, R. Schwemmer, B.\n Sciascia, A. Sciubba, M. Seco, A. Semennikov, K. Senderowska, I. Sepp, N.\n Serra, J. Serrano, P. Seyfert, M. Shapkin, I. Shapoval, Y. Shcheglov, T.\n Shears, L. Shekhtman, O. Shevchenko, V. Shevchenko, A. Shires, R. Silva\n Coutinho, M. Sirendi, N. Skidmore, T. Skwarnicki, N.A. Smith, E. Smith, E.\n Smith, J. Smith, M. Smith, M.D. Sokoloff, F.J.P. Soler, F. Soomro, D. Souza,\n B. Souza De Paula, B. Spaan, A. Sparkes, P. Spradlin, F. Stagni, S. Stahl, O.\n Steinkamp, S. Stevenson, S. Stoica, S. Stone, B. Storaci, M. Straticiuc, U.\n Straumann, V.K. Subbiah, L. Sun, W. Sutcliffe, S. Swientek, V. Syropoulos, M.\n Szczekowski, P. Szczypka, D. Szilard, T. Szumlak, S. T'Jampens, M. Teklishyn,\n E. Teodorescu, F. Teubert, C. Thomas, E. Thomas, J. van Tilburg, V.\n Tisserand, M. Tobin, S. Tolk, D. Tonelli, S. Topp-Joergensen, N. Torr, E.\n Tournefier, S. Tourneur, M.T. Tran, M. Tresch, A. Tsaregorodtsev, P.\n Tsopelas, N. Tuning, M. Ubeda Garcia, A. Ukleja, A. Ustyuzhanin, U. Uwer, V.\n Vagnoni, G. Valenti, A. Vallier, R. Vazquez Gomez, P. Vazquez Regueiro, C.\n V\\'azquez Sierra, S. Vecchi, J.J. Velthuis, M. Veltri, G. Veneziano, M.\n Vesterinen, B. Viaud, D. Vieira, X. Vilasis-Cardona, A. Vollhardt, D.\n Volyanskyy, D. Voong, A. Vorobyev, V. Vorobyev, C. Vo\\ss, H. Voss, R. Waldi,\n C. Wallace, R. Wallace, S. Wandernoth, J. Wang, D.R. Ward, N.K. Watson, A.D.\n Webber, D. Websdale, M. Whitehead, J. Wicht, J. Wiechczynski, D. Wiedner, L.\n Wiggers, G. Wilkinson, M.P. Williams, M. Williams, F.F. Wilson, J. Wimberley,\n J. Wishahi, W. Wislicki, M. Witek, G. Wormser, S.A. Wotton, S. Wright, S. Wu,\n K. Wyllie, Y. Xie, Z. Xing, Z. Yang, X. Yuan, O. Yushchenko, M. Zangoli, M.\n Zavertyaev, F. Zhang, L. Zhang, W.C. Zhang, Y. Zhang, A. Zhelezov, A.\n Zhokhov, L. Zhong, A. Zvyagin",
"submitter": "Andrea Contu",
"url": "https://arxiv.org/abs/1310.2535"
}
|
1310.2538
|
EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN)
CERN-PH-EP-2013-181 LHCb-PAPER-2013-049 January 6, 2014
Search for the doubly charmed baryon $\Xi_{cc}^{+}$
The LHCb collaboration†††Authors are listed on the following pages.
A search for the doubly charmed baryon $\Xi_{cc}^{+}$ in the decay mode
$\Xi_{cc}^{+}\\!\rightarrow\mathchar 28931\relax_{c}^{+}K^{-}\pi^{+}$ is
performed with a data sample, corresponding to an integrated luminosity of
0.65$\mbox{\,fb}^{-1}$, of $pp$ collisions recorded at a centre-of-mass energy
of 7$\mathrm{\,Te\kern-1.00006ptV}$. No significant signal is found in the
mass range 3300–3800${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$. Upper limits
at the 95% confidence level on the ratio of the $\Xi_{cc}^{+}$ production
cross-section times branching fraction to that of the $\mathchar
28931\relax_{c}^{+}$, $R$, are given as a function of the $\Xi_{cc}^{+}$ mass
and lifetime. The largest upper limits range from $R<1.5\times 10^{-2}$ for a
lifetime of 100$\rm\,fs$ to $R<3.9\times 10^{-4}$ for a lifetime of
400$\rm\,fs$.
Published in JHEP, DOI: 10.1007/JHEP12(2013)090
© CERN on behalf of the LHCb collaboration, license CC-BY-3.0.
LHCb collaboration
R. Aaij40, B. Adeva36, M. Adinolfi45, C. Adrover6, A. Affolder51, Z.
Ajaltouni5, J. Albrecht9, F. Alessio37, M. Alexander50, S. Ali40, G.
Alkhazov29, P. Alvarez Cartelle36, A.A. Alves Jr24, S. Amato2, S. Amerio21, Y.
Amhis7, L. Anderlini17,f, J. Anderson39, R. Andreassen56, J.E. Andrews57, R.B.
Appleby53, O. Aquines Gutierrez10, F. Archilli18, A. Artamonov34, M. Artuso58,
E. Aslanides6, G. Auriemma24,m, M. Baalouch5, S. Bachmann11, J.J. Back47, A.
Badalov35, C. Baesso59, V. Balagura30, W. Baldini16, R.J. Barlow53, C.
Barschel37, S. Barsuk7, W. Barter46, Th. Bauer40, A. Bay38, J. Beddow50, F.
Bedeschi22, I. Bediaga1, S. Belogurov30, K. Belous34, I. Belyaev30, E. Ben-
Haim8, G. Bencivenni18, S. Benson49, J. Benton45, A. Berezhnoy31, R. Bernet39,
M.-O. Bettler46, M. van Beuzekom40, A. Bien11, S. Bifani44, T. Bird53, A.
Bizzeti17,h, P.M. Bjørnstad53, T. Blake37, F. Blanc38, J. Blouw10, S. Blusk58,
V. Bocci24, A. Bondar33, N. Bondar29, W. Bonivento15, S. Borghi53, A.
Borgia58, T.J.V. Bowcock51, E. Bowen39, C. Bozzi16, T. Brambach9, J. van den
Brand41, J. Bressieux38, D. Brett53, M. Britsch10, T. Britton58, N.H. Brook45,
H. Brown51, A. Bursche39, G. Busetto21,q, J. Buytaert37, S. Cadeddu15, O.
Callot7, M. Calvi20,j, M. Calvo Gomez35,n, A. Camboni35, P. Campana18,37, D.
Campora Perez37, A. Carbone14,c, G. Carboni23,k, R. Cardinale19,i, A.
Cardini15, H. Carranza-Mejia49, L. Carson52, K. Carvalho Akiba2, G. Casse51,
L. Castillo Garcia37, M. Cattaneo37, Ch. Cauet9, R. Cenci57, M. Charles8, Ph.
Charpentier37, S.-F. Cheung54, N. Chiapolini39, M. Chrzaszcz39,25, K. Ciba37,
X. Cid Vidal37, G. Ciezarek52, P.E.L. Clarke49, M. Clemencic37, H.V. Cliff46,
J. Closier37, C. Coca28, V. Coco40, J. Cogan6, E. Cogneras5, P. Collins37, A.
Comerma-Montells35, A. Contu15,37, A. Cook45, M. Coombes45, S. Coquereau8, G.
Corti37, B. Couturier37, G.A. Cowan49, D.C. Craik47, M. Cruz Torres59, S.
Cunliffe52, R. Currie49, C. D’Ambrosio37, P. David8, P.N.Y. David40, A.
Davis56, I. De Bonis4, K. De Bruyn40, S. De Capua53, M. De Cian11, J.M. De
Miranda1, L. De Paula2, W. De Silva56, P. De Simone18, D. Decamp4, M.
Deckenhoff9, L. Del Buono8, N. Déléage4, D. Derkach54, O. Deschamps5, F.
Dettori41, A. Di Canto11, H. Dijkstra37, M. Dogaru28, S. Donleavy51, F.
Dordei11, A. Dosil Suárez36, D. Dossett47, A. Dovbnya42, F. Dupertuis38, P.
Durante37, R. Dzhelyadin34, A. Dziurda25, A. Dzyuba29, S. Easo48, U. Egede52,
V. Egorychev30, S. Eidelman33, D. van Eijk40, S. Eisenhardt49, U.
Eitschberger9, R. Ekelhof9, L. Eklund50,37, I. El Rifai5, Ch. Elsasser39, A.
Falabella14,e, C. Färber11, C. Farinelli40, S. Farry51, D. Ferguson49, V.
Fernandez Albor36, F. Ferreira Rodrigues1, M. Ferro-Luzzi37, S. Filippov32, M.
Fiore16,e, C. Fitzpatrick37, M. Fontana10, F. Fontanelli19,i, R. Forty37, O.
Francisco2, M. Frank37, C. Frei37, M. Frosini17,37,f, E. Furfaro23,k, A.
Gallas Torreira36, D. Galli14,c, M. Gandelman2, P. Gandini58, Y. Gao3, J.
Garofoli58, P. Garosi53, J. Garra Tico46, L. Garrido35, C. Gaspar37, R.
Gauld54, E. Gersabeck11, M. Gersabeck53, T. Gershon47, Ph. Ghez4, V. Gibson46,
L. Giubega28, V.V. Gligorov37, C. Göbel59, D. Golubkov30, A. Golutvin52,30,37,
A. Gomes2, P. Gorbounov30,37, H. Gordon37, M. Grabalosa Gándara5, R. Graciani
Diaz35, L.A. Granado Cardoso37, E. Graugés35, G. Graziani17, A. Grecu28, E.
Greening54, S. Gregson46, P. Griffith44, L. Grillo11, O. Grünberg60, B. Gui58,
E. Gushchin32, Yu. Guz34,37, T. Gys37, C. Hadjivasiliou58, G. Haefeli38, C.
Haen37, S.C. Haines46, S. Hall52, B. Hamilton57, T. Hampson45, S. Hansmann-
Menzemer11, N. Harnew54, S.T. Harnew45, J. Harrison53, T. Hartmann60, J. He37,
T. Head37, V. Heijne40, K. Hennessy51, P. Henrard5, J.A. Hernando Morata36, E.
van Herwijnen37, M. Heß60, A. Hicheur1, E. Hicks51, D. Hill54, M. Hoballah5,
C. Hombach53, W. Hulsbergen40, P. Hunt54, T. Huse51, N. Hussain54, D.
Hutchcroft51, D. Hynds50, V. Iakovenko43, M. Idzik26, P. Ilten12, R.
Jacobsson37, A. Jaeger11, E. Jans40, P. Jaton38, A. Jawahery57, F. Jing3, M.
John54, D. Johnson54, C.R. Jones46, C. Joram37, B. Jost37, M. Kaballo9, S.
Kandybei42, W. Kanso6, M. Karacson37, T.M. Karbach37, I.R. Kenyon44, T.
Ketel41, B. Khanji20, O. Kochebina7, I. Komarov38, R.F. Koopman41, P.
Koppenburg40, M. Korolev31, A. Kozlinskiy40, L. Kravchuk32, K. Kreplin11, M.
Kreps47, G. Krocker11, P. Krokovny33, F. Kruse9, M. Kucharczyk20,25,37,j, V.
Kudryavtsev33, K. Kurek27, T. Kvaratskheliya30,37, V.N. La Thi38, D.
Lacarrere37, G. Lafferty53, A. Lai15, D. Lambert49, R.W. Lambert41, E.
Lanciotti37, G. Lanfranchi18, C. Langenbruch37, T. Latham47, C. Lazzeroni44,
R. Le Gac6, J. van Leerdam40, J.-P. Lees4, R. Lefèvre5, A. Leflat31, J.
Lefrançois7, S. Leo22, O. Leroy6, T. Lesiak25, B. Leverington11, Y. Li3, L. Li
Gioi5, M. Liles51, R. Lindner37, C. Linn11, B. Liu3, G. Liu37, S. Lohn37, I.
Longstaff50, J.H. Lopes2, N. Lopez-March38, H. Lu3, D. Lucchesi21,q, J.
Luisier38, H. Luo49, O. Lupton54, F. Machefert7, I.V. Machikhiliyan30, F.
Maciuc28, O. Maev29,37, S. Malde54, G. Manca15,d, G. Mancinelli6, J. Maratas5,
U. Marconi14, P. Marino22,s, R. Märki38, J. Marks11, G. Martellotti24, A.
Martens8, A. Martín Sánchez7, M. Martinelli40, D. Martinez Santos41,37, D.
Martins Tostes2, A. Martynov31, A. Massafferri1, R. Matev37, Z. Mathe37, C.
Matteuzzi20, E. Maurice6, A. Mazurov16,37,e, J. McCarthy44, A. McNab53, R.
McNulty12, B. McSkelly51, B. Meadows56,54, F. Meier9, M. Meissner11, M.
Merk40, D.A. Milanes8, M.-N. Minard4, J. Molina Rodriguez59, S. Monteil5, D.
Moran53, P. Morawski25, A. Mordà6, M.J. Morello22,s, R. Mountain58, I. Mous40,
F. Muheim49, K. Müller39, R. Muresan28, B. Muryn26, B. Muster38, P. Naik45, T.
Nakada38, R. Nandakumar48, I. Nasteva1, M. Needham49, S. Neubert37, N.
Neufeld37, A.D. Nguyen38, T.D. Nguyen38, C. Nguyen-Mau38,o, M. Nicol7, V.
Niess5, R. Niet9, N. Nikitin31, T. Nikodem11, A. Nomerotski54, A. Novoselov34,
A. Oblakowska-Mucha26, V. Obraztsov34, S. Oggero40, S. Ogilvy50, O.
Okhrimenko43, R. Oldeman15,d, M. Orlandea28, J.M. Otalora Goicochea2, P.
Owen52, A. Oyanguren35, B.K. Pal58, A. Palano13,b, M. Palutan18, J. Panman37,
A. Papanestis48, M. Pappagallo50, C. Parkes53, C.J. Parkinson52, G.
Passaleva17, G.D. Patel51, M. Patel52, G.N. Patrick48, C. Patrignani19,i, C.
Pavel-Nicorescu28, A. Pazos Alvarez36, A. Pearce53, A. Pellegrino40, G.
Penso24,l, M. Pepe Altarelli37, S. Perazzini14,c, E. Perez Trigo36, A. Pérez-
Calero Yzquierdo35, P. Perret5, M. Perrin-Terrin6, L. Pescatore44, E. Pesen61,
G. Pessina20, K. Petridis52, A. Petrolini19,i, A. Phan58, E. Picatoste
Olloqui35, B. Pietrzyk4, T. Pilař47, D. Pinci24, S. Playfer49, M. Plo
Casasus36, F. Polci8, G. Polok25, A. Poluektov47,33, E. Polycarpo2, A.
Popov34, D. Popov10, B. Popovici28, C. Potterat35, A. Powell54, J.
Prisciandaro38, A. Pritchard51, C. Prouve7, V. Pugatch43, A. Puig Navarro38,
G. Punzi22,r, W. Qian4, B. Rachwal25, J.H. Rademacker45, B.
Rakotomiaramanana38, M.S. Rangel2, I. Raniuk42, N. Rauschmayr37, G. Raven41,
S. Redford54, S. Reichert53, M.M. Reid47, A.C. dos Reis1, S. Ricciardi48, A.
Richards52, K. Rinnert51, V. Rives Molina35, D.A. Roa Romero5, P. Robbe7, D.A.
Roberts57, A.B. Rodrigues1, E. Rodrigues53, P. Rodriguez Perez36, S. Roiser37,
V. Romanovsky34, A. Romero Vidal36, M. Rotondo21, J. Rouvinet38, T. Ruf37, F.
Ruffini22, H. Ruiz35, P. Ruiz Valls35, G. Sabatino24,k, J.J. Saborido Silva36,
N. Sagidova29, P. Sail50, B. Saitta15,d, V. Salustino Guimaraes2, B. Sanmartin
Sedes36, R. Santacesaria24, C. Santamarina Rios36, E. Santovetti23,k, M.
Sapunov6, A. Sarti18, C. Satriano24,m, A. Satta23, M. Savrie16,e, D.
Savrina30,31, M. Schiller41, H. Schindler37, M. Schlupp9, M. Schmelling10, B.
Schmidt37, O. Schneider38, A. Schopper37, M.-H. Schune7, R. Schwemmer37, B.
Sciascia18, A. Sciubba24, M. Seco36, A. Semennikov30, K. Senderowska26, I.
Sepp52, N. Serra39, J. Serrano6, P. Seyfert11, M. Shapkin34, I.
Shapoval16,42,e, Y. Shcheglov29, T. Shears51, L. Shekhtman33, O. Shevchenko42,
V. Shevchenko30, A. Shires9, R. Silva Coutinho47, M. Sirendi46, N. Skidmore45,
T. Skwarnicki58, N.A. Smith51, E. Smith54,48, E. Smith52, J. Smith46, M.
Smith53, M.D. Sokoloff56, F.J.P. Soler50, F. Soomro38, D. Souza45, B. Souza De
Paula2, B. Spaan9, A. Sparkes49, P. Spradlin50, F. Stagni37, S. Stahl11, O.
Steinkamp39, S. Stevenson54, S. Stoica28, S. Stone58, B. Storaci39, M.
Straticiuc28, U. Straumann39, V.K. Subbiah37, L. Sun56, W. Sutcliffe52, S.
Swientek9, V. Syropoulos41, M. Szczekowski27, P. Szczypka38,37, D. Szilard2,
T. Szumlak26, S. T’Jampens4, M. Teklishyn7, E. Teodorescu28, F. Teubert37, C.
Thomas54, E. Thomas37, J. van Tilburg11, V. Tisserand4, M. Tobin38, S. Tolk41,
D. Tonelli37, S. Topp-Joergensen54, N. Torr54, E. Tournefier4,52, S.
Tourneur38, M.T. Tran38, M. Tresch39, A. Tsaregorodtsev6, P. Tsopelas40, N.
Tuning40,37, M. Ubeda Garcia37, A. Ukleja27, A. Ustyuzhanin52,p, U. Uwer11, V.
Vagnoni14, G. Valenti14, A. Vallier7, R. Vazquez Gomez18, P. Vazquez
Regueiro36, C. Vázquez Sierra36, S. Vecchi16, J.J. Velthuis45, M. Veltri17,g,
G. Veneziano38, M. Vesterinen37, B. Viaud7, D. Vieira2, X. Vilasis-
Cardona35,n, A. Vollhardt39, D. Volyanskyy10, D. Voong45, A. Vorobyev29, V.
Vorobyev33, C. Voß60, H. Voss10, R. Waldi60, C. Wallace47, R. Wallace12, S.
Wandernoth11, J. Wang58, D.R. Ward46, N.K. Watson44, A.D. Webber53, D.
Websdale52, M. Whitehead47, J. Wicht37, J. Wiechczynski25, D. Wiedner11, L.
Wiggers40, G. Wilkinson54, M.P. Williams47,48, M. Williams55, F.F. Wilson48,
J. Wimberley57, J. Wishahi9, W. Wislicki27, M. Witek25, G. Wormser7, S.A.
Wotton46, S. Wright46, S. Wu3, K. Wyllie37, Y. Xie49,37, Z. Xing58, Z. Yang3,
X. Yuan3, O. Yushchenko34, M. Zangoli14, M. Zavertyaev10,a, F. Zhang3, L.
Zhang58, W.C. Zhang12, Y. Zhang3, A. Zhelezov11, A. Zhokhov30, L. Zhong3, A.
Zvyagin37.
1Centro Brasileiro de Pesquisas Físicas (CBPF), Rio de Janeiro, Brazil
2Universidade Federal do Rio de Janeiro (UFRJ), Rio de Janeiro, Brazil
3Center for High Energy Physics, Tsinghua University, Beijing, China
4LAPP, Université de Savoie, CNRS/IN2P3, Annecy-Le-Vieux, France
5Clermont Université, Université Blaise Pascal, CNRS/IN2P3, LPC, Clermont-
Ferrand, France
6CPPM, Aix-Marseille Université, CNRS/IN2P3, Marseille, France
7LAL, Université Paris-Sud, CNRS/IN2P3, Orsay, France
8LPNHE, Université Pierre et Marie Curie, Université Paris Diderot,
CNRS/IN2P3, Paris, France
9Fakultät Physik, Technische Universität Dortmund, Dortmund, Germany
10Max-Planck-Institut für Kernphysik (MPIK), Heidelberg, Germany
11Physikalisches Institut, Ruprecht-Karls-Universität Heidelberg, Heidelberg,
Germany
12School of Physics, University College Dublin, Dublin, Ireland
13Sezione INFN di Bari, Bari, Italy
14Sezione INFN di Bologna, Bologna, Italy
15Sezione INFN di Cagliari, Cagliari, Italy
16Sezione INFN di Ferrara, Ferrara, Italy
17Sezione INFN di Firenze, Firenze, Italy
18Laboratori Nazionali dell’INFN di Frascati, Frascati, Italy
19Sezione INFN di Genova, Genova, Italy
20Sezione INFN di Milano Bicocca, Milano, Italy
21Sezione INFN di Padova, Padova, Italy
22Sezione INFN di Pisa, Pisa, Italy
23Sezione INFN di Roma Tor Vergata, Roma, Italy
24Sezione INFN di Roma La Sapienza, Roma, Italy
25Henryk Niewodniczanski Institute of Nuclear Physics Polish Academy of
Sciences, Kraków, Poland
26AGH - University of Science and Technology, Faculty of Physics and Applied
Computer Science, Kraków, Poland
27National Center for Nuclear Research (NCBJ), Warsaw, Poland
28Horia Hulubei National Institute of Physics and Nuclear Engineering,
Bucharest-Magurele, Romania
29Petersburg Nuclear Physics Institute (PNPI), Gatchina, Russia
30Institute of Theoretical and Experimental Physics (ITEP), Moscow, Russia
31Institute of Nuclear Physics, Moscow State University (SINP MSU), Moscow,
Russia
32Institute for Nuclear Research of the Russian Academy of Sciences (INR RAN),
Moscow, Russia
33Budker Institute of Nuclear Physics (SB RAS) and Novosibirsk State
University, Novosibirsk, Russia
34Institute for High Energy Physics (IHEP), Protvino, Russia
35Universitat de Barcelona, Barcelona, Spain
36Universidad de Santiago de Compostela, Santiago de Compostela, Spain
37European Organization for Nuclear Research (CERN), Geneva, Switzerland
38Ecole Polytechnique Fédérale de Lausanne (EPFL), Lausanne, Switzerland
39Physik-Institut, Universität Zürich, Zürich, Switzerland
40Nikhef National Institute for Subatomic Physics, Amsterdam, The Netherlands
41Nikhef National Institute for Subatomic Physics and VU University Amsterdam,
Amsterdam, The Netherlands
42NSC Kharkiv Institute of Physics and Technology (NSC KIPT), Kharkiv, Ukraine
43Institute for Nuclear Research of the National Academy of Sciences (KINR),
Kyiv, Ukraine
44University of Birmingham, Birmingham, United Kingdom
45H.H. Wills Physics Laboratory, University of Bristol, Bristol, United
Kingdom
46Cavendish Laboratory, University of Cambridge, Cambridge, United Kingdom
47Department of Physics, University of Warwick, Coventry, United Kingdom
48STFC Rutherford Appleton Laboratory, Didcot, United Kingdom
49School of Physics and Astronomy, University of Edinburgh, Edinburgh, United
Kingdom
50School of Physics and Astronomy, University of Glasgow, Glasgow, United
Kingdom
51Oliver Lodge Laboratory, University of Liverpool, Liverpool, United Kingdom
52Imperial College London, London, United Kingdom
53School of Physics and Astronomy, University of Manchester, Manchester,
United Kingdom
54Department of Physics, University of Oxford, Oxford, United Kingdom
55Massachusetts Institute of Technology, Cambridge, MA, United States
56University of Cincinnati, Cincinnati, OH, United States
57University of Maryland, College Park, MD, United States
58Syracuse University, Syracuse, NY, United States
59Pontifícia Universidade Católica do Rio de Janeiro (PUC-Rio), Rio de
Janeiro, Brazil, associated to 2
60Institut für Physik, Universität Rostock, Rostock, Germany, associated to 11
61Celal Bayar University, Manisa, Turkey, associated to 37
aP.N. Lebedev Physical Institute, Russian Academy of Science (LPI RAS),
Moscow, Russia
bUniversità di Bari, Bari, Italy
cUniversità di Bologna, Bologna, Italy
dUniversità di Cagliari, Cagliari, Italy
eUniversità di Ferrara, Ferrara, Italy
fUniversità di Firenze, Firenze, Italy
gUniversità di Urbino, Urbino, Italy
hUniversità di Modena e Reggio Emilia, Modena, Italy
iUniversità di Genova, Genova, Italy
jUniversità di Milano Bicocca, Milano, Italy
kUniversità di Roma Tor Vergata, Roma, Italy
lUniversità di Roma La Sapienza, Roma, Italy
mUniversità della Basilicata, Potenza, Italy
nLIFAELS, La Salle, Universitat Ramon Llull, Barcelona, Spain
oHanoi University of Science, Hanoi, Viet Nam
pInstitute of Physics and Technology, Moscow, Russia
qUniversità di Padova, Padova, Italy
rUniversità di Pisa, Pisa, Italy
sScuola Normale Superiore, Pisa, Italy
## 1 Introduction
The constituent quark model [1, 2, 3] predicts the existence of multiplets of
baryon and meson states, with a structure determined by the symmetry
properties of the hadron wavefunctions. When considering $u$, $d$, $s$, and
$c$ quarks, the states form $SU(4)$ multiplets [4]. The baryon ground
states—those with no orbital or radial excitations—consist of a 20-plet with
spin-parity $J^{P}=1/2^{+}$ and a 20-plet with $J^{P}=3/2^{+}$. All of the
ground states with charm quantum number $C=0$ or $C=1$ have been discovered
[5]. Three weakly decaying $C=2$ states are expected: a $\Xi_{cc}$ isodoublet
($ccu,ccd$) and an $\Omega_{cc}$ isosinglet ($ccs$), each with
$J^{P}=1/2^{+}$. This paper reports a search for the $\Xi_{cc}^{+}$ baryon.
There are numerous predictions for the masses of these states (see, e.g., Ref.
[6] and the references therein, as well as Refs. [7, 8, 9, 10, 11]) with most
estimates for the $\Xi_{cc}^{+}$ mass in the range
3500–3700${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$. Predictions for its
lifetime range between 100 and 250$\rm\,fs$ [12, 13, 14].
Signals for the $\Xi_{cc}^{+}$ baryon were reported in the $\mathchar
28931\relax_{c}^{+}K^{-}\pi^{+}$ and $pD^{+}K^{-}$ final states by the SELEX
collaboration, using a hyperon beam (containing an admixture of $p$,
$\Sigma^{-}$, and $\pi^{-}$) on a fixed target [15, 16]. The mass was measured
to be $3519\pm 2$${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$, and the lifetime
was found to be compatible with zero within experimental resolution and less
than 33$\rm\,fs$ at the 90% confidence level (CL). SELEX estimated that 20% of
their $\mathchar 28931\relax_{c}^{+}$ yield originates from $\Xi_{cc}^{+}$
decays, in contrast to theory expectations that the production of doubly
charmed baryons would be suppressed by several orders of magnitude with
respect to singly charmed baryons [17]. Searches in different production
environments at the FOCUS, BaBar, and Belle experiments have not shown
evidence for a $\Xi_{cc}^{+}$ state with the properties reported by SELEX [18,
19, 20].
This paper presents the result of a search for the decay111The inclusion of
charge-conjugate processes is implied throughout.
$\Xi_{cc}^{+}\\!\rightarrow\mathchar 28931\relax_{c}^{+}K^{-}\pi^{+}$ with the
LHCb detector and an integrated luminosity of $0.65\mbox{\,fb}^{-1}$ of $pp$
collision data recorded at centre-of-mass energy
$\sqrt{s}=7\mathrm{\,Te\kern-1.00006ptV}$. Double charm production has been
observed previously at LHCb both in the $J/\psi\,J/\psi$ final state [21] and
in final states including one or two open charm hadrons [22]. Phenomenological
estimates of the production cross-section of $\Xi_{cc}$ in $pp$ collisions at
$\sqrt{s}=14\mathrm{\,Te\kern-1.00006ptV}$ are in the range 60–1800$\rm\,nb$
[17, 23, 24]; the cross-section at $\sqrt{s}=7\mathrm{\,Te\kern-1.00006ptV}$
is expected to be roughly a factor of two smaller. As is typical for charmed
hadrons, the production is expected to be concentrated in the low transverse
momentum ($p_{\rm T}$) and forward rapidity ($y$) kinematic region
instrumented by LHCb [24]. For comparison, the prompt $\mathchar
28931\relax_{c}^{+}$ cross-section in the range $0<\mbox{$p_{\rm
T}$}<8000$${\mathrm{\,Me\kern-1.00006ptV\\!/}c}$ and $2.0<y<4.5$ at
$\sqrt{s}=7\mathrm{\,Te\kern-1.00006ptV}$ has been measured to be $(233\pm
26\pm 71\pm 14)$$\rm\,\upmu b$ at LHCb [25], where the uncertainties are
statistical, systematic, and due to the description of the fragmentation
model, respectively. Thus, the cross-section for $\Xi_{cc}^{+}$ production at
LHCb is predicted to be smaller than that for $\mathchar 28931\relax_{c}^{+}$
by a factor of order $10^{-4}$ to $10^{-3}$.
To reduce systematic uncertainties, the $\Xi_{cc}^{+}$ cross-section is
measured relative to that of the $\mathchar 28931\relax_{c}^{+}$. This has the
further advantage that it allows a direct comparison with previous
experimental results. The production ratio $R$ that is measured is defined as
$R\equiv\frac{\sigma(\Xi_{cc}^{+})\,{\cal
B}(\mbox{$\Xi_{cc}^{+}\\!\rightarrow\mathchar
28931\relax_{c}^{+}K^{-}\pi^{+}$})}{\sigma(\mathchar
28931\relax_{c}^{+})}=\frac{N_{\text{sig}}}{N_{\text{norm}}}\frac{\varepsilon_{\text{norm}}}{\varepsilon_{\text{sig}}},$
(1)
where $N_{\text{sig}}$ and $N_{\text{norm}}$ refer to the measured yields of
the signal ($\Xi_{cc}^{+}$) and normalisation ($\mathchar
28931\relax_{c}^{+}$) modes, $\varepsilon_{\text{sig}}$ and
$\varepsilon_{\text{norm}}$ are the corresponding efficiencies, ${\cal B}$
indicates a branching fraction, and $\sigma$ indicates a cross-section.
Assuming that ${\cal B}(\mbox{$\Xi_{cc}^{+}\\!\rightarrow\mathchar
28931\relax_{c}^{+}K^{-}\pi^{+}$})\approx{\cal B}(\mbox{$\mathchar
28931\relax_{c}^{+}\\!\rightarrow pK^{-}\pi^{+}$})\approx 5\%$ [5], the
expected value of $R$ at LHCb is of order $10^{-5}$ to $10^{-4}$. By contrast,
the SELEX observation [15] reported 15.9 $\Xi_{cc}^{+}$ signal events in a
sample of 1630 $\mathchar 28931\relax_{c}^{+}$ events with an efficiency ratio
of 11%, corresponding to $R=9\%$. For convenience, the single-event
sensitivity $\alpha$ is defined as
$\alpha\equiv\frac{\varepsilon_{\text{norm}}}{N_{\text{norm}}\,\varepsilon_{\text{sig}}}$
(2)
such that $R=\alpha N_{\text{sig}}$. For each candidate the mass difference
$\delta m$ is computed as
$\delta m\equiv m([pK^{-}\pi^{+}]_{\mathchar
28931\relax_{c}}K^{-}\pi^{+})-m([pK^{-}\pi^{+}]_{\mathchar
28931\relax_{c}})-m(K^{-})-m(\pi^{+}),$ (3)
where $m([pK^{-}\pi^{+}]_{\mathchar 28931\relax_{c}}K^{-}\pi^{+})$ is the
measured invariant mass of the $\Xi_{cc}^{+}$ candidate,
$m([pK^{-}\pi^{+}]_{\mathchar 28931\relax_{c}})$ is the measured invariant
mass of the $pK^{-}\pi^{+}$ combination forming the $\mathchar
28931\relax_{c}^{+}$ candidate, and $m(K^{-})$ and $m(\pi^{+})$ are the world-
average masses of charged kaons and pions, respectively [5].
Since no assumption is made about the $\Xi_{cc}^{+}$ mass, a wide signal
window of $380<\delta m<880$${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ is used
for this search, corresponding to approximately
$3300<m(\Xi_{cc}^{+})<3800$${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$. All
aspects of the analysis procedure were fixed before the data in this signal
region were examined. Limits on $R$ are quoted as a function of the
$\Xi_{cc}^{+}$ mass and lifetime, since the measured yield depends on $\delta
m$, and $\varepsilon_{\text{sig}}$ depends on both the mass and lifetime.
## 2 Detector and software
The LHCb detector [26] is a single-arm forward spectrometer covering the
pseudorapidity range $2<\eta<5$, designed for the study of particles
containing $b$ or $c$ quarks. The detector includes a high-precision tracking
system consisting of a silicon-strip vertex detector (VELO) surrounding the
$pp$ interaction region, a large-area silicon-strip detector located upstream
of a dipole magnet with a bending power of about $4{\rm\,Tm}$, and three
stations of silicon-strip detectors and straw drift tubes placed downstream.
The combined tracking system provides a momentum measurement with relative
uncertainty that varies from 0.4% at 5${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$
to 0.6% at 100${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$, and impact parameter
(IP) resolution of 20$\,\upmu\rm m$ for tracks with large transverse momentum.
Charged hadrons are identified using two ring-imaging Cherenkov detectors
[27]. Photon, electron, and hadron candidates are identified by a calorimeter
system consisting of scintillating-pad and preshower detectors, an
electromagnetic calorimeter, and a hadronic calorimeter. Muons are identified
by a system composed of alternating layers of iron and multiwire proportional
chambers [28]. The trigger [29] consists of a hardware stage, based on
information from the calorimeter and muon systems, followed by a software
stage, which applies a full event reconstruction.
In the simulation, $pp$ collisions are generated using Pythia 6.4 [30] with a
specific LHCb configuration [31]. A dedicated generator, Genxicc v2.0, is used
to simulate $\Xi_{cc}^{+}$ baryon production [32]. Decays of hadronic
particles are described by EvtGen [33], in which final state radiation is
generated using Photos [34]. The interaction of the generated particles with
the detector and its response are implemented using the Geant4 toolkit [35,
*Agostinelli:2002hh] as described in Ref. [37]. Unless otherwise stated,
simulated events are generated with
$m(\Xi_{cc}^{+})=3500$${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$, with
$\tau_{\Xi_{cc}^{+}}=333$$\rm\,fs$, and with the $\Xi_{cc}^{+}$ and $\mathchar
28931\relax_{c}^{+}$ decay products distributed according to phase space.
## 3 Triggering, reconstruction, and selection
The procedure to trigger, reconstruct, and select candidates for the signal
and normalisation modes is designed to retain signal and to suppress three
primary sources of background. These are combinations of unrelated tracks,
especially those originating from the same primary interaction vertex (PV);
mis-reconstructed charm or beauty hadron decays, which typically occur at a
displaced vertex; and combinations of a real $\mathchar 28931\relax_{c}^{+}$
with other tracks to form a fake $\Xi_{cc}^{+}$ candidate. The first two
classes generally have a smooth distribution in both
$m([pK^{-}\pi^{+}]_{\mathchar 28931\relax_{c}})$ and $\delta m$; the third
peaks in $m([pK^{-}\pi^{+}]_{\mathchar 28931\relax_{c}})$ but is smooth in
$\delta m$.
For both the $\Xi_{cc}^{+}$ search and the normalisation mode, $\mathchar
28931\relax_{c}^{+}$ candidates are reconstructed in the final state
$pK^{-}\pi^{+}$. To minimise systematic differences in efficiency between the
signal and normalisation modes, the same trigger requirements are used for
both modes, and those requirements ensure that the event was triggered by the
$\mathchar 28931\relax_{c}^{+}$ candidate and its daughter tracks. First, at
least one of the three $\mathchar 28931\relax_{c}^{+}$ daughter tracks must
correspond to a calorimeter cluster with a measured transverse energy
$\mbox{$E_{\rm T}$}>3500$$\mathrm{\,Me\kern-1.00006ptV}$ in the hardware
trigger. Second, at least one of the three $\mathchar 28931\relax_{c}^{+}$
daughter tracks must be selected by the inclusive software trigger, which
requires that the track have $\mbox{$p_{\rm
T}$}>1700{\mathrm{\,Me\kern-1.00006ptV\\!/}c}$ and $\chi^{2}_{\rm IP}>16$ with
respect to any PV, where $\chi^{2}_{\rm IP}$ is defined as the difference in
$\chi^{2}$ of a given PV reconstructed with and without the considered track.
Third, the $\mathchar 28931\relax_{c}^{+}$ candidate must be reconstructed and
accepted by a dedicated $\mathchar 28931\relax_{c}^{+}\\!\rightarrow
pK^{-}\pi^{+}$ selection algorithm in the software trigger. This algorithm
makes several geometric and kinematic requirements, the most important of
which are as follows. The three daughter tracks are required to have
$\mbox{$p_{\rm T}$}>500$${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$, to have a
track fit $\chi^{2}/\rm{ndf}<3$, not to originate at a PV ($\chi^{2}_{\rm
IP}>16$), and to meet at a common vertex ($\chi^{2}/\rm{ndf}<15$, where
$\rm{ndf}$ is the number of degrees of freedom). The $\mathchar
28931\relax_{c}^{+}$ candidate formed from the three tracks is required to
have $\mbox{$p_{\rm T}$}>2500$${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$, to
lie within the mass window $2150<m([pK^{-}\pi^{+}]_{\mathchar
28931\relax_{c}})<2430$${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$, to be
significantly displaced from the PV (vertex separation $\chi^{2}>16$), and to
point back towards the PV (momentum and displacement vectors within
$1^{\circ}$). The software trigger also requires that the proton candidate be
inconsistent with the pion and kaon mass hypotheses. The $\mathchar
28931\relax_{c}^{+}$ trigger algorithm was only enabled for part of the data-
taking in 2011, corresponding to an integrated luminosity of
$0.65\mbox{\,fb}^{-1}$.
For events that pass the trigger, the $\mathchar 28931\relax_{c}^{+}$
selection proceeds in a similar fashion to that used in the software trigger:
three charged tracks are required to form a common vertex that is
significantly displaced from the event PV and has invariant mass in the range
$2185<m([pK^{-}\pi^{+}]_{\mathchar
28931\relax_{c}})<2385$${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$. Particle
identification (PID) requirements are imposed on all three tracks to suppress
combinatorial background and mis-identified charm meson decays. The same
$\mathchar 28931\relax_{c}^{+}$ selection is used for the signal and
normalisation modes.
The $\Xi_{cc}^{+}$ candidates are formed by combining a $\mathchar
28931\relax_{c}^{+}$ candidate with two tracks, one identified as a $K^{-}$
and one as a $\pi^{+}$. These three particles are required to form a common
vertex ($\chi^{2}/\rm{ndf}<10$) that is displaced from the PV (vertex
separation $\chi^{2}>16$). The kaon and pion daughter tracks are also required
to not originate at the PV ($\chi^{2}_{\rm IP}>16$) and to have $\mbox{$p_{\rm
T}$}>250$${\mathrm{\,Me\kern-1.00006ptV\\!/}c}$. The $\Xi_{cc}^{+}$ candidate
is required to point back to the PV and to have $\mbox{$p_{\rm
T}$}>2000$${\mathrm{\,Me\kern-1.00006ptV\\!/}c}$.
A multivariate selection is applied only to the signal mode to further improve
the purity. The selector used is an artificial neural network (ANN)
implemented in the TMVA package [38]. The input variables are chosen to have
limited dependence on the $\Xi_{cc}^{+}$ lifetime. To train the selector,
simulated $\Xi_{cc}^{+}$ decays are used as the signal sample and $3.5\%$ of
the candidates from $\delta m$ sidebands of width
200${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ adjacent to the signal region
are used as the background sample. In order to increase the available
statistics, the trigger requirements are relaxed for these samples. In
addition to the training samples, disjoint test samples of equal size are
taken from the same sources. After training, the response distribution of the
ANN is compared between the training and test samples. Good agreement is found
for both signal and background, with Kolmogorov-Smirnov test $p$-values of 80%
and 65%, respectively. A selection cut on the ANN response is applied to the
data used in the $\Xi_{cc}^{+}$ search. In the test samples, the efficiency of
this requirement is $55.7\%$ for signal and $4.2\%$ for background.
The selection has limited efficiency for short-lived $\Xi_{cc}^{+}$. This is
principally due to the requirements that the $\Xi_{cc}^{+}$ decay vertex be
significantly displaced from the PV, and that the $\Xi_{cc}^{+}$ daughter kaon
and pion have a significant impact parameter with respect to the PV. As a
consequence, the analysis is insensitive to $\Xi_{c}$ resonances that decay
strongly to the same final state, notably the $\Xi_{c}(2980)^{+}$,
$\Xi_{c}(3055)^{+}$, and $\Xi_{c}(3080)^{+}$ [20, 39].
## 4 Yield measurements
To determine the $\mathchar 28931\relax_{c}^{+}$ yield, $N_{\text{norm}}$, a
fit is performed to the $pK^{-}\pi^{+}$ mass spectrum. The signal shape is
described as the sum of two Gaussian functions with a common mean, and the
background is parameterised as a first-order polynomial. The fit is shown in
Fig. 1. The selected $\mathchar 28931\relax_{c}^{+}$ yield in the full
$0.65\mbox{\,fb}^{-1}$ sample is $N_{\text{norm}}=(818\pm 7)\times 10^{3}$,
with an invariant mass resolution of around
6${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$.
Figure 1: Invariant mass spectrum of $\mathchar
28931\relax_{c}^{+}\\!\rightarrow pK^{-}\pi^{+}$ candidates for 5$\%$ of the
data, with events chosen at random during preselection (due to bandwidth
limits for the normalisation mode). The dashed line shows the fitted
background contribution, and the solid line the sum of $\mathchar
28931\relax_{c}^{+}$ signal and background.
The $\Xi_{cc}^{+}$ signal yield is measured from the $\delta m$ distribution
under a series of different mass hypotheses. Although the methods used are
designed not to require detailed knowledge of the signal shape, it is
necessary to know the resolution with sufficient precision to define a signal
window. Since the $\Xi_{cc}^{+}$ yield may be small, its resolution cannot be
measured from data and is instead estimated with a sample of simulated events,
shown in Fig. 2. Fitting the candidates with the sum of two Gaussian
functions, the resolution is found to be approximately
4.4${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$.
Figure 2: The distribution of the invariant mass difference $\delta m$,
defined in Eq. 3, for simulated $\Xi_{cc}^{+}$ events with a $\Xi_{cc}^{+}$
mass of 3500${\mathrm{\,Me\kern-0.90005ptV\\!/}c^{2}}$. The solid line shows
the fitted signal shape. In order to increase the available statistics, the
trigger and ANN requirements are not applied in this plot.
Two complementary procedures are used to estimate the signal yield given a
mass hypothesis $\delta m_{0}$. Both follow the same general approach, but use
different methods to estimate the background. In both cases, a narrow signal
window is defined as $2273<m([pK^{-}\pi^{+}]_{\mathchar
28931\relax_{c}})<2303$${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ and
$\left|\delta m-\delta
m_{0}\right|<10$${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$, and the number of
candidates inside that window is taken as $N_{S+B}$. Candidates outside the
narrow window are used to estimate the expected background $N_{B}$ inside the
window. The signal yield is then $N_{S}=N_{S+B}-N_{B}$. This avoids any need
to model the signal shape beyond an efficiency correction for the estimated
signal fraction lost outside the window of width
20${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$.
The first method is an analytic, two-dimensional sideband subtraction in
$m([pK^{-}\pi^{+}]_{\mathchar 28931\relax_{c}})$ and $\delta m$. A two-
dimensional region of width 80${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ in
$m([pK^{-}\pi^{+}]_{\mathchar 28931\relax_{c}})$ and width
200${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ in $\delta m$ is centred around
the narrow signal window. A $5\times 5$ array of non-overlapping bins is
defined within this region, with the central bin identical to the narrow
signal window. It is assumed that the background consists of a combinatorial
component, which is described by a two-dimensional quadratic function, and a
$\mathchar 28931\relax_{c}^{+}$ component, which is described by the product
of a signal peak in $m([pK^{-}\pi^{+}]_{\mathchar 28931\relax_{c}})$ and a
quadratic function in $\delta m$. Under this assumption, the background
distribution can be fully determined from the 24 sideband bins and hence its
integral within the signal box calculated. In this way the value of $N_{B}$
and the associated statistical uncertainty are determined. This method has the
advantage that it requires only minor assumptions about the background
distribution, given that part of that distribution cannot be studied prior to
unblinding. It is adopted as the baseline approach for this reason.
The second method, used as a cross-check, imposes a narrow window on all
candidates of $2273<m([pK^{-}\pi^{+}]_{\mathchar
28931\relax_{c}})<2303$${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ to reduce
the problem to a one-dimensional distribution in $\delta m$. Based on studies
of the $m([pK^{-}\pi^{+}]_{\mathchar 28931\relax_{c}})$ and $\delta m$
sidebands, it is found that the background can be described by a function of
the form
$f(\delta m)=\left\\{\begin{array}[]{ll}\phantom{a}L(\delta
m;\mu,\sigma_{L})&\delta m\leq\mu\\\ aL(\delta m;\mu,\sigma_{R})&\delta
m\geq\mu\\\ \end{array}\right.$ (4)
where $L(\delta m;\mu,\sigma)$ is a Landau distribution, $a$ is chosen such
that $L(\mu;\mu,\sigma_{L})=aL(\mu;\mu,\sigma_{R})$, and $\mu$, $\sigma_{L}$,
and $\sigma_{R}$ are free parameters. The data are fitted with this function
across the full range, $0<\delta
m<1500$${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$, excluding the signal window
of width 20${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$. The fit function is
then integrated across the signal window to give the expected background
$N_{B}$.
## 5 Efficiency ratio
To measure $R$, it is necessary to evaluate the ratio of efficiencies for the
normalisation and signal modes,
$\varepsilon_{\text{norm}}/\varepsilon_{\text{sig}}$. The method used to
evaluate this ratio is described below. The signal efficiency depends upon the
mass and lifetime of the $\Xi_{cc}^{+}$, neither of which is known. To handle
this, simulated events are generated with
$m(\Xi_{cc}^{+})=3500$${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ and
$\tau_{\Xi_{cc}^{+}}=333$$\rm\,fs$ and the efficiency ratio is evaluated at
this working point. The variation of the efficiency ratio as a function of
$\delta m$ and $\tau_{\Xi_{cc}^{+}}$ relative to the working point is then
determined with a reweighting technique as discussed in Sec. 7. The kinematic
distribution of $\Xi_{cc}^{+}$ produced at the LHC is also unknown, but unlike
the mass and lifetime it cannot be described in a model-independent way with a
single additional parameter. Instead, the upper limits are evaluated assuming
the distributions produced by the Genxicc model.
The efficiency ratio may be factorised into several components as
$\frac{\varepsilon_{\text{norm}}}{\varepsilon_{\text{sig}}}=\frac{\varepsilon_{\text{norm}}^{\text{acc}}}{\varepsilon_{\text{sig}}^{\text{acc}}}\,\frac{\varepsilon_{\text{norm}}^{\text{sel}|\text{acc}}}{\varepsilon_{\text{sig}}^{\text{sel}|\text{acc}}}\,\frac{\varepsilon_{\text{norm}}^{\text{PID}|\text{sel}}}{\varepsilon_{\text{sig}}^{\text{PID}|\text{sel}}}\,\frac{1}{\varepsilon_{\text{sig}}^{\text{ANN}|\text{PID}}}\,\frac{\varepsilon_{\text{norm}}^{\text{trig}|\text{PID}}}{\varepsilon_{\text{sig}}^{\text{trig}|\text{ANN}}},$
(5)
where efficiencies are evaluated for the acceptance (acc), the reconstruction
and selection excluding PID and the ANN (sel), the particle identification
cuts (PID), the ANN selector (ANN) for the signal mode only, and the trigger
(trig). Each element is the efficiency relative to all previous steps in the
order given above.
In most cases the individual ratios are evaluated with simulated
$\Xi_{cc}^{+}$ and $\mathchar 28931\relax_{c}^{+}$ decays, taking the fraction
of candidates that passed the requirement in question. However, in some cases
the efficiencies need to be corrected for known differences between simulation
and data. This applies to the efficiencies for tracking, for passing PID
requirements, and for passing the calorimeter hardware trigger. Control
samples of data are used to determine these corrections as a function of track
kinematics and event charged track multiplicity, and the simulated events are
weighted accordingly. The data samples used are
$J/\psi\rightarrow\mu^{+}\mu^{-}$ for the tracking efficiency, and
$D^{*+}\rightarrow D^{0}(\rightarrow K^{-}\pi^{+})\pi^{+}$ and
$\varLambda\rightarrow p\pi^{-}$ for both the PID and calorimeter hardware
trigger requirements. The track multiplicity distribution is taken from data
for the $\mathchar 28931\relax_{c}^{+}$ sample, but for $\Xi_{cc}^{+}$ events
it is not known. It is modelled by taking a sample of events containing a
reconstructed $B^{0}_{s}$ decay, on the grounds that $B^{0}_{s}$ production
also requires two non-light quark-antiquark pairs.
The efficiency ratio obtained at this working point is
$\varepsilon_{\text{norm}}/\varepsilon_{\text{sig}}=20.4$. Together with the
value for $N_{\text{norm}}$ obtained in Sec. 4 and the definition in eq. 2,
this implies the single-event sensitivity $\alpha$ is $2.5\times 10^{-5}$ at
$m(\Xi_{cc}^{+})=3500$${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$,
$\tau_{\Xi_{cc}^{+}}=333$$\rm\,fs$.
## 6 Systematic uncertainties
The statistical uncertainty on the measured signal yield is the dominant
uncertainty in this analysis, and the systematic uncertainties on $\alpha$
have very limited effect on the expected upper limits. As in the previous
section, they will be evaluated at the working point of
$m(\Xi_{cc}^{+})=3500$${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ and
$\tau_{\Xi_{cc}^{+}}=333$$\rm\,fs$, and their variation with mass and lifetime
hypothesis considered separately. Of the systematic uncertainties, the largest
($18.0\%$) is due to the limited sample size of simulated events used to
calculate the efficiency ratio. Beyond this, there are several instances where
the simulation may not describe the signal accurately in data. These are
corrected with control samples of data, with systematic uncertainties,
outlined below, assigned to reflect uncertainties in these corrections.
The IP resolution of tracks in the VELO is found to be worse in data than in
simulated events. To estimate the impact of this effect on the signal
efficiency, a test is performed with simulated events in which the VELO
resolution is artificially degraded to the same level. This is found to change
the efficiency of the reconstruction and non-ANN selection by $6.6\%$, and
that of the ANN by $6.7\%$. Taking these effects to be fully correlated, a
systematic uncertainty of $13.3\%$ is assigned.
A track-by-track correction is applied to the PID efficiency based on control
samples of data. There are several systematic uncertainties associated with
this correction. The first is due to the limited size of the control samples,
notably for high-$p_{\rm T}$ protons from the $\varLambda$ sample. The second
is due to the assumption that the corrections factorise between the tracks,
whereas in practice there are kinematic correlations. The third is due to the
dependence on the event track multiplicity. The fourth is due to limitations
in the method (e.g. the finite kinematic binning used) and is assessed by
applying it to samples of simulated events. The sum in quadrature of the above
gives an uncertainty of $11.8\%$.
Systematic uncertainties also arise from the tracking efficiency ($4.7\%$) and
from the hardware trigger efficiency ($3.3\%$). Additional systematic
uncertainties associated with candidate multiplicity, yield measurement, and
the decay model of $\Xi_{cc}^{+}\\!\rightarrow\mathchar
28931\relax_{c}^{+}K^{-}\pi^{+}$, which may proceed through intermediate
resonances, were considered but found to be negligible in comparison with the
total systematic uncertainty. The systematic uncertainties are summarised in
Table 1. Taking their sum in quadrature, the total systematic uncertainty is
$26\%$.
Table 1: Systematic uncertainties on the single-event sensitivity $\alpha$. Source | Size
---|---
Simulated sample size | $18.0\%$
IP resolution | $13.3\%$
PID calibration | $11.8\%$
Tracking efficiency | $4.7\%$
Trigger efficiency | $3.3\%$
Total uncertainty | $26.0\%$
## 7 Variation of efficiency with mass and lifetime
The efficiency to trigger on, reconstruct, and select $\Xi_{cc}^{+}$
candidates has a strong dependence upon the $\Xi_{cc}^{+}$ lifetime. The
efficiency also depends upon the $\Xi_{cc}^{+}$ mass, since this affects the
opening angles and the $p_{\rm T}$ of the daughters.
The simulated $\Xi_{cc}^{+}$ events are generated with a proper decay time
distribution given by an exponential function of average lifetime
$\tau_{\Xi_{cc}^{+}}=333$$\rm\,fs$. To test other lifetime hypotheses, the
simulated events are reweighted to follow a different exponential distribution
and the efficiency is recomputed. Most systematic uncertainties are
unaffected, but those associated with the limited simulated sample size and
with the hardware trigger efficiency increase at shorter lifetimes (the latter
due to kinematic correlations rather than direct dependence on the decay time
distribution). The values and uncertainties of the single-event sensitivity
$\alpha$ are given for several lifetime hypotheses in Table 2.
Table 2: Single-event sensitivity $\alpha$ for different lifetime hypotheses $\tau$, assuming $m(\Xi_{cc}^{+})=3500$${\mathrm{\,Me\kern-0.90005ptV\\!/}c^{2}}$. The uncertainties quoted include statistical and systematic effects, and are correlated between different lifetime hypotheses. $\tau$ | $\alpha$ ($\times 10^{-5}$)
---|---
100$\rm\,fs$ | $63$ ± | $31$
150$\rm\,fs$ | $15$ ± | $5$
250$\rm\,fs$ | $4.1$ ± | $1.1$
333$\rm\,fs$ | $2.5$ ± | $0.6$
400$\rm\,fs$ | $1.9$ ± | $0.5$
To assess the effect of varying the $\Xi_{cc}^{+}$ mass hypothesis, large
samples of simulated events are generated for two other mass hypotheses,
$m(\Xi_{cc}^{+})=3300$${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ and
3700${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$, without running the Geant4
detector simulation. Two tests are carried out with these samples. First, the
detector acceptance efficiency is recalculated. Second, the $p_{\rm T}$
distributions of the three daughters of the $\Xi_{cc}^{+}$ in the main
$m(\Xi_{cc}^{+})=3500$${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ sample are
reweighted to match those seen at the other mass hypotheses and the remainder
of the efficiency is recalculated. In both cases the systematic uncertainties
are also recalculated, though very little change is found. Significant
variations in individual components of the efficiency are seen—notably in the
acceptance, reconstruction, non-ANN selection, and hardware trigger
efficiencies—but when combined cancel almost entirely. This is shown in Table
3, which gives the value of $\alpha$ including the mass-dependent effects
discussed above but excluding the correction for the efficiency of the $\delta
m$ signal window described in Sec. 4 ($\alpha_{u}$), the correction for the
variation in resolution, and the combined value of $\alpha$. Because the
variation of $\alpha_{u}$ with mass is extremely small, a simple first-order
correction is sufficient. A straight line is fitted to the three points in the
table and used to interpolate the fractional variation in $\alpha_{u}$ between
the mass hypotheses. The resolution correction is then applied separately. Due
to the smallness of the mass-dependence, correlations between variation with
mass and with lifetime are neglected.
Table 3: Variation in single-event sensitivity for different mass hypotheses $m(\Xi_{cc}^{+})$, assuming $\tau=333$$\rm\,fs$. The uncertainties quoted include statistical and systematic effects, and are correlated between different mass hypotheses. The variation is shown separately for all effects other than the efficiency of the $\delta m$ window ($\alpha_{u}$), for the correction due to the mass-dependent resolution, and for the combination ($\alpha$). $m(\Xi_{cc}^{+})$ | $\alpha_{u}$ ($\times 10^{-5}$) | Resolution correction | $\alpha$ ($\times 10^{-5}$)
---|---|---|---
3300${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ | $2.29$ ± | $0.61$ | 0.992 | $2.30$ ± | $0.62$
3500${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ | $2.38$ ± | $0.62$ | 0.957 | $2.49$ ± | $0.65$
3700${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ | $2.36$ ± | $0.63$ | 0.903 | $2.61$ ± | $0.70$
As explained in Sec. 1, the value of $R$ at LHCb is not well known but is
expected to be of the order $10^{-5}$ to $10^{-4}$, while the SELEX
observation corresponds to $R=9\%$. Table 4 shows the expected signal yield,
calculated according to eq. 1, for various values of $R$ and lifetime
hypotheses. From studies of the sidebands in $m([pK^{-}\pi^{+}]_{\mathchar
28931\relax_{c}})$ and $\delta m$, the expected background in the narrow
signal window is between 10 and 20 events. Thus, no significant signal excess
is expected if the value of $R$ at LHCb is in the range suggested by theory.
However, if production is greatly enhanced for baryon-baryon collisions at
high rapidity, as reported at SELEX, a large signal may be visible. The
procedure for determining the significance of a signal, or for establishing
limits on $R$, is discussed in the following section.
Table 4: Expected value of the signal yield $N_{\text{sig}}$ for different values of $R$ and lifetime hypotheses, assuming $m(\Xi_{cc}^{+})=3500$${\mathrm{\,Me\kern-0.90005ptV\\!/}c^{2}}$. The uncertainties quoted are due to the systematic uncertainty on $\alpha$. $\tau$ | $R=9\%$ | $R=10^{-4}$ | $R=10^{-5}$
---|---|---|---
100$\rm\,fs$ | $140$ ± | $70$ | $0.2$ ± | $0.1$ | $0.02$ ± | $0.01$
150$\rm\,fs$ | $600$ ± | $200$ | $0.7$ ± | $0.2$ | $0.07$ ± | $0.02$
250$\rm\,fs$ | $2200$ ± | $600$ | $2.4$ ± | $0.7$ | $0.24$ ± | $0.07$
333$\rm\,fs$ | $3600$ ± | $900$ | $4.0$ ± | $1.0$ | $0.40$ ± | $0.10$
400$\rm\,fs$ | $4800$ ± | $1200$ | $5.3$ ± | $1.4$ | $0.53$ ± | $0.14$
## 8 Tests for statistical significance and upper limit calculation
Since $m(\Xi_{cc}^{+})$ is a priori unknown, tests for the presence of a
signal are carried out at numerous mass hypotheses, between $\delta
m=380$${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ and $\delta
m=880$${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ inclusive in
1${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ steps for a total of 501 tests.
For a given value of $\delta m$, the signal and background yields and their
associated statistical uncertainties are estimated as described in Sec. 4.
From these the local significance $\mathscr{S}(\delta m)$ is calculated, where
$\mathscr{S}(\delta m)$ is defined as
$\mathscr{S}(\delta
m)\equiv\frac{N_{S+B}-N_{B}}{\sqrt{\sigma_{S+B}^{2}+\sigma_{B}^{2}}}$ (6)
and $\sigma_{S+B}$ and $\sigma_{B}$ are the estimated statistical
uncertainties on the yield in the signal window and on the expected
background, respectively. Since multiple points are sampled, the look
elsewhere effect (LEE) [40] must be taken into account. The procedure used is
to generate a large number of pseudo-experiments containing only background
events, with the amount and distribution of background chosen to match the
data (as estimated from sidebands). For each pseudo-experiment, the full
analysis procedure is applied in the same way as for data, and the local
significance is measured at all 501 values of $\delta m$. The LEE-corrected
$p$-value for a given $\mathscr{S}$ is then taken to be the fraction of the
pseudo-experiments that contain an equal or larger local significance at any
point in the $\delta m$ range.
The procedure established before unblinding is that if no signal with an LEE-
corrected significance of at least $3\sigma$ is seen, upper limits on $R$ will
be quoted. The $CL_{s}$ method [41, 42] is applied to determine upper limits
on $R$ for a particular $\delta m$ and lifetime hypothesis, given the observed
yield $N_{S+B}$ and expected background $N_{B}$ in the signal window obtained
as described in Sec. 4. The statistical uncertainty on $N_{B}$ and systematic
uncertainties on $\alpha$ are taken into account. The 95% CL upper limit is
then taken as the value of $R$ for which $CL_{s}=0.05$. Upper limits are
calculated at each of the 501 $\delta m$ hypotheses, and for five lifetime
hypotheses (100, 150, 250, 333, 400$\rm\,fs$).
## 9 Results
The $\delta m$ spectrum in data is shown in Fig. 3, and the estimated signal
yield in Fig. 4. No clear signal is found with either background subtraction
method. In both cases the largest local significance occurs at $\delta
m=513$${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$, with $\mathscr{S}=1.5\sigma$
in the baseline method and $\mathscr{S}=2.2\sigma$ in the cross-check.
Applying the LEE correction described in Sec. 8, these correspond to
$p$-values of 99% and 53%, respectively. Thus, with no significant excess
found above background, upper limits are set on $R$ at the 95% CL, shown in
Fig. 5 for the first method. These limits are tabulated in Table 9 for blocks
of $\delta m$ and the five lifetime hypotheses. The blocks are
50${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ wide, and for each block the
largest (worst) upper limit seen for a $\delta m$ point in that block is
given. Similarly, the largest upper limit seen in the entire
500${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ mass range is also given. A
strong dependence in sensitivity on the lifetime hypothesis is seen.
Figure 3: Spectrum of $\delta m$ requiring $2273<m([pK^{-}\pi^{+}]_{\mathchar
28931\relax_{c}})<2303$${\mathrm{\,Me\kern-0.90005ptV\\!/}c^{2}}$. Both plots
show the same data sample, but with different $\delta m$ ranges and binnings.
The wide signal region is shown in the right plot and indicated by the dotted
vertical lines in the left plot.
Figure 4: Measured signal yields as a function of $\delta m$. The upper two
plots show the estimated signal yield as a dark line and the $\pm 1\sigma$
statistical error bands as light grey lines for (upper left) the baseline
method and (upper right) the cross-check method. The central values of the two
methods are compared in the lower plot and found to agree well. Figure 5:
Upper limits on $R$ at the 95% CL as a function of $\delta m$, for five
$\Xi_{cc}^{+}$ lifetime hypotheses. Table 5: Largest values of the upper
limits (UL) on $R$ at the 95% CL in blocks of $\delta m$ for a range of
lifetime hypotheses, given in units of $10^{-3}$. The largest values across
the entire 500${\mathrm{\,Me\kern-0.90005ptV\\!/}c^{2}}$ range are also shown.
| $R$, largest 95% CL UL in range $\times 10^{3}$
---|---
$\delta m$ (MeV$/c^{2}$) | 100$\rm\,fs$ | 150$\rm\,fs$ | 250$\rm\,fs$ | 333$\rm\,fs$ | 400$\rm\,fs$
380–429 | 12.6 | 2.7 | 0.73 | 0.43 | 0.33
430–479 | 11.2 | 2.4 | 0.65 | 0.39 | 0.29
480–529 | 14.8 | 3.2 | 0.85 | 0.51 | 0.39
530–579 | 10.7 | 2.3 | 0.63 | 0.38 | 0.29
580–629 | 10.9 | 2.3 | 0.63 | 0.38 | 0.29
630–679 | 14.2 | 3.0 | 0.81 | 0.49 | 0.37
680–729 | 9.5 | 2.0 | 0.56 | 0.33 | 0.25
730–779 | 10.8 | 2.3 | 0.63 | 0.37 | 0.28
780–829 | 12.8 | 2.8 | 0.74 | 0.45 | 0.34
830–880 | 12.2 | 2.6 | 0.70 | 0.42 | 0.32
380–880 | 14.8 | 3.2 | 0.85 | 0.51 | 0.39
The decay $\Xi_{cc}^{+}\\!\rightarrow\mathchar
28931\relax_{c}^{+}K^{-}\pi^{+}$ may proceed through an intermediate
$\Sigma_{c}^{++}$ resonance. Such decays would be included in the yields and
limits already shown. Nonetheless, further checks are made with an explicit
requirement that the $\mathchar 28931\relax_{c}^{+}\pi^{+}$ invariant mass be
consistent with that of a $\Sigma_{c}^{++}$, since this substantially reduces
the combinatorial background. For $\Sigma_{c}(2455)^{++}$ and
$\Sigma_{c}(2520)^{++}$, the mass offsets $\left[m([pK^{-}\pi^{+}]_{\mathchar
28931\relax_{c}}\pi^{+})-m([pK^{-}\pi^{+}]_{\mathchar
28931\relax_{c}})\right]$ are required to be within
4${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ and
15${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ of the world-average value,
respectively. The resulting $\delta m$ spectra are shown in Fig. 6. No
statistically significant excess is present.
Figure 6: Mass difference spectrum requiring
$2273<m([pK^{-}\pi^{+}]_{\mathchar
28931\relax_{c}})<2303$${\mathrm{\,Me\kern-0.90005ptV\\!/}c^{2}}$. Candidates
are also required to be consistent with (left) an intermediate
$\Sigma_{c}(2455)^{++}$, (right) an intermediate $\Sigma_{c}(2520)^{++}$.
## 10 Conclusions
A search for the decay $\Xi_{cc}^{+}\\!\rightarrow\mathchar
28931\relax_{c}^{+}K^{-}\pi^{+}$ is performed at LHCb with a data sample of
$pp$ collisions, corresponding to an integrated luminosity of
0.65$\mbox{\,fb}^{-1}$, recorded at a centre-of-mass energy of
7$\mathrm{\,Te\kern-1.00006ptV}$. No significant signal is found. Upper limits
on the $\Xi_{cc}^{+}$ cross-section times branching fraction relative to the
$\mathchar 28931\relax_{c}^{+}$ cross-section are obtained for a range of mass
and lifetime hypotheses, assuming that the kinematic distributions of the
$\Xi_{cc}^{+}$ follow those of the Genxicc model. The upper limit depends
strongly on the lifetime, varying from $1.5\times 10^{-2}$ for 100$\rm\,fs$ to
$3.9\times 10^{-4}$ for 400$\rm\,fs$. These limits are significantly below the
value of $R$ found at SELEX. This may be explained by the different production
environment, or if the $\Xi_{cc}^{+}$ lifetime is indeed very short ($\ll
100$$\rm\,fs$). Future searches at LHCb with improved trigger conditions,
additional $\Xi_{cc}$ decay modes, and larger data samples should improve the
sensitivity significantly, especially at short lifetimes.
## Acknowledgements
We express our gratitude to our colleagues in the CERN accelerator departments
for the excellent performance of the LHC. We thank the technical and
administrative staff at the LHCb institutes. We acknowledge support from CERN
and from the national agencies: CAPES, CNPq, FAPERJ and FINEP (Brazil); NSFC
(China); CNRS/IN2P3 and Region Auvergne (France); BMBF, DFG, HGF and MPG
(Germany); SFI (Ireland); INFN (Italy); FOM and NWO (The Netherlands); SCSR
(Poland); MEN/IFA (Romania); MinES, Rosatom, RFBR and NRC “Kurchatov
Institute” (Russia); MinECo, XuntaGal and GENCAT (Spain); SNSF and SER
(Switzerland); NAS Ukraine (Ukraine); STFC (United Kingdom); NSF (USA). We
also acknowledge the support received from the ERC under FP7. The Tier1
computing centres are supported by IN2P3 (France), KIT and BMBF (Germany),
INFN (Italy), NWO and SURF (The Netherlands), PIC (Spain), GridPP (United
Kingdom). We are thankful for the computing resources put at our disposal by
Yandex LLC (Russia), as well as to the communities behind the multiple open
source software packages that we depend on.
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arxiv-papers
| 2013-10-09T16:31:01 |
2024-09-04T02:49:52.211946
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "LHCb collaboration: R. Aaij, B. Adeva, M. Adinolfi, C. Adrover, A.\n Affolder, Z. Ajaltouni, J. Albrecht, F. Alessio, M. Alexander, S. Ali, G.\n Alkhazov, P. Alvarez Cartelle, A.A. Alves Jr, S. Amato, S. Amerio, Y. Amhis,\n L. Anderlini, J. Anderson, R. Andreassen, J.E. Andrews, R.B. Appleby, O.\n Aquines Gutierrez, F. Archilli, A. Artamonov, M. Artuso, E. Aslanides, G.\n Auriemma, M. Baalouch, S. Bachmann, J.J. Back, A. Badalov, C. Baesso, V.\n Balagura, W. Baldini, R.J. Barlow, C. Barschel, S. Barsuk, W. Barter, Th.\n Bauer, A. Bay, J. Beddow, F. Bedeschi, I. Bediaga, S. Belogurov, K. Belous,\n I. Belyaev, E. Ben-Haim, G. Bencivenni, S. Benson, J. Benton, A. Berezhnoy,\n R. Bernet, M.-O. Bettler, M. van Beuzekom, A. Bien, S. Bifani, T. Bird, A.\n Bizzeti, P.M. Bj{\\o}rnstad, T. Blake, F. Blanc, J. Blouw, S. Blusk, V. Bocci,\n A. Bondar, N. Bondar, W. Bonivento, S. Borghi, A. Borgia, T.J.V. Bowcock, E.\n Bowen, C. Bozzi, T. Brambach, J. van den Brand, J. Bressieux, D. Brett, M.\n Britsch, T. Britton, N.H. Brook, H. Brown, A. Bursche, G. Busetto, J.\n Buytaert, S. Cadeddu, O. Callot, M. Calvi, M. Calvo Gomez, A. Camboni, P.\n Campana, D. Campora Perez, A. Carbone, G. Carboni, R. Cardinale, A. Cardini,\n H. Carranza-Mejia, L. Carson, K. Carvalho Akiba, G. Casse, L. Castillo\n Garcia, M. Cattaneo, Ch. Cauet, R. Cenci, M. Charles, Ph. Charpentier, S.-F.\n Cheung, N. Chiapolini, M. Chrzaszcz, K. Ciba, X. Cid Vidal, G. Ciezarek,\n P.E.L. Clarke, M. Clemencic, H.V. Cliff, J. Closier, C. Coca, V. Coco, J.\n Cogan, E. Cogneras, P. Collins, A. Comerma-Montells, A. Contu, A. Cook, M.\n Coombes, S. Coquereau, G. Corti, B. Couturier, G.A. Cowan, D.C. Craik, M.\n Cruz Torres, S. Cunliffe, R. Currie, C. D'Ambrosio, P. David, P.N.Y. David,\n A. Davis, I. De Bonis, K. De Bruyn, S. De Capua, M. De Cian, J.M. De Miranda,\n L. De Paula, W. De Silva, P. De Simone, D. Decamp, M. Deckenhoff, L. Del\n Buono, N. D\\'el\\'eage, D. Derkach, O. Deschamps, F. Dettori, A. Di Canto, H.\n Dijkstra, M. Dogaru, S. Donleavy, F. Dordei, A. Dosil Su\\'arez, D. Dossett,\n A. Dovbnya, F. Dupertuis, P. Durante, R. Dzhelyadin, A. Dziurda, A. Dzyuba,\n S. Easo, U. Egede, V. Egorychev, S. Eidelman, D. van Eijk, S. Eisenhardt, U.\n Eitschberger, R. Ekelhof, L. Eklund, I. El Rifai, Ch. Elsasser, A. Falabella,\n C. F\\\"arber, C. Farinelli, S. Farry, D. Ferguson, V. Fernandez Albor, F.\n Ferreira Rodrigues, M. Ferro-Luzzi, S. Filippov, M. Fiore, C. Fitzpatrick, M.\n Fontana, F. Fontanelli, R. Forty, O. Francisco, M. Frank, C. Frei, M.\n Frosini, E. Furfaro, A. Gallas Torreira, D. Galli, M. Gandelman, P. Gandini,\n Y. Gao, J. Garofoli, P. Garosi, J. Garra Tico, L. Garrido, C. Gaspar, R.\n Gauld, E. Gersabeck, M. Gersabeck, T. Gershon, Ph. Ghez, V. Gibson, L.\n Giubega, V.V. Gligorov, C. G\\\"obel, D. Golubkov, A. Golutvin, A. Gomes, P.\n Gorbounov, H. Gordon, M. Grabalosa G\\'andara, R. Graciani Diaz, L.A. Granado\n Cardoso, E. Graug\\'es, G. Graziani, A. Grecu, E. Greening, S. Gregson, P.\n Griffith, L. Grillo, O. Gr\\\"unberg, B. Gui, E. Gushchin, Yu. Guz, T. Gys, C.\n Hadjivasiliou, G. Haefeli, C. Haen, S.C. Haines, S. Hall, B. Hamilton, T.\n Hampson, S. Hansmann-Menzemer, N. Harnew, S.T. Harnew, J. Harrison, T.\n Hartmann, J. He, T. Head, V. Heijne, K. Hennessy, P. Henrard, J.A. Hernando\n Morata, E. van Herwijnen, M. He\\ss, A. Hicheur, E. Hicks, D. Hill, M.\n Hoballah, C. Hombach, W. Hulsbergen, P. Hunt, T. Huse, N. Hussain, D.\n Hutchcroft, D. Hynds, V. Iakovenko, M. Idzik, P. Ilten, R. Jacobsson, A.\n Jaeger, E. Jans, P. Jaton, A. Jawahery, F. Jing, M. John, D. Johnson, C.R.\n Jones, C. Joram, B. Jost, M. Kaballo, S. Kandybei, W. Kanso, M. Karacson,\n T.M. Karbach, I.R. Kenyon, T. Ketel, B. Khanji, O. Kochebina, I. Komarov,\n R.F. Koopman, P. Koppenburg, M. Korolev, A. Kozlinskiy, L. Kravchuk, K.\n Kreplin, M. Kreps, G. Krocker, P. Krokovny, F. Kruse, M. Kucharczyk, V.\n Kudryavtsev, K. Kurek, T. Kvaratskheliya, V.N. La Thi, D. Lacarrere, G.\n Lafferty, A. Lai, D. Lambert, R.W. Lambert, E. Lanciotti, G. Lanfranchi, C.\n Langenbruch, T. Latham, C. Lazzeroni, R. Le Gac, J. van Leerdam, J.-P. Lees,\n R. Lef\\`evre, A. Leflat, J. Lefran\\c{c}ois, S. Leo, O. Leroy, T. Lesiak, B.\n Leverington, Y. Li, L. Li Gioi, M. Liles, R. Lindner, C. Linn, B. Liu, G.\n Liu, S. Lohn, I. Longstaff, J.H. Lopes, N. Lopez-March, H. Lu, D. Lucchesi,\n J. Luisier, H. Luo, O. Lupton, F. Machefert, I.V. Machikhiliyan, F. Maciuc,\n O. Maev, S. Malde, G. Manca, G. Mancinelli, J. Maratas, U. Marconi, P.\n Marino, R. M\\\"arki, J. Marks, G. Martellotti, A. Martens, A. Mart\\'in\n S\\'anchez, M. Martinelli, D. Martinez Santos, D. Martins Tostes, A. Martynov,\n A. Massafferri, R. Matev, Z. Mathe, C. Matteuzzi, E. Maurice, A. Mazurov, J.\n McCarthy, A. McNab, R. McNulty, B. McSkelly, B. Meadows, F. Meier, M.\n Meissner, M. Merk, D.A. Milanes, M.-N. Minard, J. Molina Rodriguez, S.\n Monteil, D. Moran, P. Morawski, A. Mord\\`a, M.J. Morello, R. Mountain, I.\n Mous, F. Muheim, K. M\\\"uller, R. Muresan, B. Muryn, B. Muster, P. Naik, T.\n Nakada, R. Nandakumar, I. Nasteva, M. Needham, S. Neubert, N. Neufeld, A.D.\n Nguyen, T.D. Nguyen, C. Nguyen-Mau, M. Nicol, V. Niess, R. Niet, N. Nikitin,\n T. Nikodem, A. Nomerotski, A. Novoselov, A. Oblakowska-Mucha, V. Obraztsov,\n S. Oggero, S. Ogilvy, O. Okhrimenko, R. Oldeman, M. Orlandea, J.M. Otalora\n Goicochea, P. Owen, A. Oyanguren, B.K. Pal, A. Palano, M. Palutan, J. Panman,\n A. Papanestis, M. Pappagallo, C. Parkes, C.J. Parkinson, G. Passaleva, G.D.\n Patel, M. Patel, G.N. Patrick, C. Patrignani, C. Pavel-Nicorescu, A. Pazos\n Alvarez, A. Pearce, A. Pellegrino, G. Penso, M. Pepe Altarelli, S. Perazzini,\n E. Perez Trigo, A. P\\'erez-Calero Yzquierdo, P. Perret, M. Perrin-Terrin, L.\n Pescatore, E. Pesen, G. Pessina, K. Petridis, A. Petrolini, A. Phan, E.\n Picatoste Olloqui, B. Pietrzyk, T. Pila\\v{r}, D. Pinci, S. Playfer, M. Plo\n Casasus, F. Polci, G. Polok, A. Poluektov, E. Polycarpo, A. Popov, D. Popov,\n B. Popovici, C. Potterat, A. Powell, J. Prisciandaro, A. Pritchard, C.\n Prouve, V. Pugatch, A. Puig Navarro, G. Punzi, W. Qian, B. Rachwal, J.H.\n Rademacker, B. Rakotomiaramanana, M.S. Rangel, I. Raniuk, N. Rauschmayr, G.\n Raven, S. Redford, S. Reichert, M.M. Reid, A.C. dos Reis, S. Ricciardi, A.\n Richards, K. Rinnert, V. Rives Molina, D.A. Roa Romero, P. Robbe, D.A.\n Roberts, A.B. Rodrigues, E. Rodrigues, P. Rodriguez Perez, S. Roiser, V.\n Romanovsky, A. Romero Vidal, M. Rotondo, J. Rouvinet, T. Ruf, F. Ruffini, H.\n Ruiz, P. Ruiz Valls, G. Sabatino, J.J. Saborido Silva, N. Sagidova, P. Sail,\n B. Saitta, V. Salustino Guimaraes, B. Sanmartin Sedes, R. Santacesaria, C.\n Santamarina Rios, E. Santovetti, M. Sapunov, A. Sarti, C. Satriano, A. Satta,\n M. Savrie, D. Savrina, M. Schiller, H. Schindler, M. Schlupp, M. Schmelling,\n B. Schmidt, O. Schneider, A. Schopper, M.-H. Schune, R. Schwemmer, B.\n Sciascia, A. Sciubba, M. Seco, A. Semennikov, K. Senderowska, I. Sepp, N.\n Serra, J. Serrano, P. Seyfert, M. Shapkin, I. Shapoval, Y. Shcheglov, T.\n Shears, L. Shekhtman, O. Shevchenko, V. Shevchenko, A. Shires, R. Silva\n Coutinho, M. Sirendi, N. Skidmore, T. Skwarnicki, N.A. Smith, E. Smith, E.\n Smith, J. Smith, M. Smith, M.D. Sokoloff, F.J.P. Soler, F. Soomro, D. Souza,\n B. Souza De Paula, B. Spaan, A. Sparkes, P. Spradlin, F. Stagni, S. Stahl, O.\n Steinkamp, S. Stevenson, S. Stoica, S. Stone, B. Storaci, M. Straticiuc, U.\n Straumann, V.K. Subbiah, L. Sun, W. Sutcliffe, S. Swientek, V. Syropoulos, M.\n Szczekowski, P. Szczypka, D. Szilard, T. Szumlak, S. T'Jampens, M. Teklishyn,\n E. Teodorescu, F. Teubert, C. Thomas, E. Thomas, J. van Tilburg, V.\n Tisserand, M. Tobin, S. Tolk, D. Tonelli, S. Topp-Joergensen, N. Torr, E.\n Tournefier, S. Tourneur, M.T. Tran, M. Tresch, A. Tsaregorodtsev, P.\n Tsopelas, N. Tuning, M. Ubeda Garcia, A. Ukleja, A. Ustyuzhanin, U. Uwer, V.\n Vagnoni, G. Valenti, A. Vallier, R. Vazquez Gomez, P. Vazquez Regueiro, C.\n V\\'azquez Sierra, S. Vecchi, J.J. Velthuis, M. Veltri, G. Veneziano, M.\n Vesterinen, B. Viaud, D. Vieira, X. Vilasis-Cardona, A. Vollhardt, D.\n Volyanskyy, D. Voong, A. Vorobyev, V. Vorobyev, C. Vo\\ss, H. Voss, R. Waldi,\n C. Wallace, R. Wallace, S. Wandernoth, J. Wang, D.R. Ward, N.K. Watson, A.D.\n Webber, D. Websdale, M. Whitehead, J. Wicht, J. Wiechczynski, D. Wiedner, L.\n Wiggers, G. Wilkinson, M.P. Williams, M. Williams, F.F. Wilson, J. Wimberley,\n J. Wishahi, W. Wislicki, M. Witek, G. Wormser, S.A. Wotton, S. Wright, S. Wu,\n K. Wyllie, Y. Xie, Z. Xing, Z. Yang, X. Yuan, O. Yushchenko, M. Zangoli, M.\n Zavertyaev, F. Zhang, L. Zhang, W.C. Zhang, Y. Zhang, A. Zhelezov, A.\n Zhokhov, L. Zhong, A. Zvyagin",
"submitter": "Matthew Charles",
"url": "https://arxiv.org/abs/1310.2538"
}
|
1310.2627
|
# A Sparse and Adaptive Prior for Time-Dependent Model Parameters
Dani Yogatama Bryan R. Routledge Noah A. Smith
_draft in review; do not cite or circulate_ Dani Yogatama
Language Technologies Institute
Carnegie Mellon University
Pittsburgh, PA 15213, USA
[email protected] &Bryan R. Routledge
Tepper School of Business
Carnegie Mellon University
Pittsburgh, PA 15213, USA
[email protected] &Noah A. Smith
Language Technologies Institute
Carnegie Mellon University
Pittsburgh, PA 15213, USA
[email protected]
###### Abstract
We consider the scenario where the parameters of a probabilistic model are
expected to vary over time. We construct a novel prior distribution that
promotes sparsity and adapts the strength of correlation between parameters at
successive timesteps, based on the data. We derive approximate variational
inference procedures for learning and prediction with this prior. We test the
approach on two tasks: forecasting financial quantities from relevant text,
and modeling language contingent on time-varying financial measurements.
## 1 Introduction
When learning from streams of data to make predictions in the future, how
should we handle the timestamp associated with each instance? Ignoring
timestamps and assuming data are i.i.d. is scalable but risks distracting a
model with irrelevant “ancient history.” On the other hand, using only the
most recent portion of the data risks overfitting to current trends and
missing important time-insensitive effects. In this paper, we seek a general
approach to learning model parameters that are overall sparse, but that adapt
to variation in how different effects change over time.
Our approach is a prior over parameters of an exponential family (e.g.,
coefficients in linear or logistic regression). We assume that parameter
values shift at each timestep, with correlation between adjacent timesteps
captured using a multivariate normal distribution whose precision matrix is
restricted to a tridiagonal structure. We (approximately) marginalize the
(co)variance parameters of this normal distribution using a Jeffreys prior,
resulting in a model that allows smooth variation over time while encouraging
overall sparsity in the parameters. (The parameters themselves are not given a
fully Bayesian treatment.)
We demonstrate the usefulness of our model on two tasks, showing gains over
alternative approaches. The first is a text regression problem in which an
economic variable (volatility of returns) is forecast from financial reports
(Kogan et al., 2009). The second forecasts text by constructing a language
model that conditions on highly time-dependent economic variables.
Notation is given in §2. Our prior distribution is presented in §3. We draw
connections to related work in §4. §5 presents our inference algorithm and §6
our experimental results.
## 2 Notation
We assume data of the form $\\{(x_{n},y_{n})\\}_{n=1}^{N}$, where each $x_{n}$
includes a timestamp denoted $t\in\\{1,\ldots,T\\}$.111In this work we assume
timestamps are discretized. The aim is to learn a predictor that maps input
$x_{N+1}$, assumed to be at timestep $T$, to output $y_{N+1}$. In the
probabilistic setting we adopt here, the prediction is MAP inference over r.v.
$Y$ given $X=x$ and a model parameterized by
$\boldsymbol{\beta}\in\mathbb{R}^{I}$. Learning is parameter estimation to
solve:
$\operatorname*{argmax}_{\boldsymbol{\beta}}\log
p(\boldsymbol{\beta})+\overbrace{\sum_{n=1}^{N}\log\underbrace{p(y_{n}\mid
x_{n},\boldsymbol{\beta})}_{\mathrm{link}^{-1}(\boldsymbol{f}(x)^{\top}\boldsymbol{\beta})}}^{L(\boldsymbol{\beta})}$
(1)
The focus of the paper is on the prior distribution $p(\boldsymbol{\beta})$.
Throughout, we will denote the task-specific log-likelihood (second term) by
$L(\boldsymbol{\beta})$ and assume a generalized linear model such that a
feature vector function $\boldsymbol{f}$ maps inputs $x$ into $\mathbb{R}^{I}$
and $\boldsymbol{f}(x)^{\top}\boldsymbol{\beta}$ is “linked” to the
distribution over $Y$ using, e.g., a logit or identity. We will refer to
elements of $\boldsymbol{f}$ as “features” and to $\boldsymbol{\beta}$ as
“coefficients.” We assume $T$ discrete timesteps.
## 3 Time-Series Prior
Our time-series prior draws inspiration from the probabilistic interpretation
of the sparsity-inducing lasso (Tibshirani, 1996) and group lasso (Yuan & Lin,
2007). In non-overlapping group lasso, features are divided into groups, and
the coefficients within each group $m$ are drawn according to:
1. 1.
Variance $\sigma^{2}_{m}\sim$ an exponential distribution.222The exponential
distribution can be replaced by the (improper) Jeffreys prior, although then
the familiar Laplace distribution interpretation no longer holds (Figueiredo,
2002).
2. 2.
$\boldsymbol{\beta}_{m}\sim\mathrm{Normal}(\boldsymbol{0},\sigma^{2}_{m}\mathbf{I})$.
We seek a prior that lets each coefficient vary smoothly over time. A high-
level intuition of our prior is that we create copies of $\boldsymbol{\beta}$,
one at each timestep:
$\langle\boldsymbol{\beta}^{(1)},\boldsymbol{\beta}^{(2)},\ldots,\boldsymbol{\beta}^{(T)}\rangle$.
For each feature $i$, let the sequence
$\langle\beta^{(1)}_{i},\beta^{(2)}_{i},\ldots,\beta^{(T)}_{i}\rangle$ form a
group, denoted $\boldsymbol{\beta}_{i}$. Group lasso does not view
coefficients in a group as explicitly correlated; they are independent given
the variance parameter. Given the sequential structure of
$\boldsymbol{\beta}_{i}$, we replace the covariance matrix
$\sigma^{2}\mathbf{I}$ to capture autocorrelation. Specifically, we assume the
vector $\boldsymbol{\beta}_{i}$ is drawn from a multivariate normal
distribution with mean zero and a $T\times T$ precision matrix
$\mathbf{\Lambda}$ with the following tridiagonal form:333We suppress the
subscript $i$ for this discussion; each feature $i$ has its own
$\mathbf{\Lambda}_{i}$.
$\displaystyle\mathbf{\Lambda}$ $\displaystyle=\frac{1}{\lambda}\mathbf{A}$
$\displaystyle=\frac{1}{\lambda}\left[\begin{array}[]{cccccc}1&\alpha&0&0&\dots\\\
\alpha&1&\alpha&0&\dots\\\ 0&\alpha&1&\alpha&\dots\\\ 0&0&\alpha&1&\dots\\\
\vdots&\vdots&\vdots&\vdots&\ddots\end{array}\right]$ (7)
$\lambda\geq 0$ is a scalar multiplier whose role is to control sparsity in
the coefficients, while $\alpha$ dictates the degree of correlation between
coefficients in adjacent timesteps (autocorrelation). Importantly, $\alpha$
and $\lambda$ (and hence $\mathbf{A}$ and $\mathbf{\Lambda}$) are allowed to
be different for each group $i$.
We need to ensure that $\mathbf{A}$ is positive definite. Fortunately, it is
easy to show that for $\alpha\in(-0.5,0.5)$, the resulting $\mathbf{A}$ is
positive definite.
###### Proof sketch.
To show this, since $\mathbf{A}$ is a symmetric matrix, we verify that each of
its principal minors have strictly positive determinants. The principal minors
of $\mathbf{A}$ are uniform tridiagonal symmetric matrices, and the
determinant of a uniform tridiagonal $N\times N$ matrix can be written as
$\prod_{n=1}^{N}\left\\{1+2\alpha\cos\left(\frac{(n+1)\pi}{N+1}\right)\right\\}$
(see, e.g., Volpi (2003) for the proof). Since $\cos(x)\in[-1,1]$, if
$\alpha\in(-0.5,0.5)$, the determinant is always positive. Therefore,
$\mathbf{A}$ is always p.d. for $\alpha\in(-0.5,0.5)$. ∎
### 3.1 Generative Model
Our generative model for the group of coefficients
$\boldsymbol{\beta}_{i}=\langle\beta^{(1)}_{i},\beta^{(2)}_{i},\ldots,\beta^{(T)}_{i}\rangle$
is given by:
1. 1.
$\lambda_{i}\sim$ an improper Jeffreys prior
($p(\lambda)\propto\lambda^{-1}$).
2. 2.
$\alpha_{i}\sim$ a truncated exponential prior with parameter $\tau$. This
distribution forces $\alpha_{i}$ to fall in $(-C,0]$, so that $\mathbf{A}_{i}$
is p.d. and autocorrelations are always positive:
$p(\alpha\mid\tau)=\frac{\tau\exp(-\tau(\alpha+C))\boldsymbol{1}\\{-C<\alpha\leq
0\\}}{(1-\exp(-\tau C))}.$ (8)
We fix $C=\frac{1}{2}-10^{-5}$.
3. 3.
$\boldsymbol{\beta}_{i}\sim\mathrm{Normal}(\boldsymbol{0},\mathbf{\Lambda}_{i}^{-1})$,
with the precision matrix $\mathbf{\Lambda}_{i}$ as defined in Eq. 7.
During estimation of $\boldsymbol{\beta}$, each $\lambda_{i}$ and $\alpha_{i}$
are marginalized, giving a sparse and adaptive estimate for
$\boldsymbol{\beta}$.
### 3.2 Scalability
Our design choice of the precision matrix $\mathbf{\Lambda}_{i}$ is driven by
scalability concerns. Instead of using, e.g., a random draw from a Wishart
distribution, we specify the precision matrix to have a tridiagonal structure.
This induces dependencies between coefficients in adjacent timesteps (first-
order dependencies) and allows the prior to scale to fine-grained timesteps
more efficiently. Let $N$ denote the number of training instances, $I$ the
number of base features, and $T$ the number of timesteps. A single pass of our
variational algorithm (discussed in §5) has runtime $\mathcal{O}(I(N+T))$ and
space requirement $\mathcal{O}(I(N+T))$, instead of $\mathcal{O}(I(N+T^{2}))$
for both if each $\mathbf{\Lambda}_{i}$ is drawn from a Wishart distribution.
This can make a big difference for applications with large numbers of features
($I$). Additionally, we choose the off-diagonal entries to be uniform, so we
only need one $\alpha_{i}$ for each base feature. This design choice restricts
the expressive power of the prior but still permits flexibility in adapting to
trends for different coefficients, as we will see. The prior encourages
sparsity at the group level, essentially performing feature selection: some
feature coefficients $\boldsymbol{\beta}_{i}$ may be driven to zero across all
timesteps, while others will be allowed to vary over time, with an expectation
of smooth changes.
Note that this model introduces only one hyperparameter, $\tau$, since we
marginalize $\boldsymbol{\alpha}=\langle\alpha_{1},\ldots,\alpha_{I}\rangle$
and $\boldsymbol{\lambda}=\langle\lambda_{1},\ldots,\lambda_{I}\rangle$.
## 4 Related Work
Our model is related to autoregressive integrated moving average approaches to
time-series data (Box et al., 2008), but we never have access to _direct_
observations of the time-series. Instead, we observe data ($x$ and $y$)
assumed to have been sampled using time-series-generated variables as
_coefficients_ ($\boldsymbol{\beta}$). During learning, we therefore use
probabilistic inference to reason about the variables at all timesteps
together. In §5, we describe a scalable variational inference algorithm for
inferring coefficients at all timesteps, enabling prediction of future data
and inspection of trends.
We follow Yogatama et al. (2011) in creating time-specific copies of the base
coefficients, so that
$\boldsymbol{\beta}=\langle\boldsymbol{\beta}^{(1)},\boldsymbol{\beta}^{(2)},\ldots,\boldsymbol{\beta}^{(T)}\rangle$.
As a prior over $\boldsymbol{\beta}$, they used a multivariate Gaussian
imposing non-zero covariance between each $\beta_{i}^{(t)}$ and its time-
adjacent copies $\beta_{i}^{(t-1)}$ and $\beta_{i}^{(t+1)}$. The strength of
that covariance was set for each base feature by a global hyperparameter,
which was tuned on held-out development data along with the global variance
hyperparameter. Yogatama et al.’s model can be obtained from ours by fixing
the same $\alpha$ and $\lambda$ for all features $i$. Our approach differs in
that (i) we marginalize the hyperparameters, (ii) we allow each coefficient
its own autocorrelation, and (iii) we encourage sparsity.
There are many related Bayesian approaches for time-varying model parameters
(Belmonte et al., 2012; Nakajima & West, 2012; Caron et al., 2012), as well as
work on time-varying signal estimation (Angelosante & Giannakis, 2009;
Angelosante et al., 2009; Charles & Rozell, 2012). Each provides a different
probabilistic interpretation of parameter generation. Our model has a
distinctive generative story in that correlations between parameters of
successive timesteps are encoded in a precision matrix. Additionally, unlike
these fully Bayesian approaches that infer full posterior distributions, we
only obtain posterior mode estimates of coefficients, which has computational
advantages at prediction time (e.g., straightforward MAP inference and
sparsity) and interpretability of $\boldsymbol{\beta}$.
As noted, our grouping together of each feature’s instantiations at all
timesteps,
$\langle\beta_{i}^{(1)},\beta_{i}^{(2)},\ldots,\beta_{i}^{(T)}\rangle$ and
seeking sparsity, bears clear similarity to _group lasso_ (Yuan & Lin, 2007),
which encourages whole groups of coefficients to collectively go to zero. A
probabilistic interpretation for lasso as a two level exponential-normal
distribution that generalizes to (non-overlapping) group lasso was introduced
by Figueiredo (2002). He also showed that the exponential distribution prior
can be replaced with an improper Jeffreys prior for a parameter-free model, a
step we follow as well.
Our model is also related to the fused lasso (Tibshirani et al., 2005), which
penalizes a loss function by the $\ell_{1}$-norm of the coefficients and their
differences. Our prior has a more clear probabilistic interpretation and
adapts the degree of autocorrelation for each coefficient, based on the data.
Zhang & Yeung (2010) proposed a regularization method using a matrix-variate
normal distribution prior to model task relationships in multitask learning.
If we consider timesteps as tasks, the technique resembles our regularizer.
Their method jointly optimizes the covariance matrix with the feature
coefficients; we choose a Bayesian treatment and encode our prior belief to
the (inverse) covariance matrix, while still allowing the learned feature
coefficients to modify the matrix by posterior inference. As a result, our
method allows different base features to have different matrices.
## 5 Learning and Inference
We marginalize $\boldsymbol{\lambda}$ and $\boldsymbol{\alpha}$ and obtain a
maximum _a posteriori_ estimate for $\boldsymbol{\beta}$, which includes a
coefficient for each base feature $i$ at each timestep $t$. Specifically, we
seek to maximize:
$\displaystyle L(\boldsymbol{\beta})+\sum_{i=1}^{I}\log\int d\alpha_{i}\int
d\lambda_{i}p(\boldsymbol{\beta}_{i}\mid\alpha_{i},\lambda_{i})p(\alpha_{i}\mid\tau)p(\lambda_{i})$
(9)
Exact inference in this model is intractable. We use mean-field variational
inference to derive a lower bound on the above log-likelihood function. We
then apply a standard optimization technique to jointly optimize the
variational parameters and the coefficients $\boldsymbol{\beta}$.
We introduce fully factored variational distributions for each $\lambda_{i}$
and $\alpha_{i}$. For $\lambda_{i}$, we use a Gamma distribution with
parameters $a_{i},b_{i}$ as our variational distribution:
$\displaystyle q_{i}(\lambda_{i}\mid
a_{i},b_{i})=\frac{\lambda_{i}^{a_{i}-1}\exp(-\lambda_{i}/b_{i})}{b_{i}^{a_{i}}\Gamma(a_{i})}$
Therefore, we have $\mathbb{E}_{q_{i}}[\lambda_{i}]=a_{i}b_{i}$,
$\mathbb{E}_{q_{i}}[\lambda_{i}^{-1}]=((a_{i}-1)b_{i})^{-1}$, and
$\mathbb{E}_{q_{i}}[\log\lambda_{i}]=\Psi(a_{i})+\log b_{i}$ ($\Psi$ is the
digamma function).
For $\alpha_{i}$, we choose the form of our variational distribution to be the
same truncated exponential distribution as its prior, with parameter
$\kappa_{i}$, denoting this distribution $q_{i}(\alpha_{i}\mid\kappa_{i})$. We
have
$\displaystyle\mathbb{E}_{q_{i}}[\alpha_{i}]$
$\displaystyle=\int^{0}_{-C}\alpha_{i}\frac{\kappa_{i}\exp(-\kappa_{i}(\alpha_{i}+C))}{1-\exp(-\kappa_{i}C)}d\alpha_{i}$
$\displaystyle=\frac{1}{\kappa_{i}}-\frac{C}{1-\exp(-\kappa_{i}C)}$ (10)
We let $q$ denote the set of all variational distributions over
$\boldsymbol{\lambda}$ and $\boldsymbol{\alpha}$.
The variational bound $B$ that we seek to maximize is given in Figure 1. Our
learning algorithm involves optimizing with respect to variational parameters
$\boldsymbol{a}$, $\boldsymbol{b}$, and $\boldsymbol{\kappa}$, and the
coefficients $\boldsymbol{\beta}$. We employ the L-BFGS quasi-Newton method
(Liu & Nocedal, 1989), for which we need to compute the gradient of $B$. We
turn next to each part of this gradient.
$\displaystyle
B(\boldsymbol{a},\boldsymbol{b},\boldsymbol{\kappa},\boldsymbol{\beta})$
$\displaystyle\propto$ $\displaystyle
L(\boldsymbol{\beta})+\sum_{i=1}^{I}\left\\{\frac{1}{2}(-T\mathbb{E}_{q}[\log\lambda_{i}]\framebox{$-\mathbb{E}_{q}[\log\det{\mathbf{A}_{i}^{-1}}]$})-\mathbb{E}_{q}[\lambda_{i}^{-1}]\frac{1}{2}\boldsymbol{\beta}_{i}^{\top}\mathbb{E}_{q}[\mathbf{A}_{i}]\boldsymbol{\beta}_{i}\right\\}$
$\displaystyle+\sum_{i=1}^{I}\left\\{-(\mathbb{E}_{q}[\alpha_{i}]+C)\tau-\mathbb{E}_{q}[\log\lambda_{i}]\right\\}-\sum_{i=1}^{I}\left\\{(a_{i}-1)\mathbb{E}_{q}[\log\lambda_{i}]-\frac{\mathbb{E}_{q}[\lambda_{i}]}{b_{i}}-\log\Gamma({a_{i}})-a_{i}\log
b_{i}\right\\}$
$\displaystyle-\sum_{i=1}^{I}\left\\{\log\kappa_{i}-\kappa_{i}(\mathbb{E}_{q}[\alpha_{i}]+C)-\log(1-\exp(-\kappa_{i}C))\right\\}$
Figure 1: The variational bound on Equation 1 that is maximized to learn
$\boldsymbol{\beta}$. The boxed expression is further bounded by
$-\log\det\mathbb{E}_{q}[\mathbf{A}_{i}]$ using Jensen’s inequality, giving a
new lower bound we denote by $B^{\prime}$.
### 5.1 Coefficients $\boldsymbol{\beta}$
For $1<t<T$, the first derivative with respect to time-specific coefficient
$\beta_{i}^{(t)}$ is:
$\frac{\partial B}{\partial\beta_{i}^{(t)}}=\frac{\partial
L}{\partial\beta_{i}^{(t)}}-\frac{1}{2}\mathbb{E}[\lambda_{i}^{-1}]\left(\mathbb{E}[\alpha_{i}](\beta_{i}^{(t-1)}+\beta_{i}^{(t+1)})+2\beta_{i}^{(t)}\right)$
(11)
We can interpret the first derivative as including a penalty scaled by
$\mathbb{E}[\lambda_{i}^{-1}]$. We rewrite this penalty as:
$\displaystyle\mathbb{E}[\lambda_{i}^{-1}]\left(\vphantom{1-\mathbb{E}[\alpha_{i}])2\beta_{i}^{(t)}}\right.$
$\displaystyle(1-\mathbb{E}[\alpha_{i}])$ $\displaystyle\cdot
2\beta_{i}^{(t)}$ $\displaystyle+\mathbb{E}[\alpha_{i}]$
$\displaystyle\cdot(\beta_{i}^{(t)}-\beta^{(t-1)}_{i})$
$\displaystyle+\mathbb{E}[\alpha_{i}]$
$\displaystyle\left.\cdot(\beta_{i}^{(t)}-\beta^{(t+1)}_{i})\right)$
This form makes it clear that the penalty depends on $\beta^{(t-1)}_{i}$ and
$\beta^{(t+1)}_{i}$, penalizing the difference between $\beta^{(t)}_{i}$ and
these time-adjacent coefficients proportional to $\mathbb{E}[\alpha_{i}]$.
The form bears strong similarity to the first derivative of the time-series
(log-)prior introduced in Yogatama et al. (2011), which depends on fixed,
global hyperparameters analogous to our $\alpha$ and $\lambda$. Because our
approach does not require us to specify scalars playing the roles of
“$\mathbb{E}[\lambda_{i}^{-1}]$” and “$\mathbb{E}[\alpha_{i}]$” in advance, it
is possible for each feature to have its own autocorrelation. Obtaining the
same effect in their model would require careful tuning of $\mathcal{O}(I)$
hyperparameters, which is not practical.
It also has some similarities to the fused lasso penalty (Tibshirani et al.,
2005), which is intended to encourage sparsity in the differences between
features coefficients across timesteps. Our prior, on the other hand,
encourages smoothness in the differences, with additional sparsity at the
feature level.
### 5.2 Variational Parameters for $\boldsymbol{\alpha}$ and
$\boldsymbol{\lambda}$
Recall that the variational distribution for $\lambda_{i}$ is a Gamma
distribution with parameters $a_{i}$ and $b_{i}$.
##### Precision matrix scalar $\boldsymbol{\lambda}$.
The first derivative for variational parameters $\boldsymbol{a}$ is easy to
compute:
$\frac{\partial B}{\partial
a_{i}}=\left(-\frac{T}{2}-a_{i}\right)\Psi_{1}(a_{i})+\frac{\boldsymbol{\beta}_{i}^{\top}\mathbb{E}[\mathbf{A}_{i}]\boldsymbol{\beta}_{i}}{2b_{i}(a_{i}-1)^{2}}+1$
(12)
where $\Psi_{1}$ is the trigamma function. We can solve for $\boldsymbol{b}$
in closed form given the other free variables:
$b_{i}=\frac{\boldsymbol{\beta}_{i}^{\top}\mathbb{E}[\mathbf{A}_{i}]\boldsymbol{\beta}_{i}}{(a_{i}-1)T}$
(13)
We therefore treat $\boldsymbol{b}$ as a function of $\boldsymbol{a}$,
$\boldsymbol{\kappa}$, and $\boldsymbol{\beta}$ in optimization.
##### Off-diagonal entries $\boldsymbol{\alpha}$.
First, notice that using Jensen’s inequality:
$\mathbb{E}[\log\det{\mathbf{A}_{i}^{-1}}]=\mathbb{E}[-\log\det{\mathbf{A}_{i}}]\geq-\log\det\mathbb{E}[\mathbf{A}_{i}]$
due to the fact that $-\log\det\mathbf{A}_{i}$ is a convex function.
Furthermore, for a uniform symmetric tridiagonal matrix like $\mathbf{A}_{i}$,
the log determinant can be computed in closed form as follows (Volpi, 2003):
$\displaystyle\log\det\mathbb{E}[\mathbf{A}_{i}]=$
$\displaystyle\log\left(\prod_{t=1}^{T}1+2\mathbb{E}[\alpha_{i}]\cos\left(\frac{(t+1)\pi}{T+1}\right)\right)$
$\displaystyle=$
$\displaystyle\sum_{t=1}^{T}\log\left(1+2\mathbb{E}[\alpha_{i}]\cos\left(\frac{(t+1)\pi}{T+1}\right)\right)$
We therefore maximize a lower bound on $B$, making use of the above to
calculate first derivatives with respect to $\kappa_{i}$:
$\displaystyle\frac{\partial B^{\prime}}{\partial\kappa_{i}}=$
$\displaystyle-\tau\frac{\partial\mathbb{E}[\alpha_{i}]}{\partial\kappa_{i}}-\frac{1}{\kappa_{i}}+C+\mathbb{E}[\alpha_{i}]+\frac{\partial\mathbb{E}[\alpha_{i}]}{\partial\kappa_{i}}\kappa_{i}$
$\displaystyle+\frac{C\exp(-C\kappa_{i})}{1-\exp(-C\kappa_{i})}+\frac{1}{2}\frac{\partial\log\det\mathbb{E}[\mathbf{A}_{i}]}{\partial\kappa_{i}}$
$\displaystyle-\frac{1}{2}\mathbb{E}[\lambda_{i}^{-1}]\frac{\partial\boldsymbol{\beta}_{i}^{\top}\mathbb{E}[\mathbf{A}_{i}]\boldsymbol{\beta}_{i}}{\partial\kappa_{i}}$
The partial derivatives
$\frac{\partial\mathbb{E}[\alpha_{i}]}{\partial\kappa_{i}}$,
$\frac{\partial\log\det\mathbb{E}[\mathbf{A}_{i}]}{\partial\kappa_{i}}$, and
$\frac{\partial\boldsymbol{\beta}_{i}^{\top}\mathbb{E}[\mathbf{A}_{i}]\boldsymbol{\beta}_{i}}{\partial\kappa_{i}}$
are easy to compute. We omit them for space.
### 5.3 Implementation Details
A well-known property of numerical optimizers like the one we use (L-BFGS; Liu
& Nocedal (1989)) is the failure to reach optimal values exactly at zero.
Although theoretically strongly sparse, our prior only produces weak sparsity
in practice. Future work might consider a more principled proximal-gradient
algorithm to obtain strong sparsity (Bach et al., 2011; Liu & Ye, 2010; Duchi
& Singer, 2009).
If we expect feature coefficients at specific timesteps to be sparse as well,
it is straightforward to incorporate additional terms in the objective
function that encode this prior belief (analogous to an extension from group
lasso to _sparse_ group lasso). For the tasks we consider in our experiments,
we found that it does not substantially improve the overall performance.
Therefore, we keep the simpler bound given in Figure 1.
## 6 Experiments
We report two sets of experiments, one with a continuous $y$, the other a
language modeling application where $y$ is text. Each timestep in our
experiments is one year.
### 6.1 Baselines
On both tasks, we compare our approach to a range of baselines. Since this is
a forecasting task, at each test year, we only used training examples that
come from earlier years. Our baselines vary in how they use this earlier data
and in how they regularize.
* •
ridge-one: ridge regression (Hoerl & Kennard, 1970), trained on only examples
from the year prior to the test data (e.g., for the 2002 task, train on
examples from 2001)
* •
ridge-all: ridge regression trained on the full set of past examples (e.g.,
for the 2002 task, train on examples from 1996–2001)
* •
ridge-ts: the non-adaptive time-series ridge model of Yogatama et al. (2011)
* •
lasso-one: lasso regression (Tibshirani, 1996), trained on only examples from
the year prior to the test data444Brendan O’Connor (personal communication)
has established the superiority of the lasso to the support vector regression
method of Kogan et al. (2009) on this dataset; lasso is a strong baseline for
this problem.
* •
lasso-all: lasso regression trained on the full set of past examples
In all cases, we tuned hyperparameters on a development data. Note that, of
the above baselines, only ridge-ts replicates the coefficients at different
timesteps (i.e., $IT$ parameters); the others have only $I$ time-insensitive
coefficients.
The model with our prior always uses all training examples that are available
up to the test year (this is equivalent to a sliding window of size infinity).
Like ridge-ts, our model trusts more recent data more, allowing coefficients
farther in the past to drift farther away from those most relevant for
prediction at time $T+1$. Our model, however, adapts the “drift” of each
coefficient separately rather than setting a global hyperparameter.
### 6.2 Forecasting Risk from Text
In the first experiment, we apply our prior to a forecasting task. We consider
the task of predicting volatility of stock returns from financial reports of
publicly-traded companies, similar to Kogan et al. (2009).
Table 1: MSE on the 10-K dataset (various test sets). The first test year (2002) was used as our development data. Our model uses the sparse adaptive prior described in §3. The overall differences between our model and all competing models are statistically significant (Wilcoxon signed-rank test, $p<0.01$). year | # examples | ridge-one | ridge-all | ridge-ts | lasso-one | lasso-all | our model
---|---|---|---|---|---|---|---
2002(dev) | 2,845 | 0.182 | 0.176 | 0.171 | 0.165 | 0.156 | 0.158
2003 | 3,611 | 0.185 | 0.173 | 0.171 | 0.164 | 0.176 | 0.164
2004 | 3,558 | 0.125 | 0.137 | 0.129 | 0.116 | 0.119 | 0.113
2005 | 3,474 | 0.135 | 0.133 | 0.136 | 0.124 | 0.124 | 0.122
overall | 13,488 | 0.155 | 0.154 | 0.151 | 0.141 | 0.143 | 0.139
In finance, _volatility_ refers to a measure of variation in a quantity over
time; for stock returns, it is measured using the standard deviation during a
fixed period (here, one year). Volatility is used as a measure of financial
risk. Consider a linear regression model for predicting the log
volatility555Similar to Kogan et al. (2009) and as also the standard practice
in finance, we perform a log transformation, since log volatilities are
typically close to normally distributed. of a stock from a set of features
(see §6.2.1 for a complete description of our features). We can interpret a
linear regression model probabilistically as drawing $y\in\mathbb{R}$ from a
normal distribution with $\boldsymbol{\beta}^{\top}\boldsymbol{f}(x)$ as the
mean of the normal. Therefore, in this experiment:
$L(\boldsymbol{\beta})=-\sum_{t=1}^{T}\sum_{i=1}^{N_{t}}(y_{i}^{(t)}-\boldsymbol{\beta}^{(t)\top}\boldsymbol{f}(x_{i}^{(t)}))^{2}$.
We apply the time-series prior to the feature coefficients
$\boldsymbol{\beta}$. When making a prediction for the test data, we use
$\boldsymbol{\beta}^{(T)}$, the set of feature coefficients for the last
timestep in the training data.
#### 6.2.1 Dataset
We used a collection of Securities Exchange Commission-mandated annual reports
from 10,492 publicly traded companies in the U.S. There are 27,159 reports
over a period of ten years from 1996–2005 in the corpus. These reports are
known as “Form 10-K.”666See Kogan et al. (2009) for a complete description of
the dataset; it is available at http://www.ark.cs.cmu.edu/10K. For the feature
set, we downcased and tokenized the texts and selected the 101st–10,101st most
frequent words as binary features. The feature set was kept the same across
experiments for all models. It is widely known in the financial community that
the past history of volatility of stock returns is a good indicator of the
future volatility. Therefore, we also included the log volatility of the
stocks twelve months prior to the report as a feature. Our response variable
$y$ is the log volatility of stock returns over a period of twelve months
after the report is published.
#### 6.2.2 Results
The first test year (i.e., 2002) was used as our development data for
hyperparameter tuning ($\tau$ was selected to be $1.0$). We initialized all
the feature coefficients by the coefficients from training a lasso regression
on the last year of the training data (lasso-one). Table 1 provides a summary
of experimental results. We report the results in mean squared error on the
test set: $\frac{1}{N}\sum_{i=1}^{N}(y_{i}-\hat{y}_{i})^{2}$, where $y_{i}$ is
the true response for instance $i$ and $\hat{y}_{i}$ is the predicted
response.
Our model consistently outperformed ridge variants, including the one with a
time-series penalty (Yogatama et al., 2011). It also outperformed the lasso
variants without any time-series penalty, on average and in three out of four
test sets apiece.
One of the major challenges in working with time-series data is to choose the
right window size, in which the data is still relevant to current predictions.
Our model automates this process with a Bayesian treatment of the strength of
each feature coefficient’s autocorrelation. The results indicate that our
model was able to learn when to trust a longer history of training data, and
when to trust a shorter history of training data, demonstrating the
adaptiveness of our prior. Figure 2 shows the distribution of the expected
values of the autocorrelation paramaters under the variational distributions
$\mathbb{E}_{q_{i}}[\alpha_{i}]$ for 10,002 features, learned by our model
from the last run (test year 2005).
In future work, an empirical Bayesian treatment of the hyperprior $\tau$,
fitting it to improve the variational bound, might lead to further
improvements.
Figure 2: The distribution of expected values of the autocorrelation
paramaters under the variational distributions
$\mathbb{E}_{q_{i}}[\alpha_{i}]$ for 10,002 features used in our experiments
(10,000 unigram features, the previous year log volatility feature, and a bias
feature).
### 6.3 Text Modeling in Context
In the second experiment, we consider a hard task of modeling a collection of
texts over time conditioned on economic measurements. The goal is to predict
the probability of words appearing in a document, based on the “state of the
world” at the time the document was authored. Given a set of macroeconomic
variables in the U.S. (e.g., unemployment rate, inflation rate, average
housing prices, etc.), we want to predict what kind of texts will be produced
at a specific timestep. These documents can be written by either the
government or publicly-traded companies directly or indirectly affected by the
current economic situation.
#### 6.3.1 Model
Our text model is a sparse additive generative model (SAGE; Eisenstein et al.
(2011)). In SAGE, there is a background lexical distribution that is perturbed
additively in the log-space. When the effects are due to a (sole) feature
$f(x)$, the probability of a word is:
$\displaystyle
p(w\mid\boldsymbol{\theta},\boldsymbol{\beta},x)=\frac{\exp(\theta_{w}+\beta_{w}f(x))}{\sum_{w^{\prime}\in
V}\exp(\theta_{w^{\prime}}+\beta_{w^{\prime}}f(x))}$
where $V$ is the vocabulary, $\boldsymbol{\theta}$ (always observed) is the
vector of background log-frequencies of words in the corpus, $f(x)$ (observed)
is the feature derived from the context $x$, and $\boldsymbol{\beta}$ is the
feature-specific deviation.
Notice that the formulation above is easily extended to multiple effects with
coefficients $\boldsymbol{\beta}$. In our experiment, we have 117 effects
(features), each with its own $\boldsymbol{\beta}_{i}$. The first 50
correspond to U.S. states, plus an additional feature for the entire U.S., and
they are observed for each text since each text is associated with a known set
of states (discussed below). We assume that texts that are generated in
different states have distinct characteristics; for each state, we have a
binary indicator feature. The other 66 features depend on observed
macroeconomics variables at each timestep (e.g., unemployment rate, inflation
rate, house price index, etc.). Given an economic state of the world, we
hypothesize that there are certain words that are more likely to be used, and
each economic variable has its own (sparse) deviation from the background word
frequencies. The generative story for a word at timestep $t$ associated with
(observed) features $\boldsymbol{f}(x^{(t)})$ is:
* •
Given observed real-world observed variables $x^{(t)}$, draw word $w$ from a
multinomial distribution
$p(w\mid\boldsymbol{\theta}^{(t)},\boldsymbol{\beta}^{(t)},x^{(t)})\propto\exp(\theta_{w}^{(t)}+\boldsymbol{\beta}_{w}^{(t)\top}\boldsymbol{f}(x^{(t)}))$.
Our $L(\boldsymbol{\beta})$ is simply the negative log-loss function commonly
used in multiclass logistic regression:
$L(\boldsymbol{\beta})=\sum_{t=1}^{T}\sum_{i=1}^{N_{t}}\log
p(\boldsymbol{w}_{i}^{(t)}\mid\boldsymbol{\theta}^{(t)},\boldsymbol{\beta}^{(t)},x^{(t)}_{i})$.
We apply our time-series prior from §3 to $\boldsymbol{\beta}$.
$\boldsymbol{\theta}^{(t)}$ is fixed to be the log frequencies of words at
timestep $t$. For a single feature, coefficients over time for different
classes (words) are assumed to be generated from the same prior.
#### 6.3.2 Dataset
There is a great deal of text that is produced to describe current
macroeconomic events. We conjecture that the connection between the economy
and the text will have temporal dependencies (e.g., the amount of discussion
about housing or oil prices might vary over time). We use three sources of
text commentary on the economy. The first is a subset of the 10-K reports we
used in §6.2. We selected the 10-K reports of 200 companies chosen randomly
from the top quintile of size (measured by beginning-of-sample market
capitalization). This gives us a manageable sample of the largest U.S.
companies. Each report is associated with the state in which the company’s
head office is located. Our next two data sources come from the Federal
Reserve System, the primary body responsible for monetary policy in the
U.S.777 For an overview of the Federal Reserve System, see the Federal
Reserve’s “Purpose and Functions” document at
http://www.federalreserve.gov/pf/pf.htm. The Federal Open Market Committee
(FOMC) meets roughly eight times per year to discuss economic conditions and
set monetary policy. Prior to each meeting, each of the twelve regional banks
write an informal “anecdotal” description of economic activity in their region
as well as a national summary. This “Beige Book” is akin to a blog of economic
activity released prior to each meeting. Each FOMC meeting also produces a
transcript of the discussion. For our experiments here, we focus on text from
1996–2006.888All the text is freely available at
http://www.federalreserve.gov. The Beige Book is released to the public prior
to each meeting. The transcripts are released five years after the meetings.
As a result, we have 2,075 documents in the final corpus, consisting of 842
documents of the 10-K reports, 89 documents of the FOMC meeting transcripts,
and 1,144 documents of the Beige Book summaries.
We use the 501st–5,501st most frequent words in the dataset. We associated the
FOMC meeting transcripts with all states. The “Beige Book” texts were produced
by the Federal Reserve Banks. There are twelve Federal Reserve Banks in the
United States, each serving a collection of states. We associated texts from a
Federal Reserve Bank with the states that it serves.
Table 2: Negative log-likelihood of the documents on various test sets (lower is better). The first test year (2003) was used as our development data. Our model uses the sparse adaptive prior in §3. | # tokens | ridge-one | ridge-all | ridge-ts | lasso-one | lasso-all | our model
---|---|---|---|---|---|---|---
year | ($\times 10^{6}$) | ($\times 10^{3}$) | ($\times 10^{3}$) | ($\times 10^{3}$) | ($\times 10^{3}$) | ($\times 10^{3}$) | ($\times 10^{3}$)
2003(dev) | 1.1 | 2,736 | 2,765 | 2,735 | 2,736 | 2,765 | 2,735
2004 | 1.5 | 2,975 | 3,004 | 2,975 | 2,975 | 3,004 | 2,974
2005 | 1.9 | 2,999 | 3,027 | 2,997 | 2,998 | 3,027 | 2,997
2006 | 2.3 | 2,916 | 2,922 | 2,913 | 2,912 | 2,922 | 2,912
overall | 6.8 | 11,626 | 11,718 | 11,619 | 11,620 | 11,718 | 11,618
Quantitative U.S. macroeconomic data was obtained from the Federal Reserve
Bank of St. Louis data repository (“FRED”). We used standard measures of
economic activity focusing on output (GDP), employment, and specific markets
(e.g., housing).999For growing output series, like GDP, we calculate growth
rates as log differences. We use equity market returns for the U.S. market as
a whole and various industry and characteristic portfolios.101010Returns are
monthly, excess of the risk-free rate, and continuously compounded. The data
are from CRSP and are available for these portfolios at
http://mba.tuck.dartmouth.edu/pages/faculty/ken.french/data_library.html. They
are used as $\boldsymbol{f}(x)$ in our model; in addition to state indicator
variables, there are 66 macroeconomic variables in total.
We compare our model to the baselines in §6.1. The lasso variants are
analogous to the original formulation of SAGE (Eisenstein et al., 2011),
except that our model directly conditions on macroeconomic variables instead
of a Dirichlet-multinomial compound.
#### 6.3.3 Results
We score models by computing the negative log-likelihood on the test
dataset:111111Out-of-vocabulary items are ignored. $-\sum_{i=1}^{N}\log
p(\boldsymbol{w}_{i}^{(T+1)}\mid\boldsymbol{\theta}^{(T)},\boldsymbol{\beta}^{(T)},x^{(T+1)}_{i})$.
We initialized all the feature coefficients by the coefficients by training a
lasso regression on the last year of the training data (lasso-one). The first
test year (i.e., 2003) was used as our development data for hyperparameter
tuning ($\tau$ was selected to be $.001$). Table 2 shows the results for the
six models we compared. Similar to the forecasting experiments, at each test
year, we trained only on documents from earlier years. When we collapsed all
the training data and ignored the temporal dimension (ridge-all and lasso-
all), the background log-frequencies $\boldsymbol{\theta}^{(t)}$ are computed
using the entire training data, which is different compared to the background
log-frequencies for only the last timestep of the training data. Our model
outperformed all ridge and lasso variants, including the one with a time-
series penalty (Yogatama et al., 2011), in terms of negative log-likelihood on
unseen dataset.
In addition to improving predictive accuracy, the prior also allows us to
discover trends in the feature coefficients and gain insight. We manually
examined the model from the last run (test year 2006). Examples of temporal
trends learned by our model are shown in Figure 3. The plot illustrates
feature coefficients for words that contain the string employ. For comparison,
we also included the percentage of unemployment rate in the U.S. (which was
used as one of the features $f(x)$), scaled to fit into the plot. We can see
that there is a correlation between feature coefficients for the word
unemployment and the actual unemployment rate. On the other hand, the
correlations are less evident for other words.
Figure 3: Temporal trends learned by our model for the words that contains
employ in our dataset, as well as the actual unemployment rate (scaled by
$10^{-16}$ for ease of reading). The $y$-axis denotes coefficients and the
$x$-axis is years. See the text for explanation.
## 7 Conclusions
We presented a time-series prior for the parameters of probabilistic models;
it produces sparse models and adapts the strength of temporal effects on each
coefficient separately, based on the data, without an explosion in the number
of hyperparameters. We showed how to do inference under this prior using
variational approximations. We evaluated the prior for the task of forecasting
volatility of stock returns from financial reports, and demonstrated that it
outperforms other competing models. We also evaluated the prior for the task
of modeling a collection of texts over time, i.e., predicting the probability
of words given some observed real-world variables. We showed that the prior
achieved state-of-the-art results as well.
## Acknowledgments
The authors thank several anonymous reviewers for helpful feedback on earlier
drafts of this paper. This research was supported in part by a Google research
award to the second and third authors. This research was supported in part by
the Intelligence Advanced Research Projects Activity via Department of
Interior National Business Center contract number D12PC00347. The U.S.
Government is authorized to reproduce and distribute reprints for Governmental
purposes notwithstanding any copyright annotation thereon. The views and
conclusions contained herein are those of the authors and should not be
interpreted as necessarily representing the official policies or endorsements,
either expressed or implied, of IARPA, DoI/NBC, or the U.S. Government.
## References
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|
arxiv-papers
| 2013-10-09T20:39:08 |
2024-09-04T02:49:52.223514
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/",
"authors": "Dani Yogatama and Bryan R. Routledge and Noah A. Smith",
"submitter": "Dani Yogatama",
"url": "https://arxiv.org/abs/1310.2627"
}
|
1310.2631
|
# Saturation of Concurrent Collapsible Pushdown Systems
M. Hague
(Royal Holloway University of London, and LIGM, Marne-la-Vallée
[email protected])
###### Abstract
Multi-stack pushdown systems are a well-studied model of concurrent
computation using threads with first-order procedure calls. While, in general,
reachability is undecidable, there are numerous restrictions on stack
behaviour that lead to decidability. To model higher-order procedures calls, a
generalisation of pushdown stacks called collapsible pushdown stacks are
required. Reachability problems for multi-stack collapsible pushdown systems
have been little studied. Here, we study ordered, phase-bounded and scope-
bounded multi-stack collapsible pushdown systems using saturation techniques,
showing decidability of control state reachability and giving a regular
representation of all configurations that can reach a given control state.
## 1 Introduction
Pushdown systems augment a finite-state machine with a stack and accurately
model first-order recursion. Such systems then are ideal for the analysis of
sequential first-order programs and several successful tools, such as Moped
[25] and SLAM [3], exist for their analysis. However, the domination of multi-
and many-core machines means that programmers must be prepared to work in
concurrent environments, with several interacting execution threads.
Unfortunately, the analysis of concurrent pushdown systems is well-known to be
undecidable. However, most concurrent programs don’t interact pathologically
and many restrictions on interaction have been discovered that give
decidability (e.g. [5, 6, 26, 14, 15]).
One particularly successful approach is _context-bounding_. This
underapproximates a concurrent system by bounding the number of context
switches that may occur [24]. It is based on the observation that most real-
world bugs require only a small number of thread interactions [23].
Additionally, a number of more relaxed restrictions on stack behaviour have
been introduced. In particular phase-bounded [29], scope-bounded [30], and
ordered [7] (corrected in [2]) systems. There are also generic frameworks —
that bound the tree- [20] or split-width [10] of the interactions between
communication and storage — that give decidability for all communication
architectures that can be defined within them.
Languages such as C++, Haskell, Javascript, Python, or Scala increasingly
embrace higher-order procedure calls, which present a challenge to
verification. A popular approach to modelling higher-order languages for
verification is that of (higher-order recursion) schemes [11, 21, 16].
Collapsible pushdown systems (CPDS) are an extension of pushdown systems [13]
with a “stack-of-stacks” structure. The “collapse” operation allows a CPDS to
retrieve information about the context in which a stack character was created.
These features give CPDS equivalent modelling power to schemes [13].
These two formalisms have good model-checking properties. E.g, it is decidable
whether a $\mu$-calculus formula holds on the execution graph of a scheme [21]
(or CPDS [13]). Although, the complexity of such analyses is high, it has been
shown by Kobayashi [15] (and Broadbent et al. for CPDS [9]) that they can be
performed in practice on real code examples.
However concurrency for these models has been little studied. Work by Seth
considers phase-bounding for CPDS without collapse [27] by reduction to a
finite state parity game. Recent work by Kobayashi and Igarashi studies
context-bounded recursion schemes [17].
Here, we study global reachability problems for ordered, phase-bounded, and
scope-bounded CPDS. We use _saturation_ methods, which have been successfully
implemented by e.g. Moped [25] for pushdown systems and C-SHORe [9] for CPDS.
Saturation was first applied to model-checking by Bouajjani et al. [4] and
Finkel et al. [12]. We presented a saturation technique for CPDS in ICALP 2012
[8]. Here, we present the following advances.
1. 1.
Global reachability for ordered CPDSs (§5). This is based on Atig’s algorithm
[1] for ordered PDSs and requires a non-trivial generalisation of his notion
of _extended_ PDSs (§3). For this we introduce the notion of _transition
automata_ that encapsulate the behaviour of the saturation algorithm. In
Appendix F we show how to use the same machinery to solve the global
reachability problem for phase-bounded CPDSs.
2. 2.
Global reachability for scope-bounded CPDSs (§6). This is a backwards analysis
based upon La Torre and Napoli’s forwards analysis for scope-bounded PDSs,
requiring new insights to complete the proofs.
Because the naive encoding of a single second-order stack has an undecidable
MSO theory (we show this folklore result in Appendix A) it remains a
challenging open problem to generalise the generic frameworks above ([20, 10])
to CPDSs, since these frameworks rely on MSO decidability over graph
representations of the storage and communication structure.
## 2 Preliminaries
Before defining CPDSs, we define $2\uparrow_{0}\left({x}\right)=x$ and
$2\uparrow_{i+1}\left({x}\right)=2^{2\uparrow_{i}\left({x}\right)}$.
### 2.1 Collapsible Pushdown Systems (CPDS)
For a readable introduction to CPDS we defer to a survey by Ong [22]. Here, we
can only briefly describe higher-order collapsible stacks and their
operations. We use a notion of collapsible stacks called _annotated stacks_
(which we refer to as collapsible stacks). These were introduced in ICALP
2012, and are essentially equivalent to the classical model [8].
#### Higher-Order Collapsible Stacks
An order-$1$ stack is a stack of symbols from a stack alphabet $\Sigma$, an
order-$n$ stack is a stack of order-$(n-1)$ stacks. A collapsible stack of
order $n$ is an order-$n$ stack in which the stack symbols are annotated with
collapsible stacks which may be of any order $\leq n$. Note, often in examples
we will omit annotations for clarity. We fix the maximal order to $n$, and use
$k$ to range between $n$ and $1$. We simultaneously define for all $1\leq
k\leq n$, the set $\mathrm{Stacks}_{k}^{n}$ of order-$k$ stacks whose symbols
are annotated by stacks of order at most $n$. Note, we use subscripts to
indicate the order of a stack. Furthermore, the definition below uses a least
fixed-point. This ensures that all stacks are finite. An order-$k$ stack is a
collapsible stack in $\mathrm{Stacks}_{k}^{n}$.
###### Definition 2.1 (Collapsible Stacks)
The family of sets $(\mathrm{Stacks}_{k}^{n})_{1\leq k\leq n}$ is the smallest
family (for point-wise inclusion) such that:
1. 1.
for all $2\leq k\leq n$, $\mathrm{Stacks}_{k}^{n}$ is the set of all (possibly
empty) sequences $[{w_{1}\ldots w_{\ell}}]_{k}$ with
$w_{1},\ldots,w_{\ell}\in\mathrm{Stacks}_{k-1}^{n}$.
2. 2.
$\mathrm{Stacks}_{1}^{n}$ is all sequences
$[{{a_{1}}^{w_{1}}\ldots{a_{\ell}}^{w_{\ell}}}]_{1}$ with $\ell\geq 0$ and for
all $1\leq i\leq\ell$, $a_{i}$ is a stack symbol in $\Sigma$ and $w_{i}$ is a
collapsible stack in $\bigcup\limits_{1\leq k\leq n}\mathrm{Stacks}_{k}^{n}$.
An order-$n$ stack can be represented naturally as an edge-labelled tree over
the alphabet
$\left\\{{[_{n-1},\ldots,[_{1},]_{1},\ldots,]_{n-1}}\right\\}\uplus\Sigma$,
with $\Sigma$-labelled edges having a second target to the tree representing
the annotation. We do not use $[_{n}$ or $]_{n}$ since they would appear
uniquely at the beginning and end of the stack. An example order-$3$ stack is
given below, with only a few annotations shown (on $a$ and $c$). The
annotations are order-$3$ and order-$2$ respectively.
[nodealign=true,colsep=2ex,rowsep=2ex] $\bullet$ & $\bullet$ $\bullet$
$\bullet$ $\bullet$ $\bullet$ $\bullet$
$\bullet$ $\bullet$ $\bullet$ $\bullet$ $\bullet$ $\bullet$
$\bullet$ $\bullet$ $\bullet$ $\bullet$
^$[_{2}$^$[_{1}$^$a$^$b$^$]_{1}$^$]_{2}$
^$[_{2}$^$[_{1}$^$c$^$]_{1}$^$]_{2}$
^$[_{1}$^$d$^$]_{1}$
Given an order-$n$ stack $w=[{w_{1}\ldots w_{\ell}}]_{n}$, we define
$top_{n+1}(w)=w$ and
$\begin{array}[]{rcll}{top_{n}}\mathord{\left({[{w_{1}\ldots
w_{\ell}}]_{n}}\right)}&=&w_{1}&\text{when $\ell>0$}\\\
{top_{n}}\mathord{\left({[{}]_{n}}\right)}&=&[{}]_{n-1}&\text{otherwise}\\\
{top_{k}}\mathord{\left({[{w_{1}\ldots
w_{\ell}}]_{n}}\right)}&=&{top_{k}}\mathord{\left({w_{1}}\right)}&\text{when
$k<n$ and $\ell>0$}\end{array}$
noting that ${top_{k}}\mathord{\left({w}\right)}$ is undefined if
${top_{k^{\prime}}}\mathord{\left({w}\right)}=[{}]_{k^{\prime}-1}$ for any
$k^{\prime}>k$.
We write ${u}:_{k}{v}$ — where $u$ is order-$(k-1)$ — to denote the stack
obtained by placing $u$ on top of the $top_{k}$ stack of $v$. That is, if
$v=[{v_{1}\ldots v_{\ell}}]_{k}$ then ${u}:_{k}{v}=[{uv_{1}\ldots
v_{\ell}}]_{k}$, and if $v=[{v_{1}\ldots v_{\ell}}]_{k^{\prime}}$ with
$k^{\prime}>k$, ${u}:_{k}{v}=[{\left({{u}:_{k}{v_{1}}}\right)v_{2}\ldots
v_{\ell}}]_{k^{\prime}}$. This composition associates to the right. E.g., the
stack $[{[{[{{a}^{w}b}]_{1}}]_{2}}]_{3}$ above can be written ${u}:_{3}{v}$
where $u$ is the order-$2$ stack $[{[{{a}^{w}b}]_{1}}]_{2}$ and $v$ is the
empty order-$3$ stack $[{}]_{3}$. Then ${u}:_{3}{{u}:_{3}{v}}$ is
$[{[{[{{a}^{w}b}]_{1}}]_{2}[{[{{a}^{w}b}]_{1}}]_{2}}]_{3}$.
#### Operations on Order-$n$ Collapsible Stacks
The following operations can be performed on an order-$n$ stack where $noop$
is the null operation ${noop}\mathord{\left({w}\right)}=w$.
$\begin{array}[]{rcl}\mathcal{O}_{n}&=&\left\\{{noop,pop_{1}}\right\\}\cup\left\\{{rew_{a},push^{k}_{a},copy_{k},pop_{k}}\
\left|\ {a\in\Sigma\land 2\leq k\leq n}\right.\right\\}\end{array}$
We define each $o\in\mathcal{O}_{n}$ for an order-$n$ stack $w$. Annotations
are created by $push^{k}_{a}$, which pushes a character onto $w$ and annotates
it with
${top_{k+1}}\mathord{\left({{pop_{k}}\mathord{\left({w}\right)}}\right)}$.
This, in essence, attaches a closure to a new character.
1. 1.
We set ${pop_{k}}\mathord{\left({{u}:_{k}{v}}\right)}=v$.
2. 2.
We set ${copy_{k}}\mathord{\left({{u}:_{k}{v}}\right)}={u}:_{k}{{u}:_{k}{v}}$.
3. 3.
We set
${collapse_{k}}\mathord{\left({{{a}^{u^{\prime}}}:_{1}{{u}:_{(k+1)}{v}}}\right)}={u^{\prime}}:_{(k+1)}{v}$
when $u$ is order-$k$ and $1\leq k<n$; and
${collapse_{n}}\mathord{\left({{{a}^{u}}:_{1}{v}}\right)}=u$ when $u$ is
order-$n$.
4. 4.
We set ${push^{k}_{b}}\mathord{\left({w}\right)}={{b}^{u}}:_{1}{w}$ where
$u={top_{k+1}}\mathord{\left({{pop_{k}}\mathord{\left({w}\right)}}\right)}$.
5. 5.
We set
${rew_{b}}\mathord{\left({{{a}^{u}}:_{1}{v}}\right)}={{b}^{u}}:_{1}{v}$.
For example, beginning with $[{[{a}]_{1}[{b}]_{1}}]_{2}$ and applying
$push^{2}_{c}$ we obtain $[{[{{c}^{[{[{b}]_{1}}]_{2}}a}]_{1}[{b}]_{1}}]_{2}$.
In this setting, the order-$2$ context information for the new character $c$
is $[{[{b}]_{1}}]_{2}$. We can then apply $copy_{2};collapse_{2}$ to get
$[{[{{c}^{[{[{b}]_{1}}]_{2}}a}]_{1}[{{c}^{[{[{b}]_{1}}]_{2}}a}]_{1}[{b}]_{1}}]_{2}$
then $[{[{b}]_{1}}]_{2}$. That is, $collapse_{k}$ replaces the current
$top_{k+1}$ stack with the annotation attached to $c$.
#### Collapsible Pushdown Systems
We are now ready to define collapsible PDS.
###### Definition 2.2 (Collapsible Pushdown Systems)
An order-$n$ _collapsible pushdown system ( $n$-CPDS)_ is a tuple
$\mathcal{C}=\left({\mathcal{P},\Sigma,\mathcal{R}}\right)$ where
$\mathcal{P}$ is a finite set of control states, $\Sigma$ is a finite stack
alphabet, and
$\mathcal{R}\subseteq\left({\mathcal{P}\times\Sigma\times\mathcal{O}_{n}\times\mathcal{P}}\right)$
is a set of rules.
We write _configurations_ of a CPDS as a pair
$\langle{p},{w}\rangle\in\mathcal{P}\times\mathrm{Stacks}_{n}^{n}$. We have a
transition
$\langle{p},{w}\rangle\longrightarrow\langle{p^{\prime}},{w^{\prime}}\rangle$
via a rule $\left({{p},{a},{o},{p^{\prime}}}\right)$ when
${top_{1}}\mathord{\left({w}\right)}=a$ and
$w^{\prime}={o}\mathord{\left({w}\right)}$.
#### Consuming and Generating Rules
We distinguish two kinds of rule or operation: a rule
$\left({{p},{a},{o},{p^{\prime}}}\right)$ or operation $o$ is _consuming_ if
$o=pop_{k}$ or $o=collapse_{k}$ for some $k$. Otherwise, it is _generating_.
We write $\mathcal{R}^{{\mathcal{P}},{\Sigma}}_{\mathcal{G}_{n}}$ for the set
of generating rules of the form $\left({{p},{a},{o},{p^{\prime}}}\right)$ such
that $p,p^{\prime}\in\mathcal{P}$ and $a\in\Sigma$, and $o\in\mathcal{O}_{n}$.
We simply write $\mathcal{R}_{\mathcal{G}_{n}}$ when no confusion may arise.
### 2.2 Saturation for CPDS
Our algorithms for concurrent CPDSs build upon the saturation technique for
CPDSs [8]. In essence, we represent sets of configurations $C$ using a
$\mathcal{P}$-stack automaton $A$ reading stacks. We define such automata and
their languages ${\mathcal{L}}\mathord{\left({A}\right)}$ below. Saturation
adds new transitions to $A$ — depending on rules of the CPDS and existing
transitions in $A$ — to obtain $A^{\prime}$ representing configurations with a
path to a configuration in $C$. I.e., given a CPDS $\mathcal{C}$ with control
states $\mathcal{P}$ and a $\mathcal{P}$-stack automaton $A_{0}$, we compute
${Pre^{*}_{\mathcal{C}}}\mathord{\left({A_{0}}\right)}$ which is the smallest
set s.t.
${Pre^{*}_{\mathcal{C}}}\mathord{\left({A_{0}}\right)}\supseteq{\mathcal{L}}\mathord{\left({A_{0}}\right)}$
and
${Pre^{*}_{\mathcal{C}}}\mathord{\left({A_{0}}\right)}\supseteq\left\\{{\langle{p},{w}\rangle}\
\left|\
{\exists\langle{p},{w}\rangle\longrightarrow\langle{p^{\prime}},{w^{\prime}}\rangle\;\textrm{s.t.\;}\langle{p^{\prime}},{w^{\prime}}\rangle\in{Pre^{*}_{\mathcal{C}}}\mathord{\left({A_{0}}\right)}}\right.\right\\}$.
#### Stack Automata
Sets of stacks are represented using order-$n$ stack automata. These are
alternating automata with a nested structure that mimics the nesting in a
higher-order collapsible stack. We recall the definition below.
###### Definition 2.3 (Order-$n$ Stack Automata)
An _order- $n$ stack automaton_ is a tuple
$A=\left({\mathbb{Q}_{n},\ldots,\mathbb{Q}_{1},\Sigma,\Delta_{n},\ldots,\Delta_{1},\mathcal{F}_{n},\ldots,\mathcal{F}_{1}}\right)$
where $\Sigma$ is a finite stack alphabet,
$\mathbb{Q}_{n},\ldots,\mathbb{Q}_{1}$ are disjoint, and
1. 1.
for all $2\leq k\leq n$, we have $\mathbb{Q}_{k}$ is a finite set of states,
$\mathcal{F}_{k}\subseteq\mathbb{Q}_{k}$ is a set of accepting states, and
$\Delta_{k}\subseteq\mathbb{Q}_{k}\times\mathbb{Q}_{k-1}\times
2^{\mathbb{Q}_{k}}$ is a transition relation such that for all $q$ and $Q$
there is _at most one_ $q^{\prime}$ with
$\left({q,q^{\prime},Q}\right)\in\Delta_{k}$, and
2. 2.
$\mathbb{Q}_{1}$ is a finite set of states,
$\mathcal{F}_{1}\subseteq\mathbb{Q}_{1}$ is a set of accepting states, and the
transition relation is $\Delta_{1}\subseteq\bigcup\limits_{2\leq k\leq
n}\left({\mathbb{Q}_{1}\times\Sigma\times 2^{\mathbb{Q}_{k}}\times
2^{\mathbb{Q}_{1}}}\right)$.
States in $\mathbb{Q}_{k}$ recognise order-$k$ stacks. Stacks are read from
“top to bottom”. A stack ${u}:_{k}{v}$ is accepted from $q$ if there is a
transition $\left({q,q^{\prime},Q}\right)\in\Delta_{k}$, written
$q\xrightarrow{q^{\prime}}Q$, such that $u$ is accepted from
$q^{\prime}\in\mathbb{Q}_{(k-1)}$ and $v$ is accepted from each state in $Q$.
At order-$1$, a stack ${{a}^{u}}:_{1}{v}$ is accepted from $q$ if there is a
transition $\left({q,a,Q_{col},Q}\right)$ where $u$ is accepted from all
states in $Q_{col}$ and $v$ is accepted from all states in $Q$. An empty
order-$k$ stack is accepted by any state in $\mathcal{F}_{k}$. We write
$w\in{\mathcal{L}_{q}}\mathord{\left({A}\right)}$ to denote the set of all
stacks $w$ accepted from $q$. Note that a transition to the empty set is
distinct from having no transition.
We show a part run using $q_{3}\xrightarrow{q_{2}}Q_{3}\in\Delta_{3}$,
$q_{2}\xrightarrow{q_{1}}Q_{2}\in\Delta_{2}$,
$q_{1}\xrightarrow[Q_{col}]{a}Q_{1}\in\Delta_{1}$.
[nodealign=true,colsep=2ex,rowsep=2ex] hich return a set of long-form
transitions to be added by saturation. When $r$ is consuming,
${\Pi_{r}}\mathord{\left({A}\right)}$ returns the set of long-form transitions
to be added to $A$ due to the rule $r$. When $r$ is generating $\Pi_{r}$ also
takes as an argument a long-form transition $t$ of $A$. Thus
${\Pi_{r}}\mathord{\left({t,A}\right)}$ returns the set of long-form
transitions that should be added to $A$ as a result of the rule $r$ combined
with the transition $t$ (and possibly other transitions of $A$).
For example, if $r=\left({{p},{a},{rew_{b}},{p^{\prime}}}\right)$ and
$t={q_{p^{\prime}}}\xrightarrow[Q_{col}]{b}\left({{Q_{1},\ldots,Q_{n}}}\right)$
is a transition of $A$, then ${\Pi_{r}}\mathord{\left({t,A}\right)}$ contains
only the long-form transition
$t^{\prime}={q_{p}}\xrightarrow[Q_{col}]{a}\left({{Q_{1},\ldots,Q_{n}}}\right)$.
The idea is if $\langle{p^{\prime}},{{{b}^{u}}:_{1}{w}}\rangle$ is accepted by
$A$ via a run whose first (sequence of) transition(s) is $t$, then by adding
$t^{\prime}$ we will be able to accept $\langle{p},{{{a}^{u}}:_{1}{w}}\rangle$
via a run beginning with $t^{\prime}$ instead of $t$. We have
$\langle{p},{{{a}^{u}}:_{1}{w}}\rangle\in{Pre^{*}_{\mathcal{C}}}\mathord{\left({A}\right)}$
since it can reach $\langle{p^{\prime}},{{{b}^{u}}:_{1}{w}}\rangle$ via the
rule $r$.
###### Definition 2.4 (The Saturation Function $\Pi$)
For a CPDS with rules $\mathcal{R}$, and given an order-$n$ stack automaton
$A_{i}$ we define $A_{i+1}={\Pi}\mathord{\left({A_{i}}\right)}$. The state-
sets of $A_{i+1}$ are defined implicitly by the transitions which are those in
$A_{i}$ plus, for each
$r=\left({{p},{a},{o},{p^{\prime}}}\right)\in\mathcal{R}$, when
1. 1.
$o$ is consuming and $t\in{\Pi_{r}}\mathord{\left({A_{i}}\right)}$, then add
$t$ to $A_{i+1}$,
2. 2.
$o$ is generating, $t$ is in $A_{i}$, and
$t^{\prime}\in{\Pi_{r}}\mathord{\left({t,A}\right)}$, then add $t^{\prime}$ to
$A_{i+1}$.
In ICALP 2012 we showed that saturation adds up to
${\mathcal{O}}\mathord{\left({2\uparrow_{n}\left({{f}\mathord{\left({\left|{\mathcal{P}}\right|}\right)}}\right)}\right)}$
transitions, for some polynomial $f$, and that this can be reduced to
${\mathcal{O}}\mathord{\left({2\uparrow_{n-1}\left({{f}\mathord{\left({\left|{\mathcal{P}}\right|}\right)}}\right)}\right)}$
(which is optimal) by restricting all $Q_{n}$ to have size $1$ when $A_{0}$ is
“non-alternating at order-$n$”. Since this property holds of all $A_{0}$ used
here, we use the optimal algorithm for complexity arguments.
## 3 Extended Collapsible Pushdown Systems
To analyse concurrent systems, we extend CPDS following Atig [1]. Atig’s
extended PDSs allow words from arbitrary languages to be pushed on the stack.
Our notion of extended CPDSs allows sequences of _generating operations_ from
a language ${\mathcal{L}_{g}}$ to be applied, rather than a single operation
per rule. We can specify ${\mathcal{L}_{g}}$ by any system (e.g. a Turing
machine).
###### Definition 3.1 (Extended CPDSs)
An order-$n$ _extended CPDS ( $n$-ECPDS)_ is a tuple
$\mathcal{C}=\left({\mathcal{P},\Sigma,\mathcal{R}}\right)$ where
$\mathcal{P}$ is a finite set of control states, $\Sigma$ is a finite stack
alphabet, and
$\mathcal{R}\subseteq\left({\mathcal{P}\times\Sigma\times\mathcal{O}_{n}\times\mathcal{P}}\right)\cup\left({\mathcal{P}\times\Sigma\times
2^{\left({\mathcal{R}^{{\mathcal{P}},{\Sigma}}_{\mathcal{G}_{n}}}\right)^{\ast}}\times\mathcal{P}}\right)$
is a set of rules.
As before, we have a transition
$\langle{p},{w}\rangle\longrightarrow\langle{p^{\prime}},{w^{\prime}}\rangle$
of an $n$-ECPDS via a rule $\left({{p},{a},{o},{p^{\prime}}}\right)$ with
${top_{1}}\mathord{\left({w}\right)}=a$ and
$w^{\prime}={o}\mathord{\left({w}\right)}$. Additionally, we have a transition
$\langle{p},{w}\rangle\longrightarrow\langle{p^{\prime}},{w^{\prime}}\rangle$
when we have a rule $\left({{p},{a},{{\mathcal{L}_{g}}},{p^{\prime}}}\right)$,
a sequence
$\left({{p},{a},{o_{1}},{p_{1}}}\right)\left({{p_{1}},{a_{2}},{o_{2}},{p_{2}}}\right)\ldots\left({{p_{\ell-1}},{a_{\ell}},{o_{\ell}},{p^{\prime}}}\right)\in{\mathcal{L}_{g}}$
and
$w^{\prime}={o_{\ell}}\mathord{\left({\cdots{o_{1}}\mathord{\left({w}\right)}}\right)}$.
That is, a single extended rule may apply a sequence of stack updates in one
step. A run of an ECPDS is a sequence
$\langle{p_{0}},{w_{0}}\rangle\longrightarrow\langle{p_{1}},{w_{1}}\rangle\longrightarrow\cdots$.
### 3.1 Reachability Analysis
We adapt saturation for ECPDSs. In Atig’s algorithm, an essential property is
the decidability of
${\mathcal{L}_{g}}\cap{\mathcal{L}}\mathord{\left({A}\right)}$ for some
order-1 $\mathcal{P}$-stack automaton $A$ and a language ${\mathcal{L}_{g}}$
appearing in a rule of the extended PDS. We need analogous machinery in our
setting. For this, we first define a class of finite automata called
_transition_ automata, written $\mathcal{T}$. The states of these automata
will be long-form transitions of a stack automaton
$t={q}\xrightarrow[Q_{col}]{a}\left({{Q_{1},\ldots,Q_{n}}}\right)$.
Transitions $t\xrightarrow{r}t^{\prime}$ are labelled by rules. We write
$t\xrightarrow{\overrightarrow{r}}_{\ast}t^{\prime}$ to denote a run over
$\overrightarrow{r}\in\left({\mathcal{R}_{\mathcal{G}_{n}}}\right)^{\ast}$.
During the saturation algorithm we will build from $A_{i}$ a transition
automaton $\mathcal{T}$. Then, for each rule
$\left({{p},{a},{{\mathcal{L}_{g}}},{p^{\prime}}}\right)$ we add to $A_{i+1}$
a new long-form transition $t$ if there is a word
$\overrightarrow{r}\in{\mathcal{L}_{g}}$ such that
$t\xrightarrow{\overrightarrow{r}}_{\ast}t^{\prime}$ is a run of $\mathcal{T}$
and $t^{\prime}$ is already a transition of $A_{i}$.
For example, consider
$\left({{p},{a},{{\mathcal{L}_{g}}},{p^{\prime}}}\right)$ where
${\mathcal{L}_{g}}=\left\\{{\left({{p},{a},{rew_{b}},{p^{\prime}}}\right)}\right\\}$.
A transition
$\left({{q_{p}}\xrightarrow[Q_{col}]{a}\left({{Q_{1},\ldots,Q_{n}}}\right)}\right)\xrightarrow{\left({{p},{a},{rew_{b}},{p^{\prime}}}\right)}\left({{q_{p^{\prime}}}\xrightarrow[Q_{col}]{b}\left({{Q_{1},\ldots,Q_{n}}}\right)}\right)$
will correspond to the fact that the presence of
${q_{p^{\prime}}}\xrightarrow[Q_{col}]{b}\left({{Q_{1},\ldots,Q_{n}}}\right)$
in $A_{i}$ causes
${q_{p}}\xrightarrow[Q_{col}]{a}\left({{Q_{1},\ldots,Q_{n}}}\right)$ to be
added by $\Pi$. A run $t_{1}\xrightarrow{r_{1}}t_{2}\xrightarrow{r_{2}}t_{3}$
comes into play when e.g. ${\mathcal{L}_{g}}=\left\\{{r_{1}r_{2}}\right\\}$.
If the rule were split into two ordinary rules with intermediate control
states, $\Pi$ would first add $t_{2}$ derived from $t_{3}$, and then from
$t_{2}$ derive $t_{1}$. In the case of extended CPDSs, the intermediate
transition $t_{2}$ is not added to $A_{i+1}$, but its effect is still present
in the addition of $t_{1}$. Below, we repeat the above intuition more
formally. Fix a $n$-ECPDS
$\mathcal{C}=\left({\mathcal{P},\Sigma,\mathcal{R}}\right)$.
#### Transition Automata
We build a transition automaton from a given $\mathcal{P}$-stack automaton
$A$. Let $A$ have order-$n$ to order-$1$ state-sets $Q_{n},\ldots,Q_{1}$ and
alphabet $\Sigma$, let $T_{A}$ be the set of all
${q}\xrightarrow[Q_{col}]{a}\left({{Q_{1},\ldots,Q_{n}}}\right)$ with $q\in
Q_{n}$, for all $k$, $Q_{k}\subseteq\mathbb{Q}_{k}$, and for some $k$,
$Q_{col}\subseteq\mathbb{Q}_{k}$.
###### Definition 3.2 (Transition Automata)
Given an order-$n$ $\mathcal{P}$-stack automaton $A$ with alphabet $\Sigma$,
and $t,t^{\prime}\in T_{A}$, we define the transition automaton
$\mathcal{T}^{A}_{{t},{t^{\prime}}}=\left({T_{A},\mathcal{R}^{{\mathcal{P}},{\Sigma}}_{\mathcal{G}_{n}},\delta,t,t^{\prime}}\right)$
such that $\delta\subseteq
T_{A}\times\mathcal{R}^{{\mathcal{P}},{\Sigma}}_{\mathcal{G}_{n}}\times T_{A}$
is the smallest set such that $t_{1}\xrightarrow{r}t_{2}\in\delta$ if
$t_{1}\in{\Pi_{r}}\mathord{\left({t_{2},A}\right)}$.
We define
${\mathcal{L}}\mathord{\left({\mathcal{T}^{A}_{{t},{t^{\prime}}}}\right)}=\left\\{{\overrightarrow{r}}\
\left|\ {t\xrightarrow{\overrightarrow{r}}_{\ast}t^{\prime}}\right.\right\\}$.
#### Extended Saturation Function
We now extend the saturation function following the intuition explained above.
For $t={q_{p}}\xrightarrow[Q_{col}]{a}\left({{Q_{1},\ldots,Q_{n}}}\right)$,
let ${top_{1}}\mathord{\left({t}\right)}=a$ and
${control}\mathord{\left({t}\right)}=p$.
###### Definition 3.3 (Extended Saturation Function $\Pi$)
The extended $\Pi$ is $\Pi$ from Definition 2.4 plus for each extended rule
$\left({{p},{a},{{\mathcal{L}_{g}}},{p^{\prime}}}\right)\in\mathcal{R}$ and
$t,t^{\prime}$, we add $t$ to $A_{i+1}$ whenever 1.
${control}\mathord{\left({t}\right)}=p$and
${top_{1}}\mathord{\left({t}\right)}=a$, 2. $t^{\prime}$is a transition of
$A_{i}$ with ${control}\mathord{\left({t^{\prime}}\right)}=p^{\prime}$, and
3.
${\mathcal{L}_{g}}\cap{\mathcal{L}}\mathord{\left({\mathcal{T}^{A_{i}}_{{t},{t^{\prime}}}}\right)}\neq\emptyset$.
###### Theorem 3.1 (Global Reachability of ECPDS)
Given an ECPDS $\mathcal{C}$ and a $\mathcal{P}$-stack automaton $A_{0}$, the
fixed point $A$ of the extended saturation procedure accepts
${Pre^{*}_{\mathcal{C}}}\mathord{\left({A_{0}}\right)}$.
In order for the saturation algorithm to be effective, we need to be able to
decide
${\mathcal{L}_{g}}\cap{\mathcal{L}}\mathord{\left({\mathcal{T}^{A_{i}}_{{t},{t^{\prime}}}}\right)}\neq\emptyset$.
We argue in the appendix that number of transitions added by extended
saturation has the same upper bound as the unextended case.
## 4 Multi-Stack CPDSs
We define a general model of concurrent collapsible pushdown systems, which we
later restrict. In the sequel, assume a bottom-of-stack symbol $\perp$ and
define the “empty” stacks $\perp_{0}=\perp$ and
$\perp_{k+1}=[{\perp_{k}}]_{k+1}$. As standard, we assume that $\perp$ is
neither pushed onto, nor popped from, the stack (though may be copied by
$copy_{k}$).
###### Definition 4.1 (Multi-Stack Collapsible Pushdown Systems)
An order-$n$ _multi-stack collapsible pushdown system ( $n$-MCPDS)_ is a tuple
$\mathcal{C}=\left({\mathcal{P},\Sigma,\mathcal{R}_{1},\ldots,\mathcal{R}_{m}}\right)$
where $\mathcal{P}$ is a finite set of control states, $\Sigma$ is a finite
stack alphabet, and for each $1\leq i\leq m$ we have a set of rules
$\mathcal{R}_{i}\subseteq\mathcal{P}\times\Sigma\times\mathcal{O}_{n}\times\mathcal{P}$.
A configuration of $\mathcal{C}$ is a tuple
$\langle{p},{w_{1},\ldots,w_{m}}\rangle$. There is a transition
$\langle{p},{w_{1},\ldots,w_{m}}\rangle\longrightarrow\langle{p^{\prime}},{w_{1},\ldots,w_{i-1},w^{\prime}_{i},w_{i+1},\ldots,w_{m}}\rangle$
via $\left({{p},{a},{o},{p^{\prime}}}\right)\in\mathcal{R}_{i}$ when
$a={top_{1}}\mathord{\left({w_{i}}\right)}$ and
$w^{\prime}_{i}={o}\mathord{\left({w_{i}}\right)}$.
We also need MCPD _Automata_ , which are MCPDSs defining languages over an
input alphabet $\Gamma$. For this, we add labelling input characters to the
rules. Thus, a rule $\left({{p},{a},{\gamma},{o},{p^{\prime}}}\right)$ reads a
character $\gamma\in\Gamma$. This is defined formally in Appendix D.
We are interested in two problems for a given $n$-MCPDS $\mathcal{C}$.
###### Definition 4.2 (Control State Reachability Problem)
Given control states ${p_{\text{in}}},{p_{\text{out}}}$ of $\mathcal{C}$,
decide if there is for some $w_{1},\ldots,w_{m}$ a run
$\langle{{p_{\text{in}}}},{\perp_{n},\ldots,\perp_{n}}\rangle\longrightarrow\cdots\longrightarrow\langle{{p_{\text{out}}}},{w_{1},\ldots,w_{m}}\rangle$.
###### Definition 4.3 (Global Control State Reachability Problem)
Given a control state ${p_{\text{out}}}$ of $\mathcal{C}$, construct a
representation of the set of configurations
$\langle{p},{w_{1},\ldots,w_{m}}\rangle$ such that there exists for some
$w^{\prime}_{1},\ldots,w^{\prime}_{m}$ a run
$\langle{p},{w_{1},\ldots,w_{m}}\rangle\longrightarrow\cdots\longrightarrow\langle{{p_{\text{out}}}},{w^{\prime}_{1},\ldots,w^{\prime}_{m}}\rangle$.
We represent sets of configurations as follows. In Appendix D we show it forms
an effective boolean algebra, membership is linear time, and emptiness is in
PSPACE.
###### Definition 4.4 (Regular Set of Configurations)
A regular set $R$ of configurations of a multi-stack CPDS $\mathcal{C}$ is
definable via a finite set $\chi$ of tuples
$\left({p,A_{1},\ldots,A_{m}}\right)$ where $p$ is a control state of
$\mathcal{C}$ and $A_{i}$ is a stack automaton with designated initial state
$q_{i}$ for each $i$. We have $\langle{p},{w_{1},\ldots,w_{m}}\rangle\in R$
iff there is some $\left({p,A_{1},\ldots,A_{m}}\right)\in\chi$ such that
$w_{i}\in{\mathcal{L}_{q_{i}}}\mathord{\left({A_{i}}\right)}$ for each $i$.
Finally, we often partition runs of an MCPDS
$\sigma=\sigma_{1}\ldots\sigma_{\ell}$ where each $\sigma_{i}$ is a sequence
of configurations of the MCPDS. A transition from $c$ to $c^{\prime}$ occurs
in segment $\sigma_{i}$ if $c^{\prime}$ is a configuration in $\sigma_{i}$.
Thus, transitions from $\sigma_{i}$ to $\sigma_{i+1}$ are said to belong to
$\sigma_{i+1}$.
## 5 Ordered CPDS
We generalise _ordered multi-stack pushdown systems_ [7]. Intuitively, we can
only remove characters from stack $i$ whenever all stacks $j<i$ are empty.
###### Definition 5.1 (Ordered CPDS)
An order-$n$ _ordered CPDS_ ($n$-OCPDS) is an $n$-MCPDS
$\mathcal{C}=\left({\mathcal{P},\Sigma,\mathcal{R}_{1},\ldots,\mathcal{R}_{m}}\right)$
such that a transition from $\langle{p},{w_{1},\ldots,w_{m}}\rangle$ using the
rule $r$ on stack $i$ is permitted iff, when $r$ is consuming, for all $1\leq
j<i$ we have $w_{j}=\perp_{n}$.
###### Theorem 5.1 (Decidability of Reachability Problems)
For $n$-OCPDSs the control state reachability problem and the global control
state reachability problem are decidable.
We outline the proofs below. In Appendix E we show control state reachability
uses
${\mathcal{O}}\mathord{\left({2\uparrow_{m(n-1)}\left({\ell}\right)}\right)}$
time, where $\ell$ is polynomial in the size of the OCPDS, and we have at most
${\mathcal{O}}\mathord{\left({2\uparrow_{mn}\left({\ell}\right)}\right)}$
tuples in the solution to the global problem. First observe that reachability
can be reduced to reaching
$\langle{{p_{\text{out}}}},{\perp_{n},\ldots,\perp_{n}}\rangle$ by clearing
the stacks at the end of the run.
#### Control State Reachability
Using our notion of ECPDS, we may adapt Atig’s inductive algorithm for ordered
PDSs [1] for the control state reachability problem. The induction is over the
number of stacks. W.l.o.g. we assume that all rules
$\left({{p},{\perp},{o},{p^{\prime}}}\right)$ of $\mathcal{C}$ have
$o=push^{n}_{a}$.
In the base case, we have an $n$-OCPDS with a single stack, for which the
global reachability problem is known to be decidable (e.g. [4]).
In the inductive case, we have an $n$-OCPDS $\mathcal{C}$ with $m$ stacks. By
induction, we can decide the reachability problem for $n$-OCPDSs with fewer
than $m$ stacks. We first show how to reduce the problem to reachability
analysis of an extended CPDS, and then finally we show how to decide
${\mathcal{L}_{g}}\cap{\mathcal{L}}\mathord{\left({\mathcal{T}^{A_{i}}_{{t},{t^{\prime}}}}\right)}\neq\emptyset$
using an $n$-OCPDS with $(m-1)$ stacks.
Consider the $m$th stack of $\mathcal{C}$. A run of $\mathcal{C}$ can be split
into $\sigma_{1}\tau_{1}\sigma_{2}\tau_{2}\ldots\sigma_{\ell}\tau_{\ell}$.
During the subruns $\sigma_{i}$, the first $(m-1)$ stacks are non-empty, and
during $\tau_{i}$, the first $(m-1)$ stacks are empty. Moreover, during each
$\sigma_{i}$, only generating operations may occur on stack $m$.
We build an extended CPDS that directly models the $m$th stack during the
$\tau_{i}$ segments where the first $(m-1)$ stacks are empty, and uses rules
of the form $\left({{p},{a},{{\mathcal{L}_{g}}},{p^{\prime}}}\right)$ to
encapsulate the behaviour of the $\sigma_{i}$ sections where the first $(m-1)$
stacks are non-empty. The ${\mathcal{L}_{g}}$ attached to such a rule is the
sequence of updates applied to the $m$th stack during $\sigma_{i}$.
We begin by defining, from the OCPDS $\mathcal{C}$ with $m$ stacks, an OCPDA
$\mathcal{C}^{L}$ with $(m-1)$ stacks. This OCPDA will be used to define the
${\mathcal{L}_{g}}$ described above. $\mathcal{C}^{L}$ simulates a segment
$\sigma_{i}$. Since all updates to stack $m$ in $\sigma_{i}$ are generating,
$\mathcal{C}^{L}$ need only track its top character, hence only keeps $(m-1)$
stacks. The top character of stack $m$ is kept in the control state, and the
operations that would have occurred on stack $m$ are output.
###### Definition 5.2 ($\mathcal{C}^{L}$)
Given an $n$-OCPDS
$\mathcal{C}=\left({\mathcal{P},\Sigma,\mathcal{R}_{1},\ldots,\mathcal{R}_{m}}\right)$,
we define $\mathcal{C}^{L}$ to be an $n$-OCPDA with $(m-1)$ stacks
$\left({\mathcal{P}\times\Sigma,\Sigma,\mathcal{R}^{\prime}_{1}\cup\mathcal{R}^{\prime},\mathcal{R}^{\prime}_{2},\ldots,\mathcal{R}^{\prime}_{m-1}}\right)$
over input alphabet $\mathcal{R}_{\mathcal{G}_{n}}$ where for all $i$
$\mathcal{R}^{\prime}_{i}=\left\\{{\left({{\left({p,a}\right)},{b},{\left({{p},{a},{noop},{p^{\prime}}}\right)},{o},{\left({p^{\prime},a}\right)}}\right)}\
\left|\
{a\in\Sigma\land\left({{p},{b},{o},{p^{\prime}}}\right)\in\mathcal{R}_{i}}\right.\right\\}\text{,
and}$
$\begin{array}[]{rcl}\mathcal{R}^{\prime}&=&\left\\{{\left({{\left({p,a}\right)},{b},{r},{noop},{\left({p^{\prime},c}\right)}}\right)}\
\left|\ {b\in\Sigma\land
r=\left({{p},{a},{rew_{c}},{p^{\prime}}}\right)\in\mathcal{R}_{m}}\right.\right\\}\
\cup\\\
&&\left\\{{\left({{\left({p,a}\right)},{b},{r},{noop},{\left({p^{\prime},a}\right)}}\right)}\
\left|\ {b\in\Sigma\land
r=\left({{p},{a},{copy_{k}},{p^{\prime}}}\right)\in\mathcal{R}_{m}}\right.\right\\}\
\cup\\\
&&\left\\{{\left({{\left({p,a}\right)},{b},{r},{noop},{\left({p^{\prime},c}\right)}}\right)}\
\left|\ {b\in\Sigma\land
r=\left({{p},{a},{push^{k}_{c}},{p^{\prime}}}\right)\in\mathcal{R}_{m}}\right.\right\\}\
\cup\\\
&&\left\\{{\left({{\left({p,a}\right)},{b},{r},{noop},{\left({p^{\prime},a}\right)}}\right)}\
\left|\ {b\in\Sigma\land
r=\left({{p},{a},{noop},{p^{\prime}}}\right)\in\mathcal{R}_{m}}\right.\right\\}\
.\end{array}$
We define the language
${\mathcal{L}^{{b},{i}}_{{p},{a},{p^{\prime}}}}\mathord{\left({\mathcal{C}^{L}}\right)}$
to be the set of words $\gamma_{1}\ldots\gamma_{\ell}$ such that there exists
a run of $\mathcal{C}^{L}$ over input $\gamma_{1}\ldots\gamma_{\ell}$ from
$\langle{\left({p,a}\right)},{w_{1},\ldots,w_{m-1}}\rangle$ to
$\langle{\left({p^{\prime},c}\right)},{\perp_{n},\ldots,\perp_{n}}\rangle$ for
some $c$, where $w_{i}={push^{n}_{b}}\mathord{\left({\perp_{n}}\right)}$ and
$w_{j}=\perp_{n}$ for all $j\neq i$. This language describes the effect on
stack $m$ of a run $\sigma_{j}$ from $p$ to $p^{\prime}$. (Note, by
assumption, all $\sigma_{j}$ start with some $push^{n}_{b}$.)
We now define the extended CPDS $\mathcal{C}^{R}$ that simulates $\mathcal{C}$
by keeping track of stack $m$ in its stack and using extended rules based on
$\mathcal{C}^{L}$ to simulate parts of the run where the first $(m-1)$ stacks
are not all empty. Note, since all rules operating on $\perp$ (i.e.
$\left({{p},{\perp},{o},{p^{\prime}}}\right)$) have $o=push^{n}_{b}$, rules
from $\mathcal{R}_{1},\ldots,\mathcal{R}_{m-1}$ may only fire during (or at
the start of) the segments where the first $(m-1)$ stacks are non-empty (and
thus appear in $\mathcal{R}_{\mathcal{L}_{g}}$ below).
###### Definition 5.3 ($\mathcal{C}^{R}$)
Given an $n$-OCPDS
$\mathcal{C}=\left({\mathcal{P}\times\Sigma,\Sigma,\mathcal{R}_{1},\ldots,\mathcal{R}_{m}}\right)$
with $m$ stacks, we define $\mathcal{C}^{R}$ to be an $n$-ECPDS such that
$\mathcal{C}^{R}=\left({\mathcal{P},\Sigma,\mathcal{R}^{\prime}}\right)$ where
$\mathcal{R}^{\prime}=\mathcal{R}_{m}\cup\mathcal{R}_{\mathcal{L}_{g}}$ and
$\mathcal{R}_{\mathcal{L}_{g}}=\left\\{{\left({{p},{a},{{\mathcal{L}^{{b},{i}}_{{p_{1}},{a},{p_{2}}}}\mathord{\left({\mathcal{C}^{L}}\right)}},{p_{2}}}\right)}\
\left|\
{a\in\Sigma\land\left({{p},{\perp},{push^{n}_{b}},{p_{1}}}\right)\in\mathcal{R}_{i}\land
1\leq i<m}\right.\right\\}$
###### Lemma 5.1 ($\mathcal{C}^{R}$ simulates $\mathcal{C}$)
Given an $n$-OCPDS $\mathcal{C}$ and control states
${p_{\text{in}}},{p_{\text{out}}}$, we have
$\langle{{p_{\text{in}}}},{w}\rangle\in{Pre^{*}_{\mathcal{C}^{R}}}\mathord{\left({A}\right)}$,
where $A$ is the $\mathcal{P}$-stack automaton accepting only the
configuration $\langle{{p_{\text{out}}}},{\perp_{n}}\rangle$ iff
$\langle{{p_{\text{in}}}},{\perp_{n},\ldots,\perp_{n},w}\rangle\longrightarrow\cdots\longrightarrow\langle{{p_{\text{out}}}},{\perp_{n},\ldots,\perp_{n}}\rangle$.
Lemma 5.1 only gives an effective decision procedure if we can decide
${\mathcal{L}_{g}}\cap{\mathcal{L}}\mathord{\left({\mathcal{T}^{A_{i}}_{{t},{t^{\prime}}}}\right)}\neq\emptyset$
for all rules $\left({{p},{a},{{\mathcal{L}_{g}}},{p^{\prime}}}\right)$
appearing in $\mathcal{C}^{R}$. For this, we use a standard product
construction between the $\mathcal{C}^{L}$ associated with
${\mathcal{L}_{g}}$, and $\mathcal{T}^{A_{i}}_{{t},{t^{\prime}}}$. This gives
an ordered CPDS with $(m-1)$ stacks, for which, by induction over the number
of stacks, reachability (and emptiness) is decidable. Note, the initial
transition of the construction sets up the initial stacks of
$\mathcal{C}^{L}$.
###### Definition 5.4 ($\mathcal{C}_{\emptyset}$)
Given the non-emptiness problem
${\mathcal{L}^{{b},{i}}_{{p_{1}},{a},{p_{2}}}}\mathord{\left({\mathcal{C}^{L}}\right)}\cap{\mathcal{L}}\mathord{\left({\mathcal{T}^{A_{i}}_{{t},{t^{\prime}}}}\right)}\neq\emptyset$,
where ${top_{1}}\mathord{\left({t}\right)}=a$,
$\mathcal{C}^{L}=\left({\mathcal{P}\times\Sigma,\Sigma,\mathcal{R}_{1},\ldots,\mathcal{R}_{m-1}}\right)$
and
$\mathcal{T}^{A_{i}}_{{t},{t^{\prime}}}=\left({T_{A_{i}},\mathcal{R}_{\mathcal{G}_{n}},\delta,t,t^{\prime}}\right)$,
we define an $n$-OCPDS
$\mathcal{C}_{\emptyset}=\left({\mathcal{P}^{\emptyset},\Sigma,\mathcal{R}^{\emptyset}_{1},\ldots,\mathcal{R}^{\emptyset}_{i}\cup\mathcal{R}_{I/O},\ldots,\mathcal{R}^{\emptyset}_{m-1}}\right)$
where, for all $1\leq i\leq(m-1)$,
$\displaystyle\mathcal{P}^{\emptyset}$
$\displaystyle=\left\\{{p_{1},p_{2}}\right\\}\uplus\left\\{{\left({p,t_{1}}\right)}\
\left|\ {t_{1}\in
T_{A_{i}}\land{control}\mathord{\left({t_{1}}\right)}=p}\right.\right\\}\ ,$
$\displaystyle\mathcal{R}_{I/O}$
$\displaystyle=\left\\{{\left({{p_{1}},{\perp},{push^{n}_{b}},{\left({p_{1},t}\right)}}\right)}\right\\}\cup\left\\{{\left({{\left({p_{2},t}\right)},{\perp},{noop},{p_{2}}}\right)}\
\left|\ {t\in T_{A_{i}}}\right.\right\\}\ ,\text{ and}$
$\displaystyle\mathcal{R}^{\emptyset}_{i}$
$\displaystyle=\left\\{{\left({{\left({p,t_{1}}\right)},{c},{o},{\left({p^{\prime},t_{2}}\right)}}\right)}\
\left|\
{\left({{\left({p,{top_{1}}\mathord{\left({t_{1}}\right)}}\right)},{c},{r},{o},{\left({p^{\prime},{top_{1}}\mathord{\left({t_{2}}\right)}}\right)}}\right)\in\mathcal{R}_{i}\land\left({t_{1},r,t_{2}}\right)\in\Delta}\right.\right\\}$
###### Lemma 5.2 (Language Emptiness for OCPDS)
We have
${\mathcal{L}^{{b},{i}}_{{p_{1}},{a},{p_{2}}}}\mathord{\left({\mathcal{C}^{L}}\right)}\cap{\mathcal{L}}\mathord{\left({\mathcal{T}^{A_{i}}_{{t},{t^{\prime}}}}\right)}\neq\emptyset$
iff, in $\mathcal{C}_{\emptyset}$ from Definition 5.4, we have that
$\langle{p_{2}},{\perp_{n},\ldots\perp_{n}}\rangle$ is reachable from
$\langle{p_{1}},{\perp_{n},\ldots,\perp_{n}}\rangle$.
#### Global Reachability
We sketch a solution to the global reachability problem, giving a full proof
in Appendix E. From Lemma 5.1 ($\mathcal{C}^{R}$ simulates $\mathcal{C}$) we
gain a representation
$A_{m}={Pre^{*}_{\mathcal{C}^{R}}}\mathord{\left({A}\right)}$ of the set of
configurations $\langle{p},{\perp_{n},\ldots,\perp_{n},w_{m}}\rangle$ that
have a run to $\langle{{p_{\text{out}}}},{\perp_{n},\ldots,\perp_{n}}\rangle$.
Now take any $\langle{p},{\perp_{n},\ldots,\perp_{n},w_{m-1},w_{m}}\rangle$
that reaches $\langle{{p_{\text{out}}}},{\perp_{n},\ldots,\perp_{n}}\rangle$.
The run must pass some
$\langle{p^{\prime}},{\perp_{n},\ldots,\perp_{n},w^{\prime}_{m}}\rangle$ with
$\langle{p^{\prime}},{w^{\prime}_{m}}\rangle$ accepted by $A_{m}$. From the
product construction above, one can (though not immediately) extract a tuple
$\left({p,A_{m-1},A^{\prime}_{m}}\right)$ such that $w_{m-1}$ is accepted by
$A_{m-1}$ and $w_{m}$ is accepted by $A^{\prime}_{m}$. We repeat this
reasoning down to stack $1$ and obtain a tuple of the form
$\left({p,A_{1},\ldots,A_{m}}\right)$. We can only obtain a finite set of
tuples in this manner, giving a solution to the global reachability problem.
## 6 Scope-Bounded CPDS
Recently, scope-bounded multi-pushdown systems were introduced [30] and their
reachability problem was shown to be decidable. Furthermore, reachability for
scope- and phase-bounding was shown to be incomparable [30]. Here we consider
scope-bounded CPDS.
A run $\sigma=\sigma_{1}\ldots\sigma_{\ell}$ of an MCPDS is _context-
partitionable_ when, for each $\sigma_{i}$, if a transition in $\sigma_{i}$ is
via $r\in\mathcal{R}_{j}$ on stack $j$, then all transitions of $\sigma_{i}$
are via rules in $\mathcal{R}_{j}$ on stack $j$. A _round_ is a context-
partitioned run $\sigma_{1}\ldots\sigma_{m}$, where during $\sigma_{i}$ only
$\mathcal{R}_{i}$ is used. A _round-partitionable_ run can be partitioned
$\sigma_{1}\ldots\sigma_{\ell}$ where each $\sigma_{i}$ is a round. A run of
an SBCPDS is such that any character or stack removed from a stack must have
been created at most $\zeta$ rounds earlier. For this, we define pop- and
collapse-rounds for stacks. That is, we mark each stack and character with the
round in which it was created. When we copy a stack via $copy_{k}$, the pop-
round of the new copy of the stack is the current round. However, all stacks
and characters within the copy of $u$ keep the same pop- and collapse-round as
in the original $u$.
E.g. take $[{u}]_{2}$ where $u=[{ab}]_{1}$, $u$ and $a$ have pop-round $2$,
and $b$ has pop-round $1$. Suppose in round $3$ we use $copy_{2}$ to obtain
$[{uu}]_{2}$. The new copy of $u$ has pop-round $3$ (the current round), but
the $a$ and $b$ appearing in the copy of $u$ still have pop-rounds $2$ and $1$
respectively. If the scope-bound is $2$, the latest each $a$ and the original
$u$ could be popped is in round $4$, but the new $u$ may be popped in round
$5$.
We will write ${{}_{\mathfrak{p}}\mathord{w}}$ for a stack $w$ with pop-round
$\mathfrak{p}$ and ${{}_{\mathfrak{p},\mathfrak{c}}\mathord{a}}$ for a
character with pop-round $\mathfrak{p}$ and collapse-round $\mathfrak{c}$.
Pop- and collapse-rounds will be sometimes omitted for clarity. Note, the
outermost stack will always have pop-round $0$. In particular, for all
${u}:_{k}{v}$ in the definition below, the pop-round of $v$ is 0.
###### Definition 6.1 (Pop- and Collapse-Round)
Given a round-partitioned run $\sigma_{1}\ldots\sigma_{\ell}$ we define
inductively the pop- and collapse-rounds. The pop- and collapse-round of each
stack and character in the first configuration of $\sigma_{1}$ is $0$. Take a
transition
$\langle{p},{w}\rangle\longrightarrow\langle{p^{\prime}},{w^{\prime}}\rangle$
with $\langle{p^{\prime}},{w^{\prime}}\rangle$ in $\sigma_{z}$ via a rule
$\left({{p},{a},{o},{p^{\prime}}}\right)$. If $o=noop$ then $w=w^{\prime}$,
otherwise when
1. 1.
$o=copy_{k}$ and $w={{{}_{\mathfrak{p}}\mathord{u}}}:_{k}{v}$, then
$w^{\prime}={{{}_{z}\mathord{u}}}:_{k}{({{{}_{\mathfrak{p}}\mathord{u}}}:_{k}{v})}$
where
${{}_{z}\mathord{u}}={{}_{z}\mathord{[{{{}_{\mathfrak{p}_{1}}\mathord{u_{1}}}\ldots{{}_{\mathfrak{p}_{\ell}}\mathord{u_{\ell}}}}]_{k-1}}}$
when
${{}_{\mathfrak{p}}\mathord{u}}={{}_{\mathfrak{p}}\mathord{[{{{}_{\mathfrak{p}_{1}}\mathord{u_{1}}}\ldots{{}_{\mathfrak{p}_{\ell}}\mathord{u_{\ell}}}}]_{k-1}}}$.
2. 2.
$o=push^{k}_{b}$, then
$w^{\prime}={{{}_{z,\mathfrak{c}}\mathord{{b}^{\left({{}_{\mathfrak{p}^{\prime}}\mathord{u}}\right)}}}}:_{1}{w}$
where
${{}_{\mathfrak{p}^{\prime}}\mathord{u}}={top_{k+1}}\mathord{\left({{pop_{k}}\mathord{\left({w}\right)}}\right)}$
and $\mathfrak{c}$ is the pop-round of ${top_{k}}\mathord{\left({w}\right)}$.
(Note, when $k=n$, we know $\mathfrak{p}^{\prime}=0$ since the $top_{n+1}$
stack is outermost.)
3. 3.
$o=pop_{k}$, when $w={u}:_{k}{v}$ then $w^{\prime}=v$.
4. 4.
We set
${collapse_{k}}\mathord{\left({{{a}^{\left({{}_{\mathfrak{p}}\mathord{u^{\prime}}}\right)}}:_{1}{{u}:_{(k+1)}{v}}}\right)}={{{}_{\mathfrak{p}}\mathord{u^{\prime}}}}:_{(k+1)}{v}$
when $u$ is order-$k$ and $1\leq k<n$; and
${collapse_{n}}\mathord{\left({{{a}^{\left({{}_{0}\mathord{u}}\right)}}:_{1}{v}}\right)}={{}_{0}\mathord{u}}$
when $u$ is order-$n$.
5. 5.
$o=rew_{b}$ and
$w={{{}_{\mathfrak{p},\mathfrak{c}}\mathord{{a}^{\left({{}_{\mathfrak{p}^{\prime}}\mathord{u}}\right)}}}}:_{1}{v}$,
then
$w^{\prime}={{{}_{\mathfrak{p},\mathfrak{c}}\mathord{{b}^{\left({{}_{\mathfrak{p}^{\prime}}\mathord{u}}\right)}}}}:_{1}{v}$.
###### Definition 6.2 (Scope-Bounded CPDS)
A $\zeta$-scope-bounded $n$-CPDS ($n$-SBCPDS) $\mathcal{C}$ is an order-$n$
MCPDS whose runs are all runs of $\mathcal{C}$ that are round-partitionable,
that is $\sigma_{1}\ldots\sigma_{\ell}$, such that for all $z$, if a
transition in $\sigma_{z}$ from $\langle{p},{w}\rangle$ to
$\langle{p^{\prime}},{w^{\prime}}\rangle$ is
1. 1.
a $pop_{k}$ transition with $1<k\leq n$ and
$w={{{}_{\mathfrak{p}}\mathord{u}}}:_{k}{v}$, then $z-\zeta\leq\mathfrak{p}$,
2. 2.
a $pop_{1}$ transition with
$w={{{}_{\mathfrak{p},\mathfrak{c}}\mathord{{a}^{u}}}}:_{1}{v}$, then
$z-\zeta\leq\mathfrak{p}$, or
3. 3.
a $collapse_{k}$ transition with
$w={{{}_{\mathfrak{p},\mathfrak{c}}\mathord{{a}^{u}}}}:_{1}{v}$, then
$z-\zeta\leq\mathfrak{c}$.
La Torre and Napoli’s decidability proof for the order-$1$ case already uses
the saturation method [30]. However, while La Torre and Napoli use a forwards-
reachability analysis, we must use a backwards analysis. This is because the
forwards-reachable set of configurations is in general not regular. We thus
perform a backwards analysis for CPDS, resulting in a similar approach.
However, the proofs of correctness of the algorithm are quite different.
###### Theorem 6.1 (Decidability of Reachability Problems)
For $n$-OCPDSs the control state reachability problem and the global control
state reachability problem are decidable.
In Appendix E we show our non-global algorithm requires
${\mathcal{O}}\mathord{\left({2\uparrow_{n-1}\left({\ell}\right)}\right)}$
space, where $\ell$ is polynomial in $\zeta$ and the size of the SBCPDS, and
we have at most
${\mathcal{O}}\mathord{\left({2\uparrow_{n}\left({\ell}\right)}\right)}$
tuples in the global reachability solution. La Torre and Parlato give an
alternative control state reachability algorithm at order-$1$ using _thread
interfaces_ , which allows sequentialisation [19] and should generalise
order-$n$, but, does not solve the global reachability problem.
#### Control State Reachability
Fix initial and target control states ${p_{\text{in}}}$ and
${p_{\text{out}}}$. The algorithm first builds a _reachability graph_ , which
is a finite graph with a certain kind of path iff ${p_{\text{out}}}$ can be
reached from ${p_{\text{in}}}$. To build the graph, we define layered stack
automata. These have states $q_{p}^{i}$ for each $1\leq i\leq\zeta$ which
represent the stack contents $i$ rounds later. Thus, a layer automaton tracks
the stack across $\zeta$ rounds, which allows analysis of scope-bounded CPDSs.
###### Definition 6.3 ($\zeta$-Layered Stack Automata)
A _$\zeta$ -layered stack automaton_ is a stack automaton $A$ such that
$\mathbb{Q}_{n}=\left\\{{q_{p}^{i}}\ \left|\ {p\in\mathcal{P}\land 1\leq
i\leq\zeta}\right.\right\\}$.
A state $q_{p}^{i}$ is of layer $i$. A state $q^{\prime}$ labelling
$q\xrightarrow{q^{\prime}}Q$ has the same layer as $q$. We require that there
is no $q\xrightarrow{q^{\prime}}Q$ with $q^{\prime\prime}\in Q$ where $q$ is
of layer $i$ and $q^{\prime\prime}$ is of layer $j<i$. Similarly, there is no
$q\xrightarrow[Q_{col}]{a}Q$ with $q^{\prime}\in Q\cup Q_{col}$ where $q$ is
of layer $i$ and $q^{\prime}$ is of layer $j<i$.
Next, we define several operations from which the reachability graph is
constructed. The $\text{\tt Predecessor}_{j}$ operation connects stack $j$
between two rounds. We define for stack $j$
${\text{\tt
Predecessor}_{j}}\mathord{\left({A,q_{p},q_{p^{\prime}}}\right)}={\text{\tt
Saturate}_{j}}\mathord{\left({{\text{\tt EnvMove}}\mathord{\left({{\text{\tt
Shift}}\mathord{\left({A}\right)},q_{p_{1}}^{1},q_{p_{2}}^{2}}\right)}}\right)}$
where definitions of Shift, EnvMove and $\text{\tt Saturate}_{j}$ are given in
Appendix G. Shift moves transitions in layer $i$ to layer $(i+1)$. E.g.
$q_{p}^{1}\xrightarrow{q}\left\\{{q_{p^{\prime}}^{2}}\right\\}$ would become
$q_{p}^{2}\xrightarrow{q}\left\\{{q_{p^{\prime}}^{3}}\right\\}$. Moreover,
transitions involving states in layer $\zeta$ are removed. This is because the
stack elements in layer $\zeta$ will “go out of scope”. EnvMove adds a new
transition (analogously to a $\left({{p_{1}},{a},{rew_{a}},{p_{2}}}\right)$
rule) corresponding to the control state change from $p_{1}$ to $p_{2}$
effected by the runs over the other stacks between the current round and the
next (hence layers $1$ and $2$ in the definition above). $\text{\tt
Saturate}_{j}$ gets by saturation all configurations of stack $j$ that can
reach via $\mathcal{R}_{j}$ the stacks accepted from the layer-$1$ states of
its argument (i.e. saturation using initial states $\left\\{{q_{p}^{1}}\
\left|\ {p\in\mathcal{P}}\right.\right\\}$, which accept stacks from the next
round).
The current layer automaton represents a stack across up to $\zeta$ rounds.
The predecessor operation adds another round on to the front of this
representation. A key new insight in our proofs is that if a transition goes
to a layer $i$ state, then it represents part of a run where the stack read by
the transition is removed in $i$ rounds time. Thus, if we add a transition at
layer $0$ (were it to exist) that depends on a transition of layer $\zeta$,
then the push or copy operation would have a corresponding pop $(\zeta+1)$
scopes away. Scope-bounding forbids this.
#### The Reachability Graph
The reachability graph
$\mathcal{G}^{{p_{\text{out}}}}_{\mathcal{C}}=\left({\mathcal{V},\mathcal{E}}\right)$
has vertices $\mathcal{V}$ and edges $\mathcal{E}$. Firstly, $\mathcal{V}$
contains some _initial_ vertices
$\left({p_{0},A_{1},p_{1},\ldots,p_{m-1},A_{m},p_{m}}\right)$ where
$p_{m}={p_{\text{out}}}$, and for all $1\leq i\leq m$ we have that $A_{i}$ is
the layer automaton ${\text{\tt Saturate}_{i}}\mathord{\left({A}\right)}$
where for all $w$, $A$ accepts $\langle{p_{i}},{w}\rangle$ from
$q_{p_{i}}^{1}$. Furthermore, we require that there is some $w$ such that
$\langle{p_{i-1}},{w}\rangle$ is accepted by $A_{i}$ from $q_{p_{i}}^{1}$.
That is, there is a run from $\langle{p_{i-1}},{w}\rangle$ to $p_{i}$.
Intuitively, initial vertices model the final round of a run to
${p_{\text{out}}}$ with context switches at $p_{0},\ldots,p_{m}$.
The complete set $\mathcal{V}$ is the set of all tuples
$\left({p_{0},A_{1},p_{1},\ldots,p_{m-1},A_{m},p_{m}}\right)$ where there is
some $w$ such that $\langle{p_{i-1}},{w}\rangle$ is accepted by $A_{i}$ from
state $q_{p_{i-1}}^{1}$. To ensure finiteness, we can bound $A_{i}$ to at most
$N$ states. The value of $N$ is
${\mathcal{O}}\mathord{\left({2\uparrow_{n-1}\left({\ell}\right)}\right)}$
where $\ell$ is polynomial in $\zeta$ and the size of $\mathcal{C}$. We give a
full definition of $N$ and proof in Appendix G.
We have an edge from a vertex $\left({p_{0},A_{1},\ldots,A_{m},p_{m}}\right)$
to
$\left({p^{\prime}_{0},A^{\prime}_{1},\ldots,A^{\prime}_{m},p^{\prime}_{m}}\right)$
whenever $p_{m}=p^{\prime}_{0}$ and for all $i$ we have $A_{i}={\text{\tt
Predecessor}_{i}}\mathord{\left({A^{\prime}_{i},q_{p_{i}},q_{p^{\prime}_{i-1}}}\right)}$.
An edge means the two rounds can be concatenated into a run since the control
states and stack contents match up.
###### Lemma 6.1 (Simulation by
$\mathcal{G}^{{p_{\text{out}}}}_{\mathcal{C}}$)
Given a scope-bounded CPDS $\mathcal{C}$ and control states
${p_{\text{in}}},{p_{\text{out}}}$, there is a run of $\mathcal{C}$ from
$\langle{{p_{\text{in}}}},{w_{1},\ldots,w_{m}}\rangle$ to
$\langle{{p_{\text{out}}}},{w^{\prime}_{1},\ldots,w^{\prime}_{m}}\rangle$ for
some $w^{\prime}_{1},\ldots,w^{\prime}_{m}$ iff there is a path in
$\mathcal{G}^{{p_{\text{out}}}}_{\mathcal{C}}$ to a vertex
$\left({p_{0},A_{1},\ldots,A_{m},p_{m}}\right)$ with $p_{0}={p_{\text{in}}}$
from an initial vertex where for all $i$ we have
$\langle{p_{i-1}},{w_{i}}\rangle$ accepted from $q_{p_{i}}^{1}$ of $A_{i}$.
#### Global Reachability
The $\left({p_{0},A_{1},p_{1},\ldots,p_{m-1},A_{m},p_{m}}\right)$ in
$\mathcal{G}^{{p_{\text{out}}}}_{\mathcal{C}}$ reachable from an initial
vertex are finite in number. We know by Lemma 6.1 that there is such a vertex
accepting all $\langle{p_{i-1}},{w_{i}}\rangle$ iff
$\langle{p_{0}},{w_{1},\ldots,w_{m}}\rangle$ can reach the target control
state. Let $\chi$ be the set of tuples
$\left({p_{0},A_{1},\ldots,A_{m}}\right)$ for each reachable vertex as above,
where $A_{i}$ is restricted to the initial state $q_{p_{i-1}}^{1}$. This is a
regular solution to the global control state reachability problem.
## 7 Conclusion
We have shown decidability of global reachability for ordered and scope-
bounded collapsible pushdown systems (and phase-bounded in the appendix). This
leads to a challenge to find a general framework capturing these systems.
Furthermore, we have only shown upper-bound results. Although, in the case of
phase-bounded systems, our upper-bound matches that of Seth for CPDSs without
collapse [27], we do not know if it is optimal. Obtaining matching lower-
bounds is thus an interesting though non-obvious problem. Recently, a more
relaxed notion of scope-bounding has been studied [18]. It would be
interesting to see if we can extend our results to this notion. We are also
interested in developing and implementing algorithms that may perform well in
practice.
#### Acknowledgments
Many thanks for initial discussions with Arnaud Carayol and to the referees
for their helpful remarks. This work was supported by Fond. Sci. Math. Paris;
AMIS [ANR 2010 JCJC 0203 01 AMIS]; FREC [ANR 2010 BLAN 0202 02 FREC]; VAPF
(Région IdF); and the Engineering and Physical Sciences Research Council
[EP/K009907/1].
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## Appendix A Undecidability of MSO Over The Naive Encoding of Order-$2$
Stacks
We show that the naive graph representation of an order-$2$ stack leads to the
undecidability of MSO. By naive graph representation we mean a graph where
each node is a configuration on a run of the CPDS, and we have an edge
labelled $S$ between $c_{1}$ and $c_{2}$ if the configurations are
neighbouring on the run. We have an further edge labelled $1$ if $c_{2}$ was
obtained by popping a character via $pop_{1}$ that was first pushed on to the
stack by a $push^{k}_{a}$ at node $c_{1}$. More formally, we define the
_originating configuration_ for each character.
###### Definition A.1 (Originating Configuration)
Given a run as a sequence of configurations $c_{1},c_{2},\ldots$ we define
inductively the originating configuration of each character. The originating
configuration of each character in $c_{1}$ is $1$. Take a transition
$c_{i}\longrightarrow c_{i+1}$ via a rule
$\left({{p},{a},{o},{p^{\prime}}}\right)$. If
1. 1.
$o=copy_{k}$, then each character copied inherits its originating
configuration from the character it is a copy of. All other characters keep
the same originating configuration.
2. 2.
$o=push^{k}_{b}$, all characters maintain the same originating configuration
except the new $b$ character that has originating configuration $i$.
3. 3.
$o=rew_{b}$, all characters maintain the same originating configuration except
the new $b$ character that has the originating configuration of the $a$
character it is replacing.
4. 4.
$o=noop,pop_{k}$ or $collapse_{k}$, all originating configurations are
inherited from the previous stack.
Thus, from a run $c_{1},c_{2},\ldots$ we define a graph
$\left({\mathcal{V},\mathcal{E}_{1},\mathcal{E}_{2}}\right)$ with vertices
$\mathcal{V}=\left\\{{c_{1},c_{2},\ldots}\right\\}$ and edge sets
$\mathcal{E}_{1}$ and $\mathcal{E}_{2}$, where
$\mathcal{E}_{1}=\left\\{{\left({c_{i},c_{i+1}}\right)}\ \left|\ {1\leq
i}\right.\right\\}$ and $\mathcal{E}_{2}$ contains all pairs
$\left({c_{i},c_{j}}\right)$ where $c_{j}$ was obtained by a $pop_{1}$ from
$c_{j-1}$ and the originating configuration of the character removed is $i$.
Now, consider the CPDS generating the following run
$\begin{array}[]{l}\langle{p_{0}},{[{[{\perp}]_{1}}]_{2}}\rangle\longrightarrow\langle{p_{1}},{[{[{a\perp}]_{1}}]_{2}}\rangle\longrightarrow\langle{p_{2}},{[{[{a\perp}]_{1}[{a\perp}]_{1}}]_{2}}\rangle\longrightarrow\langle{p_{2}},{[{[{\perp}]_{1}[{a\perp}]_{1}}]_{2}}\rangle\longrightarrow\\\
\\\
\langle{p_{0}},{[{[{a\perp}]_{1}}]_{2}}\rangle\longrightarrow\langle{p_{1}},{[{[{aa\perp}]_{1}}]_{2}}\rangle\longrightarrow\langle{p_{2}},{[{[{aa\perp}]_{1}[{aa\perp}]_{1}}]_{2}}\rangle\longrightarrow\langle{p_{2}},{[{[{a\perp}]_{1}[{aa\perp}]_{1}}]_{2}}\rangle\longrightarrow\\\
\langle{p_{2}},{[{[{\perp}]_{1}[{aa\perp}]_{1}}]_{2}}\rangle\longrightarrow\\\
\\\
\langle{p_{0}},{[{[{aa\perp}]_{1}}]_{2}}\rangle\longrightarrow\langle{p_{1}},{[{[{aaa\perp}]_{1}}]_{2}}\rangle\longrightarrow\langle{p_{2}},{[{[{aaa\perp}]_{1}[{aaa\perp}]_{1}}]_{2}}\rangle\longrightarrow\langle{p_{2}},{[{[{aa\perp}]_{1}[{aaa\perp}]_{1}}]_{2}}\rangle\\\
\longrightarrow\langle{p_{2}},{[{[{a\perp}]_{1}[{aaa\perp}]_{1}}]_{2}}\rangle\longrightarrow\langle{p_{2}},{[{[{\perp}]_{1}[{aaa\perp}]_{1}}]_{2}}\rangle\longrightarrow\\\
\\\ \langle{p_{0}},{[{[{aaa\perp}]_{1}}]_{2}}\rangle\longrightarrow\cdots\
.\end{array}$
That is, beginning at $\langle{p_{0}},{\perp_{2}}\rangle$ the CPDS pushes an
$a$ character, copies the stack with a $copy_{2}$ and removes all $a$s. After
all $a$s are removed, it performs $pop_{2}$ the obtain the stack below
containing only $a$. It pushes another $a$ onto the stack and repeats this
process. After each $pop_{2}$ it adds one more $a$ character, performs a
$copy_{2}$, pops all $a$s and so on. This produces the graph shown below with
$\mathcal{E}_{1}$ represented with solid lines, and $\mathcal{E}_{2}$ with
dashed lines. Furthermore, nodes from which an $a$ is pushed are the target of
a dashed arrow, and nodes reached by popping an $a$ are the sources of dashed
arrows.
[nodealign=true,rowsep=15ex]
$c_{1}$ $c_{2}$ $c_{3}$ $c_{4}$ $c_{5}$ $c_{6}$ $c_{7}$ $c_{8}$ $c_{9}$
$c_{10}$ $c_{11}$ $c_{12}$ $c_{13}$ $c_{14}$ $c_{15}$ $\cdots$
In this graph we can interpret the infinite half-grid. We restrict the graph
to nodes that are the source of a dashed arrow. We define horizontal and
vertical edges to obtain the grid below.
[nodealign=true,rowsep=5ex,colsep=5ex]
& $\vdots$
$c_{13}$ $\cdots$
$c_{8}$ $c_{14}$ $\cdots$
$c_{4}$ $c_{9}$ $c_{15}$ $\cdots$
There is a vertical edge from $c$ to $c^{\prime}$ whenever
$\left({c^{\prime},c}\right)\in\mathcal{E}_{1}$. There is a horizontal edge
from $c$ to $c^{\prime}$ whenever we have $c^{\prime\prime}$ such that
1. 1.
$\left({c^{\prime\prime},c}\right)\in\mathcal{E}_{2}$ and
$\left({c^{\prime\prime},c^{\prime}}\right)\in\mathcal{E}_{2}$, and
2. 2.
there is a path in $\mathcal{E}_{1}$ from $c$ to $c^{\prime}$, and
3. 3.
there is no $c^{\prime\prime\prime}$ on the above path with
$\left({c^{\prime\prime},c^{\prime\prime\prime}}\right)\in\mathcal{E}_{2}$.
Thus, we can MSO-interpret the infinite half-grid, and hence MSO is
undecidable over this graph.
This naive encoding contains basic matching information about pushes and pops.
It remains an interesting open problem to obtain an encoding of CPDS that is
amenable to MSO based frameworks that give positive decidability results for
concurrent behaviours.
## Appendix B Definition of The Saturation Function
We first introduce two more short-hand notation for sets of transitions.
The first is a variant on the long-form transitions. E.g. for the run in
Section 2 we can write
${q_{3}}\xrightarrow{q_{1}}\left({{Q_{2},Q_{3}}}\right)$ to represent the use
of $q_{3}\xrightarrow{q_{2}}Q_{3}$ and $q_{2}\xrightarrow{q_{1}}Q_{2}$ as the
first two transitions in the run. That is, for a sequence
$q\xrightarrow{q_{k-1}}Q_{k},q_{k-1}\xrightarrow{q_{k-2}}Q_{k-1},\ldots,q_{k^{\prime}}\xrightarrow{q_{k^{\prime}-1}}Q_{k^{\prime}}$
in $\Delta_{k}$ to $\Delta_{k^{\prime}}$ respectively, we write
${q}\xrightarrow{q_{k^{\prime}-1}}\left({{Q_{k^{\prime}},\ldots,Q_{k}}}\right)$.
The second notation represents sets of long-form transitions. We write
${Q}\xrightarrow[Q_{col}]{a}\left({{Q_{1},\ldots,Q_{k}}}\right)$ if there is a
set $\left\\{{t_{1},\ldots,t_{\ell}}\right\\}$ of long-form transitions such
that $Q=\left\\{{q_{1},\ldots,q_{\ell}}\right\\}$ and for all $1\leq
i\leq\ell$ we have
$t_{i}={q_{i}}\xrightarrow[Q^{i}_{col}]{a}\left({{Q^{i}_{1},\ldots,Q^{i}_{k}}}\right)$
and $Q_{col}=\bigcup_{1\leq
i\leq\ell}Q^{i}_{col}\subseteq\mathbb{Q}_{k^{\prime}}$ for some $k^{\prime}$,
and for all $k^{\prime}$, $Q_{k^{\prime}}=\bigcup_{1\leq
i\leq\ell}Q^{i}_{k^{\prime}}$.
###### Definition B.1 (The Auxiliary Saturation Function $\Pi_{r}$)
For a consuming CPDS rule $r=\left({{p},{a},{o},{p^{\prime}}}\right)$ we
define for a given stack automaton $A$, the set
${\Pi_{r}}\mathord{\left({A}\right)}$ to be the smallest set such that, when
1. 1.
$o=pop_{k}$, for each
${q_{p^{\prime}}}\xrightarrow{q_{k}}\left({{Q_{k+1},\dots,Q_{n}}}\right)$ in
$A$, the set ${\Pi_{r}}\mathord{\left({A}\right)}$ contains the transition
${q_{p}}\xrightarrow[\emptyset]{a}\left({{\emptyset,\ldots,\emptyset,\left\\{{q_{k}}\right\\},Q_{k+1},\ldots,Q_{n}}}\right)$,
2. 2.
$o=collapse_{k}$, when $k=n$, the set ${\Pi_{r}}\mathord{\left({A}\right)}$
contains
${q_{p}}\xrightarrow[\left\\{{q_{p^{\prime}}}\right\\}]{a}\left({{\emptyset,\ldots,\emptyset}}\right)$,
and when $k<n$, for each transition
${q_{p^{\prime}}}\xrightarrow{q_{k}}\left({{Q_{k+1},\dots,Q_{n}}}\right)$ in
$A$, the set ${\Pi_{r}}\mathord{\left({A}\right)}$ contains the transition
${q_{p}}\xrightarrow[\left\\{{q_{k}}\right\\}]{a}\left({{\emptyset,\ldots,\emptyset,Q_{k+1},\ldots,Q_{n}}}\right)$,
For a generating CPDS rule $r=\left({{p},{a},{o},{p^{\prime}}}\right)$ we
define for a given stack automaton $A$ and long-form transition $t$ of $A$,
the set ${\Pi_{r}}\mathord{\left({t,A}\right)}$ to be the smallest set such
that, when
1. 1.
$o=copy_{k}$,
$t={q_{p^{\prime}}}\xrightarrow[Q_{col}]{a}\left({{Q_{1},\ldots,Q_{k},\ldots,Q_{n}}}\right)$
and
${Q_{k}}\xrightarrow[Q^{\prime}_{col}]{a}\left({{Q^{\prime}_{1},\ldots,Q^{\prime}_{k}}}\right)$
is in $A$, the set ${\Pi_{r}}\mathord{\left({t,A}\right)}$ contains the
transition
${q_{p}}\xrightarrow[Q_{col}\cup Q^{\prime}_{col}]{a}\left({{Q_{1}\cup
Q^{\prime}_{1},\ldots,Q_{k-1}\cup
Q^{\prime}_{k-1},Q^{\prime}_{k},Q_{k+1},\ldots,Q_{n}}}\right)\ ,$
2. 2.
$o=push^{k}_{b}$, for all transitions
$t={q_{p^{\prime}}}\xrightarrow[Q_{col}]{b}\left({{Q_{1},\ldots,Q_{n}}}\right)$
and $Q_{1}\xrightarrow[Q^{\prime}_{col}]{a}Q^{\prime}_{1}$ is in $A$ with
$Q_{col}\subseteq\mathbb{Q}_{k}$, the set
${\Pi_{r}}\mathord{\left({t,A}\right)}$ contains the transition
${q_{p}}\xrightarrow[Q^{\prime}_{col}]{a}\left({{Q^{\prime}_{1},Q_{2},\ldots,Q_{k-1},Q_{k}\cup
Q_{col},Q_{k+1},\ldots,Q_{n}}}\right)\ ,$
3. 3.
$o=rew_{b}$ or $o=noop$,
$t={q_{p^{\prime}}}\xrightarrow[Q_{col}]{b}\left({{Q_{1},\dots,Q_{n}}}\right)$
the set ${\Pi_{r}}\mathord{\left({t,A}\right)}$ contains the transition
${q_{p}}\xrightarrow[Q_{col}]{a}\left({{Q_{1},\dots,Q_{n}}}\right)$ (where
$b=a$ if $o=noop$).
As a remark, omitted from the main body of the paper, during saturation, we
add transitions
${q_{n}}\xrightarrow[Q_{col}]{a}\left({{Q_{1},\ldots,Q_{n}}}\right)$ to the
automaton. Recall this represents a sequence of transitions
$q\xrightarrow{q_{k-1}}Q_{k}\in\Delta_{k},q_{k-1}\xrightarrow{q_{k-2}}Q_{k-1}\in\Delta_{k-1},\ldots,q_{1}\xrightarrow[Q_{col}]{a}Q_{1}\in\Delta_{1}$.
Hence, we first, for each $n\geq k>1$, add $q_{k}\xrightarrow{q_{k-1}}Q_{k}$
to $\Delta_{k}$ if it does not already exist. Then, we add
$q_{1}\xrightarrow[Q_{col}]{a}Q_{1}$ to $\Delta_{1}$. Note, in particular, we
only add _at most one_ $q^{\prime}$ with
$\left({q,q^{\prime},Q}\right)\in\Delta_{k}$ for all $q$ and $Q$. This ensures
termination.
Also, we say a state is _initial_ if it is of the form $q_{p}\in Q_{n}$ for
some control state $p$ or if it is a state $q_{k}\in Q_{k}$ for $k<n$ such
that there exists a transition $q_{k+1}\xrightarrow{q_{k}}Q_{k+1}$ in
$\Delta_{k+1}$. A pre-condition (that does not sacrifice generality) of the
saturation technique is that there are no incoming transitions to initial
states.
## Appendix C Proofs for Extended CPDS
We provide the proof of Theorem 3.1 (Global Reachability of ECPDS). The proof
is via the two lemmas in the sections that follow. A large part of the proof
is identical to ICALP 2012 and hence not repeated here.
### C.1 Completeness of Saturation for ECPDS
###### Lemma C.1 (Completeness of $\Pi$)
Given an extended CPDS $\mathcal{C}$ and an order-$n$ stack automaton $A_{0}$,
the automaton $A$ constructed by saturation with $\Pi$ is such that
$\langle{p},{w}\rangle\in{Pre^{*}_{\mathcal{C}}}\mathord{\left({A_{0}}\right)}$
implies $w\in{\mathcal{L}_{q_{p}}}\mathord{\left({A}\right)}$.
_Proof._ We begin with a definition of
${Pre^{*}_{\mathcal{C}}}\mathord{\left({A_{0}}\right)}$ that permits an
inductive proof of completeness. Thus, let
${Pre^{*}_{\mathcal{C}}}\mathord{\left({A_{0}}\right)}=\bigcup\limits_{\alpha<\omega}{Pre^{\alpha}_{\mathcal{C}}}\mathord{\left({A_{0}}\right)}$
where
$\begin{array}[]{rcl}{Pre^{0}_{\mathcal{C}}}\mathord{\left({A_{0}}\right)}&=&\left\\{{\langle{p},{w}\rangle}\
\left|\
{w\in{\mathcal{L}_{q_{p}}}\mathord{\left({A_{0}}\right)}}\right.\right\\}\\\
\\\
{Pre^{\alpha+1}_{\mathcal{C}}}\mathord{\left({A_{0}}\right)}&=&\left\\{{\langle{p},{w}\rangle}\
\left|\
{\exists\langle{p},{w}\rangle\longrightarrow\langle{p^{\prime}},{w^{\prime}}\rangle\in{Pre^{\alpha}_{\mathcal{C}}}\mathord{\left({A_{0}}\right)}}\right.\right\\}\end{array}$
The proof is by induction over $\alpha$. In the base case, we have
$w\in{\mathcal{L}_{q_{p}}}\mathord{\left({A_{0}}\right)}$ and the existence of
a run of $A_{0}$, and thus a run in $A$ comes directly from the run of
$A_{0}$. Now, inductively assume
$\langle{p},{w}\rangle\longrightarrow\langle{p^{\prime}},{w^{\prime}}\rangle$
and an accepting run of $w^{\prime}$ from $q_{p^{\prime}}$ of $A$.
There are two cases depending on the rule used in the transition above. Here
we consider the case where the rule is of the form
$\left({{p},{{top_{1}}\mathord{\left({w}\right)}},{{\mathcal{L}_{g}}},{p^{\prime}}}\right)$.
The case where the rule is a standard CPDS rule is identical to ICALP 2012 and
hence we do not repeat it here (although a variation of the proof appears in
the proof of Lemma G.2).
Take the rule
$\left({{p},{{top_{1}}\mathord{\left({w}\right)}},{{\mathcal{L}_{g}}},{p^{\prime}}}\right)$
and the sequence
$\left({{p_{0}},{a_{1}},{o_{1}},{p_{1}}}\right)\ldots,\left({{p_{\ell-1}},{a_{\ell}},{o_{\ell}},{p_{\ell}}}\right)\in{\mathcal{L}_{g}}$
that witnessed the transition, observing that $p_{0}=p$ and
$p_{\ell}=p^{\prime}$. Now, let
$w_{i}={o_{\ell}}\mathord{\left({\cdots{o_{i+1}}\mathord{\left({w^{\prime}}\right)}}\right)}$
for all $0\leq i\leq\ell$. Note, $w=w_{0}$ and $w^{\prime}=w_{\ell}$.
Take
$t^{\prime}={q_{p^{\prime}}}\xrightarrow[Q_{col}]{b}\left({{Q_{1},\ldots,Q_{n}}}\right)$
to be the first transition on the accepting run of
$\langle{p^{\prime}},{w^{\prime}}\rangle$. Beginning with
$t_{\ell}=t^{\prime}$, we are going to show that there is a run of
$\langle{p_{i}},{w_{i}}\rangle$ beginning with $t_{i}$ and thereafter only
using transitions appearing in $A$. Since, by the definition of $\Pi$, we add
$t_{0}=t$ to $A$, we will obtain an accepting run of $A$ for
$\langle{p_{0}},{w_{0}}\rangle=\langle{p},{w}\rangle$ as required. We will
induct from $\ell$ down to $0$.
The base case $i=\ell$ is trivial, since $t_{\ell}=t^{\prime}$ and we already
have an accepting run of $A$ over $\langle{p_{\ell}},{w_{\ell}}\rangle$
beginning with $t_{\ell}$. Now, assume the case for
$\langle{p_{i}},{w_{i}}\rangle$ and $t_{i}$. We show the case for $i-1$. Take
$\left({p_{i-1},a_{i},o_{i},p_{i}}\right)$, we do a case split on $o_{i}$. A
reader familiar with the saturation method for CPDS will observe that the
arguments below are very similar to the arguments for ordinary CPDS rules.
1. 1.
When $o_{i}=copy_{k}$, let $w_{i-1}={u_{k-1}}:_{k}{{\cdots}:_{n}{u_{n}}}$. We
know
$w_{i}={u_{k-1}}:_{k}{{u_{k-1}}:_{k}{{u_{k}}:_{(k+1)}{{\cdots}:_{n}{u_{n}}}}}\
.$
Let
$t_{i}={q_{p_{i}}}\xrightarrow[Q_{col}]{a}\left({{Q_{1},\ldots,Q_{k},\ldots
Q_{n}}}\right)$ and
${Q_{k}}\xrightarrow[Q^{\prime}_{col}]{a}\left({{Q^{\prime}_{1},\ldots,Q^{\prime}_{k}}}\right)$
be the initial transitions used on the run of $w_{i}$ (where the transition
from $Q_{k}$ reads the second copy of $u_{k-1}$).
From the construction of $\mathcal{T}^{A}_{{t},{t^{\prime}}}$ we have have a
transition
$t_{i-1}\xrightarrow{\left({{p_{i-1}},{a_{i}},{o_{i}},{p_{i}}}\right)}t_{i}$
where
$t_{i-1}={q_{p_{i-1}}}\xrightarrow[Q_{col}\cup
Q^{\prime}_{col}]{a}\left({{Q_{1}\cup Q^{\prime}_{1},\ldots,Q_{k-1}\cup
Q^{\prime}_{k-1},Q^{\prime}_{k},Q_{k+1},\ldots,Q_{n}}}\right)\ .$
Since we know ${u_{k}}:_{(k+1)}{{\cdots}:_{n}{u_{n}}}$ is accepted from
$Q^{\prime}_{k}$ via $Q_{k+1},\ldots,Q_{n}$, and we know that $u_{k-1}$ is
accepted from $Q_{1},\ldots,Q_{k-1}$ and
$Q^{\prime}_{1},\ldots,Q^{\prime}_{k-1}$ via $a$-transitions labelling
annotations with $Q_{col}$ and $Q^{\prime}_{col}$ respectively, we obtain an
accepting run of $w_{i-1}$.
2. 2.
When $o_{i}=push^{k}_{c}$, let
$w_{i-1}={u_{k-1}}:_{k}{{u_{k}}:_{k+1}{{\cdots}:_{n}{u_{n}}}}$. We know
$w_{i}={push^{k}_{c}}\mathord{\left({w_{i-1}}\right)}$ is
${{c}^{u_{k}}}:_{1}{{u_{k-1}}:_{k}{{\cdots}:_{n}{u_{n}}}}\ .$
Let
$t_{i}={q_{p_{i}}}\xrightarrow[Q_{col}]{c}\left({{Q_{1},\ldots,Q_{n}}}\right)\quad\text{and}\quad
Q_{1}\xrightarrow[Q^{\prime}_{col}]{a}Q^{\prime}_{1}$ be the first transitions
used on the accepting run of $w_{i}$. The construction of
$\mathcal{T}^{A}_{{t},{t^{\prime}}}$ means we have a transition
$t_{i-1}\xrightarrow{\left({{p_{i-1}},{a_{i}},{o_{i}},{p_{i}}}\right)}t_{i}$
where
$t_{i-1}={q_{p_{i-1}}}\xrightarrow[Q^{\prime}_{col}]{a}\left({{Q^{\prime}_{1},Q_{2},\ldots,Q_{k}\cup
Q_{col},\ldots,Q_{n}}}\right)$. Thus we can construct an accepting run of
$w_{i-1}$ (which is $w_{i}$ without the first $c$ on top of the top order-$1$
stack). A run from $Q_{k}\cup Q_{col}$ exists since $u_{k}$ is also the stack
annotating $c$.
3. 3.
When $o_{i}=rew_{c}$ let
${q_{p_{i}}}\xrightarrow[Q_{col}]{c}\left({{Q_{1},\ldots,Q_{n}}}\right)$ be
the first transition on the accepting run of $w_{i}={{c}^{u}}:_{1}{v}$ for
some $v$ and $u$. From the construction of
$\mathcal{T}^{A}_{{t},{t^{\prime}}}$ we know we have a transition
$t_{i-1}\xrightarrow{\left({{p_{i-1}},{a_{i}},{o_{i}},{p_{i}}}\right)}t_{i}$
where
$t_{i-1}={q_{p_{i-1}}}\xrightarrow[Q_{col}]{a}\left({{Q_{1},\ldots,Q_{n}}}\right)$,
from which we get an accepting run of $w_{i-1}={{a}^{u}}:_{1}{v}$ as required.
4. 4.
When $o_{i}=noop$ let
${q_{p_{i}}}\xrightarrow[Q_{col}]{a}\left({{Q_{1},\ldots,Q_{n}}}\right)$ be
the first transition on the accepting run of $w_{i}={{a}^{u}}:_{1}{v}$ for
some $v$ and $u$. From the construction of
$\mathcal{T}^{A}_{{t},{t^{\prime}}}$ we know we have a transition
$t_{i-1}\xrightarrow{\left({{p_{i-1}},{a_{i}},{o_{i}},{p_{i}}}\right)}t_{i}$
where
$t_{i-1}={q_{p_{i-1}}}\xrightarrow[Q_{col}]{a}\left({{Q_{1},\ldots,Q_{n}}}\right)$,
from which we get an accepting run of $w_{i-1}={{a}^{u}}:_{1}{v}$ as required.
Hence, for every
$\langle{p},{w}\rangle\in{Pre^{*}_{\mathcal{C}}}\mathord{\left({A_{0}}\right)}$
we have $w\in{\mathcal{L}_{q_{p}}}\mathord{\left({A}\right)}$. $\square$
### C.2 Soundness of Saturation for ECPDS
As in the previous section, the soundness argument repeats a large part of the
proof given in ICALP 2012. We first recall the machinery used for soundness,
before giving the soundness proof.
First, assume all stack automata are such that their initial states are not
final. This is assumed for the automaton $A_{0}$ in and preserved by the
saturation function $\Gamma$.
We assign a “meaning” to each state of the automaton. For this, we define what
it means for an order-$k$ stack $w$ to satisfy a state $q\in\mathbb{Q}_{k}$,
which is denoted $w\models q$.
###### Definition C.1 ($w\models q$)
For any $Q\subseteq\mathbb{Q}_{k}$ and any order-$k$ stack $w$, we write
$w\models Q$ if $w\models q$ for all $q\in Q$, and we define $w\models q$ by a
case distinction on $q$.
1. 1.
$q$ is an initial state in $\mathbb{Q}_{n}$. Then for any order-$n$ stack $w$,
we say that $w\models q$ if
$\langle{q},{w}\rangle\in{Pre^{*}_{\mathcal{C}}}\mathord{\left({A_{0}}\right)}$.
2. 2.
$q$ is an initial state in $\mathbb{Q}_{k}$, labeling a transition
$q_{k+1}\xrightarrow{q}Q_{k+1}\in\Delta_{k+1}$. Then for any order-$k$ stack
$w$, we say that $w\models q$ if for all order-$(k+1)$ stacks s.t. $v\models
Q_{k+1}$, then ${w}:_{(k+1)}{v}\models q_{k+1}$.
3. 3.
$q$ is a non-initial state in $\mathbb{Q}_{k}$. Then for any order-$k$ stack
$w$, we say that $w\models q$ if $A_{0}$ accepts $w$ from $q$.
By unfolding the definition, we have that an order-$k$ stack $w_{k}$ satisfies
an initial state $q_{k}\in\mathbb{Q}_{k}$ with
${q}\xrightarrow{q_{k}}\left({{Q_{k+1},\dots,Q_{n}}}\right)$ if for any
order-$(k+1)$ stack $w_{k+1}\models Q_{k+1}$, …, and any order-$n$ stack
$w_{n}\models Q_{n}$, we have ${w_{k}}:_{(k+1)}{{\cdots}:_{n}{w_{n}}}\models
q$.
###### Definition C.2 (Soundness of transitions)
A transition ${q}\xrightarrow[Q_{col}]{a}\left({{Q_{1},\ldots,Q_{k}}}\right)$
is sound if for any order-$1$ stack $w_{1}\models Q_{1}$, …, and any order-$k$
stack $w_{k}\models Q_{k}$ and any stack $u\models Q_{col}$, we have
${{a}^{u}}:_{1}{{w_{1}}:_{2}{{\cdots}:_{k}{w_{k}}}}\models q$.
The proof of the following lemma can be found in ICALP 2012 [8].
###### Lemma C.2 ([8])
If ${q_{p}}\xrightarrow[Q_{col}]{a}\left({{Q_{1},\ldots,Q_{n}}}\right)$ is
sound, then any transition
${q_{k}}\xrightarrow[Q_{col}]{a}\left({{Q_{1},\ldots,Q_{k}}}\right)$ contained
within the transition from $q_{p}$ is sound.
###### Definition C.3 (Soundness of stack automata)
A stack automaton $A$ is sound if the following holds.
* •
$A$ is obtained from $A_{0}$ by adding new initial states of order $<n$ and
transitions starting in an initial state.
* •
In $A$, any transition
${q}\xrightarrow[Q_{col}]{a}\left({{Q_{1},\ldots,Q_{k}}}\right)$ for $k\leq n$
is sound.
Unsurprisingly, if some order-$n$ stack $w$ is accepted by a _sound_ stack
automaton $A$ from a state $q_{p}$ then $\langle{p},{w}\rangle$ belongs to
${Pre^{*}_{\mathcal{C}}}\mathord{\left({A_{0}}\right)}$. More generally, we
have the following lemma whose proof can be found in ICALP 2012.
###### Lemma C.3 ([8])
Let $A$ be a sound stack automaton $A$ and let $w$ be an order-$k$ stack. If
$A$ accepts $w$ from a state $q\in\mathbb{Q}_{k}$ then $w\models q$. In
particular, if $A$ accepts an order-$n$ stack $w$ from a state
$q_{p}\in\mathbb{Q}_{n}$ then $\langle{p},{w}\rangle$ belongs to
${Pre^{*}_{\mathcal{C}}}\mathord{\left({A_{0}}\right)}$.
We also recall that the initial automaton $A_{0}$ is sound.
###### Lemma C.4 (Soundness of $A_{0}$ [8])
The automaton $A_{0}$ is sound.
We are now ready to prove that the soundness of saturation for extended CPDS.
###### Lemma C.5 (Soundness of $\Pi$)
The automaton $A$ constructed by saturation with $\Pi$ and $\mathcal{C}$ from
$A_{0}$ is sound.
_Proof._ The proof is by induction on the number of iterations of $\Pi$. The
base case is the automaton $A_{0}$ and the result was established in Lemma
C.4. As in the completeness case, the argument for the ordinary CPDS rules is
identical to ICALP 2012 and not repeated here (although the arguments appear
in the proof of Lemma G.3).
We argue the case for those transitions added because of extended rules
$\left({{p},{a},{{\mathcal{L}_{g}}},{p^{\prime}}}\right)$.
Hence, we consider the inductive step for transitions introduced by extended
rules of the form $\left({{p},{c},{{\mathcal{L}_{g}}},{p^{\prime}}}\right)$.
Take the $t,t^{\prime}$ and
$\left({{p_{0}},{a_{1}},{o_{1}},{p_{1}}}\right)\left({{p_{1}},{a_{2}},{o_{2}},{p_{2}}}\right)\ldots\left({{p_{\ell-1}},{a_{\ell}},{o_{\ell}},{p_{\ell}}}\right)\in{\mathcal{L}_{g}}\cap{\mathcal{L}}\mathord{\left({\mathcal{T}^{A_{i}}_{{t},{t^{\prime}}}}\right)}$
with $t^{\prime}$ being a transition of $A_{i}$ that led to the introduction
of $t$. Note $p=p_{0}$ and $p^{\prime}=p_{\ell}$.
Let $t_{0},\ldots,t_{\ell}$ be the sequence of states on the accepting run of
$\mathcal{T}^{A_{i}}_{{t},{t^{\prime}}}$. In particular $t_{0}=t$ and
$t_{\ell}=t^{\prime}$. We will prove by induction from $i=\ell$ to $i=0$ that
for each $t_{i}$, letting
$t_{i}={q_{p_{i}}}\xrightarrow[Q_{col}]{a}\left({{Q_{1},\ldots,Q_{n}}}\right)\
,$
and for all $u\models Q_{col}$, $w_{1}\models Q_{1}$, …, $w_{n}\models Q_{n}$
that for $w^{i}={{a}^{u}}:_{1}{{w_{1}}:_{2}{\cdots{}:_{n}{w_{n}}}}$ we have
${o_{\ell}}\mathord{\left({\cdots{o_{i+1}}\mathord{\left({w^{i}}\right)}}\right)}\models
q_{p^{\prime}}$. Thus, at $t_{0}=t$ , we have
${o_{\ell}}\mathord{\left({\cdots{o_{1}}\mathord{\left({w^{0}}\right)}}\right)}\models
q_{p^{\prime}}$ and thus
$\langle{p^{\prime}},{{o_{\ell}}\mathord{\left({\cdots{o_{1}}\mathord{\left({w^{0}}\right)}}\right)}}\rangle\in{Pre^{*}_{\mathcal{C}}}\mathord{\left({A_{0}}\right)}$.
Since the above sequence
$\left({{p_{0}},{a_{1}},{o_{1}},{p_{1}}}\right)\left({{p_{1}},{a_{2}},{o_{2}},{p_{2}}}\right)\ldots\left({{p_{\ell-1}},{a_{\ell}},{o_{\ell}},{p_{\ell}}}\right)$
is in ${\mathcal{L}_{g}}$, we have
$\langle{p_{0}},{w^{0}}\rangle\in{Pre^{*}_{\mathcal{C}}}\mathord{\left({A_{0}}\right)}$
and thus $w^{0}\models q_{p}$, giving soundness of the new transition $t_{0}$.
The base case is $t_{\ell}=t^{\prime}$. Since $t^{\prime}$ appears in $A_{i}$,
we know it is sound. That gives us that $w^{\ell}\models q_{p^{\prime}}$ as
required.
Now assume that $t_{i}$ satisfies the hypothesis. We prove that $t_{i-1}$ does
also. Take the transition
$t_{i-1}\xrightarrow{\left({{p_{i-1}},{a_{i}},{o_{i}},{p_{i}}}\right)}t_{i}$.
We perform a case split on $o_{i}$. Readers familiar with ICALP 2012 will
notice that the arguments here very much follow the soundness proof for
ordinary rules.
1. 1.
Assume that $o_{i}=copy_{k}$, that we had
$t_{i}={q_{p_{i}}}\xrightarrow[Q_{col}]{a}\left({{Q_{1},\ldots,Q_{n}}}\right)\quad\text{and}\quad{Q_{k}}\xrightarrow[Q^{\prime}_{col}]{a}\left({{Q^{\prime}_{1},\ldots,Q^{\prime}_{k}}}\right)$
where the latter set of transition are in $A_{i}$ and therefore sound, and
that
$t_{i-1}={q_{p_{i-1}}}\xrightarrow[Q_{col}\cup
Q^{\prime}_{col}]{a}\left({{Q_{1}\cup Q^{\prime}_{1},\ldots,Q_{k-1}\cup
Q^{\prime}_{k-1},Q^{\prime}_{k},Q_{k+1},\ldots,Q_{n}}}\right)\ .$
To establish the property for this latter transition, we have to prove that
for any $w_{1}\models Q_{1}\cup Q^{\prime}_{1},\ldots$, any $w_{k-1}\models
Q_{k-1}\cup Q^{\prime}_{k-1}$, any $w_{k}\models Q^{\prime}_{k},$ any
$w_{k+1}\models Q_{k+1},\ldots$, any $w_{n}\models Q_{n}$ and any $u\models
Q_{col}\cup Q^{\prime}_{col}$, we have for
$w^{i-1}={{a}^{u}}:_{1}{{w_{1}}:_{2}{{\cdots}:_{n}{w_{n}}}}$ that
${o_{\ell}}\mathord{\left({\cdots{o_{i}}\mathord{\left({w^{i-1}}\right)}}\right)}\models
q_{p^{\prime}}$.
Let
$v={top_{k}}\mathord{\left({w^{i-1}}\right)}={{a}^{u}}:_{1}{{w_{1}}:_{2}{{\cdots}:_{(k-1)}{w_{k-1}}}}$.
From the soundness of
${Q_{k}}\xrightarrow[Q^{\prime}_{col}]{a}\left({{Q^{\prime}_{1},\ldots,Q^{\prime}_{k}}}\right)$
and as $u\models Q^{\prime}_{col},w_{1}\models
Q^{\prime}_{1},\ldots,w_{k}\models Q^{\prime}_{k}$, we have
${v}:_{k}{w_{k}}\models Q_{k}$.
Then, from $w_{1}\models Q_{1},\ldots,w_{k-1}\models Q_{k-1}$, and
${v}:_{k}{w_{k}}\models Q_{k}$, and $w_{k+1}\models
Q_{k+1},\ldots,w_{n}\models Q_{n}$ and $u\models Q_{col}$ and the induction
hypothesis for
$t_{i}={q_{p_{i}}}\xrightarrow[Q_{col}]{a}\left({{Q_{1},\ldots,Q_{n}}}\right)$
we get
${o_{\ell}}\mathord{\left({\cdots{o_{i+1}}\mathord{\left({{copy_{k}}\mathord{\left({w}\right)}}\right)}}\right)}={o_{\ell}}\mathord{\left({\cdots{o_{i+1}}\mathord{\left({{v}:_{k}{{v}:_{k}{{w_{k}}:_{(k+1)}{{\cdots}:_{n}{w_{n}}}}}}\right)}}\right)}\models
q_{p^{\prime}}$
as required.
2. 2.
Assume that $o_{i}=push^{k}_{b}$, that we have
$t_{i}={q_{p_{i}}}\xrightarrow[Q_{col}]{b}\left({{Q_{1},\ldots,Q_{n}}}\right)\quad\text{and}\quad
Q_{1}\xrightarrow[Q^{\prime}_{col}]{a}(Q^{\prime}_{1})$
where the latter set of transitions is sound, and that we have
$t_{i-1}={q_{p_{i-1}}}\xrightarrow[Q^{\prime}_{col}]{a}\left({{Q^{\prime}_{1},Q_{2},\ldots,Q_{k}\cup
Q_{col},\ldots,Q_{n}}}\right)\ .$
To prove the induction hypothesis for the latter transition, we have to prove
that for any $w_{1}\models Q^{\prime}_{1}$, any $w_{2}\models Q_{2},\ldots$,
any $w_{k-1}\models Q_{k-1}$, any $w_{k}\models Q_{k}\cup Q_{col}$, any
$w_{k+1}\models Q_{k+1},\ldots$, any $w_{n}\models Q_{n}$ and any $u\models
Q^{\prime}_{col}$, that we have for
$w^{i-1}={{a}^{u}}:_{1}{{w_{1}}:_{2}{{\cdots}:_{n}{w_{n}}}}$ that
${o_{\ell}}\mathord{\left({\cdots{o_{i}}\mathord{\left({w^{i-1}}\right)}}\right)}\models
q_{p^{\prime}}$.
From the soundness of
${Q_{1}}\xrightarrow[Q^{\prime}_{col}]{a}\left({{Q^{\prime}_{1}}}\right)$ and
as $u\models Q^{\prime}_{col}$ and $w_{1}\models Q^{\prime}_{1}$ we have
${{a}^{u}}:_{1}{w_{1}}\models Q_{1}$.
Then, from ${{a}^{u}}:_{1}{w_{1}}\models Q_{1},w_{2}\models
Q_{2},\ldots,w_{n}\models Q_{n}$, and
${top_{k+1}}\mathord{\left({{pop_{k}}\mathord{\left({w}\right)}}\right)}=w_{k}\models
Q_{col}$, and induction for
$t_{i}={q_{p_{i}}}\xrightarrow[Q_{col}]{b}\left({{Q_{1},\ldots,Q_{n}}}\right)$,
we get
${o_{\ell}}\mathord{\left({\cdots{o_{i+1}}\mathord{\left({{push^{k}_{b}}\mathord{\left({w^{i-1}}\right)}}\right)}}\right)}={o_{\ell}}\mathord{\left({\cdots{o_{i+1}}\mathord{\left({{{b}^{w_{k}}}:_{1}{{{a}^{u}}:_{1}{{w_{1}}:_{2}{{\cdots}:_{n}{w_{n}}}}}}\right)}}\right)}\models
q_{p^{\prime}}$
as required.
3. 3.
Assume that $o=rew_{b}$, that we have
$t_{i}={q_{p_{i}}}\xrightarrow[Q_{col}]{b}\left({{Q_{1},\dots,Q_{n}}}\right)$
and that
$t_{i-1}={q_{p}}\xrightarrow[Q_{col}]{a}\left({{Q_{1},\dots,Q_{n}}}\right)\ .$
To prove the hypothesis for this later transition, we have to prove that for
any $w_{1}\models Q_{1},\ldots,$ for any $w_{n}\models Q_{n}$ and any
$u\models Q_{col}$, we have that for
$w^{i-1}={{a}^{u}}:_{1}{{w_{1}}:_{2}{{\cdots}:_{n}{w_{n}}}}$ we have
${o_{\ell}}\mathord{\left({\cdots{o_{i}}\mathord{\left({w^{i-1}}\right)}}\right)}\models
q_{p^{\prime}}$.
From $w_{1}\models Q_{1},\ldots,w_{n}\models Q_{n}$, and $u\models Q_{col}$,
and the hypothesis for
$t_{i}={q_{p_{i}}}\xrightarrow[Q_{col}]{b}\left({{Q_{1},\dots,Q_{n}}}\right)$,
we get
${o_{\ell}}\mathord{\left({\cdots{o_{i+1}}\mathord{\left({{rew_{b}}\mathord{\left({w^{i-1}}\right)}}\right)}}\right)}={o_{\ell}}\mathord{\left({\cdots{o_{i+1}}\mathord{\left({{{b}^{u}}:_{1}{{w_{1}}:_{2}{{\cdots}:_{n}{w_{n}}}}}\right)}}\right)}\models
q_{p^{\prime}}$
as required.
4. 4.
Assume that $o=noop$, that we have
$t_{i}={q_{p_{i}}}\xrightarrow[Q_{col}]{b}\left({{Q_{1},\dots,Q_{n}}}\right)$
and that
$t_{i-1}={q_{p}}\xrightarrow[Q_{col}]{a}\left({{Q_{1},\dots,Q_{n}}}\right)\ .$
To prove the hypothesis for this later transition, we have to prove that for
any $w_{1}\models Q_{1},\ldots,$ for any $w_{n}\models Q_{n}$ and any
$u\models Q_{col}$, we have that for
$w^{i-1}={{a}^{u}}:_{1}{{w_{1}}:_{2}{{\cdots}:_{n}{w_{n}}}}$ we have
${o_{\ell}}\mathord{\left({\cdots{o_{i}}\mathord{\left({w^{i-1}}\right)}}\right)}\models
q_{p^{\prime}}$.
From $w_{1}\models Q_{1},\ldots,w_{n}\models Q_{n}$, and $u\models Q_{col}$,
and the hypothesis for
$t_{i}={q_{p_{i}}}\xrightarrow[Q_{col}]{a}\left({{Q_{1},\dots,Q_{n}}}\right)$,
we get
${o_{\ell}}\mathord{\left({\cdots{o_{i+1}}\mathord{\left({{rew_{a}}\mathord{\left({w^{i-1}}\right)}}\right)}}\right)}={o_{\ell}}\mathord{\left({\cdots{o_{i+1}}\mathord{\left({{{a}^{u}}:_{1}{{w_{1}}:_{2}{{\cdots}:_{n}{w_{n}}}}}\right)}}\right)}\models
q_{p^{\prime}}$
as required.
This completes the proof. $\square$
### C.3 Complexity of Saturation for ECPDS
We argue that saturation for ECPDS maintains the same complexity as saturation
for CPDS.
###### Proposition C.1
The saturation construction for an order-$n$ CPDS $\mathcal{C}$ and an
order-$n$ stack automaton $A_{0}$ runs in $n$-EXPTIME.
_Proof._ The number of states of $A$ is bounded by
$2\uparrow_{(n-1)}\left({\ell}\right)$ where $\ell$ is the size of
$\mathcal{C}$ and $A_{0}$: each state in $\mathbb{Q}_{k}$ was either in
$A_{0}$ or comes from a transition in $\Delta_{k+1}$. Since the automata are
alternating, there is an exponential blow up at each order except at
order-$n$. Each iteration of the algorithm adds at least one new transition.
Only $2\uparrow_{n}\left({\ell}\right)$ transitions can be added. $\square$
The complexity can be reduced by a single exponential when runs of the stack
automata are “non-alternating at order-$n$”. In this case an exponential is
avoided by only adding a transition
${q_{p}}\xrightarrow[Q_{col}]{a}\left({{Q_{1},\ldots,Q_{n}}}\right)$ when
$Q_{n}$ contains at most one element.
We refer the reader to ICALP 2012 for a full discussion of non-alternation
since it relies on the original notion of collapsible pushdown system that we
have not defined here. ICALP 2012 describes the connection between our notion
of CPDS (using annotations) and the original notion, as well as defining non-
alternation at order-$n$ and arguing completeness for the restricted
saturation step. It is straightforward to extend this proof to include ECPDS
as in the proof of Lemma C.1 (Completeness of $\Pi$) above.
## Appendix D Definitions and Proofs for Multi-Stack CPDS
### D.1 Multi-Stack Collapsible Pushdown Automata
We formally define mutli-stack collapsible pushdown automata.
###### Definition D.1 (Multi-Stack Collapsible Pushdown Automata)
An order-$n$ _multi-stack collapsible pushdown automaton ( $n$-OCPDA)_ over
input alphabet $\Gamma$ is a tuple
$\mathcal{C}=\left({\mathcal{P},\Sigma,\mathcal{R}_{1},\ldots,\mathcal{R}_{m}}\right)$
where $\mathcal{P}$ is a finite set of control states, $\Sigma$ is a finite
stack alphabet, $\Gamma$ is a finite set of output symbols, and for each
$1\leq i\leq m$ we have a set of rules
$\mathcal{R}_{i}\subseteq\mathcal{P}\times\Sigma\times\Gamma\times\mathcal{O}_{n}\times\mathcal{P}$.
Configurations of an OCPDA are defined identically to configurations for
OCPDS. We have a transition
$\langle{p},{w_{1},\ldots,w_{m}}\rangle\xrightarrow{\gamma}\langle{p^{\prime}},{w_{1},\ldots,w_{i-1},w^{\prime}_{i},w_{i+1},\ldots,w_{m}}\rangle$
whenever
$r=\left({{p},{a},{\gamma},{o},{p^{\prime}}}\right)\in\mathcal{R}_{i}$ with
$a={top_{1}}\mathord{\left({w}\right)}$,
$w^{\prime}_{i}={o}\mathord{\left({w_{i}}\right)}$.
### D.2 Regular Sets of Configurations
We prove several properties about Definition 4.4 (Regular Set of
Configurations).
###### Property D.1
Regular sets of configurations of a multi-stack CPDS
1. 1.
form an effective boolean algebra,
2. 2.
the emptiness problem is decidable in PSPACE,
3. 3.
the membership problem is decidable in linear time.
_Proof._ We first prove $(\ref{item:bool-alg})$. We recall from [8] that stack
automata form an effective boolean algebra. Given two regular sets $\chi_{1}$
and $\chi_{2}$, we can form $\chi=\chi_{1}\cup\chi_{2}$ as the simple union of
the two sets of tuples. We obtain the intersection of $\chi_{1}$ and
$\chi_{2}$ by defining $\chi=\chi_{1}\cap\chi_{2}$ via a product construction.
That is,
$\chi=\left\\{{\left({p,A_{1}\cap A^{\prime}_{1},\ldots,A_{m}\cap
A^{\prime}_{m}}\right)}\ \left|\
{\begin{array}[]{c}\left({p,A_{1},\ldots,A_{m}}\right)\in\chi_{1}\ \land\\\
\left({p,A^{\prime}_{1},\ldots,A^{\prime}_{m}}\right)\in\chi_{2}\end{array}}\right.\right\\}\
.$
It remains to define the complement $\overline{\chi}$ of a set $\chi$. Let
$\chi=\chi_{1}\cup\cdots\cup\chi_{\ell}$ where each $\chi_{i}$ is a singleton
set of tuples. Observe that
$\overline{\chi}=\overline{\chi_{1}}\cap\cdots\cap\overline{\chi_{\ell}}$.
Hence, we define for a singleton $\chi_{i}$ its complement
$\overline{\chi_{i}}$. Let $A$ be a stack automaton accepting all stacks.
Furthermore, let $\chi_{i}$ contain only
$\left({p,A_{1},\ldots,A_{\ell}}\right)$. We define
$\begin{array}[]{rcl}\overline{\chi_{i}}&=&\left\\{{\left({p^{\prime},A,\ldots,A}\right)}\
\left|\ {p\neq p^{\prime}\in\mathcal{P}}\right.\right\\}\ \cup\\\
&&\left\\{{\left({p,A,\ldots,A,\overline{A_{j}},A,\ldots,A}\right)}\ \left|\
{1\leq j\leq m}\right.\right\\}\ .\end{array}$
That is, either the control state does not match, or at least one of the $m$
stacks does not match.
We now prove $(\ref{item:emptiness})$. We know from [8] that the emptiness
problem for a stack automaton is PSPACE. By checking all tuples to find some
tuple $\left({p,A_{1},\ldots,A_{m}}\right)$ such that $A_{i}$ is non-empty for
all $i$, we have a PSPACE algorithm for determining the emptiness of a regular
set $\chi$.
Finally, we show $(\ref{item:membership})$, recalling from [8] that the
membership problem for stack automata is linear time. To check whether
$\langle{p},{w_{1},\ldots,w_{m}}\rangle$ is contained in $\chi$ we check each
tuple $\left({p,A_{1},\ldots,A_{m}}\right)\in\chi$ to see if $w_{i}$ is
contained in $A_{i}$ for all $i$. This requires linear time. $\square$
## Appendix E Proofs for Ordered CPDS
### E.1 Proofs for Simulation by $\mathcal{C}^{R}$
We prove Lemma 5.1 ($\mathcal{C}^{R}$ simulates $\mathcal{C}$) via Lemma E.1
and Lemma E.2 below.
###### Lemma E.1
Given an $n$-OCPDS $\mathcal{C}$ and control states
${p_{\text{in}}},{p_{\text{out}}}$, we have
$\langle{{p_{\text{in}}}},{\perp_{n},\ldots,\perp_{n},w}\rangle\longrightarrow\cdots\longrightarrow\langle{{p_{\text{out}}}},{\perp_{n},\ldots,\perp_{n}}\rangle\
.$
only if
$\langle{{p_{\text{in}}}},{w}\rangle\in{Pre^{*}_{\mathcal{C}^{R}}}\mathord{\left({A}\right)}$,
where $A$ is the $\mathcal{P}$-stack automaton accepting only the
configuration $\langle{{p_{\text{out}}}},{\perp_{n}}\rangle$.
_Proof._ Take such a run
$\langle{{p_{\text{in}}}},{\perp_{n},\ldots,\perp_{n},w}\rangle\longrightarrow\cdots\longrightarrow\langle{{p_{\text{out}}}},{\perp_{n},\ldots,\perp_{n}}\rangle$
of $\mathcal{C}$. Observe that the run can be partitioned into
$\tau_{0}\sigma_{1}\tau_{1}\ldots\sigma_{\ell}\tau_{\ell}$ where during each
$\tau_{i}$, the first $(m-1)$ stacks are $\perp_{n}$, and, during each
$\sigma_{i}$, there is at least one stack in the first $(m-1)$ stacks that is
not $\perp_{n}$. Let $p^{1}_{i}$ be the control state of the first
configuration of $\tau_{i}$, $p^{2}_{i}$ be the control state in the final
configuration of $\tau_{i}$, $p^{3}_{i}$ be the control state at the beginning
of each $\sigma_{i}$, and $p^{4}_{i}$ be the control state at the end of each
$\sigma_{i}$. Note, $p^{4}_{\ell}={p_{\text{out}}}$ and
$p^{1}_{1}={p_{\text{in}}}$. Next, let $r_{i}$ be the rule fired between the
final configuration of $\tau_{i-1}$ and the first configuration of
$\sigma_{i}$ (if it exists). Finally, let $w_{i}$ be the contents of stack $m$
in the final configuration of each $\tau_{i}$. Note $w_{\ell}=w$.
We proceed by backwards induction from $i=\ell$ down to $i=0$. Trivially it is
the case that
$\langle{p^{4}_{\ell}},{w_{\ell}}\rangle\in{Pre^{*}_{\mathcal{C}^{R}}}\mathord{\left({A}\right)}$.
In the inductive step, first assume
$\langle{p^{4}_{i}},{w_{i}}\rangle\in{Pre^{*}_{\mathcal{C}^{R}}}\mathord{\left({A}\right)}$.
We have the final configuration of $\tau_{i}$ is
$\langle{p^{4}_{i}},{\perp_{n},\ldots,\perp_{n},w_{i}}\rangle$. Let
$\langle{p^{3}_{i}},{\perp_{n},\ldots,\perp_{n},w^{\prime}}\rangle$ be the
first configuration of $\tau_{i}$. Note, since we assume all rules of the form
$\left({{p_{1}},{\perp},{o},{p_{2}}}\right)$ have $o=push^{n}_{a}$ for some
$a$, and during $\tau_{i}$ the first $(m-1)$ stacks are empty, we know that no
rule from $\mathcal{R}_{1},\ldots,\mathcal{R}_{m-1}$ was used during
$\tau_{i}$. Thus, $\tau_{i}$ is a run of $\mathcal{C}^{R}$ using only rules
from $\mathcal{R}_{m}$. Hence, we have
$\langle{p^{3}_{i}},{w^{\prime}}\rangle\in{Pre^{*}_{\mathcal{C}^{R}}}\mathord{\left({A}\right)}$.
Now consider $\sigma_{i}$ with
$\langle{p^{3}_{i}},{\perp_{n},\ldots,\perp_{n},w^{\prime}}\rangle$ appended
to the end. Suppose we have that
$r_{i-1}=\left({{p^{4}_{i-1}},{\perp},{push^{n}_{b}},{p^{1}_{i}}}\right)\in\mathcal{R}_{j}$.
We thus have a run
$\langle{p^{1}_{i}},{w^{\prime}_{1},\ldots,w^{\prime}_{m-1},w_{i-1}}\rangle\xrightarrow{r^{1}}\cdots\xrightarrow{r^{\ell-1}}\langle{p^{2}_{i}},{w^{\prime\prime}_{1},\ldots,w^{\prime\prime}_{m}}\rangle\xrightarrow{r^{\ell}}\langle{p^{3}_{i}},{\perp_{n},\ldots,\perp_{n},w^{\prime}}\rangle$
where $w^{\prime}_{j}={push^{n}_{b}}\mathord{\left({\perp_{n}}\right)}$ and
$w^{\prime}_{j^{\prime}}=\perp_{n}$ for all $j^{\prime}\neq j$. Since it is
not the case that the first $(m-1)$ stacks are empty, we know that only
generating rules from $\mathcal{R}_{m}$ can be used during this run. Let
${top_{1}}\mathord{\left({w_{i-1}}\right)}=a$. From this run we can
immediately project a sequence
$\left({{p^{0}},{a^{1}},{o^{1}},{p^{1}}}\right)\left({{p^{1}},{a^{2}},{o^{2}},{p^{2}}}\right)\ldots\left({{p^{\ell^{\prime}-1}},{a^{\ell}},{o^{\ell^{\prime}}},{p^{\ell^{\prime}}}}\right)\in{\mathcal{L}^{{b},{j}}_{{p^{1}_{i}},{a},{p^{3}_{i}}}}\mathord{\left({\mathcal{C}^{L}}\right)}$
such that we have
$w^{\prime}={o^{\ell^{\prime}}}\mathord{\left({\cdots{o^{1}}\mathord{\left({w_{i-1}}\right)}}\right)}$,
$p^{0}=p^{1}_{i}$ and $p^{\ell^{\prime}}=p^{3}_{i}$. Since we have
$\langle{p^{3}_{i}},{w^{\prime}}\rangle\in{Pre^{*}_{\mathcal{C}^{R}}}\mathord{\left({A}\right)}$
and a rule
$\left({{p^{4}_{i-1}},{a},{{\mathcal{L}^{{b},{j}}_{{p^{1}_{i}},{a},{p^{3}_{i}}}}\mathord{\left({\mathcal{C}^{L}}\right)}},{p^{3}_{i}}}\right)$
in $\mathcal{C}^{R}$, we thus have
$\langle{p^{4}_{i-1}},{w_{i-1}}\rangle\in{Pre^{*}_{\mathcal{C}^{R}}}\mathord{\left({A}\right)}$
as required.
Hence, when $i=0$, we have
$\langle{{p_{\text{in}}}},{w}\rangle\in{Pre^{*}_{\mathcal{C}^{R}}}\mathord{\left({A}\right)}$,
completing the proof. $\square$
###### Lemma E.2
Given an $n$-OCPDS $\mathcal{C}$ and control states
${p_{\text{in}}},{p_{\text{out}}}$, we have
$\langle{{p_{\text{in}}}},{\perp_{n},\ldots,\perp_{n},w}\rangle\longrightarrow\cdots\longrightarrow\langle{{p_{\text{out}}}},{\perp_{n},\ldots,\perp_{n}}\rangle\
.$
whenever
$\langle{{p_{\text{in}}}},{w}\rangle\in{Pre^{*}_{\mathcal{C}^{R}}}\mathord{\left({A}\right)}$,
where $A$ is the $\mathcal{P}$-stack automaton accepting only the
configuration $\langle{{p_{\text{out}}}},{\perp_{n}}\rangle$.
_Proof._ Since
$\langle{{p_{\text{in}}}},{w}\rangle\in{Pre^{*}_{\mathcal{C}^{R}}}\mathord{\left({A}\right)}$
we have a run of $\mathcal{C}^{R}$ of the form $\sigma_{1}\ldots\sigma_{\ell}$
where the rules used to connect the last configuration of $\sigma_{i}$ to
$\sigma_{i+1}$ are of the form
$\left({{p^{\prime}_{i}},{a},{{\mathcal{L}_{g}}},{p_{i+1}}}\right)$ and no
other rules of this form are used otherwise. Thus, let $p^{\prime}_{i}$ denote
the control state at the end of $\sigma_{i}$ and $p_{i}$ denote the control
state in the first configuration of $\sigma_{i}$. Similarly, let
$w^{\prime}_{i}$ denote the stack contents at the end of $\sigma_{i}$ and
$w_{i}$ the stack contents at the beginning.
We proceed by induction from $i=\ell$ down to $i=1$. In the base case, we
immediately have a run from
$\langle{p_{\ell}},{\perp_{n},\ldots,\perp_{n},w_{\ell}}\rangle$ to
$\langle{p^{\prime}_{\ell}},{\perp_{n},\ldots,\perp_{n}}\rangle$. Now, assume
the we have a run from
$\langle{p^{\prime}_{i}},{\perp_{n},\ldots,\perp_{n},w^{\prime}_{i}}\rangle$
to the final configuration. Since we have a run to this configuration from
$\langle{p_{i}},{w_{i}}\rangle$ to
$\langle{p^{\prime}_{i}},{w^{\prime}_{i}}\rangle$ in $\mathcal{C}^{R}$ that
uses only ordinary rules, we can execute the same run from
$\langle{p_{i}},{\perp_{n},\ldots,\perp_{n},w_{i}}\rangle$ to reach
$\langle{p^{\prime}_{i}},{\perp_{n},\ldots,\perp_{n},w^{\prime}_{i}}\rangle$.
Now consider the rule
$\left({{p^{\prime}_{i-1}},{a},{{\mathcal{L}_{g}}},{p_{i}}}\right)$ that
connects $\sigma_{i-1}$ and $\sigma_{i}$. We have
${\mathcal{L}_{g}}={\mathcal{L}^{{b},{j}}_{{p^{1}_{i}},{a},{p_{i}}}}\mathord{\left({\mathcal{C}^{L}}\right)}$
for some $p^{1}_{i}$, $b$ and $j$, and there is a rule
$\left({{p^{\prime}_{i-1}},{\perp},{push^{n}_{b}},{p^{1}_{i}}}\right)\in\mathcal{R}_{j}$
of $\mathcal{C}$. Furthermore, there is a sequence
$\left({{p^{0}},{a^{1}},{o^{1}},{p^{1}}}\right)\left({{p^{1}},{a^{2}},{o^{2}},{p^{2}}}\right)\ldots\left({{p^{\ell^{\prime}-1}},{a^{\ell}},{o^{\ell^{\prime}}},{p^{\ell^{\prime}}}}\right)\in{\mathcal{L}_{g}}$
such that
$w_{i}={o^{\ell^{\prime}}}\mathord{\left({\cdots{o^{1}}\mathord{\left({w^{\prime}_{i-1}}\right)}}\right)}$,
$p^{0}=p^{1}_{i}$, and $p^{\ell^{\prime}}=p_{i}$.
From the definition of $\mathcal{C}^{L}$, this sequence immediately describes
a run
$\begin{array}[]{rcl}\langle{p^{\prime}_{i-1}},{\perp_{n},\ldots,\perp_{n},w^{\prime}_{i-1}}\rangle&\longrightarrow&\langle{p^{1}_{i}},{\perp_{n},\ldots,{push^{n}_{b}}\mathord{\left({\perp_{n}}\right)},\ldots,\perp_{n},w^{\prime}_{i-1}}\rangle\\\
&\longrightarrow&\cdots\\\
&\longrightarrow&\langle{p_{i}},{\perp_{n},\ldots,\perp_{n},w_{i}}\rangle\end{array}$
of $\mathcal{C}$. Thus we have a run from
$\langle{p^{\prime}_{i-1}},{\perp_{n},\ldots,\perp_{n},w^{\prime}_{i-1}}\rangle$
to the final configuration, to complete the inductive case.
Finally, when $i=1$, we repeat the first half of the argument above to obtain
a run from $\langle{p_{1}},{\perp_{n},\ldots,\perp_{n},w_{1}}\rangle$, and
since $p_{1}={p_{\text{in}}}$ and $w_{1}=w$ we have a run of $\mathcal{C}$ as
required. $\square$
### E.2 Proofs for Language Emptiness for OCPDS
We prove Lemma 5.2 (Language Emptiness for OCPDS) below.
_Proof._ By standard product construction arguments, a run of
$\mathcal{C}_{\emptyset}$ can be projected into runs of $\mathcal{C}^{L}$ and
$\mathcal{T}^{A_{i}}_{{t},{t^{\prime}}}$ and vice-versa. We need only note
that in any control state $\left({p,t_{1}}\right)$ of
$\mathcal{C}_{\emptyset}$, the corresponding state in $\mathcal{C}^{L}$ is
always $\left({p,{top_{1}}\mathord{\left({t_{1}}\right)}}\right)$. $\square$
### E.3 Global Reachability
We provide an inductive proof of global reachability for ordered CPDS.
_Proof._ Take $A_{m}={Pre^{*}_{\mathcal{C}^{R}}}\mathord{\left({A}\right)}$
from Lemma 5.1 ($\mathcal{C}^{R}$ simulates $\mathcal{C}$). Furthermore, let
$A_{\perp}$ be the stack automaton accepting only $\perp_{n}$ from its initial
state. For each control state $p$, we have that
$\left({p,A_{\perp},\ldots,A_{\perp},A_{m}}\right)$ represents all
configurations $\langle{p},{\perp_{n},\ldots,\perp_{n},w_{m}}\rangle$ for
which there is a run to
$\langle{{p_{\text{out}}}},{\perp_{n},\ldots,\perp_{n}}\rangle$ when $A_{m}$
is restricted to have initial state $q_{p}$.
Hence, inductively assume for $i+1$ that we have a finite set of tuples $\chi$
such that for each configuration
$\langle{p},{\perp_{n},\ldots,\perp_{n},w_{i+1},\ldots,w_{m}}\rangle$ for
which there is a run to
$\langle{{p_{\text{out}}}},{\perp_{n},\ldots,\perp_{n}}\rangle$ there is a
tuple $\left({p,A_{\perp},\ldots,A_{\perp},A_{i+1},\ldots,A_{m}}\right)$ such
that $w_{j}$ is accepted by $A_{j}$ for each $j$.
Now consider any configuration
$\langle{p},{\perp_{n},\ldots,\perp_{n},w_{i},\ldots,w_{m}}\rangle$ that can
reach the final configuration. We know the run goes via some
$\langle{p^{\prime}},{\perp_{n},\ldots,\perp_{n},w^{\prime}_{i+1},\ldots,w^{\prime}_{m}}\rangle$
accepted by some tuple
$\left({p^{\prime},A_{\perp},\ldots,A_{\perp},A_{i+1},\ldots,A_{m}}\right)\in\chi$.
Furthermore, we know from the proof of correctness of the extended saturation
algorithm, that there is a run of the $i$ stack OCPDS
$\mathcal{C}_{\emptyset}$ from
$\langle{\left({p,t_{i+1},\ldots,t_{m}}\right)},{\perp_{n},\ldots,\perp_{n},w_{i}}\rangle$
to
$\langle{\left({p^{\prime},t^{\prime}_{i+1},\ldots,t^{\prime}_{m}}\right)},{\perp_{n},\ldots,\perp_{n}}\rangle$
where
1. 1.
$t^{\prime}_{j}$ is the initial transition of $A_{j}$ accepting
$w^{\prime}_{j}$, and
2. 2.
the sequence of stack operations to the $j$th stack $o_{1},\ldots,o_{\ell}$
connected to this run give
$w^{\prime}_{j}={o_{\ell}}\mathord{\left({\cdots{o_{1}}\mathord{\left({w_{j}}\right)}}\right)}$,
and
3. 3.
$w_{j}$ can be accepted by first taking transition $t_{j}$ and thereafter only
transitions in $A_{j}$.
Thus, let $A_{i}$ be
${Pre^{*}_{\mathcal{C}_{\emptyset}}}\mathord{\left({A}\right)}$ where $A$
accepts
$\langle{\left({p^{\prime},t^{\prime}_{i+1},\ldots,t^{\prime}_{m}}\right)},{\perp_{n}}\rangle$.
Restrict $A_{i}$ to have initial state
$q_{\left({p,t_{i+1},\ldots,t_{m}}\right)}$ and let $A^{t_{j}}_{j}$ be the
automaton $A_{j}$ with the transition $t_{j}$ added from a new state, which is
designated as the initial state. Thus, for each configuration
$\langle{p},{\perp_{n},\ldots,\perp_{n},w_{i},\ldots,w_{m}}\rangle$, there is
a tuple
$\left({p,A_{\perp},\ldots,A_{\perp},A_{i},A^{t_{i+1}}_{i+1},\ldots,A^{t_{m}}_{m}}\right)$
such that $w_{i}$ is accepted by $A_{i}$ and $w_{j}$ is accepted by
$A^{t_{j}}_{j}$ for all $j>i$. This results in a finite set of tuples
$\chi^{\prime}$ satisfying the induction hypothesis.
Thus, after $i=1$ we obtain a finite set of tuples $\chi$ of the form
$\left({p,A_{1},\ldots,A_{m}}\right)$ representing all configurations that can
reach $\langle{{p_{\text{out}}}},{\perp_{n},\ldots,\perp_{n}}\rangle$, as
required. $\square$
### E.4 Complexity
Assume $n>1$. Our control state reachability algorithm requires
$2\uparrow_{m(n-1)}\left({\ell}\right)$ time, where $\ell$ is polynomial in
the size of the OCPDS. Beginning with stack $m$, the saturation algorithm can
add at most
${\mathcal{O}}\mathord{\left({2\uparrow_{n-1}\left({\ell}\right)}\right)}$
transitions over the same number of iterations. Each of these iterations may
require analysis of some $\mathcal{C}_{\emptyset}$ which has
${\mathcal{O}}\mathord{\left({2\uparrow_{n-1}\left({\ell}\right)}\right)}$
control states and thus the stack-automaton constructed by saturation over
$\mathcal{C}_{\emptyset}$ may have up to
${\mathcal{O}}\mathord{\left({2\uparrow_{2(n-1)}\left({\ell}\right)}\right)}$
transitions. By continuing in this way, we have at most
${\mathcal{O}}\mathord{\left({2\uparrow_{(m-1)(n-1)}\left({\ell}\right)}\right)}$
control states when there is only one stack remaining, and thus the number of
transitions, and the total running time of the algorithm is
${\mathcal{O}}\mathord{\left({2\uparrow_{m(n-1)}\left({\ell}\right)}\right)}$.
This also gives us at most
${\mathcal{O}}\mathord{\left({2\uparrow_{mn}\left({\ell}\right)}\right)}$
tuples in the solution to the global reachability problem.
## Appendix F Phase-Bounded CPDS
Phase-bounding [29] for multi-stack pushdown systems is a restriction where
each computation can be split into a fixed number of phases. During each
phase, characters can only be removed from one stack, but push actions may
occur on any stack.
###### Definition F.1 (Phase-Bounded CPDS)
Given a fixed number $\zeta$ of phases, an order-$n$ _phase-bounded CPDS_
($n$-PBCPDS) is an $n$-MCPDS with the restriction that each run $\sigma$ can
be partitioned into $\sigma_{1}\ldots\sigma_{\zeta}$ and for all $i$, if some
transition in $\sigma_{i}$ by $r\in\mathcal{R}_{j}$ on stack $j$ for some $j$
is consuming, then all consuming transitions in $\sigma_{i}$ are by some
$r^{\prime}\in\mathcal{R}_{j}$ on stack $j$.
We give a direct111For PDS, phase-bounded reachability can be reduced to
ordered PDS. We do not know if this holds for CPDS, and prefer instead to give
a direct algorithm. algorithm for deciding the reachability problem over
phase-bounded CPDSs. We remark that Seth [28] presented a saturation technique
for order-$1$ phase-bounded pushdown systems. Our algorithm was developed
independently of Seth’s, but our product construction can be compared with
Seth’s automaton $T_{i}$.
###### Theorem F.1 (Decidability of the Reachability Problems)
For $n$-PBCPDSs the control state reachability problem and the global control
state reachability problem are decidable.
In Appendix F.3 we show that our control state reachability algorithm will
require
${\mathcal{O}}\mathord{\left({2\uparrow_{m(n-1)}\left({\ell}\right)}\right)}$
time, where $\ell$ is polynomial in the size of the PBCPDS, and we have at
most ${\mathcal{O}}\mathord{\left({2\uparrow_{mn}\left({\ell}\right)}\right)}$
tuples in the solution to the global reachability problem.
#### Control State Reachability
A run of the PBCPDS will be $\sigma_{1}\ldots\sigma_{\zeta}$, assuming
(w.l.o.g.) that all phases are used. We can guess (or enumerate) the sequence
$p_{0}p_{1}\ldots p_{\zeta}$ of control states occurring at the boundaries of
each $\sigma_{i}$. That is, $\sigma_{i}$ ends with control state $p_{i}$,
$p_{\zeta}$ is the target control state, and $p_{0}$ is the initial control
state. We also guess for each $i$, the stack $\iota_{i}$ that may perform
consuming operations between $p_{i-1}$ and $p_{i}$. Our algorithm iterates
from $i=\zeta$ down to $i=0$.
We begin with the stack automata $A^{1}_{\zeta},\ldots,A^{m}_{\zeta}$ which
each accept $\langle{p_{\zeta}},{w}\rangle$ for all stacks $w$. Note we can
vary these automata to accept any regular set of stacks we wish.
Thus, $A^{1}_{i},\ldots,A^{m}_{i}$ will characterise a possible set of stack
contents at the end of phase $i$. We show below how to construct
$A^{1}_{i-1},\ldots,A^{m}_{i-1}$ given $A^{1}_{i},\ldots,A^{m}_{i}$. This is
repeated until we have $A^{1}_{0},\ldots,A^{m}_{0}$. We then check, for each
$j$, that $\langle{p_{0}},{\perp_{n}}\rangle$ is accepted by $A^{j}_{0}$. This
is the case iff we have a positive instance of the reachability problem.
We construct $A^{1}_{i-1},\ldots,A^{m}_{i-1}$ from
$A^{1}_{i},\ldots,A^{m}_{i}$. For each $j\neq\iota_{i}$ we build $A^{j}_{i-1}$
by adding to $A^{j}_{i}$ a brand new set of initial states $q_{p}$ and a
guessed transition
$t_{j}={q_{p_{i-1}}}\xrightarrow[Q_{col}]{a}\left({{Q_{1},\ldots,Q_{n}}}\right)$
with $Q_{col},Q_{1},\ldots,Q_{n}$ being states of $A^{j}_{i}$ and
$q_{p_{i-1}}$ being one of the new states. The idea is $t_{j}$ will be the
initial transition accepting $\langle{p_{i-1}},{w}\rangle$ where $w$ is stack
$j$ at the beginning of phase $i$. By guessing an accompanying
$t^{\prime}_{j}$ of $A^{j}_{i}$ we can build
$\mathcal{T}^{A^{j}_{i}}_{{t_{j}},{t^{\prime}_{j}}}$ (by instantiating
Definition 3.2 (Transition Automata) with $A=A^{j}_{i}$, $t=t_{j}$ and
$t^{\prime}=t^{\prime}_{j}$) for which there will be an accepting run if the
updates to stack $j$ during phase $i$ are concordant with the introduction of
transition $t_{j}$.
Thus, for each $j\neq\iota_{i}$ we have $A^{j}_{i-1}$ and
$\mathcal{T}^{A^{j}_{i}}_{{t_{j}},{t^{\prime}_{j}}}$. We now consider the
$\iota_{i}$th stack. We build a CPDS $\mathcal{C}_{i}$ that accurately models
stack $\iota_{i}$ and tracks each
$\mathcal{T}^{A^{j}_{i}}_{{t_{j}},{t^{\prime}_{j}}}$ in its control state. We
ensure that $\mathcal{C}_{i}$ has a run from $\langle{p_{i-1}},{w}\rangle$ to
$\langle{p_{i}},{w^{\prime}}\rangle$ for some $w$ and $w^{\prime}$ iff there
is a corresponding run over the $\iota_{i}$th stack of $\mathcal{C}$ that
updates the remaining stacks $j$ in concordance with each guessed $t_{j}$.
Thus, we define $A^{\iota_{i}}_{i-1}$ to be the automaton recognising
${Pre^{*}_{\mathcal{C}_{i}}}\mathord{\left({A^{\iota_{i}}_{i}}\right)}$
constructed by saturation. The construction of $\mathcal{C}_{i}$ (given below)
follows the standard product construction of a CPDS with several finite-state
automata.
Note $\mathcal{C}_{i}$ is looking for a run from $p_{i-1}$ to $p_{i}$
concordant with runs of $t_{j}$ to $t^{\prime}_{j}$ for each $j$. To let
$\mathcal{C}_{i}$ start in $p_{i-1}$ and finish in $p_{i}$, we have an initial
transition from $p_{i-1}$ to $\left({p_{i-1},t_{1},\ldots,t_{m}}\right)$.
Thereafter, the components are updated as in a standard product construction.
When $\left({p_{i},t^{\prime}_{1},\ldots,t^{\prime}_{m}}\right)$ is reached,
there is a final transition to $p_{i}$. To ease notation, we use dummy
variables
$t_{\iota_{i}}=t^{\prime}_{\iota_{i}}=t^{\iota_{i}}=t^{\iota_{i}}_{1}$ for the
transition automaton component of the $\iota_{i}$th stack (for which we do not
have a $t$ and $t^{\prime}$ to track).
In the definition below, the first line of the definition of $\mathcal{R}^{i}$
gives the initial and final transitions, the second line models rules
operating on stack $\iota_{i}$, and the final line models generating
operations occurring on the $j$th stack for $j\neq\iota_{i}$.
###### Definition F.2 ($\mathcal{C}_{i}$)
Given for all $1\leq j\neq\iota_{i}\leq m$ a transition automaton
$\mathcal{T}_{j}=\mathcal{T}^{A^{j}_{i}}_{{t_{j}},{t^{\prime}_{j}}}$ and a
phase-bounded CPDS
$\mathcal{C}=\left({\mathcal{P},\Sigma,\mathcal{R}_{1},\ldots,\mathcal{R}_{m}}\right)$
and control states $p_{i-1}$, $p_{i}$, we define the CPDS
$\mathcal{C}_{i}=\left({\left\\{{p_{i-1},p_{i}}\right\\}\cup\mathcal{P}^{i},\mathcal{R}^{i},\Sigma}\right)$
where, letting
$t_{\iota_{i}}=t^{\prime}_{\iota_{i}}=t^{\iota_{i}}=t^{\iota_{i}}_{1}$ be
dummy transitions for technical convenience, and letting $t^{j}$ for all
$j\neq\iota_{i}$ range over all states of $\mathcal{T}_{j}$, we have
* •
$\mathcal{P}^{i}$ contains all states $\left({p,t^{1},\ldots,t^{m}}\right)$
where $p\in\mathcal{P}$, and
* •
the rules $\mathcal{R}^{i}$ of $\mathcal{C}_{i}$ are
$\begin{array}[]{l}\left\\{{\left({{p_{i-1}},{a},{noop},{\left({p_{i-1},t_{1},\ldots,t_{m}}\right)}}\right),\left({{\left({p_{i},t^{\prime}_{1},\ldots,t^{\prime}_{m}}\right)},{a},{noop},{p_{i}}}\right)}\
\left|\ {a\in\Sigma}\right.\right\\}\ \cup\\\
\left\\{{\left({{\left({p,t^{1},\ldots,t^{m}}\right)},{a},{o},{\left({p^{\prime},t^{1}_{1},\ldots,t^{m}_{1}}\right)}}\right)}\
\left|\
{\begin{array}[]{l}\left({{p},{a},{o},{p^{\prime}}}\right)\in\mathcal{R}_{\iota_{i}}\\\
\forall j^{\prime}\neq j\ .\
t^{j^{\prime}}\xrightarrow{\left({{p},{\\_},{noop},{p^{\prime}}}\right)}t^{j^{\prime}}_{1}\end{array}}\right.\right\\}\
\cup\\\ \left\\{{\left({{p_{1}},{a},{o},{p_{2}}}\right)}\ \left|\
{\begin{array}[]{c}p_{1}=\left({p,t^{1},\ldots,t^{j},\ldots
t^{m}}\right)\land\par\par
p_{2}=\left({p^{\prime},t^{1}_{1},\ldots,t^{j}_{1},\ldots t^{m}_{1}}\right)\\\
\land\ \left({{p},{b},{o},{p^{\prime}}}\right)\in\mathcal{R}_{j}\land\par
t^{j}\xrightarrow{\left({{p},{b},{o},{p^{\prime}}}\right)}t^{j}_{1}\ \land\\\
\forall j^{\prime}\neq j\ .\
t^{j^{\prime}}\xrightarrow{\left({p,\\_,noop,p^{\prime}}\right)}t^{j^{\prime}}_{1}\end{array}}\right.\right\\}\
.\end{array}$
We state the correctness of our reduction, deferring the proof to Appendix
F.2.
###### Lemma F.1 (Simulation of a PBCPDS)
Given a phase-bounded CPDS $\mathcal{C}$ control states $p_{0}$ and
$p_{\zeta}$, there is a run of $\mathcal{C}$ from
$\langle{p_{0}},{w_{1},\ldots,w_{m}}\rangle$ to
$\langle{p_{\zeta}},{w^{\prime}_{1},\ldots,w^{\prime}_{m}}\rangle$ iff for
each $1\leq j\leq m$, we have that $\langle{p_{0}},{w_{j}}\rangle$ is accepted
by $A^{j}_{0}$.
### F.1 Global Reachability
$A^{1}_{0},\ldots,A^{m}_{0}$ were obtained by a finite sequence of non-
deterministic choices ranging over a finite number of values. Let $\chi$ be
the therefore finite set of tuples $\left({p_{0},A_{1},\ldots,A_{m}}\right)$
for each sequence as above, where $A_{i}$ is $A^{i}_{0}$ with initial state
$q_{p_{0}}$. From Lemma F.1, we have a regular solution to the global control
state reachability problem as required.
### F.2 Proofs for Control-State Reachability
In this section we prove Lemma F.1 (Simulation of a PBCPDS) via Lemma F.2 and
Lemma F.3 below.
###### Lemma F.2
Given a phase-bounded CPDS $\mathcal{C}$ control states $p_{0}$ and
$p_{\zeta}$, there is a run of $\mathcal{C}$ from
$\langle{p_{0}},{w_{1},\ldots,w_{m}}\rangle$ to
$\langle{p_{\zeta}},{w^{\prime}_{1},\ldots,w^{\prime}_{m}}\rangle$ only if for
each $1\leq j\leq m$, we have that $\langle{p_{0}},{w_{j}}\rangle$ is accepted
by $A^{j}_{0}$.
_Proof._ Take a run of $\mathcal{C}$ from
$\langle{p_{0}},{w^{1}_{0},\ldots,w^{m}_{0}}\rangle$ to
$\langle{p_{\zeta}},{w^{1}_{\zeta},\ldots,w^{m}_{\zeta}}\rangle$ and split it
into phases $\sigma_{1}\ldots\sigma_{\zeta}$. Let $p_{i}$ be the control state
at the end of each $\sigma_{i}$, and $p_{0}$ be the control state at the
beginning of $\sigma_{1}$. Similarly, let $w^{j}_{i}$ be the stack contents of
stack $j$ at the end of $\sigma_{i}$. We include, for convenience, the
transition from the end of $\sigma_{i}$ to the beginning of $\sigma_{i+1}$ in
$\sigma_{i+1}$. Thus, the last configuration of $\sigma_{i}$ is also the first
configuration of $\sigma_{i+1}$.
We proceed by induction from $i=\zeta$ down to $i=1$. In the base case we know
by definition that $\langle{p_{\zeta}},{w^{j}_{\zeta}}\rangle$ is accepted by
$A^{j}_{\zeta}$.
Hence, assume $\langle{p_{i+1}},{w^{j}_{i+1}}\rangle$ is accepted by
$A^{j}_{i+1}$. We show the case for $i$. First consider $\iota_{i}$. Take the
run
$\langle{p_{i}},{w^{1}_{i},\ldots,w^{m}_{i}}\rangle\longrightarrow\cdots\longrightarrow\langle{p_{i+1}},{w^{1}_{i+1},\ldots,w^{m}_{i+1}}\rangle\
.$
We want to find a run
$\langle{p_{i}},{w^{\iota_{i}}_{i}}\rangle\longrightarrow\langle{\left({p_{i},t_{1},\ldots,t_{m}}\right)},{w^{\iota_{i}}_{i}}\rangle\longrightarrow\cdots\longrightarrow\langle{\left({p_{i+1},t^{\prime}_{1},\ldots,t^{\prime}_{m}}\right)},{w^{\iota_{i}}_{i+1}}\rangle\longrightarrow\langle{p_{1}},{w^{\iota_{i}}_{i+1}}\rangle$
of $\mathcal{C}_{i}$, giving us that
$\langle{p_{i}},{w^{\iota_{i}}_{i}}\rangle$ is accepted by
$A^{\iota_{i}}_{i}$. This is almost by definition, except we need to prove for
each $j\neq\iota_{i}$ that there is a sequence $t^{0},\ldots,t^{\ell}$ that is
also the projection of the run of $\mathcal{C}_{i}$ to the $(j+1)$th component
(that is, the state of the $j$th transition automaton). In particular, we
require $t^{0}=t_{j}$ and $t^{\ell}=t^{\prime}_{j}$. The proof proceeds in
exactly the same manner as the case of
$\left({{p},{a},{{\mathcal{L}_{g}}},{p^{\prime}}}\right)$ in the proof of
Lemma C.1 (Completeness of $\Pi$) for ECPDS. Namely, from the sequence of
operations $o^{0},\ldots,o^{\ell}$ taken from the run $t^{0},\ldots,t^{\ell}$,
we obtain a sequence of stacks such that at each $z$ there is an accepting run
of the $z$th stack constructed from $t^{z}$ and thereafter only transitions of
$A^{j}_{i+1}$. Thus, since $t_{j}$ is added to $A^{j}_{i+1}$ to obtain
$A^{j}_{i}$, we additionally get an accepting run of $A^{j}_{i}$ over
$\langle{p_{i}},{w^{j}_{i}}\rangle$. We do not repeat the arguments here.
Finally, then, when $i$ reaches $1$, we repeat the arguments above to conclude
$\langle{p_{0}},{w^{j}_{0}}\rangle$ is accepted by $A^{j}_{0}$ for each $j$,
giving the required lemma. $\square$
###### Lemma F.3
Given a phase-bounded CPDS $\mathcal{C}$ control states $p_{0}$ and
$p_{\zeta}$, there is a run of $\mathcal{C}$ from
$\langle{p_{0}},{w_{1},\ldots,w_{m}}\rangle$ to
$\langle{p_{\zeta}},{w^{\prime}_{1},\ldots,w^{\prime}_{m}}\rangle$ whenever
for each $1\leq j\leq m$, we have that $\langle{p_{0}},{w_{j}}\rangle$ is
accepted by $A^{j}_{0}$.
_Proof._ Assume for each $1\leq j\leq m$, we have that
$\langle{p_{0}},{w_{j}}\rangle$ is accepted by $A^{j}_{0}$.
Thus, we can inductively assume for each $j$ we have
$\langle{p_{i}},{w^{j}_{i}}\rangle$ accepted by $A^{j}_{i}$ and a run of
$\mathcal{C}$ of the form
$\langle{p_{0}},{w_{1},\ldots,w_{m}}\rangle\longrightarrow\cdots\longrightarrow\langle{p_{i}},{w^{1}_{i},\ldots,w^{m}_{i}}\rangle\
.$
Taking $w^{j}_{0}=w_{j}$ trivially gives us the base case. We prove the case
for $(i+1)$.
From the induction hypothesis, we have in particular that
$\langle{p_{i}},{w^{\iota_{i}}_{i}}\rangle$ is accepted by $A^{\iota_{i}}_{i}$
and hence we have a run of $\mathcal{C}_{i+1}$ of the form
$\langle{p_{i}},{w^{\iota_{i}}_{i}}\rangle\longrightarrow\langle{\left({p_{i},t_{1},\ldots,t_{m}}\right)},{w^{\iota_{i}}_{i}}\rangle\longrightarrow\cdots\longrightarrow\langle{\left({p_{i+1},t^{\prime}_{1},\ldots,t^{\prime}_{m}}\right)},{w^{\iota_{i}}_{i+1}}\rangle\longrightarrow\langle{p_{1}},{w^{\iota_{i}}_{i+1}}\rangle$
such that $\langle{p_{1}},{w^{\iota_{i}}_{i+1}}\rangle$ is accepted by
$A^{\iota_{i}}_{i+1}$. From this run, due to the definition of
$\mathcal{C}_{i}$ we can build a run
$\langle{p_{i}},{w^{1}_{i},\ldots,w^{m}_{i}}\rangle\longrightarrow\cdots\longrightarrow\langle{p_{i+1}},{w^{1}_{i+1},\ldots,w^{m}_{i+1}}\rangle$
of $\mathcal{C}$ where for all $j\neq\iota_{i}$, we define
$w^{j}_{i+1}={o^{\ell}}\mathord{\left({\cdots{o^{1}}\mathord{\left({w^{j}_{i}}\right)}}\right)}$
where
$\left({{p^{0}},{a^{1}},{o^{1}},{p^{1}}}\right)\left({{p^{1}},{a^{2}},{o^{2}},{p^{2}}}\right)\ldots\left({{p^{\ell-1}},{a^{\ell}},{o^{\ell}},{p^{\ell}}}\right)$
is the sequence of labels on the run of
$\mathcal{T}^{A^{j}_{i}}_{{t_{j}},{t^{\prime}_{j}}}$. We have to prove for all
$j\neq\iota_{i}$ that $\langle{p_{i+1}},{w^{j}_{i+1}}\rangle$ is accepted by
$A^{j}_{i+1}$. For the proof observe that the introduction of $t_{j}$ to
$A^{j}_{i+1}$ to form $A^{j}_{i}$ followed the saturation technique for
extended CPDS for a rule
$\left({{p_{i}},{a},{{\mathcal{L}_{g}}},{p_{i+1}}}\right)$ where
${\mathcal{L}_{g}}$ is the language of possible sequences of the form above.
Thus, from the soundness of the saturation method for extended CPDS, we have
that there must be the required run of $A^{j}_{i+1}$ over
$\langle{p_{i+1}},{w^{j}_{i+1}}\rangle$ beginning with transition
$t^{\prime}_{j}$.
Alternatively, we can argue similarly to the proof of Lemma C.1 (Completeness
of $\Pi$), but in the reverse direction. That is, we start with the
observation that the accepting run of $\langle{p_{i}},{w^{j}_{i}}\rangle$ uses
$t_{j}=t^{0}$ for the first transition, and thereafter only transitions from
$A^{j}_{i+1}$. We prove this by induction for the stack obtained by applying
$o^{1}$ and $t^{1}$, then for the stack obtained by applying $o^{2}$ and
$t^{2}$. This continues until we reach $w^{j}_{i+1}$, and since
$t^{\ell}=t^{\prime}_{j}$ with $t^{\prime}_{j}$ being a transition of
$A^{j}_{i+1}$, we get the accepting run we need. We remark that this is how
the soundness proof for the standard saturation algorithm would proceed if we
were able to assume that each new transition is only used at the head of any
new runs the transition introduces (but in general this is not the case
because new transitions may introduce loops). We leave the construction of
this proof as an exercise for the interested reader, for which they may follow
the proof of the extended rule case for Lemma C.5 (Soundness of $\Pi$).
Thus, finally, by induction, we obtain a run to
$\langle{p_{\zeta}},{w_{1},\ldots,w_{m}}\rangle$ such that
$\langle{p_{\zeta}},{w_{j}}\rangle$ is accepted by $A^{j}_{\zeta}$. $\square$
### F.3 Complexity
Assume $n>1$. Our control state reachability algorithm requires
$2\uparrow_{\zeta(n-1)}\left({\ell}\right)$ time, where $\ell$ is polynomial
in the size of the PBCPDS. Beginning with phase $\zeta$, the saturation
algorithm can add at most
${\mathcal{O}}\mathord{\left({2\uparrow_{n-1}\left({\ell}\right)}\right)}$
transitions over the same number of iterations to
$A^{\iota_{\zeta}}_{\zeta-1}$. Thus we assume each $A^{j}_{i}$ to have at most
${\mathcal{O}}\mathord{\left({2\uparrow_{(\zeta-i)(n-1)}\left({\ell}\right)}\right)}$
transitions. The largest automaton $A^{j}_{i-1}$ construction is when
$j=\iota_{i}$. For this we build a CPDS with
${\mathcal{O}}\mathord{\left({2\uparrow_{(\zeta-i)(n-1)}\left({\ell}\right)}\right)}$
control states and thus $A^{\iota_{i}}_{i-1}$ has at most
${\mathcal{O}}\mathord{\left({2\uparrow_{(\zeta-i+1)(n-1)}\left({\ell}\right)}\right)}$
transitions. Hence, when $i=0$, we have at most
${\mathcal{O}}\mathord{\left({2\uparrow_{\zeta(n-1)}\left({\ell}\right)}\right)}$
transitions, which also gives the run time of the algorithm. This also implies
we have at most ${\mathcal{O}}\mathord{\left({2\uparrow_{\zeta
n}\left({\ell}\right)}\right)}$ tuples in the solution to the global
reachability problem.
## Appendix G Proofs for Scope-Bounded CPDS
### G.1 Operations on Layer Automata
#### Shift of a Layer Automaton
The idea behind Shift is that all transitions in layer $i$ are moved up to
layer $(i+1)$ and transitions involving states in layer $\zeta$ are removed.
Intuitively this is because the stack elements in layer $\zeta$ will “go out
of scope” when the context switch corresponding to the Shift occurs. In more
detail, states of layer $i$ are renamed to become states of layer $(i+1)$,
with all states of layer $\zeta$ being deleted. Similarly, all transitions
that involved a layer $\zeta$ state are also removed.
We define ${\text{\tt Shift}}\mathord{\left({A}\right)}$ of an order-$n$
$\zeta$-layer stack automaton
$A=\left({\mathbb{Q}_{n},\ldots,\mathbb{Q}_{1},\Sigma,\Delta_{n},\ldots,\Delta_{1},\emptyset,\ldots,\emptyset}\right)$
to be
$A^{\prime}=\left({\mathbb{Q}^{\prime}_{n},\ldots,\mathbb{Q}^{\prime}_{1},\Sigma,\Delta^{\prime}_{n},\ldots,\Delta^{\prime}_{1},\emptyset,\ldots,\emptyset}\right)$
where defining
${\text{\tt Shift}}\mathord{\left({q}\right)}=\begin{cases}q&\text{if
$q\in\mathbb{Q}_{k}$, $n>k$ and $q$ is layer $i<\zeta$}\\\
q_{p}^{i+1}&\text{if $q=q_{p}^{i}\in\mathbb{Q}_{n}$ and $i<\zeta$}\\\
\text{undefined}&\text{otherwise}\end{cases}$
and extending Shift point-wise to sets of states, we have
$\Delta^{\prime}_{n}=\left\\{{{\text{\tt
Shift}}\mathord{\left({q}\right)}\xrightarrow{q^{\prime}}{\text{\tt
Shift}}\mathord{\left({Q}\right)}}\ \left|\
{q\xrightarrow{q^{\prime}}Q\in\Delta_{n}\text{ and $q$ is layer
$i<\zeta$}}\right.\right\\}$
and for all $n>k>1$
$\Delta^{\prime}_{k}=\left\\{{q\xrightarrow{q^{\prime}}{\text{\tt
Shift}}\mathord{\left({Q}\right)}}\ \left|\
{q\xrightarrow{q^{\prime}}Q\in\Delta_{k}\text{ and $q$ is layer
$i<\zeta$}}\right.\right\\}$
and
$\Delta^{\prime}_{1}=\left\\{{q\xrightarrow[{\text{\tt
Shift}}\mathord{\left({Q_{col}}\right)}]{q^{\prime}}{\text{\tt
Shift}}\mathord{\left({Q}\right)}}\ \left|\
{q\xrightarrow[Q_{col}]{q^{\prime}}Q\in\Delta_{1}\text{ and $q$ is layer
$i<\zeta$}}\right.\right\\}\ .$
In all cases above, transitions are only created if the applications of Shift
result in a defined state or set of states. This operation will erase all
layer $\zeta$ states, and all transitions that go to a layer $\zeta$ state.
All other states will be shifted up one layer. E.g. layer $1$ states become
layer $2$.
#### Environment Moves
Given an automaton $A$, define ${\text{\tt
EnvMove}}\mathord{\left({A,q,q^{\prime}}\right)}$ of an order-$n$
$\zeta$-layer stack automaton to be $A^{\prime}$ obtained from $A$ by adding
for each transition
${q^{\prime}}\xrightarrow[Q_{col}]{a}\left({{Q_{1},\ldots,Q_{n}}}\right)$ the
transition ${q}\xrightarrow[Q_{col}]{a}\left({{Q_{1},\ldots,Q_{n}}}\right)$.
This operation can be thought of as a saturation rule that captures the effect
of an external context, and could be considered as rules
$\left({{p},{a},{noop},{p^{\prime}}}\right)$ for each $a\in\Sigma$.
#### Saturating a Layer Automaton
Given a layer automaton $A$, we define ${\text{\tt
Saturate}_{j}}\mathord{\left({A}\right)}$ to be the result of applying the
saturation procedure with the CPDS
$\left({\mathcal{P},\Sigma,\mathcal{R}_{j}}\right)$ and the stack automaton
$A$ with initial state-set $\left\\{{q_{p}^{1}}\ \left|\
{p\in\mathcal{P}}\right.\right\\}$.
### G.2 Size of the Reachability Graph
We define $N$.
###### Lemma G.1
The maximum number of states in any layer automaton constructable by repeated
applications of $\text{\tt Predecessor}_{j}$ is
$2\uparrow_{n-2}\left({{f}\mathord{\left({\zeta,\left|{\mathcal{P}}\right|}\right)}}\right)$
states for some computable polynomial $f$.
_Proof._ A $\zeta$-layer automaton may have in $q_{n}$ only the states
$q_{p}^{i}$ for $1\leq i\leq\zeta$ and $p\in\mathcal{P}$, and thus at most
$\zeta\left|{\mathcal{P}}\right|=d$ states. There may be at most $d$
transitions from any state at order-$n$ using the restricted saturation
algorithm where $Q_{n}$ has cardinality $1$ for any transition added, and thus
at most $d\cdot d$ states at order-$(n-1)$ (noting that the shift operation
deletes all states that would become non-initial if they were to remain).
Next, there may be at most $2^{d\cdot d}$ transitions from any state at
order-$(n-1)$, and thus at most $d\cdot d\cdot 2^{d\cdot d}$ states at
order-$(n-2)$ (noting that the shift operation deletes all states that would
become non-initial if they were to remain).
Thus, we can repeat this argument down to order-$1$ and obtain
$2\uparrow_{n-2}\left({{f}\mathord{\left({\zeta,\left|{\mathcal{P}}\right|}\right)}}\right)$
states for some computable polynomial $f$. $\square$
Take the automaton accepting any $\langle{p_{i}},{w}\rangle$ from
$q_{p_{i}}^{1}$. This automaton has order-$n$ states of the form $q_{p}^{i}$,
and at most a single transition from each of the layer $1$ states to
$\emptyset$. Each of these transitions is labelled by a state with at most one
transition to $\emptyset$, and so on until order-$1$.
###### Definition G.1 ($N$)
Following Lemma G.1, we take
$N=2\uparrow_{n-2}\left({{f}\mathord{\left({\zeta,d}\right)}}\right)$ for some
computable polynomial $f$.
### G.3 Proofs for Control State Reachability
In this section, we prove Lemma 6.1 (Simulation by
$\mathcal{G}^{{p_{\text{out}}}}_{\mathcal{C}}$). The proof is split in to two
directions, given in Lemma G.2 and Lemma G.3 below.
###### Lemma G.2
Given a scope-bounded CPDS $\mathcal{C}$ and control states ${p_{\text{in}}}$
and ${p_{\text{out}}}$, there is a run of $\mathcal{C}$ from
$\langle{{p_{\text{in}}}},{w_{1},\ldots,w_{m}}\rangle$ to
$\langle{{p_{\text{out}}}},{w^{\prime}_{1},\ldots,w^{\prime}_{m}}\rangle$ for
some $w^{\prime}_{1},\ldots,w^{\prime}_{m}$ only if there is a path in
$\mathcal{G}^{{p_{\text{out}}}}_{\mathcal{C}}$ from an initial vertex to a
vertex
$\left({p_{0},A_{1},p_{1},\ldots,p_{m-1},A_{m},p_{m}}\right)$
where for all $i$ we have $\langle{p_{i-1}},{w_{i}}\rangle$ accepted from the
$1$st layer of $A_{i}$ and $p_{0}={p_{\text{in}}}$.
_Proof._ Take a run of the scope-bounded CPDS from
$\langle{{p_{\text{in}}}},{w_{1},\ldots,w_{m}}\rangle$ to
$\langle{{p_{\text{out}}}},{w^{\prime}_{1},\ldots,w^{\prime}_{m}}\rangle$. We
proceed by induction over the number of rounds in the run. In the following we
will override the $w_{i}$ and $w^{\prime}_{i}$ in the statement of the lemma
to ease notation.
In the base case, take a single round
$\langle{p_{0}},{w_{1},\ldots,w_{m}}\rangle\longrightarrow^{\ast}\langle{p_{1}},{w^{\prime}_{1},w_{2},\ldots,w_{m}}\rangle\longrightarrow^{\ast}\cdots\longrightarrow^{\ast}\langle{p_{m}},{w^{\prime}_{1},\ldots,w^{\prime}_{m}}\rangle$
where $p_{i}$ is the control state after the run on stack $i$, and
$w^{\prime}_{i}$ is the $i$th stack at the end of this run. Take an initial
vertex
$\left({p_{0},A_{1},p_{1},\ldots,p_{m-1},A_{m},p_{m}}\right)\ .$
We know $A_{i}$ is constructed by saturation from an automaton accepting
$\langle{p_{i}},{w^{\prime}_{i}}\rangle$ and thus
$\langle{p_{i-1}},{w_{i}}\rangle$ is accepted by $A_{i}$ from the $1$st layer.
This vertex then gives us a path in the reachability graph to a vertex where
for all $i$ we have $\langle{p_{i-1}},{w_{i}}\rangle$ accepted from the $1$st
layer of $A_{i}$.
Now consider the inductive step where we have a round
$\langle{p_{0}},{w_{1},\ldots,w_{m}}\rangle\longrightarrow^{\ast}\langle{p_{1}},{w^{\prime}_{1},w_{2},\ldots,w_{m}}\rangle\longrightarrow^{\ast}\cdots\longrightarrow^{\ast}\langle{p_{m}},{w^{\prime}_{1},\ldots,w^{\prime}_{m}}\rangle$
and a run from $\langle{p_{m}},{w^{\prime}_{1},\ldots,w^{\prime}_{m}}\rangle$
to the destination control state. By induction we have a vertex in the
reachability graph
$\left({p^{\prime}_{0},A^{\prime}_{1},p^{\prime}_{1},\ldots,p^{\prime}_{m-1},A^{\prime}_{m},p^{\prime}_{m}}\right)$
with $p_{m}=p^{\prime}_{0}$ that is reachable from an initial vertex and has
for all $i$ that $\langle{p^{\prime}_{i-1}},{w^{\prime}_{i}}\rangle$ is
accepted from the $1$st layer of $A^{\prime}_{i}$.
By definition of the reachability graph, there exists an edge to this vertex
from a vertex
$\left({p_{0},A_{1},p_{1},\ldots,p_{m-1},A_{m},p_{m}}\right)\ .$
such that $A_{i}={\text{\tt
Predecessor}_{i}}\mathord{\left({A^{\prime}_{i},q_{p_{i}},q_{p^{\prime}_{i-1}}}\right)}$.
Since the run of $\mathcal{C}$ is scope-bounded, we know there is an accepting
run of $w^{\prime}_{i}$ from $q_{p^{\prime}_{i-1}}^{1}$ in $A^{\prime}_{i}$
that does not use any layer $\zeta$ states (by the further condition described
below and since layer $\zeta$ corresponds to the round out of scope for
elements of $w^{\prime}_{i}$). Therefrom, we have an accepting run of
$w^{\prime}_{i}$ from $q_{p^{\prime}_{i-1}}^{2}$ in ${\text{\tt
Shift}}\mathord{\left({A^{\prime}_{i}}\right)}$. Thus, there is an accepting
run of $w^{\prime}_{i}$ from $p_{i}^{1}$ after the application of EnvMove.
Since there is a run over stack $i$ from $\langle{p_{i-1}},{w_{i}}\rangle$ to
$\langle{p_{i}},{w^{\prime}_{i}}\rangle$ we therefore have an accepting run of
$w_{i}$ from $q_{p_{i-1}}^{1}$ in $A_{i}$.
In addition to the above, we need a further property that reflects the scope
boundedness. In particular, if no character or stack with pop- or collapse-
round $0$ is removed during the $z$th round, then there is a run over $w_{i}$
that uses only transitions $q\xrightarrow{q^{\prime}}Q$ to read stacks $u$
such that no layer $z$ state is in $Q$ and, similarly, for characters $a$, the
run uses only transitions $q\xrightarrow[Q_{col}]{a}{Q}$ to read the instance
of $a$ where no layer $z$ state appears in $Q$ and no layer $z$ state appears
in $Q_{col}$.
Note that the base case is for the automata accepting any stack, only
containing transitions to the empty set, for which the property is trivial. In
the inductive step, we prove this property by further induction over the
length of the run from $\langle{p_{i}},{w_{i}}\rangle$ to
$\langle{p_{i+1}},{w^{\prime}_{i}}\rangle$. In the base case we have a run of
length $0$ and the property holds since, by induction, we can assume that
$A^{\prime}_{i}$ has the property (with the round numbers shifted) and it is
maintained by the Shift and EnvMove. Hence, assume we have a run beginning
$\langle{p},{w}\rangle\longrightarrow\langle{p^{\prime}},{w^{\prime}}\rangle$
and the required run over $w^{\prime}$. We do a case split on the stack
operation $o$ associated with the transition.
1. 1.
If $o=pop_{k}$ then we have $w={u}:_{k}{v}$ and $w^{\prime}=v$. If $z=1$ and
$u$ has pop-round $0$ (i.e. appears in $w_{i}$), then this case cannot occur
because the transition we’re currently analysing appears in round $1$ and by
assumption $u$ is not removed in round $1$. Hence, assume $z>1$. We had a run
over $w^{\prime}$ from
${q_{p^{\prime}}^{1}}\xrightarrow{q_{k}}\left({{Q_{k+1},\dots,Q_{n}}}\right)$
in $A_{i}$ respecting the property, and by saturation we have a run over $w$
beginning with
${q_{p}^{1}}\xrightarrow[\emptyset]{a}\left({{\emptyset,\ldots,\emptyset,\left\\{{q_{k}}\right\\},Q_{k+1},\ldots,Q_{n}}}\right)$
that also respects the property, since $q_{k}$ is layer $1$ and $z\neq 1$.
2. 2.
When $o=copy_{k}$ we have $w={u}:_{k}{v}$ and
$w^{\prime}={u}:_{k}{{u}:_{k}{v}}$. Let
${q_{p^{\prime}}^{1}}\xrightarrow[Q_{col}]{a}\left({{Q_{1},\ldots,Q_{k},\ldots
Q_{n}}}\right)$ and
${Q_{k}}\xrightarrow[Q^{\prime}_{col}]{a}\left({{Q^{\prime}_{1},\ldots,Q^{\prime}_{k}}}\right)$
be the initial transitions used on the run of $w^{\prime}$. We know neither
these transitions, nor the runs from these transitions, pass a layer $z$ state
on any component with pop- or collapse-round $0$. Furthermore, we know the
first $u$ has pop-round $1$. The second $u$ may have pop-round $0$. If it
does, we know $Q^{\prime}_{k}$ does not contain any layer $z$ states.
From the saturation algorithm, we have a transition
${q_{p}^{1}}\xrightarrow[Q_{col}\cup Q^{\prime}_{col}]{a}\left({{Q_{1}\cup
Q^{\prime}_{1},\ldots,Q_{k-1}\cup
Q^{\prime}_{k-1},Q^{\prime}_{k},Q_{k+1},\ldots,Q_{n}}}\right)\ .$
from which we have an accepting run of $w$ that satisfies the property.
3. 3.
If $o=collapse_{k}$, $w={{a}^{u^{\prime}}}:_{1}{{u}:_{(k+1)}{v}}$ and
$w^{\prime}={u^{\prime}}:_{(k+1)}{v}$. When $k=n$, we have an accepting run of
$w^{\prime}$ respecting the property, and from the saturation, an accepting
run of $w$ beginning with a transition
${q_{p}^{1}}\xrightarrow[\left\\{{q_{p^{\prime}}^{1}}\right\\}]{a}\left({{\emptyset,\ldots,\emptyset}}\right)$
and $w^{\prime}=u^{\prime}$. When $z=1$ and $a$ has collapse-round $0$, this
case cannot occur because the transition we’re currently analysing appears in
round $1$ (similarly to the $pop_{k}$ case). Otherwise $z>1$ and we have a run
over $w$ respecting the property.
When $k<n$, we have an accepting run of $w^{\prime}$ in beginning with
${q_{p^{\prime}}^{1}}\xrightarrow{q_{k}}\left({{Q_{k+1},\dots,Q_{n}}}\right)$
that respects the property. By saturation, we have an accepting run of $w$
beginning with a transition
${q_{p}^{1}}\xrightarrow[\left\\{{q_{k}}\right\\}]{a}\left({{\emptyset,\ldots,\emptyset,Q_{k+1},\ldots,Q_{n}}}\right)$.
If the collapse-round of $a$ is $0$ and $z=1$, this case cannot occur.
Otherwise, the run over $w$ satisfies the property since the run over
$w^{\prime}$ does and $q_{k}$ is layer $1$ and $z>1$.
4. 4.
When $o=push^{k}_{c}$, let
$w={u_{k-1}}:_{k}{{u_{k}}:_{k+1}{{\cdots}:_{n}{u_{n}}}}$. We know
$w^{\prime}={push^{k}_{c}}\mathord{\left({w}\right)}$ is
${{c}^{u_{k}}}:_{1}{{u_{k-1}}:_{k}{{\cdots}:_{n}{u_{n}}}}\ .$
Let
${q_{p^{\prime}}^{1}}\xrightarrow[Q_{col}]{c}\left({{Q_{1},\ldots,Q_{n}}}\right)\quad\text{and}\quad
Q_{1}\xrightarrow[Q^{\prime}_{col}]{a}Q^{\prime}_{1}$ be the first transitions
used on the accepting run of $w^{\prime}$. If the pop-round of $a$ is $0$, we
know there are no layer $z$ states in $Q^{\prime}_{1}$. Similarly if the pop-
round of $u_{k}$ is $0$ we know that there are no layer $z$ states in
$Q_{col}$. The saturation algorithm means we have
${q_{p}^{1}}\xrightarrow[Q^{\prime}_{col}]{a}\left({{Q^{\prime}_{1},Q_{2},\ldots,Q_{k}\cup
Q_{col},\ldots,Q_{n}}}\right)$ leading to an accepting run that respects the
property.
5. 5.
If $o=rew_{b}$ then $w={{a}^{u}}:_{1}{v}$ and $w^{\prime}={{b}^{u}}:_{1}{v}$.
Note none of the pop- or collapse-rounds are changed, and the run of
$w^{\prime}$ beginning
${q_{p^{\prime}}^{1}}\xrightarrow[Q_{col}]{b}\left({{Q_{1},\dots,Q_{n}}}\right)$
and satisfying the property implies a run of $w$ beginning
${q_{p}^{1}}\xrightarrow[Q_{col}]{a}\left({{Q_{1},\dots,Q_{n}}}\right)$ and
also satisfying the property.
6. 6.
If $o=noop$ then $w={{a}^{u}}:_{1}{v}$ and $w^{\prime}={{a}^{u}}:_{1}{v}$.
Note none of the pop- or collapse-rounds are changed, and the run of
$w^{\prime}$ beginning
${q_{p^{\prime}}^{1}}\xrightarrow[Q_{col}]{a}\left({{Q_{1},\dots,Q_{n}}}\right)$
and satisfying the property implies a run of $w$ beginning
${q_{p}^{1}}\xrightarrow[Q_{col}]{a}\left({{Q_{1},\dots,Q_{n}}}\right)$ and
also satisfying the property.
Finally then, by induction over the number of rounds, we reach the first round
beginning with $\langle{p_{0}},{w_{1},\ldots,w_{m}}\rangle$ and we know there
is a path from an initial vertex to a vertex
$\left({p_{0},A_{1},p_{1},\ldots,p_{m-1},A_{m},p_{m}}\right)$
with $p_{0}=p$ and for all $i$ we have $\langle{p_{i-1}},{w_{i}}\rangle$
accepted from the $1$st layer of $A_{i}$. $\square$
###### Lemma G.3
Given a scope-bounded CPDS $\mathcal{C}$ and control states ${p_{\text{in}}}$
and ${p_{\text{out}}}$, there is a run of $\mathcal{C}$ from
$\langle{{p_{\text{in}}}},{w_{1},\ldots,w_{m}}\rangle$ to
$\langle{{p_{\text{out}}}},{w^{\prime}_{1},\ldots,w^{\prime}_{m}}\rangle$ for
some $w^{\prime}_{1},\ldots,w^{\prime}_{m}$ whenever there is a path in
$\mathcal{G}^{{p_{\text{out}}}}_{\mathcal{C}}$ from an initial vertex to a
vertex
$\left({p_{0},A_{1},p_{1},\ldots,p_{m-1},A_{m},p_{m}}\right)$
with $p_{0}={p_{\text{in}}}$ and for all $i$ we have
$\langle{p_{i-1}},{w_{i}}\rangle$ accepted from the $1$st layer of $A_{i}$.
_Proof._ Note, in the following proof, we override the $w_{i}$ and
$w^{\prime}_{i}$ in the statement of the lemma. Take a path in the
reachability graph. The proof goes by induction over the length of the path.
When the path is of length $0$ we have a single vertex
$\left({p_{0},A_{1},p_{1},\ldots,p_{m-1},A_{m},p_{m}}\right)$. Take any
configuration $\langle{p_{i-1}},{w_{i}}\rangle$ accepted by $A_{i}$. We know
$A_{i}$ accepts all configurations that can reach $\langle{p_{i}},{w}\rangle$
for some $w$. Therefore, from the initial configuration
$\langle{p_{0}},{w_{1},\ldots,w_{m}}\rangle$
we first apply the run over the $1$st stack to $p_{1}$ to obtain
$\langle{p_{1}},{w^{\prime}_{1},w_{2},\ldots,w_{m}}\rangle$
for some $w^{\prime}_{1}$. Then we apply the run over the $2$nd stack to
$p_{2}$ and so on until we reach
$\langle{p_{m}},{w^{\prime}_{1},\ldots,w^{\prime}_{m}}\rangle$
for some $w^{\prime}_{1},\ldots,w^{\prime}_{m}$. This witnesses the
reachability property as required.
Now consider the inductive case where we have a path beginning with an edge of
the reachability graph from
$\left({p_{0},A_{1},p_{1},\ldots,p_{m-1},A_{m},p_{m}}\right)$
to
$\left({p^{\prime}_{0},A^{\prime}_{1},p^{\prime}_{1},\ldots,p^{\prime}_{m-1},A^{\prime}_{m},p^{\prime}_{m}}\right)\
.$
By induction we have a run from
$\langle{p_{m}},{w^{\prime}_{1},\ldots,w^{\prime}_{m}}\rangle$
to the final control state for any $w^{\prime}_{i}$ accepted by
$A^{\prime}_{i}$ from $q_{p_{i-1}}^{1}$.
Now, similarly to the base case, take any configuration
$\langle{p_{i-1}},{w_{i}}\rangle$ accepted by $A_{i}$. We know $A_{i}$ accepts
all configurations that can reach $\langle{p_{i}},{w}\rangle$ for some $w$
accepted from $q_{p^{\prime}_{i-1}}^{2}$ in ${\text{\tt
Shift}}\mathord{\left({A^{\prime}_{i}}\right)}$ and therefore, from
$q_{p^{\prime}_{i-1}}^{1}$ in $A^{\prime}_{i}$. Hence, from the initial
configuration
$\langle{p_{0}},{w_{1},\ldots,w_{m}}\rangle$
we first apply the run over the $1$st stack to $p_{1}$ to obtain
$\langle{p_{1}},{w^{\prime}_{1},w_{2},\ldots,w_{m}}\rangle$
for some $w^{\prime}_{1}$. Then we apply the run over the $2$nd stack to
$p_{2}$ and so on until we reach
$\langle{p_{m}},{w^{\prime}_{1},\ldots,w^{\prime}_{m}}\rangle$
for some $w^{\prime}_{1},\ldots,w^{\prime}_{m}$ and then, by induction, we
have a run from this configuration to the target control state as required.
We need to prove a stronger property that we can in fact build a scope-bounded
run. In particular, we show that, for all stacks $u$ in $w_{i}$, if the
accepting run of $w_{i}$ uses only transitions $q\xrightarrow{q^{\prime}}Q$ to
read $u$ such that no layer $z$ state is in $Q$, then there is a run to the
final control state such that $u$ is not popped during round $z$. Similarly,
for characters $a$, if the accepting run uses only transitions
$q\xrightarrow[Q_{col}]{a}{Q}$ to read the instance of $a$ where no layer $z$
state appears in $Q$, then $a$ is not popped in round $z$. Similarly, if no
layer $z$ state appears in $Q_{col}$, then collapse is not called on that
character during round $z$. We observe the property is trivially true for the
base case where the automata accept any stack using only transitions to
$\emptyset$. The inductive case is below.
We start from $\langle{p},{w}\rangle=\langle{p_{i}},{w_{i}}\rangle$. First
assign each stack and character in $w$ pop- and collapse-round $0$. Noting
that $A$ is obtained by saturation from $A^{\prime}$ (after a Shift and
EnvMove — call this automaton $B$), we aim to exhibit a run from
$\langle{p},{w}\rangle$ to $\langle{p_{i+1}},{w_{i+1}}\rangle$ (in fact we
choose $w_{i+1}$ via this procedure) such that all stacks and characters in
$w_{i+1}$ with pop- or collapse-round $0$ do not pass layer $z$ states in $B$.
Since we have a run over $w_{i+1}$ in $A^{\prime}_{i}$ that does not pass
layer $1$ states for parts of the stack with pop- or collapse-round $0$, we
know by induction we have a run from $\langle{p_{i+1}},{w_{i+1}}\rangle$ that
is scope bounded.
To generate such a run we follow the counter-example generation algorithm in
[9]. We refer the reader to this paper for a precise exposition of the
algorithm. Furthermore, that this routine terminates is non-trivial and
requires a subtle well-founded relation over stacks, which is also shown in
[9].
Beginning with the run over $\langle{p_{i}},{w_{i}}\rangle$ that has the
property of not passing layer $z$ states, we have our base case. Now assume we
have a run to $\langle{p},{w}\rangle$ such that the run over $w$ has no
transitions to layer $z$ states reading stacks or characters with pop- or
collapse-rounds of $0$. We take the first transition of such a run, which was
introduced by the saturation algorithm because of a rule
$\left({{p},{a},{o},{p^{\prime}}}\right)$ and certain transitions of the
partially saturated $B$. Let $\langle{p^{\prime}},{w^{\prime}}\rangle$ be the
configuration reached via this rule. We do a case split on $o$.
1. 1.
If $o=pop_{k}$, then we have $w={u}:_{k}{v}$ and the accepting run of $w$
begins with
${q_{p}^{1}}\xrightarrow[\emptyset]{a}\left({{\emptyset,\ldots,\emptyset,\left\\{{q_{k}}\right\\},Q_{k+1},\ldots,Q_{n}}}\right)$
where
${q_{p^{\prime}}^{1}}\xrightarrow{q_{k}}\left({{Q_{k+1},\dots,Q_{n}}}\right)$
was already in $B$. This gives us an accepting run of $v$ beginning with this
transition. Note that $q_{k}$ is of layer $1$. Thus, if $u$ has pop-round $0$
and $z=1$, this case cannot occur. Otherwise, we have that the run of $v$
visits a subset of the states in the run over $w$ and thus maintains the
property.
2. 2.
If $o=copy_{k}$, then we have $w={u}:_{k}{v}$ and
$w^{\prime}={u}:_{k}{{u}:_{k}{v}}$. Furthermore, we had an accepting run of
$w$ using the initial transition
${q_{p}^{1}}\xrightarrow[Q_{col}\cup Q^{\prime}_{col}]{a}\left({{Q_{1}\cup
Q^{\prime}_{1},\ldots,Q_{k-1}\cup
Q^{\prime}_{k-1},Q^{\prime}_{k},Q_{k+1},\ldots,Q_{n}}}\right)$
and an accepting run of $B$ on $w^{\prime}$ using the initial transitions
${q_{p^{\prime}}^{1}}\xrightarrow[Q_{col}]{a}\left({{Q_{1},\ldots,Q_{k},\ldots,Q_{n}}}\right)$
and
${Q_{k}}\xrightarrow[Q^{\prime}_{col}]{a}\left({{Q^{\prime}_{1},\ldots,Q^{\prime}_{k}}}\right)$
from which we have an accepting run over $w^{\prime}$. Note that, to prove the
required property, we observe that for all elements of $w^{\prime}$ obtaining
their pop- and collapse-rounds from $w$, the targets of the transitions used
to read them already appear in the run of $w$, hence the run satisfies the
property. The only new part of the run is to $Q^{\prime}_{k}$ after reading
the new copy of $u$, which has pop-round $1$. Thus the property is maintained.
3. 3.
If $o=collapse_{k}$ then we have $w={{a}^{u^{\prime}}}:_{1}{{u}:_{(k+1)}{v}}$
and $w^{\prime}={u^{\prime}}:_{(k+1)}{v}$. When $k=n$, the accepting run of
$w$ begins with a transition
${q_{p}^{1}}\xrightarrow[\left\\{{q_{p^{\prime}}^{1}}\right\\}]{a}\left({{\emptyset,\ldots,\emptyset}}\right)$
and $w^{\prime}=u^{\prime}$. When $z=1$ and $a$ has collapse-round $0$, this
case cannot occur because the initial transition goes to a layer $z$ state.
Otherwise, we have a run over $w^{\prime}$ that is a subrun of that over $w$,
and thus the property is transferred.
When $k<n$, the accepting run of $w$ begins with
${q_{p}^{1}}\xrightarrow[\left\\{{q_{k}}\right\\}]{a}\left({{\emptyset,\ldots,\emptyset,Q_{k+1},\ldots,Q_{n}}}\right)$
and we have an accepting run of $w^{\prime}$ in $B$ beginning with
${q_{p^{\prime}}^{1}}\xrightarrow{q_{k}}\left({{Q_{k+1},\dots,Q_{n}}}\right)$.
If the collapse-round of $a$ is $0$ and $z=1$, this case cannot occur because
$q_{k}$ is layer $z$. Otherwise, the run over $w^{\prime}$ is a subrun of that
over $w$ and the property is transferred.
4. 4.
If $o=push^{k}_{b}$ then $w^{\prime}={{b}^{u}}:_{1}{w}$ where
$u={top_{k+1}}\mathord{\left({{pop_{k}}\mathord{\left({w}\right)}}\right)}$
and the collapse-round of $b$ is the pop-round of
${top_{k}}\mathord{\left({w}\right)}$. The run of $w$ begins with a transition
${q_{p}^{1}}\xrightarrow[Q^{\prime}_{col}]{a}\left({{Q^{\prime}_{1},Q_{2},\ldots,Q_{k-1},Q_{k}\cup
Q_{col},Q_{k+1},\ldots,Q_{n}}}\right)$
and there is a run over $w^{\prime}$ in $B$ beginning with
${q_{p^{\prime}}^{1}}\xrightarrow[Q_{col}]{b}\left({{Q_{1},\ldots,Q_{n}}}\right)$
and $Q_{1}\xrightarrow[Q^{\prime}_{col}]{a}Q^{\prime}_{1}$. Note that, to
prove the required property, we observe that for all elements of $w^{\prime}$
obtaining their pop- and collapse-rounds from $w$, the targets of the
transitions used to read them already appear in the run of $w$, hence the run
satisfies the property. The only new parts of the run are to $Q^{\prime}_{1}$
after reading $b$, which has pop-round $1$, and the transition to $Q_{col}$ on
the collapse branch of $b$. Note, however, that $b$ has the collapse-round
equal to the pop-round of ${top_{k}}\mathord{\left({w}\right)}$ and hence we
know that $Q_{col}$ has no layer $z$ states if the collapse-round of $b$ is
$0$. Thus the property is maintained.
5. 5.
If $o=rew_{b}$ then $w={{a}^{u}}:_{1}{v}$ and $w^{\prime}={{b}^{u}}:_{1}{v}$.
Note none of the pop- or collapse-rounds are changed, and the run of $w$
beginning
${q_{p}^{1}}\xrightarrow[Q_{col}]{a}\left({{Q_{1},\dots,Q_{n}}}\right)$ and
satisfying the property implies a run of $w^{\prime}$ in $B$ beginning
${q_{p^{\prime}}^{1}}\xrightarrow[Q_{col}]{b}\left({{Q_{1},\dots,Q_{n}}}\right)$
and also satisfying the property.
6. 6.
If $o=noop$ then $w={{a}^{u}}:_{1}{v}$ and $w^{\prime}={{a}^{u}}:_{1}{v}$.
Note none of the pop- or collapse-rounds are changed, and the run of $w$
beginning
${q_{p}^{1}}\xrightarrow[Q_{col}]{a}\left({{Q_{1},\dots,Q_{n}}}\right)$ and
satisfying the property implies a run of $w^{\prime}$ in $B$ beginning
${q_{p^{\prime}}^{1}}\xrightarrow[Q_{col}]{a}\left({{Q_{1},\dots,Q_{n}}}\right)$
and also satisfying the property.
Thus we are done. $\square$
### G.4 Complexity
Solving the control state reachability problem requires finding a path in the
reachability graph. Since each vertex can be stored in
${\mathcal{O}}\mathord{\left({2\uparrow_{n-1}\left({{f}\mathord{\left({\zeta,\ell}\right)}}\right)}\right)}$
space, where $f$ is a polynomial and $\ell$ the number of control states, and
we require
${\mathcal{O}}\mathord{\left({2\uparrow_{n-1}\left({{f}\mathord{\left({\zeta,\ell}\right)}}\right)}\right)}$
time to decide the edge relation, we have via Savitch’s algorithm, a
${\mathcal{O}}\mathord{\left({2\uparrow_{n-1}\left({{f}\mathord{\left({\zeta,\ell}\right)}}\right)}\right)}$
space procedure for deciding the control state reachability problem. We also
observe that the solution to the global control state reachability problem may
contain at most
${\mathcal{O}}\mathord{\left({2\uparrow_{n}\left({{f}\mathord{\left({\zeta,\ell}\right)}}\right)}\right)}$
tuples.
|
arxiv-papers
| 2013-10-09T20:58:12 |
2024-09-04T02:49:52.235349
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Matthew Hague",
"submitter": "Matthew Hague",
"url": "https://arxiv.org/abs/1310.2631"
}
|
1310.2668
|
The Lumer-Phillips Theorem For Two–parameter $C_{0}$–semigroups
Rasoul Abazari111Corresponding Author E-mail:[email protected],
[email protected], Assadollah Niknam, Mahmoud Hassani
Department of Mathematics, Faculty of Sciences, Mashhad Branch, Islamic Azad
University, Mashhad, Iran.
> Abstract: In this paper we extend the Lumer-Phillips theorem to the context
> of two–parameter $C_{0}$–semigroup of contractions. That is, we characterize
> the infinitesimal generators of two–parameter $C_{0}$–semigroups of
> contractions. Conditions on the behavior of the resolvent of operators,
> which are necessary and sufficient for the pair of operators to be the
> infinitesimal generator of a $C_{0}$–semigroup of contractions are given.
> Keywords:Lumer-Phillips Theorem,Two–parameter $C_{0}$–semigroup, Dissipative
> operator.
## 1 Preliminaries
The semigroups of operators have several application in areas of applied
mathematics such as prediction theory and random fields. This theory is useful
to describe the time evolution of physical system in quantum field theory,
statistical mechanic and partial differential equations[2], [4].
In this section, we state some definitions and theorems as preliminaries to
describe the main results. We start by state the definition of two–parameter
semigroups.
###### Definition 1.1.
Let $X$ be a Banach space. By a two–parameter semigroup of operators we mean a
function $T:\mathbb{R}_{+}\times\mathbb{R}_{+}\longrightarrow B(X)$ with the
following properties;
i) $T(0,0)=I$
ii) $T(s+s^{\prime},t+t^{\prime})=T(s,t)T(s^{\prime},t^{\prime})$
If $(s,t)\longrightarrow T(s,t)x$ is continuous for all $x\in X$, then it is
called strongly continuous and if $(s,t)\longrightarrow T(s,t)x$ is norm
continuous, then it is called uniformly continuous.
A strongly continuous semigroup of bounded linear operators on $X$ will be
called a semigroup of class $C_{0}$ or simply $C_{0}$–semigroup.
Let $T(s,t)$ be any two–parameter semigroup, if we consider $u(s)$ and $v(t)$
as below,
$u(s)=T(s,0)\ \ \ ,\ \ \ \ v(t)=T(0,t)$
then the semigroup property of $T$ implies that $T(s,t)=u(s)v(t)$ and $T(s,t)$
is strongly (resp. uniformly) continuous if and only if $u(s)$ and $v(t)$ are
strongly (resp. uniformly) continuous as one–parameter semigroup.
If $A_{1}$ and $A_{2}$ are infinitesimal generators of $u(s)$ and $v(t)$
respectively, then we will thinks of the pair $(A_{1},A_{2})$ as infinitesimal
generator of $T(s,t)$. For more details on the such generators, see [1]
###### Theorem 1.1.
[5], Let $T(t)$ be $C_{0}$–semigroup, there exist constants $\omega\geq 0$ and
$M\geq 1$ such that
$\|T(t)\|\leq Me^{\omega t},\ \ \ \ \text{for}\ \ 0\leq t<\infty.$
Let $T(s,t)$ be a $C_{0}$–semigroup, Since $T(s,t)=u(s)v(t)$ and $u(s),v(t)$
are $C_{0}$–semigroup in the manner of one–parameter, then by the previous
theorem there exist constants $\omega_{1},\omega_{2}\geq 0$ and
$M_{1},M_{2}\geq 1$such that
$\|u(s)\|\leq M_{1}e^{\omega_{1}s},$ $\|v(t)\|\leq M_{2}e^{\omega_{2}t}.$
Let $M=M_{1}M_{2}$, then we have
$\displaystyle\begin{split}\|T(s,t)\|&=\|u(s)v(t)\|\leq\|u(s)\|\|v(t)\|\\\
&\leq
M_{1}M_{2}e^{\omega_{1}s}e^{\omega_{2}t}=Me^{\omega_{1}s+\omega_{2}t}.\end{split}$
###### Definition 1.2.
If $\omega_{1}=\omega_{2}=0,$ then $T(s,t)$ is called uniformly bounded and if
moreover $M=1$ it is called semigroup of contractions.
Recall that if $A$ is a linear, not necessarily bounded operator in Banach
space $X$, the resolvent set $\rho(A)$ of $A$ is the set of all complex
numbers $\lambda$ for which $\lambda I-A$ is invertible i.e, $(\lambda
I-A)^{-1}$ is a bounded linear operator in $X$. The family
$R(\lambda,A)=(\lambda I-A)^{-1}$, $\lambda\in\rho(A)$ of bounded linear
operators is called the resolvent of $A$.
Let $X$ be a Banach space and $X^{*}$ be its dual. $<x^{*},x>$ or $<x,x^{*}>$
denotes the value of $x^{*}\in X^{*}$ at $x\in X$. For every $x\in X$ we
define the duality set $F(x)\subset X^{*}$ by
$F(x)=\\{x^{*}:\ \ x^{*}\in X^{*}\ \ and\ \
<x^{*},x>=\|x\|^{2}=\|x^{*}\|^{2}\\}.$
From the Hahn-Banach theorem it follows that $F(x)\neq\phi$ for every $x\in
X.$
A linear operator $A$ is dissipative if for every $x\in D(A)$, the domain of
$A$, there is a $x^{*}\in F(x)$ such that $Re<Ax,x^{*}>\leq 0.$
## 2 Main Results
We state first the following useful theorems which can be found for example in
[5], [3].
###### Theorem 2.1.
A linear operator $A$ is dissipative if and only if,
$\|(\lambda I-A)x\|\geq\lambda\|x\|\ \ \text{for all}\ \ x\in D(A)\ \
\text{and}\ \ \lambda>0.$
###### Theorem 2.2.
For a dissipative operator $(A,D(A))$ the following properties hold.
i) $\lambda-A$ is injective for all $\lambda>0$ and
$\|(\lambda-A)^{-1}z\|\leq\frac{1}{\lambda}\|z\|,$
for all $z$ in the range $R(\lambda-A)=(\lambda-A)D(A)$.
ii) $\lambda-A$ is surjective for some $\lambda>0$ if and only if it is
surjective for each $\lambda>0$. In that case, one has
$(0,\infty)\subset\rho(A)$.
ii) $A$ is closed if and only if the range $R(\lambda-A)$ is closed for some
(hence all) $\lambda>0$.
iv) If $R(A)\subseteq\overline{D(A)}$ then $A$ is closable. Its closure
$\overline{A}$ is again dissipative and satisfies
$R(\lambda-\overline{A})=\overline{R(\lambda-A)}$ for all $\lambda>0$.
The following theorem can be found in [3].
###### Theorem 2.3.
A pair $(A_{1},A_{2})$ of operators with domain in $X$ is infinitesimal
generator of $C_{0}$–two–parameter semigroup $T(s,t)$ satisfying
$\|T(s,t)\|\leq M_{0}e^{\omega s+\omega^{\prime}t},$ for some $M_{0}\geq
1,\omega,\omega^{\prime}>0,$ if and only if
(i) $A_{1}$ and $A_{2}$ are closed and densely defined operators and
$R(\lambda^{\prime},A_{2})R(\lambda,A_{1})=R(\lambda,A_{1})R(\lambda^{\prime},A_{2}),$
for each $\lambda\geq\omega,\lambda^{\prime}\geq\omega^{\prime}.$
(ii) The resolvent sets $\rho(A_{1})$ and $\rho(A_{2})$ contain
$[\omega,\infty)$ and $[\omega^{\prime},\infty)$, respectively and there is
some $M\geq 1$ such that,
$\|R(\lambda,A_{1})^{n}\|\leq\frac{M}{(Re\lambda-\omega)^{n}},$
$\|R(\lambda^{\prime},A_{2})^{n}\|\leq\frac{M}{(Re\lambda^{\prime}-\omega^{\prime})^{n}},$
where $Re\lambda\geq\omega$ and $Re\lambda^{\prime}\geq\omega^{\prime}.$
Now we state extended Lumer–Phillips theorem as follows.
###### Theorem 2.4.
Let $A_{1}$ and $A_{2}$ are linear operators with dense domain in $X$.
(a) If $A_{1}$ and $A_{2}$ are dissipative and $A_{2}$ be bounded and there
exist $\lambda_{1},\lambda_{2}>0$ such that
$R(\lambda_{1}I-A_{1})=R(\lambda_{2}I-A_{2})=X$ and $A_{1}A_{2}=A_{2}A_{1}$
then $(A_{1},A_{2})$ is the infinitesimal generator of a $C_{0}$–semigroup of
contractions on $X$.
(b) If $(A_{1},A_{2})$ is the infinitesimal generator of $C_{0}$–semigroup of
contractions on $X$, then
$R(\lambda I-A_{1})=R(\lambda I-A_{2})=X,\ for\ all\ \lambda>0,$
$A_{1}$ and $A_{2}$ are dissipative. Moreover for every $x\in D(A_{1}),y\in
D(A_{2}),x^{*}\in F(x)$ and $y^{*}\in F(y)$, we have
$Re<A_{1}x,x^{*}>\leq 0,$
and
$Re<A_{2}y,y^{*}>\leq 0.$
###### Proof.
(a) Since $R(\lambda_{1}I-A_{1})=R(\lambda_{2}I-A_{2})=X,$ then by theorem
2.2, $R(\lambda I-A_{1})=R(\lambda I-A_{2})=X$ for every $\lambda>0$.
Therefore $\rho(A_{1})\supset[0,\infty),\rho(A_{2})\supset[0,\infty)$ and by
theorem 2.1 we have, $\|R(\lambda I,A_{1})\|\leq\lambda^{-1}$ and $\|R(\lambda
I,A_{2})\|\leq\lambda^{-1}.$
On the other hand, let $\lambda,\lambda^{\prime}>0$. Hence by theorem 2.1 we
have that $(\lambda I-A_{1})^{-1}$ and $(\lambda^{\prime}I-A_{2})^{-1}$ exist.
By the assumption $A_{1}A_{2}=A_{2}A_{1}$ hence
$\displaystyle(\lambda
I-A_{1})(\lambda^{\prime}I-A_{2})=(\lambda^{\prime}I-A_{2})(\lambda I-A_{1}),$
(1)
Also $(\lambda I-A_{1})D(A_{2})=X$. Since $A_{2}$ is bounded therefore
$(\lambda^{\prime}I-A_{2})(\lambda
I-A_{1})D(A_{1})=(\lambda^{\prime}I-A_{2})X=X.$
Now let $y\in X$, so there is some $x\in D(A_{1})$ such that
$\displaystyle\begin{split}y&=(\lambda^{\prime}I-A_{2})(\lambda I-A_{1})x\\\
&=(\lambda I-A_{1})(\lambda^{\prime}I-A_{2})x,\end{split}$
last equality holds from (1). Therefore we have
$R(\lambda^{\prime},A_{2})R(\lambda,A_{1})y=x=R(\lambda,A_{1})R(\lambda^{\prime},A_{2})y,$
and also,
$R(\lambda^{\prime},A_{2})R(\lambda,A_{1})=R(\lambda,A_{1})R(\lambda^{\prime},A_{2}).$
By theorem 2.3, we conclude that $(A_{1},A_{2})$ is the infinitesimal
generator of a $C_{0}$–two–parameter semigroup of contractions on $X$.
(b) If $(A_{1},A_{2})$ is the infinitesimal generator of a
$C_{0}$–two–parameter semigroup $\\{W(s,t)\\}$ of contractions on $X$. Then by
theorem 2.3 part (ii), $[0,\infty)$ is contained in $\rho(A_{1})$ and
$\rho(A_{2})$, therefore
$R(\lambda I-A_{1})=R(\lambda I-A_{2})=X,\ \ \ for\ all\ \ \lambda>0.$
For prove the dissipatedness of $A_{1}$ and $A_{2}$, following the proof of
theorem 4.3 in [5] for the case of one–parameter, let $x\in D(A_{1}),\ y\in
D(A_{2}),\ x^{*}\in F(x)$ and $y^{*}\in F(y)$. Hence
$|<W(s,0)x,x^{*}>|\leq\|W(s,0)x\|\|x^{*}\|\leq\|x\|^{2},$
$|<W(0,t)y,y^{*}>|\leq\|W(0,t)y\|\|y^{*}\|\leq\|y\|^{2},$
and therefore
$Re<W(s,0)x-y,x^{*}>=Re<W(s,0)x,x^{*}>-\|x\|^{2}\leq 0,$
$Re<W(0,t)y-y,y^{*}>=Re<W(0,t)y,y^{*}>-\|y\|^{2}\leq 0.$
Dividing above states to $s$ and $t$ respectively and letting $s$ and $t$ to
zero, yield
$Re<A_{1}x,x^{*}>\leq 0,$
and
$Re<A_{2}y,y^{*}>\leq 0.$
These hold for every $x^{*}\in F(x)$ and $y^{*}\in F(y)$ and complete the
proof.
∎
The following example shows that there is a Banach space and operators
satisfying conditions in theorem 2.4.
###### Example 2.5.
Suppose $X$ be a set of functions on $\mathbb{R}^{2}$ as below;
$X=span\\{e^{\alpha x+\beta y}:\ -\infty<\alpha,\beta<\infty\\},$
and for every $s,t\geq 0,$ define $T(s,t)$ on $X$ by,
$(T(s,t)f)(x,y)=f(x+s,y+t),$
which $f\in X.$
Then $\\{T(s,t)\\}$ is a $C_{0}$–semigroup of contractions on $X$. Its
infinitesimal generator $(A_{1},A_{2})$ has the domains $D(A_{1})$ and
$D(A_{2})$ respectively which,
$D(A_{1})=\\{f:\ f\in X,\ f_{x}\ exists\ and\ f_{x}\in X\\},$ $D(A_{2})=\\{f:\
f\in X,\ f_{y}\ exists\ and\ f_{y}\in X\\}.$
and on $D(A_{1})$ and $D(A_{2})$,
$A_{1}f=f_{x}\ \ \ \ ,\ \ \ \ A_{2}f=f_{y},$
such that $f_{x}$ and $f_{y}$ are derivatives on $x$ and $y$, respectively.
Hence $A_{1}$ and $A_{2}$ have the property such that $A_{1}A_{2}=A_{2}A_{1}$
on X.
###### Remark 2.1.
A dissipative operator $A$ for which $R(I-A)=X,$ is called m–dissipative. If
$A$ is dissipative so is $\mu A$ for all $\mu>0$ and therefore if $A$ is
$m$–dissipative then $R(\lambda I-A)=X$ for every $\lambda>0.$ In terms of
$m$–dissipative operators the theorem 2.4 can be restated as: A pair of
densely defined operators $(A_{1},A_{2})$ is the infinitesimal generator of a
two–parameter $C_{0}$–semigroup of contractions if and only if these are
$m$–dissipative with the property $A_{1}A_{2}=A_{2}A_{1}$.
## References
* [1] R. Abazari, A. Niknam, M. Hassani, Approximately Inner Two–parameter $C_{0}$–group of Tensor Product of $C^{*}$–algebras, Australian Journal of Basic and Applied Sciences. 5(9) (2011) 2120-2126.
* [2] Van. Casteren, Generators of strongly continuous semigroups, Pitman Advanced Publishing, Research Note(1985).
* [3] K. Engel, R. Nagel, A Short Course on Operator Semigroups, Springer verlag (2006).
* [4] A. Niknam, _Infinitesimal generators of $C^{\star}$–Algebras_, Potential Analysis, 6 (1997) 1–9.
* [5] A. Pazy, Semigroups of linear operators and Applications to Partial Differential Equations, Springer verlag (1984).
Rasoul Abazari∗, Assadollah Niknam, Mahmoud Hassani
Department of Mathematics, Faculty of Sciences, Mashhad Branch, Islamic Azad
University, P.O.Box 413-91735, Mashhad, Iran.
∗Corresponding E-mail: [email protected], [email protected]
|
arxiv-papers
| 2013-10-10T00:17:59 |
2024-09-04T02:49:52.253570
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Rasoul Abazari, Assadollah Niknam, Mahmoud Hassani",
"submitter": "Rasoul Abazari",
"url": "https://arxiv.org/abs/1310.2668"
}
|
1310.2944
|
LHCP 2013
11institutetext: The Enrico Fermi Institute, The University of Chicago
# Recent QCD Results From ATLAS
Christopher Meyer on behalf of the ATLAS Collaboration 11 [email protected]
###### Abstract
A survey of recent QCD results using the ATLAS detector at the LHC is
presented.
## 1 Introduction
The precision measurement of basic quantum chromodynamic (QCD) observables
provides information on various aspects of the Standard Model. Measurements of
high-$p_{\mathrm{T}}$ (hard) QCD processes involving jets and photons can be
used to constrain the gluon portion of the protons parton distribution
functions (PDFs) at high-momentum fraction. Jet physics also provides a check
of the strong coupling constant, $\alpha_{\mathrm{S}}$. The underlying event
arising from multiple-parton interactions, beam-beam remnants, and
initial/final state radiation provides an irreducible background to all
measurements. As such, a good description by Monte Carlo (MC) simulation is
essential for making precision measurements, and searches for physics beyond
the standard model. A measurement of the effective area parameter for double-
parton scattering has also been performed, which is an important background in
certain searches. This proceeding to the LHCP 2013 conference provides a brief
summary of recent results on QCD using the ATLAS Aad:2008zzm detector at the
LHC.
## 2 Hard QCD
### 2.1 Jet Physics
Inclusive jet cross sections have been measured at $\sqrt{s}=2.76{\mathrm{\
Te\kern-1.00006ptV}}$ for anti-$k_{t}$ jets with $|y|<4.4$ and
$p_{\mathrm{T}}$ up to $300{\mathrm{\ Ge\kern-1.00006ptV}}$ Aad:2013lpa .
Because the pileup conditions at $\sqrt{s}=2.76{\mathrm{\
Te\kern-1.00006ptV}}$ are similar to those of the 2010 run at
$\sqrt{s}=7{\mathrm{\ Te\kern-1.00006ptV}}$, the same jet energy calibration
is used. This provides a detailed understanding of the correlations of the jet
energy calibration uncertainty between the two measurements. The double-
differential cross section has been measured as a function of both
$p_{\mathrm{T}}$ and $y$, so that the experimental uncertainties are much
reduced when the ratio of $2.76{\mathrm{\ Te\kern-1.00006ptV}}$ is taken with
$7{\mathrm{\ Te\kern-1.00006ptV}}$. The measurement is also performed in bins
of $x_{\mathrm{T}}=2p_{\mathrm{T}}/\sqrt{s}$ and $y$, where the theoretical
uncertainties largely cancel between the two centre-of-mass energies. A PDF
fit exploiting the measurements at both $2.76{\mathrm{\ Te\kern-1.00006ptV}}$
and $7{\mathrm{\ Te\kern-1.00006ptV}}$ is performed, providing a strong
constraint on the gluon PDF at high-momentum fraction (see figure 1).
Figure 1: The gluon portion resulting from various PDF fits, including
different combinations of HERA-I and ATLAS $2.76{\mathrm{\
Te\kern-1.00006ptV}}$ and $7{\mathrm{\ Te\kern-1.00006ptV}}$ data Aad:2013lpa
.
Multijet production ATLAS-CONF-2013-041 provides a direct probe to the
dependence of the theory prediction on higher order terms. Two observables are
defined: First, the cross section of events with $\geq 3$ jets divided by the
cross section of events with $\geq 2$ jets, both as a function of highest
jet-$p_{\mathrm{T}}$. Second, the ratio of $\geq 3$-jet to $\geq 2$-jet
samples of the inclusive jet cross section as a function of jet
$p_{\mathrm{T}}$. A ratio of the two observables is taken (see figure 2) to
reduce the uncertainty on the jet energy calibration, the dominant source of
error. Because the first definition is proportional to the probability that a
two-jet event radiates a third jet (thus is proportional to
$\alpha_{\mathrm{S}}$), and is less sensitive to the choice of
renormalisation/factorisation scale, it is used in the fit for
$\alpha_{\mathrm{S}}$. The best fit for $\alpha_{\mathrm{S}}$ is determined
using NLOJet++ predictions interfaced with the MSTW 2008 PDF set, for a scan
of $\alpha_{\mathrm{S}}$ values. A best fit value of
$\alpha_{\mathrm{S}}(M_{Z})=0.111\pm
0.006\mathrm{(exp.)}^{+0.016}_{-0.003)}\mathrm{(theory)}$ is found, showing
good agreement with the global average. The fit value for
$\alpha_{\mathrm{S}}$ using different $p_{\mathrm{T}}$ bins is evolved to the
average $p_{\mathrm{T}}$ value for each bin (up to $800{\mathrm{\
Ge\kern-1.00006ptV}}$) using the two-loop approximation of the Renormalization
Group Equation, where agreement within the experimental uncertainties is seen
when compared to the world average.
Figure 2: The inclusive jet cross section taken as a ratio for events with
$\geq 3$ jets to events with $\geq 2$ jets ATLAS-CONF-2013-041 . Theory
predictions using NLOJet++ interfaced with the MSTW 2008 PDF set and including
non-perturbative corrections are shown for two separate values of
$\alpha_{\mathrm{S}}$.
### 2.2 Photon Production
The inclusive photon ATLAS-CONF-2013-022 and diphoton ATLAS-CONF-2013-023
cross sections measure prompt photon production with minimal surrounding
activity. The cross sections include direct photons (those produced by the
hard collision) as well as fragmentation photons (resulting from the
fragmentation of a high-$p_{\mathrm{T}}$ parton). In general, photons in an
acceptance $|\eta^{\gamma}|<1.37$ and $1.52\leq|\eta^{\gamma}|<2.37$ are used
to avoid uninstrumented portions of the electromagnetic calorimeter. The
inclusive photon cross sections as a function of $E_{\mathrm{T}}$ is well
described within uncertainties by next-to-leading order (NLO) theory
predictions made by Jetphox (which includes both direct and fragmentation
contributions). A slight deficit in the theory prediction is observed for
low-$E_{\mathrm{T}}$, while the data is overestimated by the theory prediction
at high-$E_{T}$.
The $p_{\mathrm{T}}$ of two-photon systems is compared to theory predictions
by DIPHOX and 2$\gamma$NNLO. DIPHOX includes both direct and fragmentation
components at NLO, as well as the NNLO diagram for $gg\to\gamma\gamma$.
2$\gamma$NNLO includes the full NNLO prediction of the direct photon
contribution, however neglects the fragmentation component. The NNLO
prediction best describes data, except at low $p_{\mathrm{T}}$ where the
fragmentation contribution is large (see figure 3).
Figure 3: The diphoton cross section as a function of the transverse momentum
of the diphoton system ATLAS-CONF-2013-023 . The black points are data, the
green bands are the DIPHOX prediction, and the yellow band is the
2$\gamma$NNLO prediction.
Measuring photon production in association with a jet Aad:2012tba provides an
interesting probe of $|\cos\theta^{\gamma j}|$, which is sensitive to the spin
of the exchange particle. Good agreement is observed compared with the
predictions of Jetphox, using multiple PDF sets. The angular distribution also
serves as a discriminating variable between photons produced directly and by
fragmentation, as seen in figure 4.
Figure 4: The cross section for photon production in association with a jet
Aad:2012tba . The solid pink circles are data, the lines represent the leading
order prediction for the direct photon contribution (blue) and the
fragmentation contribution (pink).
## 3 Underlying Event
### 3.1 Event Shape
A measurement of the underlying event has been performed in ATLAS which
focuses on inclusive jet and dijet events ATLAS-CONF-2012-164 , considering
jets of $p_{\mathrm{T}}>20{\mathrm{\ Ge\kern-1.00006ptV}}$ and $|y|<2.8$.
Distributions of charged particle multiplicity, charged and inclusive $\sum
p_{\mathrm{T}}$ densities, and mean charged-particle $p_{\mathrm{T}}$ are
studied in the “transverse region,” defined as the region
$\pi/3\leq|\Delta\phi|<2\pi/3$ from the highest-$p_{\mathrm{T}}$ jet in the
event. Activity in the transverse region is increased due to NLO emission,
such that the difference in activity between two transverse regions is also an
interesting observable. Good agreement is observed when restricting the
comparison of data and leading order MC simulation to events with exactly two
jets, as expected in a region of phase space with little emission. In general
the MC simulation shows decent agreement across a variety of variables, with
HERWIG performing slightly better describing the properties of charged
particles in underlying event (see figure 5).
Figure 5: The number of charged particles per unit area ($\eta/\phi$) in the
transverse region, for events where only two jets are present ATLAS-
CONF-2012-164 . Data (solid black points) are compared with leading order MC
predictions.
### 3.2 Double-parton Scattering
At higher $\sqrt{s}$ the low momentum-fraction region where PDFs are large is
probed, so that multiple-parton contributions can become non-negligible. This
gives rise to an important background for many single parton scattering
measurements. The ATLAS analysis of double-parton scattering Aad:2013bjm
employs a template fit to determine the fraction of events where a $W$ is
produced in association with exactly two jets arising from double-parton
interactions. Jets with $p_{\mathrm{T}}>20{\mathrm{\ Ge\kern-1.00006ptV}}$ and
$|y|<2.8$ are considered for this measurement. The double-parton production
fraction is used to derive the effective area parameter
($\sigma_{\mathrm{eff}}$) for hard double-parton scattering. As shown in
figure 6 the result of $\sigma_{\mathrm{eff}}=15\pm
3(\mathrm{stat.})^{+5}_{-3}(\mathrm{sys.})$ mb is consistent with those
measured by previous experiments.
Figure 6: The effective cross section for double parton scattering in ATLAS,
compared with previous results Aad:2013bjm .
## References
* (1) ATLAS Collaboration, JINST 3, S08003 (2008)
* (2) ATLAS Collaboration (2013), 1304.4739
* (3) ATLAS Collaboration, ATLAS-CONF-2013-041 (2013), http://cds.cern.ch/record/1543225
* (4) ATLAS Collaboration, ATLAS-CONF-2013-022 (2013), http://cds.cern.ch/record/1525723
* (5) ATLAS Collaboration, ATLAS-CONF-2013-023 (2013), http://cds.cern.ch/record/1525728
* (6) ATLAS Collaboration, JHEP 1301, 086 (2013), 1211.1913
* (7) ATLAS Collaboration, ATLAS-CONF-2012-164 (2013), http://cds.cern.ch/record/1497185
* (8) ATLAS Collaboration, New J.Phys. 15, 033038 (2013), 1301.6872
|
arxiv-papers
| 2013-10-10T20:00:09 |
2024-09-04T02:49:52.264534
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Chris Meyer (for the ATLAS Collaboration)",
"submitter": "Christopher Meyer",
"url": "https://arxiv.org/abs/1310.2944"
}
|
1310.2945
|
LHCP 2013
11institutetext: The Enrico Fermi Institute, The University of Chicago
# The ATLAS Tile Calorimeter Calibration and Performance
Christopher Meyer on behalf of the ATLAS Collaboration 11 [email protected]
###### Abstract
A brief summary of the hadronic calorimeter calibration systems and
performance results, in the ATLAS detector at the LHC is given.
## 1 Introduction
The ATLAS Aad:2008zzm tile calorimeter (TileCal) Aad:2010af is a sampling,
hadron calorimeter, located at the LHC. The central barrel portion covers
$|\eta|<0.8$, while the extended partitions on either side cover out to
$|\eta|<1.7$. It is composed of alternating layers of plastic scintillating
material and steel (see figure 1), grouped to create cells. In total, 9852
photmultiplier tubes (PMTs) read out energy deposited in the detector. The
calorimeter was designed to have a resolution of $\sigma/E=50\%/\sqrt{E}\oplus
3\%$, and enable a jet energy calibration uncertainty of $<1\%$. The PMT
signal is first shaped into a pulse with FWHM$\sim 50$ ns, then sampled by a
40 MHz analog to digital converter (ADC). To cover the full dynamic range two
gain channels are used, so that smaller signals are amplified before being
sampled by the ADC. This results in precise signal reconstruction over the
full dynamic range. The resulting seven samples of the pulse are fit using the
optimal filter method Usai:2011zz to determine the amplitude in ADC counts
and timing in ns (see figure 2). Although the digital signal processor which
performs the fit has limited resolution due to the use of fast lookup tables,
the majority of the signals produced by physics show negligible difference
when compared to the full offline reconstruction.
To convert from ADC counts to energy in MeV a series of calibrations is
applied:
* •
Charge injection system (CIS): provides a calibration $C_{\mathrm{CIS}}$ from
ADC counts to pC.
* •
Test beam: the initial calibration $C_{\mathrm{testbeam}}$ converting pC to
MeV , derived using test beam results.
* •
Cesium (${}^{137}\mathrm{Cs}$): provides a relative calibration
$C_{\mathrm{Cs}}$ to account for changes in the scintillating material,
optical fibers, and PMTs since deriving the test beam calibration
Adragna:2009zz .
* •
Laser: provides a relative calibration $C_{\mathrm{laser}}$ to account for the
drift of the PMTs and optical fibers between ${}^{137}\mathrm{Cs}$ runs.
The total calibration applied to the fitted pulse amplitude $A$ in ADC counts
to derive the measured electromagnetic scale energy $E$ is:
$E[{\mathrm{\ Me\kern-1.00006ptV}}]=C_{\mathrm{testbeam}}\times
C_{\mathrm{Cs}}\times C_{\mathrm{laser}}\times C_{\mathrm{CIS}}\times
A[\mathrm{ADC}]$
Below, the performance of the calibration systems and TileCal as a whole are
discussed in more detail.
Figure 1: One module of the tile calorimeter, showing alternating steel and
scintillating material. Aad:atlaslist Figure 2: Example pulse shape showing
the 7 sampled points, the fitted pulse shape, and the amplitude, timing, and
pedestal components.
## 2 Calibration Systems
In addition to providing up to date calibration constants for physics signals,
the calibration systems are also staggered such that issues throughout the
physics readout path can be diagnosed. For example, if an issue is found in
the calibration data from the ${}^{137}\mathrm{Cs}$ and Laser systems, but not
in CIS, the problem likely lies in the PMT (see figure 3). This setup is
particularly useful for trouble-shooting unstable high-voltage power supplies
and pathological PMTs. For this reason, as well as providing high quality
calibrations, it is important to keep track of performance for the various
calibration systems, as outlined below.
Figure 3: Overview of the calibration systems in TileCal, and which portions
of the readout electronics they are sensitive to.
### 2.1 Cesium
The cesium system Aad:2010af ; Starchenko:2002ju ; Shalanda:2003rq circulates
a radioactive ${}^{137}\mathrm{Cs}$ source through 10 km of tubes in TileCal.
Photons with energy of $0.662{\mathrm{\ Me\kern-1.00006ptV}}$ are emitted at a
known rate, and the integrated current is read out from each cell as the
source passes by. This provides a relative calibration $C_{\mathrm{Cs}}$,
which can be combined with the preliminary test beam calibration factor
$C_{\mathrm{testbeam}}$ to convert pC to MeV. Three ${}^{137}\mathrm{Cs}$
sources are circulated through TileCal approximately once per month to measure
the changing conditions of the scintillating material (resulting from
irradiation by physics runs) as well as the PMTs (changes of the PMT vacuum
due to signals flushing out impurities). The results have been cross checked
in situ using muons from cosmic rays as well as physics runs. By using
tracking information to determine the muon momentum, the expected energy
deposited in the calorimeter can be compared with the actual energy measured.
### 2.2 Laser
The laser system Aad:2010af ; Viret:2010zz sends a pulse of light with a
known amplitude through fiber-optic cables to each PMT. This provides a
relative calibration $C_{\mathrm{laser}}$ with respect to the cesium, tracking
the PMT drift over the period of a month (the time interval between
${}^{137}\mathrm{Cs}$ runs). By definition $C_{\mathrm{laser}}=1$ immediately
following a ${}^{137}\mathrm{Cs}$ run. The evolution of the laser and
${}^{137}\mathrm{Cs}$ calibrations are compared in figure 4 for E1 and E2
cells, located in the crack between the long and extended barrels. Because
these cells have the most direct exposure to the interaction point, they are
expected to show the largest change. Differences are visible because the laser
system is only sensitive to the drift of the PMTs, while the
${}^{137}\mathrm{Cs}$ calibration is also affected by changes in the
scintillating material. For the majority of the cells the PMT and optical
fiber response is much more stable, as shown in figure 5. During the last two
months of 2012 running, the laser calibration changed only $0.5\%$ on average.
Figure 4: Evolution of the ${}^{137}\mathrm{Cs}$ and laser calibration
constants during 2012 data taking. Shown for the E1 and E2 cells, located in
the crack between the long and extended barrel of TileCal Aad:tilepub1 .
Figure 5: Average variation of the laser calibration for low gain channels in
TileCal, shown for the last two months of 2012 data taking Aad:tilepub1 .
### 2.3 Charge Injection System
The charge injection system Aad:2010af ; Anderson:2005ym injects a pulse of
known charge and records the output in ADC counts. After scanning a range of
input charge, the results are used to derive the $C_{\mathrm{CIS}}$ conversion
factor, converting ADC counts to pC. CIS is generally very stable, with an
average drift of $0.4\%$ during 2012 running (see figure 6). While
instabilities in the electronics can cause shifts of the individual channel
calibrations of up to 1%, these jumps are rare and are generally corrected
within 28 days.
Figure 6: Distribution of fractional change of CIS constants over the 2012 run
period Aad:tilepub1 .
## 3 Performance
In general TileCal provided stable, good quality data throughout Run I at the
LHC. The readout electronics continue to perform well, as seen by the
comparing of a pulse shape from the readout electronics with the reference
pulse shape.The calibration systems obtain stable results, and correct for any
changes due to irradiation before it can affect physics. Good agreement is
observed between data and MC simulation when plotting the mean $E/p$ for
single particle response, shown in figure 7. The high-voltage power supplies
also performed very well, with minimal deviations from the set value. During
long shutdown 1 new radiation-hard low-voltage power supplies are being
installed, which also provide more Gaussian shaped noise with a smaller RMS
(see figure 8). This will reduce the number of tripped modules during physics
runs, as well as further improve the resolution for physics analyses which
rely on TileCal.
Figure 7: Average energy (E) over momentum (p) of single particles measured in
data (black) and MC simulation (red) Aad:tilepub2 . Figure 8: Cell noise in
TileCal readout electronics, comparing the new (red) low voltage power
supplies with the old (blue) Aad:tilepub2 .
## 4 Summary
The calibration systems (${}^{137}\mathrm{Cs}$, Laser, and CIS) provide an up-
to-date status of TileCal performance. In general, the calibrations have shown
to be stable over time. When drifts arise due to changes in the scintillating
material, optical fibers, PMTs, or readout electronics they are quickly caught
and corrected for. TileCal has consistently provided good quality, well
calibrated data for the duration of Run I at the LHC.
## References
* (1) ATLAS Collaboration, JINST 3, S08003 (2008)
* (2) ATLAS Collaboration, Eur. Phys. J. C70, 1193 (2010), 1007.5423
* (3) G. Usai (ATLAS Tile Calorimeter Collaboration), J. Phys. Conf. Ser. 293, 012056 (2011)
* (4) P. Adragna, C. Alexa, K. Anderson, A. Antonaki, A. Arabidze et al., Nucl. Instrum. Meth. A606, 362 (2009)
* (5) ATLAS Collaboration, _ATLAS Technical Paper List Of Figures_ , http://twiki.cern.ch/twiki/bin/view/AtlasPublic/AtlasTechnicalPaperListOfFigures (2008)
* (6) E. Starchenko et al. (ATLAS Collaboration), Nucl. Instrum. Meth. A494, 381 (2002)
* (7) N. Shalanda et al. (ATLAS Tile Calorimeter Collaboration), Nucl. Instrum. Meth. A508, 276 (2003)
* (8) S. Viret (LPC ATLAS Collaboration), Nucl. Instrum. Meth. A617, 120 (2010)
* (9) ATLAS Collaboration, _Approved Tile Calorimeter Plots_ , http://twiki.cern.ch/twiki/bin/view/AtlasPublic/ApprovedPlotsTile (2013)
* (10) K. Anderson, A. Gupta, F. Merritt, M. Oreglia, J. Pilcher et al., Nucl. Instrum. Meth. A551, 469 (2005)
* (11) ATLAS Collaboration, _Public Tile Calorimeter Plots for Collision Data_ , http://twiki.cern.ch/twiki/bin/view/AtlasPublic/TileCaloPublicResults (2013)
|
arxiv-papers
| 2013-10-10T20:00:17 |
2024-09-04T02:49:52.269533
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Chris Meyer (for the ATLAS Collaboration)",
"submitter": "Christopher Meyer",
"url": "https://arxiv.org/abs/1310.2945"
}
|
1310.2946
|
# Results on QCD jet production at ATLAS and CMS
Christopher J. Meyer, on behalf of the ATLAS and CMS Collaborations
The production of jets at the Large Hadron Collider (LHC) at $\sqrt{s}=7$ TeV
is summarized, including results from both the ATLAS and CMS detectors.
Current jet performance is described, followed by inclusive jet and multi-jet
measurements in various final state configurations. Finally some results on
heavy flavour and jet substructure are presented.
At both the ATLAS $\\!{}^{{\bf?}}$ and CMS $\\!{}^{{\bf?}}$ detectors in the
LHC, jets serve as a proxy to final state partons. Following the hard
collision they undergo parton showering, hadronisation, and subsequently
interact in the surrounding detector. To reconstruct and calibrate the
constituents of jets, CMS uses a particle flow method which employs several
subdetectors $\\!{}^{{\bf?}}$ while ATLAS forms jets using finely segmented
calorimeters.$\\!{}^{{\bf?}}$ In both cases a pileup offset correction is
applied to remove additional energy from multiple proton-proton collisions.
The anti-$k_{T}$ clustering algorithm $\\!{}^{{\bf?}}$ is the preferred choice
for jet reconstruction, with other methods such as Cambridge-Aachen
$\\!{}^{{\bf?}}$ used for jet substructure. For cross section measurements a
radius between $0.4\leq R\leq 0.7$ is used, while larger jet radii ($R\geq
1.0$) are used for jet substructure studies.
The 2010 jet energy calibration is derived using Monte Carlo (MC) simulation
tuned using test beam and early collision data. The response is derived by
comparing fully simulated and reconstructed jets to truth jets, with in situ
techniques such as multijet, photon-jet, and Z-jet balance used as cross
checks in data. The uncertainty on the derived jet energy calibration, shown
in Figure 1, is often the dominant source of experimental error on cross
section measurements.
The inclusive jet cross section measures the production rate of jets as a
function of both transverse momentum ($p_{T}$) and rapidity ($y$). In 2010
jets were measured with 20 GeV $<p_{T}<$ 1550 GeV out to rapidities of
$|y|=4.4$ using 37 pb-1 of integrated luminosity at ATLAS,$\\!{}^{{\bf?}}$
with similar results seen in CMS.$\\!{}^{{\bf?}}$ In 2011 the data sample of
4.8 fb-1 has extended the reach to a $p_{T}$ of almost 2 TeV in
CMS.$\\!{}^{{\bf?}}$ The large reach of this basic observable offers a
powerful test of the Standard Model over many orders of magnitude. A next-to-
leading order (NLO) calculation is performed using NLOJET++ $\\!{}^{{\bf?}}$,
with non-perturbative corrections applied to account for hadronisation and
underlying event. Monte Carlo events are also generated with POWHEG
BOX,$\\!{}^{{\bf?}}$ producing NLO matrix elements with parton showering which
are then interfaced to PYTHIA or HERWIG for hadronisation. There is generally
good agreement seen between acceptance corrected measurement and theory, as
shown in Figure 2. At high $p_{T}$, especially for large values of $y$, a
tension is observed with theory over estimating data.
Ratios of jet measurements are powerful because many systematics (jet energy
calibration uncertainty and luminosity for example) either paritally or fully
cancel when the ratio is taken. Taking the ratio of events with $N\geq 3$ jets
to $N\geq 2$ jets is an interesting probe of NLO effects. $\\!{}^{{\bf?}}$
Measured as a function of the scalar sum of jets $p_{T}$,
$H_{T}=\Sigma~{}jet~{}p_{T}$ where the sum is extended to all jets with
$p_{T}>50$ GeV and $|y|<2.5$, Figure 3 shows that for $H_{T}>500$ GeV a
variety of MC predicts the data well.
Figure 1: Fractional uncertainty on the jet energy calibration as a function
of jet $p_{T}$ in ATLAS $\\!{}^{{\bf?}}$ and CMS.$\\!{}^{{\bf?}}$
Figure 2: Measurement of the inclusive jet cross section in the ATLAS
detector.$\\!{}^{{\bf?}}$ A slight tension is observed between data and theory
at high $p_{T}$. Figure 3: Ratio of the cross section of events in CMS with
$N\geq 3$ jets to $N\geq 2$ jets, where only jets with $p_{T}>50$ GeV and
$|y|<2.5$ are considered. The ratio is shown as a function of
$H_{T}$.$\\!{}^{{\bf?}}$
The ratio of the inclusive dijet cross section (considering all combinations
of $N\geq 2$ jets in an event) to the exclusive dijet cross section (only
consider events with exactly $N=2$ jets) is sensitive to the resummation of
large $log(1/x)$ terms (BFKL evolution). All jets with $p_{T}>35$ GeV and
$|y|<4.7$ are considered, with the ratio plotted as a function of absolute
rapidity separation $|\Delta y|$ between jet pairs. $\\!{}^{{\bf?}}$ As seen
in Figure 4(a) PYTHIA gives the best agreement to data.
Heavy flavour at the LHC is important for understanding backgrounds in
searches for the Higgs boson and/or super-symmetric particles, as well as
providing a check of the hadronisation description in MC. Figure 4(b) shows
the ratio of jets containing a $D^{*\pm}$ meson to all jets as a function of
$z$, the $D^{*\pm}$ momentum along the jet axis divided by the jet energy.
$\\!{}^{{\bf?}}$ For this low $p_{T}$ slice the agreement between data and MC
is poor. At low $p_{T}$, $D^{*\pm}$ originate mostly from c-hadrons showing
that c-fragmentation in jets is not well modeled.
(a) Ratio of the inclusive dijet cross section to the exclusive dijet cross
section in CMS.$\\!{}^{{\bf?}}$
(b) Fraction of jets containing a $D^{*\pm}$ meson in ATLAS.$\\!{}^{{\bf?}}$
Figure 4: Two ratio measurements from ATLAS and CMS.
(a) $b\bar{b}$ dijet cross section as a function of dijet mass in ATLAS.
(b) $b\bar{b}$ dijet cross section as a function of $\Delta\phi$ in ATLAS.
Figure 5: Results on jets produced from $b$-hadrons.$\\!{}^{{\bf?}}$
The measurement of the dijet cross section for jets from b-hadrons tests the
production and hadronization of b-quarks. $\\!{}^{{\bf?}}$ Figure 5(a) shows
the dijet mass cross section from $b\bar{b}$ pairs, where theory is seen to
describe data well. For dijet systems which radiate a gluon, the azimuthal
angle $\Delta\phi$ between them will be reduced. Figure 5(b) shows that while
back-to-back systems (larger $\Delta\phi$) are well described, as $\Delta\phi$
decreases both POWHEG+Pythia and MC@NLO+Herwig begin to over estimate the
data.
Jet substructure is useful for identifying hadronic decays of boosted heavy
particles. Splitting/filtering using Cambridge-Aachen $R=1.2$ jets is one
example which undoes the clustering procedure until a large mass drop is
observed. This type of technique is robust against the effects of multiple
proton-proton interactions in a single bunch crossing. Figure 6 shows the
improved agreement between data and MC after splitting/filtering has been
performed,$\\!{}^{{\bf?}}$ giving confidence in the MC hadronisation
description for substructure studies.
Figure 6: Results from ATLAS on jet substructure for Cambridge-Aachen $R=1.2$
jets, before and after application of splitting/filtering.$\\!{}^{{\bf?}}$
## References
## References
* [1] ATLAS Collaboration, JINST 3 (2008) S08003.
* [2] CMS Collaboration, JINST 3 (2008) S08004.
* [3] CMS Collaboration, JINST 6 (2011) 11002.
* [4] ATLAS Collaboration, arXiv:1112.6426 [hep-ex].
* [5] M. Cacciari, G. P. Salam and G. Soyez, JHEP 0804 (2008) 063.
* [6] Y. L. Dokshitzer, G. D. Leder, S. Moretti and B. R. Webber, JHEP 9708 (1997) 001.
* [7] ATLAS Collaboration, Phys. Rev. D 86 (2012) 014022.
* [8] CMS Collaboration, Phys. Rev. Lett. 107 (2011) 132001
* [9] CMS Collaboration, CMS-PAS-QCD-11-004 (2012), https://cdsweb.cern.ch/record/1431022.
* [10] S. Catani and M. H. Seymour, Nucl. Phys. B 485 (1997) 291 [Erratum-ibid. B 510 (1998) 503].
* [11] S. Alioli, K. Hamilton, P. Nason, C. Oleari and E. Re, JHEP 1104 (2011) 081.
* [12] CMS Collaboration, Phys. Lett. B 702 (2011) 336.
* [13] CMS Collaboration, arXiv:1204.0696 [hep-ex].
* [14] ATLAS Collaboration, Phys. Rev. D 85 (2012) 052005.
* [15] ATLAS Collaboration, Eur. Phys. J. C 71 (2011) 1846.
* [16] ATLAS Collaboration, JHEP 1205 (2012) 128.
|
arxiv-papers
| 2013-10-10T20:00:22 |
2024-09-04T02:49:52.274495
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Chris Meyer (for the ATLAS and CMS Collaborations)",
"submitter": "Christopher Meyer",
"url": "https://arxiv.org/abs/1310.2946"
}
|
1310.3323
|
# It‘s Alive! Spontaneous Motion of Shear Thickening Fluids Floating on the
Air-Water Interface
Sunilkumar Khandavalli, Michael Donnell and Jonathan P. Rothstein
Department of Mechanical and Industrial Engineering
University of Massachusetts, Amherst, MA 01003, USA
###### Abstract
In this fluid dynamics video, we show the spontaneous random motion of thin
filaments of a shear-thickening colloidal dispersions floating on the surface
of water. The fluid is a dispersion of fumed silica nanoparticles in a low
molecular weight polypropylene glycol (PPG) solvent. No external field or
force is applied. The observed motion is driven by strong surface tension
gradients as the polypropylene glycol slowly diffuses from from the filaments
into water, resulting in the observed Marangoni flow. The motion is
exaggerated by the thin filament constructs by the attractive interactions
between silica nanoparticles and the PPG.
## 1 Introduction
Tears of wine, the soap boat, bubble motion are some of the commonly seen
amazing effects due to Marangoni flow - stresses generated by surface tension
gradient causing fluid motion. Here, we demonstrate a spectacular
manifestation of the Marangonic effect by using a shear-thickening fluid. The
shear-thickening fluid is hydrophilic fumed silica nanoparticles (AEROSIL@200)
in low molecular weight polypropylene glycol (PPG) (M.W 1000 g/mol). These
dispersions demonstrate strong shear and extensional thickening behavior. In
these two videos, Video1 and Video2, we show a spontaneous and intense erratic
motion of filaments of shear-thickening colloidal dispersion at air-water
interface. The profound motion which begins as a back and forth motion for
long fibres and transitions to high speed spinning as the fibres break and
become shorter is due to Marangonic effect - flow driven by stresses resulting
from surface tension gradient. The surface tension of PPG is $\sim$ 30 mN/m
and that of water is 72.8 mN/m. When PPG is added to the water, the difference
in the surface tension drives water away from PPG, due to the resulting local
stresses in the fluid. When thin filaments of shear-thickening fluid is added
to the water, the Marangoni effect is exaggeraged by the shape of the filament
constructs and the slow diffusion of PPG from the filaments which is limited
by the presence of nanoparticles and the shear-thickening rheology of the
fluid.
|
arxiv-papers
| 2013-10-12T01:23:54 |
2024-09-04T02:49:52.291794
|
{
"license": "Public Domain",
"authors": "Sunilkumar Khandavalli, Michael Donnell and Jonathan P. Rothstein",
"submitter": "Sunilkumar Khandavalli",
"url": "https://arxiv.org/abs/1310.3323"
}
|
1310.3365
|
# Electrohydrodynamically induced mixing
in immiscible multilayer flows
Radu Cimpeanu, Demetrios T. Papageorgiou
Department of Mathematics,
Imperial College London, SW7 2AZ London, United Kingdom
###### Abstract
In the present study we investigate electrostatic stabilization mechanisms
acting on stratified fluids. Electric fields have been shown to control and
even suppress the Rayleigh-Taylor instability when a heavy fluid lies above
lighter fluid. From a different perspective, similar techniques can also be
used to generate interfacial dynamics in otherwise stable systems. We aim to
identify active control protocols in confined geometries that induce time
dependent flows in small scale devices without having moving parts. This
effect has numerous applications, ranging from mixing phenomena to electric
lithography. Two-dimensional computations are carried out and several such
protocols are described. We present computational fluid dynamics videos with
different underlying mixing strategies, which show promising results.
## 1 Introduction
The field of microfluidics has been one of the most active areas in fluid
dynamics for the past few decades. With applications as diverse as microchip
design and medical/pharmaceutical devices, recent advances in theoretical,
numerical and experimental settings have had a powerful impact in the research
world.
A key process in such systems is represented by mixing of agents, which
becomes increasingly challenging as lengthscales become smaller and reach
micrometer-sized geometries. Very low Reynolds numbers generate numerous
difficulties in the control of such devices and several passive or active
mechanisms to manipulate the fluid flow in an accurate way have been explored
to date. In the following paragraphs, we focus especially on
electrohydrodynamically controlled models, which have proved successful in
reaching remarkable results with limited resource consumption.
An introduction of the model of the effects of an electric field on a flow of
immiscible fluids in a channel is introduced by Ozen et al.[1] Linear
stability theory, as well as a variety of theoretical parameter studies
centered around a Reynolds number of $1$, which is typical in microfluidic
context, are presented. The impact of the electric field, as well as other
quantities in the problem such as initial position of the interface within the
channel or viscosity ratio are discussed. An experimental setup from the same
authors [2] is constructed in order to identify features of drop formation in
a channel as a result of the influence of electric fields of various
strengths. The results reported are based on a channel of dimensions $70$ mm
(in $x$) $\times$ $0.25$ mm (in $y$) $\times$ $1.5$ mm (in $z$) with a
background Poiseuille flow at a flux which generates a
$Re=\mathcal{O}(10^{-2})$. Glycerine and corn oil have been used as the two
immiscible fluids in the system. Key findings indicate how drop size decreases
as the prescribed voltage is increased. Another extensive theoretical and
numerical study of instabilities in a channel flow that can be used for mixing
applications is shown in Ozen et al. [3]. Scenarios with both Couette and
Poiseuille flow are considered for leaky, as well as perfect dielectrics.
Computations are carried out for a large set of values of the Reynolds
numbers($0-10^{4}$), as interesting discussions can be based on parameter
regimes around known critical Reynolds numbers for the classical flows. An
important result is given by the fact that in the case of perfect dielectrics,
the electric field normal to the interface always has destabilizing effects,
which can then be exploited in the context of microfluidic mixing.
Lee et al. [4] have recently conducted a highly acclaimed review study of the
most successful mixing devices in geometries pertaining to microfluidic flow.
Key parameters in the vast majority of contemporary water-based systems are of
$Re=\mathcal{O}(10^{-1})$, with reference lengthscales of the order of $100\
\mu$m. These magnitudes provide an estimate which allows us to design a
theoretical framework, as well as a computational study with applicability to
devices presently used. The authors also indicate the experimental work of El
Moctar, Aubry and Batton [5] as representative for systems based on
electrohydrodynamic forces. El Moctar et al. use a T-type mixer with fluids of
similar properties (in this case corn oil, however dyed in a different color
and with different electric properties in each inflow channel) of sizes $30$
mm (in $x$) $\times$ $0.25$ mm (in $y$) $\times$ $0.25$ mm (in $z$) and
subject to an electric field corresponding to approximately $10^{5}$ V/m. The
setup corresponds to a Re $<0.02$ and both continuous and alternating currents
have been used with results drastically improving over scenarios with no
electric field. T-shaped mixers are in general one of the most popular choices
for mixing devices ([5],[6],[7],[8],[9],[10]) due to the their richness of
experimental and modelling possibilities and hence versatility for parameter
studies resulting in rapid advances for this application. The use of time
pulsing [6] has shown to be particularly successful in this context, leading
to high degrees of mixing on shorter timescales. Reynolds numbers are again in
the order of $10^{-1}-10^{1}$ ($0.3$ and $2.55$ for the mentioned publication)
and reference lengthscales are $\mu$m-sized. Electric fields strengths for
such geometries are of the order $10^{5}-10^{6}$ V/m, which is very common for
the relevant microdevices. Another example can be found in the experiment of
Tsouris et al.[7], where flows characterized by Re$=0.2,0.4$ and $0.9$ are
subjected to electric fields of $0-2\cdot 10^{6}$ V/m and show highly improved
mixing as the electric field strength increases.
In the present work we aim to describe a mixing mechanism that requires no
hydrodynamic forcing or a certain imposed velocity field. Instead we rely on
control protocols targeted towards the electric field only. The interfacial
dynamics achieved in response to electric excitation is then proved to be
effective in terms of reaching high degrees of mixing efficiency. Due to
simplicity and small resource consumption, the protocols we describe become an
attractive alternative to classical choices in microgeometries. Note that
similar mixing policies can then be applied to further enhance the performance
of existing devices in a very broad context.
## 2 Mathematical Description
The mathematical framework on which we construct our study is similar to the
investigation of Cimpeanu, Papageorgiou and Petropoulos[11], focused on the
electrohydrodynamic stabilization of the Rayleigh-Taylor instability in an
infinite vertical channel.
In the present work we consider two incompressible, immiscible, viscous fluids
in a two-dimensional setting as shown in Fig. 1. The flow is bounded by
horizontal parallel walls that are separated by a distance $L$, and are
unconfined in the lateral direction as shown in the figure (periodic boundary
conditions are considered). Using a Cartesian coordinate system, the interface
between the two fluids is denoted by $y=S(x,t)$, and fluids 1 and 2 occupy the
regions $y<S(x,t)$ and $y>S(x,t)$, respectively (in what follows subscripts
1,2 will refer to fluids 1 and 2). The horizontal walls at $y=\pm L/2$ are no-
slip, no-penetration boundaries and are also electrodes that can support a
voltage potential difference. The fluids are perfect dielectrics with given
permittivities $\epsilon_{1,2}$, viscosities $\mu_{1,2}$ and densities
$\rho_{1,2}$, and corresponding velocity vectors are
$\textbf{u}_{1,2}=(u_{1,2},v_{1,2})$. We denote the constant surface tension
coefficient at the interface by $\sigma$.
Figure 1: Sketch of domain
An electric field is imposed by grounding the electrode at $y=L/2$ and
imposing a constant voltage $\bar{V}^{*}$ at $y=-L/2$. The voltage potentials
$V_{1,2}$ in regions 1,2 satisfy Laplace’s equation (this follows from the
electrostatic approximation: Maxwell’s equations reduce to
$\nabla\times\textbf{E}_{1,2}=0$,
$\nabla\cdot(\epsilon_{1,2}\textbf{E}_{1,2})=0$, hence
$\textbf{E}_{1,2}=-\nabla V_{1,2}$ from the former condition with Laplace
equations following from the second condition away from the interface):
$\left(\frac{\partial^{2}}{\partial x^{2}}+\frac{\partial^{2}}{\partial
y^{2}}\right)V_{1,2}=0.$ (1)
The dimensional momentum and continuity equations are
$\displaystyle\rho_{1}(\textbf{u}_{1t}+(\textbf{u}_{1}\cdot\nabla)\textbf{u}_{1})$
$\displaystyle=$ $\displaystyle-\nabla
p_{1}+\mu_{1}\Delta\textbf{u}_{1}-\rho_{1}g\textbf{j},$ (2)
$\displaystyle\rho_{2}(\textbf{u}_{2t}+(\textbf{u}_{2}\cdot\nabla)\textbf{u}_{2})$
$\displaystyle=$ $\displaystyle-\nabla
p_{2}+\mu_{2}\Delta\textbf{u}_{2}-\rho_{2}g\textbf{j},$ (3)
$\displaystyle\nabla\cdot{{\textbf{u}}_{1,2}}$ $\displaystyle=$ $\displaystyle
0.$ (4)
We introduce the density, viscosity and permittivity ratio parameters
$\displaystyle r=\rho_{1}/\rho_{2},\ m=\mu_{2}/\mu_{1},\
\epsilon=\epsilon_{2}/\epsilon_{1},$ (5)
and non-dimensionalize the equations and boundary conditions using fluid $1$
as reference. Lengths are scaled by $L$, velocities by a reference value $U$
and pressures by $\rho_{1}U^{2}$. We list the following dimensionless
parameters
$\tilde{g}=\dfrac{gL}{U^{2}},\ \tilde{\mu}=\dfrac{\mu_{1}}{\rho_{1}UL}\
W_{e}=\dfrac{\sigma}{\rho_{1}gL^{2}},$ (6)
representing an inverse square Froude number $\tilde{g}$, an inverse Reynolds
number $\tilde{\mu}$ and an inverse Weber number denoted by $W_{e}$. Note that
since we consider $U\sim\sqrt{gL}$, dimensionless number $\tilde{g}\sim 1$ for
all cases. The effect of gravity can be artificially increased or decreased by
modifying the value of this number, however usually gravity plays a negligible
role within devices of very small physical lengthscales.
Furthermore, we scale voltage potentials by $V^{*}$ so that the dimensionless
electric parameter measuring the size of Maxwell stresses in the interfacial
stress balance equation becomes unity in fluid 1 variables. Inspection of the
stress tensor shows that this choice necessitates
$\displaystyle\rho_{1}U^{2}=\frac{\epsilon_{1}(V^{*})^{2}}{L^{2}}\Rightarrow
V^{*}=UL\sqrt{\rho_{1}/\epsilon_{1}}.$ (7)
With these scalings the Navier-Stokes equations for each fluid become
$\displaystyle\tilde{\textbf{u}}_{1t}+(\tilde{\textbf{u}}_{1}\cdot\nabla)\tilde{\textbf{u}}_{1}$
$\displaystyle=$
$\displaystyle-\nabla\tilde{p}_{1}+\tilde{\mu}\Delta\tilde{\textbf{u}}_{1}-\tilde{g}\textbf{j},$
(8)
$\displaystyle\tilde{\textbf{u}}_{2t}+(\tilde{\textbf{u}}_{2}\cdot\nabla)\tilde{\textbf{u}}_{2}$
$\displaystyle=$
$\displaystyle-r\nabla\tilde{p}_{2}+m\tilde{\mu}r\Delta\tilde{\textbf{u}}_{2}-\tilde{g}\textbf{j},$
(9)
where j is the unit vector in vertical direction and the decoration tilde is
used to refer to dimensionless quantities. The continuity equation in each
fluid is
$\nabla\cdot{\tilde{\textbf{u}}_{1,2}}=0.$ (10)
From the previously described set of equations, following a classical
linearization procedure and normal mode analysis, we identify the most
unstable wavenumbers within a certain setup. We concentrate on stably
stratified formats, where the vertical electric field can be used to generate
and enhance instabilities. Exploiting this, we construct initial perturbations
that allow for the rapid formation of high amplitudes of the disturbance and
ultimately lead to efficient mixing. This effect is achieved by imposing on-
off protocols in the electric field, which simply means oscillating between a
uniform vertical electric field to destabilize the flow, followed by an
interruption of the voltage feed. The repeated use of such a control leads to
rich dynamics of the passive tracer.
In the on-off scenario, the voltage is controlled via the boundary condition
on $\bar{V}^{*}$ at $y=-1/2$ (after non-dimensionalization). This can either
be a positive prescribed constant $\bar{V}$ for the "on"-mode, whereas the
"off"-mode is described by $\bar{V}^{*}=0$ at $y=-1/2$. We notice (see the
first segment of accompanying simulation video) that the dynamics generated by
the vertical motion of the interface is sufficient to achieve high degrees of
mixing. More interestingly however, it is possible to generate horizontal
motion as well (see right side of first segment of the attached video), since
the mechanism tries to select the most unstable mode at the expense of
breaking symmetry in the current interfacial shape. The following electric
field manipulation is geared towards controlling this particular effect.
The alternative to the on-off protocol is the imposition of a relay-type
structure, where the time-dependent voltage is now described by
$\bar{V}^{*}=\bar{V}+a((f\cdot(\textrm{atan}(x+s\cdot
t)-x_{0}))-(f\cdot(\textrm{atan}(x+s\cdot t)-x_{1}))).$ (11)
Notation $a$ is used for the amplitude of the voltage fluctuation, which is
normalized by $\pi$, $\bar{V}$ is the imposed background voltage, $s$ is a
term that gives the velocity of the leftward or rightward moving perturbation,
while $f$ is the factor that controls the arctan-smoothing. A high value of
$f$ results in a very steep slope of the disturbance, whereas a small value of
$f$ leads to well-behaved transition from $\bar{V}$ to $\bar{V}+a$ over a
larger area. This perturbation in the voltage is then contained between
regions $x_{0}$ and $x_{1}$, which need to have appropriately chosen values
within our scaling. Multiple such perturbations are constructed to mimic the
structure of the most unstable wavenumber as picked up by linear stability and
allow the generation of time-dependent flows within our confined geometry.
This is essentially a form of microfluidic pumping, which will be investigated
in more detail in the near future.
All simulations have been performed using the GERRIS [12] package, which
employs the volume-of-fluid method to discretize the multi-fluid system.
Several other features such as spatial adaptivity and parallelization, as well
as numerical techniques specialized for solving the Navier-Stokes equations,
are available in order to optimize the numerical treatment of the problem.
## 3 Key Parameters
The first segment of the attached simulation video (roughly $45$ seconds) is
composed of three simulations stacked horizontally, each representing a
separate on-off protocol scenario. All parameters related to the fluids
themselves are kept the same, the only difference is the non-dimensional time
at which the electric field is turned on or off. The imposed voltage is
constant in all computational experiments and is set to $\bar{V}=6.0$.
The domain has non-dimensional size $1\times 1$, while the relevant fluid
parameters are
* •
Density ratio $r=\rho_{1}/\rho_{2}=6/1$;
* •
Viscosity ratio $m=\mu_{2}/\mu_{1}=1/10$;
* •
Dimensionless viscosity $\tilde{\mu}=0.025$;
* •
Permittivity ratio $\epsilon=\epsilon_{2}/\epsilon_{1}=2/1$;
* •
Surface tension $\tilde{\sigma}=0.1$;
* •
Passive tracer radius $r=0.1$;
* •
Enhanced dimensionless gravity $\tilde{g}=10.0$.
We allow the simulations to run over approximately six dimensionless time
units and the spatial adaptivity is set to allow for a maximum of $2^{8}=256$
cells in the case of all variables in the problem, except for the interface
and the horizontal velocity, which can carry a maximum of $2^{9}=512$ cells,
thus resulting in a minimum $h=1/512\approx 0.002$.
The imposed electric field in each of the simulations (from left to right) is
as follows:
* •
Left: on between $t=0.0-5.0$, off between $t=5.0-6.0$;
* •
Center: on between $t=0.0-1.0$, $t=2.0-3.0$ and $t=4.0-5.0$, off between
$t=1.0-2.0$, $t=3.0-4.0$ and $t=5.0-6.0$;
* •
Right: on between $t=0.0-2.0$ and $t=4.0-6.0$, off between $t=2.0-4.0$.
The animation shows the concentration field $T$ varying from 0 (blue) to 1
(red), with a circular initial condition. The aim is to reach a homogeneous
structure inside the concentration field, as a result of the mixing procedure.
In white we show the active fluid interfacial shape, with an initial
perturbation of amplitude $0.025$ and a wavenumber of $k=6\pi$. As the
electric field is turned on, the perturbation grows and generates motion
affecting the passive tracer. The switching off of the electric field then
allows the stabilization to a flat interface. Repeating this procedure within
a certain range of appropriate parameters becomes an effective mixing
strategy.
The second segment of the attached video contains three examples of relay type
constructions. As in the previous case, the geometry and fluid properties are
kept the same, the differences lie in the amplitude of the voltage
perturbation and the imposed (horizontal) velocity of this anomaly. The fluids
are characterized by the same set of properties as before, however the
electric fields are now given by a base voltage of $\bar{V}=6.0$ with either
* •
Left: voltage perturbation amplitude $a=1.0$, velocity $s=1.0$;
* •
Center: voltage perturbation amplitude $a=1.5$, velocity $s=1.0$;
* •
Right: voltage perturbation amplitude $a=1.5$, velocity $s=0.5$.
An additional feature of the GERRIS package, droplet removal, has been used to
limit numerical artifacts in the solution. Furthermore, in black we plot
equipotential lines, which allow for the clean visualization of the imposed
voltage as a function of time. As non-dimensional time $t$ reaches $1$ unit,
the relay is switched on and this type of microfluidic pumping generates a
flux that initiates the horizontal motion of the interface. The mixing of the
passive tracer becomes highly efficient in this case and can be further
enhanced by combining this strategy with on-off protocols as described before.
All simulations are modelled to contain fictitious fluids, with properties
that are representative in the context of our study. Fully realistic values,
representing two actual fluids, as could be reproduced under experimental
conditions, will be used at a later stage. The Reynolds number in the
computational experiments is of order $\mathcal{O}(10^{1})$ and requires
further reduction as we enter the microfluidic range.
## 4 Future Work
Identifying optimal mixing protocols in a general framework is the main focus
of the project at the current stage. Once satisfactory results are obtained,
we will direct our attention to micro-scale devices, where existing strategies
will be improved to adapt to low Reynolds number flows. Realistic fluids,
frequently used in experimental contexts, will be preferred to the current
model. Further extensions to other geometries (such as a full T-mixer) and
three-dimensional generalizations are also within reach.
## References
* [1] O. Ozen, N. Aubry, D.T. Papageorgiou and P.G. Petropoulos. Electrohydrodynamic linear stability of two immiscible fluids in channel flow. Electrochimica Acta, 51:5316–5323, 2006.
* [2] O. Ozen, N. Aubry, D.T. Papageorgiou and P.G. Petropoulos. Monodisperse drop formation in square microchannels. PRL, 96:144501, 2006.
* [3] O. Ozen, N. Aubry, D.T. Papageorgiou and P.G. Petropoulos. Linear stability of a two-fluid interface for electrohydrodynamic mixing in a channel. J. Fluid Mech., 583:347–377, 2007.
* [4] C.-Y. Lee, C.-L. Chang, Y.-N. Wang and L.-M. Fu. Microfluidic mixing: A review. Int. J. Mol. Sci., 12:2911–2925, 1965.
* [5] A.O. El Moctar, N. Aubry and J. Batton. Electro-hydrodynamic microfluidic mixer. Lab Chip, 3:273–280, 2003.
* [6] A. Goullet, I. Glasgow and N. Aubry. Dynamics of microfluidic mixing using time pulsing. Discrete and Continuous Dynamical Systems, Supplement Volume:327–336, 2005.
* [7] C. Tsouris, C.T. Culbertson, D.W. DePaoli, S.C. Jacobson, V.F. de Almeida and J.M. Ramsey. Electrohydrodynamic mixing in microchannels. AIChE, 49:2181–2186, 2003.
* [8] I. Glasgow and N. Aubry. Enhancement of microfluidic mixing using time pulsing. Lab Chip, 3:114–120, 2003.
* [9] T.J. Johnson, D. Ross and L.E. Locascio. Rapid microfluidic mixing. Anal. Chem., 74:45–51, 2002.
* [10] L.-H. Lu, K.S. Ryu and C. Liu. A magnetic microstirrer and array for microfluidic mixing. Journal of Microelectromechanical Systems, 11:462–469, 2002.
* [11] R. Cimpeanu, D.T. Papageorgiou and P.G. Petropoulos. On the control and suppression of Rayleigh-Taylor instability using electric fields. Phys. Fluids, submitted for publication, 2013.
* [12] S. Popinet. Gerris: A tree-based adaptive solver for the incompressible Euler equations in complex geometries. J. Comput. Phys., 190:572, 2003.
|
arxiv-papers
| 2013-10-12T12:30:05 |
2024-09-04T02:49:52.297423
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Radu Cimpeanu and Demetrios T. Papageorgiou",
"submitter": "Radu Cimpeanu",
"url": "https://arxiv.org/abs/1310.3365"
}
|
1310.3408
|
# Poly(dA:dT)-rich DNAs are highly flexible in the context of DNA looping
Stephanie Johnson†,
Department of Biochemistry and Molecular Biophysics,
California Institute of Technology,
Pasadena, CA 91125
Present address: Department of Biochemistry and Biophysics,
University of California San Francisco,
San Francisco, CA, USA
Yi-Ju Chen†,
Department of Physics,
California Institute of Technology,
Pasadena, CA 91125
Rob Phillips111To whom correspondence should be addressed. Email:
[email protected]. †These authors contributed equally to this work.,
Departments of Applied Physics and Biology,
California Institute of Technology,
Pasadena, CA 91125
###### Abstract
Large-scale DNA deformation is ubiquitous in transcriptional regulation in
prokaryotes and eukaryotes alike. Though much is known about how transcription
factors and constellations of binding sites dictate where and how gene
regulation will occur, less is known about the role played by the intervening
DNA. In this work we explore the effect of sequence flexibility on
transcription factor-mediated DNA looping, by drawing on sequences identified
in nucleosome formation and ligase-mediated cyclization assays as being
especially favorable for or resistant to large deformations. We examine a
poly(dA:dT)-rich, nucleosome-repelling sequence that is often thought to
belong to a class of highly inflexible DNAs; two strong nucleosome positioning
sequences that share a set of particular sequence features common to
nucleosome-preferring DNAs; and a CG-rich sequence representative of high
G+C-content genomic regions that correlate with high nucleosome occupancy in
vivo. To measure the flexibility of these sequences in the context of DNA
looping, we combine the in vitro single-molecule tethered particle motion
assay, a canonical looping protein, and a statistical mechanical model that
allows us to quantitatively relate the looping probability to the looping free
energy. We show that, in contrast to the case of nucleosome occupancy, G+C
content does not positively correlate with looping probability, and that
despite sharing sequence features that are thought to determine nucleosome
affinity, the two strong nucleosome positioning sequences behave markedly
dissimilarly in the context of looping. Most surprisingly, the
poly(dA:dT)-rich DNA that is often characterized as highly inflexible in fact
exhibits one of the highest propensities for looping that we have measured.
These results argue for a need to revisit our understanding of the mechanical
properties of DNA in a way that will provide a basis for understanding DNA
deformation over the entire range of biologically relevant scenarios that are
impacted by DNA deformability.
## 1 Introduction
Although it has been known since the work of Jacob and Monod that genomes
encode special regulatory sequences in the form of binding sites for proteins
that modulate transcription, only recently has it become clear that genomes
encode other regulatory features in their sequences as well. Further, with the
advent of modern sequencing methods, it is of great interest to have a base-
pair resolution understanding of the significance of the entirety of genomes,
not just specific coding regions and putative regulatory sites.
One well-known example of other information present in genomes is the
different sequence preferences that confer nucleosome positioning [1, 2, 3],
with similar ideas at least partially relevant in the context of architectural
proteins in bacteria also [4]. It has been shown both from analyses of
sequences isolated from natural sources and from in vitro nucleosome affinity
studies with synthetic sequences that the DNA sequence can cause the relative
affinity of nucleosomes for DNA to vary over several orders of magnitude, most
likely due to the intrinsic flexibility, especially bendability, of the
particular DNA sequence in question [5, 6, 7, 8, 3]. The claim that intrinsic
DNA sequence flexibility determines nucleosome affinity has led not only to
many theoretical and experimental studies on the relationship between sequence
and flexibility [9, 10, 11, 12, 13, 14, 15], but also to the elucidation of
numerous sequence “rules” that can be used to predict the likelihood that a
nucleosome will prefer certain sequences over others (summarized recently in
[7, 2]). For example, AA/TT/AT/TA steps in phase with the helical repeat of
the DNA, with GG/CC/CG/GC steps five base pairs out of phase with the
AA/TT/AT/TA steps, are a common motif in both naturally occurring and
synthetic nucleosome-preferring sequences [7, 3]. Similarly, the G+C content
of a sequence and occurrence of poly(dA:dT) tracts have been very powerful
parameters in predicting nucleosome occupancy in vivo [16, 17, 18, 2, 19]. Our
aim here is to explore the extent to which these sequences, when taken beyond
the context of cyclization and nucleosome formation to another critical DNA
deformation motif, exhibit similar effects on a distinct kind of deformation.
There has been an especially long history of the study of these intriguing
sequence motifs known as poly(dA:dT) tracts, in the context of nucleosome
occupancy as well as many other biological contexts. Such sequences, composed
of 4 or more A bases in a row ($\mathrm{A}_{n}$ with $n\geq 4$) or two or more
A bases followed by an equal number of T bases ($\mathrm{A}_{n}\mathrm{T}_{n}$
with $n\geq 2$), strongly disfavor nucleosome formation, both in vivo [20, 21,
22, 23] and in vitro [24, 25, 26, 27, 28], and are in fact thought to be one
of the primary determinants of nucleosome positions in vivo [2, 21], with
their presence upstream of promoters and in the downstream genes correlating
with increased gene expression levels [29, 20, 30]. Poly(dA:dT) tracts show
unique structural and dynamic properties in a variety of in vitro and in vivo
assays (summarized recently in [31, 21]), with one of their hallmark
characteristics being a marked intrinsic curvature [31]. There is evidence
that poly(dA:dT) tracts may also be less flexible than other sequences [32,
33, 34], which is often given as the reason for their low affinity for
nucleosomes, though there is some evidence that poly(dA:dT) tracts might
actually be more flexible than other sequences [9]. It is clear, however, that
some special property or properties of A-tracts leads them to be especially
resistant to the deformations that are required for DNA wrapped in a
nucleosome [21, 31], and, indeed, to their important functions in several
other biological contexts as well [31].
In this work, we make use of sequences that, in the context of nucleosome
formation and cyclization assays, appear to be associated with distinct
flexibilities as a starting point for examining the question of what sequence
rules control deformations induced by a DNA-loop-forming transcription factor,
as opposed to those induced in nucleosomes. We have previously argued using
two synthetic sequences that DNA looping does not necessarily follow the same
sequence-dependent trends as do nucleosome formation and cyclization [35].
Here we expand our repertoire of sequences to specifically test the
generalizability of three sequence features known to be important in
nucleosome biology and cyclization. We focus in particular on the intriguing
class of nucleosome-repelling, poly(dA:dT)-rich DNAs that are thought to be
especially resistant to deformation, making use of a naturally occurring
poly(dA:dT)-rich sequence that forms a nucleosome-free region at a yeast
promoter [23]. We note that the poly(dA:dT)-rich DNA we use here differs from
the phased A-tracts that have been extensively characterized in the context of
DNA looping, both in vivo and in vitro [36, 37, 38, 39, 40, 41, 42, 43, 44].
Phased A-tracts contain short poly(dA:dT) tracts spaced by non-A-tract DNAs
such that the poly(dA:dT) tracts are in phase with the helical period of the
DNA, generating globally curved structures that are known to significantly
enhance DNA looping [36, 37, 38, 39, 40]. The poly(dA:dT)-rich sequence we
examine here contains unphased A-tracts that we do not anticipate to have a
sustained, global curvature.
We compare the effects on looping of this poly(dA:dT)-rich DNA not only to the
effects of two synthetic sequences we have previously studied, but also to
those of two additional naturally occurring, genomic sequences: the well-
known, strong nucleosome positioning sequence 5S from a sea urchin ribosomal
subunit [45], which, along with the 601TA sequence we previously studied,
contains the repeating AA/TT/TA/AT and offset GG/CC/CG/GC steps that are
common in nucleosome-preferring sequences; and one of the GC-rich sequences
that are abundant in the exons and regulatory regions (e.g. promoters) of
human genes, and that correlate with high nucleosome occupancy in vivo [22,
18, 19]. The 5S sequence has been examined using both in vitro cyclization and
in vitro nucleosome formation assays and, along with the two synthetic
sequences E8 and 601TA [46, 8], can be used as a standard for comparison
between our and other in vitro assays. The five sequences used in this work
and their effects on nucleosomes are summarized in Table 1.
Table 1: Naturally occurring and synthetic nucleosome-positioning or
nucleosome-repelling sequences used in this study.
Sequence | Species | Genomic Position | Nucleosome Affinity
---|---|---|---
Name | | |
poly(dA:dT) | Budding yeast | Chr III, 38745 – 39785 bp | $\sim$3-fold in vivo nucleosome
(“dA”) | (S. cerevisiae) | (Ref. [23]) | depletion relative to average
| | | genomic DNA (Fig. 2E of
| | | Ref. [22]); $\sim 2~{}k_{B}T$ increase in
| | | energy of nucleosome formation
| | | in vitro relative to 5S (Fig. 8D of
| | | Ref. [22]) (estimates based on
| | | similar sequences)
GC-rich | Human | Chr Y, 4482107 – 4481956 bp | (not determined)
(“CG”) | | (Ref. [22]) |
5S | Sea urchin | 20 bp-165 bp | 1.6 $k_{B}T$ decrease in energy of
| (L. variegatus) | from the Mbo II fragment | nucleosome formation compared
| | containing 5S rRNA gene | to E8 in vitro (Ref. [8])
| | (Ref. [45]) |
601TA | synthetic, strong | N/A | 3 $k_{B}T$ decrease in energy of
(“TA”) | nucleosome | | nucleosome formation compared
| positioning sequence | | to E8 in vitro (Ref. [8])
| (Refs. [59, 8, 60]) | |
E8 | synthetic random | N/A | (used as a reference)
| (Refs. [8, 60]) | |
The sequences described here were chosen because each has been found to have
significant effects on in vivo nucleosome positions and/or in vitro nucleosome
affinities, as shown in the rightmost column. The exception is the GC-rich
sequence from humans: although its nucleosome affinity has not been directly
determined either in vivo or in vitro, it is predicted to correlate with high
nucleosome occupancy because of its high G+C content [17] and is occupied by a
nucleosome(s) in vivo according to micrococcal nuclease digestion [22]. Two-
letter abbreviations given in parentheses under each full sequence name will
be used in figure legends in the rest of this work.
To measure the effect of these sequences on looping rather than nucleosome
formation, we made use of a combination of an in vitro single-molecule assay
for DNA looping, called tethered particle motion (TPM) [47, 48, 49, 50], with
the canonical E. coli Lac repressor to induce looping, and a statistical
mechanical model for looping that allows us to extract a quantitative measure
of DNA flexibility, called the looping J-factor, for the DNA in the loop [51,
35]. We have recently demonstrated [35] that this combined method offers a
powerful and complementary approach to established assays that have been used
to probe the mechanical properties of DNA, particularly at short length
scales, to great effect, such as ligase-mediated DNA cyclization [52, 53, 54,
8, 55, 56, 15, 57] and measured DNA end-to-end distance by fluorescence
resonance energy transfer [58, 34]. In particular, using the Lac repressor as
a tool to probe the role of DNA deformability in loop formation allows us to
examine the effect of sequence on the formation of shapes other than the
roughly circular ones formed by cyclization and nucleosome formation, which we
have argued may be an important caveat to discovering general flexibility
rules from nucleosome formation and cyclization studies alone [35].
Interestingly, we find that the poly(dA:dT)-rich sequence that strongly
excludes nucleosomes in vivo [23] and that belongs to a class of sequences
usually thought of as highly resistant to deformation is in fact the strongest
looping sequence we have studied so far. Moreover, the 5S and TA sequences,
which share sequence features important to nucleosome formation (see Figures
LABEL:fig:SIseqlist1 and LABEL:fig:SIseqlist2 in File S1 and Ref. [7]) as well
as trends in apparent flexibility in in vitro cyclization and nucleosome
formation assays [59, 8, 60], behave very differently from each other in the
context of looping. We also find that G+C content, a good predictor of
nucleosome occupancy, is not likewise positively correlated with looping, and
in fact our data suggest the G+C content and looping may be anticorrelated.
Taken together, these results strongly suggest that very different sequence
rules determine DNA looping versus cyclization and nucleosome formation,
possibly because of the protein-mediated boundary conditions that differ
between looping geometries and nucleosomal geometries, and that the
biophysical characteristics of poly(dA:dT)-rich DNAs and their biological
functions may be more diverse and context-dependent than has been previously
appreciated.
## 2 Results
Figure 1: Looping probability as a function of loop length and sequence. (A)
Schematic of the tethered particle motion (TPM) assay for measuring looping.
In TPM, a bead is tethered to the surface of a microscope coverslip by a
linear DNA. The motion of the bead serves as a readout for the state of the
tether: if the DNA tether contains two binding sites for a looping protein
such as the Lac repressor, and the looping protein is present and binds both
sites simultaneously, forming a loop, the motion of the bead is reduced in a
detectable fashion [47, 48, 49, 50]. The motion of the bead is observed over
time, and the looping probability for a particular DNA is defined as the time
spent in the looped (reduced motion) state, divided by the total observation
time. (B) Schematic of the “no promoter” (left) and “with-promoter” (right)
constructs used in this work. “Loop length” is defined as the inner edge-to-
edge distance between operators (excluding the operators themselves, but
including the promoter, if present). (C) Looping probabilities for the five
sequences described in Table 1, without the bacterial lacUV5 promoter sequence
as part of the loop. (D) Looping probabilities for the same five sequences but
with the promoter sequence in the loop. Righthand panels in (C) and (D) show
the same data as lefthand panels, except magnified around loop lengths 100-110
bp. The five sequences do not all share the same maxima of looping (colored
arrows), not even the TA and 5S sequences that share similar sequence features
(see Figures LABEL:fig:SIseqlist1 and LABEL:fig:SIseqlist2 in File S1), though
each sequence has the same maximum with and without the promoter (as far as
can be determined with the current data; note that the with-promoter maximum
for the TA sequence could be at 105 or 106, as those points are within error).
All E8 and TA data (in particular, those outside of the 101-108 bp range) were
previously described in [35].
Our experimental approach to examining the effect of DNA sequence on looping
combines an in vitro single-molecule assay for DNA looping, called tethered
particle motion (TPM) [47, 48, 49, 50], with a statistical mechanical model
that allows us to extract biological parameters from the single-molecule data
[51, 35]. As shown schematically in Fig. 1(A), in TPM, a microscopic bead is
tethered to a microscope coverslip by a linear piece of DNA, with the motion
of the bead serving as a reporter of the state of the DNA tether: the
formation of a protein-mediated DNA loop in the tether reduces the motion of
the bead in a detectable fashion [47, 48, 49, 50]. We use the canonical Lac
repressor from E. coli to induce DNA loops. Because more readily deformable
sequences allow loops to form more easily, we can quantify sequence-dependent
DNA flexibility by quantifying the looping probability, which we calculate as
the time spent in the looped state divided by the total observation time (see
Methods for details).
More precisely, our statistical mechanical model (described in the Methods
section) allows us to extract a parameter called the looping J-factor from
looping probabilities [35]. The J-factor is the effective concentration of one
end of the loop in the vicinity of the other, analogous to the J-factor
measured in ligase-mediated DNA cyclization assays [52, 61], and is
mathematically related to the energy required to deform the DNA into a loop,
$\Delta F_{\mathrm{loop}}$, according to the relationship:
$J_{\mathrm{loop}}=1~{}\mathrm{M}~{}e^{-\beta\Delta F_{\mathrm{loop}}},$ (1)
where $\beta=1/(k_{B}T)$ ($k_{B}$ being Boltzmann’s constant and $T$ the
temperature). A higher J-factor therefore corresponds to a lower free energy
of loop formation. In the case of cyclization, where the boundary conditions
of the ligated circular DNA are well understood, the J-factor can be expressed
in terms of parameters describing the twisting and bending flexibility of the
DNA, and its helical period [62, 63, 10, 15]. However, in the case of DNA
looping by the Lac repressor, where the boundary conditions are not well known
(summarized in Fig. 4 of [35]), an expression for the looping J-factor in
terms of the twist and bend flexibility parameters of the loop DNA has not
been described. Nevertheless, by measuring the J-factors for different
sequences, we can comparatively assess the effect of sequence on the energy
required to deform the DNA into a loop, and thereby gain insight into the
sequence rules that control this deformation.
### 2.1 Loop sequence affects both the looping magnitude and the position of
the looping maximum.
Given that 5S and TA share both sequence features and similar trends in
apparent flexibility in the contexts of nucleosome formation and cyclization
[59, 8, 60], we expected these two sequences to behave similarly to each other
in the context of looping. On the other hand, since poly(dA:dT)-rich sequences
are supposed to assume such unique structures as to strongly disfavor
nucleosome formation [21, 31], while high GC content is one of the strongest
predictors of high nucleosome occupancy [17, 22], we expected these two
sequences to behave very differently from each other in the context of
looping. Given the common assumption that poly(dA:dT)-rich DNAs are highly
resistant to deformation, we especially did not expect to observe much, if
any, loop formation with the poly(dA:dT)-rich, nucleosome-repelling sequence.
As shown in Fig. 1, none of these expectations were borne out. TA and 5S do
not behave similarly, nor do CG and poly(dA:dT) behave especially
dissimilarly, nor does poly(dA:dT) resist loop formation. Moreover, the
behavior of these special nucleosome-preferring or nucleosome-repelling
sequences is dependent on the larger DNA context, in that the addition of the
36-bp bacterial lacUV5 promoter sequence to these roughly 100-bp loops changes
the relative looping probabilities of the five sequences (see Methods for the
rationale behind the inclusion of this promoter). Without this promoter
sequence (Fig. 1(C)), the two synthetic sequences, E8 and TA, exhibit
comparable amounts of looping, while the three natural sequences, including
both 5S and poly(dA:dT), all loop more than either E8 or TA. With the promoter
(Fig 1(D)), however, TA loops more than E8, but 5S less than either E8 or TA.
Both with and without the promoter the supposedly very different GC-rich and
poly(dA:dT)-rich DNAs loop more than the random E8 sequence. The looping
probabilities of the poly(dA:dT) sequence are especially surprising—instead of
looping very little, as we expected, this sequence loops more than any other
sequence without the promoter and a comparable amount to TA with the promoter.
These five sequences differ not only in looping probability, but also in the
loop length at which that looping is maximal: the poly(dA:dT) sequence is
maximized at 104 bp, the 5S and CG sequences at 105 bp, and the E8 and TA
sequences at 106 bp. These different maxima could be explained by different
helical periods for these five DNAs, though without more periods of data we
cannot definitively quantify their helical periods. In the case of the
poly(dA:dT) sequence, an altered helical period would not be unexpected, as
pure poly(dA:dT) copolymers are known to have shorter helical periods (10.1
bp/turn) than random DNAs (10.6 bp/turn) [64, 65]. On the other hand, 5S
exhibits the same helical period as E8 and TA in cyclization assays [60], so
it is intriguing that its looping maximum occurs at a different length than
that of E8 and TA, perhaps suggesting a different helical period in the
context of looping than that of E8 and TA. The promoter does not appear to
alter the maximum of looping for a given sequence. As noted above, it is
difficult to use these looping data to comment further on other DNA elasticity
parameters, in particular any sequence-dependent differences in torsional
stiffness, but in Fig. LABEL:fig:SITwistStiffness in File S1 we provide
evidence that these sequences may share the same twisting flexibility, even if
they differ in helical period.
Figure 2: Looping J-factors as a function of loop length and sequence.
J-factors for sequences without (closed circles) and with (open circles) the
lacUV5 promoter were extracted from the data in Fig. 1 as described in the
Methods section. The J-factor is a measure of the free energy of loop
formation (and is related to the bending and twisting flexibility of the DNA
in the loop): the higher the J-factor, the lower the free energy required to
deform the loop region DNA into a loop. As described in [35], the addition of
the promoter to the E8 loop sequence does not significantly affect its
J-factor, so the J-factor for E8 with the promoter is shown as a reference in
all panels (black open points). In contrast to E8, the addition of the
promoter does change the J-factors for three of the four other sequences,
making the TA-containing loops more flexible, but the 5S and, to a lesser
extent, CG sequences less flexible. Interestingly, the poly(dA:dT) sequence,
like the E8 sequence, is unchanged with the inclusion of the promoter. We note
that because the no-promoter versus with-promoter constructs contain different
combinations of repressor binding sites, we can only use J-factors, not
looping probabilities, to quantitatively examine the effect of the promoter;
the statistical mechanical model of Eqn. 2 allows us to make this comparison.
The effect of the promoter on loop formation can be more clearly seen when
looping J-factors are compared across sequences, instead of the looping
probabilities. Because the no-promoter and with-promoter loops are flanked by
different combinations of operators (Fig. 1(B); see also Methods), their
looping probabilities cannot be directly compared. However, as described above
and in the Methods section, we can use the statistical mechanical model that
we have described for this system to extract J-factors from each looping
probability [35]. These J-factors are shown in Fig. 2. Loop sequence can
modulate the looping J-factor by at least an order of magnitude (compare the
poly(dA:dT) J-factors to those of 5S with promoter or E8 and TA, no-promoter).
The lacUV5 promoter has the largest effect on the TA and 5S sequences (though
of opposite sign), but appears to have little effect on poly(dA:dT)-containing
and E8-containing loops, and moderate effect on CG-containing loops. It is
intriguing how large and diverse an effect the 36-bp lacUV5 promoter has on
the roughly 100 bp loops we examine here; but one possible explanation for its
minimal effect on the poly(dA:dT)-rich sequence, at least, compared to the
others, is that the properties of A-tract structures tend to dominate over the
properties of surrounding sequences [31]. We note that our results in [35]
comparing the effect of sequence versus flanking operators on measured
J-factors preclude the possibility that the differences between the no-
promoter and with-promoter constructs are due to the difference in flanking
operators. We also note that it is possible that the effect of the promoter
stems not from the promoter sequence itself, but from the fact that the
sequences of interest that form the rest of the loop are shorter when 36 bp of
the loop are replaced by the promoter sequence. However, we consider this
explanation to be less likely, because as shown in the left-hand panels of
Fig. 1(C) and (D) above, we have measured the looping probabilities (and
J-factors; see [35]) of more than two periods of E8- and TA-containing DNAs,
allowing a direct comparison of loops that contain the same amount of E8 and
TA both with and without the promoter (compare, for example, no-promoter loop
lengths of 90 bp to with-promoter lengths of 120 bp). In this case we still
find that without the promoter the J-factors of the E8- and TA-containing
loops are indistinguishable, but with the promoter the TA sequence loops more
than the E8 sequence, indicating that it is the promoter and not a shortening
of some unique element(s) of the E8 or TA sequences that cause the difference
in J-factors with versus without the promoter for these two sequences.
### 2.2 The Lac repressor supports a range of looped-state conformational
preferences.
TPM trajectories not only provide information about the free energy of loop
formation, captured by the J-factors discussed in the previous section, but
also contain some information about the preferred loop conformation as a
function of sequence, through the observed length of the TPM tether when a
loop has formed. In fact, previous work from our group and others has shown
that the Lac repressor can support at least two observable loop conformations
for any pair of operators, with any sequence, because these conformations lead
to distinct tether lengths in TPM [35, 51, 38, 39, 66, 67, 68, 37, 39, 40].
Although the underlying molecular details of these two looped states, which we
label the “middle” (“M”) and “bottom” (“B”) states according to their tether
lengths relative to the unlooped state, are as yet unknown, they must differ
in repressor and/or DNA conformation in a way that alters the boundary
conditions of the loop, since they are distinguishable in TPM. It has been
proposed that the two states arise from the four distinct DNA binding
topologies allowed by a V-shaped Lac repressor similar to that shown in the
Lac repressor crystal structure [69, 70], and/or two repressor conformations,
the V-shape seen in the crystal structure and a more extended “E” shape [71,
72, 73, 66, 68, 39, 40]. It is likely, in fact, that the two observed looped
states are each composed of more than one microstate (that is, some
combination of V-shaped and E-shaped repressor conformation(s) and associated
binding topologies [69]). Even without knowing the details of the underlying
molecular conformation(s) of these two states, however, we can use them to
provide a window into the effect of sequence on preferred loop conformation.
In particular, by examining the relative probability of the two looped states
as a function of both loop length and loop sequence, we can assess the
contributions of sequence to the energy required to form the associated loop
conformation(s). As shown in Figure 3, which of the two looped states
predominates depends in a complicated way upon the loop sequence, the presence
versus absence of the lacUV5 promoter, and the loop length. In [35], we showed
that having E8 or TA in the loop region, over two to three helical periods,
leads to alternating preferences for the middle versus the bottom looped
state, with the middle state predominating when the operators are in-phase and
looping is maximal, but the bottom state predominating when the operators are
out-of-phase. The inclusion of the promoter in the loop increases the
preference for middle state for out-of-phase operators. These trends are
captured in the top left panel of Fig. 3.
Figure 3: Comparison of the likelihood of the “middle” (longer) versus
“bottom” (shorter) looped states. The y-axes indicate the fraction of the
total J-factor that is contributed by the middle state (as in Fig. 2, since
the with- and without-promoter constructs have different operators, J-factors
and not looping probabilities must be compared). That is, when the ratio
$J_{\mathrm{loop,M}}/J_{\mathrm{loop,tot}}$ is unity, indicated by a
horizontal black dashed line, only the middle state is observed; when this
ratio is zero, again indicated by a horizontal black dashed line, only the
bottom state is observed. Closed circles are no-promoter constructs; open
circles are with-promoter. E8 and TA data are a subset of those in [35].
Figure LABEL:fig:SIBvsM in File S1 shows the looping probabilities and
J-factors for the two states instead of the relative measures shown here.
These trends do not hold for the three genomically sourced loop sequences (CG,
dA, and 5S). For the poly(dA:dT)-rich sequence, as with E8 and TA, the
promoter increases the preference for the middle looped state for out-of-phase
operators; for 5S, however, the presence of the promoter decreases the
preference for the middle state. The preferred state of the CG sequence is
mostly insensitive to the presence versus absence of the promoter. Both with
and without the promoter, though, the middle state is generally preferred
($J_{\mathrm{loop,M}}/J_{\mathrm{loop,tot}}\geq 0.5$) at more loop lengths for
the genomically sourced DNAs than for the synthetic sequences, insofar as we
are able to determine from the lengths shown in Fig. 3. These results
demonstrate a complicated dependence of preferred loop state on sequence that
does not always follow overall trends in looping free energy: for example, 5S
and TA are the two sequences that show the largest change in J-factor with the
inclusion of the promoter, but E8 and TA are the sequences that show the
largest change in preferred looped state with the promoter. However, the trend
seen in the preceding section with CG and poly(dA:dT) having more in common
than 5S and TA holds true for preferred loop conformation as well.
A different measure of loop conformation can be derived from the TPM tether
lengths themselves—that is, from the measured root-mean-squared motion of the
bead, $\langle R\rangle$, as in the example trajectory shown in Fig. 4(A),
which exhibits three clear states, the two looped states and the unlooped
state. Because of variability in initial tether length, even in the absence of
Lac repressor, we calculate a relative measure of tether length for the
unlooped and looped states, where the motion of each bead is normalized to its
motion in the absence of repressor. We might expect, then, that in the
presence of repressor, the unlooped state would fall at a relative $\langle
R\rangle$ of zero, and the looped states at negative values. However, as can
be seen in the sample trace in Fig. 4(A) and in the lefthand panels of Fig.
4(B), the unlooped state in the presence of repressor is actually shorter than
the tether in the absence of repressor (i.e., the horizontal black dashed line
in Fig. 4(A) lies above the mean of the unlooped state in the blue data). In
[35] we present evidence for this shortening of the unlooped state in the
presence of repressor being due to the bending of the operators induced by the
Lac repressor protein that is observed in the crystal structure of the Lac
repressor complexed with DNA [70]. (We note that this is a Lac repressor-
specific result; compare, for example, the recent results from Manzo and
coworkers with the lambda repressor [74], where a similar shortening of the
unlooped state is shown to be due to nonspecific binding. For example, the Lac
repressor does not exhibit the dependence of the looped tether length on
repressor concentration that is seen with the lambda repressor [35, 74]).
As shown in Fig. 4(B), the length of the TPM tether in both the unlooped and
looped states is similar but not identical for the five sequences and eight
lengths that we examine here. The most obvious modulation of tether length
correlates with loop length, with the shortest unlooped- and looped-state
tether lengths occurring near the maxima of the looping probability. We
believe this modulation with length is due to the phasing of the bends of the
DNA tether as it exits the repressor-bound operators in the looped state, or
the phasing of the bent operators in the unlooped state. At the repressor
concentration we use here, the unlooped state should be primarily composed of
the doubly-bound state [35], meaning that the two operators are both bent by
bound repressor. As shown schematically in Fig. 4(C), when these bends are in-
phase, the tether length should be shortest (and also the looping probability
is highest, because the operators are in-phase). A similar argument can be
made for the modulation of the looped state, regarding the relative phases of
the tangents of the DNA exiting the loop.
Figure 4: Tether length as a function of loop length, sequence and J-factor.
(A) Sample TPM time trajectory showing the smoothed (i.e. Gaussian-filtered)
root-mean-squared motion, $\langle R\rangle$, of a single bead. This construct
shows an unlooped state and two looped states, the “middle” state around 130
nm, and the “bottom” state around 110 nm. Black horizontal dashed line
indicates the average $\langle R\rangle$ for this particular tether in the
absence of repressor. Due to variability in tether length even in the absence
of repressor [35], on the y-axes in (B) and (D) we plot a relative measure of
tether length, by normalizing the mean $\langle R\rangle$ value for a
particular state to the mean $\langle R\rangle$ of each tether in the absence
of repressor, and then taking the population average of this difference. (B)
Tether length as a function of loop length. We observe a modulation of tether
length with loop length, with the shortest tether lengths for both the
unlooped and looped states occurring near the maximum of looping (indicated
for each sequence by the colored arrows at the bottoms of the plots). See Fig.
LABEL:fig:SItetherlengths1 in File S1 for bottom state lengths. (C) Schematic
of our proposed model for the observed variations in unlooped tether length as
a function of loop length, which we attribute to the phasing of the bends that
the repressor creates upon binding the operators. A similar argument can be
made for the looped states. Note that to emphasize the effect of bending from
the operators, here we have for the most part represented the DNA as straight
segments. (D) Tether length as a function of J-factor. Unlooped state tether
lengths are plotted versus the total J-factor, whereas middle state tether
lengths are plotted versus the J-factor for the middle state. As in (B), in
general the length of the tether in both the unlooped and middle looped states
is shorter at larger J-factors (that is, more in-phase operators) for a
particular sequence. However, this trend is sharper for some sequences than
others (see Fig. LABEL:fig:SItetherlengths2 in File S1 for the other
sequences, which generally have more scatter than either the dA or CG
sequences).
It is interesting to consider how the sequence of the loop might influence the
length of the tether in the unlooped state, when no loop has formed; see, for
example, the CG with-promoter versus 5S with-promoter sequences, where the
latter is consistently longer than the former (Fig. 4(B)). We do not see a
sequence dependence to tether length in the absence of repressor, ruling out
the possibility of a detectable intrinsic curvature to the CG sequence. We
speculate instead that CG alters the trajectory of the DNA as it exits the
bend in the operators in the unlooped state, compared to the trajectory when
the sequence next to the operators is 5S, leading to a consistent difference
in unlooped tether lengths.
Interestingly, in contrast to its influence on preferred looped state (middle
versus bottom), the promoter does not alter the length of the tether for a
given sequence at a given loop length (see also the bottom left panel of Fig.
LABEL:fig:SItetherlengths1 and Fig. LABEL:fig:SItetherlengths2 in File S1). On
the other hand, as shown in Fig. 4(D), the poly(dA:dT)-rich sequence,
noticeably more so than the other sequences, stands out as a sequence that
does strongly affect the tether length of the loop, in that it mandates a very
narrow range of tether lengths as a function of looping J-factor (related, for
a particular sequence, to the loop length or equivalently the operator
spacing). A similar but less pronounced trend can be observed for the unlooped
state with the GC-rich sequence (Fig. 4(D)). The other sequences allow much
more variability in tether length as a function of J-factor/operator spacing
(see Figure LABEL:fig:SItetherlengths2 in File S1). This strong trend in
tether length as a function of J-factor could be evidence of the formation of
special, defined loop structures with the GC-rich and poly(dA:dT)-rich
sequences that constrain the allowed loop conformations as a function of
operator spacing more than the other sequences do.
Further computational and modeling efforts will be required to relate these
data on tether lengths and preferred loop length to loop structure, similarly
to how Towles and coworkers have used TPM tether lengths to show that
different DNA loop topologies can explain the observed tethered lengths of the
two looped states [69]. However, even without currently knowing the underlying
molecular details causing these sequence-specific trends in tether length and
preferred loop state, and therefore in loop conformation, it is clear that it
is the loop sequence, and not the Lac repressor itself, that determines the
loop conformation to a large degree. It has been shown recently that the Lac
repressor is capable of accommodating many different loop conformations [40],
which is consistent with the results we present here. We hope that
computational and modeling efforts with these data, as well as continued
efforts to use assays such as FRET to directly probe loop conformation [37,
38, 39, 40], will shed light on this complex interplay between sequence and
loop conformation.
## 3 Discussion.
In [35] we showed that the synthetic E8 and TA sequences show no sequence
dependence to looping in the absence of the lacUV5 promoter but a nucleosome-
like sequence dependence in the presence of the promoter. We hypothesized that
perhaps the promoter alters the preferred state of the loop to one whose shape
is more similar to that of DNA in a nucleosome or DNA minicircle formed by
cyclization, leading to similar sequence trends with the promoter as with
nucleosomes. We still attribute the difference in the patterns of sequence
dependence that we observe between looping and nucleosome formation to the
role of the shape of the deformation in determining the observed deformability
of a particular sequence. However, we have shown here with a broader range of
sequences that the role of the promoter in controlling loopability is more
complicated than we had previously hypothesized. Neither with nor without the
promoter does loop formation follow the sequence trends of nucleosome
formation. As shown in Figure 5, if looping J-factors did follow the same
patterns of sequence preference as do cyclization J-factors and nucleosome
formation free energies, a plot of the looping J-factors versus cyclization
J-factors for the various sequences we have studied here would fall on a line
with a positive slope. We find that this is not the case; in fact, without the
promoter there is perhaps a slight anticorrelation between looping J-factors
and cyclization J-factors (and no discernible correlation with the promoter).
Figure 5: Comparing trends in sequence flexibility for looping versus
cyclization and nucleosome formation. (A) Nucleosome formation and
cyclization share trends in sequence flexibility, with sequences that have
lower energies of nucleosome formation ($\Delta\Delta G^{0}_{nucl}$) also
having lower energies of cyclization ($\Delta\Delta G^{0}_{cyc}$). Cloutier
and Widom used this correlation to argue that the same mechanical properties,
particularly the bendability, of the DNA contributed to nucleosome formation
as to cyclization [8]. The energy of cyclization, $\Delta G^{0}_{cyc}$, is
related to the cyclization J-factor for a particular DNA, $J_{i}$, through the
relationship $\Delta G^{0}_{cyc}=-RT\ln(J_{i}/J_{ref})$, where $T$ is the
temperature, $R$ is the gas constant and $J_{ref}$ is an arbitrary reference
molecule (see Ref. [8] for details). Adapted from Refs. [8, 77]. (B) Looping
J-factors for the no-promoter data do not show the same trends in sequence
dependence as do cyclization and nucleosome formation: if anything, a higher
cyclization J-factor correlates with a lower looping J-factor. (C) Same as (B)
but for with-promoter DNAs. The cyclization J-factors of the poly(dA:dT)-rich
and GC-rich sequences that we use here have not been reported, so they are
shown as shaded regions whose height reflects the uncertainty in the looping
J-factors we measure, and whose width reflect our estimates about what their
cyclization behavior should be. In particular, the poly(dA:dT)-rich sequence
exhibits very low nucleosome occupancy in vivo [23, 22], and similar sequences
have high energies of nucleosome formation in vitro [28, 22], which, according
to the logic of (A), should correspond to a low cyclization J-factor, probably
lower than that of E8. Some poly(dA:dT)-rich DNAs were in fact recently
directly shown to cyclize less readily than random sequences [34]. In
contrast, the GC-rich sequence should be a good nucleosome former (though the
nucleosome affinity of this particular sequence has not been tested either in
vivo or in vitro), and so its cyclization J-factor is probably comparable to
that of 5S and TA, the other strong nucleosome-preferring sequences on this
plot. Additional details of how this plot was generated can be found in the
Methods section.
The strong correlation between a sequence’s ease of cyclization and of
nucleosome formation, as shown in Fig. 5(A), has been used to argue that
nucleosome sequence preferences depend largely on the intrinsic mechanical
properties of a DNA, particularly its bendability [8], though other mechanisms
have also been proposed, such as that described by Rohs and coworkers, which
depends not on sequence-dependent DNA flexibility but on sequence-dependent
minor groove shape [75]. We have shown here that three sequence features that
commonly determine nucleosome preferences, either through their effect on DNA
flexibility or on other structural aspects recognized by the nucleosome, do
not likewise determine looping, arguing for the need to identify a different
set of sequence features that determine loopability. The most striking
contrast between previously established sequence “rules” derived from
nucleosome studies and the trends in looping J-factors that we observe here is
that of the nucleosome-repelling, poly(dA:dT) sequence, which has the lowest
looping free energy that we have quantified so far. Other in vitro assays
predominantly show poly(dA:dT) copolymers to be highly resistant to
deformations; for example, Vafabakhsh and coworkers recently used a FRET-based
cyclization assay, analogous to traditional ligase-mediated cyclization
assays, to show that poly(dA:dT)-rich sequences have cyclization rates
significantly smaller than other sequences such as E8 and TA [34]. Although
ease of cyclization is often equated with bendability, it appears that such
observed bendability is more context-dependent than has been previously
appreciated: that is, the simplest model that one would write down to describe
the energetics of these different deformed DNAs would feature the persistence
length as the governing parameter that is used to characterize bendability,
and yet, the distinct responses seen in looping, nucleosomes and cyclization
belie that simplest model. It will be informative to extend this study of an
unphased poly(dA:dT) tract in DNA loops to include more sequences containing
both pure poly(dA:dT) copolymers and naturally-occuring poly(dA:dT)-rich DNAs
that exclude nucleosomes in vivo, in order to elucidate the precise role of
poly(dA:dT)-tracts in determining looping. It is clear, however, that
poly(dA:dT)-rich DNAs should not be exclusively thought of as stiff or
resistant to bending in all biological contexts.
Figure 6: Maximum looping J-factor as a function of loop G+C content. Maximum
J-factors for each of the five sequences, with (closed circles) and without
(open circles) promoter, are plotted with respect to each sequence’s G+C
content. For nucleosomes, G+C content strongly correlates with nucleosome
occupancy [17]. In contrast, it appears that G+C content and loopability are
anticorrelated. Loop lengths plotted here are the same as in Fig. 5.
A second striking contrast between our results here and previously established
rules for nucleosome formation concerns the role of G+C content in determining
loop formation. The G+C content of a DNA is one of the most powerful
parameters for predicting nucleosome occupancy in vivo [17, 19], with higher
G+C content correlating with higher occupancy. However, as shown in Fig. 6,
G+C content offers little predictive power for loopability, or is
anticorrelated with looping. We note that a recent, systematic DNA cyclization
study demonstrated a quadratic dependence of DNA bending stiffness on G+C
content [15]. In our case of protein-mediated DNA looping, the looping
J-factor contains contributions from protein elasticity in addition to those
from DNA elasticity, and our DNA sequences contain A-tracts and GGGCCC motifs
that were excluded in [15], making a direct comparison between our results and
theirs difficult; but it is possible that the looping J-factor is neither
correlated or anticorrelated with G+C content but instead depends
quadratically on G+C content, as do cyclization J-factors. More data will be
necessary to make a strong statistical statement about the anticorrelation or
lack of correlation between the looping J-factor and G+C content, and to
determine the form of the relationship between the looping J-factor and G+C
content (e.g. quadratic versus linear), but we propose low G+C content as the
starting point of a potential new sequence “rule” for predicting looping
J-factors, and a fertile realm of further investigation. Finally, we have
shown that the repeating AA/TT/TA/AT and GG/CC/GC/CG steps that characterize
the 5S and TA sequences, as well as many nucleosome-preferring sequences, do
not likewise determine looping J-factors, as these two sequences behave very
differently from each other in the context of transcription factor-mediated
DNA looping.
## 4 Conclusions
Here we have extended our previous work on the sequence dependence of loop
formation by the Lac repressor to include three naturally occurring, genomic
sequences that have either nucleosome-repelling or nucleosome-attracting
functions in vivo, in addition to the two synthetic sequences we described
previously [35]. We find that two sequences that share sequence features
important to nucleosome formation and that share trends in observed
flexibility in cyclization and nucleosome formation assays, the 601TA and 5S
sequences, behave less similarly in the context of DNA looping than the two
sequences that should have least in common, the GC-rich, nucleosome attracting
sequence and the poly(dA:dT)-rich, nucleosome repelling sequence. 5S and TA
share neither trends in looping free energy relative to the random E8
sequence, nor loop length where looping is maximal, nor preferred loop
conformation, nor their response to the larger sequence context (as evidenced
by the fact that the inclusion of the lacUV5 promoter sequence in the loop
increases the looping J-factor for TA but decreases it for 5S).
We have also shown that a poly(dA:dT)-rich DNA that forms a nucleosome-free
region in yeast [23] is actually extremely deformable in the context of
looping by a transcription factor. The rest of the sequences show a range of
J-factors that does not correlate with any observed trends in flexibility as
measured by ligase-mediated cyclization assays, nor with the observation that
high G+C content correlates with nucleosome occupancy [17]. The diversity of
the effects on DNA looping that we observe with these five sequences (ten, if
the inclusion of the promoter is considered to create a “new” sequence)
underscores the necessity of a large-scale screen for sequences that control
loop formation both in vivo and in vitro, much as has been done in the context
of nucleosome formation to help establish the sequence-dependence rules of
that field (for example, see [59, 5]).
Our work in no way undermines previous claims of the sequence dependence to
nucleosome formation and/or occupancy either in vivo or in vitro; rather, it
demonstrates that the “rules” of sequence flexibility derived from cyclization
and nucleosome formation studies are inapplicable to DNA looping, possibly due
to the difference in the boundary conditions and therefore DNA conformations
involved in forming a protein-mediated loop versus a DNA minicircle or a
nucleosome. It will be interesting to extend these studies of the role of
sequence in loop formation to other DNA looping proteins besides the Lac
repressor. As noted above, it has been shown recently that the Lac repressor
can accommodate many different loop conformations [40]. The variety in tether
lengths and preferred looped states that we observe are consistent with a
forgiving Lac repressor protein. Nucleosomes, on the other hand, have a more
fixed structure that should not be as accommodating to a range of helical
periods and DNA polymer conformations (hence the hypothesis that
poly(dA:dT)-rich DNAs disfavor nucleosome formation because they adopt
geometry that is incompatible with the structure of the DNA in a nucleosome
[21]). It would be informative to measure the looping J-factors of these same
sequences with a more rigid looping protein. It will also be interesting to
see if other bacterial promoter sequences have similar effect of altering the
looping boundary condition as the very strong and synthetic lacUV5 promoter.
In fact, the lacUV5 promoter should be a key starting point for identifying
sequences that have a strong effect on looping, since it can have significant
effects on the behavior of a loop, even when it comprises only one-third of
the loop length.
## 5 Materials and Methods
### 5.1 DNAs.
The poly(dA:dT)-rich sequence (from Fig. 4 of Ref. [23]), GC-rich sequence
(from “Human 2” at
http://genie.weizmann.ac.il/pubs/field08/field08_data.html), and 5S sequences
(from Fig. 1 of [45]) were cloned into the pZS25 plasmid used in [35], with
these eukaryotic sequences replacing the E8 or TA sequences in that plasmid.
In cases where the loop lengths used in this study were shorter than the 147
bp that are wrapped in nucleosomes, the corresponding looping sequences used
in TPM were taken from the middle of these sequences (relative to the
nucleosomal dyad); in cases where the nucleosomal sequences were shorter than
the desired loop length, they were padded at one end with the random E8
sequence [8, 60, 35]. See Figures LABEL:fig:SIseqlist1 and
LABEL:fig:SIseqlist2 in File S1 for details. As in [35], “no-promoter” loops
were flanked by the synthetic, strongest known operator (repressor binding
site) $O_{id}$ and the strongest naturally occurring operator $O_{1}$; “with-
promoter” loops were flanked by $O_{id}$ and a weaker naturally occurring
operator, $O_{2}$, because these with-promoter constructs are also used in in
vivo studies of the effect of loop architecture on YFP expression, in which
case $O_{2}$ is a more convenient choice of operator than $O_{1}$. Similarly,
the motivation to include the lacUV5 promoter in the loop stems from parallel
in vivo studies, in which the promoter is a natural part of the looping
architecture. The promoter is included in the loop between the sequence of
interest and the $O_{2}$ operator. Figures LABEL:fig:SIseqlist1 and
LABEL:fig:SIseqlist2 in File S1 gives the exact sequences used in this work;
Fig. 1(B) shows the TPM constructs schematically.
Cloning of the sequences of interest into the pZS25 plasmid was accomplished
in either one or two steps. For the 5S sequences, oligomers were first ordered
from Integrated DNA Technologies as single-stranded forward and reverse
complements, consisting of 69 bp (for the “with-promoter” constructs) or 105
bp (for the “no-promoter” constructs) of the 5S sequence, plus the $O_{id}$
and $O_{1}$/$O_{2}$ operators, and, where applicable, the lacUV5 promoter
sequence. These oligomers were annealed and then ligated into the pZS25
plasmid at the AatII and EcoRI restriction sites that fall just outside the
operators that flank the E8 or TA sequences in the original pZS25 plasmids
[35]. Second, Quik-Change mutagenesis (Agilent Technologies) was performed to
generate additional lengths (that is, to introduce insertions or deletions) of
the 5S sequence from the initial 105 bp loop lengths. However, we found that
this site-directed mutagenesis step generated distributions of products for
the poly(dA:dT) constructs, possibly due to replication slipped mispairing
over repetitive sequences [76]. Therefore all lengths of the poly(dA:dT)
sequence, as well as of the GC-rich sequence, which also have the potential to
contain such “slippery” regions, were created by ligation of synthesized
oligomers into the pZS25 plasmid. All constructs were confirmed by sequencing
(Laragen Inc.) to have clean sequence reads, and the approximately 450 bp
digoxigenin- and biotin-labeled TPM constructs were created by PCR as
described for the E8- and TA-containing constructs in [51, 35]. Sequences of
TPM constructs were again confirmed by sequencing before use.
### 5.2 TPM sample preparation, data acquisition and analysis.
Tethered particle motion assays were performed as described in [35]. Briefly,
linear DNAs, labeled on one end with digoxigenin and on the other end with
biotin, were introduced into chambers created between a microscope slide and
coverslip, with the coverslip coated nonspecifically with anti-digoxigenin.
Streptavidin-coated beads (Bangs Laboratories, Inc) were then introduced into
the chamber to complete the formation of tethered particles. The motion of the
beads was tracked using custom Matlab code that calculated each bead’s root-
mean-squared (RMS) motion in the plane of the coverslip, and looping
probabilities were extracted from these RMS-versus-time trajectories as the
time spent in the looped state (reduced RMS), divided by total observation
time. Similarly, the probabilities of the “bottom” versus “middle” states (see
Results section) were defined as the time spent in a particular state, divided
by the total observation time.
By measuring the looping probability of a construct at a particular repressor
concentration, and using the repressor-operator dissociation constants for
$O_{1}$, $O_{2}$ and $O_{id}$ in [35], we can calculate the J-factor for that
construct. All measurements in this work were carried out at 100 pM repressor,
using repressor purified in-house. The relationship between the looping
probabilities measured in TPM ($p_{\mathrm{loop}}$), the repressor-operator
dissociation constants for the two operators that flank the loop ($K_{1}$,
$K_{2}$ and $K_{id}$), and the looping J-factor of the DNA in the loop
($J_{\mathrm{loop}}$) can be described as
$p_{\mathrm{loop}}=\frac{\frac{[R]J_{\mathrm{loop}}}{2K_{A}K_{B}}}{1+\frac{[R]}{K_{A}}+\frac{[R]}{K_{B}}+\frac{[R]^{2}}{K_{A}K_{B}}+\frac{[R]J_{\mathrm{loop}}}{2K_{A}K_{B}}},$
(2)
where $[R]$ is the concentration of Lac repressor, and $K_{A}$ and $K_{B}$ are
repressor-operator dissociation constants of the two operators flanking the
loop ($K_{id}$ and $K_{1}$ or $K_{2}$). A similar expression can be derived
for the J-factors of the individual “bottom” and “middle” looped states and is
given in [35].
### 5.3 Generating the plots in Figure 5.
The J-factors plotted in Figure 5 are the maximum looping or cyclization
J-factors over a particular period. Specifically, the looping J-factors used
are those at 104 bp for dA, 105 for 5S and CG, and 106 for E8 and TA; the
cyclization J-factors are for 94 bp of the E8, 5S or TA sequences and are
taken from [60]. Although we are not directly comparing identical lengths
between cyclization and looping, the general trends hold regardless of lengths
chosen. In fact, identifying the loop length that corresponds to a particular
cyclization length is difficult, given that the flanking operators for looping
must be taken into account in some fashion. That is, for cyclization, DNA
length is easy to compute—it is simply the length of the oligomer used in the
ligation reactions. However, in the case of looping, it is unclear if the
appropriate length for comparison is just the DNA in the loop (excluding the
operators), or the length between the midpoints of the operators, or including
all of the operators. Similarly, we are not comparing identical loop lengths
across sequences; we chose to compare loop flexibility at the looping maximum
for each sequence in an attempt to compare lengths at which the operators are
most likely to be in phase, such that we are comparing only bending and not
twisting flexibility. Finally, we note that here we are interested in the same
kind of comparison that Cloutier and Widom were in Ref. [8], which was the
inspiration for this figure; in [8], Cloutier and Widom compared cyclization
and nucleosome formation free energies, even though the cyclization
experiments were performed with roughly 100 bp DNAs and the nucleosome
formation assays with roughly 150 bp DNAs. Likewise, we do not expect that the
fragments of nucleosome-preferring or nucleosome-repelling sequences that we
examine here in the context of looping will necessarily have exactly the same
characteristics as the full-length nucleosomal sequences from which they were
derived; but we are interested in comparing general trends in observed
flexibility of these roughly 110 bp loops with those of roughly 100 bp ligated
minicircles and of roughly 150 bp nucleosomal DNAs.
## 6 Acknowledgements
We are indebted to the late Jon Widom for the inspiration of this project and
for his guidance, mentorship and friendship over many years. We thank Chao
Liu, David Wu, David Van Valen, Hernan Garcia, Martin Lindén, Mattias
Rydenfelt, Yun Mou, Tsui-Fen Chou, Eugene Lee, Matthew Raab, Daniel Grilley,
Niv Antonovsky, Lior Zelcbuch, Matthew Moore, Ron Milo, Eran Segal, and the
Phillips, Mayo, Pierce and Elowitz labs for insightful discussions, equipment
and technical help; and Winston Warman at Transgenomic, Inc. (Omaha, NE, USA)
and Jin Li at Laragen, Inc (Culver City, CA, USA) for special help with
sequencing the poly(dA:dT)-rich DNAs.
## 7 Supporting Information
File S1: Supporting figures. Figure S1 “No-promoter” looping sequences used in
this work. Figure S2 “With-promoter” looping sequences used in this work.
Figure S3 Sequence-dependent twist stiffness. Figure S4 Looping probabilities
and J-factors for the two looped states separately. Figure S5 Tether lengths
of looped and unlooped states as a function of loop length and sequence.
Figure S6 Tether length as a function of J-factor.
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|
arxiv-papers
| 2013-10-12T17:31:05 |
2024-09-04T02:49:52.306077
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Stephanie Johnson, Yi-Ju Chen, Rob Phillips",
"submitter": "Stephanie Johnson",
"url": "https://arxiv.org/abs/1310.3408"
}
|
1310.3457
|
# On $hp$-Convergence of PSWFs and A New Well-Conditioned Prolate-Collocation
Scheme
Li-Lian Wang1, Jing Zhang2 and Zhimin Zhang3
###### Abstract.
The first purpose of this paper is to provide a rigorous proof for the
nonconvergence of $h$-refinement in $hp$-approximation by the PSWFs, a
surprising convergence property that was first observed by Boyd et al [3, J.
Sci. Comput., 2013]. The second purpose is to offer a new basis that leads to
spectral-collocation systems with condition numbers independent of $(c,N),$
the intrinsic bandwidth parameter and the number of collocation points. In
addition, this work gives insights into the development of effective spectral
algorithms using this non-polynomial basis. We in particular highlight that
the collocation scheme together with a very practical rule for pairing up
$(c,N)$ significantly outperforms the Legendre polynomial-based method (and
likewise other Jacobi polynomial-based method) in approximating highly
oscillatory bandlimited functions.
###### Key words and phrases:
Prolate spheroidal wave functions, collocation method, pseudospectral
differentiation matrix, condition number, $hp$-convergence, eigenvalues
###### 1991 Mathematics Subject Classification:
65N35, 65E05, 65M70, 41A05, 41A10, 41A25
1 Division of Mathematical Sciences, School of Physical and Mathematical
Sciences, Nanyang Technological University, 637371, Singapore. The research of
this author is partially supported by Singapore MOE AcRF Tier 1 Grant (RG
15/12), MOE AcRF Tier 2 Grant (2013-2016), and A∗STAR-SERC-PSF Grant
(122-PSF-007). This author would like to thank the hospitality of Beijing
Computational Science Research Center during the visit in June 2013.
2 School of Mathematics and Statistics, Huazhong Normal University, Wuhan
430079, China, and Beijing Computational Science Research Center, China. The
work of this author is supported by the National Natural Science Foundation of
China (11201166).
3 Beijing Computational Science Research Center, and Department of
Mathematics, Wayne State University, Detroit, MI 48202. This author is
supported in part by the US National Science Foundation under grant
DMS-1115530.
## 1\. Introduction
The prolate spheroidal wave functions of order zero provide an optimal tool
for approximating bandlimited functions (whose Fourier transforms are
compactly supported), and appear superior to polynomials in approximating
nearly bandlimited functions (cf. [32]). PSWFs also offer an alternative to
Chebyshev and Legendre polynomials for pseudospectral/collocation and
spectral-element algorithms, which enjoy a “plug-and-play” function by simply
swapping the cardinal basis, collocation points and differentiation matrices
(cf. [4, 7, 33, 3]). With an appropriate choice of the underlying tunable
bandwidth parameter, PSWFs exhibit some advantages: (i) Spectral accuracy can
be achieved on quasi-uniform computational grids; (ii) Spatial resolution can
be enhanced by a factor of $\pi/2;$ and (iii) The resulted method relaxes the
Courant-Friedrichs-Lewy (CFL) condition of explicit time-stepping scheme. Boyd
et al [3, Table 1] provided an up-to-date review of recent developments since
the series of seminal works by Slepian et al. [26, 17, 24].
While PSWFs enjoy some unique properties (e.g., being bandlimited and
orthogonal over both a finite and an infinite interval), they are anyhow a
non-polynomial basis, and therefore might lose certain capability of
polynomials, when they are used for solving PDEs. This can be best testified
by the nonconvergence of $h$-refinement in prolate-element methods, which was
discovered by Boyd et al [3] through simply examining $hp$-prolate
approximation of the trivial function $u(x)=1.$ Indeed, PSWFs lack some
crucial properties of polynomial spectral algorithms. A naive extension of
existing algorithms to this setting might be unsatisfactory or fail to work
sometimes, so the related numerical issues are worthy of investigation.
The purpose of this paper is to give new insights into spectral algorithms
using PSWFs. The main contributions reside in the following aspects:
* •
We establish an $hp$-error bound for a PSWF-projection. As a by-product, this
provides a rigorous proof, from an approximation theory viewpoint, for the
nonconvergence of $h$-refinement in $hp$-approximation. We also present more
numerical evidences to demonstrate this surprising convergence behavior.
* •
We offer a new PSWF basis of dual nature.
Firstly, it produces a matrix that nearly inverts the second-order prolate
pseudospectral differentiation matrix, in the sense that their product is
approximately an identity matrix for large $N$ (see (5.10)). Consequently, it
can be used as a preconditioner for the usual prolate-collocation scheme for
second-order boundary value problems, leading to well-conditioned collocation
linear systems. We remark that the idea along this line is mimic to the
integration preconditioning (see e.g., [13, 10, 28]). However, the PSWFs lack
some properties of polynomials, so the procedure here is quite different from
that for the polynomials.
Secondly, under the new basis, the matrix of the highest derivative in the
collocation linear system is an identity matrix, and the resulted linear
system is well-conditioned. In contrast with the above preconditioning
technique, this does not involve the differentiation matrices.
It is noteworthy that the non-availability of a quadrature rule exact for
products of PSWFs, makes the PSWF-Galerkin method less attractive. We believe
that the proposed well-conditioned collocation approach might be the best
choice.
* •
We propose a practical approximation to Kong-Rokhlin’s rule for pairing up
$(c,N)$ (see [15]), and demonstrate that the collocation scheme using this
rule significantly outperforms the Legendre polynomial-based method when the
involved solution is bandlimited. For example, the portion of discrete
eigenvalues of the prolate differentiation matrix that approximates the
eigenvalues of the continuous operator to $12$-digit accuracy is about $87\%$
against $25\%$ for the Legendre case (see Subsection 3.2). Similar advantages
are also observed in solving Helmholtz equations with high wave numbers in
heterogeneous media (see Subsection 5.3).
The paper is organized as follows. In Section 2, we review basic properties of
PSWFs, and the related quadrature rules, cardinal bases and differentiation
matrices. In Section 3, we introduce the Kong-Rokhlin’s rule for pairing up
$(c,N),$ and study the discrete eigenvalues of the second-order prolate
differentiation matrix. In Section 4, we establish the $hp$-error bound for a
PSWF-projection and explain the nonconvergence of $h$-refinement in prolate-
element methods. In Section 5, we introduce a new PSWF-basis which leads to
well-conditioned collocation schemes. We also propose a collocation-based
prolate-element method for solving Helmholtz equations with high wave numbers
in heterogeneous media.
## 2\. PSWFs and prolate pseudospectral differentiation
In this section, we review some relevant properties of the PSWFs, and
introduce the quadrature rules, cardinal basis and associate prolate
pseudospectral differentiation matrices.
### 2.1. Prolate spheroidal wave functions
The PSWFs arise from two contexts: (i) in solving the Helmholtz equation in
prolate spheroidal coordinates by separation of variables (see e.g., [1]), and
(ii) in studying time-frequency concentration problem (see [26]). As
highlighted in [26], “PSWFs form a complete set of bandlimited functions which
possesses the curious property of being orthogonal over a given finite
interval as well as over $(-\infty,\infty).$”
Firstly, PSWFs, denoted by $\psi_{n}(x;c),$ are eigenfunctions of the singular
Sturm-Liouville problem:
${\mathcal{D}}_{x}^{c}[\psi_{n}]:=-\partial_{x}\big{(}(1-x^{2})\partial_{x}\psi_{n}(x;c)\big{)}+c^{2}x^{2}\psi_{n}(x;c)=\chi_{n}(c)\psi_{n}(x;c),$
(2.1)
for $x\in I:=(-1,1),$ and $c\geq 0.$ Here, $\\{\chi_{n}(c)\\}_{n=0}^{\infty},$
are the corresponding eigenvalues, and the positive constant $c$ is dubbed as
the “bandwidth parameter” (see Remark 2.3). PSWFs are complete and orthogonal
in $L^{2}(I)$ (the space of square integrable functions). Hereafter, we adopt
the conventional normalization:
$\int_{-1}^{1}\psi_{n}(x;c)\psi_{m}(x;c)\,dx=\delta_{mn}:=\begin{cases}1,\quad&m=n,\\\
0,\quad&m\not=n.\end{cases}$ (2.2)
The eigenvalues $\\{\chi_{n}(c)\\}_{n=0}^{\infty}$ (arranged in ascending
order), have the property (cf. [32]):
$\chi_{n}(0)<\chi_{n}(c)<\chi_{n}(0)+c^{2},\quad n\geq 0,\;\;c>0.$ (2.3)
For fixed $c$ and large $n,$ we have (cf. [21, (64)]):
$\chi_{n}(c)=n(n+1)+\frac{c^{2}}{2}+\frac{c^{2}(4+c^{2})}{32n^{2}}\Big{(}1-\frac{1}{n}+O(n^{-2})\Big{)}.$
(2.4)
###### Remark 2.1.
Note that when $c=0,$ (2.1) reduces to the Sturm-Liouville equation of the
Legendre polynomials. Denote the Legendre polynomials by $P_{n}(x),$ and
assume that they are orthonormal. Then we have $\psi_{n}(x;0)=P_{n}(x)$ and
$\chi_{n}(0)=n(n+1).$
Secondly, D. Slepian et al (cf. [26, 25]) discovered that PSWFs luckily
appeared from the context of time-frequency concentration problem. Define the
integral operator related to the finite Fourier transform:
${\mathcal{F}}_{c}[\phi](x):=\int_{-1}^{1}e^{{\rm
i}cxt}\phi(t)\,dt,\quad\forall\,c>0.$ (2.5)
Remarkably, the differential and integral operators are commutable:
${\mathcal{D}}_{x}^{c}\circ{\mathcal{F}}_{c}={\mathcal{F}}_{c}\circ{\mathcal{D}}_{x}^{c}.$
This implies that PSWFs are also eigenfunctions of ${\mathcal{F}}_{c},$
namely,
${\rm i}^{n}\lambda_{n}(c)\psi_{n}(x;c)=\int_{-1}^{1}e^{{\rm
i}cx\tau}\psi_{n}(\tau;c)\,d\tau,\quad x\in I,\;\;c>0.$ (2.6)
The corresponding eigenvalues $\\{\lambda_{n}(c)\\}$ (modulo the factor ${\rm
i}^{n}$) are all real, positive, simple and ordered as
$\lambda_{0}(c)>\lambda_{1}(c)>\cdots>\lambda_{n}(c)>\cdots>0,\quad c>0.$
(2.7)
We have the following uniform upper bound (cf. [27, (2.14)]):
$\lambda_{n}(c)<\frac{\sqrt{\pi}c^{n}(n!)^{2}}{(2n)!\Gamma(n+3/2)},\quad n\geq
1,\;\;c>0,$ (2.8)
where $\Gamma(\cdot)$ is the Gamma function.
###### Remark 2.2.
As demonstrated in [27], the upper bound in (2.8) provided a fairly accurate
approximation to $\lambda_{n}(c)$ for a wide range of $c,n$ of interest.
###### Remark 2.3.
Recall that a function $f(x)$ defined in $(-\infty,\infty),$ is said to be
bandlimited, if its Fourier transform $F(\omega),$ defined by
$F(\omega)=\int_{-\infty}^{\infty}f(x)e^{{\rm i}\omega x}dx,$ (2.9)
has a finite support (cf. [26]), that is, $F(\omega)$ vanishes when
$|\omega|>\sigma>0$. Then $f(x)$ can be recovered by the inverse Fourier
transform
$f(x)=\frac{1}{2\pi}\int_{-\sigma}^{\sigma}F(\omega)e^{-{\rm i}\omega
x}d\omega.$ (2.10)
One verifies from (2.6) and the parity: $\psi_{n}(-x;c)=(-1)^{n}\psi_{n}(x;c)$
(see [26]) that
$\psi_{n}(x;c)=\frac{{\rm
i}^{n}}{c\lambda_{n}(c)}\int_{-c}^{c}\psi_{n}\Big{(}\frac{\omega}{c};c\Big{)}e^{-{\rm
i}\omega x}d\omega.$ (2.11)
Hence, the PSWF $\psi_{n}$ is bandlimited to $[-c,c],$ and $c$ is therefore
called the bandwidth parameter. However, its counterpart $P_{n}(x)$ is not
bandlimited. Indeed, we have the following formula (see [11, P. 213]):
$\int_{-1}^{1}P_{n}(\omega)e^{-{\rm i}\omega x}\,d\omega=(-{\rm
i})^{n}(2n+1)\sqrt{\frac{\pi}{2}}\frac{J_{n+1/2}(x)}{\sqrt{x}},$ (2.12)
where $J_{n+1/2}$ is the Bessel function (cf. [1]). This implies
$J_{n+1/2}(x)/\sqrt{x}$ is bandlimited, as its Fourier transform is
$P_{n}(\omega)\chi_{{}_{I}}(\omega)$ (up to a constant multiple), where
$\chi_{{}_{I}}$ is the indicate function of $(-1,1).$ Since a function and its
Fourier transform cannot both have finite support, $P_{n}(x)$ is not
bandlimited.
The PSWFs provide an optimal tool in approximating general bandlimited
functions (see e.g., [26, 25, 32, 15]). On the other hand, being the
eigenfunctions of a singular Sturm-Liouville problem (cf. (2.1)), the PSWFs
offer a spectral basis on quasi-uniform grids with spectral accuracy (see
e.g., [4, 7, 16, 27, 33, 29, 3]). However, the PSWFs are non-polynomials, so
they lack some important properties that make the naive extension of
polynomial algorithms to PSWFs unsatisfactory or infeasible sometimes. For
example, Boyd et al [3] demonstrated the nonconvergence of $h$-refinement in
prolate elements, which was in distinctive contrast with Legendre polynomials.
In addition, we observe that for any
$\psi_{m},\psi_{n}\in V_{N}^{c}:={\rm span}\big{\\{}\psi_{n}\,:\,0\leq n\leq
N\big{\\}},$ (2.13)
we have
$\partial_{x}\psi_{n}\not\in{V_{N-1}^{c}};\quad\int\psi_{n}\,dx\not\in
V_{N+1}^{c};\quad\psi_{n}\cdot\psi_{m}\not\in V_{2N}^{c},\;\;\;\;c>0.$ (2.14)
These will bring about some numerical issues to be addressed later.
###### Remark 2.4.
In what follows, we might drop $c$ and simply denote by $\psi_{n}(x)$ the
PSWFs and likewise for the eigenvalues, whenever no confusion might cause.
### 2.2. Quadrature rules and grid points
The conventional choice of grid points for pseudospectral and spectral-element
methods, is the Gauss-Lobatto points. The quadrature rule using such a set of
points as quadrature nodes has the highest degree of precision (DOP) for
polynomials. For example, let $\\{\xi_{j},\rho_{j}\\}_{j=0}^{N}$ (with
$\xi_{0}=-1$ and $\xi_{N}=1$) be the Legendre-Gauss-Lobatto (LGL) points
(i.e., zeros of $(1-x^{2})P_{N}^{\prime}(x)$) and quadrature weights. Then we
have
$\int_{-1}^{1}P_{n}(x)\,dx=\sum_{j=0}^{N}P_{n}(\xi_{j})\rho_{j},\quad 0\leq
n\leq 2N-1.$ (2.15)
It is also exact for all $P_{n}\cdot P_{m}\in{\mathbb{P}}_{2N-1}$ (the set of
all algebraic polynomials of degree at most $2N-1$), which plays an essential
role in spectral/spectral-element methods based on the Galerkin formulation.
The choice of computational grids for the PSWFs is controversial, largely due
to (2.14). The pursuit of the highest DOP leads to the generalized Gaussian
quadrature (see e.g., [8, 32, 4]). In particular, the generalized prolate-
Gauss-Lobatto (GPGL) quadrature in [4] is based on the fixed points:
$x_{0}=-1,x_{N}=1,$ and the interior quadrature points
$\\{x_{j}\\}_{j=1}^{N-1}$ and weights $\\{\omega_{j}\\}_{j=0}^{N}$ being
determined by
$\int_{-1}^{1}\psi_{n}(x)\,dx=\psi_{n}(-1)\,\omega_{0}+\sum_{j=1}^{N-1}\psi_{n}(x_{j})\omega_{j}+\psi_{n}(1)\,\omega_{N},\quad
0\leq n\leq 2N-1.$ (2.16)
Another choice is the prolate-Lobatto (PL) points (see [16, 5] and [32, 19]
for prolate-Gaussian case), which are zeros of
$(1-x^{2})\partial_{x}\psi_{N}(x)$ (still denoted by $\\{x_{j}\\}_{j=0}^{N}$).
Then the quadrature weights $\\{\omega_{j}\\}_{j=0}^{N}$ are determined by
$\int_{-1}^{1}\psi_{n}(x)\,dx=\sum_{j=0}^{N}\psi_{n}(x_{j})\omega_{j},\quad
0\leq n\leq N,$ (2.17)
which is exact for $\\{\psi_{n}\\}_{n=0}^{N}$.
###### Remark 2.5.
It is noteworthy that in the Legendre case (i.e., $c=0$), the quadrature rules
(2.16) and (2.17) are identical.
###### Remark 2.6.
In view of (2.14), the GPGL quadrature (2.16) is not exact for
$\psi_{n}\cdot\psi_{m}$ with $0\leq m+n\leq 2N-1.$ This makes the spectral-
Galerkin method using PSWFs less attractive. On the other hand, when it comes
to prolate pseudospectral/collocation approaches, we find there is actually
very subtle difference between two sets of points (also see [7]). Moreover,
much more effort is needed to compute the GPGL points, so in what follows, we
just use the PL points.
### 2.3. Prolate differentiation matrices
With the grid points at our disposal, we now introduce the cardinal
(synonymously, nodal or Lagrange) basis. Here, we have two different routines
to define the prolate cardinal basis once again due to (2.14).
Let $\\{x_{j}\\}_{j=0}^{N}$ be the PL points. The first approach searches for
the cardinal basis $h_{k}(x):=h_{k}(x;c)\in V_{N}^{c}$ such that
$h_{k}(x_{j})=\delta_{jk},\quad 0\leq k,j\leq N.$ (2.18)
To compute the basis functions, we write
$h_{k}(x)=\sum_{n=0}^{N}t_{nk}\,\psi_{n}(x),$ (2.19)
and find the coefficients $\\{t_{nk}\\}$ from (2.18). More precisely,
introducing the $(N+1)^{2}$ matrices:
$\boldsymbol{\Psi}_{jk}=\psi_{k}(x_{j}),\quad\boldsymbol{\Psi}_{jk}^{(m)}=\psi_{k}^{(m)}(x_{j}),\quad\boldsymbol{T}_{nk}=t_{nk},\quad{\boldsymbol{D}}^{(m)}_{jk}=h_{k}^{(m)}(x_{j}),$
(2.20)
we have $\boldsymbol{\Psi}\boldsymbol{T}=\boldsymbol{I}_{N+1},$ so
$\boldsymbol{T}=\boldsymbol{\Psi}^{-1}.$ Thus, the $m$th-order differentiation
matrix is computed by
$\quad{\boldsymbol{D}}^{(m)}=\boldsymbol{\Psi}^{(m)}\boldsymbol{\Psi}^{-1},\quad
m\geq 1.$ (2.21)
The second approach is to define
$l_{k}(x)=\frac{s(x)}{s^{\prime}(x_{k})(x-x_{k})},\;\;0\leq k\leq N\;\;{\rm
with}\;\;s(x)=(1-x^{2})\partial_{x}\psi_{N}(x).$ (2.22)
Then one verifies readily that
$l_{k}(x_{j})=\delta_{jk},\quad 0\leq k,j\leq N.$ (2.23)
Different from the previous case, the so-defined
$\\{l_{k}\\}_{k=0}^{N}\not\subseteq V_{N}^{c}$ for $c>0.$ The differentiation
matrix $\widehat{\boldsymbol{D}}^{(m)}$ with the entries
$\widehat{\boldsymbol{D}}_{jk}^{(m)}=l_{k}^{(m)}(x_{j})$ for $0\leq k,j\leq N$
can be computed by directly differentiating the cardinal basis in (2.22). We
provide in Appendix A the explicit formulas for computing the entries of
$\widehat{\boldsymbol{D}}^{(1)}$ and $\widehat{\boldsymbol{D}}^{(2)},$ which
only involve the function values $\\{\psi_{N}(x_{j})\\}_{j=0}^{N}.$
## 3\. Study of Eigenvalues of the prolate differentiation matrix
The appreciation of eigenvalue distribution of spectral differentiation
matrices is important in many applications of spectral methods (see e.g., [30,
31]). For example, for the second-order differentiation matrix, we are
interested in the answer to the question: to what extent can the discrete
eigenvalues approximate those of the continuous operator accurately?
With this in mind, we first introduce the Kong-Rokhlin’s rule in [15] for
pairing up $(c,N)$ that guarantees high accuracy in integration and
differentiation of bandlimited functions, but it requires computing
$\lambda_{N}.$ In this section, we first propose a practical mean for its
implementation. We demonstrate that with the choice of $(c,N)$ by this rule,
the portion of discrete eigenvalues of the prolate differentiation matrix that
approximates the eigenvalues of the continuous operator to $12$-digit accuracy
is about $87\%$ against $25\%$ for the Legendre case. This implies that the
polynomial interpolation can not resolve the continuous spectrum, while the
PSWF interpolation has significant higher resolution.
### 3.1. The Kong-Rokhlin’s rule
An important issue related to the PSWFs is the choice of bandlimit parameter
$c.$ As commented by [4], the so-called “transition bandwidth”:
$c_{*}(N)=\frac{\pi}{2}\Big{(}N+\frac{1}{2}\Big{)},$ (3.1)
turned out to be very crucial for asymptotic study of PSWFs and all aspects of
their applications. In fact, when $c$ is close to $c_{*}(N),$ $\psi_{N}(x;c)$
behaves like the trigonometric function $\cos([\pi/2]N(1-x)),$ so it’s nearly
uniformly oscillatory. However, when $c>c_{*}(N),$ $\psi_{N}(x;c)$ transits to
the region of the scaled Hermite function, so it vanishes near the endpoints
$x=\pm 1.$ In other words, the PSWFs with $c>c_{*}(N)$ lose the capability of
approximating general functions in $(-1,1)$. Consequently, the feasible
bandwidth parameter $c$ should fall into $[0,c_{*}(N)).$ However, this range
appears rather loose, as many numerical evidences showed the significant
degradation of accuracy when $c$ is close to $c_{*}(N).$
A conservative bound was provided in [29] (which improved that in [7]):
$0<q_{N}:=\frac{c}{\sqrt{\chi_{N}}}<\frac{1}{\sqrt[6]{2}}\approx 0.8909.$
(3.2)
Note that $q_{N}\approx 1,$ if $c=c_{*}(N).$ In practice, a quite safe choice
is $c=N/2$ (see e.g., [7, 27]).
From a different perspective, Kong and Rokhlin [15] proposed a useful rule for
pairing up $(c,N).$ The starting point is a prolate quadrature rule, say
(2.17). We know from [32] that it has the accuracy for the complex exponential
$e^{{\rm i}cax}:$
$\Big{|}\int_{-1}^{1}e^{{\rm i}cax}\,dx-\sum_{j=0}^{N}e^{{\rm
i}cax_{j}}\omega_{j}\Big{|}=O(\lambda_{N}).$ (3.3)
Furthermore, for a bandlimited function of bandwidth $c$, defined by
$f(x)=\int_{-1}^{1}\phi(t)\,e^{{\rm i}cxt}\,dt,\quad\text{for some}\;\;\phi\in
L^{2}(-1,1),$
we have (see [32, Remark 5.1])
$\Big{|}\int_{-1}^{1}f(x)\,dx-\sum_{j=0}^{N}f(x_{j})\omega_{j}\Big{|}\leq\varepsilon\|\phi\|,$
(3.4)
where $\varepsilon$ is the maximum error of integration of a single complex
exponential as in (3.3). In view of this, Kong and Rokhlin [15] suggested the
rule: given $c$ and an error tolerance $\varepsilon,$ choose the smallest
$N_{*}=N_{*}(c,\varepsilon)$ such that
$\lambda_{N_{*}}(c)\leq\varepsilon\leq\lambda_{N_{*}-1}(c).$ (3.5)
In what follows, we introduce a very practical mean to implement this rule
approximately, which does not require computing the eigenvalues
$\\{\lambda_{N}\\}.$ We start with the upper bound of $\lambda_{N}$ in (2.8):
$\frac{\sqrt{\pi}c^{N}(N!)^{2}}{(2N)!\Gamma(N+3/2)}\leq\sqrt{\frac{\pi
e}{2}}\Big{(}\frac{ec}{4}\Big{)}^{N}\Big{(}N+\frac{1}{2}\Big{)}^{-(N+1/2)}e^{1/(6N)}:=\nu_{N}(c),$
(3.6)
where we used the property $n!=\Gamma(n+1)$ and the formula (see [1,
(6.1.38)]):
$\Gamma(x+1)=\sqrt{2\pi}\,x^{x+\frac{1}{2}}{\rm
exp}\Big{(}-x+\frac{\theta}{12x}\Big{)},\quad x>0,\;\;\theta\in(0,1).$ (3.7)
We intend to replace $\lambda_{N}$ in (3.5) by its upper bound $\nu_{N}.$ For
a given tolerance $\varepsilon>0,$ we look for $N_{*}$ satisfying the
equation: $\nu_{N_{*}}(c)=\varepsilon.$ Taking the common log on both sides,
we then consider the equation: $F_{\varepsilon}(x;c)=0$ with
$F_{\varepsilon}(x;c):=x\log\frac{ec}{4}-\Big{(}x+\frac{1}{2}\Big{)}\log\Big{(}x+\frac{1}{2}\Big{)}+\frac{1}{6x}+\log\frac{1}{\varepsilon}+\frac{1}{2}\log\frac{\pi
e}{2},\quad x\geq 1.$ (3.8)
One verifies that $F_{\varepsilon}^{\prime}(x;c)<0$ for slightly large $x,$
and $F_{\varepsilon}^{\prime\prime}(x;c)<0.$ In addition,
$F_{\varepsilon}(1;c)>0$ and $F_{\varepsilon}(\infty;c)<0,$ so
$F_{\varepsilon}(x;c)=0$ has a unique root $x_{*}$. Then we set
$N_{*}=[x_{*}].$
###### Remark 3.1.
Note that $\nu_{N}(c)$ provides a fairly accurate approximation to
$\lambda_{N}(c)$ (cf. [27]) and $\lambda_{N_{*}}$ decays exponentially with
respect to $N_{*},$ so we have
$\lambda_{N_{*}}\approx\varepsilon\approx\lambda_{N_{*}-1}.$
We compare in Table 3.1 the approximate approach with the exact approach in
[15], and very similar performance is observed.
Table 3.1. A comparison of the pairs $(c,N_{*})$ obtained by the approximate approach and $(c,N)$ obtained by the Kong-Rokhlin’s rule [15], where $\varepsilon=10^{-14}.$ $c$ | $N_{*}$ | $\lambda_{N_{*}}$ | $N$ [15] | $\lambda_{N}$ | $c$ | $N_{*}$ | $\lambda_{N_{*}}$ | $N$ [15] | $\lambda_{N}$
---|---|---|---|---|---|---|---|---|---
10 | 24 | 1.77e-14 | 26 | 8.54e-16 | 100 | 94 | 2.79e-15 | 96 | 8.25e-16
20 | 34 | 5.96e-15 | 36 | 8.54e-16 | 200 | 163 | 8.00e-16 | 164 | 7.49e-16
40 | 50 | 8.79e-15 | 52 | 1.78e-15 | 400 | 299 | 5.20e-16 | 294 | 2.69e-15
80 | 79 | 1.10e-14 | 82 | 7.57e-16 | 800 | 571 | 1.57e-16 | 554 | 7.73e-16
### 3.2. Eigenvalues of the second-order prolate differentiation matrix
Consider the model eigen-problem:
$\text{Find $(\lambda,u)$ such that}\;\;u^{\prime\prime}(x)=\lambda u(x),\quad
x\in(-1,1);\quad u(\pm 1)=0,$ (3.9)
which has the eigen-pairs $(\lambda_{k},u_{k}):$
$\lambda_{k}=-\frac{k^{2}\pi^{2}}{4},\quad
u_{k}(x)=\sin\frac{k\pi(x+1)}{2},\;\;\;\;k\geq 1.$ (3.10)
The corresponding discrete eigen-problems are
$\begin{split}&\text{Find $(\tilde{\lambda},\tilde{\boldsymbol{u}})$ such
that}\;\;\boldsymbol{D}^{(2)}_{\rm
in}\tilde{\boldsymbol{u}}=\tilde{\lambda}\tilde{\boldsymbol{u}};\quad{\rm
or}\quad\text{Find $(\hat{\lambda},\hat{\boldsymbol{u}})$ such
that}\;\;\widehat{\boldsymbol{D}}^{(2)}_{\rm
in}\hat{\boldsymbol{u}}=\hat{\lambda}\hat{\boldsymbol{u}},\end{split}$ (3.11)
where ${\boldsymbol{D}}^{(2)}_{\rm in}$ and
$\widehat{\boldsymbol{D}}^{(2)}_{\rm in},$ which are obtained by deleting the
first and last rows and columns of ${\boldsymbol{D}}^{(2)}$ and
$\widehat{\boldsymbol{D}}^{(2)},$ respectively.
We examine the relative errors:
$\tilde{e}_{j}:=\frac{|\tilde{\lambda}_{j}-\lambda_{j}|}{|\lambda_{j}|},\quad\hat{e}_{j}:=\frac{|\hat{\lambda}_{j}-\lambda_{j}|}{|\lambda_{j}|},\quad
1\leq j\leq N-1.$
In the computation, $(c,N)$ is paired up by the approximate Kong-Rokhlin’s
rule with $\varepsilon=10^{-14}.$ We plot in Figure 3.1 the relative errors
between the discrete and continuous eigenvalues of the prolate differentiation
matrices with $c=120\pi$ and $N=284,$ compared with those of the Legendre
differentiation matrix at the Legendre-Gauss-Lobatto (LGL) points. Among $283$
eigenvalues of ${\boldsymbol{D}}^{(2)}_{\rm in},$ $245$ (approximately $87\%$)
are accurate to at least $12$ digits with respect to the exact eigenvalues,
while only $72$ (approximately $25\%$) of the Legendre case are of this
accuracy. A very similar number of accurate eigenvalues is also obtained from
$\widehat{\boldsymbol{D}}^{(2)}_{\rm in}.$
Figure 3.1. Behavior of the relative errors $\\{\tilde{e}_{j}\\}_{j=1}^{N-1}$
(left) and $\\{\hat{e}_{j}\\}_{j=1}^{N-1}$ (right), obtained by
$c=120\pi,\varepsilon=10^{-14}$ and $N=284.$ The prolate differentiation
matrices ${\boldsymbol{D}}^{(2)}_{\rm in}$ (left, marked by
“$\bigtriangleup$”) and $\widehat{\boldsymbol{D}}^{(2)}_{\rm in}$ (right,
marked by “$\bigtriangleup$”), against the Legendre case (marked by
“$\circ$”).
###### Remark 3.2.
Some remarks are in order.
* •
As shown in [30] for the Legendre case, a portion $2/\pi$ of the eigenvalues
approximate the eigenvalues of the continuous problem with one or two digit
accuracy (about $180$ among $283$). The errors in the remaining ones are
large, which can not be resolved by polynomial interpolation even on spectral
grids. However, the prolate interpolation significantly improves the
resolution to this portion around $95\%.$
* •
We remark that the behavior of the usual prolate differentiation scheme under
the approximate Kong-Rokhlin’s rule is very similar to the differentiation
scheme proposed by Kong and Rokhlin [15] (which was based on a Gram-Schmidt
orthogonalization of certain modal basis).
We next consider the eigen-problem involving the Bessel’s operator:
$u^{\prime\prime}(r)+\frac{1}{r}u^{\prime}(r)-\frac{1}{r^{2}}u(r)=\lambda
u(r),\;\;\;r\in(0,1);\quad u(0)=u(1)=0.$ (3.12)
The exact eigenvalues are $\lambda_{k}=-r_{k}^{2},\,k\geq 1,$ where each
$r_{k}$ is a root of the Bessel function $J_{1}(r).$ We adopt the same
computational setting as for Figure 3.1, and the relative errors are depicted
in Figure 3.2. Among $283$ (discrete) eigenvalues, $245$ are accurate to at
least $12$ digits with respect to the exact eigenvalues. In comparison, there
are only $111$ eigenvalues produced by Legendre collocation method that are
within the same accurate level.
Figure 3.2. Behavior of the relative errors $\\{\tilde{e}_{j}\\}_{j=1}^{N-1}$
(left) and $\\{\hat{e}_{j}\\}_{j=1}^{N-1}$ (right) for (3.12) with
$c=120\pi,\varepsilon=10^{-14}$ and $N=284.$ The prolate differentiation
matrices ${\boldsymbol{D}}^{(2)}_{\rm in}$ (left, marked by “$\bigstar$”) and
$\widehat{\boldsymbol{D}}^{(2)}_{\rm in}$ (right, marked by “$\bigstar$”),
against the Legendre case (marked by “$\square$”).
We demonstrate in Figure 3.3 the growth of the magnitude of the largest and
smallest eigenvalues of ${\boldsymbol{D}}^{(2)}_{\rm in}$ and
$\widehat{\boldsymbol{D}}^{(2)}_{\rm in},$ compared with the Legendre case,
where $(c,N)$ is chosen based on the approximate Kong-Rokhlin’s rule. We
observe a much slower growth of the largest eigenvalue, so the condition
number of the differentiation matrix behaves better.
Figure 3.3. Growth of the magnitude of the largest and smallest eigenvalues of
${\boldsymbol{D}}^{(2)}_{\rm in}$ (left) and
$\widehat{\boldsymbol{D}}^{(2)}_{\rm in}$ (right) at the PL points ($c\not=0)$
against the Legendre case at LGL points ($c=0$).
## 4\. Proof of nonconvergence of $h$-refinement in prolate elements
In a very recent paper [3], Boyd et al. discovered the nonconvergence of
$h$-refinement in prolate-element methods, whose argument was based on the
study of $hp$-PSWF approximation to the trivial function $u(x)=1.$ However,
the theoretical justification for general functions in Sobolev spaces is
lacking. In this section, we derive a $hp$-error bound for a PSWF-projection
and this gives a rigorous proof of the claim in [3]. We also provide more
numerical evidences to illustrate this surprising convergence property.
We first introduce the notation and setting for $hp$-approximation by the
PSWFs. Let $\Omega=(a,b).$ For simplicity, we partition it uniformly into $M$
non-overlapping subintervals, that is,
$\bar{\Omega}=\bigcup_{i=1}^{M}{\bar{I}}_{i},\quad
I_{i}:=(a_{i-1},a_{i}),\quad a_{i}=a+ih,\;\;h=\frac{b-a}{M},\;\;\;\;1\leq
i\leq M.$ (4.1)
Note that the transform between $I_{i}$ and the reference interval $I_{\rm
ref}:=(-1,1)$ is given by
$x=\frac{h}{2}y+\frac{a_{i-1}+a_{i}}{2}=\frac{hy+2a+(2i-1)h}{2},\quad x\in
I_{i},\;\;y\in I_{\rm ref}.$ (4.2)
For any $u(x)$ defined in $\Omega,$ denote
$u|_{x\in I_{i}}=u^{I_{i}}(x)=\hat{u}^{I_{i}}(y),\quad
x=\frac{hy+2a+(2i-1)h}{2}\in I_{i},\;\;\;y\in I_{\rm ref}.$ (4.3)
Let $\hat{\pi}_{N}^{c}$ be the $L^{2}(I_{\rm ref})$-orthogonal projector upon
$V_{N}^{c}={\rm span}\\{\psi_{n}\,:\,0\leq n\leq N\\},$ given by
$(\hat{\pi}_{N}^{c}\hat{u})(y)=\sum_{n=0}^{N}\hat{u}_{n}(c)\psi_{n}(y;c)\;\;\;{\rm
with}\;\;\;\hat{u}_{n}(c)=\int_{I_{\rm ref}}\hat{u}(y)\psi_{n}(y;c)\,dy.$
(4.4)
Define the approximation space
$X_{h,N}^{c}=\big{\\{}v\in
H^{1}(\Omega)\,:\,v|_{I_{i}}(x)=\hat{v}^{I_{i}}(y)\in V_{N}^{c},\;\;1\leq
i\leq M\big{\\}}.$ (4.5)
Let $\boldsymbol{\pi}_{h,N}^{c}\,:\,H^{1}(\Omega)\to X_{h,N}^{c}$ be a
mapping, assembled by
$\big{(}\boldsymbol{\pi}_{h,N}^{c}u\big{)}\big{|}_{I_{i}}(x)=\big{(}\hat{\pi}_{N}^{c}\hat{u}^{I_{i}}\big{)}(y),\quad
1\leq i\leq M,$ (4.6)
where by definition, we have
$\big{(}\boldsymbol{\pi}_{h,N}^{c}u\big{)}\big{|}_{I_{i}}(x)=\sum_{n=0}^{N}\hat{u}_{n}^{I_{i}}(c)\,\psi_{n}(y;c)\;\;\;{\rm
with}\;\;\;\hat{u}_{n}^{I_{i}}(c)=\int_{I_{\rm
ref}}\hat{u}^{I_{i}}(y)\psi_{n}(y;c)\,dy.$ (4.7)
Here, $H^{s}(I)$ with $s>0$ denotes the usual Sobolev space with the norm
$\|\cdot\|_{H^{s}(I)}$ as in Admas [2].
We introduce the broken Sobolev space:
$\widetilde{H}^{\sigma}(a,b)=\big{\\{}u\,:\,u^{I_{i}}\in
H^{\sigma}(I_{i}),\;\;1\leq i\leq M\big{\\}},\;\;\;\sigma\geq 1,$ (4.8)
equipped with the norm and semi-norm
$\|u\|_{\widetilde{H}^{\sigma}(a,b)}=\Big{(}\sum_{i=1}^{M}\|u^{I_{i}}\|^{2}_{H^{\sigma}(I_{i})}\Big{)}^{\frac{1}{2}},\quad|u|_{\widetilde{H}^{\sigma}(a,b)}=\Big{(}\sum_{i=1}^{M}\big{\|}\partial_{x}^{\sigma}u^{I_{i}}\big{\|}^{2}_{L^{2}(I_{i})}\Big{)}^{\frac{1}{2}}.$
The $hp$-approximability of $\boldsymbol{\pi}_{h,N}^{c}u$ to $u$ is stated in
the following theorem.
###### Theorem 4.1.
Let $\boldsymbol{\pi}_{h,N}^{c}$ be the projector defined as in (4.6). For any
constant $q_{*}<1,$ if
$\frac{c}{\sqrt{\chi_{N}}}\leq\frac{q_{*}}{\sqrt[6]{2}}\approx 0.8909q_{*},$
(4.9)
then for any $u\in\widetilde{H}^{\sigma}(a,b)$ with $\sigma\geq 1,$ we have
$\|\boldsymbol{\pi}_{h,N}^{c}u-u\|_{L^{2}(a,b)}\leq
D\Big{\\{}\sqrt{N}\Big{(}\frac{h}{N}\Big{)}^{\sigma}|u|_{\widetilde{H}^{\sigma}(a,b)}+\frac{1}{\sqrt{\delta\ln(1/q_{*})}}(q_{*})^{\delta
N}\|u\|_{L^{2}(a,b)}\Big{\\}},$ (4.10)
where $D$ and $\delta$ are positive constants independent of $u,N$ and $c.$
To be not distracted from the main result, we postpone its proof to Appendix
B.
###### Remark 4.1.
Some remarks are in orders.
* •
Observe from (4.10) that the second term of the upper bound is independent of
$h.$ This implies that for fixed $N,$ the refinement of $h$ does not lead to
any convergence in $h.$ For the trivial example, $u(x)=1,$ considered in [3],
the first term of the upper bound vanishes, so (4.10) indicates non
$h$-convergence, but exponential convergence in $N$.
* •
This should be in distinct contrast with the Legendre approximation (see e.g.,
[6, 14]), for which we have
$\big{\|}\boldsymbol{\pi}_{h,N}^{0}u-u\big{\|}_{L^{2}(a,b)}\leq
D\Big{(}\frac{h}{N}\Big{)}^{\sigma}|u|_{\widetilde{H}^{\sigma}(a,b)}.$
* •
For fixed $c,$ the estimate in (4.10) appears sub-optimal due to the factor
$\sqrt{N},$ which can be improved to the optimal order by applying [27,
Theorem 3.3] to (B.1).
We next provide some numerical evidences. Consider the prolate-element method
for the equation:
$\begin{split}&-(1+x^{2})u^{\prime\prime}(x)-(2x+\sin
x)u^{\prime}(x)+u(x)=f(x),\quad x\in(0,1),\\\ &u(0)=0,\quad
u(1)=u_{1},\end{split}$ (4.11)
where $u_{1}$ and $f(x)$ are computed from the exact solution:
$u(x)=(x+1)^{\alpha}\sin({\pi x}/{2})$ with $\alpha=13/3.$ The prolate-element
scheme is based on swapping the points, cardinal basis and differentiation
matrices of the standard Legendre spectral-element method (see e.g., [20, 5]).
Figure 4.1. Illustration of nonconvergence of $h$-refinement in prolate
elements. Maximum point-wise errors with $N=2$, $c=0,0.5$ (left), and with
$N=4$, $c=0,1$ (right).
In Figure 4.1, we plot the maximum point-wise errors against $h$ with fixed
$N=2,4$ for the prolate and Legendre spectral-element methods. It clearly
shows that the prolate elements do not have $h$-refinement convergence, while
its counterpart possesses.
We tabulate in Table 4.1 the maximum point-wise errors of two methods with
various $h,N.$ For fixed $N,$ nonconvergence is observed by refining $h$ for
the prolate-element method, as opposite to the Legendre spectral-element
scheme. Benefited from $h$-convergence, the Legendre approach appears more
accurate for small $h$ and fixed $N.$ However, from the viewpoint of
$p$-version (e.g., $h=1/2$), the prolate-element method slightly outperforms
its counterpart.
Table 4.1. Performance of the prolate-element method with $c=N/4$ and the Legendre spectral-element method. $h$ $N(c\not=0)$ | 2 | 3 | 4 | 6 | 8 | 16
---|---|---|---|---|---|---
$1/2$ | 8.98E-02 | 4.76E-03 | 1.98E-04 | 1.97E-06 | 4.91E-08 | 1.03E-13
$1/4$ | 6.90E-03 | 4.32E-04 | 7.27E-05 | 1.84E-06 | 4.77E-08 | 7.60E-12
$1/8$ | 2.80E-03 | 3.52E-04 | 4.47E-05 | 1.12E-06 | 2.94E-08 | 1.27E-12
$1/16$ | 3.30E-03 | 3.93E-04 | 3.21E-05 | 8.58E-07 | 2.31E-08 | 3.16E-12
$h$ $N(c=0)$ | 2 | 3 | 4 | 6 | 8 | 16
$1/2$ | 5.97E-01 | 7.17E-03 | 6.60E-04 | 1.35E-06 | 3.35E-09 | 5.91E-12
$1/4$ | 3.79E-02 | 3.00E-04 | 1.08E-05 | 5.89E-09 | 7.99E-12 | 6.26E-12
$1/8$ | 2.37E-03 | 1.06E-05 | 1.71E-07 | 8.98E-11 | 7.29E-12 | 1.52E-11
$1/16$ | 1.48E-04 | 3.45E-07 | 2.68E-09 | 4.24E-11 | 2.22E-11 | 3.26E-11
## 5\. Well-conditioned prolate-collocation methods
In this section, we propose a well-conditioned prolate-collocation methods for
second-order boundary value problems. The essential piece of the puzzle is to
construct a new basis of dual nature. Firstly, this basis generates a matrix,
denoted by $\boldsymbol{B}_{\rm in},$ such that the eigenvalues of
$\boldsymbol{B}_{\rm in}\boldsymbol{D}_{\rm in}^{(2)}$ and
$\boldsymbol{B}_{\rm in}\widehat{\boldsymbol{D}}_{\rm in}^{(2)}$ are nearly
concentrated around one. In other words, the matrix $\boldsymbol{B}_{\rm in}$
is approximately the “inverse” of the second-order differentiation matrix.
Therefore, the matrix $\boldsymbol{B}_{\rm in}$ is a nearly optimal
preconditioner, leading to a well-conditioned prolate-collocation linear
system. On the other hand, using the new basis, the matrix of the highest
derivative in the linear system of the usual collocation scheme is identity
and the condition number of the whole linear system is independent of $N$ and
$c.$ The idea can be extended to prolate-collocation methods for the first-
order and higher-order equations.
### 5.1. A new basis
Let $\\{\beta_{k}(x):=\beta_{k}(x;c)\\}_{k=0}^{N}$ be a set of functions in an
$(N+1)$-dimensional space to be specified shortly, which satisfies the
conditions:
$\begin{split}&\beta_{0}(-1)=1,\quad\beta_{0}^{\prime\prime}(x_{j})=0,\;\;1\leq
j\leq N-1,\quad\beta_{0}(1)=0;\\\
&\beta_{k}(-1)=0,\quad\beta_{k}^{\prime\prime}(x_{j})=\delta_{jk},\quad\beta_{k}(1)=0,\quad
1\leq j,k\leq N-1;\\\
&\beta_{N}(-1)=0,\quad\beta_{N}^{\prime\prime}(x_{j})=0,\;\;1\leq j\leq
N-1,\quad\beta_{N}(1)=1,\end{split}$ (5.1)
where $\\{x_{j}\\}$ are the PL points.
If we look for $\\{\beta_{k}\\}_{k=0}^{N}\subseteq V_{N}^{c}={\rm
span}\big{\\{}\psi_{n}\,:\,0\leq n\leq N\big{\\}},$ then (5.1) is associated
with a generalized Birkhoff interpolation problem: Given $u\in C^{2}(-1,1),$
find $p\in V_{N}^{c}$ such that
$p(-1)=u(-1);\quad p^{\prime\prime}(x_{j})=u^{\prime\prime}(x_{j});\;\;\;1\leq
j\leq N-1,\quad p(1)=u(1).$ (5.2)
We can express the interpolant as
$p(x)=u(-1)\beta_{0}(x)+\sum_{k=1}^{N-1}u^{\prime\prime}(x_{k})\beta_{k}(x)+u(1)\beta_{N}(x).$
(5.3)
The basis $\\{\beta_{k}\\}$ for (5.2) can be computed by writing
$\beta_{k}(x)=\sum_{k=0}^{N}\alpha_{nk}\psi_{n}(x),$ and solving the
coefficients by the interpolation conditions. However, this process requires
the inversion of a matrix as ill-conditioned as $\boldsymbol{\Psi}^{(2)}$ and
$\boldsymbol{D}^{(2)},$ which is apparently unstable even for slightly large
$N.$ However, this approach works for the Legendre and Chebyshev cases (see
[28]), thanks to some formulas (but only available for orthogonal
polynomials).
###### Remark 5.1.
The Birkhoff interpolation is typically considered in the polynomial setting
(see [18, 9, 34]). In contrast with the Lagrange and Hermite interpolation, it
does not interpolate the function and its derivative values consecutively at
every point. For example, in (5.2), the data $u(x_{j})$ and
$u^{\prime}(x_{j})$ are not interpolated at the interior point $x_{j}$.
In what follows, we search for $\\{\beta_{k}\\}$ and $p$ in a different finite
dimensional space other than $V_{N}^{c}$, which allows for stable computation
of the new basis. More precisely, we set
$\beta_{0}(x)=\frac{1-x}{2},\quad\beta_{N}(x)=\frac{1+x}{2},$ (5.4)
and for $1\leq k\leq N-1,$ we look for
$\beta_{k}\in W_{N}^{c,0}:={\rm
span}\big{\\{}\phi_{n}:\phi_{n}^{\prime\prime}(x)=\psi_{n}(x)\;{\rm
with}\;\phi_{n}(\pm 1)=0,\;0\leq n\leq N-2\big{\\}},$ (5.5)
which therefore satisfy $\beta_{k}(\pm 1)=0$ in (5.1). Solving the ordinary
differential equation in (5.5) directly leads to
$\phi_{n}(x)=x\int_{-1}^{x}\psi_{n}(t)\,dt-\int_{-1}^{x}t\,\psi_{n}(t)\,dt+\frac{1+x}{2}\int_{-1}^{1}(t-1)\psi_{n}(t)\,dt.$
(5.6)
Then we compute $\\{\beta_{k}\\}_{k=1}^{N-1},$ by writing
$\beta_{k}(x)=\sum_{n=0}^{N-2}\alpha_{nk}\phi_{n}(x),\;\;\;{\rm
so}\;\;\;\beta_{k}^{\prime\prime}(x)=\sum_{n=0}^{N-2}\alpha_{nk}\psi_{n}(x).$
(5.7)
Thus we can find the coefficients $\\{\alpha_{nk}\\}$ by
$\beta_{k}^{\prime\prime}(x_{j})=\delta_{jk}$ with $1\leq k,j\leq N-1$, that
is,
$\boldsymbol{A}=\boldsymbol{\bar{\Psi}}^{-1}\;\;\;\;{\rm
where}\;\;\;\;\boldsymbol{A}_{nk}=\alpha_{nk},\;\;\;\boldsymbol{\bar{\Psi}}_{jn}=\psi_{n}(x_{j}),$
(5.8)
for $1\leq j,k\leq N-1$ and $0\leq n\leq N-2.$
###### Remark 5.2.
Like the cardinal basis in (2.19), this process only involves inverting a
matrix of PSWF function values, rather than derivative values (if one requires
$\beta_{k}\in V_{N}^{c}$). Hence, the operations are very stable even for very
large $N.$
Introduce the matrix $\boldsymbol{B}$ with entries
$\boldsymbol{B}_{jk}=\beta_{k}(x_{j})$ for $0\leq k,j\leq N,$ and let
$\boldsymbol{B}_{\rm in}$ be the $(N-1)^{2}$ matrix obtained by deleting the
first and last rows and columns from $\boldsymbol{B}.$ Observe from
(5.5)-(5.6) that $\boldsymbol{B}_{\rm in}$ is generated from integration of
PSWFs, which is an “inverse process” of the spectral differentiation in the
sense of (5.10)-(5.11) below. For large $N$ and $c$ satisfying (3.2), we infer
from the approximability of the cardinal basis that
$\beta_{k}^{\prime\prime}(x)\approx\sum_{p=1}^{N-1}\beta_{k}(x_{p})h_{p}^{\prime\prime}(x),\quad
1\leq k\leq N-1,$ (5.9)
where the equality does not hold as $\beta_{k}\not\in V_{N}^{c}.$ Since
$\beta_{k}(x_{j})=\delta_{jk}$ (see (5.1)), letting $x=x_{j}$ in (5.9) leads
to
$\boldsymbol{I}_{N-1}\approx\boldsymbol{D}_{\rm in}^{(2)}\boldsymbol{B}_{\rm
in},$ (5.10)
where $\boldsymbol{I}_{N-1}$ is an $(N-1)^{2}$ identity matrix. Similarly, by
(5.3),
$h_{j}(x)\approx\sum_{k=1}^{N-1}h_{j}^{\prime\prime}(x_{k})\beta_{k}(x),\quad
1\leq k\leq N-1,$
which implies
$\boldsymbol{I}_{N-1}\approx\boldsymbol{B}_{\rm in}\boldsymbol{D}_{\rm
in}^{(2)}.$ (5.11)
###### Remark 5.3.
The above argument also applies to the cardinal basis $\\{l_{j}\\}$ defined in
(2.22), so one can replace $\boldsymbol{D}_{\rm in}^{(2)}$ in (5.10) and
(5.11) by $\widehat{\boldsymbol{D}}_{\rm in}^{(2)}$.
As a numerical illustration, we depict in Figure 5.1 the distribution of the
largest and smallest eigenvalues of $\boldsymbol{B}_{\rm
in}\boldsymbol{D}_{\rm in}^{(2)}$ and $\boldsymbol{B}_{\rm
in}\widehat{\boldsymbol{D}}_{\rm in}^{(2)}$ at the PL points. We see that all
their eigenvalues for various $N$ with $c=N/2$ are confined in $[\lambda_{\rm
min},\lambda_{\rm max}],$ which are concentrated around one for slightly large
$N.$
Figure 5.1. Distribution of the largest and smallest eigenvalues of
${\boldsymbol{B}}_{\rm in}{\boldsymbol{D}}^{(2)}_{\rm in}$ (left) and
${\boldsymbol{B}}_{\rm in}\widehat{\boldsymbol{D}}^{(2)}_{\rm in}$ (right) for
various $N\in[4,218]$ and $c=N/2.$
### 5.2. Well-conditioned prolate-collocation methods
To demonstrate the idea, we consider the second-order variable coefficient
problem:
$u^{\prime\prime}(x)+p(x)u^{\prime}(x)+q(x)u(x)=f(x),\quad x\in I=(-1,1);\quad
u(\pm 1)=u_{\pm},$ (5.12)
where $p,q$ and $f$ are continuous functions. Let $\\{x_{j}\\}_{j=0}^{N}$ be
the PL points as before. Then the usual collocation scheme is: Find $u_{N}\in
V_{N}^{c}$ such that
$u^{\prime\prime}_{N}(x_{j})+p(x_{j})u^{\prime}_{N}(x_{j})+q(x_{j})u_{N}(x_{j})=f(x_{j}),\quad
1\leq j\leq N-1;\quad u_{N}(\pm 1)=u_{\pm}.$ (5.13)
Under the cardinal basis $\\{h_{k}\\}$ defined in (2.18)-(2.19), the prolate-
collocation system reads
$\big{(}\boldsymbol{D}^{(2)}_{\rm
in}+\boldsymbol{\Lambda}_{p}\boldsymbol{D}_{\rm
in}^{(1)}+\boldsymbol{\Lambda}_{q}\big{)}\boldsymbol{u}=\boldsymbol{g},$
(5.14)
where $\boldsymbol{\Lambda}_{p}$ is a diagonal matrix of entries
$\\{p(x_{j})\\}_{j=1}^{N-1}$ (and likewise for $\boldsymbol{\Lambda}_{q}$),
the unknown vector $\boldsymbol{u}=(u_{N}(x_{1}),\cdots,u_{N}(x_{N-1}))^{t},$
and $\boldsymbol{g}$ is the vector with elements
$\boldsymbol{g}_{j}=f(x_{j})-u_{-}(h_{0}^{\prime\prime}(x_{j})+p(x_{j})h_{0}^{\prime}(x_{j}))-u_{+}(h_{N}^{\prime\prime}(x_{j})+p(x_{j})h_{N}^{\prime}(x_{j})),\;\;\;1\leq
j\leq N-1.$
It is known that the system (5.14) is ill-conditioned.
Thanks to (5.11), we precondition the system (5.14), leading to
$\boldsymbol{B}_{\rm in}\big{(}\boldsymbol{D}^{(2)}_{\rm
in}+\boldsymbol{\Lambda}_{p}\boldsymbol{D}_{\rm
in}^{(1)}+\boldsymbol{\Lambda}_{q}\big{)}\boldsymbol{u}=\boldsymbol{B}_{\rm
in}\boldsymbol{g},$ (5.15)
which is well-conditioned (see e.g., Table 5.1).
On the other hand, one can directly use $\\{\beta_{j}\\}$ as a basis.
Different from (5.13), the collocation scheme becomes: Find $v_{N}\in
W_{N}^{c}={\rm span}\big{\\{}\beta_{k}\,:\,0\leq k\leq N\big{\\}}$ such that
$v^{\prime\prime}_{N}(x_{j})+p(x_{j})v^{\prime}_{N}(x_{j})+q(x_{j})v_{N}(x_{j})=f(x_{j}),\quad
1\leq j\leq N-1;\quad v_{N}(\pm 1)=u_{\pm}.$ (5.16)
By writing
$v_{N}(x)=u_{-}\beta_{0}(x)+\sum_{k=1}^{N-1}w_{k}\beta_{k}(x)+u_{+}\beta_{N}(x),$
(5.17)
the collocation system becomes
$\big{(}\boldsymbol{I}_{N-1}+\boldsymbol{\Lambda}_{p}\boldsymbol{B}_{\rm
in}^{(1)}+\boldsymbol{\Lambda}_{q}\boldsymbol{B}_{\rm
in}\big{)}\boldsymbol{w}=\boldsymbol{h},$ (5.18)
where $\boldsymbol{w}$ is the vector of unknowns and $\boldsymbol{h}$ has the
components
$\boldsymbol{h}_{j}=f(x_{j})-(p(x_{j})+x_{j}q(x_{j}))\frac{u_{+}-u_{-}}{2}-q(x_{j})\frac{u_{+}+u_{-}}{2},\quad
1\leq j\leq N-1.$
Finally, we recover
$\boldsymbol{v}=(v_{N}(x_{1}),\cdots,v_{N}(x_{N-1}))^{t}$—the approximation of
the solution, from (5.17):
$\boldsymbol{v}=\boldsymbol{B}_{\rm
in}\boldsymbol{w}+u_{-}\boldsymbol{b}_{0}+u_{+}\boldsymbol{b}_{N},$ (5.19)
where $\boldsymbol{b}_{0}=(\beta_{0}(x_{1}),\cdots,\beta_{0}(x_{N-1}))^{t}$
and $\boldsymbol{b}_{N}=(\beta_{N}(x_{1}),\cdots,\beta_{N}(x_{N-1}))^{t}$ (cf.
(5.4)).
###### Remark 5.4.
Compared with (5.15), the system (5.18) does not involve differentiation
matrices. However, the unknowns are not physical values, so an additional step
(5.19) is needed to recover the physical values.
###### Remark 5.5.
Similar to the spectral-Galerkin method in [22], an essential idea is to
construct an appropriate basis so that the matrix of the highest derivative
becomes diagonal or identity. We refer to [23, P. 160] for the proof of the
well-conditioning of such spectral-Galerkin schemes. However, a rigorous
justification in this context appears challenging. Here, we just provide some
intuition for (5.12) with $p=0$ and $q=q_{0}$ (a constant). Let $\lambda_{\rm
min}$ and $\lambda_{\rm max}$ be the minimum and maximum eigenvalues of
$\boldsymbol{D}^{(2)}_{\rm in}.$ By (5.11), the eigenvalues of
$\boldsymbol{B}_{\rm in}$ in magnitude are roughly confined in $[|\lambda_{\rm
max}|^{-1},|\lambda_{\rm min}|^{-1}].$ As a result, the the eigenvalues of
$\boldsymbol{I}_{N-1}+q_{0}\boldsymbol{B}_{\rm in}$ in magnitude approximately
fall into the range $[1+q_{0}|\lambda_{\rm max}|^{-1},1+q_{0}|\lambda_{\rm
min}|^{-1}].$ Note that for large $N,$ $|\lambda_{\rm min}|$ behaves like a
constant, while $|\lambda_{\rm max}|$ grows like $O(N^{4})$ (see Figure 3.3).
This implies $\boldsymbol{I}_{N-1}+q_{0}\boldsymbol{B}_{\rm in}$ is well-
conditioned.
We now provide some numerical examples, and compare the condition numbers
between (5.14), (5.15) and (5.18). Consider
$u^{\prime\prime}(x)-xu^{\prime}(x)-u(x)=f(x)=\begin{cases}0,\quad&-1<x<0,\\\
-3x^{2}/2,\quad&0\leq x<1,\end{cases}\\\ $ (5.20)
with the exact solution
$u(x)=\begin{cases}\exp(\frac{x^{2}}{2}+1)+\exp(\frac{x^{2}}{2}),\quad&-1\leq
x<0,\\\\[5.69054pt] \exp(\frac{x^{2}}{2}+1)+\frac{x^{2}}{2}+1,\quad&0\leq
x\leq 1.\end{cases}\\\ $ (5.21)
Note that $f\in C^{1}(\bar{I})$ and $u\in C^{3}(\bar{I})$. The systems (5.14),
(5.15) and (5.18) are neither sparse nor symmetric, so we solve them by the
iterative method—biconjugated gradient stabilized method. In Table 5.1, we
tabulate the condition numbers, iteration steps, and maximum point-wise errors
between the numerical and exact solutions obtained from the prolate-
collocation scheme (5.14) (PCOL), the preconditioned scheme (5.15) (P-PCOL),
and the new collocation scheme (5.18) (N-PCOL), respectively. Here, we choose
$c=N/2.$ In Figure 5.2, we plot the maximum point-wise errors for three
schemes.
Table 5.1. Performance of PCOL, P-PCOL and N-COL methods. | PCOL | P-PCOL | N-PCOL
---|---|---|---
$N$ | Cond. | Errors | Steps | Cond. | Errors | Steps | Cond. | Errors | Steps
4 | 6.64E+00 | 1.40E-02 | 3 | 1.24 | 1.40E-02 | 3 | 1.25 | 7.71E-03 | 3
8 | 4.58E+01 | 1.29E-04 | 8 | 1.32 | 1.29E-04 | 6 | 1.59 | 1.03E-04 | 6
16 | 5.32E+02 | 6.78E-06 | 23 | 1.33 | 6.78E-06 | 6 | 1.74 | 6.78E-06 | 7
32 | 7.61E+03 | 4.80E-07 | 69 | 1.33 | 4.91E-07 | 6 | 1.82 | 4.80E-07 | 7
64 | 1.16E+05 | 3.20E-08 | 271 | 1.33 | 3.20E-08 | 6 | 1.86 | 3.20E-08 | 7
128 | 1.82E+06 | 2.14E-09 | 1037 | 1.33 | 2.07E-09 | 6 | 1.38 | 2.07E-09 | 7
256 | 2.88E+07 | 3.29E-08 | 6038 | 1.33 | 1.32E-10 | 6 | 1.88 | 1.32E-10 | 7
512 | 4.60E+08 | 8.65E-04 | 65791 | 1.33 | 1.21E-11 | 6 | 1.89 | 8.35E-12 | 7
Figure 5.2. Maximum point-wise errors for PCOL, P-PCOL and N-PCOL methods. The
slope of two lines is approximately $-3.95.$
We see that the last two schemes are well-conditioned and the iterative solver
converges in a few steps, so they significantly outperform the usual prolate-
collocation method using the cardinal basis (2.18)-(2.19). Note that the exact
solution $u\in H^{4-\epsilon}(I)$ for some $\epsilon>0,$ so the slope of the
line is approximately $-3.95$ as expected.
### 5.3. A collocation-based $p$-version prolate-element method
As already discussed, prolate-element method does not possess $h$-refinement
convergence, and the Galerkin method is less attractive due to the lack of
accurate quadrature rules for products of PSWFs. We therefore propose a
$p$-version prolate-element method using the collocation formulation and the
new basis $\\{\beta_{j}\\}$. It will be particularly applied to problems with
discontinuous variable coefficients, e.g., the Helmholtz equations with high
wave numbers in heterogeneous media.
To fix the idea, we consider the model problem:
$\begin{split}&L[u](x):=-(p(x)u^{\prime}(x))^{\prime}+q(x)u(x)=f(x),\quad
x\in\Omega=(a,b);\\\ &u(a)=u_{a},\;\;u(b)=u_{b}.\end{split}$ (5.22)
We adopt the same setting as in (4.1)-(4.3). Here, the interval $\Omega$ is
uniformly partitioned into $M$ non-overlapping subintervals
$\\{I_{i}=(a_{i-1},a_{i})\\}_{i=1}^{M}.$ Recall that the transform between
$I_{i}$ and the reference interval $I_{\rm ref}=(-1,1)$ is given by
$x=\frac{h}{2}y+\frac{a_{i-1}+a_{i}}{2}=\frac{hy+2a+(2i-1)h}{2},\quad x\in
I_{i},\;\;y\in I_{\rm ref}.$ (5.23)
As before, let $W_{N}^{c}={\rm span}\\{\beta_{k}\,:\,0\leq k\leq N\\}.$
Without loss of generality, assume that the same number of points will be used
for each subinterval. Introduce the approximation space
$Y_{h,N}^{c}:=\big{\\{}u\in H^{1}(\Omega)\,:\,u(x)|_{x\in
I_{i}}=u^{I_{i}}(x)=\hat{u}^{I_{i}}(y)|_{y\in I_{\rm ref}}\in
W_{N}^{c},\;0\leq i\leq M\big{\\}}.$ (5.24)
Define
$\phi_{k}^{I_{i}}(x)=\begin{cases}\beta_{k}(y),\quad&x=(hy+2a+(2i-1)h)/2\in
I_{i},\\\\[2.0pt] 0,\quad&{\rm otherwise},\end{cases}$ (5.25)
and at the adjoined points $a_{i},1\leq i\leq M-1,$
$\varphi^{a_{i}}(x)=\begin{cases}(1+y)/2,\quad&x=(hy+2a+(2i-1)h)/2\in
I_{i},\\\\[2.0pt] (1-y)/2,\quad&x=(hy+2a+(2i+1)h)/2\in I_{i+1},\\\\[2.0pt]
0,\quad&{\rm otherwise}.\end{cases}$ (5.26)
Then we have
$Y_{h,N}^{c}:={\rm
span}\Big{\\{}\big{\\{}\phi^{I_{1}}_{k}\big{\\}}_{k=0}^{N-1}\,,\,\big{\\{}\phi^{I_{2}}_{k}\big{\\}}_{k=1}^{N-1}\,,\,\cdots,\,\big{\\{}\phi^{I_{M-1}}_{k}\big{\\}}_{k=1}^{N-1},\,\big{\\{}\phi^{I_{M}}_{k}\big{\\}}_{k=1}^{N};\,\big{\\{}\varphi^{a_{i}}\big{\\}}_{i=1}^{M-1}\Big{\\}},$
(5.27)
and the dimension of $Y_{h,N}^{c}$ is $MN+1.$
Let $\\{y_{j}\\}$ be the PL points in the reference interval $I_{\rm ref}.$
Then the grids on each $I_{i}$ are given by
$x_{j}^{I_{i}}=\frac{hy_{j}+2a+(2i-1)h}{2},\quad 0\leq j\leq N,\;\;1\leq i\leq
M.$ (5.28)
The prolate-element method for (5.22) is: Find $v\in Y_{h,N}^{c}$ such that
$v(a)=u_{a},$ $v(b)=u_{b},$ and
$L[v](x_{j}^{I_{i}})=f(x_{j}^{I_{i}}),\quad 1\leq j\leq N-1,\;\;1\leq i\leq
M,$ (5.29)
and at the joint points $a_{i},$
$\int_{a}^{b}\big{[}p(x)v^{\prime}(x)(\varphi^{a_{i}}(x))^{\prime}+q(x)v(x)\varphi^{a_{i}}(x)\big{]}\,dx=\int_{a}^{b}f(x)\varphi^{a_{i}}(x)\,dx,\quad
1\leq i\leq M-1.$ (5.30)
We see that the scheme is collocated at the interior points in each
subinterval, and at the joint points, it is built upon the Galerkin-
formulation for ease of imposing the continuity across elements. As shown in
Subsection 5.2, the interior solvers (5.29) are well-conditioned, and the
differentiation matrices are not involved.
We next present some numerical results to show the performance of the new
scheme. We focus on the Helmholtz equation with high wave number in a
heterogeneous medium:
$\begin{split}&(c^{2}(x)u^{\prime}(x))^{\prime}+k^{2}n^{2}(x)u(x)=0,\quad
x\in\Omega=(a,b);\\\ &u(a)=u_{a},\quad(cu^{\prime}-{\rm{i}}knu)(b)=0,\\\
&u,\;\;c^{2}u\;\;\text{are continuous on}\;\;\Omega,\end{split}$ (5.31)
where the wave number $k>0,$ and $c(x),n(x)$ are piecewise smooth such that
$0<c_{0}\leq c(x)\leq c_{1},\quad 0<n_{0}\leq n(x)\leq n_{1}.$
Note that $c(x),n(x)$ represent the local speed of sound and the index of
refraction in a heterogeneous medium, respectively.
In the first example, we choose $\Omega=(0,1),$ $n(x)=1$ and $c(x)$ to be
piecewise constant:
$c(x)=\begin{cases}2,\quad&0<x<{1}/{2},\\\\[2.0pt]
1,\quad&1/2<x<1.\end{cases}$
Then the problem (5.31) admits the exact solution (cf. [12]):
$u(x)=\begin{cases}\big{(}3\exp(\frac{{\rm i}k(1+2x)}{4})+\exp(\frac{{\rm
i}k(3-2x)}{4})\big{)}/4,\quad&0<x<{1}/{2},\\\\[2.0pt] \exp({\rm
i}kx),\quad&1/2<x<1.\end{cases}$ (5.32)
In this case, we partition $\Omega=(0,1)$ into two subintervals
$I_{1}=(0,1/2)$ and $I_{2}=(1/2,1).$
In Figure 5.3, we plot the maximum point-wise errors for the usual Legendre
spectral-element method and the new $p$-version prolate-element method, where
$(c,N)$ is paired up by the approximate Kong-Rokhlin’s rule with
$\varepsilon=10^{-14}$ and samples of $c$ in $[2,52].$ From Figure 5.3, a much
rapid convergence rate of the new approach is observed for high wave numbers.
Figure 5.3. Maximum point-wise errors of Legendre spectral-element and new
prolate-element methods for the Helmholtz equation with exact solution (5.32).
Left: $k=60$ and right: $k=100$.
As a second example, we take $\Omega=(0,1),$ $f(x)=1$ and consider the problem
(5.31) with piecewise smooth coefficients (cf. [12]):
$c(x)=\begin{cases}1+x^{2},\quad&0<x<0.25,\\\\[2.0pt]
1-x^{2},\quad&0.25<x<0.5,\\\\[2.0pt] 1,\quad&0.5<x<1,\end{cases}\quad
n(x)=\begin{cases}1.75+x,\quad&0<x<0.25,\\\\[2.0pt]
1.25-x,\quad&0.25<x<0.5,\\\\[2.0pt] 2,\quad&0.5<x<1.\end{cases}$
Naturally, we partition $\Omega$ into four subintervals of equal length. In
this case, we do not have the explicit exact solution, so we generate a
reference “exact” solution using very refine grids by the new prolate-element
method $(c,N)=(177,144)$ (paired up by the approximate Kong-Rokhlin’s rule
again). In Figure 5.4, we plot the real and image parts of the “exact”
solution (where $k=160$) against the numerical solution obtained by very
coarse grids with $(c,N)=(36,48),$ which approximates the highly oscillatory
solution with an accuracy about $10^{-6}$.
Figure 5.4. Real part (left) and imaginary part (right) of the reference
“exact” solution $u$ computed by $(c,N)=(177,144)$ and $k=160,$ against the
numerical solution $u_{N}$ of the prolate-element method with $(c,N)=(36,48).$
The maximum point-wise error is $1.19E-06.$
In Figure 5.5, we make a comparison of convergence behavior similar to that in
(5.3). Here, we sample $c\in[4,52].$ One again, we observe significantly
faster convergence rate for the new approach under the approximate Kong-
Rokhlin’s rule (with $\varepsilon=10^{-14}$) of selecting $(c,N).$
Figure 5.5. Maximum point-wise errors of Legendre spectral-element and new
prolate-element methods. Left: $k=100$ and right: $k=160$.
Concluding remarks
In this paper, we provided a rigorous proof for nonconvergence of
$h$-refinement in prolate elements, which was claimed very recently by Boyd et
al. [3]. We further proposed well-conditioned collocation and collocation-
based $p$-version prolate-element methods using a new PSWF-basis. We
demonstrated that the new approach with the Kong-Rokhlin’s rule of selecting
$(c,N)$ significantly outperformed the Legendre polynomial-based method in
particular when the underlying solution is bandlimited. Advantages of our
proposals were confirmed in solving the Helmholtz equations with high wave
numbers in heterogeneous media.
## Appendix A Formulas for differentiation matrices
To this end, we derive the explicit formulas involving only function values
$\\{\psi_{N}(x_{j})\\}_{j=0}^{N}$ for computing the entries of the first-order
and second-order differentiation matrices generated from the cardinal basis
(2.22).
A direct derivation from (2.22) leads to
$l_{k}^{\prime}(x_{j})=\begin{cases}\dfrac{1}{x_{j}-x_{k}}\dfrac{s^{\prime}(x_{j})}{s^{\prime}(x_{k})},\quad&{\rm
if}\;\;j\not=k,\\\\[10.0pt]
\dfrac{s^{\prime\prime}(x_{k})}{2s^{\prime}(x_{k})},\quad&{\rm
if}\;\;j=k,\end{cases}$ (A.1)
where $s(x)=(1-x^{2})\psi_{N}^{\prime}(x).$ By (2.1),
$s^{\prime}(x)=(c^{2}x^{2}-\chi_{N})\psi_{N}(x),\quad
s^{\prime\prime}(x)=2c^{2}x\,\psi_{N}(x)+(c^{2}x^{2}-\chi_{N})\psi_{N}^{\prime}(x).$
(A.2)
As $\\{x_{k}\\}_{k=1}^{N-1}$ are zeros of $\psi_{N}^{\prime}(x),$ we have
$s^{\prime\prime}(x_{k})=2c^{2}x_{k}\,\psi_{N}(x_{k}),\quad 1\leq k\leq N-1.$
(A.3)
Again by (2.1),
$\psi_{N}^{\prime}(-1)=-\frac{1}{2}\big{(}\chi_{N}-c^{2}\big{)}\psi_{N}(-1),\quad\psi_{N}^{\prime}(1)=\frac{1}{2}\big{(}\chi_{N}-c^{2}\big{)}\psi_{N}(1),$
(A.4)
which, together with (A.2), implies
$s^{\prime\prime}(-1)=\big{(}-2c^{2}+(c^{2}-\chi_{N})^{2}/2\big{)}\psi_{N}(-1),\quad
s^{\prime\prime}(1)=\big{(}2c^{2}-(c^{2}-\chi_{N})^{2}/2\big{)}\psi_{N}(1).$
(A.5)
Then, (A.1) can be computed by
$l_{k}^{\prime}(x_{j})=\begin{cases}-\dfrac{q^{2}}{q^{2}-1}+\dfrac{\chi_{N}}{4}(q^{2}-1),\quad&{\rm
if}\;\;j=k=0,\\\\[10.0pt]
\dfrac{1}{x_{j}-x_{k}}\,\dfrac{q^{2}x_{j}^{2}-1}{q^{2}x_{k}^{2}-1}\,\dfrac{\psi_{N}(x_{j})}{\psi_{N}(x_{k})},\quad&{\rm
if}\;\;j\not=k,\;\;0\leq j,k\leq N,\\\\[10.0pt]
\dfrac{q^{2}x_{k}}{q^{2}x_{k}^{2}-1},\quad&{\rm if}\;\;1\leq j=k\leq
N-1,\\\\[10.0pt]
\dfrac{q^{2}}{q^{2}-1}-\dfrac{\chi_{N}}{4}(q^{2}-1),\quad&{\rm
if}\;\;j=k=N,\end{cases}$ (A.6)
where $q=c/\sqrt{\chi_{N}}.$
We now compute the entries of the second-order differentiation matrix. A
direct differentiation of $s(x)=s^{\prime}(x_{k})(x-x_{k})l_{k}(x)$ (cf.
(2.22)) yields
$s^{\prime\prime}(x)=s^{\prime}(x_{k})(x-x_{k})l_{k}^{\prime\prime}(x)+2s^{\prime}(x_{k})l_{k}^{\prime}(x).$
(A.7)
Therefore, for $j\not=k,$
$l_{k}^{\prime\prime}(x_{j})=\frac{1}{x_{j}-x_{k}}\Big{\\{}\frac{s^{\prime\prime}(x_{j})}{s^{\prime}(x_{k})}-2l_{k}^{\prime}(x_{j})\Big{\\}},$
(A.8)
so the off-diagonal entries of $\widehat{\boldsymbol{D}}^{(2)}$ can be
computed from (A.2)–(A.6).
It remains to compute diagonal entries of $\widehat{\boldsymbol{D}}^{(2)}.$
Differentiating (A.7) and letting $x=x_{k},$ gives
$l_{k}^{\prime\prime}(x_{k})=\frac{s^{\prime\prime\prime}(x_{k})}{3s^{\prime}(x_{k})},\quad
0\leq k\leq N.$
By (A.2),
$s^{\prime\prime\prime}(x)=(c^{2}x^{2}-\chi_{N})\psi_{N}^{\prime\prime}(x)+4c^{2}x\psi_{N}^{\prime}(x)+2c^{2}\psi_{N}(x).$
(A.9)
For $1\leq k\leq N-1,$ we find from (2.1) and the fact
$\psi_{N}^{\prime}(x_{k})=0$ that
$\psi^{\prime\prime}_{N}(x_{k})={\frac{c^{2}x_{k}^{2}-\chi_{N}}{1-x_{k}^{2}}}\psi_{N}(x_{k}),\;\;{\rm
so}\;\;s^{\prime\prime\prime}(x_{k})=\Big{\\{}2c^{2}+\frac{(c^{2}x_{k}^{2}-\chi_{N})^{2}}{1-x_{k}^{2}}\Big{\\}}\psi_{N}(x_{k}),$
which, together with (A.2), gives
$l_{k}^{\prime\prime}(x_{k})=\frac{s^{\prime\prime\prime}(x_{k})}{3s^{\prime}(x_{k})}=\frac{2}{3}\,\frac{q^{2}}{q^{2}x_{k}^{2}-1}+\frac{\chi_{N}}{3}\,\frac{q^{2}x_{k}^{2}-1}{1-x_{k}^{2}},\quad
1\leq k\leq N-1.$ (A.10)
It is seen from (A.9) that the remaining two entries
$l^{\prime\prime}_{0}(-1)$ and $l^{\prime\prime}_{N}(1)$ involve
$\psi_{N}^{\prime\prime}(\pm 1),$ which can also be represented by
$\psi_{N}(\pm 1).$ Indeed, differentiating (2.1) and letting $x=\pm 1$, leads
to
$4\psi_{N}^{\prime\prime}(\pm 1)=\pm(\chi_{N}-2-c^{2})\psi_{N}^{\prime}(\pm
1)-2c^{2}\psi_{N}(\pm 1),$
so by (A.4), $\psi_{N}^{\prime\prime}(\pm 1)$ is a multiple of $\psi_{N}(\pm
1).$ Finally, we get
$l_{0}^{\prime\prime}(-1)=l_{N}^{\prime\prime}(1)=\frac{2q^{2}}{3(q^{2}-1)}+\frac{1}{24}(c^{2}-\chi_{N}+1)^{2}-\frac{5}{6}c^{2}-\frac{1}{24},$
(A.11)
where $q=c/\sqrt{\chi_{N}}$ as before.
## Appendix B Proof of Theorem 4.1
We derive from the definition (4.6) that
$\|\boldsymbol{\pi}_{h,N}^{c}u-u\|^{2}_{L^{2}(a,b)}=\sum_{i=1}^{M}\big{\|}(\boldsymbol{\pi}_{h,N}^{c}u)|_{I_{i}}-u^{I_{i}}\big{\|}^{2}_{L^{2}(I_{i})}=\frac{h}{2}\sum_{i=1}^{M}\big{\|}\hat{\pi}_{N}^{c}\hat{u}^{I_{i}}-\hat{u}^{I_{i}}\big{\|}^{2}_{L^{2}(I_{\rm
ref})}.$ (B.1)
Thus, it suffices to estimate $L^{2}(I_{\rm ref})$-orthogonal projection error
in the reference interval $I_{\rm ref}=(-1,1).$ To do this, we recall the
estimate in [29, Theorem 2.1]: if
${c}/{\sqrt{\chi_{n}}}\leq{q_{*}}/{\sqrt[6]{2}},$ then for any
$\hat{u}\in B^{\sigma}(I_{\rm
ref}):=\big{\\{}\hat{u}\,:\,(1-y^{2})^{k/2}\partial_{y}^{k}\hat{u}(y)\in
L^{2}(I_{\rm ref}),\;0\leq k\leq\sigma\big{\\}},\quad\sigma\geq 0,$ (B.2)
we have the estimate for the PSWF expansion coefficient in (4.4):
$\big{|}\hat{u}_{n}(c)\big{|}\leq
D\big{(}n^{-\sigma}\big{\|}(1-y^{2})^{{\sigma}/{2}}\partial_{y}^{\sigma}\hat{u}\big{\|}_{L^{2}(I_{\rm
ref})}+(q_{*})^{\delta n}\|\hat{u}\|_{L^{2}(I_{\rm ref})}\big{)},\quad n\gg
1,$ (B.3)
where $D$ and $\delta$ are generic positive constants independent of
$\hat{u},n$ and $c.$ Then we have the following $L^{2}$-error estimate for the
orthogonal projection defined in (4.4):
$\|\hat{\pi}_{N}^{c}\hat{u}-\hat{u}\|_{L^{2}(I_{\rm ref})}\leq
D\Big{(}N^{1/2-\sigma}\big{\|}(1-y^{2})^{\sigma/{2}}\partial_{y}^{\sigma}\hat{u}\big{\|}_{L^{2}(I_{\rm
ref})}+\frac{1}{\sqrt{\delta\ln(1/q_{*})}}(q_{*})^{\delta
N}\|\hat{u}\|_{L^{2}(I_{\rm ref})}\Big{)},$ (B.4)
for integer $\sigma\geq 1.$ Indeed, by the orthogonality (2.2) and the bound
(B.3),
$\begin{split}\|\hat{\pi}_{N}^{c}\hat{u}-\hat{u}\|_{L^{2}(I_{\rm
ref})}^{2}=&\sum_{n=N+1}^{\infty}\big{|}\hat{u}_{n}(c)\big{|}^{2}\leq
D\bigg{\\{}\Big{(}\sum_{n=N+1}^{\infty}n^{-2\sigma}\Big{)}\big{\|}(1-y^{2})^{\sigma/{2}}\partial_{y}^{\sigma}\hat{u}\big{\|}_{L^{2}(I_{\rm
ref})}^{2}\\\ &+\Big{(}\sum_{n=N+1}^{\infty}(q_{*})^{2\delta
n}\Big{)}\|\hat{u}\|_{L^{2}(I_{\rm ref})}^{2}\bigg{\\}}.\end{split}$
Since
$\sum_{n=N+1}^{\infty}n^{-2\sigma}\leq\int_{N}^{\infty}\frac{1}{x^{2\sigma}}\,dx=\frac{1}{2\sigma-1}N^{1-2\sigma},\quad{\rm
if}\;\;\sigma>\frac{1}{2},$
and
$\sum_{n=N+1}^{\infty}(q_{*})^{2\delta
n}\leq\int_{N}^{\infty}(q_{*}^{2})^{\delta
x}dx\leq\frac{1}{2\delta\ln(1/q_{*})}(q_{*})^{2\delta N},$
we obtain (B.4).
One verifies readily from (4.3) that for $x\in I_{i}$ and $y\in I_{\rm ref},$
$\partial_{y}^{\sigma}\hat{u}^{I_{i}}(y)=\frac{h^{\sigma}}{2^{\sigma}}\partial_{x}^{\sigma}u^{I_{i}}(x),\quad(1-y^{2})^{\sigma}=2^{2\sigma}\Big{(}\frac{a_{i}-x}{h}\Big{)}^{\sigma}\Big{(}\frac{x-a_{i-1}}{h}\Big{)}^{\sigma}\leq
2^{2\sigma}.$
Then applying (B.4) to (B.1) leads to the desired result.
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|
arxiv-papers
| 2013-10-13T08:22:41 |
2024-09-04T02:49:52.319640
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Li-Lian Wang, Jing Zhang and Zhimin Zhang",
"submitter": "Li-Lian Wang Dr.",
"url": "https://arxiv.org/abs/1310.3457"
}
|
1310.3549
|
# Puzzling the $120$–cell
Saul Schleimer Department of Mathematics
University of Warwick
Coventry, UK [email protected] and Henry Segerman Department of
Mathematics
Oklahoma State University
Stillwater, OK USA [email protected]
###### Abstract.
We introduce _Quintessence_ : a new family of burr puzzles based on the
geometry and combinatorics of the $120$–cell. Written for a broad,
mathematically-minded audience, our paper discusses the quaternions, the
three-sphere, isometries of three-space, polytopes, and the construction of
the dodecahedron and its four-dimensional sibling, the $120$–cell. The design
of our puzzle pieces uses a drawing technique of Leonardo da Vinci; the paper
ends with a catalog of new puzzles.
This work is in the public domain.
## 1\. Introduction
(a)
(b)
Figure 1.1. The star burr.
A _burr puzzle_ is a collection of notched wooden sticks [2, page xi] that
must be fitted together to form a highly symmetric design, without internal
voids, often based on one of the Platonic solids. Ideally, no force is
required for the solution, which is unique. Of course, a puzzle may break
these rules in various ways and still be called a burr.
Best known is the $6$–piece burr, investigated in detail by Cutler [5]. See
also [2, Chapter 7]. Another puzzle, the star burr [2, Chapter 9], is closely
related to our work. The star burr has six sticks that are, unusually, all
identical. These are shown in Figure 1.1a. Once solved, the star burr forms
the first stellation of the rhombic dodecahedron [2, page 83]. See Figure
1.1b.
spine at 100 514 inner 6 at 310 580 outer 6 at 570 573 inner 4 at 190 60 outer
4 at 480 60 equator at 800 50
Figure 1.2. The six rib types.
The $6$–piece and star burrs are closely related to the Borromean rings. To
see this, we divide the sticks into three pairs. Each pair forms, roughly, a
loop. These three loops interlock in the fashion of the Borromean rings.
The goal of this paper is to describe _Quintessence_ : a new family of burr
puzzles based on the $120$–cell, a regular four-dimensional polytope. The
puzzles are built from collections of six kinds of sticks, shown in Figure
1.2; we call these _ribs_ as they are gently curving chains of distorted
dodecahedra.
In Section 2 we recall the definition of the quaternions and briefly discuss
stereographic projection; this allows us to translate objects from four-space,
and from the three-sphere, into our usual three-dimensional space. In Section
3 we review the basic concepts of regular polytopes in low dimensions; in
Section 4 we construct the dodecahedron and derive several trigonometric
facts.
With this preparation in hand, in Section 5 we construct the $120$–cell. Using
the binary dodecahedral group, as it lies inside of the quaternions, in
Section 6 we investigate the combinatorics of the $120$–cell, focusing on how
it decomposes into spheres and rings of dodecahedra. In Section 7 we lay out
our choice of ribs, as influenced by the cell-centered stereographic
projection. We use this to give a basic combinatorial restriction on the
possible burr puzzles in Quintessence. In Section 8 we recall Leonardo da
Vinci’s technique for drawing polytopes; we adapt his method to our 3D prints.
We end with Appendix A, a catalog of some of the burr puzzles in Quintessence.
The connection between the classic burrs and ours is left as a final exercise
for the intrigued reader.
### Acknowledgments
We thank Robert Tang and Stuart Young for their insights into the
combinatorics of the $120$–cell.
## 2\. Four-space and quaternions
In this section we review the quaternions and stereographic projection. We
recommend [4, Chapter 6], [17, Section 2.7], or [3, Part II] as references on
these topics.
### 2.1. Space and sphere
Let ${\langle{1,i,j,k}\rangle}$ be the usual orthonormal basis for
$\mathbb{R}^{4}$. We write $\mathbb{H}=\mathbb{R}\oplus\mathbb{I}$ where
$\mathbb{H}$ is the space of _quaternions_ and where
$\mathbb{I}=i\mathbb{R}\oplus j\mathbb{R}\oplus k\mathbb{R}$ is the three-
dimensional subspace of _purely imaginary_ quaternions. Following Hamilton we
endow $\mathbb{H}$ with the relations
$i^{2}=j^{2}=k^{2}=ijk=-1.$
These relations, $\mathbb{R}$–linearity, associativity, and distributivity
allow us to compute any product in $\mathbb{H}$. If
$p=a+bi+cj+dk\in\mathbb{H}$ then we call $a$ the _real part_ of $p$ and
$bi+cj+dk$ the _imaginary part_ of $p$. We call ${\overline{p}}=a-bi-cj-dk$
the _conjugate_ of $p$. Since $ij=-ji$ and so on, we deduce that
${\overline{p\cdot q}}={\overline{q}}\cdot{\overline{p}}$ for any
$p,q\in\mathbb{H}$. We define the usual norm and Euclidean distance on
$\mathbb{H}$.
$|p|=\sqrt{p{\overline{p}}}=\sqrt{a^{2}+b^{2}+c^{2}+d^{2}}\quad\mbox{and}\quad
d_{\mathbb{H}}(p,q)=|p-q|.$
Thus
$|pq|^{2}=pq{\overline{pq}}=pq\cdot{\overline{q}}\cdot{\overline{p}}=p|q|^{2}{\overline{p}}=|p|^{2}|q|^{2}$,
and so $|pq|=|p||q|$.
The _three-sphere_ is denoted by $S^{3}=\\{q\in\mathbb{H}:|q|=1\\}$. The
metric on $\mathbb{H}$ induces a metric on the sphere, namely
$d_{S}(p,q)=\arccos(\langle p,q\rangle),$
where $\langle p,q\rangle=\sum p_{i}q_{i}$ is the usual inner product. If
$L\subset\mathbb{H}$ is a linear subspace of dimension one, two, or three then
$L\cap S^{3}$ is either a pair of antipodal points, a _great circle_ , or a
_great sphere_ , respectively. We call $1$ and $-1$, as they lie in $S^{3}$,
the _south_ and _north_ poles, respectively. We call
$S^{2}_{\mathbb{I}}=S^{3}\cap\mathbb{I}$ the _equatorial_ great sphere. See
Figure 2.2 for a depiction of how several great circles among $1,i,j,k$ lie
inside of $S^{3}$.
### 2.2. Group structure
The points of the three-sphere, the _unit quaternions_ , form a group under
quaternionic multiplication. Here $1\in S^{3}$ serves as the identity,
associativity follows from the associativity of $\mathbb{H}$, and
$q^{-1}={\overline{q}}$.
###### Lemma 2.1.
The left and right actions of $S^{3}$ on $\mathbb{H}$ are via orientation-
preserving isometries. The same holds for the three-sphere’s action on itself.
###### Proof.
Fix $p\in S^{3}$ and $q,r\in\mathbb{H}$. We compute
$d_{\mathbb{H}}(pq,pr)=|pq-pr|=|p(q-r)|=|p||q-r|=|q-r|=d_{\mathbb{H}}(q,r)$,
verifying the left action is via isometry. Since $S^{3}$ is connected, and
since $1$ acts trivially, the action is orientation preserving. Also, the
action preserves the three-sphere, and so preserves the induced metric. ∎
Note the group elements $\pm 1$ are very special; they are the only elements
that are their own inverses. The sphere $S^{2}_{\mathbb{I}}$ of pure
imaginaries is much more homogeneous, as follows.
###### Lemma 2.2.
We have
$u^{2}=v^{2}=w^{2}=uvw=-1$
when ${\langle{u,v,w}\rangle}$ is a right-handed orthonormal basis for
$\mathbb{I}$. ∎
For any $u\in S^{2}_{\mathbb{I}}$ and any $\alpha\in\mathbb{R}$, define
(2.3) $\displaystyle e^{u\alpha}$ $\displaystyle=\cos\alpha+u\cdot\sin\alpha.$
This is the unit circle in the plane ${\langle{1,u}\rangle}\subset\mathbb{H}$
and thus is a great circle in $S^{3}$.
###### Lemma 2.4.
For any pure imaginary $u\in S^{2}_{\mathbb{I}}$ and for any
$\alpha,\beta\in\mathbb{R}$ we have
$e^{u\alpha}e^{u\beta}=e^{u(\alpha+\beta)}$. Thus $\\{e^{u\alpha}\\}$ is a
one-parameter subgroup of $S^{3}$. Also, $d_{S}(1,e^{u\alpha})=\alpha$ for
$\alpha\in[0,\pi]$. ∎
$\mathbb{I}$ at 97 263 2pt $u$ at 117 246 $-u$ at 120 17 $1$ at 218 133 $-1$
at -13 132 $q=e^{u\alpha}$ [Bl] at 163 227 $\alpha$ at 126 142 $\rho(q)$ at 87
196
Figure 2.1. Stereographic projection from $S^{1}-\\{-1\\}$ to $\mathbb{I}$.
This gives parameterization of $S^{3}$, as follows.
###### Lemma 2.5.
For any $q\in S^{3}-\\{\pm 1\\}$ there is a unique $u\in S^{2}_{\mathbb{I}}$
and a unique $\alpha\in(0,\pi)$ so that $q=e^{u\alpha}$. ∎
### 2.3. Stereographic projection
Throughout the paper we use stereographic projection to visualize objects in,
and motions of, the three-sphere. Recall that $\mathbb{I}$ is a copy of
$\mathbb{R}^{3}$. We define stereographic projection $\rho\colon
S^{3}-\\{-1\\}\to\mathbb{I}$ by
$\parshapelength\rho(q)=\frac{\sin(\alpha)}{1+\cos(\alpha)}\cdot u$
with $q=e^{u\alpha}$ as in Lemma 2.5. See Figure 2.1 for a cross-sectional
view. Note that $\rho$ sends the south pole to the origin, fixes the
equatorial sphere $S^{2}_{\mathbb{I}}$ pointwise, and sends the north pole to
“infinity”. The one-parameter subgroup $e^{u\theta}$ is sent to the straight
line in the direction of $u$. Figure 2.2 shows the result of applying
stereographic projection to various great circles connecting $1,i,j,k$ inside
of $S^{3}$.
2pt $-j$ at 100 183 $i$ at 174 114 $1$ at 207 166 $k$ at 212 257 $-k$ at 214
47 $-i$ at 257 186 $j$ at 307 140
Figure 2.2. Several great circles connecting $1,i,j,k$, shown after
stereographic projection to $\mathbb{R}^{3}$.
### 2.4. Mapping to $\operatorname{SO}(3)$
By definition, $\operatorname{SO}(3)$ is the group of three-by-three
orthogonal matrices with determinant one. Taking ${\langle{i,j,k}\rangle}$ as
a basis for $\mathbb{I}$, we identify $\operatorname{SO}(3)$ with
$\operatorname{Isom}^{+}_{0}(\mathbb{I})$, the group of orientation-preserving
isometries of $\mathbb{I}$ fixing the origin. Euler’s rotation theorem [8]
states that every element $A\in\operatorname{Isom}^{+}_{0}(\mathbb{I})$ is a
rotation about some _axis_ : a line through the origin fixed pointwise by $A$.
Also, when $A$ is not the identity, this axis is unique. See [13] for several
proofs and a historical discussion.
In Lemma 2.1 we discussed the left and right actions of $S^{3}$ on
$\mathbb{H}$. We combine these to obtain the _twisted action_ : for $q\in
S^{3}$ define $\phi_{q}\colon\mathbb{H}\to\mathbb{H}$ by
$\phi_{q}(p)=qpq^{-1}$. The twisted action is again via isometries. Note that
the action preserves $\mathbb{R}\subset\mathbb{H}$ pointwise. Thus it
preserves $\mathbb{I}\subset\mathbb{H}$ setwise. We define
$\psi_{q}\colon\mathbb{I}\to\mathbb{I}$ by $\psi_{q}=\phi_{q}|\mathbb{I}$ and
deduce the following.
###### Lemma 2.6.
The map $\psi_{q}$ is an element of $\operatorname{SO}(3)$. The induced map
$\psi\colon S^{3}\to\operatorname{SO}(3)$ is a group homomorphism.
###### Proof.
As remarked above, $\psi_{q}$ is an isometry of $\mathbb{I}$ that fixes the
origin. Since $S^{3}$ is connected, the isometries $\psi_{q}$ and
$\psi_{1}=\operatorname{Id}$ have the same handedness. Thus $\psi_{q}$ lies in
$\operatorname{SO}(3)$. The equality $\psi_{qr}=\psi_{q}\psi_{r}$ follows from
the associativity of $\mathbb{H}$. ∎
We need an explicit form of $\psi$, discovered independently by Gauss,
Rodrigues, Cayley, and Hamilton [15, page 21].
###### Lemma 2.7.
For $q=\pm e^{u\alpha}$ the isometry $\psi_{q}$ is a rotation of $\mathbb{I}$
about the direction $u$ through angle $2\alpha$. Thus $\psi\colon
S^{3}\to\operatorname{SO}(3)$ is a double cover.
###### Proof.
As a convenient piece of notation, we write $q=a+bu$ where $a=\cos(\alpha)$
and $b=\sin(\alpha)$. So $q^{-1}=a-bu$. We check that $\psi_{q}(u)=u$.
$\displaystyle\psi_{q}(u)$ $\displaystyle=quq^{-1}=(a+bu)u(a-bu)$
$\displaystyle=(au-b)(a-bu)$ $\displaystyle=a^{2}u+ab-ab+b^{2}u$
$\displaystyle=u$
By Euler’s rotation theorem, the line through $u$ is an axis for $\psi_{q}$.
Now suppose that $v$ is orthogonal to $u$. Let $w=uv$. Thus
${\langle{u,v,w}\rangle}$ is a right-handed orthonormal basis of $\mathbb{I}$.
We compute $\psi_{q}(v)$.
$\displaystyle\psi_{q}(v)$ $\displaystyle=(a+bu)v(a-bu)$
$\displaystyle=a^{2}v-abvu+abuv-b^{2}uvu$ $\displaystyle=a^{2}v+2abw-b^{2}uvu$
$\displaystyle=(a^{2}-b^{2})v+2abw$
$\displaystyle=\cos(2\alpha)v+\sin(2\alpha)w$
Thus $\psi_{q}$ rotates by the desired amount. It follows from the rotation
theorem that $\psi$ is surjective. Note that $\psi_{q}=\operatorname{Id}$ if
and only if $\cos(2\alpha)=1$ if and only if $\alpha\in\\{0,\pi\\}$. Thus
$\psi$ is two-to-one. We leave the proof that $\psi$ is a covering map as a
topological exercise. ∎
###### Definition 2.8.
If $\mathcal{G}\subset\operatorname{SO}(3)$ is a group, then we call
$\mathcal{G}^{*}=\psi^{-1}(\mathcal{G})$ the _binary_ group corresponding to
$\mathcal{G}$.
## 3\. Polytopes
We refer to [19] for an in-depth discussion of polytopes. Here we concentrate
on the ideas needed to understand regular polytopes.
### 3.1. Convexity
A set $C\subset\mathbb{R}^{n}$ is _convex_ if for any points $x$ and $y$ in
$C$ the line segment $[x,y]$ is also contained in $C$. For any subset
$S\subset\mathbb{R}^{n}$ the _convex hull_ of $S$, denoted by
$\operatorname{hull}(S)$, is the smallest convex set containing $S$. For
example, the convex hull of two distinct points is a line segment. The convex
hull of three points, not all in a line, is a triangle. The convex hull of
four points, not all in a plane, is a tetrahedron. In general, if $S$ is a
collection of $k+1$ points, not all in a $k$–dimensional hyperplane, then
$\operatorname{hull}(S)$ is called a _$k$ –simplex_.
When the set $S$ is finite, we call the convex hull $P=\operatorname{hull}(S)$
a _polytope_. The dimension of $P$ is the dimension of the smallest affine
subspace $H\subset\mathbb{R}^{n}$ containing $P$. We call $H$ the _affine
span_ of $P$. In the examples above the interval has dimension one, the
triangle two, and the tetrahedron three. Define ${\operatorname{interior}}(P)$
to be those points $p\in P$ where there is a relatively open set $U\subset H$
so that $p\in U\subset P$. Define $\partial P=P-{\operatorname{interior}}(P)$.
If $K\subset\mathbb{R}^{n}$ is a hyperplane and if $Q=P\cap K$ lies in
$\partial P$ then we call $Q$ a _face_ of $P$. If, in addition, the dimension
of $Q$ is one less than that of $P$ then we call $Q$ a _facet_ of $P$. The
_vertices_ of $P$ are exactly the zero-dimensional faces. For example, any
tetrahedron has four facets, all triangles; this gives the tetrahedron its
name. Note that $\partial P$ is the union of the facets of $P$.
### 3.2. Regular polytopes
Suppose that $P$ is a $k$–dimensional polytope, with affine span $H$. A
collection of faces $Q_{0}\subset Q_{1}\subset\ldots\subset Q_{k-1}\subset
Q_{k}=P$ is called a _flag_ of $P$ if $Q_{\ell}$ has dimension $\ell$. As an
example, the tetrahedron has $4\times 3\times 2\times 1=24$ flags. Fixing a
basis $\\{u_{\ell}\\}$ for $H$ we may define the _handedness_ of a flag
$\\{Q_{\ell}\\}$ as follows: for $\ell>0$, pick $v_{\ell}$ in $Q_{\ell}$,
based at $Q_{0}$ and not in the affine span of $Q_{\ell-1}$. A flag is right-
or left-handed according to the sign of the determinant of the matrix of
coefficients of $\\{v_{\ell}\\}$ written in terms of the $\\{u_{\ell}\\}$.
Let $\operatorname{Sym}(P)$ be the group of isometries of $H$ that preserve
$P$ setwise. We call elements of $\operatorname{Sym}(P)$ the _symmetries_ of
$P$.
###### Definition 3.1.
A polytope $P$ is _regular_ if for any pair of flags $F$ and $G$ of $P$ there
is a symmetry $\phi\in\operatorname{Sym}(P)$ with $\phi(F)=G$.
It follows that all facets of a regular polytope are congruent and also
regular. As an example, consider the octahedron $O\subset\mathbb{R}^{3}$: the
convex hull of the six points
$(\pm 1,0,0),\;(0,\pm 1,0),\;(0,0,\pm 1).$
The octahedron has $6\times 4\times 2\times 1=48$ flags. Any one can be sent
to any other by reflections in the coordinate planes and rotations about the
coordinate axes. Note that the facets of $O$ are all congruent equilateral
triangles, so are themselves regular two-polytopes. Note that
$\mathcal{O}=\operatorname{Sym}(O)$ acts transitively on the vertices of $O$.
This is true for any regular polytope $P$.
Define $p=\operatorname{center}(P)$ to be the average of the vertices of $P$.
Since $\operatorname{Sym}(P)$ permutes the vertices of $P$, it fixes $p$.
Since $\operatorname{Sym}(P)$ acts transitively on the vertices, they are all
the same distance from $p$. Thus $p$ is a _circumcenter_ : $P$ is
circumscribed by the sphere $S_{P}$ centered at $p$ containing the vertices of
$P$.
The sphere $S_{P}$ is crucial in our study of $P$. Typically, our first move
towards constructing an $n$–dimensional regular polytope $P$ will be to build
a spherical tiling $\mathcal{T}_{P}\subset S_{P}\cong S^{n-1}$. The tiling
$\mathcal{T}_{P}$ is the radial projection of $\partial P$, from the center
$p$, into $S_{P}$. The tiling is often more tractable, and is certainly easier
to visualize.
###### Definition 3.2.
Suppose that $P$ is regular and $F=\\{Q_{i}\\}$ is a flag in $P$. Then the
_flag polytope_ $Q_{F}$ is the convex hull of the centers of the $Q_{i}$. The
spherical flag polytope is the radial projection of $Q_{F}-p$ to $S_{P}$.
Since $P$ is regular, all of its flag polytopes are congruent, perhaps via
orientation-reversing symmetries of $P$.
###### Definition 3.3.
Suppose $P$ is a regular polytope. We form the _dual_ polytope $P^{\prime}$ by
taking the convex hull of the centers of the facets of $P$ and then rescaling
so all vertices of $P^{\prime}$ lie on $S_{P}$.
For example, the dual of the octahedron is the cube (hexahedron). Rescaling,
we may assume that the eight points
$\big{(}\pm 1,\pm 1,\pm 1\big{)}$
are the vertices of the cube.
### 3.3. Constructions
There are four infinite families of regular polytopes; each family is
associated to a topological operation. We begin in dimension two, with the
regular polygons. Let $\rho_{n}\colon\mathbb{C}\to\mathbb{C}$ be the map
$\rho_{n}(\omega)=\omega^{n}$. Restricted to $S^{1}$ this becomes an $n$–fold
covering map of the circle.
###### Definition 3.4.
The _$n$ –gon_ $P_{n}$ is the convex hull of $\rho_{n}^{-1}(1)$: that is, of
the $n^{\rm th}$ roots of unity.
Note that the interior angle at the vertex of $P_{n}$ is $\pi(1-\frac{2}{n})$,
and also that $P_{n}$ is self-dual. Already in this first example we see a
recurring theme: a regular polytope $P$ is first understood via its
circumscribing sphere, in this case the unit circle.
We now turn to the three families that exist in all dimensions: the simplex,
the cube, and the cross-polytope. Each family is defined in terms of convex
hulls and also given by its topological operation. We take
$e^{k}_{i}=(0,\ldots,0,1,0,\ldots,0)\in\mathbb{R}^{k}$ to be the point with a
single $1$ in the $i^{\rm th}$ coordinate and all other coordinates zero.
###### Definition 3.5.
The _$k$ –simplex_ is the convex hull of the $k+1$ points $\\{e_{i}\\}$ in
$\mathbb{R}^{k+1}$. Thus it is a (right) cone with base the $(k-1)$–simplex
and with height $\frac{1}{k}\sqrt{k^{2}+k}$.
###### Definition 3.6.
The _$k$ –cube_ is the convex hull of the $2^{k}$ points $\\{\pm e_{1},\pm
e_{2},\ldots\pm e_{k}\\}$ in $\mathbb{R}^{k}$. Thus it is a product between
the $(k-1)$–cube and the unit interval.
###### Definition 3.7.
The _$k$ –cross-polytope_ is the convex hull of the $2k$ points $\\{\pm
e_{i}\\}$, taken in $\mathbb{R}^{k}$. Thus it is a suspension with base the
$(k-1)$–cross-polytope and with height one. Here the _suspension_ is a double
right cone, to points lying symmetrically above and below the center of the
base.
Figure 3.1. The first four simplices, cubes, and cross-polytopes.
As shown in Figure 3.1, in dimension one all of these are intervals. In
dimension two the cube and cross-polytope give the square and diamond, which
are similar. In dimension three the simplex is the tetrahedron and the cross-
polytope is the octahedron. We collect several useful statements, which we
will not prove here. Instead see [9, page 143].
###### Lemma 3.8.
The simplex, cube, and cross-polytope are regular. The cube and the cross-
polytope are dual while the simplex is self-dual. In dimensions three and
higher, these three polytopes are distinct. ∎
###### Theorem 3.9.
There are exactly five regular polytopes not in one of the four families.
These are, in dimension three, the dodecahedron and icosahedron (dual) and, in
dimension four, the $24$–cell (self-dual), and the $120$–cell and $600$–cell
(dual). ∎
The next sections of the paper are devoted to constructing the dodecahedron
and the $120$–cell.
## 4\. Dodecahedron
### 4.1. Construction
The dodecahedron, and its dual the icosahedron, exists for a more subtle
reason than that of the simplex, cube, or cross-polytope. As such it has
several different constructions; the earliest of which we are aware is
Proposition 17 in Book 13 of Euclid’s Elements [7]. See [18] for one
historical account of the five Platonic solids.
We give an indirect construction of the dodecahedron $D$ that has two
advantages. The argument finds the symmetry group $\operatorname{Sym}(D)$
along the way. It also generalizes to all other regular tessellations of the
sphere, the Euclidean plane, and hyperbolic plane. We begin with Girard’s
formula for the area of a triangle in $S^{2}$ [4, Equation 2.11].
###### Lemma 4.1.
A spherical triangle with interior angles $A$, $B$, $C$ has area $A+B+C-\pi$.
∎
By continuity, for any angle
$\theta\in\mathopen{}\mathclose{{}\left(3\pi/5,\pi}\right)$ there is a regular
spherical pentagon $P\subset S^{2}$ with all angles equal to $\theta$. (See
[17, Figure 1.12] for a hyperbolic version.) Thus we may take $\theta$ equal
to $2\pi/3$. Adding a vertex at the center and at the midpoints of the edges,
we divide $P$ into ten flag triangles: five right-handed, five left-handed,
and all having internal angles $\pi\cdot(1/2,1/3,1/5)$. These three angles
appear at the edge, vertex, and center of $P$. Let $T_{R}$ and $T_{L}$ be
copies of the right and left handed flag triangles, and note that there are
rotations of $S^{2}$ matching the edges of $T_{R}$ and $T_{L}$ in pairs.
The celebrated Poincaré polygon theorem [6, Theorem 4.14] now implies that
copies of $T_{R}$ and $T_{L}$ give a tiling $\mathcal{T}$ of $S^{2}$. One half
of the stereographic projection of $\mathcal{T}$ is shown in Figure 4.1.
Poincaré’s theorem also implies that $\operatorname{Sym}(\mathcal{T})$ is
transitive on the triangles of $\mathcal{T}$. and that any local symmetry
extends to be an element of $\operatorname{Sym}(\mathcal{T})$.
Figure 4.1. One half of the image of $\mathcal{T}$ after stereographic
projection from $S^{2}$ to $\mathbb{R}^{2}$. The white and grey triangles are
copies of $T_{R}$ and $T_{L}$, respectively. See also [11, page 688].
Applying Lemma 4.1, the area of $T_{R}$ is
$\pi\cdot(1/2+1/3+1/5)-\pi\,=\,\pi/30.$
Since the area of $S^{2}$ is $4\pi$ deduce that the tiling $\mathcal{T}$
contains $120$ flag triangles.
###### Definition 4.2.
We partition $\mathcal{T}$ into copies of $P$ to obtain the tiling
$\mathcal{T}_{D}$; this has $12$ pentagonal faces, $12\cdot 5/2=30$ edges, and
$12\cdot 5/3=20$ vertices. We take the convex hull (in $\mathbb{R}^{3}$) of
the vertices of $\mathcal{T}_{D}$ (in $S^{2}$) to obtain $D$, the
dodecahedron.
Define $\mathcal{D}=\operatorname{Sym}^{+}(\mathcal{T})<\operatorname{SO}(3)$;
this is the group of orientation-preserving symmetries of the dodecahedron. We
end this section by examining $\mathcal{D}$.
###### Lemma 4.3.
The group $\operatorname{Sym}(\mathcal{T})$ has order $120$; the orientation-
preserving subgroup $\mathcal{D}$ has order $60$. Also, the tiling
$\mathcal{T}$ is invariant under the antipodal map.
###### Proof.
Suppose that $F\in\operatorname{SO}(3)$ is a non-trivial symmetry of
$\mathcal{T}$. By Euler’s rotation theorem $F$ fixes, and rotates about,
antipodal points $p,q\in S^{2}$. If $p$ lies in the interior of a triangle
$T$, then $F$ non-trivially permutes the vertices of $T$, contradicting the
fact that all of their internal angles are distinct. Suppose instead that $p$
lies in the interior of an edge of $T$. Then $F$ swaps the endpoints of the
edge, another contradiction. The last possibility is that $p$ is a vertex of
$T$, say of degree $2d$. In this case $F$ is one of the $d-1$ possible
rotations.
We deduce that the orientation-preserving symmetries of $\mathcal{T}$ are in
one-to-one correspondence with (say) the right-handed flag triangles. This
counts the elements of $\mathcal{D}=\operatorname{Sym}^{+}(\mathcal{T})$ and
thus of $\operatorname{Sym}(\mathcal{T})$.
It remains to prove that $\mathcal{T}$ is invariant under the antipodal map.
Suppose that $p$ is a vertex of degree $2d$ of $\mathcal{T}$. There is a local
symmetry $f$ of $\mathcal{T}$ that rotates about $p$, with order $d$. Thus $f$
extends to a global symmetry $F\in\operatorname{SO}(3)$. Since $F$ is a non-
trivial rotation, Euler again gives us a pair of antipodal fixed points for
$F$ on the unit sphere $S^{2}$. One of these is $p$; call the antipode $q$.
Restricting $F$ to a small neighborhood of $q$ yields a rotation of order $d$
(of the opposite handedness). It follows that $q$ is another vertex of
$\mathcal{T}$, also of degree $2d$. ∎
###### Corollary 4.4.
The group $\mathcal{D}$ contains:
* •
the identity,
* •
$12$ face rotations through angle $2\pi/5$,
* •
$20$ vertex rotations through angle $2\pi/3$,
* •
$12$ face rotations through angle $4\pi/5$, and
* •
$15$ edge rotations through angle $\pi$.
###### Proof.
For any vertex $p$ of $\mathcal{T}$ of degree $2d$ we obtain a cyclic subgroup
$\mathbb{Z}/d\mathbb{Z}$ in $\mathcal{D}$. By the second part of Lemma 4.3 the
vertex $p$ and its antipode $q$ give rise to the same subgroup. Thus we may
count elements of $\mathcal{D}$ by always restricting to those rotations
through an angle of $\pi$ or less. Counting the symmetries obtained this way
gives $60$; by the first part of Lemma 4.3 there are no others. ∎
### 4.2. Trigonometry
Figure 4.2. The angle between the center and the vertex of the pentagon $P$.
For the construction of the $120$–cell, in Section 5, we require some
trigonometric information about $\mathcal{T}_{D}$. Recall that $P$ is a
regular spherical pentagon with all angles equal to $2\pi/5$.
###### Lemma 4.5.
The spherical distance between the center and the vertex of $P$ is
$\parshapelength\arccos\mathopen{}\mathclose{{}\left(\frac{1}{\sqrt{3}}\cot\pi/5}\right).$
###### Proof.
Any spherical triangle with angles $A,B,C$ and opposite edge lengths $a,b,c$
satisfies the dual spherical law of cosines [17, pages 74–76]:
$\parshapelength\cos A=-\cos B\cos C+\sin B\sin C\cos a.$
Recall the pentagon $P$ is a union of 10 flags triangles; any one of these is
a spherical triangle $T$ with angles $A=\pi/2$, $B=\pi/3$, and $C=\pi/5$.
Using the law of cosines we find
$\cos a=\frac{1}{\sqrt{3}}\cot\pi/5$
as desired. ∎
###### Corollary 4.6.
The square of the Euclidean distance between the center and the vertex of $P$
is $2-\frac{2}{\sqrt{3}}\cot\pi/5$. ∎
We gather together several trigonometric facts needed to construct the
$120$–cell. For an elementary and enlightening discussion, see Langlands’
lectures [10, Part 3, pages 1-9].
$\theta$ | $\cos\theta$ | $\sin\theta$ | $\cot\theta$
---|---|---|---
$\pi/5$ | $\frac{1}{4}\mathopen{}\mathclose{{}\left(1+\sqrt{5}}\right)$ | $\frac{1}{4}\sqrt{10-2\sqrt{5}}$ | $\sqrt{1+\frac{2}{\sqrt{5}}}$
$2\pi/5$ | $\frac{1}{4}\mathopen{}\mathclose{{}\left(-1+\sqrt{5}}\right)$ | $\frac{1}{4}\sqrt{10+2\sqrt{5}}$ | $\sqrt{1-\frac{2}{\sqrt{5}}}$
Deduce the following identities.
(4.7) $\displaystyle\cot^{2}\pi/5+\cot^{2}2\pi/5$ $\displaystyle=2$ (4.8)
$\displaystyle 4\cos^{2}\pi/5-2\cos\pi/5-1$ $\displaystyle=0$
## 5\. The $120$–cell
We construct the $120$–cell. We could use a continuity argument, as in Section
4.1, to build a spherical dodecahedron in $S^{3}$ with all dihedral angles
equal to $2\pi/3$. Again, the Poincaré polyhedron theorem would produce a
tiling of $S^{3}$; regularity of the tile leads to regularity of the tiling.
Taking the convex hull of the vertices would gives the $120$–cell; however,
the number of cells and the overall symmetry group are less than clear. Since
it is crucial for us to see the symmetries of the $120$–cell we proceed along
different lines. We refer to [1, 15, 16] for very useful commentaries on the
$120$–cell.
### 5.1. Outline of the construction
2pt $i$ at 74 125 $j$ at 298 140 $k$ at 181 297 $f$ at 157 110
Figure 5.1. The tiling $\mathcal{T}_{D}$ can be positioned with one vertex at
$v=\frac{1}{\sqrt{3}}(i+j+k)$ and with one face center $f$ in the $ij$–plane.
Let $\mathcal{T}_{D}\subset S^{2}_{\mathbb{I}}$ be the tiling constructed in
Section 4.1. Let $\mathcal{D}\subset\operatorname{SO}(3)$ be its group of
orientation-preserving symmetries. As in Definition 2.8, let
$\mathcal{D}^{*}\subset S^{3}$ be the binary dodecahedral group. From Lemma
4.3 deduce that $\mathcal{D}^{*}$ has $120$ elements. Let $\mathcal{T}_{120}$
be the tiling of $S^{3}$ by Voronoi domains about the points of
$\mathcal{D}^{*}$. We show that each domain is a regular spherical
dodecahedron. Taking the convex hull of the vertices of $\mathcal{T}_{120}$
yields the $120$–cell. We now give the details.
### 5.2. Positioning the dodecahedron
As in Definition 4.2, let
$\mathcal{T}_{D}\subset\mathbb{I}\cong\mathbb{R}^{3}$ be the tiling of the
unit sphere $S^{2}_{\mathbb{I}}$ by twelve spherical pentagons. See Figure 5.1
for a picture of its one-skeleton. We rotate $\mathcal{T}_{D}$ to have one
vertex at the point $v=\frac{1}{\sqrt{3}}(i+j+k)$. With this choice of $v$,
the vertex rotation about $v$ permutes the coordinate planes. Pick
$f\in\mathcal{T}_{D}$ to be one of the three face centers closest to $v$. We
wish to rotate $\mathcal{T}_{D}$, about the line through $0$ and $v$, to bring
$f$ into the $ij$–plane. To show that this is possible, and to find the
resulting coordinates of $f$, suppose $f=xi+yj$, where $x^{2}+y^{2}=1$. We now
compute.
$\displaystyle|v-f|^{2}$ $\displaystyle=1-\frac{2}{\sqrt{3}}(x+y)+x^{2}+y^{2}$
$\displaystyle=2-\frac{2}{\sqrt{3}}(x+y).$
From Corollary 4.6 deduce that $x+y=\cot\pi/5$. Solving the resulting
quadratic in $x$, and applying Equation 4.7, yields
$\\{x,y\\}=\mathopen{}\mathclose{{}\left\\{\frac{1}{2}\mathopen{}\mathclose{{}\left(\cot\pi/5\pm\cot
2\pi/5}\right)}\right\\}.$
We choose the solution where $x>y$. The resulting position of
$\mathcal{T}_{D}$ is shown in Figure 5.1
Using the vertex rotation about $v$ deduce $f^{\prime}=xj+yk$ and
$f^{\prime\prime}=yi+xk$ are the other face centers of $\mathcal{T}_{D}$ that
are closest to $v$.
###### Remark 5.1.
Note that Figure 5.1 contains more information; the small dots on the three
axes are edge centers of $\mathcal{T}_{D}$ and also are the points $i$, $j$,
and $k$. As with Lemma 4.5, verifying this is an exercise in spherical
trigonometry.
### 5.3. Voronoi cells
Let $\mathcal{D}\subset\operatorname{SO}(3)$ be the group of orientation-
preserving symmetries of the dodecahedron $D$ given in Section 5.2. Let
$\mathcal{D}^{*}\subset S^{3}$ be the corresponding binary dodecahedral group.
For any $q\in\mathcal{D}^{*}$ we define the Voronoi cell
$V_{q}=\\{r\in S^{3}\mathbin{\mid}\mbox{for all $p\in\mathcal{D}^{*}$,
$d_{S}(q,r)\leq d_{S}(r,p)$}\\}.$
Let $\mathcal{T}_{120}$ be the resulting tiling of $S^{3}$ by Voronoi cells.
By construction $\mathcal{T}_{120}$ contains $120$ three-cells. Define
$\mathcal{C}=\operatorname{Sym}(\mathcal{T}_{120})$.
###### Lemma 5.2.
The left action of $\mathcal{D}^{*}$ on $\mathcal{T}_{120}$ is free and
transitive on the three-cells. The twisted action of $\mathcal{D}^{*}$ fixes
$V_{1}$ setwise. Both actions give homomorphisms of $\mathcal{D}^{*}$ to
$\mathcal{C}$. ∎
###### Lemma 5.3.
Each cell $V_{q}$ is a regular spherical dodecahedron.
###### Proof.
Let $1$ be the identity of $S^{3}$. By Lemma 5.2 it suffices to prove the
lemma for $V_{1}$. For any $q\in\mathcal{D}^{*}$, not equal to $1$, we define
$S(q)\subset S^{3}$ to be the great sphere of points equidistant from $1$ and
$q$. Note that $V_{1}$ is obtained by cutting $S^{3}$ along all of the $S(q)$
and taking the closure of the component that contains $1$.
By Corollary 4.4 and by Lemmas 2.7 and 2.4 there are twelve quaternions
$\\{q_{i}\\}_{i=1}^{12}$ in $\mathcal{D}^{*}$ that are distance $\pi/5$ from
$1$. Define $U$ by cutting $S^{3}$ along the spheres $S(q_{i})$ only, and then
taking the closure of the component containing $1$. By Lemma 5.2 the twisted
action of $\mathcal{D}^{*}$ preserves the $q_{i}$; we deduce $U$ is a regular
spherical dodecahedron. Also, $U$ contains $V_{1}$.
###### Claim.
$U=V_{1}$.
###### Proof.
We must show, for $p\in\mathcal{D}^{*}$, if $p$ is not one of the $q_{i}$ then
the sphere $S(p)$ misses $U$. We will only do this for a single lift of a
vertex rotation of $D$, leaving the other cases as exercises.
Take $v$, $f$, $f^{\prime}$, and $f^{\prime\prime}$ as defined in Section 5.2.
Fix the following quaternions in $\mathcal{D}^{*}$
$\displaystyle p$ $\displaystyle=\cos\pi/3+v\cdot\sin\pi/3,$ $\displaystyle q$
$\displaystyle=\cos\pi/5+f\cdot\sin\pi/5$
and define $q^{\prime}$ and $q^{\prime\prime}$ similarly with respect to
$f^{\prime}$ and $f^{\prime\prime}$. Thus $p$ is the desired lift of the
vertex rotation about $v$. Note $q,q^{\prime},q^{\prime\prime}\in\\{q_{i}\\}$
are lifts of face rotations. By Lemma 2.4 the elements $q$, $q^{\prime}$, and
$q^{\prime\prime}$ are all distance $\pi/5$ from $1$ in $S^{3}$. We compute
$\displaystyle(q^{-1})\cdot q^{\prime}$
$\displaystyle=(\cos\pi/5-f\cdot\sin\pi/5)(\cos\pi/5+f^{\prime}\cdot\sin\pi/5)$
$\displaystyle=\cos^{2}\pi/5+(-f+f^{\prime})\cos\pi/5\sin\pi/5-ff^{\prime}\cdot\sin^{2}\pi/5.$
Expanding the product $ff^{\prime}$ and applying Equation 4.8, we find the
real part of $(q^{-1})\cdot q^{\prime}$ is also equal to $\cos\pi/5$. Since
the twisted action of $p$ permutes $q,q^{\prime},q^{\prime\prime}$ cyclically,
deduce that $1,q,q^{\prime},q^{\prime\prime}$ are the vertices of a regular
spherical tetrahedron, $T$. Let $t=\operatorname{center}(T)$ be the spherical
center - the radial projection of the Euclidean center of $T$. It follows that
$t$ is a vertex of $U$. We claim $t$ is the point of $U$ closest to $p$. Note
the real part of $t$ is $\frac{1}{2}\sqrt{1+3\cos\pi/5}$. Since this is
greater than $\cos\pi/6$ deduce that $S(p)$ does not cut $t$ off of $U$, and
thus $S(p)$ misses $U$, as desired. We leave the analysis of the other point
of $\mathcal{D}^{*}$ as exercises. This proves the claim. ∎
This completes the proof of Lemma 5.3. ∎
###### Definition 5.4.
The _$120$ –cell_ $C$ is the convex hull, taken in $\mathbb{H}$, of the
vertices of $\mathcal{T}_{120}$.
This completes the construction of the $120$–cell.
Figure 5.2. The half of the one-skeleton of the tiling $\mathcal{T}_{120}$.
This is the half nearest to the south pole, after cell-centered stereographic
projection to $\mathbb{R}^{3}$. See also [16, color plate].
###### Theorem 5.5.
The $120$–cell $C$ is a regular polytope.
###### Proof.
We must show that the group
$\mathcal{C}=\operatorname{Sym}(\mathcal{T}_{120})$ acts transitively on the
flags of $C$. Now, the flags of $C$ are four-simplices with one vertex at the
origin. These are in one-to-one correspondence with the flag tetrahedra of
$\mathcal{T}_{120}$. The group $\mathcal{C}$ acts on these two sets and
preserves the correspondence. Thus it suffices to fix a right-handed flag
tetrahedron $T$ of $V_{1}$ and to prove that any other flag $T^{\prime}$ in
$\mathcal{T}_{120}$ can be taken to $T$. By Lemma 5.2 we may use the left
action of $\mathcal{D}^{*}$ to transport $T^{\prime}$ into $V_{1}$. Now, if
$T^{\prime}$ is also right handed then we may use the twisted action of
$\mathcal{D}^{*}$ to send $T^{\prime}$ to $T$. There are several ways to deal
with left-handed flags; we resort to a simple trick. The conjugation map
$a+bi+cj+dk\mapsto a-bi-cj-dk$
is the product of three reflections, so is orientation reversing in
$\mathbb{H}$. It preserves $S^{3}$ and is again orientation reversing there.
Since $\mathcal{D}^{*}$ is a group of quaternions, it is closed under
conjugation. Since the tiling $\mathcal{T}_{120}$ is metrically defined in
terms of $\mathcal{D}^{*}$, it is also invariant under conjugation. This
reverses the handedness of flags, and we are done. ∎
###### Corollary 5.6.
The spherical dodecahedra of $\mathcal{T}_{120}$ have dihedral angle $2\pi/3$.
###### Proof.
Again it suffices to check this for $V_{1}$, the cell about $1$. With notation
as in the proof of Lemma 5.3: let $1$, $q$, and $q^{\prime}$ be elements of
$\mathcal{D}^{*}$, all at distance $\pi/5$ from each other. Let $T$ be the
regular spherical triangle having $1$, $q$, and $q^{\prime}$ as vertices. The
center $c=\operatorname{center}(T)$ is equidistant from the vertices of $T$.
Also, there is a reflection symmetry of $\mathcal{T}_{120}$ that fixes $T$
pointwise. It follows that $V_{1}$, $V_{q}$, and $V_{q^{\prime}}$ share an
edge and this edge is perpendicular to $T$. As all of these cells are
isometric regular spherical dodecahedra, the corollary follows. ∎
###### Remark 5.7.
Note the $24$–cell can be constructed in the same way as the $120$–cell, by
starting with the regular tetrahedron in place of the dodecahedron. The
symmetries of the cube (equivalently, octahedron) do not give rise to a
regular four-dimensional polytope; the reason can be traced to the failure of
the inequality at the heart of Lemma 5.3.
## 6\. Combinatorics of the $120$–cell
With the $120$–cell in hand, we turn to the combinatorics of
$\mathcal{T}_{120}$, the spherical $120$–cell. By Lemma 5.3 and Corollary 5.6,
the cells of $\mathcal{T}_{120}$ are regular spherical dodecahedra with
dihedral angle $2\pi/3$.
### 6.1. Layers of dodecahedra
Recall that the centers of the cells of $\mathcal{T}_{120}$ are the elements
of the binary dodecahedral group $\mathcal{D}^{*}$. Recall also that Corollary
4.4 lists the elements of $\mathcal{D}$, ordered by their angle of rotation.
We deduce that the cells of $\mathcal{T}_{120}$ divide into spherical layers,
ordered by their distance from $1$. Figure 6.1 displays the stereographic
projections of the first five layers.
(a)
(b)
(c)
(d)
(e)
Figure 6.1. The five spheres in the southern hemisphere, starting with the
pole.
In the table below, for each layer $L$ we list the spherical distance between
$1$ and the cell-centers of $L$, the number of cells in $L$, the type of the
covered rotation in $\operatorname{SO}(3)$, and the name of $L$. See also [14,
page 176].
angle | number of cells | type of rotation | name of layer
---|---|---|---
$0$ | 1 | identity | south pole
$\pi/5$ | 12 | face | antarctic sphere
$\pi/3$ | 20 | vertex | southern temperate
$2\pi/5$ | 12 | face | tropic of Capricorn
$\pi/2$ | 30 | edge | equatorial sphere
$3\pi/5$ | 12 | face | tropic of Cancer
$2\pi/3$ | 20 | vertex | northern temperate
$4\pi/5$ | 12 | face | arctic sphere
$\pi$ | 1 | identity | north pole
### 6.2. Rings of dodecahedra
With notation as in Section 5.2, suppose that $q\in\mathcal{D}^{*}$ is the
lift of the face rotation $A\in\mathcal{D}$ of angle $2\pi/5$ about the vector
$f$. Let $R={\langle{q}\rangle}<\mathcal{D}^{*}$ be the resulting cyclic group
of order ten. Note that $R$ has twelve right cosets in $\mathcal{D}^{*}$. We
call the cosets _rings_ because each corresponding union of spherical
dodecahedra forms a solid torus in $S^{3}$. We give the rings the following
names: $R$ is the _spine_ , $R^{\operatorname{eq}}$ is the _equator_ ,
$R^{\operatorname{in}}_{0}$ to $R^{\operatorname{in}}_{4}$ are the _inner
rings_ , and $R^{\operatorname{out}}_{0}$ to $R^{\operatorname{out}}_{4}$ are
the _outer rings_. The names are justified by the following proposition.
###### Proposition 6.1.
The rings meet the spherical layers of $\mathcal{T}_{120}$ as follows.
layer | number of cells | spine | equator | remaining | inner | outer
---|---|---|---|---|---|---
south pole | 1 | 1 | 0 | 0 | 0 | 0
antarctic sphere | 12 | 2 | 0 | 10 | 2 | 0
southern temperate | 20 | 0 | 0 | 20 | 2 | 2
tropic of Capricorn | 12 | 2 | 0 | 10 | 0 | 2
equatorial sphere | 30 | 0 | 10 | 20 | 2 | 2
tropic of Cancer | 12 | 2 | 0 | 10 | 0 | 2
northern temperate | 20 | 0 | 0 | 20 | 2 | 2
arctic sphere | 12 | 2 | 0 | 10 | 2 | 0
north pole | 1 | 1 | 0 | 0 | 0 | 0
The column titled “remaining” counts the number of cells left in each layer
after the spinal and equatorial rings have been removed.
###### Proof.
Let $P$ be the pentagon of $\mathcal{T}_{D}$ with center $f$ and let $-P$ be
the antipodal pentagon to $P$, which exists by Lemma 4.3. Let
$\mathcal{D}_{P}<\mathcal{D}$ be the stabilizer of $\pm P=P\cup-P$.
###### Claim.
The stabilizer $\mathcal{D}_{P}$ is a dihedral group of order ten: it contains
the rotations of $P$, contains five edge rotations perpendicular to $f$, and
acts dihedrally on the plane $f^{\perp}$.
###### Proof.
Consider the full stabilizer $\Delta_{P}$ of $\pm P$ inside of
$\operatorname{Sym}(D)$. Counting the flags of $\pm P$ deduce that
$\Delta_{P}$ has at most $20$ elements. Also, $\Delta_{P}$ contains the
rotations of $P$, the antipodal map, and also the five reflections fixing $f$
and preserving $P$. The composition of the antipodal map with a reflection is
an edge rotation perpendicular to $f$. Since $\pm P$ has ten right-handed
flags, the claim follows. ∎
Let $\mathcal{D}_{P}^{*}$ be the lift of $\mathcal{D}_{P}$ to
$\mathcal{D}^{*}$. So $\mathcal{D}_{P}^{*}\subset S^{3}$ is a binary dihedral
group. The spinal ring $R={\langle{q}\rangle}$ is an index two subgroup of
$\mathcal{D}_{P}^{*}$. The equatorial ring $R^{\operatorname{eq}}$ is the
unique coset of $R$ inside of $\mathcal{D}_{P}^{*}$. By the claim immediately
above, every element of $R^{\operatorname{eq}}$ is a lift of an edge rotation.
This verifies the spine and equator columns in the table.
For any great circle $C$ and any great sphere $S\subset S^{3}$ the
intersection $C\cap S$ is either two antipodal points, or all of $C$. When $S$
is round, but not great, $C\cap S$ is zero, one, or two points. As noted
immediately after Equation 2.3 the elements of $R={\langle{q}\rangle}$ lie on
a great circle through the identity. Since the right action of $S^{3}$ on
itself is via isometry, the right cosets $R\cdot p$ also lie on great circles.
Since these great circles are disjoint, deduce $R^{\operatorname{eq}}$ is the
only one of them contained in the equatorial sphere. The remainder meet the
equatorial sphere in at most two points.
By counting intersections, deduce the inner and outer rings meet each of the
temperate and the equatorial spheres in exactly two elements. This accounts
for six elements of each ring; we must pin down the remaining four.
Recall the definition of $q^{\prime}$ from the proof of Lemma 5.3: the
quaternion $q^{\prime}$ is the lift of a face rotation about $f^{\prime}$,
where $f^{\prime}$ is the center of a face $P^{\prime}$ of the tiling
$\mathcal{T}_{D}$, and where $P^{\prime}$ is adjacent to the face $P$. The
inner rings are the cosets $R^{\operatorname{in}}_{i}=R\cdot
q^{\prime}q^{-i}$, for $i=0,1,2,3,4$. As shown in the proof of Lemma 5.3, the
real part of $(q^{-1})\cdot q^{\prime}$ is $\cos(\pi/5)$. Thus
$R^{\operatorname{in}}_{0}=R\cdot q^{\prime}$ meets the antarctic sphere in
exactly two elements, namely $q^{\prime}$ and $(q^{-1})\cdot q^{\prime}$.
Note also that $R^{\operatorname{in}}_{i}=R\cdot q^{\prime}q^{-i}=q^{i}(R\cdot
q^{\prime})q^{-i}=\phi_{q}^{i}(R^{\operatorname{in}}_{0})$. That is, the
$i^{\rm th}$ coset is obtained from $R^{\operatorname{in}}_{0}$ via the
twisted action. It is now an exercise to show that all of the
$R^{\operatorname{in}}_{i}$ are distinct.
Note that all cosets are invariant under the antipodal map, because $-1\in R$.
This implies $R^{\operatorname{in}}_{0}$ also meets the arctic sphere in two
points. This accounts for all ten elements of $R^{\operatorname{in}}_{0}$.
Since $\phi_{q}$ fixes each spherical layer setwise, and since
$R^{\operatorname{in}}_{i}=\phi_{q}^{i}(R^{\operatorname{in}}_{0})$, the inner
column of the table is verified.
There are only twenty elements of $D^{*}$ left to be accounted for, and these
all lie in the tropics. It follows that each outer ring (the cosets
$R^{\operatorname{out}}_{i}$) contains two elements from each of the tropics.
This verifies the outer column of the table. ∎
###### Remark 6.2.
The _Hopf fibration_ is the partition of $S^{3}$ into cosets of the one-
parameter subgroup $\\{\exp(i\alpha)\\}$. After a rotation, we see that the
cosets of $R$ give a combinatorial Hopf fibration: they divide the $120$–cell
into 12 rings of ten dodecahedra each. The centers of the rings lie on 12
great circles of the Hopf fibration. Note also that the quotient space of the
Hopf fibration is homeomorphic to $S^{2}$. In similar fashion there is a kind
of combinatorial map from the $120$–cell to the dodecahedron.
## 7\. Rings to ribs
We describe the ribs of Quintessence: a collection of puzzle pieces, in
$\mathbb{R}^{3}$, that combined in various ways to produce burr puzzles. The
puzzle pieces are based on the rings of spherical dodecahedra described in
Section 6.2. We use stereographic projection, $\rho$, defined in Section 2.3,
to move the pieces into $\mathbb{R}^{3}$, where we can 3D print the resulting
ribs.
Following the notation of Section 2.3 we have
$\frac{d\rho}{d\alpha}=\frac{1}{1+\cos(\alpha)}\cdot u.$
In particular, if $e^{u\alpha}$ is near the south pole then $\alpha$ is close
to zero and stereographic projection shrinks objects by a factor of
approximately two. If $e^{u\alpha}$ is near the equatorial sphere then
$\alpha$ is close to $\pi/2$ and stereographic projection leaves sizes
essentially unchanged. However, if $e^{u\alpha}$ approaches the north pole
then $\alpha$ approaches $\pi$ and sizes blow up. Thus a dodecahedron of the
$120$–cell close to the south pole shrinks slightly, and a dodecahedron close
to the north pole becomes much larger.
All of the calculations so far have been dimensionless. When we wish to 3D
print a rib, we have to choose a scale $\lambda$, say in millimetres or
inches, corresponding to a unit distance in the image of $\rho$. Many
considerations need to be taken into account in choosing $\lambda$; the scale
is sensitive to the design of the ribs. However, two issues are clear: a large
feature on a rib causes the cost to grow with the cube of $\lambda$ while a
very small feature may be too fragile or may fall below the resolution of the
printer.
These two issues are in tension, and lead to the general principle that
features that are identical in $S^{3}$ should have sizes in bounded ratio in
$\mathbb{R}^{3}$, after projection. In this particular case, the features of
the ribs are the dodecahedra. The principle tells us that we should not be
using dodecahedra that are too close to the north pole. However, the ratio of
two between sizes near the equator and near the south pole is acceptable.
(a) Spine.
(b) Inner 6.
(c) Outer 6.
(d) Equator.
Figure 7.1. The colouring of the cells is by layer, consistent with the
convention of Figure 6.1. We obtain the inner 4 and outer 4 ribs by deleting
the equatorial cells.
So, we remove from our rings any dodecahedra that are closer to the north pole
than the equator, giving us the _spine_ , _inner 6_ and _outer 6_ ribs.
Experimentation shows that many interesting constructions require even shorter
ribs; hence we also make the _inner 4_ and _outer 4_ ribs. These are the
result of removing the two equatorial dodecahedra from the inner 4 and outer
6\. The equatorial ring can be printed as is, but again experimentation shows
that more puzzles are possible if we break the equatorial ring into two ribs
of five dodecahedra each. See Figure 1.2 as well as Figure 7.1.
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)
(j)
(k)
Figure 7.2. Building the Dc45 Meteor: start with just the spine, in Figure
7.2a. One at a time add five copies of the inner 4 rib in Figures 7.2b through
7.2f. Then add five copies of the outer 4 rib, as in Figures 7.2g through
7.2k.
With the spine and short ribs in hand, we can build, in $\mathbb{R}^{3}$, the
stereographic projection of (almost) one-half of the $120$–cell. We call the
resulting puzzle the _Dc45 Meteor_ ; its construction is shown in Figure 7.2.
The spine and ribs are arranged according to the combinatorial Hopf fibration
(Remark 6.2). Since all dodecahedra near the south pole are retained, and all
dodecahedra near the north pole are discarded, the result looks very much like
Figure 5.2: one-half of the $120$-cell.
It is not at all obvious that the puzzle can be constructed in Euclidean space
using physical objects. However, when printed in plastic the Meteor _is_
possible to assemble. Also, when complete it holds together with no other
support. For photos see Dc45 Meteor in Appendix A. The ribs apparently need a
small amount of flexibility; we have not been able to solve a similar puzzle
(the Dc30 Ring) when printed in a steel/bronze composite.
It came as a surprise to us that there are numerous other burr puzzles based
on these ribs, and most do not derive from the combinatorial Hopf fibration.
We list many of our discoveries in Appendix A. In the remainder of this
section we derive a combinatorial restriction on the numbers of ribs that can
be used in any given burr puzzle. This theorem is sharp, as shown by the
examples in Appendix A.
###### Theorem 7.1.
1. (1)
At most six inner ribs are used in any puzzle.
2. (2)
At most six outer ribs are used in any puzzle.
3. (3)
At most ten inner and outer ribs are used in any puzzle.
###### Proof.
The stereographic projection map $\rho$ is equivariant: $\rho$ transports the
twisted action on $S^{3}$ to the $\operatorname{SO}(3)$ action on
$\mathbb{R}^{3}$. That is, $\rho$ respects the $S^{2}$ symmetry about the
identity in $S^{3}$. Thus any two cells in a given sphere (antarctic, southern
temperate, and so on) are identical, after projection, in $\mathbb{R}^{3}$.
Also, any pair of cells in different layers are different, due to the growth
of $d\rho/d\alpha$.
Examining the table in Proposition 6.1, we see that the each inner rib
contains exactly two antarctic cells. By the table in Section 6.1, there are
exactly 12 cells in the antarctic sphere. Part (1) follows. We prove part (2)
by examining the tropic of Capricorn and we prove part (3) using the southern
temperate zone. A color-coded guide is provided in Figure 7.1. ∎
## 8\. Leonardo da Vinci’s polytopes
Figure 8.1. The dodecahedron, as drawn by Leonardo da Vinci [12, Folio CV
recto].
If we use injection moulding to make the ribs, then the simplest route would
be to realise each rib as a union of solid dodecahedra. However, since we are
3D printing the ribs, we are able to reduce costs by hollowing out the
dodecahedra. Our design is closely related to Leonardo da Vinci’s technique
for drawing polytopes. See Figure 8.1.
Da Vinci’s design retains all of the symmetry of the dodecahedron itself.
Since the dodecahedron is regular, we need only determine the design inside of
a single flag tetrahedron. Then the symmetries of the dodecahedron copy this
geometry to all other flag tetrahedra, recreating the entire design. We do
something very similar, by constructing our design inside of a flag polytope
of the spherical 120-cell, $\mathcal{T}_{120}$.
We have two versions of the design in the flag tetrahedron for
$\mathcal{T}_{120}$, depending on whether or not the flag meets a boundary
pentagonal face of the rib, or meets an internal pentagon between two adjacent
dodecahedra. See Figures 8.2 and 8.3. In the former case, we add a surface in
the pentagonal face to separate the inside of the rib from the outside. This
is not necessary in the latter case. The “outer” parts of the geometry of the
ribs are identical (in $S^{3}$) for all dodecahedra in our ribs. For reasons
of cost and strength, we slightly thicken the internal geometry of the smaller
dodecahedra closer to the south pole, and thin that of those further from the
south pole. Note that Figure 5.2 is modelled similarly, using only the
internal design.
(a) Geometry for an external face of a puzzle piece within the flag
tetrahedron.
(b) Geometry for an internal face of a puzzle piece within the flag
tetrahedron.
Figure 8.2. The two versions of the flag polytope design. Here the flag
polytope is the tetrahedron drawn with a dashed line. We show only three faces
of the central dodecahedron of the stereographic projection to
$\mathbb{R}^{3}$.
(a) Twenty copies of the external design, forming two faces of a dodecahedron
drawn in the Da Vinci style.
(b) Twenty copies of the external design and twenty copies of the internal
design, forming two faces of two adjacent dodecahedra, and the face between
those dodecahedra, drawn in the Da Vinci style.
Figure 8.3. Two examples of how the external and internal face designs fit
together to form the geometry of the rib puzzle pieces.
## References
* [1] Arnaud Chéritat. Le $120$. CNRS, 2012. http://images.math.cnrs.fr/Le-120.html.
* [2] Stewart Coffin. Geometric puzzle design. A K Peters Ltd., Wellesley, MA, 2007.
* [3] John H. Conway and Derek A. Smith. On quaternions and octonions: their geometry, arithmetic, and symmetry. A K Peters Ltd., Natick, MA, 2003.
* [4] Harold S. M. Coxeter. Regular complex polytopes. Cambridge University Press, London, 1974.
* [5] Bill Cutler. A computer analysis of all $6$–piece burrs. 1994\. http://home.comcast.net/$\sim$billcutler/docs/CA6PB/index.html.
* [6] David B. A. Epstein and Carlo Petronio. An exposition of Poincaré’s polyhedron theorem. Enseign. Math. (2), 40(1-2):113–170, 1994.
* [7] Euclid. The thirteen books of Euclid’s Elements translated from the text of Heiberg. Vol. I: Introduction and Books I, II. Vol. II: Books III–IX. Vol. III: Books X–XIII and Appendix. Dover Publications Inc., New York, 1956. Translated with introduction and commentary by Thomas L. Heath, 2nd ed.
* [8] Leonhard Euler. Formulae generales pro translatione quacunque corporum rigidorum. Novi Commentarii academiae scientiarum imperialis Petropolitanae, 20:189–207, 1776. E478.
* [9] L. Fejes Tóth. Regular figures. A Pergamon Press Book. The Macmillan Co., New York, 1964.
* [10] Robert Langlands. The practice of mathematics, 1999. http://www.math.duke.edu/langlands/.
* [11] August Ferdinand Möbius. Theorie der symmetrischen figuren. In Gesammelte Werke, volume 2, pages 561–708. Hirzel, Leipzig, 1886.
* [12] Luca Pacioli. De Divina Proportione. 1498\. Manuscript held by Biblioteca Ambrosiana di Milano. Illustrations by Leonardo da Vinci.
* [13] Bob Palais, Richard Palais, and Stephen Rodi. A disorienting look at Euler’s theorem on the axis of a rotation. Amer. Math. Monthly, 116(10):892–909, 2009.
* [14] Duncan M. Y. Sommerville. An introduction to the geometry of $n$ dimensions. Dover Publications Inc., New York, 1958.
* [15] John Stillwell. The story of the 120-cell. Notices Amer. Math. Soc., 48(1):17–24, 2001.
* [16] John M. Sullivan. Generating and rendering four-dimensional polytopes. The Mathematica Journal, 1:76–85, 1991. http://torus.math.uiuc.edu/jms/Papers/dodecaplex/.
* [17] William P. Thurston. Three-dimensional geometry and topology. Vol. 1, volume 35 of Princeton Mathematical Series. Princeton University Press, Princeton, NJ, 1997. Edited by Silvio Levy.
* [18] William C. Waterhouse. The discovery of the regular solids. Arch. History Exact Sci., 9(3):212–221, 1972.
* [19] Günter M. Ziegler. Lectures on polytopes, volume 152 of Graduate Texts in Mathematics. Springer-Verlag, New York, 1995.
## Appendix A Catalog
When trying to build a puzzle out of the ribs, there is a spectrum of
possibilities. At one end there are constructions that hold together so
loosely that a small tap causes them to fall apart. At the other end there are
puzzles that hold together so tightly that there seems to be no way to
assemble them without applying large amounts of force. Below we list those
puzzles, avoiding both ends of this spectrum, that we find visually pleasing.
Please let us know of any others you find!
###### Remark.
The designation Dc$N$ at the beginning of each puzzle stands for “dodecahedral
cell-centered” and $N$ counts the number of cells. Using other polytopes, such
as the $600$–cell, would lead to puzzles with different unit cells, such as
tetrahedra. Using other projection points would lead to, say, vertex-centered
puzzles.
Dc24 Star |
---|---
$6\times\text{inner 4}$
Up to three ribs
can be replaced
by inner 6s.
Dc24 Pulsar |
$6\times\text{inner 4}$
Any number of ribs
can be replaced by
inner 6s.
Dc29 Space Invader |
$2\times\text{inner 6}$
$2\times\text{outer 6}$
$1\times\text{spine}$
Can add $2\times\text{equator}$.
Dc30 Star |
$3\times\text{outer 4}$
$3\times\text{outer 6}$
Dc30 Ring |
$5\times\text{outer 6}$
Replace all ribs with
inner 6s to get the
Inner Ring.
Dc30 Comet |
$5\times\text{outer 6}$
Add a spine and one
inner 4 to make the
Comet more rigid.
Dc36 Alien |
---|---
$3\times\text{inner 6}$
$3\times\text{outer 6}$
Either set of 6s can
be replaced by 4s.
Dc36 Pulsar |
$6\times\text{outer 6}$
Up to three ribs
can be replaced
by outer 4s.
Dc42 Alien |
$6\times\text{outer 4}$
$3\times\text{inner 6}$
Dc45 Meteor |
$5\times\text{inner 4}$
$5\times\text{outer 4}$
$1\times\text{spine}$
There are six ways
to build this.
Dc50 Galaxy |
$5\times\text{inner 4}$
$5\times\text{outer 4}$
$2\times\text{equator}$
Dc75 Meteor |
$5\times\text{inner 6}$
$5\times\text{outer 6}$
$1\times\text{spine}$
$2\times\text{equator}$
|
arxiv-papers
| 2013-10-14T02:56:46 |
2024-09-04T02:49:52.332317
|
{
"license": "Creative Commons Zero - Public Domain - https://creativecommons.org/publicdomain/zero/1.0/",
"authors": "Saul Schleimer, Henry Segerman",
"submitter": "Saul Schleimer",
"url": "https://arxiv.org/abs/1310.3549"
}
|
1310.3579
|
ON THE REGULARITY OF THE SOLUTIONS FOR CAUCHY PROBLEM
OF INCOMPRESSIBLE 3D NAVIER-STOKES EQUATION
Qun Lin School of Mathematical Sciences, Xiamen University, P.
R. China
11cm Abstract. In this paper we will prove that the vorticity
belongs to $L^\infty (0,T;\;L^2({\mathbb R}^3))$ for the Cauchy
problem of 3D incompressible Navier-Stokes equation, then the
existence of a global smooth solution is obtained. Our approach is
to construct a set of auxiliary problems to approximate the original
one for vorticity equation.
Keywords. Navier-Stokes equation; Regularity;
AMS subject classifications. 35Q30 76N10
1. Introduction
Let $\mathscr{D} ({\mathbb R}^3)$ be the space of $C^\infty $
functions with compact support contained in ${\mathbb R}^3$. Some
basic spaces will be used in this paper:
\begin{equation*}
\begin{split}
&{\cal V}=\{\,u\in \mathscr{D} ({\mathbb R}^3),\;\;\mbox{div}u=0\,\} \\
R}^3) \\
R}^3) \\
\end{split}
\end{equation*}
The velocity-pressure form for Navier- Stokes equation is
\begin{equation}
\begin{split}
&\partial _t u_1 + u_1 \partial _{x_1 } u_1 + u_2 \partial _{x_2 } u_1
+ u_3 \partial _{x_3 } u_1 + \partial _{x_1 } p=\Delta u_1 \\
&\partial _t u_2 + u_1 \partial _{x_1 } u_2 + u_2 \partial _{x_2 } u_2 + u_3
\partial _{x_3 } u_2 + \partial _{x_2 } p=\Delta u_2 \\
&\partial _t u_3 + u_1 \partial _{x_1 } u_3 + u_2 \partial _{x_2 } u_3 + u_3
\partial _{x_3 } u_3 + \partial _{x_3 } p=\Delta u_3 \\
\end{split}
\end{equation}
with the initial conditions $\left. {(u_1 ,u_2 ,u_3 )} \right|_{t=0}
=(u_{10} ,u_{20} ,u_{30} )(x)$ and the incompressible condition :
\[
\partial _{x_1 } u_1 +\partial _{x_2 } u_2 +\partial _{x_3 } u_3 =0
\]
We will here recall the global $L^2-$estimate from [4].
In the sequel, it is assumed that the initial value $u_0 $ satisfies
the following conditions:
\begin{equation}
\begin{split}
\sum\limits_{i=1}^3 {\left\| {u_{i0} } \right\|_{L^2({\mathbb
R}^3)}^2 } <+\infty ,\quad \;\sum\limits_{i=1}^3 {\left\| {\partial
_t u_{i0} } \right\|_{L^2({\mathbb R}^3)}^2 } <+\infty ,\quad
\;\sum\limits_{i=1}^3 {\left\| {\nabla u_{i0} }
\right\|_{L^2({\mathbb R}^3)}^2 } <+\infty
\end{split}
\end{equation}
For the handling the initial value problem, a weighted function is
\begin{equation*}
\begin{split}
\theta_{r} =\left\{ {{\begin{array}{*{20}c}
&{e^{-\;\frac{\left| x \right|^2 }{r^2-\left| x \right|^2}}\quad \left| x \right|<r}
\hfill \\
&{\quad \;0 \qquad \quad\; \left| x \right|\ge r} \hfill \\
\end{array} }} \right.\quad \quad (r>0)
\end{split}
\end{equation*}
which is of the properties:
\begin{equation}
\begin{split}
\theta _{r} \to 1,\quad \quad \partial _i \theta _{r} \to 0,\quad
\quad \partial _i \partial _j \theta _{r} \to 0
\end{split}
\end{equation}
as $ r\to +\infty $.
Moreover, let $v=\theta _{r} u$, we still have
\begin{equation}
\begin{split}
&\partial _i v=u\,\partial _i \theta _{r} +\theta _{r}
\,\partial _i u \\
&\partial _i^2 v=u\,\partial _i^2 \theta _{r} +2\,\partial _i
\theta _{r} \partial _i u+\theta _{r} \,\partial _i^2 u \\
&\partial _i \partial _j v=u\,\partial _i \partial _j \theta _{r}
+\partial _j \theta _{r} \partial _i u+\partial _i \theta _{r} \partial _j u+\theta _{r} \,\partial _i \partial _j u \\
\end{split}
\end{equation}
\begin{equation*}
\begin{split}
&\int_{{\mathbb R}^3} {\theta _{r} u_i (u_1 \partial _{x_1 } u_i +u_2
\partial _{x_2 } u_i +u_3 \partial _{x_3 } u_i )} =\frac{1}{2}\int_{{\mathbb
R}^3} {\theta _{r} (u_1 \partial _{x_1 } u_i^2 +u_2
\partial _{x_2
} u_i^2 +u_3 \partial _{x_3 } u_i^2 )} \\
&=-\frac{1}{2}\int_{{\mathbb R}^3} {u_i^2 (\partial _{x_1 } (\theta _{r} u_1 )+\partial _{x_2 } (\theta _{r} u_2 )+\partial
_{x_3 }
(\theta _{r} u_3 ))} \\
&=-\frac{1}{2}\int_{{\mathbb R}^3} {\theta _{r} u_i^2 (\partial _{x_1 }
u_1 +\partial _{x_2 } u_2 +\partial _{x_3 } u_3 )}
-\frac{1}{2}\int_{{\mathbb R}^3} {u_i^2 (u_1 \partial _{x_1 } \theta
_{r} +u_2 \partial _{x_2
} \theta _{r} +u_3 \partial _{x_3 } \theta _{r} )} \\
&=-\frac{1}{2}\int_{{\mathbb R}^3} {u_i^2 (u_1 \partial _{x_1 } \theta _{r} +u_2 \partial _{x_2 } \theta _{r} +u_3 \partial _{x_3 }
\theta _{r} )} ,\quad \quad i=1,2,3
\end{split}
\end{equation*}
Taking $ r\to +\infty $ we get
\[
\int_{{\mathbb R}^3} {u_i (u_1 \partial _{x_1 } u_i +u_2 \partial
_{x_2 } u_i +u_3 \partial _{x_3 } u_i )} =0,\quad \quad i=1,2,3
\]
in the same way,
\[
\int_{{\mathbb R}^3} {(u_1 \partial _{x_1 } p+u_2 \partial _{x_2 }
\partial _{x_3 } p)} =0
\]
\[
\int_{{\mathbb R}^3} {u_i \Delta u_i } =\int_{{\mathbb R}^3} {u_i
(\partial _{x_1 }^2 u_i +\partial _{x_2 }^2 u_i +\partial _{x_3 }^2
u_i )} =-\int_{{\mathbb R}^3} {((\partial _{x_1 } u_i )^2+(\partial
_{x_2 } u_i )^2+(\partial _{x_3 } u_i )^2)}
\]
\begin{equation*}
\begin{split}
&\int_{{\mathbb R}^3} {u_1 \partial _t \,u_1 } +\int_{{\mathbb R}^3} {u_1 (u_1
\partial _{x_1 } u_1 +u_2 \partial _{x_2 } u_1 +u_3 \partial _{x_3 } u_1 )}
+\int_{{\mathbb R}^3} {u_1 \partial _{x_1 } p} =\int_{{\mathbb R}^3}
{u_1 \Delta
u_1 } \\
&\int_{{\mathbb R}^3} {u_2 \partial _t \,u_2 } +\int_{{\mathbb R}^3} {u_2 (u_1
\partial _{x_1 } u_2 +u_2 \partial _{x_2 } u_2 +u_3 \partial _{x_3 } u_2 )}
+\int_{{\mathbb R}^3} {u_2 \partial _{x_2 } p} =\int_{{\mathbb R}^3}
{u_2 \Delta u_2
} \\
&\int_{{\mathbb R}^3} {u_3 \partial _t \,u_3 } +\int_{{\mathbb R}^3} {u_3 (u_1
\partial _{x_1 } u_3 +u_2 \partial _{x_2 } u_3 +u_3 \partial _{x_3 } u_3 )}
+\int_{{\mathbb R}^3} {u_3 \partial _{x_3 } p} =\int_{{\mathbb R}^3}
{u_3 \Delta
u_3 } \\
\end{split}
\end{equation*}
so that
\begin{equation*}
\begin{split}
&\frac{1}{2}\partial _t \;\int_{{\mathbb R}^3} {(u_1^2 +u_2^2 +u_3^2 )}
+\;\int_{{\mathbb R}^3} {((\partial _{x_1 } u_1 )^2+(\partial _{x_2
} u_1
)^2+(\partial _{x_3 } u_1 )^2+} \\
&+(\partial _{x_1 } u_2 )^2+(\partial _{x_2 } u_2 )^2+(\partial _{x_3 } u_2
)^2+(\partial _{x_1 } u_3 )^2+(\partial _{x_2 } u_3 )^2+(\partial _{x_3 }
u_3 )^2)=0 \\
\end{split}
\end{equation*}
it follows that
\begin{equation*}
\begin{split}
&\int_{{\mathbb R}^3} {(u_1^2 +u_2^2 +u_3^2 )} +2\;\int_0^T {(\,\left\| {\nabla
u_1 } \right\|_{L^2({\mathbb R}^3)}^2 +} \left\| {\nabla u_2 }
\right\|_{L^2({\mathbb R}^3)}^2 +\left\| {\nabla u_3 }
\right\|_{L^2({\mathbb
R}^3)}^2 ) \\
&\quad \quad =\int_{{\mathbb R}^3} {(u_{10}^2 +u_{20}^2 +u_{30}^2 )} \\
\end{split}
\end{equation*}
Hence from (2) we have
\begin{equation}
\begin{split}
&\mathop {\sup }\limits_{t\in (0,T)} \;\;\int_{{\mathbb R}^3} {(u_1^2 +u_2^2
+u_3^2 )} <+\infty \\
&\int_0^T {(\,\left\| {\nabla u_1 } \right\|_{L^2({\mathbb R}^3)}^2 +} \left\|
{\nabla u_2 } \right\|_{L^2({\mathbb R}^3)}^2 +\left\| {\nabla u_3 }
\right\|_{L^2({\mathbb R}^3)}^2 )<+\infty \\
\end{split}
\end{equation}
Above $u$ can be interpreted as the Galerkin approximation of the
solution, but (5) is also true for the solution of problem (1).
2. Auxiliary Problems
For the 3D regularity, we just need to prove that the vorticity
belongs to $L^\infty (0,T;L^2({\mathbb R}^3))$. The
vorticity-velocity form for Navier-Stokes equation is
\begin{equation}
\begin{split}
&\partial _t \omega _1 +u_1 \partial _{x_1 } \omega _1 +u_2 \partial
_{x_2 } \omega _1 +u_3 \partial _{x_3 } \omega _1 -\omega _1
\partial _{x_1 } u_1 -\omega _2 \partial _{x_2 } u_1 -\omega _3
\partial _{x_3 } u_1
=\Delta \omega _1 \\
&\partial _t \omega _2 +u_1 \partial _{x_1 } \omega _2 +u_2 \partial _{x_2 }
\omega _2 +u_3 \partial _{x_3 } \omega _2 -\omega _1 \partial _{x_1 } u_2
-\omega _2 \partial _{x_2 } u_2 -\omega _3 \partial _{x_3 } u_2 =\Delta
\omega _2 \\
&\partial _t \omega _3 +u_1 \partial _{x_1 } \omega _3 +u_2 \partial _{x_2 }
\omega _3 +u_3 \partial _{x_3 } \omega _3 -\omega _1 \partial _{x_1
} u_3 -\omega _2 \partial _{x_2 } u_3 -\omega _3 \partial _{x_3 }
u_3 =\Delta
\omega _3 \\
\end{split}
\end{equation}
with the initial conditions $\left. {(\omega _1 ,\omega _2 ,\omega _3 )}
\right|_{t=0} =(\omega _{10} ,\omega _{20} ,\omega _{30}
)=(\mbox{curl}u_{10} ,\;\mbox{curl}u_{20} ,\;\mbox{curl}u_{30} )$, and the
incompressible condition :
\begin{equation*}
\begin{split}
&\partial _{x_1 } \omega _1 +\partial _{x_2 } \omega _2 +\partial _{x_3 }
\omega _3 =0 \\
&\partial _{x_1 } u_1 \;+\partial _{x_2 } u_2 \;\,+\partial _{x_3 } u_3 =0
\\
\end{split}
\end{equation*}
Given a partition with respect to $t$ as follows:
\[
0=t_0 <t_1 <t_2 <\cdots <t_{k-1} <t_k <\cdots <t_N =T
\]
On each $t\in (t_{k-1} ,\;t_k )$, we introduce an auxiliary problem:
\begin{equation}
\begin{split}
&\partial _t \tilde {\omega }_1 \,+\bar {u}_1^k \partial _{x_1 } \bar
{\omega }_1^k +\bar {u}_2^k \partial _{x_2 } \bar {\omega }_1^k +\bar
{u}_3^k \partial _{x_3 } \bar {\omega }_1^k -\bar {\omega }_1^k \partial
_{x_1 } \bar {u}_1^k -\bar {\omega }_2^k \partial _{x_2 } \bar {u}_1^k -\bar
{\omega }_3^k \partial _{x_3 } \bar {u}_1^k +\partial _{x_1 } q=\Delta
\tilde {\omega }_1 \\
&\partial _t \tilde {\omega }_2 +\bar {u}_1^k \partial _{x_1 } \bar {\omega
}_2^k +\bar {u}_2^k \partial _{x_2 } \bar {\omega }_2^k +\bar {u}_3^k
\partial _{x_3 } \bar {\omega }_2^k -\bar {\omega }_1^k \partial _{x_1 }
\bar {u}_2^k -\bar {\omega }_2^k \partial _{x_2 } \bar {u}_2^k -\bar {\omega
}_3^k \partial _{x_3 } \bar {u}_2^k +\partial _{x_2 } q=\Delta \tilde
{\omega }_2 \\
&\partial _t \tilde {\omega }_3 +\bar {u}_1^k \partial _{x_1 } \bar {\omega
}_3^k +\bar {u}_2^k \partial _{x_2 } \bar {\omega }_3^k +\bar {u}_3^k
\partial _{x_3 } \bar {\omega }_3^k -\bar {\omega }_1^k \partial _{x_1 }
\bar {u}_3^k -\bar {\omega }_2^k \partial _{x_2 } \bar {u}_3^k -\bar {\omega
}_3^k \partial _{x_3 } \bar {u}_3^k +\partial _{x_3 } q=\Delta \tilde
{\omega }_3 \\
\end{split}
\end{equation}
where the initial value is assumed to be $\tilde {\omega }_i (x,t_{k-1}
)=\tilde {\omega }_i^{k-1} $ and
\[
\bar {\omega }_i^k (x)=\frac{1}{\Delta t_k }\int_{t_{k-1} }^{t_k } {\tilde
{\omega }_i (x,t)dt}
\]
\[
\bar {u}_i^k (x)=\frac{1}{\Delta t_k }\int_{t_{k-1} }^{t_k } {u_i (x,t)dt}
,\quad \quad i=1,2,3
\]
It is easy to check that
\begin{equation*}
\begin{split}
&\partial _{x_1 } \tilde {\omega }_1 +\partial _{x_2 } \tilde {\omega }_2
+\partial _{x_3 } \tilde {\omega }_3 =0\quad \Rightarrow \quad \partial
_{x_1 } \bar {\omega }_1^k +\partial _{x_2 } \bar {\omega }_2^k +\partial
_{x_3 } \bar {\omega }_3^k =0 \\
&\partial _{x_1 } u_1 +\partial _{x_2 } u_2 +\partial _{x_3 } u_3 =0\quad
\,\Rightarrow \quad \partial _{x_1 } \bar {u}_1^k +\partial _{x_2 }
\bar
{u}_2^k +\partial _{x_3 } \bar {u}_3^k =0 \\
\end{split}
\end{equation*}
In the section 3, by means of the Galerkin method and the
compactness imbedding theorem, we can prove the local existences of
the weak solutions of these systems for each $(t_{k-1} ,\;t_k )$
being small enough. Below we also interpret $\tilde \omega$ as the
Galerkin approximation of the solution of the problem (7), and first
prove that $\tilde \omega, t \in (0,T)$, belong to $L^\infty
(0,T;L^2({\mathbb R}^3))$. In section 4, an approach of
approximation is used to assert that the solution of (6) also
belongs to $L^\infty (0,T;L^2({\mathbb R}^3))$ as ${k}'\to \infty $,
or $\Delta t_k ^\prime \to 0$.
\begin{equation*}
\begin{split}
&\int_{{\mathbb R}^3} {\theta _{r} (\tilde {\omega }_1 (\bar {u}_1^k
\partial _{x_1 } \bar {\omega }_1^k +\bar {u}_2^k \partial _{x_2 } \bar {\omega
}_1^k +\bar {u}_3^k \partial _{x_3 } \bar {\omega }_1^k )} \\
&\;\;\qquad +\tilde {\omega }_2 (\bar {u}_1^k \partial _{x_1 } \bar
{\omega }_2^k +\bar {u}_2^k \partial _{x_2 } \bar {\omega }_2^k +\bar
{u}_3^k \partial _{x_3 } \bar {\omega }_2^k ) \\
&\;\;\qquad +\tilde {\omega }_3 (\bar {u}_1^k \partial _{x_1 } \bar
{\omega }_3^k +\bar {u}_2^k \partial _{x_2 } \bar {\omega }_3^k +\bar
{u}_3^k \partial _{x_3 } \bar {\omega }_3^k )) \\
&=-\int_{{\mathbb R}^3} {(\bar {\omega }_1^k \partial _{x_1 } (\theta _{r} \tilde {\omega }_1 \bar {u}_1^k )+\bar {\omega }_1^k \partial
_{x_2 } (\theta _{r} \tilde {\omega }_1 \bar {u}_2^k )+\bar {\omega
\partial _{x_3 } (\theta _{r} \tilde {\omega }_1 \bar {u}_3^k )} \\
&\quad \qquad +\bar {\omega }_2^k \partial _{x_1 } (\theta _{r}
\tilde {\omega }_2 \bar {u}_1^k )+\bar {\omega }_2^k \partial _{x_2
} (\theta _{r} \tilde {\omega }_2 \bar {u}_2^k )+\bar {\omega }_2^k
\partial _{x_3 } (\theta _{r} \tilde {\omega }_2 \bar {u}_3^k ) \\
&\quad \qquad +\bar {\omega }_3^k \partial _{x_1 } (\theta _{r}
\tilde {\omega }_3 \bar {u}_1^k )+\bar {\omega }_3^k \partial _{x_2
} (\theta _{r} \tilde {\omega }_3 \bar {u}_2^k )+\bar {\omega }_3^k
\partial _{x_3 } (\theta _{r} \tilde {\omega }_3 \bar {u}_3^k )) \\
&=-\int_{{\mathbb R}^3} {(\bar {\omega }_1^k \tilde {\omega }_1 \bar {u}_1^k
\partial _{x_1 } \theta _{r} + \bar {\omega }_1^k \theta _{r} \bar {u}_1^k \partial _{x_1 } \tilde {\omega }_1 + \bar {\omega }_1^k
\theta _{r} \tilde {\omega }_1 \partial _{x_1 } \bar {u}_1^k } \\
&\quad \qquad + \bar {\omega }_1^k \tilde {\omega }_1 \bar {u}_2^k \partial
_{x_2 } \theta _{r} + \bar {\omega }_1^k \theta _{r} \bar {u}_2^k
\partial _{x_2 } \tilde {\omega }_1 + \bar {\omega }_1^k \theta
_{r} \tilde {\omega }_1 \partial _{x_2 } \bar {u}_2^k \\
&\quad \qquad + \bar {\omega }_1^k \tilde {\omega }_1 \bar {u}_3^k \partial
_{x_3 } \theta _{r} + \bar {\omega }_1^k \theta _{r} \bar {u}_3^k
\partial _{x_3 } \tilde {\omega }_1 + \bar {\omega }_1^k \theta
_{r} \tilde {\omega }_1 \partial _{x_3 } \bar {u}_3^k \\
&\quad \qquad + \bar {\omega }_2^k \tilde {\omega }_2 \bar {u}_1^k \partial
_{x_1 } \theta _{r} + \bar {\omega }_2^k \theta _{r} \bar {u}_1^k
\partial _{x_1 } \tilde {\omega }_2 + \bar {\omega }_2^k \theta
_{r} \tilde {\omega }_2 \partial _{x_1 } \bar {u}_1^k \\
&\quad \qquad + \bar {\omega }_2^k \tilde {\omega }_2 \bar {u}_2^k \partial
_{x_2 } \theta _{r} + \bar {\omega }_2^k \theta _{r} \bar {u}_2^k
\partial _{x_2 } \tilde {\omega }_2 + \bar {\omega }_2^k \theta
_{r} \tilde {\omega }_2 \partial _{x_2 } \bar {u}_2^k \\
&\quad \qquad + \bar {\omega }_2^k \tilde {\omega }_2 \bar {u}_3^k \partial
_{x_3 } \theta _{r} + \bar {\omega }_2^k \theta _{r} \bar {u}_3^k
\partial _{x_3 } \tilde {\omega }_2 + \bar {\omega }_2^k \theta
_{r} \tilde {\omega }_2 \partial _{x_3 } \bar {u}_3^k \\
&\quad \qquad + \bar {\omega }_3^k \tilde {\omega }_3 \bar {u}_1^k \partial
_{x_1 } \theta _{r} + \bar {\omega }_3^k \theta _{r} \bar {u}_1^k
\partial _{x_1 } \tilde {\omega }_3 + \bar {\omega }_3^k \theta
_{r} \tilde {\omega }_3 \partial _{x_1 } \bar {u}_1^k \\
&\quad \qquad + \bar {\omega }_3^k \tilde {\omega }_3 \bar {u}_2^k \partial
_{x_2 } \theta _{r} + \bar {\omega }_3^k \theta _{r} \bar {u}_2^k
\partial _{x_2 } \tilde {\omega }_3 + \bar {\omega }_3^k \theta
_{r} \tilde {\omega }_3 \partial _{x_2 } \bar {u}_2^k \\
&\quad \qquad + \bar {\omega }_3^k \tilde {\omega }_3 \bar {u}_3^k \partial
_{x_3 } \theta _{r} + \bar {\omega }_3^k \theta _{r} \bar {u}_3^k
\partial _{x_3 } \tilde {\omega }_3 + \bar {\omega }_3^k \theta
_{r} \tilde {\omega }_3 \partial _{x_3 } \bar {u}_3^k ) \\
\end{split}
\end{equation*}
\begin{equation*}
\begin{split}
&=-\int_{{\mathbb R}^3} {[\theta _{r} (\bar {\omega }_1^k \bar {u}_1^k
\partial _{x_1 } \tilde {\omega }_1 +\bar {\omega }_1^k \bar {u}_2^k
\partial _{x_2 } \tilde {\omega }_1 +\bar {\omega }_1^k \bar {u}_3^k
\partial _{x_3 } \tilde {\omega }_1 } \\
&\qquad \quad \quad \quad +\bar {\omega }_2^k \bar {u}_1^k \partial _{x_1 }
\tilde {\omega }_2 +\bar {\omega }_2^k \bar {u}_2^k \partial _{x_2 } \tilde
{\omega }_2 +\bar {\omega }_2^k \bar {u}_3^k \partial _{x_3 } \tilde {\omega
}_2 \\
&\qquad \quad \quad \quad +\bar {\omega }_3^k \bar {u}_1^k \partial _{x_1 }
\tilde {\omega }_3 +\bar {\omega }_3^k \bar {u}_2^k \partial _{x_2 }
\tilde {\omega }_3 +\bar {\omega }_3^k \bar {u}_3^k \partial _{x_3 }
\tilde
{\omega }_3 ) \\
&\quad +(\bar {\omega }_1^k \tilde {\omega }_1 \bar {u}_1^k \partial _{x_1 }
\theta _{r} +\bar {\omega }_1^k \tilde {\omega }_1 \bar {u}_2^k
\partial _{x_2 } \theta _{r} +\bar {\omega }_1^k \tilde {\omega }_1
\bar {u}_3^k \partial _{x_3 } \theta _{r} \\
&\quad +\; \bar {\omega }_2^k \tilde {\omega }_2 \bar {u}_1^k \partial
_{x_1 } \theta _{r} +\bar {\omega }_2^k \tilde {\omega }_2 \bar
{u}_2^k \partial _{x_2 } \theta _{r} +\bar {\omega }_2^k \tilde
{\omega }_2 \bar {u}_3^k \partial _{x_3 } \theta _{r} \\
&\quad +\; \bar {\omega }_3^k \tilde {\omega }_3 \bar {u}_1^k \partial
_{x_1 } \theta _{r} +\bar {\omega }_3^k \tilde {\omega }_3 \bar
{u}_2^k \partial _{x_2 } \theta _{r} +\bar {\omega }_3^k \tilde
{\omega }_3 \bar {u}_3^k \partial _{x_3 } \theta _{r} )] \\
\end{split}
\end{equation*}
Let $ r\to +\infty $ we get
\begin{equation*}
\begin{split}
&\int_{{\mathbb R}^3} {(\tilde {\omega }_1 (\bar {u}_1^k \partial _{x_1 } \bar
{\omega }_1^k +\bar {u}_2^k \partial _{x_2 } \bar {\omega} _1^k
+\bar {u}_3^k
\partial _{x_3 } \bar {\omega} _1^k )} \\
&\;\;\;\,+\tilde {\omega }_2 (\bar {u}_1^k \partial _{x_1 } \bar {\omega
}_2^k +\bar {u}_2^k \partial _{x_2 } \bar {\omega }_2^k +\bar {u}_3^k
\partial _{x_3 } \bar {\omega }_2^k ) \\
&\;\;\;\,+\tilde {\omega }_3 (\bar {u}_1^k \partial _{x_1 } \bar {\omega
}_3^k +\bar {u}_2^k \partial _{x_2 } \bar {\omega }_3^k +\bar {u}_3^k
\partial _{x_3 } \bar {\omega }_3^k )) \\
&=-\int_{{\mathbb R}^3} {(\bar {\omega }_1^k \bar {u}_1^k \partial _{x_1 }
\tilde {\omega }_1 +\bar {\omega }_1^k \bar {u}_2^k \partial _{x_2 }
\tilde {\omega }_1 +\bar {\omega }_1^k \bar {u}_3^k \partial _{x_3 }
\tilde
{\omega }_1 } \;\, \\
&\quad \qquad +\bar {\omega }_2^k \bar {u}_1^k \partial _{x_1 } \tilde
{\omega }_2 +\bar {\omega }_2^k \bar {u}_2^k \partial _{x_2 } \tilde {\omega
}_2 +\bar {\omega }_2^k \bar {u}_3^k \partial _{x_3 } \tilde {\omega }_2 \\
&\quad \qquad +\bar {\omega }_3^k \bar {u}_1^k \partial _{x_1 } \tilde
{\omega }_3 +\bar {\omega }_3^k \bar {u}_2^k \partial _{x_2 } \tilde
{\omega }_3 +\bar {\omega }_3^k \bar {u}_3^k \partial _{x_3 } \tilde
{\omega }_3
\,) \\
\end{split}
\end{equation*}
\begin{equation*}
\begin{split}
&\int_{{\mathbb R}^3} {(\tilde {\omega }_1 (\bar {\omega }_1^k \partial _{x_1 }
\bar {u}_1^k +\bar {\omega }_2^k \partial _{x_2 } \bar {u}_1^k +\bar
}_3^k \partial _{x_3 } \bar {u}_1^k } ) \\
&\;\,\;\,+\tilde {\omega }_2 (\bar {\omega }_1^k \partial _{x_1 } \bar
{u}_2^k +\bar {\omega }_2^k \partial _{x_2 } \bar {u}_2^k +\bar {\omega
}_3^k \partial _{x_3 } \bar {u}_2^k ) \\
&\;\;\,\,+\tilde {\omega }_3 (\bar {\omega }_1^k \partial _{x_1 } \bar
{u}_3^k +\bar {\omega }_2^k \partial _{x_2 } \bar {u}_3^k +\bar {\omega
}_3^k \partial _{x_3 } \bar {u}_3^k )) \\
&=-\int_{{\mathbb R}^3} {(\bar {\omega }_1^k \bar {u}_1^k \partial _{x_1 }
\tilde {\omega }_1 +\bar {\omega }_2^k \bar {u}_1^k \partial _{x_2 }
\tilde {\omega }_1 +\bar {\omega }_3^k \bar {u}_1^k
\partial _{x_3 }
\tilde {\omega }_1 } \\
&\quad \qquad +\bar {\omega }_1^k \bar {u}_2^k \partial _{x_1 } \tilde
{\omega }_2 +\bar {\omega }_2^k \bar {u}_2^k \partial _{x_2 } \tilde
{\omega }_2 +\bar {\omega }_3^k \bar {u}_2^k \partial _{x_3 } \tilde
{\omega }_2 \\
&\quad \qquad +\bar {\omega }_1^k \bar {u}_3^k \partial _{x_1 } \tilde
{\omega }_3 +\bar {\omega }_2^k \bar {u}_3^k \partial _{x_2 } \tilde
{\omega }_3 +\bar {\omega }_3^k \bar {u}_3^k \partial _{x_3 } \tilde
{\omega }_3 ) \\
\end{split}
\end{equation*}
\[
\int_{{\mathbb R}^3} {(\tilde {\omega }_1 \partial _{x_1 } q+\tilde
{\omega }_2
\partial _{x_2 } q+\tilde {\omega }_3 \partial _{x_3 } q)} =0
\]
\[
\int_{{\mathbb R}^3} {\tilde {\omega }_i \Delta \tilde {\omega }_i }
=\int_{{\mathbb R}^3} {\tilde {\omega }_i (\partial _{x_1 }^2 \tilde
{\omega }_i +\partial _{x_2 }^2 \tilde {\omega }_i +\partial _{x_3
}^2 \tilde {\omega }_i )} =-\int_{{\mathbb R}^3} {((\partial _{x_1 }
\tilde {\omega }_i )^2+(\partial _{x_2 } \tilde {\omega }_i
)^2+(\partial _{x_3 } \tilde {\omega }_i )^2)}
\]
Thus from (7) we have
\begin{equation*}
\begin{split}
&\int_{{\mathbb R}^3} {\tilde {\omega }_1 \partial _t \tilde {\omega }_1 }
\;\,+\int_{{\mathbb R}^3} {\tilde {\omega }_1 (\bar {u}_1^k \partial
_{x_1 } \bar {\omega }_1^k +\bar {u}_2^k \partial _{x_2 } \bar
{\omega }_1^k +\bar
{u}_3^k \partial _{x_3 } \bar {\omega }_1^k )} \\
&\quad \quad \quad \quad \quad \quad \quad \quad -\int_{{\mathbb R}^3} {\tilde
{\omega }_1 (\bar {\omega }_1^k \partial _{x_1 } \bar {u}_1^k +\bar
{\omega }_2^k \partial _{x_2 } \bar {u}_1^k +\bar {\omega }_3^k
\partial _{x_3 } \bar {u}_1^k )} \,+\int_{{\mathbb R}^3} {\tilde {\omega
}_1 \partial _{x_1 } q}
=\int_{{\mathbb R}^3} {\tilde {\omega }_1 \Delta \tilde {\omega }_1 } \\
\end{split}
\end{equation*}
\begin{equation*}
\begin{split}
&\int_{{\mathbb R}^3} {\tilde {\omega }_2 \partial _t \tilde {\omega }_2 }
+\int_{{\mathbb R}^3} {\tilde {\omega }_2 (\bar {u}_1^k \partial
_{x_1 } \bar {\omega }_2^k +\bar {u}_2^k \partial _{x_2 } \bar
{\omega }_2^k +\bar
{u}_3^k \partial _{x_3 } \bar {\omega }_2^k )} \\
&\quad \quad \quad \quad \quad \quad \quad \quad -\int_{{\mathbb R}^3} {\tilde
{\omega }_2 (\bar {\omega }_1^k \partial _{x_1 } \bar {u}_2^k +\bar
{\omega }_2^k \partial _{x_2 } \bar {u}_2^k +\bar {\omega }_3^k
\partial _{x_3 } \bar {u}_2^k )} +\int_{{\mathbb R}^3} {\tilde {\omega
}_2 \partial _{x_2 } q}
=\int_{{\mathbb R}^3} {\tilde {\omega }_2 \Delta \tilde {\omega }_2 } \\
&\int_{{\mathbb R}^3} {\tilde {\omega }_3 \partial _t \tilde {\omega }_3 }
+\int_{{\mathbb R}^3} {\tilde {\omega }_3 (\bar {u}_1^k \partial
_{x_1 } \bar {\omega }_3^k +\bar {u}_2^k \partial _{x_2 } \bar
{\omega }_3^k +\bar
{u}_3^k \partial _{x_3 } \bar {\omega }_3^k )} \\
&\quad \quad \quad \quad \quad \quad \quad \quad -\int_{{\mathbb R}^3} {\tilde
{\omega }_3 (\bar {\omega }_1^k \partial _{x_1 } \bar {u}_3^k +\bar
{\omega }_2^k \partial _{x_2 } \bar {u}_3^k +\bar {\omega }_3^k
\partial _{x_3 } \bar {u}_3^k )} +\int_{{\mathbb R}^3} {\tilde {\omega
}_3 \partial _{x_3 } q}
=\int_{{\mathbb R}^3} {\tilde {\omega }_3 \Delta \tilde {\omega }_3 } \\
\end{split}
\end{equation*}
so that
\begin{equation*}
\begin{split}
&\frac{1}{2}\partial _t \int_{{\mathbb R}^3} {(\tilde {\omega }_1^2 +\tilde
{\omega }_2^2 +\tilde {\omega }_3^2 )} +\,\,\int_{{\mathbb R}^3}
{((\partial _{x_1 } \tilde {\omega }_1 )^2+(\partial _{x_2 } \tilde
{\omega }_1
)^2+(\partial _{x_3 } \tilde {\omega }_1 )^2 } \\
&\qquad \quad \quad \quad \quad \quad \quad \quad \quad \quad
\;\,\;\;\;\, +(\partial _{x_1 } \tilde {\omega }_2 )^2+(\partial
_{x_2 } \tilde
{\omega }_2 )^2+(\partial _{x_3 } \tilde {\omega }_2 )^2 \\
&\qquad \quad \quad \quad \quad \quad \quad \quad \quad \quad
\;\,\;\;\;\, +(\partial _{x_1 } \tilde {\omega }_3 )^2+(\partial
_{x_2 } \tilde
{\omega }_3 )^2+(\partial _{x_3 } \tilde {\omega }_3 )^2) \\
&\quad \quad \quad -\int_{{\mathbb R}^3} {(\bar {\omega }_1^k \bar {u}_1^k
\partial _{x_1 } \tilde {\omega }_1 +\bar {\omega }_1^k \bar {u}_2^k
\partial _{x_2 } \tilde {\omega }_1 +\bar {\omega }_1^k \bar {u}_3^k
\partial _{x_3 } \tilde {\omega }_1 } \\
&\quad \quad \quad \,\quad \;\;\; +\bar {\omega }_2^k \bar {u}_1^k \partial
_{x_1 } \tilde {\omega }_2 +\bar {\omega }_2^k \bar {u}_2^k \partial _{x_2 }
\tilde {\omega }_2 +\bar {\omega }_2^k \bar {u}_3^k \partial _{x_3 } \tilde
{\omega }_2 \\
&\quad \quad \quad \,\quad \;\;\; +\bar {\omega }_3^k \bar {u}_1^k \partial
_{x_1 } \tilde {\omega }_3 +\bar {\omega }_3^k \bar {u}_2^k
\partial _{x_2 } \tilde {\omega }_3 +\bar {\omega }_3^k \bar
{u}_3^k \partial _{x_3 }
\tilde {\omega }_3 ) \\
&\quad \quad \quad +\int_{{\mathbb R}^3} {(\bar {\omega }_1^k \bar {u}_1^k
\partial _{x_1 } \tilde {\omega }_1 +\bar {\omega }_2^k \bar {u}_1^k
\partial _{x_2 } \tilde {\omega }_1 +\bar {\omega }_3^k \bar {u}_1^k
\partial _{x_3 } \tilde {\omega }_1 } \\
&\quad \quad \quad \,\quad \;\;\; +\bar {\omega }_1^k \bar {u}_2^k \partial
_{x_1 } \tilde {\omega }_2 +\bar {\omega }_2^k \bar {u}_2^k \partial _{x_2 }
\tilde {\omega }_2 +\bar {\omega }_3^k \bar {u}_2^k \partial _{x_3 } \tilde
{\omega }_2 \\
&\quad \quad \quad \,\quad \;\;\; +\bar {\omega }_1^k \bar {u}_3^k \partial
_{x_1 } \tilde {\omega }_3 +\bar {\omega }_2^k \bar {u}_3^k
\partial _{x_2 } \tilde {\omega }_3 +\bar {\omega }_3^k \bar
{u}_3^k \partial _{x_3 }
\tilde {\omega }_3 )=0 \\
\end{split}
\end{equation*}
By using Young inequality: $uv\le \frac{1}{4}u^2+v^2$, it follows
\begin{equation*}
\begin{split}
&\int_{{\mathbb R}^3} {(\tilde {\omega }_1^2 +\tilde {\omega }_2^2 +\tilde
{\omega }_3^2 )} \;\,+\,\;2\;\int_{t_{k-1} }^t {\int_{{\mathbb R}^3}
{\;((\partial _{x_1 } \tilde {\omega }_1 )^2\;+(\partial _{x_2 }
\tilde
{\omega }_1 )^2+(\partial _{x_3 } \tilde {\omega }_1 )^2} } \\
&\qquad \qquad \qquad \quad \quad \quad \quad \quad \quad \quad
\;\;\;\;\;\;\; +(\partial _{x_1 } \tilde {\omega }_2 )^2+(\partial
_{x_2 }
\tilde {\omega }_2 )^2+(\partial _{x_3 } \tilde {\omega }_2 )^2 \\
&\qquad \qquad \qquad \quad \quad \quad \quad \quad \quad \quad
\;\;\;\;\;\;\; +(\partial _{x_1 } \tilde {\omega }_3 )^2+(\partial
_{x_2 }
\tilde {\omega }_3 )^2+(\partial _{x_3 } \tilde {\omega }_3 )^2) \\
&\le \int_{{\mathbb R}^3} {(\tilde {\omega }_1^{k-1^2} +\tilde {\omega
}_2^{k-1^2} +\tilde {\omega }_3^{k-1^2} )} \;\,+\;\,2\int_{t_{k-1}
}^t {\int_{{\mathbb R}^3} {(\bar {\omega }_1^{k^2} \bar {u}_1^{k^2}
+\bar {\omega
}_1^{k^2} \bar {u}_2^{k^2} +\bar {\omega }_1^{k^2} \bar {u}_3^{k^2} } } \\
&\qquad \qquad \qquad \qquad \qquad \quad \quad \quad \quad \quad \,\quad \,\quad
\quad \quad \;\;+ \bar {\omega }_2^{k^2} \bar {u}_1^{k^2} +\bar
}_2^{k^2} \bar {u}_2^{k^2} +\bar {\omega }_2^{k^2} \bar {u}_3^{k^2} \\
&\qquad \qquad \qquad \qquad \qquad \quad \quad \quad \quad \quad \quad \,\,\quad
\quad \quad \;\;+ \bar {\omega }_3^{k^2} \bar {u}_1^{k^2} +\bar
}_3^{k^2} \bar {u}_2^{k^2} +\bar {\omega }_3^{k^2} \bar {u}_3^{k^2} ) \\
&\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad
\;\;\;\,+\;\,2\int_{t_{k-1} }^t {\int_{{\mathbb R}^3} {(\bar {\omega
}_1^{k^2} \bar {u}_1^{k^2} +\bar {\omega }_2^{k^2} \bar {u}_1^{k^2}
+\bar {\omega
}_3^{k^2} \bar {u}_1^{k^2} } } \\
&\qquad \qquad \quad \quad \quad \quad \quad \quad \quad \quad \,\quad \;\,\quad
\quad \quad +\bar {\omega }_1^{k^2} \bar {u}_2^{k^2} +\bar {\omega }_2^{k^2}
\bar {u}_2^{k^2} +\bar {\omega }_3^{k^2} \bar {u}_2^{k^2} \\
&\qquad \qquad \quad \quad \quad \quad \quad \quad \quad \quad \quad \;\;\, \quad
\quad \quad +\bar {\omega }_1^{k^2} \bar {u}_3^{k^2} +\bar {\omega
}_2^{k^2} \bar {u}_3^{k^2} +\bar {\omega }_3^{k^2} \bar {u}_3^{k^2} ) \\
&\quad \quad \quad \quad \quad \quad \quad \le \int_{t_{k-1} }^t {\int_{{\mathbb
R}^3} {((\partial _{x_1 } \tilde {\omega }_1 )^2+(\partial _{x_2 }
\tilde
{\omega }_1 )^2+(\partial _{x_3 } \tilde {\omega }_1 )^2} } \\
\end{split}
\end{equation*}
\begin{equation*}
\begin{split}
&\qquad \qquad \quad \quad \quad \quad \quad \quad \quad \;\;+(\partial _{x_1
} \tilde {\omega }_2 )^2+(\partial _{x_2 } \tilde {\omega }_2 )^2+(\partial
_{x_3 } \tilde {\omega }_2 )^2 \\
&\qquad \qquad \quad \quad \quad \quad \quad \quad \quad \;\;+(\partial _{x_1
} \tilde {\omega }_3 )^2+(\partial _{x_2 } \tilde {\omega }_3 )^2+(\partial
_{x_3 } \tilde {\omega }_3 )^2) \\
\end{split}
\end{equation*}
According to Cauchy-Schwarz inequality on $Q_{T_k } =(t_{k-1} ,t_k
)\times {\mathbb R}^3$, we have
\begin{equation*}
\begin{split}
&\int_{{\mathbb R}^3} {(\tilde {\omega }_1^2 +\tilde {\omega }_2^2 +\tilde
{\omega }_3^2 )} +\int_{t_{k-1} }^t {\int_{{\mathbb R}^3}
{((\partial _{x_1 } \tilde {\omega }_1 )^2+(\partial _{x_2 } \tilde
{\omega }_1 )^2+(\partial
_{x_3 } \tilde {\omega }_1 )^2} } \\
&\quad \qquad \qquad \quad \quad \quad \quad \quad \quad \;\;\;\;\;\; +(\partial
_{x_1 } \tilde {\omega }_2 )^2+(\partial _{x_2 } \tilde {\omega }_2
)^2+(\partial _{x_3 } \tilde {\omega }_2 )^2 \\
&\quad \qquad \qquad \quad \quad \quad \quad \quad \quad \;\;\;\;\;\; +(\partial
_{x_1 } \tilde {\omega }_3 )^2+(\partial _{x_2 } \tilde {\omega }_3
)^2+(\partial _{x_3 } \tilde {\omega }_3 )^2) \\
&\le \int_{{\mathbb R}^3} {(\tilde {\omega }_1^{k-1^2} +\tilde {\omega
}_2^{k-1^2} +\tilde {\omega }_3^{k-1^2} )} + \\
&+4\;\{\,(\int_{t_{k-1} }^t {\int_{{\mathbb R}^3} {\bar {u}_1^{k^4} } }
)^{\frac{1}{2}}(\int_{t_{k-1} }^t {\int_{{\mathbb R}^3} {\bar
{\omega }_1^{k^4} } } )^{\frac{1}{2}}+(\int_{t_{k-1} }^t
{\int_{{\mathbb R}^3} {\bar {u}_1^{k^4} } }
)^{\frac{1}{2}}(\int_{t_{k-1} }^t {\int_{{\mathbb R}^3} {\bar
{\omega }_2^{k^4} } } )^{\frac{1}{2}}+(\int_{t_{k-1} }^t
{\int_{{\mathbb R}^3} {\bar {u}_1^{k^4} } }
)^{\frac{1}{2}}(\int_{t_{k-1} }^t {\int_{{\mathbb R}^3} {\bar
{\omega }_3^{k^4} } } )^{\frac{1}{2}} \\
&\quad \,+(\int_{t_{k-1} }^t {\int_{{\mathbb R}^3} {\bar {u}_2^{k^4} } }
)^{\frac{1}{2}}(\int_{t_{k-1} }^t {\int_{{\mathbb R}^3} {\bar
{\omega }_1^{k^4} } } )^{\frac{1}{2}}+(\int_{t_{k-1} }^t
{\int_{{\mathbb R}^3} {\bar {u}_2^{k^4} } }
)^{\frac{1}{2}}(\int_{t_{k-1} }^t {\int_{{\mathbb R}^3} {\bar
{\omega }_2^{k^4} } } )^{\frac{1}{2}}+(\int_{t_{k-1} }^t
{\int_{{\mathbb R}^3} {\bar {u}_2^{k^4} } }
)^{\frac{1}{2}}(\int_{t_{k-1} }^t {\int_{{\mathbb R}^3} {\bar
{\omega }_3^{k^4} } } )^{\frac{1}{2}} \\
&\quad \,+(\int_{t_{k-1} }^t {\int_{{\mathbb R}^3} {\bar {u}_3^{k^4} } }
)^{\frac{1}{2}}(\int_{t_{k-1} }^t {\int_{{\mathbb R}^3} {\bar
{\omega }_1^{k^4} } } )^{\frac{1}{2}}+(\int_{t_{k-1} }^t
{\int_{{\mathbb R}^3} {\bar {u}_3^{k^4} } }
)^{\frac{1}{2}}(\int_{t_{k-1} }^t {\int_{{\mathbb R}^3} {\bar
{\omega }_2^{k^4} } } )^{\frac{1}{2}}+(\int_{t_{k-1} }^t
{\int_{{\mathbb R}^3} {\bar {u}_3^{k^4} } }
)^{\frac{1}{2}}(\int_{t_{k-1} }^t {\int_{{\mathbb R}^3} {\bar
{\omega }_3^{k^4} } } )^{\frac{1}{2}}\,\} \\
&\le \int_{{\mathbb R}^3} {(\tilde {\omega }_1^{k-1^2} +\tilde {\omega
}_2^{k-1^2} +\tilde {\omega }_3^{k-1^2} )} \;+4\;\{\,\,\left\| {\bar
{u}_1^k } \right\|_{L^4(Q_{T_k } )}^2 (\,\left\| {\bar {\omega }_1^k
} \right\|_{L^4(Q_{T_k } )}^2 +\left\| {\bar {\omega }_2^k }
\right\|_{L^4(Q_{T_k } )}^2 +\left\| {\bar {\omega }_3^k }
\right\|_{L^4(Q_{T_k } )}^2 ) \\
&\qquad \qquad \qquad \qquad \quad \quad \quad \quad \quad \quad \quad
\;\;\; +\left\| {\bar {u}_2^k } \right\|_{L^4(Q_{T_k } )}^2
(\,\left\| {\bar {\omega }_1^k } \right\|_{L^4(Q_{T_k } )}^2
+\left\| {\bar {\omega }_2^k } \right\|_{L^4(Q_{T_k } )}^2 +\left\|
{\bar {\omega }_3^k }
\right\|_{L^4(Q_{T_k } )}^2 ) \\
&\qquad \qquad \qquad \qquad \quad \quad \quad \quad \quad \quad \quad
\;\;\; +\left\| {\bar {u}_3^k } \right\|_{L^4(Q_{T_k } )}^2
(\,\left\| {\bar {\omega }_1^k } \right\|_{L^4(Q_{T_k } )}^2
+\left\| {\bar {\omega }_2^k } \right\|_{L^4(Q_{T_k } )}^2 +\left\|
{\bar {\omega }_3^k }
\right\|_{L^4(Q_{T_k } )}^2 )\,\} \\
\end{split}
\end{equation*}
\begin{equation*}
\begin{split}
&=\int_{{\mathbb R}^3} {(\tilde {\omega }_1^{k-1^2} +\tilde {\omega }_2^{k-1^2}
+\tilde {\omega }_3^{k-1^2} )} + \\
&\quad +4\,(\,\left\| {\bar {u}_1^k } \right\|_{L^4(Q_{T_k } )}^2 +\left\|
{\bar {u}_2^k } \right\|_{L^4(Q_{T_k } )}^2 +\left\| {\bar {u}_3^k }
\right\|_{L^4(Q_{T_k } )}^2 )\,(\,\left\| {\bar {\omega }_1^k }
\right\|_{L^4(Q_{T_k } )}^2 +\left\| {\bar {\omega }_2^k }
\right\|_{L^4(Q_{T_k } )}^2 +\left\| {\bar {\omega }_3^k }
\right\|_{L^4(Q_{T_k } )}^2 ) \\
\end{split}
\end{equation*}
From Sobolev imbedding theorem in [1], there exists a constant $C_1
>0$ independent of $\omega $ and the size of $Q_{T_k } $ such that
\[
(\int_{t_{k-1} }^t {\left\| {\bar {\omega }_i^k }
\right\|_{L^4({\mathbb R}^3)}^4 } )^{1/2}\le C_1 \;\int_{t_{k-1} }^t
{\,\{\,\left\| {\bar {\omega }_i^k } \right\|_{L^2({\mathbb R}^3)}^2
+\left\| {\nabla \bar {\omega }_i^k } \right\|_{L^2({\mathbb
R}^3)}^2 \}} ,\quad \quad i=1,2,3
\]
it follows that
\begin{equation}
\begin{split}
&\int_{{\mathbb R}^3} {(\tilde {\omega }_1^2 +\tilde {\omega }_2^2 +\tilde
{\omega }_3^2 )} +\int_{t_{k-1} }^t {(\,\left\| {\nabla \tilde
{\omega }_1 } \right\|_{L^2({\mathbb R}^3)}^2 +\left\| {\nabla
\tilde {\omega }_2 } \right\|_{L^2({\mathbb R}^3)}^2 +\left\|
{\nabla \tilde {\omega }_3 }
\right\|_{L^2({\mathbb R}^3)}^2 )} \\
&\le \int_{{\mathbb R}^3} {(\tilde {\omega }_1^{k-1^2} +\tilde {\omega
}_2^{k-1^2} +\tilde {\omega }_3^{k-1^2} )} + \\
&+4C\int_{t_{k-1} }^t {(\,\left\| {\bar {u}_1^k } \right\|_{L^2({\mathbb
R}^3)}^2 +\left\| {\nabla \bar {u}_1^k } \right\|_{L^2({\mathbb
R}^3)}^2 +\left\| {\bar {u}_2^k } \right\|_{L^2({\mathbb R}^3)}^2
+\left\| {\nabla \bar {u}_2^k } \right\|_{L^2({\mathbb R}^3)}^2
+\left\| {\bar {u}_3^k } \right\|_{L^2({\mathbb
R}^3)}^2 +\left\| {\nabla \bar {u}_3^k } \right\|_{L^2({\mathbb R}^3)}^2 )} \, \\
&\;\;\;\;\times \int_{t_{k-1} }^t {(\,\left\| {\bar {\omega }_1^k }
\right\|_{L^2({\mathbb R}^3)}^2 +\left\| {\nabla \bar {\omega }_1^k
} \right\|_{L^2({\mathbb R}^3)}^2 +\left\| {\bar {\omega }_2^k }
\right\|_{L^2({\mathbb R}^3)}^2 +\left\| {\nabla \bar {\omega }_2^k
} \right\|_{L^2({\mathbb R}^3)}^2 +\left\| {\bar {\omega }_3^k }
\right\|_{L^2({\mathbb R}^3)}^2 +\left\| {\nabla \bar {\omega }_3^k
\right\|_{L^2({\mathbb R}^3)}^2 )} \, \\
\end{split}
\end{equation}
Noting that
\begin{equation*}
\begin{split}
&\left\| {\bar {\omega }_i^k } \right\|_{L^2({\mathbb R}^3)}^2 =\int_{{\mathbb R}^3}
{\left( {\frac{1}{\Delta t_k }\int_{t_{k-1} }^{t_k } {\tilde {\omega
}_i (x,t)dt} } \right)} ^2\le \frac{1}{\Delta t_k^2 }\int_{{\mathbb
R}^3} {\Delta
t_k \int_{t_{k-1} }^{t_k } {\tilde {\omega }_i^2 (x,t)dt} } \\
&\qquad \quad \quad \;\;\,=\frac{1}{\Delta t_k }\int_{t_{k-1} }^{t_k }
{\left\| {\tilde {\omega }_i } \right\|_{L^2({\mathbb R}^3)}^2 } \\
\end{split}
\end{equation*}
in the same way,
\[
\left\| {\bar {u}_i^k } \right\|_{L^2({\mathbb R}^3)}^2 \le
\frac{1}{\Delta t_k }\int_{t_{k-1} }^{t_k } {\left\| {u_i }
\right\|_{L^2({\mathbb R}^3)}^2 } ,\quad \quad i=1,2,3
\]
from (8) we have
\begin{equation*}
\begin{split}
&\int_{{\mathbb R}^3} {(\tilde {\omega }_1^2 +\tilde {\omega }_2^2 +\tilde
{\omega }_3^2 )} +\int_{t_{k-1} }^t {(\,\left\| {\nabla \tilde
{\omega }_1 } \right\|_{L^2({\mathbb R}^3)}^2 +\left\| {\nabla
\tilde {\omega }_2 } \right\|_{L^2({\mathbb R}^3)}^2 +\left\|
{\nabla \tilde {\omega }_3 }
\right\|_{L^2({\mathbb R}^3)}^2 )} \\
&\le \int_{{\mathbb R}^3} {(\tilde {\omega }_1^{k-1^2} +\tilde {\omega
}_2^{k-1^2} +\tilde {\omega }_3^{k-1^2} )} + \\
&+\;4C\,\left( {(t_k -t_{k-1} )\mathop {\sup }\limits_{(t_{k-1} ,\;t)}
\;\{\,\left\| {u_1 } \right\|_{L^2({\mathbb R}^3)}^2 +\left\| {u_2 }
\right\|_{L^2({\mathbb R}^3)}^2 +\left\| {u_3 } \right\|_{L^2(\Omega
R}^3)}^2 \}} \right.+ \\
&\quad \quad +\left. {\int_{t_{k-1} }^{t_k } {\{\,\left\| {\nabla u_1 }
\right\|_{L^2({\mathbb R}^3)}^2 +\left\| {\nabla u_2 }
\right\|_{L^2({\mathbb R}^3)}^2 +\left\| {\nabla u_3 }
\right\|_{L^2({\mathbb R}^3)}^2 \} } } \right) \\
&\times \int_{t_{k-1} }^{t_k } {(\,\left\| {\tilde {\omega }_1 }
\right\|_{L^2({\mathbb R}^3)}^2 +\left\| {\nabla \tilde {\omega }_1
} \right\|_{L^2({\mathbb R}^3)}^2 +\left\| {\tilde {\omega }_2 }
\right\|_{L^2({\mathbb R}^3)}^2 +\left\| {\nabla \tilde {\omega }_2
} \right\|_{L^2({\mathbb R}^3)}^2 +\left\| {\tilde {\omega }_3 }
\right\|_{L^2({\mathbb R}^3)}^2 +\left\| {\nabla \tilde {\omega }_3
\right\|_{L^2({\mathbb R}^3)}^2 )} \\
\end{split}
\end{equation*}
\begin{equation*}
\begin{split}
&K_0 =\int_{{\mathbb R}^3} {(\omega _{10}^2 +\omega _{20}^2 +\omega
_{30}^2 )} \\
&K_k^\ast =\Delta t_k \mathop {\sup }\limits_{(t_{k-1} ,t_k )} \;\{\,\left\|
{u_1 } \right\|_{L^2({\mathbb R}^3)}^2 +\left\| {u_2 }
\right\|_{L^2({\mathbb
R}^3)}^2 +\left\| {u_3 } \right\|_{L^2({\mathbb R}^3)}^2 \}\;+ \\
&\quad \quad \quad +\,\int_{t_{k-1} }^{t_k } {\{\,\left\| {\nabla u_1 }
\right\|_{L^2({\mathbb R}^3)}^2 +} \left\| {\nabla u_2 }
\right\|_{L^2({\mathbb
R}^3)}^2 +\left\| {\nabla u_3 } \right\|_{L^2({\mathbb R}^3)}^2 \} \\
\end{split}
\end{equation*}
\[
f_k (t)=\;\mathop {\sup }\limits_{(t_{k-1} ,t)} \int_{{\mathbb R}^3}
{(\tilde {\omega }_1^2 +\tilde {\omega }_2^2 +\tilde {\omega }_3^2
)} \;\;+\varepsilon _0 \int_{t_{k-1} }^t {\{\,\left\| {\nabla \tilde
{\omega }_1 } \right\|_{L^2({\mathbb R}^3)}^2 +\left\| {\nabla
\tilde {\omega }_2 } \right\|_{L^2({\mathbb R}^3)}^2 +\left\|
{\nabla \tilde {\omega }_3 } \right\|_{L^2({\mathbb R}^3)}^2 \}}
\]
where $0<\varepsilon _0 <1$ is a constant. By (5) , (8) we have
\begin{equation*}
\begin{split}
&K_k^\ast \le T\mathop {\sup }\limits_{t\in (0,T)} \;\;\int_{{\mathbb R}^3}
{(u_1^2 +u_2^2 +u_3^2 )} \;\;+\,\int_0^T {(\,\left\| {\nabla u_1 }
\right\|_{L^2({\mathbb R}^3)}^2 +} \left\| {\nabla u_2 }
\right\|_{L^2({\mathbb
R}^3)}^2 +\left\| {\nabla u_3 } \right\|_{L^2({\mathbb R}^3)}^2 ) \\
&\quad \;<+\infty \\
\end{split}
\end{equation*}
\begin{equation*}
\begin{split}
&\mathop {\sup }\limits_{t\in (t_{k-1} ,t_k )} \int_{{\mathbb R}^3} {(\tilde
{\omega }_1^2 +\tilde {\omega }_2^2 +\tilde {\omega }_3^2 )}
+(1-4CK_k^\ast )\int_{t_{k-1} }^{t_k } {(\,\left\| {\nabla \tilde
{\omega }_1 } \right\|_{L^2({\mathbb R}^3)}^2 +\left\| {\nabla
\tilde {\omega }_2 } \right\|_{L^2({\mathbb R}^3)}^2 +\left\|
{\nabla \tilde {\omega }_3 }
\right\|_{L^2({\mathbb R}^3)}^2 )} \\
&\;\,\le \int_{{\mathbb R}^3} {(\tilde {\omega }_1^{k-1^2} +\tilde {\omega
}_2^{k-1^2} +\tilde {\omega }_3^{k-1^2} )} +4CK_k^\ast \int_{t_{k-1}
} {f_k (t)} \\
\end{split}
\end{equation*}
On $(0,t_1 )$, $t_1 $ be small enough, since $\sum\limits_{k=1}^N
{K_k^\ast } <+\infty $, the partition is assumed to be fine enough
such that $1-4CK_1^\ast \ge \varepsilon _0 $, that is, $K_1^\ast \le
\frac{1-\varepsilon _0 }{4C}$ is valid because of the absolute
continuity of integration with respect to $t$, thus
\[
f_1 (t_1 )\le K_0 +4CK_1^\ast \int_0^{t_1 } {f_1 (t)}
\]
By using Gronwall inequality it follows that
\[
f_1 (t)\le K_0 \;e^{(1-\varepsilon _0 )\;t_1 }
\]
Therefore we set
\[
M_k =\mathop {\sup }\limits_{t\in T_k } \;\int_{{\mathbb R}^3}
{(\tilde {\omega }_1^2 +\tilde {\omega }_2^2 +\tilde {\omega }_3^2
)} ,\quad \quad k=1,\cdots ,N
\]
\[
M_1 \le K_0 \;e^{(1-\varepsilon _0 )\;t_1 } \qquad\qquad \mbox{on}
\quad (0,t_1 )
\]
Similar to above we further have
\begin{equation*}
\begin{split}
&\mbox{on}\quad (t_1 ,\;t_2 )\quad \Rightarrow \quad M_2 \le M_1
\;e^{(1-\varepsilon _0 )\;(t_2 -t_1 )} \\
&\cdots \;\;\cdots \;\;\cdots \\
&\mbox{on}\quad (t_{N-1} ,T)\quad \Rightarrow \quad M_N \le M_{N-1} \;e^{(1-\varepsilon
_0 )\;(T-t_{N-1} )} \\
&\qquad \qquad \qquad \qquad \qquad \quad \;\, \le K_0 \;e^{(1-\varepsilon _0 )\;[t_1 +(t_2 -t_1 )+\cdots +(T-t_{N-1} )]} = K_0 \;e^{(1-\varepsilon _0 )\;T} \\
\end{split}
\end{equation*}
Finally we get
\[
\mathop {\sup }\limits_{t\in (0,T)} \int_{{\mathbb R}^3} {(\tilde
{\omega }_1^2 +\tilde {\omega }_2^2 +\tilde {\omega }_3^2 )} \;\le
\mathop {\max }\limits_k \{M_k \}\le K_0 \;e^{(1-\varepsilon _0
\]
This conclusion is also true for the weak solution of problem (7),
by means of the result of section 3 and the lower limit of Galerkin
sequence according to the page 196 of [4].
3. Existence
In this section we have to consider the existence of solutions of
the auxiliary problems. We just need considering the following
system on $(0,\delta )$:
\begin{equation}
\begin{split}
&\partial _t \omega _1 +\bar {u}_1 \partial _{x_1 } \bar {\omega }_1 +\bar
{u}_2 \partial _{x_2 } \bar {\omega }_1 +\bar {u}_3 \partial _{x_3 }
\bar {\omega }_1 -\bar {\omega }_1 \partial _{x_1 } \bar {u}_1 -\bar
{\omega }_2 \partial _{x_2 } \bar {u}_1 -\bar {\omega }_3
\partial _{x_3 } \bar
{u}_1 +\partial _{x_1 } q=\Delta \omega _1 \\
&\partial _t \omega _2 +\bar {u}_1 \partial _{x_1 } \bar {\omega }_2 +\bar
{u}_2 \partial _{x_2 } \bar {\omega }_2 +\bar {u}_3 \partial _{x_3 } \bar
{\omega }_2 -\bar {\omega }_1 \partial _{x_1 } \bar {u}_2 -\bar {\omega }_2
\partial _{x_2 } \bar {u}_2 -\bar {\omega }_3 \partial _{x_3 } \bar {u}_2
+\partial _{x_2 } q=\Delta \omega _2 \\
&\partial _t \omega _3 +\bar {u}_1 \partial _{x_1 } \bar {\omega }_3 +\bar
{u}_2 \partial _{x_2 } \bar {\omega }_3 +\bar {u}_3 \partial _{x_3 } \bar
{\omega }_3 -\bar {\omega }_1 \partial _{x_1 } \bar {u}_3 -\bar {\omega }_2
\partial _{x_2 } \bar {u}_3 -\bar {\omega }_3 \partial _{x_3 } \bar {u}_3
+\partial _{x_3 } q=\Delta \omega _3 \\
\end{split}
\end{equation}
with the initial value $\omega _i (x,0)=\omega _{i0} \;\;(i=1,2,3)$ and
\[
\bar {\omega }_i (x)=\frac{1}{\delta }\int_0^\delta {\omega _i (x,t)dt}
\]
\[
\bar {u}_i (x)=\frac{1}{\delta }\int_0^\delta {u_i (x,t)dt}
\quad
\]
as well as the incompressible conditions:
\begin{equation*}
\begin{split}
&\partial _{x_1 } \omega _1 +\partial _{x_2 } \omega _2 +\partial _{x_3 }
\omega _3 =0\quad \Rightarrow \quad \partial _{x_1 } \bar {\omega }_1
+\partial _{x_2 } \bar {\omega }_2 +\partial _{x_3 } \bar {\omega }_3 =0 \\
&\partial _{x_1 } u_1 +\partial _{x_2 } u_2 +\partial _{x_3 } u_3 =0\quad
\, \Rightarrow \quad \partial _{x_1 } \bar {u}_1 +\partial _{x_2 }
\bar
{u}_2 +\partial _{x_3 } \bar {u}_3 =0 \\
\end{split}
\end{equation*}
(i) The Galerkin procedure is applied. For each $m$ and $i=1,2,3$ we
define an approximate solution $(\omega _{1m} ,\;\omega _{2m}
,\;\omega _{3m} )$ as follows:
\[
\omega _{im} =\sum\limits_{j=1}^m {g_{ij} (t)w_{ij} }
\]
where $\{w_{i1} ,\;\cdots ,\;w_{im} ,\cdots \}$ is the basis of $W$,
and $W$= the closure of ${\cal V}$ in the Sobolev space
$W^{2,4}({\mathbb R}^3)$, which is separable and is dense in $V$.
Thus by means of weighted function $\theta _{r} $ introduced in
Section 1,
\begin{equation}
\begin{split}
&(\theta _{r} \partial _t \omega _{im} ,\;w_{il} )+(\,\theta
_{r} \nabla \omega _{im} ,\;\nabla w_{il} )+(\,\nabla
\omega _{im} ,\;w_{il} \nabla \theta _{r} )+ \\
_{r} (\bar {u}\cdot \nabla )\bar {\omega }_{im} ,\;w_{il} )-(\theta
_{r} (\bar {\omega }_m \cdot \nabla )\bar {u}_i
,\;w_{il} )=0 \\
\end{split}
\end{equation}
let $ r\to +\infty $ we get
\begin{equation}
\begin{split}
&(\partial _t \omega _{im} ,\;w_{il} )+(\nabla \omega _{im} ,\;\nabla
w_{il} )+((\bar {u}\cdot \nabla )\bar {\omega }_{im} ,\;w_{il}
)-((\bar {\omega }_m \cdot \nabla )\bar {u}_i ,\;w_{il} )=0 \\
&\qquad \qquad t\in (0,\delta ),\quad \omega _{im} (0)=\omega _{i0}^m ,\quad l=1,\cdots ,m
\\
\end{split}
\end{equation}
where $\omega _{i0}^m $ is the orthogonal projection in $H$ of
$\omega _{i0} $ onto the space spanned by $w_{i1} ,\;\cdots
,\;w_{im} $. Therefore,
\begin{equation*}
\begin{split}
&\sum\limits_{j=1}^m {(w_{ij} ,\;w_{il} ){g}'_{ij} (t)} +\sum\limits_{j=1}^m
{(\nabla w_{ij} ,\;\nabla w_{il} )g_{ij} (t)} + \\
&\quad \quad \quad \quad +\sum\limits_{j=1}^m {\{((\bar {u}(t)\cdot \nabla
)w_{ij} ,\;w_{il} )-((w_j \cdot \nabla )w_{il} ,\;\bar {u}_i (t))\}} \;\bar
{g}_{ij} (t)=0 \\
\end{split}
\end{equation*}
where $\bar {g}_{ij} (t)=\frac{1}{\delta }\int_0^\delta {g_{ij}
(t)dt} $ and $u_i \in L^\infty (0,T;H)$ from Section 1. Inverting
the nonsigular matrix with elements $(w_{ij} ,\;w_{il} ),\;\;1\le
j,l\le m$, we can write above system in the following form
\begin{equation}
\begin{split}
{g}'_{ij} (t)+\sum\limits_{l=1}^m {\alpha _{ijl} \;g_{il} (t)}
+\sum\limits_{l=1}^m {\beta _{ijl} \;\bar {g}_{il} (t)} =0
\end{split}
\end{equation}
where $\alpha _{ijl} ,\;\,\beta _{ijl} $ are constants.
The initial conditions are equivalent to
\[
g_{ij} (0)=g_{ij}^0
=\mbox{the}\;j^{\,th}\;\mbox{component}\;\mbox{of}\;\omega _{i0}^m
\]
We construct a sequence $\{g_{ij}^k \}$ by using a successive approximation:
\begin{equation*}
\begin{split}
&{g_{ij}^1}^\prime =-\sum\limits_{l=1}^m {\alpha _{ijl} g_{il}^0 }
-\sum\limits_{l=1}^m {\beta _{ijl} \bar {g}_{il}^0 } \quad \Rightarrow \quad
g_{ij}^1 =g_{ij}^0 -\int_0^t {\left( {\sum\limits_{l=1}^m {\alpha _{ijl}
g_{il}^0 } +\sum\limits_{l=1}^m {\beta _{ijl} \bar {g}_{il}^0 } } \right)}
\\
&{g_{ij}^2}^\prime =-\sum\limits_{l=1}^m {\alpha _{ijl} g_{il}^1 }
-\sum\limits_{l=1}^m {\beta _{ijl} \bar {g}_{il}^1 } \quad \Rightarrow \quad
g_{ij}^2 =g_{ij}^0 -\int_0^t {\left( {\sum\limits_{l=1}^m {\alpha _{ijl}
g_{il}^1 } +\sum\limits_{l=1}^m {\beta _{ijl} \bar {g}_{il}^1 } } \right)}
\\
&\quad \quad \quad \cdots \cdots \cdots \cdots \\
&{g_{ij}^k}^\prime =-\sum\limits_{l=1}^m {\alpha _{ijl} g_{il}^{k-1} }
-\sum\limits_{l=1}^m {\beta _{ijl} \bar {g}_{il}^{k-1} } \quad \Rightarrow
\quad g_{ij}^k =g_{ij}^0 -\int_0^t {\left( {\sum\limits_{l=1}^m {\alpha
_{ijl} g_{il}^{k-1} } +\sum\limits_{l=1}^m {\beta _{ijl} \bar {g}_{il}^{k-1}
} } \right)} \\
\end{split}
\end{equation*}
so that
\[
\left| {g_{ij}^k (t)-g_{ij}^{k-1} (t)} \right|\le \int_0^t {\left(
{\sum\limits_{l=1}^m {\left| {\alpha _{ijl} } \right|\;\left| {g_{il}^{k-1}
(t)-g_{il}^{k-2} (t)} \right|} +\sum\limits_{l=1}^m {\left| {\beta _{ijl} }
\right|\;\left| {\bar {g}_{il}^{k-1} (t)-\bar {g}_{il}^{k-2} (t)} \right|} }
\right)}
\]
Related to the a priori estimates we shall give later on, we have
\[
\mathop {\max }\limits_{i,j} \;\mathop {\sup }\limits_t \left| {g_{ij}^k
(t)-g_{ij}^{k-1} (t)} \right|\le \mathop {\max }\limits_{i,j}
\sum\limits_{l=1}^m {\left( {\left| {\alpha _{ijl} } \right|+\left| {\beta
_{ijl} } \right|} \right)\cdot t\cdot \mathop {\max }\limits_{i,j} \;\mathop
{\sup }\limits_t \;\left| {g_{ij}^{k-1} (t)-g_{ij}^{k-2} (t)} \right|}
\]
Taking $\delta :\,=\frac{1}{\mathop {\max }\limits_{i,j} \sum\limits_{l=1}^m
{\left( {\left| {\alpha _{ijl} } \right|+\left| {\beta _{ijl} } \right|}
\right)} }$, as $t\le \delta $, then choosing $\delta ^\ast $:
\[
0<\delta ^\ast =\frac{\mathop {\max }\limits_{i,j} \sum\limits_{l=1}^m
{\left( {\left| {\alpha _{ijl} } \right|+\left| {\beta _{ijl} } \right|}
\right)} }{\mathop {\max }\limits_{i,j} \sum\limits_{l=1}^m {\left( {\left|
{\alpha _{ijl} } \right|+2\left| {\beta _{ijl} } \right|} \right)} }<1
\]
it follows that
\[
\mathop {\max }\limits_{i,j} \;\left\| {g_{ij}^k -g_{ij}^{k-1} }
\right\|_\infty \le \delta ^\ast \mathop {\max }\limits_{i,j} \;\left\|
{g_{ij}^{k-1} -g_{ij}^{k-2} } \right\|_\infty \le \cdots \le (\delta ^\ast
)^{k-1}\mathop {\max }\limits_{i,j} \;\left\| {g_{ij}^1 -g_{ij}^0 }
\right\|_\infty
\]
For any $n,\;k$ (we can set $n>k$ without loss of generality), we get
\begin{equation*}
\begin{split}
&\mathop {\max }\limits_{i,j} \;\left\| {g_{ij}^n -g_{ij}^k }
\right\|_\infty \le \mathop {\max }\limits_{i,j} \;\left\| {g_{ij}^n
-g_{ij}^{n-1} } \right\|_\infty +\cdots +\mathop {\max }\limits_{i,j}
\;\left\| {g_{ij}^{k+1} -g_{ij}^k } \right\|_\infty \\
&\le ((\delta ^\ast )^{n-1}+\cdots +(\delta ^\ast )^k)\;\,\mathop {\max
}\limits_{i,j} \;\left\| {g_{ij}^1 -g_{ij}^0 } \right\|_\infty =(\delta
^\ast )^k\frac{1-(\delta ^\ast )^{n-k}}{1-\delta ^\ast }\mathop {\max
}\limits_{i,j} \;\left\| {g_{ij}^1 -g_{ij}^0 } \right\|_\infty \\
&\to 0\quad (k\to \infty ) \\
\end{split}
\end{equation*}
Thus, for every $i=1,2,3;\;\;j=1,\cdots ,m$, $\{g_{ij}^k \}$ is a
Cauchy sequence in $L^\infty (0,\delta )$. Since $L^\infty (0,\delta
)$ is complete, then there exists a function $g_{ij}^\ast \in
L^\infty (0,\delta )$ such that $\left\| {g_{ij}^k -g_{ij}^\ast }
\right\|_\infty \to 0$ as $k\to \infty $.
\[
g_{ij}^k (t)=g_{ij}^0 -\int_0^t {\left( {\sum\limits_{l=1}^m {\alpha _{ijl}
\;g_{il}^{k-1} (t)} +\sum\limits_{l=1}^m {\beta _{ijl} \;\bar {g}_{il}^{k-1}
(t)} } \right)}
\]
let $k\to \infty $, it follows that
\[
g_{ij}^\ast (t)=g_{ij}^0 -\int_0^t {\left( {\sum\limits_{l=1}^m {\alpha
_{ijl} \;g_{il}^\ast (t)} +\sum\limits_{l=1}^m {\beta _{ijl} \;\bar
{g}_{il}^\ast (t)} } \right)}
\]
i.e., $g_{ij}^\ast $ is a solution of the system (12) on $(0,\delta )$ for
which $g_{ij}^\ast (0)=g_{ij}^0 $, $i=1,2,3;\;\;j=1,\cdots ,m$.
(ii) By means of the weighted function $\theta _{r} $:
\begin{equation*}
\begin{split}
&\sum\limits_{i=1}^3 {(\,\theta _{r} \partial _t \omega _{im}
,\;\omega _{im} )} +\sum\limits_{i=1}^3 {(\,\theta _{r} \nabla
\omega _{im} ,\;\nabla \omega _{im} )} +\sum\limits_{i=1}^3
\omega _{im} ,\;\,\omega _{im} \nabla \theta _{r} )} + \\
&\quad \quad \;+\sum\limits_{i=1}^3 {(\,\theta _{r} (\bar {u}\cdot
\nabla )\bar {\omega }_{im} ,\;\omega _{im} )} -\sum\limits_{i=1}^3
{(\,\theta _{r} (\bar {\omega }_m \cdot \nabla )\bar {u}_i
,\;\omega _{im} )} =0 \\
\end{split}
\end{equation*}
Let $ r\to +\infty $ we get
\[
\sum\limits_{i=1}^3 {(\partial _t \omega _{im} ,\;\omega _{im} )}
+\sum\limits_{i=1}^3 {(\nabla \omega _{im} ,\;\nabla \omega _{im} )}
+\sum\limits_{i=1}^3 {((\bar {u}\cdot \nabla )\bar {\omega }_{im} ,\;\omega
_{im} )} -\sum\limits_{i=1}^3 {((\bar {\omega }_m \cdot \nabla )\bar {u}_i
,\;\omega _{im} )} =0
\]
Then we write
\begin{equation*}
\begin{split}
&\frac{1}{2}\frac{d}{dt}\left( {\sum\limits_{i=1}^3 {\left\| {\omega _{im} }
\right\|_{L^2({\mathbb R}^3)}^2 } } \right)+\sum\limits_{i=1}^3
{\left\| {\nabla \omega _{im} } \right\|_{L^2({\mathbb R}^3)}^2 }
-\sum\limits_{i=1}^3 {((\bar
{u}\cdot \nabla )\omega _{im} ,\;\bar {\omega }_{im} )} + \\
&\quad \quad \quad \quad \quad +\sum\limits_{i=1}^3 {((\bar {\omega }_m
\cdot \nabla )\omega _{im} ,\;\bar {u}_i )} =0 \\
\end{split}
\end{equation*}
Similar to those in the section 2, and $\eta $ is chosen to be small
enough, we have
\[
\sum\limits_{i=1}^3 {\left\| {\omega _{im} } \right\|_{L^2({\mathbb
R}^3)}^2 } +\varepsilon _0 \int_0^\eta {\left( {\sum\limits_{i=1}^3
{\left\| {\nabla \omega _{im} } \right\|_{L^2({\mathbb R}^3)}^2 } }
\right)} \le e^{(1-\varepsilon _0 )\;\eta }\left(
{\sum\limits_{i=1}^3 {\left\| {\omega _{i0}^m }
\right\|_{L^2({\mathbb R}^3)}^2 } } \right)
\]
\begin{equation}
\begin{split}
\mathop {\sup }\limits_{t\in (0,\;\eta )} \left(
{\sum\limits_{i=1}^3 {\left\| {\omega _{im} } \right\|_{L^2({\mathbb
R}^3)}^2 } } \right)\le e^{(1-\varepsilon _0 )\;\eta }\left(
{\sum\limits_{i=1}^3 {\left\| {\omega _{i0}^m }
\right\|_{L^2({\mathbb R}^3)}^2 } } \right)
\end{split}
\end{equation}
\begin{equation}
\begin{split}
\sum\limits_{i=1}^3 {\left\| {\omega _{im} (\eta )}
\right\|_{L^2({\mathbb R}^3)}^2 } +\int_0^\eta {\left(
{\sum\limits_{i=1}^3 {\left\| {\nabla \omega _{im} }
\right\|_{L^2({\mathbb R}^3)}^2 } } \right)} \le
\frac{1}{\varepsilon _0 }e^{(1-\varepsilon _0 )\;\eta }\left(
{\sum\limits_{i=1}^3 {\left\| {\omega _{i0}^m }
\right\|_{L^2({\mathbb R}^3)}^2 } } \right)
\end{split}
\end{equation}
The inequalities (13) and (14) are valid for any fixed $\delta \le
\eta $.
(iii) Let $\tilde {\omega }_m $ denote the function from ${\mathbb
R}$ into $V$, which is equal to $\omega _m $ on $(0,\delta )$ and to
0 on the complement of this interval. The Fourier transform of
$\tilde {\omega }_m $ is denoted by $\hat {\omega }_m $. We want to
show that
\[
\int_{-\infty }^{+\infty } {\left| \tau \right|^{2\gamma }\left(
{\sum\limits_{i=1}^3 {\left\| {\hat {\omega }_{im} (\tau )}
\right\|_{L^2(\Omega )}^2 } } \right)} \,d\tau <+\infty ,\quad \quad
\forall\; \Omega \subset {\mathbb R}^3
\]
For some $\gamma >0$. Along with (14) this will imply that
$\tilde {\omega }_m $ belongs to a bounded set of $H^\gamma
({\mathbb R},\;H^1(\Omega ),\;L^2(\Omega )),\quad \forall\; \Omega $
and will enable us to apply the result of compactness.
We observe that (10) can be written as
\begin{equation*}
\begin{split}
&\frac{d}{dt}\left( {\sum\limits_{i=1}^3 {(\,\theta _{r} \tilde
{\omega }_{im} ,\;w_{ij} )} } \right)=\sum\limits_{i=1}^3 {(\,\theta
_{r} \tilde {f}_{im} ,\;w_{ij} )} +\sum\limits_{i=1}^3 {(\,\theta
_{r} \omega _{i0}^m ,\;w_{ij} )\,} \eta _0 - \\
&\qquad \qquad \qquad \quad \quad \quad \quad \quad \quad -\sum\limits_{i=1}^3
{(\,\theta _{r} \omega _{im} (\delta ),\;w_{ij} )\,} \eta _\delta
\\
\end{split}
\end{equation*}
where $\eta _0 ,\;\eta _\delta $ are Dirac distributions at 0 and
$\delta $, and
\begin{equation*}
\begin{split}
&f_{im} =-\Delta \omega _{im} +(\bar {u}\cdot \nabla )\bar {\omega
}_{im} -(\bar {\omega }_m \cdot \nabla )\;\bar {u}_i
\\
&\tilde {f}_{im} =f_{im} \;\; \mbox{on}\;\; (0,\delta ),\quad 0\; \mbox{ outside
this interval} \\
\end{split}
\end{equation*}
By the Fourier transform,
\begin{equation*}
\begin{split}
&2\mbox{i}\pi \tau \sum\limits_{i=1}^3 {(\,\theta _{r} \hat {\omega
}_{im} ,\;w_{ij} )} =\sum\limits_{i=1}^3 {(\,\theta _{r} \hat
{f}_{im} ,\;w_{ij} )} +\sum\limits_{i=1}^3 {(\,\theta _{r} \omega
_{i0}^m ,\;w_{ij} )} - \\
&\quad \quad \quad \quad \quad \quad \quad \quad \quad -\sum\limits_{i=1}^3
{(\,\theta _{r} \omega _{im} (\delta ),\;w_{ij} )\,} \exp
(-2\mbox{i}\pi \delta \tau ) \\
\end{split}
\end{equation*}
where $\hat {\omega }_{im} $ and $\hat {f}_{im} $ denote the Fourier
transforms of $\tilde {\omega }_{im} $ and $\tilde {f}_{im} $
We multiply above equalities by $\hat {g}_{ij} (\tau )=$Fourier
transform of $\tilde {g}_{ij} $ and add the resulting equations for
$j=1,\cdots ,m$, we get
\begin{equation*}
\begin{split}
&2\mbox{i}\pi \tau \sum\limits_{i=1}^3 {\left\| {\,\theta^{1/2} _{
r} \;\hat {\omega }_{im} (\tau )} \right\|_{L^2({\mathbb R}^3)}^2 }
=\sum\limits_{i=1}^3 {(\,\theta _{r} \hat {f}_{im} (\tau
),\;\hat {\omega }_{im} (\tau ))} \\
&\quad \quad +\sum\limits_{i=1}^3 {(\,\theta _{r} \omega _{i0}^m
,\;\hat {\omega }_{im} (\tau ))} -\sum\limits_{i=1}^3 {(\,\theta
_{r} \omega _{im} (\delta ),\;\hat {\omega }_{im} (\tau ))\,} \exp
(-2\mbox{i}\pi \delta \tau )
\\
\end{split}
\end{equation*}
For some $\varphi _i \in V$ and $Q_\delta =(0,\delta )\times
{\mathbb R}^3$,
\begin{equation*}
\begin{split}
&\int_0^\delta {\sum\limits_{i=1}^3 {(\,\theta _{r} f_{im}
,\;\varphi _i )} } =\int_0^\delta {\sum\limits_{i=1}^3 {(-\theta
_{r} \Delta \omega _{im} ,\;\varphi _i )} } +\int_0^\delta
{\sum\limits_{i=1}^3 {(\,\theta _{r} (\bar {u}\cdot \nabla )\bar
{\omega }_{im} ,\;\varphi _i )} } - \\
&\quad \quad \quad \quad \quad \quad {\kern 1pt}\qquad \qquad -\int_0^\delta
{\sum\limits_{i=1}^3 {(\,\theta _{r} (\bar {\omega }_m \cdot \nabla
)\;\bar {u}_i ,\;\varphi _i )} } \\
&=\int_0^\delta {\sum\limits_{i=1}^3 {(\,\theta _{r} \nabla \omega
_{im} ,\;\nabla \varphi _i )} } +\int_0^\delta {\sum\limits_{i=1}^3 {(\nabla
\omega _{im} ,\;\,\varphi _i \nabla \theta _{r} )} } \\
&\quad -\int_0^\delta {\sum\limits_{i=1}^3 {(\,\theta _{r} (\bar
{u}\cdot \nabla )\varphi _i ,\;\bar {\omega }_{im} )} } -\int_0^\delta
{\sum\limits_{i=1}^3 {(\,\varphi _i (\bar {u}\cdot \nabla )\theta _{r} ,\;\bar {\omega }_{im} )} } \\
&\quad +\int_0^\delta {\sum\limits_{i=1}^3 {(\,\theta _{r} (\bar
{\omega }_m \cdot \nabla )\varphi _i ,\;\bar {u}_i )} } +\int_0^\delta
{\sum\limits_{i=1}^3 {(\,\varphi _i (\bar {\omega }_m \cdot \nabla )\,\theta
_{r} ,\;\bar {u}_i )} } \\
\end{split}
\end{equation*}
Let $ r\to +\infty $ we get
\begin{equation*}
\begin{split}
&\int_0^\delta {\sum\limits_{i=1}^3 {(f_{im} ,\;\varphi _i )} }
=\int_0^\delta {\sum\limits_{i=1}^3 {(\nabla \omega _{im} ,\;\nabla \varphi
_i )} } -\int_0^\delta {\sum\limits_{i=1}^3 {((\bar {u}\cdot \nabla )\varphi
_i ,\;\bar {\omega }_{im} )} } +\int_0^\delta {\sum\limits_{i=1}^3 {((\bar
{\omega }_m \cdot \nabla )\varphi _i ,\;\bar {u}_i )} } \\
&\le \int_0^\delta {\sum\limits_{i=1}^3 {\left\| {\nabla \omega _{im} }
\right\|_{L^2({\mathbb R}^3)} \left\| {\nabla \varphi _i }
\right\|_{L^2({\mathbb
R}^3)} } } + \\
&\quad +2\left( {\sum\limits_{i=1}^3 {\left\| {\bar {u}_i }
\right\|_{L^4(Q_\delta )}^2 } } \right)^{1/2}\left( {\sum\limits_{i=1}^3
{\left\| {\bar {\omega }_{im} } \right\|_{L^4(Q_\delta )}^2 } }
\right)^{1/2}\left( {\sum\limits_{i=1}^3 {\left\| {\nabla \varphi _i }
\right\|_{L^2(Q_\delta )}^2 } } \right)^{1/2} \\
&\le \int_0^\delta {\left( {\sum\limits_{i=1}^3 {\left\| {\nabla \omega
_{im} } \right\|_{L^2({\mathbb R}^3)}^2 } } \right)^{1/2}} \left(
{\sum\limits_{i=1}^3 {\left\| {\nabla \varphi _i }
\right\|_{L^2({\mathbb
R}^3)}^2 } } \right)^{1/2}+ \\
\end{split}
\end{equation*}
\begin{equation*}
\begin{split}
&\quad +2C\sqrt \delta \left( {\int_0^\delta {\sum\limits_{i=1}^3
{\{\,\left\| {\bar {u}_i } \right\|_{L^2({\mathbb R}^3)}^2 +\left\|
{\nabla \bar
{u}_i } \right\|_{L^2({\mathbb R}^3)}^2 \}} } } \right)^{1/2} \\
&\quad \quad \times \left( {\int_0^\delta {\sum\limits_{i=1}^3 {\{\,\left\|
{\bar {\omega }_{im} } \right\|_{L^2({\mathbb R}^3)}^2 +\left\|
{\nabla \bar {\omega }_{im} } \right\|_{L^2({\mathbb R}^3)}^2 \}} }
} \right)^{1/2}\left( {\sum\limits_{i=1}^3 {\left\| {\nabla \varphi
_i } \right\|_{L^2({\mathbb
R}^3)}^2 } } \right)^{1/2} \\
&\le \int_0^\delta {\left( {\sum\limits_{i=1}^3 {\left\| {\nabla \omega
_{im} } \right\|_{L^2({\mathbb R}^3)}^2 } } \right)^{1/2}} \left\|
\varphi } \right\|_V + \\
&\quad +2C\sqrt \delta \left( {\delta \;\mathop {\sup }\limits_{(0,\delta )}
\;\sum\limits_{i=1}^3 {\left\| {u_i } \right\|_{L^2({\mathbb
R}^3)}^2 } +\int_0^\delta {\sum\limits_{i=1}^3 {\left\| {\nabla u_i
\right\|_{L^2({\mathbb R}^3)}^2 } } } \right)^{1/2} \\
&\quad \times \left( {\delta \;\mathop {\sup }\limits_{(0,\delta )}
\sum\limits_{i=1}^3 {\left\| {\omega _{im} } \right\|_{L^2({\mathbb
R}^3)}^2 } +\int_0^\delta {\sum\limits_{i=1}^3 {\left\| {\nabla
\omega _{im} } \right\|_{L^2({\mathbb R}^3)}^2 } } }
\right)^{1/2}\left\| {\nabla \varphi }
\right\|_V \\
\end{split}
\end{equation*}
this remains bounded according to (5) and (14). Therefore,
\[
\int_0^\delta {\left\| {f_{im} (t)} \right\|_V dt} =\int_0^\delta {\;\mathop
{\sup }\limits_{\left\| \varphi \right\|_V =1} \;\sum\limits_{i=1}^3
{(f_{im} ,\;\varphi _i )} } <+\infty
\]
it follows that
\[
\mathop {\sup }\limits_{\tau \in {\mathbb R}} \left\| {\hat {f}_{im}
(\tau )} \right\|_V <+\infty ,\quad \;\forall m
\]
Due to (13) we have
\[
\left\| {\omega _{im} (0)} \right\|_{L^2({\mathbb R}^3)} <+\infty
,\quad \quad \left\| {\omega _{im} (\delta )} \right\|_{L^2({\mathbb
R}^3)} <+\infty
\]
then by Poincare inequality,
\begin{equation*}
\begin{split}
&\left| \tau \right|\;\sum\limits_{i=1}^3 {\left\| {\,\theta^{1/2}
_{r} \;\hat {\omega }_{im} (\tau )} \right\|_{L^2({\mathbb R}^3)}^2
} \le c_1 \sum\limits_{i=1}^3 {\left\| {\hat {f}_{im} (\tau )}
\right\|_V \;\left\| {\,\theta
_{r} \hat {\omega }_{im} (\tau )} \right\|_V } \\
& \qquad \qquad \qquad \qquad \qquad \qquad + c_2
\sum\limits_{i=1}^3 {\left\| {\,\theta _{r} \hat {\omega }_{im}
(\tau )}
\right\|_{L^2({\mathbb R}^3)} } \\
&\quad \le c_3 \sum\limits_{i=1}^3 {\left\| {\nabla (\theta _{r} \hat
{\omega }_{im} (\tau ))} \right\|_{L^2({\mathbb R}^3)} } \\
&\quad \le c_4 \sum\limits_{i=1}^3 {\left( {\left\| {\,\hat {\omega }_{im} \nabla
\theta _{r} } \right\|_{L^2({\mathbb R}^3)} } \right.} +\left.
{\left\| {\,\theta _{r} \nabla \hat {\omega }_{im} }
\right\|_{L^2({\mathbb R}^3)} } \right) \\
\end{split}
\end{equation*}
Using $x^2e^{-\kappa x}\le C_1 \; (\kappa>0)$ and assuming that $ r
$ is sufficiently large, we get
\begin{equation}
\begin{split}
&\left| \tau \right|\;\sum\limits_{i=1}^3 {\left\| {\,\theta^{1/2}
_{r} \;\hat {\omega }_{im} (\tau )}
\right\|_{L^2({\mathbb R}^3)}^2 } \\
&\quad \le c_5 \sum\limits_{i=1}^3
{\left\| {\,\theta^{1/2} _{r} \hat {\omega }_{im} }
\right\|_{L^2({\mathbb R}^3)} } +\;c_6 \;\sum\limits_{i=1}^3
{\left\| {\,\theta _{r} \nabla \hat {\omega }_{im} }
\right\|_{L^2({\mathbb R}^3)} } \\
\end{split}
\end{equation}
For $\gamma $ fixed, $\gamma <1/4$, we observe that
\[
\left| \tau \right|^{2\gamma }\le c_7 (\gamma )\frac{1+\left| \tau
\right|}{1+\left| \tau \right|^{1-2\gamma }},\quad \quad \forall
\tau \in {\mathbb R}
\]
Thus by (15),
\begin{equation*}
\begin{split}
&\int_{-\infty }^{+\infty } {\left| \tau \right|^{2\gamma }\left(
{\sum\limits_{i=1}^3 {\left\| {\,\theta^{1/2} _{r} \;\hat {\omega
}_{im} (\tau )} \right\|_{L^2({\mathbb R}^3)}^2 } } \right)} \,d\tau
\le c_7 (\gamma )\int_{-\infty }^{+\infty } {\frac{1+\left| \tau
\right|}{1+\left| \tau \right|^{1-2\gamma }}\left(
{\sum\limits_{i=1}^3 {\left\| {\,\theta^{1/2} _{r} \;\hat {\omega
}_{im} (\tau )} \right\|_{L^2({\mathbb R}^3)}^2 } } \right)}
\,d\tau \\
&\le c_8 \;\int_{-\infty }^{+\infty } {\frac{1}{1+\left| \tau
\right|^{1-2\gamma }}\;\sum\limits_{i=1}^3 {\left\| {\,\theta^{1/2}
_{r} \hat {\omega }_{im} (\tau )}
\right\|_{L^2({\mathbb R}^3)} } } d\tau \;\; + \\
&+\;c_9 \int_{-\infty }^{+\infty } {\frac{1}{1+\left| \tau \right|^{1-2\gamma
}}\;\sum\limits_{i=1}^3 {\left\| {\,\theta _{r} \nabla \hat {\omega
}_{im} (\tau )} \right\|_{L^2({\mathbb R}^3)} } } d\tau +\;c_{10}
\int_{-\infty }^{+\infty } {\sum\limits_{i=1}^3 {\left\|
{\,\theta^{1/2} _{r} \;\hat {\omega }_{im} (\tau )}
\right\|_{L^2({\mathbb R}^3)}^2 } } \,d\tau \\
\end{split}
\end{equation*}
Because of the Parseval equality,
\begin{equation*}
\begin{split}
&\int_{-\infty }^{+\infty } {\sum\limits_{i=1}^3 {\left\| {\,\theta
_{r} \hat {\omega }_{im} (\tau )} \right\|_{L^2({\mathbb R}^3)}^2 }
} \,d\tau =\int_0^\delta {\sum\limits_{i=1}^3 {\left\| {\,\theta
\omega _{im} (t)} \right\|_{L^2({\mathbb R}^3)}^2 } } \,dt \\
&\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \;\,\le C_2
\delta \;\mathop {\sup }\limits_{(0,\delta )} \;\sum\limits_{i=1}^3 {\left\|
{\omega _{im} } \right\|_{L^2({\mathbb R}^3)}^2 } <+\infty \\
&\int_{-\infty }^{+\infty } {\sum\limits_{i=1}^3 {\left\| {\,\theta
_{r} \nabla \hat {\omega }_{im} (\tau )} \right\|_{L^2({\mathbb
R}^3)}^2 } } \,d\tau =\int_0^\delta {\sum\limits_{i=1}^3 {\left\|
{\,\theta _{r} \nabla \omega _{im} (t)} \right\|_{L^2({\mathbb
R}^3)}^2 } } \,dt
\\
&\quad \quad \quad \quad \quad \quad \quad \quad \quad
\quad \,\le C_3 \int_0^\delta {\sum\limits_{i=1}^3 {\left\| {\nabla \omega
_{im} } \right\|_{L^2({\mathbb R}^3)}^2 } } <+\infty \\
\end{split}
\end{equation*}
as $m\to \infty $. By Cauchy-Schwarz inequality and the Parseval
\begin{equation*}
\begin{split}
&\int_{-\infty }^{+\infty } {\frac{1}{1+\left| \tau \right|^{1-2\gamma
}}\sum\limits_{i=1}^3 {\left\| {\,\theta^{1/2} _{r} \;\hat
{\omega }_{im} (\tau )} \right\|_{L^2({\mathbb R}^3)} } } d\tau \\
&\quad \quad \le \sqrt 3 \left( {\int_{-\infty }^{+\infty }
{\frac{1}{(1+\left| \tau \right|^{1-2\gamma })^2}d\tau } }
\right)^{1/2}\left( {\int_0^\delta {\sum\limits_{i=1}^3 {\left\|
{\,\theta^{1/2} _{r} \;\omega _{im} (t)} \right\|_{L^2({\mathbb
R}^3)}^2 } }
dt} \right)^{1/2}<+\infty \\
&\int_{-\infty }^{+\infty } {\frac{1}{1+\left| \tau \right|^{1-2\gamma
}}\sum\limits_{i=1}^3 {\left\| {\,\theta _{r} \nabla \hat {\omega
}_{im} (\tau )} \right\|_{L^2({\mathbb R}^3)} } } d\tau \\
&\quad \quad \le \sqrt 3 \left( {\int_{-\infty }^{+\infty }
{\frac{1}{(1+\left| \tau \right|^{1-2\gamma })^2}d\tau } }
\right)^{1/2}\left( {\int_0^\delta {\sum\limits_{i=1}^3 {\left\|
{\,\theta _{r} \nabla \omega _{im} (t)} \right\|_{L^2({\mathbb
R}^3)}^2 } } dt}
\right)^{1/2}<+\infty \\
\end{split}
\end{equation*}
as $m\to \infty $ by $\gamma <1/4$ and (14).
(iv) The estimates (12) and (14) enable us to assert the existence
of an element $\omega ^\ast \in L^2(0,\delta ;H^1(\Omega ))\cap
L^\infty (0,\delta ;L^2(\Omega )),\quad \forall\; \Omega \subset
{\mathbb R}^3$, and a subsequence $\omega _{{m}'} $ such that
$\omega _{{m}'} \to \omega ^\ast $ in $L^2(0,\delta
;H^1(\Omega ))$ weakly, and in $L^\infty (0,\delta
;L^2(\Omega ))$ weak-star,
as ${m}'\to \infty $, for any $\Omega \subset
{\mathbb R}^3$
Due to (iii) we also have
$\omega _{{m}'} \to \omega ^\ast $ in $L^2(0,\delta
;L^2(\Omega ))$ strongly as ${m}'\to \infty $, for
any $\Omega \subset {\mathbb
which means
$\omega _{{m}'} \to \omega ^\ast $ in $L^2(0,\delta ;L_{
\mbox{
\begin{footnotesize}loc \end{footnotesize}} } ^2 (\Omega ))$ strongly
In particular, for a fixed $j$
$\left. {\omega _{{m}'} } \right|_{{\Omega }'} \to \left. {\omega
^\ast } \right|_{{\Omega }'} $ in $L^2(0,\delta
;L^2({\Omega }'))$
where ${\Omega }'$ denotes the support of $w_{ij} $. This
convergence result enable us to pass to the limit.
Let $\psi _i $ be a continuously differentiable function on $(0,\delta )$
with $\psi _i (\delta )=0$. We multiply (11) by $\psi _i (t)$ then integrate
by parts. This leads to the equation
\begin{equation*}
\begin{split}
&-\int_0^\delta {\sum\limits_{i=1}^3 {(\omega _{im} (t),\;\partial _t \psi
_i (t)w_{ij} )\,dt} } +\int_0^\delta {\sum\limits_{i=1}^3 {(\nabla \omega
_{im} ,\;\psi _i (t)\nabla w_{ij} )\,dt} } \\
&+\int_0^\delta {\sum\limits_{i=1}^3 {((\bar {u}\cdot \nabla )\bar {\omega
}_{im} ,\;w_{ij} \psi _i (t))} } -\int_0^\delta {\sum\limits_{i=1}^3 {((\bar
{\omega }_m \cdot \nabla )\,\bar {u}_i ,\;w_{ij} \psi _i (t))} }
=\sum\limits_{i=1}^3 {(\omega _{i0}^m ,\;w_{ij} )\psi _i (0)} \\
\end{split}
\end{equation*}
Since $\omega _{i{m}'} $ converges to $\omega _i^\ast $ in
$L^2(0,\delta ;L^2(\Omega ))$ strongly as ${m}'\to \infty $, then
$\bar {\omega }_{i{m}'} $ also converges strongly to $\bar {\omega
}_i^\ast $, and
\begin{equation*}
\begin{split}
&\int_0^\delta {\sum\limits_{i=1}^3 {(\omega _{i{m}'} ,\;\partial _t
\psi _i (t)w_{ij} )\,dt} } \to \int_0^\delta {\sum\limits_{i=1}^3
{(\omega _i^\ast ,\;\partial _t \psi _i (t)w_{ij} )\,dt} } \\
&\int_0^\delta {\sum\limits_{i=1}^3 {(\nabla \omega _{i{m}'} ,\;\psi _i
(t)\nabla w_{ij} )\,dt} } = -\int_0^\delta {\sum\limits_{i=1}^3
_{i{m}'} ,\;\psi _i (t)\Delta w_{ij} )\,dt} } \\
&\quad \;\quad \quad \quad \to -\int_0^\delta {\sum\limits_{i=1}^3 {(\omega
_i^\ast ,\;\psi _i (t)\Delta w_{ij} )} } =\int_0^\delta {\sum\limits_{i=1}^3
{(\nabla \omega _i^\ast ,\;\psi _i (t)\nabla w_{ij} )\,dt} } \\
&\int_0^\delta {\sum\limits_{i=1}^3 {((\bar {u}\cdot \nabla )\bar {\omega
}_{i{m}'} ,\;w_{ij} \psi _i (t))} } =-\int_0^\delta {\sum\limits_{i=1}^3
{((\bar {u}\cdot \nabla )w_{ij} \psi _i (t),\;\bar {\omega }_{i{m}'} )} } \\
&\quad \;\quad \quad \quad \to -\int_0^\delta {\sum\limits_{i=1}^3 {((\bar
{u}\cdot \nabla )w_{ij} \psi _i (t),\;\bar {\omega }_i^\ast )} }
=\int_0^\delta {\sum\limits_{i=1}^3 {((\bar {u}\cdot \nabla )\bar {\omega
}_i^\ast ,\;w_{ij} \psi _i (t))} } \\
&\int_0^\delta {\sum\limits_{i=1}^3 {((\bar {\omega }_{i{m}'} \cdot \nabla
)\,\bar {u}_i ,\;w_{ij} \psi _i (t))} } \to \int_0^\delta
{\sum\limits_{i=1}^3 {((\bar {\omega }^\ast \cdot \nabla )\,\bar
{u}_i ,\;w_{ij} \psi _i (t))} } \\
&\sum\limits_{i=1}^3 {(\omega _{i0}^{{m}'} ,\;w_{ij} )\psi _i (0)} \to
\sum\limits_{i=1}^3 {(\omega _{i0},\;w_{ij} )\psi _i (0)} \\
\end{split}
\end{equation*}
Thus, in the limit we find
\begin{equation}
\begin{split}
&-\int_0^\delta {\sum\limits_{i=1}^3 {(\omega _i^\ast ,\;\partial _t \psi _i
(t)v_i )\,dt} } +\int_0^\delta {\sum\limits_{i=1}^3 {(\nabla \omega _i^\ast
,\;\psi _i (t)\nabla v_i )\,dt} } \\
&+\int_0^\delta {\sum\limits_{i=1}^3 {((\bar {u}\cdot \nabla )\bar {\omega
}_i^\ast ,\;v_i \psi _i (t))dt} } -\int_0^\delta {\sum\limits_{i=1}^3
{((\bar {\omega }^\ast \cdot \nabla )\,\bar {u}_i ,\;v_i \psi _i (t))} }
=\sum\limits_{i=1}^3 {(\omega _{i0} ,\;v_i )\psi _i (0)} \\
\end{split}
\end{equation}
holds for $v_i =w_{i1} ,\;w_{i2} ,\cdots $; by this equation holds
for $v_i =$ any finite linear combination of the $w_{ij} $, and by a
continuity argument above equation is still true for any $v_i \in
V$. Hence we find that $\omega _i^\ast (i=1,2,3)$ is a Leray-Hopf
weak solution of the system (9).
Finally it remains to prove that $\omega _i^\ast $ satisfy the initial
conditions. For this we multiply (9) by $v_i \psi _i (t)$, after integrating
some terms by parts, we get in the same way,
\begin{equation*}
\begin{split}
&-\int_0^\delta {\sum\limits_{i=1}^3 {(\omega _i^\ast ,\;\partial _t \psi _i
(t)v_i )} } +\int_0^\delta {\sum\limits_{i=1}^3 {(\nabla \omega _i^\ast
,\;\psi _i (t)\nabla v_i )\,dt} } \\
&+\int_0^\delta {\sum\limits_{i=1}^3 {((\bar {u}\cdot \nabla )\bar {\omega
}_i^\ast ,\;v_i \psi _i (t))} } -\int_0^\delta {\sum\limits_{i=1}^3 {((\bar
{\omega }^\ast \cdot \nabla )\,\bar {u}_i ,\;v_i \psi _i (t))} }
=\sum\limits_{i=1}^3 {(\omega _i^\ast (0),\;v_i )\psi _i (0)} \\
\end{split}
\end{equation*}
By comparison with (16),
\[
\sum\limits_{i=1}^3 {(\omega _i^\ast (0)-\omega _{i0} ,\;v_i )\psi _i (0)}
\]
Therefore we can choose $\psi _i $ particularly such that
\[
(\omega _i^\ast (0)-\omega _{i0} ,\;v_i )=0,\quad \quad \forall\;
v_i \in V
\]
4. Convergence
Now the partition is refined infinitely, we will prove that there
exists some subsequence of the solutions of auxiliary problems which
converges to a weak solution of (6).
\[
\mathop {\sup }\limits_{t\in (0,T)} \;\int_{{\mathbb R}^3} {(\tilde
{\omega }_1^2 +\tilde {\omega }_2^2 +\tilde {\omega }_3^2 )}
\;<+\infty
\]
the family $(\tilde {\omega }_1 ,\tilde {\omega }_2 ,\tilde {\omega }_3 )$
is uniformly bounded in $L^2(0,T;H)\cap L^\infty (0,T;H)$, then we can
choose ${k}'\to \infty $, or $\Delta t_k ^\prime \to 0$, such that there
exists a subsequence $({\tilde {\omega }}'_1 ,{\tilde {\omega }}'_2 ,{\tilde
{\omega }}'_3 )$ converging weakly in $L^2(0,T;H)$ and weak-star in
$L^\infty (0,T;H)$ to some element $(\omega _1^\ast ,\omega _2^\ast ,\omega
_3^\ast )$. On the other hand, because $\tilde {\omega }_i (i=1,2,3)$ belong
to $L^2(0,T;H)$, we can verify that
\[
\bar {\omega }_i (x,t)=\left\{ {\frac{1}{\Delta t_k }\int_{t_{k-1} }^{t_k }
{\tilde {\omega }_i (x,t)dt} ,\;\;t\in (t_{k-1} ,t_k )\subset (0,T)}
\right\}
\]
also belongs to $L^2(0,T;H)$. In fact,
\begin{equation*}
\begin{split}
&\int_0^T {\int_{{\mathbb R}^3} {\bar {\omega }_i^2 (x,t)} } =\sum\limits_k
{\int_{t_{k-1} }^{t_k } {\int_{{\mathbb R}^3} {\left(
{\frac{1}{\Delta t_k
}\int_{t_{k-1} }^{t_k } {\tilde {\omega }_i (x,t)} } \right)} } } ^2= \\
&\quad \quad =\sum\limits_k {\frac{1}{\Delta t_k^2 }\cdot
\Delta t_k \cdot \int_{{\mathbb R}^3} {\left( {\int_{t_{k-1} }^{t_k
} {\tilde {\omega }_i (x,t)} } \right)} } ^2\le \sum\limits_k
{\frac{1}{\Delta t_k }\int_{{\mathbb R}^3} {\int_{t_{k-1} }^{t_k } 1
\cdot \int_{t_{k-1} }^{t_k }
{\tilde {\omega }_i^2 (x,t)} } } \\
&\quad \quad =\sum\limits_k {\int_{t_{k-1} }^{t_k }
{\int_{{\mathbb R}^3} {\tilde {\omega }_i^2 (x,t)} } } =\int_0^T
R}^3} {\tilde {\omega }_i^2 (x,t)} } <+\infty \\
\end{split}
\end{equation*}
In the same way, we know from (5) that the function
\[
\bar {u}_i (x,t)=\left\{ {\frac{1}{\Delta t_k }\int_{t_{k-1} }^{t_k } {u_i
(x,t)dt} ,\;\;t\in (t_{k-1} ,t_k )\subset (0,T)} \right\}
\]
belongs to $L^2(0,T;H)$.
Finally we will prove that $(\omega _1^\ast ,\omega _2^\ast ,\omega _3^\ast
)$ is a solution of the vorticity-velocity form of Navier-Stokes equation
Taking $\varphi _i \in C^\infty ((0,T)\times {\mathbb
R}^3)\;\;(i=1,2,3)$, and
\[
\partial _{x_1 } \varphi _1 +\partial _{x_2 }
\varphi _2 +\partial _{x_3 } \varphi _3 =0
\]
we have
\begin{equation*}
\begin{split}
&\sum\limits_{k=1}^N {\int_{t_{k-1} }^{t_k } {\int_{{\mathbb R}^3} {\theta
_{r} \varphi _1 (\partial _t \tilde {\omega }_1 \,+\bar {u}_1^k
\partial _{x_1 } \bar {\omega }_1^k +\bar {u}_2^k \partial _{x_2 } \bar
{\omega }_1^k +\bar {u}_3^k \partial _{x_3 } \bar {\omega }_1^k -} } } \\
&\quad \quad \quad \quad \quad \quad \quad \quad \quad -\bar {\omega }_1^k
\partial _{x_1 } \bar {u}_1^k -\bar {\omega }_2^k \partial _{x_2 } \bar
{u}_1^k -\bar {\omega }_3^k \partial _{x_3 } \bar {u}_1^k +\partial _{x_1 }
q-\Delta \tilde {\omega }_1 )=0 \\
&\sum\limits_{k=1}^N {\int_{t_{k-1} }^{t_k } {\int_{{\mathbb R}^3} {\theta
_{r} \varphi _2 (\partial _t \tilde {\omega }_2 +\bar {u}_1^k
\partial _{x_1 } \bar {\omega }_2^k +\bar {u}_2^k \partial _{x_2 } \bar
{\omega }_2^k +\bar {u}_3^k \partial _{x_3 } \bar {\omega }_2^k } } } - \\
&\quad \quad \quad \quad \quad \quad \quad \quad \quad -\bar {\omega }_1^k
\partial _{x_1 } \bar {u}_2^k -\bar {\omega }_2^k \partial _{x_2 } \bar
{u}_2^k -\bar {\omega }_3^k \partial _{x_3 } \bar {u}_2^k +\partial _{x_2 }
q-\Delta \tilde {\omega }_2 )=0 \\
&\sum\limits_{k=1}^N {\int_{t_{k-1} }^{t_k } {\int_{{\mathbb R}^3} {\theta
_{r} \varphi _3 (\partial _t \tilde {\omega }_3 +\bar {u}_1^k
\partial _{x_1 } \bar {\omega }_3^k +\bar {u}_2^k \partial _{x_2 } \bar
{\omega }_3^k +\bar {u}_3^k \partial _{x_3 } \bar {\omega }_3^k } } } - \\
&\quad \quad \quad \quad \quad \quad \quad \quad \quad -\bar {\omega }_1^k
\partial _{x_1 } \bar {u}_3^k -\bar {\omega }_2^k \partial _{x_2 } \bar
{u}_3^k -\bar {\omega }_3^k \partial _{x_3 } \bar {u}_3^k +\partial _{x_3 }
q-\Delta \tilde {\omega }_3 )=0 \\
\end{split}
\end{equation*}
Here $\tilde \omega_i \; (i=1,2,3)$ denote the collection of those
solutions of problem (7) defined on every $(t_{k-1},t_k)$.
Integrating by parts we get
\begin{equation*}
\begin{split}
&\sum\limits_{k=1}^N {\int_{t_{k-1} }^{t_k } {\int_{{\mathbb R}^3} {\theta
_{r} (\tilde {\omega }_1 \partial _t \varphi _1 \,+\bar {\omega
}_1^k ((\bar {u}_1^k \partial _{x_1 } \varphi _1 +\varphi _1
\,\partial _{x_1 } \bar {u}_1^k )+(\bar {u}_2^k \partial _{x_2 }
\varphi _1 +\varphi _1
\,\partial _{x_2 } \bar {u}_2^k )+} } } \\
&\quad +(\bar {u}_3^k \partial _{x_3 } \varphi _1 +\varphi _1 \,\partial
_{x_3 } \bar {u}_3^k ))-\bar {u}_1^k ((\bar {\omega }_1^k \partial _{x_1 }
\varphi _1 +\varphi _1 \,\partial _{x_1 } \bar {\omega }_1^k )+(\bar {\omega
}_2^k \partial _{x_2 } \varphi _1 +\varphi _1 \,\partial _{x_2 } \bar
{\omega }_2^k )+ \\
&\quad +(\bar {\omega }_3^k \partial _{x_3 } \varphi _1 +\varphi _1
\,\partial _{x_3 } \bar {\omega }_3^k ))+q\partial _{x_1 } \varphi _1
+\tilde {\omega }_1 \Delta \varphi _1 )+ \\
&+\sum\limits_{k=1}^N {\int_{t_{k-1} }^{t_k } {\int_{{\mathbb R}^3} {(\bar
{\omega }_1^k (\varphi _1 \bar {u}_1^k \partial _{x_1 } \theta _{r}
+\varphi _1 \bar {u}_2^k \partial _{x_2 } \theta _{r} +\varphi _1
\bar {u}_3^k \partial _{x_3 } \theta _{r} )-} } } \\
&\quad -\bar {u}_1^k (\varphi _1 \bar {\omega }_1^k \partial _{x_1 } \theta
_{r} +\varphi _1 \bar {\omega }_2^k \partial _{x_2 } \theta _{r}
+\varphi _1 \bar {\omega }_3^k \partial _{x_3 } \theta
_{r} ) \\
&\quad +q\varphi _1 \partial _{x_1 } \theta _{r} +\tilde {\omega
}_1 \varphi _1 \Delta \theta _{r} +2\tilde {\omega }_1 (\partial
_{x_1 } \theta _{r} \partial _{x_1 } \varphi _1 +\partial _{x_2 }
\theta _{r} \partial _{x_2 } \varphi _1 +\partial _{x_3 } \theta
_{r} \partial _{x_3 } \varphi _1 )) \\
&\quad =\sum\limits_{k=1}^N {\int_{{\mathbb R}^3} {\theta _{r} (\varphi
_1 (x,t_k )\tilde {\omega }_1 (x,t_k )-\varphi _1 (x,t_{k-1} )\tilde
}_1 (x,t_{k-1} ))} } \\
&\sum\limits_{k=1}^N {\int_{t_{k-1} }^{t_k } {\int_{{\mathbb R}^3} {\theta
_{r} (\tilde {\omega }_2 \partial _t \varphi _2 \,+\bar {\omega
}_2^k ((\bar {u}_1^k \partial _{x_1 } \varphi _2 +\varphi _2
\,\partial _{x_1 } \bar {u}_1^k )+(\bar {u}_2^k \partial _{x_2 }
\varphi _2 +\varphi _2
\,\partial _{x_2 } \bar {u}_2^k )+} } } \\
&\quad +(\bar {u}_3^k \partial _{x_3 } \varphi _2 +\varphi _2 \,\partial
_{x_3 } \bar {u}_3^k ))-\bar {u}_2^k ((\bar {\omega }_1^k \partial _{x_1 }
\varphi _2 +\varphi _2 \,\partial _{x_1 } \bar {\omega }_1^k )+(\bar {\omega
}_2^k \partial _{x_2 } \varphi _2 +\varphi _2 \,\partial _{x_2 } \bar
{\omega }_2^k )+ \\
\end{split}
\end{equation*}
\begin{equation*}
\begin{split}
&\quad +(\bar {\omega }_3^k \partial _{x_3 } \varphi _2 +\varphi _2
\,\partial _{x_3 } \bar {\omega }_3^k ))+q\partial _{x_2 } \varphi _2
+\tilde {\omega }_2 \Delta \varphi _2 )+ \\
&+\sum\limits_{k=1}^N {\int_{t_{k-1} }^{t_k } {\int_{{\mathbb R}^3} {(\bar
{\omega }_2^k (\varphi _2 \bar {u}_1^k \partial _{x_1 } \theta _{r}
+\varphi _2 \bar {u}_2^k \partial _{x_2 } \theta _{r} +\varphi _2
\bar {u}_3^k \partial _{x_3 } \theta _{r} )-} } } \\
&\quad -\bar {u}_2^k (\varphi _2 \bar {\omega }_1^k \partial _{x_1 } \theta
_{r} +\varphi _2 \bar {\omega }_2^k \partial _{x_2 } \theta _{r}
+\varphi _2 \bar {\omega }_3^k \partial _{x_3 } \theta
_{r} ) \\
&\quad +q\varphi _2 \partial _{x_2 } \theta _{r} +\tilde {\omega
}_2 \varphi _2 \Delta \theta _{r} +2\tilde {\omega }_2 (\partial
_{x_1 } \theta _{r} \partial _{x_1 } \varphi _2 +\partial _{x_2 }
\theta _{r} \partial _{x_2 } \varphi _2 +\partial _{x_3 } \theta
_{r} \partial _{x_3 } \varphi _2 )) \\
&\quad =\sum\limits_{k=1}^N {\int_{{\mathbb R}^3} {\theta _{r} (\varphi
_2 (x,t_k )\tilde {\omega }_2 (x,t_k )-\varphi _2 (x,t_{k-1} )\tilde
}_2 (x,t_{k-1} ))} } \\
&\sum\limits_{k=1}^N {\int_{t_{k-1} }^{t_k } {\int_{{\mathbb R}^3} {\theta
_{r} (\tilde {\omega }_3 \partial _t \varphi _3 \,+\bar {\omega
}_3^k ((\bar {u}_1^k \partial _{x_1 } \varphi _3 +\varphi _3
\,\partial _{x_1 } \bar {u}_1^k )+(\bar {u}_2^k \partial _{x_2 }
\varphi _3 +\varphi _3
\,\partial _{x_2 } \bar {u}_2^k )+} } } \\
&\quad +(\bar {u}_3^k \partial _{x_3 } \varphi _3 +\varphi _3 \,\partial
_{x_3 } \bar {u}_3^k ))-\bar {u}_3^k ((\bar {\omega }_1^k \partial _{x_1 }
\varphi _3 +\varphi _3 \,\partial _{x_1 } \bar {\omega }_1^k )+(\bar {\omega
}_2^k \partial _{x_2 } \varphi _3 +\varphi _3 \,\partial _{x_2 } \bar
{\omega }_2^k )+ \\
&\quad +(\bar {\omega }_3^k \partial _{x_3 } \varphi _3 +\varphi _3
\,\partial _{x_3 } \bar {\omega }_3^k ))+q\partial _{x_3 } \varphi _3
+\tilde {\omega }_3 \Delta \varphi _3 )+ \\
&+\sum\limits_{k=1}^N {\int_{t_{k-1} }^{t_k } {\int_{{\mathbb R}^3} {(\bar
{\omega }_3^k (\varphi _3 \bar {u}_1^k \partial _{x_1 } \theta _{r}
+\varphi _3 \bar {u}_2^k \partial _{x_2 } \theta _{r} +\varphi _3
\bar {u}_3^k \partial _{x_3 } \theta _{r} )-} } } \\
&\quad -\bar {u}_3^k (\varphi _3 \bar {\omega }_1^k \partial _{x_1 } \theta
_{r} +\varphi _3 \bar {\omega }_2^k \partial _{x_2 } \theta _{r}
+\varphi _3 \bar {\omega }_3^k \partial _{x_3 } \theta
_{r} ) \\
&\quad +q\varphi _3 \partial _{x_3 } \theta _{r} +\tilde {\omega
}_3 \varphi _3 \Delta \theta _{r} +2\tilde {\omega }_3 (\partial
_{x_1 } \theta _{r} \partial _{x_1 } \varphi _3 +\partial _{x_2 }
\theta _{r} \partial _{x_2 } \varphi _3 +\partial _{x_3 } \theta
_{r} \partial _{x_3 } \varphi _3 )) \\
&\quad =\sum\limits_{k=1}^N {\int_{{\mathbb R}^3} {\theta _{r} (\varphi
_3 (x,t_k )\tilde {\omega }_3 (x,t_k )-\varphi _3 (x,t_{k-1} )\tilde
}_3 (x,t_{k-1} ))} } \\
\end{split}
\end{equation*}
Let $ r\to +\infty $,
\begin{equation*}
\begin{split}
&\sum\limits_{k=1}^N {\int_{t_{k-1} }^{t_k } {\int_{{\mathbb R}^3} {(\tilde
{\omega }_1 \partial _t \varphi _1 \,+\bar {\omega }_1^k ((\bar
\partial _{x_1 } \varphi _1 +\varphi _1 \,\partial _{x_1 } \bar {u}_1^k
)+(\bar {u}_2^k \partial _{x_2 } \varphi _1 +\varphi _1 \,\partial _{x_2 }
\bar {u}_2^k )+} } } \\
&\quad +(\bar {u}_3^k \partial _{x_3 } \varphi _1 +\varphi _1 \,\partial
_{x_3 } \bar {u}_3^k ))-\bar {u}_1^k ((\bar {\omega }_1^k \partial _{x_1 }
\varphi _1 +\varphi _1 \,\partial _{x_1 } \bar {\omega }_1^k )+(\bar {\omega
}_2^k \partial _{x_2 } \varphi _1 +\varphi _1 \,\partial _{x_2 } \bar
{\omega }_2^k )+ \\
&\quad +(\bar {\omega }_3^k \partial _{x_3 } \varphi _1 +\varphi _1
\,\partial _{x_3 } \bar {\omega }_3^k ))+q\partial _{x_1 } \varphi _1
+\tilde {\omega }_1 \Delta \varphi _1 ) \\
&\quad =\sum\limits_{k=1}^N {\int_{{\mathbb R}^3} {(\varphi _1 (x,t_k )\tilde
{\omega }_1 (x,t_k )-\varphi _1 (x,t_{k-1} )\tilde {\omega }_1
))} } \\
&\sum\limits_{k=1}^N {\int_{t_{k-1} }^{t_k } {\int_{{\mathbb R}^3} {(\tilde
{\omega }_2 \partial _t \varphi _2 \,+\bar {\omega }_2^k ((\bar {u}_1^k
\partial _{x_1 } \varphi _2 +\varphi _2 \,\partial _{x_1 } \bar {u}_1^k
)+(\bar {u}_2^k \partial _{x_2 } \varphi _2 +\varphi _2 \,\partial _{x_2 }
\bar {u}_2^k )+} } } \\
&\quad +(\bar {u}_3^k \partial _{x_3 } \varphi _2 +\varphi _2 \,\partial
_{x_3 } \bar {u}_3^k ))-\bar {u}_2^k ((\bar {\omega }_1^k \partial _{x_1 }
\varphi _2 +\varphi _2 \,\partial _{x_1 } \bar {\omega }_1^k )+(\bar {\omega
}_2^k \partial _{x_2 } \varphi _2 +\varphi _2 \,\partial _{x_2 } \bar
{\omega }_2^k )+ \\
&\quad +(\bar {\omega }_3^k \partial _{x_3 } \varphi _2 +\varphi _2
\,\partial _{x_3 } \bar {\omega }_3^k ))+q\partial _{x_2 } \varphi _2
+\tilde {\omega }_2 \Delta \varphi _2 ) \\
&\quad =\sum\limits_{k=1}^N {\int_{{\mathbb R}^3} {(\varphi _2 (x,t_k )\tilde
{\omega }_2 (x,t_k )-\varphi _2 (x,t_{k-1} )\tilde {\omega }_2 (x,t_{k-1}
))} } \\
\end{split}
\end{equation*}
\begin{equation*}
\begin{split}
&\sum\limits_{k=1}^N {\int_{t_{k-1} }^{t_k } {\int_{{\mathbb R}^3} {(\tilde
{\omega }_3 \partial _t \varphi _3 \,+\bar {\omega }_3^k ((\bar {u}_1^k
\partial _{x_1 } \varphi _3 +\varphi _3 \,\partial _{x_1 } \bar {u}_1^k
)+(\bar {u}_2^k \partial _{x_2 } \varphi _3 +\varphi _3 \,\partial _{x_2 }
\bar {u}_2^k )+} } } \\
&\quad +(\bar {u}_3^k \partial _{x_3 } \varphi _3 +\varphi _3 \,\partial
_{x_3 } \bar {u}_3^k ))-\bar {u}_3^k ((\bar {\omega }_1^k \partial _{x_1 }
\varphi _3 +\varphi _3 \,\partial _{x_1 } \bar {\omega }_1^k )+(\bar {\omega
}_2^k \partial _{x_2 } \varphi _3 +\varphi _3 \,\partial _{x_2 } \bar
{\omega }_2^k )+ \\
&\quad +(\bar {\omega }_3^k \partial _{x_3 } \varphi _3 +\varphi _3
\,\partial _{x_3 } \bar {\omega }_3^k ))+q\partial _{x_3 } \varphi _3
+\tilde {\omega }_3 \Delta \varphi _3 ) \\
&\quad =\sum\limits_{k=1}^N {\int_{{\mathbb R}^3} {(\varphi _3 (x,t_k )\tilde
{\omega }_3 (x,t_k )-\varphi _3 (x,t_{k-1} )\tilde {\omega }_3 (x,t_{k-1}
))} } \\
\end{split}
\end{equation*}
From Section 2 we have the following conclusions:
$\tilde {\omega }_i \to \omega _i^\ast $ in $L^2(0,T;H)$
weakly, and in $L^\infty (0,T;H)$
$\bar {\omega }_i \to \omega _i^\ast $ in $L^2(0,T;H)$ weakly
as ${k}'\to \infty $, or $\Delta t_k ^\prime \to 0$.
In addition, for a certain solution $u$ of (1), we can prove due to (5) that
$\bar {u}_i \to u_i $ in $L^2(0,T;H)$ strongly
as $k \to \infty $, or $\Delta t_k \to 0$.
In fact, set $Q=(0,T)\times {\mathbb R}^3$, $\Delta t=\mathop {\max
}\limits_k \{\Delta t_k \}$, $\forall \varepsilon >0$, and $u_i \in
L^2(0,T;L^2({\mathbb R}^3))$, there exists a $v_i \in C^\infty
(0,T;L^2({\mathbb R}^3))$ such that
\[
\left\| {\,u_i -v_i } \right\|_{L^2(Q)} <\varepsilon
\]
By means of the same partition as that for $\bar {u}_i $ to
construct $\bar {v}_i $, since there exists a constant $C>0$ such
that $\left\| {\partial _t v_i } \right\|_{L^2({\mathbb R}^3)} \le
C$, and $\mathop {\max }\limits_t \left\| {\,\bar {v}_i -v_i }
\right\|_{L^2({\mathbb R}^3)} \le C\;\Delta t$, it follows that
\[
\left\| {\,\bar {v}_i -v_i } \right\|_{L^2(Q)} =\left( {\int_0^T
{\left\| {\,\bar {v}_i -v_i } \right\|_{L^2({\mathbb R}^3)}^2 } }
\right)^{1/2}\le C\,T^{1/2}\Delta t
\]
\[
\bar {v}_i \to v_i \quad \left( {\;L^\infty (0,T;L^2({\mathbb
R}^3))\;} \right),\quad \mbox{as}\;\,\Delta t\to 0
\]
Take $\Delta t$ such that $\left\| {\bar {v}_i -v_i } \right\|_{L^2(Q)}
<\varepsilon $. Moreover,
\begin{equation*}
\begin{split}
&\int_0^T {\left\| {\,\bar {u}_i -\bar {v}_i } \right\|_{L^2({\mathbb R}^3)}^2 }
=\sum\limits_{k=1}^N {\left\| {\frac{1}{\Delta t_k }\int_{t_{k-1}
}^{t_k }
{(u_i -v_i )} } \right\|} _{L^2({\mathbb R}^3)}^2 \Delta t_k \\
&\quad \le \sum\limits_{k=1}^N {\left\| {\;\left( {\int_{t_{k-1} }^{t_k }
{(u_i -v_i )^2} } \right)^{1/2}} \right\|} _{L^2({\mathbb R}^3)}^2
\le \int_0^T
{\left\| {\,u_i -v_i } \right\|_{L^2({\mathbb R}^3)}^2 } \\
\end{split}
\end{equation*}
so that $\left\| {\,\bar {u}_i -\bar {v}_i } \right\|_{L^2(Q)} \le
\left\| {u_i -v_i } \right\|_{L^2(Q)} <\varepsilon $. Therefore,
\[
\left\| {\,\bar {u}_i -u_i } \right\|_{L^2(Q)} \le \left\| {\,u_i -v_i }
\right\|_{L^2(Q)} +\left\| {\,v_i -\bar {v}_i } \right\|_{L^2(Q)} +\left\|
{\,\bar {v}_i -\bar {u}_i } \right\|_{L^2(Q)} <3\varepsilon
\]
Hence as $\Delta t\to 0$, we have $\left\| {\,\bar {u}_i -u_i }
\right\|_{L^2(Q)} \to 0$.
These convergence results enable us to pass the limit. That is,
\begin{equation*}
\begin{split}
&\sum\limits_{{k}'} {\int_{t_{{k}'-1} }^{t_{{k}'} } {\int_{{\mathbb R}^3}
{(\tilde {\omega }_1 \partial _t \varphi _1 \,+\bar {\omega
}_1^{{k}'} (\bar {u}_1^{{k}'} \partial _{x_1 } \varphi _1 +\bar
{u}_2^{{k}'} \partial _{x_2 }
\varphi _1 +\bar {u}_3^{{k}'} \partial _{x_3 } \varphi _1 )-} } } \\
&\quad \quad \quad \quad \quad \quad -\bar {u}_1^{{k}'} (\bar {\omega
}_1^{{k}'} \partial _{x_1 } \varphi _1 +\bar {\omega }_2^{{k}'} \partial
_{x_2 } \varphi _1 +\bar {\omega }_3^{{k}'} \partial _{x_3 } \varphi _1
)+q\partial _{x_1 } \varphi _1 +\tilde {\omega }_1 \Delta \varphi _1 ) \\
&\quad \quad \quad \quad \quad \quad =\int_{{\mathbb R}^3} {(\varphi _1
(x,T)\tilde {\omega }_1 (x,T)-\varphi _1 (x,0)\tilde {\omega }_1 (x,0))} \\
&\sum\limits_{{k}'} {\int_{t_{{k}'-1} }^{t_{{k}'} } {\int_{{\mathbb R}^3}
{(\tilde {\omega }_2 \partial _t \varphi _2 \,+\bar {\omega
}_2^{{k}'} (\bar {u}_1^{{k}'} \partial _{x_1 } \varphi _2 +\bar
{u}_2^{{k}'} \partial _{x_2 }
\varphi _2 +\bar {u}_3^{{k}'} \partial _{x_3 } \varphi _2 )-} } } \\
&\quad \quad \quad \quad \quad \quad -\bar {u}_2^{{k}'} (\bar {\omega
}_1^{{k}'} \partial _{x_1 } \varphi _2 +\bar {\omega }_2^{{k}'} \partial
_{x_2 } \varphi _2 +\bar {\omega }_3^{{k}'} \partial _{x_3 } \varphi _2
)+q\partial _{x_2 } \varphi _2 +\tilde {\omega }_2 \Delta \varphi _2 ) \\
&\quad \quad \quad \quad \quad \quad =\int_{{\mathbb R}^3} {(\varphi _2
(x,T)\tilde {\omega }_2 (x,T)-\varphi _2 (x,0)\tilde {\omega }_2 (x,0))} \\
&\sum\limits_{{k}'} {\int_{t_{{k}'-1} }^{t_{{k}'} } {\int_{{\mathbb R}^3}
{(\tilde {\omega }_3 \partial _t \varphi _3 \,+\bar {\omega
}_3^{{k}'} (\bar {u}_1^{{k}'} \partial _{x_1 } \varphi _3 +\bar
{u}_2^{{k}'} \partial _{x_2 }
\varphi _3 +\bar {u}_3^{{k}'} \partial _{x_3 } \varphi _3 )-} } } \\
&\quad \quad \quad \quad \quad \quad -\bar {u}_3^{{k}'} (\bar {\omega
}_1^{{k}'} \partial _{x_1 } \varphi _3 +\bar {\omega }_2^{{k}'} \partial
_{x_2 } \varphi _3 +\bar {\omega }_3^{{k}'} \partial _{x_3 } \varphi _3
)+q\partial _{x_3 } \varphi _3 +\tilde {\omega }_3 \Delta \varphi _3 ) \\
&\quad \quad \quad \quad \quad \quad =\int_{{\mathbb R}^3} {(\varphi _3
(x,T)\tilde {\omega }_3 (x,T)-\varphi _3 (x,0)\tilde {\omega }_3 (x,0))} \\
\end{split}
\end{equation*}
This is equivalent to
\begin{equation*}
\begin{split}
&\int_0^T {\int_{{\mathbb R}^3} {\,\{(\omega _1^\ast \partial _t \varphi _1
\,+\omega _2^\ast \partial _t \varphi _2 \,+\omega _3^\ast \partial
\varphi _3 )+} } \\
&\qquad \quad +(\omega _1^\ast \Delta \varphi _1 +\omega _2^\ast \Delta
\varphi _2 +\omega _3^\ast \Delta \varphi _3 )+ \\
&\quad +\omega _1^\ast (u_1 \partial _{x_1 } \varphi _1 +u_2 \partial _{x_2
} \varphi _1 +u_3 \partial _{x_3 } \varphi _1 )+\omega _2^\ast (u_1 \partial
_{x_1 } \varphi _2 +u_2 \partial _{x_2 } \varphi _2 +u_3 \partial _{x_3 }
\varphi _2 )+ \\
&\quad +\omega _3^\ast (u_1 \partial _{x_1 } \varphi _3 +u_2 \partial _{x_2
} \varphi _3 +u_3 \partial _{x_3 } \varphi _3 ) \\
&\quad -u_1 (\omega _1^\ast \partial _{x_1 } \varphi _1 +\omega _2^\ast
\partial _{x_2 } \varphi _1 +\omega _3^\ast \partial _{x_3 } \varphi _1
)-u_2 (\omega _1^\ast \partial _{x_1 } \varphi _2 +\omega _2^\ast \partial
_{x_2 } \varphi _2 +\omega _3^\ast \partial _{x_3 } \varphi _2 )- \\
&\quad -u_3 (\omega _1^\ast \partial _{x_1 } \varphi _3 +\omega _2^\ast
\partial _{x_2 } \varphi _3 +\omega _3^\ast \partial _{x_3 } \varphi _3 )\}
\\
\end{split}
\end{equation*}
\begin{equation*}
\begin{split}
&=\int_{{\mathbb R}^3} {\{(\varphi _1 (x,T)\omega _1^\ast (x,T)+\varphi _2
(x,T)\omega _2^\ast (x,T)+\varphi _3 (x,T)\omega _3^\ast (x,T))-} \\
&\quad \quad \;\;-(\varphi _{10} (x)\omega _{10} (x)+\varphi _{20} (x)\omega
_{20} (x)+\varphi _{30} (x)\omega _{30} (x))\} \\
\end{split}
\end{equation*}
Here we also have
\[
\omega _i^\ast (x,0)=\omega _{i0} (x),\quad \varphi _i (x,0)=\varphi _{i0}
(x),\quad i=1,2,3
\]
Hence we know that there exists some $\omega _i^\ast $ which belongs
to $L^\infty (0,T;L^2({\mathbb R}^3))$ and is a Leray-Hopf weak
solution of (6).
5. Regularity
We can still use Galerkin procedure as in Section 3. Since $V$ is separable
there exists a sequence of linearly independent elements $w_{i1} ,\;\cdots
,\;w_{im} ,\;\cdots $ which is total in $V$. For each $m$ we define an
approximate solution $u_{im} $ of (1) as follows:
\[
u_{im} =\sum\limits_{j=1}^m {g_{ij} (t)\;w_{ij} }
\]
and by means of weighted function $\theta _{r} $
\begin{equation}
\begin{split}
&\int_{{\mathbb R}^3} {\theta _{r} w_{1j} \partial _t u_{1m} }
+\int_{{\mathbb R}^3} {\theta _{r} (u_{1m} \partial _{x_1 } u_{1m}
+u_{2m} \partial _{x_2 } u_{1m} +u_{3m} \partial _{x_3 } u_{1m}
)w_{1j} } +
\\
&\quad \quad \quad \quad \quad \quad \quad +\int_{{\mathbb R}^3} {\theta
_{r} w_{1j} \partial _{x_1 } p} =\int_{{\mathbb R}^3}
{\theta _{r} w_{1j} \Delta u_{1m} } \\
&\int_{{\mathbb R}^3} {\theta _{r} w_{2j} \partial _t u_{2m} }
+\int_{{\mathbb R}^3} {\theta _{r} (u_{1m} \partial _{x_1 } u_{2m}
+u_{2m} \partial _{x_2 } u_{2m} +u_{3m} \partial _{x_3 } u_{2m}
)w_{2j} } +
\\
&\quad \quad \quad \quad \quad \quad \quad +\int_{{\mathbb R}^3} {\theta
_{r} w_{2j} \partial _{x_2 } p} =\int_{{\mathbb R}^3} {\theta
_{r} w_{2j} \Delta u_{2m} } \\
&\int_{{\mathbb R}^3} {\theta _{r} w_{3j} \partial _t u_{3m} }
+\int_{{\mathbb R}^3} {\theta _{r} (u_{1m} \partial _{x_1 } u_{3m}
+u_{2m} \partial _{x_2 } u_{3m} +u_{3m} \partial _{x_3 } u_{3m}
)w_{3j} } +
\\
&\quad \quad \quad \quad \quad \quad \quad +\int_{{\mathbb R}^3} {\theta
_{r} w_{3j} \partial _{x_3 } p} =\int_{{\mathbb R}^3} {\theta
_{r} w_{3j} \Delta u_{3m} } \\
&\quad \quad u_{im} (0)=u_{i0}^m ,\quad \quad j=1,\cdots ,m \\
\end{split}
\end{equation}
where $u_{i0}^m $ is the orthogonal projection in $H$ of $u_{i0} $
on the space spanned by $w_{i1} ,\;\cdots ,\;w_{im} $.
We now are allowed to differentiate (17) in the $t$, we get
\begin{equation}
\begin{split}
&\int_{{\mathbb R}^3} {\theta _{r} w_{1j} \partial _t^2 u_{1m} }
+\int_{{\mathbb R}^3} {\theta _{r} (\partial _t u_{1m}
\partial _{x_1 } u_{1m} +\partial _t u_{2m} \partial _{x_2 } u_{1m}
+\partial _t u_{3m}
\partial _{x_3 } u_{1m} )w_{1j} } + \\
&\quad \quad \quad \quad \quad \quad \quad +\int_{{\mathbb R}^3} {\theta
_{r} (u_{1m} \partial _{x_1 } \partial _t u_{1m} +u_{2m}
\partial _{x_2 } \partial _t u_{1m} +u_{3m} \partial _{x_3 }
\partial _t u_{1m}
)w_{1j} } + \\
&\quad \quad \quad \quad \quad \quad \quad +\int_{{\mathbb R}^3} {\theta
_{r} w_{1j} \partial _{x_1 } \partial _t p} =\int_{{\mathbb R}^3}
{\theta _{r} w_{1j} \Delta \partial _t u_{1m} } \\
&\int_{{\mathbb R}^3} {\theta _{r} w_{2j} \partial _t^2 u_{2m} }
+\int_{{\mathbb R}^3} {\theta _{r} (\partial _t u_{1m}
\partial _{x_1 } u_{2m} +\partial _t u_{2m} \partial _{x_2 } u_{2m}
+\partial _t u_{3m}
\partial _{x_3 } u_{2m} )w_{2j} } + \\
&\quad \quad \quad \quad \quad \quad \quad +\int_{{\mathbb R}^3} {\theta
_{r} (u_{1m} \partial _{x_1 } \partial _t u_{2m} +u_{2m}
\partial _{x_2 } \partial _t u_{2m} +u_{3m} \partial _{x_3 }
\partial _t u_{2m}
)w_{2j} } + \\
&\quad \quad \quad \quad \quad \quad \quad +\int_{{\mathbb R}^3} {\theta
_{r} w_{2j} \partial _{x_2 } \partial _t p} =\int_{{\mathbb R}^3}
{\theta _{r} w_{2j} \Delta \partial _t u_{2m} } \\
&\int_{{\mathbb R}^3} {\theta _{r} w_{3j} \partial _t^2 u_{3m} }
+\int_{{\mathbb R}^3} {\theta _{r} (\partial _t u_{1m}
\partial _{x_1 } u_{3m} +\partial _t u_{2m} \partial _{x_2 } u_{3m}
+\partial _t u_{3m}
\partial _{x_3 } u_{3m} )w_{3j} } + \\
&\quad \quad \quad \quad \quad \quad \quad +\int_{{\mathbb R}^3} {\theta
_{r} (u_{1m} \partial _{x_1 } \partial _t u_{3m} +u_{2m}
\partial _{x_2 } \partial _t u_{3m} +u_{3m} \partial _{x_3 }
\partial _t u_{3m}
)w_{3j} } + \\
&\quad \quad \quad \quad \quad \quad \quad +\int_{{\mathbb R}^3} {\theta
_{r} w_{3j} \partial _{x_3 } \partial _t p} =\int_{{\mathbb R}^3}
{\theta _{r} w_{3j} \Delta \partial _t u_{3m} } \\
\end{split}
\end{equation}
\begin{equation*}
\begin{split}
&\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad
\quad j=1,\cdots ,m \\
\end{split}
\end{equation*}
We multiply (18) by ${g}'_{ij} (t)$ and add the resulting equations for
$j=1,\cdots ,m$, we find
\begin{equation*}
\begin{split}
&\frac{1}{2}\partial _t \int_{{\mathbb R}^3} {\theta _{r} (\partial _t
u_{1m} )^2} +\int_{{\mathbb R}^3} {\theta _{r} \partial _t u_{1m}
(\partial _t u_{1m} \partial _{x_1 } u_{1m} +\partial _t u_{2m}
\partial
_{x_2 } u_{1m} +\partial _t u_{3m} \partial _{x_3 } u_{1m} )} + \\
&\quad \quad +\int_{{\mathbb R}^3} {\theta _{r} \partial _t u_{1m}
(u_{1m} \partial _{x_1 } \partial _t u_{1m} +u_{2m} \partial _{x_2 }
\partial _t u_{1m} +u_{3m} \partial _{x_3 } \partial _t u_{1m} )} + \\
&\quad \quad +\int_{{\mathbb R}^3} {\theta _{r} \partial _t u_{1m}
\partial _{x_1 } \partial _t p} =\int_{{\mathbb R}^3} {\theta _{r}
\partial _t u_{1m} \,\Delta \partial _t u_{1m} } \\
&\frac{1}{2}\partial _t \int_{{\mathbb R}^3} {\theta _{r} (\partial _t
u_{2m} )^2} +\int_{{\mathbb R}^3} {\theta _{r} \partial _t u_{2m}
(\partial _t u_{1m} \partial _{x_1 } u_{2m} +\partial _t u_{2m}
\partial
_{x_2 } u_{2m} +\partial _t u_{3m} \partial _{x_3 } u_{2m} )} + \\
&\quad \quad +\int_{{\mathbb R}^3} {\theta _{r} \partial _t u_{2m}
(u_{1m} \partial _{x_1 } \partial _t u_{2m} +u_{2m} \partial _{x_2 }
\partial _t u_{2m} +u_{3m} \partial _{x_3 } \partial _t u_{2m} )} + \\
&\quad \quad +\int_{{\mathbb R}^3} {\theta _{r} \partial _t u_{2m}
\partial _{x_2 } \partial _t p} =\int_{{\mathbb R}^3} {\theta _{r}
\partial _t u_{2m} \,\Delta \partial _t u_{2m} } \\
&\frac{1}{2}\partial _t \int_{{\mathbb R}^3} {\theta _{r} (\partial _t
u_{3m} )^2} +\int_{{\mathbb R}^3} {\theta _{r} \partial _t u_{3m}
(\partial _t u_{1m} \partial _{x_1 } u_{3m} +\partial _t u_{2m}
\partial
_{x_2 } u_{3m} +\partial _t u_{3m} \partial _{x_3 } u_{3m} )} + \\
&\quad \quad +\int_{{\mathbb R}^3} {\theta _{r} \partial _t u_{3m}
(u_{1m} \partial _{x_1 } \partial _t u_{3m} +u_{2m} \partial _{x_2 }
\partial _t u_{3m} +u_{3m} \partial _{x_3 } \partial _t u_{3m} )} + \\
&\quad \quad +\int_{{\mathbb R}^3} {\theta _{r} \partial _t u_{3m}
\partial _{x_3 } \partial _t p} =\int_{{\mathbb R}^3} {\theta _{r}
\partial _t u_{3m} \,\Delta \partial _t u_{3m} } \\
\end{split}
\end{equation*}
Let $ r\to +\infty $,
\begin{equation*}
\begin{split}
&\frac{1}{2}\partial _t \int_{{\mathbb R}^3} {(\partial _t u_{1m} )^2}
+\int_{{\mathbb R}^3} {\partial _t u_{1m} (\partial _t u_{1m}
\partial _{x_1 } u_{1m} +\partial _t u_{2m} \partial _{x_2 } u_{1m}
+\partial _t u_{3m}
\partial _{x_3 } u_{1m} )} + \\
&\quad \quad +\int_{{\mathbb R}^3} {\partial _t u_{1m} (u_{1m} \partial _{x_1 }
\partial _t u_{1m} +u_{2m} \partial _{x_2 } \partial _t u_{1m} +u_{3m}
\partial _{x_3 } \partial _t u_{1m} )} + \\
&\quad \quad +\int_{{\mathbb R}^3} {\partial _t u_{1m} \partial _{x_1 } \partial
_t p} =\int_{{\mathbb R}^3} {\partial _t u_{1m} \,\Delta \partial _t u_{1m} } \\
\end{split}
\end{equation*}
\begin{equation}
\begin{split}
&\frac{1}{2}\partial _t \int_{{\mathbb R}^3} {(\partial _t u_{2m} )^2}
+\int_{{\mathbb R}^3} {\partial _t u_{2m} (\partial _t u_{1m}
\partial _{x_1 } u_{2m} +\partial _t u_{2m} \partial _{x_2 } u_{2m}
+\partial _t u_{3m}
\partial _{x_3 } u_{2m} )} + \\
&\quad \quad +\int_{{\mathbb R}^3} {\partial _t u_{2m} (u_{1m} \partial _{x_1 }
\partial _t u_{2m} +u_{2m} \partial _{x_2 } \partial _t u_{2m} +u_{3m}
\partial _{x_3 } \partial _t u_{2m} )} + \\
&\quad \quad +\int_{{\mathbb R}^3} {\partial _t u_{2m} \partial _{x_2 } \partial
_t p} =\int_{{\mathbb R}^3} {\partial _t u_{2m} \,\Delta \partial _t u_{2m} } \\
&\frac{1}{2}\partial _t \int_{{\mathbb R}^3} {(\partial _t u_{3m} )^2}
+\int_{{\mathbb R}^3} {\partial _t u_{3m} (\partial _t u_{1m}
\partial _{x_1 } u_{3m} +\partial _t u_{2m} \partial _{x_2 } u_{3m}
+\partial _t u_{3m}
\partial _{x_3 } u_{3m} )} + \\
&\quad \quad +\int_{{\mathbb R}^3} {\partial _t u_{3m} (u_{1m} \partial _{x_1 }
\partial _t u_{3m} +u_{2m} \partial _{x_2 } \partial _t u_{3m} +u_{3m}
\partial _{x_3 } \partial _t u_{3m} )} + \\
&\quad \quad +\int_{{\mathbb R}^3} {\partial _t u_{3m} \partial _{x_3 } \partial
_t p} =\int_{{\mathbb R}^3} {\partial _t u_{3m} \,\Delta \partial _t u_{3m} } \\
\end{split}
\end{equation}
\begin{equation*}
\begin{split}
&\int_{{\mathbb R}^3} {\theta _{r} (\partial _t u_{1m} \partial _{x_1 }
\partial _t p+\partial _t u_{2m} \partial _{x_2 } \partial _t p+\partial _t
u_{3m} \partial _{x_3 } \partial _t p)} \\
&\quad =-\int_{{\mathbb R}^3} {\theta _{r} \partial _t p\,\;\partial _t
(\partial _{x_1 } u_{1m} +\partial _{x_2 } u_{2m} +\partial _{x_3 }
)} \\
&\quad \;\;\,-\int_{{\mathbb R}^3} {\partial _t p\,(\;\partial _t u_{1m}
\partial _{x_1 } \theta _{r} +\partial _t u_{2m} \partial _{x_2 }
\theta _{r} +\partial _t u_{3m} \partial _{x_3 } \theta _{r} )} \\
\end{split}
\end{equation*}
let $ r\to +\infty $ we get
\[
\int_{{\mathbb R}^3} {(\partial _t u_{1m} \partial _{x_1 } \partial
_t p+\partial _t u_{2m} \partial _{x_2 } \partial _t p+\partial _t
\partial _{x_3 } \partial _t p)} =0
\]
\begin{equation*}
\begin{split}
&\int_{{\mathbb R}^3} {\theta _{r} \partial _t u_{im} (u_{1m} \partial
_{x_1 } \partial _t u_{im} +u_{2m} \partial _{x_2 } \partial _t
+u_{3m} \partial _{x_3 } \partial _t u_{im} )} \\
&\quad =\frac{1}{2}\int_{{\mathbb R}^3} {\theta _{r} (u_{1m} \partial
_{x_1 } (\partial _t u_{im} )^2+u_{2m} \partial _{x_2 } (\partial _t
)^2+u_{3m} \partial _{x_3 } (\partial _t u_{im} )^2)} \\
&\quad =-\frac{1}{2}\int_{{\mathbb R}^3} {\theta _{r} (\partial _t
u_{im} )^2(\partial _{x_1 } u_{1m} +\partial _{x_2 } u_{2m}
+\partial _{x_3
} u_{3m} )} \\
&\quad \;\;\,-\frac{1}{2}\int_{{\mathbb R}^3} {(\partial _t u_{im} )^2(u_{1m}
\partial _{x_1 } \theta _{r} +u_{2m} \partial _{x_2 } \theta
_{r} +u_{3m} \partial _{x_3 } \theta _{r} )} \\
\end{split}
\end{equation*}
let $ r\to +\infty $ we get
\[
\int_{{\mathbb R}^3} {\partial _t u_{im} (u_{1m} \partial _{x_1 }
\partial _t u_{im} +u_{2m} \partial _{x_2 } \partial _t u_{im}
+u_{3m} \partial _{x_3 }
\partial _t u_{im} )} =0,\quad \quad i=1,2,3
\]
as well as
\begin{equation*}
\begin{split}
&\int_{{\mathbb R}^3} {\theta _{r} \partial _t u_{im} \,\Delta \partial
_t u_{im} } =\int_{{\mathbb R}^3} {\theta _{r} \partial _t u_{im}
(\partial _{x_1 }^2 \partial _t u_{im} +\partial _{x_2 }^2
\partial _t
u_{im} +\partial _{x_3 }^2 \partial _t u_{im} )} \\
\end{split}
\end{equation*}
\begin{equation*}
\begin{split}
&\quad =-\int_{{\mathbb R}^3} {\theta _{r} ((\partial _{x_1 } \partial
_t u_{im} )^2+(\partial _{x_2 } \partial _t u_{im} )^2+(\partial
_{x_3 }
\partial _t u_{im} )^2)} \\
&\quad \;\;\,-\int_{{\mathbb R}^3} {\partial _t u_{im} (\,\partial _{x_1 }
\theta _{r} \partial _{x_1 } \partial _t u_{im} +\partial _{x_2 }
\theta _{r} \partial _{x_2 } \partial _t u_{im} +\partial _{x_3 }
\theta _{r} \partial _{x_3 } \partial _t u_{im} )} \\
\end{split}
\end{equation*}
let $ r\to +\infty $ we get
\[
\int_{{\mathbb R}^3} {\partial _t u_{im} \,\Delta \partial _t u_{im}
} =-\int_{{\mathbb R}^3} {((\partial _{x_1 } \partial _t u_{im}
)^2+(\partial _{x_2 } \partial _t u_{im} )^2+(\partial _{x_3 }
\partial _t u_{im} )^2)} ,\quad \quad i=1,2,3
\]
it follows from (19) and above conclusions that
\begin{equation*}
\begin{split}
&\frac{1}{2}\partial _t \int_{{\mathbb R}^3} {((\partial _t u_{1m} )^2+(\partial
_t u_{2m} )^2+(\partial _t u_{3m} )^2)} \;\; + \\
&\quad \quad \quad +\left\| {\nabla \partial _t u_{1m} } \right\|_{L^2({\mathbb
R}^3)}^2 +\left\| {\nabla \partial _t u_{2m} }
\right\|_{L^2({\mathbb R}^3)}^2
+\left\| {\nabla \partial _t u_{3m} } \right\|_{L^2({\mathbb R}^3)}^2 \\
\end{split}
\end{equation*}
\begin{equation*}
\begin{split}
&\quad \le \left\| {\partial _t u_{1m} } \right\|_{L^4({\mathbb R}^3)} \left(
{\left\| {\partial _t u_{1m} } \right\|_{L^4({\mathbb R}^3)} \left\|
{\partial _{x_1 } u_{1m} } \right\|_{L^2({\mathbb R}^3)} +\left\|
{\partial _t u_{2m} } \right\|_{L^4({\mathbb R}^3)} \left\|
{\partial _{x_2 } u_{1m} }
\right\|_{L^2({\mathbb R}^3)} } \right.+ \\
&\left. {\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad
+\left\| {\partial _t u_{3m} } \right\|_{L^4({\mathbb R}^3)} \left\|
_{x_3 } u_{1m} } \right\|_{L^2({\mathbb R}^3)} } \right) \\
&\quad +\left\| {\partial _t u_{2m} } \right\|_{L^4({\mathbb R}^3)} \left(
{\left\| {\partial _t u_{1m} } \right\|_{L^4({\mathbb R}^3)} \left\|
{\partial _{x_1 } u_{2m} } \right\|_{L^2({\mathbb R}^3)} +\left\|
{\partial _t u_{2m} } \right\|_{L^4({\mathbb R}^3)} \left\|
{\partial _{x_2 } u_{2m} }
\right\|_{L^2({\mathbb R}^3)} +} \right. \\
&\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \left.
{+\left\| {\partial _t u_{3m} } \right\|_{L^4({\mathbb R}^3)}
\left\| {\partial
_{x_3 } u_{2m} } \right\|_{L^2({\mathbb R}^3)} } \right) \\
&\quad +\left\| {\partial _t u_{3m} } \right\|_{L^4({\mathbb R}^3)} \left(
{\left\| {\partial _t u_{1m} } \right\|_{L^4({\mathbb R}^3)} \left\|
{\partial _{x_1 } u_{3m} } \right\|_{L^2({\mathbb R}^3)} +\left\|
{\partial _t u_{2m} } \right\|_{L^4({\mathbb R}^3)} \left\|
{\partial _{x_2 } u_{3m} }
\right\|_{L^2({\mathbb R}^3)} +} \right. \\
&\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \left.
{+\left\| {\partial _t u_{3m} } \right\|_{L^4({\mathbb R}^3)}
\left\| {\partial
_{x_3 } u_{3m} } \right\|_{L^2({\mathbb R}^3)} } \right) \\
&\quad \le \left( {\sum\limits_{i=1}^3 {\left\| {\partial _t u_{im} }
\right\|_{L^4({\mathbb R}^3)}^2 } } \right)^{1/2}\left(
{\sum\limits_{j=1}^3 {\left\| {\partial _t u_{jm} }
\right\|_{L^4({\mathbb R}^3)}^2 } } \right)^{1/2}\left(
{\sum\limits_{i,j=1}^3 {\left\| {\partial _{x_i } u_{jm}
} \right\|_{L^2({\mathbb R}^3)}^2 } } \right)^{1/2} \\
\end{split}
\end{equation*}
\begin{equation*}
\begin{split}
&\sum\limits_{i=1}^3 {\left\| {\partial _t u_{im} } \right\|_{L^4({\mathbb
R}^3)}^2 } \le 2\sum\limits_{i=1}^3 {\left( {\left\| {\partial _t
u_{im} } \right\|_{L^2({\mathbb R}^3)}^{1/2} \left\| {\nabla
\partial _t u_{im} }
\right\|_{L^2({\mathbb R}^3)}^{3/2} } \right)} \\
&\quad \le 2\left( {\sum\limits_{i=1}^3 {\left\| {\partial _t u_{im} }
\right\|_{L^2({\mathbb R}^3)}^2 } } \right)^{1/4}\left(
{\sum\limits_{i=1}^3 {\left\| {\nabla \partial _t u_{im} }
\right\|_{L^2({\mathbb R}^3)}^2 } }
\right)^{3/4} \\
\end{split}
\end{equation*}
\begin{equation*}
\begin{split}
&\partial _t \left( {\sum\limits_{i=1}^3 {\left\| {\partial _t u_{im} }
\right\|_{L^2({\mathbb R}^3)}^2 } } \right)+2\left(
{\sum\limits_{i=1}^3 {\left\| {\nabla \partial _t u_{im} }
\right\|_{L^2({\mathbb R}^3)}^2 } }
\right) \\
&\le 2^2\left( {\sum\limits_{i=1}^3 {\left\| {\partial _t u_{im} }
\right\|_{L^2({\mathbb R}^3)}^2 } } \right)^{1/4}\left(
{\sum\limits_{i=1}^3 {\left\| {\nabla \partial _t u_{im} }
\right\|_{L^2({\mathbb R}^3)}^2 } } \right)^{3/4}\left(
{\sum\limits_{i=1}^3 {\left\| {\nabla u_{im} }
\right\|_{L^2({\mathbb R}^3)}^2 } } \right)^{1/2} \\
&\le 3^3\left( {\sum\limits_{i=1}^3 {\left\| {\partial _t u_{im} }
\right\|_{L^2({\mathbb R}^3)}^2 } } \right)\left(
{\sum\limits_{i=1}^3 {\left\| {\nabla u_{im} }
\right\|_{L^2({\mathbb R}^3)}^2 } } \right)^2+\left(
{\sum\limits_{i=1}^3 {\left\| {\nabla
\partial _t u_{im} }
\right\|_{L^2({\mathbb R}^3)}^2 } } \right) \\
\end{split}
\end{equation*}
it follows that
\[
\partial _t \left( {\sum\limits_{i=1}^3 {\left\| {\partial _t u_{im} }
\right\|_{L^2({\mathbb R}^3)}^2 } } \right)+\left(
{\sum\limits_{i=1}^3 {\left\| {\nabla \partial _t u_{im} }
\right\|_{L^2({\mathbb R}^3)}^2 } } \right)\le \phi _m (t)\left(
{\sum\limits_{i=1}^3 {\left\| {\partial _t u_{im} }
\right\|_{L^2({\mathbb R}^3)}^2 } } \right)
\]
where $\phi _m (t)=3^3\left( {\sum\limits_{i=1}^3 {\left\| {\nabla
u_{im} } \right\|_{L^2({\mathbb R}^3)}^2 } } \right)^2$.
Introducing a stream function: $\psi =(\psi _2 ,\psi _2 ,\psi _3 )$,
\[
\mbox{curl}\psi =(\partial _{x_2 } \psi _3 -\partial _{x_3 } \psi _2
,\;\,\;\partial _{x_3 } \psi _1 -\partial _{x_1 } \psi _3 ,\;\,\;\partial
_{x_1 } \psi _2 -\partial _{x_2 } \psi _1 )
\]
According to $\omega =\mbox{curl}u$, $u=\mbox{curl}\psi $ and
$\mbox{div}\psi =0$, we have
\[
\mbox{curlcurl}\psi =-\Delta \psi =\omega ,
\quad
-\Delta \mbox{curl}\psi =\mbox{curl}\omega
\]
That is, $-\Delta u=\mbox{curl}\omega $. Then $(-\Delta
u,\;\,u)=(\mbox{curl}\omega ,\;\,u)$, where
\[
(-\Delta u,\;\,\theta _{r} u)=\sum\limits_{i=1}^3 {(-\Delta u_i
,\;\,\theta _{r} u_i )} =\sum\limits_{i=1}^3 {(\nabla u_i
,\;\,\theta _{r} \nabla u_i )} +\sum\limits_{i=1}^3 {(\nabla u_i
,\;\,u_i \nabla \theta _{r} )}
\]
let $ r\to +\infty $ we get
\[
(-\Delta u,\;\,u)=\sum\limits_{i=1}^3 {(\nabla u_i ,\;\,\nabla u_i
)} =\sum\limits_{i=1}^3 {\left\| {\nabla u_i }
\right\|_{L^2({\mathbb R}^3)}^2 }
\]
in addition,
\begin{equation*}
\begin{split}
&(\mbox{curl}\omega ,\;\,\theta _{r} u)=(\partial _{x_2 } \omega _3
-\partial _{x_3 } \omega _2 ,\;\;\theta _{r} u_1 )+(\partial _{x_3
} \omega _1 -\partial _{x_1 } \omega _3 ,\;\;\theta _{r} u_2 ) \\
&\quad \quad \quad \quad \quad \quad \quad +(\partial _{x_1 } \omega _2
-\partial _{x_2 } \omega _1 ,\;\;\theta _{r} u_3 ) \\
&\quad =-(\omega _3 ,\;\theta _{r} \partial _{x_2 } u_1 )+(\omega
_2 ,\;\theta _{r} \partial _{x_3 } u_1 )-(\omega _1 ,\;\theta
_{r} \partial _{x_3 } u_2 ) \\
&\qquad +(\omega _3 ,\;\theta _{r} \partial _{x_1 } u_2
)-(\omega _2 ,\;\theta _{r} \partial _{x_1 } u_3 )+(\omega _1
,\;\theta _{r} \partial _{x_2 } u_3 ) \\
&\qquad -(\omega _3 ,\;u_1 \partial _{x_2 } \theta _{r}
)+(\omega _2 ,\;u_1 \partial _{x_3 } \theta _{r} )-(\omega _1
,\;u_2 \partial _{x_3 } \theta _{r} ) \\
&\qquad +(\omega _3 ,\;u_2 \partial _{x_1 } \theta _{r}
)-(\omega _2 ,\;u_3 \partial _{x_1 } \theta _{r} )+(\omega _1
,\;u_3 \partial _{x_2 } \theta _{r} ) \\
\end{split}
\end{equation*}
let $ r\to +\infty $ we get
\begin{equation*}
\begin{split}
&(\mbox{curl}\omega ,\;\,u)=-(\omega _3 ,\;\partial _{x_2 } u_1 )+(\omega _2
,\;\partial _{x_3 } u_1 )-(\omega _1 ,\;\partial _{x_3 } u_2 )+(\omega _3
,\;\partial _{x_1 } u_2 ) \\
&\qquad \quad \quad \quad \quad -(\omega _2 ,\;\partial _{x_1 } u_3 )+(\omega
_1 ,\;\partial _{x_2 } u_3 ) \\
&\quad =(\omega _1 ,\;\;\partial _{x_2 } u_3 -\partial _{x_3 } u_2 )+(\omega
_2 ,\;\;\partial _{x_3 } u_1 -\partial _{x_1 } u_3 )+(\omega _3
,\;\;\partial _{x_1 } u_2 -\partial _{x_2 } u_1 ) \\
&\quad =(\omega ,\;\,\mbox{curl}u)=(\omega ,\omega )=\sum\limits_{i=1}^3
{\left\| {\omega _i } \right\|_{L^2({\mathbb R}^3)}^2 } \\
\end{split}
\end{equation*}
\[
\left( {\sum\limits_{i=1}^3 {\left\| {\nabla u_i }
\right\|_{L^2({\mathbb R}^3)}^2 } } \right)^{1/2}=\left(
{\sum\limits_{i=1}^3 {\left\| {\omega _i } \right\|_{L^2({\mathbb
R}^3)}^2 } } \right)^{1/2}
\]
it follows that
\[
\phi _m (t)=3^3\left( {\sum\limits_{i=1}^3 {\left\| {\omega _{im} }
\right\|_{L^2({\mathbb R}^3)}^2 } } \right)^2<+\infty
\]
By the Gronwall inequality,
\[
\frac{d}{dt}\left\{ {\left( {\sum\limits_{i=1}^3 {\left\| {\partial
_t u_{im} } \right\|_{L^2({\mathbb R}^3)}^2 } } \right)\;\exp \left(
{-\int_0^t {\phi _m (s)ds} } \right)} \right\}\le 0
\]
\[
\mathop {\sup }\limits_{t\in (0,T)} \left( {\sum\limits_{i=1}^3
{\left\| {\partial _t u_{im} (t)} \right\|_{L^2({\mathbb R}^3)}^2 }
} \right)\le \left( {\sum\limits_{i=1}^3 {\left\| {\partial _t
u_{im} (0)} \right\|_{L^2({\mathbb R}^3)}^2 } } \right)\;\exp \left(
{\int_0^T {\phi _m (s)ds} } \right)
\]
\[
\partial _t u_{im} \in L^\infty (0,T;\;H)\cap L^\infty (0,T;\;V),\quad \quad
\]
Finally we write (1) in the form
\[
\sum\limits_{i=1}^3 {(-\Delta (\,\theta _{r} u_i ),\;v_i )}
=\sum\limits_{i=1}^3 {(-\theta _{r} \partial _t u_i -\theta _{r}
(u\cdot \nabla )u_i +g_i ,\;\;v_i )} ,\quad \quad v_i \in V
\]
\[
g_i = -\; u_i \,\Delta \theta _{r} -\;2\,(\nabla \theta _{r}
,\;\,\nabla u_i ) + p\,\partial _{x_i } \theta _{r}
\]
That is,
\[
\sum\limits_{i=1}^3 {(\,\nabla (\,\theta _{r} u_i ),\;\;\nabla v_i
)} =\sum\limits_{i=1}^3 {(-\theta _{r} \partial _t u_i -\theta _{r}
(u\cdot \nabla )u_i +g_i ,\;\;v_i )}
\]
\[
\partial _t u_i \in L^\infty (0,T;\;H),\quad \quad (u\cdot \nabla )u_i \in
L^\infty (0,T;\;H)
\]
Similar to the Theorem 3.8 in Chapter 3 of [4], and let $ r\to
+\infty $, we obtain
\[
u_i \in L^\infty (0,T;\;H^2({\mathbb R}^3)),\quad \quad i=1,2,3
\]
Remark 1. Noting that $(-\Delta u,\;v)=(-\partial _t
u-(u\cdot \nabla )u,\;v)$. If $\partial _t u$ and $(u\cdot \nabla
)u$ are of some degree of continuity, then $u$ can reach a higher
degree of continuity, based upon the smoothing effect of inverse
elliptic operator $\Delta ^{-1}$. By repeated application of this
process one can prove that the solution $u$ is in $C^\infty
((0,T)\times {\mathbb R}^3)$.
Remark 2. Based on problems separated and potential theory
of fluid flow, we may keep the same result for the general
initial-boundary value problems of 3D Navier-Stokes equation under
the assumptions of regularity on the boundary and data.
[1] R. A. Adams, and J. J. F. Fournier, Sobolev Spaces, Second ed., Pure and Applied Mathematics, Elsevier, Oxford, (2003);
[2] O.A.Ladyženskaya, V.A.Solonnikov, and N.N.Ural'ceva, Linear and Quasi-linear
Equations of Parabolic Type, American Mathematical Society, (1988);
[3] Qun Lin, and Lung-an Ying, Interval Vorticity Methods, (2009);
[4] R. Temam, Navier-Stokes equations Theory and numerical analysis, Reprint of the 1984, AMS Chelsea Publishing, Providence, R.I., (2001).
|
arxiv-papers
| 2013-10-14T07:00:16 |
2024-09-04T02:49:52.345605
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/",
"authors": "Qun Lin",
"submitter": "Qun Lin",
"url": "https://arxiv.org/abs/1310.3579"
}
|
1310.3583
|
# Fast and Scalded: Capillary Leidenfrost Droplets in micro-Ratchets
Álvaro Marí[email protected], Daniel del Cerro2, Gert-Willem Römer2,
Detlef Lohse3
_1 Institute of Fluid Mechanics, Universität der Bundeswehr München, Germany_
_2 Applied Laser Technology, University of Twente, The Netherlands_
_3 Physics of Fluids Group, University of Twente, The Netherlands_
In the Fluid Dynamics Video included in the ancillary files (a different
version is also available on http://youtu.be/CS0c05WQ_js), we illustrate the
special dynamics of Capillary self-propelled Leidenfrost droplets [1][2] and
confirm the so-called “viscous mechanism” model [3] by testing it in
micrometric ratchets with capillary droplets. In order to be able to propel
water droplets of sizes of the order of 1 mm, micro-ratchets were produced by
direct material removal using a picosecond pulsed laser source. Surface micro-
patterning with picosecond laser pulses allows creating a well controlled
topography on a variety of substrates, with a resolution typically in the
micron range[4]. The experiments yielded the surprising result that capillary
drops can be much faster, and be propelled as much as bigger droplets. Based
on the viscous mechanism model by D. Quéré and C. Clanet [3] and adapting
their scaling laws to capillary drops we obtain good agreement with the
experimental results. More information can be found in reference [5] and [6].
## References
* [1] H. Linke, B. Alemán, L. Melling, M. Taormina, M. Francis, C. Dow-Hygelund, V. Narayanan, R. Taylor, and A. Stout. Self-Propelled Leidenfrost Droplets. Physical Review Letters, 96(15), April 2006.
* [2] G. Lagubeau, M. Le Merrer, C. Clanet, and D. Quéré. Leidenfrost on a ratchet. Nature Physics, 7(5):395–398, 2011.
* [3] G. Dupeux, M. Le Merrer, G. Lagubeau, C. Clanet, S. Hardt, and D. Quéré. Viscous mechanism for Leidenfrost propulsion on a ratchet. Europhysics Letters, 96:1–7, November 2011.
* [4] D. Arnaldo del Cerro, G. Römer, and A. J. Huis In’t Veld. Erosion resistant anti-ice surfaces generated by ultra short laser pulses. Physics Procedia, 5:231–235, 2010.
* [5] A. G. Marin, Arnaldo del Cerro, D., G. W. Römer, B. Pathiraj, A. Huis in ’t Veld, and D. Lohse. Capillary droplets on leidenfrost micro-ratchets. Physics of fluids, 24(12):1–10, 2012.
* [6] A. G. Marin, Arnaldo del Cerro, D., G. W. Römer, B. Pathiraj, A. Huis in ’t Veld, and D. Lohse. Capillary droplets on leidenfrost micro-ratchets. arXiv preprint arXiv:1210.4978, 2012.
|
arxiv-papers
| 2013-10-14T07:36:37 |
2024-09-04T02:49:52.359392
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Alvaro G. Marin, Daniel Arnaldo del Cerro, Gert-Willem R\\\"omer, Detlef\n Lohse",
"submitter": "Alvaro Marin",
"url": "https://arxiv.org/abs/1310.3583"
}
|
1310.3598
|
# Search for R-Parity Violating Supersymmetry at the CMS Experiment
On behalf of the CMS Collaboration
Deutsches Elektronen-Synchrotron (DESY)
E-mail
###### Abstract:
The latest results from CMS on R-Parity violating Supersymmetry based on the
19.5/fb full dataset from the 8 TeV LHC run of 2012 are reviewed. The results
are interpreted in the context of simplified models with multilepton and
b-quark jets signatures that have low missing transverse energy arising from
light top-squark pair with R-parity-violating decays of the lightest
supersymmetric particle. In addition to simplified model, a new approach for
phenomenological MSSM interpretation is shown which demonstrates that the
obtained results from multilepton final states are valid for a wide range of
supersymmetry models.
## 1 Introduction
Searches for supersymmetry (SUSY) have taken an unexpected turn with the Higgs
discovery at 125 GeV [1]. The contributions of SUSY particle loops to the
Higgs mass is at most $(m_{h}^{tree})^{2}$ $\leq$ $m_{Z}^{2}$, implying
top/supersymmetric-top (stop) loops provide the necessary contribution to
stabilize the electroweak scale [2]. Experiments therefore show us that, if
SUSY exists, it is either tuned or extended, or it does not fullfill the
standard approaches and that more complicated models, with possibility to
additional contributions to the model, have to be taken into account. It is
well known that in most of the SUSY models, where R-parity is conserved,
superpartners can only be produced in pairs and the lightest supersymmetric
particle (LSP) is stable, and serves as a a dark matter candidate. In the last
decade R-parity violation (RPV) scenarios have been considered as unlikely
models for a supersymmetric extension of the Standard Model (SM) [3].
R-parity is a discrete symmetry, which can be defined as $R_{P}$ =
$(-1)^{(3B+L+2s)}$. Here $B$ denotes the baryon number, $L$ the lepton number
and $s$ the spin of a particle. SUSY models with RPV interactions necessarily
violate either B or L but can avoid proton decay limits. The most general RPV
superpotential terms can be written as:
$W_{RPV}=\lambda_{ijk}L_{i}L_{j}\bar{E}_{k}+\lambda^{{}^{\prime}}_{ijk}L_{i}Q_{j}\bar{D}_{k}+\lambda^{{}^{\prime\prime}}_{ijk}\bar{U}_{i}\bar{D}_{j}\bar{D}_{k}$
(1)
where $i,j$ and $k$ are generation indices; $L$ and $Q$ are the $SU(2)_{L}$
doublet superfields of the lepton and quark; and the $\bar{E}$, $\bar{D}$, and
$\bar{U}$ are the $SU(2)_{L}$ singlet superfields of the charge lepton, down
like quark, and up-like quark. The third term violates baryon number
conservation, while the first and second terms violate lepton number
conservation. In the following sections several searches for SUSY based on the
leptonic RPV in events with multilepton final states are discussed. All
analyses are performed using the full dataset collected with the Compact Muon
Solenoid (CMS) [4] in proton-proton collisions at a center-of-mass energy of 8
TeV, corresponding integrated luminosity of $19.5$/fb.
## 2 Search for top squarks in R-parity-violating supersymmetry using three
or more leptons and b-tagged jets
In this analysis, the result of a search for pair production of top squarks
with RPV decays of the lightest sparticle using multilepton events with one or
more b-quark tagged jets is presented [5]. Events with three or more leptons
(including tau leptons) are selected that satisfy a trigger requiring two
leptons, which may be electrons or muons. The invariant mass requirement,
$m_{ll}$ $\geq$ $12$ GeV, has been applied for any opposite sign same-flavor
(OSSF) pair of electrons and muons. This removes low-mass bound states and
$\gamma^{*}$ $\rightarrow$ $l^{+}l^{-}$ production. It is required that at
least one electron or muon in each event has a transverse momentum of $p_{T}$
$>$ $20$ GeV. Additional electrons and muons must have $p_{T}$ $>$ $10$ GeV
and all of them must be within in the pseudorapidity of —$\eta$— $\leq$ $2.4$.
Tau leptons decay either into a lepton (electron or muon) and neutrinos or a
hadronic final state generally made up of charged pions and neutral pions. The
hadronic decays yield either a single charged track (one-prong) or three
charged tracks (three prong) occasionally with additional electromagnetic
energy from neutral pion decays. Both one- and three-prong candidates are used
in this analysis if they have $p_{T}$ $>$ $20$ GeV. Leptonically decaying taus
are included with other electrons and muons. Jets are reconstructed from all
of the particle flow candidates using an anti-$k_{T}$ algorithm with a
distance parameter of $0.5$, that have —$\eta$— $\leq$ $2.5$ and $p_{T}$ $>$
$30$ GeV. Jets are required to have a distance $\Delta$R ¿ 0.3 away from any
isolated electron, muon, or $\tau_{h}$ candidate.
The background composition, arising from processes that produce genuine
multilepton events, can be generally divided into two main sources. The most
significant contributions to multilepton signatures are WZ and ZZ production,
but rare processes such as $t\bar{t}$W and $t\bar{t}$Z can also contribute.
The second source are misidentified leptons, which can be classified in the
following three categories: misidentified light leptons, misidentified
$\tau_{h}$ leptons, and light leptons originated from asymmetric internal
conversions, where a virtual photon decays promptly to a lepton pair and only
one lepton passes the selection criteria.
The contribution of misidentified light leptons can be estimated by measuring
the number of isolated tracks and applying a scale factor between isolated
leptons and isolated tracks. The $\tau_{h}$ misidentification rate is measured
in a jet-dominated control sample by using a ratio the number of $\tau_{h}$
candidates in the signal region defined by $E_{cone}$ $<$ 2 GeV with respect
to non-isolated $\tau_{h}$ candidates in $6$ GeV $<$ $E_{cone}$ $<$ 30 GeV.
Figure 1: The $S_{T}$ distribution for three lepton and b-quark jet events
(SR1) including observed yields and background contributions. Both statistical
and systematical uncertainties are shown in the shaded zone. The variable
$S_{T}$ is the scalar sum of missing transverse momentum over all jets and
isolated leptons.
The rate of asymmetric conversion to light leptons is measured in a control
region where no new physics expected. It is measured as the ratio of
$l^{+}l^{-}l^{\pm}$ with respect to $l^{+}l^{-}l^{\gamma}$ candidates in the Z
boson decays.
Depending on the total number of leptons, the number of $\tau_{h}$ candidates
and whether there is a b tagged jet in the event. Eight signal regions are
defined in five $S_{T}$ bins. The $S_{T}$ distribution for one of the signal
region (SR$1$) is shown in Fig. 1. Data are in good agreement with the SM
predictions in all signal regions.
Figure 2: The $95\%$ CL level limits in the stop mass and bino mass plane for
models with RPV couplings $\lambda_{122}$(a), $\lambda_{233}$(b) and
$\lambda^{`}_{233}$(d). For leptonic RPV couplings (a and b), the region to
the left of the curve is excluded. For semileptonic RPV coupling (d), the
region inside the curve is excluded. The kinematic properties of different
regions for the $\lambda^{`}_{233}$ exclusion result from different stop decay
products as explained in Table (c).
To demonstrate the sensitivity for various signal-model scenarios for RPV
couplings, the light decays to a top quark and intermediate on- or off-shell
bino ($\tilde{t_{1}}$ $\rightarrow$ $\tilde{\chi_{1}}^{0*}+t$) is discussed in
Fig. 2. The bino then decays to two leptons and a neutrino through the
leptonic RPV interactions or through the semileptonic RPV interactions. The
stop is assumed to be right-handed, and the RPV couplings are large enough
that all decays are prompt. In the leptonic RPV SUSY, where $\lambda_{ijk}$
$\neq$ $0$, the corresponding limits are approximately independent of the bino
mass and the stop mass below $1020$ GeV and $820$ GeV are excluded for
$\lambda_{122}$ and $\lambda_{233}$, respectively. For the $\lambda_{233}$
coupling there is a change kinematics at the $m_{\tilde{\chi}^{0}_{1}}$ =
$m_{\tilde{t}_{1}}$ \- $m_{t}$, which below the stop decay is two-body, while
above it is a four-body decay. In the region, around $\sim$$750$ GeV, the
$\tilde{\chi}^{0}_{1}$ and top are produced at rest, which results in soft
leptons, reducing the acceptance. For semileptonic coupling, which has non-
zero $\lambda^{{}^{\prime}}_{233}$, the kinematics of the decay are more
challenging. These different kinematic regions are shown in Fig. 2. The most
significant effects, happens where $\tilde{\chi}^{0}_{1}$ $\rightarrow$
$\mu$+$t$+$b$ is kinematically disfavoured, as can be seen in region B, where
the number of available leptons is reduced. The regions, where this effect is
pronounced drive the shape of the exclusion for $\lambda^{{}^{\prime}}_{233}$.
The observed limit is stronger than the expected one so that it allows the
observed exclusion region to reach into the regime where the bino decouples.
## 3 Search for RPV SUSY in the four-lepton final state
In this analysis, the lepton number violating term
($\lambda_{ijk}L_{i}L_{j}\bar{e}_{k}$), which causes the LSP in SUSY model to
decay into four leptons, is studied [6]. The main goal of this analysis is
that the RPV term exists on top of some underlying RPC model, with properties
which are currently barely constrained. Therefore, the results are interpreted
by exploring RPV on top of very specific RPC SUSY pMSSM model in addition to
the simplified model approach.
Events are selected with at least one electron or muon with transverse
momentum $p_{T}$$>$$17$ GeV, and another electron or muon with $p_{T}$$>$$8$
GeV which satisfies the trigger requirement. Events are reconstructed using
the particle flow algorithm approach. It is required that leading highest
electron or muon has $p_{T}$$>$$20$ GeV. Additional electrons or muons must
have $p_{T}$$>$$10$ GeV and all of them must be within —$\eta$— $\leq$ $2.4$.
In order to remove quarkonia resonances, photon conversions, and low-mass
continuum events the $m_{ll}$ $\geq$ $12$ GeV invariant mass cut, which is
discussed in the previous section, is applied. Events with exactly 4 isolated
leptons (electron and/or muons) containing at least one OSSF pair is selected.
And then all OSSF lepton pairs with an invariant mass closest to the Z mass of
$91$ GeV are determined. The invariant mass of this lepton pair and the
remaining lepton pair are defined in 2 dimensional distribution. Each mass are
then classified as ”below Z mass” (M $<$ $75$ GeV), ”in Z mass” ( $75$ GeV $<$
M $<$ $105$ GeV) and ”above Z mass” (M $>$ $105$ GeV). This provides nine
regions reflecting different kind of resonant and non-resonant $4$-lepton
production. The presence of 4 prompt leptons, which is the only selection
applied to the data in this analysis, is sufficiently discriminating on its
own. The SM processes contributing to this signature are processes producing
exactly 4 prompt or more leptons (ZZ, Z$t\bar{t}$, WW$t\bar{t}$, WWZ, WZZ and
ZZZ), processes producing 3 prompt and one non-prompt lepton (WZ and
W$t\bar{t}$) and Drell-Yan production with two extra non-prompt leptons. The
contribution of non-prompt leptons is estimated using the fake rate technique,
which is extensively explained in the public note. Consequently, the observed
number of events in different background processes are consistent with the SM
background expectations.
Figure 3: $95$$\%$ C.L. upper limit on the mass and cross section of the
simplified models (upper row) and generic SUSY models (lower row). Each band
corresponds to the isolation efficiency for each SUSY models[cite]. The middle
column shows the result for neutralino decaying exclusively to electrons or
muons. The right column shows the result for the lepton flavors mixture
corresponding to $\lambda_{121}$ and $\lambda_{122}$. A $30$$\%$ theoretical
uncertainty for NLO+NLL calculations of SUSY production cross sections is
included in the uncertainty band.
One of the features of this analysis is the determination of the lepton
efficiency for neutralino decays. The kinematics of these leptons are in
general driven by the momentum distribution of the decaying neutralinos and
their mass. In most scenarios, simplified models as well as generaic SUSY
models, the lepton momentum is well above threshold, which results in high
efficiency. However, large hadronic activity in the event can generally reduce
the isolation efficiency. Therefore, it is concluded that the reduction of the
total efficiency for this search may be up to $50$$\%$. As a result, once an
upper limit $\sigma$x$L$x$\epsilon$ is extracted from the observations, and
the efficiency is evaluated, the corresponding limit on the cross section,
$\sigma_{total}^{SUSY}$, may be calculated.
The cross section and mass exclusion limits are presented in Fig. 3 for
simplified and generic SUSY models. Using the total cross sections as a
function of the mass of the corresponding SUSY particles, the cross section
limit bands into mass exclusion bands as a function of the LSP mass is
presented. Results for neutralinos decaying exclusively to electrons and muons
and an appropriate mixture of electrons and muons in neutralino decays are
also shown. In the analysis it is discussed that the kinematic efficiency is
controlled by the neutralino mass and only weakly depends on the neutralino
momentum. For the cases, where $\lambda_{121}$ or $\lambda_{122}$ has non-zero
RPV coupling, the gluino mass is generally excluded below about $1.4$ TeV for
a neutralino mass higher than $400$ GeV in case of $\lambda$ sufficiently
large decay to prompt neutralino decays. For the benchmark point considered
with a $2.4$ TeV gluino, squarks with a mass below about $1.6$ TeV are
excluded.
## 4 Summary
Results of searches for RPV SUSY in events with multilepton final states at
the CMS experiment have been presented. The final number of events selected in
data are consistent with the predictions for SM processes and no evidence of
SUSY has been observed. The results of the leptonic RPV SUSY $\lambda_{ijk}$
and semileptonic RPV SUSY $\lambda^{{}^{\prime}}_{ijk}$ searches are discussed
in the context of the pMSSM and various simplified models. In the absence of
signal, limits on the allowed parameter space in the corresponding models are
set. In addition, a new approach for interpreting experimental observations
are discussed in the pMSSM framework, allowing for a more general conclusion
possible for SUSY searches.
## References
* [1] S. Chatrchyan et al. [CMS Collaboration], Phys. Lett. B 716 (2012) 30 [arXiv:1207.7235 [hep-ex]].
* [2] M. Papucci, J. T. Ruderman and A. Weiler, JHEP 1209 (2012) 035 [arXiv:1110.6926 [hep-ph]].
* [3] R. Barbier, C. Berat, M. Besancon, M. Chemtob, A. Deandrea, E. Dudas, P. Fayet and S. Lavignac et al., Phys. Rept. 420 (2005) 1 [hep-ph/0406039].
* [4] CMS Collaboration, JINST 3 S08004 (2008).
* [5] S. Chatrchyan et al. [CMS Collaboration], arXiv:1306.6643 [hep-ex].
* [6] The CMS Collaboration, PAS SUS-13-010.
|
arxiv-papers
| 2013-10-14T09:01:27 |
2024-09-04T02:49:52.364442
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Altan Cakir (on behalf of the CMS Collaboration)",
"submitter": "Altan Cakir",
"url": "https://arxiv.org/abs/1310.3598"
}
|
1310.3600
|
00footnotetext: The research leading to these results has received funding
from the [European Community’s] Seventh Framework Programme [FP7/2007-2013]
under grant agreement n 238381
# A Canonical partition theorem for uniform families of finite strong subtrees
VLITAS Dimitris
###### Abstract.
Extending a result of K. Milliken [Mi2], in this paper we prove a Ramsey
classification result for equivalence relations defined on uniform families of
finite strong subtrees of a finite sequence $(U_{i})_{i\in d}$ of fixed trees
$U_{i}$, $i\in d$, that have a finite uniform branching but are of infinite
length.
## 1\. Introduction
Canonical results in Ramsey theory try to describe equivalence relations in a
given Ramsey structure, based on the underlying pigeonhole principles. The
first example of them is the classical Canonization Theorem by P. Erdős and R.
Rado [Er-Ra] which can be presented as follows: Given
$\alpha\leq\beta\leq\omega$ let
$\binom{\beta}{\alpha}:=\\{f(\alpha)\,:\,f:\alpha\rightarrow\beta\text{ is
strictly increasing}\\}.$
The previous is commonly denoted by $[\beta]^{\alpha}$. Then for any
$n<\omega$ and any finite coloring of $\binom{\omega}{n}$ there is an
isomorphic copy $M$ of $\omega$ (i.e. the image of a strictly increasing
$f:\omega\rightarrow\omega$) and some $I\subseteq n(:=\\{0,1,\dots,n-1\\})$
such that any two $n$-element subsets have the same color if and only if they
agree on the corresponding relative positions given by $I$.
This was extended by P. Pudlák and V. Rödl in [Pu-Ro] for colorings of a given
_uniform_ family $\mathcal{G}$ of finite subsets of $\omega$ (see Section 3)
by showing that given any coloring of $\mathcal{G}$, there exists $A$ an
infinite subset of $\omega$, a uniform family $\mathcal{T}$ and a mapping
$f:\mathcal{G}\to\mathcal{T}$ such that $f(X)\subseteq X$ for all
$X\in\mathcal{G}$ and such that any two $X,Y\in\mathcal{G}\upharpoonright A$
have the same color if and only if $f(X)=f(Y)$.
There is a natural extension of the Erdős-Rado result, a kind of two-
dimensional result for certain trees. Let us define a $b$-branching tree as a
rooted tree $(T,<)$ of height at most $\omega$ with the properties that for
every non-terminal node $t$ the set of immediate successors $T_{t}$ has
cardinality $b$ and it is equipped with a fixed linear ordering $<_{t}$, and
such that the terminal nodes (if any) have all the same height. Examples of
them are, given $\tau\leq\omega$, the tree $(b^{<\tau},<)$ of functions
$f:i\rightarrow b$, $i<\tau$, endowed with the extension of functions ordering
$<$, and ordering the set of immediate successors of a given $f$ naturally. It
is easy to see that for any $b$-branching trees $T$ and $U$ of the same height
there is a unique lexicographical-isomorphism $i_{T,U}:T\rightarrow U$, i.e. a
tree-isomorphism preserving the corresponding orderings on sets of immediate
successors (see Section $3$). In fact $(b^{<\tau},<)$ are the only examples,
up to isomorphism, of $b$-branching trees with all terminal nodes of the same
height. Given two $b$-branching trees $T$ and $U$, a strong embedding is a
lexicographical-isomorphic embedding $i:T\rightarrow U$ which is level and
meet preserving, that is, if $s,t\in T$ have the same height then also $i(s)$
and $i(t)$ and the meet $i(s)\wedge i(t)$ of $i(s)$ and $i(t)$ is $i(s\wedge
t)$. For a definition of $s\wedge t$ see Section $3$. In this case, we say
that $i(T)$ is a strong subtree of $U$ isomorphic to $T$. Let $\binom{U}{T}$
denote the family of strong-subtrees of $U$ isomorphic to $T$. Then it is
proved by K. Milliken [Mi1] (see Section $3$) that for every finite coloring
of $\binom{b^{<\omega}}{b^{<n}}$ there is
$T\in\binom{b^{<\omega}}{b^{<\omega}}$ such that the coloring on
$\binom{T}{b^{<n}}$ is constant. Notice that when $b=1$, then the result is
exactly the Ramsey theorem for $[\omega]^{n}$. In an unpublished paper,
Milliken [Mi2] extended the Erdős-Rado canonization theorem by proving that
given $n$ and an arbitrary coloring $c:\binom{b^{<\omega}}{b^{<n}}\to\omega$,
there is $T\in\binom{b^{<\omega}}{b^{<\omega}}$ and there are a set of levels
$I\subseteq n$ and a set of nodes $J\subseteq b^{<n}$ such that for every
$T_{0},T_{1}\in\binom{T}{b^{<n}}$ one has that $c(T_{0})=c(T_{1})$ if and only
if the $i$-th level of $T_{0}$ and of $T_{1}$ sit in the same level of $T$
(equivalently of $b^{<\omega}$) for every $i\in I$, and if for every $t\in J$
the $t$th position of $T_{0}$ and of $T_{1}$ are the same, i.e.,
$i_{b^{<n},T_{0}}(t)=i_{b^{<n},T_{1}}(t)$.
In this paper we define properly the notion of uniform family of finite strong
subtrees of a given infinite $b$-branching tree $U$, and then we extend
Milliken’s result by proving the Pudlák-Rödl canonization analogue for such
uniform families. More precisely, our main result Theorem 7 in Section $6$ is
the following.
###### Theorem.
Given any coloring of a uniform family of finite strong subtrees of $U$, there
exists a strong subtree $T$ of $U$ and a family of node-level sets, so that
any two finite strong subtrees of the uniform family have the same color if
and only if they agree on one of these node-level sets.
The proof is by induction on the complexity of the given uniform family, and
Lemma $8$ is the natural version of the corresponding result used by Pudlák
and Rödl to derive their theorem. Roughly tells that given any two uniform
families $\mathcal{S}$ and $\mathcal{T}$ on $U$ and two mappings
$f:\mathcal{S}\to R$ and $g:\mathcal{T}\to R$, there is a strong subtree $T$
of $U$ such that either $\mathcal{S}\upharpoonright
T=\mathcal{T}\upharpoonright T$ and
$f\upharpoonright(\mathcal{S}\upharpoonright
T)=g\upharpoonright(\mathcal{T}\upharpoonright T)$, or else
$f(\mathcal{S}\upharpoonright T)\cap g(\mathcal{T}\upharpoonright
T)=\emptyset$.
The paper is organized as follows:
In the beginning, Section $2$, we present the results of Erdős-Rado and
Pudlák-Rödl to provide the reader with some intuition as they form particular
cases of our Main Theorem. Then, in Section $3$, we introduce the notion of a
uniform family of finite strong subtrees, given an infinite $b$-branching tree
$U$. We give all the elementary properties and then we state the results of
Milliken. Next, in Section $4$, we show that
$\mathcal{S}_{\infty}((U_{i})_{i\in\omega})$, the set of all infinite strong
subtrees of a $d$-sequence of $b$-branching trees, forms a topological Ramsey
space, a fact that is used in the proof of our Main Theorem that is stated and
proved in the last section.
## 2\. Canonical Ramsey theorems of Erdős-Rado and Pudlák-Rödl
Let $\mathcal{G}$ be a family of finite subsets of $\omega$. We say that
$\mathcal{G}$ is Ramsey when for every partition
$\mathcal{G}=\mathcal{G}_{1}\cup\mathcal{G}_{2}$, there is an infinite subset
$X\subseteq\omega$ and some $i\in\\{1,2\\}$ such that the restriction
$\mathcal{G}_{i}\upharpoonright X:=\\{s\in\mathcal{G}_{i}\,:\,s\subset X\\}$
of $\mathcal{G}_{i}$ to $X$ is empty. As one can expect, not just any family
of finite subsets is Ramsey. A trivial example of a non Ramsey family is
$[\omega]^{\leq n}:=\\{s\subset\omega\,:\,|s|\leq n\\}$ for $n>1$. Remarkably,
C. Nash-Williams intrinsically characterizes the Ramsey property as follows.
###### Theorem 1 (Nash-Williams, [Na-Wi]).
Let $\mathcal{G}$ be a family of finite subsets of $\omega$.
1. (a)
Suppose that $\mathcal{G}\upharpoonright X$ is thin; that is, there are no
$s,t\in\mathcal{G}\upharpoonright X$ such that $s$ is a proper initial segment
of $t$. Then $\mathcal{G}$ is Ramsey.
2. (b)
Suppose that $\mathcal{G}$ is Ramsey. Then there is some $X$ such that
$\mathcal{G}\upharpoonright X$ is thin.
Given a family $\mathcal{G}$ on $\omega$ and $n\in\omega$, let
$\mathcal{G}(n)=\\{A\subset\omega|\,\\{n\\}\cup A\in\mathcal{G}\text{ and
}n<\min A\\}.$
We pass now to recall the notion of $\alpha$-uniform families on some infinite
set $X$.
###### Definition 1 (Pudlák-Rödl).
Let $\mathcal{G}$ be a family of finite sets of an infinite subset $X$ of
$\omega$, and let $\alpha$ be a countable ordinal number. The family
$\mathcal{G}$ is called $\alpha$-uniform when
1. (a)
$\mathcal{G}=\\{\emptyset\\}$ if $\alpha=0$;
2. (b)
$\emptyset\notin\mathcal{G}$, $\mathcal{G}(n)$ is $\beta$-uniform on
$X\setminus(n+1)$ for every $n\in X$, if $\alpha=\beta+1$;
3. (c)
$\emptyset\notin\mathcal{G}$, there is an increasing sequence
$(\alpha_{n})_{n}$ with limit $\alpha$ such that each $\mathcal{G}(n)$ is
$\alpha_{n}$-uniform on $X\setminus(n+1)$, if $\alpha$ is a limit ordinal.
It is easy to see that the only $n$-uniform families on $X$ are
$[X]^{n}:=\\{s\subset\omega\,:\,|s|=n\\}$ for $n\in\omega$. For
$\alpha\geq\omega$ this is not the case (consider for example the two
$\omega$-uniform families on $\omega$ $\\{s\subset\omega\,:\,|s|=\min s+1\\}$
and $\\{s\subset\omega\,:\,|s|=\min s+2\\}$).
Notice that if $\mathcal{G}$ is an $\alpha$-uniform family on $X$, then for
any infinite subset $Y$ of $X$, the restriction $\mathcal{G}\upharpoonright Y$
is also an $\alpha$-uniform family on $Y$. Also if $\mathcal{G}$ is a uniform
family, then it is Nash-Williams as well. The relevance of uniform families is
given by the following.
###### Lemma 1.
[Pu-Ro] For every family $\mathcal{G}$ on $X$ there exists $Y\subseteq X$ such
that either $\mathcal{G}\upharpoonright Y=\emptyset$ or
$\mathcal{G}\upharpoonright Y$ contains a uniform family on $Y$.
To state the canonization result by Pudlák and Rödl we need the following
definition which will be later extended in Definition 16 to the context of
trees.
###### Definition 2.
Let $\mathcal{G}$ be a uniform family on some set $X$. A coloring $c$ of
$\mathcal{G}$ is called a _canonical coloring_ of $\mathcal{G}$ if there
exists a uniform family $\mathcal{T}$ on $X$ and a mapping
$f:\mathcal{G}\to\mathcal{T}$ such that
1. (a)
$f$ is _inner_ , i.e. $f(s)\subseteq s$ for every $s\in\mathcal{G}$.
2. (b)
For every $s,t\in\mathcal{G}$, $c(s)=c(t)$ if and only if $f(s)=f(t)$.
Notice that the condition (b) above is equivalent to say that there exists a
one-to-one coloring $\phi$ of $\mathcal{T}$ with the same list of colors as
that for the coloring of $\mathcal{G}$, such that $c(s)=\phi(f(s))$ for every
$s\in\mathcal{G}$.
Roughly speaking $c$ is a canonical coloring of $\mathcal{G}$ if the color of
each $s\in\mathcal{G}$ is determined by some subset $t$ of $s$ in a minimal
way.
###### Theorem 2 (Pudlák-Rödl,[Pu-Ro]).
For every coloring $c$ of a uniform family $\mathcal{G}$ on $X$, there exists
$Y\subseteq X$ such that $c\upharpoonright(\mathcal{G}\upharpoonright Y)$ is a
canonical coloring of $\mathcal{G}\upharpoonright Y$.
Given $A=(a_{0},\dots,a_{n-1)},B=(b_{0},\dots,b_{n-1})\in[\omega]^{n}$ and
$I\subseteq n$ we write $A:I=B:I$ to denote that $\\{a_{i}:i\in
I\\}=\\{b_{i}:i\in I\\}$. In particular, for uniform families of finite rank
the Erdős-Rado Theorem follows from the Pudlák-Rödl Theorem.
###### Theorem 3 (Erdős-Rado,[Er-Ra]).
Given $n\in\omega$ and a mapping $c:[\omega]^{n}\to R$, there exist an
infinite subset $X\subseteq\omega$ and a finite set $I\subseteq n$ such that
for any $A,B\in[X]^{n}$ one has $c(A)=c(B)$ if and only if $A:I=B:I$.
The proof goes as follows. Use the Pudlák-Rödl Theorem to find some subset
$X$, some $k\leq n$ and some inner $\phi:[X]^{n}\to[X]^{k}$ such that
$c(s)=c(t)$ iff $\phi(s)=\phi(t)$. Now consider the finite coloring
$d:[X]^{n}\to\mathcal{P}(n)$ defined by $d(s):=I\subseteq n$ such that
$s:I=\phi(s)$. By the Ramsey Theorem, there is a subset $Y$ of $X$ and
$I_{0}\subseteq n$ such that $d$ is constant on $[Y]^{n}$ with value $I_{0}$.
This just means that $A$ and $B$ in $[Y]^{n}$ have the same $c$-color if and
only if $A$ and $B$ agree on the relative positions given by $I_{0}$, denoted
by
$A:I_{0}=B:I_{0}.$
The Pudlák-Rödl Theorem was proved by transfinite induction on the rank of the
uniform family, and it crucially uses the following lemma, that we will use
later in our paper.
###### Lemma 2.
[Pu-Ro] Let $\mathcal{G}_{1}$ and $\mathcal{G}_{2}$ be two uniform families on
$Y\subseteq\omega$, $\phi_{1},\phi_{2}$ one- to-one mappings defined on
$\mathcal{G}_{1}$ and $\mathcal{G}_{2}$ respectively. Then there exists an
infinite subset $X\subseteq Y$ such that one of the following two statements
holds:
1. (1)
$\mathcal{G}_{1}\upharpoonright X=\mathcal{G}_{2}\upharpoonright X$ and
$\phi_{1}(A)=\phi_{2}(A)$ for every $A\in\mathcal{G}_{1}\upharpoonright X$.
2. (2)
$\phi_{1}(\mathcal{G}_{1}\upharpoonright
X)\cap\phi_{2}(\mathcal{G}_{2}\upharpoonright X)=\emptyset$.
## 3\. Uniform families of finite strong subtrees
All the trees $U$ that we consider are rooted and have height at most
$\omega$. For a given node $s\in U$ let $|s|$ be its height in $U$, and
similarly we write $|X|$ to denote the height of a subtree $X$ of $U$. Given
$n<|U|$, let $U(n)$ be the $n$th level of $U$, that is, the set of all nodes
of $U$ of height $n+1$. Given $X\subseteq U$ let
$L_{X}:=\\{|s|-1\,:\,s\in X\\}\subseteq L_{U}=\omega.$
By $L_{X}<L_{Y}$ we mean that $\max L_{X}<\min L_{Y}$. It is clear that in our
context we can identify each node $s$ with the sequence of its predecessors.
Given $s,t\in U$ we write $s\wedge t$ to denote the _meet_ of $s$ and $t$,
that is
$s\wedge t:=\max_{<}\\{u\,:\,u\leq s,t\\}.$
To simplify the terminology we introduce the following concept.
###### Definition 3.
Let $b>0$ be an integer. We call a tree $(U,<)$ a _$b$ -branching tree_ when
1. (a)
$U$ is rooted, and it has height at most $\omega$.
2. (b)
All terminal nodes (if any) have the same height.
3. (c)
For every non-terminal node $t\in U$ the set $U_{t}$ of immediate successors
of $t$ has cardinality $b$, and it is equipped with a total ordering $<_{t}$.
Notice that $b$-branching trees are naturally lexicographically well ordered
by $s<_{\mathrm{lex}}t$ if and only if one of the following two possibilities
holds.
1. (1)
The unique node $u_{s}$ in $U_{s\wedge t}$ below $s$ is $<_{s\wedge t}$ than
the unique node $u_{t}$ in $U_{s\wedge t}$ below $t$, where $<_{s\wedge t}$ is
the prescribed linear ordering on $U_{s\wedge t}$.
2. (2)
The two nodes satisfy $|s|<|t|$.
The typical $b$-branching tree is for $\tau\leq\omega$ the set $b^{<\tau}$ of
mappings $f:n\to b$, $n<\tau$, endowed with the ordering of extension of
functions.
###### Definition 4.
Given two $b$-branching trees $U$ and $T$, an isomorphic embedding
$\iota:U\rightarrow T$ is called a _strong embedding_ when
1. (1)
$\iota$ is $<_{\mathrm{lex}}$-preserving, i.e. if $s<_{\mathrm{lex}}t$ in $U$,
then $\iota(s)<_{\mathrm{lex}}\iota(t)$ in $T$;
2. (2)
$\iota$ is meet-preserving, i.e. $\iota(s\wedge t)=\iota(s)\wedge\iota(t)$;
and
3. (3)
$\iota$ is level-preserving, i.e. if $|s|=|t|$ then $|\iota(s)|=|\iota(t)|$.
$\iota$ is a strong isomorphism if it is a strong and onto embedding. In that
case we call $U$, $T$ isomorphic and we denote $\iota_{U,T}:U\to T$ the strong
isomorphism.
The following is easy to prove.
###### Proposition 1.
For every $b$-branching tree $U$ there is a unique $\tau\leq\omega$ and a
unique strong isomorphism $\iota_{b^{\tau},U}:b^{\tau}\rightarrow U$. Moreover
such $\tau$ is the height of $U$.
###### Definition 5.
Let $U$ be a $b$-branching tree and let $T\subseteq U$ be a $b$-branching
subtree of $U$. We say that $T$ is a _strong subtree_ of $U$ when the
inclusion mapping is a strong embedding.
Given $n\in\omega$, let $\mathcal{S}_{n}(U)$ be the family of all strong
subtrees of $U$ of height $n$. By $\mathcal{S}_{\infty}(U)$ we denote the
family of all strong subtrees of $U$ of infinite height.
Similarly for a $d$-sequence of $b$-branching trees $(U_{i})_{i\in d}$ we call
$(X_{i})_{i\in d}$ a strong subtree of $(U_{i})_{i\in d}$ if
$X_{i}\in\mathcal{S}_{\tau}(U_{i})$ and $L_{X_{i}}=L_{X_{j}}$ for all $i,j\in
d$ and some $\tau\leq\omega$.
Observe that nodes of $U$ are 1-strong subtrees of $U$
From now on, we fix an infinite $b$-branching tree $U$. We are going to use
letters $X,Y,Z,...$ and $F,T,V,...$ to denote finite and infinite strong
subtrees of $U$, respectively. Given strong subtrees $X,Y$ of $U$ by
$X\sqsubseteq Y$ we mean that $X$ is an initial segment of $Y$, i.e.
$X\subseteq Y$ and $Y(n)=X(n)$ for every $n<|X|$. Identical in the case of
$Y=U$. Similarly in the case of a $d$-sequence of $b$-branching trees
$(U_{i})_{i\in d}$ we call $(X_{i})_{i\in d}$ and initial segment of
$(Y_{i})_{i\in d}$ if and only if $X_{i}\sqsubseteq Y_{i}$ for all $i\in d$.
We denote the fact that $(X_{i})_{i\in d}$ is an initial segment of
$(Y_{i})_{i\in d}$ by $(X_{i})_{i\in d}\sqsubseteq(Y_{i})_{i\in d}$. We pass
now to introduce operations for producing strong subtrees of $U$.
###### Definition 6.
Given $t\in U$, let
$U[t]=\\{\,s\in U:t\leq s\,\\}.$
For $X\in\mathcal{S}_{n}(U)$ let
$U[X]=\\{s\in U:\exists t\in X,t\leq s\,\\}.$
So, $U[X]$ is the largest, under inclusion, strong subtree of $U$ that has $X$
as initial segment. Similarly for a given
$t=(t_{0},\dots,t_{n-1})\in\prod_{i\in d}U_{i}(n)$, let
$(U_{i})_{i\in d}[t]=\\{U_{i}[t_{i}]\text{ for all }i\in d\\}.$
###### Definition 7.
Let $Y$ be a finite strong subtree of $U$ of height $k$, and let
$(T_{i})_{i\in b^{k}}$ be a sequence of strong subtrees of $U$ such that
1. (a)
$L_{T_{i}}=L_{T_{j}}$ for every $i,j\in b^{k}$;
2. (b)
The root of $T_{i}$ is different from the root of $T_{j}$ for every $i\neq
j\in b^{k}$; and
3. (c)
$\\{T_{i}\\}_{i\in[j\cdot b,(j+1)\cdot b^{)}}\subseteq U[t_{j}]$ for every
$j<b^{k-1}$, where $\\{t_{j}\\}_{j\in b^{k-1}}$ is the lexicographically
ordered set of terminal nodes of $Y$.
Set
$Y^{\frown}(T_{i})_{i\in b^{k}}:=Y\cup\bigcup_{i\in b^{k}}T_{i}.$
Given a strong subtree $W$ of $U$ and given an initial part $Y$ of $W$ let
$W(Y)$ be the unique sequence $(Z_{i})_{i\in d}$ of strong subtrees of $W$
such that $Y^{\frown}(Z_{i})_{i\in d}=W$.
###### Remark 1.
Let $Y$ and $(T_{i})_{i\in b^{k}}$ be as in Definition $7$. Let
$\iota:b^{<k+\tau}\to U$ be the mapping defined by
$\iota(s):=\iota_{b^{k},Y}(s)$ for $s\in b^{<k}$ and
$\iota(f):=\iota_{b^{\tau},T_{f(k)}}(\widehat{f})$ for $f\in b^{k+l}$,
$l<\tau$, and where $\widehat{f}:l\to b$ is defined by
$\widehat{f}(j):=f(k+j)$. Then (a)-(c) above is equivalent to saying that
$\iota$ is a strong embedding.
Whenever we write $Y^{\frown}(T_{i})_{i\in b^{k}}$ we implicitly assume that
(a)-(c) above hold. For a node $t\in W$ considered as a 1-strong subtree of
$W$ we write $t^{\frown}(T_{i})_{i\in b}$ and $W[t]$ instead of
$\\{t\\}^{\frown}(T_{i})_{i\in b}$ and $W[\\{t\\}]$, respectively. For
$t=(t_{0},\dots,t_{n-1})\in\prod_{i\in d}U_{i}(n)$,
$n\in\omega=L_{(U_{i})_{i\in d}}$, and a $d\cdot b$-sequence of $b$-branching
trees $(Y_{j})_{j\in d\cdot b}$, we define
$t^{\frown}(Y_{j})_{j\in d\cdot b}=\bigcup_{i\in
d}t_{i}^{\frown}(Y_{j})_{j\in[i\cdot b,(i+1)\cdot b)}.$
Let $(Y_{i})_{i\in d}\in\mathcal{S}_{n}((U_{i})_{i\in d})$ and $(T_{j})_{j\in
d\cdot b^{n}}$. We define the $d$-sequence of trees
$((Y_{i})_{i\in d})^{\frown}(T_{j})_{j\in d\cdot
b^{n}}=(Y_{i}^{\frown}(T_{j})_{j\in[i\cdot b^{n},(i+1)\cdot b^{n})})_{i\in d}$
an infinite strong subtree of $(U_{i})_{i\in d}$.
Now for every node $t$ of $U$ we define a $b$-sequence of strong subtrees as
follows:
$U(t):=\\{(T_{i})_{i\in b}:t^{\frown}(T_{i})_{i\in b}=U[t]\\}$
Similarly for $t=(t_{0},\dots,t_{d-1})\in\prod_{i\in d}U_{i}(n)$ we define a
$d\cdot b$-sequence of strong subtrees as follows:
$(U_{i})_{i\in d}(t):=\\{(T_{i})_{i\in d\cdot b}:t^{\frown}(T_{i})_{i\in
d\cdot b}=(U_{i})_{i\in d}[t]\\}$
###### Definition 8.
Let $\mathcal{G}$ be a family of finite strong subtrees of $U$. Let $Y$ be a
finite strong subtree of $U$ of height $k$. We define
(1) $\mathcal{G}(Y):=\\{\,(Z_{i})_{i\in b^{k}}:Y^{\frown}(Z_{i})_{i\in
b^{k}}\in\mathcal{G}\,\\}.$
For a node $t$ we write $\mathcal{G}(t)$ instead of $\mathcal{G}(\\{t\\})$.
Given $t\in U$, let
$t^{\frown}\mathcal{G}(t):=\\{\,t^{\frown}(X_{i})_{i\in d}:(X_{i})_{i\in
d}\in\mathcal{G}(t)\,\\}\subset\mathcal{G}.$
and given $i\in b$,
$\pi_{i}(\mathcal{G}(t)):=\\{X\in\mathcal{S}_{<\omega}(U)\,:\,\text{there is
$(X_{j})_{j\in b}\in\mathcal{G}(t)$ and $X_{i}=X$}\\}.$
Similarly for $t=(t_{0},\dots,t_{n-1})\in\prod_{i\in d}U_{i}(n)$ and
$\mathcal{G}$ a family of finite strong subtrees of $(U_{i})_{i\in d}$, we
define
$\mathcal{G}(t):=\\{\,(X_{j})_{j\in d\cdot b}:t^{\frown}(X_{j})_{j\in d\cdot
b}\in\mathcal{G}\,\\}$
and
$\mathcal{G}(t_{i}):=\\{\,(X_{j})_{j\in[i\cdot b,(i+1)\cdot
b)}:\exists(X^{\prime}_{j})_{j\in d\cdot
b}\in\mathcal{G}(t),X^{\prime}_{j}=X_{j}\text{ for all }j\in[i\cdot
b,(i+1)\cdot b)\,\\}.$
Finally, we are ready to define uniform families of finite strong subtrees of
$U$ and of $(U_{i})_{i\in d}$.
###### Definition 9.
Let $\alpha$ be a countable ordinal number. We say that a family $\mathcal{G}$
of finite strong subtrees of $U$ is _$\alpha$ -uniform_ if the following hold.
1. (1)
If $\alpha=0$, then $\mathcal{G}=\\{\emptyset\\}$.
2. (2)
If $\alpha=\beta+1$, then $\emptyset\notin\mathcal{G}$ and
$\pi_{i}(\mathcal{G}(t))$ is $\beta$ uniform on $U[t^{\frown}i]$ for every
$t\in U$ and $i\in b$.
3. (3)
If $\alpha$ is a limit ordinal, then $\emptyset\notin\mathcal{G}$, and for all
$t\in U$ and $i\in b$, there is some $\alpha_{t}<\alpha$ such that
$\pi_{i}(\mathcal{G}(t))$ is $\alpha_{t}$ uniform on $U[t^{\frown}i]$ and
1. (3.1)
$\\{\,t\in U:\alpha_{t}=\beta\,\\}$ is finite for every $\beta<\alpha$, and
2. (3.2)
$\sup_{t\in C}\\{\alpha_{t}\\}=\alpha$ for every infinite chain $C$ of $U$.
Similarly we define _$\alpha$ -uniform families of $d$-tuples $(X_{i})_{i\in
d}$ of finite strong subtrees of $(U_{i})_{i\in d}$_ as follows:
1. (1)
If $\alpha=0$, then $\mathcal{G}=\\{\emptyset\\}$;
2. (2)
If $\alpha=\beta+1$, then $\emptyset\notin\mathcal{G}$ and for every
$t=(t_{0},\dots,t_{d-1})\in\prod_{i\in d}U_{i}(n)$ one has that:
$(\pi_{j_{i}}\mathcal{G}(t_{i}))_{i\in d}$ on
$(U_{i}[t_{i}^{\frown}j_{i}])_{i\in d}$ is $\beta$-uniform, where for every
$i\in d$, $j_{i}\in b$.
3. (3)
If $\alpha$ is a limit ordinal, then $\emptyset\notin\mathcal{G}$ and for
every $t=(t_{0},\dots,t_{d-1})\in\prod_{i\in d}U_{i}(n)$, $n\in\omega$, one
has that:
$(\pi_{j_{i}}\mathcal{G}(t_{i}))_{i\in d}$ on
$(U_{i}[t_{i}^{\frown}j_{i}])_{i\in d}$ is $\alpha_{t}$-uniform, where for
every $i\in d$, $j_{i}\in b$ and
1. (3.1)
$\\{\,t=(t_{0},\dots,t_{d-1})\in\prod_{i\in d}U_{i}(n):\alpha_{t}=\beta\,\\}$
is finite for every $\beta<\alpha$,
2. (3.2)
for any infinite chain $C$ of $\bigcup_{n\in\omega}\prod_{i\in d}U_{i}(n)$,
the tree that results by taking the level product of $(U_{i})_{i\in d}$, we
have that $(\alpha_{t})_{t\in C}\to\alpha$.
The first thing that we remark is that by an easy inductive argument, if
$\mathcal{G}$ is an $\alpha$-uniform family on $U$ and
$T\in\mathcal{S}_{\infty}(U)$, then $\mathcal{G}\upharpoonright
T=\\{X\in\mathcal{G}:X\in\mathcal{S}_{n}(T),n\in\omega\\}$ is also
$\alpha$-uniform on $T$. For $n\in\omega$ there is exactly one $n$-uniform
family on $U$, the family of all strong subtrees of height $n$, namely
$\mathcal{S}_{n}(U)$. It is easy to show that for each $\alpha\geq\omega$
there are infinitely many different $\alpha$-uniform families. A typical
example of an $\omega$-uniform family on $U$ is the family $\mathcal{F}$
defined by $X\in\mathcal{F}$ if and only if the height of $X$ is equa tol the
height of its root $r_{X}$.
### 3.1. Canonical Ramsey Theorem of Milliken.
Recall the following pigeonhole principle for $\mathcal{S}_{n}(U)$.
###### Theorem 4 (Milliken,[Mi1]).
Let $n,l$ be positive integers. For any finite coloring
$c:\mathcal{S}_{n}(U)\to l$ of the $n$-uniform family of finite strong
subtrees of $U$, there exists an infinite strong subtree $T$ of $U$ such that
$c$ restricted on $\mathcal{S}_{n}(T)$ is constant.
###### Definition 10.
Let $X$ and $Y$ be strong subtrees of $U$ of height $n$. Let $N\subseteq
b^{n}$ be a _node set_. We say that $X$ and $Y$ _agree on $N$_ when
$\iota_{b^{n},X}(s)=\iota_{b^{n},Y}(s)$ for every $s\in N$.
Let $L\subseteq n$ be a _set of levels_. We say that $X$ and $Y$ _agree on
$L$_ if for every $l\in L$ the $l$th level of $X$ and the $l$th level of $Y$
both lie on the same level of $U$.
For $N\subseteq b^{<n}$ and $L\subseteq n$ We write
$X:(N,L)=Y:(N,L)$
to denote that $X$ and $Y$ agree on the node-level set $(N,L)$.
Extending the Erdős-Rado Theorem, Milliken obtained the following:
###### Theorem 5 (Milliken,[Mi2]).
For any coloring $c$ of the $n$-uniform family of finite strong subtrees of
$U$, there exists an infinite strong subtree $T$ of $U$ and a node-level set
$(N,L)$ so that for any $X,Y\in\mathcal{S}_{n}(T)$ one has $c(X)=c(Y)$ if and
only if
$X:(N,L)=Y:(N,L)$
For the above pair it holds that $L_{N}<L$, that is, the levels of $b^{n}$ on
which the nodes of $N$ lie are strictly less than the levels appearing in $L$.
Observe that in the case of the uniform family of rank one, namely
$\mathcal{S}_{1}(U)$, the above theorem gives us an infinite strong subtree
$T$ of $U$ such that the coloring $c$ is constant $(N=L=\emptyset)$, one-to-
one $(N=b^{1}$, $L=\emptyset)$, or constant on the levels $(N=\emptyset$,
$L=\\{0\\})$, i.e. $c(t)=c(s)$ if and only if $|t|=|s|$.
We assume from now on that for any uniform family of infinite rank
$\mathcal{G}$, that we consider, the rank of each uniform family
$\mathcal{G}(t)$ on $U(t)$, for every node $t$, follows the lexicographic
ordering $(U,<_{\mathrm{lex}})$ introduced above, i.e. for
$s<_{\mathrm{lex}}t$ we have that the rank of $\mathcal{G}(s)$ on $U(s)$ is
less than or equal the rank of $\mathcal{G}(t)$ on $U(t)$. This is obvious if
$\alpha$ is a successor ordinal. If $\alpha$ is a limit ordinal, then by
Definition $9(3.1)$ we have that the set $\\{\,t\in U:\alpha_{t}=\beta\,\\}$
is finite for every $\beta<\alpha$. Consider the coloring
$c:\mathcal{S}_{1}(U)\to\alpha$ defined by $c(t)=\beta$ if $\mathcal{G}(t)$ is
of rank $\beta<\alpha$. By Theorem $5$ there exists
$T\in\mathcal{S}_{\infty}(U)$ such that $c\upharpoonright\mathcal{S}_{1}(T)$
is either one-to one, or constant on the levels. In both cases the rank of
each uniform family $\mathcal{G}(t)$ on $T(t)$, for every node $t$, follows
the lexicographic ordering $(T,<_{\mathrm{lex}})$, modulo passing to an
infinite strong subtree.
Notice that Theorem $5$ is an analog, in some sense, of the Pudlák-Rödl
theorem and extends the finite version of Milliken’s theorem. Our main theorem
of this paper is going to extend Theorem 5 to an arbitrary uniform family,
completing the analog between Erdős-Rado and Pudlák-Rödl. Before stating the
main theorem we still need some new concepts and results.
## 4\. $\mathcal{S}_{\infty}((U_{i})_{i\in\omega})$ as topological Ramsey
space
We introduce the notion of Nash-Williams on families of finite strong
subtrees. We remind the reader the notion of initial segment. Given strong
subtrees $X,Y$ of $U$ by $X\sqsubseteq Y$ we mean that $X$ is an initial
segment of $Y$, i.e. $X\subseteq Y$ and $Y(n)=X(n)$ for every $n<|X|$.
Identical in the case of $Y=U$. Similarly in the case of a $d$-sequence of
$b$-branching trees $(U_{i})_{i\in d}$ we call $(X_{i})_{i\in d}$ and initial
segment of $(Y_{i})_{i\in d}$ if and only if $X_{i}\sqsubseteq Y_{i}$ for all
$i\in d$. We denote the fact that $(X_{i})_{i\in d}$ is an initial segment of
$(Y_{i})_{i\in d}$ by $(X_{i})_{i\in d}\sqsubseteq(Y_{i})_{i\in d}$.
###### Definition 11.
A family $\mathcal{F}$ of finite strong subtrees of $U$ is Nash–Williams if
given any two $X,Y\in\mathcal{F}$, $X$ is not an initial segment of $Y$.
The first thing we notice is the following lemma:
###### Lemma 3.
If $\mathcal{G}$ is uniform of $(U_{i})_{i\in d}$, then $\mathcal{G}$ is Nash-
Williams.
###### Proof.
By induction on $\alpha$ such that $\mathcal{G}$ is $\alpha$-uniform. If
$\alpha=0$, then the assertion is trivial. Let $\alpha>0$ and assume the
assertion holds for every $\beta<\alpha$. Assume that there are $(X_{i})_{i\in
d}$, $(Y_{i})_{i\in d}\in\mathcal{G}$ and $(X_{i})_{i\in
d}\sqsubseteq(Y_{i})_{i\in d}$. Let $t=(t_{i})_{i\in d}$ be the common root of
$(X_{i})_{i\in d}$ and $(Y_{i})_{i\in d}$. By definition of uniform family for
all $i\in b$ and $j_{i}\in b$,
$(\pi_{j_{i}}\mathcal{G}(t_{i}^{\frown}j_{i}))_{i\in d}$ is a $\beta$-uniform
family on $(U_{i}[t_{i}^{\frown}j_{i}])_{i\in d}$, for $\beta<\alpha$. From
our assumption it follows that $(X_{i}[t_{i}^{\frown}j_{i}])_{i\in d}$ is an
initial segment of $(Y_{i}[t_{i}^{\frown}j_{i}])_{i\in d}$ contradicting the
inductive hypothesis. Therefore $\mathcal{G}$ has the property that for any
two $(X_{i})_{i\in d},(Y_{i})_{i\in d}\in\mathcal{G}$ is not the case that
$(X_{i})_{i\in d}$ is an initial segment of $(Y_{i})_{i\in d}$. ∎
The following lemma has an easy proof by induction on $\alpha$
###### Lemma 4.
If $\mathcal{G}$ is $\alpha$-uniform on $(U_{i})_{i\in d}$ then
$\mathcal{G}\upharpoonright(T_{i})_{i\in d}$ is also $\alpha$-uniform on
$(T_{i})_{i\in d}$, for any $(T_{i})_{i\in
d}\in\mathcal{S}_{\infty}((U_{i})_{i\in d})$
Now we introduce the notion of Ramsey on families of finite strong subtrees.
###### Definition 12.
A family of finite strong subtrees $\mathcal{G}$ on $(U_{i})_{i\in d}$ is
Ramsey if for every finite partition
$\mathcal{G}=\mathcal{G}_{0}\cup\dots\cup\mathcal{G}_{l-1}$ there exists
$(T_{i})_{i\in d}\in\mathcal{S}_{\infty}((U_{i})_{i\in d})$ such that at most
one of the sets $\mathcal{G}_{i}\upharpoonright(T_{i})_{i\in d}$ is non empty.
###### Lemma 5.
Any $\alpha$-uniform family $\mathcal{G}$ on $(U_{i})_{i\in d}$ is Ramsey.
Before proving this Lemma we show that the family
$\mathcal{S}_{\infty}((U_{i})_{i\in d})$ forms a topological Ramsey space in
the sense of [To]. The reader is assumed to be familiar with the Theory of
topological Ramsey spaces as presented in [To]. In [To] Chapter $6$, it is
shown that $\mathcal{S}_{\infty}(U)$ forms a topological Ramsey space, here we
extend that argument in the case of finite sequences of trees. For
$(X_{i})_{i\in d}\in\mathcal{S}_{n}((U_{i})_{i\in d})$, $n\in\omega$ and
$(T_{i})_{i\in d}\in\mathcal{S}_{\infty}((U_{i})_{i\in d})$ we define:
$(T_{i})_{i\in d}\upharpoonright n=\Big{(}\bigcup_{m<n}(T_{i}(m))_{i\in
d}\Big{)},\text{ and}$
$[(X_{i})_{i\in d},(T_{i})_{i\in d}]=\\{\,(T^{\prime}_{i})_{i\in
d}\in\mathcal{S}_{\infty}((T_{i})_{i\in d}):(T^{\prime}_{i})_{i\in
d}\upharpoonright n=(X_{i})_{i\in d}\,\\}.$
With that definition $\mathcal{S}_{\infty}((U_{i})_{i\in d})$ becomes a
topological space where the above sets are its basic open sets.
For $(T_{i})_{i\in d}\in\mathcal{S}_{\infty}((U_{i})_{i\in d})$ the sequence
$r_{n}((T_{i})_{i\in d})$ of finite approximations (restrictions) is defined
as follows:
$r_{n}((T_{i})_{i\in d})=(T_{i})_{i\in d}\upharpoonright n$
Thus the set of all finite approximations to elements of
$\mathcal{S}_{\infty}((U_{i})_{i\in d})$ is the set
$\mathcal{S}_{<\infty}((U_{i})_{i\in
d})=\bigcup_{n\in\omega}\mathcal{S}_{n}((U_{i})_{i\in d})$
of strong subtrees of $(U_{i})_{i\in d}$ of finite height. The inclusion order
on $\mathcal{S}_{\infty}((U_{i})_{i\in d})$ is finitized as follows:
$(X_{i})_{i\in d}\subseteq_{fin}(Y_{i})_{i\in d}$ iff $(X_{i})_{i\in
d}=(Y_{i})_{i\in d}=\emptyset$ or $(X_{i})_{i\in d}\subseteq(Y_{i})_{i\in d}$
and $(X_{i})_{i\in d}(\max)\subseteq(Y_{i})_{i\in d}(\max)$
where $(X_{i})_{i\in d}(\max)$ and $(Y_{i})_{i\in d}(\max)$ denote the maximal
levels of the strong subtrees $(X_{i})_{i\in d},(Y_{i})_{i\in d}$
respectively. Finitized in this way the space
$(\mathcal{S}_{\infty}((U_{i})_{i\in d}),\subseteq,r)$
is easily seen to satisfy the following list of axioms:
$\bf{A.1}$
1. (1)
$r_{0}((X_{i})_{i\in d})=r_{0}((Y_{i})_{i\in d})$ for all $(X_{i})_{i\in
d},(Y_{i})_{i\in d}\in\mathcal{S}_{<\infty}((U_{i})_{i\in d})$;
2. (2)
$(X_{i})_{i\in d}\neq(Y_{i})_{i\in d}$ implies that $r_{n}((X_{i})_{i\in
d})\neq r_{n}((Y_{i})_{i\in d})$ for some $n$;
3. (3)
$r_{n}((X_{i})_{i\in d})=r_{m}((Y_{i})_{i\in d})$ implies $n=m$ and
$r_{k}((X_{i})_{i\in d})=r_{k}((Y_{i})_{i\in d})$ for all $k\leq n$.
$\bf{A.2}$
1. (1)
$\\{\,(X_{i})_{i\in d}\subseteq_{fin}(Y_{i})_{i\in d}\,\\}$ is finite for all
$(Y_{i})_{i\in d}$;
2. (2)
$(T^{0}_{i})_{i\in d}\subseteq(T^{1}_{i})_{i\in d}$ iff $\forall n\,\exists m$
$r_{n}((T^{0}_{i})_{i\in d})\subseteq_{fin}r_{m}((T^{1}_{i})_{i\in d})$;
3. (3)
$\forall(X_{i})_{i\in d},(Y_{i})_{i\in d}$ $[(X_{i})_{i\in
d}\sqsubseteq(Y_{i})_{i\in d}\wedge(Y_{i})_{i\in
d}\subseteq_{fin}(Z_{i})_{i\in d}$ implies $\exists(W_{i})_{i\in
d}\sqsubseteq(Z_{i})_{i\in d}\text{ such that }(X_{i})_{i\in
d}\subseteq_{fin}(W_{i})_{i\in d}]$.
$\bf{A.3}$
1. (1)
If $[(X_{i})_{i\in d},(T_{i})_{i\in d}]\neq\emptyset$ then $[(X_{i})_{i\in
d},(T^{\prime}_{i})_{i\in d}]\neq\emptyset$ for all $(T^{\prime}_{i})_{i\in
d}\in[(X_{i})_{i\in d},(T_{i})_{i\in d}]$;
2. (2)
$(T^{0}_{i})_{i\in d}\subseteq(T^{1}_{i})_{i\in d}$ and $[(X_{i})_{i\in
d},(T^{0}_{i})_{i\in d}]\neq\emptyset$ imply that there exists
$(T^{\prime}_{i})_{i\in d}\in[(X_{i})_{i\in d},(T^{1}_{i})_{i\in d}]$ such
that
$\emptyset\neq[(X_{i})_{i\in d},(T^{\prime}_{i})_{i\in
d}]\subseteq[(X_{i})_{i\in d},(T^{0}_{i})_{i\in d}].$
The following requirement, that forms the pigeon hole principle in our case,
requires some proof.
$\bf{A.4}$
Let $\mathcal{O}\subseteq\mathcal{S}_{l+1}((U_{i})_{i\in d})$ and
$[(X_{i})_{i\in d},(T_{i})_{i\in d}]\neq\emptyset$, where the height of
$(X_{i})_{i\in d}$ is $l$ and we assume that $(T_{i})_{i\in d}\upharpoonright
l=(X_{i})_{i\in d}$. There exists $(T^{\prime}_{i})_{i\in d}\in[(X_{i})_{i\in
d},(T_{i})_{i\in d}]$ such that $r_{l+1}[(X_{i})_{i\in
d},(T^{\prime}_{i})_{i\in d}]\subseteq\mathcal{O}$ or $r_{l+1}[(X_{i})_{i\in
d},(T^{\prime}_{i})_{i\in d}]\subseteq\mathcal{O}^{c}$. Where
$\displaystyle r_{l+1}[(X_{i})_{i\in d},(T^{\prime}_{i})_{i\in
d}]=\\{(Y_{i})_{i\in d}\in\mathcal{S}_{l+1}((U_{i})_{i\in d})$
$\displaystyle:$ $\displaystyle(Y_{i})_{i\in d}=(T^{\prime\prime}_{i})_{i\in
d}\upharpoonright l+1\text{ for }$ $\displaystyle(T^{\prime\prime}_{i})_{i\in
d}\in[(X_{i})_{i\in d},(T^{\prime}_{i})_{i\in d}]\\}.$
###### Proof.
Let $u_{0},\dots,u_{p-1}$ be a one-to-one enumeration of the set of nodes of
$\bigcup_{i\in d}U_{i}$ that are immediate successors of some node of the set
$\\{\bigcup_{i\in d}X_{i}(l-1)\\}$. For $j\in p$, let: $V_{j}=\\{t\in
U_{i}:u_{j}\leq t\\}$, where $i$ is such that $u_{j}\in U_{i}$. Note that
every ${t}=(t_{0},\dots,t_{p-1})\in\prod_{j\in p}V_{j}(k)$, for some
$k\in\omega$, determines the strong subtree
$b({t})=(T_{i})_{i\in d}\upharpoonright l\cup(t_{0},\dots,t_{p-1})$
of $(T_{i})_{i\in d}$ of length $l+1$. Let
$\mathcal{O}^{\star}=\\{{t}:b({t})\in\mathcal{O}\\}$.
By the strong subtree version of Halpern Läuchli theorem ([Ha-Lau], [To]
Theorem 3.2), there is a sequence of strong subtrees $(F_{j})_{j\in
p}\in\mathcal{S}_{\infty}((U_{i}[u_{j}])_{j\in p})$, all with the same level
sets, such that: $\bigcup_{n\in\omega}\prod_{j\in p}F_{j}(n)$ is a subset of
either $\mathcal{O}^{\star}$ or its complement. Let: $(T^{\prime}_{i})_{i\in
d}=((T_{i})_{i\in d}\upharpoonright l)^{\frown}(F_{j})_{j\in p}$. Then
$(T^{\prime}_{i})_{i\in d}$ is a strong subtree of $(U_{i})_{i\in d}$ that
belongs to the basic open set $[(X_{i})_{i\in d},(T_{i})_{i\in d}]$ such that
$r_{l+1}[(X_{i})_{i\in d},(T^{\prime}_{i})_{i\in d}]$ is included either in
$\mathcal{O}$ or its complement. ∎
Therefore the space $(\mathcal{S}_{\infty}((U_{i})_{i\in d}),\subseteq,r)$
forms a topological Ramsey space. We provide to the reader a brief explanation
of what it means $(\mathcal{S}_{\infty}((U_{i})_{i\in d}),\subseteq,r)$ to be
a topological Ramsey space. We say that a subset $\mathcal{X}$ of
$\mathcal{S}_{\infty}((U_{i})_{i\in d})$ is _Ramsey_ if for every
$[(Y_{i})_{i\in d},(V_{i})_{i\in d}]\neq\emptyset$ there is a $(F_{i})_{i\in
d}\in[(Y_{i})_{i\in d},(V_{i})_{i\in d}]$ such that either $[(Y_{i})_{i\in
d},(F_{i})_{i\in d}]\subset\mathcal{X}$ or $[(Y_{i})_{i\in d},(F_{i})_{i\in
d}]\subset\mathcal{X}^{c}$, and $\mathcal{X}$ is _Ramsey null_ if for every
$[(Y_{i})_{i\in d},(V_{i})_{i\in d}]\neq\emptyset$, there is $(F_{i})_{i\in
d}$ such that $[(Y_{i})_{i\in d},(F_{i})_{i\in d}]\cap\mathcal{X}=\emptyset$.
Being a topological Ramsey space it means that Ramsey subsets of
$\mathcal{S}_{\infty}((U_{i})_{i\in d})$ are exactly those with the Baire
property and that meager sets are Ramsey null.
Recall that a mapping $f:A\to B$ between two topological spaces is Suslin
measurable, if the preimage $f^{-1}(O)$ of every open subset $O$ of $B$ belong
to the minimal $\sigma-$field of subsets of $A$ containing closed sets and
being closed under the Suslin operation, see [Ke].
As a consequence of the fact that $(\mathcal{S}_{\infty}((U_{i})_{i\in
d}),\subseteq,r)$ forms a topological Ramsey space is that its field of Baire
measurable subsets coincides with that of Ramsey and is closed under the
Suslin operation. Therefore for any finite coloring, where each color is
Suslin measurable, the assertion of the following theorem is immediate.
###### Theorem 6.
For every finite Suslin measurable coloring of the set
$\mathcal{S}_{\infty}((U_{i})_{i\in d})$, there exists a strong subtree
$(T_{i})_{i\in d}\in\mathcal{S}_{\infty}((U_{i})_{i\in d})$ such that
$\mathcal{S}_{\infty}((T_{i})_{i\in d})$ is monochromatic
The first consequence is the following:
###### Corollary 1.
For every $\mathcal{F}\subseteq\mathcal{S}_{<\infty}((U_{i})_{i\in d})$, there
is a strong subtree $(T_{i})_{i\in d}$ of $(U_{i})_{i\in d}$ such that either
1. (1)
$\mathcal{S}_{<\infty}((T_{i})_{i\in d})\cap\mathcal{F}=\emptyset$ or
2. (2)
For every $(T^{\prime}_{i})_{i\in d}\in\mathcal{S}_{\infty}((T_{i})_{i\in d})$
there is some $n$ such that $(T^{\prime}_{i})_{i\in d}\upharpoonright
n\in\mathcal{F}$.
###### Proof.
Color elements of $\mathcal{S}_{\infty}((U_{i})_{i\in d})$ according to
whether they have a restriction in $\mathcal{F}$ or not. This is a Borel
coloring. Now apply Theorem $6$.∎
We give now a proof for Lemma $5$.
###### Proof.
Let $\mathcal{G}$ be an $\alpha$-uniform family on $(U_{i})_{i\in d}$. By
Lemma $3$, $\mathcal{G}$ is Nash-Williams. Let $G_{0}\cup\dots\cup G_{l-1}$ be
a finite partition of $\mathcal{G}$. Apply the previous corollary successively
to each of the colors.
∎
Therefore, any $\alpha$-uniform family $\mathcal{G}$ on $(U_{i})_{i\in d}$ is
Ramsey.
## 5\. Strong subtree envelopes
At this point we would like to introduce a key notion of this paper, the
strong subtree envelope of a given subset of $U$. This notion is discussed in
[To].
We recall that for $s,t\in U$, we have defined:
$s\wedge t=\max\\{\,u\in U:u\leq s\text{ and }u\leq t\,\\}.$
The $\wedge$-closure of $A\subseteq U$ is the set:
$A^{\wedge}=\\{\,s\wedge t:s,t\in A\,\\}.$
We point out that in the definition of $A^{\wedge}$ $s$ can be equal to $t$.
Note that $A\subseteq A^{\wedge}$ and that $A^{\wedge}$ is a rooted tree.
Finally, for $A\subseteq U$, let
$||A||=|\\{\,|s\wedge t|:s,t\in A\,\\}|$
be the number of levels of $U$ which $A^{\wedge}$ intersects.
###### Definition 13.
The _strong subtree envelope_ of a node set $A\subseteq U$ is the following
subset of $\mathcal{S}_{||A||}(U)$ defined by:
$\mathcal{C}^{U}_{A}=\\{\,X\in\mathcal{S}_{||A||}(U):A^{\wedge}\subseteq
X\,\\}.$
Notice that if $X,Y\in\mathcal{C}^{U}_{A}$, then $L_{X}=L_{Y}$ and also
$i_{b^{||A||},X}\circ i^{-1}_{b^{||A||},Y}$ is the identity on $A$.
For a given finite level set $L\subseteq L_{U}=\omega$, its strong subtree
envelope is defined by:
$\mathcal{C}^{U}_{L}=\\{X\in\mathcal{S}_{|L|}(U):\,L_{X}=L\\}.$
If in addition $L$ is such that such that $L_{A}<L$, then we define
$\mathcal{C}^{U}_{(A,L)}=\\{\,X\in\mathcal{S}_{(||A||+|L|)}(U):A^{\wedge}\subset
X$ and the last $|L|$ many levels of $X$ lie on the levels of $U$ indicated by
$L\\},$
i.e., $\mathcal{C}^{U}_{(A,L)}$ is the set of all
$X\in\mathcal{S}_{(||A||+|L|)}(U)$ such that $A^{\wedge}\subset X$ and such
that for every $i\in|L|$ one has that $X(||A||+i)\subset U(l_{i})$, where
$\\{l_{0},\dots,l_{|L|-1}\\}$ is the increasing enumeration of $L$.
Similarly, given a finite sequence of trees $(U_{i})_{i\in d}$ we define the
strong subtree envelope of $(N_{i},L_{i})_{i\in d}$ in $(U_{i})_{i\in d}$,
where for all $i\in d$, $N_{i}\subset U_{i}$, $L_{i}\subset L_{U_{i}}$ and
$L_{N_{i}}<L_{i}$, as follows:
$\mathcal{C}^{(U_{i})_{i\in d}}_{(N_{i},L_{i})_{i\in d}}=\\{\,(X_{i})_{i\in
d}\in\mathcal{S}_{n}((U_{i})_{i\in d})\,:\,\text{ $\forall i\in d$ $\exists
Y_{i}\in\mathcal{C}^{U_{i}}_{(N_{i},L_{i})}$ with $Y_{i}\subseteq X_{i}$}\\},$
where $n=|\bigcup_{i\in d}(L_{N^{\wedge}_{i}}\cup L_{i})|$.
We make the observation that if $(X_{i})_{i\in d}\in\mathcal{C}^{(U_{i})_{i\in
d}}_{(N_{i},L_{i})_{i\in d}}$ then $X_{i}$ is not necessarily a member of
$\mathcal{C}^{U_{i}}_{(N_{i},L_{i})}$.
We introduce now the notion of a translation of a strong subtree.
###### Definition 14.
Let $X$ be a strong subtree of $U$ of finite height with root $r_{X}$, by a
$\mathrm{translation}$ of $X$ we mean a strong subtree $Y$ of $U$, with root
$r_{Y}\neq r_{X}$ such that the following two conditions hold:
1. (1)
$L_{Y}=L_{X}$;
2. (2)
for every node $t\in X$ there is a corresponding node $s\in Y$ with $|s|=|t|$,
and if $s,t$ are viewed as finite sequences of $\\{0,\dots,b-1\\}$, then
$t\upharpoonright(|t|\setminus|r_{X}|)=s\upharpoonright(|s|\setminus|r_{Y}|)$.
In other words we allow strong subtrees to be translated horizontally.
For a subset $A$ of $U$ its translation is obtained as follows: Let
$X\in\mathcal{C}^{U}_{A}$ and $Y$ be a translation of $X$. Set
$i_{b^{||A||},Y}\circ i^{-1}_{b^{||A||},X}(A)$ a translation of $A$.
Similarly we define translation in the context of a $d$-sequence of
$b$-branching trees $(U_{i})_{i\in d}$. For $(X_{i})_{i\in
d}\in\mathcal{S}_{n}((U_{i})_{i\in d})$ by a translation of $(X_{i})_{i\in d}$
we mean another $(Y_{i})_{i\in d}\in\mathcal{S}_{n}((U_{i})_{i\in d})$ such
that $Y_{i}$ is a translate of $X_{i}$ for at least one $i\in d$.
In the inductive step of the proof of Theorem $7$,we are going to consider
translations of uniform families defined on $U(t)$ at $U(s)$, for $s,t\in U$
with $s\neq t$. That is why we consider only horizontal translations of trees.
We extend now the notion of agreement of Definition $10$ on node-level sets as
follows:
###### Definition 15.
Given a finite node set $N\subset U$ we say that two finite strong subtrees
$X,Y$ of $U$ _agree_ on $N$ if $N\subseteq X$ and $N\subseteq Y$ up to
translation, i.e. either $N\subseteq X,Y$ or $N\subseteq X$ and
$N^{\prime}\subseteq Y$, where $N^{\prime}$ is a translate of $N$. We denote
that $X,Y$ _agree_ on $N$ by $X:N=Y:N$.
For a finite level set $L$ now, we say that
$X\in\mathcal{S}_{n}(U),Y\in\mathcal{S}_{n^{\prime}}(U)$ _agree_ on $L$, if
for every $m\in L$ we have $X(k),Y(k^{\prime})\subseteq U(m)$, for some $k\in
n$ and some $k^{\prime}\in n^{\prime}$. We denote that $X,Y$ agree on $L$ by
$X:L=Y:L$.
Given now a node-level set $(N,L)$ where $L_{N}<L$, we say that $X,Y$ _agree
on_ $(N,L)$, if they agree on $N$ and on $L$. We denote that $X,Y$ agree on
$(N,L)$ by $X:(N,L)=Y:(N,L)$.
Similarly $(X_{i})_{i\in d}$ and $(Y_{i})_{i\in d}$, finite strong subtrees of
$(U_{i})_{i\in d}$, $\mathrm{agree}$ on $(N_{i},L_{i})_{i\in d}$ if
$X_{i},Y_{i}$ agree on $(N_{i},L_{i})$ for every $i\in d$.
To demonstrate how Definition $10$ and $15$ relate we consider
$X^{\prime}\in\mathcal{C}^{U}_{(N,L)}$ and
$Y^{\prime}\in\mathcal{C}^{U}_{(N^{\prime},L)}$, both of height $n$.
Definition $10$ says that $X^{\prime}$ and $Y^{\prime}$ agree on $(N,L)$,
$N\subseteq b^{n},L\subseteq n$, if and only if $N=N^{\prime}$,
$\iota_{b^{n},X^{\prime}}\circ\iota^{-1}_{b^{n},Y^{\prime}}$ is the identity
on $N$ and if for every $l\in L$ the $l$th level of $X^{\prime}$ and the $l$th
level of $Y^{\prime}$ both lie on the same level of $U$. Definition $15$ says
that $X^{\prime}\in\mathcal{C}^{U}_{(N,L)}$ and
$Y^{\prime}\in\mathcal{C}^{U}_{(N^{\prime},L)}$ agree on $(N,L)$ if and only
if
$\iota^{-1}_{b^{n},X^{\prime}}(N)=\iota^{-1}_{b^{n},Y^{\prime}}(N^{\prime})$
and for every $l\in L$ the $l$th level of $X^{\prime}$ and the $l$th level of
$Y^{\prime}$ both lie on the same level of $U$. Therefore, it allows the node
set $N$ to be translated. It allows also agreement between finite strong
subtrees of different height.
For a strong subtree $X\in\mathcal{C}_{(N,L)}^{U}$, we define
$X^{in}\sqsubseteq X$ as follows: If the node-level set $(N,L)$ is a node set,
i.e. $L=\emptyset$, then $X^{in}=X$. If both $N\neq\emptyset$ and
$L\neq\emptyset$, then by $X^{in}$ we denote the initial segment of $X$ that
covers the node set $N$ and as a result $N^{\wedge}$. Consider the case of the
very first level $l_{0}$ of the level set $L=\\{l_{0},\dots,l_{m}\,\\}$ being
as $l_{0}=\max L_{N}+1$. Notice in this case we cannot choose the successors
$N^{\prime}$ of the nodes in $N$ that lie on $l_{0}-1$. They get imposed to us
by the choice of $l_{0}$. This pair gives rise to the same strong subtree
envelope as the pair with node set $N\cup N^{\prime}$ and level set
$L^{\prime}=\\{l_{1},\dots,l_{m}\\}$. Therefore we can assume from now on that
in any node-level set the level set lies further from the node set. Finally if
the node-level set is only a level set $(L)$, by $X^{in}$ we denote the
initial segment of $X$ whose level set forms an initial segment of $L_{U}$
i.e. $L_{X^{in}}\sqsubset L_{U}$ and as a result $X^{in}$ forms an initial
segment of $U$. In this case $|\\{Y:Y=X^{in},X\in\mathcal{C}^{U}_{L}\\}|=1$.
If there is not a subset $L_{X^{in}}$ of $L_{X}$ so that
$L_{X^{in}}\sqsubseteq L_{U}$, then $X^{in}$ is not defined.
In other words $X^{in}\sqsubseteq X$ is the finite strong subtree of $U$ that
is a cover of the set of nodes that are in any member of the envelope
$\mathcal{C}_{(N,L)}^{U}$ such that $X\in\mathcal{C}_{(N,L)}^{U}$. Therefore
if we eliminate one node from that set, on any of the resulting strong
subtrees $T$ of $U$ it holds that $\mathcal{C}_{(N,L)}^{T}=\emptyset$.
Consider now the $d$-sequence $(X_{i})_{i\in d}\in\mathcal{C}^{(U_{i})_{i\in
d}}_{(N_{i},L_{i})_{i\in d}}$ of strong subtree of $(U_{i})_{i\in d}$. Notice
that it might not be the case that $L_{\cup N_{i}}<\cup L_{i}$. Then let
$L_{in}=\\{l\in\cup L_{i}:l\leq\max L_{\cup N_{i}}\\}.$
The strong subtree envelope $\mathcal{C}^{U_{j}}_{(N_{i},L_{i})_{i\in d}}$ in
a fixed coordinate $j\in d$, is defined as the strong subtree envelope of the
set of nodes $N_{j}\subset U_{j}$ and the set of levels
$L^{j}=\cup_{i\in d}L_{i}\bigcup_{i\in d,i\neq j}\\{L_{{N_{i}}^{\wedge}}\\}.$
Then we set
$L^{j}_{in}=\\{l\in L^{j}:l\leq\max L_{N_{j}}\\}.$
Let $n=|L_{N_{j}^{\wedge}}\cup L^{j}|$ and $\sigma:L_{N_{j}^{\wedge}}\cup
L^{j}\to n$ is the increasing bijection witnessing that
$n=|L_{N_{j}^{\wedge}}\cup L^{j}|$. We define the strong subtree envelope
$\mathcal{C}^{U_{j}}_{(N_{i},L_{i})_{i\in d}}$ as follows:
$\mathcal{C}^{U_{j}}_{(N_{i},L_{i})_{i\in
d}}=\\{\,Y:Y\in\mathcal{S}_{n}{(U_{j})}\text{, }N_{j}^{\wedge}\subseteq
Y\text{ and for every }k\in L^{j}\text{ with }\sigma(k)=k^{\prime}\text{,
}Y(k^{\prime})\subset U_{j}(k)\,\\}.$
Then the strong subtree envelop of $(N_{i},L_{i})_{i\in d}$ in $(U_{i})_{i\in
d}$ as defined above, has another equivalent formulation:
$\mathcal{C}^{(U_{i})_{i\in d}}_{(N_{i},L_{i})_{i\in d}}=\\{(X_{i})_{i\in
d}:X_{j}\in\mathcal{C}^{U_{j}}_{(N_{i},L_{i})_{i\in d}}\text{ for }j\in d\\}$
In this case now, for $X_{j}\in\mathcal{C}^{U_{j}}_{(N_{i},L_{i})_{i\in d}}$,
$j\in d$ fixed, we define $X_{j}^{in}\sqsubseteq X_{j}$ its initial segment
that covers $N_{j}\cup L^{j}_{in}$, if it is defined. Set
(2) $n=\max\\{\,\vline X_{j}^{in}\vline:\,j\in d\,\\}.$
Then define the initial segment $((X_{i})_{i\in d})^{in}=(Z_{i})_{i\in d}$, of
$(X_{i})_{i\in d}$ so that the height of $(Z_{i})_{i\in d}$ is $n$ and for all
$i\in d$ we have $Z_{i}\sqsubseteq X_{i}$. Notice that the only possibility of
$((X_{i})_{i\in d})^{in}$ not being defined is the case that $\bigcup_{i\in
d}N_{i}=\emptyset$ and $L^{j}=\cup_{i\in d}L_{i}$ does not contain an initial
segment of $L_{U}$.
## 6\. Main theorem
To state our main theorem we need the following definition:
###### Definition 16.
A mapping $c$ defined on a uniform family $\mathcal{G}$ of finite strong
subtrees on $U$ is called a $\mathrm{canonical}$ coloring of $\mathcal{G}$ on
$U$ if there exists a family of node-level sets on $U$ denoted by
$\mathcal{T}$ and a mapping $f:\mathcal{G}\to\mathcal{T}$ such that:
1. (1)
For every $X\in\mathcal{G}$ if $f(X)=(N^{X},L^{X})$ then $N^{X}\subseteq X$,
$L^{X}\subseteq L_{X}$ and $L_{N^{X}}<L^{X}$.
2. (2)
For any $X,Y\in\mathcal{G}$, $c(X)=c(Y)$ if and only if $f(X)=f(Y)$ up to
translation of the node set.
The second condition is equivalent to the existence of a one-to-one, up to
translation, mapping $\phi$ defined on $\mathcal{T}$ such that
$\phi(f(X))=c(X)$ for all $X\in\mathcal{G}$.
Similarly for the case of a $d$-sequence of $b$-branching trees $(U_{i})_{i\in
d}$. A mapping $c$ defined on a uniform family $\mathcal{G}$ of finite strong
subtrees on $(U_{i})_{i\in d}$ is called a $\mathrm{canonical}$ coloring of
$\mathcal{G}$ on $(U_{i})_{i\in d}$ if there exists a family of $d$-sequences
of node-level sets on $(U_{i})_{i\in d}$ denoted by $\mathcal{T}$ and a
mapping $f:\mathcal{G}\to\mathcal{T}$ such that:
1. (1)
For every $(X_{i})_{i\in d}\in\mathcal{G}$ if $f((X_{i})_{i\in
d})=(N^{X_{i}},L^{X_{i}})_{i\in d}$ then $N^{X_{i}}\subseteq X_{i}$,
$L^{X_{i}}\subseteq L_{X_{i}}$ and $L_{N^{X_{i}}}<L^{X_{i}}$ for all $i\in d$.
2. (2)
For any $(X_{i})_{i\in d},(Y_{i})_{i\in d}\in\mathcal{G}$, $c((X_{i})_{i\in
d})=c((Y_{i})_{i\in d})$ if and only if $f((X_{i})_{i\in d})=f((Y_{i})_{i\in
d})$ up to translation of node set.
The second condition is equivalent to the existence of a one-to-one, up to
translation, mapping $\phi$ defined on $\mathcal{T}$ such that
$\phi(f((X_{i})_{i\in d})=c((X_{i})_{i\in d}))$ for all $(X_{i})_{i\in
d}\in\mathcal{G}$.
In other words two finite strong subtrees $X,Y$ of $U$ get mapped in the same
place by $c$ if and only if they agree on a node-level set
$(N,L)\in\mathcal{T}$ in the sense of Definition $15$, i.e.
$X:(N,L)=Y:(N,L)$
###### Remark 2.
We must remark that if we take the union of the strong subtree envelopes of
all the node-level sets in $\mathcal{T}$ and by passing to a strong subtree,
if necessary, we get another uniform family of finite strong subtrees. That
new uniform family has rank less than or equal to the rank of $\mathcal{G}$.
For a proof see at the very end of this section, Proposition $3$.
The main theorem of this paper is the following:
###### Theorem 7.
For any uniform family of finite strong subtrees $\mathcal{G}$ on $U$, and
every mapping $c$ on $\mathcal{G}$, there exists $T\in\mathcal{S}_{\infty}(U)$
such that $c\upharpoonright(\mathcal{G}\upharpoonright T)$ is a canonical
coloring of $\mathcal{G}\upharpoonright T$ on $T$.
Moreover we have also its version for finite sequences of trees:
###### Theorem 8.
For any uniform family of finite strong subtrees $\mathcal{G}$ on
$(U_{i})_{i\in d}$, and every mapping $c$ on $\mathcal{G}$, there exists
$(T_{i})_{i\in d}\in\mathcal{S}_{\infty}((U_{i})_{i\in d})$ such that
$c\upharpoonright(\mathcal{G}\upharpoonright(T_{i})_{i\in d})$ is a canonical
coloring of $\mathcal{G}\upharpoonright(T_{i})_{i\in d}$ on $(T_{i})_{i\in
d}$.
Notice that the range of $c$ in both of the above theorems is at most
countably infinite. The proofs of Theorems $7$ and $8$ are done by induction
on the rank of the uniform family. The case of a $0$-uniform family
$\mathcal{G}$ is trivially true. Now assuming that Theorems $7$ and $8$ hold
for any $\beta$-uniform family of finite strong subtrees, where
$\beta<\alpha$, we are going to show that they both hold for any
$\alpha$-uniform family $\mathcal{G}$ on $U$ and any $\alpha$-uniform family
$\mathcal{G}$ on $(U_{i})_{i\in d}$ respectively. For the inductive step we
need to establish some new results. Up to Section $6.1$ we develop the tools
that we need in order to do our inductive step.
Let us consider an $\alpha$-uniform family $\mathcal{G}$ on $U$ and an
equivalence relation $c$ on it, or equivalently a mapping. By definition
$\mathcal{G}(t)$ is a $\beta$-uniform family on $U(t)$, for some
$\beta<\alpha$. The inductive hypothesis applies for $c_{t}$ on
$\mathcal{G}(t)$ defined by $c_{t}((X_{i})_{i\in b})=c(t^{\frown}(X_{i})_{i\in
b})$ to give us a $U^{\prime}_{t}\in\mathcal{S}_{\infty}(U(t))$,
$U^{\prime}_{t}(0)=t$, where the restriction
$c_{t}\upharpoonright(\mathcal{G}(t)\upharpoonright U^{\prime}_{t}(t))$ is a
canonical coloring of $\mathcal{G}(t)\upharpoonright U^{\prime}_{t}(t)$ on
$U^{\prime}_{t}(t)$.
By a simple fusion sequence we get a $T\in\mathcal{S}_{\infty}(U)$ such that
for every $t\in T$ the restriction $c_{t}$ of $c$ on
$\mathcal{G}(t)\upharpoonright T(t)$ defined by $c_{t}((X_{i})_{i\in
b})=c(t^{\frown}(X_{i})_{i\in b})$ is canonical on $T(t)$. To see that
consider $t_{0}\in U(1)$. By the inductive hypothesis we get
$U^{\prime}_{t_{0}}\in\mathcal{S}_{\infty}(U[t_{0}])$,
$U^{\prime}_{t_{0}}(0)=t_{0}$, where
$c_{t_{0}}\upharpoonright(\mathcal{G}(t_{0})\upharpoonright
U^{\prime}_{t_{0}}(t_{0}))$ is a canonical coloring of
$\mathcal{G}(t_{0})\upharpoonright U^{\prime}_{t_{0}}(t_{0})$ on
$U^{\prime}_{t_{0}}(t_{0})$. Consider the level set $L_{U^{\prime}_{t_{0}}}$.
Proceed in $t_{1}\in U(1)$, let
$U^{\prime\prime}_{t_{1}}\in\mathcal{S}_{\infty}(U[t_{1}])$be such that
$U^{\prime\prime}_{t_{1}}(0)=t_{1}$,
$L_{U^{\prime\prime}_{t_{1}}}=L_{U^{\prime}_{t_{0}}}$. By the inductive
hypothesis we get a
$U^{\prime}_{t_{1}}\in\mathcal{S}_{\infty}(U^{\prime\prime}_{t_{1}})$,
$U^{\prime}_{t_{1}}(0)=t_{1}$ where the restriction $c_{t_{1}}$ is a canonical
coloring of $\mathcal{G}(t_{1})\upharpoonright U^{\prime}_{t_{1}}(t_{1})$ on
$U^{\prime}_{t_{1}}(t_{1})$. Repeat that for all nodes $t_{i}\in U(1)$, $i\in
b$. Consider $L_{U_{t_{b-1}}}$. Let
$U_{t_{i}}\in\mathcal{S}_{\infty}(U^{\prime}_{t_{i}})$ so that
$U_{t_{i}}(0)=t_{i}$, $L_{U_{t_{i}}}=L_{U_{t_{b-1}}}$, for all $i\in b-1$. Set
$T(0)=U(0)$, $T(1)=U(1)$ and $T(2)=\bigcup_{i\in b}U_{t_{i}}(1)$. Suppose we
have constructed $T(n)$ and we would like to choose $T(n+1)$. Let
$(s_{i})_{i\in b^{n}}$ be an enumeration of the nodes in $T(n)$. Start with
$s_{0}$. By the inductive hypothesis we get
$U^{\prime}_{s_{0}}\in\mathcal{S}_{\infty}(U[s_{0}])$,
$U^{\prime}_{s_{0}}(0)=s_{0}$ where
$c_{s_{0}}\upharpoonright(\mathcal{G}(s_{0})\upharpoonright
U^{\prime}_{s_{0}}(s_{0}))$ is a canonical coloring of
$\mathcal{G}(s_{0})\upharpoonright U^{\prime}_{s_{0}}(s_{0})$ on
$U^{\prime}_{s_{0}}(s_{0})$. Consider the level set $L_{U^{\prime}_{s_{0}}}$.
Proceed in $s_{1}\in T(n)$, let
$U^{\prime\prime}_{s_{1}}\in\mathcal{S}_{\infty}(U[s_{1}])$,
$U^{\prime\prime}_{s_{1}}(0)=s_{1}$ be such that
$L_{U^{\prime\prime}_{s_{1}}}=L_{U^{\prime}_{s_{0}}}$. By the inductive
hypothesis we get a
$U^{\prime}_{s_{1}}\in\mathcal{S}_{\infty}(U^{\prime\prime}_{s_{1}})$,
$U^{\prime}_{s_{1}}(0)=s_{1}$ where the restriction $c_{s_{1}}$ is a canonical
coloring of $\mathcal{G}(s_{1})\upharpoonright U^{\prime}_{s_{1}}(s_{1})$ on
$U^{\prime}_{s_{1}}(s_{1})$. Repeat that for all nodes $s_{i}\in T(n)$, $i\in
b^{n}$. Consider $L_{U_{s_{b^{n}-1}}}$. Let
$U_{s_{i}}\in\mathcal{S}_{\infty}(U^{\prime}_{s_{i}})$ so that
$U_{s_{i}}(0)=s_{i}$, $L_{U_{s_{i}}}=L_{U_{t_{b^{n}-1}}}$ for all $i\in
b^{n}-1$. Set $T(n+1)=\bigcup_{i\in b^{n}}U_{s_{i}}(1)$. The limit of this
fusion sequence $T\in\mathcal{S}_{\infty}(U)$ has the property that for every
$t\in T$ the restriction $c_{t}$ of $c$ on $\mathcal{G}(t)\upharpoonright
T(t)$ is a canonical coloring of $\mathcal{G}(t)\upharpoonright T(t)$ on
$T(t)$. For notational simplicity we assume that $T=U$.
Therefore we have that at each node $t$ of $U$ the restriction $c_{t}$ of $c$
on $\mathcal{G}(t)\upharpoonright U(t)$, defined by $c_{t}((X_{i})_{i\in
b})=c(t^{\frown}(X_{i})_{i\in b})$, is canonical. As a result there exists a
family of $b$-sequences of node-level sets, like $(N_{i},L_{i})_{i\in b}$,
denoted by $\mathcal{T}^{t}$ and a mapping $f_{t}$ that satisfy conditions
$(1)$ and $(2)$ of Definition $16$. The family $\mathcal{T}^{t}$, by the
Remark $2$ above, gives rise to a $\gamma$-uniform family
$\mathcal{F}(\mathcal{G})(t)$ on a strong subtree of $U(t)$. By a simple
fusion sequence identical with the one just above, we can assume that
$\mathcal{F}(\mathcal{G})(t)$ is defined on $U(t)$ for every $t\in U$. The
mappings $f_{t}$ are defined on $\mathcal{G}(t)\upharpoonright U(t)$ and the
one-to-one mappings $\phi_{t}$ are defined on $\mathcal{T}^{t}$ by
$\phi_{t}((N_{i},L_{i})_{i\in b}=f_{t}((X_{i})_{i\in b}))=c_{t}((X_{i})_{i\in
b})$
where $\mathcal{C}^{U(t)}_{(N_{i},L_{i})_{i\in
d}}\subset\mathcal{F}(\mathcal{G})(t)$ and $t^{\frown}(X_{i})_{i\in
b}\in\mathcal{G}$.
In that way we can think of $\mathcal{F}$ as a functor defined on the set of
all pairs $(\mathcal{G},c)$ of a uniform family of finite strong subtrees on a
tree $U$ with a fixed branching number and an equivalence relation $c$ on that
family. For every $t\in U$, $\mathcal{F}(\mathcal{G})(t)$ is a uniform family
on a strong subtree of $U(t)$ with rank less than or equal to that of
$\mathcal{G}(t)$. By $\mathcal{F}(\mathcal{G})$ we denote the uniform family
that results from the union of $t^{\frown}\mathcal{F}(\mathcal{G})(t)$, for
all nodes $t$ of $U$.
From now on we work with the uniform family $\mathcal{F}(\mathcal{G})$ and not
with the original uniform family $\mathcal{G}$ that we started with. So all
the definitions and notation developed so far apply to the resulting uniform
family $\mathcal{F}(\mathcal{G})$. For simplicity reasons from this point up
to the end of the paper, we will assume that $\mathcal{F}(\mathcal{G})$ is
directly defined on $U$ instead of one of its infinite strong subtrees. As a
consequence, $\mathcal{F}(\mathcal{G})(t)$ is assumed to be defined directly
on $U(t)$, for all $t\in U$. In particular we consider the pair
$(\mathcal{F}(\mathcal{G}),c^{\prime})$ with $c^{\prime}$ defined on
$\mathcal{F}(\mathcal{G})$ by $c^{\prime}(t^{\frown}(Y_{i})_{i\in
d})=\phi_{t}((N_{i},L_{i})_{i\in d})$, where $(Y_{i})_{i\in
d}\in\mathcal{C}^{U(t)}_{(N_{i},L_{i})_{i\in d}}\subset\mathcal{F}(G)(t)$ and
$(N_{i},L_{i})_{i\in d}=f_{t}((X_{i})_{i\in d})$ for a $(X_{i})_{i\in
d}\in\mathcal{G}(t)$, $t\in U$. We make identical assumptions in the case of
$(U_{i})_{i\in d}$.
The last thing to notice is that given any mapping on the $n$-uniform family
$\mathcal{S}_{n}((U_{i})_{i\in d})$, by the inductive hypothesis of Theorem
$8$, we can assume that the mapping is canonical. There is a family of node-
level sets $\mathcal{T}$ that satisfies conditions $(1)$ and $(2)$ of the
Definition $16$ and a mapping $f$. Consider the mapping
$c^{\star}:\mathcal{S}_{n}((U_{i})_{i\in d})\to n$ defined by
$c^{\star}((X_{i})_{i\in d})=i$ if $\mathcal{C}^{(U_{i})_{i\in
d}}_{f((X_{i})_{i\in d})}$ contains strong subtrees of height equal to $i\in
n$. By Theorem $4$ we get a strong subtree $(V_{i})_{i\in
d}\in\mathcal{S}_{\infty}((U_{i})_{i\in d})$ on which $c^{\star}$ is constant
and equal to some fixed $i_{0}$. Let $k$ be the cardinality of the set of
node-level sets $\\{(N^{j}_{i},L^{j}_{i})_{i\in d,j\in k}\\}$ such that for
any $(Y_{i})_{i\in d}\in\mathcal{C}^{(V_{i})_{i\in
d}}_{(N^{j}_{i},L^{j}_{i})_{i}}$ we have that its height is equal to $i_{0}$.
Consider the coloring $\tilde{c}:\mathcal{S}_{n}((V_{i})_{i\in d})\to k$
defined by $\tilde{c}((X_{i})_{i\in d})=j\in k$ if and only if
$f((X_{i})_{i\in d})=(N^{j}_{i},L^{j}_{i})_{i\in d}$. By an application of
Theorem $4$ we get a $(V^{\prime}_{i})_{i\in
d}\in\mathcal{S}_{\infty}((V_{i})_{i\in d})$, so that
$\tilde{c}\upharpoonright(V^{\prime}_{i})_{i\in d}$ is constant. Therefore we
can assume that for any two node-level sets $(N_{i},L_{i})_{i\in
d},(N^{\prime}_{i},L^{\prime}_{i})_{i\in d}$ and any two members of their
strong subtree envelopes $(X_{i})_{i\in d}\in\mathcal{C}^{(U_{i})_{i\in
d}}_{(N_{i},L_{i})_{i\in d}}$ and $(Y_{i})_{i\in
d}\in\mathcal{C}^{(U_{i})_{i\in d}}_{(N^{\prime}_{i},L^{\prime}_{i})_{i\in
d}}$ one has:
$\iota_{b^{i_{0}},X_{i}}\circ\iota^{-1}_{b^{i_{0}},Y_{i}}(N^{\prime}_{i})=(N_{i})$
and $|L_{i}|=|L^{\prime}_{i}|$ for all $i\in d$. Therefore any two members of
$\mathcal{F}(\mathcal{G})$ are isomorphic in the sense of Definition $4$.
We need to obtain some results that they are going to give us the inductive
step. The first thing we notice is the following lemma:
###### Lemma 6.
Let $d,d^{\prime}\in\omega$, $\mathcal{G}$ an $\alpha$-uniform family on
$(U_{i})_{i\in d}$ and $\lambda:\mathcal{G}\to\bigcup_{j\in d^{\prime}}F_{j}$,
where $F_{j}\neq U_{i}$ for all $i\in d$, $j\in d^{\prime}$ are also
$b$-branching trees of infinite length. There exists for all $i\in d$,
$T_{i}\in\mathcal{S}_{\infty}(U_{i})$, and for all $j\in d^{\prime}$,
$V_{j}\in\mathcal{S}_{\infty}(F_{j})$, all having the same level sets, such
that
$\lambda(\mathcal{G}\upharpoonright(T_{i})_{i\in d})\bigcap(\cup_{j\in
d^{\prime}}V_{j})=\emptyset$
###### Proof.
We are giving a proof by induction on the rank of $\mathcal{G}$. The case of a
$0$-uniform family is vacuously true. Consider a $1$-uniform family
$\mathcal{G}$ and a mapping $\lambda:\mathcal{G}\to\bigcup_{j\in
d^{\prime}}F_{j}$. By the inductive hypothesis of Theorem $8$ we can assume
that $\lambda$ is canonical i.e. there exists a family $\mathcal{T}$ of
$d$-sequences of node-level sets and a one-to-one mapping $\phi$ on
$\mathcal{T}$. That family $\mathcal{T}$ gives rise to a uniform family
$\mathcal{F}(\mathcal{G})$ on $(U_{i})_{i\in d}$. If $\mathcal{T}=\emptyset$,
so the rank of $\mathcal{F}(\mathcal{G})$ is zero, then the mapping $\lambda$
is constant and the assertion of our lemma is trivial. Let
$\mathcal{T}\neq\emptyset$. By Remark $2$ observe that the rank of
$\mathcal{F}(\mathcal{G})$ is equal to one because the rank of $\mathcal{G}$
is equal to one. As a result the set $\mathcal{T}$ contains $d$-sequences of
either node or level sets, if otherwise by taking the strong subtree envelop
of $(N_{i},L_{i})_{i\in d}\in\mathcal{T}$ we would get finite strong subtrees
of height greater than $1$ contradicting that the rank of
$\mathcal{F}(\mathcal{G})$ is equal to one. Therefore for $(N_{i},L_{i})_{i\in
d}\in\mathcal{T}$ we have that either $N_{i}=\emptyset$ or $L_{i}=\emptyset$,
for all $i\in d$. Pick strong subtrees $(X^{1}_{i})_{i\in
d}\in\mathcal{S}_{1}((U_{i})_{i\in d})$and $(Y^{1}_{j})_{j\in
d^{\prime}}\in\mathcal{S}_{1}((V_{j})_{j\in d^{\prime}})$ so that
$L_{(X^{1}_{i})_{i\in d}}=L_{(Y^{1}_{j})_{j\in d^{\prime}}}=n\in
L_{(U_{i})_{i\in d}}=\omega$ and such that:
$\lambda((X^{1}_{i})_{i\in d})\notin\cup_{j\in d^{\prime}}Y^{1}_{j}.$
For every $t\in\bigcup_{j\in d^{\prime}}Y_{j}$, look at the level set, if non
empty, of $\lambda^{-1}(t)$. Then for each such a $t$ subtract the level
$L_{\lambda^{-1}(t)}$ from both level sets $L_{(U_{i})_{i\in d}}$ and
$L_{(V_{j})_{j\in d^{\prime}}}$. Having done that for all $t\in\bigcup_{j\in
d^{\prime}}Y^{1}_{j}$ we get strong subtrees $(T^{1}_{i})_{i\in
d}\sqsupseteq(X^{1}_{i})_{i\in d}$ and $(V^{1}_{j})_{j\in
d^{\prime}}\sqsupseteq(Y^{1}_{j})_{j\in d^{\prime}}$ with the same levels
sets. To be precise $L_{(T^{1}_{i})_{i\in d}}=L_{(V_{j})_{j\in
d^{\prime}}}=L_{(U_{i})_{i\in
d}}\setminus\\{L_{\lambda^{-1}(t)}:t\in\bigcup_{j\in d^{\prime}}Y_{j}\\}$.
These two strong subtrees have the property that for any $(Z_{i})_{i\in
d}\in\mathcal{S}_{1}((T^{1}_{i})_{i\in d})$, $\lambda((Z_{i})_{i\in
d})\notin\bigcup_{j\in d^{\prime}}Y^{1}_{j}$. To see that notice that for any
$t\in\bigcup_{j\in d^{\prime}}Y^{1}_{j}$ if there exists $(Z_{i})_{i\in
d}\in\mathcal{C}^{(U_{i})_{i\in d}}_{(N_{i},L_{i})_{i}}$ so that
$\lambda((Z_{i})_{i\in d})=t$, then $\mathcal{C}^{(T^{1}_{i})_{i\in
d}}_{(N_{i},L_{i})_{i}}=\emptyset$. This is because we have removed the level
$L_{\lambda^{-1}(t)}=L_{(N_{i},L_{i})_{i}}$.
Set
$(T_{i})_{i\in d}\upharpoonright 1=(T^{1}_{i})_{i\in d}\upharpoonright
1=(X^{1}_{i})_{i\in d}\text{ and }(V_{j})_{j\in d^{\prime}}\upharpoonright
1=(V^{1}_{j})_{j\in d^{\prime}}\upharpoonright 1=(Y_{j}^{1})_{i\in
d^{\prime}}.$
Suppose we have chosen the restrictions $(T_{i})_{i\in d}\upharpoonright
n=(X^{n}_{i})_{i\in d}\sqsubseteq(T^{n}_{i})_{i\in d}$ and $(V_{j})_{j\in
d^{\prime}}\upharpoonright n=(Y_{j}^{n})_{j\in
d^{\prime}}\sqsubseteq(V^{n}_{j})_{j\in d^{\prime}}$. We would like to decide
the $(T_{i})_{i\in d}\upharpoonright n+1$ and $(V_{j})_{j\in
d^{\prime}}\upharpoonright n+1$. Then pick a level $m^{\prime}\in
L_{(V^{n}_{j})_{j\in d^{\prime}}}$ such that the successors of each node in
$Y^{n}_{j}(n-1)$ on $V^{n}_{j}(m^{\prime})$ are more than $b^{n\cdot d}$. Now
for any choice of successors $\bigcup_{i\in d}X^{n}_{i}(n-1)$ on $\cup_{i\in
d}T^{n}_{i}(m^{\prime})$ we can always choose successors of $\bigcup_{j\in
d^{\prime}}Y^{n}_{j}(n-1)$, that lie on $\bigcup_{j\in
d^{\prime}}V^{n}_{j}(m^{\prime})$, so that the resulting strong subtrees
$(X^{n+1}_{i})_{i\in d}$ and $(Y^{n+1}_{j})_{j\in d^{\prime}}$, both of length
$n+1$, satisfy: $\lambda((X^{\prime}_{i})_{i\in d})\notin\bigcup_{j\in
d^{\prime}}Y^{n+1}_{j}(n)$, for all $(X^{\prime}_{i})_{i\in
d}\in\mathcal{S}_{1}(X^{n+1}_{i})_{i\in d}$. For any $t\in\bigcup_{j\in
d^{\prime}}Y^{n+1}_{j}(n)$ subtract the level $L_{\lambda^{-1}(t)}$ from both
level sets $L_{(T^{n}_{i}[X^{n+1}_{i}])_{i\in d}}$ and
$L_{(V^{n}_{j}[Y^{n+1}_{j}])_{j\in d^{\prime}}}$. Having done that for all
$t\in\bigcup_{j\in d^{\prime}}Y^{n+1}_{j}(n)$ we get strong subtrees
$(T^{n+1}_{i})_{i\in d}\in\mathcal{S}_{\infty}((T^{n}_{i}[X^{n+1}_{i}])_{i\in
d})$ and $(V^{n+1}_{j})_{j\in
d^{\prime}}\in\mathcal{S}_{\infty}((V^{n}_{j}[Y^{n+1}_{j}])_{j\in
d^{\prime}})$ such that $(X^{n+1}_{i})_{i\in d}\sqsubseteq(T^{n+1}_{i})_{i\in
d}$ and $(Y^{n+1}_{j})_{j\in d^{\prime}}\sqsubseteq(V^{n+1}_{j})_{j\in
d^{\prime}}$. These strong subtrees satisfy that for any $(X_{i})_{i\in
d}\in\mathcal{S}_{1}((T^{n+1}_{i})_{i\in d})$ it holds that
$\lambda((X_{i})_{i\in d})\cap(\cup_{j\in d^{\prime}}Y^{n+1}_{j})=\emptyset$.
To see that notice that for any $t\in\bigcup_{j\in d^{\prime}}Y^{n+1}_{j}$ if
there exists $(X_{i})_{i\in d}\in\mathcal{C}^{(T^{n}_{i})_{i\in
d}}_{(N_{i},L_{i})_{i}}$ so that $\lambda((X_{i})_{i\in d})=t$, then
$\mathcal{C}^{(T^{n+1}_{i}[X^{n+1}_{i}])_{i\in
d}}_{(N_{i},L_{i})_{i}}=\emptyset$. This is cause we have removed the level
$L_{\lambda^{-1}(t)}=L_{(N_{i},L_{i})_{i}}$.
Set
$(T_{i})_{i\in d}\upharpoonright n+1=(X^{n+1}_{i})_{i\in d}\text{ and
}(V_{j})_{j\in d^{\prime}}\upharpoonright n+1=(Y_{j}^{n+1})_{i\in
d^{\prime}}.$
Let $(T_{i})_{i\in d}$ be such that $(T_{i})_{i\in d}\upharpoonright
n=(X^{n}_{i})_{i\in d}\text{ and }(V_{j})_{j\in d^{\prime}}\upharpoonright
n=(Y_{j}^{n})_{i\in d^{\prime}}$ for all $n\in\omega$. $(T_{i})_{i\in d}$ and
$(V_{j})_{j\in d^{\prime}}$ satisfy the conclusions of our lemma. Suppose not,
let $(X_{i})_{i\in d}\in\mathcal{S}_{1}((T_{i})_{i\in d})$, $s\in\bigcup_{j\in
d^{\prime}}V_{j}$ with $|s|=k$, be so that $\lambda((X_{i})_{i\in d})=s$. Then
$s\in(Y_{j}^{k+1})_{j\in d^{\prime}}$. By construction we have that
$\lambda((X_{i})_{i\in d})\cap(\bigcup_{j\in
d^{\prime}}Y_{j}^{k+1})=\emptyset$, a contradiction.
So far we have shown that the statement of our lemma holds in the case of a
uniform family of rank $0$ and of rank $1$. Assume now that our lemma holds
for any $\beta$-uniform family, $\beta<\alpha$ and consider an
$\alpha$-uniform family $\mathcal{G}$ on $(U_{i})_{i\in d}$. Pick an arbitrary
$t=(t_{0},\dots,t_{d-1})\in\prod_{i\in d}U_{i}(n)$, for some $n$, and
$s=(s_{0},\dots,s_{d^{\prime}-1})\in\prod_{i\in d^{\prime}}V_{i}(n)$. By
definition $\mathcal{G}(t)$ is a $\beta$-uniform family, $\beta<\alpha$, on
$(U_{i})_{i\in d}(t)$, a $d\cdot b$ sequence of trees. The inductive
hypothesis applies on
$\lambda_{t}:(U_{i})_{i\in d}(t)\to\bigcup_{i\in d^{\prime}}F_{i}(s_{i})$
defined by
$\lambda_{t}((X_{k})_{k\in d\cdot b})=\lambda(t^{\frown}(X_{k})_{k\in d\cdot
b})$
to give us strong subtrees $(T^{1}_{k})_{k\in d\cdot b}$ and
$(V^{1}_{m})_{m\in d^{\prime}\cdot b}$ that satisfy
$\lambda(t^{\frown}(X_{k})_{k\in d\cdot b})\notin\bigcup_{m\in d^{\prime}\cdot
b}V^{1}_{m}$, for all $(X_{k})_{k\in d\cdot
b}\in\mathcal{G}(t)\upharpoonright(T^{1}_{k})_{k\in d\cdot b}$. Set
$(T^{2}_{i})_{i\in d}=t^{\frown}(T^{1}_{i})_{i\in d}\text{ and
}(V^{2}_{j})_{j\in d^{\prime}}=s^{\frown}(V^{1}_{j})_{j\in d^{\prime}},$
and
$(T_{i})_{i\in d}\upharpoonright 2=(T^{2}_{i})_{i\in d}\upharpoonright 2\text{
and }(V_{j})_{j\in d^{\prime}}\upharpoonright 2=(V^{2}_{j})_{j\in
d^{\prime}}\upharpoonright 2.$
We can assume that
$\\{s_{0},\dots,s_{d^{\prime}-1}\\}\cap\lambda(t^{\frown}(T^{1}_{k})_{k\in
d\cdot b})=\emptyset$. To see that consider the level set of
$\lambda_{t}^{-1}(s_{j})$ and subtract a level $l_{s_{j}}$ in
$L_{\lambda_{t}^{-1}(s_{j})}$ from both level sets $L_{(U_{i}(t_{i}))_{i\in
d}}$ and $L_{(F_{j}(s_{j}))_{j\in d^{\prime}}}$. Having done that for all
$s_{j}$, $j\in d^{\prime}$ we get strong subtrees $(T^{\prime 1}_{i})_{i\in
d}\sqsupseteq(t_{0},\dots,t_{d-1})=t$ and $(V^{\prime 1}_{j})_{j\in
d^{\prime}}\sqsupseteq(s_{0},\dots,s_{d^{\prime}-1})=s$ with the same levels
sets. Namely $L_{(T^{\prime 1}_{i})_{i\in d}}=L_{(V^{\prime 1}_{j})_{j\in
d^{\prime}}}=L_{(U_{i}(t_{i}))_{i\in d}}\setminus\\{l_{s_{j}}:s_{j}\in
s=(s_{0},\dots,s_{d^{\prime}-1})\\}$. These two strong subtrees have the
property that for any $(Z_{i})_{i\in d\cdot
b}\in\mathcal{G}(t)\upharpoonright(T^{\prime 1}_{i})_{i\in d}$,
$\lambda_{t}((Z_{i})_{i\in d})\notin\\{s_{0},\dots,s_{d^{\prime}-1}\\}$. To
see that suppose there exist $(Z_{i})_{i\in d\cdot
b}\in\mathcal{G}(t)\upharpoonright(T^{\prime 1}_{i})_{i\in d}$, $(Z_{i})_{i\in
d\cdot b}\in\mathcal{C}^{(T^{\prime 1}_{i})_{i\in d}}_{(N_{i},L_{i})_{i}}$ and
$s_{j}\in s=(s_{0},\dots,s_{d^{\prime}-1})$ such that
$\lambda_{t}((Z_{i})_{i\in d})=s_{j}$. There exists $l_{s_{j}}\in
L_{(N_{i},L_{i})_{i}}$ so that $l_{s_{j}}\notin L_{(T^{\prime 1}_{i})_{i\in
d}}$. As a result $\mathcal{C}^{(T^{\prime 1}_{i})_{i\in
d}}_{(N_{i},L_{i})_{i}}=\emptyset$, a contradiction.
Suppose we have constructed $(T_{i})_{i\in d}\upharpoonright
n=(T^{n}_{i})_{i\in d}\upharpoonright n$ and $(V_{j})_{j\in
d^{\prime}}\upharpoonright n=(V^{n}_{j})_{j\in d^{\prime}}\upharpoonright n$
so that for any $(X_{i})_{i\in d}\in\mathcal{G}(t^{\prime})$,
$t^{\prime}\in\prod_{i\in d}T^{n}_{i}(k)$ for $k<n$ it holds that
$\lambda(t^{\prime\frown}(X_{i})_{i\in d})\cap(\bigcup_{j\in
d^{\prime}}V^{n}_{j}\upharpoonright n)=\emptyset$. We wish to decide
$(T_{i})_{i\in d}\upharpoonright n+1$ and $(V_{j})_{j\in
d^{\prime}}\upharpoonright n+1$.
Let $\\{r_{0},\dots,r_{d\cdot b^{n-1}-1}\\}$ be a one-to-one enumeration of
the nodes $\bigcup_{i\in d}T^{n}_{i}(n-1)$ and
$\\{s^{\prime}_{0},\dots,s^{\prime}_{d^{\prime}\cdot b^{n-1}-1}\\}$ a one-to-
one enumeration of the nodes $\bigcup_{j\in d^{\prime}}V^{n}_{j}(n-1)$.
For any $r=(r_{k_{i}})_{i\in d}$, where for all $i\in d$, $r_{k_{i}}\in
T^{n}_{i}$, consider the uniform family
$\mathcal{G}(r)\upharpoonright(T^{n}_{i})_{i\in d}(r)$. Apply once more the
inductive hypothesis on $(T^{n}_{i})_{i\in d}(r)$ and
$(F^{n}_{j}(s^{\prime}_{m})_{m\in[j\cdot b^{n-1},(j+1)\cdot b^{n-1})})_{j\in
d^{\prime}}$ to get strong subtrees $(T^{\prime n}_{l})_{l\in d\cdot
b}\in\mathcal{S}_{\infty}((T^{n}_{i})_{i\in d}(r))$ and $(F^{\prime
n}_{f})_{f\in d^{\prime}\cdot
b^{n}}\in\mathcal{S}_{\infty}((F^{n}_{j}(s^{\prime}_{m})_{m\in[j\cdot
b^{n-1},(j+1)\cdot b^{n-1})})_{j\in d^{\prime}})$ that satisfy the conclusions
of our lemma. At this point we can assume that
$\\{s^{\prime}_{0},\dots,s^{\prime}_{d^{\prime}\cdot
b^{n-1}-1}\\}\cap\lambda_{r}(\mathcal{G}(r)\upharpoonright(T^{n}_{i})_{i\in
d}(r))=\emptyset$. That can be guaranteed by the fact that $\lambda_{r}$ on
$\mathcal{G}(r)\upharpoonright(T^{n}_{i})_{i\in d}(r)$ is a canonical coloring
on a uniform family of rank $\beta<\alpha$. The argument is identical with the
one just above. Having done that for all possible $r$ as above, we get strong
subtrees
$(T^{n+1}_{g})_{g\in d\cdot
b^{n}}\in\mathcal{S}_{\infty}((T^{n}_{i}(r_{k})_{k\in[i\cdot
b^{n-1},(i+1)\cdot b^{n-1}})_{i\in d})$
and
$(V^{n+1}_{f})_{f\in d^{\prime}\cdot
b^{n}}\in\mathcal{S}_{\infty}((V^{n}_{j}(s^{\prime}_{m})_{m\in[j\cdot
b^{n-1},(j+1)\cdot b^{n-1})})_{j\in d^{\prime}})$
all with the same level sets. Let $(T^{n+1}_{i})_{i\in d}=((T^{n}_{i})_{i\in
d}\upharpoonright n)^{\frown}(T^{n+1}_{g})_{g\in d\cdot b^{n}}$ and
$(V^{n+1}_{j})_{j\in d^{\prime}}=((V^{n}_{j})_{j\in d^{\prime}}\upharpoonright
n)^{\frown}(V^{n+1}_{f})_{f\in d^{\prime}\cdot b^{n}}$. Set
$(T_{i})_{i\in d}\upharpoonright n+1=(T^{n+1}_{i})_{i\in d}\upharpoonright
n+1\text{ and }(V_{j})_{j\in d^{\prime}}\upharpoonright
n+1=(V^{n+1}_{j})_{j\in d^{\prime}}\upharpoonright n+1$
For any $(X_{i})_{i\in d}\in\mathcal{G}(t^{\prime})$,
$t^{\prime}\in\prod_{i\in d}T^{n+1}_{i}(k)$ where $k<n+1$, it holds that
$\lambda(t^{\prime\frown}(X_{i})_{i\in d})\cap(\cup_{j\in
d^{\prime}}V^{n+1}_{j}\upharpoonright n+1)=\emptyset$. The resulting strong
subtrees $(T_{i})_{i\in d}$ such that $(T_{i})_{i\in d}\upharpoonright
n=(T^{n}_{i})_{i\in d}\upharpoonright n\text{ and }(V_{j})_{j\in
d^{\prime}}\upharpoonright n=(V_{j}^{n})_{i\in d}\upharpoonright n$ for all
$n\in\omega$, satisfy the conclusions of our lemma, with an argument identical
with that of the case of rank equal to $1$.
∎
Having established the previous lemma, we prove the following:
###### Lemma 7.
Let $d\in\omega$, $\mathcal{G}$ an $\alpha$-uniform family on $(U_{i})_{i\in
d}$ and $\lambda:\mathcal{G}\to(U_{i})_{i\in d}$ be a mapping with the
property that: $\lambda(X_{0},\dots,X_{d-1})\notin\bigcup_{i\in d}X_{i}$, for
all $(X_{0},\dots,X_{d-1})\in\mathcal{G}$. There exists a strong subtree
$(T_{i})_{i\in d}\in\mathcal{S}_{\infty}((U_{i})_{i\in d})$ such that
$\lambda(\mathcal{G}\upharpoonright(T_{i})_{i\in d})\bigcap(\cup_{i\in
d}T_{i})=\emptyset$
###### Proof.
We give a proof by induction on the rank of $\mathcal{G}$. For a $0$-uniform
family the assertion of the lemma is vacuously true. Let $\mathcal{G}$ be a
$1$-uniform family and $\lambda:\mathcal{G}\to(U_{i})_{i\in d}$ be a mapping
with the property that $\lambda((X_{i})_{i\in d})\notin\bigcup_{i\in d}X_{i}$.
By the inductive hypothesis of Theorem $8$ we can assume that $\lambda$ is
canonical i.e. there exists a non empty family $\mathcal{T}$ of node-level
sets, which gives rise to a uniform family $\mathcal{F}(\mathcal{G})$ on
$(U_{i})_{i\in d}$.
Pick now $(X^{1}_{i})_{i\in d}\in\mathcal{S}_{1}((U_{i})_{i\in d})$ and for
every $t\in\bigcup_{i\in d}X^{1}_{i}$ consider the level set, if non empty,
$L_{\lambda^{-1}(t)}$. Then subtract for each $t\in\bigcup_{i\in d}X^{1}_{i}$
the level $L_{\lambda^{-1}(t)}$ from $L_{(U_{i})_{i\in d}}$ so that a
resulting strong subtree $(T^{1}_{i})_{i\in d}$ of $(U_{i})_{i\in d}$ with
$(X^{1}_{i})_{i\in d}\sqsubseteq(T^{1}_{i})_{i\in d}$, has the property that
for any $(X^{\prime}_{i})_{i\in d}\in\mathcal{S}_{1}((T^{1}_{i})_{i\in d})$,
$\lambda((X^{\prime}_{i})_{i\in d})\notin\bigcup_{i\in d}X^{1}_{i}$. Set
$(T_{i})_{i\in d}\upharpoonright 1=(T^{1}_{i})_{i\in d}\upharpoonright
1=(X^{1})_{i\in d}$. Suppose we have constructed $(T_{i})_{i\in
d}\upharpoonright n=(T^{n}_{i})_{i\in d}\upharpoonright n=(X^{n}_{i})_{i\in
d}$ and we have to decide $(T_{i})_{i\in d}\upharpoonright(n+1)$.
Let $\\{t_{0},\dots,t_{b^{n-1}-1}\\}$ be a one-to-one enumeration of the nodes
$\bigcup_{i\in d}T^{n}_{i}(n-1)$. Pick $m\in L_{(T^{n}_{i})_{i\in d}}$ such
that any $t\in\\{t_{0},\dots,t_{b^{n-1}-1}\\}$ has more than $b^{n\cdot d}$
successors on $\bigcup_{i\in d}T^{n}_{i}(m)$. Choose successors of
$\\{t_{0},\dots,t_{b^{n-1}-1}\\}$ on $\cup_{i\in d}T^{n}_{i}(m)$ so that the
resulting strong subtree $(X^{n+1}_{i})_{i\in d}$, of length $n+1$, where
$(X^{n+1}_{i})_{i\in d}\sqsupseteq(X^{n}_{i})_{i\in d}$ has the following
property: $\lambda((Z_{i})_{i\in d})\notin\cup_{i\in d}X^{n+1}_{i}(n)$ for any
$(Z_{i})_{i\in d}\in\mathcal{S}_{1}((X^{n+1}_{i})_{i\in d})$. Consider the
strong subtree $(T^{n}_{i}[X^{n+1}_{i}])_{i\in d}$. Now for any
$t\in\cup_{i\in d}X^{n+1}_{i}(n)$ subtract the level $L_{\lambda^{-1}(t)}$
from the level set $L_{(T^{n}_{i}[X^{n+1}_{i}])_{i\in d}}$. Let
$(T^{n+1}_{i})_{i\in d}$ be a resulting strong subtree with
$(X^{n+1}_{i})_{i\in d}\sqsubseteq(T^{n+1}_{i})_{i\in d}$. For every
$(Z_{i})_{i\in d}\in\mathcal{S}_{1}((T^{n+1}_{i})_{i\in d})$ we have that
$\lambda((Z_{i})_{i\in d})\notin\bigcup_{i\in d}X^{n+1}_{i}$. Set
$(T_{i})_{i\in d}\upharpoonright n+1=(T^{n+1}_{i})_{i\in d}\upharpoonright
n+1=(X^{n+1}_{i})_{i\in d}.$
Let $(T_{i})_{i\in d}$ be such that $(T_{i})_{i\in d}\upharpoonright
n=(T^{n}_{i})_{i\in d}$ for all $n\in\omega$. We claim that it satisfies the
conclusions of our lemma. Suppose that $(X_{i})_{i\in
d}\in\mathcal{S}_{1}((T_{i})_{i\in d})$ and $\lambda((X_{i})_{i\in
d})=t\in\bigcup_{i\in d}T_{i}$ with $|t|=k$. By our construction we have that
$\lambda((X_{i})_{i\in d})\notin\bigcup_{i\in d}T^{k+1}_{i}\upharpoonright
k+1=\bigcup_{i\in d}T_{i}\upharpoonright k+1$, a contradiction.
Assume now the lemma holds for any $\beta$-uniform family, $\beta<\alpha$ and
consider an $\alpha$-uniform family $\mathcal{G}$ on $(U_{i})_{i\in d}$. Pick
$t=(t_{0},\dots,t_{d-1})\in\prod_{i\in d}U_{i}(n)$, for some $n\in\omega$. By
definition $\mathcal{G}(t)$ is a $\beta$-uniform family on $(U_{i})_{i\in
d}(t)$. Apply our assumption to the canonical mapping
$\lambda_{t}:\mathcal{G}(t)\to\bigcup_{i\in d}U_{i}(t_{i})$, defined by
$\lambda_{t}((X_{m})_{m\in d\cdot b})=\lambda(t^{\frown}(X_{m})_{m\in d\cdot
b})$, to get strong subtrees $(T^{1}_{m})_{m\in d\cdot
b}\in\mathcal{S}_{\infty}((U_{i}(t_{i}))_{i\in d})$ such that for any
$(X_{m})_{m\in d\cdot b}\in\mathcal{G}(t)$ one has $\lambda_{t}((X_{m})_{m\in
d\cdot b})\notin\bigcup_{m\in d\cdot b}T^{1}_{m}$. We can also assume, as in
Lemma $6$ above, that
$t=(t_{0},\dots,t_{d-1})\cap\lambda_{t}(\mathcal{G}(t)\upharpoonright(T^{1}_{m})_{m\in
d\cdot b})=\emptyset$ since $\lambda_{t}$ is a coloring on a uniform family or
rank $\beta<\alpha$. Set
$(T^{2}_{i})_{i\in d}=t^{\frown}(T^{1}_{m})_{m\in d\cdot b}\text{ and
}(T_{i})_{i\in d}\upharpoonright 2=(T^{2}_{i})_{i\in d}\upharpoonright 2.$
Suppose we have constructed $(T_{i})_{i\in d}\upharpoonright
n=(T^{n}_{i})_{i\in d}\upharpoonright n$ and we have to decide $(T_{i})_{i\in
d}\upharpoonright(n+1)=(T^{n+1}_{i})_{i\in d}\upharpoonright(n+1)$. Let
$\\{r_{0},\dots,r_{d\cdot b^{n-1}-1}\\}$ be a one-to-one enumeration of the
terminal nodes of $(T^{n}_{i})_{i\in d}\upharpoonright n$. Let
$r=(r_{k_{i}})_{i\in d}$, where for all $i\in d$, $r_{k_{i}}\in T^{n}_{i}$.
Let also $w=d\cdot b^{n-1}$, $w_{r}=\\{j\in d\cdot b^{n-1}:r_{j}\in r\\}$ and
$w_{r}^{c}=w\setminus w_{r}$. Consider the mappings
$\lambda_{r}:\mathcal{G}(r)\upharpoonright(T^{n}_{i})_{i\in d}(r)\to\cup_{j\in
w_{r}^{c}}T^{n}_{i}(r_{j})$, where $r_{j}\in T^{n}_{i}$, defined by
$\lambda_{r}((X_{k})_{k\in w_{r}})=\lambda(r^{\frown}(X_{k})_{k\in w_{r}})$.
By the inductive hypothesis we assume that
$\lambda_{r}(\mathcal{G}(r)\upharpoonright(T^{n}_{i})_{i\in
d}(r))\bigcap(\cup(T^{n}_{i})_{i\in d}(r))=\emptyset$. Now by Lemma $6$ we get
strong subtrees $(T^{\prime n}_{j})_{j\in w_{r}}$ of
$(T^{n}_{i}[r_{k_{i}}])_{i\in d}$ and $(T^{\prime n}_{j})_{j\in w^{c}_{r}}$ of
$(T^{n}_{i}[r_{j}])_{j\in w_{r}^{c}}$, all with the same levels sets, that
satisfy
$\lambda_{r}((T^{\prime n}_{j})_{j\in w_{r}})\bigcap(\bigcup_{j\in
w^{c}_{r}}T^{\prime n}_{j})=\emptyset.$
At this point we can assume that $\\{r_{j}:j\in
w_{r}\\}\bigcap\lambda_{r}((T^{n}_{i})_{i\in d}(r))=\emptyset$ by the fact
that $\lambda_{r}$ is a canonical coloring restricted on a uniform family of
rank $\beta<\alpha$, as we did in Lemma $6$ above. Repeat this last step for
all possible such a $r$ to get strong subtrees $(T^{\prime n+1}_{f})_{f\in
d\cdot b^{n}}\in\mathcal{S}_{\infty}((T^{\prime n}_{j})_{j\in w})$. Set
$(T^{n+1}_{i})_{i\in d}=((T^{n}_{i})_{i\in d}\upharpoonright
n)^{\frown}(T^{\prime n+1}_{f})_{f\in d\cdot b^{n}}\text{ and }(T_{i})_{i\in
d}\upharpoonright(n+1)=(T^{n+1}_{i})_{i\in d}\upharpoonright(n+1).$
Let $(T_{i})_{i\in d}$ be such that $(T_{i})_{i\in d}\upharpoonright
n=(T^{n}_{i})_{i\in d}$ for all $n\in\omega$. It satisfies the conclusions of
our lemma with an argument identical with that in the case of rank equal to
one.
∎
We would like to establish a result that will give us the possibility of
comparing two uniform families and two canonical colorings defined on them. We
use Lemma $7$ to prove the following:
###### Lemma 8.
Let $\mathcal{T}_{1}$ and $\mathcal{T}_{2}$ be two families of node-level sets
so that they generate two uniform families $\mathcal{F}(\mathcal{G}_{1})$ and
$\mathcal{F}(\mathcal{G}_{2})$, on $(U_{i})_{i\in d}$, by taking the union of
all strong subtree envelopes of all members of $\mathcal{T}_{1}$ and
$\mathcal{T}_{2}$ respectively . Let $c^{\prime}_{1}$ a mapping on
$\mathcal{F}(\mathcal{G}_{1})$ with the property that
$c^{\prime}_{1}((X^{1}_{i})_{i\in d})=c^{\prime}_{1}((X^{2}_{i})_{i\in d})$ if
and only if $(X^{1}_{i})_{i\in d}:(N^{1}_{i},L^{1}_{i})_{i\in
d}=(X^{2}_{i})_{i\in d}:(N^{1}_{i},L^{1}_{i})_{i\in d}$ for
$(N^{1}_{i},L^{1}_{i})_{i\in d}\in\mathcal{T}_{1}$. Let also $c_{2}$ a mapping
on $\mathcal{F}(\mathcal{G}_{2})$ such that $c_{2}((Y^{1}_{i})_{i\in
d})=c_{2}((Y^{2}_{i})_{i\in d})$ if and only if $(Y^{1}_{i})_{i\in
d}:(N^{2}_{i},L^{2}_{i})_{i\in d}=(Y^{2}_{i})_{i\in
d}:(N^{2}_{i},L^{2}_{i})_{i\in d}$ for $(N^{2}_{i},L^{2}_{i})_{i\in
d}\in\mathcal{T}_{2}$. There exists $(T_{i})_{i\in
d}\in\mathcal{S}_{\infty}((U_{i})_{i\in d})$ such that one of the following
two statements holds.
1. (1)
$\mathcal{F}(\mathcal{G}_{1})\upharpoonright(T_{i})_{i\in
d}=\mathcal{F}(\mathcal{G}_{2})\upharpoonright(T_{i})_{i\in d}$ and
$c^{\prime}_{1}((X_{i})_{i\in d})=c^{\prime}_{2}((X_{i})_{i\in d})$ for every
$(X_{i})_{i\in d}\in\mathcal{F}(\mathcal{G}_{1})\upharpoonright(T_{i})_{i\in
d}=\mathcal{F}(\mathcal{G}_{2})\upharpoonright(T_{i})_{i\in d}$.
2. (2)
The image of $c^{\prime}_{1}$ on
$\mathcal{F}(\mathcal{G}_{1})\upharpoonright(T_{i})_{i\in d}$ and the image of
$c^{\prime}_{2}$ on $\mathcal{F}(\mathcal{G}_{2})\upharpoonright(T_{i})_{i\in
d}$ are disjoint.
###### Proof.
Partition $\mathcal{F}(\mathcal{G}_{1})$ into two pieces $\mathcal{S}_{1,1}$
and $\mathcal{S}_{1,2}$ as follows: $(X_{i})_{i\in d}\in\mathcal{S}_{1,1}$ if
and only if $(X_{i})_{i\in
d}\in\mathcal{F}(\mathcal{G}_{2}),\,c^{\prime}_{1}((X_{i})_{i\in
d})=c^{\prime}_{2}((X_{i})_{i\in d})$ and $(X_{i})_{i\in
d}\in\mathcal{S}_{1,2}$ if and only if $(X_{i})_{i\in
d}\notin\mathcal{S}_{1,1}$. Since $\mathcal{F}(\mathcal{G}_{1})$ is Ramsey, we
get $(T^{0}_{i})_{i\in d}\in\mathcal{S}_{\infty}((U_{i})_{i\in d})$ such that
either $\mathcal{F}(\mathcal{G}_{1})\upharpoonright(T^{0}_{i})_{i\in
d}\subseteq\mathcal{S}_{1,1}$, in which case we have the first statement
holding, or $\mathcal{F}(\mathcal{G}_{1})\upharpoonright(T^{0}_{i})_{i\in
d}\subseteq\mathcal{S}_{1,2}$, in which case we have to show that the second
statement is on hold.
Therefore we assume that
$\mathcal{F}(\mathcal{G}_{1})\upharpoonright(T^{0}_{i})_{i\in
d}\subseteq\mathcal{S}_{1,2}$ and we show that the second statement is true.
Note that for $(X_{i})_{i\in
d}\in\mathcal{F}(\mathcal{G}_{1})\upharpoonright(T^{0}_{i})_{i\in d}$ to be a
member of $\mathcal{S}_{1,2}$ it is either the case that $(X_{i})_{i\in
d}\notin\mathcal{F}(\mathcal{G}_{2})\upharpoonright(T^{0}_{i})_{i\in d}$ or if
$(X_{i})_{i\in
d}\in\mathcal{F}(\mathcal{G}_{2})\upharpoonright(T^{0}_{i})_{i\in d}$ then one
must have $c^{\prime}_{1}((X_{i})_{i\in d})\neq c^{\prime}_{2}((X_{i})_{i\in
d})$.
Let $(X_{i})_{i\in
d}\in\mathcal{F}(\mathcal{G}_{1})\upharpoonright(T^{0}_{i})_{i\in d}$ and
pick, if it exists, a $(Y_{i})_{i\in
d}\in\mathcal{F}(\mathcal{G}_{2})\upharpoonright(T^{0}_{i})_{i\in d}$ such
that
$c^{\prime}_{1}((X_{i})_{i\in d})=c^{\prime}_{2}((Y_{i})_{i\in d})$
This would imply that $(X_{i})_{i\in d}\neq(Y_{i})_{i\in d}$ and that will be
true not only for $(X_{i})_{i\in d}$, $(Y_{i})_{i\in d}$ but for all members
of the strong subtree envelope of $(N^{1}_{i},L^{1}_{i})_{i\in
d}\in\mathcal{T}_{1}$, $(N^{2}_{i},L^{2}_{i})_{i\in d}\in\mathcal{T}_{2}$,
where $(X_{i})_{i\in d}\in\mathcal{C}^{(T^{0}_{i})_{i\in
d}}_{(N^{1}_{i},L^{1}_{i})_{i\in d}}$ and $(Y_{i})_{i\in
d}\in\mathcal{C}^{(T^{0}_{i})_{i\in d}}_{(N^{2}_{i},L^{2}_{i})_{i\in d}}$. To
see this observe that if we had $(X^{\prime}_{i})_{i\in
d}=(Y^{\prime}_{i})_{i\in d}$ for some $(X^{\prime}_{i})_{i\in
d}\in\mathcal{C}^{(T^{0}_{i})_{i\in d}}_{(N^{1}_{i},L^{1}_{i})_{i\in d}}$ and
some $(Y^{\prime}_{i})_{i\in d}\in\mathcal{C}^{(T^{0}_{i})_{i\in
d}}_{(N^{2}_{i},L^{2}_{i})_{i\in d}}$ i.e. $(X^{\prime}_{i})_{i\in
d}\in\mathcal{F}(\mathcal{G}_{2})\upharpoonright(T^{0}_{i})_{i\in d}$, then we
would get a contradiction because $c^{\prime}_{1}((X^{\prime}_{i})_{i\in
d})=c^{\prime}_{1}((X_{i})_{i\in d})=c^{\prime}_{2}((Y_{i})_{i\in
d})=c^{\prime}_{2}((Y^{\prime}_{i})_{i\in
d})=c^{\prime}_{2}((X^{\prime}_{i})_{i\in d})$ and we have assumed that
$\mathcal{F}(\mathcal{G}_{1})\upharpoonright(T^{0}_{i})_{i\in
d}\subseteq\mathcal{S}_{1,2}$ .
To proceed further, we need the following lemma:
###### Lemma 9.
In the above context, i.e.
$\mathcal{F}(\mathcal{G}_{1})\upharpoonright(T^{0}_{i})_{i\in
d}\subseteq\mathcal{S}_{1,2}$, by passing to a strong subtree if necessarily,
we can assume that there are not
$(X_{i})_{i\in
d}\in\mathcal{F}(\mathcal{G}_{1})\upharpoonright(T^{0}_{i})_{i\in d}\text{ and
}(Y_{i})_{i\in
d}\in\mathcal{F}(\mathcal{G}_{2})\upharpoonright(T^{0}_{i})_{i\in d}$
such that $c^{\prime}_{1}((X_{i})_{i\in d})=c^{\prime}_{2}((Y_{i})_{i\in d})$
and $((X_{i})_{i\in d})^{in}=((Y_{i})_{i\in d})^{in}$.
###### Proof.
For simplicity reasons in the proof we write $\mathcal{F}(\mathcal{G}_{j})$,
$j\in\\{1,2\\}$ instead of
$\mathcal{F}(\mathcal{G}_{j})\upharpoonright(T^{0}_{i})_{i\in d}$. Suppose now
that $c^{\prime}_{1}((X_{i})_{i\in d})=c^{\prime}_{2}((Y_{i})_{i\in d})$ for
$(X_{i})_{i\in d}\in\mathcal{F}(\mathcal{G}_{1})$, $(Y_{i})_{i\in
d}\in\mathcal{F}(\mathcal{G}_{2})$ and $L_{(X_{i})_{i\in d}}=L_{(Y_{i})_{i\in
d}}$. If one has $((X_{i})_{i\in d})^{in}=((Y_{i})_{i\in
d})^{in}=(Z_{i})_{i\in d}$, this would imply that there exist
$(X^{\prime}_{i})_{i\in d}\in\mathcal{C}^{(T^{0}_{i})_{i\in
d}}_{(Z_{i},L^{1}_{i})_{i\in d}}$ and $(Y^{\prime}_{i})_{i\in
d}\in\mathcal{C}^{(T^{0}_{i})_{i\in d}}_{(Z_{i},L^{2}_{i})_{i\in d}}$ so that
$X^{\prime}_{i}=Y^{\prime}_{i}$ for all $i\in d$ and
$c^{\prime}_{1}((X^{\prime}_{i})_{i\in
d})=c^{\prime}_{2}((Y^{\prime}_{i})_{i\in d}$ a contradiction. From now on we
consider the case of $L_{(X_{i})_{i\in d}}\neq L_{(Y_{i})_{i\in d}}$.
The proof is by induction on the countable ordinals $\alpha,\beta$ the ranks
of $\mathcal{F}(\mathcal{G}_{1})$ and $\mathcal{F}(\mathcal{G}_{2})$
respectively. Let both $\alpha,\beta$ be finite. Let $(X_{i})_{i\in
d}\in\mathcal{F}(\mathcal{G}_{1})$, $(Y_{i})_{i\in
d}\in\mathcal{F}(\mathcal{G}_{2})$ and consider $((X_{i})_{i\in d})^{in}$,
$((Y_{i})_{i\in d})^{in}$. Let $n<|L_{(X_{i})_{i\in d}}|=\alpha$ be the length
of $((X_{i})_{i\in d})^{in}$ and $k<|L_{(Y_{i})_{i\in d}}|=\beta$ the length
of $((Y_{i})_{i\in d})^{in}$. Assume that $k=n$. We distinguish the following
three cases:
$\bf{Case\,1:}$ Let both sets $L_{(X_{i})_{i\in d}}\setminus L_{((X_{i})_{i\in
d})^{in}}$ and $L_{(Y_{i})_{i\in d}}\setminus L_{((Y_{i})_{i\in d})^{in}}$ be
non empty. Pick a finite strong subtree $(Z^{1}_{i})_{i\in d}$ of
$(T^{0}_{i})_{i\in d}$ with height $n$. Then by applying Lemma $2$ on the
$\alpha-n$, $\beta-n$ uniform families on $L_{(T^{0}_{i}[Z^{1}_{i}])_{i\in
d}}\setminus n$, and the mappings $c^{\prime\prime}_{j},j\in\\{1,2\\}$,
defined by
$c^{\prime\prime}_{j}(L_{j})=c^{\prime}_{j}(\mathcal{C}^{(T^{0}_{i})_{i\in
d}}_{(Z^{1}_{i})_{i\in d},L_{j}})$ we get $(T^{1}_{i})_{i\in
d}\in\mathcal{S}_{\infty}((T^{1}_{i})_{i\in d})$, where $(T^{1}_{i})_{i\in
d}\upharpoonright n=(Z^{1}_{i})_{i\in d}$. $(T^{1}_{i})_{i\in d}$ satisfies
the second alternative of Lemma $2$, because we have assumed that
$\mathcal{F}(\mathcal{G}_{1})\upharpoonright(T^{0}_{i})_{i\in
d}\subseteq\mathcal{S}_{1,2}$. On $(T^{1}_{i})_{i\in d}$ for $(X_{i})_{i\in
d}\in\mathcal{F}(\mathcal{G}_{1})$ and $(Y_{i})_{i\in
d}\in\mathcal{F}(\mathcal{G}_{2})$ we have that
$\text{If }((X_{i})_{i\in d})^{in}=((Y_{i})_{i\in d})^{in}=(Z^{1}_{i})_{i\in
d}\text{, then }c((X_{i})_{i\in d})\neq c((Y_{i})_{i\in d}).$
Set $(T_{i})_{i\in d}\upharpoonright n=(T^{1}_{i})_{i\in d}\upharpoonright
n=(Z^{1}_{i})_{i\in d}$.
Suppose we have constructed $(T_{i})_{i\in
d}\upharpoonright(n+m)=(T^{m}_{i})_{i\in
d}\upharpoonright(n+m)=(Z^{m}_{i})_{i\in d}$ and we have to decide
$(T_{i})_{i\in d}\upharpoonright(n+m+1)=(T^{m+1}_{i})_{i\in
d}\upharpoonright(n+m+1)$. Let now
$(Z^{m}_{i})_{i\in d}\sqsubset(Z^{m+1}_{i})_{i\in d}\text{ and
}(Z^{m+1}_{i})_{i\in d}\in\mathcal{S}_{n+m+1}((T^{m}_{i})_{i\in d}).$
Consider the finite set $A_{m+1}=\\{\,(Z^{\prime}_{i})_{i\in
d}\in\mathcal{S}_{n}((Z^{m+1}_{i})_{i\in d})\,\\}$. For each
$(Z^{\prime}_{i})_{i\in d}\in A_{m+1}$ apply Lemma $2$ on
$L_{(T^{m}_{i}[Z^{\prime}_{i}])_{i\in d}}\setminus n$ and the mappings
$c^{\prime\prime}_{j},j\in\\{1,2\\}$, defined by
$c^{\prime\prime}_{j}(L_{j})=c^{\prime}_{j}(\mathcal{C}^{(T^{m}_{i})_{i\in
d}}_{(Z^{\prime}_{i})_{i\in d},L_{j}})$. That gives us $(T^{\prime
m}_{i})_{i\in d}\in\mathcal{S}_{\infty}((T^{m}_{i})_{i\in d})$, where
$(T^{\prime m}_{i})_{i\in d}\upharpoonright(n+m+1)=(Z^{m+1}_{i})_{i\in d}$,
that satisfies the second alternative of Lemma $2$. On $(T^{\prime
m}_{i})_{i\in d}$ for $(X_{i})_{i\in d}\in\mathcal{F}(\mathcal{G}_{1})$ and
$(Y_{i})_{i\in d}\in\mathcal{F}(\mathcal{G}_{2})$ we have that
$\text{If }((X_{i})_{i\in d})^{in}=((Y_{i})_{i\in
d})^{in}=(Z^{\prime}_{i})_{i\in d}\text{, then }c((X_{i})_{i\in d})\neq
c((Y_{i})_{i\in d}).$
Repeat this step for all the elements of $A_{m+1}$, to get
$(T^{m+1}_{i})_{i\in d}\in\mathcal{S}_{\infty}((T^{m}_{i})_{i\in d})$ where
$T^{m+1}_{i}\upharpoonright(n+m+1)=Z^{m+1}_{i}$, for all $i\in d$. On
$(T^{m+1}_{i})_{i\in d}$ it holds that
$c^{\prime}_{1}((X_{i})_{i\in d})\neq c^{\prime}_{2}((Y_{i})_{i\in d})$ for
all $(X_{i})_{i\in d}\in\mathcal{F}(\mathcal{G}_{1})$ and $(Y_{i})_{i\in
d}\in\mathcal{F}(\mathcal{G}_{2})$ with $((X_{i})_{i\in
d})^{in}=((Y_{i})_{i\in d})^{in}=(Z^{\prime}_{i})_{i\in d}\in A_{m+1}$. Set
$(T_{i})_{i\in d}\upharpoonright(n+m+1)=(T^{m+1}_{i})_{i\in
d}\upharpoonright(n+m+1)=(Z^{m+1}_{i})_{i\in d}.$
Let $(T_{i})_{i\in d}$ be such that $(T_{i})_{i\in d}\upharpoonright
n=(T^{1}_{i})_{i\in d}\upharpoonright n=(Z^{1}_{i})_{i\in d}$ and
$(T_{i})_{i\in d}\upharpoonright n+m=(T^{m}_{i})_{i\in d}\upharpoonright n+m$
for all $m\in\omega$. $(T_{i})_{i\in d}$ satisfies the conclusions of Lemma
$9$, in our case. Suppose not. Let $(X_{i})_{i\in
d}\in\mathcal{F}(\mathcal{G}_{1})\upharpoonright(T_{i})_{i\in d}$ and
$(Y_{i})_{i\in d}\in\mathcal{F}(\mathcal{G}_{2})\upharpoonright(T_{i})_{i\in
d}$ with $((X_{i})_{i\in d})^{in}=((Y_{i})_{i\in
d})^{in}=(Z^{\prime}_{i})_{i\in d}\in A_{m^{\prime}}$ and
$c^{\prime}_{1}((X_{i})_{i\in d})=c^{\prime}_{2}((Y_{i})_{i\in d})$. But we
have that $c^{\prime}_{1}((X_{i})_{i\in d})\neq c^{\prime}_{2}((Y_{i})_{i\in
d})$ for all $(X_{i})_{i\in
d}\in\mathcal{F}(\mathcal{G}_{1})\upharpoonright(T_{i})_{i\in d}$ and
$(Y_{i})_{i\in d}\in\mathcal{F}(\mathcal{G}_{2})\upharpoonright(T_{i})_{i\in
d}$ with $((X_{i})_{i\in d})^{in}=((Y_{i})_{i\in
d})^{in}=(Z^{\prime}_{i})_{i\in d}\in A_{m^{\prime}}$, a contradiction.
$\bf{Case\,2:}$ If now $L_{(Y_{i})_{i\in d}}\setminus L_{((Y_{i})_{i\in
d})^{in}}=\emptyset$. Pick a finite strong subtree $(Z^{1}_{i})_{i\in d}$ of
$(T^{0}_{i})_{i\in d}$ with height $n$. Let $(X_{i})_{i\in
d}\in\mathcal{F}(\mathcal{G}_{1})$, $(X_{i})_{i\in
d}\in\mathcal{C}^{(T^{0}_{i})_{i\in d}}_{(N^{1}_{i},L^{1}_{i})_{i\in
d}}\subset\mathcal{F}(\mathcal{G}_{1})$ with $((X_{i})_{i\in
d})^{in}=(Z^{1}_{i})_{i\in d}$, and consider the level set
$L^{X}=L_{(X_{i})_{i\in d}}\setminus L_{((X_{i})_{i\in d})^{in}}\subset
L_{(T^{0}_{i}[Z^{1}_{i}])_{i\in d}}$. If there exists $(Y_{i})_{i\in d}$ such
that $((Y_{i})_{i\in d})^{in}=(Y_{i})_{i\in d}=(Z^{1}_{i})_{i\in d}$ and
$c^{\prime}_{1}((X_{i})_{i\in d})=c^{\prime}_{2}((Y_{i})_{i\in d})$, then
subtract a level $l$ from $L_{(T^{0}_{i}[Z^{1}_{i}])_{i\in d}}$ where $l\in
L^{X}$. Let $(T^{1}_{i})_{i\in d}\sqsupseteq(Z^{1}_{i})_{i\in d}$ be a strong
subtree of $(T^{0}_{i}[Z^{1}_{i}])_{i\in d}$ with level set equal to
$L_{(T^{0}_{i}[Z^{1}_{i}])_{i\in d}}\setminus\\{l\\}$. Then
$\mathcal{C}^{(T^{1}_{i})_{i\in d}}_{(N^{1}_{i},L^{1}_{i})_{i\in
d}}=\emptyset$. Set $(T_{i})_{i\in d}\upharpoonright n=(T^{1}_{i})_{i\in
d}\upharpoonright n=(Z^{1}_{i})_{i\in d}$. Suppose we have constructed
$(T_{i})_{i\in d}\upharpoonright m=(T^{m}_{i})_{i\in d}\upharpoonright
m=(Z^{m}_{i})_{i\in d}$, $m>n$, and we have to decide $(T_{i})_{i\in
d}\upharpoonright m+1=(T^{m+1}_{i})_{i\in d}\upharpoonright m+1$. Let
$(Z^{m+1}_{i})_{i\in d}\sqsupset(Z^{m}_{i})_{i\in d}$ and $(Z^{m+1}_{i})_{i\in
d}\in\mathcal{S}_{m+1}((T^{m}_{i})_{i\in d})$. Let
$A=\\{\,(Z^{\prime}_{i})_{i\in d}\in\mathcal{S}_{n}((Z^{m+1}_{i})_{i\in
d})\,\\}$. For each $(Z^{\prime}_{i})_{i\in d}\in A$ if there exists
$(Y_{i})_{i\in d}\in\mathcal{F}(\mathcal{G}_{2})$, $(X_{i})_{i\in
d}\in\mathcal{F}(\mathcal{G}_{1})$ so that $((X_{i})_{i\in
d})^{in}=(Y_{i})_{i\in d}=(Z^{\prime}_{i})_{i\in d}$ and
$c^{\prime}_{1}((X_{i})_{i\in d})=c^{\prime}_{2}((Y_{i})_{i\in d})$, then
subtract a level $l^{\prime}$ from $L_{(T^{m}_{i}[Z^{\prime}_{i}])_{i\in d}}$
where $l^{\prime}\in L^{X}=L_{(X_{i})_{i\in d}}\setminus L_{((X_{i})_{i\in
d})^{in}}$. Repeat this step for every element of $A$, to get
$(T^{m+1}_{i})_{i\in d}\in\mathcal{S}_{\infty}((T^{m}_{i})_{i\in d})$ and
$T^{m+1}_{i}\upharpoonright(m+1)=Z^{m+1}_{i}$ for all $i\in d$. We have that
$c^{\prime}_{1}((X_{i})_{i\in d})\neq c^{\prime}_{2}((Y_{i})_{i\in d})$ for
all $(X_{i})_{i\in
d}\in\mathcal{F}(\mathcal{G}_{1})\upharpoonright(T^{m}_{i})_{i\in d}$,
$(Y_{i})_{i\in
d}\in\mathcal{F}(\mathcal{G}_{2})\upharpoonright(T^{m}_{i})_{i\in d}$ with
$((X_{i})_{i\in d})^{in}=((Y_{i})_{i\in d})=(Z^{\prime}_{i})_{i\in d}\in A$.
Set
$(T_{i})_{i\in d}\upharpoonright(m+1)=(T^{m+1}_{i})_{i\in
d}\upharpoonright(m+1)=(Z^{m+1}_{i})_{i\in d}.$
Let $(T_{i})_{i\in d}$ be such that $(T_{i})_{i\in d}\upharpoonright
n=(T^{1}_{i})_{i\in d}\upharpoonright n=(Z^{1}_{i})_{i\in d}$ and
$(T_{i})_{i\in d}\upharpoonright m=(T^{m}_{i})_{i\in d}\upharpoonright m$ for
all $n<m\in\omega$. We claim that it satisfies the conclusions of our lemma in
this case. Suppose not. Let $(X_{i})_{i\in
d}\in\mathcal{F}(\mathcal{G}_{1})\upharpoonright(T_{i})_{i\in d}$,
$(Y_{i})_{i\in d}\in\mathcal{F}(\mathcal{G}_{2})\upharpoonright(T_{i})_{i\in
d}$ with $((X_{i})_{i\in d})^{in}=(Y_{i})_{i\in
d}=(Z^{\prime\prime}_{i})_{i\in d}\in\\{\,(Z^{\prime}_{i})_{i\in
d}\in\mathcal{S}_{n}((Z^{m^{\prime}}_{i})_{i\in d})\,\\}$ and
$c^{\prime}_{1}((X_{i})_{i\in d})=c^{\prime}_{2}((Y_{i})_{i\in d})$. By
definition we have that $(T_{i})_{i\in d}\upharpoonright
m^{\prime}=(T^{m^{\prime}}_{i})_{i\in d}\upharpoonright
m^{\prime}=(Z^{m^{\prime}}_{i})_{i\in d}$. For all $(X_{i})_{i\in
d}\in\mathcal{F}(\mathcal{G}_{1})\upharpoonright(T_{i})_{i\in d}$,
$(Y_{i})_{i\in d}\in\mathcal{F}(\mathcal{G}_{2})\upharpoonright(T_{i})_{i\in
d}$ with $((X_{i})_{i\in d})^{in}=((Y_{i})_{i\in
d})=(Z^{\prime\prime}_{i})_{i\in d}\in\\{\,(Z^{\prime}_{i})_{i\in
d}\in\mathcal{S}_{n}((Z^{m^{\prime}}_{i})_{i\in d})\,\\}$, it holds that
$c^{\prime}_{1}((X_{i})_{i\in d})\neq c^{\prime}_{2}((Y_{i})_{i\in d})$, a
contradiction.
$\bf{Case\,3:}$ If $L_{(X_{i})_{i\in d}}\setminus L_{((X_{i})_{i\in
d})^{in}}=\emptyset$ and $L_{(Y_{i})_{i\in d}}\setminus L_{((Y_{i})_{i\in
d})^{in}}=\emptyset$. In the beginning of our lemma we have assumed that
$L_{(X_{i})_{i\in d}}\neq L_{(Y_{i})_{i\in d}}$. The assumption of our case
implies that $((X_{i})_{i\in d})^{in}=(X_{i})_{i\in d},((Y_{i})_{i\in
d})^{in}=(Y_{i})_{i\in d}$. We cannot have $((X_{i})_{i\in
d})^{in}=((Y_{i})_{i\in d})^{in}$ because it implies that $(X_{i})_{i\in
d}=(Y_{i})_{i\in d}$ contradicting
$\mathcal{F}(\mathcal{G}_{1})\upharpoonright(T^{0}_{i})_{i\in
d}\subseteq\mathcal{S}_{1,2}$ and the assumption that $L_{(X_{i})_{i\in
d}}\neq L_{(Y_{i})_{i\in d}}$.
Suppose now that $\alpha$ and $\beta$ are arbitrary and assume that our lemma
holds for any $\gamma$-uniform and $\delta$-uniform families, where
$\gamma<\alpha$, $\delta<\beta$. Pick a $t=(t_{i})_{i\in d}\in\prod_{i\in
d}U_{i}(n)$. Apply the above assumption on the uniform families
$\mathcal{F}(\mathcal{G}_{1})(t)$ and $\mathcal{F}(\mathcal{G}_{2})(t)$ to get
strong subtrees $(T^{0}_{p})_{p\in d\cdot b}$ that satisfy the following
property: for $(X^{t}_{j})_{j\in d\cdot
b}\in\mathcal{F}(\mathcal{G}_{1})(t)\upharpoonright(T^{0}_{p})_{p\in d\cdot
b}$ and $(Y^{t}_{j})_{j\in d\cdot
b}\in\mathcal{F}(\mathcal{G}_{2})(t)\upharpoonright(T^{0}_{p})_{p\in d\cdot
b}$ with $(t^{\frown}(X^{t}_{j})_{j\in d\cdot
b})^{in}=(t^{\frown}(Y^{t}_{j})_{j\in d\cdot b})^{in}$ we have
$c^{\prime}_{1}(t^{\frown}(X^{t}_{j})_{j\in d\cdot b})\neq
c^{\prime}_{2}(t^{\frown}(Y^{t}_{j})_{j\in d\cdot b})$. Let
$(T^{1}_{i})_{i\in d}=t^{\frown}(T^{0}_{p})_{p\in d\cdot b}\text{ and
}(T_{i})_{i\in d}\upharpoonright 1=(T^{1}_{i})_{i\in d}\upharpoonright 1=t.$
Suppose we have constructed $(T_{i})_{i\in d}\upharpoonright
n=(T^{n}_{i})_{i\in d}\upharpoonright n$ and we have to decide $(T_{i})_{i\in
d}\upharpoonright(n+1)$. Let $\\{r_{0},\dots,r_{(d\cdot b^{n-1})-1}\\}$ be the
lexicographically increasing enumeration of the set $\bigcup_{i\in
d}T^{n}(n-1)$. Let $r=(r_{k_{i}})_{i\in d}$ be so that $r_{k_{i}}\in
T^{n}_{i}$ for all $i\in d$. Apply once more our assumption to the uniform
families $\mathcal{F}(\mathcal{G}_{1})(r)\upharpoonright(T^{n}_{i})_{i\in d}$
and $\mathcal{F}(\mathcal{G}_{2})(r)\upharpoonright(T^{n}_{i})_{i\in d}$.
After considering all possible such a $r$ we get strong subtrees $(T^{\prime
n+1}_{i})_{i\in d\cdot b^{n}}$. Let $(T^{n+1}_{i})_{i\in d}=((T^{n}_{i})_{i\in
d}\upharpoonright n)^{\frown}(T^{\prime n+1}_{i})_{i\in d\cdot b^{n}}$. Set
$(T_{i})_{i\in d}\upharpoonright(n+1)=(T^{n+1}_{i})_{i\in
d}\upharpoonright(n+1)$.
Let $(T_{i})_{i\in d}$ be such that $(T_{i})_{i\in d}\upharpoonright
n=(T^{n}_{i})_{i\in d}\upharpoonright n$ for all $n\in\omega$. We claim that
it satisfies the conclusion of our lemma. Suppose not. Let $(X_{i})_{i\in
d}\in\mathcal{F}(\mathcal{G}_{1})\upharpoonright(T_{i})_{i\in d}$,
$(Y_{i})_{i\in d}\in\mathcal{F}(\mathcal{G}_{2})\upharpoonright(T_{i})_{i\in
d}$ with $((X_{i})_{i\in d})^{in}=((Y_{i})_{i\in
d})^{in}=(Z^{\prime}_{i})_{i\in d}$ and $c^{\prime}_{1}((X_{i})_{i\in
d})=c^{\prime}_{2}((Y_{i})_{i\in d})$. Let $t=(t_{0},\dots,t_{d-1})$ be the
common root of $(X_{i})_{i\in d}$ and $(Y_{i})_{i\in d}$. By the definition of
$(T_{i})_{i\in d}$, for $(X^{t}_{j})_{j\in d\cdot
b}\in\mathcal{F}(\mathcal{G}_{1})(t)\upharpoonright(T_{i})_{i\in d}$ and
$(Y^{t}_{j})_{j\in d\cdot
b}\in\mathcal{F}(\mathcal{G}_{2})(t)\upharpoonright(T_{i})_{i\in d}$ with
$(t^{\frown}(X^{t}_{j})_{j\in d\cdot b})^{in}=(t^{\frown}(Y^{t}_{j})_{j\in
d\cdot b})^{in}$ we have that $c^{\prime}_{1}(t^{\frown}(X^{t}_{j})_{j\in
d\cdot b})\neq c^{\prime}_{2}(t^{\frown}(Y^{t}_{j})_{j\in d\cdot b})$, a
contradiction.
∎
Now we return to the proof of Lemma $8$. The idea is to use the above lemma to
construct mappings $\lambda_{1},\lambda_{2}$ that have the following property:
for every $(X_{i})_{i\in
d}\in\mathcal{F}(\mathcal{G}_{1})\upharpoonright(T^{0}_{i})_{i\in d}$, if
there exists $(Y_{i})_{i\in
d}\in\mathcal{F}(\mathcal{G}_{2})\upharpoonright(T^{0}_{i})_{i\in d}$ with
$c^{\prime}_{1}((X_{i})_{i\in d})=c^{\prime}_{2}((Y_{i})_{i\in d})$ then we
would like to pick appropriately a $y\in((Y_{i})_{i\in d})^{in}$ so that
$y\notin\bigcup_{i}X_{i}$ and $y$ is a node of any element of
$\mathcal{C}^{(T_{i})_{i\in d}}_{(N^{2}_{i},L^{2}_{i})_{i\in d}}$. Then set
$\lambda_{1}((X_{i})_{i\in d})=y$. By an application of Lemma $7$ we eliminate
the possibility of the strong subtree envelope $\mathcal{C}^{(T^{0}_{i})_{i\in
d}}_{(N^{2}_{i},L^{2}_{i})_{i\in d}}$ to occur on the resulting infinite
strong subtree $(T_{i})_{i\in d}$. In other words $\mathcal{C}^{(T_{i})_{i\in
d}}_{(N^{2}_{i},L^{2}_{i})_{i\in d}}=\emptyset$.
Let $(X_{i})_{i\in
d}\in\mathcal{F}(\mathcal{G}_{1})\upharpoonright(T^{0}_{i})_{i\in d}$ and
$(Y_{i})_{i\in
d}\in\mathcal{F}(\mathcal{G}_{2})\upharpoonright(T^{0}_{i})_{i\in d}$ such
that
$c^{\prime}_{1}((X_{i})_{i\in d})=c^{\prime}_{2}((Y_{i})_{i\in d})$
If $(X_{i})_{i\in d}\in\mathcal{C}^{(T^{0}_{i})_{i\in d}}_{L_{1}}$ and
$(Y_{i})_{i\in d}\in\mathcal{C}^{(T^{0}_{i})_{i\in d}}_{L_{2}}$, where
$L_{1}=\cup_{i\in d}L^{1}_{i}$ and $L_{2}=\cup_{i\in d}L^{2}_{i}$, then
$L_{1}\neq L_{2}$. If $L_{1}=L_{2}$ we will have $(X^{\prime}_{i})_{i\in
d}\in\mathcal{C}^{(T^{0}_{i})_{i\in d}}_{L_{1}}$ and $(X^{\prime}_{i})_{i\in
d}\in\mathcal{C}^{(T^{0}_{i})_{i\in d}}_{L_{2}}$ such that
$c^{\prime}_{1}((X^{\prime}_{i})_{i\in
d})=c^{\prime}_{2}((X^{\prime}_{i})_{i\in d})$ contradicting that
$\mathcal{F}(\mathcal{G}_{1})\upharpoonright(T^{0}_{i})_{i\in
d}\subseteq\mathcal{S}_{1,2}$. Assume that $L_{1}$ is not a proper initial
segment of $L_{2}$, or vice versa. If now $L_{1}\neq L_{2}$ then
$\mathcal{C}^{(T^{0}_{i})_{i\in d}}_{L_{1}}\neq\mathcal{C}^{(T^{0}_{i})_{i\in
d}}_{L_{2}}$. Let $l=\min\\{(L_{2}\setminus L_{1})\cup(L_{1}\setminus
L_{2})\\}$ and assume that $l\in L_{2}\setminus L_{1}$. Then for every
$(X^{\prime}_{i})_{i\in d}\in\mathcal{C}^{(T^{0}_{i})_{i\in d}}_{L_{1}}$ ,
pick
$y\in
D=\\{\cup_{i}Y^{\prime}_{i}(k):k\in|Y^{\prime}_{i}|,(Y^{\prime}_{i})_{i\in
d}\in\mathcal{C}^{(T^{0}_{i})_{i\in d}}_{L_{2}}\\}=\cup_{i\in d}T^{0}_{i}(l)$
Set $\lambda_{1}((X^{\prime}_{i})_{i\in d})=y$. All the members of $D$ are in
the image of $\mathcal{C}^{(T^{0}_{i})_{i\in d}}_{L_{1}}$ under $\lambda_{1}$.
Then by an application of Lemma $7$ we get a strong subtree $(T_{i})_{i\in d}$
of $(T^{0}_{i})_{i\in d}$ so that $\mathcal{C}^{(T_{i})_{i\in
d}}_{L_{2}}=\emptyset$. The possibility of $L_{1}\sqsubseteq L_{2}$, or vice
versa, is eliminated by the following lemma.
###### Lemma 10.
By passing to a strong subtree, if necessary, we can assume that on
$(T^{0}_{i})_{i\in d}$ there are not two strong subtrees $(X_{i})_{i\in
d}\in\mathcal{C}^{(T^{0}_{i})_{i\in d}}_{L_{1}}$, $(Y_{i})_{i\in
d}\in\mathcal{C}^{(T^{0}_{i})_{i\in d}}_{L_{2}}$, where $L_{1}=\bigcup_{i\in
d}L^{1}_{i}$, $L_{2}=\bigcup_{i\in d}L^{2}_{i}$, such that $L_{1}\sqsubseteq
L_{2}$ and $c^{\prime}_{1}((X_{i})_{i\in d})=c^{\prime}_{2}((Y_{i})_{i\in
d})$.
###### Proof.
We prove the lemma by induction on the countable ordinals $\alpha$, $\beta$
the ranks of $\mathcal{F}(\mathcal{G}_{1})$ and $\mathcal{F}(\mathcal{G}_{2})$
respectively. If both are finite then for every $t\in\prod_{i\in
d}T^{0}_{i}(n)$, $n\in\omega$, $\mathcal{T}^{t}_{1}$ contains only level sets
of a fixed cardinality equal to $\alpha-1$ and $\mathcal{T}^{t}_{2}$ contains
also level sets of fixed cardinality $\beta-1$. Suppose that
$\alpha-1<\beta-1$. Notice that the case of $\alpha=\beta$ is not possible in
the above context, since we have assumed that
$\mathcal{F}(\mathcal{G}_{1})\upharpoonright(T^{0}_{i})_{i\in
d}\subseteq\mathcal{S}_{1,2}$. Let $t\in\prod_{i\in d}T^{0}_{i}(n)$, for some
$n\in\omega$. Pick $(X_{i})_{i\in d}\in\mathcal{F}(\mathcal{G}_{1})$ with
$(X_{i}(0))_{i\in d}=t$. If there is a $(Y_{i})_{i\in
d}\in\mathcal{F}(\mathcal{G}_{2})$, $(Y_{i})_{i\in
d}\in\mathcal{C}^{(T^{0}_{i})_{i\in d}}_{(N^{2}_{i},L^{2}_{i})_{i\in d}}$ such
that $(X_{i})_{i\in d}\sqsubseteq(Y_{i})_{i\in d}$ and
$c^{\prime}_{1}((X_{i})_{i\in d})=c^{\prime}_{2}((Y_{i})_{i\in d})$, then
subtract a level $l$ from the level set of $(T^{0}_{i})_{i\in d}$ that is in
the level set of $(Y_{i})_{i\in d}$ as well, so that $L_{(X_{i})_{i\in d}}<l$.
Let $(T^{\prime 0}_{i})_{i\in d}$ a resulting strong subtree of
$(T^{0}_{i})_{i\in d}$ with $(X_{i})_{i\in d}\sqsubseteq(T^{\prime
0}_{i})_{i\in d}$. Then $\mathcal{C}^{(T^{\prime 0}_{i})_{i\in
d}}_{(N^{2}_{i},L^{2}_{i})_{i\in d}}=\emptyset$. Set
$(T_{i})_{i\in d}\upharpoonright\alpha=(T^{\prime 0}_{i})_{i\in
d}\upharpoonright\alpha=(X_{i})_{i\in d}.$
Suppose we have constructed $(T_{i})_{i\in
d}\upharpoonright(\alpha+n)=(T^{\prime n}_{i})_{i\in
d}\upharpoonright(\alpha+n)=(X^{n}_{i})_{i\in d}$ and we have to decide
$(T_{i})_{i\in d}\upharpoonright(\alpha+n+1)$. Let $(X^{n+1}_{i})_{i\in
d}\sqsupset(X^{n}_{i})_{i\in d}$, where $(X^{n+1}_{i})_{i\in
d}\in\mathcal{S}_{\alpha+n+1}((T^{\prime n}_{i})_{i\in d})$. Let
$B_{n+1}=\\{(X^{\prime}_{i})_{i\in
d}\in\mathcal{S}_{\alpha}((X^{n+1}_{i})_{i\in d})\\}$. For every element
$(X^{\prime}_{i})_{i\in d}\in B_{n+1}$, if there exists $(Y_{i})_{i\in
d}\in\mathcal{F}(\mathcal{G}_{2})$, so that $(X^{\prime}_{i})_{i\in
d}\sqsubseteq(Y_{i})_{i\in d}$ and $c^{\prime}_{1}((X^{\prime}_{i})_{i\in
d})=c^{\prime}_{2}((Y_{i})_{i\in d})$ then subtract a level $l$ from the level
set of $(T^{\prime n}_{i})_{i\in d}$ that is in $L_{(Y_{i})_{i\in d}}$ as
well, so that $L_{(X^{\prime}_{i})_{i\in d}}<l$. Having done that for all
elements of $B_{n+1}$ we get $(T^{\prime n+1}_{i})_{i\in
d}\in\mathcal{S}_{\infty}((T^{\prime n}_{i})_{i\in d})$ so that
$(X^{n+1}_{i})_{i\in d}\sqsubset(T^{\prime n+1}_{i})_{i\in d}$. This strong
subtree $(T^{\prime n+1}_{i})_{i\in d}$ has the property that for any element
$(Y_{i})_{i\in d}$ of $\mathcal{F}(\mathcal{G}_{2})\upharpoonright(T^{\prime
n+1}_{i})_{i\in d}$ and $(X^{\prime}_{i})_{i\in d}\in B_{n+1}$, one has that
if $c^{\prime}_{1}((X^{\prime}_{i})_{i\in d})=c^{\prime}_{2}((Y_{i})_{i\in
d})$ then $L_{(X^{\prime}_{i})_{i\in d}}$ is not an initial segment of
$L_{(Y_{i})_{i\in d}}$. Set
$(T^{\prime}_{i})_{i\in d}\upharpoonright(\alpha+n+1)=(T^{\prime
n+1}_{i})_{i\in d}\upharpoonright(\alpha+n+1)=(X^{n+1}_{i})_{i\in d}.$
Let $(T_{i})_{i\in d}$ be such that $(T_{i})_{i\in
d}\upharpoonright\alpha=(T^{\prime 0}_{i})_{i\in d}\upharpoonright\alpha$ and
$(T_{i})_{i\in d}\upharpoonright(\alpha+n)=(T^{\prime n}_{i})_{i\in
d}\upharpoonright(\alpha+n)$ for all $n\in\omega$. We claim that it satisfies
the conclusions of our lemma. Suppose not. Let $(X_{i})_{i\in
d}\in\mathcal{F}(\mathcal{G}_{1})\upharpoonright(T_{i})_{i\in d}$,
$(Y_{i})_{i\in d}\in\mathcal{F}(\mathcal{G}_{2})\upharpoonright(T_{i})_{i\in
d}$ with $(X_{i})_{i\in d}\sqsubseteq(Y_{i})_{i\in d}$ and
$c^{\prime}_{1}((X_{i})_{i\in d})=c^{\prime}_{2}((Y_{i})_{i\in d})$. Then
$(X_{i})_{i\in d}\in B_{n^{\prime}}$ for some $n^{\prime}\in\omega$. This
implies that $L_{(X_{i})_{i\in d}}$ is not an initial segment of
$L_{(Y_{i})_{i\in d}}$, a contradiction.
Consider arbitrary countable ordinals $\alpha$ and $\beta$ and assume that our
lemma holds for every $\delta<\alpha$ and $\gamma<\beta$ uniform families.
Pick once more $t=(t_{0},\dots,t_{d-1})\in\prod_{i\in d}T^{0}_{i}(n)$. By
definition $\mathcal{F}(\mathcal{G}_{1})(t)$ and
$\mathcal{F}(\mathcal{G}_{2})(t)$ are of ranks $\delta$ and $\gamma$ so the
inductive hypothesis gives us $(T^{1}_{i})_{i\in d\cdot b}$ strong subtree of
$(T^{0}_{i})_{i\in d}(t)$ that satisfies the following property: For
$(X_{i})_{i\in d\cdot
b}\in\mathcal{F}(\mathcal{G}_{1})(t)\upharpoonright(T^{1}_{i})_{i\in d\cdot
b}$ and $(Y_{i})_{i\in d\cdot
b}\in\mathcal{F}(\mathcal{G}_{2})(t)\upharpoonright(T^{1}_{i})_{i\in d\cdot
b}$ if we have $c^{\prime}_{1}(t^{\frown}(X_{i})_{i\in d\cdot
b})=c^{\prime}_{2}(t^{\frown}(Y_{i})_{i\in d\cdot b})$ then $L_{(X_{i})_{i\in
d\cdot b}}$ is not an initial segment of $L_{(Y_{i})_{i\in d\cdot b}}$. Set
$(T^{\prime 2}_{i})_{i\in d}=t^{\frown}(T^{1}_{i})_{i\in d\cdot b}$ and
$(T_{i})_{i\in d}\upharpoonright 2=(T^{\prime 2}_{i})_{i\in d}\upharpoonright
2$. Suppose we have constructed $(T_{i})_{i\in d}\upharpoonright n=(T^{\prime
n}_{i})_{i\in d}\upharpoonright n$ and we have to decide $(T_{i})_{i\in
d}\upharpoonright(n+1)$.
Consider the set $H=\\{\bigcup_{i\in d}T^{\prime n}_{i}(n-1)\\}$. For any
$r=(r_{0},\dots,r_{d-1})\subset H$, where $r_{i}\in T^{\prime n}_{i}(n-1)$ for
all $i\in d$, $\mathcal{F}(\mathcal{G}_{1})(r)\upharpoonright(T^{\prime
n}_{i})_{i\in d}(r)$ and
$\mathcal{F}(\mathcal{G}_{2})(r)\upharpoonright(T^{\prime n}_{i})_{i\in d}(r)$
are of ranks $\delta$ and $\gamma$. The inductive hypothesis gives us strong
subtrees $(T^{\prime r}_{i})_{i\in d\cdot b}\in\mathcal{S}_{\infty}((T^{\prime
n}_{i})_{i\in d}(r))$ that satisfy the following: For any $(Z_{i})_{i\in
d\cdot b}\in\mathcal{F}(\mathcal{G}_{1})(r)\upharpoonright(T^{\prime
r}_{i})_{i\in d\cdot b}$ and $(Y_{i})_{i\in d\cdot
b}\in\mathcal{F}(\mathcal{G}_{2})(r)\upharpoonright(T^{\prime r}_{i})_{i\in
d\cdot b}$,
$\text{if }c^{\prime}_{1}(r^{\frown}(Z_{i})_{i\in d\cdot
b})=c^{\prime}_{2}(r^{\frown}(Y_{i})_{i\in d\cdot b})\text{, then }L_{1}\text{
is not an initial segment of }L_{2}$
for $L_{1}$ being the level set of $(Z_{i})_{i\in d\cdot b}$ and $L_{2}$ the
one of $(Y_{i})_{i\in d\cdot b}$. Repeat the above step for any such an $r$ to
get strong subtrees $(T^{\prime\prime}_{i})_{i\in d\cdot b^{n}}$. Set
$(T^{\prime n+1}_{i})_{i\in d}=((T^{\prime n}_{i})_{i\in d}\upharpoonright
n)^{\frown}(T^{\prime\prime}_{i})_{i\in d\cdot b^{n}}\text{ and }(T_{i})_{i\in
d}\upharpoonright(n+1)=(T^{\prime n+1}_{i})_{i\in d}\upharpoonright(n+1).$
Let $(T_{i})_{i\in d}\upharpoonright n=(T^{\prime n}_{i})_{i\in
d}\upharpoonright n$, for all $n\in\omega$. $(T_{i})_{i\in
d}\in\mathcal{S}_{\infty}((T^{0}_{i})_{i\in d})$ satisfies the conclusions of
our lemma. Suppose not. Let $(X_{i})_{i\in
d}\in\mathcal{F}(\mathcal{G}_{1})\upharpoonright(T_{i})_{i\in d}$,
$(Y_{i})_{i\in d}\in\mathcal{F}(\mathcal{G}_{2})\upharpoonright(T_{i})_{i\in
d}$ with $(X_{i})_{i\in d}\sqsubseteq(Y_{i})_{i\in d}$ and
$c^{\prime}_{1}((X_{i})_{i\in d})=c^{\prime}_{2}((Y_{i})_{i\in d})$. Let
$t=(X_{i}(0))_{i\in d}=(Y_{i}(0))_{i\in d}$. On $(T_{i})_{i\in d}(t)$ we have
that if $c^{\prime}_{1}((X_{i})_{i\in d}=t^{\frown}(X^{\prime}_{i})_{i\in
d\cdot b})=c^{\prime}_{2}(t^{\frown}(Y^{\prime}_{i})_{i\in d\cdot
b}=(Y_{i})_{i\in d})$, then $L_{(X^{\prime}_{i})_{i\in d\cdot b}}$ is not an
initial segment of $L_{(Y^{\prime}_{i})_{i\in d\cdot b}}$, a contradiction.
∎
Now we return to the proof of Lemma $8$. Let $(X_{i})_{i\in
d}\in\mathcal{F}(\mathcal{G}_{1})\upharpoonright(T^{0}_{i})_{i\in d}$ and
$(Y_{i})_{i\in
d}\in\mathcal{F}(\mathcal{G}_{2})\upharpoonright(T^{0}_{i})_{i\in d}$, with
$(X_{i})_{i\in d}\in\mathcal{C}^{(T^{0}_{i})_{i\in
d}}_{(N^{1}_{i},L^{1}_{i})_{i\in d}}$ and $(Y_{i})_{i\in
d}\in\mathcal{C}^{(T^{0}_{i})_{i\in d}}_{(N^{2}_{i},L^{2}_{i})_{i\in d}}$. Let
also
$c^{\prime}_{1}((X_{i})_{i\in d})=c^{\prime}_{2}((Y_{i})_{i\in d})$
If both sets $\bigcup_{i\in d}N^{1}_{i}$ and $\bigcup_{i\in d}N^{2}_{i}$ are
nonempty there are the following possibilities. Firstly
$\cup_{i\in d}N^{1}_{i}\neq\cup_{i\in d}N^{2}_{i}\text{ and
}L^{1}_{in}\neq\emptyset\text{ or }L^{2}_{in}\neq\emptyset$
If there exists either $y\in\bigcup_{i\in d}N^{2}_{i}$ so that
$y\notin((X^{\prime}_{i})_{i\in d})$, for a $(X^{\prime}_{i})_{i\in
d}\in\mathcal{C}^{(T^{0}_{i})_{i\in d}}_{(N_{i}^{1},L^{1}_{i})_{i}}$, or
$x\in\bigcup_{i\in d}N^{1}_{i}$ so that $x\notin((Y^{\prime}_{i})_{i\in d})$,
for a $(Y^{\prime}_{i})_{i\in d}\in\mathcal{C}^{(T^{0}_{i})_{i\in
d}}_{(N^{2}_{i},L^{2}_{i})_{i}}$, then set
$\lambda_{1}((X^{\prime}_{i})_{i\in d})=y\text{ or
}\lambda_{2}((Y^{\prime}_{i})_{i\in d})=x$
respectively. If no such an $y$ or $x$ are possible to be found and since by
Lemma $9$ we have that $((X_{i})_{i\in d})^{in}\neq((Y_{i})_{i\in d})^{in}$,
we conclude that $L_{in}^{1}\neq L_{in}^{2}$.
Suppose that $L_{in}^{1}\neq L_{in}^{2}$. In this case let
$l=\min\\{(L^{1}_{in}\setminus L^{2}_{in})\cup(L^{2}_{in}\setminus
L^{1}_{in})\\}$
Suppose that $l\in L^{1}_{in}$. Identical argument holds if $l\in L^{2}_{in}$.
Consider the set
$D=\\{x\in\cup_{i\in d}T^{0}_{i}(l):(\exists x^{\prime}\in\cup_{i\in
d}N^{1}_{i})x\leq x^{\prime}\\}$
Then for every $(Y^{\prime}_{i})_{i\in d}\in\mathcal{C}^{(T^{0}_{i})_{i\in
d}}_{(N^{2}_{i},L^{2}_{i})_{i\in d}}$ pick an $x\in D$ and set
$\lambda_{2}((Y^{\prime}_{i})_{i\in d})=x$. Notice that every element of $D$
is a node of any strong subtree of the strong subtree envelope
$\mathcal{C}^{(T^{0}_{i})_{i\in d}}_{(N^{2}_{i},L^{2}_{i})_{i\in d}}$.
Identical argument applies in the case of $\bigcup_{i\in
d}N^{1}_{i}=\bigcup_{i\in d}N^{2}_{i}$. In this case by Lemma $9$ we must have
$L_{in}^{1}\neq L_{in}^{2}$.
Consider the case that $\bigcup_{i\in d}N^{1}_{i}\neq\bigcup_{i\in
d}N^{2}_{i}$ and $L^{1}_{in}=L^{2}_{in}=\emptyset$. $((X_{i})_{i\in
d})^{in}\neq((Y_{i})_{i\in d})^{in}$ implies that there exists either
$y\in\bigcup_{i\in d}N^{2}_{i}$ so that $y\notin(X^{\prime}_{i})_{i\in d}$, or
$x\in\cup_{i\in d}N^{1}_{i}$ so that $x\notin(Y^{\prime}_{i})_{i\in d}$, for
$(X^{\prime}_{i})_{i\in d}$ a member of $\mathcal{C}^{(T^{0}_{i})_{i\in
d}}_{(N_{i},L_{i})_{i\in d}}$ and $(Y^{\prime}_{i})_{i\in d}$ a member of
$\mathcal{C}^{(T^{0}_{i})_{i\in d}}_{(N_{i},L_{i})_{i\in d}}$. We set
$\lambda_{1}((X^{\prime}_{i})_{i\in d})=y$ or
$\lambda_{2}((Y^{\prime}_{i})_{i\in d})=x$. If not such an $x$ or $y$ is
possible to be fund, then $\bigcup_{i\in d}N^{1}_{i}$ is in any strong subtree
of $\mathcal{C}^{(T^{0}_{i})_{i\in d}}_{(N^{2}_{i},L^{2}_{i})_{i}}$ and
$\bigcup_{i\in d}N^{2}_{i}$ is in any strong subtree of
$\mathcal{C}^{(T^{0}_{i})_{i\in d}}_{(N^{1}_{i},L^{1}_{i})_{i}}$. This implies
that $((X_{i})_{i\in d})^{in}=((Y_{i})_{i\in d})^{in}$, a contradiction with
Lemma $9$.
Lastly if $\bigcup_{i\in d}N^{1}_{i}\neq\emptyset$ and $\bigcup_{i\in
d}N^{2}_{i}=\emptyset$. In the case that $((Y_{i})_{i\in d})^{in}$ is not
defined, there will be an $x\in\bigcup_{i\in d}N^{1}_{i}$ so that
$x\notin(Y^{\prime}_{i})_{i\in d}$ for some $(Y^{\prime}_{i})_{i\in
d}\in\mathcal{C}^{(T^{0}_{i})_{i\in d}}_{\cup_{i\in d}L^{2}_{i}}$. To see this
notice that the strong subtree envelope $\mathcal{C}^{(T^{0}_{i})_{i\in
d}}_{\cup_{i\in d}L^{2}_{i}}$ is taken over the level set $\cup_{i\in
d}L^{2}_{i}$. As a result we can choose a $(Y^{\prime}_{i})_{i\in
d}\in\mathcal{C}^{(T^{0}_{i})_{i\in d}}_{\cup_{i\in d}L^{2}_{i}}$ so that
$x\notin\cup_{i\in d}Y^{\prime}_{i}$. Set $\lambda_{2}((Y^{\prime}_{i})_{i\in
d})=x$. If now $((Y_{i})_{i\in d})^{in}$ is defined, since $((X_{i})_{i\in
d})^{in}\neq((Y_{i})_{i\in d})^{in}$, if we cannot choose such an $x$, then we
will be able to choose $y\in((Y_{i})_{i\in d})^{in}$ and set
$\lambda_{1}((X_{i})_{i\in d})=y$.
The above show that we can construct mappings $\lambda_{1},\lambda_{2}$ such
that by two consecutive applications of Lemma $7$ we get $(T_{i})_{i\in d}\in
S_{\infty}((U_{i})_{i\in d})$ that
$\lambda_{j}(\mathcal{F}(\mathcal{G}_{j})\upharpoonright(T_{i})_{i\in
d}))\cap(T_{i})_{i\in d}=\emptyset\text{, }j\in\\{1,2\\}$
Suppose that $c_{1}((X_{i})_{i\in d})=c_{2}((Y_{i})_{i\in d})$ for some
$(X_{i})_{i\in d}\in\mathcal{F}(\mathcal{G}_{1})\upharpoonright(T_{i})_{i\in
d}$ and a $(Y_{i})_{i\in
d}\in\mathcal{F}(\mathcal{G}_{2})\upharpoonright(T_{i})_{i\in d}$, where
$(X_{i})_{i\in d}\in\mathcal{C}^{(T_{i})_{i\in
d}}_{(N^{1}_{i},L^{1}_{i})_{i\in d}}$ and $(Y_{i})_{i\in
d}\in\mathcal{C}^{(T_{i})_{i\in d}}_{(N^{2}_{i},L^{2}_{i})_{i\in d}}$.
This contradicts the way that $\lambda_{1},\lambda_{2}$ are defined. We must
have either $\mathcal{C}^{(T_{i})_{i\in
d}}_{(N^{1}_{i},L^{1}_{i})_{i}}=\emptyset$ or $\mathcal{C}^{(T_{i})_{i\in
d}}_{(N^{2}_{i},L^{2}_{i})_{i}}=\emptyset$. Therefore $(T_{i})_{i\in d}$
satisfies the second alternative of our lemma. ∎
We make the following observation:
###### Lemma 11.
Under the assumptions of Lemma $8$, if $\mathcal{F}(\mathcal{G}_{1})$ is an
$\alpha$-uniform family, $\mathcal{F}(\mathcal{G}_{2})$ is a $\beta$-uniform,
with $\alpha\neq\beta$,then the first statement of the lemma is excluded.
###### Proof.
It is an easy inductive argument that if $\mathcal{G}$ is an $\alpha$-uniform
cannot be $\beta$-uniform, for any $\beta\neq\alpha$. ∎
Finally we are able do the inductive step of Theorem $7$ for any
$\alpha$-uniform family on $U$.
### 6.1. Inductive step
Let $\mathcal{G}$ be an $\alpha$-uniform family of finite strong subtrees of
$U$. For any $t\in U$, $\mathcal{G}(t)$ is a $\beta$-uniform family on $U(t)$
for some $\beta<\alpha$. Therefore by the inductive hypothesis we can assume
that the coloring $c_{t}$ defined on $\mathcal{G}(t)$ by $c_{t}((X_{i})_{i\in
b})=c(t^{\frown}(X_{i})_{i\in b})$, is canonical. As a consequence at each
node $t$ of $U$ we have a uniform family $\mathcal{F}(\mathcal{G})(t)$, that
results by taking the union of the strong subtree envelopes of all members of
$\mathcal{T}^{t}$, together with $f_{t}$ and a one-to-one mapping $\phi_{t}$
that witness the coloring $c_{t}$ being canonical on $U(t)$. As we have
mentioned above $c_{t}$ is defined on $\mathcal{G}(t)$ by $c_{t}((X_{i})_{i\in
b})=\phi_{t}(f_{t}((X_{i})_{i\in b})=(N_{i},L_{i})_{i\in b})$ where
$(X_{i})_{i\in b}\in\mathcal{G}(t)$, $(N_{i},L_{i})_{i\in
b}\in\mathcal{T}^{t}$ and $\mathcal{C}^{U(t)}_{(N_{i},L_{i})_{i\in
d}}\subset\mathcal{F}(\mathcal{G})(t)$.
We will construct the strong subtree $T$ that satisfies the conclusion of the
Theorem $7$, by applying continuously Lemma $8$. Pick a node $r\in U$ and set
$T(0)=r$. Let $(r^{\frown}i)_{i\in b}$ be the set of the immediate successors
of $r$ in $U$ and let $T^{2}=U[r]$. Set $T(1)=(r^{\frown}i)_{i\in b}$.
Equivalently $T\upharpoonright 2=T^{2}\upharpoonright 2$. Suppose we have
constructed $T\upharpoonright n=T^{n}\upharpoonright n$ and we have to decide
$T\upharpoonright(n+1)$.
Let $T^{n}(n-1)=(r_{p})_{p\in b^{n-1}}$. Consider the uniform families
$\mathcal{F}(\mathcal{G})(r_{p})$ on $T^{n}(r_{p})$, for all $p\in b^{n-1}$.
For any pair $\mathcal{F}(\mathcal{G})(r_{i})$, on $T^{n}(r_{i})$ and
$\mathcal{F}(\mathcal{G})(r_{j})$ on $T^{n}(r_{j})$, $i,j\in b^{n-1}$, apply
Lemma $8$ up to translation. Having done that for all possible such pairs, we
get strong subtrees $(T^{\prime 1}_{m})_{m\in
b^{n}}\in\mathcal{S}_{\infty}((T^{n}(r_{p}))_{p\in b^{n-1}})$ that satisfy
either the first or the second alternative of Lemma $8$. Consider the uniform
families $\mathcal{F}(\mathcal{G})(r_{0})\upharpoonright(T^{\prime
1}_{m})_{m\in b^{n}}$ and
$\mathcal{F}(\mathcal{G})(s)\upharpoonright(T^{\prime 1}_{m})_{m\in b^{n}}$,
for $s\in T^{n}(n^{\prime})$, $n^{\prime}<n-1$. There exists a
$k=b^{n-1-n^{\prime}}$ and $l\in\omega$, so that $\\{(r_{p})_{p\in[l\cdot
b,(l\cdot b)+k)}\\}=T^{n}(n-1)\cap T^{n}(s)$. In other words $s^{\frown}i$ has
$k/b$ many successors on $T^{n}(n-1)$. These successors are precisely:
$(r_{p})_{p\in[(l\cdot b)+(i\cdot k/b),(l\cdot b)+(i\cdot k/b)+k/b)}$. As a
result $(T^{\prime 1}_{m})_{m\in[l\cdot b^{2},(l\cdot b^{2})+k\cdot
b)}\in\mathcal{S}_{\infty}(T^{n}(s))$.
Apply Lemma $8$ on the uniform family
$\mathcal{F}(\mathcal{G})(r_{0})\upharpoonright(T^{\prime 1}_{m})_{m\in[0,b)}$
and the family $\pi_{m}(\mathcal{F}(\mathcal{G})(s)\upharpoonright T^{\prime
1}_{(l\cdot b^{2})+(m\cdot k)})$ translated on $T^{\prime 1}_{m}$, for all
$m\in b$. If we have the first alternative of Lemma $8$ holding, we proceed to
the node $r_{1}$. Otherwise we consider the uniform families
$\mathcal{F}(\mathcal{G})(r_{0})\upharpoonright(T^{\prime 1}_{m})_{m\in[0,b)}$
and $\pi_{m}(\mathcal{F}(\mathcal{G})(s)\upharpoonright(T^{\prime 1}_{(l\cdot
b^{2}+1)+(m\cdot k)})$ translated on $T^{\prime 1}_{m}$, for all $m\in b$.
Once again if we get the first statement of Lemma $8$, we proceed to the node
$r_{1}$, otherwise we apply again Lemma $8$ to the uniform families
$\mathcal{F}(\mathcal{G})(r_{0})\upharpoonright(T^{\prime 1}_{m})_{m\in b}$
and $\pi_{m}(\mathcal{F}(\mathcal{G})(s)\upharpoonright T^{\prime 1}_{(l\cdot
b^{2}+2)+(m\cdot k)}$ translated on $T^{\prime 1}_{m}$, for all $m\in b$, etc.
Having done that for the finite set of all possible pairs of nodes $r_{p}$ and
$s$, we get strong subtrees $(T^{\prime n}_{m})_{m\in
b^{n}}\in\mathcal{S}_{\infty}((T^{\prime 1}_{m})_{m\in b^{n}})$ such that for
any two uniform families $\mathcal{F}(\mathcal{G})(r_{p})$ and
$\mathcal{F}(\mathcal{G})(s)$ we have either the first or the second statement
of Lemma $8$ holding.
Suppose that we get always the first statement of Lemma $8$. In this case let
$T^{n+1}=(T^{n}\upharpoonright n)^{\frown}(T^{\prime n}_{m})_{m\in b^{n}}$.
Set $T\upharpoonright n+1=T^{n+1}\upharpoonright n+1$.
If the second statement of Lemma $8$ occurs, we distinguish two cases: first
if it occurs on an application of Lemma $8$ on
$\mathcal{F}(\mathcal{G})(r_{i})$ and $\mathcal{F}(\mathcal{G})(r_{j})$,
$i,j\in b^{n-1}$. This case has no impact on the argument, since
$c(\mathcal{F}(\mathcal{G})(r_{i})\upharpoonright\tilde{T}^{n})\cap
c(\mathcal{F}(\mathcal{G})(r_{j}\upharpoonright\tilde{T}^{n})=\emptyset$,
where $\tilde{T}_{n}=(T^{n}\upharpoonright n)^{\frown}(T^{\prime n}_{m})_{m\in
b^{n}}$.
Secondly if it occurs on an application of Lemma $8$ on the uniform families
$\mathcal{F}(\mathcal{G})(r_{p})$ and $\mathcal{F}(\mathcal{G})(s)$. In this
case we have to reassure that if
$c(\mathcal{F}(\mathcal{G})(r_{p})\upharpoonright\tilde{T}^{n})\cap
c(\mathcal{F}(\mathcal{G})(s)\upharpoonright(T^{\prime 1}_{m})_{m\in[l\cdot
b^{2},(l\cdot b^{2})+k\cdot b)})=\emptyset$, then
$c(\mathcal{F}(\mathcal{G})(r_{p}))\cap
c(\mathcal{F}(\mathcal{G})(s))=\emptyset$ on an infinite strong subtree of
$\tilde{T}^{n}$.
At first notice that there are at most finitely many strong subtrees
$X_{s}=(X^{\prime}_{i})_{i\in b}$ members of
$\mathcal{F}(\mathcal{G})(s)\upharpoonright(T^{\prime 1}_{m})_{m\in[l\cdot
b^{2},(l\cdot b^{2})+k\cdot b)}$ with $L_{X_{s}}<n$. We can eliminate the
possibility of any strong subtree $X_{s}=(X^{\prime}_{i})_{i\in b}$, with
$L_{X_{s}}<n$, that corresponds to the uniform family
$\mathcal{F}(\mathcal{G})(s)$, having the same color with a strong subtree
$X_{r_{p}}=(Y^{\prime}_{i})_{i\in b}\in\mathcal{F}(\mathcal{G})(r_{p})$. We do
that by simply eliminating a level $l$ from the level set
$L_{\tilde{T}^{n}[r_{p}]}$ so that $l\in
L_{(N^{r_{p}}_{i},L^{r_{p}}_{i})_{i}}\cap L_{\tilde{T}^{n}[r_{p}]}$ where
$(Y^{\prime}_{i})_{i\in
b}\in\mathcal{C}^{\tilde{T}^{n}}_{(N^{r_{p}}_{i},L^{r_{p}}_{i})_{i}}$. In any
of the resulting strong subtrees $T^{\prime}$ of $\tilde{T}^{n}[r_{p}]$, with
$L_{T^{\prime}}=L_{\tilde{T}^{n}[r_{p}]}\setminus\\{l\\}$, we have that
$\mathcal{C}^{T^{\prime}}_{(N^{r_{p}}_{i},L^{r_{p}}_{i})_{i}}=\emptyset$. For
notational simplicity we are going to use $X_{s},X_{r_{p}}$ instead of
$(X^{\prime}_{i})_{i\in b}$ and $(Y^{\prime}_{i})_{i\in b}$ respectively.
There may be a strong subtree $X_{s}$ with a level set that contains both
levels smaller than $n$ and bigger as well. In that case we restrict on $Y$
the initial segment of $X_{s}$ with level set that lies below $n$ i.e.
$Y\sqsubset X$ and $L_{Y}<n$. Observe that $\mathcal{F}(\mathcal{G})(s)(Y)$
contains $d>b$ sequences of finite strong subtrees. Notice that $d$ is a
multiple of $b$. In that case we need an extended version of Lemma $8$ as
follows:
###### Lemma 12.
Let $(U_{i})_{i\in d}$, where $d=kb$ is a multiple of $b$, the branching
number of $U_{i}$ for all $i$. Let $\mathcal{T}_{1}$ be a family of node-level
sets on $(U_{i})_{i\in d^{\prime}}$ where $d^{\prime}\subset d$, that
generates an $\beta$-uniform family $\mathcal{F}(\mathcal{G}_{1})$ on
$(U_{i})_{i\in d^{\prime}}$. Let $\mathcal{T}_{2}$ be a family of node-level
sets on $(U_{i})_{i\in d}$, that generates an $\alpha$-uniform family
$\mathcal{F}(\mathcal{G}_{2})$ on $(U_{i})_{i\in d}$, for $\alpha>\beta$. Let
$c^{\prime}_{1}$ a mapping on $\mathcal{F}(\mathcal{G}_{1})$ with the property
that $c^{\prime}_{1}((X^{1}_{i})_{i\in
d^{\prime}})=c^{\prime}_{1}((X^{2}_{i})_{i\in d^{\prime}})$ if and only if
$(X^{1}_{i})_{i\in d^{\prime}}:(N^{1}_{i},L^{1}_{i})_{i\in
d^{\prime}}=(X^{2}_{i})_{i\in d^{\prime}}:(N^{1}_{i},L^{1}_{i})_{i\in
d^{\prime}}$ for $(N^{1}_{i},L^{1}_{i})_{i\in d^{\prime}}\in\mathcal{T}_{1}$.
Let also $c_{2}$ a mapping on $\mathcal{F}(\mathcal{G}_{2})$ such that
$c_{2}((Y^{1}_{i})_{i\in d})=c_{2}((Y^{2}_{i})_{i\in d})$ if and only if
$(Y^{1}_{i})_{i\in d}:(N^{2}_{i},L^{2}_{i})_{i\in d}=(Y^{2}_{i})_{i\in
d}:(N^{2}_{i},L^{2}_{i})_{i\in d}$ for $(N^{2}_{i},L^{2}_{i})_{i\in
d}\in\mathcal{T}_{2}$.
There exists a strong subtree $(T_{i})_{i\in d}$ of $(U_{i})_{i\in d}$ such
that the following holds:
$c^{\prime}_{1}(\mathcal{F}(\mathcal{G}_{1})\upharpoonright(T_{i})_{i\in
d^{\prime}})\cap
c^{\prime}_{2}(\mathcal{F}(\mathcal{G}_{2})\upharpoonright(T_{i})_{i\in
d})=\emptyset.$
###### Proof.
Notice that we can extend $(\mathcal{F}(\mathcal{G}_{1}),c^{\prime}_{1})$ on
$(U_{i})_{i\in d}$ by $c^{\prime}_{1}((X_{i})_{i\in
d})=c^{\prime}_{1}((X_{j})_{j\in d^{\prime}})$ and $X_{i}=X_{j}$ for $j\in
d^{\prime}$. Then apply Lemma $8$ and Lemma $11$.
∎
We can consider now the corresponding uniform families
$\mathcal{F}(\mathcal{G})(s)(Y)$ on $\tilde{T}^{n}(Y)$ and
$\mathcal{F}(\mathcal{G})(r_{p})$ on $\tilde{T}^{n}(r_{p})$. Then apply Lemma
$12$ to get a strong subtree that satisfies its conclusion. Repeating that for
the finite set of all
$X_{s}\in\mathcal{F}(\mathcal{G})(s)\upharpoonright\tilde{T^{n}}$ whose set of
levels intersects $[n,\infty)$, we succeed in getting a strong subtree
$T^{n+1}$ of $\tilde{T}^{n}$ such that
$c(\mathcal{F}(\mathcal{G})(s)\upharpoonright T^{n+1})\cap
c(\mathcal{F}(\mathcal{G})(r_{p})\upharpoonright T^{n+1})=\emptyset$
Set $T\upharpoonright n+1=T^{n+1}\upharpoonright n+1$.
Proceeding in that manner we construct $T\in\mathcal{S}_{\infty}(U)$, where
$T\upharpoonright n=T^{n}\upharpoonright n$, for all $n\in\omega$, such that
for any two nodes $s_{0},s_{1}\in T$, with $|s_{0}|\leq|s_{1}|$, we have one
of the two following alternatives.
1. (1)
There exists $(T^{s_{0}}_{i})_{i\in b}\in\mathcal{S}_{\infty}(T(s_{0}))$ such
that $\mathcal{F}(\mathcal{G})(s_{0})\upharpoonright(T^{s_{0}}_{i})_{i\in
b}=\mathcal{F}(\mathcal{G})(s_{1})$, up to translation. Also for every
$X\in\mathcal{F}(\mathcal{G})(s_{0})\upharpoonright(T^{s_{0}}_{i})_{i\in
b},Y\in\mathcal{F}(\mathcal{G})(s_{1})$, with $Y$ a translate of $X$, it holds
that $c(X)=c(Y)$.
2. (2)
$c(\mathcal{F}(\mathcal{G})(s_{0}))\cap
c(\mathcal{F}(\mathcal{G})(s_{1}))=\emptyset$.
To define precisely the family of node-level sets $\mathcal{T}$ that will
satisfy the conclusions of Theorem $7$ we need the following result.
###### Proposition 2.
Let $\mathcal{T}_{1}$ and $\mathcal{T}_{2}$ be two families of node-level sets
that generate two uniform families $\mathcal{F}(\mathcal{G}_{1})$ and
$\mathcal{F}(\mathcal{G}_{2})$ on $U$ by taking the union of all strong
subtree envelopes of all node-level sets of $\mathcal{T}_{1}$ and
$\mathcal{T}_{2}$ respectively. Let $c_{1}$ a mapping on
$\mathcal{F}(\mathcal{G}_{1})$ with the property that
$c_{1}(X_{1})=c_{1}(X_{2})$ if and only if
$X_{1}:(N_{1},L_{1})=X_{2}:(N_{1},L_{1})$ for
$(N_{1},L_{1})\in\mathcal{T}_{1}$. Let also $c_{2}$ a mapping on
$\mathcal{F}(\mathcal{G}_{2})$ so that $c_{2}(Y_{1})=c_{2}(Y_{2})$ if and only
if $Y_{1}:(N_{2},L_{2})=Y_{2}:(N_{2},L_{2})$ for
$(N_{2},L_{2})\in\mathcal{T}_{2}$. If by an application of Lemma $8$ we get a
$T\in\mathcal{S}_{\infty}(U)$ such that the first alternative holds, then we
have that $\mathcal{T}_{1}\upharpoonright T=\mathcal{T}_{2}\upharpoonright T$.
###### Proof.
The proof is by induction on the rank $\alpha$ of the uniform families
$\mathcal{F}(\mathcal{G}_{1})$ and $\mathcal{F}(\mathcal{G}_{2})$, which is
identical by Lemma $11$. If $\alpha\in\omega$, then by the discussion before
Lemma $6$ we have that for any $(N_{0},L_{0}),(N_{1},L_{1})\in\mathcal{T}_{1}$
and any two members of their strong subtree envelopes
$X_{0}\in\mathcal{C}^{T}_{(N_{0},L_{0})}$ and
$X_{1}\in\mathcal{C}^{T}_{(N_{1},L_{1})}$ one has:
$\iota_{b^{\alpha},X_{0}}\circ\iota^{-1}_{b^{\alpha},X_{1}}(N_{1})=(N_{0})$
and $|L_{0}|=|L_{1}|$. Similarly for $\mathcal{T}_{2}$. Suppose that
$\mathcal{T}_{1}\upharpoonright T\neq\mathcal{T}_{2}\upharpoonright T$. Let
$(N_{1},L_{1})\in\mathcal{T}_{1}\upharpoonright T$, so that if $|N_{1}|>1$
then for every $t,t^{\prime}\in N_{1}$, the absolute value of the difference
$|t|-|t^{\prime}|$ is greater than $1$. For any
$X\in\mathcal{C}^{T}_{(N_{1},L_{1})}$ consider $c_{1}(X)$. Since we have the
first alternative of Lemma $8$ on hold, we must have that $c_{2}(X)=c_{1}(X)$
for $X\in\mathcal{C}^{T}_{(N_{2},L_{2})}$,
$(N_{2},L_{2})\in\mathcal{T}_{2}\upharpoonright T$ as well. That must be true
for all the members of $\mathcal{C}^{T}_{(N_{1},L_{1})}$, which implies that
$\mathcal{C}^{T}_{(N_{1},L_{1})}=\mathcal{C}^{T}_{(N_{2},L_{2})}$. As a result
for $X\in\mathcal{C}^{T}_{(N_{1},L_{1})}$ and
$Y\in\mathcal{C}^{T}_{(N_{2},L_{2})}$ we have that $L_{X}=L_{Y}$. If
$N_{1}=\\{t\\}$, then $N_{2}=\\{t\\}$ as well, otherwise if $N_{2}=\\{s\\}$,
then $X(0)=t\neq s=X(0)$, a contradiction. Suppose that $|N_{1}|>1$ and let
$t\in\\{(N_{1}\setminus N_{2})\cup(N_{2}\setminus N_{1})\\}$ is of minimal
height. Suppose that $t\in N_{1}$. Since $X\in\mathcal{C}^{T}_{(N_{1},L_{1})}$
there exists $n\in|X|$ such that $t\in X(n)$. Choose a
$Y\in\mathcal{C}^{T}_{(N_{2},L_{2})}$ such that $t\notin Y(n)$. Notice that
$Y\notin\mathcal{C}^{T}_{(N_{1},L_{1})}$, a contradiction. If now
$N_{1}=N_{2}=\emptyset$ then we must have $L_{1}=L_{2}$, other wise for every
$X^{\prime}\in\mathcal{C}^{T}_{(N_{1},L_{1})}$ and
$Y^{\prime}\in\mathcal{C}^{T}_{(N_{2},L_{2})}$ we would have that
$L_{X}^{\prime}\neq L_{Y}^{\prime}$ contradicting that
$\mathcal{C}^{T}_{(N_{1},L_{1})}=\mathcal{C}^{T}_{(N_{2},L_{2})}$. Finally if
$N_{2}=\emptyset$ and $N_{1}\neq\emptyset$ pick $t\in N_{1}$ so that for any
other $t^{\prime}\in N_{1}$, we have that $l=|t|\geq|t^{\prime}|$. Pick a
$Y\in\mathcal{C}^{T}_{L_{2}}$ so that $t\notin Y$. This is always possible
since our node-level set $(N_{2},L_{2})$ is only a level set. Then
$Y\notin\mathcal{C}^{T}_{(N_{1},L_{1})}$, a contradiction of
$\mathcal{C}^{T}_{(N_{1},L_{1})}=\mathcal{C}^{T}_{(N_{2},L_{2})}$. As a
consequence $\mathcal{C}^{T}_{(N_{1},L_{1})}=\mathcal{C}^{T}_{(N_{2},L_{2})}$
implies that for any $X^{\prime}\in\mathcal{C}^{T}_{(N_{1},L_{1})}$,
$Y^{\prime}\in\mathcal{C}^{T}_{(N_{2},L_{2})}$ both finite strong subtrees of
height $\alpha<\omega$, we have that
$\iota_{b^{\alpha},X^{\prime}}\circ\iota^{-1}_{b^{\alpha},Y^{\prime}}(N_{2})=(N_{1})$
and $|L_{1}|=|L_{2}|$. Therefore $\mathcal{T}_{1}\upharpoonright
T=\mathcal{T}_{2}\upharpoonright T$.
Assume that the assertion of our proposition holds for $\beta<\alpha$ uniform
families and consider the case of $\alpha\geq\omega$ uniform families
$\mathcal{F}(\mathcal{G}_{1})$ and $\mathcal{F}(\mathcal{G}_{2})$. For any
node $t\in T$, $\mathcal{F}(\mathcal{G}_{1})(t)\upharpoonright T$ and
$\mathcal{F}(\mathcal{G}_{2})(t)\upharpoonright T$ are both uniform families
of rank less than $\alpha$. The inductive hypothesis applies to give us
$\mathcal{T}^{t}_{1}\upharpoonright T=\mathcal{T}^{t}_{2}\upharpoonright T$.
That being true for every $t\in T$ implies that
$\mathcal{T}_{1}\upharpoonright T=\mathcal{T}_{2}\upharpoonright T$. ∎
Now the family of node-level sets $\mathcal{T}$ that will satisfy the
conditions of Definition $16$ is defined as follows: For a node $s_{0}\in T$
if there exists a node $s_{1}\in T$ so that the first alternative of the above
statement holds, then $\mathcal{T}^{s_{0}}\subset\mathcal{T}$. If for all
$s_{1}\in T$ we have the second alternative holding then
$s_{0}\cup\mathcal{T}^{s_{0}}:=\\{(s_{0}\cup
N,L):(N,L)\in\mathcal{T}_{s_{0}}\\}\subset\mathcal{T}$. Similarly for $\phi$
i.e. if $\mathcal{T}^{s_{0}}\subset\mathcal{T}$ then
$\phi\upharpoonright\mathcal{T}^{s_{0}}=\phi_{s_{0}}$. If now
$s_{0}\cup\mathcal{T}^{s_{0}}\subset\mathcal{T}$, then
$\phi\upharpoonright(s_{0}\cup\mathcal{T}^{s_{0}})=\phi_{s_{0}}\upharpoonright\mathcal{T}^{s_{0}}$.
This completes the inductive step and the proof of Theorem $7$.
We give a proof now of our second remark.
###### Proposition 3.
In the contact of Definition $16$, by taking the union of all the strong
subtree envelopes of all node-level sets of the family $\mathcal{T}$ and by
passing to an infinite strong subtree if necessary, we obtain a uniform family
$\mathcal{F}(G)$ of rank less than or equal to the rank of $\mathcal{G}$.
###### Proof.
We give a proof by induction on the rank of $\mathcal{G}$. Suppose that the
rank of $\mathcal{G}$ is finite. As we have seen above from the discussion
before Lemma $6$, if $(N_{1},L_{1})$, $(N_{2},L_{2})\in\mathcal{T}$, then
$X_{1}\in\mathcal{C}^{U}_{(N_{1},L_{1})}$ and
$X_{2}\in\mathcal{C}^{U}_{(N_{2},L_{2})}$ are isomorphic and have height equal
to $n$. By taking the union of all the strong subtree envelopes of all members
of $\mathcal{T}$, we get a family $\mathcal{F}(\mathcal{G})$ of finite strong
subtrees of $U$ with height equal to $n$. By applying Corollary $1$ we get a
strong subtree $T$ of $U$ so that the second statement of this corollary
holds. To see that suppose we get $T\in\mathcal{S}_{\infty}(U)$ such that
$\mathcal{S}_{n}(T)\cap\mathcal{F}(\mathcal{G})=\emptyset$. But
$\mathcal{G}\upharpoonright T$ is also a uniform family and the mapping $c$
restricted on that family is canonical. Pick an
$X\in\mathcal{G}\upharpoonright T$ and consider $Y\in\mathcal{C}^{T}_{f(X)}$.
Note that $Y\in\mathcal{F}(\mathcal{G})\cap\mathcal{S}_{n}(T)$, a
contradiction. Notice that the elements of any strong subtree envelop have
height $n$. Therefore we get a uniform family $\mathcal{F}(G)$ of rank $n$. In
fact we get the unique uniform family of rank $n$ on $T$.
Assume now that the rank of $\mathcal{G}$ is $\omega$. By definition
$\mathcal{G}(t)$ is of rank $n$, for some $n\in\omega$. By above
$\mathcal{F}(\mathcal{G})(t)$ is of rank less than or equal to $n$. Consider
the coloring $c^{\prime}:\mathcal{S}_{1}(U)\to\omega$ defined by $c(t)=n$ if
and only if the rank of $\mathcal{F}(\mathcal{G})(t)$ is $n$. By Theorem $5$
we get a strong subtree $T$ of $U$ such that either the coloring is constant
and equal to $n_{0}\in\omega$, one-to-one, or is constant on each level, i.e.
$c(t)=c(s)$ if and only if $|t|=|s|$. In the first case the rank of
$\mathcal{G}_{0}$ is $n_{0}$. In the last two cases the rank of
$\mathcal{F}(G)$ is $\omega$.
Suppose now that the rank of $\mathcal{G}$ is $\alpha$, for $\alpha>\omega$
and for all $\beta<\alpha$ our proposition holds. By definition
$\mathcal{G}(t)$ is of rank $\beta<\alpha$, so the inductive hypothesis
applies and we proceed as in the above paragraph. ∎
Finally we show that our definition of a canonical coloring is the appropriate
one.
###### Proposition 4.
Let $c$ be a canonical coloring of a uniform family $\mathcal{G}$ on $U$ and
let $(\mathcal{T}_{0},f_{0})$, $(\mathcal{T}_{1},f_{1})$ be two pairs that
satisfy the conditions $1$ and $2$ of the Definition $16$. Then there exists
$T\in\mathcal{S}_{\infty}(U)$ so that:
$\mathcal{T}_{0}\upharpoonright T=\mathcal{T}_{1}\upharpoonright T$ and
$f_{0}=f_{1}$ on $\mathcal{G}\upharpoonright T$
###### Proof.
By definition $f_{i}:\mathcal{G}\to\mathcal{T}_{i}$ is such that
$c(X_{0})=c(X_{1})$ if and only if $f_{i}(X_{0})=f_{i}(X_{1})$, for $i\in 2$.
Let $\mathcal{G}_{0}$ be the uniform family resulting by taking the union of
all the strong subtree envelopes of the node-level sets in $\mathcal{T}_{0}$
and $\mathcal{G}_{1}$ the one resulting from $\mathcal{T}_{1}$. We remind the
reader here that both uniform families are assumed to be defined on $U$
instead of one of its infinite strong subtrees. By an application of Lemma $8$
on $(\mathcal{G}_{0},c_{0})$ and $(\mathcal{G}_{1},c_{1})$, we get
$T\in\mathcal{S}_{\infty}(U)$ such that the first statement of the lemma
holds. We also notice that both ranks of $\mathcal{G}_{0}$ and
$\mathcal{G}_{1}$ must be equal by Lemma $11$. By Proposition $2$ we have that
$\mathcal{T}_{0}\upharpoonright T=\mathcal{T}_{1}\upharpoonright T$.
We claim that $\mathcal{T}_{0}\upharpoonright T=\mathcal{T}_{1}\upharpoonright
T$, implies that $f_{0}$ agree with $f_{1}$ on $\mathcal{G}\upharpoonright T$.
To see this suppose that for $X\in\mathcal{G}$ we have that
$f_{0}(X)=(N_{0},L_{0})\neq f_{1}(X)=(N_{1},L_{1})$. Let
$X_{0}\in\mathcal{C}^{T}_{(N_{0},L_{0})}$ and
$X_{1}\in\mathcal{C}^{T}_{(N_{1},L_{1})}$. Then $c_{0}(X_{0})\neq
c_{0}(X_{1})$ and $c_{1}(X_{0})\neq c_{1}(X_{1})$. But then $c(X)\neq c(X)$, a
contradiction.
∎
The inductive step of Theorem $8$ is identical with the inductive step of
Theorem $7$. Therefore we extended the result of Milliken completing the
research along the line of P.Erdös and R. Rado.
Next we mention a possible application of our canonical result. Suppose that
$\mathcal{U}$ and $\mathcal{V}$ are ultrafilters on index-sets $X$ and $Y,$
respectively. Let $\mathcal{V}\leq_{RK}\mathcal{U}$ denote the fact that there
is a map $F:X\rightarrow Y$ such that $\mathcal{V}=\\{M\subseteq
Y:F^{-1}(M)\in\mathcal{U}\\}.$ Put $\mathcal{U}\equiv_{RK}\mathcal{V}$
whenever $\mathcal{V}\leq_{RK}\mathcal{U}$ and
$\mathcal{U}\leq_{RK}\mathcal{V}.$ This is equivalent to saying that there is
a bijection between a set in $\mathcal{U}$ and a set in $\mathcal{V}$ that
transfers one ultrafilter into the other. There is a coarser pre-ordering
between ultrafilters that is of a considerable recent interest. This is the
_Tukey ordering_ which says that $\mathcal{V}\leq_{T}\mathcal{U}$ if there is
a monotone map $F^{\prime}:\mathcal{U}\rightarrow\mathcal{V}$ whose range
generates $\mathcal{V},$ i.e., every element of $\mathcal{V}$ is refined by
$F^{\prime}(M)$ for $M\in\mathcal{U}.$ Recall that a (non-principal)
ultrafilter $\mathcal{U}$ on $\mathbb{N}$ is _selective_ if for every map
$f:\mathbb{N}\rightarrow\mathbb{N}$ there is $M\in\mathcal{U}$ such that the
restriction $f\upharpoonright M$ is either one-to-one or constant.
In [Ra-To], S. Todorcevic has used the Theorem $2$ to prove the following
result.
###### Theorem 9 ([Ra-To]).
Tukey predecessors of a selective ultrafilter on $\mathbb{N}$ are exactly its
countable transfinite Fubini powers modulo, of course, the Rudin-Keisler
equivalence.
In section $4$ we established that $(\mathcal{S}_{\infty}((U_{i})_{i\in
d}),\subseteq,r)$ forms a topological Ramsey space. It turns out that every
topological Ramsey space has the corresponding notion of a selective
ultrafilter (see [Mij]). Since we proved the analogue of the Pudlák-Rödl
result for the space of $\mathcal{S}_{\infty}(U)$ of strong subtrees, we
really believe that our Theorem $7$ can be used to characterize the Tukey
predecessors of ultrafilters on $\mathcal{S}_{1}(U)$ that are selective
relative to the space $\mathcal{S}_{\infty}(U)$.
## References
* [Er-Ra] P.Erdös and R. Rado, A combinatorial theorem, J. London Math. Soc. 25 (1950) 249-255.
* [Ha-Lau] J.D. Halpern and H. Lauchli, A partition theorem. Trans. Amer. Math. Soc., 124:360-367, 1963.
* [Ke] Alexander S. Kechris. Classical descriptive set theory, volume 156 of Graduate Texts in Mathematics. Springer-Verlag, New York, 1995.
* [Mi1] K.R. Milliken, A Ramsey Theorem for trees, Journal of Combinatorial Theory, Series A. 26(1979) pp. 215-237.
* [Mi2] K.R. Milliken, Canonical partition theorems for strongly embedded subtrees of regular trees. Unpublished note, 1980’s.
* [Mij] J.G. Mijares, A notion of selective ultrafilter corresponding to topological Ramsey spaces, Math. logic Q. 53 (2007), 255-267
* [Na-Wi] C. St. J. A. Nash-Williams. On well-quasi-ordering infinite trees. Proc. Cambridge Philos. Soc., 61:697-720, 1965.
* [Pu-Ro] P. Pudlák and V. Rödl, Partition theorems for systems of finite subsets of integers. Discrete Math. , 39(1):67-73, 1982.
* [Ra-To] D. Raghavan and S. Todorcevic, Cofinal types of ultrafilters, Ann. Pure Appl. logic, 163(2012), 185-199.
* [To] S. Todorcevic, Introduction to Ramsey Spaces, Annals of Mathematics Studies, No.174, Princeton Univ. Press, 2010.
|
arxiv-papers
| 2013-10-14T09:20:19 |
2024-09-04T02:49:52.373418
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Dimitris Vlitas",
"submitter": "Dimitris Vlitas",
"url": "https://arxiv.org/abs/1310.3600"
}
|
1310.3627
|
# Transverse momentum distribution of charged particles and identified hadrons
in p–Pb collisions at the LHC with ALICE
for the ALICE Collaboration
Centro Studi e Ricerche e Museo Storico della Fisica “Enrico Fermi”, Rome,
Italy
Sezione INFN, Bologna, Italy
E-mail
###### Abstract:
Hadron production has been measured at mid-rapidity by the ALICE experiment at
the LHC in proton-lead (p–Pb) collisions at $\sqrt{s_{\rm NN}}$ = 5.02 TeV.
The transverse momentum ($p_{\rm T}$) distribution of primary charged
particles and of identified light-flavoured hadrons ($\pi^{\pm}$, K±,
K${}^{0}_{\rm S}$, p, $\bar{\rm p}$, $\Lambda$, $\bar{\Lambda}$) are presented
in this report. Charged-particle tracks are reconstructed in the central
barrel over a wide momentum range. Furthermore they can be identified by
exploiting specific energy loss (d$E$/d$x$), time-of-flight and topological
particle-identification techniques. Particle-production yields, spectral
shapes and particle ratios are measured in several multiplicity classes and
are compared with results obtained in Pb–Pb collisions at the LHC.
The measurement of charged-particle transverse momentum spectra and nuclear
modification factor RpPb indicates that the strong suppression of high-$p_{\rm
T}$ hadrons observed in Pb–Pb collisions is not due to initial-state effects,
but it is rather a fingerprint of jet quenching in hot QCD matter. The
systematic study of the hadronic spectral shapes as a function of the particle
mass and of particle ratios as a function of charged-particle density provides
insights into collective phenomena, as observed in Pb–Pb collisions. Similar
features, that could be present in high-multiplicity p–Pb collisions, will
also be discussed.
## 1 Introduction
High-energy heavy-ion (AA) collisions offer a unique possibility to study
hadronic matter under extreme conditions, in particular the deconfined quark-
gluon plasma which has been predicted by quantum chromodynamics (QCD) [1, 2,
3, 4]. The interpretation of the results depends crucially on the comparison
with results from smaller collision systems such as proton-proton (pp) or
proton-nucleus (pA). Proton-nucleus (pA) collisions are intermediate between
proton-proton (pp) and nucleus-nucleus (AA) collisions both in terms of system
size and number of produced particles. Comparing particle production in pp,
pA, and AA reactions is frequently used to separate initial state effects,
connected to the use of nuclear beams or targets, from final state effects,
connected to the presence of hot and dense matter. Moreover, pA collisions
allow for the investigation of fundamental properties of QCD; the $p_{\rm T}$
distributions and yields of particles of different mass at low and
intermediate momenta of $p_{\rm T}$ $\lesssim$ 3 ${\rm GeV}/c$ (where the vast
majority of particles is produced) can provide important information about the
system created in high-energy hadron reactions.
Previous results on identified particle production in pp and Pb–Pb collisions
at the LHC have been reported in [5, 6, 7, 8, 9, 10, 11]. Results on
transverse momentum distribution and nuclear modification factor of charged
particles in p–Pb collisions at $\sqrt{s_{\rm NN}}$ = 5.02 TeV have been
reported in [12]. In this paper we report on the measurement of $\pi^{\pm}$,
K±, ${\rm K}^{0}_{S}$, p($\rm\overline{p}$) and $\Lambda$($\bar{\Lambda}$)
production in p–Pb collisions at a nucleon-nucleon center-of-mass energy
$\sqrt{s_{\rm NN}}$ = 5.02 TeV.
## 2 Sample and Data analysis
The results presented here were obtained from a sample of the data collected
during the LHC p–Pb run at $\sqrt{s_{\rm NN}}$ = 5.02 TeV in the beginning of
2013. Due to the asymmetric beam energies for the proton and lead beams, the
nucleon-nucleon center-of-mass system was moving in the laboratory frame with
a rapidity of $y_{\rm NN}$ = $-0.465$ in the direction of the proton beam. A
detailed description of the ALICE apparatus can be found in [13] and a
description of the data-taking and trigger setup in minimum-bias trigger in
[14]. In order to study the multiplicity dependence, the selected event sample
was divided into seven event classes, based on cuts applied on the total
charge deposited in the VZERO-A scintillator hodoscope ($2.8<\eta_{\rm
lab}<5.1$, Pb beam direction).
The ALICE central-barrel tracking covers the full azimuth within $|\eta_{\rm
lab}|<0.9$. The tracking detectors are located inside a solenoidal magnet
providing a magnetic field of 0.5 T. The innermost barrel detector is the
Inner Tracking System (ITS). The Time Projection Chamber (TPC), the main
central-barrel tracking device, follows outwards. Finally the Transition
Radiation Detector (TRD) extends the tracking farther away from the beam axis.
Charged-hadron identification in the central barrel was performed with the
ITS, TPC [15] and Time-Of-Flight (TOF) [16] detectors [17]. Three approaches
were used for the identification of $\pi^{\pm}$, K±, and p($\bar{\rm p}$),
called “ITS standalone”, “TPC/TOF” and “TOF fits” and are described in details
in [8, 9]. Contamination from secondary particles was subtracted with a data-
driven approach, based on the fit to the transverse distance-of-closest
approach to the primary vertex (DCAxy) distribution with the expected shapes
for primary and secondary particles [8, 9].
The ${\rm K}^{0}_{S}$ and $\Lambda$($\bar{\Lambda}$) particles were identified
exploiting their “${\rm V}^{0}$” weak decay topology in the channels ${\rm
K}^{0}_{S}\to\pi^{+}\pi^{-}$ and
$\Lambda(\bar{\Lambda})\to\rm{p}\pi^{-}(\rm{\bar{p}}\pi^{+})$. The selection
criteria used to define two tracks as ${\rm V}^{0}$ decay candidates are
detailed in [6, 18]. The contribution from weak decays of the charged and
neutral $\Xi$ to the $\Lambda$($\bar{\Lambda}$) yield has been corrected
following a data-driven approach.
The study of systematic uncertainties follows the analysis described in [8, 9,
6, 18] and was repeated for the different multiplicity bins in order to
separate the sources of uncertainty which are dependent on multiplicity and
uncorrelated across different bins (depicted as shaded boxes in the figures).
## 3 Results
Figure 1: Ratios p/$\pi$ (left) and $\Lambda$/K${}_{\rm S}^{0}$ (right) as a
function of $p_{\rm T}$ in two multiplicity bins compared to results in Pb–Pb
collisions. The empty boxes show the total systematic uncertainty; the shaded
boxes indicate the contribution uncorrelated across multiplicity bins (not
estimated in Pb–Pb).
Figure 2: Ratios p/$\pi$ (left) and $\Lambda$/K${}_{\rm S}^{0}$ (right) as a
function of the charged-particle density d$N_{\rm ch}$/d$\eta$ in three
$p_{\rm T}$ intervals in p–Pb, Pb–Pb and pp collisions (pp only shown for
$\Lambda$/K${}_{\rm S}^{0}$). The dashed lines show the corresponding power-
law fit.
Figure 3: Exponent of the p/$\pi$ (left) and $\Lambda$/K${}_{\rm S}^{0}$
(right) power-law fit as a function of $p_{\rm T}$ in p–Pb, Pb–Pb and pp
collisions (pp only shown for $\Lambda$/K${}_{\rm S}^{0}$).
The $p_{\rm T}$ distributions of $\pi^{\pm}$, K±, ${\rm K}^{0}_{S}$,
p($\rm\overline{p}$) and $\Lambda$($\bar{\Lambda}$) in $0<y_{\rm CMS}\ <0.5$
are reported in [19] for different multiplicity intervals.
Particle/antiparticle as well as charged/neutral kaon transverse momentum
distributions are identical within systematic uncertainties. The $p_{\rm T}$
distributions show a clear evolution, becoming harder as the multiplicity
increases. The multiplicity dependence of the $p_{\rm T}$ spectral shape is
stronger for heavier particles, as evident when looking at the ratios ${\rm
K}/\pi$ = (K+\+ K-)/($\pi^{+}$\+ $\pi^{-}$), ${\rm p}/\pi$ = (p +
$\rm\overline{p}$)/($\pi^{+}$\+ $\pi^{-}$) and $\Lambda$/${\rm K}^{0}_{S}$ as
functions of $p_{\rm T}$, shown in Fig. 1 for the 0–5% and 60–80% event
classes. The ratios ${\rm p}/\pi$ and $\Lambda$/${\rm K}^{0}_{S}$ show a
significant enhancement at intermediate $p_{\rm T}$ $\sim 3$ ${\rm GeV}/c$,
qualitatively reminiscent of the one measured in Pb–Pb collisions [8, 9, 18].
The latter is generally discussed in terms of collective flow or quark
recombination [20, 21, 22]. A similar enhancement of the ${\rm p}/\pi$ ratio
in high-multiplicity d–Au collisions has also been reported for RHIC energies
[23].
It is worth noticing that the ratio ${\rm p}/\pi$ as a function of
$\mathrm{d}N_{\mathrm{ch}}/\mathrm{d}\eta$ in a given $p_{\rm T}$-bin follows
a power-law behavior: $\frac{\rm p}{\pi}\left(p_{\rm T}\right)=A(p_{\rm
T})\times\left[\mathrm{d}N_{\mathrm{ch}}/\mathrm{d}\eta\right]^{B(p_{\rm
T})}$. As shown in Fig. 2, the same trend is also observed in Pb–Pb
collisions. The exponent of the power-law function exhibits the same value in
both collision systems (Fig. 3, left). The same feature is also observed in
the $\Lambda$/${\rm K}^{0}_{S}$ ratio and this also holds in pp collisions
(Fig. 3, right).
## 4 Discussion
Figure 4: Results of blast-wave fits, compared to Pb–Pb data, pp data and MC
simulations from PYTHIA8 with and without color reconnection. Charged-particle
multiplicity increases from left to right. Uncertainties from the global fit
are shown as correlation ellipses for p–Pb and Pb–Pb data and with errors bars
for pp data. Figure 5: Pion, kaon, and proton transverse momentum
distributions in the 5-10% multiplicity class compared to the several models
(see text for details).
In heavy-ion collisions, the flattening of transverse momentum distribution
and its mass ordering find their natural explanation in the collective radial
expansion of the system [24]. This picture can be tested in a blast-wave model
[25] with a simultaneous fit to all particles. This parameterization assumes a
locally thermalized medium, expanding collectively with a common velocity
field and undergoing an instantaneous common freeze-out. The fit presented
here is performed in the same range as in [8, 9], also including ${\rm
K}^{0}_{S}$ and $\Lambda$($\bar{\Lambda}$). The results are reported Fig. 4.
Variations of the fit range lead to large shifts ($\sim 10\%$) of the fit
results (correlated across centralities), as discussed for Pb–Pb data in [8,
9]. As can be seen in Fig. 4, the parameters show a similar dependency with
event multiplicity as observed with the Pb–Pb data. Within the limitations of
the blast-wave model, this observation is consistent with the presence of
radial flow in p–Pb collisions. Under the assumptions of a collective
hydrodynamic expansion, a larger radial velocity in p–Pb collisions has been
suggested as a consequence of stronger radial gradients in [26]. On the other
hand it is worth noticing that very similar results are obtained when
performing the same study on pp spectra measured as a function of the event
multiplicity. Other processes not related to hydrodynamic collectivity could
also be responsible for the observed results. This is illustrated in Fig. 4,
which shows the results obtained by applying the same fitting procedure to
transverse momentum distributions from the simulation of pp collisions at
$\sqrt{s}$ = 7 TeV with the PYTHIA8 event generator (tune 4C) [27], a model
not including any collective system expansion. The fit results are shown for
PYTHIA8 simulations performed both with and without the color reconnection
mechanism [28, 29]. With color reconnection the evolution of PYTHIA8
transverse momentum distributions follows a similar trend as the one observed
for p–Pb, pp and Pb–Pb collisions at the LHC, while without color reconnection
it is not as strong. This generator study shows that other final state
mechanisms, such as color reconnection, can mimic the effects of radial flow
[30].
The $p_{\rm T}$ distributions in the 5-10% bin are compared in Fig. 5 with
calculations from the DPMJET [31], Kraków [32] and EPOS LHC 1.99 v3400 [33]
models. The transverse momentum distributions in the 5-10% multiplicity class
are compared to the predictions by Kraków for $11\leq N_{\rm part}\leq 17$,
since the $\mathrm{d}N_{\mathrm{ch}}/\mathrm{d}\eta$ from the model matches
best with the measured value in this class. DPMJET and EPOS events have been
selected according to the charged particle multiplicity in the VZERO-A
acceptance in order to match the experimental selection. DPMJET distributions
are softer than the measured ones and the model overpredicts the production of
all particles for $p_{\rm T}$ lower than about 0.5–0.7 ${\rm GeV}/c$ and
underpredicts it at higher momenta. At high-$p_{\rm T}$, the $p_{\rm T}$
spectra shapes of pions and kaons are rather well reproduced for momenta above
1 and 1.5 ${\rm GeV}/c$ respectively. Final state effects may be needed in
order to reproduce the data. In fact, The Kraków model reproduces reasonably
well the shape of pions and kaons below transverse momenta of 1 ${\rm GeV}/c$
where hydrodynamic effects are expected to dominate. For higher momenta, the
observed deviations for pions and kaons could be explained in a hydrodynamic
framework as due to the onset of a non-thermal component. EPOS can reproduce
the pion and proton distributions within 20% over the full measured range,
while larger deviations are seen for kaons and lambdas. It is interesting to
notice that when final state interactions are disabled in EPOS, the
description of many pp and p–Pb observables worsens significantly [33].
## 5 Conclusions
We presented a comprehensive measurement of $\pi^{\pm}$, K±, ${\rm
K}^{0}_{S}$, p($\rm\overline{p}$) and $\Lambda$($\bar{\Lambda}$) in p–Pb
collisions at $\sqrt{s_{\rm NN}}$ = 5.02 TeV at the LHC. The transverse
momentum distributions show a clear evolution with multiplicity, similar to
the pattern observed in high-energy pp and heavy-ion collisions, where in the
latter case the effect is usually attributed to collective radial expansion.
Models incorporating final state effects give a better description of the
data.
## References
* [1] N. Cabibbo and G. Parisi, Phys. Lett. B59, 67 (1975).
* [2] E. V. Shuryak, Phys. Lett. B78, 150 (1978).
* [3] L. D. McLerran and B. Svetitsky, Phys. Lett. B98, 195 (1981).
* [4] E. Laermann and O. Philipsen, Ann. Rev. Nucl. Part. Sci. 53, 163 (2003).
* [5] ALICE Collaboration, K. Aamodt et al., Eur. Phys. J C71, 1655 (2011).
* [6] ALICE Collaboration, K. Aamodt et al., Eur. Phys. J. C71, 1594 (2011).
* [7] ALICE Collaboration, B. Abelev et al., Phys. Lett. B712, 309 (2012).
* [8] ALICE Collaboration, B. Abelev et al., Phys. Rev. Lett. 109, 252301 (2012).
* [9] ALICE Collaboration, B. Abelev et al., (2013), hep-ex/1303.0737.
* [10] CMS Collaboration, S. Chatrchyan et al., Eur. Phys. J. C72, 2164 (2012).
* [11] CMS Collaboration, V. Khachatryan et al., JHEP 1105, 064 (2011).
* [12] ALICE Collaboration, B. Abelev et al., Phys. Rev. Lett. 110, 082302 (2012).
* [13] ALICE Collaboration, K. Aamodt et al., JINST 3, S08002 (2008).
* [14] ALICE Collaboration, B. Abelev et al., Phys. Rev. Lett. 110, 032301 (2013).
* [15] J. Alme et al., Nucl. Instrum. Meth. A622, 316 (2010).
* [16] A. Akindinov et al., Eur. Phys. J. Plus 128, 44 (2013).
* [17] ALICE Collaboration, Performance of the ALICE Experiment at CERN LHC, in preparation.
* [18] ALICE Collaboration, (2013), nucl-ex/1307.5530.
* [19] ALICE Collaboration, B. B. Abelev et al., (2013), nucl-ex/1307.6796.
* [20] R. Fries, B. Muller, C. Nonaka, and S. Bass, Phys. Rev. Lett. 90, 202303 (2003).
* [21] P. Bozek, (2011), nucl-th/1111.4398.
* [22] B. Muller, J. Schukraft, and B. Wyslouch, Ann. Rev. Nucl. Part. Sci. 62, 361 (2012).
* [23] PHENIX Collaboration, A. Adare et al., (2013), nucl-ex/1304.3410.
* [24] U. W. Heinz, Concepts of heavy ion physics, CERN-2004-001-D, 2004.
* [25] E. Schnedermann, J. Sollfrank, and U. W. Heinz, Phys. Rev. C48, 2462 (1993).
* [26] E. Shuryak and I. Zahed, (2013), hep-ph/1301.4470.
* [27] R. Corke and T. Sjostrand, JHEP 1103, 032 (2011).
* [28] P. Z. Skands and D. Wicke, Eur. Phys. J. C52, 133 (2007).
* [29] H. Schulz and P. Skands, Eur. Phys. J. C71, 1644 (2011).
* [30] A. Ortiz, P. Christiansen, E. Cuautle, I. Maldonado, and G. Paic, Phys. Rev. Lett. 111, 042001 (2013).
* [31] S. Roesler, R. Engel, and J. Ranft, p. 1033 (2000), hep-ph/0012252.
* [32] P. Bozek, Phys. Rev. C85, 014911 (2012).
* [33] T. Pierog, I. Karpenko, J. Katzy, E. Yatsenko, and K. Werner, (2013), hep-ph/1306.0121.
|
arxiv-papers
| 2013-10-14T11:08:25 |
2024-09-04T02:49:52.392911
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Roberto Preghenella (for the ALICE Collaboration)",
"submitter": "Roberto Preghenella",
"url": "https://arxiv.org/abs/1310.3627"
}
|
1310.3749
|
Christian H. [email protected]
# In-situ growth optimization in focused electron-beam induced deposition
Paul M. Weirich Marcel Winhold Michael Huth Physikalisches Institut, Goethe
Universität, Max-von-Laue-Str. 1, 60438 Frankfurt am Main, Germany
###### Abstract
We present the application of an evolutionary genetic algorithm for the in-
situ optimization of nanostructures prepared by focused electron-beam-induced
deposition. It allows us to tune the properties of the deposits towards
highest conductivity by using the time gradient of the measured in-situ rate
of change of conductance as fitness parameter for the algorithm. The
effectiveness of the procedure is presented for the precursor $\rm W(CO)_{6}$
as well as for post-treatment of Pt-C deposits obtained by dissociation of
$\rm MeCpPt(Me)_{3}$. For $\rm W(CO)_{6}$-based structures an increase of
conductivity by one order of magnitude can be achieved, whereas the effect for
$\rm MeCpPt(Me)_{3}$ is largely suppressed. The presented technique can be
applied to all beam-induced deposition processes and has great potential for
further optimization or tuning of parameters for nanostrucures prepared by
FEBID or related techniques.
###### keywords:
electron beam induced deposition; genetic algorithm; nanotechnology; tungsten
## 1 Introduction
In focused electron-beam-induced deposition, FEBID in short, a (metal-)
organic or inorganic volatile precursor gas, previously adsorbed on a
substrate surface, is dissociated in the focus of an electron beam provided by
a scanning (SEM) or transmission electron microscope (TEM). During the last
decade FEBID has developed from a highly specialized nanofabrication method
with a limited selection of application fields to one of the most flexible
approaches for functional nanostructure fabrication with true 3D patterning
capabilities. By now FEBID-based nanostructures are used in highly
miniaturized magnetic field [2, 3], strain/force [4, 5] and gas sensing [6]
applications, as well as in micromagnetic studies on domain wall nucleation
and propagation [7, 8]. On the basis of the in-situ, electron irradiation-
induced tunability of metallic FEBID- structures significant progress could be
made in understanding the charge transport regimes in nanogranular metals [9,
10, 11]. In addition, by the controlled combination of two precursors it has
become possible to prepare amorphous binary alloys [12, 13], as well as
nanogranular intermetallic compounds [14].
As the FEBID-immanent parameter space becomes larger, the identification of
the parameters for an optimized deposition protocol is becoming a very
challenging problem. In fact, even for a single organometallic precursor,
finding the deposition parameters for, e.g., obtaining the maximum possible
metal content, can be a difficult task. This can be exemplified for the
commonly used precursor $\rm W(CO)_{6}$. Rosenberg and co-workers recently
studied the electron-dose and substrate-temperature dependence on the final
deposit in electron-induced dissociation experiments with 500 eV electron
energy for this precursor [15, 16]. They showed that the initial dissociation
at electron doses below about 100 $\rm pC/\mu m^{2}$ leads to the release
(i.e., dissociation and desorption) of two CO ligands from the parent
molecule. The decarbonylated residual species is then subject to electron-
stimulated decomposition rather than desorption resulting in an average
composition of the deposit of [W]/[C] $\sim$ 1/4. By increasing the electron
dose and/or the substrate temperature, which causes changes in the coverage
and average residence time of the precursor molecules on the surface, the
metal content can be increased to above 40 at% [17]. Changes of the precursor
flux and the partial pressure of water in the residual gas also influence the
final composition and increases the extend of tungsten oxidation in the
deposit [16].
With regard to the electrical conductivity of the deposits, a key quantity in
many applications of FEBID structures, no reliable prediction can be made
concerning its value for different deposition parameters and conditions. This
is due to the fact that metal content alone is not a sufficient indicator
since in most instances transport is of the hopping type, so that the matrix
composition and the oxidation state of the metal are also important but
a-priori unknown quantities [5, 9]. From this one can conclude that the
optimization of any FEBID process towards the largest possible conductivity
should ideally monitor the conductance as the growth proceeds [11] and use
this information in adaptively changing the deposition parameters. Here, we
present a first implementation of such a feedback control mechanism and employ
an evolutionary genetic algorithm (GA) for the in-situ optimization of the
electrical conductivity of nanostructures prepared by FEBID [18]. By using the
time gradient of the measured in-situ conductance as a fitness parameter for
the GA we are able to tune the properties of the deposits towards highest
conductivity. In order to demonstrate the efficiency of this method, we chose
$\rm W(CO)_{6}$. Our study reveals that an increase of conductivity by two
orders of magnitude can be achieved with the GA by solely varying the process
parameters pitch p and dwell-time $t_{D}$ in the deposition process. The
precursor-specific limitations of the approach are also exemplified for
another precursor, $\rm MeCpPt(Me)_{3}$, which is known to show only one bond-
cleavage in the initial step [19]. This results in a largely deposition
parameter independent Pt/C ratio. Furthermore, in contrast to tungsten,
platinum is not susceptible to oxidation or carbide formation, which results
in a nano-granular rather than amorphous microstructure.
## 2 Experimental
The FEBID process takes place in a dual-beam SEM/FIB microscope (FEI, Nova
Nanolab 600) equipped with a Schottky electron emitter. The precursor gases
are introduced into the high-vacuum chamber via a gas injection system through
a thin capillary (Ø = 0.5 mm) in close proximity to the focus of the electron
beam. As a substrate material n-doped Si(100) (350 $\mu$m)/LPCVD Si3N4 (300
nm) was used equipped with 10/200 nm thick Cr/Au contacts with a separation of
3 $\mu$m that were prepared using UV-lithography and a lift-off process.
The optimization process using the GA in combination with in-situ electrical
conductance measurements is schematically displayed in Figure 1a. At first a
seed-layer is deposited ensuring that all optimization processes start with
the same initial conditions. On top of the seed layer subsequent layers with
different deposition parameters are added.
Figure1.png
Figure 1: Schematic representation of the optimization process: a) Layer
structure of FEBID deposits: m optimization cycles, each consisting of n
parameter sets except for the parent optimization cycle with 2n parameter
sets, are deposited onto a seed-layer between two Cr/Au electrodes. During the
deposition process the conductance of the whole layer structure is measured.
b) Representative $S(t)$-graph for layer structure of a). Altering background
colors indicate the deposition of different optimization cycles. The inset
depicts $S(t)$ during the deposition of one layer. The $S(t)$-curve shows a
sharp increase when the FEBID process is started and decreases when the
deposition process is stopped, respectively.
With regard to a GA-based optimization process, the set of parameters used for
the deposition of one specific layer consists of {x- and y-size of the
deposit, dwell time ($t_{D}$), pitch in x ($p_{x}$) and y ($p_{y}$) direction,
beam current ($I$), acceleration voltage ($U$), temperature ($T$), refresh-
time ($t_{r}$), scan-type (raster or serpentine), dose ($D$) and passes
($p$)}. However, not all parameters are independent, e.g. in order to keep $D$
fixed, $P$ has to be adapted according to the specific combination of {x- and
y-size of the deposit, $p_{x}$ and $p_{y}$, $I$ and $t_{D}$ }. The aim of the
GA’s search is to find parameter sets leading to an enhancement of conductance
due to an increasing growth rate of the deposit and/or intrinsic effects e.g.
the increase of the metal content and/or a change of the dielectric matrix.
The GA allows for the optimization of deposition parameters for an arbitrary
precursor, without having any additional information about the deposition
process. Therefore, the following procedure is performed:
The parent optimization cycle based on the first 2n parameter sets with
randomly generated parameters is deposited onto the seed layer. After the
deposition of each layer a fitness evaluation is carried out for each
parameter set according to the following principle. During the optimization
process the conductance S is measured and the rate of change of conductance
over time $\overline{S}^{\prime}$ = $S/t$ is calculated. Assuming a parallel
circuited resistance is added, once another layer is deposited on top of the
existing structure, $\overline{S}^{\prime}$ is constant if the growth rate and
the conductivity do not change. However, if either the conductivity or the
growth rate is altered by the variation of deposition parameters, the gradient
of $\overline{S}^{\prime}$ is a suitable variable to describe the influence of
deposition parameters on the conductance of the deposit. Hence, the gradient
of $\overline{S}^{\prime}$ is chosen as the fitness parameter for the GA in
order to detect effects leading to a change of the growth rate and/or the
conductivity. Layers with the highest fitness values are selected to generate
the next optimization cycle of n parameter sets using genetic operators such
as crossover and mutation. For the next optimization cycle a number of new
parameter sets are created, according to half the size of the initial parent
optimization cycle. One half of the next optimization cycle is created with
the crossover method, the other half with the mutation method. The parents of
the new parameter sets are chosen via an uniform distributed random choice.
The crossover method is performed by exchanging parameters of the parents. For
the mutation method parameters are chosen randomly within the given parameter-
range. A representative time-dependent development of the conductance during
the optimization process is shown in Figure 1b. The GA is stopped after a
predefined number of m optimization cycles yielding a set of FEBID deposition
parameters for each precursor for a deposit of optimized conductance. A flow-
chart of the GA optimization process is shown in Figure 2.
Figure2.png
Figure 2: Logical flow representation for the in-situ optimization of
conductance of FEBID deposits with a GA. After the initialization of the
program, the GA optimizes the conductance of the deposits by using the
measured gradient of $\overline{S}^{\prime}$ to evaluate the fitness of the
parameter sets used for deposition. Selection, recombination and mutation of
parameter sets are carried out after the fitness evaluation to obtain the next
optimization cycle with optimized parameter sets. The process is stopped after
the deposition of a pre-defined number of optimization cycles.
## 3 Results
In order to check for the proper operation of the GA we first applied it for
the optimization of deposition parameters for the widely used precursor $\rm
W(CO)_{6}$ [11, 20, 21, 22]. For $\rm W(CO)_{6}$ it is well known that the
metal content and, respectively, the conductivity strongly depend on the
deposition parameters during the FEBID process. At the beginning a reference
sample was deposited using standard deposition parameters
($U=5\leavevmode\nobreak\ kV$, $I=6.3\leavevmode\nobreak\ nA$,
$t_{D}=100\leavevmode\nobreak\ \mu s$, $p_{x}=40\leavevmode\nobreak\ nm$,
$p_{y}=40\leavevmode\nobreak\ nm$). For the reference the GA protocol was
used, meaning that the process was paused after the deposition of each layer,
indicated by drops in the curves of Figure 3a. However, for the reference
sample the parameters were kept fixed for the complete deposition process. For
each parameter set a dose of $3\leavevmode\nobreak\ nC/\mu m^{2}$ was used.
The GA was carried out for 6 optimization cycles with a population size of 8
parameter sets. The measured rate of change of conductance during the FEBID
process for the reference sample is displayed in Figure 3a (Sample 1).
Subsequently the GA was applied for finding the optimized parameters for
deposition using $\rm W(CO)_{6}$ as a precursor. First, only the dwell time
$t_{D}$ was used as optimization parameter and was allowed to vary in the
range of $0.2-1500\leavevmode\nobreak\ \mu s$. The corresponding rate of
change of conductance is displayed in Figure 3a (Sample 2). In addition, we
let the GA search for deposition parameters leading to minimum conductance.
Dwell time $t_{D}$ and pitch $p_{x}$, $p_{y}$ were allowed to vary in the
range of $0.2-1500\leavevmode\nobreak\ \mu s$ and $30-200\leavevmode\nobreak\
nm$, respectively (Figure 3a, Sample 3). The highest conductance for W-C-O
deposits was obtained for short dwell times ($t_{D}=0.5\leavevmode\nobreak\
\mu s$) whereas a low conductance was observed for long dwell times
($t_{D}=831\leavevmode\nobreak\ \mu s$) and a larger y-pitch
($p_{y}=150\leavevmode\nobreak\ nm$). In order to study the success of the GA
procedure the optimized parameter sets returned by the GA for highest and
lowest conductance as well as for the reference sample were used for a
standard FEBID process and the conductance during deposition was measured (see
Figure 3b). As can be clearly seen, sample 2 (optimized for highest
conductance) shows by far the highest value of conductance, whereas for sample
3 (optimized for lowest conductance) the lowest value is achieved.
Figure3.png
Figure 3: a) Rate of change of conductance during the GA optimization for
W-C-O reference (green), GA optimized deposit for highest conductance (black)
and GA optimized deposit for lowest conductance (red). For each parameter set
a dose of $3\leavevmode\nobreak\ nC/\mu m^{2}$ was used. The population size
amounted to 8 parameter sets and 6 optimization cycles which were deposited
for the GA optimization. b) Conductance of $A=3\times 7\leavevmode\nobreak\
\mu m^{2}$ W-C-O structures deposited with parameters derived from the
optimization processes in a) as well as for the W-C-O reference using a dose
of $27\leavevmode\nobreak\ {nC}/{\mu m^{2}}$.
For the purpose of characterizing the chemical composition of the different
samples energy dispersive x-ray spectroscopy (EDX) was performed. EDX
measurements were carried out on $2\times 2\leavevmode\nobreak\ \mu m^{2}$
reference structures deposited with the identical parameters used for the
conductance measurements. In Figure 4a the results of the EDX measurements are
displayed. Sample 2 has the highest metal content of 39.2 at% W, whereas the
metal content decreases for reference sample 1 (32.7 at% W) and sample 3 (26.0
at% W). Apparently a difference of more than 13 at% between the intentionally
optimal and the worst parameter set can be observed. In addition the carbon
content in the deposits increases from sample 2 to sample 3, whereas the
oxygen content is reduced. The corresponding resistivity of the different
samples was calculated from the conductance measurements in Figure 3b in
combination with AFM measurements of the deposits. As already indicated by the
result of the EDX measurements the resistivity of the tungsten deposits is
reduced by one order of magnitude for the optimized GA parameters compared to
the GA parameters for lowest conductance. The results for the GA optimization
for the $\rm W(CO)_{6}$ deposits are summarized in Table 1.
Figure4.png
Figure 4: Chemical composition of sample 1 ($t_{D}=100\leavevmode\nobreak\ \mu s$), sample 2 ($t_{D}=0.5\leavevmode\nobreak\ \mu s$) and sample 3 ($t_{D}=831\leavevmode\nobreak\ \mu s$). EDX measurements were performed on separate $2\times 2\leavevmode\nobreak\ \mu m^{2}$ samples b) Resistivity of samples 1, 2 and 3: By solely varying the deposition parameters dwell-time and pitch as obtained from GA experiments, resistivity of W-C-O samples can be tuned by one order of magnitude. Table 1: Summary of parameters used for deposition of samples 1 (reference), 2 (GA optimization for highest conductance) and 3 (GA optimization for lowest conductance). The reference sample was deposited with fixed values for dwell-time and pitch whereas the dwell-time for sample 2 was varied by the GA in the range of $t_{D}=0.2-1500\leavevmode\nobreak\ \mu s$ at fixed pitch of $p_{x}=p_{y}=40\leavevmode\nobreak\ nm$. For sample 3, dwell-time and pitch were both allowed to vary in the range of $t_{D}=0.2-1500\leavevmode\nobreak\ \mu s$ and $p_{x}$, $p_{y}=30-200\leavevmode\nobreak\ nm$. The GA optimization was performed for 6 optimization cycles each comprising 8 parameter sets which were deposited between Cr/Au electrodes using a dose of $3\leavevmode\nobreak\ \frac{nC}{\mu m^{2}}$ per parameter set. The parameters obtained from the in-situ experimental GA analysis were used to deposit another set of samples with a dose of $27\leavevmode\nobreak\ \frac{nC}{\mu m^{2}}$ and $A=3\times 7\leavevmode\nobreak\ \mu m^{2}$, which were analyzed by means of AFM and electrical I(V)-measurements to obtain resisitivity of the samples. The chemical composition was determined by EDX-measurements which were performed on separate $2\times 2\leavevmode\nobreak\ \mu m^{2}$ samples to prevent changing the conductivity of the samples for further electrical measurements. All other deposition parameters were kept fixed: $U=5\leavevmode\nobreak\ kV$, $I_{nominal}=6.3\leavevmode\nobreak\ nA$ Sample | Parameters varied | Parameters used | Chemical | Resistivity | Height
---|---|---|---|---|---
Nr. | by GA | for deposition | composition
# | $t_{D}$ | $p_{x}$ | $p_{y}$ | $t_{D}$ | $p_{x}$ | $p_{y}$ | W | C | O | $\rho$ | h
($\mu s$) | (nm) | (nm) | ($\mu s$) | (nm) | (nm) | (at%) | (at%) | (at%) | (m$\Omega$cm) | (nm)
1 | - | - | - | 100 | 40 | 40 | 32.7 | 43.8 | 23.5 | 87.7 | 32
2 | 0.2 - 1500 | - | - | 0.5 | 40 | 40 | 39.2 | 27.0 | 33.8 | 16.5 | 36
3 | 0.2 - 1500 | 30 - 200 | 30 - 200 | 831 | 35 | 150 | 26.0 | 55.4 | 18.6 | 133.3 | 25
## 4 Discussion
For the thus far presented case of $\rm W(CO)_{6}$, the great success of the
GA optimization process is due to the fact that the metal content of the
deposits can be tuned over a wide range and strongly depends on the deposition
parameters which is known to be the case for many carbonyl-based precursors
(e.g. $\rm W(CO)_{6}$ [11, 20, 22], $\rm Co_{2}(CO)_{8}$ [3, 23] and $\rm
Fe(CO)_{5}$ [24, 25]). With regard to the two process parameters dwell-time
and pitch the FEBID process can in general be divided into two deposition
regimes. For small dwell-times and larger pitches the electron induced
dissociation reactions are locally limited by the number of incident electrons
(reaction rate limited regime (RRL)). However, if the dwell-time is large and
a small pitch is used the reactions are limited by the number of available
precursor molecules (mass transport limited regime (MTL)). In most cases the
electron-induced complete dissociation of a precursor molecule is not a
single-step process but requires several electron-precursor interactions [26,
27]. Therefore in the RRL regime precursor molecules are not dissociated
completely leading either to an implantation of non-dissociated precursor
molecules or reaction by-products into the deposit but also allowing reaction
by-products such as, e.g., CO groups to diffuse away from the electron impact
area, desorb and finally be removed from the vacuum chamber. In the MTL regime
due to the large number of locally available electrons, precursor molecules
are rapidly depleted leaving enough electrons to dissociate reaction by-
products which can be incorporated as non-metallic impurities into the
deposit. With regard to our GA experiments RRL-like conditions [28] were
fulfilled for sample 2 which was optimized by the GA for maximum conductance.
As it is evident from the ratio of W:C:O = 1:0.69:0.86 obtained by the EDX
measurements, for a short dwell-time of $0.5\leavevmode\nobreak\ \mu s$ the
electron stimulated decomposition of the W-precursor and its surrounding CO
ligands is very efficient as only $14.3\%$ and $11.5\%$ of oxygen and carbon
atoms, respectively, of the original $W(CO)_{6}$ molecules are incorporated
into the deposit. These findings suggest that due to the limited number of
electrons available in the RRL regime the majority of volatile CO by-products
can be removed during the FEBID process leading to a deposit with a high metal
content. On the contrary, for a dwell-time of $831\leavevmode\nobreak\ \mu s$
the growth regime shifts to MTL regime where the replenishment rate of
precursor molecules is too low leading to further electron stimulated
dissociation of CO. The result is a strongly enhanced carbon content of
$55.4\leavevmode\nobreak\ at\%$ in the deposit accompanied by a decrease of
tungsten and oxygen to $26.0\leavevmode\nobreak\ at\%$ and
$18.6\leavevmode\nobreak\ at\%$, respectively. Furthermore, the oxygen content
of the deposits is coupled to the amount of tungsten indicating that tungsten-
oxide is formed (Figure 4b). The strong increase of carbon in the deposits
with decreasing oxygen content can be explained by the electron-induced
decomposition of CO groups, which is in accordance with several studies on
electron-induced dissociation of adsorbed and gaseous CO molecules [29, 30].
Furthermore the studies show that carbon remains at the surface whereas oxygen
is liberated which is in agreement with our measurements. In order to describe
the observed increase of conductance it is not sufficient to only regard the
metal content alone as the growth rate can also have a significant impact.
However, as depicted in Table 1 AFM measurements reveal that the height of
samples 1-3 varies by a factor of 1.44 corresponding to a monotonic increase
of height with decreasing dwell time from 25 nm to 36 nm for samples 3 and 2,
respectively. Thus, for the presented case of $W(CO)_{6}$ the growth rate only
has a minor impact on conductance of the different samples.
The results of the GA optimization presented in this work for a precursor
sensitive to the deposition parameters are extremely promising. Nevertheless,
there are precursors known for the FEBID process for which the chemical
composition is almost independent of the deposition parameters dwell-time and
pitch. A prominent example is $\rm MeCpPt(Me)_{3}$. However, in this case it
could be shown that the resulting Pt-C deposits are very sensitive to post-
treatment either by annealing [31, 32, 33] or electron-beam irradiation [4, 9,
10, 34], which can result in an increase of conductivity of many orders of
magnitude. In order to investigate the influence of the GA for such a post-
treatment process of FEBID deposits several Pt-C test-structures were
fabricated via FEBID using identical depostion parameters
($U=5\leavevmode\nobreak\ kV$, $I=1.6\leavevmode\nobreak\ nA$,
$t_{D}=1\leavevmode\nobreak\ \mu s$, $p_{x}=40\leavevmode\nobreak\ nm$,
$p_{y}=40\leavevmode\nobreak\ nm$) and an electron dose of
$30\leavevmode\nobreak\ nC/\mu m^{2}$. This results in a height of
approximately 120 nm of the deposits, ensuring a complete penetration of the
deposit by electrons. As proposed by Plank et al. [34] RRL-like conditions as
best initial conditions for e-beam curing were used for the deposition of Pt-C
deposits, as non-dissociated precursor molecules are incorporated in the
deposit. Afterwards each of the identical deposits was irradiated with the
electron-beam of the SEM using: (1) standard parameters serving as a reference
sample ($t_{D}=1\leavevmode\nobreak\ \mu s$,
$p_{x}=p_{y}=40\leavevmode\nobreak\ nm$), (2) GA for dwell-time optimization
($t_{D}=0.2-1500\leavevmode\nobreak\ \mu s$,
$p_{x}=p_{y}=40\leavevmode\nobreak\ nm$), and (3) GA for pitch optimization
$t_{D}=1\leavevmode\nobreak\ \mu s$, $p_{x},p_{y}=10-100\leavevmode\nobreak\
nm$ (Figure 5). As can be seen in Figure 5, in contrast to the previous
experiments for the deposition of $\rm W(CO)_{6}$, the variation of the
irradiation parameters for dwell-time and pitch does not influence the rate of
change of conductance over time which is in all cases very strong. Therefore,
for electron post-treatment of samples deposited with the Pt-based precursor
$\rm MeCpPt(Me)_{3}$ no parameter sets resulting in a significantly faster
enhancement of the conductance could be identified with the GA.
Figure5.png
Figure 5: Time-dependent rate of change of conductance for Pt-C deposits - The
GA is applied for the optimization of conductance during post-irradiation with
electrons ($U=5\leavevmode\nobreak\ kV$, $I_{nominal}=6.3\leavevmode\nobreak\
nA$). Reference sample (blue): $t_{D}=1\leavevmode\nobreak\ \mu s$,
$p_{x}=p_{y}=40\leavevmode\nobreak\ nm$, sample for GA dwell-time optimization
(red): $t_{D}=0.2-1500\leavevmode\nobreak\ \mu s$,
$p_{x}=p_{y}=40\leavevmode\nobreak\ nm$, sample for GA pitch optimization
(black): $t_{D}=1\leavevmode\nobreak\ \mu s$,
$p_{x},p_{y}=10-100\leavevmode\nobreak\ nm$. A variation of the beam-
parameters dwell-time and pitch during post-growth electron treatment does not
influence the rate of change of conductance during e-beam irradiation for Pt-C
deposits compared to the reference (inset). The offsets in the conductance
data result from small variations of conductance of the seed layer.
According to Plank et al. the post-growth irradiation-induced dissociation of
incorporated molecules leads to the creation of small Pt-crystallites between
existing Pt-crystals in the nanogranular structure of Pt-C or to a growth of
the previously present Pt crystallites leading to a reduction of the
intergrain distance and therefore to decreasing resistivity [34]. We found
that, as already shown in previous experiments [10], the resistivity could be
reduced during e-beam curing, however, independent of dwell-time and pitch.
This can be expected because precursor depletion as the dominant factor during
deposition does not play a role during e-beam curing. Furthermore, effects
like the growth of existing Pt crystals should depend on the electron dose
rather than on parameters such as dwell-time or pitch for post-irradiation of
samples at fixed dose.
## 5 Conclusion
In this work we presented the application of an evolutionary GA for the in-
situ optimization of FEBID nanostructures with regard to their electrical
conductivity. By using the gradient of the measured in-situ rate of change of
conductance as a fitness parameter the GA was able to tune the metal content
of tungsten deposits created from $\rm W(CO)_{6}$ over a large range by either
targeting the highest or lowest conductance, respectively. This resulted in a
difference in conductivity of one order of magnitude. This experiment
highlights the effectiveness of the procedure for precursors for which the
chemical composition of the deposit is sensitive to the deposition parameters.
In a second experiment the GA was applied for post-treatment of Pt-C deposits
obtained from the precursor $\rm MeCpPt(Me)_{3}$ by electron-beam irradiation.
For this system the GA revealed that solely the applied electron dose and not
specific irradiation parameters leads to the observed strong increase of
conductance over time.
The presented technique can be applied to all beam-induced deposition
processes and has great potential for further optimization or tuning of
parameters for nanostrucures prepared by FEBID or related techniques. In
particular finding optimized deposition parameters for new precursor
materials, which in general is a very time-consuming and often an arbitrary
process, can be achieved in a fast and efficient way. The GA’s independence of
the mechanism responsible for the enhancement of conductance (e.g. increase of
metal content, changes of height of the deposit, structural or phase changes,
etc.) and its adaption to every experimental circumstance with direct feedback
promises significant potential for future FEBID research. Furthermore, the
application of the GA is not restricted to the optimization of conductance but
can also be applied to e.g. optimize dielectric properties of FEBID deposits
using capacative measurements or optical reflectivity. Especially it will play
a major role for the analysis and optimization of FEBID binary systems that
have been recently adressed [12, 13, 14]. Some of us were able to stabilize an
amorphous, metastable $\rm Pt_{2}Si_{3}$ phase showing a maximum of
conductivity compared to other Pt-Si samples with different stoichiometric
proportions of platinum and silicon [12]. In follow-up experiments it will be
shown, that the GA can be applied to obtain deposition parameters for binary
systems e.g. Pt-Si or Co-Pt, which automatically lead to the formation of
binary phases with highest conductivity.
###### Acknowledgements.
The authors acknowledge financial support by the Beilstein-Institut,
Frankfurt/Main, Germany within the research collaboration NanoBiC.
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|
arxiv-papers
| 2013-10-14T17:07:53 |
2024-09-04T02:49:52.401247
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Paul M. Weirich, Marcel Winhold, Christian H. Schwalb and Michael Huth",
"submitter": "Paul M. Weirich",
"url": "https://arxiv.org/abs/1310.3749"
}
|
1310.3933
|
.
# Examples of quasitoric manifolds as special unitary manifolds
Zhi Lü and Wei Wang School of Mathematical Sciences, Fudan University,
Shanghai, 200433, P.R. China. [email protected] College of Information
Technology, Shanghai Ocean University, 999 Hucheng Huan Road, 201306,
Shanghai, P.R. China [email protected]
###### Abstract.
This note shows that for each $n\geq 5$ with only $n\not=6$, there exists a
$2n$-dimensional specially omnioriented quasitoric manifold $M^{2n}$ which
represents a nonzero element in $\Omega_{*}^{U}$. This provides the
counterexamples of Buchstaber–Panov–Ray conjecture.
###### Key words and phrases:
Unitary bordism, special unitary manifold, quasitoric manifold, small cover,
Stong manifold.
###### 2010 Mathematics Subject Classification:
57S10, 57R85, 14M25, 52B70.
Supported by grants from NSFC (No. 11371093, No. 11301335 and No. 10931005)
and RFDP (No. 20100071110001).
## 1\. Introduction
Let $\Omega_{*}^{U}$ denote the ring formed by the unitary bordism classes of
all unitary manifolds, where a unitary manifold is an oriented closed smooth
manifold whose tangent bundle admits a stably complex structure. In [5], Davis
and Januszkiewicz introduced and studied a class of nicely behaved manifolds
$M^{2n}$, so-called the quasitoric manifolds (as the topological versions of
toric varieties), each of which admits a locally standard $T^{n}$-action such
that the orbit space of action is homeomorphic to a simple convex polytope.
Buchstaber and Ray showed in [3] that each quasitoric manifold with an
omniorientation always admits a compatible tangential stably complex
structure, so omnioriented quasitoric manifolds provide abundant examples of
unitary manifolds. In particular, Buchstaber and Ray also showed in [3] that
each class of $\Omega_{2n}^{U}$ contains an omnioriented quasitoric
$2n$-manifold as its representative. In [2], Buchstaber, Panov and Ray
investigated the property of specially omnioriented quasitoric manifolds, and
proved that if $n<5$, then each $2n$-dimensional specially omnioriented
quasitoric manifold represents the zero element in $\Omega_{2n}^{U}$, where
the word “specially” for a specially omnioriented quasitoric manifold means
that the first Chern class vanishes. Furthermore, they posed the following
conjecture.
Conjecture ($\star$):Let $M^{2n}$ be a specially omnioriented quasitoric
manifold. Then $M^{2n}$ represents the zero element in $\Omega_{2n}^{U}$.
The purpose of this note is to construct some examples of specially
omnioriented quasitoric manifolds that are not bordant to zero in
$\Omega_{*}^{U}$, which give the negative answer to the above conjecture in
almost all possible dimensional cases. Our main result is stated as follows.
###### Theorem 1.1.
For each $n\geq 5$ with only $n\not=6$, there exists a $2n$-dimensional
specially omnioriented quasitoric manifold $M^{2n}$ which represents a nonzero
element in $\Omega_{*}^{U}$.
Our strategy is related to the unoriented bordism theory. Milnor’s work tells
us in [7] (see also [8]) that there is an epimorphism
$\begin{CD}F_{*}:\Omega^{U}_{*}@ >>>\mathfrak{N}^{2}_{*}\end{CD}$
where $\mathfrak{N}_{*}$ denotes the ring produced by the unoriented bordism
classes of all smooth closed manifolds, and
$\mathfrak{N}^{2}_{*}=\\{\alpha^{2}|\alpha\in\mathfrak{N}_{*}\\}$. This
actually implies that there is a covering homomorphism
$\begin{CD}H_{n}:\Omega^{U}_{2n}@ >>>\mathfrak{N}_{n}\end{CD}$
which is induced by $\theta_{n}\circ F_{n}$, where
$\theta_{n}:\mathfrak{N}^{2}_{n}\longrightarrow\mathfrak{N}_{n}$ is defined by
mapping $\alpha^{2}\longmapsto\alpha$. On the other hand, Buchstaber and Ray
showed in [3] that each class of $\mathfrak{N}_{n}$ contains an
$n$-dimensional small cover as its representative, where a small cover is also
introduced by Davis and Januszkiewicz in [5], and it is the real analogue of a
quasitoric manifold. In addition, Davis and Januszkiewicz tell us in [5] that
each quasitoric manifold $M^{2n}$ over a simple convex polytope $P^{n}$ always
admits a natural conjugation involution $\tau$ whose fixed point set
$M^{\tau}$ is just a small cover over $P^{n}$. In particular, this conjugation
involution $\tau$ is independent of the choices of omniorientations on
$M^{2n}$, and by [5, Corollaries 6.7–6.8], one has that the mod 2 reductions
of all Chern numbers of $M^{2n}$ with an omniorientation determine all
Stiefel–Whitney numbers of $M^{\tau}$, and in particular,
$\\{M^{2n}\\}=\\{M^{\tau}\\}^{2}$ as unoriented bordism classes in
$\mathfrak{N}_{*}$. Thus, $\tau$ induces a homomorphism
$\phi_{n}^{\tau}:\Omega^{U}_{2n}\longrightarrow\mathfrak{N}_{n}$, which
exactly agrees with the above homomorphism
$H_{n}:\Omega^{U}_{2n}\longrightarrow\mathfrak{N}_{n}$.
With the above understood, to obtain the counterexamples of
Buchstaber–Panov–Ray conjecture, an approach is to construct the examples of
specially omnioriented quasitoric manifolds whose images under $\phi^{\tau}$
are nonzero in $\mathfrak{N}_{*}$. We shall see that Stong manifolds play an
important role in our argument.
This note is organized as follows. We shall review the notions and basic
properties of quasitoric manifolds and small covers, and state the related
result of Buchstaber–Panov–Ray on specially omnioriented quasitoric manifolds
in Section 2. We shall review the Stong’s work on Stong manifolds and
construct some nonbounding orientable Stong manifolds in Section 3. In
addition, we also calculate the characteristic matrices of Stong manifolds
there. In Section 4 we shall construct required examples of omnioriented
quasitoric manifolds as special unitary manifolds and complete the proof of
our main result.
## 2\. Quasitoric manifolds and small covers
Davis and Januszkiewicz in [5] introduced and studied two kinds of equivariant
manifolds–quasitoric manifolds and small covers, whose geometric and algebraic
topology has a strong link to the combinatorics of polytopes. Following [5],
let
$G_{d}^{n}=\begin{cases}({\mathbb{Z}}_{2})^{n}&\text{ if }d=1\\\ T^{n}&\text{
if }d=2\end{cases}\ \ \text{and }\ \
R_{d}=\begin{cases}{\mathbb{Z}}_{2}&\text{ if }d=1\\\ {\mathbb{Z}}&\text{ if
}d=2.\end{cases}$
A $G_{d}^{n}$-manifold $\pi_{d}:M^{dn}\longrightarrow P^{n}$ $(d=1,2)$ is a
smooth closed $(dn)$-dimensional $G_{d}^{n}$-manifold admitting a locally
standard $G_{d}^{n}$-action such that its orbit space is a simple convex
$n$-polytope $P^{n}$. Such a $G_{d}^{n}$-manifold is called a small cover if
$d=1$ and a quasitoric manifold if $d=2$.
For a simple convex polytope $P^{n}$, let $\mathcal{F}(P^{n})$ denote the set
of all facets (i.e., $(n-1)$-dimensional faces) of $P^{n}$. We know from [5]
that each $G_{d}^{n}$-manifold $\pi_{d}:M^{dn}\longrightarrow P^{n}$
determines a characteristic function $\lambda_{d}$ on $P^{n}$
$\lambda_{d}:\mathcal{F}(P^{n})\longrightarrow R_{d}^{n}$
defined by mapping each facet in $\mathcal{F}(P^{n})$ to nonzero elements of
$R_{d}^{n}$ such that $n$ facets meeting at each vertex are mapped to a basis
of $R_{d}^{n}$. Conversely, the pair $(P^{n},\lambda_{d})$ can be used to
reconstruct $M^{dn}$ as follows: first $\lambda_{d}$ gives the following
equivalence relation $\sim_{\lambda_{d}}$ on $P^{n}\times G_{d}^{n}$
(2.1)
$(x,g)\sim_{\lambda_{d}}(y,h)\Longleftrightarrow\begin{cases}x=y,g=h&\text{ if
}x\in\text{\rm int}(P^{n})\\\ x=y,g^{-1}h\in G_{F}&\text{ if }x\in\text{\rm
int}F\subset\partial P^{n}\end{cases}$
then the quotient space $P^{n}\times G_{d}^{n}/\sim_{\lambda_{d}}$, denoted by
$M(P^{n},\lambda_{d})$, is the reconstruction of $M^{dn}$, where $G_{F}$ is
explained as follows: for each point $x\in\partial P^{n}$, there exists a
unique face $F$ of $P^{n}$ such that $x$ is in its relative interior. If $\dim
F=k$, then there are $n-k$ facets, say $F_{i_{1}},...,F_{i_{n-k}}$, such that
$F=F_{i_{1}}\cap\cdots\cap F_{i_{n-k}}$, and furthermore,
$\lambda_{d}(F_{i_{1}}),...,\lambda_{d}(F_{i_{n-k}})$ determine a subgroup of
rank $n-k$ in $G_{d}^{n}$, denoted by $G_{F}$. This reconstruction of $M^{dn}$
tells us that the topology of $\pi_{d}:M^{dn}\longrightarrow P^{n}$ can be
determined by $(P^{n},\lambda_{d})$.
###### Remark 1.
If we fix an ordering for all facets in $\mathcal{F}(P)$ (e.g., say
$F_{1},...,F_{m}$) , then the characteristic function
$\lambda_{d}:\mathcal{F}(P^{n})\longrightarrow R_{d}^{n}$ uniquely determines
a matrix of size $n\times m$ over $R_{d}$
$\Lambda_{d}=(\lambda_{d}(F_{1}),\cdots,\lambda_{d}(F_{m}))$
with $\lambda_{d}(F_{i})$ as columns, which is called the characteristic
matrix of $(P^{n},\lambda_{d})$ or $M(P^{n},\lambda_{d})$.
We may see from this reconstruction of $G_{d}^{n}$-manifolds that there is
also an essential relation between small covers and quasitoric manifolds over
a simple polytope. In fact, given a quasitoric manifold $M(P^{n},\lambda_{2})$
over $P^{n}$, as shown in [5, Corollary 1.9], there is a natural conjugation
involution on $P^{n}\times T^{n}$ defined by $(p,g)\longmapsto(p,g^{-1})$,
which fixes $P^{n}\times({\mathbb{Z}}_{2})^{n}$. Then this involution descends
an involution $\tau$ on $M(P^{n},\lambda_{2})$ whose fixed point set is
exactly a small cover $M(P^{n},\lambda_{1})$ over $P^{n}$, where $\lambda_{1}$
is the mod 2 reduction of $\lambda_{2}$.
As shown in [3], an omniorientation of a quasitoric manifold
$\pi:M(P^{n},\lambda_{2})\longrightarrow P^{n}$ is just one choice of
orientations of $M(P^{n},\lambda_{2})$ and submanifolds
$\pi^{-1}(F),F\in\mathcal{F}(P^{n})$. Thus, a quasitoric manifold
$\pi:M(P^{n},\lambda_{2})\longrightarrow P^{n}$ has $2^{m+1}$
omniorientations, where $m$ is the number of all facets of $P^{n}$. Clearly,
the conjugation involution $\tau$ on $M(P^{n},\lambda_{2})$ is independent of
the choices of omniorientations of $M(P^{n},\lambda_{2})$. Now let
$\mathcal{O}(M(P^{n},\lambda_{2}))$ denote the set of all $2^{m+1}$
omniorientations. Buchstaber and Ray showed in [3] (also see [2]) that for
each omniorientation $\mathfrak{o}\in\mathcal{O}(M(P^{n},\lambda_{2}))$,
$M(P^{n},\lambda_{2})$ with this omniorientation $\mathfrak{o}$ always admits
a tangential stably complex structure, so it is a unitary manifold. In [2],
Buchstaber, Panov and Ray gave a characterization for $M(P^{n},\lambda_{2})$
with $\mathfrak{o}\in\mathcal{O}(M(P^{n},\lambda_{2}))$ to be a special
unitary manifold in terms of $\lambda_{2}$, which is stated as follows.
###### Proposition 2.1 ([2]).
Let $M(P^{n},\lambda_{2})$ be a quasitoric manifold. Then
$M(P^{n},\lambda_{2})$ with an omniorientation
$\mathfrak{o}\in\mathcal{O}(M(P^{n},\lambda_{2}))$ is a special unitary
manifold if and only if for each facet $F\in\mathcal{F}(P^{n})$, the sum of
all entries of $\lambda_{2}(F)$ is exactly $1$.
## 3\. Stong manifolds
### 3.1. Stong manifolds
In [9], Stong introduced the Stong manifolds, from which all generators of the
unoriented bordism ring $\mathfrak{N}_{*}$ can be chosen. A Stong manifold is
defined as the real projective space bundle denoted by
${\mathbb{R}}P(n_{1},...,n_{k})$ of the bundle
$\gamma_{1}\oplus\cdots\oplus\gamma_{k}$ over
${\mathbb{R}}P^{n_{1}}\times\cdots\times{\mathbb{R}}P^{n_{k}}$, where
$\gamma_{i}$ is the pullback of the canonical bundle over the $i$-th factor
${\mathbb{R}}P^{n_{i}}$. The Stong manifold ${\mathbb{R}}P(n_{1},...,n_{k})$
has dimension $n_{1}+\cdots+n_{k}+k-1$.
As shown in [9], the cohomology with ${\mathbb{Z}}_{2}$ coefficients of
${\mathbb{R}}P(n_{1},...,n_{k})$ is the free module over the cohomology of
${\mathbb{R}}P^{n_{1}}\times\cdots\times{\mathbb{R}}P^{n_{k}}$ on
$1,e,...,e^{k-1}$, where $e$ is the first Stiefel-Whitney class of the
canonical line bundle over ${\mathbb{R}}P(n_{1},...,n_{k})$, with the relation
$e^{k}=w_{1}e^{k-1}+\cdots+w_{r}e^{k-r}+\cdots+w_{k}$
where $w_{i}$ is the $i$-th Sitefel-Whitney class of
$\gamma_{1}\oplus\cdots\oplus\gamma_{k}$. Then the total Stiefel-Whitney class
of ${\mathbb{R}}P(n_{1},...,n_{k})$ is
(3.1) $\prod_{i=1}^{k}(1+a_{i})^{n_{i}+1}(1+a_{i}+e)$
where $a_{i}$ is the pullback of the nonzero class in
$H^{1}({\mathbb{R}}P^{n_{i}};{\mathbb{Z}}_{2})$.
###### Remark 2.
In fact, it is easy to see that the total Stiefel-Whitney class of
$\gamma_{1}\oplus\cdots\oplus\gamma_{k}$ is exactly
$w(\gamma_{1}\oplus\cdots\oplus\gamma_{k})=\prod_{i=1}^{k}(1+a_{i}).$
So the cohomology with ${\mathbb{Z}}_{2}$ coefficients of
${\mathbb{R}}P(n_{1},...,n_{k})$ may be written as
${\mathbb{Z}}_{2}[a_{1},...,a_{k},e]/A$
where $A$ is the ideal generated by $a_{1}^{n_{1}+1},...,a_{k}^{n_{k}+1}$, and
$\prod_{i=1}^{k}(a_{i}+e)$.
Stong further showed in [9] that
###### Proposition 3.1 ([9]).
For $k>1$, ${\mathbb{R}}P(n_{1},...,n_{k})$ is indecomposable in
$\mathfrak{N}_{*}$ if and only if
${{\ell+k-2}\choose{n_{1}}}+\cdots+{{\ell+k-2}\choose{n_{k}}}\equiv 1\mod 2$
where $\ell=n_{1}+\cdots+n_{k}$.
It is not difficult to see from the expression (3.1) of the total Stiefel-
Whitney class of ${\mathbb{R}}P(n_{1},...,n_{k})$ that
###### Corollary 3.2.
${\mathbb{R}}P(n_{1},...,n_{k})$ is orientable if and only if $k$ and all
$n_{i}$ are even.
By Proposition 3.1 and Corollary 3.2, we may choose the following examples of
indecomposable, orientable Stong manifolds. For $l\geq 0$,
${\mathbb{R}}P(2,\underbrace{0,...,0}_{4l+3})$ and
${\mathbb{R}}P(4,2,\underbrace{0,...,0}_{8l+4})$ are indecomposable and
orientable, so they represent nonzero elements in $\mathfrak{N}_{*}$. Let
$\alpha_{4l+5}$ and $\alpha_{8l+11}$ denote the unoriented bordism classes of
${\mathbb{R}}P(2,\underbrace{0,...,0}_{4l+3})$ and
${\mathbb{R}}P(4,2,\underbrace{0,...,0}_{8l+4})$, respectively. Then we have
that
###### Lemma 3.3.
All $\alpha_{4l+5}$ and $\alpha_{8l+11}$ with $l\geq 0$ form a polynomial
subring
${\mathbb{Z}}_{2}[\alpha_{4l+5},\alpha_{8l+11}|l\geq 0]$
of $\mathfrak{N}_{*}$, which contains nonzero classes of dimension
$\not=1,2,3,4,6,7,8,12$.
### 3.2. Characteristic matrices of Stong manifolds
We see that ${\mathbb{R}}P(n_{1},...,n_{k})$ is a ${\mathbb{R}}P^{k-1}$-bundle
over ${\mathbb{R}}P^{n_{1}}\times\cdots\times{\mathbb{R}}P^{n_{k}}$, so it is
a special generalized real Bott manifold, and in particular, it is also a
small cover over
$\Delta^{n_{1}}\times\cdots\times\Delta^{n_{k}}\times\Delta^{k-1}$, where
$\Delta^{l}$ denotes a $l$-dimensional simplex.
###### Remark 3.
A generalized real Bott manifold of is the total space $B^{\mathbb{R}}_{k+1}$
of an iterated fiber bundle:
$\begin{CD}B^{\mathbb{R}}_{k+1}@
>{\pi_{k+1}}>>B^{\mathbb{R}}_{k}@>{\pi_{k}}>{}>\cdots
@>{\pi_{2}}>{}>B^{\mathbb{R}}_{1}@ >{\pi_{1}}>>B^{\mathbb{R}}_{0}=\\{\text{a
point}\\}\end{CD}$
where each $\pi_{i}:B^{\mathbb{R}}_{i}\longrightarrow B^{\mathbb{R}}_{i-1}$ is
the projectivization of a Whitney sum of $n_{i}+1$ real line bundles over
$B^{\mathbb{R}}_{i}$. It is well-known that the generalized real Bott manifold
$B^{\mathbb{R}}_{k+1}$ is a small cover over
$\Delta^{n_{1}}\times\cdots\times\Delta^{n_{k+1}}$. Conversely, we also know
from [4] that a small cover over a product of simplices is a generalized real
Bott manifold.
Now let us look at the characteristic matrix of
${\mathbb{R}}P(n_{1},...,n_{k})$ as a small cover over the product
$P=\Delta^{n_{1}}\times\cdots\times\Delta^{n_{k}}\times\Delta^{k-1}$ with
$k>1$ and $n_{1}\geq n_{2}\geq\cdots\geq n_{k}>0$. Clearly $P$ has
$n_{1}+\cdots+n_{k}+2k$ facets, which are listed as follows:
$F_{n_{i},j}=\Delta^{n_{1}}\times\cdots\times\Delta^{n_{i-1}}\times\Delta^{(n_{i})}_{j}\times\Delta^{n_{i+1}}\times\cdots\times\Delta^{n_{k}}\times\Delta^{k-1},1\leq
j\leq n_{i}+1,1\leq i\leq k$
and
$F_{k-1,j}=\Delta^{n_{1}}\times\cdots\times\Delta^{n_{k}}\times\Delta^{(k-1)}_{j},1\leq
j\leq k$
where $\Delta^{(l)}_{j},j=1,...,l+1$, denote $l+1$ facets of $\Delta^{l}$.
Throughout the following, we shall carry out our work on a fixed ordering of
all facets of
$P=\Delta^{n_{1}}\times\cdots\times\Delta^{n_{k}}\times\Delta^{k-1}$ as
follows:
$F_{n_{1},1},...,F_{n_{1},n_{1}+1},...,F_{n_{k},1},...,F_{n_{k},n_{k}+1},F_{k-1,1},...,F_{k-1,k}.$
###### Proposition 3.4.
Up to automorphisms of $({\mathbb{Z}}_{2})^{n_{1}+\cdots+n_{k}+k-1}$, the
characteristic matrix $\Lambda_{1}^{(n_{1},...,n_{k})}$ of
${\mathbb{R}}P(n_{1},...,n_{k})$ may be written as
$\displaystyle\left(\begin{array}[]{ccccccccc}I_{n_{1}}&\textbf{1}_{n_{1}}&&&&&&\\\
&&\ddots&&&&&&\\\ &&&I_{n_{k-1}}&\textbf{1}_{n_{k-1}}&&&&\\\
&&&&&I_{n_{k}}&\textbf{1}_{n_{k}}&&\\\
&J_{1}&\cdots&&J_{k-1}&&\textbf{1}_{k-1}&I_{k-1}&\textbf{1}_{k-1}\\\
\end{array}\right)$
with only blocks $I_{i}$, $\textbf{1}_{i}$ $(i=n_{1},...,n_{k},k-1)$ and
$J_{j}(j=1,...,k-1)$ being nonzero, and $0$ otherwise, where $I_{i}$ denotes
the identity matrix of size $i\times i$, $J_{j}$ denotes the matrix of size
$(k-1)\times 1$ with only $(j,1)$-entry being $1$ and $0$ otherwise, and
$\textbf{1}_{i}$ denotes the matrix of size $i\times 1$ with all entries being
$1$.
###### Proof.
Without the loss of generality, assume that the values of the characteristic
function $\lambda_{1}^{(n_{1},...,n_{k})}$ on the following
$n_{1}+\cdots+n_{k}+k-1$ facets
$F_{n_{1},1},...,F_{n_{1},n_{1}},...,F_{n_{k},1},...,F_{n_{k},n_{k}},F_{k-1,1},...,F_{k-1,k-1}$
meeting at a vertex are all columns with an ordering from the first column to
the last column in $I_{n_{1}+\cdots+n_{k}+k-1}$, respectively. It suffices to
determine the values of $\lambda_{1}^{(n_{1},...,n_{k})}$ on the $k+1$ facets
$F_{n_{1},n_{1}+1},F_{n_{2},n_{2}+1},...,F_{n_{k},n_{k}+1},F_{k-1,k}$. By [6,
Lemma 6.2], we have that for $1\leq i\leq k$
$\lambda_{1}^{(n_{1},...,n_{k})}(F_{n_{i},n_{i}+1})=\sum_{j=1}^{n_{i}}\lambda_{1}^{(n_{1},...,n_{k})}(F_{n_{i},j})+\beta_{i}$
and
$\lambda_{1}^{(n_{1},...,n_{k})}(F_{k-1,k})=\sum_{j=1}^{k-1}\lambda_{1}^{(n_{1},...,n_{k})}(F_{k-1,j})+\beta_{k+1}.$
In particular, we also know by [6, Lemma 6.3] that there is at least one
$\beta_{i}$ such that $\beta_{i}=0$ in
$({\mathbb{Z}}_{2})^{n_{1}+\cdots+n_{k}+k-1}$.
Now by [5, Theorem 4.14], we may write
$H^{*}({\mathbb{R}}P(n_{1},...,n_{k});{\mathbb{Z}}_{2})$ as
${\mathbb{Z}}_{2}[F_{n_{1},1},...,F_{n_{1},n_{1}+1},...,F_{n_{k},1},...,F_{n_{k},n_{k}+1},F_{k-1,1},...,F_{k-1,k}]/I_{P}+J_{\lambda_{1}^{(n_{1},...,n_{k})}}$
where the $F_{i,j}$ are used as indeterminants of degree 1, $I_{P}$ is the
Stanley-Reisner ideal generated by
$\prod_{j=1}^{n_{i}+1}F_{n_{i},j}(i=1,...,k)$ and $\prod_{i=1}^{k}F_{k-1,i}$,
and $J_{\lambda_{1}^{(n_{1},...,n_{k})}}$ is the ideal determined by
$\lambda_{1}^{(n_{1},...,n_{k})}$. Furthermore, we have by [5, Corollary 6.8]
that the total Stiefel-Whitney class of ${\mathbb{R}}P(n_{1},...,n_{k})$ is
$\prod_{i=1}^{k}\Big{(}\prod_{j=1}^{n_{i}+1}(1+F_{n_{i},j})\Big{)}(1+F_{k-1,i}).$
Comparing with the formula (3.1) or by Remark 2, we see that for each $1\leq
i\leq k$,
$F_{n_{i},1}=\cdots=F_{n_{i},n_{i}+1}\text{ (denoted by }a_{i})$
so $a_{i}^{n_{i}+1}=\prod_{j=1}^{n_{i}+1}F_{n_{i},j}=0$. This implies that
$\beta_{k+1}$ must be the zero element, and for $1\leq i\leq k$, each
$\beta_{i}$ is of the form
$(\underbrace{0,...,0}_{n_{1}+\cdots+n_{k}},\beta_{i,1},...,\beta_{i,k-1})^{\top}$
in $({\mathbb{Z}}_{2})^{n_{1}+\cdots+n_{k}+k-1}$. Moreover, one has that
(3.3)
$\begin{cases}F_{k-1,1}=F_{k-1,k}+\beta_{1,1}F_{n_{1},n_{1}+1}+\cdots+\beta_{k,1}F_{n_{k},n_{k}+1}\\\
\cdots\\\
F_{k-1,k-1}=F_{k-1,k}+\beta_{1,k-1}F_{n_{1},n_{1}+1}+\cdots+\beta_{k,k-1}F_{n_{k},n_{k}+1}\end{cases}$
Comparing with the formula (3.1) again, one should have that
$\prod_{i=1}^{k}(1+F_{k-1,i})=\prod_{i=1}^{k}(1+a_{i}+e)=\prod_{i=1}^{k}(1+F_{n_{i},n_{i}+1}+e).$
Without the loss of generality, assume that
$1+F_{k-1,i}=1+F_{n_{i},n_{i}+1}+e$ for $1\leq i\leq k$. Then for $i=k$, one
has that $e=F_{k-1,k}+F_{n_{k},n_{k}+1}$, and for $1\leq i<k$, one has by
(3.3) that
$\beta_{1,i}F_{n_{1},n_{1}+1}+\cdots+\beta_{k,i}F_{n_{k},n_{k}+1}=F_{n_{i},n_{i}+1}+F_{n_{k},n_{k}+1}$
so $\beta_{i,i}=\beta_{k,i}=1$ and $\beta_{j,i}=0$ if $j\not=i,k$ since
$F_{n_{1},n_{1}+1},...,F_{n_{k},n_{k}+1}$ are linearly independent in
$H^{1}({\mathbb{R}}P(n_{1},...,n_{k});{\mathbb{Z}}_{2})$. This completes the
proof. ∎
If there is a minimal integer $i$ with $1\leq i<k$ such that $n_{i}>0$ but
$n_{i+1}=0$ (so $n_{j}=0$ for $j\geq i+1$), then a similar argument as above
gives
###### Proposition 3.5.
Suppose that there is some $i$ with $1\leq i<k$ such that $n_{1}\geq\cdots\geq
n_{i}>0$ and $n_{i+1}=\cdots=n_{k}=0$. Up to automorphisms of
$({\mathbb{Z}}_{2})^{n_{1}+\cdots+n_{i}+k-1}$, the characteristic matrix
$\Lambda_{1}^{(n_{1},...,n_{i},0,...,0)}$ of
${\mathbb{R}}P(n_{1},...,n_{i},0,...,0)$ may be written as
$\displaystyle\left(\begin{array}[]{ccccccc}I_{n_{1}}&\textbf{1}_{n_{1}}&&&&\\\
&&\ddots&&&&\\\ &&&I_{n_{i}}&\textbf{1}_{n_{i}}&&\\\
&J_{1}&\cdots&&J_{i}&I_{k-1}&\textbf{1}_{k-1}\\\ \end{array}\right)$
with only blocks $I_{j}$, $\textbf{1}_{j}$ $(j=n_{1},...,n_{i},k-1)$ and
$J_{l}(l=1,...,i)$ being nonzero, and $0$ otherwise, where $I_{j}$, $J_{l}$
and $\textbf{1}_{j}$ represent the same meanings as stated in Proposition 3.4.
## 4\. Proof of Main Result
### 4.1. Examples of specially omnioriented quasitoric manifolds
Throughout the following, for a $k$-dimensional simplex $\Delta^{k}$,
$\Delta^{(k)}_{i},i=1,...,k+1$ mean the $k+1$ facets of $\Delta^{k}$, and for
a product $P=\Delta^{k_{1}}\times\cdots\times\Delta^{k_{r}}$ of simplices,
$F_{k_{i},j}$ means that the facet
$\Delta^{k_{1}}\times\cdots\times\Delta^{k_{i-1}}\times\Delta^{(k_{i})}_{j}\times\Delta^{k_{i+1}}\times\cdots\times\Delta^{k_{r}}$
of $P$. Then let us construct some required examples.
###### Example 4.1.
Let $P^{4l+5}=\Delta^{2}\times\Delta^{4l+3}$ with $l\geq 0$. Define a
characteristic function $\lambda_{2}^{(2,0,...,0)}$ on $P^{4l+5}$ in the
following way. First let us fix an ordering of all facets of $P^{4l+5}$ as
follows
$F_{2,1},F_{2,2},F_{2,3},F_{4l+3,1},...,F_{4l+3,4l+3},F_{4l+3,4l+4}.$
Then we construct the characteristic matrix $\Lambda_{2}^{(2,0,...,0)}$ of the
required characteristic function $\lambda_{2}^{(2,0,...,0)}$ on the above
ordered facets as follows:
$\displaystyle\Lambda_{2}^{(2,0,...,0)}=\left(\begin{array}[]{cccc}I_{2}&\widetilde{\textbf{1}}_{2}&&\\\
&J_{1}&I_{4l+3}&\widetilde{\textbf{1}}_{4l+3}\\\ \end{array}\right)$
with only blocks $I_{j}$, $\widetilde{\textbf{1}}_{j}$ $(j=2,4l+3)$ and
$J_{1}$ being nonzero, and $0$ otherwise, where $I_{j}$ and $J_{1}$ denote the
same meanings as in Proposition 3.4, and $\widetilde{\textbf{1}}_{j}$ denotes
the matrix of size $j\times 1$ with $(2i,1)$-entries being $-1$ and other
entries being $1$. We see that the sum of all entries of each column in the
characteristic matrix $\Lambda_{2}^{(2,0,...,0)}$ is always 1. Thus, by
Proposition 2.1, one has that the quasitoric manifold
$M(P^{4l+5},\lambda_{2}^{(2,0,...,0)})$ with any omniorientation is a special
unitary manifold.
###### Example 4.2.
Let $P^{8l+11}=\Delta^{4}\times\Delta^{2}\times\Delta^{8l+5}$ with $l\geq 0$.
In a similar way as above, fix an ordering of all facets of $P^{8l+11}$ as
follows:
$F_{4,1},F_{4,2},F_{4,3},F_{4,4},F_{4,5},F_{2,1},F_{2,2},F_{2,3},F_{8l+5,1},...,F_{8l+5,8l+5},F_{8l+5,8l+6}.$
Then we define a characteristic function $\lambda_{2}^{(4,2,0,...,0)}$ on the
above ordered facets of $P^{8l+11}$ by the following characteristic matrix
$\displaystyle\Lambda_{2}^{(4,2,0,...,0)}=\left(\begin{array}[]{ccccccc}I_{4}&\widetilde{\textbf{1}}_{4}&&&&\\\
&&I_{2}&\widetilde{\textbf{1}}_{2}&&&\\\
&J_{1}&&J_{2}&&I_{8l+5}&\widetilde{\textbf{1}}_{8l+5}\\\ \end{array}\right)$
with only blocks $I_{i}$, $\widetilde{\textbf{1}}_{i}$ $(i=2,4,8l+5)$ and
$J_{j}(j=1,2)$ being nonzero, and $0$ otherwise, where $I_{i}$, $J_{j}$ and
$\widetilde{\textbf{1}}_{i}$ denote the same meanings as above. By Proposition
2.1, $M(P^{8l+11},\lambda_{2}^{(4,2,0,...,0)})$ with any omniorientation is a
special unitary manifold.
###### Example 4.3.
The case in which $n=7$. Consider the polytope
$P^{7}=\Delta^{4}\times\Delta^{3}$ with the following ordered facets
$F_{4,1},F_{4,2},F_{4,3},F_{4,4},F_{4,5},F_{3,1},F_{3,2},F_{3,3},F_{3,4}.$
Then we may define a characteristic function $\lambda_{2}^{<7>}$ on the
ordered facets of $P^{7}$ by the following characteristic matrix
$\displaystyle\left(\begin{array}[]{ccccccccc}1&&&&1&&&&\\\ &1&&&-1&&&&\\\
&&1&&1&&&&\\\ &&&1&-1&&&&\\\ &&&&1&1&&&1\\\ &&&&&&1&&-1\\\ &&&&&&&1&1\\\
\end{array}\right),$
which gives a special unitary manifold $M(P^{7},\lambda_{2}^{<7>})$. Moreover,
by the Davis–Januszkiewicz theory, we may read off the cohomology of
$M(P^{7},\lambda_{2}^{<7>})$ as follows:
$H^{*}(M(P^{7},\lambda_{2}^{<7>}))={\mathbb{Z}}[x,y]/<x^{5},y^{4}+xy^{3}>$
with $\deg x=\deg y=2$, and by [5, Theorem 4.8] and [2], the total Chern class
of $M(P^{7},\lambda_{2}^{<7>})$ may be written as
$c(M(P^{7},\lambda_{2}^{<7>}))=(1-x^{2})^{2}(1+x)(1-x-y)(1-y^{2})(1+y).$
A direct calculation gives the Chern number $\langle
c_{3}c_{4},[M(P^{7},\lambda_{2}^{<7>})]\rangle=-2\not=0$, which implies that
this specially omnioriented quasitoric manifold $M(P^{7},\lambda_{2}^{<7>})$
is not bordant to zero in $\Omega_{*}^{U}$.
###### Example 4.4.
The case in which $n=8$. Consider the polytope
$P^{8}=\Delta^{3}\times\Delta^{5}$ with the ordered facets as follows:
$F_{3,1},F_{3,2},F_{3,3},F_{3,4},F_{5,1},F_{5,2},F_{5,3},F_{5,4},F_{5,5},F_{5,6}.$
Then we may define a characteristic function $\lambda_{2}^{<8>}$ on the
ordered facets of $P^{8}$ by
$\displaystyle\left(\begin{array}[]{cccccccccc}1&&&1&&&&&&\\\ &1&&-1&&&&&&\\\
&&1&1&&&&&&\\\ &&&-1&1&&&&&1\\\ &&&1&&1&&&&-1\\\ &&&&&&1&&&1\\\
&&&&&&&1&&-1\\\ &&&&&&&&1&1\\\ \end{array}\right),$
which also gives a special unitary manifold $M(P^{8},\lambda_{2}^{<8>})$.
Similarly, one has the cohomology of $M(P^{8},\lambda_{2}^{<8>})$
$H^{*}(M(P^{8},\lambda_{2}^{<8>}))={\mathbb{Z}}[x,y]/<x^{4},y^{4}(x-y)^{2}>$
with $\deg x=\deg y=2$, and the total Chern class of
$M(P^{8},\lambda_{2}^{<8>})$
$c(M(P^{8},\lambda_{2}^{<8>}))=(1-x^{2})^{2}(1-y^{2})^{2}[1-(x-y)^{2}].$
Furthermore, one has the Chern number $\langle
c_{4}^{2},[M(P^{8},\lambda_{2}^{<8>})]\rangle=4\not=0$. So
$M(P^{8},\lambda_{2}^{<8>})$ is not bordant to zero in $\Omega_{*}^{U}$.
###### Example 4.5.
The case in which $n=12$. Consider the polytope
$P^{12}=\Delta^{3}\times\Delta^{9}$ with the ordered facets as follows:
$F_{3,1},F_{3,2},F_{3,3},F_{3,4},F_{9,1},F_{9,2},F_{9,3},F_{9,4},F_{9,5},F_{9,6},F_{9,7},F_{9,8},F_{9,9},F_{9,10},$
and define a characteristic function $\lambda_{2}^{<12>}$ on the ordered
facets of $P^{12}$ by the matrix
$\displaystyle\left(\begin{array}[]{cccccccccccccc}1&&&1&&&&&&&&&&\\\
&1&&-1&&&&&&&&&&\\\ &&1&1&&&&&&&&&&\\\ &&&-1&1&&&&&&&&&1\\\
&&&1&&1&&&&&&&&-1\\\ &&&&&&1&&&&&&&1\\\ &&&&&&&1&&&&&&-1\\\ &&&&&&&&1&&&&&1\\\
&&&&&&&&&1&&&&-1\\\ &&&&&&&&&&1&&&1\\\ &&&&&&&&&&&1&&-1\\\ &&&&&&&&&&&&1&1\\\
\end{array}\right),$
from which one obtains a special unitary manifold
$M(P^{12},\lambda_{2}^{<12>})$ with its cohomology
$H^{*}(M(P^{12},\lambda_{2}^{<12>}))={\mathbb{Z}}[x,y]/<x^{4},y^{8}(x-y)^{2}>\text{\rm
with }\deg x=\deg y=2$
and with its total Chern class
$c(M(P^{12},\lambda_{2}^{<12>}))=(1-x^{2})^{2}(1-y^{2})^{4}[1-(x-y)^{2}].$
Then one has that the 6-th Chern class
$c_{6}=-10y^{6}+12xy^{5}-26x^{2}y^{4}+16x^{3}y^{3}$, so the Chern number
$\langle c_{6}^{2},[M(P^{12},\lambda_{2}^{<12>})]\rangle=64\not=0$. Thus
$M(P^{12},\lambda_{2}^{<12>})$ is not bordant to zero in $\Omega_{*}^{U}$.
### 4.2. Proof of Theorem 1.1
Obviously, the mod 2 reductions of the characteristic matrices
$\Lambda_{2}^{(2,0,...,0)}$ and $\Lambda_{2}^{(4,2,0,...,0)}$ of
$M(P^{4l+5},\lambda_{2}^{(2,0,...,0)})$ and
$M(P^{8l+11},\lambda_{2}^{(4,2,0,...,0)})$ are
$\displaystyle\left(\begin{array}[]{cccc}I_{2}&\textbf{1}_{2}&&\\\
&J_{1}&I_{4l+3}&\textbf{1}_{4l+3}\\\ \end{array}\right)$
and
$\displaystyle\left(\begin{array}[]{ccccccc}I_{4}&\textbf{1}_{4}&&&&\\\
&&I_{2}&\textbf{1}_{2}&&&\\\ &J_{1}&&J_{2}&&I_{8l+5}&\textbf{1}_{8l+5}\\\
\end{array}\right)$
respectively. Thus, by Proposition 3.5, one has that the fixed point sets of
the conjugation involutions on $M(P^{4l+5},\lambda_{2}^{(2,0,...,0)})$ and
$M(P^{8l+11},\lambda_{2}^{(4,2,0,...,0)})$ are homeomorphic to the Stong
manifolds ${\mathbb{R}}P(2,\underbrace{0,...,0}_{4l+3})$ and
${\mathbb{R}}P(4,2,\underbrace{0,...,0}_{8l+4})$, respectively. Thus, the
subring of $\Omega_{*}^{U}$ generated by the unitary bordism classes of
$M(P^{4l+5},\lambda_{2}^{(2,0,...,0)})$ and
$M(P^{8l+11},\lambda_{2}^{(4,2,0,...,0)})$ is mapped onto the subring
${\mathbb{Z}}_{2}[\alpha_{4l+5},\alpha_{8l+11}|l\geq 0]$ of $\mathfrak{N}_{*}$
in Lemma 3.3 via $H_{*}:\Omega^{U}_{*}\longrightarrow\mathfrak{N}_{*}$. Then
Theorem 1.1 follows from this and Examples 4.3–4.5. $\Box$
###### Remark 4.
We have done many tries to find a counterexample in the case $n=6$, but
failed. It seems to be reasonable to the assertion as in the
Buchstaber–Panov–Ray conjecture that each 12-dimensional specially
omnioriented quasitoric manifold is bordant to zero in $\Omega_{*}^{U}$ since
each 6-dimensional orientable smooth closed manifold is always bordant to zero
in $\mathfrak{N}_{*}$.
## References
* [1] V. M. Buchstaber and T.E. Panov, Torus actions and their applications in topology and combinatorics, University Lecture Series, 24. American Mathematical Society, Providence, RI, 2002.
* [2] V. M. Buchstaber, T.E. Panov and N. Ray, Toric Genera, Internat. Math. Res. Notices 2010, No. 16, 3207–3262.
* [3] V. M. Buchstaber and N. Ray, Toric manifolds and complex cobordisms, Uspekhi Mat. Nauk 53 (1998), 139–140. In Russian; translated in Russ. Math. Surv. 53 (1998), 371–373.
* [4] S. Y. Choi, M. Masuda and D. Y. Suh, Quasitoric manifolds over a product of simplices, Osaka J. Math. 47 (2010), 109–129.
* [5] M. Davis and T. Januszkiewicz, Convex polytopes, Coxeter orbifolds and torus actions, Duke Math. J. 61 (1991), 417-451.
* [6] Z. Lü and Q. B. Tan, Small covers and the equivariant bordism classification of 2-torus manifolds, Int. Math. Res. Notices (First published online: September 3, 2013), doi: 10.1093/imrn/rnt183. arXiv:1008.2166
* [7] J.W. Milnor, On the Stiefel–Whitney numbers of complex manifolds and of spin manifolds, Topology, 3 (1965), 223–230.
* [8] R.E. Stong, Notes on cobordism theory, Princeton University Press, 1968.
* [9] R.E. Stong, On Fibering of Cobordism Classes, Trans. Amer. Math. Soc. 178 (1973), 431–447.
|
arxiv-papers
| 2013-10-15T07:06:17 |
2024-09-04T02:49:52.412419
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Zhi L\\\"u and Wei Wang",
"submitter": "Zhi L\\\"u",
"url": "https://arxiv.org/abs/1310.3933"
}
|
1310.3954
|
# Sparse Solution of Underdetermined Linear Equations via Adaptively Iterative
Thresholding
Jinshan Zeng111 E-mail: [email protected] Shaobo Lin222 E-mail:
[email protected] Zongben Xu333Corresponding author,E-mail:
[email protected] School of Mathematics and Statistics, Xi’an Jiaotong
University, Xi’an, 710049, China
###### Abstract
Finding the sparset solution of an underdetermined system of linear equations
$y=Ax$ has attracted considerable attention in recent years. Among a large
number of algorithms, iterative thresholding algorithms are recognized as one
of the most efficient and important classes of algorithms. This is mainly due
to their low computational complexities, especially for large scale
applications. The aim of this paper is to provide guarantees on the global
convergence of a wide class of iterative thresholding algorithms. Since the
thresholds of the considered algorithms are set adaptively at each iteration,
we call them adaptively iterative thresholding (AIT) algorithms. As the main
result, we show that as long as $A$ satisfies a certain coherence property,
AIT algorithms can find the correct support set within finite iterations, and
then converge to the original sparse solution exponentially fast once the
correct support set has been identified. Meanwhile, we also demonstrate that
AIT algorithms are robust to the algorithmic parameters. In addition, it
should be pointed out that most of the existing iterative thresholding
algorithms such as hard, soft, half and smoothly clipped absolute deviation
(SCAD) algorithms are included in the class of AIT algorithms studied in this
paper.
###### keywords:
Iterative thresholding algorithm; global convergence; underdetermined linear
equations; sparse solution.
††journal: Signal Processing
## 1 Introduction
Finding the sparsest solution of an undertermined system of linear equations
is an important problem emerged in many applications (especially, in
compressed sensing [1], [2]). Generally, an undertermined system of linear
equations can be described as
$y=Ax,$ (1.1)
where $y\in\mathbf{R}^{M}$ and $A\in\mathbf{R}^{M\times N}$ ($M<N$) are known,
$x=(x_{1},\dots,x_{N})^{T}\in\mathbf{R}^{N}$ is unknown. Thus, finding the
sparsest solution of the equations (1.1) can be mathematically modeled as the
following $l_{0}$-minimization, that is,
$\min_{x\in\mathbf{R}^{N}}{\|x\|_{0}}\ \ \text{s.t.}\ y=Ax,$ (1.2)
where $\|x\|_{0}$ denotes the number of the nonzero components of $x$ and is
formally called the $l_{0}$-norm. However, the problem (1.2) is NP-hard and
generally intractable for computing.
Instead, there are mainly two classes of methods, that is, the greedy and
relaxed methods for approximately solving the problem (1.2). The basic idea of
the greedy method is that a sparse solution is refined iteratively by
successively identifying one or more components that yield the greatest
improvement in quality [3]. There are many commonly used greedy algorithms
such as orthogonal matching pursuit (OMP) [4], [5], stagewise OMP (StOMP) [6],
regularized OMP (ROMP) [7], compressive sampling matching pursuit (CoSaMP) [8]
and subspace pursuit [9]. The greedy algorithms can be quite fast, especially
in the ultra-sparse case, and also may be very efficient at certain situations
such as the dictionary contains a continuum of elements [10]. However, the
performance of the greedy algorithms can not be guaranteed when the signal is
not very sparse or the level of the observational noise is relatively high.
The relaxed method converts the combinatorial $l_{0}$-minimization into a more
tractable model via replacing the $l_{0}$-norm with a certain nonnegative and
continuous function $P(\cdot)$, that is,
$\min_{x\in\mathbf{R}^{N}}{P(x)}\ \ \text{s.t.}\ y=Ax.$ (1.3)
One of the most important cases is the $l_{1}$-minimization (also known as
Basis Pursuit (BP) [11]) with $P(x)=\|x\|_{1}$, where
$\|x\|_{1}=\sum_{i=1}^{N}|x_{i}|$ is called the $l_{1}$-norm. The
$l_{1}$-minimization is a convex optimization problem and thus can be
efficiently solved. Because of this, the $l_{1}$-minimization gets its
popularity and has been accepted as a very useful tool for solution to
sparsity problems. Nevertheless, it cannot promote further sparsity when
applied to compressed sensing [12], [13], [14], [15], [16]. Moreover, many
nonconvex functions were proposed as relaxations of the $l_{0}$-norm. Some
typical nonconvex examples are the $l_{q}$-norm ($0<q<1$) [12], [13], [14],
[15], smoothly clipped absolute deviation (SCAD) [17] and minimax concave
penalty (MCP) [18]. As compared with the $l_{1}$-minimization, the nonconvex
relaxed models can usually induce better sparsity, however, they are generally
more difficult to be solved.
There are mainly two kinds of algorithms to solve the constrained optimization
problem (1.3). The first one is the iteratively reweighted algorithm. Two of
the most important iteratively reweighted algorithms are the reweighted
$l_{1}$-minimization [16] and iteratively reweighted least squares (IRLS)
[19], [20] algorithms. One of the main advantages of this kind of algorithms
is that they can be used to solve a general model (1.3). However, the
computational complexities of these algorithms are usually relatively high.
The other one is commonly called the regularization method, which transforms
the constrained optimization problem (1.3) into the following unconstrained
optimization problem via introducing a regularization parameter
$\min_{x\in\mathbf{R}^{N}}\\{\|Ax-y\|_{2}^{2}+\lambda P(x)\\},$ (1.4)
where $\lambda>0$ is a regularization parameter. There are many algorithms for
solving the regularization model (1.4). Particularly, for some special $P(x)$
such as the $l_{0}$-norm, $l_{q}$-norms ($q=1,2/3,1/2$), SCAD and MCP, the
regularization models (1.4) can permit the thresholding representations and
thus yield the corresponding iterative thresholding algorithms [15], [21],
[22], [23]. Intuitively, an iterative thresholding algorithm can be seen as a
procedure of Landweber iteration projected by a certain thresholding operator.
Compared to the aforementioned algorithms including the greedy, BP and
iteratively reweighted algorithms, iterative thresholding algorithms can be
implemented fast and have almost the least computational complexity for large
scale problems [24], [25], [26]. So far, most of theoretical results of the
iterative thresholding algorithms were developed for the regularization model
(1.4) with fixed $\lambda$. However, it is in general difficult to determine
an appropriate regularization parameter $\lambda$, especially when $P$ is
nonconvex.
Alternatively, some adaptive strategies for setting the regularization
parameters were proposed for iterative thresholding algorithms. One of the
commonly used strategies is to set the regularization parameter adaptively
according to a specified sparsity level at each iteration. Once the specified
sparsity level is given, the number of nonzero components of vector at each
iteration is also determined. In practice, the specified sparsity level is
desired to be a good estimation of the true sparsity level. This strategy was
first adopted to the iterative hard thresholding algorithm (called hard
algorithm for short) in [27], and later the iterative soft [28] (called soft
algorithm for short) and half [15] (called half algorithm for short)
thresholding algorithms. The convergence of hard algorithm was justified when
$A$ satisfies a certain restricted isometry property (RIP) [27]. Later, Maleki
investigated the convergence of both hard and soft algorithms in terms of the
coherence [28]. Both in the analysis of [27] and [28], the specified sparsity
levels of AIT algorithms are set to be the true sparsity level of the original
sparse solution, however, which is commonly unknown in practice. Therefore,
the robustness of AIT algorithms to the specified sparsity levels is very
important in practice and worth of investigation. Moreover, besides the hard
and soft algorithms, there are many other AIT algorithms such as half, SCAD,
MCP algorithms which are widely used in signal processing, variable selection
and feature extraction. However, as far as we know, there are lack of the
corresponding theoretical guarantees on the global convergence of these
algorithms for sparse solution to the underdetermined linear equations. Thus,
the theoretical performance of these AIT algorithms should be further studied.
In this paper, we consider the global convergence a wide class of adaptively
iterative thresholding (AIT) algorithms for sparse solution to an
underdetermined system of linear equations. The associated thresholding
functions satisfy some basic assumptions including odevity, monotonicity and
boundedness. We show that if $A$ satisfies a certain coherence property and
the specified sparsity level is set in an appropriate range, then AIT
algorithms can find the correct support set within finite iterations.
Moreover, once the correct support set has been identified, then AIT
algorithms converge to the original sparse solution exponentially fast. In
other words, the asymptotic convergence rates of AIT algorithms are linear. It
should be pointed out that the linear rates of asymptotic convergence of AIT
algorithms are not trivial since most of the thresholding operators studied in
this paper are expansive. Thus, the classical theoretical results of the
Landweber iteration can not be straightly applicable to these algorithms.
The reminder of this paper is organized as follows. In section 2, we introduce
the adaptively iterative thresholding (AIT) algorithms. In section 3, we
present the main theoretical results of AIT algorithms. In section 4, we give
the proof of the main theorem. In section 5, we discuss some related work. We
conclude the paper in section 6.
## 2 Adaptively Iterative Thresholding Algorithms
In this section, we first give some notations used in this paper, and then
introduce the adaptively iterative thresholding algorithms.
### 2.1 Notion and Notation
For any $x\in\mathbf{R}^{N}$, $x_{i}$ represents its $i$-th component. Given a
positive integer $k<N$, $|x_{[k]}|$ represents its $k$-th largest component of
$x$ in magnitude. For any $A\in\mathbf{R}^{M\times N}$,
$A_{i}\in\mathbf{R}^{M}$ denots its $i$th column, $A^{T}$ represents its
transposition. For any index set $S$, $|S|$ denotes its cardinality, $S^{c}$
represents its complementary set. Moreover, we denote by $A_{S}$ the submatrix
of $A$ with the columns restricted to $S$.
### 2.2 AIT Algorithms
The adaptively iterative thresholding algorithm for sparse solution to (1.1)
can be generally expressed as the following iterative form:
$z^{(t+1)}=x^{(t)}-A^{T}(Ax^{(t)}-y),\\\ $ (2.1)
$x^{(t+1)}=H_{\tau^{(t+1)}}(z^{(t+1)}),$ (2.2)
where
$H_{\tau^{(t+1)}}(x)=(h_{\tau^{(t+1)}}(x_{1}),\cdots,h_{\tau^{(t+1)}}(x_{N}))^{T}$
(2.3)
is a componentwise thresholding operator associated with a thresholding
function $h_{\tau^{(t+1)}}$, $\tau^{(t+1)}$ is the threshold value at
$(t+1)$-th iteration. More specifically, a thresholding function $h_{\tau}$ is
commonly defined as
$h_{\tau}(u)=\left\\{\begin{array}[]{cc}f_{\tau}(u),&|u|>\tau\\\ 0,&{\rm
otherwise}\end{array}\right.$ (2.4)
where $f_{\tau}(u)$ is formally called the defining function for any
$u\in\mathbf{R}$. We give some basic assumptions of the defining function as
follows:
1. 1.
Odevity. $f_{\tau}$ is an odd function.
2. 2.
Monotonicity. $f_{\tau}(u)\geq f_{\tau}(v)$ for any $u\geq v\geq 0$.
3. 3.
Boundedness. There exists a constant $0\leq c\leq 1$ such that $u-c\tau\leq
f_{\tau}(u)\leq u$ for $u\geq\tau$.
The odevity and monotonicity are two regular assumptions for the defining
function, while the boundedness confines $h_{\tau}$ to be an appropriate
thresholding function. It can be noted that most of the commonly used
thresholding functions satisfy these assumptions. We list some typical
examples as follows.
Example 1. Hard thresholding function for $L_{0}$ regularization ([23])
$h_{\tau,0}(u)=\left\\{\begin{array}[]{cc}u,&|u|>\tau\\\ 0,&{\rm
otherwise}\end{array}\right..$ (2.5)
Example 2. Half thresholding function for $L_{1/2}$ regularization ([15])
$h_{\tau,1/2}(u)=\left\\{\begin{array}[]{cc}{\frac{2}{3}}u\left(1+\cos\left({\frac{2{\pi}}{3}}-{\frac{2}{3}}\arccos\left({\frac{\sqrt{2}}{2}}{(\frac{\tau}{|u|})}^{\frac{3}{2}}\right)\right)\right),&|u|>\tau\\\
0,&{\rm otherwise}\end{array}\right..$ (2.6)
Example 3. $2/3$-thresholding function for $L_{2/3}$ regularization ([22])
$h_{\tau,2/3}(u)=\left\\{\begin{array}[]{cc}sign(u)\left(\frac{\phi_{\tau}(u)+\sqrt{\frac{2|u|}{\phi_{\tau}(u)}-\phi_{\tau}(u)^{2}}}{2}\right)^{3},&|u|>\tau\\\
0,&{\rm otherwise}\end{array}\right.,$ (2.7)
where $sign(u)$ denotes as the sign function of $u$ henceforth,
$\phi_{\tau}(u)=\frac{2^{13/16}}{4\sqrt{3}}\tau^{3/16}(\cosh(\frac{\theta_{\tau}(u)}{3}))^{1/2}$
with $\theta_{\tau}(u)=arccosh(\frac{3\sqrt{3}u^{2}}{2^{7/4}(2\tau)^{9/8}})$.
Example 4. Soft thresholding function for $L_{1}$ regularization ([21])
$h_{\tau,1}(u)=\left\\{\begin{array}[]{cc}u-sign(u)\tau,&|u|>\tau\\\ 0,&{\rm
otherwise}\end{array}\right..$ (2.8)
Example 5. $SCAD$-thresholding function for nonconvex likelihood model ($a>2$)
([17])
$h_{\tau,SCAD}(u)=\left\\{\begin{array}[]{cc}u,&|u|>a\tau\\\
\frac{(a-1)u-sign(u)a\tau}{a-2},&2\tau<|u|\leq a\tau\\\
u-sign(u)\tau,&\tau<|u|\leq 2\tau\\\ 0,&{\rm otherwise}\end{array}\right..$
(2.9)
The plots of these thresholding functions and their corresponding boundedness
parameters $c$ are shown in Figure 1 and Table 1, respectively.
It can be observed that the tuning strategies of the threshold value
$\tau^{(t)}$ are crucial for AIT algorithms. In this paper, we consider a
heuristic way for setting the threshold value, i.e., the threshold value is
set to the $(k+1)$-th largest coefficient of $z$ in magnitude at each
iteration, where $k$ is the unique algorithmic parameter and called the
specified sparsity level. Therefore, the adaptively iterative thresholding
algorithms can be formulated as Algorithm 1.
It should be noticed that at $(t+1)$-th iteration, the AIT algorithm yields a
sparse solution with $k$ nonzero components by setting
$\tau^{(t+1)}=|z^{(t+1)}|_{[k+1]}$ in step 4 of Algorithm 1. To guarantee the
performance of the AIT algorithm, the specified sparsity level is very
critical. Assume that the true sparsity level of the original sparse solution
is $k^{*}$. On one hand, when $k\geq k^{*}$, the results will get better as
$k$ approaching to $k^{*}$. On the other hand, once $k<k^{*}$, then the AIT
algorithm fails to find the original sparse solution. Thus, $k$ should be
specified as an upper bound estimation of $k^{*}$.
## 3 Convergence Analysis of AIT Algorithms
In this section, we provide the convergence analysis of AIT algorithms for
sparse solution to (1.1). For simplicity, we assume that the normalization
step has been done before the analysis, that is, $\|A_{j}\|_{2}=1$ for
$j=1,\ldots,N$. We use $x^{*}=(x_{1}^{*},\cdots,x_{N}^{*})^{T}$ to denote the
original sparse solution with $k^{*}$ nonzeros components. Without loss of
generality, we further assume that
$|x_{1}^{*}|\geq|x_{2}^{*}|\geq\cdots\geq|x_{k^{*}}^{*}|>0$ and $x_{j}^{*}=0$
for $j>k^{*}$. Moreover, we denote by $I^{*}$ and $I^{(t)}$ the support sets
of $x^{*}$ and $x^{(t)}$, respectively. Furthermore, we denote
$I_{r}=\\{1,\ldots,r\\}$ for $1\leq r\leq k^{*}$ as the set formed by the
first $r$ largest components of $x^{*}$ in magnitude. Thus, we have
$I^{*}=I_{k^{*}}$.
To investigate the convergence of AIT algorithms, we introduce the coherence
of a matrix $A$, which is defined as follows [29]
$\mu(A)=\max_{i\neq j}|\langle A_{i},A_{j}\rangle|\quad\mbox{for}\
i,j\in\\{1,\ldots,N\\}.$
The coherence measures the maximal correlation between two different columns
of $A$. For simplicity, we use $\mu$ instead of $\mu(A)$ henceforth if there
is no confusion. In [29], it was shown that if
$k^{*}\leq\frac{1}{2}(1+\frac{1}{\mu})$, then $x^{*}$ is the unique sparsest
solution of (1.1). Next, we define the dynamic range of the original sparse
solution as
$Dr=\frac{\min_{i\in I^{*}}|x_{i}^{*}|}{\min_{i\in I^{*}}|x_{i}^{*}|},$
which measures the diversity of the nonzero components of $x^{*}$. Moreover,
we define two positive constants in the following
$T_{k^{*}}=k^{*}+(k^{*}-1)\log_{(1+c)k\mu}\frac{1-(3+c)k\mu}{(3+c)-(c^{2}+4c+3+2/Dr)k\mu}-\log_{(1+c)k\mu}Dr,$
(3.1)
and
$T_{k^{*}}^{*}=k^{*}+(k^{*}-1)\log_{(1+c)k^{*}\mu}\frac{1-(3+c)k^{*}\mu}{(3+c)-(c^{2}+4c+3+2/Dr)k^{*}\mu}-\log_{(1+c)k^{*}\mu}Dr.$
(3.2)
With these notations, we present the main result as follows.
Theorem 1. Suppose that $0<\mu<\frac{1}{(3+c)k^{*}}$ and $k^{*}\leq
k<\frac{1}{(3+c)\mu}$. Then there exists a positive integer $t^{*}\leq
T_{k^{*}}$ such that when $t\geq t^{*}$, it holds
$I^{*}\subset I^{(t)},$ (3.3)
and
$\|x^{(t)}-x^{*}\|_{\infty}\leq\frac{3+c}{2}\min_{i\in
I^{*}}|x_{i}^{*}|\rho^{t-t^{*}+1}$ (3.4)
with $\rho=(1+c)k\mu<1/2.$
In Theorem 1, we justify the global convergence of AIT algorithms. It shows
that as long as $A$ satisfies a certain coherence property and the specified
sparsity level $k$ is chosen in an appropriate range, AIT algorithms can find
the correct support set within finite iterations. Furthermore, once the
correct support set has been identified, then AIT algorithms converge to the
original sparse solution exponentially fast.
As shown by Theorem 1 and (3.1), the upper bound on the number of iterations
required for identifying the correct support set is mainly determined by
several parameters, i.e., $k^{*}$, $Dr$ and $k$. On one hand, according to
(3.1), $T_{k^{*}}$ is monotonic increasing with respective to both $k^{*}$ and
$Dr$. In other words, if the original sparse solution has more nonzero
components and its dynamic range is larger, then more iterations are commonly
required to identify the correct support set. These coincide with the common
senses. As we all known, it is generally more difficult to find a denser
solution. Also, if the dynamic range of the original solution is larger, more
efforts are usually required to detect the smallest nonzero component. On the
other hand, we can easily verify that $T_{k^{*}}$ is monotonically increasing
with respective to $k$. Therefore, if the specified sparsity level $k$ is
estimated more precisely, the number of iterations required for finding the
correct support set may get fewer. Moreover, according to (3.4), it can be
seen that AIT algorithms converge faster with smaller $\rho$ when $k$ is
closer to $k^{*}$. Thus, in practice, $k$ is desired to be estimated more
precisely in terms of computational efficiency and convergence speed.
As analysed in the previous, a tighter upper bound estimation of the true
sparsity level is more desired for the AIT algorithm in the perspectives of
both theory and practice. However, the upper bound is commonly unknown in
practice. In applications, we may conduct an empirical study or based on some
known priors to yield a reasonable upper bound. Moreover, there are several
efficient ways inspired by some theoretical analysis. In [30], it suggested
that an upper bound can be estimated by the undersampling-sparsity tradeoff,
or “phase-transition curve”. However, it is generally very time-consuming to
obtain the “phase-transition curve”. According to [31], it was shown that the
coherence satisfies $\mu\in\left[\sqrt{\frac{N-M}{M(N-1)}},1\right]$. The
lower bound is known as the Welch bound [32]. Particularly, when $N\gg M$, the
lower bound is approximately $\mu\geq\frac{1}{\sqrt{M}}$. Together with
Theorem 1, we can suggest $\mathcal{O}(\sqrt{M})$ as a reasonable upper bound
estimation of $k^{*}$.
In the following, we give a corollary to show the special case with $k=k^{*}$.
Corollary 1. Suppose that $0<\mu<\frac{1}{(3+c)k^{*}}$ and $k=k^{*}$. Then
there exists a positive integer $\hat{t}^{*}\leq T^{*}_{k^{*}}$ such that when
$t\geq\hat{t}^{*}$, it holds
$I^{*}=I^{(t)},$ (3.5)
and
$\|x^{(t)}-x^{*}\|_{\infty}\leq\frac{3+c}{2}\min_{i\in
I^{*}}|x_{i}^{*}|\hat{\rho}^{t-\hat{t}^{*}+1}$ (3.6)
with $\hat{\rho}=(1+c)k^{*}\mu<1/2.$
From Corollary 1, when $k=k^{*}$, the AIT algorithm can recover the support
set of $x^{*}$ exactly within finite iterations. According to (3.2), it can be
observed that if $k^{*}\mu$ is not sufficient close to $\frac{1}{3+c}$ and the
dynamic range of the original sparse solutio is not too large, then the log
term about $k^{*}\mu$ and $Dr$ in the second and third terms of (3.2)
respectively are relatively small constants. In this case, the number of
iterations required for the AIT algorithm is about several times of $k^{*}$.
For an instance, assume that $k^{*}=9$, $\mu=\frac{1}{40}$ and $Dr=10$,
according to (3.2), the number of iterations required for $hard$, $soft$ and
$half$ algorithms are 20, 42 and 25, which are about 2, 5 and 3 times of
$k^{*}$, respectively. Motivated by this observation, we can suggest an
efficient halting rule for AIT algorithms through setting the number of
maximum iterations according to the true sparsity level.
It can be observed from Corollary 1 that the boundedness parameter $c$ plays
an important role in the guarantees of the convergence of AIT algorithms. The
restriction of the matrix $A$ gets stricter as $c$ increasing. As shown in
Table 1, among these AIT algorithms, hard algorithm permits the weakest
requirement of $A$ with $\mu<\frac{1}{3k^{*}}$, while soft algorithm requires
the strictest restriction of $A$ with $\mu<\frac{1}{4k^{*}}$. It should be
noticed that the restriction on $\mu$ is relatively loose and can be attained
in practice. In fact, it was shown that the coherence $\mu$ is in the order of
$\sqrt{\log N/M}$ for the random matrix where entries of $A$ are independently
and identically gaussian distributed [33]. This implies that
$k^{*}=O(M^{\xi_{1}})$ might suffice for the AIT algorithm when $\log
N=O(M^{\xi_{2}})$ for some positive constants $\xi^{1}$ and $\xi^{2}$
satisfying $2\xi^{1}+\xi^{2}<1$.
Remark 1. As shown by the proof of Theorem 1 in Section 4, it is interested
that the procedure of identifying the correct support set is a sequential
recruitment process. That is, the supports are sequentially recruited in a
descending order of the values of their coefficients with the larger one being
identified not later than the smaller one. This procedure may be very useful
to certain applications such as feature screening problem in statistics.
## 4 Proof of Theorem 1
We denote
$i_{[k+1]}^{(t)}=\arg\min_{i\in\\{1,2,\cdots,N\\}}\left\\{i:\left|z_{i}^{(t)}\right|=\left|z^{(t)}\right|_{[k+1]}\right\\}$
and then let
$\Lambda_{[k+1]}^{(t)}=I^{(t)}\cup\left\\{i_{[k+1]}^{(t)}\right\\}$. To prove
Theorem 1, we need the following lemmas. First, we give a lemma to bound the
gap between the components of $x^{(t)}$ and $z^{(t)}$ at $t$-th iteration,
which is served as the basis of the other lemmas.
Lemma 1. At any $t$-th iteration ($t\geq 1$), there exists an
$i_{0}^{(t)}\in\Lambda_{[k+1]}^{(t)}\setminus I^{*}$, such that
(i) for any $i\in I^{(t)}$,
$\left|z_{i}^{(t)}-x_{i}^{(t)}\right|\leq
c\left|z_{i_{0}^{(t)}}^{(t)}-x_{i_{0}^{(t)}}^{*}\right|,$ (4.1)
where $c$ is the boundedness parameter of the associated thresholding
function;
(ii) for any $i\notin I^{(t)}$,
$\left|z_{i}^{(t)}-x_{i}^{(t)}\right|\leq\left|z_{i_{0}^{(t)}}^{(t)}-x_{i_{0}^{(t)}}^{*}\right|.$
(4.2)
Here, it should be mentioned that $x_{i_{0}^{(t)}}^{*}=0$ and we keep it in
(4.1) and (4.2) only for better formats.
Proof. (i) For $i\in I^{(t)}$, by the definition of the thresholding function
$H_{\tau}$ and the boundness assumption of $f_{\tau}$, it holds
$\left|z_{i}^{(t)}-x_{i}^{(t)}\right|=\left|z_{i}^{(t)}-f_{\tau^{(t)}}(z_{i}^{(t)})\right|\leq
c\tau^{(t)}=c\left|z^{(t)}\right|_{[k+1]}.$ (4.3)
Since $i_{[k+1]}^{(t)}\notin I^{(t)}$, then the cardinality of
$\Lambda_{[k+1]}^{(t)}$ is $k+1$. Moreover, by $|I^{*}|=k^{*}<k+1$, then there
exists an index $i_{0}^{(t)}$ such that
$i_{0}^{(t)}\in\Lambda_{[k+1]}^{(t)}\setminus I^{*}$. Thus, (4.3) becomes
$\left|z_{i}^{(t)}-x_{i}^{(t)}\right|\leq c\left|z^{(t)}\right|_{[k+1]}\leq
c\left|z_{i_{0}^{(t)}}^{(t)}\right|=c\left|z_{i_{0}^{(t)}}^{(t)}-x_{i_{0}^{(t)}}^{*}\right|.$
(4.4)
(ii) Similarly, for any $i\notin I^{(t)}$, it holds
$\left|z_{i}^{(t)}-x_{i}^{(t)}\right|=\left|z_{i}^{(t)}\right|\leq\left|z^{(t)}\right|_{[k+1]}\leq\left|z_{i_{0}^{(t)}}^{(t)}-x_{i_{0}^{(t)}}^{*}\right|.$
(4.5)
Thus, we end the proof of this lemma.
In the next, we give a lemma to show that the largest component (in magnitude)
of $x^{*}$ will be detected at the first iteration.
Lemma 2. Suppose that $0<\mu<\frac{1}{2k^{*}-1}$ and $k^{*}\leq
k<\frac{1}{2}(1+\frac{1}{\mu})$. Then at the first iteration, it holds:
(i) $\\{1\\}\subset I^{(1)}$;
(ii) for any $j\in I^{(1)}$,
$\left|x_{j}^{(1)}-x_{j}^{*}\right|\leq\frac{(1+c)(3+c)}{2}k\mu\left|x_{1}^{*}\right|.$
Proof. First, we show that $\\{1\\}\subset I^{(1)}$. On one hand, we observe
that
$\left|z_{1}^{(1)}\right|=\left|x_{1}^{*}+\sum_{i\in
I^{*}\setminus{\\{1\\}}}\langle A_{1},A_{i}\rangle
x_{i}^{*}\right|\geq|x_{1}^{*}|-\mu\sum_{i=2}^{k^{*}}|x_{i}^{*}|\geq|x_{1}^{*}|-(k-1)\mu|x_{1}^{*}|.$
On the other hand, for any $i\notin I^{*}$, it holds
$\left|z_{i}^{(1)}\right|=\left|\sum_{j=1}^{k^{*}}\langle A_{i},A_{j}\rangle
x_{j}^{*}\right|\leq{k^{*}}\mu\left|x_{1}^{*}\right|\leq
k\mu\left|x_{1}^{*}\right|.$
Since $k<\frac{1}{2}(1+\frac{1}{\mu})$, then
$k\mu\left|x_{1}^{*}\right|<\left|x_{1}^{*}\right|-(k-1)\mu\left|x_{1}^{*}\right|,$
which implies that
$\left|z_{1}^{(1)}\right|>\max_{i\notin I^{*}}\left|z_{i}^{(1)}\right|.$
Thus, $\\{1\\}\subset I^{(1)}$.
Next, we give the error bound. For any $j\in I^{(1)}$, we observe that
$\left|x_{j}^{(1)}-x_{j}^{*}\right|\leq\left|x_{j}^{(1)}-z_{j}^{(1)}\right|+\left|z_{j}^{(1)}-x_{j}^{*}\right|\leq
c\left|x_{i_{0}^{(1)}}^{*}-z_{i_{0}^{(1)}}^{(1)}\right|+\left|z_{j}^{(1)}-x_{j}^{*}\right|,$
(4.6)
where the second inequality holds for Lemma 1. Furthermore, for any $i$, it
holds
$\left|z_{i}^{(1)}-x_{i}^{*}\right|=\left|\sum_{j\in
I^{*}\setminus{\\{i\\}}}\langle A_{i},A_{j}\rangle x_{j}^{*}\right|\leq
k^{*}\mu\left|x_{1}^{*}\right|\leq k\mu\left|x_{1}^{*}\right|.$ (4.7)
Combining (4.6) with (4.7), for any $j\in I^{(1)}$, it holds
$\left|x_{j}^{(1)}-x_{j}^{*}\right|\leq(1+c)k\mu\left|x_{i}^{*}\right|\leq\frac{(1+c)(3+c)}{2}k\mu\left|x_{1}^{*}\right|.$
Thus, we end the proof of this lemma.
Lemma 3. Suppose that $0<\mu<\frac{1}{(3+c)k^{*}}$ and $k^{*}\leq
k<\frac{1}{(3+c)\mu}$. Moreover, assume that at $m$-th iteration,
$I_{r}\subset I^{(m)}$ ($0<r\leq k^{*}$) and for any $j\in I^{(m)}$, it holds
$\left|x_{j}^{(m)}-x_{j}^{*}\right|\leq\frac{(1+c)(3+c)}{2}k\mu\left|x_{r}^{*}\right|$.
Then at $(m+s)$-th iteration ($s\geq 1$), it holds
(i) for any $j$,
$\left|z_{j}^{(m+s)}-x_{j}^{*}\right|\leq\frac{(3+c)}{2}k\mu\left((1+c)k\mu\right)^{s}\left|x_{r}^{*}\right|+k\mu\left|x_{r+1}^{*}\right|\left[1+(1+c)k\mu+\cdots+((1+c)k\mu)^{s-1}\right];$
(ii) for any $i\in I^{(m+s)}$,
$\left|x_{i}^{(m+s)}-x_{i}^{*}\right|\leq\frac{(3+c)}{2}((1+c)k\mu)^{s+1}\left|x_{r}^{*}\right|+k\mu\left|x_{r+1}^{*}\right|\left[(1+c)k\mu+\cdots+((1+c)k\mu)^{s}\right];$
(iii) $I_{r}\subset I^{(m+s)}$.
Proof. We prove this lemma by induction. First, when $s=1$, for any $i\in
I^{(m+1)}$, it holds
$\left|x_{i}^{(m+1)}-x_{i}^{*}\right|\leq\left|x_{i}^{(m+1)}-z_{i}^{(m+1)}\right|+\left|z_{i}^{(m+1)}-x_{i}^{*}\right|.$
By Lemma 1, there exists an $i_{0}^{(m+1)}\in\Lambda_{[k+1]}^{(m+1)}\setminus
I^{*}$ such that
$\left|x_{i}^{(m+1)}-z_{i}^{(m+1)}\right|\leq
c\left|z_{i_{0}^{(m+1)}}^{(m+1)}-x_{i_{0}^{(m+1)}}^{*}\right|,$
then it holds
$\left|x_{i}^{(m+1)}-x_{i}^{*}\right|\leq
c\left|z_{i_{0}^{(m+1)}}^{(m+1)}-x_{i_{0}^{(m+1)}}^{*}\right|+\left|z_{i}^{(m+1)}-x_{i}^{*}\right|.$
(4.8)
Moreover, for any $j$, it holds
$\displaystyle\left|z_{j}^{(m+1)}-x_{j}^{*}\right|$ $\displaystyle=$
$\displaystyle\left|\sum_{i\in I^{(m)}\cup I^{*}\setminus{\\{j\\}}}\langle
A_{j},A_{i}\rangle(x_{i}^{*}-x_{i}^{(m)})\right|$ (4.9) $\displaystyle=$
$\displaystyle\left|\sum_{i\in I^{(m)}\setminus{\\{j\\}}}\langle
A_{j},A_{i}\rangle(x_{i}^{*}-x_{i}^{(m)})+\sum_{i\in
I^{*}\setminus(I^{(m)}\cup{\\{j\\}})}\langle A_{j},A_{i}\rangle
x_{i}^{*}\right|$ $\displaystyle\leq$ $\displaystyle
k\mu\left(\frac{(1+c)(3+c)}{2}k\mu\left|x_{r}^{*}\right|\right)+(k^{*}-r)\mu\left|x_{r+1}^{*}\right|$
$\displaystyle\leq$
$\displaystyle\frac{(3+c)}{2}k\mu\left((1+c)k\mu\left|x_{r}^{*}\right|\right)+k\mu\left|x_{r+1}^{*}\right|.$
Combining (4.8) with (4.9), for any $i\in I^{(m+1)}$, it holds
$\displaystyle\left|x_{i}^{(m+1)}-x_{i}^{*}\right|$ $\displaystyle\leq$
$\displaystyle(1+c)\left(\frac{(3+c)}{2}k\mu((1+c)k\mu\left|x_{r}^{*}\right|)+k\mu\left|x_{r+1}^{*}\right|\right)$
(4.10) $\displaystyle=$
$\displaystyle\frac{(3+c)}{2}((1+c)k\mu)^{2}\left|x_{r}^{*}\right|+(1+c)k\mu\left|x_{r+1}^{*}\right|.$
Then we need to prove that $I_{r}\subset I^{(m+1)}$. According to (4.9), for
any $j$, it holds
$|z_{j}^{(m+1)}-x_{j}^{*}|\leq\left(1+\frac{(3+c)(1+c)}{2}k\mu\right)k\mu|x_{r}^{*}|.$
Since $k<\frac{1}{(3+c)\mu}$, it holds
$\left(1+\frac{(3+c)(1+c)}{2}k\mu\right)k\mu<\frac{1}{2}.$
Then for any $j$, it holds
$\left|z_{j}^{(m+1)}-x_{j}^{*}\right|<\frac{1}{2}\left|x_{r}^{*}\right|.$
(4.11)
According to (4.11), we observe that, for any $i\in I_{r}$,
$\left|z_{i}^{(m+1)}\right|\geq\left|x_{i}^{*}\right|-\left|z_{i}^{(m+1)}-x_{i}^{*}\right|\geq\left|x_{r}^{*}\right|-\frac{1}{2}\left|x_{r}^{*}\right|>\frac{1}{2}\left|x_{r}^{*}\right|.$
(4.12)
While for $i\notin I^{*}$,
$\left|z_{i}^{(m+1)}\right|=\left|z_{i}^{(m+1)}-x_{i}^{*}\right|<\frac{1}{2}\left|x_{r}^{*}\right|.$
(4.13)
With (4.12) and (4.13), it follows that $I_{r}\subset I^{(m+1)}$. Therefore,
the conclusion holds for $s=1$.
Second, assume that the conclusion holds for $s$ ($s\geq 1$), then we need to
check it holds for $s+1$. The proof is similar to the case $s=1$ and we omit
it here.
Lemma 4. Suppose that $0<\mu<\frac{1}{(3+c)k^{*}}$ and $k^{*}\leq
k<\frac{1}{(3+c)\mu}$. Moreover, assume that at $m$-th iteration,
$I_{r}\subset I^{(m)}$ ($r<k^{*}$) and for any $j\in I^{(m)}$,
$|x_{j}^{(m)}-x_{j}^{*}|\leq\frac{(1+c)(3+c)}{2}k\mu|x_{r}^{*}|$. Then it
holds:
(i) the index $\\{r+1\\}$ will be detected after at most $l_{r}$ iterations
with
$l_{r}=\left\lfloor\log_{(1+c)k\mu}\frac{1-(3+c)k\mu}{(3+c)(1-(1+c)k\mu)|x_{r}^{*}|/|x_{r+1}^{*}|-2k\mu}\right\rfloor,$
where the function $\lfloor u\rfloor$ denotes the smallest integer not less
than $u$ for any $u\in\mathbb{R}$.
(ii) for any $j\in I^{(m+l_{r}+1)}$,
$\left|x_{j}^{(m+l_{r}+1)}-x_{j}^{*}\right|<\frac{(1+c)(3+c)}{2}k\mu\left|x_{r+1}^{*}\right|.$
Proof. We first show that the index $\\{r+1\\}$ will be detected after at most
$l_{r}$ iterations, and then give the error bound. According to Lemma 3, at
$(m+l_{r})$-th iteration, for any $j$, it holds
$\displaystyle\left|z_{j}^{(m+l_{r})}-x_{j}^{*}\right|$ $\displaystyle\leq$
$\displaystyle\frac{(3+c)}{2}((1+c)k\mu)^{l_{r}}\left|x_{r}^{*}\right|+k\mu\left|x_{r+1}^{*}\right|\left(1+\cdots+((1+c)k\mu)^{l_{r}-1}\right)$
$\displaystyle<$
$\displaystyle\frac{(3+c)}{2}((1+c)k\mu)^{l_{r}}\left|x_{r}^{*}\right|+k\mu\left|x_{r+1}^{*}\right|\frac{1-((1+c)k\mu)^{l_{r}}}{1-(1+c)k\mu}$
$\displaystyle=$
$\displaystyle\left|x_{r+1}^{*}\right|\left(\frac{(3+c)}{2}((1+c)k\mu)^{l_{r}}\frac{\left|x_{r}^{*}\right|}{\left|x_{r+1}^{*}\right|}+k\mu\frac{1-((1+c)k\mu)^{l_{r}}}{1-(1+c)k\mu}\right)$
$\displaystyle\leq$
$\displaystyle\left|x_{r+1}^{*}\right|\left(\frac{(3+c)}{2}((1+c)k\mu)^{l_{r}}\frac{\left|x_{r}^{*}\right|}{\left|x_{r+1}^{*}\right|}+k\mu\frac{1-((1+c)k\mu)^{l_{r}}}{1-(1+c)k\mu}\right).$
Since
$l_{r}\geq\log_{(1+c)k\mu}\frac{1-(3+c)k\mu}{(3+c)(1-(1+c)k\mu)|x_{r}^{*}|/|x_{r+1}^{*}|-2k\mu},$
then
$\frac{(3+c)}{2}((1+c)k\mu)^{l_{r}}\frac{|x_{r}^{*}|}{|x_{r+1}^{*}|}+k\mu\frac{1-((1+ck\mu)^{l_{r}}}{1-(1+ck\mu)}\leq\frac{1}{2}.$
Thus, for any $j$, it holds
$\left|z_{j}^{(m+l_{r})}-x_{j}^{*}\right|<\frac{1}{2}\left|x_{r+1}^{*}\right|.$
(4.14)
By (4.14), on one hand
$\left|z_{r+1}^{(m+l_{r})}\right|\geq\left|x_{r+1}^{*}\right|-\left|z_{r+1}^{(m+l_{r})}-x_{r+1}^{*}\right|>\frac{1}{2}\left|x_{r+1}^{*}\right|,$
(4.15)
and on the other hand, for any $j\notin I^{*}$,
$|z_{j}^{(m+l_{r})}|=|z_{j}^{(m+l_{r})}-x_{j}^{*}|<\frac{1}{2}|x_{r+1}^{*}|.$
(4.16)
With (4.15) and (4.16), it shows that $\\{r+1\\}$ will be detected at
$(m+l_{r})$-th iteration, that is, $\\{r+1\\}\subset I^{(m+l_{r})}$.
Next, we give the upper bound of the error. For any $i\in I^{(m+l_{r}+1)}$, it
holds
$\displaystyle\left|x_{i}^{(m+l_{r}+1)}-x_{i}^{*}\right|$ $\displaystyle=$
$\displaystyle\left|\sum_{j\in I^{(m+l_{r})}\setminus{\\{i\\}}}\langle
A_{i},A_{j}\rangle(x_{j}^{*}-x_{j}^{(m+l_{r})})+\sum_{j\in
I^{*}\setminus(I^{(m+l_{r})}\cup{\\{i\\}})}\langle
A_{i},A_{j}\rangle\beta_{j}^{*}\right|$ (4.17) $\displaystyle\leq$
$\displaystyle\mu\sum_{j\in
I^{(m+l_{r})}\setminus{\\{i\\}}}\left|x_{j}^{*}-x_{j}^{(m+l_{r})}\right|+(k^{*}-r-1)\mu\left|x_{r+1}^{*}\right|.$
Moreover, for any $j\in I^{(m+l_{r})}$, it holds
$\left|x_{j}^{*}-x_{j}^{(m+l_{r})}\right|\leq\left|x_{j}^{*}-z_{j}^{(m+l_{r})}\right|+\left|z_{j}^{(m+l_{r})}-x_{j}^{(m+l_{r})}\right|.$
(4.18)
According to Lemma 1 and (4.14), then (4.18) becomes
$\left|x_{j}^{*}-x_{j}^{(m+l_{r})}\right|<\frac{1}{2}\left|x_{r+1}^{*}\right|+c\left|z_{i_{0}^{(m+l_{r})}}^{(m+l_{r})}-x_{i_{0}^{(m+l_{r})}}^{*}\right|<\frac{1+c}{2}\left|x_{r+1}^{*}\right|.$
(4.19)
Combining (4.17) and (4.19), for any $i\in I^{(m+l_{r}+1)}$, it holds
$\displaystyle\left|x_{i}^{(m+l_{r}+1)}-x_{i}^{*}\right|$ $\displaystyle\leq$
$\displaystyle\frac{(1+c)}{2}k\mu\left|x_{r+1}^{*}\right|+(k^{*}-r-1)\mu\left|x_{r+1}^{*}\right|$
$\displaystyle=$
$\displaystyle\left(\frac{1+c}{2}+\frac{k^{*}-r-1}{k}\right)k\mu\left|x_{r+1}^{*}\right|$
$\displaystyle\leq$
$\displaystyle\frac{(1+c)(3+c)}{2}k\mu\left|x_{r+1}^{*}\right|.$
Therefore, for any $i\in I^{(m+l_{r}+1)}$, it holds
$\left|x_{i}^{(m+l_{r}+1)}-x_{i}^{*}\right|\leq\frac{(1+c)(3+c)}{2}k\mu\left|x_{r+1}^{*}\right|.$
Thus, we end the proof of Lemma 4.
Proof of Theorem 1. With these lemmas, we prove Theorem 1 inductively. For
$i=1$, by Lemma 2, the largest component (in magnitude) will be detected at
the first iteration, that is, $I_{1}=\\{1\\}\subset I^{(1)}$. By Lemma 3, once
the first largest index is identified, then it remains in the support set
forever. Furthermore, by Lemma 4, the second largest component will be
identified after at most $l_{1}$ iterations, i.e., $I_{2}\subset I^{(t)}$ when
$t\geq 1+l_{1}$. In order to obtain the required error bound for the inductive
procedure, one more iteration should be implemented. When this procedure is
repeated for $r$ times with $0<r\leq k^{*}-1$, it holds $I_{r+1}\subset
I^{(t)}$ when $t\geq r+\sum_{i=1}^{r-1}l_{i}$. Furthermore, by Lemma 3, once
all the correct indices are detected, they remains in the support set and the
error estimation of the iteration can be obtained. Therefore, there exists an
integer constant $t^{*}\leq k^{*}+\sum_{i=1}^{k^{*}-1}l_{i}$ such that when
$t\geq t^{*}$, it holds $I^{*}\subset I^{(t)}$ and the error estimation of the
iteration can be achieved. Moreover, by the definition of $l_{i}$ in Lemma 4
and the fact that $|x^{*}_{i}|/|x^{*}_{i+1}|\leq Dr$, it holds
$\displaystyle l_{i}$ $\displaystyle\leq$
$\displaystyle\log_{(1+c)k\mu}\frac{1-(3+c)k\mu}{(3+c)(1-(1+c)k\mu)|x_{i}^{*}|/|x_{i+1}^{*}|-2k\mu}$
(4.20) $\displaystyle\leq$
$\displaystyle\log_{(1+c)k\mu}\frac{1-(3+c)k\mu}{(3+c)-(c^{2}+4c+3+2/Dr)k\mu}-\log_{(1+c)k\mu}\frac{|x_{i}^{*}|}{|x_{i+1}^{*}|}$
for $i=1,\cdots,k^{*}-1$. Therefore,
$k^{*}+\sum_{i=1}^{k^{*}-1}l_{i}\leq
k^{*}+(k^{*}-1)\log_{(1+c)k\mu}\frac{1-(3+c)k\mu}{(3+c)-(c^{2}+4c+3+2/Dr)k\mu}-\log_{(1+c)k\mu}\frac{|x_{1}^{*}|}{|x_{k^{*}}^{*}|}=T_{k^{*}}.$
Thus, we obtain the proof of Theorem 1.
## 5 Related Work
In this section, we first discuss some related work of AIT algorithms, and
then give some comparisons with other typical algorithms including BP, OMP,
CoSaMP in terms of the sufficient condition for convergence and computational
complexity.
(i) On related work of AIT algorithms. In [28], Maleki provided some similar
results for two special AIT algorithms, i.e., the hard and soft algorithms
with $k=k^{*}$. The sufficient conditions for convergence are
$\mu<\frac{1}{3.1k^{*}}$ and $\mu<\frac{1}{4.1k^{*}}$ for hard and soft
algorithms, respectively. As shown by Corollary 1, our conditions for both
algorithms are slightly weaker than Maleki’s conditions. Moreover, from
Theorem 1, we show the robustness of AIT algorithms to the specified sparsity
levels, which is very important in practice. Except the hard and soft
algorithms, as far as we know, there are no similar results on the global
convergence of other AIT algorithms such as half, SCAD and MCP algorithms for
sparse solution to the underdetermined linear equations.
Besides the coherence property, another important property called the
restricted isometry property (RIP) is commonly used to characterize the
performance of an algorithm for sparse solution to (1.1). The $s$-order
restricted isometry constant (RIC), $\delta_{s}$ of $A$ is defined as the
smallest constant $0<\delta<1$ such that
$(1-\delta)\|x\|_{2}^{2}\leq\|Ax\|_{2}^{2}\leq(1+\delta)\|x\|_{2}^{2},~{}\forall\|x\|_{0}\leq
s.$ (5.1)
In [34], it was demonstrated that if $A$ has unit-norm columns and coherence
$\mu$, then $A$ has the $(s,\delta_{s})$-RIP with
$\delta_{s}\leq(s-1)\mu.$ (5.2)
In terms of RIP, Blumensath and Davies justified the performance of the hard
algorithm when applied to signal recovery problem [27]. It was shown that if
$A$ satisfies a certain RIP with $\delta_{3k^{*}}<\frac{1}{8\sqrt{2}-1}$, then
the global convergence of the hard algorithm can be guaranteed. Later, this
condition was significantly improved to by Foucart [38], i.e.,
$\delta_{3k^{*}}<\frac{1}{2}$. Together with (5.2), we can easily deduce a
coherence based sufficient condition of convergence, that is,
$\mu<\frac{1}{2(3k^{*}-1)}$. As compared with the existing RIP based
conditions, it is hard to claim whether our conditions are better. Instead, we
can give some useful remarks on these conditions. On one hand, the sufficient
conditions based on coherence can be in general verified much easier than
those based on RIP. On the other hand, the RIP based conditions can be
generalized and improved usually easier than those based on coherence.
(ii) On comparison with other algorithms. For better comparison, we list the
state-of-the-art results on sufficient conditions of some typical algorithms
including BP, OMP, CoSaMP, hard, soft, half and other AIT algorithms in Table
2.
From Table 2, in the perspective of coherence, the sufficient conditions of
AIT algorithms are slightly stricter than those of BP and OMP algorithms.
However, AIT algorithms are generally faster than both algorithms with lower
computational complexities, especially for large scale applications. As
analyzed in Section 3, the number of iterations required for the convergence
of the AIT algorithm is empirically of the same order of the original sparsity
level $k^{*}$, that is, $\mathcal{O}(k^{*})$. At each iteration of the AIT
algorithm, only some simple matrix-vector multiplications and a projection on
the vector need to be done, and thus the computational complexity per
iteration is $\mathcal{O}(MN)$. Therefore, the total computational complexity
of the AIT algorithm is $\mathcal{O}(k^{*}MN)$. While the total computational
complexities of BP and OMP algorithms are generally $\mathcal{O}(M^{2}N)$ and
$\max\\{\mathcal{O}(k^{*}MN),\mathcal{O}(\frac{(k^{*})^{2}(k^{*}+1)^{2}}{4})\\}$,
respectively. It should be pointed out that the computational complexity of
OMP algorithm is related to the commonly used halting rule of OMP algorithm,
that is, the number of maximal iterations is set to be the true sparsity level
$k^{*}$.
As another important greedy algorithm, CoSaMP algorithm identifies
multicomponents (commonly $2k^{*}$) at each iteration. From Table 2, the RIP
based sufficient condition of CoSaMP is $\delta_{4k^{*}}<0.384$ and a deduced
coherence based sufficient condition is $\mu<\frac{0.384}{4k^{*}-1}$. In the
perspective of coherence, our conditions for AIT algorithms are better than
CoSaMP, though this comparison is not very reasonable. At each iteration of
CoSaMP algorithm, some simple matrix-vector multiplications and a least
squares problem should be considered. Thus, the computational complexity per
iteration of CoSaMP algorithm is generally
$\max\\{\mathcal{O}(MN),\mathcal{O}((3k^{*})^{3})\\}$, which is higher than
those of AIT algorithms, especially when $k^{*}$ is very large. However, the
number of iterations required for CoSaMP algorithm is commonly fewer than
those of AIT algorithms, since the speed of convergence of CoSaMP algorithm is
exponential while those of AIT algorithms are asymptotically exponential, that
is, AIT algorithms converge exponentially fast after certain iterations.
Therefore, as claimed in the introduction, when applied to very sparse case,
both OMP and CoSaMP algorithms may be more efficient than AIT algorithms.
While AIT algorithms may be better when applied to more general cases.
## 6 Conclusion
In this paper, we provide the convergence analysis of a wide class of
adaptively iterative thresholding (AIT) algorithms for sparse solution to an
underdetermined system of linear equations $y=Ax$. We prove that as long as
$A$ satisfies a certain coherence property and the specified sparsity level is
set in an appropriate range, AIT algorithms can identify the correct support
set within finite steps. Furthermore, we demonstrate that the asymptotic
convergence rates of AIT algorithms are linear, that is, once the correct
support set has been identified, AIT algorithms converge to the original
sparse solution exponentially fast. It is interested that the procedure of
finding the correct support set is a sequential recruitment process, i.e., the
supports are sequentially recruited into the support set in the descending
order of the magnitudes of their coefficients. This property may be very
useful to certain applications such as feature screening problem. It should be
noted that most of the commonly used iterative thresholding algorithms (say,
hard, soft, half and SCAD algorithms) are included in the class of iterative
thresholding algorithms studied in this paper. Besides the hard and soft
algorithms, we provide some fundamental guarantees on the performance of the
other AIT algorithms for sparse solution to an underdetermined linear
equations.
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* [6] D. L. Donoho, Y. Tsaig, O. Drori and J.-L. Starck, Sparse solution of underdetermined systems of linear equations by stagewise orthogonal matching pursuit, IEEE Transactions on Information Theory, 58 (2): 1094 - 1121, 2012.
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* [27] T. Blumensath and M. E. Davies, Iterative hard thresholding for compressed sensing, Applied and Computational Harmonic Analysis, 27: 265-274, 2008.
* [28] A. Maleki, Coherence analysis of iteative thresholding algorithms, in Forty-Seventh Annual Allerton Conference, Allerton House, UIUC, Illinois, USA, 2009.
* [29] D. L. Donoho and M. Elad, Optimally sparse representation in general (nonorthogonal) dictionaries via $l_{1}$ minimization, Proceedings of the National Academy of Sciences, 100 (5): 2197-2202, 2003.
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* [38] S. Foucart, Sparse recovery algorithms: Sufficient conditions in terms of restricted isometry constants, in Proceedings of the 13th International Conference on Approximation Theory, M. Neantu and L. Schumaker, eds., San Antonio, TX, 2010, Springer.
Figure 1: Typical thresholding functions $h_{\tau}(u)$ with $\tau=1$.
Table 1: Boundedness parameters $c$ for different thersholding functions $f_{\tau,*}$ | $f_{\tau,0}$ | $f_{\tau,1/2}$ | $f_{\tau,2/3}$ | $f_{\tau,1}$ | $f_{\tau,SCAD}$
---|---|---|---|---|---
$c$ | 0 | $\frac{1}{3}$ | $\frac{1}{2}$ | 1 | 1
$\frac{1}{3+c}$ | $\frac{1}{3}$ | $\frac{3}{10}$ | $\frac{2}{7}$ | $\frac{1}{4}$ | $\frac{1}{4}$
Algorithm 1: Adaptively Iterative Thresholding Algorithm
Step 1. Normalize $A$ such that $\|A_{j}\|_{2}=1$ for $j=1,\ldots,N$;
---
Step 2. Choose a specified sparsity level $k$ and begin with $x^{(0)}=0$;
Step 3. Compute $z^{(t+1)}=x^{(t)}+A^{T}(y-Ax^{(t)})$;
Step 4. Set $\tau^{(t+1)}=|z^{(t+1)}|_{[k+1]}$;
Step 5. Update $x^{(t+1)}=H_{\tau^{(t+1)}}(z^{(t+1)})$;
Step 6. Repeat steps 3-5 until the stop rule being satisfied;
Table 2: Sufficient Conditions for Different Algorithms Algorithm | BP | OMP | CoSaMP | hard | soft | half | Other AIT
---|---|---|---|---|---|---|---
$\mu$ | $\frac{1}{2k^{*}-1}^{[28]}$ | $\frac{1}{2k^{*}-1}^{[32]}$ | $\frac{0.384}{4k^{*}-1}^{\star}$ | $\frac{1}{3k^{*}}$ | $\frac{1}{4k^{*}}$ | $\frac{3}{10k^{*}}$ | $\frac{1}{(3+c)k^{*}}$
$(s,\delta_{s})$ | $(2k^{*},0.465)^{[31]}$ | $(k^{*}+1,\frac{1}{3\sqrt{k^{*}}})^{[33]}$ | $(4k^{*},0.384)^{[34]}$ | $(3k^{*},0.5)^{[34]}$ | – | – | –
$\star$: a coherence based sufficient condition for CoSaMP derived directly by
the fact that $\delta_{4k^{*}}<0.384$ and $\delta_{s}\leq(s-1)\mu$; –:
represents no related theoretical result as far as we know.
|
arxiv-papers
| 2013-10-15T08:30:59 |
2024-09-04T02:49:52.420563
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Jinshan Zeng, Shaobo Lin, Zongben Xu",
"submitter": "Jinshan Zeng",
"url": "https://arxiv.org/abs/1310.3954"
}
|
1310.4230
|
# ABOUT THE GLOBAL MAGNETIC FIELDS OF STARS
V.D. Bychkov1, L.V. Bychkova1, J. Madej2
1 Special Astronomical Observatory of RAS, Nizhnij Arkhyz, Russia,
[email protected] 2 Warsaw University Observatory, Warsaw, Poland [email protected]
ABSTRACT. We present a review of observations of the stellar longitudinal
(effective) magnetic field ($B_{e}$) and its properties. This paper also
discusses contemporary views on the origin, evolution and structure of
$B_{e}$.
Key words: Stars: magnetic field
1\. Introduction
At present there are collected direct measurements of the longitudinal
(effective) magnetic fields in 1873 stars of various spectral types. The total
number of the magnetic field $B_{e}$ measurements amounts to 24124. In the
following text we shall refer to $B_{e}$ as the magnetic field for brevity.
Figure 1: Number distribution of stars with measured longitudinal magnetic
fields $B_{e}$ vs. spectral type.
The dominant part of existing observations (for over 900 objects) was obtained
for CP stars.
2\. Observational data
We list here the most obvious advantages of the above progress:
1\. There is accumulated a large set of $B_{e}$ measurements.
2\. In some cases new magnetic measurements were obtained from spectra of
relatively low resolution.
3\. Those data were accumulated during a long time period (over 60 years),
which actually allows one to study the long-period magnetic behavior of some
objects.
Table 1: Principal methods of $B_{e}$ measurements: Method | N measurements
---|---
Phot. | 5375
Elc. | 6991
LSD and WDLS | 4083
BS | 1544
FORS1/2 | 2540
“Phot.” stands for the photographic method (Babcock 1947a,b, 1958 and many
others). This method is now obsolete and is not used.
The “Elc.” method is an analogue of the photographic method, but a CCD matrix
is used as the receiver of light. Previously CCD matrix replaced a
photographic plate in classical spectrometers. Currently echelle spectrometers
are routinely used due to limited size of CCD matrices. This method is still
sometimes applied.
“LSD and WDLS”: It is a well known method, cf. Donati et al. (1997), Wade et
al. (2000) and many other papers. This is a precise method, which was actively
in recent years and has yielded many new results.
“BS” denotes the average surface field of stars. Such a number of measurements
does not imply that “BS” was measured for high number of stars. For some
slowly rotating CP stars BS was measured many times.
FORS1/2 stands for the low-resolution spectropolarimeter at the ESO Very Large
Telescope.
“H-line” denotes $B_{e}$ measurements observed in hydrogen lines (Borra and
Landstreet 1980, and many other papers).
Figure 2: Number of individual $B_{e}$ measurements.
Figure 3: Distribution of magnetic stars vs. apparent stellar magnitude.
Figure 4: Number of $B_{e}$ measurements obtained in various years.
3\. Stars with known magnetic phase curves.
There exist 218 stars with measured phase curves of their longitudinal
(effective) magnetic field $B_{e}$. In that group, 172 objects are classified
as magnetic chemically peculiar stars. Remaining objects are stars of various
spectral types, from the most massive hot Of?p supergiants to low-mass red
dwarfs and stars with planets.
Table 2: Number of stars for which magnetic phase curves were determined vs. the most important types. All stars with mag. phase curves | 218
---|---
mCP stars | 172
Ae/Be Herbig stars | 7
Be stars | 7
Supermassive Of? | 3
Normal early B stars | 5
Flare stars | 3
TTS (T Tau type) | 2
var. Beta Cep type | 6
SPBS | 3
var. BY Dra type | 4
var. RS CVn type | 1
Semi-regular var. | 1
DA | 1
var.pulsating stars | 2
HPMS (high proper motions stars) | 3
var.Ori type | 2
Some stars were simultaneously put into two different classes. For example, HD
96446 belongs to both the He-r and $\beta$ Cep classes and HD 97048 belongs to
both the TTS and Ae/Be Herbig classes. The binary system DT Vir consists of
two companions: UV+RS (Flare + RS CVn type stars). Therefore, the distribution
of stars between classes had to be arbitrary or redundant in some cases.
For example, Fig.5 shows the magnetic phase curve for mCp stars $\beta$ CrB.
Periodic variability of the magnetic field of stars was described in more
detail by Bychkov et al. (2005, 2013).
Figure 5: Magnetic rotational phase curve of the mCp star $\beta$ CrB (HD
137909) for the accurate rotational period derived by Wade et al. (2000).
We selected the following most important conclusions about the magnetic
activity among stars of various types.
* •
1\. New class of magnetised objects was recently discovered – supermassive hot
stars, type Ofp?. These stars show periodic variations of the longitudinal
magnetic field. Amplitudes of magnetic phase curves (MPC) reach several
hundred G. Of?p stars apparently are slow rotators. Configuration of their
magnetic field is represented by an oblique rotator.
* •
2\. Magnetic fields were found among chemically normal early B stars. MPC’s
were obtained for 3 stars of this type. In one object, HD 149438, MPC shows
complicated double wave shape, displayed also by some mCP stars.
* •
3\. Magnetic field and its behaviour was best investigated in the group of mCP
stars. Longitudinal magnetic fields $B_{e}$ have simple dipole configuration
in majority of mCP stars (in 86 % objects). Rotational magnetic phase curves
often display simple harmonic shape with amplitudes reaching 10 kG.
Remaining 14 % of investigated mCP stars display more complex phase curves
being a superposition of two sine waves and have either dipole or more complex
structure of their global magnetic fields. Amplitudes of rotational $B_{e}$
variation essentially do not differ from those in “sine-wave” mCP stars.
* •
4\. Solar-type stars have global magnetic fields of low strength, seldom
approaching few dozens of G. Measuring of such low-intensity fields meets with
many methodologicacl difficulties. Therefore, we can only suppose, that in
some investigated stars (in $\xi$ Boo A, for example) magnetic phase curves
appear as simple harmonic waves. Very significant progress in measuring of
magnetic fields in stars was achieved using the ZDI method (magnetic
cartography of the surface). More credible considerations require higher
number of investigated stars and still higher accuracy of magnetic field
observations. Moreover, it is known that magnetic properties of solar-type
stars vary periodically in time scale from few years to several dozens of
years.
* •
5\. Ae/Be Herbig stars usually exhibit magnetic rotational phase curves of a
purely harmonic shape with amplitudes reaching several hundred G.
* •
6\. Magnetic phase curves of pulsating $\beta$ Cep stars vary with the period
of rotation. MPC show a complicated structure with low amplitudes of dozens G.
Closely related slowly pulsating B stars (SPB) also display longitudinal
magnetic field varying with the period of rotation. MPC show a simple harmonic
shape with amplitudes reaching several dozens G.
* •
7\. T Tau stars have magnetic fields of complex structure, display also
complex magnetic phase curves with amplitudes approaching several hundred G.
Undoubtedly, fields of such a strength have to strongly influence accretion of
matter onto stars.
* •
8\. Late-type stars – M dwarfs have global magnetic fields of complex
structure. Magnetic rotational phase curves only roughly can be approximated
by a superposition of two waves. This was also directly confirmed by recent
observations using the ZDI method. Amplitudes of variations of the integrated
longitudinal magnetic fields reach several hundred G. Some stars present an
amazing feature, stepwise creation or anihilation of the global magnetic field
and related $B_{e}$ variations.
* •
9\. HD 189733 – this is a typical dwarf of spectral class K2V, where a giant
planet, “hot Jupiter” was found. Central star in the system is a solar-like
object. The star possesses magnetic field which is typical for its spectral
class, and its longitudinal component varies with the amplitude of several G.
4\. mCp stars.
Magnetic fields of stars are best studied for mCp stars. One of major problems
for these stars is the relations between their magnetic field and the chemical
composition. We proposed a way to clarify this problem (Bychkov et al. 2009).
We defined relative magnetization (MA) for different types of chemically
peculiarity comparing distributions of their occurrence with the observed
$<B_{e}>$. Example of such a distribution for stars of Si peculiarity is shown
in Fig. 6. Number distribution of CP stars vs. $T_{\rm eff}$ for all different
types of chemical peculiarity was shown in Fig. 7. Magnetization “MA” for
various subclasses of CP stars vs. $T_{\rm eff}$ was shown in Fig. 8.
Reduction of “MA” with the reduction of $T_{\rm eff}$ is apparent there for
H-r, He-w and Si stars. Such a reduction of “MA” supports the fossil theory of
the magnetic field origin in those stars. If the age of a star is high, then
its mass is lower and “MA” also is lower. But we see sharp rise of “MA” about
$T_{\rm eff}=$ 10000 $K^{o}$. Therefore, we raise the assumption that the
dynamo mechanism joins at this point on the $T_{\rm eff}$ scale.
Figure 6: Integrated distribution function $N_{Int}(B)$ in percent (upper
panel), and the number distribution function $N(B)$ (lower panel) for stars of
Si peculiarity type.
Figure 7: Number distribution of CP stars vs. $T_{\rm eff}$ for various types
of chemical peculiarity.
Figure 8: Magnetization (MA) for various subclasses of CP stars. Bars define
the range of $T_{\rm eff}$ and MA occupied by a given subclass.
Summary.
In recent years significant progress was attained in the study of stellar
magnetism. While previously one could measure and discuss behaviour of the
stellar magnetic field only in mCP stars, white dwarfs and the Sun, currently
we can measure and collect data on the magnetic field for many more types of
stars ranging from supermassive hot giants to fully convective cold dwarfs of
low mass. One can note significant contribution of the MiMeS collaboration
which has discovered a new class of magnetic objects, supermassive hot giants
Ofp? type and other magnetised hot stars. These discoveries significantly
extended our knowledge about magnetism of hot stars and in future will give
rise to our understanding of processes in stellar atmospheres and
circumstellar space.
One can expect that rapid accumulation of new observational data will allow
one to study in detail the variability of stellar magnetic field in stars both
of different spectral types and evolutionary stages. We share the conviction
that the magnetic field and its evolution is a crucial agent of stellar
physics.
Acknowledgements. We acknowledge support from the Polish Ministry of Science
and Higher Education grant No. N N203 511638 and the Russian grant “Leading
Scientific Schools” N4308-2012.2.
References
Babcock H.W.: 1958, Ap.J.Suppl.Ser., 30, 141.
Borra E.F., Landstreet J.D.: 1980, Ap.J.Suppl.Ser., 42, 421.
Donati J.F., Semel M., Carter B.D., Rees D.E., Cameron A.C.: 1997, MNRAS, 291,
658.
Wade G. A., Donati J.-F., Landstreet J. D., Shorlin S. L. S.: 2000, MNRAS,
313, 823.
Babcock H. W.: 1947a, ApJ, 105, 105.
Babcock H. W.: 1947b, PASP, 59, 260.
Bychkov V.D., Bychkova L.V. and Madej J.: 2005, A&A, 430, 1143.
Bychkov V.D., Bychkova L.V. and Madej J.: 2013, AJ, 146:74, 10pp.
|
arxiv-papers
| 2013-10-16T00:28:56 |
2024-09-04T02:49:52.437368
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "V.D. Bychkov, L.V. Bychkova, J. Madej",
"submitter": "Jerzy Madej",
"url": "https://arxiv.org/abs/1310.4230"
}
|
1310.4283
|
11institutetext: Inria Paris-Rocquencourt
Ens, 45 rue d’Ulm, 75230 Paris Cedex 05, France
11email: [email protected]
# Abstract interpretation as anti-refinement
Arnaud Spiwack
###### Abstract
This article shows a correspondence between abstract interpretation of
imperative programs and the refinement calculus: in the refinement calculus,
an abstract interpretation of a program is a specification which is a
function.
This correspondence can be used to guide the design of mechanically verified
static analyses, keeping the correctness proof well separated from the
heuristic parts of the algorithms.
## 1 Introduction
A mathematical way to describe a static analysis is to see it as a program
which tries to prove a theorem about programs. It may fail to do so, but if it
succeeds, it effectively acts as a proof of the said theorem. The proof,
however, is essentially impossible to check by a human.
To increase the level of trust in a static analysis tool, the tool can be
mechanically verified, for instance in Coq [1], thus ensuring that the
produced proof is always correct. In the design of a static analysis tool,
some parts are crucial for correctness, while other are heuristic. For
instance, a static analysis can choose to lose precision to gain performance.
Hence, from the point of view of he who wants to ensure the correctness, a
static analysis can be seen as an interplay between a correctness enforcer and
an heuristic-providing oracle. The question addressed in this article is how
to formalise this interplay.
To that end, we use the refinement calculus [2, 3]. The refinement calculus is
a well-established method for proving program properties. It comes with a
natural notion of interaction, generally used to model the interaction between
the implementer of a unit of code and its user. In the context of this
article, the correctness enforcer plays the role of the implementer while the
oracle is the user.
Specifically, this article shows the connection between static analysis by
abstract interpretation [4] and the refinement calculus. Namely, it shows that
an abstract domain constructs a _specification_ of the analysed program, which
happens to be given by a function. This correspondence is instrumental in the
design of Cosa [5], a Coq formalisation of a shape analysis.
The two subjects have some notation overlap, hence some unconventional
notations will be used. The author apologises, but hopes that practitioners of
both subjects will not find the notations too surprising or confusing.
## 2 Predicate transformers
Edsger Dijkstra introduced the idea of using predicate transformers as
semantics of imperative programs [6]. The idea is to associate to each program
$p$ a function $\mathsf{wlp}{\left(p\right)}$, its _weakest liberal
precondition_ operator, such that for a property $P$ of program states,
$\mathsf{wlp}{\left(p\right)}{\left(P\right)}$ is the weakest condition on the
initial state, such that after running $p$, if $p$ terminates, then $P$ holds.
Weakest liberal precondition accounts for partial correctness. Alternatively,
one could use the weakest precondition operator (which additionally imposes
that $p$ terminates) to account for total correctness. Termination is not our
purpose here, and we will identify programs with their weakest liberal
precondition operator.
Predicate transformer semantics is the starting point of refinement calculus
[2], and is also commonly used in abstract interpretation – see [7] for a
discussion of weakest liberal precondition in relation to abstract
interpretation.
### 2.1 Basic definitions
We will call _predicate transformers_ monotone functions in
$\mathcal{P}{\left(A\right)}\rightarrow\mathcal{P}{\left(B\right)}$ for some
sets $A$ and $B$, and write
$\mathcal{P}{\left(A\right)}{\rightarrow}^{+}\mathcal{P}{\left(B\right)}$ for
the set of predicate transformers. The set
$\mathcal{P}{\left(A\right)}\rightarrow\mathcal{P}{\left(B\right)}$ inherits
the complete lattice structure of $\mathcal{P}{\left(B\right)}$ and
$\mathcal{P}{\left(A\right)}{\rightarrow}^{+}\mathcal{P}{\left(B\right)}$,
equipped with the lattice operations of
$\mathcal{P}{\left(A\right)}\rightarrow\mathcal{P}{\left(B\right)}$, is also a
complete lattice. We write $a\sqsubseteq
b\iff\forall{X}^{\in\mathcal{P}{\left(A\right)}}.\,\,a{\left(X\right)}\subseteq
b{\left(X\right)}$ for the inherited order.
We shall call the following operations of predicate transformers _regular
operations_. They have a direct interpretation as program constructs. Programs
will be interpreted as homogeneous predicate transformers
$\mathcal{P}{\left(A\right)}{\rightarrow}^{+}\mathcal{P}{\left(A\right)}$,
however the regular operations also work with general predicate transformers
$\mathcal{P}{\left(A\right)}{\rightarrow}^{+}\mathcal{P}{\left(B\right)}$.
Sequence
$\left(a;b\right){\left(X\right)}=a{\left(b{\left(X\right)}\right)}$
Reads as “do $a$ then do $b$”. The definition of sequence emphasises the fact
that the weakest liberal precondition semantics is a _backward_ semantics.
Sequence is associative, and monotone:
* •
$\left(a;b\right);c=a;\left(b;c\right)$
* •
$a\sqsubseteq a^{\prime}\land b\sqsubseteq b^{\prime}\implies a;b\sqsubseteq
a^{\prime};b^{\prime}$
Skip
$1{\left(X\right)}=X$
Does not do anything. Skip is neutral for sequence:
* •
$1;a=a=a;1$
Choice
$\left(a+b\right){\left(X\right)}=a{\left(X\right)}\cap b{\left(X\right)}$
Non-deterministic choice. Choice is associative, commutative and monotone.
Moreover sequence distributes on the right over choice:
* •
$\left(a+b\right)+c=a+\left(b+c\right)$
* •
$a+b=b+a$
* •
$\left(a+b\right);c=a;c+b;c$
* •
$p\sqsubseteq\left(a+b\right);q\iff p\sqsubseteq a;q\land p\sqsubseteq b;q$
Hang
$0{\left(X\right)}=\top$
Hang loops indefinitely. It is neutral for choice, sequence distributes on the
right over it, and it is the largest predicate transformer:
* •
$0+a=a=a+0$
* •
$0;a=0$
* •
$a\sqsubseteq 0$
Iteration
${a}^{*}$, for
$a\in\mathcal{P}{\left(A\right)}{\rightarrow}^{+}\mathcal{P}{\left(A\right)}$,
is the largest fixed point of the (monotone) function which maps $p$ to
$1+a;p$. It runs $a$ in sequence a non-deterministic number of times
(including none, and infinitely many). It has the following properties [2,
Chapter 21]:
* •
${a}^{*};q=q+a;{a}^{*};q$
* •
$p\sqsubseteq q+a;p\implies p\sqsubseteq{a}^{*};q$
It should be noted that despite the name “regular operations”, predicate
transformers do not form a Kleene algebra under these operations. Indeed the
left distributivity laws are missing: $a;\left(b+c\right)=a;b+a;c$ and $a;0=0$
do not hold in general.
### 2.2 Programs
In this setting, a programming language consists in a set $\mathcal{S}$ of
states together with a set
$\mathcal{I}\subseteq\mathcal{P}{\left(\mathcal{S}\right)}{\rightarrow}^{+}\mathcal{P}{\left(\mathcal{S}\right)}$
of _basic instructions_. A program in the language
$\left(\mathcal{S},\mathcal{I}\right)$ is an element of the subset of
$\mathcal{P}{\left(\mathcal{S}\right)}{\rightarrow}^{+}\mathcal{P}{\left(\mathcal{S}\right)}$
generated by $\mathcal{I}$ and the regular operations.
The use of non-deterministic choice and iterations make the programs non-
deterministic. This is a natural setting for both program refinement and
abstract interpretation. However, a typical programming language will feature
a set of tests $\mathcal{B}$ such that for all $b\in B$, there is $\llbracket
b\rrbracket\in\mathcal{P}{\left(\mathcal{S}\right)}$, and
$\mathsf{guard}{\left(b\right)}$ is an instruction, such that
$s\in\mathsf{guard}{\left(b\right)}{\left(X\right)}\iff s\in\llbracket
b\rrbracket\implies s\in X$.
With this assumption, the usual deterministic programming constructs can be
recovered:
$\mathsf{if}~{}b~{}\mathsf{then}~{}u~{}\mathsf{else}~{}v=\left(\mathsf{guard}{\left(b\right)};u\right)+\left(\mathsf{guard}{\left(\neg
b\right)};v\right)$, and
$\mathsf{while}~{}b~{}\mathsf{do}~{}u={\left(\mathsf{guard}{\left(b\right)};u\right)}^{*};\mathsf{guard}{\left(\neg
b\right)}$.
###### Example 1
As an example, let us consider a language with a single memory cell containing
an integer. In other words, $\mathcal{S}=\mathbb{Z}$. It has two tests,
$\mathsf{pos}$ and $\mathsf{npos}$, whose semantics are given by:
* •
$\llbracket\mathsf{pos}\rrbracket\iff\left\\{n\,{\in}\,\mathbb{Z}\mid
n>0\right\\}$
* •
$\llbracket\mathsf{npos}\rrbracket\iff\left\\{n\,{\in}\,\mathbb{Z}\mid
n\leqslant 0\right\\}$
and a operation $\mathsf{dec}$, which decrements the integer held in the
state. Its semantics is given by:
* •
$\mathsf{dec}{\left(X\right)}=\left\\{n\,{\in}\,\mathbb{Z}\mid n-1\in
X\right\\}$
This language expresses, for example, the simple program whose effect is to
decrease the integer held in the state until it is non-positive. We shall call
this program $d$:
* •
$d=\mathsf{while}~{}\mathsf{pos}~{}\mathsf{do}~{}\mathsf{dec}={\left(\mathsf{guard}{\left(\mathsf{pos}\right)};\mathsf{dec}\right)}^{*};\mathsf{guard}{\left(\mathsf{npos}\right)}$
### 2.3 Relations
A relation is usually seen as a subset of $A\times B$, however, it will be
more convenient to see them, equivalently, as functions of
$A\rightarrow\mathcal{P}{\left(B\right)}$.
Given a relation $r\in A\rightarrow\mathcal{P}{\left(B\right)}$, we can extend
it to a predicate transformer in two ways:
* •
$\left\langle
r\right\rangle\in\mathcal{P}{\left(A\right)}{\rightarrow}^{+}\mathcal{P}{\left(B\right)}$
defined by $\left\langle
r\right\rangle{\left(X\right)}=\bigcup_{\mbox{\scriptsize{$x$${\in}$$X$}}}r{\left(x\right)}$
* •
$\left[r\right]\in\mathcal{P}{\left(B\right)}{\rightarrow}^{+}\mathcal{P}{\left(A\right)}$
defined by $\left[r\right]{\left(Y\right)}=\left\\{x\,{\in}\,A\mid
r{\left(x\right)}\subseteq Y\right\\}$
The predicate transformers $\left\langle r\right\rangle$ and $\left[r\right]$
form a Galois connection _i.e._ :
* •
$\forall{X}^{\in\mathcal{P}{\left(A\right)}},{Y}^{\in\mathcal{P}{\left(B\right)}}.\,\,\left\langle
r\right\rangle{\left(X\right)}\subseteq Y\iff
X\subseteq\left[r\right]{\left(Y\right)}$
or equivalently:
* •
$\forall{X}^{\in\mathcal{P}{\left(A\right)}}.\,\,X\subseteq\left[r\right]{\left(\left\langle
r\right\rangle{\left(X\right)}\right)}$
* •
$\forall{Y}^{\in\mathcal{P}{\left(B\right)}}.\,\,\left\langle
r\right\rangle{\left(\left[r\right]{\left(Y\right)}\right)}\subseteq Y$
In fact, every Galois connection between powersets is of that form. This is
due to the general fact about complete lattices that a left adjoint – like
$\left\langle r\right\rangle$ – is the same thing as a function which
preserves joins. In the case of powersets, a function which preserves joins is
characterised by its action on singletons, hence is of the form $\left\langle
r\right\rangle$.
Identifying a function $f$ to its graph, we hence have a Galois connection
between $\left\langle f\right\rangle$ and $\left[f\right]$. These are better
known as the direct image and the inverse image of $f$, which we will write
${f}_{*}$ and ${f}^{-1}$ respectively. We shall make use of the following
consequence of their being a Galois connection:
* •
$x\in{f}^{-1}{\left(X\right)}\iff f{\left(x\right)}\in X$
The properties of Galois connections can also be read directly in terms of the
predicate transformer lattice:
* •
$\left\langle r\right\rangle;p\sqsubseteq q\iff p\sqsubseteq\left[r\right];q$
* •
$p;\left[r\right]\sqsubseteq q\iff p\sqsubseteq q;\left\langle r\right\rangle$
* •
${f}_{*};p\sqsubseteq q\iff p\sqsubseteq{f}^{-1};q$
* •
$p;{f}^{-1}\sqsubseteq q\iff p\sqsubseteq q;{f}_{*}$
## 3 Abstract interpretation
Abstract interpretation [4] is a framework for static analysis in which the
objects of study are called _domains_. As general as the definitions in this
section are, they fail to capture the full generality of abstract
interpretation. However, they are sufficient for most purposes – at least for
imperative languages.
Fixing a programming language $\left(\mathcal{S},\mathcal{I}\right)$, the
powerset $\mathcal{P}{\left(\mathcal{S}\right)}$ is called the concrete domain
and the interpretation of a program as a predicate transformer
$\mathcal{P}{\left(A\right)}{\rightarrow}^{+}\mathcal{P}{\left(A\right)}$ is
called the concrete semantics.
A departure from common practice is that the concrete semantics, the weakest
liberal precondition, is backward – _i.e._ a function from a set of final
states to corresponding initial states – whereas often the concrete semantics
is chosen to be forward. This choice has been made to stay closer to the
practice in refinement calculus. Having a backward concrete semantics does
not, however, constrain the analysis to be backward too. In the rest of the
paper we will mainly consider forward analysis. Moreover, forward semantics
are usually constructed from a relational semantics, _i.e._ they are of the
form $\left\langle r\right\rangle$, in which case $\left[r\right]$ will be our
backward semantics.
An abstract domain is a set ${\mathcal{S}}^{\sharp}$ together with a
concretisation function
$\gamma:{\mathcal{S}}^{\sharp}\rightarrow\mathcal{P}{\left(\mathcal{S}\right)}$
and extra material to construct an _abstract semantics_ to each program. The
abstract semantics of a program is a forward function
${p}^{\sharp}:{\mathcal{S}}^{\sharp}\rightarrow{\mathcal{S}}^{\sharp}$ which
has the following correctness property:
* •
$\forall{{s}^{\sharp}}^{\in{\mathcal{S}}^{\sharp}}.\,\,\forall{S}^{\in\mathcal{P}{\left(\mathcal{S}\right)}}.\,\,S\subseteq\gamma{\left({s}^{\sharp}\right)}\implies
S\subseteq
p{\left(\gamma{\left({p}^{\sharp}{\left({s}^{\sharp}\right)}\right)}\right)}$
Which can, equivalently be stated as:
* •
$\forall{{s}^{\sharp}}^{\in{\mathcal{S}}^{\sharp}}.\,\,\gamma{\left({s}^{\sharp}\right)}\subseteq
p{\left(\gamma{\left({p}^{\sharp}{\left({s}^{\sharp}\right)}\right)}\right)}$
This phrasing of the correctness property may look a bit contorted to the
practitioner of abstract interpretation. It is the consequence of having a
backward concrete semantics and a forward abstract semantics. When the
concrete semantics is of the form $p=\left[{p}_{0}\right]$, then this
correctness property coincides with the more familiar one:
* •
$\forall{{s}^{\sharp}}^{\in{\mathcal{S}}^{\sharp}}.\,\,\left\langle{p}_{0}\right\rangle{\left(\gamma{\left({s}^{\sharp}\right)}\right)}\subseteq\gamma{\left({p}^{\sharp}{\left({s}^{\sharp}\right)}\right)}$
Abstract domains are meant to be composed. For that reason, the abstract
semantics ${p}^{\sharp}$ is computed out of more atomic functions, which are,
in particular, stable by Cartesian product. Writing $s\leqslant
s^{\prime}\iff\gamma{\left(s\right)}\subseteq\gamma{\left(s^{\prime}\right)}$
for the order induced on ${\mathcal{S}}^{\sharp}$ by the concretisation
function, the abstract domain comes equipped with the following:
Join
An operator
$\sqcup\in{\mathcal{S}}^{\sharp}\times{\mathcal{S}}^{\sharp}\rightarrow{\mathcal{S}}^{\sharp}$
such that:
* •
${s}^{\sharp}\leqslant{s}^{\sharp}\sqcup{t}^{\sharp}$
* •
${t}^{\sharp}\leqslant{s}^{\sharp}\sqcup{t}^{\sharp}$
Post-fixed point
An operator
$\mathsf{pfp}\in\left({\mathcal{S}}^{\sharp}\rightarrow{\mathcal{S}}^{\sharp}\right)\rightarrow\left({\mathcal{S}}^{\sharp}\rightarrow{\mathcal{S}}^{\sharp}\right)$
such that:
* •
$\forall{f}^{\in{\mathcal{S}}^{\sharp}\rightarrow{\mathcal{S}}^{\sharp}},{{s}^{\sharp}}^{\in{\mathcal{S}}^{\sharp}}.\,\,{s}^{\sharp}\leqslant\mathsf{pfp}{\left(f\right)}{\left({s}^{\sharp}\right)}$
* •
$\forall{f}^{\in{\mathcal{S}}^{\sharp}\rightarrow{\mathcal{S}}^{\sharp}},{{s}^{\sharp}}^{\in{\mathcal{S}}^{\sharp}}.\,\,f{\left(\mathsf{pfp}{\left(f\right)}{\left({s}^{\sharp}\right)}\right)}\leqslant\mathsf{pfp}{\left(f\right)}{\left({s}^{\sharp}\right)}$
Typically, the post-fixed point operator is derived from a widening operator
$\nabla\in{\mathcal{S}}^{\sharp}\times{\mathcal{S}}^{\sharp}\rightarrow{\mathcal{S}}^{\sharp}$,
which has the following properties:
* •
${s}^{\sharp}\leqslant{s}^{\sharp}\nabla{t}^{\sharp}$
* •
${t}^{\sharp}\leqslant{s}^{\sharp}\nabla{t}^{\sharp}$
* •
For every increasing sequence ${\left({x}_{n}\right)}_{n\in\mathbb{N}}$, the
sequence ${\left({y}_{n}\right)}_{n\in\mathbb{N}}$ defined by
${y}_{0}={x}_{0}$ and ${y}_{n+1}={y}_{n}\nabla{x}_{n+1}$ verifies
$\exists{n}^{\in\mathbb{N}}.\,\,{y}_{n+1}\leqslant{y}_{n}$.
Then, taking, mutually recursively, ${x}_{0}={s}^{\sharp}$,
${x}_{n+1}=f{\left({y}_{n}\right)}$, and ${y}_{n}$ such as above, we can then
define $\mathsf{pfp}{\left(f\right)}{\left({s}^{\sharp}\right)}$ as any
${y}_{n}$ such that ${y}_{n+1}\leqslant{y}_{n}$.
Transfer functions
An abstract semantics ${i}^{\sharp}$ of the instruction $i\in\mathcal{I}$
The abstract semantics ${p}^{\sharp}$ of the program $p$ is defined by
induction on $p$ where the base case is given by the transfer functions. The
correction of ${p}^{\sharp}$ follows from the properties stated above.
* •
${\left(a;b\right)}^{\sharp}{\left({s}^{\sharp}\right)}={b}^{\sharp}{\left({a}^{\sharp}{\left({s}^{\sharp}\right)}\right)}$
* •
${\left(a+b\right)}^{\sharp}{\left({s}^{\sharp}\right)}=\left({a}^{\sharp}{\left({s}^{\sharp}\right)}\right)\sqcup\left({b}^{\sharp}{\left({s}^{\sharp}\right)}\right)$
* •
${1}^{\sharp}{\left({s}^{\sharp}\right)}={s}^{\sharp}$
* •
${0}^{\sharp}{\left({s}^{\sharp}\right)}$ can be chosen arbitrarily
* •
${\left({a}^{*}\right)}^{\sharp}{\left({s}^{\sharp}\right)}=\mathsf{pfp}{\left({a}^{\sharp}\right)}{\left({s}^{\sharp}\right)}$
###### Example 2
Let us define an abstract domain for the example language of Section 2.2: we
shall abstract the state – a single integer – by the signs it may take. More
precisely, we take for ${S}^{\sharp}$ the non-empty sets in
$\mathcal{P}{\left(\left\\{-,0,+\right\\}\right)}$ and the concretisation is
defined as:
* •
$\gamma{\left({s}^{\sharp}\right)}=\left\\{n\,{\in}\,\mathbb{Z}\mid\mathsf{sign}{\left(n\right)}\in{s}^{\sharp}\right\\}$
The abstract transfer function for guard instructions constrain the abstract
state to the relevant signs.
* •
${\mathsf{guard}}^{\sharp}{\left(\mathsf{pos}\right)}{\left({s}^{\sharp}\right)}={s}^{\sharp}\cap\left\\{+\right\\}$
* •
${\mathsf{guard}}^{\sharp}{\left(\mathsf{npos}\right)}{\left({s}^{\sharp}\right)}={s}^{\sharp}\cap\left\\{-,0\right\\}$
The abstract transfer function for the decrementing command maps positive to
non-negative and non-positive to negative:
* •
${\mathsf{dec}}_{0}{\left(+\right)}=\left\\{0,+\right\\}$
* •
${\mathsf{dec}}_{0}{\left(0\right)}=\left\\{-\right\\}$
* •
${\mathsf{dec}}_{0}{\left(-\right)}=\left\\{-\right\\}$
* •
${\mathsf{dec}}^{\sharp}{\left({s}^{\sharp}\right)}=\bigcup_{\mbox{\scriptsize{$x$${\in}$${s}^{\sharp}$}}}{\mathsf{dec}}_{0}{\left(x\right)}$
Since the abstract state space is a powerset, we can use union as the abstract
join, and since it is finite, union is also a widening:
* •
${s}^{\sharp}\nabla{t}^{\sharp}={s}^{\sharp}\sqcup{t}^{\sharp}={s}^{\sharp}\cup{t}^{\sharp}$
Now that the abstract domain is set up, let us run the abstract interpretation
on the program $d$ from Section 2.2 with the input state
$\left\\{0,+\right\\}$:
1. 1.
Entering the loop with initial state $\left\\{0,+\right\\}$
2. 2.
Applying ${\mathsf{guard}}^{\sharp}{\left(\mathsf{pos}\right)}$: state becomes
$\left\\{+\right\\}$
3. 3.
Applying ${\mathsf{dec}}^{\sharp}$: state becomes $\left\\{0,+\right\\}$
4. 4.
Invariant found after one iteration:
$\left\\{0,+\right\\}\cup\left\\{0,+\right\\}=\left\\{0,+\right\\}$
5. 5.
Applying ${\mathsf{guard}}^{\sharp}{\left(\mathsf{npos}\right)}$: final state
is $\left\\{0\right\\}$
## 4 Data refinement
Refinement calculus [2, 3] is a discipline to prove the correctness of
imperative programs, in a spirit close to Hoare logic. It arises from the
remark that, if most predicate transformers do not represent programs, they
still represent program _specifications_. Specifications are then _refined_
into more precise specifications, and eventually into programs.
A key point of the refinement calculus is that the refined specification need
not act on the same state as the abstract one. It is typical to use ideal
objects – like multisets – on the abstract side, and more concrete datatypes –
like linked lists – on the refined side.
We say [3] that
$a\in\mathcal{P}{\left(A\right)}{\rightarrow}^{+}\mathcal{P}{\left(A\right)}$
is refined by
$b\in\mathcal{P}{\left(B\right)}{\rightarrow}^{+}\mathcal{P}{\left(B\right)}$
through the _coupling invariant_
$\iota\in\mathcal{P}{\left(A\right)}{\rightarrow}^{+}\mathcal{P}{\left(B\right)}$,
written $a\sqsubseteq_{\iota}b$, when $\iota;a\sqsubseteq b;\iota$.
Intuitively $\iota$ is an action which transforms concrete states into
abstract states, so $\iota;a\sqsubseteq b;\iota$ reads “doing $b$ then
abstracting the state is more precise than abstracting the state then doing
$a$”. To emphasise that the type of the state has changed, this relation is
often called a _data refinement_.
###### Example 3
Specifications of imperative programs are typically given as pairs of a
precondition and a postcondition. For instance: under the precondition that
the initial state is a non-positive integer, the postcondition that the state
is $0$ holds after the program has been run. Both preconditions and
postconditions can be expressed systematically as (backward) predicate
transformers; they can be paired up into a full specification using sequence:
* •
${F}_{\mathsf{post}}{\left(X\right)}=\left\\{p\,{\in}\,\mathbb{Z}\mid 0\in
X\right\\}$
* •
${F}_{\mathsf{pre}}{\left(X\right)}=\left\\{n\,{\in}\,\mathbb{Z}\mid
n\leqslant 0\land n\in X\right\\}$
* •
$F={F}_{\mathsf{pre}};{F}_{\mathsf{post}}$
So that ${F}_{\mathsf{post}}{\left(X\right)}$ is either all of $\mathbb{Z}$ if
$0\in X$ or the empty set otherwise, and ${F}_{\mathsf{pre}}{\left(X\right)}$
simply ignores the states in $X$ which do not verify the precondition.
The program $d$ from Section 2.2 meets the specification $F$, however, the
state is represented as the _opposite_ integer. Hence we have an $\iota$ which
reflects this representation:
* •
${\iota}_{0}{\left(n\right)}=-n$
* •
$\iota={{\iota}_{0}}_{*}$
As per the definition of refinement, the statement that the program $d$
implements the specification reads
* •
$F\sqsubseteq_{\iota}\mathsf{while}~{}\mathsf{pos}~{}\mathsf{do}~{}\mathsf{decr}$
It is equivalent to the statement that the precondition entails the weakest
liberal precondition of
$d=\mathsf{while}~{}\mathsf{pos}~{}\mathsf{do}~{}\mathsf{decr}$:
* •
$\forall{n}^{\in\mathbb{Z}}.\,\,n\geqslant 0\implies
n\in\mathsf{wlp}{\left(d\right)}{\left(\left\\{0\right\\}\right)}$
which is the typical proof obligation in a Hoare logic setting.
The take away from data refinement is that it does not matter what coupling
invariant is used, as long as _all the function use the same coupling
invariant_. Or, more realistically, under some separation property, if all the
function _which have access to some part $A$ of the state_ all have coupling
invariants which agree on $A$.
In practice there are two reasons to refine the type of (a part of) the state:
it may be that it is an ideal type, say finite sets of integer, which may be
refined into an actual concrete data type, for instance list of integers. Or
it may be that the proposed data type is not efficient, and will be refined
into a more efficient representation – list of integers could be refined into
binary trees.
## 5 Abstract interpretation in refinement calculus
The main result of this article is that abstract interpretation can be
characterised in the language of the refinement calculus: an abstract
interpretation of a program $p$ is a _specification_ verified by $p$ which is
also a function.
###### Theorem 5.1
The soundness condition of abstract interpretation is a refinement condition:
${{p}^{\sharp}}^{-1}\sqsubseteq_{\left\langle\gamma\right\rangle}p\iff\forall{{s}^{\sharp}}^{\in{\mathcal{S}}^{\sharp}}.\,\,\gamma{\left({s}^{\sharp}\right)}\subseteq
p{\left(\gamma{\left({p}^{\sharp}{\left({s}^{\sharp}\right)}\right)}\right)}$
###### Proof
We have the following equivalent characterisation, thanks to the Galois
connection properties:
* •
${{p}^{\sharp}}^{-1}\sqsubseteq_{\left\langle\gamma\right\rangle}p\iff{{p}^{\sharp}}^{-1};\left[\gamma\right]\sqsubseteq\left[\gamma\right];p$
From which it follows that:
* ${{p}^{\sharp}}^{-1}\sqsubseteq_{\left\langle\gamma\right\rangle}p$
* ${\Longleftrightarrow}$ (Definition of sequence)
* $\forall{Y}^{\in\mathcal{P}{\left(\mathcal{S}\right)}}.\,\,{{p}^{\sharp}}^{-1}{\left(\left[\gamma\right]{\left(Y\right)}\right)}\subseteq\left[\gamma\right]{\left(p{\left(Y\right)}\right)}$
* ${\Longleftrightarrow}$ (Definition of inclusion)
* $\forall{Y}^{\in\mathcal{P}{\left(\mathcal{S}\right)}},{{s}^{\sharp}}^{\in{\mathcal{S}}^{\sharp}}.\,\,{s}^{\sharp}\in{{p}^{\sharp}}^{-1}{\left(\left[\gamma\right]{\left(Y\right)}\right)}\implies{s}^{\sharp}\in\left[\gamma\right]{\left(p{\left(Y\right)}\right)}$
* ${\Longleftrightarrow}$ (Definition of $\left[\gamma\right]$ and basic property of ${{p}^{\sharp}}^{-1}$)
* $\forall{Y}^{\in\mathcal{P}{\left(\mathcal{S}\right)}},{{s}^{\sharp}}^{\in{\mathcal{S}}^{\sharp}}.\,\,{p}^{\sharp}{\left({s}^{\sharp}\right)}\in\left[\gamma\right]{\left(Y\right)}\implies\gamma{\left({s}^{\sharp}\right)}\subseteq p{\left(Y\right)}$
* ${\Longleftrightarrow}$ (Definition of $\left[\gamma\right]$)
* $\forall{Y}^{\in\mathcal{P}{\left(\mathcal{S}\right)}},{{s}^{\sharp}}^{\in{\mathcal{S}}^{\sharp}}.\,\,\gamma{\left({p}^{\sharp}{\left({s}^{\sharp}\right)}\right)}\subseteq Y\implies\gamma{\left({s}^{\sharp}\right)}\subseteq p{\left(Y\right)}$
* ${\Longleftrightarrow}$ (${\Rightarrow}$ by $Y=\gamma{\left({p}^{\sharp}{\left({s}^{\sharp}\right)}\right)}$ and ${\Leftarrow}$ by monotonicity of $p$)
* $\forall{{s}^{\sharp}}^{\in{\mathcal{S}}^{\sharp}}.\,\,\gamma{\left({s}^{\sharp}\right)}\subseteq p{\left(\gamma{\left({p}^{\sharp}{\left({s}^{\sharp}\right)}\right)}\right)}$
In [8], Cousot & Cousot describe abstract interpretation of inference rule
systems. Their approach to defining abstract interpretation resembles
refinement calculus, they use, in particular, the remark that inference rule
systems can be represented as predicate transformers. Theorem 5.1 further
illuminates the connection.
Although so far we have mostly considered forward analyses, a similar
characterisation to Theorem 5.1 holds for backward analysis:
###### Theorem 5.2
${p}_{*}^{\sharp}\sqsubseteq_{\left\langle\gamma\right\rangle}p\iff\forall{{s}^{\sharp}}^{\in{\mathcal{S}}^{\sharp}}.\,\,\gamma{\left({p}^{\sharp}{\left({s}^{\sharp}\right)}\right)}\subseteq
p{\left(\gamma{\left({s}^{\sharp}\right)}\right)}$
In traditional refinement calculus, the process consists in starting with an
abstract definition, and refine it towards a more concrete definition,
weakening the preconditions, strengthening the postconditions while making the
state more suitable for execution. In static analysis, refinement calculus is
used somewhat backwards: starting from a concrete implementation, it is
refined into a more abstract definition, in effect strengthening the
precondition and weakening the postconditions, while still making the state
more suitable for execution.
## 6 Conclusion
A previous work by Sylvain Boulmé and Michaël Périn [9] uses refinement
calculus as a mean to check, in Coq, the correctness of a certificate
validation procedure for certificate meant to be output by an abstract
interpreter. Although this work is at the intersection of abstract
interpretation and refinement calculus, it does not try to establish a
connection between refinement calculus and the correctness condition of the
abstract interpretation procedure.
The present article shows that the language of abstract interpretation can be
recast in terms of the refinement calculus. This has been used in the
formalisation of Cosa [5], a Coq verified implementation of an abstract domain
for shape analysis. Cosa targets Compcert C [10], and uses numerical domains
by David Pichardie & _al_ [11].
Cosa relies on a variant of the refinement calculus introduced by Peter
Hancock based not on predicate transformers but on so-called _interaction
structures_ [12]. Compared to predicate transformers, interaction structures
carry more information: the set of predicate transformers can be seen as a
quotient of the set of interaction structures. The additional information
contained in interaction structures can be used to derive a datatype of
_strategies_ which the oracle is charged with providing, hence formalising the
separation between the oracle, which has no bearing on the correctness and
does not need to be mechanically verified, and the rules constituting the
domain which ensure correctness.
Interaction structures were initially developed as a variant of refinement
calculus suitable for type theory. Thanks to the results of this article,
interaction structures can be also leveraged for abstract interpretation.
## References
* [1] The Coq development team: The Coq Proof Assistant
* [2] Back, R.J., von Wright, J.: Refinement calculus: a systematic introduction. (1998)
* [3] von Wright, J.: The lattice of data refinement. Acta Informatica 135 (1994) 105–135
* [4] Cousot, P., Cousot, R.: Abstract interpretation frameworks. Journal of logic and computation 2(4) (1992) 511–547
* [5] Spiwack, A.: Cosa (2013)
* [6] Dijkstra, E.W.: Guarded commands, nondeterminacy and formal derivation of programs. Communications of the ACM 18(8) (August 1975) 453–457
* [7] Cousot, P.: Constructive Design of a Hierarchy of Semantics of a Transition System by Abstract Interpretation (Extended Abstract). Electronic Notes in Theoretical Computer Science 6 (January 1997) 77–102
* [8] Cousot, P., Cousot, R.: Inductive definitions, semantics and abstract interpretations. Proceedings of the 19th ACM SIGPLAN-SIGACT …(1992)
* [9] Boulmé, S., Périn, M.: Refinement calculus for a simple certification of static polyhedral analysis with code transformations. Technical report, Verimag (2013)
* [10] Leroy, X., Blazy, S., Dargaye, Z., Tristan, J.B.: CompCert
* [11] Blazy, S., Laporte, V., Maroneze, A., Pichardie, D.: Formal verification of a C value analysis based on abstract interpretation. Static Analysis (2013)
* [12] Hancock, P., Hyvernat, P.: Programming interfaces and basic topology. Annals of Pure and Applied Logic 137(1-3) (May 2009) 1–55
|
arxiv-papers
| 2013-10-16T07:16:49 |
2024-09-04T02:49:52.443875
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Arnaud Spiwack",
"submitter": "Arnaud Spiwack",
"url": "https://arxiv.org/abs/1310.4283"
}
|
1310.4486
|
# Strong Gravitational Lensing in a Charged Squashed Kaluza- Klein Gödel Black
hole
J. Sadeghi a and H. Vaez a
a _Physics Department, Mazandaran University_ ,
_P.O.Box 47416-95447, Babolsar, Iran_
Email: [email protected]: [email protected]
###### Abstract
In this paper we investigate the strong gravitational lansing in a charged
squashed Kaluza-Klein Gödel black hole. The deflection angle is considered by
the logarithmic term proposed by Bozza et al. Then we study the variation of
deflection angle and its parameters $\bar{a}$ and $\bar{b}$ . We suppose that
the supermassive black hole in the galaxy center can be considered by a
charged squashed Kaluza-Klein black hole in a Gödel background and by relation
between lensing parameters and observables, we estimate the observables for
different values of charge, extra dimension and Gödel parameters.
PACS numbers: 95.30.sf, 04.70.-s, 98.62.sb
Keywords: Gravitational lensing; Charged Squashed Kaluza-Klein Gödel Black
hole
## 1 Introduction
As we know the light rays or photons would be deviated from their straight way
when they pass close to the massive object such as black holes. This
deflection of light rays is known as gravitational lensing. This gravitational
lensing is one of the applications and results of general relativity [1] and
is used as an instrument in astrophysics, because it can help us to extract
the information about stars. In 1924 Chwolson pointed out that when a
star(source), a deflector(lens) and an observer are perfectly aligned, a ring-
shape image of the star appears which is called ’Einstein ring’. Other studies
have been led by Klimov, Liebes, refsdal and Bourassa and Kantowski [2].
Klimov investigated the lensing of galaxies by galaxies [3], but Liebes
studied the lensing of stars by stars and also stars by clusters in our galaxy
[4]. Refsdal showed that the geometrical optics can be used for investigating
the gravitational lenses properties and time delay resulting from it [5, 6].
The gravitational lensing has been presented in details in [7] and reviewed by
some papers (see for examples [8]-[11]). At this stage, the gravitational
lensing is developed for weak field limit and could not describe some
phenomena such as looping of light rays near the massive objects. Hence,
scientists started to study these phenomena from another point of view and
they proposed gravitational lensing in a strong field limit. When the source
is highly aligned with lens and the observer, one set of infinitive
relativistic ”ghost” images would be produce on each side of black hole. These
images are produced when the light rays that pass very close to black hole,
wind one or several times around the black hole before reaching to observer.
At first, this phenomenon was proposed by Darwin [12] and revived in Refs.
[13]-[15]. Darwin proposed a surprisingly easy formula for the positions of
the relativistic images generated by a Schwarzschild black hole. Afterward
several studies of null geodesics in strong gravitational fields have been led
in literatures: a semi-analytical investigation about geodesics in Kerr
geometry has been made in [16], also the appearance of a black hole in front
of a uniform background was studied in Refs. [17, 18]. Recently, Virbhadra and
Ellis formulated lensing in the ”strong field limit” and obtain the position
and magnification of these images for the Schwarzschild black hole [19, 20].
In Ref [21], by an alternative formulation, Frittelli, Kiling and Newman
attained an exact lens equation, integral expressions for its solutions, and
compared their result with Virbhadra and Ellis. Afterwards, the new method was
proposed by Bozza et al. in which they revisited the schwarzschild black hole
lensing by retaining the first two leading order terms [22]. This technic was
used by Eiroa, Romero and Torres to study a Reissner-Nordstrom black hole [23]
and Petters to calculate relativistic effects on microlensing events [24].
Finally, the generalization of Bozza’s method for spherically symetric metric
was developed in [25]. Bozza compared the image patterns for several
interesting backgrounds and showed that by the separation of the first two
relativistic images we can distinguish two different collapsed objects.
Further development for other black holes can be found in [26]-[33]. Several
interesting studeies are devoted to lensing by naked singularities [34, 35],
Janis-Newman-Winicour metric [25] and role of scaler field in gravitational
lensing [36]. In Ref [36], Virbhadra et al. have considered a static and
circularly symmetric lens characterized by mass and scalar charge parameter
and investigated the lensing for different values of charge parameter.
The gravitational lenses are important tools for probing the universe.
Narasimha and Chitre predicted that the gravitational lening of dark matter
can give the useful data about of position of dark matter in the universe [37,
39]. Also in some papers gravitational lens is used to detect exotic objects
in the universe, such as cosmic strings [40]-[42].
Recently, the idea of large extra dimensions has attracted much attention to
construct theories in which gravity is unified with other forces [43]. One of
the most interesting problems is the verification of extra dimensions by
physical phenomena. For this purpose higherdimensional black holes in
accelerators [44, 45] and in cosmic rays [46]-[49] and gravitational waves
from higher-dimensional black holes [50] are studied. The five-dimensional
Einstein-Maxwell theory with a Chern- Simons term predicted five-dimensional
charged black holes [51]. Such a higher-dimensional black holes would reside
in a spacetime that is approximately isotropic in the vicinity of the black
holes, but effectively four-dimensional far from the black holes [52]. These
higher dimensional black holes are called Kaluza-Klein black holes. The
presence of extra dimension is tested by quasinormal modes from the
perturbation around the higher dimensional black hole [53]-[59] and the
spectrum of Hawking radiation [60]-[63]. The gravitational lensing is another
method to investigate the extra dimension. Thus, the study of strong
gravitational lensing by higher dimensional black hole can help us to extract
information about the extra dimension in astronomical observations in the
future. The Kaluza-Klein black holes with squashed horizon [64] is one of the
extra dimensional black holes and it’s Hawking radiation and quasinormal modes
have been investigated in some papers [65]-[68]. Also, the gravitational
lensing of these black holes is studied in several papers. Liu et al. have
studied the gravitational lensing by squashed Kaluza-Klein black holes in Refs
[69, 70] and Sadeghi et al. investigated the charged type of this black hole
[71].
One the other hands we know that our universe is rotational and it is
reasonable to consider Gödel background for our universe. An exact solution
for rotative universe was obtained by Gödel. He solved Einstein equation with
pressureless matter and negative cosmological constant [72]. The solutions
representing the generalization of the Gödel universe in the minimal five
dimensional gauged supergravity are considered in many studies [74]-[80]. The
properties of various black holes in the Gödel background are investigated in
many works [gghghghhg]. The strong gravitational lensing in a Squashed Kaluza-
Klein Black hole in a Gödel universe is investigated in Ref. [70].
In this paper, we tudy the strong gravitational lensing in a charged squashed
Kaluza-Klein Gödel black hole. In that case, we see the effects of the scale
of the extra dimension, charge of black hole and Gödel parameter on the
coefficients and observables of strong gravitational lensing.
So, this paper is organized as follows: Section 2 is briefly devoted to
charged squashed Kaluza-Klein Gödel black hole background. In section 3 we use
the Bozza’s method [26, 27] to obtain the deflection angle and other
parameters of strong gravitational lensing as well as variation of them with
extra dimension, Gödel parameter and charge of black hole . In section 4, we
suppose that the supermassive object at the center of our galaxy can be
considered by the metric of charged squashed Kaluza-Klein Gödel black hole.
Then, we evaluate the numerical results for the coefficients and observables
in the strong gravitational lensing . In the last Section, we present a
summary of our work.
## 2 The charged squashed Kaluza- Klein Gödel black hole metric
The charged squashed Kaluza- Klein Gödel black hole spacetime is given by
[74],
$ds^{2}=-f(r)dt^{2}+\frac{k^{2}(r)}{V(r)}dr^{2}-2g(r)\sigma_{3}dt+h(r)\sigma^{2}_{3}+\frac{r^{2}}{4}[k(r)(\sigma^{2}_{1}+\sigma^{2}_{2})+\sigma^{2}_{3}],$
(1)
where
$\displaystyle\sigma_{1}=\cos\psi\,d\theta+\sin\psi\,\sin\theta\,d\phi,$
$\displaystyle\sigma_{2}=-\sin\psi\,d\theta+\cos\psi\,\sin\theta\,d\phi,$
$\displaystyle\sigma_{3}=d\psi+\cos\theta\,d\phi.$ (2) $\displaystyle
f(r)=1-\frac{2M}{r^{2}}+\frac{q^{2}}{r^{4}},\,\,\,\,\,\,\,g(r)=j(r^{2}+3q),\,\,\,\,\,h(r)=-j^{2}r^{2}(r^{2}+2M+6q),\,\,\,\,\,\,$
$\displaystyle
V(r)=1-\frac{2M}{r^{2}}+\frac{16j^{2}(M+q)(M+2q)}{r^{2}}+\frac{q^{2}(1-8j^{2}(M+3q))}{r^{4}},\,\,\,\,\,\,k(r)=\frac{V(r_{\infty})r_{\infty}^{4}}{(r^{2}-r_{\infty}^{2})^{2}}.$
(3)
and $0\leq\theta<\pi$, $0\leq\phi<2\pi$, $0\leq\psi<4\pi$ and
$0<r<r_{\infty}$. Here $M$ and $q$ are the mass and charge of the black hole
respectively and $j$ is the parameter of Gödel background. The killing horizon
of the black hole is given by equation $V(r)=0$ , where
$\displaystyle
r_{h}^{2}=M-8j^{2}(M+q)(M+2q)\pm\sqrt{[M+q-8j^{2}(M+2q)^{2}][M-q-8j^{2}(M+q)^{2}]}.$
(4)
We see that the black hole has two horizons. As $q\longrightarrow 0$ the
horizon of the squashed Kaluza- Klein Gödel black hole is recovered [70] and
when $q$ and $j$ tend to zero, we have $r_{h}^{2}=2M$, which is the horizon of
five-dimensional Schwarzschild black hole. Here we note that the argument of
square root constraints the mass, charge and Gödel parameter values. When
$r_{\infty}\longrightarrow\infty$, we have $k(r)\longrightarrow 1$, which
means that the squashing effect disappears and the five-dimensional charged
black hole is recovered.
By using the transformations,
$\rho=\rho_{0}\frac{r^{2}}{r^{2}_{\infty}-r^{2}}$,
$\tau=\sqrt{\frac{\rho_{0}(1+\alpha)}{\rho_{0}+\rho_{M}}}t$ and
$\alpha=\frac{\rho_{q}^{2}(\rho_{0}+\rho_{M})}{\rho_{0}(\rho_{0}+\rho_{q})^{2}}$,
the metric (1) can be written in the following form,
$ds^{2}=-\mathcal{F}(\rho)d\tau^{2}+\frac{K(\rho)}{\mathcal{G}(\rho)}d\rho^{2}+\mathcal{C}(\rho)(d\theta^{2}+sin^{2}\theta\,d\phi^{2})-2H(\rho)\sigma_{3}d\tau+\mathcal{D}(\rho)\sigma_{3}^{2},$
(5)
$\displaystyle\mathcal{F}(\rho)=1-\frac{\rho_{M}-2\alpha\rho_{0}}{(1+\alpha)\rho_{M}}(\frac{\rho_{M}}{\rho})+\frac{\alpha}{1+\alpha}(\frac{\rho_{0}}{\rho_{M}})^{2}(\frac{\rho_{M}}{\rho})^{2},$
$\displaystyle
K(\rho)=1+\frac{\rho_{0}}{\rho}\,,\,\,\,\,\,\mathcal{G}=(1-\frac{\rho_{h+}}{\rho})(1-\frac{\rho_{h-}}{\rho}),$
$\displaystyle\mathcal{C}(\rho)=\rho^{2}K(\rho),\,\,\,\,H(\rho)=jr_{\infty}^{2}\left(\frac{1}{K(\rho)}+\frac{3\rho_{q}}{\rho_{0}+\rho_{q}}\right)\sqrt{\frac{\rho_{0}+\rho_{M}}{\rho_{0}(1+\alpha)}},$
$\displaystyle\mathcal{D}(\rho)=\frac{r^{2}_{\infty}}{4K(\rho)}-\frac{j^{2}\rho
r_{\infty}^{2}}{(\rho+\rho_{0})^{2}(\rho_{M}+\rho_{0})(\rho_{q}+\rho_{0})}\times$
$\displaystyle\left\\{\rho[\rho_{0}(\rho_{0}+2\rho_{M})+7\rho_{0}\rho_{q}+8\rho_{M}\rho_{q}]+\rho_{0}[\rho_{0}(\rho_{M}+6\rho_{q})+7\rho_{M}\rho_{q}]\right\\},$
(6)
with
$\displaystyle\rho_{M}=\rho_{0}\frac{2M}{r^{2}_{\infty}-2M}\,\,,\,\,\,\,\,\,\,\rho_{q}=\rho_{0}\frac{q}{r^{2}_{\infty}-q},\,\,\,\,\,\,\rho_{h\pm}=\rho_{0}\frac{r_{h\pm}^{2}}{r^{2}_{\infty}-r_{h\pm}^{2}}\,.$
(7)
Figure 1: The plots show the variation of horizon radiuses with respect to
$j$, $\rho_{0}$ and $\rho_{q}$ (Note that in each figure, for $\rho_{q}\neq
0$, two horizons merge at a point. This point has been shown for one of
figures. )
Where $\rho_{h+}$ and $\rho_{h-}$ denote the outer and inner horizons of the
black hole in the new coordinate and $\rho_{0}$ is a scale of transition from
five-dimensional spacetime to an effective four-dimensional one. Here
$\rho_{0}^{2}=\frac{r^{2}_{\infty}}{4}V(r_{\infty})$, so that
$r^{2}_{\infty}=4(\rho_{0}+\rho_{h+})(\rho_{0}+\rho_{h-})$. The Komar mass of
black hole is related to $\rho_{M}$ with $\rho_{M}=2G_{4}M$, where $G_{4}$ is
the four dimensional gravitational constant. By using relations (4) and (7) we
can obtain $\rho_{h\pm}$ in the following coupled equations,
$\displaystyle
2\left[\rho_{0}(\rho_{h+}+\rho_{h-})+2\rho_{h+}\rho_{h-}\right]={a}(\rho_{h+},\rho_{h-}),$
$\displaystyle
2\left[\rho_{0}(\rho_{h+}-\rho_{h-})\right]={b}(\rho_{h+},\rho_{h-}),$ (8)
where
$\displaystyle
a=\frac{\rho_{M}r_{\infty}^{2}}{2(\rho_{0}+\rho_{M})}-2j^{2}r_{\infty}^{4}\frac{\left(\rho_{M}\rho_{0}+3\rho_{M}\rho_{0}+2\rho_{q}\rho_{0}\right)\left(\rho_{M}\rho_{0}+5\rho_{M}\rho_{q}+4\rho_{q}\rho_{0}\right)}{(\rho_{0}+\rho_{M})^{2}(\rho_{0}+\rho_{q})^{2}},$
$\displaystyle
b=\left\\{a^{2}-4\frac{\rho_{q}^{2}r_{\infty}^{4}}{(\rho_{0}+\rho_{q})^{2}}\left(1-4j^{2}r_{\infty}^{2}\frac{(\rho_{0}\rho_{M}+7\rho_{M}\rho_{q}+6\rho_{q}\rho_{0})}{(\rho_{0}+\rho_{M})(\rho_{0}+\rho_{q})}\right)\right\\}^{\frac{1}{2}}.$
(9)
when $\rho_{q}\longrightarrow 0$, the horizon of black hole becomes
$\rho_{h}=\frac{2(\rho_{0}+\rho_{M})}{\sqrt{1+64j^{2}\rho_{M}^{2}}+}-\rho_{0}$,
as obtained in [70]. In case of $\rho_{q}\longrightarrow 0$ and
$j\longrightarrow 0$, we have $\rho_{h}=\rho_{M}$ which is consistent with
neutral squashed Kaluza-Klein black hole [69]. You note that the square root
in relation (2) constrains the values of $\rho_{0}$, $\rho_{q}$ and $j$. In
the case $j=0$, the permissive regime is shown in Ref.[71]. For any value of
$\rho_{q}$ there is allowed rang for $\rho_{0}$. Hence these parameters can
not select any value and when $j$ increases from zero, permissive regime for
$\rho_{q}$ and $\rho_{0}$ becomes more confined. We solved the above coupled
equations numerically and results are shown in figure 1. We see that the outer
horizon of black hole increases with the size of extra dimension, $\rho_{0}$
and decreases with $j$ and $\rho_{q}$. Note that two horizons of black hole
coincide in especial values of parameters, which in that case we have an
extremal black hole.
## 3 Geodesic equations, Deflection angle
In this section, we are going to investigate the deflection angle of light
rays when they pass close to a charged squashed Kaluza-Klein Gödel black hole.
We also study the effect of the charge parameter $\rho_{q}$, the scale of
extra dimension $\rho_{0}$ and Gödel parameter on the deflection angle and
it’s coefficients in the equatorial plane ($\theta=\pi/2$).
In this plane, the squashed Kaluza-Klein Gödel metric reduces to
$ds^{2}=-\mathcal{F}(\rho)d\tau^{2}+\mathcal{B}(\rho)d\rho^{2}+\mathcal{C}(\rho)\,d\phi^{2}+\mathcal{D}(\rho)d\psi^{2}-2H(\rho)dtd\psi,$
(10)
where
$\mathcal{B}(\rho)=\frac{K(\rho)}{\mathcal{G}(\rho)}.$ (11)
The null geodesic equations are,
${\ddot{x}_{i}}+\Gamma_{jk}^{i}\,\dot{x}^{j}\,\dot{x}^{k}=0,$ (12)
where
$g_{ij}\dot{x}^{i}\,\dot{x}^{j}=0,$ (13)
where $\dot{x}$ is the tangent vector to the null geodesics and the dote
denotes derivative with respect to affine parameter. We use equation (12) and
obtain the following equations,
$\displaystyle\dot{t}=\frac{\mathcal{D}(\rho)E-H(\rho)L_{\psi}}{H^{2}(\rho)+\mathcal{F}(\rho)\mathcal{D}(\rho)},$
$\displaystyle\dot{\phi}=\frac{L_{\phi}}{\mathcal{C}(\rho)},$
$\displaystyle\dot{\psi}=\frac{H(\rho)E+\mathcal{F}(\rho)L_{\psi}}{H^{2}(\rho)+\mathcal{F}(\rho)\mathcal{D}(\rho)},$
(14)
$(\dot{\rho})^{2}=\frac{1}{\mathcal{B}(\rho)}\left[\frac{\mathcal{D}(\rho)E-2H(\rho)EL_{\psi}-\mathcal{F}(\rho)L^{2}_{\psi}}{H^{2}(\rho)+\mathcal{F}(\rho)\mathcal{D}(\rho)}-\frac{L^{2}_{\phi}}{\mathcal{C}(\rho)}\right].$
(15)
where $E$, $L_{\phi}$ and $L_{\psi}$ are constants of motion. Also, the
$\theta$-component of equation (12) in equatorial plane $\theta=\pi/2$, is
given by,
$\displaystyle\dot{\phi}\left[\mathcal{D}(\rho)\dot{\psi}-H(\rho)\dot{t}\right]=0.$
(16)
If $\dot{\phi}=0$, then deflection angle will be zero and this is illegal, So
we set $L_{\psi}=\mathcal{D}(\rho)\dot{\psi}-H(\rho)\dot{t}=0$. By using
equation (15) one can obtain following expression for the impact parameter,
$L_{\phi}=u=\sqrt{\frac{\mathcal{C}(\rho_{s})\mathcal{D}(\rho_{s})}{H^{2}(\rho_{s})+\mathcal{F}(\rho_{s})\mathcal{D}(\rho_{s})}},$
(17)
and the minimum of impact parameter takes place in photon sphere radius
$r_{ps}$, that is given by the root of following equation [81],
$\mathcal{D}(\rho_{s})\,\left[H(\rho_{s})^{2}+\mathcal{F}(\rho_{s})\mathcal{D}(\rho_{s})\right]\mathcal{C}^{\prime}(\rho_{s})-\mathcal{C}(\rho_{s})\,\left[\mathcal{D}(\rho_{s})^{2}\mathcal{F}^{\prime}(\rho_{s})+2\mathcal{D}(\rho_{s})H(\rho_{s})H^{\prime}(\rho_{s})-H(\rho_{s})^{2}\mathcal{D}^{\prime}(\rho_{s})\right]=0.$
(18)
Figure 2: The variation of photon sphere radius with respect to $j$,
$\rho_{0}$ and $\rho_{q}$.
Figure 3: The variation of impact parameter as a function of $j$, $\rho_{0}$
and $\rho_{q}$.
Figure 4: The variation of $\bar{a}$ with respect to $j$, $\rho_{0}$ and
$\rho_{q}$.
Here $\rho_{s}$ is the closet approach for light ray and the prime is
derivative with respect to $\rho_{s}$. The analytical solution for the above
equation is very complicated, so we calculated the equation (18) numerically.
Variations of r photon sphere radius are plotted with respect to the charge
$\rho_{q}$, the scale of extra dimension $\rho_{0}$ and Gödel parameter in the
figure 2. Also figure 3 shows variations of impact parameter in it’s minimum
value (at radius of photon sphere). These figures show that by adding the
charge to the black hole, the behavior of the photon sphere radius and minimum
of impact parameter is different compare with the neutral black hole [69]. As
$\rho_{0}$ approaches to it’s minimum values the radius of photon sphere and
impact parameter become divergent.
By using the chain derivative and equation (15), the deflection angle in the
charged squashed Kaluza-Klein Gödel black hole can be written as,
$\displaystyle\alpha_{\varphi}(\rho_{s})=I_{\varphi}(\rho_{s})-\pi,$
$\displaystyle\alpha_{\psi}(\rho_{s})=I_{\psi}(\rho_{s})-\pi,$ (19)
$\displaystyle
I_{\varphi}(\rho_{s})=2\int^{\infty}_{\rho_{s}}\frac{\sqrt{\mathcal{B}(\rho)\mathcal{A}(\rho)\mathcal{C}(\rho_{s})}}{\mathcal{C}(\rho)}\frac{1}{\sqrt{\mathcal{F}(\rho_{s})-\mathcal{F}(\rho)\frac{\mathcal{C}(\rho_{s})}{\mathcal{C}(\rho)}}}\,\,\,d\rho,$
(20) $\displaystyle
I_{\psi}(\rho_{s})=2\int^{\infty}_{\rho_{s}}\frac{H(\rho)}{\mathcal{D}(\rho)}\sqrt{\frac{\mathcal{B}(\rho)\mathcal{A}(\rho_{s})}{\mathcal{A}(\rho)}}\frac{1}{\sqrt{\mathcal{F}(\rho_{s})-\mathcal{F}(\rho)\frac{\mathcal{C}(\rho_{s})}{\mathcal{C}(\rho)}}}\,\,\,d\rho,$
(21)
with
$\mathcal{A}(\rho)=\frac{H^{2}(\rho)+\mathcal{F}(\rho)\mathcal{D}(\rho)}{\mathcal{D}(\rho)}.$
(22)
When we decrease the $\rho_{s}$ (and consequently $u$) the deflection angle
increases. At some points, the deflection angle exceeds from $2\pi$ so that
the light ray will make a complete loop around the compact object before
reaching at the observer. By decreasing $\rho_{s}$ further, the photon will
wind several times around the black hole before emerging. Finally, for
$\rho_{s}=\rho_{sp}$ the deflection angle diverges and the photon is captured
by the black hole. Moreover, from equations (20) and (21), we can find that in
the Charged Squashed Kaluza-Klein Gödel black hole, both of the deflection
angles depend on the parameters $j$, $\rho_{0}$ and $\rho_{q}$, which implies
that we could detect the rotation of universe, the extra dimension and charge
of black hole in theory by gravitational lens. Note that the
$I_{\phi}(\rho_{s})$ depend on $j^{2}$, not $j$. It shows that the deflection
angle is independent of the direction of rotation of universe. But, from
equation (21) we find that the integral $I_{\psi}(\rho)$ contains the factor
$j$, then the deflection angle $\alpha_{\psi}(\rho_{s})$ for the photon
traveling around the lens in two opposite directions is different .The main
reason is that the equatorial plan is parallel with Gödel rotation plan [70].
When $j$ vanishes, the deflection angle of $\psi$ tends zero [69, 71]. We
focus on the deflection angle in the $\phi$ direction,
So we can rewrite the equation (20) as,
$I(\rho_{s})=\int^{1}_{0}R(z,\rho_{s})f(z,\rho_{s})\,dz,$ (23)
with
$R(z,\rho_{s})=2\frac{\rho}{\rho_{s}\mathcal{C}(\rho)}\sqrt{\mathcal{B}(\rho_{s})\mathcal{A}(\rho)\mathcal{C}(\rho_{s})},$
(24)
and
$f(z,\rho_{s})=\frac{1}{\sqrt{\mathcal{A}(\rho_{s})-\mathcal{A}(\rho)\mathcal{C}(\rho_{s})/\mathcal{C}(\rho)}},$
(25)
where $z=1-\frac{\rho_{s}}{\rho}$. The function $R(z,\rho_{s})$ is regular for
all values of $z$ and $\rho_{s}$, while $f(z,\rho_{s})$ diverges as $z$
approaches to zero. Therefore, we can split the integral (23) in two parts,
the divergent part $I_{D}(\rho_{s})$ and the regular one $I_{R}(\rho_{s})$,
which are given by,
$I_{D}(\rho_{s})=\int^{1}_{0}R(0,\rho_{ps})f_{0}(z,\rho_{s})\,dz,$ (26)
$I_{R}(\rho_{s})=\int^{1}_{0}\left[R(z,\rho_{s})f(z,\rho_{s})-R(0,\rho_{ps})f_{0}(z,\rho_{s})\right]\,dz.$
(27)
Here, we expand the argument of the square root in $f(z,\rho_{s})$ up to the
second order in $z$ [70],
$f_{0}(z,\rho_{s})=\frac{1}{\sqrt{p(\rho_{s})z+q(\rho_{s})z^{2}}},$ (28)
where
$p(\rho_{s})=\frac{\rho_{s}}{\mathcal{C}(\rho_{s})}\left[\mathcal{C}^{\prime}(\rho_{s})\mathcal{A}(\rho_{s})-\mathcal{C}(\rho_{s})\mathcal{A}^{\prime}(\rho_{s})\right],$
(29)
${q}(\rho_{s})=\frac{\rho_{s}^{2}}{2\mathcal{C}(\rho_{s})}\left[2\mathcal{C}^{\prime}(\rho_{s})\mathcal{C}(\rho_{s})\mathcal{A}^{\prime}(\rho_{s})-2\mathcal{C}^{\prime}(\rho_{s})^{2}\mathcal{A}(\rho_{s})+\mathcal{A}(\rho_{s})\mathcal{C}(\rho_{s})\mathcal{C}^{\prime\prime}(\rho_{s})-\mathcal{C}^{2}(\rho_{s})\mathcal{A}^{\prime\prime}(\rho_{s})\right].$
(30)
For $\rho_{s}>\rho_{ps}$, $p(\rho_{s})$ is nonzero and the leading order of
the divergence in $f_{0}$ is $z^{-1/2}$, which have a finite result. As
$\rho_{s}\longrightarrow\rho_{ps}$, $p(\rho_{s})$ approaches zero and
divergence is of order $z^{-1}$, that makes the integral divergent. Therefor,
the deflection angle can be approximated in the following form [25],
$\alpha=-\bar{a}\,log\left(\frac{u}{u_{sp}}-1\right)+\bar{b}+O(u-u_{sp}),$
(31)
where
$\displaystyle\bar{a}=\frac{R(0,\rho_{ps})}{2\sqrt{q(\rho_{ps})}}\,,$
$\displaystyle\bar{b}=-\pi+b_{R}+\bar{a}\,log\frac{\rho_{ps}^{2}\left[\mathcal{C}^{\prime\prime}(\rho_{ps})\mathcal{A}(\rho_{ps})-\mathcal{C}(\rho_{ps})\mathcal{A}^{\prime\prime}(\rho_{ps})\right]}{u_{ps}\sqrt{\mathcal{A}^{3}(\rho_{ps})\mathcal{C}(\rho_{ps})}}\,,$
$\displaystyle
b_{R}=I_{R}(\rho_{ps}),\,\,\,\,\,\,u_{ps}=\sqrt{\frac{\mathcal{C}(\rho_{ps})}{\mathcal{A}(\rho_{ps})}}\,.$
(32)
By using (31) and (3), we can investigate the properties of strong
gravitational lensing in the charged squashed Kaluza- Klein Gödel black hole.
In this case, variations of the coefficients $\bar{a}$ and $\bar{b}$, and the
deflection angle $\alpha$ have been plotted with respect to the extra
dimension $\rho_{0}$, charge of the black hole $\rho_{q}$, and Gödel parameter
$j$ in figures 4-6.
As $j$ tends to zero, these quantities reduce to charged squashed Kaluza-klein
black hole [71] and with $\rho_{q}=0$ the squashed Kaluza-klein black hole
recovers [69]. One can see that the deflection angle increases with extra
dimension and decreases with $\rho_{q}$. By comparing these parameters with
those in four-dimensional Schwarzschild and Reissner-Nordström black holes ,
we could extract information about the size of extra dimension as well as the
charge of the black hole by using strong field gravitational lensing.
Figure 5: The variation of $\bar{b}$ with respect to $j$, $\rho_{0}$ and
$\rho_{q}$.
Figure 6: Deflection angle as a function of $j$, $\rho_{0}$ and $\rho_{q}$ at
$x_{s}=x_{ps}+0.05$. Note that $\alpha$ is given in Radian.
Figure 7: The variation of $s$ with respect to $j$, $\rho_{0}$ and $\rho_{q}$ . The angular separation is expressed in $\mu$arcseconds. $\rho_{q}$ | $\rho_{0}$ | | $\theta_{\infty}$ | | | $s$ | | | $r_{m}$ |
---|---|---|---|---|---|---|---|---|---|---
| | $j=0$ | $j=0.03$ | $j=0.06$ | $j=0$ | $j=0.03$ | $j=0.06$ | $j=0$ | $j=0.03$ | $j=0.06$
| $0$ | 26.007 | 25.473 | 24.013 | 0.0325 | 0.0319 | 0.0300 | 6.8219 | 6.8219 | 6.8219
| $0.2$ | 27.669 | 26.962 | 25.080 | 0.0339 | 0.0390 | 0.0365 | 6.6838 | 6.8212 | 6.6736
| $0.4$ | 29.214 | 28.327 | 25.987 | 0.0476 | 0.0465 | 0.0434 | 6.5678 | 6.5617 | 6.5440
0 | $0.6$ | 30.662 | 29.583 | 26.735 | 0.0556 | 0.0543 | 0.0507 | 6.4681 | 6.4583 | 6.4269
| $0.8$ | 32.032 | 30.749 | 27.370 | 0.0639 | 0.0625 | 0.0587 | 6.3830 | 6.3660 | 6.3188
| $1$ | 33.335 | 31.836 | 27.899 | 0.0726 | 0.0710 | 0.0672 | 6.3046 | 6.2835 | 6.2167
| $0.2$ | 29.389 | 28.121 | 25.005 | 0.0171 | 0.0164 | 0.0150 | 7.7688 | 7.7606 | 7.7378
| $0.4$ | 29.260 | 27.914 | 24/574 | 0.0411 | 0.0397 | 0.0361 | 6.7556 | 6.7450 | 6.7142
0.03 | $0.6$ | 30.594 | 29.093 | 25.339 | 0.0531 | 0.0512 | 0.0477 | 6.5281 | 6.5136 | 6.4706
| $0.8$ | 31.953 | 30.258 | 26.000 | 0.0627 | 0.0607 | 0.0557 | 6.4063 | 6.3872 | 6.3270
| $1$ | 33.261 | 31.344 | 26.525 | 0.0719 | 0.0697 | 0.0647 | 6.3163 | 6.2907 | 6.2107
| $0.2$ | 34.882 | 33.270 | 29.337 | 0.0048 | 0.0047 | 0.0042 | 9.3012 | 9.2919 | 9.2658
| $0.4$ | 29.395 | 27.677 | 23.623 | 0.0292 | 0.0281 | 0.0251 | 7.1808 | 7.1680 | 7.1155
0.06 | $0.6$ | 30.412 | 28.527 | 24.031 | 0.0469 | 0.0450 | 0.0404 | 6.6806 | 6.6598 | 6.5995
| $0.8$ | 31.738 | 29.648 | 24.632 | 0.0593 | 0.0575 | 0.0515 | 6.4727 | 6.4470 | 6.3704
| $1$ | 33.052 | 30.736 | 25.132 | 0.0699 | 0.0674 | 0.0616 | 6.3480 | 6.3159 | 6.2190
| $0.2$ | 48.924 | 46.807 | 41.546 | 0.0021 | 0.0020 | 0.0018 | 10.476 | 10.470 | 10.452
| $0.4$ | 29.635 | 27.645 | 23.073 | 0.0191 | 0.0183 | 0.0163 | 7.6851 | 7.6624 | 7.6058
0.09 | $0.6$ | 30.144 | 27.932 | 22.858 | 0.0403 | 0.0338 | 0.0337 | 6.8883 | 6.8601 | 6.7795
| $0.8$ | 31.408 | 28.961 | 23.318 | 0.0548 | 0.0523 | 0.0468 | 6.5684 | 6.5351 | 6.4372
| $1$ | 32.732 | 30.027 | 23.761 | 0.0672 | 0.0643 | 0.0582 | 6.3948 | 6.3555 | 6.2362
| $0.2$ | 111.910 | 107.525 | 96.386 | 0.0019 | 0.0018 | 0.0017 | 11.362 | 11.360 | 11.353
| $0.4$ | 30.003 | 27.821 | 22.893 | 0.0124 | 0.0118 | 0.0105 | 8.1843 | 8.1580 | 8.0848
0.12 | $0.6$ | 29.809 | 27.331 | 21.824 | 0.0324 | 0.0308 | 0.0274 | 7.1226 | 7.0867 | 6.9846
| $0.8$ | 30.984 | 28.221 | 22.075 | 0.0497 | 0.0473 | 0.0422 | 6.6823 | 6.6418 | 6.5175
| $1$ | 32.311 | 29.255 | 22.433 | 0.0639 | 0.0610 | 0.0550 | 6.4516 | 6.4031 | 6.2563
Table 1: Numerical estimations for the coefficients and observables of strong
gravitational lensing by considering the supermmasive object of galactic
center be a charged squashed Kaluza-Klein Gödel black hole. (Not that the
numerical values for $\theta_{\infty}$ and $s$ are of order microarcsec)
## 4 Observables estimation
In the previous section, we investigated the strong gravitational lensing by
using a simple and reliable logarithmic formula for deflection angle, which
was obtain by Bozza et al. Now, by using relations between the parameters of
the strong gravitational lensing and observables, estimat the position and
magnification of the relativistic images. By comparing these observables with
the data from the astronomical observation, we could detect properties of an
massive object. We suppose that the spacetime of the supermassive object at
the galaxy center of Milky Way can be considered as a charged squashed Kaluza-
Klein Gödel black hole, then we can estimate the numerical values for
observables.
We can write the lens equation in strong gravitational lensing, as the source,
lens, and observer are highly aligned as follows [22],
$\beta=\theta-\frac{D_{LS}}{D_{OS}}\Delta\alpha_{n},$ (33)
where $D_{LS}$ is the distance between the lens and source. $D_{OS}$ is the
distance between the observer and the source so that, $D_{OS}=D_{LS}+D_{OL}$.
$\beta$ and $\theta$ are the angular position of the source and the image with
respect to lens, respectively. $\Delta\alpha_{n}=\alpha-2n\pi$ is the offset
of deflection angle with integer $n$ which indicates the $n$-th image.
The $n$-th image position $\theta_{n}$ and the $n$-th image magnification
$\mu_{n}$ can be approximated as follows [22, 25],
Figure 8: The variation of $r_{m}$ with $j$, $\rho_{0}$ and $\rho_{q}$.
Figure 9: The variation of angular position $\theta_{\infty}$ with respect to
$j$, $\rho_{0}$ and $\rho_{q}$ that is given in $\mu$arcseconds.
$\theta_{n}=\theta^{0}_{n}+\frac{u_{ps}(\beta-\theta_{n}^{0})e^{\frac{\bar{b}-2n\pi}{\bar{a}}}D_{OS}}{\bar{a}D_{LS}D_{OL}},$
(34)
$\mu_{n}=\frac{u_{ps}^{2}(1+e^{\frac{\bar{b}-2n\pi}{\bar{a}}})e^{\frac{\bar{b}-2n\pi}{\bar{a}}}D_{OS}}{\bar{a}\beta
D_{LS}D_{OL}^{2}}.$ (35)
$\theta_{n}^{0}$ is the angular position of $\alpha=2n\pi$. In the limit
$n\longrightarrow\infty$, the relation between the minimum of impact parameter
$u_{ps}$ and asymptotic position of a set of images $\theta_{\infty}$ can be
expressed by $u_{ps}=D_{OL}\theta_{\infty}$. In order to obtain the
coefficients $\bar{a}$ and $\bar{b}$, in the simplest case, we separate the
outermost image $\theta_{1}$ and all the remaining ones which are packed
together at $\theta_{\infty}$, as done in Refs [22, 25]. Thus
$s=\theta_{1}-\theta_{\infty}$ is considered as the angular separation between
the first image and other ones and the ratio of the flux of them is given by,
$\mathcal{R}=\frac{\mu_{1}}{\sum_{n=2}^{\infty}\mu_{n}}.$ (36)
We can simplify the observables and rewrite them in the following form [22,
25],
$\displaystyle
s=\theta_{\infty}e^{\frac{\bar{b}}{\bar{a}}-\frac{2\pi}{\bar{a}}},$ (37)
$\displaystyle\mathcal{R}=e^{\frac{2\pi}{\bar{a}}}.$
Thus, by measuring the $s$, $\mathcal{R}$ and $\theta_{\infty}$, one can
obtain the values of the coefficients $\bar{a}$, $\bar{b}$ and $u_{sp}$. If we
compare these values by those obtained in the previous section, we could
detect the size of the extra dimension, charge of black hole and rotation of
universe. Another observable for gravitational lensing is relative
magnification of the outermost relativistic image with the other ones. This
observable is shown by $r_{m}$ which is related to $\mathcal{R}$ by,
$\displaystyle r_{m}=2.5\,\log\mathcal{R}.$ (38)
Using $\theta_{\infty}=\frac{u_{sp}}{D_{OL}}$ and equations (3), (37) and (38)
we can estimate the values of the observable in the strong field gravitational
lensing. The variation of the observables $\theta_{\infty}$, $s$ and $r_{m}$
are plotted in figures 7-9. Note that the mass of the central object of our
galaxy is estimated to be $4.31\times 10^{6}M_{\odot}$ and the distance
between the sun and the center of galaxy is $D_{OL}=8.5\,kpc$ [82].
For different $\rho_{0}$, $\rho_{q}$ and $j$, the numerical values for the
observables are listed in Table 1. One can see that our results reduce to
those in the four-dimensional Schwarzschild black hole as
$\rho_{0}\longrightarrow 0$. Also our results are in agreement with the
results of Ref. [70] in the limit $\rho_{q}\longrightarrow 0$ and in the limit
$j\longrightarrow 0$, the results of Ref. [71] are recovered.
## 5 Summary
The light rays can be deviated from the straight way in the gravitational
field as predicted by General Relativity. This deflection of light rays is
known as gravitational lensing. In the strong field limit, the deflection
anglethe of the light rays which pass very close to the black hole, becomes so
large that, it winds several times around the black hole before appearing at
the observer. Therefore the observer would detect two infinite set of faint
relativistic images produced on each side of the black hole. On the other hand
the extra dimension is one of the important predictions in the string theory
which is believed to be a promising candidate for the unified theory. Also it
is reasonable to consider a rotative universe with global rotation. Hence the
five-dimensional Einstein-Maxwell theory with a Chern- Simons term in string
theory predicted five-dimensional charged black holes in the Gödel background.
In our study, we considered the charged squashed Kaluza-Klein Gödel black hole
spacetime and investigated the strong gravitational lensing by this metric. We
obtained theoretically the deflection angle and other parameters of strong
gravitational lensing . Finally, we suppose that the supermassive black hole
at the galaxy center of Milky Way can be considered by this spacetime and we
estimated numerically the values of observables that are realated to the
lensing parameters. Theses observable parameters are $\theta_{\infty}$, $s$
and $R$, where $\theta_{\infty}$ is the position of relativistic images, $s$
angular separation between the first image $\theta_{1}$ and other ones
$\theta_{\infty}$ and $R$ is the ratio of the flux from the first image and
those from all the other images. Our results are presented in figures 1-9 and
Table 1. By comparatione observable parameters with observational data
measured by the astronomical instruments in the future, we can discuss the
properties of the massive object in the center of our galaxy.
### Acknowledgments
H. Vaez would like to thank Afshin. Ghari for helpful comments.
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|
arxiv-papers
| 2013-10-16T19:40:26 |
2024-09-04T02:49:52.458409
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "J. Sadeghi and H. Vaez",
"submitter": "Hassan Vaez",
"url": "https://arxiv.org/abs/1310.4486"
}
|
1310.4507
|
# Bifurcation in entanglement renormalization group flow of a gapped spin
model
Jeongwan Haah Department of Physics, Massachusetts Institute of Technology,
Cambridge, MA 02139 Institute for Quantum Information and Matter, California
Institute of Technology, Pasadena, CA 91125
(5 February 2014)
###### Abstract
We study entanglement renormalization group transformations for the ground
states of a spin model, called cubic code model $H_{A}$ in three dimensions,
in order to understand long-range entanglement structure. The cubic code model
has degenerate and locally indistinguishable ground states under periodic
boundary conditions. In the entanglement renormalization, one applies local
unitary transformations on a state, called disentangling transformations,
after which some of the spins are completely disentangled from the rest and
then discarded. We find a disentangling unitary to establish equivalence of
the ground state of $H_{A}$ on a lattice of lattice spacing $a$ to the tensor
product of ground spaces of two independent Hamiltonians $H_{A}$ and $H_{B}$
on lattices of lattice spacing $2a$. We further find a disentangling unitary
for the ground space of $H_{B}$ with the lattice spacing $a$ to show that it
decomposes into two copies of itself on the lattice of the lattice spacing
$2a$. The disentangling transformations yield a tensor network description for
the ground state of the cubic code model. Using exact formulas for the
degeneracy as a function of system size, we show that the two Hamiltonians
$H_{A}$ and $H_{B}$ represent distinct phases of matter.
## I Introduction
The renormalization group (RG) is a collection of transformations that select
out quantities relevant to long-distance physics. Wilson1975RG It generally
consists of averaging out short-distance fluctuations and rescaling of the
system in order to recover the original picture. In practice, however, details
of RG transformations are context-dependent. When an action is given and the
corresponding partition function is of interest, the RG transformation
concerns the effective parameters (e.g., coupling constants, temperature) of
the theory as a function of probing length/energy scale. When a wave-function
is of interest, the RG transformation takes place in a parametrization space
of the wave functions such that the transformed wave-function recovers
correlations at long distance.
This paper is on the wave function renormalization, focusing on long-range
entanglement structure. As the entanglement of many body system is not
characterized by a single number, our general goal is to compare states with
well-known states or to classify them under a suitable RG scheme.
VerstraeteCiracLatorreEtAl2005Renormalization ; Vidal2007ER ;
ChenGuWen2010transformation The entanglement between any adjacent pair of
spins can be arbitrary since it can be changed simply by applying a local
unitary operator, which will certainly not affect the long-range behavior in
any possible way. This means that we should allow local unitary
transformations in our definition of equivalence of long-range entanglement,
and the block of spins on which the local unitary is acting should generally
be regarded as a single degree of freedom; the long-range entanglement will
only depend on the entanglement among the coarse-grained blocks. In the case
where the state is represented by some fixed network of tensors,
VerstraeteCirac2004PEPS this observation has been used to choose the most
relevant part of the tensors VerstraeteCiracLatorreEtAl2005Renormalization ;
ChenGuWen2010transformation and to speed up certain numerical calculations.
LevinNave2007Tensor
Here, we study long-range entanglement of the ground states of a particular
three-dimensional gapped spin model, via local unitary transformations that
simplify the entanglement pattern. This model, called the cubic code model,
Haah2011Local shares an important property with intrinsically topologically
ordered systems, Wen1991SpinLiquid namely the _local indistinguishability_
BravyiHastings2011short of ground states. However, there are two crucial
differences: One is that the degeneracy under periodic boundary conditions is
a very sensitive function of the system size. The other is that it only admits
point-like excitations whose hopping amplitude is exactly zero in presence of
any small perturbation. Although the cubic code model as presented is exactly
solvable, it is important to ask for a corresponding continuum theory. This is
one of the main motivations of this work.
Our result is as follows. Let $H_{A}(a)$ be the Hamiltonian of the cubic code
model. (See Eq. (4).) $H_{A}(a)$ lives on a simple cubic lattice with two
qubits per site where the lattice spacing is $a$. (We will mostly use the term
‘qubit’ in place of ‘spin-$1/2$’ from now on, since only the fact that each
local degree of freedom is two-dimensional is important.) Let $H_{B}(a)$
denote another gapped spin Hamiltonian on a three-dimensional simple cubic
lattice with four qubits per site where the lattice spacing is $a$. $H_{B}(a)$
will be given explicitly later in Eq. (14). We find a constant number of
layers of local unitary transformations (finite-depth quantum circuit) $U$
such that for any ground state $\left|\psi_{A}(a)\right\rangle$ of $H_{A}(a)$,
we have
$U\left|\psi_{A}(a)\right\rangle=\sum_{i}c_{i}\left|\psi_{A}^{i}(2a)\right\rangle\otimes\left|\psi_{B}^{i}(2a)\right\rangle\otimes\left|\uparrow\cdots\uparrow\right\rangle$
(1)
where $c_{i}$ are complex numbers that depend on
$\left|\psi_{A}(a)\right\rangle$, and
$\left|\psi_{A}(2a)^{i}\right\rangle,~{}\left|\psi_{B}(2a)^{i}\right\rangle$
are ground states of $H_{A}(2a),~{}H_{B}(2a)$, respectively. Note that on the
right-hand side the wave function lives on the coarser lattice with lattice
spacing $2a$. The coarser lattice is depicted in Fig. 1. The unit cell of the
coarser lattice has 16 qubits per Bravais lattice point. 10 qubits in each
unit cell are in the trivial state, disentangled from the rest. The
Hamiltonian $H_{A}$ and $H_{B}$ live on the disjoint systems of qubits
designated by $A$ and $B$ in Fig. 1, respectively.
Figure 1: Simple cubic lattice of lattice spacing of $2a$ and the unit cell.
There are 16 qubits labeled by $0$, $A$, or $B$ in the unit cell. Those that
are labeled by $0$ are in the trivial product state.
$\left|\psi_{A}\right\rangle$ and $\left|\psi_{B}\right\rangle$ in Eq. (1) are
states of the system of the qubits labeled by $A$ and $B$, respectively.
Furthermore, we find another finite-depth quantum circuit $V$ such that for
any ground state $\psi_{B}(a)$ of $H_{B}(a)$,
$V\left|\psi_{B}(a)\right\rangle=\sum_{i}c^{\prime}_{i}\left|\psi_{B}^{i}(2a)\right\rangle\otimes\left|\psi_{B}^{i}(2a)\right\rangle\otimes\left|\uparrow\cdots\uparrow\right\rangle$
(2)
for some numbers $c^{\prime}_{i}$. Again, on the right-hand side the wave
functions live on the coarser lattice. The qubits in the trivial state in Eq.
(2) are uniformly distributed throughout the lattice, similar to Fig. 1. The
first and second $\left|\psi_{B}\right\rangle$ in Eq. (2) are states of
disjoint systems of qubits, similar to $A$ and $B$ of Fig. 1.
The result can be written suggestively as
$\mathcal{R}(H_{A})=H_{A}\oplus H_{B},\quad\mathcal{R}(H_{B})=H_{B}\oplus
H_{B}$ (3)
where $\mathcal{R}$ denotes the disentangling transformation followed by the
scaling transformation by a factor of 2. This is rather unexpected and should
be contrasted with the previous results. AguadoVidal2007Entanglement ;
ChenGuWen2010transformation ; Aguado2011 ;
BuerschaperMombelliChristandlEtAl2013 ; SchuchCiracPerez-Garcia2010G-injective
It has been known that Levin-Wen string-net model LevinWen2005String-net and
Kitaev quantum double model Kitaev2003Fault-tolerant are entanglement RG
fixed points. Those results would have been summarized as $\mathcal{R}(H)=H$.
The ground-state subspace is retained at the coarse-grained lattice. There was
no splitting. We will comment further on it later.
The present paper is organized as follows. We begin by defining the model and
reviewing its properties in Sec. II. We give details on the entanglement RG in
Sec. III. The actual unitary operators appearing in Eqs. (1),(2) will not be
displayed in the text, but in a Mathematica script in Supplementary Material.
SM Next, we argue in Sec. IV that the newly found Hamiltonian $H_{B}$
represents a different phase of matter, based on the degeneracy formulas of
the models on periodic lattices. In Sec. V, we point out the relevance of so-
called branching MERA EvenblyVidal2012RG description for the ground states of
the cubic code model. In Sec. VI, we describe a special representation of the
models, exploiting the translation symmetry and properties of Pauli matrices.
The special representation simplifies the calculation of the unitaries of Eqs.
(1),(2) significantly. Sec. VII builds on the preceeding section, giving an
algebro-geometric criterion and some intuition behind the entanglement RG
calculations. We conclude with a short discussion in Sec. VIII. Appendix A
contains a direct bound EvenblyVidal2013bMERAentropy on the entanglement
entropy of a branching MERA state for a box region.
## II Model
The spin model primarily considered in this paper is described by an
unfrustrated translation-invariant Hamiltonian on the simple cubic lattice
$\Lambda=\mathbb{Z}^{3}$ with two qubits per lattice site. Haah2011Local
$\displaystyle H_{A}=-J\sum_{i\in\Lambda}\left(G^{x}_{i}+G^{z}_{i}\right)$ (4)
where $J>0$ and
$\displaystyle G^{x}_{i}$
$\displaystyle=\sigma^{x}_{i,1}\sigma^{x}_{i,2}\sigma^{x}_{i+\hat{x},1}\sigma^{x}_{i+\hat{y},1}\sigma^{x}_{i+\hat{z},1}\sigma^{x}_{i+\hat{y}+\hat{z},2}\sigma^{x}_{i+\hat{z}+\hat{x},2}\sigma^{x}_{i+\hat{x}+\hat{y},2}$
(5) $\displaystyle G^{z}_{i}$
$\displaystyle=\sigma^{z}_{i,1}\sigma^{z}_{i,2}\sigma^{z}_{i-\hat{x},2}\sigma^{z}_{i-\hat{y},2}\sigma^{z}_{i-\hat{z},2}\sigma^{z}_{i-\hat{y}-\hat{z},1}\sigma^{z}_{i-\hat{z}-\hat{x},1}\sigma^{z}_{i-\hat{x}-\hat{y},1}$
(6)
are eight-qubit interaction terms consisted of Pauli matrices. The index $i$
runs over all elementary cubes. The terms $G^{x}_{i}$ and $G^{z}_{i}$ are
visually depicted as
---
$\textstyle{XI\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{IX\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{IX\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{II\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{XX\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{XI\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{XI\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{IX\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$
---
$\textstyle{ZI\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{IZ\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{IZ\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{ZZ\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{II\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{ZI\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{ZI\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{IZ\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\circ\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\hat{z}}$$\scriptstyle{\hat{x}}$$\scriptstyle{\hat{y}}$
For the arrangement of the Pauli matrices on the vertices of the unit cube,
this is called cubic code model. (It is a quantum error correcting code, but
we will not discuss the theory of quantum error correction.) One can easily
verify that each term $G^{x}_{i}$ or $G^{z}_{i}$ commutes with any other term
$G^{x}_{j}$ or $G^{z}_{j}$ in the Hamiltonian $H_{A}$. A ground state
$\left|\psi\right\rangle$ of $H_{A}$ can be written as
$\displaystyle\left|\psi\right\rangle=\sum_{G\in\mathcal{G}}G\left|\uparrow\uparrow\cdots\uparrow\right\rangle$
(7)
where $\mathcal{G}$ is the abelian group generated multiplicatively by terms
$G^{x}_{i}$’s and $G^{z}_{i}$’s. Since
$\left|\uparrow\uparrow\cdots\uparrow\right\rangle$ is an eigenstate of
$G^{z}_{i}$ for any $i$ with eigenvalue $+1$, the group $\mathcal{G}$ can be
replaced by a smaller group consisting all products of $G^{x}_{i}$’s. The
ground state is degenerate (ground space). This will not concern us.
The energy spectrum can be understood by commutation relations among Pauli
matrices, since the Hamiltonian Eq. (4) is a sum of commuting tensor products
of Pauli matrices. Let us call a tensor product of Pauli matrices
$\sigma^{x},\sigma^{y},\sigma^{z}$ a Pauli operator. If
$\left|\psi\right\rangle$ is a ground state and $P$ is any Pauli operator,
then $P\left|\psi\right\rangle$ is also an energy eigenstate. In fact, it is a
common eigenstate for $G^{x}_{i}$ and $G^{z}_{i}$. This is because any term
$G^{x}_{i}$ or $G^{z}_{i}$ in the Hamiltonian, being a Pauli operator, either
commutes or anticommutes with $P$ ($PG^{x,z}=\pm G^{x,z}P$) and the ground
state $\left|\psi\right\rangle$ is stabilized by any $G^{x}$ and $G^{z}$
($G^{x,z}\left|\psi\right\rangle=\left|\psi\right\rangle$).
To understand the (excited) state $P\left|\psi\right\rangle$ better, imagine
that we measure all $G^{x}$ and $G^{z}$ simultaneously. This is possible since
they are pairwise commuting. The measurement outcomes on
$P\left|\psi\right\rangle$ are definite and take values $\pm 1$. Let us say
that there is a $X$-type _defect_ at $i$ if the expectation value of
$G^{x}_{i}$ is $-1$. Likewise, we define $Z$-type defects. Each defect has
energy $2J$, and a state with no defect is a ground state. A configuration of
the defects characterizes an excited state effectively, but not uniquely due
to the ground state degeneracy; for orthogonal ground states
$\left|\psi\right\rangle$ and $\left|\phi\right\rangle$, orthogonal states
$P\left|\psi\right\rangle$ and $P\left|\phi\right\rangle$ give the same
configuration of defects. Note that the whole Hilbert space is spanned by
states of form $P\left|\psi\right\rangle$ for some Pauli operator $P$ and some
ground state $\left|\psi\right\rangle$.
An exotic property of the cubic code model is that the excitations are
_pointlike_ but _immobile_. They are pointlike because a single isolated
defect is a valid configuration, but are immobile because they are not allowed
to hop to other position by a local operator. Here, the locality is important.
There indeed exists a non-local operator that annihilates a defect and create
another at a different place. The statement remains true even if we loosen our
restriction that there be exactly one defect at $p$. In a general case, one
should distinguish a cluster of defects that is locally created, in which case
we call the cluster _neutral_ , from a cluster that is not locally created, in
which case we call the cluster _charged_. (Since the charged cluster has
nothing to do with any symmetry, it is termed “topologically charged.”) The
immobility asserts that any charged cluster cannot be transported by any
operator of finite support.
Rigorously, the immobility is stated as follows. Suppose
$\left|\psi\right\rangle$ is a state with a single defect, or more generally
any charged cluster of defects, contained in a box of linear dimension $w$.
Let $\mathbb{T}$ denote a translation operator by one unit length in the
lattice along arbitrary direction. Then, for any operator $O$ of finite
support, (i.e., $O$ is local,) one has
$\langle\psi|O\mathbb{T}^{n}|\psi\rangle=0$ whenever $n>15w$. The number $15$
is merely a convenient number to make an argument smooth. Important is that
there is some finite $n=n(w)$ such that the transition amplitude becomes
_exactly_ zero. See Ref. Haah2011Local, for proofs.
The cubic code model is _topologically ordered_ Wen1991SpinLiquid in the
sense that the ground state subspace is degenerate and no local operator is
capable of distinguishing any two ground states; Haah2011Local if $O$ is an
arbitrary local operator and $\left|\psi_{1}\right\rangle$ and
$\left|\psi_{2}\right\rangle$ are two arbitrary ground states, then one has
$\displaystyle\langle\psi_{1}|O|\psi_{1}\rangle=c(O)\langle\psi_{1}|\psi_{2}\rangle$
(8)
for some number $c(O)$ that only depends on the operator $O$ but not on the
states $|\psi_{1,2}\rangle$. In addition, the model satisfies the so-called
“local topological order” condition, MichalakisZwolak2013Stability which
implies that the degeneracy is exact up to an error that is exponentially
small in the system size. BravyiHastings2011short In other words, all ground
states have exactly the same local reduced density matrices, and this property
does not require a fine-tuning. For an application of the model in robust
quantum memory, see Ref. BravyiHaah2011Memory, .
The actual degeneracy and questions on non-local operators that distinguish
different ground states are fairly technical. One can show Haah2012PauliModule
that the degeneracy of the cubic code model on a $L\times L\times L$ lattice
with periodic boundary conditions is equal to $2^{k}$ where
$\displaystyle\frac{k+2}{4}$
$\displaystyle=\mathrm{deg}_{x}~{}\gcd\begin{bmatrix}1+(1+x)^{L},\\\
1+(1+\omega x)^{L},\\\ 1+(1+\omega^{2}x)^{L}\end{bmatrix}_{\mathbb{F}_{4}}$
(9) $\displaystyle=\begin{cases}1&\text{if $L=2^{p}+1~{}(p\geq 1)$},\\\
L&\text{if $L=2^{p}~{}(p\geq 1)$}\end{cases}$ (10)
That is, one computes three polynomials over the field of four elements
$\mathbb{F}_{4}=\\{0,1,\omega,\omega^{2}\\}$ and takes the greatest common
divisor polynomial and reads off the degree in $x$. The proof of this formula
contained in Ref. Haah2012PauliModule, is based on an algebraic
representation of the Hamiltonian Eq. (4), which will be reviewed in Sec. VI
below.
The cubic code Hamiltonian Eq. (4) belongs to a class of so-called stabilizer
(code) Hamiltonians, as it is defined as a sum of commuting Pauli operators.
The Kitaev toric code model Kitaev2003Fault-tolerant and the Wen plaquette
model Wen2003Plaquette are well-known examples of stabilizer Hamiltonians.
The ground states in these models have a nice geometric interpretation in
terms of string-nets, LevinWen2005String-net whereas, unfortunately, there is
no known geometric interpretation for the ground state of the cubic code
model, other than the trivial expression Eq. (7).
## III Entanglement renormalization and bifurcation
It will be useful to recall the notion of _finite depth quantum circuit_. A
depth-1 quantum circuit is a product of local unitary operators of disjoint
support. We do not restrict the number of the unitary operators participating
in the product, but each unitary operator must be local, that is, its support
can be covered by some ball of fixed radius. This radius is referred to as the
range of the circuit. A finite depth quantum circuit is a finite product of
depth-1 quantum circuits. The number of layers must be independent of system
size. The finite depth quantum circuit is a discrete version of the unitary
evolution $e^{-it\mathcal{H}}$ by a sum $\mathcal{H}$ of local Hermitian
operators for $t=O(1)$.
The _entanglement renormalization group_ transformation is a procedure where
one disentangles some of degrees of freedom by local unitary transformations,
and compares the transformed state to the original state. The purpose is to
understand “long range” entanglement. Given a many-qubit quantum state
$\left|\psi\right\rangle$ and a finite depth quantum circuit $U$ such that
$U\left|\psi\right\rangle=\left|\phi\right\rangle\otimes\left|\uparrow\right\rangle\otimes\cdots\otimes\left|\uparrow\right\rangle$,
we discard the qubits in the trivial state $\left|\uparrow\right\rangle$ from
$U\left|\psi\right\rangle$. Then we proceed with $\left|\phi\right\rangle$ in
the next stage of entanglement RG transformations.
The entanglement RG analysis can be done in the Heisenberg picture when we are
interested in a state that is a common eigenstate of a set of operators.
Suppose $\left|\psi\right\rangle$ is defined by equations
$\displaystyle
G_{i}\left|\psi\right\rangle=\left|\psi\right\rangle\quad\text{for any }i$
(11)
where $i$ is some index. Then, the transformed state
$U\left|\psi\right\rangle$ is described by equations
$(UG_{i}U^{\dagger})U\left|\psi\right\rangle=U\left|\psi\right\rangle.$
If $UG_{i}U^{\dagger}$ happens to be an operator, say $\sigma^{z}$ on a single
qubit, then that qubit must be in the state $\left|\uparrow\right\rangle$,
disentangled from the others. This is the criterion by which we identify
disentangled qubits in the calculation below. In addition, we can use this
information to restrict other $G_{j}$ in the next stage of entanglement
renormalization.
The ground state subspace of our model Eq. (4) is described by the stabilizer
equation (11) where the stabilizers $G_{i}$ are just $G^{x}_{i}$ and
$G^{z}_{i}$. Here, observe that the stabilizers $G_{i}$ in Eq. (11) are
invertible operators; $G_{i}$’s form an abelian group $\mathcal{G}=\langle
G_{i}\rangle$, called the stabilizer group. Then, the disentangling criterion
is that for some element $G$ of the stablizer group $\mathcal{G}$,
$UGU^{\dagger}$ acts on a single qubit, where $G$ can be a product of several
$G_{i}$’s.
In fact, only the group $\mathcal{G}$ is important. Consider two gapped
Hamitonians
$H=-J\sum_{i}G_{i},\quad\quad H^{\prime}=-J\sum_{j}G^{\prime}_{j}$
where the terms $G_{i}$ and $G^{\prime}_{j}$ generate the same multiplicative
group $\mathcal{G}=\langle G_{i}\rangle=\langle G^{\prime}_{i}\rangle$. The
ground-state subspace of the two gapped Hamiltonians are identical and they
represent the same quantum phase of matter, in which case we will write
$H\cong H^{\prime}.$ (12)
One can say that $H^{\prime}$ is another parent gapped Hamiltonian of the
ground-state subspace of $H$.
Since the ground state is degenerate, the stabilizer equation (11) does not
pick out a particular state. Nevertheless, the disentanglement criterion in
the Heisenberg picture determines a qubit in the trivial state unambiguously
for any ground state. Thus, even after discarding disentangled qubits, the
transformed Hamiltonian $UHU^{\dagger}$ has a ground-state subspace that is
isomorphic to that of $H$. Our entanglement RG transformation preserves the
ground-state subspace.
We can now state our main result. Let $H_{A}(a)$ be the cubic code Hamiltonian
defined in Eq. (4). Here, the lattice spacing constant $a$ is specified for
notational clarity. We find a finite depth quantum circuit $U$ such that
$\displaystyle UH_{A}(a)U^{\dagger}\cong H_{A}(2a)+H_{B}(2a)$ (13)
where no qubit is involved in both $H_{A}(2a)$ and $H_{B}(2a)$. In Eq. (13),
we have suppressed disentangled qubits; single $\sigma^{z}$ terms are dropped.
The new model $H_{B}$ is defined on a simple cubic lattice with four qubits
per site, with the Hamiltonian
$\displaystyle
H_{B}=-J\sum_{i\in\Lambda}\left(S^{x,1}_{i}+S^{x,2}_{i}+S^{z,1}_{i}+S^{z,2}_{i}\right)$
(14)
where
$\displaystyle S^{x,1}_{i}$
$\displaystyle=\sigma^{x}_{i+\hat{x},1}\sigma^{x}_{i+\hat{z},1}\sigma^{x}_{i,2}\sigma^{x}_{i+\hat{x},2}\sigma^{x}_{i+\hat{x},3}\sigma^{x}_{i+\hat{y},3}\sigma^{x}_{i,4}\sigma^{x}_{i+\hat{y},4}$
$\displaystyle S^{x,2}_{i}$
$\displaystyle=\sigma^{x}_{i,1}\sigma^{x}_{i+\hat{x},1}\sigma^{x}_{i,2}\sigma^{x}_{i+\hat{z},2}\sigma^{x}_{i,3}\sigma^{x}_{i+\hat{y},3}\sigma^{x}_{i,4}\sigma^{x}_{i+\hat{x},4}$
$\displaystyle S^{z,1}_{i}$
$\displaystyle=\sigma^{z}_{i,1}\sigma^{z}_{i-\hat{y},1}\sigma^{z}_{i,2}\sigma^{z}_{i-\hat{x},2}\sigma^{z}_{i,3}\sigma^{z}_{i-\hat{x},3}\sigma^{z}_{i,4}\sigma^{z}_{i-\hat{z},4}$
$\displaystyle S^{z,2}_{i}$
$\displaystyle=\sigma^{z}_{i-\hat{x},1}\sigma^{z}_{i-\hat{y},1}\sigma^{z}_{i,2}\sigma^{z}_{i-\hat{y},2}\sigma^{z}_{i-\hat{x},3}\sigma^{z}_{i-\hat{z},3}\sigma^{z}_{i,4}\sigma^{z}_{i-\hat{x},4}$
are eight-qubit interactions. The interaction terms are visually depicted as
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We perform a similar entanglement RG transformation for $H_{B}$, and find a
finite depth quantum circuit $V$ such that
$\displaystyle VH_{B}(a)V^{\dagger}\cong H_{B}(2a)+H_{B}^{\prime}(2a)$ (15)
with no qubit is involved in both $H_{B}(2a)$ and $H_{B}^{\prime}(2a)$.
$H_{B}$ and $H_{B}^{\prime}$ are the _same_ but act on disjoint sets of
qubits. We have dropped single qubits in the disentangled states on the right-
hand side of Eq. (15). The proof of these formulas and a compact
representation of the models are given in Sec. VI.
The new model $H_{B}$ is different from the original cubic code model $H_{A}$.
We will argue in the next section that they represent different quantum phases
of matter. However, they resemble each other in many ways because they are
related by the finite depth quantum circuit of Eq. (13). Recall that under a
finite depth quantum circuit, any local operator is mapped to a local
operator, and the corresponding operator algebras are isomorphic. In
particular, the two models admit pointlike excitations, which are immobile in
both cases. They have degenerate ground states that are locally
indistinguishable.
## IV Model A and B are different
By the quantum phase of matter, we mean an equivalence class of gapped
Hamiltonians where the equivalence is defined by adiabatic paths in the space
of gapped Hamiltonians and finite depth quantum circuits assisted with some
ancillary qubits. ChenGuWen2010transformation The equivalence may be observed
at some different length scale, so one might have to coarse-grain the lattice
in order to see the equivalence. The nonequivalence, on the other hand, must
be proved by contrasting some invariants. We focus on the degeneracy of the
ground states for this purpose.
Suppose the two models $A$ and $B$ represent the same quantum phase of matter.
They must have the same ground-state subspace structure, and in particular the
dimension of the ground-state subspace must be the same. In view of the
fluctuating degeneracy as in Eqs. (9),(10), it means that the degeneracy is
given by the same function of the system size under the same boundary
conditions. Let $k_{A}(L)$ be $\log_{2}$ of the ground-state subspace
dimension of the model $A$, the original cubic code model, on $L\times L\times
L$ periodic lattice, and let $k_{B}(L)$ be that of the model $B$, the new
model discovered by the entanglement RG transformation. From Eq. (13), we have
$k_{A}(2L)=k_{A}(L)+k_{B}(L).$
Eq. (10) implies that
$k_{A}(2L)=2k_{A}(L)+2.$ (16)
Then, it follows that
$k_{B}(2L)=2k_{B}(L),$ (17)
which can also be shown by Eq. (15). It is clear that the function $L\mapsto
k_{A}(L)$ is _different_ from the function $L\mapsto k_{B}(L)$. This is the
basis of the argument for distinctness of the two phases.
We need to take into account the possibility of the equivalence at different
length scales or on distorted lattices. For example, we know that the Wen
plaquette model Wen2003Plaquette exhibits the same phases of matter as the
toric code model. Kitaev2003Fault-tolerant However, the Wen plaquette model
$H_{\text{Wen}}=-\sum_{i}\sigma^{z}_{i}\sigma^{x}_{i+\hat{x}}\sigma^{x}_{i+\hat{y}}\sigma^{z}_{i+\hat{x}+\hat{y}}$
has one qubit per lattice site, whereas the toric code model has two. The
degeneracies as functions of system size are different, too.
$k_{\text{toric}}(L)=2,\quad k_{\text{Wen}}(L)=\begin{cases}1&\text{if $L$ is
odd,}\\\ 2&\text{if $L$ is even.}\end{cases}$
To see the equivalence, one has to take a new Bravais lattice for the Wen
plaquette model such that the new unit cell now contains two qubits, and the
unit vectors for the coarser lattice are in the diagonal directions of the
original lattice. The toric code model is recovered once we make local unitary
transformations $\sigma^{z}\leftrightarrow\sigma^{x}$ on every, say, first
qubit in each new unit cell. See Fig. 2.
Figure 2: Equivalence between the Wen plaquette model and the toric code model
can be observed, only when a unit cell is properly chosen.
For the most general choice of new Bravais lattice (smaller translation group)
in the cubic lattice, the new unit translation vectors have integer
coordinates such that the $3\times 3$ matrix $M$ of the cooridnates in the
columns is nonsingular. The unit vectors define a rhombohedron unit cell.
Conversely, given a $3\times 3$ nonsingular integer matrix $M$, one can
introduce a new Bravais lattice to the original cubic lattice by declaring the
columns of $M$ to be new unit translation vectors. Imposing periodic boundary
conditions amounts to specifying the number of translations in each new
direction $\vec{L}^{\prime}=(L_{x}^{\prime},L_{y}^{\prime},L_{z}^{\prime})$
before the translations become the identity translation. Hence, the degeneracy
under periodic boundary conditions is a function of $M$ and the lattice
dimension vector $\vec{L}^{\prime}$; $k=k(M,\vec{L}^{\prime})$.
Suppose now that two models $H_{A}$ and $H_{B}$ are equivalent, and the
equivalence is made explicit at coarser lattices $\Lambda^{\prime}_{A}$ and
$\Lambda^{\prime}_{B}$ defined by nonsingular integer matrices $M_{A}$ and
$M_{B}$, respectively, with respect to the original cubic lattice $\Lambda$.
In particular, we must have
$k_{A}\left(M_{A},\vec{L^{\prime}}\right)=k_{B}\left(M_{B},\vec{L^{\prime}}\right)$
for any lattice dimension vector $\vec{L^{\prime}}$. Consider an even coarser
lattice $\Lambda^{\prime\prime}_{A}$ defined by a nonsingular integer matrix
$N$ with respect to $\Lambda^{\prime}_{A}$, and $\Lambda^{\prime\prime}_{B}$
defined by the same $N$ with respect to $\Lambda^{\prime}_{B}$. We must have
$k_{A}\left(M_{A}N,\vec{L^{\prime\prime}}\right)=k_{B}\left(M_{B}N,\vec{L^{\prime\prime}}\right)$
for any lattice dimension vector $\vec{L^{\prime\prime}}$. Note that $N$ was
arbitrary.
Set the matrix $N$ to be the adjugate matrix of $M_{A}$ so that
$M_{A}N=\det(M_{A})I_{3\times 3}$. $N$ is nonsingular and integral. For
$\vec{L^{\prime\prime}}=(\ell,\ell,\ell)$, we have
$\displaystyle k_{A}\left(\det(M_{A})I_{3\times 3},(\ell,\ell,\ell)\right)$
$\displaystyle=k_{B}\left(M_{B}N,(\ell,\ell,\ell)\right)$
$\displaystyle=k_{A}\left(\det(M_{A})\ell\right),$
where the last function is one that has appeared in Eq. (16). The function
$\phi_{B}:\ell\mapsto k_{B}\left(M_{B}N,(\ell,\ell,\ell)\right)$ has a
property that $\phi_{B}(2\ell)=2\phi_{B}(\ell)$ because of Eq. (15),
regardless of how $M_{B}$ or $N$ is chosen. However, we know from Eq. (16)
that the function $\phi_{A}:\ell\mapsto k_{A}\left(\det(M_{A})\ell\right)$ has
a property that $\phi_{A}(2\ell)=2\phi_{A}(\ell)+2$. This is a contradiction,
and therefore the model $H_{A}$ and $H_{B}$ represents different phases of
matter.
## V A tensor network description: Branching MERA
The entanglement RG transformation yields a tensor network description for the
state. If one reverses the transformation starting from, say, a state on
$L^{3}$ lattice, one gets a state on $(2L)^{3}$ lattice. After many iterations
one obtains a state on an infinite lattice. It will be an exact description
since our finite depth quantum circuits $U,V$ are exact. In this section we
will refer to local degrees of freedom as qudits.
Let us review Multi-scale Entanglement Renormalization Ansatz (MERA) states.
Vidal2007ER ; Vidal2008MERA The MERA state is a many-qudit state that is
obtained by reversing the entanglement RG transformations as follows. One
starts with a qudit system on some lattice. (Step 1) Apply a finite depth
quantum circuit with some ancillary qudits in a fixed state
$\left|\uparrow\right\rangle$. Due to the insertion of the ancillary qudits,
the number density of qudits is increased. In order to retain the number
density, (Step 2) one expands the lattice. Then, (Step 3) Iterate Step 1 and
2. In a scale invariant system, one expects that the quantum circuit in Step 1
is the same for every level of the iterations. The class of states that can be
written as a MERA is proposed to describe ground states of some critical
systems, and is shown to admit efficient classical algorithms.
Since the ground state of the toric code model for example is an entanglement
RG fixed point, it naturally has a scale-invariant MERA description. On the
other hand, the cubic code model is not a usual fixed point. At a coarse-
grained level, the ground-state subspace is a tensor product of two
independent ground-state subspaces (Eq. (13)), each of which is again a tensor
product of two independent ground-state subspaces (Eqs. (13),(15)). Reversing
the entanglement RG flow, we see that the final state is obtained by
entangling two states, each of which is again obtained by entangling two
states, and so on.
The “branching MERA,” recently introduced by Evenbly and Vidal,
EvenblyVidal2012RG is a variant of MERA that captures this scenario. In a
branching MERA, the ancillary trivial qudits in the Step 1 of the usual MERA
are allowed to be in branching MERA states. The self-referential nature is
essential. The total number of branches would grow exponentially with the
coarse-graining level.
The branching structure usually yields very highly entangled states. For
example, in a 1D spin chain, a typical branching MERA state with the total
number of branches being $b_{n}=2^{n}$ at coarse-graining level $n$, obeys a
“volume” law of entanglement entropy. In general, the entanglement entropy of
a ball-like region of linear dimension $L$, for a branching MERA state in a
$D$-dimensional lattice scales like
$S\leq O(1)\sum^{\log_{2}L}_{n=0}b_{n}\left(\frac{L}{2^{n}}\right)^{D-1}$ (18)
where $b_{n}$ is the total number of branches at RG level $n$.
EvenblyVidal2013bMERAentropy A proof of the formula is given in Appendix A.
In case of our cubic code model where $b_{n}=2^{n}$, it gives an area law. It
is consistent with the fact that it is a stabilizer code Hamiltonian.
HammaIonicioiuZanardi2005
It should be noted that the entanglement entropy scaling alone does not
necessitate the branching structure; it does not nullify the possibility of a
description by the usual unbranched MERA. Our bound in Eq. (18) merely
illustrates that the branching MERA description of the cubic code model is
consistent in view of the entanglement entropy scaling, despite the intuition
that the branching MERA yields much more entanglement.
Rather, the necessity of the branching structure relies on the ground state
degeneracy. If a usual MERA description were possible, the ground space of the
cubic code model on $L^{3}$ (with $L=2^{n}$) lattice would have a one-to-one
correspondence with the Hilbert space of $O(1)=O(L^{0})$ qubits in the top
level of the MERA, and therefore would be of a constant dimension. This would
contradict Eq. (10).
## VI Calculation method
The finite depth quantum circuits $U$ and $V$ are complicated and not very
enlightening. Explicit circuits and calculation can be found in a Mathematica
script in Supplementary Material. SM In this section, we explain a machinary
to compute $U$ and $V$. It heavily depends on a special structure of the
Hamiltonians $H_{A}$ and $H_{B}$. The content here is essentially presented in
Ref. Haah2012PauliModule, , so we will be brief.
### VI.1 Laurent polynomial matrix description
The Pauli $2\times 2$ matrices $\sigma^{x},\sigma^{y},\sigma^{z}$ have a
special property that (i) they square to identity, (ii) the product of any
pair of the matrices results in the third up to a phase factor $\pm 1,\pm i$,
and (iii) they anticommute with one another. In other words, they form an
abelian group under multiplication up to the phase factors. This group,
ignoring the phase factors, is just $\mathbb{Z}_{2}\times\mathbb{Z}_{2}$. A
conventional correspondence is given by
$\displaystyle(\sigma^{x})^{n}(\sigma^{z})^{m}$
$\displaystyle\in\langle\sigma^{x},\sigma^{y},\sigma^{z}\rangle/\\{\pm 1,\pm
i\\}$ $\displaystyle\Updownarrow$ (19) $\displaystyle(n,m)$
$\displaystyle\in\mathbb{Z}_{2}\times\mathbb{Z}_{2}$
The correspondence easily generalizes to Pauli operators (tensor products of
Pauli matrices). An $n$-qubit Pauli operator corresponds to a bit $\\{0,1\\}$
string of length $2n$: The first half of the bit string expresses
$\sigma^{x}$, while the second half expresses $\sigma^{z}$.
If a qubit system admits translations, e.g. one-dimensional spin chain, the
corresponding bit string can be written in a compact way: Write the bits in
the coefficients of the translation group elements in a formal linear
combination. For example,
$\displaystyle\cdots\otimes\sigma^{x}\otimes\sigma^{z}\otimes
I\otimes\sigma^{y}\otimes\cdots$
$\displaystyle\Leftrightarrow\begin{pmatrix}\cdots&1&0&0&1&\cdots\\\
\cdots&0&1&0&1&\cdots\end{pmatrix}$ (20)
$\displaystyle\Leftrightarrow\begin{pmatrix}\cdots+1t^{-1}+0t^{0}+0t^{1}+1t^{2}+\cdots\\\
\cdots+0t^{-1}+1t^{0}+0t^{1}+1t^{2}+\cdots\end{pmatrix}$
$\displaystyle=\begin{pmatrix}\cdots+t^{-1}+t^{2}+\cdots\\\
\cdots+1+t^{2}+\cdots\end{pmatrix}$
where $t$ denotes the translation by one unit length to the right. This is
merely a change of notation. It yields a particularly simple expression for
translation-invariant Hamiltonians whose terms are Pauli operators, because
one only has to keep a few polynomials that express different types of local
terms. Local terms are expressed not by an infinite Laurent series, but by a
finite linear combination of the translation group elements. Summarizing, we
have introduced a notation for Hamiltonians of Pauli operators using the
translation group algebra with coefficients in $\mathbb{Z}_{2}$.
The cubic code model $H_{A}$ in Eq. (4) can now be written as
$\displaystyle G^{x}=\begin{pmatrix}1+x+y+z\\\ 1+xy+yz+zx\\\ 0\\\
0\end{pmatrix},~{}~{}G^{z}=\begin{pmatrix}0\\\ 0\\\
1+\bar{x}\bar{y}+\bar{y}\bar{z}+\bar{z}\bar{x}\\\
1+\bar{x}+\bar{y}+\bar{z}\end{pmatrix}.$ (21)
where $x,y,z$ are translations along $+\hat{x},+\hat{y},+\hat{z}$-direction,
respectively, and $\bar{x}=x^{-1}$, etc. Since the unit cell of the cubic code
model contains two qubits, we need $2\times 2=4$ rows in the matrix. The first
row expresses $\sigma^{x}$ in the first qubit at each site, the second row
$\sigma^{x}$ the second qubit, the third row $\sigma^{z}$ in the first qubit,
and the fourth row $\sigma^{z}$ in the second qubit. It is the most convenient
to write two matrices in a single matrix where each type of term is written in
each column.
$\displaystyle\sigma=\begin{pmatrix}1+x+y+z&0\\\ 1+xy+yz+zx&0\\\
0&1+\bar{x}\bar{y}+\bar{y}\bar{z}+\bar{z}\bar{x}\\\
0&1+\bar{x}+\bar{y}+\bar{z}\end{pmatrix}$ (22)
We refer to this matrix $\sigma$ as a _generating matrix_ of $H_{A}$.
### VI.2 Applying periodic local unitary operators
A subclass of finite depth quantum circuits is effectively implemented using
this Laurent polynomial description. It consists of unitaries that respect the
translation symmetry and map Pauli operators to Pauli operators. More
specifically, they are compositions of so-called CNOT, Hadamard, and Phase
gates. For example, Hadamard gate
$U_{\mathrm{Hadamard}}=\frac{1}{\sqrt{2}}\begin{pmatrix}1&1\\\
1&-1\end{pmatrix}\begin{matrix}\left|\uparrow\right\rangle\\\
\left|\downarrow\right\rangle\end{matrix}$
swaps $\sigma^{x}$ and $\sigma^{z}$:
$U_{H}\sigma^{x}U_{H}^{\dagger}=\sigma^{z},~{}~{}U_{H}\sigma^{z}U_{H}^{\dagger}=\sigma^{x}$
If the Hadamard is applied for every qubit on the lattice, then the upper half
and the lower half of the Laurent polynomial matrix will be interchanged.
Similarly, one can work out the action of the CNOT gate
$U_{\mathrm{CNOT}}=\begin{pmatrix}1&0&0&0\\\ 0&1&0&0\\\ 0&0&0&1\\\
0&0&1&0\end{pmatrix}\begin{matrix}\left|\uparrow\uparrow\right\rangle\\\
\left|\uparrow\downarrow\right\rangle\\\
\left|\downarrow\uparrow\right\rangle\\\
\left|\downarrow\downarrow\right\rangle\end{matrix}$
and Phase gate
$U_{\mathrm{Phase}}=\begin{pmatrix}1&0\\\
0&i\end{pmatrix}\begin{matrix}\left|\uparrow\right\rangle\\\
\left|\downarrow\right\rangle\end{matrix}$
on the Laurent polynomial matrix. The result is that they correspond to _row
operations_ on the Laurent polynomial matrix. That is, any elementary row
operation $E$, viewed as a left matrix multiplication $\sigma\mapsto E\sigma$,
is admissible as long as $E$ satisfies the symplectic condition
$\bar{E}^{T}\begin{pmatrix}0&I_{q}\\\
I_{q}&0\end{pmatrix}E=\begin{pmatrix}0&I_{q}\\\ I_{q}&0\end{pmatrix}\mod 2.$
(23)
where the bar means the antipode map under which $x\mapsto x^{-1}$, $y\mapsto
y^{-1}$, and $z\mapsto z^{-1}$. Here, $q$ is the number of qubits per unit
cell. $I_{q}$ is the $q\times q$ identity matrix. For a proof, see Ref.
Haah2012PauliModule, .
Note that when the two-qubit unitary operator CNOT above acts within a unit
cell, the antipode map is trivial since $E$ in Eq. (23) will not involve any
variable $x,y,z$, etc; the antipode map does not do anything to coefficients.
When the CNOT acts on a pair of qubits across the unit cells, which is allowed
only if the unit cell contains two or more qubits, the antipode map is
nontrivial. Of course, in any case, the overall unitary must have the same
periodicity with the lattice.
Using the above row operations, one can only generate a finite depth quantum
circuit whose periodicity is $1$. If one wishes to apply, say, Hadamard gates
on every other qubits (periodicity 2), one has to choose a subgroup
$\mathcal{T}^{\prime}$ of the original translation group $\mathcal{T}$, so
that one unit of translation under $\mathcal{T}^{\prime}$ is the translation
by two units under $\mathcal{T}$. Then, one can implement the periodicity $2$
quantum circuit, using the prescription above. Under such a coarse translation
group, our matrix representation of the Hamiltonian must be different.
Computing a new representation is easy, and a prescription is as follows. If
one wishes to take the coarse translation group to be
$\mathcal{T}^{\prime}=\langle x^{\prime},y,z\rangle\leq\langle
x,y,z\rangle=\mathcal{T}$
where $x^{\prime}=x^{2}$, one simply replaces each Laurent polynomial
$f(x,y,z)$ of $\sigma$ with the matrix
$\displaystyle f\left(\begin{pmatrix}0&x^{\prime}\\\
1&0\end{pmatrix},\begin{pmatrix}y&0\\\ 0&y\end{pmatrix},\begin{pmatrix}z&0\\\
0&z\end{pmatrix}\right)$ (24)
If the old generating matrix $\sigma$ was $2q\times m$, then the new
generating matrix is $4q\times 2m$. Again, a proof of this claim can be found
in Ref. Haah2012PauliModule, .
### VI.3 Example: Toric code model
Let us perform an entanglement RG for the toric code model (Ising gauge
theory). Kitaev2003Fault-tolerant As we call for strict translation-
invariance, we take the square lattice with the unit cell at a vertex
consisting of one horizontal edge on the east (1) and one vertical edge on the
north (2). The Hamiltonian is
$\displaystyle H_{\mathrm{toric}}=$
$\displaystyle-\sum_{i}\sigma^{x}_{i,1}\sigma^{x}_{i-\hat{x},1}\sigma^{x}_{i,2}\sigma^{x}_{i-\hat{y},2}$
$\displaystyle-\sum_{i}\sigma^{z}_{i,1}\sigma^{z}_{i+\hat{y},1}\sigma^{z}_{i,2}\sigma^{z}_{i+\hat{x},2}$
Following the correspondence Eq. (20), the generating matrix is
$\displaystyle\sigma_{\mathrm{toric}}=\begin{pmatrix}1+\bar{x}&0\\\
1+\bar{y}&0\\\ 0&1+y\\\ 0&1+x\end{pmatrix}.$ (25)
Let us take a smaller translation group $\mathcal{T}^{\prime}=\langle
x^{\prime},y\rangle\leq\langle x,y\rangle$ where $x^{\prime}=x^{2}$.
Accordingto the prescription Eq. (24), the new generating matrix with respect
to $\mathcal{T}^{\prime}$ becomes
$\displaystyle\sigma^{\prime}_{\mathrm{toric}}=\begin{pmatrix}1&1&&\\\
\bar{x}^{\prime}&1&&\\\ 1+\bar{y}&0&&\\\ 0&1+\bar{y}&&\\\ &&1+y&0\\\
&&0&1+y\\\ &&1&x^{\prime}\\\ &&1&1\end{pmatrix}$ (26)
Some zeros are not shown. Now we apply row operations that satisfy Eq. (23).
$\displaystyle\left(\begin{array}[]{cccccccc}1&0&0&0&&&&\\\
\bar{x}^{\prime}&1&0&0&&&&\\\ \bar{y}+1&0&1&0&&&&\\\ \bar{y}+1&0&1&1&&&&\\\
&&&&1&x^{\prime}&1+y&0\\\ &&&&0&1&0&0\\\ &&&&0&0&1&1\\\ &&&&0&0&0&1\\\
\end{array}\right)\sigma^{\prime}_{\mathrm{toric}}$ (35)
$\displaystyle=\left(\begin{array}[]{cccc}1&1&&\\\ 0&\bar{x}^{\prime}+1&&\\\
0&\bar{y}+1&&\\\ 0&0&&\\\ &&0&0\\\ &&0&1+y\\\ &&0&1+x^{\prime}\\\ &&1&1\\\
\end{array}\right)$ (44)
Let us recover the Hamiltonian. We have found a finite depth quantum circuit
$U$ from Eq. (44) such that
$\displaystyle UH_{\mathrm{toric}}U^{\dagger}$
$\displaystyle=-\sum_{i^{\prime}}\sigma^{x}_{i^{\prime},1}-\sum_{i^{\prime}}\sigma^{x}_{i^{\prime},1}\sigma^{x}_{i^{\prime},2}\sigma^{x}_{i^{\prime}-\hat{x}^{\prime},2}\sigma^{x}_{i^{\prime},3}\sigma^{x}_{i^{\prime}-\hat{y},3}$
$\displaystyle-\sum_{i^{\prime}}\sigma^{z}_{i^{\prime},4}-\sum_{i^{\prime}}\sigma^{z}_{i^{\prime},2}\sigma^{z}_{i^{\prime}+\hat{y},2}\sigma^{z}_{i^{\prime},3}\sigma^{z}_{i^{\prime}+\hat{x}^{\prime},3}\sigma^{z}_{i^{\prime},4}.$
Since the Hamiltonian is frustration-free, it is clear that the first and
fourth qubits in each unit cell are in a trivial state and are disentangled
from the rest. As noted above in Sec. III, only the multiplicative group
generated by the terms in the Hamiltonian is important, and we recover
$H_{\mathrm{toric}}$ we started with at a coarse-grained lattice
$\mathcal{T}^{\prime}$. The example demonstrates that _any column operation on
the generating matrix $\sigma$ is allowed_ in view of equivalence Eq. (12).
This shows that the ground state of the toric code model is a fixed point in
an entanglement RG flow. AguadoVidal2007Entanglement
In Supplementary Material, we perform similar calculations for 3D and 4D toric
code models. (3D toric code model is also known as 3D Ising gauge theory.
Wegner1971IsingGauge 4D toric code is similar; qubits live on plaquettes, and
the gauge transformation flips qubits around an edge.
DennisKitaevLandahlEtAl2002Topological ) We verify that they are all
entanglement RG fixed points.
## VII An algebro-geometric test on entanglement RG
Our example of the bifurcation is very specific to the cubic code model, and
general criteria for the bifurcation to happen are not well understood.
However, we can rule out certain possibilities as follows. We have found an
equivalence by a finite depth quantum circuit between the ground space of
$H_{A}(a)$, where $a$ in the parentheses is the lattice spacing, and that of
$H_{A}(2a)\oplus H_{B}(2a)$. Can we find a similar relation between the ground
space of $H_{A}(a)$ and that of, say, $H_{A}(3a)\oplus H^{\prime}$ for some
Hamiltonian $H^{\prime}$? Put differently, how coarse should a new Bravais
lattice be, if one wishes to find a copy of $H_{A}$ on the new Bravais lattice
by a finite depth quantum circuit?
In this section, we give a _necessary_ condition for this question to be
answered positively by exploiting our Laurent polynomial matrix descriptions.
The condition will detect cases when one will not find a copy of the original
model one started with on a coarser lattice. Our choice of new Bravais lattice
of lattice spacing $2a$ for the cubic code model and the toric code model
satisfies the condition, as it must do.
Let us restrict ourselves to the simplest situation where the generating
matrix $\sigma$ is $2q\times q$, where $q$ is even, and block-diagonal, as in
Eq. (22) and Eq. (25). This is the case when the number of qubits in the unit
cell is the same as the number of interaction types in the Hamiltonian. Note
that in either Eq. (22) or Eq. (25), the upper-left block is described by two
polynomials $f,g$: For the cubic code model, they are $1+x+y+z$ and
$1+xy+yz+zx$. For the toric code model, they are $1+x^{-1}$ and $1+y^{-1}$.
The lower-right blocks in both cases are related to the upper-left blocks by
the antipode map, so we can focus only on the upper-left blocks.
Consider all $q/2\times q/2$ submatrices of the upper-left block of the
generating matrix $\sigma$, and take the determinants of them. Let
$I(\sigma)=\\{f_{i}\\}$ be the set of all such determinants. For example,
$I(\sigma_{\text{toric}})=\\{1+x^{-1},1+y^{-1}\\}$, and
$I(\sigma_{\text{cubic}})=\\{1+x+y+z,1+xy+yz+zx\\}$. Let $V(\sigma)$ be the
set of solutions of the polynomial equations $f_{i}=0$. For example,
$V(\sigma_{\text{toric}})=\\{(x,y)|1+x^{-1}=0,~{}1+y^{-1}=0\\}=\\{(1,1)\\}$.
It is shown in Ref. Haah2012PauliModule, that $V(\sigma)$ is invariant under
a class of local unitary transformations such that the transformed Hamiltonian
still admits a description by a Laurent polynomial matrix. $V(\sigma)$ is the
object for our algebro-geometric test.
$V(\sigma)$ is a variety, a rather abstract geometric set. In our Laurent
polynomial matrix description, the variables $x,y$, etc. were directly related
to translations. But, now we are treating them as unknown variables and
furthermore equating the polynomials in those variables with zero! Indeed, it
requires good deal of preparation before defining the variety properly, which
is out of the scope of the present paper. We will state facts that are useful
for our purpose. Interested readers are referred to Ref. Haah2012PauliModule,
.
We have seen in Sec. VI.3 that the generating matrix $\sigma$ takes a
different form $\sigma\to\sigma^{\prime}$ depending on our choice of
translation group. Upon taking a coarse translation group, the variety is
changed to $V(\sigma)\to V(\sigma^{\prime})$. Interestingly, one can show that
the change is again given by a nice algebraic map. For example, if we take
$\mathcal{T}^{\prime}=\langle
x^{\prime},y^{\prime},z^{\prime}\rangle\leq\langle x,y,z\rangle=\mathcal{T}$
where $x^{\prime}=x^{n}$, $y^{\prime}=y^{n}$, and $z^{\prime}=z^{n}$ in three-
dimensional lattice, which means $n^{3}$ sites are blocked to form a single
new site, then the change is given by an almost surjective map111Rigorously
speaking, the image of the map is dense in the target variety under Zariski
topology. See e.g. Hartshorne, Algebraic Geometry, Springer
$V(\sigma)\ni(a,b,c)\mapsto(a^{n},b^{n},c^{n})\in V(\sigma^{\prime}).$ (45)
The variety $V(\sigma_{1}\oplus\sigma_{2})$ for the juxtaposition of two
independent systems $\sigma_{1}$ and $\sigma_{2}$ as in Eq. (13), is given by
the union $V(\sigma_{1})\cup V(\sigma_{2})$ of respective varieties.
We have noted that $V(\sigma)$ is invariant under local unitary
transformations. The entanglement RG is a combination of local unitary
transformations after a choice of a smaller translation group. Hence, if a
copy of the original model is to be found in the coarse lattice, _the new
variety $V(\sigma^{\prime})$ must contain the original $V(\sigma)$._ This is a
criterion by which the bifurcation, or an occurrence of the original model at
a coarse lattice _may_ happen. It is unknown if the criterion is a sufficient
condition.
### VII.1 Examples
Let us apply the criterion to the toric code model and the cubic code model.
As we have seen above, $V(\sigma_{\text{toric}})=\\{(1,1)\\}$. Upon a choice
of a coarser lattice, blocking $2\times 2$ sites as a new one site, the
variety is transformed by the map $x\mapsto x^{2}$ and $y\mapsto y^{2}$.
Obviously, the point $(1,1)$ is invariant under this map, which is consistent
with the fact that the toric code is a RG fixed point.
AguadoVidal2007Entanglement (See Sec. VI.3.) The readers are encouraged to
compute $V(\sigma^{\prime}_{\text{toric}})$ from Eq. (26) and Eq. (44):
Compute the determinants of all possible $2\times 2$ submatrices of the upper-
left block of $\sigma^{\prime}_{\text{toric}}$, equate them with zero, and
decide the set of solutions.
For the cubic code, the variety is also simple. It consists of two lines each
of which is parametrized by an auxiliary variable $s$:
$\displaystyle\begin{cases}x&=1+s\\\ y&=1+\omega s\\\
z&=1+\omega^{2}s\end{cases},\quad\begin{cases}x&=1+s\\\ y&=1+\omega^{2}s\\\
z&=1+\omega s\end{cases}.$
where $\omega$ is a third root of unity satisfying $\omega^{2}+\omega+1=0$.
(It should be noted that the numbers are not complex numbers; they belong to
extension fields of the binary field $\mathbb{F}_{2}$.) On a coarser lattice
blocking $2^{3}$ sites together, the variety is transformed by the squaring
map. See Eq. (45). Over the binary field,
$(a+b)^{2}=a^{2}+2ab+b^{2}=a^{2}+b^{2}$ for any $a,b$. Hence, the image of the
squaring map is the union of two lines
$\displaystyle\begin{cases}x&=1+s^{2}\\\ y&=1+\omega^{2}s^{2}\\\ z&=1+\omega
s^{2}\end{cases}\quad\begin{cases}x&=1+s^{2}\\\ y&=1+\omega s^{2}\\\
z&=1+\omega^{2}s^{2}\end{cases}$
This is indeed the original variety, although the two lines are interchanged
by the squaring map. This is consistent with the fact that we have found the
original copy $H_{A}$ in the coarse lattice.
Note that the varieties for $H_{A}$ and $H_{B}$ are the same. They do not
distinguish two different phases of matter; the variety is a crude algebro-
geometric object associated to the Hamiltonian.
Before concluding the section, we illustrate an example where the test helps
to choose a correct new unit cell. The color code model,
BombinMartinDelgado2006ColorCode which is known to be equivalent to two
copies of the toric code model, BombinDuclosCianciPoulin2012 lives on a
honeycomb lattice with one qubit at each vertex. Being a hexagon, any
plaquette $p$ has six vertices $v$. The color code model is defined by the
Hamiltonian
$H=-J\sum_{p}\left(\prod_{v\in p}\sigma^{z}_{v}+\prod_{v\in
p}\sigma^{x}_{v}\right),$
where the sum is over all hexagons. This is expressed with Pauli matrices and
each term commutes with any other, and thus our Laurent polynomial matrix
description is applicable. Since the honeycomb lattice has two vertices in the
conventional unit cell (Fig. 3), our generating matrix $\sigma_{\text{color}}$
is $4\times 2$, as in the toric code model. Explicitly,
$\sigma_{\text{color}}=\begin{pmatrix}1+x+y&0\\\ x+y+xy&0\\\ 0&1+x+y\\\
0&x+y+xy\end{pmatrix}.$
The associated variety is
$\displaystyle V(\sigma_{\text{color}})$
$\displaystyle=\\{(x,y)~{}|~{}1+x+y=0,~{}x+y+xy=0\\}$
$\displaystyle=\\{(\omega,\omega^{2}),~{}(\omega^{2},\omega)\\},$
where $\omega$ is a third root of unity over the binary field.
Figure 3: Honeycomb lattice with qubits numbered within a unit cell.
Suppose one tries to find a copy of itself at a coarser lattice, to see if the
model is an entanglement RG fixed point. One could choose a new Bravais
lattice $\Lambda^{\prime}$ by saying that $x^{\prime}=x^{3}$ and
$y^{\prime}=y^{3}$ are new unit translations. According to Eq. (45), the new
variety $V(\sigma^{\prime}_{\text{color}})$ would be a single point $(1,1)$
since $\omega^{3}=(\omega^{2})^{3}=1$. The original variety is not contained
in the new variety, and therefore one will not find a copy of the original
model on the coarse Bravais lattice $\Lambda^{\prime}$.
On the other hand, if one tried to show the equivalence of the color code
model and the toric code model, then one should take the mentioned Bravais
lattice $\Lambda^{\prime}$; otherwise, the variety of the transformed color
code model would not match that of the toric code model, and the equivalence
would never be explicit.
## VIII Discussion
We have shown that under the entanglement renormalization group flow the cubic
code model bifurcates. The cubic code model $A$ does not simply produce
exactly the same two copies of itself, but yields a different model $B$. In
order to complete the entanglement RG, we have further shown that the model
$B$ bifurcates into two copies of itself.
The bifurcation alone, as seen in phase B, can be observed in a trivial and
rather ad hoc example: An infinite stack of toric codes. We need to be a
little formal because the example is too trivial. Let $H_{\text{toric}}(a)$ be
the Hamiltonian of the toric code model on a 2D square lattice with qubits on
edges, where lattice spacing is $a$. The entanglement RG transformation
reveals that there is a finite depth quantum circuit $U$ such that
$UH_{\text{toric}}(a)U^{\dagger}\cong H_{\text{toric}}(2a)$
Consider an infinite stack of 2D square lattices with qubits on the edges.
Suppose each layer is parallel to $xy$-plane, and the total system is stacked
in $z$-direction. Our ad hoc Hamiltonian is
$H_{\text{stack}}(a)=\sum_{z=-\infty}^{\infty}H_{\text{toric}}(a)_{z},$
where the subscript $z$ designates the layer that $H_{\text{toric}}(a)$ lives
on. Choosing a new Bravais lattice such that $(0,0,2)$ is a new unit
translation vector, we have
$H_{\text{stack}}(a)=\sum_{z^{\prime}=-\infty}^{\infty}H_{\text{toric}}(a)_{2z^{\prime}}+H_{\text{toric}}(a)_{2z^{\prime}+1}.$
Let $V=\bigotimes_{z=-\infty}^{\infty}U_{z}$ be a finite depth quantum circuit
where $U_{z}$ is just $U$ acting on the layer $z$. Then,
$\displaystyle VH_{\text{stack}}(a)V^{\dagger}$
$\displaystyle=\sum_{z^{\prime}=-\infty}^{\infty}U_{2z^{\prime}}H_{\text{toric}}(a)_{2z^{\prime}}U_{2z^{\prime}}^{\dagger}$
$\displaystyle~{}~{}+\sum_{z^{\prime}=-\infty}^{\infty}U_{2z^{\prime}+1}H_{\text{toric}}(a)_{2z^{\prime}+1}U_{2z^{\prime}+1}^{\dagger}$
$\displaystyle\cong\sum_{z^{\prime}=-\infty}^{\infty}H_{\text{toric}}(2a)_{2z^{\prime}}$
$\displaystyle~{}~{}+\sum_{z^{\prime}=-\infty}^{\infty}H_{\text{toric}}(2a)_{2z^{\prime}+1}$
$\displaystyle=H_{\text{stack}}(2a)_{\text{even}}+H_{\text{stack}}(2a)_{\text{odd}}.$
In contrast, our model cannot be written as a stack of lower dimensional
systems. If it were possible, the ground state degeneracy could not have such
complicated dependence on the system size; at least one parameter, say $L_{z}$
must be factored out from Eq. (9). The fact that the model A and the model B
are different gives a more direct proof that the model $A$ cannot be described
in terms of 2D systems. If the model $A$ was a stack of lower dimensional
ones, the entanglement RG would have yielded the same two copies of itself.
In our tensor network description, the branching MERA, one parametrizes states
by a network of tensors. The topology of the network is fixed and the
entanglement RG changes the values of components of the tensors — It is the
space of tensors where the entanglement RG flows. It should be pointed out,
however, that in our calculation of entanglement RG the disentangling
transformations are obtained accidentally. The calculation was not guided by
any equation, but we just tried to disentangle as many qubits as possible and
discovered that the state belongs to the ground space of two independent
systems. (In fact, the only guide was the consistent behavior of the algebraic
variety under a choice of a new Bravais lattice.) This motivates us to
establish RG equations that incorporates the branching structure. In previous
studies in this direction, VerstraeteCiracLatorreEtAl2005Renormalization ;
GuLevinWen2008Tensor-entanglement it was implicitly assumed that there is no
branching at the coarse-grained level.
Recently, Swingle Swingle2013 has shown several examples where entanglement
entropy does not decrease under renormalization group transformations, and
argued that the so-called $c$-theorem Zamolodchikov1986cthm and its higher
dimensional analogs JafferisKlebanovPufuEtAl2011 ; KomargodskiSchwimmer2011
can be violated if Lorentz symmetry is broken. In other words, he argues that
the entanglement entropy is not a RG-monotone in non-Lorentz-invariant
theories. Our example is a yet different (counter)example to those RG-monotone
theorems. The picture that the number density of effective degrees of freedom
should decrease under RG, is manifestly broken. Although it is not
straightforward to directly relate our entanglement RG and the field-theoretic
RG, it will not be the case that in any renormalizable field theory the number
of distinct fields increases as the probing energy scale decreases. This
suggests that the model admits no conventional field theory description that
gives the correct ground space.
###### Acknowledgements.
The author would like to thank Guifre Vidal for raising a question that has
resulted in this work. The author also thanks John Preskill and Glen Evenbly
for numerous helpful discussions. A part of this work was done at IBM Watson
Research Center, Yorktown Heights, New York, where the author was a summer
research intern. The author is supported in part by Caltech Institute for
Quantum Information and Matter, an NSF Physics Frontier Center with support
from Gordon and Betty Moore Foundation, and by MIT Pappalardo Fellowship in
Physics.
## Appendix A Entanglement entropy of branching MERA states
In this section, we bound the entanglement entropy of a branching MERA
EvenblyVidal2012RG state between some ball-like region and its complement by
a function of the region’s size. The proof here will be a simplified version
of Ref. EvenblyVidal2013bMERAentropy, . We will relate the entropy scaling
with spatial dimension and the number of branches. A simple lemma will be
useful. Each qudit has Hilbert space dimension $\chi$.
Lemma. Let $A,B,C,D$ be disjoint sets of qudits of dimension $\chi$, and $U$
be a unitary operator acting on $B$ and $C$. Let
$S_{AB}(\rho)=S(\mathop{\mathrm{Tr}}\nolimits_{(AB)^{c}}\rho)$ be the von
Neumann entropy. Then, we have
$\displaystyle|S_{AB}(U\rho U^{\dagger})-S_{AB}(\rho)|\leq(2\log\chi)|C|$ (46)
where $|C|$ is the number of qudits in $C$.
###### Proof.
Let $\rho^{\prime}=U\rho U^{\dagger}$.
$\displaystyle|S_{AB}(\rho^{\prime})-S_{AB}(\rho)|$
$\displaystyle=|S_{AB}(\rho^{\prime})-S_{ABC}(\rho^{\prime})+S_{ABC}(\rho^{\prime})-S_{AB}(\rho)|$
$\displaystyle=|S_{AB}(\rho^{\prime})-S_{ABC}(\rho^{\prime})+S_{ABC}(\rho)-S_{AB}(\rho)|$
$\displaystyle\leq|S_{AB}(\rho^{\prime})-S_{ABC}(\rho^{\prime})|+|S_{ABC}(\rho)-S_{AB}(\rho)|$
$\displaystyle\leq S_{C}(\rho^{\prime})+S_{C}(\rho)\leq(2\log\chi)|C|$
In the second inequality, we used the subadditivity of entropy. ∎
The inequality is saturated by the swap operator. If $A,B,C,D$ are single
qubits, respectively, and $\psi$ consists of two pairs of singlets in $AB$ and
$CD$, then $S_{AB}(\psi)=0$. Swapping $B$ and $C$, we have
$S_{AB}(\psi^{\prime})=2\log 2$. The lemma implies that a finite depth quantum
circuit can only generate entanglement between two regions along the boundary.
We wish to consider the entanglement entropy
$S_{0}(\left|\psi\right\rangle)=S(\rho)$, where
$\rho=\mathop{\mathrm{Tr}}\nolimits_{B^{c}}(\left|\psi\right\rangle\left\langle\psi\right|)$,
between a (hyper)cubic region $B$ of linear size $L$ and its complement of a
branching MERA state $\left|\psi\right\rangle$.
By definition, $\left|\psi\right\rangle$ accompanies entanglement RG
transformations $U_{\tau}$ ($\tau=1,2,\ldots$). $U_{1}\left|\psi\right\rangle$
is either a tensor product of one or more states
$\left|\psi_{1}^{1}\right\rangle,\left|\psi_{1}^{2}\right\rangle,\ldots,\left|\psi_{1}^{b}\right\rangle$
($b\geq 1$) each of which is living on a coarser lattice (branch), or some
entangled state of those. To be concrete, suppose the density of degrees of
freedom decreases by a factor of $2^{D}$ on the coarser lattice. The number
$b$ of branches should be $\leq 2^{D}$.
Let $\rho_{1}^{(1)},\ldots,\rho_{1}^{(b)}$ be reduced density matrices of
$U_{1}\left|\psi\right\rangle$ for the corresponding region $B_{1}^{i}$ on
each branch. Each $B_{1}^{i}$ contains $(L/2)^{D}$ qudits. By the lemma and
the subadditivity of entropy, we have
$\displaystyle S(\rho)$ $\displaystyle\leq
S(\mathop{\mathrm{Tr}}\nolimits_{B^{c}}U_{1}\left|\psi\right\rangle\left\langle\psi\right|U_{1}^{\dagger})+c|\partial
B|$ $\displaystyle\leq S(\rho_{1}^{(1)})+\cdots+S(\rho_{1}^{(b)})+c|\partial
B|$ (47)
where $c$ is a constant depending only on the detail of the circuit $U_{1}$’s
locality property. Here, $|\partial B|$ is the number of qudits outside $B$
but within the range of $U_{1}$ from $B$. So, $c|\partial
B|\leq(2\log\chi)2D(L+2)^{D-1}$ if $U_{1}$ is of depth 1 and range 2. One can
iterate the inequality Eq. (47) with $B_{1}^{i}$ in place of $B$.
$S(\rho)\leq\sum_{i=1}^{b_{N}}S(\rho_{N}^{(i)})+c^{\prime}\sum_{n=0}^{N-1}b_{n}\left(\frac{L}{2^{n}}\right)^{D-1}$
(48)
for any $N\geq 0$ where $b_{n}$ is the total number of all branches, and
$\rho_{N}^{(i)}$ is the reduced density matrix of $U_{N}U_{N-1}\cdots
U_{1}\left|\psi\right\rangle$ for the region $B_{N}^{(i)}$ of linear size
$L/2^{N}$ on branch $i$. In particular, $b_{0}=1$ and $b_{1}=b$ above. In a
usual MERA, we have $b_{n}=1$ for all $n$. The constant $c^{\prime}$ only
depends on $\chi$ and the details of the depth and range of circuits
$U_{1},\ldots,U_{N}$.
An appropriate $N$ must be chosen in order for Eq. (48) to be useful. A
straightforward choice is such that $B_{N}^{(i)}$ contains a constant number
of qudits, i.e., $N=\lfloor\log_{2}L\rfloor$. Then, $\rho_{N}^{(i)}$ is a
density matrix of a constant number of qudits, so
$S(\rho_{N}^{(i)})=O(\log\chi)$. Eq. (48) finally implies
$\displaystyle S(\rho)$ $\displaystyle\leq
O(\log\chi)\sum_{n=0}^{\lfloor\log_{2}L\rfloor}b_{n}\left(\frac{L}{2^{n}}\right)^{D-1}.$
(49)
Specializing, we get
$\displaystyle S(\rho)$ $\displaystyle=\begin{cases}O(L^{D-1})&\text{if
$b_{n}=b^{n}<(2^{D-1})^{n}$},\\\ O(L^{D-1}\log L)&\text{if
$b_{n}=(2^{D-1})^{n}$},\\\ O\left(L^{\log_{2}(b/2^{D-1})}\right)&\text{if
$b_{n}=b^{n}>(2^{D-1})^{n}$}.\end{cases}$ (50)
The number $2$ is of course the linear size of a superblock, and can be
replaced by any positive integer.
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|
arxiv-papers
| 2013-10-16T20:01:25 |
2024-09-04T02:49:52.468431
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Jeongwan Haah",
"submitter": "Jeongwan Haah",
"url": "https://arxiv.org/abs/1310.4507"
}
|
1310.4564
|
# Strichartz estimates and nonlinear wave equation on nontrapping
asymptotically conic manifolds
Junyong Zhang Department of Mathematics, Beijing Institute of Technology,
Beijing 100081 China, and Department of Mathematics, Australian National
University, Canberra ACT 0200, Australia [email protected]
###### Abstract.
We prove the global-in-time Strichartz estimates for wave equations on the
nontrapping asymptotically conic manifolds. We obtain estimates for the full
set of wave admissible indices, including the endpoint. The key points are the
properties of the microlocalized spectral measure of Laplacian on this setting
showed in [20] and a Littlewood-Paley squarefunction estimate. As
applications, we prove the global existence and scattering for a family of
nonlinear wave equations on this setting.
Key Words: Strichartz estimate, Asymptotically conic manifold, Spectral
measure, Global existence, Scattering theory
AMS Classification: 35Q40, 35S30, 47J35.
## 1\. Introduction and Statement of Main Results
Let $(M^{\circ},g)$ be a Riemannian manifold of dimension $n\geq 2$, and let
$I\subset\R$ be a time interval. Suppose $u(t,z)$: $I\times
M^{\circ}\rightarrow\mathbb{C}$ to be the solutions of the wave equation
$\partial_{t}^{2}u+\mathrm{H}u=0,\quad
u(0)=u_{0}(z),~{}\partial_{t}u(0)=u_{1}(z)$
where $\mathrm{H}=-\Delta_{g}$ denotes the minus Laplace-Beltrami operator on
$(M^{\circ},g)$. The general homogeneous Strichartz estimates read
$\|u(t,z)\|_{L^{q}_{t}L^{r}_{z}(I\times M^{\circ})}\leq
C\big{(}\|u_{0}\|_{H^{s}(M^{\circ})}+\|u_{1}\|_{H^{s-1}(M^{\circ})}\big{)},$
where $H^{s}$ denotes the $L^{2}$-Sobolev space over $M^{\circ}$, and $2\leq
q,r\leq\infty$ satisfy
$s=n(\frac{1}{2}-\frac{1}{r})-\frac{1}{q},\quad\frac{2}{q}+\frac{n-1}{r}\leq\frac{n-1}{2},\quad(q,r,n)\neq(2,\infty,3).$
In the flat Euclidean space, where $M^{\circ}=\R^{n}$ and
$g_{jk}=\delta_{jk}$, one can take $I=\R$; see Strichartz [30], Ginibre and
Velo [10], Keel and Tao [22], and references therein. In general manifolds,
for instance the compact manifold with or without boundary, most of the
Strichartz estimates are local in time. If $M^{\circ}$ is a compact manifold
without boundary, due to finite speed of propagation one usually works in
coordinate charts and establishes local Strichartz estimates for variable
coefficient wave operators on $\R^{n}$. See for examples [21, 26, 32].
Strichartz estimates also are considered on compact manifold with boundary,
see [6], [2] and references therein. When we consider the noncompact manifold
with nontrapping condition, one can obtain global-in-time Strichartz
estimates. For instance, when $M^{\circ}$ is a exterior manifold in
$\mathbb{R}^{n}$ to a convex obstacle, for metrics $g$ which agree with the
Euclidean metric outside a compact set with nontrapping assumption, the global
Strichartz estimates are obtained by Smith-Sogge [27] for odd dimension, and
Burq [5] and Metcalfe [25] for even dimension. Blair-Ford-Marzuola [3]
established global Strichartz estimates for the wave equation on flat cones
$C(\mathbb{S}_{\rho}^{1})$ by using the explicit representation of the
fundamental solution.
In this paper, we consider the establishment of global-in-time Strichartz
estimates on asymptotically conic manifolds satisfying a nontrapping
condition. Here, ‘asymptotically conic’ is meant in the sense that $M^{\circ}$
can be compactified to a manifold with boundary $M$ such that $g$ becomes a
scattering metric on $M$. On the nontrapping asymptotically conic manifolds,
Hassell, Tao, and Wunsch first established an $L^{4}_{t,z}$-Strichartz
estimate for Schrödinger equation in [14] and then they [15] extended the
estimate to full admissible Strichartz exponents except endpoint $q=2$. More
precisely, they obtained the local-in-time Strichartz inequalities for non-
endpoint Schrödinger admissible pairs $(q,r)$
$\|e^{it\Delta_{g}}u_{0}\|_{L^{q}_{t}L^{r}_{z}([0,1]\times M^{\circ})}\leq
C\|u_{0}\|_{L^{2}(M^{\circ})}.$
Recently, Hassell and the author [20] improved the Strichartz inequalities by
replacing the interval $[0,1]$ by $\R$. The purpose of this article is to
extend the above investigations carried out for Schrödinger to wave equations.
Let us recall the asymptotically conic geometric setting (i.e. scattering
manifold), which is the same as in [12, 13, 17, 15, 20]. Let $(M^{\circ},g)$
be a complete noncompact Riemannian manifold of dimension $n\geq 2$ with one
end, diffeomorphic to $(0,\infty)\times Y$ where $Y$ is a smooth compact
connected manifold without boundary. Moreover, we assume $(M^{\circ},g)$ is
asymptotically conic which means that $M^{\circ}$ allows a compactification
$M$ with boundary, with $\partial M=Y$, such that the metric $g$ becomes an
asymptotically conic metric on $M$. In details, the metric $g$ in a collar
neighborhood $[0,\epsilon)_{x}\times\partial M$ near $Y$ takes the form of
(1.1)
$g=\frac{\mathrm{d}x^{2}}{x^{4}}+\frac{h(x)}{x^{2}}=\frac{\mathrm{d}x^{2}}{x^{4}}+\frac{\sum
h_{jk}(x,y)dy^{j}dy^{k}}{x^{2}},$
where $x\in C^{\infty}(M)$ is a boundary defining function for $\partial M$
and $h$ is a smooth family of metrics on $Y$. Here we use
$y=(y_{1},\cdots,y_{n-1})$ for local coordinates on $Y=\partial M$, and the
local coordinates $(x,y)$ on $M$ near $\partial M$. Away from $\partial M$, we
use $z=(z_{1},\cdots,z_{n})$ to denote the local coordinates. If
$h_{jk}(x,y)=h_{jk}(y)$ is independent of $x$, we say $M$ is perfectly conic
near infinity. Moreover if every geodesic $z(s)$ in $M$ reaches $Y$ as
$s\rightarrow\pm\infty$, we say $M$ is nontrapping. The function $r:=1/x$ near
$x=0$ can be thought of as a “radial” variable near infinity and $y$ can be
regarded as the $n-1$ “angular” variables; the metric is asymptotic to the
exact conic metric $((0,\infty)_{r}\times Y,dr^{2}+r^{2}h(0))$ as
$r\rightarrow\infty$. The Euclidean space $M^{\circ}=\mathbb{R}^{n}$ is an
example of an asymptotically conic manifold with $Y=\mathbb{S}^{n-1}$ and the
standard metric. However a metric cone itself is not an asymptotically conic
manifold because of its cone point. We remark that the Euclidean space is a
perfectly metric nontrapping cone, where the cone point is a removable
singularity.
Let $\dot{H}^{s}(M^{\circ})={(-\Delta_{g})}^{-\frac{s}{2}}L^{2}(M^{\circ})$ be
the homogeneous Sobolev space over $M^{\circ}$. Throughout this paper, pairs
of conjugate indices are written as $r,r^{\prime}$, where
$\frac{1}{r}+\frac{1}{r^{\prime}}=1$ with $1\leq r\leq\infty$. Our main result
concerning Strichartz estimates is the following.
###### Theorem 1.1 (Global-in-time Strichartz estimate).
Let $(M^{\circ},g)$ be an asymptotically conic non-trapping manifold of
dimension $n\geq 3$. Let $\mathrm{H}=-\Delta_{g}$ and suppose that $u$ is the
solution to the Cauchy problem
(1.2) $\begin{cases}\partial_{t}^{2}u+\mathrm{H}u=F(t,z),\quad(t,z)\in I\times
M^{\circ};\\\ u(0)=u_{0}(z),~{}\partial_{t}u(0)=u_{1}(z),\end{cases}$
for some initial data $u_{0}\in\dot{H}^{s},u_{1}\in\dot{H}^{s-1}$, and the
time interval $I\subseteq\R$, then
(1.3)
$\begin{split}&\|u(t,z)\|_{L^{q}_{t}(I;L^{r}_{z}(M^{\circ}))}+\|u(t,z)\|_{C(I;\dot{H}^{s}(M^{\circ}))}\\\
&\qquad\lesssim\|u_{0}\|_{\dot{H}^{s}(M^{\circ})}+\|u_{1}\|_{\dot{H}^{s-1}(M^{\circ})}+\|F\|_{L^{\tilde{q}^{\prime}}_{t}(I;L^{\tilde{r}^{\prime}}_{z}(M^{\circ}))},\end{split}$
where the pairs $(q,r),(\tilde{q},\tilde{r})\in[2,\infty]^{2}$ satisfy the
wave-admissible condition
(1.4)
$\frac{2}{q}+\frac{n-1}{r}\leq\frac{n-1}{2},\quad(q,r,n)\neq(2,\infty,3).$
and the gap condition
(1.5)
$\frac{1}{q}+\frac{n}{r}=\frac{n}{2}-s=\frac{1}{\tilde{q}^{\prime}}+\frac{n}{\tilde{r}^{\prime}}-2.$
###### Remark 1.2.
We remark that the estimates are sharp from the sharpness in [22] for the
Euclidean space. There is no loss of derivatives. We can take the interval
$I=\R$ which means the estimates are global in time.
We sketch the proof as follows. Our strategy is to use the abstract Strichartz
estimate proved in Keel-Tao [22] and our previous argument [20] for
Schrödinger. Thus, with $U(t)$ denoting the (abstract) propagator, we need to
show uniform $L^{2}\rightarrow L^{2}$ estimate for $U(t)$, and
$L^{1}\rightarrow L^{\infty}$ type dispersive estimate on the $U(t)U(s)^{*}$
with a bound of the form $O(|t-s|^{-(n-1)/2})$. In the flat Euclidean setting,
the estimates are considerably simpler because of the explicit formula of the
spectral measure. But in our general setting, the estimates turn out to be
more complicated. It follows from [17] that the Schrödinger propagator
$e^{it\Delta_{g}}$ fails to satisfy such a dispersive estimate at any pair of
conjugate points $(z,z^{\prime})\in M^{\circ}\times M^{\circ}$ (i.e. pairs
$(z,z^{\prime})$ where a geodesic emanating from $z$ has a conjugate point at
$z^{\prime}$), so we need localize the propagator such that the conjugating
points are separated. One may avoid the conjugated points in a sufficiently
short time by using the finite speed of propagation $U(t)(z,z^{\prime})$. If
we do this, we would only obtain the local-in-time Strichartz estimates. We
instead overcome the difficulties caused by conjugate points by
microlocalizing the spectral measure [20], which is in the same spirit of the
proof in [13] of a _restriction estimate_ for the spectral measure, that is,
an estimate of the form
$\big{\|}dE_{\sqrt{\mathbf{H}}}(\lambda)\big{\|}_{L^{p}(M^{\circ})\to
L^{p^{\prime}}(M^{\circ})}\leq
C\lambda^{n(\frac{1}{p}-\frac{1}{p^{\prime}})-1},\quad 1\leq
p\leq\frac{2(n+1)}{n+3}.$
However, the microlocalized spectral measure
$Q_{i}(\lambda)dE_{\sqrt{\mathbf{H}}}(\lambda)Q_{i}(\lambda)^{*}$ only has a
size estimate in [13], where $Q_{i}(\lambda)$ is a member of a partition of
the identity operator in $L^{2}(M^{\circ})$. To obtain the dispersive
estimate, the authors [20] refined the microlocalized spectral measure by
capturing its oscillatory behavior. Thus we efficiently exploit the
oscillation of the ‘spectral multiplier’ $e^{it\lambda^{2}}$ and
microlocalized spectral measure to prove the dispersive estimate for
Schrödinger. However, the multiplier $e^{it\lambda}$ corresponding to the wave
equation has much less oscillation than the Schrödinger multiplier
$e^{it\lambda^{2}}$ at high frequency, so we need to modify the argument.
Because of this, we have to resort to a Littlewood-Paley squarefunction
estimate on this setting. We remark that the authors [20] avoid using the
Littlewood-Paley squarefunction estimate in the Schrödinger case. We prove the
Littlewood-Paley squarefunction estimate on this setting by using a spectral
multiplier estimate in Alexopoulos [1] and Stein’s [28] classical argument
involving Rademacher functions. The crucial ingredient is to obtain the
Gaussian upper bounds on the heat kernel on this setting. We show the Gaussian
upper bounds on the heat kernel by using the local-in-time heat kernel bounds
in Cheng-Li-Yau [7], and Guillarmou-Hassell-Sikora’s [13] restriction estimate
for low frequency which implies the long-time bounds. Having the
squarefunction estimate, we reduce Theorem 1.1 to prove a frequency-localized
estimate. To do this, we define a microlocalized half-wave propagator and
prove that it satisfies $L^{2}\rightarrow L^{2}$-bounded and dispersive
estimate. We prove the homogeneous Strichartz estimates for the microlocalized
half-wave propagator by using a semiclassical version of Keel-Tao’s argument.
The Strichartz estimate for $e^{it\sqrt{\mathrm{H}}}$ then follows by summing
each microlocalizing piece. The inhomogeneous Strichartz estimates follow from
the homogeneous estimates and the Christ-Kiselev lemma. Compared with the
establishment of Schrödinger inhomogeneous Strichartz estimate in [20], we do
not require additional argument since one must have $q>\tilde{q}^{\prime}$ if
both $(q,r)$ and $(\tilde{q},\tilde{r})$ satisfy (1.4) and (1.5).
As an application of the Strichartz estimates, we note that these inequalities
can be utilized to generalize a theorem of Lindblad-Sogge [24] on the
asymptotically conic non-trapping manifolds. More precisely, we prove the
well-posedness and scattering of the following semi-linear wave equation,
(1.6)
$\begin{cases}\partial_{t}^{2}u+\mathrm{H}u=\gamma|u|^{p-1}u,\qquad(t,z)\in\R\times
M^{\circ},\gamma\in\\{1,-1\\},\\\
u(t,z)|_{t=0}=u_{0}(z),\quad\partial_{t}u(t,z)|_{t=0}=u_{1}(z).\end{cases}$
In the case of flat Euclidean space, there are many results on the
understanding of the global existence and scattering. We refer the readers to
[24, 29] and references therein. Blair-Ford-Marzuola [3] also considered
similar results for the wave equation on flat cones
$C(\mathbb{S}_{\rho}^{1})$. Due to better understanding the spectral measure,
we can extend the result to high dimension. We here are mostly interested in
the range of exponents $p\in[p_{\text{conf}},1+\frac{4}{n-2}]$ and the initial
data is in $\dot{H}^{s_{c}}(M^{\circ})\times\dot{H}^{s_{c}-1}(M^{\circ})$,
where $p_{\text{conf}}=1+\frac{4}{n-1}$ and $s_{c}=\frac{n}{2}-\frac{2}{p-1}$.
Our main result concerning well-posedness and scattering is the following.
###### Theorem 1.3.
Let $(M^{\circ},g)$ be a non-trapping asymptotically conic manifold of
dimension $n\geq 3$. Suppose $p\in[p_{\mathrm{conf}},1+\frac{4}{n-2}]$ and
$(u_{0},u_{1})\in\dot{H}^{s_{c}}(M^{\circ})\times\dot{H}^{s_{c}-1}(M^{\circ})$,
then there exist $T>0$ and a unique solution $u$ to (1.6) satisfying
(1.7) $u\in C_{t}([0,T];\dot{H}^{s_{c}}(M^{\circ}))\cap
L^{q_{0}}([0,T];L^{q_{0}}(M^{\circ})),$
where $q_{0}=(p-1)(n+1)/2$. In addition, if there is a small constant
$\epsilon(p)$ such that
(1.8) $\|u_{0}\|_{\dot{H}^{s_{c}}}+\|u_{1}\|_{\dot{H}^{s_{c}-1}}<\epsilon(p),$
then there is a unique global and scattering solution $u$ to (1.6) satisfying
(1.9) $u\in C_{t}(\R;H^{s_{c}}(M^{\circ}))\cap
L^{q_{0}}(\R;L^{q_{0}}(M^{\circ})).$
This paper is organized as follows. In Section 2 we review the results of the
microlocalized spectral measure and prove the square function inequalities on
this setting. Section 3 is devoted to the proofs of the microlocalized
dispersive estimates and $L^{2}$-estimates. In Section 4, we prove the
homogeneous and inhomogeneous Strichartz estimates. Finally, we apply the
Strichartz estimates to show Theorem 1.3.
Acknowledgments: The author would like to thank Jean-Marc Bouclet, Andrew
Hassell and Changxing Miao for their helpful discussions and encouragement. He
also would like to thank the anonymous referee for careful reading the
manuscript and for giving useful comments. This research was supported by
PFMEC(20121101120044), Beijing Natural Science Foundation(1144014), National
Natural Science Foundation of China (11401024) and Discovery Grant DP120102019
from the Australian Research Council.
## 2\. The microlocalized spectral measure and Littlewood-Paley
squarefunction estimate
In this section, we briefly recall the key elements of the microlocalized
spectral measure, which was constructed by Hassell and the author [20] to
capture both its size and the oscillatory behavior. We also prove the
Littlewood-Paley squarefunction estimates on this setting that we require in
subsequence section.
### 2.1. The microlocalized spectral measure
In the free Euclidean space, the half wave propagator has an explicit formula
by using the Fourier transform, but in the asymptotically conical manifold it
turns out to be quite complicated. From the results of [12, 16], we have known
that the Schwartz kernel of the spectral measure can be described as a
Legendrian distribution on the compactification of the space $M\times M$
uniformly with respect to the spectral parameter $\lambda$. As pointed out in
introduction, we really need to choose an operator partition of unity to
microlocalize the spectral measure such that the spectral measure can be
expressed in a formula capturing not only the size also the oscillatory
behavior. This was constructed and proved in [20]. For convenience, we recall
and slightly modify the statement to adapt our following application.
###### Proposition 2.1.
Let $(M^{\circ},g)$ and $\mathrm{H}$ be in Theorem 1.1. For fixed
$\lambda_{0}>0$, then there exists an operator partition of unity on
$L^{2}(M)$
(2.1)
$\begin{split}\mathrm{Id}=\sum_{i=0}^{N_{l}}Q^{\mathrm{low}}_{i}(\lambda)\quad\text{for}~{}0<\lambda\leq
2\lambda_{0};\\\
\mathrm{Id}=\big{(}\sum_{i=1}^{N^{\prime}}+\sum_{i=N^{\prime}+1}^{N_{h}}\big{)}Q^{\mathrm{high}}_{i}(\lambda)\quad\text{for}~{}\lambda\geq\lambda_{0}/2,\end{split}$
where the $Q^{\mathrm{low}}_{i}$ and $Q^{\mathrm{high}}_{i}$ are uniformly
bounded as operators on $L^{2}$ and $N_{l}$ and $N_{h}$ are bounded
independent of $\lambda$, such that
$\bullet$ when $Q(\lambda)$ is equal to either $Q^{\mathrm{low}}_{0}(\lambda)$
or $Q^{\mathrm{low}}_{1}(\lambda)$; or $Q(\lambda)$ is equal to
$Q^{\mathrm{high}}_{1}(\lambda)$, we have
(2.2)
$\begin{split}\Big{|}\big{(}\frac{d}{d\lambda}\big{)}^{\alpha}\big{(}Q(\lambda)dE_{\sqrt{\mathrm{H}}}(\lambda){Q}^{*}(\lambda)\big{)}\Big{|}\leq
C_{\alpha}\lambda^{n-1-\alpha}\quad\forall\alpha\in\mathbb{N}.\end{split}$
$\bullet$ when $Q(\lambda)$ is equal to $Q^{\mathrm{low}}_{i}(\lambda)$ or
$Q^{\mathrm{high}}_{i}(\lambda)$ for $i\geq 2$, we have
(2.3)
$(Q(\lambda)dE_{\sqrt{\mathrm{H}}}(\lambda)Q^{*}(\lambda))(z,z^{\prime})=\lambda^{n-1}e^{\pm
i\lambda d(z,z^{\prime})}a(\lambda,z,z^{\prime}).$
Here $d(\cdot,\cdot)$ is the Riemannian distance on $M^{\circ}$, and $a$
satisfies
(2.4) $|\partial_{\lambda}^{\alpha}a(\lambda,z,z^{\prime})|\leq
C_{\alpha}\lambda^{-\alpha}(1+\lambda d(z,z^{\prime}))^{-\frac{n-1}{2}}.$
Having this result, we can exploit the oscillations both in the multiplier
$e^{i(t-s)\lambda}$ and in $e^{\pm i\lambda d(z,z^{\prime})}$ to obtain the
required dispersive estimate for the $TT^{*}$ version of the microlocalized
propagator.
### 2.2. The Littlewood-Paley squarefunction estimate
In this subsection, we prove the Littlewood-Paley squarefunction estimate for
the asymptotically conic manifold, which allows us to reduce Theorem 1.1 to a
frequency-localized estimate (see Proposition 4.2).
Let $\varphi\in C_{0}^{\infty}(\mathbb{R}\setminus\\{0\\})$ take values in
$[0,1]$ and be supported in $[1/2,2]$ such that
(2.5) $1=\sum_{j\in\Z}\varphi(2^{-j}\lambda),\quad\lambda>0.$
Define $\varphi_{0}(\lambda)=\sum_{j\leq 0}\varphi(2^{-j}\lambda)$. Then the
result about the Littlewood-Paley squarefunction estimate reads as follows:
###### Proposition 2.2.
Let $(M^{\circ},g)$ be an asymptotically conic manifold, trapping or not, and
$\mathrm{H}=-\Delta_{g}$ is the Laplace-Beltrami operator on $(M^{\circ},g)$.
Then for $1<p<\infty$, there exist constants $c_{p}$ and $C_{p}$ depending on
$p$ such that
(2.6)
$c_{p}\|f\|_{L^{p}(M^{\circ})}\leq\big{\|}\big{(}\sum_{j\in\Z}|\varphi(2^{-j}\sqrt{\mathrm{H}})f|^{2}\big{)}^{\frac{1}{2}}\big{\|}_{L^{p}(M^{\circ})}\leq
C_{p}\|f\|_{L^{p}(M^{\circ})}.$
###### Remark 2.3.
To our knowledge, such squarefunction estimates are new in the case of
asymptotically conic manifolds, though the proof is considerably simpler due
to the heat kernel bounds in Cheng-Li-Yau [7], Guillarmou-Hassell-Sikora’s
[13] restriction estimate for low frequency and the spectral multiplier
estimates in Alexopoulos [1]. In the general noncompact manifolds with ends,
Bouclet [4] proved a weak version square function inequality which was given
by for $1<p<\infty$
(2.7) $\|f\|_{L^{p}}\lesssim\big{\|}\big{(}\sum_{j\geq
0}|\varphi(2^{-2j}\mathrm{H})f|^{2}\big{)}^{\frac{1}{2}}\big{\|}_{L^{p}}+\|f\|_{L^{2}}.$
Bouclet also pointed out that the usual square function inequalities may fail
on asymptotically hyperbolic manifolds and improved (2.7) for asymptotically
conic manifolds by showing
(2.8) $\|\varphi_{0}(\mathrm{H})f\|_{L^{p}}+\big{\|}\big{(}\sum_{j\geq
0}|\varphi(2^{-2j}\mathrm{H})f|^{2}\big{)}^{\frac{1}{2}}\big{\|}_{L^{p}}\sim\|f\|_{L^{p}}.$
One can see that the squarefunction estimate in (2.6) involves the low
frequency in contrast to (2.8).
###### Proof.
This proof follows from the Stein’s [28] classical argument (in $\R^{n}$)
involving Rademacher functions and an appropriate Mikhlin-Hörmander multiplier
theorem. Now we provide details as follows. We notice that the asymptotically
conic manifolds are a relatively well-behaved class of manifolds. In
particular, all section curvatures of $(M^{\circ},g)$ approach zero as $x$
goes to zero, and thus $(M^{\circ},g)$ has bounded sectional curvature and has
low bounds for the injectivity radius. Now we need a theorem in Cheng-Li-Yau
[7] and recall it for convenience. For complete Riemannian manifolds
$M^{\circ}$ of bounded sectional curvature and injectivity radius bounded
below, Cheng-Li-Yau’s theorem gives the following local-in-time Gaussian upper
bound for the heat kernel
###### Lemma 2.4.
There exist nonzero constants $c$ and $C$ such that the heat kernel on
$M^{\circ}$, denoted $H(t,z,z^{\prime})$, satisfies the Gaussian upper bound
of the form for $t\in[0,T]$
(2.9) $H(t,z,z^{\prime})\leq
Ct^{-\frac{n}{2}}\exp\Big{(}-\frac{d(z,z^{\prime})^{2}}{ct}\Big{)},$
where $d(z,z^{\prime})$ is the distance between $z$ and $z^{\prime}$ on
$M^{\circ}$.
We claim that the global-in-time Gaussian upper bound for the heat kernel also
holds, that is
(2.10)
$H(t,z,z^{\prime})\lesssim\frac{1}{|B(z,\sqrt{t})|}\exp\Big{(}-\frac{d(z,z^{\prime})^{2}}{ct}\Big{)}$
holds for all $t>0$, where $|B(z,\sqrt{t})|$ is the volume of the ball of
radius $\sqrt{t}$ at $z$. By (2.9), we only consider the case $t\geq 1$. To
prove this, we write
$H(t,z,z^{\prime})=e^{-t\mathrm{H}}(z,z^{\prime})=\int_{0}^{\infty}e^{-t\lambda^{2}}dE_{\sqrt{\mathrm{H}}}(\lambda).$
Choose $\chi\in C_{c}^{\infty}(\R)$, such that $\chi(\lambda)=1$ for
$\lambda\leq 1$, we decompose
$\begin{split}&H(t,z,z^{\prime})\\\
&=\int_{0}^{\infty}e^{-t\lambda^{2}}\chi(\lambda)dE_{\sqrt{\mathrm{H}}}(\lambda)+\int_{0}^{\infty}e^{-t\lambda^{2}}(1-\chi)(\lambda)dE_{\sqrt{\mathrm{H}}}(\lambda)\\\
&=:I+II.\end{split}$
By using [13, Theorem 1.3], we see for $\lambda\leq 1$
$\begin{split}|dE_{\sqrt{\mathrm{H}}}(\lambda)(z,z^{\prime})|\leq
C\lambda^{n-1}.\end{split}$
Hence $I\leq Ct^{-\frac{n}{2}}$. To treat $II$, we need the following lemma
###### Lemma 2.5.
If the local-in-time heat kernel bound $\|e^{-t\mathrm{H}}\|_{L^{1}\rightarrow
L^{2}}\leq Ct^{-\frac{n}{4}}$ holds for $t\leq 1$, then the following spectral
projection estimate holds for $\mu\geq 1$,
$\|E_{\sqrt{\mathrm{H}}}([0,\mu])\|_{L^{1}\rightarrow L^{2}}\leq C\mu^{n/2}.$
###### Proof.
Let $t=\mu^{-2}$. Notice $1_{[0,\mu]}(s)\leq e\exp(-\frac{s^{2}}{\mu^{2}})$,
then spectral projection estimate is proved by writing
$E_{\sqrt{\mathrm{H}}}([0,\mu])=E_{\sqrt{\mathrm{H}}}([0,\mu])e^{\mathrm{H}/\mu^{2}}e^{-\mathrm{H}/\mu^{2}}$.
Indeed, we have
$\begin{split}\|E_{\sqrt{\mathrm{H}}}([0,\mu])\|_{L^{1}\rightarrow
L^{2}}\leq\|E_{\sqrt{\mathrm{H}}}([0,\mu])e^{\mathrm{H}/\mu^{2}}\|_{L^{2}\rightarrow
L^{2}}\|e^{-\mathrm{H}/\mu^{2}}\|_{L^{1}\rightarrow L^{2}}\leq
C\mu^{n/2}.\end{split}$
∎
Now we turn to estimate $II$. From the local-in-time heat kernel estimate
(2.9), one has $\|e^{-t\mathrm{H}}\|_{L^{1}\rightarrow L^{\infty}}\leq
Ct^{-\frac{n}{2}}$ for $t\leq 1$. By using a $TT^{*}$ argument,
$\|e^{-t\mathrm{H}}\|_{L^{1}\rightarrow L^{2}}\leq Ct^{-\frac{n}{4}}$ for
$t\leq 1$. Hence by Lemma 2.5
$\|E_{\sqrt{\mathrm{H}}}([0,\lambda])\|_{L^{1}\rightarrow L^{2}}\leq
C\lambda^{n/2}$ for $\lambda\geq 1$, which implies
$\|E_{\sqrt{\mathrm{H}}}([0,\lambda])\|_{L^{1}\rightarrow L^{\infty}}\leq
C\lambda^{n}$. Therefore we have for $t\geq 1$
$\begin{split}\|II\|_{L^{1}\rightarrow L^{\infty}}&\leq\sum_{k\geq
0}\int_{0}^{\infty}\frac{d}{d\lambda}\left(e^{-t\lambda^{2}}\phi_{k}\left(\lambda\right)(1-\chi)(\lambda)\right)\left\|E_{\sqrt{\mathrm{H}}}(\lambda)\right\|_{L^{1}\rightarrow
L^{\infty}}d\lambda\\\ &\leq Ce^{-t/2}\leq Ct^{-\frac{n}{2}}.\end{split}$
Hence we have proved for all $t>0$
$H(t,z,z^{\prime})\lesssim t^{-\frac{n}{2}}.$
We use a theorem of Grigor’yan [11, Theorem 1.1] that establishes Gaussian
upper bounds for arbitrary Riemannian manifolds. His conclusion implies that
if $H(t,z,z^{\prime})$ satisfies on-diagonal bounds
$H(t,z,z)\lesssim t^{-\frac{n}{2}},\quad H(t,z^{\prime},z^{\prime})\lesssim
t^{-\frac{n}{2}},$
then we have
$H(t,z,z^{\prime})\lesssim
t^{-\frac{n}{2}}\exp\Big{(}-\frac{d(z,z^{\prime})^{2}}{ct}\Big{)}.$
Since $|B(z,\sqrt{t})|\sim t^{\frac{n}{2}}$, this gives
(2.11)
$H(t,z,z^{\prime})\lesssim\frac{1}{|B(z,\sqrt{t})|}\exp\Big{(}-\frac{d(z,z^{\prime})^{2}}{ct}\Big{)}.$
Now we need a result of Alexopoulos [1, Theorem 6.1], which outlines how his
results on Markov chains can be extended to treat differential operators on
manifolds where the associated heat kernel satisfies Gaussian upper bounds. We
remark here that the asymptotically conic manifold satisfies the doubling
condition in contrast to the hyperbolic case. Given (2.11), Alexopoulos’
theorem implies that any spectral multiplier $m(\sqrt{\mathrm{H}})$ satisfying
the usual Hörmander condition maps $L^{p}(M)\rightarrow L^{p}(M)$ for any
$p\in(1,\infty)$. Furthermore, this boundedness holds true for function $m\in
C^{N}(\R)$ which satisfies the weaker Mihlin-type condition for
$N\geq\frac{n}{2}+1$
(2.12) $\sup_{0\leq k\leq
N}\sup_{\lambda\in\R}\Big{|}\big{(}\lambda\partial_{\lambda}\big{)}^{k}m(\lambda)\Big{|}\leq
C<\infty.$
We now want to apply this result to a family of multipliers
$m^{\pm}(s,\sqrt{\mathrm{H}}),0\leq s\leq 1$ defined using the Rademacher
functions. Let us introduce the Rademacher functions defined as follows:
(i) the function $r_{0}(s)$ is defined by $r_{0}(s)=1$ on $[0,1/2]$ and
$r_{0}(s)=-1$ on $(1/2,1)$, and then extended to the real line by periodicity,
i.e. $r_{0}(s+1)=r_{0}(s)$;
(ii) for $k\in\N\setminus\\{0\\}$, $r_{k}(s)=r_{0}(2^{k}s)$.
Given any square integrable sequence of scalars $\\{a_{k}\\}_{k\geq 0}$,
consider the function $m(s)=\sum_{k\geq 0}a_{k}r_{k}(s)$. By a lemma in [28,
Appendix D], for any $p\in(1,\infty)$ there exist constants $c_{p}$ and
$C_{p}$ such that
(2.13)
$c_{p}\|m(s)\|_{L^{p}([0,1])}\leq\|m(s)\|_{L^{2}([0,1])}=\Big{(}\sum_{k\geq
0}|a_{k}|^{2}\Big{)}^{\frac{1}{2}}\leq C_{p}\|m(s)\|_{L^{p}([0,1])}.$
Now define
$m^{\pm}(s,\lambda)=\sum_{j\geq 0}r_{j}(s)\varphi_{\pm j}(\lambda)$
where $\varphi_{\pm j}(\lambda)=\varphi(2^{\mp j}\lambda)$. Then we define the
operator $m^{\pm}(s,\sqrt{\mathrm{H}})$ through the spectral measure
$dE_{\sqrt{\mathrm{H}}}(\lambda)$:
(2.14)
$m^{\pm}(s,\sqrt{\mathrm{H}})=\int_{0}^{\infty}m^{\pm}(s,\lambda)dE_{\sqrt{\mathrm{H}}}(\lambda).$
We note that this is well-defined by the spectral theory. It can be verified
that $m^{\pm}(s,\lambda)$ satisfies the condition (2.12), and we can take the
constant $C$ independent of $s$. Therefore we have that for $1<p<\infty$ and
$f$ in $L^{p}$ by (2.13)
$\begin{split}&\Big{\|}\Big{(}\sum_{j\geq 0}\big{|}\varphi_{\pm
j}(\sqrt{\mathrm{H}})f\big{|}^{2}\Big{)}^{\frac{1}{2}}\Big{\|}^{p}_{L^{p}}\lesssim\Big{\|}\sum_{j\geq
0}\varphi_{\pm
j}(\sqrt{\mathrm{H}})f(z)r_{k}(s)\Big{\|}^{p}_{L^{p}(M;L^{p}([0,1]))}\\\
&\lesssim\int_{M^{\circ}}\int_{0}^{1}\Big{|}m^{\pm}(s,\sqrt{\mathrm{H}})f(z)\Big{|}^{p}dsdg(z)\lesssim\|f\|^{p}_{L^{p}}.\end{split}$
Therefore we prove
(2.15)
$\begin{split}&\Big{\|}\Big{(}\sum_{j\in\Z}\big{|}\varphi_{j}(\sqrt{\mathrm{H}})f\big{|}^{2}\Big{)}^{\frac{1}{2}}\Big{\|}_{L^{p}}\lesssim\|f\|_{L^{p}}.\end{split}$
To see the other inequality, we first define
$\widetilde{\varphi}_{j}(\lambda)=\sum_{i=j-1}^{j+1}\varphi_{i}(\lambda)$,
then the above also is true when $\varphi_{j}(\lambda)$ is replaced by
$\widetilde{\varphi}_{j}(\lambda)$. Let $f_{1}\in L^{p}$ and $f_{2}\in
L^{p^{\prime}}$, we see by Hölder’s inequality and (2.15)
$\begin{split}\Big{|}\int_{M^{\circ}}f_{1}(z)\overline{f_{2}(z)}dg(z)\Big{|}&=\Big{|}\int_{M^{\circ}}\sum_{j\in\Z}\big{(}\widetilde{\varphi}_{j}(\sqrt{\mathrm{H}})f_{1}\big{)}(z)\overline{\big{(}\varphi_{j}(\sqrt{\mathrm{H}})f_{2}\big{)}(z)}dg(z)\Big{|}\\\
&\lesssim\Big{\|}\big{(}\sum_{j\in\Z}\big{|}\widetilde{\varphi}_{j}(\sqrt{\mathrm{H}})f_{1}\big{|}^{2}\big{)}^{\frac{1}{2}}\Big{\|}_{L^{p}}\Big{\|}\big{(}\sum_{j\in\Z}\big{|}\varphi_{j}(\sqrt{\mathrm{H}})f_{2}\big{|}^{2}\big{)}^{\frac{1}{2}}\Big{\|}_{L^{p^{\prime}}}\\\
&\lesssim\|f_{1}\|_{L^{p}}\Big{\|}\big{(}\sum_{j\in\Z}\big{|}\varphi_{j}(\sqrt{\mathrm{H}})f_{2}\big{|}^{2}\big{)}^{\frac{1}{2}}\Big{\|}_{L^{p^{\prime}}}.\end{split}$
By duality, we hence prove (2.6).
∎
## 3\. $L^{2}$-estimates and dispersive estimates
In this section, we prove the $L^{2}$-estimates and dispersive estimates
needed for the abstract Keel-Tao argument. We begin by defining microlocalized
propagators and then show the definition makes sense. We do this by showing
that each microlocalized propagator is a bounded operator on $L^{2}$. This
serves both to make the definition of each microlocalized propagator
allowable, and to establish the $L^{2}\to L^{2}$ estimate needed for the
abstract Keel-Tao argument. We point out here that the microlocalized
propagators are different from the ones defined in [20], which allow us to
easily show the $L^{2}\to L^{2}$ estimate by spectral theory on Hilbert space
but we need a square function inequality in the establishment of the
Strichartz estimate. Since the microlocalized propagators avoid the conjugate
points, we can prove the $TT^{*}$ version dispersive estimates.
### 3.1. Microlocalized propagator and $L^{2}$-estimates
We start by dividing the half wave propagator into a low-energy piece and a
high-energy piece. Choose $\chi\in C_{c}^{\infty}(\R)$, such that $\chi(t)=1$
for $t\leq 1$. We define
(3.1)
$U^{\mathrm{low}}(t)=\int_{0}^{\infty}e^{it\lambda}\chi(\lambda)dE_{\sqrt{\mathrm{H}}}(\lambda),\quad
U^{\mathrm{high}}(t)=\int_{0}^{\infty}e^{it\lambda}(1-\chi)(\lambda)dE_{\sqrt{\mathrm{H}}}(\lambda).$
Using the partition of unity $1=\sum_{j\in\Z}\varphi(2^{-j}\lambda)$ we define
(3.2)
$\begin{split}U^{\mathrm{low}}_{j}(t)&=\int_{0}^{\infty}e^{it\lambda}\varphi(2^{-j}\lambda)\chi(\lambda)dE_{\sqrt{\mathrm{H}}}(\lambda),\\\
U^{\mathrm{high}}_{j}(t)&=\int_{0}^{\infty}e^{it\lambda}\varphi(2^{-j}\lambda)(1-\chi)(\lambda)dE_{\sqrt{\mathrm{H}}}(\lambda).\end{split}$
Further using the low-energy and high-energy operator partition of identity
operator in Proposition 2.1, we define
(3.3)
$\begin{gathered}U_{i,j}(t)=\int_{0}^{\infty}e^{it\lambda}\varphi(2^{-j}\lambda)\chi(\lambda)Q_{i}^{\mathrm{low}}(\lambda)dE_{\sqrt{\mathrm{H}}}(\lambda),\quad
0\leq i\leq N_{l};\\\
U_{i,j}(t)=\int_{0}^{\infty}e^{it\lambda}\varphi(2^{-j}\lambda)(1-\chi)(\lambda)Q_{i-N_{l}}^{\mathrm{high}}(\lambda)dE_{\sqrt{\mathrm{H}}}(\lambda),~{}N_{l}+1\leq
i\leq N:=N_{l}+N_{h}.\end{gathered}$
Now we show this definition is unambiguous. To do so, it suffices to show the
above integrals are well defined over any compact interval in $(0,\infty)$.
Suppose that $A(\lambda)$ is a family of bounded operators on
$L^{2}(M^{\circ})$, compactly supported in $[a,b]$ and $C^{1}$ in
$\lambda\in(0,\infty)$. Integrating by parts, the integral of
$\int_{a}^{b}A(\lambda)dE_{\sqrt{\mathrm{H}}}(\lambda)$
is given by
(3.4)
$E_{\mathrm{\sqrt{H}}}(b)A(b)-E_{\mathrm{\sqrt{\mathrm{H}}}}(b)A(a)-\int_{a}^{b}\frac{d}{d\lambda}A(\lambda)E_{\sqrt{\mathrm{H}}}(\lambda)\,d\lambda.$
Now we need the following lemma which is the consequence of [20, Lemma 2.3,
Lemma 3.1].
###### Lemma 3.1.
Each $Q^{\mathrm{low}}_{i}(\lambda)$ and each operator
$\lambda\partial_{\lambda}Q^{\mathrm{low}}_{i}(\lambda)$ is bounded on
$L^{2}(M^{\circ})$ uniformly in $\lambda$. The same statements are true for
the high energy operators $Q^{\mathrm{high}}_{i}(\lambda)$.
###### Proof.
We use the notation in [12, 20, 16]. The uniform boundedness of the scattering
pseudodifferential operator
$Q^{\mathrm{low}}_{i}(\lambda)\in\Psi^{-\infty}_{k}(M,M^{2}_{k,b})$ is
straightforward to prove using the fact that the order is $-\infty$. This
implies that the kernel is smooth and uniformly bounded on iterated blowup
space $M^{2}_{k,\mathrm{sc}}$, as a multiple of the half density bundle
$\Omega_{k,b}^{\frac{1}{2}}$. This bundle has a nonzero section given, in the
region where $x\leq C\lambda$, by
$\lambda^{n}|dgdg^{\prime}|^{1/2}|d\lambda/\lambda|^{1/2}$, where the
$|d\lambda/\lambda|^{1/2}$ is a purely formal factor, included to make a half-
density on the whole space $M^{2}_{k,b}$, including in the
$\lambda$-direction. On the other hand, the kernels are chosen to have support
in a neighborhood of the diagonal, which is equivalent to the region where
$d(z,z^{\prime})\leq C\lambda^{-1}$. It follows that the kernel is bounded by
a multiple of the characteristic function of the set $\\{(z,z^{\prime})\mid
d(z,z^{\prime})\leq C\lambda^{-1}\\}$ times the Riemannian half-density.
Moreover, the same is true for $\lambda
d_{\lambda}Q_{i}^{\mathrm{low}}(\lambda)$, due to the smoothness of the kernel
on $M^{2}_{k,\mathrm{sc}}$. Since the volume of each ball of radius $r$ on
$M^{\circ}$ is between $cr^{n}$ and $Cr^{n}$, Schur’s test shows that such
kernels are bounded on $L^{2}(M^{\circ})$ uniformly in $\lambda$.
The high energy operators $Q^{\mathrm{high}}(\lambda)$ are semiclassical
pseudodifferential operators of semiclassical order 0 and differential order
$-\infty$. Therefore, they take the form
$\lambda^{n}\int
e^{i\lambda(z-z^{\prime})\cdot\zeta}a(z,\zeta,\lambda^{-1})\,d\zeta$
in the interior, or
$\lambda^{n}\int
e^{i\lambda((y-y^{\prime})\cdot\eta+(\sigma-1)\nu/x}a(x,y,\eta,\nu,\lambda^{-1})\,d\eta\,d\nu$
near the boundary. Here $a$ is smooth and compactly supported in its
arguments. Integration by parts in $\zeta$, or in $\eta,\nu$, shows that the
kernel is rapidly decreasing in $\lambda|z-z^{\prime}|$, respectively
$\lambda\sqrt{|y-y^{\prime}|^{2}/x^{2}+(\sigma-1)^{2}/x^{2}}$. Equivalently,
the kernel is rapidly decreasing in $\lambda d(z,z^{\prime})$. We see that the
kernel is point-wise bounded by $C\lambda^{n}(1+\lambda d(z,z^{\prime}))^{-N}$
for any $N$. The same is true for $\lambda
d_{\lambda}Q_{i}^{\mathrm{high}}(\lambda)$. Again Schur’s test shows that such
kernels are bounded on $L^{2}(M^{\circ})$ uniformly in $\lambda$. ∎
In view of this lemma, we can take
$A(\lambda)=e^{it\lambda}\chi(\lambda)\varphi(2^{-j})Q^{\mathrm{low}}_{i}(\lambda)$
(for $0\leq i\leq N_{l}$), or
$e^{it\lambda}\varphi(2^{-j})(1-\chi)(\lambda)Q^{\mathrm{high}}_{i-N_{l}}(\lambda)$
(for $N_{l}+1\leq i\leq N$), this means that the integrals are well-defined
over any compact interval in $(0,\infty)$, hence the operators $U_{i,j}(t)$
are well-defined. Now we see these operators are bounded on $L^{2}$. We only
consider the low frequency part since a similar argument also gives the
boundedness on $L^{2}$ for high energy part. We have for $0\leq i\leq N_{l}$,
by [20, Lemma 5.3],
(3.5)
$\begin{gathered}U_{i,j}(t)U_{i,j}(t)^{*}=\int\chi(\lambda)^{2}\varphi\big{(}\frac{\lambda}{2^{j}}\big{)}\varphi\big{(}\frac{\lambda}{2^{j}}\big{)}Q^{\mathrm{low}}_{i}(\lambda)dE_{\sqrt{\mathrm{H}}}(\lambda)Q^{\mathrm{low}}_{i}(\lambda)^{*}\\\
=-\int\frac{d}{d\lambda}\Big{(}\chi(\lambda)^{2}\varphi\big{(}\frac{\lambda}{2^{j}}\big{)}\varphi\big{(}\frac{\lambda}{2^{j}}\big{)}Q^{\mathrm{low}}_{i}(\lambda)\Big{)}E_{\sqrt{\mathrm{H}}}(\lambda)Q^{\mathrm{low}}_{i}(\lambda)^{*}\\\
-\int\chi(\lambda)^{2}\varphi\big{(}\frac{\lambda}{2^{j}}\big{)}\varphi\big{(}\frac{\lambda}{2^{j}}\big{)}Q^{\mathrm{low}}_{i}(\lambda)E_{\sqrt{\mathrm{H}}}(\lambda)\frac{d}{d\lambda}Q^{\mathrm{low}}_{i}(\lambda)^{*}.\end{gathered}$
We observe that this is independent of $t$ and we also note that the integrand
is a bounded operator on $L^{2}$, with an operator bound of the form
$C/\lambda$ where $C$ is uniform, as we see from Lemma 3.1 and the support
property of $\varphi$. The integral is therefore uniformly bounded, as we are
integrating over a dyadic interval in $\lambda$. Hence we have shown that
###### Proposition 3.2 ($L^{2}$-estimates).
Let $U_{i,j}(t)$ be defined in (3.3). Then there exists a constant $C$
independent of $t,z,z^{\prime}$ such that $\|U_{i,j}(t)\|_{L^{2}\rightarrow
L^{2}}\leq C$ for all $i\geq 0,j\in\Z$.
### 3.2. Dispersive estimates
Next we aim to establish the dispersive estimates for the microlocalized
$U_{i,j}(t)U^{*}_{i,j}(s)$. We need the following proposition.
###### Proposition 3.3 (Microlocalized dispersive estimates).
Let $Q(\lambda)$ be the operator $Q_{i}^{\mathrm{low}}$ or
$Q_{i}^{\mathrm{high}}$ constructed as in Proposition 2.1 and suppose $\phi\in
C_{c}^{\infty}([1/2,2])$ and takes value in $[0,1]$. Then the kernel estimate
(3.6)
$\begin{split}\Big{|}\int_{0}^{\infty}e^{it\lambda}\phi(2^{-j}\lambda)\big{(}Q(\lambda)&dE_{\sqrt{\mathrm{H}}}(\lambda)Q^{*}(\lambda)\big{)}(z,z^{\prime})d\lambda\Big{|}\\\
&\leq C2^{j(n+1)/2}(2^{-j}+|t|)^{-(n-1)/2}\end{split}$
holds for a constant $C$ independent of $j\in\Z$ and points $z,z^{\prime}\in
M^{\circ}$.
###### Proof.
The key to the proof is to apply Proposition 2.1. For $Q=Q_{i}^{\mathrm{low}}$
for $i=0,1$, or $Q=Q_{1}^{\mathrm{high}}$, we have by Proposition 2.1
$\Big{|}\int_{0}^{\infty}e^{it\lambda}\phi(2^{-j}\lambda)\big{(}Q(\lambda)dE_{\sqrt{\mathrm{H}}}(\lambda)Q^{*}(\lambda)\big{)}(z,z^{\prime})d\lambda\Big{|}\leq
C2^{jn}.$
We use the $N$-times integration by parts to obtain by (2.2)
$\begin{split}&\Big{|}\int_{0}^{\infty}e^{it\lambda}\phi(2^{-j}\lambda)\big{(}Q(\lambda)dE_{\sqrt{\mathrm{H}}}(\lambda)Q^{*}(\lambda)\big{)}(z,z^{\prime})d\lambda\Big{|}\\\
&\leq\Big{|}\int_{0}^{\infty}\big{(}\frac{1}{it}\frac{\partial}{\partial\lambda}\big{)}^{N}\big{(}e^{it\lambda}\big{)}\phi(2^{-j}\lambda)\big{(}Q(\lambda)dE_{\sqrt{\mathrm{H}}}(\lambda)Q^{*}(\lambda)\big{)}(z,z^{\prime})d\lambda\Big{|}\\\
&\leq C_{N}|t|^{-N}\int_{2^{j-1}}^{2^{j+1}}\lambda^{n-1-N}d\lambda\leq
C_{N}|t|^{-N}2^{j(n-N)}.\end{split}$
Therefore we obtain
(3.7)
$\begin{split}&\Big{|}\int_{0}^{\infty}e^{it\lambda}\phi(2^{-j}\lambda)\big{(}Q(\lambda)dE_{\sqrt{\mathrm{H}}}(\lambda)Q^{*}(\lambda)\big{)}(z,z^{\prime})d\lambda\Big{|}\leq
C_{N}2^{jn}(1+2^{j}|t|)^{-N}.\end{split}$
By choosing $N=(n-1)/2$, we prove (3.6). When $Q$ is equal to
$Q_{i}^{\mathrm{low}}$ or $Q_{i}^{\mathrm{high}}$ for $i\geq 2$, we see by
Proposition 2.1
$\begin{split}&\Big{|}\int_{0}^{\infty}e^{it\lambda}\phi(2^{-j}\lambda)\big{(}Q(\lambda)dE_{\sqrt{\mathrm{H}}}(\lambda)Q^{*}(\lambda)\big{)}(z,z^{\prime})d\lambda\Big{|}\\\
&=\Big{|}\int_{0}^{\infty}\left(\frac{1}{i(t-d(z,z^{\prime}))}\frac{\partial}{\partial\lambda}\right)^{N}\big{(}e^{i(t-d(z,z^{\prime}))\lambda}\big{)}\phi(2^{-j}\lambda)\lambda^{n-1}a(\lambda,z,z^{\prime})d\lambda\Big{|}\\\
&\leq
C_{N}|t-d(z,z^{\prime})|^{-N}\int_{2^{j-1}}^{2^{j+1}}\lambda^{n-1-N}(1+\lambda
d(z,z^{\prime}))^{-\frac{n-1}{2}}d\lambda\\\ &\leq
C_{N}2^{j(n-N)}|t-d(z,z^{\prime})|^{-N}(1+2^{j}d(z,z^{\prime}))^{-(n-1)/2}.\end{split}$
It follows that
(3.8)
$\begin{split}&\Big{|}\int_{0}^{\infty}e^{it\lambda}\phi(2^{-j}\lambda)\big{(}Q(\lambda)dE_{\sqrt{\mathrm{H}}}(\lambda)Q^{*}(\lambda)\big{)}(z,z^{\prime})d\lambda\Big{|}\\\
&\leq
C_{N}2^{jn}\big{(}1+2^{j}|t-d(z,z^{\prime})|\big{)}^{-N}(1+2^{j}d(z,z^{\prime}))^{-(n-1)/2}.\end{split}$
If $|t|\sim d(z,z^{\prime})$, it is clear to see (3.6). Otherwise, we have
$|t-d(z,z^{\prime})|\geq c|t|$ for some small constant $c$, then choose
$N=(n-1)/2$ to prove (3.6). ∎
###### Remark 3.4.
If $N=\frac{n-1}{2}$ is not an integer, one may need geometric mean argument
to modify the proof.
As a consequence of Proposition 3.3, we immediately have
###### Proposition 3.5.
Let $U_{i,j}(t)$ be defined in (3.3). Then there exists a constant $C$
independent of $t,z,z^{\prime}$ for all $i\geq 0,j\in\Z$ such that
(3.9) $\|U_{i,j}(t)U^{*}_{i,j}(s)\|_{L^{1}\rightarrow L^{\infty}}\leq
C2^{j(n+1)/2}(2^{-j}+|t-s|)^{-(n-1)/2}.$
## 4\. Strichartz estimates
In this section, we show the Strichartz estimates in Theorem 1.1. To obtain
the Strichartz estimates, we need a variant of Keel-Tao’s abstract Strichartz
estimate for wave equation.
### 4.1. Semiclassical Strichartz estimates
We need a variety of the abstract Keel-Tao’s Strichartz estimates theorem.
This is an analogue of the semiclassical Strichartz estimates for Schrödinger
in [23, 33].
###### Proposition 4.1.
Let $(X,\mathcal{M},\mu)$ be a $\sigma$-finite measured space and
$U:\mathbb{R}\rightarrow B(L^{2}(X,\mathcal{M},\mu))$ be a weakly measurable
map satisfying, for some constants $C$, $\alpha\geq 0$, $\sigma,h>0$,
(4.1) $\begin{split}\|U(t)\|_{L^{2}\rightarrow L^{2}}&\leq C,\quad
t\in\mathbb{R},\\\ \|U(t)U(s)^{*}f\|_{L^{\infty}}&\leq
Ch^{-\alpha}(h+|t-s|)^{-\sigma}\|f\|_{L^{1}}.\end{split}$
Then for every pair $q,r\in[1,\infty]$ such that
$(q,r,\sigma)\neq(2,\infty,1)$ and
$\frac{1}{q}+\frac{\sigma}{r}\leq\frac{\sigma}{2},\quad q\geq 2,$
there exists a constant $\tilde{C}$ only depending on $C$, $\sigma$, $q$ and
$r$ such that
(4.2)
$\Big{(}\int_{\R}\|U(t)u_{0}\|_{L^{r}}^{q}dt\Big{)}^{\frac{1}{q}}\leq\tilde{C}\Lambda(h)\|u_{0}\|_{L^{2}}$
where $\Lambda(h)=h^{-(\alpha+\sigma)(\frac{1}{2}-\frac{1}{r})+\frac{1}{q}}$.
###### Proof.
If $(q,r,\sigma)\neq(2,\infty,1)$ is on the line
$\frac{1}{q}+\frac{\sigma}{r}=\frac{\sigma}{2}$, we replace
$(|t-s|+h)^{-\sigma}$ by $|t-s|^{-\sigma}$ and then we closely follow Keel-
Tao’s argument [22, Sections 3-7] to show (4.2). So we only consider
$\frac{1}{q}+\frac{\sigma}{r}<\frac{\sigma}{2}$. By the $TT^{*}$ argument, it
suffices to show
$\begin{split}\Big{|}\iint\langle U(s)^{*}f(s),U(t)^{*}g(t)\rangle
dsdt\Big{|}\lesssim\Lambda(h)^{2}\|f\|_{L^{q^{\prime}}_{t}L^{r^{\prime}}}\|g\|_{L^{q^{\prime}}_{t}L^{r^{\prime}}}.\end{split}$
By the interpolation of the bilinear form of (4.1), we have
$\begin{split}\langle U(s)^{*}f(s),U(t)^{*}g(t)\rangle&\leq
Ch^{-\alpha(1-\frac{2}{r})}(h+|t-s|)^{-\sigma(1-\frac{2}{r})}\|f\|_{L^{r^{\prime}}}\|g\|_{L^{r^{\prime}}}.\end{split}$
Therefore we see by Hölder’s and Young’s inequalities for
$\frac{1}{q}+\frac{\sigma}{r}<\frac{\sigma}{2}$
$\begin{split}\Big{|}\iint\langle U(s)^{*}f(s),&U(t)^{*}g(t)\rangle
dsdt\Big{|}\\\ &\lesssim
h^{-\alpha(1-\frac{2}{r})}\iint(h+|t-s|)^{-\sigma(1-\frac{2}{r})}\|f(t)\|_{L^{r^{\prime}}}\|g(s)\|_{L^{r^{\prime}}}dtds\\\
&\lesssim
h^{-\alpha(1-\frac{2}{r})}h^{-\sigma(1-\frac{2}{r})+\frac{2}{q}}\|f\|_{L^{q^{\prime}}_{t}L^{r^{\prime}}}\|g\|_{L^{q^{\prime}}_{t}L^{r^{\prime}}}.\end{split}$
This proves (4.2). ∎
### 4.2. Homogeneous Strichartz estimates
To prove the homogeneous Strichartz estimates, we first reduce the estimates
to frequency localized estimates. Using the Littlewood-Paley frequency cutoff
$\varphi_{k}(\sqrt{\mathrm{H}})$, we define
(4.3) $u_{k}(t,\cdot)=\varphi_{k}(\sqrt{\mathrm{H}})u(t,\cdot).$
Notice the frequency cutoffs commute with the operator
$\mathrm{H}=-\Delta_{g}$, the frequency localized solutions
$\\{u_{k}\\}_{k\in\Z}$ satisfy the family of Cauchy problems
(4.4) $\partial_{t}^{2}u_{k}+\mathrm{H}u_{k}=0,\quad
u_{k}(0)=f_{k}(z),~{}\partial_{t}u_{k}(0)=g_{k}(z),$
where $f_{k}=\varphi_{k}(\sqrt{\mathrm{H}})u_{0}$ and
$g_{k}=\varphi_{k}(\sqrt{\mathrm{H}})u_{1}$. By the squarefunction estimates
(2.6) and Minkowski’s inequality, we obtain for $q,r\geq 2$
(4.5)
$\|u\|_{L^{q}(\R;L^{r}(M^{\circ}))}\lesssim\Big{(}\sum_{k\in\Z}\|u_{k}\|^{2}_{L^{q}(\R;L^{r}(M^{\circ}))}\Big{)}^{\frac{1}{2}}.$
Let $U(t)=e^{it\sqrt{\mathrm{H}}}$ be the half wave operator, then we write
(4.6)
$\begin{split}u_{k}(t,z)=\frac{U(t)+U(-t)}{2}f_{k}+\frac{U(t)-U(-t)}{2i\sqrt{\mathrm{H}}}g_{k}.\end{split}$
To prove the homogeneous estimates in Theorem 1.1, that is $F=0$, it suffices
to show by (4.5) and (4.6)
###### Proposition 4.2.
Let $f=\varphi_{k}(\sqrt{\mathrm{H}})f$ for $k\in\Z$, we have
(4.7) $\|U(t)f\|_{L^{q}_{t}L^{r}_{z}(\mathbb{R}\times M^{\circ})}\lesssim
2^{ks}\|f\|_{L^{2}(M^{\circ})},$
where the admissible pair $(q,r)\in[2,\infty]^{2}$ and $s$ satisfy (1.4) and
(1.5).
Now we prove this proposition. By using Proposition 3.2 and Proposition 3.5,
we have the estimates (4.1) for $U_{i,j}(t)$, where $\alpha=(n+1)/2$,
$\sigma=(n-1)/2$ and $h=2^{-j}$. Then it follows from Proposition 4.1 that
$\|U_{i,j}(t)f\|_{L^{q}_{t}(\R:L^{r}(M^{\circ}))}\lesssim
2^{j[n(\frac{1}{2}-\frac{1}{r})-\frac{1}{q}]}\|f\|_{L^{2}(M^{\circ})}.$
Notice that
$U(t)=\sum_{i=0}^{N}\sum_{j\in\Z}U_{i,j}(t),$
we can write
$U(t)f=\sum_{i}\sum_{j\in\mathbb{Z}}\int_{0}^{\infty}e^{it\lambda}\varphi(2^{-j}\lambda)Q_{i}(\lambda)dE_{\sqrt{\mathrm{H}}}(\lambda)\widetilde{\varphi}(2^{-j}\sqrt{\mathrm{H}})f$
where $\widetilde{\varphi}\in C_{0}^{\infty}(\R\setminus\\{0\\})$ takes values
in $[0,1]$ such that $\widetilde{\varphi}\varphi=\varphi$. In view of the
condition $f=\varphi(2^{-k}\sqrt{\mathrm{H}})f$, then
$\widetilde{\varphi}(2^{-j}\sqrt{\mathrm{H}})f$ vanishes if $|j-k|\gg 1$.
Hence we obtain
$\|U(t)f\|_{L^{q}_{t}(\R:L^{r}(M^{\circ}))}\lesssim
2^{k[n(\frac{1}{2}-\frac{1}{r})-\frac{1}{q}]}\|f\|_{L^{2}(M^{\circ})},$
which implies (4.7).
### 4.3. Inhomogeneous Strichartz estimates
In this subsection, we prove the inhomogeneous Strichartz estimates including
the endpoint $q=2$ for $n\geq 4$. Let
$U(t)=e^{it\sqrt{\mathrm{H}}}:L^{2}\rightarrow L^{2}$. We have already proved
that
(4.8) $\|U(t)u_{0}\|_{L^{q}_{t}L^{r}_{z}}\lesssim\|u_{0}\|_{\dot{H}^{s}}$
holds for all $(q,r,s)$ satisfying (1.4) and (1.5). For $s\in\R$ and $(q,r)$
satisfying (1.4) and (1.5), we define the operator $T_{s}$ by
(4.9) $\begin{split}T_{s}:L^{2}_{z}&\rightarrow L^{q}_{t}L^{r}_{z},\quad
f\mapsto\mathrm{H}^{-\frac{s}{2}}e^{it\sqrt{\mathrm{H}}}f.\end{split}$
Then we have by duality
(4.10)
$\begin{split}T^{*}_{1-s}:L^{\tilde{q}^{\prime}}_{t}L^{\tilde{r}^{\prime}}_{z}\rightarrow
L^{2},\quad
F(\tau,z)&\mapsto\int_{\R}\mathrm{H}^{\frac{s-1}{2}}e^{-i\tau\sqrt{\mathrm{H}}}F(\tau)d\tau,\end{split}$
where $1-s=n(\frac{1}{2}-\frac{1}{\tilde{r}})-\frac{1}{\tilde{q}}$. Therefore
we obtain
$\Big{\|}\int_{\R}U(t)U^{*}(\tau)\mathrm{H}^{-\frac{1}{2}}F(\tau)d\tau\Big{\|}_{L^{q}_{t}L^{r}_{z}}=\big{\|}T_{s}T^{*}_{1-s}F\big{\|}_{L^{q}_{t}L^{r}_{z}}\lesssim\|F\|_{L^{\tilde{q}^{\prime}}_{t}L^{\tilde{r}^{\prime}}_{z}}.$
Since $s=n(\frac{1}{2}-\frac{1}{r})-\frac{1}{q}$ and
$1-s=n(\frac{1}{2}-\frac{1}{\tilde{r}})-\frac{1}{\tilde{q}}$, thus
$(q,r),(\tilde{q},\tilde{r})$ satisfy (1.5). By the Christ-Kiselev lemma [8],
we thus obtain for $q>\tilde{q}^{\prime}$,
(4.11)
$\begin{split}\Big{\|}\int_{\tau<t}\frac{\sin{(t-\tau)\sqrt{\mathrm{H}}}}{\sqrt{\mathrm{H}}}F(\tau)d\tau\Big{\|}_{L^{q}_{t}L^{r}_{z}}\lesssim\|F\|_{L^{\tilde{q}^{\prime}}_{t}{L}^{\tilde{r}^{\prime}}_{z}}.\end{split}$
Notice that for all $(q,r),(\tilde{q},\tilde{r})$ satisfy (1.4) and (1.5), we
must have $q>\tilde{q}^{\prime}$. Therefore we have proved all inhomogeneous
Strichartz estimates including the endpoint $q=2$.
## 5\. Wellposedness and scattering
In this section, we prove Theorem 1.3. We prove the result by a contraction
mapping argument. The key point is the application of Strichartz estimates.
Let $q_{0}=(n+1)(p-1)/2$, $q_{1}=2(n+1)/(n-1)$ and $\alpha=s_{c}-\frac{1}{2}$.
For any small constant $\epsilon>0$ such that $2\epsilon<\epsilon(p)$ given by
(1.8), there exists $T>0$ such that
(5.1) $\begin{split}X:=\Big{\\{}u:~{}&u\in C_{t}(\dot{H}^{s_{c}})\cap
L^{q_{0}}([0,T];L^{q_{0}}(M^{\circ}))\cap
L^{q_{1}}([0,T];\dot{H}^{\alpha}_{q_{1}}(M^{\circ})),\\\
&\|u\|_{L^{q_{0}}([0,T];L^{q_{0}}(M^{\circ}))}+\|u\|_{L^{q_{1}}([0,T];\dot{H}^{\alpha}_{q_{1}}(M^{\circ}))}\leq
C\epsilon\Big{\\}}.\end{split}$
Consider the solution map $\Phi$ defined by
$\begin{split}\Phi(u)&=\cos(t\sqrt{\mathrm{H}})u_{0}(z)+\frac{\sin(t\sqrt{\mathrm{H}})}{\sqrt{\mathrm{H}}}u_{1}(z)+\int_{0}^{t}\frac{\sin\big{(}(t-s)\sqrt{\mathrm{H}}\big{)}}{\sqrt{\mathrm{H}}}F(u(s,z))\mathrm{d}s\\\
&=:u_{\text{hom}}+u_{\text{inh}},\end{split}$
where $F(u)=\gamma|u|^{p-1}u$. We claim the map $\Phi:X\rightarrow X$ is
contracting. Indeed, by Theorem 1.1, we obtain
(5.2) $\begin{split}\|u_{\text{hom}}\|_{C_{t}(\dot{H}^{s_{c}})\cap
L^{q_{0}}(\R;L^{q_{0}}(M^{\circ}))\cap
L^{q_{1}}(\R;\dot{H}^{\alpha}_{q_{1}}(M^{\circ}))}\leq
C\big{(}\|u_{0}\|_{\dot{H}^{s_{c}}}+\|u_{1}\|_{\dot{H}^{s_{c}-1}}\big{)}.\end{split}$
Hence we must have
(5.3)
$\begin{split}\|u_{\text{hom}}\|_{L^{q_{0}}([0,T];L^{q_{0}}(M^{\circ}))\cap
L^{q_{1}}([0,T];\dot{H}^{\alpha}_{q_{1}}(M^{\circ}))}\leq\frac{1}{2}C\epsilon\end{split}$
for $T=\infty$ if the initial data has small norm $\epsilon(p)$, or, if not,
this inequality will be satisfied for some $T>0$ by the dominated convergence
theorem. We first note that the Sobolev embedding
$L^{q_{0}}_{t}\dot{H}^{\alpha}_{r_{0}}\hookrightarrow L_{t,z}^{q_{0}}$ where
$r_{0}=2n(n+1)(p-1)/[(n^{2}-1)(p-1)-4]$. Under the condition
$p\in[p_{\mathrm{conf}},1+\frac{4}{n-2}]$, it is easy to check that the pairs
$(q_{0},r_{0}),(q_{1},q_{1})$ satisfy (1.4) and (1.5) with $s=1/2$. Applying
Theorem 1.1 with $\tilde{q}^{\prime}=\tilde{r}^{\prime}=\frac{2(n+1)}{n+3}$,
one has
(5.4) $\begin{split}\|u_{\text{inh}}\|_{C_{t}(\dot{H}^{s_{c}})\cap
L^{q_{0}}([0,T];L^{q_{0}}(M^{\circ}))\cap
L^{q_{1}}([0,T];\dot{H}^{\alpha}_{q_{1}}(M^{\circ}))}\leq
C\|F(u)\|_{L^{\tilde{q}^{\prime}}_{t}\dot{H}^{\alpha}_{\tilde{r}^{\prime}}}.\end{split}$
By the assumption on $p$, we have $0\leq\alpha\leq 1$. By using the fraction
Liebniz rule for Sobolev spaces on the asymptotically conic manifold [9,
Theorem 27], we have
(5.5)
$\begin{split}\|F(u)\|_{L^{\tilde{q}^{\prime}}_{t}\dot{H}^{\alpha}_{\tilde{r}^{\prime}}}\leq
C\|u\|^{p-1}_{L^{q_{0}}_{t,z}}\|u\|_{L^{q_{1}}_{t}\dot{H}^{\alpha}_{q_{1}}}\leq
C^{2}(C\epsilon)^{p-1}\epsilon\leq\frac{C\epsilon}{2}.\end{split}$
A similar argument as above leads to
(5.6)
$\begin{split}&\|\Phi(u_{1})-\Phi(u_{2})\|_{L^{q_{1}}([0,T];\dot{H}^{\alpha}_{q_{1}}(M^{\circ}))\cap
L^{q_{0}}([0,T];L^{q_{0}}(M^{\circ}))}\\\ &\leq
C\|F(u_{1})-F(u_{2})\|_{L^{\tilde{q}^{\prime}}_{t}\dot{H}^{\alpha}_{\tilde{r}^{\prime}}}\\\
&\leq
C^{2}(C\epsilon)^{p-1}\|u_{1}-u_{2}\|_{L^{q_{1}}([0,T];\dot{H}^{\alpha}_{q_{1}}(M^{\circ}))\cap
L^{q_{0}}([0,T];L^{q_{0}}(M^{\circ}))}\\\
&\leq\frac{1}{2}\|u_{1}-u_{2}\|_{L^{q_{1}}([0,T];\dot{H}^{\alpha}_{q_{1}}(M^{\circ}))\cap
L^{q_{0}}([0,T];L^{q_{0}}(M^{\circ}))}.\end{split}$
Therefore the solution map $\Phi$ is a contraction map on $X$ under the metric
$d(u_{1},u_{2})=\|u_{1}-u_{2}\|_{{L^{q_{1}}([0,T];\dot{H}^{\alpha}_{q_{1}}(M^{\circ}))}\cap
L^{q_{0}}([0,T];L^{q_{0}}(M^{\circ}))}$. The standard contraction argument
completes the proof of Theorem 1.3.
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* [28] E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton University Press, Princeton (1970).
* [29] C. D. Sogge, Lectures on Nonlinear Wave Equations, International Press, Cambridge, MA, 1995.
* [30] R. Strichartz, Restrictions of Fourier transforms to quadratic surfaces and decay of solutions of wave equations, Duke. Math. J., 44(1977), 705-714.
* [31] G. Staffilani, D. Tataru, Strichartz estimates for a Schrödinger operator with nonsmooth coefficients, Comm. in PDE, 27(2002), 1337-1372.
* [32] D. Tataru, Strichartz estimates for second order hyperbolic operators with nonsmooth coefficients III, J. Amer. Math. Soc., 15(2002), 419-442.
* [33] M. Zworski, Semiclassical Analysis, Graduate Studies in Mathematics 138, AMS 2012.
|
arxiv-papers
| 2013-10-17T02:27:34 |
2024-09-04T02:49:52.481345
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Junyong Zhang",
"submitter": "Junyong Zhang",
"url": "https://arxiv.org/abs/1310.4564"
}
|
1310.4637
|
# Higher-Order Daehee numbers and polynomials
Dae San Kim and Taekyun Kim
###### Abstract.
Recently, Daehee numbers and polynomials are introduced by the authors. In
this paper, we consider the Daehee numbers and polynomials of order
$k\left(\in\mathbb{N}\right)$ and give some relation between Daehee
polynomials of order $k$$\left(\in\mathbb{N}\right)$ and special polynomials.
## 1\. Introduction
For $\alpha\in\mathbb{N}$, as is well known, the Bernoulli polynomials of
order $\alpha$ are defined by the generating function to be
(1)
$\left(\frac{t}{e^{t}-1}\right)^{\alpha}e^{xt}=\sum_{n=0}^{\infty}B_{n}^{\left(\alpha\right)}\left(x\right)\frac{t^{n}}{n!},$
(see [1-14]).
When $x=0$,
$B_{n}^{\left(\alpha\right)}=B_{n}^{\left(\alpha\right)}\left(0\right)$ are
the Bernoulli numbers of order $\alpha$. In [2013DSKIM, MR2390695, MR2479746],
the Daehee polynomials are defined by the generating function to be
(2)
$\left(\frac{\log\left(1+t\right)}{t}\right)\left(1+t\right)^{x}=\sum_{n=0}^{\infty}D_{n}\left(x\right)\frac{t^{n}}{n!}.$
When $x=0$, $D_{n}=D_{n}\left(0\right)$ are called the Daehee numbers.
Throughout this paper, $\mathbb{Z}_{p}$, $\mathbb{Q}_{p}$ and $\mathbb{C}_{p}$
will denote the ring of $p$-adic integers, the field of $p$-adic numbers and
the completion of algebraic closure of $\mathbb{Q}_{p}$. The $p$-adic norm
$\left|\cdot\right|_{p}$ is normalized as $\left|p\right|_{p}=\frac{1}{p}$.
Let $\textnormal{UD}\left(\mathbb{Z}_{p}\right)$ be the space of uniformly
differentiable functions on $\mathbb{Z}_{p}$. For
$f\in\textnormal{UD}\left(\mathbb{Z}_{p}\right)$, the $p$-adic invariant
integral on $\mathbb{Z}_{p}$ is defined by
(3)
$I\left(f\right)=\int_{\mathbb{Z}_{p}}f\left(x\right)d\mu\left(x\right)=\lim_{n\rightarrow\infty}\frac{1}{p^{n}}\sum_{x=0}^{p^{n}-1}f\left(x\right),$
(see [MR2845943]).
Let $f_{1}\left(x\right)=f\left(x+1\right).$ Then, by (3), we get
(4) $I\left(f_{1}\right)-I\left(f\right)=f^{\prime}\left(0\right),\textrm{
where
}f^{\prime}\left(0\right)=\left.\frac{df\left(x\right)}{dx}\right|_{x=0}.$
The signed Stirling numbers of the first kind $S_{1}(n,l)$ are defined by
$\displaystyle\left(x\right)_{n}$ $\displaystyle=$ $\displaystyle
x\left(x-1\right)\cdots\left(x-n+1\right)$ $\displaystyle=$
$\displaystyle\sum_{l=0}^{\infty}S_{1}\left(n,l\right)x^{l},$
(see [MR2508979, MR2597988, 2013DSKIM]).
From (1), we note that
$\displaystyle x^{\left(n\right)}$ $\displaystyle=$ $\displaystyle
x\left(x+1\right)\cdots\left(x+n-1\right)=\left(-1\right)^{n}\left(-x\right)_{n}$
$\displaystyle=$
$\displaystyle\sum_{l=0}^{n}\left(-1\right)^{n-l}S_{1}\left(n,l\right)x^{l},$
(see [2013DSKIM, MR2931605, MR2479746]).
The Stirling numbers of the second kind $S_{2}(l,n)$ are defined by the
generating function to be
$\displaystyle\left(e^{t}-1\right)^{n}$ $\displaystyle=$ $\displaystyle
n!\sum_{l=n}^{\infty}S_{2}\left(l,n\right)\frac{t^{l}}{l!}$ $\displaystyle=$
$\displaystyle\sum_{l=0}^{\infty}\frac{n!}{\left(l+n\right)!}S_{2}\left(l+n,n\right)t^{l+n}.$
In this paper, we study the higher-order Daehee numbers and polynomials and
give some relations between Daehee polynomials and special polynomials.
## 2\. Higher-order Daehee polynomials
In this section, we assume that $t\in\mathbb{C}_{p}$ with
$\left|t\right|_{p}<p^{\frac{-1}{p-1}}$.
For $k\in\mathbb{N}$, let us consider the Daehee numbers of the first kind of
order $k$ :
(7)
$D_{n}^{\left(k\right)}=\underset{k-\textrm{times}}{\underbrace{\int_{\mathbb{Z}_{p}}\cdots\int_{\mathbb{Z}_{p}}}}\left(x_{1}+x_{2}+\cdots+x_{k}\right)_{n}d\mu\left(x_{1}\right)\cdots
d\mu\left(x_{k}\right),$
where $n\in\mathbb{Z}_{\geq 0}$.
From (7), we can derive the generating function of $D_{n}^{\left(k\right)}$ as
follows :
(8) $\displaystyle\sum_{n=0}^{\infty}D_{n}^{\left(k\right)}\frac{t^{n}}{n!}$
$\displaystyle=$
$\displaystyle\int_{\mathbb{Z}_{p}}\cdots\int_{\mathbb{Z}_{p}}\sum_{n=0}^{\infty}\dbinom{x_{1}+\cdots+x_{k}}{n}t^{n}d\mu\left(x_{1}\right)\cdots
d\mu\left(x_{k}\right)$ $\displaystyle=$
$\displaystyle\int_{\mathbb{Z}_{p}}\cdots\int_{\mathbb{Z}_{p}}\left(1+t\right)^{x_{1}+\cdots+x_{k}}d\mu\left(x_{1}\right)\cdots
d\mu\left(x_{k}\right).$
By (4), we easily see that
(9)
$\int_{\mathbb{Z}_{p}}\left(1+t\right)^{x}d\mu\left(x\right)=\frac{\log\left(1+t\right)}{t}.$
Thus, by (8) and (9), we get
(10)
$\sum_{n=0}^{\infty}D_{n}^{\left(k\right)}\frac{t^{n}}{n!}=\left(\frac{\log\left(1+t\right)}{t}\right)^{k}.$
Now, we observe that
$\displaystyle\left(\frac{\log\left(1+t\right)}{t}\right)^{k}$
$\displaystyle=$
$\displaystyle\frac{k!}{t^{k}}\sum_{l=k}^{\infty}S_{1}\left(t,k\right)\frac{t^{l}}{l!}$
$\displaystyle=$
$\displaystyle\sum_{n=0}^{\infty}S_{1}\left(n+k,k\right)\frac{k!}{\left(n+k\right)!}t^{n}$
$\displaystyle=$
$\displaystyle\sum_{n=0}^{\infty}\frac{S_{1}\left(n+k,k\right)}{\tbinom{n+k}{k}}\frac{t^{n}}{n!}.$
Therefore, by (10) and (2), we obtain the following theorem.
###### Theorem 1.
For $n\in\mathbb{Z}_{\geq 0}$, $k\in\mathbb{N}$, we have
$D_{n}^{\left(k\right)}=\frac{S_{1}\left(n+k,k\right)}{\tbinom{n+k}{k}}.$
It is easy to show that
(12)
$\left(\frac{\log\left(1+t\right)}{t}\right)^{k}=\sum_{n=0}^{\infty}B_{n}^{\left(n+k+1\right)}\left(1\right)\frac{t^{n}}{n!}.$
Threfore, we obtain the following corollary.
###### Corollary 2.
For $n\in\mathbb{Z}_{\geq 0}$, $k\in\mathbb{N}$, we have
$D_{n}^{\left(k\right)}=\frac{S_{1}\left(n+k,k\right)}{\tbinom{n+k}{k}}=B_{n}^{\left(n+k+1\right)}\left(1\right).$
From (7), we note that
$\displaystyle D_{n}^{\left(k\right)}$ $\displaystyle=$
$\displaystyle\int_{\mathbb{Z}_{p}}\cdots\int_{\mathbb{Z}_{p}}\left(x_{1}+\cdots+x_{k}\right)_{n}d\mu\left(x_{1}\right)\cdots
d\mu\left(x_{k}\right)$ $\displaystyle=$
$\displaystyle\sum_{l=0}^{n}S_{1}\left(n,l\right)\int_{\mathbb{Z}_{p}}\cdots\int_{\mathbb{Z}_{p}}\left(x_{1}+\cdots+x_{k}\right)^{l}d\mu\left(x_{1}\right)\cdots
d\mu\left(x_{k}\right)$ $\displaystyle=$
$\displaystyle\sum_{l=0}^{n}S_{1}\left(n,l\right)B_{l}^{\left(k\right)}.$
Therefore, by (2), we obtain the following theorem.
###### Theorem 3.
For $n\in\mathbb{Z}_{\geq 0}$, $k\in\mathbb{N}$, we have
$\displaystyle D_{n}^{\left(k\right)}$ $\displaystyle=$
$\displaystyle\sum_{l_{1}+\cdots+l_{k}=n}\dbinom{n}{l_{1},\cdots,l_{k}}D_{l_{1}}\cdots
D_{l_{k}}$ $\displaystyle=$
$\displaystyle\sum_{l=0}^{n}S_{1}\left(n,l\right)B_{l}^{\left(k\right)}.$
From (10), we can derive
(14)
$\sum_{n=0}^{\infty}D_{n}^{\left(k\right)}\frac{\left(e^{t}-1\right)^{n}}{n!}=\left(\frac{t}{e^{t}-1}\right)^{k}=\sum_{n=0}^{\infty}B_{n}^{\left(k\right)}\frac{t^{n}}{n!},$
and
(15)
$\sum_{n=0}^{\infty}D_{n}^{\left(k\right)}\frac{\left(e^{t}-1\right)^{n}}{n!}=\sum_{m=0}^{\infty}\left(\sum_{n=0}^{m}D_{n}^{\left(k\right)}S_{2}\left(n,m\right)\right)\frac{t^{n}}{m!}.$
Therefore, by (14) and (15), we obtain the following theorem.
###### Theorem 4.
For $m\in\mathbb{Z}_{\geq 0}$, $k\in\mathbb{N}$, we have
$B_{m}^{\left(k\right)}=\sum_{n=0}^{m}D_{n}^{\left(k\right)}S_{2}\left(m,n\right).$
Now, we consider the higher-order Daehee polynomials as follows :
(16) $\displaystyle D_{n}^{\left(k\right)}\left(x\right)$ $\displaystyle=$
$\displaystyle\int_{\mathbb{Z}_{p}}\cdots\int_{\mathbb{Z}_{p}}\left(x_{1}+\cdots+x_{k}+x\right)_{n}d\mu\left(x_{1}\right)\cdots
d\mu\left(x_{k}\right).$
Thus, by (16), we get
(17) $\displaystyle D_{n}^{\left(k\right)}\left(x\right)$ $\displaystyle=$
$\displaystyle\sum_{l=0}^{n}S_{1}\left(n,l\right)\int_{\mathbb{Z}_{p}}\cdots\int_{\mathbb{Z}_{p}}\left(x_{1}+\cdots+x_{k}+x\right)^{l}d\mu\left(x_{1}\right)\cdots
d\mu\left(x_{k}\right)$ $\displaystyle=$
$\displaystyle\sum_{l=0}^{n}S_{1}\left(n,l\right)B_{l}^{\left(k\right)}\left(x\right).$
Therefore, by (17), we obtain the following theorem.
###### Theorem 5.
For $n\in\mathbb{Z}_{\geq 0}$, $k\in\mathbb{N}$, we have
$D_{n}^{\left(k\right)}\left(x\right)=\sum_{l=0}^{n}S_{1}\left(n,l\right)B_{l}^{\left(k\right)}\left(x\right).$
From (16), we derive the generating function of
$D_{n}^{\left(k\right)}\left(x\right)$:
(18)
$\displaystyle\sum_{n=0}^{\infty}D_{n}^{(k)}\left(x\right)\frac{t^{n}}{n!}$
$\displaystyle=$
$\displaystyle\int_{\mathbb{Z}_{p}}\cdots\int_{\mathbb{Z}_{p}}\sum_{n=0}^{\infty}\dbinom{x_{1}+\cdots+x_{k}+x}{n}t^{n}d\mu\left(x_{1}\right)\cdots
d\mu\left(x_{k}\right)$ $\displaystyle=$
$\displaystyle\int_{\mathbb{Z}_{p}}\cdots\int_{\mathbb{Z}_{p}}\left(1+t\right)^{x_{1}+\cdots+x_{k}+x}d\mu\left(x_{1}\right)\cdots
d\mu\left(x_{k}\right)$ $\displaystyle=$
$\displaystyle\left(\frac{\log\left(1+t\right)}{t}\right)^{k}\left(1+t\right)^{x}.$
It is easy to show that
(19)
$\left(\frac{\log\left(1+t\right)}{t}\right)^{k}\left(1+t\right)^{x}=\sum_{n=0}^{\infty}B_{n}^{\left(n+k+1\right)}\left(x+1\right)\frac{t^{n}}{n!}.$
Therefore, by (18) and (19), we obtain the following theorem.
###### Theorem 6.
For $n\in\mathbb{Z}_{\geq 0}$, $k\in\mathbb{N}$,
$\displaystyle D_{n}^{\left(k\right)}\left(x\right)$ $\displaystyle=$
$\displaystyle B_{n}^{\left(n+k+1\right)}\left(x+1\right)$ $\displaystyle=$
$\displaystyle\sum_{l=0}^{n}\dbinom{n}{l}B_{l}^{\left(n+k+1\right)}\left(x+1\right)^{n-l}.$
In (18), we note that
(20)
$\sum_{n=0}^{\infty}D_{n}^{\left(k\right)}\left(x\right)\frac{\left(e^{t}-1\right)^{n}}{n!}=\sum_{m=0}^{\infty}\left(\sum_{n=0}^{m}S_{2}\left(n,m\right)D_{n}^{\left(k\right)}\left(x\right)\right)\frac{t^{m}}{m!}$
and
$\displaystyle\sum_{n=0}^{\infty}D_{n}^{\left(k\right)}\left(x\right)\frac{\left(e^{t}-1\right)^{n}}{n!}$
$\displaystyle=$ $\displaystyle\left(\frac{t}{e^{t}-1}\right)^{k}e^{xt}$
$\displaystyle=$
$\displaystyle\sum_{m=0}^{\infty}B_{m}^{\left(k\right)}\left(x\right)\frac{t^{m}}{m!}.$
Therefore, by (20) and (2), we obtain the following theorem.
###### Theorem 7.
For $m\in\mathbb{Z}_{\geq 0}$, $k\in\mathbb{N}$, we have
$B_{m}^{\left(k\right)}\left(x\right)=\sum_{n=0}^{m}S_{2}\left(m,n\right)D_{n}^{\left(k\right)}\left(x\right).$
Now, we define Daehee numbers of the second kind of order
$k$$\left(\in\mathbb{N}\right)$ :
(22) $\displaystyle\widehat{D}_{n}^{\left(k\right)}$ $\displaystyle=$
$\displaystyle\left(-1\right)^{n}\int_{\mathbb{Z}_{p}}\cdots\int_{\mathbb{Z}_{p}}\left(-x_{1}-x_{2}-\cdots-
x_{k}\right)_{n}d\mu\left(x_{1}\right)\cdots d\mu\left(x_{k}\right)$
$\displaystyle=$
$\displaystyle\left(-1\right)^{n}\sum_{l=0}^{n}\left(-1\right)^{n-l}S_{1}\left(n,l\right)B_{l}^{\left(k\right)}=\sum_{l=0}^{n}\begin{bmatrix}n\\\
l\end{bmatrix}B_{l}^{\left(k\right)},$
where $\begin{bmatrix}n\\\
l\end{bmatrix}=\left(-1\right)^{n-l}S_{1}$$\left(n,l\right)$.
Thus, by (22), we get
(23) $\displaystyle\widehat{D}_{n}^{\left(k\right)}$ $\displaystyle=$
$\displaystyle\left(-1\right)^{n}\int_{\mathbb{Z}_{p}}\cdots\int_{\mathbb{Z}_{p}}\left(-x_{1}-x_{2}-\cdots-
x_{k}\right)_{n}d\mu\left(x_{1}\right)\cdots d\mu\left(x_{k}\right)$
$\displaystyle=$
$\displaystyle\left(-1\right)^{n}\sum_{l=0}^{n}S_{1}\left(n,l\right)\left(-1\right)^{l}\int_{\mathbb{Z}_{p}}\cdots\int_{\mathbb{Z}_{p}}\left(x_{1}+x_{2}+\cdots+x_{k}\right)^{l}d\mu\left(x_{1}\right)\cdots
d\mu\left(x_{k}\right)$ $\displaystyle=$
$\displaystyle\sum_{l=0}^{n}\left(-1\right)^{n-l}S_{1}\left(n,l\right)B_{l}^{\left(k\right)}=\sum_{l=0}^{n}\begin{bmatrix}n\\\
l\end{bmatrix}B_{l}^{\left(k\right)},$
where $\begin{bmatrix}n\\\
l\end{bmatrix}=\left(-1\right)^{n-l}S_{1}$$\left(n,l\right)$.
Therefore, by (23), we obtain the following theorem.
###### Theorem 8.
For $n\in\mathbb{Z}_{\geq 0}$, $k\in\mathbb{N}$, we have
$\widehat{D}_{n}^{\left(k\right)}=\sum_{l=0}^{n}\begin{bmatrix}n\\\
l\end{bmatrix}B_{l}^{\left(k\right)}.$
From (22), we derive the generating function of
$\widehat{D}_{n}^{\left(k\right)}$:
(24)
$\displaystyle\sum_{n=0}^{\infty}\widehat{D}_{n}^{\left(k\right)}\frac{t^{n}}{n!}$
$\displaystyle=$
$\displaystyle\int_{\mathbb{Z}_{p}}\cdots\int_{\mathbb{Z}_{p}}\sum_{n=0}^{\infty}\dbinom{x_{1}+\cdots+x_{k}+n-1}{n}t^{n}d\mu\left(x_{1}\right)\cdots
d\mu\left(x_{k}\right)$ $\displaystyle=$
$\displaystyle\int_{\mathbb{Z}_{p}}\cdots\int_{\mathbb{Z}_{p}}\left(1-t\right)^{-x_{1}-\cdots-
x_{k}}d\mu\left(x_{1}\right)\cdots d\mu\left(x_{k}\right)$ $\displaystyle=$
$\displaystyle\left(\frac{\left(1-t\right)\log\left(1-t\right)}{-t}\right)^{k}.$
By (24), we get
$\displaystyle\sum_{n=0}^{\infty}\widehat{D}_{n}^{\left(k\right)}\frac{\left(1-e^{-t}\right)^{n}}{n!}$
$\displaystyle=$
$\displaystyle\left(\frac{e^{-t}\left(-t\right)}{e^{-t}-1}\right)^{k}=\left(\frac{t}{e^{t}-1}\right)^{k}$
$\displaystyle=$
$\displaystyle\sum_{m=0}^{\infty}B_{m}^{\left(k\right)}\frac{t^{m}}{m!},$
and
(26)
$\sum_{n=0}^{\infty}\widehat{D}_{n}^{\left(k\right)}\frac{\left(1-e^{-t}\right)^{n}}{n!}=\sum_{m=0}^{\infty}\left(\sum_{n=0}^{m}\widehat{D}_{n}^{\left(k\right)}\left(-1\right)^{m-n}S_{2}\left(m,n\right)\right)\frac{t^{m}}{m!}.$
Thererfore, by (2) and (26), we obtain the following theorem.
###### Theorem 9.
For $m\in\mathbb{Z}_{\geq 0}$, $k\in\mathbb{N}$, we have
$B_{m}^{\left(k\right)}=\sum_{n=0}^{m}\widehat{D}_{n}^{\left(k\right)}\left(-1\right)^{n-m}S_{2}\left(m,n\right).$
Now, we consider the higher-order Daehee polynomials of the second kind :
(27)
$\widehat{D}_{n}^{\left(k\right)}\left(x\right)=\int_{\mathbb{Z}_{p}}\cdots\int_{\mathbb{Z}_{p}}\left(x_{1}+x_{2}+\cdots+x_{k}-x\right)^{(n)}d\mu\left(x_{1}\right)\cdots
d\mu\left(x_{k}\right).$
Thus, by (27), we get
(28) $\displaystyle\widehat{D}_{n}^{\left(k\right)}\left(x\right)$
$\displaystyle=$
$\displaystyle\left(-1\right)^{n}\int_{\mathbb{Z}_{p}}\cdots\int_{\mathbb{Z}_{p}}\left(-x_{1}-x_{2}-\cdots-
x_{k}+x\right)_{n}d\mu\left(x_{1}\right)\cdots d\mu\left(x_{k}\right)$
$\displaystyle=$
$\displaystyle\left(-1\right)^{n}\sum_{l=0}^{n}S_{1}\left(n,l\right)\int_{\mathbb{Z}_{p}}\cdots\int_{\mathbb{Z}_{p}}\left(-x_{1}-x_{2}-\cdots-
x_{k}+x\right)^{l}d\mu\left(x_{1}\right)\cdots d\mu\left(x_{k}\right)$
$\displaystyle=$
$\displaystyle\left(-1\right)^{n}\sum_{l=0}^{n}S_{1}\left(n,l\right)\sum_{m=0}^{l}\dbinom{l}{m}x^{l-m}\int_{\mathbb{Z}_{p}}\cdots\int_{\mathbb{Z}_{p}}\left(-x_{1}-x_{2}-\cdots-
x_{k}\right)^{m}d\mu\left(x_{1}\right)\cdots d\mu\left(x_{k}\right)$
$\displaystyle=$
$\displaystyle\left(-1\right)^{n}\sum_{l=0}^{n}S_{1}\left(n,l\right)\sum_{m=0}^{l}\dbinom{l}{m}\left(-1\right)^{m}x^{l-m}B_{m}^{\left(k\right)}$
$\displaystyle=$
$\displaystyle\sum_{l=0}^{n}\left(-1\right)^{n-l}S_{1}\left(n,l\right)B_{l}^{\left(k\right)}\left(-x\right).$
Thus, by (28), we get
(29)
$\widehat{D}_{n}^{\left(k\right)}\left(x\right)=\sum_{l=0}^{n}\left(-1\right)^{n-l}S_{1}\left(n,l\right)B_{l}^{\left(k\right)}\left(-x\right).$
Let us consider the generating function of
$D_{n}^{\left(k\right)}\left(x\right)$ as follows :
(30)
$\displaystyle\sum_{n=0}^{\infty}\widehat{D}_{n}^{\left(k\right)}\left(x\right)\frac{t^{n}}{n!}$
$\displaystyle=$
$\displaystyle\int_{\mathbb{Z}_{p}}\cdots\int_{\mathbb{Z}_{p}}\sum_{n=0}^{\infty}\dbinom{x_{1}+\cdots+x_{k}-x+n-1}{n}t^{n}d\mu\left(x_{1}\right)\cdots
d\mu\left(x_{k}\right)$ $\displaystyle=$
$\displaystyle\int_{\mathbb{Z}_{p}}\cdots\int_{\mathbb{Z}_{p}}\left(1-t\right)^{-x_{1}-\cdots-
x_{k}+x}d\mu\left(x_{1}\right)\cdots d\mu\left(x_{k}\right)$ $\displaystyle=$
$\displaystyle\left(\frac{\left(1-t\right)\log\left(1-t\right)}{-t}\right)^{k}\left(1-t\right)^{x}.$
From (30), we have
(31)
$\displaystyle\sum_{n=0}^{\infty}\widehat{D}_{n}^{\left(k\right)}\left(x\right)\left(-1\right)^{n}\frac{t^{n}}{n!}$
$\displaystyle=$
$\displaystyle\left(\frac{\log\left(1+t\right)}{t}\right)^{k}\left(1+t\right)^{x+k}$
$\displaystyle=$
$\displaystyle\sum_{n=0}^{\infty}B_{n}^{\left(n+k+1\right)}\left(x+k+1\right)\frac{t^{n}}{n!}.$
Therefore, by (31), we obtain the following theorem.
###### Theorem 10.
For $n\in\mathbb{Z}_{\geq 0}$, $k\in\mathbb{N}$, we have
$\left(-1\right)^{n}\widehat{D}_{n}^{\left(k\right)}\left(x\right)=B_{n}^{\left(n+k+1\right)}\left(x+k+1\right).$
By (30), we get
$\displaystyle\sum_{n=0}^{\infty}\widehat{D}_{n}^{\left(k\right)}\left(x\right)\frac{\left(1-e^{-t}\right)^{n}}{n!}$
$\displaystyle=$ $\displaystyle e^{-tx}\left(\frac{t}{e^{t}-1}\right)^{k}$
$\displaystyle=$
$\displaystyle\sum_{m=0}^{\infty}B_{m}^{\left(k\right)}\left(-x\right)\frac{t^{m}}{m!},$
and
(33)
$\displaystyle\sum_{n=0}^{\infty}\widehat{D}_{n}^{\left(k\right)}\left(x\right)\frac{1}{n!}\left(1-e^{-t}\right)^{n}$
$\displaystyle=$
$\displaystyle\sum_{m=0}^{\infty}\left(\sum_{n=0}^{m}\widehat{D}_{n}^{\left(k\right)}\left(x\right)\left(-1\right)^{m-n}S_{2}\left(m,n\right)\right)\frac{t^{m}}{m!}.$
Therefore, by (2) and (12), we obtain the following theorem.
###### Theorem 11.
For $m\in\mathbb{Z}_{\geq 0}$, $k\in\mathbb{N}$, we have
$B_{m}^{\left(k\right)}\left(-x\right)=\sum_{n=0}^{m}\widehat{D}_{n}^{\left(k\right)}\left(x\right)\left(-1\right)^{m-n}S_{2}\left(m,n\right).$
Now, we observe that
(34)
$\displaystyle\left(-1\right)^{n}\frac{D_{n}^{\left(k\right)}\left(x\right)}{n!}$
$\displaystyle=$
$\displaystyle\left(-1\right)^{n}\int_{\mathbb{Z}_{p}}\cdots\int_{\mathbb{Z}_{p}}\dbinom{x_{1}+\cdots+x_{k}+x}{n}d\mu\left(x_{1}\right)\cdots
d\mu\left(x_{k}\right)$ $\displaystyle=$
$\displaystyle\int_{\mathbb{Z}_{p}}\cdots\int_{\mathbb{Z}_{p}}\dbinom{-(x_{1}+\cdots+x_{k})-x+n-1}{n}d\mu\left(x_{1}\right)\cdots
d\mu\left(x_{k}\right)$ $\displaystyle=$
$\displaystyle\sum_{m=0}^{n}\dbinom{n-1}{n-m}\int_{\mathbb{Z}_{p}}\cdots\int_{\mathbb{Z}_{p}}\dbinom{-(x_{1}+\cdots+x_{k})-x}{m}d\mu\left(x_{1}\right)\cdots
d\mu\left(x_{k}\right)$ $\displaystyle=$
$\displaystyle\sum_{m=0}^{n}\frac{\tbinom{n-1}{n-m}}{m!}m!\int_{\mathbb{Z}_{p}}\cdots\int_{\mathbb{Z}_{p}}\dbinom{-(x_{1}+\cdots+x_{k})-x}{m}d\mu\left(x_{1}\right)\cdots
d\mu\left(x_{k}\right)$ $\displaystyle=$
$\displaystyle\sum_{m=1}^{n}\frac{\tbinom{n-1}{n-m}}{m!}\left(-1\right)^{m}\widehat{D}_{m}^{\left(k\right)}\left(-x\right).$
Therefore, by (34), we obtain the following theorem.
###### Theorem 12.
For $n\in\mathbb{Z}_{\geq 0}$, $k\in\mathbb{N}$, we have
$\left(-1\right)^{n}\frac{D_{n}^{\left(k\right)}\left(x\right)}{n!}=\sum_{m=1}^{n}\frac{\tbinom{n-1}{n-m}}{m!}\left(-1\right)^{m}\widehat{D}_{m}^{\left(k\right)}\left(-x\right).$
By the same method as Theorem 12, we get
(35) $\displaystyle\frac{\widehat{D}_{n}^{\left(k\right)}\left(x\right)}{n!}$
$\displaystyle=$
$\displaystyle\int_{\mathbb{Z}_{p}}\cdots\int_{\mathbb{Z}_{p}}\dbinom{x_{1}+\cdots+x_{k}-x+n-1}{n}d\mu\left(x_{1}\right)\cdots
d\mu\left(x_{k}\right)$ $\displaystyle=$
$\displaystyle\sum_{m=0}^{n}\dbinom{n-1}{n-m}\int_{\mathbb{Z}_{p}}\cdots\int_{\mathbb{Z}_{p}}\dbinom{x_{1}+\cdots+x_{k}-x}{m}d\mu\left(x_{1}\right)\cdots
d\mu\left(x_{k}\right)$ $\displaystyle=$
$\displaystyle\sum_{m=0}^{n}\frac{\tbinom{n-1}{n-m}}{m!}m!\int_{\mathbb{Z}_{p}}\cdots\int_{\mathbb{Z}_{p}}\dbinom{x_{1}+\cdots+x_{k}-x}{m}d\mu\left(x_{1}\right)\cdots
d\mu\left(x_{k}\right)$ $\displaystyle=$
$\displaystyle\sum_{m=1}^{n}\frac{\tbinom{n-1}{n-m}}{m!}D_{m}^{\left(k\right)}\left(-x\right).$
Thus, by (35), we get
(36)
$\frac{\widehat{D}_{n}^{\left(k\right)}\left(x\right)}{n!}=\sum_{m=1}^{n}\frac{\tbinom{n-1}{n-m}}{m!}D_{m}^{\left(k\right)}\left(-x\right).$
Department of Mathematics, Sogang University, Seoul 121-742, Republic of Korea
_E-mail_ _address :_ [email protected]
Department of Mathematics, Kwangwoon University, Seoul 139-701, Republic of
Korea
_E-mail_ _address :_ [email protected]
|
arxiv-papers
| 2013-10-17T09:45:45 |
2024-09-04T02:49:52.493091
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Dae San Kim, Taekyun Kim",
"submitter": "Taekyun Kim",
"url": "https://arxiv.org/abs/1310.4637"
}
|
1310.4643
|
# Large time behavior of the on-diagonal heat kernel for minimal submanifolds
with polynomial volume growth
Vicent Gimeno Department of Mathematics-INIT, Universitat Jaume I, Castelló
de la Plana, Spain [email protected]
###### Abstract.
In this paper we provide a lower bound for the long time on-diagonal heat
kernel of minimal submanifolds in a Cartan-hadamard ambient manifold assuming
that the submanifold is of polynomial volume growth. In particular cases, that
lower bound is related with the number of ends of the submanifold.
###### Key words and phrases:
heat kernel and minimal submanifold and Cartan-Hadamard and volume growth and
number of ends
###### 1991 Mathematics Subject Classification:
35P15
Work partially supported by DGI grant MTM2010-21206-C02-02.
## 1\. Introduction
Let $\displaystyle M^{m}$ be a $\displaystyle m$-dimensional minimally
immersed submanifold into a simply connected ambient manifold $\displaystyle
N^{n}$ with sectional curvatures $\displaystyle K_{N}$ bounded from above by
$\displaystyle K_{N}\leq 0$. S. Markvorsen proved in [Mar86] -following
[CLY84]\- that the heat kernel $\displaystyle\mathcal{H}$ of $\displaystyle
M^{m}$ is bounded from above by the heat kernel
$\displaystyle\mathcal{H}^{m,0}$ of the Euclidean space
$\displaystyle\mathbb{R}^{m}$, namely:
(1.1) $\mathcal{H}(t,x,y)\leq\mathcal{H}^{m,0}(t,r_{x}(y))=\frac{1}{\left(4\pi
t\right)^{\frac{m}{2}}}e^{-\frac{\left(r_{x}(y)\right)^{2}}{4t}},$
being $\displaystyle r_{x}(y)$ the distance in $\displaystyle N$ from
$\displaystyle x$ to $\displaystyle y$. In particular for the on-diagonal heat
kernel $\displaystyle\mathcal{H}(t,x,x)$ of $\displaystyle M^{m}$ one can
state that
(1.2) $\left(4\pi t\right)^{\frac{m}{2}}\mathcal{H}(t,x,x)\leq 1.$
This paper deals with lower bounds to the on-diagonal heat kernel assuming
certain restriction on the volume growth. In order to define that appropriate
behavior on the growth of the extrinsic volume, recall that given a minimal
submanifold $\displaystyle M^{m}$ properly immersed in a Cartan-Hadamard
manifold $\displaystyle N$ with sectional curvatures $\displaystyle K_{N}$
bounded from above by $\displaystyle K_{N}\leq 0$ and denoting by
$\displaystyle\omega_{m}$ the volume of a radius one geodesic ball in
$\displaystyle\mathbb{R}^{m}$ and by $\displaystyle B_{R}^{N}(p)$ the geodesic
ball in $\displaystyle N$ of radius $\displaystyle R$ centered at
$\displaystyle p$, by the monotonicity formula (see for instance [MP12,
theorem 2.6.9] and [Pal99]) for any point $\displaystyle p\in M^{m}$ the
function
(1.3) $\mathcal{Q}(R)=\frac{\operatorname{Vol}(M^{m}\cap
B_{R}^{N}(p))}{\omega_{m}R^{m}},$
is a non decreasing function. Throughout this paper a complete minimal
submanifold properly immersed in a Cartan-hadamard ambient manifold is called
a minimal submanifold of _polynomial volume growth_ if there exists a constant
$\displaystyle\mathcal{E}$ depending on $\displaystyle M^{m}$ such that:
(1.4) $\lim_{R\to\infty}\mathcal{Q}(R)\leq\mathcal{E}<\infty.$
Under such volume growth behavior we can state the behavior of the long time
asymptotic for the on-diagonal heat kernel by the main theorem of this paper.
The main theorem makes use of the following constant $\displaystyle C_{m}$
depending only on the dimension $\displaystyle m$ of the submanifold
(1.5)
$C_{m}:=\frac{\Gamma\left(\frac{m}{2},2\left(\frac{m}{2}\Gamma\left(\frac{m}{2}\right)\right)^{\frac{2}{m}}\right)}{\Gamma(\frac{m}{2})},$
where $\displaystyle\Gamma(z)$ and $\displaystyle\Gamma(z_{1},z_{2})$ in the
above expression denote the gamma function and the incomplete gamma function
respectively, i.e,
$\displaystyle\Gamma(z):=$ $\displaystyle\int_{0}^{\infty}t^{z-1}e^{-t}dt.$
$\displaystyle\Gamma(z_{1},z_{2}):=$
$\displaystyle\int_{z_{2}}^{\infty}t^{z_{1}-1}e^{-t}dt.$
For minimal submanifolds with an extrinsic volume growth controlled by the
above constant $\displaystyle C_{m}$ we can state the main result of this
paper:
###### Main Theorem.
Let $\displaystyle M^{m}$ be a complete $\displaystyle m$-dimensional
submanifold properly immersed in a simply connected ambient manifold
$\displaystyle N$ with sectional curvatures $\displaystyle K_{N}$ bounded from
above by $\displaystyle K_{N}\leq 0$. Suppose that $\displaystyle M^{m}$ is of
polynomial volume growth, and that
(1.6) $\mathcal{E}<\frac{1}{C_{m}},$
Then, the heat kernel $\displaystyle\mathcal{H}$ of $\displaystyle M^{m}$
satisfies
(1.7)
$\frac{\left(1-\mathcal{E}C_{m}\right)^{2}}{\mathcal{E}}\leq\limsup_{t\to\infty}\left(4\pi
t\right)^{\frac{m}{2}}\mathcal{H}(t,x,x)\leq 1.$
Figure 1. The catenoid, the Costa surface and the Scherk singly periodic
surface are examples of minimal surfaces immersed in
$\displaystyle\mathbb{R}^{3}$ with polynomial volume growth which is
equivalent to quadratic area growth when the submanifold is a surface.
It is not hard to find examples of complete minimal submanifolds properly and
minimally immersed in a Cartan-Hadamard ambient manifold with polynomial
volume growth. Indeed, for a complete minimal surface embedded in
$\displaystyle\mathbb{R}^{3}$, by a well known result (see [Oss86, JM83] and
introduction in [GP13]), if the surface has finite total curvature then the
surface has polynomial volume growth (quadratic area growth) and the constant
$\displaystyle\mathcal{E}$ given in equation (1.4) is equal to the number of
ends of the surface. This is the case of the catenoid or the Costa surface
(with $\displaystyle\mathcal{E}=2$ for the catenoid and
$\displaystyle\mathcal{E}=3$ for the Costa surface). It is also known that
there exist other surfaces with quadratic area growth but without finite total
curvature and even without finite topological type. An example of that kind of
surface is the Scherk singly periodic surface (see introduction in [MW07])
which has $\displaystyle\mathcal{E}=2$.
Since
$\displaystyle C_{2}\sim 0.14\quad\frac{1}{C_{2}}\sim 7.39,$
we can apply the main theorem to the catenoid, the Costa and the Scherk
surface, obtaining
$\displaystyle\frac{\left(1-0.28\right)^{2}}{2}\leq\limsup_{t\to\infty}\left(4\pi
t\right)\mathcal{H}(t,x,x)\leq 1,$
for the catenoid and the Scherk singly periodic surface, and
$\displaystyle\frac{\left(1-0.41\right)^{2}}{3}\leq\limsup_{t\to\infty}\left(4\pi
t\right)\mathcal{H}(t,x,x)\leq 1,$
for the Costa surface.
As we have shown, there are several examples where the volume growth is
related with the number of ends of the submanifold. In fact, the following
theorem allow us to achieve inequality (1.4) under certain decay of the norm
of the second fundamental form and to give a topological meaning to
$\displaystyle\lim_{R\to\infty}\mathcal{Q}(R)$
###### Theorem 1.1 (see theorem 2.2 of [Qin95] and [GP12]).
Let $\displaystyle M^{m}$ be an $\displaystyle m-$dimensional complete
immersed minimal submanifold in $\displaystyle\mathbb{R}^{n}$ which satisfies
(1.8) $\lim_{R\to\infty}\underset{r(x)\geq R}{\sup_{x\in
M^{m}}}r(x)\|A\|(x)=0,$
where $\displaystyle A$ denotes the second fundamental form. Then, the number
of ends $\displaystyle\mathcal{E}\left(M^{m}\right)$ of $\displaystyle M^{m}$
is given by:
(1.9) $\lim_{R\to\infty}\mathcal{Q}(R)=\mathcal{E}(M^{m})$
provided either of the following two conditions is satisfied:
1. (1)
$\displaystyle m=2$, $\displaystyle n=3$ and each end of $\displaystyle M^{m}$
is embedded.
2. (2)
$\displaystyle m\geq 3$.
Hence, we can state the following corollary showing the relation between the
number of ends and the lower bound for the heat kernel under the assumptions
of the above theorem (see introduction of [GSC09] for a complete overview on
the two sides estimates for the heat kernel on manifolds with ends):
###### Corollary 1.2.
Let $\displaystyle M^{m}$ be an $\displaystyle m-$dimensional complete
immersed minimal submanifold in $\displaystyle\mathbb{R}^{n}$ which satisfies
(1.10) $\lim_{R\to\infty}\underset{r(x)\geq R}{\sup_{x\in
M^{m}}}r(x)\|A\|(x)=0,$
and
1. (1)
if $\displaystyle m=2$ and $\displaystyle n=3$, each end of $\displaystyle
M^{m}$ is embedded. Or,
2. (2)
$\displaystyle m\geq 3$.
Then, if the number of ends $\displaystyle\mathcal{E}(M^{m})$ of
$\displaystyle M^{m}$ is bounded from above by
(1.11) $\mathcal{E}(M^{m})<\frac{1}{C_{m}},$
the heat kernel $\displaystyle\mathcal{H}$ of $\displaystyle M^{m}$ satisfies
(1.12)
$\frac{\left(1-\mathcal{E}(M^{m})C_{m}\right)^{2}}{\mathcal{E}(M^{m})}\leq\limsup_{t\to\infty}\left(4\pi
t\right)^{\frac{m}{2}}\mathcal{H}(t,x,x)\leq 1.$
If $\displaystyle M^{2}$ is a minimal surface in
$\displaystyle\mathbb{R}^{3}$, by the Gauss formula the second fundamental
form is related with the Gaussian curvature $\displaystyle K_{G}$ of
$\displaystyle M^{2}$ by
(1.13) $K_{G}=-\frac{1}{2}|A|^{2},$
in view of [MPR13, theorem 1.2] it seems that in the particular case of
complete embedded minimal surfaces in $\displaystyle\mathbb{R}^{3}$ if there
exists a constant $\displaystyle C$ such that $\displaystyle|K_{G}|R^{2}\leq
C$, then:
$\displaystyle|K_{G}|R^{2}\leq
C\quad\rightarrow\int_{M^{2}}|K_{G}|<\infty\rightarrow\lim_{R\to\infty}\underset{r(x)\geq
R}{\sup_{x\in M^{m}}}r(x)|A|(x)=0.$
Hence, the condition given in equation (1.10) in the above corollary can be
replaced in the particular case of complete embedded minimal surfaces in
$\displaystyle\mathbb{R}^{3}$ by
$\displaystyle|K_{G}|R^{2}\leq C.$
Recall also that a particular case when equality (1.10) holds is (see [Qin95])
when
$\displaystyle\int_{M^{m}}|A|^{m}dV<\infty$
i.e,. when the submanifold has finite scalar curvature (see also [And84]).
Let us finally remark that
###### Remark a.
Given a manifold $\displaystyle M^{n}$ with non-negative Ricci curvature
$\displaystyle\text{Rc}>0$, Bishop-Gromov volume comparison theorem asserts
that for any $\displaystyle o\in M^{n}$ the relative volume quotient
$\displaystyle\frac{\operatorname{Vol}(B_{R}^{M^{n}}(o))}{\omega_{n}R^{n}}$ is
non-increasing in the radius $\displaystyle R$ (being $\displaystyle
B_{R}^{M^{n}}(o)$ the geodesic ball of radius $\displaystyle R$ centered at
$\displaystyle o$). The relative volume quotient converges to a non-negative
number $\displaystyle\Theta$:
$\displaystyle\lim_{R\to\infty}\frac{\operatorname{Vol}(B_{R}^{M^{n}}(o))}{\omega_{n}R^{n}}=\Theta\geq
0.$
If $\displaystyle\Theta>0$, one says that the manifold $\displaystyle M^{n}$
has _maximal volume growth_.
P. Li proved in [Li86] (see also [Xu13]) that if $\displaystyle M^{n}$ has
$\displaystyle\text{Rc}>0$ and maximal volume growth, then
(1.14)
$\lim_{t\to\infty}\operatorname{Vol}\left(B^{M^{n}}_{\sqrt{t}}\left(y\right)\right)\mathcal{H}\left(t,x,y\right)=\omega_{n}\left(4\pi\right)^{-\frac{n}{2}}.$
Therefore
(1.15) $\lim_{t\to\infty}\left(4\pi
t\right)^{\frac{n}{2}}\mathcal{H}(t,x,y)=\frac{1}{\Theta}.$
In some sense, our main theorem can be understood (partially) as a reverse of
the Li’s theorem because at least on dimension $\displaystyle 2$, by the Gauss
formula (equation (1.13)), a submanifold properly and minimally immersed in a
Cartan-Hadamard ambient manifold has non-positive sectional curvature (instead
of $\displaystyle\text{Rc}>0$) and because, by the monotonicity formula, the
extrinsic quotient given in equation (1.3) is non-decreasing (instead of non-
increasing like the relative volume quotient).
Despite of the weakness of the inequalities (1.7) in comparison to equality
(1.15) observe, however, that a non-negatively Ricci-curved manifold with
maximal volume growth must have finite fundamental group (see [Li86]) but that
is not true for minimal submanifolds of a Cartan-Hadamard with polynomial
volume growth (see for instance the singly periodic Scherk surface (figure
1)).
The most well known examples of heat kernels of minimal submanifolds
$\displaystyle M^{m}$ in the Euclidean space $\displaystyle\mathbb{R}^{n}$ are
when $\displaystyle M^{m}$ is a totally geodesic submanifold
$\displaystyle\mathbb{R}^{m}$ in $\displaystyle\mathbb{R}^{n}$. Observe that
in that case $\displaystyle\mathcal{E}=1$ if $\displaystyle C_{m}$ were
$\displaystyle 0$ the inequality (1.7) would be an exact equality. Therefore,
it is a natural question to ask the following open question
###### Open question.
Is it possible to improve the main theorem changing $\displaystyle C_{m}$ by
$\displaystyle 0$?
The structure of the paper is as follows
In §2 we recall the definition and several properties of the heat kernel on a
Riemannian manifold and provide proposition 2.1 which states that every
complete minimal submanifold with polynomial volume growth is stochastically
complete. With those previous requirements we can, in §3, to prove the main
theorem.
## 2\. Preliminaries
Let $\displaystyle M$ be a Riemannian manifold with (possibly empty) smooth
boundary $\displaystyle\partial M$, and denote by $\displaystyle\Delta$ the
Laplacian on $\displaystyle M$. The heat kernel on $\displaystyle M$ is a
function $\displaystyle\mathcal{H}(t,x,y)$ on $\displaystyle(0,\infty)\times
M\times M$ which is the minimal positive fundamental solution to the heat
equation
(2.1) $\frac{\partial v}{\partial t}=\Delta v\quad.$
In other words, the Cauchy problem with Dirichlet boundary conditions
(2.2) $\begin{cases}\frac{\partial v}{\partial t}=\Delta v\quad,\\\
v|_{t=0}=v_{0}(x)\quad,\end{cases}$
has a solution
(2.3) $v(x,t)=\int_{M}\mathcal{H}(t,x,y)v_{0}(y)d\mu_{y}\quad,$
provided that $\displaystyle v_{0}$ is a bounded continuous positive function.
Moreover the heat kernel has the following properties:
1. (1)
Symmetry in $\displaystyle x,y$ that is
$\displaystyle\mathcal{H}(t,x,y)=\mathcal{H}(t,y,x)$.
2. (2)
The semigroup identity: for any $\displaystyle s\in(0,t)$
(2.4)
$\mathcal{H}(t,x,y)=\int_{M}\mathcal{H}(s,x,z)\mathcal{H}(t-s,z,y)d\text{V}(z).$
3. (3)
For all $\displaystyle t>0$ and $\displaystyle x\in M$,
(2.5) $\int_{M}\mathcal{H}(t,x,y)d\text{V}(y)\leq 1.$
If $\displaystyle M$ is the Euclidean space $\displaystyle\mathbb{R}^{n}$
then, due to the homogeneity and isotropy of the Euclidean space, the heat
kernel $\displaystyle\mathcal{H}^{n,0}(t,x,y)$ depends only on $\displaystyle
t$ and $\displaystyle\rho(x,y)=\text{dist}(x,y)$, and is given by the
classical formula
(2.6) $\mathcal{H}^{n,0}(t,\rho(x,y))=\frac{1}{(4\pi
t)^{\frac{n}{2}}}e^{-\frac{\rho^{2}(x,y)}{4t}}\quad.$
A manifold $\displaystyle M$ satisfying for all $\displaystyle x\in M$ and all
$\displaystyle t>0$
(2.7) $\int_{M}\mathcal{H}(t,x,y)d\text{V}(y)=1,$
is said to be stochastically complete.
In the following proposition is proved that a complete submanifold of
polynomial volume growth is stochastically complete
###### Proposition 2.1.
Let $\displaystyle M^{m}$ be a $\displaystyle m$-dimensional complete minimal
submanifold properly immersed in a Cartan-Hadamard ambient manifold. Suppose
that $\displaystyle M^{m}$ is of polynomial volume growth, then $\displaystyle
M^{m}$ is stochastically complete
###### Proof.
Since $\displaystyle M^{m}$ has polynomial volume growth by equation (1.4),
for any $\displaystyle o\in M$ and any $\displaystyle R\in\mathbb{R}_{+}$ we
have
(2.8) $\operatorname{Vol}(M^{m}\cap
B_{R}^{N}(o))\leq\mathcal{E}\omega_{m}R^{m}.$
But since the geodesic ball $\displaystyle B_{R}^{M^{m}}(o)$ of radius
$\displaystyle R$ in $\displaystyle M^{m}$ is a subset of the extrinsic ball
$\displaystyle M^{m}\cap B_{R}^{N}(o)$, one obtains that
(2.9)
$\displaystyle\int^{\infty}\frac{rdr}{\log\left(\operatorname{Vol}(B_{r}^{M^{m}}(o))\right)}$
$\displaystyle\geq\int^{\infty}\frac{rdr}{\log\left(\operatorname{Vol}(M^{m}\cap
B_{r}^{N}(o))\right)}$
$\displaystyle\geq\int^{\infty}\frac{rdr}{\log\left(\mathcal{E}\omega_{m}r^{m}\right)}=\infty.$
Hence, by [Gri99, theorem 9.1] $\displaystyle M^{m}$ is stochastically
complete. ∎
Finally in order to conclude this preliminary section let us recall here the
coarea formula
###### Theorem 2.2 (Coarea formula, see [Sak96, Cha84]).
Let $\displaystyle f$ be a proper $\displaystyle C^{\infty}$ function defined
on a Riemannian manifold $\displaystyle(M^{n},g)$. Now we set
(2.10) $\displaystyle\Omega_{t}:=\left\\{p\in M;\,f(p)<t\right\\},$
$\displaystyle\quad\text{V}_{t}:=\operatorname{Vol}(\Omega_{t}),$
$\displaystyle\Gamma_{t}:=\left\\{p\in M;\,f(p)=t\right\\},$
$\displaystyle\quad\text{A}_{t}:=\operatorname{Vol}_{n-1}(\Gamma_{t}).$
Then for an integrable function $\displaystyle u$ on $\displaystyle M^{n}$ the
following hold:
1. (1)
Let $\displaystyle g_{t}$ be the induced metric on $\displaystyle\Gamma_{t}$
from $\displaystyle g$. Then
(2.11) $\int_{M^{n}}u|\nabla
f|d\nu_{g}=\int_{-\infty}^{\infty}dt\int_{\Gamma_{t}}ud\nu_{g_{t}}.$
2. (2)
$\displaystyle t\to\text{V}_{t}$ is of class $\displaystyle C^{\infty}$ at a
regular value $\displaystyle t$ of $\displaystyle f$ such that
$\displaystyle\text{V}_{t}<+\infty$, and
(2.12) $\frac{d}{dt}\text{V}_{t}=\int_{\Gamma_{t}}\frac{1}{|\nabla
f|}d\nu_{g_{t}}.$
## 3\. Proof of the main theorem
First of all, let us denote by $\displaystyle D_{R}(x)$ the extrinsic ball of
radius $\displaystyle R$ cantered at $\displaystyle x$, i.e.,
$\displaystyle D_{R}(x):=M^{m}\cap B_{R}^{N}(x),$
therefore $\displaystyle\mathcal{Q}(R)$ is given by
$\displaystyle\mathcal{Q}(R)=\frac{\operatorname{Vol}(D_{R}(x))}{\omega_{m}R^{m}}.$
Note that $\displaystyle D_{R}(x)$ is the sublevel set of the extrinsic
distance function $\displaystyle r_{x}$:
(3.1) $D_{R}(x)=\left\\{p\in M^{m};\,r_{x}(p)<R\right\\}.$
Making use of the upper bounds for the heat kernel (equation 1.2) and the
semigroup property of the heat kernel (equation 2.4)
(3.2) $\displaystyle 1\geq\left(4\pi t\right)^{\frac{m}{2}}\mathcal{H}(t,x,x)$
$\displaystyle=\left(4\pi
t\right)^{\frac{m}{2}}\int_{M^{m}}\mathcal{H}(t/2,x,y)^{2}d\text{V}(y)$
$\displaystyle\geq\left(4\pi
t\right)^{\frac{m}{2}}\int_{D_{R}(x)}\mathcal{H}(t/2,x,y)^{2}d\text{V}(y),$
for any extrinsic ball $\displaystyle D_{R}(x)$. Applying now the
Cauchy–Schwarz inequality
(3.3) $\displaystyle 1\geq\left(4\pi t\right)^{\frac{m}{2}}\mathcal{H}(t,x,x)$
$\displaystyle\geq\left(4\pi
t\right)^{\frac{m}{2}}\frac{\left(\int_{D_{R}(x)}\mathcal{H}(t/2,x,y)d\text{V}(y)\right)^{2}}{\operatorname{Vol}(D_{R}(x))},$
Since by proposition 2.1 $\displaystyle M^{m}$ is stochastically complete
(3.4) $\displaystyle 1\geq\left(4\pi t\right)^{\frac{m}{2}}\mathcal{H}(t,x,x)$
$\displaystyle\geq\left(4\pi
t\right)^{\frac{m}{2}}\frac{\left(1-\int_{M^{m}\setminus
D_{R}(x)}\mathcal{H}(t/2,x,y)d\text{V}(y)\right)^{2}}{\operatorname{Vol}(D_{R}(x))},$
Applying the polynomial volume growth property
(3.5) $\displaystyle 1\geq\left(4\pi t\right)^{\frac{m}{2}}\mathcal{H}(t,x,x)$
$\displaystyle\geq\left(4\pi
t\right)^{\frac{m}{2}}\frac{\left(1-\int_{M^{m}\setminus
D_{R}(x)}\mathcal{H}(t/2,x,y)d\text{V}(y)\right)^{2}}{\mathcal{E}\omega_{m}R^{m}},$
for all $\displaystyle R>0$. If we choose
(3.6)
$R=R_{t}:=\frac{\left(4\pi\right)^{\frac{1}{2}}}{\omega_{m}^{\frac{1}{m}}}t^{\frac{1}{2}}=2\left[\frac{m}{2}\Gamma\left(\frac{m}{2}\right)\right]^{\frac{1}{m}}t^{\frac{1}{2}},$
we obtain
(3.7) $\displaystyle 1\geq\left(4\pi t\right)^{\frac{m}{2}}\mathcal{H}(t,x,x)$
$\displaystyle\geq\frac{\left(1-\int_{M^{m}\setminus
D_{R_{t}}(x)}\mathcal{H}(t/2,x,y)d\text{V}(y)\right)^{2}}{\mathcal{E}},$
We need now the following proposition
###### Proposition 3.1.
Suppose that
$\displaystyle\lim_{R\to\infty}\mathcal{Q}(R)=\mathcal{E}$
then
(3.8) $\int_{M^{m}\setminus
D_{R_{t}}(x)}\mathcal{H}(t/2,x,y)d\text{V}(y)\leq\mathcal{E}\left(C_{m}+\delta(t)\right),$
being $\displaystyle\delta$ a smooth function with $\displaystyle\delta\to 0$
when $\displaystyle t\to\infty$.
###### Proof.
By inequality (1.1)
(3.9) $\displaystyle\int_{M^{m}\setminus
D_{R_{t}}(x)}\mathcal{H}(t/2,x,y)d\text{V}(y)$
$\displaystyle\leq\int_{M^{m}\setminus
D_{R_{t}}(x)}\mathcal{H}^{m,0}(t/2,r_{x}(y))d\text{V}(y)$
by coarea formula (theorem 2.2)
(3.10) $\displaystyle\int_{M^{m}\setminus
D_{R_{t}}(x)}\mathcal{H}(t/2,x,y)d\text{V}(y)\leq$
$\displaystyle\int_{R_{t}}^{\infty}\int_{\partial
D_{S}(x)}\frac{\mathcal{H}^{m,0}(t/2,r_{x}(y))}{|\nabla
r_{x}|}d\text{V}_{s}(y)ds$ $\displaystyle\leq$
$\displaystyle\int_{R_{t}}^{\infty}\mathcal{H}^{m,0}(t/2,s)\left(\operatorname{Vol}(D_{s}(x)\right)^{\prime}ds.$
The derivative
$\displaystyle\frac{d}{dR}\operatorname{Vol}(D_{R}(o))=\left(\operatorname{Vol}(D_{R})\right)^{\prime}$
satisfies
(3.11)
$\left(\operatorname{Vol}(D_{R})\right)^{\prime}=m\omega_{m}\mathcal{Q}(R)R^{m-1}+\omega_{m}R^{m}\mathcal{Q}(R)\left(\log(\mathcal{Q}(R)\right)^{\prime}.$
Therefore,
(3.12) $\displaystyle\int_{{M^{m}\setminus
D_{R_{t}}(x)}}\mathcal{H}(t/2,x,y)d\text{V}(y)\leq$
$\displaystyle\frac{\omega_{m}}{(2\pi
t)^{\frac{m}{2}}}\int_{R_{t}}^{\infty}e^{-\frac{s^{2}}{2t}}\left[m\mathcal{Q}(s)s^{m-1}+s^{m}\mathcal{Q}(s)\left(\log(\mathcal{Q}(s)\right)^{\prime}\right]ds\leq$
$\displaystyle\frac{\omega_{m}\mathcal{E}}{(2\pi
t)^{\frac{m}{2}}}\int_{R_{t}}^{\infty}e^{-\frac{s^{2}}{2t}}\left[ms^{m-1}+s^{m}\left(\log(\mathcal{Q}(s)\right)^{\prime}\right]ds\leq$
$\displaystyle\frac{\omega_{m}\mathcal{E}}{(2\pi
t)^{\frac{m}{2}}}\left[\int_{R_{t}}^{\infty}me^{-\frac{s^{2}}{2t}}s^{m-1}ds+\left(\sup_{s\in[0,\infty)}e^{-\frac{s^{2}}{2t}}s^{m}\right)\log\left(\frac{\mathcal{E}}{\mathcal{Q}(R_{t})}\right)\right]=$
$\displaystyle\frac{\omega_{m}\mathcal{E}}{(2\pi)^{\frac{m}{2}}}\left[m2^{\frac{m}{2}-1}\Gamma(\frac{m}{2},\frac{{R_{t}}^{2}}{2t})+e^{-\frac{m}{2}}m^{\frac{m}{2}}\log\left(\frac{\mathcal{E}}{\mathcal{Q}(R_{t})}\right)\right].$
Taking into account the definition of $\displaystyle R_{t}$ (equation (3.6))
and that
$\displaystyle\omega_{m}=\frac{2\pi^{\frac{m}{2}}}{m\Gamma(\frac{m}{2})}$,
(3.13) $\displaystyle\int_{M^{m}\setminus
D_{R_{t}}(x)}\mathcal{H}(t/2,x,y)d\text{V}(y)\leq\mathcal{E}\left[C_{m}+\delta(t)\right],$
where
$\displaystyle\delta(t):=\frac{e^{-\frac{m}{2}}(\frac{m}{2})^{\frac{m}{2}-1}}{\Gamma(\frac{m}{2})}\log\left(\frac{\mathcal{E}}{\mathcal{Q}\left(2\left(\frac{m}{2}\Gamma\left(\frac{m}{2}\right)\right)^{\frac{1}{m}}t^{\frac{1}{2}}\right)}\right).$
Making use that $\displaystyle\mathcal{Q}(s)=\mathcal{E}$ when $\displaystyle
s\to\infty$ the proposition is proven. ∎
Hence for $\displaystyle t$ large enough we can apply the above proposition in
equation (3.7)
(3.14) $\displaystyle 1\geq\left(4\pi
t\right)^{\frac{m}{2}}\mathcal{H}(t,x,x)$
$\displaystyle\geq\frac{\left[1-\mathcal{E}\left(C_{m}+\delta(t)\right)\right]^{2}}{\mathcal{E}}.$
Therefore, taking limits the theorem follows.
## References
* [And84] Michael T. Anderson, _The compactification of a minimal submanifold in euclidean space by the gauss map_ , unpublished preprint, 1984.
* [Cha84] Isaac Chavel, _Eigenvalues in Riemannian geometry_ , Pure and Applied Mathematics, vol. 115, Academic Press Inc., Orlando, FL, 1984, Including a chapter by Burton Randol, With an appendix by Jozef Dodziuk. MR 768584 (86g:58140)
* [CLY84] Shiu Yuen Cheng, Peter Li, and Shing-Tung Yau, _Heat equations on minimal submanifolds and their applications_ , Amer. J. Math. 106 (1984), no. 5, 1033–1065. MR 761578 (85m:58171)
* [GP12] Vicent Gimeno and Vicente Palmer, _Volume growth, number of ends, and the topology of a complete submanifold_ , Journal of Geometric Analysis (2012), 1–22 (English).
* [GP13] by same author, _Extrinsic isoperimetry and compactification of minimal surfaces in Euclidean and hyperbolic spaces_ , Israel J. Math. 194 (2013), no. 2, 539–553. MR 3047082
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* [MP12] William H. Meeks, III and Joaquín Pérez, _A survey on classical minimal surface theory_ , University Lecture Series, vol. 60, American Mathematical Society, Providence, RI, 2012. MR 3012474
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* [Pal99] Vicente Palmer, _Isoperimetric inequalities for extrinsic balls in minimal submanifolds and their applications_ , J. London Math. Soc. (2) 60 (1999), no. 2, 607–616. MR 1724821 (2000j:53050)
* [Qin95] Chen Qing, _On the volume growth and the topology of complete minimal submanifolds of a euclidean space_ , J. Math. Sci. Univ. Tokyo 2 (1995), 657–669.
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|
arxiv-papers
| 2013-10-17T10:04:40 |
2024-09-04T02:49:52.499555
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Vicent Gimeno",
"submitter": "Vicent Gimeno",
"url": "https://arxiv.org/abs/1310.4643"
}
|
1310.4648
|
# LOW ENERGY $\alpha-\alpha$ SEMIMICROSCOPIC POTENTIALS
M. Lassaut1,2, F. Carstoiu2, and V. Balanica2
1 Institut de Physique Nucléaire
IN2P3-CNRS, Université Paris-Sud 11
F-91406 Orsay Cedex, France
2 National Institute for Nuclear Physics and Engineering,
P.O.Box MG-6, RO-077125 Bucharest-Magurele, Romania
###### Abstract
The $\alpha-\alpha$ interaction potential is obtained within the double
folding model with density-dependent Gogny effective interactions as input.
The one nucleon knock-on exchange kernel including recoil effects is localized
using the Perey-Saxon prescription at zero energy. The Pauli forbidden states
are removed thanks to successive supersymmetric transformations. Low energy
experimental phase shifts, calculated from the variable phase approach, as
well as the energy and width of the first $0^{+}$ resonance in 8Be are
reproduced with high accuracy.
(Received )
Key words: Gogny interaction, knock-on nonlocal kernel, variable phase
equation, SUSY potential.
## 1 INTRODUCTION
In last time there is an increasing interest in understanding the properties
of $\alpha$-matter mainly due to the believe that this type of hadronic matter
occurs in astrophysical environment in unconfined form. In the debris of a
supernova explosion, a substantial fraction of hot and dense matter resides in
$\alpha$-particles and therefore the equation of state of subnuclear matter is
essential in simulating the supernova collapse and explosions and is also
important for the formation of the supernova neutrino signal [1].
The basic ingredient in the calculation of the ground state alpha matter [2]
as well in the $\alpha$-cluster model of nuclei [3] is the $\alpha-\alpha$
interaction potential. This has been studied extensively using both local and
nonlocal interactions. Among the most important are those using the resonating
group model (RGM) [4, 5], the energy and angular momentum independent
potential model of Buck, Friedrich and Wheatley [6] and the phenomenological
potential of Ali and Bodmer [7]. There have been proposed several versions of
the Ali-Bodmer potential: a Gaussian potential with a stronger repulsive
component by Langanke and Müller [8], as well a version with a softer
repulsive component by Yamada and Schuck [9]. All these models predict
potentials quite different in strength and range but all are claimed to
reproduce experimental data up the the breakup threshold.
Microscopic RGM calculations by Schmid and Wildermuth [5] lead to the
important conclusion that due to the compact structure and the large binding
energy the radius of the $\alpha$-particle stays essentially the same during
the compound system formation and therefore the polarization effects could be
neglected. This observation substantiates the idea of calculation of a
$\alpha-\alpha$ potential from the double folding model.
We propose in this paper to generate the $\alpha-\alpha$ potential within the
double folding model using the Gogny force as input. Previously Sofianos et
al.[10] derived the $\alpha-\alpha$ potential using the energy density
formalism based on Skyrme effective interaction.
However, the potential issued from double-folding calculation, even corrected
by knock-on exchange terms, is generally too deep due to the presence of
forbidden bound states. These states have a clear interpretation within the
RGM model: they are redundant solutions giving fully antisymmetrized wave
functions that vanish identically. These latter bound states are eliminated
thanks to successive supersymmetric transformations as given in [11], which
preserves the continuous spectrum (phase-shift) and resonances [12].
In section 2 we present the derivation of the $\alpha-\alpha$ interaction. In
section 3 the derivation and the properties of supersymmetric partner are
presented. Our conclusions are given in section 4.
## 2 Bare $\alpha-\alpha$ interaction : double-folding with Gogny forces
Since the potentials providing saturation at lower densities of the alpha
matter are highly schematic (infinite repulsive short-range interactions) we
turn to a calculation of the bare $\alpha-\alpha$ interaction based on the
double-folding method for two ions at energies around the barrier, starting
with realistic densities of the $\alpha$-particle and modern effective
nucleon-nucleon interactions.
Within the double-folding model [13] the interaction between two alpha
clusters is calculated as a convolution of a local two-body potential $v_{nn}$
and the single particle densities of the two clusters, namely
$v_{\alpha\alpha}(\vec{r})=\int d\vec{r}_{1}\int
d\vec{r}_{2}\rho_{\alpha}({r}_{1})\rho_{\alpha}({r}_{2})v_{nn}(\rho,\vec{r}-\vec{r}_{1}+\vec{r}_{2})$
(1)
The effective $n-n$ interaction $v_{nn}$ is taken to be density-dependent as
expected from a realistic interaction. It depends on the density $\rho$ of the
nuclear matter where the two interacting nucleons are embedded. For the sake
of simplicity, we choose Gaussians interactions in order to have the most
tractable analytical calculations. A candidate satisfying this requirement is
provided by the Gogny forces [14]. In this paper, we will report results using
three main parametrizations of the Gogny interaction [14], denoted D1 and D1S
[15] as well as the most recent variant, labeled D1N [16].
We remind that the standard form of the Gogny interactions is,
$\displaystyle v_{nn}(r)$
$\displaystyle=\sum_{i=1}^{2}(W+BP_{\sigma}-HP_{\tau}-MP_{\sigma}P_{\tau})e^{-r^{2}/\mu_{i}^{2}}$
(2)
$\displaystyle+t_{3}(1+x_{0}P_{\sigma})\rho^{\gamma}\left(\frac{\vec{r}_{1}+\vec{r}_{2}}{2}\right)\delta(\vec{r}_{1}-\vec{r}_{2})$
where $\vec{r}=\vec{r_{1}}-\vec{r_{2}}$, and the coefficients $W,B,H,M$ refer
to the usual notations for the spin/isospin mixtures and $P_{\sigma,\tau}$ are
the spin/isospin exchange operators. The spin-orbit component, present in the
original formulation, is ignored here as it is not material for the
$\alpha-\alpha$ system.
For the sake of consistency, i.e. working with Gaussian interactions, we
consider Gaussian one-body density for the $\alpha$-particle
$\rho_{\alpha}(r)=4\left(\frac{1}{\pi b^{2}}\right)^{3/2}e^{-r^{2}/b^{2}}\ .$
(3)
In Eq.(3) the oscillator parameter $b$ is adjusted on the root mean square
radius of the $\alpha$-particle (r.m.s.) given by $<r^{2}>^{1/2}=b\sqrt{3/2}$
which has to be compared to the value 1.58 $\pm$0.002 fm, extracted from a
Glauber analysis of experimental interaction cross sections [17].
A more involved density matrix was derived by Bohigas and Stringari [18] who
included short range correlations starting from a Jastrow wave function and
evaluated the one-body density matrix by using the perturbation expansion of
[19] at a low order. The diagonal component of the density matrix so far
obtained is not far from our density (3), and since we want to keep the
results as simply as possible we use Eq. (3). We have checked that the density
Eq. (3) reproduces the experimental charge form factor [20] up to $q^{2}\sim
2fm^{-2}$ momentum transfer.
Antisymmetrization of the density dependent term in the Gogny force is
obtained at follows. Consider the operator,
$\cal{O}=\it{\left(1+x_{0}P^{\sigma}\right)(1-P^{\sigma}P^{\tau}P^{x})}\\\ $
(4)
Since $\delta$ acts only in S-states, one can take safely $P^{x}=1$ and using
the usual algebra of the exchange operators one obtains,
$v_{d}^{\rho}(r_{12})=t_{3}\left(1+\frac{x_{0}}{2}\right)\rho^{\gamma}\delta(r_{12}),$
(5)
and,
$v_{ex}^{\rho}(r_{12})=-\frac{t_{3}}{4}(1+2x_{0})\rho^{\gamma}\delta(r_{12})\
.$ (6)
The interest is that the total contribution from the density dependence, is
calculated from
$v^{\rho}(r_{12})=\frac{3}{4}t_{3}\rho^{\gamma}\delta(r_{12})$ (7)
and is independent of the value of the spin mixture $x_{0}$. Therefore we take
$x_{0}=1$. The direct spin-isospin independent effective $n-n$ force in the
Gogny parametrization [2] reads:
$v_{00}^{\rm
d}(\vec{r}_{1}-\vec{r}_{2})={\frac{1}{2}}\sum_{i=1}^{2}(4W_{i}+2B_{i}-2H_{i}-M_{i})e^{-|\vec{r}_{1}-\vec{r}_{2}|^{2}/\mu_{i}^{2}}+\frac{3}{2}t_{3}\rho^{\gamma}\delta(\vec{r}_{1}-\vec{r}_{2})$
(8)
Inserting the Gaussian density distribution (3) in the double folding integral
(1) and using a generalization of the Campi-Sprung prescription [21] for the
overlap density similar to the one proposed in [22] for $\alpha$-nucleus
scattering
$\rho(1,2)=\left(\rho_{\alpha}(\vec{r}_{1}-\frac{1}{2}\vec{s})\rho_{\alpha}(\vec{r}_{2}+\frac{1}{2}\vec{s})\right)^{\frac{1}{2}},$
(9)
where $\vec{s}=\vec{r}_{1}+\vec{r}-\vec{r}_{2}$ is the $n-n$ separation in the
heavy-ion coordinate system [13]. With this approximation, the overlap density
does not exceeds the density of the normal nuclear matter at complete overlap
and goes to zero when one of the interacting nucleon is far from the other. We
obtain the local $\alpha-\alpha$ potential,
$\displaystyle v_{\alpha\alpha}(r)$
$\displaystyle=4\sum_{i=1}^{2}(4W_{i}+2B_{i}-2H_{i}-M_{i})\left(\frac{\mu_{i}^{2}}{\mu_{i}^{2}+2b^{2}}\right)^{3/2}e^{-{r^{2}}/{(\mu_{i}^{2}+2b^{2})}}$
(10)
$\displaystyle+\frac{3}{2}t_{3}\frac{4^{\gamma+2}}{(\gamma+2)^{3/2}(\sqrt{\pi}b)^{3(\gamma+1)}}e^{-\frac{\gamma+2}{4b^{2}}{r^{2}}}$
which includes both direct and exchange arising from the density dependent
component of the force.
The derivation of the knock-on exchange component corresponding to the finite
range component of the effective interaction is more involved. It is
convenient to start from the DWBA matrix element of the exchange operator :
$\hat{U}_{ex}\chi=\sum_{\alpha\beta}<\phi_{\alpha}(\vec{r}_{1})\phi_{\beta}(\vec{r}_{2})|v_{ex}(s)P_{12}^{x}|\phi_{\alpha}(\vec{r}_{1})\phi_{\beta}(\vec{r}_{2})\chi(\vec{R})>$
(11)
where the sum runs over the single-particle wave functions of occupied states
in the projectile (target) and $\chi(\vec{R})$ is the wave function for
relative motion. After some algebra (see details in [23]), we arrive at,
$\hat{U}_{ex}\chi=\int
U_{ex}(\vec{R},\vec{R}^{\prime})\chi(\vec{R}^{\prime})d\vec{R}^{\prime}$
where the kernel $U_{ex}(\vec{R},\vec{R}^{\prime})$ is given by,
$\displaystyle
U_{ex}(\vec{R},\vec{R}^{\prime})=U_{ex}(\vec{R}^{+},\vec{R}^{-})=\mu^{3}v_{ex}(\mu
R^{-})\int\rho_{1}(\vec{X}+\delta_{1}\mu\vec{R}^{-},\vec{X}-\delta_{1}\mu\vec{R}^{-})$
$\displaystyle\times\rho_{2}(\vec{X}-\vec{R}^{+}-\delta_{2}\mu\vec{R}^{-},\vec{X}-\vec{R}^{+}+\delta_{2}\mu\vec{R}^{-})d\vec{X}$
(12)
where
$\vec{R}^{+}=(\vec{R}+\vec{R}^{\prime})/2,~{}~{}\vec{R}^{-}=\vec{R}-\vec{R}^{\prime}$
and $\rho(\vec{r},\vec{r}^{\prime})$ is the one-body matrix density. The
$\delta_{i}=1-\frac{1}{A_{i}}$ accounts for recoil effects. The equation (12)
already tells us that the range of non-locality $\vec{R}^{-}$ is
$\sim\mu^{-1}$ . In the case of the $\alpha-\alpha$ interaction we have
$\displaystyle U_{\alpha\alpha}^{\rm ex}(\vec{R},\vec{R}^{\prime})$
$\displaystyle=8v_{00}^{\rm
ex}(2R^{-})\int\rho_{\alpha}(\vec{X}+\frac{3}{2}\vec{R}^{-},\vec{X}-\frac{3}{2}\vec{R}^{-})$
(13)
$\displaystyle\times\rho_{\alpha}(\vec{X}-\vec{R}^{+}-\frac{3}{2}\vec{R}^{-},\vec{X}-\vec{R}^{+}+\frac{3}{2}\vec{R}^{-})d\vec{X}$
The local equivalent potential is well approximated [24] by the lowest order
term of the Perey-Saxon approximation. For high energy and a heavy target the
$\alpha$-nucleus potential reads,
$\displaystyle U_{L}(R)$ $\displaystyle=\int
e^{i\vec{K}\vec{R}^{-}}U_{\alpha\alpha}^{\rm
ex}(\vec{R}+\frac{1}{2}\vec{R}^{-},\vec{R}^{-})d\vec{R}^{-}$ (14)
$\displaystyle=4\pi\int\rho_{\alpha}(X)\rho_{\alpha}(|\vec{R}-\vec{X}|)d\vec{X}$
$\displaystyle\times\int v_{00}^{\rm
ex}(s)\hat{j}_{1}(\hat{k}_{1}(X)\frac{3}{4}s)\cdot\hat{j}_{1}(\hat{k}_{2}(|\vec{R}-\vec{X}|)\frac{3}{4}s)$
$\displaystyle\times j_{0}(K(R)s/2)s^{2}ds$
where $K(R)$ is the usual WKB local momentum for the relative motion,
$K^{2}(R)=\frac{2\mu}{\hbar^{2}}(E_{c.m.}-U_{D}(R)-U_{L}(R))$ (15)
and $U_{D}$ is the direct term including the nuclear and Coulomb potentials.
Truly speaking, the classical momentum is defined only for energies where
$K^{2}(R)\geq 0$. At under-barrier energies, $K(R)$ is imaginary in the region
$R_{1}<R<R_{2}$, where $R_{1,2}$ are the classical turning points of the total
potential, and the Bessel function $j_{0}$ above should be replaced by
$j_{0}(ix)=\sinh(|x|)/|x|$. In Eq. (14) the function
$\hat{j}_{1}(x)=3j_{0}(x)/x$ arises from the Slater approximation of the mixed
density.
Figure 1: Folding $\alpha-\alpha$ potentials (including knock-on exchange)
obtained from three parametrizations of the Gogny effective interaction. The
Coulomb component is omitted. The phenomenological BFW potential is plotted
for comparison.
In the particular case of the $\alpha-\alpha$ system the one body density
matrix can be evaluated exactly from $0S$ HO orbitals,
$\rho_{\alpha}(\vec{r},\vec{r}^{\prime})=4\left(\frac{1}{\pi
b^{2}}\right)^{3/2}e^{-(r_{+}^{2}+\frac{1}{4}r_{-}^{2})/b^{2}}$ (16)
where
$\vec{r}_{+}={\frac{1}{2}}(\vec{r}+\vec{r}^{\prime}),~{}~{}~{}\vec{r}_{-}=\vec{r}-\vec{r}^{\prime}$
(17)
Explicitly we have,
$\displaystyle\rho_{\alpha}(\vec{X}+\frac{3}{2}\vec{R}_{-},\vec{X}-\frac{3}{2}\vec{R}_{-})$
$\displaystyle=4\left(\frac{1}{\pi
b^{2}}\right)^{3/2}e^{-(\vec{X}+\frac{9}{4}\vec{R}_{-}^{2})/b^{2}}$ (18)
$\displaystyle\rho_{\alpha}(\vec{X}-\vec{R}_{+}-\frac{3}{2}\vec{R}_{-},\vec{X}-\vec{R}_{+}+\frac{3}{2}\vec{R}_{-})$
$\displaystyle=4\left(\frac{1}{\pi
b^{2}}\right)^{3/2}e^{-\left[(\vec{X}-\vec{R}_{+}^{2})+\frac{9}{4}\vec{R}_{-}^{2}\right]/b^{2}}$
Using the convolution techniques we obtain the compact expression of the non-
local kernel,
$U_{\alpha\alpha}^{\rm ex}(\vec{R},\vec{R}^{\prime})=-4\left(\frac{2}{\pi
b^{2}}\right)^{3/2}\sum_{i}^{2}(W_{i}+2B_{i}-2H_{i}-4M_{i})e^{-{\frac{1}{2}}\left(\frac{8}{\mu_{i}^{2}}+\frac{9}{b^{2}}\right)R_{-}^{2}}e^{-\frac{1}{2b^{2}}R_{+}^{2}}$
(19)
Adopting the short-hand notation
$\frac{1}{\beta_{i}^{2}}=\frac{8}{\mu_{i}^{2}}+\frac{9+\frac{1}{4}}{b^{2}}$
(20)
and using the integral identity
$\int
d\vec{s}e^{-\alpha^{2}s^{2}}e^{i\beta\vec{s}\cdot\vec{K}}=\left(\frac{\pi}{\alpha^{2}}\right)^{3/2}e^{-(\beta
K/2\alpha)^{2}}$ (21)
the local equivalent of the nonlocal kernel in the lowest order of the Perey-
Saxon procedure is obtained as, [25],
$\displaystyle v_{\alpha\alpha}^{\rm ex}(r)=$
$\displaystyle-32\sum_{i}(W_{i}+2B_{i}-2H_{i}-4M_{i})\left(\frac{\beta_{i}}{b}\right)^{3}e^{-\frac{1}{2b^{2}}\left[1-\frac{1}{4}\left(\frac{\beta_{i}}{b}\right)^{2}\right]r^{2}}$
(22) $\displaystyle\times
e^{\pm{\frac{1}{2}}|K|^{2}\beta_{i}^{2}}\left\\{\begin{array}[]{ccc}e^{-{\frac{1}{2}}\left(\frac{\beta_{i}}{b}\right)^{2}|{K}|{r}}&{\rm
for}&K^{2}<0\\\
\cos\left[{\frac{1}{2}}\left(\frac{\beta_{i}}{b}\right)^{2}|{K}|{r}\right]&{\rm
for}&K^{2}\geq 0\end{array}\right.$
Thus we have a sub-barrier branch ($K^{2}<0$) and an over-barrier one
($K^{2}>0$) for the real part of the local exchange potential. The potentials
depicted in Fig. (1) are obtained by applying the localization procedure at
$E_{c.m.}=0$. The deep potential of Buck et al.(BFW) [6] which has has two
$\ell=0$ bound states located at -72.79 MeV and -25.88 MeV respectively is
displayed for comparison. We notice that all Gogny forces give very close
potentials at the surface.
(a)
(b)
Figure 2: Test of the heavy ion potential calculated with the D1
parametrization of the Gogny effective interaction on the high energy $\alpha$
scattering. The results are comparable with those obtained with the zero range
and finite range versions of the well known M3Y interaction.
The potentials are tested against high energy experimental data in Figure 2.
The results with the Gogny force $D1$, are labeled Gogny1 on the figure.
Curves labeled F/N are the far side/near side components of the scattering
amplitude. The real and imaginary form factors calculated with Eq. (14) are
slightly renormalized to match the experimental data.
## 3 Supersymmetric partners of the bare interactions
Once with have obtained the bare interactions by folding including the local
equivalent of the knock-on exchange kernel we notice that the resultant deep
potential has two non-physical bound states. Also, there are several
candidates reproducing qualitatively well the experimental data (see Figure
2). Therefore, the question of the uniqueness of the potential is raised. The
question of forbidden states is well-known and has been studied in the
supersymmetry approach in [12]. These states should be removed in order to
obtain a physically meaningful $\alpha-\alpha$ potential.
In this section we describe the method used to remove two bound states using
the formalism of Baye [11, 26] and of Baye and Sparenberg [27] see also refs.
[28, 29]. We give the straightforward generalization of equations (3.3) and
(3.5) of [26] to the case where two bound states are removed simultaneously.
Our potential is expected to be energy-dependent because of the Perey-Saxon
approximation. Generally this latter energy dependence is linear and we should
apply the derivation of Sparenberg, Baye and Leeb [30] for linearly energy-
dependent potentials. For the sake of simplicity we take the Perey-Saxon at
zero energy and consider the standard derivation of supersymmetric partners
[27].
Here we consider the case in absence of Coulomb potential. In fact, we will
see further, the results are not, in a certain measure, affected by the
presence of the Coulomb potential.
### 3.1 Notations
We consider the Schrödinger equation for the $\ell$-wave
$\left(\frac{{\rm d}^{2}}{{\rm
d}r^{2}}+\frac{2\mu}{\hbar^{2}}(E-V(r))-\frac{\ell(\ell+1)}{r^{2}}\right)\psi_{\ell}(E,r)=0$
(23)
where $\psi_{\ell}(E,r)$ is called the regular solution which is uniquely
defined, as usual [31, 32], by the Cauchy condition $\lim_{r\to
0}\psi_{\ell}(E,r)r^{-\ell-1}=1$. It behaves for positive values of $E$ as
$\psi_{\ell}\propto\sin(kr-\ell\pi/2+\delta_{\ell}(k))$ when $r\to\infty$
($k=\sqrt{2\mu E/\hbar^{2}}$), provided that $V(r)$ satisfies the
integrability condition [32]
$\int_{b}^{+\infty}|V(r)|{\rm d}r<\infty,\quad
b>0,\qquad\int_{0}^{\infty}r|V(r)|{\rm d}r<\infty$ (24)
Here, the $\delta_{\ell}(k)$’s are the phase shifts. In all equations $\mu$
denotes the reduced mass of the system and $E$ the c.m. energy. When the
potential possesses bound states labeled $E_{0}<E_{1}<\ldots<E_{N}\leq 0$ (the
number of which is finite when the potential satisfies the integrability
condition Eq.(24)) we can define their normalization $C_{j}$ (relative to
$E_{j}$) constant as
$\frac{1}{C_{j}}=\int_{0}^{\infty}dr\ \psi_{\ell}(E_{j},r)^{2}\ .$ (25)
Note that the integrability condition (24) discards the Coulomb potential. In
fact, we will see further, the results are not, in a certain measure, affected
by the presence of the Coulomb potential.
It is worth to recall that the exact phase $\delta_{\ell}$ can be calculated
by using the variable phase method of Calogero [33]. With this method, the
phase-shift is obtained by solving a first order differential equation
$\frac{\partial}{\partial r}\delta_{\ell}(k,r)=-\frac{v(r)}{k}\
(u_{\ell}(kr)\cos(\delta_{\ell}(k,r))+w_{\ell}(kr)\sin(\delta_{\ell}(k,r)))^{2}\
,$ (26)
with $\delta_{\ell}(k,0)=0$ as boundary condition. In equation (26) $v(r)=2\mu
V(r)/\hbar^{2}$ is the reduced potential. The phase-shift is given by the
limit $\delta_{\ell}(k)=\lim_{r\to\infty}\delta_{\ell}(k,r)$.
The regular $u_{\ell}(kr)$ and irregular $w_{\ell}(kr)$ solutions of Eq.(23)
for $v\equiv 0$ are denoted, respectively,
$\displaystyle u_{\ell}(x)$ $\displaystyle=\sqrt{\frac{\pi
x}{2}}J_{\ell+1/2}(x)$ $\displaystyle w_{\ell}(x)$
$\displaystyle=-\sqrt{\frac{\pi x}{2}}Y_{\ell+1/2}(x)$
in terms of the Bessel functions $J_{\nu},Y_{\nu}$ of order $\nu$, given in
[34]. We have $u_{\ell}(x)=xj_{\ell}(x)$ where $j_{\ell}$ is the spherical
Bessel function of order $\ell$. For $\ell=0$ we have $u_{0}(x)=\sin(x)$ and
$w_{0}(x)=\cos(x)$.
Note that for potentials in the class (24) the Levinson theorem, ( see [31,
32] and its extension to singular potentials in [35] ) applies. We have,
except for a bound state at zero energy,
$\delta_{\ell}(k=0)-\delta_{\ell}(k=\infty)=n_{\ell}\pi$ where $\delta_{\ell}$
is the exact phase (26) and $n_{\ell}$ denotes the number of bound states, in
the $\ell$-wave.
### 3.2 Phase-equivalent potentials
In this subsection we remind the method used to remove two bound states using
the formalism of Baye [11, 26] and of Baye and Sparenberg [27]. We follow
closely the derivation given in refs. [28, 29].
Starting with the bare potential $v(r)=(2\mu V(r)/\hbar^{2})$ then the phase
equivalent potential $v^{(1)}(r)$, with the ground state removed is given by,
$v^{(1)}(r)=v(r)-2\frac{{\rm d}^{2}}{{\rm d}r^{2}}\ln\int_{0}^{r}{\rm d}t\
\psi_{\ell}(E_{0},t)^{2}\ $ (27)
and the corresponding regular solution for $v^{(1)}$ is,
$\psi_{\ell}^{(1)}(E,r)=\psi_{\ell}(E,r)-\psi_{\ell}(E_{0},r)\frac{\int_{0}^{r}{\rm
d}t\ \psi_{\ell}(E,t)\ \psi_{\ell}(E_{0},t)}{\int_{0}^{r}{\rm d}t\
\psi_{\ell}(E_{0},t)^{2}}$ (28)
The potential $v^{(1)}(r)$ behaves near $r=0$ like $2(2\ell+3)/r^{2}$. This is
due to its definition Eq.(27) taking into account that
$\psi_{\ell}(E_{0},r)\simeq r^{\ell+1}$ at the vicinity of zero.
Removing the next bound state at $E_{1}$ we have,
$v^{(2)}(r)=v(r)-2\frac{{\rm d}^{2}}{{\rm d}r^{2}}\ln det(M(r))$ (29)
where $M$ is the $2\times 2$ matrix
$M=\left[\begin{matrix}L_{E_{0},E_{0}}(\ell,r)&L_{E_{0},E_{1}}(\ell,r)\cr\
L_{E_{1},E_{0}}(\ell,r)&L_{E_{1},E_{1}}(\ell,r)\end{matrix}\right]$ (30)
with
$L_{E_{i},E_{j}}(\ell,r)=L_{E_{j},E_{i}}(\ell,r)=\int_{0}^{r}{\rm d}t\
\psi_{\ell}(E_{i},t)\ \psi_{\ell}(E_{j},t)\ .$ (31)
Clearly the determinant of the matrix $M$ behaves like $r^{4\ell+10}$ at the
vicinity of zero and the resulting potential has a singularity
$(8\ell+20)/r^{2}$ at the vicinity of zero.
On the other hand, the regular solution can be written in a compact form [28,
29]
$\psi_{\ell}^{(2)}(E,r)=\frac{det(\tilde{M}(r))}{det(M(r)}$ (32)
where we have defined
$\tilde{M}=\left[\begin{matrix}\psi_{\ell}(E,r)&L_{E,E_{0}}(\ell,r)&L_{E,E_{1}}(\ell,r)\cr\psi_{\ell}(E_{0},r)&L_{E_{0},E_{0}}(\ell,r)&L_{E_{0},E_{1}}(\ell,r)\cr\psi_{\ell}(E_{1},r)&\
L_{E_{1},E_{0}}(\ell,r)&L_{E_{1},E_{1}}(\ell,r)\end{matrix}\right]$ (33)
Figure 3: The supersymmetric partners of the renormalized bare BFW and Gogny
interactions are compared with Ali-Bodmer phenomenological interaction. We
have checked that original phase shifts and the 0+ resonance properties are
conserved
### 3.3 Uniqueness of the potential
The present discussion is made discarding the Coulomb potential. But we expect
that our conclusions remain true as well. The experimental $\alpha-\alpha$
phase-shift $\ell=0$ are known at discrete energies up to the breakup
threshold [36]. Also the properties of the first $0^{+}$ resonance in 8Be have
been measured by Benn et al.[37]. If the experimental S-wave phase-shifts
satisfy the condition $\delta_{\rm exp}(k=0)-\delta_{\rm exp}(k=\infty)=0$,
where $k^{2}=2\mu E_{cm}/\hbar^{2}$, then the underlying potential, satisfying
the integrability condition (24), has no bound state. It is a consequence of
the Levinson theorem (see above). Consequently, the potential is uniquely
determined from the phase-shift $\delta_{\ell=0}(k)$, given for all positive
energies [31, 32]. The resonance should be at the right place without any fit.
In practical cases, a serious source of uncertainty comes from the fact that
the phase shifts are known at a limited number of discrete energies. Also, the
bare potentials constructed in the above section are too deep and have two
non-physical bound S-states. Such deep potentials are not unique: indeed their
reconstruction from Gelfand-Levitan or Marchenko procedure [31, 32] includes
the S-wave phase-shifts at all positive energies, the bound states and the
corresponding normalization constants Eq.(25).
We dispose of four free parameters namely, the bound state energies
$E_{1},E_{2}$ and the associated normalization constants $C_{1},C_{2}$. We
have to adjust them on the position and width of the resonance which
eliminates two free parameters. The potential is not unique and we have a two-
parameters family of solutions. The supersymmetric transformation described
above implies a singularity at the origin which is that of a centrifugal
barrier of angular momentum $L=2N$, $N$ corresponding to the number of removed
bound states, here $L=4$ as two bound states are removed.
In a recent paper [38] it was advocated that the supersymmetric transformation
increases the angular momentum by a factor of two in the sense that the Jost
function $F_{\ell}(k)$, of the starting potential, becomes after removing the
bound state $E_{j}=k_{j}^{2}\hbar^{2}/(2\mu)$,
$\tilde{F}_{\ell+2}(k)=\frac{k^{2}}{k^{2}+k_{j}^{2}}\ F_{\ell}(k)$ (34)
This latter study was made in absence of Coulomb potential. This implies that
the $S_{\ell}$ matrix of the primitive potential is exactly the $S_{\ell+2}$
matrix of the SUSY partner (the potential obtained by removing one bound
state) We then expect that the Calogero phase of the SUSY partner, calculated
for the $\ell$-wave is $-2\pi/2=-\pi$. For two bound state we will have
$-4(\pi/2)=-2\pi$.
We stress the fact that the S-wave Calogero equation (26), used to calculate
the phase shift for a potential having a singularity at the origin starts from
a modified boundary condition. Let be $\nu(\nu+1)/r^{2}\ ,\nu\neq-1/2$ the
behavior of the singular potential at small distances, the Cauchy condition
$\delta_{\ell=0}(k,r)=0$ at small $r$ is changed. This comes from the fact
that the Calogero variable phase $\delta_{0}(k,r)$ is defined by
$\delta_{0}(k,r)=-kr+\arctan\left(\frac{\psi_{0}(k,r)}{\psi^{\prime}_{0}(k,r)}\right)\simeq-
kr+\arctan\left(\frac{kr}{\nu+1}\right)\simeq-kr\frac{\nu}{\nu+1}$
so that we start from $\delta_{0}(k,r)=-kr\nu/(\nu+1)$.
We have calculated the difference of phase between our deep potentials and the
supersymmetric partners when two bound states are removed and found $-2\pi$,
even in the presence of the Coulomb potential. To conclude our deep potentials
supposed to reproduce the experimental phase have all the same S matrix. This
latter is preserved by the supersymmetric transformations (and then the
resonance) and the resulting SUSY partners have the same S matrix but for the
angular momentum L = 4. However, when all bound states of the deep potential
are removed thanks to supersymmetry the resulting potential is expected to be
unique in the following sense. If the deep potential supports N bound states
of fixed angular momentum $\ell$, then the supersymmetric partner, obtained by
setting all normalization constants $C_{j}$ , j = 1, 2, …N to infinity [38],
is unique, depending only on the number N of bound states, which determine the
singularity at the origin of the SUSY partner.
### 3.4 Numerical details
Consider the physical potentials discussed in section 2. These potentials
reproduce reasonably well the experimental phase-shift but fail to reproduce
the properties of the first $0^{+}$ resonance in 8Be. This is true also for
the Ali-Bodmer and BFW potentials. We correct this deficiency by adjusting a
global multiplicative factor $\lambda$ and judge the success of our model if
$\lambda\approx 1$.
We first calculate the S-wave phase shift $\delta_{0}(k)$ for the effective
potential
$V_{\rm eff}(r)=V(r)+4e^{2}\ \frac{erf(3r/4)}{r}$ (35)
with $e^{2}=1.43998$ MeV fm. The screened Coulomb potential arises from the
finite size charge distributions in the $\alpha$-particle. We calculate also
the phase $\delta_{0C}(k)$ for the pure Coulomb potential $V_{c}(r)=4e^{2}/r$
and assume that the difference
$\tilde{\delta}_{0}(k)=\delta_{0}(k)-\delta_{0,C}(k)$ is the nuclear phase
shift in the presence of Coulomb potential.
We integrate Eq.(26) up to 500 fm in steps $h=0.001$ fm and reproduce the
exact value of the phase $\delta_{0C}$
($\delta_{0C}^{exact}=arg\Gamma(1+i\eta)$) with high precision. The optimum
value of the parameter $\lambda$ is obtained from a grid search around unity
with a continuous refinement of the grid step $h_{\lambda}=10^{-3}-10^{-5}$
and keep the value for which $\sin^{2}\tilde{\delta}_{0}(k)=1$, near the
required energy of 0.092 MeV. Note that varying the third decimal of $\lambda$
varies the position of the resonance by $5.10^{-4}$. We found values of
$\lambda$ close to unity (see Figs.(4) and (5)).
Figure 4: The S-state phase shift calculated with bare folding potentials
including direct and exchange components (DEX) are compared with the BFW
results. The parameter $\lambda$ indicate the renormalization constant.
Using henceforth the renormalized potential by the multiplicative factor
$\lambda$, the bound state wave functions for the redundant 0S and 1S states
are calculated using a high precision Numerov scheme. The SUSY potentials are
then calculated using Eq.(29) and shown already in Fig (3).
Figure 5: The S-state resonance in 8Be calculated with the bare folding
potentials. The BFW results are shown for comparison. Figure 6: Gaussian
expansion of the SUSY potentials. The fit was performed in a restricted radial
range $r\sim 1.5-10fm$ Figure 7: The S-state phase shift calculated with
fitted SUSY potentials. Figure 8: The S-state resonance in 8Be calculated with
fitted SUSY potentials. Resonance parameters lie in the experimental range
[37] for all interactions.
In order to facilitate the calculation for $\alpha$-matter we expand the SUSY
potentials in Gaussian form factors, similar to the Ali-Bodmer interaction,
$V_{fit}(r)=V_{r}e^{-(\mu_{r}r)^{2}}-V_{a}e^{-(\mu_{a}r)^{2}}$ (36)
with $V_{r},\mu_{r},V_{a},\mu_{a}$ fitting parameters. Since it is impossible
to obtain meaningful parameters in the whole radial range, we restrict the fit
in the relevant $r=(1.5-10)$ fm. The result is given in the Table 1. We obtain
almost perfect fits, Fig (6), but comparison with experimental data require to
repeat the renormalization procedure described above. The correction is of the
order of $1\%$ in all cases.
Table 1: Parameters for the fitted SUSY potentials. The parameter $\lambda$ is a renormalization constant which gives the best fit for the experimental S-state phase shift and the $0^{+}$ resonance in 8Be. Int | $V_{r}$(MeV) | $\mu_{r}$(fm-1) | $V_{a}$(MeV) | $\mu_{a}$(fm-1) | $\lambda$
---|---|---|---|---|---
BFW | 254.8000031 | 0.6470000 | 101.9716263 | 0.4600000 | 0.9920
D1 | 255.8999939 | 0.6049346 | 103.6447830 | 0.4370000 | 0.9891
D1N | 265.0000000 | 0.6266215 | 102.5655823 | 0.4459522 | 0.9873
D1S | 262.0000000 | 0.6194427 | 103.4447250 | 0.4437624 | 0.9906
## 4 Concluding remarks
We have calculated the $\alpha-\alpha$ interaction potential within the double
folding model using finite range density dependent NN effective interactions.
The knock-on nonlocal kernel corresponding to the finite range components of
the effective interaction is localized within the lowest order of the Perey-
Saxon approximation at zero energy. The resulted folding potentials are deep
with an average strength of $78\pm 7$ MeV very close to the value of Schmid
and Wildermuth [5] in their RGM calculation. The $\it{rms}$ radius of these
potentials is somewhat larger than the corresponding value of the
phenomenological BFW potential (see Fig. 1). Our deep folding potentials
reproduce quite well the experimental values of the S-state phase shift and
the properties of the first $0^{+}$ resonance in 8Be. The maximum deviation
from unity of the usual renormalization factor $\lambda$ is $9\%$.
Successive supersymmetric transformations which preserve the continuous
spectrum are used to remove the redundant 0S and 1S states in order to obtain
physically relevant potentials. The phase shift and the properties of the
$0^{+}$ resonance are calculated with the variable phase equation of Calogero
with proper boundary condition for singular potentials. A Gaussian expansion
of the resulted SUSY potentials shows a well known molecular pocket with an
almost unique long range attractive component with $\mu_{a}=0.442\pm 0.005$
fm-1. The potential minimum is located at about r=3 fm, which corresponds to a
touching configuration and therefore implies a very small overlap of the
single particle densities.
We believe that our potentials are physically meaningful in the energy range
$E_{lab}=0-5$ MeV. Beyond this range high $\ell$-order phase shift starts to
have significant values.
## 5 Acknoledgements
This work was supported by UEFISCDI-ROMANIA under program PN-II contract No.
55/2011 and by French-Romanian collaboration IN2P3/IFIN-HH. M. L. thanks to
the staff of DFT/IFIN-HH for the kind hospitality during the preparation of
this work.
## References
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* [11] D. Baye, Phys. Rev. Lett. 58, 2738 (1987).
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* [13] F. Carstoiu and R. J. Lombard, Ann. Phys. (N.Y.) 217, 279 (1992).
* [14] D. Gogny, in Proc. Int. Conf. on Nuclear Physics, eds. J. De Boer and H. Mang (North-Holland, Amsterdam, 1973).
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* [16] F. Chappert, M. Girod and S. Hilaire, Phys. Lett. B668, 420 (2009).
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* [25] S.Misicu, private comm.
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* [30] J. -M Sparenberg, D. Baye and H. Leeb, Phys. Rev. C 61, 024605 (2000)
* [31] R. G. Newton, Scattering Theory of Waves and Particles, (Springer, Berlin, 1982) 2nd ed.
* [32] K. Chadan and P.C. Sabatier, Inverse Problems in Quantum Scattering Theory (Springer, Berlin, 1989) 2nd ed.
* [33] F. Calogero, Variable Phase Approach to Potential Scattering, Academic Press, New York and London, (1967).
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|
arxiv-papers
| 2013-10-17T10:24:43 |
2024-09-04T02:49:52.507372
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "M. Lassaut, F. Carstoiu, V. Balanica",
"submitter": "Valentin Balanica",
"url": "https://arxiv.org/abs/1310.4648"
}
|
1310.4664
|
# SVD Factorization for Tall-and-Fat Matrices on Map/Reduce Architectures
Burak Bayramlı
(October 15, 2013)
###### Abstract
We demonstrate an implementation for an approximate rank-k SVD factorization,
combining well-known randomized projection techniques with previously
implemented map/reduce solutions in order to compute steps of the random
projection based SVD procedure, such QR and SVD. We structure the problem in a
way that it reduces to Cholesky and SVD factorizations on $k\times k$ matrices
computed on a single machine, greatly easing the computability of the problem.
## 1 Introduction
[1] presents many excellent techniques for utilizing map/reduce architectures
to compute QR and SVD for the so-called tall-and-skinny matrices. QR
factorization is turned into an $A^{T}A$ computation problem to be computed in
parallel using map/reduce, and its key element the Cholesky decomposition, can
be performed on a single machine. Let’s use $C=A^{T}A$ and, since
$C=A^{T}A=(QR)^{T}(QR)=R^{T}Q^{T}QR=R^{T}R$
and because Cholesky factorization of an $n\times n$ symmetric positive
definite matrix is
$C=LL^{T}$
where $L$ is an $n\times n$ lower triangular matrix, and R is upper
triangular, we can conclude if we factorize $C$ into $L$ and $L^{T}$, this
implies $C=LL^{T}=RR^{T}$, we have a method of calculating $R$ of QR using
Cholesky factorization on $A^{T}A$. The key observation here is $A^{T}A$
computation results an $n\times n$ matrix and if $A$ is “skinny” then $n$ is
relatively small (in the thousands), then Cholesky decomposition can be
executed on a small $n\times n$ matrix on a single computer utilizing an
already available function in a scientific computing library. $Q$ is computed
simply as $Q=AR^{-1}$. This again is relatively cheap because R is $n\times
n$, the inverse is computed locallly, matrix multiplication with $A$ can be
performed through map/reduce.
SVD is an additional step. SVD decomposition is
$A=U\Sigma V^{T}$
If we expand it with $A=QR$
$QR=U\Sigma V^{T}$
$R=Q^{T}U\Sigma V^{T}$
Let’s call $\tilde{U}=Q^{T}U$
$R=\tilde{U}\Sigma V^{T}$
This means if we run a local SVD on $R$ (we just calculated above with
Cholesky) which is an $n\times n$ matrix, we will have calculated $\tilde{U}$,
the real $\Sigma$, and real $V^{T}$.
Now we have a map/reduce way of calculating QR and SVD on $m\times n$ matrices
where $n$ is small.
### 1.1 Approximate rank-k SVD
Switching gears, we look at another method for calculating SVD. The motivation
is while computing SVD, if $n$ is large, creating a “fat” matrix which might
have columns in the billions would require reducing the dimensionality of the
problem. According to [2], one way to achieve is through random projection.
First we draw an $n\times k$ Gaussian random matrix $\Omega$. Then we
calculate
$Y=A\Omega$
We perform QR decomposition on $Y$
$Y=QR$
Then form $k\times n$ matrix
$B=Q^{T}A$
Then we can calculate SVD on this small matrix
$B=\hat{U}\Sigma V^{T}$
Then form the matrix
$U=Q\hat{U}$
The main idea is based on
$A=QQ^{T}A$
if replace $Q$ which comes from random projection $Y$,
$A\approx\tilde{Q}\tilde{Q}^{T}A$
$Q$ and $R$ of the projection are close to that of $A$. In the multiplication
above $R$ is called $B$ where $B=\tilde{Q}^{T}A$, and,
$A\approx\tilde{Q}B$
then, as in [1], we can take SVD of $B$ and apply the same transition rules to
obtain an approximate $U$ of $A$.
This approximation works because of the fact that projecting points to a
random subspace preserves distances between points, or in detail, projecting
the n-point subset onto a random subspace of $O(\log n/\epsilon^{2})$
dimensions only changes the interpoint distances by $(1\pm\epsilon)$ with
positive probability [3]. It is also said that $Y$ is a good representation of
the span of $A$.
### 1.2 Combining Both Methods
Our idea was using approximate k-rank SVD calculation steps where $k<<n$, and
using map/reduce based QR and SVD methods to implement those steps. By
utilizing random projection, we would be able to work in a smaller dimension
which would translate to local Cholesky, and SVD calls on $k\times k$ matrices
that can be performed in a speedy manner. Below we outline each map/reduce
job.
$\mbox{{random\\_projection\\_map}}(key,value)$
---
1 | input $A$
2 | returns $Y$
3 | Tokenize $value$ and pick out id value pairs
4 | result = zeros(1,$k$)
5 | for each $j^{th}$ $token$ $\in value$
6 | | Initialize seed with $j$
7 | | $j$ = generate $k$ random numbers
8 | | $result=result+r\cdot token[j]$
9 | emit key, result
First random projection job (whose reduce is a no-op). Each value of $A$ will
arrive to the algorithm as a key and value pair. Key is line number or other
identifier per row of $A$. Value is a collection of id value pairs where id is
column id this time, and value is the value for that column. Sparsity is
handled through this format, if an id for a column does not appear in a row of
A, it is assumed to be zero. The resulting $Y$ matrix has dimensions $m\times
k$.
$A^{T}A\mbox{{cholesky\\_job\\_map(key k,value a)}}$
---
1 | for $i,row$ in $\mbox{{enumerate}}a^{T}a$
2 | | emit $i,row$
$\mbox{{cholesky\\_job\\_reduce}}(key,value)$
---
1 | emit $k,\mbox{{sum}}(v_{j}^{k})$
$\mbox{{cholesky\\_job\\_final\\_local\\_reduce}}(key,value)$
---
1 | $result=\mbox{{cholesky}}(A_{sum})$
2 | $\mbox{{emit }}result$
The cholesky_job_final_local_reduce step is a function provided in most
map/reduce frameworks, it is a central point that collects the output of all
reducers, naturally a single machine which makes it ideal to execute the final
Cholesky call on by now a very small ($k\times k$) matrix. The output is $R$.
$\mbox{{Q\\_job\\_map}}(key,value)$
---
1 | During initialization, $R_{inv}=R^{-1}$, store it once for each mapper
2 | for $row$ in $Y$
3 | | $\mbox{{emit }}key,row\cdot R_{inv}$
There is no reducer in the $Q\mbox{{\\_job}}$, it is a very simple procedure,
it merely computes multiplication between row of $Y$ and a local matrix $R$.
Matrix $R$ is very small, $k\times k$, hence it can be kept locally in every
node. The initialiation is used to store the inverse of $R$ locally, once the
mapper is initialized, it will always use the same $R^{-1}$ for every
multiplication.
$A^{T}Q\mbox{{\\_job\\_map}}(key,value)$
---
1 | $left=row$ from $A$
2 | $right=row$ from $Q$
3 | for each non-zero $j^{th}$ cell in $left$
4 | | $\mbox{{emit }}j,left[j]\cdot right$
$A^{T}Q\mbox{{\\_job\\_reduce}}(key,value)$
---
1 | returs $B^{T}$
2 | $result=\mbox{{zeros}}(1,k)$
3 | $\mbox{{for }}row$ in $value$
4 | | $result=result+row$
5 | $\mbox{{emit }}key,result$
The job above takes an $AQ$ matrix which is assumed to be a join between $A$
and $Q$, per row, based on key. We split the row and deduce the $A$ part and
the $Q$ part, then iterate cells of $A$ one by one, which is assumed to be
sparse, and multiply the entire row of $Q$. Then for each $j^{th}$ non-zero
cell of $A$, we multiply this value with the row from $Q$ and emit the
multiplication result with key $j$.
The $Q^{T}A$ job’s formula can be seen at 1.1. For implementation purposes we
changed this formula into
$B^{T}=A^{T}Q$
because as output we needed to have a $n\times k$ matrix instead of a $k\times
n$ one, which would allow us to use map/reduce SVD that translates into a
local Cholesky and SVD on $k\times k$ matrices. Since we take SVD of $B^{T}$
instead of $B$, that changes the output as well,
$B=U\Sigma V^{T}$
becomes
$B^{T}=V\Sigma U^{T}$
In other words, in order to obtain $U$ of $B$, we need to take
$(U_{BT}^{T})^{T}$ from the SVD of $B^{T}$. This is how $A^{T}A$ Cholesky Job
is called, this time with $B^{T}$ as its input data.
$Q\tilde{U}\mbox{{\\_job\\_map}}(key,value)$
---
1 | input $Q,R$
2 | returns $U$
3 | initialization $\tilde{U}$ = svd of $R$
4 | for row in $Q$
5 | | $\mbox{{emit }}key,row\cdot\tilde{U}$
map_reduce_svd
---
1 | $Y$ = $\mbox{{random\\_projection\\_map}}(A)$
2 | $R_{Y}$ = $A^{T}A\mbox{{\\_cholesky\\_job}}(Y)$
3 | $Q_{Y}$ = $Q\mbox{{\\_job}}$
4 | $R_{BT}$ = $A^{T}A\mbox{{\\_cholesky\\_job}}(B^{T})$
5 | $U$ = $Q\tilde{U}\mbox{{\\_job}}(R_{BT},Q)$
### 1.3 Discussion
We performed our experiments on the Netflix dataset which has about 100
million from over 480,000 customers on 17770 movies. The implementation was
programmed on Sasha distributed framework [5], and SVD calculation on the full
dataset with $k=7$ on two notebook computers, utilizing in total 6 cores took
20 minutes. Scipy SVD calculation on the same dataset is much faster, however,
we need to stress our algorithms are prepared for cases where $N$ is very
large, i.e. in the billions. As such, for example during projection we did not
simply create and pre-store a $n\times k$ random matrix and multiply multiple
rows of $A$ with this matrix. This would certainly be possible for Netflix
data where $n$ is relatively small, but would not work well in cases where $A$
is “fat”. All code relevant for this paper can be found here [6].
There are only two passes necessary on the full dataset, and three passes on
$m$ rows but with reduced $k$ dimensions this time. Perhaps predictably, the
procedure spends most of its time at $A^{T}Q$ Job. This step performs not only
a join between $A$ and $Q$, it also emits $k$ cells per non-zero value of
$A$’s rows, then creates partial sums these $k$ vectors creating $n\times k$
result. If for simplicity we assume $k$ non-zero cells in each $A$ row, the
complexity of this step would be $O(mk)$.
## References
* [1] Gleich, Benson, Demmel, _Direct QR factorizations for tall-and-skinny matrices in MapReduce architectures_ , arXiv:1301.1071 [cs.DC], 2013
* [2] N. Halko, _Randomized methods for computing low-rank approximations of matrices_ , University of Colorado, Boulder, 2010
* [3] S. Dangupta, A. Gupta _An Elementary Proof of a Theorem of Johnson and Lindenstrauss_ , Wiley Periodicals, 2002
* [4] M. Kurucz, A. A. Benczúr, K. Csalogány, _Methods for large scale SVD with missing values_ , ACM, 2007
* [5] B. Bayramli, _Sasha Framework_ , [email protected]:burakbayramli/sasha.git Github, 2013
* [6] B. Bayramli, _Map/Reduce Code for Netflix SVD Analysis_ , http://github.com/burakbayramli/classnotes/tree/master/stat/stat_mr_rnd_svd/sasha, Github, 2013
|
arxiv-papers
| 2013-10-17T11:52:26 |
2024-09-04T02:49:52.515724
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/",
"authors": "Burak Bayramli",
"submitter": "Burak Bayramli",
"url": "https://arxiv.org/abs/1310.4664"
}
|
1310.4740
|
# Measurement of $C\\!P$ violation in the phase space of
$B^{\pm}\rightarrow K^{+}K^{-}\pi^{\pm}$ and
$B^{\pm}\rightarrow\pi^{+}\pi^{-}\pi^{\pm}$ decays
LHCb collaboration
R. Aaij40, B. Adeva36, M. Adinolfi45, C. Adrover6, A. Affolder51, Z.
Ajaltouni5, J. Albrecht9, F. Alessio37, M. Alexander50, S. Ali40, G.
Alkhazov29, P. Alvarez Cartelle36, A.A. Alves Jr24, S. Amato2, S. Amerio21, Y.
Amhis7, L. Anderlini17,f, J. Anderson39, R. Andreassen56, J.E. Andrews57, R.B.
Appleby53, O. Aquines Gutierrez10, F. Archilli18, A. Artamonov34, M. Artuso58,
E. Aslanides6, G. Auriemma24,m, M. Baalouch5, S. Bachmann11, J.J. Back47, A.
Badalov35, C. Baesso59, V. Balagura30, W. Baldini16, R.J. Barlow53, C.
Barschel37, S. Barsuk7, W. Barter46, Th. Bauer40, A. Bay38, J. Beddow50, F.
Bedeschi22, I. Bediaga1, S. Belogurov30, K. Belous34, I. Belyaev30, E. Ben-
Haim8, G. Bencivenni18, S. Benson49, J. Benton45, A. Berezhnoy31, R. Bernet39,
M.-O. Bettler46, M. van Beuzekom40, A. Bien11, S. Bifani44, T. Bird53, A.
Bizzeti17,h, P.M. Bjørnstad53, T. Blake37, F. Blanc38, J. Blouw10, S. Blusk58,
V. Bocci24, A. Bondar33, N. Bondar29, W. Bonivento15, S. Borghi53, A.
Borgia58, T.J.V. Bowcock51, E. Bowen39, C. Bozzi16, T. Brambach9, J. van den
Brand41, J. Bressieux38, D. Brett53, M. Britsch10, T. Britton58, N.H. Brook45,
H. Brown51, A. Bursche39, G. Busetto21,q, J. Buytaert37, S. Cadeddu15, O.
Callot7, M. Calvi20,j, M. Calvo Gomez35,n, A. Camboni35, P. Campana18,37, D.
Campora Perez37, A. Carbone14,c, G. Carboni23,k, R. Cardinale19,i, A.
Cardini15, H. Carranza-Mejia49, L. Carson52, K. Carvalho Akiba2, G. Casse51,
L. Castillo Garcia37, M. Cattaneo37, Ch. Cauet9, R. Cenci57, M. Charles54, Ph.
Charpentier37, S.-F. Cheung54, N. Chiapolini39, M. Chrzaszcz39,25, K. Ciba37,
X. Cid Vidal37, G. Ciezarek52, P.E.L. Clarke49, M. Clemencic37, H.V. Cliff46,
J. Closier37, C. Coca28, V. Coco40, J. Cogan6, E. Cogneras5, P. Collins37, A.
Comerma-Montells35, A. Contu15,37, A. Cook45, M. Coombes45, S. Coquereau8, G.
Corti37, B. Couturier37, G.A. Cowan49, D.C. Craik47, M. Cruz Torres59, S.
Cunliffe52, R. Currie49, C. D’Ambrosio37, P. David8, P.N.Y. David40, A.
Davis56, I. De Bonis4, K. De Bruyn40, S. De Capua53, M. De Cian11, J.M. De
Miranda1, L. De Paula2, W. De Silva56, P. De Simone18, D. Decamp4, M.
Deckenhoff9, L. Del Buono8, N. Déléage4, D. Derkach54, O. Deschamps5, F.
Dettori41, A. Di Canto11, H. Dijkstra37, M. Dogaru28, S. Donleavy51, F.
Dordei11, A. Dosil Suárez36, D. Dossett47, A. Dovbnya42, F. Dupertuis38, P.
Durante37, R. Dzhelyadin34, A. Dziurda25, A. Dzyuba29, S. Easo48, U. Egede52,
V. Egorychev30, S. Eidelman33, D. van Eijk40, S. Eisenhardt49, U.
Eitschberger9, R. Ekelhof9, L. Eklund50,37, I. El Rifai5, Ch. Elsasser39, A.
Falabella14,e, C. Färber11, C. Farinelli40, S. Farry51, D. Ferguson49, V.
Fernandez Albor36, F. Ferreira Rodrigues1, M. Ferro-Luzzi37, S. Filippov32, M.
Fiore16,e, C. Fitzpatrick37, M. Fontana10, F. Fontanelli19,i, R. Forty37, O.
Francisco2, M. Frank37, C. Frei37, M. Frosini17,37,f, E. Furfaro23,k, A.
Gallas Torreira36, D. Galli14,c, M. Gandelman2, P. Gandini58, Y. Gao3, J.
Garofoli58, P. Garosi53, J. Garra Tico46, L. Garrido35, C. Gaspar37, R.
Gauld54, E. Gersabeck11, M. Gersabeck53, T. Gershon47, Ph. Ghez4, V. Gibson46,
L. Giubega28, V.V. Gligorov37, C. Göbel59, D. Golubkov30, A. Golutvin52,30,37,
A. Gomes2, P. Gorbounov30,37, H. Gordon37, M. Grabalosa Gándara5, R. Graciani
Diaz35, L.A. Granado Cardoso37, E. Graugés35, G. Graziani17, A. Grecu28, E.
Greening54, S. Gregson46, P. Griffith44, L. Grillo11, O. Grünberg60, B. Gui58,
E. Gushchin32, Yu. Guz34,37, T. Gys37, C. Hadjivasiliou58, G. Haefeli38, C.
Haen37, S.C. Haines46, S. Hall52, B. Hamilton57, T. Hampson45, S. Hansmann-
Menzemer11, N. Harnew54, S.T. Harnew45, J. Harrison53, T. Hartmann60, J. He37,
T. Head37, V. Heijne40, K. Hennessy51, P. Henrard5, J.A. Hernando Morata36, E.
van Herwijnen37, M. Heß60, A. Hicheur1, E. Hicks51, D. Hill54, M. Hoballah5,
C. Hombach53, W. Hulsbergen40, P. Hunt54, T. Huse51, N. Hussain54, D.
Hutchcroft51, D. Hynds50, V. Iakovenko43, M. Idzik26, P. Ilten12, R.
Jacobsson37, A. Jaeger11, E. Jans40, P. Jaton38, A. Jawahery57, F. Jing3, M.
John54, D. Johnson54, C.R. Jones46, C. Joram37, B. Jost37, M. Kaballo9, S.
Kandybei42, W. Kanso6, M. Karacson37, T.M. Karbach37, I.R. Kenyon44, T.
Ketel41, B. Khanji20, O. Kochebina7, I. Komarov38, R.F. Koopman41, P.
Koppenburg40, M. Korolev31, A. Kozlinskiy40, L. Kravchuk32, K. Kreplin11, M.
Kreps47, G. Krocker11, P. Krokovny33, F. Kruse9, M. Kucharczyk20,25,37,j, V.
Kudryavtsev33, K. Kurek27, T. Kvaratskheliya30,37, V.N. La Thi38, D.
Lacarrere37, G. Lafferty53, A. Lai15, D. Lambert49, R.W. Lambert41, E.
Lanciotti37, G. Lanfranchi18, C. Langenbruch37, T. Latham47, C. Lazzeroni44,
R. Le Gac6, J. van Leerdam40, J.-P. Lees4, R. Lefèvre5, A. Leflat31, J.
Lefrançois7, S. Leo22, O. Leroy6, T. Lesiak25, B. Leverington11, Y. Li3, L. Li
Gioi5, M. Liles51, R. Lindner37, C. Linn11, B. Liu3, G. Liu37, S. Lohn37, I.
Longstaff50, J.H. Lopes2, N. Lopez-March38, H. Lu3, D. Lucchesi21,q, J.
Luisier38, H. Luo49, O. Lupton54, F. Machefert7, I.V. Machikhiliyan30, F.
Maciuc28, O. Maev29,37, S. Malde54, G. Manca15,d, G. Mancinelli6, J. Maratas5,
U. Marconi14, P. Marino22,s, R. Märki38, J. Marks11, G. Martellotti24, A.
Martens8, A. Martín Sánchez7, M. Martinelli40, D. Martinez Santos41,37, D.
Martins Tostes2, A. Martynov31, A. Massafferri1, R. Matev37, Z. Mathe37, C.
Matteuzzi20, E. Maurice6, A. Mazurov16,37,e, J. McCarthy44, A. McNab53, R.
McNulty12, B. McSkelly51, B. Meadows56,54, F. Meier9, M. Meissner11, M.
Merk40, D.A. Milanes8, M.-N. Minard4, J. Molina Rodriguez59, S. Monteil5, D.
Moran53, P. Morawski25, A. Mordà6, M.J. Morello22,s, R. Mountain58, I. Mous40,
F. Muheim49, K. Müller39, R. Muresan28, B. Muryn26, B. Muster38, P. Naik45, T.
Nakada38, R. Nandakumar48, I. Nasteva1, M. Needham49, S. Neubert37, N.
Neufeld37, A.D. Nguyen38, T.D. Nguyen38, C. Nguyen-Mau38,o, M. Nicol7, V.
Niess5, R. Niet9, N. Nikitin31, T. Nikodem11, A. Nomerotski54, A. Novoselov34,
A. Oblakowska-Mucha26, V. Obraztsov34, S. Oggero40, S. Ogilvy50, O.
Okhrimenko43, R. Oldeman15,d, M. Orlandea28, J.M. Otalora Goicochea2, P.
Owen52, A. Oyanguren35, B.K. Pal58, A. Palano13,b, M. Palutan18, J. Panman37,
A. Papanestis48, M. Pappagallo50, C. Parkes53, C.J. Parkinson52, G.
Passaleva17, G.D. Patel51, M. Patel52, G.N. Patrick48, C. Patrignani19,i, C.
Pavel-Nicorescu28, A. Pazos Alvarez36, A. Pearce53, A. Pellegrino40, G.
Penso24,l, M. Pepe Altarelli37, S. Perazzini14,c, E. Perez Trigo36, A. Pérez-
Calero Yzquierdo35, P. Perret5, M. Perrin-Terrin6, L. Pescatore44, E. Pesen61,
G. Pessina20, K. Petridis52, A. Petrolini19,i, A. Phan58, E. Picatoste
Olloqui35, B. Pietrzyk4, T. Pilař47, D. Pinci24, S. Playfer49, M. Plo
Casasus36, F. Polci8, G. Polok25, A. Poluektov47,33, E. Polycarpo2, A.
Popov34, D. Popov10, B. Popovici28, C. Potterat35, A. Powell54, J.
Prisciandaro38, A. Pritchard51, C. Prouve7, V. Pugatch43, A. Puig Navarro38,
G. Punzi22,r, W. Qian4, B. Rachwal25, J.H. Rademacker45, B.
Rakotomiaramanana38, M.S. Rangel2, I. Raniuk42, N. Rauschmayr37, G. Raven41,
S. Redford54, S. Reichert53, M.M. Reid47, A.C. dos Reis1, S. Ricciardi48, A.
Richards52, K. Rinnert51, V. Rives Molina35, D.A. Roa Romero5, P. Robbe7, D.A.
Roberts57, A.B. Rodrigues1, E. Rodrigues53, P. Rodriguez Perez36, S. Roiser37,
V. Romanovsky34, A. Romero Vidal36, M. Rotondo21, J. Rouvinet38, T. Ruf37, F.
Ruffini22, H. Ruiz35, P. Ruiz Valls35, G. Sabatino24,k, J.J. Saborido Silva36,
N. Sagidova29, P. Sail50, B. Saitta15,d, V. Salustino Guimaraes2, B. Sanmartin
Sedes36, R. Santacesaria24, C. Santamarina Rios36, E. Santovetti23,k, M.
Sapunov6, A. Sarti18, C. Satriano24,m, A. Satta23, M. Savrie16,e, D.
Savrina30,31, M. Schiller41, H. Schindler37, M. Schlupp9, M. Schmelling10, B.
Schmidt37, O. Schneider38, A. Schopper37, M.-H. Schune7, R. Schwemmer37, B.
Sciascia18, A. Sciubba24, M. Seco36, A. Semennikov30, K. Senderowska26, I.
Sepp52, N. Serra39, J. Serrano6, P. Seyfert11, M. Shapkin34, I.
Shapoval16,42,e, Y. Shcheglov29, T. Shears51, L. Shekhtman33, O. Shevchenko42,
V. Shevchenko30, A. Shires9, R. Silva Coutinho47, M. Sirendi46, N. Skidmore45,
T. Skwarnicki58, N.A. Smith51, E. Smith54,48, E. Smith52, J. Smith46, M.
Smith53, M.D. Sokoloff56, F.J.P. Soler50, F. Soomro38, D. Souza45, B. Souza De
Paula2, B. Spaan9, A. Sparkes49, P. Spradlin50, F. Stagni37, S. Stahl11, O.
Steinkamp39, S. Stevenson54, S. Stoica28, S. Stone58, B. Storaci39, M.
Straticiuc28, U. Straumann39, V.K. Subbiah37, L. Sun56, W. Sutcliffe52, S.
Swientek9, V. Syropoulos41, M. Szczekowski27, P. Szczypka38,37, D. Szilard2,
T. Szumlak26, S. T’Jampens4, M. Teklishyn7, E. Teodorescu28, F. Teubert37, C.
Thomas54, E. Thomas37, J. van Tilburg11, V. Tisserand4, M. Tobin38, S. Tolk41,
D. Tonelli37, S. Topp-Joergensen54, N. Torr54, E. Tournefier4,52, S.
Tourneur38, M.T. Tran38, M. Tresch39, A. Tsaregorodtsev6, P. Tsopelas40, N.
Tuning40,37, M. Ubeda Garcia37, A. Ukleja27, A. Ustyuzhanin52,p, U. Uwer11, V.
Vagnoni14, G. Valenti14, A. Vallier7, R. Vazquez Gomez18, P. Vazquez
Regueiro36, C. Vázquez Sierra36, S. Vecchi16, J.J. Velthuis45, M. Veltri17,g,
G. Veneziano38, M. Vesterinen37, B. Viaud7, D. Vieira2, X. Vilasis-
Cardona35,n, A. Vollhardt39, D. Volyanskyy10, D. Voong45, A. Vorobyev29, V.
Vorobyev33, C. Voß60, H. Voss10, R. Waldi60, C. Wallace47, R. Wallace12, S.
Wandernoth11, J. Wang58, D.R. Ward46, N.K. Watson44, A.D. Webber53, D.
Websdale52, M. Whitehead47, J. Wicht37, J. Wiechczynski25, D. Wiedner11, L.
Wiggers40, G. Wilkinson54, M.P. Williams47,48, M. Williams55, F.F. Wilson48,
J. Wimberley57, J. Wishahi9, W. Wislicki27, M. Witek25, G. Wormser7, S.A.
Wotton46, S. Wright46, S. Wu3, K. Wyllie37, Y. Xie49,37, Z. Xing58, Z. Yang3,
X. Yuan3, O. Yushchenko34, M. Zangoli14, M. Zavertyaev10,a, F. Zhang3, L.
Zhang58, W.C. Zhang12, Y. Zhang3, A. Zhelezov11, A. Zhokhov30, L. Zhong3, A.
Zvyagin37.
1Centro Brasileiro de Pesquisas Físicas (CBPF), Rio de Janeiro, Brazil
2Universidade Federal do Rio de Janeiro (UFRJ), Rio de Janeiro, Brazil
3Center for High Energy Physics, Tsinghua University, Beijing, China
4LAPP, Université de Savoie, CNRS/IN2P3, Annecy-Le-Vieux, France
5Clermont Université, Université Blaise Pascal, CNRS/IN2P3, LPC, Clermont-
Ferrand, France
6CPPM, Aix-Marseille Université, CNRS/IN2P3, Marseille, France
7LAL, Université Paris-Sud, CNRS/IN2P3, Orsay, France
8LPNHE, Université Pierre et Marie Curie, Université Paris Diderot,
CNRS/IN2P3, Paris, France
9Fakultät Physik, Technische Universität Dortmund, Dortmund, Germany
10Max-Planck-Institut für Kernphysik (MPIK), Heidelberg, Germany
11Physikalisches Institut, Ruprecht-Karls-Universität Heidelberg, Heidelberg,
Germany
12School of Physics, University College Dublin, Dublin, Ireland
13Sezione INFN di Bari, Bari, Italy
14Sezione INFN di Bologna, Bologna, Italy
15Sezione INFN di Cagliari, Cagliari, Italy
16Sezione INFN di Ferrara, Ferrara, Italy
17Sezione INFN di Firenze, Firenze, Italy
18Laboratori Nazionali dell’INFN di Frascati, Frascati, Italy
19Sezione INFN di Genova, Genova, Italy
20Sezione INFN di Milano Bicocca, Milano, Italy
21Sezione INFN di Padova, Padova, Italy
22Sezione INFN di Pisa, Pisa, Italy
23Sezione INFN di Roma Tor Vergata, Roma, Italy
24Sezione INFN di Roma La Sapienza, Roma, Italy
25Henryk Niewodniczanski Institute of Nuclear Physics Polish Academy of
Sciences, Kraków, Poland
26AGH - University of Science and Technology, Faculty of Physics and Applied
Computer Science, Kraków, Poland
27National Center for Nuclear Research (NCBJ), Warsaw, Poland
28Horia Hulubei National Institute of Physics and Nuclear Engineering,
Bucharest-Magurele, Romania
29Petersburg Nuclear Physics Institute (PNPI), Gatchina, Russia
30Institute of Theoretical and Experimental Physics (ITEP), Moscow, Russia
31Institute of Nuclear Physics, Moscow State University (SINP MSU), Moscow,
Russia
32Institute for Nuclear Research of the Russian Academy of Sciences (INR RAN),
Moscow, Russia
33Budker Institute of Nuclear Physics (SB RAS) and Novosibirsk State
University, Novosibirsk, Russia
34Institute for High Energy Physics (IHEP), Protvino, Russia
35Universitat de Barcelona, Barcelona, Spain
36Universidad de Santiago de Compostela, Santiago de Compostela, Spain
37European Organization for Nuclear Research (CERN), Geneva, Switzerland
38Ecole Polytechnique Fédérale de Lausanne (EPFL), Lausanne, Switzerland
39Physik-Institut, Universität Zürich, Zürich, Switzerland
40Nikhef National Institute for Subatomic Physics, Amsterdam, The Netherlands
41Nikhef National Institute for Subatomic Physics and VU University Amsterdam,
Amsterdam, The Netherlands
42NSC Kharkiv Institute of Physics and Technology (NSC KIPT), Kharkiv, Ukraine
43Institute for Nuclear Research of the National Academy of Sciences (KINR),
Kyiv, Ukraine
44University of Birmingham, Birmingham, United Kingdom
45H.H. Wills Physics Laboratory, University of Bristol, Bristol, United
Kingdom
46Cavendish Laboratory, University of Cambridge, Cambridge, United Kingdom
47Department of Physics, University of Warwick, Coventry, United Kingdom
48STFC Rutherford Appleton Laboratory, Didcot, United Kingdom
49School of Physics and Astronomy, University of Edinburgh, Edinburgh, United
Kingdom
50School of Physics and Astronomy, University of Glasgow, Glasgow, United
Kingdom
51Oliver Lodge Laboratory, University of Liverpool, Liverpool, United Kingdom
52Imperial College London, London, United Kingdom
53School of Physics and Astronomy, University of Manchester, Manchester,
United Kingdom
54Department of Physics, University of Oxford, Oxford, United Kingdom
55Massachusetts Institute of Technology, Cambridge, MA, United States
56University of Cincinnati, Cincinnati, OH, United States
57University of Maryland, College Park, MD, United States
58Syracuse University, Syracuse, NY, United States
59Pontifícia Universidade Católica do Rio de Janeiro (PUC-Rio), Rio de
Janeiro, Brazil, associated to 2
60Institut für Physik, Universität Rostock, Rostock, Germany, associated to 11
61Celal Bayar University, Manisa, Turkey, associated to 37
aP.N. Lebedev Physical Institute, Russian Academy of Science (LPI RAS),
Moscow, Russia
bUniversità di Bari, Bari, Italy
cUniversità di Bologna, Bologna, Italy
dUniversità di Cagliari, Cagliari, Italy
eUniversità di Ferrara, Ferrara, Italy
fUniversità di Firenze, Firenze, Italy
gUniversità di Urbino, Urbino, Italy
hUniversità di Modena e Reggio Emilia, Modena, Italy
iUniversità di Genova, Genova, Italy
jUniversità di Milano Bicocca, Milano, Italy
kUniversità di Roma Tor Vergata, Roma, Italy
lUniversità di Roma La Sapienza, Roma, Italy
mUniversità della Basilicata, Potenza, Italy
nLIFAELS, La Salle, Universitat Ramon Llull, Barcelona, Spain
oHanoi University of Science, Hanoi, Viet Nam
pInstitute of Physics and Technology, Moscow, Russia
qUniversità di Padova, Padova, Italy
rUniversità di Pisa, Pisa, Italy
sScuola Normale Superiore, Pisa, Italy
###### Abstract
The charmless decays $B^{\pm}\rightarrow K^{+}K^{-}\pi^{\pm}$ and
$B^{\pm}\rightarrow\pi^{+}\pi^{-}\pi^{\pm}$ are reconstructed in a data set,
corresponding to an integrated luminosity of 1.0 fb-1 of $pp$ collisions at a
center-of-mass energy of 7 TeV, collected by LHCb in 2011. The inclusive
charge asymmetries of these modes are measured to be
$A_{C\\!P}(B^{\pm}\rightarrow K^{+}K^{-}\pi^{\pm})=-0.141\pm
0.040\mathrm{\,(stat)}\pm 0.018\mathrm{\,(syst)}\pm
0.007({J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{\pm})$ and
$A_{C\\!P}(B^{\pm}\rightarrow\pi^{+}\pi^{-}\pi^{\pm})=0.117\pm
0.021\mathrm{\,(stat)}\pm 0.009\mathrm{\,(syst)}\pm
0.007({J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{\pm})$, where the third
uncertainty is due to the $C\\!P$ asymmetry of the
${B^{\pm}\to{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{\pm}}$ reference
mode. In addition to the inclusive $C\\!P$ asymmetries, larger asymmetries are
observed in localized regions of phase space.
###### pacs:
13.25.Hw,11.30.Er
The LHCb collaboration
EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN)
| |
---|---|---
| | CERN-PH-EP-2013-190
| | LHCb-PAPER-2013-051
| | 17 October 2013
© CERN on behalf of the LHCb collaboration, license CC-BY-3.0.
Submitted to Phys. Rev. Lett.
Charmless decays of $B$ mesons to three hadrons are dominated by quasi-two-
body processes involving intermediate resonant states. The rich interference
pattern present in such decays makes them favorable for the investigation of
charge asymmetries that are localized in the phase space Miranda1 ; Miranda2 .
The large samples of charmless $B$ decays collected by the LHCb experiment
allow direct $C\\!P$ violation to be measured in regions of phase space. In
previous measurements of this type, the phase spaces of ${B^{\pm}\to
K^{\pm}K^{+}K^{-}}$ and ${B^{\pm}\to K^{\pm}\pi^{+}\pi^{-}}$ decays were
observed to have regions of large local asymmetries LHCb-PAPER-2013-027 .
Concerning baryonic modes, no significant effects have been observed in either
${B^{\pm}\to p\bar{p}K^{\pm}}$ or ${B^{\pm}\to p\bar{p}\pi^{\pm}}$ decays
LHCB-PAPER-2013-031 . Large $C\\!P$-violating asymmetries have also been
observed in charmless two-body $B$ meson decays such as $B^{0}\to
K^{+}\pi^{-}$ and $B^{0}_{s}\to K^{-}\pi^{+}$ (and the corresponding $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ and $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ decays) LHCb-PAPER-2013-018 .
Some recent efforts have been made to understand the origin of the large
asymmetries. For direct $C\\!P$ violation to occur, two interfering amplitudes
with different weak and strong phases must be involved in the decay process
BSS1979 . Interference between intermediate states of the decay can introduce
large strong phase differences, and is one mechanism for explaining local
asymmetries in the phase space PhysRevD.87.076007 ; Bhattacharya:2013cvn .
Another explanation focuses on final-state $KK\leftrightarrow\pi\pi$
rescattering, which can occur between decay channels with the same flavor
quantum numbers LHCb-PAPER-2013-027 ; Bhattacharya:2013cvn ; IgnacioCPT .
Invariance of $C\\!PT$ symmetry constrains hadron rescattering so that the sum
of the partial decay widths, for all channels with the same final-state
quantum numbers related by the S matrix, must be equal for charge-conjugated
decays. Effects of SU(3) flavor symmetry breaking have also been investigated
and partially explain the observed patterns Xu:2013dta ; Bhattacharya:2013cvn
; Gronau:2013mda .
The ${B^{\pm}\to K^{+}K^{-}\pi^{\pm}}$ decay is interesting because $s\bar{s}$
resonant contributions are strongly suppressed ozzi1 ; ozzi2 ; ozzi4 .
Recently, LHCb reported an upper limit on the $\phi$ contribution to be
$\mathcal{B}(B^{\pm}\to\phi\pi^{\pm})<1.5\times 10^{-7}$ at the 90% confidence
level LHCB-PAPER-2013-048 . The lack of $K^{+}K^{-}$ resonant contributions
makes the ${B^{\pm}\to K^{+}K^{-}\pi^{\pm}}$ decay a good probe for
rescattering from decays with pions. The ${B^{\pm}\to\pi^{+}\pi^{-}\pi^{\pm}}$
mode, on the other hand, has large resonant contributions, as shown in an
amplitude analysis by the BaBar collaboration, which measured the inclusive
$C\\!P$ asymmetry to be $(0.03\pm 0.06)$ BaBarpipipi . For ${B^{\pm}\to
K^{+}K^{-}\pi^{\pm}}$ decays, the inclusive $C\\!P$-violating asymmetry was
measured by the BaBar collaboration to be ($0.00\pm 0.10$) BaBarkkpi , from a
comparison of $B^{+}$ and $B^{-}$ sample fits. Both results are compatible
with the no $C\\!P$-violation hypothesis.
In this Letter we report measurements of the inclusive $C\\!P$-violating
asymmetries for ${B^{\pm}\to\pi^{+}\pi^{-}\pi^{\pm}}$ and ${B^{\pm}\to
K^{+}K^{-}\pi^{\pm}}$ decays. The $C\\!P$ asymmetry in $B^{\pm}$ decays to a
final state $f^{\pm}$ is defined as
$A_{C\\!P}(B^{\pm}\to f^{\pm})\equiv\Phi[\Gamma(B^{-}\to
f^{-}),\Gamma(B^{+}\to f^{+})],$ (1)
where $\Phi[X,Y]\equiv(X-Y)/(X+Y)$ is the asymmetry function, $\Gamma$ is the
decay width, and the final states $f^{\pm}$ are $\pi^{+}\pi^{-}\pi^{\pm}$ or
$K^{+}K^{-}\pi^{\pm}$. The asymmetry distributions across the phase space are
also investigated.
The LHCb detector Alves:2008zz is a single-arm forward spectrometer covering
the pseudorapidity range $2<\eta<5$, designed for the study of particles
containing $b$ or $c$ quarks. The analysis is based on $pp$ collision data,
corresponding to an integrated luminosity of 1.0 fb-1, collected in 2011 at a
center-of-mass energy of 7 TeV.
Figure 1: Invariant mass spectra of (a) ${B^{\pm}\to\pi^{+}\pi^{-}\pi^{\pm}}$
decays and (b) ${B^{\pm}\to K^{+}K^{-}\pi^{\pm}}$ decays. The left panel in
each figure shows the $B^{-}$ modes and the right panel shows the $B^{+}$
modes. The results of the unbinned maximum likelihood fits are overlaid. The
main components of the fit are also shown.
Events are selected by a trigger LHCb-DP-2012-004 that consists of a hardware
stage, based on information from a calorimeter system and five muon stations,
followed by a software stage, which applies a full event reconstruction.
Candidate events are first required to pass the hardware trigger, which
selects particles with a large transverse energy. The software trigger
requires a two-, three- or four-track secondary vertex with a high sum of the
transverse momenta, $p_{\rm T}$, of the tracks and significant displacement
from the primary $pp$ interaction vertices (PVs). At least one track should
have $\mbox{$p_{\rm T}$}>1.7{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ and
$\chi^{2}_{\rm IP}$ with respect to any primary interaction greater than 16,
where $\chi^{2}_{\rm IP}$ is defined as the difference between the $\chi^{2}$
of a given PV reconstructed with and without the considered track, and IP is
the impact parameter. A multivariate algorithm BBDT is used for the
identification of secondary vertices consistent with the decay of a $b$
hadron.
Further criteria are applied offline to select $B$ mesons and suppress the
combinatorial background. The $B^{\pm}$ decay products are required to satisfy
a set of selection criteria on their momenta, their $p_{\rm T}$, the
$\chi^{2}_{\rm IP}$ of the final-state tracks, and the distance of closest
approach between any two tracks. The $B$ candidates are required to have
$\mbox{$p_{\rm T}$}>1.7{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$, $\chi^{2}_{\rm
IP}<10$ (defined by projecting the $B$ candidate trajectory backwards from its
decay vertex), decay vertex $\chi^{2}<12$, and decay vertex displacement from
any PV greater than 3 mm. Additional requirements are applied to variables
related to the $B$-meson production and decay, such as the angle $\theta$
between the $B$-candidate momentum and the direction of flight from the
primary vertex to the decay vertex, $\cos(\theta)>0.99998$. Final-state kaons
and pions are further selected using particle identification information,
provided by two ring-imaging Cherenkov detectors LHCb-DP-2012-003 , and are
required to be incompatible with a muon LHCb-DP-2013-001 . The kinematic
selection is common to both decay channels, while the particle identification
selection is specific to each final state. Charm contributions are removed by
excluding the regions of $\pm 30{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$
around the world average value of the $D^{0}$ mass PDG2012 in the two-body
invariant masses $m_{\pi^{+}\pi^{-}}$, $m_{K^{\pm}\pi^{\mp}}$ and
$m_{K^{+}K^{-}}$.
The simulated events used in this analysis are generated using Pythia 6.4
Sjostrand:2006za with a specific LHCb configuration LHCb-PROC-2010-056 .
Decays of hadronic particles are produced by EvtGen Lange:2001uf , in which
final-state radiation is generated using Photos Golonka:2005pn . The
interaction of the generated particles with the detector and its response are
implemented using the Geant4 toolkit Allison:2006ve ; *Agostinelli:2002hh as
described in Ref. LHCb-PROC-2011-006 .
Unbinned extended maximum likelihood fits to the mass spectra of the selected
$B^{\pm}$ candidates are performed to obtain the signal yields and raw
asymmetries. The ${B^{\pm}\to K^{+}K^{-}\pi^{\pm}}$ and
${B^{\pm}\to\pi^{+}\pi^{-}\pi^{\pm}}$ signal components are parametrized by a
Cruijff function Cruijff with equal left and right widths and different
radiative tails to account for the asymmetric effect of final-state radiation
on the signal shape. The means and widths are left to float in the fit, while
the tail parameters are fixed to the values obtained from simulation. The
combinatorial background is described by an exponential distribution whose
parameter is left free in the fit. The backgrounds due to partially
reconstructed four-body $B$ decays are parametrized by an ARGUS distribution
Argus convolved with a Gaussian resolution function. For
${B^{\pm}\to\pi^{+}\pi^{-}\pi^{\pm}}$ decays the shape and yield parameters
describing the backgrounds are varied in the fit, while for ${B^{\pm}\to
K^{+}K^{-}\pi^{\pm}}$ decays they are taken from simulation, due to a further
contribution from four-body $B^{0}_{s}$ decays such as $B^{0}_{s}\to
D^{-}_{s}(K^{+}K^{-}\pi^{-})\pi^{+}$. We define peaking backgrounds as decay
modes with one misidentified particle, namely the channels ${B^{\pm}\to
K^{\pm}\pi^{+}\pi^{-}}$ for the ${B^{\pm}\to\pi^{+}\pi^{-}\pi^{\pm}}$ mode,
and ${B^{\pm}\to K^{\pm}\pi^{+}\pi^{-}}$ and ${B^{\pm}\to K^{\pm}K^{+}K^{-}}$
for the ${B^{\pm}\to K^{+}K^{-}\pi^{\pm}}$ mode. The shapes and yields of the
peaking backgrounds are obtained from simulation. The yields of the peaking
and partially reconstructed background components are constrained to be equal
for $B^{+}$ and $B^{-}$ decays. The invariant mass spectra of the ${B^{\pm}\to
K^{+}K^{-}\pi^{\pm}}$ and ${B^{\pm}\to\pi^{+}\pi^{-}\pi^{\pm}}$ candidates are
shown in Fig. 1.
The signal yields obtained are $N(KK\pi)=1870\pm 133$ and
$N(\pi\pi\pi)=4904\pm 148$, and the raw asymmetries are $A_{\rm
raw}(K\\!K\pi)=-0.143\pm 0.040$ and $A_{\rm raw}(\pi\pi\pi)=0.124\pm 0.020$,
where the uncertainties are statistical. The $C\\!P$ asymmetries are expressed
in terms of the measured raw asymmetries, corrected for effects induced by the
detector acceptance and interactions of final-state pions with matter $A_{\rm
D}(\pi^{\pm})$, as well as for a possible $B$-meson production asymmetry
$A_{\rm P}(B^{\pm})$,
$\\!\\!\\!A_{C\\!P}\\!=\\!A_{\rm raw}\\!-\\!A_{\rm D}(\pi^{\pm})\\!-\\!A_{\rm
P}(B^{\pm}).$ (2)
The pion detection asymmetry, $A_{\rm D}(\pi^{\pm})=0.0000\pm 0.0025$, has
been previously measured by LHCb LHCb-PAPER-2012-009 . The production
asymmetry $A_{\rm P}(B^{\pm})$ is measured from a data sample of approximately
$6.3\times 10^{4}$ $B^{\pm}\to{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}(\mu^{+}\mu^{-})K^{\pm}$ decays. The
${B^{\pm}\to{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{\pm}}$ sample
passes the same trigger, kinematic, and kaon particle identification selection
criteria as the signal samples, and it has a similar event topology. The
$A_{\rm P}(B^{\pm})$ term is obtained from the raw asymmetry of the
${B^{\pm}\to{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{\pm}}$ mode as
$A_{\rm P}(B^{\pm})=A_{\rm raw}({J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K)-A_{C\\!P}({J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K)-A_{\rm
D}(K^{\pm}),$ (3)
where $A_{C\\!P}({J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K)=0.001\pm
0.007$ PDG2012 is the world average $C\\!P$ asymmetry of
${B^{\pm}\to{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{\pm}}$ decays, and
$A_{\rm D}(K^{\pm})=-0.010\pm 0.003$ is the kaon interaction asymmetry
obtained from $D^{0}\to K^{\pm}\pi^{\mp}$ and $D^{0}\to K^{+}K^{-}$ decays
LHCb-PAPER-2011-029 , and corrected for $A_{\rm D}(\pi^{\pm})$. The $C\\!P$
asymmetries of the ${B^{\pm}\to K^{+}K^{-}\pi^{\pm}}$ and
${B^{\pm}\to\pi^{+}\pi^{-}\pi^{\pm}}$ channels are then determined using Eqs.
2 and 3.
Figure 2: Asymmetries of the number of events (including signal and
background) in bins of the Dalitz plot, $A_{\rm raw}^{N}$, for (a)
${B^{\pm}\to\pi^{+}\pi^{-}\pi^{\pm}}$ and (b) ${B^{\pm}\to
K^{+}K^{-}\pi^{\pm}}$ decays. The inset figures show the projections of the
number of events in bins of (a) the $m^{2}_{\pi^{+}\pi^{-}\,{\rm low}}$
variable for $m^{2}_{\pi^{+}\pi^{-}\,{\rm
high}}>15{\mathrm{\,Ge\kern-1.00006ptV^{2}\\!/}c^{4}}$ and (b) the
$m^{2}_{K^{+}K^{-}}$ variable. The distributions are not corrected for
efficiency.
Since the detector efficiencies for the signal modes are not uniform across
the Dalitz plot, and the raw asymmetries are also not uniformly distributed,
an acceptance correction is applied to the integrated raw asymmetries. It is
determined by the ratio between the $B^{-}$ and $B^{+}$ average efficiencies
in simulated events, reweighted to reproduce the population of signal data
over the phase space. Furthermore, the detector acceptance and reconstruction
depend on the trigger selection. The efficiency of the hadronic hardware
trigger is found to have a small charge asymmetry for kaons. Therefore, the
data are divided into two samples: events with candidates selected by the
hadronic trigger and events selected by other triggers independently of the
signal candidate. The acceptance correction and subtraction of the $A_{\rm
P}(B^{\pm})$ term is performed separately for each trigger configuration. The
trigger-averaged value of the production asymmetry is $A_{\rm
P}(B^{\pm})=-0.004\pm 0.004$, where the uncertainty is statistical only. The
integrated $C\\!P$ asymmetries are then the weighted averages of the $C\\!P$
asymmetries for the two trigger samples.
The methods used in estimating the systematic uncertainties of the signal
model, combinatorial background, peaking background, and acceptance correction
are the same as those used in Ref. LHCb-PAPER-2013-027 . For ${B^{\pm}\to
K^{+}K^{-}\pi^{\pm}}$ decays, we also evaluate a systematic uncertainty due to
the partially reconstructed background model by varying the mean and
resolution according to the difference between simulation and data obtained
from the signal component. The $A_{\rm D}(\pi^{\pm})$ and $A_{\rm D}(K^{\pm})$
uncertainties are included as systematic uncertainties related to the
procedure. A systematic uncertainty is also evaluated to account for the
difference in kaon kinematics between the $B^{\pm}$ and $D^{0}$ decays. The
systematic uncertainties for the measurements of $A_{C\\!P}({B^{\pm}\to
K^{+}K^{-}\pi^{\pm}})$ and $A_{C\\!P}({B^{\pm}\to\pi^{+}\pi^{-}\pi^{\pm}})$
are summarized in Table 1.
The results obtained for the inclusive $C\\!P$ asymmetries of the ${B^{\pm}\to
K^{+}K^{-}\pi^{\pm}}$ and ${B^{\pm}\to\pi^{+}\pi^{-}\pi^{\pm}}$ decays are
$\displaystyle A_{C\\!P}({B^{\pm}\to K^{+}K^{-}\pi^{\pm}})\\!$
$\displaystyle=$ $\displaystyle\\!-0.141\pm 0.040\pm 0.018\pm 0.007,$
$\displaystyle A_{C\\!P}({B^{\pm}\to\pi^{+}\pi^{-}\pi^{\pm}})\\!$
$\displaystyle=$ $\displaystyle\\!0.117\pm 0.021\pm 0.009\pm 0.007,$
where the first uncertainty is statistical, the second is the experimental
systematic, and the third is due to the $C\\!P$ asymmetry of the
${B^{\pm}\to{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{\pm}}$ reference
mode PDG2012 . The significances of the inclusive charge asymmetries,
calculated by dividing the central values by the sum in quadrature of the
statistical and both systematic uncertainties, are 3.2 standard deviations
($\sigma$) for ${B^{\pm}\to K^{+}K^{-}\pi^{\pm}}$ and $4.9\sigma$ for
${B^{\pm}\to\pi^{+}\pi^{-}\pi^{\pm}}$ decays.
Table 1: Systematic uncertainties on $A_{C\\!P}({B^{\pm}\to K^{+}K^{-}\pi^{\pm}})$ and $A_{C\\!P}({B^{\pm}\to\pi^{+}\pi^{-}\pi^{\pm}})$. The total systematic uncertainties are the sum in quadrature of the individual contributions. Systematic uncertainty | $A_{C\\!P}(K\\!K\pi)$ | $A_{C\\!P}(\pi\pi\pi)$
---|---|---
Signal model | 0.001 | 0.0005
Combinatorial background | 0.003 | 0.0008
Peaking background | $\;\;\;\,0.001$ | $\;\;\;\,0.0025$
Acceptance | 0.014 | 0.0032
Part. rec. background | 0.005 | –
$A_{\rm D}(\pi^{\pm})$ uncertainty | 0.003 | 0.0025
$A_{\rm D}(K^{\pm})$ uncertainty | 0.003 | 0.0032
$A_{\rm D}(K^{\pm})$ kaon kinematics | 0.008 | 0.0075
Total | 0.018 | 0.0094
Figure 3: Invariant mass spectra of (a) ${B^{\pm}\to\pi^{+}\pi^{-}\pi^{\pm}}$
decays in the region $m^{2}_{\pi^{+}\pi^{-}\,{\rm
low}}<0.4{\mathrm{\,Ge\kern-1.00006ptV^{2}\\!/}c^{4}}$ and
$m^{2}_{\pi^{+}\pi^{-}\,{\rm
high}}>15{\mathrm{\,Ge\kern-1.00006ptV^{2}\\!/}c^{4}}$, and (b) ${B^{\pm}\to
K^{+}K^{-}\pi^{\pm}}$ decays in the region
$m^{2}_{K^{+}K^{-}}<1.5{\mathrm{\,Ge\kern-1.00006ptV^{2}\\!/}c^{4}}$. The left
panel in each figure shows the $B^{-}$ modes and the right panel shows the
$B^{+}$ modes. The results of the unbinned maximum likelihood fits are
overlaid.
In addition to the inclusive charge asymmetries, we study the asymmetry
distributions in the two-dimensional phase space of two-body invariant masses.
The Dalitz plot distributions in the signal region, defined as the three-body
invariant mass region within two Gaussian widths from the signal peak, are
divided into bins with approximately equal numbers of events in the combined
$B^{-}$ and $B^{+}$ samples. Figure 2 shows the raw asymmetries (not corrected
for efficiency), $A_{\rm raw}^{N}=\Phi[N^{-},N^{+}]$, computed using the
number of negative ($N^{-}$) and positive ($N^{+}$) entries in each bin of the
${B^{\pm}\to\pi^{+}\pi^{-}\pi^{\pm}}$ and ${B^{\pm}\to K^{+}K^{-}\pi^{\pm}}$
Dalitz plots. The ${B^{\pm}\to\pi^{+}\pi^{-}\pi^{\pm}}$ Dalitz plot is
symmetrized and its two-body invariant mass squared variables are defined as
$m^{2}_{\pi^{+}\pi^{-}\,{\rm low}}<m^{2}_{\pi^{+}\pi^{-}\,{\rm high}}$. The
$A_{\rm raw}^{N}$ distribution in the Dalitz plot of the
${B^{\pm}\to\pi^{+}\pi^{-}\pi^{\pm}}$ mode reveals an asymmetry concentrated
at low values of $m^{2}_{\pi^{+}\pi^{-}\,{\rm low}}$ and high values of
$m^{2}_{\pi^{+}\pi^{-}\,{\rm high}}$. The distribution of the projection of
the number of events onto the $m^{2}_{\pi^{+}\pi^{-}\,{\rm low}}$ invariant
mass (inset in Fig. 2(a)) shows that this asymmetry is located in the region
$m^{2}_{\pi^{+}\pi^{-}\,{\rm
low}}<0.4{\mathrm{\,Ge\kern-1.00006ptV^{2}\\!/}c^{4}}$ and
$m^{2}_{\pi^{+}\pi^{-}\,{\rm
high}}>15{\mathrm{\,Ge\kern-1.00006ptV^{2}\\!/}c^{4}}$. For ${B^{\pm}\to
K^{+}K^{-}\pi^{\pm}}$ we identify a negative asymmetry located in the low
$K^{+}K^{-}$ invariant mass region. This can be seen also in the inset figure
of the $K^{+}K^{-}$ invariant mass projection, where there is an excess of
$B^{+}$ candidates for
$m^{2}_{K^{+}K^{-}}<1.5{\mathrm{\,Ge\kern-1.00006ptV^{2}\\!/}c^{4}}$. Although
${B^{\pm}\to K^{+}K^{-}\pi^{\pm}}$ has no $\phi(1020)$ contribution LHCB-
PAPER-2013-048 ; phiBR , a clear structure is observed. This structure was
also seen by the BaBar collaboration BaBarkkpi but was not studied separately
for $B^{-}$ and $B^{+}$ components. No significant asymmetry is present in the
low-mass region of the ${K^{\pm}\pi^{\mp}}$ invariant mass projection.
The $C\\!P$ asymmetries are further studied in the regions where large raw
asymmetries are found. The regions are defined as $m^{2}_{\pi^{+}\pi^{-}\,{\rm
high}}>15{\mathrm{\,Ge\kern-1.00006ptV^{2}\\!/}c^{4}}$ and
$m^{2}_{\pi^{+}\pi^{-}\,{\rm
low}}<0.4{\mathrm{\,Ge\kern-1.00006ptV^{2}\\!/}c^{4}}$ for the
${B^{\pm}\to\pi^{+}\pi^{-}\pi^{\pm}}$ mode, and
$m^{2}_{K^{+}K^{-}}<1.5{\mathrm{\,Ge\kern-1.00006ptV^{2}\\!/}c^{4}}$ for the
${B^{\pm}\to K^{+}K^{-}\pi^{\pm}}$ mode. Unbinned extended maximum likelihood
fits are performed to the mass spectra of the candidates in these regions,
using the same models as for the global fits. The spectra are shown in Fig. 3.
The resulting signal yields and raw asymmetries for the two regions are
${N^{\mathrm{reg}}(K\\!K\pi)=342\pm 28}$ and ${A_{\rm
raw}^{\mathrm{reg}}(K\\!K\pi)=-0.658\pm 0.070}$ for the ${B^{\pm}\to
K^{+}K^{-}\pi^{\pm}}$ mode, and ${N^{\mathrm{reg}}(\pi\pi\pi)=229\pm 20}$ and
${A_{\rm raw}^{\mathrm{reg}}(\pi\pi\pi)=0.555\pm 0.082}$ for the
${B^{\pm}\to\pi^{+}\pi^{-}\pi^{\pm}}$ mode. The $C\\!P$ asymmetries are
obtained from the raw asymmetries using Eqs. 2 and 3 and applying an
acceptance correction. Systematic uncertainties are estimated due to the
signal models, acceptance correction and binning choice in the region, the
$A_{\rm D}(\pi^{\pm})$ and $A_{\rm P}(B^{\pm})$ statistical uncertainties and
the $A_{\rm D}(K^{\pm})$ kaon kinematics. The local charge asymmetries for the
two regions are measured to be
$\displaystyle A_{C\\!P}^{\mathrm{reg}}({B^{\pm}\to K^{+}K^{-}\pi^{\pm}})\\!$
$\displaystyle=$ $\displaystyle\\!-0.648\pm 0.070\pm 0.013\pm 0.007,$
$\displaystyle
A_{C\\!P}^{\mathrm{reg}}({B^{\pm}\to\pi^{+}\pi^{-}\pi^{\pm}})\\!$
$\displaystyle=$ $\displaystyle\\!0.584\pm 0.082\pm 0.027\pm 0.007,$
where the first uncertainty is statistical, the second is the experimental
systematic and the third is due to the $C\\!P$ asymmetry of the
${B^{\pm}\to{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{\pm}}$ reference
mode PDG2012 .
In conclusion, we have found the first evidence of inclusive $C\\!P$
asymmetries of the ${B^{\pm}\to K^{+}K^{-}\pi^{\pm}}$ and
${B^{\pm}\to\pi^{+}\pi^{-}\pi^{\pm}}$ modes with significances of $3.2\sigma$
and $4.9\sigma$, respectively. The results are consistent with those measured
by the BaBar collaboration BaBarkkpi ; BaBarpipipi . These charge asymmetries
are not uniformly distributed in the phase space. For ${B^{\pm}\to
K^{+}K^{-}\pi^{\pm}}$ decays, where no significant resonant contribution is
expected, we observe a very large negative asymmetry concentrated in a
restricted region of the phase space in the low $K^{+}K^{-}$ invariant mass.
For ${B^{\pm}\to\pi^{+}\pi^{-}\pi^{\pm}}$ decays, a large positive asymmetry
is measured in the low $m^{2}_{\pi^{+}\pi^{-}\,{\rm low}}$ and high
$m^{2}_{\pi^{+}\pi^{-}\,{\rm high}}$ phase-space region, not clearly
associated to a resonant state. The evidence presented here for $C\\!P$
violation in ${B^{\pm}\to K^{+}K^{-}\pi^{\pm}}$ and
${B^{\pm}\to\pi^{+}\pi^{-}\pi^{\pm}}$ decays, along with the recent evidence
for $C\\!P$ violation in ${B^{\pm}\to K^{\pm}\pi^{+}\pi^{-}}$ and ${B^{\pm}\to
K^{\pm}K^{+}K^{-}}$ decays LHCb-PAPER-2013-027 and recent theoretical
developments Bhattacharya:2013cvn ; IgnacioCPT ; Xu:2013dta ;
PhysRevD.87.076007 , indicate new mechanisms for $C\\!P$ asymmetries, which
should be incorporated in models for future amplitude analyses of charmless
three-body $B$ decays.
## Acknowledgements
We express our gratitude to our colleagues in the CERN accelerator departments
for the excellent performance of the LHC. We thank the technical and
administrative staff at the LHCb institutes. We acknowledge support from CERN
and from the national agencies: CAPES, CNPq, FAPERJ and FINEP (Brazil); NSFC
(China); CNRS/IN2P3 and Region Auvergne (France); BMBF, DFG, HGF and MPG
(Germany); SFI (Ireland); INFN (Italy); FOM and NWO (The Netherlands); SCSR
(Poland); MEN/IFA (Romania); MinES, Rosatom, RFBR and NRC “Kurchatov
Institute” (Russia); MinECo, XuntaGal and GENCAT (Spain); SNSF and SER
(Switzerland); NAS Ukraine (Ukraine); STFC (United Kingdom); NSF (USA). We
also acknowledge the support received from the ERC under FP7. The Tier1
computing centres are supported by IN2P3 (France), KIT and BMBF (Germany),
INFN (Italy), NWO and SURF (The Netherlands), PIC (Spain), GridPP (United
Kingdom). We are thankful for the computing resources put at our disposal by
Yandex LLC (Russia), as well as to the communities behind the multiple open
source software packages that we depend on.
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| 2013-10-17T15:12:26 |
2024-09-04T02:49:52.523246
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "LHCb collaboration: R. Aaij, B. Adeva, M. Adinolfi, C. Adrover, A.\n Affolder, Z. Ajaltouni, J. Albrecht, F. Alessio, M. Alexander, S. Ali, G.\n Alkhazov, P. Alvarez Cartelle, A.A. Alves Jr, S. Amato, S. Amerio, Y. Amhis,\n L. Anderlini, J. Anderson, R. Andreassen, J.E. Andrews, R.B. Appleby, O.\n Aquines Gutierrez, F. Archilli, A. Artamonov, M. Artuso, E. Aslanides, G.\n Auriemma, M. Baalouch, S. Bachmann, J.J. Back, A. Badalov, C. Baesso, V.\n Balagura, W. Baldini, R.J. Barlow, C. Barschel, S. Barsuk, W. Barter, Th.\n Bauer, A. Bay, J. Beddow, F. Bedeschi, I. Bediaga, S. Belogurov, K. Belous,\n I. Belyaev, E. Ben-Haim, G. Bencivenni, S. Benson, J. Benton, A. Berezhnoy,\n R. Bernet, M.-O. Bettler, M. van Beuzekom, A. Bien, S. Bifani, T. Bird, A.\n Bizzeti, P.M. Bj{\\o}rnstad, T. Blake, F. Blanc, J. Blouw, S. Blusk, V. Bocci,\n A. Bondar, N. Bondar, W. Bonivento, S. Borghi, A. Borgia, T.J.V. Bowcock, E.\n Bowen, C. Bozzi, T. Brambach, J. van den Brand, J. Bressieux, D. Brett, M.\n Britsch, T. Britton, N.H. Brook, H. Brown, A. Bursche, G. Busetto, J.\n Buytaert, S. Cadeddu, O. Callot, M. Calvi, M. Calvo Gomez, A. Camboni, P.\n Campana, D. Campora Perez, A. Carbone, G. Carboni, R. Cardinale, A. Cardini,\n H. Carranza-Mejia, L. Carson, K. Carvalho Akiba, G. Casse, L. Castillo\n Garcia, M. Cattaneo, Ch. Cauet, R. Cenci, M. Charles, Ph. Charpentier, S.-F.\n Cheung, N. Chiapolini, M. Chrzaszcz, K. Ciba, X. Cid Vidal, G. Ciezarek,\n P.E.L. Clarke, M. Clemencic, H.V. Cliff, J. Closier, C. Coca, V. Coco, J.\n Cogan, E. Cogneras, P. Collins, A. Comerma-Montells, A. Contu, A. Cook, M.\n Coombes, S. Coquereau, G. Corti, B. Couturier, G.A. Cowan, D.C. Craik, M.\n Cruz Torres, S. Cunliffe, R. Currie, C. D'Ambrosio, P. David, P.N.Y. David,\n A. Davis, I. De Bonis, K. De Bruyn, S. De Capua, M. De Cian, J.M. De Miranda,\n L. De Paula, W. De Silva, P. De Simone, D. Decamp, M. Deckenhoff, L. Del\n Buono, N. D\\'el\\'eage, D. Derkach, O. Deschamps, F. Dettori, A. Di Canto, H.\n Dijkstra, M. Dogaru, S. Donleavy, F. Dordei, A. Dosil Su\\'arez, D. Dossett,\n A. Dovbnya, F. Dupertuis, P. Durante, R. Dzhelyadin, A. Dziurda, A. Dzyuba,\n S. Easo, U. Egede, V. Egorychev, S. Eidelman, D. van Eijk, S. Eisenhardt, U.\n Eitschberger, R. Ekelhof, L. Eklund, I. El Rifai, Ch. Elsasser, A. Falabella,\n C. F\\\"arber, C. Farinelli, S. Farry, D. Ferguson, V. Fernandez Albor, F.\n Ferreira Rodrigues, M. Ferro-Luzzi, S. Filippov, M. Fiore, C. Fitzpatrick, M.\n Fontana, F. Fontanelli, R. Forty, O. Francisco, M. Frank, C. Frei, M.\n Frosini, E. Furfaro, A. Gallas Torreira, D. Galli, M. Gandelman, P. Gandini,\n Y. Gao, J. Garofoli, P. Garosi, J. Garra Tico, L. Garrido, C. Gaspar, R.\n Gauld, E. Gersabeck, M. Gersabeck, T. Gershon, Ph. Ghez, V. Gibson, L.\n Giubega, V.V. Gligorov, C. G\\\"obel, D. Golubkov, A. Golutvin, A. Gomes, P.\n Gorbounov, H. Gordon, M. Grabalosa G\\'andara, R. Graciani Diaz, L.A. Granado\n Cardoso, E. Graug\\'es, G. Graziani, A. Grecu, E. Greening, S. Gregson, P.\n Griffith, L. Grillo, O. Gr\\\"unberg, B. Gui, E. Gushchin, Yu. Guz, T. Gys, C.\n Hadjivasiliou, G. Haefeli, C. Haen, S.C. Haines, S. Hall, B. Hamilton, T.\n Hampson, S. Hansmann-Menzemer, N. Harnew, S.T. Harnew, J. Harrison, T.\n Hartmann, J. He, T. Head, V. Heijne, K. Hennessy, P. Henrard, J.A. Hernando\n Morata, E. van Herwijnen, M. He\\ss, A. Hicheur, E. Hicks, D. Hill, M.\n Hoballah, C. Hombach, W. Hulsbergen, P. Hunt, T. Huse, N. Hussain, D.\n Hutchcroft, D. Hynds, V. Iakovenko, M. Idzik, P. Ilten, R. Jacobsson, A.\n Jaeger, E. Jans, P. Jaton, A. Jawahery, F. Jing, M. John, D. Johnson, C.R.\n Jones, C. Joram, B. Jost, M. Kaballo, S. Kandybei, W. Kanso, M. Karacson,\n T.M. Karbach, I.R. Kenyon, T. Ketel, B. Khanji, O. Kochebina, I. Komarov,\n R.F. Koopman, P. Koppenburg, M. Korolev, A. Kozlinskiy, L. Kravchuk, K.\n Kreplin, M. Kreps, G. Krocker, P. Krokovny, F. Kruse, M. Kucharczyk, V.\n Kudryavtsev, K. Kurek, T. Kvaratskheliya, V.N. La Thi, D. Lacarrere, G.\n Lafferty, A. Lai, D. Lambert, R.W. Lambert, E. Lanciotti, G. Lanfranchi, C.\n Langenbruch, T. Latham, C. Lazzeroni, R. Le Gac, J. van Leerdam, J.-P. Lees,\n R. Lef\\`evre, A. Leflat, J. Lefran\\c{c}ois, S. Leo, O. Leroy, T. Lesiak, B.\n Leverington, Y. Li, L. Li Gioi, M. Liles, R. Lindner, C. Linn, B. Liu, G.\n Liu, S. Lohn, I. Longstaff, J.H. Lopes, N. Lopez-March, H. Lu, D. Lucchesi,\n J. Luisier, H. Luo, O. Lupton, F. Machefert, I.V. Machikhiliyan, F. Maciuc,\n O. Maev, S. Malde, G. Manca, G. Mancinelli, J. Maratas, U. Marconi, P.\n Marino, R. M\\\"arki, J. Marks, G. Martellotti, A. Martens, A. Mart\\'in\n S\\'anchez, M. Martinelli, D. Martinez Santos, D. Martins Tostes, A. Martynov,\n A. Massafferri, R. Matev, Z. Mathe, C. Matteuzzi, E. Maurice, A. Mazurov, J.\n McCarthy, A. McNab, R. McNulty, B. McSkelly, B. Meadows, F. Meier, M.\n Meissner, M. Merk, D.A. Milanes, M.-N. Minard, J. Molina Rodriguez, S.\n Monteil, D. Moran, P. Morawski, A. Mord\\`a, M.J. Morello, R. Mountain, I.\n Mous, F. Muheim, K. M\\\"uller, R. Muresan, B. Muryn, B. Muster, P. Naik, T.\n Nakada, R. Nandakumar, I. Nasteva, M. Needham, S. Neubert, N. Neufeld, A.D.\n Nguyen, T.D. Nguyen, C. Nguyen-Mau, M. Nicol, V. Niess, R. Niet, N. Nikitin,\n T. Nikodem, A. Nomerotski, A. Novoselov, A. Oblakowska-Mucha, V. Obraztsov,\n S. Oggero, S. Ogilvy, O. Okhrimenko, R. Oldeman, M. Orlandea, J.M. Otalora\n Goicochea, P. Owen, A. Oyanguren, B.K. Pal, A. Palano, M. Palutan, J. Panman,\n A. Papanestis, M. Pappagallo, C. Parkes, C.J. Parkinson, G. Passaleva, G.D.\n Patel, M. Patel, G.N. Patrick, C. Patrignani, C. Pavel-Nicorescu, A. Pazos\n Alvarez, A. Pearce, A. Pellegrino, G. Penso, M. Pepe Altarelli, S. Perazzini,\n E. Perez Trigo, A. P\\'erez-Calero Yzquierdo, P. Perret, M. Perrin-Terrin, L.\n Pescatore, E. Pesen, G. Pessina, K. Petridis, A. Petrolini, A. Phan, E.\n Picatoste Olloqui, B. Pietrzyk, T. Pila\\v{r}, D. Pinci, S. Playfer, M. Plo\n Casasus, F. Polci, G. Polok, A. Poluektov, E. Polycarpo, A. Popov, D. Popov,\n B. Popovici, C. Potterat, A. Powell, J. Prisciandaro, A. Pritchard, C.\n Prouve, V. Pugatch, A. Puig Navarro, G. Punzi, W. Qian, B. Rachwal, J.H.\n Rademacker, B. Rakotomiaramanana, M.S. Rangel, I. Raniuk, N. Rauschmayr, G.\n Raven, S. Redford, S. Reichert, M.M. Reid, A.C. dos Reis, S. Ricciardi, A.\n Richards, K. Rinnert, V. Rives Molina, D.A. Roa Romero, P. Robbe, D.A.\n Roberts, A.B. Rodrigues, E. Rodrigues, P. Rodriguez Perez, S. Roiser, V.\n Romanovsky, A. Romero Vidal, M. Rotondo, J. Rouvinet, T. Ruf, F. Ruffini, H.\n Ruiz, P. Ruiz Valls, G. Sabatino, J.J. Saborido Silva, N. Sagidova, P. Sail,\n B. Saitta, V. Salustino Guimaraes, B. Sanmartin Sedes, R. Santacesaria, C.\n Santamarina Rios, E. Santovetti, M. Sapunov, A. Sarti, C. Satriano, A. Satta,\n M. Savrie, D. Savrina, M. Schiller, H. Schindler, M. Schlupp, M. Schmelling,\n B. Schmidt, O. Schneider, A. Schopper, M.-H. Schune, R. Schwemmer, B.\n Sciascia, A. Sciubba, M. Seco, A. Semennikov, K. Senderowska, I. Sepp, N.\n Serra, J. Serrano, P. Seyfert, M. Shapkin, I. Shapoval, Y. Shcheglov, T.\n Shears, L. Shekhtman, O. Shevchenko, V. Shevchenko, A. Shires, R. Silva\n Coutinho, M. Sirendi, N. Skidmore, T. Skwarnicki, N.A. Smith, E. Smith, E.\n Smith, J. Smith, M. Smith, M.D. Sokoloff, F.J.P. Soler, F. Soomro, D. Souza,\n B. Souza De Paula, B. Spaan, A. Sparkes, P. Spradlin, F. Stagni, S. Stahl, O.\n Steinkamp, S. Stevenson, S. Stoica, S. Stone, B. Storaci, M. Straticiuc, U.\n Straumann, V.K. Subbiah, L. Sun, W. Sutcliffe, S. Swientek, V. Syropoulos, M.\n Szczekowski, P. Szczypka, D. Szilard, T. Szumlak, S. T'Jampens, M. Teklishyn,\n E. Teodorescu, F. Teubert, C. Thomas, E. Thomas, J. van Tilburg, V.\n Tisserand, M. Tobin, S. Tolk, D. Tonelli, S. Topp-Joergensen, N. Torr, E.\n Tournefier, S. Tourneur, M.T. Tran, M. Tresch, A. Tsaregorodtsev, P.\n Tsopelas, N. Tuning, M. Ubeda Garcia, A. Ukleja, A. Ustyuzhanin, U. Uwer, V.\n Vagnoni, G. Valenti, A. Vallier, R. Vazquez Gomez, P. Vazquez Regueiro, C.\n V\\'azquez Sierra, S. Vecchi, J.J. Velthuis, M. Veltri, G. Veneziano, M.\n Vesterinen, B. Viaud, D. Vieira, X. Vilasis-Cardona, A. Vollhardt, D.\n Volyanskyy, D. Voong, A. Vorobyev, V. Vorobyev, C. Vo\\ss, H. Voss, R. Waldi,\n C. Wallace, R. Wallace, S. Wandernoth, J. Wang, D.R. Ward, N.K. Watson, A.D.\n Webber, D. Websdale, M. Whitehead, J. Wicht, J. Wiechczynski, D. Wiedner, L.\n Wiggers, G. Wilkinson, M.P. Williams, M. Williams, F.F. Wilson, J. Wimberley,\n J. Wishahi, W. Wislicki, M. Witek, G. Wormser, S.A. Wotton, S. Wright, S. Wu,\n K. Wyllie, Y. Xie, Z. Xing, Z. Yang, X. Yuan, O. Yushchenko, M. Zangoli, M.\n Zavertyaev, F. Zhang, L. Zhang, W.C. Zhang, Y. Zhang, A. Zhelezov, A.\n Zhokhov, L. Zhong, A. Zvyagin",
"submitter": "Irina Nasteva",
"url": "https://arxiv.org/abs/1310.4740"
}
|
1310.4744
|
# Massless Wigner particles in conformal field theory are free
Yoh Tanimoto 111Supported by Alexander von Humboldt Stiftung until March 2013.
e-mail: [email protected]
Graduate School of Mathematical Sciences, The University of Tokyo
and Institut für Theoretische Physik, Göttingen University
3-8-1 Komaba Meguro-ku Tokyo 153-8914, Japan.
JSPS SPD postdoctoral fellow
###### Abstract
We show that in a four dimensional conformal Haag-Kastler net, its massless
particle spectrum is generated by a free field subnet. If the massless
particle spectrum is scalar, then the free field subnet decouples as a tensor
product component.
## 1 Introduction
Conformal field theories have been extensively studied in two-dimensional
spacetime. There are many examples, certain exact computations are available
and they provide also interesting mathematical structures. On the other hand,
from a mathematical point of view, no nonperturbative construction of a single
interacting quantum field theory in four dimensional spacetime is available
today. In this paper, instead of constructing models, we try to understand
general restrictions on models with a large spacetime symmetry. We prove that
if a conformal field theory in four spacetime dimensions in the operator-
algebraic approach (Haag-Kastler net) contains massless particles, then there
is a free subnet generating the massless particles. Furthermore, if the
massless particles are scalar, then they decouple as a tensor product
component. Therefore, massless particles in conformal field theory cannot
interact.
Actually Buchholz and Fredenhagen have already proved more than 30 years ago
that the S-matrix of a dilation-invariant theory is trivial [12]. Based on
this result, Baumann [3] has shown that any dilation-invariant scalar field
(in the sense of Wightman) where a complete particle interpretation is
available (asymptotic completeness with respect to massless particles) is the
Wick product of the free field. Compared to these, our results are not
necessarily stronger because we assume conformal invariance. On the other
hand, there are more general aspects: our framework is Haag-Kastler nets and
we do not assume neither the existence of Wightman fields, nor asymptotic
completeness. In two-dimensional spacetime, triviality of S-matrix does not
necessarily imply that the net is free (second quantized). Indeed, in our
previous work [36], we have seen that a two-dimensional conformal net is
asymptotically complete with respect to massless waves if and only if it is
the tensor product of its chiral components. Hence one may consider the tensor
product subnet as the “particle-like” (or “wave-like”) part. However, chiral
components can be highly nontrivial (different from the second quantized net,
the ${\rm U(1)}$-current net). In comparison, in four dimensions, we prove
that the particle spectrum is generated by the free, second quantized net. In
particular, if the particles are scalar, the free field subnet which we
construct cannot have any nontrivial extension, hence it must decouple in the
full net. This is the operator-algebraic version of the argument given in [2,
Section 1]. Relaxing the assumption of asymptotic completeness (with respect
to massless particles) is important, because while there are many physical
arguments that dilation-invariance should imply conformal invariance [28, 16],
conformal field theory may contain massive spectrum (the meaning of “massive”
will be clarified in Section 2.1.4), as one would expect from the maximally
supersymmetric Yang-Mills theory, which should be conformal [26].
We stress that our approach is nonperturbative. We make an assumption that
there is a nonperturbatively given model as a conformal Haag-Kastler net. The
existence of massless particles à la Wigner is defined in the sense that the
representation of the spacetime translations has nontrivial spectral
projection on the surface of the positive lightcone. In this case, Buchholz
has established the existence of asymptotic fields [10]. Besides, operator-
algebraic scattering theory has been successfully applied to many massive
models in low dimensions. The theory was able to reconstruct the factorizing
S-matrix as an invariant of the net [23, 37].
There are more claims that conformal fields with massless particles are free
with different assumptions [38, 39]. An advantage of our approach is to avoid
any field-theoretic calculation. One of the main tools is the Tomita-Takesaki
modular theory applied to conformal nets [7]: Brunetti, Guido and Longo have
shown that the modular group of a double cone is certain conformal
transformations which preserve the double cone. This renders the central idea
of our arguments geometric, combined with the construction of asymptotic
fields by Buchholz [10].
Let us recall a technical conjecture in [10]. In order to obtain asymptotic
fields, one had to choose local operators with a certain regularity condition
in the momentum space, although Buchholz conjectured that this construction
should extend to any local operator. In our application, this restriction is a
problem because the regularity condition is not stable under conformal
transformations. We remove this restriction and show that the asymptotic
fields are covariant under the conformal transformation of the given net.
This paper is organized as follows. In Section 2 we summarize the foundations
of conformal nets and the massless scattering theory. The technical conjecture
above is proved there. We first state and prove our results on the existence
of free subnet for globally conformal nets in Section 3. This additional
assumption greatly reduces the problem and emphasizes the geometric nature of
our proof. Section 4 treats the general case, not necessarily globally
conformal but conformal. We also prove the decoupling of the free scalar
subnet. Finally we discuss open problems and future directions in Section 5.
## 2 Preliminaries
### 2.1 Conformal field theory
A model of quantum field theory is realized as a net of von Neumann algebras.
A conformal field theory is a net with the conformal symmetry. We collect here
the definitions and results necessary for our analysis.
#### 2.1.1 The conformal group and the extended Minkowski space
We consider ${\mathbb{R}}^{4}$, the Minkowski space. A conformal symmetry is a
transformation of ${\mathbb{R}}^{4}$ which preserves the Lorentz metric
$a\cdot b=a_{0}b_{0}-\sum a_{k}b_{k}$ up to a function. Actually we allow a
symmetry to take a meager set out of ${\mathbb{R}}^{4}$. Hence we need to
consider local actions, following the work by Brunetti-Guido-Longo [7].
Let $G$ be a Lie group and $M$ be a manifold. We say that $G$ acts locally on
$M$ if there is an open nonempty set $B\subset G\times M$ and a smooth map
$T:B\to M$ such that
1. ( 1 )
For any $a\in M$ , $V_{a}:=\\{g\in G:(g,a)\in B\\}$ is an open connected
neighborhood of the unit element $e$ of $G$.
2. ( 2 )
$T_{e}a=a$ for any $a\in M$.
3. ( 3 )
For $(g,a)\in B$, it holds that $V_{T_{g}a}=V_{a}g^{-1}$ and for $h\in G$ such
that $hg\in V_{a}$, one has $T_{h}T_{g}a=T_{hg}a$.
In the following, we only consider $M={\mathbb{R}}^{4}$. The conformal group
${\mathscr{C}}$ is generated by the Poincaré group, dilations and the special
conformal transformations: a special conformal transformation is of the form
$\rho\tau(a)\rho$, where $\tau(a)$ is a translation by $a\in{\mathbb{R}}^{4}$
and $\rho$ is the relativistic ray inversion
$\rho a=-\frac{a}{a\cdot a}.$
This action is quasi global in the sense that for any $g\in{\mathscr{C}}$ the
open set $\\{a\in M:(g,a)\in B\\}$ is the complement of a meager set $S_{g}$
and it holds for $a_{0}\in S_{g}$ that $\lim_{a\to a_{0}}T_{g}a=\infty$. In
other words, the set of points in $M$ which are taken out of $M$ by $g$ is
meager. This action $T$ is transitive. It has been shown [7, Propositions 1.1,
1.2] that there is a manifold ${\bar{M}}$ such that $M$ is a dense open subset
of ${\bar{M}}$ and the action $T$ extends to a transitive global action on
${\bar{M}}$. Furthermore, the action of $T$ lifts to a transitive global
action $\widetilde{T}$ of the universal covering group $\widetilde{G}$ of $G$
on the universal covering ${\widetilde{M}}$ of ${\bar{M}}$.
Figure 1: The global space ${\widetilde{M}}$ projected on the two-dimensional
cylinder. The region surrounded by thick lines is a copy of the Minkowski
space.
We can realize ${\bar{M}}$ concretely in ${\mathbb{R}}^{6}$ as follows:
$N:=\\{(\xi_{0},\cdots,\xi_{5})\in{\mathbb{R}}^{6}\setminus\\{0\\}:\xi_{0}^{2}-\xi_{1}^{2}-\cdots-\xi_{4}^{2}+\xi_{5}^{2}=0\\}/{\mathbb{R}}^{*},$
where ${\mathbb{R}}^{*}={\mathbb{R}}\setminus\\{0\\}$ acts on
${\mathbb{R}}^{6}$ by multiplication. For $a\in M={\mathbb{R}}^{4}$, we define
the embedding by $\xi_{k}=a_{k}$ for $k=0,1,2,3$ and $\xi_{4}=\frac{1-a\cdot
a}{2},\xi_{5}=\frac{1+a\cdot a}{2}$. The group $\mathrm{PSO}(4,2)$ acts on $N$
and this corresponds to the action of the conformal group ${\mathscr{C}}$.
Since the image of $M$ in $N$ is dense, it follows that $N={\bar{M}}$ [7]. One
observes that $N$ is diffeomorphic to $(S^{3}\times S^{1})/{\mathbb{Z}}_{2}$,
hence its universal covering is $S^{3}\times{\mathbb{R}}$.
#### 2.1.2 Conformal nets
An operator-algebraic conformal field theory, or a conformal net, is a triple
$({\mathcal{A}},U,\Omega)$ of a map ${\mathcal{A}}$ from the family of open
double cones in $M$ into the family of von Neumann algebras on
${\mathcal{H}}$, a local unitary representation (the group structure is
respected only locally) $U$ of the conformal group ${\mathscr{C}}$ and a unit
vector $\Omega\in{\mathcal{H}}$ such that
1. (1)
Isotony. If $O_{1}\subset O_{2}$, then
${\mathcal{A}}(O_{1})\subset{\mathcal{A}}(O_{2})$.
2. (2)
Locality. If $O_{1}$ and $O_{2}$ are spacelike separated, then
${\mathcal{A}}(O_{1})$ and ${\mathcal{A}}(O_{2})$ commute.
3. (3)
Local conformal covariance. For each double cone $O\subset M$, there is a
neighborhood $V_{O}$ of the identity of ${\mathscr{C}}$ such that $V_{O}\times
O\subset B$, where $B$ is the domain of the local action of ${\mathscr{C}}$ on
$M$, such that ${\hbox{\rm Ad\,}}U(g)({\mathcal{A}}(O))={\mathcal{A}}(gO)$.
4. (4)
Positivity of energy. The spectrum of the subgroup of translations in
${\mathscr{C}}$ in the representation $U$ (this is well-defined although the
action $U$ is local, since the group of translations is simply connected) is
included in the closed positive lightcone
$\overline{V}_{+}:=\\{a\in{\mathbb{R}}^{4}:a_{0}\geq 0,\;a\cdot a\geq 0\\}$.
5. (5)
Vacuum. The vector $\Omega$ is invariant under the action of $U$. Such a
vector is unique up to a scalar.
6. (6)
Reeh-Schlieder property. The vector $\Omega$ is cyclic and separating for each
local algebra ${\mathcal{A}}(O)$.
Note that Reeh-Schlieder property is usually proved under additivity. We take
it here as an assumption for simplicity (see the discussion in [40, Section
2]).
A conformal net can be extended to ${\widetilde{M}}$ with the action of
${\widetilde{\mathscr{C}}}$ [7, Proposition 1.9]. Indeed, the representation
$U$ lifts to ${\widetilde{\mathscr{C}}}$ and the local algebra
${\mathcal{A}}(O)$ for $O$ which is not included in ${\widetilde{M}}$ is
defined by covariance.
A (conformal) subnet ${\mathcal{A}}_{0}$ of a net $({\mathcal{A}},U,\Omega)$
is a family of von Neumann subalgebras
${\mathcal{A}}_{0}(O)\subset{\mathcal{A}}(O)$ such that isotony and covariance
with respect to the same $U$ hold. In this case,
$\overline{{\mathcal{A}}_{0}(O)\Omega}$ is a Hilbert subspace of
${\mathcal{H}}$ independent of $O$.
#### 2.1.3 Bisognano-Wichmann property
Certain regions play a special role in the study of conformal field theory.
Here we pick the standard wedge in the $a_{1}$-direction, the unit double cone
and the future lightcone:
* •
$W_{1}:=\\{a\in M:a_{1}>|a_{0}|\\}$,
* •
$O_{1}:=\\{a\in M:|a_{0}|+\sqrt{a_{1}^{2}+a_{2}^{2}+a_{3}^{2}}<1\\}$,
* •
$V_{+}:=\\{a\in M:a_{0}>0,\;a\cdot a>0\\}$
To each of these regions $O$ in ${\widetilde{M}}$ we associate a one-parameter
group $\Lambda^{O}_{t}$ in ${\widetilde{\mathscr{C}}}$ which preserve $O$ and
commute with all $O$-preserving conformal transformations:
* •
For the wedge $W_{1}$, we take the boosts in $a_{1}$-direction. They are
linear transformations and their actions on $(a_{0},a_{1})$ components can be
written, in a matrix form, as
$\Lambda^{W_{1}}_{t}=\left(\begin{array}[]{cc}\cosh 2\pi t&-\sinh 2\pi t\\\
-\sinh 2\pi t&\cosh 2\pi t\end{array}\right)$.
* •
For the unit double cone, by rotation invariance the action is determined by
the action on $(a_{0},a_{1})$-plane:
$\Lambda^{O_{1}}_{t}a_{\pm}=\frac{(1+a_{\pm})-e^{-2\pi
t}(1-a_{\pm})}{(1+a_{\pm})-e^{-2\pi t}(1+a_{\pm})},$
where $a_{\pm}=a_{0}\pm a_{1}$.
* •
For the future lightcone $V_{+}$, we take the dilation:
$\Lambda^{V_{+}}_{t}a=e^{2\pi t}\cdot a$.
These regions are mapped to each other by conformal transformations (on
${\widetilde{M}}$) and the associated transformations are coherent, in the
sense that $\Lambda^{O}_{t}=g^{-1}\Lambda^{O^{\prime}}_{t}g$ where
$O=gO^{\prime}$, $g\in{\widetilde{\mathscr{C}}}$ and
$O,O^{\prime}=W_{1},O_{1},V_{+}$. One can define $\Lambda^{O}_{t}$ for any
other double cone, wedge or lightcone by coherence.
For a conformal net, the modular group of a local algebra with respect to the
vacuum has been completely determined [7].
###### Theorem 2.1 (Bisognano-Wichmann property).
Let $({\mathcal{A}},U,\Omega)$ be a conformal net and consider its natural
extension to ${\widetilde{M}}$. Then for any image $O$ of a double cone by a
conformal transformation in ${\widetilde{\mathscr{C}}}$, one has
$\Delta_{O}^{it}=U(\Lambda^{O}_{t})$, where $\Delta_{O}$ is the modular
operator of ${\mathcal{A}}(O)$ with respect to $\Omega$.
The following duality has been also proved [7].
###### Theorem 2.2 (Haag duality on ${\widetilde{M}}$).
Let $({\mathcal{A}},U,\Omega)$ be a conformal net and consider its natural
extension to ${\widetilde{M}}$. Then for a wedge $W$, it holds that
${\mathcal{A}}(W)^{\prime}={\mathcal{A}}(W^{\prime})$.
Since a conformal transformation can bring a wedge to a double cone $O$, a
similar duality holds for double cones. In that case, we need the causal
complement $O^{\mathrm{c}}$ on ${\widetilde{M}}$ rather than the usual
spacelike complement $O^{\prime}$.
Figure 2: Regions in the global space ${\widetilde{M}}$. The left and right
sides are identified. The white square: a copy of the Minkowski space. Black:
a double cone $O$. Dark gray: the spacelike complement $O^{\prime}$ of the
double cone in the Minkowski space. Light gray + dark gray: the causal
complement $O^{\mathrm{c}}$ in ${\widetilde{M}}$.
#### 2.1.4 Representation theory of the conformal group
The conformal group is locally isomorphic to $\mathrm{SU}(2,2)$ and its
unitary positive-energy irreducible representations have been classified [25].
Using the dimension $d\geq 0$ and half-integers $j_{1},j_{2}\geq 0$, they are
parametrized as follows. When restricted to the Poincaré group, one can
consider the mass parameter $m$ and spin $s$ or helicity.
* •
trivial representation. $d=j_{1}=j_{2}=0$.
* •
$j_{1}\neq 0\neq j_{2}$, $d>j_{1}+j_{2}+2$. In this case, $m>0$ and
$s=|j_{1}-j_{2}|,\cdots j_{1}+j_{2}$ (integer steps).
* •
$j_{1}j_{2}=0$, $d>j_{1}+j_{2}+1$. $m>0$ and $s=j_{1}+j_{2}$.
* •
$j_{1}\neq 0\neq j_{2}$, $d=j_{1}+j_{2}+2$. $m>0$ and $s=j_{1}+j_{2}$.
* •
$j_{1}j_{2}=0$, $d=j_{1}+j_{2}+1$. $m=0$ and helicity $s=j_{1}-j_{2}$.
Hence, the only massless representations are the last family. In this paper,
when we say that a conformal net contains massless particles, it means that
the representation $U$ has a subrepresentation in this family.
In [39] the following has been proved: if there is a quantum field (an
operator-valued distribution) which transforms as a vector in one of the above
massless representations, then it is free. It implicitly assumes that the
massless particles are generated by such a field. This is apparently a
stronger assumption than the one in the operator-algebraic approach (see
Section 2.2) that local observables generate states which contain massless
particles.
The other nontrivial representations have mass $m>0$. One can call them
massive, although there is no mass gap because of the action of dilations.
### 2.2 Massless scattering theory
In the operator-algebraic approach, the concept of particle is not given a
priori, but to be defined through operational process. Such a theory for
massless particles has been established in [10] for a Poincaré covariant net
under the assumption that the representation of the translation has nontrivial
spectral projection corresponding to the cone $m=0$. In such a case, we say
that the net contains massless particles (following Wigner).
#### 2.2.1 Convergence of asymptotic fields for regular operators
Let $({\mathcal{A}},U,\Omega)$ be a Poincaré covariant net (a net for which
the covariance is only assumed for the Poincaré group). Let $x$ be an operator
in ${\mathcal{A}}(O)$ which is smooth in norm under the group action
$g\mapsto{\hbox{\rm Ad\,}}U(g)(x)$. There are sufficiently many such
operators. Indeed, if $x$ is localized in a slightly smaller region than $O$,
then one can smear $x$ with a smooth function with compact support in the
group (note that the conformal group ${\mathscr{C}}$ is finite-dimensional).
For a vector $a\in M$, we denote $x(a)={\hbox{\rm Ad\,}}U(\tau(a))(x)$. For
$t\in{\mathbb{R}}$, we define
$\Phi^{t}(x):=-2t\int_{S^{2}}d\omega(\mathbf{n})\;\partial_{0}x(t,t\mathbf{n}),$
where $d\omega$ is the normalized rotation-invariant measure on $S^{2}$ and
$\partial_{0}$ is the derivative with respect to the time translation (which
is independent from $t$). By a straightforward calculation, one finds that
$\Phi^{t}(x)\Omega=\frac{1}{|\mathbf{P}|}(e^{it(H-|\mathbf{P}|)}-e^{it(H+|\mathbf{P}|)})Hx\Omega,$
where $P=(H,\mathbf{P})$ is the generator of translation:
$U(\tau(a))=e^{itP\cdot a}$. Furthermore, we need to take suitable time-
averages. We fix a positive, smooth and compactly supported function $h$ with
$\int_{{\mathbb{R}}}h(t)dt=1$ and
$h_{T}(t)=\frac{1}{\log|T|}\,h\left(\frac{t-T}{\log|T|}\right)$. We set
$\Phi^{h_{T}}(x)=\int_{{\mathbb{R}}}dt\;h_{T}(t)\Phi^{t}(x).$
Then by the mean ergodic theorem one obtains [11]
$\underset{T\to\infty}{{{\mathrm{s}\textrm{-}\lim}\,}}\Phi^{h_{T}}(x)\Omega=P_{1}x\Omega,$
where $P_{1}$ is the projection onto the massless one-particle space, where
$H=|\mathbf{P}|$ holds.
For any double cone $O$, we denote by $V_{O,+}$ the future tangent of $O$, the
set of all points separated by a future-timelike vector from any point of $O$.
For a fixed double cone $O_{+}$ in $V_{O,+}$, there is a sufficiently large
$T$ such that $\Phi^{h_{T}}(x)$ is contained in the causal complement of
$O_{+}$. In particular, for sufficiently large $T$, there is a large commutant
for $\Phi^{h_{T}}(x)$ and one can define the operator $\Phi^{\mathrm{out}}(x)$
by
$\Phi^{\mathrm{out}}(x)y\Omega=\underset{T\to\infty}{{{\mathrm{s}\textrm{-}\lim}\,}}y\Phi^{h_{T}}(x)\Omega=yP_{1}x\Omega$,
where $y\in{\mathcal{A}}(O_{+})$. Let us denote
${\mathcal{F}}(V_{O,+})=\bigcup_{O_{+}\subset V_{O,+}}{\mathcal{A}}(O_{+})$
(the union, not the weak closure and $O_{+}$ are bounded). The choice of
$O_{+}$ was arbitrary in $V_{O,+}$, hence $\Phi^{\mathrm{out}}(x)$ can be
defined on ${\mathcal{F}}(V_{O,+})\Omega$. It is easy to see that
$\Phi^{\mathrm{out}}(x)$ is closable. We denote the closure by the same symbol
and its domain by ${\mathcal{D}}(\Phi^{\mathrm{out}}(x))$.
For $N\in{\mathbb{N}}$, let ${\mathcal{A}}_{N}(O)$ be the linear span of the
operators
$\int_{{\mathbb{R}}}dt\;\varphi(t){\hbox{\rm Ad\,}}U(\tau(ta))(x),$
where $x\in{\mathcal{A}}(\check{O})$, $a$ is a timelike vector and $\varphi$
is a test function with compact support which has a Fourier transform
$\tilde{\varphi}(p)$ with an $N$-fold zero at $p=0$, and $\check{O}+({\rm
supp\,}\varphi)a\subset O$.
Figure 3: How asymptotic fields are constructed. A local observable in a dark
gray region is taken in the region between the cones indicated by dotted
lines.
The following has been proved [10, Lemma 1, Lemma 6, Theorems 7, 8, 9].
###### Theorem 2.3 (Buchholz).
Let $x=x^{*}$ be an element of ${\mathcal{A}}_{N_{0}}(O)$, where $N_{0}\geq
15$, $O$ is a double cone and $V_{O,+}$ be the future tangent of $O$. Then the
following hold.
1. (1)
For an arbitrary $y\in{\mathcal{A}}(O_{+})$, where $O_{+}\subset V_{O,+}$ is
bounded,
$y\cdot{\mathcal{D}}(\Phi^{\mathrm{out}}(x))\subset{\mathcal{D}}(\Phi^{\mathrm{out}}(x))$
and one has $[\Phi^{\mathrm{out}}(x),y]=0$ on
${\mathcal{D}}(\Phi^{\mathrm{out}}(x))$.
2. (2)
The operator $\Phi^{\mathrm{out}}(x)$ is self-adjoint and depends only on
$P_{1}x\Omega$. The subspace ${\mathcal{F}}(V_{O,+})\Omega$ is a core of
$\Phi^{\mathrm{out}}(x)$.
3. (3)
The sequence $\Phi^{h_{T}}(x)$ is convergent to $\Phi^{\mathrm{out}}(x)$ in
the strong resolvent sense.
4. (4)
The operator $\Phi^{\mathrm{out}}(x)$ can be applied to the vacuum $\Omega$
arbitrarily many times. We denote the vectors generated in this way
recursively (the first term in the right-hand side which contains $n+1$
product is defined in this way):
$\Phi^{\mathrm{out}}(x)\cdot\xi_{1}{\overset{\mathrm{out}}{\times}}\xi_{2}{\overset{\mathrm{out}}{\times}}\cdots{\overset{\mathrm{out}}{\times}}\xi_{n}=\xi{\overset{\mathrm{out}}{\times}}\xi_{1}{\overset{\mathrm{out}}{\times}}\xi_{2}{\overset{\mathrm{out}}{\times}}\cdots{\overset{\mathrm{out}}{\times}}\xi_{n}+\sum_{k=1}^{n}\langle\xi,\xi_{k}\rangle\xi_{1}{\overset{\mathrm{out}}{\times}}\cdots\check{\xi}_{k}\cdots{\overset{\mathrm{out}}{\times}}\xi_{n},$
where $\xi=P_{1}x\Omega=P_{1}x^{*}\Omega$ and $\check{\xi}_{k}$ means the
omission of the $k$-th element. Then the symbol
${\overset{\mathrm{out}}{\times}}$ is compatible (unitarily equivalent) with
the normalized symmetric tensor product on the Fock space with the one
particle space $P_{1}{\mathcal{H}}$. The domain of $\Phi^{\mathrm{out}}(x)$
includes the set ${\mathcal{H}}^{\mathrm{out}}_{\mathrm{prod}}$ of all linear
combinations (without closure) of product states
$\xi_{1}{\overset{\mathrm{out}}{\times}}\xi_{2}{\overset{\mathrm{out}}{\times}}\cdots{\overset{\mathrm{out}}{\times}}\xi_{n}$,
where $\xi_{k}$ is an arbitrary vector in $P_{1}{\mathcal{H}}$.
5. (5)
It holds that ${\hbox{\rm
Ad\,}}U(g)(\Phi^{\mathrm{out}}(x))=\Phi^{\mathrm{out}}({\hbox{\rm
Ad\,}}U(g)(x))$ if $g$ is a Poincaré transformation.
6. (6)
For the resolvent $R_{\pm i}(y)=(y\pm i)^{-1}$ of $y$, it holds that
$\displaystyle[R_{\pm i}(\Phi^{\mathrm{out}}(x_{1})),R_{\pm
i}(\Phi^{\mathrm{out}}(x_{2}))]$
$\displaystyle=\langle\Omega,[\Phi^{\mathrm{out}}(x_{1}),\Phi^{\mathrm{out}}(x_{2})]\Omega\rangle\cdot
R_{\pm i}(\Phi^{\mathrm{out}}(x_{1}))R_{\pm
i}(\Phi^{\mathrm{out}}(x_{2}))^{2}R_{\pm i}(\Phi^{\mathrm{out}}(x_{1}))$
$\displaystyle=\mathrm{Re}\,\langle P_{1}x\Omega,P_{1}x_{2}\Omega\rangle\cdot
R_{\pm i}(\Phi^{\mathrm{out}}(x_{1}))R_{\pm
i}(\Phi^{\mathrm{out}}(x_{2}))^{2}R_{\pm i}(\Phi^{\mathrm{out}}(x_{1})),$
where $\mathrm{Re}\,$ denotes the real part of the following number.
7. (7)
For $x\in{\mathcal{A}}_{N_{0}}(O)$ and $y\in{\mathcal{F}}(V_{O,+})$, it holds
that $[R_{\pm i}(\Phi^{\mathrm{out}}(x)),y]=0$.
We note that by Claims (1) and (4), the domain of $\Phi^{\mathrm{out}}(x)$
includes ${\mathcal{F}}(V_{O,+}){\mathcal{H}}^{\mathrm{out}}_{\mathrm{prod}}$.
The restriction to ${\mathcal{A}}_{N_{0}}$ is essential in the original proof
[10]. The technical issue is that the set ${\mathcal{A}}_{N_{0}}(O)$ is
covariant under Poincaré transformations and dilations but not under conformal
transformations. We will extend these results to each smooth operator in a
local algebra ${\mathcal{A}}(O)$. This has been expected by Buchholz himself
in the same paper [10, P. 157, footnote].
#### 2.2.2 Extension to general smooth operators
We exploit the arguments of [32, Chapter VIII.7] and [31, Chapter X.10]. Let
$\\{A_{n}\\}$ be a sequence of (unbounded) operators. The following is an
adaptation of [31, Theorem X.63] to the case of our interest.
###### Lemma 2.4.
Let $\\{A_{n}\\}$ be a sequence of self-adjoint operators on ${\mathcal{H}}$,
whose domains have a dense intersection ${\mathcal{D}}$ and suppose that their
resolvents $R_{\pm i}(A_{n})$ are strongly convergent, whose limits we denote
by $R_{\pm}$ and that for each $\xi\in{\mathcal{D}}$, $A_{n}\xi$ is convergent
in norm, whose limit we denote by $A\xi$. Then there is a self-adjoint
extension $\tilde{A}$ of $A$ and $A_{n}$ are convergent to $\tilde{A}$ in the
strong resolvent sense.
###### Proof.
We claim that $\ker R_{\pm}=\\{0\\}$. Let $\xi\in\ker R_{+}$ and
$\eta\in{\mathcal{D}}$. It is clear that $R_{+}^{*}=R_{-}$. It holds that
$\displaystyle\langle\xi,\eta\rangle$ $\displaystyle=$
$\displaystyle\langle\xi,R_{-i}(A_{n})(A_{n}-i)\eta\rangle$ $\displaystyle=$
$\displaystyle\langle R_{+i}(A_{n})\xi,(A_{n}-i)\eta\rangle$ $\displaystyle=$
$\displaystyle\lim_{n}\,\langle R_{+i}(A_{n})\xi,(A_{n}-i)\eta\rangle$
$\displaystyle=$ $\displaystyle\langle R_{+}\xi,(A-i)\eta\rangle$
$\displaystyle=$ $\displaystyle 0.$
As ${\mathcal{D}}$ is dense, $\xi=0$. Similarly $\ker R_{-}=\\{0\\}$ and it
follows that ${\rm Ran}\,R_{\pm}$ are dense in ${\mathcal{H}}$ since
$R_{\pm}=R_{\mp}^{*}$. Then by the Trotter-Kato theorem [32, Theorem
VIII.22] there is a self-adjoint operator $\tilde{A}$ and
$A_{n}\to\tilde{A}$ in the strong resolvent sense.
The domain of $\tilde{A}$ is exactly $R_{\pm}{\mathcal{H}}$ and for
$\xi\in{\mathcal{D}}$ it holds that
$R_{\pm}\cdot(A\pm i)\xi=\lim_{n}R_{\pm i}(A_{n})(A_{n}\pm i)\xi=\xi,$
by the uniform boundedness of $R_{\pm i}(A_{n})$, hence $\xi$ is in the range
of $R_{\pm}$ and ${\mathcal{D}}$ is included in the domain of $\tilde{A}$. ∎
We do not know whether ${\mathcal{D}}$ is a core of $\tilde{A}$ in general. We
will prove this in the case of asymptotic fields.
Let $N_{0}\geq 15$. For a smooth $x\in{\mathcal{A}}(O)$, where $O$ is a double
cone, there is a sequence $x_{n}\in{\mathcal{A}}_{N_{0}}(O_{n})$ such that
$P_{1}x\Omega=\lim P_{1}x_{n}\Omega$ and $P_{1}x^{*}\Omega=\lim
P_{1}x_{n}^{*}\Omega$ by the argument of [10, Remark, p.155], where $O_{n}$ is
growing to the past of $O$. Namely, for $n\in{\mathbb{N}}$ one can take
$\varphi_{n}(t)$ whose Fourier transform is
$\tilde{\varphi}_{n}(\omega)=(1+(e^{-i\omega n}-1)/i\omega
n)^{N_{0}}\cdot\tilde{\varphi}(\omega/n),$
where $\varphi$ is a test function which vanishes for $t\geq 0$ and $\int
dt\,\varphi(t)=1$. We define $x_{n}=\int dt\,\varphi_{n}(t){\hbox{\rm
Ad\,}}U(\tau(t,0))(x)$, where $\tau$ denotes the translation. If $x$ is self-
adjoint, we may consider $x_{n}+x^{*}_{n}$ and assume that $x_{n}$ are self-
adjoint as well. It is clear that $x_{n}$ are contained in the union of past
translations of $O$. Let $O_{n}$ be their localization regions. Let $V_{O,+}$
be the future tangent of $O$, then it is the future tangent of the finite
union $O\cup O_{1}\cup\cdots\cup O_{n}$. By [10, Theorem 7] cited above, all
$\\{\Phi^{\mathrm{out}}(x_{n})\\}$ are self-adjoint. In addition,
${\mathcal{F}}(V_{O,+})\Omega$, and accordingly
${\mathcal{F}}(V_{O,+}){\mathcal{H}}^{\mathrm{out}}_{\mathrm{prod}}$, are
common cores.
###### Lemma 2.5.
The sequence $\\{\Phi^{\mathrm{out}}(x_{n})\\}$ is convergent in the strong
resolvent sense.
###### Proof.
Let us denote $R_{\pm,n}=R_{\pm i}(\Phi^{\mathrm{out}}(x_{n}))$. On the
subspace $\\{y\Omega:y\in{\mathcal{F}}(V_{O,+})\\}$, which is a common core
for $\\{\Phi^{\mathrm{out}}(x_{n})\\}$, it holds that
$R_{\pm,n}y\Omega=yR_{\pm,n}\Omega$ and $y\in{\mathcal{F}}(V_{O,+})$ is
bounded. Since $\\{R_{\pm,n}\\}$ is uniformly bounded, it is enough to show
that $R_{\pm,n}\Omega$ is convergent.
We know from [10] that $\Phi^{\mathrm{out}}(x_{n})$ acts on
${\mathcal{H}}^{\mathrm{out}}_{\mathrm{prod}}$ like the free field. Since the
problem is now reduced to the vacuum $\Omega$ and the free fields, we can
restrict ourselves to ${\mathcal{H}}^{\mathrm{out}}_{\mathrm{prod}}$ and its
closure, namely the Fock space generated from $\Omega$ by the fields. Let us
denote $\xi_{n}:=P_{1}x_{n}\Omega$. The action of the exponentiated field
$e^{i\Phi^{\mathrm{out}}(x_{n})}$ on the vacuum $\Omega$ is given by
$e^{i\Phi^{\mathrm{out}}(x_{n})}\Omega=e^{-\frac{1}{2}\langle\xi_{n},\xi_{n}\rangle}e^{\xi_{n}}$,
where we introduced a vector (cf. [24])
$e^{\eta}:=\Omega\bigoplus_{k}\frac{1}{\sqrt{k!}}\eta^{\otimes k}.$
It is easy to see that $\langle
e^{\eta},e^{\zeta}\rangle=e^{\langle\eta,\zeta\rangle}$. Now it is obvious
that $\eta\mapsto e^{\eta}$ is continuous. This implies the convergence
$e^{\xi_{n}}\to e^{\xi}$ when $\xi_{n}\to\xi$. The exponentiated field acts by
$e^{i\Phi^{\mathrm{out}}(x_{n})}e^{\eta}=e^{-\frac{1}{2}\langle\xi_{n},\xi_{n}\rangle}e^{-\langle\xi_{n},\eta\rangle}e^{\xi_{n}+\eta}$
and $\\{e^{\eta}\\}$ is total in the Fock space. The whole argument applies to
$t\xi_{n}$ for arbitrary $t\in{\mathbb{R}}$, hence
$e^{it\Phi^{\mathrm{out}}(x_{n})}$ are strongly convergent to $W(t\xi)$ on the
Fock space (because this sequence is uniformly bounded), where $W(\xi)$ is an
operator which acts by
$W(\xi)\eta=e^{-\frac{1}{2}\langle\xi,\xi\rangle}e^{-\langle\xi,\eta\rangle}e^{\xi+\eta}$.
Hence we obtain the convergence in the strong resolvent sense [31, Theorem
VIII.21], in particular $R_{\pm,n}\Omega$ is convergent. ∎
As seen from Theorem 2.3(4), $\Phi^{\mathrm{out}}(x_{n})$ is convergent on
${\mathcal{H}}^{\mathrm{out}}_{\mathrm{prod}}$, hence on
${\mathcal{F}}(V_{O,+}){\mathcal{H}}^{\mathrm{out}}_{\mathrm{prod}}$.
By Lemma 2.4, there is a self-adjoint operator, which we denote by
$\Upsilon(\xi)$, such that $\Upsilon(\xi)$ is the limit of
$\\{\Phi^{\mathrm{out}}(x_{n})\\}$ in the strong resolvent sense. Accordingly,
$\Upsilon(\xi)$ commutes with ${\mathcal{F}}(V_{O,+})$ on its domain.
Importantly, we have shown that $\Upsilon(\xi)$ is a self-adjoint extension of
the limit of the sequence $\\{\Phi^{\mathrm{out}}(x_{n})\\}$ on a common
domain ${\mathcal{F}}(V_{O,+}){\mathcal{H}}^{\mathrm{out}}_{\mathrm{prod}}$.
Furthermore, the action of $\Upsilon(\xi)$ is determined by $\xi$ as in
Theorem 2.3(4). This implies that $\Omega$ is in the domain of
$\Upsilon(\xi)^{m}$ for any $m\in{\mathbb{N}}$.
###### Lemma 2.6.
Any vector $y\Omega\in{\mathcal{F}}(V_{O,+})\Omega$ is an analytic vector for
$\Upsilon(\xi)$. In particular,
${\mathcal{F}}(V_{O,+}){\mathcal{H}}^{\mathrm{out}}_{\mathrm{prod}}$ is a core
of $\Upsilon(\xi)$.
###### Proof.
We have to estimate $\Upsilon(\xi)^{k}y\Omega$. The operator $\Upsilon(\xi)$
commutes with $y$ and acts on $\Omega$ as the free field, hence we have
$\|\Upsilon(\xi)^{m}y\Omega\|\leq\|y\|\cdot\left(\sqrt{(2m)!\,2^{-m}(m!)^{-1}}\right)\cdot\|\xi\|^{m}.$
Then it is easy to see that
$\sum_{m}\|\Upsilon(\xi)^{m}y\Omega\|\frac{t^{m}}{m!}$ is finite for any $t$
and since the subspace
${\mathcal{F}}(V_{O,+}){\mathcal{H}}^{\mathrm{out}}_{\mathrm{prod}}$ of the
domain is stable under $\Phi^{\mathrm{out}}(\xi)$, by Nelson’s analytic vector
theorem [31, Theorem X.39, Corollary 2] (the stability of the domain is
important, see the reference222We thank D. Buchholz for pointing out this
assumption.),
${\mathcal{F}}(V_{O,+}){\mathcal{H}}^{\mathrm{out}}_{\mathrm{prod}}$ is a core
of $\Upsilon(\xi)$. ∎
###### Lemma 2.7.
The subspace ${\mathcal{F}}(V_{O,+})\Omega$ is a core of $\Upsilon(\xi)$.
###### Proof.
In [10, Lemma 6], it was shown that if $x_{0}\in{\mathcal{A}}_{N_{0}}(O)$,
$N_{0}\geq 15$, then the domain ${\mathcal{D}}(\Phi^{\mathrm{out}}(x_{0}))$ of
$\Phi^{\mathrm{out}}(x_{0})$, which is defined as the closure of the operator
on ${\mathcal{F}}(V_{O,+})\Omega$, includes
${\mathcal{H}}^{\mathrm{out}}_{\mathrm{prod}}$ and the action of
$\Phi^{\mathrm{out}}(x_{0})$ on ${\mathcal{H}}^{\mathrm{out}}_{\mathrm{prod}}$
is exactly same as that of the free fields. Actually the only properties of
$\Phi^{\mathrm{out}}(x_{0})$ used there are those that $\Omega$ is in the
domain of $\Phi^{\mathrm{out}}(x_{0})^{*}\Phi^{\mathrm{out}}(x_{0})$ and
$\Phi^{\mathrm{out}}(x_{0})$ commute with ${\mathcal{F}}(V_{O,+})$, which are
true also for $\Upsilon(\xi)$ as we have seen.
For the reader’s convenience, we review the proof of [10, Lemma 6]. Let
$x_{0}\in{\mathcal{A}}_{N_{0}}(O)$. There is an $N$ (depending on $n$ which
appears later) such that there is a sequence $\\{y_{k}\\}$ which belongs to
${\mathcal{A}}_{N}(O_{k})$, where $O_{k}\subset V_{O,+}$ (the localization
region $O_{k}$ depends on $k$),
$y_{k}\Omega\to\xi_{1}{\overset{\mathrm{out}}{\times}}\cdots{\overset{\mathrm{out}}{\times}}\xi_{n}$
weakly and $y_{k}^{*}y_{k}\Omega$ is uniformly bounded. To see that
$\xi_{1}{\overset{\mathrm{out}}{\times}}\cdots{\overset{\mathrm{out}}{\times}}\xi_{n}$
is in the domain of $\Phi^{\mathrm{out}}(x_{0})$, one needs to estimate
$\langle\Phi^{\mathrm{out}}(x_{0})^{*}\eta,y_{k}\Omega\rangle$ for an
arbitrary vector $\eta\in{\mathcal{D}}(\Phi^{\mathrm{out}}(x_{0})^{*})$. By
using the fact that $\Phi^{\mathrm{out}}(x_{0})$ commutes with $y_{k}$, (which
is also valid for $\Upsilon(\xi)$), one obtains
$|\langle\Phi^{\mathrm{out}}(x_{0})^{*}\eta,y_{k}\Omega\rangle|^{2}\leq\|\eta\|^{2}\cdot\|\Phi^{\mathrm{out}}(x_{0})y_{k}\Omega\|^{2}\leq\|\eta\|^{2}\cdot\|y_{k}^{*}y_{k}\Omega\|\cdot\|\Phi^{\mathrm{out}}(x_{0})^{*}\Phi^{\mathrm{out}}(x_{0})\Omega\|,$
if $\Phi^{\mathrm{out}}(x_{0})\Omega$ is in the domain of
$\Phi^{\mathrm{out}}(x_{0})^{*}$ (this follows in the original proof from the
assumption that $x_{0}\in{\mathcal{A}}_{N_{0}}(O)$ and this is the only point
where $N_{0}\geq 15$ is required. For $\Upsilon(\xi)$ we already know that
that one can repeat its action on $\Omega$ arbitrarily many times). This
expression is uniformly bounded by the choice of $y_{k}$, hence
$\langle\Phi^{\mathrm{out}}(x_{0})^{*}\eta,\xi_{1}{\overset{\mathrm{out}}{\times}}\cdots{\overset{\mathrm{out}}{\times}}\xi_{n}\rangle$
is bounded by $\|\eta\|$ times a constant and
$\xi_{1}{\overset{\mathrm{out}}{\times}}\cdots{\overset{\mathrm{out}}{\times}}\xi_{n}$
belongs to ${\mathcal{D}}(\Phi^{\mathrm{out}}(x_{0}))$.
In order to get the explicit action of $\Phi^{\mathrm{out}}(x_{0})$ on
$\xi_{1}{\overset{\mathrm{out}}{\times}}\cdots{\overset{\mathrm{out}}{\times}}\xi_{n}$
(see Theorem 2.3), one takes a sequence $\\{x^{(m)}\\}$, where each member
belongs to ${\mathcal{A}}_{N}(O^{(m)})$, double cones growing to the past of
$O$ as in the construction before Lemma 2.5 (it is not explicitly written in
the original proof, but $N$ must be chosen corresponding to $2(n+1)$, see also
[10, Lemmas 2, 3]). In this computation, the only point is that
$\\{Px^{(m)}\Omega\\}$ can approximate $Px_{0}\Omega$, which is true also for
$\xi$.
Although $\\{\xi_{k}\\}$ are not completely arbitrary since
$\xi_{1}{\overset{\mathrm{out}}{\times}}\cdots{\overset{\mathrm{out}}{\times}}\xi_{n}$
must be the limit of $y_{k}\Omega$, they form a total set in the free Fock
space. Once one obtained the action of $\Phi^{\mathrm{out}}(x_{0})$ on a dense
subspace, an arbitrary $n$-particle vector can be approximated in the
$n$-particle subspace and the action of $\Phi^{\mathrm{out}}(x_{0})$ is
continuous there, hence by the closedness of $\Phi^{\mathrm{out}}(x_{0})$ it
follows that any vector in ${\mathcal{H}}^{\mathrm{out}}_{\mathrm{prod}}$ is
in the domain of $\Phi^{\mathrm{out}}(x_{0})$. The same argument is valid for
$\Upsilon(\xi)$.
Altogether, the closure of the restriction of $\Upsilon(\xi)$ to
${\mathcal{F}}(V_{O,+})\Omega$ includes
${\mathcal{F}}(V_{O,+}){\mathcal{H}}^{\mathrm{out}}_{\mathrm{prod}}$, hence
the full domain of $\Upsilon(\xi)$ by Lemma 2.6. This was what we had to
prove. ∎
As $\Phi^{\mathrm{out}}(x)$ is defined as the closure of the operator
${\mathcal{F}}(V_{O,+})y\Omega\ni\eta\longmapsto yP_{1}x\Omega$, we can infer
that $\Phi^{\mathrm{out}}(x)=\Upsilon(\xi)$.
###### Theorem 2.8.
For any $x=x^{*}\in{\mathcal{A}}(O)$ smooth, $\Phi^{\mathrm{out}}(x)$ is self-
adjoint with a core ${\mathcal{F}}(V_{O,+})\Omega$ where $V_{O,+}$ is the
future tangent of $O$. The sequence $\Phi^{h_{T}}(x)$ is convergent to
$\Phi^{\mathrm{out}}(x)$ in the strong resolvent sense.
###### Proof.
By definition, $\Phi^{\mathrm{out}}(x)$ is the closure of the operator
$y\Omega\mapsto yP_{1}x\Omega$ on ${\mathcal{F}}(V_{O,+})\Omega$. But since
$\Upsilon(\xi)(=\Upsilon(P_{1}x\Omega))$ is self-adjoint and
${\mathcal{F}}(V_{O,+})\Omega$ is its core, it follows that
$\Upsilon(\xi)=\Phi^{\mathrm{out}}(x)$, as their actions coincide on their
cores.
As for the convergence, we follow the proof of [10, Theorem 9]. We know that
${\mathcal{F}}(V_{O,+})\Omega$ is a core for $\Phi^{\mathrm{out}}(x)$ and it
is self-adjoint. For $y\in{\mathcal{F}}(V_{O,+})$,
$\underset{T\to\infty}{{{\mathrm{s}\textrm{-}\lim}\,}}(\Phi^{h_{T}}(x)+\lambda)^{-1}(\Phi^{\mathrm{out}}(x)+\lambda)y\Omega=\underset{T\to\infty}{{{\mathrm{s}\textrm{-}\lim}\,}}(\Phi^{h_{T}}(x)+\lambda)^{-1}(\Phi^{h_{T}}(x)+\lambda)y\Omega=y\Omega$
by the uniform boundedness of $(\Phi^{h_{T}}(x)+\lambda)^{-1}$ for a fixed
$\lambda\notin{\mathbb{R}}$. By the self-adjointness of
$\Phi^{\mathrm{out}}(x)$,
$\\{(\Phi^{\mathrm{out}}(x)+\lambda)y\Omega,y\in{\mathcal{F}}(V_{O,+})\\}$ is
dense in ${\mathcal{H}}$ and we obtain the convergence in the strong resolvent
sense, again by the uniform boundedness of the sequence. ∎
###### Lemma 2.9.
Let $({\mathcal{A}},U,\Omega)$ be a conformal net. For
$x=x^{*}\in{\mathcal{A}}(O)$ smooth, there is a $O_{+}$ whose closure is
contained in the future tangent $V_{O,+}$ of $O$ such that
${\mathcal{A}}({\mathcal{O}}_{+})\Omega$ is a core for
$\Phi^{\mathrm{out}}(x)$.
###### Proof.
We work on the extension of ${\mathcal{A}}$ on ${\widetilde{M}}$ and the lift
of $U$ to ${\widetilde{\mathscr{C}}}$.
Recall that $V_{O,+}$ is a translation of the future lightcone, then there is
a region $D$ in ${\widetilde{M}}$ such that the inclusion $V_{O,+}\subset D$
is conformally equivalent to $O_{+}\subset V_{+}$, where $O_{+}$ is a double
cone whose past apex is the point of origin. Then the conformal
transformations associated to $V_{+}$, dilations, shrink $O_{+}$. Accordingly
the conformal transformations associated to $D$ shrink $V_{O,+}$ to double
cones whose past apex is the apex of $V_{O,+}$ (see Figure 2). In this
situation, such a transformation shrinks also $O$.
Let $g$ be a conformal transformation as in the previous paragraph. Now the
operator $\Phi^{\mathrm{out}}({\hbox{\rm Ad\,}}U(g)(x))$ has a core
${\mathcal{F}}(V_{O,+})\Omega$ and ${\hbox{\rm
Ad\,}}U(g)(\Phi^{\mathrm{out}}(x))$ has a core
$U(g){\mathcal{F}}(V_{O,+})\Omega={\mathcal{F}}(gV_{O,+})\Omega$, where
${\mathcal{F}}(gV_{O,+})$ is analogously defined as ${\mathcal{F}}(V_{O,+})$.
Their actions coincide on ${\mathcal{F}}(gV_{O,+})\Omega$, namely for
$y\in{\mathcal{F}}(gV_{O,+})$ they give $y\Omega\mapsto
yU(g)P_{1}x\Omega=yP_{1}U(g)x\Omega$ (the conformal group preserves
$P_{1}{\mathcal{H}}$ from the classification of unitary positive-energy
representations, Section 2.1.4). The operator $\Phi^{\mathrm{out}}({\hbox{\rm
Ad\,}}U(g)(x))$ is a self-adjoint extension of ${\hbox{\rm
Ad\,}}U(g)(\Phi^{\mathrm{out}}(x))$ which is also self-adjoint, hence they
must coincide.
In the discussion above, the domain of $\Phi^{\mathrm{out}}({\hbox{\rm
Ad\,}}U(g)(x))$ naturally includes ${\mathcal{A}}(gV_{O,+})\Omega$ (note that
${\mathcal{A}}(gV_{O,+})$ is a von Neumann algebra). Reversing the argument,
for any $x\in{\mathcal{A}}(O)$ there is a sufficiently large double cone
$O_{+}$ in $V_{O,+}$, whose past apex is the future apex of $O$, such that
${\mathcal{A}}(O_{+})\Omega$ is a core of $\Phi^{\mathrm{out}}(x)$.
Until now in this proof and in Theorem 2.8, regarding the localization, we
used only the assumption that $x$ is localized in $O$, a double cone in the
past tangent of $V_{O,+}$. By considering ${\hbox{\rm Ad\,}}U(\tau(-a))(x)$
which is localized in $O-a$ for a future-timelike vector $a$ and translating
everything by $a$ after the argument, we see actually that
${\mathcal{A}}(O_{+}+a)\Omega$ is a core of $\Phi^{\mathrm{out}}(x)$. In other
words, if $x$ is localized in a double cone, then there is another double cone
in the future tangent, separated by a nontrivial timelike vector, whose local
operators can generate a core for $\Phi^{\mathrm{out}}(x)$. ∎
###### Corollary 2.10.
Let $({\mathcal{A}},U,\Omega)$ be a conformal net. For
$x=x^{*}\in{\mathcal{A}}(O)$ smooth and $g\in{\widetilde{\mathscr{C}}}$
sufficiently near to the unit element such that $gO$ is still a double cone in
the Minkowski space $M$, it holds that ${\hbox{\rm
Ad\,}}U(g)(\Phi^{\mathrm{out}}(x))=\Phi^{\mathrm{out}}({\hbox{\rm
Ad\,}}U(g)(x))$.
###### Proof.
We may assume that $x$ is localized in $\check{O}$, whose closure is still in
$O$. Let $O_{+}+a$ be a double cone in $V_{O,+}$ separated from the future
apex of $O$ such that ${\mathcal{A}}(O_{+}+a)\Omega$ is a core for
$\Phi^{\mathrm{out}}(x)$ (Lemma 2.9). If $g\in{\widetilde{\mathscr{C}}}$ is
sufficiently near to the unit, we may assume the following:
* •
$g\check{O}\subset O$,
* •
$gO$ and $g(O_{+}+a)$ are included in ${\mathbb{R}}^{4}$,
* •
there is a double cone $\widehat{O}_{+}$ which include $(O_{+}+a)\cup
g(O_{+}+a)$ such that $\widehat{O}_{+}$ and $g^{-1}\widehat{O}_{+}$ are in the
future tangent $V_{O,+}$ of $O$.
The set ${\mathcal{A}}(\widehat{O}_{+})\Omega$ is a core of ${\hbox{\rm
Ad\,}}U(g)(\Phi^{\mathrm{out}}(x))$ and $\Phi^{\mathrm{out}}({\hbox{\rm
Ad\,}}U(g)(x))$. But their actions on $\Omega$ coincide and they commute with
${\mathcal{A}}(\widehat{O}_{+})$, hence the operators must coincide. This
concludes the desired local covariance of $\Phi^{\mathrm{out}}(x)$ with
respect to $U$. ∎
We can now define the outgoing free field net by
${\mathcal{A}}^{\mathrm{out}}(O):=\\{R_{\lambda}(\Phi^{\mathrm{out}}(x)):x=x^{*}\in{\mathcal{A}}(O)\mbox{
smooth},\lambda\notin{\mathbb{R}}\\}^{\prime\prime}.$
By Corollary 2.10, this net ${\mathcal{A}}^{\mathrm{out}}$ is covariant with
respect to the unitary representation $U$ for the original net
${\mathcal{A}}$. The vacuum $\Omega$ is in general not cyclic for
${\mathcal{A}}^{\mathrm{out}}$.
This free field net can be defined for any given net which contains massless
particles. We will show that it is a subnet for a given conformal net, namely
${\mathcal{A}}^{\mathrm{out}}(O)\subset{\mathcal{A}}(O)$.
## 3 A proof under global conformal invariance
In this Section we show that a globally conformal net (defined below) contains
the second quantization (free) net if it has nontrivial massless particle
spectrum. Of course these two assumptions are very strong. We can actually
drop global conformal invariance as we will see in Section 4 but here we
present a simpler proof in order to clarify the involved ideas. This result
should thus be considered as a simplification in operator-algebraic
formulation of [3] with an additional assumption, the global conformal
invariance (GCI). It is a strong property, under which there are indications
that the stress-energy tensor is the same as that of the free field [33].
A conformal net $({\mathcal{A}},U,\Omega)$ is said to be globally conformal if
the extension to ${\bar{M}}$ (the compactified Minkowski space, see Section
2.1.1) already admits a global action of ${\widetilde{\mathscr{C}}}$ (cf.
[30, 29], where GCI is defined in terms of Wightman functions). Namely, the
action of ${\widetilde{\mathscr{C}}}$ factors through the action of
${\mathscr{C}}$. For example, the massless free fields with odd integer
helicity are globally conformal, while other free fields are not [19,
Corollary 3.12].
In this case, any two operators $x,y$ localized in timelike-separated regions
commute. Indeed, any pair of timelike-separated regions can be brought into
spacelike-separated regions by an action of ${\mathscr{C}}$.
The first consequence of GCI is the following.
###### Proposition 3.1.
For a net ${\mathcal{A}}$ with GCI, it holds that
${\mathcal{A}}(V_{+})={\mathcal{A}}(V_{-})^{\prime}$, where $V_{\pm}$ are the
future and past lightcones.
###### Proof.
As remarked above, it holds that
${\mathcal{A}}(V_{+})\subset{\mathcal{A}}(V_{-})^{\prime}$ by GCI. The modular
group for ${\mathcal{A}}(V_{-})$ with respect to $\Omega$ is the dilation [7]
(see Section 2.1.3), thus the modular group for
${\mathcal{A}}(V_{-})^{\prime}$ with respect to $\Omega$ is again dilation (up
to a reparametrization). It is clear that ${\mathcal{A}}(V_{+})$ is invariant
under dilation.
Let us recall the simple variant of Takesaki’s theorem [34, Theorem IX.4.2].
Assume that ${\mathcal{N}}\subset{\mathcal{M}}$ is an inclusion of von Neumann
algebras, $\Omega$ is a cyclic separating vector for ${\mathcal{M}}$ and the
modular group ${\hbox{\rm Ad\,}}\Delta^{it}$ for ${\mathcal{M}}$ with respect
to $\Omega$ preserves ${\mathcal{N}}$. Then there is a conditional expectation
$E:{\mathcal{M}}\to{\mathcal{N}}$ which preserves the state
$\langle\Omega,\cdot\,\Omega\rangle$ and this is implemented by the projection
$P$ onto the subspace $\overline{{\mathcal{N}}\Omega}$: $E(x)\Omega=Px\Omega$.
In particular, $E(x)=x$ if and only if $x\in{\mathcal{N}}$.
In our situation, from Takesaki’s theorem it follows that
${\mathcal{A}}(V_{+})={\mathcal{A}}(V_{-})^{\prime}$ because $\Omega$ is
cyclic for the both algebras by Reeh-Schlieder property (cf. [36, Appendix
A]). Therefore the projection above is trivial and the two von Neumann
algebras must coincide. ∎
###### Lemma 3.2.
For a net ${\mathcal{A}}$ with GCI, the outgoing free field net
${\mathcal{A}}^{\mathrm{out}}$ is a subnet of ${\mathcal{A}}$.
###### Proof.
Let $O\subset V_{-}$ and $O_{+}\subset V_{+}$. In particular, $O_{+}$ is in
the future tangent of $O$. By the construction of asymptotic fields,
$\Phi^{h_{T}}(x)$ is in the spacelike complement of ${\mathcal{A}}(O_{+})$ if
$x\in{\mathcal{A}}(O)$, hence we have
$R_{\lambda}(\Phi^{\mathrm{out}}(x))\in{\mathcal{A}}(V_{+})^{\prime}$ by the
convergence in the strong resolvent sense and by Proposition 3.1 this is equal
to ${\mathcal{A}}(V_{-})$. This implies that
${\mathcal{A}}^{\mathrm{out}}(V_{-})\subset{\mathcal{A}}(V_{-})$.
By conformal covariance with respect to the same representation $U$ (see the
end of Section 2.2.2), with the conformal group ${\mathscr{C}}$ which takes
$V_{-}$ to any double cone $O$, we obtain
${\mathcal{A}}^{\mathrm{out}}(O)\subset{\mathcal{A}}(O)$. ∎
We summarize the result.
###### Theorem 3.3.
Let $({\mathcal{A}},U,\Omega)$ be a globally conformal net and assume that the
massless particle spectrum of $U$ is nontrivial. Then there is a subnet
${\mathcal{A}}^{\mathrm{out}}$ of ${\mathcal{A}}$, which is isomorphic to the
free field net associated to the massless representation. The free subnet
${\mathcal{A}}^{\mathrm{out}}$ generates the whole massless particle spectrum
of $U$.
###### Proof.
Almost all statements have been proved above. The whole massless particle
spectrum of $U$ is generated by ${\mathcal{A}}^{\mathrm{out}}$ since
$\\{P_{1}x\Omega:x\in{\mathcal{A}}(O)\\}$ is dense in $P_{1}{\mathcal{H}}$ by
the Reeh-Schlieder property of ${\mathcal{A}}$ and we only have to consider
the asymptotic fields for self-adjoint elements $x_{+}=(x+x^{*})/2$ and
$x_{-}=(x-x^{*})/2i$. The exponentiated fields
$e^{i\Phi^{\mathrm{out}}(x_{\pm})}$ are localized in
${\mathcal{A}}^{\mathrm{out}}(O)$ and the one-particle vectors are obtained by
$\frac{d}{dt}e^{it\Phi^{\mathrm{out}}(x_{\pm})}\Omega$. ∎
One can analogously define ${\mathcal{A}}^{\mathrm{in}}$ by taking the limit
$T\to-\infty$. Now that we know that the net ${\mathcal{A}}$ includes a free
field subnet, it follows that
${\mathcal{A}}^{\mathrm{out}}={\mathcal{A}}^{\mathrm{in}}$ because we can
choose local operators $x$ which creates one-particle states from the free
subnet. For the free field net, the asymptotic field net is of course itself,
so we obtain ${\mathcal{A}}^{\mathrm{out}}={\mathcal{A}}^{\mathrm{in}}$.
Accordingly, although one can define S-matrix on the subspace generated by
${\mathcal{A}}^{\mathrm{out}}={\mathcal{A}}^{\mathrm{in}}$, roughly as the
difference between
$\xi_{1}{\overset{\mathrm{out}}{\times}}\cdots{\overset{\mathrm{out}}{\times}}\xi_{n}$
and
$\xi_{1}{\overset{\mathrm{in}}{\times}}\cdots{\overset{\mathrm{in}}{\times}}\xi_{n}$.
(see [12], and [9] for its two-dimensional variant), it is trivial.
## 4 A general proof
Finally let us prove the existence of a free subnet under conformal invariance
but not necessarily under global conformal invariance. If a net is not
globally conformal, it does not necessarily hold that
${\mathcal{A}}(V_{+})^{\prime}={\mathcal{A}}(V_{-})$ and our previous argument
does not work. Instead, here we use directed asymptotic fields defined below.
As already suggested by Buchholz himself [11, Section 4], Theorem 2.3 can be
extended for asymptotic fields with a function $f$ which specifies a direction
in which a local observable proceeds asymptotically. Such a directed
asymptotic field still has a certain local property and we can construct
subnet.
### 4.1 Directed asymptotic fields
For a smooth function $f$ on the unit sphere $S^{2}$ such that
$f(\mathbf{n})\geq 0$ and $\int_{S^{2}}d\omega(\mathbf{n})\;f(\mathbf{n})=1$,
we define
$\Phi^{t}_{f}(x):=-2t\int_{S^{2}}d\omega(\mathbf{n})\;f(\mathbf{n})\partial_{0}x(t,t\mathbf{n}),\;\;\;\Phi^{h_{T}}_{f}(x)=\int_{{\mathbb{R}}}dt\;h_{T}(t)\Phi^{t}_{f}(x).$
where notations are as in Section 2.2.1. In [10] the case where $f=1$ has been
worked out and it has been suggested in [11] that the whole theory works for a
general $f$. As we need certain extended results, let us discuss the proofs
and how they should be modified when $f$ is nontrivial.
First, we explain the claim [11, Equation (4.3)]:
$\underset{T\to\infty}{{{\mathrm{s}\textrm{-}\lim}\,}}\Phi^{h_{T}}_{f}(x)\Omega=P_{1}f\left(\frac{\mathbf{P}}{|\mathbf{P}|}\right)x\Omega,$
where $\mathbf{P}$ is the 3-momentum operator of the given representation $U$
of the net (see Section 2.2.1) and
$f\left(\frac{\mathbf{P}}{|\mathbf{P}|}\right)$ is defined by functional
calculus. This follows from the mean ergodic theorem analogously as in [10,
Section 2]. Indeed, this time we have
$\Phi^{t}_{f}(x)\Omega=-\frac{it}{2\pi}\int dE_{P}\int_{0}^{\pi}\sin\theta
d\theta\int_{0}^{2\pi}d\varphi\;f(\theta,\varphi)e^{it(H-\mathbf{n}\cdot\mathbf{P})}H(x\Omega)_{P}$
where
$P=(H,\mathbf{P}),\mathbf{n}=(\sin\theta\cos\varphi,\sin\theta\sin\varphi,\cos\varphi)$
and the integral is about $\mathbf{n}$ (on the unit sphere) and the joint
spectral decomposition with respect to $P$ and accordingly $(x\Omega)_{P}$ is
the $P$-component with respect to it. Since the support of $P$ is included in
the closed positive lightcone $\overline{V}_{+}$, the $t$-dependent phase
vanishes $e^{it(H-\mathbf{n}\cdot\mathbf{P})}$ only on the surface of the cone
$H=|\mathbf{P}|$. Instead, on this surface the integral with respect to
$\theta,\varphi$ gives
$\frac{2\pi}{-it|\mathbf{P}|}f\left(\frac{\mathbf{P}}{|\mathbf{P}|}\right)e^{it(H-|\mathbf{P}|)}$
with additional terms which tend to zero when the limit in the mean ergodic
theorem is taken (this can be explicitly demonstrated by considering a
function $f$ which is $z$-rotation symmetric. A general function can be
approximated by sums of such functions with different axis of symmetry in
$L^{1}$-norm). Hence we obtain the formula above.
Only in this paragraph, the propositions and sections refer to those in [10].
Now, Lemma 1 can be modified straightforwardly. Lemma 2 is the main technical
ingredient and has been proved in the Appendix. Now, among the statements in
the Appendix, the only one in which the spherical integral matters is the
Lemma, in which commutators of spherically smeared operators are estimated.
Here the only property essentially used in the estimate is locality of
operators and the integrand gets bounded by norm. This means, if one has to
smear the integrand with $f$, it changes the weight of localization. However,
as the integrand is bounded by norm and no other technique is required, one
can simply bound $f$ by a constant in order to adapt the proof. By this bound,
the estimate gets simply multiplied by a constant depending on $f$. This does
not affect the rest of the arguments at all. Indeed, this Lemma is used later
in Corollary, and indirectly in Proposition II, where the overall constant is
unimportant. Finally, Lemma 2 is proved in Section d) and the overall constant
in the estimate does not play any role, hence we obtain the modified Lemma 2.
In the rest of the paper, the spherical integral appear only through the
correspondence from $x$ to
$P_{1}f\left(\frac{\mathbf{P}}{|\mathbf{P}|}\right)x\Omega$. Accordingly, one
can modify all the propositions of the paper.
Thereafter one can repeat our argument in order to extend the results from
${\mathcal{A}}_{N_{0}}(O)$ to ${\mathcal{A}}(O)$. In summary, we obtain the
following.
###### Theorem 4.1.
Let $x=x^{*},x_{1}=x_{1}^{*},x_{2}=x_{2}^{*}$ be smooth elements (with respect
to ${\widetilde{\mathscr{C}}}$) of ${\mathcal{A}}(O)$, $O$ be a double cone
and $f,f_{1},f_{2}$ be smooth functions on $S^{2}$.
1. (1)
For arbitrary $y\in{\mathcal{A}}(O_{+})$, where $O_{+}\subset V_{O,+}$ is
bounded,
$y\cdot{\mathcal{D}}(\Phi^{\mathrm{out}}_{f}(x))\subset{\mathcal{D}}(\Phi^{\mathrm{out}}_{f}(x))$
and one has $[\Phi^{\mathrm{out}}_{f}(x),y]=0$ on
${\mathcal{D}}(\Phi^{\mathrm{out}}_{f}(x))$.
2. (2)
The operator $\Phi^{\mathrm{out}}_{f}(x)$ is self-adjoint and depends only on
$P_{1}f\left(\frac{\mathbf{P}}{|\mathbf{P}|}\right)x\Omega$. The subspace
${\mathcal{F}}(V_{O,+})\Omega$ is a core of $\Phi^{\mathrm{out}}_{f}(x)$.
3. (3)
The sequence $\Phi^{h_{T}}_{f}(x)$ is convergent to
$\Phi^{\mathrm{out}}_{f}(x)$ in the strong resolvent sense.
4. (4)
The domain ${\mathcal{D}}(\Phi^{\mathrm{out}}_{f}(x))$ includes the set
${\mathcal{H}}^{\mathrm{out}}_{\mathrm{prod}}$ of all product states
$\xi_{1}{\overset{\mathrm{out}}{\times}}\xi_{2}{\overset{\mathrm{out}}{\times}}\cdots{\overset{\mathrm{out}}{\times}}\xi_{n}$
and its action is
$\Phi^{\mathrm{out}}_{f}(x)\cdot\xi_{1}{\overset{\mathrm{out}}{\times}}\xi_{2}{\overset{\mathrm{out}}{\times}}\cdots{\overset{\mathrm{out}}{\times}}\xi_{n}=\xi{\overset{\mathrm{out}}{\times}}\xi_{1}{\overset{\mathrm{out}}{\times}}\xi_{2}{\overset{\mathrm{out}}{\times}}\cdots{\overset{\mathrm{out}}{\times}}\xi_{n}+\sum_{k=1}^{n}\langle\xi,\xi_{k}\rangle\xi_{1}{\overset{\mathrm{out}}{\times}}\cdots\check{\xi}_{k}\cdots{\overset{\mathrm{out}}{\times}}\xi_{n},$
where
$\xi=P_{1}f\left(\frac{\mathbf{P}}{|\mathbf{P}|}\right)x\Omega=P_{1}f\left(\frac{\mathbf{P}}{|\mathbf{P}|}\right)x^{*}\Omega$.
5. (5)
For the resolvent $R_{\pm i}(y)=(y\pm i)^{-1}$ of $y$, it holds that
$\displaystyle[R_{\pm i}(\Phi^{\mathrm{out}}_{f_{1}}(x_{1})),R_{\pm
i}(\Phi^{\mathrm{out}}_{f_{2}}(x_{2}))]$
$\displaystyle=\langle\Omega,[\Phi^{\mathrm{out}}_{f_{1}}(x_{1}),\Phi^{\mathrm{out}}_{f_{2}}(x_{2})]\Omega\rangle\cdot
R_{\pm i}(\Phi^{\mathrm{out}}_{f_{1}}(x_{1}))R_{\pm
i}(\Phi^{\mathrm{out}}_{f_{2}}(x_{2}))^{2}R_{\pm
i}(\Phi^{\mathrm{out}}_{f_{1}}(x_{1}))$
$\displaystyle=\mathrm{Re}\,\left\langle
P_{1}f_{1}\left(\frac{\mathbf{P}}{|\mathbf{P}|}\right)x_{1}\Omega,P_{1}f_{2}\left(\frac{\mathbf{P}}{|\mathbf{P}|}\right)x_{2}\Omega\right\rangle\cdot
R_{\pm i}(\Phi^{\mathrm{out}}_{f_{1}}(x_{1}))R_{\pm
i}(\Phi^{\mathrm{out}}_{f_{2}}(x_{2}))^{2}R_{\pm
i}(\Phi^{\mathrm{out}}_{f_{1}}(x_{1})).$
6. (6)
For $x\in{\mathcal{A}}(O)$ and $y\in{\mathcal{F}}(V_{O,+})$, it holds that
$[R_{\pm i}(\Phi^{\mathrm{out}}_{f}(x)),y]=0$.
Other propositions in [10, Section 4] can be appropriately modified but we
state here only what we need.
### 4.2 Conformal free subnet
Let ${\mathcal{A}}$ be a conformal net with massless particles. We consider
the standard double cone $O_{1}$. The following is an easy geometric
observation (c.f. [11, P.60]).
###### Lemma 4.2.
For a double cone $O$ which is sufficiently spacelike separated from $O_{1}$,
there is a compact set $\Sigma$ in $S^{2}$ such that
$\\{a+(t,t\mathbf{n}):a\in O,\mathbf{n}\in\Sigma,t\mbox{ sufficiently
large}\\}$ is spacelike separated from $O_{1}$.
Let us explain what “sufficiently separated” means. First, we consider for
simplicity the point of origin and a spacelike vector $v$. We may assume that
$v=(v_{0},0,0,v_{3})$, where $|v_{0}|<v_{3}$. The vectors in question are of
the form
$\\{(v_{0}+t,t\sin\theta\,\cos\phi,t\sin\theta\,\sin\phi,v_{3}+t\cos\theta),t\geq
0\\}.$
As one can check easily, these are spacelike for sufficiently large $t$ if
$\cos\theta>\frac{v_{0}}{v_{3}}$. In general, even if $O$ and $O_{1}$ are open
regions, if the difference $O_{1}-O$ is almost in one direction, then the
above arguments works.
From this, we see that certain directed asymptotic fields still have certain
locality.
###### Lemma 4.3.
For $x\in{\mathcal{A}}(O)$ where $O\perp O_{1}$ (spacelike separated) and a
smooth function $f$ such that $O$ and the support of $f$ satisfy the situation
of Lemma 4.2, $\Phi^{\mathrm{out}}_{f}(x)$ is affiliated to
${\mathcal{A}}(O_{1})^{\prime}={\mathcal{A}}(O_{1}^{\mathrm{c}})$.
###### Proof.
This follows immediately from the localization of approximants
$\Phi^{h_{T}}_{f}(x)$ and their convergence to $\Phi^{\mathrm{out}}_{f}(x)$ in
the strong resolvent sense. ∎
We construct a subnet of ${\mathcal{A}}$ as follows. First, consider the
following:
$\displaystyle{\mathcal{A}}^{\mathrm{dir}}(O_{1}^{\mathrm{c}}):=$
$\displaystyle\\{{\hbox{\rm
Ad\,}}U(g)(R_{\lambda}(\Phi^{\mathrm{out}}_{f}(x))):\mathrm{Im}\,\lambda\neq
0,g\in{\widetilde{\mathscr{C}}}(O_{1}),$ $\displaystyle
x\in{\mathcal{A}}(O),O\perp O_{1},f\mbox{ as Lemma
\ref{lm:directed}}\\}^{\prime\prime},$
where ${\widetilde{\mathscr{C}}}(O_{1})$ is the stabilizer group of $O_{1}$ in
${\widetilde{\mathscr{C}}}$. This is clearly a subalgebra of
${\mathcal{A}}(O_{1}^{\mathrm{c}})={\mathcal{A}}(O_{1})^{\prime}$. For any
other double cone $O$ in the global space ${\widetilde{M}}$, we can find
$g\in{\widetilde{\mathscr{C}}}$ such that $O=gO_{1}^{\mathrm{c}}$. With this
$g$, we define ${\mathcal{A}}^{\mathrm{dir}}(O)={\hbox{\rm
Ad\,}}U(g)({\mathcal{A}}^{\mathrm{dir}}(O_{1}^{\mathrm{c}}))$. This is well-
defined, because in the definition of
${\mathcal{A}}^{\mathrm{dir}}(O_{1}^{\mathrm{c}})$ above $g$ runs in the
stability group ${\widetilde{\mathscr{C}}}(O_{1})$.
###### Lemma 4.4.
The family $\\{{\mathcal{A}}^{\mathrm{dir}}(O)\\}$ is a conformal subnet of
${\mathcal{A}}$ and generates ${\mathcal{H}}^{\mathrm{out}}$ from the vacuum
$\Omega$.
###### Proof.
Covariance of ${\mathcal{A}}^{\mathrm{dir}}$ holds by definition (and well-
definedness). ${\mathcal{A}}^{\mathrm{dir}}(O)$ is a subalgebra of
${\mathcal{A}}(O)$, hence locality follows. Positivity of energy and the
properties of vacuum are inherited from those of $U$ and $\Omega$.
Note that the closed subspace ${\mathcal{H}}^{\mathrm{out}}$ =
$\overline{{\mathcal{H}}^{\mathrm{out}}_{\mathrm{prod}}}$ is invariant under
$U(g)$. Indeed, we know already that ${\mathcal{A}}^{\mathrm{out}}$ is a net
whose restriction to the Minkowski space $M$ generates the subspace
${\mathcal{H}}^{\mathrm{out}}$. Any local algebra
${\mathcal{A}}^{\mathrm{out}}(O)$, where $O$ is a double cone in $M$, produces
a dense subspace of ${\mathcal{H}}^{\mathrm{out}}$ from $\Omega$ and if $g$ is
in a small neighborhood of the unit element of ${\widetilde{\mathscr{C}}}$,
then ${\mathcal{A}}^{\mathrm{out}}(gO)$ is again a local algebra in $M$ and
generate another dense subspace of ${\mathcal{H}}^{\mathrm{out}}$, thus
${\mathcal{H}}^{\mathrm{out}}$ is invariant under such $U(g)$. A general
element $g$ can be reached as a finite product of such elements, and the
invariance follows.
For $O\perp O_{1}$, the fields
$\Phi^{\mathrm{out}}_{f}(x),x\in{\mathcal{A}}(O)$ can generate
$P_{1}\chi_{\Sigma}\left(\frac{\mathbf{P}}{|\mathbf{P}|}\right){\mathcal{H}}$
where $\Sigma$ is the compact set in Lemma 4.2 and $\chi_{\Sigma}$ denotes the
characteristic function of $\Sigma$. One can patch such $\Sigma$ to see that
the whole one particle space is spanned by $\Phi^{\mathrm{out}}_{f}(x)$ which
are affiliated to ${\mathcal{A}}^{\mathrm{dir}}(O_{1}^{\mathrm{c}})$. Since
the second quantization structure is the same,
$\overline{{\mathcal{A}}^{\mathrm{dir}}(O_{1}^{\mathrm{c}})\Omega}$ includes
the whole free Fock space ${\mathcal{H}}^{\mathrm{out}}$. As
${\mathcal{H}}^{\mathrm{out}}$ is invariant under $U(g)$, by the construction
of ${\mathcal{A}}^{\mathrm{dir}}(O_{1}^{\mathrm{c}})$,
${\mathcal{H}}^{\mathrm{out}}$ is the Hilbert subspace generated by
${\mathcal{A}}^{\mathrm{dir}}(O_{1}^{\mathrm{c}})$ from $\Omega$. Then the
same holds for an arbitrary double cone by the covariance of
${\mathcal{A}}^{\mathrm{dir}}$ and the invariance of
${\mathcal{H}}^{\mathrm{out}}$. This is Reeh-Schlieder property of
${\mathcal{A}}^{\mathrm{dir}}$ (as a subnet).
Now we consider the isotony of ${\mathcal{A}}^{\mathrm{dir}}$. The modular
group of ${\mathcal{A}}(O)$ acts geometrically and
${\mathcal{A}}^{\mathrm{dir}}(O)$ is invariant under that by construction. By
Takesaki’s theorem, there is a conditional expectation $E^{\mathrm{dir}}$ from
${\mathcal{A}}(O)$ to ${\mathcal{A}}^{\mathrm{dir}}(O)$ implemented by the
projection $P^{\mathrm{out}}$ onto ${\mathcal{H}}^{\mathrm{out}}$. It is
immediate that this defines a coherent family of conditional expectations in
the sense that $E^{\mathrm{dir}}$ does not depend on $O$, because it is
implemented by the same projection $P^{\mathrm{out}}$. With this, the isotony
of ${\mathcal{A}}^{\mathrm{dir}}$ follows from the isotony of ${\mathcal{A}}$.
∎
###### Proposition 4.5.
Two nets ${\mathcal{A}}^{\mathrm{dir}}(O)$ and
${\mathcal{A}}^{\mathrm{out}}(O)$ coincide, the latter being defined in
Section 2.2.2.
###### Proof.
If $x\in{\mathcal{A}}(O)$ and $y\in{\mathcal{A}}(O_{1})$, where $O\perp O_{1}$
and $f$ is chosen for the pair $O,O_{1}$ as in Lemma 4.2, then
$\Phi^{\mathrm{out}}_{f}(x)$ and $\Phi^{\mathrm{out}}(y)$, or their
resolvents, commute by the techniques of Jost-Lehmann-Dyson representation as
in [22, Section 4][10, Theorem 9]. We know that ${\mathcal{A}}^{\mathrm{out}}$
is covariant with respect to $U$. Especially,
${\mathcal{A}}^{\mathrm{out}}(O_{1})$ is invariant under ${\hbox{\rm
Ad\,}}U(g)$ where $g\in{\widetilde{\mathscr{C}}}(O_{1})$. By definition of
${\mathcal{A}}^{\mathrm{dir}}$, the two nets ${\mathcal{A}}^{\mathrm{dir}}$
and ${\mathcal{A}}^{\mathrm{out}}$ are relatively local.
We saw also that they generate the same Hilbert subspace
${\mathcal{H}}^{\mathrm{out}}$ in Lemma 4.4. Both nets
${\mathcal{A}}^{\mathrm{out}}$, ${\mathcal{A}}^{\mathrm{dir}}$ are conformal
with respect to $U$, relatively local and span the same Hilbert subspace. By
the standard application of Takesaki’s theorem as in Proposition 3.1, these
local algebras coincide. ∎
This concludes our construction. Any conformal net, global or not, contains a
free subnet ${\mathcal{A}}^{\mathrm{out}}={\mathcal{A}}^{\mathrm{dir}}$ which
generates the massless particle spectrum.
#### Decoupling of the free field subnet
The next Proposition works with Haag dual (for double cones in $M$) nets with
covariance with respect to the Poincaré group. A net has split property if for
each pair $O_{1}\subset O_{2}$ such that $\overline{O}_{1}\subset O_{2}$,
there is a type I factor ${\mathcal{R}}$ such that
${\mathcal{A}}(O_{1})\subset{\mathcal{R}}\subset A(O_{2})$. A DHR sector of
the net ${\mathcal{A}}$ is the equivalence class of a representation $\pi$ of
the global $C^{*}$-algebra
$\overline{\bigcup_{O}{\mathcal{A}}(O)}^{\|\cdot\|}$ where $O$ are double
cones under certain conditions [18]. Among others, the most important one is
that there is a double cone $O$ such that the restriction of $\pi$ to
$\overline{\bigcup_{O^{\prime}\perp O}{\mathcal{A}}(O^{\prime})}^{\|\cdot\|}$
($\perp$ denotes the spacelike separation) is unitarily equivalent to the
identity representation (the vacuum representation).
###### Proposition 4.6.
Let ${\mathcal{A}}$ be a Haag dual subnet of a Haag dual net ${\mathcal{F}}$
on a separable Hilbert space and assume that ${\mathcal{A}}$ has split
property and has no nontrivial irreducible DHR sector (if
${\mathcal{A}}\subset{\mathcal{F}}$ is an inclusion of conformal nets, we have
the Haag duality on ${\widetilde{M}}$ and we do not need the Haag duality on
$M$). Then ${\mathcal{F}}$ decouples, namely
${\mathcal{F}}(O)=\tilde{\pi}_{0}({\mathcal{A}}(O))\otimes{\mathcal{C}}_{0}(O)$
where ${\mathcal{C}}(O)={\mathcal{A}}(O)^{\prime}\cap{\mathcal{F}}(O)$ is the
coset net, ${\mathcal{C}}_{0}$ is the irreducible vacuum representation of
${\mathcal{C}}$ and $\tilde{\pi}_{0}$ is the vacuum representation of
${\mathcal{A}}$ (the restriction of ${\mathcal{A}}$ to its cyclic subspace).
###### Proof.
The argument here is essentially contained in the proof of [14, Theorem 3.4]
and has been suggested to apply to globally conformal nets in [2].
The representation of ${\mathcal{A}}$ on the vacuum Hilbert space of
${\mathcal{F}}$ is a DHR representation of ${\mathcal{A}}$ [14, Lemma 3.1]
(this can be proved under split property of ${\mathcal{A}}$ only, from which
it follows that local algebras are properly infinite, and separability of the
Hilbert space), hence by split property it is the direct integral of
irreducible representations (see [21, Proposition 56], which is written for
nets on $S^{1}$ but the arguments apply to nets on $M$), and by assumption it
is the direct sum of copies of the vacuum representation. Hence on the Hilbert
space of ${\mathcal{F}}$, an element $x\in{\mathcal{A}}(O)$ is of the form
$\tilde{\pi}_{0}(x)\otimes{\mathbb{C}}{\mathbbm{1}}$ with an appropriate
decomposition ${\mathcal{H}}={\mathcal{H}}_{\mathcal{A}}\otimes{\mathcal{K}}$.
Since ${\mathcal{A}}$ is Haag dual on its vacuum representation
$\tilde{\pi}_{0}$, we have
${\mathcal{A}}(O^{\prime})=\tilde{\pi}_{0}({\mathcal{A}}(O^{\prime}))\otimes{\mathbb{C}}{\mathbbm{1}}=\tilde{\pi}_{0}({\mathcal{A}}(O))^{\prime}\otimes{\mathbb{C}}{\mathbbm{1}}$.
By the relative locality of ${\mathcal{F}}$ to ${\mathcal{A}}$, we have
${\mathcal{F}}(O)\subset{\mathcal{A}}(O^{\prime})^{\prime}=\tilde{\pi}_{0}({\mathcal{A}}(O))\otimes{\mathcal{B}}({\mathcal{K}})$.
Now we have an inclusion
${\mathcal{A}}(O)=\tilde{\pi}_{0}({\mathcal{A}}(O))\otimes{\mathbb{C}}{\mathbbm{1}}\subset{\mathcal{F}}(O)\subset\tilde{\pi}_{0}({\mathcal{A}}(O))\otimes{\mathcal{B}}({\mathcal{K}}).$
This relation holds also for a wedge $W$,
${\mathcal{A}}(W)=\tilde{\pi}_{0}({\mathcal{A}}(W))\otimes{\mathbb{C}}{\mathbbm{1}}\subset{\mathcal{F}}(W)\subset\tilde{\pi}_{0}({\mathcal{A}}(W))\otimes{\mathcal{B}}({\mathcal{K}})$
but the wedge algebra $\tilde{\pi}_{0}({\mathcal{A}}(W))$ in the vacuum
representation is a factor [4, 1.10.9 Corollary]. Now by [17, Theorem A],
there is ${\mathcal{C}}_{0}(W)\subset{\mathcal{B}}({\mathcal{K}})$ such that
${\mathcal{F}}(W)=\tilde{\pi}_{0}({\mathcal{A}}(W))\otimes{\mathcal{C}}_{0}(W)$.
It is clear that ${\mathcal{F}}(W)={\mathcal{A}}(W)\vee{\mathcal{C}}(W)$,
where ${\mathcal{C}}(W)={\mathcal{F}}(W)\cap{\mathcal{A}}(W)^{\prime}$
By Haag duality of the both nets ${\mathcal{F}}$ and ${\mathcal{A}}$, we have
${\mathcal{F}}(O)=\bigcap_{O\subset W}{\mathcal{F}}(W)=\bigcap_{O\subset
W}\tilde{\pi}_{0}({\mathcal{A}}(W))\otimes{\mathcal{C}}_{0}(W)=\tilde{\pi}_{0}({\mathcal{A}}(O))\otimes\bigcap_{O\subset
W}{\mathcal{C}}_{0}(W).$
By defining
${\mathcal{C}}(O):={\mathcal{F}}(O)\cap{\mathcal{A}}(O)^{\prime}={\mathbb{C}}{\mathbbm{1}}\otimes\bigcap_{O\subset
W}{\mathcal{C}}_{0}(W)$ and ${\mathcal{C}}_{0}(O)=\bigcap_{O\subset
W}{\mathcal{C}}_{0}(W)$, we obtain
${\mathcal{F}}(O)=\tilde{\pi}_{0}({\mathcal{A}}(O))\otimes{\mathcal{C}}_{0}(O)={\mathcal{A}}(O)\vee{\mathcal{C}}(O)$.
If ${\mathcal{A}}\subset{\mathcal{F}}$ is an inclusion of conformal nets, we
can directly argue with double cones $O$. Each ${\mathcal{A}}(O)$ is a factor,
the modular group acts geometrically and Haag duality holds on
${\widetilde{M}}$ (one should simply transplant the duality argument to
${\widetilde{M}}$) [7]. ∎
###### Corollary 4.7.
Let $({\mathcal{A}},U,\Omega)$ be a conformal net and assume that the massless
particle subspace $P_{1}{\mathcal{H}}$ of $U$ consists only of the scalar
representation with finite multiplicity. Then the free subnet
${\mathcal{A}}^{\mathrm{out}}$ decouples in ${\mathcal{A}}$, namely
${\mathcal{A}}(O)={\mathcal{A}}^{\mathrm{out}}(O)\vee{\mathcal{C}}(O)$, where
${\mathcal{C}}(O):={\mathcal{A}}(O)\cap{\mathcal{A}}^{\mathrm{out}}(O)^{\prime}$
is the coset subnet.
###### Proof.
The scalar free field net has no nontrivial DHR sector [1, 15] and has split
property [8, 13]. These properties are inherited by any finite tensor product.
Thus the claim follows from Proposition 4.6. ∎
## 5 Open problems
We have shown that massless particles in a conformal net are free. However,
massless representations are only one of the families of the irreducible
representations of the conformal group. Unfortunately, at the moment the
scattering theory, which extracts free fields, is not applicable to the rest
of the family. It would be interesting if one could extract other fields by a
different device. This would not be very easy because in general they are
expected to be interacting (e.g. the super Yang-Mills theory [26]).
As for decoupling, it relies on the split property and the absence of DHR
sector of the scalar free field. As the proofs in the scalar case are based on
the arguments in the one particle space and the second quantization, we expect
that similar results should hold for each massless finite-helicity
representation of the conformal group.
Another interesting question is whether it is possible to prove conformal
covariance from scale invariance (under certain additional conditions). Some
results have been obtained in this direction [28, 16]. An operator-algebraic
proof is unknown (if we do not assume asymptotic completeness, c.f. [36]).
By comparing with the result that any massless asymptotically complete model
in two dimensions can be obtained by “twisting” a tensor product net [35,
Section 3] [5, Proposition 2.2], one may wonder whether such a structure is
available in four dimensions, too. This is not straightforward, because wedges
are not suited for the scattering theory in four dimensions. Neither are
lightcones, because the intersection of the shifted future and past lightcones
does not give back the algebra for a double cone even in the free field net
[20]. Related to this issue is whether the S-matrix is a complete invariant of
a net under asymptotic completeness. This is open also for massive theories,
although partial results are available [6, 27].
#### Acknowledgement
I am grateful to Detlev Buchholz and Karl-Henning Rehren for pointing out
serious technical issues in early versions of this paper. I thank Marcel
Bischoff, Wojciech Dybalski and Nikolay Nikolov for interesting discussions,
Luca Giorgetti and Vincenzo Morinelli for useful comments and the referees of
Forum of Mathematics, Sigma for careful reading of the manuscript and
suggestions. I appreciate the support by Hausdorff Institut für Mathematik,
where a part of this work has been done.
This work was supported by Grant-in-Aid for JSPS fellows 25-205.
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|
arxiv-papers
| 2013-10-17T15:23:09 |
2024-09-04T02:49:52.533926
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Yoh Tanimoto",
"submitter": "Yoh Tanimoto",
"url": "https://arxiv.org/abs/1310.4744"
}
|
1310.4856
|
11institutetext: Univ Paris Diderot, Sorbonne Paris Cité, LIAFA, UMR 7089
CNRS, Paris, France
11email: {klimann,mairesse,picantin}@liafa.univ-paris-diderot.fr
# Implementing Computations
in Automaton (Semi)groups
Ines Klimann Jean Mairesse Matthieu Picantin
###### Abstract
We consider the growth, order, and finiteness problems for automaton
(semi)groups. We propose new implementations and compare them with the
existing ones. As a result of extensive experimentations, we propose some
conjectures on the order of finite automaton (semi)groups.
###### Keywords:
a
utomaton (semi)groups, growth, order, finiteness, minimization
## 1 Introduction
_Automaton (semi)groups_ — short for semigroups generated by Mealy automata or
groups generated by invertible Mealy automata — were formally introduced a
half century ago (for details, see [10, 7] and references therein). Over the
years, important results have started revealing their full potential. For
instance, the article [9] constructs simple Mealy automata generating infinite
torsion groups and so contributes to the Burnside problem, and, the article
[5] produces Mealy automata generating the first examples of (semi)groups with
intermediate growth and so answers the Milnor problem.
The classical decision problems have been investigated for such (semi)groups.
The word problem is solvable using standard minimization techniques, while the
conjugacy problem is undecidable [16]. Here we concentrate on the problems
related to growth, order, and finiteness.
(-3,-1)(5,4) [1](-4,0.5)a [3](4,0.5)b [2](0,3)c ac32 cb32 c1123 ab1121
[.7]ba112232
(-4,-2.5)(4.5,1) [1](-4,0)a [2](0,0)b [3](4,0)c a132231 b31 ba13 bc22 cb122331
Figure 1: A Mealy automaton and its dual
To illustrate, consider the two Mealy automata of Fig. 1. They are dual, that
is, they can be obtained one from the other by exchanging the roles of
stateset and alphabet. A (semi)group is associated in a natural way with each
automaton (formally defined below). The two Mealy automata of Fig. 1 are
associated with finite (semi)groups. Their orders are respectively: on the
left a semigroup of order 238, on the right a group of order
$\numprint{1494186269970473680896}=2^{64}\cdot 3^{4}\approx 1.5\times
10^{21}$.
Several points are illustrated by this example:
* •
An automaton and its dual generate (semi)groups which are either both finite
or both infinite (see [12, 2]).
* •
The order of a finite automaton (semi)group can be amazingly large. It makes a
priori difficult to decide whether an automaton (semi)group is finite or not.
Actually, the decidability of this question is open (see [10, 2]).
* •
The order of the (semi)groups generated by a Mealy automaton and its dual can
be strikingly different. It suggests to work with both automata together.
The contributions of the present paper are three-fold:
* •
We propose new implementations (in GAP [8]) of classical algorithms for the
computation of the growth function; the computation of the order (if finite);
the semidecision procedure for the finiteness.
* •
We compare the new implementations with the existing ones. Indeed, there exist
two GAP packages dedicated to Mealy automata and their associated
(semi)groups: FR by Bartholdi [4] and automgrp by Muntyan and Savchuk [11].
* •
We realize systematic experimentations on small Mealy automata as well as
randomly chosen large Mealy automata. These serve as testbeds to some
conjectures on the growth types of the associated (semi)groups, as well as on
the order of a (semi)group.
The structure of the paper is the following. In Section 2, we present basic
notions on Mealy automata and automaton (semi)groups. In Section 3, we give
new implementations and compare them with the existing ones. Section 4 is
dedicated to experimentations and to the resulting conjectures.
## 2 Automaton (Semi)groups
### 2.1 Mealy Automaton
If one forgets initial and final states, a (finite, deterministic, and
complete) automaton ${\mathcal{A}}$ is a triple
$\bigl{(}A,\Sigma,\delta=(\delta_{i}:A\rightarrow A)_{i\in\Sigma}\bigr{)}$,
where the _set of states_ $A$ and the _alphabet_ $\Sigma$ are non-empty finite
sets, and where the $\delta_{i}$’s are functions.
A _Mealy automaton_ is a quadruple
$\bigl{(}A,\Sigma,\delta=(\delta_{i}:A\rightarrow
A)_{i\in\Sigma},\rho=(\rho_{x}:\Sigma\rightarrow\Sigma)_{x\in A}\bigr{)}\>,$
such that both $(A,\Sigma,\delta)$ and $(\Sigma,A,\rho)$ are automata. In
other terms, a Mealy automaton is a letter-to-letter transducer with the same
input and output alphabets. The transitions of a Mealy automaton are
$x\xrightarrow{i|\rho_{x}(i)}\delta_{i}(x)\>.$
The graphical representation of a Mealy automaton is standard, see Fig. 1.
The notation $x\xrightarrow{{\mathbf{u}}|{\mathbf{v}}}y$ with
${\mathbf{u}}=u_{1}\cdots u_{n}$, ${\mathbf{v}}=v_{1}\cdots v_{n}$ is a
shorthand for the existence of a path
$x\xrightarrow{u_{1}|v_{1}}x_{1}\xrightarrow{u_{2}|v_{2}}x_{2}\longrightarrow\cdots\longrightarrow
x_{n-1}\xrightarrow{u_{n}|v_{n}}y$ in ${\mathcal{A}}$.
In a Mealy automaton $(A,\Sigma,\delta,\rho)$, the sets $A$ and $\Sigma$ play
dual roles. So we may consider the _dual (Mealy) automaton_ defined by
${\mathfrak{d}}({\mathcal{A}})=(\Sigma,A,\rho,\delta)$, that is:
$i\xrightarrow{x\mid y}j\ \in{\mathfrak{d}}({\mathcal{A}})\quad\iff\quad
x\xrightarrow{i\mid j}y\ \in{\mathcal{A}}\>.$
It is pertinent to consider a Mealy automaton and its dual together, that is
to work with the pair $\\{{\mathcal{A}},{\mathfrak{d}}({\mathcal{A}})\\}$, see
an example in Fig. 1.
Let ${\mathcal{A}}=(A,\Sigma,\delta,\rho)$ and
${\mathcal{B}}=(B,\Sigma,\gamma,\pi)$ be two Mealy automata acting on the same
alphabet; their _product_ ${\mathcal{A}}\times{\mathcal{B}}$ is defined as the
Mealy automaton with stateset $A\times B$, alphabet $\Sigma$, and transitions:
$xy\xrightarrow{i|\pi_{y}(\rho_{x}(i))}\delta_{i}(x)\gamma_{\rho_{x}(i)}(y)\>.$
### 2.2 Generating (Semi)groups
Let ${\mathcal{A}}=(A,\Sigma,\delta,\rho)$ be a Mealy automaton. We view
${\mathcal{A}}$ as an automaton with an input and an output tape, thus
defining mappings from input words over $\Sigma$ to output words over
$\Sigma$. Formally, for $x\in A$, the map
$\rho_{x}:\Sigma^{*}\rightarrow\Sigma^{*}$, extending
$\rho_{x}:\Sigma\rightarrow\Sigma$, is defined by:
$\rho_{x}({\mathbf{u}})={\mathbf{v}}\quad\textrm{if}\quad\exists y,\
x\xrightarrow{{\mathbf{u}}|{\mathbf{v}}}y\>.$
By convention, the image of the empty word is itself. The mapping $\rho_{x}$
is length-preserving and prefix-preserving. It satisfies
$\forall u\in\Sigma,\
\forall{\mathbf{v}}\in\Sigma^{*},\qquad\rho_{x}(u{\mathbf{v}})=\rho_{x}(u)\rho_{\delta_{u}(x)}({\mathbf{v}})\>.$
We say that $\rho_{x}$ is the _production function_ associated with
$({\mathcal{A}},x)$. For ${\mathbf{x}}=x_{1}\cdots x_{n}\in A^{n}$ with $n>0$,
set
$\rho_{\mathbf{x}}:\Sigma^{*}\rightarrow\Sigma^{*},\rho_{\mathbf{x}}=\rho_{x_{n}}\circ\cdots\circ\rho_{x_{1}}\>$.
Denote dually by $\delta_{i}:A^{*}\rightarrow A^{*},i\in\Sigma$, the
production mappings associated with the dual Mealy automaton
${\mathfrak{d}}({\mathcal{A}})$. For ${\mathbf{v}}=v_{1}\cdots
v_{n}\in\Sigma^{n}$ with $n>0$, set $\delta_{\mathbf{v}}:A^{*}\rightarrow
A^{*},\ \delta_{\mathbf{v}}=\delta_{v_{n}}\circ\cdots\circ\delta_{v_{1}}$.
###### Definition 1
Consider a Mealy automaton ${\mathcal{A}}$. The semigroup of mappings from
$\Sigma^{*}$ to $\Sigma^{*}$ generated by $\rho_{x},x\in A$, is called the
_semigroup generated by ${\mathcal{A}}$_ and is denoted by
$\langle{{{\mathcal{A}}}}\rangle_{+}$. A semigroup $G$ is an _automaton
semigroup_ if there exists a Mealy automaton ${\mathcal{A}}$ such that
$G=\langle{{{\mathcal{A}}}}\rangle_{+}$.
A Mealy automaton ${\mathcal{A}}=(A,\Sigma,\delta,\rho)$ is _invertible_ if
all the mappings $\rho_{x}:\Sigma\to\Sigma$ are permutations. Then the
production functions $\rho_{x}:\Sigma^{*}\to\Sigma^{*}$ are invertible.
###### Definition 2
Let ${\mathcal{A}}=(A,\Sigma,\delta,\rho)$ be invertible. The _group generated
by ${\mathcal{A}}$_ is the group generated by the mappings
$\rho_{x}:\Sigma^{*}\to\Sigma^{*}$, $x\in A$. It is denoted by
$\langle{{\mathcal{A}}}\rangle$.
Let ${\mathcal{A}}=(A,\Sigma,\delta,\rho)$ be an invertible Mealy automaton.
Its _inverse_ is the Mealy automaton ${{\mathcal{A}}}^{-1}$ with stateset
$A^{-1}=\\{x^{-1},x\in A\\}$ and set of transitions
$x^{-1}\xrightarrow{j\mid i}y^{-1}\ \in{\mathcal{A}}^{-1}\quad\iff\quad
x\xrightarrow{i\mid j}y\ \in{\mathcal{A}}\>.$
A Mealy automaton is _reversible_ if its dual is invertible. A Mealy automaton
${\mathcal{A}}$ is _bireversible_ if both ${\mathcal{A}}$ and
${{\mathcal{A}}}^{-1}$ are invertible and reversible.
###### Theorem 2.1 ([2, 12, 13])
The (semi)group generated by a Mealy automaton is finite if and only if the
(semi)group generated by its dual is finite.
### 2.3 Minimization and the Word Problem
Let $\mathcal{A}=(A,\Sigma,\delta,\rho)$ be a Mealy automaton. The _Nerode
equivalence on $A$_ is the limit of the sequence of increasingly finer
equivalences $(\equiv_{k})$ recursively defined by:
$\displaystyle\forall x,y\in A,\qquad\qquad x\equiv_{0}y$ $\displaystyle\
\Longleftrightarrow\ \rho_{x}=\rho_{y}\>,$ $\displaystyle\forall k\geqslant
0,\,x\equiv_{k+1}y$ $\displaystyle\ \Longleftrightarrow\
x\equiv_{k}y\quad\text{and}\quad\forall i\in\Sigma,\
\delta_{i}(x)\equiv_{k}\delta_{i}(y)\>.$
Since the set $A$ is finite, this sequence is ultimately constant; moreover if
two consecutive equivalences are equal, the sequence remains constant from
this point. The limit is therefore computable. For every element $x$ in $A$,
we denote by $[x]$ the class of $x$ w.r.t. the Nerode equivalence.
###### Definition 3
Let $\mathcal{A}=(A,\Sigma,\delta,\rho)$ be a Mealy automaton and let $\equiv$
be the Nerode equivalence on $A$. The _minimization_ of $\mathcal{A}$ is the
Mealy automaton
${\mathfrak{m}}(\mathcal{A})=(A/\mathord{\equiv},\Sigma,\tilde{\delta},\tilde{\rho})$,
where for every $(x,i)$ in $A\times\Sigma$,
$\tilde{\delta}_{i}([x])=[\delta_{i}(x)]$ and $\tilde{\rho}_{[x]}=\rho_{x}$.
This definition is consistent with the standard minimization of “deterministic
finite automata” where instead of considering the mappings
$(\rho_{x}:\Sigma\to\Sigma)_{x}$, the computation is initiated by the
separation between terminal and non-terminal states. Using Hopcroft algorithm,
the time complexity of minization is ${\cal O}(\Sigma A\log{A})$, see [1].
By construction, a Mealy automaton and its minimization generate the same
semigroup. Indeed, two states of a Mealy automaton belong to the same class
w.r.t the Nerode equivalence if and only if they represent the same element in
the generated (semi)group.
Consider the _word problem_ :
Input: a Mealy automaton $(A,\Sigma,\delta,\rho)$;
${\mathbf{x}},{\mathbf{y}}\in A^{*}$.
Question:
$(\rho_{{\mathbf{x}}}:\Sigma^{*}\to\Sigma^{*})=(\rho_{{\mathbf{y}}}:\Sigma^{*}\to\Sigma^{*})$?
The word problem is solvable by extending the above minimization procedure. FR
uses this approach, while automgrp uses a method based on the wreath recursion
[7].
## 3 Fully Exploiting the Minimization
Consider the following problems for the (semi)group given by a Mealy
automaton: compute the growth function, compute the order (if finite), detect
the finiteness. The packages FR and automgrp provide implementations of the
three problems. Here we propose new implementations based on a simple idea
which fully uses the automaton structure.
### 3.1 Growth
Consider a Mealy automaton ${\mathcal{A}}=(A,\Sigma,\delta,\rho)$ and an
element ${\mathbf{x}}\in A^{*}$. The _length_ of $\rho_{{\mathbf{x}}}$,
denoted by $|\rho_{{\mathbf{x}}}|$, is defined as follows:
$|\rho_{{\mathbf{x}}}|=\min\\{n\mid\exists{\mathbf{y}}\in
A^{n},\,\rho_{{\mathbf{x}}}=\rho_{{\mathbf{y}}}\\}\>.$
The _growth series_ of ${\mathcal{A}}$ is the formal power series given by
$\sum\limits_{g\in\langle{{{\mathcal{A}}}}\rangle_{+}}t^{|g|}=\sum\limits_{n\in\mathbb{N}}\\#\\{g\in\langle{{{\mathcal{A}}}}\rangle_{+}\,;\,|g|=n\\}\>t^{n}\>.$
In words, the growth series enumerates the semigroup elements according to
their length. This is an instanciation of the notion of spherical growth
series for a finitely generated semigroup. Observe that the series is a
polynomial if and only if the semigroup is finite.
##### Using the Generic Algorithm.
Since the word problem is solvable, it is possible to compute an arbitrary but
finite number of coefficients of the growth series. Indeed for each $n$,
generate the set of elements of length $n$ by multiplying elements of length
$n-1$ with generators and detecting-deleting duplicated elements by solving
the word problem. The functions Growth from automgrp and WordGrowth from FR
both follow this pattern. Therefore the structure of the underlying Mealy
automaton is used only to get a solution to the word problem (in fact, both
Growth and WordGrowth are generic, in the sense that they are applicable for
any (semi)group with an implemented solution to the word problem).
##### New Implementation.
We propose a new implementation based on a simple observation. Knowing the
elements of length $n-1$, Nerode minimization can be used in a global manner
to obtain simultaneously the elements of length $n$. Concretely, with each
integer $n\geq 1$ is associated a new Mealy automaton ${{\mathcal{A}}}_{n}$
defined recursively as follows:
${{\mathcal{A}}}_{n}={\mathfrak{m}}({{\mathcal{A}}}_{n-1}\times{\mathfrak{m}}({\mathcal{A}}))\qquad\hbox{and}\qquad{{\mathcal{A}}}_{1}={\mathfrak{m}}({\mathcal{A}})\>.$
Here, we assume, without real loss of generality, that the identity element is
one of the generators (otherwise simply add a new state to the Mealy automaton
coding the identity). This way, the elements of ${\mathcal{A}}_{n}$ are
exactly the elements of length at most $n$.
⬇
AutomatonGrowth:= function(arg)
local aut, radius, growth, sph, curr, next, r;
aut:=arg[1]; # Mealy automaton
if Length(arg)>1 then radius:=arg[2];
else radius:=infinity;
fi;
r := 0; curr := TrivialMealyMachine([1]);
next := Minimized(aut);
aut := Minimized(next+TrivialMealyMachine(Alphabet(aut)));
sph := aut!.nrstates - 1; # number of non-trivial states
growth := [next!.nrstates-sph];
while sph>0 and r<radius
do Add(growth,sph);
r := r+1; curr := next;
next := Minimized(next*aut);
sph := next!.nrstates-curr!.nrstates;
od;
return growth;
end;
Note that AutomatonGrowth(aut) computes the growth of the semigroup
$\langle{\tt aut}\rangle_{+}$, while AutomatonGrowth(aut+aut^-1) computes the
growth of the group $\langle{\tt aut}\rangle$.
##### Experimental Results.
First we run AutomatonGrowth and FR’s WordGrowth on the Grigorchuk automaton,
a famous Mealy automaton generating an infinite group. For radius 10,
AutomatonGrowth is much faster, 76 ms as opposed to 9 912 ms111All timings
displayed in this paper have been obtained on an Intel Core 2 Duo computer
with clock speed 3,06 GHz.. The explanation is simple: WordGrowth calls the
minimization procedure 57 577 times while AutomatonGrowth calls it only 12
times. Here are the details.
⬇
gap> aut := GrigorchukMachine;; radius:= 10;;
gap> ProfileFunctions([Minimized]);
gap> WordGrowth(SCSemigroupNC(aut), radius); time;
[ 1, 4, 6, 12, 17, 28, 40, 68, 95, 156, 216 ]
9912
gap> DisplayProfile();
count self/ms chld/ms function
57577 7712 0 Minimized
7712 TOTAL
gap> ProfileFunctions([Minimized]);
gap> AutomatonGrowth(aut, radius); time;
[ 1, 4, 6, 12, 17, 28, 40, 68, 95, 156, 216 ]
76
gap> DisplayProfile();
count self/ms chld/ms function
12 72 0 Minimized
72 TOTAL
Now we compare the running times of the implementations for the computation of
the first terms of the growth series for all 335 bireversible 3-letter 3-state
Mealy automata (up to equivalence). In Tab. 1, some computations with FR’s
WordGrowth or with automgrp’s Growth could not be completed in reasonable time
for radius 7.
Table 1: Average time (in ms) radius | 1 | 2 | 3 | 4 | 5 | 6 | 7
---|---|---|---|---|---|---|---
FR’s WordGrowth | 3.4 | 29.0 | 555.0 | 8 616.5 | 131 091.4 | 2 530 170.3 | ?
automgrp’s Growth | 0.7 | 2.8 | 16.9 | 158.9 | 1 909.0 | 22 952.8 | ?
AutomatonGrowth | 0.6 | 1.8 | 5.9 | 28.9 | 187.3 | 1 005.9 | 7 131.4
### 3.2 Order of the (Semi)group
Although the finiteness problem is still open, some semidecision procedures
enable to find the order of an expected finite (semi)group. FR and automgrp
use orthogonal approaches. Our new implementation refines the one of FR and
remains orthogonal to the one of automgrp.
##### automgrp’s Implementation.
The GAP package automgrp provides the function LevelOfFaithfulAction, which
allows to compute—very efficiently in some cases—the order of the generated
group. The principle is the following. Let
${\mathcal{A}}=(A,\Sigma,\delta,\rho)$ be an invertible Mealy automaton and
let $G_{k}$ be the group generated by the restrictions of the production
functions to $\Sigma^{k}$. If $\\#G_{k}=\\#G_{k+1}$ for some $k$, then
$\langle{{\mathcal{A}}}\rangle$ is finite of order $\\#G_{k}$. This function
can be easily adapted to a non-invertible Mealy automaton.
Observe that LevelOfFaithfulAction cannot be used to compute the growth
series. Indeed at each step a quotient of the (semi)group is computed. On the
other hand LevelOfFaithfulAction is a good bypass strategy for the order
computation. Furthermore, it takes advantage from the special ability of GAP
to manipulate permutation groups.
##### FR’s Implementation and the New Implementation.
Any algorithm computing the growth series can be used to compute the order of
the generated (semi)group if finite. It suffices to compute the growth series
until finding a coefficient equal to zero. This is the approach followed by
FR. Since we proposed, in the previous section, a new implementation to
compute the growth series, we obtain as a byproduct a new procedure to compute
the order. We call it AutomSGrOrder.
##### Experimental Results.
The orthogonality of the two previous approaches can be simply illustrated by
recalling the introductory example of Fig. 1. Neither FR’s Order nor
AutomSGrOrder are able to compute the order of the large group, while automgrp
via LevelOfFaithfulAction succeeds in only 14 338 ms. Conversely,
AutomSGrOrder computes the order of the small semigroup in 17 ms, while an
adaptation of LevelOfFaithfulAction (to non-invertible Mealy automata) takes 2
193 ms.
### 3.3 Finiteness
There exist several criteria to detect the finiteness of an automaton
(semi)group, see [2, 3, 6, 14, 15, …]. But the decidability of the finiteness
is still an open question. Each procedure to compute the order of a
(semi)group yields a semidecision procedure for the finiteness problem. Both
packages FR and automgrp apply a number of previously known criteria of
(in)finiteness and then intend to conclude by ultimately using an order
computation.
We propose an additional ingredient which uses minimization in a subtle way.
Here, the semigroup to be tested is successively replaced by new ones which
are finite if and only if the original one is finite. It is possible to
incorporate this ingredient to get two new implementations, one in the spirit
of FR and one in the spirit of automgrp. The new implementations are order of
magnitudes better than the old ones. Both are useful since the fastest one
depends on the cases.
#### 3.3.1 ${\mathfrak{m}}{\mathfrak{d}}$-reduction of Mealy Automata and
Finiteness
The ${\mathfrak{m}}{\mathfrak{d}}$-reduction was introduced in [2] to give a
sufficient condition of finiteness. The new semidecision procedures start with
this reduction.
###### Definition 4
A pair of dual Mealy automata is _reduced_ if both automata are minimal.
Recall that ${\mathfrak{m}}$ (resp. ${\mathfrak{d}}$) is the operation of
minimization (resp. dualization). The _${\mathfrak{m}}{\mathfrak{d}}$
-reduction_ of a Mealy automaton ${\mathcal{A}}$ consists in minimizing the
automaton or its dual until the resulting pair of dual Mealy automata is
reduced.
The ${\mathfrak{m}}{\mathfrak{d}}$-reduction is well-defined: if both a Mealy
automaton and its dual automaton are non-minimal, the reduction is confluent
[2]. An example of ${\mathfrak{m}}{\mathfrak{d}}$-reduction is given in Fig.
2.
.7 (-3,-19)(26,-5) [a](0,-8)AA [b](6,-8)BB (8,-9)${\mathcal{A}}$ AABB0123
BBAA0321 [.2]AA1032 [.8]BB1032 (10,-8)E2 (13,-8)F2 E2F2d [0](17,-5)A0
[1](23,-5)A1 [3](17,-9)A3 [2](23,-9)A2 [.3]A1A0aabb A0A1ab [.3]A3A2aabb A2A3ab
A0A3ba A2A1ba (20,-11)E3 (20,-13)F3 E3F3m [13](17,-16)A13 [02](23,-16)A02
A13A02aabb A02A13abba (13,-16)E4 (10,-16)F4 dashed E4F4dmdmd solid
[ab](3,-16)Z (7,-17)${\mathfrak{m}}{\mathfrak{d}}^{*}{({\mathcal{A}})}$
[.2]Z01230123
Figure 2: The ${\mathfrak{m}}{\mathfrak{d}}$-reduction of a pair of dual Mealy
automata
The sequence of minimization-dualization can be arbitrarily long: the
minimization of a Mealy automaton with a minimal dual can make the dual
automaton non-minimal.
If ${\mathcal{A}}$ is a Mealy automaton, we denote by
${\mathfrak{m}}{\mathfrak{d}}^{*}{({\mathcal{A}})}$ the corresponding Mealy
automaton after ${\mathfrak{m}}{\mathfrak{d}}$-reduction.
###### Theorem 3.1 ([2])
A Mealy automaton ${\mathcal{A}}$ generates a finite (semi)group if and only
if ${\mathfrak{m}}{\mathfrak{d}}^{*}{({\mathcal{A}})}$ generates a finite
(semi)group.
This is the starting point of the new implementations. We use an additional
fact. We can prune a Mealy automaton by deleting the states which are not
accessible from a cycle. This does not change the finiteness or infiniteness
of the generated (semi)group [3].
#### 3.3.2 The New Implementations
The design of procedure IsFinite1 is consistent with the one of
AutomatonGrowth. Hence IsFinite1 is much closer to FR than to automgrp. Here
we propose a version that works with the automaton and its dual in parallel.
⬇
IsFinite1 := function (aut, limit)
local radius, dual, curr1, next1, curr2, next2;
radius := 0;
aut := MDReduced(Prune(aut)); dual := DualMachine(aut);
curr1 := MealyMachine([[1]],[()]); curr2 := curr1;
next1 := aut; next2 := dual;
while curr2!.nrstates<>next2!.nrstates and radius<limit
do radius := radius + 1; curr1 := next1;
next1 := Minimized(next1*aut);
if curr1!.nrstates<>next1!.nrstates
then curr2 := next2;
next2 := Minimized(next2*dual);
else return true;
fi;
od;
if curr2!.nrstates = next2!.nrstates then return true; fi;
return fail;
end;
The procedure IsFinite2 is a refinement of automgrp’s LevelOfFaithfulAction:
the minimization is called on the dual and can be enhanced again to work in
parallel on the Mealy automaton and its dual.
⬇
IsFinite2 := function(aut,limit)
local f1, f2, next, cs, ns, lev;
aut := MDReduced(Prune(aut));
if IsInvertible(aut) then f1:=Group; f2:=PermList;
else f1:=Semigroup; f2:=Transformation;
fi;
lev := 0; cs := 1; ns := Size(f1(List(aut!.output,f2)));
aut := DualMachine(aut); next := aut;
while cs<ns and lev<limit
do lev := lev+1; cs := ns; next := Minimized(next*aut);
ns := Size(f1(List(DualMachine(next)!.output,f2)));
od;
if cs=ns then return true; else return fail; fi;
end;
##### Experimental Results.
Tab. 2 presents the average time to detect finiteness of (semi)groups
generated by $p$-letter $q$-state invertible or reversible Mealy automata with
$p+q\in\\{5,6\\}$. To get a fair comparison of the implementations, what is
given is the minimum of the running times for an automaton and its dual (see
Theorem 2.1).
Table 2: Average time (in ms) to detect finiteness of (semi)groups
2- 3- | 2- 4- | 3- 3-
---|---|---
FR | aut | Fin1 | Fin2 | FR | aut | Fin1 | Fin2 | FR | aut | Fin1 | Fin2
0.68 | 0.81 | 0.49 | 0.49 | 36.36 | 1.79 | 0.52 | 0.62 | 1 342.12 | 3.78 | 0.61 | 0.70
FR: FR’s IsFinite; aut: automgrp’s IsFinite; Fin1: IsFinite1; Fin2: IsFinite2
## 4 Conjectures
The efficiency of the new implementations enables to carry out extensive
experimentations. We propose several conjectures supported by these
experiments.
Recall the example given in the introduction. The (semi)groups generated by
the Mealy automaton and its dual were strikingly different, with a very large
one and a rather small one. This seems to be a general fact that we can state
as an informal conjecture:
_Whenever a Mealy automaton generates a finite (semi)group which is very large
with respect to the number of states and letters of the automaton, then its
dual generates a small one._
_Observation:_ Any pair of finite (semi)groups can be generated by a pair of
dual Mealy automata, see [2, Prop. 9]. The standard construction leads to
automata whose sizes are related to the orders of the (semi)groups. Therefore
it does not contradict the informal conjecture.
$\\#\langle{{{\mathcal{A}}}}\rangle_{+}$$\\#\langle{{{\mathfrak{d}}({{\mathcal{A}}})}}\rangle_{+}$4
000$10^{2}$$10^{4}$$10^{6}$$10^{8}$$10^{10}$$10^{12}$$10^{14}$$10^{16}$$10^{18}$$10^{20}$$10^{22}$
Figure 3: Size of $\langle{{{\mathcal{A}}}}\rangle_{+}$ vs. size of
$\langle{{{\mathfrak{d}}({{\mathcal{A}}})}}\rangle_{+}$
Fig. 3 illustrates this informal conjecture: for ${\mathcal{A}}$ covering the
set of all 3-letter 3-state invertible Mealy automata, the endpoints of each
segment represent respectively the order of
$\langle{{{\mathcal{A}}}}\rangle_{+}$ and of
$\langle{{{\mathfrak{d}}({{\mathcal{A}}})}}\rangle_{+}$, for all pairs
detected as being finite.
To assess finiteness, the procedures IsFinite1 and IsFinite2 have been used.
If the tested Mealy automaton and its dual were both found to have more than
4000 elements, the procedures were stopped, and the (semi)groups were supposed
to be infinite. Based on the informal conjecture, we believe to have captured
all finite groups. If true:
* •
There are 14 089 Mealy automata generating finite (semi)groups among the 233
339 invertible or reversible 3-letter 3-state Mealy automata;
* •
The group generated by Fig. 1-right is the largest finite group.
.4 2.2 (-4.5,-1.3)(1.5,.5) (-.62,.78)A1 [x](.22,.97)AQ (-1,0)A2 (0,0)A0
(.9,-.43)A5(.9,.43)A6 (-.62,-.78)A3 (.22,-.97)A4 A1A2$\footnotesize 2$
A2A3$\footnotesize 2$ A3A4$\footnotesize 2$ A4A5$\footnotesize 2$
A6AQ$\footnotesize 2$ [.3]A1A0$\footnotesize 1$ 135A1 75A0 AQA1 A2A0 A3A0 A4A0
A5A0 A4A0 A5A0 A6A0 dotted A5A6 (-3.3,-.2)$\rho_{x}=(1,2,\dots,p)$
(-3.3,-.6)$\forall y\neq x,\,\rho_{y}=(1,3,\dots,p)$
4.1 among invertible automata: ${\mathcal{M}}_{p,q}$
.4 2.2 (-1.5,-1.3)(5,.5) (-.62,.78)A1 [x](.22,.97)AQ (-1,0)A2 (0,0)A0
(.9,-.43)A5(.9,.43)A6 (-.62,-.78)A3 (.22,-.97)A4 A1A2 A2A3 A3A4 A5A6 A6AQ
[.3]A6A0$\footnotesize 1$ 75A0 AQA1 dotted A4A5 (3.2,-.2)$\rho_{x}=(1,2)$
(3.2,-.6)$\forall z\neq x,\,\rho_{z}=()$
4.2 among 2-letter invertible automata: ${\mathcal{M}}_{2,q}$
.4 2.2 (-5,-1.3)(1.5,1.1) [y](-0.82,.28)A1 [¯x](.22,.28)AQ AQA1 250A1
[.4]290AQ (.4,-.45)1,2 (.4,-.7)(plus $p$ if even)
(-3.05,.2)$\rho_{\bar{x}}=t(1,2,\dots,p)t^{-1}$
(-3.2,-.2)$\rho_{y}=(1,3,\dots,p)$
(-3.2,-.75)$t=\begin{cases}()&\hbox{for~{}$p$ even}\\\
(p,\frac{p+1}{2})&\hbox{for~{}$p$ odd}\end{cases}$
4.3 among 2-state invertible automata: ${\mathcal{M}}_{p,2}$
.4 2.2 (-1.5,-1.3)(5,1.1) [x](.22,.97)AQ [y](-.62,.78)A1 (-1,0)A2
(.9,-.43)A5(.9,.43)A6 (-.62,-.78)A3 (.22,-.97)A4 [.7]A1A2 [.3]A2A3 A3A4 A4A5
A6AQ AQA1 dotted A5A6 (3.2,.2)$\rho_{x}=(1,2,\dots,p)$
(3.2,-.2)$\rho_{y}=(1,3,\dots,p)$ (3.2,-.6)$\forall
z\not\in\\{x,y\\},\,\rho_{z}=(\,)$
4.4 among bireversible automata: ${\mathcal{B}}_{p,q}$
Figure 4: Automata conjectured to generate the largest finite automaton groups
Our next conjectures are concerned with the largest finite groups that can be
generated by automata of a given size.
Consider the family of $p$-letter $q$-state Mealy automata
$({\mathcal{M}}_{p,q})_{p+q>5}$ displayed on Fig. 4.1 for $p>2$ and $q>2$,
while the specializations for $p=2$ and $q=2$ are displayed on Fig. 4.2 and
Fig. 4.3. The example of Fig. 1-right is ${\mathcal{M}}_{3,3}$.
###### Conjecture 1
The group $\langle{{\mathcal{M}}_{p,q}}\rangle$ is finite. Every $p$-letter
$q$-state invertible Mealy automaton generates a group which is either
infinite or has an order smaller than
$\\#\langle{{\mathcal{M}}_{p,q}}\rangle$.
If true, Conjecture 1 implies the decidability of the finiteness problem for
automaton groups. Without entering into the details of the experimentations,
we consider that Conj. 1 is reasonably well supported for $p+q<9$. As for
actually computing $\\#\langle{{\mathcal{M}}_{p,q}}\rangle$, here are the only
cases with $q>2$ for which we succeeded:
$\displaystyle\forall q,\,4\leq q\leq
8,\qquad\\#\langle{{\mathcal{M}}_{2,q}}\rangle=2^{2^{q-1}+\frac{(q-2)(q-1)}{2}-2}\>,\qquad\qquad$
$\displaystyle\\#\langle{{\mathcal{M}}_{3,3}}\rangle=2^{64}\cdot
3^{4},\qquad\\#\langle{{\mathcal{M}}_{3,4}}\rangle=2^{325}\cdot
3^{13},\qquad\\#\langle{{\mathcal{M}}_{4,3}}\rangle=2^{288}\cdot 3^{422}\>.$
These groups are indeed huge. Incidentally, the finiteness of
$\langle{{\mathcal{M}}_{p,q}}\rangle$ is checked for $p+q<11$ and the informal
conjecture is supported further by computing the order of the much smaller
semigroups generated by the duals:
$\\#\langle{{\mathfrak{d}}({{\mathcal{M}}_{p,q}})}\rangle_{+}$ | 2 | 3 | 4 | 5 | 6 | 7 | 8
---|---|---|---|---|---|---|---
2 | - | - | 219 | 1 759 | 13 135 | 94 143 | 656 831
3 | - | 238 | 1 552 | 8 140 | 37 786 | 162 202 | $\cdots$
4 | 89 | 1 381 | 12 309 | 87 125 | 543 061 | $\cdots$ | $\cdots$
5 | 131 | 6 056 | 67 906 | 602 656 | $\cdots$ | $\cdots$ | $\cdots$
6 | 337 | 22 399 | 302 011 | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$
7 | 351 | 74 194 | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$
Experimentally, the finite groups generated by bireversible Mealy automata
seem to be much smaller. Consider the family of bireversible automata
$({\mathcal{B}}_{p,q})_{p,q}$ of Fig. 4.4. The group
$\langle{{\mathcal{B}}_{p,q}}\rangle$ is isomorphic to
${\mathfrak{S}}_{p}^{q}$, while the group
$\langle{{\mathfrak{d}}({{\mathcal{B}}_{p,q}})}\rangle$ is isomorphic to
$\mathbb{Z}_{q}$. Again, the following is reasonably well supported for
$p+q<9$:
###### Conjecture 2
Every $p$-letter $q$-state bireversible Mealy automaton generates a group
which is either infinite or has an order smaller than
$\\#\langle{{\mathcal{B}}_{p,q}}\rangle=p!^{q}$.
Our last conjecture is of a different nature and deals with the structure of
infinite automaton semigroups.
###### Conjecture 3
Every $2$-state reversible Mealy automaton generates a semigroup which is
either finite or free of rank 2.
The conjecture has been tested and seems correct for reversible 2-state Mealy
automata up to 6 letters. In the experiments, a semigroup generated by a
$p$-letter automaton is conjectured to be free if its growth series coincides
with $(2t)^{n}$ up to radius $p^{2}/2$ and if its dual generates a seemingly
infinite group.
## References
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* [6] I. V. Bondarenko, N. V. Bondarenko, S. N. Sidki, and F. R. Zapata. On the conjugacy problem for finite-state automorphisms of regular rooted trees. arXiv:math.GR/1011.2227.
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|
arxiv-papers
| 2013-10-17T21:01:37 |
2024-09-04T02:49:52.548335
|
{
"license": "Public Domain",
"authors": "Ines Klimann and Jean Mairesse and Matthieu Picantin",
"submitter": "Matthieu Picantin",
"url": "https://arxiv.org/abs/1310.4856"
}
|
1310.4883
|
# Proposed Method for Distinguishing Majorana Peak from Other Peaks: Tunneling
Spectroscopy with Ohmic Dissipation using Resistive Electrodes
Dong E. Liu Department of Physics and Astronomy, Michigan State University,
East Lansing, Michigan 48824, USA
###### Abstract
We propose a scheme to distinguish zero-energy peaks due to Majorana from
those due to other effects at finite temperature by simply replacing the
normal metallic lead with a resistive lead (large $R\sim k\Omega$) in the
tunneling spectroscopy. The dissipation effects due to the large resistance
change the tunneling conductance significantly in different ways. The Majorana
peak remains increase as temperature decreases $G\sim T^{2r-1}$ for
$r=e^{2}R/h<1/2$. The zero-energy peak due to other effects splits into two
peaks at finite temperature and the conductance at zero voltage bias varies
with temperature by a power law. The dissipative tunneling with a Majorana
mode belongs to a same universal class as the unstable critical point of the
case with a non-Majorana mode.
###### pacs:
72.10.Fk, 74.78.Na, 74.78.Fk, 03.67.Lx
Introduction — Majorana fermions (MFs), proposed to exist in solid state
systems Fu and Kane (2008); Sau et al. (2010); Alicea (2010); Lutchyn et al.
(2010); Oreg et al. (2010), cold atomic systems Sato et al. (2009); Zhu et al.
(2011); Jiang et al. (2011a), and periodic driving systems Jiang et al.
(2011a); Reynoso and Frustaglia (2013); Liu et al. (2013a), attract a great
deal of attention. A variety of signatures Das Sarma et al. (2006); Fu and
Kane (2009a, b); Akhmerov et al. (2009); Law et al. (2009); Akhmerov et al.
(2011); M. et al. (2011); Liu and Baranger (2011); Jiang et al. (2011b);
Fidkowski et al. (2012); San-Jose et al. (2012) are predicted to detect
Majorana fermion (MF) zero mode; among them, tunneling spectroscopy may
provide one of the simplest and direct tests for MF— The observation of the
zero-bias peak (ZBP) with quantized conductance $G=2e^{2}/h$ Law et al.
(2009); Akhmerov et al. (2011) at sufficiently low temperature (smaller than
intrinsic width of the Majorana peak). Recently, several groups Mourik et al.
(2012); Deng et al. (2012); Das et al. (2012) reported the observation of a
non-quantized ZBP at higher temperature in semiconductor nanowires, which is
possibly coming from MF. However, the ZBP may originate from other effects,
e.g. zero-energy impurity bound state. In addition, recent works Bagrets and
Altland (2012); Liu et al. (2012); Neven et al. (2013) show that, in a
superconducting system with both spin-rotation and time-reversal symmetry
breaking, the disorder can induce a cluster of mid-gap states around zero-
energy and thus a ZBP at finite temperature. Especially, the disorder ZBP
appears in the conditions highly similar to Majorana ZBP Bagrets and Altland
(2012); Liu et al. (2012); Neven et al. (2013). These alternative
possibilities lead to debates about the validity of the tunneling spectroscopy
methods.
In this work, we introduce a scheme by simply replacing the normal metal lead
in the tunneling spectroscopy with a resistive lead (with large resistance
$R\sim k\Omega$). In this case, electrons couple to an ohmic environmental
bath Feynman and Vernon (1963) in the tunneling process; the coupling to the
bath usually suppresses the tunneling rate and leads to dissipative tunneling
Leggett et al. (1987); Ingold and Yu.V. (1992). Dissipation effects can also
cause non-trivial phase diagrams and transitions, which was recently observed
in a simple resonant level system Mebrahtu et al. (2012, 2013); Liu et al.
(2013b). We investigate how the dissipation influences the tunneling into MFs,
zero-energy impurity bound states in superconductor, and other states causing
ZBP at finite temperature. The ways that the dissipation effects renormalize
the tunneling strength and the tunneling conductance is significantly
different for MFs and other cases. If the lead is connected to a MF, the zero-
bias conductance scales as $G\sim T^{2r-1}$ near a weak tunneling fixed point
(high $T$) and will go to perfect transmission $G=2e^{2}/h$ at $T=0$ for
$r=e^{2}R/h<1/2$. If the lead is connected to a superconductor (SC) with a
zero-energy impurity bound state (non-MF), the system can be divided into four
stable phases and an unstable symmetric point (i.e. critical point). Away from
the symmetric point, the system will flow to one of the four stable fixed
points, near which the zero-bias conductance scales as $G\sim T^{2r}$ and the
peak splits into two at finite temperature. The critical point belongs to the
same universal class as the case for dissipative tunneling into a Majorana
mode. We also consider the conductance for the dissipative tunneling into a
cluster of mid-gap states. Without dissipation, the finite temperature
conductance shows ZBP; with dissipation, the single peak splits into two as
temperature decreases. The splitting occurs at higher temperature for larger
resistance, but $r<1/2$ is required in the experiment so that Majorana ZBP
does not split. Therefore, the dissipation effect induced by the resistive
lead provides a way to distinguish Majorana ZBP and other ZBP, and serves as a
“ Majorana signature filter”.
Figure 1: (color online) Proposed experimental setup.
Model — We consider the tunneling spectroscopy from a resistive lead into the
end of a superconducting nanowire (SCNW) with Rashba spin-orbit coupling and
proximity induced superconductivity $\Delta$ as shown in Fig. 1. A magnetic
field is applied perpendicular to the direction of the Rashba spin-orbit
coupling. In this case, MFs are predicted to exist at the two ends of the wire
if $V_{z}>\sqrt{\Delta^{2}+\mu^{2}}$, where $V_{z}$ is Zeeman splitting and
$\mu$ is wire chemical potential Lutchyn et al. (2010); Oreg et al. (2010).
Unlike conventional setup, we replace the normal metallic lead with a
resistive lead. A gate is applied to control the tunneling barrier between the
lead and SCNW. We assume that the barrier is high and wide, so that the
tunneling has only a single channel, and the cooper pair tunneling can be
assisted only by the mid-gap states localized near the end of the wire. Note
that our setup is not limited only to SC wire, but also any other MF setups
with a resistive lead.
The Hamiltonian of the system can be written as
$H=\sum_{k}(\epsilon_{k}+\mu_{1})c^{\dagger}_{k}c_{k}+H_{\rm SCNW}+H_{\rm
T}+H_{\rm ENV},$ (1)
where the first term describes the lead, with the electron creation
(annihilation) operator $c^{\dagger}_{k}$ ($c_{k}$) . The second term
represents the states near the end of the nanowire:
$\displaystyle H_{\rm SCNW}$ $\displaystyle=$
$\displaystyle\sum_{\nu}(\varepsilon_{\nu}+\mu_{2})b^{\dagger}_{\nu}b_{\nu}+\text{SC
Pairing}+\text{Disorder}$ (2) $\displaystyle=$ $\displaystyle\mu_{2}N_{\rm
SCNW}+\sum_{q}\xi_{q}\gamma^{\dagger}_{q}\gamma_{q},$
where $b^{\dagger}$ ($b$) is the creation (annihilation) operator for
electrons. Including the cooper pairing terms and disorders, one can
diagonalize the Hamiltonian and reach the bogoliubov quasi-particle states
$\gamma_{q}$, which includes the MF and the disorder induced mid-gap states.
$\mu_{1}$ and $\mu_{2}$ are chemical potentials for the lead and
superconductor, respectively. The voltage bias is $V=\mu_{1}-\mu_{2}$. The
tunneling Hamiltonian in the presence of dissipation Ingold and Yu.V. (1992)
is
$H_{\rm
T}=\sum_{k,\nu}\Big{(}y_{k,\nu}c^{\dagger}_{k}b_{\nu}e^{-i\phi}+y_{k,\nu}^{*}b^{\dagger}_{\nu}c_{k}e^{i\phi}\Big{)},$
(3)
where $y_{k,\nu}$ is the tunneling strength between lead and SCNW. The
operator $\phi=(e/h)\int_{-\infty}^{t}dt^{\prime}U(t^{\prime})$ represents the
phase fluctuation across the tunneling junction, where $U(t)$ is the voltage
fluctuation across the junction. Define $Q$ as the charge fluctuation of the
junction capacitance such that $[\phi,Q]=i\,e$. The operator $e^{-i\phi}$
removes one electron from the junction capacitance, and thus represents the
single electron tunneling. Following Caldeira and Leggett Caldeira and Leggett
(1981), one can represent the dissipative environment by a set of harmonic
oscillators (i.e. $\\{q_{n},\phi_{n}\\}$ with oscillator frequency
$\omega_{n}=1/\sqrt{L_{n}C_{n}}$) bilinearly coupled to the phase $\phi$. The
last term of Eq. (1) is then Caldeira and Leggett (1981); Leggett et al.
(1987); Ingold and Yu.V. (1992)
$H_{\rm
ENV}=\frac{Q^{2}}{2C}+\sum_{n=1}^{N}\Big{[}\frac{q_{n}^{2}}{2C_{n}}+\big{(}\frac{\hbar}{e}\big{)}^{2}\frac{1}{2L_{n}}(\phi-\phi_{n})^{2}\Big{]},$
(4)
where $C$ is the capacitance of the junction. $H_{\rm ENV}$ describes the
coupling between the system and the environment.
Tunneling into Majorana Fermion — Consider the tunneling between the lead and
a MF zero-energy state, one arrives at the following Hamiltonian
$H_{\rm
T}=\sum_{k}\Big{(}y_{k}c^{\dagger}_{k}\gamma_{1}e^{-i\phi}+y_{k}^{*}\gamma_{1}c_{k}e^{i\phi}\Big{)},$
(5)
where $\gamma_{1}=\gamma_{1}^{\dagger}$ is the MF operator. Note that, even
for a spinful lead, MF couples to only a single channel, which is the linear
combination of the spin up and down channels Law et al. (2009). It is helpful
to introduce a Dirac fermion $f$: $\gamma_{1}=(f+f^{\dagger})/\sqrt{2}$. The
tunneling Hamiltonian becomes
$\displaystyle H_{\rm T}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}\sum_{k}\Big{(}y_{k}c^{\dagger}_{k}fe^{-i\phi}+y_{k}^{*}f^{\dagger}c_{k}e^{i\phi}\Big{)}$
(6)
$\displaystyle+\frac{1}{\sqrt{2}}\sum_{k}\Big{(}y_{k}c^{\dagger}_{k}f^{\dagger}e^{-i\phi}+y_{k}^{*}fc_{k}e^{i\phi}\Big{)}.$
Now, a scaling analysis is in order to see how the tunneling strength $y$
scales in the renormalization group (RG) picture. Because MF couples to the
lead at a single point, the metallic lead can be reduced to a semi-infinite
one dimensional free fermion bath Hewson (1997). Therefore, the scaling
dimension of this fermion operator is $[c]=1/2$. The localized MF operator or
operator $f$ does not contribute to the scaling dimension. To study the phase
part $e^{-i\phi}$, we consider an ideal ohmic dissipative environment with the
lead resistance $R$. If we are interested in the scaling dimension, one only
need the $T=0$ correlation function in the long time limit $\langle
e^{i\phi(t)}e^{-i\phi(0)}\rangle\sim t^{-2r}$ Ingold and Yu.V. (1992), where
$r=R/R_{K}$ with quantum resistance $R_{K}=h/e^{2}$. We choose $\hbar=k_{B}=1$
throughout the paper. Therefore, the scaling dimension of the dissipative part
is $[e^{-i\phi}]=r$, and the RG equation for the tunneling strength yields
$\frac{dy}{d\ln l}=\big{(}1-\frac{1}{2}-r\big{)}y,$ (7)
where $l$ is a time cutoff. For very large resistance $r>1/2$, the tunneling
is an irrelevant perturbation and will flow to zero at zero energy. However,
for $r<1/2$, the tunneling is relevant and will increase with reducing energy.
Near a weak tunneling fixed point (large $V$ or $T$) , the conductance scales
as $G\sim V^{-2(1-1/2-r)}=V^{2r-1}$ at $T=0$, and as $G\sim T^{2r-1}$ at
$V=0$. As energy (i.e. $\rm{max}[V,T]$) approaches zero, the system will enter
into a perfect transmission case with quantum conductance $G=2e^{2}/h$ Law et
al. (2009).
Tunneling into Zero-Energy impurity Bound States (ZEIBS) — We assume a (non-
MF) ZEIBS localized near the end of the wire as shown in Fig. 2 (a). Suppose
the ZEIBS and SC states consist of both spin up and down components, both spin
channels in the lead couple to them. These tunneling processes can be
categorized as two mechanisms shown in Fig. 2: 1) direct tunneling between the
lead and ZEIBS, 2) tunneling into SC assisted by ZEIBS-SC tunneling with a
cooper pair. The corresponding Hamiltonian is
$H_{T}=\sum_{\sigma}y_{\rm d,\sigma}\Psi_{\rm
L,\sigma}^{\dagger}(0)\,d\,e^{-i\phi}+y_{\rm c,\sigma}\Psi_{\rm
L,\sigma}^{\dagger}(0)d^{\dagger}e^{-i\phi}e^{-i\chi}+h.c.,$ (8)
where $y_{d,\sigma}$ and $y_{c,\sigma}$ are the tunneling strength for the
lead-ZEIBS and lead-SC continuum ($\sigma$ represents the spin), $\Psi_{\rm
L,\sigma}(0)=\sum_{k}\psi_{k,\sigma}(0)c_{k,\sigma}$ is the electron
annihilation operator of the lead at the point ($x=0$) coupled to SCNW, where
$\psi_{k}$ is the wavefunction amplitude for state $k$. $\chi$ is the
superconducting phase, and $e^{\pm i\chi}$ creates or annihilates a cooper
pair. We assume the SCNW is large enough to neglect the Coulomb charging
energy, and the superconducting phase does not couple to any dissipative
environment. Under these assumptions, we can neglect the superconducting phase
$\chi$, and then, the tunneling Hamiltonian is equivalent to the case with MF
shown in Eq. (6) if and only if $y_{d,\sigma}=y_{c,\sigma}$.
Figure 2: (color online) (a) Demonstration of tunneling into a Zero-energy
impurity bound states (non-Majorana). $d$ and $CP$ represent ZEIBS and cooper
pair, respectively. (b) Schematic representation of the flow diagram based on
Eq. (11). The arrows indicate the direction of the flow as energy decreases.
The red dot in the center is the symmetric fixed point
($y_{d\uparrow}=y_{c\uparrow}$ and $y_{d\downarrow}=y_{c\downarrow}$), which
is unstable. The edges of the parallelogram correspond to four stable fixed
points. 1) ($y_{d\uparrow}$ perfect transmission,
$y_{c\uparrow}=y_{d\downarrow}=y_{c\downarrow}=0$) at right edge. Note that
$y_{d\downarrow}$ and $y_{c\downarrow}$ have the same power law decay rate,
and therefore $\ln(y_{d\downarrow}/y_{c\downarrow})=\rm{constant}$ near the
Fixed point. Other three fixed points are: 2) ($y_{c\uparrow}$ perfect
transmission, $y_{d\uparrow}=y_{d\downarrow}=y_{c\downarrow}=0$); 3)
($y_{d\downarrow}$ perfect transmission,
$y_{c\uparrow}=y_{d\uparrow}=y_{c\downarrow}=0$); 4) ($y_{c\downarrow}$
perfect transmission, $y_{c\uparrow}=y_{d\uparrow}=y_{d\downarrow}=0$).
Since the tunneling has only a single channel, the lead can be reduced to a
semi-infinite free fermion field, which then can be unfolded to form a chiral
free fermionic field Affleck (1995); we take the coupling to the SCNW to be
$x=0$. Then, this field can be bosonized in a standard way Senechal (2003);
Giamarchi (2004): $\Psi_{\rm
L\sigma}(x)=F_{\sigma}\,e^{i\Phi_{\sigma}(x)}/\sqrt{2\pi}$, where
$\Phi_{\sigma}(x)$ is a chiral bosonic field with
$[\Phi_{\sigma}(x),\Phi_{\sigma^{\prime}}(x^{\prime})]=i\delta_{\sigma\sigma^{\prime}}\pi\,\rm{sgn}(x-x^{\prime})$,
$F_{\sigma}$ is Klein factor. For a spinful lead, the Hamiltonian becomes
$\displaystyle H$ $\displaystyle=$
$\displaystyle\sum_{\sigma}\frac{v_{F}}{4\pi}\int_{-\infty}^{\infty}dx\big{(}\partial_{x}\Phi_{\sigma}(x)\big{)}^{2}$
(9) $\displaystyle+\Big{[}y_{\rm
d,\sigma}\frac{F_{\sigma}e^{-i\Phi_{\sigma}(0)}}{\sqrt{2\pi}}d\,e^{-i\phi}+y_{\rm
c,\sigma}\frac{F_{\sigma}e^{-i\Phi_{\sigma}(0)}}{\sqrt{2\pi}}d^{\dagger}e^{-i\phi}$
$\displaystyle+h.c.\Big{]}+K_{\sigma}(d^{\dagger}d-1/2)\partial_{x}\Phi_{\sigma}(x=0)/\pi.$
The last term represents the density interaction between the lead (i.e.
$\Psi_{\rm L\sigma}^{\dagger}(x)\Psi_{\rm
L\sigma}(x)=-\partial_{x}\Phi_{\sigma}(x)/\pi$) and the localized ZEIBS, and
this interaction is initially very small and can be enhanced in the RG
processes. Since the correlation function of the phase $\phi$ shows the
similar power law decay to the chiral bosonic field : $\langle
e^{-i\phi(t)}e^{i\phi(0)}\rangle\sim t^{-2r}$ and $\langle
e^{-i\Phi_{\sigma}(x=0,t)}e^{i\Phi_{\sigma}(x=0,0)}\rangle\sim t^{-1}$, we can
combine the two bosonic field and introduce a new field Florens et al. (2007);
Le Hur and Li (2005); Mebrahtu et al. (2012):
$\widetilde{\Phi}_{\sigma}(x)=\sqrt{g}(\Phi_{\sigma}(x)+\phi(x))$ with $\quad
g=1/(1+2r)$, which satisfies $\langle
e^{-i\widetilde{\Phi}_{\sigma}(x=0,t)}e^{i\widetilde{\Phi}_{\sigma}(x=0,0)}\rangle\sim
t^{-1}$. Note that only $\phi(x=0)=\phi$ has the physical meaning (i.e. phase
fluctuation), and $\phi(x\neq 0)$ are auxiliary fields. Overall, we have
$[\phi(x),\phi(x^{\prime})]=2ir\pi\,\rm{sgn}(x-x^{\prime})$. Since the
tunneling involves only the phase $\phi(x=0)$, the conductance will not be
affected by the auxiliary fields. Then, the Hamiltonian becomes
$\displaystyle H$
$\displaystyle=\sum_{\sigma}\frac{v_{F}}{4\pi}\int_{-\infty}^{\infty}dx\big{(}\partial_{x}\widetilde{\Phi}_{\sigma}(x)\big{)}^{2}$
(10) $\displaystyle+\Big{[}y_{\rm
d,\sigma}\frac{F_{\sigma}}{\sqrt{2\pi}}e^{-i\widetilde{\Phi}_{\sigma}(0)/\sqrt{g}}\,d+y_{\rm
c,\sigma}\frac{F_{\sigma}}{\sqrt{2\pi}}e^{-i\widetilde{\Phi}_{\sigma}(0)/\sqrt{g}}d^{\dagger}$
$\displaystyle+h.c.\Big{]}+\frac{K_{\sigma}}{\sqrt{g}\pi}(d^{\dagger}d-1/2)\partial_{x}\widetilde{\Phi}_{\sigma}(0).$
One can define a set of dimensionless parameters:
$\widetilde{y}_{d,\sigma}=y_{d,\sigma}l/\sqrt{2\pi}$,
$\widetilde{y}_{c,\sigma}=y_{c,\sigma}l/\sqrt{2\pi}$, and
$\widetilde{K}_{\sigma}=2K_{\sigma}/(\pi v_{F})$, where $l$ is a short time
cutoff in the scaling process. Following the dimension analysis and operator
product expansion Cardy (1996); Senechal (2003); sup , one can simply obtain
the RG equations in the weak tunneling limit
$\displaystyle\frac{dy_{d,\sigma}}{d\ln l}$ $\displaystyle=$
$\displaystyle\Big{(}1-\frac{(1-\widetilde{K}_{\sigma})^{2}}{2g}-\frac{(\widetilde{K}_{-\sigma})^{2}}{2g}\Big{)}y_{d,\sigma},$
$\displaystyle\frac{dy_{c,\sigma}}{d\ln l}$ $\displaystyle=$
$\displaystyle\Big{(}1-\frac{(1+\widetilde{K}_{\sigma})^{2}}{2g}-\frac{(\widetilde{K}_{-\sigma})^{2}}{2g}\Big{)}y_{c,\sigma},$
$\displaystyle\frac{d\widetilde{K}_{\sigma}}{d\ln l}$ $\displaystyle=$
$\displaystyle
2(1-\widetilde{K}_{\sigma})\widetilde{y}_{d,\sigma}^{2}-2(1+\widetilde{K}_{\sigma})\widetilde{y}_{c,\sigma}^{2}$
(11)
$\displaystyle-2\widetilde{K}_{\sigma}\widetilde{y}_{d,-\sigma}^{2}-2\widetilde{K}_{\sigma}\widetilde{y}_{c,-\sigma}^{2}.$
Five fixed points are obtained and shown in Fig. 2 (b). The first one
corresponds to $\widetilde{K}_{\uparrow}=0$, $\widetilde{K}_{\downarrow}=-1$,
$y_{d,\uparrow}=y_{d,\downarrow}=y_{c,\uparrow}=0$. In this case,
$y_{c,\downarrow}$ will flow to perfect transmission, $dy_{d,\uparrow}/d\ln
l=-2ry_{d,\uparrow}$, $dy_{d,\downarrow}/d\ln l=(-1-4r)y_{d,\downarrow}$, and
$dy_{c,\uparrow}/d\ln l=-2ry_{c,\uparrow}$. The leading tunneling process
corresponds to $y_{d,\uparrow}\cdot y_{c,\downarrow}$, i.e. a spin-up electron
entering the ZEIBS from the lead, then hopping out to form a cooper pair with
another spin-down electron from the lead. Therefore, the zero-voltage
conductance shows a power law decay $G\sim T^{2r}$ near $T=0$. The finite
voltage bias will cut off the scaling, and thus the ZBP will split at low $T$.
Conductance shows the same power law decay for three other similar fixed
points. Unless the initial condition $y_{d,\sigma}=y_{c,\sigma}$ is satisfied,
the system will flow to one of these four fixed points.
If the bare parameters reach a symmetric point: $K_{\sigma}=0$ and
$y_{d,\sigma}=y_{c,\sigma}$, all the tunneling strength $y_{d(c),\sigma}$ is
relevant and will flow to perfect transmission (i.e. perfect Andreev
reflection); this condition leads to an unstable critical point which belongs
to the same universal class as the case of tunneling into a MF. By noting the
similarity between our model (i.e. Eq. 10) and the case with a Luttinger
liquid lead sup , one can obtain the $V=0$ conductance for this symmetric
point (or for MF) in the strong coupling limit (low $T$) sup ; Fidkowski et
al. (2012): $2e^{2}/h-G\sim T^{(2-4r)/(1+2r)}$. For ZEIBS, the condition
$y_{d,\sigma}=y_{c,\sigma}$ requires fine tuning both the tunneling barrier
and spin components, and thus its realization is extremely difficult.
Tunneling into a cluster of mid-gap states — If both the spin rotation and
time reversal symmetries are broken in SCNW, disorder can induce a cluster of
mid-gap states around zero energy localized near the end of the wire Bagrets
and Altland (2012); Liu et al. (2012); Neven et al. (2013). Therefore, even
without a zero energy state (either MF or ZEIBS), the tunneling conductance
shows a zero-energy peak at finite $T$ without dissipation effect. To study
the dissipation effects for those cases, we consider the tunneling Hamiltonian
in Eq. (3), and treat the tunneling strength $y$ as a small parameter such
that the perturbation theory can be applied. This assumption is valid for
tunneling into any non-MF state (with a small bare tunneling strength) except
at the highly symmetric situation shown in the previous section.
Figure 3: (color online) Differential conductance $dI/dV$ (tunneling into a
cluster of mid-gap states around zero energy) as a function of applied voltage
$V$. (a) An arbitrary choice of the DOS for a cluster of states, which is also
the $T=0$ conductance for $r=0$. (b) The $r=0$ finite temperature conductance.
The conductance with dissipation effect, i.e. $r=0.2$ (c) and $r=0.4$ (d), for
different temperatures. The single peak splits into two as $T$ decreases.
The current operator for the junction is
$\hat{I}=i[H_{T},\sum_{k\sigma}c^{\dagger}_{k\sigma}c_{k\sigma}]=-i\sum_{k\sigma,\nu}(y_{k\sigma,\nu}c^{\dagger}_{k\sigma}b_{\nu}e^{-i\phi}-h.c.)$
Then, the current through the junction up to the leading order in tunneling
strength is given by Kubo formula (this can also be obtained by golden rule
Ingold and Yu.V. (1992))
$\displaystyle I(t)$ $\displaystyle=$
$\displaystyle-i\int_{-\infty}^{\infty}dt^{\prime}\,\theta(t-t^{\prime})\;\langle[\hat{I}(t),H_{T}(t^{\prime})]\rangle_{0}$
(12) $\displaystyle=$
$\displaystyle\int_{-\infty}^{\infty}\frac{d\omega_{1}}{2\pi}\int_{-\infty}^{\infty}\frac{d\omega_{2}}{2\pi}\sum_{k\sigma,\nu}|y_{k\sigma,\nu}|^{2}A_{k\sigma}^{L}(\omega_{1})A_{\nu}^{SCNW}(\omega_{2})$
$\displaystyle\times\\{[1-f(\omega_{1}-eV)]f(\omega_{2})P(\omega_{2}-\omega_{1})$
$\displaystyle-f(\omega_{1}-eV)[1-f(\omega_{2})]P(\omega_{1}-\omega_{2})\\}.$
with
$P(\omega)=\frac{1}{2\pi}\int_{-\infty}^{\infty}dt\exp[i\omega t+J(t)]$ (13)
where $J(t)=\langle\phi(t)\phi(0)\rangle-\langle\phi^{2}\rangle$ (see Ingold
and Yu.V. (1992); sup for more details) and $\langle\cdots\rangle_{0}$
indicates the average without the tunneling term. $P(\omega)$ describes the
energy emission and absorption in the electron tunneling processes due to
dissipation effects. $A_{k\sigma}^{L}(\omega_{1})$ is the spectral function of
the lead, and we assume a constant density of state (DOS):
$\sum_{k\sigma}|y_{k\sigma,\nu}|^{2}A_{k\sigma}^{L}(\omega_{1})=1/(eR_{T})$,
where $R_{T}$ can be viewed as the tunneling resistance. $f$ is the Fermi-
distribution function. Without dissipation, i.e. $r=0$, at zero temperature
one obtain $dI/dV\propto\sum_{\nu}A_{\nu}^{SCNW}(\omega_{2})$ which gives the
DOS of the wire. A realization of the DOS (i.e. $T=0$ conductance for $r=0$),
is shown in Fig. 3 (a). For finite temperature, this cluster of states results
in a ZBP as shown in Fig. 3 (b). As temperature decreases (still larger than
the level spacing of the mid-gap states), the ZBP height increases for $r=0$,
which is similar to Majorana ZBP. This feature changes dramatically when the
dissipation effect is included. As shown in Fig. 3 (c) $r=0.2$ and (d) $r=0.4$
($R\sim k\Omega$), the single conductance peak splits into two peaks and zero
bias conductance decreases as temperature goes down; and this feature is
contrary to that of Majorana ZBP : The zero bias conductance for $r<1/2$
increases as $T$ goes down and finally approaches $2e^{2}/h$ at $T=0$. Fig. 3
(c) and (d) also show that the peak splitting occurs at higher $T$ for larger
$r$.
Discussion — Tunneling into a MF is equivalent to the resonant tunneling
between an electron lead and a hole lead Law et al. (2009) (also see Eq. (6))
with exactly the symmetric tunneling barriers due to the topological
properities of MF. With ohmic dissipation, the resonant tunneling shows non-
trivial phase diagrams Mebrahtu et al. (2012); Liu et al. (2013b): 1) any
asymmetry in the barriers induces a relevant backscattering which destroys the
resonant tunneling; 2) this backscattering vanishes for a special symmetric
point, and the next leading term is irrelevant for small $r$ ($r<1/2$ for our
case). This symmetry, which results in dissipative resonant tunneling, is
topologically protected by MF; it is not protected for other cases, and
requires fine tuning. In the experiments Mourik et al. (2012); Deng et al.
(2012); Das et al. (2012), the metal lead can be made rather resistive ($R\sim
k\Omega$, but need $r<1/2$), by using e.g. $\rm{Cr/Au}$ film Mebrahtu et al.
(2012, 2013). When coupling to a MF zero mode, the height of ZBP increases as
$T$ goes down: $2e^{2}/h-G\sim T^{(2-4r)/(1+2r)}$ near $T=0$, and $G\sim
T^{2r-1}$ for high $T$. When coupling to a non-MF mode causing a ZBP, however,
its height shows a power law suppression at low $T$: $G\sim T^{2r}$.
D.E.L. is grateful to H.U.Baranger and A. Levchenko for valuable discussions
and suggestions. The author acknowledges support from US DOE, Division of
Materials Sciences and Engineering, under Grant No. DE-SC0005237, Michigan
state university, and ARO through contract W911NF-12-1-023.
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* (42) See supplementary materials for further details.
|
arxiv-papers
| 2013-10-18T02:02:34 |
2024-09-04T02:49:52.556486
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Dong E. Liu",
"submitter": "Dong Liu",
"url": "https://arxiv.org/abs/1310.4883"
}
|
1310.5096
|
# Opinion Dynamic with agents immigration
Zhong-Lin Han Department of Physics, University of Science and Technology of
China, Hefei 230026, China Yu-Jian Li Department of Modern Physics,
University of Science and Technology of China, Hefei 230026, China Bing-Hong
Wang Department of Modern Physics, University of Science and Technology of
China, Hefei 230026, China
###### Abstract
Abstract
###### pacs:
89.65.-s, 02.50.Le, 07.05.Tp, 87.23.Kg
## I Introduction
Recent years, a large class of interdisciplinary problems has been
successfully studied with statistical physics methods. Statistical physics
establishes the bridge from microscopic characteristics to macroscopic
behaviors, for systems containing a large number of interacting components.
Using both analytical and numerical tools, it has contributed greatly to our
understanding of various complex systems. In this paper, we are motivated by
the statistical physics of a sociological problem, namely, opinion dynamics.
As one of the classical and traditional research areas in both social science
and theory physics, opinion dynamics has attracted much attention. A lot of
models concerning the process of opinion formation, such as voter model,
bounded confidence model, have been proposed previously. Meanwhile, some of
recent studies discussed and described the opinion dynamics on both common
conditions and various complex networks.
The issue of individual mobility has become increasingly fundamental due to
the Human migration and human dynamic. The issue is also important in other
contexts such as the emergence of Cooperation among individuals [20] and
species coexistence in cyclic competing games [21]. Recently, some empirical
data of human movements have been collected and analyzed [22,23]. From the
standpoint for dynamic of complex systems, when individuals (nodes, agents)
are mobile, the edges in the topological structures are no longer fixed,
yielding more different results on that than before.
In our paper, we try to propose a new model combining conventional opinion
dynamics with agents immigration according to information transmission and
evolution. In our simulation, we finally find a series of results reflecting
special and different features of opinion dynamics with immigration. By
introducing a parameter $\alpha$ to control the weight of influence of
individual opinions, according to a recent study considering weight influence,
we find that there also exist an optimal value of $\alpha$ leading to the
shortest consensus time for all individuals on a isotropic plane we concern.
After presenting the results of simulation in different situations, we also
analysis the results of our model in mathematical way, which leads us finding
out what are the exactly direct factors impacting the exponent of weight of
individual opinions.
Figure 1: (Color online) For $N=2000$ and $\langle k\rangle=4$, the density
$\rho_{c}$ as a function of $\alpha$ for different values of $r$ in the case
where all cooperators contribute the same cost c per game. Every cooperator
contributes a cost $c=1$ in every neighborhood that it plays.
In this paper we found up a new type of model for information dynamics with
immigration and we state related parameters and rules of our model. In order
to demonstrate the rationality of it, we presented both computational
simulation and mathematical analysis. Another most significant thing is that
we have designed a new mathematical method for model with linear
algebra.Compared with the previous method,we finally finish a complete model
for opinion and information dynamics with immigration.
## II Model
As previous classical model focusing on the material process of spread and
formation of opinions, we spend more efforts on finding special results when
individuals carry opinions with immigration. To focus on a more efficient
situation, we just confine our discussion on an isotropic plane, without
special network effects. On the other hand, the individuals we concern are
just holding two kinds of opinion, the positive opinion $\psi_{+}>0$ and the
negative opinion $\psi_{-}<0$. According to one model on opinion dynamics
proposed before (.), we introduce the weight exponent to control the weight of
influence of each individual. We describe that all of the opinions of
individuals evolve simultaneously completely rely on its neighbors’ opinion
and neighbors’ weight. Here we describe the evolution process of the whole
individuals on the plane in mathematical way, which could be denoted as follow
(yang han xing)
$\displaystyle
p_{+}=\frac{\sum_{i}^{u}\omega_{i}^{\alpha}}{\sum_{i}^{u}\omega_{i}^{\alpha}+\sum_{j}^{v}\omega_{j}^{\alpha}},$
(1) $\displaystyle
p_{-}=\frac{\sum_{i}^{v}\omega_{i}^{\alpha}}{\sum_{i}^{u}\omega_{i}^{\alpha}+\sum_{j}^{v}\omega_{j}^{\alpha}}$
(2)
where $p_{+}$ and $p_{-}$ denote the probability of choosing positive opinion
and negative opinion,and the number of neighbors holding positive opinion is
$u$ while the number of negative ones is $v$. Here the model considers the
agents with weight impact $\omega_{i}$, which is controlled by weight exponent
$\alpha$. If the probability $p_{+}$ is lager than $p_{-}$, the agent we
concern will choose positive opinion at the next step. And it will be same as
choosing negative opinion.
Figure 2: (Color online) Cumulative payoff distribution for different values
of $\alpha$. The distribution is obtained after the cooperation density
becomes stable. The multiplication factor is set to be $r=1.6$. Solid curves
are theoretical predictions from Eq. (LABEL:eq:wealthdis).
In this model, the individual we concern evolves its opinion at $t+1$
according to its neighbors’ opinion in its view radius $r$, which is shown in
Figure.1. In Figure.1, the red agents hold positive opinions and black agents
hold negative opinions. There are $u+v$ neighbors in the view range of
individual we concern, while here are $u$ individuals hold positive opinion
and $v$ individuals hold negative opinion at $t$ step. After comparing the
weight of positive opinion and negative opinion, the individual we concern
evolves its opinion at $t+1$ step as this
$\psi_{i}^{(t+1)}=\sum_{r}\psi_{j}^{(t)}$ (3)
where $\psi_{j}^{(t)}$ denotes the opinion state of the $j$th neighbor of the
$i$th agent we concern at the $t$ step.
Figure 3: (Color online) Times series of cooperator density in hubs’
neighborhoods for (a) The multiplication factor is $r=1.2$ and each data point
is obtained by averaging over 50 runs.
After changing their opinion in the way above, all the individuals immigrate
on the plane. All the agents would be confined in the plane by periodic
boundary condition. The velocity and direction angle of each agent are
randomly distributed, which are kept by each agent all the time. After enough
period of time, number of the individuals holding positive opinions $N_{+}$
and the number of other individuals holding negative ones $N_{-}$ reach a
plateaus and dynamic equilibrium. At that certain point, we believe that the
process of opinion dynamics would be terminated. And we could find that if all
of the individuals enter the plateaus, the total number of individuals who
hold positive opinion at $t$ step $\psi_{+}^{t}$ would be approximately equal
to the number of individuals who also hold that at $t+1$ step
$\psi_{+}^{t+1}$. And we could carry on this description with mathematical
language,
$\sum_{i}^{N}\psi_{i}^{(t+1)}=\sum_{i}^{N}\psi_{i}^{(t)}$ (4)
The total time steps the system took could be defined as $T_{c}$ for
convergent time.
### II.1 Results and analysis
In the following discussion and simulation, we confines our individuals on an
isotropic plane $(L\times L)$. The length of the boundary of this plane $L$ is
20, and the total number of individuals on the plane would be $N$. Here we
simulate the individuals have their initial velocity under Gauss distribution,
which would be more rational and close to facts. Each individual hold their
opinions (positive one or negative one) and their fixed weight of opinion with
random probability. The distribution of agents’ weight was established in a
random way at the beginning of evolution. The exponent $\alpha$ in equations
(1) and (2) controls the evolution process. And here we define $\rho$ and
$\Delta\rho$ to describe the changing process of the individuals holding
positive opinion. They are denoted in equations as follow
$\rho_{c}=\frac{N_{+}}{N}$ (5) $\Delta\rho=\frac{\Delta N_{+}}{N}$ (6)
In Figure 2,we show $\rho_{c}$ as function of evolution time $t$ for different
view radius $r$, both $r$=1.2 and $r$=1.5. The most interesting thing we could
find in this figure is that when evolution time $t$ is around 6500, the value
of $\Delta\rho$ plummets obviously, which finally reach the level under 0.1.
In fact, when changes of $\Delta\rho$ has lower amplitude of variation, it
also means that the individuals holding positive opinion enter the period of
dynamic equilibrium.
Figure 4: (Color online) For $r=1.6$, cooperator density $\rho_{c}$ as a
function of degree for different values of $\alpha$. Figure 5: (Color online)
For $r=1.6$, cooperator density $\rho_{c}$ as a function of degree for
different values of $\alpha$. Figure 6: (Color online) (a)[Initial
distribution of agents with two kinds of opinions,$N_{+}$:$N_{-}$=1.19:1.
(b)Final distribution of agents with two kinds of opinions,
$N_{+}$:$N_{-}$=2.91:1.
In this situation, we could find the consensus time in Figure 3. If $t<4000$
or $t>6500$, the $\Delta\rho$ changes in a very small range which would also
be shown in the figure. But the consensus time is not directly impacted by
only view radius $r$ of each individual. In Figure 4, here converge time
$T_{c}$ is taken as a function of average velocity $\alpha$ , while the view
radius $r$ is equal to 1.2. Interestingly, we show that would reach a minimum
value when $\alpha$ is around 2 under different values of total number of
individuals on the plane $N$. Here we present that consensus time $T_{c}$
changes with $\alpha$ in a ”smile curve”. And certainly the value of would be
higher if the $N$ is more. In fact, it is obvious to explain that when there
are more individuals holding different opinions, they would take more time to
reach consensus or dynamic equilibrium. In that we show that $N$ and weight
exponent $\alpha$ could both directly determine the consensus time $T_{c}$.
The more cogent demonstration is shown in Figure 2, which presents as a
function of $\alpha$. Here $\rho_{c}$ is the density of individuals holding
positive opinions at the consensus time.
In the Figure 2, we present that $\rho_{c}$ will reach a maximum when is
around 2. Meanwhile, the value of is greater if view radius is lager. To show
the result in a more intuitive way, we present the distribution map in Figure
6. In Figure 6, the red points are the ones holding positive opinions while
the black points present negative ones. In this figure, we present the
specific distribution.
To support former results, we mainly focus on the results that positive
opinions take dominant rate. In order to discuss the parameters reflecting
immigration of individuals, we present consensus time $T_{c}$ as a function of
average velocity of individuals with different $N$ in Figure 5. In this
figure, we find that consensus time $T_{c}$ would increase approximately in a
linear way when $v$ is less than 1.5. After that , it decreases in a certain
range without sharp changes.
To discuss the model in a more reliable way, we try to analysis the process by
founding up a series of equations for $m$ agents in total as follow. In
equations, we define that the $i$th individual we concern has a view radius
$r$, and at the t step there are $s_{m}$ individuals in its view range as its
neighbors,and here we define that $s_{ij}$ as the $i$th neighbor of the $j$th
agent we concern. In that, it is $j$th opinion state of agent’s neighbor at
$t$ step that determine the opinion updating of this individuals at next time
step $t+1$. If $\psi$ is positive, individuals who holding positive opinions
would have greater weight than those who hold negative opinions. As a
consequence, the equations could be formed as follow:
$\displaystyle\left\\{\begin{array}[]{c}\psi_{1}^{(t+1)}=\psi^{(t)}(r,s_{11})\cdot\omega_{s_{11}}^{\alpha}+\ldots+\psi^{(t)}(r,s_{m-1,1})\cdot\omega_{s_{m-1,1}}^{\alpha}+\psi^{(t)}(r,s_{m,1})\cdot\omega_{s_{m,1}}^{\alpha}\\\
\vdots\\\
\psi_{m-1}^{(t+1)}=\psi^{(t)}(r,s_{1,m-1})\cdot\omega_{s_{11}}^{\alpha}+\ldots+\psi^{(t)}(r,s_{m-1,m-1})\cdot\omega_{s_{m-1,m-1}}^{\alpha}+\psi^{(t)}(r,s_{m,m-1})\cdot\omega_{s_{m,m-1}}^{\alpha}\\\
\psi_{1}^{(t+1)}=\psi^{(t)}(r,s_{1,m})\cdot\omega_{s_{1,m}}^{\alpha}+\ldots+\psi^{(t)}(r,s_{m-1,m})\cdot\omega_{s_{m-1,m}}^{\alpha}+\psi^{(t)}(r,s_{m,m})\cdot\omega_{s_{m,m}}^{\alpha}\\\
\end{array}\right.$ (11)
To describe the model in a simpler way, we try to apply linear algebra instead
of these traditional equations. In order to write in that way, we also
introduce a new parameter $n_{ij}^{(t)}$ into this matrix description. Here
$n_{ij}^{(t)}$ reflects that the times of opinion exchanging or sharing of
$j$th individual we concern at $t$ step. In other word, $n_{ij}^{(t)}$ is a
standard that concerns how many times the $i$th individual impacts others
opinion updating choice of next time step at $t$ step. When the whole
individuals get into the plateaus of dynamic equilibrium, we discussed in
Sec.2, the opinions individuals holding would be described as where
$\omega(i)$ is the weight of the ith agent, which is
$\left(\begin{array}[]{c}\psi_{1}^{(t+1)}\\\ \psi_{2}^{(t+1)}\\\ \vdots\\\
\psi_{m-1}^{(t+1)}\\\ \psi_{m}^{(t+1)}\\\
\end{array}\right)=\left(\begin{array}[]{ccccc}\psi_{1}^{(t)}&\psi_{2}^{(t)}&\ldots&\psi_{m-1}^{(t)}&\psi_{m}^{(t)}\\\
\psi_{1}^{(t)}&\vdots&\vdots&\vdots&\vdots\\\
\vdots&\vdots&\vdots&\vdots&\vdots\\\
\vdots&\vdots&\vdots&\vdots&\psi_{m}^{(t)}\\\
\psi_{1}^{(t)}&\psi_{2}^{(t)}&\ldots&\ldots&\psi_{m}^{(t)}\end{array}\right)\times\left(\begin{array}[]{cccc}n_{11}^{(t)}\cdot\omega_{1}^{\alpha}&n_{12}^{(t)}\cdot\omega_{1}^{\alpha}&\ldots&n_{1m}^{(t)}\cdot\omega_{1}^{\alpha}\\\
n_{21}^{(t)}\cdot\omega_{2}^{\alpha}&n_{22}^{(t)}\cdot\omega_{2}^{\alpha}&\ldots&n_{2m}^{(t)}\cdot\omega_{2}^{\alpha}\\\
\vdots&\vdots&\ldots&\vdots\\\
n_{m-1,1}^{(t)}\cdot\omega_{m-1}^{\alpha}&n_{m-1,2}^{(t)}\cdot\omega_{m-1}^{\alpha}&\ldots&n_{m-1,m}^{(t)}\cdot\omega_{m-1}^{\alpha}\\\
n_{m,1}^{(t)}\cdot\omega_{m}^{\alpha}&n_{m,2}^{(t)}\cdot\omega_{m}^{\alpha}&\ldots&n_{m,m}^{(t)}\cdot\omega_{m}^{\alpha}\end{array}\right)$
(12)
Here we denote that $n_{ij}^{(t)}$ as the times for opinion exchanges between
the $i$th and $j$th agents at the $t$ time step. At that certain situation, we
would find that When the whole system has entered the final homeostasis, the
whole opinions of agents we concern would be invariable, which means that .
And we could finally find that the time of opinion sharing or exchanging is
related to the weight of the agent and exponent in the equation. After
simplified such matrix equation, we finally get a direct function for
$n_{ij}^{(t)}$ and $\alpha$
$\sum_{j=1}^{m}n_{ij}^{(t)}\omega_{i}^{\alpha}=1$ (13)
where $\omega_{i}$ is weight of the $i$th agent we concern.
By choosing five different groups $n_{ij}^{(t)}$ when we fixed $\alpha=2$, we
finally calculate the $\alpha$ with the function for support that, from which
we get $\alpha=1.98,1.93,1.82,1.88,1.91$.By the calculation, the function
reflects a very important and simple relationship between $n_{ij}^{(t)}$ and
weight $\omega_{i}$
## III Conclusion and Discussions
The new mode of opinion evolution with immigration is different from the
conventional opinion dynamics. It presents that density of positive opinion
agents would be maximum when the weight exponent $\alpha$ is around 2. In
summary, we found up a new model for opinion exchange and communication among
agents with immigration. The state matrix we present for analysis and
quantitative simulation could also be widely used for more complex situation.
The opinion carried by agents represent a kind of state or parameter of agents
in motion.So more application and analysis could be carried on with this model
and method in future. Discussion we present above is not only to demonstrate
our model, but also open up a new combination between opinion communication
and agent-based motion. State consensus time is also a very important
parameter to describe a system or a group of agents, which could also be one
certain standard for different situations.
###### Acknowledgements.
This work is funded by the National Basic Research Program of China (973
Program No.2006CB705500), the National Natural Science Foundation of China
(Grant Nos. 60744003, 10635040, 10532060) and by the Special Research Funds
for Theoretical Physics Frontier Problems (NSFC No.10547004 and A0524701). WXW
and YCL are supported by AFOSR under Grant No. FA9550-07-1-0045.
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|
arxiv-papers
| 2013-10-18T17:00:40 |
2024-09-04T02:49:52.568304
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Zhong-Lin Han Yu-Jian Li and Bing-Hong Wang",
"submitter": "Yu-jian Li",
"url": "https://arxiv.org/abs/1310.5096"
}
|
1310.5147
|
# A Non-radial Oscillation Mode in an Accreting Millisecond Pulsar?
Tod Strohmayer1 and Simin Mahmoodifar2
1Astrophysics Science Division and Joint Space-Science Institute, NASA’s
Goddard Space Flight Center, Greenbelt, MD 20771, USA
2Department of Physics and Joint Space-Science Institute, University of
Maryland College Park, MD 20742, USA
###### Abstract
We present results of targeted searches for signatures of non-radial
oscillation modes (such as r- and g-modes) in neutron stars using RXTE data
from several accreting millisecond X-ray pulsars (AMXPs). We search for
potentially coherent signals in the neutron star rest frame by first removing
the phase delays associated with the star’s binary motion and computing FFT
power spectra of continuous light curves with up to $2^{30}$ time bins. We
search a range of frequencies in which both r- and g-modes are theoretically
expected to reside. Using data from the discovery outburst of the 435 Hz
pulsar XTE J1751$-$305 we find a single candidate, coherent oscillation with a
frequency of $0.5727597\times\nu_{spin}=249.332609$ Hz, and a fractional
Fourier amplitude of $7.46\times 10^{-4}$. We estimate the significance of
this feature at the $1.6\times 10^{-3}$ level, slightly better than a
$3\sigma$ detection. Based on the observed frequency we argue that possible
mode identifications include rotationally-modified g-modes associated with
either a helium-rich surface layer or a density discontinuity due to electron
captures on hydrogen in the accreted ocean. In the latter case the presence of
sufficient hydrogen in this ultracompact system with a likely helium-rich
donor would present an interesting puzzle. Alternatively, the frequency could
be identified with that of an inertial mode or a core r-mode modified by the
presence of a solid crust, however, the r-mode amplitude required to account
for the observed modulation amplitude would induce a large spin-down rate
inconsistent with the observed pulse timing measurements. For the AMXPs XTE
J1814$-$338 and NGC 6440 X-2 we do not find any candidate oscillation signals,
and we place upper limits on the fractional Fourier amplitude of any coherent
oscillations in our frequency search range of $7.8\times 10^{-4}$ and
$5.6\times 10^{-3}$, respectively. We briefly discuss the prospects and
sensitivity for similar searches with future, larger X-ray collecting area
missions.
###### Subject headings:
stars: neutron – stars: oscillations – stars: rotation – X-rays: binaries –
X-rays: individual (XTE J1751$-$305, XTE J1814$-$338, NGC 6440 X-2) – methods:
data analysis
## 1\. Introduction
The study of global stellar oscillations can provide a powerful probe of the
interior properties of stars. A prime example of this is the rich field of
helioseismology. By comparison, efforts to probe the exotic interiors of
neutron stars via similar methods are still in their infancy, but recent
observational results have provided new impetus to further explore
asteroseismology of neutron stars. For example, observations of quasiperiodic
oscillations (QPOs) in the X-ray flux of highly magnetized neutron stars,
“magnetars” (Duncan, 1998; Israel et al., 2005; Strohmayer & Watts, 2005,
2006; Watts & Strohmayer, 2006; Woods & Thompson, 2006), which have been
linked to global torsional vibrations within the star’s crust, may ultimately
provide a promising new probe of a neutron star’s internal composition and
structure. In addition to the magnetar QPOs, burst oscillations, pulsations
seen at or near the neutron star spin frequency during thermonuclear X-ray
bursts from accreting, low mass X-ray binary (LMXB) neutron stars (see
Strohmayer & Bildsten (2006); Watts (2012) for reviews on burst oscillations),
may also be linked to stellar pulsations. Although a comprehensive
understanding of the physics of these oscillations is still being developed,
one of the models that has been proposed to explain them is the Rossby wave
(r-mode) model, which assumes that the oscillation is produced by a low-
frequency r-mode (Rossby wave) propagating in the neutron star surface
“ocean.” In this case the r-mode modulates the temperature distribution across
the neutron star surface and the resulting angular variations of the surface
thermal emission–combined with the spin of the star–produce pulsations in the
X-ray flux observed from the stellar surface. For example, Lee & Strohmayer
(2005) and Heyl (2005) have explored this model, and computed light curves for
small azimuthal wavenumber, $m$, surface r-modes on rotating neutron stars.
Accretion-powered millisecond X-ray pulsars (AMXPs) also show small-amplitude
X-ray oscillations with periods equal to their spin periods. To explain the
low modulation amplitudes and nearly sinusoidal waveforms in these sources,
Lamb et al. (2009) proposed a model in which the X-rays are emitted from a
hot-spot that is located at or near a magnetic pole of the star, and the
magnetic pole is assumed to be close to the spin axis of the star. When the
emitting region is close to the spin axis, a small variation in its position
can produce relatively large changes in the amplitude and phase of the X-ray
variations. Lee (2010) and Numata & Lee (2010) later suggested that global
oscillations of neutron stars (for example, r-modes) can periodically perturb
such a hot-spot and therefore the oscillation mode periods might potentially
be observable as X-ray flux oscillations from these sources (we discuss this
in more detail below).
The global oscillation spectrum of neutron stars is rich, and has been
classified according to the restoring force relevant to each particular mode
(McDermott et al., 1988). For example, pressure modes (p-modes and the f-mode)
are primarily supported by internal pressure fluctuations (essentially sound
waves) in the star and have frequencies in the $10$ kHz range that scale as
$(\bar{\rho})^{1/2}$, where $\bar{\rho}$ is the stellar mean density. The
successive overtones of these modes have higher frequencies. By overtones we
mean modes with an increasing number of nodes (zero crossings) in their radial
displacement eigenfunctions. Gravity modes (g-modes) confined primarily to the
region above the solid crust have buoyancy as their restoring force and
frequencies in the $1-100$ Hz range (in the slow-rotation limit). The
overtones of these surface g-modes have decreasing frequencies. The finite
shear modulus of the neutron star crust leads to additional, shear-dominated
modes. These include the purely transverse torsional modes (t-modes), briefly
mentioned above in the context of magnetar QPOs, which have frequencies larger
than about $30$ Hz, and the s-modes, which possess both radial and transverse
displacements. For both classes of torsional modes the overtones–whose radial
eigenfunctions have at least one node in the crust–can be thought of as shear
waves traveling vertically through the crust. They have frequencies in the kHz
range that scale inversely with the thickness of the neutron star crust.
In the case of rotating neutron stars another important class of oscillations
are the so-called inertial modes for which the restoring force of the
pulsations is provided by the Coriolis force (Yoshida & Lee, 2000a, b). A well
known sub-set of these are the r-modes that couple to gravitational radiation
and can be driven unstable by the Chandrasekhar-Friedman-Schutz (CFS)
mechanism (Friedman & Schutz, 1978; Andersson, 1998; Friedman & Morsink,
1998). Whether they are excited or not is a competition between the driving
due to the coupling to gravitational radiation and the various mechanisms–such
as bulk and shear viscosity–that can damp the oscillations. The damping and
transport properties, such as viscosity, heat conductivity and neutrino
emissivity, depend significantly on the phase of dense matter present in the
star, and since r-modes can both brake the star’s rotation and heat its
interior, study of the spin and thermal evolution of neutron stars can be a
potentially important probe of the dense matter interior (Mahmoodifar &
Strohmayer, 2013; Haskell et al., 2012). Moreover, the co-rotating frame
r-mode frequencies depend on the stellar spin rate and the internal
composition and structure of the star (Lindblom et al., 1999; Yoshida & Lee,
2000b; Alford et al., 2012). Thus, observations of the frequencies of non-
radial oscillation modes of neutron stars would be very useful in probing
their internal structure, but except for the magnetar QPOs linked to crustal
vibrations and perhaps the surface r-modes linked with burst oscillations,
there have been no other direct observations of these oscillations.
It is relevant to ask the question of how the presence of non-radial
oscillations might be inferred from observations. As noted briefly above in
the context of burst oscillations, if an r-mode modulates the temperature
distribution across the neutron star surface, then this may be revealed as a
pulsation in the X-ray flux from the star. Another possibility is that surface
motions induced by a particular oscillation mode perturb the X-ray emitting
hot-spot that is present during the outbursts of accreting millisecond X-ray
pulsars (AMXPs). This mechanism seems most relevant for quasi-toroidal modes
(such as the r- and g-modes) in which the dominant motions are
transverse–locally parallel to the stellar surface–as opposed to radial. Such
transverse motions can deform an emitting region in a periodic fashion and
thus imprint the periodic deformation on the observed light curve from the
source. Indeed, Numata & Lee (2010) explored this mechanism, and computed the
resulting light curves from such a perturbed hot-spot on a rotating neutron
star. Since the hot-spot rotates with the star it is periodically deformed at
the oscillation frequency of the mode as measured in the co-rotating frame of
the star. They computed the modulation that would be produced by a hot-spot
that is perturbed by the surface motions associated with a global r-mode, and
showed that the r-mode frequency specified in the co-rotating frame is
imprinted on the light curve seen by a distant observer. We note that surface
g-modes also have dominant horizontal displacements and could also be relevant
in this context. Using this model they also demonstrated that the observed
modulation amplitude of the light curve could be used to infer or constrain
the mode amplitude.
In the limit of slow rotation it is well known that the r-mode frequency in
the co-rotating frame is given by $\omega=2m\Omega/l(l+1)$, where $m$ and $l$
are the spherical harmonic indices that describe the angular distribution of
the dominant toroidal displacement vector, and $\Omega$ is the stellar spin
frequency. The most unstable r-mode is that associated with $l=m=2$, which has
the familiar frequency $\omega=2\Omega/3$ in the co-rotating frame. For more
rapidly rotating neutron stars, like the AMXPs, the r-mode frequency deviates
from the above limit and is typically calculated in an expansion in powers of
the angular rotation frequency (see, for example, Lockitch & Friedman (1999);
Yoshida & Lee (2000b); Lindblom et al. (1998); Alford et al. (2012)). This
leads to an expression for $\omega$ of the form,
$\omega=\Omega(\kappa_{0}+\kappa_{2}\bar{\Omega}^{2})$, where for the $l=m=2$
r-mode, $\kappa_{0}=2/3$, $\bar{\Omega}^{2}=\Omega^{2}(R^{3}/GM)$ and
$\kappa_{2}$, which represents the next-order correction to the r-mode
frequency, depends on the properties of the unperturbed stellar model, such as
its equation of state (EOS) and entropy stratification (Yoshida & Lee, 2000b;
Alford et al., 2012). Thus, if an r-mode frequency is detected it can
potentially provide interesting information about the stellar interior, and
perhaps be used to identify the dense matter phase present in the core (see,
for example, Figure 3 in Alford et al. 2012).
The above discussion ignores the effects that the solid crust of the neutron
star may have in modifying the r-modes and their surface displacements. For
example, Yoshida & Lee (2001) have investigated the r-modes for neutron star
models including a solid crust and show that they are strongly influenced by
mode coupling with the crustal torsional modes (t-modes). They found that this
mode coupling can reduce the r-mode frequency from $2\Omega/3$ to values as
low as $\Omega/2$ to $2\Omega/5$ (see their Figure 2), and the reduction
occurs at and above a critical rotational frequency that is close to the
fundamental torsional mode frequency (we discuss this in more detail below).
Since the spin frequency of outbursting AMXPs can be tracked with high
precision, and the r-mode frequencies are computed as a series expansion in
powers of the spin frequency, it is possible to carry out coherent, targeted
searches in such sources for r-modes in a specific range of frequencies both
above and below its “expected” ($\Omega\rightarrow 0$ limit) value,
$2\Omega/3$. Similar arguments apply for other modes as well, such as the
surface g-modes, some of which have frequencies that overlap the expected
frequency range for the r-modes. Here we present the results of power spectral
searches for the signatures of such modes using data from several AMXPs
obtained with the Rossi X-ray Timing Explorer (RXTE). It is not our intent
here to present an exhaustive search of all known AMXPs, rather, we illustrate
the methods and present results for three sources; XTE J1751$-$305 (hereafter
J1751), XTE J1814$-$338 (hereafter J1814), and NGC 6440 X-2 (hereafter X-2),
all of which are within the nominal r-mode instability window computed for
hadronic matter, and which had the highest inferred r-mode amplitude upper
limits in our recent study (Mahmoodifar & Strohmayer, 2013). We will present a
study of additional sources, including SAX J1808.4$-$3658, in a sequel. The
paper is organized as follows. In §2 we illustrate in some detail our search
analysis procedures using data from the 435 Hz AMXP J1751. We also present the
search results for this source and describe our best detection candidate,
which is at a frequency of $0.57276\nu_{spin}$ (249.33 Hz). In §3 we summarize
our search results for the additional targets, the 206 Hz pulsar X-2, and the
314 Hz pulsar J1814. In §4 we discuss possible mode identifications for the
best candidate frequency in J1751. We also briefly discuss how future
observations with larger collecting area missions, such as ESA’s Large
Observatory for X-ray Timing (LOFT, Feroci et al. (2012)), and the Advanced
X-ray Timing Array (AXTAR, Ray et al. (2010)) can improve the sensitivity of
such searches. We conclude with a brief summary of our findings in §5.
## 2\. A Coherent Search in XTE J1751$-$305
The most sensitive search procedure for a particular timing signature, such as
a coherent pulsation, depends on the nature of that signature. In the context
of searches employing Fourier power spectra the greatest sensitivity is
achieved by matching the frequency resolution of the power spectrum to the
expected frequency bandwidth of the signal. Thus, for a highly coherent signal
the greatest sensitivity is achieved by maximizing the frequency resolution.
This effectively means that one should compute a single Fourier power spectrum
of the longest time series obtainable from the available data. While the exact
frequency bandwidth of a candidate signal is often not known precisely, the
work of Numata & Lee (2010) suggests that a signal produced by perturbation of
a hot-spot by an r-mode (or some other non-radial mode) may be quite coherent.
On the other hand, conditions in the neutron star surface layers can evolve as
accretion continues during an outburst and these sources are known to exhibit
timing noise that is likely associated with variations in the latitude and
azimuth of the accretion hot-spot (Patruno et al., 2009), so such processes
are likely to limit the effective coherence of such signals. Because of this,
as well as computational constraints, we restrict the size of the longest
light curves for Fourier analysis in this work to $N=2^{30}$ time bins. For a
sample rate of 2048 Hz this corresponds to a time interval of 524,288 s, or
about 6 days. Depending on the amount of data present for a given source, one
can then average several independent power spectra and/or adjacent Fourier
frequency bins to search for signals with broader frequency bandwidths (such
as quasi-periodic oscillations, QPOs).
In order to carry out searches at the highest frequency resolution it is
necessary to remove as best as possible the frequency drifts associated with
the binary motion of the neutron star about the center of mass of the system
in which it resides. This effectively places the observer at the center of
mass of the binary system, a point from which the neutron star is neither
approaching nor receding. These considerations lead to the following basic
steps we use to carry out a search. First, the X-ray event arrival times are
corrected to the Solar System barycenter. Next, we fit a model to the observed
orbit-induced phase variations. This orbit model is used to convert each
photon event arrival time to a neutron star rotation phase. These phases are
then converted back to fiducial times using the best-determined spin frequency
of the neutron star. Finally, these orbit corrected times can be used to
compute a single light curve which can then be Fourier analyzed using Fast
Fourier Transform (FFT) power spectral methods.
To illustrate the procedure in some detail we step through our analysis for
J1751. This source was discovered in early April, 2002 during regular
monitoring observations of the Galactic center region using the RXTE
Proportional Counter Array (PCA, Markwardt et al. (2002)). The outburst was
relatively short, lasting only about 10 days. Timing of the X-ray pulsations
revealed an ultra-compact system with an orbital period of 42.4 min (Markwardt
et al., 2002). For our coherent search we used data spanning about 6 days
during the peak of the outburst. Figure 1 shows the source light curve sampled
in 2 s bins. We used PCA event mode data with a resolution of 125 $\mu$-sec
for our study and included all events in the full energy band-pass of the PCA
and from all operating detectors. We used the FTOOL faxbary to correct the
photon arrival times to the Solar System barycenter. We then applied the orbit
timing solution from Markwardt et al. (2002, 2007) (see Table 1 in their 2007
paper) to convert the arrival times to neutron star rotational phases. Figure
2 shows a dynamic power spectrum from a single RXTE orbit, which reveals the
time evolution of the pulsar frequency due to the neutron star’s orbital
motion. The best fitting orbit model for this time interval (thick solid
curve) is also plotted, showing that it accurately predicts the observed
evolution. Figure 3 shows the resulting phase residuals after application of
the orbit model to the light curve used for our coherent search. The remaining
variations are consistent with poisson errors in the phases.
We then used the orbit model to convert each arrival time to a rotational
phase. These phases can then be expressed as fiducial times by multiplying by
the best-fit pulsar spin period. We use the resulting times to produce a light
curve sampled at 2048 Hz that contains $2^{30}=1,073,741,824$ time bins.
Finally, we compute an FFT power spectrum of this light curve. The resulting
power spectrum has a little more than half a billion frequency bins and a
Nyquist frequency of 1024 Hz, thus, simply from file size considerations it is
not practical to present a plot of the entire spectrum. However, to
demonstrate that the coherent pulsar signal is strongly detected we show in
Figure 4 the power spectrum in a narrow frequency band centered on the pulsar
signal. Here, the units on the x-axis are $(\sigma/\Omega-1)\times 10^{5}$,
where $\sigma$ is a Fourier frequency. Thus, the pulsar signal appears at zero
in these units. Moreover, in order to enable direct comparison with the light
curve computations of Numata & Lee (2010), see for example their Figure 6, we
plot the power spectrum in units of fractional Fourier amplitudes
$\sqrt{(a_{j}^{*}a_{j})/N_{tot}}$, where the $a_{j}$ are the complex Fourier
amplitudes at Fourier frequency $\nu_{j}=j/(524,288\;s)$, $j$ ranges from $0$
to $2^{29}$, $N_{tot}=44,316,997$ is the total number of events in the light
curve, and the $*$ symbol indicates complex conjugation. To convert the
fractional Fourier amplitudes to the commonly used Leahy normalization one
simply squares the fractional amplitudes, and then multiplies by $2\times
N_{tot}$. The commonly employed fractional rms amplitude is simply $\sqrt{2}$
times the fractional Fourier amplitude defined above.
### 2.1. Search for Co-rotating Frame Frequencies Consistent with r- and
g-modes
As noted in §1 above, when a pulsation mode periodically perturbs an X-ray
emitting hot-spot that is fixed in the rotating frame of the star, the co-
rotating frame mode frequency is imprinted on the light curve seen by a
distant observer. Further, rapid rotation tends to increase the co-rotating
frame frequency of the $l=m=2$ r-mode from the slow-rotation limit of
$\omega=2\Omega/3$, while the influence of a solid crust may decrease it.
Based on the discussion above, a reasonable frequency range to search is then
$2/3-k_{1}\leq\omega/\Omega\leq(2/3+k_{2})$, where $k_{1}$ represents a
plausible reduction in the frequency based on the possible crustal effects to
the r-mode, and $k_{2}$ represents a reasonable maximum increase for
$\kappa_{2}\bar{\Omega}$ given the observed spin frequency of J1751 and
various possible masses, equations of state and interior compositions for the
neutron star. Based on the calculations of Yoshida & Lee (2001) and Alford et
al. (2012, see their Figure 3), plausible values for $k_{1}$ and $k_{2}$ are
0.25 and 0.09, respectively. This defines a search range from
$0.4166\leq\omega/\Omega\leq 0.75667$. A search in that range reveals one
candidate peak in slightly more than 77.59 million independent Fourier
frequency bins. Figure 5 shows a portion of the full-resolution spectrum in
the vicinity of this peak. It appears at a frequency of
$0.5727597\times\nu_{spin}=249.332609$ Hz, and has a fractional Fourier
amplitude of $7.455\times 10^{-4}$.
To assess the significance of this peak we first convert its fractional
Fourier amplitude to a Leahy-normalized power and then estimate its single-
trial probability using the expected noise power distribution, which for a
single power spectrum is the $\chi^{2}$ distribution with 2 degrees of
freedom. The peak Leahy-normalized power is then 49.26, which corresponds to a
single-trial probability of $2\times 10^{-11}$. Accounting for the number of
trials by multiplying by the number of independent Fourier frequencies in the
search range, $77.6\times 10^{6}$, gives a significance of $1.6\times
10^{-3}$, which is a little better than a $3\sigma$ detection. We then used a
portion of the power spectrum at higher frequencies (from 1.6 to 2.2 times the
pulsar spin frequency) to investigate how accurately the distribution of noise
powers follows the expected $\chi^{2}$ distribution. The result is shown in
Figure 6, where the red dashed line denotes the probability to exceed a given
Fourier power for the $\chi^{2}$ distribution with 2 degrees of freedom, and
the Leahy-normalized power spectral data are plotted as a histogram. Over the
range of Fourier powers present in the data the power spectral values show a
good match to the expected distribution. The Fourier power of the candidate
peak is marked by the vertical dashed-dot line, and as indicated above, has a
single trial probability of $2\times 10^{-11}$. Based on this we think our
significance estimate is reasonable.
We next averaged the full resolution power spectrum in order to search for any
broader bandwidth signals that might be present. Figure 7 shows two such
averaged power spectra over the full frequency range. The black and green
histograms have frequency resolutions of 1/2048, and 1/128 Hz, respectively.
The pulsar signal is still easily detected in each case, but we do not find
any other significant features at these or other frequency resolutions. The
horizontal dashed line marks the amplitude of the candidate signal at $249.33$
Hz discussed above. We can place upper limits on any signal power in our
defined search range at these frequency resolutions of $1.64\times 10^{-4}$
and $1.42\times 10^{-4}$, respectively. The horizontal, red dashed line in
Figure 7 marks an amplitude given by $1/(N_{tot})^{1/2}=1.50\times 10^{-4}$,
which gives a reasonably close approximation to the quoted upper limits for
broader band signals.
### 2.2. Search for Modulation at the Inertial Frame r-mode Frequency
As discussed in §1, if an oscillation mode modulates emission over the entire
neutron star surface rather than simply perturbing a hot-spot fixed in the co-
rotating frame, then one would expect a pulsation signal at the mode’s
inertial frame frequency, $\omega_{i}=2\Omega-\omega$, where $\omega$ is the
co-rotating frame frequency. Thus, to search the range of inertial frame
frequencies corresponding to the range of co-rotating frame frequencies just
discussed in §2.1 we need to search the frequency range
$2-(2/3+0.09)<\sigma/\Omega<2-(2/3-0.25)$, which reduces to
$1.243<\sigma/\Omega<1.583$. A search reveals no significant peaks in this
range. The highest peak appears at a frequency of $1.565327\nu_{spin}$, with a
fractional Fourier amplitude limit of $6.6\times 10^{-4}$.
## 3\. Coherent Searches in XTE J1814$-$338 and NGC 6440 X-2
Here we briefly summarize search results for J1814 and X-2.
### 3.1. Results for XTE J1814$-$338
J1814 was discovered by RXTE in June 2003 using data obtained with the
Galactic bulge monitoring program then being conducted with the PCA onboard
RXTE. The pulsar has a 314.36 Hz spin frequency and an orbital period of 4.275
hr (Markwardt et al., 2003; Papitto et al., 2007). The discovery outburst
lasted for $\approx 50$ days. This object was the first neutron star to
exhibit burst oscillations with a significant first harmonic (Strohmayer et
al., 2003), and indeed, the persistent pulse profile also shows substantial
harmonic content. This source is also known to exhibit significant timing
noise, that is, systematic timing residuals remain after modeling the binary
Doppler delays (Papitto et al., 2007; Watts et al., 2008b). This noise is
still not completely understood, but may represent movement of the accretion
hot-spot relative to the stellar spin axis as the accretion rate changes
during an outburst (Patruno, 2010). Here we use data from the first 12 days
for which such variations were less significant (Watts et al., 2008b). We used
data beginning on June 5, 2003 at 02:34:20 UTC and constructed two light
curves, each sampled at 2048 Hz and with $2^{30}$ time bins. There are a total
of $31,361,962$ X-ray events in the two light curves. We first modeled the
orbital variations in a similar manner as described above for J1751. Our orbit
parameters are consistent with those of Papitto et al. (2007). Figures 8 and 9
show the resulting orbit-corrected phase residuals for the two data segments
used to construct our light curves. One can see that the second interval
(Figure 9) shows more systematic timing noise than the first interval (Figure
8). We then computed power spectra for each interval in the same manner as
described for J1751. We searched the power spectra in the same frequency
ranges as described above for J1751 and for each data interval separately as
well as the average power spectrum computed from both intervals. We did not
find any significant features in the power spectra. Figure 10 shows two
average power spectra computed from both intervals, the black and green
histograms have been averaged to frequency resolutions of 1/2048, and 1/128
Hz, respectively. The pulsar fundamental and first harmonic are clearly
evident (at 1 and 2 in these units). The horizontal dashed line (black) marks
the upper limit of $7.8\times 10^{-4}$ on any signal power at the full
frequency resolution of the power spectrum. The horizontal dashed (red) line
marks an amplitude given by $1/(N_{tot}/2)^{1/2}\approx 2.5\times 10^{-4}$,
which again gives a reasonably close approximation to the upper limits for
broader band signals.
### 3.2. Results for NGC 6440 X-2
Pulsations at 205.89 Hz were detected with RXTE from the globular cluster
source NGC 6440 X-2 on 30 August, 2009 (Altamirano et al., 2009). On this date
the source was observed for a single RXTE orbit, yielding $\approx 3000$ s of
exposure, revealing a 57 min orbital period (Altamirano et al., 2010). A
subsequent outburst with detectable pulsations was observed with RXTE on 21
March, 2010, for an additional 3 RXTE orbits and 6600 s of exposure. We used
all these data in our search. As for J1751, we first barycentered the data
using the best determined position from Heinke et al. (2010). Because the
available data for this source are too sparse to enable calculation of a
single, coherent Fourier power spectrum, we separately modeled the orbital
variations in each of the four data segments. We then generated light curves
using the orbit-corrected arrival times, computed a Fourier power spectrum for
each, and then averaged them. The light curves were sampled at 8192 Hz,
yielding a Nyquist frequency of 4096 Hz. The resulting averaged power spectrum
is shown in Figure 11. Since there are many fewer Fourier frequencies compared
to either J1751 or J1814, we show the spectrum at the full frequency
resolution. Two spectra are shown in Figure 11, the black curve is plotted at
the full frequency resolution ($3.125\times 10^{-4}$ Hz), and the green curve
has been averaged by a factor of 32 to a resolution of 0.01 Hz. The pulsar
fundamental is clearly evident (at 1 in these units), but there are no other
candidate detections. At these resolutions we can place upper limits on any
signal power in our search ranges of $5.6\times 10^{-3}$ (at $3.125\times
10^{-4}$ Hz resolution), and $2.8\times 10^{-3}$ (at $0.02$ Hz). Because much
less data is available for X-2 than for either J1751 or J1814, the limits are
not as constraining as for those sources.
## 4\. Discussion
As discussed in §2, we found a candidate oscillation at a frequency
$\omega=0.5727\Omega$ with an estimated significance of $1.6\times 10^{-3}$ in
data from the discovery outburst of J$1751$. Here we discuss possible mode
identifications for this candidate oscillation. As mentioned in the
introduction, AMXPs show small-amplitude X-ray oscillations with periods equal
to the spin period of the star. To explain their low modulation amplitudes and
nearly sinusoidal waveforms Lamb et al. (2009) proposed a model in which
X-rays are emitted from a hot-spot at the stellar surface and near a magnetic
pole that is assumed to be close to the rotation axis of the star. If we
assume that this model is correct, then transverse motions induced by the non-
radial oscillations at the surface of the star can perturb the hot-spot
periodically, and these periodicities might be observable in the radiation
flux from the star (Numata & Lee, 2010). In addition to producing X-ray
variations by perturbing the hot-spot, if the amplitude of the oscillations at
the surface of the star are large enough they might also generate X-ray
variations by modulating the surface temperature of the star (Lee &
Strohmayer, 2005). In the former case–where the surface oscillation perturbs
the hot-spot–since it is co-moving with the star, a distant observer will
detect the oscillation frequency of the mode as measured in the co-rotating
frame of the star (we refer to this as the “co-rotating frame scenario”). This
has been shown by Numata & Lee (2010) for the case of r-modes. On the other
hand, oscillation-induced temperature perturbations will produce X-ray
variations with the same periodicity as the oscillation frequency of the mode
as measured in an inertial frame (we call this the “inertial frame scenario”).
For fast rotating, accreting neutron stars such as J1751 the most relevant
restoring forces that can produce stellar pulsations with frequencies that
might be consistent with the candidate frequency in J1751 are the coriolis
force–due to the star’s rotation–and buoyancy associated with thermal and
composition gradients. As briefly summarized in §1, the corresponding
oscillation modes associated with these forces are the inertial modes (which
includes the r-modes) and the gravity modes (g-modes). In general, both forces
are present and the nature of the resulting modes will depend on their
relative strength. For example, at high rotation rates the coriolis force will
almost certainly dominate–except perhaps within a very small band around the
rotational equator–and the resulting pulsation modes are expected to be
inertial in character. At the other extreme of slow rotation buoyancy can
eventually prevail resulting in essentially pure g-modes (Yoshida & Lee,
2000b; Passamonti et al., 2009). Other oscillation modes such as crustal
toroidal modes associated with the finite shear modulus of the crust, or f-
and p-modes due to pressure forces are either confined to the crust or core of
the star and may not be able to induce motions at the surface, or they have
higher frequencies which are inconsistent with the candidate oscillation. In
what follows we discuss how the candidate frequency in J$1751$ might be
identified as a surface g-mode, a core r-mode, or perhaps an inertial mode in
a fast rotating star.
### 4.1. g-modes
As briefly mentioned above, the g-modes are low frequency non-radial
oscillation modes of neutron stars with buoyancy as their restoring force. In
a 3 component NS model composed of a fluid core, a solid crust and a fluid
ocean, g-modes might be excited in the core and/or in the ocean, but the
finite shear modulus excludes them from the crust (Bildsten & Cutler, 1995).
As a result core g-modes are unlikely to have observable effects on the
radiation observed from the surface of the star. Therefore, here we focus on
the surface g-modes that are confined to a thin layer at the surface of the
star. There have been many studies on g-modes in neutron stars (McDermott &
Taam, 1987; Strohmayer & Lee, 1996; Bildsten & Cutler, 1995; Bildsten et al.,
1996; Bildsten & Cumming, 1998). We are particularly interested in the surface
g-modes in AMXPs. The conditions at the surface of these objects evolve slowly
due to the accretion and their g-mode spectrum is different from that of the
isolated and non-accreting neutron stars. The g-modes at the surface of an
accreting NS can be divided into several different categories according to
their different sources of buoyancy, such as an entropy gradient or density
discontinuity. Bildsten & Cutler (1995) studied surface g-modes in accreting
systems with thermal buoyancy as the restoring force. They obtained an
analytic result for the mode frequency in the non-rotating limit
($\Omega\rightarrow 0$)
$\displaystyle f_{th}$
$\displaystyle=6.26Hz\left(T_{8}\frac{16}{A}\right)^{\frac{1}{2}}\left(1+\left(\frac{3n\pi}{2\ln(\rho_{b}/\rho_{t})}\right)^{2}\right)^{-1/2}$
$\displaystyle\times\left(\frac{10km}{R}\right)\left(\frac{l(l+1)}{2}\right)^{\frac{1}{2}},$
(1)
where $T_{8}=\frac{T}{10^{8}K}$, $A$ is the mass number, $l$ is the spherical
harmonic index, $n$ is the number of radial nodes in the displacement
eigenfunction, $R$ is the stellar radius, and $\rho_{b}$ and $\rho_{t}$ are
densities at the bottom and top of the ocean, respectively. Strohmayer & Lee
(1996) studied thermal g-modes in a steady state accreting and nuclear burning
atmosphere, and found some modes can be excited by the $\epsilon$ mechanism
(perturbations in the nuclear burning). For example, see their Table 3 for
results on the oscillation periods of $g_{1}$ and $g_{2}$ modes. Bildsten &
Cumming (1998) studied the effect of hydrogen electron captures on g-modes in
the ocean of accreting neutron stars. They found that the sudden increase in
the density at the hydrogen electron capture layer supports a density
discontinuity mode with a non-rotating limit frequency
$f_{d}\approx
35Hz\left(\frac{X_{r}}{0.1}\right)^{\frac{1}{2}}\left(1-\frac{\Delta Z}{\Delta
A}\right)^{\frac{1}{2}}\left(\frac{10km}{R}\right)\left(\frac{l(l+1)}{2}\right)^{\frac{1}{2}},$
(2)
where $X_{r}$ is the residual mass fraction of hydrogen, and $\Delta Z$ and
$\Delta A$ are the changes in the charge and mass of the nuclei from one side
of the discontinuity to the other.
Piro & Bildsten (2004) studied non-radial oscillations at the surface of a
helium burning neutron star. They found one unstable mode that resides in the
helium atmosphere and is supported by the buoyancy of the helium/carbon
interface. The frequency of this mode in the non-rotating limit is given by
$f_{th-He}\approx(20-30Hz)\sqrt{\frac{l(l+1)}{2}},$ (3)
which depends on the accretion rate. Similarly to the results of Strohmayer &
Lee (1996), they find that this mode can also be driven unstable by the
$\epsilon$-mechanism, and they also compute results for higher accretion
rates. Now, the orbital period of J$1751$ is very short ($\sim 42$ min)
(Markwardt et al., 2002), which means that it is a very compact system and
thus the donor star is likely a helium white dwarf. This suggests that the
accreted material is helium-rich and it therefore seems plausible that the
system might show the unstable shallow surface waves that are obtained for a
helium atmosphere.
It is important to note that the g-mode frequencies just discussed are
obtained in the non-rotating limit $(\Omega\rightarrow 0)$ and, as alluded to
above, they will be modified by rapid rotation of the star. Bildsten et al.
(1996) studied the effect of high spin frequencies on the g-mode spectrum in
the so-called “traditional approximation,” in which mode propagation is
confined to a thin shell, the radial component of the coriolis force is
neglected, and the radial displacements produced by the modes are assumed to
be much less than the horizontal displacements. These approximations are
reasonable for surface g-modes as long as the coriolis force remains less than
the buoyant force (see §2 of Bildsten et al. (1996)). This condition can be
expressed as, $N^{2}\gg(\Omega\omega R)/h$, where $N$, $R$ and $h$ are the
Brunt Väisällä frequency (which sets the strength of buoyancy), stellar
radius, and characteristic scale height in the surface envelope, respectively.
Assuming the candidate frequency in J1751 represents the mode frequency in the
co-rotating frame (and using the 435 Hz spin frequency of J1751), then
$\omega=0.573\Omega$, and we would require $N^{2}\gg 4.3\times 10^{6}(R/h)$
Hz2.
From this analysis Bildsten et al. (1996) found that stellar rotation
“squeezes” the eigenfunctions toward the equator within an angle
$\cos\theta<\frac{1}{q}$ where $\theta$ is measured from the pole,
$q=\frac{2\Omega}{\omega}$, and the oscillation frequency of the mode (in the
co-rotating frame) in a non-rotating star, $\omega_{l,0}$, is related to the
mode frequency at arbitrary spin frequencies by the following equation
$\omega^{2}=2\Omega\omega_{l,0}\left[\frac{(2l_{\mu}-1)^{2}}{l(l+1)}\right]^{1/2}$
(4)
where $l_{\mu}$ is the number of zero crossings in the angular displacement
between $\cos\theta=-\frac{1}{q}$ and $+\frac{1}{q}$. Thus, the surface
displacements of the rotationally modified g-modes are strongly confined to
the equatorial region at high spin frequencies and the modes are exponentially
damped for $\cos\theta\geq 1/q$ (Bildsten et al., 1996). For the spin
frequency of J1751, $q\simeq 3.5$ (assuming that the 249.33 Hz candidate
frequency is associated with a co-rotating frame mode frequency). This leaves
open the question of what mode amplitudes would be needed to effectively
perturb a hot-spot located near the rotational pole.
Moreover, the relevant scale height, $h$, will depend on details of the
surface envelopes in question, however, a typical value for the He-rich
envelopes of Piro & Bildsten (2004) is $h\approx 200$ cm, thus, for a 10 km
radius neutron star we would require $N^{2}\gg 2.2\times 10^{10}$ Hz2 in order
to satisfy the assumptions associated with the traditional approximation. We
note that this condition appears to be technically violated for these
envelopes as $N$ is everywhere less than about $1\times 10^{5}$ Hz (see their
Figures 2 and 3). This suggests the need for more theoretical work in order to
more accurately determine the surface g-mode properties for rotation rates
appropriate to the faster spinning AMXPs (such as J1751). In addition, more
work similar to that done in the context of r-modes by Numata & Lee (2010)
should be done for the rotationally modified g-modes to determine how
efficiently these modes can perturb a hot-spot located near the spin axis of
the star and the resulting light curves.
Keeping in mind these caveats, we can nevertheless rearrange Eq. 4 to express
the frequency of the mode in a non-rotating star, $f_{l,0}$, in terms of the
observed mode co-rotating frame frequency, $f_{obs}$, stellar spin frequency,
$\nu_{spin}$ and the mode indices $l$, $m$ and $l_{\mu}$. If the observed
frequency is directly related to the modes co-rotating frame frequency (the
“co-rotating frame scenario”), we find,
$f_{l,0}=(f_{obs}^{2}/(2\nu_{spin}))*\sqrt{(l(l+1)/(2l_{\mu}-1)^{2}}\;.$ (5)
However, if the observed oscillation frequency is the modes inertial frame
frequency (the “inertial frame scenario”), then we must first relate this to
the co-rotating frame via $f_{obs}=m\nu_{spin}-f_{obs,i}$, where $f_{obs,i}$
is the (observed) inertial frame mode frequency, yielding,
$f_{l,0}=((m\nu_{spin}-f_{obs,i})^{2}/(2\nu_{spin}))*\sqrt{(l(l+1)/(2l_{\mu}-1)^{2}}\;.$
(6)
We can then find plausible non-rotating g-modes that can be consistent with
the candidate frequency. Possible identifications are summarized in Table 1
and discussed in more detail below.
From the discussion above we can see that the candidate peak at
$0.5727\times\nu_{spin}=249.33$ Hz in J$1751$ may be identified as an
$l=2,\;m=1\;(l_{\mu}=3)$ g-mode that resides in a helium atmosphere and has a
non-rotating frequency of $\sim 35$ Hz as observed in the co-rotating frame.
This is based on the assumption that this surface mode perturbs the hot-spot
periodically and the candidate frequency is related to the frequency of the
mode in the co-rotating frame. This mode is consistent with the $g_{2}$ mode
given in Table 3 of Strohmayer & Lee (1996) with a period of 29.04 ms and
$\dot{M}/\dot{M}_{Edd}=0.7$ in a pure helium shell. The thermal g-mode
computations discussed above have been done under the assumption of steady-
state nuclear burning in a thermally stable envelope. Now, stable burning of
the accreted material in the envelope requires a high, near-Eddington
accretion rate (Piro & Bildsten, 2004), however, the average accretion rate of
J1751 was about $2.1\times 10^{-11}M_{\odot}$ yr-1 (Markwardt et al., 2002),
which is low relative to the Eddington rate, and therefore the assumption of
steady-state nuclear burning may not be applicable in this case. However, we
note that the relevant accretion rate for the thermal stability calculation is
the local value (per unit area) which might be higher depending on the
accretion geometry, for example, if accretion is restricted to a portion of
the neutron star’s surface.
If we assume that the amplitude of this mode is high enough that it can modify
the temperature distribution at the surface of the star and produce observable
X-ray variations then the inertial frame scenario is relevant (see Eq. 6
above). In this case the candidate oscillation in J1751 may be consistent with
an $l=m=1$ shallow surface wave in the helium layer with a non-rotating
frequency of 18.7 Hz (this is slightly less than the lower limit of 20 Hz
given in Piro & Bildsten (2004)).
Another possibility for the candidate at $249.33$ Hz would be an $l=l_{\mu}=2$
(with $m=0$ or $m=2$) density discontinuity g-mode due to hydrogen electron
captures in the ocean of the star with a non-rotating limit frequency of
$f_{d}\simeq 58.34$ (see Eq. 2) as measured in the co-rotating frame. However,
whether or not sufficient hydrogen is present to support a density
discontinuity mode in such a compact and presumably helium-rich system as
J1751 remains an open question.
Carroll et al. (1986) showed that in the presence of strong magnetic fields
the frequencies and displacements of modes that reside in the ocean, in
particular g-modes, will be modified. For magnetic fields $B_{0}>10^{5}$ G,
these modified g-modes (magneto-gravity modes) change with increasing $B$ from
predominantly g-modes with constant periods to predominantly magnetic modes
with periods proportional to $B_{0}^{-1}$ (see their Eq. 42 and Figure 4).
Piro & Bildsten (2004) estimated the maximum magnetic field before the shallow
surface mode would be dynamically affected to be $B_{dyn}\approx 5\times
10^{7}\rm{G}(\frac{\omega/2\pi}{21.4Hz})$ which is about $6\times 10^{8}$ G
for a rotationally modified shallow surface wave with a frequency of 249.33
Hz. This is close to the estimated value of the magnetic field of J1751
obtained from spin-down measurements due to magneto-dipole radiation which is
about $4\times 10^{8}$ G (Riggio et al., 2011). However, Heng & Spitkovsky
(2009) also explored the effect of a vertical magnetic field on shallow
surface waves, and their results suggest that for the spin rate and likely
magnetic field strength appropriate to J175, the field does not strongly
modify the mode frequencies (see the “magneto-Poincare modes” in their Figure
2). The above results support the conclusion that the magnetic field likely
does not exert a dramatic influence on these g-mode frequencies.
### 4.2. r-modes and inertial modes
As we discussed earlier, another class of non-radial oscillation modes that
may have frequencies consistent with the candidate signal in J$1751$ are the
r-modes. A 3-component neutron star model may have unstable r-modes in the
ocean and/or in the core. The frequency of the r-modes in the slow-rotation
limit ($\bar{\Omega}\equiv\Omega/(GM/R^{3})^{1/2}\rightarrow 0$) is given by
$\omega_{0}=2m\Omega/[l(l+1)]$. As the rotation frequency of the star
increases, the co-rotating frame frequency of the r-modes in the surface layer
of the star deviates appreciably from this asymptotic form and becomes almost
insensitive to $\Omega$ (see Figure 4 in Lee (2004)). According to Table 2 in
Lee (2004) the frequencies of the surface r-modes are always less than 200 Hz
for the spin frequencies and mass accretion rates that are relevant to LMXBs.
For example, for the $l=|m|=2$ fundamental r-modes of radiative envelopes Lee
(2004) found that the co-rotating frequency is in the range of 101 to 173 Hz
for the stellar spin frequencies of 300 to 600 Hz and
$\dot{M}=0.02\dot{M}_{Edd}$ to $0.1\dot{M}_{Edd}$. Lee (2010) also studied the
low frequency oscillations of rotating and magnetized neutron stars and found
no r-modes confined in the ocean in the presence of a magnetic field even as
low as $B_{0}\sim 10^{7}$ G in a 3 component NS model. Thus, the candidate
frequency at 249.33 Hz in J$1751$ doesn’t appear to be consistent with that of
a surface r-mode.
Although the amplitudes of the ocean g- and r-modes tend to be confined to the
equatorial regions, this is not the case for $l=|m|$ r-modes in the fluid
core. In fact Lee (2010) showed that the displacement vector of these core
r-modes have large amplitudes around the rotation axis at the stellar surface
even in the presence of a surface magnetic field $B_{0}\sim 10^{10}$ G.
As we briefly mentioned in the previous sections, the co-rotating frame
frequency of $l=m=2$ core r-modes (which are the most unstable ones) in the
$\bar{\Omega}\rightarrow 0$ limit is equal to $\omega_{0}=\frac{2}{3}\Omega$
which is larger than the frequency of the candidate peak at
$\omega=0.5727\Omega$ and adding the corrections due to high spin rates only
slightly increases the slow-rotation limit value. Yoshida & Lee (2001) studied
the effect of a solid crust on the r-mode oscillations of a three component NS
model. At sufficiently small values of $\Omega$ the coupling between r-modes
and crustal toroidal modes is negligibly weak, but at higher spin frequencies
they found that the core r-modes are strongly affected by the mode coupling
with crustal toroidal modes, and because of the avoided crossings with the
crustal toroidal modes, the core r-modes will lose their simple form of
eigenfrequency and eigenfunction. The r-mode frequency increases as the spin
frequency of the star increases, and at some point it meets the frequency of
the crustal modes which results in avoided crossings. Depending on the
thickness of the crust and therefore the number of modes in the crust with
relevant frequencies, there might be several avoided crossings between core
r-modes and crustal toroidal modes (Levin & Ushomirsky, 2001; Glampedakis &
Andersson, 2006). The spin frequencies at which the avoided crossings occur
are given by $\Omega_{cross}\approx\frac{l(l+1)}{m}\omega_{t}(0)$ where
$\omega_{t}(0)$ is the oscillation frequency of the toroidal mode at
$\Omega=0$ and it is a function of the shear modulus of the crust. As shown in
Figure 3 of Yoshida & Lee (2001), in the presence of a solid crust and at high
rotation frequencies, the r-mode frequency in the co-rotating frame deviates
from its simple form in the $\bar{\Omega}\rightarrow 0$ limit. For fundamental
r-modes with $l=m=2$ they showed that $\kappa$ can decrease from its slow-
rotation limit and span a range of values from $\frac{2}{3}$ to less than
$0.4$ depending on the spin frequency of the star and the properties of the
solid crust, such as its shear modulus. We note that the value of $\mu/\rho$
is almost constant in the crust of a neutron star, $\mu/\rho\simeq 1-6\times
10^{16}$ cm2 s-2 (see for example Figure 1 in Glampedakis & Andersson (2006)).
The results of Yoshida & Lee (2001) given in their Figure 3 and Table 1
suggest that for $\kappa\sim 0.57$ at $\bar{\Omega}\simeq 0.2$ (relevant for
J1751) one needs the shear modulus of the crust to be a few times higher than
the standard values given by Strohmayer et al. (1991) for a bcc crystal at the
higher densities in the crust. This suggests that observations of r-mode
induced oscillations in the X-ray flux of neutron stars could be useful in
probing the structure and properties of the crust.
In addition to the r-modes the Coriolis force also supports the more general
class of inertial modes which have both significant toroidal and spheroidal
angular displacements, whereas the r-modes are principally toroidal. A number
of authors have studied the properties of inertial modes, and in particular
their relationship to other low-frequency modes such as the g-modes (Yoshida &
Lee, 2000a, b; Passamonti et al., 2009; Lee, 2010). For example, Passamonti et
al. (2009) have computed time evolutions of the linear perturbation equations
in order to explore the oscillations of rapidly rotating, stratified (non-
isentropic) neutron stars. They find that the g-modes in stratified stars
become strongly modified by rapid rotation, with each g-mode frequency
approaching that of a particular inertial mode associated with the
corresponding isentropic (ie. no bouyancy) stellar model. Earlier work by
Yoshida & Lee (2000b) reached a similar conclusion, but the more recent
results of Passamonti et al. (2009) have explored the connection to much
higher rotation rates. These studies, as well as the recent calculations of
Lee (2010), all find some inertial modes with co-rotating frame frequencies
that appear at least qualitatively consistent with the candidate oscillation
in J1751. For example, the ${}^{3}i_{1}$ and ${}^{4}i_{2}$ modes of Passamonti
et al. (2009) have frequencies near $\omega=0.573\Omega$ (see their Table 2
and Figures 3 and 11). Note that for their stellar models
$\Omega/(G\rho_{c})^{1/2}\approx 0.5$ is appropriate for the 435 Hz spin
frequency of J1751. Similarly, the $l_{0}-|m|=2$, $m=2$ prograde, isentropic
inertial mode of Yoshida & Lee (2000a), and the non-isentropic modes labelled
$g_{-1}(2)<\-->i_{-1}(2)$ in Figure 9a of Yoshida & Lee (2000b) have
frequencies near to that of our candidate oscillation. It should be noted,
however, that all these calculations have significant simplifications that
likely make detailed quantitative comparisons with our observed frequency
problematic. For example, they all employ rather simplistic stellar models,
such as the use of polytropic equations of state, and the models do not have a
solid crust. Additionally, the calculations of Yoshida & Lee (2000a,b) were
for relatively modest spin rates, and extrapolation to the higher spin rate
appropriate for J1751 is perhaps risky.
In addition, Lee (2010) has presented oscillation mode calculations for
rotating, and magnetized neutron stars using 3-component (ocean, crust, core)
models. He also finds prograde inertial modes with frequencies approximately
consistent with our candidate oscillation (see, for example, the $|m|=2$ modes
for $B_{0}=10^{10}$ G near the lower right corner of Figure 5). These
calculations were for a low mass, $0.5M_{\odot}$, neutron star and are also
only strictly valid for modest rotation rates, so, again, caution should be
exercised when making quantitative comparisons with observed frequencies. We
emphasize that all of the above calculations were for global stellar modes,
and not restricted to only surface displacements. Similarly to the global
r-modes these inertial modes will likely have appreciable surface amplitudes
closer to the rotational poles than the surface-based, rotationally modified
g-modes investigated by Bildsten et al. (1996) and Piro & Bildsten (2004).
Based on the above discussion it seems possible that inertial modes could be
relevant to our candidate oscillation in J1751, but clearly new theoretical
work is needed to explore such modes in more realistic, rapidly rotating
neutron star models before any firm conclusion should be drawn. Further, new
calculations to determine how effectively inertial modes can perturb an X-ray
emitting hot-spot, and the resulting light curves, are certainly warranted.
### 4.3. Coherence of the Candidate Oscillation
The candidate power spectral peak in J$1751$ is narrow, which means that the
oscillation frequency has to be steady over most of the time span used to
compute the power spectrum, which is about 6 days. Thus, if the candidate peak
is due to some non-radial oscillation of the star, its frequency has to be
almost constant during that time span. Between surface g-modes and core
r-modes which might be consistent with the observed candidate peak as
discussed above, r-modes are expected to have steady frequencies over such a
short time span because they reside in the core and conditions there are not
expected to change over such timescales. Among surface g-modes that are
consistent with the candidate oscillation in J$1751$, thermal g-modes of a
helium burning neutron star reside in the shallow layers close to the surface
of the star, but the density discontinuity g-modes due to hydrogen electron
capture reside in deeper layers close to the ocean-crust interface. If the
temperature and elemental composition of the ocean doesn’t change during the
time span used to compute the light curve, the frequency would be steady which
is expected to be the case if the accretion rate varies little. In fact, it
has been shown by Piro & Bildsten (2004) that the g-mode frequency scales
approximately as $\dot{m}^{1/8}$ where $\dot{m}$ is the local accretion rate,
and therefore a small change in the accretion rate will not have a large
effect on the g-mode frequencies.
The light curve of J1751 (see Figure 1) shows variation in the count rate at
the level of 30-40 counts s-1, which likely suggests some variation in the
accretion rate. Although we note that X-ray flux (or count rate) is known to
not always correlate linearly with the accretion rate. While this suggests the
mode frequency may change, a second effect likely limits the rate at which it
can vary, and that is set by the time, $t_{acc}$, required to change
conditions in the surface layers at a column depth where the mode frequency is
set. This can be roughly approximated as $t_{acc}\approx y/\dot{m}$, where $y$
is the relevant column depth in g cm-2, and $\dot{m}$ is the accretion rate
per unit surface area. For an accretion rate of $2\times 10^{-11}M_{\odot}$
yr-1, and a characteristic column depth of $10^{8}$ g cm-2, $t_{acc}$ is about
11.6 days. So, while accretion rate variations can, in principle, change the
g-mode frequencies, for timescales much less than $t_{acc}$ the frequency is
likely reasonably stable.
### 4.4. Future Capabilities and Sensitivities
As can be seen in several of our power spectra (see for example, Figure 7), an
upper limit on the modulation amplitude is approximately given by
$1/(N_{tot})^{1/2}$, where $N_{tot}$ is simply the total number of X-ray
events in the light curve from which the power spectrum is computed. The
approximation is better as one averages more frequency bins, meaning it is a
more precise limit in the context of broader band-width signals. For the full
resolution spectra presented here the derived upper limits are reasonably
approximated as $\approx 4/(N_{tot})^{1/2}$.
This is not too surprising, as the fractional Poisson error on the average
count rate within a time interval is just $1/(N_{tot})^{1/2}$. Thus, this
limit is simply a statement that one cannot measure a fractional modulation
amplitude of the X-ray count rate that is smaller than the precision with
which that rate can be determined. Assuming that other necessary capabilities
are present in future observatories—such as adequate high frequency time
resolution—then a simple way to estimate the amplitude sensitivity for future
detectors is just to scale up the expected count rates appropriately. The
above considerations are valid in the case that the source count rate
dominates any background rate.
The largest effective area for fast X-ray timing presently being planned is
ESA’s Large Observatory for X-ray Timing (LOFT, Feroci et al. 2012). The Large
Area Detector (LAD) on LOFT would consist of $\approx 12$ m2 of silicon
detectors and due to the larger collecting area and better (flatter) response
above 6-7 keV would provide an increase in source count rate compared to the
PCA on RXTE of about a factor of 30 (though the exact scaling would depend on
the X-ray spectrum of the source being considered). The other way to increase
the total counts that can be included in a light curve is to more densely
sample an outburst, and to Fourier analyse longer continuous time intervals.
For the sake of argument, if we scale based on the most sensitive observation
reported here, that is, the single $\approx 6$ day interval for J1751, and
assume that a LOFT observation has twice the duty cycle and extends for twice
as long, then we might expect to reach an amplitude limit of $a_{amp}\approx
1/(2*2*30*44\times 10^{6})^{1/2}=1.4\times 10^{-5}$.
While this represents a limit on the Fourier amplitude of X-ray flux
modulations that could be detected, the corresponding amplitude of an
oscillation mode would depend on the details of how the oscillation mode
perturbs the X-ray emission. Numata & Lee (2010) show from their light curve
modeling that the observed Fourier amplitude is proportional to the normalized
amplitude of the stellar oscillation (see their Figure 6). The details of the
scaling depends on the particular oscillation mode and other details, but a
rough estimate indicates that the Fourier amplitude $a_{amp}\approx 1-2\times
A$, where $A$ represents the maximum horizontal displacement produced by a
mode divided by the stellar radius ($A={\rm
max}(|\xi_{\theta}|/R,|\xi_{\phi}|/R)$). Based on this simple scaling one can
expect that future sensitivities with LOFT would be such that $A\approx
1\times 10^{-5}$ could be probed. We note that this corresponds to a 10 cm
maximum surface displacement for a 10 km neutron star.
In the case of r-mode oscillations, $A$ is approximately equal to $\alpha/2$,
where $\alpha$ is the dimensionless amplitude of the mode, defined in Eq. 1 of
Lindblom et al. (1998). We note that for the candidate oscillation in J$1751$,
$A\approx 7\times 10^{-4}$, and $\alpha\sim 10^{-3}$. This is much larger than
the upper limits on $\alpha$ given in Mahmoodifar & Strohmayer (2013), which
is less than $10^{-7}$ for J1751 (see also Haskell et al. (2012)). A global
r-mode with an amplitude of the order of $10^{-3}$ would cause a rapid spin-
down of the star. Using the corresponding equation for spin-down due to
gravitational wave emission from unstable r-modes (Owen et al., 1998),
$d\Omega/dt\simeq-(2\Omega/\tau_{V})Q\alpha^{2}$, where
$\tau_{V}(T,\Omega,\alpha)$ is the viscous damping timescale of the mode, and
$Q\equiv\frac{3\tilde{J}}{2\tilde{I}}$ (Mahmoodifar & Strohmayer, 2013), gives
a spin-down rate of $\sim-1.3\times 10^{-9}$ Hz s-1 for J1751 assuming a core
temperature of $\sim 3\times 10^{7}K$ (Mahmoodifar & Strohmayer, 2013). We
note that even with a higher core temperature of $\sim 3\times 10^{8}K$ the
spin-down rate would be $\sim-2.8\times 10^{-11}$ Hz s-1, which would still
dominate the accretion spin-up rate and therefore is inconsistent with the
observations (Patruno & Watts, 2012). Further, if the amplitude of the mode is
saturated at $\alpha_{s}\sim 10^{-3}$, $\tau_{V}$ in the spin evolution
equation should be replaced by $\tau_{G}$, where $\tau_{G}$ is the
gravitational radiation timescale. This would cause an even larger spin-down
rate of $\approx-1.5\times 10^{-7}$ Hz s-1. Such a large amplitude for a
global r-mode, even assuming that it is large only during the outburst and
would be damped in quiescence, would cause a large change in the frequency of
J1751 which would be easily detectable in the data. In addition, the maximum
saturation amplitude due to nonlinear mode coupling, computed by Arras et al.
(2003), $\alpha_{s}\approx 8\times 10^{-3}(\nu_{s}/1kHz)^{5/2}$, that in the
case of J1751 is $\sim 6\times 10^{-5}$ (see also Watts et al. (2008a) and
Bondarescu et al. (2007)), and the upper limits on $\alpha$ ($\sim 10^{-4}$)
from gravitational wave searches with LIGO (Owen, 2010; Aasi et al., 2013)
further support the notion that the candidate oscillation is unlikely to be a
global r-mode. This argues that a g-mode or inertial mode interpretation is
more likely. While we think the present evidence is strongly suggestive,
future, more sensitive observations will likely be needed to confirm the
presence of non-radial oscillation modes in J$1751$ and/or other AMXPs.
## 5\. Summary and Conclusions
We have carried out searches for X-ray modulations that could be produced by
global non-radial oscillation modes in several AMXPs. A likely mechanism for
generating X-ray flux modulations is that due to perturbations to the X-ray
emitting hot-spot produced by surface motions associated with the oscillation
modes (see for example, Numata & Lee 2010). In this regard the most relevant
non-radial modes are those with predominantly horizontal displacements at the
stellar surface, such as the inertial modes (which includes the r-modes), and
the g-modes. In order to search most sensitively for nearly coherent
modulations we first remove the Doppler delays due to the binary motion of the
neutron star. We search a range of frequencies–scaled to the stellar spin
frequency–that are theoretically consistent with those expected for the global
r-modes in neutron stars, and this range also encompasses the frequencies
expected for some surface g-modes. We find one plausible candidate signal in
J1751 with an estimated significance of $1.6\times 10^{-3}$, and upper limits
for the two other sources we studied, X-2, and J1814.
Our candidate signal in J1751 appears at a frequency of
$0.5727597\times\nu_{spin}=249.332609$ Hz, has a fractional Fourier amplitude
of $7.455\times 10^{-4}$, and is effectively coherent over the entire light
curve in which it was found. Based on its observed frequency it appears at
least plausible that it could be related to a surface g-mode associated with a
helium-rich layer on the neutron star surface (Piro & Bildsten, 2004). Other
possibilities include a g-mode associated with density discontinuities in the
surface layers (Bildsten & Cumming, 1998), an inertial mode (Passamonti et
al., 2009), or perhaps an r-mode modified by the presence of the neutron star
crust (Yoshida & Lee, 2001).
For J1814 we find an amplitude upper limit to any signal of $\approx 7.8\times
10^{-4}$ (for a coherent signal). For broader bandwidth signals the limit
approaches $\approx 2.2\times 10^{-4}$. In the case of X-2, because less data
is available for this source, the limits are less constraining, and we find
values of $5.6\times 10^{-3}$ and $2.8\times 10^{-3}$ at frequency resolutions
of $3.125\times 10^{-4}$ and $0.01$ Hz, respectively.
We thank Tony Piro, Andrew Cumming, Jean in ’t Zand, Cole Miller, and Diego
Altamirano for many helpful comments and discussions. We thank the anonymous
referee for valuable comments that helped us improve this paper. TS
acknowledges NASA’s support for high energy astrophysics. SM acknowledges the
support of the U.S. Department of Energy through grant number DEFG02-
93ER-40762.
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Table 1 Possible g-mode identifications
$l$ | $m$ | $l_{\mu}$ | $\omega_{l,0}/(2\pi)$ | Consistent with | $\omega_{l,0}/(2\pi)$ | Consistent with
---|---|---|---|---|---|---
| | | rotating frame | | inertial frame |
| | | scenario | | scenario |
1 | 0 | 1 | $101.1$ | | 101.1 |
1 | 1 | 2 | $33.7$ | Thermal g-mode (helium atmosphere) | 18.7 | Thermal g-mode (helium atmosphere)
2 | -1 | $\sim 1$ | $175.0$ | | 1318.5 |
2 | 0 | 2 | $58.3$ | Density discontinuity g-mode | 58.3 | Density discontinuity g-mode
2 | 1 | 3 | $35.0$ | Thermal g-mode (helium atmosphere) | 19.4 |
| | | | Density discontinuity g-mode | |
2 | 2 | 2 | $58.3$ | Density discontinuity g-mode | 361.5 |
Figure 1.— Light curve in the 2 - 60 keV band from XTE J1751-305 used in our
pulsation search. These data span the brightest portion of the outburst onset.
Time zero is 2002 Apr 05 at 15:29:03.422 UTC. Note that the background level
of $\approx 15$ counts s-1 PCU-1 has not been subtracted. Figure 2.— Dynamic
power spectrum (Leahy-normalized) of XTE J1751-305 as a function of time and
barycentric frequency in a single RXTE orbit. The contours show levels of
Leahy-normalized Fourier power and track the binary Doppler-shifted pulsar
spin frequency. The solid curve is the best-fitting orbit model for this data
interval. The origin for the time axis has the same reference as Figure 1.
Figure 3.— Pulse timing phase residuals (in cycles) for XTE J1751-305 after
application of the best fitting circular orbit model. The remaining phase
residuals are poisson dominated. Time zero is the same as in Figure 1. Figure
4.— A portion of the full frequency resolution, coherent power spectrum for
XTE J1751-305 in the vicinity of the pulsar spin frequency (at 0 in these
units). The power spectrum is shown in units of Fourier amplitudes (see the
text in §2 for further details). The side-lobe pattern of peaks results from
the uneven temporal sampling (the window function). Figure 5.— A portion of
the full frequency resolution, coherent power spectrum for XTE J1751-305 in
the vicinity of the candidate signal peak at $0.57276\times\nu_{spin}$. The
spectrum is plotted in units of fractional Fourier amplitude. Figure 6.—
Probability to exceed a given Leahy-normalized Fourier power in a single
trial. The red squares show the expected noise-power distribution (in this
case a $\chi^{2}$ distribution with 2 degrees of freedom). The solid histogram
shows the observed power-spectral distribution for XTE J1751-305 in the
frequency range from 1.6 to 2.2 $\times$ the pulsar spin frequency. The
vertical dashed line marks the power value of the candidate signal peak. The
data track the expected distribution over a broad range of power values. An
exact match at the highest power values is not expected simply due to
statistical fluctuations. Figure 7.— Frequency-averaged power spectra of XTE
J1751-305 plotted in units of fractional Fourier amplitude. Spectra averaged
to 1/2048 (black) and 1/128 (green) Hz are shown. The X-axis shows frequency
scaled by the pulsar spin frequency. The pulsar signal at 1 is clearly
evident, but there are no other significant features evident at either
resolution. The horizontal dashed red line marks the amplitude given by
$1/\sqrt{N_{tot}}$, where $N_{tot}$ is the total number of counts in the light
curve. The horizontal dashed line marks the amplitude of the candidate signal
peak at $0.57276\times\nu_{spin}$. Figure 8.— Pulse timing phase residuals
(in cycles) for XTE J1814-338 (first time interval) after application of the
best fitting circular orbit model. Time zero is 2003 June 5 at 02:34:20 UTC.
Figure 9.— Pulse timing phase residuals (in cycles) for XTE J1814-338 (second
time interval) after application of the best fitting circular orbit model.
Time zero is 2003 June 11 at 04:12:28 UTC. Figure 10.— Frequency-averaged
power spectra of XTE J1814-338 (average of both data intervals analyzed)
plotted in units of fractional Fourier amplitude. Spectra averaged to 1/2048
(black) and 1/128 (green) Hz are shown. The x-axis shows frequency scaled by
the pulsar spin frequency. The pulsar fundamental and first harmonic are
clearly evident, but there are no other significant features evident at either
resolution. The horizontal dashed red line marks the amplitude given by
$1/\sqrt{N_{tot}/2}$, where $N_{tot}$ is the total number of counts in the
light curve. The horizontal dashed line (black) marks the upper limit on the
amplitude at the full frequency resolution. Figure 11.— Frequency-averaged
power spectra of NGC 6440 X-2 plotted in units of fractional Fourier
amplitude. Spectra at the full frequency resolution (1/3200 Hz, black) and
0.01 Hz (red) are shown. The x-axis shows frequency scaled by the pulsar spin
frequency. The pulsar signal at 1 is clearly evident, but there are no other
significant features evident at either resolution. The horizontal dashed red
line marks the amplitude given by $1/\sqrt{N_{tot}/4}$, where $N_{tot}$ is the
total number of counts in the four light curves analyzed.
|
arxiv-papers
| 2013-10-18T20:00:02 |
2024-09-04T02:49:52.576788
|
{
"license": "Public Domain",
"authors": "Tod Strohmayer and Simin Mahmoodifar",
"submitter": "Tod E. Strohmayer",
"url": "https://arxiv.org/abs/1310.5147"
}
|
1310.5190
|
# Neutral gas sympathetic cooling of an ion in a Paul trap
Kuang Chen, Scott T. Sullivan, and Eric R. Hudson
###### Abstract
A single ion immersed in a neutral buffer gas is studied. An analytical model
is developed that gives a complete description of the dynamics and steady-
state properties of the ions. An extension of this model, using techniques
borrowed from the mathematics of finance, is used to explain the recent
observation of non-Maxwellian statistics for these systems. Taken together,
these results offer an explanation of the longstanding issues associated with
sympathetic cooling of an ion by a neutral buffer gas.
The fact that two isolated objects in thermal contact tend to the same
temperature is the most basic tenet of thermodynamics. It is also the essence
of the technique of sympathetic cooling, where a sample is prepared at a
desired temperature by bringing it into thermal contact with a much larger
body already at the desired temperature. It is difficult to overstate the
importance of this technique as it underpins applications ranging from basic
refrigeration to quantum information science.
It may be considered surprising then that a gas of ions trapped in a radio-
frequency Paul trap and immersed in a reservoir of neutral atoms, does not
equilibrate to the same temperature as the neutral atoms. Instead, the ions
are found to have a higher temperature than the neutral gas, and in some cases
are heated so much that they escape the trap. Since the early work of Major
and Dehmelt Major and Dehmelt (1968) it has been known that this apparent
contradiction with the laws of thermodynamics is due to the fact that ions are
subject to a time-dependent confining potential and are therefore not an
isolated system. However, despite pioneering work by Dehmelt and others
Moriwaki et al. (1992); Vedel et al. (1983), an accurate analytical
description of the relaxation process has not yet been achieved. Given the
recent surge in interest in hybrid atom-ion systems Grier et al. (2009);
Zipkes and otheres (2010); Zipkes et al. (2010); Hall et al. (2011);
Rellergert et al. (2011, 2013); Sullivan et al. (2012); Schmid et al. (2010);
Ratschbacher et al. (2012, 2013), where ions are immersed in baths of
ultracold atoms, there is currently a strong need for such a description so
that these systems can be understood and optimized.
Building upon the important work of Moriwaki et al. Moriwaki et al. (1992),
here we present a simple kinematic model, which accurately describes the ion
relaxation process. This model, which has been verified by detailed molecular
dynamics simulations, provides a simple and accurate means to calculate both
the relaxation dynamics and the properties of the ion steady state. This model
also provides significant physical intuition for the problem and as such
suggests several ways for optimizing ongoing and planned experiments in fields
as diverse as quantum chemistry Grier et al. (2009); Zipkes and otheres
(2010); Zipkes et al. (2010); Hall et al. (2011); Rellergert et al. (2011,
2013); Sullivan et al. (2012); Schmid et al. (2010); Ratschbacher et al.
(2012, 2013), mass spectrometry Drewsen et al. (2004), and quantum information
Hudson (2009).
In the remainder of this work, we first review the basics of ion trapping and
introduce the time-averaged ion kinetic energy. We then consider the effect of
a collision with a neutral particle on the evolution of the kinetic energy of
a single ion in a Paul trap and show that due to the presence of the time-
dependent potential the collision center-of-mass frame energy is not
conserved. Following this result, we develop a rate equation model, which
accounts for the relaxation and exchange of the ion energy in all three
dimensions. We then present simple formulae for the calculation of the ion
temperature relaxation rate and steady-state value, as well as the dependency
of these values on the ion trapping parameters and particle masses. We
establish the validity of these results by comparing them to a detailed
molecular dynamics simulation. We conclude with an explanation for the recent
observation DeVoe (2009) of non-Maxwellian distribution functions for these
systems.
Ion trap dynamics – The trajectory, $r_{j}$, and velocity, $v_{j}$, of an ion
in a linear Paul trap can be expanded as a linear superposition of two
orthogonal Mathieu functions $c(a_{j},q_{j};\tau)$ and $s(a_{j},q_{j};\tau)$
with coefficients $A_{j}$ and $B_{j}$,
$\displaystyle r_{j}(\tau)$
$\displaystyle=A_{j}~{}c_{j}(\tau)+B_{j}~{}s_{j}(\tau)$ (1) $\displaystyle
v_{j}(\tau)$
$\displaystyle=A_{j}~{}\dot{c}_{j}(\tau)+B_{j}~{}\dot{s}_{j}(\tau)$
where $j=x,y,z$ and the dependence on the Mathieu parameters
($\\{a_{x},a_{y},a_{z}\\}=\\{-a,-a,2a\\}$ and
$\\{q_{x},q_{y},q_{z}\\}=\\{q,-q,0\\}$ with
$q=\frac{4eV_{rf}}{mr_{0}^{2}\Omega^{2}}$ and $a=\frac{4\alpha
eU_{ec}}{mz_{0}^{2}\Omega^{2}}$) is suppressed Major and Dehmelt (1968). The
Fourier transform of $c_{j}(\tau)$ and $s_{j}(\tau)$ is a discrete spectrum,
$c_{j}(\tau)+\imath
s_{j}(\tau)=\sum_{n=-\infty}^{\infty}C_{2n}e^{\imath(\beta_{j}+2n)\tau}.$ (2)
The $n=0$ term corresponds to the ‘typical’ motion of a harmonic oscillator –
i.e. the secular ion motion. The remaining terms with $n\neq 0$ represent the
components of the ion motion driven by the rf field – i.e. the so-called
micromotion.
As a result of this spectrum, the instantaneous kinetic energy is not a
conserved quantity. Instead, energy coherently flows back and forth between
the kinetic energy of the ion and the confining electric field at frequency
$\Omega$. Therefore, it is useful to define the time-averaged kinetic energy
$W_{j}=\frac{m}{2}\lim_{T\rightarrow\infty}\frac{1}{2T}\int_{-T}^{T}v_{j}^{2}d\tau=\frac{m}{2}\overline{\dot{c}_{j}^{2}}(A_{j}^{2}+B_{j}^{2}),$
(3)
where the bar denotes the time average. $W_{j}$ includes contributions from
both the random thermal motion of the ion, i.e. the secular energy, and the
micromotion. The ratio of the secular energy, $U_{j}$, to the total average
kinetic energy is simply
$\eta_{j}\equiv\frac{U_{j}}{W_{j}}=\frac{|C_{0}|^{2}}{\sum_{n=-\infty}^{\infty}|C_{2n}|^{2}}.$
(4)
In the $x$ and $y$ directions, $\eta_{x,y}\approx\frac{1}{2}$ for $q<0.4$ and
the micromotion energy is given by $W_{mm,j}=W_{j}-U_{j}$. In the $z$
direction where the trapping field is time-independent ($q=0$), $c_{z}(\tau)$
and $s_{z}(\tau)$ simply become the cosine and sine functions. Thus, all
micromotion sidebands vanish and $\eta_{z}=1$.
Modeling the collision process – When a trapped ion is immersed in a buffer
gas of neutral atoms, the Mathieu trajectory of the ion is modified by
interactions with the neutral atoms. The ion-neutral interaction potential is
comprised of a long-range attraction $V(r)=-C_{4}/2r^{4}$ and short-range
repulsion, where $C_{4}$ is given by $C_{4}=\alpha
e^{2}/(4\pi\epsilon_{0})^{2}$, and $\alpha$ is the polarizability of the
neutral atom. Recent work Cetina et al. (2012), has explored effects of this
potential at ultracold temperatures, showing that the perturbations of the ion
trajectory by the $C_{4}$ potential can lead to heating of the ion. Here we do
not consider this effect, but given that the characteristic length of the
$C_{4}$ interaction Gao (2010) is small compared to the trap dimension we
treat the collision as a point-like interaction. As will be seen, this
approximation is justified, despite the important result of Ref. Cetina et al.
(2012), as the effects considered here typically lead to temperatures that
preclude the observation of the effects considered in Ref. Cetina et al.
(2012). We also make the additional simplifying assumptions that the density
of the neutral atoms is constant and that inelastic processes, such as charge
exchange, do not occur.
Because the motion of the ion differs significantly in the radial and axial
directions of a linear Paul trap, the relaxation and redistribution of energy
is significantly more complicated than in a time-independent harmonic trap
DeCarvalho et al. (1999). We therefore describe the statistically-averaged
evolution of ion kinetic energy $\mathbf{W}=[W_{x},W_{y},W_{z}]^{\mathrm{T}}$
by a three-dimensional rate equation,
$\frac{\text{d}\langle\mathbf{W}(t)\rangle}{\text{d}t}=-\Gamma\mathbf{M}(\langle\mathbf{W}(t)\rangle-\mathbf{W}_{st})$
(5)
where $\Gamma$ is an average collision rate (which may depend on energy),
$\mathbf{M}$ is a 3$\times$3 “relaxation matrix” that accounts for energy
damping and redistribution among the three trap directions, and
$\mathbf{W}_{st}$ is the steady-state kinetic energy. The angled bracket
denotes the statistical average after the sympathetic cooling experiment is
repeated multiple times.
In order to calculate both $\Gamma$ and $\mathbf{M}$ it is necessary to know
the neutral-ion differential elastic scattering cross-section
$\text{d}\sigma_{el}/\text{d}\Omega$, which, given an interaction potential,
is a straightforward quantum scattering calculation Friedrich (2005).
Regardless of the specific atom-ion potential, however, several generic
arguments can be made. First, the differential cross-section always exhibits a
large forward scattering peak at all energy scales Zhang et al. (2009). Thus,
the majority of atom-ion collisions lead to only slightly deflected
trajectories, resulting in a very small change in $\mathbf{W}$. Therefore, as
originally argued by Dalgarno and co-workers Dalgarno et al. (1958), to
prevent an overestimate of the energy redistribution due to collisions the
momentum transfer (diffusion) differential cross-section, i.e.
$\frac{d\sigma_{d}}{d\Omega}=\frac{d\sigma_{el}}{d\Omega}(1-\cos\theta)$
should be used to calculate the total atom-ion collision rate. Second (and
fortuitously), the diffusion differential cross-section is approximately
isotropic in scattering angle, especially after thermal averaging, and agrees
quite well with the simple Langevin cross-section Langevin (1905)
$\sigma_{d}\approx\sigma_{L}=\pi\sqrt{\frac{2C_{4}}{E}}$ – see Appendix A for
a comparison of a quantum scattering calculation to the Langevin differential
cross section. Therefore, we replace the cross-section by an isotropic profile
which integrates to $\sigma_{L}$. Under this approximation, the average
collision rate $\Gamma=2\pi\rho\sqrt{\frac{C_{4}}{\mu}}$ becomes energy
independent and the calculation of $\mathbf{M}$ is greatly simplified. As
demonstrated below, the validity of this approximation is confirmed by
comparison to a detailed molecular dynamics simulation, which uses the full
quantum differential cross-section. The resulting error in the relaxation rate
is smaller than 25% for collision energies down to 1 mK.
With the collision rate in hand, the relaxation matrix $\mathbf{M}$ is
calculated by considering the kinematics of a collision between an ion and
neutral atom as follows. Suppose that at time $\tau_{c}$ an ion undergoes an
elastic collision with an incoming neutral atom of mass $m_{n}$ and velocity
$\mathbf{v}_{n}$. Conservation of momentum and energy for the collision
dictates that the velocity of the ion after the collision with neutral atom is
given by the sum of center-of-mass velocity and the scattered relative
velocity Zipkes et al. (2011),
$\mathbf{v}^{\prime}=\frac{1}{1+\tilde{m}}\mathbf{v}+\frac{\tilde{m}}{1+\tilde{m}}\mathbf{v}_{n}+\frac{\tilde{m}}{1+\tilde{m}}\mathcal{R}(\mathbf{v}-\mathbf{v}_{n})$
(6)
where $\tilde{m}=\frac{m_{n}}{m_{i}}$ is the mass ratio and $\mathcal{R}$ is
the collision rotation matrix, which following the above discussion is
isotropic. Likewise, because the characteristic length of the $C_{4}$
interaction Gao (2010) is small compared to the trap dimension, the position
of the ion is assumed to be unchanged during the collision, i.e.
$\mathbf{r}^{\prime}=\mathbf{r}$. By requiring that $\mathbf{r}^{\prime}$ and
$\mathbf{v}^{\prime}$ also correspond to a Mathieu solution through Eq. 1, a
new set of oscillation amplitude $(A_{j}^{\prime},B_{j}^{\prime})$ and thus,
the average kinetic energy after the collision $\mathbf{W}^{\prime}$ can be
found.
This last step is the critical difference between sympathetic cooling in
static and time-dependent traps, which is illustrated with the following one-
dimensional example. In a static trap, like that in Ref. Campbell et al.
(2007), if a collision happens at position $x=a$ that reduces the velocity
such that $v_{x}^{\prime}=0$, a trapped particle of mass $m$ begins a ‘new’
oscillation trajectory, $x^{\prime}=a\cos(2\pi\sqrt{k/m}~{}t)$, where $k$ is
the trap spring constant. This collision always reduces the total energy of
the particle. By contrast in the time-dependent potential of a linear Paul
trap, because of the terms in Eq. 2 with $n\neq 0$, it is possible that even
though the collision brings the particle to rest, the particle may have a
higher energy after the collision.
This can be seen by again considering a collision that leads to
$v_{x}^{\prime}=0$, which depending on the rf phase could be accomplished by
having large and opposite contributions to the velocity from the $n=0$
(secular) mode and $n\neq 0$ (micromotion) modes. Thus, even though the
particle is momentarily stopped, it could leave the collision on a trajectory
of higher amplitude.
With this prescription the calculation of $\mathbf{M}$ is straightforward and
proceeds as follows (see Appendix B for full details). First we rewrite Eq. 3
in terms of the instantaneous coordinates for the $x$ direction and find the
change in $W_{x}$ per collision as:
$\displaystyle W_{x}^{\prime}-W_{x}$
$\displaystyle=-\frac{m\overline{\dot{c}_{x}^{2}}}{w_{0x}^{2}}(c_{x}\dot{c}_{x}+s_{x}\dot{s}_{x})(x(v_{x}^{\prime}-v_{x}))$
(7)
$\displaystyle\;\;\;\;+\frac{m\overline{\dot{c}_{x}^{2}}}{2w_{0x}^{2}}(c_{x}^{2}+s_{x}^{2})(v_{x}^{\prime
2}-v_{x}^{2})$ $\displaystyle\equiv\Delta W_{x,1}+\Delta W_{x,2}.$
Then we take the statistical average of Eq. 7 over
$\mathbf{v_{n}},\mathcal{R}$ and collision time $\tau_{c}$. Since both
$\langle\mathbf{v_{n}}\rangle$ and
$\langle\mathcal{R}(\mathbf{v}-\mathbf{v_{n}})\rangle$ vanish, $\langle
v_{x}^{\prime}\rangle=\frac{1}{1+\tilde{m}}v_{x}$, and $\langle\Delta
W_{x,1}\rangle=\frac{\tilde{m}}{1+\tilde{m}}\epsilon_{x}\langle W_{x}\rangle$,
where
$\epsilon_{x}=\frac{\overline{(c_{x}\dot{c}_{x}+s_{x}\dot{s}_{x})^{2}}}{w_{0x}^{2}}$.
Likewise, noting that since $\mathbf{v_{n}}$, $\mathbf{v}$ and $\mathcal{R}$
are uncorrelated the average value of cross-correlation terms between them
vanish and that $\mathcal{R}$ is random rotation,
$\langle[\mathcal{R}(\mathbf{v}-\mathbf{v_{n}})]_{x}^{2}\rangle=\frac{1}{3}\langle(\mathbf{v}-\mathbf{v_{n}})^{2}\rangle$,
we have
$\displaystyle\langle\Delta
W_{x,2}\rangle=\frac{\tilde{m}}{(1+\tilde{m})^{2}}\Big{(}\Big{(}-\frac{2\tilde{m}+2}{3}\Big{)}(1+\epsilon_{x})\langle
W_{x}\rangle$ $\displaystyle+\frac{\tilde{m}\alpha_{x}}{6}\langle
W_{y}\rangle+\frac{\tilde{m}\alpha_{x}}{6}\langle
W_{z}\rangle+\alpha_{x}\langle W_{n}\rangle\Big{)},$
where
$\alpha_{x}=\frac{\overline{(c_{x}^{2}+s_{x}^{2})}\cdot\overline{(\dot{c}_{x}^{2}+\dot{s}_{x}^{2})}}{w_{0,x}^{2}}$,
and $\langle W_{n}\rangle$ is the average kinetic energy of neutral atom in
each direction.
Combining the results of $\langle\Delta W_{x,1}\rangle$ and $\langle\Delta
W_{x,2}\rangle$, and the results for the $y$ and $z$ directions, finally we
have
$\displaystyle\langle\mathbf{W^{\prime}}\rangle-\langle\mathbf{W}\rangle$
$\displaystyle=-\mathbf{M}\langle\mathbf{W}\rangle+\mathbf{N}$
$\displaystyle=-\mathbf{M}(\langle\mathbf{W}\rangle-\mathbf{W}_{st})$ (8)
where
$\mathbf{M}=-\frac{\tilde{m}^{2}}{(1+\tilde{m})^{2}}\begin{bmatrix}\frac{2\epsilon-1}{3}-\frac{1}{\tilde{m}}&\frac{\alpha}{6}&\frac{\alpha}{6}\\\
\frac{\alpha}{6}&\frac{2\epsilon-1}{3}-\frac{1}{\tilde{m}}&\frac{\alpha}{6}\\\
\frac{1}{6}&\frac{1}{6}&-\frac{1}{3}-\frac{1}{\tilde{m}}\\\ \end{bmatrix}$ (9)
and
$\mathbf{N}=\frac{\tilde{m}}{(1+\tilde{m})^{2}}\begin{bmatrix}\alpha\langle
W_{n}\rangle\\\ \alpha\langle W_{n}\rangle\\\ \langle
W_{n}\rangle\end{bmatrix}.$ (10)
And, the components of the steady-state kinetic energy
$\mathbf{W}_{st}=-\mathbf{M}^{-1}\mathbf{N}$ reduce to,
$\displaystyle\frac{W_{st,x}}{\langle W_{n}\rangle}=\frac{W_{st,y}}{\langle
W_{n}\rangle}$
$\displaystyle=\frac{9(2+\tilde{m})\alpha}{18-3\tilde{m}(\alpha+4\epsilon-4)-2\tilde{m}^{2}(\alpha+2\epsilon-1)}$
(11) $\displaystyle\frac{W_{st,z}}{\langle W_{n}\rangle}$
$\displaystyle=\frac{3(6+\tilde{m}(2+\alpha-4\epsilon))}{18-3\tilde{m}(\alpha+4\epsilon-4)-2\tilde{m}^{2}(\alpha+2\epsilon-1)}$
where $\alpha\equiv\alpha_{x}=\alpha_{y}$ and
$\epsilon\equiv\epsilon_{x}=\epsilon_{y}$. Because in the $z$ direction the
trapping field is time-independent, $\alpha_{z}=1$ and $\epsilon_{z}=0$. For
low values of $q$ and $a$, the numerical values of $\alpha$ and $\epsilon$ are
approximated by Chen and otheres (2013),
$\displaystyle\alpha\approx 2+2q^{2.24}$ (12) $\displaystyle\epsilon\approx
1+2.4q^{2.4}$ (13)
Figure 1: (a) $\mathbf{W}_{st}$ as a function of $\tilde{m}$ for $q=0.14$
(red) and $q=0.42$ (blue). The axial and radial components of
$\mathbf{W}_{st}$ are denoted by dashed and solid lines (theory) and dots
(simulation). (b) Eigenvalues of $\mathbf{M}$ as a function of $\tilde{m}$ for
fixed $q=0.14$ and $a=0$. Black dots are asymptotic relaxation rates
(normalized by $\Gamma$) from numerical simulations. Lines are three
calculated eigenvalues of $\mathbf{M}$. The smallest one (blue line)
intersects $\lambda=0$ line at $\tilde{m}=\tilde{m}_{c}$, which separates
cooling from heating. (c) Simulated (dots) and calculated (blue line) critical
mass ratio $\tilde{m}_{c}$ as a function of trap $q$ parameter, as compared to
previous results in Ref. Moriwaki et al. (1992); Major and Dehmelt (1968);
DeVoe (2009).
Model results – First, shown in Fig. 1(a) are the components of
$\mathbf{W}_{st}$ normalized by $\langle W_{n}\rangle$ obtained from Eq. 11.
Also, shown in this figure are the results of a detailed molecular dynamics
simulation, described in Appendix C. In the limit of a light neutral atom
($\tilde{m}\approx 0$) and $q\rightarrow 0$, $\alpha\approx 2$,
$\mathbf{W}_{st}/\langle W_{n}\rangle\approx[2,2,1]^{\mathrm{T}}$. Thus, at
steady state,
$\langle U_{x}\rangle=\langle U_{y}\rangle=\langle U_{z}\rangle=\langle
W_{mm,x}\rangle=\langle W_{mm,y}\rangle=\langle W_{n}\rangle,$ (14)
a result often referred to as the “equipartition” Baba et al. (2002) of
kinetic energy between secular motion and micro-motion. As $\tilde{m}$
increases, the steady-state secular energy deviates from equipartition and
becomes much higher than $W_{n}$. As $q$ increases, this deviation becomes
significant more quickly.
Second, the solution to Eq. 5 is linear combination of three fundamental
relaxation processes, whose rates are determined by the three eigenvalues of
$\mathbf{M}$. The asymptotic behavior of the energy evolution is governed by
the slowest relaxation rate, $\Gamma\lambda$, where $\lambda$, the smallest
eigenvalue of $\mathbf{M}$, is
$\lambda=\frac{\tilde{m}}{(1+\tilde{m})^{2}}\left(1-\frac{\tilde{m}}{\tilde{m}_{c}}\right)$
(15)
and $\tilde{m}_{c}$ is the critical mass ratio given in terms of trap
parameters as,
$\tilde{m}_{c}=\frac{3(4-\alpha-4\epsilon+\sqrt{\alpha^{2}+8\alpha(1+\epsilon)+16\epsilon^{2}})}{4(2\epsilon+\alpha-1)}$
(16)
The eigenvalues of $\mathbf{M}$ are shown in Fig. 1(b) and are compared to the
asymptotic relaxation rates observed in the simulation. For
$\tilde{m}\ll\tilde{m}_{c}$, the cooling rate from Eq. 15 is similar to the
traditional sympathetic cooling result up to a numerical factor DeCarvalho et
al. (1999). In this regime, the initial positive slope of $\lambda$ results
from enhanced energy transfer efficiency through collisions with neutral atoms
of similar mass. However, the additional factor
$1-\frac{\tilde{m}}{\tilde{m}_{c}}$ causes $\lambda$ to reach a maximum and
decrease to negative values once $\tilde{m}$ exceeds $\tilde{m}_{c}$. At this
point, it is observed in the simulation that oscillation amplitude of the ion
grows with collisions, until the ion becomes too energetic to be trapped,
regardless of the energy of the buffer gas.
The transition from sympathetic cooling to heating by a buffer gas is thus
defined by $\tilde{m}=\tilde{m}_{c}$ and is shown in Fig. 1(c) as a function
of $q$ along with the results of the molecular dynamics simulations and
previous results from other models of the process Major and Dehmelt (1968);
Moriwaki et al. (1992); DeVoe (2009). Taken together the results of Figs.
1(a)-1(c), make the case for using as small a buffer gas mass and as low $q$
as possible, if significant sympathetic cooling is desired.
Non-Maxwellian statistics in an ion trap – As originally observed in the
seminal work of DeVoe DeVoe (2009), the peculiarity of sympathetic cooling in
ion trap is also manifested in the steady-state energy distribution of the
ion, which features a heavy power-law tail due to the random amplifications of
the ion energy by collisions. To gain a quantitative understanding of how this
distribution arises, consider a simplified model, in which the motion of the
ion and neutral atom’ are restricted to one dimension, and $\mathcal{R}=-1$ in
Eq. 6. In $(A,B)$ space, collisions result in a random walk given by
$\begin{bmatrix}A_{N+1}\\\
B_{N+1}\end{bmatrix}=\left(\mathbf{I}+\frac{\zeta}{w_{0}}\begin{bmatrix}s\dot{c}&s\dot{s}\\\
-c\dot{c}&c\dot{s}\end{bmatrix}_{\tau_{N}}\right)\begin{bmatrix}A_{N}\\\
B_{N}\end{bmatrix}+\frac{\zeta v_{n}}{w_{0}}\begin{bmatrix}s\\\
c\end{bmatrix}_{\tau_{N}}$ (17)
where $\zeta=\frac{2\tilde{m}}{1+\tilde{m}}$, $[A_{N+1},B_{N+1}]^{\mathrm{T}}$
are the coordinates after the $N$-th collision which occurs at $\tau=\tau_{N}$
($N=1,2,\cdots,\infty$). The $\tau_{N}$ constitute an array of Poissonian
variables, with average interval equal to $\Gamma^{-1}$. As can be seen from
Eq. 17, the random walk in $(A,B)$ space has both additive and multiplicative
terms. As is well known in finance Sornette et al. (2001), the multiplicative
terms in the random walk give rise to the power law distribution as follows.
A recurrence relation for $W_{N}$ can be derived from Eq. 3, and if only the
distribution of high energy ions, i.e. $W_{N}\gg W_{n}$ is considered, this
relation reduces to
$W_{N+1}=CW_{N},$ (18)
where the multiplicative coefficient $C$ is given by,
$\displaystyle C(\tau_{N},$
$\displaystyle\theta_{N})=\cos^{2}\theta_{N}\left(\left(1+\zeta\frac{s\dot{c}}{w_{0}}\right)^{2}+\zeta^{2}\frac{c^{2}\dot{c}^{2}}{w_{0}^{2}}\right)_{\tau_{N}}$
(19) $\displaystyle+$
$\displaystyle\sin^{2}\theta_{N}\left(\left(1-\zeta\frac{c\dot{s}}{w_{0}}\right)^{2}+\zeta^{2}\frac{s^{2}\dot{s}^{2}}{w_{0}^{2}}\right)_{\tau_{N}}$
$\displaystyle+$ $\displaystyle
2\sin\theta_{N}\cos\theta_{N}\left(\zeta\frac{s\dot{s}-c\dot{c}}{w_{0}}+\zeta^{2}\frac{\dot{c}\dot{s}(c^{2}+s^{2})}{w_{0}^{2}}\right)_{\tau_{N}}$
and $\theta_{N}=\arctan(B_{N}/A_{N})$. Because $W$ only depends on
$A^{2}+B^{2}$, it is expected that as $N\rightarrow\infty$, $\theta_{N}$
becomes uniformly distributed in the range of $[0,2\pi]$ and uncorrelated with
$\tau_{N}$. $Q(C)$, the probability density of $C$, is calculated from Eq. 19
and exhibits random amplification of the ion energy, i.e. $C>1$, as shown in
Fig. 2 panels $(a)$ and $(c)$ for different values of $\tilde{m}$ and $q$.
Due to this random amplification, $W$ develops a power-law tail in its
probability density at steady state, i.e. $P(W)\propto W^{-(\nu+1)}$ Takayasu
et al. (1997). Self-consistency requires that $P(W_{N+1})$, is equal to the
product of $P(W_{N})$ and $Q(C)$, under the constraint of Eq. 18, namely,
$\displaystyle P(W_{N+1})$ $\displaystyle=\iint
Q(C)P(W_{N})\delta(W_{N+1}-CW_{N})\;\mathrm{d}C\;\mathrm{d}W_{N}$ (20)
$\displaystyle=\int
Q(C)P\left(\frac{W_{N+1}}{C}\right)\frac{1}{C}\;\mathrm{d}C.$
Assuming $P(W)\propto W^{-(\nu+1)}$ then the power $\nu$ must satisfy
$\langle C^{\nu}\rangle=1.$ (21)
From this condition, $\nu$ can be found numerically and Fig. 2, panels $(b)$
and $(d)$, compare the prediction to the energy distribution extracted from a
molecular dynamics simulation, which subjects the ion to $10^{6}$ trials, in
each of which the ion undergoes $10^{4}$ collisions, for each $\tilde{m}$ and
$q$ parameter. As $\tilde{m}$ and $q$ increase random amplification becomes
more likely, causing the energy distribution to become more non-Maxwellian. In
comparison, there is no such random amplification from collisions in a static
trap (see Appendix D for details).
By considering the value of $\nu$ as $\tilde{m}\rightarrow 0$ and
$\tilde{m}\rightarrow\infty$, we find that the power can be approximated as
$\nu_{\text{1D}}\approx 1.67/\tilde{m}-0.67$ in 1D (see Appendix E). To extend
the above discussion to a full 3D model, $C$ necessarily becomes a $3\times 3$
stochastic matrix, and the theory of stochastic matrix products Kesten (1973),
which is beyond the current scope, must be considered. Nonetheless, one
expects $\zeta_{\text{3D}}\approx\frac{1}{2}\zeta_{\text{1D}}$ because in 3D
$\mathcal{R}$ average to zero, thus $\nu_{\text{3D}}\approx 2\nu_{\text{1D}}$,
which agrees reasonably well with the empirically extracted power law of
DeVoe, $\nu_{\text{emp}}\approx 4/\tilde{m}-1$.
Figure 2: Probability density of the multiplicative noise $Q(C)$ and
corresponding ion’s energy $P(W)$ for 1D model from simulations for fixed
$q=0.23$ (lines in panel $a$ and $b$), and fixed $\tilde{m}=0.23$ (dots in
panel $c$ and $d$). The tail of $P(W)$ is fitted to the power-law form of
$W^{-(\nu+1)}$ (solid line in panel $c$ and $d$), where $\nu$ is given by Eq.
21.
In summary, we have developed an analytical model that accurately predicts the
steady state value and dynamics of the kinetic energy of a singe ion immersed
in a neutral buffer gas. The transition from sympathetic cooling to heating,
and its dependence on trap parameters and masses of the particles have also
been explained. Finally, we have confirmed that the recent observation of non-
Maxwellian statistics DeVoe (2009) for a trapped ion can be attributed to
random heating collisions and provided a means to approximate the expected
power law of the energy distribution. Taken together, these results solve the
longstanding issues and questions that have existed since Dehmelt first
considered this problem over forty years ago. We believe that these results
will be critical for the design and interpretation of experiments in the
rapidly growing field of hybrid atom-ion physics Grier et al. (2009); Zipkes
and otheres (2010); Zipkes et al. (2010); Hall et al. (2011); Rellergert et
al. (2011, 2013); Sullivan et al. (2012); Schmid et al. (2010); Ratschbacher
et al. (2012, 2013).
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## Appendix A Appendix A: Comparison of a quantum scattering calculation and
$\frac{d\sigma_{L}}{d\Omega}$
Figure 3: The elastic (red solid line), diffusion (blue solid line) and
isotropic Langevin cross-section (black dashed line) for three different
collision energy for the Yb+ \+ Ca system.
In this section, we perform a quantum scattering calculation of the
differential cross-section for the Yb+ \+ Ca system, as the necessary
interaction potential was available to us Rellergert et al. (2011), and
compare the results to an isotropic Langevin differential cross-section.
Given the spherical symmetry of the atom-ion interaction potential, the
differential cross-section can be calculated from Friedrich (2005)
$\frac{d\sigma_{el}}{d\Omega}(\theta,E)=\left|\sum_{\ell}(2\ell+1)P_{\ell}(\cos\theta)\frac{e^{2\imath\eta_{\ell}}-1}{2\imath
k}\right|^{2}$ (A.1)
where $E$ is the collision energy and $\eta_{\ell}$ is the phase shift of the
$\ell$-th partial wave induced by the interaction potential Johnson (1999).
For a specific atom-ion combination, and thus for a specific interaction
potential, it is straightforward to calculate this differential cross-section
numerically.
The results are shown in Fig. 3 for the Yb+ \+ Ca system at three different
energies and are expected to be similar for other atom-ion combinations Zhang
et al. (2009).
As can be seen in Fig. 3 the differential cross-section exhibits a large
forward scattering peak at all energy scales. Thus, the majority of atom-ion
collisions lead to only slightly deflected trajectories, resulting in a very
small change in $\mathbf{W}$. Therefore, as originally argued by Dalgarno and
co-workers Dalgarno et al. (1958), to prevent an overestimate of the energy
redistribution due to collisions the momentum transfer (diffusion)
differential cross-section, i.e.
$\frac{d\sigma_{d}}{d\Omega}=\frac{d\sigma_{el}}{d\Omega}(1-\cos\theta)$, also
shown in Fig. 3, should be used to calculate the total atom-ion collision
rate. Fortuitously, the diffusion differential cross-section is approximately
isotropic in scattering angle, especially after thermal averaging, and agrees
quite well with the simple Langevin cross-section Langevin (1905)
$\sigma_{d}\approx\sigma_{L}=\pi\sqrt{\frac{2C_{4}}{E}}$, as seen in Fig. 3.
Therefore, we replace the cross-section by an isotropic profile which
integrates to $\sigma_{L}$. Under this approximation, the average collision
rate $\Gamma=2\pi\rho\sqrt{\frac{C_{4}}{\mu}}$ becomes energy independent and
the calculation of $\mathbf{M}$ is greatly simplified. As demonstrated in Fig.
4, the validity of this approximation is confirmed by comparison to a detailed
molecular dynamics simulation, which uses the full quantum differential cross-
section. The resulting error in the relaxation rate is smaller than 25% for
collision energies down to 1 mK.
Figure 4: Simulation of the kinetic energy of a Yb+ ion being sympathetically
cooled by Ca atom ($T=5\;\text{mK}$, $\rho=8\times 10^{11}\;\text{cm}^{-3}$)
using $\sigma_{el}$, compared to the prediction from Eq. 5 using $\sigma_{L}$
(lines). Inset: the ratio between aymptotic relaxation rates calculated using
$\sigma_{L}$, and $\sigma_{el}$. To save simulation time, neutral atom’s
density $\rho=8\times 10^{11}\mathrm{cm}^{-3}$.
## Appendix B Appendix B: Determination of M
In this section, we provide more explicit details of the derivation of the
relaxation matrix, $\mathbf{M}$.
First we rewrite, Eq. 3 in terms of the instantaneous coordinates for the $x$
direction,
$\displaystyle
W_{x}=\frac{m\overline{\dot{c}_{x}^{2}}}{2w_{0,x}^{2}}\Big{(}(\dot{c}_{x}^{2}+\dot{s}_{x}^{2})x^{2}+(c_{x}^{2}+s_{x}^{2})v_{x}^{2}$
(B.1) $\displaystyle-2(c_{x}\dot{c}_{x}+s_{x}\dot{s}_{x})xv_{x}\Big{)},$ (B.2)
where $w_{0,x}=c_{x}\dot{s}_{x}-s_{x}\dot{c}_{x}$ is the Wronskian and
constant in time. The change in average energy with a collision is then
$\displaystyle W_{x}^{\prime}-W_{x}$
$\displaystyle=-\frac{m\overline{\dot{c}_{x}^{2}}}{w_{0,x}^{2}}(c_{x}\dot{c}_{x}+s_{x}\dot{s}_{x})(x(v_{x}^{\prime}-v_{x}))$
(B.3)
$\displaystyle\;\;\;\;+\frac{m\overline{\dot{c}_{x}^{2}}}{2w_{0x}^{2}}(c_{x}^{2}+s_{x}^{2})(v_{x}^{\prime
2}-v_{x}^{2})$ $\displaystyle\equiv\Delta W_{x,1}+\Delta W_{x,2}.$
For $\Delta W_{x,1}$, since both $\langle\mathbf{v_{n}}\rangle$ and
$\langle\mathcal{R}(\mathbf{v}-\mathbf{v_{n}})\rangle$ vanish, $\langle
v_{x}^{\prime}\rangle=\frac{1}{1+\tilde{m}}v_{x}$. Therefore, using Eq. 1 and
3 we obtain,
$\displaystyle\langle\Delta W_{x,1}\rangle$
$\displaystyle=\frac{\tilde{m}}{1+\tilde{m}}\frac{m\overline{\dot{c}_{x}^{2}}}{w_{0,x}^{2}}\overline{(c_{x}\dot{c}_{x}+s_{x}\dot{s}_{x})(xv_{x})}$
(B.4)
$\displaystyle=\frac{\tilde{m}}{1+\tilde{m}}\frac{\overline{(c_{x}\dot{c}_{x}+s_{x}\dot{s}_{x})^{2}}}{w_{0,x}^{2}}\frac{m}{2}\overline{\dot{c}_{x}^{2}}(a_{x}^{2}+b_{x}^{2})$
$\displaystyle=\frac{\tilde{m}}{1+\tilde{m}}\epsilon_{x}\langle W_{x}\rangle,$
where
$\epsilon_{x}=\frac{\overline{(c_{x}\dot{c}_{x}+s_{x}\dot{s}_{x})^{2}}}{w_{0,x}^{2}}$.
To evaluate $\Delta W_{x,2}$, $\mathbf{v_{n}}$, $\mathbf{v}$ and $\mathcal{R}$
are uncorrelated, the average value of cross-correlation terms between them
vanish. Furthermore, since $\mathcal{R}$ is a random rotation,
$\langle[\mathcal{R}(\mathbf{v}-\mathbf{v_{n}})]_{x}^{2}\rangle=\frac{1}{3}\langle(\mathbf{v}-\mathbf{v_{n}})^{2}\rangle$.
Rearranging terms we obtain,
$\displaystyle\langle v_{x}^{\prime 2}\rangle-
v_{x}^{2}=\frac{\tilde{m}^{2}}{(1+\tilde{m})^{2}}\Big{(}\left(-\frac{2}{3}-\frac{2}{\tilde{m}}\right)v_{x}^{2}+\frac{1}{3}v_{y}^{2}$
$\displaystyle+\frac{1}{3}v_{z}^{2}+2\sigma_{v_{n}}^{2}\Big{)}$ (B.5)
where $\sigma_{vn}^{2}=2\langle W_{n}\rangle/m_{n}$ is the thermal width of
neutral atom velocity distribution. Thus, we find
$\displaystyle\langle\Delta
W_{x,2}\rangle=\frac{\tilde{m}}{(1+\tilde{m})^{2}}\Big{(}\Big{(}-\frac{2\tilde{m}+2}{3}\Big{)}(1+\epsilon_{x})\langle
W_{x}\rangle+$ (B.6) $\displaystyle\frac{\tilde{m}\alpha_{x}}{6}\langle
W_{y}\rangle+\frac{\tilde{m}\alpha_{x}}{6}\langle
W_{z}\rangle+\alpha_{x}\langle W_{n}\rangle\Big{)},$
where
$\alpha_{x}=\frac{\overline{(c_{x}^{2}+s_{x}^{2})}\cdot\overline{(\dot{c}_{x}^{2}+\dot{s}_{x}^{2})}}{w_{0,x}^{2}}$.
Combining the results of $\Delta W_{x,1}$ and $\Delta W_{x,2}$, and the
results for the $y$ and $z$ directions, finally we have
$\displaystyle\langle\mathbf{W^{\prime}}\rangle-\langle\mathbf{W}\rangle$
$\displaystyle=-\mathbf{M}\langle\mathbf{W}\rangle+\mathbf{N}$
$\displaystyle=-\mathbf{M}(\langle\mathbf{W}\rangle-\mathbf{W}_{st})$ (B.7)
where
$\mathbf{M}=-\frac{\tilde{m}^{2}}{(1+\tilde{m})^{2}}\begin{bmatrix}\frac{2\epsilon-1}{3}-\frac{1}{\tilde{m}}&\frac{\alpha}{6}&\frac{\alpha}{6}\\\
\frac{\alpha}{6}&\frac{2\epsilon-1}{3}-\frac{1}{\tilde{m}}&\frac{\alpha}{6}\\\
\frac{1}{6}&\frac{1}{6}&-\frac{1}{3}-\frac{1}{\tilde{m}}\\\ \end{bmatrix}$
(B.8)
and
$\mathbf{N}=\frac{\tilde{m}}{(1+\tilde{m})^{2}}\begin{bmatrix}\alpha\langle
W_{n}\rangle\\\ \alpha\langle W_{n}\rangle\\\ \langle
W_{n}\rangle\end{bmatrix}$ (B.9)
And, the steady-state kinetic energy is given by,
$\displaystyle\mathbf{W}_{st}$ $\displaystyle=-\mathbf{M}^{-1}\mathbf{N}$
$\displaystyle=\left(\mathbf{I}-\tilde{m}\begin{bmatrix}\frac{2\epsilon-1}{3}&\frac{\alpha}{6}&\frac{\alpha}{6}\\\
\frac{\alpha}{6}&\frac{2\epsilon-1}{3}&\frac{\alpha}{6}\\\
\frac{1}{6}&\frac{1}{6}&-\frac{1}{3}\\\
\end{bmatrix}\right)^{-1}\begin{bmatrix}\alpha\langle W_{n}\rangle\\\
\alpha\langle W_{n}\rangle\\\ \langle W_{n}\rangle\\\ \end{bmatrix}$ (B.10)
where $\alpha\equiv\alpha_{x}=\alpha_{y}$ and
$\epsilon\equiv\epsilon_{x}=\epsilon_{y}$. Because in the $z$ direction the
trapping field is time-independent, $\alpha_{z}=1$ and $\epsilon_{z}=0$. For
low values of $q$ and $a$, the numerical values of $\alpha$ and $\epsilon$ are
approximated by Chen and otheres (2013),
$\displaystyle\alpha\approx 2+2q^{2.24}$ (B.11) $\displaystyle\epsilon\approx
1+2.4q^{2.4}$ (B.12)
## Appendix C Appendix C: Procedures of Numerical Simulation
We perform two types of Monte Carlo simulations to verify the analytical
theory. Their simulation details are described below respectively. Type I
simulations were initially carried out to verify that approximation of the
differential scattering cross-section by an isotropic Langevin cross-section
was valid (Fig. 4). Following the verification of the approximation, Type II
simulations were used to make the simulations more computational efficient and
resulted in the data for Figs. 1(a)-(c).
### Type I
In Type I simulations, the ion trajectory is found numerically by integrating
the equations of motion with fixed time step $\Delta t$ using a custom
modified version of the ProtoMol software Matthey et al. (2004), where $\Delta
t$ is chosen to be much smaller than the rf period $\Omega^{-1}$. The
differential elastic collision cross-section $\frac{d\sigma_{el}}{d\theta}$
obtained from a quantum scattering calculation is used in every collision. The
simulation consists of following four steps:
1. S1.
A single ion is initialized at the origin with zero velocity, i.e.
$\mathbf{r}_{0}=\mathbf{0}$, and $\mathbf{v}_{0}=\mathbf{0}$. The simulation
step index $N$ is set to 0.
2. S2.
The position and velocity of the ion, $\mathbf{r}_{N+1}$ and
$\mathbf{v}_{N+1}$, at the next step $N+1$ are calculated by leapfrog
integration of the equations of motion.
3. S3.
To determine if a collision should happen during $\Delta t$, an atom is
generated with velocity $\mathbf{v_{n}}$ sampled from thermal distribution
characterized by $W_{n}$. The associated collision rate $\Gamma$ is given by
$\rho\sigma_{el}|\mathbf{v_{rel}}|$, where $\rho$ is the density of ultracold
atoms, $\mathbf{v_{rel}}$ is the relative velocity, and $\sigma_{el}$ depends
implicitly on the collision energy $\frac{\mu}{2}\mathbf{v_{rel}}^{2}$ in the
center-of-mass frame. A collision happens during $\Delta t$ if
$1-\exp(-\Gamma\Delta t)<d$, where $d$ is the value of a random number chosen
from a uniform distribution over$[0,1]$. If this condition is met the
simulation then proceeds to S4, otherwise it returns to S2.
4. S4.
The velocity of the ion after the collision is updated according to Eq. 6. The
rotation matrix $\mathcal{R}$ is specified by polar angle $\theta$ and
azimuthal angle $\phi$, defined with respect to $\mathbf{v_{rel}}$. $\theta$
is sampled from the probability distribution function
$\frac{d\sigma_{el}}{d\theta}\sin\theta$ defined on $[0,\pi]$, and $\phi$ is
sampled from uniform distribution on $[0,2\pi]$. The simulation then loops
back to S2 until the prescribed number of collisions have been reached.
### Type II
In Type II simulations, the isotropic Langevin differential cross-section
$\frac{d\sigma_{L}}{d\Omega}$ is used to calculate scattering process. The
collision rate $\Gamma$ thus does not depend on collision energy, allowing for
a much faster integration method based on a transfer matrix similar to Ref.
DeVoe (2009). The simulation consists of the following four steps,
1. S1.
A single ion is initialized at the origin with zero velocity, i.e.
$\mathbf{r}_{0}=\mathbf{0}$, and $\mathbf{v}_{0}=\mathbf{0}$, and, a series of
collision times $\tau_{j}~{}(j=1,2,3,\cdots)$ are pre-determined, which follow
a Poissonian distribution with average interval equal to $\Gamma^{-1}$.
2. S2.
The new coordinate of the ion
$\mathbf{P}_{i+1}=[x_{i},v_{x,i},y_{i+1},v_{y,i+1},z_{i+1},v_{z,i+1}]^{\mathrm{T}}$
at $\tau=\tau_{i+1}$ is obtained by multiplication of $\mathbf{P}_{i}$ by the
transfer matrix $\mathbf{T}(\tau_{i+1},\tau_{i})$ DeVoe (2009). $\mathbf{T}$
consists of three $2\times 2$ submatrices,
$\mathbf{T}=\begin{bmatrix}\mathbf{T}_{x}&\mathbf{0}&\mathbf{0}\\\
\mathbf{0}&\mathbf{T}_{y}&\mathbf{0}\\\
\mathbf{0}&\mathbf{0}&\mathbf{T}_{z}\end{bmatrix}$ (C.1)
where each submatrix $\mathbf{T}_{j}$ ($j=x,y,z$) is given by
$\mathbf{T}_{j}(\tau_{2},\tau_{1})=\frac{1}{w_{0,j}}\begin{bmatrix}c_{j}(\tau_{2})\dot{s}_{j}(\tau_{1})-s_{j}(\tau_{2})\dot{c}_{j}(\tau_{1})&-c_{j}(\tau_{2})s_{j}(\tau_{1})+s_{j}(\tau_{2})c_{j}(\tau_{1})\\\
\dot{c}_{j}(\tau_{2})\dot{s}_{j}(\tau_{1})-\dot{s}_{j}(\tau_{2})\dot{c}_{j}(\tau_{1})&-\dot{c}_{j}(\tau_{2})s_{j}(\tau_{1})+\dot{s}_{j}(\tau_{2})c_{j}(\tau_{1})\end{bmatrix}$
(C.2)
3. S3.
A collision then modifies the velocity of the ion according to Eq. 6, where
$\mathcal{R}$ now represents a rotation with equal probability into a $4\pi$
solid angle. The simulation then loops back to S2, until the prescribed number
of collisions has been reached.
## Appendix D Appendix D: The lack of multiplicative noise in a static trap
In sharp contrast to the ion trap case, a particle confined in a static
potential $V(x)=\frac{m}{2}\omega^{2}x^{2}$ and in contact with a reservoir at
temperature $T$ would have the same thermal distribution, regardless of the
reservoir particle’s mass $m_{n}$, or the trapping frequency $\omega$. This is
because for static traps the Mathieu functions $c(\tau)$ and $s(\tau)$ are
replaced by $\cos(\omega\tau)$ and $\sin(\omega\tau)$, which simplifies Eq. 19
into
$C=1-(2-\zeta)\zeta\sin^{2}(\omega\tau-\theta)$ (D.1)
Since $0\leq\zeta\leq 2$, $C\leq 1$, thus the energy of such system is never
amplified. From a mathematical perspective, the solution for Eq. 21 is
$\nu\rightarrow\infty$, meaning the predicted energy distribution falls faster
than any power-law tail of finite $\nu$, consistent with the thermal
distribution $\exp(-E/k_{B}T)$.
## Appendix E Appendix E: Determination of $\nu_{\text{1D}}$
To determine $\nu_{\text{1D}}$, first consider the light buffer-gas mass limit
i.e. $\tilde{m}\rightarrow 0$, and $\zeta\approx 2\tilde{m}$. Ignoring
$O(\zeta^{2})$, $C$ in Eq. 19 is simplified to:
$C=1-\zeta+\zeta\delta$ (E.1)
where $\delta=\left(\frac{c\dot{s}+s\dot{c}}{w_{0}}\right)_{\tau+\theta}$.
The analytical form of $P(C)$ is difficult to calculate. Instead, we
approximate it by a uniform distribution $\tilde{P}(C)$ in the range of
$[C_{-},C_{+}]$, which preserves the value of first and second moment of $C$,
namely $\langle C\rangle_{P}=\langle C\rangle_{\tilde{P}}$, and $\langle
C^{2}\rangle_{P}=\langle C^{2}\rangle_{\tilde{P}}$, where the subscript
denotes the distribution for which the average value is calculated. With this
requirement, $C_{\pm}$ is given by
$C_{\pm}=1-\zeta\pm\zeta\delta_{m}$ (E.2)
where $\delta_{m}=\sqrt{3\langle\delta^{2}\rangle}\approx\sqrt{3}$ for
$q<0.4$. Thus,
$\tilde{P}(C)=\begin{cases}\frac{1}{2\zeta\delta_{m}}&\text{if
}C\in[C_{-},C_{+}]\\\ 0&\text{otherwise}\end{cases}$ (E.3)
An example of $\tilde{P}(C)$ is shown in Fig. 5.
With $\tilde{P}(C)$, we solve for $\nu$ with a straightforward calculation of
$\langle C^{\nu}\rangle$,
$\langle
C^{\nu}\rangle_{\tilde{P}}=\int^{C_{+}}_{C_{-}}\frac{C^{\nu}}{2\zeta\delta_{m}}\;\text{d}C=\frac{1}{2\zeta\delta_{m}}\frac{(1+\zeta(\delta_{m}-1))^{\nu+1}}{\nu+1}=1$
(E.4)
Note $C_{-}^{\nu+1}$ vanishes because $C_{-}<1$ and $\nu\gg 1$. Introducing
$k=\zeta(\nu+1)$, we get
$\frac{1}{2\delta_{m}}\frac{(1+\zeta(\delta_{m}-1))^{k/\zeta}}{k}\approx\frac{e^{(\delta_{m}-1)k}}{2\delta_{m}k}=1$
(E.5)
with the value of $k\approx 3.35$ solved numerically. Thus we have the scaling
relation for $\tilde{m}\rightarrow 0$,
$\nu_{\text{1D}}\approx\frac{3.35}{\zeta}\approx\frac{1.67}{\tilde{m}}$ (E.6)
Figure 5: Exact value of $P(C)$ (blue solid line) sampled from Eq. 19 for
$\zeta=0.2,q=0.1$, and the uniform approximation $\tilde{P}(C)$ (red dashed
line).
020406080100120140160$\displaystyle\tilde{m}^{-1}$050100150200250$\displaystyle\nu$$\displaystyle\nu_{\mathrm{1D}}$,
exact result$\displaystyle\nu_{\mathrm{1D}}$, given by Eq.
[E.8]$\displaystyle\nu_{\mathrm{3D}}$, from Ref. DeVoe
(2009)$\displaystyle\nu_{\mathrm{3D}}=2\nu_{\mathrm{1D}}$0123402468
Figure 6: Comparison of exact solution (red dots) of $\nu_{\text{1D}}$ with
result calculated by Eq. E.8 (red dashed line). For reference, also shown are
$\nu_{\text{3D}}$ from Ref. DeVoe (2009) (blue dots) and an estimation of
$\nu_{\text{3D}}=2\nu_{\text{1D}}$ (blue dashed line).
Now consider the heavy buffer gas limit where
$\tilde{m}\rightarrow\tilde{m}_{c}$. Clearly we must have,
$\nu_{\text{1D}}(\tilde{m}=\tilde{m}_{c})=1$ (E.7)
since when $\tilde{m}=\tilde{m}_{c}$, $\langle C\rangle=1$ and the variances
of $A$ and $B$ diverge Takayasu et al. (1997).
Our previous approxmations break down because the $\zeta^{2}$ term cannot be
ignored. Thus, we do not seek to carry out further analysis, but instead add
an intercept to Eq. E.6 such that the requirement Eq. E.7 is met. For $q<0.4$
we find
$\nu_{\text{1D}}=\frac{1.67}{\tilde{m}}-0.67,$ (E.8)
which agrees surprisingly well with the exact value of $\nu_{\text{1D}}$
(shown in Fig. 6).
|
arxiv-papers
| 2013-10-19T00:56:42 |
2024-09-04T02:49:52.590357
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Kuang Chen, Scott T. Sullivan, Eric R. Hudson",
"submitter": "Kuang Chen",
"url": "https://arxiv.org/abs/1310.5190"
}
|
1310.5211
|
# Searching for a preferred direction with Union2.1 data
Xiaofeng Yang1,2,3, F. Y. Wang1,2, Zhe Chu4
$1$ School of Astronomy and Space Science, Nanjing University, Nanjing,
210093, China
$2$ Key Laboratory of Modern Astronomy and Astrophysics (Nanjing University),
Ministry of Education, Nanjing 210093, China
$3$ State Key Laboratory of Frontiers in Theoretical Physics, Institute of
Theoretical Physics, Chinese Academy of Sciences, Beijing, 100190, China
$4$ Key Laboratory for Research in Galaxies and Cosmology, Shanghai
Astronomical Observatory, Chinese Academy of Sciences,
Nandan Road 80, Shanghai, 200030, China E-mail:[email protected]
mail:[email protected]
###### Abstract
A cosmological preferred direction was reported from the type Ia supernovae
(SNe Ia) data in recent years. We use the Union2.1 data to give a simple
classification of such studies for the first time. Because the maximum
anisotropic direction is independent of isotropic dark energy models, we adopt
two cosmological models ($\Lambda$CDM, $w$CDM) for the hemisphere comparison
analysis and $\Lambda$CDM model for dipole fit approach. In hemisphere
comparison method, the matter density and the equation of state of dark energy
are adopted as the diagnostic qualities in the $\Lambda$CDM model and $w$CDM
model, respectively. In dipole fit approach, we fit the fluctuation of
distance modulus. We find that there is a null signal for the hemisphere
comparison method, while a preferred direction ($b=-14.3^{\circ}\pm
10.1^{\circ},l=307.1^{\circ}\pm 16.2^{\circ}$) for the dipole fit method. This
result indicates that the dipole fit is more sensitive than the hemisphere
comparison method.
###### keywords:
cosmology: theory - dark energy, Type Ia supernovae
## 1 Introduction
Einstein’s general relativity and the cosmological principle are the two key
foundations in modern cosmology. Cosmologists usually assumed that the general
relativity is the perfect law of gravity from small to large scales, which has
been tested by many tests in solar system and a few cosmological tests
(e.g.[35]). The cosmological principle [32] assumes that the universe is
homogeneous and isotropic on a sufficiently large scale. In practice, the
homogeneity and isotropy are confirmed by a variety of cosmological
observations, such as cosmic microwave background radiation (CMBR) [17], the
secondary effect of CMB [36], galaxy pairs [21, 30] and the large scale
structure (LSS) [25]. So far there is no any conclusive evidence for an
anisotropic cosmological model.
However, a possible challenge to the cosmological principle was reported in
recent years. Schwarz & Weinhorst (2007) claimed that a statistically
significant anisotropy of the Hubble diagram was found at 2$\sigma$ level at
$z<0.2$ by using SNe Ia data. SNe Ia data has been examined previously to test
the isotropy of the universe [19, 5, 14, 24]. For comparison, we can divide
the previous studies into two approaches as follows. (i) Local Universe
Constraint is defined as searching for preferred direction work with low
redshift astronomical probes (e.g.[11, 18]). (ii) Non-local Universe
Constraint is defined as the study with intermediate and high redshift data
(e.g.[1, 6]), which includes the redshift tomography analysis (dividing full
sample into different redshift bins).
Since there is no well accepted upper limit value of redshift about how large
the local universe is currently, we simply choose $z_{local}\leq 0.2$ in our
classification. It is obvious that if a preferred direction or any other kind
of anisotropy really exists, the physical origins may be different between
local and non-local universe. Various local effect can lead to anisotropy in
local universe, such as the bulk flow towards the Shapley supercluster [11].
Thus, the explanation of local universe anisotropy is complicate and subtle.
But if the non-local universe anisotropy was confirmed by observation, the
standard cosmological model ($\Lambda$CDM) based on cosmological principle
must be modified. So there are two merits of our classification. First, it
provides the difference of the probing scale. Second, it implicates the
different theoretical origins. At the same time, one can easily take the study
by model-independent manner in local universe constraint [18] and by model-
dependent way in non-local universe constraint [6]. In previous works, there
are serious differences and disagreements among the studies on the possible
cosmological anisotropy. Some works found no statistically significant
evidence for anisotropy using the SNe Ia data [28, 3, 15, 16, 4]. However,
many studies found that there is a statistically significant anisotropy [24,
8, 11] or a cosmological preferred direction [1, 6, 20, 7, 33]. A few works
either gave no distinct results [9, 12] or argued that the anisotropic result
of local universe constraint is not contradiction to the $\Lambda$CDM model
[18].
In this work, we search for a cosmological preferred direction from the latest
Union2.1 data for the first time. For the anisotropic analysis, we adopt two
typical and sophisticated approaches which are hemisphere comparison [1] and
dipole fit [20]. Since the preferred direction is almost independent of
isotropic dark energy models [6], we choose two simple cosmological models,
$\Lambda$CDM and $w$CDM for the hemisphere comparison approach, and
$\Lambda$CDM for the dipole fit. In the first approach, we use the matter
density and the equation of state of dark energy as the diagnostic qualities
in the $\Lambda$CDM and $w$CDM, respectively. In the second method, we employ
distance modulus as the diagnostic quality in $\Lambda$CDM model.
The paper is organized as follows.We present the Union2.1 data and the two
methods in section 2. Section 3 gives the numerical results. We compare and
discuss our results with other works in section 4. Section 5 is a brief
summary.
## 2 THE DATA AND METHODS
### 2.1 The Union2.1 data and preliminary formulae
SNe Ia are important probes of the evolution of the universe. In this work, we
use the Union2.1 sample which is a compilation consisting of 580 SNe Ia. The
redshift range is from 0.015 to 1.414 [26]. Comparing to the Union2 data, the
updated Union2.1 data consists other 23 SNe Ia. Here we get the directions of
Union2 data in the equatorial coordinates (right ascension and declination) to
each SN Ia from Blomqvist et al. (2010). We get the directions of additional
23 datapoints from NED website111http://ned.ipac.caltech.edu.We also use the
Union2.1 table from the SCP website222http://supernova.lbl.gov, which includes
each SN Ia’s name, redshift, distance modulus and uncertainties. We translated
the equatorial coordinates of SNe Ia to galactic coordinates $(l,b)$ in the
galactic systems [27].
In Figure.1, we show the angular distributions of the Union 2.1 datapoints in
galactic coordinates. The color represents the value of redshift according to
the legend on the right. The figures are viewed above the north galactic
equator and south galactic equator in the left panel and right panel,
respectively. For avoiding confusion, we do’t show Union2 data and additional
23 data on the same sphere. We show Union2 data in top panels and additional
23 data in bottom panels, respectively. Some of datapoints are nearly overlap
in different redshift because of the similar angular direction. It is obvious
that the distribution of additional 23 datapoints are slightly more isotropic
than the distribution of Union2 data.
Figure 1: (color online) The Union2 data (up panels) and additional 23 data
(bottom panels) in galactic coordinates. They are shown with viewpoint above
the north galactic equator and south galactic equator in the left panel and
right panel, respectively. The color of each point corresponds to the redshift
of each SN Ia.
We study the SNe Ia data in the classical way by applying the maximum
likelihood method. In a flat FLRW cosmological model, the luminosity distance
is
$D_{L}(z)=(1+z)\int_{0}^{z}\frac{d{z}^{\prime}{}}{E({z}^{\prime}{})}.$ (1)
In the flat $\Lambda$CDM model, $E({z})$ can be parameterized by
$E^{2}(z)=\Omega_{m0}(1+z)^{3}+(1-\Omega_{m0}),$ (2)
where $\Omega_{m0}$ is the matter density. For the $w$CDM model, $E({z})$ is
$E^{2}(z)=\Omega_{m0}(1+z)^{3}+(1-\Omega_{m0})(1+z)^{3+3w},$ (3)
where $w$ is the equation of state of dark energy.
We use the distance modulus of SN Ia data by minimizing the $\chi^{2}$. The
$\chi^{2}$ for SNe Ia is obtained by comparing theoretical distance modulus
$\mu_{th}(z)=5\log_{10}\big{(}D_{L}(z)\big{)}+\mu_{0},$ (4)
here,
$\mu_{0}=42.38-5\log_{10}h$ (5)
is a nuisance parameter. The theoretical model parameter ($\Omega_{m0}$ or
$w$) is determined by minimizing the value of $\chi^{2}$ with observed
$\mu_{obs}$ of SNe Ia:
$\chi_{\bf
SN}^{2}(\Omega_{m0},\mu_{0})=\sum_{i=1}^{580}\frac{\Big{(}\mu_{obs}(z_{i})-\mu_{th}(\Omega_{m0},\mu_{0},z_{i})\Big{)}^{2}}{\sigma_{\mu}^{2}(z_{i})}.$
(6)
Since the nuisance parameter $\mu_{0}$ is independent of the dataset, we can
expand $\chi_{\bf SN}^{2}$ with respect to $\mu_{0}$ [22]:
$\chi_{\bf SN}^{2}=A-2\mu_{0}B+\mu_{0}^{2}C,$ (7)
here
$\displaystyle A$ $\displaystyle=$
$\displaystyle\sum_{i=1}^{580}\frac{\big{(}\mu_{obs}(z_{i})-\mu_{th}(z_{i},\mu_{0}=0)\big{)}^{2}}{\sigma_{\mu}^{2}(z_{i})},$
$\displaystyle B$ $\displaystyle=$
$\displaystyle\sum_{i=1}^{580}\frac{\mu_{obs}(z_{i})-\mu_{th}(z_{i},\mu_{0}=0)}{\sigma_{\mu}^{2}(z_{i})},$
$\displaystyle C$ $\displaystyle=$
$\displaystyle\sum_{i=1}^{580}\frac{1}{\sigma_{\mu}^{2}(z_{i})}.$
The value of Eq. (7) is minimum for $\mu_{0}=B/C$ at
$\widetilde{\chi}_{\bf SN}^{2}=\chi_{\bf SN,min}^{2}=A-B^{2}/C,$ (8)
which is not rely on $\mu_{0}$.
### 2.2 The hemisphere comparison approach
Currently, it is not easy to find the angular dependence of anisotropy at
small scale with significant confidence level using SNe Ia. The reason is that
the number density of SNe Ia is relatively low, particular in the tomography
analysis. Thus, we firstly employ the hemisphere comparison for searching the
largest possible anisotropy in the largest scale of $\pi/2$. An early similar
research has been done to a CMB sky map analysis [13]. The subsequent studies
found one of the several anomalies in the WMAP data (e.g.[10]). The hemisphere
comparison method was firstly proposed for searching largest possible
anisotropy with SNe Ia by Schwarz & Weinhorst (2007). It was further developed
and used for finding the possibly preferred direction [1, 6].
In recent works, different cosmological parameters are chosen as the
diagnostic qualities, such as $\Omega_{m0}$ [1], $q_{0}$ [6] and $H_{0}$ [18].
Since the preferred direction is weakly depended on dark energy models [6], we
simply consider two cosmological models, such as $\Lambda$CDM and $w$CDM
models. We also adopt $\Omega_{m0}$ and $w$ for $\Lambda$CDM and $w$CDM as the
diagnostic qualities, respectively. It could be convenient to compare previous
results [1] with ours.
We review the procedure of hemisphere comparison method in short [1]. (i)
Generate a random direction with the same probability in unit sphere. (ii)
Divide the dataset into two subsets according to the sign of the product
between the vector generated in the step (i) and the unit vector describing
the direction of each SN Ia in the dataset. We can split the data in two
opposite hemispheres, denoted by up and down. (iii) Calculate the best fit
value of cosmological parameter on each hemisphere. (iv) Repeat a large times
from step (i) to step (iii), and search the maximum normalized difference for
the full data, thus one can get the preferred direction of maximum anisotropy.
One can get more details of this method from the two references [1, 6]. Here,
we just describe the third step of this method, which estimates the best
parameter in each hemisphere. The subscripts $u$ and $d$ represent the best
parameter fitting value in the ‘up’ and ‘down’ hemispheres, respectively. For
estimating $\Omega_{m0}$ in $\Lambda$CDM model, we can define [1]
$\delta=\frac{\Delta\Omega_{m0}}{\bar{\Omega}_{m0}}=\frac{\Omega_{m0,u}-\Omega_{m0,d}}{(\Omega_{m0,u}+\Omega_{m0,d})/2}.$
(9)
For fitting $w$ in the $w$CDM model, we define the relative anisotropic level
with the equation of state of dark energy as
$\delta^{\prime}{}=\frac{\Delta
w}{\bar{w}}=\frac{w_{u}-w_{d}}{(w_{u}+w_{d})/2},$ (10)
where $w_{u}$ and $w_{d}$ are the best fitting equation of state in the ‘up’
and ‘down’ hemispheres, respectively. The number of random axes should be more
than the number of SNe Ia on each hemisphere. For Union2.1 sample, the number
of data points per hemisphere is approximate 290, we choose 400 axes in this
works. Since the hemisphere comparison approach is not pretty fine and
sensitive enough to particular types of anisotropy [20], it is just a rough
estimation for global property. We only implement the non-local universe
constraint without redshift tomography if there is no any anisotropic signal
in global constraint with the full sample in all redshift ranges.
### 2.3 The dipole fit approach
Dipole anisotropic fit method has been used for searching the anisotropy of
fine structure constant with quasars on cosmological scale. Mariano &
Perivolaropoulos (2012) firstly applied this method to anisotropic study using
SNe Ia [20]. The main steps of the dipole fit method are shown as follows:
* •
Convert the equatorial coordinates of SNe Ia to galactic coordinates.
* •
Give the Cartesian coordinates of unit vectors $\hat{n}_{i}$ corresponding
each SN Ia with galactic coordinates $(l,b)$. So, we obtain
$\hat{n}_{i}=\cos(b_{i})\cos(l_{i})\hat{i}+\cos(b_{i})\sin(l_{i})\hat{j}+\sin(b_{i})\hat{k}.$
(11)
* •
Define the angular distribution model with dipole and monopole
$(\frac{\Delta\mu}{\bar{\mu}})=d\cos\theta+m,$ (12)
where $\mu$ is distance modulus, $m$ and $d$ denote the monopole and dipole
magnitude, respectively, $\cos\theta$ is the angle with the dipole axis
defined by the vector
$\vec{D}\equiv c_{1}\hat{i}+c_{2}\hat{j}+c_{3}\hat{k}.$ (13)
So
$\hat{n}_{i}\cdot\vec{D}=d\cos\theta_{i}.$ (14)
Then, we can fit the SNe Ia data to a dipole anisotropy model (12) using the
maximum likelihood method by minimizing
$\chi^{2}({\vec{D}},m)=\sum_{i=1}^{580}\frac{\left[(\frac{\Delta\mu}{\bar{\mu}})_{i}-d\cos\theta_{i}-m\right]^{2}}{\sigma_{i}^{2}}.$
(15)
* •
At last, we can obtain the magnitude and direction of the best fit dipole in
galactic coordinates from the best fit $c_{i}$ coordinates (e.g.
$d=\sqrt{c_{1}^{2}+c_{2}^{2}+c_{3}^{2}}$). The corresponding $1\sigma$ errors
are obtained using the covariance matrix approach.
## 3 THE RESULTS
### 3.1 Results of hemisphere comparison method
We apply the hemisphere comparison method using the latest Union2.1 dataset.
Generally, one can expect that we should get the similar results with recent
works from Union2 sample [1, 6]. It is surprised that we get different results
compared with previous works.
Table 1 shows our numerical results with the Union2.1 dataset, which could be
clearly compared with previous results shown in the second row [1]. The 1
$\sigma$ error is propagated from the uncertainties of the SNe Ia distance
moduli. The superscript $Real$ and $Sim$ denote the maximum anisotropic values
which are obtained from real SNe Ia dataset and a typical isotropic simulated
dataset, respectively. The simulated isotropic dataset has been constructed by
replacing each real data distance modulus to a random number from the normal
distribution with mean and standard deviation obtained by the best fitting
value of $\mu_{th}(z_{i})$ and by uncertainties of the corresponding real data
point, respectively. Comparing to the result derived from Union2 dataset, it
is clear that the maximum anisotropy level is $0.31\pm 0.05$ for the Union2.1
dataset. However, the value is $0.35\pm 0.05$ for simulation data, which is
larger than the one of real data. In this calculation, the same parameter and
cosmology model ($\Lambda$CDM) are used for the two different datasets.
The maximum anisotropic value will convergence in calculations with real data
by enlarging the random selected axes, whereas it’s precise value will be
fluctuated in repeated estimations with simulated data for random selected
effect. In $w$CDM model calculation, the value of Eq.(10) is $0.27\pm 0.07$
and $0.37\pm 0.07$ in real data and simulated data, respectively. The value of
real data is smaller than the one in $\Lambda$CDM fitting ($0.31\pm 0.05$) for
different cosmological parameter and model. The value ($0.37\pm 0.07$) in this
simulation dataset is still larger than the one in real data ($0.27\pm 0.07$).
Although our results show that the maximum anisotropy level is lower than
simulation isotropic dataset from Union2.1 dataset, we still process the same
numerical experiments as shown in the Antoniou & Perivolaropoulos (2010). The
purpose is to answer whether the maximum anisotropy level for real data is
higher or lower than statistical isotropy. This kind of numerical experiments
is not intend to identify the maximum anisotropic direction in standard 400
axes searching procedure. We only want to compare the real data with the
isotropic simulation data. It is important to repeat the comparison many times
(40 in our case) for acceptable statistics. Because of the limitations of
searching time, we adopt 10 axes for employing fast-speed Monte Carlo
experiments (Antoniou & Perivolaropoulos 2010). This numerical experiment is
important because of more fluctuated values with maximum anisotropy level in
the simulation data.
From a set of numerical experiments, we get different results from Union2.1
dataset comparing with Union2 dataset in Table 2. The $Real$ or $Sim$ denotes
the number of cases which maximum anisotropic value of real data is larger or
smaller than that of simulated data, respectively. For Union2 dataset, there
is about $1/3$ of the numerical experiments with
$\delta_{max}^{Sim}>\delta_{max}^{Real}$, which means that the anisotropy
level was larger than the one of the isotropic simulation data [1]. However,
we find that the possibility of real data and simulation data which have a
larger maximum anisotropic value is nearly equal. The results in Table 2 are
not consistent with the work of Antoniou & Perivolaropoulos (2010). In order
to test the dependence on the number of axes, we increase the random axes from
10 to 50. As shown in Table 2, our conclusion is unchanged.
Table 1: The value of maximum anisotropy for Union2.1 dataset and isotropic simulation dataset. The second row is the value calculated from Union2 dataset [1]. Model(Sample) | Diagnostic | $\delta_{max}^{Real}$ | $\delta_{max}^{Sim}$
---|---|---|---
$\Lambda$CDM(Union2) | $\Omega_{m0}$ | $0.43\pm 0.06$ | $0.36\pm 0.06$
$\Lambda$CDM(Union2.1) | $\Omega_{m0}$ | $0.31\pm 0.05$ | $0.35\pm 0.05$
$w$CDM (Union2.1) | $w$ | $0.27\pm 0.07$ | $0.37\pm 0.07$
Table 2: The results of 40 times numerical experiments for the value of maximum anisotropy with Union2.1 dataset and isotropic simulation datasets. The second row is the result from Union2 dataset [1]. The Real or Sim denotes the number of cases which maximum anisotropic value of real data is larger or smaller than that of simulated data, respectively. Model(Sample) | Axes$\times$All times | Real | Sim
---|---|---|---
$\Lambda$CDM(Union2) | $10\times 40$ | 26 | 14
$\Lambda$CDM(Union2.1) | $10\times 40$ | 19 | 21
| $20\times 40$ | 17 | 23
| $30\times 40$ | 20 | 20
| $40\times 40$ | 22 | 18
| $50\times 40$ | 18 | 22
$w$CDM (Union2.1) | $10\times 40$ | 18 | 22
| $20\times 40$ | 19 | 21
| $30\times 40$ | 17 | 23
| $40\times 40$ | 21 | 19
| $50\times 40$ | 20 | 20
Since there is no anisotropic signal in global constraint with full Union2.1
data, we will not apply the redshift tomography analysis in this work. We use
tomography analysis in next subsection which implements a more sensitive
searching approach.
### 3.2 Results of dipole fit method
We study the latest Union2.1 dataset using the dipole fit method, which
includes non-local universe constraint and tomography constraint. First, we
report the result of non-local universe constraint with full Union2.1 data.
Then, we will show the local universe constraint and tomography results.
We find the direction of the dark energy dipole with full data
$b=-14.3^{\circ}\pm 10.1^{\circ},l=307.1^{\circ}\pm 16.2^{\circ}.$ (16)
The values of the dipole and monopole magnitudes are
$\displaystyle d_{Union2.1}$ $\displaystyle=$ $\displaystyle(1.2\pm 0.5)\times
10^{-3},$ (17) $\displaystyle m_{Union2.1}$ $\displaystyle=$
$\displaystyle(1.9\pm 2.1)\times 10^{-4}.$ (18)
The statistical significance of the dark energy dipole is about at the
$2\sigma$ level. The direction of Union2 dipole is ($b=-15.1^{\circ}\pm
11.5^{\circ}$, $l=309.4^{\circ}\pm 18.0^{\circ}$)[20], and the dipole and
monopole magnitudes are
$\displaystyle d_{Union2}$ $\displaystyle=$ $\displaystyle(1.3\pm 0.6)\times
10^{-3},$ (19) $\displaystyle m_{Union2}$ $\displaystyle=$
$\displaystyle(2.0\pm 2.2)\times 10^{-4}.$ (20)
The statistical significance of the dark energy dipole is also at the
$2\sigma$ level using Union2, thus, our results are consistent with Mariano &
Perivolaropoulos 2012.
According to the dipole fit approach [20], we determine the likelihood of the
observed dark energy dipole magnitude with performing a Monte Carlo simulation
consisting of $10^{4}$ Union2.1 datasets constructed under the assumption of
isotropic $\Lambda$CDM. The distance modulus of point $i$ is defined as
$\mu_{MC}(z_{i})=g(\bar{\mu}(z_{i}),\sigma_{i}),$ (21)
where $g$ is the Gaussian random selection function [20], and
$\bar{\mu}(z_{i})$ is the best fit distance modulus of the Union2.1 full data
in $\Lambda$CDM model at redshift $z_{i}$. It is convenient to construct
$\left(\frac{\Delta\mu(z_{i})}{\bar{\mu}(z_{i})}\right)_{MC}$ for each Monte
Carlo dataset and search its best fit dipole direction and magnitude. In
Figure. 2 we show the probability distribution of the dark energy dipole
magnitude along with the observed dipole magnitude represented by an arrow. As
expected from Equation. (17) merely $4.55\%$ of the simulations had a dark
energy dipole magnitude bigger than the value in real dataset. The result is
consistent with Equation. (17) which indicates that the statistical
significance of the dark energy dipole is about $2\sigma$. For the number of
Monte Carlo simulation, previous work proved that $10^{4}$ adopted as the
number of simulated datasets is enough to obtain a significant results [20].
Figure 2: (color online). Histogram of distribution indicates the dark Energy
dipole magnitudes from the Monte Carlo simulation. The arrow position is the
observed best fit value. The deeper green region shows fraction of the Monte
Carlo datasets that give a dipole magnitude larger than the observed best fit
one.
We also take the redshift tomography analysis for indicating these effects in
different redshift ranges. We adopt two subsample allocations similar as
previous work based on Union2 [20], one is partitioning full sample with three
redshift bins and the other is changing the redshift upper limit. In the first
method, we divide full dataset into three redshift bins which have nearly the
same number of SNe Ia. Then we perform the similar works as above in each bin
and compare the results of each bin with respect to the quality of data with
errors, the dipole magnitudes and the dipole directions. For the second
method, we set first and second subsample with an redshift upper limit
consisting of about a half of the full datapoints. Then we enlarge the
redshift upper limit properly so that the largest subsample almost includes
full dataset. We study each subsample of the six cumulative dataset parts with
the same procedure as above.
Table 3 shows our results in different redshift ranges with each subsample of
Union2.1, which includes our above non-local universe constraint in the second
line. It also shows the deviled method of redshift bins and the datapoints
number of each redshift bin. In 2nd to 5th columns, the results in brackets
are from Union2 data [20]. In the last column, the number in and out brackets
is the datapoints of Union2 and Union2.1, respectively. There is no additional
datapoint from Union2 to Union2.1 in the redshift $0.14<z\leq 0.43$. In each
redshift bin or range, we show the corresponding best fit monopole magnitude,
the dipole magnitude and the direction of the best fit dipole in galactic
coordinates. The uncertainties shown in Table 3 are calculated via the
covariance matrix approach. We have checked and confirmed that they are
consistent with the corresponding $1\sigma$ errors calculated from the Monte
Carlo simulations. All the results we reported here in the Table 3 are
consistent with the results from Union2 [20]. We also find that the “best”
redshift bin with the smallest errors for the Union2.1 data is the lowest
redshift bin ($0.015<z\leq 0.14$), which is also similar to previous work from
Union2 [20].
Table 3: The results of diploe fit approach including non-local, local constraints and tomography analysis. In the 2nd to 5th columns, we show the monopole magnitude, dipole magnitude and direction from the estimation with Union2.1 (Union2) data in different redshift ranges (1st column). Expect for the last column, the results in brackets are the calculations from Union2 data [20] for comparison. In the last column, the number in and out brackets represents the datapoints of Union2 and Union2.1, respectively. There are the same datapoints between Union2.1 and Union2 in the redshift $0.14<z\leq 0.43$. | $\frac{m_{U2.1}(m_{U2})}{10^{-4}}$ | $\frac{d_{U2.1}(d_{U2})}{10^{-3}}$ | $b_{d_{U2.1}}(b_{d_{U2}})$ | $l_{d_{U2.1}}(l_{d_{U2}})$ | U2.1(U2)
---|---|---|---|---|---
$0.015\leq z\leq 1.414$ | $1.9\pm 2.1(2.0\pm 2.2)$ | 1.2 $\pm$ 0.5(1.3 $\pm$ 0.6) | $-14.3$ $\pm$ 10.1( $-15.1$ $\pm$ 11.5) | 307.1 $\pm$ 16.2(309.4 $\pm$ 18.0) | 580(577)
$0.015<z\leq 0.14$ | $2.5\pm 3.1$($2.6\pm 3.4$) | 1.5 $\pm$ 0.7(1.7 $\pm$ 0.8) | $-9.8$ $\pm$ 14.6($-10.1$ $\pm$ 15.1) | 304.3 $\pm$ 21.4(308.8 $\pm$ 22.8) | 193(184)
$0.14<z\leq 0.43$ | $2.6\pm 5.6$ | 1.2 $\pm$ 1.9 | $-10.7$ $\pm$ 28.7 | 291.4 $\pm$ 37.2 | 186(186)
$0.43<z\leq 1.414$ | $0.6\pm 3.7$($0.7\pm 4.3$) | 0.7 $\pm$ 0.7(0.9 $\pm$ 0.8) | $-25.9$ $\pm$ 29.7($-25.1$ $\pm$ 30.6) | 35.7 $\pm$ 73.1(34.3 $\pm$ 75.7) | 201(187)
$0.015\leq z\leq 0.23$ | $3.2\pm 2.7$($3.3\pm 2.9$) | 1.6 $\pm$ 0.6(1.8 $\pm$ 0.7) | $-7.8$ $\pm$ 11.9($-8.5$ $\pm$ 12.4) | 300.3 $\pm$ 16.1(302.2 $\pm$ 16.6) | 248(239)
$0.015\leq z\leq 0.31$ | $3.5\pm 2.7$($3.8\pm 2.9$) | 1.7 $\pm$ 0.6(1.9 $\pm$ 0.7) | $-6.8$ $\pm$ 11.1($-7.6$ $\pm$ 11.6) | 304.5 $\pm$ 13.6(307.0 $\pm$ 14.7) | 301(292)
$0.015\leq z\leq 0.41$ | $2.8\pm 2.6$($3.0\pm 2.7$) | 1.6 $\pm$ 0.6(1.8 $\pm$ 0.7) | $-13.8$ $\pm$ 9.7($-14.4$ $\pm$ 10.3) | 301.5 $\pm$ 13.5(303.6 $\pm$ 14.4) | 361(352)
$0.015\leq z\leq 0.51$ | $2.1\pm 2.5$($2.2\pm 2.6$) | 1.3 $\pm$ 0.6(1.4 $\pm$ 0.7) | $-14.1$ $\pm$ 12.1($-14.9$ $\pm$ 12.7) | 298.8 $\pm$ 17.8(301.3 $\pm$ 18.8) | 415(406)
$0.015\leq z\leq 0.64$ | $2.0\pm 2.2$($2.1\pm 2.4$) | 1.3 $\pm$ 0.5(1.4 $\pm$ 0.6) | $-15.5$ $\pm$ 10.7($-16.0$ $\pm$ 11.0) | 302.4 $\pm$ 16.1(305.3 $\pm$ 16.9) | 474(464)
$0.015\leq z\leq 0.89$ | $1.8\pm 2.1$($2.2\pm 2.3$) | 1.3 $\pm$ 0.5(1.4 $\pm$ 0.6) | $-14.8$ $\pm$ 10.0($-15.6$ $\pm$ 10.4) | 307.3 $\pm$ 15.2(309.8 $\pm$ 16.0) | 531(519)
## 4 DISCUSSION
If a preferred direction or any other anisotropy could be really confirmed at
high significant level, particular in non-local universe ($z>0.2$), we should
abandon cosmological principle and study the anisotropic cosmological models,
e.g. vector field model, Bianchi type I model or extended topological
quintessence model [20]. A comprehensive introduction of various observational
probes on the preferred axis could be found in the paper [23]. So far, the
largest anisotropic value ($>0.7$) is given by Cai & Tuo (2010)’s work from
Union2 data, which adopted the deceleration parameter $q_{0}$ for estimation
via hemisphere comparison method. However, in all of previous works, the
significance of the violation to isotropic assumption of cosmological
principle are not high. In fact, most of them are no more than 2 $\sigma$
confidence level. Although people have proved that the significance could be
improved by correlations with other preferred axes from different observations
[1], none of them has been confirmed or has acceptable fundamental physical
theory. Since there are tensions in cosmological constraints with different
observations, maybe it need more works on this issue with different probes.
There are merely adding 23 data points in this paper, thus it is not
reasonable that we get the different results compared with previous works
based on Union2. Interesting, we have the different results by the hemisphere
comparison method but obtain the same results by the dipole fit method. There
are three potential reasons for such differences. The first is the possible
tension between Union2 and Union2.1. Second, the different space distribution
is another factor. The third reason is the different method’s sensitivity. For
the data tension, recently, some other independent works focused on
constraining the dark energy model point out the tension in datasets between
Union2 and Union2.1 (e.g.[34]). For the different distribution, we show that
the distribution of Union2.1 dataset is slightly better-distributed than the
one of Union2, this hint could be found in Figure 1. However, Kalus et
al.(2012) argued that the non-uniform distribution has no significant impact
on such anisotropic estimation. Since their work is just local universe
constraint whereas our and the two other hemisphere comparison works [1, 6]
are non-local universe constraints, the detailed analysis of the different
anisotropic searching results by hemisphere comparison method and other
methods beyond the scope of this work. The third aspect may be the main point,
which is caused by that the dipole fit method is more sensitive and effective
than hemisphere comparison method [20]. Generally, our results confirm this
idea with Union2.1 data. On the other hand, although hemisphere comparison
method is neither precise nor perfect, it is really a model-independent
approach. Since we should define the angular distribution model as a fiducial
model in diploe fit method, it is much more model-dependent than hemisphere
comparison method. This situation is similar to the studies on dark energy
reconstruction. Many dark energy parameterizations could enhance the precision
of dark energy parameters constraint, but the parameterizations also impose
some bias on the exact evolution of dynamical dark energy. Correspondingly, if
we adopt any specific angular distribution model in dipole fit method, such as
the Equation.12 or the parameterization in Cai et al’s work [7], it may affect
the result of the potential unbias anisotropy of the universe. We will
investigate this interesting issue in future works. The high-redshift data,
such as gamma-ray bursts will be included [2, 31, 29].
## 5 SUMMARY
In this paper, we search for a preferred direction of acceleration using the
Union2.1 SNe Ia sample. At the beginning of this paper we simply specify and
classify previous searching works into two types according to their sample’s
redshift ranges. Many authors found that a maximum (minimum) expansion
(acceleration) in a preferred direction by applying the hemisphere comparison
method and dipole fit method to SNe Ia sample. We use the latest Union2.1
sample on this study for the first time. We adopt two cosmological models
($\Lambda$CDM, $w$CDM) for hemisphere comparison method and $\Lambda$CDM model
for dipole fit. In hemisphere comparison approach, we use matter density and
the equation of state of dark energy as the diagnostic qualities in
$\Lambda$CDM and $w$CDM models, respectively. In dipole fit approach, we study
the fluctuation of distance modulus and take the tomography analysis with
different redshift ranges. Comparing with Union2, we find a null signal for
cosmological preferred direction by hemisphere comparison method. But there is
a preferred direction ($b=-14.3^{\circ}\pm 10.1^{\circ},l=307.1^{\circ}\pm
16.2^{\circ}$) by dipole fit approach. Our results confirm that the dipole fit
method is more sensitive than the hemisphere comparison method for the
searching of a cosmological preferred direction with SNe Ia.
## ACKNOWLEDGMENTS
We thank the referee Prof. Perivolaropoulos very much for the detail and
constructive suggestions which helped to improve the manuscript significantly.
We have benefited from reading the publicly available codes of Antoniou &
Perivolaropoulos (2010) and Mariano & Perivolaropoulos (2012). Xiaofeng Yang
gratefully acknowledges the collaborating, long visiting and open fund
provided by State Key Laboratory of Theoretical Physics, Institute of
Theoretical Physics, Chinese Academy of Sciences, where the last revision of
this manuscript was completed in. This work is supported by the National Basic
Research Program of China (973 Program, grant 2014CB845800) and the National
Natural Science Foundation of China (grants 11373022, 11103007, and 11033002).
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|
arxiv-papers
| 2013-10-19T09:34:34 |
2024-09-04T02:49:52.599908
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Xiaofeng Yang, F. Y. Wang, Zhe Chu",
"submitter": "Xiaofeng Yang",
"url": "https://arxiv.org/abs/1310.5211"
}
|
1310.5351
|
# A brief remark on the topological entropy for linear switched systems
Getachew K. Befekadu G. K. Befekadu is with the Department of Electrical
Engineering, University of Notre Dame, Notre Dame, IN 46556, USA.
E-mail: [email protected]
Version - February 25, 2013.
###### Abstract
In this brief note, we investigate the topological entropy for linear switched
systems. Specifically, we use the Levi-Malcev decomposition of Lie-algebra to
establish a connection between the basic properties of the topological entropy
and the stability of switched linear systems. For such systems, we show that
the topological entropy for the evolution operator corresponding to a semi-
simple subalgebra is always bounded from above by the negative of the largest
real part of the eigenvalue that corresponds to the evolution operator of a
maximal solvable ideal part.
###### Index Terms:
Asymptotic stability; topological entropy; Lie-algebra; stability of switched
systems.
## I Introduction
In the past, the notions of measure-theoretic entropy and topological entropy
have been intensively studied in the context of measure-preserving
transformations or continuous maps (e.g., see references [1], [2] and [3] for
the review of entropy in ergodic theory as well as in dynamical systems). For
instance, Adler et al. (in the paper [4]) introduced the notion of topological
entropy as an invariant conjugacy, which is an analogue to the notion of
measure-theoretic entropy, for measuring the rate at which a continuous map in
a compact topological space generates initial-state information. Subsequently,
Bowen and Dinaburg, in the papers [5] and [6] respectively, gave a weak, but
equivalent, definition of topological entropy for continuous maps that later
led to proofs for connecting this notion of entropy with that of measure-
theoretic entropy (e.g., see also [7] or [8] for additional discussions).
With the emergence of networked control systems (e.g., see [9]), these notions
of entropy have found renewed interest in the research community (e.g., see
[10], [11] and [12]). Notably, Nair et al. [10] have introduced the notion of
topological feedback entropy, which is based on the ideas of [4], to quantify
the minimum rate at which deterministic discrete-time dynamical systems
generate information relevant to the control objective of set-invariance. More
recently, the notion of (controlled)-invariance entropy has been studied for
continuous-time control systems in [12] and [13] based on the metric-space
technique of [6]. It is noted that such an invariance entropy provides a
measure of the smallest growth rate for the number of open-loop control
functions that are needed to confine the states within an arbitrarily small
distance from a given compact subset of the system state space.
On the other hand, several results have been established to characterize the
stability and/or the performance of switched systems using Lie-brackets –
where feasible, one can consider the Lie-algebra generated by the constituent
systems (a matrix Lie-algebra in the linear case or a Lie-algebra of vector
fields in the general nonlinear case) and use this Lie-algebra to verify or
establish the stability of switched systems (e.g., see [14], [15], [16] and
references therein for a review of switched systems). Note that if the
constituent systems are linear and stable, it should be noted that the
switched system remains stable under an arbitrary switching if the Lie-algebra
is nilpotent (e.g., see [17]) or a compact semi-simple subalgebra (e.g., see
[15]). Moreover, it should be noted that each of these classes of Lie-algebras
strictly contains the others (e.g., see [18], [16]).
## II Preliminaries
### II-A Switched systems
Consider a Lie-algebra (over a real $\mathbb{R}$) that is generated by the
matrices $A_{p}\in\mathbb{R}^{n\times n}$,
$p\in\mathcal{P}\equiv\\{1,2,\ldots,N\\}$ and further identified with
$\displaystyle\mathfrak{g}=\bigl{\\{}A_{p}\colon
p\in\mathcal{P}\bigr{\\}}_{LA}.$ (1)
Let $\mathfrak{g}=\mathfrak{m}\oplus\mathfrak{h}$ be the Levi-Malcev
decomposition of the Lie-algebra, where $\mathfrak{m}$ is the radical (i.e.,
the maximal solvable ideal part) and $\mathfrak{h}$ is the semi-simple
subalgebra part. Then, we can rewrite the matrices $A_{p}$ as
$\displaystyle A_{p}=A_{p}^{\mathfrak{m}}+A_{p}^{\mathfrak{h}},$ (2)
with $A_{p}^{\mathfrak{m}}\in\mathfrak{m}$ and
$A_{p}^{\mathfrak{h}}\in\mathfrak{h}$ for $p\in\mathcal{P}$.
Next, we consider the following family of systems
$\displaystyle\dot{x}(t)=A_{\sigma(t)}x(t),\quad x(0)=x_{0},$ (3)
where $\sigma\colon[0,\,\infty)\to\mathcal{P}$ is a piecewise constant
switching function.111In this brief note, switching that is infinitely fast
(i.e., chattering) is not considered. Moreover, we assume that the matrices
$A_{p}\in\mathbb{R}^{n\times n}$, $\forall p\in\mathcal{P}$, are stable.
Let $\Phi(t,\,0)$ or simply $\Phi(t)$ (assuming that the initial time $t_{0}$
is zero) be the evolution operator for the family of systems in (3) and
observe that
$\displaystyle\dot{\Phi}(t)$ $\displaystyle=A_{\sigma(t)}\Phi(t),$
$\displaystyle=\bigl{(}A_{\sigma(t)}^{\mathfrak{m}}+A_{\sigma(t)}^{\mathfrak{h}}\bigr{)}\Phi(t),$
(4)
with $\sigma\colon[0,\,\infty)\to\mathcal{P}$.
Then, we state the following well-known result that will be useful in the
sequel.
###### Lemma 1 (Levi-Malcev Decomposition)
The evolution operator $\Phi(t)$ can be decomposed as follow
$\displaystyle\Phi(t)=\Phi^{\mathfrak{h}}(t)\Phi^{\mathfrak{m}}(t),$ (5)
where
$\displaystyle\dot{\Phi}^{\mathfrak{h}}(t)=A_{\sigma(t)}^{\mathfrak{h}}\Phi^{\mathfrak{h}}(t),\quad\Phi^{\mathfrak{h}}(0)=I,$
(6)
and
$\displaystyle\dot{\Phi}^{\mathfrak{m}}(t)=\biggm{(}\bigl{(}\Phi^{\mathfrak{h}}(t)\bigr{)}^{-1}A_{\sigma(t)}^{\mathfrak{m}}\Phi^{\mathfrak{h}}(t)\biggm{)}\Phi^{\mathfrak{m}}(t),\quad\Phi^{\mathfrak{m}}(0)=I.$
(7)
The proof follows the same lines of argument as that of [18].
Proof: Note that if we differentiate equation (5), i.e., the evolution
operator $\Phi(t)$, with respect to time and also make use of the relations in
(6) and (7), then we have
$\displaystyle\frac{d}{dt}\bigm{(}\Phi^{\mathfrak{h}}(t)\Phi^{\mathfrak{m}}(t)\bigm{)}$
$\displaystyle=\frac{d}{dt}\bigm{(}\Phi^{\mathfrak{h}}(t)\bigm{)}\Phi^{\mathfrak{m}}(t)+\Phi^{\mathfrak{h}}(t)\frac{d}{dt}\bigm{(}\Phi^{\mathfrak{m}}(t)\bigm{)},$
$\displaystyle=A_{\sigma(t)}^{\mathfrak{h}}\Phi^{\mathfrak{h}}(t)\Phi^{\mathfrak{m}}(t)+\Phi^{\mathfrak{h}}(t)\frac{d}{dt}\bigm{(}\Phi^{\mathfrak{m}}(t)\bigm{)},$
$\displaystyle=A_{\sigma(t)}^{\mathfrak{h}}\Phi^{\mathfrak{h}}(t)\Phi^{\mathfrak{m}}(t)+\Phi^{\mathfrak{h}}(t)\biggm{(}\bigm{(}\Phi^{\mathfrak{h}}(t)\bigm{)}^{-1}A_{\sigma(t)}^{\mathfrak{m}}\Phi^{\mathfrak{h}}(t)\biggm{)}\Phi^{\mathfrak{m}}(t),$
$\displaystyle=\bigm{(}A_{\sigma(t)}^{\mathfrak{h}}+A_{\sigma(t)}^{\mathfrak{m}}\bigm{)}\Phi^{\mathfrak{h}}(t)\Phi^{\mathfrak{m}}(t),$
$\displaystyle=A_{\sigma(t)}\Phi^{\mathfrak{h}}(t)\Phi^{\mathfrak{m}}(t).$ (8)
$\Box$
Next, we introduce the following notion of stability for the family of systems
in (3).
###### Definition 1
The switched system in (3) is said to be globally uniformly exponentially
stable (GUES), if there exist positive numbers $M$ and $\lambda$ such that the
solutions of (3) satisfy
$\displaystyle|x(t)|\leq M\exp(-\lambda t)|x(0)|,\quad\forall t\geq 0.$ (9)
### II-B Topological entropy for switched systems
We start by providing the definition of topological entropy for switched
linear systems that corresponds to the semi-simple subalgebra part (e.g., see
[6] or [1] for additional discussions on the topological entropy for
continuous transformations).
###### Definition 2
A set $\mathscr{F}$ is $(T,\,\epsilon)$-spanning another set $\mathscr{K}$
(with respect to $\Phi^{\mathfrak{h}}(t)\equiv\Phi_{t}^{\mathfrak{h}}$) if
there exists $y\in\mathscr{F}$ for each $x\in\mathscr{K}$ such that
$\displaystyle\sup_{\begin{subarray}{c}t\in[0,\,T]\end{subarray}}\bigl{\|}\Phi_{t}^{\mathfrak{h}}x-\Phi_{t}^{\mathfrak{h}}y\bigr{\|}\leq\epsilon,$
(10)
where $\epsilon$ is a positive real number.
For a compact subset $\mathscr{K}\subset\mathscr{X}$ (where $\mathscr{X}$ is a
compact $n$-dimensional $C^{\infty}$ manifold), let
$r(T,\epsilon,\mathscr{K},\Phi_{t}^{\mathfrak{h}})$ be the smallest
cardinality of any subset $\mathscr{F}\subset\mathscr{X}$ that
$(T,\,\epsilon)$-spans the set $\mathscr{K}$.222Note that the compactness of
$\mathscr{X}$ implies that there exit finite $(T,\,\epsilon)$-spanning sets.
Then, we have the following properties for
$r(T,\epsilon,\mathscr{K},\Phi_{t}^{\mathfrak{h}})$.
1. (i)
Clearly $r(T,\epsilon,\mathscr{K},\Phi_{t}^{\mathfrak{h}})\in[0,\,\infty)$.
2. (ii)
If $\epsilon_{1}<\epsilon_{2}$, then
$r(T,\epsilon_{1},\mathscr{K},\Phi_{t}^{\mathfrak{h}})\geq
r(T,\epsilon_{2},\mathscr{K},\Phi_{t}^{\mathfrak{h}})$.
###### Definition 3
The topological entropy for the switched linear system that corresponds to the
semi-simple subalgebra is given by
$\displaystyle
h(\mathscr{K},\Phi_{t}^{\mathfrak{h}})=\,\lim_{\begin{subarray}{c}\epsilon\searrow
0\end{subarray}}\biggm{\\{}\limsup_{\begin{subarray}{c}T\to\infty\end{subarray}}\frac{1}{T}\log
r(T,\epsilon,\mathscr{K},\Phi_{t}^{\mathfrak{h}})\biggm{\\}}.$ (11)
Then, we have the following additional properties for
$h(\mathscr{K},\Phi_{t}^{\mathfrak{h}})$.
1. (i)
$h(\mathscr{K},\Phi_{t}^{\mathfrak{h}})\in[0,\,\infty)\cup\\{\infty\\}$.
2. (ii)
If $\mathscr{K}=\bigcup_{l\in\\{1,2,\ldots,L\\}}\mathscr{K}_{l}$ with compact
$\mathscr{K}_{l}$, then
$h(\mathscr{K},\Phi_{t}^{\mathfrak{h}})=\max_{\begin{subarray}{c}l\in\\{1,2,\ldots,L\\}\end{subarray}}h(\mathscr{K}_{l},\Phi_{t}^{\mathfrak{h}})$.
## III Main result
In the following, using the Levi-Malcev decomposition of the Lie-algebra, we
establish a connection between the topological entropy for the evolution
operator (that corresponds to the semi-simple subalgebra part of the Lie-
algebra) and the stability of switched linear systems.
###### Proposition 1
Let $\mathscr{K}\subset\mathscr{X}$ be a compact subset. Suppose that the
Levi-Malcev decomposition of the Lie-algebra corresponding to the matrices
$A_{p}$ with $p\in\mathcal{P}$ is given by (5). Furthermore, if the
topological entropy for the switched linear system corresponding to the semi-
simple subalgebra part satisfies the following
$\displaystyle
h(\mathscr{K},\Phi_{t}^{\mathfrak{h}})<-\bar{\lambda}_{p}^{\mathfrak{m}},\quad\forall
p\in\mathcal{P},$ (12)
where
$\displaystyle\bar{\lambda}_{p}^{\mathfrak{m}}=\max\biggm{\\{}\operatorname{Re}\\{\lambda\\}\colon\lambda\in\operatorname{Sp}\bigl{(}A_{p}^{\mathfrak{m}}\bigr{)}\biggm{\\}},\quad
p\in\mathcal{P}.$ (13)
Then, the family of systems in (3) are
GUES.333$\operatorname{Sp}(A_{p}^{\mathfrak{m}})$ denotes the spectrum for the
matrix $A_{p}^{\mathfrak{m}}\in\mathbb{R}^{n\times n}$.
Proof: Note that if the evolution operator $\Phi(t)$ in (2) admits a
decomposition of the form in (5), then any $\mathfrak{m}$-valued solution
$x_{\mathfrak{m}}(t)$ corresponding to $\Phi^{\mathfrak{m}}(t)$ satisfies the
following
$\displaystyle|x_{\mathfrak{m}}(t)|\leq\exp(\bar{\lambda}_{p}^{\mathfrak{m}}t)|x_{\mathfrak{m}}(0)|,\quad\forall
t\in(0,\,T],\quad\forall p\in\mathcal{P},$ (14)
with
$\bar{\lambda}_{p}^{\mathfrak{m}}=\max\bigl{\\{}\operatorname{Re}\\{\lambda\\}\colon\lambda\in\operatorname{Sp}\bigl{(}A_{p}^{\mathfrak{m}}\bigr{)}\bigr{\\}}$
for $p\in\mathcal{P}$.
On the other hand, the characteristic Lyapunov exponent
${\lambda_{p}^{\mathfrak{h}}}^{*}$ for the evolution operator
$\Phi_{t}^{\mathfrak{h}}$ is given by
$\displaystyle{\lambda_{p}^{\mathfrak{h}}}^{*}=\,\lim_{\begin{subarray}{c}t\to\infty\end{subarray}}\sup\frac{1}{t}\log\bigm{|}\operatorname{det}\bigl{(}\Phi^{\mathfrak{h}}(t)\bigr{)}\bigm{|},\quad
p\in\mathcal{P},$ (15)
where such information provides a lower bound for the topological entropy
$h(\mathscr{K},\Phi_{t}^{\mathfrak{h}})$ that corresponds to the semi-simple
subalgebra part (e.g., see [19], [20] or [21] for details on the relationships
between Lyapunov exponents and entropy).444We remark that the topological
entropy of a measure-preserving transformation always majorizes the measure-
theoretic entropy with respect to any of its invariant probability measures
(see also [1]).
Then, the set of solutions for the family of systems in (3) is exponentially
bounded (i.e., the family of systems in (3) are GUES), if the following
condition holds (e.g., see also [22])
$\displaystyle\bar{\lambda}_{p}^{\mathfrak{m}}+{\lambda_{p}^{\mathfrak{h}}}^{*}<0,\quad\forall
p\in\mathcal{P}.$ (16)
Hence, this further implies the following
$\displaystyle\bar{\lambda}_{p}^{\mathfrak{m}}+h(\mathscr{K},\Phi_{t}^{\mathfrak{h}})<0,\quad\forall
p\in\mathcal{P},$
which completes the proof. $\Box$
## References
* [1] Walters, P. (1982). An introduction to ergodic theory. New York, Springer.
* [2] Sinai, Y. G. (1994). Topics in ergodic theory. Princeton, N.J., Princeton University Press.
* [3] Downarowicz, T. (2011). Entropy in dynamical systems. Cambridge, Cambridge Press.
* [4] Adler, R. Konheim, A. & McAndrew, M. (1965). Topological entropy. Trans. Amer. Math. Soc., 114, 309–319.
* [5] Dinaburg, E. I. (1970). The relation between topological entropy and metric entropy. Dokl. Akad. Nauk SSSR, 190, 19–22.
* [6] Bowen, R. (1971). Entropy for group endomophisms and homogenous spaces. Trans. Amer. Math. Soc., 153, 401–414.
* [7] Goodman, T. N. T. (1971). Relating topological entropy and measure entropy. Bull. Lond. Math. Soc., 3, 176–180.
* [8] Goodman, T. N. T. (1972). Comparing topological entropy with measure-theoretic entropy. Amer. J. Math., 94(2), 366–388.
* [9] Antsaklis, P. & Baillieul, J. (2007). Special issue on the technology of networked control systems. Proceedings of IEEE, 95(1).
* [10] Nair, G. N. Evans, R. J. Mareels, I. M. Y. & Moran, W. (2004). Topological feedback entropy and nonlinear stabilization. IEEE Trans. Automat. Contr., 49(9), 1585–1597.
* [11] Savkin, A. V. (2006). Analysis and synthesis of networked control systems: topological entropy, observability, robustness, and optimal control. Automatica, 42(1), 51–62.
* [12] Colonius, F. & Kawan, C. (2009). Invariance entropy for control systems. SIAM J. Control Optim., 48(3), 1701–1721.
* [13] Colonius, F. & Kawan, C. (2011). Invariance entropy with outputs. Math. Control Sig. Syst., 22(3), 203–227.
* [14] Liberzon, D. Hespanha, J. P. & Morse, A. S. (1999). Stability of switched systems: a Lie-algebraic condition. Syst. Contr. Lett., 37(3), 117–122.
* [15] Agrachev, A. A. & Liberzon, D. (2001). Lie-algebraic stability criteria for switched systems. SIAM J. Control Optim., 40(1), 253–269.
* [16] Liberzon, D. (2003). Switching in systems and control. Systems & Control: Foundations & Applications, Boston, Birkhäuser.
* [17] Gurvits, L. (1995). Stability of discrete linear inclusion. Linear Alg. Appl., 231, 47–85.
* [18] Chen, K. T. (1962). Decomposition of differential equations. Math. Ann., 146, 263–278.
* [19] Yomdin, Y. (1987). Volume growth and entropy. Isr. J. Math., 57, 285–300.
* [20] Pesin, Ya. B. (1977). Lyapunov characteristic exponents and smooth ergodic theory. Russ. Math. Surv., 32(4), 55–114.
* [21] Katok, A. B. (1980). Lyapunov exponents, entropy and periodic orbits for diffeomorphisms. Publ. Math. IHES, 51(1), 137–173.
* [22] Agrachev, A. A. Baryshnikov, Y. & Liberzon, D. (2012). On robust Lie-algebraic stability conditions for switched linear systems. Syst. Contr. Lett., 61(2), 347–353.
|
arxiv-papers
| 2013-10-20T17:11:46 |
2024-09-04T02:49:52.610953
|
{
"license": "Public Domain",
"authors": "Getachew K. Befekadu",
"submitter": "Getachew Befekadu",
"url": "https://arxiv.org/abs/1310.5351"
}
|
1310.5371
|
# Intrinsic scaling properties for nonlocal operators
Moritz Kassmann and Ante Mimica Fakultät für Mathematik
Universität Bielefeld
Postfach 100131
D-33501 Bielefeld
Germany [email protected] Department of Mathematics
University of Zagreb
Bijenička cesta 30
10000 Zagreb
Croatia [email protected]
###### Abstract.
We study growth lemmas and questions of regularity for generators of Markov
processes. The generators are allowed to have an arbitrary order of
differentiability less than $2$. In general, this order is represented by a
function and not by a number. The approach enables a careful study of
regularity issues up to the phase boundary between integro-differential
(positive order of differentiability) and integral operators (nonnegative
order of differentiability). The proof is based on intrinsic scaling
properties of the underlying operators and stochastic processes.
###### 2010 Mathematics Subject Classification:
Primary 35B65; Secondary: 60J75, 47G20, 31B05
Research supported by German Science Foundation (DFG) via SFB 701\. Research
supported by MZOS grant 037-0372790-2801.
## 1\. Introduction
One key argument in the regularity theory of differential equations of second
order is the so called growth lemma. Here is an example which is by now
classical. Let $A$ be an elliptic operator of second order, e.g.
$Au=\sum_{i,j}a_{ij}(\cdot)\tfrac{\partial}{\partial
x_{i}}\tfrac{\partial}{\partial x_{j}}u$ for
$u:{\mathbb{R}}^{d}\to{\mathbb{R}}$ where $(a_{ij}(\cdot))_{i,j}$ is uniformly
positive definite and bounded. One could also consider nonlinear examples. The
following growth lemma holds true in many cases:
###### Lemma 1.1.
There is a constant $\theta\in(0,1)$ such that, if $R>0$ and
$u:{\mathbb{R}}^{d}\to{\mathbb{R}}$ with
$\displaystyle-Au\leq 0\text{ in }B_{2R}\,,\qquad u\leq 1\text{ in
}B_{2R},\qquad|(B_{2R}\\!\setminus\\!B_{R})\cap\\{u\leq
0\\}|\geq\tfrac{1}{2}|B_{2R}\\!\setminus\\!B_{R}|\,,$
then $u\leq 1-\theta$ in $B_{R}$.
Such lemmas are systematically studied and applied in [Lan71]. Their
importance is underlined in the article [KS79], in which the authors establish
a priori bounds for elliptic equations of second order with bounded measurable
coefficents. Nowadays they form a standard tool for the study of various
questions of nonlinear partial differential equations of second order, cf.
[CC95] and [DGV12]. Note that the property formulated in 1.1 is also referred
to as expansion of positivity which describes the corresponding property for
$1-u$.
In the case of a linear differential operator $A$ the above lemma can be
established with the help of the Markov process it generates. Let $X$ be the
strong Markov process associated with the operator $A$, i.e. we assume that
the martingale problem has a unique solution. Denote by $T_{A},\tau_{A}$ the
hitting resp. exit time for a measurable set $A\subset{\mathbb{R}}^{d}$ and by
$\mathbb{P}_{x}$ the measure on the path space with
$\mathbb{P}_{x}(X_{0}=x)=1$. The following property is a key to the above
growth lemma.
###### Proposition 1.2.
There is a constant $c\in(0,1)$ such that for every $R>0$ and every measurable
set $A\subset B_{2R}\\!\setminus\\!B_{R}$ with
$|(B_{2R}\\!\setminus\\!B_{R})\cap
A|\geq\tfrac{1}{2}|B_{2R}\\!\setminus\\!B_{R}|$ and $x\in B_{R}$
$\displaystyle\mathbb{P}_{x}(T_{A}<\tau_{B_{2R}})\geq c\,.$ (1.1)
The aim of this work is to establish a result like 1.2 and regularity
estimates for a general class of operators and stochastic processes. The
article [KS79] deals with a very specific case: operators of second order.
Another very specific case, operators of fractional order $\alpha\in(0,2)$, is
treated in [BL02]. Therein it is shown that 1.2 holds true for jump processes
$X$ generated by integral operators ${\mathcal{L}}\colon
C^{2}_{b}({\mathbb{R}}^{d})\rightarrow C({\mathbb{R}}^{d})$ of the form
$\displaystyle\mathcal{L}u(x)$
$\displaystyle=\int\limits_{{\mathbb{R}}^{d}\setminus\\{0\\}}\big{(}u(x+h)-u(x)-\langle\nabla
u(x),h\rangle\mathbbm{1}_{B_{1}}(h)\big{)}K(x,h)\,dh$ (1.2)
$\displaystyle=\frac{1}{2}\int\limits_{{\mathbb{R}}^{d}\setminus\\{0\\}}\big{(}u(x+h)-2u(x)+u(x-h)\big{)}K(x,h)\,dh\,,$
(1.3)
under the assumption $K(x,h)=K(x,-h)$ and $K(x,h)\asymp|h|^{-d-\alpha}$ for
all $x$ and $h$ where $\alpha\in(0,2)$ is fixed. Note that this class includes
the case $\mathcal{L}u=-(-\Delta)^{\alpha/2}u$ and versions with bounded
measurable coefficients. As [KS79] does, the article [BL02] establishes a
priori estimates in Hölder spaces. Results like 1.1 have been obtained for
operators in the case $K(x,h)\asymp|h|^{-d-\alpha}$ also for nonlinear
problems, cf. [Sil06], [CS09] and [GS12].
The starting point of our research is the observation that 1.2 fails to hold
for several interesting cases. One example is given by $\mathcal{L}$ as in
(1.2) with $K(x,h)=k(h)\asymp|h|^{-d}$ for $|h|\leq 1$ and some appropriate
condition for $|h|>1$. For example, the geometric stable process with its
generator $-\ln(1+(-\Delta)^{\alpha/2})$, $0<\alpha\leq 2$, can be represented
by (1.2) with a kernel $K(x,h)=k(h)$ with such a behaviour for $|h|$ close to
zero. The operator resp. the corresponding stochastic process can be shown not
to satisfy a uniformly hitting estimate like (1.1). This leads to the question
whether a priori estimates can be obtained by this approach at all.
Given a linear operator with bounded measurable coefficients of the form
(1.2), the main idea of this article is to determine an intrinsic scale which
allows to establish a modification of (1.1). We choose a measure different
from the Lebesgue measure for the assumption
$|(B_{2R}\\!\setminus\\!B_{R})\cap
A|\geq\tfrac{1}{2}|B_{2R}\\!\setminus\\!B_{R}|$.
Let us formulate our assumptions and results. Assume $0\leq\alpha<2$ and let
$K\colon{\mathbb{R}}^{d}\times({\mathbb{R}}^{d}\setminus\\{0\\})\rightarrow[0,\infty)$
be a measurable function satisfying the following conditions:
$\displaystyle\ \ \
\sup\limits_{x\in{\mathbb{R}}^{d}}\int\limits_{{\mathbb{R}}^{d}\setminus\\{0\\}}(1\wedge|h|^{2})K(x,h)\,dh\leq
K_{0}\,,$ ($K_{1}$) $\displaystyle\ \ \
K(x,h)=K(x,-h)\qquad(x\in{\mathbb{R}}^{d},\,h\in{\mathbb{R}}^{d})\,,$
($K_{2}$) $\displaystyle\ \ \ \kappa^{-1}\,\frac{\ell(|h|)}{|h|^{d}}\leq
K(x,h)\leq\kappa\,\frac{\ell(|h|)}{|h|^{d}}\qquad(0<|h|\leq 1)$ ($K_{3}$)
for some numbers $K_{0}>0$, $\kappa>1$ and some function
$\ell\colon(0,1)\rightarrow(0,\infty)$ which is locally bounded and varies
regularly at zero with index $-\alpha\in(-2,0]$. Possible examples could be
$\ell(s)=1$, $\ell(s)=s^{-3/2}$ and $\ell(s)=s^{-\beta}\ln(\tfrac{2}{s})^{2}$
for some $\beta\in(0,2)$, see Appendix A for a more detailed discussion.
Suppose that there exists a strong Markov process $X=(X_{t},\mathbb{P}_{x})$
with trajectories that are right continous with left limits associated with
${\mathcal{L}}$ in the sense that for every $x\in{\mathbb{R}}^{d}$
* (i)
$\mathbb{P}_{x}(X_{0}=x)=1$;
* (ii)
for any $f\in C_{b}^{2}({\mathbb{R}}^{d})$ the process
$\big{\\{}f(X_{t})-f(X_{0})-\int_{0}^{t}{\mathcal{L}}f(X_{s})\,ds|\,t\geq
0\big{\\}}$ is a martingale under $\mathbb{P}_{x}$.
Note that the existence of such a Markov process comes for free in the case
when $K(x,h)$ is independent of $x$, see Section 2. In the general case it has
been established by many authors in different general contexts, see the
discussion in [AK09]. Denote by $\tau_{A}=\inf\\{t>0|\,X_{t}\not\in A\\}$,
$T_{A}=\inf\\{t>0|\,X_{t}\in A\\}$ the first exit time resp. hitting time of
the process $X$ for a measurable set $A\subset{\mathbb{R}}^{d}$.
###### Definition 1.3.
A bounded function $u\colon{\mathbb{R}}^{d}\rightarrow{\mathbb{R}}$ is said to
be harmonic in an open subset $D\subset{\mathbb{R}}^{d}$ with respect to $X$
(and ${\mathcal{L}}$) if for any bounded open set $B\subset\overline{B}\subset
D$ the stochastic process $\\{u(X_{\tau_{B}\wedge t})|\,t\geq 0\\}$ is a
$\mathbb{P}_{x}$-martingale for every $x\in{\mathbb{R}}^{d}$ .
Before we can formulate our results we need to introduce an additional
quantity. Note that ($K_{1}$) and ($K_{3}$) imply that
$\int_{0}^{1}s\,\ell(s)\,\,\textnormal{d}s\leq c$ holds for some constant
$c>0$. Let $L\colon(0,1)\rightarrow(0,\infty)$ be defined by
$L(r)=\int_{r}^{1}\frac{\ell(s)}{s}\,\,\textnormal{d}s$. The function $L$ is
well defined because $L(r)\leq
r^{-2}\int_{r}^{1}s^{2}\frac{\ell(s)}{s}\,\,\textnormal{d}s\leq cr^{-2}$. See
Appendix A for several examples. We note that the function $L$ is always
decreasing. Our main result concerning regularity is the following result:
###### Theorem 1.4.
There exist constants $c>0$ and $\gamma\in(0,1)$ so that for all
$r\in(0,\frac{1}{2})$ and $x_{0}\in{\mathbb{R}}^{d}$
$\displaystyle|u(x)-u(y)|\leq
c\|u\|_{\infty}\frac{L(|x-y|)^{-\gamma}}{L(r)^{-\gamma}},\ \ x,y\in
B_{r/4}(x_{0})$ (1.4)
for all bounded functions $u\colon{\mathbb{R}}^{d}\rightarrow{\mathbb{R}}$
that are harmonic in $B_{r}(x_{0})$ with respect to $\mathcal{L}$.
Let us comment on this result. It is important to note that the result
trivially holds if the function $L:(0,1)\to(0,\infty)$ satisfies
$\lim\limits_{r\to 0+}L(r)<+\infty$. This is equivalent to the condition
$\displaystyle\int\limits_{B_{1}}\frac{\ell(|h|)}{|h|^{d}}\,\,\textnormal{d}h<+\infty\,,$
(1.5)
which, in the case $K(x,h)=k(h)$, means that the Lévy measure is finite. Thus,
for the proof, we can concentrate on cases where (1.5) does not hold. One
feature of this article is that our result holds true up to and across the
phase boundary determined by whether the kernel $K(x,\cdot)$ is integrable
(finite Lévy measure) or not.
Furthermore, note that the main result of [BL02] is implied by 1.4 since the
choice $\ell(s)=s^{-\alpha}$, $\alpha\in(0,2)$, leads to $L(r)\asymp
r^{-\alpha}$. Given the whole spectrum of possible operators covered by our
approach, this choice is a very specific one. It allows to use scaling methods
in the usual way which are not at our disposal here. Table 1 in Appendix A
contains several admissible examples one of which leads to $L(0)<+\infty$
which means, as explained above, that (1.4) becomes pointless.
The main ingredient in the proof of 1.4 is a new version of 1.2 which we
provide now. For $r\in(0,1)$ we define a measure $\mu_{r}$ by
$\mu_{r}(dx)=\frac{\ell(|x|)}{L(|x|)|x|^{d}}\,\mathbbm{1}_{B_{1}\\!\setminus\\!B_{r}}(x)\,dx\,.$
(1.6)
Moreover, for $a>1$, we define a function $\varphi_{a}:(0,1)\to(0,1)$ by
$\varphi_{a}(r)=L^{-1}(\frac{1}{a}L(r))$. The following result is our
modification of 1.2.
###### Proposition 1.5.
There exists a constant $c>0$ such that for all $a>1$, $r\in(0,\frac{1}{2})$
and measurable sets $A\subset B_{\varphi_{a}(r)}\\!\setminus\\!B_{r}$ with
$\mu_{r}(A)\geq\frac{1}{2}\mu_{r}(B_{\varphi_{a}(r)}\\!\setminus\\!B_{r})$
$\mathbb{P}_{x}(T_{A}<\tau_{B_{\varphi_{a}(r)}})\geq\mathbb{P}_{x}(X_{\tau_{B_{r}}}\in
A)\geq c\,\tfrac{\ln{a}}{a}$
holds true for all $x\in B_{r/2}$.
The main novelties of 1.5 are that the measure $\mu_{r}$ depends on $r$ and
that its density carries the factor $|x|^{-d}$. These two changes allow us to
deal with the classical cases as well as with critical cases, e.g. given by
$K(x,h)\asymp|h|^{-d}\mathbbm{1}_{B_{1}}(h)$.
The article is organised as follows: In Section 2 we review the relation
between translation invariant nonlocal operators and semigroups/Lévy
processes. Presumably, 2.1 is interesting to many readers since it establishes
a one-to-one relation between the behavior of a Lévy measure at zero and the
multiplier of the corresponding generator for large values of $|\xi|$. In
Section 3 we establish all tools needed to prove 1.5 which is a special case
of 3.4. Section 4 contains the proof of 1.4. The last section is Appendix A in
which we collect important properties of regularly resp. slowly varying
functions. Moreover, the appendix contains a table with six examples which
illustrate the range of applicability of our approach.
Throughout the paper we use the notation $f(r)\asymp g(r)$ to denote that the
ration $f(r)/g(r)$ stays between two positive constants as $r$ converges to
some value of interest.
## 2\. Translation invariant operators
The aim of this section is to discuss properties of the operator $\mathcal{L}$
from (1.2) in the translation invariant case, i.e. when $K(x,h)$ does not
depend on $x\in{\mathbb{R}}^{d}$. In this case there is a one-to-one
correspondence between $\mathcal{L}$ and multipliers, semigroups and
stochastic processes. One aim is to prove how the behavior of $\ell(|h|)$ for
small values of $|h|$ translates into properties of the multiplier or
characteristic exponent $\psi(|\xi|)$ for large values of $|\xi|$. This is
acheived in 2.1. We add a subsection where we discuss which regularity results
are known in critical cases of the (much simpler) translation invariant case.
Note that our set-up, although allowing for a irregular dependence of $K(x,h)$
on $x\in{\mathbb{R}}^{d}$, leads to new results in these critical cases.
### 2.1. Generators of convolution semigroups and Lévy processes
In this section we consider space homogeneous kernels of the form
$K(x,h)=k(h)$ satisfing ($K_{1}$)–($K_{3}$). As we will see, the underlying
stochastic process belongs to the class of Lévy processes .
A stochastic process $X=(X_{t})_{t\geq 0}$ on a probability space
$(\Omega,\mathcal{F},\mathbb{P})$ is called a Lévy process if it has
stationary and independent increments, $\mathbb{P}(X_{0}=0)=1$ and its paths
are $\mathbb{P}$-a.s. right continous with left limits . For
$x\in{\mathbb{R}}^{d}$ we define a $\mathbb{P}_{x}$ to be the law of the
process $X+x$ . In particular, $\mathbb{P}_{x}(X_{t}\in B)=\mathbb{P}(X_{t}\in
B-x)$ for $t\geq 0$ and measurable sets $B\subset{\mathbb{R}}^{d}$ .
Due to stationarity and independence of increments, the characteristic
function of $X_{t}$ is given by
$\mathbb{E}[e^{i\langle\xi,X_{t}\rangle}]=e^{-t\psi(\xi)},$
where $\psi$ is called characteristic exponent of $X$. It has the following
Lévy-Khintchine representation
$\psi(\xi)=\frac{1}{2}\langle A\xi,\xi\rangle+\langle
b,\xi\rangle+\int_{{\mathbb{R}}^{d}\setminus\\{0\\}}(1-e^{i\langle\xi,h\rangle}+i\langle\xi,h\rangle\mathbbm{1}_{B_{1}}(h))\nu(dh)\,,$
(2.1)
where $A$ is a symmetric non-negative definite matrix , $b\in{\mathbb{R}}^{d}$
and $\nu$ is a measure on ${\mathbb{R}}^{d}\setminus\\{0\\}$ satisfying
$\int_{{\mathbb{R}}^{d}\setminus\\{0\\}}(1\wedge|y|^{2})\nu(dy)<\infty$ called
the Lévy measure of $X$.
The converse also holds; that is, given $\psi$ as in the Lévy-Khintchine
representation (2.1), there exists a Lévy process $X=\\{X_{t}\\}_{t\geq 0}$
with the characteristic exponent $\psi$ . Details about Lévy processes can be
found in [Ber96, Sat99] .
To make a connection with our set-up, let $\nu$ be a measure defined by
$\nu(dh)=k(h)\,dh$. It follows from ($K_{1}$)–($K_{3}$) that $\nu$ is a
symmetric Lévy measure. Let $X=\\{X_{t}\\}_{t\geq 0}$ be a Lévy process
corresponding to the characteristic exponent $\psi$ as in (2.1) with $A=0$,
$b=0$ and the Lévy measure $\nu(dh)=k(h)\,dh$ .
Now, $P_{t}f(x):=\mathbb{E}_{x}[f(X_{t})]$ defines a strongly continuous
contraction semigoup of operators $(P_{t})_{t\geq 0}$ on the space
$L^{\infty}({\mathbb{R}}^{d})$ equipped with the essential-supremum norm.
Moreover, it is a convolution semigroup, since
$\mathbb{P}_{t}f(x)=\mathbb{E}_{0}[f(x+X_{t})]=\int_{{\mathbb{R}}^{d}}f(x+y)\mu_{t}(dy)\,,$
where $(\mu_{t})_{t\geq 0}$ is a convolution semigroup of (probability)
measures defined by $\mu_{t}(B):=\mathbb{P}(X_{t}\in B)$.
The infinitesimal generator ${\mathcal{L}}$ of the semigroup $(P_{t})_{t\geq
0}$ is given by
$\displaystyle{\mathcal{L}}u(x)=\int_{{\mathbb{R}}^{d}\setminus\\{0\\}}\big{(}u(x+h)-u(x)-\langle\nabla
u(x),h\rangle\mathbbm{1}_{B_{1}}(h)\big{)}k(h)\,dh$ (2.2)
(cf. proof of [Sat99, Theorem 31.5]).
Since $\left\\{u(X_{t})-u(X_{0})-\int_{0}^{t}{\mathcal{L}}u(X_{s})\,ds:t\geq
0\right\\}$ is a martingale (with respect to the natural filtration) for every
$u\in C_{b}^{2}({\mathbb{R}}^{d})$ (cf. proof of [RY05, Proposition VII.1.6]),
it follows that $X$ is the process which corresponds to the kernel
$K(x,h)=k(h)$ in our set-up.
It is worth of mentioning that there is a connection between the
characteristic exponent and the symbol of the operator ${\mathcal{L}}$. To be
more precise, if $\hat{f}(\xi)=\int_{{\mathbb{R}}^{d}}e^{i\xi\cdot x}f(x)\,dx$
denotes the Fourier transform of a function $f\in L^{1}({\mathbb{R}}^{d})$,
then
$\widehat{{\mathcal{L}}f}(\xi)=-\psi(-\xi)\hat{f}(\xi)$
for any $f\in\mathcal{S}({\mathbb{R}}^{d})$, where
$\mathcal{S}({\mathbb{R}}^{d})$ is the Schwartz space (cf. [Ber96, Proposition
I.2.9]). Hence $-\psi(-\xi)$ is the symbol (multiplier) of the operator
${\mathcal{L}}$ .
We finish this section with the result that reveals connection between the
characteristic exponent $\psi$ and the function $L$ .
###### Proposition 2.1.
Let $\mathcal{L}:\mathcal{S}\to\mathcal{S}$ be given by (2.2). Assume
$K(x,h):=k(h)$ satisfies ($K_{1}$)-($K_{3}$). There is a constant $c>0$ such
that
$c^{-1}L(|\xi|^{-1})\leq\psi(\xi)\leq cL(|\xi|^{-1})\quad\text{ for
}\xi\in{\mathbb{R}}^{d},\ |\xi|\geq 5\,.$
###### Proof.
Note first that, by ($K_{3}$),
$\kappa^{-1}j(|h|)\leq k(h)\leq\kappa j(|h|),\quad|h|\leq 1\,,$
where $j(s):=s^{-d}\ell(s)\,,\ s\in(0,1)$ .
Since $1-\cos{x}\leq\frac{1}{2}x^{2}$, it follows from ($K_{1}$) and ($K_{3}$)
that
$\displaystyle\psi(\xi)$
$\displaystyle\leq\tfrac{1}{2}|\xi|^{2}\int_{|h|\leq|\xi|^{-1}}|h|^{2}j(|h|)\,dh+2\int_{|\xi|^{-1}<|h|\leq
1}j(|h|)\,dh+2\int_{|h|>1}j(|h|)\,dh$ $\displaystyle\leq
c_{1}\left[|\xi|^{2}\int_{0}^{|\xi|^{-1}}s\ell(s)\,ds+L(|\xi|^{-1})+1\right]$
$\displaystyle\leq c_{2}(\ell(|\xi|^{-1})+L(|\xi|^{-1}))\leq
c_{3}L(|\xi|^{-1})\,,$
where in the first integral of the penultimate inequality Karamata’s theorem
has been used, while in the last inequality we have used that $\ell(s)\leq
c_{3}L(s)$ for $s\in(0,1)$, cf. property (1) in Appendix A.
To prove the lower bound first we choose an orthogonal transformation of the
form $Oe_{1}=|\xi|^{-1}\xi$, where $e_{1}:=(1,0,\ldots,0)\in{\mathbb{R}}^{d}$.
Then a change of variable yields
$\displaystyle\psi(\xi)$
$\displaystyle=\int_{{\mathbb{R}}^{d}\setminus\\{0\\}}(1-\cos(\xi\cdot
h))j(|h|)\,dh=\int_{{\mathbb{R}}^{d}\setminus\\{0\\}}(1-\cos{(|\xi|h_{1})})j(|h|)\,dh$
$\displaystyle\geq\int_{[-1,1]^{d}}(1-\cos{(|\xi|h_{1})})j(|h|)\,dh$
By the Fubini theorem,
$\psi(\xi)\geq 2\int_{0}^{1}(1-\cos{(|\xi|r)})F(r)\,dr,$
where $F(r):=\int_{[-1,1]^{d-1}}j(\sqrt{|z|^{2}+r^{2}})\,dz,\quad
r\in(0,\tfrac{1}{2})$ . It follows from Potter’s theorem (cf. property (4) in
Appendix A) that there is a constant $c_{4}>0$ so that $j(r)\geq c_{4}j(s)$
for all $0<r\leq s<1$. This implies
$F(r)\geq c_{4}F(s),\qquad 0<r\leq s<1\,.$
Hence,
$\displaystyle\psi(\xi)$ $\displaystyle\geq
2\sum_{k=0}^{\lfloor\frac{\pi^{-1}|\xi|-\frac{3}{2}}{2}\rfloor}\int_{|\xi|^{-1}(\frac{\pi}{2}+2k\pi)}^{|\xi|^{-1}(\frac{3\pi}{2}+2k\pi)}(1-\cos{(|\xi|r)})F(r)\,dr\geq\frac{c_{4}\pi}{|\xi|}\sum_{k=0}^{\lfloor\frac{\pi^{-1}|\xi|-\frac{3}{2}}{2}\rfloor}F(|\xi|^{-1}(\tfrac{3\pi}{2}+2k\pi))$
$\displaystyle\geq
c_{4}^{2}\sum_{k=0}^{\lfloor\frac{\pi^{-1}|\xi|-\frac{3}{2}}{2}\rfloor}\int_{|\xi|^{-1}(\frac{3\pi}{2}+2k\pi)}^{|\xi|^{-1}(\frac{3\pi}{2}+(2k+1)\pi)}F(r)\,dr\geq
c_{4}^{2}\int_{\frac{3\pi}{2}|\xi|^{-1}}^{1}F(r)\,dr$ $\displaystyle\geq
c_{5}\int_{\frac{3\pi}{2}|\xi|^{-1}\leq|h|\leq
1}j(|h|)\,dh=c_{6}L(\tfrac{3\pi}{2}|\xi|^{-1})\geq c_{7}L(|\xi|^{-1})\,,$
where, in the last inequality, we have used property (4) from Appendix A. Note
that [Grz13] uses a similar trick to bound $\psi$ from below. ∎
### 2.2. Known results in the translation invariant case
Let us explain which results, related to Theorem 1.4, have been obtained in
the case where $K(x,h)$ is independent of $x\in{\mathbb{R}}^{d}$.
Hölder estimates of harmonic functions are obtained for the Lévy process with
the characteristic exponent $\psi(\xi)=\frac{|\xi|^{2}}{\ln(1+|\xi|^{2})}-1$
in [Mim13a] by establishing a Krylov-Safonov type estimate replacing the
Lebesgue measure with the capacity of the sets involved. Recently, regularity
estimates have been obtained in [Grz13] for a class of isotropic unimodal Lévy
processes which is quite general but does not include Lévy processes with
slowly varying Lévy exponents such as geometric stable processes. Regularity
of harmonic functions for such processes is investigated in [Mim13b], where it
is shown that a result like 1.2 fails. Using the Green function, logarithmic
bounds for the modulus of continuity are obtained. At this point it is worth
mentioning that the transition density $p_{t}(x,y)$ of the geometric stable
process satisfies $p_{1}(x,x)=\infty$, cf. [ŠSV06]. This illustrates that
regularity results like 1.4 in the case $\ell(s)=1$ are quite delicate.
## 3\. Probabilistic estimates
###### Proposition 3.1.
There exists a constant $C_{1}>0$ such that for $x_{0}\in{\mathbb{R}}^{d}$,
$r\in(0,1)$ and $t>0$
$\mathbb{P}_{x_{0}}(\tau_{B_{r}(x_{0})}\leq t)\leq C_{1}t\,L(r)\,.$
###### Proof.
Let $x_{0}\in{\mathbb{R}}^{d}$, $0<r<1$ and let $f\in C^{2}({\mathbb{R}}^{d})$
be a positive function such that
$f(x)=\left\\{\begin{array}[]{cl}|x-x_{0}|^{2},&|x-x_{0}|\leq\frac{r}{2}\\\
r^{2},&|x-x_{0}|\geq r\end{array}\right.$
and for some $c_{1}>0$
$|f(x)|\leq c_{1}r^{2},\ \ \left|\frac{\partial f}{\partial
x_{i}}(x)\right|\leq c_{1}r\ \ \textrm{ and }\ \
\left|\frac{\partial^{2}f}{\partial x_{i}\partial x_{j}}(x)\right|\leq c_{1}.$
By the optional stopping theorem we get
$\displaystyle\mathbb{E}_{x}f(X_{t\wedge\tau_{B_{r}(x_{0})}})-f(x_{0})=\mathbb{E}^{x}\int_{0}^{t\wedge\tau_{B_{r}(x_{0})}}\mathcal{L}f(X_{s})\,ds,\
\ t>0.$ (3.1)
Let $x\in B_{r}(x_{0})$. We estimate $\mathcal{L}f(x)$ by splitting the
integral in (1.2) into three parts.
$\displaystyle\int_{B_{r}}$ $\displaystyle(f(x+h)-f(x)-\nabla f(x)\cdot
h\mathbbm{1}_{\\{|h|\leq 1\\}})K(x,h)\,dh$ $\displaystyle\leq
c_{2}\int_{B_{r}}|h|^{2}K(x,h)\,dh\leq
c_{2}\kappa\int_{B_{r}}|h|^{2-d}\ell(|h|)\,dh\leq c_{3}r^{2}\ell(r),$
where in the last line we have used Karamata’s theorem, cf. property (2) in
Appendix A. On the other hand, on $B_{r}^{c}$ we have
$\displaystyle\int_{B_{r}^{c}}$ $\displaystyle(f(x+h)-f(x))K(x,h)\,dh\leq
2\|f\|_{\infty}\int_{B_{r}^{c}}K(x,h)\,dh$ $\displaystyle\leq
2\|f\|_{\infty}\left(\kappa\int_{B_{1}\setminus
B_{r}}|h|^{-d}\ell(|h|)\,dh+\int_{B_{1}^{c}}K(x,h)\,dh\right)\leq
c_{4}r^{2}L(r)\,dr\,,$
where we applied property (5) from Appendix A. Last, we estimate
$\displaystyle\left|\int_{B_{1}\setminus B_{r}}h\cdot\nabla
f(x)K(x,h)\,dh\right|$ $\displaystyle\leq c_{1}r\int_{B_{1}\setminus
B_{r}}|h|K(x,h)\,dh$ $\displaystyle\leq c_{1}\kappa r\int_{B_{1}\setminus
B_{r}}|h|^{-d+1}\ell(|h|)\,dh\leq c_{5}r^{2}\ell(r),$
by Karamata’s theorem again. Therefore, by property (1) from Appendix A we
conclude that there is a constant $c_{6}>0$ such that for all $x\in
B_{r}(x_{0})$ and $r\in(0,1)$ we have
$\mathcal{L}f(x)\leq c_{6}r^{2}L(r).$ (3.2)
Let us look again at (3.1). On $\\{\tau_{B_{r}(x_{0})}\leq t\\}$ we have
$X_{t\wedge\tau_{B_{r}(x_{0})}}\in B_{r}(x_{0})^{c}$ and so
$f(X_{t\wedge\tau_{B_{r}(x_{0})}})\geq r^{2}$. Thus, by (3.2) and (3.1) we get
$\mathbb{P}_{x_{0}}(\tau_{B_{r}(x_{0})}\leq t)\leq c_{6}L(r)t.$
∎
###### Proposition 3.2.
There are constants $C_{2}>0$ and $C_{3}>0$ such that for
$x_{0}\in{\mathbb{R}}^{d}$
$\sup_{x\in{\mathbb{R}}^{d}}\mathbb{E}_{x}\tau_{B_{r}(x_{0})}\leq\frac{C_{2}}{L(r)}\,,\quad
r\in(0,1/2)$
and
$\inf_{x\in
B_{r/2}(x_{0})}\mathbb{E}_{x}\tau_{B_{r}(x_{0})}\geq\frac{C_{3}}{L(r)}\,,\quad
r\in(0,1)$
###### Proof.
The proof is similar to the proof of the exit time estimates in [BL02].
(a) First we prove the upper estimate for the exit time. Let
$x\in{\mathbb{R}}^{d}$, $r\in(0,1/2)$ and let
$S=\inf\\{t>0|\,|X_{t}-X_{t-}|>2r\\}$
be the first time of a jump larger than $2r$. With the help of the Lévy system
formula (cf. [BL02, Proposition 2.3]) and ($K_{3}$) we can deduce
$\displaystyle\mathbb{P}_{x}(S\leq L(r)^{-1})$
$\displaystyle=\mathbb{E}_{x}\sum_{t\leq L(r)^{-1}\wedge
S}\mathbbm{1}_{\\{|X_{t}-X_{t-}|>2r\\}}=\mathbb{E}_{x}\int\limits_{0}^{L(r)^{-1}\wedge
S}\int\limits_{B_{2r}^{c}}K(X_{s},h)\,dh\,ds$ $\displaystyle\geq
c_{1}\mathbb{E}_{x}[L(r)^{-1}\wedge
S]\int\limits_{2r}^{1}\frac{\ell(t)}{t}\,dt\,.$ (3.3)
Since $L$ is regularly varying at zero,
$\displaystyle\mathbb{E}_{x}[L(r)^{-1}\wedge S]$ $\displaystyle\geq
L(r)^{-1}\mathbb{P}_{x}(S>L(r)^{-1})\geq
c_{2}L(2r)^{-1}\big{(}1-\mathbb{P}_{x}(S\leq L(r)^{-1})\big{)}\,$
and so it follows from (3.3) that
$\mathbb{P}_{x}(S\leq L(r)^{-1})\geq c_{3}$ (3.4)
with $c_{3}=\frac{c_{1}c_{2}}{c_{1}c_{2}+1}\in(0,1)$. The strong Markov
property and (3.3) lead to
$\mathbb{P}_{x}(S>mL(r)^{-1})\leq(1-c_{3})^{m},\ \ m\in{\mathbb{N}}\,.$
Since $\tau_{B_{r}(x_{0})}\leq S$,
$\displaystyle\mathbb{E}_{x}\tau_{B_{r}(x_{0})}\leq\mathbb{E}_{x}S$
$\displaystyle\leq
L(r)^{-1}\sum_{m=0}^{\infty}(m+1)\mathbb{P}_{x}(S>L(r)^{-1}m)$
$\displaystyle\leq L(r)^{-1}\sum_{m=0}^{\infty}(m+1)(1-c_{3})^{m}\,.$
(b) Now we prove the lower estimate of the exit time. Let $r\in(0,1)$ and
$y\in B_{r/2}(x_{0})$. By 3.1,
$\mathbb{P}_{y}(\tau_{B_{r}(x_{0})}\leq
t)\leq\mathbb{P}_{y}(\tau_{B_{r/2}(y)}\leq t)\leq C_{1}tL(r/2),\quad t>0\,,$
since $B_{r/2}(y)\subset B_{r}(x_{0})$ . Choose $t=\frac{1}{2C_{1}L(r/2)}$.
Then
$\displaystyle\mathbb{E}_{y}\tau_{B_{r}(x_{0})}$
$\displaystyle\geq\mathbb{E}_{y}[\tau_{B_{r}(x_{0})};\tau_{B_{r}(x_{0})}>t]\geq
t\mathbb{P}_{y}(\tau_{B_{r}(x_{0})}>t)$ $\displaystyle\geq
t(1-C_{1}L(r/2)t)=\frac{1}{4C_{1}L(r/2)}\,.$
By (3) from Appendix A we know that $L$ is regularly varying at zero. Hence
there is a constant $c_{1}>0$ such that $L(r/2)\leq c_{1}L(r)$ for all
$r\in(0,1/2)$. Therefore
$\mathbb{E}_{y}\tau_{B_{r}(x_{0})}\geq\frac{1}{4C_{1}c_{1}L(r)}$ .
∎
###### Proposition 3.3.
There is a constant $C_{4}>0$ such that for all $x_{0}\in{\mathbb{R}}^{d}$ and
$r,s\in(0,1)$ satisfying $2r<s$
$\sup_{x\in B_{r}(x_{0})}\mathbb{P}_{x}(X_{\tau_{B_{r}(x_{0})}}\not\in
B_{s}(x_{0}))\leq C_{4}\frac{L(s)}{L(r)}\,.$
###### Proof.
Let $x_{0}\in{\mathbb{R}}^{d}$, $r,s\in(0,1)$ and $x\in B_{r}(x_{0})$. Set
$B_{r}:=B_{r}(x_{0})$. By the Lévy system formula, for $t>0$
$\displaystyle\mathbb{P}_{x}(X_{\tau_{B_{r}}\wedge t}\not\in B_{s})$
$\displaystyle=\mathbb{E}_{x}\sum\limits_{s\leq\tau_{B_{r}}\wedge
t}\mathbbm{1}_{\\{X_{s-}\in B_{r},X_{s}\in
B_{s}^{c}\\}}=\mathbb{E}_{x}\int\limits_{0}^{\tau_{B_{r}}\wedge
t}\int\limits_{B_{s}^{c}}K(X_{s},z-X_{s})\,dz\,ds\,.$
Let $y\in B_{r}$. Since $s\geq 2r$, it follows that $B_{s/2}(y)\subset B_{s}$
and hence
$\displaystyle\int\limits_{B_{s}^{c}}K(y,z-y)\,dz$
$\displaystyle\leq\int\limits_{B_{s/2}(y)^{c}}K(y,z-y)\,dz\leq
c_{1}\int_{s/2}^{1}\frac{\ell(u)}{u}\,du+c_{2}\leq c_{3}L(s)\,.$
where in the last inequality we have used that $L$ varies regularly at zero
and that $\lim\limits_{r\to 0+}L(r)>0$, cf. (5) in Appendix A.
The above considerations together with 3.2 imply
$\mathbb{P}_{x}(X_{\tau_{B_{r}}\wedge t}\not\in B_{s})\leq
c_{3}L(s)\mathbb{E}_{x}\tau_{B_{r}}\leq c_{4}\frac{L(s)}{L(r)}\,.$
Letting $t\to\infty$ we obtain the desired estimate. ∎
For $x_{0}\in{\mathbb{R}}^{d}$ and $r\in(0,1)$ we define the following measure
$\mu_{x_{0},r}(dx)=\frac{\ell(|x-x_{0}|)}{L(|x-x_{0}|)}\,|x-x_{0}|^{-d}\mathbbm{1}_{\\{r\leq|x-x_{0}|<1\\}}\,dx\,.$
(3.5)
Define $\varphi_{a}(r)=L^{-1}(\frac{1}{a}L(r))$ for $r\in(0,1)$ and $a>1$. The
following property is important for the construction below:
$\displaystyle r=L^{-1}(L(r))\leq L^{-1}(\tfrac{1}{a}L(r))=\varphi_{a}(r)\,.$
(3.6)
Now we can prove a Krylov-Safonov type hitting estimate which includes 1.5 as
a special case.
###### Proposition 3.4.
There exists a constant $C_{5}>0$ such that for all
$x_{0}\in{\mathbb{R}}^{d}$, $a>1$, $r\in(0,\frac{1}{2})$ and $A\subset
B_{\varphi_{a}(r)}(x_{0})\setminus B_{r}(x_{0})$ satisfying
$\mu_{x_{0},r}(A)\geq\frac{1}{2}\mu_{x_{0},r}(B_{\varphi_{a}(r)}(x_{0})\setminus
B_{r}(x_{0}))$
$\mathbb{P}_{y}(T_{A}<\tau_{B_{\varphi_{a}(r)}(x_{0})})\geq\mathbb{P}_{y}(X_{\tau_{B_{r}(x_{0})}}\in
A)\geq C_{5}\frac{\ln{a}}{a}\,,\quad y\in B_{r/2}(x_{0})\,.$
###### Proof.
Consider $x_{0}\in{\mathbb{R}}^{d}$, $a>1$, $r\in(0,\frac{1}{2})$ and a set
$A\subset B_{\varphi_{a}(r)}(x_{0})\setminus B_{r}(x_{0})$ satisfying
$\mu_{x_{0},r}(A)\geq\frac{1}{2}\mu_{x_{0},r}(B_{\varphi_{a}(r)}(x_{0})\setminus
B_{r}(x_{0}))$. Set $\mu:=\mu_{x_{0},r}$, $\varphi:=\varphi_{a}$,
$B_{s}:=B_{s}(x_{0})$ and let $y\in B_{r/2}$. The first inequality follows
from $\\{X_{\tau_{B_{r}}}\in A\\}\subset\\{T_{A}<\tau_{B_{\varphi(r)}}\\}$
since $A\subset B_{\varphi(r)}\setminus B_{r}$ .
By the Lévy system formula, for $t>0$,
$\displaystyle\mathbb{P}_{y}(X_{\tau_{B_{r}}\wedge t}\in A)$
$\displaystyle=\mathbb{E}_{y}\sum\limits_{s\leq\tau_{B_{r}}\wedge
t}\mathbbm{1}_{\\{X_{s-}\in B_{r},X_{s}\in
A\\}}=\mathbb{E}_{y}\int\limits_{0}^{\tau_{B_{r}}\wedge
t}\int\limits_{A}K(X_{s},z-X_{s})\,dz\,ds\,.$ (3.7)
Since $|z-x|\leq|z-x_{0}|+|x_{0}-x|\leq|z-x_{0}|+r\leq 2|z-x_{0}|$ for $x\in
B_{r}$ and $z\in B_{r}^{c}$,
$\mathbb{E}_{y}\int\limits_{0}^{\tau_{B_{r}}\wedge
t}\int\limits_{A}K(X_{s},z-X_{s})\,dz\,ds\geq
c_{1}\mathbb{E}_{y}[\tau_{B_{r}}\wedge
t]\int_{A}\frac{\ell(|z-x_{0}|)}{|z-x_{0}|^{d}}\,dz\,,$ (3.8)
where we have used property (4) given in Appendix A.
Since $L$ is decreasing,
$\displaystyle\int_{A}\frac{\ell(|z-x_{0}|)}{|z-x_{0}|^{d}}\,dz$
$\displaystyle=\int_{A}L(|z-x_{0}|)\mu(dz)\geq
L(\varphi(r))\mu(A)\geq\frac{L(r)}{2a}\mu(B_{\varphi(r)}\setminus B_{r})\,.$
(3.9)
Noting that
$\mu(B_{\varphi(r)}\setminus
B_{r})=c_{2}\int_{r}^{\varphi(r)}\frac{1}{L(s)}\frac{\ell(s)\,ds}{s}=-c_{2}\ln
L(s)|_{r}^{\varphi(r)}=c_{2}\ln a\,,$
we conclude from (3.7)–(3.9) that
$\mathbb{P}_{y}(T_{A}<\tau_{B_{\varphi_{a}(r)}(x_{0})})\geq
c_{3}L(r)\frac{\ln{a}}{a}\mathbb{E}_{y}[\tau_{B_{r}}\wedge t]\,.$
Letting $t\to\infty$ and using the lower bound in 3.2 we get
$\displaystyle\mathbb{P}_{y}(T_{A}<\tau_{B_{\varphi_{a}(r)}(x_{0})})\geq
c_{3}L(r)\,\frac{\ln{a}}{a}\,\mathbb{E}_{y}\tau_{B_{r}}\geq
c_{3}L(r)\,\frac{\ln{a}}{a}\,C_{3}L(r)^{-1}=c_{3}C_{3}\frac{\ln{a}}{a}\,.$
∎
## 4\. Reglarity of harmonic functions
###### Proof of 1.4.
Let $x_{0}\in{\mathbb{R}}^{d}$, $r\in(0,\frac{1}{2})$, $x\in B_{r/4}(x_{0})$.
Using (4) from Appendix A with $\delta=1$, we see that there is a constant
$c_{0}\geq 1$ so that
$\displaystyle\frac{L(s)}{L(s^{\prime})}\leq
c_{0}\left(\frac{s}{s^{\prime}}\right)^{-\alpha-1},\quad 0<s<s^{\prime}<1\,.$
(4.1)
Define for $n\in{\mathbb{N}}$
$r_{n}:=L^{-1}(L(\tfrac{r}{2})a^{n-1})\quad\text{ and }\quad
s_{n}:=3\|u\|_{\infty}b^{-(n-1)}$
for some constants $b\in(1,\frac{3}{2})$ and $a>c_{0}2^{\alpha+1}$ that will
be chosen in the proof independently of $n$, $r$ and $u$. As we explained in
the introduction, 1.4 trivially holds true of $\lim\limits_{r\to 0+}L(r)$ is
finite. Thus, we can assume $\lim\limits_{r\to 0+}L(r)$ to be infinite. This
implies that $r_{n}\to 0$ for $n\to\infty$ as it should be.
We will use the following abbreviations:
$B_{n}:=B_{r_{n}}(x),\quad\tau_{n}:=\tau_{B_{n}},\quad
m_{n}:=\inf_{B_{n}}u,\quad M_{n}:=\sup_{B_{n}}u\,.$
We are going to prove
$M_{k}-m_{k}\leq s_{k}$ (4.2)
for all $k\geq 1$.
Assume for a moment that (4.2) is proved. Then, for any $r\in(0,\frac{1}{2})$
and $y\in B_{r/4}(x_{0})\subset B_{r/2}(x)$ we can find $n\in{\mathbb{N}}$ so
that
$r_{n+1}\leq|y-x|<r_{n}\,.$
Furthermore, since $L$ is decreasing, we obtain with
$\gamma=\frac{\ln{b}}{\ln{a}}\in(0,1)$
$\displaystyle|u(y)-u(x)|$ $\displaystyle\leq
s_{n}=3b\|u\|_{\infty}a^{-n\frac{\ln b}{\ln
a}}=3b\|u\|_{\infty}\left[\frac{L(r_{n+1})}{L(\frac{r}{2})}\right]^{-\frac{\ln{b}}{\ln{a}}}\leq
3b\|u\|_{\infty}\left[\frac{L(|x-y|)}{L(\frac{r}{2})}\right]^{-\gamma}\,,$
which proves our assertion. Thus it remains to prove (4.2).
We are going to prove (4.2) by an inductive argument. Obviously,
$M_{1}-m_{1}\leq 2\|u\|_{\infty}\leq s_{1}$. Since $1<b<\frac{3}{2}$, it
follows that
$M_{2}-m_{2}\leq 2\|u\|_{\infty}\leq 3\|u\|_{\infty}b^{-1}=s_{2}\,.$
Assume now that (4.2) is true for all $k\in\\{1,2,\ldots,n\\}$ for some $n\geq
2$.
Let $\varepsilon>0$ and take ${z_{1}},z_{2}\in B_{n+1}$ so that
$u({z_{1}})\leq m_{n+1}+\frac{\varepsilon}{2}\ \ \ \ \ \ \ \ u(z_{2})\geq
M_{n+1}-\frac{\varepsilon}{2}\,.$
It is enough to show that
$u(z_{2})-u({z_{1}})\leq s_{n+1},$ (4.3)
since then
$M_{n+1}-m_{m+1}-\varepsilon\leq s_{n+1},$
which implies (4.2) for $k=n+1$, since $\varepsilon>0$ was arbitrary.
By the optional stopping theorem,
$\displaystyle u(z_{2})-u({z_{1}})=$ $\displaystyle\
\mathbb{E}_{z_{2}}[u(X_{\tau_{n}})-u({z_{1}})]$ $\displaystyle=$
$\displaystyle\ \mathbb{E}_{z_{2}}[u(X_{\tau_{n}})-u({z_{1}});X_{\tau_{n}}\in
B_{n-1}]$
$\displaystyle+\sum\limits_{i=1}^{n-2}\mathbb{E}_{z_{2}}[u(X_{\tau_{n}})-u({z_{1}});X_{\tau_{n}}\in
B_{n-i-1}\setminus B_{n-i}]$
$\displaystyle+\mathbb{E}_{z_{2}}[u(X_{\tau_{n}})-u({z_{1}});X_{\tau_{n}}\in
B_{1}^{c}]=I_{1}+I_{2}+I_{3}\,.$
Let $A=\\{z\in B_{n-1}\setminus B_{n}|\,u(z)\leq\frac{m_{n}+M_{n}}{2}\\}$. It
is sufficient to consider the case
$\mu_{x,r_{n}}(A)\geq\frac{1}{2}\mu_{x,r_{n}}(B_{n-1}\setminus B_{n})$, where
$\mu_{x,r}$ is the measure defined by (3.5). In the remaining case we would
use $\mu_{x,r_{n}}((B_{n-1}\setminus B_{n})\setminus
A)\geq\frac{1}{2}\mu_{x,r_{n}}(B_{n-1}\setminus B_{n})$ and could continue the
proof with $\|u\|_{\infty}-u$ and
$(B_{n-1}\setminus B_{n})\setminus A=\left\\{z\in B_{n-1}\setminus
B_{n}|\,\|u\|_{\infty}-u(z)\leq\frac{\|u\|_{\infty}-m_{n}+\|u\|_{\infty}-M_{n}}{2}\right\\}$
instead of $u$ and $A$.
The estimate (4.1) implies $a=\tfrac{L(r_{n+1})}{L(r_{n})}\leq
c_{0}(\tfrac{r_{n+1}}{r_{n}})^{-\alpha-1}$, from where we deduce $r_{n+1}\leq
r_{n}(c_{0}a^{-1})^{\frac{1}{\alpha+1}}\leq\frac{r_{n}}{2}$ because of
$a>c_{0}2^{\alpha+1}$. Next, we make use of the following property:
$\displaystyle
r_{n-1}=L^{-1}(L(\tfrac{r}{2})a^{n-2})=L^{-1}(\tfrac{1}{a}L(\tfrac{r}{2})a^{n-1})=L^{-1}(\tfrac{1}{a}L(r_{n}))=\varphi_{a}(r_{n})\,.$
(4.4)
Then by 3.4 (with $r=r_{n}$ and $x_{0}=x$) we get
$p_{n}:=\mathbb{P}_{z_{2}}(X_{\tau_{n}}\in A)\geq C_{5}\frac{\ln{a}}{a}\,.$
Hence,
$\displaystyle I_{1}$
$\displaystyle=\mathbb{E}_{z_{2}}[u(X_{\tau_{n}})-u({z_{1}});X_{\tau_{n}}\in
B_{n-1}]$
$\displaystyle=\mathbb{E}_{z_{2}}[u(X_{\tau_{n}})-u({z_{1}});X_{\tau_{n}}\in
A]+\mathbb{E}_{z_{2}}[u(X_{\tau_{n}})-u({z_{1}});X_{\tau_{n}}\in
B_{n-1}\setminus A]$
$\displaystyle\leq\left(\tfrac{m_{n}+M_{n}}{2}-m_{n}\right)p_{n}+s_{n-1}(1-p_{n})$
$\displaystyle\leq\tfrac{1}{2}s_{n}p_{n}+s_{n-1}(1-p_{n})\leq
s_{n-1}(1-\tfrac{1}{2}p_{n})\leq s_{n-1}(1-\tfrac{C_{5}\ln{a}}{2a})\,.$
By 3.3,
$\displaystyle I_{2}$
$\displaystyle\leq\sum\limits_{i=1}^{n-2}s_{n-i-1}\mathbb{P}_{z_{2}}(X_{\tau_{n}}\not\in
B_{n-i})\leq
C_{4}\sum\limits_{i=1}^{n-2}s_{n-i-1}\tfrac{L(r_{n-i})}{L(r_{n})}$
$\displaystyle\leq
3C_{4}\|u\|_{\infty}\sum\limits_{i=1}^{n-2}b^{-(n-i-2)}\tfrac{a^{n-i-1}}{a^{n-1}}\leq
3C_{4}\|u\|_{\infty}\tfrac{b^{-n+3}}{a-b}$ $\displaystyle\leq
C_{4}\tfrac{b^{3}}{a-b}s_{n+1}\,.$
Similarly, by 3.3,
$I_{3}\leq 2\|u\|_{\infty}\mathbb{P}_{z_{2}}(X_{\tau_{n}}\not\in B_{1})\leq
2C_{4}\|u\|_{\infty}\tfrac{L(r_{1})}{L(r_{n})}=\tfrac{2C_{4}}{3}b\left(\tfrac{b}{a}\right)^{n-1}s_{n+1}\leq
C_{4}\tfrac{b^{2}}{a}s_{n+1}\,.$
Hence,
$u(z_{2})-u(z_{1})\leq
s_{n+1}b^{2}\left[1-\tfrac{C_{5}\ln{a}}{2a}+\tfrac{C_{4}b}{a-b}+\tfrac{C_{4}}{a}\right]\,.$
Since $a-b\geq\frac{a}{4}$ for $b\in(1,\frac{3}{2})$ and
$a>c_{0}2^{\alpha+1}\geq 2$, it follows that
$q:=1-\tfrac{C_{5}\ln{a}}{2a}+\tfrac{C_{4}b}{a-b}+\tfrac{C_{4}}{a}\leq
1-\tfrac{C_{5}\ln{a}}{2a}+\tfrac{7C_{4}}{a}=1-\tfrac{C_{5}\ln
a-14C_{4}}{2a}\,.$
Next, we choose $a>c_{0}2^{\alpha+1}$ so large that $C_{5}\ln{a}-14C_{4}>0$.
Thus $q<1$. Finally, we choose $b\in(1,\frac{3}{2})$ sufficiently small so
that $b^{2}q<1$ .
Hence, (4.3) holds, which finishes the proof of the inductive step and the
theorem . ∎
## Appendix A Slow and Regular Variation
In this section we collect some properties of slowly resp. regularly varying
functions that are used in our main arguments. Moreover we list several
examples which illustrate the range of application of our approach.
###### Definition A.1.
A measurable and positive function $\ell\colon(0,1)\rightarrow(0,\infty)$ is
said to vary regularly at zero with index $\rho\in{\mathbb{R}}$ if for every
$\lambda>0$
$\lim_{r\to 0+}\frac{\ell(\lambda r)}{\ell(r)}=\lambda^{\rho}\,.$
If a function varies regularly at zero with index $0$ it is said to vary
slowly at zero. For simplicity, we call such functions _regularly varying_
resp. _slowly varying_ functions.
Note that slowly resp. regularly varying functions include functions which are
neither increasing nor decreasing. By [BGT87, Theorem 1.4.1 (iii)] it follows
that any function $\ell$ that varies regularly with index
$\rho\in{\mathbb{R}}$ is of the form $\ell(r)=r^{\rho}\ell_{0}(r)$ for some
function $\ell_{0}$ that varies slowly.
Assume $\int_{0}^{1}s\,\ell(s)\,\,\textnormal{d}s\leq c$ for some $c>0$. Let
$L\colon(0,1)\rightarrow(0,\infty)$ be defined by
$L(r)=\int\limits_{r}^{1}\frac{\ell(s)}{s}\,\,\textnormal{d}s\ .$
The function $L$ is well defined because
$L(r)=r^{-2}\int_{r}^{1}r^{2}\frac{\ell(s)}{s}\,\,\textnormal{d}s\leq
r^{-2}\int_{r}^{1}s\ell(s)\,\,\textnormal{d}s\leq cr^{-2}$. Note that
($K_{1}$) and ($K_{3}$) imply that
$\int_{0}^{1}s\,\ell(s)\,\,\textnormal{d}s\leq c$ does hold in our setting. We
note that the function $L$ is always decreasing.
Let us list further properties which are making use of in our proofs. Note
that they are established [BGT87] for functions which are slowly resp.
regularly varying at the point $+\infty$. By a simple inversion we adopt the
results to functions which are slowly resp. regularly varying at the point
$0$.
1. (1)
If $\ell$ is slowly varying, then [BGT87, Proposition 1.5.9a] $L$ is slowly
varying with
$\lim\limits_{r\to 0+}L(r)=+\infty\qquad\text{ and }\qquad\lim\limits_{r\to
0+}\frac{\ell(r)}{L(r)}=0\,.$
2. (2)
If $\ell$ is slowly varying and $\rho>-1$, then Karamata’s theorem [BGT87,
Proposition 1.5.8] ensures
$\lim_{r\to
0+}\frac{\int_{0}^{r}s^{\rho}\ell(s)\,ds}{r^{\rho+1}\ell(r)}=(\rho+1)^{-1}\,.$
3. (3)
If $\ell$ is regularly varying of order $-\alpha<0$ (in our case
$0<\alpha<2$), then [BGT87, Theorem 1.5.11]
$\lim_{r\to 0+}\frac{L(r)}{\ell(r)}=\alpha^{-1}\,.$
In particular, if $\ell$ is regularly varying of order $-\alpha<0$, then so is
$L$.
4. (4)
Assume $\ell$ is regularly varying of order $-\alpha\leq 0$ and stays bounded
away from $0$ and $+\infty$ on every compact subset of $(0,1)$. Then Potter’s
theorem [BGT87, Theorem 1.5.6 (ii)] implies that for every $\delta>0$ there is
a constant $C=C(\delta)\geq 1$ such that for $r,s\in(0,1)$
$\displaystyle\frac{\ell(r)}{\ell(s)}\leq
C\max\left\\{\left(\frac{r}{s}\right)^{-\alpha-\delta},\left(\frac{r}{s}\right)^{-\alpha+\delta}\right\\}\,.$
5. (5)
Since $L$ is nonincreasing, we observe $\lim\limits_{r\to
0+}L(r)\in(0,+\infty]$.
Table 1. Different choices for the function $\ell$ when $\beta\in(0,2)$, $a>1$. No. (i) | $\ell_{i}(s)$ | $L_{i}(s)$ | $\varphi_{a}(s)=L_{i}^{-1}(\frac{1}{a}L_{i}(s))$
---|---|---|---
$1$ | $s^{-\beta}\,\ln(\frac{2}{s})^{2}$ | $\asymp s^{-\beta}\,\ln(\frac{2}{s})^{2}$ | $\asymp s$
$2$ | $s^{-\beta}$ | $\frac{1}{\beta}(s^{-\beta}-1)$ | $\asymp s$
$3$ | $\ln(\frac{2}{s})$ | $\asymp\ln^{2}(\frac{2}{s})$ | $\asymp s^{1/\sqrt{a}}$
$4$ | $1$ | $\ln(\frac{1}{s})$ | $s^{1/a}$
$5$ | $\ln(\frac{2}{s})^{-1}$ | $\asymp\ln(\ln(\frac{2}{s}))$ | $\asymp\exp(-(\ln(\frac{2}{s}))^{1/a})$
$6$ | $\ln(\frac{2}{s})^{-2}$ | $\ln(2)^{-1}-\ln(\tfrac{2}{s})^{-1}$ | $\asymp\exp(-(\frac{a-1}{a\ln(2)}+\frac{1}{a\ln(2/s)})^{-1})$
Let us look at different choices for the function $\ell$, given in Table 1.
Here $\beta\in(0,2)$, $a>1$ are fixed. We list six examples of a function
$s\mapsto\ell_{i}(s)$ together with $s\mapsto L_{i}(s)$ and
$s\mapsto\varphi_{a}(s)=L_{i}^{-1}(\frac{1}{a}L_{i}(s))$. Recall that the
function $\varphi_{a}$ appears in 1.5 and determines the scaling that we are
using, see also property (4.4) and the definition of $r_{n}$ in the proof of
1.4. Note that case No. 6 is significantly different from the other cases.
Both, the integral $\int_{B_{1}}|h|^{-d}\ell_{6}(|h|)\,\,\textnormal{d}h$ and
the expression $\lim\limits_{s\to 0+}L_{6}(s)$ are finite. Moreover, the limit
$\lim\limits_{s\to 0+}L_{6}^{-1}(\frac{1}{a}L_{6}(s))$ is not equal to zero.
These differences reflect the fact that the corresponding operator in (1.2)
has an integrable kernel. Recall that 2.1 relates the behavior of the function
$L$ close to the origin to the behaviour of the multiplier of the operator (in
the case of constant coefficents) for large values of $|\xi|$. In the case No.
6 the multiplier stays bounded.
Acknowledgements: We thank T. Grzywny for a helpful comment on the limit case
$\alpha=2$.
## References
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* [Grz13] T. Grzywny, _On Harnack inequality and Hölder regularity for isotropic unimodal Lévy processes_ , Potential Anal. (2013), to appear.
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|
arxiv-papers
| 2013-10-20T20:54:51 |
2024-09-04T02:49:52.617768
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Moritz Kassmann, Ante Mimica",
"submitter": "Moritz Kassmann",
"url": "https://arxiv.org/abs/1310.5371"
}
|
1310.5389
|
# THz-radiation production using dispersively-selected flat electron bunches
J. Thangaraj [email protected] Accelerator Physics Center, Fermi National
Accelerator Laboratory, Batavia, IL 60510, USA P. Piot Accelerator Physics
Center, Fermi National Accelerator Laboratory, Batavia, IL 60510, USA
Northern Illinois Center for Accelerator & Detector Development and Department
of Physics, Northern Illinois University, DeKalb IL 60115, USA
###### Abstract
We propose an alternative scheme for a tunable THz radiation source generated
by relativistic electron bunches. This technique relies on the combination of
dispersive selection and flat electron bunch. The dispersive selection uses a
slit mask inside a bunch compressor to transform the energy-chirped electron
beam into a bunch train. The flat beam transformation boosts the frequency
range of the THz source by reducing the beam emittance in one plane. This
technique generates narrow-band THz radiation with a tuning range between 0.2
- 4 THz. Single frequency THz spectrum can also be generated by properly
choosing the slit spacing, slit width, and the energy chirp.
Accelerator-driven terahertz (THz) sources have attracted immense interest
over a broad range of disciplines due to their ability to produce a high
power, tunable radiation within compact footprintWen _et al._ (2013).
Accelerator-based THz sources combine a sub-picosecond relativistic electron
bunch with an electromagnetic radiative process, e.g., the beam could either
pass through a foil radiator to emit coherent transition radiation (CTR) or
travel through a dipole to emit coherent synchrotron radiation (CSR)Wu _et
al._ (2013); Carr _et al._ (2002); Casalbuoni _et al._ (2009). The total
spectral intensity of the emitted radiation from an electron bunch consisting
of N electrons through a radiative electromagnetic process is given byNodvick
and Saxon (1954):$\displaystyle\left(\frac{d^{2}I}{d\omega
d\Omega}\right)_{t}=\left(\frac{d^{2}I}{d\omega
d\Omega}\right)_{e}[N+N(N-1)|B_{0}(\omega)|^{2}]$, where $\omega=2\pi f$ ($f$
is the frequency), $\displaystyle\left(\frac{d^{2}I}{d\omega
d\Omega}\right)_{e}$ is the single electron spectral intensity and
$B_{0}(\omega)=\sum\limits_{k=1}^{N}e^{i\omega t_{k}}$ is the bunching factor,
where $t_{k}$ is the the longitudinal time coordinate of the $k^{th}$ electron
inside the bunch. Other broad-band THz schemes include advanced acceleration
schemes such as laser-driven plasma acceleration and ion-driven
accelerationGopal _et al._ (2013); Leemans _et al._ (2003). Narrow-band THz
sources uses a variety of techniques such as corrugated waveguideBane and
Stupakov (2012), emittance exchangerPiot _et al._ (2011), modulating the
drive laser Shen _et al._ (2011); Boscolo _et al._ (2007), echo-basedDunning
_et al._ (2012), or dielectric based Antipov _et al._ (2013) schemes. In this
letter, we propose a simple scheme for THz generation using a slit mask in an
dispersive region of a linear accelerator to generate up to 4 THz using a 50
MeV beam. The achievable frequency range span 0.2 - 4 THz. All this scheme
requires is a photoinjector and a bunch compressor both of which are a
standard components at almost all modern and planned future linear
accelerators.
Magnetic bunch compressor are commonly incorporated in accelerators that drive
free-electron lasers (FEL) to enhance the electron bunch peak current.
Generating a train of sub-picosecond bunches using dispersive scraping in a
chicane (four dipoles bending angle +,-,-,+) or in a dogleg (two dipoles
separated by a drift) bunch compressor has been developed elsewhereNguyen and
Carlsten (1996); Muggli _et al._ (2008).
Figure 1: Schematic of the THz beamline: The RF photoinjector consist of a gun
and two solenoid lenses (L1, L2). After existing the gun, the electron beam is
acceleration off-crest in the RF cavity. This energy-chirped beam is focussed
using the quadrupoles (Q1, Q2, Q3) and then enters a chicane and is
intercepted by a set of slit mask (MS) at the center. After the slit mask,
some electrons are scattered while other pass through the chicane. At the end
of the chicane transversely separated electron beam are transformed into
longitudinally separated train of bunches. Blue (head) is higher energy and
red (tail) is lower energy. The beam is focussed on the CTR aluminium foil (Z)
using the quadrupole doublet (QX, QY)to extract the THz. The round to flat
beam transformer (RTFB) section of the linac has three skew quadrupoles
representd by diamond to generate a flat beam for multi-THz. There is another
skewquad (SQ) close to the center of the chicane for diagnostics.
Figure 1 illustrate the principle of the proposed method. An electron beam is
generated from a photoinjector and is then accelerated by a radio-frequency
(RF) cavity. During acceleration, the electron beam gets an energy chirp - a
time-dependent energy variation. The energy-chirped beam is then sent through
a straight section of the linac that includes quadrupole magnets (Q1, Q2, Q3
in Fig. 1) and then to the bunch compressor. At the center of the bunch
compressor, the bunch is intercepted by a slit mask (MS) which selectively
scatters some of the electrons while other electrons are transmitted through
the rest of the chicane. At the end of the chicane, such transversely
separated beamlets are transformed into a train of short bunches
longitudinally. The spacing between the bunches and the length of each bunch
is determined by several factors such as the dispersion of the chicane
($\eta$), the transverse betatron spot size of the beam at the mask, the width
of the slit mask ($w$), the uncorrelated relative beam energy spread
($\sigma_{u}$) and the RF-energy chirp on the beam ($h$). The formula that
relates the length of the bunch at the exit of chicane to the width of the
slit is given by Emma _et al._ (2004): $\sigma_{z}=\frac{1}{|\eta
h|}{\sqrt{\eta^{2}\sigma_{u}^{2}+(1+hR_{56})^{2}[\Delta
X^{2}+\varepsilon\beta]}}$, where $\sigma_{z}$ is the output bunch length,
$R_{56}$ is the longitudinal dispersion of the chicane, $\Delta
X=\frac{w}{2\sqrt{3}}$ is the rms width of the mask, $\varepsilon$ is the
natural beam emittance, and $\beta$ is the betatron function at the mask. It
can be seen that when $hR_{56}$ is large, the output bunch profile follows the
mask profile ($\Delta X$). This can be done by making $|1+hR_{56}|>>1$. For a
chicane in our convention $R_{56}<0$ ($z>0$ corresponds to the tail) ,
therefore by setting $h<0$, the output bunch profile can be made to follow the
mask profile. This technique is limited by the initial slice energy spread and
emittance of the beam. We note that same function can also be reached by
setting $h<<\frac{-2}{R_{56}}$, which can become very large and impractical
and in certain cases lead to overcompression. The above equation also
indicates that to get a bunch train one should ensure that the betatron spot
size at the slit mask is less than the slit width ($\varepsilon\beta<<(\Delta
X)^{2}$). This can be done by properly setting the quadrupole magnet triplet
(Q1, Q2, Q3) located upstream of the chicane to the right current setting. In
order to reveal the longitudinal structure, the skew quadrupole (SQ) can be
powered on that couples the x-dispersion into the y-plane and therefore the
vertical ($y$) axis on the screen downstream is transformed into a time
axisEmma _et al._ (2012). Finally, we note that this scheme allows for pulse
shaping other than a train of pulses: for e.g. a triangular wedge shaped
collimator can be used to generate ramped bunches that have application in
advanced accelerator-type applications.
Hence, in our scheme the magnetic chicane effectively acts to decompress the
bunch. By dispersing the beam inside the chicane, an $x-z$ correlation is
introduced at the center of the chicane, where $x$ is the transverse position
of the particle and $z$ its longitudinal position of the particle. Due to this
high correlation, any variation in $x$ is then mapped onto $z$. This scheme is
different from Emma _et al._ (2004) where differential spoiling is used at
high energy (few GeV) to generate femtosecond x-rays. Our scheme differs from
it in two aspects: the low energy of our beam allows us to stop or scatter
much of the beam using metallic slits and the bunch compressor is set to
decompression. Also, our intrinsic relative energy spread is fairly high
compared to that scheme because of the low energy of the beam. As mentioned
above, our scheme differs from Muggli _et al._ (2008) by using a chicane
instead of a dogleg and using the RF chirp as the tuning variable instead of
using quadrupoles and an energy slit. We note that our scheme is more
efficient since there is already an energy-chirp imparted naturally due to the
longitudinal space-charge forces when the bunch exists the photoinjector that
is favorable to our scheme (head is at high energy and tail is at lower
energy) before it enters the RF cavity.
Figure 2: Normalized current profile of the electron bunch (top) and
associated bunch form factor (bottom) with (red) and without (blue) the slits
inserted. When the slits are inserted, the beam is bunched at sub-THz
frequencies and hence the resonant enhancement in the frequency domain at
harmonics of the bunching frequencies.
We show through tracking simulation that our scheme can generate tunable,
coherent sub-THz (i.e around or less than 1 THz) radiation. The particle
tracking program ELEGANTBorland (2000) was used for simulating the beam line.
All the bending magnets are rectangular magnets. In all the simulation shown
in this paper, CSR is taken into account. Nominal values for slit width and
slit spacing along with the beam and chicane parameters are shown in Table. 1.
The initial phase-space distributions are assumed to be Gaussian. A linear
energy chirp is assumed to be imparted by the RF-cavity. This is a fairly good
approximation considering we are operating far from the off-crest with a
decompressing phase. We note that in a laboratory beam the phase-space out of
the photoinjector might still be distorted and further simulations are planned
to understand such effects.
Table 1: Simulation parameters Parameter | Value | Units
---|---|---
Initial emittance (x,y) | 0.5 | $\mu m$
Beam energy | 50 | MeV
Initial slice energy spread | 5 | keV
Initial bunch length | 0.8 | mm
$\delta-z$ correlation (chirp) | [-10 … -4] | 1/m
Charge | 100 | pC
Slit spacing (center to center) | 1 | mm
Slit width | 50 | $\mu m$
Number of particles | 106 | n/a.
Dipole bending radius | 0.958 | m
Dipole length | 0.301 | m
Dipole angle | 18 | degrees
$R_{56}$ | -18 | cm
$\eta$ | -30 | cm
Figure 2 shows the current profile and the corresponding frequency spectrum
from tracking simulation with and without the slits inside the beam line. When
the slits are out, we get a single long, decompressed Gaussian bunch and the
frequency spectrum obtained does not extend into the THz frequencies and is
limited by the long bunch length. However when the slits are inserted, we
obtain a train of short bunches and the frequency spectrum has a fundamental
and its harmonics with a narrow bandwidth. The relationship between the number
of bunches in a train, the period of the bunch train, the rms width of the
bunch and the frequency spectrum is given in Piot _et al._ (2011). By tuning
the RF-chirp on the electron beam prior to the chicane, the fundamental THz
frequency can be tuned. The upper limit of the THz frequency is limited by the
uncorrelated relative energy spread and the normalized emittance of the beam.
Figure 3: Effect of emittance on bunch train formation. Microbunch period
$\Delta T$ and rms duration $\sigma_{t}$ as a function of RF chirp. For
$\sigma_{t}$ two cases of emittance $\varepsilon_{n}=1\ \mu m$ and $0.1\ \mu
m$ are considered. Above the solid-circled line region ($\varepsilon_{n}=1\
\mu m$), the $\Delta T$ (solid line) is close to $\sigma_{t}$ and thus smears
train formation but in the region above the solid-square line region
($\varepsilon_{n}=0.1\ \mu m$), the lower emittance resolves the individual
bunches because $\Delta T>4\sigma_{t}$. Slit-width ($\Delta X=50\ \mu m$),
slit spacing ($D=100\ \mu m$) and $\beta=0.5\ m$. Figure 4: Boosted THz
spectrum due to the flat-beam transformation showing the bunch form factor
(top) and the bandwidth (bottom) extending well above 1 THz upto 4 THz
compared with no flat-beam generation.
While the slit-based technique is capable of generating sub-THz frequencies,
it is non-trivial to go above 1 THz without additional complexity. In order to
go above the THz barrier, one needs smaller mask width but then the emittance
requirement becomes challenging ($\varepsilon\beta<<(\Delta X)^{2}$). Figure 3
illustrates the effect of the normalized emittance on the formation of the
bunch train for a given slit spacing. In order to get a bunch train, the
spacing between the bunches ($\Delta T$) must be larger than the bunch
duration of the individual bunches ($\sigma_{t}=\frac{\sigma_{z}}{c}$)
(typically, $\Delta T>4\sigma_{t}$). The microbunch period is $\Delta
T=\frac{D}{\eta|hC|c}$, where $D$ is the slits spacing, $C$ is the compression
factor given by $C=(1+hR_{56})^{-1}$ and $c$ is the speed of light. As shown
in Fig. 3, lower emittance beam allows bunch train formation by producing
shorter individual bunches for a fixed $\Delta T$. One way to achieve low
emittance would be to operate the linac at a lower charge (10 pC) but when
going through the slits most of charge (upto 90%) could be lost. Another way
to achieve low emittance in one plane only for e.g. in the horizontal plane is
through flat-beam transformation. In order to generate a flat beam, the
photocathode is immersed in an axial magnetic field which generates a
magnetized electron beam. After acceleration, a set of three skew quadrupoles
(RFBT in Fig. 1), is used to transform the magnetized beam into a flat beam.
Such flat-beam transformation have been studied theoretically and demonstrated
experimentally Brinkmann _et al._ (2001); Piot _et al._ (2006). A flat beam
ratio of $\varepsilon_{x}:\varepsilon_{y}$ of 100 has been experimentally
demonstrated at low energies using the Fermilab A0 photoinjector. Note the
product of the emittances $\varepsilon_{x}\varepsilon_{y}$ remains constant
before and after the flat-beam transformation. Therefore, to achieve the
required boost in the THz frequency and break the sub-THz barrier, we use
flat-beam transformation in the linac. In order to demonstrate this, we use
ELEGANT simulation. We use an emittance ratio of 100 and 400 which is
consistent with simulationPiot _et al._ (2013a). The results shown in Fig. 4
indicates that the use of flat beam transformations helps to generate higher
THz frequencies for a given slit spacing and width. The flat-beam
transformation not only extends the maximum THz frequency but also improves
the bunch form factor at lower frequencies as well. A scan over various
emittance ratio and RF-chirp shows that frequencies as high as 4 THz can be
obtained. In a superconducting linac, the RF-chirp can be controlled in a very
precise manner with longitudinal feedback systems. Thus combining flat-beam
technique, which can be done in any modern photoinjector linac using
appropriate skew quadrupole magnets and a chicane equipped with a transverse
mask, we can generate tunable multi-THz frequencies. In order to extract the
THz radiation outside the beam pipe, we use a quadrupole doublet (QX, QY)
followed by a CTR aluminium foil (Z shown in Fig. 1). Our simulation shows
that a rms (round) spot size of $\sigma_{r}$=0.2 mm on both planes can be
obtained at the screen using the doublet. This implies an upper cut-off
frequency due to the transverse spot size of $f_{u}\sim\frac{\gamma
c}{2\pi\sigma_{r}}$ of 23 THz which is well above our highest frequency of our
scheme ($\gamma$=100 at 50 MeV ).
Figure 5: The bunching factor (above) of the single spike THz spectrum along
with the required RF-chirp (below) as a function of the spacing of the slits.
By picking a specific slit spacing and appropriate RF-chirp, a narrow-band
single frequency THz spectrum can be generated.
While both the fundamental frequency and its harmonics are present in the
bunch due to the flat-beam transformation, sometimes only a single THz
frequency might be preferred by users. This can be done by choosing the
appropriate slit spacing and the width and supplying the correct RF-chirp.
Figure 5 shows the effect of varying the slit-spacing (D) by choosing smaller-
width slits (20 $\mu m$) and RF-chirp. For this simulation, all other
parameters remaining constant (Table. 1), a flat beam ratio of
$\varepsilon_{x}:\varepsilon_{y}$=1:400 was used Piot _et al._ (2013a).
Proper choice of slit-spacing and RF-chirp allows a tunable range of 1-4 THz
with a single frequency THz spectrum. A movable plate mounted with slits of
different width and different spacing can easily be accommodated in a stepper
motor controlled actuator to add this useful feature to the machine.
In summary, we have proposed and investigated via computer simulations a THz
generation scheme that combines dispersive selection with flat electron beams.
The advantage of this technique is its simplicity, tunability and low cost.
The scheme does not require any additional hardware such as lasers, undulator,
transverse deflecting cavity. Our scheme can be readily deployed in any linac
that uses low energy compression such as ASTA Piot _et al._ (2013b),
FLUTENasse _et al._ (2013). By using low emittance beam via flat-beam
transformation in only one plane, tunable THz source covering 0.2 - 4 THz can
be achieved. This scheme is also scalable to any superconducting linac as the
only requirement is that the slit material should be able to withstand the
heat load due to the multi-pulse structure of the electron bunch. Currently,
experiments are planned at Fermilab’s ASTA facility using this scheme and we
anticipate this technique to be useful for other accelerators.
We would like to thank M. Borland for his support in ELEGANT simulation. One
of us (J. T.) would like to thank Randy-Thurman Keup for clarifying issues on
THz detection. The work was supported by the Fermi Research Alliance, LLC
under the U.S. Department of Energy.
## References
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* Leemans _et al._ (2003) W. P. Leemans, C. G. R. Geddes, J. Faure, C. Tóth, J. van Tilborg, C. B. Schroeder, E. Esarey, G. Fubiani, D. Auerbach, B. Marcelis, M. A. Carnahan, R. A. Kaindl, J. Byrd, and M. C. Martin, Phys. Rev. Lett. 91, 074802 (2003).
* Bane and Stupakov (2012) K. Bane and G. Stupakov, Nucl. Instrum. Methods. Phys. Res. A 677, 67 (2012).
* Piot _et al._ (2011) P. Piot, Y.-E. Sun, T. J. Maxwell, J. Ruan, A. H. Lumpkin, M. M. Rihaoui, and R. Thurman-Keup, Applied Physics Letters 98, 261501 (2011).
* Shen _et al._ (2011) Y. Shen, X. Yang, G. L. Carr, Y. Hidaka, J. B. Murphy, and X. Wang, Phys. Rev. Lett. 107, 204801 (2011).
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* Antipov _et al._ (2013) S. Antipov, C. Jing, P. Schoessow, A. Kanareykin, V. Yakimenko, A. Zholents, and W. Gai, Review of Scientific Instruments 84, 022706 (2013).
* Nguyen and Carlsten (1996) D. Nguyen and B. Carlsten, Nucl. Instrum. Methods. Phys. Res. A 375, 597 (1996), proceedings of the 17th International Free Electron Laser Conference.
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* Emma _et al._ (2004) P. Emma, K. Bane, M. Cornacchia, Z. Huang, H. Schlarb, G. Stupakov, and D. Walz, Phys. Rev. Lett. 92, 074801 (2004).
* Emma _et al._ (2012) P. Emma, F. Zhou, Z. Huang, and C. Behrens, Proceedings of the Free Electron Laser Conference (2012).
* Borland (2000) M. Borland, Advanced Photon Source LS-287 (2000).
* Brinkmann _et al._ (2001) R. Brinkmann, Y. Derbenev, and K. Flöttmann, Phys. Rev. ST Accel. Beams 4, 053501 (2001).
* Piot _et al._ (2006) P. Piot, Y.-E. Sun, and K.-J. Kim, Phys. Rev. ST Accel. Beams 9, 031001 (2006).
* Piot _et al._ (2013a) P. Piot, C. Prokop, B. Carlsten, D. Mihalcea, and Y. Sun, Proceedings of the International Particle Accelerator Conference (2013a).
* Piot _et al._ (2013b) P. Piot, V. Shiltsev, S. Nagaitsev, M. Church, P. Garbincius, _et al._ , (2013b), arXiv:1304.0311 [physics.acc-ph] .
* Nasse _et al._ (2013) M. J. Nasse, M. Schuh, S. Naknaimueang, M. Schwarz, A. Plech, Y.-L. Mathis, R. Rossmanith, P. Wesolowski, E. Huttel, M. Schmelling, and A.-S. Muller, Review of Scientific Instruments 84, 022705 (2013).
|
arxiv-papers
| 2013-10-21T00:24:55 |
2024-09-04T02:49:52.629524
|
{
"license": "Public Domain",
"authors": "Jayakar Thangaraj, Philippe Piot",
"submitter": "Jayakar Thangaraj",
"url": "https://arxiv.org/abs/1310.5389"
}
|
1310.5399
|
# Simultaneous analysis of three-dimensional percolation models
Xiao Xu Hefei National Laboratory for Physical Sciences at Microscale and
Department of Modern Physics, University of Science and Technology of China,
Hefei, Anhui 230026, China Junfeng Wang Hefei National Laboratory for
Physical Sciences at Microscale and Department of Modern Physics, University
of Science and Technology of China, Hefei, Anhui 230026, China School of
Electronic Science and Applied Physics, Hefei University of Technology, Hefei,
Anhui 230009, China Jian-Ping Lv [email protected] Department of Physics,
China University of Mining and Technology, Xuzhou 221116, China Youjin Deng
[email protected] Hefei National Laboratory for Physical Sciences at
Microscale and Department of Modern Physics, University of Science and
Technology of China, Hefei, Anhui 230026, China
###### Abstract
We simulate the bond and site percolation models on several three-dimensional
lattices, including the diamond, body-centered cubic, and face-centered cubic
lattices. As on the simple-cubic lattice [Phys. Rev. E, 87 052107 (2013)], it
is observed that in comparison with dimensionless ratios based on cluster-size
distribution, certain wrapping probabilities exhibit weaker finite-size
corrections and are more sensitive to the deviation from percolation threshold
$p_{c}$, and thus provide a powerful means for determining $p_{c}$. We analyze
the numerical data of the wrapping probabilities simultaneously such that
universal parameters are shared by the aforementioned models, and thus
significantly improved estimates of $p_{c}$ are obtained.
###### pacs:
05.50.+q (lattice theory and statistics), 05.70.Jk (critical point phenomena),
64.60.ah (percolation), 64.60.F- (equilibrium properties near critical points,
critical exponents)
## I Introduction
Percolation is a geometric model which involves the random occupation of sites
or edges of a regular lattice, and was first introduced by Broadbent and
Harmmersley Broadbent and Hammersley (1957). As a cornerstone of the theory of
critical phenomena Stauffer and Aharony (1994) and a central topic in
probability theory Grimmett (1999); Bollobás and Riordan (2006), percolation
attracts much attention.
The two-dimensional (2D) case has been studied extensively, and several exact
results are known. Coulomb gas arguments Nienhuis (1987) and conformal field
theory Cardy (1987) predict the exact values of the bulk critical exponents
$\beta=5/36$ and $\nu=4/3$, which have been confirmed rigorously in the
specific case of site percolation on the triangular lattice Smirnov and Werner
(2001). Moreover, percolation thresholds $p_{c}$ on many 2D lattices are
exactly known Essam (1972), or known to very high precision Feng et al.
(2009); Ding (2010). For $d>2$, estimates of $p_{c}$ have to rely on numerical
methods such as series expansions and Monte Carlo simulations, while the
critical exponents $\beta=1$ and $d\nu=3$ for $d\geq d_{c}=6$ can be predicted
by mean-field theory Toulouse (1974) and even proved rigorously Aizenman and
Newman (1984); Hara and Slade (1990) for $d\geq 19$. A more or less thorough
list of percolation thresholds for $d\in[2,13]$ is summarized on the Wikipedia
webpage: http://en.wikipedia.org/wiki/Percolation_threshold.
Very recently, two of the authors and coworkers carried out an extensive
simulation of bond and site percolation on the simple-cubic (SC) lattice up to
system size $512\times 512\times 512$ Wang et al. (2013), and determined the
percolation thresholds and critical exponents to high precision. It was
observed that in comparison with dimensionless ratios based on cluster-size
moments, the wrapping probabilities suffer from weaker finite-size corrections
and are more sensitive to the deviation $p-p_{c}$ from the percolation
threshold. As an extension of Ref. Wang et al. (2013), the present work
studies percolation on other common three-dimensional (3D) lattices, and shows
that such an observation generally holds in 3D percolation. Meanwhile, with
the employment of a simultaneous fitting procedure developed in Ref. Deng and
Blöte (2003) and the help of the accurate data reported in Ref. Wang et al.
(2013), we also provide high-precision estimates of $p_{c}$ for the site and
bond percolation on the diamond (DM), body-centered cubic (BCC), and face-
centered cubic (FCC) lattices.
The remainder of this paper is organized as follows. Section II defines the
sampled quantities of interest. In Sec. III, the numerical data of the
dimensionless ratios and the wrapping probabilities are analyzed separately
for each percolation model. Then, a simultaneous fitting of the wrapping
probabilities is carried out to determine percolation threshold $p_{c}$.
Section IV presents the analyses for other quantities at criticality $p_{c}$,
and a brief discussion is given in Sec. V.
## II Sampled quantities
We study bond and site percolation on three-dimensional lattices including the
DM, SC, BCC, and FCC lattices, illustrated in Fig. 1. The simulations follow
the standard method: each edge/site is occupied with probability $p$ and
clusters are constructed by the breadth-first search.
Figure 1: Three-dimensional lattices: (left-top), SC; (righttop), DM; (left-
bottom), BCC; (right-bottom), FCC.
The sampled quantities are the same as in Ref. Wang et al. (2013). For
completeness, they are described in the following.
* •
The number of occupied bonds ${\mathcal{N}}_{b}$ or sites ${\mathcal{N}}_{s}$.
* •
The number of clusters ${\mathcal{N}}_{c}$.
* •
The largest-cluster size ${\mathcal{C}}_{1}$.
* •
The cluster-size moments ${\mathcal{S}}_{m}=\sum_{C}|C|^{m}$ with $m=2,4$,
where the sum runs over all clusters $C$ and $|C|$ denotes cluster size.
* •
An observable ${\mathcal{S}}:=\max\limits_{C}\,\max\limits_{y\in
C}\,d(x_{C},y)$ used to determine the shortest-path exponent. Here $d(x,y)$
denotes the graph distance from vertex $x$ to vertex $y$, and $x_{C}$ is the
vertex in cluster $C$ with the smallest vertex label, according to some fixed
(but arbitrary) vertex labeling.
* •
The indicators ${\mathcal{R}}^{(x)}$, ${\mathcal{R}}^{(y)}$, and
${\mathcal{R}}^{(z)}$, for the event that a cluster wraps around the lattice
in the $x$, $y$, or $z$ directions, respectively.
From these observables we calculated the following quantities:
* •
The mean size of the largest cluster $C_{1}=\langle{\mathcal{C}}_{1}\rangle$,
which scales as $C_{1}\sim L^{y_{h}}$ at $p_{c}$, with $L$ the linear system
size and $y_{h}=d-\beta/\nu$.
* •
The cluster density $\rho=\langle{\mathcal{N}}_{c}\rangle/V$, where $V=gL^{3}$
is the number of lattice sites, with $g=1$ for the SC and DM lattices, $g=2$
for the BCC lattice, and $g=4$ for the FCC lattice.
* •
The dimensionless ratios
$Q_{1}=\frac{\langle{{\mathcal{C}}_{1}}^{2}\rangle}{\langle{\mathcal{C}}_{1}\rangle^{2}}\;,\;\;\;Q_{2}=\frac{\langle{{\mathcal{S}}_{2}}^{2}\rangle}{\langle
3{{\mathcal{S}}_{2}}^{2}-2{\mathcal{S}}_{4}\rangle}\;.$ (1)
In the case of the Ising model, $Q_{2}$ is identical to the dimensionless
ratio $Q_{M}=\langle M^{2}\rangle^{2}/\langle M^{4}\rangle$, where $M$
represents the magnetization.
* •
The mean shortest-path length $S=\langle{\mathcal{S}}\rangle$, which at
$p_{c}$ scales like $S\sim L^{d_{\rm min}}$ with $d_{\rm min}$ the shortest-
path fractal dimension.
* •
The wrapping probabilities
$\displaystyle R^{(x)}=$
$\displaystyle\langle{\mathcal{R}}^{(x)}\rangle=\langle{\mathcal{R}}^{(y)}\rangle=\langle{\mathcal{R}}^{(z)}\rangle\;,$
(2) $\displaystyle R^{(a)}=$ $\displaystyle
1-\langle(1-{\mathcal{R}}^{(x)})(1-{\mathcal{R}}^{(y)})(1-{\mathcal{R}}^{(z)})\rangle\;,$
$\displaystyle R^{(3)}=$
$\displaystyle\langle{\mathcal{R}}^{(x)}{\mathcal{R}}^{(y)}{\mathcal{R}}^{(z)}\rangle\;.$
Here $R^{(x)}$, $R^{(a)}$ and $R^{(3)}$ give the probability that a winding
exists in the $x$ direction, in at least one of the three possible directions,
and simultaneously in the three directions, respectively. At $p_{c}$, these
wrapping probabilities take non-zero universal values in the thermodynamic
limit $L\rightarrow\infty$.
* •
The covariance of ${\mathcal{R}}^{(x)}$ and ${\mathcal{N}}_{b}$
$g^{(x)}_{bR}=\langle{\mathcal{R}}^{(x)}{\mathcal{N}}_{b}\rangle-\langle{\mathcal{R}}^{(x)}\rangle\langle{\mathcal{N}}_{b}\rangle\;,$
(3)
which scales as $g^{(x)}_{bR}\sim L^{y_{t}}=L^{1/\nu}$ at criticality $p_{c}$.
Analogously, one defines $g^{(x)}_{sR}$ for site percolation, with
${\mathcal{N}}_{b}$ being replaced with ${\mathcal{N}}_{s}$.
Figure 2: Quantities $Q_{2}$ and $R^{(x)}$ as a function of $p$ for the site
percolation on the DM lattice with various sizes. In comparison with $Q_{2}$,
the plot of $R^{(x)}$ has a finer vertical scale, but still displays a clearer
intersection. This suggests that $R^{(x)}$ suffers weaker finite-size
corrections and provides a better estimator for $p_{c}$.
## III Percolation threshold
The simulation on the SC lattice is up to linear size $L_{\rm max}=512$, and
the number of samples is about $5\times 10^{8}$ for $L\leq 128$, $6\times
10^{7}$ for $L=256$, and $3\times 10^{7}$ for $L=512$. The Monte Carlo data
and the analysis have been reported in Ref. Wang et al. (2013). For the other
lattices, the simulation is less extensive with $L_{\rm max}=128$. The number
of samples is about $10^{8}$ for lattice $L<128$ and $4\times 10^{7}$ for
$L=128$.
Table 1: Percolation thresholds from the separate fits of the wrapping
probabilities and the dimensionless ratios.
| $Q_{1}$ | $Q_{2}$ | $R^{(x)}$ | $R^{(a)}$ | $R^{(3)}$
---|---|---|---|---|---
$\rm{DM}^{b}$ | 0.389 591(2) | 0.389 592(1) | 0.389 589 2(5) | 0.389 588 9(4) | 0.389 590 0(5)
$\rm{DM}^{s}$ | 0.429 987(2) | 0.429 985(1) | 0.429 987 7(9) | 0.429 987 5(6) | 0.429 987 3(4)
$\rm{SC}^{b}$ | 0.248 811 96(6) | 0.248 811 92(6) | 0.248 811 85(3) | 0.248 811 80(4) | 0.248 811 81(9)
$\rm{SC}^{s}$ | 0.311 606 9(2) | 0.311 607 1(2) | 0.311 607 68(7) | 0.311 607 74(6) | 0.311 607 7(1)
$\rm{BCC}^{b}$ | 0.180 287 8(9) | 0.180 288 3(6) | 0.180 287 5(2) | 0.180 287 4(2) | 0.180 287 9(2)
$\rm{BCC}^{s}$ | 0.245 961 7(3) | 0.245 961 5(2) | 0.245 961 7(2) | 0.245 961 70(11) | 0.245 961 7(3)
$\rm{FCC}^{b}$ | 0.120 163 9(5) | 0.120 163 3(3) | 0.120 163 6(2) | 0.120 163 6(2) | 0.120 163 7(3)
$\rm{FCC}^{s}$ | 0.199 235 3(3) | 0.199 235 2(2) | 0.199 235 2(2) | 0.199 235 14(11) | 0.199 235 0(2)
Table 2: Value of the amplitudes $q_{1}$ obtained from the separate fits of
the wrapping probabilities and the dimensionless ratios.
| $Q_{1}$ | $Q_{2}$ | $R^{(x)}$ | $R^{(a)}$ | $R^{(3)}$
---|---|---|---|---|---
$\rm{DM}^{b}$ | 0.277(2) | 0.642(3) | 0.906(4) | 1.236(5) | 0.484(3)
$\rm{DM}^{s}$ | 0.193(2) | 0.458(4) | 0.652(3) | 0.894(4) | 0.341(3)
$\rm{SC}^{b}$ | 0.30(3) | 0.90(7) | 1.20(7) | 1.80(9) | 0.65(7)
$\rm{SC}^{s}$ | 0.22(2) | 0.52(4) | 0.70(4) | 1.00(3) | 0.36(3)
$\rm{BCC}^{b}$ | 0.644(3) | 1.46(2) | 2.084(8) | 2.82(3) | 1.12(2)
$\rm{BCC}^{s}$ | 0.30(1) | 0.72(3) | 1.04(2) | 1.42(2) | 0.56(1)
$\rm{FCC}^{b}$ | 1.19(9) | 2.77(4) | 3.91(3) | 5.29(2) | 2.08(2)
$\rm{FCC}^{s}$ | 0.449(3) | 1.044(9) | 1.507(5) | 2.080(6) | 0.794(4)
Table 3: Percolation thresholds and other non-universal parameters from the
simultaneous fits of the wrapping probabilities. For all the fits, we set
$L_{\rm min}=32$ for $R^{(x)}$ and $R^{(3)}$ and $L_{\rm min}=24$ for
$R^{(a)}$, $a$, $b_{1}$ and $b_{2}$ are defined in Eq. (5).
M. | Obs. | $p_{c}$ | $a$ | $b_{1}$ | $b_{2}$ | M. | Obs. | $p_{c}$ | $a$ | $b_{1}$ | $b_{2}$
---|---|---|---|---|---|---|---|---|---|---|---
$\rm{DM}^{b}$ | $R^{(x)}$ | 0.389 589 22(18) | $0.901(4)$ | $0.012(13)$ | $0.04(17)$ | $\rm{DM}^{s}$ | $R^{(x)}$ | 0.429 986 96(19) | $0.653(4)$ | $0.023(9)$ | $-0.53(11)$
$R^{(a)}$ | 0.389 589 1(1) | $1.236(2)$ | $-0.006(6)$ | $0.08(7)$ | $R^{(a)}$ | 0.429 987 15(12) | $0.895(2)$ | $0.043(5)$ | $-0.73(6)$
$R^{(3)}$ | 0.389 589 40(20) | $0.480(1)$ | $0.011(7)$ | $0.1(1)$ | $R^{(3)}$ | 0.429 986 81(24) | $0.3463(8)$ | $-0.001(6)$ | $-0.27(7)$
$\rm{SC}^{b}$ | $R^{(x)}$ | 0.248 811 84(3) | $1.25(2)$ | $0.001(5)$ | $0.29(6)$ | $\rm{SC}^{s}$ | $R^{(x)}$ | 0.311 607 65(5) | $0.721(5)$ | $0.024(4)$ | $-0.44(5)$
$R^{(a)}$ | 0.248 811 85(3) | $1.69(1)$ | $-0.011(4)$ | $0.78(4)$ | $R^{(a)}$ | 0.311 607 69(4) | $0.992(4)$ | $0.036(3)$ | $0.02(3)$
$R^{(3)}$ | 0.248 811 94(5) | $0.651(8)$ | $0.004(4)$ | $0.03(5)$ | $R^{(3)}$ | 0.311 607 70(8) | $0.384(4)$ | $0.002(4)$ | $-0.46(4)$
$\rm{BCC}^{b}$ | $R^{(x)}$ | 0.180 287 6(1) | $2.069(8)$ | $-0.006(7)$ | $0.2(1)$ | $\rm{BCC}^{s}$ | $R^{(x)}$ | 0.245 961 48(6) | $1.032(7)$ | $0.020(4)$ | $-0.41(5)$
$R^{(a)}$ | 0.180 287 57(9) | $2.839(4)$ | $-0.016(4)$ | $~{}~{}0.01(5)$ | $R^{(a)}$ | 0.245 961 51(6) | $1.407(3)$ | $0.027(3)$ | $-0.46(3)$
$R^{(3)}$ | 0.180 287 65(9) | $1.102(3)$ | $-0.004(4)$ | $0.27(6)$ | $R^{(3)}$ | 0.245 961 46(9) | $0.543(2)$ | 0.001(3) | $-0.19(4)$
$\rm{FCC}^{b}$ | $R^{(x)}$ | 0.120 163 79(7) | $3.87(2)$ | $0.004(8)$ | $0.05(12)$ | $\rm{FCC}^{s}$ | $R^{(x)}$ | 0.199 235 17(6) | $1.48(3)$ | $0.011(6)$ | $-0.13(8)$
$R^{(a)}$ | 0.120 163 80(5) | $5.311(6)$ | $-0.008(4)$ | $0.04(5)$ | $R^{(a)}$ | 0.199 235 22(5) | $2.077(3)$ | $0.018(4)$ | $-0.13(4)$
$R^{(3)}$ | 0.120 163 72(18) | $2.059(5)$ | $0.014(6)$ | $-0.1(1)$ | $R^{(3)}$ | 0.199 235 12(9) | $0.804(2)$ | $0.002(5)$ | $0.09(6)$
### III.1 Separate fits
In numerical study of phase transitions, dimensionless ratios like $Q_{1}$ and
$Q_{2}$ are known to provide powerful tools for locating critical points
$p_{c}$. The wrapping probabilities have analogous finite-size scaling
behaviors as the dimensionless ratios, and thus should also provide a useful
method for estimating $p_{c}$. This is demonstrated in Fig. 2 for site
percolation on the DM lattice. The intersections of the $Q_{2}$ data for
different sizes $L$ would approximately give the percolation threshold
$p_{c}\approx 0.429\,95$, with uncertainty at the fourth or fifth decimal
place. Due to their faster convergence as $L$ increases, the intersections of
the $R^{(x)}$ data would yield $p_{c}\approx 0.429\,99$. Similar phenomena are
observed in all the percolation models studied in this work. Thus, it clearly
suggests that the wrapping probabilities are more powerful tools for
estimating $p_{c}$ than the dimensionless ratios $Q_{1}$ and $Q_{2}$.
According to the least-squares criterion, we fit Monte Carlo data for the
quantities $R^{(x)}$, $R^{(a)}$, $R^{(3)}$, $Q_{1}$ and $Q_{2}$ separately for
each percolation model to the following scaling ansatz
$\displaystyle U(p,L)$ $\displaystyle=$ $\displaystyle
U_{0}+\sum_{k=1}^{3}q_{k}(p-p_{c})^{k}L^{ky_{t}}$ (4)
$\displaystyle+b_{1}L^{-1.2}+b_{2}L^{-2}\;,$
where $y_{t}$ is the thermal exponent, $U_{0}$ is a universal value depending
on the quantity studied, and the $q_{k}$ ($k=1,2,3$) and $b_{j}$ ($j=1,2$) are
non-universal constants. A correction exponent of $-1.2$ is taken from the
existing literature Wang et al. (2013). To evaluate the systematic errors
caused by the scaling terms which are not included in the fitting ansatz, we
set a lower cutoff $L\geq L_{\rm min}$ on the data and study the effect on the
residual $\chi^{2}$ as $L_{\rm min}$ increases. Generally, we prefer the fit
which produces $\chi^{2}/DF\sim O(1)$ ($DF$ is the degree of freedom), and in
which the subsequent increases of $L_{\rm min}$ do not drop $\chi^{2}$ by
vastly more than one unit per degree of freedom. These principles apply in all
the fits we carry out.
In the fits, we try different combinations of corrections to scaling: (1) both
$b_{1}$ and $b_{2}$ are free to be determined by the data; (2), $b_{1}$ is set
to $0$ and $b_{2}$ is free; and (3), $b_{1}$ is free and $b_{2}$ is fixed at
$0$. We find that the correction amplitudes $b_{1}$ for the wrapping
probabilities are rather small and in many cases are statistically consistent
with zero. In contrast, for the dimensionless ratios one clearly observes a
non-zero correction amplitude $b_{1}$. Moreover, the amplitudes $q_{1}$ of the
term $q_{1}(p-p_{c})L^{y_{t}}$ in Eq. (4) for $R^{(x)}$ and $R^{(a)}$ are
larger than those for $Q_{1}$ and $Q_{2}$. This suggests that the wrapping
probabilities are more sensitive to the deviation from criticality $p-p_{c}$
than the dimensionless ratios. These observations in the fits are reflected by
Fig. 2.
Tables 1 and 2 summarize the percolation thresholds and the amplitudes $q_{1}$
from our preferred fits with combination (1), where the uncertainties are just
the statistical errors. It can be seen that the estimates of $p_{c}$ from
different quantities are consistent with each other within the combined error
margins. Further, the wrapping probabilities yield more accurate estimate of
$p_{c}$ than the dimensionless ratios by a factor of two or three.
Table 4: Simultaneous fits of the wrapping probabilities
$R^{(x)},R^{(a)},R^{(3)}$ for all models.
Obs. | $y_{t}$ | $U_{0}$ | $U_{2}$ | $U_{3}$
---|---|---|---|---
$R^{(x)}$ | 1.1424(11) | $0.257\,80(6)$ | $1.23(1)$ | $-0.9(6)$
$R^{(a)}$ | 1.1418(4) | $0.460\,02(2)$ | $0.311(2)$ | $-0.99(4)$
$R^{(3)}$ | 1.1413(6) | $0.080\,46(4)$ | $4.98(1)$ | $9.7(4)$
Figure 3: $R^{(x)}(p,L)$ versus $L^{y_{t}}$ at given $p$ values which are in close to the estimated percolation thresholds for the site and bond percolation on the BCC (top), FCC (middle) and DM (bottom) respectively. Table 5: Final estimates of percolation thresholds for the three-dimensional percolation models. The error bars include both statistical and systematic errors. Lattice | Bond | | | Site |
---|---|---|---|---|---
| $p_{c}$(Present) | $p_{c}$(Previous) | | $p_{c}$(Present) | $p_{c}$(Previous)
DM | 0.389 589 2(5) | 0.389 3(2) Marck (1998) | | 0.429 987 0(4) | 0.430 1(4) Marck (1998)
| | 0.390(11) Vyssotsky et al. (1961) | | | 0.426(+0.08,-0.02) Silverman and Adler (1990)
SC | 0.248 811 85(10) | 0.248 811 82(10) Wang et al. (2013) | | 0.311 607 68(15) | 0.311 607 7(2) Wang et al. (2013)
| | 0.248 812 6(5) Lorenz and Ziff (1998b) | | | 0.311 607 4(4) Deng and Blöte (2005)
BCC | 0.180 287 62(20) | 0.180 287 5(10) Lorenz and Ziff (1998a) | | 0.245 961 5(2) | 0.245 961 5(10) Lorenz and Ziff (1998b)
| | | | | 0.246 0(3) Bradley et al. (1991), 0.246 4(7) Gaunt and Sykes (1983)
FCC | 0.120 163 77(15) | 0.120 163 5(10) Lorenz and Ziff (1998a) | | 0.199 235 17(20) | 0.199 236 5(10) Lorenz and Ziff (1998b)
### III.2 Simultaneous fits
As described above, the Monte Carlo simulations for the SC lattice are much
more extensive and are performed on larger system sizes than those on the
other lattices. This leads to the more precise estimates of $p_{c}$ and other
parameters on the SC lattices. It is noted that for a given wrapping
probability or dimensionless ratio, the value of $U_{0}$ in Eq. (4) is
universal. To make use of the extensive simulation for the SC lattice, we
carry out a simultaneous analysis of the Monte Carlo data for all the
percolation systems studied in this work. More precisely, we choose the
wrapping probabilities $R^{(x)}$, $R^{(a)}$, and $R^{(3)}$, and for each of
them, the data is fitted by
$\displaystyle U(p_{j},L)$ $\displaystyle=$ $\displaystyle
U_{0}+\sum_{k=1}^{3}U_{k}a_{j}^{k}(p_{j}-p_{c,j})^{k}L^{ky_{t}}$ (5)
$\displaystyle+b_{1,j}L^{-1.2}+b_{2,j}L^{-2}\;,$
where $U_{k}\;(k=0,1,2,3)$ and $y_{t}$ are universal; $j\in\\{1,2,...,8\\}$
refer to the site and bond percolation models on DM, SC, BCC and FCC lattice,
and the parameters with subscript $j$ are model-dependent. In other words, Eq.
(5) can be regarded as a set of equations in which the universal parameters
$U_{k}$ and $y_{t}$ are shared by all the percolation models. We expect that
an accurate estimation of these universal parameters will be mainly achieved
by the high-precision Monte Carlo data on the SC lattice, and as in return,
this will help to improve the accuracy of $p_{c}$ for the other models. Such a
simultaneous analysis has been applied to the 3D Ising model, and the
derivation of Eq. (5) can be found in Ref. Deng and Blöte (2003).
The simultaneous fits by Eq. (5) follow the same procedure as that in the
above subsection. We first note that among $U_{1}$ and $a_{j}$ with
$j=1,\cdots,8$, there is one redundant parameter, and we thus set $U_{1}=1$.
Tables 3 and 4 summarize the results for the universal parameters, the
percolation thresholds, and other non-universal constants, taken from the
preferred fits with $L_{\rm min}=24$ or $32$. In these fits, both the
correction amplitudes $b_{1,j}$ and $b_{2,j}$ are left free. It can be seen
from Tab. 3 that the leading correction amplitudes $b_{1,j}$ are rather small.
In the cases that $b_{1,j}$ cannot be distinguished from zero within the
statistical uncertainties, one can in principle exclude the leading correction
term in the fits, which will further decrease the error margins.
In comparison with the results in Tab. 1 from the separate fits, the
simultaneous analyses do significantly improve the estimates of $p_{c}$. By
taking into account the results from different wrapping probabilities and from
fits with different $L_{\rm min}$, we obtain the final estimates of $p_{c}$,
as summarized in Tab. 5. To check the reliability of the final quoted error
margins in Tab. 5, we plot the $R^{(x)}$ data at $p_{c}$ and two other $p$
values which are away from $p_{c}$ about four or five times of final error
bars. Precisely at $p=p_{c}$, the $R^{(x)}$ data should tend to a horizontal
line as $L\rightarrow\infty$, whereas the data at $p\neq p_{c}$ will bend
upward or downward. This is indeed clearly seen in these plots, some of which
are shown in Fig. 3, confirming the reliability of our final results in Tab.
5.
Also presented in Tab. 5 are existing estimates of $p_{c}$ from the
literature. It can be seen that this work does provide the percolation
thresholds $p_{c}$ with higher precision. For the bond and site percolation
models on the DM lattices, such improvement is significant.
## IV Results at $p_{c}$
By fixing $p$ at or very close to the estimated thresholds $p_{c}$ in Tab. 5,
we study the covariances $g^{(x)}_{bR}$ and $g^{(x)}_{sR}$, the largest-
cluster size $C_{1}$, the shortest-path length $S$, and the cluster-number
density $\rho$. From their finite-size-scaling behaviors, one can determine
the thermal and magnetic renormalization exponent $y_{t}$ and $y_{h}$, the
shortest-path fractal dimension $d_{\rm min}$, and the universal excess
cluster number $b$. In addition, we also obtain the thermodynamic cluster-
number densities $\rho_{c}$ for the studied percolation models.
Figure 4: Log-log plot of $C_{1}$ , $g^{(x)}_{b(s)R}$ and $S$ versus the
rescaled linear size $L^{*}$ for all the $8$ percolation models. We set
$L=L^{*}$ for the bond percolation on the SC lattice, and rescale $L$ by a
constant factor (model-dependent) to collapse the numerical data.
### IV.1 Exponents $y_{t}$, $y_{h}$ and $d_{\rm min}$
Following an analogous simultaneous analysis procedure, we fit the data of
$g^{(x)}_{bR}$ and $g^{(x)}_{sR}$, $C_{1}$, and $S$ by the ansatz
${\mathcal{A}}=L^{y_{\mathcal{A}}}(a_{0,j}+b_{1,j}L^{-1.2}+b_{2,j}L^{-2})\;,$
(6)
where ${y_{\mathcal{A}}}$ is the universal scaling exponent. It is $y_{t}$ for
covariance $g^{(x)}_{bR}$ and $g^{(x)}_{sR}$, $y_{h}$ for the largest-cluster
size $C_{1}$, and $d_{\rm min}$ for the shortest-path length $S$. We obtain
$y_{t}=1.141\,3(15)$, $y_{h}=2.522\,93(10)$, and $d_{\rm min}=1.375\,5(3)$,
which are consistent with the estimates in Ref. Wang et al. (2013), with
comparable or slightly better precision. For an illustration of these
universal exponents, we plot in the log-log scale the data of these quantities
versus the rescaled linear size $L^{*}=wL$, with constant $w=1$ for the bond
percolation on the SC lattice.
### IV.2 Excess number of clusters
The numerical data of the cluster-number density at percolation thershold for
all the studied percolation models are simultaneous fitted by the scaling
ansatz
$\displaystyle\rho=\rho_{c}+V^{-1}(b+b_{1,j}L^{-2})\;,$ (7)
where $V$ is the number of lattice sites, and the correction amplitude $b$ is
known to be also universal and is referred to as the excess cluster number in
Ref. Ziff et al. (1997). The subleading correction is taken to be $-2$. Due to
the rapid decay of the correction term, the finite-$L$ data of $\rho$ quickly
converges to the thermodynamic value $\rho_{c}$; the well-determined values of
$\rho_{c}$ then aids in estimating the correction amplitude $b$ from the
small-$L$ data. The fitting results of $\rho_{c}$ and $b$ are shown in Table
6. Taking into account some potential systematic errors–e.g., due to the small
deviation of the simulated $p$ value from $p_{c}$, we have the final estimate
$b=0.675(1)$.
Table 6: Simultaneous fits of $\rho$ at the thresholds. Fitting parameter
$L_{\rm min}=16$ is set for all the models.
M. | $\rho_{c}$ | $b$ | $b_{1}$
---|---|---|---
$\rm DM^{b}$ | 0.231 953 78(4) | $0.674\,7(4)$ | $-0.6(5)$
$\rm DM^{s}$ | 0.075 519 45(2) | $-1.1(4)$
$\rm SC^{b}$ | 0.272 932 836(9) | 1.1(2)
$\rm SC^{s}$ | 0.052 438 217(3) | $-0.02(12)$
$\rm BCC^{b}$ | 0.298 343 834(12) | $0.3(3)$
$\rm BCC^{s}$ | 0.040 045 144(3) | $-0.76(9)$
$\rm FCC^{b}$ | 0.307 691 25(2) | $0.1(2)$
$\rm FCC^{s}$ | 0.026 526 453(4) | $-0.3(2)$
Figure 5: Excess cluster number $V(\rho-\rho_{c})$ ($\equiv b$) versus
$L^{-1}$(left) and $L^{3}$(right) for SC site (top) and BCC site (bottom)
percolation models. The dashed straight lines represent constant $0.675$.
An illustration of the excess cluster number $b$ is shown in Fig. 5, where the
values of $V(\rho-\rho_{c})$ are plotted versus $1/L$ for the site percolation
on the SC and the BCC lattices. It can be seen that the $V(\rho-\rho_{c})$
values at $p=p_{c}$ quickly converge to $b=0.675$, while those for $p\neq
p_{c}$ are either bending downward or upward. However, this does not imply
that the cluster-number density $\rho$ provides a good quantity for locating
$p_{c}$. Near $p_{c}$, the finite-size behavior of $\rho(p,L)$ near threshold
$p_{c}$ can be described by
$\displaystyle\rho(p,L)$ $\displaystyle=$
$\displaystyle\rho_{c}+f_{1}(p-p_{c})+f_{2}(p-p_{c})^{2}+V^{-1}[b+$ (8)
$\displaystyle
h_{1}(p-p_{c})L^{y_{t}}+h_{2}(p-p_{c})^{2}L^{2y_{t}}+\cdots]\;,$
where $f_{i}$ and $h_{i}$ ($i=1,2$) are non-universal parameters. The critical
density $\rho_{c}$ and the terms with $f_{i}$ arise from the analytical part
of $\rho(p,L)$ and do not depend on size $L$. They dominate the finite-size
scaling of $\rho(p,L)$ but cannot be used to determine $p_{c}$. This is
illustrated in Fig. 5. The critical singularity is reflected in the subleading
terms with $L$-dependence. For the site percolation on the SC lattice, the fit
yields $p_{c}=0.311\,604(2)$, with much larger error margin than those from
wrapping probabilities.
## V Summary
We present a Monte Carlo study of the bond and site percolation on several
three-dimensional lattices, and obtain high-precision estimates of the
percolation thresholds (Tab. 5), the cluster density (Tab. 6), the wrapping
probabilities (Tab. 4) and the excess cluster number b = 0.675(1). These
accurate scientific data can serve as a testing ground for future study of
systems in the percolation universality class. More importantly, it is
observed that the wrapping probabilities can be a useful and reliable approach
for locating phase transitions. It is very plausible that this observation
generally holds in other statistical-mechanical systems that have suitable
graphical representations.
## VI Acknowledgments
We thank R. M. Ziff and T. M. Garoni for helpful suggestions. This research
was supported in part by NSFC under Grant No. 91024026, 11275185 and 11147013,
and the Chinese Academy of Science. We also acknowledge the Specialized
Research Fund for the Doctoral Program of Higher Education under Grant No.
20103402110053. The simulations were carried out on the NYU-ITS cluster, which
is partly supported by NSF Grant No. PHY-0424082.
## References
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* Bollobás and Riordan (2006) B. Bollobás and O. Riordan, _Percolation_ (Cambridge University Press, 2006).
* Nienhuis (1987) B. Nienhuis, in _Phase Transition and Critical Phenomena_ , edited by C. Domb, M. Green, and J. L. Lebowitz (Academic Press, London, 1987), vol. 11.
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* Ding (2010) C. Ding, Z. Fu, W. Guo and F. Y. Wu, Phys. Rev. E 81, 061111 (2010), and references therein.
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* Hara and Slade (1990) T. Hara and G. Slade, Commun. Math. Phys. 128, 333 (1990).
* Wang et al. (2013) J. Wang, Z. Zhou, W. Zhang, T. M. Garoni, and Y. Deng, Phys. Rev. E 87, 052107 (2013).
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* Deng and Blöte (2005) Y. Deng and H. W. J. Blöte, Phys. Rev. E 72, 016126 (2005).
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* Silverman and Adler (1990) A. Silverman and J. Adler, Phys. Rev. B 42, 1369 (1990).
* Vyssotsky et al. (1961) V. A. Vyssotsky, S. B. Gordon, H. L. Frisch, and
* Lorenz and Ziff (1998b) C. D. Lorenz and R. M. Ziff, J. Phys. A 31, 8147 (1998b).
* Lorenz and Ziff (1998a) C. D. Lorenz and R. M. Ziff, Phys. Rev. E 57, 230 (1998a).
* Bradley et al. (1991) R. M. Bradley, P. N. Strenski, and J. M. Debierre, Phys. Rev. B 44, 76 (1991).
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|
arxiv-papers
| 2013-10-21T02:07:33 |
2024-09-04T02:49:52.636755
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Xiao Xu, Junfeng Wang, Jian-Ping Lv and Youjin Deng",
"submitter": "Junfeng Wang",
"url": "https://arxiv.org/abs/1310.5399"
}
|
1310.5441
|
# The influence of the magnetic field on the spectral properties of blazars
J. M. Rueda-Becerril1, P. Mimica1, and M. A. Aloy1
1Departamento de Astronomía y Astrofísica, Universidad de Valencia, 46100,
Burjassot, Spain E-mail: [email protected]
###### Abstract
We explore the signature imprinted by dynamically relevant magnetic fields on
the spectral energy distribution (SED) of blazars. It is assumed that the
emission from these sources originates from the collision of cold plasma
shells, whose magnetohydrodynamic evolution we compute by numerically solving
Riemann problems. We compute the SEDs including the most relevant radiative
processes and scan a broad parameter space that encompasses a significant
fraction of the commonly accepted values of not directly measurable physical
properties. We reproduce the standard double hump SED found in blazar
observations for unmagnetized shells, but show that the prototype double hump
structure of blazars can also be reproduced if the dynamical source of the
radiation field is very ultrarelativistic both, in a kinematically sense
(namely, if it has Lorentz factors $\gtrsim 50$) and regarding its
magnetization (e.g., with flow magnetizations $\sigma\simeq 0.1$). A fair
fraction of the blazar sequence could be explained as a consequence of shell
magnetization: negligible magnetization in FSRQs, and moderate or large (and
uniform) magnetization in BL Lacs. The predicted photon spectral indices
($\Gamma_{\rm ph}$) in the $\gamma-$ray band are above the observed values
($\Gamma_{\rm ph,obs}\lesssim 2.6$ for sources with redshifts $0.4\leq z\leq
0.6$) if the magnetization of the sources is moderate ($\sigma\simeq
10^{-2}$).
###### keywords:
BL Lacertae objects: general – Magnetohydrodynamics (MHD) – Shock waves –
radiation mechanisms: non-thermal – radiative transfer
## 1 Introduction
Blazars are a type of radio-loud active galactic nuclei (AGN) whose jets are
pointing very close to the line of sight towards the observer (e.g., Urry &
Padovani, 1995). They can be subdivided in two main groups: BL Lac objects,
whose spectrum is featureless or shows only weak absorption lines and flat-
spectrum radio quasars (FSRQs), which show broad emission lines in the optical
spectrum (e.g., Giommi et al., 2012). Blazars are commonly classified
according to the relative strength of their observed spectral components.
Those spectral components are associated to the contribution of a relativistic
jet (non-thermal emission), the accretion disk and the broad-line region
(thermal radiation), and the light from the host, usually a giant elliptical
galaxy. The broadest component of the spectrum is the non-thermal one, and it
spans the whole electromagnetic frequency range, usually displaying two broad
peaks. The lower-frequency part is due to the synchrotron emission (it usually
peaks in the range $10^{12}$-$10^{17}$ Hz), while the high-frequency region is
believed to be due to the inverse-Compton scattering (e.g., Fossati et al.,
1998).
In this work we concentrate exclusively on the contribution from the
relativistic jet. The internal shock (IS) scenario (e.g., Rees & Meszaros,
1994; Spada et al., 2001; Mimica et al., 2004) has been successful in
explaining many of the features of the blazar variability. At the core of the
IS scenario is the idea that the presence of relative motions in the
relativistic jet will produce ‘collisions’ of cold and dense blobs of plasma
(shells). In the course of the shell collision the plasma is shocked and part
of the jet kinetic energy is dissipated at relatively weak internal shocks,
which shall account for the observed flares in the light curves of these
events. In the past two decades this scenario has been thoroughly explored
using analytic and (simplified) numerical modeling (Kobayashi et al., 1997;
Daigne & Mochkovitch, 1998; Spada et al., 2001; Bošnjak et al., 2009; Daigne
et al., 2011) and by means of numerical hydrodynamics simulations (Kino et
al., 2004; Mimica et al., 2004, 2005, 2007).
More recently, the effects of strong magnetic fields on the shell collisions
have been investigated. The shocked plasma is believed to be magnetized, to
some extent, since we observe radiation that can be best fit as synchrotron
emission of particles accelerated in internal plasma collisions. However, we
do not really know the degree of magnetization of the jet flow, and whether
its magnetic energy is being dissipated in addition to its kinetic energy. In
the case of moderate or strong magnetic fields the IS scenario has to be
modified to account for the differences in dynamics (e.g., the suppression of
one of the two shocks resulting in a binary collision Fan et al., 2004; Mimica
& Aloy, 2010) and the emission properties of the flares (Mimica et al., 2007;
Mimica & Aloy, 2012).
This work continues along the lines sketched in our previous paper (Mimica &
Aloy, 2012, MA12 in the rest of the text). MA12 extends the work on the
dissipation (dynamic efficiency) of magnetized IS (Mimica & Aloy, 2010) by
including radiative processes in a manner similar to that of the recent
detailed models for the computation of the IS emission (Böttcher & Dermer,
2010; Joshi & Böttcher, 2011; Chen et al., 2011). In MA12 we assume a constant
flow luminosity, but vary the degree of the shell magnetization in order to
investigate the consequences of that variation for the observed spectra and
light curves. The radiative efficiency of a single shell collision is found to
be largest when one of the colliding shells is very magnetized, while the
other one has weak or no magnetic field. We proposed a way to distinguish
observationally between weakly and strongly magnetized shell collisions
through the comparison of the inverse-Compton and synchrotron maximum
frequencies and fluences111Note that the ratio of fluences $F_{\rm IC}/F_{\rm
syn}$ (a redshift-independent quantity) is related to the Compton-dominance
parameter $A_{C}$ (ratio of IC and synchrotron luminosity, see e.g., Finke,
2013). For more details see Appendix B..
One of the limitations of MA12 is that only shell magnetization is varied
(albeit with a relatively dense coverage of the potential parameter space),
leaving the rest of the parameters unchanged. In this work we present results
of a more systematic parametric study where we consider three combinations of
the shell magnetizations, which MA12 found to be of interest, but vary both
kinematical (shell Lorentz factors and relative velocity) and extrinsic
parameters (jet viewing angle), while the microphysical parameters are fixed
to typically accepted values.
In Section 2 we discuss the method and list the models considered in the
present work. Section 3 presents the results which are discussed and
summarized in Section 4.
## 2 Modeling dynamics and emission from internal shocks
In this section we summarize the method of MA12, which is used to model the
dynamics of shell collisions and the resulting non-thermal emission (we follow
Sections 2, 3 and 4 of MA12). We also discuss the three families of numerical
models used in this work.
### 2.1 Dynamics of shell collisions
Assuming a cylindrical outflow and neglecting the jet lateral expansion (e.g.,
Mimica et al., 2004) we can simplify the problem of colliding shells to a one-
dimensional interaction of two cylindrical shells with cross-sectional radius
$R$ and thickness $\Delta r$. We fix the luminosity $L$ of the outflow to a
constant value and allow the shell Lorentz factor $\Gamma$ and the
magnetization $\sigma$ (see Eq. 2 in Appendix A for definition) to vary. This
allows us to compute the number density in an unshocked shell (see Eq. 3 of
MA12):
$n=\displaystyle{\frac{L}{\pi
R^{2}m_{p}c^{3}\left[\Gamma^{2}(1+\epsilon+\chi+\sigma)-\Gamma\right]\sqrt{1-\Gamma^{-2}}}}\
,$ (1)
where $m_{p}$ and $c$ are the proton mass and the speed of light,
$\chi:=P/\rho c^{2}\ll 1$ is the ratio between the thermal pressure $P$ and
the rest-mass energy density, and $\epsilon$ is the specific internal energy
(see Eq. 2 of MA12).
Once the number density, the thermal pressure, the magnetization, and the
Lorentz factor of the faster (left) and the slower (right) shell have been
determined, we use the exact Riemann solver of Romero et al. (2005) to compute
the evolution of the shell collision. In particular, we compute the properties
of the shocked shell fluid (shock velocity, compression factor, magnetic
field) which we then use to obtain the synthetic observational signature (see
the following section).
### 2.2 Non-thermal particles and emission
For the readers benefit, we briefly summarize Sections 3.1 and 3.2 of MA12 on
the assumptions about the distribution of the dissipated unshocked shell
kinetic energy among the electrons and the magnetic fields.
We assume that a stochastic magnetic field $B_{S,st}$ is created at shocks.
The strength of this field is parametrized by assuming that the magnetic field
energy density is a fraction $\epsilon_{B}$ of the dissipated kinetic energy,
i.e. $B_{S,st}=\sqrt{8\pi\epsilon_{B}u_{S}}$, where $u_{S}$ is the internal
energy density in the shocked shell, obtained by the exact Riemann solver.
Since we study the evolution of plasma shells with arbitrary degrees of
magnetization carried out by macroscopic fields $B_{S,mac}$, the _total_
magnetic field in the shell is defined as
$B_{S}:=\sqrt{B_{S,mac}^{2}+B_{S,st}^{2}}$. $B_{S}$ is the field in which
electrons are assumed to gyrate and emit synchrotron radiation. In practice,
this means that the value of $\epsilon_{B}$ is irrelevant for models in which
the macroscopic magnetization is large, since in such a case, $B_{S}\simeq
B_{S,mac}$. The parameter $\epsilon_{B}$ only shapes the spectral properties
of _weakly magnetized_ models. In such models an increase in $\epsilon_{B}$
may modify (though not significantly) the spectral shape (e.g., Böttcher &
Dermer, 2010, Fig. 9).
We assume that a fraction $\epsilon_{e}$ of the dissipated kinetic energy is
used to accelerate electrons in the vicinity of shock fronts. We keep
$\epsilon_{e}$ fixed in this work aiming to reduce the number of free
parameters. We do not expect its possible variation to influence our results
qualitatively (e.g., Böttcher & Dermer, 2010, show in Fig. 7 that a change in
$\epsilon_{e}$ does not change the Compton dominance $A_{C}$).
In order to compute synthetic time-dependent multi-wavelength spectra and
light curves, we assume that the dominant emission processes resulting from
the shocked plasma are synchrotron, external inverse-Compton (EIC) and
synchrotron self-Compton (SSC). The EIC component is the result of the up-
scattering of near infrared photons (likely emitted from a dusty torus around
the central engine of the blazar or from the broad line region) by the non-
thermal electrons existing in the jet. We further consider that the observer’s
line of sight makes an angle $\theta$ with the jet axis. A detailed
description of how the integration of the radiative transfer equation along
the line of sight is performed can be found in Section 4 of MA12.
### 2.3 Models
The main difference between this work and MA12 is that we allow for shell
Lorentz factors and the viewing angle $\theta$ to vary. Table 3 shows the
spectrum of model parameters that we consider in the next sections. In order
to group our models according to the initial shell magnetizations we denote by
letters W, M, S, S1 and S2 the following families of models:
* W:
weakly magnetized, $\sigma_{L}=10^{-6},\sigma_{R}=10^{-6}$,
* M:
moderately magnetized, $\sigma_{L}=10^{-2},\sigma_{R}=10^{-2}$,
* S:
strongly magnetized, $\sigma_{L}=1,\sigma_{R}=10^{-1}$,
* S1:
strongly and equally magnetized, $\sigma_{L}=10^{-1},\sigma_{R}=10^{-1}$, and
* S2:
strongly magnetized, $\sigma_{L}=10^{-1},\sigma_{R}=1$.
The remaining three parameters, $\Gamma_{R}$, $\Delta g$ and $\theta$ can take
any of the values shown in Table 3. We have considered three families of
strongly magnetized models (S, S1 and S2), which differ in the distribution of
the magnetization of the interacting shells. Our reference strongly magnetized
model family is the S, since in MA12 we found that these models have the
maximum dynamical efficiency. This set of models is supplemented with two
additional families of models: S1, which accounts for shells having the same
(high) magnetization, and S2, with parameters complementary of the S-family,
and having the peculiarity that the colliding shells do not develop a forward
shock (instead they form a forward rarefaction; see MA12) if $\Delta g\lesssim
1.5$, so that they only emit because of the presence of a reverse shock. For
clarity, when we refer to a particular model we label it by appending values
of each of these parameters to the model letter. For instance, S-G10-D1.0-T3
is the strongly magnetized model with $\Gamma_{R}=10$ (G10), $\Delta g=1.0$
(D1.0) and $\theta=3^{\circ}$ (T3). If we refer to a subset of models with one
or two parameters fixed we use an abbreviated notation, where we skip any
reference to the varying parameters in the family name. As an example of this
abbreviated notation, in order to refer to all weakly magnetized models with
$\Gamma_{R}=10$ and $\theta=5^{o}$ we use W-G10-T5, while all moderately
magnetized models with $\Delta g=1.5$ are M-D1.5. We perform a systematic
variation of parameters in order to find the dependence of the radiative
signature on each of them separately, as well as their combinations by fixing,
e.g. the Doppler factor ${\cal D}:=[\Gamma(1-\beta\cos{\theta})]^{-1}$ of the
shocked fluid. We perform such a parametric scan for a typical source located
at redshift $z=0.5$.
Parameter | value
---|---
$\Gamma_{R}$ | $10,\ 12,\ 17,\ 20,\ 22,\ 25,\ 50,\ 100$
$\Delta g$ | $0.5,\ 0.7,\ 1.0,\ 1.5,\ 2.0$
$\sigma_{L}$ | $10^{-6},\ 10^{-2},\ 10^{-1},\ 1$
$\sigma_{R}$ | $10^{-6},\ 10^{-2},\ 10^{-1},\ 1$
$\epsilon_{B}$ | $10^{-3}$
$\epsilon_{e}$ | $10^{-1}$
$\zeta_{e}$ | $10^{-2}$
$\Delta_{\rm acc}$ | $10$
$a_{\rm acc}$ | $10^{6}$
$R$ | $3\times 10^{16}$ cm
$\Delta r$ | $6\times 10^{13}$ cm
$q$ | $2.6$
$L$ | $5\times 10^{48}$ erg s-1
$u_{\rm ext}$ | $5\times 10^{-4}$ erg cm-3
$\nu_{\rm ext}$ | $10^{14}$ Hz
$z$ | $0.5$
$\theta$ | $1^{\circ},\ 3^{\circ},\ 5^{\circ},8^{\circ}\,10^{\circ}$
Table 1: Parameters of the models. $\Gamma_{R}$ is the Lorentz factor of the
slow shell, $\Delta g:=\Gamma_{L}/\Gamma_{R}-1$ ($\Gamma_{L}$ is the Lorentz
factor of the fast shell), $\sigma_{L}$ and $\sigma_{R}$ are the fast and slow
shell magnetizations, $\zeta_{e}$ and $q$ are the fraction of electrons
accelerated into power-law Lorentz factor (or energy) distribution and its
corresponding power-law index33footnotemark: 3, $\Delta_{\rm acc}$ and $a_{\rm
acc}$ are the parameters controlling the shock acceleration efficiency (see
Section 3.2 of MA12 for details), $L$, $R$ and $\Delta r$ are the jet
luminosity, jet radius and the initial width of the shells, $u_{\rm ext}$ and
$\nu_{\rm ext}$ are the energy density and the frequency of the external
radiation field (see Section 4.2 of MA12 for details), $z$ is the redshift of
the source and $\theta$ is the viewing angle. Note that $\Gamma_{R}$, $\Delta
g$, $\sigma_{L}$, $\sigma_{R}$ and $\theta$ can take any of the values
indicated.
## 3 Results
Here we present the main results of the parameter study, grouping them
according to the families defined in Sec. 2.3, so that the results for the
weakly, moderately and strongly magnetized shell collisions are given in Sec.
3.1, 3.2 and 3.3, respectively. To characterize the difference between models
we resort to compute their light curves, average spectra, and their spectral
slope $\Gamma_{\rm ph}$ and photon flux $F_{\rm ph}$ (assuming a relation
$F_{\nu}\propto\nu^{-\Gamma_{\rm ph}+1}$) in the band where the observed
photon energy is above $200$ MeV. In the rest of the text we will refer to
this band as $\gamma$-ray band.
### 3.1 Weakly magnetized models
In Fig. 1 we show the light curves at optical (R-band), X-ray ($1$-$10$ keV)
and $\gamma$-ray ($1$ GeV) energies for two different values of the relative
shell Lorentz factor, i.e., for two values of the parameter $\Delta g$ while
keeping the rest fixed. The duration of the light curve depends moderately on
$\Delta g$, as can be seen from the difference in peak times for optical and
$\gamma$-ray light curves. The time of the peak of the light curve in each
band depends on the dominant emission process in that band: synchrotron and
EIC dominate the R-band and the $1$ GeV emission and peak soon after the
shocks cross the shells. The SSC emission dominates the X-rays (dashed lines
in Fig. 1), and its peak is related to the physical length of the emission
regions. The X-ray peak occurs later due to the fact that synchrotron photons
from one shocked shell have to propagate across a substantial part of the
shell volume before being scattered by the electrons in the other shell (see
Sec. 6.2 of MA12 for more details). The corresponding average flare spectra
are shown in the left panel of Fig. 2, where we also display (inset)
$\Gamma_{\rm ph}$ as a function of the photon flux $F_{\rm ph}$ in the
$\gamma$-ray band.
Figure 1: Light curves for the weakly magnetized models W-G10-D0.5-T5 (black
lines) and W-G10-D2.0-T5 (orange lines). The light curves in R-band, hard
X-ray band (1-10 keV) and at 1 GeV are shown as full, dashed and dot-dashed
lines, respectively. The time of the peaks of the R-band and 1 GeV light
curves correspond to the moment the shocks cross the respective shells (first
the RS, and then the FS). A steep decline after the peak is partly due to the
assumed cylindrical geometry, since in a conical jet the high-latitude
emission would smooth out the decline.
As can be seen from Fig. 2, the parameter $\Delta g$ has a very strong
influence on both peak frequencies and peak fluxes (see also Sec. 5.8 of
Böttcher & Dermer, 2010). In particular, the synchrotron peak shifts steadily
to ever higher frequencies (from $\simeq 10^{12}$ Hz for $\Delta g=0.5$ to
$\simeq 10^{15}$ Hz for $\Delta g=2.0$), with a similar trend for the IC peak.
$F_{\rm ph}$ has a maximum for $\Delta g=0.7$, and then it decreases
monotonically. The reason for this non monotonic behavior is that in the model
with the smallest $\Delta g$, W-G10-D0.5-T5, the SSC and EIC components (black
dot-dashed and dot-dot-dashed lines in the left panel of Fig. 2, respectively)
are of equal importance in the $\gamma$-ray band, but increasing $\Delta g$
leads to the domination of the spectrum by SSC (e.g., orange dot-dashed and
dot-dot-dashed lines in Fig. 2 show the SSC and EIC components of
W-G10-D2.0-T5, respectively). For the parameters and observational frequencies
of blazars, the Klein-Nishina cutoff affects the EIC, but does not affect the
SSC peak (see Sec. 4.2 of MA12 or Sec. 3.1 of Aloy & Mimica 2008). Therefore,
the SSC peak can increase with $\Delta g$, while EIC cannot. In the model
W-G10-D2.0-T5 the SSC peak enters the $\gamma$-ray band, thus causing the
flattening of the spectrum. Finally, the appearance of a non-smooth IC hump in
the spectrum happens when $\Delta g$ is low (see the case of $\Delta g=0.5$ in
Fig. 2). This result suggests that flares with a smooth IC spectrum in weakly
magnetized blazars are likely produced by shells whose $\Delta
g\mathbin{\lower 3.0pt\hbox{$\hbox to0.0pt{\raise 5.0pt\hbox{$\char
62\relax$}\hss}\mathchar 29208\relax$}}0.5$ (i.e. relative Lorentz factor is
larger than $\simeq 1.1$).
Table 2 lists a number of physical parameters in the shocked regions of the
models shown in the left panel of Fig. 2. As can be seen, the increase in
$\Delta g$ has as a consequence a moderate increase in the compression ratio
and the magnetic field in the shocked regions, as well as an increase in the
number of injected electrons in the both shocks (FS and RS).
The non-thermal electrons in weakly magnetized models are in a slow-cooling
regime, as inferred from the fact that $\gamma_{c}/\gamma_{1}\mathbin{\lower
3.0pt\hbox{$\hbox to0.0pt{\raise 5.0pt\hbox{$\char 62\relax$}\hss}\mathchar
29208\relax$}}1$. The typical magnetic field is of the order of $1$ G and is
of the same order of magnitude, though slightly larger in the reverse than in
the forward shocked region. The difference becomes larger for higher $\Delta
g$ (see Sec. 3.3 for a more detailed discussion of this point).
$\Delta g$ | $\Gamma$ | $r_{r}$ | $\displaystyle{\frac{B_{r}}{1{\rm G}}}$ | $\displaystyle{\frac{Q_{r,11}}{{\rm cm}^{-3}{\rm s}^{-1}}}$ | $\displaystyle{\frac{\gamma_{1r}}{10^{2}}}$ | $\displaystyle{\frac{\gamma_{2r}}{10^{4}}}$ | $\displaystyle{\frac{t^{\prime}_{crr}}{10^{3}{\rm s}}}$ | $\displaystyle{\frac{\gamma_{cr}}{\gamma_{1r}}}$ | $r_{f}$ | $\displaystyle{\frac{B_{f}}{1{\rm G}}}$ | $\displaystyle{\frac{Q_{f,11}}{{\rm cm}^{-3}{\rm s}^{-1}}}$ | $\displaystyle{\frac{\gamma_{1f}}{10^{2}}}$ | $\displaystyle{\frac{\gamma_{2f}}{10^{4}}}$ | $\displaystyle{\frac{t^{\prime}_{crf}}{10^{3}{\rm s}}}$ | $\displaystyle{\frac{\gamma_{cf}}{\gamma_{1f}}}$
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
$0.5$ | $11.8$ | $4.10$ | $0.95$ | $0.06$ | $2.90$ | $4.77$ | $91.2$ | $23.77$ | $4.01$ | $0.95$ | $0.02$ | $1.28$ | $4.78$ | $91.3$ | $54.21$
$0.7$ | $12.2$ | $4.21$ | $1.17$ | $0.22$ | $5.60$ | $4.31$ | $74.9$ | $10.53$ | $4.05$ | $1.16$ | $0.07$ | $1.91$ | $4.33$ | $75.0$ | $31.38$
$1.0$ | $12.6$ | $4.42$ | $1.40$ | $0.76$ | $11.19$ | $3.93$ | $63.0$ | $4.50$ | $4.09$ | $1.38$ | $0.17$ | $2.71$ | $3.97$ | $63.1$ | $19.17$
$1.5$ | $13.1$ | $4.86$ | $1.66$ | $2.71$ | $24.45$ | $3.61$ | $54.2$ | $1.75$ | $4.13$ | $1.60$ | $0.37$ | $3.68$ | $3.68$ | $54.3$ | $12.40$
$2.0$ | $13.4$ | $5.37$ | $1.84$ | $6.08$ | $42.66$ | $3.43$ | $50.1$ | $0.90$ | $4.16$ | $1.74$ | $0.55$ | $4.32$ | $3.53$ | $50.2$ | $9.86$
Table 2: Physical parameters in the forward and reverse shocked regions for
the family of models W-G10-T5, in which the Lorentz factor of the slower shell
as well as the viewing angle are fixed to $\Gamma_{R}=10$ and
$\theta=5^{\circ}$, respectively. Subscripts $r$ and $f$ denote the reverse
and forward regions, respectively. The bulk Lorentz factor of both shocked
regions is denoted by $\Gamma$. In each region $r$, $B$, $Q$, $\gamma_{1}$ and
$\gamma_{2}$ denote its compression ratio, comoving magnetic field, comoving
number of electrons injected per unit volume and unit time, and lower and
upper cutoffs of the injected electrons (see Eq. 11 of MA12). In the table we
show $Q_{r,11}=Q_{r}\times 10^{-11}$ and $Q_{f,11}=Q_{f}\times 10^{-11}$.
$t^{\prime}_{cr}:=\Delta r^{\prime}/(c|\beta^{\prime}|)$ is the shock crossing
time, where $\Delta r^{\prime}$ and $\beta^{\prime}$ are the shell width and
the shock velocity in the frame moving with the contact discontinuity
separating both shocks (section 2 of MA12).
$\gamma_{c}:=\gamma_{2}/(1+\nu_{0}\gamma_{2}t^{\prime}_{cr})$ is the cooling
Lorentz factor of an electron after a dynamical time scale (shock crossing
time). $\nu_{0}:=(4/3)c\sigma_{T}(u^{\prime}_{B}+u^{\prime}_{\rm
ext})/(m_{e}c^{2})$ is the cooling term, where $\sigma_{T}$ is the Thomson
cross section and the primed quantities are measured in the comoving frame.
When $\gamma_{c}/\gamma_{1}\gg(\ll)1$ the electrons are slow (fast) cooling.
Figure 2: Left panel: average spectra for weakly magnetized models W-G10-T5
(i.e. with fixed $\Gamma_{R}=10$ and $\theta=5$). The spectrum of each model
has been averaged over the time interval $0-1000$ ks. In addition, for the
models W-G10-D0.5-T5 and W-G10-D2.0-T5 we show the synchrotron, SSC and EIC
contributions (dashed, dot-dashed and dot-dot-dashed lines, respectively). The
blue line shows the spectrum of the model
$(\sigma_{L},\sigma_{R})=(10^{-6},10^{-6})$ of MA12. The inset shows the
spectral slope $\Gamma_{\rm ph}$ as a function of the photon flux $F_{\rm ph}$
in the $\gamma$-ray band. We use the same band and the spectral slope
definition as in Abdo et al. (2009). Right panel: same as left panel, but for
the models W-D1.0-T5.
Next we consider the case in which $\Gamma_{R}$ is increased, and repeat the
previous experiments, but fixing $\Delta g=1$, i.e., we consider the series of
models W-D1.0-T5 (right panel of Fig. 2). We note that increasing the Lorentz
factor of the slower shell yields a reduced flare luminosity. This behavior
results because, for the fixed viewing angle ($\theta=5^{\circ}$) and $\Delta
g$, increasing the Lorentz factor of the slower shell implies that both shells
move faster, and the resulting shocked regions are Doppler dimmed (for an
illustration of the case when both $\Gamma_{R}$ and $\Delta g$ are varied see
Fig. 6 of Joshi & Böttcher, 2011). However, the most remarkable effect is that
for values $\Gamma_{R}\mathbin{\lower 3.0pt\hbox{$\hbox to0.0pt{\raise
5.0pt\hbox{$\char 62\relax$}\hss}\mathchar 29208\relax$}}17$, we note a
qualitative change in the IC part of the spectrum. The EIC begins to dominate
in $\gamma$-rays. Since, as discussed above, the peak of the EIC spectrum is
shaped by the Klein-Nishina cut-off, for frequencies $\mathbin{\lower
3.0pt\hbox{$\hbox to0.0pt{\raise 5.0pt\hbox{$\char 62\relax$}\hss}\mathchar
29208\relax$}}10^{23}$ Hz there is no dependence on $\Gamma_{R}$. However,
since the synchrotron peak flux decreases with increasing $\Gamma_{R}$, this
means that the IC-to-synchrotron ratio of peak fluxes increases with
$\Gamma_{R}$. The weak dependence of the $\gamma$-ray spectrum on $\Gamma_{R}$
can also be seen in the inset of the right panel of Fig. 2, where the points
for $\Gamma_{R}\gtrsim 17$ accumulate around $\Gamma_{\rm ph}\lesssim 2.35$
and $F_{\rm ph}\simeq 3\times 10^{-8}$ cm-2 s-1.
### 3.2 Moderately magnetized models
The second family of models contains cases of intermediate magnetization
$\sigma_{L}=\sigma_{R}=10^{-2}$. The left panel of Fig. 3 shows the effect of
the variation of $\Delta g$ on the average spectra for the models M-G10-T5\.
The blue line corresponds to the moderately magnetized model in MA12. It can
be seen that for $\Delta g\mathbin{\lower 3.0pt\hbox{$\hbox to0.0pt{\raise
5.0pt\hbox{$\char 62\relax$}\hss}\mathchar 29208\relax$}}\ 1$, a flattening of
the spectrum below the synchrotron peak starts to become noticeable. This
effect becomes even more pronounced for the strongly magnetized models (see
next section). Low values of $\Delta g$ tend to reduce much more the IC
spectral components than the synchrotron ones. This trend is also noticeable
in weakly and strongly magnetized models. Thus, regardless of the
magnetization, very small values of $\Delta g$ may not be compatible with
observations. In the $\gamma$-ray band, an increase in $\Delta g$ causes an
increase in $F_{\rm ph}$ and a variation in $\Gamma_{\rm ph}$ characterized by
a maximum, where $\Gamma_{\rm ph}\simeq 2.9$, for $\Delta g=1$.
Figure 3: Left panel: same as left panel of Fig. 2, but for the moderately
magnetized models M-G10-T5, i.e., $\sigma_{L}=10^{-2}$ and
$\sigma_{R}=10^{-2}$. Right panel: same as right panel of Fig. 3, but for
variable $\Gamma_{R}$ while keeping fixed $\Delta g=1$ and $\theta=5^{o}$
(models M-D1.0-T5). For models M-G10-D1.0-T5 and M-G25-D1.0-T5 (i.e., models
with $\Gamma_{R}=10,25$) dashed, dot-dashed and dot-dot-dashed lines show the
synchrotron, SSC and EIC contributions, respectively.
Table 3 shows the microphysical parameters of the shocked regions in these
models. As $\Delta g$ grows, the magnetic field and the number of injected
particles increase at the region swept by the forward shock, while the
electrons transition from a moderate or intermediate-cooling regime to fast-
cooling one. A noticeable difference with respect to the weakly magnetized
models is that now the comoving magnetic field in the region swept by the
reverse shock decreases as $\Gamma_{L}$ increases with increasing $\Delta g$
(or, equivalently, $\Gamma$). This is a consequence of keeping the jet
luminosity and the shell magnetization constant while increasing the Lorentz
factor of the faster shell.
$\Delta g$ | $\Gamma$ | $r_{r}$ | $\displaystyle{\frac{B_{r}}{1{\rm G}}}$ | $\displaystyle{\frac{Q_{r,11}}{{\rm cm}^{-3}{\rm s}^{-1}}}$ | $\displaystyle{\frac{\gamma_{1r}}{10^{2}}}$ | $\displaystyle{\frac{\gamma_{2r}}{10^{4}}}$ | $\displaystyle{\frac{t^{\prime}_{crr}}{10^{3}{\rm s}}}$ | $\displaystyle{\frac{\gamma_{cr}}{\gamma_{1r}}}$ | $r_{f}$ | $\displaystyle{\frac{B_{f}}{1{\rm G}}}$ | $\displaystyle{\frac{Q_{f,11}}{{\rm cm}^{-3}{\rm s}^{-1}}}$ | $\displaystyle{\frac{\gamma_{1f}}{10^{2}}}$ | $\displaystyle{\frac{\gamma_{2f}}{10^{4}}}$ | $\displaystyle{\frac{t^{\prime}_{crf}}{10^{3}{\rm s}}}$ | $\displaystyle{\frac{\gamma_{cf}}{\gamma_{1f}}}$
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
$0.5$ | $11.7$ | $3.17$ | $19.07$ | $1.20$ | $2.88$ | $1.07$ | $79.3$ | $0.09$ | $2.55$ | $23.09$ | $0.32$ | $0.91$ | $0.97$ | $77.8$ | $0.20$
$0.7$ | $12.1$ | $3.55$ | $18.88$ | $4.05$ | $6.03$ | $1.07$ | $68.2$ | $0.05$ | $2.80$ | $25.35$ | $0.93$ | $1.47$ | $0.92$ | $66.9$ | $0.12$
$1.0$ | $12.5$ | $3.97$ | $17.94$ | $13.14$ | $13.15$ | $1.10$ | $59.1$ | $0.03$ | $3.02$ | $27.36$ | $2.32$ | $2.24$ | $0.89$ | $58.1$ | $0.08$
$1.5$ | $13.0$ | $4.55$ | $16.44$ | $48.22$ | $32.57$ | $1.15$ | $51.9$ | $0.02$ | $3.22$ | $29.09$ | $5.14$ | $3.23$ | $0.86$ | $51.1$ | $0.06$
$2.0$ | $13.3$ | $5.12$ | $15.41$ | $155.20$ | $64.93$ | $1.19$ | $48.4$ | $0.01$ | $3.31$ | $29.98$ | $7.75$ | $3.90$ | $0.85$ | $47.7$ | $0.05$
Table 3: Same as Table 2, but for models M-G10-T5.
Let us consider now the spectral variations induced by a changing $\Gamma_{R}$
and fixed $\Delta g$ (right panel of Fig. 3). In contrast to what has been
seen in weakly magnetized models (Sec. 3.1; Fig. 2), for $\Gamma_{R}\gtrsim
20$, the two IC contributions are comparable (for smaller values of
$\Gamma_{R}$ the SSC component dominates the IC spectrum). For $\Gamma_{R}=10$
the maximum of the EIC emission is 100 times smaller than the corresponding
SSC maximum, while for $\Gamma_{R}=25$ the EIC peak is higher than the SSC
peak, and indeed it is expected to keep growing as the bulk Lorentz factor
goes further into the ultrarelativistic regime. Similar to the right panel of
Fig. 2, the Klein-Nishina cut-off causes the coincidence of EIC spectra at
$\simeq 10^{23}$ Hz. This effect is also seen in the $F_{\rm ph}$-$\Gamma_{\rm
ph}$ plot, where for $\Gamma_{R}\mathbin{\lower 3.0pt\hbox{$\hbox
to0.0pt{\raise 5.0pt\hbox{$\char 62\relax$}\hss}\mathchar 29208\relax$}}17$
the photon flux is approximately constant333We point out that differences
smaller than $\lesssim 0.1$ in $\Gamma_{\rm ph}$ are probably not
distinguishable from an observational point of view., with a slight decrease
in $\Gamma_{\rm ph}$ as $\Gamma_{R}$ grows.
Shell magnetization, $\Delta g$ and $\Gamma_{R}$ are related to the intrinsic
properties of the emitting regions. It is also interesting to explore the
effects on the SED of varying extrinsic properties of the models, such as the
viewing angle $\theta$, while keeping the intrinsic ones constant. Figure 4
shows the result of changing the jet orientation. With increasing $\theta$
both the synchrotron and IC maxima decrease. As it can be noticed looking at
the brown lines, the maxima drop almost in a straight line with positive
slope. To illustrate this fact, we show the spectrum normalized to the Doppler
factor ${\cal D}^{3}$ in the left panel of Fig. 5.444We note that the
normalization in e.g. left panel of Fig. 5 is equivalent to the ${\cal
D}^{3+\alpha}$ of Dermer (1995) if we take into account that we do not only
normalize the SED by the Doppler factor but also the frequencies. As can be
seen, the synchrotron spectra coincide for all models (assuming the frequency
is normalized by ${\cal D}$), while the IC spectral fluxes decrease with
increasing $\theta$. For comparison, in the right panel of Fig. 5 we normalize
the spectra by ${\cal D}^{4}$. In this case the IC spectra below the peak
(cooling break) coincide, while the synchrotron part gets less luminous with
decreasing angle. Thus, we find a remarkable agreement among the normalized
spectra obtained from the same source but with different viewing angles, if we
scale all the spectra by ${\cal D}^{3}$.
Figure 4: Same as Fig. 3, but for variable $\theta$. $\Gamma_{R}=10$ and
$\Delta g=1.0$ have been fixed, i.e. models M-G10-D1.0 are shown. For easier
visualization the synchrotron and IC spectral maxima of different models have
been marked by boxes and connected by brown lines.
Figure 5: Left panel: same as left panel of Fig. 4, but dividing the
frequencies by ${\cal D}$ and the SED by ${\cal D}^{3}$. Right panel: same as
right panel of Fig. 4, but normalizing the SED by ${\cal D}^{4}$.
### 3.3 Strongly magnetized models
The third model family considers the strongly magnetized models where
$\sigma_{L}=1$ and $\sigma_{R}=0.1$. The left panel of Fig. 6 shows the
dependence of the average spectra on $\Delta g$. Strongly magnetized models in
moderately relativistic flows (i.e., having moderate values of $\Gamma_{R}$)
dramatically suppress the IC spectral component. However, with increasing
values of $\Delta g$ the IC component broadens in frequency range and grows
moderately. Another remarkable fact of strongly magnetized models is that for
$\Delta g>1.0$ the synchrotron spectrum ceases to be a parabolic, single-
peaked curve and becomes a more complex curve where the contributions from the
FS and the RS are separated, since the peak frequencies of the synchrotron
radiation produced at the FS and at the RS differ by two or three orders of
magnitude. The reason is the strong magnetic field in the emitting regions:
magnetization in the shocked regions increases proportionally to their
compression factors $r_{f}$ and $r_{r}$, respectively (see Eq. 3 in Appendix
A), i.e. the shocked regions are even more magnetically dominated than the
initial shells. In Table 4 we see that the electrons in the reverse shock of
the strongly magnetized models are fast-cooling. In fact, for $\Delta g\gtrsim
1.5$ the injected electron spectrum is almost mono-energetic. In these models
the lower cutoff $\gamma_{1r}$ is about a factor of $30$ larger than
$\gamma_{1f}$. Since the synchrotron maximum of the fast-cooling electrons is
determined by the lower cutoff, the synchrotron spectrum of the RS peaks at a
frequency which is $(\gamma_{1f}/\gamma_{1r})^{2}\approx 10^{3}$ times higher
than that of the FS. This can be seen in left panel of Fig. 6, where dashed
and dot-dashed lines show the respective spectra of the RS and FS of the model
S-G10-D2.0-T5.
The dominance of the EIC component for $\Gamma_{R}\gtrsim 20$ and $\nu\gtrsim
10^{21}$ Hz appears to be a property tightly related to the increment of
$\Gamma_{R}$ (right panel of Fig. 6). In this case, the EIC component
“replicates” the synchrotron peak associated to the forward shock of the
collision, modulated by the Klein-Nishina cut-off for large values of
$\Gamma_{R}$. Because of this effect, progressively larger values of
$\Gamma_{R}$ increase the Compton dominance, i.e. the trend is to recover the
standard double-hump structure of the SED as $\Gamma_{R}$ rises. We have
tested that for $\Gamma_{R}=50\mbox{ and }100$, the IC spectral component
becomes almost monotonic and concave (Fig. 7). For $\Gamma_{R}\gtrsim 50$, the
SED becomes akin to that of models with moderate or low shell magnetization,
but the IC spectrum displays a plateau rather than a maximum. As the Lorentz
factor increases ($\Gamma_{R}\gtrsim 50$), our models form a flat spectrum in
the soft X-ray band rather than a minimum between two concave regions. We note
that the spectrum of the $\Gamma_{R}=100$ model displays very steep rising
spectrum flanking the IC contribution because we have fixed a value of the
microphysical parameter $a_{\rm acc}=10^{6}$. Smaller values of such parameter
tend to broaden significantly both the IC and the synchrotron peak (Böttcher &
Dermer, 2010, see e.g.,). Hence, we foresee that a suitable combination of
microphysical and kinematical parameters would recover a more “standard”
double-hump structure.
Figure 6: Left panel: same as left panel of Fig. 2, but for the strongly magnetized models S-G10-T5, i.e., $\sigma_{L}=1$ and $\sigma_{R}=0.1$. For the cases $\Delta g=0.5,2.0$ we show the reverse and forward shock contributions to their spectra in dashed and dot-dashed lines, respectively. While at small values of $\Delta g$ the contribution of the RS dominates fully the spectrum, at larger values of $\Delta g$ the FS contribution has increased relative to the RS one, and is an order of magnitude stronger than the former one in the case of the model with $\Delta g=0.5$. This also explains a second (higher) peak in the synchrotron domain, as well as a flattening in the $\gamma$-ray band. Right panel: same as right panel of Fig. 3, but for strongly magnetized models S-D1.0-T5. $\Delta g$ | $\Gamma$ | $r_{r}$ | $\displaystyle{\frac{B_{r}}{1{\rm G}}}$ | $\displaystyle{\frac{Q_{r,11}}{{\rm cm}^{-3}{\rm s}^{-1}}}$ | $\displaystyle{\frac{\gamma_{1r}}{10^{2}}}$ | $\displaystyle{\frac{\gamma_{2r}}{10^{4}}}$ | $\displaystyle{\frac{t^{\prime}_{crr}}{10^{3}{\rm s}}}$ | $\displaystyle{\frac{\gamma_{cr}}{\gamma_{1r}}}$ | $r_{f}$ | $\displaystyle{\frac{B_{f}}{1{\rm G}}}$ | $\displaystyle{\frac{Q_{f,11}}{{\rm cm}^{-3}{\rm s}^{-1}}}$ | $\displaystyle{\frac{\gamma_{1f}}{10^{2}}}$ | $\displaystyle{\frac{\gamma_{2f}}{10^{4}}}$ | $\displaystyle{\frac{t^{\prime}_{crf}}{10^{3}{\rm s}}}$ | $\displaystyle{\frac{\gamma_{cf}}{\gamma_{1f}}}$
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
$0.5$ | $12.7$ | $1.26$ | $53.51$ | $0.11$ | $0.66$ | $0.64$ | $34.6$ | $0.12$ | $1.89$ | $51.57$ | $3.30$ | $1.91$ | $0.65$ | $37.5$ | $0.04$
$0.7$ | $12.8$ | $1.46$ | $54.72$ | $1.03$ | $2.29$ | $0.63$ | $34.1$ | $0.03$ | $1.93$ | $52.68$ | $4.20$ | $2.14$ | $0.64$ | $36.7$ | $0.04$
$1.0$ | $13.0$ | $1.75$ | $55.84$ | $7.33$ | $7.25$ | $0.62$ | $33.6$ | $0.01$ | $1.98$ | $53.90$ | $5.45$ | $2.41$ | $0.63$ | $35.8$ | $0.03$
$1.5$ | $13.2$ | $2.22$ | $56.63$ | $68.00$ | $26.38$ | $0.62$ | $32.9$ | $0.003$ | $2.02$ | $55.22$ | $7.14$ | $2.73$ | $0.63$ | $34.8$ | $0.03$
$2.0$ | $13.3$ | $2.67$ | $56.82$ | $112900.75$ | $61.68$ | $0.62$ | $32.5$ | $0.001$ | $2.05$ | $56.03$ | $8.39$ | $2.94$ | $0.62$ | $34.3$ | $0.02$
Table 4: Same as Table 2, but for models S-G10-T5\. Note that the $Q_{r,11}$
for $\Delta g=2.0$ is much larger than $Q_{r,11}$ of the other models because
$\gamma_{1r}$ is very close to $\gamma_{2r}$.
We also find that the SED of strongly magnetized models is very sensitive to
relatively small variations of magnetization between colliding shells. To show
such a variety of phenomenologies, we display in Fig. 8 the SEDs of the
families S1-G10-T5 (left panel) and S2-G10-T5, right panel, i.e., considering
only the variations in the SED induced by a change in $\Delta g$. The three
families of strongly magnetized models only have differences in magnetization
within a factor 10\. Clearly, when the faster shell is less magnetized than
the slower one (the case of the S2-family), the models recover a more typical
double-hump structure, closer to that found in actual observations. We note
that for contribution to the SED of the forward shock in the S2-family is
either non-existing, because these models do not form a FS or, if a FS forms,
it is very weak (see dashed lines in the right panel of Fig. 8.
For completeness, we consider how the SED changes when varying the viewing
angle (Fig. 9). In these models, increasing $\theta$ lowers the total emitted
flux all over the spectral range under consideration. The Compton dominance
for $\theta\mathbin{\lower 3.0pt\hbox{$\hbox to0.0pt{\raise 5.0pt\hbox{$\char
60\relax$}\hss}\mathchar 29208\relax$}}8^{\circ}$ remains constant. To explain
this behavior, we shall note that fixing both $\Gamma_{R}$ and $\Delta g$,
increasing $\theta$ is equivalent to decrease the Doppler factor ${\cal D}$.
Theoretically, it is known that the beaming pattern of a relativistically
moving blob of electrons that Thompson-scatters photons from an external
isotropic radiation field changes as ${\cal D}^{4+\alpha}$ ($\alpha$ being the
spectral index of the radiation), while the beaming pattern of radiation
emitted isotropically in the blob frame (e.g., by synchrotron and SSC
processes), changes as ${\cal D}^{3+\alpha}$ (Dermer, 1995). Left and right
panels in Fig. 10 show the spectra from Fig. 9 normalized to ${\cal D}^{3}$
and ${\cal D}^{4}$, respectively. Thus, we expect that the reduction of the
Doppler factor results in a larger suppression of the IC part of the SED, only
if it is dominated by the EIC contribution, as compared with the dimming of
the synchrotron component. In the models at hand (S-G10-D1.0), the IC spectrum
is dominated by the SSC component, and thus, reducing $\theta$ simply
decreases the overall luminosity.
Figure 7: Same as Fig. 6, bur for high $\Gamma_{R}$ cases. For each model the
synchrotron, SSC and EIC contributions are shown using dashed, dot-dashed and
dot-dot-dashed lines, respectively.
Figure 8: Left: Same as the left panel of Fig. 6 for the family S1-G10-T5\.
Right: Same as the left panel of Fig. 6 for the family S2-G10-T5\. In the
S2-family, the forward shock is either non-existing (for $\Delta g\lesssim
1.5$) or extremely weak. We add in the figure the contribution to the spectrum
of the forward shocks of the models with $\Delta g=1.5,2$. Note the difference
in the stencil of the vertical axis with respect to the left panel. Figure 9:
Same as Fig. 4, but for strongly magnetized models S-G10-D1.0.
Figure 10: Left panel: same as Fig. 9, but normalizing the SED by ${\cal
D}^{3}$. Right panel: same as Fig. 4, but normalizing the SED by ${\cal
D}^{4}$.
## 4 Discussion and conclusions
We have extended the survey of parameters started in MA12 for the internal
shocks scenario by computing the multi-wavelength, time-dependent emission for
several model families chiefly characterized by the magnetization of the
colliding shells. In this section we provide a discussion and a summary of our
results.
### 4.1 Intrinsic parameters and emission
In what follows, we consider the effect that changes in intrinsic jet
parameters (magnetization, $\Delta g$ and $\Gamma_{R}$) have on the observed
emission.
#### 4.1.1 Influence of the magnetic field
As was discussed in Sec. 6.1 of MA12, the main signature of high magnetization
is a drastic decrease of the SSC emission due to a much smaller number density
of scattering electrons (Eq. 1). As will be stated in Sec. 4.1.3, this
decrease can be offset by increasing the bulk Lorentz factor (at a cost of
decreasing the overall luminosity). However, extremely relativistic models
(from a kinematical point of view), tend to form plateaus rather than clear
maxima in the synchrotron and IC regimes, and display relatively small values
of $\Gamma_{\rm ph}$. Indeed, the photon spectral index manifest itself as a
good indicator of the flow magnetization. Values of $\Gamma_{\rm ph}\gtrsim
2.6$ result in models where the flow magnetization is $\sigma\simeq 10^{-2}$,
while either strongly or weakly magnetized shell collisions yield $\Gamma_{\rm
ph}\lesssim 2.5$. The observed degeneracy we have found in the case of
strongly magnetized and very high Lorentz factor shells is a consequence of
the fact that either raising the magnetization or the bulk Lorentz factor, the
emitting plasma enters in the ultrarelativistic regime. Which of the two
parameters determines most the final SED, depends on the precise magnitudes of
$\sigma$ and $\Gamma$.
Another way to correlate magnetization with observed properties can be found
representing the Compton dominance $A_{C}$ as a function of the ratio of IC-
to-synchrotron peak frequencies $\nu_{IC}/\nu_{syn}$ (see App. B). Models with
intermediate or low magnetization occupate a range of $A_{C}$ roughly
compatible with observations, while the strongly magnetized models tend to
have values of $A_{C}$ hardly compatible with those observed in actual
sources, unless collisions in blazars happen at much larger Lorentz factors
than currently inferred (see Sect. 4.3).
#### 4.1.2 Influence of $\Delta g$
$\Delta g$ is a parameter which indicates the magnitude of the velocity
variations in the jet. From the average spectra shown in the left panels of
Figs. 2, 3 and 6 we see that the increase of $\Delta g$ leads to the increase
of the Compton dominance parameter (see also Fig. 11), the effect being more
important for either weakly or moderately magnetized models than for strongly
magnetized ones (for which the Compton dominance is almost independent of
$\Delta g$, or even $A_{C}$ decreases for large values of that parameter).
Furthermore, the total amount of emitted radiation also increases with
increasing $\Delta g$, as is expected from the dynamic efficiency study
(Mimica & Aloy, 2010), and confirmed by the radiative efficiency study of
MA12. Finally, for low values of $\Delta g$ the EIC emission is either
dominant or comparable to the SSC one, while SSC becomes dominant at higher
$\Delta g$.
Looking at the physical parameters in the emitting regions (Tables 2, 3 and
4), we see that the increase in $\Delta g$ leads to the increase in the
compression factor $r_{f}$ and $r_{r}$ of the FS and RS. The effect is
strongest for the weakly magnetized models. This increase has as a consequence
the increase in the number density of electrons injected at both, the FS and
the RS. A similar argument can be made for the magnetic fields in the emitting
regions, since the magnetic field undergoes the shock compression as well (see
Appendix A).
In the insets of left panels of Figs. 2, 3 and 6 we see that in $\gamma$-rays
the increase of $\Delta g$ generally reflects in the increase of the photon
flux and a decrease of the spectral slope $\Gamma_{\rm ph}$. Because of the
sensitivity of the photon spectral index in the $\gamma-$ray band, we foresee
that the change in $\Gamma_{\rm ph}$ can be a powerful observational proxy for
the actual values of $\Delta g$ and a distinctive feature of magnetized flows.
Comparing equivalent weakly (Fig. 2; left) and moderately magnetized models
(Fig. 3; left), we observe that the maximum $\Gamma_{\rm ph}$ as a function of
$\Delta g$ increases by $\sim 15\%$ due to the increase in magnetization, and
the value of $\Delta g$ for which the maximum $\Gamma_{\rm ph}$ occurs also
grows, at the same time that $F_{\rm ph}$ decreases by a factor of 50.
We have also found that sufficiently large values of $\Delta g$ tend to
produce a double-peaked structure in the synchrotron dominated part of the
SED. When the relative difference of Lorentz factors grows above $\sim 1.5$,
the contributions arising from the FS and the RS shocks peak at different
times, the RS contribution lagging behind the FS contribution and being more
intense, and occurring at larger frequencies than the latter. The reason for
this phenomenology can be found looking at Tab. 4 and noting that
$\gamma_{1r}$ becomes very large and comparable to $\gamma_{2r}$ for $\Delta
g\mathbin{\lower 3.0pt\hbox{$\hbox to0.0pt{\raise 5.0pt\hbox{$\char
62\relax$}\hss}\mathchar 29208\relax$}}1.5$. For these models
$\gamma_{1r}\gg\gamma_{1f}$ and the frequency of the RS spectral peak is
almost $10^{3}$ times larger than the frequency of the FS spectral peak. The
effect is the flattening of the synchrotron spectrum, or even an appearance of
a second peak. This trend is even more clear when the magnetization of the
shells is increased, so that the most obvious peak in the UV domain happens
for strongly magnetized models (compare the left panels of Figs. 2, 3 and 6).
The observational consequences of the appearance of this peak are discussed
below (Sect. 4.3).
#### 4.1.3 Influence of $\Gamma_{R}$
$\Gamma_{R}$ is the parameter which determines the bulk Lorentz factor of the
jet flow, to a large extent. From Eq. 1 we see that the increase in
$\Gamma_{R}$ leads to a decrease of the number density in the shells, a trend
which is seen in the right panels of Figs. 2, 3 and 6, since it reduces the
emitted flux. Another effect is the decrease in dominance of SSC over EIC as
$\Gamma_{R}$ increases. A related feature is the flattening of the
$\gamma$-ray spectrum (see figure insets). A consequence of the increasing
importance of the EIC is the shifting of the IC spectral maximum to higher
frequencies, until the Klein-Nishina limit is reached. For moderately
magnetized models (right panel of Fig. 3) the IC maximum becomes independent
of $\Gamma_{R}$.
The IC emission in the strongly magnetized models (right panel of Fig. 6) is
dominated by SSC for low values of $\Gamma_{R}$. However, as $\Gamma_{R}$ is
increased, the higher-frequency EIC component becomes ever more luminous.
While none of the models in Fig. 6 reproduces the prototype double-peaked
structure of blazar spectra, the increase of the EIC component with
$\Gamma_{R}$ indicates that perhaps larger values of $\Gamma_{R}$ might
produce a blazar-like spectrum. We have shown in Fig. 7 that the average
spectra for strongly magnetized models where $\Gamma_{R}$ is allowed to grow
up to $100$ display again a double-peaked spectrum, albeit with a much lower
luminosity than the models with lower bulk Lorentz factors.
#### 4.1.4 External radiation field
In this work we did not consider the sources of external radiation in such a
detail as was recently done by e.g. Ghisellini & Tavecchio (2009). These
authors show that, for a more realistic modeling of the external radiation
field, the IC component might be dominating the emission even for a jet with
$\sigma\simeq 0.1$. We note, however, that the difference between their and
our approach is that we model the magnetohydrodynamics of the shell collision,
while they concentrate on more accurately describing the external fields. In
our model the magnetic field not only influences the cooling timescales of the
emitting particles, but also the shock crossing timescales, making direct
comparison difficult, especially for $\sigma\gtrsim 1$ where the dynamics
changes substantially (see, e.g., MA12).
In our models, we take a monochromatic external radiation field with a
frequency $\nu_{\rm ext}$ in the near infrared band, and with an energy
density $u_{\rm ext}$ that tries to mimic, in a simple manner, the emission
from a dusty torus or the emission from the broad line region. More complex
modeling, such as that introduced by Giommi et al. (2012) can be incorporated
in our analysis, at the cost of increasing the number of parameters in our set
up.
### 4.2 The effect of the observing angle
Increasing $\theta$ results in a Doppler deboosting of the collision region
and a significant reduction of the observed flux. The decrease of the flux
comes along with a moderate decrease of $\Gamma_{\rm ph}$ explained by the
different scaling properties with the Doppler factor of the SSC and EIC
contributions to the SED. From theoretical grounds, one expects that the
synchrotron and SSC contributions to the SED scale as ${\cal D}^{3}$ for,
while ${\cal D}^{4}$ is the correct scaling for the EIC spectral component.
Such a theoretical inference is based on assuming a moving spherical blob of
relativistic particles. In our case, instead a blob we have a pair of distinct
cylindrical regions moving towards the observer. The practical consequence of
such a morphological difference is that the synchrotron radiation is roughly
emitted isotropically, and thus, it scales as ${\cal D}^{3}$ (left panels of
Figs. 5 and 10), but the IC contributions are no longer isotropic and thus do
not scale either as ${\cal D}^{3}$ nor as ${\cal D}^{4}$. The effect is
exacerbated when strong magnetizations are considered (compare the right
panels of Figs. 5 and 10).
### 4.3 Comparison with observations
It has been found in several blazar sources that their SEDs have more than two
peaks. Particularly, in some cases a peak frequency of $\sim
10^{15}~{}\mathrm{Hz}$ (e.g., Lichti et al., 1995; Pian et al., 1999) is seen
(a UV bump), which is assumed to come purely from the optically thick
accretion disk (OTAD) and from the Broad Line Region (BLR). In recent works,
thermal radiation from both OTAD and BLR are considered separately in order to
classify blazars (Giommi et al., 2012; Giommi et al., 2013). In the present
work, we have shown that a peak in the UV band can arise by means of non-
thermal and purely internal jet dynamics. This “non-thermal” blue bump is due
to the contribution to the SED of the _synchrotron_ radiation from the reverse
shock in a collision of shells with a sufficiently large relative Lorentz
factor (see left panels of Figs. 2, 3 and 6). We suggest that such a secondary
peak in the UV domain is an alternative explanation for the thermal origin of
the UV bump. In Giommi et al. (2012), the prototype sources displayed in their
Fig. 1 all have synchrotron and IC components of comparable luminosity. In our
case, the strength of the UV peak is larger for the models possessing the
strongest magnetic fields. In such models, the IC part of the spectrum is
strongly suppressed and, thus, they are not compatible with observations.
However, moderate magnetization models display synchrotron and IC components
of similar luminosity. In addition, an increase in the relative Lorentz factor
of the interacting shells produces UV bumps which are more obvious and with
peaks shifted to the far UV. According to Giommi et al. (2012), the spectral
slope at frequencies below the UV-bump ranges from $\alpha_{\rm
r-BlueBump}\sim 0.4$ to $\sim 0.95$. We cannot directly compute such slope
from our data, since we have limited ourselves to compute the SED above
$10^{12}\,$Hz. However, we find compatibility between our models and
observations from comparison of the spectral slope at optical frequencies,
where it is smaller than in the whole range $[5\,{\rm GHz},\nu_{\rm
BlueBump}]$. Extrapolating the data from our models, values $\Delta g\gtrsim
1.5$ combined with shell magnetizations $\sigma\simeq 10^{-3}$ could
accomodate UV bumps with peak frequencies and luminosities in the range
pointed out by current blazar observations.
It has to be noted that the intergalactic medium absorption at frequencies
between $\sim 3\times 10^{15}\,$Hz and $\sim 3\times 10^{17}\,$Hz is extremely
strong, and is not incorporated into our models. Such an extrinsic suppression
of the emitted radiation will impose a (redshift-dependent) upper limit to the
position of the observed UV peak, below the intrinsic reverse shock
synchrotron peaks of our moderately and strongly magnetized models (see e.g.,
orange line in the left panel of Fig. 6 which peaks at $\sim 10^{17}\,$Hz). In
other words, due to the absorption we expect the observed RS synchrotron peak
of such a spectrum to appear at UV frequencies (instead of in X-rays), thus
providing an alternative explanation for the UV bump.
The current observational picture shows that there are two types of blazar
populations with notably different properties. Among other, type defining,
properties that are different in BL Lacs and in FSRQ objects we find that
their respective synchrotron peak frequencies $\nu_{syn}$ are substantially
different. BL Lacs have synchrotron peaks shifted to high frequencies, in some
cases above $10^{18}\,$Hz (e.g., Mkn 501). In contrast, FSRQs are strongly
peaked at low energies (the mean synchrotron frequency peak is
$\bar{\nu}_{syn}\simeq 10^{13.1}$; Giommi et al. 2012).
For the typically assumed or inferred values of the Lorentz factor in blazars
(namely, $\Gamma<30$), the locus of models with different magnetizations is
different in the $A_{C}$ vs $\nu_{syn}$ graph (Fig. 11). While weakly
magnetized models display $A_{C}\gtrsim 3$, the most magnetized ones occupy a
region $A_{C}\lesssim 0.1$. In between ($0.1\lesssim A_{C}\lesssim 3$) we find
the models with moderate magnetizations ($\sigma\simeq 10^{-2}$). Moreover, we
can classify the weakly magnetized models as IC dominated with synchrotron
peak in the IR band. According to observations (Finke, 2013; Giommi et al.,
2012), this region is occupied by FSRQs, while the moderately magnetized cases
fall into the area compatible with data from BL Lacs.
Strongly magnetized models are outside of the observational regime. However,
the quite obvious separation of the locus of sources with different
magnetizations is challenged when very large values of the slowest shell
Lorentz factor ($\Gamma_{R}\gtrsim 30$) are considered. The path followed by
models of the family S-D1.0-T5 (red dash-dotted line in the lower part of Fig.
11), heads towards the region of the graph filled by the weakly magnetized
models as $\Gamma_{R}$ is increased. This increase of $A_{C}$ corresponds to
the fact we have already pointed before: there is a degeneracy between
increasing magnetization and increasing Lorentz factor (Fig. 7). Higher values
of $\Gamma_{R}$ yield more luminous EC components, making that strongly
magnetized models recover the typical SED of blazars, tough with a much
smaller flux than unmagnetized models.
Comparing our Fig. 11 with Fig. 5 of Finke (2013), we find that the Compton
dominance is a good measurable parameter to correlate the magnetization of the
shells with the observed spectra. Moderately magnetized models are located in
the region where some BL Lacs are found, namely, with $0.1\lesssim
A_{C}\lesssim 1$ and $10^{14}\,{\rm Hz}\lesssim\nu_{syn}\lesssim 10^{16}\,$Hz.
We also find that models with high and uniform magnetization
($\sigma_{L}=\sigma_{R}=0.1$; S1-G10-T5 family), and large values of the
relative Lorentz factor $\Delta g\gtrsim 1$ (dot-dot-dashed lines in Fig. 11
and orange lines and symbols in Fig. 12), may account for BL Lacs having peak
synchrotron frequencies in excess of $10^{16}\,$Hz and $A_{C}\lesssim 0.1$.
There is, however, a region of the parameter space which is filled by X-ray
peaked synchrotron blazars with $0.1\lesssim A_{C}\lesssim 1$ that we cannot
easily explain unless seemingly extreme values $\Delta g\gtrsim 2$ are
considered. We point out that the most efficient way of shifting $\nu_{syn}$
towards larger values is increasing $\Delta g$. Such a growth of $\nu_{syn}$
comes with an increase in the Compton dominance, as is found observationally
for FSRQ sources (Finke, 2013). Comparatively, varying $\Gamma_{R}$ drives
moderate changes in $\nu_{syn}$, unless extreme values $\Gamma_{R}\gtrsim 50$
are considered. We must also take into account that the synchrotron peak
frequency is determined by the high-Lorentz factor cut-off $\gamma_{2}$. Most
of our models display values $\gamma_{2}\gtrsim 10^{4}$ in the emitting
(shocked) regions. For comparison, in Finke (2013) $\gamma_{2}=10^{6}$ is
fixed for all his models. The small values of $\gamma_{2}$ in our shell
collisions are due to the microphysical parameters we are using, in
particular, our choice of the shock acceleration efficiency $a_{\rm acc}$,
which was motivated by Böttcher & Dermer (2010). For the models and parameters
picked up by Böttcher & Dermer (2010), they find that neither the peak
synchrotron frequency, nor the peak flux were sensitively dependent on the
choice of $a_{\rm acc}$ (if the power-law Lorentz factor index $q>2$).
However, $\gamma_{2}$ shows the same dependence on $a_{\rm acc}$ than on the
magnetic field strength: $\gamma_{2}\simeq 4.6\times 10^{7}(a_{\rm acc}B[{\rm
G}])^{-0.5}$. In practice, thus, we find a degeneracy in the dependence on
both $a_{\rm acc}$ and $B$ for our models.
Figure 11: Compton dominance $A_{C}$ as a function of the synchrotron peak
frequency $\nu_{\rm syn}$ for the three families of models corresponding to
collisions of the three kinds of magnetized shells. We also display the
Compton dominance for the families of strongly magnetized models S1 and S2.
The different lines are drawn to show the various trends when considering
models where we vary a single parameter and keep the rest constant. The
variation induced by the change in $\Delta g$, $\Gamma_{R}$ and $\theta$ is
shown with black, red and blue lines, respectively. The numbers denote the
value of the varied parameter and the line type is associated to the
magnetization, corresponding the solid, dashed and dot-dashed lines to weakly,
moderately and strongly magnetized shells, respectively. Double-dotted-dashed
and dotted-double-dashed lines correspond to the additional models of the
families S1-G10-T5 and S2-G10-T5, respectively.
Considering the location of the strongly magnetized models with
$\sigma_{L}=1$, and $\sigma_{R}=0.1$ in the $A_{C}$ vs $\nu_{syn}$ graph (Fig.
11), they appear as only marginally compatible with the observations of Finke
(2013) , where almost all sources have $A_{C}>10^{-2}$. since in such models
is difficult to obtain $A_{C}>10^{-2}$, unless the microphysical parameters of
the emitting region are changed substantially (e.g., lowering $a_{\rm acc}$).
This seems to indicate that strongly magnetized models with sensitively
different magnetizations of the colliding shells (in our case there is a
factor 10 difference between the magnetization of the faster and of the slower
shell) are in the limit of compatibility with observations, and that even
larger magnetizations are banned by data of actual sources. MA12 found that
the combination $\sigma_{L}=1$, $\sigma_{R}=0.1$, brings the maximum dynamical
efficiency in shell collisions ($\sim 13\%$), and that has been the reason to
explore the properties of such models here. Models with large and uniform
magnetization $\sigma_{L}=\sigma_{R}=0.1$ display a dynamical efficiency $\sim
10\%$, quite close to the maximum one for a single shell collision, and
clearly bracket better the observations in the $A_{C}$ vs $\nu_{syn}$ plane.
The family of S2-models with $\sigma_{L}=0.1$, $\sigma_{R}=1$ is complementary
to the S-family, but in the former case, only a RS exists, since the FS turns
into a forward rarefaction (MA12), if $\Delta g\lesssim 1.5$. These models
possess a larger Compton dominance ($10^{-2}\lesssim A_{C}\lesssim 4\times
10^{-2}$) than those of the S-family (Fig. 11), and their locus in the $F_{\rm
ph}$ vs $\Gamma_{\rm ph}$ plane (Fig.12; green line and symbols) is much more
compatible with observations. Since the synchrotron emission of the S2-family
is only determined by the RS, if $\Delta g\lesssim 1.5$, or dominated by the
RS emission if $\Delta g\gtrsim 1.5$, the synchrotron peak tends to be at
higher frequencies than in the S and S1 families.
The value of $\Gamma_{\rm ph}$ has also been useful to differentiate
observationally between BL Lacs and FSRQs. According to Abdo et al. (2010) the
photon index, provides a convenient mean to study the spectral hardness, which
is the ratio between the _hard_ sub-band and the _soft_ sub-band (Abdo et al.,
2009). In Fig. 12 we compare the values of $\Gamma_{\rm ph}$ computed for our
three families of models with actual observations of FSRQs and BL Lacs from
the 2LAG catalog (Ackermann et al., 2011). We only represent values of such
catalog corresponding to sources with redshifts $0.4\leq z\leq 0.6$, since our
models have been computed assuming $z=0.5$. We note that the values of
$\Gamma_{\rm ph}$ calculated from fits of the $\gamma-$ray spectra in our
models with moderate magnetization (red colored in the figure) fall just above
the observed maximum values attained in FSRQs ($\Gamma_{\rm ph,obs}^{\rm
FSRQ}\lesssim 2.6$), if the Lorentz factor of the slower shell is
$\Gamma_{R}\sim 10$. However, models with moderate magnetization and larger
Lorentz factors $\Gamma_{R}\gtrsim 15$ display photon indices fully compatible
with FSRQs and photon fluxes in the lower limit set by the technical threshold
that prevents Fermi to detect sources with $F_{\rm ph}\lesssim 2\times
10^{-10}\,$photons cm-2 s-1. BL Lacs exhibit even flatter $\gamma-$ray spectra
than FSRQs, with observed values of the photon index $\Gamma_{\rm ph,obs}^{\rm
BLLac}\lesssim 2.4$. Values $\Gamma_{\rm ph}\gtrsim 2$ are on reach of both
strongly or weakly magnetized models. Nevertheless, the photon flux of
strongly magnetized models falls below the current technical threshold. Being
conservative, this under-prediction of the gamma-photon flux could be taken as
a hint indicating that only models with small or negligible magnetization can
reproduce properly the properties of FSRQs, LBL, and perhaps IBL sources,
while HBL and BL Lacs have microphysical properties which differ from the ones
parametrized in this work. According to Abdo et al. (2009), the photon index
is a quantity that could constrain the emission and acceleration processes
that may be occurring within the jet that produce the flares at hand.
Particularly, we have fixed a number of microphysical parameters
($\epsilon_{B}$, $\epsilon_{e}$, $a_{\rm acc}$, etc.) to typically accepted
values, but we shall not disregard that X-ray, synchrotron-peaked sources have
different values of the aforementioned microphysical parameters. On the other
hand, our values of $\Gamma_{\rm ph}$ are not fully precise, the reason being
the approximated treatment of the Klein-Nishina cutoff. Being not so
conservative, we may speculate that our current gamma ray detectors cannot
observe sources with sufficiently small flux ($F_{\rm ph}\lesssim 3\times
10^{-11}\,$photons cm-2 s-1) to discard or confirm that strongly magnetized
blazars may exist.
Figure 12: Comparison between our numerical models and those sources (FRSQs
and BL Lacs) whose redshift is $0.4\leq z\leq 0.6$ in the 2LAG catalog
(Ackermann et al., 2011). The size of the symbols associated to our models
grows as the parameter which is varied does. For instance, in the case of
models M-G10-D1.0, the smaller values of $\theta$ correspond to the smaller
red circles in the plot.
### 4.4 Conclusions and future work
In the standard model, the SEDs of FSRQs and BL Lacs can be fit by a double
parabolic component with maxima corresponding to the synchrotron and to the
inverse Compton peaks. We have shown that the SEDs of FSRQs and BL Lacs
strongly depends on the magnetization of the emitting plasma. Our models
predict a more complex phenomenology than is currently supported by the
observational data. In a conservative approach this would imply that the
observations restrict the probable magnetization of the colliding shells that
take place in actual sources to, at most, moderate values (i.e.,
$\sigma\lesssim 10^{-1}$), and if the magnetization is large, with variations
in magnetization between colliding shells which are smaller than a factor
$\sim 10$. However, we have also demonstrated that if the shells Lorentz
factor is sufficiently large (e.g., $\Gamma_{R}\gtrsim 50$), magnetizations
$\sigma\simeq 1$ (Fig. 7) are also compatible with a doble hump. Therefore, we
cannot completely discard the possibility that some sources are very
ultrarelativistic both in a kinematically sense and regarding its
magnetization.
We find that FSRQs have observational properties on reach of models with
negligible or moderate magnetic fields. The scattering of the observed FSRQs
in the $A_{C}$ vs $\nu_{syn}$ plane, can be explained by both variations of
the intrinsic shell parameters ($\Delta g$ and $\Gamma_{R}$ most likely), and
of the extrinsic ones (the orientation of the source). BL Lacs with moderate
peak synchrotron frequencies $\nu_{syn}\lesssim 10^{16}\,$Hz and Compton
dominance parameter $0.1\gtrsim A_{C}\gtrsim 1$ display properties that can be
reproduced with models with moderate and uniform magnetization
($\sigma_{L}=\sigma_{R}=10^{-2}$). HBL sources can be partly accommodated
within our model if the magnetization is relatively large and uniform
($\sigma_{L}=\sigma_{R}=10^{-1}$) or if the magnetization of the faster
colliding shell is a bit smaller than that of the slower one
($\sigma_{L}=10^{-1},\sigma_{R}=1$). We therefore find that a fair fraction of
the blazar sequence can be explained in terms of the intrinsically different
magnetization of the colliding shells.
We observe that the change in the photon spectral index ($\Gamma_{\rm ph}$) in
the $\gamma-$ray band can be a powerful observational proxy for the actual
values of the magnetization and of the relative Lorentz factor of the
colliding shells. Values $\Gamma_{\rm ph}\gtrsim 2.6$ result in models where
the flow magnetization is $\sigma\sim 10^{-2}$, whereas strongly magnetized
shell collisions ($\sigma>0.1$) as well as weakly magnetized models may yield
$\Gamma_{\rm ph}\lesssim 2.6$.
The EIC contribution to the SED has been included in a very simplified way in
this paper. We plan to improve on this item by considering more realistic
background field photons as in, e.g., Giommi et al. (2012). We expect that
including seed photons in a wider frequency range will modify the IC spectrum
of strongly magnetized models or of models with low-to-moderate magnetization,
but large bulk Lorentz factor. Finally, the microphysical parameters
characterizing the emitting plasma have been fixed in this manuscript. In a
follow up paper, we will explore the sensitivity of the results (particularly
in moderately to highly magnetized models) to variations of the most
significant microphysical parameters (e.g., $a_{\rm
acc},\epsilon_{B},\epsilon_{e}$, etc).
## Acknowledgments
We acknowledge the support from the European Research Council (grant StG-
CAMAP-259276), and the partial support of grants AYA2010-21097-C03-01,
CSD2007-00050, and PROMETEO-2009-103.
## Appendix A Magnetization in the shocked regions
In an one-dimensional Riemann problem in RMHD the quantity ${\cal
B}:=B^{\prime}/\rho$ is constant across shocks and rarefactions (e.g., Romero
et al., 2005), where $B^{\prime}$ and $\rho$ are the comoving magnetic field
and the fluid density, respectively. The magnetization is defined as
$\sigma:=\displaystyle{\frac{B^{{}^{\prime}2}}{4\pi\rho c^{2}}}\,,$ (2)
and can also be written as $\sigma={\cal B}^{2}\rho/(4\pi c^{2})$.
We point out that the inertial mass-density in a cold magnetized plasma is
$\rho(1+\sigma)\Gamma^{2}$. This means that the plasma can become
ultrarelativistic if either $\sigma\gg 1$ or $\Gamma\gg 1$, since in both
cases the inertial mass-density becomes much larger than the rest-mass density
$\rho$.
The density in the shocked region can be written as $\rho_{s}=r\rho_{0}$,
where $r$ is the compression ratio and $\rho_{0}$ is the density in the
unshocked region. Assuming that in the unshocked region the magnetization is
$\sigma_{0}$ and using the fact that ${\cal B}$ is a constant we have for the
magnetization in the shocked region:
$\sigma_{S}=\displaystyle{\frac{{\cal B}^{2}\rho_{S}}{4\pi
c^{2}}}=\displaystyle{\frac{{\cal B}^{2}r\rho_{0}}{4\pi
c^{2}}}=r\sigma_{0}\,.$ (3)
As can be seen from Eq. 3, the magnetization increases linearly with the shock
compression factor.
## Appendix B Relation between Compton dominance and $F_{\rm IC}/F_{\rm syn}$
Figure 13: Left: Compton dominance, $A_{C}$, as a function of
$\nu_{IC}/\nu_{syn}$. Right: Same as the left panel, but replacing
$\nu_{IC}/\nu_{syn}$ by the ratio of peak fluxes $F_{\rm IC}/F_{\rm syn}$. The
models and the lines in this figure as the same as in Fig. 11.
In Fig. 13 (left) we present a plot of the Compton dominance parameter as a
function of the ratio of peak frequencies $\nu_{IC}/\nu_{syn}$, since these
properties can be directly measured from observations. The models under
consideration in this work separate according to their respective
magnetization. As expected, the lower Compton dominance happens for strongly
magnetized models (dot-dashed lines in the figure), while the weakly
magnetized shell collisions display the larger $A_{C}$. According to $A_{C}$,
there is a factor of more than ten in Compton dominance when considering
shells with magnetizations $\sigma\sim 10^{-2}$, as compared with basically
unmagnetized models. We also note that models with varying orientation are
shifted along diagonal lines in the plot (blue lines in Fig. 13). This is also
the case for families of models in which we vary $\Gamma_{R}$ above a
threshold (magnetization dependent) such that the IC spectrum is dominated by
the EIC contribution (red lines in Fig. 13). If the IC spectrum is dominated
by the SSC contribution, changing $\Gamma_{R}$ yields a horizontal
displacement in the plot. Models with varying $\Delta g$ display a similar
drift as those in which $\theta$ is changed in the case of the moderately
magnetized shell collisions. The trend is not so well defined in case of
weakly magnetized models, and for strongly magnetized models (S-G10-T5), the
Compton dominance is rather insensitive to $\Delta g$, though lower values of
$\Delta g$ yield larger values of $\nu_{IC}/\nu_{syn}$.
To study the global trends of the models, MA12 studied the parameter space
spanned by the ratio of the IC and synchrotron peak frequencies and the ratio
of the IC and synchrotron fluences. In this section we show that the latter
ratio, which we denote by $F_{\rm IC}/F_{\rm syn}$ has a very tight
correlation with the Compton dominance parameter $A_{C}$, defined as the ratio
of the peak IC and peak synchrotron luminosity, as can be seen from Figure 13
(right). This means that either $A_{C}$ or $F_{\rm IC}/F_{\rm syn}$ can be
used interchangeably for the purpose of our parametric study.
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|
arxiv-papers
| 2013-10-21T07:18:32 |
2024-09-04T02:49:52.645551
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "J. M. Rueda-Becerril, P. Mimica, M. A. Aloy",
"submitter": "Jes\\'us Misr\\'ayim Rueda-Becerril",
"url": "https://arxiv.org/abs/1310.5441"
}
|
1310.5620
|
###### Abstract
The small medium large system (SMLsystem) is a house built at the Universidad
CEU Cardenal Herrera (CEU-UCH) for participation in the Solar Decathlon 2013
competition. Several technologies have been integrated to reduce power
consumption. One of these is a forecasting system based on artificial neural
networks (ANNs), which is able to predict indoor temperature in the near
future using captured data by a complex monitoring system as the input. A
study of the impact on forecasting performance of different covariate
combinations is presented in this paper. Additionally, a comparison of ANNs
with the standard statistical forecasting methods is shown. The research in
this paper has been focused on forecasting the indoor temperature of a house,
as it is directly related to HVAC—heating, ventilation and air
conditioning—system consumption. HVAC systems at the SMLsystem house represent
$53.89\%$ of the overall power consumption. The energy used to maintain
temperature was measured to be $30\%$–$38.9\%$ of the energy needed to lower
it. Hence, these forecasting measures allow the house to adapt itself to
future temperature conditions by using home automation in an energy-efficient
manner. Experimental results show a high forecasting accuracy and therefore,
they might be used to efficiently control an HVAC system.
###### keywords:
energy efficiency; time series forecasting; artificial neural networks
10.3390/—— 6 Received: 1 July 2013; in revised form: 17 August 2013 /
Accepted: 21 August 2013 / Published: xx Towards Energy Efficiency:
Forecasting Indoor Temperature via Multivariate Analysis Francisco Zamora-
Martínez *, Pablo Romeu, Paloma Botella-Rocamora and Juan Pardo E-Mail:
[email protected]; Tel.: +34-961-36-90-00 (ext. 2361).
## 1 Introduction
Nowadays, as the Spanish Institute for Diversification and Saving of Energy
(IDAE) Instituto para la diversificación y ahorro de la energía (2011) (IDAE)
of the Spanish Government says, energy is becoming a precious asset of
incalculable value, which converted from electricity, heat or fuel, makes the
everyday life of people easier and more comfortable. Moreover, it is also a
key factor to make the progress of industry and business feasible.
Spanish households consume $30\%$ of the total energy expenditure of the
country Instituto para la diversificación y ahorro de la energía (2011)
(IDAE). In the European Union (EU), primary energy consumption in buildings
represents about $40\%$ of the total Ferreira et al. (2012). In the whole
world, recent studies say that energy in buildings also represents a $40\%$
rate of the total consumed energy, where more than half is used by heating,
ventilation and air conditioning (HVAC) systems Álvarez et al. (2012).
Energy is a scarce resource in nature, which has an important cost, is finite
and must be shared. Hence, there is a need to design and implement new systems
at home, which should be able to produce and use energy efficiently and
wisely, reaching a balance between consumption and streamlined comfort. A
person could realize his activities much easier if his comfort is ensured and
there are no negative factors (e.g., cold, heat, low light, noise, low air
quality, etc.) to disturb him. With the evolution of technology, new
parameters have become more controllable, and the requirements for people’s
comfort level have increased.
Systems that let us monitor and control such aspects make it necessary to
refer to what in reference Arroyo et al. (2006) is called “Ambient
Intelligence” (AmI). This refers to the set of user-centered applications that
integrate ubiquitous and transparent technology to implement intelligent
environments with natural interaction. The result is a system that shows an
active behavior (intelligent), anticipating possible solutions adapted to the
context in which such a system is located. The term, home automation, can be
defined as it is mentioned in reference Sierra et al. (2005), as the set of
services provided by integrated technology systems to meet the basic needs of
security, communication, energy management and comfort of a person and his
immediate environment. Thus, home automation can be understood as the
discipline which studies the development of intelligent infrastructures and
information technologies in buildings. In this paper, the concept of smart
buildings is used in this way, as constructions that involve this kind of
solution.
In this sense, the School of Technical Sciences at the University CEU-UCH has
built a solar-powered house, known as the Small Medium Large System
(SMLsystem), which integrates a whole range of different technologies to
improve energy efficiency, allowing it to be a near-zero energy house. The
house has been constructed to participate in the 2012 Solar Decathlon Europe
competition. Solar Decathlon Europe United States Department of Energy (2012)
is an international competition among universities, which promotes research in
the development of energy-efficient houses. The objective of the participating
teams is to design and build houses that consume as few natural resources as
possible and produce minimum waste products during their lifecycle. Special
emphasis is placed on reducing energy consumption and on obtaining all the
needed energy from the sun. The SMLsystem house includes a Computer-Aided
Energy Saving System (CAES). The CAES is the system that has been developed
for the contest, which aims to improve energy efficiency using home automation
devices. This system has different intelligent modules in order to make
predictions about energy consumption and production.
To implement such intelligent systems, forecasting techniques in the area of
artificial intelligence can be applied. Soft computing is widely used in real-
life applications Wu et al. (2009); Taormina et al. (2012). In fact,
artificial neural networks (ANNs) have been widely used for a range of
applications in the area of energy systems modeling Karatasou et al. (2006);
Ruano et al. (2006); Ferreira et al. (2012); Zamora-Martínez et al. (2012).
The literature demonstrates their capabilities to work with time series or
regression, over other conventional methods, on non-linear process modeling,
such as energy consumption in buildings. Of special interest to this area is
the use of ANNs for forecasting the room air temperature as a function of
forecasted weather parameters (mainly solar radiation and air temperature) and
the actuator (heating, ventilating, cooling) state or manipulated variables,
and the subsequent use of these mid-/long-range prediction models for a more
efficient temperature control, both in terms of regulation and energy
consumption, as can be read in reference Ruano et al. (2006).
Depending on the type of building, location and other factors, HVAC systems
may represent up to $40\%$ of the total energy consumption of a building
Ferreira et al. (2012); Álvarez et al. (2012). The activation/deactivation of
such systems depends on the comfort parameters that have been established, one
of the most being indoor temperature, directly related to the notion of
comfort. Several authors have been working on this idea; in reference Ferreira
et al. (2012), an excellent state-of-the-art system can be found. This is why
the development of an ANN to predict such values could help to improve overall
energy consumption, balanced with the minimum affordable comfort of a home, in
the case that these values are well anticipated in order to define efficient
energy control actions.
This paper is focused on the development of an ANN module to predict the
behavior of indoor temperature, in order to use its prediction to reduce
energy consumption values of an HVAC system. The architecture of the overall
system and the variables being monitored and controlled are presented. Next,
how to tackle the problem of time series forecasting for the indoor
temperature is depicted. Finally, the ANN experimental results are presented
and compared to standard statistical techniques. Indoor temperature
forecasting is an interesting problem which has been widely studied in the
literature, for example, in Neto and Fiorelli (2008); Ferreira et al. (2012);
Álvarez et al. (2012); Oldewurtel et al. (2012); Mateo et al. (2013). We focus
this work in multivariate forecasting using different weather indicators as
input features. In addition, two combinations of forecast models have been
compared.
In the conclusion, it is studied how the predicted results are integrated with
the energy consumption parameters and comfort levels of the SMLsystem.
## 2 SMLhouse and SMLsystem Environment Setup
The Small Medium Large House (SMLhouse) and SMLsystem solar houses (more info
about both projects can be found here: http://sdeurope.uch.ceu.es/) have been
built to participate in the Solar Decathlon 2010 and 2012 United States
Department of Energy (2012), respectively, and aim to serve as prototypes for
improving energy efficiency. The competition focus on reproducing the normal
behavior of the inhabitants of a house, requiring competitors to maintain
comfortable conditions inside the house—to maintain temperature, CO2 and
humidity within a range, performing common tasks like using the oven cooking,
watching television (TV), shower, etc., while using as little electrical power
as possible.
As stated in reference Pan et al. (2012), due to thermal inertia, it is more
efficient to maintain a temperature of a room or building than cooling/heating
it. Therefore, predicting indoor temperature in the SMLsystem could reduce
HVAC system consumption using future values of temperature, and then deciding
whether to activate the heat pump or not to maintain the current temperature,
regardless of its present value. To build an indoor temperature prediction
module, a minimum of several weeks of sensing data are needed. Hence, the
prediction module was trained using historical sensing data from the SMLhouse,
2010, in order to be applied in the SMLsystem.
The SMLhouse monitoring database is large enough to estimate forecasting
models, therefore its database has been used to tune and analyze forecasting
methods for indoor temperature, and to show how they could be improved using
different sensing data as covariates for the models. This training data was
used for the SMLsystem prediction module.
The SMLsystem is a modular house built basically using wood. It was designed
to be an energy self-sufficient house, using passive strategies and water
heating systems to reduce the amount of electrical power needed to operate the
house.
The energy supply of the SMLsystem is divided into solar power generation and
a domestic hot water (DHW) system. The photovoltaic solar system is
responsible for generating electric power by using twenty-one solar panels.
These panels are installed on the roof and at the east and west facades. The
energy generated by this system is managed by a device to inject energy into
the house, or in case there is an excess of power, to the grid or a battery
system. The thermal power generation is performed using a solar panel that
produces DHW for electric energy savings.
The energy demand of the SMLsystem house is divided into three main groups:
HVAC, house appliances and lighting and home electronics (HE). The HVAC system
consists of a heat pump, which is capable of heating or cooling water, in
addition to a rejector fan. Water pipes are installed inside the house, and a
fan coil system distributes the heat/cold using ventilation. As shown in
reference World Business Council for Sustainable Development (2009), the HVAC
system is the main contributor to residential energy consumption, using $43\%$
of total power in U.S. households or $70\%$ of total power in European
residential buildings. In the SMLsystem, the HVAC had a peak consumption of up
to $3.6$ kW when the heat pump was activated and, as shown in Table 1, it was
the highest power consumption element of the SMLsystem in the contest with
$53.89\%$ of total consumption. This is consistent with data from studies
mentioned as the competition was held in Madrid (Spain) at the end of
September. The house has several energy-efficient appliances that are used
during the competition. Among them, there is a washing machine, refrigerator
with freezer, an induction hob/vitroceramic and a conventional oven. Regarding
the consumption of the washing machine and dishwasher, they can reduce the
SMLsystem energy demand due to the DHW system. The DHW system is capable of
heating water to high temperatures. Then, when water enters into these
appliances, the resistor must be activated for a short time only to reach the
desired temperature. The last energy-demanding group consists of several
electrical outlets (e.g., TV, computer, Internet router and others).
System | Power peak (kW) | Total power (Wh) | Percentage
---|---|---|---
HVAC | $3.544$ | $37987.92$ | $53.89\%$
Home appliances | - | $24749.10$ | $35.11\%$
Lighting & HE | $0.300$ | $7755.83$ | $11.00\%$
Table 1: Energy consumption per subsystem. HVAC: heating, ventilation and air
conditioning; HE: home electronics.
Although the energy consumption of the house could be improved, the installed
systems let the SMLsystem house be a near-zero energy building, producing
almost all the energy at the time the inhabitants need it. This performance
won the second place at the energy balance contest of the Solar Decathlon
competition. The classification of the Energy Balance contest can be found
here: http://monitoring.sdeurope.org/index.php?action=scoring&scoring=S4 .
A sensor and control framework shown in Figure 1 has been used in the
SMLsystem. It is operated by a Master Control Server (MCS) and the European
home automation standard protocol known as Konnex (KNX) (neither KNX nor
Konnex are acronyms: http://ask.aboutknx.com/questions/430/abbreviation-knx)
has been chosen for monitoring and sensing. KNX modules are grouped by
functionality: analog or binary inputs/outputs, gateways between transmission
media, weather stations, CO2 detectors, etc. The whole system provides $88$
sensor values and $49$ actuators. In the proposed system, the immediate
execution actions had been programmed to operate without the involvement of
the MCS, such as controlling ventilation, the HVAC system and the DHW system.
Beyond this basic level, the MCS can read the status of sensors and actuators
at any time and can perform actions on them via an Ethernet gateway.
Figure 1: SMLsystem sensors and actuators map.
A monitoring and control software was developed following a three-layered
scheme. In the first layer, data is acquired from the KNX bus using a KNX-IP
(Internet Protocol) bridge device. The Open Home Automation Bus (openHAB)
Kreuzer and Eichstädt-Engelen (2011) software performs the communication
between KNX and our software. In the second layer, it is possible to find a
data persistence module that has been developed to collect the values offered
by openHAB with a sampling period of 60 s. Finally, the third layer is
composed of different software applications that are able to intercommunicate:
a mobile application has been developed to let the user watch and control the
current state of domotic devices; and different intelligence modules are being
developed also, for instance, the ANN-based indoor temperature forecasting
module.
The energy power generation systems described previously are monitored by a
software controller. It includes multiple measurement sensors, including the
voltage and current measurements of photovoltaic panels and batteries.
Furthermore, the current, voltage and power of the grid is available. The
system power consumption of the house has sensors for measuring power energy
values for each group element. The climate system has power consumption
sensors for the whole system, and specifically for the heat pump. The HVAC
system is composed of several actuators and sensors used for operation. Among
them are the inlet and outlet temperatures of the heat rejector and the inlet
and outlet temperatures of the HVAC water in the SMLsystem. In addition, there
are fourteen switches for internal function valves, for the fan coil system,
for the heat pump and the heat rejector. The DHW system uses a valve and a
pump to control water temperature. Some appliances have temperature sensors
which are also monitored. The lighting system has sixteen binary actuators
that can be operated manually by using the wall-mounted switches or by the
MCS. The SMLsystem has indoor sensors for temperature, humidity and CO2.
Outdoor sensors are also available for lighting measurements, wind speed,
rain, irradiance and temperature.
## 3 Time Series Forecasting
Forecasting techniques are useful in terms of energy efficiency, because they
help to develop predictive control systems. This section introduces formal
aspects and forecasting modeling done for this work. Time series are data
series with trend and pattern repetition through time. They can be formalized
as a sequence of scalars from a variable $x$, obtained as the output of the
observed process:
$\bar{s}(x)=s_{0}(x),s_{1}(x),\ldots,s_{i-1}(x),s_{i}(x),s_{i+1}(x)\,$ (1)
a fragment beginning at position $i$ and ending at position $j$ will be
denoted by $s_{i}^{j}(x)$.
Time series forecasting could be grouped as _univariate forecasting_ when the
system forecasts variable $x$ using only past values of $x$, and _multivariate
forecasting_ when the system forecasts variable $x$ using past values of $x$
plus additional values of other variables. Multivariate approaches could
perform better than univariate when additional variables cause variations on
the predicted variable $x$, as is shown in the experimental section.
Forecasting models are estimated given different parameters: the number of
past values, the size of the future window, and the position in the future of
the prediction (future horizon). Depending on the size of the future window
and how it is produced Ben Taieb et al. (2012), forecasting approaches are
denoted as: _single-step-ahead forecasting_ if the model forecasts only the
next time step; _multi-step-ahead iterative forecasting_ if the model
forecasts only the next time step, producing longer windows by an iterative
process; and _multi-step-ahead direct forecasting_ Cheng et al. (2006) if the
model forecasts in one step a large future window of size $Z$. Following this
last approach, two different major model types exist:
* •
_Pure direct_ , which uses $Z$ forecasting models, one for each possible
future horizon.
* •
_Multiple input multiple output_ (MIMO), which uses one model to compute the
full $Z$ future window. This approach has several advantages due to the joint
learning of inputs and outputs, which allows the model to learn the stochastic
dependency between predicted values. Discriminative models, as ANNs, profit
greatly from this input/output mapping. Additionally, ANNs are able to learn
non-linear dependencies.
### 3.1 Forecast Model Formalization
A forecast model could be formalized as a function $F$, which receives as
inputs the interest variable ($x_{0}$) with its past values until current time
$t$ and a number $C$ of covariates ($x_{1},x_{2},\ldots,x_{C}$), also with its
past values, until current time $t$ and produces a future window of size $Z$
for the given $x_{0}$ variable:
$\langle\hat{s}_{t+1}(x_{0}),\hat{s}_{t+2}(x_{0}),\ldots,\hat{s}_{t+Z}(x_{0})\rangle=F(\Omega(x_{0}),\Omega(x_{1}),\ldots,\Omega(x_{C}))\,$
(2)
$\Omega(x)=s_{t-I(x)+1}^{t}(x)$ being the $I(x)$ past values of
variable/covariate $x$.
The number of past values $I(x)$ is important to ensure good performance of
the model, however, it is not easy to estimate this number exactly. In this
work, it is proposed to estimate models for several values of $I(x)$ and use
the model that achieves better performance, denoted as BEST. It is known in
the machine learning community that ensemble methods achieve better
generalization Jacobs et al. (1991); Raudys and Zliobaite (2006); Yu et al.
(2008). Several possibilities could be found in the literature, such as vote
combination, linear combination (for which a special case is the uniform or
mean combination), or in a more complicated way, modular neural networks
Happel and Murre (1994). Hence, it is also proposed to combine the outputs of
all estimated models for each different value of $I(x)$, following a linear
combination scheme (the linear combination is also known as ensemble
averaging), which is a simple, but effective method of combination, greatly
extended to the machine learning community. Its major benefit is the reduction
of overfitting problems and therefore, it could achieve better performance
than a unique ANN. The quality of the combination depends on the correlation
of the ANNs, theoretically, as the more decorrelated the models are, the
better the combination is. In this way, different input size $I(x)$ ANNs were
combined, with the expectation that they will be less correlated between
themselves than other kinds of combinations, as modifying hidden layer size or
other hyper-parameters.
A linear combination of forecasts models, given a set
$F_{\theta_{1}},F_{\theta_{2}},\dots,F_{\theta_{M}}$ of $M$ forecast models,
with the same future window size ($Z$), follows this equation:
$\langle\hat{s}_{t+1}(x_{0}),\hat{s}_{t+2}(x_{0}),\ldots,\hat{s}_{t+Z}(x_{0})\rangle=\sum_{i=1}^{M}\alpha_{i}F_{\theta_{i}}(\Omega_{i}(x_{0}),\Omega_{i}(x_{1}),\ldots,\Omega_{i}(x_{C}))\,$
(3)
where $\alpha_{i}$ is the combination weight given to the model $\theta_{i}$;
and $\Omega_{i}(x)$ is its corresponding $\Omega$ function, as described in
Section 3.1. The weights are constrained to sum one,
$\sum_{i=1}^{M}\alpha_{i}=1$. This formulation allows one to combine forecast
models with different input window sizes for each covariate, but all of them
using the same covariate inputs. Each weight $\alpha_{i}$ will be estimated
following two approaches:
* •
Uniform linear combination: $\alpha_{i}=\frac{1}{M}$ for $1\leq i\leq M$.
Models following this approach will be denoted as COMB-EQ.
* •
Exponential linear combination (softmax):
$\alpha_{i}=\frac{exp(L^{-1}(\theta_{i},\mathcal{D}))}{\sum_{i=1}^{M}exp(L^{-1}(\theta_{i},\mathcal{D}))}$
(4)
for $1\leq i\leq M$,
$L^{-1}(\theta_{i},\mathcal{D})=1/L(\theta_{i},\mathcal{D})$ being an inverted
loss-function (error function) value for the model $\theta_{i}$, given the
dataset $\mathcal{D}$. It will be computed using a validation dataset. In this
paper, the loss-function will be the mean absolute error (MAE), defined in
Section 3.2, because it is more robust on outlier errors than other quadratic
error measures. This approach will be denoted as COMB-EXP.
### 3.2 Evaluation Measures
The performance of forecasting methods over one time series could be assessed
by several different evaluation functions, which measure the empirical error
of the model. In this work, for a deep analysis of the results, three
different error functions are used: MAE, root mean square error (RMSE) and
symmetric mean absolute percentage of error (SMAPE). The error is computed
comparing target values for the time series $s_{t+1},s_{t+2},\ldots,s_{t+Z}$,
and its corresponding time series prediction
$\hat{s}_{t+1},\hat{s}_{t+2},\ldots,\hat{s}_{t+Z}$, using the model $\theta$:
$\displaystyle\text{MAE}(\theta,t)$ $\displaystyle=$
$\displaystyle\frac{1}{Z}\sum_{z=1}^{Z}|\hat{s}_{t+z}(x_{0})-s_{t+z}(x_{0})|$
(5) $\displaystyle\vspace{6pt}\text{RMSE}(\theta,t)$ $\displaystyle=$
$\displaystyle\sqrt{\frac{1}{Z}\sum_{z=1}^{Z}(\hat{s}_{t+z}(x_{0})-s_{t+z}(x_{0}))^{2}}$
(6) $\displaystyle\vspace{6pt}\text{SMAPE}(\theta,t)$ $\displaystyle=$
$\displaystyle\frac{1}{Z}\sum_{z=1}^{Z}\frac{|\hat{s}_{t+z}-s_{t+z}|}{(|\hat{s}_{t+z}|+|s_{t+z}|)/2}\times
100\,$ (7)
The results could be measured over all time series in a given dataset
$\mathcal{D}$ as:
$L^{\star}(\theta,\mathcal{D})=\frac{1}{|\mathcal{D}|}\sum_{t=1}^{|\mathcal{D}|}L(\theta,t)\,$
(8)
$|\mathcal{D}|$ being the size of the dataset and
$L=\\{\text{MAE},\text{RMSE},\text{SMAPE}\\}$, the loss-function defining
MAE⋆, RMSE⋆, and SMAPE⋆.
### 3.3 Forecasting Data Description
One aim of this work is to compare different statistical methods to forecast
indoor temperature given previous indoor temperature values. The correlation
between different weather signals and indoor temperature will also be
analyzed.
In our database, time series are measured with a sampling period of $T=1$ min.
However, in order to compute better forecasting models, each time series is
sub-sampled with a period of $T^{\prime}=15$ min, computing the mean of the
last $T^{\prime}$ values (for each hour, this mean is computed at 0 min, 15
min, 30 min and 45 min). The output of this preprocessing is the data series
$s^{\prime}(x)$, where:
$s^{\prime}_{i}(x)=\displaystyle{\frac{\displaystyle{\sum_{j=(i-1)T^{\prime}+1}^{iT^{\prime}}s_{j}(x)}}{T^{\prime}}}\,$
(9)
One time feature and five sensor signals were taken into consideration:
* •
Indoor temperature in degrees Celsius, denoted by variable $x=d$. This is the
interesting forecasted variable.
* •
Hour feature in Universal Time Coordinated (UTC), extracted from the time-
stamp of each pattern, denoted by variable $x=h$. The hour of the day is
important for estimating the Sun’s position.
* •
Sun irradiance in $W/m^{2}$, denoted by variable $x=W$. It is correlated with
temperature, because more irradiance will mean more heat.
* •
Indoor relative humidity percentage, denoted by variable $x=H$. The humidity
modifies the inertia of the temperature.
* •
Indoor air quality in CO2 ppm (parts per million), denoted by variable $x=Q$.
The air quality is related to the number of persons in the house, and a higher
number of persons means an increase in temperature.
* •
Raining Boolean status, denoted by variable $x=R$. The result of sub-sampling
this variable is the proportion of minutes in sub-sampling period
$T^{\prime}$, where raining sensor was activated with `True`.
To evaluate the forecasting models’ performance, three partitions of our
dataset were prepared: a _training partition_ composed of $2017$ time series
over $21$ days—the model parameters are estimated to reduce the error in this
data; a _validation partition_ composed of $672$ time series over seven
days—this is needed to avoid over-fitting during training, and also to compare
and study the models between themselves; training and validation were
performed in March 2011; a _test partition_ composed of $672$ time series over
seven days in June 2011. At the end, the forecasting error in this partition
will be provided, evaluating the generalization ability of this methodology.
The validation partition is sequential with the training partition. The test
partition is one week ahead of the last validation point.
## 4 Forecasting Methods
### 4.1 Standard Statistical Methods
Exponential smoothing and auto-regressive integrated moving average models
(ARIMA) are the two most widely-used methods for time series forecasting.
These methods provide complementary approaches to the time series forecasting
problems. Therefore, exponential smoothing models are based on a description
of trend and seasonality in the data, while ARIMA models aim to describe its
autocorrelations. Their results have been considered as a reference to compare
to the ANN results.
On the one hand, exponential smoothing methods are applied for forecasting.
These methods were originally classified by Pegels (1969) according to their
taxonomy. This was later extended by Gardner (1985), modified by Hyndman et
al. (2002) and extended by Taylor (2003), giving a total of fifteen methods.
These methods could have different behavior depending on their error component
[_A_ (additive) and M (multiplicative)], trend component [_N_ (none), A
(additive), Ad (additive damped), M (multiplicative) and Md (multiplicative
damped)] and seasonal component [_N_ (none), A (additive) and M
(multiplicative)]. To select the best-fitting models within this framework,
each possible model was estimated for the training partition, and the two best
models were selected. To carry out this selection, Akaike’s Information
Criterion (AIC) was used as suggested by some works in the literature Billah
et al. (2006); Snyder and Ord (2009). The selected models were: the first
model with multiplicative error, multiplicative damped trend and without the
seasonal component (MMdN model), and the second model with additive error,
additive damped trend and without the seasonal component (AAdN model). The
MMdN model was chosen for the validation partition in order to minimize the
MSE.
On the other hand, ARIMA models were estimated. The widely known ARIMA
approach was first introduced by Box and Jenkins Box and Jenkins (1976) and
provides a comprehensive set of tools for univariate time series modeling and
forecasting. These models were estimated for our data with and without
covariates. The last value of variable hour ($x=h$), codified as a
factor—using 24 categories (0 to 23), —and the hour as a continuous variable
were used as covariates.
Either linear and quadratic form of this quantity were used, but linear
performs worst. Therefore, three model groups are used: ARIMA without
covariates (ARIMA), with covariate $x=h$ as a factor (ARIMAF) and with
covariate $x=h$ as a quadratic form (ARIMAQ). The best models for each group
were estimated for the training partition, and in all cases, the non-seasonal
ARIMA(2,1,0) model was selected for the ARIMA part of each model using AIC.
The best results, in terms of MSE, were obtained in models with covariate time
as a factor and covariate time as a quadratic form.
The forecast library in the statistical package R R Development Core Team
(2005) was used for these analyses.
### 4.2 ANNs
Estimation of ANN forecast models needs data preprocessing and normalization
of input/output values in order to ensure better performance results.
#### 4.2.1 Preprocessing of Time Series for ANNs
The indoor temperature variable ($x=d$) is the interesting forecasted
variable. In order to increase model generalization, this variable is
differentiated, and a new $\bar{s}^{\prime\prime}(x=d)$ signal sequence is
obtained following this equation:
$s^{\prime\prime}_{i}(x=d)=s^{\prime}_{i}(x)-s^{\prime}_{i-1}(x)\,$ (10)
The differentiation of indoor temperature shows that is important to achieve
good generalization results, and it is based on previous work where
undifferentiated data has been used Zamora-Martínez et al. (2012).
The time series corresponding to sun irradiance ($x=W$), indoor relative
humidity ($x=H$), air quality ($x=Q$) and rain ($x=R$) are normalized,
subtracting the mean and dividing by the standard deviation, computing new
signal sequences, $\bar{s}^{\prime\prime}(x\in\\{W,H,Q,R\\})$:
$s^{\prime\prime}_{i}(x\in\\{W,H,Q,R\\})=\displaystyle{\frac{s^{\prime}_{i}(x)-\mathbb{E}[\bar{s}^{\prime}(x)]}{\sigma(\bar{s}^{\prime}(x))}}\,$
(11)
where $\mathbb{E}[\bar{s}^{\prime}(x)]$ is the mean value of the sequence;
$\bar{s}^{\prime}(x)$ and $\sigma(\bar{s}^{\prime}(x))$ is the standard
deviation. These two parameters may be computed over the training dataset. For
the hour component ($x=h$), a different approach is followed. It is
represented as a locally-encoded category, which consists of using a vector
with $24$ components, where $23$ components are set to 0, and the component
that indicates the hour value is set to 1. This kind of encoding avoids the
big jump between 23 and 0 at midnight, but forces the model to learn the
relationship between adjacent hours. Other approaches for hour encoding could
be done in future work.
#### 4.2.2 ANN Description
ANNs has an impressive ability to learn complex mapping functions, as they are
universal function approximators Bishop (1995) and are widely used in
forecasting Zhang et al. (1998); Ruano et al. (2006); Yu et al. (2008);
Escrivá-Escrivá et al. (2011).
ANNs are formed by one input layer, an output layer, and a few numbers of
hidden layers. Figure 2 is a schematic representation of an ANN with two
hidden layers for time series forecasting. The inputs of the ANN are past
values of covariates, and the output layer is formed by the $Z$ future window
predicted values, following the MIMO approach described in Section 3, which
has obtained better accuracy in previous experimentation Zamora-Martínez et
al. (2012).
Figure 2: Artificial neural network (ANN) topology for time series
forecasting.
The well-known error-backpropagation (BP) algorithm Rumelhart et al. (1988)
has been used in its on-line version to estimate the ANN weights, adding a
momentum term and an L2 regularization term (weight decay). Despite that
theoretically algorithms more advanced than BP exists nowadays, BP is easier
to implement at the empirical level, and a correct adjustment of momentum and
weight decay helps to avoid bad local minima. The BP minimizes the mean square
error (MSE) function with the addition of the regularization term weight
decay, denoted by $\epsilon$, useful for avoiding over-fitting and improving
generalization:
$E=\frac{1}{2Z}\sum_{i=1}^{Z}\left(\hat{s}_{t+i}(x_{0})-s_{t+i}(x_{0})\right)^{2}+\frac{\epsilon}{2}\sum_{w_{i}\in\mathbf{\theta_{missing}}}\displaystyle{w_{i}^{2}}\,$
(12)
where $\mathbf{\theta_{missing}}$ is a set of all weights of the ANN (without
the bias); and $w_{i}$ is the value of the $i$-th weight.
## 5 Experimental Results
Using the data acquired during the normal functioning of the house,
experiments were performed to obtain the best forecasting model for indoor
temperature. First, an exhaustive search of model hyper-parameters was done
for each covariate combination. Second, different models were trained for
different values of past size for indoor temperature $(x=d)$, and a comparison
among different covariate combinations and ANN vs. standard statistical
methods has been performed. A comparison of a combination of forecasting
models has also been performed. In all cases, the future window size $Z$ was
set to $12$, corresponding to a three-hour forecast.
A grid search exploration was done to set the best hyper-parameters of the
system and ANN topology, fixing covariates $x\in\\{d,W,H,Q,R\\}$ to a past
size, $I(x)=5$ and $I(x=h)=1$, searching combinations of:
* •
different covariates of the model input;
* •
different values for ANN hidden layer sizes;
* •
learning rate, momentum term and weight decay values.
Table 2 shows the best model parameters found by this grid search. For
illustrative purposes, Figures 3 and 4 show box-and-whisker plots of the
hyper-parameter grid search performed to optimize the ANN model, $d+h$. They
show big differences between one- and two-hidden layer ANNs, two-layered ANNs
being more difficult to train for this particular model. The learning rate
shows a big impact in performance, while momentum and weight decay seems to be
less important. This grid search was repeated for all the tested covariate
combinations, and the hyper-parameters that optimize MAE⋆ were selected in the
rest of the paper.
Covariates | $\eta$ | $\mu$ | $\epsilon$ | Hidden layers
---|---|---|---|---
$d$ | $0.005\phantom{0}$ | $0.001$ | $1\times 10^{-6}$ | $8$ tanh–$8$ tanh
$d+W$ | $0.001\phantom{0}$ | $0.005$ | $1\times 10^{-6}$ | $24$ tanh–$8$ tanh
$d+h$ | $0.005\phantom{0}$ | $0.005$ | $1\times 10^{-6}$ | $8$ tanh
$d+h+W$ | $0.005\phantom{0}$ | $0.005$ | $1\times 10^{-5}$ | $24$ tanh–$16$ tanh
$d+h+H$ | $0.005\phantom{0}$ | $0.005$ | $1\times 10^{-5}$ | $16$ tanh
$d+h+R$ | $0.005\phantom{0}$ | $0.005$ | $1\times 10^{-6}$ | $16$ logistic–$8$ logistic
$d+h+Q$ | $0.0005$ | $0.005$ | $1\times 10^{-4}$ | $24$ logistic
$d+h+W+H$ | $0.005\phantom{0}$ | $0.005$ | $1\times 10^{-5}$ | $16$ tanh
$d+h+W+R$ | $0.005\phantom{0}$ | $0.005$ | $1\times 10^{-6}$ | $16$ logistic–$8$ logistic
$d+h+W+Q$ | $0.005\phantom{0}$ | $0.005$ | $1\times 10^{-4}$ | $8$ tanh–$8$ tanh
$d+h+W+Q+R$ | $0.005\phantom{0}$ | $0.005$ | $1\times 10^{-4}$ | $24$ tanh–$8$ tanh
Table 2: Training parameters depending on the input covariates combination
($\eta$ is the learning rate, $\mu$ is the momentum term, and $\epsilon$ is
weight decay). Figure 3: Mean absolute error (MAE)⋆ box-and-whisker plots for
ANNs with one hidden layer and the hyper-parameters of the grid search
performed to optimize the ANN model, $d+h$. The x-axis of the learning rate,
momentum and weight decay are log-scaled. Figure 4: MAE⋆ box-and-whisker plots
for ANNs with two hidden layers and the hyper-parameters of the grid search
performed to optimize the ANN model, $d+h$. The x-axis of the learning rate,
momentum and weight decay are log-scaled.
### 5.1 Covariate Analysis and Comparison between Different Forecasting
Strategies
For each covariate combination, and using the best model parameters obtained
previously, different model comparison has been performed. Note that the input
past size of covariates is set to $I(x\in\\{W,H,Q,R\\})$= 5 time steps, that
is, $60$ min, and to $I(x=h)=1$. For forecasted variable $x=d$, models with
sizes $I(x=d)\in\\{1,3,5,7,9,11,13,15,17,19,21\\}$ were trained.
A comparison between BEST, COMB-EQ and COMB-EXP approaches was performed and
shown in Table 3. Figure 5 plots the same results for a better confidence
interval comparison. Table 4 shows COMB-EQ weights used in experimentation,
obtained following Equation 4 and using MAE⋆ as the loss-function. From all
these results, the superiority of ANNs vs. standard statistical methods is
clear, with clear statistical significance and with a confidence greater than
$99\%$. Different covariate combinations for ANN models show that the indoor
temperature correlates well with the hour ($d+h$) and sun irradiance ($d+W$),
and the combination of these two covariates ($d+h+W$) improves the model in a
significant way ($99\%$ confidence) with input $d+W$. The addition of more
covariates is slightly better in two cases ($d+h+W+R$ and $d+h+W+Q$), but the
differences are not important. With only the hour and sun irradiance, the ANN
model has enough information to perform good forecasting. Regarding the
combination of models, in some cases, the COMB-EXP approach obtains
consistently better results than COMB-EQ and BEST, but the differences are not
important.
A deeper analysis could be done if comparing the SMAPE values for each
possible future horizon, as Figure 6 shows. A clear trend exists: error
increases with the enlargement of the future horizon. Furthermore, an
enlargement of the confidence interval is observed with the enlargement of the
future horizon. In all cases, ANN models outperform statistical methods. For
shorter horizons (less than or equal to $90$ min), the differences between all
ANN models are insignificant. For longer horizons (greater than $90$ min), a
combination of covariates $d+h+W$ achieve a significant result (for a
confidence of $99\%$) compared with the $d+W$ combination. As was shown in
these results, the addition of covariates is useful when the future horizon
increases, probably because the impact of covariates into indoor temperature
becomes stronger over time.
Finally, to compare the generalization abilities of the proposed best models,
the error measures for the test partition are shown in Table 5 and Figure 7.
All error measures show better performance in the test partition, even when
this partition is two weeks ahead of training and contains hotter days than
the training and validation partitions. The reason for this better performance
might be that the test series has increasing/decreasing temperature cycles
that are more similar to the training partition than the cycles in the
validation partition. The differences between models are similar, and the most
significant combination of covariates is time hour and sun irradiance
($d+h+W$) following the COMB-EXP strategy, achieving a SMAPE${}^{\star}\approx
0.45\%$, MAE${}^{\star}\approx 0.11$, and RMSE${}^{\star}\approx 0.13$.
Model | SMAPE${}^{\star}(\%){[lower,upper]}$ | MAE${}^{\star}[lower,upper]$ | RMSE${}^{\star}[lower,upper]$
---|---|---|---
Standard statistical models
ARIMA-$d$ | $1.5856$ | $[1.4528,$ | $1.7183]$ | $0.3099$ | $[0.2851,$ | $0.3348]$ | $0.3715$ | $[0.3413,$ | $0.4016]$
ARIMAQ-$d+h^{2}$ | $1.5932$ | $[1.4607,$ | $1.7257]$ | $0.3113$ | $[0.2865,$ | $0.3362]$ | $0.3729$ | $[0.3428,$ | $0.4029]$
ARIMAF-$d+h$ | $1.5888$ | $[1.4558,$ | $1.7219]$ | $0.3105$ | $[0.2857,$ | $0.3352]$ | $0.3721$ | $[0.3420,$ | $0.4022]$
ETS-$d$ | $1.5277$ | $[1.3946,$ | $1.6607]$ | $0.3004$ | $[0.2753,$ | $0.3255]$ | $0.3648$ | $[0.3340,$ | $0.3957]$
ANN models
BEST-$d$ | $0.8687$ | $[0.7856,$ | $0.9517]$ | $0.1682$ | $[0.1524,$ | $0.1840]$ | $0.2109$ | $[0.1911,$ | $0.2306]$
CEQ-$d$ | $0.9315$ | $[0.8545,$ | $1.0085]$ | $0.1802$ | $[0.1661,$ | $0.1944]$ | $0.2248$ | $[0.2072,$ | $0.2423]$
CEXP-$d$ | $0.8695$ | $[0.7938,$ | $0.9452]$ | $0.1680$ | $[0.1541,$ | $0.1818]$ | $0.2109$ | $[0.1937,$ | $0.2280]$
BEST-$d+W$ | $0.7296$ | $[0.6311,$ | $0.8281]$ | $0.1418$ | $[0.1228,$ | $0.1608]$ | $0.1777$ | $[0.1544,$ | $0.2010]$
CEQ-$d+W$ | $0.7792$ | $[0.6959,$ | $0.8625]$ | $0.1510$ | $[0.1353,$ | $0.1667]$ | $0.1888$ | $[0.1695,$ | $0.2082]$
CEXP-$d+W$ | $0.7387$ | $[0.6576,$ | $0.8199]$ | $0.1430$ | $[0.1277,$ | $0.1582]$ | $0.1788$ | $[0.1601,$ | $0.1975]$
BEST-$d+h$ | $0.6593$ | $[0.5889,$ | $0.7298]$ | $0.1275$ | $[0.1143,$ | $0.1406]$ | $0.1549$ | $[0.1389,$ | $0.1708]$
CEQ-$d+h$ | $0.6787$ | $[0.6055,$ | $0.7519]$ | $0.1312$ | $[0.1175,$ | $0.1449]$ | $0.1590$ | $[0.1425,$ | $0.1754]$
CEXP-$d+h$ | $0.6768$ | $[0.6037,$ | $0.7498]$ | $0.1308$ | $[0.1172,$ | $0.1445]$ | $0.1586$ | $[0.1422,$ | $0.1750]$
BEST-$d+h+W$ | $0.5737$ | $[0.5058,$ | $0.6416]$ | $0.1121$ | $[0.0994,$ | $0.1248]$ | $0.1379$ | $[0.1222,$ | $0.1536]$
CEQ-$d+h+W$ | $0.5625$ | $[0.4944,$ | $0.6306]$ | $0.1094$ | $[0.0966,$ | $0.1222]$ | $0.1348$ | $[0.1189,$ | $0.1506]$
CEXP-$d+h+W$ | $0.5608$ | $[0.4927,$ | $0.6289]$ | $0.1091$ | $[0.0963,$ | $0.1218]$ | $0.1344$ | $[0.1187,$ | $0.1501]$
BEST-$d+h+H$ | $0.6006$ | $[0.5369,$ | $0.6642]$ | $0.1169$ | $[0.1050,$ | $0.1288]$ | $0.1429$ | $[0.1285,$ | $0.1573]$
CEQ-$d+h+H$ | $0.5897$ | $[0.5240,$ | $0.6553]$ | $0.1142$ | $[0.1019,$ | $0.1264]$ | $0.1399$ | $[0.1250,$ | $0.1548]$
CEXP-$d+h+H$ | $0.5864$ | $[0.5207,$ | $0.6521]$ | $0.1137$ | $[0.1014,$ | $0.1259]$ | $0.1393$ | $[0.1244,$ | $0.1543]$
BEST-$d+h+R$ | $0.6042$ | $[0.5292,$ | $0.6792]$ | $0.1170$ | $[0.1031,$ | $0.1309]$ | $0.1424$ | $[0.1255,$ | $0.1593]$
CEQ-$d+h+R$ | $0.5947$ | $[0.5214,$ | $0.6680]$ | $0.1149$ | $[0.1014,$ | $0.1284]$ | $0.1410$ | $[0.1245,$ | $0.1575]$
CEXP-$d+h+R$ | $0.5933$ | $[0.5196,$ | $0.6670]$ | $0.1146$ | $[0.1009,$ | $0.1282]$ | $0.1407$ | $[0.1241,$ | $0.1574]$
BEST-$d+h+Q$ | $0.6189$ | $[0.5526,$ | $0.6852]$ | $0.1200$ | $[0.1075,$ | $0.1325]$ | $0.1463$ | $[0.1311,$ | $0.1614]$
CEQ-$d+h+Q$ | $0.6219$ | $[0.5539,$ | $0.6899]$ | $0.1208$ | $[0.1080,$ | $0.1336]$ | $0.1479$ | $[0.1324,$ | $0.1633]$
CEXP-$d+h+Q$ | $0.6196$ | $[0.5518,$ | $0.6873]$ | $0.1203$ | $[0.1076,$ | $0.1331]$ | $0.1473$ | $[0.1319,$ | $0.1627]$
BEST-$d+h+W+H$ | $0.5977$ | $[0.5309,$ | $0.6646]$ | $0.1163$ | $[0.1037,$ | $0.1289]$ | $0.1434$ | $[0.1280,$ | $0.1588]$
CEQ-$d+h+W+H$ | $0.5943$ | $[0.5304,$ | $0.6583]$ | $0.1155$ | $[0.1034,$ | $0.1275]$ | $0.1424$ | $[0.1276,$ | $0.1571]$
CEXP-$d+h+W+H$ | $0.5899$ | $[0.5257,$ | $0.6540]$ | $0.1146$ | $[0.1025,$ | $0.1267]$ | $0.1413$ | $[0.1265,$ | $0.1561]$
BEST-$d+h+W+R$ | $0.5600$ | $[0.4935,$ | $0.6266]$ | $0.1090$ | $[0.0966,$ | $0.1214]$ | $0.1335$ | $[0.1183,$ | $0.1486]$
CEQ-$d+h+W+R$ | $0.5568$ | $[0.4895,$ | $0.6240]$ | $0.1080$ | $[0.0955,$ | $0.1205]$ | $0.1328$ | $[0.1174,$ | $0.1482]$
CEXP-$d+h+W+R$ | $0.5541$ | $[0.4872,$ | $0.6210]$ | $\mathbf{0.1076}$ | $[0.0951,$ | $0.1200]$ | $\mathbf{0.1323}$ | $[0.1169,$ | $0.1476]$
BEST-$d+h+W+Q$ | $0.5732$ | $[0.5111,$ | $0.6353]$ | $0.1118$ | $[0.1000,$ | $0.1236]$ | $0.1376$ | $[0.1231,$ | $0.1521]$
CEQ-$d+h+W+Q$ | $0.5537$ | $[0.4921,$ | $0.6153]$ | $0.1079$ | $[0.0962,$ | $0.1196]$ | $0.1328$ | $[0.1184,$ | $0.1472]$
CEXP-$d+h+W+Q$ | $\mathbf{0.5532}$ | $[0.4916,$ | $0.6148]$ | $0.1079$ | $[0.0962,$ | $0.1196]$ | $0.1328$ | $[0.1184,$ | $0.1472]$
BEST-$d+h+W+Q+R$ | $0.5704$ | $[0.5040,$ | $0.6369]$ | $0.1110$ | $[0.0984,$ | $0.1235]$ | $0.1363$ | $[0.1210,$ | $0.1517]$
CEQ-$d+h+W+Q+R$ | $0.5615$ | $[0.4945,$ | $0.6285]$ | $0.1088$ | $[0.0964,$ | $0.1212]$ | $0.1340$ | $[0.1187,$ | $0.1492]$
CEXP-$d+h+W+Q+R$ | $0.5606$ | $[0.4937,$ | $0.6275]$ | $0.1087$ | $[0.0963,$ | $0.1211]$ | $0.1337$ | $[0.1185,$ | $0.1490]$
Table 3: Symmetric mean absolute percentage of error (SMAPE)⋆, MAE⋆ and root
mean square error (RMSE)⋆ results on the validation partition comparing
different models, input features and combination schemes with the $99\%$
confidence interval. BEST refers to the best past size ANN, CEQ refers to
COMB-EQ ANNs, and CEXP refers to COMB-EXP ANNs. Bolded face numbers are the
best results, and the gray marked row is the most significant combination of
covariates. ARIMA: auto-regressive integrated moving average models; ARIMAQ:
ARIMA with covariate $x=h$ as a quadratic form (ARIMAQ); ARIMAF: ARIMA with
covariate $x=h$ as a factor.
Figure 5: SMAPE⋆ error plot with $99\%$ confidence interval for models of Table 3 on the validation partition. | COMB-EXP combination weights for every $d$ variable input size (min)
---|---
Input covariates | $1(15)$ | $3(45)$ | $5(75)$ | $7(105)$ | $9(135)$ | $11(165)$ | $13(195)$ | $15(225)$ | $17(255)$ | $19(285)$ | $21(315)$
$d$ | $0.002$ | $0.044$ | $0.098$ | $\mathbf{0.142}$ | $0.092$ | $0.095$ | $0.082$ | $0.103$ | $0.100$ | $0.106$ | $0.135$
$d+W$ | $0.026$ | $0.020$ | $\mathbf{0.185}$ | $0.046$ | $0.069$ | $0.075$ | $0.104$ | $0.103$ | $0.124$ | $0.117$ | $0.131$
$d+h$ | $\mathbf{0.123}$ | $0.066$ | $0.099$ | $0.085$ | $0.092$ | $0.091$ | $0.091$ | $0.097$ | $0.084$ | $0.084$ | $0.088$
$d+h+W$ | $0.040$ | $0.112$ | $\mathbf{0.137}$ | $0.072$ | $0.078$ | $0.100$ | $0.107$ | $0.120$ | $0.083$ | $0.075$ | $0.075$
$d+h+H$ | $0.049$ | $0.058$ | $0.121$ | $\mathbf{0.127}$ | $0.095$ | $0.105$ | $0.114$ | $0.068$ | $0.100$ | $0.074$ | $0.090$
$d+h+R$ | $0.049$ | $0.052$ | $\mathbf{0.126}$ | $0.099$ | $0.078$ | $0.113$ | $0.104$ | $0.114$ | $0.102$ | $0.089$ | $0.076$
$d+h+Q$ | $0.084$ | $0.089$ | $0.105$ | $\mathbf{0.123}$ | $0.115$ | $0.103$ | $0.086$ | $0.077$ | $0.069$ | $0.073$ | $0.076$
$d+h+W+H$ | $0.062$ | $0.085$ | $0.071$ | $0.091$ | $0.123$ | $\mathbf{0.134}$ | $0.094$ | $0.082$ | $0.067$ | $0.121$ | $0.069$
$d+h+W+R$ | $0.048$ | $0.089$ | $\mathbf{0.142}$ | $0.078$ | $0.062$ | $0.116$ | $0.121$ | $0.092$ | $0.109$ | $0.087$ | $0.056$
$d+h+W+Q$ | $0.064$ | $0.101$ | $0.112$ | $0.097$ | $0.068$ | $0.088$ | $\mathbf{0.115}$ | $0.090$ | $0.085$ | $0.079$ | $0.101$
$d+h+W+Q+R$ | $0.042$ | $0.090$ | $\mathbf{0.136}$ | $0.098$ | $0.111$ | $0.089$ | $0.101$ | $0.072$ | $0.090$ | $0.085$ | $0.087$
Table 4: Combination weights of every input size of $d$ for the COMB-EXP
models given tested covariates combinations. All co-variables have an input
size of $5$ ($75$ min). Bold numbers are the best input sizes.
Figure 6: SMAPE⋆ error plot with $99\%$ confidence interval of each of the $Z=12$ future horizon predicted values (from 15 min forecast to 180 min forecast.) Model | SMAPE${}^{\star}(\%)[lower,upper]$ | MAE${}^{\star}[lower,upper]$ | RMSE${}^{\star}[lower,upper]$
---|---|---|---
ETS-d | $1.3669$ | $[1.2649,$ | $1.4688]$ | $0.3254$ | $[0.3023,$ | $0.3485]$ | $0.3930$ | $[0.3643,$ | $0.4218]$
BEST-$d$ | $0.6736$ | $[0.6128,$ | $0.7343]$ | $0.1604$ | $[0.1460,$ | $0.1748]$ | $0.2022$ | $[0.1844,$ | $0.2199]$
CEQ-$d$ | $0.7462$ | $[0.6907,$ | $0.8016]$ | $0.1767$ | $[0.1638,$ | $0.1895]$ | $0.2203$ | $[0.2046,$ | $0.2360]$
CEXP-$d$ | $0.6630$ | $[0.6101,$ | $0.7159]$ | $0.1572$ | $[0.1450,$ | $0.1694]$ | $0.1976$ | $[0.1824,$ | $0.2127]$
BEST-$d+h+W$ | $0.4802$ | $[0.4339,$ | $0.5266]$ | $0.1143$ | $[0.1035,$ | $0.1252]$ | $0.1382$ | $[0.1252,$ | $0.1512]$
CEQ-$d+h+W$ | $0.4569$ | $[0.4127,$ | $0.5012]$ | $0.1090$ | $[0.0985,$ | $0.1195]$ | $0.1318$ | $[0.1193,$ | $0.1443]$
CEXP-$d+h+W$ | $0.4546$ | $[0.4111,$ | $0.4982]$ | $0.1085$ | $[0.0981,$ | $0.1189]$ | $0.1312$ | $[0.1188,$ | $0.1437]$
BEST-$d+h+W+R$ | $0.4350$ | $[0.3925,$ | $0.4774]$ | $0.1034$ | $[0.0935,$ | $0.1132]$ | $0.1255$ | $[0.1136,$ | $0.1374]$
CEQ-$d+h+W+R$ | $0.4271$ | $[0.3854,$ | $0.4688]$ | $0.1013$ | $[0.0916,$ | $0.1111]$ | $0.1225$ | $[0.1109,$ | $0.1341]$
CEXP-$d+h+W+R$ | $0.4253$ | $[0.3837,$ | $0.4670]$ | $0.1010$ | $[0.0913,$ | $0.1108]$ | $0.1223$ | $[0.1107,$ | $0.1339]$
BEST-$d+h+W+Q$ | $0.4727$ | $[0.4258,$ | $0.5196]$ | $0.1127$ | $[0.1015,$ | $0.1238]$ | $0.1353$ | $[0.1223,$ | $0.1483]$
CEQ-$d+h+W+Q$ | $0.4565$ | $[0.4136,$ | $0.4994]$ | $0.1092$ | $[0.0988,$ | $0.1195]$ | $0.1314$ | $[0.1192,$ | $0.1436]$
CEXP-$d+h+W+Q$ | $0.4565$ | $[0.4134,$ | $0.4995]$ | $0.1091$ | $[0.0988,$ | $0.1195]$ | $0.1313$ | $[0.1192,$ | $0.1435]$
BEST-$d+h+W+Q+R$ | $0.4434$ | $[0.3997,$ | $0.4872]$ | $0.1051$ | $[0.0949,$ | $0.1153]$ | $0.1268$ | $[0.1147,$ | $0.1388]$
CEQ-$d+h+W+Q+R$ | $0.4195$ | $[0.3792,$ | $0.4597]$ | $0.0996$ | $[0.0903,$ | $0.1090]$ | $0.1201$ | $[0.1090,$ | $0.1312]$
CEXP-$d+h+W+Q+R$ | $\mathbf{0.4192}$ | $[0.3790,$ | $0.4595]$ | $\mathbf{0.0994}$ | $[0.0902,$ | $0.1087]$ | $\mathbf{0.1200}$ | $[0.1089,$ | $0.1311]$
Table 5: SMAPE⋆, MAE⋆ and RMSE⋆ results on test partition comparing the best
models with the $99\%$ confidence interval. Bolded face numbers are the best
results, and the gray marked row is the most significant combination of
covariates.
Figure 7: SMAPE⋆ error plot with the $99\%$ confidence interval for the models
of Table 5 in the test partition.
In order to perform a better evaluation, the conclusions above are compared
with mutual information (MI), shown in Table 6. Probability densities have
been estimated with histograms, making the assumption of independence between
time points, which is not true for time series Papana and Kugiumtzis (2009),
but is enough for our contrasting purpose. The behavior of the ANNs is similar
to the MI study. Sun irradiance ($W$) covariates show high MI with indoor
temperature ($d$), which is consistent with our results. Humidity ($H$) and
air quality ($Q$) MI with indoor temperature ($d$) is higher than sun
irradiance, which seems contradictory with our expectations. However, if we
compute MI only during the day (removing the night data points), the sun
irradiance shows higher MI with indoor temperature than other covariates.
Regarding the hour covariate, it shows lower MI than expected, probably due to
the cyclical shape of the hour, which breaks abruptly with the jump between 23
and 0, affecting the computation of histograms.
Data | Algorithm | $d$ | $h$ | $W$ | $H$ | $R$ | $Q$
---|---|---|---|---|---|---|---
| MI (for $d$) | $9.24$ | $4.44$ | $6.06$ | $8.95$ | $0.51$ | $7.70$
Validation set | Normalized MI (for $d$) | $2.00$ | $1.48$ | $1.65$ | $1.95$ | $1.06$ | $1.82$
Validation set, | MI (for $d$) | $8.23$ | $3.50$ | $8.11$ | $8.09$ | $0.58$ | $7.41$
removing night data points | Normalized MI (for $d$) | $2.00$ | $1.42$ | $1.98$ | $1.97$ | $1.07$ | $1.89$
Table 6: Mutual Information (MI) and normalized MI between considered
covariates and the indoor temperature, for the validation set.
## 6 Conclusions
An overview of the monitoring and sensing system developed for the SMLsystem
solar powered house has been described. This system was employed during the
participation at the Solar Decathlon Europe 2012 competition. The research in
this paper has been focused on how to predict the indoor temperature of a
house, as this is directly related to HVAC system consumption. HVAC systems
represent $53.89\%$ of the overall power consumption of the SMLsystem house.
Furthermore, performing a preliminary exploration of the SMLsystem competition
data, the energy used to maintain temperature was found to be $30\%$–$38.9\%$
of the energy needed to lower it. Therefore, an accurate forecasting of indoor
temperature could yield an energy-efficient control.
An analysis of time series forecasting methods for prediction of indoor
temperature has been performed. A multivariate approach was followed, showing
encouraging results by using ANN models. Several combinations of covariates,
forecasting model combinations, comparison with standard statistical methods
and a study of covariate MI has been performed. Significant improvements were
found by combining indoor temperature with the hour categorical variable and
sun irradiance, achieving a MAE${}^{\star}\approx 0.11$ degrees Celsius
(SMAPE${}^{\star}\approx 0.45\%$). The addition of more covariates different
from hour and sun irradiance slightly improves the results. The MI study shows
that humidity and air quality share important information with indoor
temperature, but probably, the addition of these covariates does not add
different information from which is indicated by hour and sun irradiance. The
combination of ANN models following the softmax approach (COMB-EXP) produce
consistently better forecasts, but the differences are not important. The data
available for this study was restricted to one month and a week of a Southern
Europe house. It might be interesting to perform experiments using several
months of data in other houses, as weather conditions may vary among seasons
and locations.
As future work, different techniques for the combination of forecasting models
could be performed. A deeper MI study to understand the relationship between
covariates better would also be interesting. The use of second order methods
to train the ANN needs to be studied. In this work, for the ANN models, the
hour covariate is encoded using 24 neurons; other encoding methods will be
studied, for example, using splines, sinusoidal functions or a neuron with
values between 0 and 23.
Following these results, it is intended to design a predictive control based
on the data acquired from ANNs, for example, from this one that is devoted to
calculating the indoor temperature, extrapolating this methodology to other
energy subsystems that can be found in a home.
###### Acknowledgements.
Acknowledgments This work has been supported by Banco Santander and CEU
Cardenal Herrera University through the project Santander-PRCEU-UCH07/12.
Conflicts of Interest The authors declared not conflict of interest.
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|
arxiv-papers
| 2013-10-21T16:07:08 |
2024-09-04T02:49:52.669343
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Francisco Zamora-Martinez, Pablo Romeu, Paloma Botella-Rocamora and\n Juan Pardo",
"submitter": "Francisco Zamora-Martinez",
"url": "https://arxiv.org/abs/1310.5620"
}
|
1310.5723
|
# A family of steady two-phase generalized Forchheimer flows and their linear
stability analysis
Luan T. Hoang, Akif Ibragimov and Thinh T. Kieu† Department of Mathematics and
Statistics, Texas Tech University, Box 41042 Lubbock, TX 79409–1042, U.S.A.
[email protected] [email protected] [email protected] † Corresponding
author
###### Abstract.
We model multi-dimensional two-phase flows of incompressible fluids in porous
media using generalized Forchheimer equations and the capillary pressure.
Firstly, we find a family of steady state solutions whose saturation and
pressure are radially symmetric and velocities are rotation-invariant. Their
properties are investigated based on relations between the capillary pressure,
each phase’s relative permeability and Forchheimer polynomial. Secondly, we
analyze the linear stability of those steady states. The linearized system is
derived and reduced to a parabolic equation for the saturation. This equation
has a special structure depending on the steady states which we exploit to
prove two new forms of the lemma of growth of Landis-type in both bounded and
unbounded domains. Using these lemmas, qualitative properties of the solution
of the linearized equation are studied in details. In bounded domains, we show
that the solution decays exponentially in time. In unbounded domains, in
addition to their stability, the solution decays to zero as the spatial
variables tend to infinity. The Bernstein technique is also used in estimating
the velocities. All results have a clear physical interpretation.
Dedicated to the Memory of Evgenii Mikhailovich Landis (1921–1997)
###### Contents
1. 1 Introduction
2. 2 Special steady states
3. 3 Linearization
4. 4 Case of bounded domain
5. 5 Case of unbounded domain
1. 5.1 Maximum principle for unbounded domain
2. 5.2 Lemma of growth in spatial variables
6. A
## 1\. Introduction
In this paper, we study two-phase flows of incompressible fluids in porous
media with each phase subjected to a Forchheimer equation. Forchheimer
equations are often used by engineers to take into account the deviation from
Darcy’s law in case of high velocity, see e.g. [4, 20]. The standard
Forchheimer equations are two-term law with quadratic nonlinearity, three-term
law with cubic nonlinearity, and power law with a non-integer power less than
two (see again [4, 20]). These models are extended to the generalized
Forchheimer equation of the form
$g(|\mathbf{u}|)\mathbf{u}=-\nabla p,$ (1.1)
where $\mathbf{u}(\mathbf{x},t)$ is the velocity field, $p(\mathbf{x},t)$ is
the pressure, and $g(s)$ is a generalized polynomial of arbitrary order
(integer or non-integer) with positive coefficients. This equation was
intensively analyzed for single-phase flows from mathematical and applied
point of view in [3, 11, 12, 13, 15]. Its study for two-phase flows was later
initiated in [14]. Regarding two-phase flows in porous media, it is always a
challenging subject even for Darcy’s law. Their models involve a complicated
system of nonlinear partial differential equations (PDE) for pressures,
velocities, densities and saturations with many parameters such as porosity,
relative permeability functions and capillary pressure function. Current
analysis of two-phase Darcy flows in literature is mainly focused on the
existence of weak solutions [8, 7, 6] and their regularity [17, 18, 1, 2, 9].
However, questions about the stability and dynamics are not answered. The
nonlinearity of the relative permeabilities and capillary pressure and their
imprecise characteristics near the extreme values make it hard to analyze the
modeling PDE system. The two-phase generalized Forchheimer flows are even more
difficult due to the additional nonlinearity in the momentum equation. For
example, unlike the Darcy flows, there is no Kruzkov-Sukorjanski
transformation [17] to convert the system to a convenient form for the total
velocity. Therefore, new methods are needed for the Forchheimer flows. In
[14], we study the one-dimensional case using a novel approach. We will
develop the techniques in [14] further to investigate the multi-dimensional
case in this article.
We consider $n$-dimensional two-phase flows in porous media with constant
porosity $\phi$ between $0$ and $1$. Here the dimension $n$ is greater or
equal to $2$, even though in practice we only need $n=2,3$. Each position
$\mathbf{x}=(x_{1},x_{2},\ldots,x_{n})\in\mathbb{R}^{n}$ in the medium is
considered to be occupied by two fluids called phase 1 (for example, water)
and phase 2 (for example, oil).
Saturation, density, velocity, and pressure for each $i$th-phase ($i=1,2$) are
$S_{i}\in[0,1]$, $\rho_{i}\geq 0$, $\bf u_{i}\in\mathbb{R}^{n}$ and
$p_{i}\in\mathbb{R}$, respectively. The saturation functions naturally satisfy
$S_{1}+S_{2}=1.$ (1.2)
Each phase’s velocity is assumed to obey the generalized Forchheimer equation:
$g_{i}(|\mathbf{u}_{i}|)\mathbf{u}_{i}=-\tilde{f}_{i}(S_{i})\nabla p_{i},\quad
i=1,2,$ (1.3)
where $\tilde{f}_{i}(S_{i})$ is the relative permeability for the $i$th phase,
and $g_{i}$ is of the form
$g_{i}(s)=a_{0}s^{\alpha_{0}}+a_{1}s^{\alpha_{1}}+\ldots+a_{N}s^{\alpha_{N}},\quad
s\geq 0,$ (1.4)
with $N\geq 0$, $a_{0}>0,$ $a_{1},\ldots a_{N}\geq 0$,
$\alpha_{0}=0<\alpha_{1}<\ldots\alpha_{N}$, all $\alpha_{1},\ldots\alpha_{N}$
are real numbers. The above $N$, $a_{j}$, $\alpha_{j}$ in (1.4) depend on each
$i$. We call $g_{i}(s)$ in (1.4) the Forchheimer polynomial of (1.3).
Conservation of mass commonly holds for each of the phases:
$\partial_{t}(\phi\rho_{i}S_{i})+{\rm div}(\rho_{i}\mathbf{u}_{i})=0,\quad
i=1,2.$ (1.5)
Due to incompressibility of the phases, i.e. $\rho_{i}=const.>0$, Eq. (1.5) is
reduced to
$\phi\partial_{t}S_{i}+{\rm div}\,\mathbf{u}_{i}=0,\quad i=1,2.$ (1.6)
Let $p_{c}$ be the capillary pressure between two phases, more specifically,
$p_{1}-p_{2}=p_{c}.$ (1.7)
Hereafterward, we denote $S=S_{1}$. The relative permeabilities and capillary
pressure are re-denoted as functions of $S$, that is,
$\tilde{f}_{1}(S_{1})=f_{1}(S)$, $\tilde{f}_{2}(S_{2})=f_{2}(S)$ and
$p_{c}=p_{c}(S)$. Then (1.3) and (1.7) become
$g_{i}(|{\mathbf{u}_{i}}|){\mathbf{u}_{i}}=-f_{i}(S)\nabla p_{i},\quad i=1,2,$
(1.8) $p_{1}-p_{2}=p_{c}(S).$ (1.9)
By scaling time, we can mathematically consider, without loss of generality,
$\phi=1$.
By (1.2) and (1.6):
$S_{t}=-{\rm div}\,{\bf u}_{1},\quad S_{t}={\rm div}\,{\bf u}_{2}.$ (1.10)
For $i=1,2$, define the function $\mathbf{G}_{i}({\bf u})=g_{i}(|{\bf u}|){\bf
u}$ for ${\bf u}\in\mathbb{R}^{n}$. Then by (1.8),
$\mathbf{G}_{i}({\bf u}_{i})=-f_{i}(S)\nabla p_{i},\quad\text{or,}\quad\nabla
p_{i}=-\frac{\mathbf{G}_{i}({\bf u}_{i})}{f_{i}(S)}.$ (1.11)
Taking gradient of the equation (1.9) we have
$\nabla p_{1}-\nabla p_{2}=p_{c}^{\prime}(S)\nabla S.$ (1.12)
Substituting (1.11) into (1.12) yields
$\frac{g_{2}(|\mathbf{u}_{2}|)\mathbf{u}_{2}}{f_{2}(S)}-\frac{g_{1}(|\mathbf{u}_{1}|)\mathbf{u}_{1}}{f_{1}(S)}=p_{c}^{\prime}(S)\nabla
S,$
hence
$F_{2}(S)g_{2}(|{\bf
u}_{2}|)\mathbf{u}_{2}-F_{1}(S)g_{1}(|\mathbf{u}_{1}|)\mathbf{u}_{1}=\nabla
S,$
where
$F_{i}(S)=\frac{1}{p_{c}^{\prime}(S)f_{i}(S)},\quad i=1,2.$ (1.13)
In summary we study the following PDE system for $\mathbf{x}\in\mathbb{R}^{n}$
and $t\in\mathbb{R}$:
$\displaystyle 0\leq S=S(\mathbf{x},t)\leq 1,$ (1.14a) $\displaystyle
S_{t}=-{\rm div}\,{\mathbf{u}}_{1},$ (1.14b) $\displaystyle S_{t}={\rm
div}\,{\mathbf{u}}_{2},$ (1.14c) $\displaystyle\nabla
S=F_{2}(S)\mathbf{G}_{2}(\mathbf{u}_{2})-F_{1}(S)\mathbf{G}_{1}(\mathbf{u}_{1}).$
(1.14d)
This paper is devoted to studying system (1.14). We will obtain a family of
non-constant steady states with particular geometric properties. Specifically,
the saturation and pressure are functions of $|\mathbf{x}|$, while each
phase’s velocity is $\mathbf{x}$ multiplied by a radial scalar function. Their
properties, particularly, the behavior as $|\mathbf{x}|\to\infty$, will be
obtained. For the stability study, we linearize system (1.14) at these steady
states. We deduce from this linearized system a parabolic equation for the
saturation. In bounded domains, we establish the lemma of growth in time and
prove the exponential decay of its solutions in sup-norm as time $t\to\infty$.
In unbounded domains, we prove the maximum principle and the stability.
Furthermore, we show that the solutions go to zero as the spatial variables
tend to infinity.
The paper is organized as follows. In section 2 we find the family of non-
constant steady states described above. Various sufficient conditions are
given for their existence in unbounded domains (Theorems 2.2). Their
asymptotic behavior as $|\mathbf{x}|\to\infty$ is studied in details. In
section 3, we linearize the originally system at the obtained steady states.
We derive a parabolic equation for the saturation which will become the focus
of our study. It is then converted to a convenient form for the study of sup-
norm of solutions. Such a conversion is possible thanks to the special
structure of the equation and of the steady states. Preliminary properties of
the coefficient functions of this linearized equation are presented. Section 4
is focused on the study of the linearized equation for saturation in bounded
domains. We prove the asymptotic stability results (Theorems 4.8 and 4.9) by
utilizing a variation of Landis’s lemma of growth in time variable (Lemma
4.3). The Bernstein’s a priori estimate technique is used in proving interior
continuous dependence of the velocities on the initial and boundary data
(Proposition 4.7). In section 5, we study the linearized equation in an
(unbounded) outer domain. The maximum principle (Theorem 5.2) is proved and
used to obtain the stability of the zero solution (Theorems 5.10 and 5.11,
part (ii)). We also prove a lemma of growth in the spatial variables (Lemma
5.5) by constructing particular barriers (super-solutions) using the specific
structure of the linearized equation for saturation (Lemma 5.4). Using this,
we prove a dichotomy theorem on the solution’s behavior (Lemma 5.6), and
ultimately show that the solution, on any finite time interval, decays to zero
as $|\mathbf{x}|\to\infty$. For time tending to infinity, we find an
increasing, continuous function $r(t)>0$ with $r(t)\to\infty$ as $t\to\infty$
such that along any curve $\mathbf{x}(t)$ with $|\mathbf{x}(t)|\geq r(t)$, the
solution goes to zero. (See Theorems 5.10 and 5.11, part (iii).) It is worth
mentioning that the asymptotic stability in sup-norm in section 4 and behavior
of the solution at spatial infinity have their own merits in the qualitative
theory of linear parabolic equations.
## 2\. Special steady states
In this section we find and study steady states which processes some symmetry.
Assume $p_{i}$ and $S$ are radial functions. We can write
$p_{i}({\bf x},t)=p_{i}(r,t),\quad S({\bf x},t)=S(r,t),\quad\text{where
}r=|\mathbf{x}|=\big{(}\sum_{i=1}^{n}x_{i}^{2}\big{)}^{1/2}.$ (2.1)
Denote ${\bf e}_{r}=\mathbf{x}/|\mathbf{x}|$. By (1.8),
$g_{i}(|\mathbf{u}_{i}|){\mathbf{u}_{i}}=-f_{i}(S)\frac{\partial
p_{i}}{\partial r}\cdot\frac{\bf x}{r}=-f_{i}(S)\frac{\partial p_{i}}{\partial
r}{\bf e}_{r}.$ (2.2)
Noting in (2.2) that $f_{i}(S)\frac{\partial p_{i}}{\partial r}$ is radial,
then clearly $|{\bf u}_{i}|$ is also radial and we have
${\bf u}_{i}=u_{ir}{\bf e}_{r},\quad\text{where }u_{ir}={\bf u}_{i}\cdot{\bf
e}_{r}=u_{ir}(r,t).$ (2.3)
Therefore
${\rm div}\,\mathbf{u}_{i}=\frac{1}{r^{n-1}}\frac{\partial}{\partial
r}(r^{n-1}u_{ir})$ (2.4)
and, from (1.14d),
$F_{2}(S)g_{2}(|\mathbf{u}_{2}|)\mathbf{u}_{2}-F_{1}(S)g_{1}(|\mathbf{u}_{1}|){\mathbf{u}_{1}}=\nabla
S=\frac{\partial S}{\partial r}{\bf e}_{r}.$ (2.5)
Taking the scalar product of both sides of (2.5) with ${\bf e}_{r}$ we obtain
$G_{2}(u_{2r})F_{2}(S)-G_{1}(u_{1r})F_{1}(S)=\frac{\partial S}{\partial r},$
(2.6)
where
$G_{i}(u)=g_{i}(|u|)u\quad\text{for }u\in\mathbb{R}.$ (2.7)
We will study $S(r,t)$ and $u_{i}(r,t)\mathbin{\buildrel\rm
def\over{\mathbin{=\kern-2.0pt=}}}u_{ir}$ ($i=1,2$) as functions of
independent variables $(r,t)\in(0,\infty)\times\mathbb{R}$. The system (1.14)
becomes
$\displaystyle 0\leq S\leq 1,$ (2.8a) $\displaystyle\frac{\partial S}{\partial
t}=-r^{1-n}\frac{\partial}{\partial r}(r^{n-1}u_{1}),$ (2.8b)
$\displaystyle\frac{\partial S}{\partial t}=r^{1-n}\frac{\partial}{\partial
r}(r^{n-1}u_{2}),$ (2.8c) $\displaystyle\frac{\partial S}{\partial
r}=G_{2}(u_{2})F_{2}(S)-G_{1}(u_{1})F_{1}(S).$ (2.8d)
We make basic assumptions on the relative permeabilities and capillary
pressure.
Assumption A.
$f_{1},f_{2}\in C([0,1])\cap C^{1}((0,1)),$ (2.9a) $f_{1}(0)=0,\quad
f_{2}(1)=0,$ (2.9b) $f_{1}^{\prime}(S)>0,\quad f_{2}^{\prime}(S)<0\text{ on
}(0,1).$ (2.9c)
Assumption B.
$p_{c}^{\prime}\in C^{1}((0,1)),\quad p^{\prime}_{c}(S)>0\text{ on }(0,1).$
(2.10)
We find steady state solutions $(S,u_{1},u_{2})=(S(r),u_{1}(r),u_{2}(r))$ for
system (2.8) in the domain $[r_{0},\infty)$ for a fixed $r_{0}>0$.
From (2.8b), we have $\frac{d}{dr}(r^{n-1}u_{i})=0$, hence
$u_{i}(r)=c_{i}r^{1-n},\quad\text{where }c_{i}=const.,\quad i=1,2.$ (2.11)
Substituting (2.11) into (1.14d) yields
$S^{\prime}=G_{2}(c_{2}r^{1-n})F_{2}(S)-G_{1}(c_{1}r^{1-n})F_{1}(S)\quad\text{for
}r>r_{0}.$ (2.12)
The rest of this section is devoted to studying the following initial value
problem with constraints:
$S^{\prime}=F(r,S(r))\quad\text{for }r>r_{0},\quad S(r_{0})=s_{0},\quad
0<S(r)<1.$ (2.13)
where $s_{0}$ is always a number in $(0,1)$ and
$F(r,S(r))=G_{2}(c_{2}r^{1-n})F_{2}(S)-G_{1}(c_{1}r^{1-n})F_{1}(S).$
First we state a standard local existence theorem.
###### Theorem 2.1.
There exist a maximal interval of existence $[r_{0},R_{\rm max})$, where
$R_{\rm max}\in(r_{0},\infty]$, and a unique solution $S\in
C^{1}([r_{0},R_{\rm max}))$ of (2.13) on $(r_{0},R_{\rm max})$. Moreover, if
$R_{\rm max}$ is finite then either
$\lim_{r\to R_{\rm max}^{-}}S(r)=0\quad\text{ or}\lim_{r\to R_{\rm
max}^{-}}S(r)=1.$ (2.14)
###### Proof.
Under Assumption B, $F(r,S)$ is continuous and locally Lipschitz for the
second variable for all $r\in(r_{0},\infty)$, $S\in(0,1)$. The existence of
the unique solution $S\in C^{1}([r_{0},R_{\rm max});(0,1))$ on the maximal
interval $[0,R_{\rm max})$ is classical.
Assume $R_{\max}<\infty$. For given
$0<\varepsilon\leq\varepsilon_{0}\mathbin{\buildrel\rm
def\over{\mathbin{=\kern-2.0pt=}}}\min\\{1/4,R_{\rm max}/2\\}$, let
$K=[r_{0},R_{\max}]\times[\varepsilon,1-\varepsilon]$. We claim that there is
$R_{\varepsilon}\geq r_{0}$ such that $(r,S(r))\notin K$ for all
$r\in(R_{\varepsilon},R_{\max})$. Suppose not, then there is the sequence
$r_{i}\to R_{\max}$ as $i\to\infty$ such that $(r_{i},S(r_{i}))\in K$ for all
$i$. Choose $N>0$ such that for all $i\geq N$,
$\\{(r,S):|r-r_{i}|\leq\varepsilon/2\text{ and
}|S-S(r_{i})|\leq\varepsilon/2\\}\subset K^{\prime},$
where
$K^{\prime}=[r_{0},R_{\max}+\varepsilon/2]\times[\varepsilon/2,1-\varepsilon/2]$.
According to the local Existence and Uniqueness theorem (Theorem 3.1 p. 18
in[10]) the solution starting at point $(r_{i},S(r_{i}))$ exists on the
interval $[r_{i},r_{i}+d)$, where
$d=\min\\{\frac{1}{L},\frac{\varepsilon}{2},\frac{\varepsilon}{2M}\\}$ with
$M=\max_{K^{\prime}}|F(r,S)|$ and $L$ being the Lipschitz constant for $F$ in
$K^{\prime}$. Note that $d$ is independent of $i$. Let $i$ be sufficiently
large such that $r_{i}+d>R_{\max}$, then solution $S(r)$ exists beyond
$R_{\max}$ which is a contradiction to maximality of $R_{\max}$. Hence our
claim is true. Now using the continuity of $S(r)$ we have
$\text{either }S(r)>1-\varepsilon,\forall r\in(R_{\varepsilon},R_{\max})\text{
or }S(r)<\varepsilon,\forall r\in(R_{\varepsilon},R_{\max}).$ (2.15)
In particular, for $\varepsilon=\varepsilon_{0}$ we have either (a)
$S(r)>1-\varepsilon_{0},\forall r\in(R_{\varepsilon_{0}},R_{\max})$, or (b)
$S(r)<\varepsilon_{0},\forall r\in(R_{\varepsilon_{0}},R_{\max})$. In case
(a), it is easy to see from (2.15) that for $0<\varepsilon<\varepsilon_{0}$,
$S(r)>1-\varepsilon,\forall r\in(R^{\prime}_{\varepsilon},R_{\max})$ where
$R^{\prime}_{\varepsilon}=\max\\{R_{\varepsilon_{0}},R_{\varepsilon}\\}$.
Thus, $\lim_{r\to R_{\max}^{-}}S(r)=1$. Similarly, for the case (b) we have
$\lim_{r\to R_{\max}^{-}}S(r)=0$. The proof is complete. ∎
Next, we are interested in the case $R_{\rm max}=\infty$. First, we find
sufficient conditions for that. We need to make the following assumptions on
the relative permeabilities and capillary pressure:
$\lim_{S\to 0}p^{\prime}_{c}(S)f_{1}(S)=\lim_{S\to
1}p^{\prime}_{c}(S)f_{2}(S)=+\infty.$ (2.16)
These are our interpretation of experimental data (c.f. [4]), especially of
those obtained in [5]. They cover certain scenarios of two-phase fluids in
reality.
By (1.13) and (2.16), $F_{1}$ and $F_{2}$ can now be extended to functions of
class $C([0,1])\cap C^{1}((0,1))$ and satisfy
$F_{1}(0)=F_{1}(1)=F_{2}(0)=F_{2}(1)=0.$ (2.17)
Therefore the right hand side of (1.14d) is well-defined for all $S\in[0,1]$.
Note that
$\lim_{S\to 0^{+}}\frac{F_{1}(S)}{F_{2}(S)}=\lim_{S\to
1^{-}}\frac{F_{2}(S)}{F_{1}(S)}=\infty.$ (2.18)
The following additional conditions on $F_{1}$ and $F_{2}$ will be referred to
in our considerations:
$\limsup_{S\to 0^{+}}F_{1}^{\prime}(S)<\infty,$ (2.19) $\liminf_{S\to
1^{-}}F^{\prime}_{1}(S)>-\infty.$ (2.20) $\liminf_{S\to
1^{-}}F^{\prime}_{2}(S)>-\infty.$ (2.21) $\limsup_{S\to
0^{+}}F_{2}^{\prime}(S)<\infty,$ (2.22)
###### Theorem 2.2.
Assume (2.16) and $c_{1}^{2}+c_{2}^{2}>0$. Then $R_{\max}$ in Theorem 2.1 is
infinity, that is, the solution $S(r)$ of (2.13) exists on $[r_{0},\infty)$,
in the following cases
Case 1a. $c_{2}\leq 0<c_{1}$ and (2.19). Case 1b. $c_{1}=0>c_{2}$ and (2.22).
Case 2a. $c_{1}\leq 0<c_{2}$ and (2.21). Case 2b. $c_{2}=0>c_{1}$ and (2.20).
Case 3. $c_{1},c_{2}>0$ and (2.19), (2.21).
Case 4. $c_{1},c_{2}<0$.
###### Proof.
Suppose $R_{\max}<\infty$. We consider the following four cases.
Case 1. $c_{2}\leq 0\leq c_{1}$. We provide the proof of Case 1a, while Case
1b can be proved similarly. We have $F(r,S)<0$ for all $r\in[r_{0},R_{\max})$.
Thus $S^{\prime}<0$ for all $r\in[r_{0},R_{\max})$. By Theorem 2.1,
$\lim_{r\to R_{\rm max}^{-}}S(r)=0.$ (2.23)
Note that $G_{1}(c_{1}r^{1-n})$ and $G_{2}(c_{2}r^{1-n})$ are bounded, and
$G_{1}(c_{1}r^{1-n})$ is bounded below by a positive number on $[r_{0},R_{\rm
max}]$. Combining these facts with relation (2.18), we infer that there are
$\delta>0$ and $C_{1},C_{2}>0$ such that for $r\in[0,R_{\rm max})$ and
$S\in(0,\delta)$,
$-C_{1}F_{1}(S)\leq F(r,S)\leq-C_{2}F_{1}(S).$ (2.24)
By (2.23), there is $r_{1}\in(0,R_{\rm max})$ such that $S(r)<\delta$ for all
$r\in[r_{1},R_{\rm\max})$. Define $Y(r)=F_{1}(S(r))$. By (2.19), there are
$\tilde{r}\in(r_{1},R_{\max})$ and $C_{3}>0$
$F^{\prime}_{1}(S(r))<C_{3}\text{ for all }r\in(\tilde{r},R_{\rm max}).$
(2.25)
For $r\in(\tilde{r},R_{\max})$, using (2.24) we have
$Y^{\prime}(r)=F^{\prime}_{1}(S)S^{\prime}=F^{\prime}_{1}(S)F(r,S)\geq-
CF^{\prime}_{1}(S)F_{1}(S)>-C_{4}F_{1}(S)=-C_{4}Y(r),$ (2.26)
where $C>0$, $C_{4}=CC_{3}>0.$ Thus (2.26) gives
$Y(r)\geq Y(\tilde{r})e^{-C_{4}(r-\tilde{r})},\quad r\in[\tilde{r},R_{\max}).$
(2.27)
We have from (2.23) and (2.17) that
$\lim_{r\to R_{\max}^{-}}Y(r)=0.$ (2.28)
Let $r\to R_{\max}^{-}$ in (2.27) and using (2.28), we obtain $0\geq
Y(\tilde{r})e^{-C_{4}(R_{\max}-\tilde{r})}>0$ which is a contradiction.
Case 2. $c_{1}\leq 0\leq c_{2}$. Both Case 2a and 2b are proved similarly.
Consider Case 2a. Since $F(r,S)>0$ for all $r\in[r_{0},R_{\max})$,
$S^{\prime}>0$ for all $r\in[r_{0},R_{\max})$ therefore by Theorem 2.1,
$\lim_{r\to R_{\rm max}^{-}}S(r)=1$.
Let $X=1-S$. Then $\lim_{r\to R_{\rm max}^{-}}X(r)=0$ and
$X^{\prime}=-S^{\prime}=-F(r,1-X)=\tilde{F}(r,X)=G_{1}(c_{1}r^{1-n})\tilde{F}_{1}(X)-G_{2}(c_{2}r^{1-n})\tilde{F}_{2}(X),$
(2.29)
where $\tilde{F}_{i}(X)=F_{i}(1-X)$. Similar to the proof of Case 1a, there
are $\delta>0$ and $C_{1},C_{2}>0$ such that
$-C_{1}\tilde{F}_{2}(X)\leq\tilde{F}(r,X)\leq-C_{2}\tilde{F}_{2}(X),$ (2.30)
for all $r\in[r_{0},R_{\max}]$ and $X\in(0,\delta)$. Note that condition
(2.21) is equivalent to $\limsup_{X\to
0^{+}}\tilde{F}_{2}^{\prime}(X)<\infty$. Repeating the proof in Case 1a with
$\tilde{F}_{2}$ instead of $F_{1}$ leads to a contradiction.
Case 3. According to Theorem 2.1 we have two cases.
(i) Case $\lim_{r\to R_{\rm max}^{-}}S(r)=0$. By (2.18) there are constants
$C_{1},C_{2}>0$ and $\delta>0$ such that
$-C_{1}F_{1}(S)\leq F(r,S)\leq-C_{2}F_{1}(S).$
for all $r\in[r_{0},R_{\rm max}]$ and $S\in(0,\delta)$. Also, there is
$r_{1}\in(0,R_{\max})$ such that $S(r)<\delta$ for all $r\in(r_{1},R_{\rm
max})$. Then the exact argument for Case 1a yields a contradiction.
(ii) Case $\lim_{r\to R_{\rm max}^{-}}S(r)=1$. By (2.18), there $\delta>0$ and
$C_{1},C_{2}>0$ such that
$C_{1}F_{2}(S)\leq F(r,S)\leq C_{2}F_{2}(S)$
for all $r\in[r_{0},R_{\rm max}]$ and $S\in(1-\delta,1)$. Then the proof is
proceeded similar to Case 2a under condition (2.21) to obtain a contradiction.
Case 4. Again, according to Theorem 2.1 we have two cases.
(i) Case $\lim_{r\to R_{\rm max}^{-}}S(r)=0$. By (2.18), there are $\delta>0$
and $C_{1},C_{2}>0$ such that
$0<C_{1}F_{1}(S)\leq F(r,S)\leq C_{2}F_{1}(S)$
for all $r\in[r_{0},R_{\rm max}]$ and $S\in(0,\delta)$. Let $r_{1}$ be as in
Case 3(i). Then for $r\in(r_{1},R_{\rm max})$ we have $S^{\prime}(r)>0$, and
hence $S(r)\geq S(r_{1})>0$ which contradicts the fact $\lim_{r\to R_{\rm
max}^{-}}S(r)=0$.
(ii) Case $\lim_{r\to R_{\rm max}^{-}}S(r)=1$. By (2.18), there are $\delta>0$
and $C_{1},C_{2}>0$ such that
$-C_{1}F_{2}(S)\leq F(r,S)\leq-C_{2}F_{2}(S)<0$
for all $r\in[r_{0},R_{\rm max}]$ and $S\in(1-\delta,1)$. There is
$r_{1}\in(r_{0},R_{\max})$ such that $S(r)\in(1-\delta,1)$ for all
$r\in(r_{1},R_{\max})$. Thus $S^{\prime}(r)<0$ for all $r\in(r_{1},R_{\max})$
which gives $S(r)\leq S(r_{1})$. Letting $r\to R_{\max}$ yields $1\leq
S(r_{1})<1$. This is a contradiction.
From all the above contradictions, we must have $R_{\max}=\infty$ and the
proof is complete. ∎
To study $S(r)$ as $r\to\infty$, for the solution $S(r)$ in the Theorem 2.2 we
will need function $h(r)\in(0,1)$ such that
$G_{2}(c_{2}r^{1-n})F_{2}(h(r))-G_{1}(c_{1}r^{1-n})F_{1}(h(r))=0.$ (2.31)
To prove existence of such function consider $c_{1}c_{2}\neq 0$. Then (2.31)
is equivalent to
$\frac{f_{1}(h(r))}{f_{2}(h(r))}=\frac{c_{1}g_{1}(|c_{1}|r^{1-n})}{c_{2}g_{2}(|c_{2}|r^{1-n})}.$
Since $f\mathbin{\buildrel\rm def\over{\mathbin{=\kern-2.0pt=}}}f_{1}/f_{2}$
is strictly increasing and maps $(0,1)$ onto $(0,\infty)$, we can solve
$h(r)=f^{-1}\Big{(}\frac{c_{1}g_{1}(|c_{1}|r^{1-n})}{c_{2}g_{2}(|c_{2}|r^{1-n})}\Big{)}\quad\text{provided}\quad
c_{1}c_{2}>0.$ (2.32)
Note that
$\lim_{r\to\infty}h(r)=s^{*}\mathbin{\buildrel\rm
def\over{\mathbin{=\kern-2.0pt=}}}f^{-1}\Big{(}\frac{c_{1}a_{1}^{0}}{c_{2}a_{2}^{0}}\Big{)}\in(0,1).$
(2.33)
Let $\xi(r)=r^{1-n}\in(0,\infty)$. We rewrite $h(r)$ as
$h(r)=f^{-1}\Big{(}Q(\xi(r))\Big{)}\quad\text{ where }\quad
Q(\xi)=\frac{c_{1}g_{1}(|c_{1}|\xi)}{c_{2}g_{2}(|c_{2}|\xi)}\quad\text{for
}\xi>0.$ (2.34)
###### Theorem 2.3.
If solution $S(r)$ of (2.13) exists in $[r_{0},\infty)$, then there exists
$R>r_{0}$ such that solution $S(r)$ is monotone on $(R,\infty)$, and,
consequently, $\lim_{r\to\infty}S(r)$ exists.
###### Proof.
If $c_{1}c_{2}\leq 0$ then all $r\geq r_{0}$ either $S^{\prime}\geq 0$ or
$S^{\prime}\leq 0$. Thus $S(r)$ is monotone on $[r_{0},\infty)$.
Consider the case $c_{1}c_{2}>0$. Then $h(r)$ in (2.34) exists. We rewrite
$Q(\xi)$ as
$Q(\xi)=\frac{c_{1}}{c_{2}}\cdot\frac{\sum_{i=0}^{m_{1}}a_{i}\xi^{\alpha_{i}}}{\sum_{i=0}^{m_{2}}b_{i}\xi^{\beta_{i}}}.$
(2.35)
where $a_{i},b_{i}>0$, $0=\alpha_{0}<\alpha_{1}<\cdots<\alpha_{m_{1}}$,
$0=\beta_{0}<\beta_{1}<\cdots<\beta_{m_{2}}$ .
If $Q^{\prime}\equiv 0$ then $h(r)\equiv s^{*}$ is an equilibrium. It is easy
to see that if $s_{0}>(<)s^{*}$ then $S(r)>(<)s^{*}$ for all $r$, hence $S(r)$
is monotone on $r\in[r_{0},\infty)$.
Now we consider $Q^{\prime}\neq 0$. A simple calculation gives
$\displaystyle Q^{\prime}(\xi)=\frac{c_{1}}{\xi
c_{2}}(\sum_{i=0}^{m_{2}}b_{i}\xi^{\beta_{i}})^{-2}\sum_{i=1}^{m_{3}}A_{i}\xi^{\gamma_{i}},$
where $m_{3}\geq 1$, $A_{i}\neq
0,0<\gamma_{1}<\gamma_{2}<\cdots<\gamma_{m_{3}}$. Note that $Q^{\prime}(\xi)$
has the same sign as $A_{1}$ for $\xi>0$ sufficiently small. Combining this
with the fact $f^{\prime}>0$, we have that $A_{1}h^{\prime}(r)<0$ for all
$r>R$, where $R>0$ is a sufficiently large number.
Claim 1. There is $\tilde{R}>R$ such that $S^{\prime}(r)\geq 0$ on
$(\tilde{R},\infty)$ or $S^{\prime}(r)\leq 0$ on $(\tilde{R},\infty)$.
Then the theorem’s statements obviously follow Claim 1.
To prove Claim 1 we consider the following cases.
Case 1: $A_{1}<0$. Then $h(r)$ is increasing in $[R,\infty)$ and, hence,
$h(r)<s^{*}$ for all $r\geq R$.
Case 1A: $S(r)\geq h(r)$ for all $r>R$. Then $S^{\prime}\geq 0$ for all $r>R$
or $S^{\prime}\leq 0$ for all $r>R$.
Case 1B: There exists $R_{1}>R$ such that $S(R_{1})<h(R_{1})$.
\+ Case 1B(i): $F(r,S)>0\Leftrightarrow S>h(r)$. Then $S^{\prime}>0$ if
$S(r)>h(r)$ and $S^{\prime}<0$ if $S(r)<h(r)$. It is easy to see that
$S(r)<h(R_{1})\leq h(r)$ for all $r>R_{1}$. Therefore $S^{\prime}(r)<0$ for
all $r>R_{1}$.
\+ Case 1B(ii): $F(r,S)<0\Leftrightarrow S>h(r)$. Then $S^{\prime}<0$ if
$S(r)>h(r)$ and $S^{\prime}>0$ if $S(r)<h(r)$.
Claim 2. $S(r)\leq h(r)$ for all $r\geq R_{1}$ and hence $S^{\prime}(r)\geq 0$
for all $r>R_{1}$.
Suppose Claim 2 is false. Then there is $R_{2}>R_{1}$ such that
$S(R_{2})>h(R_{2})$. There is $\tilde{r}\in(R_{1},R_{2})$ such that
$S(\tilde{r})=h(\tilde{r})$. Hence, $S$ is decreasing on $(\tilde{r},R_{2})$,
$S(R_{2})\leq S(\tilde{r})=h(\tilde{r})\leq h(R_{2})$. This is a
contradiction.
Case 2: $A_{1}>0$. Then $h(r)$ is decreasing in $[R,\infty)$ and $h(r)>s^{*}$
for all $r\geq R$ .
Case 2A: $S(r)\leq h(r)$ for all $r>R$. Then $S^{\prime}\leq 0$ for all $r>R$
or $S^{\prime}\geq 0$ for all $r>R$.
Case 2B: There exists $R_{1}>R$ such that $S(R_{1})>h(R_{1})$.
\+ Case 2B(i): $F(r,S)>0\Leftrightarrow S>h(r)$. Then $S^{\prime}>0$ if
$S(r)>h(r)$ and $S^{\prime}<0$ if $S(r)<h(r)$. Similar to Case 1B(i),
$h(r)<h(R_{1})<S(r)$ for all $r>R_{1}$. Therefore $S^{\prime}(r)>0$ for all
$r>R_{1}$.
\+ Case 2B(ii): $F(r,S)<0\Leftrightarrow S>h(r)$. Then $S^{\prime}<0$ if
$S(r)>h(r)$ and $S^{\prime}>0$ if $S(r)<h(r)$. Similar to Case 1B(ii),
$S(r)\geq h(r)$ for all $r\geq R_{1}$. Therefore $S^{\prime}(r)\leq 0$ for all
$r>R_{1}$.
From the above considerations, we see that Claim 1 holds true and the proof is
complete. ∎
Let $s_{\infty}=\lim_{r\to\infty}S(r)$ in Theorem 2.3. Note that
$s_{\infty}\in[0,1]$.
###### Lemma 2.4.
For $n=2$ and $c_{1}^{2}+c_{2}^{2}>0$, if $s_{\infty}$ is neither $0$ nor $1$
then $s_{\infty}$ must be $s^{*}$.
###### Proof.
Assume $s_{\infty}\neq 0,1$. We prove by contradiction. Suppose
$s_{\infty}\not=s^{*}$. Then
$c_{3}\mathbin{\buildrel\rm
def\over{\mathbin{=\kern-2.0pt=}}}|F_{2}(s_{\infty})a_{2}^{0}c_{2}-F_{1}(s_{\infty})a_{1}^{0}c_{1}|>0.$
(2.36)
For any $R>r_{0}$, We write $S(r)=I_{1}(R)+I_{2}(R)$ where
$I_{1}(R)=s_{0}+\int_{r_{0}}^{R}F(z,S(z))dz\quad\text{and}\quad
I_{2}(R)=\int_{R}^{r}F(z,S(z))dz.$
For sufficiently large $R$ and $r>R$
$|I_{2}(R)|=\int_{R}^{r}F(z,S(z))dz\geq\frac{c_{3}}{2}\int_{R}^{r}z^{-1}dz=\frac{c_{3}}{2}(\ln
r-\ln R).$
Therefore
$|S(r)|\geq\frac{c_{3}}{2}(\ln r-\ln R)-I_{1}(R)\to\infty\text{ as
}r\to\infty.$
Thus $S(r)$ is unbounded which contradicts the fact $S(r)\in(0,1)$. Hence
$s_{\infty}=s^{*}$. ∎
Using Lemma 2.4 we can drastically reduce the range of $s_{\infty}$ in case
$n=2$.
###### Theorem 2.5.
Let $n=2$ and $c_{1}^{2}+c_{2}^{2}>0$. Suppose $S(r)$ is a solution of (2.13)
on $[r_{0},\infty)$.
(i) If $c_{1}\leq 0$ and $c_{2}\geq 0$ then $s_{\infty}=1$.
(ii) If $c_{1}\geq 0$ and $c_{2}\leq 0$ then $s_{\infty}=0$.
(iii) If $c_{1},c_{2}<0$ then $s_{\infty}=s^{*}$.
(iv) If $c_{1},c_{2}>0$ then $s_{\infty}\in\\{0,1,s^{*}\\}$.
###### Proof.
(i) In this case, $S^{\prime}(r)>0$ for all r, hence $S(r)>s_{0}$. This
implies $s_{\infty}\neq 0$. In addition, $s^{*}$ does not exist. Therefore, by
Lemma 2.4, $s_{\infty}$ must be $1$.
(ii) The proof is similar to that of (i).
(iii) We have $F(r,S)<0$ for $S<h(r)$ and $F(r,S)>0$ for $S<h(r)$. Thus, it is
easy to see that $s_{\infty}$ cannot be $0,1$. By Lemma 2.4, $s_{\infty}$ must
be $s^{*}$.
(iv) This is a direct consequence of Lemma 2.4. ∎
In general, we do not know the value of $s_{\infty}$ based on $s_{0}$.
However, in some particular cases, we can determine the range of $s_{\infty}$.
###### Example 2.6.
We consider the following special $g_{i}$’s:
$g_{i}(u)=a_{i}+b_{i}u^{\alpha}\quad\text{ where }a_{i}>0,\ b_{i}>0,\text{ for
}i=1,2\text{ and }\alpha>0.$ (2.37)
We have from (2.34) when $c_{1}c_{2}>0$ that
$Q^{\prime}(\xi)=\frac{c_{1}\Delta}{c_{2}(a_{2}+b_{2}|c_{2}|^{\alpha}\xi)^{2}}\text{
with }\Delta=a_{2}b_{1}|c_{1}|^{\alpha}-a_{1}b_{2}|c_{2}|^{\alpha}.$
We now detail the range of $s_{\infty}$ case by case.
Case $n>2$.
* A.
$c_{1},c_{2}>0$.
* A1.
$\Delta<0$.
(i) $s_{0}>s^{*}$. Then $s_{\infty}\in(s_{0},1]$.
(ii) $h(r_{0})\leq s_{0}\leq s^{*}$. Then $s_{\infty}\in[0,1]$.
(iii) $s_{0}<h(r_{0})$. Then $s_{\infty}\in[0,s_{0})$.
* A2.
$\Delta>0$.
(i) $s_{0}>h(r_{0})$. Then $s_{\infty}\in(s_{0},1]$.
(ii) $s^{*}\leq s_{0}\leq h(r_{0})$. Then $s_{\infty}\in[0,1]$.
(iii) $s_{0}<s^{*}$. Then $s_{\infty}\in[0,s_{0})$.
* A3.
$\Delta=0$.
(i) $s_{0}>s^{*}$. Then $s_{\infty}\in(s_{0},1]$.
(ii) $s_{0}=s^{*}$. Then $s_{\infty}=s^{*}$.
(iii) $s_{0}<s^{*}$. Then $s_{\infty}\in[0,s_{0})$.
* B.
$c_{1},c_{2}<0$.
* B1.
$\Delta<0$.
(i) $s_{0}>s^{*}$. Then $s_{\infty}\in(h(r_{0}),s_{0})$.
(ii) $h(r_{0})\leq s_{0}\leq s^{*}$. Then $s_{\infty}\in(h(r_{0}),s^{*}]$.
(iii) $s_{0}<h(r_{0})$. Then $s_{\infty}\in(s_{0},s^{*}]$.
* B2.
$\Delta>0$.
(i) $s_{0}>h(r_{0})$. Then $s_{\infty}\in[s^{*},s_{0})$.
(ii) $s^{*}\leq s_{0}\leq h(r_{0})$. Then $s_{\infty}\in[s^{*},h(r_{0}))$.
(iii) $s_{0}<s^{*}$. Then $s_{\infty}\in(s_{0},h(r_{0}))$.
* B3.
$\Delta=0$.
(i) $s_{0}>s^{*}$. Then $s_{\infty}\in[s^{*},s_{0})$.
(ii) $s_{0}=s^{*}$. Then $s_{\infty}=s^{*}$.
(iii) $s_{0}<s^{*}$. Then $s_{\infty}\in(s_{0},s^{*}]$.
* C.
$c_{1}\leq 0<c_{2}$ or $c_{1}<0=c_{2}$. Then $s_{\infty}\in(s_{0},1]$.
* D.
$c_{2}\leq 0<c_{1}$ or $c_{1}=0>c_{2}$. Then $s_{\infty}\in[0,s_{0})$.
Verifications of the cases above are presented in the Appendix.
Case $n=2$. We use the analysis in A, which is still valid for $n=2$, to
explicate the case $c_{1},c_{2}>0$ in Theorem 2.5. Let
$s_{m}=\min\\{h(r_{0}),s^{*}\\}$ and $s_{M}=\max\\{h(r_{0}),s^{*}\\}$.
* (i)
$s_{0}>s_{M}$. Then $s_{\infty}=1$.
* (ii)
$s_{m}\leq s_{0}\leq s_{M}$. Then $s_{\infty}\in\\{0,1,s^{*}\\}$.
* (iii)
$s_{0}<s_{m}$. Then $s_{\infty}=0$.
## 3\. Linearization
We study the linear stability of a steady state solution
$(\mathbf{u}_{1}^{*}(\mathbf{x}),\mathbf{u}_{2}^{*}(\mathbf{x}),S_{*}(\mathbf{x}))$
of system (1.14). The formal linearizion of system (1.14) at
$(\mathbf{u}_{1}^{*}(\mathbf{x}),\mathbf{u}_{2}^{*}(\mathbf{x}),S_{*}(\mathbf{x}))$
is
$\displaystyle\sigma_{t}$ $\displaystyle=-{\rm div}\ {\mathbf{v}}_{1},$ (3.1a)
$\displaystyle\sigma_{t}$ $\displaystyle={\rm div}\ {\mathbf{v}}_{2},$ (3.1b)
$\displaystyle\nabla\sigma$
$\displaystyle=F_{2}(S_{*})\mathbf{G}^{\prime}_{2}({\mathbf{u}}_{2}^{*}){\mathbf{v}}_{2}+F^{\prime}_{2}(S_{*})\sigma\mathbf{G}_{2}({\mathbf{u}}_{2}^{*})-\Big{(}F_{1}(S_{*})\mathbf{G}^{\prime}_{1}({\mathbf{u}}_{1}^{*}){\mathbf{v}}_{1}+F^{\prime}_{1}(S_{*})\sigma\mathbf{G}_{1}({\mathbf{u}}_{1}^{*})\Big{)}.$
(3.1c)
Above, the unknowns are $\sigma(\mathbf{x},t)\in\mathbb{R}$,
$\mathbf{v}_{1}(\mathbf{x},t),\mathbf{v}_{2}(\mathbf{x},t)\in\mathbb{R}^{n}$.
A solution $(\sigma,\mathbf{v}_{1},\mathbf{v}_{2})$ of (3.1) is considered as
an approximation of the difference between a solution
$(S(\mathbf{x},t),\mathbf{u}_{1}(\mathbf{x},t),\mathbf{u}_{2}(\mathbf{x},t))$
of (1.14) and the steady state
$(\mathbf{u}_{1}^{*}(\mathbf{x}),\mathbf{u}_{2}^{*}(\mathbf{x}),S_{*}(\mathbf{x}))$
in (3.2). The system (3.1) is obtained by utilizing Taylor expansions in
(1.14) at $(\mathbf{u}_{1}^{*},\mathbf{u}_{2}^{*},S_{*})$ with respect to
variables $\mathbf{u}_{1},\mathbf{u}_{2},S$ and then neglecting non-linear
terms. In theory of ordinary differential equations, linearizion has direct
connections with the stability of steady states. In PDE theory, this is not
always the case. Nonetheless, in many scenarios, stability of the linearized
equations lead to the stability of the original ones. In this article we only
focus on the stability for the linearized system (3.1).
We consider, particularly, the steady states obtained in the previous section,
that is,
$\mathbf{u}_{1}^{*}(\mathbf{x})=c_{1}|\mathbf{x}|^{-n}\mathbf{x},\quad\mathbf{u}_{2}^{*}(\mathbf{x})=c_{2}|\mathbf{x}|^{-n}\mathbf{x},\quad
S_{*}(\mathbf{x})=\hat{S}(|\mathbf{x}|),$ (3.2)
where $c_{1},c_{2}$ are constants and $\hat{S}(r)$ is a solution of (2.13).
Let ${\mathbf{v}}={\mathbf{v}}_{1}+{\mathbf{v}}_{2}.$ Adding equation (3.1a)
to (3.1b) gives
${\rm div}\ {\mathbf{v}}=0.$ (3.3)
Assume ${\mathbf{v}}=\mathbf{V}(\mathbf{x},t)\in\mathbb{R}^{n}$, where
$\mathbf{V}(\mathbf{x},t)$ is a given function. We have
$\mathbf{v}_{1}=\mathbf{V}-\mathbf{v}_{2},$ (3.4)
hence (3.1c) provides
$\nabla\sigma=\sigma\mathbf{b}+\underline{\mathbf{B}}\mathbf{v}_{2}-\mathbf{c},$
(3.5)
where
$\displaystyle\underline{\mathbf{B}}$
$\displaystyle=\underline{\mathbf{B}}(\mathbf{x})=F_{2}(S_{*})\mathbf{G}^{\prime}_{2}({\mathbf{u}}_{2}^{*})+F_{1}(S_{*})\mathbf{G}^{\prime}_{1}({\mathbf{u}}_{1}^{*}),$
(3.6) $\displaystyle\mathbf{b}$
$\displaystyle=\mathbf{b}(\mathbf{x})=F^{\prime}_{2}(S_{*})\mathbf{G}_{2}({\mathbf{u}}_{2}^{*})-F^{\prime}_{1}(S_{*})\mathbf{G}_{1}({\mathbf{u}}_{1}^{*}),$
(3.7) $\displaystyle\mathbf{c}$
$\displaystyle=\mathbf{c}(\mathbf{x},t)=F_{1}(S_{*})\mathbf{G}^{\prime}_{1}({\mathbf{u}}_{1}^{*})\mathbf{V}(\mathbf{x},t).$
(3.8)
The $n\times n$ matrix $\underline{\mathbf{B}}$ is invertible (see Lemma 3.2
below), and we denote its inverse by
$\underline{\mathbf{A}}=\underline{\mathbf{A}}(\mathbf{x})=\underline{\mathbf{B}}^{-1}(\mathbf{x}).$
(3.9)
Solving for $\mathbf{v}_{2}$ from (3.5) we obtain
$\mathbf{v}_{2}=\underline{\mathbf{A}}(\nabla\sigma-\sigma\mathbf{b})+\underline{\mathbf{A}}\mathbf{c}.$
(3.10)
Substituting (3.10) into (3.1b) gives
$\displaystyle\sigma_{t}$
$\displaystyle=\nabla\cdot\Big{[}\underline{\mathbf{A}}(\nabla\sigma-\sigma\mathbf{b})\Big{]}+\nabla\cdot(\underline{\mathbf{A}}\mathbf{c}).$
(3.11)
Then (3.11), (3.4) and (3.10) is our linearized system for (1.14) at the
steady state
$(\mathbf{u}_{1}^{*}(\mathbf{x}),\mathbf{u}_{2}^{*}(\mathbf{x}),S_{*}(\mathbf{x}))$.
###### Remark 3.1.
In our approach, the total velocity $\bf V$ and hence the vector function $\bf
c$ are supposed to be known, whereas the phase velocities $\mathbf{v}_{i}$
($i=1,2$) are the unknowns. Therefore, our results below can be considered as
the qualitative study of the flow depending on the property of the total
velocity. Such restriction, however, is justified in practice or in case $\bf
V$, as a perturbation, itself is radial. In the latter consideration, by
(3.3), $\mathbf{V}=\mathbf{V}(t)$ is totally determined by its boundary
values.
We will focus on studying classical solutions of (3.11). For such purpose, the
maximum principle plays an important role. Although there is not an obvious
maximum principle for (3.11), we can convert it to an equation for which there
is one. We proceed as follows. Rewrite vector function
$\mathbf{b}(\mathbf{x})$ explicitly as
$\mathbf{b}(\mathbf{x})=\Big{(}F^{\prime}_{2}(S_{*}(\mathbf{x}))g_{2}(\frac{|c_{2}|}{|\mathbf{x}|^{n-1}})\frac{c_{2}}{|\mathbf{x}|^{n}}-F^{\prime}_{1}(S_{*}(\mathbf{x}))g_{1}(\frac{|c_{1}|}{|\mathbf{x}|^{n-1}})\frac{c_{1}}{|\mathbf{x}|^{n}}\Big{)}\mathbf{x}=\lambda(|\mathbf{x}|)\mathbf{x},$
(3.12)
where
$\lambda(r)=F^{\prime}_{2}(\hat{S}(r))g_{2}(\frac{|c_{2}|}{r^{n-1}})\frac{c_{2}}{r^{n}}-F^{\prime}_{1}(\hat{S}(r))g_{1}(\frac{|c_{1}|}{r^{n-1}})\frac{c_{1}}{r^{n}}.$
(3.13)
By defining
$\Lambda(\mathbf{x})=\frac{1}{2}\int_{r_{0}^{2}}^{|\mathbf{x}|^{2}}\lambda(\sqrt{\xi})d\xi=\int_{r_{0}}^{|\mathbf{x}|}r\lambda(r)dr,$
(3.14)
we have for $\mathbf{x}\neq 0$ that
$\mathbf{b}(\mathbf{x})=\nabla\Lambda(\mathbf{x}).$ (3.15)
Substituting this relation into (3.11) we obtain
$\displaystyle\sigma_{t}$
$\displaystyle=\nabla\cdot\Big{[}\underline{\mathbf{A}}(\nabla\sigma-\sigma\nabla\Lambda)\Big{]}+\nabla\cdot(\underline{\mathbf{A}}\mathbf{c})=\nabla\cdot\Big{[}e^{\Lambda}\underline{\mathbf{A}}\nabla(e^{-\Lambda}\sigma)\Big{]}+\nabla\cdot(\underline{\mathbf{A}}\mathbf{c})$
$\displaystyle=e^{\Lambda}\nabla\cdot\Big{[}\underline{\mathbf{A}}\nabla(e^{-\Lambda}\sigma)\Big{]}+e^{\Lambda}\nabla\Lambda\cdot\Big{[}\underline{\mathbf{A}}\nabla(e^{-\Lambda}\sigma)\Big{]}+\nabla\cdot(\underline{\mathbf{A}}\mathbf{c}).$
Let
$w(\mathbf{x},t)=e^{-\Lambda(\mathbf{x})}\sigma(\mathbf{x},t).$ (3.16)
Then $w$ satisfies
$w_{t}=e^{-\Lambda}\sigma_{t}=\nabla\cdot\Big{(}\underline{\mathbf{A}}\nabla
w\Big{)}+\nabla\Lambda\cdot\underline{\mathbf{A}}\nabla
w+e^{-\Lambda}\nabla\cdot(\underline{\mathbf{A}}\mathbf{c}).$ (3.17)
Using relation (3.15) again yields
$w_{t}-\nabla\cdot(\underline{\mathbf{A}}\nabla
w)-\mathbf{b}\cdot\underline{\mathbf{A}}\nabla
w=e^{-\Lambda}\nabla\cdot(\underline{\mathbf{A}}\mathbf{c}).$ (3.18)
For the velocities, we have from (3.10) and (3.16) that
$\displaystyle\mathbf{v}_{2}=\underline{\mathbf{A}}\big{[}\nabla(e^{\Lambda}w)-e^{\Lambda}w{\bf
b}\big{]}+\underline{\mathbf{A}}{\bf
c}=\underline{\mathbf{A}}\big{[}e^{\Lambda}\nabla
w+we^{\Lambda}\nabla\Lambda-e^{\Lambda}w{\bf
b}\big{]}+\underline{\mathbf{A}}{\bf c}.$
Thus,
$\mathbf{v}_{2}=e^{\Lambda}\underline{\mathbf{A}}\nabla
w+\underline{\mathbf{A}}{\bf c}.$ (3.19)
We will proceed by studying (3.18) first and then drawing conclusions for
$\sigma,\mathbf{v}_{1},\mathbf{v}_{2}$ via the relations (3.16), (3.19) and
(3.4).
In the following, we present some properties of $\underline{\mathbf{B}}$,
$\underline{\mathbf{A}}$ and $\bf b$. They have some structures and estimates
which are crucial for our next sections. These are based on the special form
of the steady state $(\mathbf{u}^{*}_{1},\mathbf{u}^{*}_{2},S_{*})$.
Denote by $\underline{\mathbf{I}}_{n}$ the $n\times n$ identity matrix.
Consider $c_{1}^{2}+c_{2}^{2}>0$ and $\mathbf{x}\neq 0$. We have for $i=1,2$
that
$\mathbf{G}^{\prime}_{i}(\mathbf{u}_{i}^{*})=g_{i}(|\mathbf{u}_{i}^{*}|)\underline{\mathbf{I}}_{n}+g^{\prime}_{i}(|\mathbf{u}_{i}^{*}|)\frac{\mathbf{u}_{i}^{*}(\mathbf{u}_{i}^{*})^{T}}{|\mathbf{u}_{i}^{*}|}=g_{i}(|c_{i}|\,|\mathbf{x}|^{1-n})\underline{\mathbf{I}}_{n}+g^{\prime}_{i}(|c_{i}|\,|\mathbf{x}|^{1-n})|c_{i}|\,|\mathbf{x}|^{-1-n}\mathbf{x}\mathbf{x}^{T}.$
(3.20)
Since these matrices are symmetric, so is $\underline{\mathbf{B}}$. For each
$i=1,2$ and arbitrary $\mathbf{z}\in\mathbb{R}^{n}$,
$\displaystyle\mathbf{z}^{T}\mathbf{G}^{\prime}_{i}(\mathbf{u}_{i}^{*})\mathbf{z}$
$\displaystyle=g_{i}(|c_{i}|\,|\mathbf{x}|^{1-n})|\mathbf{z}|^{2}+g^{\prime}_{i}(|c_{i}|\,|\mathbf{x}|^{1-n})|c_{i}|\,|\mathbf{x}|^{-1-n}|\mathbf{x}\cdot{\mathbf{z}}|^{2}.$
Define
$\displaystyle\beta$ $\displaystyle=\beta(\mathbf{x})\mathbin{\buildrel\rm
def\over{\mathbin{=\kern-2.0pt=}}}\sum_{i=1}^{2}F_{i}(S_{*}(\mathbf{x}))g_{i}(|c_{i}|\,|\mathbf{x}|^{1-n}),$
(3.21) $\displaystyle\gamma$
$\displaystyle=\gamma(\mathbf{x})\mathbin{\buildrel\rm
def\over{\mathbin{=\kern-2.0pt=}}}\sum_{i=1}^{2}F_{i}(S_{*}(\mathbf{x}))g^{\prime}_{i}(|c_{i}|\,|\mathbf{x}|^{1-n})|c_{i}|\,|\mathbf{x}|^{1-n}.$
(3.22)
Then
$\displaystyle\beta|\mathbf{z}|^{2}\leq\mathbf{z}^{T}\underline{\mathbf{B}}\mathbf{z}$
$\displaystyle\leq(\beta+\gamma)|\mathbf{z}|^{2}.$ (3.23)
The first inequality in (3.23) proves that
$\mathbf{z}^{T}\underline{\mathbf{B}}\mathbf{z}>0$ for all $\mathbf{z}\neq 0$.
Therefore, $\underline{\mathbf{B}}$ is positive definite and hence it is
invertible. Since $\underline{\mathbf{B}}$ is symmetric, so is its inverse
$\underline{\mathbf{A}}$. Thus, we have:
###### Lemma 3.2.
For any $c_{1}^{2}+c_{2}^{2}>0$ and $\mathbf{x}\neq 0$, matrices
$\underline{\mathbf{B}}(\mathbf{x})$ and $\underline{\mathbf{A}}(\mathbf{x})$
are symmetric, invertible and positive definite.
Since matrix $\underline{\mathbf{B}}$ is symmetric and positive definite, it
has positive eigenvalues
$\lambda_{1}(\underline{\mathbf{B}})\leq\lambda_{2}(\underline{\mathbf{B}})\leq\cdots\leq\lambda_{n}(\underline{\mathbf{B}})$.
We have
$\lambda_{1}(\underline{\mathbf{B}})=\min_{\mathbf{z}\neq
0}\frac{\mathbf{z}^{T}\underline{\mathbf{B}}\mathbf{z}}{|\mathbf{z}|^{2}}\quad\text{and}\quad\lambda_{n}(\underline{\mathbf{B}})=\max_{\mathbf{z}\neq
0}\frac{\mathbf{z}^{T}\underline{\mathbf{B}}\mathbf{z}}{|\mathbf{z}|^{2}}.$
(3.24)
It follows from (3.24) and (3.23) that
$\beta\leq\lambda_{1}(\underline{\mathbf{B}})\leq\lambda_{n}(\underline{\mathbf{B}})\leq\beta+\gamma.$
(3.25)
By the Spectral Theorem,
$\lambda_{1}(\underline{\mathbf{A}})=\frac{1}{\lambda_{n}(\underline{\mathbf{B}})}\geq\frac{1}{\beta+\gamma}\quad\text{and}\quad\lambda_{n}(\underline{\mathbf{A}})=\frac{1}{\lambda_{1}(\underline{\mathbf{B}})}\leq\frac{1}{\beta}.$
(3.26)
We now consider $0<r_{0}\leq|\mathbf{x}|<R_{\rm max}$. Let
$a_{0}^{(i)}=g_{i}(0)$ for $i=1,2$, and define
$\displaystyle d_{0}$ $\displaystyle=\min\\{a_{0}^{(1)},a_{0}^{(2)}\\},\quad
d_{1}=d_{1}(r_{0})=\sum_{i=1}^{2}g_{i}(|c_{i}|r_{0}^{1-n}),$ (3.27)
$\displaystyle d_{2}$
$\displaystyle=d_{2}(r_{0})=\sum_{i=1}^{2}g_{i}(|c_{i}|r_{0}^{1-n})|c_{i}|r_{0}^{1-n},\quad
d_{3}=d_{3}(r_{0})=\sum_{i=1}^{2}g^{\prime}_{i}(|c_{i}|r_{0}^{1-n})|c_{i}|r_{0}^{1-n},$
(3.28) $\displaystyle d_{4}$ $\displaystyle=d_{4}(r_{0})=d_{1}+d_{3}.$ (3.29)
Then
$d_{0}\sum_{i=1}^{2}F_{i}(S_{*}(\mathbf{x}))\leq\beta(\mathbf{x})\leq
d_{1}\sum_{i=1}^{2}F_{i}(S_{*}(\mathbf{x}))\quad\text{and}\quad\gamma(\mathbf{x})\leq
d_{3}\sum_{i=1}^{2}F_{i}(S_{*}(\mathbf{x})).$ (3.30)
By (3.23), (3.26) and (3.30),
$d_{0}|\mathbf{z}|^{2}\sum_{i=1}^{2}F_{i}(S_{*}(\mathbf{x}))\leq\mathbf{z}^{T}\underline{\mathbf{B}}(\mathbf{x})\mathbf{z}\leq
d_{4}|\mathbf{z}|^{2}\sum_{i=1}^{2}F_{i}(S_{*}(\mathbf{x})),$ (3.31)
$\frac{1}{d_{4}\sum_{i=1}^{2}F_{i}(S_{*}(\mathbf{x}))}\leq\lambda_{1}(\underline{\mathbf{A}})\leq\lambda_{n}(\underline{\mathbf{A}})\leq\frac{1}{d_{0}\sum_{i=1}^{2}F_{i}(S_{*}(\mathbf{x}))}.$
(3.32)
Applying (3.24) to matrix $\underline{\mathbf{A}}$, we have
$\mathbf{z}^{T}\underline{\mathbf{A}}(\mathbf{x})\mathbf{z}\geq\lambda_{1}(\underline{\mathbf{A}})|\mathbf{z}|^{2}\geq\frac{|{\bf
z}|^{2}}{d_{4}\sum_{i=1}^{2}F_{i}(S_{*}(\mathbf{x}))}\quad\forall\bf
z\in\mathbb{R}^{n}.$ (3.33)
Denote by $|\underline{\mathbf{A}}|$ and $\|\underline{\mathbf{A}}\|_{\rm op}$
the Euclidean and operator norms of matrix $\underline{\mathbf{A}}$,
respectively. Then
$|\underline{\mathbf{A}}|\leq c_{0}\|\underline{\mathbf{A}}\|_{\rm
op}=c_{0}\lambda_{n}(\underline{\mathbf{A}}),$ (3.34)
for some constant $c_{0}>0$. Thus,
$|\underline{\mathbf{A}}(\mathbf{x})|\leq\frac{c_{0}}{d_{0}\sum_{i=1}^{2}F_{i}(S_{*}(\mathbf{x}))}\quad\forall|\mathbf{x}|\in[r_{0},R_{\rm
max}).$ (3.35)
For the boundedness of $\mathbf{b}$, we have
$|\mathbf{b}(\mathbf{x})|\leq\sum_{i=1}^{2}\Big{[}|F^{\prime}_{i}(\hat{S}(|\mathbf{x}|))|g_{i}(|c_{i}||\mathbf{x}|^{1-n})|c_{i}||\mathbf{x}|^{1-n}\Big{]}\leq
d_{2}\sum_{i=1}^{2}|F^{\prime}_{i}(\hat{S}(|\mathbf{x}|))|\quad\forall|\mathbf{x}|\in[r_{0},R_{\rm
max}).$ (3.36)
From (3.14) and (3.13),
$\Lambda(\mathbf{x})=\int_{r_{0}}^{|\mathbf{x}|}r\lambda(r)dr=\int_{r_{0}}^{|\mathbf{x}|}\Big{[}F^{\prime}_{2}(\hat{S}(r))G_{2}(c_{2}r^{1-n})-F^{\prime}_{1}(\hat{S}(r))G_{1}(c_{1}r^{1-n})\Big{]}dr.$
(3.37)
Then
$|\Lambda(\mathbf{x})|\leq
d_{2}\int_{r_{0}}^{|\mathbf{x}|}\Big{[}|F^{\prime}_{1}(\hat{S}(r))|+|F^{\prime}_{2}(\hat{S}(r))|\Big{]}dr\quad\forall|\mathbf{x}|\in[r_{0},R_{\rm
max}).$ (3.38)
Also, matrix $\underline{\mathbf{B}}$ has the following special property:
$\underline{\mathbf{B}}(\mathbf{x})\mathbf{x}=\sum_{i=1}^{2}\Big{\\{}F_{i}(\hat{S}(|\mathbf{x}|))\left[g_{i}(|c_{i}||\mathbf{x}|^{1-n})+g^{\prime}_{i}(|c_{i}||\mathbf{x}|^{1-n})|c_{i}||\mathbf{x}|^{1-n}\right]\Big{\\}}\mathbf{x}=\phi(|\mathbf{x}|)\mathbf{x},$
(3.39)
where
$\phi(r)=\sum_{i=1}^{2}F_{i}(\hat{S}(r))\left[g_{i}(|c_{i}|r^{1-n})+g^{\prime}_{i}(|c_{i}|r^{1-n})|c_{i}|r^{1-n}\right].$
(3.40)
Since $g^{\prime}_{i}\geq 0$,
$\phi(r)\geq d_{0}[F_{1}(\hat{S}(r))+F_{2}(\hat{S}(r))]\quad\forall
r\in[r_{0},R_{\rm max}).$ (3.41)
Since $g_{i}(s)$ and $g_{i}^{\prime}(s)s$ are increasing on $[0,\infty)$, we
have
$\phi(r)\leq d_{4}[F_{1}(\hat{S}(r))+F_{2}(\hat{S}(r))]\quad\forall
r\in[r_{0},R_{\rm max}).$ (3.42)
We now discuss the regularity of the involved functions. For
$D\subset\mathbb{R}^{n}\times\mathbb{R}$, we define class
$C_{\mathbf{x}}^{m}(D)$ as the set of functions $f(\mathbf{x},t)\in C(D)$
whose partial derivatives with respect to $\mathbf{x}$ up to order $m$ are
continuous in $D$. The class $C_{t}^{m}$ is defined similarly and
$C_{\mathbf{x},t}^{m,m^{\prime}}=C_{\mathbf{x}}^{m}\cap C_{t}^{m^{\prime}}$.
Note that
$\frac{\partial\underline{\mathbf{A}}}{\partial
x_{i}}=-\underline{\mathbf{A}}\frac{\partial\underline{\mathbf{B}}}{\partial
x_{i}}\underline{\mathbf{A}}.$ (3.43)
By definitions (3.6), (3.7), (3.8) and relation (3.43), we easily obtain:
###### Lemma 3.3.
Assume $F_{1},F_{2}\in C^{m}((0,1))$ for some $m\geq 1$. Let
$R\in(r_{0},R_{\rm max})$ and denote
$\mathcal{O}=\\{\mathbf{x}:r_{0}<|\mathbf{x}|<R\\}.$
(i) Then $\underline{\mathbf{B}},\underline{\mathbf{A}}\in
C^{m}(\bar{\mathcal{O}})$, $\mathbf{b}\in C^{m-1}(\bar{\mathcal{O}})$ and
$\Lambda\in C^{m}(\bar{\mathcal{O}})$.
(ii) If, in addition, $\mathbf{V}\in X(\mathcal{O}\times(0,\infty))$ then
$\mathbf{c}\in X(\mathcal{O}\times(0,\infty))$, where $X$ can be $C^{m}$ or
$C_{\mathbf{x}}^{m}$ or $C_{t}^{m}$.
## 4\. Case of bounded domain
In this section, we study the linear stability of the obtained steady flows in
section 2 on bounded domains. More specifically, we investigate the stability
of the trivial solution for the linearized system (3.1). The key instrument in
proving the asymptotic stability is a Landis-type lemma of growth (see [19]).
To prove such a lemma we use specific structures of the coefficients of
equation (3.18) to construct singular sub-parabolic functions. These are
motivated by the so-called $F_{s,\beta}$ functions introduced in [19].
Let $r_{0}>0$ be fixed throughout. We consider in this section an open,
bounded set $U$ in $\mathbb{R}^{n}\setminus\bar{B}_{r_{0}}$. We fix $R>0$ such
that $U\subset\mathscr{U}\mathbin{\buildrel\rm
def\over{\mathbin{=\kern-2.0pt=}}}B_{R}\setminus\bar{B}_{r_{0}}$. Denote
$\Gamma=\partial U$, $D=U\times(0,\infty)$ and
$\mathscr{D}=\mathscr{U}\times(0,\infty)$.
We consider a steady state
$(u^{*}_{1}(\mathbf{x}),u^{*}_{2}(\mathbf{x}),S_{*}(\mathbf{x}))$ as in (3.2)
with $c_{1}^{2}+c_{2}^{2}>0$. Recall that (3.11), (3.4) and (3.10) is our
linearized system for (1.14). We study the equation for $\sigma(\mathbf{x},t)$
first. More specifically, we study the following initial-boundary value
problem (IBVP):
$\begin{cases}\sigma_{t}=\nabla\cdot\Big{[}\underline{\mathbf{A}}(\nabla\sigma-\sigma{\bf
b})\Big{]}+\nabla\cdot(\underline{\mathbf{A}}\mathbf{c})&\text{ on
}U\times(0,\infty),\\\ \sigma=g(\mathbf{x},t)&\text{ on
}\Gamma\times(0,\infty),\\\ \sigma=\sigma_{0}(\mathbf{x})&\text{ on
}U\times\\{t=0\\}.\end{cases}$ (4.1)
Regarding the initial and boundary data in (4.1), we always assume that
$\sigma_{0}\in C(\bar{U}),\ g\in C(\Gamma\times[0,\infty))\text{ and
}\sigma_{0}(\mathbf{x})=g(\mathbf{x},0)\text{ on }\Gamma.$ (4.2)
Assume that
$0<\underline{s}\leq\hat{S}(r)\leq\bar{s}<1\quad\forall
r\in[r_{0},R],\quad\text{where }\underline{s}\text{ and }\bar{s}\text{ are
constants.}$ (4.3)
Assumption (4.3) is valid for any solution $\hat{S}$ in Theorem 2.1 with
$R_{\rm max}>R$, in particular, when $R_{\rm max}=\infty$ as in Theorem 2.2.
Under constraint (4.3) and Assumptions A and B, we easily see the following
facts. Let
$\mu_{1}=\sum_{i=1}^{2}\max_{\underline{s}\leq
s\leq\bar{s}}F_{i}(s),\quad\mu_{2}=\sum_{i=1}^{2}\min_{\underline{s}\leq
s\leq\bar{s}}F_{i}(s),\quad\mu_{3}=\sum_{i=1}^{2}\max_{\underline{s}\leq
s\leq\bar{s}}|F^{\prime}_{i}(s)|.$ (4.4)
Then $\mu_{1}$, $\mu_{2}$ and $\mu_{3}$ are positive numbers.
From (3.33) and (4.3) follows that
${\bf z}^{T}\underline{\mathbf{A}}(\mathbf{x}){\bf z}\geq\frac{|{\bf
z}|^{2}}{C_{0}}\quad\forall\mathbf{x}\in\bar{\mathscr{U}},\
\mathbf{z}\in\mathbb{R}^{n},$ (4.5)
where $C_{0}=d_{4}\mu_{1}$.
From (3.35), (3.36) and (4.3), we get
$|\underline{\mathbf{A}}(\mathbf{x})|\leq\frac{c_{0}}{C_{1}}\quad\text{and}\quad\quad|\mathbf{b}(\mathbf{x})|\leq
C_{2}\quad\forall\mathbf{x}\in\bar{\mathscr{U}},$ (4.6)
where $c_{0}$ is in (3.34), $C_{1}=d_{0}\mu_{2}$ and $C_{2}=d_{2}\mu_{3}$.
For the smoothness, by Lemma 3.3,
$\underline{\mathbf{B}},\underline{\mathbf{A}}\in
C^{1}(\bar{\mathscr{U}})\quad\text{and}\quad\mathbf{b}\in
C(\bar{\mathscr{U}}).$ (4.7)
First, we consider the the existence of classical solutions of (4.1). We use
the known result from theory of linear parabolic equations in [16]. This will
require certain regularity of the coefficients of (4.1). Those requirements,
in turn, can be formulated in terms of functions $F_{1}$ and $F_{2}$, thanks
to Lemma 3.3.
Condition (E1). $F_{1},F_{2}\in C^{7}((0,1))$ and $V\in
C_{\mathbf{x}}^{6}(\bar{D})$; $V_{t}\in C_{\mathbf{x}}^{3}(\bar{D})$.
###### Theorem 4.1 ([16]).
Assume (E1), then there exists a unique solution $\sigma\in C(\bar{D})\cap
C^{2,1}_{\mathbf{x},t}(D)$ of problem (4.1).
Note that we did not attempt to optimize Condition (E1). As seen below, the
study of qualitative properties of solution $\sigma$ will require much less
stringent conditions than (E1).
Now we turn to the stability, asymptotic stability and structural stability
issues. Our main tool is the maximum principle. As discussed in the previous
section, we use the transformation (3.16) to convert the PDE in (4.1) to a
more convenient form (3.18). Define the differential operator on the left-hand
side of (3.18) by
$\mathcal{L}w=\partial_{t}w-\nabla\cdot(\underline{\mathbf{A}}\nabla w)-{\bf
b}\cdot\underline{\mathbf{A}}\nabla w.$ (4.8)
Corresponding to (4.1), the IBVP for $w(\mathbf{x},t)$ is
$\begin{cases}\mathcal{L}w=f_{0}&\text{in }U\times(0,\infty),\\\
w(\mathbf{x},0)=w_{0}(\mathbf{x})&\text{in }U,\\\
w(\mathbf{x},t)=G(\mathbf{x},t)&\text{on }\Gamma\times(0,\infty),\end{cases}$
(4.9)
where $w_{0}(\mathbf{x})$ and $G(\mathbf{x},t)$ are given initial data and
boundary data, respectively, and $f_{0}(\mathbf{x},t)$ is a known function. We
will obtain results for solution $w$ of (4.9) and then reformulate them in
terms of solution $\sigma$ of the original problem (4.1).
Since the existence and uniqueness issues are settled in Theorem 4.1, our main
focus now is the qualitative properties of solution $w$ of (4.9). For these,
we only need properties (4.5), (4.6), the special structure of equation (4.1),
and the assumption that the classical solution in $C(\bar{D})\cap
C^{2,1}_{\mathbf{x},t}(D)$ already exists. The fine properties of the
solutions obtained below have their own merit in the theory of linear
parabolic equations.
It follows from (4.5) and (4.6) that the maximum principle holds for any
classical solution of $\mathcal{L}w\leq(\geq)0$ in $D$. To obtain better
estimates for solutions, especially as $t\to\infty$, we use the following
barrier function. Define
$W(\mathbf{x},t)=\begin{cases}t^{-s}e^{-\frac{\varphi(\mathbf{x})}{t}}&\text{if
}t>0,\\\ 0&\text{if }t\leq 0,\end{cases}$ (4.10)
where the number $s>0$ and the function $\varphi(\mathbf{x})>0$ will be
decided later. Then
$\mathcal{L}W=t^{-s-2}e^{-\frac{\varphi}{t}}\Big{\\{}t\big{(}-s+\nabla\cdot(\underline{\mathbf{A}}\nabla\varphi)+{\bf
b}\cdot\underline{\mathbf{A}}\nabla\varphi\big{)}+\varphi-(\underline{\mathbf{A}}\nabla\varphi)\cdot\nabla\varphi\Big{\\}}.$
Thus, $\mathcal{L}W\leq 0$ if
$s\geq\nabla\cdot(\underline{\mathbf{A}}\nabla\varphi)+{\bf
b}\cdot\underline{\mathbf{A}}\nabla\varphi\quad\text{and}\quad\varphi\leq(\underline{\mathbf{A}}\nabla\varphi)\cdot\nabla\varphi.$
(4.11)
We will choose $\varphi$ to satisfy
$\underline{\mathbf{A}}\nabla\varphi=\kappa_{0}\mathbf{x},$ (4.12)
where $\kappa_{0}$ is a positive constant selected later. Equivalently, with
the use of (3.39),
$\nabla\varphi=\kappa_{0}\underline{\mathbf{A}}^{-1}\mathbf{x}=\kappa_{0}\underline{\mathbf{B}}\mathbf{x}=\kappa_{0}\phi(|\mathbf{x}|)\mathbf{x},$
(4.13)
where $\phi(r)$ is defined by (3.40). By (3.41), (4.3) and (4.4),
$\phi(r)\geq d_{0}\mu_{2}=C_{1}\quad\text{for }r_{0}\leq r\leq R.$ (4.14)
By (3.42), (4.3) and (4.4),
$\phi(r)\leq d_{4}\mu_{1}=C_{0}\quad\text{for }r_{0}\leq r\leq R.$ (4.15)
Define for $\mathbf{x}\in\bar{\mathscr{U}}$ the function
$\varphi(\mathbf{x})=\kappa_{0}\Big{(}\varphi_{0}+\int_{r_{0}}^{|\mathbf{x}|}r\phi(r)dr\Big{)},\quad\text{where
}\varphi_{0}=\frac{C_{0}r_{0}^{2}}{2}\text{ and
}\kappa_{0}=\frac{C_{0}}{2C_{1}}.$ (4.16)
Then $\varphi(\mathbf{x})$ satisfies both equations (4.12) and (4.13). We have
for $\mathbf{x}\in\bar{\mathscr{U}}$ that
$0<\varphi(\mathbf{x})\leq\kappa_{0}\Big{(}\varphi_{0}+C_{0}\int_{r_{0}}^{|\mathbf{x}|}rdr\Big{)}=\frac{\kappa_{0}C_{0}}{2}|\mathbf{x}|^{2}.$
(4.17)
Applying (4.12), (4.13), and then (3.31) and (4.4) we obtain
$(\underline{\mathbf{A}}\nabla\varphi)\cdot\nabla\varphi=\kappa_{0}^{2}\mathbf{x}^{T}\underline{\mathbf{B}}\mathbf{x}\geq
d_{0}\kappa_{0}^{2}\Big{(}\sum_{i=1}^{2}F_{i}(\hat{S}(|\mathbf{x}|))\Big{)}|\mathbf{x}|^{2}\geq
d_{0}\kappa_{0}^{2}\mu_{2}|\mathbf{x}|^{2}=\kappa_{0}^{2}C_{1}|\mathbf{x}|^{2}=\frac{\kappa_{0}C_{0}}{2}|\mathbf{x}|^{2}$
(4.18)
Then we have from (4.17) and (4.18) that
$\varphi\leq(\underline{\mathbf{A}}\nabla\varphi)\cdot\nabla\varphi$ in
$\mathscr{U}$, which is the second requirement in (4.11). On the other hand,
by (4.12) and (4.6),
$\nabla\cdot(\underline{\mathbf{A}}\nabla\varphi)+{\bf
b}\cdot\underline{\mathbf{A}}\nabla\varphi=\kappa_{0}(\nabla\cdot\mathbf{x}+{\bf
b}\cdot\mathbf{x})\leq\kappa_{0}(n+C_{2}R).$ (4.19)
Select
$s=s_{R}\mathbin{\buildrel\rm
def\over{\mathbin{=\kern-2.0pt=}}}\kappa_{0}(n+C_{2}R).$ (4.20)
Then we have
$s\geq\nabla\cdot(\underline{\mathbf{A}}\nabla\varphi)+\mathbf{b}\cdot(\underline{\mathbf{A}}\nabla\varphi)$
in $\mathscr{U}$, which is the first requirement in (4.11). Thus, we obtain
$\mathcal{L}W\leq 0$ in $\mathscr{U}\times(0,\infty)$. For further references,
we formulate this as a lemma.
###### Lemma 4.2.
With parameter $s=s_{R}$ selected as in (4.20) and function $\varphi$ defined
by (4.16), the function $W(\mathbf{x},t)$ in (4.10) belongs to
$C_{\mathbf{x},t}^{2,1}(\mathscr{D})\cap C(\bar{\mathscr{D}})$ and satisfies
$\mathcal{L}W\leq 0$ in $\mathscr{D}$.
Above, the regularity of $W(\mathbf{x},t)$ follows the fact that
$\varphi(\mathbf{x})\geq\kappa_{0}\varphi_{0}>0$ for
$\mathbf{x}\in\bar{\mathscr{U}}$.
We now establish this section’s key lemma of growth. We fix $s=s_{R}$ by
(4.20) and also the following two parameters
$q=\frac{\kappa_{0}C_{0}}{2s}\quad\text{and}\quad\eta_{0}=\Big{(}\frac{r_{0}}{R}\Big{)}^{2s},$
(4.21)
and denote $D_{1}=U\times(0,qR^{2}]$.
###### Lemma 4.3 (Lemma of growth in time).
Assume $w(\mathbf{x},t)\in C_{\mathbf{x},t}^{2,1}(D_{1})\cap C(\bar{D}_{1})$.
If
$\mathcal{L}w\leq 0\text{ on }D_{1}\quad\text{ and }\quad w\leq 0\text{ on
}\Gamma\times(0,qR^{2}),$ (4.22)
then
$\max\\{0,\sup_{U}w(\mathbf{x},qR^{2})\\}\leq\frac{1}{1+\eta_{0}}\max\\{0,\sup_{U}w(\mathbf{x},0)\\}.$
(4.23)
###### Proof.
(i) Let $M=\max\\{0,\sup_{\bar{D}_{1}}w\\}$. By (4.22) and maximum principle,
we have
$M=\max\\{0,\sup_{\bar{U}}w(\mathbf{x},0)\\}.$ (4.24)
Let $W(\mathbf{x},t)$ be as in (4.10) and define the auxiliary function
$\tilde{W}(\mathbf{x},t)=M[1-\eta W(\mathbf{x},t)],$
where constant $\eta>0$ will be specified later. Our intention is to prove
that
$\tilde{W}(\mathbf{x},t)\geq w(\mathbf{x},t)\quad\text{for all
}(\mathbf{x},t)\in\bar{D}_{1}.$ (4.25)
By Lemma 4.2, $\mathcal{L}W\leq 0$ in $D_{1}$, hence,
$\mathcal{L}\tilde{W}\geq 0$ in $D_{1}.$ By maximum principle, it suffices to
show that
$\tilde{W}(\mathbf{x},t)\geq w(\mathbf{x},t)\quad\text{for all
}(\mathbf{x},t)\in\partial_{p}D_{1}=\big{[}\bar{U}\times\\{0\\}\big{]}\cup\big{[}\Gamma\times(0,qR^{2}]\big{]}.$
(4.26)
On the base $\bar{U}\times\left\\{0\right\\}$, function $W(\mathbf{x},0)$
vanishes, hence,
$\tilde{W}(\mathbf{x},0)=M\geq w(\mathbf{x},0).$
On the side boundary $\Gamma\times(0,qR^{2}]$, additional analysis is
required. First observe for $\mathbf{x}\in\bar{\mathscr{U}}$ that
$\varphi(\mathbf{x})\geq\kappa_{0}\varphi_{0}=\frac{\kappa_{0}C_{0}r_{0}^{2}}{2}$.
Therefore,
$\tilde{W}(\mathbf{x},t)=M\left[1-\eta
t^{-s}e^{-\frac{\varphi(\mathbf{x})}{t}}\right]\geq M\left[1-\eta
t^{-s}e^{-\frac{\kappa_{0}C_{0}r_{0}^{2}}{2t}}\right]\quad\text{in
}\bar{\mathscr{U}}\times[0,\infty).$ (4.27)
Let $h_{0}(t)=t^{-s}e^{-\frac{\kappa_{0}C_{0}r_{0}^{2}}{2t}}$ for $t\geq 0$.
By elementary calculations, the maximum of $h_{0}(t)$ is attained at
$t_{0}=\frac{\kappa_{0}C_{0}r_{0}^{2}}{2s}$. By letting
$\eta=\frac{1}{\max_{[0,\infty)}h_{0}(t)}=\frac{1}{h_{0}(t_{0})}=\Big{(}\frac{e\kappa_{0}C_{0}r_{0}^{2}}{2s}\Big{)}^{s},$
(4.28)
we get from (4.27) that $\tilde{W}(\mathbf{x},t)\geq M[1-\eta h_{0}(t_{0})]=0$
in $\bar{\mathscr{U}}\times[0,\infty)$. Particularly,
$\tilde{W}(\mathbf{x},t)\geq 0\geq w(\mathbf{x},t)\quad\text{on
}\Gamma\times(0,qR^{2}].$
Thus, the comparison in (4.26) holds and, therefore, (4.25) is proved.
We now estimate $\tilde{W}(\mathbf{x},t)$. By (4.17), for $(\mathbf{x},t)\in
D$ we have
$\displaystyle\tilde{W}(\mathbf{x},t)\leq M\left[1-\eta
t^{-s}e^{-\frac{\kappa_{0}C_{0}|\mathbf{x}|^{2}}{2t}}\right]\leq M\left[1-\eta
t^{-s}e^{-\frac{\kappa_{0}C_{0}R^{2}}{2t}}\right].$
Let $h_{1}(t)=t^{-s}e^{-\frac{\kappa_{0}C_{0}R^{2}}{2t}}$ for $t>0$. Then
$t_{1}=\frac{\kappa_{0}C_{0}R^{2}}{2s}=qR^{2}$ is the critical point and
$h_{1}(t_{1})=(qR^{2})^{-s}e^{-\frac{\kappa_{0}C_{0}}{2q}}\geq\Big{(}\frac{2s}{e\kappa_{0}C_{0}R^{2}}\Big{)}^{s}.$
Letting $t=t_{1}$ in (4.25), we have
$w(\mathbf{x},t_{1})\leq\tilde{W}(\mathbf{x},t_{1})\leq
M\left[1-\eta\Big{(}\frac{2s}{e\kappa_{0}C_{0}R^{2}}\Big{)}^{s}\right]=M(1-\eta_{0})\leq\frac{M}{1+\eta_{0}},$
(4.29)
and, hence, (4.23) follows. ∎
Using Lemma 4.3, we show the decay, as $t\to\infty$, of solution
$w(\mathbf{x},t)$ of the IBVP (4.9) in the homogeneous case, i.e., when
$f_{0}\equiv 0$ and $G\equiv 0$.
###### Proposition 4.4 (Homogeneous problem).
Assume $w(\mathbf{x},t)\in C_{\mathbf{x},t}^{2,1}(D)\cap C(\bar{D})$ satisfies
$\mathcal{L}w=0\text{ in }D\quad\text{and}\quad w=0\text{ on
}\Gamma\times(0,\infty).$ (4.30)
Then
$-e^{-\eta_{1}t}\inf_{U}|w(\mathbf{x},0)|\leq
w(\mathbf{x},t)\leq(1+\eta_{0})e^{-\eta_{1}t}\sup_{U}|w(\mathbf{x},0)|\quad\forall(\mathbf{x},t)\in
D,$ (4.31)
where $\eta_{1}=\frac{\ln(1+\eta_{0})}{qR^{2}}$.
###### Proof.
Let $k\in\mathbb{N}$. Applying Lemma 4.3 with $D_{1}$ being replaced by
$U\times(T_{k-1},T_{k}]$ gives
$\max\\{0,\sup_{U}w(\mathbf{x},kqR^{2})\\}\leq\frac{1}{1+\eta_{0}}\max\\{0,\sup_{U}w(\mathbf{x},(k-1)qR^{2})\\}.$
By induction in $k$, we obtain
$\max\\{0,\sup_{U}w(\mathbf{x},kqR^{2})\\}\leq\frac{1}{(1+\eta_{0})^{k}}\max\\{0,\sup_{U}w(\mathbf{x},0)\\}.$
(4.32)
Now applying (4.32) to function $-w$ instead of $w$, we obtain
$\min\\{0,\inf_{U}w(\mathbf{x},kqR^{2})\\}\geq\frac{1}{(1+\eta_{0})^{k}}\min\\{0,\inf_{U}w(\mathbf{x},0)\\}.$
(4.33)
For any $t>0$, there is an integer $k\geq 0$ such that $t\in(T_{k},T_{k+1}]$
where $T_{k}=kqT^{2}$. By (4.30) and maximum principle for domain
$U\times(T_{k},T_{k+1}]$, and then using (4.32) we have
$\displaystyle w(\mathbf{x},t)$
$\displaystyle\leq\max\\{0,\sup_{U}w(\mathbf{x},T_{k})\\}\leq(1+\eta_{0})^{-k}\max\\{0,\sup_{U}w(\mathbf{x},0)\\}$
$\displaystyle=(1+\eta_{0})e^{-\eta_{1}T_{k+1}}\sup_{U}|w(\mathbf{x},0)|\leq(1+\eta_{0})e^{-\eta_{1}t}\sup_{U}|w(\mathbf{x},0)|.$
(4.34)
Similarly, using (4.33) instead of (4.32) we have
$\displaystyle w(\mathbf{x},t)$
$\displaystyle\geq\min\\{0,\inf_{U}w(\mathbf{x},T_{k})\\}\geq(1+\eta_{0})^{-k}\min\\{0,\inf_{U}w(\mathbf{x},0)\\}$
$\displaystyle\geq-e^{-\eta_{1}T_{k}}\inf_{U}|w(\mathbf{x},0)|\geq-e^{-\eta_{1}t}\inf_{U}|w(\mathbf{x},0)|.$
(4.35)
Therefore, (4.31) follows (4) and (4). ∎
Next, we consider the non-homogeneous case for the IBVP (4.9). Similar to
(4.2), we always consider
$w_{0}\in C(\bar{U}),\ G\in C(\Gamma\times[0,\infty))\text{ and
}w_{0}(\mathbf{x})=G(\mathbf{x},0)\text{ on }\Gamma.$ (4.36)
###### Proposition 4.5 (Non-homogeneous problem).
Assume $f_{0}\in C(\bar{D})$ and
$\Delta_{1}\mathbin{\buildrel\rm
def\over{\mathbin{=\kern-2.0pt=}}}\sup_{U\times(0,\infty)}|f_{0}(\mathbf{x},t)|+\sup_{\Gamma\times(0,\infty)}|G(\mathbf{x},t)|<\infty$
(4.37)
There is a positive constant $C$ such that if $w(\mathbf{x},t)\in
C_{\mathbf{x},t}^{2,1}(D)\cap C(\bar{D})$ is a solution of (4.9), then
$|w(\mathbf{x},t)|\leq
C\big{[}e^{-\eta_{1}t}\sup_{U}|w_{0}(\mathbf{x})|+\Delta_{1}\big{]}\quad\forall(\mathbf{x},t)\in
D,$ (4.38)
where $\eta_{1}>0$ is defined in Proposition 4.4.
###### Proof.
Denote $T_{k}=kqR^{2}$ for any integer $k\geq 0$. Let $k\in\mathbb{N}$ and
$v_{k}(\mathbf{x},t)=w(\mathbf{x},t)-\Delta_{1}(t-T_{k-1}+1)\quad\text{ for
}(\mathbf{x},t)\in\bar{U}\times[T_{k-1},T_{k}].$ (4.39)
Then $v_{k}$ satisfies
$\mathcal{L}v_{k}=\mathcal{L}w-\Delta_{1}\mathcal{=}f_{0}-\Delta_{1}\leq
0\quad\text{in }U\times(T_{k-1},T_{k}],$
and
$v_{k}(\mathbf{x},t)\leq 0\text{ on }\Gamma\times(T_{k-1},T_{k}).$
Applying Lemma 4.3 to function $v_{k}$, we have
$\max\\{0,\sup_{U}v_{k}(\mathbf{x},T_{k})\\}\leq\frac{1}{1+\eta_{0}}\max\\{0,\sup_{U}v_{k}(\mathbf{x},T_{k-1})\\}.$
(4.40)
Note that $v_{k}(\mathbf{x},T_{k})=w(\mathbf{x},T_{k})-\Delta_{1}(qR^{2}+1)$
and $v_{k}(\mathbf{x},T_{k-1})=w(\mathbf{x},T_{k-1})-\Delta_{1}\leq
w(\mathbf{x},T_{k-1})$. Hence,
$\displaystyle\max\\{0,\sup_{U}w(\mathbf{x},T_{k})\\}$
$\displaystyle\leq\max\\{0,\sup_{U}v_{k}(\mathbf{x},T_{k})\\}+\Delta_{1}(qR^{2}+1)$
$\displaystyle\leq\frac{1}{1+\eta_{0}}\max\\{0,\sup_{U}w(\mathbf{x},T_{k-1})\\}+\Delta_{1}(qR^{2}+1).$
Iterating this inequality gives
$\displaystyle\max\\{0,\sup_{U}w(\mathbf{x},T_{k})\\}$
$\displaystyle\leq\frac{1}{(1+\eta_{0})^{k}}\max\\{0,\sup_{U}w(\mathbf{x},0)\\}+\Delta_{1}(qR^{2}+1)\sum_{j=0}^{k-1}\frac{1}{(1+\eta_{0})^{j}}$
(4.41)
$\displaystyle\leq\frac{1}{(1+\eta_{0})^{k}}\max\\{0,\sup_{U}w(\mathbf{x},0)\\}+\frac{\Delta_{1}(1+qR^{2})(1+\eta_{0})}{\eta_{0}}.$
By using the relation (4.39) between $v_{k}(\mathbf{x},t)$ and
$w(\mathbf{x},t)$, maximum principle for function $v_{k}(\mathbf{x},t)$, and
estimate (4.41), we have for any $t\in[T_{k-1},T_{k}]$ with $k\geq 1$ that
$\displaystyle w(\mathbf{x},t)$ $\displaystyle\leq
v_{k}(\mathbf{x},t)+\Delta_{1}(1+qR^{2})\leq\max\\{0,\sup_{U}w(\mathbf{x},T_{k-1})\\}+\Delta_{1}(1+qR^{2})$
$\displaystyle\leq(1+\eta_{0})^{-k+1}\max\\{0,\sup_{U}w(\mathbf{x},0)\\}+\frac{\Delta_{1}(1+qR^{2})(1+\eta_{0})}{\eta_{0}}+\Delta_{1}(1+qR^{2})$
$\displaystyle\leq(1+\eta_{0})^{-\frac{t}{qR^{2}}+1}\sup_{U}|w(\mathbf{x},0)|+\frac{2\Delta_{1}(1+qR^{2})(1+\eta_{0})}{\eta_{0}}.$
Therefore,
$w(\mathbf{x},t)\leq
C\big{[}e^{-\eta_{1}t}\sup_{U}|w(\mathbf{x},0)|+\Delta_{1}\big{]}.$ (4.42)
Similarly, we obtain the same estimate for $(-w)$ and hence, (4.38) follows. ∎
For the asymptotic behavior of $w(\mathbf{x},t)$ as $t\to\infty$, we have the
following.
###### Corollary 4.6.
Assume $f_{0}\in C(\bar{D})$ is bounded and
$\Delta_{2}\mathbin{\buildrel\rm
def\over{\mathbin{=\kern-2.0pt=}}}\limsup_{t\to\infty}\left[\sup_{\mathbf{x}\in
U}|f_{0}(\mathbf{x},t))|+\sup_{\mathbf{x}\in\Gamma}|G(\mathbf{x},t)|\right]<\infty.$
(4.43)
There exists $C=C(\eta_{0},q,R,M)>0$ such that if $w(\mathbf{x},t)\in
C_{\mathbf{x},t}^{2,1}(D)\cap C(\bar{D})$ solves (4.9), then
$\limsup_{t\to\infty}\left[\sup_{\mathbf{x}\in U}|w(\mathbf{x},t)|\right]\leq
C\Delta_{2}.$ (4.44)
###### Proof.
Note that
$\sup_{U}|w_{0}(\mathbf{x})|+\sup_{D}|f_{0}(\mathbf{x},t)|+\sup_{\Gamma\times(0,\infty)}|G(\mathbf{x},t)|<\infty.$
Then by Proposition 4.5, $w(\mathbf{x},t)$ is bounded on $\bar{D}$. Let
$\varepsilon>0$. From our assumption there is $t_{0}>0$ such that
$\sup_{U\times[t_{0},\infty)}|f_{0}(\mathbf{x},t))|+\sup_{\Gamma\times[t_{0},\infty)}|G(\mathbf{x},t)|<\Delta_{2}+\varepsilon.$
Applying Lemma 4.5 to the domain $U\times(t_{0},\infty)$ we obtain
$|w(\mathbf{x},t)|\leq C[e^{-\eta_{1}(t-t_{0})}\sup_{\mathbf{x}\in
U}|w(\mathbf{x},t_{0})|+\Delta_{2}+\varepsilon].$ (4.45)
Therefore, passing $t\to\infty$ and then $\varepsilon\to 0$ in (4.45) yields
(4.44). ∎
Next, we estimate $|\nabla w(\mathbf{x},t)|$ by using Bernstein’s technique
(c.f. [16]).
###### Proposition 4.7.
Assume $f_{0}\in C(\bar{D})$, $\nabla f_{0}\in C(D)$, (4.37) and
$\Delta_{3}\mathbin{\buildrel\rm
def\over{\mathbin{=\kern-2.0pt=}}}\sup_{D}|\nabla f_{0}|<\infty.$ (4.46)
For any $U^{\prime}\Subset U$ there is $\tilde{M}>0$ such that if
$w(\mathbf{x},t)\in C_{\mathbf{x},t}^{2,1}(D)\cap C(\bar{D})$ is a solution of
(4.9) that also satisfies $w\in C_{\mathbf{x}}^{3}(D)$ and $w_{t}\in
C_{\mathbf{x}}^{1}(D)$, then
$|\nabla
w(\mathbf{x},t)|\leq\tilde{M}\Big{[}1+\frac{1}{\sqrt{t}}\Big{]}\Big{[}e^{-\eta_{1}t}\sup_{U}|w(\mathbf{x},0)|+\Delta_{1}+\sqrt{\Delta}_{3}\Big{]}\quad\forall(\mathbf{x},t)\in
U^{\prime}\times(0,\infty).$ (4.47)
###### Proof.
Note that $\nabla w\in C_{\mathbf{x},t}^{2,1}(D)$. By using finite covering of
compact set $U^{\prime}$, it suffices to prove (4.47) for $\mathbf{x}$ in some
ball inside $U$. Consider a ball
$B_{{\delta}}(\mathbf{x}_{*})=\\{\mathbf{x}:|\mathbf{x}-\mathbf{x}^{*}|\leq\delta\\}\Subset
U$ with some $\mathbf{x}_{*}\in U$ and $\delta>0$. Let $t_{0}>0$, define in
the cylinder $G_{\delta}\mathbin{\buildrel\rm
def\over{\mathbin{=\kern-2.0pt=}}}B_{\delta}(\mathbf{x}_{*})\times(t_{0},1+t_{0}]$
the following auxiliary function
$\tilde{w}(\mathbf{x},t)=\tau\Phi(\mathbf{x})|\nabla
w|^{2}+Nw^{2}+N_{1}(1+t_{0}-t),$ (4.48)
where
$\tau=t-t_{0}\in(0,1],\quad\Phi(\mathbf{x})=(\delta^{2}-|\mathbf{x}-\mathbf{x}_{*}|^{2})^{2}.$
(4.49)
The numbers $N,N_{1}\geq 0$ will be chosen later. We rewrite the operator
$\mathcal{L}$ as
$\mathcal{L}w=w_{t}-\sum_{i,j=1}^{n}a_{ij}(\mathbf{x})\partial_{i}\partial_{j}w-\mathbf{\tilde{b}}\cdot\nabla
w,$ (4.50)
where
$\mathbf{\tilde{b}}(\mathbf{x})=(\tilde{b}_{1},\tilde{b}_{2},\ldots,\tilde{b}_{n})\mathbin{\buildrel\rm
def\over{\mathbin{=\kern-2.0pt=}}}\nabla\cdot\underline{\mathbf{A}}+\underline{\mathbf{A}}\mathbf{b}$.
Then following the calculations in Theorem 1 of section 2 on page 450 in [16]
we have
$\displaystyle\mathcal{L}\tilde{w}$ $\displaystyle\leq
2\tau\Phi\Big{\\{}\sum_{i,j,k=1}^{n}\frac{\partial{a_{ij}}}{\partial{x_{k}}}\frac{\partial{w}}{\partial{x_{k}}}\frac{\partial^{2}{w}}{\partial{x_{i}}\partial{x_{j}}}+\sum_{i,k=1}^{n}\frac{\partial{\tilde{b}_{i}}}{\partial{x_{k}}}\frac{\partial{w}}{\partial{x_{k}}}\frac{\partial{w}}{\partial{x_{i}}}-\sum_{i,j,k=1}^{n}a_{ij}\frac{\partial^{2}{w}}{\partial{x_{k}}\partial{x_{i}}}\frac{\partial^{2}{w}}{\partial{x_{k}}\partial{x_{j}}}\Big{\\}}$
(4.51) $\displaystyle\quad-(\tau\mathcal{L}(\Phi)-\Phi)|\nabla
w|^{2}-4\tau\sum_{i,j,k=1}^{n}a_{ij}\frac{\partial{\Phi}}{\partial{x_{i}}}\frac{\partial{w}}{\partial{x_{k}}}\frac{\partial^{2}{w}}{\partial{x_{k}}\partial{x_{j}}}-2N\sum_{i,j=1}^{n}a_{ij}\frac{\partial{w}}{\partial{x_{i}}}\frac{\partial{w}}{\partial{x_{j}}}$
$\displaystyle\quad-2\tau\Phi\sum_{k=1}^{n}\frac{\partial f_{0}}{\partial
x_{k}}-2Nwf_{0}-N_{1}.$
We estimate the right-hand side of (4.51) term by term. Let $\varepsilon>0$.
The numbers $K_{i}$, for $i=1,2,3\ldots$, used in the calculations below are
all positive and independent of $w$. We denote the matrix of second
derivatives of $w$ by $\nabla^{2}w$, and denote its Euclidean norm by
$|\nabla^{2}w|$. Note that $\underline{\mathbf{A}}$, $\mathbf{b}$ and
$\mathbf{\tilde{b}}$ are bounded in $B_{\delta}(\mathbf{x}^{*})$. This and
Cauchy-Schwarz inequality imply
$\displaystyle
2\tau\Phi\sum_{i,j,k=1}^{n}\frac{\partial{a_{ij}}}{\partial{x_{k}}}\frac{\partial{w}}{\partial{x_{k}}}\frac{\partial^{2}{w}}{\partial{x_{i}}\partial{x_{j}}}\leq
2C\tau\Phi|\nabla w||\nabla^{2}w|^{2}\leq\varepsilon^{-1}K_{1}|\nabla
w|^{2}+2\varepsilon\tau\Phi|\nabla^{2}w|^{2},$
$\displaystyle-(\tau\mathcal{L}(\Phi)-\Phi)|\nabla
w|^{2}+2\tau\Phi\sum_{i,k=1}^{n}\frac{\partial{\tilde{b}_{i}}}{\partial{x_{k}}}\frac{\partial{w}}{\partial{x_{k}}}\frac{\partial{w}}{\partial{x_{i}}}\leq
K_{2}|\nabla w|^{2}.$
Since $\underline{\mathbf{A}}$ is positive definite,
$\sum_{i,j,k=1}^{n}a_{ij}\frac{\partial^{2}{w}}{\partial{x_{k}}\partial{x_{i}}}\frac{\partial^{2}{w}}{\partial{x_{k}}\partial{x_{j}}}\geq
K_{3}|\nabla^{2}w|^{2},\quad\sum_{i,j=1}^{n}a_{ij}\frac{\partial{w}}{\partial{x_{i}}}\frac{\partial{w}}{\partial{x_{j}}}\geq
K_{3}|\nabla w|^{2}.$
Also, we have
$-4\tau\sum_{i,j,k=1}^{n}a_{ij}\frac{\partial{\Phi}}{\partial{x_{i}}}\frac{\partial{w}}{\partial{x_{k}}}\frac{\partial^{2}{w}}{\partial{x_{k}}\partial{x_{j}}}\leq\varepsilon^{-1}K_{4}|\nabla
w|^{2}+2\varepsilon\tau|\nabla\Phi|^{2}|\nabla^{2}w|^{2},$
$-2\tau\Phi\sum_{k=1}^{n}\frac{\partial f_{0}}{\partial x_{k}}\leq
K_{5}\Delta_{3},$
and by using estimate (4.38) for $w$,
$-2Nwf_{0}\leq
K_{6}\Delta_{1}N\big{[}e^{-\eta_{1}t_{0}}\sup_{U}|w(\mathbf{x},0)|+\Delta_{1}\big{]}.$
Combining the above estimates, we obtain from (4.51) that
$\displaystyle\mathcal{L}\tilde{w}$ $\displaystyle\leq
2\tau\Phi\Big{(}2\varepsilon+\varepsilon\frac{|\nabla\Phi|^{2}}{\Phi}-K_{3}\Big{)}|\nabla^{2}w|^{2}+\Big{(}K_{2}+\varepsilon^{-1}(K_{1}+K_{4})-2NK_{3}\Big{)}|\nabla
w|^{2}$
$\displaystyle\quad+K_{5}\Delta_{3}+K_{6}\Delta_{1}N\big{[}e^{-\eta_{1}t_{0}}\sup_{U}|w(\mathbf{x},0)|+\Delta_{1}\big{]}-N_{1}.$
Since $|\nabla\Phi|^{2}/\Phi\leq 16\delta^{2}$, we have
$\displaystyle\mathcal{L}\tilde{w}$ $\displaystyle\leq
2\tau\Phi\Big{(}K_{7}\varepsilon-
K_{3}\Big{)}|\nabla^{2}w|^{2}+\Big{(}K_{2}+K_{8}\varepsilon^{-1}-2NK_{3}\Big{)}|\nabla
w|^{2}$ (4.52)
$\displaystyle\quad+(K_{5}+K_{6}N)\big{[}\Delta_{3}+\Delta_{1}e^{-\eta_{1}t_{0}}\sup_{U}|w(\mathbf{x},0)|+\Delta_{1}^{2}\big{]}-N_{1}.$
In (4.52), choose $\varepsilon=K_{3}/K_{7}$ and
$N=[K_{2}+K_{8}\varepsilon^{-1}]/(2K_{3})$, then take
$N_{1}=(K_{5}+K_{6}N)(\Delta_{1}e^{-\eta_{1}t_{0}}\sup_{U}|w(\mathbf{x},0)|+\Delta_{1}^{2}+\Delta_{3}).$
We find that $\mathcal{L}\tilde{w}\leq 0$ in $G_{\delta}$. Applying the
maximum principle gives
$\max_{\bar{G}_{\delta}}\tilde{w}=\max\big{\\{}\tilde{w}(\mathbf{x},t):(\mathbf{x},t)\in
B_{\delta}(\mathbf{x}_{*})\times\\{t_{0}\\}\cup\partial
B_{\delta}(\mathbf{x}_{*})\times[t_{0},t_{0}+1]\big{\\}}.$ (4.53)
Note that $\tau\Phi(\mathbf{x})=0$ when $t=t_{0}$ or $\mathbf{x}\in\partial
B_{\delta}(\mathbf{x}_{*})$. Hence (4.53) implies,
$\max_{\bar{G}_{\delta}}\tilde{w}\leq
N\max_{B_{\delta}(\mathbf{x}_{*})}w^{2}(\mathbf{x},t_{0})+N\max_{\partial
B_{\delta}(\mathbf{x}_{*})\times[t_{0},t_{0}+1]}w^{2}(\mathbf{x},t)+N_{1}.$
(4.54)
Using estimate (4.38) for the first two terms on the right-hand side of (4.54)
we obtain
$\displaystyle\max_{\bar{G}_{\delta}}\tilde{w}$ $\displaystyle\leq
2K_{9}N\Big{[}e^{-\eta_{1}t_{0}}\sup_{U}|w(\mathbf{x},0)|+\Delta_{1}\Big{]}^{2}+N_{1}\leq
K_{10}\Big{[}e^{-2\eta_{1}t_{0}}\sup_{U}|w(\mathbf{x},0)|^{2}+\Delta_{1}^{2}+\Delta_{3}\Big{]}$
$\displaystyle\leq
C\Big{[}e^{-2\eta_{1}t}\sup_{U}|w(\mathbf{x},0)|^{2}+\Delta_{1}^{2}+\Delta_{3}\Big{]}.$
Now, we consider $\mathbf{x}\in B_{\delta/2}(\mathbf{x}_{*})$. If $t\in(0,1]$
let $t_{0}=t/2$, then $t=2t_{0}\in[t_{0},1+t_{0}]$ and hence
$\frac{t}{2}|\nabla
w(\mathbf{x},t)|^{2}\min_{B_{\delta/2}(\mathbf{x}_{*})}\Phi(\mathbf{x})\leq(t-t_{0})\Phi(\mathbf{x})|\nabla
w(\mathbf{x},t)|^{2}\\\ \leq\tilde{w}(\mathbf{x},t)\leq
C\Big{[}e^{-2\eta_{1}t}\sup_{U}|w(\mathbf{x},0)|+\Delta_{1}^{2}+\Delta_{3}\Big{]}.$
(4.55)
If $t>1$ let $t_{0}=t-1/2$, then $t\in[t_{0},1+t_{0}]$ and hence
$\frac{1}{2}|\nabla
w(\mathbf{x},t)|^{2}\min_{B_{\delta/2}(\mathbf{x}_{*})}\Phi(\mathbf{x})\leq(t-t_{0})\Phi(\mathbf{x})|\nabla
w(\mathbf{x},t)|^{2}\\\ \leq\tilde{w}(\mathbf{x},t)\leq
C\Big{[}e^{-2\eta_{1}t}\sup_{U}|w(\mathbf{x},0)|+\Delta_{1}^{2}+\Delta_{3}\Big{]}.$
(4.56)
Since $\min_{B_{\delta/2}(\mathbf{x}_{*})}\Phi(\mathbf{x})>0$, it follows
(4.55) and (4.56) that
$|\nabla w(\mathbf{x},t)|\leq
M(\delta)\Big{[}1+\frac{1}{\sqrt{t}}\Big{]}\Big{[}e^{-\eta_{1}t}\sup_{U}|w(\mathbf{x},0)|+\Delta_{1}+\sqrt{\Delta_{3}}\Big{]}$
(4.57)
for $\mathbf{x}\in B_{\delta/2}(\mathbf{x}_{*})$ and $t>0$. Then using a
finite covering of $U^{\prime}$, we obtain (4.47) from (4.57). ∎
We return to the IBVP (4.1) for $\sigma(\mathbf{x},t)$ now. Recall that the
existence and uniqueness of the solution $\sigma$ were already addressed in
Theorem 4.1.
###### Theorem 4.8.
Assume (E1) and
$\Delta_{4}\mathbin{\buildrel\rm
def\over{\mathbin{=\kern-2.0pt=}}}\sup_{D}(|\mathbf{V}(\mathbf{x},t)|+|\nabla\mathbf{V}(\mathbf{x},t)|)+\sup_{\Gamma\times[0,\infty)}|g(\mathbf{x},t)|<\infty.$
(4.58)
Then the solution $\sigma(\mathbf{x},t)$ of the IBVP (4.1) satisfies
$\sup_{\mathbf{x}\in U}|\sigma(\mathbf{x},t)|\leq
C\Big{[}e^{-\eta_{1}t}\sup_{U}|\sigma_{0}(\mathbf{x})|+\Delta_{4}\Big{]}\quad\text{for
all }t>0.$ (4.59)
Moreover,
$\limsup_{t\to\infty}\left[\sup_{\mathbf{x}\in
U}|\sigma(\mathbf{x},t)|\right]\leq C\Delta_{5},$ (4.60)
where
$\Delta_{5}=\limsup_{t\to\infty}\Big{[}\sup_{\mathbf{x}\in
U}(|\mathbf{V}(\mathbf{x},t)|+|\nabla\mathbf{V}(\mathbf{x},t)|)+\sup_{\mathbf{x}\in\Gamma}|g(\mathbf{x},t)|\Big{]}.$
(4.61)
###### Proof.
Let $w(\mathbf{x},t)=\sigma(\mathbf{x},t)e^{-\Lambda(\mathbf{x})}$,
$f_{0}(\mathbf{x},t)=e^{-\Lambda(\mathbf{x})}\nabla\cdot(\underline{\mathbf{A}}(\mathbf{x})\mathbf{c}(\mathbf{x},t))$,
$G(\mathbf{x},t)=e^{-\Lambda(\mathbf{x})}g(\mathbf{x},t)$ and
$w_{0}(\mathbf{x})=e^{-\Lambda(\mathbf{x})}\sigma_{0}(\mathbf{x})$. Then
$w(\mathbf{x},t)$ solves (4.9). We observe from (3.38) that
$|\Lambda(\mathbf{x})|\leq
d_{2}\mu_{3}(R-r_{0})\quad\forall\mathbf{x}\in\mathscr{U}.$ (4.62)
Combining with the boundedness of
$\|\underline{\mathbf{A}}\|_{C^{1}(\mathscr{U})}$, we have
$|f_{0}(\mathbf{x},t)|\leq
C(|\mathbf{V}(\mathbf{x},t)|+|\nabla\mathbf{V}(\mathbf{x},t)|)\quad\forall(\mathbf{x},t)\in
D.$ (4.63)
Thanks to these relations, the assumptions in Proposition 4.5 hold, thus, the
assertions (4.59) and (4.60) follow directly from (4.38) and (4.44). ∎
For the velocities, we have the following result.
###### Theorem 4.9.
Assume (E1) and
$\Delta_{6}\mathbin{\buildrel\rm
def\over{\mathbin{=\kern-2.0pt=}}}\sup_{D}(|\mathbf{V}(\mathbf{x},t)|+|\nabla\mathbf{V}(\mathbf{x},t)|+|\nabla^{2}\mathbf{V}(\mathbf{x},t)|)<\infty\text{
and }\Delta_{7}\mathbin{\buildrel\rm
def\over{\mathbin{=\kern-2.0pt=}}}\sup_{\Gamma\times[0,\infty)}|g(\mathbf{x},t)|<\infty.$
(4.64)
Then for any $U^{\prime}\Subset U$, there is a positive number $\tilde{M}$
such that for $i=1,2$, and $t>0$,
$\sup_{\mathbf{x}\in
U^{\prime}}|\mathbf{v}_{i}(\mathbf{x},t)|\leq\tilde{M}\Big{(}1+\frac{1}{\sqrt{t}}\Big{)}\Big{[}e^{-\eta_{1}t}\sup_{U}|\sigma_{0}(\mathbf{x})|+\Delta_{6}+\sqrt{\Delta}_{6}+\Delta_{7}\Big{]}.$
(4.65)
Consequently, if
$\lim_{t\to\infty}\Big{\\{}\sup_{\mathbf{x}\in
U}(|\mathbf{V}(\mathbf{x},t)|+|\nabla\mathbf{V}(\mathbf{x},t)|+|\nabla^{2}\mathbf{V}(\mathbf{x},t)|)+\sup_{\mathbf{x}\in\Gamma}|g(\mathbf{x},t)|\Big{\\}}=0,$
(4.66)
then for any $\mathbf{x}\in U$,
$\lim_{t\to\infty}\mathbf{v}_{1}(\mathbf{x},t)=\lim_{t\to\infty}\mathbf{v}_{2}(\mathbf{x},t)=0.$
(4.67)
###### Proof.
Note that solution $\sigma(\mathbf{x},t)$ of (4.1) satisfies $\sigma\in
C_{\mathbf{x}}^{3}(D)$ and $\sigma_{t}\in C_{\mathbf{x}}^{2}(D)$. Let
$w,f_{0},G,w_{0}$ be the same as in Theorem 4.8. Using the estimate of $\nabla
w$ in Lemma 4.7 and formula (3.19), we easily obtain estimate (4.65) for
$\mathbf{v}_{2}$. Then the estimate for $\mathbf{v}_{1}$ follows this and
(3.4). The proof of (4.67) is similar to that of (4.44). We take
$U^{\prime}=B_{\delta}(\mathbf{x})$ such that $U^{\prime}\Subset U$. For
$T>0$, let
$\Delta_{6,T}=\sup_{U\times[T,\infty)}(|\mathbf{V}(\mathbf{x},t)|+|\nabla\mathbf{V}(\mathbf{x},t)|+|\nabla^{2}\mathbf{V}(\mathbf{x},t)|)\text{
and }\Delta_{7,T}=\sup_{\Gamma\times[T,\infty)}|g(\mathbf{x},t)|.$
Use (4.65) for all $t>T$ and $\Delta_{6,T}$, $\Delta_{7,T}$ in place of
$\Delta_{6}$, $\Delta_{7}$. Then let $T\to\infty$ noting that $\Delta_{6,T}\to
0$ and $\Delta_{7,T}\to 0$. ∎
###### Remark 4.10.
The key ingredient of the above asymptotic results is Lemma 4.3, the lemma of
growth in time. It is worth mentioning that this result can be extended to
more general parabolic equations in more general domains $D$ in
$\mathbb{R}^{n+1}$ rather than just cylindrical-in-time domains
$D=U\times(0,\infty)$.
## 5\. Case of unbounded domain
We will analyze the linear stability of the steady flows from section 2 in an
unbounded, outer domain $U=\mathbb{R}^{n}\setminus\bar{\Omega}$, where
$\Omega$ is a simply connected, open, bounded set containing the origin. To
emphasize the ideas and techniques, we consider the simple case
$\Omega=B_{r_{0}}$ for some $r_{0}>0$.
For $R>r>0$, denote $\mathcal{O}_{r}=\mathbb{R}^{n}\setminus\bar{B}_{r}$,
$\mathcal{O}_{r,R}=B_{R}\setminus\bar{B}_{r}$, and denote their closures by
$\bar{\mathcal{O}}_{r}$ and $\bar{\mathcal{O}}_{r,R}$, respectively. Then
$U=\mathcal{O}_{r_{0}}$. Let $\Gamma=\partial
U=\\{\mathbf{x}:|\mathbf{x}|=r_{0}\\}$ and $D=U\times(0,\infty)$.
For $T>0$ we denote $U_{T}=U\times(0,T]$, then its closure is
$\bar{U}_{T}=\bar{U}\times[0,T]$ and its parabolic boundary is
$\partial_{p}U_{T}=[\bar{U}\times\\{0\\}]\cup[\Gamma\times(0,T]]$.
Same as in section 4, we consider a steady state
$(u^{*}_{1}(\mathbf{x}),u^{*}_{2}(\mathbf{x}),S_{*}(\mathbf{x}))$ in (3.2)
with $c_{1}^{2}+c_{2}^{2}>0$ and $\hat{S}(r)$ exists for all $r\geq r_{0}$. We
assume throughout this section that
$0<\underline{s}\leq\hat{S}(r)\leq\bar{s}<1\quad\forall r\geq r_{0},\text{
where }\underline{s},\bar{s}=const.$ (5.1)
For instance, in one of the cases in Theorem 2.2 if the limit
$s_{\infty}\mathbin{\buildrel\rm
def\over{\mathbin{=\kern-2.0pt=}}}\lim_{r\to\infty}\hat{S}(r)$, which exists
according to Theorem 2.3, belongs to the interval $(0,1)$ then (5.1) holds.
The problems of our interest are (4.1) and its transformed form (4.9).
Let $\mu_{i}$, for $i=1,2,3$, and $C_{j}$, for $j=0,1,2$, be defined as in
section 4 (see (4.4), (4.5) and (4.6)). Thanks to condition (5.1), which plays
the role of (4.3) in section 4, the main properties (4.5), (4.6) and (4.7)
still hold with $\mathscr{U}=\mathcal{O}_{r_{0},R}$ being replaced by
$\mathscr{U}=U=\mathcal{O}_{r_{0}}$.
### 5.1. Maximum principle for unbounded domain
We establish the maximum principle for equation $\mathcal{L}w=0$ in the domain
$U$ with operator $\mathcal{L}$ defined by (4.8). For $T>0$, we construct a
barrier function $W(\mathbf{x},t)$ of the form:
$W(\mathbf{x},t)\mathbin{\buildrel\rm
def\over{\mathbin{=\kern-2.0pt=}}}(T-t)^{-s}e^{\frac{\varphi(\mathbf{x})}{T-t}}\quad\text{for
}(\mathbf{x},t)\in\mathcal{O}_{r_{0},R}\times(0,T),$ (5.2)
where constant $s>0$ and function $\varphi(\mathbf{x})>0$ will be decided
later. Elementary calculations give
$\mathcal{L}W=(T-t)^{-s-2}e^{\frac{\varphi}{T-t}}\Big{\\{}(T-t)\big{(}s-\nabla\cdot(\underline{\mathbf{A}}\nabla\varphi)-{\bf
b}\cdot\underline{\mathbf{A}}\nabla\varphi\big{)}+\varphi-(\underline{\mathbf{A}}\nabla\varphi)\cdot\nabla\varphi\Big{\\}}.$
Then $\mathcal{L}W\geq 0$ if
$s\geq\nabla\cdot(A\nabla\varphi)+{\bf
b}\cdot\underline{\mathbf{A}}\nabla\varphi\quad\text{and}\quad\varphi\geq(\underline{\mathbf{A}}\nabla\varphi)\cdot\nabla\varphi.$
(5.3)
Similar to section 4, we choose
$\varphi(\mathbf{x})=\kappa_{1}\Big{(}\varphi_{1}+\int_{r_{0}}^{|\mathbf{x}|}r\phi(r)dr\Big{)},\quad\text{where
}\varphi_{1}=\frac{C_{1}r_{0}^{2}}{2}>0\text{ and
}\kappa_{1}=\frac{C_{1}}{2C_{0}},$ (5.4)
and function $\phi$ is defined by (3.40). As in Lemma 4.2, we have
$\underline{\mathbf{A}}\nabla\varphi=\kappa_{1}\mathbf{x}\quad\text{and}\quad\nabla\varphi=\kappa_{1}\phi(|\mathbf{x}|)\mathbf{x}.$
(5.5)
By (3.41), $\phi(r)\geq d_{0}\mu_{2}=C_{1}>0$. Then
$\varphi(\mathbf{x})\geq\kappa_{1}\Big{(}\varphi_{1}+C_{1}\int_{r_{0}}^{|\mathbf{x}|}rdr\Big{)}=\frac{\kappa_{1}C_{1}}{2}|\mathbf{x}|^{2}.$
(5.6)
Also, we see from (5.5) and (3.31) that
$(\underline{\mathbf{A}}\nabla\varphi)\cdot\nabla\varphi=\kappa_{1}^{2}\mathbf{x}^{T}\underline{\mathbf{B}}\mathbf{x}\leq
d_{4}\kappa_{1}^{2}|\mathbf{x}|^{2}\sum_{i=1}^{2}F_{i}(S_{*}(\mathbf{x}))\leq\kappa_{1}^{2}d_{4}\mu_{1}|\mathbf{x}|^{2}=\kappa_{1}^{2}C_{0}|\mathbf{x}|^{2}=\frac{\kappa_{1}C_{1}}{2}|\mathbf{x}|^{2}.$
(5.7)
Then we have from (5.6) and (5.7) that
$\varphi(\mathbf{x})\geq(A\nabla\varphi)\cdot\nabla\varphi.$ (5.8)
By (4.6) and (5.5), we have
$\nabla\cdot(\underline{\mathbf{A}}\nabla\varphi)+{\bf
b}\cdot\underline{\mathbf{A}}\nabla\varphi\leq\kappa_{1}(n+C_{2}|\mathbf{x}|)\leq
C_{3}(1+|\mathbf{x}|),\quad\text{where }C_{3}=\kappa_{1}(n+C_{2}).$
Select
$s=s_{R}\mathbin{\buildrel\rm def\over{\mathbin{=\kern-2.0pt=}}}C_{3}(1+R),$
(5.9)
then
$s\geq\nabla\cdot(\underline{\mathbf{A}}\nabla\varphi)+{\bf
b}\cdot\underline{\mathbf{A}}\nabla\varphi\quad\text{in
}\mathcal{O}_{r_{0},R}.$ (5.10)
Therefore $\mathcal{L}W\geq 0$ in $\mathcal{O}_{r_{0},R}\times(0,T)$. We
summarize the above arguments in the following lemma.
###### Lemma 5.1.
Let $T>0$, $R>r_{0}$ and let the function $\varphi$ be defined by (5.4). Then
for $s=s_{R}$ in (5.9), the function $W(\mathbf{x},t)$ in (5.2) belongs to
$C_{\mathbf{x},t}^{2,1}(D)\cap C(\bar{D})$ and satisfies $\mathcal{L}W\geq 0$
in $\mathcal{O}_{r_{0},R}\times(0,T)$.
Using the above barrier function $W(x,t)$, we have the following maximum
principle.
###### Theorem 5.2.
Let $T>0$ and $w(\mathbf{x},t)$ be a bounded function in
$C^{2,1}_{\mathbf{x},t}(U_{T})\cap C(\bar{U}_{T})$ that solves
$\mathcal{L}w=f_{0}$ in $U_{T}$, where $f_{0}\in C(\bar{U}_{T})$. Then
$\sup_{\bar{U}_{T}}|w(\mathbf{x},t)|\leq\sup_{\partial_{p}U_{T}}|w(\mathbf{x},t)|+(T+1)\sup_{\bar{U}_{T}}|f_{0}|.$
(5.11)
###### Proof.
Given any $(\mathbf{x}_{0},t_{0})\in U\times(0,T)$. Let $\delta>0$ such that
$t_{0}<T-\delta$. Let $M=\sup_{\bar{U}_{T}}|w(\mathbf{x},t)|$ and
$N=\sup_{\bar{U}_{T}}|f_{0}|$ which are finite numbers. Let $\mu>0$ be
arbitrary. Select $R>0$ sufficiently large such that
$T^{-C_{3}(1+R)}e^{\frac{\kappa_{1}C_{1}R^{2}}{2T}}>M/\mu.$ (5.12)
Denote $\mathcal{C}=\mathcal{O}_{r_{0},R}\times(0,T-\delta]$. Then
$(\mathbf{x}_{0},t_{0})\in\mathcal{C}$. Let $W(\mathbf{x},t)$ be as in Lemma
5.1. We define the auxiliary function
$u(\mathbf{x},t)=w(\mathbf{x},t)-N(t+1)-\mu
W(\mathbf{x},t),\quad(\mathbf{x},t)\in\mathcal{C}.$ (5.13)
We have $u\in C^{2,1}_{\mathbf{x},t}(\mathcal{C})\cap C(\mathcal{C})$ and,
thanks to Lemma 5.1, function $u$ satisfies
$\mathcal{L}u=f_{0}-N-\mu\mathcal{L}W\leq 0\quad\text{in }\mathcal{C}.$
By the maximum principle,
$\max_{\bar{\mathcal{C}}}u=\max_{\partial_{p}\mathcal{C}}u.$ (5.14)
Let us evaluate $u(x,t)$ on the parabolic boundary $\partial_{p}\mathcal{C}$.
For any $\mathbf{x}\in\mathcal{O}_{r_{0},R}$,
$u(\mathbf{x},0)\leq w(\mathbf{x},0)-\mu W(\mathbf{x},0)=w(\mathbf{x},0)-\mu
T^{-s}e^{\frac{\varphi(\mathbf{x})}{T}}\leq w(\mathbf{x},0).$ (5.15)
For $|\mathbf{x}|=r_{0}$ and $0\leq t\leq T-\delta$,
$u(\mathbf{x},t)\leq w(\mathbf{x},t)-\mu W(\mathbf{x},t)\leq w(\mathbf{x},t).$
(5.16)
For $|\mathbf{x}|=R$ and $0\leq t\leq T-\delta$, we have from (5.6), (5.9) and
(5.12) that
$u(\mathbf{x},t)\leq
w(\mathbf{x},t)-\mu(T-t)^{-s}e^{\frac{\varphi(\mathbf{x})}{T-t}}\leq M-\mu
T^{-C_{3}(1+R)}e^{\frac{\kappa_{1}C_{1}R^{2}}{2T}}\leq 0.$ (5.17)
Hence, we have from (5.14), (5.15),(5.16) and (5.17) that
$\max_{\bar{\mathcal{C}}}u(\mathbf{x},t)\leq\max\\{0,\sup_{U}w(\mathbf{x},0),\sup_{\Gamma\times[0,T]}w(\mathbf{x},t)\\}.$
(5.18)
In particular, it follows from (5.18) that
$u(\mathbf{x}_{0},t_{0})\leq\max\\{0,\sup_{\partial_{p}U_{T}}w\\}.$ (5.19)
Now, letting $\mu\to 0$ in (5.13) yields
$w(\mathbf{x}_{0},t_{0})-N(t_{0}+1)\leq\max\\{0,\sup_{\partial_{p}U_{T}}w\\}\leq\sup_{\partial_{p}U_{T}}|w|.$
Hence,
$w(\mathbf{x}_{0},t_{0})\leq\sup_{\partial_{p}U_{T}}|w|+N(T+1).$
Repeating the above arguments for $(-w)$ gives
$|w(\mathbf{x}_{0},t_{0})|\leq\sup_{\partial_{p}U_{T}}|w|+N(T+1)$ (5.20)
for any $(\mathbf{x}_{0},t_{0})\in U\times(0,T)$. Therefore, (5.11) follows. ∎
We study the following IBVP (4.9) for $w(\mathbf{x},t)$.
Condition (E2). $F_{1},F_{2}\in C^{7}((0,1))$, $w_{0}\in C(\bar{U})$, $G\in
C(\Gamma\times[0,\infty))$ and $G(\mathbf{x},0)=w_{0}(\mathbf{x})$ on
$\Gamma$.
###### Theorem 5.3.
Assume (E2), $f_{0}\in C_{\mathbf{x}}^{5}(\bar{D})$, $\partial_{t}f_{0}\in
C_{\mathbf{x}}^{3}(\bar{D})$. Suppose $w_{0}(\mathbf{x})$, $G(\mathbf{x},t)$
and $f_{0}(\mathbf{x},t)$ are bounded functions. Then,
(i) There exists a solution $w(\mathbf{x},t)\in C^{2,1}_{\mathbf{x},t}(D)\cap
C(\bar{D})$ of (4.9) .
(ii) This solution is unique in class of locally (in time) bounded solutions,
i.e., the class of solutions $w(\mathbf{x},t)$ such that
$\sup_{U\times[0,T]}|w(\mathbf{x},t)|<\infty\quad\text{for any }T>0.$ (5.21)
(iii) Furthermore, for $(\mathbf{x},t)\in D$,
$|w(\mathbf{x},t)|\leq\Delta_{8}+\Delta_{9}(t+1),$ (5.22)
where
$\Delta_{8}=\max\\{\sup_{U}|w_{0}(\mathbf{x})|,\sup_{\Gamma\times[0,\infty)}|G(\mathbf{x},t)|\\}\quad\text{and}\quad\Delta_{9}=\sup_{D}|f_{0}|.$
(5.23)
###### Proof.
We rewrite equation in the non-divergent form
$\mathcal{L}w=\frac{\partial w}{\partial
t}-\sum_{i,j=1}^{n}a_{ij}\frac{\partial^{2}w}{\partial x_{i}\partial
x_{j}}-\sum_{i,j=1}^{n}\big{[}(a_{ij})_{x_{i}}+a_{ij}b_{i}\big{]}\frac{\partial
w}{\partial x_{j}}=0.$
Thanks to Theorem 4 p.474 of [16] and the maximum principle in Theorem 5.2,
one can prove (i), (ii) and (iii) using similar arguments presented in Theorem
4.6 of [14]. We omit the details. ∎
### 5.2. Lemma of growth in spatial variables
We now study the behavior of the solutions as $|\mathbf{x}|\to\infty$. This
requires a different type of lemma of growth and a new barrier function.
Let $R>0$ and $\ell\geq R+r_{0}$. Denote
$\mathcal{O}_{R}(\ell)=\mathcal{O}_{\ell-R,\ell+R}=\\{\mathbf{x}\in\mathbb{R}^{n}:||\mathbf{x}|-\ell|<R\\}\quad\text{and}\quad\mathcal{S}_{\ell}=\\{\mathbf{x}\in\mathbb{R}^{n}:|\mathbf{x}|=\ell\\}.$
(5.24)
Define the barrier function
$\mathcal{W}(\mathbf{x},t)=\frac{1}{(t+1)^{s}}e^{-\frac{\psi(\mathbf{x})}{t+1}}\quad\text{for
}|\mathbf{x}|\geq r_{0},\ t\geq 0,$ (5.25)
where parameter $s>0$ and function $\psi>0$. Then
$\mathcal{L}\mathcal{W}=(t+1)^{-s-2}e^{-\frac{\psi(\mathbf{x})}{t+1}}\Big{\\{}(t+1)\big{[}-s+\nabla\cdot(\underline{\mathbf{A}}\nabla\psi)+{\bf
b}\cdot\underline{\mathbf{A}}\nabla\psi\big{]}+\psi-(\underline{\mathbf{A}}\nabla\psi)\cdot\nabla\psi\Big{\\}}.$
(5.26)
Hence, $\mathcal{L}\mathcal{W}\leq 0$ if
$s\geq\nabla\cdot(\underline{\mathbf{A}}\nabla\psi)+{\bf
b}\cdot\underline{\mathbf{A}}\nabla\psi\quad\text{and}\quad\psi\leq(\underline{\mathbf{A}}\nabla\psi)\cdot\nabla\psi.$
(5.27)
Denote ${\boldsymbol{\xi}}(\mathbf{x})=\ell\mathbf{x}/|\mathbf{x}|$. We will
choose $\psi$ such that
$\underline{\mathbf{A}}\nabla\psi=\kappa_{2}(\mathbf{x}-\boldsymbol{\xi})\quad\text{for
some }\kappa_{2}>0.$
By (3.39) and (3.40),
$\nabla\psi=\kappa_{2}\underline{\mathbf{A}}^{-1}(\mathbf{x}-\boldsymbol{\xi})=\kappa_{2}\underline{\mathbf{B}}\mathbf{x}(|\mathbf{x}|-\ell)/|\mathbf{x}|=\kappa_{2}\phi(|\mathbf{x}|)(|\mathbf{x}|-\ell)\mathbf{x}/|\mathbf{x}|.$
(5.28)
Select
$\psi(\mathbf{x})=\kappa_{2}\int_{\ell}^{|\mathbf{x}|}(r-\ell)\phi(r)dr,\quad\text{where
}\kappa_{2}=\frac{C_{0}}{2C_{1}}$ (5.29)
and function $\phi$ is defined by (3.40). For all
$\mathbf{x}\in\mathcal{O}_{R}(\ell)$, we have from (3.42) that
$\psi(\mathbf{x})\leq\kappa_{2}C_{0}\int_{\ell}^{|\mathbf{x}|}(r-\ell)dr=\frac{\kappa_{2}C_{0}}{2}(|\mathbf{x}|-\ell)^{2}.$
(5.30)
By (3.31),
$\displaystyle(\underline{\mathbf{A}}\nabla\psi)\cdot\nabla\psi$
$\displaystyle=\kappa_{2}^{2}(\mathbf{x}-\boldsymbol{\xi})^{T}\underline{\mathbf{B}}(\mathbf{x})(\mathbf{x}-\boldsymbol{\xi})\geq
d_{0}\kappa_{2}^{2}|\mathbf{x}-\boldsymbol{\xi}|^{2}\sum_{j=1}^{2}F_{j}(S_{*}(\mathbf{x}))$
$\displaystyle\geq\kappa_{2}^{2}C_{1}(|\mathbf{x}|-\ell)^{2}=\frac{\kappa_{2}C_{0}}{2}(|\mathbf{x}|-\ell)^{2}.$
Hence this and (5.30) give
$\psi\leq(\underline{\mathbf{A}}\nabla\psi)\cdot\nabla\psi$, that is, the
second condition in (5.27). Also,
$\displaystyle\nabla\cdot(\underline{\mathbf{A}}\nabla\psi)+{\bf
b}\cdot(\underline{\mathbf{A}}\nabla\psi)$
$\displaystyle=\kappa_{2}\Big{[}\nabla\cdot(\mathbf{x}-\boldsymbol{\xi})+{\bf
b}\cdot(\mathbf{x}-\boldsymbol{\xi})\Big{]}=\kappa_{2}\Big{[}n-(n-1)\frac{\ell}{|\mathbf{x}|}+{\bf
b}\cdot(\mathbf{x}-\boldsymbol{\xi})\Big{]}$
$\displaystyle\leq\kappa_{2}(n+|{\bf b}|R).$
Then by (4.6),
$\nabla\cdot(\underline{\mathbf{A}}\nabla\psi)+{\bf
b}\cdot(\underline{\mathbf{A}}\nabla\psi)\leq\kappa_{2}(n+C_{2}R)\leq
C_{3}(1+R),$ (5.31)
where $C_{3}=\kappa_{2}(n+C_{2})$. By selecting
$s=s_{R}\mathbin{\buildrel\rm def\over{\mathbin{=\kern-2.0pt=}}}C_{3}(1+R),$
(5.32)
we have $s\geq\nabla\cdot(\underline{\mathbf{A}}\nabla\psi)+{\bf
b}\cdot(\underline{\mathbf{A}}\nabla\psi)$ which is the first condition in
(5.27). Therefore $\mathcal{L}W\leq 0$ in
$\mathcal{O}_{R}(\ell)\times(0,\infty)$. We have proved:
###### Lemma 5.4.
Given any $R>0$ and $\ell\geq R+r_{0}$. Let $s=s_{R}$ be defined by (5.32) and
the function $\psi$ be defined by (5.29). Then the function
$\mathcal{W}(\mathbf{x},t)$ in (5.25) belongs to
$C_{\mathbf{x},t}^{2,1}(D)\cap C(\bar{D})$ and satisfies
$\mathcal{L}\mathcal{W}\leq 0$ on $\mathcal{O}_{R}(\ell)\times(0,\infty)$.
Next is the lemma of growth in the spatial variables.
###### Lemma 5.5.
Given $T>0$, let
$\displaystyle R$ $\displaystyle=R(T)=C_{4}(1+T),$ (5.33)
$\displaystyle\eta_{0}$
$\displaystyle=\eta_{0}(T)=\Big{(}1-\frac{1}{2^{C_{5}(T+1)}}\Big{)}\frac{1}{(T+1)^{2C_{5}(T+1)}},$
(5.34)
where $C_{4}=\max\\{1,\frac{8C_{3}}{\kappa_{2}eC_{0}}\\}$ and
$C_{5}=C_{3}C_{4}$. Suppose $w(\mathbf{x},t)\in
C_{\mathbf{x},t}^{2,1}(U_{T})\cap C(\bar{U}_{T})$ satisfies $\mathcal{L}w\leq
0$ on $U_{T}$ and $w(\mathbf{x},0)\leq 0$ on $\bar{U}$. Let $\ell$ be any
number such that $\ell\geq R+r_{0}$, then
$\max\big{\\{}0,\sup_{\mathcal{S}_{\ell}\times[0,T]}w(\mathbf{x},t)\big{\\}}\leq\frac{1}{1+\eta_{0}}\max\big{\\{}0,\sup_{\bar{\mathcal{O}}_{R}(\ell)\times[0,T]}w(\mathbf{x},t)\big{\\}}.$
(5.35)
###### Proof.
Denote
$\displaystyle M_{\ell}$
$\displaystyle=\max\big{\\{}0,\sup_{\bar{\mathcal{O}}_{R}(\ell)\times[0,T]}w(\mathbf{x},t)|\big{\\}}\quad\text{and}\quad
m_{\ell}=\max\big{\\{}0,\sup_{\mathcal{S}_{\ell}\times[0,T]}w(\mathbf{x},t)\big{\\}}.$
Let $\mathcal{W}$ be defined as in Lemma 5.4. Let $\eta>0$ chosen later and
define
$\widetilde{W}(\mathbf{x},t)=M_{\ell}(1-\mathcal{W}(\mathbf{x},t)+\eta),$
then $\mathcal{L}\widetilde{\mathcal{W}}\geq 0$ in
$\mathcal{O}_{R}(\ell)\times(0,T]$. We have
$\widetilde{W}(\mathbf{x},0)=M_{\ell}(1-e^{-\psi(\mathbf{x})}+\eta)\geq 0\geq
w(\mathbf{x},0).$ (5.36)
By (5.30), $\psi(\mathbf{x})\leq\kappa_{2}C_{0}R^{2}/2$ when
$|\mathbf{x}|=\ell\pm R$, hence
$\widetilde{W}(\mathbf{x},t)|_{|\mathbf{x}|=\ell\pm R}\geq
M_{\ell}\Big{(}1-(t+1)^{-s}e^{-\frac{\kappa_{2}C_{0}R^{2}}{2(t+1)}}+\eta\Big{)}.$
(5.37)
Let $f(z)=z^{-s}e^{-\frac{\kappa_{2}C_{0}R^{2}}{2z}}$ for $z\geq 0$. Select
$\eta=\max_{[0,\infty)}f(z)$. Elementary calculations show
$\eta=(\frac{2s}{\kappa_{2}eC_{0}R^{2}})^{s}$. Then $t\in[0,T]$, it follows
(5.37) that
$\widetilde{W}(\mathbf{x},t)|_{|\mathbf{x}|=\ell\pm R}\geq
M_{\ell}\geq\max\\{0,w(\mathbf{x},t)|_{|\mathbf{x}|=\ell\pm R}\\}.$ (5.38)
From (5.36), (5.38) and maximum principle we obtain
$\widetilde{W}(\mathbf{x},t)\geq w(\mathbf{x},t)\quad\text{on
}\bar{\mathcal{O}}_{R}(\ell)\times(0,T).$
Particularly,
$\widetilde{W}(\mathbf{x},t)\geq w(\mathbf{x},t)\quad\text{on
}\mathcal{S}_{\ell}\times(0,T).$ (5.39)
Moreover, since $\psi(\mathbf{x})=0$ when $|\mathbf{x}|=\ell$,
$\mathcal{W}(\mathbf{x},t)\geq\frac{1}{(T+1)^{s}}$ thus
$\widetilde{W}(\mathbf{x},t)|_{|\mathbf{x}|=\ell}\leq
M_{\ell}\Big{[}1-\frac{1}{(T+1)^{s}}+\eta\Big{]}.$ (5.40)
Since $R\geq 1$, we easily estimate
$\eta=\Big{[}\frac{2C_{3}(1+R)}{\kappa_{2}eC_{0}R^{2}}\Big{]}^{C_{3}(1+R)}\leq\Big{(}\frac{4C_{3}R}{\kappa_{2}eC_{0}R^{2}}\Big{)}^{C_{3}(1+R)}\leq\Big{(}\frac{C_{4}}{2R}\Big{)}^{C_{3}(1+R)}.$
Hence
$\displaystyle\frac{1}{(T+1)^{s}}-\eta$
$\displaystyle\geq\frac{1}{(T+1)^{C_{3}(1+R)}}-\Big{(}\frac{C_{4}}{2R}\Big{)}^{C_{3}(1+R)}=\Big{(}1-\frac{1}{2^{C_{3}(R+1)}}\Big{)}\frac{1}{(T+1)^{C_{3}(1+R)}}$
(5.41)
$\displaystyle\geq\Big{(}1-\frac{1}{2^{C_{3}R}}\Big{)}\frac{1}{(T+1)^{2C_{3}R}}=\Big{(}1-\frac{1}{2^{C_{5}(T+1)}}\Big{)}\frac{1}{(T+1)^{2C_{5}(T+1)}}=\eta_{0}.$
From (5.39), (5.40) and (5.41) we obtain $(1-\eta_{0})M_{\ell}\geq m_{\ell}$,
thus, $M_{\ell}\geq\frac{m_{\ell}}{1-\eta_{0}}\geq(1+\eta_{0})m_{\ell}$, which
gives (5.35). ∎
###### Lemma 5.6.
Let $T>0$ and $R$, $\eta_{0}$ and $w(\mathbf{x},t)$ be as in Lemma 5.5. For
$i\geq 1$, let
$\bar{m}_{i}=\max\big{\\{}0,\sup_{\mathcal{S}_{r_{0}+iR}\times[0,T]}w(\mathbf{x},t)\big{\\}}.$
(5.42)
Part A (Dichotomy for one cylinder). Then for any $i\geq 1$, we have either of
the following cases.
* (a)
If $\bar{m}_{i+1}\geq\bar{m}_{i-1}$, then
$\bar{m}_{i+1}\geq(1+\eta_{0})\bar{m}_{i}$.
* (b)
If $\bar{m}_{i-1}\geq\bar{m}_{i+1}$, then
$\bar{m}_{i-1}\geq(1+\eta_{0})\bar{m}_{i}$.
Part B (Dichotomy for many cylinders). For any $k\geq 0$, we have the
following two possibilities:
* (i)
There is $i_{0}\geq k+1$ such that
$\bar{m}_{i_{0}+j}\geq(1+\eta_{0})^{j}\bar{m}_{i_{0}}$ for all $j\geq 0$.
* (ii)
For all $j\geq 0$, $\bar{m}_{k+j}\leq(1+\eta_{0})^{-j}\bar{m}_{k}$.
###### Proof.
Part A. By maximum principle,
$\displaystyle\sup_{\bar{\mathcal{O}}_{R}(r_{0}+iR)\times[0,T]}w(\mathbf{x},t)$
$\displaystyle\leq\max\big{\\{}\sup_{\mathcal{S}_{r_{0}+(i\pm
1)R}\times[0,T]}w(\mathbf{x},t),\sup_{\bar{\mathcal{O}}_{R}(r_{0}+iR)}w(\mathbf{x},0)\big{\\}}$
$\displaystyle\leq\max\big{\\{}\sup_{\mathcal{S}_{r_{0}+(i\pm
1)R}\times[0,T]}w(\mathbf{x},t),0\big{\\}}\leq\max\\{\bar{m}_{i-1},\bar{m}_{i+1}\\}.$
Hence,
$\sup_{\bar{\mathcal{O}}_{R}(r_{0}+iR)\times[0,T]}w(\mathbf{x},t)\leq\max\\{\bar{m}_{i-1},\bar{m}_{i+1}\\}.$
(5.43)
Let $\ell=r_{0}+iR$. Applying Lemma 5.5 and (5.43), we obtain
$\bar{m}_{i}\leq\frac{1}{1+\eta_{0}}\max\big{\\{}0,\sup_{\bar{\mathcal{O}}_{R}(r_{0}+iR)\times[0,T]}w(\mathbf{x},t)\big{\\}}\leq\frac{1}{1+\eta_{0}}\max\\{\bar{m}_{i-1},\bar{m}_{i+1}\\}.$
Then the statements (a) and (b) obviously follow.
Part B. For $i<j$, define the cylinder
$\mathcal{C}_{i,j}=\mathcal{O}_{r_{0}+iR,r_{0}+jR}\times(0,T)=\\{(\mathbf{x},t):r_{0}+iR<|\mathbf{x}|<r_{0}+jR,\
t\in(0,T)\\}.$
We say that (a) and (b) above are two cases for cylinder
$\mathcal{C}_{i-1,i+1}$.
Let $k\geq 0$. By Part A, we have either of the following two cases.
Case 1. There is $i_{0}\geq k$ such that Case (a) holds for
$\mathcal{C}_{i_{0},i_{0}+2}$, that is,
$\bar{m}_{i_{0}+2}\geq\bar{m}_{i_{0}}\quad\text{ and
}\quad\bar{m}_{i_{0}+2}\geq(1+\eta_{0})\bar{m}_{i_{0}+1}.$ (5.44)
Then applying Part A to $\mathcal{C}_{i_{0}+1,i_{0}+3}$ we have either
$\text{Case (a) holds for $\mathcal{C}_{i_{0}+1,i_{0}+3}$, which gives
}\bar{m}_{i_{0}+3}\geq\bar{m}_{i_{0}+1}\text{ and
}\bar{m}_{i_{0}+3}\geq(1+\eta_{0})\bar{m}_{i_{0}+2},$ (5.45)
or
$\text{Case (b) holds for $\mathcal{C}_{i_{0}+1,i_{0}+3}$, which gives
}\bar{m}_{i_{0}+1}\geq\bar{m}_{i_{0}+3}\text{ and
}\bar{m}_{i_{0}+1}\geq(1+\eta_{0})\bar{m}_{i_{0}+2}.$ (5.46)
Observe that (5.44) and (5.46) hold simultaneously if only if
$\bar{m}_{i_{0}}=\bar{m}_{i_{0}+1}=\bar{m}_{i_{0}+2}=\bar{m}_{i_{0}+3}=0,$
(5.47)
which is a special case of (5.45). Hence we always have Case (a) for the next
cylinder $\mathcal{C}_{i_{0}+1,i_{0}+3}$. Then by induction, Case (a) holds
for the cylinders $\mathcal{C}_{i_{0}+j-1,i_{0}+j+1}$ for all $j\geq 1$. Thus,
$\bar{m}_{i_{0}+j+1}\geq(1+\eta_{0})\bar{m}_{i_{0}+j}\geq(1+\eta_{0})^{2}\bar{m}_{i_{0}+j-1}\geq\ldots\geq(1+\eta_{0})^{j}\bar{m}_{i_{0}+1}.$
(5.48)
Re-indexing $i_{0}+1$ by $i_{0}$ in (5.48), we obtain (i).
Case 2. For all $i\geq k$, Case (b) holds for $\mathcal{C}_{i,i+2}$, that is,
$\bar{m}_{i}\geq(1+\eta_{0})\bar{m}_{i+1}$ for all $i\geq k$. Therefore,
$\bar{m}_{k}\geq(1+\eta_{0})\bar{m}_{k+1}\geq(1+\eta_{0})^{2}\bar{m}_{k+2}\geq\ldots\geq(1+\eta_{0})^{j}\bar{m}_{k+j},$
(5.49)
which implies (ii). ∎
Using the above dichotomy, we obtain the behavior of a sub-solution $w$ as
$|\mathbf{x}|\to\infty$.
###### Proposition 5.7.
Assume $w\in C_{\mathbf{x},t}^{2,1}(U_{T})\cap C(\bar{U}_{T})$ satisfies
$w(\mathbf{x},0)\leq 0$ in $U$, $\mathcal{L}w\leq 0$ on $U_{T}$, and
$w(\mathbf{x},t)$ is bounded on $\bar{U}_{T}$. Then
$\limsup_{r\to\infty}(\sup_{\mathcal{S}_{r}\times[0,T]}w(\mathbf{x},t))\leq
0.$ (5.50)
###### Proof.
Let $\bar{m}_{i}$ be defined as in Lemma 5.6.
Case 1: There are infinitely many $i$ such that $\bar{m}_{i}=0$. Then there is
a sequence $\\{i_{l}\\}$ increasing to $\infty$ as $l\to\infty$ such that
$\bar{m}_{i_{l}}=0$ for all $l\geq 1$. Then by maximum principle for cylinder
$\mathcal{C}_{i_{l},i_{l+1}}$ we have $w(\mathbf{x},t)\leq 0$ on
$\mathcal{C}_{i_{l},i_{l+1}}$ for all $l\geq 1$. Therefore
$w(\mathbf{x},t)\leq 0$ in $\\{|\mathbf{x}|\geq r_{0}+i_{1}R\\}\times[0,T]$.
This gives (5.50).
Case 2: There are only finitely many $i$ such that $\bar{m}_{i}=0$. Then there
is $N>0$ such that $\bar{m}_{i}>0$ for all $i\geq N$. We apply part B of Lemma
5.6 to $k=N$. If (i) holds, then there is $i_{0}\geq N+1$ such that
$\bar{m}_{i_{0}+j}\geq(1+\eta_{0})^{j}\bar{m}_{i_{0}}>0$ for all $j\geq 0$;
thus, $\lim_{j\to\infty}\bar{m}_{i_{0}+j}=\infty$ which contradicts
$w(\mathbf{x},t)$ being bounded on $U_{T}$. Hence we must have (ii), that is,
for all $j\geq 0$, $\bar{m}_{N+j}\leq(1+\eta_{0})^{-j}\bar{m}_{N}$. Therefore,
$\lim_{j\to\infty}\bar{m}_{N+j}=0$ which, in combining with (5.43), proves
(5.50). ∎
As for solutions of the IBVP (4.9) in a finite time interval, we have the
following.
###### Theorem 5.8.
Let $w\in C_{\mathbf{x},t}^{2,1}(U_{T})\cap C(\bar{U}_{T})$ be a bounded
solution of (4.9) on $U_{T}$ with $f_{0}\in C(\bar{U}_{T})$. If
$\lim_{|\mathbf{x}|\to\infty}w_{0}(\mathbf{x})=0,$ (5.51)
$\lim_{|\mathbf{x}|\to\infty}\sup_{0\leq t\leq T}|f_{0}(\mathbf{x},t))|=0,$
(5.52)
then
$\lim_{r\to\infty}\Big{(}\sup_{\mathcal{S}_{r}\times[0,T]}|w(\mathbf{x},t)|\Big{)}=0.$
(5.53)
###### Proof.
Note that $w_{0}\in C(\bar{U})$, $G\in C(\Gamma\times[0,T])$. By Theorem 5.2,
$w(\mathbf{x},t)$ is bounded on $\bar{U}_{T}$. Let $\varepsilon$ be an
arbitrary positive number. There is $\tilde{r}_{0}>0$ such that for
$|\mathbf{x}|>\tilde{r}_{0}$ we have
$|w_{0}(\mathbf{x})|<\varepsilon\quad\text{ and }\quad\sup_{0\leq t\leq
T}|f_{0}(\mathbf{x},t)|<\varepsilon.$ (5.54)
Let $\tilde{w}=\pm w-\varepsilon(t+1)$ then $\tilde{w}$ is bounded on
$\bar{U}_{T}$ and $\mathcal{L}\tilde{w}<0$ on
$\mathcal{O}_{\tilde{r}_{0}}\times(0,T]$, and $\tilde{w}(\mathbf{x},0)\leq 0$
on $\bar{\mathcal{O}}_{\tilde{r}_{0}}$. Applying Proposition 5.7 to
$\tilde{w}$ with $r_{0}$ being replaced by $\tilde{r}_{0}$ gives
$\limsup_{r\to\infty}(\sup_{\mathcal{S}_{r}\times[0,T]}\tilde{w}(\mathbf{x},t))\leq
0.$
This implies
$\limsup_{r\to\infty}(\sup_{\mathcal{S}_{r}\times[0,T]}[\pm
w(\mathbf{x},t)])\leq\varepsilon(T+1).$
Therefore,
$\limsup_{r\to\infty}(\sup_{\mathcal{S}_{r}\times[0,T]}|w(\mathbf{x},t)|)\leq\varepsilon(T+1).$
Letting $\varepsilon\to 0$ we obtain (5.53). ∎
We now consider problem (4.9) for all $t>0$ under condition (5.51). Although
it is not known whether $\lim_{t\to\infty}w(\mathbf{x},t)$ exists for each
$\mathbf{x}$, we prove in the corollary below that such limit is zero along
some curve $\mathbf{x}(t)$ which goes to infinity as $t\to\infty$.
###### Corollary 5.9.
Let $w(\mathbf{x},t)\in C_{\mathbf{x},t}^{2,1}(D)\cap C(\bar{D})$ be a bounded
solution of (4.9) on $D$ with $f_{0}\in C(\bar{D})$. Assume $w_{0}\in
C(\bar{U})$ satisfies (5.51), $G\in C(\Gamma\times[0,\infty))$ is bounded, and
(5.52) holds for each $T>0$. Then there exists an increasing, continuous
function $r(t)>0$ satisfying $\lim_{t\to\infty}r(t)=\infty$ such that
$\lim_{t\to\infty}\Big{(}\sup_{\mathbf{x}\in\bar{\mathcal{O}}_{r(t)}}|w(\mathbf{x},t)|\Big{)}=0.$
(5.55)
###### Proof.
By Theorem 5.8, there exists a strictly increasing sequence
$\\{r_{k}\\}_{k=1}^{\infty}$ of positive numbers such that
$\lim_{k\to\infty}r_{k}=\infty$ and
$\sup_{\\{\mathbf{x}:|\mathbf{x}|\geq
r_{k}\\}\times[0,k]}|w(\mathbf{x},t)|<\frac{1}{k}.$ (5.56)
Let $r(t)$ be the piecewise linear function passing through the points
$(k,r_{k+1})$ then $r(t)$ is increasing and $r(t)\to\infty$ as $t\to\infty$.
By (5.56), for each $k$ we have
$\sup\\{|w(\mathbf{x},t)|:k\leq t\leq k+1,|\mathbf{x}|\geq
r(t)\\}\leq\sup_{\\{\mathbf{x}:|\mathbf{x}|\geq
r_{k+1}\\}\times[0,k+1]}|w(\mathbf{x},t)|<\frac{1}{k+1}.$
Taking $k\to\infty$ we obtain (5.55). ∎
We now return to the IBVP (4.1) for $\sigma$. We will use the transformation
$\sigma=we^{\Lambda}$. To compare $\sigma$ and $w$, we need to estimate
$\Lambda(\mathbf{x})$. Recall from (3.37) that
$\displaystyle\Lambda(\mathbf{x})$
$\displaystyle=\int_{r_{0}}^{|\mathbf{x}|}\tilde{F}(r)dr,\text{ where
}\tilde{F}(r)=F^{\prime}_{2}(\hat{S}(r))g_{2}(\frac{|c_{2}|}{r^{n-1}})\frac{c_{2}}{r^{n-1}}-F^{\prime}_{1}(\hat{S}(r))g_{1}(\frac{|c_{1}|}{r^{n-1}})\frac{c_{1}}{r^{n-1}}.$
For $R$ sufficiently large and $r\geq R$, we have $|\tilde{F}(r)|\leq
Cr^{1-n}$. Then we have in the case $n\geq 3$ that $|\tilde{F}(r)|\leq
Cr^{-2}$, hence $|\Lambda(\mathbf{x})|\leq C_{6}$ for all $|\mathbf{x}|\geq
r_{0}$, and
$0<C_{7}^{-1}\leq e^{\Lambda(\mathbf{x})}\leq
C_{7}\quad\forall|\mathbf{x}|\geq r_{0}.$ (5.57)
###### Theorem 5.10.
Let $n\geq 3$. Assume (E1) and
$\Delta_{10}\mathbin{\buildrel\rm
def\over{\mathbin{=\kern-2.0pt=}}}\max\\{\sup_{U}|\sigma_{0}(\mathbf{x})|,\sup_{\Gamma\times[0,\infty)}|g(\mathbf{x},t)|\\}<\infty,$
(5.58) $\Delta_{11}\mathbin{\buildrel\rm
def\over{\mathbin{=\kern-2.0pt=}}}\sup_{D}|\nabla\cdot(\underline{\mathbf{A}}(\mathbf{x})\mathbf{c}(\mathbf{x},t))|<\infty.$
(5.59)
Then,
(i) There exists a solution $\sigma(\mathbf{x},t)\in
C^{2,1}_{\mathbf{x},t}(D)\cap C(\bar{D})$ of problem (4.1). This solution is
unique in class of solutions $\sigma(\mathbf{x},t)$ that satisfy
$\sup_{U\times[0,T]}|\sigma(\mathbf{x},t)|<\infty\quad\text{for any }T>0.$
(5.60)
(ii) There is $C>0$ such that for $(\mathbf{x},t)\in D$,
$|\sigma(\mathbf{x},t)|\leq C\big{[}\Delta_{10}+\Delta_{11}(t+1)\big{]}.$
(5.61)
(iii) In addition, if
$\lim_{|\mathbf{x}|\to\infty}\sigma_{0}(\mathbf{x})=0\quad\text{and}\quad\lim_{|\mathbf{x}|\to\infty}\sup_{0\leq
t\leq
T}|\nabla\cdot(\underline{\mathbf{A}}(\mathbf{x})\mathbf{c}(\mathbf{x},t))|=0\text{
for each }T>0,$ (5.62)
then
$\lim_{r\to\infty}\Big{(}\sup_{\mathcal{S}_{r}\times[0,T]}|\sigma(\mathbf{x},t)|\Big{)}=0\quad\text{for
any }T>0,$ (5.63)
and furthermore, there is a continuous, increasing function $r(t)>0$ with
$\lim_{t\to\infty}r(t)=\infty$ such that
$\lim_{t\to\infty}\Big{(}\sup_{\mathbf{x}\in\bar{\mathcal{O}}_{r(t)}}|\sigma(\mathbf{x},t)|\Big{)}=0.$
(5.64)
###### Proof.
Let $w_{0}(\mathbf{x})=\sigma_{0}(\mathbf{x})e^{-\Lambda(\mathbf{x})}$,
$G(\mathbf{x},t)=g(\mathbf{x},t)e^{-\Lambda(\mathbf{x})}$ and
$f_{0}(x,t)=e^{-\Lambda(\mathbf{x})}|\nabla\cdot(\underline{\mathbf{A}}(\mathbf{x})\mathbf{c}(\mathbf{x},t))|$.
Thanks to (5.57) and (5.58), we have
$\max\\{\sup_{U}|w_{0}(\mathbf{x})|,\sup_{\Gamma\times[0,\infty)}|w(\mathbf{x},t|\\}\leq
C\Delta_{10},$ $\sup_{D}|f_{0}|\leq C\Delta_{11}.$
Then statements in (i), (ii) and (iii) follow directly from Theorems 5.3 and
5.8, and Corollary 5.9 for problem (4.9), the relation
$\sigma(\mathbf{x},t)=w(\mathbf{x},t)e^{\Lambda(\mathbf{x})}$ and the
boundedness of $e^{\Lambda(\mathbf{x})}$ in (5.57). We omit the details. ∎
As a consequence of (5.64), for any continuous curve $\mathbf{x}(t)$ with
$|\mathbf{x}(t)|\geq r(t)$, one has
$\lim_{t\to\infty}\sigma(\mathbf{x}(t),t)=0.$ (5.65)
The case $n=2$ is treated next with some restriction on the steady state.
###### Theorem 5.11.
Let $n=2$ and $\hat{S}(r)$ be a solution of (2.13) with $c_{1},c_{2}<0$.
Assume (E1) and
$\Delta_{12}\mathbin{\buildrel\rm
def\over{\mathbin{=\kern-2.0pt=}}}\max\\{\sup_{U}e^{-\Lambda(\mathbf{x})}|\sigma_{0}(\mathbf{x})|,\sup_{\Gamma\times[0,\infty)}|g(\mathbf{x},t)|\\}<\infty,$
(5.66) $\Delta_{13}\mathbin{\buildrel\rm
def\over{\mathbin{=\kern-2.0pt=}}}\sup_{D}e^{-\Lambda{(\mathbf{x})}}|\nabla\cdot(\underline{\mathbf{A}}(\mathbf{x})\mathbf{c}(\mathbf{x},t))|<\infty.$
(5.67)
Then the following statements hold true.
(i) There exists a solution $\sigma(\mathbf{x},t)\in
C^{2,1}_{\mathbf{x},t}(D)\cap C(\bar{D})$ of problem (4.1). This solution is
unique in class of solutions $\sigma(\mathbf{x},t)$ that satisfy
$\sup_{U\times[0,T]}e^{-\Lambda(\mathbf{x})}|\sigma(\mathbf{x},t)|<\infty\quad\text{for
any }T>0.$ (5.68)
(ii) There is $C>0$ such that for $(\mathbf{x},t)\in D$,
$|\sigma(\mathbf{x},t)|\leq C\big{[}\Delta_{12}+\Delta_{13}(t+1)\big{]}.$
(iii) Statement (iii) of Theorem 5.10 holds true if condition (5.62) is
replaced by
$\lim_{|\mathbf{x}|\to\infty}e^{-\Lambda(\mathbf{x})}\sigma_{0}(\mathbf{x})=0\quad\text{and}\quad\lim_{|\mathbf{x}|\to\infty}\sup_{0\leq
t\leq
T}e^{-\Lambda(\mathbf{x})}|\nabla\cdot(\underline{\mathbf{A}}(\mathbf{x})\mathbf{c}(\mathbf{x},t))|=0\text{
for each }T>0.$ (5.69)
###### Proof.
According to Theorem 2.5, $\lim_{r\to\infty}\hat{S}(r)=s^{*}\in(0,1),$ where
$s^{*}$ is defined in (2.33). The proof consists of two steps.
Step 1. We show that statements (i)–(iii) hold true under the following
condition
$F_{2}^{\prime}(s^{*})a_{2}^{0}c_{2}-F_{1}^{\prime}(s^{*})a_{1}^{0}c_{1}<0.$
(5.70)
Let
$c_{4}=-(F_{2}^{\prime}(s^{*})a_{2}^{0}c_{2}-F_{1}^{\prime}(s^{*})a_{1}^{0}c_{1})>0$.
We have for any $R>r_{0}$ and $|\mathbf{x}|>R$ that
$\Lambda(\mathbf{x})=\int_{r_{0}}^{R}\tilde{F}(r)dr+\int_{R}^{|\mathbf{x}|}\tilde{F}(r)dr=I_{1}(R)+I_{2}(R).$
For sufficiently large $R_{0}>r_{0}$, we have for $|\mathbf{x}|>R_{0}$ that
$I_{2}(R_{0})\leq\frac{1}{2}\int_{R_{0}}^{|\mathbf{x}|}\big{(}F^{\prime}_{2}(\hat{S}(r))a_{2}^{0}c_{2}-F^{\prime}_{1}(\hat{S}(r))a_{1}^{0}c_{1}\big{)}r^{-1}dr\leq-\frac{1}{4}\int_{R_{0}}^{|\mathbf{x}|}c_{4}r^{-1}d\xi\leq
0.$
Obviously, $I_{1}(R_{0})$ is finite. This gives $e^{\Lambda(\mathbf{x})}\leq
C_{8}<\infty$ for all $|\mathbf{x}|\geq r_{0}$. Thus,
$|\sigma|\leq C_{9}|w|\quad\text{with constant }C_{9}>0.$ (5.71)
Setting $w(\mathbf{x},t)=\sigma(\mathbf{x},t)e^{-\Lambda(\mathbf{x})}$, we
have $\mathcal{L}w=f_{0}$, where $f_{0}$ is as in Theorem 5.10. Then (i)–(iii)
easily follow Theorems 5.3, 5.8, Corollary 5.9 and relation (5.71).
Step 2. Now, it suffices to show that condition (5.70) is satisfied with
$c_{1},c_{2}<0$. On the one hand, we have from (2.33) that
$\frac{a_{1}^{0}c_{1}}{a_{2}^{0}c_{2}}=f(s^{*})=\frac{f_{1}}{f_{2}}(s^{*})=\frac{F_{2}(s^{*})}{F_{1}(s^{*})}.$
Then
$a_{1}^{0}c_{1}F_{1}(s^{*})=a_{2}^{0}c_{2}F_{2}(s^{*})\mathbin{\buildrel\rm
def\over{\mathbin{=\kern-2.0pt=}}}\mathcal{A}\neq 0$. On the other hand,
$\displaystyle
F_{2}^{\prime}(s^{*})a_{2}^{0}c_{2}-F_{1}^{\prime}(s^{*})a_{1}^{0}c_{1}=\mathcal{A}\Big{[}\frac{F_{2}^{\prime}(s^{*})}{F_{2}(s^{*})}-\frac{F_{1}^{\prime}(s^{*})}{F_{1}(s^{*})}\Big{]}=\mathcal{A}\frac{F_{1}(s^{*})}{F_{2}(s^{*})}\Big{(}\frac{F_{2}}{F_{1}}\Big{)}^{\prime}(s^{*}).$
The assumptions on $f_{1}$ and $f_{2}$ provide
$(F_{2}/F_{1})^{\prime}(s^{*})=(f_{1}/f_{2})^{\prime}(s^{*})>0$ and
$F_{1}(s^{*}),F_{2}(s^{*})>0$. Since $c_{1},c_{2}<0$, we have $\mathcal{A}<0$
and, hence,
$F_{2}^{\prime}(s^{*})a_{2}^{0}c_{2}-F_{1}^{\prime}(s^{*})a_{1}^{0}c_{1}<0$.
The proof is complete. ∎
###### Remark 5.12.
Similar to Theorem 4.9, we can use Bernstein’s technique to estimate
$\mathbf{v}_{1}(\mathbf{x},t)$ and $\mathbf{v}_{2}(\mathbf{x},t)$ uniformly in
$\mathbf{x}\in U^{\prime}\Subset U$. We do not provide details here.
## Appendix A
We give proof to the statements on the range of $s_{\infty}$ in Example 2.6.
Recall that $s_{\infty}\in[0,1]$.
In the case $\Delta=0$ of A and B, $h(r)\equiv s^{*}$ is the equilibrium and
the conclusions are clear. Also, for C and D, $S(r)$ is monotone and the
statements easily follow. We focus on the remaining cases.
A. $c_{1},c_{2}>0$. Note that $F(r,S)>0$ iff $S>h(r)$, hence $S^{\prime}(r)>0$
iff $S(r)>h(r)$.
* •
$\Delta<0$. Then $h(r)$ increases and $h(r)<s^{*}$ for all $r$. Consider
$s_{0}>s^{*}$. Then $S(r)>s^{*}>h(r)$ for all $r$. It follows that $S(r)$ is
strictly increasing which implies $s_{\infty}>s_{0}$. Now, consider
$s_{0}<h(r_{0})$. Then $S(r)<h(r)$ for all $r$, thus $S(r)$ is strictly
decreasing and, therefore, $s_{\infty}<s_{0}$.
* •
$\Delta>0$. In this case, $h(r)$ is decreasing, and $h(r)>s^{*}$ for all $r$.
Then the arguments are the same as in the case $\Delta<0$.
B. $c_{1},c_{2}<0$. Observe that $F(r,S)>0$ iff $S<h(r)$, hence
$S^{\prime}(r)>0$ iff $S(r)<h(r)$.
* •
$\Delta<0$. Then $h(r)$ is increasing and $h(r)<s^{*}$ for all $r$.
We prove (iii) first when $s_{0}<h(r_{0})$. Exactly the same as Claim 2 in the
proof of Theorem 2.3, we have $S(r)\leq h(r)<s^{*}$ for all $r$. Thus $S(r)$
is increasing on $[r_{0},\infty)$. Hence $s_{\infty}\in[s_{0},s^{*}]$. Since
$S(r)$ is strictly increasing for $r$ near $r_{0}$, we have
$s_{\infty}>s_{0}$.
We prove (ii). Consider the subcase $h(r_{0})<s_{0}\leq s^{*}$. Then there
exists $r_{1}>r_{0}$ such that $S(r)>h(r)$ for $r<r_{1}$ and
$S(r_{1})=h(r_{1})$. Similar arguments to (iii), we have $S(r_{1})\leq
S(r)\leq h(r)$ for all $r<r_{1}$. Hence $s_{\infty}\leq s^{*}$ and
$s_{\infty}\geq h(r_{1})>h(r_{0})$.
In the particular case $s_{0}=h(r_{0})$, one can show that $h(r_{0})\leq
S(r)\leq h(r)$ for all $r>r_{0}$. If $S(r)\equiv h(r)$ then
$s_{\infty}=s^{*}$. Otherwise, there is $r_{1}>r_{0}$ and such that
$h(r_{0})\leq S(r_{1})<h(r_{1})$. Similar to (iii) with $r_{0}$ playing the
role of $r_{1}$, we have $s_{\infty}\in(S(r_{1}),s^{*}]$. Hence
$s_{0}\in(h(r_{0}),s^{*}]$.
Finally, we prove (i) when $s_{0}>s^{*}$. Clearly, $S(r)<s_{0}$ for all
$r>r_{0}$. If $s_{0}>S(r)>s^{*}$ for all $r>r_{0}$ then we have $S(r)$
strictly deceasing and $s_{\infty}\in[s^{*},s_{0})$. Otherwise, there is
$r_{1}$ such that $S(r_{1})=s^{*}$. Then using (ii) we obtain
$s_{\infty}\in(h(r_{0}),s_{*}]$.
* •
$\Delta>0$. Then $h(r)$ is decreasing, and $h(r)>s^{*}$ for all $r$. The proof
is similar to the case $\Delta<0$.
## References
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* [2] H. W. Alt and E. DiBenedetto. Nonsteady flow of water and oil through inhomogeneous porous media. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 12(3):335–392, 1985.
* [3] E. Aulisa, L. Bloshanskaya, L. Hoang, and A. Ibragimov. Analysis of generalized Forchheimer flows of compressible fluids in porous media. J. Math. Phys., 50(10):103102, 44, 2009.
* [4] J. Bear. Dynamics of Fluids in Porous Media. Dover, New York, 1972.
* [5] R. H. Brooks and A. Corey. Hydraulic properties of porous media. Hydrol. Pap, Colo. State Univ., Forst Collins, (3), 1964.
* [6] S. Brull. Two compressible immiscible fluids in porous media: the case where the porosity depends on the pressure. Adv. Differential Equations, 13(7-8):781–800, 2008.
* [7] C. Cancès. Finite volume scheme for two-phase flows in heterogeneous porous media involving capillary pressure discontinuities. M2AN Math. Model. Numer. Anal., 43(5):973–1001, 2009.
* [8] C. Cancès, T. Gallouët, and A. Porretta. Two-phase flows involving capillary barriers in heterogeneous porous media. Interfaces Free Bound., 11(2):239–258, 2009.
* [9] E. DiBenedetto, U. Gianazza, and V. Vespri. Continuity of the saturation in the flow of two immiscible fluids in a porous medium. Indiana Univ. Math. J., 59(6):2041–2076, 2010.
* [10] J. K. Hale. Ordinary differential equations. Robert E. Krieger Publishing Co. Inc., Huntington, N.Y., second edition, 1980.
* [11] L. Hoang and A. Ibragimov. Structural stability of generalized Forchheimer equations for compressible fluids in porous media. Nonlinearity, 24(1):1–41, 2011.
* [12] L. Hoang and A. Ibragimov. Qualitative study of generalized Forchheimer flows with the flux boundary condition. Adv. Diff. Eq., 17(5–6):511–556, 2012.
* [13] L. Hoang, A. Ibragimov, T. Kieu, and Z. Sobol. Stability of solutions to generalized Forchheimer equations of any degree. 2012\. submitted.
* [14] L. T. Hoang, A. Ibragimov, and T. T. Kieu. One-dimensional two-phase generalized Forchheimer flows of incompressible fluids. J. Math. Anal. Appln., 401(2):921–938, 5 2013.
* [15] L. T. Hoang, T. T. Kieu, and T. V. Phan. Properties of generalized Forchheimer flows in porous media. 2013\. submitted.
* [16] A. M. Il′in, A. S. Kalashnikov, and O. A. Oleĭnik. Second-order linear equations of parabolic type. Tr. Semin. im. I. G. Petrovskogo, (21):9–193, 341, 2001.
* [17] S. N. Kružkov and S. M. Sukorjanskiĭ. Boundary value problems for systems of equations of two-phase filtration type; formulation of problems, questions of solvability, justification of approximate methods. Mat. Sb. (N.S.), 104(146)(1):69–88, 175–176, 1977.
* [18] S. N. Kruzhkov. Uniqueness of the solutions of mixed problems for a degenerate system of the theory of two-phase filtration. Vestnik Moskov. Univ. Ser. I Mat. Mekh., (2):28–33, 95, 1985.
* [19] E. M. Landis. Second order equations of elliptic and parabolic type, volume 171 of Translations of Mathematical Monographs. American Mathematical Society, Providence, RI, 1998. Translated from the 1971 Russian original by Tamara Rozhkovskaya, With a preface by Nina Ural′tseva.
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|
arxiv-papers
| 2013-10-21T20:29:54 |
2024-09-04T02:49:52.686489
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Luan T. Hoang, Akif Ibragimov, Thinh T. Kieu",
"submitter": "Thinh Kieu",
"url": "https://arxiv.org/abs/1310.5723"
}
|
1310.5769
|
# Quad-equations and auto-Bäcklund transformations of NLS-type systems
D.K. Demskoi School of Computing and Mathematics,
Charles Sturt University, NSW 2678, Australia
###### Abstract
Treating an integrable quad-equation along with its two generalised symmetries
as a compatible system allows one to construct an auto-Bäcklund transformation
for solutions of the related NLS-type system. A fixed periodic reduction of
the quad-equation yields a quasi-periodic reduction of its generalised
symmetries that turn them into differential constraints compatible with the
NLS-type system.
## 1 Introduction
Integrable differential-difference equations with one continuous and one
discrete variable, subsequently referred to as chains, are known to be closely
connected with (systems of) integrable partial differential equations (PDEs).
In particular, many integrable chains can be interpreted as Bäcklund
transformations of some PDEs [1]. The integrability of a chain assumes the
existence of a formal recursion operator and infinitely many commuting flows.
This property has been used to classify both integrable chains and PDEs [2]. A
pair of commuting flows from the same hierarchy is called compatible. Shabat
and Yamilov demonstrated that a pair of compatible chains with some
restrictions on their form always yields a system of PDEs through the
construction often referred to as elimination of shifts [3]. A by-product of
this construction is an invertible auto-transformation of the resulting system
of PDEs.
As far as construction of exact solutions is concerned, a more important class
of transformations is non-invertible auto-transformations containing an
arbitrary parameter (auto-Bäcklund transformations). A direct calculation of
such transformations is a tedious task. The knowledge of other structures
associated with integrability, e.g. a Lax pair or Painlevé structure, may
significantly simplify the calculation of such transformations [4]. In this
paper we show how an auto-Bäcklund transformation can be constructed when a
system of PDEs is obtainable through the elimination of shifts from a
compatible system of two integrable chains. The necessary ingredient in this
construction is that the chains should represent the generalised symmetries of
an integrable quad-equation.
To illustrate the idea we consider the integrable chain
$\displaystyle\partial_{x}u_{k,l}=\frac{1}{u_{k+1,l}-u_{k-1,l}},$ (1)
where $u_{k,l}=u(k,l;x,y)$ is a function that simultaneously depends on
discrete and continuous variables:
$(k,l)\in\mathbb{Z}^{2},\,(t,x)\in\mathbb{C}^{2}$. Throughout the article the
subscripts $k$ and $l$ indicate dependence on discrete variables, while the
subscripts $t$ and $x$ indicate partial derivatives. Equation (1) is related
to the famous Volterra equation
$\displaystyle\partial_{x}w_{k,l}=w_{k,l}(w_{k+1,l}-w_{k-1,l})$
via the substitution [5]
$w_{k,l}=-\frac{1}{(u_{k+1,l}-u_{k-1,l})(u_{k+2,l}-u_{k,l})}.$
The complete classification of the Volterra-type equations can be found in
[6].
The simplest commuting flow, i.e. an equation $\partial_{t}u_{k,l}=G$ of the
lowest order that satisfies
$\partial_{t}\partial_{x}u_{k,l}=\partial_{x}\partial_{t}u_{k,l}$, of (1) is
given by
$\partial_{t}u_{k,l}=\frac{u_{k+2,l}-u_{k-2,l}}{(u_{k+1,l}-u_{k-1,l})^{2}(u_{k+2,l}-u_{k,l})(u_{k,l}-u_{k-2,l})}.$
(2)
It can be computed by using the standard tools, such as master symmetry [5] or
recursion operator [7]. On the other hand the whole hierarchy of (1) can be
represented by a single formula (see formula (9) of [8]). Note that neither of
chains (1) or (2) depends on shifts with respect to variable $l$. Nevertheless
it is indicated here in order to make possible a connection with a quad-
equation (see below).
In order to obtain a system of PDEs satisfied by $u_{k,l}$ and $u_{k+1,l}$, we
use (1) and its shifted versions to express variables $u_{k-2,l},\,u_{k-1,l}$
and $u_{k+2,l}$:
$u_{k-2,l}=u_{k,l}-\frac{1}{\partial_{x}u_{k-1,l}},\ \
u_{k-1,l}=u_{k+1,l}-\frac{1}{\partial_{x}u_{k,l}},\ \
u_{k+2,l}=u_{k,l}+\frac{1}{\partial_{x}u_{k+1,l}}.$ (3)
The substitution of (3) into (2) yields the derivative NLS system [9] in the
potential form:
$\begin{array}[]{l}u_{t}=u_{xx}+2u_{x}^{2}v_{x},\\\\[2.84526pt]
v_{t}=-v_{xx}+2v_{x}^{2}u_{x},\end{array}$ (4)
where $u_{k,l}=u,\,u_{k+1,l}=v.$ The shifts along chain (1)
$(u_{k,l},u_{k+1,l})\to(u_{k+1,l},u_{k+2,l}),\ \
(u_{k-1,l},u_{k,l})\to(u_{k,l},u_{k+1,l}),$
can now be interpreted as the auto-transformation of (4)
$\left(\begin{array}[]{c}u\\\
v\end{array}\right)\to\left(\begin{array}[]{c}v\\\
u+1/v_{x}\end{array}\right)$ (5)
and its inverse
$\left(\begin{array}[]{c}u\\\
v\end{array}\right)\to\left(\begin{array}[]{c}v-1/u_{x}\\\
u\end{array}\right)$ (6)
correspondingly.
It is known that integrable quad-equations possess hierarchies of generalised
symmetries (see e.g. [10, 11]). For instance, the hierarchy of equations (1)
and (2) is related to the quad-equation
$(u_{k,l}-u_{k+1,l+1})(u_{k+1,l}-u_{k,l+1})-\lambda+\mu=0,$ (7)
where $\lambda,\mu=\mbox{const}$. This equation is often referred to as
$H_{1}$ due to the labeling it received in the classification [12] of
equations consistent around the cube. The $H_{1}$ equation is also well known
in the context of the potential KdV equation where it serves as a
superposition formula for solutions related by the auto-Bäcklund
transformation [13]. Moreover, equation (7) reduces to pKdV in the continuum
limit [14, 15]. This example therefore highlights the link between the classes
of NLS and KdV-type equations.
In what follows we are concerned with implications of the mentioned connection
between integrable chains, NLS-type systems and quad-equations. We show that
it automatically yields an auto-Bäcklund transformation for the related NLS-
system. A formula of superposition can then be derived from the assumption of
commutativity of the auto-Bäcklund transformations. In general the
compatibility of a PDE and a superposition formula needs to be verified
separately, and is not always guaranteed. One of the corollaries of the
presented construction is that a traveling wave reduction of an integrable
quad-equation generates the quasi-periodic closure of the related chains,
which turn them into differential constraints compatible with the NLS-type
system.
## 2 Auto-Bäcklund transformations of NLS-type systems
The statement that (1) and (2) are generalised symmetries of (7) implies that
the relations
$\partial_{t}F=0,\ \ \partial_{x}F=0,$ (8)
where $F$ is the left hand side of (7), are identically satisfied on solutions
of the system consisting of (1), (2) and (7). In other words, (8) become
identities when partial derivatives are eliminated by using (1) and (2), and
mixed shifts by using (7).
Note that due to the symmetry $(k,l)\to(l,k)$, equation (7) possesses the
generalised symmetries of the form (1) and (2), where $k$ and $l$ are
interchanged. However, the corresponding system of PDEs will still be the same
(potential dNLS). The construction being considered here can be applied to
non-symmetrical quad-equations to show that one quad-equation can generate
auto-Bäcklund transformations for two different NLS-type systems. However, for
the sake of simplicity we will consider only the example of the $H_{1}$
equation.
Since equations (1) and (2) do not involve shifts with respect to the variable
$l$, the quantities
$p=u_{k,l+1},\ \ q=u_{k+1,l+1}$
must satisfy a system of form (4) with $(u,v)$ being replaced by $(p,q)$:
$\begin{array}[]{l}p_{t}=p_{xx}+2p_{x}^{2}q_{x},\\\\[2.84526pt]
q_{t}=-q_{xx}+2q_{x}^{2}p_{x}.\end{array}$ (9)
This observation implies that quad-equation (7) when re-written as
$(u-q)(v-p)=\kappa,$ (10)
where $\kappa=\lambda-\mu$, is a part of a certain auto-transformation for the
potential dNLS system. Importantly, the constant $\kappa$ is not present in
(9); hence it can play the role of the Bäcklund parameter. Another part of the
auto-transformation can be found the following way.
Consider the up- and down-shifted versions of (7):
$(u_{k+1,l}-u_{k+2,l+1})(u_{k+2,l}-u_{k+1,l+1})=\kappa,$ (11)
$(u_{k-1,l}-u_{k,l+1})(u_{k,l}-u_{k-1,l+1})=\kappa.$ (12)
It follows from (1) that
$\displaystyle u_{k+2,l}=u+\frac{1}{v_{x}},\ \ u_{k+2,l+1}=p+\frac{1}{q_{x}},\
\ \displaystyle u_{k-1,l}=v-\frac{1}{u_{x}},\ \
u_{k-1,l+1}=q-\frac{1}{p_{x}}.$
Substituting these expressions into (11) and (12) we obtain the additional
relations
$\left(v-p-\tfrac{1}{q_{x}}\right)\left(u-q+\tfrac{1}{v_{x}}\right)=\kappa,$
(13)
$\left(v-p-\tfrac{1}{u_{x}}\right)\left(u-q+\tfrac{1}{p_{x}}\right)=\kappa.$
(14)
One can verify that the combination of (10) and (13) implies formula (14).
Therefore any combination of two relations from the list of (10), (13) and
(14) constitutes an auto-Bäcklund transformation for (4). The analogous
transformations for the dNLS system were previously constructed in [16, 17] by
using different approaches.
### Superposition formula and construction of solutions
Now we turn to constructing a superposition formula based on the auto-Bäcklund
transformation found previously, i.e. the combination of relations (10) and
(13). To this end we look at implications of commutativity of a few
transformations (10) which can be schematically represented by the Bianchi
diagram:
$\begin{diagram}$
The relation (13) is used to obtain the new solution from a seed solution. The
diagram yields the following relations
$\begin{array}[]{l}(u-q)(v-p)=\kappa,\\\ (u-n)(v-m)=\nu,\end{array}\ \
\begin{array}[]{l}(p-s)(q-r)=\nu,\\\ (m-s)(n-r)=\kappa\end{array}$
which in turn give rise to the possible expressions for $r$ and $s$:
$\displaystyle r=u+\frac{\kappa-\nu}{p-m},\ \ \displaystyle
s=v+\frac{\kappa-\nu}{q-n}$ (15)
and
$r=n+q-u,\ \ s=v-\frac{\nu}{u-n}-\frac{\kappa}{u-q}.$ (16)
One can check that the second relation is not compatible with the dNLS system,
whereas the first one is! The compatibility is verified by differentiating
(15) (or (16)) with respect to the time variable and then making use of the
potential dNLS system itself, and also of (10), (13) and (15) (or (16)).
Obviously (15) is nothing but the two copies of the standard potential KdV
superposition formula relating the corresponding components in the Bianchi
diagram. Note that (15) is not the only possible form of the superposition
formula since $m$ and $p$ could be eliminated from the formula.
By iterating formula (15) we obtain rational expressions in terms of a seed
solution and the solution obtained through the dNLS system (9), (10) and (13).
Example. If we start with the exponential solution
$u=\exp(x-t),\ \ v=\exp(-x+t),$
then it follows that $q$ satisfies the system
$q_{x}=\frac{q(1-qv)}{\kappa},\ \ q_{t}=\frac{(1+\kappa)q^{2}v-q}{\kappa^{2}}$
(17)
while $p$ is given explicitly by
$p=v+\frac{\kappa}{q}-\frac{1}{q_{x}}.$
Integrating equations (17), we obtain
$q=\frac{1-\kappa}{v+c\exp(-\tfrac{x}{\kappa}+\tfrac{t}{\kappa^{2}})},$
where $c$ is the constant of integration.
A more intricate solution is then obtained through superposition formula (15).
Note that expressions for $m$ and $n$ coincide with $p$ and $q$
correspondingly, where the parameter $\kappa$ is replaced by $\nu$. A common
feature of the solutions obtained from the exponential seed solution is that
the individual components grow/decay exponentially while their product has the
shape of a multi-soliton solution. Such solutions are called dissipatons [18].
For instance, for the values of parameters $\kappa=2,\,\nu=1/2,\,c=1$ the plot
for the product of $r$ and $s$ is
Remark. The fact that equation (7) serves two different hierarchies suggests
the presence of a common member in the KdV and potential dNLS hierarchies.
Indeed, the hierarchy of chains (1) and (2) also contains the “negative” flow
$\partial_{z}u_{k,l}=-\partial_{z}u_{k+1,l}+(u_{k,l}-u_{k+1,l})^{2}+\lambda.$
(18)
By differentiating (1) and (18) with respect to $z$ and $x$ correspondingly
and then eliminating shifts from the obtained expressions, we get the
hyperbolic system
$\begin{array}[]{l}u_{xz}=2(u-v)u_{x}+1,\\\\[2.84526pt]
v_{xz}=-2(u-v)v_{x}-1.\end{array}$ (19)
It is not difficult to verify that (19) commutes with the potential dNLS
system. On the other hand, the compatibility of chains (1) and (18) can be
written as one scalar equation [19]
$u_{xzz}=\frac{1}{2}\frac{u_{xz}^{2}-1}{u_{x}}+2u_{x}(2u_{z}-\lambda),$ (20)
which commutes with the potential KdV equation
$u_{t}=u_{zzz}-6u_{z}^{2}.$
### Reductions
Here we discuss the connections of periodic reductions of quad-equations and
quasi-periodic closures of the integrable chains. In fact we could have come
to the same construction of auto-Bäcklund transformations by considering the
reductions $u_{k,l}\to u_{\alpha k+\beta l}$, where $\alpha$ and $\beta$ are
some integers, which induce the periodicity constraint
$u_{k,l}=u_{k-\beta,l+\alpha}$. The simplest reduction of this type is when
$\alpha=1$. This reduction, being applied to equation (7), brings it to the
form
$(u_{k}-u_{k+\beta+1})(u_{k+1}-u_{k+\beta})=\kappa.$ (21)
It is important that chains (1) and (2) survive this reduction for an
arbitrary $\beta$ and become the symmetries of (21) upon the substitution
$u_{k+i,l}\to u_{k+i}$. Moreover, the same procedure of elimination of shifts
yields the potential dNLS system with unknowns $u_{k}=u$ and $u_{k+1}=v$.
Since $\beta$ is arbitrary, the quantities
$u_{k+\beta}=p,\ \ u_{k+\beta+1}=q$
should be treated as algebraically independent from $u_{k}$ and $u_{k+1}$.
Thus equation (21) yields the auto-transformation
$(u-q)(v-p)=\kappa$
of (4) into itself. Relations (13) and (14) can be derived in exactly the same
way as before.
In the case when $\beta$ is fixed, the quantities $u_{k+\beta}$ and
$u_{k+\beta+1}$ can no longer be treated as independent because we can express
them in terms of $u_{k}$ and $u_{k+1}$ by using the reduction of (1). As a
result we obtain a differential constraint in the form of a dynamical system
compatible with the potential dNLS equation. On the other hand, the
periodicity constraint transforms the quad-equation into an ordinary
difference equation which can be interpreted as a mapping acting in a finite-
dimensional space. By construction this mapping will preserve the differential
constraint.
Example. Consider the case $\alpha=1,\,\beta=2$. Equation (7) turns into the
ordinary difference equation
$(u_{k}-u_{k+3})(u_{k+1}-u_{k+2})=\kappa,$ (22)
while chain (1) becomes
$\displaystyle\partial_{x}u_{k}=\frac{1}{u_{k+1}-u_{k-1}}.$ (23)
Writing (23) for $k=0\dots 2$ and eliminating $u_{-1}$ and $u_{3}$ using (22),
we obtain the system
$\partial_{x}u_{0}=\frac{(u_{1}-u_{0})(u_{2}-u_{0})}{f},\ \
\partial_{x}{u_{1}}=\frac{1}{u_{2}-u_{0}},\ \
\partial_{x}u_{2}=\frac{(u_{2}-u_{1})(u_{2}-u_{0})}{f},$ (24)
where
$f=\big{(}(u_{2}-u_{1})(u_{0}-u_{1})+\kappa\big{)}(u_{2}-u_{0}),$ (25)
which can be interpreted as a differential constraint compatible with the
potential dNLS system. In order to verify this, one has to eliminate the
$x-$derivatives in the two copies ($(u_{0},u_{1})$ and $(u_{1},u_{2})$) of the
potential dNLS systems using (24), and check that derivatives $\partial_{t}$
and $\partial_{x}$ commute. By construction, (24) is invariant with respect to
the mapping defined by equation (22):
$M:(u_{0},u_{1},u_{2})\to\left(u_{1},u_{2},\displaystyle
u_{0}+\frac{\kappa}{u_{2}-u_{1}}\right).$ (26)
This implies, in particular, that derivative $\partial_{x}$ preserves the
integral(s) of mapping $M$. One can check that $M$ has only one integral given
by (25) - it is also the integral of (24). This integral can be obtained by
means of the staircase method [20, 21] (see also [22]).
### Concluding remarks
Integrable quad-equations provide us with auto-transformations for solutions
of some NLS-type systems. Although we used only one example of $H_{1}$ – dNLS
equations, the presented construction is not specific to this case. It can be
applied to other integrable quad-equations as well – this will be the subject
of further research.
The author is grateful to V.E. Adler and W.K. Schief for clarifying comments
and indicating some relevant references.
## References
* [1] Levi D 1981 Nonlinear differential-difference equations as Bäcklund transformations J. Phys. A 14, 5 1083-1098
* [2] Adler V É, Shabat A B, Yamilov R I 2000 The symmetry approach to the integrability problem Theoret. and Math. Phys., 125:3 1603-1661
* [3] Shabat A B, Yamilov R I 1991 Symmetries of nonlinear lattices Leningrad Math. J., 2 377-400
* [4] Weiss J 1983 The Painlevé property for partial differential equations. II: Bäcklund transformation, Lax pairs, and the Schwarzian derivative J. Math. Phys., 24, 1405
* [5] Cherdantsev I Yu, Yamilov R I 1995 Master symmetries for differential-difference equations of the Volterra type Physica D 87 140-144
* [6] Yamilov R 2006 Symmetries as integrability criteria for differential difference equations J. Phys. A: Math. Gen. 39 541 623
* [7] Mikhailov A V, Wang J P and Xenitidis P 2011 Recursion operators, conservation laws, and integrability conditions for difference equations Theoret. and Math. Phys., 167:1 421 443
* [8] Svinin A K 2011 On some integrable lattice related by the Miura-type transformation to the Itoh-Narita-Bogoyavlenskii lattice J. Phys. A: Math. Theor. 44 465210\.
* [9] Kaup D J, Newell A C 1978 An exact solution for a derivative nonlinear Schr dinger equation J. Math. Phys. 19:4 798-801
* [10] Rasin O G, Hydon P E, 2007 Symmetries of integrable difference equations on the quad-graph Stud. Appl. Math. 119:3 253-269
* [11] Levi D, Yamilov R I 2009 The generalized symmetry method for discrete equations J. Phys. A: Math. Theor. 42 454012
* [12] Adler V E, Bobenko A I, and Suris Y B 2003 Classification of integrable equations on quad-graphs. The consistency approach Comm. Math. Phys., 233:3 513-543
* [13] Wahlquist H D, Estabrook F B 1973 Bäcklund transformation for solutions of the Korteweg-de Vries equation Phys. Rev. Lett., 31 1386-1390
* [14] Quispel G R W, Nijhoff F W, Capel H W and van der Linden J 1984 Linear integral equations and nonlinear difference-difference equations. Physica A 125, 344 380
* [15] Nijhoff F W and Capel H W 1995 The discrete Korteweg-de Vries equation. Acta Appl. Math. 39, 133 158
* [16] Kundu A 1987 Explicit auto-Bäcklund relation through gauge transformation J. Phys. A: Math. Gen. 20 1107
* [17] Steudel H, 2003 The hierarchy of multi-soliton solutions of the derivative nonlinear Schrödinger equation, J. Phys. A: Math. Gen., 36:7 1931-1946
* [18] Pashaev O K 1997 Integrable models as constrained topological gauge theory Nucl. Phys. B 57, 338 341
* [19] Adler V E, Shabat A B 2006 A dressing chain for the acoustic spectral problem Theoret. and Math. Phys. 149
* [20] Papageorgiou V G, Nijhoff F W and Capel H W 1990 Integrable mappings and nonlinear integrable lattice equations Phys. Lett. A 147 106-114
* [21] Quispel G R W, Capel H W, Papageorgiou V G and Nijhoff F W 1991 Integrable mappings derived from soliton equations Physica A 173 243-266
* [22] Tran D T, van der Kamp P H and Quispel G R W 2009 Closed-form expressions for integrals of traveling wave reductions of integrable lattice equations, J. Phys. A: Math. Theor. 42 225201\.
|
arxiv-papers
| 2013-10-22T00:58:39 |
2024-09-04T02:49:52.703384
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Dmitry K Demskoi",
"submitter": "Dmitry Demskoi K",
"url": "https://arxiv.org/abs/1310.5769"
}
|
1310.5783
|
# Supernova Early Warning in Daya Bay Reactor Neutrino Experiment
Hanyu Wei for the Daya Bay Collaboration Department of Engineering Physics,
Tsinghua University, Beijing, China [email protected]
###### Abstract
Providing an early warning of a galactic supernova using neutrino signals is
of importance in studying both supernova dynamics and neutrino physics. The
Daya Bay reactor neutrino experiment, with a unique feature of multiple liquid
scintillator detectors separated in space, is sensitive to the full energy
spectrum of supernova burst electron-antineutrinos. By deploying 8
Antineutrino Detectors (ADs) in three different experimental halls, we obtain
a more powerful and prompt rejection of muon spallation background than
single-detector experiments. A dedicated supernova online trigger system
embedded in the data acquisition system has been installed to allow the
detection of a coincidence of neutrino signals within a 10-second window, thus
providing a robust early warning of a supernova occurrence within the Milky
Way.
## 1 Motivation
The Daya Bay reactor neutrino experiment is specifically designed for
measuring the neutrino mixing angle $\theta_{13}$ with a sensitivity down to
the 1% level [1]. However, the deployment of 8 electron-antineutrino detectors
(8 ADs) in three different experimental halls (Daya Bay near site, Ling Ao
near site, and Far site) motivates studies for a supernova online trigger
without complicated reconstruction and offline analysis. The three
experimental halls are more than 1 km apart from each other, which enables a
powerful and prompt rejection of muon-induced and accidental backgrounds
superior to that of a single-detector. In addition, a relatively low energy
threshold of 0.7 MeV enhances the detection of the full energy spectrum of
supernova burst neutrinos (SN$\nu$) since the energy spectrum may vary
according to the supernova core collapse model. A supernova online trigger
system is installed and in preparation to join the Supernova Early Warning
System (an international organization abbreviated to SNEWS), providing the
astronomical community with a prompt alert of the occurrence of a galactic
core collapse event [2] with the false alarm rate $<$ 1/month.
## 2 Detection of electron-antineutrinos in Daya Bay
The ADs in Daya Bay are designed to detect the $\bar{\nu}_{e}$ via inverse
beta decay (IBD) interactions $\bar{\nu}_{e}+p\rightarrow n+e^{+}$. Each
single AD has 22 tons of liquid scintillator (LS) and 20 tons of liquid
scintillator doped with gadolinium (Gd-LS), giving a total target mass of
$\sim$330 tons in 8 ADs. The coincidence of the prompt scintillation from the
$e^{+}$ and the delayed gamma emission of the neutron capture provides a
distinctive $\bar{\nu}_{e}$ signature against the background. The dominant
backgrounds are accidentals, cosmogenically produced fast neutrons, 9Li/8He
decays and the neutrons from the retracted 241Am-13C calibration source. The
average delay of the gamma emission of the neutron capture is 28 $\mu$s for
gadolinium and 200 $\mu$s for hydrogen. [3, 4]
## 3 Neutrino emission from supernovae
Supernova burst neutrinos (consisting of
$\nu_{e},\bar{\nu}_{e},\nu_{\mu},\bar{\nu}_{\mu},\nu_{\tau},\bar{\nu}_{\tau}$)
play a role in the study of both supernova dynamics and neutrino physics,
because
* •
$\sim$99% of the stellar collapse gravitational binding energy is converted to
neutrinos which arrive at the earth a few hours before the visual supernova
explosion (SNe). So, it is believed that neutrino emission and interaction are
a key diagnostic for the dynamics of core collapse and supernova explosion
[5];
* •
Supernova burst neutrinos can serve as probes of neutrino properties, e.g.
neutrino mixing, neutrino mass, neutrino lifetime, magnetic moment of
neutrino, electric charge of neutrino, radiative decay of neutrino, etc. [6]
and also the mass hierarchy [7] ;
* •
Joint analysis with gravitational waves can provide deep insight into the core
collapse of supernovae [8].
The expected SN explosion rate is $\sim$0.01/year [9] within kilo-parsec (kpc)
distances and is around once per 50 years in the Milky Way. Within Mpc
distances, the rate is $\sim$1/year [9], but the neutrino flux is much
smaller. The SN$\nu$ energy spectrum within the first 10 seconds of the
supernova exlosion [10] for different flavor components implies the energy
range of electron-antineutrinos is up to $\sim$60 MeV with average energy
12$\sim$15 MeV. The explosion timescale is $\sim$10 s with $\sim$98% of the
$\bar{\nu}_{e}$ luminosity emitted [11]. This timing feature is exploited to
form an online trigger for SN$\nu$ in all the experiments listed in Tab. 1,
where the main features are summarized. Based on the target mass, Tab. 1 shows
that about 12 SN$\nu$ events in one AD and 100 events in all for 8-ADs are
expected at Daya Bay and a SN$\nu$ event is defined as the detection of one
neutrino from a single SN explosion. Even though other experiments may have
higher expected SN$\nu$ events mainly due to the target mass, it is emphasized
that the unique feature of Daya Bay in contrast is that it is not a single-
detector. This paper explains this advantage and shows that the Daya Bay
experiment is sensitive to all the 1987A-type (referring to the luminosity and
average energy of $\bar{\nu}_{e}$) SNe in the Milky Way which can be seen in
Fig. 4.
Table 1: SN$\nu$ sensitive detectors and expected events for a SN at 10 kpc, emission of $5\times 10^{52}$ erg in $\bar{\nu}_{e}$, average energy 12 MeV, compatible with SN1987A. [5] Detector | Type | Location | Mass[kt] | Events | Status
---|---|---|---|---|---
IceCube | Ice Cherenkov | South Pole | 0.6/OM | $10^{6}$ | Running
Super-K IV | Water | Japan | 32 | 7000 | Running
LVD | Scintillator | Italy | 1 | 300 | Running
KamLAND | Scintillator | Japan | 1 | 300 | Running
SNO+ | Scintillator | Canada | 1 | 300 | Commissioning 2013
MiniBOONE | Scintillator | USA | 0.7 | 200 | Running
Daya Bay | Scintillator | China | 0.33 | 100 | Running
Borexino | Scintillator | Italy | 0.3 | 80 | Running
BST | Scintillator | Russia | 0.2 | 50 | Running
HALO | Lead | Canada | 0.079 | tens | Almost ready
ICARUS | Liquid argon | Italy | 0.6 | 200 | Running
| | | | |
## 4 Background sources and the supernova burst neutrino event
The supernova online trigger system in Daya Bay is embedded in the Data
Acquisition System (DAQ), online looking for increase in multi-AD signals in
10s-time-window and sending prompt alarms. According to this task, all the
study of supernova online trigger is for online prompt trigger judgment and
not so precise as offline analysis. The purpose of the background study on one
hand is to have a good understanding of the background coincidences in multi-
AD, thus allowing to set a precise false alarm rate threshold. The false alarm
happens frequently as the detectable SN explosion to the earth is so rare and
the selected events are always backgrounds. On the other hand, the background
study contributes to the event selection criteria establishment which has to
be simpler than that of the offline analysis so as to be prompt and not to
bring much workload to DAQ. A data sample from Dec. 24, 2011 to Jul. 28, 2012
is used to train our online trigger algorithm to give the event selection
criteria and study the backgrounds since no observation of SN$\nu$ was
declared during the period of the data sample by all detectors including Daya
Bay. In addition, the supernova burst neutrinos that undergo an IBD in the
detector volume are simulated aiming to study the detection efficiency of
SN$\nu$.
### 4.1 Background sources
In the Daya Bay ADs (Section 2), referring to Fig. 1, the delayed signal of an
IBD event is either an 8 MeV $\gamma$ cascade from neutron capture on Gd, or a
2.2 MeV $\gamma$ from neutron capture on H. It is observed that the large
amount of accidental backgrounds in the low energy region significantly affect
the background event rate, therefore we set the online energy threshold at 2
MeV for the prompt signal associated with the 8 MeV $\gamma$ cascade and 8 MeV
for that associated with single 2.2 MeV $\gamma$ in which case the majority of
accidental backgrounds are removed. Using optimized selection criteria for
SN$\nu$, the prompt vs. delayed signal energy plot is shown in the red box in
Fig. 1. Along the Y-axis, the Gd neutron capture peak is seen around 8 MeV and
the hydrogen neutron capture peak is around 2.2 MeV where many fast neutrons
are present in the high energy range along the X-axis and reactor neutrino
signals are present below 10 MeV.
Notice that the data for trigger algorithm training are offline reconstructed
while the supernova online trigger can only access the raw data. A simple but
relatively effective reconstruction is applied online for real SN$\nu$
selection in which the average PMT gain and energy scale calibration constants
are used for energy reconstruction and a charge-weighted method is used for
prompt vertex reconstruction. The resulting online, measured single AD event
rates are 0.019, 0.013 and 0.0013 Hz/AD at the Daya Bay near site, Ling Ao
near site and far site, respectively.
Figure 1: Prompt signal energy vs. delayed signal energy 2-D plot for
backgrounds. In the red box is the selection region for SN$\nu$, suggesting
the prompt and delayed energy cut.
### 4.2 Supernova burst neutrino event
Assuming that the spectrum of supernova burst neutrinos follows a quasithermal
distribution [13]
$f_{\nu}(E)\propto E^{\alpha}e^{-(\alpha+1)E/E_{av}}$
where $E_{av}$ is the average energy and $\alpha$ a numerical parameter
describing the amount of spectral pinching. The value $\alpha$ = 2.30
corresponds to a Fermi-Dirac distribution with zero chemical potential. In our
simulation $\alpha$ is $>$ 2.30 and varies with the three main phases of the
detectable supernova neutrino signals: prompt $\nu_{e}$ burst phase, accretion
phase and cooling phase [5].
SN$\nu$s have been simulated separately in both the Gd-LS and LS region of a
single AD, whose results after selection cuts are shown in Fig. 2. With these
simulation results, the detection efficiency of SN$\nu$ that undergo an IBD in
the detector volume is estimated to be $\sim 70\%$ and used to determine the
expected number of SN$\nu$ events in each AD at Daya Bay.
Figure 2: The plots are after selection cut with respect to one single AD.
Top: Simulation 2-D plot of supernova neutrino selection for delayed signal
against prompt signal. Bottom: Prompt signal energy projection of the
corresponding 2-D plot above, which indicates the shape of the supernova burst
neutrinos. Left: For Gd-LS volume. Right: For LS volume.
## 5 Supernova online trigger judgment
An approach is developed to investigate the background coincidence rate, e.g.
false alarm rate, by combining all 8 ADs’ SN$\nu$ candidate events in the
10s-time-window. The SN$\nu$ candidates are always backgrounds as so rare SN
explosion can be observed by neutrino detection in the earth. Every one
second, the SN$\nu$ candidates in the previous 10s-time-window are counted in
each AD, forming a combination to judge whether to trigger or not.
### 5.1 Trigger table and trigger cut
A trigger table is generated to list the AD background combination cases in
order of their corresponding false alarm rate for the sliding 10s-time-window.
Utilizing this table, it is convenient to set the cut for the combination
cases due to a certain false alarm rate threshold according to SNEWS
requirement. Below (Tab. 2), part of the trigger table for online test is
shown as an example where the contents are all for backgrounds.
In Tab. 2, the number under each AD is the background event number counted in
the 10s-time-window. The first two columns correspond to the detectors in the
Daya Bay near site, the next two columns correspond to the Ling Ao near site
and the four remaining columns correspond to the Far site. The column “False
Alarm Rate” is defines not as the trigger rate relative to the combination in
that row but as the total trigger rate of all the AD background combination
cases below. Before the false alarm rate calculation, the AD background
combination cases are firstly in descending order with respect to their
trigger rates. Then for each combination case, the total trigger rate of those
below it and itself is calculated serving as the corresponding quantity “False
Alarm Rate”. Obviously, a trigger cut can be determined easily due to the
false alarm rate threshold. For example, a 1/34s (0.0293111 Hz) false alarm
rate threshold is required and then the last row of Tab. 2 is where to place
the cut below which all the AD background combination cases have a smaller
“False Alarm Rate” and are supposed to trigger a supernova early warning.
This table is for background false alarm control and SN$\nu$ events are
expected to have higher probability for coincidence in 8-ADs than muon-induced
fast neutrons or reactor neutrinos, etc. In detailed detection probability for
SN explosion, please see Section 6. It is also emphasized here that the “False
Alarm Rate” in Tab. 2 is predicted rather than measured. This will be
explained in the next subsection.
Table 2: Part of the trigger table for supernova online judgment. AD1 to AD8 indicates the 8 antineutrino detectors in the three experimental halls in Daya Bay. AD1 | AD2 | | AD3 | AD4 | | AD5 | AD6 | AD7 | AD8 | | False Alarm Rate (Hz)
---|---|---|---|---|---|---|---|---|---|---|---
0 | 0 | | 0 | 0 | | 0 | 0 | 0 | 0 | | 1
0 | 1 | | 0 | 0 | | 0 | 0 | 0 | 0 | | 0.499092
1 | 0 | | 0 | 0 | | 0 | 0 | 0 | 0 | | 0.404098
⋮ | ⋮ | | ⋮ | ⋮ | | ⋮ | ⋮ | ⋮ | ⋮ | | ⋮
0 | 0 | | 0 | 1 | | 0 | 1 | 0 | 0 | | 0.0317780
0 | 0 | | 1 | 0 | | 0 | 1 | 0 | 0 | | 0.0305445
0 | 1 | | 0 | 2 | | 0 | 0 | 0 | 0 | | 0.0293111
⋮ | ⋮ | | ⋮ | ⋮ | | ⋮ | ⋮ | ⋮ | ⋮ | | ⋮
| | | | | | | | | | |
### 5.2 Background rate prediction
The reason we use the predicted background rate is that the data sample used
for the supernova online trigger study is only about 120 days, which provides
insufficient statistics to set a false alarm rate threshold like 1/year.
However, the prediction has a challenge – the overlap in the sliding 10s-time-
window – every one second, the SN$\nu$ candidates of each AD are combined for
judgment and the 10s-time-window is overlapped by a few adjacent ones.
For a single AD, it is verified with the numerical simulation that the rate
(Hz) (here the probability is numerically equal to the rate as every one
second there is a combination) of the event count in the sliding 10s-time-
window still follows the Poisson distribution with the mean value
$10~{}seconds~{}\times~{}single~{}AD~{}event~{}rate$. This is the fundament of
the combination calculation.
In terms of the combination of multiple ADs, assuming different experimental
halls are mutually independent for backgrounds, the correlation between ADs in
the same site is considered and measured using the data. The correlation
between ADs in the same site originates from the muon-induced fast neutrons
which cause several consecutive signals in detectors of the same experiment
hall. The trigger rate for each combination case is predicted using several
unknown independent Poisson variables that formulate the event rate of each AD
and some of which are shared by the correlated ADs in the same experimental
halls representing the correlation part. These unknown Poisson variables can
be calculated eventually based on the measured single AD event rates and
correlation between ADs.
In addition, given the trigger rate of each combination case, the statistical
error can be derived utilizing some statistic skills in which case the data
sample has to be split into 10 parts according to the time and each of the 10
parts is 1s delay or earlier than the adjacent one. To verify the prediction,
the rates measured on data are compared to the prediction and 82% are within
1$\sigma$, 98.4% are within 2$\sigma$, and 99.7% are within 3$\sigma$
consistent with the prediction. Therefore, the prediction of background
combination rate is plausible to replace the measured one. Notice that the
systematic error is negligible compared with the statistical error when the
threshold is set too small such as 1/year, or even 1/month.
### 5.3 Supernova online trigger diagram
The scheme of the supernova online trigger system in Daya Bay is shown in Fig.
3. It includes several software applications implemented in the DAQ of Daya
Bay.
The online part is able to get access to all the raw data and make a simple
reconstruction. The IBD selection program for each AD provides the information
of SN$\nu$ candidates to a combination server with the function of combination
and trigger judgment according to the trigger table mentioned above. There are
two levels of trigger, silent trigger (1/month) and golden trigger (1/year),
which are related to different offline responses. In case of a golden trigger,
an e-mail alert is immediately sent and information of those SN$\nu$
candidates is written into an offline database with about 10 seconds time
latency. A pure offline analysis would cross check both the golden and silent
triggers with less than 40 min latency. The shaded area in the diagram has
been tested and officially installed, while the offline analysis/cross-check
is being developed based on the Performance Quality Monitoring System (PQM) of
Daya Bay. Daya Bay is negotiating to join the SNEWS and the e-mail alert is
presently sent to Daya Bay collaborators who are interested.
To exclude unexpected trigger bursts (e.g. electronic noise) in one detector
or one experimental hall, a simple but effective uniformity cut based on the
$\chi^{2}$ method is applied with less than 1% detection probability lost for
supernova explosions. This $\chi^{2}$ is the minimum value of
$\sum_{i}\frac{(n_{i}-\lambda)^{2}}{n_{i}}$ where $n_{i}$ is the event counts
in the combination for each AD and $\lambda$ is the best fit value of event
counts for all ADs considering SN$\nu$ events are distributed uniformly among
ADs. Detection probability of a supernova explosion is explained next section.
Figure 3: Diagram of supernova online trigger system in Daya Bay. It is the
framework of the software applications on the basis of the existing DAQ system
and on-site host.
## 6 Detection probability of a supernova explosion
According to the target mass of the Daya Bay detectors, the detection
efficiency of SN$\nu$ obtained based on MC and the relation between supernova
neutrino time-integrated flux and distance to the earth [5], single AD’s
expected SN$\nu$ event counts can be determined below,
$N_{AD}=N_{0}\times\frac{L_{\bar{\nu}_{e}}}{5\times
10^{52}erg}\times(\frac{10kpc}{D})^{2}$
where $L_{\bar{\nu}_{e}}$ is the luminosity of electron-antineutrino emission
and $D$ is the SN explosion distance to earth. $N_{0}$ is the single AD’s
expected SN$\nu$ event number in 10s-time-window corresponding to $5\times
10^{52}erg$ luminosity and $10~{}kpc$ distance. Detection efficiency of
SN$\nu$ is considered in $N_{0}$.
Here, the supernova model for the detection probability calculation is set to
SN1987A-type and a typical value for $N_{AD}$ is $\sim$8 at a distance of 10
kpc to earth. Based on the expected SN$\nu$ events of each AD, the detection
probability of a supernova explosion is calculated by summing up the
probabilities of the combination cases that pass the trigger threshold. Notice
that the single AD event rate increases simultaneously during a supernova
explosion and coincidence signals in multiple ADs occur more frequently. As a
result, the detection probability of the SN explosion has been calculated as a
function of distance to the earth. The result is shown in Fig. 4. From the
“8-AD Golden Trigger” line, the Milky Way center is around 8.5 kpc from the
earth with a 100% detection probability and the most distant edge of the Milky
Way is 23.5 kpc from the earth with a 94% detection probability. Moreover, the
silent trigger will add a potential 5% to 10% detection probability of SN
explosion.
Particularly, the “Single Detector” line is comparable to the “8-AD Golden
Trigger” which obviously indicates the gain in sensitivity of the 8-AD
configuration over a single detector. A rough estimation implies the Daya Bay
is equivalent to a single 0.7 kton liquid scintillator detector with respect
to the detection probability of SN explosion as a consequence of the multi-AD
configuration. The background rate level per target mass is the average
background rate per unit of the target mass of Daya Bay ADs.
Figure 4: The X-axis is SN explosion distance from the earth and the Y-axis is
the corresponding detection probability. “8-AD golden trigger” corresponds to
the result with false alarm rate $<$1/year, and “8-AD silent trigger”
corresponds to that with false alarm rate $<$1/month. “Single Detector” is the
scenario also with false alarm rate $<$1/year in which the 8-AD target mass is
combined into a single detector with the background rate level per target mass
of Daya Bay ADs.
The detection probability has two elements in reality: one is the probability
for a SN explosion can be detected, the other one is the corresponding “false
alarm rate” threshold (defined in Subsection 5.1), for example, 1/month or
1/year here which is for background false alarm control. Based on this, the
difference between single-detector and multi-detector can be explained. In the
scenario of single-detector, the total number of SN$\nu$ events is exploited
for trigger cut setting, for example, 10 SN$\nu$ events in 10s-time-window
corresponding to 1/month false alarm rate threshold. While in the scenario of
multi-detector, the background combination case is exploited for trigger cut
setting, for example, combination 0-0-2-3-1-1-0-0 corresponding to 1/month
false alarm rate threshold. Obviously, the total number of events in multi-
detector here is 7 which is smaller than the single-detector, thus providing a
higher detection probability.
## 7 Summary
The supernova online trigger system in Daya Bay has been officially installed
after several pretests. The extra workload to the current CPU consumption of
DAQ is around 8% and is far from the computing maximum workload online.
Moreover, the time latency from electronics triggers to an alarm is around 10
s (20 s considering the duration of 10s-time-window). In the future, the pure
offline cross check will be added and joining the SNEWS is underway. With a
relatively low energy threshold, superior energy resolution and separated 8-AD
deployment, the online detection probability for a SN1987A-type SN explosion
could be larger than 94% within the Milky Way.
This work is supported in part by the Ministry of Science and Technology of
China and the National Natural Science Foundation of China (Grants
No.11235006). In addition, the author also wishes to acknowledge the Daya Bay
Reactor Neutrino Experiment Collaboration, particularly Shaomin Chen, Zhe
Wang, Logan Lebanowski and Fei Li for precious information, useful discussion
and selfless help.
## References
## References
* [1] Daya Bay Collaboration, arXiv: hep-ex/0701029
* [2] SuperNova Early Warning System, http://snews.bnl.gov
* [3] Daya Bay Collaboration 2012 Phys. Rev. Lett. 108 171803
* [4] Daya Bay Collaboration 2013 Chinese Phys. C 37 011001
* [5] Raffelt G, arXiv: 1201.1637v2 [astro-ph. SR]
* [6] Mohapatra R and Pal P 2004 Massive Neutrinos in Physics and Astrophysics (Singapore: World Scientific Printers) Chapter 17
* [7] Serpico P, Chakraborty S, Fischer T, Hudepohl L, Janka H and Mirizzi A 2012 Phys. Rev. D 85 085031
* [8] Ott C, O’Connor E, Gossan S, Abdikamalov E, Gamma U and Drasco S 2013 Nucl. Phys. Proc. Suppl. 235-236 381
* [9] Ando S, Beacom F, Yüksel H 2005 Phys. Rev. Lett. 95 171101
* [10] Scholberg K 2012 Annual Review of Nuclear and Particle Science 62 81
* [11] Fischer T et al 2010 Astron. Astrophys. 517 A80
* [12] Antonioli P, Fienberg T et al 2010 New Journal of Physics 6 114
* [13] Tamborra I, Muller B, Hudepohl L, Janka H and Raffelt G 2012 Phys. Rev. D 86 125031
|
arxiv-papers
| 2013-10-22T02:46:59 |
2024-09-04T02:49:52.709766
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Hanyu Wei (for the Daya Bay collaboration)",
"submitter": "Hanyu Wei",
"url": "https://arxiv.org/abs/1310.5783"
}
|
1310.5835
|
The Innermost Regions of Relativistic Jets and Their Magnetic Fields.
11institutetext: Instituto de Astrofísica de Andalucía, CSIC, Glorieta de la
Astronomía s/n, 1808 Granada, Spain. 22institutetext: Institute for
Astrophysical Research, Boston University, 725 Commonwealth Avenue, Boston, MA
02215-1401, USA. 33institutetext: Current Address: Joint Institute for VLBI in
Europe, Postbus 2, NL-7990 AA, Dwingeloo, the Netherlands.
# The innermost regions of the jet in NRAO 150
Wobbling or internal rotation?
Sol N. Molina 11 [email protected] I. Agudo 112233 [email protected] J. L. Gómez 11
[email protected]
###### Abstract
NRAO 150 is a very bright millimeter to radio quasar at redshift $z$=1.52 for
which ultra-high-resolution VLBI monitoring has revealed a counter-clockwise
jet-position-angle wobbling at an angular speed $\sim$11∘/yr in the innermost
regions of the jet. In this paper we present new total and linearly polarized
VLBA images at 43 GHz extending previous studies to cover the evolution of the
jet in NRAO 150 between 2006 and early 2009. We propose a new scenario to
explain the counter-clockwise rotation of the jet position angle based on a
helical motion of the components in a jet viewed faced-on. This alternative
scenario is compatible with the interpretation suggested in previous works
once the indetermination of the absolute position of the self-calibrated VLBI
images is taken into account. Fitting of the jet components motion to a simple
internal rotation kinematical model shows that this scenario is a likely
alternative explanation for the behavior of the innermost regions in the jet
of NRAO 150.
## 1 Introduction
The ultra-high angular resolution provided by current Very Long Baseline
Interferometry (VLBI) instruments has revealed an increasing number of cases
where the innermost regions of jets in powerful blazars wobble in the plane of
the sky key1 ; key2 . Blazar jet curved structures (key3 ), and helical paths
of jet features (key4 ) are also thought to be related to the same phenomenon.
However, the physical origin of blazar jet wobbling is still far from being
understood. Current jet models indicate that the magnetic field plays a
relevant role in the dynamics of the innermost regions of relativistic jets,
although there are still uncertainties on what is the actual configuration of
the magnetic field in such regions. A possibility is that the magnetic field
is organized in a helical geometry and the jet material traces a spiral path
following the field streamlines in the magnetically dominatet jet region key5
; key6 . However, there is no direct observational evidence showing the jet
plasma describing trajectories consistent with helical paths so far, which is
one of the main motivations behind the study of jet wobbling, as it may be
tied to magnetic processes in the inner regions of relativistic jets in active
galactic nuclei (AGN).
NRAO 150 is an ideal source for this kind of studies. It is a powerful quasar
at $z$=1.52 (key7 ) showing a misalignment by more than 100∘ between the
innermost jet regions (on sub-milliarcsecond scales) and those at larger
distances from the central engine (on milliarcsecond and arcsecons scales)
key2 . This suggests a bent structure of the inner jet oriented within a very
small angle to the line of sight.
The most intriguing process shown by NRAO 150 is the fast rotation of the jet
position angle at an angular rate of up to $\sim$11∘/yr within the inner
$\sim$0.5 mas of the jet structure, as reported by key2 from 43 GHz VLBA
monitoring observations. Such angular speed was estimated by assuming that the
brightest innermost jet feature in the VLBI images remains stationary, from
which the remaining components were observed to move with superluminal speeds
both, in the radial and non-radial directions.
Some scenarios proposed to explain the physical origin of the jet wobbling
phenomenon involve either the orbital motion of the accretion disk or orbital
motion of the jet nozzle, both induced by a companion supermassive compact
object (e.g., key8 ; key9 ). These scenarios may be useful when the jet source
shows periodic jet wobbling (i.e. jet precession), as reported for some well
known blazars (e.g. 3C 273 key10 , 3C 345 key11 ). However, there are other
different cases where the wobbling behavior is far from periodic, as for BL
Lac key12 and OJ287 key1 , hence suggesting that other kinds of jet
instabilities may play a relevant role in the phenomenon.
In this paper, we present a new set of VLBA 43 GHz images of NRAO 150. We use
the new data to follow the trajectories of jet features with the aim to obtain
a better understanding of the jet wobbling phenomenon in this source. In
particular, we revisit the kinematic scenario previously proposed for NRAO 150
in key2 and we present an alternative model to explain it, which is based on
the idea that we are seeing the internal rotation of the jet material.
## 2 Observations
Here we present a set of six new total and linearly polarized intensity 43 GHz
VLBA images of NRAO 150 obtained in May 2006, November 2006, May 2007, January
2008, July 2008, and January 2009. Calibration of the data was performed
within the AIPS software following the standard procedure for polarimetric
observations (e.g. key13 ; key14 ). After the initial phase and amplitude
calibration, the data were edited, self-calibrated in phase and amplitude and
imaged both in total and polarized intensity with a combination of the AIPS
and Difmap key15 software packages. Calibration of the electric vector
position angle (EVPA) was performed by comparison of the integrated
polarization measured from the VLBA images and three polarization calibrators
(BL Lac, DA193, and OJ287) that were observed contemporaneously with the Very
Large Array (VLA). The EVPA calibration obtained was consistent in all cases
with instrumental polarization (D-terms) across epochs key16 . Estimated
uncertainties in the final calibration of the EVPA lie in the range of 5∘ to
10∘.
Figure 1: Sequence of the new 43 GHz VLBA total flux and polarization maps of
NRAO 150 from 2006 to 2009. Contours symbolize the observed total intensity,
the gray scale represents the linearly polarized intensity, whereas the short
sticks indicate the EVPA distribution for every image. The common convolving
beam is 0.17 $\times$ 0.123 mas2 with major-axis position angle at
$-14.85^{\circ}$. The black circles represent the circular Gaussians that fit
the total flux brightness distribution of the source in each epoch. The
distance between different images is proportional to their observing time,
which is indicated to the right of each image.
Figure 1 shows the sequence of new images. To have a simpler representation of
the source, we fitted the total flux brightness distribution of every image
with a set of four circular-Gaussians emission components. For the naming of
components Q1, Q2, and Q3 we used the nomenclature by key2 , while the
northern component is named Q0 here, instead of the ”Core” as in key2 . In the
images the contours symbolize the observed total intensity, the gray scale
represents the linearly polarized intensity, whereas the short sticks indicate
the EVPA distribution for every image. We have used a common convolving beam
of FWHM equal to 0.17 $\times$ 0.123 mas2 with major-axis position angle at
$-14.85^{\circ}$.
Components Q0, Q1 and Q2 are present in all six new observing epochs. In May
2006 we start observing a new component called Qn. This component shows a
peculiar trajectory, traveling very fast from the south to the north of the
jet structure (region represented by Q0) in a few years. It shows a peak in
total intensity in May 2007, while the maximum in linearly polarized emission
is reached in January 2008. After this epoch we cannot distinguish Qn from Q0.
In the last two epochs (July 2008 and January 2009) we can detect again Q3
(see key2 ) because this region of the jet is not strongly disturbed by Qn
after Januray 2008.
## 3 Discussion
Figure 2: Position of 43 GHz model-fit components as observed in the plain of
the sky. Positions are indicated by crosses, whereas curved lines represent
the fits to the trajectory of every jet feature. Like in key2 , this kinematic
representation assumes that the position of Q0 is stationary.
To increase the time span in our study of the kinematical behavior in NRAO 150
we also use the data from the 34 VLBA images at 43 GHz presented by key2 . In
this work Q0, the brightest emission feature in most of the epochs reported by
key2 , was assumed to be the core of the jet. Hence Q0 was considered to
remain stationary, so that the motion of all the other jet components was
referred to its position. An assumption about the reference position on a time
sequence of images is needed to be done when such images are obtained through
phase self-calibration, which removes the actual phase reference, and hence
the absolute position imposed through the VLBI correlation process.
In this paper we add the position of components fitted in the new images (Fig.
1), which –under the above mentioned assumption– gives the kinematical
behavior showed in Fig. 2. To obtain Fig. 2 we did not use the data
corresponding to the observing epochs in 2008 because the region of the jet
represented by Q0 is strongly perturbed by the nearby Qn component, so that
the position of components are not reliable for a kinematical study.
Qn is a rather peculiar component when compared to Q0, Q1, Q2, and Q3. First
Qn has a drastically different speed, with a mean proper motion – measured
considering Q0 as reference – of 0.09$\pm$0.02 mas/yr, (6.29$\pm$1.16 c) while
the velocities measured for Q1, Q2 and Q3 are 3.26$\pm$0.14 c, 2.85$\pm$0.07
c, and 2.29$\pm$0.14 c, respectively key2 . Secondly, the degree of
polarization in the Q0 region increases when Qn approaches.
By looking at Fig. 2 it is evident that if Q0 is taken as the kinematic
reference of the source the jet wobbling in NRAO 150 did not change its
counter-clockwise swing reported previously key2 . This implies that if there
is any periodicity in the behavior of the source (which cannot be assessed by
the data we have compiled so far), it cannot have a period smaller than around
12 years.
While the work presented in key2 helped to understand some key properties of
the relativistic jet in NRAO 150 not studied before, and to identify an
extreme case of jet wobbling (even involving non-radial superluminal speeds),
the lack of a position reference for the images allows to explore other
plausible kinematic scenarios. In this context, we present in the next section
an alternative model to explain the kinematic behavior of jet features in NRAO
150 where none of the fitted positions of emission components is assumed to
remain stationary in the jet.
Figure 3: Fit to the trajectories of model fit components of NRAO 150 under
the assumptions made in our new kinematic model presented in Section 3.1.
### 3.1 A new alternative model: internal rotation of the jet
The extreme misalignment shown by the jet in NRAO 150 from the sub-
milliarcsecond to the arcsecond scale (see key2 and Section 1) needs a
slightly bent jet-structure and an extremely small orientation of the jet axis
with regard to the line of sight to explain the phenomenon. Also, the jet
shows a very small degree of polarization, less that 10 percent in all epochs,
which is also consistent with the geometry where the jet is seen under a very
small angle.
In contrast to the image sequence presented in key2 , Fig. 1 shows that there
is not an emission region in our new 43 GHz images that could be considered
the core of NRAO 150 by its dominance in the brightness distribution. Hence,
since there are no evidence to assure that any of the emission features in the
jet is fixed in the plane of the sky, we consider here a new simple
kinematical model in which no emission feature is fixed in position.
We assume that the innermost jet emission regions move following a bent
trajectory rotating around the jet axis when the jet is seen face on –which is
approximately the case of NRAO 150. This kind of trajectories may be produced
by a helical or quasi-helical magnetic field threading the innermost,
magnetically dominated regions of the jet. If this is the case, the material
has to follow the field lines, hence also tracing bent trajectories around the
jet axis. If the jet is seen almost face-on, during the evolution of the main
emission features traveling outwards the innermost regions, they should be
observed rotating around a fixed point –the actual jet axis–, as seen in
projection in the plane of the sky.
Figure 4 shows a conceptual scheme of this kind of kinematic scenario, in
which the z axis points towards the observer within a very small (assumed
negligible) angle from the line of sight. The equations used to describe this
kinematic scenario are
$\centering r_{(t)}=r_{0}+v_{r}\hskip 2.84544ptt\@add@centering$ (1)
$\centering\phi_{(t)}=\omega\hskip 2.84544ptt,\@add@centering$ (2)
where $r_{(t)}$ is proportional to the radial velocity $v_{r}$ (that we assume
constant but different for each component) and $r_{0}$ is the distance from
the jet axis at time $t=0$. $\phi_{(t)}$ is the angle measured in the x$-$y
plane and varies in time depending on the angular velocity, $\omega$, which is
also assumed constant, but different for every emission feature. In cartesian
coordinates this is
$x_{(t)}=r_{(t)}\hskip 2.84544pt\cos(\phi_{0}+\omega\hskip 2.84544ptt)\\\ $
(3) $y_{(t)}=r_{(t)}\hskip 2.84544pt\sin(\phi_{0}+\omega\hskip 2.84544ptt),$
(4)
where $\phi_{0}$ is the initial angle at $t=0$.
We used this simple model to fit the kinematical behavior represented in Fig.
2, but contrary to was assumed previously, we are not considering any of the
components to remain stationary. We used a $\chi^{2}$ minimization scheme to
look for the best fit values of $r_{0}$, $v_{r}$, $\phi_{0}$ and $\omega$ for
every one of the emission features under study. The fitted trajectories of Q0,
Q1, Q2, and Q3 are graphically represented in Fig. 3. The corresponding
fitting parameters are shown in Table 1. Component Q1 has a small angular
speed, while the remaining emission features rotate around the jet axis –the
(0,0) position in Fig. 3– with a considerably larger angular speed.
To analyze the proper motion of each emission feature we fitted their
trajectories, as given by our rotation model, with a second order polynomial
(as in key17 ; key18 ; key2 ). The mean measured proper motions are
0.0253$\pm$0.0015 mas/yr, 0.030$\pm$0.002 mas/yr, 0.0420$\pm$0.0007 mas/yr,
and 0.043$\pm$0.003 mas/yr for Q0, Q1, Q2 and Q3, respectively. These values
correspond to superluminal apparent speeds of 1.75$\pm$0.10 c, 2.08$\pm$0.13
c, 2.91$\pm$0.05 c, and 2.98$\pm$0.19 c. Fitting the straight trajectory of Qn
with a first order polynomial yields a mean proper motion of 0.09$\pm$0.02
mas/yr, which corresponds to 6.29$\pm$1.16 c. The larger speed of this
emission feature clearly distinguishes it from the remaining components.
By decomposing the mean projected speed into their radial and non-radial
directions we obtain non-radial speeds of 1.54$\pm$0.18 c, 0.129$\pm$3.05 c,
1.54$\pm$0.12 c, and 2.59$\pm$0.25 c for Q0, Q1, Q2 and Q3, respectively.
Therefore, as under the assumptions for the stationary position of Q0 made in
key2 , our new kinematic model yields superluminal apparent velocities in the
non-radial direction of propagation of emission features. This points out the
remarkable non-ballistic properties of the emission regions in NRAO 150.
Figure 4: Conceptual representation of the new model proposed to explain the bent trajectories of emission features in the 43 GHz images of NRAO150. The plot to the right represents the trajectory of an emission feature when the z axis points towards observer within a very small angle from the line of sight. Table 1: Best-fit parameters. Comp | $r_{0}(mas)$ | $v_{r}(mas/yr)$ | $\phi_{0}$(o) | $\omega$(o/yr)
---|---|---|---|---
Q0 | 0.17 | 0.012 | 279 | 5.60
Q1 | 0.02 | 0.030 | 246 | 0.73
Q2 | 0.21 | 0.036 | 252 | 3.29
Q3 | 0.20 | 0.022 | 242 | 5.85
## 4 Summary and Conclusions
We present six new total intensity and polarimetric 43 GHz VLBA images of NRAO
150 covering a time period of three years between mid of 2006 and the
beginning of 2009. We fitted the total flux brightness distribution of each of
these images with sets of circular Gaussians in order to analyze the
kinematics of the jet. We also used the data presented in previous work to
revisit the kinematic behavior of the source in a time span of 12 years since
1997. As in previous work, we report that all emission features follow a
counter-clockwise rotation as measured in the plane of the sky without changes
of the sense of rotation, which sets a lower limit for the time scale of the
jet wobbling phenomenon in NRAO 150 of 12 years. We present an alternative
kinematic scenario to explain the observations of NRAO 150 and to characterize
the structure of this source. By assuming the jet as being observed at a
negligible angle from the line of sight –which is consistent with previous
studies– the motion of the jet emission regions is consistent with an scenario
driven by internal rotation of the jet material around its axis. To test this
idea we developed a $\chi^{2}$ minimization fit scheme to find the best
kinematic parameters to fit data. Our results show that this new model is able
to fit reasonably well the trajectories of the individual emission features,
which sets this new scenario as a likely possibility to explain the kinematics
of the jet in NRAO 150. This work also opens the possibility to interpret the
behavior of both NRAO 150 and other jet wobbling sources in terms of internal
rotation in the innermost regions of relativistic jets.
This research has been supported by the Spanish Ministry of Economy and
Competitiveness grant AYA2010-14844 and by the Regional Government of
Andalucía (Spain) grant P09-FQM-4784. The VLBA is an instrument of the
National Radio Astronomy Observatory, a facility of the National Science
Foundation operated under cooperative agreement by Associated Universities,
Inc.
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* (13) Agudo, I., Gómez, J. L., Gabuzda, D. C., Marscher, A. P., Jorstad, S. G., & Alberdi, A., A&A, 453, 477 , 2006.
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* (15) Shepherd, M. C., ASP Conf. Ser. 125, Astronomical Data Analysis Software and Systems VI, 77, 1997.
* (16) Gómez, J. L., Marscher, A. P., Alberdi, A., Jorstad, S. G., & Agudo, I., VLBA Scientific Memo, 30, 2002.
* (17) Homan, D. C., Ojha, R., Wardle, J. F. C., et al., ApJ, 549, 840, 2001.
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|
arxiv-papers
| 2013-10-22T08:34:49 |
2024-09-04T02:49:52.720376
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Sol N. Molina (1), I. Agudo (1,2,3) and J. L. G\\'omez (1) ((1)\n Instituto de Astrof\\'isica de Andaluc\\'ia, CSIC. (2) Institute for\n Astrophysical Research, Boston University. (3) Joint Institute for VLBI in\n Europe (JIVE))",
"submitter": "Sol Molina",
"url": "https://arxiv.org/abs/1310.5835"
}
|
1310.6118
|
# Circuit QED with a graphene double quantum dot and a reflection-line
resonator
Guang-Wei Deng Key Laboratory of Quantum Information, University of Science
and Technology of China, Chinese Academy of Sciences, Hefei 230026, China Da
Wei Key Laboratory of Quantum Information, University of Science and
Technology of China, Chinese Academy of Sciences, Hefei 230026, China J.R.
Johansson iTHES research group, RIKEN, Wako-shi, Saitama, 351-0198 Japan
Miao-Lei Zhang Key Laboratory of Quantum Information, University of Science
and Technology of China, Chinese Academy of Sciences, Hefei 230026, China
Shu-Xiao Li Key Laboratory of Quantum Information, University of Science and
Technology of China, Chinese Academy of Sciences, Hefei 230026, China Hai-Ou
Li Key Laboratory of Quantum Information, University of Science and
Technology of China, Chinese Academy of Sciences, Hefei 230026, China Gang
Cao Key Laboratory of Quantum Information, University of Science and
Technology of China, Chinese Academy of Sciences, Hefei 230026, China Ming
Xiao Key Laboratory of Quantum Information, University of Science and
Technology of China, Chinese Academy of Sciences, Hefei 230026, China Tao Tu
Key Laboratory of Quantum Information, University of Science and Technology of
China, Chinese Academy of Sciences, Hefei 230026, China Guang-Can Guo Key
Laboratory of Quantum Information, University of Science and Technology of
China, Chinese Academy of Sciences, Hefei 230026, China Hong-Wen Jiang
Department of Physics and Astronomy, University of California at Los Angeles,
California 90095, USA Franco Nori CEMS, RIKEN, Wako-shi, Saitama, 351-0198
Japan Physics Department, The University of Michigan, Ann Arbor, Michigan
48109-1040, USA Guo-Ping Guo Corresponding author: [email protected] Key
Laboratory of Quantum Information, University of Science and Technology of
China, Chinese Academy of Sciences, Hefei 230026, China
Graphene has attracted considerable attention in recent years due to its
unique physical properties and potential applications. Graphene quantum dots
have been proposed as quantum bits, and their excited-state relaxation rates
have been studied experimentally. However, their dephasing rates remain
unexplored. In addition, it is still not clear how to implement long-range
interaction among qubits for future scalable graphene quantum computing
architectures. Here we report a circuit quantum electrodynamics (cQED)
experiment using a graphene double quantum dot (DQD) charge qubit and a
superconducting reflection-line resonator (RLR). The demonstration of this
capacitive coupling between a graphene qubit and a resonator provides a
possible approach for mediating interactions between spatially-separated
graphene qubits. Furthermore, taking advantage of sensitive microwave readout
measurements using the resonator, we measure the charge-state dephasing rates
in our hybrid graphene nanostructure, which is found to be of the order of
GHz. A spectral analysis method is also developed to simultaneously extract:
the DQD-resonator coupling strength, the tunneling rate between the DQD charge
states, and the charge-state dephasing rate. Our results show that this
graphene cQED architecture can be a compelling platform for both graphene
physics research and potential applications.
## I Introduction
Circuit QED provides a platform for studying microwave photons and artificial
atoms in electrical circuits Xiang _et al._ (2013); You and Nori (2011).
Fundamental physical phenomena and quantum algorithms have been demonstrated
using spatially-separated Josephson-junction qubits coupled via
superconducting microwave resonators Xiang _et al._ (2013); You and Nori
(2011). Extending this idea, theoretical works Childress _et al._ (2004); Guo
_et al._ (2008); Lin _et al._ (2008); Cottet and Kontos (2010); Jin _et al._
(2012); Bergenfeldt and Samuelsson (2013); Contreras-Pulido _et al._ (2013);
Lambert _et al._ (2013) on hybrid systems using superconducting transmission-
line resonators (TLRs) and semiconducting artificial atoms have been made and
there have been experiments on qubits based on carbon nanotubes Delbecq _et
al._ (2011, 2013); Viennot _et al._ (2013), GaAs/AlGaAs Frey _et al._
(2012); Toida _et al._ (2013); Basset _et al._ (2013), and InAs nanowires
Petersson _et al._ (2012). Recently, graphene has attracted considerable
attention for its particular properties and variety of applications Geim and
Novoselov (2007); Geim (2009); Guo _et al._ (2009). Due to its gapless
electronic band structure and the Klein tunneling phenomena Rozhkov _et al._
(2011); Castro Neto _et al._ (2009), most graphene quantum dots are formed by
the shape-effect of etched nanostructures. This etching procedure introduces
new physics related to edge states and further increases the difficulties in
fabricating and manipulating the graphene nanostructures. With advanced device
designs, researchers have recently realized pulsed-gate transient spectroscopy
and a relaxation time of 100 ns has been measured Volk _et al._ (2013) in a
graphene quantum dot device. However, the dephasing times, which may be
significantly shorter than the relaxation times, remain unexplored.
Furthermore, there are no proposals on how to couple multiple graphene qubits
of etched quantum dots.
Here we report a circuit-QED experiment with a hybrid device using a graphene
etched DQD and a superconductor reflection-line resonator Zhang _et al._
(2013). This provides a platform for investigating the physics of graphene
nanostructures interacting with microwave photons and for exploring potential
applications. A DQD-resonator coupling strength of the order of tens of MHz is
demonstrated in this hybrid architecture, which is consistent with coupling
strengths reported in cQED experiments using GaAs Frey _et al._ (2012); Toida
_et al._ (2013) and InAs Petersson _et al._ (2012) quantum dots. In addition,
this DQD-resonator architecture provides access to a sensitive dispersive
microwave readout Blais _et al._ (2004) mechanism for the graphene
nanostructures. Previously, graphene quantum dots have only been studied using
direct current (DC) transport measurements Ponomarenko _et al._ (2008) or
quantum-point contacts for charge sensing Wang _et al._ (2010); Güttinger
_et al._ (2008). Using a dispersive readout via the resonator, we can
simultaneously extract the tunneling rate between graphene DQD charge states,
the DQD-resonator coupling strength, and the dephasing rate, by measuring the
resonators phase response as a function of the DQD bias at multiple probe
frequencies. We find that the charge-state dephasing rates in our graphene DQD
varies between 0.5 GHz and 2 GHz for different charge states.
## II The device
Our hybrid graphene-DQD/superconducting resonator device is shown in Fig. 1.
The coupling of cavity to randomly located graphene flasks is a technical
challenge. To meet this challenge, we have designed and fabricated a half-
wavelength superconducting reflection-line resonator consisting of two
differential microstrip lines which does not require the ground plane that is
indispensable in traditional transmission-line designs. The microwave field is
mostly confined between the two strips, which at each point along the line has
an electrical potential with opposite sign (180 degree phase shift). The RLR
is coupled to a regular transmission line via a 180 degree hybrid, which
splits the microwave signal into two opposite phases [Fig. 1(a)]. The
reflected microwave signal is measured using a network analyzer (NA). This RLR
structure is a flexible design that could accommodate the coupling of multiple
qubits (see supplementary materials).
We couple the RLR to the DQD by connecting the two strips at one end of the
RLR to the two Ti/Au plunger gates LP and RP of the graphene quantum dot, see
Fig. 1(b-c). This design of the RLR allows us to apply bias voltages through
the two strips to facilitate the needed electrostatic confinement of the
graphene DQD. The basic structure of the DQD along with an adjacent quantum
point contact channel is defined by plasma etching of a large graphene flake.
The electron numbers $(M,N)$ in the left and right dots are well defined by
the confinement potential induced by the LP and RP gates, respectively. An
electric dipole moment of $d\sim 1000$ $ea_{0}$ is formed by the change in
charge distribution as one trapped electron moves between the two potential
wells of the DQD (see the supplementary materials). Here $a_{0}$ is the Bohr
radius and $e$ is the electron charge. The DQD couples to the microwave field
generated by the superconducting resonator via this dipole moment. The sample
is mounted in a dry dilution refrigerator with a base temperature of about
$26$ mK. The resonance frequency of the RLR is $6.23896$ GHz and the quality
factor is about 1600 with all the gates of the DQD grounded.
## III Measurement of the DQD through the QPC
We first demonstrate a gate-defined graphene DQD with a QPC charge sensing
measurement. In order to study electron tunneling between the two dots, and to
form a dipole coupling to the microwave field, the tunneling barriers of the
DQD must be made large. This also makes the resistance through the DQD large,
which makes it difficult to detect DC transport through the DQD. We therefore
use a nearby QPC as a charge sensor to probe the DQD. By recording the
transconductance $dI_{\rm QPC}/dV_{\rm LP}$ as a function of the LP-RP gate
voltages using a standard lock-in amplifier technique, we can measure the
hexagon-like charge-stability diagram in a very large range of gate voltages.
The result demonstrates that a graphene double quantum dot is formed in our
device [see Fig 2(a)]. We also measured the full width at half maximum (FWHM)
of the QPC signal across the $(M+1,N)\leftrightarrow(M,N+1)$ interdot
transition line as a function of temperature, and we extracted the electron
temperature $T_{e}$, the gate lever arm $\alpha$, and interdot tunneling rate
$2t_{C}$ from the experimental data Wei _et al._ (2013), see Fig. 2(d,e,f)
and the supplementary materials.
## IV Measurement of the DQD through the resonator
We also probe the DQD using the RLR by applying a coherent microwave signal to
the resonator and analyzing the reflected signal. We fix the probe frequency
at 6.2385 GHz and record the amplitude $A$ and phase $\phi$ of the reflected
signal $S_{11}$, as a function of the DQD bias voltages $V_{\rm LP}$ and
$V_{\rm RP}$. Phase shifts $\Delta\phi$ and amplitude changes $\Delta A$ are
observed at the triple points and on the interdot transition lines, where the
charge states of the left and right dots are degenerate [see Fig. 2(b,c)]. On
the cotunneling lines, no phase shift or amplitude change is detected because
the charging energy (about 10 meV) of a single quantum dot is much larger than
the RLR photon energy (26 $\mu$eV). However, the RLR photon energy is close to
the interdot transition energy, and the electron transitions between the dots
can therefore be assisted by and detected through the RLR. Using the same LP-
RP gate voltage biases as in our previous QPC measurements, we can again
measure the charge-stability diagram using the phase shift and amplitude
change [see Fig. 2(a-c)]. The phase shift and amplitude change are caused by a
dispersive shift of the resonance frequency shift due to the interaction with
the off-resonant DQD. Keeping the probe frequency $\omega_{R}/2\pi$ fixed,
when $\omega_{0}/2\pi$ is shifted to lower frequencies, produces a change in
$\Delta A$ and $\Delta\phi$ [see Fig. 3(e,f)]. In order to study the dipole
coupling of this hybrid system, we record the phase and amplitude response
while we sweep the DQD gate voltages across the
$(M+1,N)\leftrightarrow(M,N+1)$ interdot transition line, corresponding to the
DQD qubit energy bias $\epsilon$ being swept from negative to positive values.
A two-level artificial atom is formed with an energy splitting of
$\Omega=\sqrt{\epsilon^{2}+4t_{C}^{2}}$, where $2t_{C}$ is the tunneling
splitting caused by the interdot coupling. The charge states hybridizes around
$\epsilon=0$ [see Fig. 3(a)]. The effective interaction strength is
characterized by the AC susceptibility ${\rm Re}(\chi)$ [see Fig. 3(b)]. We
find experimentally that the resonator and the DQD can be successfully
coupled. This is encouraging as it was not obvious previously to us that the
coupling strength between this cavity and an atomic layered material can be
sufficiently strong. We find that the phase and amplitude response sensitively
depend on the graphene DQD parameters, and these relations are analyzed
theoretically in the supplementary material. Although $2t_{C}$ generally can
be tuned in this kind of etched graphene structure using a middle gate Wei
_et al._ (2013), our setup lacks of this middle gate because the QPC is
fabricated in its place, and we therefore cannot tune $2t_{C}$ in-situ.
$2t_{C}$ is measured for a large region [see Fig. 2(a)] in our sample and is
found to be larger than $\omega_{0}$. In previous work Frey _et al._ (2012);
Petersson _et al._ (2012); Toida _et al._ (2013); Basset _et al._ (2013),
single-peak and double-peak structures in the response of the phase and
amplitude as a function of $\epsilon$ for different values of $2t_{C}$ have
been demonstrated. The observed double-peak response is due to the changing
sign of the dispersive shift when the qubit energy transition from larger to
smaller than the cavity frequency, which can occur if $2t_{C}<\omega_{0}$ when
$|\epsilon|$ is swept.
## V Device parameters
The measured phase shift $\Delta\phi=-{\rm arg}(S_{11})$ depends on the
resonance frequency $\omega_{0}$, the internal and external resonator
dissipation rates $\kappa_{\rm i}$ and $\kappa_{\rm e}$, the DQD-resonator
coupling strength $g_{C}$, the DQD interdot tunneling rate $2t_{C}$, energy
bias $\epsilon$, energy relaxation rate $\gamma_{1}$, and dephasing rate
$\gamma_{2}$. Here $\omega_{0}$, $\kappa_{\rm i}$, and $\kappa_{\rm e}$ can be
obtained by fitting the phase response as a function of probe frequency [see
Fig. 4(a)] (see the supplementary materials), $g_{C}$ can be calculated using
a capacitance model, $t_{C}$ can be extracted from measurements at varying
temperature Wei _et al._ (2013), and $\epsilon$ can be calibrated from the
gate voltage lever arm measurements (6%) that is also obtained from the
measurements when varying the temperature. Previous work on graphene quantum
dots has reported $\gamma_{1}$ to be about 100 MHz Volk _et al._ (2013). This
leaves $\gamma_{2}$ as the only remaining unknown parameter. As a example,
near the DQD bias region $V_{\rm LP}=325$ mV and $V_{\rm RP}=268$ mV, where
$g_{C}=15$ MHz and $2t_{C}=8$ GHz, by fitting the phase shift as a function of
$\epsilon$, we obtain the $\gamma_{2}$ for these charge states to be about
$1.7\pm 0.1$ GHz. Actually, $g_{C}$ and $2t_{C}$ can also be extracted from
this fitting. Previous work Basset _et al._ (2013) has proven that using the
resonator is more precise to measure the tunneling rates when they approach
the resonator eigenfrequency $\omega_{0}$. $2t_{C}$ in our device is larger
than $\omega_{0}$ so that double peak in phase response is not observed.
Moreover, $g_{C}$ is found to be different for various DQD bias regions
Viennot _et al._ (2013). Depending on the setting of the two plunger gates,
we get $g_{C}$ ranging from 6.5 MHz to 20 MHz. It is particularly worth noting
that the experimentally discovered $g_{C}$ of the hybrid structures for
graphene qubits is comparable to that for superconducting qubits and
semiconductor qubits. Here the probe frequency is fixed at $\omega_{0}$. Later
we will discuss a method to extract $\gamma_{2}$ more precisely with varying
probe frequency.
## VI Measurements at multiple probe frequencies
From the theoretical analysis, we find that across the interdot transition
line where $2t_{C}>\omega_{0}$, a double-peak response can also be observed
with a suitable choice of probe frequency (see the supplementary materials).
The narrower structures of the double-peak response, compared to the single-
peak response, are more sensitive to a variety of device parameters, and
therefore more suitable for parameter extraction. We therefore developed a
method where mutiple probe frequencies are applied to the RLR (Fig. 4(c)),
which spans across the region where the double-peaked phase-shift response is
observed, as a function of $\epsilon$ and for $2t_{C}>\omega_{0}$ (Fig 3). We
would like to point out here this multiple probe frequency technique is
particularly useful for our graphene DQD, and other systems, where the qubit
parameters cannot be varied in a broad range. We extracted the phase error of
our measurement setup from the measurement data. Based on this error, a
simulation shows the extraction of $\gamma_{2}$ at the double-peak region is
more precise than single-peak region for $2t_{C}>\omega_{0}$ (see the
supplementary materials). We therefor fit the DQD parameters at double-peak
region as $2t_{C}$ in our device is larger than $\omega_{0}$. For example,
near the DQD bias region $V_{\rm LP}=302$ mV and $V_{\rm RP}=244$ mV, we
obtain $g_{C}=16.4\pm 0.4$ MHz, $2t_{C}=10.3\pm 0.1$ GHz and
$\gamma_{2}=1.6\pm 0.1$ GHz. In another DQD bias regime, $V_{\rm LP}=283$ mV
and $V_{\rm RP}=212$ mV, we obtain $g_{C}=6.7\pm 0.4$ MHz, $2t_{C}=7.3\pm 0.1$
GHz, and $\gamma_{2}=0.65\pm 0.1$ GHz. Errors of the fitted results are small,
as data was the measured and averaged until smooth curves were obtained. The
variances we use here are obtained from the least-square fit, and are subject
to the assumption that the model used correctly describes the measured data.
The main error of this fitting comes from the converting from gate voltage to
$\epsilon$ Wei _et al._ (2013); DiCarlo _et al._ (2004), this may cause an
error of about 20 percent for $\epsilon$.
## VII Discussion
In Ref. Basset _et al._ (2013), the $\gamma_{2}$ in a GaAs DQD system was
found to depend strongly on $2t_{C}$. Here we have measured different $2t_{C}$
in different charge-state regions and found that both $g_{C}$ and $\gamma_{2}$
depend on the bias conditions, using a mutiple-probe-frequency method. In the
supplementary materials, we have analyzed the double peak region for the phase
response as a function of $\epsilon$, and we found that it is a sensitive
region for extracting parameters. In contrast to previous work Frey _et al._
(2012); Basset _et al._ (2013), where the resonantor response as a function
of $\epsilon$ has been used to extract parameters by varying $2t_{C}$, in our
method we only tune the probe frequency. However, theoretically we find that
tuning any of the free parameters, for example $2t_{C}$, $g_{C}$, $\gamma_{2}$
or $\omega$, can result in a double-peaked phase response. In our experiment
we tune the probe frequency $\omega$ because it is easy to control and can be
tuned much more accurately than other parameters. Also, our method of
measuring the phase shift at multiple probe frequencies could have an
advantage since it does not induce variations in the DQD parameters due to
changes in the DQD bias conditions (while, for example, tuning $2t_{C}$ might
Frey _et al._ (2012); Basset _et al._ (2013)). In our hybrid DQD-resonator
device, we demonstrate that $g_{C}$ varies from 6.5 MHz to 20 MHz, and
$\gamma_{2}$ from 0.5 GHz to 2 GHz, as the DQD bias conditions are changed.
Since $g_{C}\ll\gamma_{2}$ in our device, we do not reach the strong coupling
regime and we therefore do not observe vacuum Rabi splitting Wallraff _et
al._ (2004). It is therefore important to analyze dephasing time of the
graphene DQD. The dephasing time however cannot be easily obtained by normal
means because paddles and edge states Wang _et al._ (2010); Molitor _et al._
(2010); Evaldsson _et al._ (2008); Gallagher _et al._ (2010) in graphene can
mask-off its determination in charge transport based measurements. The
resonant cavity, on the other hand, is primarily sensitive to the electrical
dipole of the DQD and is affected substantially less by the electrostatic
disorders. Indeed, the previously unknown $\gamma_{2}$ has been extracted for
the first time in our experiment. Reducing $\gamma_{2}$ and reaching the
strong coupling regime remains an important goal for future work.
In conclusion, we have designed and fabricated a superconducting reflection-
line resonator, and for the first time coupled a resonator to a graphene
double quantum dot. This provides a platform for studying the physics of
light-matter interaction with graphene devices in the microwave regime. In the
future, long-distance and scalable quantum information processing with
graphene qubits may be possible using this circuit quantum electrodynamics
architecture. We demonstrate a graphene-qubit/resonator coupling rate of
around tens of MHz in this hybrid device. This is consistent with results
obtained in previous experiments using semiconducting quantum dots and
transmission-line resonators Frey _et al._ (2012); Toida _et al._ (2013). By
fitting the phase shift as a function of the graphene-qubit energy splitting,
we have accurately extracted device parameters and dephasing rates of the
hybrid nanostructure using multiple probe-frequency measurements. For the
first time, we have measured the dephasing rate in a graphene double quantum
dot, which was observed to range from 0.5 GHz to 2 GHz depending on the
graphene-qubit bias conditions.
## Acknowledgements
This work was supported by the National Fundamental Research Programme (Grant
No. 2011CBA00200), and National Natural Science Foundation (Grant Nos.
11222438, 10934006, 11274294, 11074243, 11174267 and 91121014).
## VIII contributions
G.W.D., D.W., S.X.L., M.L.Z. and H.W.J. fabricated the samples and performed
the measurements. J.R.J., F.N., H.O.L., G.C., T.T. and G.C.G. provided
theoretical support and analysed the data. G.P.G. supervised the project. All
authors contributed to the writing of this paper.
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Figure 1: Hybrid graphene DQD/superconducting RLR device. (a) Circuit
schematic and micrograph of the hybrid device. The half-wavelength reflection-
line resonator is connected to a graphene DQD at the end of its two
striplines. A microwave signal is applied to the other end of the resonator,
and the reflected signal is detected using a network analyzer. The DC voltage
used to control the electron numbers in the DQD is applied via the two DC pads
directly connected to the resonator striplines. (b) Micrograph of the DQD gate
structure. (c) Scanning-electron micrograph of a typical sample of our device.
Figure 2: Measurements of the graphene DQD charge-stability diagram. (a) The
charge-stability diagram measured using a quantum-point contact. (b-c) The
charge-stability diagram measured by the amplitude (b) and phase (c) response
of the reflection-line resonator. The three charge-stability diagrams show a
close correspondence. (d) A charge-stability diagram of a weak tunnel coupling
region, used to measure the full width at half maximum (FWHM) of the QPC
signal. (e) FWHM measured at the base temperature. We fit the data to a
Lorentzian. (f) FWHM as a function of the lattice temperature, measured by
varying the temperature of the mixing chamber. The high temperature region
shows a linear dependence and $2t_{C}$, $\alpha$, and $T_{e}$ can be extracted
by fitting the FWHM as a function of the lattice temperature.
Figure 3: Measurements of the DQD-resonator coupling. [note: inaccurate
figure title] (a) The DQD energy levels. (b) AC susceptibility, ${\rm
Re}(\chi)$, as functions of the DQD detuning $\epsilon$. (c) The RLR resonance
frequency $f_{0}=\omega_{0}/2\pi$, compared to the DQD qubit transition
frequency, $\Omega$, for different interdot tunneling rates $2t_{C}$. (d) The
phase response of the RLR as a function of gate voltages $V_{\rm LP}$ and
$V_{\rm RP}$ near the $(M+1,N)\leftrightarrow(M,N+1)$ interdot transition
line, measured at a fixed probe frequency $\omega_{R}/2\pi=6.2385$ GHz. (e-f)
The spectrum of the phase (e) and amplitude (f) response for $\epsilon=0$
(blue) and for very large $\epsilon$ (red). (g) The phase response as a
function of DQD detuning $\epsilon$, in the signel-peak region (upper panel)
and the double-peak region (lower panel). Figure 4: Phase response. (a)
Best-fit of the phase vs frequency curve. Quality factor and resonance center
can be obtained. (b) $\gamma_{2}$ sensitivity to the fitting. Blue dot line is
the measured data, red line shows the best-fit curve, green and yellow lines
are the results with changing $\gamma_{2}$ at a small value
$\Delta\gamma_{2}$. (c) Experimental data of the phase shift $\Delta\phi$, as
a function of the DQD detuning $\epsilon$, collected for the same interdot
transition line as shown in Fig. 3(d). Each measurement is taken at a
different probe frequency $f_{R}$, which has a detuning of the cavity $\Delta
f=f_{R}-f_{0}$. The theoretical model used in the fitting is described in the
supplementary materials. The free fitting parameters were $2t_{C}$, $g_{C}$
and $\gamma_{2}$, and other DQD and resonator parameters were assumed to be
known from other measurements and calibrations. The extracted parameters are
given in the text.
|
arxiv-papers
| 2013-10-23T06:12:15 |
2024-09-04T02:49:52.739084
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/",
"authors": "Guang-Wei Deng, Da Wei, J.R. Johansson, Miao-Lei Zhang, Shu-Xiao Li,\n Hai-Ou Li, Gang Cao, Ming Xiao, Tao Tu, Guang-Can Guo, Hong-Wen Jiang, Franco\n Nori and Guo-Ping Guo",
"submitter": "Guo-Ping Guo",
"url": "https://arxiv.org/abs/1310.6118"
}
|
1310.6164
|
# Bright gamma-rays from betatron resonance acceleration in near critical
density plasma
B. Liu Institute of Applied Physics and Computational Mathematics, Beijing,
China, 100088 H. Y. Wang Key Laboratory of HEDP of the Ministry of
Education, CAPT,and State Key Laboratory of Nuclear Physics and Technology,
Peking University, Beijing, China, 100871 D. Wu Key Laboratory of HEDP of
the Ministry of Education, CAPT,and State Key Laboratory of Nuclear Physics
and Technology, Peking University, Beijing, China, 100871 J. Liu Institute
of Applied Physics and Computational Mathematics, Beijing, China, 100088
C.E.Chen Key Laboratory of HEDP of the Ministry of Education, CAPT,and State
Key Laboratory of Nuclear Physics and Technology, Peking University, Beijing,
China, 100871 X. Q. Yan [email protected] Key Laboratory of HEDP of the
Ministry of Education, CAPT,and State Key Laboratory of Nuclear Physics and
Technology, Peking University, Beijing, China, 100871 X. T. He
[email protected] Institute of Applied Physics and Computational Mathematics,
Beijing, China, 100088 Key Laboratory of HEDP of the Ministry of Education,
CAPT,and State Key Laboratory of Nuclear Physics and Technology, Peking
University, Beijing, China, 100871
###### Abstract
We show that electron betatron resonance acceleration by an ultra-intense
ultra-short laser pulse in a near critical density plasma works as a high-
brightness gamma-ray source. Compared with laser plasma X-ray sources in
under-dense plasma, near critical density plasma provides three benefits for
electron radiation: more radiation electrons, larger transverse amplitude, and
higher betatron oscillation frequency. Three-dimensional particle-in-cell
simulations show that, by using a 7.4J laser pulse, 8.3mJ radiation with
critical photon energy 1MeV is emitted. The critical photon energy $E_{c}$
increases with the incident laser energy $W_{I}$ as $E_{c}\propto
W_{I}^{1.5}$, and the corresponding photon number is proportional to $W_{I}$.
A simple analytical synchrotron-like radiation model is built, which can
explain the simulation results.
###### pacs:
52.38.Kd, 52.38.Fz, 52.27.Ny, 52.59.-f
High-brightness high-speed X-ray pulses have become powerful tools for a wide
variety of scientific applications in physics, chemistry, biology, and
material science, etc. X-ray pulses can be generated when relativistic
electrons experience transverse oscillations. The traditional X-ray sources,
such as synchrotron radiation sources and Compton scattering sources, are
usually based on the conventional particle accelerators, which are very large
and expensive. Recently, with the rapid development of laser-driven
acceleration technology, all optical X-ray sources, which are compact and
cost-effective, attract many interests rmp_x-ray .
When a relativistic electron experiences transverse oscillation, with Lorentz
factor $\gamma$, transverse velocity $v_{\perp}$, and transverse oscillating
frequency $\omega_{\beta}$, X-ray pulse will be radiated, with critical photon
energy textbook
$E_{c}\sim\hbar\omega_{\beta}\gamma^{3}v_{\perp}/c,$ (1)
radiation power $P\sim 2\alpha E_{c}\omega_{\beta}\gamma v_{\perp}/(3c)$, and
confined in a narrow angle $\Delta\theta\sim 1/\gamma$ along the electron
motion direction, where $\alpha$ is the fine-structure constant, $\hbar$ is
the plank constant, and $c$ denotes the velocity of light. It is shown that,
both the critical photon energy and the radiation power can be enhanced by
increasing the values of electron energy, transverse velocity, and transverse
oscillation frequency. Laser wake field in under-dense plasma is a promising
medium for compact high-brightness source of keV x-rays puk_04a ; puk_04b ;
kneip_np . State-of-the-art laser plasma electron accelerators can now
accelerate electrons to GeV energies in centi-metres gev . However, it is very
difficult to increase the energy more than one order of magnitude.
Fortunately, there are still some ways to increase the other two values. The
transverse betatron velocity can be enhanced more than one order of magnitude
by resonance between the electron betatron motion and the laser pulse. By
irradiating a petawatt laser pulse on a gas target, in the direct laser
acceleration dominated regime puk_dla ; gahn , high-brightness synchrotron
X-ray can be generated kneip_prl . In laser wake field, the betatron
oscillation amplitude of GeV electrons can be dramatically enhanced when
resonance occur. By interacting the relativistic electrons with the rear of
the driven laser pulse, $10^{8}$ gamma-ray photons with spectra peaking
between $20$ and $150keV$ have been observed in experiment cipi_np . On the
other hand, by colliding high energy electrons with a laser pulse, the
transverse oscillation frequency can be an order of magnitude as the laser
frequency, which is usually two orders of magnitude higher than the betatron
frequency in the wake field. With the combination of a laser-wake-field
accelerator and a plasma mirror, $10^{8}$ X-ray photons with photon energy
ranging from $50keV$ to $200keV$ have been generated in experiment phuoc .
With further optimizing, $10^{7}$ $MeV$ gamma-rays have been emitted chen .
Figure 1: (color online). Isosurface plot of electron energy density
distribution with isosurface value $190n_{c}m_{e}c^{2}$ at time $t=233fs$.
In this letter, we investigate betatron radiation of electrons by propagating
a ultra-intense ultra-short laser pulse in near critical density plasma. We
found that, both the transverse velocity $v_{\perp}$ and the betatron
frequency $\omega_{\beta}$ can be enhanced dramatically. In this condition,
when the transverse betatron frequency is close to the laser frequency in the
electron frame, relativistic electrons can undergo acceleration and betatron
oscillation simultaneously, and then a helical electron beam can be generated
smra , as illustrated in Fig. 1, by propagating a 7.4J laser pulse in a near
critical density plasma. The relativistic electrons experience transverse
oscillations with very high energy and very high frequency, can emit high
energy photons along electron motion direction. In simulation, 8.3mJ
electromagnetic radiation with critical photon energy $E_{c}\sim 1.17MeV$ is
emitted. Simulation results at different laser plasma parameters show that,
$E_{c}$ can increase with the initial laser energy $W_{I}$ as $E_{c}\propto
W_{I}^{1.5}$, and meanwhile the photon number $N_{\gamma}$ can be proportional
to $W_{I}$.
Here we normalized the betatron oscillation frequency and transverse velocity
by $\nu=\omega_{\beta}/\omega_{0}$, and $\beta=v_{\perp}/c$, where
$\omega_{0}$ is the initial incident laser frequency. According to the self-
matching resonance acceleration regime smra , for a resonance electron, we
have $\beta=\sqrt{\nu/2}$, and $\nu=1-v_{z}/v_{ph}$, where $v_{z}$ is the
electron velocity along laser propagation direction, $v_{ph}=\omega_{0}/k$ is
the phase velocity of the laser pulse, and $k$ is the wave number which
satisfies $\omega_{0}^{2}=\omega_{p}^{2}+c^{2}k^{2}$. The relativistic self-
transparent plasma frequency $\omega_{p}$ can be written as
$\omega_{p}=\sqrt{4\pi e^{2}n_{e}^{2}/am_{e}}$, where
$a=eE_{L}/m_{e}c\omega_{0}^{2}$ is the normalized vector potential for a laser
pulse with electric field $E_{L}$ and laser frequency $\omega_{0}$, and
$n_{e}$ is the density of electron beam in the center of the laser channel.
The betatron frequency under azimuthal quasi-static transverse magnetic field
$B_{\theta}$ is smra $\omega_{\beta}=\sqrt{(ev_{z}/\gamma m_{e})(\partial
B_{\theta}/\partial r)}=\sqrt{\mu_{0}n_{e}e^{2}v_{z}^{2}/\gamma m_{e}}$. Then
the maximum value of $\gamma$ accelerated by resonance is
$\gamma_{r}=\mu_{0}n_{e}e^{2}v_{z}^{2}/\omega_{\beta}^{2}m_{e}$. At the limit
of $n_{e}/a\ll 1$ and $v_{z}\to c$, one can get
$\nu=\frac{n_{e}}{2an_{c}},\quad\beta=\frac{1}{2}\sqrt{\frac{n_{e}}{an_{c}}},\quad\gamma_{r}=\frac{4a^{2}n_{c}}{n_{e}}.$
(2)
Then we can get
$E_{c}\sim 16\hbar\omega_{0}a^{9/2}\left(n_{e}/n_{c}\right)^{-3/2}.$ (3)
It is appropriate to assume that every one electron experience one whole
period to radiate. Then the radiation energy per electron become
$w_{r}=P\times 2\pi/\omega_{\beta}=4\pi\alpha E_{c}\gamma_{r}\beta/3$. The
total energy of the betatron electrons can be written as
$W_{ele}=N_{\beta}\gamma_{r}m_{e}c^{2},$ where $N_{\beta}$ denotes the total
number of betatron resonance electrons. Then we can get the total radiation
energy
$W_{r}=N_{\beta}w_{r}=\frac{4\pi\alpha\beta}{3}\frac{W_{ele}}{m_{e}c^{2}}E_{c},$
(4)
and the number of radiation photons with photon energy around $E_{c}$
$N_{\gamma}=W_{r}/E_{c}=\frac{4\pi\alpha\beta}{3}\frac{W_{ele}}{m_{e}c^{2}},$
(5)
Further more, we can investigate the angle distribution of the radiation. The
peak of the angular distribution is at
$\theta_{p}\sim\arctan\beta\sim\beta,$ (6)
and the divergence angle (full angle) is smra
$\Delta\theta\sim\frac{\beta
a}{\pi(R/\lambda)\left[(B_{Sz}/B_{0})^{2}+2(B_{Sz}/B_{0})\right]},$ (7)
where $B_{Sz}$ denotes the axial magnetic field, and $R$ is the spot size of
the field.
Now we present the details of the 3D simulations. In our condition, the
electromagnetic radiation is dominated by synchrotron-like radiation regime.
When the pair generation can be ignored, and radiation coherence is neglected,
the synchrotron-like radiation can be evaluated by calculating the Lorentz-
Abraham-Dirac equation. However, the equation is very difficult to solve.
There are many modified methods to simplify the calculation nau_pop ; chen_min
. Here we extended a fully relativistic three-dimensional (3D) particle-in-
cell (PIC) code (KLAP) klap1 ; klap2 by using the calculation method in Ref.
nau_pop , in which the radiation process and the recoil force are both
considered consistently. A circularly polarized (CP) laser pulse, with central
wavelength $\lambda_{0}=1~{}\mathrm{\mu m}$, wave period
$T_{0}=\lambda_{0}/c$, rising time $2T_{0}$, duration time $15T_{0}$, ramping
time $2T_{0}$, and a Gaussian transverse (X,Y) envelope
$a=a_{0}\exp\left(-r^{2}/\sigma^{2}\right)$, here $\sigma=3\mu m$, $a_{0}=13$
corresponding to a peak laser intensity $I=4.6\times
10^{20}~{}\mathrm{W/cm^{2}}$, is normally incident from the left boundary
($z=0$) of a $100\times 12\times 12~{}\mathrm{\mu m^{3}}$ simulation box with
a grid of $1200\times 144\times 144$ cells. A near-critical density plasma
target consisting of electrons and protons is located in $6~{}\mathrm{\mu
m}<z<97~{}\mathrm{\mu m}$. In the laser propagation direction, the plasma
density rises linearly from $0$ to $n_{0}=0.8n_{c}$ in a distance of
$5~{}\mathrm{\mu m}$, and then remains constant, where
$n_{c}=m_{e}\omega_{0}^{2}\epsilon_{0}/e^{2}$ is the critical plasma density,
$m_{e}$ is the electron mass, and $\epsilon_{0}$ is the vacuum permittivity.
In the radial direction, the density is uniform. The number of super-particles
used in the simulation is about $1.8\times 10^{8}$ for each species (8
particles per cell for each species corresponds to $n_{0}$). An initial
electron temperature $T_{e}$ of $150~{}~{}\mathrm{keV}$ is used to resolve the
initial Debye length ( $T_{i}=10~{}~{}\mathrm{eV}$ initially).
Figure 2: (color online). Longitudinal (Z, X) cuts along the laser pulse axis
at $t=70T_{0}$, (a), instantaneous laser intensity distribution $I$,
normalized by the initial intensity $I_{0}=4.6\times
10^{20}~{}\mathrm{W/cm}^{2}$; (b), electron density distribution $n_{e}$,
normalized by the critical density $n_{c}$; (c), electron energy density
distribution, normalized by $n_{c}m_{e}c^{2}$; (d)(e), self-generated quasi-
static azimuthal and axial magnetic fields $B_{S\theta}$ and $B_{Sz}$,
averaged over $4$ laser periods, normalized by $m_{e}\omega_{0}/e$.
Figure 2 presents snapshots of simulation results at $t=70T_{0}$. After a
stage of filamentary and self-channelling, about $3/4$ of the laser energy has
been exhausted by the plasma. The laser pulse is slightly self-focused, and
the laser intensity is close to the initial intensity, i.e., $a\sim a_{0}$, as
shown in Fig. 2(a). Both electrons and ions are expelled by the self-focused
laser pulse, and a laser channel is formed. A strong current of relativistic
electrons is driven by the laser pulse in the direction of light propagation,
and confined in the laser channel. A helical high density electron beam is
formed in the center of the laser channel. In the longitudinal (Z, X) cut of
the electron density, the helical beam shows a zigzag profile, as shown in
Fig. 2(b), labeled by a white dashed box. The density of the beam is about
$n_{e}\sim 2n_{0}$. Then according to Eq.(2,3), we can get that,
$\nu=0.062,\quad\beta=0.175,\quad\gamma=422,\quad E_{c}=1MeV.$ (8)
The energy density distribution is shown in Fig. 2(c). It is shown that, most
of the electron energy is localized in the beam in the selected box. The total
energy of the electrons in the selected box is $0.9J$, which is $12\%$ of the
initial laser energy. Then we can get
$W_{r}=9mJ,\quad N_{\gamma}=6\times 10^{10},$ (9)
according to Eq. (4,5). The isosurface of the energy density with isosurface
value $190n_{c}m_{e}c^{2}$ in 3D is shown in Fig.1, which shows a helical
structure clearly. A strong quasi-static azimuthal magnetic field up to 0.5GG
is generated by the strong electron current, as shown in Fig. 2(d). Meanwhile,
a strong axial magnetic field up to 0.12GG, with spot size $R\sim 1\mu m$ is
generated, as shown in Fig. 2(e). Then we can get $\Delta\theta\sim 0.18rad$.
In this condition, electron acceleration is dominated by the self-matching
resonance acceleration regime smra . The accelerated relativistic electrons
are executing collective circularly betatron motion.
Figure 3: (color online). (a) Energy angular distribution of electrons in the
selected box in Fig. 2(b) at $t=70T_{0}$. (b) Energy spectra of electrons in
the selected box (solid line), and all electrons (dashed line) at time
$t=70T_{0}$. Inset figure shows time evolution of the maximum electron energy.
The spectra property of electrons in the selected box at $t=70T_{0}$ is shown
in Fig. 3. The energy angular distribution shows that, most of the high energy
electrons is distributed at a same angle of $\theta\sim 0.18rad$, with a
divergence angle (full angle) of $\Delta\theta\sim 0.15rad$, although the
energy is ranging from $50MeV$ to $290MeV$, as plotted in Fig. 3(a). This
means that the high energy electrons are executing a collective circularly
betatron motion, with a transverse velocity $\beta=0.18$, and a Lorentz factor
$\gamma$ ranging from $100$ to $550$. The simulation results coincide with the
theoretical estimation. The energy spectrum of electrons in the selected box
exhibits a plateau profile distribution, as shown in Fig. 3(b) by a solid
line. The inset figure plots time evolution of the maximum energy of
electrons. The electron energy increases dramatically at the begin, then
reaches the maximum value $300MeV$ at $t=70T_{0}$, and then decreases slowly,
since the driven laser pulse is exhausting. The energy spectrum of all
electrons is shown in Fig. 3(b) by a dashed line. It is shown that, most of
the high energy electrons are included in the selected box in Fig. 2(b).
Figure 4: (color online). Angular distribution of radiation energy with photon
energies above $100keV$. The radial coordinate and the angular coordinate,
labels the the polar angle $\theta$ and the azimuthal angle $\phi$ along the
laser propagation direction, respectively.
The angular distribution of the final radiation with photon energies above
$100keV$ is shown in Figure 3(a). The distribution is approximately azimuthal
symmetric about the laser propagation direction, and most of the radiation
energy is distributed in a polar angle ranging from $0.12rad$ to $0.35rad$,
with a peak value $3.7\times 10^{4}MeV/mrad^{2}$ at about $0.2rad$. The final
radiation distribution is a result of the energy angular distribution of high
energy electrons, and confirms that most of the high energy electrons are
executing collective circularly betatron motion. The total radiation energy
calculated by integrating all the angles is about $8.3mJ$, which is $0.1\%$ of
the incident laser energy. The corresponding photon number is $6.6\times
10^{10}$. The simulation results close to above theoretical estimation. It is
noticed that, most of the radiation energy emitted with a finite polar angle,
rather than that in most cases along the laser propagation direction. This is
because that the synchrotron-like radiation of relativistic electrons is
emitted almost along the electron motion direction, and the resonance
electrons have a large transverse velocity. The duration time of the gamma-ray
pulse is close to the length of the electron beam, is about $17fs$. Since the
radius of the radiation source is less than $1\mu m$, then we can get the
brightness of the gamma-ray emission with energies above $0.1E_{c}$ is
$1.5\times 10^{22}\rm{photons/s/mm^{2}/mrad^{2}/0.1\%bandwidth}$.
Figure 5: (color online). (a) Polar-angularly and spectrally resolved
radiation energy. (b) Radiation spectrum (radiation energy per $0.1\%$ band
width (BW)). (c)(d), Time evolution of critical photon energy, and total
radiation power, respectively.
More details of the radiation is shown in Fig. 5. Since the radiation is
azimuthal symmetric, we can plot the polar-angularly and spectrally resolved
radiation energy, as shown in Fig. 5(a). It is shown that, most of the
radiation energy is distributed at a peak angle $\sim 0.2rad$, and a
divergence angle (full angle) about $\sim 0.2rad$, with photon energy ranging
from $100keV$ to $20MeV$. The radiation energy spectrum by integrating the
polar angle is shown in Fig. 5(b). It is shown that, The peak of the spectrum
is located at $1.3MeV$. Since the spectrum is synchrotron-like, we can define
a critical photon energy, divided by which the integration of the two parts
are equal. Here the critical photon energy is $1.2MeV$ close to the peak
value, and agree well with above theoretical estimation. Fig. 5(c)(d) show
time evolution of the critical photon energy and the radiation power,
respectively. They are calculated by analyzing the radiation every per $10$
laser periods. It is shown that, the critical photon energy and the radiation
power show similar evolution in time. After a fast increasing, both the
critical photon energy and the radiation power reach peak values At
$t=70T_{0}$.
Figure 6: (color online). Variation of (a) critical photon energy, (b) total
radiation energy, with the incident laser energy $W_{I}$, by keeping
$l_{s}=\sqrt{an_{c}/n_{e}}$ fixed.
Above investigation can be extended to a large range of laser energies. We
simulated different laser plasma parameters, by keeping the dimensionless
plasma skin length $l_{s}=\sqrt{an_{c}/n_{e}}$ fixed, with initial laser
energy ranging from $1J$ to $21J$. We found that, the laser plasma
interactions exhibit a scaling property on $l_{s}$, especially, by keeping
$l_{s}$ fixed, the values of $n_{e}/n_{0}$ and $W_{I}/W_{ele}$ nearly keep
constant. Many other works also show that there is a scaling on $l_{s}$
scale_puk ; wang_scale . Since the energy of the laser pulse
$W_{I}=2\pi\sigma^{2}T_{L}I=2\pi\sigma^{2}T_{L}I_{1}a^{2}$, where
$T_{L}=17T_{0}$ is the effective laser duration time, and $I_{1}=1.37\times
10^{18}W/cm^{2}$ is the laser intensity when $a=1$, then we can get the
critical photon energy as a power function of the initial laser energy as
$E_{c}=16\hbar\omega_{0}l_{s}^{3}\left(\frac{W_{I}}{2\pi\sigma^{2}T_{L}I_{1}}\right)^{3/2}=5\times
10^{4}W_{I}[J]^{1.5}(eV).$ (10)
And the number of the gamma photons is proportional to $W_{I}$,
$N_{\gamma}=\frac{4\pi\alpha\beta}{3m_{e}c^{2}}\frac{W_{ele}}{W_{I}}W_{I}=8\times
10^{9}W_{I}[J].$ (11)
The simulation results of the critical photon energy $E_{c}$ and the photon
number with photon energies above $0.1E_{c}$ are shown in Fig. 5(a),(b),
respectively. The dashed lines are the theoretical estimation of Eq.
(10),(11). The simulation results agree well with the theoretical estimation.
The critical photon energy is increasing with the initial laser energy much
faster than a linear relation, which is the upper limit of the X-ray radiation
in under-dense plasma rmp_x-ray . It is noticed that the critical photon
energy, gamma photon number and radiation spectrum are similar in case of
Linear Polarized laser pulse, only the Angular distribution of radiation
energy is little different.
In conclusion, we have investigated electromagnetic emission by propagating an
$7.4J$ ultra-intense ultra-short laser pulse in a near critical density
plasma. $6.6\times 10^{10}$ gamma-ray photons with critical photon energy
$1MeV$ are emitted when electrons experience betatron resonance acceleration.
With the initial incident laser energy $W_{I}$ increasing, the critical photon
energy $E_{c}$ and the photon number $N_{\gamma}$ increase as $E_{c}\propto
W_{I}^{1.5}$, and $N_{\gamma}\propto W_{I}$, respectively.
This work was supported by National Basic Research Program of China (Grant No.
2013CBA01502),National Natural Science Foundation of China (Grant Nos.
11025523,10935002,10835003,J1103206) and National Grand Instrument
Projetc(2012YQ030142).
## References
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|
arxiv-papers
| 2013-10-23T09:39:46 |
2024-09-04T02:49:52.749752
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "B. Liu, H. Y. Wang, D. Wu, J. Liu, C.E.Chen, X. Q. Yan, X. T. He",
"submitter": "Xueqing Yan Dr",
"url": "https://arxiv.org/abs/1310.6164"
}
|
1310.6243
|
# Effect of the generalized uncertainty principle on Galilean and Lorentz
transformations
V. M. Tkachuk
Department for Theoretical Physics,
Ivan Franko National University of Lviv,
12 Drahomanov St., Lviv, UA-79005, Ukraine
e-mail: [email protected]
[email protected]
###### Abstract
Generalized Uncertainty Principle (GUP) was obtained in string theory and
quantum gravity and suggested the existence of a fundamental minimal length
which, as was established, can be obtained within the deformed Heisenberg
algebra. We use the deformed commutation relations or in classical case
(studied in this paper) the deformed Poisson brackets, which are invariant
with respect to the translation in configurational space. We have found
transformations relating coordinates and times of moving and rest frames of
reference in the space with GUP in the first order over parameter of
deformation. For the non-relativistic case we find the deformed Galilean
transformation which is similar to the Lorentz one written for Euclidean space
with signature $(+,+,+,+)$. The role of the speed of light here plays some
velocity $u$ related to the parameter of deformation, which as we estimate is
many order of magnitude larger than the speed of light $u\simeq 1.2\times
10^{22}c$. The coordinates of the rest and moving frames of reference for
relativistic particle in the space with GUP satisfy the Lorentz transformation
with some effective speed of light. We estimate that the relative deviation of
this effective speed of light $\tilde{c}$ from $c$ is ${(\tilde{c}-c)/c}\simeq
3.5\times 10^{-45}$. The influence of GUP on the motion of particle and the
Lorentz transformation in the first order over parameter of deformation is
hidden in $1/c^{2}$ relativistic effects.
## 1 Introduction
The investigations in string theory and quantum gravity (see, e.g., [1, 2, 3])
lead to the Generalized Uncertainty Principle (GUP)
$\displaystyle\Delta X\geq{\hbar\over 2}\left({1\over\Delta P}+\beta\Delta
P\right),$ (1)
from which follows the existence of the fundamental minimal length $\Delta
X_{\rm min}=\hbar\sqrt{\beta}$, which, as it is supposed, is of order of
Planck’s length $l_{p}=\sqrt{\hbar G/c^{3}}\simeq 1.6\times 10^{-35}\rm m$. A
broad recent review on this subject can be found in paper [4]. We would like
also to point out the recent discussion around the question whether we can
measure structures with precision better than the Planck’s length, which can
be found in [5].
It was established that minimal length can be obtained in the frame of small
quadratic modification (deformation) of the Heisenberg algebra [6, 7]
$\displaystyle[X,P]=i\hbar(1+\beta P^{2}).$ (2)
In the classical limit $\hbar\to 0$ the quantum-mechanical commutator for
operators is replaced by the Poisson bracket for corresponding classical
variables
$\displaystyle{1\over i\hbar}[X,P]\to\\{X,P\\},$ (3)
which in the deformed case (2) reads
$\displaystyle\\{X,P\\}=(1+\beta P^{2}).$ (4)
We would like to note that historically the first algebra of that kind in the
relativistic case was proposed by Snyder in 1947 [8]. But only investigations
in string theory and quantum gravity renewed the interest in the studies of
physical properties of classical and quantum systems in spaces with deformed
algebras. The observation that GUP can be obtained from the deformed
Heisenberg algebra opens the possibility to study the influence of minimal
length on properties of physical systems on the quantum level as well as on
the classical one.
Deformed commutation relations bring new difficulties in the quantum mechanics
as well as in the classical one. There are known only a few problems, which
can be solved exactly. Namely, one-dimensional harmonic oscillator with
minimal uncertainty in position [6] and also with minimal uncertainty in
position and momentum [9, 10], $D$-dimensional isotropic harmonic oscillator
[11, 12], three-dimensional Dirac oscillator [13], (1+1)-dimensional Dirac
oscillator within Lorentz-covariant deformed algebra [14], one-dimensional
Coulomb problem [15], the singular inverse square potential with a minimal
length [16, 17], the (2+1) dimensional Dirac equation in a constant magnetic
field in the presence of a minimal length [18]. Three-dimensional Coulomb
problem with the deformed Heisenberg algebra was studied within the
perturbation theory in [19, 20], where it was found that common perturbation
theory does not work for $ns$-levels. In [21, 22, 23] the modified
perturbation theory was proposed, which allows to obtain an explicit
expression for corrections to $ns$-levels for hydrogen atom caused by the
deformation of the Heisenberg algebra. In [24] the scattering problem in the
deformed space with minimal length was studied. The ultra-cold neutrons in
gravitational field with minimal length were considered in [25, 26, 27]. The
influence of minimal length on Lamb’s shift, Landau levels, and tunneling
current in scanning tunneling microscope was studied in [28, 29]. The Casimir
effect in a space with minimal length was examined in [30]. In paper [31] the
effect of noncommutativity and of the existence of a minimal length on the
phase space of cosmological model was investigated. The authors of paper [32]
studied various physical consequences, which follow from the noncommutative
Snyder space-time geometry. The gauge invariancy in space with GUP was
considered in [33]. In paper [34] the GUP and localization of a particle in a
discrete space was studied. Some consequences of the GUP-induced ultraviolet
wave-vector cutoff in one-dimensional quantum mechanics was studied in recent
paper [35]. The classical mechanics in a space with deformed Poisson brackets
was studied in [36, 37, 38]. The composite quantum and classical system
($N$-particle system) in the deformed space with minimal length was studied in
[39, 40].
The study of deviation from standard quantum mechanics as well as from
classical one caused by GUP gives a possibility to estimate the upper bound
for minimal length. The collection of upper boundes for minimal length
obtained form the investigation of different properties of different systems
can be found in recent paper [41]. The authors of this paper propose to use
the gravitational bar detectors to place an upper limit for a possible Planck-
scale modifications on the ground-state energy of an oscillator. In [42] the
authors propose to use the quantum-optical control of the mechanical system to
probe a possible deviation from the quantum commutation relation at the Planck
scale.
Note that deformation of the Heisenberg algebra and in classical case
respectively Poisson brackets bring not only technical difficulties in solving
of corresponding equations but also bring problems of a fundamental nature.
One of them is the violation of the equivalence principle in the space with
minimal length [43]. This is the result of assumption that the parameter of
deformation for macroscopic bodies of different mass is unique. In paper [39]
we showed that the center of mass of a macroscopic body in deformed space is
described by an effective parameter of deformation, which is essentially
smaller than the parameters of deformation for particles constituting the
body. Using the result of [39] for the effective parameter of deformation in
[45] we showed that the equivalence principle in the space with minimal length
can be recovered.
In this paper we study the Galilean and Lorentz transformations in space with
deformed Poisson brackets which correspond to the space with minimal length or
GUP. This paper organized as follows. In section 2 starting from a non-
relativistic Hamiltonian we find the Lagrangian of a particle in the space
with deformed Poisson brackets. In section 3 we study the invariancy of action
with the Lagrangian obtained in section 2 and find the deformed Galilean
transformation for coordinates of a non-relativistic particle in one-
dimensional space with GUP. In section 4 this result is generalized for the
three-dimensional case. The Lorentz transformation for coordinates of
relativistic particle in the space with GUP is studied in section 5. And
finally, in section 6 we conclude the results.
## 2 Hamiltonian and Lagrangian of a particle in deformed space
In this section we find the Lagrangian of a classical particle in space with
minimal length starting from the Hamiltonian formalism. It is commonly
supposed that Hamiltonian in deformed case has the form of Hamiltonian in non-
deformed case where instead of canonical variables of non-deformed phase space
are written variables of deformed phase space. So, the Hamiltonian of a
particle (a macroscopic body which we consider as a point particle) of mass
$m$ in the potential $U(X)$ moving in one-dimensional configurational space
reads
$\displaystyle H={P^{2}\over 2m}+U(X),$ (5)
where $X$ and $P$ satisfy deformed Poisson bracket (4). This Poisson bracket
allows the following coordinate representation
$\displaystyle P={1\over\sqrt{\beta}}\tan({\sqrt{\beta}p}),\ \ X=x,$ (6)
where small variables satisfy canonical Poisson bracket
$\displaystyle\\{x,p\\}=1$ (7)
and represent the non-deformed phase space. The Hamiltonian in this
representation reads
$\displaystyle H={\tan^{2}({\sqrt{\beta}p})\over 2m\beta}+U(x).$ (8)
As we see, the deformation of the Poisson bracket in representation (6) is
equivalent to the deformation of kinetic energy.
We consider the linear approximation over the parameter of deformation
$\beta$. In this approximation the Hamiltonian reads
$\displaystyle H={p^{2}\over 2m}+{1\over 3}{\beta\over m}p^{4}+U(x).$ (9)
This Hamiltonian is similar to the relativistic one written in the first order
over $1/c^{2}$
$\displaystyle H_{r}=mc^{2}\sqrt{1+{p^{2}\over
m^{2}c^{2}}}+U(x)=mc^{2}+{p^{2}\over 2m}-{1\over
8m^{3}c^{2}}p^{4}+U(x)+O(1/c^{4}).$ (10)
Introducing effective velocity
$\displaystyle u^{2}={3\over 8\beta m^{2}}.$ (11)
Hamiltonian (9) in the first order over $\beta$ or $1/u^{2}$ can be obtained
from the following one
$\displaystyle H=-mu^{2}\sqrt{1-{p^{2}\over m^{2}u^{2}}}+mu^{2}+U(x).$ (12)
This suggests that corrections to all properties related with deformations
will be similar to relativistic ones in the first order over $1/c^{2}$ but
with an opposite sign before $1/c^{2}$. In particular it suggests that the
Galilean transformations in the first order over $\beta$ will be similar to
the Lorentz one but with an opposite sign before $1/c^{2}$. Let us show it
subsequently.
Because $x$ and $p$ represent the non-deformed canonical space, the Lagrangian
can be found in the traditional way
$\displaystyle L=\dot{x}p-H(x,p),$ (13)
where $p$ is the function of $x$, $\dot{x}$ and can be found from equation
$\displaystyle\dot{x}={\partial H\over\partial p}={p\over m}+{4\over
3}{\beta\over m}p^{3}.$ (14)
In linear over $\beta$ approximation we find
$\displaystyle p=m\dot{x}\left(1-{4\over 3}\beta m^{2}\dot{x}^{2}\right).$
(15)
Substituting it into (13) we finally find the Lagrangian in the linear
approximation over $\beta$
$\displaystyle L={m\dot{x}^{2}\over 2}-{1\over 3}\beta m^{3}\dot{x}^{4}-U(x).$
(16)
Similarly as Hamiltonian (9) this Lagrangian is very similar to the Lagrangian
of a relativistic particle in first order over $1/c^{2}$, namely
$\displaystyle L_{r}=-mc^{2}\sqrt{1-{\dot{x}^{2}\over
c^{2}}}-U(x)=-mc^{2}+{m\dot{x}^{2}\over 2}+{m\over 8c^{2}}\dot{x}^{4}-U(x).$
(17)
The difference is only in constant $mc^{2}$ and opposite sing in the last
term. Thus, we rewrite Lagrangian (16) as follows
$\displaystyle L=mu^{2}\sqrt{1+{\dot{x}^{2}\over u^{2}}}-mu^{2}-U(x),$ (18)
where the effective velocity $u$ is the same as in (11). Of course, Lagrangian
(18) corresponds to (16) only in the first order over $1/u^{2}$ or $\beta$.
The constant $-mu^{2}$ does not influence the equation of motion and can be
omitted.
## 3 Galilean transformation in deformed space
To establish the Galilean transformation it is enough to consider free
particle with Lagrangian (18), where $U=0$. Omitting constant $-mu^{2}$ the
Lagrangian for free particle in first order over $\beta$ reads
$\displaystyle L=mu^{2}\sqrt{1+{\dot{x}^{2}\over u^{2}}}.$ (19)
So, in the first order over parameter of deformation $\beta$ the action reads
$\displaystyle S=mu^{2}\int_{t_{1}}^{t_{2}}\sqrt{1+{\dot{x}^{2}\over
u^{2}}}dt=mu^{2}\int_{(1)}^{(2)}ds,$ (20)
where
$\displaystyle ds^{2}=u^{2}(dt)^{2}+(dx)^{2}$ (21)
is squared interval in the Euclidean space whereas in relativistic case the
second term has an opposite sign and space is pseudo-Euclidean.
Interval (21) is invariant under rotation in plane ($ut,x$). So, symmetry
transformation reads
$\displaystyle x=x^{\prime}\cos\phi+ut^{\prime}\sin\phi,$ (22) $\displaystyle
ut=-x^{\prime}\sin\phi+ut^{\prime}\cos\phi.$ (23)
The angle $\phi$ is related with the velocity $V$ of motion of the point
$x^{\prime}=0$ with respect to the rest frame of reference
$\displaystyle{V\over u}={x\over ut}=\tan\phi.$ (24)
Then Galilean transformation reads
$\displaystyle x={x^{\prime}+Vt^{\prime}\over\sqrt{1+V^{2}/u^{2}}},\ \
t={t^{\prime}-x^{\prime}V/u^{2}\over\sqrt{1+V^{2}/u^{2}}}.$ (25)
We call it the deformed Galilean transformation. This transformation is very
similar to the Lorenz one. The important difference is that here we have an
opposite sign before $1/u^{2}$ that is the result of positive $\beta$, for
which just a minimal length exists. For negative $\beta$ the minimal length is
zero and according to (11) $1/u^{2}$ must be changed to $-1/u^{2}$. In this
case we have common Lorentz transformations where instead of speed of light
$c$ an effective velocity $u$ appears. Note that in fact this transformations
are correct only in the first order over the parameter of deformation $\beta$,
which is related with $1/u^{2}$ [see (11)]. So, in first order over parameter
of deformation we find
$\displaystyle x=(x^{\prime}+Vt^{\prime})\left(1-{V^{2}\over 2u^{2}}\right),\
\ t=t^{\prime}\left(1-{V^{2}\over 2u^{2}}\right)-x^{\prime}{V\over u^{2}}.$
(26)
In the limit $\beta\to 0$ or according to (11) $u\to\infty$ transformation
(26) recover ordinary Galilean transformation. Here it is interesting to note
that transformation (25) or (26) is one of the possible transformations, which
can be obtained in the frame of the following question asked in Special
Relativity Theory: what the most general transformations of spacetime were
that implemented the relativity principle, without making use of the
requirement of the constancy of the speed of light? For details, see section
“Algebraic and Geometric Structures in Special Relativity” in review [44].
Here it is worth to mention the result of paper [45] where we showed that for
a body of mass $m$ the parameter of deformation reads
$\displaystyle\beta={\gamma^{2}\over m^{2}},$ (27)
where $\gamma$ is the same constant for bodies of different mass. It is
interesting to note that constant $c\gamma$ is dimensional. Stress that that
only the relation (27) as was showed in paper [45] leads to recovering of the
equivalence principle in the deformed case. As a result of (27) we have
$\displaystyle u^{2}={3\over 8\gamma^{2}}.$ (28)
and thus the effective velocity does not depend on mass of a body. It means
that Galilean transformation is the same for coordinates of particles of
different mass as everybody feels it must be.
## 4 Three-dimensional case
The generalization of obtained Galilean transformation on three dimensional
case is straightforward. We consider deformed algebra, which is invariant with
respect to translations in configurational space. Different algebras of this
type can be found in [37] (see also references therein). One of the possible
algebra of this type reads
$\displaystyle[X_{i},P_{j}]=i\hbar\sqrt{1+\beta P^{2}}\left(\delta_{i,j}+\beta
P_{i}P_{j}\right),$ (29) $\displaystyle{}[X_{i},X_{j}]=[P_{i},P_{j}]=0.$ (30)
and can be obtained using the representation
$\displaystyle X_{i}=x_{i},\ \ P_{i}={p_{i}\over\sqrt{1-\beta p^{2}}},$ (31)
where ${\bf x}=(x_{1},x_{2},x_{3})$, ${\bf p}=(p_{1},p_{2},p_{3})$ represent
the coordinates and momentum in non-deformed space with canonical commutation
relations
$\displaystyle[x_{i},p_{j}]=\hbar\delta_{i,j},\ \
[x_{i},x_{j}]=[p_{i},p_{j}]=0.$ (32)
Note that in the momentum representation as follows from (31) $p^{2}<1/\beta$
and as a result there is nonzero minimal uncertainty in position or minimal
length. The algebra given by (29) and (30) is invariant with respect to the
transformation $\bf X=\bf X^{\prime}+\bf a$ and thus is translation-invariant
in configurational space. It means that the space is uniform.
Now we consider the classical limit $\hbar\to 0$. Then the deformed Poisson
brackets corresponding to algebra (29), (30) read
$\displaystyle\\{X_{i},P_{j}\\}=\sqrt{1+\beta P^{2}}\left(\delta_{i,j}+\beta
P_{i}P_{j}\right),$ (33)
$\displaystyle{}\\{X_{i},X_{j}\\}=\\{P_{i},P_{j}\\}=0.$ (34)
The Hamiltonian in representation (31) is the following
$\displaystyle H={1\over 2m}{p^{2}\over 1-\beta p^{2}}+U({\bf x})={p^{2}\over
2m}+{\beta\over 2m}p^{4}+U({\bf x})+O(\beta^{2}),$ (35)
where in our consideration we restrict oneself up to to the first order over
$\beta$.
Similarly as in one-dimensional case we find the Lagrangian corresponding to
Hamiltonian (35). First, we find the relation between the velocity and
momentum of the particle
$\displaystyle\dot{x}_{i}={1\over m}{p_{i}\over(1-\beta
p^{2})^{4}}={p_{i}\over m}(1+2\beta p^{2})+O(\beta^{2})$ (36)
and in first order over $\beta$ we obtain
$\displaystyle p_{i}=m\dot{x}_{i}(1-2\beta\dot{x}^{2}).$ (37)
The Lagrangian in this approximation reads
$\displaystyle L={m\dot{\bf x}^{2}\over 2}-{\beta m^{3}\over 2}\dot{\bf
x}^{4}-U({\bf x}).$ (38)
Similarly as in one-dimensional case this Lagrangian in the first order over
$\beta$ can be written in the form (18) and the action of free particle with
$U=0$ for three dimensional case takes form (20) where
$\displaystyle ds^{2}=u^{2}(dt)^{2}+(dx_{1})^{2}+(dx_{2})^{2}+(dx_{3})^{2},$
(39)
here
$\displaystyle u^{2}={1\over 4\beta m^{2}}={1\over 4\gamma^{2}}.$ (40)
In paper [45] from the suggestion that minimal length for electron is of order
of Planck’s length we estimate $\gamma$. Doing similarly we suggest that for
electron $\hbar\sqrt{\beta}=l_{p}$. Then taking into account relation (27) and
substituting for $m$ the mass of electron we find $c\gamma\simeq 4.2\times
10^{-23}$ that reproduce the result of paper [45]. Using this result we find
that $u\simeq 1.2\times 10^{22}c$ which is many order of magnitude large than
the speed of light.
Thus, when the second frame of reference $(t^{\prime},{\bf x^{\prime}})$ moves
with respect to the first one $(t,{\bf x})$ with velocity $V$ along axis
$x_{1}$ then Galilean transformation of coordinate $x^{\prime}_{1}$ and time
$t^{\prime}$ to $x_{1}$ and time $t$ satisfies (26), other coordinates are not
changed $x_{2}=x^{\prime}_{2},\ \ x_{3}=x^{\prime}_{3}$.
## 5 Lorentz transformation in deformed space
In this section we generalize the above consideration for the relativistic
case. Let us start from the one-dimensional relativistic Hamiltonian for free
particle
$\displaystyle H=mc^{2}\sqrt{1+{P^{2}\over m^{2}c^{2}}},$ (41)
where position and momentum satisfy deformed Poisson bracket (4). Using
representation (6) in the first order over $\beta$ and $1/c^{2}$ this
Hamiltonian reads
$\displaystyle H=m^{2}c^{2}+{p^{2}\over 2m}-\left({1\over
8m^{2}c^{2}}-{\beta\over 3}\right){p^{4}\over m}.$ (42)
Introducing notation
$\displaystyle{1\over 8m^{2}\tilde{c}^{2}}={1\over 8m^{2}c^{2}}-{\beta\over
3}$ (43)
we find that this Hamiltonian can be obtained in first order over
$1/\tilde{c}^{2}$ from the following one
$\displaystyle H=m\tilde{c}^{2}\sqrt{1+{p^{2}\over
m^{2}\tilde{c}^{2}}}-m\tilde{c}^{2}+mc^{2}.$ (44)
We suppose that $\beta$ is much smaller than $1/m^{2}c^{2}$. Then this
Hamiltonian corresponds to the relativistic one but with an effective velocity
$\tilde{c}$, which is defined by (43). Note that $\tilde{c}>c$ and
$\tilde{c}\to c$ when $\beta\to 0$. Thus transformation relating coordinates
and time of two reference frames is the Lorentz transformation which contains
instead of speed of light $c$ the effective speed $\tilde{c}$.
Taking into account (27) we find that (43) reads
$\displaystyle{1\over\tilde{c}^{2}}={1\over c^{2}}-{8\over 3}\gamma^{2}$ (45)
and thus the effective speed of light does not depend on the mass of a body.
It means that the Lorentz transformation is the same for particles of
different mass as it must be.
The generalization on three-dimensional case is straitforward. For the
deformed algebra given by (29), (30) we obtain the Hamiltonian in form (44)
where effective velocity is defined by
$\displaystyle{1\over\tilde{c}^{2}}={1\over c^{2}}-4\gamma^{2}.$ (46)
So, similarly as in the one-dimensional case the Lorentz transformation
contains instead of speed of light the effective speed of light. Note that for
different deformed algebras we obtain the same result, only the factor before
$\gamma^{2}$ will be different. In general we can write
$\displaystyle{1\over\tilde{c}^{2}}={1\over c^{2}}-{1\over u^{2}},$ (47)
where $u=\alpha c/\gamma$ and $\alpha$ is a multiplier different for different
algebras. The relative deviation of the effective speed of light $\tilde{c}$
from $c$ in the first order over the parameter of deformation $\beta$ or
$\gamma$ reads
$\displaystyle{\tilde{c}-c\over c}=2c^{2}\gamma^{2}\simeq 3.5\times 10^{-45},$
(48)
here we use that $c\gamma\simeq 4.2\times 10^{-23}$ [see explanation after eq.
(40)].
## 6 Conclusions
In the present paper we have found the transformations relating coordinates
and times of particle in moving and rest frames of reference in the space with
GUP or minimal length in the first order over the parameter of deformation.
For the description of the space with GUP we used the deformed algebra which
is invariant with respect to translation in the configurational space. In the
classical case considered in this paper we have corresponding deformed Poisson
brackets.
For the non-relativistic case we find that this transformation is similar to
the Lorentz one but for space with signature $(+,+,+,+)$. We call it the
deformed Galilean transformation and it is rotation in Euclidian space. The
role of the speed of light here plays some velocity $u$ which is inverse to
${\sqrt{\beta}}m$. It is important to note that, as we shown in our previous
paper [45], the equivalence principle and independence of kinetic energy on
composition of a body require that ${\sqrt{\beta}}m=\gamma$ is constant and
does not depend on the mass of the body. Doing similarly as in paper [45] we
suggest that minimal length for electron is of order of Planck’s length and
set $\hbar\sqrt{\beta}=l_{p}$, then $c\gamma\simeq 4.2\times 10^{-23}$.
Applying this result to the deformed Galilean transformation we find that this
transformation is the same for bodies of different mass as everybody feels it
must be and also estimate the effective velocity which is many orders of
magnitude larger than the speed of light $u\simeq 1.2\times 10^{22}c$.
Therefore, the effect of GUP on the motion of particle and Galilean
transformation is much order smaller the relativistic one. Note that the
deformed Galilean transformation in contrary to ordinal one contains also the
transformation of time which thus is not absolute in space with GUP. In the
limit $\beta\to 0$ or $\gamma\to 0$ the deformed Galilean transformation
recovers the ordinary one. Let us now explain qualitatively why the deformed
Galilean transformation is similar to the Lorentz one. Considering non-
relativistic case we start from the common non-relativistic Hamiltonian
written in deformed variables. But using the representation of deformed
variables over non-deformed ones we find that the Hamiltonian in the first
order over the parameter of deformation contains an additional term
proportional to $p^{4}$. This Hamiltonian is similar to the relativistic one
written in the first order over $1/c^{2}$ but with an opposite sign before
$p^{4}$. This very sign leads to a four-dimensional Euclidean space with
signature $(+,+,+,+)$ in contrary to the ordinary relativistic case with
pseudo-Euclidean space with signature $(+,-,-,-)$. It is interesting to note
that deformed Galilean transformation obtained here for space with GUP is
Euclidian rotation and it is one of the possible transformations, which can be
obtained in the frame of the following question asked in Special Relativity
Theory: what the most general transformations of spacetime were that
implemented the relativity principle, without making use of the requirement of
the constancy of the speed of light? For details, see section “Algebraic and
Geometric Structures in Special Relativity” in review [44].
The similarity of the deformed Galilean transformation to the Lorentz one
forced us to study the relativistic particle in a space with GUP predicting
that the effect of GUP can be hidden in the relativistic effect. We describe
the relativistic particle in space with minimal length or GUP by the
relativistic Hamiltonian which contains deformed variables instead of non-
deformed ones. Using the representation of deformed variables over non-
deformed ones we find that the Hamiltonian in the first order over the
parameter of deformation and first order over $1/c^{2}$ has also a
relativistic form in non-deformed variables with some effective speed of light
$\tilde{c}$. Therefore, coordinates of a relativistic particle in the rest and
moving frames of reference in space with minimal length satisfy the Lorentz
transformation with an effective speed of light. Similarly as in the non-
relativistic case the effective speed of light does not contain the mass of
particle when condition ${\sqrt{\beta}}m=\gamma$ holds and thus in this case
the Lorentz transformation is the same for coordinates and time of particles
of different masses. We estimate that the relative deviation of effective
speed of light $\tilde{c}$ from $c$ is very small ${(\tilde{c}-c)/c}\simeq
3.5\times 10^{-45}$. Finally let us note that the influence of GUP on the
motion of particle and the Lorentz transformation in the first order over the
parameter of deformation is hidden in $1/c^{2}$ relativistic effects.
## Acknowledgment
I am grateful to Dr. T. Masłowski for drawing my attention to review [44].
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|
arxiv-papers
| 2013-10-23T14:46:48 |
2024-09-04T02:49:52.758544
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "V. M. Tkachuk",
"submitter": "Volodymyr Tkachuk",
"url": "https://arxiv.org/abs/1310.6243"
}
|
1310.6303
|
# Simulation Over One-counter Nets is PSPACE-Complete ††thanks: Technical
Report EDI-INF-RR-1418 of the School of Informatics at the University of
Edinburgh, UK. (http://www.inf.ed.ac.uk/publications/report/). Extended
version of material presented at FST&TCS 2013. Made available at arXiv.org -
Creative Commons License CC-BY. This work was partially supported by Polish
NCN grant 2012/05/NST6/03226 and Polish MNiSW grant N N206 567840.
Piotr Hofman University of Warsaw, Poland Sławomir Lasota University of
Warsaw, Poland Richard Mayr University of Edinburgh, UK Patrick Totzke
University of Edinburgh, UK
###### Abstract
One-counter nets (OCN) are Petri nets with exactly one unbounded place. They
are equivalent to a subclass of one-counter automata with just a weak test for
zero. Unlike many other semantic equivalences, strong and weak simulation
preorder are decidable for OCN, but the computational complexity was an open
problem. We show that both strong and weak simulation preorder on OCN are
PSPACE-complete.
## 1 Introduction
The model. One-counter automata (OCA) are Minsky counter automata with only
one counter, and they can also be seen as a subclass of pushdown automata with
just one stack symbol (plus a bottom symbol). One-counter nets (OCN) are Petri
nets with exactly one unbounded place, and they correspond to a subclass of
OCA where the counter cannot be fully tested for zero, because transitions
enabled at counter value zero are also enabled at nonzero values. OCN are
arguably the simplest model of discrete infinite-state systems, except for
those that do not have a global finite control.
##### Previous results on semantic equivalence checking.
Notions of behavioral semantic equivalences have been classified in Van
Glabbeek’s linear time - branching time spectrum [3]. The most common ones
are, in order from finer to coarser, bisimulation, simulation and trace
equivalence. Each of these have their standard (called strong) variant, and a
weak variant that abstracts from arbitrarily long sequences of internal
actions.
For OCA/OCN, strong bisimulation is PSPACE-complete [2], while weak
bisimulation is undecidable [9]. Strong trace inclusion is undecidable for OCA
[11], and even for OCN [4], and this trivially carries over to weak trace
inclusion.
The picture is more complicated for simulation preorders. While strong and
weak simulation are undecidable for OCA [7], they are decidable for OCN.
Decidability of strong simulation on OCN was first proven in [1], by
establishing that the simulation relation follows a certain regular pattern.
This idea was made more graphically explicit in later proofs [6, 5], which
established the so-called Belt Theorem, that states that the simulation
preorder relation on OCN can be described by finitely many partitionings of
the grid $\mathbb{N}\times\mathbb{N}$, each induced by two parallel lines. In
particular, this implies that the simulation relation is semilinear. However,
the proofs in [1, 6, 5] did not yield any upper complexity bounds, since the
first was based on two semi-decision procedures and the later proof of the
Belt Theorem was non-constructive. A PSPACE lower bound for strong simulation
on OCN follows from [10].
Decidability of weak simulation on OCN was shown in [4], using a converging
series of semilinear approximants. This proof used the decidability of strong
simulation on OCN as an oracle, and thus did not immediately yield any upper
complexity bound.
##### Our contribution.
We provide a new constructive proof of the Belt Theorem and derive a PSPACE
algorithm for checking strong simulation preorder on OCN. Together with the
lower bound from [10], this shows PSPACE-completeness of the problem.
Via a technical adaption of the algorithm for weak simulation in [4], and the
new PSPACE algorithm for strong simulation, we also obtain a PSPACE algorithm
for weak simulation preorder on OCN. Thus even weak simulation preorder on OCN
is PSPACE-complete.
| simulation | bisimulation | weak sim. | weak bis. | trace inclusion
---|---|---|---|---|---
OCN | PSPACE | PSPACE [2] | PSPACE | undecidable [9] | undecidable [11]
OCA | undecidable [7] | PSPACE [2] | undecidable [7] | undecidable [9] | undecidable [4]
## 2 Problem Statement
A labelled transition system (LTS) over a finite alphabet $A$ of actions
consists of a set of configurations and, for every action $a\in A$, a binary
relation $\,{\stackrel{{\scriptstyle a}}{{\longrightarrow}}}\\!\,$ between
configurations.
Given two LTS $S$ and $S^{\prime}$, a relation $R$ between the configurations
of $S$ and $S^{\prime}$ is a _simulation_ if for every pair of configurations
$(c,c^{\prime})\in R$ and every step $c\,{\stackrel{{\scriptstyle
a}}{{\longrightarrow}}}\\!\,d$ there exists a step
$c^{\prime}\,{\stackrel{{\scriptstyle a}}{{\longrightarrow}}}\\!\,d^{\prime}$
such that $(d,d^{\prime})\in R$. Simulations are closed under union, so there
exists a unique maximal simulation. If $S=S^{\prime}$ then this maximal
simulation is a preorder, called simulation preorder, and denoted by
$\preccurlyeq$. If $c\preccurlyeq c^{\prime}$ then one says that $c^{\prime}$
_simulates_ $c$.
Simulation preorder can also be characterized by a _Simulation Game_ as
follows. The _positions_ are all pairs $(c,c^{\prime})$ of configurations of
$S$ and $S^{\prime}$ respectively. The game is played by two players called
_Spoiler_ and _Duplicator_ and proceeds in rounds. In every round, starting in
a position $(c,c^{\prime})$, Spoiler chooses some $a\in A$ and some
configuration $d$ with $c\,{\stackrel{{\scriptstyle
a}}{{\longrightarrow}}}\\!\,d$. Then Duplicator responds by choosing a
configuration $d^{\prime}$ with $c^{\prime}\,{\stackrel{{\scriptstyle
a}}{{\longrightarrow}}}\\!\,d^{\prime}$, and the next round continues from
position $(d,d^{\prime})$. If one of the players cannot move then the other
player wins, and Duplicator wins every infinite play. It is well known that
the Simulation Game is determined: for every initial position
$(c,c^{\prime})$, exactly one of players has a winning strategy. Configuration
$c^{\prime}$ simulates $c$ iff Duplicator has a strategy to win the Simulation
Game from position $(c,c^{\prime})$.
###### Definition 1 (One-Counter Nets).
A _one-counter net_ (OCN) is a triple ${\cal N}=(Q,A,\delta)$ given by finite
sets of control-states $Q$, action labels $A$ and transitions $\delta\subseteq
Q\times A\times\\{-1,0,1\\}\times Q$. It induces an infinite-state labelled
transition system over the state set $Q\times\mathbb{N}$, whose elements will
be written as $pm$, where $pm\,{\stackrel{{\scriptstyle
a}}{{\longrightarrow}}}\\!\,qn$ iff $(p,a,d,q)\in\delta\text{ and }n=m+d\geq
0$.
We study the computational complexity of the following decision problem.
Simulation Checking for OCN
---
Input: | Two OCN ${\cal N}$ and ${\cal N}^{\prime}$ together with configurations $qn$ and $q^{\prime}n^{\prime}$
| of ${\cal N}$ and ${\cal N}^{\prime}$ respectively, where $n$ and
$n^{\prime}$ are given in binary.
Question: | $qn\preccurlyeq q^{\prime}n^{\prime}$ ?
###### Theorem 2.
The Simulation Checking Problem for OCN is in PSPACE.
Combined with the PSPACE-hardness result of [10], this yields PSPACE-
completeness of the problem.
###### Remark 3.
Our construction can also be used to compute the simulation relation as a
semilinear set, but its description requires exponential space. However,
checking a point instance $qn\preccurlyeq q^{\prime}n^{\prime}$ of the
simulation problem can be done in polynomial space by stepwise guessing and
verifying only a polynomialy bounded part of the relation; cf. Section 5.
Without restriction (see [1] for a justification) we assume that both OCN are
_normalised_ :
1. 1.
In Spoiler’s net ${\cal N}$, every control-state has some outgoing transition
with a non-negative change of counter value.
2. 2.
Duplicator’s net ${\cal N}^{\prime}$ is _complete_ , i.e., every control-state
has an outgoing transition for every action (though the change in counter
value may be negative).
Thus Spoiler cannot get stuck and only loses the game if it is infinite.
Moreover, Duplicator can only be stuck (and lose the game) when his counter
equals zero.
##### Outline of the proof.
One easily observes that the Simulation Game is monotone for both players. If
Duplicator wins the Simulation Game from a position
$(qn,q^{\prime}n^{\prime})$ then he also wins from $(qn,q^{\prime}m)$ for
$m>n^{\prime}$. Similarly, if Spoiler wins from $(qn,q^{\prime}n^{\prime})$
then she also wins from $(qm,q^{\prime}n^{\prime})$ for $m>n$. For a fixed
pair $(q,q^{\prime})$ of control-states, both players winning regions
therefore split the grid $\mathbb{N}\times\mathbb{N}$ into two connected
subsets. It is known [6, 5] that the _frontier_ between these subsets is
contained in a _belt_ , i.e., it lays between two parallel lines with rational
slope.
For the proof of our main result we analyse a symbolic _Slope Game_. This new
game is similar to the Simulation Game but necessarily ends after a small
number of rounds. We show that given sufficiently high excess of counter-
values, both players can re-use winning strategies for the Slope Game also in
the Simulation Game. As a by-product of this characterization, we obtain
polynomial bounds on widths and slopes of the belts. Once the belt-
coefficients are known, one can compute the frontiers exactly because every
frontier necessarily adheres to a regular pattern.
## 3 Polynomially Bounded Belts
Let us fix two OCN ${\cal N}$ and ${\cal N}^{\prime}$, with sets of control-
states $Q$ and $Q^{\prime}$, respectively. Following [5], we interpret
$\,\preccurlyeq\,$ as 2-colouring of $K=|Q\times Q^{\prime}|$ Euclidean
planes, one for each pair of control-states $(q,q^{\prime})\in Q\times
Q^{\prime}$.
The main combinatorial insight of [5] (this was also present in [1], albeit
less explicitly) is the so-called _Belt Theorem_ , that states that each such
plane can be cut into segments by two parallel lines such that the colouring
of $\,\preccurlyeq\,$ in the outer two segments is constant; see Figure 1. We
provide a new constructive proof of this theorem, stated as Theorem 5 below,
that allows us to derive polynomial bounds on the coefficients of all belts.
###### Definition 4 (Positive vectors, direction, c-above, c-below).
A vector $(\rho,\rho^{\prime})\in\mathbb{Z}\times\mathbb{Z}$ of integers is
called _positive_ if $(\rho,\rho^{\prime})\in\mathbb{N}\times\mathbb{N}$ and
$(\rho,\rho^{\prime})\neq(0,0)$. Its _direction_ is the half-line
$\mathbb{R}^{+}\cdot(\rho,\rho^{\prime})$. For a positive vector
$(\rho,\rho^{\prime})$ and a number $c\in\mathbb{N}$ we say that the point
$(n,n^{\prime})\in\mathbb{Z}\times\mathbb{Z}$ is _$c$ -above_
$(\rho,\rho^{\prime})$ iff there exists some point
$(r,r^{\prime})\in\mathbb{R}^{+}\cdot(\rho,\rho^{\prime})$ in the direction of
$(\rho,\rho^{\prime})$ such that
$n<r-c\qquad\text{and}\qquad n^{\prime}>r^{\prime}+c.$ (1)
Symmetrically, $(n,n^{\prime})$ is _$c$ -below_ $(\rho,\rho^{\prime})$ if is a
point $(r,r^{\prime})\in\mathbb{R}^{+}\cdot(\rho,\rho^{\prime})$ with
$n>r+c\qquad\text{and}\qquad n^{\prime}<r^{\prime}-c.$ (2)
###### Theorem 5 (Belt Theorem).
For every two one-counter nets ${\cal N}$ and ${\cal N}^{\prime}$ with sets of
control-states $Q$ and $Q^{\prime}$ respectively, there is a bound
$c\in\mathbb{N}$ such that for every pair $(q,q^{\prime})\in Q\times
Q^{\prime}$ of control-states there is a positive vector
$(\rho,\rho^{\prime})$ such that
1. 1.
if $(n,n^{\prime})$ is $c$-above $(\rho,\rho^{\prime})$ then
$qn\,\preccurlyeq\,q^{\prime}n^{\prime}$, and
2. 2.
if $(n,n^{\prime})$ is $c$-below $(\rho,\rho^{\prime})$ then
$qn\,\not\preccurlyeq\,q^{\prime}n^{\prime}$.
Moreover, $c$ and all $\rho,\rho^{\prime}$ are bounded polynomially w.r.t. the
sizes of ${\cal N}$ and ${\cal N}^{\prime}$.
Duplicator $n^{\prime}$Spoiler
$n$$(\rho,\rho^{\prime})$$c$$\preceq$$\not\preceq$ Figure 1: A belt with slope
$\frac{\rho}{\rho^{\prime}}$. The dashed half-line is the direction of
$(\rho,\rho^{\prime})$.
## 4 Proof of the Belt Theorem
We consider OCN ${\cal N}$ and ${\cal N}^{\prime}$ with sets of control-states
$Q$ and $Q^{\prime}$, resp., and define the constant $K=|Q\times Q^{\prime}|$.
Abdulla and Cerans [1] showed that, above a certain level, the simulation
relation has a regular structure. An important parameter for this structure is
the ratio $n/n^{\prime}$ of the respective counter values $n$ in Spoiler’s
configuration $qn$ of ${\cal N}$ and $n^{\prime}$ in Duplicator’s
configuration $q^{\prime}n^{\prime}$ of ${\cal N}^{\prime}$.
We further develop this intuition by defining a new finitary game (called the
Slope Game; cf. Section 4.1) that is played directly on the control graphs of
the nets, and in which the objective of the players is to minimize (resp.
maximize) the ratio of the effects of recently observed minimal cycles. Then
we show how to transform winning strategies in the Slope Game into winning
strategies in the original simulation game. First we need to define some
properties of vectors.
###### Definition 6 (Behind, Steeper).
Let $(\rho,\rho^{\prime})$ be a positive and
$(\alpha,\alpha^{\prime})\in\mathbb{Z}^{2}$ an arbitrary vector. We place the
two on the plane with a common starting point and consider the clockwise
oriented angle from $(\rho,\rho^{\prime})$ to $(\alpha,\alpha^{\prime})$. We
say that $(\alpha,\alpha^{\prime})$ is _behind_ $(\rho,\rho^{\prime})$ if the
oriented angle is strictly between $0^{\circ}$ and $180^{\circ}$. See Figure 3
for an illustration.
Positive vectors may be naturally ordered: We will call $(\rho,\rho^{\prime})$
_steeper_ than $(\alpha,\alpha^{\prime})$, written
$(\alpha,\alpha^{\prime})\prec(\rho,\rho^{\prime})$, if
$(\alpha,\alpha^{\prime})$ is behind $(\rho,\rho^{\prime})$.
Note that the property of one vector being behind another only depends on
their directions. The following simple lemma will be useful in the sequel.
###### Lemma 7.
Let $(\rho,\rho^{\prime})$ be a positive vector and
$c,n,n^{\prime}\in\mathbb{N}$.
1. 1.
If $(n,n^{\prime})$ is $c$-below $(\rho,\rho^{\prime})$ then
$(n,n^{\prime})+(\alpha,\alpha^{\prime})$ is $c$-below $(\rho,\rho^{\prime})$
for any vector $(\alpha,\alpha^{\prime})$ which is behind
$(\rho,\rho^{\prime})$.
2. 2.
If $(n,n^{\prime})$ is $c$-above $(\rho,\rho^{\prime})$ then
$(n,n^{\prime})+(\alpha,\alpha^{\prime})$ is $c$-above $(\rho,\rho^{\prime})$
for any vector $(\alpha,\alpha^{\prime})$ which is not behind
$(\rho,\rho^{\prime})$.
Figure 2: Vectors $(\alpha,\alpha^{\prime})$ and $(\beta,\beta^{\prime})$ are
behind $(\rho,\rho^{\prime})$, but $(\delta,\delta^{\prime})$ is not. Also,
$(\alpha,\alpha^{\prime})\prec(\rho,\rho^{\prime})$.
Figure 3: Evaluating the winning condition in position
$(\pi,(\rho,\rho^{\prime}))$ after a phase of the Slope Game.
### 4.1 Slope Game
###### Definition 8 (Product Control Graph, Lasso, Effect of a path).
Given two OCN ${\cal N}=(Q,A,\delta)$ and ${\cal
N}^{\prime}=(Q^{\prime},A,\delta^{\prime})$, their _product control graph_ is
the finite, edge-labelled graph with nodes $Q\times Q^{\prime}$ and
$(A\times\mathbb{N}\times\mathbb{N})$-labelled edges $E$ given by
$(p,p^{\prime})\,{\stackrel{{\scriptstyle
a,d,d^{\prime}}}{{\longrightarrow}}}\\!\,(q,q^{\prime})\in E\text{ iff
}p\,{\stackrel{{\scriptstyle a,d}}{{\longrightarrow}}}\\!\,q\in\delta\text{
and }p^{\prime}\,{\stackrel{{\scriptstyle
a,d^{\prime}}}{{\longrightarrow}}}\\!\,q^{\prime}\in\delta^{\prime}.$ (3)
A _path_
$\pi=(q_{0},q^{\prime}_{0})\,{\stackrel{{\scriptstyle
a_{0},d_{0},d_{0}^{\prime}}}{{\longrightarrow}}}\\!\,(q_{1},q^{\prime}_{1})\,{\stackrel{{\scriptstyle
a_{1},d_{1},d^{\prime}_{1}}}{{\longrightarrow}}}\\!\,\dots\,{\stackrel{{\scriptstyle
a_{k-1},d_{k-1},d^{\prime}_{k-1}}}{{\longrightarrow}}}\\!\,(q_{k},q^{\prime}_{k})$
(4)
from $(q_{0},q^{\prime}_{0})$ to $(q_{k},q^{\prime}_{k})$ in this graph is
called _lasso_ if it contains a cycle while none of its strict prefixes does.
That is, if there exist $i<k$ such that
$(q_{k},q^{\prime}_{k})=(q_{i},q^{\prime}_{i})$ and for all $0\leq i<j<k$,
$(q_{i},q^{\prime}_{i})\neq(q_{j},q^{\prime}_{j})$. The lasso $\pi$ splits
into $\text{\sc prefix}(\pi)=(q_{0},q^{\prime}_{0})\,{\stackrel{{\scriptstyle
a_{0},d_{0},d_{0}^{\prime}}}{{\longrightarrow}}}\\!\,\dots\,{\stackrel{{\scriptstyle
a_{i-1},d_{i-1},d^{\prime}_{i-1}}}{{\longrightarrow}}}\\!\,(q_{i},q^{\prime}_{i})$
and $\text{\sc cycle}(\pi)=(q_{i},q^{\prime}_{i})\,{\stackrel{{\scriptstyle
a_{i},d_{i},d_{i}^{\prime}}}{{\longrightarrow}}}\\!\,\dots\,{\stackrel{{\scriptstyle
a_{k-1},d_{k-1},d^{\prime}_{k-1}}}{{\longrightarrow}}}\\!\,(q_{k},q^{\prime}_{k})$.
The _effect_ of a path is the cumulative sum of the effects of its
transitions:
$\Delta(\pi)=\sum_{i=0}^{k-1}(d_{i},d_{i}^{\prime})\in\mathbb{Z}\times\mathbb{Z}.$
(5)
The effects of cycles will play a central role in our further construction.
The intuition is that if a play of a Simulation Game describes a lasso then
the players “agree” on the chosen cycle. Repeating this cycle will change the
ratio of the counter values towards its effect.
To formalize this intuition, we define a finitary Slope Game which proceeds in
phases. In each phase, the players alternatingly move on the control graphs of
their original nets, ignoring the counter, and thereby determine the next
lasso that occurs. After such a phase, a winning condition is evaluated that
compares the effect of the chosen lasso’s cycle with that of previous phases.
Now either one player immediately wins or the next phase starts, but then the
steepness of the observed effect must have strictly decreased. The number of
different effects of simple cycles thus bounds the maximal length of a game.
###### Definition 9 (Slope Game).
A _Slope Game_ is a strictly alternating two player game played on a pair
${\cal N},{\cal N}^{\prime}$ of one-counter nets. The game positions are pairs
$(\pi,(\rho,\rho^{\prime}))$, where $\pi$ is an acyclic path in the product
control graph of ${\cal N}$ and ${\cal N}^{\prime}$, and
$(\rho,\rho^{\prime})$ is a positive vector which we call _slope_.
The game is divided into _phases_ , each starting with a path
$\pi=(q_{0},q^{\prime}_{0})$ of length $0$. Until a phase ends, the game
proceeds in rounds like a Simulation Game, but the players pick transition
rules instead of transitions: in a position $(\pi,(\rho,\rho^{\prime}))$ where
$\pi$ ends in states $(q,q^{\prime})$, Spoiler chooses a transition rule
$q\,{\stackrel{{\scriptstyle a,d}}{{\longrightarrow}}}\\!\,p$, then Duplicator
responds with a transition rule $q^{\prime}\,{\stackrel{{\scriptstyle
a,d}}{{\longrightarrow}}}\\!\,p^{\prime}$. If the extended path
$\pi^{\prime}=\pi\,{\stackrel{{\scriptstyle
a,d,d^{\prime}}}{{\longrightarrow}}}\\!\,(p,p^{\prime})$ is still not a lasso,
the next round continues from the updated position
$(\pi^{\prime},(\rho,\rho^{\prime}))$; otherwise the phase ends with _outcome_
$(\pi^{\prime},(\rho,\rho^{\prime}))$. The slope $(\rho,\rho^{\prime})$ does
not restrict the possible moves of either player, nor changes during a phase.
We thus speak of _the slope of a phase_.
If a round ends in position $(\pi,(\rho,\rho^{\prime}))$ where $\pi$ is a
lasso, then the winning condition is evaluated. We distinguish three non-
intersecting cases depending on how the effect $\Delta(\text{\sc
cycle}(\pi))=(\alpha,\alpha^{\prime})$ of the lasso’s cycle relates to
$(\rho,\rho^{\prime})$:
1. 1.
If $(\alpha,\alpha^{\prime})$ is not behind $(\rho,\rho^{\prime})$, Duplicator
wins immediately.
2. 2.
If $(\alpha,\alpha^{\prime})$ is behind $(\rho,\rho^{\prime})$ but not
positive, Spoiler wins immediately.
3. 3.
If $(\alpha,\alpha^{\prime})$ is behind $(\rho,\rho^{\prime})$ and positive,
the game continues with a new phase from position
$(\pi^{\prime},(\alpha,\alpha^{\prime}))$, where $\pi^{\prime}$ is the path of
length $0$ consisting of the pair of ending states of $\pi$.
Figure 3 illustrates the winning condition. Note that if there is no immediate
winner it is guaranteed that $(\alpha,\alpha^{\prime})$ is a positive vector.
The fundamental intuition for the connection between the Slope Game and the
Simulation Game is as follows. The Slope Game from initial position
$((q,q^{\prime}),(\rho,\rho^{\prime}))$ determines how the initial slope
$(\rho,\rho^{\prime})$ relates to the belt in the plane for $(q,q^{\prime})$
in the simulation relation. Roughly speaking, if $(\rho,\rho^{\prime})$ is
less steep than the belt then Spoiler wins; if $(\rho,\rho^{\prime})$ is
steeper then Duplicator wins. Finally, when the initial slope
$(\rho,\rho^{\prime})$ is exactly as steep as the belt, any player may win the
Slope Game.
Consider a Simulation Game in which the ratio $n/n^{\prime}$ of the counter
values of Spoiler and Duplicator is the same as the ratio
$\rho/\rho^{\prime}$, i.e. suppose $(n,n^{\prime})$ is contained in the
direction of $(\rho,\rho^{\prime})$. Suppose also that the values
$(n,n^{\prime})$ are sufficiently large. By monotonicity, we know that the
steeper the slope $(\rho,\rho^{\prime})$, the better for Duplicator. Hence if
the effect $(\alpha,\alpha^{\prime})$ of some cycle is behind
$(\rho,\rho^{\prime})$ and positive, then it is beneficial for Spoiler to
repeat this cycle. With more and more repetitions, the ratio of the counter
values will get arbitrarily close to $(\alpha,\alpha^{\prime})$. On the other
hand, if $(\alpha,\alpha^{\prime})$ is behind $(\rho,\rho^{\prime})$ but not
positive then Spoiler wins by repeating the cycle until the Duplicator’s
counter decreases to $0$. Finally, if the effect of the cycle is not behind
$(\rho,\rho^{\prime})$ then repeating this cycle leads to Duplicator’s win.
The next lemma follows from the observation that in Slope Games, the slope of
a phase must be strictly less steep than those of all previous phases.
###### Lemma 10.
For a fixed pair ${\cal N},{\cal N}^{\prime}$ of OCN,
1. 1.
any Slope Game ends after at most $(\text{\sc K}+1)^{2}$ phases, and
2. 2.
Slope Games are effectively solvable in PSPACE.
###### Proof.
After every phase, the slope $(\rho,\rho^{\prime})$ is equal to the effect of
a simple cycle, which must be a positive vector. Thus the absolute values of
both numbers $\rho$ and $\rho^{\prime}$ are bounded by $\text{\sc K}=|Q\times
Q^{\prime}|$. It follows that the total number of different possible values
for $(\rho,\rho^{\prime})$, and therefore the maximal number of phases played,
is at most $(\text{\sc K}+1)^{2}$. This proves the first part of the claim.
Point 2 is a direct consequence as one can find and verify winning strategies
by an exhaustive search. ∎
##### Strategies in Slope Games.
Consider one phase of a Slope Game, starting from a position
$(\pi,(\rho,\rho^{\prime}))$. The phase ends with a lasso whose cycle effect
$(\alpha,\alpha^{\prime})$ satisfies exactly one of three conditions, as
examined by the evaluating function. Accordingly, depending on its initial
position, every phase falls into exactly one of three disjoint cases:
1. 1.
Spoiler has a strategy to win the Slope Game immediately,
2. 2.
Duplicator has a strategy to win the Slope Game immediately or
3. 3.
neither Spoiler nor Duplicator have a strategy to win immediately.
In case 1. or 2. we call the phase _final_ , and in case 3. we call it _non-
final_. The non-final phases are the most interesting ones because in those,
both players have a strategy that at least prevents an immediate loss.
##### Strategy Trees.
Both in final and non-final phases, a strategy for Spoiler or Duplicator is a
tree as described below. For the definition of strategy trees we need to
consider, not only Spoiler’s positions $(\pi,(\rho,\rho^{\prime}))$ but also
Duplicator’s positions, the intermediate positions within a single round.
These intermediate positions may be modelled as triples
$(\pi,(\rho,\rho^{\prime}),t)$ where $t$ is a transition rule in ${\cal N}$
from the last state of $\pi$. Observe that the bipartite directed graph, with
positions of a phase as vertices and edges determined by the single-move
relation, is actually a tree, call it $T$. Thus a Spoiler-strategy, i.e. a
subgraph of $T$ containing exactly one successor of every Spoiler’s position
and all successors of every Duplicator’s position, is a tree as well; and so
is any strategy for Duplicator.
Such a strategy (tree) in the Slope Game naturally splits into _segments_ ,
each segment being a strategy (tree) in one phase. The segments themselves are
also arranged into a tree, which we call _segment tree_. Irrespectively which
player wins a Slope Game, according to the above observations, this player’s
winning strategy contains segments of two kinds:
* •
non-leaf segments are strategies to either win immediately or continue the
Slope Game (these are strategies for non-final phases);
* •
leaf segments are strategies to win the Slope Game immediately (these are
strategies in final phases).
By the _segment depth_ of a strategy we mean the depth of its segment tree. By
Lemma 10, Point 1, we know that a Slope Game ends after at most
$d_{\text{max}}=(\text{\sc K}+1)^{2}$ phases. Consequently, the segment depths
of strategies are at most $d_{\text{max}}$ as well.
A value of $c=\text{\sc K}\cdot d_{\text{max}}$ is sufficient for the claim of
Theorem 5. The intuition behind this value is that for a winning player in the
Slope Game, an excess of K per phase is sufficient to be able to safely
“replay” a winning strategy in the Simulation Game. Formally, this is stated
by the following two crucial lemmas, proofs of which can be found in Appendix
A.
###### Lemma 11.
Suppose Spoiler has a winning strategy of segment depth $d$ in the Slope Game
from a position $((q,q^{\prime}),(\rho,\rho^{\prime}))$. Then Spoiler wins the
Simulation Game from every position $(qn,q^{\prime}n^{\prime})$ which is
$(\text{\sc K}\cdot d)$-below $(\rho,\rho^{\prime})$.
###### Lemma 12.
Suppose Duplicator has a winning strategy of segment depth $d$ in the Slope
Game from a position $((q,q^{\prime}),(\rho,\rho^{\prime}))$. Then Duplicator
wins the Simulation Game from every position $(qn,q^{\prime}n^{\prime})$ which
is $(\text{\sc K}\cdot d)$-above $(\rho,\rho^{\prime})$.
### 4.2 Proof of Theorem 5
Let $c=\text{\sc K}\cdot d_{max}$. For any two states $q\in Q$ and
$q^{\prime}\in Q^{\prime}$ of the nets ${\cal N}$ and ${\cal N}^{\prime}$ we
will determine the ratio $(\rho,\rho^{\prime})$ that, together with $c$,
characterises the belt of the plane $(q,q^{\prime})$. First observe the
following monotonicity property of the Slope Game.
###### Lemma 13.
If Spoiler wins the Slope Game from a position
$((q,q^{\prime}),(\rho,\rho^{\prime}))$ and $(\sigma,\sigma^{\prime})$ is less
steep than $(\rho,\rho^{\prime})$ then Spoiler also wins the Slope Game from
$((q,q^{\prime}),(\sigma,\sigma^{\prime}))$.
###### Proof.
Assume that Spoiler wins the Slope Game from
$((q,q^{\prime}),(\rho,\rho^{\prime}))$ while Duplicator wins from
$((q,q^{\prime}),(\sigma,\sigma^{\prime}))$, for some
$(\sigma,\sigma^{\prime})\prec(\rho,\rho^{\prime})$. Observe that in both
cases, winning strategies of segment depth $\leq d_{\text{max}}$ exist. As
$(\sigma,\sigma^{\prime})$ is less steep than $(\rho,\rho^{\prime})$, there is
a point $(n,n^{\prime})\in\mathbb{N}\times\mathbb{N}$ which is both $c$-above
$(\sigma,\sigma^{\prime})$ and $c$-below $(\rho,\rho^{\prime})$. Applying both
Lemma 11 and 12 immediately yields a contradiction. ∎
Equivalently, if Duplicator wins the Slope Game from
$((q,q^{\prime}),(\rho,\rho^{\prime}))$ and $(\sigma,\sigma^{\prime})$ is
steeper than $(\rho,\rho^{\prime})$ then Duplicator also wins the Slope Game
from $((q,q^{\prime}),(\sigma,\sigma^{\prime}))$. We conclude that for every
pair $(q,q^{\prime})$ of states, there is a _boundary slope_
$(\beta,\beta^{\prime})$ such that
1. 1.
Spoiler wins the Slope Game from $((q,q^{\prime}),(\sigma,\sigma^{\prime}))$
for every $(\sigma,\sigma^{\prime})$ less steep than $(\beta,\beta^{\prime})$;
2. 2.
Duplicator wins the Slope Game from
$((q,q^{\prime}),(\sigma,\sigma^{\prime}))$ for every
$(\sigma,\sigma^{\prime})$ steeper than $(\beta,\beta^{\prime})$.
Note that we claim nothing about the winner from the position
$((q,q^{\prime}),(\beta,\beta^{\prime}))$ itself. Applying Lemmas 11 and 12 we
see that this boundary slope $(\beta,\beta^{\prime})$ satisfies the claims 1
and 2 of Theorem 5. Indeed, consider a pair
$(n,n^{\prime})\in\mathbb{N}\times\mathbb{N}$ of counter values. If
$(n,n^{\prime})$ is $c$-below $(\beta,\beta^{\prime})$, then there is
certainly a line $(\bar{\beta},\bar{\beta}^{\prime})$ less steep than
$(\beta,\beta^{\prime})$ such that $(n,n^{\prime})$ is $c$-below
$(\bar{\beta},\bar{\beta}^{\prime})$. By point 1 above, Spoiler wins the Slope
Game from $((q,q^{\prime}),(\bar{\beta},\bar{\beta}^{\prime}))$. By Lemma 11,
Spoiler wins the Simulation Game from $(qn,q^{\prime}n^{\prime})$.
Analogously, one can use point 2 above together with Lemma 12 to show Point 2
of Theorem 5.
It remains to show that the boundary slope $(\beta,\beta^{\prime})$ is
polynomial in the sizes of ${\cal N}$ and ${\cal N}^{\prime}$. We show that
$(\beta,\beta^{\prime})$ must in fact be the effect of a simple cycle. Because
such cycles are no longer than $K=|Q\times Q^{\prime}|$ and because along a
path of length $K$ the counter values cannot change by more than $K$, we
conclude that $-K\leq\beta,\beta^{\prime}\leq K$.
###### Definition 14 (Equivalent vectors).
Consider all the non-zero effects $(\alpha,\alpha^{\prime})$ of all cycles
together with their opposite vectors $(-\alpha,-\alpha^{\prime})$ and denote
the set of all these vectors by $V$. Call two positive vectors
$(\rho,\rho^{\prime})$ and $(\sigma,\sigma^{\prime})$ _equivalent_ if for all
$(\alpha,\alpha^{\prime})\in V$,
$(\alpha,\alpha^{\prime})\text{ is behind
}(\rho,\rho^{\prime})\iff(\alpha,\alpha^{\prime})\text{ is behind
}(\sigma,\sigma^{\prime}).$ (6)
In other words, equivalent vectors lie in the same angle determined by a pair
of vectors from $V$ that are neighbours angle-wise. We claim that equivalent
slopes have the same winner in the Slope Game:
###### Lemma 15.
If $(\rho,\rho^{\prime})$ and $(\sigma,\sigma^{\prime})$ are equivalent then
the same player wins the Slope Game from
$((q,q^{\prime}),(\rho,\rho^{\prime}))$ and
$((q,q^{\prime}),(\sigma,\sigma^{\prime}))$.
###### Proof.
A winning strategy in the Slope Game from
$((q,q^{\prime}),(\rho,\rho^{\prime}))$ may be literally used in the Slope
Game from $((q,q^{\prime}),(\sigma,\sigma^{\prime}))$. This holds because the
assumption that $(\rho,\rho^{\prime})$ and $(\sigma,\sigma^{\prime})$ are
equivalent implies that all possible outcomes of the initial phase of the
Slope Game are evaluated equally. ∎
Lemma 15 implies that the boundary slope is in $V$. This concludes the proof
of Theorem 5.∎
### 4.3 A Sharper Estimation
Theorem 5 provides a polynomial bound on the constant $c$ and the slopes of
all belts, with respect to the sizes of ${\cal N}$ and ${\cal N}^{\prime}$.
However, the proof of Theorem 5 reveals that a slightly stronger result
actually holds, which will be useful in proving the complexity bound for weak
simulation in Section 6. We can estimate a bound on $c$ in terms of the
following two parameters of the product control graph ${\cal N}\times{\cal
N}^{\prime}$:
* •
scc, the size of the largest strongly connected component, and
* •
acyc, the length of the longest acyclic path.
In particular, we claim that Theorem 5 still holds with the constant $c$
bounded by
$c\leq poly(\text{\sc scc})+\text{\sc acyc}.$ (7)
Intuitively, $c$ is the excess of counter value needed to replay a Slope Game
strategy in the Simulation Game. This directly corresponds to the maximal
number of alternations in a play of the Slope Game. Every phase ends in a
cycle, which must be contained in some strongly connected component and is
thus no longer than scc. So the segment depth of Slope Game strategies is
bounded by $(\text{\sc scc}+1)^{2}$.
We can decompose plays of the Slope Game by separating subpaths that contain
at least one cycle and stay in one strongly connected component, and the
remaining subpaths. One can now show that in fact, a counter value of scc
suffices to enable subpaths of the first kind. The segment depth bounds the
number of such subpaths in any play. Secondly, by definition, the subpaths of
the second kind cannot share any points. The sum of their lengths is hence
bounded by acyc. We conclude that a value of $c=(\text{\sc
scc}+1)^{2}\cdot\text{\sc scc}+\text{\sc acyc}$ is sufficient.
## 5 Strong Simulation is PSPACE-complete
Using our stronger version of the Belt Theorem from Section 4, we derive an
algorithm for checking simulation preorder, similarly as in [1, 6, 5].
As before we fix two OCN ${\cal N}$ and ${\cal N}^{\prime}$, with sets of
control-states $Q$ and $Q^{\prime}$, respectively. By Lemma 10, Point 2, we
can compute in PSPACE, for every pair $(q,q^{\prime})\in Q\times Q^{\prime}$,
the positive vector $(\rho,\rho^{\prime})$ satisfying Theorem 5; we denote
this vector by $\text{\sc slope}(q,q^{\prime})$. We define $\text{\sc
belt}(q,q^{\prime})$ to be the set of points $(n,n^{\prime})\in\mathbb{N}^{2}$
that are neither $c$-above nor $c$-below $\text{\sc slope}(q,q^{\prime})$. As
all vectors $\text{\sc slope}(q,q^{\prime})$ and the widths of all belts are
polynomially bounded (by Theorem 5), we observe that every two non-parallel
belts are disjoint outside a polynomially bounded _initial rectangle_ ,
denoted $L_{0}$, between corners $(0,0)$ and $(l_{0},l_{0}^{\prime})$ (see
Figure 4).
$l_{0}$$l_{0}^{\prime}$periodicaperiodic$A$$P_{1}$$P_{2}$$L_{0}$Duplicator
$n^{\prime}$Spoiler $n$ Figure 4: The initial rectangle $L_{0}$ (blue) and two
belts. Outside $L_{0}$, the colouring of a belt consists of some exponentially
bounded block (red), and another exponentially bounded non-trivial block
(green) which repeats ad infinitum along the rest of the belt.
Recall that the simulation preorder on the configurations with the pair of
control-states $(q,q^{\prime})$ is trivial outside of $\text{\sc
belt}(q,q^{\prime})$: it contains all pairs $(qn,q^{\prime}n^{\prime})$ s.t.
$(n,n^{\prime})$ is $c$-above $\text{\sc slope}(q,q^{\prime})$, and contains
no pairs $(qn,q^{\prime}n^{\prime})$ s.t. $(n,n^{\prime})$ is $c$-below
$\text{\sc slope}(q,q^{\prime})$. We show that inside a belt, the points
corresponding to configurations in simulation are ultimately periodic in the
sense defined below.
By the definition of belts, $(n,n^{\prime})\in\text{\sc
belt}(q,q^{\prime})\iff(n,n^{\prime})+\text{\sc
slope}(q,q^{\prime})\in\text{\sc belt}(q,q^{\prime})$, i.e., translation via
the vector $\text{\sc slope}(q,q^{\prime})$ preserves membership in $\text{\sc
belt}(q,q^{\prime})$. This is why we restrict our focus to multiples of
vectors $\text{\sc slope}(q,q^{\prime})$. We write $\text{\sc
rect}(q,q^{\prime},j)$ for the rectangle between corners $(0,0)$ and
$(l_{0},l_{0}^{\prime})+j\cdot\text{\sc slope}(q,q^{\prime})$.
###### Definition 16 (ultimately-periodic).
For a fixed pair $(q,q^{\prime})\in Q\times Q^{\prime}$ and
$j,k\in\mathbb{N}$, a subset $R\subseteq\text{\sc belt}(q,q^{\prime})$ is
called _$(j,k)$ -ultimately-periodic_ if for all
$(n,n^{\prime})\in\mathbb{N}^{2}\setminus\text{\sc rect}(q,q^{\prime},j)$,
$\displaystyle\begin{aligned} (n,n^{\prime})\in
R\iff(n,n^{\prime})+k\cdot\text{\sc slope}(q,q^{\prime})\in R.\end{aligned}$
(8)
###### Remark 17.
Observe that for fixed $q$ and $q^{\prime}$, every $(j,k)$-ultimately-periodic
set $R$ can be represented by the numbers $j$ and $k$, and two sets
$R\ \cap\ \text{\sc rect}(q,q^{\prime},j)\qquad\text{ and
}\qquad(R\setminus\text{\sc rect}(q,q^{\prime},j))\ \cap\ \text{\sc
rect}(q,q^{\prime},j+k).$
The following lemma states a property which is crucial for our algorithm. It
is actually a sharpening of the result of [5], with additional effective
bounds on periods inside belts.
###### Lemma 18.
For every pair $(q,q^{\prime})\in Q\times Q^{\prime}$, the set
$\displaystyle\preccurlyeq_{q,q^{\prime}}\ \ =\ \
\\{(n,n^{\prime})\in\text{\sc belt}(q,q^{\prime}):qn\preccurlyeq
q^{\prime}n^{\prime}\\}$
is $(j,k)$-ultimately periodic for some $j,k\in\mathbb{N}$ exponentially
bounded w.r.t. the sizes of ${\cal N}$, ${\cal N}^{\prime}$.
Thus, when searching for a simulation relation inside belts, we may safely
restrict ourselves to $(j,k)$-ultimately-periodic relations, for exponentially
bounded $j$ and $k$. According to the remark above, every such simulation
admits the EXPSPACE description that consists, for every pair of states
$(q,q^{\prime})$, of:
* •
a polynomially bounded vector $(\rho,\rho^{\prime})=\text{\sc
slope}(q,q^{\prime})$;
* •
a polynomially bounded relation $\text{\sc init}(q,q^{\prime})\subseteq L_{0}$
inside the initial rectangle $L_{0}$;
* •
exponentially bounded natural numbers
$j_{q,q^{\prime}},k_{q,q^{\prime}}\in\mathbb{N}$; and
* •
two exponentially bounded relations:
$\displaystyle\text{\sc aperiodic}(q,q^{\prime})$ $\displaystyle\ \subseteq\
\text{\sc belt}(q,q^{\prime})\ \cap\ \text{\sc
rect}(q,q^{\prime},j_{q,q^{\prime}})$ $\displaystyle\text{\sc
periodic}(q,q^{\prime})$ $\displaystyle\ \subseteq\ (\text{\sc
belt}(q,q^{\prime})\setminus\text{\sc rect}(q,q^{\prime},j_{q,q^{\prime}}))\
\cap\ \text{\sc rect}(q,q^{\prime},j_{q,q^{\prime}}+k_{q,q^{\prime}}).$
The above characterization leads to the following naive decision procedure,
which works in EXPSPACE: Guess the description of a candidate relation $R$ for
the simulation relation, verify that it is a simulation and check if it
contains the input pair of configurations.
Checking whether the input pair is in the (semilinear) relation $R$ is
trivial. To verify that the relation $R$ is a simulation, one needs to check
the _simulation condition_ for every pair of configurations
$(qn,q^{\prime}n^{\prime})$ in $R$, i.e., Duplicator can ensure that after
playing one round of the Simulation Game, the resulting pair of configurations
is still in $R$.
The simulation condition is local in the sense that it refers only to
positions with neighbouring counter values (plus/minus $1$). This, together
with the fact that belts are disjoint outside $L_{0}$, implies that the
complete one-neighbourhoods of points in the periodic part repeats along the
belt. It therefore suffices to examine those elements which are in the
EXPSPACE description to check if the simulation condition holds.
##### A PSPACE procedure.
The naive algorithm outlined above may easily be turned into a PSPACE
algorithm by a standard shifting window trick. Instead of guessing the
complete exponential-size description upfront, we start by guessing the
polynomially bounded relation inside $L_{0}$ and verifying it locally. Next,
the procedure stepwise guesses parts of the relations $\text{\sc
aperiodic}(q,q^{\prime})$ and later $\text{\sc periodic}(q,q^{\prime})$,
inside a polynomially bounded rectangle window through the belt and shifts
this window along the belt, checking the simulation condition for all
contained points on the way. Since the simulation condition is local,
everything outside this window may be forgotten, save for the first repetitive
window that is used as a certificate for successfully having guessed a
consistent periodic set, once it repeats. Because this repetition needs to
occur after an exponentially bounded number of shifts, polynomial space is
sufficient to store a binary counter that counts the number of shifts and
allows to terminate unsuccessfully once the limit is reached. ∎
## 6 Application to Weak Simulation Checking
A natural extension of simulation is _weak simulation_ , that abstracts from
internal steps.
###### Definition 19.
For a LTS over actions $A\cup\\{\tau\\}$ define _weak_ step relations by
$\,{\stackrel{{\scriptstyle\tau}}{{\Longrightarrow}}}\\!\,=\,{\stackrel{{\scriptstyle\tau}}{{\longrightarrow}}}\\!{}^{\scriptstyle{*}}\,$
and $\,{\stackrel{{\scriptstyle
a}}{{\Longrightarrow}}}\\!\,=\,{\stackrel{{\scriptstyle\tau}}{{\longrightarrow}}}\\!{}^{\scriptstyle{*}}\,\,{\stackrel{{\scriptstyle
a}}{{\longrightarrow}}}\\!\,\,{\stackrel{{\scriptstyle\tau}}{{\longrightarrow}}}\\!{}^{\scriptstyle{*}}\,$
for $a\neq\tau$. Weak simulation ($\curlyeqprec{}{}$) is now defined just like
$\,\preccurlyeq\,$, using Simulation Games, in which Duplicator moves along
weak steps.
For systems without $\tau$-labelled transitions, $\,{\stackrel{{\scriptstyle
a}}{{\longrightarrow}}}\\!\,=\,{\stackrel{{\scriptstyle
a}}{{\Longrightarrow}}}\\!\,$ and therefore strong and weak simulation
coincide. The PSPACE lower bound from [10] for checking strong simulation thus
also holds for weak simulation checking over OCN.
Weak simulation has recently been shown to be decidable for OCN [4]. The main
obstacle was that Duplicator’s system is infinitely branching w.r.t. the weak
$\,{\stackrel{{\scriptstyle a}}{{\Longrightarrow}}}\\!\,$ steps, which implies
that non-simulation does not necessarily manifest itself locally.
In [4], this problem is resolved by constructing a monotone decreasing
sequence of semilinear _approximant relations_ that converges to weak
simulation at a finite index. The approximant relations are derived from a
symbolic characterization of Duplicator’s infinitely-branching system. They
can be computed inductively by characterizing them in terms of strong
simulation over suitably modified OCN. The fact that one can effectively
compute semilinear descriptions of $\,\preccurlyeq\,$ over OCN [5] allows to
successively compute the approximant relations and to detect convergence of
the sequence.
Here we show that the polynomial bounds from Theorem 5, together with the
technique from [4], imply a PSPACE upper bound even for checking _weak_
simulation on OCN. In particular, we claim that the sizes of the “suitably
modified OCN” mentioned above, which characterize the approximants, are in
fact polynomial for every index $i\in\mathbb{N}$ in the sequence. A more
detailed analysis can be found in Appendix B.
###### Theorem 20.
Checking weak simulation preorder on OCN is PSPACE-complete.
## 7 Conclusion
We have shown that both strong and weak simulation preorder checking between
two given OCN processes is PSPACE-complete. Moreover, it is possible to
compute representations of the entire simulation relations as semilinear sets,
but these require exponential space. One cannot expect polynomial-size
representations of the relations as semilinear sets, because otherwise one
could first guess the representation and then verify in ${\it coNP}^{\it NP}$
(for strong simulation) that there are no counterexamples to the local
simulation condition. This would yield an algorithm in $\Sigma_{p}^{3}$ in the
polynomial hierarchy, which (under standard assumptions in complexity theory)
contradicts the PSPACE-hardness of the problem.
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## Appendix A Missing Proofs from Sections 4 and 5
### A.1 Proof of Lemma 11
Suppose Spoiler wins the Slope Game from
$((q,q^{\prime}),(\rho,\rho^{\prime}))$ using a strategy of segment depth $d$.
A position in the Slope Game contains a positive vector
$(\rho,\rho^{\prime})$, while a position in the Simulation Game contains a
pair $(n,n^{\prime})\in\mathbb{N}\times\mathbb{N}$ of counter values, that can
also be interpreted as a positive vector. We will derive a strategy for
Spoiler in the Simulation Game that is winning from all positions
$(qn,q^{\prime}n^{\prime})$ where $(n,n^{\prime})$ is $(\text{\sc K}\cdot
d)$-below $(\rho,\rho^{\prime})$. The crucial idea of the proof is to consider
the segments of the supposed winning strategy in the Slope Game separately.
Each such segment is a strategy for one phase and as such, describes how to
move in the Simulation Game until the next lasso is observed. Afterwards,
Spoiler can chose to continue playing according to the next lower segment, or
“roll back” the cycle and continue playing according to the current segment.
By the rules of the Slope Game we observe that after sufficiently many such
rollbacks the difference between the ratio $n/n^{\prime}$ of the actual
counters and the slope of the next lower segment is negligible, i.e., these
vectors are equivalent in the sense of Definition 14 in Section 4.2. Then,
Spoiler can safely continue to play according to the next lower segment at
level $d-1$.
To safely play such a strategy in the Simulation Game, Spoiler needs to ensure
that her own counter does not decrease too much as that could restrict her
ability to move. We observe however, that any partial play that “stays in some
segment” at height $d$, can be decomposed into a single acyclic prefix plus a
number of cycles. Such a play therefore preserves the invariant that all
visited points are $\text{\sc K}\cdot(d-1)$-below the slope of the phase. In
particular, this means that Spoiler’s counter is always $\geq\text{\sc
K}\cdot(d-1)$.
Formally, the proof of Lemma 11 proceeds by induction on the segment depth
$d$.
##### Case $d=1$.
This means that Spoiler has a strategy to win the Slope Game in the first
phase, and hence to enforce that the effect of all cycles is behind
$(\rho,\rho^{\prime})$ but not positive. Denote this strategy by $T$. In the
Simulation Game, Spoiler will re-use this strategy as we describe below. At
every position $(qn,q^{\prime}n^{\prime})$ in the Simulation Game, Spoiler
keeps a record of the _corresponding position_ $(\pi,(\rho,\rho^{\prime}))$ in
the Slope Game, enforcing the invariant that $(q,q^{\prime})$ are the ending
states of the path $\pi$.
From the initial position $(qn,q^{\prime}n^{\prime})$ with corresponding
position $((q,q^{\prime}),(\rho,\rho^{\prime}))$, Spoiler starts playing the
Simulation Game according to $T$, until the path in the corresponding position
of the Slope Game, say $\pi_{1}$, describes a lasso (this must happen after at
most K rounds). Thus $\pi_{1}$ splits into:
$\pi_{1}=\widetilde{\pi}_{1}\,\bar{\pi}_{1}$ (9)
where the suffix $\bar{\pi}_{1}$ is a cycle. Denote by
$(\widetilde{\alpha}_{1},\widetilde{\alpha}^{\prime}_{1})$ and
$(\bar{\alpha}_{1},\bar{\alpha}^{\prime}_{1})$ the effects of
$\widetilde{\pi}_{1}$ and $\bar{\pi}_{1}$, respectively. The current values of
counters are clearly
$n+\widetilde{\alpha}_{1}+\bar{\alpha}_{1}\qquad\text{and }\quad
n^{\prime}+\widetilde{\alpha}^{\prime}_{1}+\bar{\alpha}^{\prime}_{1}$ (10)
assuming that the play did not end by now with Spoiler’s win. As the length of
path $\pi_{1}$ is at most K and $(n,n^{\prime})$ is assumed to be K-below
$(\rho,\rho^{\prime})$, we know that all positions visited by now in the
Simulation Game were below $(\rho,\rho^{\prime})$. In particular, Spoiler’s
counter value was surely non-negative by now.
Now Spoiler _rolls back_ the cycle $\bar{\pi}_{1}$, namely changes the
corresponding position in the Slope Game from $(\pi_{1},(\rho,\rho^{\prime}))$
to $(\widetilde{\pi}_{1},(\rho,\rho^{\prime}))$ and continues playing
according to $T$. The play continues until Spoiler wins or the path in the
corresponding position of the Slope Game, say $\pi_{2}$, is a lasso again.
Again, we split the path into an acyclic prefix and a cycle:
$\pi_{2}=\widetilde{\pi}_{2}\,\bar{\pi}_{2}.$ (11)
Denote the respective effects by
$(\widetilde{\alpha}_{2},\widetilde{\alpha}^{\prime}_{2})$ and
$(\bar{\alpha}_{2},\bar{\alpha}^{\prime}_{2})$. A crucial but simple
observation is that, assuming that the play did not end by now with Spoiler’s
win, the current values of counters are now
$n+\widetilde{\alpha}_{2}+\bar{\alpha}_{1}+\bar{\alpha}_{2}\qquad\text{and
}\quad
n^{\prime}+\widetilde{\alpha}^{\prime}_{2}+\bar{\alpha}^{\prime}_{1}+\bar{\alpha}^{\prime}_{2},$
(12)
i.e. the effect $(\widetilde{\alpha}_{1},\widetilde{\alpha}^{\prime}_{1})$ of
$\widetilde{\pi}_{1}$ does not contribute any more. As
$(\bar{\alpha}_{1},\bar{\alpha}^{\prime}_{1})$ is behind
$(\rho,\rho^{\prime})$ we may apply Lemma 7 to
$(\bar{\alpha}_{1},\bar{\alpha}^{\prime}_{1})$ and $c=0$ in order to deduce,
similarly as before, that all positions by now were below
$(\rho,\rho^{\prime})$. Now Spoiler rolls back $\bar{\pi}_{2}$ by establishing
$(\widetilde{\pi}_{2},(\rho,\rho^{\prime}))$ as the new corresponding position
in the Slope Game. Continuing in this way, after $k$ rollbacks the counter
values are:
$\displaystyle\begin{aligned} &n\
+\widetilde{\alpha}_{k}+(\bar{\alpha}_{1}+\bar{\alpha}_{2}+\ldots\
+\bar{\alpha}_{k-1})+\bar{\alpha}_{k}\qquad\text{and}\\\
&n^{\prime}+\widetilde{\alpha}^{\prime}_{k}+(\bar{\alpha}^{\prime}_{1}+\bar{\alpha}^{\prime}_{2}+\ldots+\bar{\alpha}^{\prime}_{k-1})+\bar{\alpha}^{\prime}_{k},\end{aligned}$
(13)
assuming that Spoiler did not win earlier. All the vectors
$(\bar{\alpha}_{i},\bar{\alpha}^{\prime}_{i})$, and thus also the sum
$(\bar{\alpha}_{1}+\bar{\alpha}_{2}+\ldots\
+\bar{\alpha}_{k-1},\bar{\alpha}^{\prime}_{1}+\bar{\alpha}^{\prime}_{2}+\ldots+\bar{\alpha}^{\prime}_{k-1})$
(14)
are behind $(\rho,\rho^{\prime})$, hence similarly as before all positions by
now have been below $(\rho,\rho^{\prime})$, by Lemma 7 applied to the vector
(14) above.
This in particular means that Spoiler’s counter remains above value $c$.
However, as by assumption all observed cycles come from a final segment in her
Slope Game strategy, the vector (14) cannot be positive for any $k$. Thus,
every rollback strictly decreases Duplicator’s counter value. We conclude that
after sufficiently many rollbacks, Duplicator’s counter must eventually drop
below $0$ and hence Spoiler eventually wins.
##### Case $d>1$.
By assumption, Spoiler has a strategy $T$ for the Slope Game, which has
segment depth $d>0$. As before, Spoiler’s strategy in the Simulation Game will
re-use the strategy $T$ from the Slope Game, using rollbacks.
Spoiler plays according to the initial segment of this strategy, that allows
her to win or at least guarantee that the effect of the first observed lasso’s
circle is less steep than $(\rho,\rho^{\prime})$. After $l$ rollbacks, the
counter values will be of the form:
$\displaystyle\begin{aligned} &n+\widetilde{\alpha}+(\bar{\alpha}_{1}+\ldots\
+\bar{\alpha}_{m})+(\bar{\gamma}_{1}+\ldots\
+\bar{\gamma}_{l})\quad\text{and}\\\
&n^{\prime}+\widetilde{\alpha}^{\prime}+(\bar{\alpha}^{\prime}_{1}+\ldots+\bar{\alpha}^{\prime}_{m})+(\bar{\gamma}^{\prime}_{1}+\ldots+\bar{\gamma}^{\prime}_{l}),\end{aligned}$
(15)
where the absolute values of $\widetilde{\alpha}$ and
$\widetilde{\alpha}^{\prime}$ are at most K, the vectors
$(\bar{\gamma}_{i},\bar{\gamma}^{\prime}_{i})$ are behind
$(\rho,\rho^{\prime})$ and positive, and the vectors
$(\bar{\alpha}_{i},\bar{\alpha}^{\prime}_{i})$ are behind
$(\rho,\rho^{\prime})$ and non-positive. We apply Lemma 7 to $c=\text{\sc
K}\cdot(d-1)$ and learn that all the positions by now have been $(\text{\sc
K}\cdot(d-1))$-below $(\rho,\rho^{\prime})$.
In general Spoiler has no power to choose whether a effect of a cycle at a
next rollback is positive or not. However, if from some point on all effects
are non-positive then Duplicator’s counter eventually drops below $0$ and
Spoiler wins. Thus w.l.o.g,̇ we focus on positions in the Simulation Game
immediately after a rollback of a cycle with positive effect. Using the
notation from (15), suppose $(\gamma_{l},\gamma^{\prime}_{l})$ is the effect
of the last rolled back cycle. We need the following claim in order to apply
the induction assumption:
###### Claim 1.
After sufficiently many rollbacks the vector $(\bar{n},\bar{n}^{\prime})$ of
current counter values in the Simulation Game is $(\text{\sc
K}\cdot(d-1))$-below some vector $(\gamma,\gamma^{\prime})$ which is
equivalent to the positive effect $(\gamma_{l},\gamma^{\prime}_{l})$ of the
last rolled back cycle.
###### Proof.
By an easy geometric argument. Ignore vectors
$(\alpha_{i},\alpha^{\prime}_{i})$ as they preserve being $(\text{\sc
K}\cdot(d-1))$-below all positive vectors that are less steep than
$(\rho,\rho^{\prime})$. If Duplicator wants to falsify the condition, he would
need to increase the steepness of the rolled back cycle infinitely often,
which is clearly impossible as there are only finitely many simple cycles. ∎
Let $(\bar{q}\bar{n},\bar{q}^{\prime}\bar{n}^{\prime})$ be a position of the
Simulation Game satisfying the claim. We know that Spoiler has a winning
strategy in the Slope Game from
$((\bar{q},\bar{q}^{\prime}),(\gamma_{l},\gamma^{\prime}_{l}))$, of segment
depth at most $d-1$. Because $(\gamma_{l},\gamma^{\prime}_{l})$ is equivalent
to $(\gamma,\gamma^{\prime})$, we can apply Lemma 15 and know that the same
strategy is winning in the Slope Game from
$((\bar{q},\bar{q}^{\prime}),(\gamma,\gamma^{\prime}))$. By the induction
assumption we conclude that Spoiler wins the Simulation Game from
$(\bar{q}\bar{n},\bar{q}^{\prime}\bar{n}^{\prime})$, which completes the proof
of Lemma 11.∎
### A.2 Proof of Lemma 12
Suppose Duplicator wins the Slope Game from
$((q,q^{\prime}),(\rho,\rho^{\prime}))$ using a strategy of segment depth $d$.
We will show that Duplicator wins the Simulation Game from every position
$(qn,q^{\prime}n^{\prime})$ where $(n,n^{\prime})$ is $(\text{\sc K}\cdot
d)$-above $(\rho,\rho^{\prime})$. We will again build on the concept of
rollbacks and proceed by induction on $d$.
##### Case $d=1$.
In this case, Duplicator has a strategy to win the Slope Game immediately
after the first phase. This means he can enforce that the effects of the
cycles of all observed lassos are not behind $(\rho,\rho^{\prime})$. By a
straightforward induction using part 2 of Lemma 7 one can show that Duplicator
can preserve the invariant that all visited points are $K$-above
$(\rho,\rho^{\prime})$. This in particular means that his counter value stays
positive and he wins by enforcing an infinite play.
##### Case $d>1$.
Let $T$ denote the initial segment of Duplicator’s strategy in the Slope Game.
Every effect of a cycle in $T$ is either not behind $(\rho,\rho^{\prime})$ or
behind $(\rho,\rho^{\prime})$, but positive.
In the Simulation Game, Duplicator will play according to this initial segment
$T$, using rollbacks, as long as the effect of the rolled back cycle is not
behind $(\rho,\rho^{\prime})$. Just as in the previous case, we can apply part
2 of Lemma 7 for $c=\text{\sc K}\cdot d$ and derive that in this way,
Duplicator is able to keep the current counter values $(\text{\sc K}\cdot
d)$-above $(\rho,\rho^{\prime})$.
Suppose that after a few iterations, the effect $(\alpha,\alpha^{\prime})$ of
the last cycle _is_ behind $(\rho,\rho^{\prime})$ and let
$(\bar{q}\bar{n},\bar{q}^{\prime}\bar{n}^{\prime})$ be the position in the
Simulation Game directly afterwards. In this case, $(\alpha,\alpha^{\prime})$
is clearly positive and less steep than $(\rho,\rho^{\prime})$. Now the point
described by the counter values before this last cycle was $(\text{\sc K}\cdot
d)$-above $(\rho,\rho^{\prime})$ and because the cycle is no longer than $K$,
we know that the point $(\bar{n},\bar{n}^{\prime})$ of current counter values
(after the cycle) is still $(\text{\sc K}\cdot(d-1))$-above
$(\rho,\rho^{\prime})$. This means, as
$(\alpha,\alpha^{\prime})\prec(\rho,\rho^{\prime})$, that
$(\bar{n},\bar{n}^{\prime})$ is also $(\text{\sc K}\cdot(d-1))$-above
$(\alpha,\alpha^{\prime})$.
Knowing that Duplicator has a winning strategy in the Slope Game from
$((\bar{q},\bar{q}^{\prime}),(\alpha,\alpha^{\prime}))$ of segment depth at
most $d-1$, by induction assumption we obtain a winning strategy for
Duplicator in the Simulation Game from
$(\bar{q}\bar{n},\bar{q}^{\prime}\bar{n}^{\prime})$. This completes the
description of Duplicator’s winning strategy from $(qn,q^{\prime}n^{\prime})$
and hence also the proof of Lemma 12.∎
### A.3 Proof of Lemma 18
For technical convenience we assume w.l.o.g. that no belt contains the upper
right corner of $L_{0}$ (this can always be achieved by minimally extending
$L_{0}$, if necessary.) Thus every belt intersects either the horizontal, or
the vertical border of $L_{0}$, but not both.
Recall that the non-parallel belts do not overlap/interfere with each other
outside $L_{0}$, hence we can consider them separately. For the rest of the
proof fix states $q,q^{\prime}$ and let $(\rho,\rho^{\prime})=\text{\sc
slope}(q,q^{\prime})$. W.l.o.g. suppose that $\text{\sc belt}(q,q^{\prime})$
intersects the horizontal border of $L_{0}$ (if it intersects the vertical
border of $L_{0}$ the proof is analogous).
For simplicity we assume that no other belt is parallel to $\text{\sc
belt}(q,q^{\prime})$. The proof below may be easily adapted to the general
case by considering a bunch of parallel belts jointly, instead of just the
single one $\text{\sc belt}(q,q^{\prime})$.
By a _cross-section_ at level $n^{\prime}$ we mean the intersection of
$\preccurlyeq_{q,q^{\prime}}$ with two consecutive horizontal lines at that
level, i.e. with $\\{(n,n^{\prime}),(n,n^{\prime}+1):n\in\mathbb{N}\\}$. We
may assume that cross-sections are always non-empty (this can always be
ensured by slightly widening $\text{\sc belt}(q,q^{\prime})$ if necessary). We
say that two cross-sections $s_{1}$ and $s_{2}$ are _equal_ if one of them is
obtained by a shift of the other by a multiple of $(\rho,\rho^{\prime})$;
formally, we require for some $k\in\mathbb{N}$,
$\displaystyle s_{1}+k\cdot(\rho,\rho^{\prime})\ \ =\ \ s_{2}.$ (16)
Choose two cross-sections $s_{1},s_{2}$ at levels $n^{\prime}_{1}$ and
$n^{\prime}_{2}$ respectively, and $k>0$ that satisfies (16). Let $P$ be the
restriction of $\preccurlyeq_{q,q^{\prime}}$ to the area between $s_{1}$ and
$s_{2}$, and $A$ be the restriction of $\preccurlyeq_{q,q^{\prime}}$ to the
area below $s_{1}$:
$\displaystyle A\ $ $\displaystyle=\ \\{(n,n^{\prime})\in\
\preccurlyeq_{q,q^{\prime}}\ :\ n^{\prime}<n^{\prime}_{1}\\}$ $\displaystyle
P\ $ $\displaystyle=\ \\{(n,n^{\prime})\in\ \preccurlyeq_{q,q^{\prime}}\ :\
n^{\prime}_{1}\leq n^{\prime}<n^{\prime}_{2}\\}.$
Recall that $A$ and $P$, similarly as $\preccurlyeq_{q,q^{\prime}}$, are
subsets of $\text{\sc belt}(q,q^{\prime})$. We claim:
###### Lemma 21.
For every $s_{1},s_{2}$ and $k>0$ satisfying (16),
$\preccurlyeq_{q,q^{\prime}}\ =\ A\ \cup\ P^{*},\quad\text{ where }P^{*}\ =\
\bigcup_{i\in\mathbb{N}}(P+i\cdot k\cdot(\rho,\rho^{\prime})).$
Before proving this lemma note that it implies Lemma 18. Indeed, by Theorem 5,
a cross-section contains polynomially many points, and therefore there are at
most exponentially many non-equal cross sections. Thus, by the pigeonhole
principle, there are surely two equal cross-sections at exponentially bounded
levels $n^{\prime}_{1}$ and $n^{\prime}_{2}$.
Now we prove Lemma 21. The proof strongly relies on the locality of the
simulation condition. We first claim one inclusion of Lemma 21, namely:
###### Claim 2.
$A\ \cup\ P^{*}\subseteq\ \preccurlyeq_{q,q^{\prime}}$.
###### Proof.
We show that the following relation is a simulation:
$R\ \ =\ \ \preccurlyeq\ \setminus\
\\{(qn,q^{\prime}n^{\prime}):(n,n^{\prime})\in\text{\sc
belt}(q,q^{\prime})\\}\ \cup\ \\{(qn,q^{\prime}n^{\prime}):(n,n^{\prime})\in
A\ \cup\ P^{*}\\}.$
(Roughly speaking, $R$ is obtained from $\preccurlyeq$ by replacing
$\preccurlyeq_{q,q^{\prime}}$ with $A\ \cup\ P^{*}$.) We claim that $R$ is a
simulation, relying on the locality of the simulation condition. Formally, we
define the _relative $R$-neighborhood_ of a point $(n,n^{\prime})$ as
$\\{(pl,p^{\prime}l^{\prime}):(p(n+l),p^{\prime}(n^{\prime}+l^{\prime}))\in
R,\ (p,p^{\prime})\in Q\times Q^{\prime},\ l,l^{\prime}\in\\{-1,0,1\\}\\}.$
Note that the simulation condition for a pair of configurations
$(qn,q^{\prime}n^{\prime})$ with respect to the relation $R$ only depends on
the relative $R$-neighborhood of $(n,n^{\prime})$. Similarly, one defines the
relative $\preccurlyeq$-neighborhood of a point $(n,n^{\prime})$.
By the definition of cross-section and of the sets $A$ and $P$, the relative
$R$-neighborhood of a point $(n,n^{\prime})\in R$ equals the relative
$\preccurlyeq$-neighborhood of some (possibly other) point in
$\preccurlyeq_{q,q^{\prime}}$. Thus we deduce that every pair in $R$ satisfies
the simulation condition wrt. $R$, i.e. $R$ is a simulation. As $\preccurlyeq$
is the largest simulation, the claim follows. ∎
In order to show the other inclusion of Lemma 21, extend $n^{\prime}_{1}$ and
$n^{\prime}_{2}$ to an infinite arithmetic progression
$n^{\prime}_{1},\ n^{\prime}_{2},\ n^{\prime}_{3},\ \ldots,$
i.e. $n_{i+1}=n^{\prime}_{i}+k\cdot\rho^{\prime}$ for $i\geq 1$, and consider
the “segments” $P_{i}$ of $\preccurlyeq_{q,q^{\prime}}$ defined by the
corresponding cross-sections:
$P_{i}\ =\ \\{(n,n^{\prime})\in\ \preccurlyeq_{q,q^{\prime}}\ :\
n^{\prime}_{i}\leq n^{\prime}<n^{\prime}_{i+1}\\}\qquad\text{ for }i\geq 1.$
Clearly, $P=P_{1}$ and $\preccurlyeq_{q,q^{\prime}}\ =\ A\ \cup\
\bigcup_{i\geq 1}P_{i}$. By Claim 2 it follows that
$P_{1}+k\cdot(\rho,\rho^{\prime})\subseteq P_{2}$, or equivalently
$P_{1}\subseteq P_{2}-k\cdot(\rho,\rho^{\prime})$. Analogously one shows:
$\displaystyle P_{i}\ \subseteq\
P_{i+1}-k\cdot(\rho,\rho^{\prime})\qquad\text{ for every }i\geq 1.$ (17)
We claim that the inclusions are actually equalities:
###### Claim 3.
$P_{i}\ =\ P_{i+1}-k\cdot(\rho,\rho^{\prime})$, for every $i\geq 1$.
###### Proof.
Due to Equation (17), it suffices to show the inclusions
$P_{i+1}-k\cdot(\rho,\rho^{\prime})\ \subseteq\ P_{i}$. The inclusions follow,
similarly as in the proof of Claim 2, from the observation that the following
relation is a simulation:
$R\ \ =\ \ \preccurlyeq\ \setminus\
\\{(qn,q^{\prime}n^{\prime}):(n,n^{\prime})\in\text{\sc
belt}(q,q^{\prime})\\}\ \ \cup\ \
\\{(qn,q^{\prime}n^{\prime}):(n,n^{\prime})\in A\ \cup\ \bigcup_{i\geq
2}P_{i}-k\cdot(\rho,\rho^{\prime})\\}.$
The relation $R$ is obtained from $\preccurlyeq$, roughly speaking, by
removing the first segment $P_{1}$ and shifting all other segments $P_{i}$ by
vector $-k\cdot(\rho,\rho^{\prime})$. To prove that $R$ is a simulation, we
exploit locality of the simulation condition exactly as before. Additionally,
we use the observation that the simulation condition is monotonic with respect
to inclusion of relative neighborhoods, together with the inclusions (17). ∎
Claim 3 immediately implies Lemma 21 and thus Lemma 18.
## Appendix B Weak Simulation Checking
We show that the bounds on the coefficients of the Belt Theorem, as derived in
Section 4, imply that the construction from [4] for checking _weak_ simulation
uses only polynomial space.
In order to avoid repeating the involved construction from [4], we refer the
reader to the original paper for technical details and recover here only those
notions and properties which suffice to provide some intuition and derive the
claimed PSPACE bound.
We aim to compute a description of $\curlyeqprec{}{}$, the largest weak
simulation over a given pair of OCN. First we reduce this weak simulation game
to a strong simulation game between two modified systems.
###### Definition 22 ($\omega$-Nets).
An _$\omega$ -net_ ${\cal M}=(Q,A,\delta)$ is given by a finite set of
control-states $Q$, a finite set of actions $A$ and transitions
$\delta\subseteq Q\times A\times\\{-1,0,1,\omega\\}\times Q$. It induces a
transition system over the stateset $Q\times\mathbb{N}$ that allows a step
$pm\,{\stackrel{{\scriptstyle
a}}{{\longrightarrow}}}\\!\,p^{\prime}m^{\prime}$ if either
$(p,a,d,p^{\prime})\in\delta$ and $m^{\prime}=m+d\in\mathbb{N}$ or if
$(p,a,\omega,p^{\prime})\in\delta$ and $m^{\prime}>m$.
###### Lemma 23 ([4]).
For two OCN ${\cal N}$ and ${\cal N}^{\prime}$ with sets of control-states $Q$
and $Q^{\prime}$ resp., one can construct a OCN ${\cal M}$ with states
$Q_{{\cal M}}\supseteq Q$ and an $\omega$-net ${\cal M}^{\prime}$ with states
$Q_{{\cal M}^{\prime}}\supseteq Q^{\prime}$, such that for each pair
$(q,q^{\prime})\in Q\times Q^{\prime}$ of original control states,
$qn\curlyeqprec{}{}q^{\prime}n^{\prime}\text{ w.r.t. }{\cal N},{\cal
N}^{\prime}\text{ iff }qn\,\preccurlyeq\,q^{\prime}n^{\prime}\text{ w.r.t.
}{\cal M},{\cal M}^{\prime}.$ (18)
Moreover, the sizes of ${\cal M}$ and ${\cal M}^{\prime}$ are polynomial in
the size of ${\cal N}$ and ${\cal N}^{\prime}$.
Thus, it suffices to compute a description of the strong simulation relation
relative to a given OCN ${\cal M}=(Q,A,\delta)$ and an $\omega$-net ${\cal
M}^{\prime}=(Q^{\prime},A,\delta^{\prime})$. To do that, we construct a
sequence of successively decreasing (w.r.t. set inclusion) approximant
relations $\,\preccurlyeq^{i}\,$ and show that 1) for all $i\in\mathbb{N}$,
$\,\preccurlyeq^{i}\,$ is effectively semilinear and 2) there is some
$k\in\mathbb{N}$ with
$\,\preccurlyeq^{k}\,=\,\preccurlyeq^{k+1}\,=\,\preccurlyeq\,$, i.e., the
sequence converges to simulation preorder at some finite level $k$.
Intuitively, $\,\preccurlyeq^{i}\,$ is given by a _parameterized simulation
game_ that keeps track of how often Duplicator uses $\omega$-labelled
transitions and in which Duplicator immediately wins if he plays such a step
the $i$th time. It is easy to see that this game favours Duplicator due to the
additional winning condition. With growing index $i$, this advantage becomes
less important and the game increasingly resembles a standard simulation game.
Hence, $\forall
i\in\mathbb{N},\,\preccurlyeq^{i}\,\supseteq\,\preccurlyeq^{i+1}\,$.
In [4], it is shown that these approximants $\,\preccurlyeq^{i}\,$ can in fact
be characterized by equivalent (in the sense of Lemma 24 below) ordinary
strong simulation relations between suitably extended OCN.
###### Lemma 24.
There is a sequence $({\cal S}_{i},{\cal S}_{i}^{\prime})$ of pairs of OCN
such that for all indices $i\in\mathbb{N}$:
1. 1.
${\cal S}_{i}$ and ${\cal S}_{i}^{\prime}$ contain all states of ${\cal M}$
and ${\cal M}^{\prime}$ respectively.
2. 2.
For all configurations $qn\in(Q\times\mathbb{N})$ and
$q^{\prime}n^{\prime}\in(Q^{\prime}\times\mathbb{N})$ of ${\cal M}$ and ${\cal
M}^{\prime}$ it holds that $qn\,\preccurlyeq^{i}\,q^{\prime}n^{\prime}$ w.r.t.
${\cal M},{\cal M}^{\prime}$ iff $qn\,\preccurlyeq\,q^{\prime}n^{\prime}$
w.r.t. $S_{i},S_{i}^{\prime}$.
3. 3.
${\cal S}_{i+1}$ and ${\cal S}_{i+1}^{\prime}$ can be computed from ${\cal
S}_{i}$ and ${\cal S}_{i}^{\prime}$ alone.
The above conditions ensure decidability of weak simulation as they allow to
iteratively compute the approximants and detect convergence, by the effective
semilinearity of strong simulation over OCN [5].
To obtain an upper bound for the complexity of this procedure, we will bound
the sizes of all $(S_{i},S_{i}^{\prime})$ polynomially in the sizes of ${\cal
M}$ and ${\cal M}^{\prime}$. To do that, we recall some more properties of the
construction, starting by describing how the nets ${\cal S}_{i}$ and ${\cal
S}_{i}^{\prime}$ look like.
#### The nets ${\cal S}_{i}$ and ${\cal S}_{i}^{\prime}$
These nets are constructed using the notion of _minimal sufficient values_ :
###### Definition 25.
Consider the approximant $\,\preccurlyeq^{i}\,$ for some parameter $i$, which
is characterised by nets ${\cal S}_{i},{\cal S}_{i}^{\prime}$ (cf. point 2 of
Lemma 24 above) and let $(q,q^{\prime})\in(Q\times Q^{\prime})$ be a pair of
states. By monotonicity, there is a minimal value ${\it
suf}({q,q^{\prime},i})\in\mathbb{N}\cup\\{\omega\\}$ satisfying
$\forall n^{\prime}\in\mathbb{N}.\ q({\it
suf}({q,q^{\prime},i}))\,\not\preccurlyeq^{i}\,q^{\prime}n^{\prime}.$ (19)
Let ${\it suf}({q,q^{\prime},i})$ be $\omega$ if no finite value satisfies
this condition.
The idea behind the construction of nets for parameter $i+1$ is as follows. A
Simulation Game played on the arena ${\cal S}_{i+1},{\cal S}_{i+1}^{\prime}$
mimics the $(i+1)$-parameterized simulation game played on ${\cal M},{\cal
M}^{\prime}$ until Duplicator uses an $\omega$-labelled transition, leading to
some game position $qn$ vs. $q^{\prime}n^{\prime}$. Afterwards, the
parameterized game would continue with the next lower parameter $i$.
By induction assumption, we can compute a representation of
$\,\preccurlyeq^{i}\,$ and hence ${\it suf}({q,q^{\prime},i})$ for every pair
$(q,q^{\prime})$. Given these values, the nets ${\cal S}_{i+1}$ and ${\cal
S}_{i+1}^{\prime}$ are constructed so that instead of making steps that are
due to $\omega$-labelled transitions, Duplicator can enforce the play to
continue in some subgame that he wins iff Spoiler’s counter is smaller than
the hard-wired value ${\it suf}({q,q^{\prime},i})$.
This “forcing” of the play can be implemented for OCN simulation using a
standard technique called _defender’s forcing_ (see e.g. [8]). So, the nets
${\cal S}_{i}$ and ${\cal S}_{i}^{\prime}$ consist of the original nets ${\cal
M},{\cal M}^{\prime}$ where all $\omega$-transitions in Duplicator’s net
${\cal M}^{\prime}$ are replaced by a small constant defenders-forcing script,
leading to the corresponding testing gadgets that test if Spoiler’s counter is
at least as large as the pre-computed sufficient value and let Spoiler win
only if that is the case.
The actual test-gadgets are not very complicated: On Duplicator’s side, all
gadgets are the same simple loop over a newly introduced symbol, say $e$.
Hence, ${\cal S}_{i}^{\prime}={\cal S}_{1}^{\prime}$ for every $i$ and this
new net is polynomial in the size of ${\cal M}^{\prime}$ and ${\cal M}$.
In Spoiler’s net ${\cal S}_{i}$, the gadgets $G(q,q^{\prime},i)$ for states
$(q,q^{\prime})$ and index $i$ solely depend on the value ${\it
suf}({q,q^{\prime},i})$: If ${\it suf}({q,q^{\prime},i})$ is finite, it
suffices to have a counter-decreasing chain of $e$-steps of length ${\it
suf}({q,q^{\prime},i})$, leading to some state which enables an action that
cannot be replied to by Duplicator. Otherwise, if ${\it
suf}({q,q^{\prime},i})=\omega$ (no counter finite value satisfies Equation
19), Spoiler should always lose, so a simple $e$-labelled loop can be used as
gadget. To conclude, each ${\cal S}_{i}$ essentially consists of ${\cal M}$
plus chains $G(q,q^{\prime},i)$, one for every pair of states
$(q,q^{\prime})$. We summarize the crucial properties of this construction
below.
###### Lemma 26.
1. 1.
${\it suf}({q,q^{\prime},1})=\omega$ for every pair $(q,q^{\prime})\in Q\times
Q^{\prime}$.
2. 2.
${\it suf}({q,q^{\prime},i})\geq{\it suf}({q,q^{\prime},i+1})$.
3. 3.
$({\cal S}_{i},{\cal S}_{i}^{\prime})$ contains precisely $|Q\times
Q^{\prime}|$ many gadgets, each.
4. 4.
If ${\it suf}({q,q^{\prime},i})\in\mathbb{N}$ then the size of gadget
$G(q,q^{\prime},i)$ is ${\it suf}({q,q^{\prime},i})+2$.
5. 5.
No chain $G(q,q^{\prime},i)$ contains transitions leading back to ${\cal M}$.
Using properties 2 and 3 we derive that indeed $({\cal S}_{k},{\cal
S}_{k}^{\prime})=({\cal S}_{k+1},{\cal S}_{k+1}^{\prime})$, and hence
$\,\preccurlyeq^{k}\,=\,\preccurlyeq^{k+1}\,=\,\preccurlyeq\,$ for some finite
$k\in\mathbb{N}$.
Our goal is to bound the sizes of the nets ${\cal S}_{i},{\cal
S}_{i}^{\prime}$ polynomially in the sizes of ${\cal M},{\cal M}^{\prime}$ and
to show that they can indeed be constructed in polynomial space. From point 1
and the fact whenever ${\it suf}({q,q^{\prime},i})=\omega$, the gadget
$G(q,q^{\prime},i)$ is a trivial loop, we already know that the sizes of
${\cal S}_{1}$ and ${\cal S}_{1}^{\prime}$ are polynomial in ${\cal M},{\cal
M}^{\prime}$. Due to the particular shape of the nets $({\cal S}_{i+1},{\cal
S}_{i+1}^{\prime})$, it suffices to bound the values ${\it
suf}({q,q^{\prime},i})$.
#### Bounding ${\it suf}({q,q^{\prime},i})$
Observe that ${\it suf}({q,q^{\prime},i})$ is defined in terms of the
approximant $\,\preccurlyeq^{i}\,$, which is characterized as the strong
simulation $\,\preccurlyeq\,$ relative to the nets ${\cal S}_{i},{\cal
S}_{i}^{\prime}$ by Lemma 24, Point 2. In fact, if we consider the colouring
of $\,\preccurlyeq\,$ w.r.t. ${\cal S}_{i},{\cal S}_{i}^{\prime}$, the value
${\it suf}({q,q^{\prime},i})$ is the width of the belt for $(q,q^{\prime})$ if
this belt is vertical and $\omega$ otherwise. Therefore, the value $c$ in the
Belt Theorem applied to this colouring bounds all finite ${\it
suf}({q,q^{\prime},i})$.
We show how to bound $c$ using the sharper estimation as formulated in Section
4.3 in terms $\it scc$, the maximal size of any strongly connected component
and $\it acyc$, the length of the longest acyclic path in the product ${\cal
S}_{i}\times{\cal S}_{i}^{\prime}$:
$c\leq\it poly(\it scc)+\it acyc.$ (20)
This allows us to bound all values ${\it suf}({q,q^{\prime},i})$ and hence the
size of the nets for index $i+1$.
First, observe that the shape of all ${\cal S}_{i},{\cal S}_{i}^{\prime}$
(particularly Point 5 of Lemma 26, and the fact that
$\forall_{i\in\mathbb{N}}S_{i}^{\prime}=S_{1}^{\prime}$) implies that the
strongly connected components are unchanged from index $i=1$ onward. Thus,
$\it scc$ is in fact polynomial in ${\cal M},{\cal M}^{\prime}$. Secondly, any
path in the product ${\cal S}_{i}\times{\cal S}_{i}^{\prime}$ can be split
into two (possibly empty) parts: the part that remains in ${\cal M}\times{\cal
M}^{\prime}$ and a suffix that moves into at most one gadget
$G(q,q^{\prime},i)$. Since the maximal length of paths in $G(q,q^{\prime},i)$
is bounded by ${\it suf}({q,q^{\prime},i})$, we can bound $\it acyc$ as
follows.
$\it acyc\leq|{\cal M}\times{\cal M}^{\prime}|+\max\\{{\it
suf}({q,q^{\prime},i})\in\mathbb{N}\ |\ (q,q^{\prime})\in Q\times
Q^{\prime}\\}.$ (21)
Let $W_{i}$ denote the maximal width of all vertical belts at level $i$, i.e.,
the largest finite value ${\it suf}({q,q^{\prime},i})$ over all
$(q,q^{\prime})$. By the argument above, we get for all indices
$i\in\mathbb{N}$,
$W_{i+1}\leq\it poly({\cal M},{\cal M}^{\prime})+W_{i}.$ (22)
Now, from properties 2 and 4 of Lemma 26 we can deduce that there are no more
than $K=|Q\times Q^{\prime}|$ indices $i$ such that $W_{i+1}\geq W_{i}$. This
is because the size of the value ${\it suf}({q,q^{\prime},i})$ for a
particular pair $(q,q^{\prime})$ can only increase once, going from index $i$
to $i+1$ if ${\it suf}({q,q^{\prime},i})=\omega>{\it
suf}({q,q^{\prime},i+1})$. Therefore, we can bound $W_{i}$, and thus values
${\it suf}({q,q^{\prime},i})$, for all indices $i\in\mathbb{N}$ by
$W_{i}\leq K\cdot\ (poly(|{\cal M}\times{\cal M}^{\prime}|)+1).$ (23)
We conclude that the sizes of all ${\cal S}_{i},{\cal S}_{i}^{\prime}$ are
polynomial in the sizes of ${\cal N}$ and ${\cal N}^{\prime}$. It remains to
show that we can compute these values in polynomial space, because this allows
us to effectively construct the nets for the next parameter $i+1$.
#### Computing ${\it suf}({q,q^{\prime},i})$
We analyse the colouring of the simulation $\,\preccurlyeq\,$ relative to the
one-counter nets ${\cal S}_{i}$ and ${\cal S}_{i}^{\prime}$. In particular we
need to answer the following questions, for each given pair of states
$(q,q^{\prime})\in Q\times Q^{\prime}$,
1. 1.
Is the belt for $(q,q^{\prime})$ vertical? And if yes,
2. 2.
What is its exact width?
By Theorem 5, we can bound all ratios $(\rho,\rho^{\prime})$, which are the
slopes of belts polynomially. Let $(\rho,\rho^{\prime})$ be the ratio of the
steepest belt with $\rho^{\prime}>0$. Recall that $c$ bounds the width of all
vertical belts. To answer the first question, it suffices to check the colour
of some point $(n,n^{\prime})$ that is both $c$-above $(\rho,\rho^{\prime})$
and $c$-below of $(0,1)$, i.e., $n>c$. For instance, $n=c+1$ and
$n^{\prime}=2(c+1)(\frac{\rho}{\rho^{\prime}})$ is surely such a point.
If the belt for $(q,q^{\prime})$ is vertical, then by Theorem 5, Point 2, we
have $qn,\,\not\preccurlyeq\,q^{\prime}n^{\prime}$. Otherwise, if the belt is
not vertical, then by point 1 of Theorem 5, we must have
$qn,\,\preccurlyeq\,q^{\prime}n^{\prime}$.
To answer the second question, we consider the periodicity description of the
colouring (cf. Section 5). Although this description is of exponential size
and we thus cannot fully keep it in memory, we can, in polynomial space,
compute point queries. Moreover, we know that the colouring in any belt is
described by some non-trivial initial colouring and is repetitive from some
exponentially bounded level onwards. Thus, if we consider the vertical belt
for states $(q,q^{\prime})$, from some level $n^{\prime}_{0}$, the colouring
stabilizes so that for all $n^{\prime}\geq n^{\prime}_{0}$, we have
$qn\,\preccurlyeq\,q^{\prime}n^{\prime}$ iff $n<{\it suf}({q,q^{\prime},i})$.
We can now iteratively check the colour of the point $(n,n^{\prime}_{0})$ for
decreasing values $n=c$ to $0$ and some fixed, but sufficiently high
$n^{\prime}_{0}$. By Theorem 2, this can surely be done in polynomial space.
${\it suf}({q,q^{\prime},i})$ must be the largest considered $n<c$ where
$qn\,\not\preccurlyeq\,q^{\prime}n^{\prime}_{0}$ still holds.
|
arxiv-papers
| 2013-10-23T17:32:05 |
2024-09-04T02:49:52.768017
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Piotr Hofman, Slawomir Lasota, Richard Mayr, Patrick Totzke",
"submitter": "Richard Mayr",
"url": "https://arxiv.org/abs/1310.6303"
}
|
1310.6339
|
# Disconnected quark loop contributions to nucleon observables in lattice QCD
A. Abdel-Rehim(a), C. Alexandrou (a,b), M. Constantinou (b), V. Drach (c), K.
Hadjiyiannakou (b), K. Jansen (b,c), G. Koutsou (a), A. Vaquero (a) (a)
Computation-based Science and Technology Research Center, Cyprus Institute, 20
Kavafi Street, Nicosia 2121, Cyprus
(b) Department of Physics, University of Cyprus, P.O. Box 20537, 1678 Nicosia,
Cyprus
(c) NIC, DESY, Platanenallee 6, D-15738 Zeuthen, Germany
###### Abstract
We perform a high statistics calculation of disconnected fermion loops on
Graphics Processing Units for a range of nucleon matrix elements extracted
using lattice QCD. The isoscalar electromagnetic and axial vector form
factors, the sigma terms and the momentum fraction and helicity are among the
quantities we evaluate. We compare the disconnected contributions to the
connected ones and give the physical implications on nucleon observables that
probe its structure.
###### pacs:
11.15.Ha, 12.38.Gc, 12.38.Aw, 12.38.-t, 14.70.Dj
## I Introduction
Lattice QCD simulations are currently performed near or at the physical value
of the light quark mass. This allows a study of hadron structure that can
provide valuable information for phenomenology and experiment. However, a
number of important observables are computed neglecting disconnected quark
loop contributions. The evaluation of disconnected quark loops is therefore of
paramount importance if we want to eliminate a systematic error inherent in
the determination of hadron matrix elements in lattice QCD. The computation of
disconnected quark loops within the lattice QCD formulation requires the
calculation of the so-called all-to-all or time-slice-to-all propagators, for
which the computational resources required to estimate them with, e.g.
stochastic methods, are much larger than those required for the corresponding
connected contributions. In addition, they are prone to large gauge noise. It
is for these reasons that in most hadron structure studies up to now the
disconnected contributions were neglected, introducing an uncontrolled
systematic uncertainty Alexandrou (2012).
Recent progress in algorithms, however, combined with the increase in
computational power, have made such calculations feasible. On the algorithmic
side, a number of improvements like the one-end trick Boucaud et al. (2008);
Michael and Urbach (2007); Dinter et al. (2012), dilution Bernardson et al.
(1994); Viehoff et al. (1998); O’Cais et al. (2005); Foley et al. (2005);
Alexandrou et al. (2012a), the Truncated Solver Method (TSM) Alexandrou et al.
(2012a); Collins et al. (2007); Bali et al. (2010) and the Hopping Parameter
Expansion (HPE) Boucaud et al. (2008); McNeile and Michael (2001) have led to
a significant reduction in both stochastic and gauge noise associated with
disconnected quark loops. Moreover, using special properties of the twisted
mass fermion Lagrangian Frezzotti et al. (2001); Frezzotti and Rossi (2004a,
b, c) one can further enhance the signal-to-noise ratio by taking the
appropriate combination of flavors. On the hardware side, graphics cards
(GPGPUs or GPUs) can provide a large speedup in the evaluation of quark
propagators and contractions. In particular, for the TSM, which relies on a
large number of inversions of the Dirac matrix in single or half precision,
GPUs provide an optimal platform.
In this paper, the aim is to use our findings on the performance of recently
developed methods Alexandrou et al. (2013a) to compute to high accuracy the
disconnected contributions that enter in the determination of nucleon form
factors, sigma terms and first moments of parton distributions. The evaluation
will be performed using one ensemble generated with two light degenerate
quarks and a strange and charm quark with masses fixed to their physical
values ($N_{f}=2+1+1$) using the twisted mass fermion discretization. The
lattice size is $32^{3}\times 64$, the lattice spacing extracted from the
nucleon mass Alexandrou et al. (2013b) $a=0.082(1)(4)$ fm and the pion mass
about 370 MeV. This ensemble will be hereafter referred to as the B55.32
ensemble. The aim is to compare the disconnected contributions computed using
${\cal O}(10^{5})$ measurements to the connected ones and assess the
importance of the disconnected contributions to nucleon observables computed
in lattice QCD for this given ensemble. The paper is organized as follows: in
Section II we summarize the algorithms and variance reduction techniques
employed, and in Section III we present the main numerical results of this
paper, namely the disconnected contributions to nucleon generalized form
factors. In Section IV we compare the disconnected contributions with the
corresponding connected ones. In Section V we give our conclusions and
outlook.
## II Methods for disconnected calculations
### II.1 Truncated Solver Method
The exact computation of all-to-all (or time-slice-to-all) propagators on a
lattice volume of physical interest is outside our current computer power,
since this would require volume (or spatial volume) times inversions of the
Dirac matrix, whose size ranges from $\sim 10^{7}$ for a $24^{3}\times 48$
lattice to $\sim 10^{9}$ for the largest volumes of $96^{3}\times 192$
considered nowadays. We will use the Truncated Solver Method (TSM) combined
with the one-end trick to evaluate the disconnected contributions. This method
was shown to be optimal for most observables involved in nucleon structure
computations Alexandrou et al. (2013a). For completeness we summarize here the
methods and refer the reader to Ref. Alexandrou et al. (2013a) for a more
detailed description and the comparison against other methods.
The usual approach to evaluate disconnected contributions is to compute an
unbiased stochastic estimate of the all-to-all propagator Bitar et al. (1989)
by generating a set of $N_{r}$ sources $\left|\eta_{r}\right\rangle$ randomly
drawn from e.g. $\mathbb{Z}_{2}\otimes i\mathbb{Z}_{2}$. Solving for
$\left|s_{r}\right\rangle$ in
$M\left|s_{r}\right\rangle=\left|\eta_{r}\right\rangle$ (1)
and calculating
$M_{E}^{-1}:=\frac{1}{N_{r}}\sum_{r=1}^{N_{r}}\left|s_{r}\right\rangle\left\langle\eta_{r}\right|\approx
M^{-1}$ (2)
provides an unbiased estimate of the all-to-all propagator as
$N_{r}\rightarrow\infty$. Since, in general, the number of noise vectors
$N_{r}$ required is much smaller than the lattice volume $V$, the computation
becomes feasible. How large $N_{r}$ should be depends on the observable.
Figure 1: The error on the isoscalar momentum fraction $\delta\langle
x\rangle_{u+d}$ as a function of $N_{\rm HP}+N_{\rm LP}$ for 68000
measurements. The three leftmost points (red squares) correspond to $N_{\rm
LP}=0$ and the three rightmost to $N_{\rm HP}=24$. The dotted line is the
result of fitting to the Ansatz $1/\sqrt{a+\frac{b}{N_{\rm HP}+N_{\rm LP}}}$.
The TSM is a way to increase $N_{r}$ at a reduced computational cost. The idea
behind the method is the following: instead of inverting to high precision the
stochastic sources in Eq. (1), we can aim at a low precision (LP) estimate
$\left|s_{r}\right\rangle_{LP}=\left(M^{-1}\right)_{LP}\left|\eta_{r}\right\rangle,$
(3)
where the number of inversions of the Conjugate Gradient (CG) used is
truncated. The criterion for the low precision inversions can be selected by
specifying a relaxed stopping condition in the CG e.g. by allowing a
relatively large value of the residual, which in turn determines the number of
iterations required to invert a source to low precision. Following Refs.
Alexandrou et al. (2012a, 2013a), we choose a stopping condition at fixed
value of the residual $|\hat{r}|_{\rm LP}\sim 10^{-2}$. $N_{\rm HP}$ is then
selected by requiring that the bias introduced when using $N_{\rm LP}$ low
precision vectors is corrected. We estimate the correction $C_{E}$ to the bias
stochastically by inverting a number of sources to high and low precision, and
calculating the difference,
$C_{E}:=\frac{1}{N_{\rm HP}}\sum_{r=1}^{N_{\rm
HP}}\left[\left|s_{r}\right\rangle_{\rm HP}-\left|s_{r}\right\rangle_{\rm
LP}\right]\left\langle\eta_{r}\right|,$ (4)
where the $\left|s_{r}\right\rangle_{\rm HP}$ are calculated by solving Eq.
(1) up to high precision, so our final estimate becomes
$\displaystyle M_{E_{TSM}}^{-1}:=$ $\displaystyle\frac{1}{N_{\rm
HP}}\sum_{r=1}^{N_{\rm HP}}\left[\left|s_{r}\right\rangle_{\rm
HP}-\left|s_{r}\right\rangle_{\rm LP}\right]\left\langle\eta_{r}\right|$ (5)
$\displaystyle+$ $\displaystyle\frac{1}{N_{\rm LP}}\sum_{j=N_{\rm HP}}^{N_{\rm
HP}+N_{\rm LP}}\left|s_{r}\right\rangle_{\rm LP}\left\langle\eta_{r}\right|,$
which requires $N_{\rm HP}$ high precision (HP) inversions and $N_{\rm
HP}+N_{\rm LP}$ low precision inversions. The ratio of the number of HP
inversions to the LP ones is determined with the criterion of choosing as
large a ratio as possible while still ensuring that the final result is
unbiased. In this work, we will compute fermion loops with the complete set of
$\Gamma$-matrices up to one-derivative operators. The tuning is, thus,
performed using an operator that requires a large number of stochastic noise
vectors, such as the nucleon isoscalar momentum fraction $\langle
x\rangle_{u+d}$ and we optimize $N_{\rm HP}$ and $N_{\rm LP}$ so as to get the
smallest error at the lowest computational cost. In Fig. 1 we show the error
on $\langle x\rangle_{u+d}$ as one varies $N_{\rm HP}$ and $N_{\rm LP}$. As
can be seen, the error decreases like $1/\sqrt{a+\frac{b}{N_{\rm HP}+N_{\rm
LP}}}$ with $a$ and $b$ positive parameters. Fixing $N_{\rm HP}=24$ and
increasing $N_{\rm LP}$ reduces the error rapidly until $N_{\rm LP}$ reaches
about $N_{\rm LP}\sim~{}300$. In Ref. Alexandrou et al. (2013a) we showed that
a ratio of $N_{\rm LP}$ to $N_{\rm HP}$ of about 20 can be considered
sufficient to produce an unbiased estimate for the class of observables
considered here. Therefore, in this work we take $N_{\rm HP}=24$ and choose
$N_{\rm LP}=500$ for the light quark sector. For the strange and charm quarks
we take $N_{\rm LP}=300$. These values were shown to also be optimal for the
isoscalar axial charge Alexandrou et al. (2013a).
### II.2 The one-end trick
The twisted mass fermion (TMF) formulation allows the use of a very powerful
method to reduce the variance of the stochastic estimate of the disconnected
diagrams. From the discussion given in section II.1, the standard way to
proceed with the computation of disconnected diagrams would be to generate
$N_{r}$ stochastic sources $\eta_{r}$, invert them as indicated in Eq. (1),
and compute the disconnected diagram corresponding to an operator $X$ as
$\displaystyle\frac{1}{N_{r}}\sum_{r=1}^{N_{r}}\left\langle\eta^{\dagger}_{r}Xs_{r}\right\rangle$
$\displaystyle=$ $\displaystyle\textrm{Tr}\left(M^{-1}X\right)$ (6)
$\displaystyle+$ $\displaystyle O\left(\frac{1}{\sqrt{N_{r}}}\right),$
where the operator $X$ is expressed in the twisted basis. However, if the
operator $X$ involves a $\tau_{3}$ acting in flavor space, one can utilize the
following identity of the twisted mass Dirac operator with $+\mu$ denoted by
$M_{u}$ and $-\mu$ denoted by $M_{d}$:
$M_{u}-M_{d}=2i\mu a\gamma_{5}.$ (7)
Inverting this equation we obtain
$M^{-1}_{u}-M^{-1}_{d}=-2i\mu aM_{d}^{-1}\gamma_{5}M_{u}^{-1}.$ (8)
Therefore, instead of using Eq. (6) for the operator $X\tau_{3}$, we can
alternatively write
$\displaystyle\frac{2i\mu a}{N_{r}}\sum_{r=1}^{N_{r}}\left\langle
s^{\dagger}_{r}\gamma_{5}Xs_{r}\right\rangle=$
$\displaystyle\textrm{Tr}\left(M_{u}^{-1}X\right)-\textrm{Tr}\left(M_{d}^{-1}X\right)$
$\displaystyle+O\left(\frac{1}{\sqrt{N_{r}}}\right)=$ $\displaystyle-2i\mu
a\textrm{Tr}\left(M_{d}^{-1}\gamma_{5}M_{u}^{-1}X\right)$
$\displaystyle+O\left(\frac{1}{\sqrt{N_{r}}}\right).$ (9)
Two main advantages result due to this substitution: i) the fluctuations are
effectively reduced by the $\mu$ factor, which is small in current
simulations, and ii) an implicit sum of $V$ terms appears in the right hand
side (rhs) of Eq. (8). The trace of the left hand side (lhs) of the same
equation develops a signal-to-noise ratio of $1/\sqrt{V}$, but thanks to this
implicit sum, the signal-to-noise ratio of the rhs becomes $V/\sqrt{V^{2}}$.
In fact, using the one-end trick yields for the same operator a large
reduction in the errors for the same computational cost as compared to not
using it Boucaud et al. (2008); Michael and Urbach (2007); Dinter et al.
(2012). A similar approach proved to be very successful in the determination
of the $\eta^{\prime}$ mass Jansen et al. (2008); Ottnad et al. (2012);
Michael et al. (2013). The identity given in Eq. (8) can only be applied when
a $\tau_{3}$ flavor matrix appears in the operator expressed in the twisted
basis. For other operators one can use the identity
$M_{u}+M_{d}=2D_{W},$ (10)
where $D_{W}$ is the Dirac-Wilson operator without a twisted mass term. After
some algebra, one finds
$\displaystyle\frac{2}{N_{r}}\sum_{r=1}^{N_{r}}\left\langle
s^{\dagger}_{r}\gamma_{5}X\gamma_{5}D_{W}s_{r}\right\rangle$ $\displaystyle=$
$\displaystyle\textrm{Tr}\left(M_{u}^{-1}X\right)+\textrm{Tr}\left(M_{d}^{-1}X\right)$
(11) $\displaystyle+$ $\displaystyle O\left(\frac{1}{\sqrt{N_{r}}}\right).$
This lacks the $\mu$-suppression factor, which, as we will see in the
following sections and as discussed in more detail in Ref. Alexandrou et al.
(2013a), introduces a considerable penalty in the signal-to-noise ratio.
Because of the volume sum that appears in Eq. (8) and Eq. (11), the sources
must have entries on all sites, which in turn means that we can compute the
fermion loop at all time slices where the operator is inserted in a single
inversion. This allows us to evaluate the three-point function for all
combinations of source-sink time separation and insertion time slices, which
will prove essential in identifying the contribution of excited state effects
for the different operators.
## III Results
In this section we present results from a high statistics evaluation of all
the disconnected contributions involved in the evaluation of nucleon form
factors and first moments of generalized parton distributions as well as sigma
terms. As already mentioned, the analysis is performed using an ensemble of
$N_{f}=2+1+1$ twisted mass configurations simulated with pion mass of
$am_{\pi}=0.15518(21)(33)$ and strange and charm quark masses fixed to
approximately their physical values (B55.32 ensemble) Baron et al. (2010). The
lattice size is $32^{3}\times 64$ giving $m_{\pi}L\sim 5$. We use the one-end
trick method combined with the TSM with $N_{\rm HP}=24$ and $N_{\rm LP}=500$
noise vectors for the light quark loops. For the strange and charm quark
sector we use $N_{\rm HP}=24$ and $N_{\rm LP}=300$. Using 2,300 gauge-field
configurations, with 16 source positions for the two-point function and by
averaging results for the proton/neutron and forward/backward propagating
nucleons we effectively have $\sim 150,000$ measurements.
An advantage of the one-end trick is that, having the loop at all time slices,
we can combine with two-point functions produced at any source time slice.
Furthermore, since the noise sources are defined on all sites, we obtain the
fermion loops at all insertion time slices. We can thus compute all possible
combinations of source-sink time separations and insertion times in the three-
point function. This feature enables us to use the summation method, in
addition to the plateau method, with no extra computational effort.
The summation method has been known for a long time Maiani et al. (1987);
Gusken (1999) and has been revisited in the study of $g_{A}$ Capitani et al.
(2010). In both the plateau and summation approaches, one constructs ratios of
three- to two-point functions in order to cancel unknown overlaps and
exponentials in the leading contribution such that the matrix element of the
ground state is isolated. For general momentum transfer we consider the ratio
$R(t_{\rm ins},t_{s}){=}\frac{G^{3pt}(\Gamma^{\nu},{\vec{p}},{\vec{q}},t_{\rm
ins},t_{s})}{G^{2pt}(\vec{p}^{\prime},t_{s})}\sqrt{\frac{G^{2pt}(\vec{p},t_{s}{-}t_{\rm
ins})G^{2pt}(\vec{p}^{\prime},t_{\rm
ins})G^{2pt}(\vec{p}^{\prime},t_{s})}{G^{2pt}(\vec{p}^{\prime},t_{s}{-}t_{\rm
ins})G^{2pt}(\vec{p},t_{\rm ins})G^{2pt}(\vec{p},t_{s})}}$ (12)
where the two- and three-point functions are given respectively by
$\displaystyle G^{2pt}(\vec{q},t_{s})=$
$\displaystyle\sum_{\vec{x}_{s}}\,e^{-ix_{s}\cdot\vec{q}}\,{\Gamma^{0}_{\beta\alpha}}\,\langle{J_{\alpha}(t_{s},\vec{x}_{s})}{\overline{J}_{\beta}(0,\vec{0})}\rangle$
(13) $\displaystyle G^{3pt}(\Gamma^{\nu},\vec{p},\vec{q},t_{\rm ins},t_{s})=$
$\displaystyle\sum_{\vec{x}_{\rm ins},\vec{x}_{s}}\,e^{i\vec{x}_{\rm
ins}\cdot\vec{q}}\,e^{-i\vec{x}_{s}\cdot\vec{p}}\,\Gamma^{\nu}_{\beta\alpha}\,\langle{J_{\alpha}(t_{s},\vec{x}_{s})}\mathcal{O}^{\mu_{1}\cdots\mu_{n}}(t_{\rm
ins},\vec{x}_{\rm ins}){\overline{J}_{\beta}(0,\vec{0})}\rangle\,.$ (14)
$q=p^{\prime}-p$ is the momentum transfer, $t_{s}$ is the time separation
between the sink and the source with the source taken at zero, and $t_{\rm
ins}$ the time separation between the current insertion and the source. We
consider the complete set of operators $\mathcal{O}^{\mu_{1},\cdots,\mu_{n}}$
up to one derivative, namely the scalar $\bar{\psi}\,\psi$, vector
$\bar{\psi}\,\gamma^{\mu}\psi$, axial-vector
$\bar{\psi}\,\gamma^{5}\,\gamma^{\mu}\psi$ and the tensor
$\bar{\psi}\sigma^{\mu\nu}\psi$ currents, and the one-derivative vector
$\bar{\psi}\,\gamma^{\\{\mu_{1}}D^{\mu_{2}\\}}\psi$ and axial-vector
$\bar{\psi}\,\gamma_{5}\,\gamma^{\\{\mu_{1}}D^{\mu_{2}\\}}\psi$ operators. We
consider kinematics for which the final momentum $\vec{p}^{\prime}=0$ when we
consider the connected contributions. For the evaluation of disconnected
contributions we use kinematics where $\vec{p}=\vec{p}^{\prime}\neq 0$ as well
as $\vec{p}^{\prime}=0$. The projection matrices ${\Gamma^{0}}$ and
${\Gamma^{k}}$ are given by:
${\Gamma^{0}}=\frac{1}{4}(\mathds{1}+\gamma_{0})\,,\quad{\Gamma^{k}}={\Gamma^{0}}i\gamma_{5}\sum_{k=1}^{3}\gamma_{k}\,.$
(15)
For zero momentum transfer the ratio simplifies to
$R(t_{ins},t_{s})=\frac{G^{3pt}(\Gamma^{\nu},\vec{p},t_{ins},t_{s})}{G^{2pt}(t_{,}\vec{p})}$
(16)
The leading time dependence of the ratio $R(t_{\rm ins},t_{s})$ is given by
$R(t_{ins},t_{s})=R_{GS}+O(e^{-\Delta E_{K}t_{ins}})+O(e^{-\Delta
E_{K}(t_{s}-t_{ins})}),$ (17)
where $R_{GS}$ is the matrix element of interest, and the other contributions
come from the undesired excited states of energy difference $\Delta E_{K}$. In
the plateau method, one plots $R(t_{ins},t_{s})$ as a function of $t_{\rm
ins}$. For large time separations $t_{\rm ins}$ and $t_{s}-t_{\rm ins}$ when
excited state effects are negligible this ratio becomes a constant (plateau
region) and therefore fitting it to a constant yields $R_{GS}$. In the
alternative summation method, one performs a sum over $t_{\rm ins}$ to obtain:
$R_{\rm sum}(t_{s})=\sum_{t_{\rm ins}=0}^{t_{\rm ins}=t_{s}}R(t_{\rm
ins},t_{s})=t_{s}R_{GS}+a+O(e^{-\Delta E_{K}t_{s}})$ (18)
where $a$ is a constant and the exponential contributions coming from the
excited states decay as $e^{-\Delta E_{K}t_{s}}$ as opposed to the plateau
method where excited states are suppressed like $e^{-\Delta
E_{K}(t_{s}-t_{ins})}$, with $0\leq t_{\rm ins}\leq t_{s}$. Therefore, we
expect a better suppression of the excited states for the same $t_{s}$. Note
that one can exclude from the summation the initial and final time slices
$t_{s}$ and $0$ without affecting the dependence on $t_{s}$ in Eq. (18). The
results given in this work are obtained excluding these contact terms from the
summation. The drawback of the summation method is that one requires knowledge
of the three point function for all insertion times and multiple sink times
and one needs to fit to a straight line with two fitting parameters instead of
one.
$Z_{A}$ | $Z_{T}$ | $Z_{P}$ | $Z_{DV}^{\mu\mu}$ | $Z_{DV}^{\mu\neq\nu}$ | $Z_{DA}^{\mu\mu}$ | $Z_{DA}^{\mu\neq\nu}$
---|---|---|---|---|---|---
0.757(3) | 0.769(1) | 0.506(4) | 1.019(4) | 1.053(11) | 1.086(3) | 1.105(2)
Table 1: Renormalization constants in the chiral limit at $\beta=1.95$ in the
$\overline{\rm MS}$-scheme at $\mu=2$ GeV. $Z_{A}$, $Z_{T}$ and $Z_{P}$ are
the renormalization constants for the axial-vector, tensor and scalar
currents, and $Z_{DV}$ and $Z_{DA}$ for the one-derivative vector and axial-
vector operators ${\cal O}^{\mu\nu}$. The errors given are statistical.
Figure 2: The disconnected contribution to the ratio from which $\sigma_{\pi
N}$ is extracted. On the upper panel we show the ratio as a function of the
insertion time slice with respect to the mid-time separation ($t_{\rm
ins}-t_{s}/2$) for source-sink time separations, $t_{\rm s}=$14$a$ (red filled
circles), $t_{\rm s}=16a$ (blue filled squares), $t_{\rm s}=18a$ (green open
squares) and $t_{\rm s}=20a$ (yellow filled triangles). In the central panel
we show the summed ratio, for which the fitted slope yields the desired matrix
element. On the lower panel we show the results obtained for the fitted slope
of the summation method for various choices of the initial and final fit time
slices. The star shows the choice for which the gray bands are plotted in the
upper and central panels.
Figure 3: The ratio from which the strange quark content of the nucleon,
$\sigma_{s}$, is extracted. The notation is the same as that of Fig. 2.
Figure 4: The ratio from which the charm quark content of the nucleon,
$\sigma_{c}$, is extracted. The notation is the same as that of Fig. 2.
Figure 5: The disconnected contribution to the renormalized ratio which yields
the isoscalar axial charge of the nucleon, $g_{A}^{u+d}$. The upper panel
shows the ratio as a function of the insertion time slice with respect to the
mid-time separation ($t_{\rm ins}-t_{s}/2$) for source-sink separations
$t_{\rm s}=8a$ (red filled circles), $t_{\rm s}=10a$ (blue filled squares),
$t_{\rm s}=12a$ (green open squares) and $t_{\rm s}=14a$ (yellow filled
triangles). The central panel shows the summed ratio and the lower panel the
results obtained for the fitted slope of the summation method for various
choices of the initial and final fit time slices as explained in the text. The
star shows the choice of $t_{i}$, which yields the gray bands shown in the
upper and central plots.
Figure 6: The strange-quark contribution to the renormalized ratio yielding
the nucleon axial charge $g_{A}^{s}$. The notation is the same as that of Fig.
5.
Figure 7: The charm-quark contribution to the renormalized ratio yielding the
nucleon axial charge $g_{A}^{c}$. The notation is the same as that of Fig. 5.
Figure 8: The disconnected contribution to the renormalized ratio yielding the
nucleon isoscalar tensor charge $g_{T}^{u+d}$. The notation is the same as
that of Fig. 2.
Figure 9: The disconnected contribution to the renormalized ratio yielding the
nucleon isoscalar momentum fraction $\langle x\rangle_{u+d}$. The notation is
the same as that of Fig. 5.
Figure 10: The disconnected contribution to the renormalized ratio yielding
nucleon isoscalar helicity moment $\langle x\rangle_{\Delta u+\Delta d}$. The
notation is the same as that of Fig. 5.
Figure 11: Disconnected contributions to the renormalized ratio yielding the
isoscalar axial-vector and pseudo-scalar form-factors $G_{A}$ and $G_{p}$
(upper), the electric form-factor $G_{E}$ (center) and the magnetic form-
factor $G_{M}$ (lower) at the lowest non-zero momentum transfer allowed for
this lattice size. Figure 12: The renormalized ratio which yields the
strange-quark contribution to the axial charge of the nucleon, $g_{A}^{s}$. In
the left panel, the plateau method is used on the first half of the ensemble
(A-set), while the summation method is used on the second half of the ensemble
(B-set). In the right panel, he plateau method is used on the A-set, while the
summation method is used on the B-set.
Before comparing the lattice matrix elements $R_{\rm GS}$ with experiment we
need to renormalize them. We denote the renormalized ratio by
$\tilde{R}(t_{\rm ins},t_{s})$. Regarding the renormalization of the sigma
terms, the twisted mass formulation has the additional advantage of avoiding
any mixing, even though we are using Wilson-type fermions Dinter et al.
(2012). For the case of the axial charge, renormalization involves mixing from
the three quark sectors. For the tree-level Symanzik improved gauge action
this mixing was shown to be a small effect of a few percent Skouroupathis and
Panagopoulos (2009). We expect this to hold also for the Iwasaki action used
in this work and for the other isoscalar quantities. In this work, we neglect
the small difference in the renormalization constant between connected and
disconnected contributions and we use the same renormalization constants as
for the connected piece. They are given in Table 1. The value of $Z_{P}$ needs
a pole subtraction and is taken from Ref. Blossier et al. (2011); ETMC , while
all the others have been calculated using the approach given in Refs.
Alexandrou et al. (2011); Alexandrou et al. (2012b). All the renormalization
constants, except $Z_{A}$ which is scheme and scale independent, are converted
from RI-MOM to $\overline{\rm MS}$ at a scale of $\mu=2$ GeV. The conversion
factors for $Z_{T}$ are taken from Ref. Gracey (2003), and for the one-
derivative operators from Ref. Alexandrou et al. (2011), computed to three-
loops. We remark that in the twisted basis the scalar charge is renormalized
with $Z_{P}$.
In Fig. 2 we show the results for the disconnected contribution to the ratio
from which the $\sigma_{\pi N}$-term is extracted. The ratio is plotted versus
the time separation of the current insertion $t_{\rm ins}$ from the source,
shifted by $t_{s}/2$. When this ratio becomes time independent (plateau
region) fitting to a constant yields $\sigma_{\pi N}$. As can be seen,
however, increasing the source-sink time separation increases the value
extracted from fitting to the plateau (plateau value). We observe that one
requires a source-sink time separation of at least 18 to 20 time slices in
order for the plateau value to stabilize. This is a distance of $\gtrsim 1.5$
fm, which is significantly larger than the nominal source-sink separation of
1.0 fm-1.2 fm typically used in nucleon matrix element calculations. In the
central panel we show the ratio summed over the insertion time slice as given
in Eq. (18) referred to as summation method (SM) as a function of the source-
sink time separation time. As explained earlier, by fitting the ratio to a
straight line one obtains the desired matrix element as the slope. This is
done for several choices of the initial and final fit time slices ($t_{i}$ and
$t_{f}$ respectively) with the results displayed in the lower panel of the
figure. As one increases the initial fit time slice the excited state
contributions are expected to become smaller and thus the fitted value
stabilizes. Note, however, that the slope changes and one needs to vary the
fit range until the slope converges. Therefore, if one has only a small number
of source-sink time separations one may miss the variation of the slope. As in
the case of the plateau method where we take the smallest $t_{s}$ for which
excited states are sufficiently suppressed, it is desirable to take the
smallest $t_{i}$ for which the excited states no longer contribute
significantly, since the error to signal ratio increases with $t_{i}$. Taking
the value of the slope to be the one given by the star yields the value of
$\sigma_{\pi N}$ shown by the gray band in the upper panel of the figure. As
can be seen, the resulting value is in agreement with the (colored) band
obtained from the plateau method for $t_{s}/a=20$.
A similar analysis is undertaken for the strange- and charm-quark sigma terms,
shown in Figs. 3 and 4 respectively. For $\sigma_{s}$, similar remarks can be
made as in the case of $\sigma_{\pi N}$, most notably concerning the large
source-sink separation required for the plateau method to converge. As
expected, the results between the summation and the plateau method are
consistent also in this case, when excited states are suppressed. Non-zero
results for $\sigma_{s}$ were also obtained in Ref. Gong et al. (2013) using
optimal noise sources and low-mode substitution techniques. For the case of
the charm content, our results are consistent with zero both when using the
plateau method as well as when using the summation method allowing us only to
obtain an upper bound to its value. In Ref. Gong et al. (2013) a non-zero
result was obtained as one approaches the chiral limit. Since our aim in this
work is to compute quark loops using high statistics for one ensemble we will
address the quark mass dependence in a follow-up work.
Similar analyses are carried out for the disconnected contributions entering
the ratios determining the nucleon axial charge. For observables like $g_{A}$
where one does not have the $\tau^{3}$ flavor combination in the twisted basis
it is advantageous to use the discrete symmetries of the twisted mass
formulation Frezzotti and Rossi (2004b, c), namely parity combined with
isospin flip $u\leftrightarrow d$, $\gamma_{5}$-isospin hermiticity, and
charge-$\gamma_{5}$-isospin hermiticity, in order to reduce gauge noise.
Considering the properties of the quark loops and of the nucleon two-point
functions that enter in the computation of the disconnected three-point
function under these symmetries one can derive appropriate products taking
their real or imaginary parts thus suppressing gauge noise. This was shown to
be advantageous in the calculation of the first moments of the unpolarized
momentum distribution in Ref. Deka et al. (2009). These symmetries are used
for the results shown from now on. In Figs. 5, 6 and 7 we show, respectively,
results for the ratio from which the nucleon matrix elements of the axial-
vector current yielding the isoscalar $g_{A}$,the strange $g_{A}^{s}$ and the
charm $g_{A}^{c}$ are extracted. We first note that for the case of
$g_{A}^{u+d}$ we observe less contamination from excited states than in the
case of the sigma terms. This is evident from the smaller source-sink time
separations required in order for the plateau or summation method to converge.
Furthermore, we clearly observe a non-zero value for the case of the
disconnected contributions to the isoscalar $g_{A}$ as well as for
$g_{A}^{s}$. For $g_{A}^{c}$ our results are consistent with zero and we can
only give an upper bound to its value. The nucleon tensor charge $g_{T}^{u+d}$
is also computed and the ratio from which is extracted is shown in Fig 8. We
observe a very small value for the disconnected contribution, with an error of
about 90%. For the summation method the statistical uncertainty does not allow
a meaningful fit.
The nucleon matrix elements involving derivative operators probe moments of
parton distributions, which are extracted from deep inelastic scattering
measurements. In this work we compute the disconnected contributions to the
isoscalar nucleon momentum fraction $\langle x\rangle_{u+d}$, which involves
the vector derivative operator and the isoscalar nucleon polarized moment
$\langle x\rangle_{\Delta u+\Delta d}$ involving the axial-vector derivative
operator. We apply the symmetries of the twisted mass action discussed above
as well as consider a moving frame and thus have the nucleon carrying non-zero
equal initial and final momentum for three-point functions with zero momentum
transfer. We find that, when the nucleon carries the lowest momentum allowed
for this lattice, the statistical error is reduced. The disconnected
contributions to the ratios, from which the matrix elements of the vector and
axial-vector derivative operators, are extracted are shown in Figs. 9 and 10
respectively. For $\langle x\rangle_{u+d}$ we find a value consistent with
zero both with the plateau and summation method. Having one unit of momentum
improves the signal enabling us to deduce an upper bound on the value of this
matrix element. For $\langle x\rangle_{\Delta u+\Delta d}$ the statistical
errors remain large but nevertheless we obtain a non-zero value. Considering a
moving nucleon leads in this particular case to a substantial reduction in the
error. We note that increasing the sink-source time separation is crucial in
order for this observable to develop a non-zero result. This is clearly seen
in the slope which becomes non-zero for $t_{s}/a>8$. Since a large $t_{s}$
also leads to larger errors it is no surprise that such a large number of
statistics is needed to obtain a meaningful signal. This may also indicate
that even larger number of statistics are needed to stabilize further the
signal.
Apart from matrix elements for zero momentum transfer presented so far,
disconnected contributions arise in the isoscalar electromagnetic and axial
form factors at finite momentum. Computationally, these are straightforward to
extract, since one takes the Fourier transform of the insertion coordinate of
the loop to obtain the matrix element at all momenta. The finite momentum
matrix elements, however, are expected to be nosier than the zero-momentum
ones, since the energy factors appearing in the exponents of the signal are
larger. The disconnected contributions to the axial form-factors, electric
form-factor and magnetic form-factor are shown in Fig. 11 for a single unit of
momentum transfer. Due to the structure of the matrix elements and the way
these are computed on the lattice, for the case of the axial form factors
$G_{A}$ and $G_{p}$, the plot shows the ratio of a linear combination from
which these form factors are extracted after the plateau fit. $G_{E}$ and
$G_{M}$, on the other hand, can be extracted from different ratios allowing us
to plot them separately. We note that we perform a similar analysis for these
quantities as for the zero-momentum case where both plateau and summation
methods are investigated for the optimal fit ranges. For the axial form-
factors we obtain a clearly non-zero value. For the electromagnetic case, the
disconnected contributions for both the isoscalar electric and magnetic form
factors are statistically consistent with zero.
Figure 13: Connected contributions to the ratio yielding $\sigma_{\pi N}$
(upper) and nucleon isoscalar axial charge (lower), for various source-sink
time separations are shown. Results obtained from a fit to a constant to the
ratio (colored band) and from a linear fit to the summed ratio (gray band) are
also displayed.
.
Figure 14: Connected contributions to the renormalized ratio yielding the
isoscalar nucleon momentum fraction (upper), the isoscalar nucleon helicity
moment (center) and the axial and pseudo-scalar form factors $G_{A}(Q^{2})$
and $G_{p}(Q^{2})$ at a single unit of momentum (lower) are shown. For the
momentum fraction and helicity, we show the results obtained from a fit to a
constant to the renormalized ratio (colored band) and from a linear fit to the
summed renormalized ratio (gray band).
.
Finally we comment on the issue of correlations. The summation and plateau
methods for various quantities are compared using the same set of gauge
configurations and found to be consistent. Since these results can be
correlated, the difference between the results of the two methods maybe
underestimated. Thus, it is worthwhile to investigate the two methods using
different sets of configurations. To perform this check we split our ensemble
into two equal sets, which we will refer to as A-set and B-set, and redo our
analysis on these two sets separately. We show the result for the case of the
strange-quark contribution to the axial charge in Fig. 12. As can seen, the
values computed in each set both using the plateau and summation methods are
in agreement. Furthermore, the plateau computed using the A-set is consistent
with the summation method computed using the B-set and vice versa. This
agreement indicates that the consistency between the results extracted using
the summation and plateau methods on the full ensemble is not due to possible
correlations.
## IV Comparison with connected contribution
The main motivation for calculating disconnected fermion loops is to eliminate
the systematic uncertainty, which arises when these are omitted from
calculations of hadronic matrix elements. For instance, the nucleon axial
charge is typically computed in the isovector combination, where the fermion
loops of the up- and down- quarks cancel. However, if one is interested in the
intrinsic spin fraction carried by the individual quarks, one needs, in
addition to the isovector, the isoscalar combination, which involves
disconnected diagrams. Typically, in lattice QCD calculations up to now, the
disconnected contributions have been omitted. It is, therefore, important to
identify how large the contributions of disconnected diagrams are, in order to
bound the systematic error introduced when these are neglected.
Observable | connected | disconnected | total
---|---|---|---
Results at zero momentum transfer ($Q^{2}=0$)
$\sigma_{\pi N}$ | [MeV] | 164.6(7.2) | 16.6(2.4) | 181.3(7.6)
$\sigma_{s}$ | [MeV] | | 21.7(3.6) | 21.7(3.6)
$\sigma_{c}$ | [MeV] | | 16(30) | 16(30)
$g_{S}^{u+d}$ | | 6.30(27) | 0.639(95) | 6.94(29)
$g_{S}^{s}$ | | | 0.246(41) | 0.246(41)
$g_{A}^{u+d}$ | | 0.576(13) | -0.0699(89) | 0.506(15)
$g_{A}^{s}$ | | | -0.0227(34) | -0.0227(34)
$g_{T}^{u+d}$ | | 0.673(13) | -0.0016(14) | 0.671(13)
$\langle x\rangle_{u+d}$ | | 0.586(22) | 0.027(76) | 0.614(80)
$\langle x\rangle_{\Delta u+\Delta d}$ | | 0.1948(51) | -0.058(22) | 0.136(23)
$J^{u}$ | | 0.2781(94) | -0.076(77) | 0.202(78)
$J^{d}$ | | -0.0029(94) | -0.076(77) | -0.078(78)
$\Delta\Sigma^{u}/2$ | | 0.4273(50) | -0.0174(75) | 0.4098(55)
$\Delta\Sigma^{d}/2$ | | -0.1389(50) | -0.0174(75) | -0.1564(55)
Results for $\vec{q}^{2}=(2\pi/L)^{2}$ or $Q^{2}\simeq$0.19 GeV2
$G^{u+d}_{E}$ | | 2.2698(78) | 0.024(21) | 2.293(22)
$G^{u+d}_{M}$ | | 2.088(49) | -0.066(75) | 2.022(89)
$G^{u+d}_{A}$ | | 0.5155(94) | -0.0564(72) | 0.459(11)
$G^{u+d}_{p}$ | | 9.81(65) | -1.90(35) | 7.90(74)
$B^{u+d}_{20}$ | | -0.035(16) | -0.33(29) | -0.36(29)
$G^{p}_{E}$ | | 0.7453(32) | 0.0040(58) | 0.7493(47)
$G^{n}_{E}$ | | 0.0113(32) | 0.0040(58) | 0.0153(47)
$G^{p}_{M}$ | | 1.847(28) | -0.011(42) | 1.836(31)
$G^{n}_{M}$ | | -1.151(28) | -0.011(42) | -1.162(31)
Table 2: The connected and disconnected contributions to the various nucleon
observables for the B55.32 ensemble are given in column two and three, whereas
column four has the total contribution. The form factors $G_{E}$, $G_{M}$,
$G_{A}$ and $G_{p}$, and generalized form factor $B_{20}$ are given for
$\vec{q}=2\pi/L$. The disconnected contributions were obtained using about
150,000 measurements.
In order to assess the importance of disconnected contributions, we evaluate
the connected contributions to the isoscalar matrix elements of the operators
discussed in the previous section. In Figs. 13 and 14 we show the renormalized
ratios from which the connected part of the isoscalar matrix elements are
extracted. These results are obtained using 1200 gauge field configurations
and inverted for multiple source-sink time separations to allow applying the
summation method. We stress that, for the evaluation of the connected
contributions unlike the case of the disconnected, to obtain multiple source-
sink time separations one needs to do a new set of inversions for each sink-
source time separation.
The multiple source-sink time separations are computed more efficiently by
using the EigCG Stathopoulos and Orginos (2010); Stathopoulos et al. (2009)
method to deflate the lowest eigenvalues with every new right-hand-side. For
the connected contributions shown here, we compute the sequential propagators
for eight source-sink time separations, namely from $t_{s}=4a$ to $t_{s}=18a$
for every even time separation. In addition, the sequential propagators are
computed for both unpolarized and polarized nucleon sinks, meaning in total 16
sequential propagators per configuration, or 16$\times$12=192 right-hand-sides
are needed, one for each color-spin component. Our EigCG is set up such that
ten eigenvalues per right-hand-side are deflated, stopping after a total of 24
right-hand-sides, after which the deflated space is kept constant at 240
eigenvalues for the remaining 168 right-hand-sides. With this setup, and at
this pion mass, we observe a speedup of more than 3 times, i.e. the 192 right-
hand-sides are computed for the same computational cost needed to compute 64
right-hand-sides when not using EigCG.
The ratios yielding the connected contribution to $\sigma_{\pi N}$, and the
isoscalar $g_{A}$ are shown in Fig. 13. These can be compared with the
corresponding ratios yielding the disconnected contributions to $\sigma_{\pi
N}$ and isoscalar $g_{A}$ shown in Figs. 2 and 5, respectively. As can be
seen, the behavior of the connected contributions is similar to the
disconnected ones, namely the sigma term shows large excited state
contamination requiring large sink-source separations whereas in the case of
$g_{A}^{u+d}$ the excited states are negligible even for $t_{s}/a=10$. For a
better comparison between connected and disconnected contributions we collect
the results extracted from the plateau method for all nucleon observables in
Table 2. The disconnected contribution to the $\sigma_{\pi N}$ and isoscalar
$g_{A}$ are found to be larger than 10% of the connected contribution at this
quark mass. Clearly for both $\sigma_{\pi N}$ and $g_{A}^{u+d}$ these are
sizable effects and have to be taken into account. The scalar charge derives
from the same matrix element as the sigma term and therefore it also requires
inclusion of disconnected contributions. For the case of the momentum
fraction, the disconnected contribution is found to be consistent with zero as
can be seen in Fig. 9, and therefore we can only give an upper bound to its
size to be included in the systematic error of $\langle x\rangle_{u+d}$. For
the polarized moment $\langle x\rangle_{\Delta u+\Delta d}$, on the other
hand, one obtains a sizable non-zero result. Note that the disconnected
contribution is negative decreasing the value of $\langle x\rangle_{\Delta
u+\Delta d}$ quite substantially. The disconnected contribution to the tensor
charge is essentially zero not affecting its total value.
A comment can also be made for the case of the disconnected contributions to
the nucleon form factors computed at non-zero momentum shown in Fig. 11 at a
single unit of momentum transfer squared. For the electromagnetic form-factors
$G_{E}$ and $G_{M}$, we find that the disconnected contributions are
consistent with zero and with magnitude less than 1%. With the connected
contributions at this momentum transfer being of $O(1)$, this means that the
disconnected contributions will, at most, be at the 1% level. For the case of
the axial form factor $G_{A}^{u+d}$, the disconnected contribution is about
10% that of the connected and thus, it must be included. In the case of the
pseudo-scalar form factor $G_{p}$, we find that the disconnected contribution
is of similar magnitude as the connected one and thus it is crucial in order
to get reliable results for this observable to include the disconnected part.
Having the complete set of isoscalar matrix elements with both connected and
disconnected contributions, one can combine with the corresponding isovector
matrix elements, which do not depend on disconnected contributions, to obtain
the separate quark contributions to nucleon matrix elements. This is done in
Table 2 for all the various quantities considered in this work. Namely, the
up- and down-quark contributions to the nucleon spin $\Delta\Sigma^{u}/2$ and
$\Delta\Sigma^{d}/2$ are obtained by combining the isovector and isoscalar
axial charges. Including the disconnected contributions affects the values of
the intrinsic spin in particular in the case of the d-quark. In contrast, the
values of the nucleon total spin $J^{u}$ and $J^{d}$, obtained by combining
the isoscalar and isovector vector generalized form-factors $A_{20}$ and
$B_{20}$, are not affected and the disconnected contributions only contribute
an upper bound to the error. Finally, the proton/neutron electric and magnetic
form factors $G^{p/n}_{E}$ and $G^{p/n}_{M}$ at a single unit of momentum
transfer squared, which for this lattice size and quark mass corresponds to
$Q^{2}\simeq 0.19$ GeV2, are obtained from the isovector and isoscalar proton
electric and proton magnetic form-factors assuming flavor-SU(2) isospin
symmetry between up- and down-quarks. Only the value of $G_{E}^{n}$ is
affected although, within error bars, it is still consistent with the
connected value.
## V Conclusions
The computation of disconnected contributions for flavor singlet quantities
has become feasible, due to the development of new techniques to reduce the
gauge and stochastic noise, and due to the increase in computational
resources. In this work, we use the truncated solver method and the one-end
trick on GPUs for the determination of disconnected contributions to the
nucleon matrix elements. The usage of GPUs is particularly important, due to
its efficiency in the evaluation of disconnected diagrams using the TSM, since
GPUs can yield a large speedup when employing single- and half-precision for
the computation of the LP inversions and contractions. The calculation is
performed for one ensemble of $N_{f}=2+1+1$ twisted mass fermions using very
high statistics. This is necessary in order to reduce the gauge noise and
obtain statistically significant results.
The results for all observables are analyzed using both the plateau and the
summation methods. A careful analysis of excited states is performed and we
find that the methods yield results that are compatible, as expected when
excited states contributions are negligible and identification of the fitting
ranges in both methods are well selected. Therefore, agreement of the values
extracted with the plateau and summation methods provides a good consistency
check. Since the one-end trick provides results for all sink-source
separations at no additional computational cost, such a check can be always
carried out.
Comparison of the connected to the disconnected contributions reveals clearly
that the latter are important for a number of observables related to nucleon
structure. For the sigma terms and scalar charge the disconnected
contributions amount to 10% the total value and thus they must be taken into
account. Similarly for the isoscalar axial charge we find more than 10%
contributions that must be taken into account in the discussion of the spin
carried by quarks in the proton. The disconnected contribution reduces the
value of $\Sigma^{d}$ by more than 10%, an effect that is important if we aim
at a few % accuracy. On the other hand, we find that the disconnected
contributions to the electromagnetic form factors at low $q^{2}$-values are
less than 1% at this pion mass. For the axial form factor $G_{A}$ the
disconnected contributions are sizable and persist at the level of 10% of the
value of the connected contribution even at non-zero momentum-transfer. For
$G_{p}$ the disconnected contribution is even larger reaching 20%.
In the future we plan to compute the disconnected contributions to these
quantities using simulations at physical pion mass. Such a computation will
require very large computational resources in order to obtain results with
meaningful statistical errors.
## Acknowledgments
A. V. and M. C. are supported by funding received from the Cyprus Research
Promotion Foundation under contracts EPYAN/0506/08 and and
TECHNOLOGY/$\Theta$E$\Pi$I$\Sigma$/0311(BE)/16 respectively. K. J. is partly
supported by RPF under contract
$\Pi$PO$\Sigma$E$\Lambda$KY$\Sigma$H/EM$\Pi$EIPO$\Sigma$/0311/16. This
research was in part supported by the Research Executive Agency of the
European Union under Grant Agreement number PITN-GA-2009-238353 (ITN
STRONGnet) and the infrastructure project INFRA-2011-1.1.20 number 283286
(HadronPhysics3), and the Cyprus Research Promotion Foundation under contracts
KY-$\Gamma$A/0310/02 and NEA Y$\Pi$O$\Delta$OMH/$\Sigma$TPATH/0308/31
(infrastructure project Cy-Tera, co-funded by the European Regional
Development Fund and the Republic of Cyprus through the Research Promotion
Foundation). Computational resources were provided by the Cy-Tera machine and
Prometheus (partly funded by the EU FP7 project PRACE-2IP under grant
agreement number: RI-283493) of CaSToRC, Forge at NCSA Illinois (USA),
Minotauro at BSC (Spain), and by the Jugene Blue Gene/P machine of the Jülich
Supercomputing Center awarded under PRACE.
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|
arxiv-papers
| 2013-10-23T19:33:50 |
2024-09-04T02:49:52.783305
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "A. Abdel-Rehim (The Cyprus Inst.), C. Alexandrou (Univ. of Cyprus &\n The Cyprus Inst.), M. Constantinou (Univ. of Cyprus), V. Drach\n (DESY-Zeuthen), K. Hadjiyiannakou (Univ. of Cyprus), K. Jansen\n (DESY-Zeuthen), G. Koutsou (The Cyprus Inst.), A. Vaquero (The Cyprus Inst.)",
"submitter": "Constantia Alexandrou",
"url": "https://arxiv.org/abs/1310.6339"
}
|
1310.6340
|
Initial work at proposal time is discussed here. Also included are some glimpses into the theory that motivates
this work. This section enumerates the work borrowed from previous work done in this field notably at Novosibirsk, Russia.
Also presented are some recent developments in Ionization Chamber Technology.
Ground Work
In this section, the work is presented with additional studies resulting from discussion with experts. It includes establishing the idea with a "back of the envelop calculation"
backed by a full fledged GEANT-4 simulation. Also, related effects such as beam motion were studied and their effects that impact the polarimeter negatively were shown to be minimal.
Even though the GEANT-4 simulation is still in the making, a skeletal code is briefly explained here.
Project Work
|
arxiv-papers
| 2013-10-22T13:22:59 |
2024-09-04T02:49:52.792748
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Prajwal Mohanmurthy",
"submitter": "Prajwal Mohanmurthy",
"url": "https://arxiv.org/abs/1310.6340"
}
|
1310.6491
|
# Development of a sub-milimeter position sensitive gas detector ††thanks:
Supported by NSFC (11075095) and Shandong Province Science Foundation
(ZR2010AM015)
DU Yanyan1 XU Tongye1 SHAO Ruobin1
WANG Xu1 ZHU Chengguang1;1) [email protected] 1 MOE key lab on particle physics
and particle irradiation,
Shandong University, Ji’nan 250100, China
###### Abstract
A position sensitive thin gap chamber has been developed. The position
resolution was measured using the cosmic muons. This paper presents the
structure of this detector, position resolution measurement method and
results.
###### keywords:
Thin gap chamber, Position resolution
###### pacs:
0
7.77.Ka, 29.40.Gx
## 1 Introduction
TGC( Thin Gap Chamber) used in ATLAS experiment [1] shows good performance in
the fast response and time resolution, but with limited position resolution.
The improvement of the position resolution with the timing performance
retained is straightforward for its flexibility to be used in the future
experiments and radiation measurement, for example, the upgrade of the trigger
system of ATLAS experiment. The main goal of the study described in this paper
aimed to build a prototype detector based on TGC, which can have a position
resolution better than 300${\mu}m$, while keeping timing performance not
deteriorated
The TGC detector operates in saturated mode by using a highly quenching gas
mixture of carbon dioxide and n-pentane, $55\%$ :$45\%$, which has many
advantages, such as small sensitivity to mechanical deformations, small
parallax, small Landau tails and good time resolution, but a position
sensitivity of around 1$cm$, decided by the geometrical width of the readout
channel and the strength of the induced signal. To improve the position
resolution, we concentrated on the improving of the method of signal readout
by fine tuning of the structure of the detector.
The new detector, named as pTGC(precision Thin Gap Chamber) based on the ATLAS
TGC, is constructed and tested. We found the position resolution can be
improved to be less than $300{\mu}m$, which meets the requirements.
In section 2, the structure of pTGC detector is described. Section 3 is
devoted to pTGC’s position resolution measurement. Results of the measurement
is summarized in section 4.
Figure 1: The schematic structure of pTGC-I chamber. Anode wires are placed in
the middle, with copper strip etched on the inner surface of the PCB board,
perpendicular to the wire direction.
Figure 2: The schematic structure of pTGC-II chamber. Compared to pTGC-I,
additional isolation layer and graphite layer cover the etched copper strips.
## 2 Construction of pTGC
In the pTGC development, two versions of detectors are constructed and tested,
which are referred as $pTGC-I$ in the first stage and $pTGC-II$ in the second
stage, respectively.
The schematic structure of $pTGC-I$ is shown in Fig. 2, similar to the
structure of ATLAS TGC, except that the position of the strips for signal
collection are modified. 48 copper strips of $0.8mm$ wide and $0.2mm$ spaced
are etched on the inner surface of the 2 parallel PCB boards, which form a
thin spaced chamber. The wires, segmented at $1.8mm$ interval and
perpendicular to the strip direction, are sandwiched in between the two PCB
boards. The resulted size of the detector is defined by the number of wires
and strips, which are $290mm{\times}50mm$.
In the test of $pTGC-I$, the discharge happened between wires and strips
resulted in fatal damage on the frontend electronics, even though we have
designed a protection circuit to insert between detector and frontend
electronics board. This means an instability for big detector and for long
time running. Besides, the induced charge on strips spread roughly $5$ to
$6mm$, which leaves limited rooms for reducing the quantity of the channels by
enlarging the width of strips. Based on $pTGC-I$, the $pTGC-II$ is developed
to deal with these problems.
The schematic structure of $pTGC-II$ is shown in the Fig. 2. The strip width
is enlarged to $3.8mm$ ($0.2mm$ spaced), and a thin ($100{\mu}m$) insolation
layer is pasted on the strip layer. The isolation layer is then coated with a
thin ( $30{\mu}m$) graphite layer as the electric ground to form the electric
field with wires. This graphite layer acts as the protector of the frontend
electronics from discharge and can enlarge the spreading size of the induced
charge on the strip layer. We tune the resistivity of the graphite layer to be
around $100k\Omega$, considering the diffusion speed of the charge, as well.
the resulted size of $pTGC-II$ is $290mm*200mm$.
Both detector use gas mixture of carbon dioxide and n-pentane, $55\%$ :$45\%$,
as working gas, and the anode wire is set to high voltage of 2900v, which are
all the same configuration as the ATLAS TGC detector to maintain the its
features relative to the time measurement of the detector.
## 3 Position resolution measurement
With 3 layers of identical chambers placed in parallel, and 2 layers of
scintillator detectors to build a muon hodoscope(see Fig 3), the $pTGC-I$ and
$pTGC-II$ detectors are tested. The induced charge on each strip is integrated
for the the position calculation based on the charge center-of-gravity
algorithms. The measured hitting position on the 3 layers of chamber are
supposed to be aligned into a straight line concerning the penetration power
of muons. The residue of the position relative to the straight line is then
used to calculate the position resolution of the detectors.
### 3.1 Signal definition
Using oscilloscope, we first observed the induced signal in one wire group and
3 adjacent strips (limited by channels of oscilloscope), as shown in Fig. 4.
It’s apparent that the signals are great significant above the noise and the
signals on the strip are in different magnitudes as expected.
For position resolution measurement, we designed a much more complicated
DAQ(data acquisition) system based on gassiplex frontend electronics [2] to
readout and digitize the induced charge from a quantity of channels of the 3
chambers in a more complex hodoscope [3]. Once the two scintillator detector
of the hodoscope are both fired, the DAQ is triggered. The trigger signal is
sent to the detector front end electronics, which then close the gate for the
discharge of capacitance which has integrated the signal charge on. The charge
on the capacitance are then read out one by one controlled by the clock
distributed from the DAQ system. The charge are then digitized and saved into
computer.
The digitized charge, denoted by $Q_{i}$ where $i$ is the channel number,
consists of three parts: electronic pedestal, noise, and charge induced by
muon hit.
First of all, we need to figure out the pedestal and noise for each channel.
The method is to histogram the integrated charge for each channel using a soft
trigger where no real muon induced signal appear in the data. Fitting the
histogram with a gaussian function to get the pedestal and the noise, denoted
by $P_{i}$ and $\sigma_{i}$, as shown in Fig. 5, where the height of the
histogram represents the pedestal and the error bar represents the noise of
that channel.
In the analysis, if $Q_{i}>P_{i}+3\sigma_{i}$, the channel is considered to be
fired by real muon hit, and the signal charge is calculated as:
$S_{i}=Q_{i}-P_{i},$ (1)
Figure 3: The comic muon hodoscope used for the chamber testing. Plastic
scintillator detector are used for trigger. 3 identical pTGC chambers placed
in parallel in between the 2 scintillator detectors. Figure 4: The observed
signals on wires and several copper strips induced by the same incident cosmic
ray. The signal on wire are negative, and the signal on strips are positive.
Figure 5: The noise and pedestal distribution of 96 signal strips in one
chamber (The x-axis is the signal strip number, the vertical coordinate is the
pedestal value and the error bars presents the noise of that channel.)
The signal magnitude distribution of the largest signal in each cluster
(cluster definition is in next section), named as the peak signal, is shown in
Fig. 6. The distribution of the second largest signal in each cluster, named
as second peak signal, is shown in Fig. 7. The distribution of the sum of all
charge in one cluster is shown in Fig. 8. The correspondence between the
magnitude of the signal and the charge is $1fC/3.6bits$. We can then calculate
that the maximum probable charge of the largest signal in one cluster is
$69fC$, the maximum probable total charge of one cluster is $470fC$, which is
consistent with the measurement in [1]
### 3.2 Cluster definition
The induced charge by the incident muons are distributed on several adjacent
strips, which are grouped in ”cluster” in the analysis and used for the hit
position calculation. In one event, we search all the channels of one
detector, and define group of fired adjacent strips without space as a
cluster. To suppress the fake signals from noise, if the cluster contains only
one strip, the cluster is dropped. The cluster size and number of cluster per
detector per event are shown in Fig. 9 for $pTGC-I$ and Fig. 10 for $pTGC-II$.
It can be seen that in both cases one cluster contain average six strips and
almost every event contains one cluster, which is consistent with the
expected. The hit position is then calculated for each cluster by
$x=\sum_{i}(S_{i}*x_{i})/\sum_{i}(S_{i}),$ (2)
where $x_{i}$ is the center coordination of the $i-th$ strip.
Figure 6: The distribution of the largest signal in one cluster. The x-axis is
the digitized charge collected.
Figure 7: The distribution of the second largest signal in one cluster
Figure 8: The distribution of total charge induced in one cluster
Figure 9: For pTGC-I: (Left) The distribution of cluster size (quantity of
strips in one cluster). (Right) The quantity of cluster in one chamber per
triggered event.
Figure 10: For pTGC-II: (Left) The distribution of cluster size (quantity of
strips in one cluster). (Right) The quantity of cluster in one chamber per
triggered event.
### 3.3 Position resolution
As redundant design, the strips are etched on both inner surface of the PCB
boards. Signals will be induced by the same avalanche on the 2 face-to-face
strips, which corresponds to an double measurements of a single hit. To
compare the two measurements, denoted as $x_{1}$ and $x_{1}^{\prime}$, we fill
$x_{1}-x_{1}^{\prime}$ into histogram to see the broadness of the
distribution. From a simple gaussian function fit, we observed a narrow width
of around $36{\mu}m$, which means that the electronics noise effect on the
resolution is much small. This is consistent with the expectation when to
compare Fig. 5 and Fig. 6, where it shows the signal is great significant
compared to the noise.
After the three hit positions $x_{1}$, $x_{2}$, $x_{3}$ are calculated for the
3 parallel chambers, to simplify the calculation, we first use $x_{1}$ and
$x_{3}$ to calculate the expected hit position on the second layer $x_{2c}$:
$x_{2c}=x_{1}\frac{L_{23}}{L_{12}+L_{23}}+x_{3}\frac{L_{12}}{L_{12}+L_{23}},$
(3)
where $L_{12}$ and $L_{23}$ are the vertical distance between the detector 1,2
and 2,3. To assume the same position resolution $\sigma$ for the 3 identical
detectors, we know the resolution of $x_{2c}$, with the error propagation, is:
$\sigma_{2c}=\sqrt{\frac{L_{23}^{2}}{(L_{12}+L_{23})^{2}}+\frac{L_{12}^{2}}{(L_{12}+L_{23})^{2}}}\sigma{\equiv}k\sigma,$
(4)
Filling $x_{2}-x_{2c}$ into the histogram and then fit with gaussian function,
the width is $w=\sqrt{1+k^{2}}\sigma$. So we can directly calculate the
position resolution of the detector as
$\sigma=\frac{w}{\sqrt{1+k^{2}}}.$ (5)
From Fig. 11 and Fig. 12, we can obtain that the position resolution are
$359um$ for $pTGC-I$ and $233um$ for $pTGC-II$. In both of the cases, the
detector resolution has reach our design requirement. In test, we see that
$pTGC-II$ are more stable with the graphite layer protection and achieve a
better resolution even with less channels.
Figure 11: The distribution of $x_{2}-x_{2c}$ for pTGC-I. The corresponding
position resolution of the chamber is
$\sigma=\frac{w}{\sqrt{1+k^{2}}}=\frac{439{\mu}m}{1.22}=359{\mu}m$.
Figure 12: The distribution of $x_{2}-x_{2c}$ for pTGC-II. The corresponding
position resolution of the chamber is
$\sigma=\frac{w}{\sqrt{1+k^{2}}}=\frac{286{\mu}m}{1.22}=233{\mu}m$.
To look at the dependence of the position sensitivity of the detector to the
incident angle of the muon, we divide the data into groups. Each group of data
contains the events of muon with specific incident angle. To redo the analysis
above, the result is shown in Fig. 13, which shows that the position
resolution of $pTGC-II$ is insensitive to the incident angle of muons.
To check the effect of the electronic noise, we use part of the top highest
signals in one cluster to calculate the position resolution. The result is
shown in Fig. 14, which shows that the resolution are similar and the
electronic noise doesn’t affect much.
Figure 13: The position resolution variance relative to the incident angle of
the cosmic rays. The x-axis is the incident angle of cosmic rays.
Figure 14: The position resolution variance relative to the quantity of strips
in one cluster used for position calculation. The x-axis is the quantity of
strips in one cluster used for position calculation.
## 4 Summary
Two pTGC version $pTGC-I$ and $pTGC-II$, have been constructed and tested.
With the basic structure and working gas unchanged, the detector can attains
the exiting features like good time resolution and fast response, which are
essential for trigger. By revising the signal collecting structure and method,
the position resolution is improved from the level of centimeter to be less
than $300{\mu}m$, which meet the requirement of design. To be noticed that the
resolution measured is a global resolution of the detector, which include the
effect of the non-uniformity of the detector all over the sensitive area. The
3 detectors are placed in parallel with mechanical method, the relative
rotation of the 3 detectors will deteriorate the final measured resolution,
which means that the measured resolution is much conservative.
## References
* [1] Atlas Collaboration, ATLAS muon spectrometer: Technical design report, 1997. CEAN/LHCC/97-22.
* [2] Liu Minghui $etal.$ Nuclear electronics and detection technology, 2008(5)
* [3] Xu Tongye $etal.$ arXiv:1308.5751
|
arxiv-papers
| 2013-10-24T05:59:26 |
2024-09-04T02:49:52.799992
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Yanyan Du, Tongye Xu, Ruobin Shao, Xu Wang, Chengguang Zhu",
"submitter": "Chengguang Zhu",
"url": "https://arxiv.org/abs/1310.6491"
}
|
1310.6638
|
# Community Detection in Quantum Complex Networks
Mauro Faccin [email protected] ISI Foundation, Via Alassio 11/c, 10126
Torino, Italy Piotr Migdał ICFO–Institut de Ciències Fotòniques, 08860
Castelldefels (Barcelona), Spain ISI Foundation, Via Alassio 11/c, 10126
Torino, Italy Tomi H. Johnson Centre for Quantum Technologies, National
University of Singapore, 3 Science Drive 2, 117543, Singapore Clarendon
Laboratory, University of Oxford, Parks Road, Oxford OX1 3PU, United Kingdom
Keble College, University of Oxford, Parks Road, Oxford OX1 3PG, United
Kingdom ISI Foundation, Via Alassio 11/c, 10126 Torino, Italy Ville Bergholm
ISI Foundation, Via Alassio 11/c, 10126 Torino, Italy Jacob D. Biamonte ISI
Foundation, Via Alassio 11/c, 10126 Torino, Italy
###### Abstract
Determining community structure is a central topic in the study of complex
networks, be it technological, social, biological or chemical, in static or
interacting systems. In this paper, we extend the concept of community
detection from classical to quantum systems — a crucial missing component of a
theory of complex networks based on quantum mechanics. We demonstrate that
certain quantum mechanical effects cannot be captured using current classical
complex network tools, and provide new methods that overcome these problems.
Our approaches are based on defining closeness measures between nodes, and
then maximizing modularity with hierarchical clustering. Our closeness
functions are based on quantum transport probability and state fidelity, two
important quantities in quantum information theory. To illustrate the
effectiveness of our approach in detecting community structure in quantum
systems, we provide several examples, including a naturally occurring light
harvesting complex, LHCII. The prediction of our simplest algorithm, semi-
classical in nature, mostly agrees with a proposed partitioning for the LHCII
found in quantum chemistry literature, whereas our fully quantum treatment of
the problem of uncover a new, consistent and appropriately quantum community
structure.
The identification of the community structure within a network addresses the
problem of characterizing the mesoscopic boundary between the microscopic
scale of basic network components (herein called nodes) and the macroscopic
scale of the whole network [1, 2, 3]. In non-quantum networks, the detection
of community structures dates back to Rice [4]. Such analysis has revealed
countless important hierarchies of community groupings within real-world
complex networks. Salient examples can be found in social networks such as
human [5] or animal relationships [6], biological [7, 8, 9, 10], biochemical
[11] and technological [12, 13] networks, as well as numerous others (see Ref.
[1]). In quantum networks, as researchers explore networks of an increasingly
non-trivial geometry and large size [14, 15, 16, 17], their analysis and
understanding will involve identifying non-trivial community structures. In
this article, we devise methods to perform this task, providing an important
missing component in the recent drive to unite quantum physics and complex
network science [18, 19].
For quantum systems, beyond being merely a tool for analysis following
simulations, community partitioning is closely related to performing the
simulations themselves. Simulation is generally a difficult task [20], e.g.
simulating exciton transport in dissipative quantum biological networks [21,
22, 23, 24, 25, 26]. The amount of resources required to exactly simulate such
processes scales exponentially with the number of nodes. To overcome this, one
must in general seek to describe only limited correlations between certain
parts of the network [27]. Mean-field [28, 29, 30, 31] and tensor-network
methods [32, 33] assume correlations between bi-partitions of the system along
some node-structure to be zero or limited by an area law. Hartree-Fock methods
assume limited correlations between particles [34, 35]. Thus planning a
simulation involves identifying a partitioning of a system for which it is
appropriate to limit inter-community correlations, i.e. is a type of community
detection.
We apply our detection methods to artificial networks and the real-world light
harvesting complex II (LHCII) network. In past works, researchers have divided
the LHCII _by hand_ in order to gain more insight into the system dynamics
[36, 37, 38]. Meanwhile, our methods optimize the task of identifying
communities within a quantum network _ab initio_ and, as we will show, the
resulting communities consistently point towards a structure that is different
to those previously identified for the LHCII. For larger networks, as with our
artifical examples, automatic methods would appear to be the only feasible
option.
Our specific approach is to generate a hierarchical community structure [39]
by defining both inter-node and inter-community closeness. The optimum level
in the hierarchy is determined by a modularity-based measure, which quantifies
how good a choice of communities is for the quantum network on average
relative to an appropriately randomized version of the network. Although
modularity-based methods are known to struggle with large sparse networks [40,
41], this work focuses on quantum systems whose size remains much smaller than
the usual targets of classical community detection algorithms.
While the backbone of our quantum community method is shared with classical
methods, the physical properties used to characterize a good community in a
quantum must necessarily be very different to the properties used for a
classical system. Here we show how two quantum properties are used to obtain
closeness and modularity functions: the first is the coherent transport
between communities and the second is the change in the states of individual
communities during a coherent evolution.
Figure 1: Hierarchical community structure arising from a quantum evolution.
On the left are the closenesses $c(i,j)$ between $n=60$ nodes. On the right is
the dendrogram showing the resulting hierarchical community structure. The
dashed line shows the optimum level within this hierarchy, according to the
modularity. The particular example shown here is the one corresponding to Fig.
3d.
In Section I we will begin by recalling several common notions from classical
community detection that we rely on in this work. This sets the stage for the
development of a quantum treatment of community detection in Section II. We
then turn to several examples in Section III including the LHCII complex
mentioned previously, before concluding in Section IV.
## I Community detection
Community detection is the partitioning of a set of nodes $\mathcal{N}$ into
non-overlapping 111We do not consider generalizations to overlapping
communities here. and non-empty subsets $\mathcal{A},\leavevmode\nobreak\
\mathcal{B},\leavevmode\nobreak\ \mathcal{C},\ldots\leavevmode\nobreak\
\subseteq\leavevmode\nobreak\ \mathcal{N}$, called communities, that together
cover $\mathcal{N}$.
There is usually no agreed upon optimal partitioning of nodes into
communities. Instead there is an array of approaches that differ in both the
definition of optimality and the method used to achieve, exactly or
approximately, this optimality (see Ref. [3] for a recent review). In
classical networks optimality is, for example, defined statistically [42],
e.g. in terms of connectivity [1] or communicability [43, 44], or
increasingly, and sometimes relatedly [45], in terms of stochastic random
walks [46, 47, 48]. Our particular focus is on the latter, since the concept
of transport (e.g. a quantum walk) is central to nearly all studies conducted
in quantum physics. As for achieving optimality, methods include direct
maximization via simulated annealing [40, 10] or, usually faster, iterative
division or agglomeration of communities [49]. We focus on the latter since it
provides a simple and effective way of revealing a full hierarchical structure
of the network, requiring only the definition of the closeness of a pair of
communities.
Formally, hierarchical community structure detection methods are based on a
(symmetric) closeness function
$c(\mathcal{A},\mathcal{B})=c(\mathcal{B},\mathcal{A})$ of two communities
$\mathcal{A}\neq\mathcal{B}$. In the agglomerative approach, at the lowest
level of the hierarchy, the nodes are each assigned their own communities. An
iterative procedure then follows, in each step of which the closest pair of
communities (maximum closeness $c$) are merged. This procedure ends at the
highest level, where all nodes are in the same community. To avoid
instabilities in this agglomerative procedure, the closeness function is
required to be non-increasing under the merging of two communities,
$c(\mathcal{A}\cup\mathcal{B},\mathcal{C})\leq\max(c(\mathcal{A},\mathcal{C}),c(\mathcal{B},\mathcal{C}))$,
which allows the representation of the community structure as a linear
hierarchy indexed by the merging closeness. The resulting structure is often
represented as a dendrogram (as shown in Fig. 1) 222In general it may happen
that more than one pair of communities are at the maximum closeness. In this
case the decision on which pair merges first can influence the structure of
the dendrogram, see [69, 39]. In [39] a permutation invariant formulation of
the agglomerative algorithm is given, where more than two clusters can be
merged at once. In our work we use this formulation unless stated otherwise. .
This leaves open the question of which level of the hierarchy yields the
optimal community partitioning. If a partitioning is desired for simulation,
for example, then there may be a desired maximum size or minimum number of
communities. However, without such constraints, one can still ask what is the
best choice of communities within those given by the hierarchical structure.
A type of measure that is often used to quantify the quality of a community
partitioning choice for this purpose is modularity [50, 51, 52], denoted $Q$.
It was originally introduced in the classical network setting, in which a
network is specified by a (symmetric) adjacency matrix of (non-negative)
elements $A_{ij}=A_{ji}\geq 0$ ($A_{ii}=0$), each off-diagonal element giving
the weight of connections between nodes $i$ and $j\neq i$ 333As will become
apparent, we need only consider undirected networks without self-loops.. The
modularity attempts to measure the fraction of weights connecting elements in
the same community, relative to what might be expected. Specifically, one
takes the fraction of intra-community weights and subtracts the average
fraction obtained when the start and end points of the connections are
reassigned randomly, subject to the constraint that the total connectivity
$k_{i}=\sum_{j}A_{ij}$ of each node is fixed. The modularity is then given by
$\displaystyle
Q=\frac{1}{2m}\operatorname{tr}\left\\{C^{\mathrm{T}}BC\right\\},$ (1)
where $m=\mbox{$\textstyle\frac{1}{2}$}\sum_{i}k_{i}$ is the total weight of
connections, $B$ is the modularity matrix with elements
$B_{ij}=A_{ij}-k_{i}k_{j}/2m$, and $C$ is the community matrix, with elements
$C_{i\mathcal{A}}$ equal to unity if $i\in\mathcal{A}$, otherwise zero. The
modularity then takes values strictly less than one, possibly negative, and
exactly zero in the case that the nodes form a single community.
As we will see, there is no natural adjacency matrix associated with the
quantum network and so for the purposes of modularity we use $A_{ij}=c(i,j)$
for $i\neq j$. The modularity $Q$ thus measures the fraction of the closeness
that is intra-community, relative to what would occur if the inter-node
closeness $c(i,j)$ were randomly mixed while fixing the total closeness
$k_{i}=\sum_{j\neq i}c(i,j)$ of each node to all others. Thus both the
community structure and optimum partitioning depend solely on the choice of
the closeness function.
Modularity-based methods such as above are intuitive, fast and on the most
part effective, yet we must note that for classical systems it has been shown
that modularity-based methods suffer from a number of flaws that influence the
overall efficacy of those approaches. In Refs. [40, 41] modularity-based
methods show a poor performance in large, sparse real-world and model
networks. This is due mainly to the resolution limit problem [53], where small
communities can be overlooked, and modularity landscape degeneracy, which
strongly influence accuracy in large networks. Another modularity-related
problem is the so-called detectability/undetectability threshold [54, 55, 56]
where an approximate bi-partition of the system becomes undetectable in some
cases, in particular in presence of degree homogeneity. However, in the
present work we focus on quantum networks whose size typically remains small
compared to classical targets of community detection algorithms, and for which
the derived adjacency matrices are not sparse. These characteristics help to
limit the known flaws of our modularity-based approach, making it adequate for
our purposes.
Finally, once a community partitioning is obtained it is often desired to
compare it against another. Here we use the common normalized mutual
information (NMI) [57, 58, 59] as a measure of the mutual dependence of two
community partitionings. Each partitioning
$X=\\{\mathcal{A},\mathcal{B},\dots\\}$ is represented by a probability
distribution ${P_{X}=\\{|\mathcal{A}|/|\mathcal{N}|\\}_{\mathcal{A}\in X}}$,
where $|\mathcal{A}|=\sum_{i}C_{i\mathcal{A}}$ is the number of nodes in
community $\mathcal{A}$. The similarity of two community partitionings $X$ and
$X^{\prime}$ depends on the joint distribution
$P_{XX^{\prime}}=\\{|\mathcal{A}\cap\mathcal{A}^{\prime}|/|\mathcal{N}|\\}_{\mathcal{A}\in
X,\mathcal{A}^{\prime}\in X^{\prime}}$, where
$|\mathcal{A}\cap\mathcal{A}^{\prime}|=\sum_{i}C_{i\mathcal{A}}C_{i\mathcal{A}^{\prime}}$
is the number of nodes that belong to both communities $\mathcal{A}$ and
$\mathcal{A}^{\prime}$. Specifically, NMI is defined as
$\operatorname{NMI}(X,X^{\prime})=\frac{2\,I(X,X^{\prime})}{H(X)+H(X^{\prime})}.$
(2)
Here $H(X)$ is the Shannon entropy of $P_{X}$, and the mutual information
$I(X,X^{\prime})=H(X)+H(X^{\prime})-H(X,X^{\prime})$ depends on the entropy
$H(X,X^{\prime})$ of the joint distribution $P_{XX^{\prime}}$. The mutual
information is the average of the amount of information about the community of
a node in $X$ obtained by learning its community in $X^{\prime}$. The
normalization ensures that the NMI has a minimum value of zero and takes its
maximum value of unity for two identical community partitionings. The symmetry
of the definition of NMI follows from that of mutual information and Eq. (2).
## II Quantum community detection
The task of community detection has a particular interpretation in a quantum
setting. The state of a quantum system is described in terms of a Hilbert
space $\mathcal{H}$, spanned by a complete orthonormal set of basis states
$\\{|i\rangle\\}_{i\in\mathcal{N}}$. Each basis state $|i\rangle$ can be
associated with a node $i$ in a network and often, as in the case of single
exciton transport, there is a clear choice of basis states that makes this
abstraction to a spatially distributed network natural.
The partitioning of nodes into communities then corresponds to the
partitioning of the Hilbert space $\mathcal{H}=\bigoplus_{\mathcal{A}\in
X}\mathcal{V}_{\mathcal{A}}$ into mutually orthogonal subspaces
$\mathcal{V}_{\mathcal{A}}=\operatorname{span}_{i\in\mathcal{A}}\\{|i\rangle\\}$.
As with classical networks, one can then imagine an assortment of optimality
objectives for community detection, for example, to identify a partitioning
into subspaces in which inter-subspace transport is small, or in which the
state of the system remains relatively unchanged within each subspace. In the
next two subsections we introduce two classes of community closeness measures
that correspond to these objectives. Technical details can be found in the
Supplemental Material.
In what follows, we focus our analysis one an isolated quantum system governed
by Hamiltonian $H$, which enables us to derive convenient closed-form
expressions for the closeness measures. We may expand $H$ in the node basis
$\\{|i\rangle\\}_{i\in\mathcal{N}}$:
$H=\sum_{ij}H_{ij}|i\rangle\langle j|.$ (3)
A diagonal element $H_{ii}$ is a real value denoting the energy of state
$|i\rangle$, whilst an off-diagonal element $H_{ij}$, $i\neq j$, is a complex
weight denoting the change in the amplitude of the wave function during a
transition from state $|j\rangle$ to $|i\rangle$. The matrix formed by these
elements can be thought of as a $|\mathcal{N}|\times|\mathcal{N}|$ complex,
hermitian adjacency matrix. In quantum mechanics, complex elements in the
Hamiltonian lead to a range of phenomena not captured by real matrices, such
as time-reversal symmetry breaking [19, 60]. In the case where each state
$|i\rangle$ corresponds to a particle being localised at a spatially distinct
node $i$, the Hamiltonian describes a spinless single-particle walk with an
energy landcape given by the diagonal elements, and transition amplitudes by
the off-diagonal elements. Any quantum evolution can be viewed in this
picture, making the single particle spiness walk scenario rather general.
A community partitioning based on a Hamiltonian $H$ could be used, among other
things, to guide the simulation or analysis of a more complete model in the
presence of an environment, where this more complete model may be much more
difficult to describe. Additionally, our method could be generalized to use
closeness measures based on open-system dynamics obtained numerically.
### II.1 Inter-community transport
Several approaches to detecting communities in classical networks are based on
the flow of probability through the network during a classical random walk
[45, 61, 48, 62, 46, 47]. In particular, many of these methods seek
communities for which the inter-community probability flow or transport is
small. A natural approach to quantum community detection is thus to consider
the flow of probability during a continuous-time quantum walk, and to
investigate the _change_ in the probability of observing the walker within
each community:
$\displaystyle T_{X}(t)$ $\displaystyle=\sum_{\mathcal{A}\in
X}T_{\mathcal{A}}(t)=\sum_{\mathcal{A}\in
X}\frac{1}{2}\left|p_{\mathcal{A}}\left\\{\rho(t)\right\\}-p_{\mathcal{A}}\left\\{\rho(0)\right\\}\right|,$
(4)
where $\rho(t)=\mathrm{e}^{-\mathrm{i}Ht}\rho(0)\mathrm{e}^{\mathrm{i}Ht}$ is
the state of the walker, at time $t$, during the walk generated by $H$, and
$\displaystyle
p_{\mathcal{A}}\left\\{\rho\right\\}=\operatorname{tr}\left\\{\Pi_{\mathcal{A}}\rho\right\\},$
(5)
is the probability of a walker in state $\rho$ being found in community
$\mathcal{A}$ upon a von Neumann-type measurement 444Equivalently,
$p_{\mathcal{A}}\left\\{\rho\right\\}$ is the norm of the projection
(performed by projector $\Pi_{\mathcal{A}}$) of the state $\rho$ onto the
community subspace $\mathcal{V}_{\mathcal{A}}$..
$\Pi_{\mathcal{A}}=\sum_{i\in\mathcal{A}}|i\rangle\langle i|$ denotes the
projector to the $\mathcal{A}$ subspace.
The initial state $\rho(0)$ can be chosen freely. The change in inter-
community transport is clearest when the process begins either entirely inside
or entirely outside each community. Because of this, we choose the walker to
be initially localized at a single node $\rho(0)=|i\rangle\langle i|$ and
then, for symmetry, sum $T_{X}(t)$ over all $i\in\mathcal{N}$. This results in
the particularly simple expression
$\displaystyle
T_{\mathcal{A}}(t)=\sum_{i\in\mathcal{A},j\notin\mathcal{A}}\frac{R_{ij}(t)+R_{ji}(t)}{2}=\sum_{i\in\mathcal{A},j\notin\mathcal{A}}\widetilde{R}_{ij}(t),$
(6)
where $R(t)$ is the doubly stochastic transfer matrix whose elements
$R_{ij}(t)=|\langle i|\mathrm{e}^{-\mathrm{i}Ht}|j\rangle|^{2}$ give the
probability of transport from node $j$ to node $i$, and $\widetilde{R}(t)$ its
symmetrization. This is reminiscent of classical community detection methods,
e.g. [48], using closeness measures based on the transfer matrix of a
classical random walk.
We can thus build a community structure that seeks to reduce $T_{X}(t)$ at
each hierarchical level by using the closeness function
$\displaystyle c^{T}_{t}(\mathcal{A},\mathcal{B})$
$\displaystyle=\frac{T_{\mathcal{A}}(t)+T_{\mathcal{B}}(t)-T_{\mathcal{A}\cup\mathcal{B}}(t)}{|\mathcal{A}||\mathcal{B}|}$
$\displaystyle=\frac{2}{|\mathcal{A}||\mathcal{B}|}\sum_{i\in\mathcal{A},j\in\mathcal{B}}\widetilde{R}_{ij}(t),$
(7)
where the numerator is the decrement in $T_{X}(t)$ caused by merging
communities $\mathcal{A}$ and $\mathcal{B}$. The normalizing factor in Eq.
(II.1) avoids the effects due to the uninteresting scaling of the numerator
with the community size.
Since a quantum walk does not converge to a stationary state, a time-average
of the closeness defined in Eq. (II.1) is needed to obtain a quantity that
eventually converges with increasing time. Given the linearity of the
formulation, this corresponds to replacing the transport probability
$R_{ij}(t)$ in Eq. (II.1) with its time-average
$\displaystyle\widehat{R}_{ij}(t)=\frac{1}{t}\int_{0}^{t}R_{ij}(t^{\prime})\>\mathrm{d}t^{\prime}.$
(8)
It follows that, as with similar classical community detection methods [46],
our method is in fact a class of approaches, each corresponding to a different
time $t$. The appropriate value of $t$ will depend on the specific
application, for example, a natural time-scale might be the decoherence time.
Not wishing to lose generality and focus on a particular system, we focus here
on the short and long time limits.
In the short time limit $t\to 0$, relevant if $tH_{ij}\ll 1$ for $i\neq j$,
the averaged transfer matrix $\widehat{T}_{ij}(t)$ is simply proportional to
$|H_{ij}|^{2}$. Note that in the short time limit there is no interference
between different paths from $|i\rangle$ to $|j\rangle$, and therefore for
short times $c^{T}_{t}(i,j)$ does not depend on the on-site energies $H_{ii}$
or the phases of the hopping elements $H_{ij}$. This is because, to leading
order in time, interference does not play a role in the transport out of a
single node. For this reason we can refer to this approach as “semi-
classical”.
In the long time limit $t\to\infty$, relevant if $t$ is much larger than the
inverse of the smallest gap between distinct eigenvalues of $H$, the
probabilities are elements of the mixing matrix [63],
$\displaystyle\lim_{t\to\infty}\widehat{R}_{ij}(t)=\sum_{k}|\left\langle
i\left|\Lambda_{k}\right|j\right\rangle|^{2},$ (9)
where $\Lambda_{k}$ is the projector onto the $k$-th eigenspace of $H$. This
thus provides a simple spectral method for building the community structure.
Note that, unlike in a classical infinitesimal stochastic walk where each
$\widehat{R}_{ij}(t)$ eventually becomes proportional to the connectivity
$k_{j}$ of the final node $j$, the long time limit in the quantum setting is
non-trivial and, as we will see, $\widehat{R}_{ij}(t)$ retains a strong
impression of the community structure for large $t$ 555Note that, apart from
small or large times $t$, there is no guarantee of symmetry
$R_{ij}(t)=R_{ji}(t)$ in the transfer matrix for a given Hamiltonian. See
[19]. Hamiltonians featuring this symmetry, e.g., those with real $H_{ij}$,
are called time-symmetric..
### II.2 Intra-community fidelity
Classical walks, and the community detection methods based on them, are fully
described by the evolution of the probabilities of the walker occupying each
node. The previous quantum community detection approach is based on the
evolution of the same probabilities but for a quantum walker. However, quantum
walks are richer than this, they are not fully described by the evolution of
the node-occupation probabilities. We therefore introduce another community
detection method that captures the full quantum dynamics within each community
subspace.
Instead of reducing merely the change in probability within the community
subspaces, we reduce the change in the projection of the quantum state in the
community subspaces. This change is measured using (squared) fidelity, a
common measure of distance between two quantum states. For a walk beginning in
state $\rho(0)$ we therefore focus on the quantity
$\displaystyle F_{X}(t)$ $\displaystyle=\sum_{\mathcal{A}\in
X}F_{\mathcal{A}}(t)=\sum_{\mathcal{A}\in
X}F^{2}\left\\{\Pi_{\mathcal{A}}\rho(t)\Pi_{\mathcal{A}},\Pi_{\mathcal{A}}\rho(0)\Pi_{\mathcal{A}}\right\\},$
(10)
where $\Pi_{\mathcal{A}}\rho\Pi_{\mathcal{A}}$ is the projection of the state
$\rho$ onto the subspace $\mathcal{V}_{\mathcal{A}}$ and
$\displaystyle
F\left\\{\rho,\sigma\right\\}=\operatorname{tr}\left\\{\sqrt{\sqrt{\rho}\sigma\sqrt{\rho}}\right\\}\in[0,\sqrt{\operatorname{tr}\\{\rho\\}\operatorname{tr}\\{\sigma\\}}]$
(11)
is the fidelity, which is symmetric between $\rho$ and $\sigma$.
We build a community structure that seeks to maximize the increase in
$F_{X}(t)$ at each hierarchical level by using the closeness measure
$\displaystyle
c^{F}_{t}(\mathcal{A},\mathcal{B})=\frac{F_{\mathcal{A}\cup\mathcal{B}}(t)-F_{\mathcal{A}}(t)-F_{\mathcal{B}}(t)}{|\mathcal{A}||\mathcal{B}|}\in[-1,1],$
(12)
i.e. the change in $F_{X}(t)$ caused by merging communities $\mathcal{A}$ and
$\mathcal{B}$. Our choice for the denominator prevents uninteresting size
scaling, as in Eq. (II.1).
The initial state $\rho(0)$ can be chosen freely. Here we choose the pure
uniform superposition state $\rho(0)=|\psi_{0}\rangle\langle\psi_{0}|$
satisfying $\langle\,i\,|\,\psi_{0}\,\rangle=1/\sqrt{n}$ for all $i$. This
state was used to investigate the effects of the connectivity on the dynamics
of a quantum walker in Ref. [18].
As for our other community detection approach, we consider the time-average of
Eq. (12), which yields
$\displaystyle
c_{t}^{F}(\mathcal{A},\mathcal{B})=\frac{2}{|\mathcal{A}||\mathcal{B}|}\sum_{i\in\mathcal{A},j\in\mathcal{B}}\operatorname{Re}(\widehat{\rho}_{ij}(t)\rho_{ji}(0)),$
(13)
where
$\widehat{\rho}_{ij}(t)=\frac{1}{t}\int_{0}^{t}\mathrm{d}t^{\prime}\rho_{ij}(t^{\prime})$.
In the long time limit, the time-average of the density matrix takes a
particularly simple expression:
$\displaystyle\lim_{t\to\infty}\widehat{\rho}_{ij}(t)=\sum_{k}\Lambda_{k}\rho_{ij}(0)\Lambda_{k},$
(14)
where $\Lambda_{k}$ is as in the previous Sec. II.1.
The definition of community closeness given in Eq. (12) can exhibit negative
values. In this case the usual definition of modularity fails [64] and one
must extended it. In this work we use the definition of modularity proposed in
Ref. [64], which coincides with Eq. (1) in the case of non-negative closeness.
The extended definition treats negative and positive links separately, and
tries to minimize intra-community negative links while maximizing intra-
community positive links.
## III Performance analysis
To analyze the performance of our quantum community detection methods we apply
them to three different networks. The first one (Sec. III.1) is a simple
quantum network, which we use to highlight how some intuitive notions in
classical community detection do not necessarily transfer over to quantum
systems. The second example (Sec. III.2) is an artificial quantum network
designed to exhibit a clear classical community structure, which we show is
different from the quantum community structure obtained and fails to capture
significant changes in this structure induced by quantum mechanical phases on
the hopping elements of the Hamiltonian. The final network (Sec. III.3) is a
real world quantum biological network, describing the LHCII light harvesting
complex, for which we find a consistent quantum community structure differing
from the community structure cited in the literature. These findings confirm
that a quantum mechanical treatment of community detection is necessary as
classical and semi-classical methods cannot be reproduce the structures that
appropriately capture quantum effects.
Below we will compare quantum community structures against more classical
community structures, such the one given by the semi-classical method based on
the short time transport and, in the case of the example of Sec. III.2, the
classical network from which the quantum network is constructed. Additionally
we use a traditional classical community detection algorithm, OSLOM [42], an
algorithm based on the maximization of the statistical significance of the
proposed partitioning, whose input adjacency matrix $A$ must be real. For this
purpose we use the absolute values of the Hamiltonian elements in the site
basis: $A_{ij}=|H_{ij}|$.
### III.1 Simple quantum network
Figure 2: Simple quantum network — a graph with six nodes. Each solid line
represents transition amplitude $H_{ij}=1$. For dashed and dotted lines the
transition amplitude can be either zero (a, b and c) or the absolute value is
the same $|H_{ij}|=1$ but phase is (d and g) coherent (all ones), (e and h)
random $\exp(i\varphi_{k})$ for each link, (f and i) canceling (ones for
dashed red and minus one for dotted green). Plots show the node closeness for
both methods based on transport and fidelity (only the long-time-averages are
considered, in plots (g), (h) and (i) we used a perturbed Hamiltonian to solve
the eigenvalues degeneracy, this explains the non-symmetric closeness in (i)).
Here we use a simple six-site network model to study ways in which quantum
effects lead to non-intuitive results, and how methods based on different
quantum properties can, accordingly, lead to very different choices of
communities.
We begin with two disconnected cliques of three nodes each, where all
Hamiltonian matrix elements within the groups are identical and real. Fig. 2
illustrates this highly symmetric topology. The community detection method
based on quantum transport identifies the two fully-connected groups as two
separate communities (Fig. 2a), as is expected. Contrastingly, the methods
based on fidelity predict counter-intuitively only a single community; two
disconnected nodes can retain coherence and, by this measure, be considered
part of the same community (Fig. 2b).
This symmetry captured by the fidelity-based community structure breaks down
if we introduce random perturbations into the Hamiltonian. Specifically, the
fidelity-based closeness $c_{t}^{F}$ is sensitive to perturbations of the
order $t^{-1}$, above which the community structure is divided into the two
groups of three (Fig. 2c) expected from transport considerations. Thus we may
tune the resolution of this community structure method to asymmetric
perturbations by varying $t$.
Due to quantum interference we expect that the Hamiltonian phases should
significantly affect the quantum community partitioning. The same toy model
can be used to demonstrate this effect. For example, consider adding four
elements to the Hamiltonian corresponding to hopping from nodes 2 and 3 to 4
and 5 (see diagram in Fig. 2). If these hopping elements are all identical to
the others, it is the two nodes, 1 and 6, that are not directly connected for
which the inter-node transport is largest (and thus their inter-node closeness
is the largest). However, when the phases of the four additional elements are
randomized, this transport is decreased. Moreover, when the phases are
canceling, the transport between nodes 1 and 6 is reduced to zero, and the
closeness between them is minimized (see Figs. 2d–2f).
The fidelity method has an equally strong dependence on the phases (see Figs.
2g–2i), with variations in the phases breaking up the network from a large
central community (with nodes 1 and 6 alone) into the two previously
identified communities.
### III.2 Artificial quantum network
Figure 3: Artificial community structure. (a) Classical community structure
used in creating the network. (b–e) Community partitionings found using the
three quantum methods and OSLOM. (f,g) Behavior of the approaches as the
phases of the Hamiltonian elements are randomly sampled from a Gaussian
distribution of width $\sigma$. The mean NMI, compared with zero phase
partitioning (f) and the classical model data (g), over 200 samplings of the
phase distribution is plotted. The standard deviation is indicated by the
shading. Both OSLOM and $c^{T}_{0}$ are insensitive to phases and thus do not
respond to the changes in the Hamiltonian.
The Hamiltonian of our second quantum network is constructed from the
adjacency matrix $A$ of a classical unweighted, undirected network exhibiting
a clear classical partitioning, using the relation $H_{ij}=A_{ij}$. We
construct $A$ using the algorithm proposed by Lancichinetti et al. in Ref.
[65], which provides a method to construct a network with heterogeneous
distribution both for the node degree and for the communities dimension and a
controllable inter-community connection. We start with a rather small network
of 60 nodes with average intra-community connectivity $\langle k\rangle=6$,
and only 5% of the edges are rewired to join communities. The network is
depicted in Fig. 3a. To confirm the expected, the known classical community
structure is indeed obtained by the semi-classical short-time-transport
algorithm 666In the case of short-time transport, a small perturbation was
also added to the closeness function in order to break the symmetries of the
system. and the OSLOM algorithm (see Figs. 3b–3e), achieving
$\text{NMI}=0.953$ and $\text{NMI}=0.975$ with the known structure,
respectively.
The quantum methods based on the long-time average of both transport and
fidelity reproduce the main features of the original community structure while
unveiling new characteristics. The transport-based long-time average method
($\text{NMI}=0.82$ relative to the classical partitioning) exhibits
disconnected communities, i.e. the corresponding subgraph is disconnected.
This behavior can be explained by interference-enhanced quantum walker
dynamics, as exhibited by the toy model in the previous subsection. The long-
time average fidelity method ($\text{NMI}=0.85$) returns the four main
classical communities plus a number of single-node communities. Both methods
demonstrate that the quantum and classical community structures are
unsurprisingly different, with the quantum community structure clearly
dependent on the quantum property being optimized, more so than the different
classical partitionings.
#### Adjusted phases
As shown in Sec. III.1, due to interference the dynamics of the quantum system
can change drastically if the phases of the Hamiltonian elements are non-zero.
This is known as a chiral quantum walk [19]. Such walks exhibit, for example,
time-reversal symmetry breaking of transport between sites [19] and it has
been proposed that nature might actually make use of phase controlled
interference in transport processes [66]. OSLOM, our semi-classical short-time
transport algorithm and other classical community partitioning methods are
insensitive to changes in the hopping phases. Thus, by establishing that the
quantum community structure is sensitive to such changes in phase, as expected
from above, we show that classical methods are inadequate for finding quantum
community structure.
To analyze this effect we take the previous network and adjust the phases of
the Hamiltonian terms while preserving their absolute values. Specifically,
the phases are sampled randomly from a normal distribution with mean zero and
standard deviation $\sigma$. We find that, typically, as the standard
deviation $\sigma$ increases, when comparing quantum communities and the
corresponding communities without phases the NMI between them decreases, as
shown in Fig. 3f. A similar deviation reflects on the comparison with the
classical communities used to construct the system, shown in Fig. 3g. This
sensitivity of the quantum community structures to phases, as revealed by the
NMI, confirms the expected inadequacy of classical methods. The partitioning
based on long-time average fidelity seems to be the most sensitive to phases.
### III.3 Light-harvesting complex
Figure 4: Light harvesting complex II (LHCII). (top left) Monomeric subunit of
the LHCII complex with pigments Chl-a (red) and Chl-b (green) packed in the
protein matrix (gray). (top center) Schematic representation of Chl-a and
Chl-b in the monomeric subunit, here the labeling follows the usual
nomenclature (b601, a602…). (top right) Network representation of the pigments
in circular layout, colors represent the typical partitioning of the pigments
into communities. The widths of the links represent the strength of the
couplings $|H_{ij}|$ between nodes. Here the labels maintain only the ordering
(b601$\to$1, a602$\to$2,…). (a,b,c) Quantum communities as found by the
different quantum community detection methods. Link width denotes the pairwise
closeness of the nodes.
An increasing number of biological networks of non-trivial topology are being
described using quantum mechanics. For example, light harvesting complexes
have drawn significant attention in the quantum information community.
One of these is the LHCII, a two-layer 14-chromophore complex embedded into a
protein matrix (see Fig. 4 for a sketch) that collects light energy and
directs it toward the reaction center where it is transformed into chemical
energy. The system can be described as a network of 14 sites connected with a
non-trivial topology. The single-exciton subspace is spanned by 14 basis
states, each corresponding to a node in the network, and the Hamiltonian in
this basis was found in Ref. [38].
In a widely adopted chromophore community structure [37], the sites are
partitioned _by hand_ into communities according to their physical closeness
(e.g. there are no communities spanning the two layers of the complex), and
the strength of Hamiltonian couplings (see the top right of Fig. 4). Here, we
apply our _ab initio_ automated quantum community detection algorithms to the
same Hamiltonian.
All of our approaches predict a modified partitioning to that commonly used in
the literature. The method based on short-time transport returns communities
that do not connect the two layers. This semi-classical approach relies only
on the coupling strength of the system, without considering interference
effects, and provides the closest partitioning to the one provided by the
literature (also relying only on the coupling strengths). Meanwhile, the
methods based on the long-time transport and fidelity return very similar
community partitionings, in which node 6 on one layer and node 9 on the other
are in the same community. These two long-time community partitionings are
identical, except one of the communities predicted by the fidelity based
method is split when using the transport based method. It is therefore a
difference in modularity only.
The classical OSLOM algorithm fails spectacularly: it gives only one
significant community involving nodes 11 and 12 which exhibit the highest
coupling strength. If assigning a community to each node is forced, a unique
community with all nodes is provided.
Note that here we have used the LHCII closed-system dynamics, valid only for
short times, to partition it. As explained in Sec. II, for the purpose of
analysis one could alternatively use the less tractable open-system dynamics
to obtain a partitioning that reflects the environment of the LHCII [26].
However, we argue that community partitioning, e.g. that based on the closed-
system dynamics, is essential in devising approaches to simulating the full
open-system dynamics.
## IV Discussion
We have developed methods to detect community structure in quantum systems,
thereby extending the purview of community detection from classical networks
to include quantum networks. Our approach involves the development of a number
of methods that focus on different characteristics of the system and return a
community structure reflecting that specific characteristic. The variation of
the quantum community structure with the property on which this structure is
based seems greater than for classical community structures.
All our methods are based on the full unitary dynamics of the system, as
described by the Hamiltonian, and account for quantum effects such as coherent
evolution and interference. In fact phases are often fundamental to
characterizing the system evolution. For example, Harel et al. [66] have shown
that in light harvesting complexes interference between pathways is important
even at room temperature. In our light harvesting complex example (see
Sec.III.3), the _ab-initio_ community structures provided by the long-time
measures propose consistent communities that stretch across the lumenal and
stromal layers of the complex, absent in the structure proposed by the
community.
Since we consider time evolution, the averaging time $t$ acts as a tuning
parameter for the partitioning methods. In the case of transport it transforms
the method from a semi-classical approach ($t\to 0$) to a fully quantum-aware
measure ($t\to\infty$), For all times, the complexity of our algorithms scales
polynomially in the number of nodes $|\mathcal{N}|$, at worst
$O(|\mathcal{N}|^{3})$ if the diagonalization of $H$ is required. This allows
the study of networks with node numbers up the thousands and tens of
thousands, which is appropriate for the real-world quantum networks currently
being considered.
As with classical community structure, there are many possible definitions of
a quantum community. We restricted ourselves to two broad classes based on
transport and fidelity under coherent evolution, both based on dynamics,
though in the limits considered in this paper the closenesses and thus quantum
community structure can be expressed purely in terms of static properties. We
end by briefly discussing some other possible definitions based on statics
(the earliest classical community definitions were based on statics [67]). The
first type is based on some quantum state $|\psi\rangle$, e.g. the ground
state of $H$. We might wish to partition the network by repeatedly diving the
network in two based on minimally entangled bipartitions. This could be viewed
as identifying optimum communities for some cluster-based mean-field-like
simulation [31] whose entanglement structure is expected to be similar to
$|\psi\rangle$. The second type is based directly on the spectrum of the
Hamiltonian $H$. We might partition the Hilbert space into unions of the
eigenspaces of $H$ by treating the corresponding eigenvalues as 1D coordinates
and applying a traditional agglomerative or divisive clustering algorithm on
them. Note that the resulting partitioning would normally not be in the
position basis.
The use of community detection in quantum systems addresses an open challenge
in the drive to unite quantum physics and complex network science, and we
expect such partitioning, based on our definitions or extensions such as
above, to be used extensively in making the large quantum systems currently
being targeted by quantum physicists tractable to numerical analysis.
Conversely, quantum measures have also been shown to add novel perspectives to
classical network analysis [68].
###### Acknowledgements.
We thank Michele Allegra, Leonardo Banchi, Giovanni Petri and Zoltan Zimboras
for fruitful discussions. MF, THJ and JDB completed part of this study while
visiting the Institute for Quantum Computing, at the University of Waterloo.
PM acknowledges the Spanish MINCIN/MINECO project TOQATA (FIS2008-00784), EU
Integrated Projects AQUTE and SIQS, and HISTERA project DIQUIP. THJ
acknowledges the European Research Council under the European Union’s Seventh
Framework Programme (FP7/2007-2013) / ERC Grant Agreement No. 319286, and the
National Research Foundation and the Ministry of Education of Singapore for
support. JDB acknowledges the Foundational Questions Institute (under grant
FQXi-RFP3-1322) for financial support. All authors acknowledge the Q-ARACNE
project funded by the Fondazione Compagnia di San Paolo.
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## Appendix A Definitions
### A.1 Modularity
Assume we have a directed, weighted graph (with possibly negative weights) and
self-links, described by a real adjacency matrix $A$. The element $A_{ij}$ is
the weight of the link from node $i$ to node $j$.
The in- and outdegrees of node $i$ are defined as
$k^{\text{in}}_{i}=\sum_{j}A_{ji},\qquad k^{\text{out}}_{i}=\sum_{j}A_{ij}.$
(15)
For a symmetric graph $A$ is symmetric and the indegree is equal to the
outdegree. The total connection weight is
$m=\sum_{i}k^{\text{in}}_{i}=\sum_{i}k^{\text{out}}_{i}=\sum_{ij}A_{ij}$.
The community matrix $C$ defines the membership of the nodes in different
communities. The element $C_{i\mathcal{A}}$ is equal to unity if
$i\in\mathcal{A}$, otherwise zero.777For a fuzzy definition of membership we
could require $C_{i\mathcal{A}}\geq 0$ and
$\sum_{\mathcal{A}}C_{i\mathcal{A}}=1$ instead. The size of a community is
given by $|\mathcal{A}|=\sum_{i}C_{i\mathcal{A}}$. For strict (non-fuzzy)
communities we can define $C$ using an assignment vector $\sigma$ (the entries
being the communities of each node):
$C_{i\mathcal{A}}=\delta_{\mathcal{A},\sigma_{i}}$. This yields
$(CC^{T})_{ij}=\delta_{\sigma_{i},\sigma_{j}}$.
There are many different ways of partitioning a graph into communities. A
simple approach is to minimize the _frustration_ of the partition, defined as
the sum of the absolute weight of positive links between communities and
negative links within them:
$F=-\sum_{ij}A_{ij}\delta_{\sigma_{i},\sigma_{j}}=-\operatorname{tr}\left(C^{T}AC\right).$
(16)
Frustration is inadequate as a goodness measure for partitioning nonnegative
graphs (in which a single community containing all the nodes minimizes it).
For nonnegative graphs we can instead maximize another measure called
_modularity_ :
$Q=\frac{1}{m}\sum_{\mathcal{A},ij}(A_{ij}-p_{ij})C_{i\mathcal{A}}C_{j\mathcal{A}}=\frac{1}{m}\operatorname{tr}\left(C^{T}(A-p)C\right),$
(17)
where $p_{ij}$ is the “expected” link weight from $i$ to $j$, with
$\sum_{ij}p_{ij}=m$, and is what separates modularity from plain frustration.
Different choices of the “null model” $p$ give different modularities. Using
degrees, we can define $p_{ij}=k^{\text{out}}_{i}k^{\text{in}}_{j}/m$.
For graphs with both positive and negative weights the usual definitions of
degrees do not make much sense, since usually negative and positive links
should not simply cancel each other out. Also, plain modularity will fail e.g.
when $m=0$. This can be solved by treating positive and negative links
separately [64].
### A.2 Hierarchical clustering
All our community detection approaches share a common theme. For each
(proposed) community $\mathcal{A}$ we have a goodness measure
$M_{\mathcal{A}}(t)$ that depends on the system Hamiltonian, the initial
state, and $t$. This induces a corresponding measure for a partition $X$:
$\displaystyle M_{X}(t)=\sum_{\mathcal{A}\in X}M_{\mathcal{A}}(t).$ (18)
Using this, we define a function for comparing two partitions, $X$ and
$X^{\prime}$, which only differ in a single merge that combines $\mathcal{A}$
and $\mathcal{B}$:
$\displaystyle
M_{\mathcal{A},\mathcal{B}}(t)=M_{X^{\prime}}(t)-M_{X}(t)=M_{\mathcal{A}\cup\mathcal{B}}(t)-M_{\mathcal{A}}(t)-M_{\mathcal{B}}(t).$
(19)
We can make $M_{\mathcal{A},\mathcal{B}}(t)$ into a symmetric closeness
measure $c(\mathcal{A},\mathcal{B})$ by fixing the time $t$ and normalizing it
with $|\mathcal{A}||\mathcal{B}|$. Using this closeness measure together with
the agglomerative hierarchical clustering algorithm (as explained in Sec. I)
we then obtain a community hierarchy. The goodness of a specific partition in
the hierarchy is given by its modularity, obtained using the adjacency matrix
given by $A_{ij}=c(i,j)$.
The standard hierarchical clustering algorithm requires closeness to fulfill
the _monotonicity property_
$\displaystyle\min(c(\mathcal{A},\mathcal{C}),c(\mathcal{B},\mathcal{C}))\leq
c(\mathcal{A}\cup\mathcal{B},\mathcal{C})\leq\max(c(\mathcal{A},\mathcal{C}),c(\mathcal{B},\mathcal{C})).$
(20)
for any communities $\mathcal{A},\mathcal{B},\mathcal{C}$. If this does not
hold, we may encounter a situation where the merging closeness sometimes
increases, which in turn means that the results cannot be presented as a
dendrogram indexed by decreasing closeness. The real downside of not having
the monotonicity property, however, is stability-related. The clustering
algorithm should be stable, i.e. a small change in the system should not
dramatically change the resulting hierarchy. Assume we encounter a situation
where all the pairwise closenesses between a subset of clusters
$S=\\{\mathcal{A}_{i}\\}_{i}$ are within a given tolerance. A small
perturbation can now change the pair $\\{\mathcal{A},\mathcal{B}\\}$ chosen
for the merge. If Eq. (20) is fulfilled, then the rest of $S$ is merged into
the same new cluster during subsequent rounds, and hence their relative
merging order does not matter.
### A.3 Notation
Let the Hamiltonian of the system have the spectral decomposition
$H=\sum_{k}E_{k}\Lambda_{k}$. The unitary propagator of the system decomposes
as $U(t)=\mathrm{e}^{-\mathrm{i}Ht}=\sum_{k}e^{-iE_{k}t}\Lambda_{k}$. We
denote the state of the system at time $t$ by
$\displaystyle\rho(t)=U(t)\rho(0)U(t)^{\dagger}.$ (21)
Sometimes we make use of the state obtained by measuring in which community
subspace $\mathcal{V}_{\mathcal{A}}$ the quantum state is located, and then
discarding the result. The resulting state is
$\displaystyle\rho_{X}(t)$ $\displaystyle=\sum_{\mathcal{A}\in
X}\Pi_{\mathcal{A}}\rho(t)\Pi_{\mathcal{A}}.$ (22)
This state is normally not pure even if $\rho(t)$ is.
The probability of transport from node $b$ to node $a$, the transfer matrix,
is given by the elements
$\displaystyle R_{ab}(t)=|\langle a|U(t)|b\rangle|^{2}.$ (23)
$R(t)$ is doubly stochastic, i.e. its rows and columns all sum up to unity. We
use $\widetilde{R}=(R+R^{T})/2$ to denote its symmetrization.
The time average of a function $f(t)$ is denoted using $\widehat{f}(t)$:
$\displaystyle\widehat{f}(t)=\frac{1}{t}\int_{0}^{t}f(t^{\prime})\>\mathrm{d}t^{\prime}.$
(24)
Now we have
$\displaystyle\widehat{R}_{ab}(t)$
$\displaystyle=\sum_{jk}\frac{1}{t}\int_{0}^{t}e^{-i(E_{j}-E_{k})t^{\prime}}\>\mathrm{d}t^{\prime}\langle
a|\Lambda_{j}|b\rangle\langle b|\Lambda_{k}|a\rangle.$ (25)
The $tH\ll 1$ and $t\to\infty$ limits of this average are
$\displaystyle\widehat{R}_{ab}(t\to 0)$
$\displaystyle=\delta_{ab}\left(1-\frac{t^{2}}{3}(H^{2})_{aa}\right)+\frac{t^{2}}{3}|H_{ab}|^{2}+O(t^{3}),$
$\displaystyle\widehat{R}_{ab}(t\to\infty)$
$\displaystyle=\sum_{jk}\delta_{jk}\langle a|\Lambda_{j}|b\rangle\langle
b|\Lambda_{k}|a\rangle=\sum_{k}|\langle a|\Lambda_{k}|b\rangle|^{2}.$ (26)
The time average of the state of the system is given by
$\displaystyle\widehat{\rho}(t)=\sum_{jk}\frac{1}{t}\int_{0}^{t}e^{-i(E_{j}-E_{k})t^{\prime}}\>\mathrm{d}t^{\prime}\Lambda_{j}\rho(0)\Lambda_{k}.$
(27)
It can be interpreted as the density matrix of a system that has evolved for a
random time, sampled from the uniform distribution on the interval $[0,t]$.
Again, in the short- and infinite-time limits this yields
$\displaystyle\widehat{\rho}(t\to 0)=$
$\displaystyle\rho(0)-\frac{it}{2}\left[H,\rho(0)\right]+\frac{t^{2}}{3}\left(H\rho(0)H-\frac{1}{2}\left\\{H^{2},\rho(0)\right\\}\right)+O(t^{3}),$
$\displaystyle\widehat{\rho}(t\to\infty)=$
$\displaystyle\sum_{k}\Lambda_{k}\rho(0)\Lambda_{k}.$ (28)
## Appendix B Closeness measures
### B.1 Inter-community transport
Considering the flow of probability during a continuous-time quantum walk, let
us investigate the _change_ in the probability of observing the walker within
a community:
$\displaystyle
T_{\mathcal{A}}(t)=\frac{1}{2}\left|p_{\mathcal{A}}\left\\{\rho(t)\right\\}-p_{\mathcal{A}}\left\\{\rho(0)\right\\}\right|,$
(29)
where
$p_{\mathcal{A}}\left\\{\rho\right\\}=\operatorname{tr}\left(\Pi_{\mathcal{A}}\rho\right)$
is the probability of a walker in state $\rho$ being found in community
$\mathcal{A}$ upon a von Neumann-type measurement.888 Equivalently,
$p_{\mathcal{A}}\left\\{\rho\right\\}$ is the norm of the projection
(performed by projector $\Pi_{\mathcal{A}}$) of the state $\rho$ onto the
community subspace $\mathcal{V}_{\mathcal{A}}$. A good partition should
intuitively minimize this change, keeping the walkers as localized to the
communities as possible. $T_{X}=\sum_{\mathcal{A}\in X}T_{\mathcal{A}}$ is of
course minimized by the trivial choice of a single community,
$X=\\{\mathcal{A}\\}$, and any merging of communities can only decrease
$T_{X}$. Therefore we have $T_{\mathcal{A}\cup\mathcal{B}}(t)\leq
T_{\mathcal{A}}(t)+T_{\mathcal{B}}(t)$.
The initial state $\rho(0)$ can be chosen freely. For a pure initial state
$\rho(0)=|\psi\rangle\langle\psi|$ we obtain
$\displaystyle
T_{\mathcal{A}}(t)=\frac{1}{2}\left|\left\langle\psi\left|U^{\dagger}(t)\Pi_{\mathcal{A}}U(t)\right|\psi\right\rangle-\left\langle\psi\left|\Pi_{\mathcal{A}}\right|\psi\right\rangle\right|.$
(30)
The change in inter-community transport is clearest when the process begins
either entirely inside or entirely outside each community. Because of this, we
choose the walker to be initially localized at a single node
$\rho(0)=|b\rangle\langle b|$ and then, for symmetry, sum (or average)
$T_{\mathcal{A}}(t)$ over all $b\in\mathcal{N}$:
$\displaystyle T_{\mathcal{A}}(t)$
$\displaystyle=\frac{1}{2}\sum_{b}\left|\left\langle
b\left|U(t)^{\dagger}\Pi_{\mathcal{A}}U(t)\right|b\right\rangle-\left\langle
b\left|\Pi_{\mathcal{A}}\right|b\right\rangle\right|$
$\displaystyle=\frac{1}{2}\sum_{b}\left|\sum_{a\in\mathcal{A}}(R_{ab}(t)-\delta_{ab})\right|$
$\displaystyle=\frac{1}{2}\left(\sum_{b\in\mathcal{A}}\left|1-\sum_{a\in\mathcal{A}}R_{ab}(t)\right|+\sum_{b\notin\mathcal{A}}\left|\sum_{a\in\mathcal{A}}R_{ab}(t)\right|\right)$
$\displaystyle=\frac{1}{2}\left(\sum_{a\notin\mathcal{A},b\in\mathcal{A}}R_{ab}(t)+\sum_{a\in\mathcal{A},b\notin\mathcal{A}}R_{ab}(t)\right)$
$\displaystyle=\sum_{a\in\mathcal{A},b\notin\mathcal{A}}\frac{R_{ab}(t)+R_{ba}(t)}{2}=\sum_{a\in\mathcal{A},b\notin\mathcal{A}}\widetilde{R}_{ab}(t),$
(31)
since $R(t)$ is doubly stochastic. Now we have
$\displaystyle
T_{\mathcal{A},\mathcal{B}}(t)=T_{\mathcal{A}}(t)+T_{\mathcal{B}}(t)-T_{\mathcal{A}\cup\mathcal{B}}(t)=2\sum_{a\in\mathcal{A},b\in\mathcal{B}}\widetilde{R}_{ab}(t)$
(32)
with $0\leq T_{\mathcal{A},\mathcal{B}}(t)\leq
2\min(|\mathcal{A}|,|\mathcal{B}|)$. The short- and long-time limits of the
time-averaged $T_{\mathcal{A},\mathcal{B}}(t)$ can be found using Eqs. (A.3):
$\displaystyle T_{\mathcal{A},\mathcal{B}}^{t\to 0}$
$\displaystyle=2\sum_{a\in\mathcal{A},b\in\mathcal{B}}\left(\delta_{ab}+\frac{t^{2}}{3}\left(|H_{ab}|^{2}-\delta_{ab}(H^{2})_{aa}\right)+O(t^{3})\right),$
(33) $\displaystyle T_{\mathcal{A},\mathcal{B}}^{t\to\infty}$
$\displaystyle=2\sum_{a\in\mathcal{A},b\in\mathcal{B}}\sum_{k}|(\Lambda_{k})_{ab}|^{2}.$
(34)
### B.2 Intra-community fidelity
Our next measure aims to maximize the “similarity” between the evolved and
initial states when projected to a community subspace. We do this using the
squared fidelity
$\displaystyle
F_{\mathcal{A}}(t)=F^{2}\left\\{\Pi_{\mathcal{A}}\rho(t)\Pi_{\mathcal{A}},\Pi_{\mathcal{A}}\rho(0)\Pi_{\mathcal{A}}\right\\},$
(35)
where $\Pi_{\mathcal{A}}\rho\Pi_{\mathcal{A}}$ is the projection of the state
$\rho$ onto the subspace $\mathcal{V}_{\mathcal{A}}$ and
$\displaystyle
F\left\\{\rho,\sigma\right\\}=\operatorname{tr}\left\\{\sqrt{\sqrt{\rho}\sigma\sqrt{\rho}}\right\\}\in[0,\sqrt{\operatorname{tr}\\{\rho\\}\operatorname{tr}\\{\sigma\\}}],$
(36)
is the fidelity, which is symmetric between $\rho$ and $\sigma$. If either
$\rho$ or $\sigma$ is rank-1, their fidelity reduces to
$F\left\\{\rho,\sigma\right\\}=\sqrt{\operatorname{tr}\\{\rho\sigma\\}}$.
Thus, if the initial state $\rho(0)$ is pure, we have
$\displaystyle
F_{\mathcal{A}}(t)=\operatorname{tr}\left(\Pi_{\mathcal{A}}\rho(t)\Pi_{\mathcal{A}}\rho(0)\right).$
(37)
This assumption makes $F_{X}(t)$ equivalent to the squared fidelity between
$\rho_{X}(t)$ and a pure $\rho(0)$:
$\displaystyle F_{X}(t)$ $\displaystyle=\sum_{\mathcal{A}\in
X}\operatorname{tr}\left(\Pi_{\mathcal{A}}\rho(t)\Pi_{\mathcal{A}}\rho(0)\right)=\operatorname{tr}\left(\rho_{X}(t)\rho(0)\right)$
$\displaystyle=F^{2}\\{\rho_{X}(t),\rho(0)\\}=F^{2}\\{\rho(t),\rho_{X}(0)\\},$
(38)
and yields
$\displaystyle F_{\mathcal{A},\mathcal{B}}(t)$
$\displaystyle=F_{\mathcal{A}\cup\mathcal{B}}(t)-F_{\mathcal{A}}(t)-F_{\mathcal{B}}(t)$
$\displaystyle=2\operatorname{Re}\operatorname{tr}\left(\Pi_{\mathcal{A}}\rho(t)\Pi_{\mathcal{B}}\rho(0)\right)$
$\displaystyle=2\sum_{a\in\mathcal{A},b\in\mathcal{B}}\operatorname{Re}\left(\rho_{ab}(t)\rho_{ba}(0)\right).$
(39)
We use as the initial state the uniform superposition of all the basis states
with arbitrary phases:
$|\psi\rangle=\frac{1}{\sqrt{n}}\sum_{k}e^{i\theta_{k}}|k\rangle$, which gives
$\displaystyle F_{\mathcal{A},\mathcal{B}}(t)$
$\displaystyle=\frac{2}{n^{2}}\sum_{a\in\mathcal{A},b\in\mathcal{B}}\sum_{xy}\operatorname{Re}\left(e^{i(\theta_{x}-\theta_{y}+\theta_{b}-\theta_{a})}U_{ax}\overline{U_{by}}\right).$
(40)
In this case the short-term limit does not yield anything interesting. The
long-time limit of the time-average of $F_{\mathcal{A},\mathcal{B}}(t)$ is
$\displaystyle F_{\mathcal{A},\mathcal{B}}^{t\to\infty}$
$\displaystyle=\frac{2}{n^{2}}\sum_{a\in\mathcal{A},b\in\mathcal{B}}\sum_{xy,k}\operatorname{Re}\left(e^{i(\theta_{x}-\theta_{y}+\theta_{b}-\theta_{a})}(\Lambda_{k})_{ax}(\Lambda_{k})_{yb}\right).$
We may now (somewhat arbitrarily) choose all the phases $\theta_{k}$ to be the
same, or average the closeness measure over all possible phases
$\theta_{k}\in[0,2\pi]$.
### B.3 Purity
The coherence between any communities $X=\\{\mathcal{A},\mathcal{B},\dots\\}$
is completely destroyed by measuring in which community subspace
$\mathcal{V}_{\mathcal{A}}$ the quantum state is located, see Eq. (22). If the
measurement outcome is not revealed, the purity of the measured state
$\rho_{X}(t)$ is, due to the orthogonality of the projectors,
$\displaystyle P_{X}(t)$
$\displaystyle=\operatorname{tr}\left(\rho_{X}^{2}(t)\right)=\sum_{\mathcal{A}\in
X}\operatorname{tr}\left((\Pi_{\mathcal{A}}\rho(t))^{2}\right)=\sum_{\mathcal{A}\in
X}P_{\mathcal{A}}(t),$
where
$\displaystyle P_{\mathcal{A}}(t)$
$\displaystyle=\operatorname{tr}\left((\Pi_{\mathcal{A}}\rho(t)\Pi_{\mathcal{A}})^{2}\right)=\operatorname{tr}\left((\Pi_{\mathcal{A}}\rho(t))^{2}\right).$
(41)
If $\rho(t)$ is pure, we have (cf. Eq. (B.2))
$\displaystyle P_{X}(t)=\sum_{\mathcal{A}\in
X}\operatorname{tr}(\Pi_{\mathcal{A}}\rho(t)\Pi_{\mathcal{A}}\rho(t))=F^{2}\\{\rho_{X}(t),\rho(t)\\}.$
(42)
The change in purity of the state after a projective measurement locating the
walker into one of the communities is
$\displaystyle P_{\mathcal{A},\mathcal{B}}(t)$
$\displaystyle=P_{\mathcal{A}\cup\mathcal{B}}(t)-P_{\mathcal{A}}(t)-P_{\mathcal{B}}(t)$
$\displaystyle=2\operatorname{tr}\left(\Pi_{\mathcal{A}}\rho(t)\Pi_{\mathcal{B}}\rho(t)\right)$
$\displaystyle=2\sum_{a\in\mathcal{A},b\in\mathcal{B}}|\rho_{ab}(t)|^{2}\geq
0.$ (43)
Again, we will use the initial state $|\psi\rangle\leavevmode\nobreak\
=\leavevmode\nobreak\ \frac{1}{\sqrt{n}}\sum_{k}e^{i\theta_{k}}|k\rangle$:
$\displaystyle P_{\mathcal{A},\mathcal{B}}(t)$
$\displaystyle=\frac{2}{n^{2}}\sum_{a\in\mathcal{A},b\in\mathcal{B}}\left|\sum_{xy}e^{i(\theta_{x}-\theta_{y})}U_{ax}(t)\overline{U_{by}(t)}\right|^{2}.$
(44)
As with the fidelity-based measure, the short-time limit is uninteresting. The
long-time limit of the time-average of $P_{\mathcal{A},\mathcal{B}}(t)$ is
$\displaystyle P_{\mathcal{A},\mathcal{B}}^{t\to\infty}$
$\displaystyle=2\sum_{a\in\mathcal{A},b\in\mathcal{B}}\left(|\left\langle
a\left|\widehat{\rho}(\infty)\right|b\right\rangle|^{2}+\sum_{k\neq
m}|\left\langle
a\left|\Lambda_{k}\rho_{0}\Lambda_{m}\right|b\right\rangle|^{2}\right)$
$\displaystyle=2\sum_{a\in\mathcal{A},b\in\mathcal{B}}\left(|\sum_{kxy}e^{i(\theta_{x}-\theta_{y})}(\Lambda_{k})_{ax}(\Lambda_{k})_{yb}|^{2}+\sum_{k\neq
m}|\sum_{xy}e^{i(\theta_{x}-\theta_{y})}(\Lambda_{k})_{ax}(\Lambda_{m})_{yb}|^{2}\right).$
(45)
method | initial state | limit | $A_{ab}$
---|---|---|---
T | $|j\rangle$, summed over | before $t$-average | $\frac{1}{2}\left(|U_{ab}(t)|^{2}+|U_{ba}(t)|^{2}\right)$
| | $t\to 0$ | $\delta_{ab}+\frac{t^{2}}{3}\left(|H_{ab}|^{2}-\delta_{ab}(H^{2})_{aa}\right)+O(t^{3})$
| | $t\to\infty$ | $\sum_{k}|(\Lambda_{k})_{ab}|^{2}$
F | $\sum_{j}|j\rangle$ | before $t$-average | $\sum_{xy}\operatorname{Re}\left(e^{i(\theta_{x}-\theta_{y}+\theta_{b}-\theta_{a})}U_{ax}(t)\overline{U_{by}(t)}\right)$
| | $t\to\infty$ | $\sum_{x,y,k}\operatorname{Re}\left((\Lambda_{k})_{ax}(\Lambda_{k})_{yb}\right)$
F${}^{\text{ph}}$ | $\sum_{j}e^{i\theta_{j}}|j\rangle$, | before $t$-average | $\operatorname{Re}\left(U_{aa}(t)\overline{U_{bb}(t)}\right)+\delta_{ab}\left(1-|U_{aa}(t)|^{2}\right)$
| phase-averaged | $t\to\infty$ | $\sum_{k}(\Lambda_{k})_{aa}(\Lambda_{k})_{bb}+\delta_{ab}\left(1-\sum_{k}((\Lambda_{k})_{aa})^{2}\right)$
P | $\sum_{j}|j\rangle$ | before $t$-average | $|\sum_{xy}e^{i(\theta_{x}-\theta_{y})}U_{ax}(t)\overline{U_{by}(t)}|^{2}$
| | $t\to\infty$ | $\left|\sum_{k,x,y}(\Lambda_{k})_{ax}(\Lambda_{k})_{yb}\right|^{2}+\sum_{k\neq m}\left|\sum_{x,y}(\Lambda_{k})_{ax}(\Lambda_{m})_{yb}\right|^{2}$
P${}^{\text{ph}}$ | $\sum_{j}e^{i\theta_{j}}|j\rangle$, | before $t$-average | $1+\delta_{ab}-\sum_{x}|U_{ax}(t)|^{2}|U_{bx}(t)|^{2}$
| phase-averaged | $t\to\infty$ | $1+\delta_{ab}-\sum_{x}\left(\sum_{km}|(\Lambda_{k})_{ax}|^{2}|(\Lambda_{m})_{bx}|^{2}+\sum_{k\neq m}(\Lambda_{k})_{ax}(\Lambda_{k})_{xb}(\Lambda_{m})_{bx}(\Lambda_{m})_{xa}\right)$
Table 1: Adjacency matrices, based on time-averaged measures. The closeness
measure in each case is
$c(\mathcal{A},\mathcal{B})=\frac{2}{n^{2}|\mathcal{A}||\mathcal{B}|}\sum_{a\in\mathcal{A},b\in\mathcal{B}}A_{ab}$.
Note that if $H$ has purely nondegenerate eigenvalues, then all the projectors
are of the form $\Lambda_{k}=|\psi_{k}\rangle\langle\psi_{k}|$, which makes
some of the $t\to\infty$ measures above identical. For example $T^{\infty}$
becomes the same as $F^{\infty,\text{ph}}$ outside the diagonal. This type of
nondegeneracy occurs e.g. when a small random perturbation is used to break
the symmetries of $H$.
|
arxiv-papers
| 2013-10-24T15:02:31 |
2024-09-04T02:49:52.812836
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Mauro Faccin, Piotr Migda{\\l}, Tomi H. Johnson, Ville Bergholm and\n Jacob D. Biamonte",
"submitter": "Mauro Faccin",
"url": "https://arxiv.org/abs/1310.6638"
}
|
1310.6680
|
# Quantitative Comparison of Methods for Predicting the Arrival of Coronal
Mass Ejections at Earth based on multi-view imaging
R. C. Colaninno, A. Vourlidas, C.-C. Wu
Space Science Division, Naval Research Laboratory, Washington, District of
Columbia, USA.
[email protected]
###### Abstract
We investigate the performance of six methods for predicting the CME time of
arrival (ToA) and velocity at Earth using a sample of nine Earth-impacting
CMEs between May 2010 and June 2011. The CMEs were tracked continuously from
the Sun to near Earth in multi-viewpoint imaging data from STEREO SECCHI and
SOHO LASCO. We use the Graduate Cylindrical Shell (GCS) model to estimate the
three-dimensional direction and height of the CMEs in every image out to
$\sim$200 R⊙. We fit the derived three-dimensional (deprojected) height and
time data with six different methods to extrapolate the CME ToA and velocity
at Earth. We compare the fitting results with the in situ data from the WIND
spacecraft. We find that a simple linear fit after a height of 50$R_{\odot}$
gives the best ToA with a total error $\pm$13 hours. For seven (78%) of the
CMEs, we are able to predict the ToA to within $\pm$6 hours. These results are
a full day improvement over past CME arrival time methods that only used SOHO
LASCO data. We conclude that heliographic measurements, beyond the
coronagraphic field of view, of the CME front made away from the Sun-Earth
line are essential for accurate predictions of their time of arrival.
## 1\. Introduction
In the last few decades, we have discovered that the space environment around
our planet is as dynamic as terrestrial weather. The source of this space
weather is the Sun which produces winds and storms that affect modern
infrastructure. The most geo-effective aspects of space weather are coronal
mass ejections (CMEs) which are analogous to terrestrial hurricanes. These
powerful storms, comprised of plasma and magnetic fields ejected from the
solar corona, can significantly disrupt Earth’s magnetosphere and cause a
range of terrestrial effects from the aurora to ground induced currents. CMEs
were first observed in visible-light coronagraphs (Tousey and Koomen, 1972) as
bright large-scale density enhancements propagating outwards from the Sun.
Signatures of CMEs are also seen _in situ_ plasma and magnetic field data
(Cane et al., 2000). _In situ_ measurements at Earth give us the real-time
arrival and physical properties of geo-effective CMEs. Throughout the paper,
we use the term CME to describe both the imaging and _in situ_ observations.
Since mid-2007, we are able to continuously monitor the propagation of CMEs
from the Sun to Earth using the observations from the Sun-Earth Connection
Coronal and Heliospheric Investigation (SECCHI; Howard et al. 2008) instrument
suite aboard the _Solar TErrestrial RElations Observatory_ mission (_STEREO_ ;
Kaiser et al. 2008). However, even with complete coverage of the Sun-Earth
line with visible-light imaging data, it remains difficult to accurately
predict the arrival of CMEs at Earth. CMEs are detectable in visible-light due
to Thomson scattering of photospheric sunlight by the electrons within the
CME. This emission is optically thin, thus, the observations are integrations
along the line of sight (LOS) making it difficult to identify individual
features of the diffuse CME structure. The emission from the CME electrons
drops off quickly as the CME expands away from the Sun causing a decrease in
density and scattering efficiency with the viewing angle. By the time CMEs
reach Earth, they are large ($\sim$ 0.5 AU width) with correspondingly long
LOS integration paths. All these elements of CME observations at large
heliocentric distances make it difficult to know the time of arrival (ToA)
even when both the Earth and the CME are visible simultaneously in the same
field of view (FOV).
With careful treatment of the visible-light image data, the position of the
CME front can be measured and the resulting height and time (HT) data points
can be fitted by a curve (e.g. polynomial or spline) to model its motion. If
the height of the CME is not measured up to 1 AU, the fit can then be
extrapolated to predict the ToA of the CME at Earth. This is one of the most
basic space weather prediction techniques that can be applied to imaging
observations.
Even prior to the availability of heliographic data from SECCHI, several
methods had been proposed in the literature to predict the ToA using
coronagraphic data from the Large Angle and Spectroscopic Coronagraph (LASCO;
Brueckner et al. 1995) aboard the _SOlar and Heliospheric Observatory_ mission
(_SOHO_ ; Domingo et al. 1995). Empirical CME propagation models (Gopalswamy
et al., 2000, 2001; Schwenn et al., 2005) were developed using the LASCO data
with limited success. These models use the projected velocity of Earth-
directed CMEs observed in the LASCO FOV. Owens and Cargill (2004) evaluated
the predicted ToA for three empirical models: Gopalswamy et al. (2000, 2001)
and Vršnak and Gopalswamy (2002). They found little difference between the
three models with an average ToA error of 0.46 days. Schwenn et al. (2005)
reported similar results for their method which used the expansion speed of
the CME as a proxy for the CME radial speed. All these models can predict the
arrival of the CME at Earth within a $\pm$24 hours window with a 95% error
margin.
At the time, the inaccuracy of these models was attributed to the inability to
measure the true (deprojected) speed of Earth-directed CMEs since LASCO is
located along the Sun-Earth line offering a head-on view of the propagation.
Lindsay et al. (1999) compiled a data set of CMEs observed in quadrature using
Solar Maximum Mission and Solwind coronagraphs, and _Helios 1_ and _Pioneer
Venus Orbiter_ _in situ_ magnetic field and plasma measurements. Owens and
Cargill (2004) found that using these data where the projection effects are
minimized did not improve the results of the three studied ToA models.
Physics-based shock propagation models, using the speeds from the metric type-
II radio bursts which are not affected by projection effects, do not fare any
better with ToA predictions (Cho et al., 2003).
Kilpua et al. (2012) fitted the stereoscopic data from SECCHI and LASCO with
the Graduated Cylindrical Shell (GCS; Thernisien et al. 2009) model to derive
the three-dimensional (3D) position, direction and speed of 30 CMEs within the
LASCO FOV ($<$ 30 R⊙) between 2008 and 2010. Kilpua et al. (2012) did not
extend their analysis into the SECCHI HI FOV. They applied these deprojected
CME speeds in the models of Gopalswamy et al. (2000, 2001). They compared the
predicted ToA to the _in situ_ measurements and found an error of $\pm$30
hours with a 95% error margin. This result is actually worse than the
predictions obtained using the same model with projected CME speeds. It is
unclear what caused this unexpected result. It may be due to the low average
speeds of the Kilpua et al. (2012) sample, taken during the most unusual
minimum of the space age, compared to samples used in the past. For all
empirical models, the error in ToA prediction is largest for slower CMEs.
Given these uncertainties and apparent insensitivity of empirical models to
deprojected speeds, we will not consider them further.
Instead, we focus on models that make certain assumptions about the CME shape
and/or direction to estimate the 3D speed from heliographic image data. Models
using single-viewpoint heliographic imaging data in this way are Fixed-$\phi$
(Sheeley et al., 1999; Rouillard, 2011), harmonic mean (Lugaz, 2010) and self-
similar expansion (Davies et al., 2012). These models use geometric arguments
to derive the longitude and speed of the CME assuming that these values are
constant throughout the range of the observations. The Fixed-$\phi$ method
simplifies the CME to a single point (or rather a very narrow LOS extension).
The harmonic mean method simplifies the CME to a circle which intersects the
Sun and CME front. The self-similar expansion method is an extension of the
harmonic-mean method in that the CME is no longer anchored at the Sun but is
expanding with a constant angular extent. Lugaz et al. (2012) compared the
predicted ToA from the Fixed-$\phi$ and harmonic mean methods for 20 CMEs
which impacted _STEREO_. They found ToA errors of $\pm$33 hours for the
Fixed-$\phi$ and $\pm$20 hours for the harmonic mean method both with 95%
error margin. The self-similar expansion method has not been applied to CME
data, at this point.
Howard et al. (2006) used Solar Mass Ejection Imager (SMEI) heliospheric
imaging data to predict the ToA of 15 CMEs in 2003 and 2004. Their results are
within a range of -24 to 20 hours for all CMEs. Howard and Tappin (2010) used
a 3D model to predict the ToA of three CMEs also using SMEI data at many
different elongations of the front. Their best predictions were within an hour
of the CME ToA.
Methods such as triangulation (Liu et al., 2010; Liewer et al., 2011; Liu et
al., 2013) and constrained harmonic mean (Lugaz et al., 2010) have been
developed to take advantage of the stereoscopic data from _STEREO_ -SECCHI.
Liu et al. (2010) was able to predict the arrival of the front within 12 hours
for their single CME. Liu et al. (2013) used both triangulation and the
constrained harmonic mean method to study the kinematics of three Earth-
impacting CMEs. The constrained harmonic mean method gave the best results
with an error between -2.3 to 8.4 hours in the ToA of the CME driven shock.
Another approach is to use the _in situ_ detection of the CME to constrain the
imaging observations (Wood et al., 2009a; Rollett et al., 2012; Temmer et al.,
2011). Wood et al. (2009a) used a multiple-function fit to the HT data to
describe the kinematics of a CME at different heliocentric distances. Rollett
et al. (2012) and Temmer et al. (2011) have used spline fits to derive the
velocity and acceleration profiles of the CMEs studied in these papers. This
approach may provide some insight into the CME kinematics but cannot be used
for operational space weather predictions because the ToA is no longer a free
variable.
In this paper, we attempt a more operational approach based on a sample of
nine Earth-impacting CMEs. We use the GCS model to fit the multi-viewpoint
observations from SECCHI and LASCO similar to Kilpua et al. (2012). However,
unlike Kilpua et al. (2012), we extend our fits into the heliospheric
observations as far as $\sim$1 AU, in some cases. We then fit the derived 3D
positions using a variety of models, such as constant speed or accelerating
profiles, restricting the fits to certain HT ranges, taking into account the
geometry of the impact and finally comparing with the _in situ_ measurements.
Our aim is to find the simplest and most reliable model for a set of HT
observations than can lead to better operational ToA predictions.
## 2\. Observations of Earth-impacting CMEs
Our primary data comes from the coronagraphic and heliospheric imaging
observations of _STEREO_ -SECCHI and _SOHO_ -LASCO from March 2010 to June
2011. This data set allows us to continuously track Earth-impacting CMEs from
2 or 3 viewpoints at all times. _STEREO_ is comprised of two spacecraft with
nearly identical instrumentation; the _STEREO_ -Ahead (STA) spacecraft orbits
slightly faster than the Earth and the _STEREO_ -Behind (STB) spacecraft
slightly slower. The two spacecraft separate from Earth at a rate of 22.5o per
year since their launch on 25 October 2006. In this study, we use _STEREO_
-SECCHI observations from the outer coronagraph, COR2, which has a FOV from
2.5 to 15 $R_{\odot}$ (Howard et al., 2008). SECCHI also includes two
heliospheric imagers (HI-1, HI-2) which are similar to coronagraphs but have
no occulter and a FOV off-pointed from the center of the Sun. The heliospheric
imagers view the Sun-Earth line from opposite sides of the heliosphere. HI-1
and HI-2 have square FOVs centered on the elliptic plane from 15 to 84
$R_{\odot}$ (20o) and 66 $R_{\odot}$ to 1 AU (70o), respectively (Howard et
al., 2008). We also use the data from the _SOHO_ -LASCO C2 (FOV 2.2–6
$R_{\odot}$) and C3 (FOV 3.8–32 $R_{\odot}$) coronagraphs (Brueckner et al.,
1995).
When studying CMEs, especially Earth-impacting CMEs, it is advantageous to
combine the data from LASCO and SECCHI since LASCO has a head-on view of the
CME and provides a view of the extent of the CME while SECCHI has a side view
and provides information on the location of the CME front. During the time
period of our study, March 2010 - June 2011, the _STEREO_ spacecraft were
separated from each other by 132o to 190o. On 1 March 2010, STB and STA were
-71∘ and 66o, respectively, from Earth. The spacecraft reached opposition on 6
February 2011 and began moving closer to each other on the far side of the
Sun. On 30 June 2011, STB and STA were -93o and 97o from Earth, respectively.
In this configuration, an Earth-directed CME appears on the West limb in STB
and on the East limb in STA.
Table 1: Studied CME | LASCO Detection | Halo or | Lon | _Wind_ Detection | Velocity | Detection
---|---|---|---|---|---|---
CME | Date | Time (UT) | Partial | (deg) | Date | Time (UT) | ($kms^{-1}$) | Type
1 | 19-Mar-2010 | 10:30 | | 27 | 23-Mar-2010 | 23:02 | 284 | CME
2 | 03-Apr-2010 | 10:33 | H | 6 | 05-Apr-2010 | 06:43 | 755 | MC
3 | 08-Apr-2010 | 01:31 | PH | -2 | 11-Apr-2010 | 11:59 | 430 | MC
4 | 16-Jun-2010 | 14:54 | PH | -18 | 20-Jun-2010 | 23:59 | 400 | CME
5 | 11-Sep-2010 | 02:00 | PH | -21 | 14-Sep-2010 | 14:24 | 368 | MC
6 | 26-Oct-2010 | 01:36 | | 22 | 31-Oct-2010 | 04:48 | 365 | MC
7 | 15-Feb-2011 | 02:24 | H | 2 | 18-Feb-2011 | 00:00 | 510 | MC
8 | 25-Mar-2011aaThe CME was listed as two events in the _SOHO_ LASCO CME Catalog. | 14:36 | PHbbSecond detection. | -27 | 29-Mar-2011 | 14:38 | 378 | MC
9 | 2-Jun-2011 | 8:12 | H | -22 | 04-Jun-2011 | 00:00 | 482 | CME
To determine the ToA and speed of the CME at Earth, we use the _in situ_
plasma data from the _Wind_ spacecraft. The _Wind_ spacecraft, like _SOHO_ ,
orbits the L-1 Lagrange point and is ideally situated for monitoring near-
Earth space weather. In this study, we will use data from the _Wind_ Magnetic
Field Investigation (MFI; Lepping et al. (1995)) and Solar Wind Experiment
(SWE; Ogilvie et al. (1995)). The MFI instrument is a triaxial magnetometer
which provides the magnitude and direction of the solar wind’s magnetic field.
The SWE instrument provides the density, velocity and temperature of the ions
of the solar wind. We use the magnetic field data to confirm the passage of a
CME-like magnetic structure, an increase in magnetic field and smooth rotation
in one of the field components (Cane et al., 2000). With the data from plasma
instrument, we can determine the ToA and velocity of a CME to compare with our
results derived from the imaging data.
### 2.1. Description of CME Event Sample
We analyze nine Earth-impacting CMEs observed between March 2010 and June 2011
in both imaging and _in situ_ data. This time range corresponds to the rising
phase of Solar Cycle 24 and is quite advantageous. CMEs during this period are
more energetic but not so numerous as to result in many CME-CME interaction
which confuse measurements of individual features. The CMEs are identified in
Table 1 by the date and time of their first appearance in the LASCO C2
coronagraph taken from the _SOHO_ LASCO CME Catalog
(http://cdaw.gsfc.nasa.gov/CME_list, Yashiro et al. 2004). We denote each CME
with a number in chronological order in Table 1. We will refer to the CMEs by
these numbers, throughout this paper. In Table 1, we also list whether the CME
was identified as a halo (H) or partial halo (PH) in the catalog. Despite all
nine CMEs being Earth-directed, only three were identified as halos and four
were identified as partial halos. Therefore, a CME can impact the Earth even
if it is not identified as a partial halo in the LASCO catalog. With complete
imaging coverage of the Sun-Earth line, we are able to show that all the
studied CMEs are detected at Earth in the _Wind_ spacecraft data. The
Heliocentric Earth Ecliptic (HEE) longitude of the CME derived from GSC model
fitting is listed in column 5 of Table 1.
We search the SECCHI data set beginning in January 2009, when the spacecraft
were separated by $\sim$88.5o and ending in June 2011. To be included in our
sample, the CMEs must be observed in all the imaging data (SECCHI, LASCO,
eight instruments in total) without a significant period ($<$ 1 hour) of
missing data. The CME must be easily tracked between instruments. Thus the
structure of the CME had to be visible out to nearly the edge of the FOV of
each instrument (with the exception of SECCHI HI-2). Due to the effects of
Thomson scattering, the CME emission is dimmest from the LASCO viewpoint for
Earth-impacting CMEs. Thus the visibility of the CME in the LASCO coronagraphs
is usually the limiting factor for selection. To ensure we properly fit the
CME envelope, we rejected any CMEs that expanded outside the upper or lower
edges of the HI-1 FOV. These restrictions are severe and eliminate many CMEs
from study but are required for robust fitting of the GCS model.
To identify the CME region in the _Wind_ data, we used the criteria of Lepping
et al. (2005) automatic detection technique. The technique was developed to
detect potential magnetic clouds (MC) in the data based on the definition from
Burlaga et al. (1981). The technique can also identify possible CMEs in the
_in situ_ data. The detection requirements for a potential MC are higher than
for a CME detection. The minimum requirements for potential CME detection are;
the proton plasma beta must be $<\beta_{p}>\leq 0.3$, the field directions
must change smoothly, and these two conditions must persist continuously for a
minimum of eight hours. For possible MC detection, a period of data must meet
the minimum criteria above and have (i) a high average magnetic field strength
(B $>$ 7 nT), (ii) a low proton thermal velocity (vth=30 $kms^{-1}$) and,
(iii) a minimum change in the magnetic field latitude ($\Delta\theta=35^{o}$).
All nine CMEs meet the minimum detection criteria of Lepping et al. (2005);
seven of them also met the criteria for MC detection. In Table 1, we list the
_in situ_ detection type for each CME. The detection type only indicates if
the _in situ_ data met the outlined criteria. To determine the presence of a
MC or a MC-like structure in the data further analysis would be needed (Wu and
Lepping, 2007).
In Table 1, we list the time when the CME is detected at the _Wind_
spacecraft. There is no consensus in the literature as to which parameter of
the _in situ_ data marks the arrival of the CME (see the discussion in
Gopalswamy et al. 2003 and Cane and Richardson 2003). Since CMEs are large
structures which can persist in the _Wind_ data for days, the selection of the
CME arrival criteria can affect the ToA by several hours. Two parameters
commonly used for the _in situ_ ToA of a CME are the time of the peak magnetic
field intensity of the shock or the beginning of the MC. To properly compare
the imaging to _in situ_ data, we determine the ToA of the CME at _Wind_ based
on the density since it is the common physical parameter between _in situ_ and
imaging measurements. We do not use the peak of the shock magnetic field, if
present, since it arrives before the CME density front. Similarly, we do not
use the arrival of MC because it occurs after the CME density front.
Therefore, we propose that the ToA of the density increase is the most
appropriate for comparison to the imaging data. In Table 2, we list the
duration of the density front, the time between the density increase and the
region of low plasma beta, and the mean of the velocity detected by _Wind_
during the passage of the density increase.
## 3\. Graduated Cylindrical Shell (GCS) Model
To locate the CME front in 3D space from the imaging data, we use the GCS
model. The graduated cylindrical shell model (GCS) was developed by Thernisien
et al. (2006, 2009). It is a forward modeling method for estimating the 3D
properties and position of CMEs in white-light observations. Unlike the
methods discussed earlier (Sheeley et al., 1999; Rouillard, 2011; Liu et al.,
2010; Liewer et al., 2011; Lugaz, 2010; Möstl and Davies, 2012) that use only
the front of the CME, the GCS model is a complete 3D reconstruction of the CME
envelope. Other such 3D reconstruction models as well as the GCS model are
reviewed in Mierla et al. (2010).
The GCS modeling software allows the user to fit a geometric representation of
the CME envelope to all simultaneous imaging observations. The shape of the
GCS model is designed to mimic that of a cylindrical magnetic flux rope. The
CME is described by a curved hallow body with a circular cross-section
connected by two conical legs anchored at the Sun’s centers. It is important
to note that the GCS model is purely geometric and does not provide any
information about the magnetic field. Complete details of the model geometry
can be found in Thernisien (2011).
The model is fit by overplotting the projection of the cylindrical shell
structure onto each image. The observer then adjusts six parameters of the
model until a best visual fit with the data is achieved. The model is
positioned using the longitude, latitude and the rotation parameters. The
origin of the model remains fixed at the center of the Sun. The size of the
flux rope model is controlled using three parameters which define the apex
height, footpoint separation and the radius of the shell. Figure 1 shows
simultaneous images from three viewpoints, STA and STB HI-1 and LASCO C3, as
well as the GCS model fit to the data. In each image, the model is projected
onto the plane of the image using a grid of points (green) that represent the
surface of the model.
### 3.1. Application of the GCS Model to Remote Sensing Data
We fit the GCS model to all available images from all nine CMEs starting at
the CME’s first appearance in the SECCHI COR2 and LASCO C2 FOVs until the
SECCHI HI-2 FOV. When the CME is visible in the LASCO data, we use all three
viewpoints to make the GCS fit. The LASCO viewpoint is essential for a robust
fit because the projection of an Earth-directed CME is usually quite symmetric
between STA and STB. The LASCO viewpoint can give essential information about
the orientation and dimensions of the CME that is ambiguous in the SECCHI data
for Earth-directed CMEs (Vourlidas et al., 2011).
Once the front of the CME is no longer visible in the LASCO FOV, we must make
some assumptions about its evolution. We assume that it expands self-
similarly. This assumption is implemented by keeping constant all parameters
of the GCS model except height. We believe self-similar expansion is a good
assumption, since for most CMEs the model parameters vary only slightly when
fitted using the LASCO view. A notable exception, CME 4 has a rapid change in
its rotation angle in the LASCO FOV (Vourlidas et al., 2011). The effects of
the rotation on the GCS model fit to this CME are discussed in Nieves-
Chinchilla et al. (2012).
Figure 1: A sample of the remote sensing data used in the study. The panels
are data from STA HI-1, LASCO C3 and STB HI-1 from left to right. The data in
the top and bottom panels are the same. The images in the bottom panel have
been over plotted with the GCS model. The GCS model is represented by a grid
of points on the surface of the model.
The GCS fitting provides measurements of various physical aspects of the CME,
such as size, direction, orientation, etc. In this paper, we concentrate our
analysis only on the measurements of the CME 3D height versus time (HT). In
Figures 2-4, the HT measurements and _in situ_ data are plotted on the same
time axis for each of the nine CMEs. The HT data are plotted in the top panel
for each CME with plus signs. We fit the GCS model at a maximum height of 211
R⊙ (0.98 AU) for CME 2. The average maximum height for all the studied CMEs is
179 R⊙ (0.83 AU). The bottom three panels for each CME in Figures 2-4 show the
magnetic field magnitude, proton density and proton velocity measured _in
situ_ from the _Wind_ spacecraft. The first vertical green dashed line marks
the ToA of the density increase. The second green dashed line is the backend
of the density front and the beginning of the low beta plasma and smooth
magnetic field rotation. The mean of the plasma velocity is also plotted as a
horizontal green solid line in each bottom panel. We will discuss the fits to
the HT data in section 4.
### 3.2. Error Estimation in Stereoscopic Localization
To properly assess the various HT fitting methods for deriving the CME
velocity and extrapolating the ToA, we need to assign an error to our height
measurements. Thernisien et al. (2009) estimated the error associated with the
six GCS model parameters when applied to a CME in the SECCHI COR2 views only.
They found an error of $\pm 0.48$ R⊙ in the height. Since we are using LASCO
data in addition to the SECCHI COR2 data, we consider the errors from
Thernisien et al. (2009) as an upper limit
Figure 2: HT measurements and _in situ_ data plotted on the same temporal
axis. The HT data are plotted in the top panel with plus signs. The bottom
three panels show the magnetic field magnitude, proton density and proton
velocity _in situ_ data from the _Wind_ spacecraft. The vertical green dashed
line marks the width of the density increase (ToA). Fit 1 and 5 and the their
velocities are plotted with blue and orange solid lines, respectively.
Figure 3: Same as Figure 2 for CMEs 4-6.
Figure 4: Same as Figure 2 for CMEs 7-9.
for the height measurements in these FOV. Thus we need to estimate the error
for heights measured in the HI images without the LASCO images. As mentioned
in the previous section, once the CME in no longer visible in the LASCO FOV,
we fit the GCS model to the data by only adjusting the height parameter. Thus
in the HI-1 and HI-2 images, the accuracy of the GCS model fit is primarily
driven by the proper localization of the CME front from the two viewpoints.
To estimate this error, we consider the simplified problem of stereo
triangulation (Hartley and Zisserman, 2004). In a digital image, there is
always an error associated with the localization of a feature in the image.
The error can be represented by a cone of uncertainty around the line-of-sight
(LOS) from each viewpoint. In Figure 5, we represent the triangulation
geometry between two points, $P$ and $P^{\prime}$, near the Sun-Earth line
with the _STEREO_ spacecraft. The LOS from each spacecraft is drawn with
dashed lines and the cone of uncertainty is drawn with solid lines. The
intersection of the uncertainty in the LOS from STA and STB creates a region
of uncertainty around the feature. This region is a trapezoid defined by the
angle between the two LOS, $\alpha$, and the uncertainty in locating the
feature in the image. Thus $\alpha$ is given by
$\alpha=2\pi-(\theta_{A}+\theta_{B}+\varepsilon_{A}+\varepsilon_{B})$ (1)
where $\theta_{A}$ and $\theta_{B}$ are the longitudes of the spacecraft
relative to the Sun-Earth line and $\varepsilon_{A}$ and $\varepsilon_{B}$ are
the solar elongation of the feature in each instrument. The insert in Figure 5
shows a close up of the geometry for the region of uncertainty for
$P^{\prime}$. Since the LOS are large and the error in locating the feature is
small for the _STEREO_ case, we can assume that the sides of the trapezoid are
separated by a constant distance $w$. The length of the trapezoid axes, $dx$
and $dy$, are given by the equations,
$dx=\frac{w}{2\cos{\frac{\alpha}{2}}},\quad
dy=\frac{w}{2\sin{\frac{\alpha}{2}}}$ (2)
where $dx$ and $dy$ are parallel and perpendicular to the longitude of the
feature, respectively. Based on our experience, we estimate the error in
locating the CME front in HI-1 to be $\pm$ 5 pixels and in HI-2 is $\pm$ 10
pixels, thus, $w$ is 0.2
Figure 5: The error in fitting of the GCS model in the HI-1 and HI-2 images
can be simplified to the error in triangulating a feature in stereoscopic
images. The LOS from each spacecraft is drawn with dashed lines and the cone
of uncertainty is drawn with solid lines. The intersection of the uncertainty
in the LOS from STA and STB creates a region of uncertainty around the
feature. The insert shows a close up of the geometry for the region of
uncertainty for point $P^{\prime}$ assuming a long LOS.
$R_{\odot}$ and 1.4 $R_{\odot}$ for HI-1 and HI1-2, respectively.
Equations 2 require careful consideration despite their simplicity. For
example, the error $dx$ goes to infinity for $\alpha=\pi$. We can see in
Figure 5 that as the CME front moves between point $P$ and $P^{\prime}$, that
the longitude of the CME will be unconstrained. From equation 1, before the
spacecraft reach opposition ($\theta_{A}+\theta_{B}>2\pi$) the range of values
of angle $\alpha$ includes $\alpha=\pi$. This uncertainty in the CME longitude
is part of the reason why once the CME is no longer visible in the LASCO FOV,
we keep the longitude of the model fixed. Since we can fit the GCS model for
all the HI-1 and HI-2 images without changing the longitude, the error in the
longitude must be within the minimum value of $dx$ for all measurements. If
the error in the longitude is bounded by the minimum of $dx$, then the error
in the height is simply $dy$ for each measurement. The maximum error in the
height measurements for each CME varies between 7.4 and 12.9 $R_{\odot}$ in
the HI-2 FOV. In Figures 2-4, the error in the height is too small to be
visible in the plot. The error bars for the HT measurements are shown in
Figures 7 for the case of CME 9.
## 4\. Height and Time Data Fitting Methods
To find the best HT fitting procedure for predicting the ToA and velocity of
the CME at Earth, we explore six methods that assume various kinematic
profiles for the CME front. It is not possible to measure the front height all
the way to Earth for all the CMEs in our sample. The six fitting methods are
described below in approximately the order of their complexity. We assign a
color to each fit type which is used throughout.
#### Fit 1 - Linear (blue)
We fit a first-order polynomial to the HT measurements above a height of
50$R_{\odot}$ (0.23 AU). We selected the lower cutoff of 50$R_{\odot}$ because
for most of the CMEs the HT measurements appear to be approximately linear
after this height. Also 50$R_{\odot}$ is approximately the mean height at
which the CME front is no longer visible in the LASCO data. Although the LASCO
C3 FOV is 32$R_{\odot}$, the 3D front height for Earth-directed CME usually
reaches 50$R_{\odot}$ within the image. Also we remind the reader that we
assume self-similar expansion of the CMEs after the CME front is no longer
visible in LASCO. Thus the longitude of the GSC model is fixed after
50$R_{\odot}$. We extrapolate the linear fit to 1 AU to find the ToA and
velocity at Earth.
#### Fit 2 - Quadratic (purple)
We fit a second-order polynomial to the HT measurements above a height of
50$R_{\odot}$. While most of the CMEs appear to be well described by Fit 1,
some of the CMEs, notably CMEs 2 and 9, have an obviously curved HT profile.
This fit assumes that the CME continues to Earth with a constant acceleration.
We extrapolate the function and take the first derivative at 1 AU to find the
ToA and velocity at Earth.
#### Fit 3 - Multiple Polynomials (red)
We fit all available HT measurements for a given event with multiple first-
and second-order polynomial functions for different time ranges. The HT
measurements are fit by an initial first-order polynomial and then two second-
order polynomials. The boundaries of the three functions are determined by the
best fit while keeping the function and its first derivative continuous. We
extrapolate the ToA by assuming a constant velocity after the final data
point, again keeping the velocity continuous. This multi-function polynomial
fit method is similar to that used by Wood et al. (2009a, b) to fit the
kinematics of two CMEs observed in _STEREO_. However, Wood et al. (2009a, b)
used the ToA of the CME as a final data point for their fit.
#### Fit 4 - Spline (magenta)
We fit all HT measurements with a ridged spline. Again, we extrapolate the ToA
by assuming the CME continues with a constant velocity after the final data
point. The shape of the ridged spline fit is similar to Fit 3. These two
methods provide similar velocity profiles. The spline fit velocity is,
however, a smoothly varying curve throughout the CME trajectory which seems
more physical than the velocity profiles from Fit 3 which are piecewise
continuous with a discontinuous acceleration. This fit is similar to the
method used by Rollett et al. (2012) and Temmer et al. (2011).
#### Fit 5 - LASCO FOV (orange)
With this fit we try to compare coronagraphic analyses of the past against the
heliospheric data available with _STEREO_. We fit only those data points where
the CME was visible in the LASCO images which is the opposite approach of Fits
1 and 2 where we use HT measurements after the CME front leaves the LASCO FOV.
We fit the LASCO measurements with a second-order polynomial. We then
extrapolate the ToA using a first-order polynomial with the velocity derived
from the final LASCO data point. This method is similar to Kilpua et al.
(2012). However, we use a simple linear extrapolation instead of the empirical
propagration models of Gopalswamy et al. (2000, 2001).
#### Fit 6 - Geometric Correction (light blue)
With this fit, we attempt to take into account the effect of the curvature of
the CME front on the ToA and velocity. So we use the height of the GCS model
along the Sun-Earth line instead of the apex height. These heights take into
account the curvature of the GCS model front and the longitude of the CME
propagation. As an example, Figure 6 shows an ecliptic cut through all GCS
model fits for CME 8 where the central line of the plot is the Sun-Earth line
and the dashed line is the longitudinal direction of the model. In Table 1, we
Figure 6: Ecliptic cut through the GCS model fits for CME 8 where the central
line of the plot is the Sun-Earth line and the dashed line is the longitudinal
direction of the model.
list the HEE longitude from each fit GCS. Obviously, the front height along
the Sun-Earth line is less then the apex height. Thus the curvature of the CME
front delays the arrival of the CME and reduces the velocity. We fit these
curvature corrected HT data in the same way as Fit 1.
In Figure 7, we have plotted all the fit methods for the HT measurements of
CME 9. The CME 9 HT measurements of the apex are plotted with black crosses.
The error for each measurement is plotted in gray. Fits 1 (blue), 2 (purple),
3 (red), 4 (magenta), 5 (orange), and 6 (light blue) and their ToA are plotted
with solid and vertical dashed lines, respectively. The green dashed lines
mark the time of the _in situ_ density front. The light blue squares represent
the HT measurements corrected for the front curvature (Fit 6) as derived from
the GCS fit. In Figures 2-4, we have plotted in the top panel Fit 1 and Fit 5
for each of the CMEs. Again the solid and vertical dashed lines represent the
fits and ToA, respectively.
## 5\. Results
To quantify the accuracy of the various HT fits in predicting the ToA and CME
velocity at 1 AU, we calculate the difference $\Delta$T = ToApredicted-
ToA${}_{\emph{Wind}}$. A negative $\Delta$T implies an early arrival and
conversely, a positive $\Delta$T implies a late arrival. The $\Delta$T in
hours are listed in Table 2 for each fitting method. In the first column, we
list the duration of the _in situ_ density front in hours. In Figures 2-4, the
boundaries of the _in situ_ density front are marked with vertical dashed
green lines. The velocity listed in Table 1, is the mean of the measured
proton velocity during the passage of the _in situ_ density front. The _Wind_
proton velocity is plotted in the bottom panels of Figures 2-4. The mean
velocity, listed in Table 1, is plotted over the _Wind_ measurements with a
horizontal solid green line between the dashed lines of the density front. We
calculate the velocity error by finding the difference between the predicted
velocity and the mean of the plasma velocity within the _in situ_ density
front ($\Delta$V = Vpredicted-$\overline{V}_{\emph{Wind}}$). In Table 2, we
list the range of the measured _in situ_ velocities. We have included the
duration and velocity variability of the _in situ_ density front in our
discussion because they may provide a sense of scale for the prediction
errors.
We visually represent the results from Table 2 in Figure 8. In the left panel,
the $\Delta$T for each fit method is plotted with plus signs by CME number on
the vertical axis. The results for the various fits follow the color code in
section 4. The green line represents the duration of the CME _in situ_ density
front. In the right panel, we plot $\Delta$V using the same scheme. The green
lines in the right panel represent the range of velocities measured within the
_in situ_ density front.
For Fit 1, the $\Delta$T is within $\pm$ 6 hours, for seven of the CMEs. For 6
out of 9 events, the predicted ToA are either 2 hours before or within the
density front. The two events (CMEs 2, 4) with $\Delta$T$\pm$13 hour are
possibly violating the self-similar expansion assumption (Nieves-Chinchilla et
al., 2012; Wood et al., 2011). It is unclear how the violation of this
assumption could affect the ToA, furthermore, the $\Delta$T error is in the
opposite sense for these two events. The CME 2 ToA is late while CME 4
Figure 7: Comparison of the six HT fitting methods for CME 9. The green dashed
lines mark the time of _in situ_ density front. Fits 1 (blue), 2 (purple), 3
(red), 4 (magenta), 5 (orange), and 6 (light blue) and their ToA are plotted
with solid and vertical dashed lines, respectively. Black crosses represent
the deprojected HT measurements and light blue squares represent the same
points corrected for the front curvature as derived from the GCS fit. See
Section 4 for details.
is early. The predicted velocities from Fit 1 do not compare as well as the
ToA. For only two CMEs (6 and 7), $\Delta$V is within $\pm$ 50 $kms^{-1}$ of
the mean _in situ_ velocity. For four of the CMEs (1, 2, 5 and 9) the
$\Delta$V is greater than $\pm$ 100 $kms^{-1}$. Almost all the predicted
velocities are too fast with the exception of CME 2. Clearly, all CMEs in our
sample decelerate on the way to 1 AU.
The increased complexity of Fit 2 (quadratic), improves the $\Delta$T for CMEs
4, 5, and 7. Yet, the improvements to the ToA of CMEs 5 and 7 are trivial and
only vary the $\Delta$T of the CME within the density front. The $\Delta$T of
CME 4 is improved significantly from -12.74 to -2.94 hours. We cannot predict
the ToA for CME 6 because the quadratic fit fails to intersect with 1 AU,
_i.e_ , the CME does not make it to the Earth. The ToA for the remainder of
the events is not improved with Fit 2. This is true even for CMEs 2 and 9
which are not fit well with a constant velocity and hence Fit 2 was expected
to improve $\Delta$T. Overall, $\Delta$V is also not improved with Fit 2. Only
two of the CMEs are within $\pm$ 100 $kms^{-1}$ of the mean _in situ_
velocity. While the ToA of CME 4 is significantly improved with Fit 2, the
predicted velocity is worse. Clearly, the quadratic fit overestimates the CME
deceleration to 1 AU.
Table 2: Error in Predicted Arrival and Velocity at Earth | CME | Duration11The duration of the CME density front. | Fit 1 | Fit 2 | Fit 3 | Fit 4 | Fit 5 | Fit 6
---|---|---|---|---|---|---|---|---
$\Delta$T (hrs) | 1 | 6.00 | -0.94 | -6.17 | -2.47 | -4.07 | -28.93 | 56.42
2 | 6.42 | 12.41 | 15.90 | 9.52 | 15.28 | 3.59 | 13.00
3 | 13.17 | -1.58 | -3.41 | -4.03 | -2.86 | 6.21 | 8.09
4 | 9.83 | -12.74 | -2.94 | -13.97 | -9.39 | -27.45 | 6.83
5 | 11.67 | 2.47 | -0.70 | 9.97 | 0.29 | -25.69 | 30.76
6 | 20.17 | 2.18 | | 9.30 | 11.48 | -29.82 | 37.32
7 | 10.03 | 3.97 | 1.87 | 0.83 | 1.90 | 8.66 | 4.23
8 | 9.33 | -5.69 | -5.81 | -4.81 | -5.64 | -19.68 | 0.04
9 | 5.95 | -0.10 | 5.60 | 2.50 | 3.53 | -36.92 | 8.34
$\Delta$V ($kms^{-1}$) | | Velocity Range22The absolute range of in situ speeds detected within the CME density front. | | | | | |
1 | 10 | 129 | 243 | 153 | 166 | 308 | -13
2 | 78 | -137 | -273 | -131 | -326 | 55 | -138
3 | 37 | 84 | 174 | 107 | 120 | 23 | 18
4 | 25 | 83 | -136 | 143 | 34 | 190 | -8
5 | 40 | 102 | 183 | 32 | 135 | 390 | -13
6 | 31 | 35 | | -13 | -27 | 296 | -65
7 | 55 | 6 | 89 | 39 | 65 | -31 | 4
8 | 25 | 76 | 82 | 38 | 74 | 187 | 50
| 9 | 61 | 115 | -169 | -138 | -52 | 1426 | 49
Fit 3 does not improve the ToA predictions despite having more free parameters
than the pervious fits. Only the ToA for CME 7 is improved over Fits 1 and 2.
For only four CMEs, $\Delta$V is within $\pm$ 100 $kms^{-1}$ of the mean _in
situ_ velocity. Similarly, Fit 4 with the most free parameters fails to
provide an overall improvement in the predictions.
The most important finding from this exercise may be the disappointing
performance of Fit 5. Similar to Kilpua et al. (2012), we are investigating
whether accurate 3D HT measurements in coronagraphic FOVs can be used to
reliably predict the ToA of CMEs. Our results and Kilpua et al. (2012) suggest
that restricting the measurements to these heights dramatically increases the
ToA error compared to using the inner heliospheric measurements. Our fit uses
the fewest HT measurements but these measurements are based on images from
three viewpoints and are thus the most constrained. The $\Delta$T for only
three CMEs is within $\pm$12 hour. These results should be of interest to the
operational Space Weather community since most CME ToA prediction methods use
measurements from LASCO coronagraph along the Sun-Earth line. For this reason,
we explore this fit and the influence of the final height in the ToA accuracy
in the next Section.
Interestingly, this fit has the best prediction for the ToA of CME 2 (3.59
hours error) of all methods and leads us to two conclusions: 1) CME 2
underwent most of its kinematic evolution before $\sim 50R_{\odot}$, and 2)
the heliospheric measurements for this event are likely inaccurate. As we
mentioned earlier, this is a peculiar event with an undetermined orientation
which may not conform to the GCS model fitting. The six remaining CMEs are
predicted to arrive $>14$ hours early and the predicted velocities are $>$100
$kms^{-1}$ higher than the _in situ_ velocities. The results from Fits 1-5
confirm
Figure 8: A visual representation of the results in Table 2. The $\Delta$T for
each fit method has been plotted with plus signs in the left panel by CME
number. The green line represents the duration of the CME density front. In
the right panel, we have plotted $\Delta$V. The green lines represent the
range of velocities detected _in situ_ within the density front. The results
for the various fits (described in Section 4) are plotted in the following
color scheme: Fit 1 (blue), Fit 2 (purple), Fit 3(red), Fit 4 (magenta), Fit 5
(orange), Fit 6 (light blue).
past findings that CMEs undergo significant deceleration above 50 R⊙, on
average. Our $\Delta$T results are similar to those of Kilpua et al. (2012).
With Fit 6, we investigate the effect of the CME geometry predicted by the GCS
model on the ToA. Since the front the GCS model, and presumably of the actual
CME, is curved, the intersection of the CME with Earth will be delayed
relative to the 1 AU arrival of the CME apex. Möstl and Davies (2012) found
that for a hypothetical circular CME front, the flank can be delayed by up to
2 days compared to the apex arrival at 1 AU. Our model is a bit more realistic
since the front of the GCS model is not circular but slightly oblate depending
on the footpoint separation. Since all ToAs are based on the CME apex height,
the geometric correction of Fit 6 can only delay the ToA. Hence, only the ToA
errors for CMEs 1, 3, 4, and 8 can be improved by considering the CME front
geometry. With Fit 6, the ToA of CMEs 1 and 3 are "overcorrected"; the ToA is
too late. The correction lowers the $\Delta$T for CMEs 4 and 8 by 6.79 and
5.65 hours, respectively. We discuss the implications in the next section.
Interesting, the geometric correction improves $\Delta$V compared to Fit 1 for
all CMEs, even for CMEs 1, 3, and 9, where the correction increases $\Delta$T.
The $\Delta$V error is within $\pm$100 $kms^{-1}$ for eight CMEs. For CME 2,
the velocity is unchanged.
## 6\. The Effect of Final Height in Fit 5 on the ToA Accuracy
We repeat Fit 5 but instead of using the last LASCO FOV measurement as the
limit for the quadratic fit, we include measurements at larger heights within
the HI FOV. In Figure 9, we plot the resulting $\Delta$T versus the final
height of the second-order fit. The curves trace the errors for a given CME
and the best prediction for each event is highlighted with a red square. The
CME number is given on the right of the plot. For most of the CMEs the fit is
nearly linear and as more points are added, the function become more and more
linear.
It is clear, and generally expected, that the
Figure 9: Time of arrival error, $\Delta$T$=ToA_{fit5}-ToA_{\emph{Wind}}$, for
Fit 5 as a function of the final height used for the fit. The curves trace the
error for a given CME and the best result is shown by the red square. The
event number is shown on the right end of the corresponding curve.
addition of HT measurements beyond 50 R⊙ improve the ToA accuracy, sometimes
considerably (ie., by 40 hours for CME 9). Interestingly, it seems that most
of the gain lies in just extending the measurements to 60 R⊙. Additional
heights do not improve the ToA or can even make it worse (e.g., CME 4).
However, this improvement does not occur for the events with the best ToAs in
Fit 5. In the case of CME 7, the additional HT measurements decrease the ToA
accuracy threefold. If we ignore CMEs 4 and 8 for the moment, we see that the
addition of higher HT points tends to result in later arrival times; namely,
it gives slower velocities at the final point used for the quadratic fit. This
is another indication that CMEs decelerate above 50 R⊙. However, the lower
velocity bias strongly affects the events that have already undergone the
majority of their deceleration (events with $\Delta T>0$, CMEs 3, 5, 7). We do
not have an obvious explanation for this at the moment. Larger sample studies
are needed.
However, we can reach a couple of interesting conclusions from this exercise:
(1) ToA predictions can be improved considerably with a few HT measurements in
the HI FOV ($>50R_{\odot}$) especially for events without strong deceleration
within the LASCO FOV range. (2) ToA predictions for strongly decelerating
events may be better if based on HT measurements below $50R_{\odot}$. (3)
There does not seem to be a “standard” distance range where CMEs undergo most
of their deceleration, as may be suggested by the multi-polynomial plots in
Wood et al. (2009a), for example. CMEs 3, 5, 7 seem to have decelerated by
50$R_{\odot}$ and to have picked up speed after this height; CMEs 1, 6, 9 seem
to decelerate in the $50-60R_{\odot}$ range while CME 2 or 5 seem to propagate
more or less with a constant speed.
## 7\. Discussion
In this paper, we investigate methods for predicting the ToA of Earth-
impacting CMEs based on de-projected HT measurements from multi-viewpoint
coronagraphic and heliospheric images. From the comparison of six methods, we
conclude that a simple linear fit of the HT measurements above 50 R⊙ can
significantly reduce the ToA error. The predicted ToA from the linear fit (Fit
1) is within $\pm$6 hours of the arrival of the density front at the _Wind_
spacecraft for 78% of CMEs. This result is a 9 hour improvement over the
results of Gopalswamy et al. (2001) that reports an accuracy of $\pm$15 hour
for 72% of CMEs studied. If we include all events in our study, we can predict
the arrival of CMEs at Earth with $\pm$13 hours which is almost a half day
improvement over the $\pm$24 hour window with a 95% error margin previously
reported in Schwenn et al. (2005). Our results are also an improvement over
the Fixed-$\phi$ and harmonic mean methods, $\pm$33 and $\pm$20 hours,
respectively, which use heliospheric data without taking advantage of the two
_STEREO_ viewpoints (Lugaz et al., 2012). Even our worst case results are a
significant improvement in predicting CME ToA. Therefore, deprojected HT
measurements using images of the CME front obtained from outside the Sun-Earth
line can improve the accuracy of the ToA prediction of Earth-impacting CMEs by
a half day compared to single-view coronagraphic observations obtained along
the Sun-Earth line.
The CMEs with the poorest ToA results (2 and 4) are peculiar. They may violate
the self-similar expansion assumption used to fit the GCS model to the HI-1
and HI-2 images. Nieves-Chinchilla et al. (2012) found that CME 4 is rotating
between 0.5 AU and 1 AU and that its appearance is subject to considerable
projection effects. The CME 2 orientation is ambiguous despite being the
subject of several studies. Wood et al. (2011), for example, found that the
cross section of the CME is significantly elliptical irrespective of the
actual orientation. An elliptical cross-section may indicate that the
expansion of the CME was not self-similar; rotation is also likely. In any
case, the heliospheric HT measurements for this CME are suspect as it is the
only event with an improved ToA from Fit 5. Given the small sample of CMEs,
and the even smaller number of discrepancies, we cannot reach a firm
conclusion on whether CME rotation or other projection effects may be
responsible for the poor ToA predictions.
We are not aware of any previous studies of the CME ToA that report the
predicted velocity at 1 AU as well. We think that this is a serious omission,
since a reliable prediction of the CME velocity at Earth can, in turn, provide
reliable estimates of the CME ram pressure and hence help predict one more
geo-effective parameter. We also use the predicted velocity as a diagnostic of
our fit methods. Since the distance traveled by the CMEs is fixed, we would
assume a correlation with $\Delta$T and $\Delta$V. In other words, if the
fitted velocity is too fast, we would expect the CME to arrive early and vice
versa. In Figure 10, we have plotted $\Delta$T versus $\Delta$V where the
results are plotted using the CME number and the color scheme from section 4.
It is clear that while $\Delta$T is evenly distributed around zero (with the
exception of Fit 5), $\Delta$V is largely positive. More precisely,
$\overline{\Delta T}=1.1$ hours and $\overline{\Delta V}$ = 53 $kms^{-2}$.
There is no obvious trend or correlation, between $\Delta$T and $\Delta$V,
within a particular fit or among the fitting methods with the exception of Fit
5. For Fit 5, the faster velocities are somewhat correlated with early ToA, as
one would expect. Fit 6 has the smallest velocity error but it has the three
largest ToA errors. The geometric correction of Fit 6 systematically decreases
the predicted velocity, as expected, but it does not increase the ToA
accuracy. We conclude that _a linear fit to the HT measurements above 50
$R_{\odot}$ is sufficient for predicting the ToA but fails to capture the true
kinematics of the CME. _
This result would seem to suggest that the CMEs are traveling at a constant
speed between 50 $R_{\odot}$ and 1 AU. However, closer analysis of our of
results does not support this claim. First, if the CME reached a constant
velocity by 50$R_{\odot}$, we would expect the results from Fit 5 (LASCO FOV
only) to be as accurate as Fit 1. But Fit 5 results in early arrivals which
implies that the velocity derived at 50$R_{\odot}$ with the quadratic fit is
too high and hence the velocity of the CME must decrease after 50 $R_{\odot}$.
This deceleration, however, must occur very gradually otherwise Fit 2
(quadratic) would perform better than Fit 1 (linear). It is well known that
the velocities measured in the LASCO FOV have a broader range compared the
velocities at Earth which converge around the average solar wind speed
(Gopalswamy et al., 2000). However, it is not known at what heights CMEs reach
a constant velocity. We assert a CME should reach a constant velocity only
after its velocity matches the ambient solar wind velocity. If there is a
difference in the velocity of the CME and the ambient solar wind, the CME will
be effected by a drag force (Vršnak and Gopalswamy, 2002). Six of our CMEs (2,
3, 4, 6, 7, 8, 9) exhibit an abrupt increase in the _in situ_ velocity
coincident with the density front. Thus, they are still traveling faster than
the solar wind and are still decelerating. For the two CMEs that are traveling
with the solar wind velocity (1 and 5), the $\Delta$V from Fit 5 is 308
$kms^{-2}$ and 390 $kms^{-2}$, respectively. Therefore, these CME decelerated
sometime after 50$R_{\odot}$ but before reaching 1 AU and did so smoothly
since the linear fit gives the best ToA for those events. We conclude that
_all CMEs in our sample are decelerated between 50 $R_{\odot}$ and 1 AU._
Figure 10: The error in the arrival time, $\Delta$T versus the error in the 1
AU velocity, $\Delta$V. The color scheme is the same as in Figure 8.
Counterintuitively, there is no obvious correlation between the two variables,
with the exception of Fit 5 (orange). There is a slight tendency to
overestimate the arrival velocity by about 50 $kms^{-1}$.
We assume that the primary cause of the CME deceleration is the drag force due
to the interaction with the ambient solar wind. While the drag force could
also accelerate a CME between between 50$R_{\odot}$ and 1 AU, we did not
measure such a CME in our sample. While the drag force varies as
$|V_{CME}-V_{SW}|^{2}$, the effect on the velocity would not be quadratic. The
drag force is degenerate; as the velocity of the CME decreases so does the
drag force. Thus the deceleration due to the drag would be very gradual and
occur smoothly as we see in our HT measurements. Also the transition of the
CME into equilibrium with the solar wind would also occur smoothly. This would
explain why the HT profiles are not well fit by Fit 2 (quadratic) and Fit 3
(multiple polynomials) but are better represented by Fit 1 (linear). We
believe that Fit 4 (rigid spline) failed because there is too much error in
the HT measurements.
But we have to reconcile our two conclusions: (1) A linear fit to the HT data
is the best method for predicting the ToA; (2) All measured CME are
decelerating. The obvious suggestion is that the linear fit provides the mean
velocity of the gradually decelerating CME front between 50 $R_{\odot}$ and 1
AU. This also explains the systematic overestimation of the CME velocity with
Fit 1. The mean velocity of a gradually decelerating function will always be
higher than the final velocity. Thus we somewhat alter our original
conclusion. _The mean velocity of a CME between 50 $R_{\odot}$ and 1 AU is the
best parameter for predicting the ToA._ The linear fit is a simple method for
calculating the average velocity from the HT data.
We compare the results from Fit 1 and 6 since the HT measurements were fitted
in the same way. For Fit 6 we used the height of the GCS model along the Sun-
Earth line as opposed to the apex height in Fit 1 (see Figure 6). The
corrected height is less than the apex height depending on the width and
longitude of the GCS model. We find that the apex height is a better predictor
of the CME arrival. We interpret this result as evidence for flattening of the
CME fronts during Earth propagation. The flattening of the CME front has been
theorized in the past (Riley and Crooker, 2004, and references therein) and
seems to occur in the HI-1 and HI-2 images perpendicular to the ecliptic (but
see discussion in Nieves-Chinchilla et al. (2012)). In the heliographic
images, we do not have reliable information about the extent or curvature of
the CME in the ecliptic plane. However, if the curvature of the CME was the
dominant factor in the CME ToA error, we would expect the results of Fit 1 to
be systematically early ($-\Delta$T). We do not see this. Only CMEs 1, 3, 6,
and 8 have early predicted ToA and, therefore, could benefit from the
correction. However, all four "corrected" arrivals result in much later ToA,
i.e., they are overcorrected. Thus we have to assume that the front of these
CMEs is not as curved the GCS model predicts. Therefore, we have indications
of flattening of the CME front in the ecliptic beyond 50 R⊙, for some events.
Further investigations on the role of projection effects (e.g, Nieves-
Chinchilla et al., 2012) and on the proper identification of the CME
substructures (e.g., Vourlidas et al., 2013) is needed.
## 8\. Conclusions
With Fits 1 to 4, we add complexity with each fit by increasing the number of
free parameters in an attempt to capture the kinematics of the CME in the
heliosphere. We assume that the increased number of free parameters would
result in better fits to the HT measurements and that the ToA and velocity
prediction would correspondingly improve. Surprisingly, Fit 1 while having the
fewest free parameters, gives the best results. We find that the best results
are obtained by ignoring complex fitting functions to the full data range,
even discarding the coronal observations, and fitting a simple straight line
to the HT measurements above 50 R⊙ only. We show that measurements close to
the Sun, as those provided by coronagraphs, are not sufficiently robust for
ToA predictions even if those HT measurements are deprojected somehow.
Furthermore, we find that being able to follow a CME front all the way to
Earth (e.g., CMEs 2 and 8 but see CME 9 for a counterexample) does not
actually improve the ToA. Correcting for the CME curvature does not improve
the ToA. Imaging observations integrate along a long LOS, which becomes longer
with increasing heliocentric distance. Therefore, the location of a CME
feature can be subject to considerable uncertainty, including a bias towards
the location of the Thompson sphere (Vourlidas and Howard, 2006), if the CME
is undergoing rotation or other interaction with the ambient environment. Such
evolution is likely to affect the derived CME longitude and its curvature.
These results have important implications for Space Weather and CME
propagation studies:
1. 1.
A simple linear fit to deprojected HT measurements of the CME front only above
50 $R_{\odot}$ is sufficient to predict the ToA within $\pm 6$ hours (for 7/9
events) and the 1 AU velocity within $\pm$ 140 $kms^{-1}$.
2. 2.
Deprojected HT measurements of CMEs made using imaging from outside the Sun-
Earth line can improve the Earth ToA prediction of CMEs by a half day compared
to single-view coronagraphic observations along the Sun-Earth line.
3. 3.
CMEs decelerate slowly and smoothly between 50$R_{\odot}$ and 1 AU.
4. 4.
HT measurements within coronagraphs FOVs (30 $R_{\odot}$) even if they are
deprojected, are insufficient for accurate Earth ToA or CME velocity
predictions.
5. 5.
Despite the improvements in CME size and direction, achieved using _STEREO_
data, there remain several open issues in the interpretation of the images
such as the precise localization of the Earth-impacting part of the CME.
#### Acknowledgments
R.C and A.V are supported by NASA contract S-136361-Y to the Naval Research
Laboratory. C.W. is supported by Navy ONR 6.1 program. The SECCHI data are
produced by an international consortium of the NRL, LMSAL and NASA GSFC (USA),
RAL and Univ. Bham (UK), MPS (Germany), CSL (Belgium), IOTA and IAS (France).
LASCO was constructed by a consortium of institutions: NRL, MPIA (Germany),
LAM (France), and Univ. of Birmingham (UK). The LASCO CME catalog is generated
and maintained at the CDAW Data Center by NASA and The Catholic University of
America in cooperation with the Naval Research Laboratory.
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|
arxiv-papers
| 2013-10-24T17:34:50 |
2024-09-04T02:49:52.826514
|
{
"license": "Public Domain",
"authors": "R. C. Colaninno, A. Vourlidas, C.-C. Wu",
"submitter": "Robin Colaninno",
"url": "https://arxiv.org/abs/1310.6680"
}
|
1310.6684
|
definition definition
††thanks: Research is supported in part by the grant 159240 of the Swiss
National Science Foundation as well as by the National Center of Competence in
Research SwissMAP of the Swiss National Science Foundation.
# TROPICAL APPROACH TO NAGATA’S CONJECTURE IN POSITIVE CHARACTERISTIC
Nikita Kalinin Université de Genève, Mathématiques, Villa Battelle, 1227
Carouge, Suisse St. Petersburg Department of the Steklov Mathematical
Institute, Russian Academy of Sciences, Fontanka 27, St. Petersburg, 191023
Russia. [email protected]
(August 27, 2024)
###### Abstract
Suppose that there exists a hypersurface with the Newton polytope $\Delta$,
which passes through a given set of subvarieties. Tropical geometry provides a
tool for visualising the subsets of $\Delta$, “influenced” by these
subvarieties. We prove that a weighted sum of the volumes of these subsets
estimates the volume of $\Delta$ from below.
As an particular application of this method we consider a planar algebraic
curve $C$ which passes through generic points $p_{1},\dots,p_{n}$ with
prescribed multiplicities $m_{1},\dots,m_{n}$. Suppose that the minimal
lattice width $\omega(\Delta)$ of the Newton polygon $\Delta$ of $C$ is at
least $\max(m_{i})$. Using tropical floor diagrams (i.e. degeneration of
$p_{1},\dots,p_{n}$ on a line) we prove that
$\mathrm{area}(\Delta)\geq\frac{1}{2}\sum_{i=1}^{n}m_{i}^{2}-\frac{1}{2}\max(\sum_{i=1}^{n}s_{i}^{2})$
where $s_{i}\leq m_{i},\sum_{i=1}^{n}s_{i}\leq\omega(\Delta)$.
In the case $m_{1}=m_{2}=\dots=m\leq\omega(\Delta)$ this estimate becomes
$\mathrm{area}(\Delta)\geq\frac{1}{2}(n-\frac{\omega(\Delta)}{m}-1)m^{2}$.
That gives $d\geq(\sqrt{n}-\frac{1}{2}-\frac{1}{\sqrt{n}})m$ for the curves of
degree $d$, if $n\geq 4$.
It is not clear what is a collection of generic points in the case of a finite
field. We construct such collections for fields big enough, what may be also
interesting for code theory.
## 0 Introduction
It is simple to find a polynomial in one variable with prescribed values at
given points. A bit more involved is to find a polynomial in many variables
with prescribed values at given points or to find a polynomial in one variable
with prescribed higher derivatives at given points. Each of the conditions
appeared above imposes one linear constraint on the polynomial’s coefficients.
Therefore the only difficulty is to prove the linear independence of these
constraints.
One can generalize this question: given natural numbers
$m_{1},m_{2},\dots,m_{n}$ and a set of varieties
$X_{1},X_{2},\dots,X_{n}\subset{\mathbb{F}}^{k}$ (where ${\mathbb{F}}$ is an
infinite field of any characteristic), we are wondering if there exists a
hypersurface $Y\subset{\mathbb{F}}^{k}$ (with a given Newton polytope
$\Delta$) which passes through each of $X_{i}$ with multiplicity
$m_{i}\in{\mathbb{N}}$ respectively. That is not just arbitrary chosen
problem: once discussed smooth varieties we inevitably fall into the realm of
singular varieties, where a rather important area concerns constructing
explicit examples. A particular way to pick a variety is to prescribe it by
the above incidence relations.
This paper promotes the tropical viewpoint on singularities. We define the
subsets $\mathfrak{Infl}(X_{i})$ of $\Delta$, “influenced” by each of $X_{i}$.
These subsets can overlap but no more than $k$ at once. Consider the case
$k=\dim Y+1=2$, i.e. $Y$ is an algebraic curve and each of $X_{i}$ is a point.
###### Definition 0.1
The lattice width $\omega_{u}(\Delta)$ of a polygon
$\Delta\subset{\mathbb{Z}}^{2}$ in a direction $u\in P({\mathbb{Z}}^{2})$ is
$\max\limits_{x,y,\in\Delta}(u_{1},u_{2})\cdot(x-y),$ where
$(u_{1},u_{2})\in{\mathbb{Z}}^{2}$ is any representative of the direction $u$.
###### Definition 0.2
The minimal lattice width $\omega(\Delta)$ of a polygon
$\Delta\subset{\mathbb{Z}}^{2}$ is $\min\limits_{u\in
P({\mathbb{Z}}^{2})}\omega_{u}(\Delta).$
If $X_{i}$ is a point of multiplicity $m_{i}$ for $Y$ and $\omega(\Delta)\geq
m_{i}$, then the estimate
$\mathrm{area}(\mathfrak{Infl}(X_{i}))\geq\frac{m_{i}^{2}}{2}$ holds [15].
This gives an estimate (Lemma 2.1) for the area of $\Delta$ in terms of
$m_{i}$:
$\mathrm{area}(\Delta)\geq\frac{1}{4}\sum_{i=1}^{n}m_{i}^{2}.$
###### Theorem 1
If $\omega(D)\geq\max(m_{i})$ and for each set of points
$p_{1},p_{2},\dots,p_{n}\in{\mathbb{F}}^{2}$ there is an algebraic curve
$C\subset{\mathbb{F}}^{2}$ with the Newton polygon $\Delta$, passing through
$p_{1},p_{2},\dots,p_{n}$ with multiplicities $m_{1},m_{2},\dots,m_{n}$
correspondingly, then
$\mathrm{area}(\Delta)\geq\frac{1}{2}\sum_{i=1}^{n}m_{i}^{2}-\frac{1}{2}\max(\sum_{i=1}^{n}s_{i}^{2})$
where we maximize by all sets of numbers $\\{s_{i}\\}_{i=1}^{n}$ with
$s_{i}\leq m_{i},\sum_{i=1}^{n}s_{i}\leq\omega(\Delta)$.
Let ${\mathbb{K}}$ be the field ${\mathbb{F}}\\{\\{t\\}\\}$ of Puiseux series.
That means, ${\mathbb{F}}\\{\\{t\\}\\}=\\{\sum\limits_{\alpha\in
I}c_{\alpha}t^{\alpha}|c_{\alpha}\in({\mathbb{F}}^{*}),I\subset{\mathbb{Q}}\\}$,
where $t$ is a formal variable and $I$ is a well-ordered set (each its
nonempty subset has a least element). Define a valuation map
$\mathrm{val}:{\mathbb{K}}\to{\mathbb{T}}$ by the rule
$\mathrm{val}(\sum_{\alpha\in
I}c_{\alpha}t^{\alpha}):=-\min\\{\alpha|\alpha\in I,c_{\alpha}\neq 0\\}$ and
$\mathrm{val}(0):=-\infty$. Different versions of Puiseux series are listed in
[18, 22].
We will prove that the above theorem holds over the valuation field
${\mathbb{K}}$. We use the nature of a singular point’s influence on the
Newton polygon of a curve [15] and tropical floor diagrams [6, 7]. Tropical
floor diagrams illustrate the process of a degeneration of the points
$p_{1},\dots,p_{n}$ on a line, in a sense it is a tropical version of the
Horace method [12]. The idea of the proof is the following. While degenerating
$p_{1},p_{2},\dots,p_{n}$ onto a line, on the tropical picture we see the
following behavior of the points (Figure 2). Each point of the multiplicity
$m_{i}$ splits into two parts $m_{i}=s_{i}+r_{i}$, such that
$\sum_{i=1}^{n}s_{i}\leq\omega(\Delta)$. Furthermore, we choose a part of
$\mathfrak{Infl}(p_{i})$ for each $i=1,\dots,n$ , these parts do not intersect
and the area of such a part for a point $p_{i}$ is at least
$\frac{1}{2}(m_{i}^{2}-s_{i}^{2})$.
Then, using a substitution $t\to a\in{\mathbb{F}},{\mathbb{K}}\to{\mathbb{F}}$
we prove (Detropicalization lemma) that there is a constant $N\in{\mathbb{N}}$
such that if the cardinality of ${\mathbb{F}}$ is at least $N$ (which is
always the case if ${\mathbb{F}}$ is infinite), then Theorem 1 holds for
${\mathbb{F}}$. In small fields we can not find a sufficiently generic
collection of points. The constant $N$, then, depends on $\max(m_{i}),\Delta$
and $char({\mathbb{F}})$. This reasoning could be of a particular interest to
code theory, see Section 4.
###### Corollary 0.3
Suppose that $m_{1}=m_{2}=\dots=m_{n}=m\leq\omega(\Delta)$. Therefore, under
the conditions of Theorem 1 we have
$\mathrm{area}(\Delta)\geq\frac{1}{2}(n-\frac{\omega(\Delta)}{m}-1)m^{2}$.
definition
###### Proof 0.1
Seeking for the minimum of $\sum_{i=1}^{n}(m^{2}-s_{i}^{2})$ under conditions
$\sum s_{i}=\omega(\Delta),s_{i}\leq m$ we see that the minimum is attained
when
$s_{i}=m,\text{if }1\leq i\leq k,\text{and }0\leq s_{k+1}<m,\text{and
}s_{>k+1}=0.$
In our case, write $\omega(\Delta)=mk+k^{\prime},0\leq k^{\prime}<m$. Then,
$\sum_{i=1}^{n}(m_{i}^{2}-s_{i}^{2})\geq(n-k-1)m^{2}+(m-k^{\prime})^{2}$.
Therefore,
$\mathrm{area}(\Delta)\geq\frac{1}{2}((n-k-1)m^{2}+(m-k^{\prime})^{2})\geq\frac{1}{2}(n-\omega(\Delta)/m-1)m^{2}.$
###### Corollary 4
For the curves of degree $d$, the above corollary gives
$d\geq(\sqrt{n}-\frac{1}{2}-\frac{1}{\sqrt{n}})m$ if $n\geq 4$.
###### Proof 0.2
Indeed, the Newton polygon of such a curve is the triangle
$\mathrm{ConvHull}(\\{(0,0),(d,0),(0,d)\\})$
and its area is $\frac{d^{2}}{2}$. So, we have $d^{2}\geq((n-d/m-1)m^{2}$. If
$d\geq m\sqrt{n}$, then we are done. Suppose that $d<m\sqrt{n}$, then
$d^{2}\geq(n-d/m-1)m^{2}\geq(n-\sqrt{n}-1)m^{2}\geq(\sqrt{n}-\frac{1}{2}-\frac{1}{\sqrt{n}})^{2}m^{2}$
if $n\geq 4$.
### 0.1 Nagata’s conjecture.
Let us fix a field ${\mathbb{F}}$. For a point
$p=(p_{1},p_{2})\in{\mathbb{F}}^{2}$ we denote by $I_{p}$ the ideal of the
point $p$, namely $I_{p}=\langle x-p_{1},y-p_{2}\rangle$.
###### Definition 5
Consider an algebraic curve $C$ given by an equation
$F(x,y)=0,F\in{\mathbb{F}}[x,y]$. We say that $p$ is of multiplicity at least
$m$ for $C$ ($\mu_{p}(C)\geq m$), if $F\in(I_{p})^{m}$ in the local ring of
$p$.
In the most non-degenerate case $p$ being a point of multiplicity $m$ on $C$
means that there are at least $m$ branches of $C$ passing through $p$. For the
fields of zero characteristic $F\in(I_{p})^{m}$ is equivalent to the fact that
all the partial derivatives of $F$ up to order $m-1$ vanish at $p$.
###### Example 6
Consider an affine algebraic curve $C$ of degree $d$ given by an equation
$F(x,y)=0$, where
$F(x,y)=\sum\limits_{i,j\geq 0,i+j\leq d}a_{ij}x^{i}y^{j}$
The point $p=(0,0)$ is of multiplicity $m$ for $C$ if and only if for all
$i,j\geq 0$ with $i+j<m$ we have $a_{ij}=0$. As a consequence, for each point
$p\in{\mathbb{F}}^{2}$ the condition “$p$ is a point of multiplicity at least
$m$ for $C$” can be rewritten as a system of $\frac{m(m+1)}{2}$ linear
equations in the coefficients $\\{a_{ij}\\}$ of $F$.
Let $p_{1},\dots,p_{n}$ be a collection of $n>9$ points in ${\mathbb{F}}^{2}$
and $m_{1},\dots,m_{n}\in{\mathbb{N}}$. We are looking for the minimal degree
$d_{min}$ of an algebraic curve passing through $p_{1},\dots,p_{n}$ with
multiplicities at least $m_{1},\dots,m_{n}$ respectively.
One can naively calculate the expected dimension
$\mathfrak{edim}(d,m_{1},\dots,m_{n})$ of the space $\mathfrak{S}$ of the
curves of degree $d$ satisfying the hypothesis above: each singular point
freezes $\frac{m(m+1)}{2}$ degrees of freedom, i.e. imposes $\frac{m(m+1)}{2}$
constraints on the coefficients of the curve equation. Therefore,
$\mathfrak{edim}(d,m_{1},\dots,m_{n})=\max\left(-1,\frac{d(d+3)}{2}-\sum\limits_{i=1}^{n}\frac{m_{i}(m_{i}+1)}{2}\right).$
The actual dimension of $\mathfrak{S}$ is always at least the expected one,
because all the constraints are linear. However, sometimes even for a
“generic” choice of a set of points the actual dimension is strictly greater
than the expected.
###### Example 7
Let us consider two points $p_{1},p_{2}$. The minimal degree of a curve
passing through $p_{1},p_{2}$ with multiplicities $m_{1},m_{2}$ is $m_{1}$, if
$m_{1}\geq m_{2}$: it is the line passing through $p_{1}$ and $p_{2}$ taken
with multiplicity $m_{1}$. So the inequality
$d_{min}\geq\frac{m_{1}+m_{2}}{\sqrt{2}}$ in the Nagata’s conjecture is not
satisfied if $m_{2}>m_{1}(\sqrt{2}-1)$. We see a similar situation for five
points: one can draw a non-reduced conic through them.
As a reasonable estimate for $d_{min}$, Nagata’s conjecture claims:
###### Conjecture 8
If $d\leq\frac{\sum\limits_{i=1}^{n}m_{i}}{\sqrt{n}}$ and points
$p_{1},\dots,p_{n},n>9$ are chosen generically then $\dim\mathfrak{S}=-1$. In
other words, $d_{min}>\frac{\sum\limits_{i=1}^{n}m_{i}}{\sqrt{n}}$.
The case $n=l^{2}$ had been proven by Nagata himself [20]. Now, even the case
$n=10$ and $m_{1}=m_{2}=\dots=m_{10}=m$ is under exhaustive study [10] but has
not yet been proven. The similar questions in higher dimensions are widely
open (cf. [3],[11]). The pictures appeared in our approach are somewhat
similar to those in [21], though the relation is not direct.
Historically Nagata’s conjecture appeared as a tool (with $n=16$) to disprove
Hilbert 14th problem. There also exists Segre-Harbourne-Hirschowitz conjecture
which basically says that if the expected dimension $\mathfrak{edim}$ of
$\mathfrak{S}$ is not equal to the actual one, then the linear system
$\mathfrak{S}$ contains a rational curve in its base locus. The reader is
kindly referred to look into surveys [8, 9, 14, 19] for an introduction to
Nagata’s conjecture and related topics.
In view of Theorem 1 the following three results should be mentioned:
Theorem ([27], Xu). If $C$ is a reduced and irreducible curve passing through
generically chosen points $p_{1},p_{2},\dots,p_{n}\in{\mathbb{C}}P^{2}$ with
multiplicities $m_{1},m_{2},\dots,m_{n}$ respectively, then the estimate
$d^{2}\geq\sum_{i=1..n}m_{i}^{2}-\min(m_{i})$ holds.
Unlike Xu’s theorem we consider arbitrary Newton polygons and fields of any
characteristic. Furthermore, our curves are allowed to be reducible and non-
reduced.
Theorem ([1], Alexander, Hirschowitz). The dimension of the space of degree
$d>2$ hypersurfaces in ${\mathbb{C}}P^{k},k\geq 3$ passing through generic
points $p_{1},p_{2},\dots,p_{n}$ with multiplicities $m_{1}=\dots=m_{n}=2$ is
the expected one except the cases $(k,d,n)=(2,4,5),(3,4,9),(4,4,14),(4,3,7)$.
Using the methods of this article and classification in [17], we can prove
that the volume $V$ of the Newton polytope of a surface in ${\mathbb{C}}P^{3}$
with $n$ $2$-fold points in general position satisfies $n\leq 2V$. Using the
above theorem we can obtain a better estimate. Indeed, for the case of
hypersurfaces of degree $d$ in ${\mathbb{C}}P^{3}$ the above theorem gives
$4n\leq(d+1)(d+2)(d+3)/6$, i.e. $n\sim V/4$.
Theorem ([2], Alexander, Hirschowitz). For each field ${\mathbb{F}}$, the
dimension of degree $d$ hypersurfaces in ${\mathbb{F}}P^{k}$ passing through
generic points $p_{1},p_{2},\dots,p_{n}$ with multiplicities
$m_{1},m_{2},\dots,m_{n}$ is the expected one if $d\gg\max m_{i}$.
We expect that our approach can be extended to the cases $k\geq 3$ and
$m_{i}>2$. Such an extension would lead to explicit degree estimates.
## 1 Tropical geometry
In this section we recall some definitions and set up the notation. We discuss
the notion of a set of points in ${\mathbb{Z}}^{k}$ in tropical general
position with respect to a polytope $\Delta$. We use this construction in the
following sections. We refer the reader to [5],[16] for a general introduction
to tropical geometry.
Let ${\mathbb{T}}$ denote ${\mathbb{Q}}\cup\\{-\infty\\}$, and ${\mathbb{K}}$
be a field with a valuation map $\mathrm{val}:{\mathbb{K}}\to{\mathbb{T}}$. We
use the convention
$\mathrm{val}(a+b)\leq\mathrm{val}(a)+\mathrm{val}(b),\mathrm{val}(0)=-\infty$.
Usually ${\mathbb{T}}$ is called tropical semi-ring.
Consider a hypersurface $Y\subset{\mathbb{K}}^{k}$. Let $Y$ be given by an
equation
$F(x_{1},x_{2},\dots,x_{k})=0$
where $F=\sum_{I\in{\mathcal{A}}}c_{I}x^{I}$,
$I=(i_{1},i_{2},\dots,i_{k}),c_{I}\neq 0$. In such a case
$\Delta=\mathrm{ConvexHull}({\mathcal{A}})$ is called the Newton polytope of
$Y$.
The Newton polytope of $F$ is provided with a subdivision defined by $F$.
Indeed, consider the extended Newton polytope of $Y$,
$\widetilde{\Delta}=\mathrm{ConvexHull}\\{(I,x)\in{\mathbb{Z}}^{k}\times{\mathbb{T}}|I\in{\mathcal{A}},x\leq\mathrm{val}(c_{I}))\\}.$
Projection of the faces of the extended Newton polytope $\widetilde{\Delta}$
onto the Newton polytope $\Delta$ defines a subdivision of $\Delta$.
We give a definition of the tropicalization of $Y$, based on its equation
$F(x)=\sum_{I\in{\mathcal{A}}}c_{I}x^{I}$. For a weight
$\omega=(w_{1},w_{2},\dots,w_{k})\in{\mathbb{T}}^{k}$ we consider the weight
function $\omega(cx_{1}^{i_{1}}x_{2}^{i_{2}}\dots
x_{k}^{i_{k}}):=\mathrm{val}(c)+i_{1}w_{1}+i_{2}w_{2}+\dots+i_{k}w_{k}$. Then
we define initial part $\mathrm{in}_{\omega}(F)$ as the $\omega$-maximal part
of $F$. Now we define $\mathrm{Trop}(Y)$ to be the set of all weights $\omega$
such that $\mathrm{in}_{\omega}(F)$ is not a monomial.
We can describe the subdivision of $\Delta$: a point $I\in\Delta$ is a vertex
of the subdivision if there is such a weight $\omega\in{\mathbb{T}}^{k}$ that
$\mathrm{in}_{\omega}(F)=c_{I}x^{I}$. An interval $I_{1}I_{2}$ between two
vertices $I_{1},I_{2}\in\Delta$ is an edge of the subdivision if there is a
weight $\omega$ such that $\mathrm{in}_{\omega}(F)=\sum_{I\in J}c_{I}x^{I}$
where the convex hull of $J$ is the interval $I_{1}I_{2}$, etc. In general,
each cell of the subdivision of $\Delta$ is of the type
$\Delta_{\omega}=\mathrm{ConvexHull(support(}\mathrm{in}_{\omega}(F)))$ for
some $\omega\in{\mathbb{T}}^{k}$.
###### Remark 1
If $Y$ is a hypersurface, then $\mathrm{Trop}(Y)\subset{\mathbb{T}}^{k}$ is a
polyhedral complex of codimension one. For each cell
$\Delta_{\omega}\subset\Delta$ we define
$d(\Delta_{\omega})=\\{\omega^{\prime}\in{\mathbb{T}}^{l}|\Delta_{\omega}=\Delta_{\omega^{\prime}}\\}$.
This map $d$ provides the following correspondence: the vertices of the
subdivision of $\Delta$ correspond to the connected components of the
complement of $\mathrm{Trop}(Y)$, the edges of the subdivision correspond to
the faces of $\mathrm{Trop}(Y)$ of maximal codimension, 2-cells of the
subdivision correspond to faces of codimension 1 in $\mathrm{Trop}(Y)$, etc.
###### Remark 2
If $X\subset{\mathbb{K}}^{n}$ is a variety of higher codimension, we define
its tropicalization $\mathrm{Trop}(X)$ as follows. Let $I$ be the ideal of
$X$. Let $\mathrm{in}_{\omega}(I)$ be the ideal generated by the elements
$\mathrm{in}_{\omega}(f),f\in I$. Then, by definition,
$\omega\in\mathrm{Trop}(X)$ if and only if $\mathrm{in}_{\omega}(I)$ is
monomial free.
### 1.1 Influenced subsets in the Newton polytope
In this subsection, for a given subvariety $X\subset Y$, we define the set
$\mathfrak{I}(X)$ of vertices of $\mathrm{Trop}(Y)$.
###### Remark 3
The set $\mathfrak{I}(X)$ depends only on $\mathrm{Trop}(X)$, so we will write
$\mathfrak{I}(\mathrm{Trop}(X))$.
The distinguished domain in $\Delta$, corresponding to $X$, is
$\mathfrak{Infl}(X)=\bigcup_{V\in\mathfrak{I}(\mathrm{Trop}(X))}d(V),$
where $d(V)$ is the cell (of the maximal dimension) of $\Delta$, dual to the
vertex $V$ of $\mathrm{Trop}(Y)$. These definitions generalize definitions
given in [15].
Let $Q$ be some polyhedral (i.e. defined by a set of linear inequalities)
subset of ${\mathbb{T}}^{k}$.
###### Definition 4
We denote by $P({\mathbb{Z}}^{k})$ the set of all directions in
${\mathbb{Z}}^{k}$. Let $l_{Q}(u)$ be the hyperplane with the normal direction
$u\in P({\mathbb{Z}}^{k})$, passing through $Q$, if exists, and
$l_{Q}(u)=\varnothing$, otherwise.
We call $TC(Q)=\bigcup_{u\in P({\mathbb{Z}}^{k})}l_{Q}(u)$ the tangent cone at
$Q$.
###### Definition 5
Let $\mathfrak{I}(Q)$ be the set of the vertices of $\mathrm{Trop}(Y)$ in the
connected component of $Q$ in the intersection $\mathrm{Trop}(Y)\cap TC(Q)$.
Let $P(\Delta)\subset P({\mathbb{Z}}^{k})$ be the set of the directions
generated by the vectors $\\{\overline{IJ}|I,J\in\Delta\\}$ between the
lattice points in $\Delta$. Instead of $TC(Q)=\bigcup_{u\in
P({\mathbb{Z}}^{k})}l_{Q}(u)$ we will consider $TC^{\Delta}(Q)=\bigcup_{u\in
P(\Delta)}l_{Q}(u)$. Indeed, $\mathfrak{I}(Q)$ is contained in the connected
component of $Q$ in the intersection $\mathrm{Trop}(Y)\cap TC^{\Delta}(Q)$.
The cone $TC^{\Delta}(Q)$ is naturally stratified on cells, we provide each
point in $TC^{\Delta}(Q)$ with multiplicity corresponding to the codimension
of its stratum. Namely, for a point $V\in TC^{\Delta}(Q)$ we define
$\mathrm{mult}_{Q}(V)$ as the dimension of the linear span of the directions
$u\in P(\Delta)$ such that the hyperplane through $V$ with the normal
direction $u$ contains $Q$.
###### Example 6
If $\Delta\subset{\mathbb{Z}}^{2}$ and $Q$ is a point, then $TC^{\Delta}(Q)$
is a union or rays emanating from $Q$. In this case $\mathrm{mult}_{Q}(Q)=2$
and $\mathrm{mult}_{Q}(V)=1$ for $V\in TC^{\Delta}(Q),V\neq Q$.
Each tropical variety $\mathrm{Trop}(X)$ is naturally decomposed into
vertices, edges, faces, etc, $\mathrm{Trop}(X)=\bigcup X^{p,q}$ where $p$ is
the dimension of the cell $X^{p,q}$ and $q$ is its number. Each cell is an
equivalence class of some $\omega\in\mathrm{Trop}(X)$, with the equivalence
relation $\omega\sim\omega^{\prime}$ iff
$\Delta_{\omega}=\Delta_{\omega^{\prime}}$.
###### Definition 7
Define $\mathfrak{I}(\mathrm{Trop}(X))=\bigcup\mathfrak{I}(X^{p,q})$. Also,
define
$TC^{\Delta}(\mathrm{Trop}(X))=\bigcup TC^{\Delta}(X^{p,q}).$
For a vertex $V\in\mathfrak{I}(\mathrm{Trop}(X))$ we define its multiplicity
$\mathrm{mult}_{\mathrm{Trop}(X)}(V)$ as
$\max_{X^{p,q}}\mathrm{mult}_{X^{p,q}}(V)$, i.e. we take the maximum of the
multiplicities of $V$ with respect to the cells in the natural cell
decomposition of $\mathrm{Trop}(X)$.
###### Definition 8
By $\mathrm{volume}(\mathfrak{Infl}(\mathrm{Trop}(X)))$ we denote the sum of
volumes (with multiplicities) of the cells in the subdivision of $\Delta$,
dual to the vertices in $\mathfrak{I}(\mathrm{Trop}(X))$, i.e.
$\mathrm{volume}(\mathfrak{Infl}(\mathrm{Trop}(X)))=\sum\limits_{V\in\mathfrak{I}(\mathrm{Trop}(X))}\mathrm{mult}_{\mathrm{Trop}(X)}(V)\cdot\mathrm{volume}(d(V)).$
###### Example 9
In the two dimensional case this means that if
$X=(x_{1},x_{2})\in{\mathbb{K}}^{2}$ is a point such that
$\mathrm{Trop}(X)=P=(\mathrm{val}(x_{1}),\mathrm{val}(x_{2}))\in{\mathbb{T}}^{2}$
is a vertex of $\mathrm{Trop}(Y)$, then
$\mathrm{area}(\mathfrak{Infl}(P))=\sum\limits_{\begin{subarray}{c}V\in\mathfrak{I}(P),\\\
V\neq P\end{subarray}}1\cdot\mathrm{area}(d(V))+2\cdot\mathrm{area}(d(P)),$
cf. with the definition of $area(\mathfrak{Infl}(P))$ in [15].
###### Remark 10
The dual object for a hypersurface is its Newton polytope. The dual objects
for the varieties of higher codimension are so-called generalized Newton
polytopes or valuations in the McMullen polytope algebra [4, 23]. In fact,
$\mathfrak{Infl}$ for a variety $Y$ of any codimension can be defined in a
similar way, but it is not clear what is the right substitute for
$\mathrm{volume}(\mathfrak{Infl}(P))$ in this case.
### 1.2 General position of points with respect to the Newton polygon
###### Definition 11
A collection of tropical subvarieties
$Z_{1},Z_{2},\dots,Z_{n}\in{\mathbb{T}}^{k}$ is in general position with
respect to a polytope $\Delta$ if for each collection of indices
$i_{1}<i_{2}<\dots<i_{k+1}$ the intersection $TC^{\Delta}(Z_{i_{1}})\cap
TC^{\Delta}(Z_{i_{2}})\cap\dots\cap TC^{\Delta}(Z_{i_{k+1}})$ is empty.
Let $T_{v}$ be the translation ${\mathbb{T}}^{k}\to{\mathbb{T}}^{k}$ by the
vector $v$.
###### Proposition 12
For a polytope $\Delta$ and given set
$Z_{1},Z_{2},\dots,Z_{n}\in{\mathbb{T}}^{k}$ of tropical varieties there
exists a set of vectors $v_{1},v_{2},\dots,v_{n}\in{\mathbb{Z}}^{k}$ such that
the tropical varieties $T_{v_{i}}(Z_{i})$ are in general position with
$\Delta$.
###### Proof 1.1
Indeed, each tangent cone $TC^{\Delta}(Z_{i})$ consists of a finite union of
hyperplanes. Therefore, we can choose a vector $v_{1}=0$ and
$v_{2}\in{\mathbb{Z}}^{k}$ such that the intersection of each two hyperplanes
$L_{1},L_{2}$ from the collections $TC^{\Delta}(Z_{1})$ and
$TC^{\Delta}(T_{v_{2}}(Z_{2}))$ respectively is a linear subspace of dimension
at most $k-2$. Then we choose a vector $v_{3}\in{\mathbb{Z}}^{k}$ such that
the intersection of each pair of hyperplanes from different collections
$TC^{\Delta}(T_{v_{i}}(Z_{i})),i=1,2,3$ is of dimension at most $k-2$ and the
intersection of a triple of hyperplanes from different collections is of
dimension at most $k-3$, etc.
###### Corollary 13
There exists a constant $N$ depending on $\Delta,n,k$ and the total number of
cells in the natural subdivisions of $Z_{1},Z_{2},\dots,Z_{n}$ such that the
vectors $v_{1},\dots,v_{n}$ can be chosen in such a way that $|v_{i}|\leq N$
for each $i$.
###### Corollary 14
For each $n,k\in\mathbb{N},\Delta$ there exists a set of points
$P_{1},P_{2}\dots,$ $P_{n}\in\mathbb{Z}^{k}\subset{\mathbb{T}}^{k}$ in general
position with respect to $\Delta$.
###### Proof 1.2
We start from $P_{1}=P_{2}=\dots=P_{n}=0\in{\mathbb{Z}}^{n}$. Then we use the
fact that $\mathbb{Z}^{k}$ is not coverable by a finite number of linear
spaces of dimension $k-1$ and proceed as in Proposition 12.
###### Corollary 15
For a generic for $\Delta$ collection of tropical varieties $Z_{1},Z_{2},$
$\dots,Z_{n}\in{\mathbb{T}}^{k}$ the sum
$\sum_{i=1}^{n}\mathrm{volume}(\mathfrak{Infl}(Z_{i}))$ is at most
$k\cdot\mathrm{Volume}(\Delta)$.
###### Proof 1.3
This follows from the definitions of a general position and multiplicities in
the volume of $\mathfrak{Infl}$.
## 2 An estimate of a singular points’ influence of the Newton polygon of a
curve
Let $C$ be a curve over ${\mathbb{K}}$ with the Newton polygon $\Delta$ such
that $\omega(\Delta)\geq m$.
###### Theorem 2 ([15], Lemma 2.8, Theorems 2,3)
Suppose that a point $p=(p_{1},p_{2})\in({\mathbb{K}}^{*})^{2}$ is of
multiplicity $m$ for this curve $C$,
$P=(\mathrm{val}(p_{1}),\mathrm{val}(p_{2}))$. Then
$\mathrm{area}(\mathfrak{Infl}(P))\geq\frac{m^{2}}{2}$.
###### Example 1
Consider a curve $C$ given by the equation $(x-1)^{k}(y-1)^{m-k}=0$, take
$p=(1,1)$. Clearly, $\mu_{p}(C)=m$ but the Newton polygon $\Delta$ of $C$
violates the condition $\omega(\Delta)\geq m$, and the inequality
$\mathrm{area}(\mathfrak{Infl}(\mathrm{val}(p)))=2k(m-k))\geq\frac{m^{2}}{2}$
does not hold except the case $k=m/2$.
Consider now a curve $C$ passing through
$p_{1},p_{2},\dots,p_{n}\in{\mathbb{K}}^{2},n\geq 2$ with multiplicities
$m_{1},m_{2},\dots,m_{n}$ respectively. Suppose that the Newton polygon
$\Delta$ of $C$ has the minimal lattice width $\omega(\Delta)$ at least
$\max(m_{i})$.
###### Lemma 2.1
If the points $\mathrm{val}(p_{i})\in{\mathbb{Z}}^{2},i=1,\dots,n$ are in
general position with respect to $\Delta$ (see Lemma 12 and its corollaries),
then the area of $\Delta$ satisfies the inequality
$\mathrm{area}(\Delta)\geq\frac{1}{4}\sum_{i=1}^{n}m_{i}^{2}$.
###### Proof 2.2
Theorem 2 and Corollary 15 imply that
$\sum_{i=1}^{n}\frac{m_{i}^{2}}{2}\leq\sum\mathrm{area}(\mathfrak{Infl}(P_{i}))\leq
2\cdot\mathrm{area}(\Delta).$
###### Corollary 2
Consider curves of degree $d$, in lieu of fixing the Newton polygon. Then, we
have $d^{2}\geq\frac{1}{2}\sum_{i=1}^{n}m_{i}^{2}$ if $d\geq\max(m_{i})$.
###### Proof 2.3
Indeed, consider any curve under the above conditions. The equation of a curve
of degree $d$ may contain some monomials with zero coefficients. So, if the
minimal lattice width of the Newton polygon of $C$ is at least $\max(m_{i})$,
then we are done. If it is not the case, we apply the following lemma.
###### Lemma 2.4
(Lemma 1.25, [15]) If $\mu_{(1,1)}(C)=m$ and $\omega_{u}({\mathcal{A}})=m-a$
for some $a>0,u\sim(u_{1},u_{2})$, then $C$ contains a rational component
parametrized as $(s^{u_{1}},s^{u_{2}})$.
If $C$ has a rational component of this given type, then $C$ is reducible, and
we can perturb this component. After that this component is no longer of the
type $(as^{k},bs^{l})$, and this perturbation does not change the degree of
the curve.
Let $P$ be a vertex of $\mathrm{Trop}(C)$ and the edge $E$ through $P$ is
horizontal. Suppose that $\omega_{(1,0)}(d(P))=a\leq m$, i.e. $a$ is the
length of the projection of $d(P)$ onto the $x$-axis.
###### Lemma 2.5 ([15], Lemma 2.10, Lemma 5.19)
If $\mu_{p}(C)\geq m$, $P=\mathrm{val}(p)$ is a vertex of $\mathrm{Trop}(C)$,
and $u=(1,0)$, then
$\sum\limits_{V\in\mathfrak{I}_{P}(u),V\neq
P}\mathrm{area}(d(V))\geq\frac{1}{2}(m-a)^{2}.$ (1)
We use this lemma for the horizontal direction $(1,0)$ (in [15] $u\in
P({\mathbb{Z}}^{2})$). In our case $\mathfrak{I}(u)$ is the set of vertices of
$\mathrm{Trop}(C)$, lying in the connected component of $P$ in the
intersection of $\mathrm{Trop}(C)$ with the straight horizontal line through
$P$, see Figure 1.
$a$$\geq m-a$$\geq m-a$$d(P)$$L$$M$$N$$K$ Figure 1: Dual picture to a singular
point $P$ on an edge. Since $\omega_{(1,0)}(d(P))=a$, the lengths of $LM$ and
$NK$ are at least $m-a$. The set $\bigcup d(Q)$ for
$Q\in\mathfrak{I}_{P}((1,0)),Q\neq P$ is colored. The sum of the areas of the
colored faces is at least $\frac{1}{2}(m-a)^{2}$.
###### Remark 3
Using the classification of combinatorial neighborhood of $2$-fold point $P$
of a tropical surface in ${\mathbb{T}}^{3}$ ([17]) we can prove that
$\mathrm{volume}(\mathfrak{Infl}(P))\geq 2$ in such a case. With a few work
that gives an estimate $n\leq\frac{d^{3}}{3}$ for the degree $d$ of a surface
with $n$ $2$-fold points, but the theorem of Alexander and Hirschowitz
provides a better estimate $n\leq\frac{(d+1)(d+2)(d+3)}{24}$.
###### Remark 4
We expect that for a line $L$ of multiplicity $m$ inside a surface of degree
$d$ in ${\mathbb{C}}P^{3}$ the estimate
$\mathrm{volume}(\mathfrak{Infl}(\mathrm{Trop}(L)))\geq cm^{2}d$ holds with
some constant $c$. This would give an estimate for the degree of a surface
with multiple $2$-fold points and $m$-fold lines.
### 2.1 Detropicalization Lemma
An algebraic statement over an algebraically closed field sometimes implies
the same statement over all fields of the same characteristic. Tropical
geometry may help in such a situation, see [25]. This section describes a
particular application of this principle to our estimate.
We use the field ${\mathbb{K}}={\mathbb{F}}\\{\\{t\\}\\}$. Note that each
element $a\in{\mathbb{F}}$ defines a map $\nu_{a}:{\mathbb{K}}\to{\mathbb{F}}$
by means of the substitution $t=a$. However, $\nu_{a}$ is not well-defined on
the whole ${\mathbb{K}}$ but we can compute it on the elements of the type
$\frac{f(t)}{g(t)}$ where $f,g\in{\mathbb{F}}[t]$ and $g(a)\neq 0$.
Let us recall how to tropicalize the problem of curves’ counting. We would
like to count plane complex algebraic curves of given genus and degree, these
curves are required to pass through a number of generic points
$q_{1},q_{2},\dots,q_{l}\in{\mathbb{C}}P^{2}$ ($l$ is chosen in such a way
that the number of curves becomes finite). Since the points are generic we can
force them to go to infinity with some asymptotics, say
$q_{i}=(t^{x_{i}},t^{y_{i}})$. Then we consider the limits of the constructed
curves $C_{t}$ under the function
$\log_{t}(|z|):{\mathbb{C}}^{2}\to{\mathbb{R}}^{2}$. This is more or less the
same as if we considered a curve over ${\mathbb{C}}\\{\\{t\\}\\}$ passing
through $(t^{x_{i}},t^{y_{i}})\in{\mathbb{C}}\\{\\{t\\}\\}$ and then have
taken its non-Archimedean amoeba. Hence we started from ${\mathbb{C}}$, lifted
to ${\mathbb{C}}\\{\\{t\\}\\}$, and finally descended to ${\mathbb{T}}$.
Detropicalization is the opposite process: firstly, we prove something in
${\mathbb{T}}$, then lift the construction to ${\mathbb{F}}\\{\\{t\\}\\}$, and
finally return to ${\mathbb{F}}$ using $\nu_{a}$.
Here we establish the following lemma.
###### Lemma 2.6
Let $m_{1},m_{2},\dots,m_{n}$ be non-negative integers. Let $\Delta$ be a
lattice polygon such that
$\mathrm{area}(\Delta)<\sum_{i}^{n}\frac{m_{i}^{2}}{4}$. Then, if the set of
points $(x_{i},y_{i})\in{\mathbb{T}}^{2}$ is in general position with respect
to $\Delta$, then for each valuation field ${\mathbb{K}}$ and points
$p_{1},p_{2},\dots,p_{n}\in({\mathbb{K}}^{*})^{2}$ such that
$\mathrm{val}(p_{i})=(x_{i},y_{i})$ there is no curve $C$ over ${\mathbb{K}}$
with the Newton polygon $\Delta$, with $\mu_{p_{i}}(C)\geq m_{i},i=1,\dots,n$.
###### Proof 2.7
Suppose that such a curve $C$ exists. Then, consider $\mathrm{Trop}(C)$. We
know that in this case
$\mathrm{area}(\mathfrak{Infl}((x_{i},y_{i})))\geq\frac{m_{i}^{2}}{2}$
for $i=1,\dots,n$ and
$\sum_{i=1}^{n}\mathrm{area}(\mathfrak{Infl}(x_{i},y_{i}))\leq
2\cdot\mathrm{area}(\Delta)$. So, we arrived to a contradiction.
###### Lemma 2.8 (Detropicalization lemma)
Let ${\mathbb{K}}={\mathbb{F}}\\{\\{t\\}\\}$. Suppose that there is no curve
$C$ over ${\mathbb{K}}$ with the Newton polygon $\Delta$ such that
$\mu_{(t^{-x_{i}},t^{-y_{i}})}(C)\geq m_{i}.$
Then, there exists a constant $N$ depending on
$m_{1},m_{2},\dots,m_{n},\Delta,\max x_{i},\max y_{i}$ with the following
property. If $|{\mathbb{F}}|\geq N$, then there exists $a\in{\mathbb{F}}$ such
that there is no curve over ${\mathbb{F}}$ with the Newton polygon $\Delta$
and $\mu_{{(a^{-x_{i}},a^{-y_{i}})}}(C)\geq m_{i}$ for each $i=1,\dots,n.$
###### Proof 2.9
Indeed, all the constraints imposed by the fact $\mu_{p}(C)\geq m$ are linear
equations in the coefficients of the equation of $C$. Therefore the only
reason why there is no solution for this system over Puiseux series and there
is a solution over ${\mathbb{F}}$ is that some minor of the matrix of the
equations becomes 0 after substituting $t=a$. Thus, let us compute all needed
minors before, they reveal to be polynomials in $t$ with degrees depending on
our data. Therefore the only condition for $a$ is that $a$ is not a root of
some fixed polynomial of some bounded degree. Obviously, if $|{\mathbb{F}}|$
is big enough, then there exists such an $a$.
###### Remark 5
In a similar way we can detropicalize in other situations, if the conditions
imposed on $C$ reveal to be algebraic conditions on the coefficients of the
equation of $C$.
## 3 Degeneration of tropical points to a line.
In this section, using tropical floor diagrams (see [5, 7]), we construct a
special collection of tropical points which are in general position with
respect to the Newton polygon $\Delta$; this construction gives another
estimate for $\mathrm{area}(\Delta)$.
Consider a tropical curve $H$ given by
$\mathrm{Trop}(F)=\max_{(i,j)}(ix+jy+\mathrm{val}(a_{ij}))$ where $(i,j)$ runs
over lattice points in a fixed Newton polygon $\Delta$. We may assume that the
minimal lattice width $\omega(\Delta)$ of $\Delta$ is attained in the
horizontal direction. Let $\Delta$ is contained in the strip $\\{(x,y)|0\leq
y\leq N\\}$. Let us choose points $P_{1},P_{2},\dots,P_{n}$ on the line
$l=\\{(x,y)|y=\frac{1}{N+1}x\\}$ which is almost horizontal, i.e. its slope
$\frac{1}{N+1}$ is less than any possible slope of non-horizontal edges of a
curve with the given Newton polygon $\Delta$.
###### Proposition 1
Suppose that each of the points $P_{1},P_{2},\dots,P_{n}$ is not a vertex of
$H$, and each $P_{i}$ is lying on a horizontal edge $E_{i}$ of $H$. In this
case, for each $1\leq i<j\leq n$ we have
$\mathfrak{Infl}(P_{i})\cap\mathfrak{Infl}(P_{j})=\emptyset$.
###### Proof 3.1
Indeed, in this case the vertices in $\mathfrak{I}(P_{i})$ are lying on the
horizontal lines through $P_{i}$, and all $P_{i}$ have different
$y$-coordinates.
###### Corollary 2
In the above case,
$\sum_{i=1}^{n}\mathrm{area}(\mathfrak{Infl}(P_{i}))\leq\mathrm{area}(\Delta)$.
In general, the situation is not much worse than in the hypothesis of the
above proposition. The line $l$ is subdivided by intersections with $H$, each
connected component of $l\setminus H$ corresponds to a monomial in
$\mathrm{Trop}(F)$, i.e. to a lattice point in $\Delta$. Moving by $l$ from
left to right and marking corresponding lattice points in $\Delta$ we obtain a
lattice path in $\Delta$, which possesses the following property: each edge in
this path is either vertical or has positive projection on the horizontal
line.
If $P_{i}$ is not a vertex of $\mathrm{Trop}(C)$, and $P_{i}$ belongs to an
edge $E_{i}$ of $\mathrm{Trop}(C)$, then denote by $s_{i}$ the length of the
horizontal projection of $d(E_{i})$. If $P_{i}$ is a vertex of
$\mathrm{Trop}(C)$, then denote by $s_{i}$ the length of the horizontal
projection of $d(P_{i})$.
Previous considerations shows that
$\sum\limits_{i=1}^{n}s_{i}\leq\omega(\Delta)$.
$\bullet$$P_{1}$$\bullet$$P_{2}$$\bullet$$P_{3}$
$s_{2}$$\mathfrak{Infl}(P_{1})$$\mathfrak{Infl}(P_{2})$$\mathfrak{Infl}(P_{3})$$1$$2$$3$$4$$\bullet$$\bullet$$\bullet$$\bullet$
Figure 2: The first(top) picture represents a part of a tropical curve through
points $P_{1},P_{2},P_{3}$ on an almost horizontal line. The second picture is
dual to the first picture, we see the regions of influence of the points
$P_{1},P_{2},P_{3}$. The marked points $1,2,3,4$ represent the monomials which
are maximal on the parts of the dotted line on the left picture. The lattice
path $1,2,3,4$ is non-decreasing by the $x$-coordinate, therefore
$\sum_{i=1}^{n}s_{i}\leq\omega_{(1,0)(\Delta)}$.
###### Proposition 3
In the above notation,
$\frac{1}{2}\sum\limits_{i=1}^{n}(m_{i}^{2}-s_{i}^{2})\leq\sum\limits_{i=1}^{n}\left(\sum\limits_{V\in\mathfrak{I}_{P_{i}}((0,1))}\mathrm{area}(d(V))\right)\leq\mathrm{area}(\Delta).$
###### Proof 3.2
The right inequality is trivial, because the sets
$\mathfrak{I}_{P_{i}}((1,0))$ do not intersect each other. The left inequality
follows from the estimate
$\sum\limits_{V\in\mathfrak{I}_{P_{i}}((0,1))}\mathrm{area}(d(P_{i}))\geq\frac{1}{2}(m_{i}^{2}-s_{i}^{2})$
for each $i=1,\dots,n$. Indeed, if $P_{i}$ is not a vertex of $H$, then
$P_{i}$ belongs to an edge $E_{i}$.
If $E_{i}$ is horizontal, then $s_{i}=0$ and
$\sum\limits_{V\in\mathfrak{I}_{P_{i}}((0,1))}\mathrm{area}(d(P_{i}))\geq\frac{1}{2}m_{i}^{2}$
by Lemma 2.5. If $E_{i}$ is not horizontal, then $s_{i}\geq m_{i}$ and the
inequality becomes trivial. If $P_{i}$ is a vertex of $H$, then the inequality
follows from Lemma 2.5, because in this case
$\sum\limits_{V\in\mathfrak{I}_{P_{i}}((0,1))}\mathrm{area}(d(P_{i}))\geq(m_{i}-s_{i})\cdot
s_{i}+\frac{1}{2}(m_{i}-s_{i})^{2}=\frac{m_{i}^{2}-s_{i}^{2}}{2}.$
###### Proof 3.3 (Proof of Theorem 1)
By Corollary 13 there exists $N$ such that there exists a generic with respect
to $\Delta$ collection of points on the line $y=\frac{1}{\omega(\Delta)+1}$
with $|x_{i}|,|y_{i}|<N$. Then, Proposition 2.5 and Lemma 2.8 conclude the
proof.
## 4 Code theory
In informatics, (error-correcting) code-theory deals with subsets $C\subset
A^{n}$ ($A$ is a finite set) which are as big as possible, and the Hamming
distance $d$ between the elements in $C$ is also as big as possible, i.e. we
maximize $\delta=\min_{a,b\in C,a\neq b}d(a,b)$. Such a subset $C$ is called a
code and it is suitable for the following problem. We transmit a message which
is an element of $C$. If, during the transmission procedure, the message does
change in at most $\frac{\delta}{2}-1$ positions, then we can uniquely repare
it back, that is why this is called an error-correcting code. As an
introductory book, which relates this subject to algebraic geometry, see [24].
Studying of singular varieties is related with code-theory ([26]), for the
relation of this topic with Seshadri constants (which is a relative of
Nagata’s conjecture), see [13].
Finding such subsets $C$ is a hard combinatorial problem. A particular source
for codes is the set of linear subspaces of ${\mathbb{F}}_{q}^{n}$ (linear
codes), mostly because they have comparatively simple description. A common
construction is the following. We chose points
$p_{1},p_{2},\dots,p_{n}\subset{\mathbb{F}}_{q}^{m}$ and consider the set
$V_{d}\subset{\mathbb{F}}_{q}[x_{1},x_{2},\dots,x_{m}]$ of the polynomials of
degree no more than $d$ (or we can take any linear system on a toric variety).
Then we take the evaluation map:
$ev_{p}:V_{d}\to{\mathbb{F}}_{q}^{n},ev_{p}(f)=(f(p_{1}),f(p_{2}),\dots,f(p_{n}))$.
The image of $ev_{p}$ is a linear code, it is quite simple to calculate it,
but the problem is how to chose points $p_{i}$ such that there is no
polynomials which vanish at chosen points (otherwise we need to deal with the
kernel of $ev_{p}$) and how to estimate the minimal distance $\delta$. For
example, one may take all the points with all non-zero coordinates.
Thanks to Joaquim Roé suggestion, we mention here the way we can exploit the
main ideas of this article to construct a linear code, which uses not too much
points and provides a map, similar to $ev_{p}$, without kernel.
In the previous sections, for a given polygon $\Delta$ and numbers
$m_{1},m_{2},\dots,m_{n}$ we constructed the set of points
$p_{1},p_{2},\dots,p_{n}\in{\mathbb{F}}_{q}$ such that there is no curve $C$
with the Newton polygon $\Delta$, possessing the property $\mu_{p_{i}}(C)\geq
m_{i}$ for each $i$. Recall, that for this construction we should carefully
chose points $(x_{i},y_{i})\in{\mathbb{Z}}^{2}$, then, for $q$ big enough
there is $t\in{\mathbb{F}}_{q}$, such that the points
$p_{i}=(t^{x_{i}},t^{y_{i}})$ possess the required properties.
###### Example 1
Consider $\Delta=[0,1,\dots,d]\times[0,1\dots,N]\subset{\mathbb{Z}}^{2}$. If
we put $n$ points $p_{1},p_{2},\dots,p_{n}$ of multiplicity $m\leq\min(N,d)$
along an almost horizontal line, then there is no algebraic curve $C$ with the
Newton polygon $\Delta$ and $\mu_{p_{i}}(C)\geq m$ if
$dN<\frac{1}{2}(n-d/m-1)m^{2}$.
Therefore, taking $N<\frac{(n-d/m-1)m^{2}}{2d}$ we construct the evaluation
map ${\mathbb{F}}_{q}^{dN}\to{\mathbb{F}}_{q}^{\frac{nm(m+1)}{2}}$ with a
trivial kernel. For this map, we take any polynomial $F$ with the Newton
polygon $\Delta$, then take the coefficients of $F\pmod{I_{p_{i}}^{m}}$ for
each $i=1,\dots,n$.
## References
* [1] J. Alexander and A. Hirschowitz. Polynomial interpolation in several variables. J. Algebraic Geom., 4(2):201–222, 1995.
* [2] J. Alexander and A. Hirschowitz. An asymptotic vanishing theorem for generic unions of multiple points. Inventiones mathematicae, 140(2):303–325, 2000.
* [3] C. Bocci. Special effect varieties in higher dimension. Collect. Math., 56(3):299–326, 2005.
* [4] M. Brion. Piecewise polynomial functions, convex polytopes and enumerative geometry. In Parameter spaces (Warsaw, 1994), volume 36 of Banach Center Publ., pages 25–44. Polish Acad. Sci., Warsaw, 1996.
* [5] E. Brugallé, I. Itenberg, G. Mikhalkin, and K. Shaw. Brief introduction to tropical geometry. Proceedings of 21st Gökova Geometry-Topology Conference, arXiv:1502.05950, 2015.
* [6] E. Brugallé and G. Mikhalkin. Enumeration of curves via floor diagrams. C. R. Math. Acad. Sci. Paris, 345(6):329–334, 2007.
* [7] E. Brugallé and G. Mikhalkin. Floor decompositions of tropical curves: the planar case. In Proceedings of Gökova Geometry-Topology Conference 2008, pages 64–90. Gökova Geometry/Topology Conference (GGT), Gökova, 2009.
* [8] C. Ciliberto. Geometric aspects of polynomial interpolation in more variables and of Waring’s problem. In European Congress of Mathematics, Vol. I (Barcelona, 2000), volume 201 of Progr. Math., pages 289–316. Birkhäuser, Basel, 2001.
* [9] C. Ciliberto, B. Harbourne, R. Miranda, and J. Roé. Variations of Nagata’s conjecture. In A celebration of algebraic geometry, volume 18 of Clay Math. Proc., pages 185–203. Amer. Math. Soc., Providence, RI, 2013.
* [10] C. Ciliberto and R. Miranda. Homogeneous interpolation on ten points. J. Algebraic Geom., 20(4):685–726, 2011.
* [11] M. Dumnicki, B. Harbourne, T. Szemberg, and H. Tutaj-Gasińska. Linear subspaces, symbolic powers and Nagata type conjectures. Adv. Math., 252:471–491, 2014.
* [12] L. Evain. Computing limit linear series with infinitesimal methods. Ann. Inst. Fourier (Grenoble), 57(6):1947–1974, 2007.
* [13] S. H. Hansen. Error-correcting codes from higher-dimensional varieties. Finite fields and their applications, 7(4):530–552, 2001.
* [14] B. Harbourne. Problems and progress: a survey on fat points in $\mathbb{P}^{2}$. In Zero-dimensional schemes and applications (Naples, 2000), volume 123 of Queen’s Papers in Pure and Appl. Math., pages 85–132. Queen’s Univ., Kingston, ON, 2002.
* [15] N. Kalinin. The Newton polygon of a planar singular curve and its subdivision (under third round of revision in Combinatorial Series A). ArXiv e-prints, June 2013.
* [16] D. Maclagan and B. Sturmfels. Introduction to tropical geometry, volume 161 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, 2015.
* [17] H. Markwig, T. Markwig, and E. Shustin. Tropical surface singularities. Discrete Comput. Geom., 48(4):879–914, 2012.
* [18] T. Markwig. A field of generalised Puiseux series for tropical geometry. Rend. Semin. Mat. Univ. Politec. Torino, 68(1):79–92, 2010.
* [19] R. Miranda. Linear systems of plane curves. Notices AMS, 46(2):192–202, 1999.
* [20] M. Nagata. On the 14-th problem of hilbert. American Journal of Mathematics, pages 766–772, 1959.
* [21] S. Paul. New methods for determining speciality of linear systems based at fat points in $\mathbb{P}^{n}$. J. Pure Appl. Algebra, 217(5):927–945, 2013.
* [22] J. M. Ruiz. The basic theory of power series. Advanced Lectures in Mathematics. Friedr. Vieweg & Sohn, Braunschweig, 1993.
* [23] R. Steffens and T. Theobald. Combinatorics and genus of tropical intersections and ehrhart theory. SIAM Journal on Discrete Mathematics, 24(1):17–32, 2010.
* [24] M. Tsfasman, S. Vlăduţ, and D. Nogin. Algebraic geometric codes: basic notions, volume 139 of Mathematical Surveys and Monographs. American Mathematical Society, Providence, RI, 2007.
* [25] I. Tyomkin. On Zariski’s theorem in positive characteristic. J. Eur. Math. Soc. (JEMS), 15(5):1783–1803, 2013.
* [26] J. Wahl. Nodes on sextic hypersurfaces in ${\bf P}^{3}$. J. Differential Geom., 48(3):439–444, 1998.
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|
arxiv-papers
| 2013-10-24T17:53:27 |
2024-09-04T02:49:52.838924
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/",
"authors": "Nikita Kalinin",
"submitter": "Nikita Kalinin",
"url": "https://arxiv.org/abs/1310.6684"
}
|
1310.6698
|
###### Abstract
We mainly establish a monotonicity property between some special Riemann sums
of a convex function $f$ on $[a,b]$, which in particular yields that
$\frac{b-a}{n+1}\sum_{i=0}^{n}f\left(a+i\frac{b-a}{n}\right)$ is decreasing
while $\frac{b-a}{n-1}\sum_{i=1}^{n-1}f\left(a+i\frac{b-a}{n}\right)$ is an
increasing sequence. These give us a new refinement of the Hermitt-Hadamard
inequality. Moreover, we give a refinement of the classical Alzer’s inequality
together with a suitable converse to it. Applications regarding to some
important convex functions are also included.
Some Monotonicity Properties of Convex
Functions with Applications
Jamal Rooin and Hossein Dehghan
Department of Mathematics
Institute for Advanced Studies in Basic Sciences
Zanjan 45137-66731, Iran
[email protected]
[email protected]
2010 Mathematics Subject Classification: 26D15, 26A51, 26A06.
_Keywords and phrases: Convexity, Mean, Hermitt-Hadamard inequality, Alzer
inequality, Bennett inequality._
## 1 Introduction
Throughout this paper, we suppose that $a<b$ are two real numbers and $f$ is a
real-valued function on the closed interval $[a,b]$. We put
$\displaystyle
A_{n}:=\frac{b-a}{n+1}\sum_{i=0}^{n}f\left(x_{i}^{(n)}\right),\hskip
56.9055ptB_{n}:=\frac{b-a}{n-1}\sum_{i=1}^{n-1}f\left(x_{i}^{(n)}\right),$
and
$\displaystyle
S_{n}=\frac{b-a}{n}\sum_{i=1}^{n}f\left(x_{i}^{(n)}\right),\hskip
28.45274ptT_{n}=\frac{b-a}{n}\sum_{i=0}^{n-1}f\left(x_{i}^{(n)}\right),$
where
$\displaystyle x_{i}^{(n)}=a+i\frac{b-a}{n}\hskip
42.67912pt(i=0,1,\ldots,n;\leavevmode\nobreak\ n=1,2,\ldots).$
($B_{n}$ is defined for $n\geq 2$.)
It is known (see, e.g., [1, p. 565]) that if $f$ is increasing convex or
increasing concave on $[0,1]$, then the sequence $S_{n}$ is decreasing while
$T_{n}$ is increasing, i.e.
$\displaystyle S_{n+1}\leq S_{n},\hskip 42.67912ptT_{n}\leq T_{n+1}.$ (1.1)
The inequalities in (1.1) are strict if $f$ is strictly increasing and convex
or strictly increasing and concave.
In 1964, H. Minc and L. Sathre [11] proved that
$\displaystyle\frac{n}{n+1}\leq\frac{\sqrt[n]{n!}}{\sqrt[n+1]{(n+1)!}}\hskip
42.67912pt(n=1,2,\ldots).$ (1.2)
In 1988, J.S. Martins [10] established that for each $r>0$,
$\displaystyle\left((n+1)\sum_{i=1}^{n}i^{r}\left/n\sum_{i=1}^{n+1}i^{r}\right.\right)^{1/r}\leq\frac{\sqrt[n]{n!}}{\sqrt[n+1]{(n+1)!}}\hskip
42.67912pt(n=1,2,\ldots).$ (1.3)
In 1992, G. Bennett [4] proved the following inequality
$\displaystyle\left((n+1)\sum_{i=1}^{n}i^{r}\left/n\sum_{i=1}^{n+1}i^{r}\right.\right)^{1/r}\leq\frac{n+1}{n+2}\hskip
42.67912pt(r>1;\ n=1,2,\ldots),$ (1.4)
which is reversed if $r<1$.
In 1993, H. Alzer [2] came into comparing the left-hand sides of (1.2) and
(1.3) and proved that for each $r>0$,
$\displaystyle\frac{n}{n+1}\leq\left((n+1)\sum_{i=1}^{n}i^{r}\left/n\sum_{i=1}^{n+1}i^{r}\right.\right)^{1/r}\hskip
42.67912pt(n=1,2,\ldots).$ (1.5)
The proof of Alzer is technical, but quite complicated. So, in several
articles Alzer’s proof has been simplified, and also in many others, this
inequality has been extended; see e.g. [6, 5, 9, 15, 16], and see also [1] for
some historical notes.
Obviously, the Alzer inequality (1.5) and Martins inequality (1.3)
simultaneously give us a refinement of Minc-Sathre inequality (1.2).
Note that if $r\rightarrow 0+$ in (1.5), we get (1.2) without appealing to
(1.3).
Clearly, for $r>1$ the Alzer inequality (1.5) gives us a reverse of Bennet
inequality (1.4), while as considering $n/(n+1)<(n+1)/(n+2)$, the Bennet
inequality for $0<r<1$ is a refinement of Alzer inequality.
In 1994, H. Alzer [3] showed that if $r<0$, the Martins inequality (1.3) is
reversed. This result is reobtained by C.P. Chen et al. [7] in 2005, too.
Recently, J. Rooin et al. [13], using some technics of convexity, generalized
the Alzer and Bennett inequalities to operators when $-1\leq r\leq 2$.
Let $f$ be convex on $[a,b]$. The main purpose of this paper is to prove the
inequality (2.2) regarding some Riemann sums of $f$. This inequality yields
that the sequence $A_{n}$ is decreasing while $B_{n}$ is increasing, without
any monotonicity assumptions on $f$. As a consequence, we give an extension
and a refinement to the well-known Hermitt-Hadamard inequality [12]:
$\displaystyle
f\left(\frac{a+b}{2}\right)\leq\frac{1}{b-a}\int_{a}^{b}f(t)dt\leq\frac{f(a)+f(b)}{2}.$
(1.6)
For more details see [8]. If in addition $f$ is increasing, we get some
refinements and converses to (1.1).
Applying these results to the power function $x^{r}$, we get the Bennett
inequality (1.4) and refinements and converses of the classical Alzer
inequality (1.5) in the case of $-\infty<r<+\infty$. These extend the
numerical results of [13]. Also, we obtain new inequalities concerning
$p$-logarithmic means. Finally, we give applications regarding to some other
important convex functions, which in particular, yield us new rational
approximations of trigonometric functions.
## 2 Main results
In this section, we prove some monotonicity properties of convex functions.
The following theorem is the main source of all results in this paper.
###### Theorem 2.1.
Let $a=x_{0}<x_{1}<\cdots<x_{n}=b$ and $a=y_{0}<y_{1}<\cdots<y_{n+1}=b$ be two
partitions of $[a,b]$ such that $x_{i-1}\leq y_{i}\leq x_{i}$
($i=1,2,\ldots,n$). If $f$ is convex on $[a,b]$, then
$\displaystyle\sum_{i=1}^{n}(x_{i}-x_{i-1})f(y_{i})\leq\sum_{i=0}^{n}(x_{i+1}-x_{i-1}+y_{i}-y_{i+1})f(x_{i})$
(2.1)
and
$\displaystyle\sum_{i=0}^{n}(y_{i+1}-y_{i})f(x_{i})\leq\sum_{i=0}^{n+1}(y_{i+1}-y_{i-1}+x_{i-1}-x_{i})f(y_{i}),$
(2.2)
where $x_{-1}=y_{-1}=a$ and $x_{n+1}=y_{n+2}=b$. If $f$ is strictly convex,
then inequality (2.1) (respectively (2.2)) is strict whenever
$x_{i-1}<y_{i}<x_{i}$ for some $i\in\\{1,2,\ldots,n\\}$ (respectively
$y_{i}<x_{i}<y_{i+1}$ for some $i\in\\{1,2,\ldots,n-1\\}$).
Proof. Since $x_{i-1}\leq y_{i}\leq x_{i}$ ($i=1,2,\ldots,n$), using
$y_{i}=\frac{x_{i}-y_{i}}{x_{i}-x_{i-1}}\
x_{i-1}+\frac{y_{i}-x_{i-1}}{x_{i}-x_{i-1}}\ x_{i}$
and convexity of $f$ we have
$\displaystyle(x_{i}-x_{i-1})f(y_{i})\leq(x_{i}-y_{i})f(x_{i-1})+(y_{i}-x_{i-1})f(x_{i})\hskip
42.67912pt(i=1,2,\ldots,n).$ (2.3)
Now summing up (2.3) from $1$ to $n$, we get
$\displaystyle\sum_{i=1}^{n}(x_{i}-x_{i-1})f(y_{i})$ $\displaystyle\leq$
$\displaystyle\sum_{i=1}^{n}(x_{i}-y_{i})f(x_{i-1})+\sum_{i=1}^{n}(y_{i}-x_{i-1})f(x_{i})$
$\displaystyle=$
$\displaystyle\sum_{i=0}^{n-1}(x_{i+1}-y_{i+1})f(x_{i})+\sum_{i=1}^{n}(y_{i}-x_{i-1})f(x_{i})$
$\displaystyle=$
$\displaystyle\sum_{i=0}^{n}(x_{i+1}-y_{i+1})f(x_{i})+\sum_{i=0}^{n}(y_{i}-x_{i-1})f(x_{i})$
$\displaystyle=$
$\displaystyle\sum_{i=0}^{n}(x_{i+1}-x_{i-1}+y_{i}-y_{i+1})f(x_{i}).$
The inequality (2.2) follows in a similar manner by considering $y_{i}\leq
x_{i}\leq y_{i+1}$ ($i=0,1,\ldots,n$). The rest is clear. $\Box$
###### Remark 2.2.
With the assumptions of Theorem 2.1 we may write the inequalities (2.1) and
(2.2) in following single form
$\displaystyle\sum_{i=0}^{n}$
$\displaystyle(y_{i+1}-y_{i})f(x_{i})+\sum_{i=0}^{n-1}(x_{i+1}-x_{i})f(y_{i+1})$
$\displaystyle\leq\min\left\\{\sum_{i=0}^{n}(y_{i+1}-y_{i})(f(y_{i})+f(y_{i+1})),\sum_{i=0}^{n-1}(x_{i+1}-x_{i})(f(x_{i})+f(x_{i+1}))\right\\},$
(2.4)
which is a monotonicity property between some special Riemann sums.
###### Corollary 2.3.
With the above assumptions, if $f$ is convex on $[a,b]$, then we have
$\displaystyle A_{n+1}\leq A_{n}\hskip 28.45274pt(n=1,2,\ldots)\hskip
28.45274pt\mbox{and}\hskip 28.45274ptB_{n}\leq B_{n+1}\hskip
28.45274pt(n=2,3,\ldots).$ (2.5)
Both inequalities are strict if $f$ is strictly convex.
Proof. Take $x_{i}=x_{i}^{(n)}$ ($i=0,1,\ldots,n$) and $y_{i}=x_{i}^{(n+1)}$
($i=0,1,\ldots,n+1$) in Theorem 2.1. $\Box$
###### Corollary 2.4.
If $f$ is convex on $[a,b]$, then
$\displaystyle\frac{1}{m-1}\sum_{i=1}^{m-1}f\left(x_{i}^{(m)}\right)\leq\frac{1}{b-a}\int_{a}^{b}f(t)dt\leq\frac{1}{n+1}\sum_{i=0}^{n}f\left(x_{i}^{(n)}\right)\hskip
14.22636pt(m=2,3,\ldots;\leavevmode\nobreak\ n=1,2,\ldots),$ (2.6)
which is a refinement and extension of Hermitt-Hadamard inequality (1.6).
Both inequalities in (2.6) are strict, if $f$ is strictly convex.
Proof. Clearly $f$ is Riemann integrable on $[a,b]$ and
$\displaystyle\lim_{n\to\infty}A_{n}=\lim_{n\to\infty}B_{n}=\int_{a}^{b}f(t)dt.$
(2.7)
Now, (2.6) follows from Corollary 2.3. $\Box$
###### Corollary 2.5.
If $f>0$ is logarithmically convex on $[a,b]$, then
$\displaystyle\frac{\sqrt[m-1]{\prod_{i=1}^{m-1}f\left(\frac{(m-i)a+ib}{m}\right)}}{\sqrt[m]{\prod_{i=1}^{m}f\left(\frac{(m+1-i)a+ib}{m+1}\right)}}\leq
1\leq\frac{\sqrt[n+1]{\prod_{i=0}^{n}f\left(\frac{(n-i)a+ib}{n}\right)}}{\sqrt[n+2]{\prod_{i=0}^{n+1}f\left(\frac{(n+1-i)a+ib}{n+1}\right)}}$
(2.8)
and
$\displaystyle\sqrt[m-1]{\prod_{i=1}^{m-1}f\left(\frac{(m-i)a+ib}{m}\right)}$
$\displaystyle\leq$ $\displaystyle\exp\left(\frac{1}{b-a}\int_{a}^{b}\ln
f(t)dt\right)$ (2.9) $\displaystyle\leq$
$\displaystyle\sqrt[n+1]{\prod_{i=0}^{n}f\left(\frac{(n-i)a+ib}{n}\right)},$
where $m=2,3,\ldots$ and $n=1,2,\ldots$.
All inequalities are strict if $f$ is strictly logarithmically convex.
Proof. Take $\ln f$ instead of $f$ in (2.5) and (2.6). $\Box$
###### Corollary 2.6.
With the above assumptions, if $f$ is convex on $[a,b]$, then we have
$\displaystyle\frac{1}{n(n+2)}\left[S_{n+1}-(b-a)f(a))\right]\leq
S_{n}-S_{n+1}\leq\frac{1}{n^{2}}\left[(b-a)f(b)-S_{n+1}\right]$ (2.10)
and
$\displaystyle\frac{1}{n^{2}}\left[T_{n+1}-(b-a)f(a)\right]\leq
T_{n+1}-T_{n}\leq\frac{1}{n(n+2)}\left[(b-a)f(b)-T_{n+1}\right].$ (2.11)
Moreover, except than the case $n=1$ in which equality always holds in the
right of (2.10) and left hand of (2.11), all inequalities are strict if $f$ is
strictly convex.
If $f$ is concave, all inequalities reverse.
Proof. The left inequality of (2.10) and right inequality of (2.11) follow
from the left hand of (2.5), by considering
$A_{n}=\frac{n}{n+1}S_{n}+\frac{b-a}{n+1}f(a)\hskip 14.22636pt\mbox{and}\hskip
14.22636ptA_{n}=\frac{n}{n+1}T_{n}+\frac{b-a}{n+1}f(b)\hskip
42.67912pt(n=1,2,\ldots).$
Obviously, equality holds in right hand of (2.10) and left hand of (2.11) if
$n=1$. Now, if $n\geq 2$, the right inequality of (2.10) and the left
inequality of (2.11) follow from the right hand of (2.5), by considering
$B_{n}=\frac{n}{n-1}S_{n}-\frac{b-a}{n-1}f(b)\hskip 14.22636pt\mbox{and}\hskip
14.22636ptB_{n}=\frac{n}{n-1}T_{n}-\frac{b-a}{n-1}f(a)\hskip
42.67912pt(n=2,3,\ldots).$
If $f$ is strictly convex, the strictness of all inequalities follow from
strictness of inequalities in (2.5).
$\Box$
###### Remark 2.7.
If $f$ is increasing and convex (concave) on $[a,b]$, the inequalities in
(2.10) and (2.11) (the reversed forms of the inequalities in (2.10) and
(2.11)) give us a refinement and converse to the inequalities in (1.1).
## 3 Applications
In this section, using the results of the preceding one, we give several nice
applications regarding some important convex functions.
### 3.1 Applications to normed spaces
Let $X$ be a real normed linear space, $x,y\in X$ and $p\geq 1$. It is clear
that
$\varphi(t)=\|(1-t)x+ty\|^{p}\hskip 42.67912pt(t\in\mathbb{R})$
is a convex function on the real line. If $X$ is strictly convex and $x,y$ are
linearly independent, then using $\|u+v\|<\|u\|+\|v\|$ for any linearly
independent vectors $u$ and $v$, we see that $t\rightarrow\|(1-t)x+ty\|$ is
strictly convex on $\mathbb{R}$. Now, since the function $t\rightarrow t^{p}$
is convex and strictly increasing on $[0,\infty)$, we conclude that $\varphi$
is strictly convex on $\mathbb{R}$.
###### Theorem 3.1.
Let $x,y$ be two vectors in a real normed linear space $X$, not both of them
zero, and $p\geq 1$. Then
$\displaystyle\left(\frac{n\sum_{i=1}^{n-1}\|(n-i)x+iy\|^{p}}{(n-1)\sum_{i=1}^{n}\|(n+1-i)x+iy\|^{p}}\right)^{1/p}\leq\frac{n}{n+1}\leq\left(\frac{(n+2)\sum_{i=0}^{n}\|(n-i)x+iy\|^{p}}{(n+1)\sum_{i=0}^{n+1}\|(n+1-i)x+iy\|^{p}}\right)^{1/p}$
and
$\displaystyle\frac{\sum_{i=1}^{n-1}\|(n-i)x+iy\|^{p}}{n^{p}(n-1)}\leq\int_{0}^{1}\|(1-t)x+ty\|^{p}dt\leq\frac{\sum_{i=0}^{n}\|(n-i)x+iy\|^{p}}{n^{p}(n+1)},$
(3.1)
where in the left hands $n\geq 2$ and in the right hands $n\geq 1$.
Note that (3.1) is a generalization and refinement of the well-known chain
inequalities [12]
$\displaystyle\left\|\frac{x+y}{2}\right\|^{p}\leq\int_{0}^{1}\|(1-t)x+ty\|^{p}dt\leq\frac{\|x\|^{p}+\|y\|^{p}}{2}.$
If $X$ is strictly convex, then all inequalities are strict if $x$ and $y$ are
linearly independent.
Proof. Apply (2.5) and (2.6) to the convex function $\varphi$ on $[0,1]$.
$\Box$
### 3.2 Applications to power and logarithmic functions
We recall that the $p$-logarithmic, identric and logarithmic means of $a,b>0$
are defined respectively by
$\displaystyle L_{p}(a,b)=\left\\{\begin{array}[]{cl}a&\mbox{if}\hskip
5.69054pta=b\\\
\left[\frac{b^{p+1}-a^{p+1}}{(p+1)(b-a)}\right]^{1/p}&\mbox{if}\hskip
5.69054pta\not=b\end{array}\right.,\hskip
42.67912ptp\in\mathbb{R}\setminus\\{0,-1\\},$ $\displaystyle
I(a,b)=\left\\{\begin{array}[]{cl}a&{\rm if\hskip 5.69054pt}a=b\\\
\frac{1}{e}\left(\frac{b^{b}}{a^{a}}\right)^{\frac{1}{b-a}}&{\rm if}\hskip
5.69054pta\not=b\end{array}\right.$
and
$\displaystyle L(a,b)=\left\\{\begin{array}[]{cl}a&\mbox{if}\hskip
5.69054pta=b\\\ \frac{b-a}{\ln b-\ln a}&\mbox{if}\hskip
5.69054pta\not=b\end{array}\right..$
Note that
$\displaystyle\lim_{p\rightarrow 0}L_{p}(a,b)=I(a,b)\hskip
42.67912pt\mbox{and}\hskip 42.67912pt\lim_{p\rightarrow-1}L_{p}(a,b)=L(a,b).$
So, we can take $L_{0}=I$ and $L_{-1}=L$. Note that $L_{p}(a,b)$ is also
defined if $0\not=p>-1$ and $a,b\geq 0$.
###### Theorem 3.2.
Let $0\leq a<b$. If $r>1$, then
$\displaystyle\left(\frac{n\sum_{i=1}^{n-1}[(n-i)a+ib]^{r}}{(n-1)\sum_{i=1}^{n}[(n+1-i)a+ib]^{r}}\right)^{1/r}<\frac{n}{n+1}<\left(\frac{(n+2)\sum_{i=0}^{n}[(n-i)a+ib]^{r}}{(n+1)\sum_{i=0}^{n+1}[(n+1-i)a+ib]^{r}}\right)^{1/r}$
(3.5)
and
$\displaystyle\left(\frac{\sum_{i=1}^{n-1}[(n-i)a+ib]^{r}}{n^{r}(n-1)}\right)^{1/r}<L_{r}(a,b)<\left(\frac{\sum_{i=0}^{n}[(n-i)a+ib]^{r}}{n^{r}(n+1)}\right)^{1/r},$
(3.6)
where in the left hand inequalities, we have $n\geq 2$, and in the right hand
ones, $n\geq 1$.
If $r<0$ with $a>0$ or $0<r<1$, all inequalities in (3.5) and (3.6) reverse.
Proof. For $r>1$ and $r<0$ the function $f(x)=x^{r}$ is strictly convex on
$[0,\infty)$ and $(0,\infty)$ respectively. So if we apply (2.5) and (2.6) for
$f$ on $[a,b]$, we achieve the results.
If $0<r<1$, the function $f$ is strictly concave on $[0,\infty)$, and so both
inequalities in (3.5) and (3.6) reverse. $\Box$
###### Theorem 3.3.
If $0<a<b$, then
$\displaystyle\frac{\sqrt[n+1]{\prod_{i=0}^{n}[(n-i)a+ib]}}{\sqrt[n+2]{\prod_{i=0}^{n+1}[(n+1-i)a+ib]}}<\frac{n}{n+1}<\frac{\sqrt[n-1]{\prod_{i=1}^{n-1}[(n-i)a+ib]}}{\sqrt[n]{\prod_{i=1}^{n}[(n+1-i)a+ib]}}$
(3.7)
and
$\displaystyle\frac{\sqrt[n+1]{\prod_{i=0}^{n}[(n-i)a+ib]}}{n}<I(a,b)<\frac{\sqrt[n-1]{\prod_{i=1}^{n-1}[(n-i)a+ib]}}{n},$
(3.8)
where in the left hand inequalities, we have $n\geq 1$, and in the right hand
one $n\geq 2$.
Proof. Applying (2.8) and (2.9) for the strictly logarithmically convex
function $f(x)=1/x$ on $[a,b]$, we get (3.7) and (3.8). $\Box$
###### Remark 3.4.
(i) If we set $a=0$ and $b=1$ in (3.2), we get for $r>1$,
$\displaystyle\left(\frac{n\sum_{i=1}^{n-1}i^{r}}{(n-1)\sum_{i=1}^{n}i^{r}}\right)^{1/r}<\frac{n}{n+1}<\left(\frac{(n+2)\sum_{i=1}^{n}i^{r}}{(n+1)\sum_{i=1}^{n+1}i^{r}}\right)^{1/r},$
(3.9)
where in the left hand inequality, we have $n\geq 2$, and in the right hand
one $n\geq 1$. It can be seen that (3.9) in turn is equivalent to
$\displaystyle\frac{n}{n+1}\left(1+\frac{1}{n(n+2)}\right)^{1/r}<\left((n+1)\sum_{i=1}^{n}i^{r}\left/n\sum_{i=1}^{n+1}i^{r}\right.\right)^{1/r}<\frac{n+1}{n+2}\hskip
28.45274pt(n\geq 1).$ (3.10)
Similarly, if $0<r<1$, all inequalities in (3.9) and so in (3.10) reverse.
The inequalities in (3.10) and their reversed forms in the case of $0<r<1$,
give us Bennett inequality (1.4) in the case of $r>0$ and a refinement and
converse of the classical Alzer’s inequality (1.5) which are stronger than the
result in [6, Corollary 1].
(ii) If we take $b=1$ and let $a\to 0+$ in the right hand inequality of (3.8),
we get
$\displaystyle\sqrt[n]{n!}\geq\frac{n+1}{e}\hskip 42.67912pt(n=1,2,\ldots).$
(iii) If $r<0$, letting $a\rightarrow 0+$ and changing $n$ by $n+1$ in the
reversed form of the left hand inequality of (3.2), we obtain
$\displaystyle\frac{n+1}{n+2}\leq\left(\frac{(n+1)\sum_{i=1}^{n}i^{r}}{n\sum_{i=1}^{n+1}i^{r}}\right)^{1/r}\hskip
42.67912pt(r<0;\leavevmode\nobreak\ n=1,2,\ldots),$
which is the Bennett inequality (1.4) for $r<0$.
(iv) If $r>1$, setting $a=0$ and $b=1$ in (3.6), we get
$\displaystyle\frac{(n+1)n^{r}}{r+1}<\sum_{i=1}^{n}i^{r}<\frac{n(n+1)^{r}}{r+1}\hskip
42.67912pt(n=1,2,\ldots).$ (3.11)
If $0<r<1$, then inequalities in (3.11) reverse.
Also, if in the reversed form of the left hand of (3.6), we take $b=1$, let
$a\to 0+$ and change $n$ by $n+1$, we obtain
$\displaystyle\sum_{i=1}^{n}i^{r}\leq\frac{n(n+1)^{r}}{r+1}\hskip
42.67912pt(-1<r<0;\ n=1,2,\ldots).$
(v) If $0<a<b$, letting $r\rightarrow 0$ in the reversed form of (3.5) and
(3.6), we get a weaker form of (3.7) and (3.8), loosing the strictness of
inequalities.
(vi) If $a\rightarrow 0+$ in (3.7), changing $n$ by $n+1$, we get [3, Lemma
2.1]
$\displaystyle\frac{n+1}{n+2}\leq\frac{\sqrt[n]{n!}}{\sqrt[n+1]{(n+1)!}}\hskip
42.67912pt(n=1,2,\ldots),$
which is a refinement of the inequality of H. Minc and L. Sathre (1.2).
###### Theorem 3.5.
If $0<a<b\leq\frac{1}{2}$, then
$\displaystyle\frac{\sqrt[m-1]{\prod_{i=1}^{m-1}\frac{m-(m-i)a-ib}{(m-i)a+ib}}}{\sqrt[m]{\prod_{i=1}^{m}\frac{(m+1)-(m+1-i)a-ib}{(m+1-i)a+ib}}}<1<\frac{\sqrt[n+1]{\prod_{i=0}^{n}\frac{n-(n-i)a-ib}{(n-i)a+ib}}}{\sqrt[n+2]{\prod_{i=0}^{n+1}\frac{(n+1)-(n+1-i)a-ib}{(n+1-i)a+ib}}}$
(3.12)
and
$\displaystyle\sqrt[m-1]{\prod_{i=1}^{m-1}\frac{m-(m-i)a-ib}{(m-i)a+ib}}<\frac{I(1-a,1-b)}{I(a,b)}<\sqrt[n+1]{\prod_{i=0}^{n}\frac{n-(n-i)a-ib}{(n-i)a+ib}}$
(3.13) $(m=2,3,\ldots;n=1,2,\ldots).$
In particular,
$\displaystyle{2m-1\choose m-1}^{\frac{1}{m-1}}\leq{2m+1\choose
m}^{\frac{1}{m}}\hskip 42.67912pt(m=2,3,\ldots),$ (3.14)
and so ${2m+1\choose m}^{\frac{1}{m}}$ is an increasing sequence which tends
to $4$. Also, we have [14, Theorem 3.4]
$\displaystyle\frac{2-a-b}{a+b}<\frac{I(1-a,1-b)}{I(a,b)}<\sqrt{\frac{(1-a)(1-b)}{ab}}.$
(3.15)
Proof. The function $f(t)=\frac{1-t}{t}$ is strictly logarithmically convex on
$(0,1/2]$. So, employing (2.8) and (2.9) for the function $f$ on $[a,b]$ we
yield (3.12) and (3.13). The inequality (3.14) follows from the left hand
inequality in (3.12) by taking $b=1/2$ and letting $a\rightarrow 0+$. Set
$\displaystyle
u(m,a)=\sqrt[m-1]{\prod_{i=1}^{m-1}\frac{m-(m-i)a-\frac{i}{2}}{(m-i)a+\frac{i}{2}}}\hskip
42.67912pt(m=2,3,\ldots;\ 0<a<1/2).$
Since $B_{m}$ is increasing and $f$ is decreasing, $u(m,a)$ is increasing with
respect to $m$ and decreasing with respect to $a$. So considering (2.7) we get
$\displaystyle\lim_{m\to\infty}{2m+1\choose m}^{\frac{1}{m}}$ $\displaystyle=$
$\displaystyle\lim_{m\to\infty}\lim_{a\to
0+}u(m,a)=\sup_{m}\sup_{0<a<1/2}u(m,a)=\sup_{0<a<1/2}\sup_{m}u(m,a)$
$\displaystyle=$ $\displaystyle\lim_{a\to
0+}\lim_{m\to\infty}u(m,a)=\lim_{a\to 0+}\frac{I(1-a,1/2)}{I(a,1/2)}=4.$
Finally, (3.15) is an special case of (3.13) for the choices $m=2$ and $n=1$.
$\Box$
### 3.3 Applications to trigonometric functions
We conclude this section with the following trigonometric estimations.
###### Theorem 3.6.
If $0<x\leq\pi/2$, then
$\displaystyle\frac{n-1}{n}\cot\frac{x}{n+1}+\frac{1}{n}\cot
x<\cot\frac{x}{n}<\frac{n+1}{n+2}\cot\frac{x}{n+1}-\frac{1}{n+2}\cot x,$
(3.16) $\displaystyle\frac{1}{n+1}(\sin x\cot\frac{x}{n}+\cos x)<\frac{\sin
x}{x}<\frac{1}{n-1}(\sin x\cot\frac{x}{n}-\cos x),$ (3.17)
and in particular,
$\displaystyle\frac{n-1}{n}\cot\frac{\pi}{2(n+1)}<\cot\frac{\pi}{2n}<\frac{n+1}{n+2}\cot\frac{\pi}{2(n+1)}.$
(3.18)
where in the left hands of (3.16) and (3.18) and in the right hand of (3.17)
we have $n\geq 2$ and in the others $n\geq 1$.
Proof. The function $f(t)=\sin t$ is strictly concave on
$[0,2x]\subseteq[0,\pi]$. Now, applying (2.5) and (2.6) in the reversed order
to $f$, and considering
$\displaystyle\frac{n+1}{2x}A_{n}=\sum_{i=0}^{n}\sin\frac{2ix}{n}=\frac{\sin(\frac{n+1}{n}x)\sin
x}{\sin\frac{x}{n}}=\sin^{2}x\cot\frac{x}{n}+\sin x\cos x$
and
$\displaystyle\frac{n-1}{2x}B_{n}=\sum_{i=1}^{n-1}\sin\frac{2ix}{n}=\sum_{i=1}^{n}\sin\frac{2ix}{n}-\sin
2x=\sin^{2}x\cot\frac{x}{n}-\sin x\cos x$
we obtain (3.16) and (3.17). The inequalities in (3.18) follow from (3.16) by
taking $x=\pi/2$. $\Box$
###### Remark 3.7.
From (3.18), we have
$\displaystyle\frac{k-1}{k}<\frac{\tan\frac{\pi}{2(k+1)}}{\tan\frac{\pi}{2k}}<\frac{k+1}{k+2}\hskip
42.67912pt(k=2,3,\ldots),$ (3.19)
which by multiplying each side of (3.19) from $k=2$ to $k=n-1$, we obtain
$\displaystyle\frac{1}{n-1}<\tan\frac{\pi}{2n}<\frac{3}{n+1}\hskip
42.67912pt(n=3,4,\ldots).$ (3.20)
Now using the representations of $\tan 2x$, $\cos 2x$ and $\sin 2x$ in terms
of $\tan x$, and applying (3.20), we get for $n=3,4,\ldots$, the following
rational approximations
$\displaystyle\frac{2(n-1)}{n(n-2)}<\tan\frac{\pi}{n}<\frac{6(n+1)}{(n+4)(n-2)},$
(3.21)
$\displaystyle\frac{(n-2)(n+4)}{n^{2}+2n+10}<\cos\frac{\pi}{n}<\frac{n(n-2)}{n^{2}-2n+2},$
(3.22)
and
$\displaystyle\frac{2(n+1)^{2}}{(n-1)(n^{2}+2n+10)}<\sin\frac{\pi}{n}<\frac{6(n-1)^{2}}{(n+1)(n^{2}-2n+2)}.$
(3.23)
But since
$\displaystyle\frac{6(n+1)}{(n+4)(n-2)}-\frac{2(n-1)}{n(n-2)}=\frac{1}{n}\left(\frac{4n^{2}+8}{n^{2}+2n-8}\right),$
$\displaystyle\frac{n(n-2)}{n^{2}-2n+2}-\frac{(n-2)(n+4)}{n^{2}+2n+10}=\frac{1}{n^{2}}\left(\frac{16n^{4}-40n^{3}+16n^{2}}{n^{4}+8n^{2}-16n+20}\right)$
and
$\displaystyle\frac{6(n-1)^{2}}{(n+1)(n^{2}-2n+2)}$ $\displaystyle-$
$\displaystyle\frac{2(n+1)^{2}}{(n-1)(n^{2}+2n+10)}$ $\displaystyle=$
$\displaystyle\frac{1}{n}\left(\frac{4n^{6}-8n^{5}+44n^{4}-152n^{3}+160n^{2}-64n}{n^{6}+7n^{4}-16n^{3}+12n^{2}+16n-20}\right),$
we may write
$\displaystyle\tan\frac{\pi}{n}=\frac{2(n-1)}{n(n-2)}+O\left(\frac{1}{n}\right),$
$\displaystyle\cos\frac{\pi}{n}=\frac{(n-2)(n+4)}{n^{2}+2n+10}+O\left(\frac{1}{n^{2}}\right)$
and
$\displaystyle\sin\frac{\pi}{n}=\frac{2(n+1)^{2}}{(n-1)(n^{2}+2n+10)}+O\left(\frac{1}{n}\right).$
At the end of this paper we express the following conjecture.
Conjecture. It seems that (3.21)-(3.23) to be true for all reals $x>2$ instead
of integers $n\geq 3$. The graphs of functions obtained by replacing $n$ by
$x$ in (3.21)-(3.23) drawn in Figure 1 (a)-(c) respectively strengthen our
conjecture.
(a) Related to (3.21)
(b) Related to (3.22)
(c) Related to (3.23)
Figure 1:
## References
* [1] S. Abramovich, J. Baric, M. Matic, J. Pecaric, On Van de Lune-Alzer’s inequality, J. Math. Inequal. 1 (2007) 563-587.
* [2] H. Alzer, On an inequality of H. Minc and L. Sathre, J. Math. Anal. Appl. 179 (1993) 396-402.
* [3] H. Alzer, Refinement of an inequality of G. Bennett, Discrete Math. 135 (1994) 39-46.
* [4] G. Bennett, Lower bounds for matrices II, Canad. J. Math. 44(1) (1992) 54-74.
* [5] C.P. Chen, F. Qi, Extension of an inequality of H. Alzer for negative powers. Tamkang J. Math. 36(1) (2005) 69-72.
* [6] C.P. Chen, F. Qi, P. Cerone, S.S. Dragomir, Monotonicity of sequences involving convex and concave functions, Math. Inequal. Appl. 6(2) (2003) 229-239.
* [7] C.P. Chen, F. Qi, S.S. Dragomir, Reverse of Martins’ inequality, Aust. J. Math. Anal. Appl. 2(1) (2005) Art. 2, 5 pp.
* [8] S.S. Dragomir, C.E.M. Pearce, Selected Topics on Hermite-Hadamard Inequalities and Applications, RGMIA Monographs, Victoria University, 2000.
http://rgmia.vu.edu.au/monographs.html
* [9] N. Elezović, J. Pečarić, On Alzer’s inequality. J. Math. Anal. Appl. 223(1) (1998) 366-369.
* [10] J.S. Martins, Arithmetic and geometric means, an application to Lorentz sequence spaces, Math. Nachr. 139 (1988) 281-288.
* [11] H. Minc, L. Sathre, Some inequalities involving $(r!)^{1/r}$, Proc. Edinburgh Math. Soc. 14 (1964/65) 41-46.
* [12] D.S. Mitrinović, J.E. Pečarić, A.M. Fink, Classical and New Inequalities in Analysis, Kluwer Academic Publishers, Dordrecht, 1993.
* [13] J. Rooin, A. Alikhani, M. S. Moslehian, Riemann sums for self-adjoint operators, Math. Inequal. Appl. to appear.
* [14] J. Rooin, M. Hassani, Some new inequalities between important means and applications to Ky Fan types Inequalities, Math. Inequal. Appl. 10(3) (2007) 517-527.
* [15] F. Qi, Generalization of H. Alzer’s inequality, J. Math. Anal. Appl. 240(1) (1999) 294-297.
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|
arxiv-papers
| 2013-10-23T06:52:54 |
2024-09-04T02:49:52.848050
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Jamal Rooin and Hossein Dehghan",
"submitter": "Hossein Dehghan",
"url": "https://arxiv.org/abs/1310.6698"
}
|
1310.6722
|
# Theory of excitations of dipolar Bose-Einstein condensate at finite
temperature
Abdelâali Boudjemâa Department of Physics, Faculty of Sciences, Hassiba
Benbouali University of Chlef P.O. Box 151, 02000, Ouled Fares, Chlef,
Algeria.
###### Abstract
We present a systematic study of dilute three-dimensional dipolar Bose gas
employing a finite temperature perturbation theory (beyond the mean field). We
analyze in particular the behavior of the anomalous density, we find that this
quantity has a finite value in the limit of weak interactions at both zero and
finite temperatures. We show that the presence of the dipole-dipole
interaction (DDI) enhances fluctuations, the second order correlation function
and thermodynamic quantities such as the chemical potential, the ground state
energy, the compressibility and the superfluid fraction. We identify the
validity criterion of the small parameter of the theory for Bose-condensed
dipolar gases.
###### pacs:
03.75.Nt, 05.30.Jp
## I Introduction
The recent experimental realization of Bose-Einstein condensate (BEC) of 52Cr
Pfau , 164Dy ming , 168Er erbium and more recently with a degenerate Fermi
gas of 161Dy lu atoms with large magnetic dipolar interaction (6 $\mu_{B}$,
$10\mu_{B}$ and $7\mu_{B}$, respectively) has opened fascinating prospects for
the observation of novel quantum phases and many-body phenomenaBaranov ; Pfau
; Carr ; Pupillo2012 . Polar molecules which have much larger electric dipole
moments than those of the atomic gases have been also produced in their ground
rovibrational state Aik ; kk . The most important feature of these systems is
that the atoms interact via a DDI that is both long ranged and anisotropic.
The anisotropy introduced by the dipolar interactions manifests in the
expansion dynamics Pfau , the excitation spectrum bism , superfluid properties
Tic ; Odell , solitons and soliton-moleculeTik ; santos2 ; Adhi .
Additionally, the DDI is partially attractive and exhibits a roton-maxon
structure in the spectrum santos1 and enhances fluctuations abdougora ; Blak2
. On the other hand, the long range character of the dipolar interaction leads
to scattering properties that are radically different from those found on the
usual short-ranged potentials of quantum gases and therefore, all of the
higher-order partial waves contribute equally to the scattering at low energy
Baranov .
The majority of theoretical investigation of BEC with DDI has often been
focused on zero temperature case described by either the Gross-Pitaevskii
equation or the standard Bogoliubov approximation Santos ; Eberlein ; santos1
; dell ; lime . These works have studied in particular, excitations, ground-
state properties and the stability of dipolar BECs. In contrast, few are the
attempts directed towards the finite temperature behavior of dipolar Bose
gases, for instance we can quote path-integral Monte Carlo simulations Nho ;
Filin and mean field Hartree-Fock-Bogoliubov (HFB) theory within numerical
calculationsRon ; Blak ; Hut ; Blak1 .
The above theories, although being satisfactory, they suffer from several
drawbacks. First, the HFB approximation is not able in principle to describe
the condensed state with broken gauge symmetry, since the breaking of gauge
symmetry is a necessary and sufficient condition for BEC Lieb . Also, the HFB
approximation leads to an unphysical gap in the excitation spectrum, which
causes a violation of the Hugenholtz-Pines (HP) theorem HP . The standard
Bogoliubov approximation Bog by its construction is applicable only at zero
or at very low temperatures, where the Bose-condensed fraction is dominant.
Furthermore, in many approaches, the anomalous density is neglected under the
claim that it is an unmeasured quantity, as well as its contribution being
very small compared to the noncondensed density. In fact, it is not obvious
that it is consistent to calculate the noncondensed density self-consistently
and ignore the anomalous density, since the lowest order interaction
contributions to both quantities are found to be of the same order in three-
and two-dimensional Bose gas Griffin ; Burnet ; Yuk ; boudj2010 ; boudj2011 ;
boudj2012 with contact interactions. Moreover, it has been proved that this
quantity plays a crucial role on the stability of the system. By definition,
the anomalous average arises of the symmetry breaking assumption Yuk ;
boudj2012 and it quantifies the correlations between pairs of condensed atoms
with pairs of noncondensed atoms.
In this paper we present a full self-consistent theory to study the properties
of three-dimensional homogeneous dipolar Bose gas at finite temperature. This
method which based on Beliaev’s higher order (finite temperature) of
perturbation theory being gapless and conserving. For homogeneous gases,
Beliaev Bel was developed the theory beyond the mean field approach by
constructing the zero-temperature diagram technique which allows one to find
corrections to the energies of Bogoliubov excitations, proportional to
$\sqrt{n_{c}a^{3}}$, where $n_{c}$ is the condensate density. For BECs with
contact interactions, Beliaev’s work was extended by several authors pop ; Fed
; Griffin at finite temperatures.
The rest of the paper is organized as follows. In sectionII, we review the
main steps, which the model is based on. In section III, we study the quantum
fluctuations and their effects on the thermodynamics of the system. We examine
in particular the behavior of the anomalous density and its effects on the
second order correlation function at zero temperature. Moreover, we show that
the DDIs enhance the condensate depletion, the anomalous density and
thermodynamic quantities such as the chemical potential, the ground state
energy and the compressibility. The universal small parameter of the theory is
also established . In section IV, we extend our results to the finite
temperature case. Finally, our conclusions are drawn in sectionV.
## II The model
We consider a dilute Bose gas with $N$ dipoles aligned along the $z$ axis, in
this case the interaction potential has a contact component related to the
s-wave scattering length $a$ as
$V_{c}(\vec{r})=g\delta(\vec{r})=(4\pi\hbar^{2}a/m)\delta(\vec{r})$, and the
dipole-dipole component which reads
$V_{d}(\vec{r})=\frac{C_{dd}}{4\pi}\frac{1-3\cos^{2}\theta}{r^{3}},$ (1)
where the coupling constant $C_{dd}$ is $M_{0}M^{2}$ for particles having a
permanent magnetic dipole moment $M$ ($M_{0}$ is the magnetic permeability in
vacuum) and $d^{2}/\epsilon_{0}$ for particles having a permanent electric
dipole $d$ ($\epsilon_{0}$ is the permittivity of vacuum), $m$ is the particle
mass, and $\theta$ is the angle between the relative position of the particles
$\vec{r}$ and the direction of the dipole.
The characteristic dipole-dipole distance can be defined as
$r_{*}=mC_{dd}/4\pi\hbar^{2}$ abdougora . For most polar molecules $r_{*}$
ranges from 10 to $10^{4}$ Å.
In the ultracold limit where the particle momenta satisfy the inequality
$kr_{*}\ll 1$, the scattering amplitude is given by Baranov
$f(\vec{k})=g[1+\epsilon_{dd}(3\cos^{2}\theta{{}_{k}}-1)],$ (2)
where the vector $\vec{k}$ represents the momentum transfer imparted by the
collision, and $\epsilon_{dd}=C_{dd}/3g$ is the dimensionless relative
strength which describes the interplay between the DDI and short-range
interactions. The expression (2) can be obtained also using the Fourier
transfromPfau ; Baranov ; Carr . Employing this result in the second quantized
Hamiltonian, we obtain in the uniform case
$\displaystyle\\!\\!\\!\\!\hat{H}\\!\\!=\\!\\!\sum_{\vec{k}}\\!\frac{\hbar^{2}k^{2}}{2m}\hat{a}^{\dagger}_{\vec{k}}\hat{a}_{\vec{k}}\\!+\\!\frac{1}{2V}\\!\\!\sum_{\vec{k},\vec{q},\vec{p}}\\!\\!f(\vec{p})\hat{a}^{\dagger}_{\vec{k}+\vec{p}}\hat{a}^{\dagger}_{\vec{q}-\vec{p}}\hat{a}_{\vec{q}}\hat{a}_{\vec{k}},$
(3)
where $V$ is a quantization volume, and $\hat{a}_{\vec{k}}^{\dagger}$,
$\hat{a}_{\vec{k}}$ are the creation and annihilation operators of particles.
In Hamiltonian (3), the first term in the single-particle part corresponds to
the kinetic energy of particles and the second term describes the two-body
interaction Hamiltonian of the dipolar force.
Assuming the weakly interacting regime where $r_{*}\ll\xi_{c}$ with
$\xi_{c}=\hbar/\sqrt{mgn_{c}}$ being the corrected healing length, we may use
the Bogoliubov approach up to the fourth order of perturbation theory.
Employing the canonical Bogoliubov transformations:
$\hat{a}^{\dagger}_{\vec{k}}=u_{k}\hat{b}^{\dagger}_{\vec{k}}-v_{k}\hat{b}_{-\vec{k}},\qquad\hat{a}_{\vec{k}}=u_{k}\hat{b}_{\vec{k}}-v_{k}\hat{b}^{\dagger}_{-\vec{k}},$
(4)
where $\hat{b}^{\dagger}_{\vec{k}}$ and $\hat{b}_{\vec{k}}$ are operators of
elementary excitations. Thus, the Hamiltonian (3) reduces to the diagonal form
$\hat{H}=E_{0}+\sum\limits_{\vec{k}}\varepsilon_{k}\hat{b}^{\dagger}_{\vec{k}}\hat{b}_{\vec{k}}$.
The Bogoliubov functions $u_{k},v_{k}$ are expressed in a standard way:
$u_{k},v_{k}=(\sqrt{\varepsilon_{k}/E_{k}}\pm\sqrt{E_{k}/\varepsilon_{k}})/2$
with $E_{k}=\hbar^{2}k^{2}/2m$ is the energy of free particle, and the higher
order Bogoliubov excitations energy is given by
$\varepsilon_{k}=\sqrt{[E_{k}-f(\vec{k})n_{c}+\Sigma_{11}(\vec{k})]^{2}-\Sigma_{12}(\vec{k})^{2}},$
(5)
where $\Sigma_{11}(\vec{k})=2f(\vec{k})n_{c}$ and
$\Sigma_{12}(\vec{k})=f(\vec{k})(n_{c}+\tilde{m})$ are respectively, the first
order normal and anomalous self-energies, $\tilde{m}$ is the anomalous
density.
The spectrum (5) in principle cannot be used as it stands since it does not
guarantee to give the best excitation frequencies due to the inclusion of the
anomalous average which leads to the appearance of a gap in the excitation
spectrum Burnet ; Yuk ; boudj2011 . One way to overcome this problem is to use
the condition $\tilde{m}/n_{c}\ll 1$, which is valid at low temperature and
necessary to ensure the diluteness of the system. Otherwise, the gas becomes
strongly correlated and, thus, the Bogoliubov approach fails for
$\tilde{m}/n_{c}\gg 1$.
Assuming now the limit $\tilde{m}/n_{c}\ll 1$, the normal and anomalous self
energies simplify to $\Sigma_{12}(\theta_{k})=\mu_{0}(\theta_{k})$ and
$\Sigma_{11}(\theta_{k})=2\mu_{0}(\theta_{k})$ where
$\mu_{0}=n_{c}\lim\limits_{k\rightarrow 0}f({\vec{k}})$ is the chemical
potential defined in the first order of perturbation theory Bel . Therefore,
the excitation frequency (5) reduces to
$\varepsilon_{k}=\sqrt{E_{k}^{2}+2\mu_{0}(\theta_{k})E_{k}},$ (6)
which is a gapless specrtum.
It is also easy to check that the HP theorem HP
$\Sigma_{11}(\theta_{k})-\Sigma_{12}(\theta_{k})=\mu_{0}(\theta_{k})$, is well
satisfied.
For $k\rightarrow 0$, the excitations are sound waves $\varepsilon_{k}=\hbar
c_{sd}(\theta_{k})k$, where
$c_{sd}(\theta_{k})=c_{s}\sqrt{1+\epsilon_{dd}(3\cos^{2}\theta_{k}-1)}$ with
$c_{s}=\sqrt{gn_{c}/m}$ is the sound velocity without DDI.
Due to the anisotropy of the dipolar interaction, the self energies and the
sound velocity acquire a dependence on the propagation direction, which is
fixed by the angle $\theta_{k}$ between the propagation direction and the
dipolar orientation. This angular dependence of the sound velocity has been
confirmed experimentally bism .
The noncondensed and the anomalous densities are defined as
$\tilde{n}=\sum_{\vec{k}}\langle\hat{a}^{\dagger}_{\vec{k}}\hat{a}_{\vec{k}}\rangle$
and
$\tilde{m}=\sum_{\vec{k}}\langle\hat{a}_{\vec{k}}\hat{a}_{-\vec{k}}\rangle$,
respectively. Then invoking for the operators $a_{k}$ the transformation (4),
setting
$\langle\hat{b}^{\dagger}_{\vec{k}}\hat{b}_{\vec{k}}\rangle=\delta_{\vec{k}^{\prime}\vec{k}}N_{k}$
and putting the rest of the expectation values equal to zero, where
$N_{k}=[\exp(\varepsilon_{k}/T)-1]^{-1}$ are occupation numbers for the
excitations. As we work in the thermodynamic limit, the sum over $\vec{k}$ can
be replaced by the integral $\sum_{\vec{k}}=V\int d^{3}k/(2\pi)^{3}$ and using
the fact that $2N(x)+1=\coth(x/2)$, we obtain:
$\displaystyle\tilde{n}=\frac{1}{2}\int\frac{d^{3}k}{(2\pi)^{3}}\left[\frac{E_{k}+\Sigma_{12}(\theta_{k})}{\varepsilon_{k}}-1\right]$
(7)
$\displaystyle+\frac{1}{2}\int\frac{d^{3}k}{(2\pi)^{3}}\frac{E_{k}+\Sigma_{12}(\theta_{k})}{\varepsilon_{k}}\left[\coth\left(\frac{\varepsilon_{k}}{2T}\right)-1\right],$
and
$\tilde{m}=-\frac{1}{2}\int\frac{d^{3}k}{(2\pi)^{3}}\frac{\Sigma_{12}(\theta_{k})}{\varepsilon_{k}}-\frac{1}{2}\int\frac{d^{3}k}{(2\pi)^{3}}\frac{\Sigma_{12}(\theta_{k})}{\varepsilon_{k}}\coth\left(\frac{\varepsilon_{k}}{2T}\right).$
(8)
First terms in Eqs.(7) and (8) are the zero-temperature contribution to the
noncondensed $\tilde{n}_{0}$ and anomalous $\tilde{m}_{0}$ densities,
respectively. Second terms represent the contribution of the so-called thermal
fluctuations and we denote them as $\tilde{n}_{T}$ and $\tilde{m}_{T}$,
respectively.
Expressions (7) and (8) must satisfy the equality boudj2010 ; boudj2011 ;
boudj2012
$I_{k}=(2\tilde{n}_{k}+1)^{2}-|2\tilde{m}_{k}|^{2}=\coth^{2}\left(\frac{\varepsilon_{k}}{2T}\right).$
(9)
Equation (9) clearly shows that $\tilde{m}$ is larger than $\tilde{n}$ at low
temperature, so the omission of the anomalous density in this situation is
principally unjustified approximation and wrong from the mathematical point of
view.
The expression of $I$ allows us to calculate in a very useful way the
dissipated heat $Q=(1/n)\int E_{k}I_{k}d^{d}k/(2\pi)^{d}$ for $d$-dimensional
Bose gasYuk ; boudj2012 , where $n=n_{c}+\tilde{n}$ is the total density.
Indeed, the dissipated heat or the superfluid fraction (see below) are defined
through the dispersion of the total momentum operator of the whole system.
This definition is valid for any system, including nonequilibrium and
nonuniform systems of arbitrary statistics. In an equilibrium system, the
average total momentum is zero. Hence, the corresponding heat becomes just the
average total kinetic energy per particle.
## III Fluctuations at zero temperature
In this section we restrict ourselves to study the quantum fluctuations and
their effects on the thermodynamics of the system.
Let us start by calculating the quantum depletion. At zero temperature
($\tilde{n}=\tilde{n}_{0}$), the integral in Eq.(7) gives
$\frac{\tilde{n}}{n_{c}}=\frac{8}{3}\sqrt{\frac{n_{c}a^{3}}{\pi}}{\cal
Q}_{3}(\epsilon_{dd}).$ (10)
The contribution of the DDI is expressed by the function ${\cal
Q}_{3}(\epsilon_{dd})$, which is special case $j=3$ of ${\cal
Q}_{j}(\epsilon_{dd})=(1-\epsilon_{dd})^{j/2}{}_{2}\\!F_{1}\left(-\frac{j}{2},\frac{1}{2};\frac{3}{2};\frac{3\epsilon_{dd}}{\epsilon_{dd}-1}\right)$,
where ${}_{2}\\!F_{1}$ is the hypergeometric function. Note that functions
${\cal Q}_{j}(\epsilon_{dd})$ attain their maximal values for
$\epsilon_{dd}\approx 1$ and become imaginary for $\epsilon_{dd}>1$.
Equation (10) is formally similar to the that obtained from the zeroth order
of perturbation theory lime . The density $n_{c}$ of condensed particles which
constitutes our corrections, appears as a key parameter instead of the total
density $n$.
Now if we use the integral in Eq.(8) directly by summing over all states, we
find that the expression for $\tilde{m}$ diverges as we take the sum over
higher and higher states i.e. the so called ultraviolet divergence. The price
to be paid to circumvent this divergence is to introduce the Beliaev-type
second order coupling constant lime ; peth
$g_{R}(\vec{k})=f(\vec{k})-\frac{m}{\hbar^{2}}\int\frac{d^{3}q}{(2\pi)^{3}}\frac{f(-\vec{q})f(\vec{q})}{2E_{k}}.$
(11)
After the subtraction of the ultraviolet divergent part, the renormalized
anomalous density is given Griffin
$\tilde{m}_{R}=-n_{c}\int\frac{d^{3}k}{(2\pi)^{3}}f(\vec{k})\left[\frac{1}{2\varepsilon_{k}}\coth\left(\frac{\varepsilon_{k}}{2T}\right)-\frac{1}{2E_{k}}\right].$
(12)
In contrast to $\tilde{m}$ in (8), $\tilde{m}_{R}$ has no ultraviolet
divergence from large $k$ contributions. The authors of Burnet1 have pointed
out that the self-consistent ladder diagram approximation for the $T$-matrix
can be expressed in terms of $\tilde{m}_{R}$.
To obtain an estimate value of $\tilde{m}$, we note that the quasi-particle
energy goes over to the free particle energy for $\varepsilon_{k}>gn_{c}$. At
zero temperature ($\tilde{m}=\tilde{m}_{0}$), we find
$\frac{\tilde{m}}{n_{c}}=8\sqrt{\frac{n_{c}a^{3}}{\pi}}{\cal
Q}_{3}(\epsilon_{dd}).$ (13)
One should mention at this level that this expression has never been obtained
before in the literature.
Equation (13) is important in several respects: first of all, it shows that
the anomalous density is three times larger than the noncondensed density
whatever the type of the interaction. Second, $\tilde{m}$ has a positive value
in argreement with the case of uniform Bose gas with pure contact interaction
Yuk ; boudj2012 . Likewise, the anomalous density obtained in Eq.(13) leads us
to reproduce exactly the Lee-Huang-Yang (LHY) corrected equation of state LHY
(see below).
Remarkably, we see from expressions (10) and (13) that the noncondensed and
the anomalous densities increase monotocally with $\epsilon_{dd}$. For a
condensate with pure contact interactions (${\cal Q}_{3}(\epsilon_{dd}=0)=1$),
$\tilde{n}$ and $\tilde{m}$ reduce to their usual expressions. While, for
maximal value of DDI i.e. $\epsilon_{dd}\approx 1$, they are 1.3 larger than
their values of pure contact interactions which means that the DDI may enhance
fluctuations of the condensate at zero temperature.
The anomalous density manifests itself into the second-order correlation
function as Glaub
$\displaystyle
G^{(2)}(r)=\langle\hat{\psi}^{\dagger}(r)\hat{\psi}^{\dagger}(r)\hat{\psi}(r)\hat{\psi}(r)\rangle$
$\displaystyle=n_{c}^{2}+\tilde{m}^{2}+2\tilde{n}^{2}+4\tilde{n}n_{c}+2\tilde{m}n_{c}.$
(14)
Equation (III) is obtained using Wick’s theorem. Inserting then Eqs.(10) and
(13) into (III), we obtain
$\frac{G^{(2)}}{n^{2}}=1+\frac{64}{3}\sqrt{\frac{n_{c}a^{3}}{\pi}}{\cal
Q}_{3}(\epsilon_{dd}).$ (15)
This equation is accurate to the first order in $\tilde{n}/n_{c}$ and
$\tilde{m}/n_{c}$ and shows how the correlation function depends to the
interaction parameter $\epsilon_{dd}$.
The presence of quantum fluctuations leads also to corrections of the chemical
potential which are given by
$\delta\mu=\sum\limits_{\vec{k}}f(\vec{k})[v_{k}(v_{k}-u_{k})]=\sum\limits_{\vec{k}}f(\vec{k})(\tilde{n}+\tilde{m})$
Griffin ; boudj2012 ; abdougora .
Inserting the definitions (7) and (8) into the expression of $\delta\mu$, we
find after integration:
$\delta\mu=\frac{32}{3}gn_{c}\sqrt{\frac{n_{c}a^{3}}{\pi}}{\cal
Q}_{5}(\epsilon_{dd}).$ (16)
The total chemical potential is then written as
$\mu=\mu_{0}(\theta_{k})+\delta\mu$. For $n_{c}\approx n$ and for a condensate
with pure contact interaction (${\cal Q}_{5}(\epsilon_{dd}=0)=1$), the
obtained chemical potential excellently agrees with the famous LHY quantum
corrected equation of state LHY .
By integrating the chemical potential correction with respect to the density,
one obtains beyond mean field the ground state energy as
$E=E_{0}(\theta_{k})+\frac{64}{15}Vgn_{c}^{2}\sqrt{\frac{n_{c}a^{3}}{\pi}}{\cal
Q}_{5}(\epsilon_{dd}),$ (17)
where $E_{0}(\theta_{k})=\mu_{0}(\theta_{k})N_{c}/2$ with $N_{c}$ is the
number of condensed particles.
Note that our formulas of the equation of state (16) and the ground state
energy (17) constitute a natural extension of those obtained in Ref lime .
At $T=0$, the inverse compressibility is equal to
$\kappa^{-1}=n^{2}\partial\mu/\partial n$. Then, using Eq.(16), we get
$\frac{\kappa^{-1}}{n^{2}}=\frac{\mu_{0}(\theta_{k})}{n_{c}}+16g\sqrt{\frac{n_{c}a^{3}}{\pi}}{\cal
Q}_{5}(\epsilon_{dd}).$ (18)
One can also show that the shift of the sound velocity is
$16g\sqrt{n_{c}a^{3}/\pi}{\cal Q}_{5}(\epsilon_{dd})$, which is consistent
with the change in the compressibility $mc_{s}^{2}=n\partial\mu/\partial n$
Lev associated with the LHY correction in the equation of state (16).
Expanding the square root of the obtained formula with $\epsilon_{dd}=0$ in
powers of the gas parameter $n_{c}a^{3}$, we recover easily the Beliaev sound
velocity of Bose gas with pure contact interaction $\delta c_{s}/c_{s}\approx
8\sqrt{n_{c}a^{3}/\pi}$ Bel ; Lev .
What is noticeable is that the chemical potential, the energy and the
compressibility are increasing with dipole interaction parameter. For
$\epsilon_{dd}\approx 1$, these quantities are 2.6 larger than their values of
pure contact interaction which means that DDI effects are more significant for
thermodynamic quantities than for the condensate depletion and the anomalous
density.
The Bogoliubov approach assumes that fluctuations should be small. We thus
conclude from Eqs. (10) and (13) that at $T=0$, the validity of the Bogoliubov
theory requires the inequality
$\sqrt{n_{c}a^{3}}{\cal Q}_{3}(\epsilon_{dd})\ll 1.$ (19)
For $n_{c}=n$, this parameter differs only by the factor ${\cal
Q}_{3}(\epsilon_{dd})$ from the universal small parameter of the theory,
$\sqrt{na^{3}}\ll 1$, in the absence of DDI.
## IV Fluctuations at finite temperature
We now generalize the above obtained results for the case of a spatially
homogeneous dipolar Bose-condensed gas at finite temperature.
At temperatures $T\ll gn_{c}$, the main contribution to integrals (7) and (8)
comes from the region of small momentum where $\varepsilon_{k}=\hbar c_{sd}k$.
After some algebra, we obtain the following expressions for the thermal
contribution of the noncondensed and anomalous densities:
$\frac{\tilde{n}_{T}}{n_{c}}=-\frac{\tilde{m}_{T}}{n_{c}}=\frac{2}{3}\sqrt{\frac{n_{c}a^{3}}{\pi}}\left(\frac{\pi
T}{gn_{c}}\right)^{2}{\cal Q}_{-1}(\epsilon_{dd}).$ (20)
Equation (20) shows clearly that $\tilde{n}$ and $\tilde{m}$ are of the same
order of magnitude at low temperature and only their signs are opposite.
Comparing the result of Eq. (20) with the zero-temperature noncondensed
$\tilde{n}_{0}$ and anomalous $\tilde{m}_{0}$ densities following from Eqs.
(10) and (13) we see that at temperatures $T\ll gn_{c}$, thermal contributions
$\tilde{n}_{T}$ and $\tilde{m}_{T}$ are small and can be omitted when
calculating the total fractions. The situation is quite different at
temperatures $T\gg gn_{c}$, where the main contribution to integrals (7) and
(8) comes from the single particle excitations. Hence,
$\tilde{n}_{T}\approx(mT/2\pi\hbar^{2})^{3/2}\zeta(3/2)$, where $\zeta(3/2)$
is the Riemann Zeta function. The obtained $\tilde{n}_{T}$ is nothing else
than the density of noncondensed atoms in ideal Bose gas. Moreover, the
anomalous density being proportional to the condensed density, tend to zero
together and hence their contribution becomes automatically small.
Another important remark is that for $\epsilon_{dd}\approx 1$, thermal
fluctuations (20) are 10.7 greater than their values of pure short range
interaction. This reflects that the DDIs may strongly enhance fluctuations of
the condensate at finite temperature than at zero temperature (see figure.1).
Figure 1: Functions ${\cal Q}_{3}$ (solide line), ${\cal Q}_{-1}$ (red dashed
line) and ${\cal Q}_{-5}$ (blue dotted line), which govern the dependence of
the condensate depletion, the anomalous fraction correction and superfluid
fraction vs. the dipolar interaction parameter $\epsilon_{dd}$.
The same factor of Eq. (20) appears in the correction to the second order
correlation function due to thermal fluctuations:
$\frac{G^{(2)}}{n^{2}}=\frac{8}{3}\sqrt{\frac{n_{c}a^{3}}{\pi}}\left(\frac{\pi
T}{gn_{c}}\right)^{2}{\cal Q}_{-1}(\epsilon_{dd}).$ (21)
Thermal fluctuations corrections to the chemical potential and the energy can
be also obtained easily through expressions (20).
The Bogoliubov approach requires the conditions $\tilde{n}_{T}\ll n_{c}$ and
$\tilde{m}_{T}\ll n_{c}$. Therefore, at temperatures $T\ll gn_{c}$, the small
parameter of the theory turns out to be given as
$\frac{T}{gn_{c}}\sqrt{n_{c}a^{3}}{\cal Q}_{-1}(\epsilon_{dd})\ll 1.$ (22)
The appearance of the extra factor ($T/gn_{c}$) originates from the thermal
fluctuations corrections.
The superfluid fraction can be given as (c.f. LL9 ; abdougora ; boudj2012 )
$\frac{n_{s}}{n}=1-\int E_{k}\frac{\partial
N_{k}}{\partial\varepsilon_{k}}\frac{d^{3}k}{(2\pi)^{3}}=1-\frac{2Q}{3T},$
(23)
where the quantity $2Q/3T$ represents the normal fraction of the Bose-
condensed gas (liquid).
It is worth stressing that if in expression (23) $\tilde{m}$ were omitted,
then the related integral would be divergent leading to the meaningless value
$n_{s}\rightarrow−\infty$. This indicates that the presence of the anomalous
density is crucial for the occurrence of the superfluidity in Bose gases
boudj2013 ; Yuk1 which is in fact understandable since both quantities are
caused by atomic correlations.
Again at $T\ll gn_{c}$, a straightforward calculation leads to
$\frac{n_{s}}{n}=1-\frac{2\pi^{2}T^{4}}{45mn\hbar^{3}c_{s}^{5}}{\cal
Q}_{-5}(\epsilon_{dd}).$ (24)
Remarkably, the normal density is $\propto T^{4}$, whereas the noncondensed
density $\propto T^{2}$ as shown in (20). One can see also from Eq.(24) that
at $T=0$, the whole liquid is superfluid and $n_{s}=n$. This shows that the
normal density does not coincide with the noncondensed density $\tilde{n}$,
and the superfluid density $n_{s}$ does not coincide with the condensed
density $n_{c}$ of dipolar Bose gas. At $T\gg gn_{c}$, there is copious
evidence that the normal density agrees with the noncondensed density of an
ideal Bose gas. Additionally, the normal density is rapidly increasing with
the dipolar interaction as is depicted in figure.1.
The system pressure can be expressed through $P=-(\partial F/\partial S)_{T}$,
with the free energy given by
$F=E_{0}+T\int\ln[1-\exp(-\varepsilon_{k}/T)](d^{3}k/(2\pi)^{3})$. At
temperatures $T\ll gn_{c}$, the thermal pressure can be calculated as
$P_{T}=\frac{\pi^{2}T^{4}}{90(\hbar c_{s})^{3}}{\cal Q}_{-3}(\epsilon_{dd}).$
(25)
The inverse isothermal compressibility is proportional to $(\partial
P/\partial n)_{T}$
$\left(\frac{\partial P}{\partial
n}\right)_{T}=-\frac{\pi^{2}T^{4}}{60m\hbar^{3}c_{s}^{5}}{\cal
Q}_{-3}(\epsilon_{dd})+\cdots,$ (26)
where the zero temperature contribution to the compressibility is given by the
expression (18).
## V Conclusion
In this paper, we have derived the first corrections to the elementary
excitations of homogeneous dipolar BEC gases arising from effects of finite
temperature perturbation theory (beyond mean field theory). Useful analytic
expressions for the noncondensed and the anomalous densities are obtained. We
find that these fluctuations are angular independence at zero and finite
temperatures. We have shown that the anomalous density is larger than the
noncondensed density at zero temperature while both quantities are comparable
at $T\ll gn_{c}$. Our results show that the anomalous density changes its sign
with increasing temperature in agreement with uniform Bose gas with pure
contact interaction Yuk ; boudj2012 . It was also shown that the roton modes
of trapped dipolar BEC (pancake geometry) serve to change the sign of the
anomalous density near the trap center for largre values of
$\epsilon_{dd}$Blak2 . Indeed, the importance of the anomalous density is
ascribed rather to its modulus but not to its sign. It is worth stressing that
the qunatum depletion and the anomalous fraction are not yet observed
experimentally and remain challenging even for a condensate with pure contact
interaction. Effects of dipolar interactions on quantum fluctuations and on
thermodynamic quantities such as the chemical potential, the ground state
energy and the compressibility are profoundly discussed. Although these
effects are not considerable at zero temperature and there is almost no
difference with the short-range interaction case, we believe that our results
are important from the theoretical point of view since they clarify how the
condensate fluctuations and thermodynamic quantities depend on the relative
interaction strength on the one hand and they show how the anisotropy of the
DDI involves these quantities on the other. Moreover, we have pointed out that
at finite temperature, the DDI may significantly enhance thermal fluctuations
and the thermodynamics of the system supplying a real opportunity for a future
experimental realization. The validity criterion of the Bogoliubov approach is
precisely determined at both zero and finite temperatures.
## VI Acknowledgements
We would like to thank the LPTMS-France for a visit, during which part of this
work was conceived. Gora Shlyapnikov and Axel Pelster are acknowledged for
valuable discussions and comments.
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|
arxiv-papers
| 2013-10-24T19:37:35 |
2024-09-04T02:49:52.857553
|
{
"license": "Public Domain",
"authors": "Abdelaali Boudjemaa",
"submitter": "Abdel\\^aali Boudjem\\^aa abdou abdel aalim",
"url": "https://arxiv.org/abs/1310.6722"
}
|
1310.6772
|
# Sockpuppet Detection in Wikipedia: A Corpus of Real-World Deceptive Writing
for Linking Identities
###### Abstract
This paper describes the corpus of sockpuppet cases we gathered from
Wikipedia. A sockpuppet is an online user account created with a fake identity
for the purpose of covering abusive behavior and/or subverting the editing
regulation process. We used a semi-automated method for crawling and curating
a dataset of real sockpuppet investigation cases. To the best of our
knowledge, this is the first corpus available on real-world deceptive writing.
We describe the process for crawling the data and some preliminary results
that can be used as baseline for benchmarking research. The dataset will be
released under a Creative Commons license from our project website:
http://docsig.cis.uab.edu.
Keywords: sockpuppet detection, authorship identification, deceptive language
Sockpuppet Detection in Wikipedia: A Corpus of Real-World Deceptive Writing
for Linking Identities
Thamar Solorio, Ragib Hasan, Mainul Mizan
---
University of Alabama at Birmingham
Birmingham, Alabama
[email protected], [email protected], [email protected]
Abstract content
## 1\. Introduction
In Wikipedia, users can create multiple accounts for many different purposes.
According to Wikipedia’s policies, each user is supposed to create only one
user account. However, Wikipedia does not enforce the one-user-one-account
rule through technical means. As a result, users are free to create multiple
accounts if they want to. A secondary account created by a user for malicious
purposes is called a sockpuppet. This ease of creating an identity has led
malicious users to create multiple identities and use them for various
purposes, ranging from block evasion, false majority opinion claims, and vote
stacking.
One of the main applications of the sockpuppet dataset is to develop an
automated tool for sockpuppet detection in Wikipedia. Currently, the process
for detecting sockpuppets is manual and involves significant experience from
the administrators. In many cases, the user IP addresses have to be accessed
by special Wikipedia administrators with IP-address viewing privileges
(“checkusers”). This violates user privacy. Without accessing the IP
addresses, the administrators need to depend on their experience in dealing
with sockpuppets to detect similarities in writing style and behavior
manually. That leaves a lot of room for error. In contrast, an automated tool
trained using our sockpuppet dataset can be used to identify the sockpuppets
without requiring IP address information or expert administrator knowledge. In
practice, the automated tool can be used to assist administrators to more
accurately identify malicious sockpuppets.
Besides the use in development of tools for automated detection of sockpuppets
in Wikipedia, the sockpuppet dataset has many other potential applications. In
particular, this corpus can be used by researchers working on authorship
attribution problems. The sockpuppet corpus provides a real world data set of
short messages from real malicious users. The sockpuppet cases involve text
from actual users who are intentionally creating multiple identities and
actively trying to hide their connections to the sockpuppet master. Therefore,
using this corpus, researchers can test their work in a real life setting.
This type of authorship attribution of short text has potential applications
in identifying terrorists in web forums, online discussion boards, phone text
messages, tweets and other social media interactions where comments and text
tend to be brief and short in length.
## 2\. Related Work
Authorship analysis has received a great deal of attention in recent years
[Stamatatos, 2008]. The field has grown from a pure manual stylistic analysis
to machine learning approaches that combine stylistic features with richer
representations of writing preferences, such as n-grams of syntactic features
[Sidorov et al., 2013] and local histograms of character n-grams [Escalante et
al., 2011]. Recent work started exploring the limits of automated approaches
to the problem of authorship analysis by looking at extremely short documents
[Layton et al., 2010], very large candidate sets [Koppel et al., 2011], and
cross-domain scenarios [Goldstein-Stewart et al., 2008].
Less work has been devoted to authorship analysis on deceptive writing. Some
of the exceptions include the work in [Brennan et al., 2012, Novak et al.,
2004]. The main barrier to study attribution in adversarial scenarios is the
lack of suitable data. This is understandable as the nature of the problem
makes it difficult to have readily available data where subjects have been
intentionally trying to deceive humans. To solve this barrier researchers have
turn to the generation of artificial data sets. For instance Novak et al.
generated sub aliases from message boards by randomly splitting data from the
same alias [Novak et al., 2004]. Then they evaluated performance of their
method on linking the two sub aliases. The Brennan-Greenstadt adversarial
stylometry corpus was collected from volunteers [Brennan et al., 2012]. The
authors instructed the subjects to submit original writings of an academic
nature. Then the subjects were asked to obfuscate their writing style during
the creation of a topic specific writing of 500 words. In addition, subjects
were also requested to submit an imitation writing excerpt, where they were
instructed to imitate the writing of Cormac McCarthy in The Road. Here again,
the topic of the imitation writing was controlled by the corpus developers.
Both resources are valuable in that they enabled researchers to explore
attribution approaches and allowed them to show that in adversarial scenarios
state of the art approaches will degrade performance. This gap in performance
calls for more research in deceptive writing. However, these two data sets
still have an artificial flavor to them since the authors were not self
motivated and it is not clear whether this will cause major differences in the
final stylistic markers of their writings. The sockpuppet corpus we created is
a real-world alternative to the study of deceptive writing in social media.
The authors were not aware of someone collecting their writings to study
attribution, thus this new data set will allow the study of deceptive writing
in the wild.
## 3\. Sockpuppet Investigations (SPI) in Wikipedia
Wikipedia allows any editor to request investigation of suspected
sockpuppetry. The requester needs to include any evidence of the abusive
behavior. Typical evidence includes information about the editing patterns
related to those accounts, such as the articles, the topics, vandalism
patterns, timing of account creation, timing of edits, and voting pattern in
disagreements.
Once a case is filed, an administrator will investigate the case. An
administrator is an editor with privileges to make account management
decisions, such as banning an editor. The administrator performs a behavioral
evidence investigation and will try to determine whether the two accounts are
related and will then issue a decision confirming or rejecting the
sockpuppetry case, or request involvement of a check user. Check users are
higher privileged editors, who have access to private information regarding
editors and edits, such as the IP address from which an editor has logged in.
Check users perform a technical evidence investigation. But as explained in
Wikipedia SPI description, these users will be involved in the investigation,
if needed, only after strong behavioral evidence has been collected.
When an SPI concludes with a confirmed sock puppetry verdict, the sockpuppet
account will be banned indefinitely. The administrators have the discretion to
establish bans or to block the main account as well.
The process to resolve SPI described above is time consuming and expensive.
The last time we checked the list of current cases, on 10/23/13, there were
30$+$ unique SPI cases listed for the month of October. This high rate of
cases filed in a single month show the need for a streamlined process to
handle SPIs. The data set we created is a first step on this direction.
## 4\. Data Collection Process
All the data we collected from Wikipedia is readily available from the
Wikipedia website. Wikipedia archives all information related to each
sockpuppet case filed, and once a verdict is issued, that too is stored in the
archives. However, because of the lack of a standard format in the archives,
our process for data collection is semi-automated. The sockpuppet cases we
collected were crawled from the following urls:
* •
https://en.wikipedia.org/wiki/Wikipedia:Sockpuppet_investigations/SPI/Closed/2009
* •
https://en.wikipedia.org/wiki/Wikipedia:Sockpuppet_investigations/SPI/Closed/2010
* •
http://en.wikipedia.org/w/index.php?title=Wikipedia:Sockpuppet_investigations/Cases/Overview&offset=&limit=500&action=history
For each case selected for addition to our corpus we collect all data from the
talk pages of each editor involved in the SPI case. This step is done
automatically by crawling the corresponding Wikipedia archives. We only
collect data from discussion pages since these are free form discussions among
editors that give editors more freedom to show their stylistic writing
markers. In contrast, the basic namespaces in Wikipedia, and in particular the
articles the editors contribute to, have a more restrictive format that can
make difficult the identification of editors. Moreover, some of the edits in
the main Wikipedia articles include things like reverts, or typo corrections,
that are related to the user behavior and not necessarily to editors writing
styles. Our main goal to develop this corpus is to support research in
deceptive writing, and thus the behavior treats mentioned above fall outside
this goal. However, this information could still be crawled at a later stage
and be leveraged to perform a persona identification.
The manual process for this task involves retrieving the final decision
reached by the investigative administrator or check user. There is no fixed
format for recording decisions on SPI cases and therefore parsing the data
with regular expressions will not work for most cases. We were required to
visit each SPI case and read the discussion of any administrators
investigating the case and check users involved. This was the bottle neck for
the process and what prevented us from having a larger sample. Although we
continue to add cases to our data set as feasible.
The majority of the SPI cases in Wikipedia end up being confirmed as sock
puppets. This is reasonable since editors file cases after they have already
seen some suspicious behavior. Therefore, to provide a larger number of non-
sock puppet cases, we crawled pairs of editors that have not been involved in
SPI before but that have participated in the same talk pages as editors
involved in SPI cases.
## 5\. The Sockpuppet Corpus
Comment from the sockpuppeteer: -Inanna-
---
Mine was original and i have worked on it more than 4 hours.I have changed it
many times by opinions.Last one was accepted by all the users(except for
khokhoi).I have never used sockpuppets.Please dont care Khokhoi,Tombseye and
Latinus.They are changing all the articles about Turks.The most important and
famous people are on my picture.
Comment from the sockpuppet: Altau
Hello.I am trying to correct uncited numbers in Battle of Sarikamis and
Crimean War by resources but khoikhoi and tombseye always try to revert
them.Could you explain them there is no place for hatred and propagandas,
please?
Comment from another editor: Khoikhoi
Actually, my version WAS the original image. Ask any other user. Inanna’s
image was uploaded later, and was snuck into the page by Inanna’s sockpuppet
before the page got protected. The image has been talked about, and people
have rejected Inanna’s image (see above).
Table 1: Sample excerpt from a single sockpuppet case. We show in boldface
some of the stylistic features shared between the sockpuppeter and the
sockpuppet.
We originally collected around 700 cases, but after manual inspection we
removed about 80 cases where editors did not have content on the talk pages.
These were editors that just made contributions directly to Wikipedia pages
but did not engage in any side discussions about them. The resulting corpus
currently has 623 cases where 305 of them were confirmed SPI cases by
Wikipedia administrators or check users. The remaining 318 are non-sockpuppet
cases that combine 105 SPI cases where the administrators verdict was
negative, and 213 cases we created from other editors.
Examples from a couple of cases are shown in Table 1. In that table we show a
comment from the editor named Inanna that was accused of being the puppeteer
of editor Altau. For comparison purposes we show as well a comment made by
another editor, not involved in the SPI case on the same talk pages. A
noticeable feature in the table is the omission of a white space after the
periods.
The table also shows that the comments resemble what we would see in web forum
data. For our corpus we found out that the average length in characters is
529. While texts are short, previous work has carried out author
identification from tweets [Layton et al., 2010], and many researchers,
ourselves included, have reached good prediction performance on social media
data that is very similar to the data of this corpus. Some statistics about
this dataset are shown in Table 2.
Confirmed SPI cases | 305
---|---
Denied SPI cases | 105
Created non-sock puppet cases | 213
Average number of comments per case | $\sim$ 180
Average number of comments per editors | $\sim$ 83
Table 2: The sockpuppet data set
## 6\. A Machine Learning Approach to Sockpuppet Detection
Figure 1: The bars show average F-measures when testing support vector machine
removing one feature group at a time in a 10 fold cross-validation setting.
Earlier this year we did a case study of adapting a standard machine-learning
authorship attribution approach to predict sockpuppet cases [Solorio et al.,
2013]. This preliminary study shows some promising results for this task. But
it was based on a smaller set of cases, only 77. These 77 cases are a subset
of the editors included in the new version of the corpus.
Here we present new results using all 623 cases in a ten-fold cross-validation
setting. We hope these results can be used as a sort of baseline comparison
for other researchers using this data set.
For these experiments we also changed the underlying framework for the task.
Here we assume any pair of editors can be considered an instance of the
classification problem, a SPI, and the learner has to decide whether to
declare the editors as belonging to the same person or not based on
observations from the comments made by each editor involved. The features used
in this problem are then the pairwise normalized differences of the feature
vectors representing each comment. A complete list of features can be found at
the following link: https://www.dropbox.com/s/15tztqd48jrbr2h/features.list
and a detailed description is in our previous paper [Solorio et al., 2013].
Figure 1 shows the results of training a support vector machine (SVM)
classifier removing one feature group at a time. We used Weka’s implementation
of SVMs with default parameters. The best results (F-measure 73%) are achieved
using all features. These results are very similar to the results attained on
our case study (F-measure 72%).
## 7\. Conclusion
This paper presents a new dataset that will enable research in authorship
attribution under real-world adversarial conditions. The nature of the data is
very similar to what can be found in social media, which makes it an even more
attractive resource as security and privacy concerns in social media data will
continue to grow. The prediction results reported here will also be a good
baseline for future research.
The data set will be available from the project website under a Creative
Commons license. Our goal is to continue adding SPI cases on a regular basis
to maintain an updated resource.
## References
* Brennan et al., 2012 Michael Brennan, Sadia Afroz, and Rachel Greenstadt. 2012\. Adversarial stylometry: Circumventing authorship recognition to preserve privacy and anonymity. ACM Trans. Inf. Syst. Secur., 15(3):12:1–12:22, Nov.
* Escalante et al., 2011 Hugo J. Escalante, Thamar Solorio, and Manuel Montes. 2011\. Local histograms of character n-grams for authorship attribution. In Proceedings of the 49th Annual Meeting of the Association for Computational Linguistics, pages 288–298. Association for Computational Linguistics (ACL).
* Goldstein-Stewart et al., 2008 Jade Goldstein-Stewart, Kerri A. Goodwin, Roberta E. Sabin, and Ransom K. Winder. 2008\. Creating and using a correlated corpora to glean communicative commonalities. In Proceedings of LREC 2008, pages 3029–3035, Marrakech, Morocco, June.
* Koppel et al., 2011 Moshe Koppel, Jonathan Schler, and Shlomo Argamon. 2011\. Authorship attribution in the wild. Language Resources and Evaluation, 45:83–94.
* Layton et al., 2010 Robert Layton, Paul Watters, and Richard Dazeley. 2010\. Authorship attribution for twitter in 140 characters or less. In Second Cybercrime and Trustworthy Computing Workshop, CTC 2010, pages 1–8, Ballart, VIC, Australia, July.
* Novak et al., 2004 Jasmine Novak, Prabhakar Raghavan, and Andrew Tomkins. 2004\. Anti-aliasing on the web. In Proceedings of the 13th international conference on World Wide Web, WWW ’04, pages 30–39, New York, NY, USA. ACM.
* Sidorov et al., 2013 G. Sidorov, F. Velasquez, E. Stamatatos, A. Gelbukh, and L. Chanona-Hérnandez. 2013\. Syntactic n-grams as machine learning features for natural language processing. Expert Systems with Applications.
* Solorio et al., 2013 Thamar Solorio, Ragib Hasan, and Mainul Mizan. 2013\. A case study of sockpuppet detection in wikipedia. In Proceedings of the Workshop on Language Analysis in Social Media, pages 59–68, Atlanta, Georgia, June. Association for Computational Linguistics.
* Stamatatos, 2008 Efstathios Stamatatos. 2008\. A survey on modern authorship attribution methods. Journal of the American Society for Information Science and Technology, 60(3):538–556.
|
arxiv-papers
| 2013-10-24T20:59:27 |
2024-09-04T02:49:52.866294
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Thamar Solorio and Ragib Hasan and Mainul Mizan",
"submitter": "Ragib Hasan",
"url": "https://arxiv.org/abs/1310.6772"
}
|
1310.6934
|
# Generalized activity equations for spiking neural network dynamics
Michael A. Buice1 Carson C. Chow2 1Allen Institute for Brain Science
2Laboratory of Biological Modeling, NIDDK, NIH, Bethesda, MD
###### Abstract
Much progress has been made in uncovering the computational capabilities of
spiking neural networks. However, spiking neurons will always be more
expensive to simulate compared to rate neurons because of the inherent
disparity in time scales - the spike duration time is much shorter than the
inter-spike time, which is much shorter than any learning time scale. In
numerical analysis, this is a classic stiff problem. Spiking neurons are also
much more difficult to study analytically. One possible approach to making
spiking networks more tractable is to augment mean field activity models with
some information about spiking correlations. For example, such a generalized
activity model could carry information about spiking rates and correlations
between spikes self-consistently. Here, we will show how this can be
accomplished by constructing a complete formal probabilistic description of
the network and then expanding around a small parameter such as the inverse of
the number of neurons in the network. The mean field theory of the system
gives a rate-like description. The first order terms in the perturbation
expansion keep track of covariances.
## Introduction
Even with the rapid increase in computing power due to Moore’s law and
proposals to simulate the entire human brain notwithstanding Markram and
collaborators (2012), a realistic simulation of a functioning human brain
performing nontrivial tasks is remote. While it is plausible that a network
the size of the human brain could be simulated in real time Izhikevich and
Edelman (2008); Eliasmith et al. (2012) there are no systematic ways to
explore the parameter space. Technology to experimentally determine all the
parameters in a single brain simultaneously does not exist and any attempt to
infer parameters by fitting to data would require exponentially more computing
power than a single simulation. We also have no idea how much detail is
required. Is it sufficient to simulate a large number of single compartment
neurons or do we need multiple-compartments? How much molecular detail is
required? Do we even know all the important biochemical and biophysical
mechanisms? There are an exponential number of ways a simulation would not
work and figuring out which remains computationally intractable. Hence, an
alternative means to provide appropriate prior distributions for parameter
values and model detail is desirable. Current theoretical explorations of the
brain utilize either abstract mean field models or small numbers of more
biophysical spiking models. The regime of large but finite numbers of spiking
neurons remains largely unexplored. It is not fully known what role spike time
correlations play in the brain. It would thus be very useful if mean field
models could be augmented with some spike correlation information.
This paper outlines a scheme to derive generalized activity equations for the
mean and correlation dynamics of a fully deterministic system of coupled
spiking neurons. It synthesizes methods we have developed to solve two
different types of problems. The first problem was how to compute finite
system size effects in a network of coupled oscillators. We adapted the
methods of the kinetic theory of gases and plasmas Ichimaru (1973); Nicholson
(1993) to solve this problem. The method exploits the exchange symmetry of the
oscillators and characterizes the phases of all the oscillators in terms of a
phase density function $\eta(\theta,t)$, where each oscillator is represented
as a point mass in this density. We then write down a formal flux conservation
equation of this density, called the Klimontovich equation, which completely
characterizes the system. However, because the density is not differentiable,
the Klimontovich equation only exists in the weak or distributional sense.
Previously, e.g Desai and Zwanzig (1978); Strogatz and Mirollo (1991); Abbott
and van Vreeswijk (1993); Treves (1993) the equations were made usable by
taking the “mean field limit” of $N\rightarrow\infty$ and assuming that the
density is differentiable in that limit, resulting in what is called the
Vlasov equation. Instead of immediately taking the mean field limit, we
regularize the density by averaging over initial conditions and parameters and
then expand in the inverse system size $N^{-1}$ around the mean field limit.
This results in a system of coupled moment equations known as the BBGKY moment
hierarchy. In Hildebrand et al. (2007), we solved the moment equations for the
Kuramoto model perturbatively to compute the pair correlation function between
oscillators. However, the procedure was somewhat ad hoc and complicated. We
then subsequently showed in Buice and Chow (2007), that the BBGKY moment
hierarchy could be recast in terms of a density functional of the phase
density. This density functional could be written down explicitly as an
integral over all possible phase histories, i.e. a Feynman-Kac path integral.
The advantage of using this density functional formalism is that the moments
to arbitrary order in $1/N$ could be computed as a steepest-descent expansion
of the path integral, which can be expressed in terms of Feynman diagrams.
This made the calculation more systematic and mechanical. We later applied the
same formalism to synaptically coupled spiking models Buice and Chow (2013a).
Concurrently with this line of research, we also explored the question of how
to generalize population activity equations, such as the Wilson-Cowan
equations, to include the effects of correlations. The motivation for this
question is that the Wilson-Cowan equations are mean field equations and do
not capture the effects of spike-time correlations. For example, the gain in
the Wilson-Cowan equations is fixed, (which is a valid approximation when the
neurons fire asynchronously), but correlations in the firing times can change
the gain Salinas and Sejnowski (2000). Thus, it would be useful to develop a
systematic procedure to augment population activity equations to include spike
correlation effects. The approach we took was to posit plausible microscopic
stochastic dynamics, dubbed the spike model, that reduced to the Wilson-Cowan
equations in the mean field limit and compute the self-consistent moment
equations from that microscopic theory. Buice and Cowan Buice and Cowan (2009)
showed that the solution of the master equation of the spike model could be
expressed formally in terms of a path integral over all possible spiking
histories. The random variable in the path integral is a spike count whereas
in the path integral for the deterministic phase model we described above, the
random variable is a phase density. To generate a system of moment equations
for the microscopic stochastic system, we transformed the random spike count
variable in the path integral into moment variables Buice et al. (2010). This
is accomplished using the effective action approach of field theory, where the
exponent of the cumulant generating functional, called the action, which is a
function of the random variable is Legendre transformed into an effective
action of the cumulants. The desired generalized Wilson-Cowan activity
equations are then the equations of motion of the effective action. This is
analogous to the transformation from Lagrangian variables of position and
velocity to Hamiltonian variables of position and momentum. Here, we show how
to apply the effective action approach to a deterministic system of
synaptically coupled spiking neurons to derive a set of moment equations.
## Approach
Consider a network of single compartment conductance-based neurons
$\displaystyle C\frac{dV_{i}}{dt}$ $\displaystyle=$
$\displaystyle-\sum_{r=1}^{n}g_{r}(x_{i}^{r})(V_{i}-v_{r})+\sum_{j=1}^{N}g_{ij}s_{j}(t)$
$\displaystyle\tau_{i}^{r}\frac{dx_{i}^{r}}{dt}$ $\displaystyle=$
$\displaystyle f(V_{i},x_{i})$ $\displaystyle\tau_{j}\frac{ds_{j}}{dt}$
$\displaystyle=$ $\displaystyle h(V_{j},s_{j})$
$\displaystyle\tau_{g}\frac{dg_{ij}}{dt}$ $\displaystyle=$
$\displaystyle\phi(g_{ij},V)$
The equations are remarkably stiff with time scales spanning orders of
magnitude from milliseconds for ion channels, to seconds for adaptation, and
from hours to years for changes in synaptic weights and connections. Parameter
values must be assigned for $10^{11}$ neurons with $10^{4}$ connections each.
Here, we present a formalism to derive a set of reduced activity equations
directly from a network of deterministic spiking neurons that capture the
spike rate and spike correlation dynamics. The formalism first constructs a
density functional for the firing dynamics of all the neurons in a network. It
then systematically marginalizes the unwanted degrees of freedom to isolate a
set of self-consistent equations for the desired quantities. For heuristic
reasons, we derive an example set of generalized activity equations for the
first and second cumulants of the firing dynamics of a simple spiking model
but the method can be applied to any spiking model.
A convenient form to express spiking dynamics is with a phase oscillator.
Consider the quadratic integrate-and-fire neuron
$\frac{dV_{i}}{dt}=I_{i}+V_{i}^{2}+\alpha_{i}u(t)$ (1)
where $I$ is an external current and $u(t)$ are the synaptic currents with
some weight $\alpha_{i}$. The spike is said to occur when $V$ goes to infinity
whereupon it is reset to minus infinity. The quadratic nonlinearity ensures
that this transit will occur in a finite amount of time. The substitution
$V=\tan(\theta/2)$ yields the theta model Ermentrout and Kopell (1986):
$\frac{d\theta_{i}}{dt}=1-\cos\theta_{i}+(1+\cos\theta_{i})(I_{i}+\alpha_{i}u)$
(2)
which is the normal form of a Type I neuron near the bifurcation to firing
Ermentrout (1996). The phase neuron is an adequate approximation to spiking
dynamics provided the inputs are not overly strong as to disturb the limit
cycle. The phase neuron also includes realistic dynamics such as not firing
when the input is below threshold. Coupled phase models arise naturally in
weakly coupled neural networks Ermentrout and Kopell (1991); Golomb and Hansel
(2000); Hoppensteadt and Izhikevich (1997). They include the Kuramoto model
Kuramoto (1984), which we have previously analyzed Hildebrand et al. (2007);
Buice and Chow (2007).
Here, we consider the phase dynamics of a set of $N$ coupled phase neurons
obeying
$\displaystyle\dot{\theta}_{i}$ $\displaystyle=$ $\displaystyle
F(\theta,\gamma_{i},u(t))$ (3) $\displaystyle\dot{u}(t)$ $\displaystyle=$
$\displaystyle-\beta u(t)+\beta\nu(t)$ (4) $\displaystyle\nu(t)$
$\displaystyle=$
$\displaystyle\frac{1}{N}\sum_{j=1}^{N}\sum_{l}\delta(t-t^{l}_{j})$ (5)
where each neuron has a phase $\theta_{i}$ that is indexed by $i$, $u$ is a
global synaptic drive, $F(\theta,\gamma,u)$ is the phase and synaptic drive
dependent frequency, $\gamma_{i}$ represents all the parameters for neuron $i$
drawn from a distribution with density $g(\gamma)$, $\nu$ is the population
firing rate of the network,$t^{l}_{j}$ is the $l$th firing time of neuron $j$
and a neuron fires when its phase crosses $\pi$. In the present paper, we
consider all-to-all or global coupling through a synaptic drive variable
$u(t)$. However, our basic approach is not restricted to global coupling.
We can encapsulate the phase information of all the neurons into a neuron
density function Hildebrand et al. (2007); Buice and Chow (2007, 2011, 2013b,
2013a).
$\eta(\theta,\gamma,t)=\frac{1}{N}\sum_{i=1}^{N}\delta(\theta-\theta_{i}(t))\delta(\gamma-\gamma_{i})$
(6)
where $\delta(\cdot)$ is the Dirac delta functional, and $\theta_{i}(t)$ is a
solution to system (3)-(5). The neuron density gives a count of the number of
neurons with phase $\theta$ and synaptic strength $\gamma$ at time $t$. Using
the fact that the Dirac delta functional in (5) can be expressed as
$\sum_{l}\delta(t-t_{j}^{l})=\dot{\theta_{j}}\delta(\pi-\theta_{j})$, the
population firing rate can be rewritten as
$\nu(t)=\int d\gamma\,F(\pi,\gamma,u(t))\eta(\pi,\gamma,t)$ (7)
The neuron density formally obeys the conservation equation
$\frac{\partial}{\partial
t}\eta(\theta,\gamma,t)+\frac{\partial}{\partial\theta}\left[F\eta(\theta,\gamma,t)\right]=0$
(8)
with initial condition $\eta(\theta,\gamma,t_{0})=\eta_{0}(\theta,\gamma)$ and
$u(t_{0})=u_{0}$. Equation (8) is known as the Klimontovich equation Ichimaru
(1973); Liboff (2003). The Klimontovich equation, the equation for the
synaptic drive (4), and the firing rate expressed in terms of the neuron
density (7), fully define the system. The system is still fully deterministic
but is now in a form where various sets of reduced descriptions can be
derived. Here, we will produce an example of a set of reduced equations or
generalized activity equations that capture some aspects of the spiking
dynamics. The path we take towards the end will require the introduction of
some formal machinery that may obscure the intuition around the
approximations. However, we feel that it is useful because it provides a
systematic and controlled way of generating averaged quantities that can be
easily generalized.
For finite $N$, (8) is only valid in the weak or distributional sense since
$\eta$ is not differentiable. In the $N\rightarrow\infty$ limit, it has been
argued that $\eta$ will approach a smooth density $\rho$ that evolves
according to the Vlasov equation that has the same form as (8) but with $\eta$
replaced by $\rho$ Ichimaru (1973); Nicholson (1993); Desai and Zwanzig
(1978); Strogatz and Mirollo (1991); Hildebrand et al. (2007). This has been
proved rigorously in the case where noise is added using the theory of coupled
diffusions McKean Jr (1966); Faugeras et al. (2009); Touboul (2012); Baladron
et al. (2012). This $N\rightarrow\infty$ limit is called mean field theory. In
mean field theory, the original microscopic many body neuronal network is
represented by a smooth macroscopic density function. In other words, the
ensemble of networks prepared with different microscopic initial conditions is
sharply peaked at the mean field solution. For large but finite $N$, there
will be deviations away from mean field Hildebrand et al. (2007); Buice and
Chow (2007, 2013a, 2013b). These deviations can be characterized in terms of a
distribution over an ensemble of coupled networks that are all prepared with
different initial conditions and parameter values. Here, we show how a
perturbation theory in $N^{-1}$ can be developed to expand around the mean
field solution. This requires the construction of the probability density
functional over the ensemble of spiking neural networks. We adapt the tools of
statistical field theory to perform such a construction.
### Formalism
The complete description of the system given by equations (4), (7), and (8)
can be written as
$\displaystyle\dot{u}(t)+\beta u(t)-\beta\int
d\gamma\,F(\pi,\gamma,u(t))\eta(\pi,\gamma,t)=0$ (9)
$\displaystyle\frac{\partial}{\partial
t}\eta(\theta,\gamma,t)+\frac{\partial}{\partial\theta}\left[F(\theta,\gamma,u(t))\eta(\theta,\gamma,t)\right]\equiv{\cal
L}\eta=0$ (10)
The probability density functional governing the system specified by the
synaptic drive and Klimontovich equations (9) and (10) given initial
conditions $(\eta_{0},u_{0})$ can be written as
$\displaystyle P[\eta,u]=$ $\displaystyle\int{\cal D}u_{0}(t)\,{\cal
D}\eta_{0}(\theta,\gamma)\,P[\eta,u|\eta_{0},u_{0}]\,P_{0}[\eta_{0},u_{0},\gamma]$
(11)
where $P[\eta,u|\eta_{0},u_{0}]$ is the conditional probability density
functional of the functions $(\eta,u)$, and $P_{0}[\eta_{0},u_{0}]$ is the
density functional over initial conditions of the system. The integral is a
Feynman-Kac path integral over all allowed initial condition functions.
Formally we can write $P[\eta,u|\eta_{0},u_{0}]$ as a point mass (Dirac delta)
located at the solutions of (9) and (10) given the initial conditions:
$\displaystyle\delta\left[{\cal
L}\eta-\eta_{0}\delta(t-t_{0})\right]\delta\left[\dot{u}+\beta u-\beta\int
d\gamma\,F(\pi,\gamma,u(t))\eta(\pi,\gamma,t)-u_{0}\delta(t-t_{0})\right]$
The probability density functional (11) is then
$\displaystyle P[\eta,u]=$ $\displaystyle\int{\cal D}u_{0}(t)\,{\cal
D}\eta_{0}(\theta,\gamma)\,\delta\left[{\cal
L}\eta-\eta_{0}\delta(t-t_{0})\right]$
$\displaystyle\times\delta\left[\dot{u}+\beta u-\beta\int
d\gamma\,F(\pi,\gamma,u(t))\eta(\pi,\gamma,t)-u_{0}\delta(t-t_{0})\right]P_{0}[\eta_{0},u_{0},\gamma]$
(12)
Equation (12) can be made useful by noting that the Fourier representation of
a Dirac delta is given by $\delta(x)\propto\int dk\,e^{ikx}$. Using the
infinite dimensional Fourier functional transform then gives
$P[\eta,u]=\int{\cal D}\tilde{\eta}{\cal
D}\tilde{u}\,e^{-NS[\eta,\tilde{\eta},u,\tilde{u}]}.$
The exponent $S[\eta,u]$ in the probability density functional is called the
action and has the form
$S=S_{u}+S_{\varphi}+S_{0}$ (13)
where
$\displaystyle S_{\varphi}=\int d\theta d\gamma
dt\,\tilde{\varphi}(x)\left[\partial_{t}\varphi(x)+\partial_{\theta}F(\theta,\gamma,u(t))\varphi(x)\right]$
(14)
represents the contribution of the transformed neuron density to the action,
$\displaystyle S_{u}=\frac{1}{N}\int dt\,\tilde{u}(t)\left(\dot{u}(t)+\beta
u(t)-\beta\int d\gamma
F(\pi,\gamma,u(t))[\tilde{\varphi}(\pi,\gamma,t)+1]\varphi(\pi,\gamma,t)\right)$
(15)
represents the global synaptic drive,
$S_{0}[\tilde{\varphi}_{0}(x_{0}),u_{0}(t_{0})]$ represents the initial
conditions, and $x=(\theta,\gamma,t)$. For the case where the neurons are
considered to be independent in the initial state, we have
$\displaystyle S_{0}[\tilde{\varphi}_{0}(x_{0}),u_{0}(t_{0})]$
$\displaystyle=-\frac{1}{N}\tilde{u}(t_{0})u_{0}-\ln\left(1+\int d\theta
d\gamma\tilde{\varphi}_{0}(\theta,\gamma,t_{0})\rho_{0}(\theta,\gamma,t_{0})\right)$
(16)
where $u_{0}$ is the initial value of the coupling variable and
$\rho_{0}(\theta,\gamma,t)$ is the distribution from which the initial
configuration is drawn for each neuron. The action includes two imaginary
auxiliary response fields (indicated with a tilde), which are the infinite
dimensional Fourier transform variables. The factor of $1/N$ appears to ensure
correct scaling between the $u$ and $\varphi$ variables since $u$ applies to a
single neuron while $\varphi$ applies to the entire population. The full
derivation is given in Buice and Chow (2013a) and a review of path integral
methods applied to differential equations is given in Buice and Chow (2010).
In the course of the derivation we have made a Doi-Peliti-Jannsen
transformation Janssen and Täuber (2005); Buice and Chow (2013a), given by
$\displaystyle\varphi(x)$ $\displaystyle=$
$\displaystyle\eta(x)e^{-\tilde{\eta}(x)}$ $\displaystyle\tilde{\varphi}(x)$
$\displaystyle=$ $\displaystyle e^{\tilde{\eta}(x)}-1$
In deriving the action, we have explicitly chosen the Ito convention so that
the auxiliary variables only depend on variables in the past. The action (13)
contains all the information about the statistics of the network.
The moments for this distribution can be obtained by taking functional
derivatives of a moment generating functional. Generally, the moment
generating function for a random variable is given by the expectation value of
the exponential of that variable with a single parameter. Because our goal is
to transform to new variables for the first and second cumulants, we form a
“two-field” moment generating functional, which includes a second parameter
for pairs of random variables,
$\displaystyle\exp(N\,$ $\displaystyle W[J,K])=$ $\displaystyle\int{\cal
D}\xi\,\exp\left[-NS[\xi]+N\int dx\,J^{i}(x)\xi_{i}(x)+\frac{N}{2}\int
dxdx^{\prime}\xi_{i}(x)K^{ij}(x,x^{\prime})\xi_{j}(x^{\prime})\right]$ (17)
where $J$ and $K$ are moment generating fields, $\xi_{1}(x)=u(t)$,
$\xi_{2}(x)=\tilde{u}(t)$, $\xi_{3}(x)=\varphi(x)$,
$\xi_{4}(x)=\tilde{\varphi}(x)$, and $x=(\theta,\gamma,t)$. Einstein summation
convention is observed beween upper and lower indices. Unindexed variables
represent vectors. The integration measure $dx$ is assumed to be $dt$ when
involving indices 1 and 2. Covariances between an odd and even index
corresponds to a covariance between a field and an auxiliary field. Based on
the structure of the action $S$ and (17) we see that this represents a linear
propagator and by causality and the choice of the Ito convention is only
nonzero if the time of the auxiliary field is evaluated at an earlier time
than the field. Covariances between two even indices correspond to that
between two auxiliary fields and are always zero because of the Ito
convention.
The mean and covariances of $\xi$ can be obtained by taking derivatives of the
action $W[J,K]$ in (17), with respect to $J$ and $K$ and setting $J$ and $K$
to zero:
$\displaystyle\frac{\delta W}{\delta J^{i}}$
$\displaystyle=\left.\langle\xi_{i}\rangle\right|_{J,K=0}$
$\displaystyle\frac{\delta W}{\delta K^{ij}}$
$\displaystyle=\left.\frac{1}{2}\langle\xi_{i}\xi_{j}\rangle\right|_{J,K=0}$
Expressions for these moments can be computed by expanding the path integral
in (17) perturbatively around some mean field solution. However, this can be
unwieldy if closed form expressions for the mean field equations do not exist.
Alternatively, the moments at any order can be expressed as self-consistent
dynamical equations that can be analyzed or simulated numerically. Such
equations form a set of generalized activity equations for the means
$a_{i}=\langle\xi_{i}\rangle$, and covariances
$C_{ij}=N[\langle\xi_{i}\xi_{j}\rangle-a_{i}a_{j}]$.
We derive the generalized activity equations by Legendre transforming the
action $W$, which is a function of $J$ and $K$, to an effective action
$\Gamma$ that is a function of $a$ and $C$. Just as a Fourier transform
expresses a function in terms of its frequencies, a Legendre transform
expresses a convex function in terms of its derivatives. This is appropriate
for our case because the moments are derivatives of the action. The Legendre
transform of $W[J,K]$ is
$\Gamma[a,C]=-W[J,K]+\int dxJ^{i}a_{i}+\frac{1}{2}\int
dxdx^{\prime}\left[a_{i}a_{j}+\frac{1}{N}C_{ij}\right]K^{ij}$ (18)
which must obey the constraints
$\displaystyle\frac{\delta W}{\delta J^{i}}$ $\displaystyle=a_{i}$
$\displaystyle\frac{\delta W}{\delta K^{ij}}$
$\displaystyle=\frac{1}{2}\left[a_{i}a_{j}+\frac{1}{N}C_{ij}\right]$
and
$\displaystyle\frac{\delta\Gamma}{\delta a_{i}}$
$\displaystyle\equiv\Gamma^{i,00}=J^{i}+\frac{1}{2}a_{j}\left[K^{ij}+K^{ji}\right]$
$\displaystyle\frac{\delta\Gamma}{\delta C_{ij}}$
$\displaystyle\equiv\Gamma^{0,ij}=\frac{1}{2N}K^{ij}$ (19)
The generalized activity equations are given by the equations of motion of the
effective action, in direct analogy to the Euler-Lagrange equations of
classical mechanics, and are obtained by setting $J^{i}=0$ and $K^{ij}=0$ in
(19).
In essence, what the effective action does is to take a probabilistic
(statistical mechanical) system in the variables $\xi$ with action $S$ and
transform them to a deterministic (classical mechanical) system with an action
$\Gamma$. Our approach here follows that used in Buice et al. (2010) to
construct generalized activity equations for the Wilson Cowan model. However,
there are major differences between that system and this one. In Buice et al.
(2010), the microscopic equations were for the spike counts of an inherently
probabilistic model so the effective action and ensuing generalized activity
equations could be constructed directly from the Markovian spike count
dynamics. Here, we start from deterministically firing individual neurons and
get to a probabilistic description through the Klimontovich equation. It would
be straightforward to include stochastic effects into the spiking dynamics.
Using (18) in (17) gives
$\displaystyle\exp(-N\,\Gamma[a,C])=\int{\cal D}\psi\,\exp\left[-NS[\xi]+N\int
dx\,J^{i}(\xi_{i}-a_{i})\right.$ $\displaystyle+\left.\frac{N}{2}\int
dxdx^{\prime}\left[\xi_{i}\xi_{j}-a_{i}a_{j}-\frac{1}{N}C_{ij}\right]K^{ij}\right]$
(20)
where $J$ and $K$ are constrainted by (19). We cannot compute the effective
action explicitly but we can compute it perturbatively in $N^{-1}$. We first
perform a shift $\xi_{i}=a_{i}+\psi_{i}$, expand the action as
$S[a+\psi]=S[a]+\int dx(L^{i}[a]\psi_{i}+(1/2)\int
dx^{\prime}L^{ij}[a]\psi_{i}\psi_{j})+\cdots$ and substitute for $J$ and $K$
with the constraints (19) to obtain
$\displaystyle\exp(-N\,\Gamma[a,C])=$
$\displaystyle\exp(-NS[a]-N\operatorname{Tr}\Gamma^{0,ij}C_{ij})\int{\cal
D}\psi\,\exp\left[-N\int dx\bigg{(}L^{i}[a]\psi_{i}\right.$
$\displaystyle\left.+\frac{1}{2}\int
dx^{\prime}L^{ij}[a]\psi_{i}\psi_{j}\bigg{)}+N\int
dx\,\Gamma^{i,00}\psi_{i}+N^{2}\int
dxdx^{\prime}\psi_{i}\psi_{j}\Gamma^{0,ij}\right]$ (21)
where
$\operatorname{Tr}A^{ij}B_{ij}=\int
dxdx^{\prime}A^{ij}(x,x^{\prime})B_{ij}(x,x^{\prime})$ (22)
Our goal is to construct an expansion for $\Gamma$ by collecting terms in
successive orders of $N^{-1}$ in the path integral of (21). Expanding $\Gamma$
as $\Gamma[a,C]=\Gamma_{0}+N^{-1}\Gamma_{1}+N^{-2}\Gamma_{2}$ and equating
coefficients of $N$ in (21) immediately leads to the conclusion that
$\Gamma_{0}=S[a]$, which gives
$\displaystyle\exp(-N\,\Gamma[a,C])=\exp\left(-NS[a]-\operatorname{Tr}\Gamma_{1}^{0,ij}C_{ij}\right)\int{\cal
D}\psi\,\exp\left[-\frac{N}{2}\int dxL^{ij}[a]\psi_{i}\psi_{j}\right.$
$\displaystyle+\left.N\int dx\,\Gamma_{1}^{0,ij}\psi_{i}\psi_{j}\right]$
where higher order terms in $N^{-1}$ are not included. To lowest nonzero order
$\Gamma^{0,ij}=N^{-1}\Gamma_{1}^{0,ij}$ since $\Gamma_{0}$ is only a function
of $a$ and not $C$. If we set
$\Gamma_{1}^{0,ij}=(1/2)L^{ij}-(1/2)Q^{ij},$ (23)
we obtain
$\displaystyle\exp(-N\,\Gamma[a,C])=\exp\left(-NS[a]-\frac{1}{2}\operatorname{Tr}L^{ij}C_{ij}+\frac{1}{2}\operatorname{Tr}Q^{ij}C_{ij}\right)$
$\displaystyle\times\int{\cal D}\psi\,\exp\left[-\frac{N}{2}\int
dx\,Q^{ij}[a]\psi_{i}\psi_{j}\right]$ (24)
to order $N^{-1}$. $Q^{ij}$ is an unknown function of $a$ and $C$, which we
will deduce using self-consistency. The path integral in (24), which is an
infinite dimensional Gaussian that can be explicitly integrated, is
proportional to $1/\sqrt{\det Q^{ij}}=\exp(-(1/2)\ln\det
Q^{ij})=\exp(-(1/2)\operatorname{Tr}\ln Q^{ij})$, using properties of
matrices. Hence, (24) becomes
$\displaystyle\exp(-N\,\Gamma[a,C])=\exp\left(-NS[a]-\frac{1}{2}\operatorname{Tr}L^{ij}C_{ij}-\frac{1}{2}\operatorname{Tr}Q^{ij}C_{ij}+\frac{1}{2}\operatorname{Tr}\ln
Q^{ij}\right)$
and
$\displaystyle\Gamma[a,C]=S[a]+\frac{1}{2N}\operatorname{Tr}L^{ij}C_{ij}+\frac{1}{2N}\operatorname{Tr}\ln
Q^{ij}-\frac{1}{2N}\operatorname{Tr}Q^{ij}C_{ij}$
Taking the derivative of $\Gamma$ with respect to $C_{ij}$ yields
$\displaystyle\Gamma^{0,ij}=\frac{1}{2N}\left(L^{ij}+(Q^{-1})^{kl}\frac{\partial}{\partial
C_{ij}}Q^{lk}-\frac{\partial}{\partial C_{ij}}(Q^{kl}C_{lk})\right)$
Self consistency with (23) then requires that $Q^{ij}=(C^{-1})^{ij}$ which
leads to the effective action
$\Gamma[a,C]=S[a]+\frac{1}{2N}\operatorname{Tr}\ln(C^{-1})^{ij}+\frac{1}{2N}\operatorname{Tr}L^{ij}C_{ij}$
(25)
where
$\int
dx^{\prime}\,(C^{-1})^{ik}(x,x^{\prime})C_{kj}(x^{\prime},x_{0})=\delta_{ij}\delta(x-x_{0})$
and we have dropped the irrelevant constant terms.
The equations of motion to order $N^{-1}$ are obtained from (19) with $J^{i}$
and $K^{ij}$ set to zero:
$\displaystyle\frac{\delta S[a]}{\delta
a_{i}}+\frac{1}{2N}\frac{\delta}{\delta a_{i}}\operatorname{Tr}L^{ij}C_{ij}=0$
(26) $\displaystyle\frac{1}{2N}[-(C^{-1})^{ij}+L^{ij}]=0$ (27)
and (27) can be rewritten as
$\displaystyle\int
dx^{\prime}L^{ik}(x,x^{\prime})C_{kj}(x^{\prime},x_{0})=\delta_{ij}\delta(x-x_{0})$
(28)
Hence, given any network of spiking neurons, we can write down the a set of
generalized activity equations for the mean and covariance functions by 1)
constructing a neuron density function, 2) writing down the conservation law
(Klimontovich equation), 3) constructing the action and 4) using formulas (26)
and (28). We could have constructed these equations directly by multiplying
the Klimontovich and synaptic drive equations by various factors of $u$ and
$\eta$ and recombining. However, as we saw in Buice et al. (2010) this is not
a straightforward calculation. The effective action approach makes this much
more systematic and mechanical.
### Phase Model Example
We now present a simple example to demonstrate the concepts and approximations
involved in our expansion. Our goal is not to analyze the system per se but
only to demonstrate the application of our method in a heuristic setting. We
begin with a simple nonleaky integrate-and-fire neuron model, which responds
to a global coupling variable. This is a special case of the dynamics given
above, with $F$ given by
$\displaystyle F[\theta,\gamma,u]=I(t)+\gamma u$ (29)
The action from (14) and (15) is
$\displaystyle S[a]=$ $\displaystyle\int d\theta d\gamma
dt\,a_{4}(x)\left[\partial_{t}a_{3}(x)+\partial_{\theta}(I+\gamma
a_{1}(t))a_{3}(x)\right]$ $\displaystyle+\frac{1}{N}\int
dt\,a_{2}(t)\left(\dot{a}_{1}(t)+\beta a_{1}(t)-\beta\int d\gamma\,(I+\gamma
a_{1}(t))[a_{4}(\pi,\gamma,t)+1]a_{3}(\pi,\gamma,t)\right)$ (30)
and we ignore initial conditions for now.
In order to construct the generalized activity equations we need to compute
the first and second derivatives of the action $L^{i}$ and $L^{ij}$. Taking
the first derivative of (30) gives
$\displaystyle L^{1}[a](x,x^{\prime})=\frac{\delta S[a(x)]}{\delta
a_{1}(t^{\prime})}$ $\displaystyle=\int d\theta d\gamma\,dt\gamma
a_{4}(x)\partial_{\theta}a_{3}(x)\delta(t-t^{\prime})$
$\displaystyle+\frac{1}{N}\left[\int
dt\,a_{2}(t)\frac{d}{dt}\delta(t-t^{\prime})+\beta
a_{2}(t^{\prime})-a_{2}(t^{\prime})\beta\int
d\gamma\,\gamma[a_{4}(\pi,\gamma,t^{\prime})+1]a_{3}(\pi,\gamma,t^{\prime})\right]$
$\displaystyle L^{2}[a](x,x^{\prime})=\frac{\delta S[a(x)]}{\delta
a_{2}(t^{\prime})}$
$\displaystyle=\frac{1}{N}\left[\frac{da_{1}}{dt^{\prime}}+\beta
a_{1}(t^{\prime})-\beta\int d\gamma(I+\gamma
a_{1}(t^{\prime}))[a_{4}(\pi,\gamma,t^{\prime})+1]a_{3}(\pi,\gamma,t^{\prime})\right]$
$\displaystyle L^{3}[a](x,x^{\prime})=\frac{\delta S[a(x)]}{\delta
a_{3}(x^{\prime})}$ $\displaystyle=\int
dt\,a_{4}(\theta^{\prime},\gamma^{\prime},t)\partial_{t}\delta(t-t^{\prime})+\int
d\theta
a_{4}(\theta,\gamma^{\prime},t^{\prime})\partial_{\theta}(I+\gamma^{\prime}a_{1}(t^{\prime}))\delta(\theta-\theta^{\prime})$
$\displaystyle-\frac{\beta}{N}a_{2}(t^{\prime})(I+\gamma^{\prime}a_{1}(t^{\prime}))(a_{4}(\pi,\gamma^{\prime},t^{\prime})+1)\delta(\pi-\theta^{\prime})$
$\displaystyle L^{4}[a](x,x^{\prime})=\frac{\delta S[a(x)]}{\delta
a_{4}(x^{\prime})}$
$\displaystyle=\partial_{t^{\prime}}a_{3}(x^{\prime})+\partial_{\theta^{\prime}}(I+\gamma^{\prime}a_{1}(t^{\prime}))a_{3}(x^{\prime})-\frac{\beta}{N}a_{2}(t^{\prime})(I+\gamma^{\prime}a_{1}(t^{\prime}))a_{3}(\pi,\gamma^{\prime},t^{\prime})\delta(\pi-\theta^{\prime})$
The mean field equations are obtained by solving $L^{i}=0$ using
(LABEL:firstderivs). We immediately see that $a_{2}=a_{4}=0$ are solutions,
which leaves us with
$\displaystyle\dot{a_{1}}+\beta a_{1}-\beta\int d\gamma(I+\gamma
a_{1})a_{3}(\pi,\gamma,t)=0$ (32) $\displaystyle\partial_{t}a_{3}+(I+\gamma
a_{1})\partial_{\theta}a_{3}=0$ (33)
The mean field equations should be compared to those of the spike response
model Gerstner (1995, 2000). We can also solve (33) directly to obtain
$a_{3}(x,t)=\rho_{0}\left(\theta-\int_{t_{0}}^{t}dt^{\prime}\left[I_{\Omega}(t^{\prime})+\gamma
a_{1}(t^{\prime})\right],\gamma,\Omega\right)$
where $\rho_{0}$ is the initial distribution. If the neurons are initially
distributed uniformly in phase, then $\rho_{0}=g(\gamma)/2\pi$ and the mean
field equations reduce to
$\dot{a_{1}}(t)+\beta
a_{1}(t)-\frac{\beta}{2\pi}\left(I+\bar{\gamma}a_{1}(t)\right)=0$ (34)
which has the form of the Wilson-Cowan equation, with
$(\beta/2\pi)\left(I+\bar{\gamma}a_{1}\right)$ acting as a gain function.
Hence, the Wilson-Cowan equation is a full description of the infinitely large
system limit of a network of globally coupled simple phase oscillators in the
asynchronous state. For all other initial conditions, the one-neuron
conservation equation (called the Vlasov equation in kinetic theory) must be
included in mean field theory.
To go beyond mean field theory we need to compute
$L^{ij}(x,x^{\prime},x^{\prime\prime})=\delta L^{i}(x,x^{\prime})/\delta
a_{j}(x^{\prime\prime})$:
$\displaystyle L^{11}[a]$ $\displaystyle=0$ $\displaystyle L^{12}[a]$
$\displaystyle=\frac{1}{N}\left[-\frac{d}{dt^{\prime\prime}}+\beta-\beta\int
d\gamma\,\gamma[a_{4}(\pi,\gamma,t^{\prime\prime})+1]a_{3}(\pi,\gamma,t^{\prime\prime})]\right]\delta(t^{\prime\prime}-t^{\prime})$
$\displaystyle L^{13}[a]$ $\displaystyle=\left[\gamma^{\prime\prime}\int
d\theta\,a_{4}(x)\delta(\gamma-\gamma^{\prime\prime})\partial_{\theta}\delta(\theta-\theta^{\prime\prime})-\frac{\beta}{N}\gamma^{\prime\prime}a_{2}(t^{\prime})[a_{4}(\pi,\gamma^{\prime\prime},t^{\prime\prime})+1]\delta(\pi-\theta^{\prime\prime})\right]\delta(t^{\prime}-t^{\prime\prime})$
$\displaystyle L^{14}[a]$
$\displaystyle=\left[\gamma^{\prime\prime}\partial_{\theta^{\prime\prime}}a_{3}(x^{\prime\prime})-\frac{\beta}{N}\gamma^{\prime\prime}a_{2}(t^{\prime\prime})a_{3}(\pi,\gamma^{\prime\prime},t^{\prime\prime})\delta(\pi-\theta^{\prime\prime})\right]\delta(t^{\prime}-t^{\prime\prime})$
$\displaystyle L^{21}[a]$
$\displaystyle=\frac{1}{N}\left[\frac{d}{dt^{\prime}}+\beta-\beta\int
d\gamma\,\gamma[a_{4}(\pi,\gamma,t^{\prime})+1]a_{3}(\pi,\gamma,t^{\prime})\right]\delta(t^{\prime}-t^{\prime\prime})$
$\displaystyle L^{22}[a]$ $\displaystyle=0$ $\displaystyle L^{23}[a]$
$\displaystyle=-\frac{\beta}{N}(I+\gamma^{\prime\prime}a_{1}(t^{\prime}))[a_{4}(\pi,\gamma^{\prime\prime},t^{\prime}))+1]\delta(\pi-\theta^{\prime})\delta(t^{\prime}-t^{\prime\prime})$
$\displaystyle L^{24}[a]$
$\displaystyle=-\frac{\beta}{N}(I+\gamma^{\prime\prime}a_{1}(t^{\prime}))a_{3}(\pi,\gamma^{\prime\prime},t^{\prime})]\delta(\pi-\theta^{\prime\prime})\delta(t^{\prime}-t^{\prime\prime})$
$\displaystyle L^{31}[a]$ $\displaystyle=\left[\int
d\theta\,a_{4}(\theta,\gamma^{\prime},t^{\prime})\gamma^{\prime}\partial_{\theta}\delta(\theta-\theta^{\prime})-\frac{\beta}{N}a_{2}(t^{\prime})\gamma^{\prime}[a_{4}(\pi,\gamma^{\prime},t^{\prime})+1]\delta(\pi-\theta^{\prime})\right]\delta(t^{\prime}-t^{\prime\prime})$
$\displaystyle L^{32}[a]$
$\displaystyle=-\frac{\beta}{N}(I+\gamma^{\prime}a_{1}(t^{\prime}))(a_{4}(\pi,\gamma^{\prime},t^{\prime})+1)\delta(\pi-\theta^{\prime})\delta(t^{\prime}-t^{\prime\prime})$
$\displaystyle L^{33}[a]$ $\displaystyle=0$ $\displaystyle L^{34}[a]$
$\displaystyle=\left[\delta(\theta^{\prime}-\theta^{\prime\prime})\partial_{t^{\prime\prime}}-\partial_{\theta^{\prime\prime}}(I+\gamma^{\prime}a_{1}(t^{\prime}))\delta(\theta^{\prime\prime}-\theta^{\prime})\right.$
$\displaystyle\left.-\frac{\beta}{N}a_{2}(t^{\prime})(I+\gamma^{\prime}a_{1}(t^{\prime}))\delta(\pi-\theta^{\prime})\delta(\pi-\theta^{\prime\prime})\right]\delta(\gamma^{\prime}-\gamma^{\prime\prime})\delta(t^{\prime\prime}-t^{\prime})$
$\displaystyle L^{41}[a]$
$\displaystyle=\left[\partial_{\theta^{\prime}}\gamma^{\prime}a_{3}(x^{\prime})-\frac{\beta}{N}a_{2}(t^{\prime})\gamma^{\prime}a_{3}(\pi,\gamma^{\prime},t^{\prime})\delta(\pi-\theta^{\prime})\right]\delta(t^{\prime}-t^{\prime\prime})$
$\displaystyle L^{42}[a]$
$\displaystyle=-\frac{\beta}{N}(I+\gamma^{\prime}a_{1}(t^{\prime}))a_{3}(\pi,\gamma^{\prime},t^{\prime})\delta(\pi-\theta^{\prime})\delta(t^{\prime}-t^{\prime\prime})$
$\displaystyle L^{43}[a]$
$\displaystyle=\partial_{t^{\prime}}\delta(x^{\prime}-x^{\prime\prime})+\partial_{\theta^{\prime}}(I+\gamma^{\prime}a_{1}(t^{\prime}))\delta(x^{\prime}-x^{\prime\prime})$
$\displaystyle-\frac{\beta}{N}a_{2}(t^{\prime})(I+\gamma
a_{1}(t^{\prime}))\delta(\pi-\theta^{\prime})\delta(\pi-\theta^{\prime\prime})\delta(\gamma^{\prime}-\gamma^{\prime\prime})\delta(t^{\prime}-t^{\prime\prime})$
$\displaystyle L^{44}[a]$ $\displaystyle=0$
The activity equations for the means to order $N^{-1}$ are given by (26). The
only nonzero contributions are given by $L^{13}$ and $L^{31}$ resulting in
$\displaystyle L^{2}+\frac{1}{2N}\frac{\delta}{\delta a_{2}}\int
dxdx^{\prime}(L^{13}C_{13}+L^{31}C_{31})=0$ $\displaystyle
L^{4}+\frac{1}{2N}\frac{\delta}{\delta a_{4}}\int
dxdx^{\prime}(L^{13}C_{13}+L^{31}C_{31})=0$
since $a_{2}=a_{4}=0$ and correlations involving response variables (even
indices) will be zero for equal times. The full activity equations for the
means are thus
$\displaystyle\dot{a_{1}}+\beta a_{1}-\beta\int d\gamma(I+\gamma
a_{1})a_{3}(\pi,\gamma,t)-\frac{\beta}{N}\int d\gamma\,\gamma
C(\pi,\gamma,t)=0$ (35) $\displaystyle\partial_{t}a_{3}+(I+\gamma
a_{1})\partial_{\theta}a_{3}+\frac{1}{N}\gamma\partial_{\theta}C(\theta,\gamma,t)=0$
(36)
where
$C(\theta,\gamma,t)=C_{13}(t;\theta,\gamma,t)=C_{31}(\theta,\gamma,t;t)$.
We can now use the $L^{ij}$ in (28) to obtain activity equations for $C_{ij}$.
There will be sixteen coupled equations in total but the applicable nonzero
ones are
$\displaystyle\left[\frac{d}{dt}+\beta-\beta\int d\gamma\,\gamma
a_{3}(\pi,\gamma,t)]\right]$ $\displaystyle C_{11}(t;t_{0})-\beta\int
d\gamma\,(I+\gamma a_{1})C_{31}(\pi,\gamma,t;t_{0})$ $\displaystyle-\beta\int
d\gamma\,(I+\gamma a_{1}(t))a_{3}(\pi,\gamma,t)C_{41}(\pi,\gamma,t;t_{0})=0$
(37) $\displaystyle\left[\frac{d}{dt}+\beta-\beta\int d\gamma\,\gamma
a_{3}(\pi,\gamma,t)\right]$ $\displaystyle C_{13}(t;x_{0})-\beta\int
d\gamma\,(I+\gamma a_{1})C_{33}(\pi,\gamma,t;x_{0})$ $\displaystyle-\beta\int
d\gamma\,(I+\gamma a_{1}(t))a_{3}(\pi,\gamma,t)C_{43}(\pi,\gamma,t;x_{0})=0$
(38)
$\displaystyle\gamma\partial_{\theta}a_{3}(x)C_{11}(t;t_{0})+[\partial_{t}+$
$\displaystyle(I+\gamma a_{1})\partial_{\theta}]C_{31}(x;t_{0})$
$\displaystyle-\frac{\beta}{N}(I+\gamma
a_{1}(t))a_{3}(\pi,\gamma,t)\delta(\pi-\theta)C_{21}(t,t_{0})=0$ (39)
$\displaystyle\gamma\partial_{\theta}a_{3}(x)C_{13}(t;x_{0})+[\partial_{t}+$
$\displaystyle(I+\gamma a_{1}(t))\partial_{\theta}]C_{33}(x,x_{0})$
$\displaystyle-\frac{\beta}{N}(I+\gamma
a_{1}(t))a_{3}(\pi,\gamma,t)\delta(\pi-\theta)C_{23}(t,x_{0})=0$ (40)
Adding (38) and (39) and taking the limit $t_{0}\rightarrow t$ and setting
$\theta_{0}=\theta$, $\gamma_{0}=\gamma$ gives
$\displaystyle\partial_{t}$ $\displaystyle
C(\theta,\gamma,t)+\left[\beta-\beta\int
d\gamma^{\prime}\,\gamma^{\prime}a_{3}(\pi,\gamma^{\prime},t)+(I+\gamma
a_{1})\partial_{\theta}\right]C(\theta,\gamma,t)-\beta\int
d\gamma^{\prime}\,(I+\gamma^{\prime}a_{1})C_{33}(\pi,\gamma^{\prime},t;x)$
$\displaystyle-2\beta(I+\gamma
a_{1}(t))a_{3}(\pi,\gamma,t)\delta(\pi-\theta)+\gamma\partial_{\theta}a_{3}(x)C_{11}(t;t)=0$
where we use the fact that $C_{21}(t,t^{\prime})=N$ and
$C_{43}(x;x^{\prime})=\delta(\theta-\theta^{\prime})\delta(\gamma-\gamma^{\prime})$
in the limit of $t^{\prime}$ approaching $t$ from below and equal to zero when
approaching from above. Adding (37) and (40) to themselves with $t$ and
$t_{0}$ interchanged and taking the limit of $t_{0}$ approaching $t$ gives
$\displaystyle\left[\frac{d}{dt}+2\beta-2\beta\int d\gamma\,\gamma
a_{3}(\pi,\gamma,t)]\right]C_{11}(t;t)-2\beta\int d\gamma\,(I+\gamma
a_{1})C(\pi,\gamma,t)=0$ $\displaystyle\left[\partial_{t}+(I+\gamma
a_{1}(t))\partial_{\theta}\right]C_{33}(x;x)+2\gamma[\partial_{\theta}a_{3}(x)]C(x)=0$
because $C_{41}(x;t)=0$ and $C_{23}(t;x)=0$. Putting this all together, we get
the generalized activity equations
$\displaystyle\frac{{da_{1}}}{dt}+\beta a_{1}(t)-\beta\int d\gamma(I+\gamma
a_{1}(t))a_{3}(\pi,\gamma,t)-\frac{\beta}{N}\int d\gamma\,\gamma
C(\pi,\gamma,t)=0$ (41)
$\displaystyle\partial_{t}a_{3}(\theta,\gamma,t)+(I+\gamma
a_{1})\partial_{\theta}a_{3}(\theta,\gamma,t)+\frac{1}{N}\gamma\partial_{\theta}C(\theta,\gamma,t)=0$
(42) $\displaystyle\partial_{t}C(\theta,\gamma,t)+\left[\beta-\beta\int
d\gamma^{\prime}\,\gamma^{\prime}a_{3}(\pi,\gamma^{\prime},t)+(I+\gamma
a_{1})\partial_{\theta}\right]C(\theta,\gamma,t)$ $\displaystyle-\beta\int
d\gamma^{\prime}\,(I+\gamma^{\prime}a_{1})C_{33}(\pi,\gamma^{\prime},t;\theta,\gamma,t)-2\beta(I+\gamma
a_{1}(t))a_{3}(\theta,\gamma,t)\delta(\pi-\theta)$
$\displaystyle+\gamma\partial_{\theta}a_{3}(\theta,\gamma,t)C_{11}(t;t)=0$
(43) $\displaystyle\left[\frac{d}{dt}+2\beta-2\beta\int d\gamma\,\gamma
a_{3}(\pi,\gamma,t)]\right]C_{11}(t;t)-2\beta\int d\gamma\,(I+\gamma
a_{1})C(\pi,\gamma,t)=0$ (44) $\displaystyle\left[\partial_{t}+(I+\gamma
a_{1}(t))\partial_{\theta}\right]C_{33}(\theta,\gamma,t;\theta,\gamma,t)+2\gamma\partial_{\theta}a_{3}(\theta,\gamma,t)C(\theta,\gamma,t)=0$
(45)
Initial conditions, which are specified in the action, are required for each
of these equations. The derivation of these equations using classical means
require careful consideration for each particular model. Our method provides a
blanket mechanistic algorithm. We propose that these equations represent a new
scheme for studying neural networks.
Equations (41)-(45) are the complete self-consistent generalized activity
equations for the mean and correlations to order $N^{-1}$. It is a system of
partial differential equations in $t$ and $\theta$. These equations can be
directly analyzed or numerically simulated. Although the equations seem
complicated, one must bear in mind that they represent the dynamics of the
system averaged over initial conditions and unknown parameters. Hence, the
solution of this PDE system replaces multiple simulations of the original
system. In previous work, we required over a million simulations of the
original system to obtained adequate statistics Buice and Chow (2013a). There
is also a possibility that simplifying approximations can be applied to such
systems. The system has complete phase memory because the original system was
fully deterministic. However, the inclusion of stochastic effects will shorten
the memory and possibly simplify the dynamics. It will pose no problem to
include such stochastic effects. In fact, the formalism is actually more
suited for stochastic systems Buice et al. (2010).
## Discussion
The main goal of this paper was to show how to systematically derive
generalized activity equations for the ensemble averaged moments of a
deterministically coupled network of spiking neurons. Our method utilizes a
path integral formalism that makes the process algorithmic. The resulting
equations could be derived using more conventional perturbative methods
although possibly with more calculational difficulty as we found before Buice
et al. (2010). For example, for the case of the stochastic spike model, Buice
et al. Buice et al. (2010) presumed that the Wilson-Cowan activity variable
was the rate of a Poisson process and derived a system of generalized activity
equations that corresponded to deviations around Poisson firing. Bressloff
Bressloff (2010), on the other hand, assumed that the Wilson-Cowan activity
variable was a mean density and used a system-size expansion to derive an
alternative set of generalized activity equations for the spike model. The
classical derivations of these two interpretations look quite different and
the differences and similarities between them are not readily apparent.
However, the connections between the two types of expansions are very
transparent using the path integral formalism.
Here, we derived equations for the rate and covariances (first and second
cumulants) of a deterministic synaptically coupled spiking network as a system
size expansion to first order. However, our method is not restricted to these
choices. What is particularly advantageous about the path integral formalism
is that it is straightforward to generalize to include higher order cumulants,
extend to higher orders in the inverse system size, or to expand in other
small parameters such as the inverse of a slow time scale. The action fully
specifies the system and all questions regarding the system can be addressed
with it.
To give a concrete illustration of the method, we derived the self-consistent
generalized activity equations for the rates and covariances to order $N^{-1}$
for a simple phase model. The resulting equations consist of ordinary and
partial differential equations. This is to be expected since the original
system was fully deterministic and memory cannot be lost. Even mean field
theory requires the solution of an advective partial differential equation.
The properties of these and similar equations remain to be explored
computationally and analytically. The system is possibly simpler near the
asynchronous state, which is marginally stable in mean field theory like the
Kuramoto model Strogatz and Mirollo (1991) and like the Kuramoto model, we
conjecture that the finite size effects will stabilize the asynchronous state
Hildebrand et al. (2007); Buice and Chow (2007). The addition of noise will
also stabilize the asynchronous state. Near asynchrony could be exploited to
generate simplified versions of the asynchronous state.
We considered a globally connected network, which allowed us to assume that
networks for different parameter values and initial conditions converge
towards a “typical” system in the large $N$ limit. However, this property may
not hold for more realistic networks. While the formalism describing the
ensemble average will hold regardless of this assumption, the utility of the
equations as descriptions of a particular network behavior may suffer. For
example, heterogeneity in the connectivity (as opposed to the global
connectivity we consider here) may threaten this assumption. This is the case
with so called “chaotic random networks” Sompolinsky et al. (1988) in which
there is a spin-glass transition owing to the variance of the connectivity
crossing a critical threshold. This results in the loss of a “typical” system
in the large $N$ limit requiring an effective stochastic equation which
incorporates the noise induced by the network heterogeneity. Whether the
expansion we present here is useful without further consideration depends upon
whether the network heterogeneity induces this sort of effect. This is an area
for future work. A simpler issue arises when there are a small discrete number
of “typical” systems (such as with bistable solutions to the continuity
equation). In this case, there are noise induced transitions between states.
While the formalism has a means of computing this transition Elgart and
Kamenev (2004), we do not consider this case here.
An alternative means to incorporate heterogeneous connections is to consider a
network of coupled systems. In such a network, a set of generalized activity
equations, such as those derived here or simplified versions, would be derived
for each local system, together with equations governing the covariances
between the local systems. Correlation based learning dynamics could then be
imposed on the connections between the local systems. Such a network could
serve as a generalization of current rate based neural networks to include the
effects of spike correlations with applications to both neuroscience and
machine learning.
## Acknowledgments
This work was supported by the Intramural Research Program of the NIH, NIDDK.
## References
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|
arxiv-papers
| 2013-10-25T14:32:44 |
2024-09-04T02:49:52.875920
|
{
"license": "Public Domain",
"authors": "Michael A. Buice and Carson C. Chow",
"submitter": "Carson C. Chow",
"url": "https://arxiv.org/abs/1310.6934"
}
|
1310.6994
|
aainstitutetext: Harish-Chandra Research Institute, Allahabad 211019,
Indiabbinstitutetext: Regional Centre for Accelerator-based Particle Physics
Harish-Chandra Research Institute, Allahabad 211019, India
# Non-minimal Universal Extra Dimensions with Brane Local Terms: The Top Quark
Sector
AseshKrishna Datta b Kenji Nishiwaki b Saurabh Niyogi [email protected],
[email protected], [email protected]
###### Abstract
We study the physics of Kaluza-Klein (KK) top quarks in the framework of a
non-minimal Universal Extra Dimension (nmUED) with an orbifolded
($S^{1}/Z_{2}$) flat extra spatial dimension in the presence of brane-
localized terms (BLTs). In general, BLTs affect the masses and the couplings
of the KK excitations in a non-trivial way including those for the KK top
quarks. On top of that, BLTs also influence the mixing of the top quark chiral
states at each KK level and trigger mixings among excitations from different
levels with identical KK parity (even or odd). The latter phenomenon of mixing
of KK levels is not present in the popular UED scenario known as the minimal
UED (mUED) at the tree level. Of particular interest are the mixings among the
KK top quarks from level ‘0’ and level ‘2’ (driven by the mass of the Standard
Model (SM) top quark). These open up new production modes in the form of
single production of a KK top quark and the possibility of its direct decays
to SM particles leading to rather characteristic signals at the colliders.
Experimental constraints and the restrictions they impose on the nmUED
parameter space are discussed. The scenario is implemented in MadGraph 5 by
including the quark, lepton, the gauge-boson and the Higgs sectors up to the
second KK level. A few benchmark scenarios are chosen for preliminary studies
of the decay patterns of the KK top quarks and their production rates at the
LHC in various different modes. Recast of existing experimental analyzes in
scenarios having similar states is found to be not so straightforward for the
KK top quarks of the nmUED scenario under consideration.
††preprint: HRI-P-13-10-001
RECAPP-HRI-2013-021
## 1 Introduction
The top quark is altogether a different kind of a fermion in the realm of the
Standard Model (SM) sheerly because of its large mass or equivalently, its
large (Yukawa) coupling to the Higgs boson. Even when the discovery of the
Higgs boson was eagerly awaited, the implications of such a large Yukawa
coupling was already much appreciated. Many new physics scenarios beyond the
SM (BSM), which have extended top quark sectors offering top quark partners,
derive theoretically nontrivial and phenomenologically rich attributes from
this aspect. At colliders, they warrant dedicated searches which generically
result in weaker bounds on them when compared to their peers from the first
two generations.
Naturally, ever since the confirmation of the recent discovery of a Higgs-like
scalar particle came in, the top quark sectors of different new physics
scenarios have been in the spotlight triggering a spur of focussed activities.
While popular supersymmetric (SUSY) scenarios are excellent hunting grounds
for such possibilities and have taken the center stage during the recent past
and at a time of renewed drives, there exist other physics scenarios that
offer interesting signatures at the colliders with phenomenologically rich top
quark sectors. Scenarios with Universal Extra Dimensions (UEDs) are also no
exceptions even though the setups are not necessarily tied to and/or address
the ‘naturalness’ issue of the Higgs sector like many of the competing
scenarios do thus requiring relatively light ‘top partner’ (${\cal O}(1)$
TeV). However, on a somewhat different track, attempts to understand the
hierarchy of masses and mixings of the (4D) SM fermions while conforming with
the strong FCNC constraints for the first two generations often adopt
mechanisms that distinguish the third generation from the first two Del
Aguila:2001pu . This could also lead to lighter states for the former. Thus,
in the absence of a robust principle that prohibits them and until the
experiments exclude them specifically, it is important that these should make
a necessary part of the search programme at the colliders. This is further
appropriate while being under the cloak of the so-called ‘SUSY-UED’ confusion
Cheng:2002ab which may not allow us understand immediately the nature of such
a newly-discovered state.
Thus, there has been a reasonable amount of activity involving comparatively
light KK top quarks of the UED scenarios in the past Petriello:2002uu ;
Rai:2005vy ; Maru:2009cu ; Nishiwaki:2011vi ; Nishiwaki:2011gk ;
Nishiwaki:2011gm and also from recent times post Higgs-discovery
Belanger:2012mc ; Kakuda:2013kba ; Dey:2013cqa ; Flacke:2013nta . The latter
set of works have constrained the respective scenarios discussed to varying
degrees by analyzing the Higgs results. In this work, we study the structure
of the top quark sector of the so-called non-minimal universal extra
dimensions (nmUED), the nontrivial features it is endowed with and their
implications for the LHC.
The particular nmUED scenario we deal with in this work is different from the
popular minimal UED (mUED) scenario Appelquist:2000nn ; Cheng:2002iz (an
incarnation of the so-called generic TeV-scale extra dimensions
Antoniadis:1990ew ) in the fact that the former takes into consideration the
effect of brane-local terms (BLTs) which are already non-vanishing at the tree
level111Note that BLTs get renormalized and thus cannot be set to zero at all
scales. delAguila:2003bh ; delAguila:2003gu ; del Aguila:2006kj and that
develop at the two fixed points222A possibility with multiple fixed points
(branes) are helpful for explaining the fermion flavor structure
Fujimoto:2012wv ; Fujimoto:2013ki . of $S^{1}/Z_{2}$ orbifold on which the
extra space dimension of such a 5-dimensional scenario is compactified. As is
well-known, BLTs affect both properties of the KK modes (corresponding to the
fields present in the bulk) that crucially govern their phenomenology: they
modify the masses of these KK modes and alter their wavefunctions thus
affecting their physical couplings in four dimensions.
The phenomenology of such a scenario at the LHC has recently been discussed in
Datta:2012tv with reference to strong productions of the KK gluons and
(vector-like) KK quarks from the first excited level333Phenomenology of KK-
parity violating BLTs are discussed in Datta:2012xy ; Datta:2013lja .. It was
demonstrated how such processes could closely mimic the corresponding SUSY
processes. There, such a scenario was also contrasted against the popular mUED
scenario. Tentative bounds on these excitations were derived from recent LHC
results. However, for the KK quarks, such bounds referred only to the first
two generation quarks.
The top quark sector of the mUED had earlier been studied at the LHC in ref.
Choudhury:2009kz . In the present work we take up the case of KK top quarks in
the nmUED scenario. These are ‘vector-like’ states and can be lighter than the
KK quarks from the first two generations. This is exactly the reason behind
the current surge in studies on ‘top-partners’ at the LHC
AguilarSaavedra:2009es ; Cacciapaglia:2010vn ; Cacciapaglia:2011fx ;
Berger:2012ec ; DeSimone:2012fs ; Kearney:2013oia ; Buchkremer:2013bha ;
Aguilar-Saavedra:2013qpa . From phenomenological considerations, the nmUED
scenario under consideration is different from the mUED scenario in the
following important aspects: (i) the KK masses for these excitations and their
couplings derived form the compactification of the extra dimension can be very
different444An extreme example of decoupling the mass scale of new physics
form the compactification scale can be found in ref. Del Aguila:2001pu . from
their mUED counterparts for a given value of the inverse compactification
radius $R{{}^{-1}}$ and (ii) the mixing between the (chiral) top quark states
driven by the top quark mass (which is a generic feature of scenarios with
extended top quark sector) can be essentially different. Further, we highlight
a rather characteristic feature of such an nmUED scenario which triggers
mixing of excitations from similar KK levels of similar parities (even or
odd). Such _level-mixings_ are triggered by BLTs delAguila:2003kd ;
delAguila:2003gv due to non-vanishing overlap integrals and arise from the
Yukawa sector. Hence, such effects depend on the corresponding brane-local
parameter. These induce tree level couplings among the resulting states
(mixtures of corresponding states from different KK levels). Note that in
mUED, such couplings are only present beyond Born-level and are thus
suppressed. Also, as we will see later in this work, such mixings can be
interesting only for the KK fermions from the third generation and in
particular, for the top quark sector thanks to the large top quark mass.
Moreover, in the context of the LHC, the only relevant mixings are going to be
those involving the SM (level ‘0’) and the level ‘2’ KK states.
In the nmUED scenario, the general setup for the quark sector involves BLTs of
both kinetic and Yukawa type. This was discussed in appropriate details in
Datta:2012tv for the level ‘1’ KK excitations including the third generation
quarks. In this work, we extend the scheme to include the level ‘2’
excitations as well with particular emphasis on the top quark sector. It is
demonstrated how presence of level mixing may potentially open up interesting
phenomenological possibilities at the LHC in the form of new modes of their
production and decay some of which would necessarily involve KK excitations of
the gauge and the Higgs bosons in crucial ways. This would no doubt have
significant phenomenological implications at the LHC and could provide us with
an understanding of how the same can be contrasted against other scenarios
having similar signatures and/or can be deciphered from experimental data.
The paper is organized as follows. In section 2 we discuss the theoretical
framework of the top quark sector at higher KK levels along with those of the
gauge and the Higgs sectors which are intimately connected to the theory and
phenomenology of the KK top quarks. The resulting mass spectra and the form of
the relevant couplings are discussed in section 3. In section 4 we discuss in
some details the experimental constraints that potentially restrict the
parameter space of the scenario under consideration. A few benchmark points,
which satisfy all these constraints, are also chosen for further studies.
Section 5 is devoted to the basic phenomenology of the KK top quarks at the
LHC by outlining their production and decay patterns. In section 6 we
conclude.
## 2 Theoretical framework
We consider the top quark sector of a 5D nmUED scenario compactified on
$S^{1}/Z_{2}$ in the presence of tree-level BLTs that develop at the orbifold
fixed points. The compactification is characterized by the length parameter
$L$ where $L=\pi R/2$, $R$ being the radius of the orbifolded extra space
dimension. The two fixed points of the $S^{1}/Z_{2}$ geometry are taken to be
at $y=\pm L$. The derivations broadly follow the notations, the conventions
and the treatments adopted in reference Datta:2012tv . The phenomenological
relevance of the KK gauge and Higgs sectors prompts us to incorporate them
thoroughly in the present analysis, including even the level ‘2’ KK
excitations in some of these cases. In the following we outline the necessary
theoretical setup involving these sectors. We start with the gauge and the
Higgs sectors first since the issue of Higgs vacuum expectation value (VEV) is
relevant for the top quark (Yukawa) sector.
### 2.1 The gauge boson and the Higgs sectors
The gauge boson and the Higgs sectors of the nmUED scenario had been discussed
in some detail in ref. Flacke:2008ne with due stress on their mutual
relationship and the implications thereof for possible dark matter candidates
of such a scenario. We closely follow the approach there and summarize the
aspects that are relevant for our present study.
We consider the following 5D action Flacke:2008ne describing the gauge and
the Higgs sectors of the nmUED scenario under study:
$\displaystyle S$ $\displaystyle=\int
d^{4}x\int_{-L}^{L}dy\Bigg{\\{}-\frac{1}{4}G_{MN}^{a}G^{aMN}-\frac{1}{4}W_{MN}^{i}W^{iMN}-\frac{1}{4}B_{MN}B^{MN}$
$\displaystyle\phantom{S=\int
d^{4}x\int_{-L}^{L}dy\,}+(D_{M}\Phi)^{\dagger}(D^{M}\Phi)+\hat{\mu}^{2}\Phi^{\dagger}\Phi-\frac{\hat{\lambda}}{4}(\Phi^{\dagger}\Phi)^{2}$
$\displaystyle\quad+\Big{(}\delta(y-L)+\delta(y+L)\Big{)}\Big{[}-\frac{r_{G}}{4}G_{\mu\nu}^{a}G^{a\mu\nu}-\frac{r_{W}}{4}W_{\mu\nu}^{i}W^{i\mu\nu}-\frac{r_{B}}{4}B_{\mu\nu}B^{\mu\nu}$
$\displaystyle\phantom{\quad+\Big{(}\delta(y-L)+\delta(y+L)\Big{)}\Big{[}\,\,\,}+r_{H}(D_{\mu}\Phi)^{\dagger}(D^{\mu}\Phi)+{\mu}_{b}^{2}\Phi^{\dagger}\Phi-\frac{\lambda_{b}}{4}(\Phi^{\dagger}\Phi)^{2}\Big{]}\Bigg{\\}},$
(1)
where $y$ represents the compact extra spatial direction, the Lorentz indices
$M$ and $N$ run over $0,1,2,3,y$ while $\mu$ and $\nu$ run over $0,1,2,3$.
$G_{MN}^{a}$, $W_{MN}^{i}$ and $B_{MN}$ are the 5D field-strengths associated
with the gauge groups $SU(3)_{C}$, $SU(2)_{W}$ and $U(1)_{Y}$ respectively
with the corresponding 5D gauge bosons $G_{M}^{a}$, $W_{M}^{i}$ and $B_{M}$.
$a$ and $i$ are the adjoint indices for the groups $SU(3)_{C}$ and
$SU(2)_{W}$, respectively. The 5D Higgs doublet is represented by $\Phi$ with
its components given by
$\displaystyle\Phi=\begin{pmatrix}\phi^{+}\\\
\frac{1}{\sqrt{2}}\left(\hat{v}(y)+H+i\chi\right)\end{pmatrix}$ (2)
where $\phi^{+}$ is the charged component, $H$ and $\chi$ are the neutral
components and $\hat{v}(y)$ is the 5D bulk Higgs VEV. $D_{M}$ stands for the
5D covariant derivatives and $\hat{\mu}$ and $\hat{\lambda}$ represent the 5D
bulk Higgs mass and the Higgs self-coupling, respectively.
We take $Z_{2}$ eigenvalues for the fields
$G_{\mu}^{a},\,W_{\mu}^{i},\,B_{\mu},\,H,\ \chi,\,\phi^{+}$ to be even at both
the fixed points to realize the zero modes (that correspond to the SM degrees
of freedom) have vanishing KK-masses from compactification. This automatically
renders the eigenvalues of $G_{y}^{a},\,W_{y}^{i},\,B_{y}$ to be odd because
of 5D gauge symmetry for which there are no corresponding zero modes.
As can be seen in equation 1, the BLTs (proportional to the
$\delta$-functions) are introduced at the orbifold fixed points for both the
gauge and the Higgs sectors. The bulk mass term and the Higgs self-interaction
term are considered only for the latter for preserving the 4D gauge
invariance. The six coefficients $r_{G}$, $r_{W}$, $r_{B}$, $r_{H}$, $\mu_{b}$
and $\lambda_{b}$ influence the masses of the KK excitations and the effective
couplings involving them. As is well-known, due to the existence of the BLTs,
momentum conservation along the $y$ direction is violated even at the tree
level (in contrast to the mUED where this could happen only beyond the tree
level), but a discrete symmetry, called the KK-parity, under the reflection
$y\to-y$ is still preserved. KK-parity ensures the existence of a stable dark
matter candidate which is the lightest KK particle (LKP) at level ‘1’ obtained
on compactification.
In this work, for simplicity, we focus on the following situation:
$\displaystyle\sqrt{\frac{4\hat{\mu}^{2}}{\hat{\lambda}}}=\sqrt{\frac{4{\mu_{b}}^{2}}{{\lambda_{b}}}}\quad\text{and}\quad
r_{W}=r_{B}\,\equiv\,r_{\text{EW}}.$ (3)
The first condition ensures a constant profile of the Higgs VEV over the whole
space, _i.e._ ,
$\displaystyle\hat{v}(y)\to\sqrt{\frac{4\hat{\mu}^{2}}{\hat{\lambda}}}=\sqrt{\frac{4{\mu_{b}}^{2}}{{\lambda_{b}}}}\,\equiv\,\hat{v},$
(4)
while with the second condition555For $r_{W}\not=r_{B}$, obtaining the correct
value of the Weinberg angle in the SM sector is nontrivial. We, thus, do not
consider this possibility in the present work although the same could have
interesting phenomenological implications both at colliders or otherwise (see
ref. Flacke:2008ne that discusses its implication for possible KK dark matter
candidates). we can continue to relate the 5D $W$, $Z$ and the photon
($\gamma$) states (at tree level) via the usual Weinberg angle $\theta_{W}$ at
all KK levels, _i.e._ ,
$\displaystyle W_{M}^{\pm}=\frac{W^{1}_{M}\mp
iW^{2}_{M}}{\sqrt{2}},\quad\begin{pmatrix}Z_{M}\\\
\gamma_{M}\end{pmatrix}=\begin{pmatrix}\cos{\theta_{W}}&\sin{\theta_{W}}\\\
-\sin{\theta_{W}}&\cos{\theta_{W}}\end{pmatrix}\begin{pmatrix}W^{3}_{M}\\\
B_{M}\end{pmatrix}.$ (5)
The gauge-fixing conditions along with their consequences are discussed
briefly in appendix A. We choose the unitary gauge. For the fields
$G_{\mu}^{a},\,W_{\mu}^{+},\,Z_{\mu},\,H,\,\chi,\,\phi^{+}$ and for the ones
like $\partial_{y}W_{y}^{+},\,\partial_{y}Z_{y}$, the mode functions for KK
decomposition and the conditions that determine their KK-masses are summarized
below.
$\displaystyle f_{F_{(n)}}(y)$
$\displaystyle=N_{F_{(n)}}\times\begin{cases}\displaystyle\frac{\cos(M_{F_{(n)}}y)}{C_{F_{(n)}}}&\text{for
even }n,\\\ -\displaystyle\frac{\sin(M_{F_{(n)}}y)}{S_{F_{(n)}}}&\text{for odd
}n,\\\ \end{cases}$ (6) $\displaystyle m_{F_{(n)}}^{2}$
$\displaystyle=m_{F}^{2}+M_{F_{(n)}}^{2},$ (7)
$\displaystyle\frac{(r_{F}m_{F_{(n)}}^{2}-m_{F,b}^{2})}{M_{F_{(n)}}}$
$\displaystyle=\begin{cases}-T_{F_{(n)}}&\text{for even }n,\\\
+1/T_{F_{(n)}}&\text{for odd }n\end{cases}$ (8)
with the following short-hand notations:
$\displaystyle C_{F_{(n)}}=\cos\left(\frac{M_{F_{(n)}}\pi R}{2}\right),\quad
S_{F_{(n)}}=\sin\left(\frac{M_{F_{(n)}}\pi R}{2}\right),\quad
T_{F_{(n)}}=\tan\left(\frac{M_{F_{(n)}}\pi R}{2}\right).$ (9)
The normalization factors $N_{F_{(n)}}$ for the mode functions
$f_{F_{(n)}}(y)$ are given by
$\displaystyle N_{F_{(n)}}^{-2}=\begin{cases}\displaystyle
2r_{F}+\frac{1}{C_{F_{(n)}}^{2}}\left[\frac{\pi
R}{2}+\frac{1}{2M_{F_{(n)}}}\sin(M_{F_{(n)}}\pi R)\right]&\text{for even
}n,\\\ \displaystyle 2r_{F}+\frac{1}{S_{F_{(n)}}^{2}}\left[\frac{\pi
R}{2}-\frac{1}{2M_{F_{(n)}}}\sin(M_{F_{(n)}}\pi R)\right]&\text{for odd
}n.\end{cases}$ (10)
Here $m_{F_{(n)}}$, $m_{F}$, $M_{F_{(n)}}$, $r_{F}$ and $m_{F,b}$ stand for
the physical mass, the bulk mass, the KK mass, the coefficient of the
corresponding brane-local kinetic term (BLKT) and brane mass term of the field
$F$, respectively. Inputs for the mass-determining conditions for all these
fields are presented in appendix A. Further, following conditions must hold to
ensure the zero-mode (SM) profiles to be flat which help evade severe
constraints from electroweak observables like the Z-boson mass,
$\sin^{2}{\theta_{W}}$ etc.
$\displaystyle r_{\text{EW}}=r_{H}\qquad$ $\displaystyle\text{for
}W_{\mu}^{+},\,Z_{\mu},$ $\displaystyle
r_{H}(2\hat{\mu}^{2})=2\mu_{b}^{2}\qquad$ $\displaystyle\text{for }H.$ (11)
Non-compliance of the above relations could result in unacceptable
modifications in the level-‘0’ (SM) Lagrangian Flacke:2008ne .
Also, with the above two conditions, equation 8 reduces to the following
simple form:
$\displaystyle r_{F}{M_{F_{(n)}}}=\begin{cases}-T_{F_{(n)}}&\text{for $n$
even,}\\\ 1/T_{F_{(n)}}&\text{for $n$ odd}\end{cases}$ (12)
where $M_{F_{(0)}}$ $=$ $0$ (thus ensuring vanishing KK masses for the level
‘0’ (SM) fields). A theoretical lower bound of $r_{F}>-\frac{\pi R}{2}$ must
hold to circumvent tachyonic zero modes. In figure 1, we illustrate the
generic profile of the variation of $M_{F_{(n)}}/R^{-1}$ as a function of
$r^{\prime}_{F}\,(=r_{F}R^{-1})$ for the cases $n=1$ and $n=2$.
Figure 1: The generic profile of the variation of $M_{F_{(n)}}/R^{-1}$ as a
function of $r^{\prime}_{F}\,(=r_{F}R^{-1})$ for the cases $n=1$ and $n=2$.
On the other hand, vanishing KK masses at level ‘0’ are always realized for
$\phi^{+}$ and $\chi$ which are eventually “eaten up” by the massless level
‘0’ $W_{\mu}^{+},\,Z_{\mu}$ states respectively as they become massive.
However, no zero mode appears for $W_{y}^{+},\,Z_{y}$ since they are projected
out by the $Z_{2}$-odd condition. The mode functions for the fields
$W_{y}^{+},\,Z_{y}$ are given by
$\displaystyle f_{F_{(n)}}(y)$
$\displaystyle=N_{F_{(n)}}\times\begin{cases}\displaystyle\frac{\sin(M_{F_{(n)}}y)}{C_{F_{(n)}}}&\text{for
even }n,\\\ \displaystyle\frac{\cos(M_{F_{(n)}}y)}{S_{F_{(n)}}}&\text{for odd
}n\\\ \end{cases}$ (13)
with the mass-determination condition as given in equation 8. Use of equation
47 allows one to eliminate $\chi$ in favor of $Z_{y}$ and $\phi^{+}$ in favor
of $W_{y}^{+}$. Correct normalization of the kinetic terms requires $Z_{y}$
and $W_{y}^{+}$ to be renormalized in the following way:
$\displaystyle
Z_{y}^{(n)}\to\left(1+\frac{M_{{Z_{y}}_{(n)}}^{2}}{M_{Z}^{2}}\right)^{-1/2}Z_{y}^{(n)}\,,\qquad\qquad
W_{y}^{{(n)}{+}}\to\left(1+\frac{M_{{W_{y}}_{(n)}}^{2}}{M_{W}^{2}}\right)^{-1/2}W_{y}^{{(n)}{+}}.$
(14)
Note that $Z_{y}^{(n)}$ is the pseudoscalar Higgs state and $W_{y}^{{(n)}{+}}$
is the charged Higgs boson from the $n$-th KK level which has no level ‘0’
counterpart. In subsequent phenomenological discussions we use the more
transparent notations $A^{(n)^{0}}$ and $H^{(n)^{+}}$ for ${Z_{y}}^{(n)}$ and
$W_{y}^{{(n)}{+}}$, respectively. Thus, up to KK level ‘1’, the Higgs spectrum
consists of the following five Higgs bosons: the SM (level ‘0’) Higgs boson
($H$) and four Higgs states from level ‘1’, _i.e._ , the neutral $CP$-even
Higgs boson ($H^{(1)^{0}}$) which is the level ‘1’ excitation of the SM Higgs
boson, the neutral $CP$-odd Higgs boson ($A^{(1)^{0}}$) and the two charged
Higgs bosons $H^{(1)^{\pm}}$. For the rest of the paper, we use a modified
convention for the (KK) gluon to be $g^{(n)}$ instead of $G^{(n)}$ for
convenience.
### 2.2 The top quark sector
We start with the following general framework for the fermion sector where, in
addition to fermion BLKTs, we incorporate brane-local Yukawa terms (BLYTs):
$\displaystyle S_{\text{quark}}$ $\displaystyle=\int
d^{4}x\int_{-L}^{L}dy\sum_{i=1}^{3}\Bigg{\\{}+i\overline{U^{\prime}_{i}}\Gamma^{M}\mathcal{D}_{M}U^{\prime}_{i}+r_{U_{i}}\Big{(}\delta(y-L)+\delta(y+L)\Big{)}\Big{[}i\overline{U^{\prime}_{i}}\Gamma^{\mu}\mathcal{D}_{\mu}P_{L}U^{\prime}_{i}\Big{]}$
$\displaystyle\phantom{=\int
d^{4}x\int_{-L}^{L}dy\sum_{i=1}^{3}\Bigg{\\{}\,\,}+i\overline{D^{\prime}_{i}}\Gamma^{M}\mathcal{D}_{M}D^{\prime}_{i}+r_{D_{i}}\Big{(}\delta(y-L)+\delta(y+L)\Big{)}\Big{[}i\overline{D^{\prime}_{i}}\Gamma^{\mu}\mathcal{D}_{\mu}P_{L}D^{\prime}_{i}\Big{]}$
$\displaystyle\phantom{=\int
d^{4}x\int_{-L}^{L}dy\sum_{i=1}^{3}\Bigg{\\{}\,\,}+i\overline{u^{\prime}_{i}}\Gamma^{M}\mathcal{D}_{M}u^{\prime}_{i}+r_{u_{i}}\Big{(}\delta(y-L)+\delta(y+L)\Big{)}\Big{[}i\overline{u^{\prime}_{i}}\Gamma^{\mu}\mathcal{D}_{\mu}P_{R}u^{\prime}_{i}\Big{]}$
$\displaystyle\phantom{=\int
d^{4}x\int_{-L}^{L}dy\sum_{i=1}^{3}\Bigg{\\{}\,\,}+i\overline{d^{\prime}_{i}}\Gamma^{M}\mathcal{D}_{M}d^{\prime}_{i}+r_{d_{i}}\Big{(}\delta(y-L)+\delta(y+L)\Big{)}\Big{[}i\overline{d^{\prime}_{i}}\Gamma^{\mu}\mathcal{D}_{\mu}P_{R}d^{\prime}_{i}\Big{]}\Bigg{\\}},$
(15) $\displaystyle S_{\text{Yukawa}}$ $\displaystyle=\int
d^{4}x\int_{-L}^{L}dy\sum_{i,j=1}^{3}\Bigg{\\{}-\Big{(}1+r_{Y}(\delta(y-L)+\delta(y+L))\Big{)}\Big{[}\hat{Y}^{u}_{ij}\overline{Q^{\prime}_{i}}u^{\prime}_{j}\tilde{\Phi}+\hat{Y}^{d}_{ij}\overline{Q^{\prime}_{i}}d^{\prime}_{j}\Phi+\text{h.c.}\Big{]}\Bigg{\\}},$
(16)
where $U^{\prime}_{i},D^{\prime}_{i},u^{\prime}_{i},d^{\prime}_{i}$ correspond
to the 5D $SU(2)_{W}$ up-doublet, down-doublet, up-singlet and down-singlet of
the $i$-th generation, respectively and
$Q^{\prime}_{i}\,\equiv\,(U^{\prime}_{i},D^{\prime}_{i})^{\text{T}}$ is the
compact notation used for the $i$-th 5D doublet. $r_{U_{i}}$ and $r_{u_{i}}$
are the coefficients of the corresponding BLKTs. The field $\Phi$ represents
the 5D Higgs scalar with $\tilde{\Phi}\,\equiv\,i\sigma_{2}\Phi^{\ast}$,
$\sigma_{2}$ being the second Pauli matrix. $r_{Y}$ is the universal
coefficient for the brane-local Yukawa term. We adopt the 5D Minkowski metric
to be $\eta_{MN}=\text{diag}(1,-1,-1,-1,-1)$ and the representation of the
Clifford algebra is chosen to be $\Gamma^{M}=\\{\gamma^{\mu},i\gamma_{5}\\}$.
The 4D chiral projectors for (4D) right/left-handed states are defined
following the standard convention _i.e._ , $P_{R\atop L}=(1\pm\gamma_{5})/2$.
$\mathcal{D}_{M}$ stands for the 5D covariant derivative.
In the presence of non-vanishing BLKT in the gauge sector (see equation 2),
the 5D VEV of $\Phi$ is given by
$\displaystyle\langle\Phi\rangle=\begin{pmatrix}0\\\
\frac{\hat{v}}{\sqrt{2}}\end{pmatrix}=\begin{pmatrix}0\\\
\frac{v}{\sqrt{2}}\frac{1}{\sqrt{\pi R+2r_{\text{EW}}}}\end{pmatrix}$ (17)
where $v=246$ GeV is the usual 4D Higgs VEV associated with the breaking of
electroweak symmetry. The 5D Yukawa couplings
$\hat{Y}^{u}_{ij},\hat{Y}^{d}_{ij}$ are related to their 4D counterparts
${Y}^{u}_{ij},{Y}^{d}_{ij}$ as
$\displaystyle{Y}^{u/d}_{ij}=\frac{\hat{Y}^{u/d}_{ij}}{\sqrt{\pi
R+2r_{\text{EW}}}}.$ (18)
The free part of $S_{\text{quark}}$ has already been discussed in Datta:2012tv
and hence we skip the details. Using that we can KK-expand the mass terms in
$S_{\text{Yukawa}}$ as follows:
$\displaystyle-\int
d^{4}x\sum_{i,j=1}^{3}\frac{v}{\sqrt{2}}\bigg{\\{}Y^{u}_{ij}F^{u,(0,0)}_{ij}\overline{u^{\prime(0)}_{iL}}u^{\prime(0)}_{jR}+Y^{d}_{ij}F^{d,(0,0)}_{ij}\overline{d^{\prime(0)}_{iL}}d^{\prime(0)}_{jR}+\text{h.c.}\bigg{\\}},$
(19)
where, for simplicity, we only present the zero-mode part with fields
redefined (to make them appear more conventional) as
$u^{\prime(0)}_{iL}\,\equiv\,U^{\prime(0)}_{iL}$,
$d^{\prime(0)}_{iL}\,\equiv\,D^{\prime(0)}_{iL}$. The fermionic mode functions
for KK decomposition are described in an appropriate context in section 3. The
concrete forms of the factors $F^{u/d,(0,0)}_{ij}$ (which arise from the mode
functions of the $L$, $R$ type fields participating in equation 19) are
$\displaystyle F^{u,(0,0)}_{ij}=\frac{2r_{Y}+\pi R}{\sqrt{2r_{U_{i}}+\pi
R}\sqrt{2r_{u_{i}}+\pi R}},\quad F^{d,(0,0)}_{ij}=\frac{2r_{Y}+\pi
R}{\sqrt{2r_{D_{i}}+\pi R}\sqrt{2r_{d_{i}}+\pi R}}.$ (20)
The $3\times 3$ matrices $Y^{u}_{ij}F^{u,(0,0)}_{ij}$ and
$Y^{d}_{ij}F^{d,(0,0)}_{ij}$ are diagonalized by the following bi-unitary
transformations
$\displaystyle q^{\prime(0)}_{iR}=(U_{qR})_{ij}q^{(0)}_{jR},\quad
q^{\prime(0)}_{iL}=(U_{qL})_{ij}q^{(0)}_{jL}\qquad(\text{for}\ q=u,d),$ (21)
as follows:
$\displaystyle-\int
d^{4}x\sum_{i=1}^{3}\frac{v}{\sqrt{2}}\bigg{\\{}\mathcal{Y}^{u}_{ii}\overline{u^{(0)}_{iL}}u^{(0)}_{iR}+\mathcal{Y}^{d}_{ii}\overline{d^{(0)}_{iL}}d^{(0)}_{iR}+\text{h.c.}\;\text{(+
KK excitations)}\bigg{\\}},$ (22)
where $\mathcal{Y}^{u}_{ii}$ and $\mathcal{Y}^{d}_{ii}$ are the diagonalized
Yukawa couplings for up and down quarks, respectively. We discuss later in
this paper that the diagonalized values do not directly correspond to those in
the SM due to level mixing effects. Also, from now on, we would consider
universal values of the BLKT parameters $r_{Q}$ for the quarks from the first
two generations and $r_{T}$ for those from the third generation replacing the
many different ones appearing in equation 15. We will see later, this provides
us with a separate handle (modulo some constraints from experiments) on the
top quark sector of the nmUED scenario under consideration. Further, this
simplifies the expressions in equation 20.
## 3 Mixings, masses and effective couplings
Mixings in the fermion sector, quite generically, could have interesting
implications as these affect both couplings and the spectra of the concerned
excitations. Fermions with a certain flavor from a given KK level and
belonging to $SU(2)_{W}$ doublet and singlet representations always mix once
the electroweak symmetry is broken. Presence of BLTs affects such a mixing at
every KK level. On top of this, the dynamics driven by the BLTs allows for
mixing of fermions from different KK levels that have the same KK-parity. Both
kinds of mixings are proportional to the Yukawa mass of the fermion in
reference and thus, are pronounced for the top quark sector.
As pointed out in the introduction, since _level-mixing_ among the even KK-
parity top quarks involves the SM top quark (from level ‘0’), this naturally
evokes a reasonable curiosity about its consequences and it is indeed found to
give rise to interesting phenomenological possibilities. However, the
phenomenon draws significant constraints from experiments which we will
discuss in some detail. We restrict ourselves to the mixing of level ‘0’-level
’2’ KK top quarks ignoring all higher even KK states the effects of which
would be suppressed by their increasing masses. Also, we do not consider the
effects of level-mixings among KK states from levels with $n>0$, including
say, those among the excitations from levels with odd KK-parity. Generally,
these could be appreciable. However, in contrast to the case where SM
excitations mix with higher KK levels, these would only entail details within
a sector yet to be discovered.
### 3.1 Mixing in level ‘1’ top quark sector
We first briefly recount Datta:2012tv the mixing of the top quarks at KK
level ‘1’. In presence of BLTs, the Yukawa part of the action embodying the
mass-matrix is of the form
$\displaystyle-\int d^{4}x\Bigg{\\{}\begin{bmatrix}\overline{T}^{(1)},\
\overline{t}^{(1)}\end{bmatrix}_{L}\begin{bmatrix}M_{T_{(1)}}&r^{\prime}_{T11}m_{t}^{\text{in}}\\\
-R^{\prime}_{T11}m_{t}^{\text{in}}&M_{T_{(1)}}\end{bmatrix}\begin{bmatrix}T^{(1)}\\\
t^{(1)}\end{bmatrix}_{R}+\text{h.c.}\Bigg{\\}},$ (23)
with “input” top mass $m_{t}^{\text{in}}$ (which is an additional free
parameter in our scenario) and
$\displaystyle r^{\prime}_{T11}$
$\displaystyle=\frac{1}{R_{T00}}\int_{-L}^{L}dy\Big{(}1+r_{Y}\left(\delta(y-L)+\delta(y+L)\right)\Big{)}f_{T_{(1)}}^{2}$
$\displaystyle=\frac{2r_{T}+\pi R}{2r_{Y}+\pi
R}\times\frac{2r_{Y}+\frac{1}{S_{T_{(1)}}^{2}}\left[\frac{\pi
R}{2}-\frac{1}{2M_{T_{(1)}}}\sin(M_{T_{(1)}}\pi
R)\right]}{2r_{T}+\frac{1}{S_{T_{(1)}}^{2}}\left[\frac{\pi
R}{2}-\frac{1}{2M_{T_{(1)}}}\sin(M_{T_{(1)}}\pi R)\right]},$ (24)
$\displaystyle R^{\prime}_{T11}$
$\displaystyle=\frac{1}{R_{T00}}\int_{-L}^{L}dy\Big{(}1+r_{Y}\left(\delta(y-L)+\delta(y+L)\right)\Big{)}g_{T_{(1)}}^{2}$
$\displaystyle=\frac{2r_{T}+\pi R}{2r_{Y}+\pi
R}\times\frac{2r_{Y}(C_{T_{(1)}}/S_{T_{(1)}})^{2}+\frac{1}{S_{T_{(1)}}^{2}}\left[\frac{\pi
R}{2}+\frac{1}{2M_{T_{(1)}}}\sin(M_{T_{(1)}}\pi
R)\right]}{\frac{1}{S_{T_{(1)}}^{2}}\left[\frac{\pi
R}{2}+\frac{1}{2M_{T_{(1)}}}\sin(M_{T_{(1)}}\pi R)\right]}$ (25)
where $R_{T00}$ is given by
$\displaystyle R_{T00}$
$\displaystyle=\int_{-L}^{L}dy\Big{(}1+r_{Y}\left(\delta(y-L)+\delta(y+L)\right)\Big{)}f_{T_{(0)}}^{2}=\frac{2r_{Y}+\pi
R}{2r_{T}+\pi R}.$ (26)
$f_{T_{(n)}}$ and $g_{{}_{T_{(n)}}}$ represent the mode functions for $n$-th
KK level and are given by Datta:2012tv :
$\displaystyle f_{T_{(n)}}\;\equiv\;f_{T_{(n)L}}=f_{t_{(n)R}}$
$\displaystyle=N_{T_{(n)}}\times\begin{cases}\displaystyle\frac{\cos(M_{T_{(n)}}y)}{C_{T_{(n)}}}&\text{for
$n$ even,}\\\ \displaystyle\frac{{-}\sin(M_{T_{(n)}}y)}{S_{T_{(n)}}}&\text{for
$n$ odd,}\end{cases}$ (27) $\displaystyle
g_{{}_{T_{(n)}}}\;\equiv\;f_{T_{(n)R}}=-f_{t_{(n)L}}$
$\displaystyle=N_{T_{(n)}}\times\begin{cases}\displaystyle\frac{\sin(M_{T_{(n)}}y)}{C_{T_{(n)}}}&\text{for
$n$ even,}\\\ \displaystyle\frac{\cos(M_{T_{(n)}}y)}{S_{T_{(n)}}}&\text{for
$n$ odd}\end{cases}$ (28)
with
$\displaystyle C_{T_{(n)}}=\cos\left(\frac{M_{T_{(n)}}\pi R}{2}\right),\quad
S_{T_{(n)}}=\sin\left(\frac{M_{T_{(n)}}\pi R}{2}\right)$ (29)
and the normalization factors $N_{T_{(n)}}$ for the mode functions are given
by
$\displaystyle N_{T_{(n)}}^{-2}=\begin{cases}\displaystyle
2r_{T}+\frac{1}{C_{T_{(n)}}^{2}}\left[\frac{\pi
R}{2}+\frac{1}{2M_{T_{(n)}}}\sin(M_{T_{(n)}}\pi R)\right]&\text{for $n$
even,}\\\ \displaystyle 2r_{T}+\frac{1}{S_{T_{(n)}}^{2}}\left[\frac{\pi
R}{2}-\frac{1}{2M_{T_{(n)}}}\sin(M_{T_{(n)}}\pi R)\right]&\text{for $n$
odd.}\end{cases}$ (30)
The KK mass $M_{T_{(n)}}$ for the ‘$n$’-th level top quark excitation follows
from equation 12 where chiral zero modes occur.666Here, we consider a
situation where the fields $T^{(1)}_{L,R}$ and $t^{(1)}_{L,R}$ are rotated by
the same matrices $U_{qR}$ and $U_{qL}$ (of equation 21) from the basis used
in equations 15 and 16. We ignore the diagonal and non-diagonal modifications
in the boundary conditions. In our scenario, these modifications are Cabibbo-
suppressed (see equation 52) and hence such a treatment is justified. Note
that the off-diagonal terms are asymmetric and pick up nontrivial
multiplicative factors. This is because two different mode functions,
$f_{T_{(n)}}$ and $g_{{}_{T_{(n)}}}$ (associated with the specific states with
particular chiralities and gauge quantum numbers), contribute to them. On the
other hand, the diagonal KK mass terms are now solutions of the appropriate
transcendental equations. When expanded, the diagonal entries of the mixing
matrix involve the $L$ and $R$ components of the same gauge multiplet ($T$
from $SU(2)_{W}$ doublet or $t$ from $SU(2)_{W}$ singlet). In contrast, the
off-diagonal entries are of Yukawa-origin (signalled by the presence of
$m_{t}^{\text{in}}$) and involve both $r_{T}$ and $r_{Y}$. These terms
represent the conventional Dirac mass-terms as they connect the $L$ and the
$R$ components belonging to two different multiplets. It may be noted that
even when either $r_{T}$ or $r_{Y}$ vanishes, the mixing remains nontrivial.
Only the case with $r_{T}$ $=$ $r_{Y}$ $=$ $0$ trivially reduces to the (tree-
level) mUED.
The mass matrix of equation 23 can be diagonalized by bi-unitary
transformation with the matrices $V_{tL}^{(1)}$ and $V_{tR}^{(1)}$ where
$\displaystyle\begin{bmatrix}T^{(1)}\\\
t^{(1)}\end{bmatrix}_{L}=V^{(1)}_{tL}\begin{bmatrix}t^{(1)}_{l}\\\
t^{(1)}_{h}\end{bmatrix}_{L},\quad\begin{bmatrix}T^{(1)}\\\
t^{(1)}\end{bmatrix}_{R}=V^{(1)}_{tR}\begin{bmatrix}t^{(1)}_{l}\\\
t^{(1)}_{h}\end{bmatrix}_{R}.$ (31)
Then, equation 23 takes the diagonal form
$\displaystyle-\int d^{4}x\begin{bmatrix}\overline{t}^{(1)}_{l},\
\overline{t}^{(1)}_{h}\end{bmatrix}\begin{bmatrix}m_{t^{(1)}_{l}}&\\\
&m_{t^{(1)}_{h}}\end{bmatrix}\begin{bmatrix}t^{(1)}_{l}\\\
t^{(1)}_{h}\end{bmatrix}$ (32)
where $t^{(1)}_{l},\;t^{(1)}_{h}$ are the level ‘1’ top quark mass eigenstates
and $(m_{t^{(1)}_{l}})^{2}$ and $(m_{t^{(1)}_{l}})^{2}$ are the mass-
eigenvalues of the squared mass-matrix with $m_{t^{(1)}_{h}}>m_{t^{(1)}_{l}}$.
Note that, for clarity and convenience, we have modified the notations and the
ordering of the states in the presentations above from what appear in ref.
Datta:2012tv .
### 3.2 Mixing among level ‘0’ and level ‘2’ top quark states
The formulation described above can be extended in a straight-forward manner
for the level ‘2’ KK top quarks when this sector is augmented by the level ‘0’
(SM) top quark. Thus, the mass-matrix for the even KK parity top quark sector
(keeping only level ‘0’ and level ‘2’ KK excitations) takes the following
form:
$\displaystyle-\int d^{4}x\Bigg{\\{}\begin{bmatrix}\overline{t^{(0)}},\
\overline{T}^{(2)},\
\overline{t}^{(2)}\end{bmatrix}_{L}\begin{bmatrix}m_{t}^{\text{in}}&0&m_{t}^{\text{in}}R^{\prime}_{T02}\\\
m_{t}^{\text{in}}R^{\prime}_{T02}&M_{T_{(2)}}&m_{t}^{\text{in}}r^{\prime}_{T22}\\\
0&-m_{t}^{\text{in}}R^{\prime}_{T22}&M_{T_{(2)}}\end{bmatrix}\begin{bmatrix}t^{(0)}\\\
T^{(2)}\\\ t^{(2)}\end{bmatrix}_{R}+\text{h.c.}\Bigg{\\}}$ (33)
where $r^{\prime}_{T22}$, $R^{\prime}_{T22}$, $R^{\prime}_{T02}$ are defined
as follows, in a way similar to the case for level ‘1’ top quarks:
$\displaystyle r^{\prime}_{T22}$
$\displaystyle=\frac{1}{R_{T00}}\int_{-L}^{L}dy\Big{(}1+r_{Y}\left(\delta(y-L)+\delta(y+L)\right)\Big{)}f_{T_{(2)}}^{2}$
$\displaystyle=\frac{2r_{T}+\pi R}{2r_{Y}+\pi
R}\times\frac{2r_{Y}+\frac{1}{C_{T_{(2)}}^{2}}\left[\frac{\pi
R}{2}+\frac{1}{2M_{T_{(2)}}}\sin(M_{T_{(2)}}\pi
R)\right]}{2r_{T}+\frac{1}{C_{T_{(2)}}^{2}}\left[\frac{\pi
R}{2}+\frac{1}{2M_{T_{(2)}}}\sin(M_{T_{(2)}}\pi R)\right]},$ (34)
$\displaystyle R^{\prime}_{T22}$
$\displaystyle=\frac{1}{R_{T00}}\int_{-L}^{L}dy\Big{(}1+r_{Y}\left(\delta(y-L)+\delta(y+L)\right)\Big{)}g_{T_{(2)}}^{2}$
$\displaystyle=\frac{2r_{T}+\pi R}{2r_{Y}+\pi
R}\times\frac{2r_{Y}(S_{T_{(2)}}/C_{T_{(2)}})^{2}+\frac{1}{C_{T_{(2)}}^{2}}\left[\frac{\pi
R}{2}-\frac{1}{2M_{T_{(2)}}}\sin(M_{T_{(2)}}\pi
R)\right]}{\frac{1}{C_{T_{(2)}}^{2}}\left[\frac{\pi
R}{2}-\frac{1}{2M_{T_{(2)}}}\sin(M_{T_{(2)}}\pi R)\right]},$ (35)
$\displaystyle R^{\prime}_{T02}$
$\displaystyle=\frac{1}{R_{T00}}\int_{-L}^{L}dy\Big{(}1+r_{Y}\left(\delta(y-L)+\delta(y+L)\right)\Big{)}f_{T_{(0)}}f_{T_{(2)}}$
$\displaystyle=\frac{2r_{T}+\pi R}{2r_{Y}+\pi
R}\times\frac{2r_{Y}+2(S_{T_{(2)}}/M_{T_{(2)}}C_{T_{(2)}})}{\sqrt{2r_{T}+\pi
R}\sqrt{2r_{T}+\frac{1}{C_{T_{(2)}}^{2}}\left[\frac{\pi
R}{2}+\frac{1}{2M_{T_{(2)}}}\sin(M_{T_{(2)}}\pi R)\right]}},$ (36)
with $R_{T00}$ given by equation 26. The lower $2\times 2$ block of the mass-
matrix in equation 33 is reminiscent of the level ‘1’ top quark mass-matrix of
equation 23. Beyond this, the mass-matrix contains as the first diagonal
element the ‘input’ top quark mass, $m_{t}^{\text{in}}$ and two other non-
vanishing off-diagonal elements as the 13 and 21 elements. Obviously, the
latter two play direct roles in the mixings of the level ‘0’ and level ‘2’ top
quarks. Note that all the off-diagonal terms of the mass-matrix are
proportional to $m_{t}^{\text{in}}$ which is clearly indicative of their
origins in the Yukawa sector. The zeros in turn reflect $SU(2)_{W}$
invariance.
Diagonalization of this $3\times 3$ mass-matrix yields the physical states (3
of them) along with their mass-eigenvalues. Thus, the level ‘0’ top quark
(_i.e._ , the SM top quark) ceases to be a physical state and mixes with the
level ‘2’ top states. Given the rather involved structure of the mass-matrix,
neither is it possible to express the eigenvalues analytically in a compact
way nor they would be much illuminating theoretically. We, thus, diagonalize
the mass-matrix numerically. Similar to the case of the level ‘1’ states, we
adopt the following conventions:
$\displaystyle\begin{bmatrix}t^{(0)}\\\ T^{(2)}\\\
t^{(2)}\end{bmatrix}_{L}=V_{tL}^{(2)}\begin{bmatrix}t\\\ t^{(2)}_{l}\\\
t^{(2)}_{h}\end{bmatrix}_{L},\quad\begin{bmatrix}t^{(0)}\\\ T^{(2)}\\\
t^{(2)}\end{bmatrix}_{R}=V_{tR}^{(2)}\begin{bmatrix}t\\\ t^{(2)}_{l}\\\
t^{(2)}_{h}\end{bmatrix}_{R}$ (37)
with the physical masses $m_{t}^{\text{phys}}$, $m_{t^{(2)}_{l}}$ and
$m_{t^{(2)}_{h}}$ and with the ordering
$m_{t}^{\text{phys}}<m_{t^{(2)}_{l}}<m_{t^{(2)}_{h}}$.
### 3.3 Quantitative estimates
As can be seen from the equations above, the free parameters of the top-quark
sector in the nmUED scenario under consideration are $R$, $r_{T}$ and $r_{Y}$.
For the latter two, we use Datta:2012tv the dimensionless quantities
$r_{T}^{\prime}$ and $r_{Y}^{\prime}$ where $r_{T}^{\prime}=r_{T}R{{}^{-1}}$
and $r_{Y}^{\prime}=r_{Y}R{{}^{-1}}$. In addition, $m_{t}^{\text{in}}$ serves
as an extra free parameter from the SM sector.
#### 3.3.1 Top quark masses
Figure 2: Masses of level ‘1’ and level ‘2’ KK top quarks as functions of
$r_{T}^{\prime}$ for given $r_{Y}^{\prime}$ and $R{{}^{-1}}$ with
$m_{t}^{\text{in}}=173\,\text{GeV}$.
In figure 2 we illustrate the variations of the masses, as functions of
$r_{T}^{\prime}$, of the two KK top quark eigenstates from level ‘1’ and the
two heavier mass eigenstates that result from the mixing of level ‘0’ and
level ‘2’. The plot in the middle, when compared to the one in the left,
demonstrates how the spectrum changes as $r_{Y}^{\prime}$ varies with
$R{{}^{-1}}$ held fixed. We set the input top mass $m_{t}^{\text{in}}$ to
$173\,\text{GeV}$ in all the plots of figure 2. In turn, the effect of
changing $R{{}^{-1}}$ can be seen as one goes from the plot in the middle
($R{{}^{-1}}=1$ TeV) to the one on the right ($R{{}^{-1}}=2$ TeV). An
interesting feature common to all these plots is that there is a cross-over of
the curves for $m_{t^{(1)}_{h}}$ and $m_{t^{(2)}_{l}}$, _i.e._ , as a function
of $r_{T}^{\prime}$, at some point, the lighter of the mixed level ‘2’ state
top quark eigenstates becomes less massive compared to the heavier of the
level ‘1’ KK top quark eigenstate. The cross-overs take place at smaller
values of $r_{T}^{\prime}$ when $r_{Y}^{\prime}$ is increased for a given
$R{{}^{-1}}$ and at larger values of $r_{T}^{\prime}$ when $R{{}^{-1}}$ is
increased with $r_{Y}^{\prime}$ held fixed. Accordingly, the mass-values at
those flipping points also go down or up, respectively. Here, the dominant
role is being played by the ‘chiral mixing’ while _level-mixing_ is unlikely
to have much bearing. These plots also reveal that achieving a ‘flipped-
spectrum’ (in the above sense) is difficult if one requires the light level
‘1’ KK top quark to be heavier than about 400 GeV. Nonetheless, the overall
trend could provide easier reach for a KK top quark from level ‘2’ at the LHC.
Thus, it may be possible for up to three excited top quark states
($m_{t^{(1)}_{l}},\,m_{t^{(1)}_{h}},\,m_{t^{(2)}_{l}}$) to pop up at the LHC.
#### 3.3.2 Top quark mixings
In this subsection we take a quantitative look at the mixings in the top quark
sector from the first KK level discussed earlier in section 3.1. The mixing is
known to be near-maximal in the case of quarks (fermions) from the lighter
generations Datta:2012tv . Deviations from such maximal mixings occur in the
top quark sector due to its nontrivial structure777This is in direct contrast
with competing SUSY scenarios where mixings in the light sfermion sector are
always negligible while for top squark sector it could attain the maximal
value.. Such mixings are expected to follow similar trends at level ‘2’ (and
higher) KK levels and hence we do not present them separately. However, some
deviations are expected in the presence of _level-mixings_ which can at best
be modest for the case of $t^{(0)}-t^{(2)}$ system that we focus on in this
work.
Figure 3: Variations of the (1,1) elements of the matrices $V^{(1)}_{tL}$
(left) and $V^{(1)}_{tL}$ (right) as functions of $r_{T}^{\prime}$ for fixed
set of values of $R{{}^{-1}}$ and $r_{Y}^{\prime}$. Conventions used for
different sets of $R{{}^{-1}}$ and $r_{Y}^{\prime}$ values are: bold red for
$R{{}^{-1}}=1$ TeV and $r_{Y}^{\prime}=1$, dashed black for $R{{}^{-1}}=1$ TeV
and $r_{Y}^{\prime}=10$, bold green for $R{{}^{-1}}=2$ TeV and
$r_{Y}^{\prime}=1$ and dashed blue for $R{{}^{-1}}=2$ TeV and
$r_{Y}^{\prime}=10$.
Figure 4: Same as in figure 3 but for the variations of the (1,2) elements of
the matrices $V^{(1)}_{tL}$ (left) and $V^{(1)}_{tL}$ (right). The respective
(2,1) elements can be obtained from the orthogonality of these matrices.
The elements of the $V$-matrices in equation 31 give the admixtures of
different participating states in the KK top quark eigenstates. To be precise,
$V^{(1)}_{{tL}_{(1,1)}}$ and $V^{(1)}_{{tL}_{(2,2)}}$ represent the admixture
of $T_{L}^{(1)}$ in $t^{(1)}_{lL}$ and $t_{L}^{(1)}$ in $t^{(1)}_{hL}$
respectively while $V^{(1)}_{{tL}_{(1,2)}}$ and $V^{(1)}_{{tL}_{(2,1)}}$
indicate the same for $t_{L}^{(1)}$ in $t^{(1)}_{lL}$ and $T_{L}^{(1)}$ in
$t^{(1)}_{hL}$ in that order. Similar descriptions hold for the $V_{R}^{(1)}$
matrix. In figures 3 and 4 we illustrate the deviations from maximal mixing in
the level ‘1’ top quark sector in terms of these components of the $V$
matrices as functions of $r_{T}^{\prime}$. Each figure contains multiples
curves which present situations for different combinations of $R{{}^{-1}}$ and
$r_{Y}^{\prime}$ (see the captions for details). Note that the abrupt changes
in sign of the mixings that happen between $-1<r_{T}^{\prime}<2$ can be
understood in terms of the trends of the red and blue curves in figure 2 (the
blue curves smoothly evolve to the red ones and vice-versa).
The flat, broken magenta lines indicate maximal mixing
($|V^{(1)}_{{tL}_{(1,1)}}|=|V^{(1)}_{{tL}_{(1,2)}}|=1/\sqrt{2}$). It is clear
from these figures that there can be appreciable deviations from maximal
mixing in all these cases. As can be seen, the effects are bigger for larger
values of $r_{T}^{\prime}$ and smaller $R{{}^{-1}}$. Some dependence on
$r_{Y}^{\prime}$ is observed for smaller values of $r_{T}^{\prime}$. However,
it is to be kept in mind that the effective deviations arise from the
interplay of these elements which is again neither easy to present nor much
illuminating.
### 3.4 Effective couplings
As mentioned earlier, not only masses undergo modifications in the presence of
BLTs but also the wavefunctions get distorted. The latter affects the
couplings through the overlap integrals. These are integrals over the extra
dimension of a product of mode functions of the states that appear at a given
interaction vertex. In this section we briefly discuss the generic properties
of some of these overlap integrals which play roles in the present study.
Assuming the wavefunctions to be real, the general form of the multiplicative
factor that scales the corresponding SM coupling strengths is given by
$\displaystyle
g_{f_{i}^{(l)}f_{j}^{(m)}f_{k}^{(n)}}=\mathcal{N}_{ijk}\int_{-L}^{L}dy\Big{[}1+r_{ijk}^{(l,m,n)}\left(\delta(y-L)+\delta(y+L)\right)\Big{]}f_{i}^{(l)}(y)f_{j}^{(m)}(y)f_{k}^{(n)}(y)$
(38)
where $i,j,k$ represent different interacting fields and
$f_{i}^{(l)},f_{j}^{(m)},f_{k}^{(n)}$ are the corresponding mode functions
with the KK indices $l,m,n$, respectively, as defined in sections 2.1, 3.1 and
3.2. The factor $r_{ijk}^{(l,m,n)}$ stands for relevant BLT parameter(s) while
the normalization factor $\mathcal{N}_{ijk}$ is suitably chosen to recover the
SM vertices when $l$=$m$=$n$=0 (except for the Yukawa sector of the nmUED
scenario under consideration).
The key to understand the general structure is the flatness of the zero-mode
($n=0$) profiles in our minimal configuration. For these, the factor takes the
following form:
$\displaystyle
g_{f_{i}^{(l)}f_{j}^{(m)}f_{k}^{(0)}}=\mathcal{N}_{ijk}f_{k}^{(0)}\int_{-L}^{L}dy\Big{[}1+r_{ijk}^{(l,m,0)}\left(\delta(y-L)+\delta(y+L)\right)\Big{]}f_{i}^{(l)}(y)f_{j}^{(m)}(y),$
(39)
where we see the zero-mode field has been taken out of the integral in
equation 38. For $i=j$, the overlap integral reduces to Kronecker’s delta
function, $\delta_{l,m}$ and the overall strength turns out to be identically
equal to 1. Orthonormality of the involved states constrains the
possibilities. In table 1 we collect some of these interactions and group them
in terms of their effective strengths (given by equation 39). This list, in
particular, the set of couplings in the third column, is not exhaustive and
presented for demonstrative purposes only.
In addition to these, mixings in the top quark sector in the form of both
chiral mixing and _level-mixing_ play roles in determining the effective
couplings. In this subsection we briefly discuss such effects on some of the
important interaction-vertices involving the top quarks, the gauge and the
Higgs bosons from different KK levels. As in section 3.3, we further introduce
the dimensionless parameters $r_{\text{EW}}^{\prime}\,(=R^{-1}r_{\text{EW}})$,
$r_{Q}^{\prime}\,(=R^{-1}r_{Q})$ and $r_{G}^{\prime}\,(=R^{-1}r_{G})$
replacing $r_{\text{EW}}\,(=r_{H})$, $r_{Q}$ and $r_{G}$, the BLKT parameters
for the electroweak gauge boson and Higgs sectors, the first two generation
quark sector and the gluon sector, respectively. In addition, we also
introduce a corresponding universal parameter $r_{L}$ for the lepton sector
which we will use in section 4.3. Later, in section 5, we will refer back to
this discussion in the context of phenomenological analyses of the scenario.
| | $Q_{R/L}^{(1)}-{V}^{(1)}-Q_{R/L}^{(0)}$
---|---|---
| $V^{(2)}-V^{(2)}-V^{(0)}$ | $q_{R/L}^{(1)}-{V}^{(1)}-q_{R/L}^{(0)}$
| $V^{(1)}-V^{(1)}-V^{(0)}$ | $Q_{R/L}^{(0)}-{V}^{(2)}-Q_{R/L}^{(0)}$
| $Q_{R/L}^{(1)}-{V}^{(0)}-Q_{R/L}^{(1)}$ | $Q_{L}^{(1)}-{H}^{(0)}-q_{R}^{(1)}$
$Q_{R/L}^{(2)}-{V}^{(0)}-Q_{R/L}^{(0)}$ | $q_{R/L}^{(1)}-{V}^{(0)}-q_{R/L}^{(1)}$ | $Q_{L}^{(2)}-{H}^{(0)}-q_{R}^{(0)}$
$q_{R/L}^{(2)}-{V}^{(0)}-q_{R/L}^{(0)}$ | $Q_{R/L}^{(2)}-{V}^{(0)}-Q_{R/L}^{(2)}$ | $Q_{L}^{(0)}-{H}^{(2)}-q_{R}^{(0)}$
${V}^{(2)}-{V}^{(0)}-{V}^{(0)}$ | $q_{R/L}^{(2)}-{V}^{(0)}-q_{R/L}^{(2)}$ | $Q_{L}^{(0)}-{H}^{(0)}-q_{R}^{(2)}$
0 | 1 | non-zero
Table 1: Classes of different effective (tree level) couplings (given by
equation 39) involving the gauge boson ($V$), Higgs ($H$) and the left- and
right-handed, $SU(2)_{W}$ doublet ($Q$) and singlet ($q$) quark excitations
and their relative strengths (shown in the last row) compared to the
corresponding SM cases.
#### 3.4.1 Effective couplings involving the gauge bosons
The set of couplings that we briefly discuss here are those that would appear
in the production of the KK top quarks at the LHC and their decays. In figure
5 we illustrate the coupling-deviation (a multiplicative factor of the
corresponding SM value at the tree level) $g^{(2)}$-$q^{(0)}$-$q^{(0)}$ (left)
and $g^{(2)}$-$q^{(2)}$-$q^{(0)}$ (right) in the generic
$r^{\prime}_{V}-r^{\prime}_{Q/T/L}$ plane. In both of these plots, the mUED
case is realized along the diagonals over which
$r_{G}^{\prime}=r_{Q}^{\prime}$. In the first case, the mUED value is known to
be vanishing at the tree level since KK number is violated. Hence, the
diagonal appears with the contour-value of zero. For vertices involving the
top quarks, $r_{T}^{\prime}$ replaces $r_{Q}^{\prime}$. For a process like
$pp\to\bar{t}^{(2)}_{l}t$ \+ h.c., the former kind of coupling appears at the
parton-fusion (initial state) vertex while the latter shows up at the
production vertex. The combined strength of these two couplings controls the
production rate for the mentioned process. Further, the situation is not much
different for the level ‘2’ electroweak gauge bosons except for some
modifications due to mixings present in the electroweak sector. In general, it
can be seen from the first plot of figure 5 that the coupling
$g^{(2)}$-$q^{(0)}$-$q^{(0)}$ picks up a negative sign for
$r_{G}^{\prime}>r_{Q}^{\prime}$. This could have nontrivial phenomenological
implications for processes in which interfering Feynman diagrams are present.
On the other hand, $g^{(2)}$-$q^{(2)}$-$q^{(0)}$ remains always positive as is
clear from the second plot of figure 5. Note that the three-point vertex
$V^{(0)}$-$V^{(0)}$-$V^{(2)}$ and the generic ones of the form
$V^{(0)}$-$f^{(0)}$-$f^{(2)}$ are absent because the corresponding overlap
integrals vanish due to orthogonality of the involved mode functions.
Figure 5: Contours of deviation for the generic couplings
$V^{(2)}$-$F^{(0)}$-$F^{(0)}$ (or $V^{(2)}$-$f^{(0)}$-$f^{(0)}$) (left) and
$V^{(2)}$-$F^{(2)}$-$F^{(0)}$ (or $V^{(2)}$-$f^{(2)}$-$f^{(0)}$) (right) from
the corresponding SM values in the $r^{\prime}_{V}-r^{\prime}_{Q/T/L}$ plane.
$V$, $F$ and $f$ stand for generic gauge boson, $SU(2)_{W}$ doublet and
singlet fermion fields (with corresponding chiralities), respectively. Note
that when $V$ is the (KK) $W$ boson, types of the two fermions involved at a
given vertex are different.
Figure 6: Same as in figure 5 but for the generic couplings
$V^{(2)}$-$F^{(1)}_{L}$-$F^{(1)}_{L}$ or $V^{(2)}$-$f^{(1)}_{R}$-$f^{(1)}_{R}$
(left) and $V^{(2)}$-$f^{(1)}_{L}$-$f^{(1)}_{L}$ or
$V^{(2)}$-$F^{(1)}_{R}$-$F^{(1)}_{R}$ (right).
In figure 6 we present the corresponding contours of similar deviations in the
couplings involving the level ‘2’ KK gauge bosons and the level ‘1’ KK quarks.
The plot on left shows the situation for the left- (right-) chiral component
of the $SU(2)_{W}$ doublet (singlet) quarks while the plot on right
illustrates the case for left- (right-) chiral component of the $SU(2)_{W}$
singlet (doublet) quarks. These are in conformity with the mode functions for
these individual components of the level ‘1’ KK quarks. However, it should be
noted that the KK quarks being vector-like states, each of the $SU(2)_{W}$
doublet and singlet partners have both left- and right-chiral components.
Thus, the effective couplings are obtained only by suitably combining (with
appropriate weights) the strengths as given by the two plots. In the case of
KK top quarks, the situation would be further complicated because of
significant mixing between the two gauge eigenstates. For brevity, a list of
relevant couplings is presented in table 1 with mentions of the kind of
modifications they undergo in the nmUED scenario. It is clear from these
figures that these (component) couplings involving level ‘2’ KK states are in
general suppressed compared to the relevant SM couplings except over a small
region with $r^{\prime}_{Q/T/L}<0$.
#### 3.4.2 Effective couplings involving the Higgs bosons
Figure 7: Contours of deviation in the $r_{T}^{\prime}-r_{Y}^{\prime}$ plane
for the generic couplings $H^{(0)}$-$T^{(0)}_{L}$-$t^{(0)}_{R}$ (left) and
$H^{(0)}$-$T^{(2)}_{L}$-$t^{(0)}_{R}$ or $H^{(0)}$-$T^{(0)}_{L}$-$t^{(2)}_{R}$
compared to the corresponding SM cases.
The association of the Higgs sector with the third SM family is rather
intricate and has deep implications which unfold themselves in many scenarios
beyond the SM. SUSY scenarios provide very good examples of this, some
analyses have been done in the mUED Bandyopadhyay:2009gd and the nmUED
scenario is also no exception. The couplings among the Higgs bosons and the KK
top quarks of the nmUED scenario can deviate significantly from the
corresponding SM Yukawa coupling. However, the zero-mode (SM) Higgs Yukawa
couplings do not depend upon $r_{H}^{\prime}\,(=r_{\text{EW}}^{\prime})$. In
the left panel of figure 7 we illustrate the possible deviation in the SM
Yukawa coupling itself in the $r_{T}^{\prime}-r_{Y}^{\prime}$ plane. Along the
diagonal of this figure (with $r_{T}^{\prime}=r_{Y}^{\prime}$) the SM value of
the Yukawa coupling is preserved. Note that the latest LHC data still allows
for significant deviations in the $H$-$t$-$t$ coupling Chatrchyan:2013yea ;
cms-tth-gamma ; atlas-tth-gamma ; Nishiwaki:2013cma .
In the right panel we show deviations of the generic $H$-$t^{(2)}$-$t$ which
appears at the tree level in nmUED. Unlike in the case of the interaction
vertex $V^{(0)}$-$f^{(2)}$-$f^{(0)}$ (where $V^{(0)}$ is a massive SM gauge
boson) where the involved coupling vanishes in the absence of _level-mixing_
between $f^{(2)}$ and $f^{(0)}$, the analogous Higgs vertex remains non-
vanishing even in the absence of _level-mixing_ between the fermions. However,
in this case, for $r_{T}^{\prime}=r_{Y}^{\prime}$ the coupling vanishes. This
implies that the more the Yukawa coupling involving the level ‘0’ fields
appears to agree with the SM expectation (from future experimental analyses),
the weaker the coupling $H$-$t^{(2)}$-$t$ in such a scenario would get to be.
In both cases, however, we find that the coupling strengths get enhanced for
smaller values of $r_{T}^{\prime}$ with $r_{T}^{\prime}<r_{Y}^{\prime}$. All
these indicate that production of the SM Higgs boson via gluon-fusion and its
decay to di-photon final state can receive non-trivial contributions from such
couplings and thus might get constrained from the LHC data. The issue is
currently under study.
## 4 Experimental constraints and benchmark scenarios
Several different experimental observations put constraints of varying degrees
on the parameters (like $R{{}^{-1}}$, $r_{T}^{\prime}$, $r_{Q}^{\prime}$,
$r_{Y}^{\prime}$ and the input top quark mass ($m_{t}^{\text{in}}$)) that
control the KK top quark sector. First and foremost, $R{{}^{-1}}$ is expected
to be constrained from the searches for level ‘1’ KK quarks and KK gluon at
the LHC. In the absence of any such dedicated search, a rough estimate of
$R{{}^{-1}}>1$ TeV has been derived in ref. Datta:2012tv by appropriate
recast of the LHC constraints obtained for the squarks and the gluino in SUSY
scenarios.
As discussed in the previous subsection, observed mass of the top quark
restricts the parameter space in a nontrivial way. Also, important constraints
come from the experimental bounds on flavor changing neutral currents (FCNC),
electroweak precision bounds in terms of the Peskin–Takeuchi parameters
($S,\,T$ and $U$) and bounds on effective four-fermion interactions. In this
section we discuss these constraints briefly and choose a few benchmark
scenarios that satisfy them and are phenomenologically interesting.
### 4.1 Constraints from the observed mass of the SM-like top quark
In figure 8 we show the allowed regions in the $r_{T}^{\prime}-r_{Y}^{\prime}$
plane that result in top quark pole mass within the range 171-175 GeV
Alekhin:2012py (which is argued to be a more appropriate range than what the
experiments actually quote CDF:2013jga ) for given values of $R{{}^{-1}}$ and
input top quark masses.
Figure 8: Regions (in green) in the $r_{T}^{\prime}-r_{Y}^{\prime}$ plane for
three $R{{}^{-1}}$ values of (1.5, 2 and 3 TeV, varying along the rows) and
for different suitable values of $m_{t}^{\text{in}}$ (indicated on top of each
plot) that are consistent with physical (SM-like) top quark mass
($m_{t}^{\text{phys}}$) being within the range $m_{t}^{\text{phys}}=173\pm 2$
GeV.
Some general observations are that the physical top quark mass
($m_{t}^{\text{phys}}$) rarely becomes larger than the input top quark mass
($m_{t}^{\text{in}}$). This means, to have $m_{t}^{\text{phys}}$ at least of
171 GeV, $m_{t}^{\text{in}}$ has to be larger than 171 GeV. Further,
increasing $m_{t}^{\text{in}}$ beyond around 175 GeV, as we go over to the
second row of figure 8, opens up disjoint sets of allowed islands in the
$r_{T}^{\prime}-r_{Y}^{\prime}$ plane with increasing region allowed for
negative $r_{Y}^{\prime}$ (and extending to larger $r_{T}^{\prime}$ values) at
the expense of the same with positive $r_{Y}^{\prime}$. Increasing
$m_{t}^{\text{in}}$ further (beyond say, 180 GeV) results in allowed regions
diminishing to an insignificant level. These features remain more or less
unaltered as $R{{}^{-1}}$ is increased, as we go from left to right along a
horizontal panel. A palpable direct effect that can be attributed to
increasing $R{{}^{-1}}$ is in the moderate increase of the region in the
$r_{T}^{\prime}-r_{Y}^{\prime}$ plane consistent with $m_{t}^{\text{phys}}$,
in particular, for negative $r_{Y}^{\prime}$ values and when
$m_{t}^{\text{in}}$ is not terminally large (_i.e._ , below $190$ GeV, say)
for the purpose.
Although a moderate range of input top quark mass
$171<m_{t}^{\text{in}}\lesssim 190$ is consistent with
$171<m_{t}^{\text{phys}}<175$ GeV in the space of
$R{{}^{-1}}-r_{T}^{\prime}-r_{Y}^{\prime}$, the allowed region there is rather
sensitive to the variation in $m_{t}^{\text{in}}$. Thus, the allowed range of
the $m_{t}^{\text{phys}}$ restricts the nmUED parameter space in a significant
way which, in turn, influences the masses and the couplings of the KK top
quarks. An important point is to be noted here. The level ‘1’ top quark
sector, though does not talk to either level ‘0’ or level ‘2’ sector directly
(because of conserved KK-parity), is influenced by these constraints since
$r_{T}^{\prime}$, $r_{Y}^{\prime}$ and $R{{}^{-1}}$ also govern the same.
### 4.2 Flavor constraints
The BLKTs ($r_{Q}^{\prime}$) and the BLYTs ($r_{Y}^{\prime}$) are matrices in
the flavor space. Hence, their generic choices may induce large FCNCs at the
tree level. It is possible to choose a basis in which the BLKT matrix is
diagonal. This ensures no mixing among fermions of different flavors or from
different KK levels arising from the gauge kinetic terms. However, with the
Yukawa sector included, off-diagonal terms (mixings) appear in the gauge
sector on rotating the gauge kinetic terms into a basis where the quark mass
matrices are diagonal. These terms could induce unacceptable FCNCs at the tree
levels and thus, would be constrained by experiments. In figure 9 we present
the tree level diagram that could give rise to unwanted FCNC effects.
Figure 9: Feynman diagram showing the induced FCNC vertex.
A rather high compactification scale ($R{{}^{-1}}\sim{\cal{O}}(10^{5})$ TeV;
the so-called decoupling mechanism) or a near-perfect mass-degeneracy among
the KK quarks at a given level (${\Delta m\over m^{(1)}}\lesssim 10^{-6}$;
across all three generations) could suppress the FCNCs to the desired level
gerstenlauer . While the first option immediately renders all the KK particles
rather too massive, the second one makes the KK top quarks as heavy as the KK
quarks from the first two generations thus making them quite difficult to be
accessed at the LHC. A third option in the form of “alignment” (of the
rotation matrices) gerstenlauer can make way for significant lifting of
degeneracy thus allowing for light enough quarks from the third generation
that are within the reach of the LHC. In such a setup, FCNC occurs in the
$up$-type doublet sector. Hence, the strongest of the bounds in terms of the
relevant Wilson coefficient ($C^{1}_{D}$) comes from the recent observation of
$D^{0}-\overline{D^{0}}$ mixing Aaij:2012nva (and not from the $K$ or the $B$
meson systems) and the requirement is $|C^{1}_{D}|<7.2\times
10^{-7}\,\text{TeV}^{-2}$ Bona:2007vi , attributed solely to the gluonic
current which is by far the dominant contribution. The essential contents of
the setup are summarized in appendix B.
In the left-most panel of figure 10 we demonstrate the allowed/disallowed
region in the $r_{T}^{\prime}-r_{Q}^{\prime}$ plane for $r_{G}^{\prime}=1$
with $R=1\,\text{TeV}$. The panel in the middle demonstrates the corresponding
regions in the $r_{T}^{\prime}-r_{G}^{\prime}$ plane for $r_{Q}^{\prime}=+1$.
It is seen that some region with $r_{T}^{\prime}<0$ is disallowed when
$r_{G}^{\prime}$ is large, _i.e._ , when the level ‘2’ KK gluon is relatively
light. The right-most panel illustrates the region allowed in the same plane
but for $r_{Q}^{\prime}=-1$. The bearing of the FCNC constraint is most
pronounced in this case. It can be noted that the smaller the value of
$r_{G}^{\prime}$ is, the heavier is the mass of the level ‘2’ gluon and hence,
the stronger is the suppression of the dangerous FCNC contribution. Such a
suppression could then allow $r_{T}^{\prime}$ to be significantly different
from $r_{Q}^{\prime}$ but still satisfying the FCNC bounds. This feature is
apparent from the rightmost panel of figure 10. Note that a rather minimal
value for $R{{}^{-1}}$ (=1 TeV) is chosen for this demonstration. A larger
$R{{}^{-1}}$ results in a more efficient suppression of FCNC effects and
hence, leads to a larger allowed region. In summary, it appears that FCNC
constraints do not seriously restrict the third generation sector as yet.
Figure 10: Regions in the $r_{T}^{\prime}-r_{Q}^{\prime}$ (for fixed
$r_{G}^{\prime}$; the left-most plot) and $r_{T}^{\prime}-r_{G}^{\prime}$ (for
fixed $r_{Q}^{\prime}$; the middle and the right-most plot) planes for
$R{{}^{-1}}=1$ TeV that are allowed (in green) by FCNC constraints. For the
first two figures, thin strip(s) of the disallowed regions (in red) are
highlighted for better visibility.
### 4.3 Precision constraints
It is well known that the Peskin–Takeuchi parameters $S$, $T$ and $U$ that
parametrize the so-called oblique corrections to the electroweak gauge boson
propagators Peskin:1990zt ; Peskin:1991sw put rather strong constraints on
the mUED scenario. These observables are affected by the modification in the
Fermi constant $G_{F}$ (determined experimentally by studying muon decay) due
to induced effective 4-fermion vertices originating from exchange of
electroweak gauge bosons from even KK levels. These were first calculated in
refs. Kakuda:2013kba ; Appelquist:2002wb ; Flacke:2005hb ; Gogoladze:2006br ;
Baak:2011ze assuming mUED tree-level spectrum while ref. Flacke:2013pla
expressed them in terms of the actual (corrected) masses of the KK modes.
As discussed in refs. Rizzo:1999br ; Davoudiasl:1999tf ; Csaki:2002gy ;
Carena:2002dz ; Flacke:2011nb , the correction to $G_{F}$ can be incorporated
in the electroweak fit via the modifications it induces in the Peskin–Takeuchi
parameters and contrasting them with the experimentally determined values of
the latter. Note that in the nmUED scenario we consider, level ‘2’ electroweak
gauge bosons have tree-level couplings to the SM fermions and these modify the
effective 4-fermion couplings. These effects are over and above what mUED
induces888To be precise, in general, the mUED type higher-order contributions
(usual one-loop-induced oblique corrections) would not be exactly the same as
that from the actual mUED scenario. However, as pointed out in ref.
Flacke:2011nb , in the “minimal” case of $r_{W}=r_{B}=r_{H}$ along with the
requirements on the relations involving $\mu$-s and $\lambda$-s as given in
equations 3 and 11, exact mUED limits for the couplings are restored while
departures in the KK masses (from the corresponding mUED values) still remain.
where such KK number violating couplings appear only at higher orders. It is
thus natural to expect that usual oblique corrections to $S$, $T$ and $U$
induced at one-loop level would be sub-dominant when compared to the above
nmUED tree-level contributions. Thus, in our present analysis, we neglect the
one-loop contributions but otherwise follow the approach originally adopted in
ref. Flacke:2011nb and which was later used in ref. Flacke:2013pla . The
nmUED effects are thus parametrized as:
$\displaystyle S_{\text{nmUED}}=0,\qquad\quad
T_{\text{nmUED}}=-\frac{1}{\alpha}\frac{\delta G_{F}}{G_{F}},\qquad\quad
U_{\text{nmUED}}=\frac{4\sin^{2}{\theta_{W}}}{\alpha}\frac{\delta
G_{F}}{G_{F}}$ (40)
where $\alpha$ is the electromagnetic coupling strength, $\theta_{W}$ is the
$\overline{MS}$ Weinberg angle, both given at the scale $M_{Z}$ and $G_{F}$ is
given by
$\displaystyle G_{F}=G_{F}^{0}+\delta G_{F}$ (41)
with $G_{F}^{0}$ ($\delta G_{F}$) originating from the $s$-channel SM (even
KK) $W^{\pm}$ boson exchange. The concrete forms of these effects are
calculated in our model following ref. Flacke:2011nb . Using our notations,
these are given by:
$\displaystyle G_{F}^{0}$
$\displaystyle=\frac{g_{2}^{2}}{4\sqrt{2}}\frac{1}{M_{W}^{2}},\quad\delta
G_{F}=\sum_{n\geq
2:\text{even}}\frac{g_{2}^{2}}{4\sqrt{2}}\frac{1}{m_{W_{(n)}}^{2}}\left(g_{{}_{L_{(0)}W_{(n)}L_{(0)}}}\right)^{2},$
(42)
$\displaystyle\left.g_{{}_{L_{(0)}W_{(n)}L_{(0)}}}\right|_{n\text{:even}}$
$\displaystyle\equiv\frac{1}{f_{W^{(0)}}}\int_{-L}^{L}dy\left(1+r_{\text{EW}}\left[\delta(y-L)+\delta(y+L)\right]\right)f_{L_{(0)}}f_{W_{(n)}}f_{L_{(0)}}$
$\displaystyle=\frac{2\sqrt{4r_{\text{EW}}+2\pi
R}\left(M_{W_{(n)}}r_{L}+\tan\left(\frac{M_{W_{(n)}}\pi
R}{2}\right)\right)}{M_{W_{(n)}}\left(2r_{L}+\pi
R\right)\sqrt{4r_{\text{EW}}+\pi R\sec^{2}\left(\frac{M_{W_{(n)}}\pi
R}{2}\right)}+2\tan\left(\frac{M_{W_{(n)}}\pi R}{2}\right)/M_{W_{(n)}}}$ (43)
where $M_{W_{(n)}}$ is determined by equation 12. Even though the KK leptons
do not appear in the process, the BLKT parameter $r_{L}$ in the lepton sector
(to be precise, the one for the 5D muon doublet) inevitably influences the
coupling-strength given in equation 43. We, however, assume a flavor-universal
BLKT parameter $r_{L}$ (just like what we do in the quark sector when we take
$r_{Q}=r_{T}$) which help trivially circumvent tree-level contributions to
lepton-flavor-violating processes.
Figure 11: Regions (in green) in the $r_{\text{EW}}^{\prime}-R{{}^{-1}}$ plane
allowed by electroweak precision data at $95\%$ C.L. The black asterisks
represent the global minimum in each one of them:
$\chi^{2}_{\text{min}}=8.8\times 10^{-9}$ at $(r_{\text{EW}}^{\prime},R^{-1})$
= $(6.11\times 10^{-3},1229\,\text{GeV})$ when $r_{L}^{\prime}=0$,
$\chi^{2}_{\text{min}}=3.9\times 10^{-9}$ at $(r_{\text{EW}}^{\prime},R^{-1})$
= $(0.505,1029\,\text{GeV})$ when $r_{L}^{\prime}=0.5$,
$\chi^{2}_{\text{min}}=1.5\times 10^{-8}$ at $(r_{\text{EW}}^{\prime},R^{-1})$
= $(2.02,1306\,\text{GeV})$ when $r_{L}^{\prime}=2$.
We perform a $\chi^{2}$ fit of the parameters $S_{\text{nmUED}}$,
$T_{\text{nmUED}}$ and $U_{\text{nmUED}}$ (with $\delta G_{F}$ evaluated for
$n=2$ only) for three fixed values of $r_{L}^{\prime}$
($r_{L}^{\prime}=r_{L}R{{}^{-1}}=0$, $0.5$ and $2$) to the experimentally
fitted values of the allowed new physics (NP) components in these respective
observables as reported by the GFitter group Baak:2012kk which are given by
$\displaystyle S_{\text{NP}}=0.03\pm 0.10,$ $\displaystyle
T_{\text{NP}}=0.05\pm 0.12,$ $\displaystyle U_{\text{NP}}=0.03\pm 0.10,$
the correlation coefficients being
$\displaystyle\rho_{ST}=+0.89,$ $\displaystyle\rho_{SU}=-0.54,$
$\displaystyle\rho_{TU}=-0.83,$
and the reference input masses of the SM top quark and the Higgs boson being
$m_{t}=173$ GeV and $m_{H}=126$ GeV, respectively.
In figure 11 we show the $95\%$ C.L. allowed region in the
$r_{\text{EW}}^{\prime}-R{{}^{-1}}$ plane as a result of the fit performed. As
can be expected, the bound refers to $r_{\text{EW}}^{\prime}$ as the only
brane-local parameter which, unlike in ref. Flacke:2013pla , can be different
from the corresponding parameters governing other sectors of the theory. Such
a constraint is going to restrict the mass-spectrum and the couplings in the
electroweak sector which is relevant for our present study. It is not
unexpected that for larger values of $r_{\text{EW}}$ which result in
decreasing masses for the electroweak gauge bosons, only larger values of
$R{{}^{-1}}$ (which compensates for the former effect) remain allowed thus
rendering these excitations (appearing in the propagators) massive enough to
evade the precision bounds. Interestingly, it is possible to relax the bounds
by introducing a positive $r_{L}^{\prime}$ as shown in figure 11, a feature
that can be taken advantage of as we explore the nmUED parameter space
further. This is since the coupling involved $g_{{}_{L_{(0)}W_{(n)}L_{(0)}}}$
gets reduced in the process (see the left plot in figure 5).
### 4.4 Benchmark scenarios
For our present analysis, we now choose some benchmark scenarios which satisfy
the constraints discussed in the previous subsection. The parameter space of
these scenarios mainly spans over
$r_{T}^{\prime},\,r_{Y}^{\prime},\,R{{}^{-1}}$ and, as a minimal choice,
$r_{\text{EW}}^{\prime}=r_{H}^{\prime}$999Departure from this assumption makes
the gauge boson zero modes non-flat and hence correct values (within
experimental errors) of the SM parameters like $\alpha_{em},G_{f},m_{W},m_{Z}$
can only be reproduced in a constrained region of
$r_{\text{EW}}^{\prime}-r_{H}^{\prime}$ parameter space Flacke:2008ne .. We
also include $r_{G}^{\prime}$, $r_{Q}^{\prime}$ and $r_{L}^{\prime}$ which are
the BLKT parameters for the KK gluon, the KK quark and the KK lepton sectors,
respectively. $r_{G}^{\prime}$ has some non-trivial implications for the
couplings of the KK top quarks to the gluonic excitations as discussed in
section 3.4. The parameter $r_{Q}^{\prime}$, though enters our discussion
primarily through FCNC considerations (see section 4.2 and appendix B),
governs the couplings $V^{(2)}$-$q^{(0)}$-$q^{(0)}$ (as shown in figure 5)
that control KK top quark production processes. Both $r_{G}^{\prime}$ and
$r_{Q}^{\prime}$ serve as key handles on the masses of the KK gluon and the KK
quarks from the first two generations, respectively. Similar is the status of
$r_{L}^{\prime}$ which enters through the oblique parameters and controls the
masses and couplings in the lepton sector.
In search for suitable benchmark scenarios, we require the following
conditions to be satisfied. We require the approximate lower bound on
$R{{}^{-1}}$ to hover around 1 TeV which is obtained by recasting the LHC
bounds on squarks (from the first two generations) and the gluino in terms of
level ‘1’ KK quarks and KK gluons in the nmUED scenario Datta:2012tv .
Further, the lighter of the level ‘1’ KK top quark ($t^{(1)}_{l}$) is required
to be at least about 500 GeV. This safely evades current LHC-bounds on similar
excitations while lower values may still be allowed given that these bounds
result from model-dependent assumptions.
The above requirements together calls for a non-minimal sector for the
electroweak gauge bosons ($r_{\text{EW}}^{\prime}\neq 0$) such that the
lightest KK gauge boson, the KK photon ($\gamma^{(1)}$) is the lightest KK
particle (LKP, a possible dark matter candidate)101010This is a possible
choice for the dark matter candidate in the nmUED scenario. Ref. Flacke:2008ne
explores other possible candidates in such a scenario.. Incorporation of a
non-minimal gauge sector affects the couplings of the gauge bosons which, as
we will see, could be phenomenologically non-trivial. The choice
$r_{\text{EW}}^{\prime}=r_{H}^{\prime}$ renders the KK excitations of the
gauge and the Higgs boson very close in mass thus allowing them to take part
in the phenomenology of the KK top quarks. In the present scenario, other BLT
parameters in the Higgs sector, $\mu_{b}$ and $\lambda_{b}$, are constrained
by equations 3 and 11 in addition to the measured Higgs mass as an input.
Therefore, these are not independent degrees of freedom.
In table 2 we present the spectra for three such benchmark scenarios: two of
them with $R{{}^{-1}}=1$ TeV and the other with $R{{}^{-1}}=1.5$ TeV. The BLKT
parameters $r_{G}^{\prime}$ and $r_{Q}^{\prime}$ are so chosen such that the
masses of the level ‘1’ KK gluon are in the range 1.6-1.7 TeV (_i.e._ ,
somewhat above the current LHC lower bounds on similar (SUSY) excitations)
while the KK quarks from the first two generations are heavier111111Such a
hierarchy of masses opens up the possibility of level ‘1’ KK top quarks being
produced in the cascade decays of the KK gluon and the KK quarks.. Note that
in both cases we are having negative $r_{G}^{\prime}$ and $r_{Q}^{\prime}$. In
the top quark sector, the BLKT parameter $r_{T}^{\prime}$ are fixed at values
for which both light and heavy level ‘1’ KK top quarks have sub-TeV masses and
hence expected to be within the LHC reach. Also, $r_{Y}^{\prime}$, the BLT
parameter for the Yukawa sector, has been tuned in the process to end up with
such spectra. Note that the choices of values for $r_{T}^{\prime}$ and
$r_{Y}^{\prime}$ are consistent with the constraints from the physical top
quark mass as discussed in section 4.1 and the flavor constraints discussed in
section 4.2. Larger values of $R{{}^{-1}}$ would tend to make the level ‘2’ KK
top quark a little too heavy (${\lesssim}\,1.5$ TeV) to be explored at the LHC
while if one requires the lighter level ‘1’ KK top quark not too light
($\lesssim 300$ GeV) which can be quickly ruled out by the LHC experiments
even in an nmUED scenario which we consider. Nonetheless, the lighter of the
level ‘2’ top quark may anyway be heavy and only the level ‘1’ top quarks
remain to be relevant at the LHC. In that case, larger values of $R{{}^{-1}}$
also remain relevant. Values of $r_{\text{EW}}^{\prime}$ are so chosen as to
have $\gamma^{(1)}$ as the LKP with masses around half a TeV. This renders the
level ‘2’ electroweak gauge bosons to have masses around 1.5 TeV thus making
them possibly sensitive to searches for gauge boson resonances at the LHC
Flacke:2012ke ; Chatrchyan:2012su 121212The caveats are that these level ‘2’
gauge bosons could have very large decay widths (exceptionally fat) due to
enhanced $V^{(2)}$-$f^{(0)}$-$f^{(0)}$ couplings as opposed to narrow-width
approximation for the resonances assumed in the experimental analysis
Chatrchyan:2012su and hence need dedicated studies for them at the LHC
Kelley:2010ap . Further, the involved assumption of a 100% branching fraction
for the resonance decaying to quarks may also not hold. These two issues would
invariably relax the mentioned bounds on level ‘2’ gauge bosons..
In table 2 we also indicate the masses of the level ‘2’ KK excitations. It is
to be noted that the lighter of the level ‘2’ KK top quark may not be that
heavy ($\lesssim 1.5$ TeV). Level ‘2’ gluon, for our choices of parameters, is
pushed to around 3 TeV and hence, unless their couplings to quarks (SM ones or
from level ‘1’) are enhanced, LHC may be barely sensitive to their presence.
This is a rather involved issue which again warrants dedicated studies and is
beyond the scope of the present work.
For the first benchmark point (BM1) with $R{{}^{-1}}=1$ TeV, the mass-
splitting between the two level ‘1’ top quark states is much smaller ($\sim
100$ GeV) with a somewhat heavier $t^{(1)}_{l}$ when compared to the second
case (BM2) for which $R{{}^{-1}}=1.5$ TeV. We will see in section 5 that such
mass-splittings and the absolute masses themselves for the KK top quarks have
interesting bearing on their phenomenology at the LHC. Further, the relevant
couplings do change (see figures 5, 6 and 7) in going from one point to the
other. The third benchmark point BM3 is just BM1 but with different
$r_{Y}^{\prime}$ and $m_{t}^{\text{in}}$. BM3 demonstrates a situation with
enhanced Higgs-sector couplings and its ramifications at the LHC. It is found
that for all the three benchmark points, the coupling
$V^{(2)}$-$f^{(0)}$-$f^{(0)}$ get enhanced when level ‘2’ $W$ or $Z$ boson is
involved.
Note that the KK bottom quark masses are also governed by $r_{T}^{\prime}$ and
$r_{Y}^{\prime}$ for a given $R{{}^{-1}}$. However, since the splitting
between the two physical states at a given KK level is proportional to the SM
bottom quark mass, the KK bottom quarks at each given level are almost
degenerate (just as it is for the KK quark flavors from the first two
generations) in mass unlike their top quark counterparts. Thus, some of the KK
bottom quarks can have masses comparable to those of the corresponding KK top
quark states and hence would eventually enter a collider study otherwise
dedicated for the latter. A detailed discussion on the involved issues are out
of the scope of the present work.
BM1 | $R{{}^{-1}}=1$ TeV, $r_{G}^{\prime}=-1$, $r_{Q}^{\prime}=-1.2$, $r_{T}^{\prime}=1$, $r_{Y}^{\prime}=0.5$, $r_{\text{EW}}^{\prime}=1.5$, $r_{L}^{\prime}=0.4$, $m_{t}^{\text{in}}=173$ GeV
---|---
Gauge | $m_{\gamma^{(1)}}=556.9$, $m_{Z^{(1)}}={m_{A^{(1)^{0}}}}=564.4$, $m_{W^{(1)^{\pm}}}={m_{H^{(1)^{\pm}}}}=562.7$, $m_{g^{(1)}}=1653.8$
bosons | $m_{\gamma^{(2)}}=1301.4$, $m_{Z^{(2)}}={m_{A^{(2)^{0}}}}=1304.6$, $m_{W^{(2)^{\pm}}}={m_{H^{(2)^{\pm}}}}=1303.9$, $m_{g^{(2)}}=2780.2$
& Higgs | $m_{H^{(1)^{0}}}=570.8,m_{H^{(2)^{0}}}=1307.4$
| $m_{q^{(1)}}=1711.5$, $m_{q^{(2)}}=2816.9$
Quarks | $m_{t}^{\text{phys}}=172.6$, $m_{t^{(1)}_{l}}=620.4$, $m_{t^{(1)}_{h}}=714.5$
& | $m_{t^{(2)}_{l}}=1359.6$, $m_{t^{(2)}_{h}}=1471.7$
Leptons | $m_{b^{(1)}}=638.3$, $m_{b^{(2)}}=1395.8$
| $m_{l^{(1)}}=802.3$, $m_{l^{(2)}}=1631.8$
BM2 | $R{{}^{-1}}=1.5$ TeV, $r_{G}^{\prime}=-0.1$, $r_{Q}^{\prime}=-1.1$, $r_{T}^{\prime}=4$, $r_{Y}^{\prime}=8$, $r_{\text{EW}}^{\prime}=5.5$, $r_{L}^{\prime}=2$, $m_{t}^{\text{in}}=173$ GeV
Gauge | $m_{\gamma^{(1)}}=487.3$, $m_{Z^{(1)}}={m_{A^{(1)^{0}}}}=495.7$, $m_{W^{(1)^{\pm}}}={m_{H^{(1)^{\pm}}}}=493.9$, $m_{g^{(1)}}=1601.6$
bosons | $m_{\gamma^{(2)}}=1655.9$, $m_{Z^{(2)}}={m_{A^{(2)^{0}}}}=1658.4$, $m_{W^{(2)^{\pm}}}={m_{H^{(2)^{\pm}}}}=1657.8$, $m_{g^{(2)}}=3200.8$
& Higgs | $m_{H^{(1)^{0}}}=503.0,m_{H^{(2)^{0}}}=1660.6$
| $m_{q^{(1)}}=2527.5$, $m_{q^{(2)}}=4200.2$
Quarks | $m_{t}^{\text{phys}}=172.4$, $m_{t^{(1)}_{l}}=504.2$, $m_{t^{(1)}_{h}}=813.3$
& | $m_{t^{(2)}_{l}}=1366.3$, $m_{t^{(2)}_{h}}=2220.2$
Leptons | $m_{b^{(1)}}=561.9$, $m_{b^{(2)}}=1706.6$
| $m_{l^{(1)}}=750.0$, $m_{l^{(2)}}=1865.1$
BM3 | Input values same as in BM1 except for $r_{Y}^{\prime}=5$ and $m_{t}^{\text{in}}=176$ GeV
Gauge |
bosons | Masses same as in BM1
& Higgs |
Quarks | Masses same as in BM1 except for $m_{t}^{\text{phys}}=173.4$ and
& | $m_{t^{(1)}_{l}}={626.3}$, $m_{t^{(1)}_{h}}={710.5}$
Leptons | $m_{t^{(2)}_{l}}={1350.7}$, $m_{t^{(2)}_{h}}={1488.6}$
Table 2: Masses (in GeV) of different KK excitations in three benchmark
scenarios. With $r_{H}^{\prime}=r_{\text{EW}}^{\prime}$, the level ‘1’ Higgs
boson masses are very much similar to the masses of the level ‘1’ electroweak
gauge bosons. Choices of the input parameters satisfy the experimental bounds
discussed earlier.
## 5 Phenomenology at the LHC
Given the nontrivial structure of the top quark sector of the nmUED it is
expected that the same would have a rich phenomenology at the LHC. A good
understanding of the same requires a thorough study of the decay patterns of
the KK top quarks and their production rates. In this section we discuss these
issues at the lowest order in perturbation theory.
Towards this we implement the scenario in MadGraph 5 Alwall:2011uj using
Feynrules version 1 Christensen:2008py via its UFO (Univeral Feynrules
Output) Degrande:2011ua ; deAquino:2011ub interface. This now contains the KK
gluons, quarks (including the top and the bottom quarks), leptons131313 The KK
leptons would eventually get into the cascades of the KK gauge bosons. and the
electroweak gauge bosons up to KK level ‘2’. Level ‘1’ and level ‘2’ KK Higgs
bosons are also incorporated. The mixings in the quark sector, including
‘level-mixing’ between KK level ‘2’ and level ‘0’, have now been incorporated
in a generic way. In this section we discuss these with the help of the
benchmark scenarios discussed in section 4.4. We then consolidate the
information to summarize the important issues in the search for such
excitations at the LHC.
### 5.1 Decays of the KK top quarks
BM1 | $t^{(1)}_{l}\to bW^{(1)^{+}}={0.597}$ | $t^{(1)}_{h}\to bW^{(1)^{+}}={0.615}$ | $t^{(2)}_{l}\to b^{(1)}_{h}W^{(1)^{+}}={0.351}$
---|---|---|---
| $bH^{(1)^{+}}={0.403}$ | $bH^{(1)^{+}}={0.370}$ | $t^{(1)}_{h}A^{(1)^{0}}={0.177}$
| | ${t^{(1)}_{l}Z=0.016}$ | $\boldsymbol{bW^{+}={0.062}}$
| | | $\boldsymbol{{tH=0.062}}$
| | | ${b^{(1)}_{h}H^{(1)^{+}}=0.057}$
| | | ${b^{(1)}_{l}H^{(1)^{+}}=0.055}$
| | | $\boldsymbol{{tZ=0.031}}$
BM2 | $t^{(1)}_{l}\to bH^{(1)^{+}}={0.842}$ | $t^{(1)}_{h}\to b^{(1)}_{h}W^{+}={0.305}$ | $t^{(2)}_{l}\to t^{(1)}_{h}A^{(1)^{0}}={0.377}$
| $bW^{(1)^{+}}={0.158}$ | $t^{(1)}_{l}Z={0.180}$ | $b^{(1)}_{h}H^{(1)^{+}}={0.208}$
| | $b^{(1)}_{l}W^{+}={0.141}$ | $b^{(1)}_{l}H^{(1)^{+}}={0.200}$
| | $tA^{(1)^{0}}={0.130}$ | $t^{(1)}_{l}H^{(1)^{0}}={0.109}$
| | $t^{(1)}_{l}H={0.126}$ | ${t^{(1)}_{l}A^{(1)^{0}}=0.055}$
| | ${bH^{(1)^{+}}=0.069}$ | $\boldsymbol{{tH=0.014}}$
| | ${bW^{(1)^{+}}=0.020}$ | $\boldsymbol{{bW^{+}=0.0022}}$
| | ${tH^{(1)^{0}}=0.015}$ | $\boldsymbol{{tZ=0.00058}}$
BM3 | $t^{(1)}_{l}\to bH^{(1)^{+}}={0.946}$ | $t^{(1)}_{h}\to bH^{(1)^{+}}={0.941}$ | $\boldsymbol{t^{(2)}_{l}\to tH={0.448}}$
| $bW^{(1)^{+}}={0.054}$ | $bW^{(1)^{+}}={0.060}$ | ${t^{(1)}_{l}A^{(1)^{0}}=0.102}$
| | | $t^{(1)}_{h}A^{(1)^{0}}={0.092}$
| | | ${t^{(1)}_{l}H^{(1)^{0}}=0.082}$
| | | ${t^{(1)}_{h}H^{(1)^{0}}=0.063}$
| | | $\boldsymbol{{bW^{+}=0.046}}$
| | | $\boldsymbol{{tZ=0.022}}$
Table 3: Decay branching fractions of different KK top quarks for the three
benchmark points presented in table 2. Modes having branching fractions less
than about a percent are not presented except for the ones with a pair of SM
particles in the final state. Tree level decays of $t^{(2)}_{l}$ to SM states
are shown in bold in the right-most column.
Decays of the KK top quarks are mainly governed by the two input parameters,
$r_{T}^{\prime}$ and $r_{\text{EW}}^{\prime}$, for a given value of
$R{{}^{-1}}$.141414In the present analysis, the level ‘1’ KK gluon is taken to
be heavier than all three KK top quark states that are relevant for our
present work, _i.e._ , the two level ‘1’ and the lighter level ‘2’ KK top
quarks. The dependence is rather involved since these two parameters not only
determine the spectra of the KK top quarks and the KK electroweak gauge bosons
but also the involved couplings. The latter, in turn, are complicated
functions of the input parameters as given by equation 39 and as illustrated
in figures 5, 6 and 7. In the following, we briefly discuss the possible decay
modes of the KK top quarks and the significance of some of them at the LHC. In
table 3 we list the branching fractions for the three benchmark points
presented earlier in table 2.
For our choices of input parameters, two decay modes are possible for
$t^{(1)}_{l}$: $t^{(1)}_{l}\to bW^{(1)^{+}}$ and $t^{(1)}_{l}\to
bH^{(1)^{+}}$. Decays to $tZ^{(1)}/t\gamma^{(1)}/tH^{(1)^{0}}/tA^{(1)^{0}}$
are also possible when the mass-splitting between $t^{(1)}_{l}$ and
$Z^{(1)}/\gamma^{(1)}/H^{(1)^{0}}/A^{(1)^{0}}$ is larger than the mass of the
SM-like top quark. In our scenario, its decays to $b^{(1)}_{l}$ and
$b^{(1)}_{h}$ are prohibited on kinematic grounds. Unlike in some competing
scenarios (like the MSSM) where channels like, say, $\tilde{t}_{1}\to
b\chi^{+}_{1}$ and $\tilde{t}_{1}\to t\chi^{0}_{1}$) could attain a 100%
branching fraction, the spectra of the involved KK excitations in our scenario
would not allow $t^{(1)}_{l}$ decaying exclusively to either $bW^{(1)^{\pm}}$
or $t\gamma^{(1)}$. The reason behind this is that $W^{(1)^{\pm}}$ and
$\gamma^{(1)}$ are rather close in mass and hence if decays to $t\gamma^{(1)}$
is allowed, the same to $bW^{(1)^{+}}$ is also kinematically possible.
Further, even the latter mode has to compete with $t^{(1)}_{l}\to
bH^{(1)^{+}}$ as $m_{W^{(1)^{\pm}}}\approx m_{H^{(1)^{\pm}}}$. Translating
constraints on such KK top quarks from those obtained in the LHC-studies of,
say, the top squarks is not at all straight-forward since the latter
explicitly assume either $\tilde{t}_{1}\to b\chi^{+}_{1}=100\%$ Aad:2013ija ;
Chatrchyan:2013xna or $\tilde{t}_{1}\to t\chi^{0}_{1}=100\%$
Chatrchyan:2013xna . Further, $W^{(1)^{\pm}}$ (and also $Z^{(1)}$), being
among the lighter most ones of all the level ‘1’ KK excitations, would only
undergo three-body decays to LKP ($\gamma^{(1)}$) accompanied by leptons or
jets that would be rather soft because of the near-degeneracy of the masses of
the level ‘1’ KK gauge bosons. This would lead to loss of experimental
sensitivity for final states with more number of hard leptons and jets
Aad:2013ija .
The situation with $t^{(1)}_{h}$ is not qualitatively much different as long
as decay modes similar to $t^{(1)}_{l}$ are the dominant ones. This is the
case with BM1. Under such circumstances, they could turn out to be reasonable
backgrounds to each other (if their production rates are comparable) and
dedicated studies would be required to disentangle them. In any case (even in
the absence of good discriminators), simultaneous productions of both
$t^{(1)}_{l}$ and $t^{(1)}_{h}$ would enhance the new-physics signal. On the
other hand, in a situation like BM2, more decay modes may be available to
$t^{(1)}_{h}$ although decays to level ‘1’ bottom and top quarks along with SM
$W^{\pm}$ and $Z$ are the dominant ones. The ensuing cascades of these states
would inevitably make the analysis challenging. However, under favorable
circumstances, reconstructions of the $W^{\pm}$ and/or $Z$ bosons along with
$b$\- and/or _top-tagging_ could help disentangle the signals. Thus, it
appears that search for level ‘1’ KK top quarks involves complicated issues
(some of which are common to top squark searches in SUSY scenarios) and a
multi-channel analysis could turn out to be very effective.
We now turn to the case of level ‘2’ top KK top quarks. The lighter of the two
states, $t^{(2)}_{l}$ can have substantial rates at the LHC which is discussed
in some detail in section 5.2. This motivates us to study the decay patterns
of $t^{(2)}_{l}$. In the last column of table 3 we present the decay branching
fractions of $t^{(2)}_{l}$. As can be seen, the decay modes that are usually
enhanced involve a pair of level ‘1’ KK excitations which would cascade to the
LKP. We, however, strive to understand to what extent $t^{(2)}_{l}$, being an
even KK-parity state, could decay directly to a pair of comparatively light
(level ‘0’) particles (and hence, boosted) comprising of an SM fermion and an
SM gauge/Higgs boson151515These may be contrasted with the popular SUSY
scenarios (sparticles carrying odd $R$-parity) where such possibilities are
absent.. Thus, in the one hand, these decay products are unlikely to be missed
in an experiment while on the other hand, new techniques to reconstruct (like
the study of jet substructure Altheimer:2012mn ; Dasgupta:2013ihk etc.) some
of them have to be employed.
In scenario BM1, the total decay branching fraction to SM states (shown in
bold) is just about 15% while in scenario BM2 such decays are practically
absent. Given the large phase space available, such small (or non-existent)
decay rates to SM particles can only be justified in terms of rather feeble
(effective) couplings among the involved states. The couplings of
$t^{(2)}_{l}$ to the SM gauge bosons and an SM fermion would have vanished
(due to the orthogonality of the mode functions involved) had $t^{(2)}_{l}$
been a pure level ‘2’ state. The smallness of these couplings thus readily
follows from the tiny admixture of the SM top quark in the physical
$t^{(2)}_{l}$ state and thus, results in its small branching fractions to SM
gauge bosons. The same argument does not hold for the corresponding coupling
$t^{(2)}_{l}$-$t$-$H$ that controls the other SM decay mode of $t^{(2)}_{l}$,
_i.e._ , $t^{(2)}_{l}\to tH$. However, it is clear from figure 7 that this
coupling is going to be small for both the benchmark points BM1 and BM2.
Since direct decays of $t^{(2)}_{l}$ to SM states could provide the ‘smoking
guns’ at the LHC in the form of rather boosted objects (top and bottom quarks,
$Z$, $W^{\pm}$ and Higgs boson) that could eventually be reconstructed to
their parent, this motivates us to study if such decays can ever become
appreciable. We find that the coupling $t^{(2)}_{l}$-$t$-$H$ gets
significantly enhanced with a slight modification in the parameters of BM1
(called BM3 in table 2) by setting $r_{Y}^{\prime}=5$ (see figure 7) and
$m_{t}^{\text{in}}=176$ GeV while keeping other parameters untouched and still
satisfying all the experimental constraints that we discussed. As we can see,
the branching fraction to $tH$ final state could attain a level of 50% which
should be healthy for the purpose. Efficient tagging of boosted top quarks
Kaplan:2008ie ; CMS:2009lxa ; Plehn:2011tg ; Schaetzel:2013vka and boosted
Higgs bosons Butterworth:2008iy would hold the key in such a situation. Some
such techniques have already been proposed in recent literature Berger:2012ec
, in particular, in the context of vector-like top quarks or more generally,
in the study of ‘top-partners’.
On the other hand, since the $t^{(2)}_{l}$-$t$-$Z$ and
$t^{(2)}_{l}$-$b$-$W^{\pm}$ are dynamically constrained, these could only get
enhanced if the competing modes (decays to a pair of level ‘1’ KK states) face
closure. As the couplings involved in the latter cases are generically of SM
strength, these could only be effectively suppressed by having them
kinematically forbidden. From figure 12 we find that, by itself, this is not
very difficult to achieve (in yellow shade) over the nmUED parameter space.
However, rather conspicuously, the simultaneous demands for the KK photon to
be the LKP with $m_{\gamma^{(1)}}>400$ GeV (the red-shaded region) and that of
$m_{t^{(2)}_{l}}<1.5$ TeV (in blue shade) leave no overlapping region in the
nmUED parameter space. It may appear that one simple way to find some overlap
is by moving down in $r_{T}^{\prime}$. However, this implies $t^{(2)}_{l}$
becomes more massive thus loosing in its production cross section in the first
place. Although the interplay of events that leads to this kind of a situation
is not an easy thing to follow, the issue that is broadly conspiring is the
similarity in the basic evolution-pattern of the masses of the KK excitations
as functions of the BLKT parameters (see figure 2 and ref. Datta:2012tv ).
Figure 12: Region in $r_{T}^{\prime}-r_{\text{EW}}^{\prime}$ plane where the
decays $t^{(2)}_{l}\to
t^{(1)}_{l}\gamma^{(1)},\,t^{(1)}_{l}Z^{(1)},\,b^{(1)}_{l}W^{(1)^{+}}$ are
kinematically prohibited (in yellow), $\gamma^{(1)}$ is the LKP with
$m_{\gamma^{(1)}}>400$ GeV (in red) and $m_{t^{(2)}_{l}}<1.5$ TeV (in blue).
The entire region shown is compatible with the acceptable range of the mass of
the top quark and other precision constraints.
### 5.2 Production processes
In this section we discuss different production modes of the KK top quarks at
the 14 TeV (the design energy) LHC with reference to the nmUED parameter
space. These are of following four broad types (in line with top squark
phenomenology in SUSY scenarios):
* •
the generic mode with two top quark excitations in the final state that
receives contributions from processes involving both strong and electroweak
interactions,
* •
exclusively electroweak processes leading to a single top quark excitation
* •
the associated production of a pair of KK top quarks and the (SM) Higgs boson
and
* •
production from the cascades of KK gluons and KK quarks.
#### 5.2.1 Final states with a pair of top quark excitations
These are the processes where two similar or different kind of top quark
excitations are produced in the final state. The interesting modes in this
category are pair-production of $t^{(1)}_{l}$ and $t^{(1)}_{h}$ along with the
associated productions of $t^{(1)}_{l}t^{(1)}_{h}$ and $t^{(2)}_{l}t$. The
latter two processes are possible in an nmUED scenario and the corresponding
Feynman diagrams161616All the Feynman diagrams in this paper are drawn by use
of Jaxodraw Binosi:2003yf , based on Axodraw Vermaseren:1994je . are presented
in figure 13. Note that the requirement of current conservation does not allow
the massless SM gauge bosons (gluon and photon) to mediate these processes
while the pair-productions receive contributions from all possible mediations.
Also, these two associated production modes have no counter-parts in a
competing SUSY scenario like the MSSM.
In figure 14 we illustrate the variations of the rates for these processes
with $r_{T}^{\prime}$ for $R{{}^{-1}}$=1 TeV (left) and 2 TeV (right). As can
be seen, pair production of $t^{(1)}_{l}$, has by far the largest cross
section for $r_{T}^{\prime}\gtrsim 3$ reaching up to 10 (1) pb for
$R{{}^{-1}}$ = 1.5 (2) TeV. This is not unexpected since $t^{(1)}_{l}$ is the
lightest of the KK top quarks. In this regime, the yields for
$t^{(1)}_{h}$-pair and $t^{(1)}_{l}t^{(1)}_{h}$ associated productions are
very similar touching 1 (0.1) pb for $R{{}^{-1}}$ = 1.5 (2) TeV. The
corresponding rates for $t^{(2)}_{l}t$ associated production do not lag much
notching 0.5 (0.05) pb, respectively. Further, the $t^{(2)}_{l}$-pair has a
trend similar to that of the $t^{(1)}_{l}$-pair in this respect but, rate-
wise, falls out of the competition.
Figure 13: Feynman diagrams for the associated $t^{(2)}_{l}-t^{(0)}$ (left)
and $t^{(1)}_{l}-t^{(1)}_{h}$ productions at the LHC. The gluon-initiated
processes are only mediated by $g^{(2)}$ while the quark-initiated processes
are mediated by both $g^{(2)}$ and other electroweak gauge bosons from level
‘0’ ($Z$) and level ‘2’ ($\gamma^{(2)},\,Z^{(2)}$).
Figure 14: Cross sections (in picobarns, at tree level) for different
production processes involving the KK top quarks as functions of
$r_{T}^{\prime}$ at the 14 TeV LHC for $R{{}^{-1}}=1.5$ TeV (left) and
$R{{}^{-1}}=2$ TeV (right), $r_{Y}^{\prime}=3$, $r_{G}^{\prime}=0.5$ and the
other parameters are chosen as in the BM2. CTEQ6L1 parton distributions
Nadolsky:2008zw are used and the factorization/renormalization scale is set
at the sum of the masses in the final state.
Note that with increasing $r_{T}^{\prime}$ masses of all the KK states
decrease. Interestingly enough, this effect is reflected in a straight-forward
manner only in the case of $t^{(1)}_{l}$-pair for which the rates increase
with growing $r_{T}^{\prime}$. For other competing processes mentioned above,
the curves flatten out. This behavior signals non-trivial interplays of the
intricate couplings involved. These have much to do with when all these rates
become comparable for $r_{T}^{\prime}\lesssim 3$.171717It may be noted in this
context that an effective $SU(3)$ coupling involving a set of KK excitations
is not necessarily stronger than the effective electroweak coupling among them
and these might even have relative signs between them (see figures 5, 6 and
7). Thus, contributions from different mediating processes heavily depend on
the nmUED parameters. In the process, the rate for usual $t\bar{t}$ pair
production also gets affected to some extent. However, our estimates are all
being at the tree level, these do not pose any immediate concern while facing
the measured $t\bar{t}$ cross section which is much larger and agrees with its
estimation at higher orders in perturbation theory. Also, in table 4 we
present the cross sections for the three benchmark points.
The bottom-line is that the production rates of three different KK top quark
excitations remain moderately healthy over favorable region of the nmUED
parameter space at a future LHC run. With the knowledge of their decay
patterns (see table 3) and the associated features discussed in section 5.1 it
is required to chalk out a strategy to reach out to these excitations.
Benchmark | $t^{(1)}_{l}\bar{t}^{(1)}_{l}$ | $t^{(1)}_{l}\bar{t}^{(1)}_{h}$ | $t^{(1)}_{h}\bar{t}^{(1)}_{h}$ | $t\bar{t}^{(2)}_{l}$
---|---|---|---|---
| (pb) | (pb) | (pb) | (pb)
BM1 | 0.63 | 0.10 | 0.35 | 0.07
BM2 | 2.24 | 0.35 | 0.76 | 0.21
BM3 | 0.76 | 0.11 | 0.30 | 0.07
Table 4: Production cross sections (in picobarns, at tree level) for different
pairs of KK top quarks for the benchmark points. Contributions from the
Hermitian conjugate processes are taken into account wherever applicable. The
choices for the parton distribution and the scheme for determining the
factorization/renormalization scale are the same as in figure 14.
Figure 15: Generic Feynman diagrams for the single production of a KK top
quark along with KK excitations of $W^{\pm}$ boson (upper panel) and KK bottom
quark (lower panel) at the LHC. Superscripts $m$ and $n$ standing for the KK
levels can be different (like ‘0’ and ‘2’) but should ensure KK-parity
conservation.
#### 5.2.2 Single production processes
We consider two broad categories of single production of KK top quarks which
are closely analogous to single top production in the SM once the issue of KK-
parity conservation is taken into account. In the first case, a level ‘1’ KK
top quark is produced in association with level $W^{(1)^{\pm}}$ or $b^{(1)}$
quark. The second one involves the lighter of the level ‘2’ KK top quarks
along with an SM $W^{\pm}$ boson or an SM bottom quark. The generic, tree-
level Feynman diagrams that contribute to the processes are presented in
figure 15.
Benchmark | $t^{(1)}_{l}W^{(1)^{-}}$ | $t^{(1)}_{l}\bar{b}^{(1)}_{l}$ | $t^{(2)}_{l}b$ | $t^{(1)}_{l}\bar{t}^{(1)}_{l}H$ | $t^{(1)}_{l}\bar{t}^{(1)}_{h}H$ | $t^{(1)}_{h}\bar{t}^{(1)}_{h}H$ | $t\bar{t}^{(2)}_{l}H$ | $t\bar{t}H$
---|---|---|---|---|---|---|---|---
| (pb) | (pb) | (pb) | (pb) | (pb) | (pb) | (pb) | (pb)
BM1 | 0.01 | 0.11 | 0.11 | $\sim 10^{-5}$ | $\sim 10^{-4}$ | $\sim 10^{-3}$ | 0.03 | 0.24
BM2 | 0.04 | 0.21 | 0.13 | 0.73 | 5.39 | 0.17 | 0.11 | 1.25
BM3 | $\sim 10^{-3}$ | 0.23 | 0.11 | $\sim 10^{-4}$ | $\sim 10^{-3}$ | 0.01 | 0.04 | 2.21
Table 5: Cross sections (in picobarns, at tree level) for single and (SM)
Higgs-associated KK top quark productions for the benchmark points. The mass
of the SM Higgs boson is taken to be 125 GeV. Contributions from the Hermitian
conjugate processes are taken into account wherever applicable. The choices
for the parton distribution and the scheme for determining the
factorization/renormalization scale are the same as in figure 14.
##### Single production of level ‘1’ top quarks:
Single production of level ‘1’ top quarks along with a level ‘1’ $W^{\pm}$
boson proceeds via $gb$ fusion in $s$-channel and $gb$ scattering in
$t$-channel. The rates are at best a few tens of femtobarns as can be seen
from table 5. On the other hand, the mode in which a level ‘1’ bottom quark is
produced in association proceeds through $s$-channel fusion of light quarks
and propagated by $W^{\pm}$ and $W^{(2)^{\pm}}$ bosons. The cross sections are
found to be rather healthy ranging from 110 fb to 230 fb. The observed rates
for $t^{(1)}_{l}W^{(1)^{\pm}}$ production appear to be consistently lower than
that for $t^{(1)}_{l}b^{(1)}_{l}$ production. This can be traced back to the
presence of enhanced $q$-$q^{\prime}$-$W^{(2)^{\pm}}$ coupling. Moreover,
cross sections for other combinations involving heavier states of $t^{(1)}$
and $b^{(1)}$ in the final state could have comparable strengths because of
such enhanced couplings.
##### Single production of level ‘2’ top quark:
The associated $t^{(2)}_{l}W^{-}$ production involves the vertex
$t^{(2)}_{l}$-$W^{\pm}$-$b$ which, as we discussed earlier (see sections 3.4.1
and 5.1), vanishes but for a small admixture of level ‘0’ top in the physical
state $t^{(2)}$. Hence, the rates in this mode turn out to be insignificant.
Further, the $W^{\pm}$-mediated diagram in the associated $t^{(2)}_{l}b$
production also has the same vertex and thus contributes negligibly. The only
contribution here comes from the diagram mediated by $W^{(2)^{\pm}}$ which is
somewhat massive. Thus, the prospect of having healthy rates for the single
production of $t^{(2)}$ depends entirely on the coupling strength
$t^{(2)}_{l}$-$W^{(2)^{\pm}}$-$b$ and $W^{(2)^{\pm}}$-$q$-$q$ (see figure 5).
Fortunately, this is the case here and the cross sections for all three
benchmark points, as can be seen from table 5, are above and around 100 fb.
We also looked into the production of $t^{(2)}_{l}$ along with light quark
jets which is analogous to, by far the most dominant, ‘$t$-channel’ single top
production process (the so-called $W$-gluon fusion process) in the SM.
However, in our scenario, such a process with somewhat heavy $t^{(2)}_{l}$
yields a few tens of a femtobarn for all the three benchmark points.
For both the categories mentioned above, the new-physics contributions to the
corresponding SM processes are systematically small. This is since these
contain the couplings that involve level-mixing effect in the top-quark sector
which is not large.
#### 5.2.3 Associated production of KK top quarks with the SM Higgs boson
The associated Higgs production processes we consider involve both light and
heavy level ‘1’ top quarks in pairs and the level ‘2’ lighter top quark along
with the SM top quark. The generic tree level Feynman diagrams are presented
in figure 16. Given that the study of the SM $t\bar{t}H$ production is by
itself complicated enough, it is only natural to expect that the same with its
KK counterparts would not be any simpler.
Figure 16: Generic Feynman diagrams for the associated (SM) Higgs production
along with a pair of KK excitations of the top quark. Superscripts $k$, $m$
and $n$ can be different (like ‘0’ and ‘2’) but should ensure KK-parity
conservation.
Cross sections for such processes are listed in table 5 for the benchmark
points we consider. To have a feel about the their phenomenological prospects,
these can be compared with similar processes in the SM and a SUSY scenario
like the MSSM. In the MSSM, the lowest order cross section is around a few
tens of a fb for the process $\tilde{t}_{1}\tilde{t}^{*}_{1}H$ with
$m_{\tilde{t}_{1}}\approx 300$ GeV and for the most favorable values of the
involved couplings Djouadi:1997xx ; Djouadi:2005gj while for the SM the
corresponding rate is about 430 fb Beenakker:2001rj ; Djouadi:2005gi . It is
encouraging to find that the yield for $tt^{(2)}_{l}H$ is either comparable
(for BM1 and BM3) or larger (BM2) than what can at best be expected in MSSM.
Note that the level ‘1’ lighter KK top quark is somewhat heavier (with mass
around or above 500 GeV) for our benchmark points when compared to the mass of
the top squark as indicated above. For other processes, BM2 consistently leads
to larger cross sections. The interplay of different Feynman diagrams (see
figure 16) along with the modified strengths of the participating gauge and
Yukawa interactions play roles in some such enhancements.
In the last column of table 5 we indicate the lowest order cross sections for
the SM process $t\bar{t}H$ which now gets affected in an nmUED scenario. Note
that for BM1 the cross section is smaller than the SM value of $\approx 430$
fb while for BM2 and BM3 the same is about 3 and 5 times as large,
respectively. Such deviations can be expected if we refer back to the left
panel of figure 7 that illustrates how the $t$-$\bar{t}$-$H$ coupling gets
modified over the nmUED parameter space. Note that, non-observation of such a
process at the LHC, till recently, could only restrict the rate up to around
five times the SM rate Chatrchyan:2013yea ; cms-tth-gamma ; atlas-tth-gamma .
Thus, benchmark point BM3, as such, can be considered as a borderline case.
But given that $t\bar{t}H$ cross section depends on other nmUED parameters
like $r_{G}^{\prime}$, $r_{Q}^{\prime}$ etc., one could easily circumvent this
restriction without requiring a compromise with the parameters like
$r_{T}^{\prime}$ and $r_{Y}^{\prime}$ that define the essential feature of
BM3, _i.e._ , the enhanced couplings among the top quark excitations and the
SM Higgs boson. It is interesting to find that in favorable regions of
parameter space, the cross section for Higgs production in association with a
pair of rather heavy KK top quarks could compare with or even exceed the
$t\bar{t}H$ cross section. Note that in the MSSM, such enhancement only
happens for large mixing in the stop sector and when $m_{\tilde{t}_{1}}<m_{t}$
Djouadi:2005gj .
Further, once the level ‘1’ KK Higgs bosons are taken up for studies, the
associated production of a charged KK Higgs boson (from level ‘1’) in the
final state $bt^{(1)}_{l}H^{{(1)}^{\pm}}$ would become rather relevant and may
turn out to be interesting as the total mass involved in this final state can
be comparatively much lower. The prospect there depends crucially on the
strength of the involved 3-point vertex though.
#### 5.2.4 Production of KK top quarks under cascades
KK gluon(s) and quarks, once produced, can cascade to KK top quarks. This
would result in multiple top quarks (upto four of them) in the final state at
the LHC. In our benchmark scenarios where $m_{g^{(1)}}<m_{q^{(1)}}$, KK gluons
would directly decay to KK top quarks while KK quarks from the first two
generations would undergo a two-step decay via KK gluon to yield a KK top
quark. The latter one has thus suppressed contribution. We find that the
branching fraction for $g^{(1)}\to t^{(1)}t$ is around 50% for all three
benchmark points (the rest 50% is to level ‘1’ bottom quark states). With
strong production rates for the $g^{(1)}$-pair, $g^{(1)}q^{(1)}$ and
$q^{(1)}$-pair ranging between 0.01 pb to 2.6 pb (in increasing order), the
yield of a single level ‘1’ KK top final state could be anywhere between 10 fb
to a few pb. These seem quite healthy. However, one has to cope with
backgrounds which now have enhanced level of jet activity.
## 6 Conclusions and outlook
We discuss the structure and the phenomenology of the top quark sector in a
scenario with one flat extra spatial dimension orbifolded on $S^{1}/Z_{2}$ and
containing non-vanishing BLTs. The discussion inevitably draws reference to
the gauge and the Higgs sectors. The scenario, by construct, preserves KK-
parity.
The main purpose of the present work is to organize and work out (following
ref. Flacke:2008ne ) the necessary details in the involved sectors and explore
the salient features with their broad phenomenological implications in terms
of a few benchmark scenarios. This lay down the basis for future, detailed
studies of such a top quark sector at the LHC.
The masses and the couplings of the Kaluza-Klein excitations are estimated at
the lowest order in perturbation theory as functions of $R{{}^{-1}}$ and the
BLT parameters. For the KK top quarks, the extended mixing scheme (originating
in the Yukawa sector) is thoroughly worked out by incorporating _level-mixing_
among the level ‘0’ and the level ‘2’ KK top quark states, a phenomenon that
is not present in the popular mUED scenario. In addition, unlike in the mUED,
tree-level couplings that violate KK-number (but conserve KK-parity) are
possible. We demonstrate how all these new effects, together, attract
constraints from different precision experiments and shape the phenomenology
of such a scenario.
The nmUED scenario we consider has eight free parameters: $R{{}^{-1}}$ and the
scaled (by $R{{}^{-1}}$) BLT coefficients $r_{Q}^{\prime}$, $r_{L}^{\prime}$,
$r_{T}^{\prime}$, $r_{Y}^{\prime}$, $r_{G}^{\prime}$,
$r_{\text{EW}}^{\prime}\,(=r_{W}^{\prime}=r_{B}^{\prime}=r_{H}^{\prime})$ and
$m_{t}^{\text{in}}$. However, in the present study, the most direct roles are
played by $r_{T}^{\prime}$, $r_{Y}^{\prime}$ and $r_{\text{EW}}^{\prime}$
(=$r_{H}^{\prime}$) in conjunction with $R{{}^{-1}}$. $r_{Q}^{\prime}$ and
$r_{G}^{\prime}$ play roles in the production processes by determining some
relevant gauge-fermion couplings beside controlling the KK quark and gluon
masses, respectively. On the other hand, $r_{L}^{\prime}$ and
$m_{t}^{\text{in}}$ only play some indirect roles through their influence on
the experimentally measured effects that determine the allowed region of the
parameter space.
The scenario has been thoroughly implemented in MadGraph 5\. Three benchmark
scenarios that satisfy all the relevant experimental constraints are chosen
for our study. These give conservatively light KK spectra with sub-TeV masses
for both level ‘1’ electroweak KK gauge bosons (with $\gamma^{(1)}$ as the
LKP) and the KK top quarks while having the lighter level ‘2’ top quark below
1.5 TeV thus making them all relevant at the LHC. Level ‘1’ KK quarks from the
first two generations and the KK gluon are taken to be heavier than 1.6 TeV.
Near mass-degeneracy of the electroweak KK gauge bosons and the KK Higgs
bosons (at a given KK level) is a feature. This influences the decays of the
KK top quarks. The lighter of the level ‘1’ KK top quark can never decay 100%
of the time to a top quark and the LKP photon. This is in sharp contrast to a
similar possibility in a SUSY scenario like the MSSM when a top squark can
decay 100% of the time to a top quark and the LSP neutralino, an assumption
that is frequently made by the LHC collaborations. Instead, such a KK top
quark has significant branching fractions to both charged KK Higgs boson and
to KK $W$ bosons at the same time. Further, split between the KK top quark and
the KK electroweak gauge bosons that is attainable in the nmUED scenario would
generically lead to hard primary jets in the decays of the former. This is
again in clear contrast to the mUED scenario. However, near mass-degeneracy
prevailing in the gauge and the Higgs sector would still result in rather soft
leptons/secondary jets. Limited mass-splitting among the KK gauge and Higgs
bosons is a possibility that has non-trivial ramifications and hence needs
closer scrutiny.
The level ‘2’ KK top quark we consider can decay directly to much lighter SM
particles like the $W$, the $Z$, the Higgs boson and the top quark. These
would then be boosted and hence may serve as ‘smoking guns’. Recent studies of
the vector-like top partners CMS:2012ab ; ATLAS:2012qe ; atlas:heavytop are
in context. However, these studies mainly bank on their pair-production and
decays that comprise only of pairs of SM particles like $bW^{\pm}$ and/or $tZ$
and/or $tH$. In the nmUED model that we consider, these are _always_
accompanied by other modes that may be dominant as well. The level ‘2’ top
quark decaying to a pair of level ‘1’ KK states is one such example.
Thus, phenomenology of the KK top quarks could turn out to be rather rich (and
complex) at the LHC. Clearly, strategies tailor-made for searches of similar
excitations under different scenarios could at best be of very limited use.
Even recasting the analyses for some of them to the nmUED scenario is not at
all straight-forward. This calls for a dedicated strategy that incorporates
optimal triggers and employs advanced techniques like analysis of jet-
substructures etc. to tag the boosted objects in the final states.
In any case, viability of a dedicated hunt depends crucially on optimal
production rates. We study these for the 14 TeV run of the LHC. For all the
possible modes in which KK top quarks can be produced (like the pair-
production, the single production and the associated production with the SM
Higgs boson), the rates are found to be rather encouraging and may even exceed
the corresponding MSSM processes, a yard-stick that can perhaps be used safely
(with a broad brush, though) for the purpose.
The LHC experiments are either already sensitive or will be achieving the same
soon in the next run for all the generic processes discussed in this work.
Given that the nmUED provides several top quark KK excitations with different
characteristic decays and production rates, the sensitivity to them can only
be increased if multi-channel searches are carried out. It is thus possible
that the LHC, running at its design energy of 14 TeV (or even a little less),
finds some of these states. However, concrete studies with rigorous detector-
level simulations are prerequisites to chalking out a robust strategy.
Last but not the least, the intimate connection between the top quark and the
Higgs sectors raises genuine curiosity in the phenomenology for the KK Higgs
bosons as well. The nmUED Higgs sector holds good promise for a rather rich
phenomenology at the LHC which has become further relevant after the discovery
of the ‘SM-like’ Higgs boson and hence can turn out to be a fertile area to
embark upon.
Acknowledgments KN and SN are partially supported by funding available from
the Department of Atomic Energy, Government of India for the Regional Centre
for Accelerator-based Particle Physics (RECAPP), Harish-Chandra Research
Institute. The authors like to thank Benjamin Fuks for very helpful
discussions on issues with FeynRules and SN thanks Ujjal Kumar Dey for many
helpful discussions. The authors acknowledge the use of computational facility
available at RECAPP and thank Joyanto Mitra for technical help.
## Appendix A Gauge and the Higgs sector of the nmUED: some relevant details
In this appendix we briefly supplement our discussion in section 2.1 with some
necessary details pertaining to the gauge fixing conditions, the inputs that
go into the mass-determining conditions.
### A.1 Gauge fixing conditions
We introduce the gauge-fixing terms in the bulk and at the boundaries in the
following way to obtain the physical states:
$\displaystyle S_{\text{gf}}$ $\displaystyle=\int
d^{4}x\int_{-L}^{L}dy\Bigg{\\{}-\frac{1}{2\xi_{A}}\left[\partial_{\mu}A^{\mu}-\xi_{A}\partial_{y}A_{y}\right]^{2}-\frac{1}{\xi_{W}}\left|\partial_{\mu}W^{+\mu}-\xi_{W}\left(\partial_{y}W_{y}^{+}+iM_{W}\phi^{+}\right)\right|^{2}$
$\displaystyle\phantom{=\int
d^{4}x\int_{-L}^{L}dy\Bigg{\\{}\,\,}-\frac{1}{2\xi_{Z}}\left[\partial_{\mu}Z^{\mu}-\xi_{Z}\left(\partial_{y}Z_{y}+M_{Z}\chi\right)\right]^{2}-\frac{1}{2\xi_{G}}\left[\partial_{\mu}G^{a\mu}-\xi_{G}\partial_{y}G_{y}^{a}\right]^{2}$
$\displaystyle-\frac{1}{2\xi_{A,b}}\Big{\\{}\left[\partial_{\mu}A^{\mu}+\xi_{A,b}A_{y}\right]^{2}\delta(y-L)+\left[\partial_{\mu}A^{\mu}-\xi_{A,b}A_{y}\right]^{2}\delta(y+L)\Big{\\}}$
$\displaystyle-\frac{1}{\xi_{W,b}}\Big{\\{}\left|\partial_{\mu}W^{+\mu}+\xi_{W,b}\left(W_{y}^{+}-ir_{H}M_{W}\phi^{+}\right)\right|^{2}\delta(y-L)+\left|\partial_{\mu}W^{+\mu}-\xi_{W,b}\left(W_{y}^{+}+ir_{H}M_{W}\phi^{+}\right)\right|^{2}\delta(y+L)\Big{\\}}$
$\displaystyle-\frac{1}{2\xi_{Z,b}}\Big{\\{}\left[\partial_{\mu}Z^{\mu}+\xi_{Z,b}\left(Z_{y}-r_{H}M_{Z}\chi\right)\right]^{2}\delta(y-L)+\left[\partial_{\mu}Z^{\mu}-\xi_{Z,b}\left(Z_{y}+r_{H}M_{Z}\chi\right)\right]^{2}\delta(y+L)\Big{\\}}$
$\displaystyle-\frac{1}{2\xi_{G,b}}\Big{\\{}\left[\partial_{\mu}G^{a\mu}+\xi_{G,b}G_{y}^{a}\right]^{2}\delta(y-L)+\left[\partial_{\mu}G^{a\mu}-\xi_{G,b}G_{y}^{a}\right]^{2}\delta(y+L)\Big{\\}}\Bigg{\\}}$
(44)
where the eight gauge-fixing parameters are
$\xi_{A},\,\xi_{W},\,\xi_{Z},\,\xi_{G}$ (in the bulk),
$\xi_{A,b},\,\xi_{W,b},\,\xi_{Z,b},\,\xi_{G,b}$ (at the boundary) and
$M_{W},\,M_{Z}$ are the masses of the $W$ and $Z$ bosons181818This part of the
action is also symmetric under the reflection $y\to-y$..
Imposing the unitary gauge in both the bulk and at the boundaries by setting
$\displaystyle\xi_{A},\,\xi_{W},\,\xi_{Z},\,\xi_{G},\,\xi_{A,b},\,\xi_{W,b},\,\xi_{Z,b},\,\xi_{G,b}\to\infty$
(45)
we obtain the following relations:
$\displaystyle A_{y}$ $\displaystyle=0,$ $\displaystyle Z_{y}\mp
r_{H}M_{Z}\chi$ $\displaystyle=0,$ $\displaystyle W_{y}^{+}\mp
ir_{H}M_{W}\phi^{+}$ $\displaystyle=0,$ $\displaystyle G_{y}^{a}$
$\displaystyle=0,\qquad\text{at }y=\pm L,$ (46)
$\displaystyle\partial_{y}A_{y}$ $\displaystyle=0,$
$\displaystyle\partial_{y}W_{y}^{+}+iM_{W}\phi^{+}$ $\displaystyle=0,$
$\displaystyle\partial_{y}Z_{y}+M_{Z}\chi$ $\displaystyle=0,$
$\displaystyle\partial_{y}G_{y}^{a}$ $\displaystyle=0,\qquad\text{in the
bulk}.$ (47)
As we see, $A_{y}$ and $G_{y}^{a}$ are totally gauged away from the theory as
would-be Nambu-Goldstone bosons. The two mixed boundary conditions in equation
46 can be cast into a set containing the individual fields with the help of
equation 47 as
$\displaystyle\chi\pm r_{H}\partial_{y}\chi$ $\displaystyle=0,$
$\displaystyle\phi^{+}\pm r_{H}\partial_{y}\phi^{+}$ $\displaystyle=0,$
$\displaystyle Z_{y}\pm r_{H}\partial_{y}Z_{y}$ $\displaystyle=0,$
$\displaystyle W_{y}^{+}\pm r_{H}\partial_{y}W_{y}^{+}$
$\displaystyle=0,\qquad\text{at }y=\pm L.$ (48)
### A.2 Input parameters for masses of the the KK gauge and Higgs bosons
Input parameters that determine the masses of the KK gauge and the Higgs
bosons of the nmUED Flacke:2008ne (as solutions for the conditions given in
equation (8)) are presented in table 6.
Type | $m_{F}^{2}$ | $m_{F,b}^{2}$ | $r_{F}$
---|---|---|---
$W_{\mu}^{+}$ | $M_{W}^{2}$ | $r_{H}M_{W}^{2}$ | $r_{\text{EW}}$
$Z_{\mu}$ | $M_{Z}^{2}$ | $r_{H}M_{Z}^{2}$ | $r_{\text{EW}}$
$H$ | $(\sqrt{2}\hat{\mu})^{2}$ | $(\sqrt{2}\mu_{b})^{2}$ | $r_{H}$
$\phi^{+},\,\partial_{y}W_{y}^{+}$ | $M_{W}^{2}$ | $r_{H}M_{W}^{2}$ | $r_{H}$
$\chi,\,\partial_{y}Z_{y}$ | $M_{Z}^{2}$ | $r_{H}M_{Z}^{2}$ | $r_{H}$
Table 6: Input parameters that determine the masses of the KK gauge and Higgs
bosons. See section 2.1 for notations and conventions.
## Appendix B Tree-level FCNCs, the “aligned” scenario and constraints from
$D^{0}-\overline{D^{0}}$ mixing
It has been demonstrated in ref. gerstenlauer that an appropriate short-
distance description for a $\Delta F$=2 FCNC process like
$D^{0}-\overline{D^{0}}$ can be found in processes involving only the even KK
modes (starting at level ‘2’) of the gauge bosons and the ‘0’ mode fermions.
In an effective Hamiltonian approach, such a process would reduce to a four-
Fermi interaction whose strength is suppressed by the mass of the exchanged KK
gauge boson. The effective FCNC Hamiltonian can be expressed in terms of
suitable fermionic operators and their associated Wilson coefficients. The
latter involve the overlap matrices in the gauge kinetic terms (by now,
suitably rotated to the basis where the quark mass matrix is diagonal) which
are functions of the BLKT parameter, $r_{Q}^{\prime}$ and $r_{T}^{\prime}$.
Thus, any constraint on the Wilson coefficients can be translated into
constraints in the $r_{Q}^{\prime}$-$r_{T}^{\prime}$ plane.
The gauge interactions in the diagonalized basis involving the level ‘0’
quarks and the KK gluons $g^{(k)}$, with the KK index $k$ being even and
$k\geq 2$, are given by:
$\displaystyle
g_{s}\sum_{i,j,l=1}^{3}\bigg{(}\overline{q^{(0)}_{iL}}\gamma^{\mu}T^{a}\Big{[}(U_{qL}^{\dagger})_{il}F^{Q,[k]}_{g,ll}(U_{qL})_{lj}\Big{]}q^{(0)}_{jL}+\overline{q^{(0)}_{iR}}\gamma^{\mu}T^{a}\Big{[}(U_{qR}^{\dagger})_{il}F^{q,[k]}_{g,ll}(U_{qR})_{lj}\Big{]}q^{(0)}_{jR}\bigg{)}g_{\mu}^{(k)},$
(49)
where the 4D and the 5D (the ‘hatted’ one) gauge couplings are related by
$g_{s}\,\equiv\,\hat{g}_{s}/\sqrt{2r_{G}+\pi R}$. $T^{a}$ represents the
$SU(3)$ generators, $a$ being the color index. $U_{q(L,R)}$ are the matrices
that diagonalize the $q_{L,R}$ fields in the Yukawa sector. $F^{Q,[k]}_{g,ll}$
and $F^{q,[k]}_{g,ll}$ are the diagonal overlap matrices (in the original
bases)
$\displaystyle F^{Q,[k]}_{g,ll}$
$\displaystyle=\frac{1}{f_{g^{(0)}}}\int_{-L}^{L}dy\left(1+r_{Q_{l}}\left[\delta(y-L)+\delta(y+L)\right]\right)f_{Q^{(0)}_{l}}f_{g^{(k)}}f_{Q^{(0)}_{l}},$
(50) $\displaystyle F^{q,[k]}_{g,ll}$
$\displaystyle=\frac{1}{f_{g^{(0)}}}\int_{-L}^{L}dy\left(1+r_{q_{l}}\left[\delta(y-L)+\delta(y+L)\right]\right)f_{q^{(0)}_{l}}f_{g^{(k)}}f_{q^{(0)}_{l}}$
(51)
while the explicit form is shown in equation 43. Similar FCNC processes are
also induced by the KK photons and the KK $Z$ bosons. However, because of
weaker couplings their contributions are only sub-leading and henceforth
neglected in the present work.
The so-called “aligned” scenario in which the rotation matrices for the left-
and the right-handed quark fields are tuned to avoid as many flavor
constraints as possible can be summarized as
$\displaystyle U_{uR}$ $\displaystyle=U_{dR}=U_{dL}={\bf 1}_{3},$
$\displaystyle U_{uL}$ $\displaystyle=V_{\text{CKM}}^{\dagger}$ (52)
along with universal BLKT parameters $r_{Q}^{\prime}$ and $r_{T}^{\prime}$,
for the first two and the third quark generations respectively, irrespective
of their chiralities. In such a scenario, by construct, dominant tree-level
FCNC is induced via KK gluon exchange and only through the doublet up-quark
sector. Note that no FCNC appears at the up-quark singlet part and the down-
quark sector. The latter helps evade severe bounds from the $K$ and $B$ meson
sectors. The forms of the 4D Yukawa couplings, before diagonalization, are
determined simultaneously as:
$\displaystyle
Y^{u}_{ij}=\sum_{l=1}^{3}\frac{\left(V_{\text{CKM}}^{\dagger}\right)_{il}\mathcal{Y}^{u}_{lj}}{F^{d,(0,0)}_{ij}},\qquad
Y^{d}_{ij}=\begin{cases}\displaystyle\frac{\mathcal{Y}^{d}_{ii}}{F^{d,(0,0)}_{ii}}&\text{for}\
i=j,\\\ \displaystyle 0&\text{for}\ i\not=j.\end{cases}$ (53)
In this configuration, the structure of the vertex
$\overline{u^{(0)}_{iL}}-d^{(0)}_{jL}-W^{+(0)}_{\mu}$ is reduced to that of
the SM. The overlap matrices in the gauge kinetic sector receive bi-unitary
transformations when these terms are rotated to a basis where the quark mass
matrices in the Yukawa sector are diagonal. These rotated overlap matrices are
given by
$\displaystyle\sum_{l=1}^{3}(U_{uL}^{\dagger})_{il}F^{U,[k]}_{g,ll}(U_{uL})_{lj}$
$\displaystyle=\left\\{F^{U,[k]}_{g,11}{\bf
1}_{3}+V_{\text{CKM}}\begin{pmatrix}0&&\\\ &0&\\\
&&\underbrace{F^{U,[k]}_{g,33}-F^{U,[k]}_{g,11}}_{=:\Delta
F^{U,[k]}_{g}}\end{pmatrix}V_{\text{CKM}}^{\dagger}\right\\}_{ij}$
$\displaystyle\simeq\left\\{F^{U,[k]}_{g,11}{\bf 1}_{3}+\Delta
F^{U,[k]}_{g}\begin{pmatrix}A^{2}\lambda^{6}&-A^{2}\lambda^{5}&A\lambda^{3}\\\
-A^{2}\lambda^{5}&A^{2}\lambda^{4}&-A\lambda^{2}\\\
A\lambda^{3}&-A\lambda^{2}&1\end{pmatrix}\right\\}_{ij}$ (54)
where $A(=0.814)$ and $\lambda(=0.23)$ are the usual Wolfenstein parameters
and we use the relation $F^{U,[k]}_{g,11}=F^{U,[k]}_{g,22}$. Clearly, the
difference of the two overlap matrices in that diagonal term governs the FCNC
contribution and thus, in turn, relative values of the corresponding BLKT
parameters, $r_{Q}^{\prime}$ and $r_{T}^{\prime}$ that shape the overlap
matrices, get constrained.
To exploit the model independent constraints provided by the UTfit
collaboration Bona:2007vi , the effective Hamiltonian for the $t$-channel KK
gluon exchange process (that describes the $D^{0}-\overline{D^{0}}$ mixing
effect) needs to be written down in terms of the following quark operators and
the associated Wilson coefficient:
$\displaystyle\Delta\mathcal{H}_{\text{eff}}^{\Delta
C=2}=C_{D}^{1}(\overline{u}^{a}_{L}\gamma_{\mu}c^{a}_{L})(\overline{u}^{b}_{L}\gamma^{\mu}c^{b}_{L})$
(55)
where $a$ and $b$ are the color indices and we use $SU(3)$ algebra and
appropriate Fierz transformation to obtain
$\displaystyle C_{D}^{1}=\sum_{k\geq
2:\text{even}}\frac{g_{s}^{2}(\mu_{D})}{6}\frac{1}{m_{g^{(2)}}^{2}}(-A^{2}\lambda^{5}\Delta
F^{U,[k]}_{g})^{2}\simeq\frac{2\pi\alpha_{s}(\mu_{D})}{3m_{g^{(2)}}^{2}}A^{4}\lambda^{10}(\Delta
F^{U,[k]}_{g})^{2}.$ (56)
As it appears, the value of $C_{D}^{1}$ is highly Cabibbo-suppressed. Heavier
KK gluons (except the one from level ‘2’) effectively decouples. The QCD
coupling at the $D^{0}$-meson scale $(\mu_{D}\simeq 2.8\,\text{GeV})$ is
estimated by the relation,
$\displaystyle\alpha^{-1}_{s}(\mu_{D})=\alpha^{-1}_{s}(M_{Z})-\frac{1}{6\pi}\left(23\ln{\frac{M_{Z}}{m_{b}}}+25\ln{\frac{m_{b}}{\mu_{D}}}\right)\simeq
1/0.240$ (57)
with $\alpha_{s}(M_{Z})=0.1184$ Adachi:2011tn . One would now be able to put
bounds on the parameter space by use of the result by the UTfit collaboration
Bona:2007vi ,
$\displaystyle|C^{1}_{D}|<7.2\times 10^{-7}\,\text{TeV}^{-2}$ (58)
which, for a given set of values for $R{{}^{-1}}$ and $r_{G}^{\prime}$,
actually exploits the dependence of $\Delta F^{U,[k]}_{g}$ (appearing in
equation (B.6)) on the BLKT parameters $r_{Q}^{\prime}$ and $r_{T}^{\prime}$.
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|
arxiv-papers
| 2013-10-25T18:02:44 |
2024-09-04T02:49:52.889956
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "AseshKrishna Datta, Kenji Nishiwaki and Saurabh Niyogi",
"submitter": "Kenji Nishiwaki",
"url": "https://arxiv.org/abs/1310.6994"
}
|
1310.7113
|
Singleton sets random attractor for stochastic FitzHugh-Nagumo lattice
equations driven by fractional Brownian motions 111This work has been
partially supported by NSFC Grants 11071199, NSF of Guangxi Grants
2013GXNSFBA019008 and Guangxi Provincial Department of Research Project Grants
2013YB102.
Anhui Gu, Yangrong Li
School of Mathematics and Statistics, Southwest University, Chongqing 400715,
China
Abstract: The paper is devoted to the study of the dynamical behavior of the
solutions of stochastic FitzHugh-Nagumo lattice equations, driven by
fractional Brownian motions, with Hurst parameter greater than $1/2$. Under
some usual dissipativity conditions, the system considered here features
different dynamics from the same one perturbed by Brownian motion. In our
case, the random dynamical system has a unique random equilibrium, which
constitutes a singleton sets random attractor.
Keywords: Stochastic FitzHugh-Nagumo lattice equations; fractional Brownian
motion; random dynamical systems; random attractor.
## 1 Introduction
Recently, the dynamics of deterministic lattice dynamical systems have drawn
much attention of mathematicians and physicists, see e.g. [1]-[5] and the
references therein. As we know, most of the realistic systems involve noises
which may play an important role as intrinsic phenomena rather than just
compensation of defects in deterministic models. Stochastic lattice dynamical
systems (SLDS) arise naturally while these random influences or uncertainties
are taken into account. Since Bates et al. [6] initiated the study of SLDS,
many works have been done regarding the existence of global random attractors
for SLDS with white noises on infinite lattices (see e.g. [7]-[11]). Later,
the existence of global random attractors was extended to other SLDS with
additive white noises, for example, first-order SLDS on $\mathbb{Z}^{k}$ [8],
stochastic Ginzburg-Landau lattice equations [9], stochastic FitzHugh-Nagumo
lattice equations [10], second-order stochastic lattice systems [11] and first
(or second)-order SLDS with a multiplicative white noise [7, 12]. Zhao and
Zhou [13] gave some sufficient conditions for the existence of a global random
attractor for general SLDS in the non-weighted space $\mathbb{R}$ of infinite
sequences and provided an application to damped sine-Gordon lattice systems
with additive noises. Very recently, Han et al. [14] provided some sufficient
conditions for the existence of global compact random attractors for general
SLDS in the weighted space $\ell_{\rho}^{p}$ $(p\geqslant 1)$ of infinite
sequences, and their results are applied to second-order SLDS in [15] and
[16].
However, as can be seen that all the work above are considered in the
framework of the classical It$\ddot{\rm o}$ theory of Brownian motion. Note
that fractional Brownian motion (fBm) does not possess independent increments
and stochastic differential equations driven by fBm do not define the Markov
process as in the case of usual white noises. Therefore, it is not possible to
apply standard methods (see e.g. Theorem 3.1 in [14]) to deal with these
questions. Fortunately, the theory of random dynamical systems still works for
non-Markovian processes (see [17, 18]). In [19], we consider the first-order
lattice dynamical system perturbed by fractional Brownian motions.
FBm appears naturally in the modeling of many complex phenomena in
applications when the systems are subject to “rough” external forcing. An fBm
is a stochastic process which differs significantly from the standard Brownian
motion and semi-martingales, and other classically used processes in the
theory of stochastic process. As a centered Gaussian process, it is
characterized by the stationarity of its increments and a medium- or long-
memory property. It also exhibits power scaling with exponent $H$. Its paths
are H$\ddot{\rm o}$lder continuous of any order $H^{\prime}\in(0,H)$. An fBm
is not a semi-martingale nor a Markov process. Especially, when the Hurst
parameter $H\in(1/2,1)$, the fBm has the properties of self-similarity and
long-range dependence. So, fBm is the good candidate to model random long term
influences in climate systems, hydrology, medicine and physical phenomena. For
more details on fBm, we can refer to the monographs [20, 21].
Motivated by [10, 19], we investigate the long-term behavior of the following
stochastic FitzHugh-Nagumo lattice equations:
$\left\\{\begin{array}[]{l}\frac{du_{i}}{dt}=u_{i-1}-2u_{i}+u_{i+1}-\lambda
u_{i}+f_{i}(u_{i})-v_{i}+a_{i}\frac{d\beta_{i}^{H}(t)}{dt},\\\
\frac{dv_{i}}{dt}=\varrho u_{i}-\sigma
v_{i}+b_{i}\frac{d\beta_{i}^{H}(t)}{dt},\\\
u(0)=u_{0}=(u_{i0})_{i\in\mathbb{Z}},\quad
v(0)=v_{0}=(v_{i0})_{i\in\mathbb{Z}},\end{array}\right.$ (1.1)
where $\mathbb{Z}$ denotes the integer set, $u_{i}\in\mathbb{R}$,
$\lambda,\varrho$ and $\sigma$ are positive constants, $f_{i}$ are smooth
functions satisfying some dissipative conditions,
$(a_{i})_{i\in\mathbb{Z}}\in\ell^{2}$, $(b_{i})_{i\in\mathbb{Z}}\in\ell^{2}$
and $\\{\beta_{i}^{H}:i\in\mathbb{Z}\\}$ are independent two-sided fractional
Brownian motions with Hurst parameter $H\in(1/2,1)$,
$\ell^{2}=(\ell^{2},(\cdot,\cdot),\|\cdot\|)$ denotes the regular space of
infinite sequences. When there are no noises terms, form similar to (1.1) is
the discrete of the FitzHugh-Nagumo system which arose as modeling the signal
transmission across axons in neurobiology (see [22]). FitzHugh-Nagumo lattice
system was used to stimulate the propagation of action potentials in
myelinated nerve axons (see [23]). The stochastic FitzHugh-Nagumo lattice
equations were first proposed in [10]. The existence of random attractors of
(similar) stochastic FitzHugh-Nagumo lattice equations with white noises were
established in [10, 24] and [25].
The goal of this article is to establish the existence of a random attractor
for stochastic FitzHugh-Nagumo lattice equations with the nonlinear $f$ under
some dissipative conditions and driven by fractional Brownian motions with
Hurst parameter $H\in(1/2,1)$. By borrowing the main ideas of [26], we first
define a random dynamical system by using a pathwise interpretation of the
stochastic integral with respect to the fractional Brownian motions. This
method is based on the fact that a stochastic integral with respect to an fBm
with Hurst parameter $H\in(1/2,1)$ can be defined by a generalized pathwise
Riemann-Stieltjes integral (see e.g. [27]–[30]). And then we show the
existence of a pullback absorbing set for the random dynamical system achieved
by means of a fractional Ornstein-Uhlenbeck transformation and Gronwall lemma.
Since every trajectory of the solutions of system (1.1) cannot be
differentiated, we have to consider the difference between any two solutions
among them, which is pathwise differentiable (see [26]). Due to the
stationarity of the fractional Ornstein-Uhlenbeck solution, we get a unique
random equilibrium finally. All solutions converge pathwise to each other, so
the random attractor, which consists of a unique random equilibrium, is proven
to be a singleton sets random attractor.
The paper is organized as follows. In Sec. 2, we recall some basic concepts on
random dynamical systems. In Sec. 3, we give a unique solution to system (1.1)
and make sure that the solution generates a random dynamical system. We
establish the main result, that is, the random dynamical system generated by
equation (1.1) has a unique random equilibrium, which constitutes a singleton
sets random attractor in Sec. 4.
## 2 Preliminaries
In this section, we introduce some basic concepts related to random dynamical
systems and random attractors, which are taken from [31]-[33].
Let $(\mathbb{E},\|\cdot\|_{\mathbb{E}})$ be a separable Hilbert space and
$(\Omega,\mathcal{F},\mathbb{P})$ be a probability space.
###### Definition 2.1.
A metric dynamical system $(\Omega,\mathcal{F},\mathbb{P},\theta)$ with two-
sided continuous time $\mathbb{R}$ consists of a measurable flow
$\theta:(\mathbb{R}\times\Omega,\mathcal{B}(\mathbb{R})\otimes\mathcal{F})\rightarrow(\Omega,\mathcal{F}),$
where the flow property for the mapping $\theta$ holds for the partial
mappings $\theta_{t}=\theta(t,\cdot)$:
$\theta_{t}\circ\theta_{s}=\theta_{t}\theta_{s}=\theta_{t+s},\ \
\theta_{0}={\rm id}_{\Omega}$
for all $s,t\in\mathbb{R}$, and $\theta\mathbb{P}=\mathbb{P}$ for all
$t\in\mathbb{R}$.
###### Definition 2.2.
A continuous random dynamical system (RDS) $\varphi$ on $\mathbb{E}$ over
$(\Omega,\mathcal{F},\mathbb{P},(\theta_{t})_{t\in\mathbb{R}})$ is a
$(\mathcal{B}(\mathbb{R}^{+})\times\mathcal{F}\times\mathcal{B}(\mathbb{E}),\mathcal{B}(\mathbb{E}))$-measurable
mapping and satisfies
(i) $\varphi(0,\omega)$ is the identity on $\mathbb{E}$;
(ii) $\varphi(t+s,\omega)=\varphi(t,\theta_{s}\omega)\circ\varphi(s,\omega)$
for all $s,$ $t\in\mathbb{R}^{+}$, $\omega\in\Omega$;
(iii) $\varphi(t,\omega)$ is continuous on $\mathbb{E}$ for all
$(t,\omega)\in\mathbb{R}^{+}\times\Omega$.
A universe $\mathcal{D}=\\{D(\omega),\omega\in\Omega\\}$ is a collection of
nonempty subsets $D(\omega)$ of $\mathbb{E}$ satisfying the following
inclusion property: if $D\in\mathcal{D}$ and $D^{\prime}(\omega)\subset
D(\omega)$ for all $\omega\in\Omega$, then $D^{\prime}\in\mathcal{D}$.
###### Definition 2.3.
A family $\mathcal{A}=\\{A(\omega),\omega\in\Omega\\}$ of nonempty measurable
compact subsets $\mathcal{A}(\omega)$ of $\mathbb{E}$ is called $\varphi$\-
invariant if
$\varphi(t,\omega,\mathcal{A}(\omega))=\mathcal{A}(\theta_{t}\omega)$ for all
$t\in\mathbb{R^{+}}$ and is called a random attractor if in addition it is
pathwise pullback attracting in the sense that
$H_{d}^{*}(\varphi(t,\theta_{-t}\omega,D(\theta_{-t}\omega)),\mathcal{A}(\omega))\rightarrow
0\ \ \mbox{as}\ \ t\rightarrow\infty$
for all $D\in\mathcal{D}$. Here $H_{d}^{*}$ is the Hausdorff semi-distance on
$\mathbb{E}$.
###### Definition 2.4.
A random variable $u:\Omega\mapsto\mathbb{E}$ is said to be a random
equilibrium of the RDS $\varphi$ if it is invariant under $\varphi$, i.e. if
$\varphi(t,\omega)u(\omega)=u(\theta_{t}\omega)\quad\mbox{for all}\quad t\geq
0\quad\mbox{and all}\quad\omega\in\Omega.$
###### Definition 2.5.
A random variable $r:\Omega\rightarrow\mathbb{R}$ is called tempered if
$\lim_{t\rightarrow\pm\infty}\frac{\log|r(\theta_{t}\omega)|}{|t|}=0\ \
\mathbb{P}-a.s.$
and a random set $\\{D(\omega),\omega\in\Omega\\}$ with
$D(\omega)\subset\mathbb{E}$ is called tempered if it is contained in the ball
$\\{x\in\mathbb{R}:|x|\leq r(\omega)\\}$, where $r$ is a tempered random
variable.
Here we will always work with the attracting universe given by the tempered
random sets.
###### Definition 2.6.
A family $\hat{B}=\\{B\mathcal{(\omega)},\omega\in\Omega\\}$ is said to be
pullback absorbing if for every $D(\omega)\in\mathcal{D}$, there exists
$T_{D}(\omega)\geq 0$ such that
$\varphi(t,\theta_{-t}\omega,D(\theta_{-t}\omega))\subset B(\omega)\ \ \forall
t\geq T_{D}(\omega).$ (2.1)
The following result (cf. Proposition 9.3.2 in [31], Theorem 2.2 in [33])
guarantees the existence of a random attractor.
###### Theorem 2.7.
Let $(\theta,\varphi)$ be a continuous RDS on $\Omega\times\mathbb{E}$. If
there exists a pullback absorbing family
$\hat{B}=\\{B\mathcal{(\omega)},\omega\in\Omega\\}$ such that, for every
$\omega\in\Omega$, $B(\omega)$ is compact and $B(\omega)\in\mathcal{D}$, then
the RDS $(\theta,\varphi)$ has a random attractor
$\mathcal{A}(\omega)=\bigcap_{\tau>0}\overline{\bigcup_{t\geqslant\tau}\varphi(t,\theta_{-t}\omega)B(\theta_{-t}\omega)}.$
Note that if the random attractor consists of singleton sets, i.e.
$\mathcal{A}(\omega)=\\{u^{*}(\omega)\\}$ for some random variable $u^{*}$,
then $u^{*}(t)(\omega)=u^{*}(t)(\theta_{t}\omega)$ is a stationary stochastic
process.
## 3 FitzHugh-Nagumo Lattice Equations with Fractional Brownian Motions
We now recall the definition of a fractional Brownian motion. Given
$H\in(0,1)$, a continuous centered Gaussian process
$\beta^{H}(t),t\in\mathbb{R}$, with the covariance function
$\mathbf{E}\beta^{H}(t)\beta^{H}(s)=\frac{1}{2}(|t|^{2H}+|s|^{2H}-|t-s|^{2H}),\
\ t,s\in\mathbb{R}$
is called a two-sided one-dimensional fBm, and $H$ is the Hurst parameter. For
$H=1/2$, $\beta$ is a standard Brownian motion, while for $H\neq 1/2$, it is
neither a semimartingale nor a Markov process. Moreover,
$\mathbf{E}|\beta^{H}(t)-\beta^{H}(s)|^{2}=|t-s|^{2H},\ \ \mbox{for all}\ \
s,t\in\mathbb{R}.$
Here, we assume that $H\in(1/2,1)$ throughout the paper. When $H\in(0,1/2)$ we
cannot define the stochastic integral by a generalized Stieljes integral and,
therefore, dealing with such values of the Hurst parameter seems to be much
more complicated. It is worth mentioning that when $H=1/2$ the fBm becomes the
standard Wiener process, the random dynamical system generated by the
(similar) stochastic FitzHugh-Nagumo lattice equations has been studied in
[10, 24].
Using the definition of $\beta^{H}(t)$, Kolmogorov’s theorem ensures that
$\beta^{H}$ has a continuous version, and almost all the paths are H$\ddot{\rm
o}$lder continuous of any order $H^{\prime}\in(0,H)$ (see [34]). Thus, let
$\mathbb{E}=\ell^{2}\times\ell^{2}$ and norm $\|\cdot\|_{\mathbb{E}}$, we can
consider the canonical interpretation of an fBm: denote
$\Omega=C_{0}(\mathbb{R},\ell^{2})$, the space of continuous functions on
$\mathbb{R}$ with values in $\ell^{2}$ such that $\omega(0)=0$, equipped with
the compact open topology. Let $\mathcal{F}$ be the associated
Borel-$\sigma$-algebra and $\mathbb{P}$ the distribution of the fBm
$\beta^{H}$, and $\\{\theta_{t}\\}_{t\in\mathbb{R}}$ be the flow of Wiener
shifts such that
$\theta_{t}\omega(\cdot)=\omega(\cdot+t)-\omega(t),\ \ t\in\mathbb{R}.$
Due to [17]-[35], we know that the quadruple
$(\Omega,\mathcal{F},\mathbb{P},\theta)$ is an ergodic metric dynamical
system. Furthermore, it holds that
$\displaystyle\begin{split}&\beta^{H}(\cdot,\omega)=\omega(\cdot),\\\
\beta^{H}(\cdot,\theta_{s}\omega)&=\beta^{H}(\cdot+s,\omega)-\beta^{H}(s,\omega)\\\
&=\omega(\cdot+s)-\omega(s).\end{split}$ (3.1)
For $u=(u_{i})_{i\in\mathbb{Z}}\in\ell^{2}$, define
$\mathbb{A},\mathbb{B},\mathbb{B}^{*}$ to be linear operators from $\ell^{2}$
to $\ell^{2}$ as follows:
$\displaystyle\begin{split}(\mathbb{A}u)_{i}&=-u_{i-1}+2u_{i}-u_{i+1},\\\
(\mathbb{B}u)_{i}&=u_{i+1}-u_{i},\ \ (\mathbb{B}^{*}u)_{i}=u_{i-1}-u_{i},\ \
i\in\mathbb{Z}.\end{split}$
It is easy to show that
$\mathbb{A}=\mathbb{B}\mathbb{B}^{*}=\mathbb{B}^{*}\mathbb{B}$,
$(\mathbb{B}^{*}u,u^{\prime})=(u,\mathbb{B}u^{\prime})$ for all
$u,u^{\prime}\in\ell^{2}$, which implies that $(\mathbb{A}u,u)\geq 0$.
Let $W_{1}(t)\equiv
W_{1}(t,\omega)=\sum_{i\in\mathbb{Z}}a_{i}\omega_{i}(t)e^{i}$ and
$W_{2}(t)\equiv W_{2}(t,\omega)=\sum_{i\in\mathbb{Z}}b_{i}\omega_{i}(t)e^{i}$,
here $(e^{i})_{i\in\mathbb{Z}}\in\ell^{2}$ denote the element having $1$ at
position $i$ and the other components $0$. Then SLDS (1.1) with initial
conditions can be rewritten as pathwise Riemann-Stieltjes integral equations
in $\mathbb{E}$
$\displaystyle\left\\{\begin{array}[]{l}u(t)=u(0)+\int_{0}^{t}(-\mathbb{A}u(s)-\lambda
u(s)+f(u(s))-v(s))ds+W_{1}(t),\\\ v(t)=v(0)+\int_{0}^{t}(\varrho u(s)-\sigma
v(s))ds+W_{2}(t),\\\ u(0)=u_{0}=(u_{i0})_{i\in\mathbb{Z}},\quad
v(0)=v_{0}=(v_{i0})_{i\in\mathbb{Z}},\end{array}\right.$ (3.5)
where $u=(u_{i})_{i\in\mathbb{Z}}$, $\lambda,\varrho$ and $\sigma$ are
positive constants, $a=(a_{i})_{i\in\mathbb{Z}}\in\ell^{2}$,
$b=(b_{i})_{i\in\mathbb{Z}}\in\ell^{2}$ and
$\\{\omega_{i}=\beta^{H}_{i}:i\in\mathbb{Z}\\}$ are independent two-sided
fractional Brownian motions with Hurst parameter $H\in(1/2,1)$,
$f(u)=(f_{i}(u_{i}))_{i\in\mathbb{Z}}$ is a nonlinear smooth function
satisfies a one-sided dissipative Lipschitz condition
$(f(u)-f(v),u-v)\leq-\gamma\|u-v\|^{2}\ \ \text{for all}\ u,v\in\mathbb{R}$
(3.6)
and the polynomial growth condition
$|f(u)|\leq c_{f}(|u|^{2p+1}+1)\ \text{for all}\ \ u\in\mathbb{R},$ (3.7)
where $\gamma$ is a positive constant, $p$ is a positive integer.
In addition we could consider a more general dissipativity condition, which
would lead to nontrivial setvalued random attractors, we will restrict here to
the dissipativity condition (3.6). When system (1.1) with Hurst parameter
$H=1/2$ and under conditions (3.6) and (3.7), we can apply the result of
Theorem 3.1 in [14], i.e. the combination of the existence of a bounded closed
random absorbing set and the property of random asymptotic nullity to get the
existence of a compact random attractor. Moreover, we have the following
results:
###### Lemma 3.1.
There exists positive random constants
$(\tilde{\rho}_{i}(\omega))_{i\in\mathbb{Z}}\in\ell^{2}$ and
$\rho(\omega)=\|\tilde{\rho}(\omega)\|$ such that for every
$\omega\in\bar{\Omega}$, where $\bar{\Omega}\in\mathcal{F}$ is a
$(\theta_{t})_{t\in\mathbb{R}}$-invariant set of full measure, the fractional
Brownian motions are well defined for $t\in\mathbb{R}$ in $\ell^{2}$
satisfying
$\displaystyle\|W_{j}(t)\|^{2}\leq
2\max\\{\|a\|^{2},\|b\|^{2}\\}\rho^{2}(\omega)(1+|t|^{4}),\ \ j=1,2.$
###### Proof.
Obviously. ∎
###### Proposition 3.2.
Let the above assumptions on $f$ be satisfied and $T>0$. Then system (3.5) has
a unique pathwise solution $\Psi=(\Psi(t))_{t\geq 0}=(u(t),v(t))_{t\geq 0}$.
Furthermore, the solution satisfies
$\displaystyle\sup_{t\in[0,T]}\|\Psi(t)\|_{\mathbb{E}}^{2}\leq
M[\|\Psi_{0}\|_{\mathbb{E}}^{2}+\sup_{t\in[0,T]}(\|W_{1}(t)\|^{2}+\|W_{2}(t)\|^{2})$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+\int_{0}^{T}(\|W_{1}(s)\|^{4p+2}+\|W_{1}(s)\|^{2}+\|W_{2}(s)\|^{2}+1)ds],$
where $M$ is a positive constant independent of $T$.
###### Proof.
Let $\tilde{u}(t)=u(t)-W_{1}(t)$ and $\tilde{v}(t)=v(t)-W_{2}(t)$, system
(3.5) has a solution $\Psi=(\Psi(t))_{t\geq 0}$ for all $\omega\in\Omega$ if
and only if the following system
$\displaystyle\left\\{\begin{array}[]{l}\tilde{u}(t)=u(0)+\int_{0}^{t}(-\mathbb{A}\tilde{u}(s)-\lambda\tilde{u}(s)+f(\tilde{u}(s)+W_{1}(s))\\\
~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}-\tilde{v}(s)-\mathbb{A}W_{1}(s)-\lambda
W_{1}(s)-W_{2}(s))ds,\\\
\tilde{v}(t)=v(0)+\int_{0}^{t}(\varrho\tilde{u}(s)-\sigma\tilde{v}(s)-\varrho
W_{1}(s)-\sigma W_{2}(s))ds,\\\
u(0)=u_{0}=(\tilde{u}_{i0})_{i\in\mathbb{Z}},\quad
v(0)=v_{0}=(\tilde{v}_{i0})_{i\in\mathbb{Z}}\end{array}\right.$ (3.12)
has a unique pathwise solution for $t\in[0,T]$. However, since the integrand
is pathwise continuous, the fundamental theorem of calculus says that the left
hand side of (3.12) is pathwise differentiable. Thus, for a fixed
$\omega\in\Omega$, system (3.12) is the pathwise system of random ODEs
$\displaystyle\left\\{\begin{array}[]{l}\frac{d\tilde{u}(t)}{dt}=-\mathbb{A}\tilde{u}(t)-\lambda\tilde{u}(t)+f(\tilde{u}(t)+W_{1}(t))\\\
~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}-\tilde{v}(t)-\mathbb{A}W_{1}(t)-\lambda
W_{1}(t)-W_{2}(t),\\\
\frac{d\tilde{v}(t)}{dt}=\varrho\tilde{u}(t)-\sigma\tilde{v}(t)+\varrho
W_{1}(t)-\sigma W_{2}(t),\\\
u(0)=u_{0}=(\tilde{u}_{i0})_{i\in\mathbb{Z}},\quad
v(0)=v_{0}=(\tilde{v}_{i0})_{i\in\mathbb{Z}}.\end{array}\right.$ (3.17)
Since $f(u)$ is a continuous function, and the assumptions on $f$ are
satisfied, by the standard argument on existence theorem for ODEs, it follows
that system (3.17) possesses a local solution in a small interval
$[0,\tau(\omega)]$, which means system (3.5) has a unique local solution in
the same small interval $[0,\tau(\omega)]$. Here, we remain to show that the
local solution is a global one.
For a fixed $\omega\in\Omega$, by taking the inner product of (3.17) with
$(\tilde{u},\tilde{v})$ in $\mathbb{E}$, it follows that
$\displaystyle\|\tilde{u}(t)\|^{2}+\frac{1}{\varrho}\|\tilde{v}(t)\|^{2}=\|\tilde{u}_{0}\|^{2}+\frac{1}{\varrho}\|\tilde{v}_{0}\|^{2}+2\int_{0}^{t}(-\mathbb{A}\tilde{u}(s),\tilde{u}(s))ds$
(3.18)
$\displaystyle~{}~{}+2\int_{0}^{t}(f(\tilde{u}(s)+W_{1}(s)),\tilde{u}(s))ds+2\int_{0}^{t}(-\mathbb{A}W_{1}(s),\tilde{u}(s))ds$
$\displaystyle~{}~{}~{}-2\int_{0}^{t}(\lambda
W_{1}(s),\tilde{u}(s))ds-2\int_{0}^{t}(W_{2}(s),\tilde{u}(s))ds$
$\displaystyle~{}~{}~{}~{}-2\lambda\int_{0}^{t}\|\tilde{u}(s)\|^{2}ds-\frac{2\sigma}{\varrho}\int_{0}^{t}\|\tilde{v}(s)\|^{2}ds$
$\displaystyle~{}~{}~{}~{}~{}~{}-\frac{2\sigma}{\varrho}\int_{0}^{t}(W_{2}(s),\tilde{v}(s))ds+2\int_{0}^{t}(W_{1}(s),\tilde{v}(s))ds.$
By (3.6) and (3.7), we obtain that
$\displaystyle 2(f(\tilde{u}(s)+W_{1}(s)),\tilde{u}(s))$ (3.19)
$\displaystyle=$ $\displaystyle
2(f(\tilde{u}(s)+W_{1}(s)),\tilde{u}(s)+W_{1}(s))-2(f(\tilde{u}(s)+W_{1}(s),W_{1}(s))$
$\displaystyle\leq$
$\displaystyle-\gamma\|\tilde{u}(s)+W_{1}(s)\|^{2}+2|f(\tilde{u}(s)+W_{1}(s)||W_{1}(s)|$
$\displaystyle\leq$ $\displaystyle
c_{1}(\|W_{1}(s)\|^{4p+2}+\|W_{1}(s)\|^{2}+1),$
where $c_{1}$ is a positive constant depends on $\gamma,c_{f}$ and $p$. By
Young’s inequality, it yields that
$\displaystyle
2(-\mathbb{A}\tilde{u}(s),\tilde{u}(s))+2(f(\tilde{u}(s)+W_{1}(s)),\tilde{u}(s))+2(-\mathbb{A}W_{1}(s),\tilde{u}(s))$
(3.20) $\displaystyle\leq$
$\displaystyle\lambda\|\tilde{u}(s)\|^{2}+c_{2}(\|W_{1}(s)\|^{2}+\|W_{2}(s)\|^{2}),$
where $c_{2}$ is a positive constant depends on $\lambda$, and
$\displaystyle\frac{2\sigma}{\varrho}(W_{2}(s),\tilde{v}(s))+2(W_{1}(s),\tilde{v}(s))$
(3.21) $\displaystyle\leq$
$\displaystyle\frac{\sigma}{\varrho}\|\tilde{v}(s)\|^{2}+c_{3}(\|W_{1}(s)\|^{2}+\|W_{2}(s)\|^{2}),$
where $c_{3}$ is a positive constant depends on $\varrho$ and $\sigma$. Let
$\alpha=\min\\{\lambda,\sigma\\}$ and combine (3.19)-(3.21)with (3.18), for
$t\geq 0$, we get
$\displaystyle\|\tilde{u}(t)\|^{2}+\frac{1}{\varrho}\|\tilde{v}(t)\|^{2}\leq\|\tilde{u}_{0}\|^{2}+\frac{1}{\varrho}\|\tilde{v}_{0}\|^{2}-\alpha\int_{0}^{t}(\|\tilde{u}(s)\|^{2}+\frac{1}{\varrho}\|\tilde{v}(s)\|^{2})ds$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+c\int_{0}^{t}(\|W_{1}(s)\|^{4p+2}+\|W_{1}(s)\|^{2}+\|W_{2}(s)\|^{2}+1)ds$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\leq\|\tilde{u}_{0}\|^{2}+\frac{1}{\varrho}\|\tilde{v}_{0}\|^{2}$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+c\int_{0}^{t}(\|W_{1}(s)\|^{4p+2}+\|W_{1}(s)\|^{2}+\|W_{2}(s)\|^{2}+1)ds,$
(3.22)
where $c$ is a positive constant depends on
$\varrho,\sigma,\lambda,\gamma,c_{f}$ and $p$. Hence, from (3), we know that
$\|\tilde{u}(t)\|^{2}+\frac{1}{\varrho}\|\tilde{v}(t)\|^{2}$ is bounded by a
continuous function, which implies the global existence of a solution on
interval $[0,T]$. Furthermore, for all $\omega\in\Omega$, it follows that
$\displaystyle\sup_{t\in[0,T]}(\|u(t)\|^{2}+\frac{1}{\varrho}\|v(t)\|^{2})=\sup_{t\in[0,T]}(\|\tilde{u}(t)+W_{1}(t)\|^{2}+\frac{1}{\varrho}\|\tilde{v}(t)+W_{2}(t)\|^{2})$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\leq
2(\|u_{0}\|^{2}+\frac{1}{\varrho}\|v_{0}\|^{2})+2\sup_{t\in[0,T]}(\|W_{1}(t)\|^{2}+\frac{1}{\varrho}\|W_{2}(t)\|^{2})$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+2c\int_{0}^{T}(\|W_{1}(s)\|^{4p+2}+\|W_{1}(s)\|^{2}+\|W_{2}(s)\|^{2}+1)ds.$
(3.23)
According to Lemma 3.1, we know that the right side of (3) is well defined.
Let
$\tilde{\alpha}=\frac{\max\\{1,\frac{1}{\varrho}\\}}{\min\\{1,\frac{1}{\varrho}\\}}$,
the proof is complete. ∎
###### Proposition 3.3.
The solution of (3.5) determinants a continuous random dynamical system
$\varphi:\mathbb{R^{+}}\times\Omega\times\mathbb{E}\rightarrow\mathbb{E}$,
which is given by
$\displaystyle\varphi(t,\omega,\Psi_{0})=\Psi_{0}+\int_{0}^{t}G(\Psi(s))ds+\eta(t,\omega)\
\ \mbox{for}\ \ t\geq 0,$ (3.24)
where $G(\Psi(t))=L\Psi(t)+F(\Psi(t))$ and
$\displaystyle L=\left(\begin{array}[]{cc}-\mathbb{A}-\lambda&\ \ -1\\\
\varrho&\ \ -\sigma\end{array}\right),\
F(\Psi)=\left(\begin{array}[]{c}f(u)\\\ 0\end{array}\right),\
\eta(t,\omega)=\left(\begin{array}[]{c}W_{1}(t,\omega)\\\
W_{2}(t,\omega)\end{array}\right).$
###### Proof.
For the need of making the relations clear between $G(\Psi(t))(\cdot)$ and
$\omega$, we write $G(\Psi(t))(\omega)$ instead if necessary. Note that (3.1)
is satisfied for $\omega\in\Omega$ and by the definition of
$(\theta_{t})_{t\in\mathbb{R}}$, we have the property
$\eta(\tau+t,\omega)=\eta(\tau,\theta_{t}\omega)+\eta(t,\omega)\ \ \mbox{for
all}\ \ t,\tau\in\mathbb{R}.$
By Proposition 3.2 we know that $\varphi$ solves (3.5), thus $\varphi$ is
measurable and satisfies $\varphi(0,\omega,\cdot)=\rm id_{\mathbb{E}}$. It
remains to verify that the cocycle property in Definition 2.2. Let
$t,\tau\in\mathbb{R^{+}},\omega\in\Omega$ and $\Psi_{0}\in\mathbb{E}$, it
yields from (3.1) that
$\displaystyle\varphi(t+\tau,\omega,\Psi_{0})$ $\displaystyle=$
$\displaystyle\Psi_{0}+\int_{0}^{t+\tau}G(\Psi(s))(\omega)ds+\eta(t+\tau,\omega)$
$\displaystyle=$
$\displaystyle\Psi_{0}+\int_{0}^{t}G(\Psi(s))(\omega)ds+\eta(t,\omega)+\int_{t}^{t+\tau}G(\Psi(s))(\omega)ds+\eta(\tau,\theta_{t}\omega)$
$\displaystyle=$
$\displaystyle\Psi(t)+\int_{0}^{\tau}G(\Psi(s))(\theta_{t}\omega)ds+\eta(\tau,\theta_{t}\omega)$
$\displaystyle=$
$\displaystyle\varphi(\tau,\theta_{t}\omega,\cdot)\circ\varphi(t,\omega,\Psi_{0}),$
which completes the proof. ∎
## 4 Existence of a Random Attractor
In this section, we will prove the existence of a random attractor for the RDS
defined in Proposition 3.3. Sometimes, for the need of making the relations
between $\bar{u}(\cdot)$ (or $\bar{v}$, $\Psi$, $\bar{\Phi}$) and $\omega$
more explicitly, we will write $\bar{u}(\omega)$ (or $\bar{v}(\omega)$,
$\Psi(\omega)$, $\bar{\Phi}(\omega)$) instead if necessary.
Consider the following fractional Ornstein-Uhlenbeck processes
$du(t)=-\lambda u(t)dt+dW_{1}(t),\ dv(t)=-\sigma v(t)dt+dW_{2}(t),$ (4.1)
where $\lambda,\sigma$ defined in (3.5) and $W_{1}(t),W_{2}(t)$ denote one-
dimensional fractional Brownian motions. They have the explicit solutions
$u(t)=u_{0}e^{-\lambda t}+e^{-\lambda t}\int_{0}^{t}e^{\lambda s}dW(s),\
v(t)=v_{0}e^{-\sigma t}+e^{-\sigma t}\int_{0}^{t}e^{\sigma s}dW(s).$ (4.2)
Take the pathwise pullback limits, we get the stochastic stationary solutions
$\bar{u}(t)=e^{-\lambda t}\int_{-\infty}^{t}e^{\lambda s}dW(s),\
\bar{v}(t)=e^{-\sigma t}\int_{-\infty}^{t}e^{\sigma s}dW(s),\ \
t\in\mathbb{R},$ (4.3)
which are called the fractional Ornstein-Uhlenbeck solutions. We have the
following properties:
###### Lemma 4.1.
There exists positive random constants
$(\check{\rho}_{i}(\omega))_{i\in\mathbb{Z}},(\hat{\rho}_{i}(\omega))_{i\in\mathbb{Z}}\in\ell^{2}$
and
$\check{\rho}^{2}(\omega)=16\sum_{i\in\mathbb{Z}}a_{i}^{2}\check{\rho}_{i}^{2}(\omega),\hat{\rho}^{2}(\omega)=16\sum_{i\in\mathbb{Z}}b_{i}^{2}\hat{\rho}_{i}^{2}(\omega)$
for all $\omega\in\Omega$, the Riemann-Stieltjes integrals in (4.3) are well
defined in $\ell^{2}$. Moreover, for all $\omega\in\Omega,t\in\mathbb{R}$, we
have
$\|e^{-\lambda t}\int_{-\infty}^{t}e^{\lambda
s}dW_{1}(s)\|\leq\check{\rho}(\omega)(1+|t|)^{2},\ \|e^{-\sigma
t}\int_{-\infty}^{t}e^{\sigma s}dW_{2}(s)\|\leq\hat{\rho}(\omega)(1+|t|)^{2}.$
###### Proof.
By the Lemma 1 in [26], we can easily get the conclusion. ∎
Now, we are in the position to state the main result.
###### Theorem 4.2.
Assume that the conditions on $f$ are satisfied. Then the random dynamical
system $\varphi$ has a unique random equilibrium, which constitutes a
singleton sets random attractor.
###### Proof.
Let $\Psi(t)=(u(t),v(t)),~{}\Phi(t)=(\tilde{u}(t),\tilde{v}(t))$ be any two
solutions of system (1.1). Their sample paths are not differentiable, but the
difference satisfies pathwise for $t\geq 0$,
$\displaystyle\Psi(t)-\Phi(t)=\Psi_{0}-\Phi_{0}+\int_{0}^{t}(L(\Psi(s)-\Phi(s))+(F(\Psi(s))-F(\Phi(s)))ds,$
and again, since the integrand is pathwise continuous, the fundamental theorem
of calculus indicates that the left hand side is pathwise differentiable and
satisfies
$\displaystyle\frac{d}{dt}(\Psi(t)-\Phi(t))=L(\Psi(t)-\Phi(t))+F(\Psi(t))-F(\Phi(t)),\
t\geq 0.$ (4.4)
Recall that $\alpha=\min\\{\lambda,\sigma\\}$, we obtain from (4.4) that
$\displaystyle\begin{split}\frac{d}{dt}\|\Psi(t)-\Phi(t)\|_{\mathbb{E}}^{2}&=2(\Psi(t)-\Phi(t),L(\Psi(t)-\Phi(t)))_{\mathbb{E}}\\\
&\quad\quad+2(\Psi(t)-\Phi(t),F(\Psi(t))-F(\Phi(t)))_{\mathbb{E}}\\\
&\leq-2\alpha\|\Psi(t)-\Phi(t)\|_{\mathbb{E}}^{2}.\end{split}$
Thus pathwise we have
$\|\Psi(t)-\Phi(t)\|_{\mathbb{E}}^{2}\leq\|\Psi_{0}-\Phi_{0}\|_{\mathbb{E}}^{2}e^{-2\alpha
t}\rightarrow 0,\ \ \mbox{as}\ \ t\rightarrow\infty.$
That is to say that all solutions converge pathwise forward to each other in
time.
Now, we want to know where the solution will converge to. Let
$\bar{\Phi}(t)=(\bar{u}(t),\bar{v}(t))$. We consider the difference
$\Psi(t)-\bar{\Phi}(t)$. Since their paths are continuous, the difference is
pathwise differentiable and satisfies the integral equation for $t\geq 0$,
$\displaystyle\Psi(t)-\bar{\Phi}(t)=\Psi_{0}-\bar{\Phi}_{0}+\int_{0}^{t}(L(\Psi(s)-\bar{\Phi}(s))+(F(\Psi(s))-F(\bar{\Phi}(s)))ds,$
which is equivalent to the pathwise differential equation
$\displaystyle\frac{d}{dt}(\Psi(t)-\bar{\Phi}(t))=L(\Psi(s)-\bar{\Phi}(s))+(F(\Psi(s))-F(\bar{\Phi}(s)),\
\ t\geq 0.$
That is to consider the following system
$\displaystyle\left\\{\begin{array}[]{l}\frac{d}{dt}(u(t)-\bar{u}(t))=-\mathbb{A}u(t)-\lambda(u(t)-\bar{u}(t))+f(u(t))-v(t),\\\
\frac{d}{dt}(v(t)-\bar{v}(t))=\varrho
u(t)-\sigma(v(t)-\bar{v}(t)).\end{array}\right.$ (4.7)
By taking the inner product in $\mathbb{E}$, we get
$\displaystyle\frac{d}{dt}(\|u(t)-\bar{u}(t)\|^{2}+\frac{1}{\varrho}\|v(t)-\bar{v}(t)\|^{2})$
(4.8) $\displaystyle=$ $\displaystyle
2(-\mathbb{A}u(t),u(t)-\bar{u}(t))+2(f(u),u(t)-\bar{u}(t))$
$\displaystyle\quad-2(v(t),u(t)-\bar{u}(t))+2(u(t),v(t)-\bar{v}(t))$
$\displaystyle\quad\quad-2\lambda\|u(t)-\bar{u}(t)\|^{2}-\frac{2\sigma}{\varrho}\|v(t)-\bar{v}(t)\|^{2}.$
We know that
$\displaystyle
2(-\mathbb{A}u(t),u(t)-\bar{u}(t))=2(-\mathbb{A}(u(t)-\bar{u}(t)),u(t)-\bar{u}(t))$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+2(\mathbb{A}\bar{u}(t),u(t)-\bar{u}(t))$
$\displaystyle\leq\frac{\lambda}{2}\|u(t)-\bar{u}(t)\|^{2}+\frac{32}{\lambda}\|\bar{u}(t)\|^{2},$
$\displaystyle 2(f(u),u(t)-\bar{u}(t))=2(f(u)-f(\bar{u}),u(t)-\bar{u}(t))$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+2(f(\bar{u}),u(t)-\bar{u}(t))$
$\displaystyle\leq-\gamma\|u(t)-\bar{u}(t)\|^{2}+\frac{\lambda}{2}\|u(t)-\bar{u}(t)\|^{2}+\frac{8}{\lambda}\|f(\bar{u})\|^{2},$
$\displaystyle-2(v(t),u(t)-\bar{u}(t))+2(u(t),v(t)-\bar{v}(t))$
$\displaystyle\leq$
$\displaystyle\gamma\|u(t)-\bar{u}(t)\|^{2}+\frac{4}{\gamma}\|\bar{v}(t)\|^{2}+\frac{\sigma}{\varrho}\|v(t)-\bar{v}(t)\|^{2}+\frac{4\varrho}{\sigma}\|\bar{u}(t)\|^{2}.$
Combine the three inequalities above with (4.8), we have
$\displaystyle\frac{d}{dt}(\|u(t)-\bar{u}(t)\|^{2}+\frac{1}{\varrho}\|v(t)-\bar{v}(t)\|^{2})$
(4.9) $\displaystyle\leq$
$\displaystyle-\lambda\|u(t)-\bar{u}(t)\|^{2}-\frac{\sigma}{\varrho}\|v(t)-\bar{v}(t)\|^{2}$
$\displaystyle\quad+c_{4}(\|\bar{u}(t)\|^{2}+\|\bar{v}(t)\|^{2}+\|f(\bar{u})\|^{2}),$
where $c_{4}$ is a positive constant depends on $\lambda,\varrho$ and
$\sigma$. Then we obtain
$\displaystyle\frac{d}{dt}\|\Psi(t)-\bar{\Phi}(t)\|_{\mathbb{E}}^{2}\leq-\alpha\|\Psi(t)-\bar{\Phi}(t)\|_{\mathbb{E}}^{2}+c_{4}(\|\bar{u}(t)\|^{2}+\|\bar{v}(t)\|^{2}+\|f(\bar{u})\|^{2}),$
and hence
$\displaystyle\|\Psi(t)-\bar{\Phi}(t)\|_{\mathbb{E}}^{2}\leq\|\Psi_{0}(\omega)-\bar{\Phi}_{0}(\omega)\|^{2}e^{-\alpha
t}$ $\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+c_{4}e^{-\alpha
t}\int_{0}^{t}e^{\alpha
s}(\|\bar{u}(s)\|^{2}+\|\bar{v}(s)\|^{2}+\|f(\bar{u}(s))\|^{2})ds.$ (4.10)
Let us check that the family of balls centered on $\bar{\Phi}_{0}(\omega)$
with the random radius
$R(\omega):=\sqrt{1+c_{4}\int_{-\infty}^{0}e^{\alpha
s}(\|\bar{u}(s)(\omega)\|^{2}+\|\bar{v}(s)(\omega)\|^{2}+\|f(\bar{u}(s)(\omega))\|^{2})ds}$
(4.11)
is a pullback absorbing family for the random dynamical system generated by
system (1.1).
Due to the assumptions on $f$ and Lemma 4.1, the radius defined in (4.11) is
well defined. Now, by replacing $\omega$ by $\theta_{-t}\omega$ in (4), we get
$\displaystyle\|\Psi(\theta_{-t}\omega)-\bar{\Phi}(\theta_{-t}\omega)\|_{\mathbb{E}}^{2}$
(4.12) $\displaystyle\leq$
$\displaystyle\|\Psi_{0}(\theta_{-t}\omega)-\bar{\Phi}_{0}(\theta_{-t}\omega)\|_{\mathbb{E}}^{2}e^{-\alpha
t}$
$\displaystyle+c_{4}\int_{0}^{t}e^{\alpha(s-t)}(\|\bar{u}(s)(\theta_{-t}\omega)\|^{2}+\|\bar{v}(s)(\theta_{-t}\omega)\|^{2}+\|f(\bar{u}(s)(\theta_{-t}\omega))\|^{2})ds$
$\displaystyle=$
$\displaystyle\|\Psi_{0}(\theta_{-t}\omega)-\bar{\Phi}_{0}(\theta_{-t}\omega)\|_{\mathbb{E}}^{2}e^{-\alpha
t}$ $\displaystyle+c_{4}\int_{-t}^{0}e^{\alpha
s}(\|\bar{u}(s)(\omega)\|^{2}+\|\bar{v}(s)(\omega)\|^{2}+\|f(\bar{u}(s)(\omega))\|^{2})ds.$
The last term in (4.12) due to
$\bar{u}(s)(\theta_{-t}\omega)=\bar{u}_{0}(\theta_{s-t}\omega)=\bar{u}(s-t)(\omega)$
and
$\bar{v}(s)(\theta_{-t}\omega)=\bar{v}_{0}(\theta_{s-t}\omega)=\bar{v}(s-t)(\omega)$
which deduced from that $(\bar{u}(t))_{t\in\mathbb{R}}$ and
$(\bar{v}(t))_{t\in\mathbb{R}}$ are stationary processes. The conclusion now
follows as $t\rightarrow\infty$.
Because of the stationarity and Lemma 4.1, we have $e^{-\alpha
t}\|\bar{\Phi}_{0}(\theta_{-t}\omega)\|_{\mathbb{E}}^{2}=e^{-\alpha
t}\|\bar{\Phi}(-t)(\omega)\|_{\mathbb{E}}^{2}\rightarrow 0$ as
$t\rightarrow\infty$. Then we have the pullback absorption
$\|\Psi(\theta_{-t}\omega)\|_{\mathbb{E}}^{2}\leq\|\bar{\Phi}_{0}(\omega)\|_{\mathbb{E}}^{2}+R^{2}(\omega),\
\ \forall t\geq T_{\mathcal{D}(\omega)}.$ (4.13)
So, we have the stationary random process
$\tilde{\Phi}(t)(\omega):=\tilde{\Phi}_{0}(\theta_{t}\omega)$, which pathwise
attracts all other solutions in both forward and pullback senses, is a random
equilibrium. Now, we define a singleton sets
$\mathcal{A}=\\{A(\omega),\omega\in\Omega\\}=\\{\tilde{\Phi}_{0}(\omega)\\}$,
i.e. the singleton sets is formed by the random equilibrium. Here, we remain
to show that the singleton sets turns out to be a random attractor. According
to Definition 2.3, we can easily get the compactness, invariance and
attraction (implied by absorbtion). The proof is complete.
∎
## 5 Conclusions
We studied the stochastic FitzHugh-Nagumo equations driven by fractional
Brownian motion. The existence of the random attractor formed by the unique
random equilibrium turns out to be a single sets random attractor, which
differs from the results obtained in [10] and [24] where the same system is
driven by white noises. The methodology can be used to deal with other
stochastic lattice systems, which is a topic that will be the focus of further
research.
## Acknowledgements
The authors would like to express their sincere thanks to the anonymous
referees for their time and helpful comments and suggestions, which have
largely improved the presentation of this paper.
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|
arxiv-papers
| 2013-10-26T14:20:07 |
2024-09-04T02:49:52.910483
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Anhui Gu and Yangrong Li",
"submitter": "Anhui Gu Dr.",
"url": "https://arxiv.org/abs/1310.7113"
}
|
1310.7201
|
EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN)
CERN-PH-EP-2013-180 LHCb-PAPER-2013-054
Measurements of indirect $C\\!P$ asymmetries in $D^{0}\\!\rightarrow
K^{-}K^{+}$ and $D^{0}\\!\rightarrow\pi^{-}\pi^{+}$ decays
The LHCb collaboration†††Authors are listed on the following pages.
A study of indirect $C\\!P$ violation in $D^{0}$ mesons through the
determination of the parameter $A_{\Gamma}$ is presented using a data sample
of $pp$ collisions, corresponding to an integrated luminosity of
$1.0\mbox{\,fb}^{-1}$, collected with the LHCb detector and recorded at the
centre-of-mass energy of $7\mathrm{\,Te\kern-1.00006ptV}$ at the LHC. The
parameter $A_{\Gamma}$ is the asymmetry of the effective lifetimes measured in
decays of $D^{0}$ and $\kern 1.99997pt\overline{\kern-1.99997ptD}{}^{0}$
mesons to the $C\\!P$ eigenstates $K^{-}K^{+}$ and $\pi^{-}\pi^{+}$. Fits to
the data sample yield $A_{\Gamma}(KK)=(-0.35\pm 0.62\pm 0.12)\times 10^{-3}$
and $A_{\Gamma}(\pi\pi)=(0.33\pm 1.06\pm 0.14)\times 10^{-3}$, where the first
uncertainties are statistical and the second systematic. The results represent
the world’s best measurements of these quantities. They show no difference in
$A_{\Gamma}$ between the two final states and no indication of $C\\!P$
violation.
Accepted for publication in Phys. Rev. Lett.
© CERN on behalf of the LHCb collaboration, license CC-BY-3.0.
LHCb collaboration
R. Aaij40, B. Adeva36, M. Adinolfi45, C. Adrover6, A. Affolder51, Z.
Ajaltouni5, J. Albrecht9, F. Alessio37, M. Alexander50, S. Ali40, G.
Alkhazov29, P. Alvarez Cartelle36, A.A. Alves Jr24, S. Amato2, S. Amerio21, Y.
Amhis7, L. Anderlini17,f, J. Anderson39, R. Andreassen56, J.E. Andrews57, R.B.
Appleby53, O. Aquines Gutierrez10, F. Archilli18, A. Artamonov34, M. Artuso58,
E. Aslanides6, G. Auriemma24,m, M. Baalouch5, S. Bachmann11, J.J. Back47, A.
Badalov35, C. Baesso59, V. Balagura30, W. Baldini16, R.J. Barlow53, C.
Barschel37, S. Barsuk7, W. Barter46, Th. Bauer40, A. Bay38, J. Beddow50, F.
Bedeschi22, I. Bediaga1, S. Belogurov30, K. Belous34, I. Belyaev30, E. Ben-
Haim8, G. Bencivenni18, S. Benson49, J. Benton45, A. Berezhnoy31, R. Bernet39,
M.-O. Bettler46, M. van Beuzekom40, A. Bien11, S. Bifani44, T. Bird53, A.
Bizzeti17,h, P.M. Bjørnstad53, T. Blake37, F. Blanc38, J. Blouw10, S. Blusk58,
V. Bocci24, A. Bondar33, N. Bondar29, W. Bonivento15, S. Borghi53, A.
Borgia58, T.J.V. Bowcock51, E. Bowen39, C. Bozzi16, T. Brambach9, J. van den
Brand41, J. Bressieux38, D. Brett53, M. Britsch10, T. Britton58, N.H. Brook45,
H. Brown51, A. Bursche39, G. Busetto21,q, J. Buytaert37, S. Cadeddu15, O.
Callot7, M. Calvi20,j, M. Calvo Gomez35,n, A. Camboni35, P. Campana18,37, D.
Campora Perez37, A. Carbone14,c, G. Carboni23,k, R. Cardinale19,i, A.
Cardini15, H. Carranza-Mejia49, L. Carson52, K. Carvalho Akiba2, G. Casse51,
L. Castillo Garcia37, M. Cattaneo37, Ch. Cauet9, R. Cenci57, M. Charles54, Ph.
Charpentier37, S.-F. Cheung54, N. Chiapolini39, M. Chrzaszcz39,25, K. Ciba37,
X. Cid Vidal37, G. Ciezarek52, P.E.L. Clarke49, M. Clemencic37, H.V. Cliff46,
J. Closier37, C. Coca28, V. Coco40, J. Cogan6, E. Cogneras5, P. Collins37, A.
Comerma-Montells35, A. Contu15,37, A. Cook45, M. Coombes45, S. Coquereau8, G.
Corti37, B. Couturier37, G.A. Cowan49, D.C. Craik47, M. Cruz Torres59, S.
Cunliffe52, R. Currie49, C. D’Ambrosio37, P. David8, P.N.Y. David40, A.
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Zvyagin37.
1Centro Brasileiro de Pesquisas Físicas (CBPF), Rio de Janeiro, Brazil
2Universidade Federal do Rio de Janeiro (UFRJ), Rio de Janeiro, Brazil
3Center for High Energy Physics, Tsinghua University, Beijing, China
4LAPP, Université de Savoie, CNRS/IN2P3, Annecy-Le-Vieux, France
5Clermont Université, Université Blaise Pascal, CNRS/IN2P3, LPC, Clermont-
Ferrand, France
6CPPM, Aix-Marseille Université, CNRS/IN2P3, Marseille, France
7LAL, Université Paris-Sud, CNRS/IN2P3, Orsay, France
8LPNHE, Université Pierre et Marie Curie, Université Paris Diderot,
CNRS/IN2P3, Paris, France
9Fakultät Physik, Technische Universität Dortmund, Dortmund, Germany
10Max-Planck-Institut für Kernphysik (MPIK), Heidelberg, Germany
11Physikalisches Institut, Ruprecht-Karls-Universität Heidelberg, Heidelberg,
Germany
12School of Physics, University College Dublin, Dublin, Ireland
13Sezione INFN di Bari, Bari, Italy
14Sezione INFN di Bologna, Bologna, Italy
15Sezione INFN di Cagliari, Cagliari, Italy
16Sezione INFN di Ferrara, Ferrara, Italy
17Sezione INFN di Firenze, Firenze, Italy
18Laboratori Nazionali dell’INFN di Frascati, Frascati, Italy
19Sezione INFN di Genova, Genova, Italy
20Sezione INFN di Milano Bicocca, Milano, Italy
21Sezione INFN di Padova, Padova, Italy
22Sezione INFN di Pisa, Pisa, Italy
23Sezione INFN di Roma Tor Vergata, Roma, Italy
24Sezione INFN di Roma La Sapienza, Roma, Italy
25Henryk Niewodniczanski Institute of Nuclear Physics Polish Academy of
Sciences, Kraków, Poland
26AGH - University of Science and Technology, Faculty of Physics and Applied
Computer Science, Kraków, Poland
27National Center for Nuclear Research (NCBJ), Warsaw, Poland
28Horia Hulubei National Institute of Physics and Nuclear Engineering,
Bucharest-Magurele, Romania
29Petersburg Nuclear Physics Institute (PNPI), Gatchina, Russia
30Institute of Theoretical and Experimental Physics (ITEP), Moscow, Russia
31Institute of Nuclear Physics, Moscow State University (SINP MSU), Moscow,
Russia
32Institute for Nuclear Research of the Russian Academy of Sciences (INR RAN),
Moscow, Russia
33Budker Institute of Nuclear Physics (SB RAS) and Novosibirsk State
University, Novosibirsk, Russia
34Institute for High Energy Physics (IHEP), Protvino, Russia
35Universitat de Barcelona, Barcelona, Spain
36Universidad de Santiago de Compostela, Santiago de Compostela, Spain
37European Organization for Nuclear Research (CERN), Geneva, Switzerland
38Ecole Polytechnique Fédérale de Lausanne (EPFL), Lausanne, Switzerland
39Physik-Institut, Universität Zürich, Zürich, Switzerland
40Nikhef National Institute for Subatomic Physics, Amsterdam, The Netherlands
41Nikhef National Institute for Subatomic Physics and VU University Amsterdam,
Amsterdam, The Netherlands
42NSC Kharkiv Institute of Physics and Technology (NSC KIPT), Kharkiv, Ukraine
43Institute for Nuclear Research of the National Academy of Sciences (KINR),
Kyiv, Ukraine
44University of Birmingham, Birmingham, United Kingdom
45H.H. Wills Physics Laboratory, University of Bristol, Bristol, United
Kingdom
46Cavendish Laboratory, University of Cambridge, Cambridge, United Kingdom
47Department of Physics, University of Warwick, Coventry, United Kingdom
48STFC Rutherford Appleton Laboratory, Didcot, United Kingdom
49School of Physics and Astronomy, University of Edinburgh, Edinburgh, United
Kingdom
50School of Physics and Astronomy, University of Glasgow, Glasgow, United
Kingdom
51Oliver Lodge Laboratory, University of Liverpool, Liverpool, United Kingdom
52Imperial College London, London, United Kingdom
53School of Physics and Astronomy, University of Manchester, Manchester,
United Kingdom
54Department of Physics, University of Oxford, Oxford, United Kingdom
55Massachusetts Institute of Technology, Cambridge, MA, United States
56University of Cincinnati, Cincinnati, OH, United States
57University of Maryland, College Park, MD, United States
58Syracuse University, Syracuse, NY, United States
59Pontifícia Universidade Católica do Rio de Janeiro (PUC-Rio), Rio de
Janeiro, Brazil, associated to 2
60Institut für Physik, Universität Rostock, Rostock, Germany, associated to 11
61Celal Bayar University, Manisa, Turkey, associated to 37
aP.N. Lebedev Physical Institute, Russian Academy of Science (LPI RAS),
Moscow, Russia
bUniversità di Bari, Bari, Italy
cUniversità di Bologna, Bologna, Italy
dUniversità di Cagliari, Cagliari, Italy
eUniversità di Ferrara, Ferrara, Italy
fUniversità di Firenze, Firenze, Italy
gUniversità di Urbino, Urbino, Italy
hUniversità di Modena e Reggio Emilia, Modena, Italy
iUniversità di Genova, Genova, Italy
jUniversità di Milano Bicocca, Milano, Italy
kUniversità di Roma Tor Vergata, Roma, Italy
lUniversità di Roma La Sapienza, Roma, Italy
mUniversità della Basilicata, Potenza, Italy
nLIFAELS, La Salle, Universitat Ramon Llull, Barcelona, Spain
oHanoi University of Science, Hanoi, Viet Nam
pInstitute of Physics and Technology, Moscow, Russia
qUniversità di Padova, Padova, Italy
rUniversità di Pisa, Pisa, Italy
sScuola Normale Superiore, Pisa, Italy
The asymmetry under simultaneous charge and parity transformation ($C\\!P$
violation) has driven the understanding of electroweak interactions since its
discovery in the kaon system [1]. $C\\!P$ violation was subsequently
discovered in the $B^{0}$ and $B^{0}_{s}$ systems [2, 3, 4]. Charmed mesons
form the only neutral meson-antimeson system in which $C\\!P$ violation has
yet to be observed unambiguously. This system is the only one in which mesons
of up-type quarks participate in matter-antimatter transitions, a loop-level
process in the Standard Model (SM). This charm mixing process has recently
been observed for the first time unambiguously in single measurements [5, 6,
7]. The theoretical calculation of charm mixing and $C\\!P$ violation is
challenging for the charm quark [8, 9, 10, 11, 12]. Significant enhancement of
mixing or $C\\!P$ violation would be an indication of physics beyond the SM.
The mass eigenstates of the neutral charm meson system, $|D_{1,2}\rangle$,
with masses $m_{1,2}$ and decay widths $\Gamma_{1,2}$, can be expressed as
linear combinations of the flavour eigenstates, $|D^{0}\rangle$ and $|\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}\rangle$, as
$|D_{1,2}\rangle=p|D^{0}\rangle\pm{}q|\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}\rangle$ with complex coefficients
satisfying $|p|^{2}+|q|^{2}=1$. This allows the definition of the mixing
parameters $x\equiv 2(m_{2}-m_{1})/(\Gamma_{1}+\Gamma_{2})$ and
$y\equiv(\Gamma_{2}-\Gamma_{1})/(\Gamma_{1}+\Gamma_{2})$.
Non-conservation of $C\\!P$ symmetry enters as a deviation from unity of
$\lambda_{f}$, defined as
$\lambda_{f}\equiv\frac{q\bar{A}_{f}}{pA_{f}}=-\eta_{C\\!P}\left|\frac{q}{p}\right|\left|\frac{\bar{A}_{f}}{A_{f}}\right|e^{i\phi},$
(1)
where $A_{f}$ ($\bar{A}_{f}$) is the amplitude for a $D^{0}$ ($\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}$) meson decaying into a $C\\!P$
eigenstate $f$ with eigenvalue $\eta_{C\\!P}$, and $\phi$ is the
$C\\!P$-violating relative phase between $q/p$ and $\bar{A}_{f}/A_{f}$. Direct
$C\\!P$ violation occurs when the asymmetry
$A_{d}\equiv(|A_{f}|^{2}-|\bar{A}_{f}|^{2})/(|A_{f}|^{2}+|\bar{A}_{f}|^{2})$
is different from zero. Indirect $C\\!P$ violation comprises non-zero $C\\!P$
asymmetry in mixing, $A_{m}\equiv(|q/p|^{2}-|p/q|^{2})/(|q/p|^{2}+|p/q|^{2})$
and $C\\!P$ violation through a non-zero phase $\phi$. The phase convention of
$\phi$ is chosen such that, in the limit of no $C\\!P$ violation,
$C\\!P|D^{0}\rangle=-|\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}\rangle$. In this convention $C\\!P$
conservation leads to $\phi=0$ and $|D_{1}\rangle$ being $C\\!P$-odd.
The asymmetry of the inverse of effective lifetimes in decays of $D^{0}$
($\kern 1.99997pt\overline{\kern-1.99997ptD}{}^{0}$) mesons into $C\\!P$-even
final states, $\hat{\Gamma}$ ($\hat{\bar{\Gamma}}$), leads to the observable
$A_{\Gamma}$ defined as
$A_{\Gamma}\equiv\frac{\hat{\Gamma}-\hat{\bar{\Gamma}}}{\hat{\Gamma}+\hat{\bar{\Gamma}}}\approx\eta_{C\\!P}\left(\frac{A_{m}+A_{d}}{2}y\cos\phi-x\sin\phi\right).$
(2)
This makes $A_{\Gamma}$ a measurement of indirect $C\\!P$ violation, as the
contributions from direct $C\\!P$ violation are measured to be small [13]
compared to the precision on $A_{\Gamma}$ available so far [14]. Here,
effective lifetimes refer to lifetimes measured using a single-exponential
model in a specific decay mode. Currently available measurements of
$A_{\Gamma}$ [15, 16] are in agreement with no $C\\!P$ violation at the per
mille level [13].
This Letter reports measurements of $A_{\Gamma}$ in the $C\\!P$-even final
states $K^{-}K^{+}$ and $\pi^{-}\pi^{+}$ using $1.0\mbox{\,fb}^{-1}$ of $pp$
collisions at $7\mathrm{\,Te\kern-1.00006ptV}$ centre-of-mass energy at the
LHC recorded with the LHCb detector in 2011. In the SM, the phase $\phi$ is
final-state independent and thus measurements in the two final states are
expected to yield the same results. At the level of precision of the
measurements presented here, differences due to direct $C\\!P$ violation are
negligible. However, contributions to $\phi$ from physics beyond the SM may
lead to different results. Even small final-state differences in the phase,
$\Delta\phi$, can lead to measurable effects in the observables of the order
of $x\Delta\phi$, for sufficiently small phases $\phi$ in both final states
[17]. In addition, the measurements of $A_{\Gamma}$ in both final states are
important to quantify the contribution of indirect $C\\!P$ violation to the
observable $\Delta A_{C\\!P}$, which measures the difference in decay-time
integrated $C\\!P$ asymmetry of $D^{0}\\!\rightarrow K^{-}K^{+}$ to
$\pi^{-}\pi^{+}$ decays [18, 19].
The LHCb detector [20] is a single-arm forward spectrometer covering the
pseudorapidity range $2<\eta<5$, designed for the study of particles
containing $b$ or $c$ quarks. The spectrometer dipole magnet is operated in
either one of two polarities, the magnetic field vector points either up or
down. The trigger [21] consists of a hardware stage, based on information from
the calorimeter and muon systems, followed by a software stage, which performs
a full event reconstruction. The software trigger applies two sequential
selections. The first selection requires at least one track to have momentum
transverse to the beamline, $p_{\rm T}$, greater than
$1.7{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ and an impact parameter $\chi^{2}$,
$\chi^{2}_{\rm IP}$, greater than $16$. The $\chi^{2}_{\rm IP}$ is defined as
the difference in $\chi^{2}$ of a given primary interaction vertex
reconstructed with and without the considered track. This $\chi^{2}_{\rm IP}$
requirement introduces the largest effect on the observed decay-time
distribution compared to other selection criteria. In the second selection
this track is combined with a second track to form a candidate for a $D^{0}$
decay into two hadrons (charge conjugate states are included unless stated
otherwise). The second track must have $\mbox{$p_{\rm
T}$}>0.8{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ and $\chi^{2}_{\rm IP}>2$. The
decay vertex is required to have a flight distance $\chi^{2}$ per degree of
freedom greater than $25$ and the $D^{0}$ invariant mass, assuming kaons or
pions as final state particles, has to lie within
$50{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ (or within
$120{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ for a trigger whose rate is
scaled down by a factor of $10$) around
$1865{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$. The momentum vector of two-
body system is required to point back to the $pp$ interaction region.
The event selection applies a set of criteria that are closely aligned to
those applied at the trigger stage. The final-state particles have to match
particle identification criteria to separate kaons from pions [22] according
to their mass hypothesis and must not be identified as muons using combined
information from the tracking and particle identification systems.
Flavour tagging is performed through the measurement of the charge of the pion
in the decay $D^{*+}\\!\rightarrow D^{0}\pi^{+}$ (soft pion). Additional
criteria are applied to the track quality of the soft pion as well as to the
vertex quality of the $D^{*+}$ meson. Using a fit constraining the soft pion
to the $pp$ interaction vertex, the invariant mass difference of the $D^{*+}$
and $D^{0}$ candidates, $\Delta m$, is required to be less than
$152{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$.
About $10\,\%$ of the selected events have more than one candidate passing the
selections, mostly due to one $D^{0}$ candidate being associated with several
soft pions. One candidate per event is selected at random to reduce the
background from randomly associated soft pions. The $D^{0}$ decay-time range
is restricted to $0.25{\rm\,ps}$ to $10{\rm\,ps}$ such that there are
sufficient amounts of data in all decay-time regions included in the fit to
ensure its stability.
The whole dataset is split into four subsets, identified by the magnet
polarity, and two separate data-taking periods to account for known
differences in the detector alignment and calibration. The smallest subset
contains about $20\%$ of the total data sample. Results of the four subsets
are combined in a weighted average.
The selected events contain about $3.11\times 10^{6}$ $D^{0}\\!\rightarrow
K^{-}K^{+}$ and $1.03\times 10^{6}$ $D^{0}\\!\rightarrow\pi^{-}\pi^{+}$ signal
candidates, where the $D^{*+}$ meson is produced at the $pp$-interaction
vertex, with purities of $93.6\%$ and $91.2\%$, respectively, as measured in a
region of two standard deviations of the signal peaks in $D^{0}$ mass, $m(hh)$
(with $h=K,\pi$), and $\Delta m$.
Figure 1: Fit of $\Delta m$ for one of the eight subsets, containing the
$\kern 1.79997pt\overline{\kern-1.79997ptD}{}^{0}\\!\rightarrow K^{-}K^{+}$
candidates with magnet polarity down for the earlier run period.
The effective lifetimes are extracted by eight independent multivariate
unbinned maximum likelihood fits to the four subsamples, separated by the
$D^{0}$ flavour as determined by the charge of the soft pion. The fits are
carried out in two stages, a fit to $m(hh)$ and $\Delta m$ to extract the
signal yield and a fit to the decay time and $\ln(\chi^{2}_{\rm IP})$ of the
$D^{0}$ candidate to extract the effective lifetime. The first stage is used
to distinguish the following candidate classes: correctly tagged signal
candidates, which peak in both variables; correctly reconstructed $D^{0}$
candidates associated with a random soft pion (labelled “rnd.
$\pi_{\mathrm{s}}$” in figures), which peak in $m(hh)$ but follow a threshold
function in $\Delta m$; and combinatorial background. The threshold functions
are polynomials in $\sqrt{\Delta m-m_{\pi^{+}}}$. The signal peaks in $m(hh)$
and $\Delta m$ are described by the sum of three Gaussian functions. For the
$\pi^{-}\pi^{+}$ final state a power-law tail is added to the $m(hh)$
distribution to describe the radiative tail [23]. The combinatorial background
is described by an exponential function in $m(hh)$ and a threshold function in
$\Delta m$.
Partially reconstructed decays constitute additional background sources. The
channels that give significant contributions are the decays
$D^{0}\\!\rightarrow K^{-}\pi^{+}\pi^{0}$, with the charged pion reconstructed
as a kaon and the $\pi^{0}$ meson not reconstructed, and
$D^{+}_{s}\\!\rightarrow K^{-}K^{+}\pi^{+}$, with the pion not reconstructed.
The former peaks broadly in $\Delta m$ while the latter follows a threshold
function and both are described by an exponential in $m(hh)$. Reflections due
to incorrect mass assignment of the tracks are well separated in mass and are
suppressed by particle identification and are not taken into account. An
example fit projection is shown in Fig. 1.
Charm mesons originating from long-lived $b$ hadrons (secondary candidates)
form a large background that cannot be separated in the mass fit. They do not
come from the interaction point leading to a biased decay-time measurement.
The flight distance of the $b$ hadrons causes the $D^{0}$ candidates into
which they decay to have large $\chi^{2}_{\rm IP}$ on average. This is
therefore used as a separating variable.
Candidates for signal decays, where the $D^{*+}$ is produced at the
$pp$-interaction vertex, are modelled by an exponential function in decay
time, whose decay constant determines the effective lifetime, and by a
modified $\chi^{2}$ function in $\ln(\chi^{2}_{\rm IP})$ of the form
$f(x)\equiv\begin{cases}e^{\alpha x-e^{\alpha(x-\mu)}}&x\leq\mu\\\
e^{\alpha\mu+\beta(x-\mu)-e^{\beta(x-\mu)}}&x>\mu,\\\ \end{cases}$ (3)
where all parameters are allowed to have a linear variation with decay time.
The parameters $\alpha$ and $\beta$ describe the left and right width of the
distribution, respectively, and $\mu$ is the peak position. Secondary
candidates are described by the convolution of two exponential probability
density functions in decay time. Since there can be several sources of
secondary candidates, the sum of two such convolutions is used with one of the
decay constants shared, apart from the smaller $\pi^{-}\pi^{+}$ dataset where
a single convolution is sufficient to describe the data. The
$\ln(\chi^{2}_{\rm IP})$ distribution of secondary decays is also given by Eq.
3, however, the three parameters are replaced by functions of decay time
$\alpha(t)=A+B\,t+C\,\arctan(D\,t),$ (4)
and similarly for $\beta$ and $\mu$, where the parametrisations are motivated
by studies on highly enriched samples of secondary decays and where $A$, $B$,
$C$, and $D$ describe the decay-time dependence.
The background from correctly reconstructed $D^{0}$ mesons associated to a
random soft pion share the same $\ln(\chi^{2}_{\rm IP})$ shape as the signal.
Other combinatorial backgrounds and partially reconstructed decays for the
$K^{-}K^{+}$ final state are described by non-parametric distributions. The
shapes are obtained by applying an unfolding technique described in Ref. [24]
to the result of the $m(hh)$, $\Delta m$ fit. Gaussian kernel density
estimators are applied to create smooth distributions [25].
Figure 2: (Top) Fit of decay time to $\kern
1.79997pt\overline{\kern-1.79997ptD}{}^{0}\\!\rightarrow K^{-}K^{+}$ and
corresponding pull plot for candidates with magnet polarity down for the
earlier run period, where pull is defined as $({\rm data}-{\rm model})/{\rm
uncertainty}$, and (middle and bottom) ratio of $\kern
1.79997pt\overline{\kern-1.79997ptD}{}^{0}$ to $D^{0}$ data and fit model for
decays to $K^{-}K^{+}$ and $\pi^{-}\pi^{+}$ for all data, respectively.
The detector resolution is accounted for by the convolution of a Gaussian
function with the decay-time function. The Gaussian width is $50\rm\,fs$, an
effective value extracted from studies of
$B\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}X$ decays [26],
which has negligible effect on the measurement. Biases introduced by the
selection criteria are accounted for through per-candidate acceptance
functions which are determined in a data-driven way. The acceptance functions,
which take values of 1 for all decay-time intervals in which the candidate
would have been accepted and 0 otherwise, enter the fit in the normalisation
of the decay-time parametrisations. The procedure for determination and
application of these functions is described in detail in Refs. [15, 27].
Additional geometric detector acceptance effects are also included in the
procedure. An example decay-time fit projection is shown in Fig. 2. The fit
yields $A_{\Gamma}(KK)=(-0.35\pm 0.62)\times 10^{-3}$ and
$A_{\Gamma}(\pi\pi)=(0.33\pm 1.06)\times 10^{-3}$, with statistical
uncertainties only. The results of the four subsets are found to be in
agreement with each other.
Figure 3: Fits of $\ln(\chi^{2}_{\rm IP})$ for $\kern
1.79997pt\overline{\kern-1.79997ptD}{}^{0}\\!\rightarrow K^{-}K^{+}$
candidates for decay-time bins (left to right) $0.25-0.37{\rm\,ps}$,
$0.74-0.78{\rm\,ps}$, and $1.55-1.80{\rm\,ps}$.
The fit has regions where the model fails to describe the data accurately,
particularly at small decay times and intermediate values of
$\ln(\chi^{2}_{\rm IP})$ as shown in the pull plot in Fig. 2. The same
deviations are observed in pseudo-experiment studies, and are reproduced in
several independent parametrisations, indicating that the origin is related to
the non-parametric treatment of backgrounds in connection with non-ideal
parametrisations of the $\ln(\chi^{2}_{\rm IP})$ distributions. They do not
significantly affect the central value of $A_{\Gamma}$ due to the low
correlations between the effective lifetime and other fit parameters. The
deviations are very similar for fits to $D^{0}$ and $\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}$ samples leading to their
cancellations in the final asymmetry calculations as shown in Fig. 2.
In addition to the nominal procedure an alternative method is used, in which
the data are binned in equally-populated regions of the decay-time
distribution and the ratio of $\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}$ to $D^{0}$ yields calculated in
each bin. This avoids the need to model the decay-time acceptance. The time
dependence of this ratio, $R$, allows the calculation of $A_{\Gamma}$ from a
simple linear $\chi^{2}$ minimisation, with
$R(t)\approx\frac{N_{\kern
1.39998pt\overline{\kern-1.39998ptD}{}^{0}}}{N_{D^{0}}}\left(1+\frac{2A_{\Gamma}}{\tau_{KK}}t\right)\frac{1-e^{-\Delta
t/\tau_{\kern 0.99998pt\overline{\kern-0.99998ptD}{}^{0}}}}{1-e^{-\Delta
t/\tau_{D^{0}}}},$ (5)
where $\tau_{KK}=\tau_{K\pi}/(1+y_{C\\!P})$ is used as an external input based
on current world averages [28, 13], $N_{\kern
1.39998pt\overline{\kern-1.39998ptD}{}^{0}}/N_{D^{0}}$ is the signal yield
ratio integrated over all decay times and $\Delta{}t$ is the bin width. The
dependence on $\tau_{D^{0}}$ and $\tau_{\kern
1.39998pt\overline{\kern-1.39998ptD}{}^{0}}$ cancels in the extraction of
$A_{\Gamma}$. For this method the signal yields for decays, where the $D^{*+}$
is produced at the $pp$-interaction vertex, for each decay-time bin are
extracted by simultaneous unbinned maximum likelihood fits to $m(hh)$, $\Delta
m$, and $\ln(\chi^{2}_{\rm IP})$. Each bin is chosen to contain about $4\times
10^{4}$ candidates, leading to $118$ and $40$ bins for $K^{-}K^{+}$ and
$\pi^{-}\pi^{+}$, respectively. In general, the binned fit uses similar
parametrisations to the unbinned fit, though a few simplifications are
required to account for the smaller sample size per bin. The evolution of the
fit projections in $\ln(\chi^{2}_{\rm IP})$ with decay time is shown in Fig.
3.
The fits for both methods are verified by randomising the flavour tags and
checking that the results for $A_{\Gamma}$ are in agreement with zero.
Similarly, the measurement techniques for $A_{\Gamma}$ are applied to the
Cabibbo-favoured $K^{-}\pi^{+}$ final state for which they also yield results
in agreement with zero. The unbinned fit is further checked by comparing the
extracted lifetime using the $K^{-}\pi^{+}$ final state to the world average
$D^{0}$ lifetime, $(410.1\pm 1.5)\rm\,fs$ [28]. The result of $(412.88\pm
0.08)\rm\,fs$, where the uncertainty is only statistical, is found to be in
reasonable agreement. If the full difference to the world average were taken
as a relative systematic bias it would lead to an absolute bias of less than
$10^{-4}$ on $A_{\Gamma}$. Large numbers of pseudo-experiments, with both zero
and non-zero input values for $A_{\Gamma}$, are used to confirm the accuracy
of the results and their uncertainties. Finally, dependencies on $D^{0}$
kinematics and flight direction, the selection at the hardware trigger stage,
and the track and vertex multiplicity, are found to be negligible.
The binned fit yields $A_{\Gamma}(KK)=(0.50\pm 0.65)\times 10^{-3}$ and
$A_{\Gamma}(\pi\pi)=(0.85\pm 1.22)\times 10^{-3}$. Considering the statistical
variation between the two methods and the uncorrelated systematic
uncertainties the results from both methods yield consistent results.
The systematic uncertainties assessed are summarised in Table 1. The effect of
shortcomings in the description of the partially reconstructed background
component in the $K^{-}K^{+}$ final state is estimated by fixing the
respective distributions to those obtained in fits to simulated data. The
imperfect knowledge of the length scale of the vertex detector as well as
decay-time resolution effects are found to be negligible. Potential
inaccuracies in the description of combinatorial background and background
from signal candidates originating from $b$-hadron decays are assessed through
pseudo-experiments with varied background levels and varied generated
distributions while leaving the fit model unchanged. The impact of imperfect
treatment of background from $D^{0}$ candidates associated to random soft
pions is evaluated by testing several fit configurations with fewer
assumptions on the shape of this background.
Table 1: Systematic uncertainties, given as multiples of $10^{-3}$. The first column for each final state refers to the unbinned fit method and the second column to the binned fit method. Source | $A_{\Gamma}^{\rm unb}(KK)$ | $A_{\Gamma}^{\rm bin}(KK)$ | $A_{\Gamma}^{\rm unb}(\pi\pi)$ | $A_{\Gamma}^{\rm bin}(\pi\pi)$
---|---|---|---|---
Partially reconstructed backgrounds | $\pm 0.02$ | $\pm 0.09$ | $\pm 0.00$ | $\pm 0.00$
Charm from $b$ decays | $\pm 0.07$ | $\pm 0.55$ | $\pm 0.07$ | $\pm 0.53$
Other backgrounds | $\pm 0.02$ | $\pm 0.40$ | $\pm 0.04$ | $\pm 0.57$
Acceptance function | $\pm 0.09$ | — | $\pm 0.11$ | —
Magnet polarity | — | $\pm 0.58$ | — | $\pm 0.82$
Total syst. uncertainty | $\pm 0.12$ | $\pm 0.89$ | $\pm 0.14$ | $\pm 1.13$
The accuracy of the decay-time acceptance correction in the unbinned fit
method is assessed by testing the sensitivity to artificial biases applied to
the per-event acceptance functions. The overall systematic uncertainties of
the two final states for the unbinned method have a correlation of $0.31$.
A significant difference between results for the two magnet polarities is
observed in the binned method. As this cannot be guaranteed to cancel, a
systematic uncertainty is assigned. The unbinned method is not affected by
this as it is not sensitive to the overall normalisation of the $D^{0}$ and
$\kern 1.99997pt\overline{\kern-1.99997ptD}{}^{0}$ samples. In general the two
methods are subject to different sets of systematic effects due to the
different ways in which they extract the results. The systematic uncertainties
for the binned method are larger due to the fact that the fits are performed
independently in each decay-time bin. This can lead to instabilities in the
behaviour of particular fit components with time, an effect which is minimised
in the unbinned fit. The effects of such instabilities are determined by
running simulated pseudo-experiments.
The use of the external input for $\tau_{KK}$ in the binned fit method does
not yield a significant systematic uncertainty. A potential bias in this
method due to inaccurate parametrisations of other background is tested by
replacing the probability density functions by different models and a
corresponding systematic uncertainty is assigned.
In summary, the $C\\!P$-violating observable $A_{\Gamma}$ is measured using
the decays of neutral charm mesons into $K^{-}K^{+}$ and $\pi^{-}\pi^{+}$. The
results of $A_{\Gamma}(KK)=(-0.35\pm 0.62\pm 0.12)\times 10^{-3}$ and
$A_{\Gamma}(\pi\pi)=(0.33\pm 1.06\pm 0.14)\times 10^{-3}$, where the first
uncertainties are statistical and the second are systematic, represent the
world’s best measurements of these quantities. The result for the $K^{-}K^{+}$
final state is obtained based on an independent data set to the previous LHCb
measurement [15], with which it agrees well. The results show no significant
difference between the two final states and both results are in agreement with
zero, thus indicating the absence of indirect $C\\!P$ violation at this level
of precision.
## Acknowledgements
We express our gratitude to our colleagues in the CERN accelerator departments
for the excellent performance of the LHC. We thank the technical and
administrative staff at the LHCb institutes. We acknowledge support from CERN
and from the national agencies: CAPES, CNPq, FAPERJ and FINEP (Brazil); NSFC
(China); CNRS/IN2P3 and Region Auvergne (France); BMBF, DFG, HGF and MPG
(Germany); SFI (Ireland); INFN (Italy); FOM and NWO (The Netherlands); SCSR
(Poland); MEN/IFA (Romania); MinES, Rosatom, RFBR and NRC “Kurchatov
Institute” (Russia); MinECo, XuntaGal and GENCAT (Spain); SNSF and SER
(Switzerland); NAS Ukraine (Ukraine); STFC (United Kingdom); NSF (USA). We
also acknowledge the support received from the ERC under FP7. The Tier1
computing centres are supported by IN2P3 (France), KIT and BMBF (Germany),
INFN (Italy), NWO and SURF (The Netherlands), PIC (Spain), GridPP (United
Kingdom). We are thankful for the computing resources put at our disposal by
Yandex LLC (Russia), as well as to the communities behind the multiple open
source software packages on which we depend.
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|
arxiv-papers
| 2013-10-27T14:31:38 |
2024-09-04T02:49:52.922870
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "LHCb collaboration: R. Aaij, B. Adeva, M. Adinolfi, C. Adrover, A.\n Affolder, Z. Ajaltouni, J. Albrecht, F. Alessio, M. Alexander, S. Ali, G.\n Alkhazov, P. Alvarez Cartelle, A.A. Alves Jr, S. Amato, S. Amerio, Y. Amhis,\n L. Anderlini, J. Anderson, R. Andreassen, J.E. Andrews, R.B. Appleby, O.\n Aquines Gutierrez, F. Archilli, A. Artamonov, M. Artuso, E. Aslanides, G.\n Auriemma, M. Baalouch, S. Bachmann, J.J. Back, A. Badalov, C. Baesso, V.\n Balagura, W. Baldini, R.J. Barlow, C. Barschel, S. Barsuk, W. Barter, Th.\n Bauer, A. Bay, J. Beddow, F. Bedeschi, I. Bediaga, S. Belogurov, K. Belous,\n I. Belyaev, E. Ben-Haim, G. Bencivenni, S. Benson, J. Benton, A. Berezhnoy,\n R. Bernet, M.-O. Bettler, M. van Beuzekom, A. Bien, S. Bifani, T. Bird, A.\n Bizzeti, P.M. Bj{\\o}rnstad, T. Blake, F. Blanc, J. Blouw, S. Blusk, V. Bocci,\n A. Bondar, N. Bondar, W. Bonivento, S. Borghi, A. Borgia, T.J.V. Bowcock, E.\n Bowen, C. Bozzi, T. Brambach, J. van den Brand, J. Bressieux, D. Brett, M.\n Britsch, T. Britton, N.H. Brook, H. Brown, A. Bursche, G. Busetto, J.\n Buytaert, S. Cadeddu, O. Callot, M. Calvi, M. Calvo Gomez, A. Camboni, P.\n Campana, D. Campora Perez, A. Carbone, G. Carboni, R. Cardinale, A. Cardini,\n H. Carranza-Mejia, L. Carson, K. Carvalho Akiba, G. Casse, L. Castillo\n Garcia, M. Cattaneo, Ch. Cauet, R. Cenci, M. Charles, Ph. Charpentier, S.-F.\n Cheung, N. Chiapolini, M. Chrzaszcz, K. Ciba, X. Cid Vidal, G. Ciezarek,\n P.E.L. Clarke, M. Clemencic, H.V. Cliff, J. Closier, C. Coca, V. Coco, J.\n Cogan, E. Cogneras, P. Collins, A. Comerma-Montells, A. Contu, A. Cook, M.\n Coombes, S. Coquereau, G. Corti, B. Couturier, G.A. Cowan, D.C. Craik, M.\n Cruz Torres, S. Cunliffe, R. Currie, C. D'Ambrosio, P. David, P.N.Y. David,\n A. Davis, I. De Bonis, K. De Bruyn, S. De Capua, M. De Cian, J.M. De Miranda,\n L. De Paula, W. De Silva, P. De Simone, D. Decamp, M. Deckenhoff, L. Del\n Buono, N. D\\'el\\'eage, D. Derkach, O. Deschamps, F. Dettori, A. Di Canto, H.\n Dijkstra, M. Dogaru, S. Donleavy, F. Dordei, A. Dosil Su\\'arez, D. Dossett,\n A. Dovbnya, F. Dupertuis, P. Durante, R. Dzhelyadin, A. Dziurda, A. Dzyuba,\n S. Easo, U. Egede, V. Egorychev, S. Eidelman, D. van Eijk, S. Eisenhardt, U.\n Eitschberger, R. Ekelhof, L. Eklund, I. El Rifai, Ch. Elsasser, A. Falabella,\n C. F\\\"arber, C. Farinelli, S. Farry, D. Ferguson, V. Fernandez Albor, F.\n Ferreira Rodrigues, M. Ferro-Luzzi, S. Filippov, M. Fiore, C. Fitzpatrick, M.\n Fontana, F. Fontanelli, R. Forty, O. Francisco, M. Frank, C. Frei, M.\n Frosini, E. Furfaro, A. Gallas Torreira, D. Galli, M. Gandelman, P. Gandini,\n Y. Gao, J. Garofoli, P. Garosi, J. Garra Tico, L. Garrido, C. Gaspar, R.\n Gauld, E. Gersabeck, M. Gersabeck, T. Gershon, Ph. Ghez, V. Gibson, L.\n Giubega, V.V. Gligorov, C. G\\\"obel, D. Golubkov, A. Golutvin, A. Gomes, P.\n Gorbounov, H. Gordon, M. Grabalosa G\\'andara, R. Graciani Diaz, L.A. Granado\n Cardoso, E. Graug\\'es, G. Graziani, A. Grecu, E. Greening, S. Gregson, P.\n Griffith, L. Grillo, O. Gr\\\"unberg, B. Gui, E. Gushchin, Yu. Guz, T. Gys, C.\n Hadjivasiliou, G. Haefeli, C. Haen, S.C. Haines, S. Hall, B. Hamilton, T.\n Hampson, S. Hansmann-Menzemer, N. Harnew, S.T. Harnew, J. Harrison, T.\n Hartmann, J. He, T. Head, V. Heijne, K. Hennessy, P. Henrard, J.A. Hernando\n Morata, E. van Herwijnen, M. He\\ss, A. Hicheur, E. Hicks, D. Hill, M.\n Hoballah, C. Hombach, W. Hulsbergen, P. Hunt, T. Huse, N. Hussain, D.\n Hutchcroft, D. Hynds, V. Iakovenko, M. Idzik, P. Ilten, R. Jacobsson, A.\n Jaeger, E. Jans, P. Jaton, A. Jawahery, F. Jing, M. John, D. Johnson, C.R.\n Jones, C. Joram, B. Jost, M. Kaballo, S. Kandybei, W. Kanso, M. Karacson,\n T.M. Karbach, I.R. Kenyon, T. Ketel, B. Khanji, O. Kochebina, I. Komarov,\n R.F. Koopman, P. Koppenburg, M. Korolev, A. Kozlinskiy, L. Kravchuk, K.\n Kreplin, M. Kreps, G. Krocker, P. Krokovny, F. Kruse, M. Kucharczyk, V.\n Kudryavtsev, K. Kurek, T. Kvaratskheliya, V.N. La Thi, D. Lacarrere, G.\n Lafferty, A. Lai, D. Lambert, R.W. Lambert, E. Lanciotti, G. Lanfranchi, C.\n Langenbruch, T. Latham, C. Lazzeroni, R. Le Gac, J. van Leerdam, J.-P. Lees,\n R. Lef\\`evre, A. Leflat, J. Lefran\\c{c}ois, S. Leo, O. Leroy, T. Lesiak, B.\n Leverington, Y. Li, L. Li Gioi, M. Liles, R. Lindner, C. Linn, B. Liu, G.\n Liu, S. Lohn, I. Longstaff, J.H. Lopes, N. Lopez-March, H. Lu, D. Lucchesi,\n J. Luisier, H. Luo, O. Lupton, F. Machefert, I.V. Machikhiliyan, F. Maciuc,\n O. Maev, S. Malde, G. Manca, G. Mancinelli, J. Maratas, U. Marconi, P.\n Marino, R. M\\\"arki, J. Marks, G. Martellotti, A. Martens, A. Mart\\'in\n S\\'anchez, M. Martinelli, D. Martinez Santos, D. Martins Tostes, A. Martynov,\n A. Massafferri, R. Matev, Z. Mathe, C. Matteuzzi, E. Maurice, A. Mazurov, J.\n McCarthy, A. McNab, R. McNulty, B. McSkelly, B. Meadows, F. Meier, M.\n Meissner, M. Merk, D.A. Milanes, M.-N. Minard, J. Molina Rodriguez, S.\n Monteil, D. Moran, P. Morawski, A. Mord\\`a, M.J. Morello, R. Mountain, I.\n Mous, F. Muheim, K. M\\\"uller, R. Muresan, B. Muryn, B. Muster, P. Naik, T.\n Nakada, R. Nandakumar, I. Nasteva, M. Needham, S. Neubert, N. Neufeld, A.D.\n Nguyen, T.D. Nguyen, C. Nguyen-Mau, M. Nicol, V. Niess, R. Niet, N. Nikitin,\n T. Nikodem, A. Nomerotski, A. Novoselov, A. Oblakowska-Mucha, V. Obraztsov,\n S. Oggero, S. Ogilvy, O. Okhrimenko, R. Oldeman, M. Orlandea, J.M. Otalora\n Goicochea, P. Owen, A. Oyanguren, B.K. Pal, A. Palano, M. Palutan, J. Panman,\n A. Papanestis, M. Pappagallo, C. Parkes, C.J. Parkinson, G. Passaleva, G.D.\n Patel, M. Patel, G.N. Patrick, C. Patrignani, C. Pavel-Nicorescu, A. Pazos\n Alvarez, A. Pearce, A. Pellegrino, G. Penso, M. Pepe Altarelli, S. Perazzini,\n E. Perez Trigo, A. P\\'erez-Calero Yzquierdo, P. Perret, M. Perrin-Terrin, L.\n Pescatore, E. Pesen, G. Pessina, K. Petridis, A. Petrolini, A. Phan, E.\n Picatoste Olloqui, B. Pietrzyk, T. Pila\\v{r}, D. Pinci, S. Playfer, M. Plo\n Casasus, F. Polci, G. Polok, A. Poluektov, E. Polycarpo, A. Popov, D. Popov,\n B. Popovici, C. Potterat, A. Powell, J. Prisciandaro, A. Pritchard, C.\n Prouve, V. Pugatch, A. Puig Navarro, G. Punzi, W. Qian, B. Rachwal, J.H.\n Rademacker, B. Rakotomiaramanana, M.S. Rangel, I. Raniuk, N. Rauschmayr, G.\n Raven, S. Redford, S. Reichert, M.M. Reid, A.C. dos Reis, S. Ricciardi, A.\n Richards, K. Rinnert, V. Rives Molina, D.A. Roa Romero, P. Robbe, D.A.\n Roberts, A.B. Rodrigues, E. Rodrigues, P. Rodriguez Perez, S. Roiser, V.\n Romanovsky, A. Romero Vidal, M. Rotondo, J. Rouvinet, T. Ruf, F. Ruffini, H.\n Ruiz, P. Ruiz Valls, G. Sabatino, J.J. Saborido Silva, N. Sagidova, P. Sail,\n B. Saitta, V. Salustino Guimaraes, B. Sanmartin Sedes, R. Santacesaria, C.\n Santamarina Rios, E. Santovetti, M. Sapunov, A. Sarti, C. Satriano, A. Satta,\n M. Savrie, D. Savrina, M. Schiller, H. Schindler, M. Schlupp, M. Schmelling,\n B. Schmidt, O. Schneider, A. Schopper, M.-H. Schune, R. Schwemmer, B.\n Sciascia, A. Sciubba, M. Seco, A. Semennikov, K. Senderowska, I. Sepp, N.\n Serra, J. Serrano, P. Seyfert, M. Shapkin, I. Shapoval, Y. Shcheglov, T.\n Shears, L. Shekhtman, O. Shevchenko, V. Shevchenko, A. Shires, R. Silva\n Coutinho, M. Sirendi, N. Skidmore, T. Skwarnicki, N.A. Smith, E. Smith, E.\n Smith, J. Smith, M. Smith, M.D. Sokoloff, F.J.P. Soler, F. Soomro, D. Souza,\n B. Souza De Paula, B. Spaan, A. Sparkes, P. Spradlin, F. Stagni, S. Stahl, O.\n Steinkamp, S. Stevenson, S. Stoica, S. Stone, B. Storaci, M. Straticiuc, U.\n Straumann, V.K. Subbiah, L. Sun, W. Sutcliffe, S. Swientek, V. Syropoulos, M.\n Szczekowski, P. Szczypka, D. Szilard, T. Szumlak, S. T'Jampens, M. Teklishyn,\n E. Teodorescu, F. Teubert, C. Thomas, E. Thomas, J. van Tilburg, V.\n Tisserand, M. Tobin, S. Tolk, D. Tonelli, S. Topp-Joergensen, N. Torr, E.\n Tournefier, S. Tourneur, M.T. Tran, M. Tresch, A. Tsaregorodtsev, P.\n Tsopelas, N. Tuning, M. Ubeda Garcia, A. Ukleja, A. Ustyuzhanin, U. Uwer, V.\n Vagnoni, G. Valenti, A. Vallier, R. Vazquez Gomez, P. Vazquez Regueiro, C.\n V\\'azquez Sierra, S. Vecchi, J.J. Velthuis, M. Veltri, G. Veneziano, M.\n Vesterinen, B. Viaud, D. Vieira, X. Vilasis-Cardona, A. Vollhardt, D.\n Volyanskyy, D. Voong, A. Vorobyev, V. Vorobyev, C. Vo\\ss, H. Voss, R. Waldi,\n C. Wallace, R. Wallace, S. Wandernoth, J. Wang, D.R. Ward, N.K. Watson, A.D.\n Webber, D. Websdale, M. Whitehead, J. Wicht, J. Wiechczynski, D. Wiedner, L.\n Wiggers, G. Wilkinson, M.P. Williams, M. Williams, F.F. Wilson, J. Wimberley,\n J. Wishahi, W. Wislicki, M. Witek, G. Wormser, S.A. Wotton, S. Wright, S. Wu,\n K. Wyllie, Y. Xie, Z. Xing, Z. Yang, X. Yuan, O. Yushchenko, M. Zangoli, M.\n Zavertyaev, F. Zhang, L. Zhang, W.C. Zhang, Y. Zhang, A. Zhelezov, A.\n Zhokhov, L. Zhong, A. Zvyagin",
"submitter": "Marco Gersabeck",
"url": "https://arxiv.org/abs/1310.7201"
}
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