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1211.2921
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1201220129999999
11institutetext: Astronomical Institute, University of Wrocław, Kopernika 11,
51-622 Wrocław, Poland
November 2012
# Variability survey in the young open cluster IC 1805
D. Moździerski Corresponding author:
[email protected]
(October 2012; October 2012)
###### Abstract
We present preliminary results of the photometric variability survey in the
very young open cluster IC 1805. We found more than 70 variable stars in the
field, including pulsating stars and a large sample of most likely pre-main
sequence stars.
###### keywords:
The Galaxy: open clusters and associations: IC 1805 - Stars: oscillations -
Stars: pre-main sequence - Stars: variables: general
## 1 Introduction
The variability survey in the young open cluster IC 1805 is a part of the
ongoing program of searching for early-type variable stars in open clusters of
the northern sky conducted in Wrocław (see Jerzykiewicz et al. 2011 and
references therein). We are particularly interested in pulsating stars such as
$\beta$ Cephei and SPB. These stars are very promising targets for
asteroseismology. In particular, open clusters rich in these variables can be
used for so called ensemble asteroseismology (Saesen et al. 2010).
The open cluster IC 1805 is located in the Perseus spiral arm of our Galaxy in
the centre of Cas OB6 association and the molecular cloud W4. The surrounding
nebula, named also IC 1805, has a nickname Heart Nebula (Fig. 1). The cluster
is very young; its age is estimated for only several Myr and distance, for
about 2 – 2.4 kpc (Ishida 1968; Joshi & Sagar 1983; Guetter & Vrba 1989;
Massey, Johnson & DeGioia-Eastwood 1995; Sung & Lee 1995). It is also heavily
reddened and the reddening is not uniform across the cluster. There was no
thorough variability survey in the cluster up to date. The NSV catalog (Samus
et al. 2010) lists only eight suspected variables falling into the field of
view of our observations.
Figure 1: DSS image of the Heart Nebula IC 1805 and its central open cluster.
The observed field is marked by the cyan square.
## 2 Observations and reductions
All observations were carried out in Białków Observatory (University of
Wrocław) between 2007 and 2010. We used a 60-cm Cassegrain telescope equipped
with the Andor Tech. DW 432-BV CCD camera covering a 13${}^{\prime}\times$ 12′
field of view. We have acquired almost ten thousands CCD frames during 44
observing nights. They were taken through three filters, $B$, $V$ and
$I_{\rm{C}}$. The exposure times ranged from 10 to 100 s. The observed field
of IC 1805 is shown in Fig. 1.
The observations were calibrated in a standard way which included bias and
dark subtraction and correction for inhomogeneous sensitivity using flat-field
frames. The aperture and profile magnitudes calculated by means of the DAOPHOT
II package (Stetson 1987) were used to derive differential magnitudes that
were subsequently used in the variability search.
The instrumental $BVI_{\rm C}$ photometry for all 1511 detected stars was
transformed to the standard system using photometry published by Sung & Lee
(1995). The resulting colour-magnitude diagram is shown in Fig. 2. As can be
seen, the main sequence of the cluster is slightly smeared due to variable
reddening.
Figure 2: $V$ vs. $(V-I_{\rm{C}})$ colour-magnitude diagram for stars in the
observed field. All detected variables are divided into five groups that are
shown with different symbols and labeled.
## 3 Analysis and results
The search for variable stars was based on the analysis of differential
magnitudes by means of a Fourier amplitude spectrum and eye inspection of
light curves and calculation of phase diagrams. The search has been done using
the most numerous $I_{\rm C}$-filter observations. The periodograms were
calculated in the range between 0 and 80 d-1. In the first step, the
identification of variable stars was made automatically. The light curves and
Fourier spectra of variable candidates were then inspected by eye. In this
way, some non-periodic variables were also identified. For multiperiodic
stars, we also carried out a prewhitening procedure in order to derive
frequencies and amplitudes of all significant signals.
The search resulted in the detection of 71 variables. Only four of them were
suspected to be variable before our study, the remaining 67 are new
discoveries. We found eight pulsating stars in the observed field: two $\beta$
Cephei and six $\delta$ Scuti type stars. The two $\beta$ Cephei stars and one
$\delta$ Scuti star are likely members of IC 1805. If this is the case, the
$\delta$ Scuti star might be still in the pre-main sequence phase of
evolution. The periodogram of one (low-amplitude) $\beta$ Cephei star is shown
in Fig. 3. In addition, we found 8 stars showing variability caused by
eclipses or ellipsoidal effects, 24 irregular and 31 monoperiodic variables.
The monoperiodic variables occupy mostly the region where pre-main sequence
members of IC 1805 are expected (Fig. 2). Their variablity is probably related
to rotation. The light curves of some of the irregular variables resemble
those of UX Orionis stars in which the variability is attributed to the
changes in the circumstellar environment. An example light curve for one of IC
1805 stars is shown in Fig. 4.
Figure 3: Top: Fourier periodogram of one of the two $\beta$ Cephei variables
we found. Bottom: the same, after prewhitening with the dominant frequency.
Figure 4: The $I_{\rm C}$-filter light curve of one of UX Ori-type candidates
in IC 1805.
###### Acknowledgements.
We are indebted to Prof. A. Pigulski and Dr. Z. Kołaczkowski for their help
and discussions. We want to thank D. Drobek, M. Jurecki, G. Kopacki, G.
Michalska, P. Nowak, and M. Stȩślicki for making some observations of the
cluster. The work was supported by the NCN grant 2011/03/B/ST9/02667.
## References
* [1] Guetter, H.H., Vraba, F.J.: 1989, AJ 98, 611
* [2] Ishida K.: 1968, MNRAS 144, 55
* [3] Jerzykiewicz, M., Kopacki, G., Pigulski, A., Kołaczkowski, Z., Kim, S.-L.: 2011, Acta Astron. 61, 247
* [4] Joshi, U.C., Sagar, M.: 1983, JRASC 77, 40
* [5] Massey, P., Johnson, K.E., DeGioia-Eastwood, K.: 1995, ApJ 454, 151
* [6] Saesen, S., Pigulski, A., Carrier, F., et al.: 2010, AN 331, 1080
* [7] Samus, N. N., Goranskii, V. P., Durlevich, O. V., et al.: 2010, VizieR Online Data Catalog
* [8] Stetson, P.B.: 1987, PASP 99, 191
* [9] Sung, H., Lee, S.-W.: 1995, JKAS 28, 119
|
arxiv-papers
| 2012-11-13T09:05:59 |
2024-09-04T02:49:37.913882
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "D. Mo\\'zdzierski",
"submitter": "Dawid Mo\\'zdzierski",
"url": "https://arxiv.org/abs/1211.2921"
}
|
1211.2938
|
# Thermal Rectification in the Nonequilibrium Quantum-Dot-System
Tian Chen 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
Quantum thermal transport in two-quantum-dot system with Dzyaloshinskii-Moriya
interaction (DM interaction) has been studied. The sign of thermal
rectification can be controlled through changing the energy splitting or the
DM interaction strength. The anisotropic term in the system can also affect
the sign of rectification. Compared with other proposals [L Zhang et al, Phys.
Rev. B 80, 172301 (2009)], our model can offer larger rectification efficiency
and show the potential application in designing the polarity-controllable
thermal diode with a small size system (N=2). Moreover, quantum correlations
of two-quantum-dots are investigated. We find that almost perfect quantum
correlations can be obtained in the large temperature bias region, and quantum
entanglement is more sensitive to the change of the DM interaction strength
than quantum discord.
## I Introduction
Thermal diode is a very important thermal devicesA. Nitzan ; B. Li ; N. B. Li
. It has extensive application in phonon informationL. Wang . The heat current
flow can be induced in a unidirectional way, and can be regarded as the basic
part in controlling the large-scale heat flow. So far, many proposals have
been raised to design a thermal diode by different models, such as Frenkel-
Kontorova lattice and spin chainsB. Li ; Y. Yan ; L. Zhang . In all these
designs, the different lattice sizes will influence the heat current flow
significantly between the two reservoirs. In the spin chain model, the
polarity of the thermal diode can be controlled by modulating the spin energy
gap at different sites or changing the anisotropic interaction strength
between the adjacent spins. The efficiency of the thermal diode model can be
measured by thermal rectification ($\mathcal{R}$). In the six-spin-chain
system, the thermal rectification $\mathcal{R}$ can reach the value from -0.5
to 0.3 by adjusting the system variables, and the different signs of
$\mathcal{R}$ means the change of the thermal diode model polarity. However,
as studied in Ref. Y. Yan , it is not likely to control the thermal diode
model polarity in the small system with the spin number $N\leq 4$.
Recently, the problems of the interaction between the quantum-dot system and
the thermal environment have attracted a lot of attentionR. Hanson ; F.
Kheirandisha ; H. Mohammadia ; L. A. Wu_PRA03 , for the quantum-dot system
display its advantage in its scalability and long coherence timeR. Hanson .
Due to the quantum tunnel effect in the quantum-dot system, the spin-spin
interaction and the spin-orbit interaction will play an important roleF.
Kheirandisha ; S. J. Akhtarshenas . Moreover, when the material possesses the
reversal asymmetry, the coexistence of the superexchange interaction and the
spin-orbit interaction will induce a new interaction so called Dzyaloshinskii-
Moriya (DM) interactionT. Mo ; T. Moriya ; S. Chutia ; A. Pfund ; S. Herzog .
It has been verifiedT. Mo ; T. Moriya that DM interaction can result in the
change of the spin alignment in the material, and many problems of the
quantum-dot system containing the DM interaction have been investigated
alreadyF. Kheirandisha ; H. Mohammadia ; S. J. Akhtarshenas .
In this paper, we study extensively the heat transport problem in the two-
quantum-dot system containing the DM interaction. We calculate the heat
current in the steady state limit and analyze the heat current contributions
of the different system states. We propose a polarity-controllable thermal
diode in this two-quantum-dot system. Thermal rectification of this diode
model is studied. We find that, in this two-quantum-dot system, thermal
rectification $\mathcal{R}$ can reach the value from -0.3 to 0.35, through
changing the DM interaction strength. Compared with the six-spin-chain
proposal, thermal rectification $\mathcal{R}$ in our proposal is larger than
several cases in the spin-chain proposal. Based on this two-quantum-dot
system, we can realize the reversal of thermal rectification in small system
case. Besides, the quantum correlations are calculated for the steady state.
We find that, compared with the quantum discord, the quantum entanglement is
more sensitive to the DM interaction. Perfect quantum entanglement and quantum
discord can appear in the large temperature bias region, not only in the two
zero reservoirsF. Kheirandisha .
## II Thermal rectification
### II.1 Model
Consider a two-quantum-dot system and two thermal reservoirs as shown in Fig.
1. The left quantum dot interacts with the left reservoir only, and the right
quantum dot interacts with the right reservoir only. The total Hamiltonian
isF. Kheirandisha ; H. Mohammadia ,
$\hat{H}=\hat{H}_{s}+\hat{H}_{BL}+\hat{H}_{BR}+\hat{H}_{SBL}+\hat{H}_{SBR}$.
Here, $H_{s}$ is the two-quantum-dot system Hamiltonian, $H_{B\nu}$ is the
reservoir Hamiltonian, and $H_{SB\nu}$ is the interaction between the
reservoir $\nu$ and the system. We consider the anisotropic interaction and
the DM interaction between the two-quantum-dot system, and neglect the second
order spin-orbit interaction. The system Hamiltonian isF. Kheirandisha ,
Figure 1: (color online): A schematic representation of the two-quantum-dot
system containing DM interaction. The two quantum dots interact with the
thermal reservoirs at different temperatures $T_{L}$ and $T_{R}$,
respectively.
$\begin{split}H_{s}=&J\chi(\sigma_{L}^{+}\sigma_{R}^{+}+\sigma_{L}^{-}\sigma_{R}^{-})+J(1+iD)\sigma_{L}^{+}\sigma_{R}^{-}\\\
&+J(1-iD)\sigma_{L}^{-}\sigma_{R}^{+}+\frac{B+b}{2}\sigma_{L}^{z}+\frac{B-b}{2}\sigma_{R}^{z}\\\
\end{split}$ (1)
where, the parameter $\chi$ describes the anisotropy term in the system spin
xy plane, the parameter D stands for the DM interaction between the two-
quantum-dot system, and the energy gaps of the left spin and the right spin
are $\frac{B+b}{2}$, $\frac{B-b}{2}$ respectively.
In our case, the reservoir consists of a collection of non-interacting
oscillators. The parameter $\omega_{n}$ labels different oscillator frequency.
The system-reservoir interaction strength $g_{n}^{\nu}$ denotes the
interaction between the spin $\nu$ and the oscillator of frequency
$\omega_{n}$ in reservoir $\nu$. The reservoir Hamiltonian $H_{B\nu}$ and the
system-reservoir interaction Hamiltonian $H_{SB_{\nu}}$ can be written asF.
Kheirandisha ,
$\begin{split}H_{B\nu}&=\sum_{n}\omega_{n}b_{n\nu}^{+}b_{n\nu}\quad\\\
H_{SB\nu}&=S_{\nu}B_{\nu}=(\sigma_{\nu}^{+}+\sigma_{\nu}^{-})(\sum_{n}g_{n}^{(\nu)}b_{n\nu}+g_{n}^{(\nu)*}b_{n\nu}^{+})\end{split}$
(2)
We want to investigate the system properties in the steady state limit. The
eigenvalues and the corresponding eigenvectors of the system Hamiltonian areF.
Kheirandisha ,
$\begin{split}|\varepsilon_{1}\rangle&=N^{+}((\frac{b+\xi}{J(1-iD)})|01\rangle+|10\rangle),E_{1}=+\xi\\\
|\varepsilon_{2}\rangle&=N^{-}((\frac{b-\xi}{J(1-iD)})|01\rangle+|10\rangle),E_{2}=-\xi\\\
|\varepsilon_{3}\rangle&=M^{+}((\frac{B+\eta}{J\chi})|00\rangle+|11\rangle),E_{3}=+\eta\\\
|\varepsilon_{4}\rangle&=M^{-}((\frac{B-\eta}{J\chi})|00\rangle+|11\rangle),E_{4}=-\eta\\\
\end{split}$ (3)
Here, the state $|0\rangle$ and $|1\rangle$ are the spin of upper level energy
and spin of lower level energy, respectively. The coefficients $N^{\pm}$,
$M^{\pm}$, $\xi$, and $\eta$ satisfy the relation
$N^{\pm}=(1+\frac{(b\pm\xi)}{J^{2}+(JD)^{2}})^{-1/2}$, and
$M^{\pm}=(1+(\frac{B\pm\eta}{J\chi})^{2})^{-1/2}$,
$\xi=(b^{2}+J^{2}+(JD)^{2})^{1/2}$, $\eta=(B^{2}+(J\chi)^{2})^{1/2}$. We now
rewrite the system components in the system-reservoir interaction Hamiltonian
$H_{SB\nu}$ in the new basis
$\\{|\varepsilon_{1}\rangle,|\varepsilon_{2}\rangle,|\varepsilon_{3}\rangle,|\varepsilon_{4}\rangle\\}$,
$\begin{split}\sigma_{L}^{x}=&S_{3,1}^{L}|\varepsilon_{3}\rangle\langle\varepsilon_{1}|+S_{4,1}^{L}|\varepsilon_{4}\rangle\langle\varepsilon_{1}|\\\
&+S_{3,2}^{L}|\varepsilon_{3}\rangle\langle\varepsilon_{2}|+S_{4,2}^{L}|\varepsilon_{4}\rangle\langle\varepsilon_{2}|+H.C.\\\
\sigma_{R}^{x}=&S_{3,1}^{R}|\varepsilon_{3}\rangle\langle\varepsilon_{1}|+S_{4,1}^{R}|\varepsilon_{4}\rangle\langle\varepsilon_{1}|\\\
&+S_{3,2}^{R}|\varepsilon_{3}\rangle\langle\varepsilon_{2}|+S_{4,2}^{R}|\varepsilon_{4}\rangle\langle\varepsilon_{2}|+H.C.\\\
\end{split}$ (4)
The parameters $|S_{m,n}^{\nu}|$ is defined as,
$|S_{3,1}^{L}|^{2}=|S_{4,2}^{L}|^{2}=\frac{\eta\xi-
Bb}{2\eta\xi}+\frac{J^{2}\chi}{2\eta\xi},|S_{4,1}^{L}|^{2}=|S_{3,2}^{L}|^{2}=\frac{\eta\xi+Bb}{2\eta\xi}-\frac{J^{2}\chi}{2\eta\xi},|S_{3,1}^{R}|^{2}=|S_{4,2}^{R}|^{2}=\frac{\eta\xi+Bb}{2\eta\xi}+\frac{J^{2}\chi}{2\eta\xi},\quad|S_{4,1}^{R}|^{2}=|S_{3,2}^{R}|^{2}=\frac{\eta\xi-
Bb}{2\eta\xi}-\frac{J^{2}\chi}{2\eta\xi}$.
We apply the master equation method to solve the system dynamics. We obtain
the system time evolution function by the second order approximation in the
interaction pictureL. A. Wu ; L. A ,
$\begin{split}\frac{d\rho_{m,n}}{dt}=&-i[H_{SBj}(t),\rho(0)]_{m,n}\\\
&-\int_{0}^{t}d\tau[H_{SBj}(t),[H_{SBj}(\tau),\rho(\tau)]]_{m,n}\end{split}$
(5)
In our calculation, we assume the weak interaction of the system and the
reservoirs. The Markov approximation is applied, and the reservoirs keep in
its thermal equilibrium throughout the evolution. Our goal is to see the
system steady state behavior in the long time limit, we can neglect the
coherence term in the system. Under these conditions mentioned above, we can
get the Pauli master equation related to the system state population
$\rho_{nn}$ (n=1, 2, 3, 4),
$\dot{\rho_{nn}}^{s}=-\sum_{m}W_{nm}\rho_{nn}^{s}+\sum_{m}W_{mn}\rho_{mm}^{s}\\\
$ (6)
Where $W_{nm}$ is $W_{nm}=k_{n\rightarrow
m}^{L}|S_{m,n}^{L}|^{2}+k_{n\rightarrow m}^{R}|S_{m,n}^{R}|^{2}$, and the
rates $k_{n\rightarrow m}$ are,
$\begin{split}k_{n\rightarrow
m}^{\nu}&=\Gamma_{B,\nu}(\omega_{nm})(n_{B}^{\nu}(\omega_{nm})+1),\quad n>m\\\
k_{n\rightarrow
m}^{\nu}&=\Gamma_{B,\nu}(\omega_{mn})n_{B}^{\nu}(\omega_{mn}),\qquad
n<m\end{split}$ (7)
Here, $n_{B}^{\nu}(\omega)$ denotes thermal distribution of the reservoir
$\nu$ at temperature $T_{\nu}$, i.e.
$n_{B}^{\nu}(\omega_{pq})=\frac{1}{e^{\omega_{pq}/T_{\nu}}-1}$.
$\Gamma_{B,\nu}(\omega_{pq})=2\pi\sum_{n}g_{n,\nu}^{2}\delta(\omega_{pq}-\omega_{n})$
is the interaction strength between the reservoir $\nu$ and the system. The
system steady state population are,
$\begin{split}\rho_{11}^{s}=\frac{W_{41}W_{31}}{(W_{14}+W_{41})(W_{13}+W_{31})}\\\
\rho_{22}^{s}=\frac{W_{13}W_{14}}{(W_{14}+W_{41})(W_{13}+W_{31})}\\\
\end{split}$
$\begin{split}\rho_{33}^{s}=\frac{W_{41}W_{13}}{(W_{14}+W_{41})(W_{13}+W_{31})}\\\
\rho_{44}^{s}=\frac{W_{31}W_{14}}{(W_{14}+W_{41})(W_{13}+W_{31})}\end{split}$
(8)
with the basis
$\\{|\varepsilon_{1}\rangle,|\varepsilon_{2}\rangle,|\varepsilon_{3}\rangle,|\varepsilon_{4}\rangle\\}$.
Expand the density matrix $\rho^{s}$ in the system basis
$\\{|00\rangle,|01\rangle,|10\rangle,|11\rangle\\}$, the system reduced
density matrix $\rho^{s}$ is,
$\left(\begin{array}[]{cccc}\rho_{33}^{s}\frac{\eta+B}{2\eta}+\rho_{44}^{s}\frac{\eta-B}{2\eta}&0&0&\rho_{33}^{s}\frac{J\chi}{2\eta}-\rho_{44}^{s}\frac{J\chi}{2\eta}\\\
0&\rho_{11}^{s}\frac{\xi+b}{2\xi}+\rho_{22}^{s}\frac{\xi-b}{2\xi}&\rho_{11}^{s}\frac{J(1+iD)}{2\xi}-\rho_{22}^{s}\frac{J(1+iD)}{2\xi}&0\\\
0&\rho_{11}^{s}\frac{J(1-iD)}{2\xi}-\rho_{22}^{s}\frac{J(1-iD)}{2\xi}&\rho_{11}^{s}\frac{\xi-b}{2\xi}+\rho_{22}^{s}\frac{\xi+b}{2\xi}&0\\\
\rho_{33}^{s}\frac{J\chi}{2\eta}-\rho_{44}^{s}\frac{J\chi}{2\eta}&0&0&\rho_{33}^{s}\frac{\eta-B}{2\eta}+\rho_{44}^{s}\frac{\eta+B}{2\eta}\end{array}\right)$
(9)
Given the system steady states obtained above, we can now formulate the
expression of the heat current.
### II.2 Results
Heat current. The symmetric expression of the heat current $\hat{J}$ is
defined asL. A. Wu
$\begin{split}J=Tr[\hat{J}\rho]\quad\hat{J}=\frac{i}{2}[H_{SBL},H_{s}]+\frac{i}{2}[H_{s},H_{SBR}]\end{split}$
(10)
in our case, the heat current can be written as
$\begin{split}J=&\frac{1}{2}\sum_{m,n}\omega_{m,n}|S_{n,m}^{L}|^{2}P_{n}[k_{n\rightarrow
m}^{L}(T_{L})]\\\
&-\frac{1}{2}\sum_{m,n}\omega_{m,n}|S_{n,m}^{R}|^{2}P_{n}[k_{n\rightarrow
m}^{R}(T_{R})]\end{split}$ (11)
and $P_{n}=Tr_{Bj}[\rho_{n,n}(t)]$.
Thermal rectification control by DM interaction. Under different temperature
bias ($\Delta T=T_{L}-T_{R}$) case, the control of the thermal rectification
with system DM interaction strength is shown in the Fig. 2.(a). The mean
temperature of two reservoirs is, $TM=1/2(T_{L}+T_{R})$. As defined by Ref.L.
Zhang , thermal rectification $\mathcal{R}$ satisfies the relation,
Figure 2: Thermal rectification (a) Thermal rectification with the DM
interaction strength. The sign changes at critical value $D_{cr1}$. Dashed,
solid and dot-dashed correspond to $|\Delta|T=0.4$, $|\Delta|T=1$ and
$|\Delta|T=1.6$. The mean temperature of two reservoirs is $TM=1$. (b) The
population of four system eigenstates with the change of DM interaction
strength. (c) Values of $J_{+,n}$ and $J_{-,n}$ with DM interaction strength.
In (b) and (c), solid lines are for the case $\Delta T>0$; (i.e., the
temperature of left reservoir is higher than the right one); dashed line are
for $\Delta T<0$. The red, blue, green and brown lines correspond to the state
$|\varepsilon_{1}\rangle$, $|\varepsilon_{2}\rangle$,
$|\varepsilon_{3}\rangle$ and $|\varepsilon_{4}\rangle$ respectively. (d)
Values of $J_{net,n}$. The red dotted, blue dashed, green dot-dashed and brown
solid line correspond to the state $|\varepsilon_{1}\rangle$,
$|\varepsilon_{2}\rangle$, $|\varepsilon_{3}\rangle$, and
$|\varepsilon_{4}\rangle$ respectively. $\Delta T=0.4$ and $TM=1$ in (b), (c)
and (d). Other parameters are: $\chi=0.3$, B=4, b=2, J=1, $\Gamma_{L}=1$, and
$\Gamma_{R}=0.25$.
$\mathcal{R}=(J_{+}-J_{-})/\max(J_{+},J_{-})$ (12)
where the forward heat current $J_{+}$ is the heat current (from left to
right) when the reservoir at higher temperature is connected to the left
quantum dot, and the backward current $J_{-}$ is the heat cuurent (from right
to left) when the left quantum dot is in contact with the reservoir at lower
temperature. The energy gaps of the two quantum dots are not equal. From Fig.
2. (a), the thermal rectification $\mathcal{R}$ will change from the positive
value to the negative value after the DM interaction strength reaches a
critical value $D_{cr1}$. This means one can obtain rather different heat
current through adjusting the DM interaction strength. The different system
states population with the change of the DM interaction strength are shown in
Fig. 2. (b). When the DM interaction strength is not strong, the system
populates mainly at the lowest eigenstate $|\varepsilon_{4}\rangle$. The
population of the eigenstate $|\varepsilon_{2}\rangle$ rises with the DM
interaction strength. Values of $J_{+,n}$ and $J_{-,n}$ are shown in Fig. 2.
(c). To illustrate the contribution of four different system states to the
thermal rectification, Define $J_{net}$ as,
$J_{net,n}=J_{+,n}-J_{-,n}$ (13)
where $J_{+,n}$ is the heat current contributed by the system state
$|\varepsilon_{n}\rangle$ (from left to right) when the left quantum dot is in
contact with the reservoir at higher temperature, and $J_{-,n}$ is the heat
current contributed by the system state $|\varepsilon_{n}\rangle$ (from right
to left) when the left quantum dot is connected with the reservoir at lower
temperature. We display the values of $J_{net}$ in Fig. 2. (d), in the region
of the weak DM interaction strength, state $|\varepsilon_{2}\rangle$ provides
the negative net heat current, and the net heat current contributions from
state $|\varepsilon_{4}\rangle$ is positive. The heat current contribution
from state $|\varepsilon_{4}\rangle$ is more notable than that from state
$|\varepsilon_{2}\rangle$. The heat current contributed by state
$|\varepsilon_{1}\rangle$ and $|\varepsilon_{3}\rangle$ are negligence. So
according to the formula of the thermal rectification above, the thermal
rectification value is positive when the DM interaction strength is small.
After the DM interaction strength reaches the critical value $D_{cr1}$, the
contribution of $|\varepsilon_{2}\rangle$ $J_{net,2}$ is positive, and the
heat current of $|\varepsilon_{4}\rangle$ $J_{net,4}$ is negative. The major
contribution is from $|\varepsilon_{4}\rangle$, and the sign of thermal
rectification will change to the negative sign. In our two-quantum-dot system,
the lowest system eigenstate changes from state $|\varepsilon_{4}\rangle$ to
state $|\varepsilon_{2}\rangle$ when the DM interaction strength reaches the
critical value $D_{cr2}$. We find that the net heat current contribution of
state $|\varepsilon_{2}\rangle$ is negative, and state
$|\varepsilon_{4}\rangle$ provides a positive contribution. When DM
interaction exceeds critical value $D_{cr2}$, $J_{net}$ is mainly contributed
by state $|\varepsilon_{2}\rangle$. From the analysis above, we find that the
lowest eigenstate of the two-quantum-dot system makes the major contribution
on the thermal rectification. Through modulating the DM interaction strength,
we can change the sign of thermal rectification to control the polarity of
thermal diode.
Magnetic field effect. In our model, the magnetic field affect the energy gap
of the quantum dot.
Figure 3: (color online): The relation between thermal rectification and the
magnetic field strength in the four different anisotropic situations. Blue
solid, red dashed, green dotted and orange dot-dashed correspond to
$\chi=0.9$, $\chi=0.3$, $\chi=0.0$ and $\chi=-0.3$ respectively. By setting
b=B in Eq.(1), the energy gap of the right dot is zero. The system parameters:
the mean temperature $TM=1$, the temperature bias $\Delta T=1$, J=1,
$\Gamma_{L}=1$ and $\Gamma_{R}=0.25$. (a) The DM interaction strength is 0.
(b) The DM interaction strength is 4.
We choose the parameters that the energy gap of the left dot is not zero, and
the energy gap of the right one is zero. Different anisotropic system
conditions are shown in Fig. 3. The two different situations are described in
Fig. 3. (a) and Fig. 3. (b) with the DM interaction strength being zero and
four respectively. From the figure, we find that the sign of the thermal
rectification changes around certain value of external magnetic field. In
different anisotropic situations, the sign of thermal rectification will
change at different critical value of external magnetic field. So, we can
realize the reversal of the thermal diode model polarity by adjusting the
external magnetic field in our two-quantum-dot system, also.
## III Quantum correlation
Quantum entanglement. To qualify the quantum properties in the steady state,
we use the concurrence to measure the quantum entanglement.
Figure 4: (color online): Quantum correlations with DM interaction. Solid
line: quantum entanglement; dashed line: quantum discord. Under the same
parameter conditions, quantum entanglement rises with DM interaction more
rapidly than quantum discord does. $\Delta T=0.4$, $TM=1$, $\chi=0.3$, B=4,
b=2, J=1, $\Gamma_{L}=1$, and $\Gamma_{R}=0.25$.
Given the final X states, we have the analytic result of the concurrence
$C(\rho^{AB})$ asW. K. Wooters ,
$\begin{split}C(\rho^{AB})=2\max\\{0,&\frac{J}{2\xi}\sqrt{1+D^{2}}|\rho^{s}_{11}-\rho^{s}_{22}|\\\
&-\frac{1}{2}\sqrt{(\rho_{33}^{s}+\rho_{44}^{s})^{2}-\frac{B^{2}}{\eta^{2}}(\rho_{33}^{s}-\rho_{44}^{s})^{2}},\\\
&\frac{J\chi}{2\eta}|\rho^{s}_{33}-\rho^{s}_{44}|\\\
&-\frac{1}{2}\sqrt{(\rho_{11}^{s}+\rho_{22}^{s})^{2}-\frac{b^{2}}{\xi^{2}}(\rho_{11}^{s}-\rho_{22}^{s})^{2}}\\}\end{split}$
(14)
Quantum discord. Correlation in the quantum aspects of the bipartite system
can also be measured by quantum discord. The system quantum discord is defined
asH. Ollivier ,
$D(\rho^{AB})=I(A:B)-C_{cor}(\rho)$ (15)
The quantum mutual information $I(A:B)$ has the form as,
$I(A:B)=S(\rho_{A})+S(\rho_{B})-S(\rho_{AB})$. The expression $S(\rho)$ is the
Von Neumann entropy of the destiny matrix $\rho$, satisfy
$S(\rho)=-Tr(\rho\log\rho)$. $\rho^{A}$ and $\rho^{B}$ are the reduced destiny
matrix of the system X state density matrix. The Classical correlation
$C_{cor}(\rho)$ satisfies the equation
$C_{cor}(\rho)=S(\rho^{A})-\min_{B_{i}}[S(\rho|\\{B_{i}\\})]$, and the last
term $\min_{B_{i}}[S(\rho|\\{B_{i}\\})]$ is the condition entropy.
Figure 5: (color online): Quantum correlation with temperature difference of
two reservoirs. TR: temperature of right reservoir, D: DM interaction
strength. Almost perfect quantum correlation can be achieved even though the
temperature difference of two reservoirs is large. Parameters set in
calculation: $T_{L}=0.5$, $\chi=0.9$, b=0, B=0.2, J=1, $\Gamma_{L}=1$ and
$\Gamma_{R}=0.25$. Figure 6: (color online): Population of different system
eigenstates $|\varepsilon_{1}\rangle$(a), $|\varepsilon_{2}\rangle$(b),
$|\varepsilon_{3}\rangle(c)$ and $|\varepsilon_{4}\rangle$(d) with different
right reservoir temperatures and DM interaction strengths. The vertical axis
labels the population. Parameters set: $T_{L}=0.5$, $\chi=0.9$, b=0, B=0.2,
J=1, $\Gamma_{L}=1$ and $\Gamma_{R}=0.25$.
We study the relation between the quantum correlation and the DM interaction
strength. From Fig. 4, it is seen that quantum entanglement is more sensitive
to the DM interaction strength than quantum discord, and perfect quantum
correlation can be got within the small increment of the DM interaction
strength.
Almost perfect quantum entanglement and quantum discord can be obtained in the
large temperature bias region in our model, see in Fig. 5. The corresponding
four system states population are shown in the Fig. 6.The reason for the high
value of the quantum correlations is that almost all system population are at
the system state $|\varepsilon_{2}\rangle$.
## IV Concluding Remarks
We study the nonequilibrium two-quantum-dot system with the DM interaction
extensively. The behavior of the heat current, quantum entanglement and
quantum discord have been investigated. We find that we can control the
thermal rectification by two types of modulations, one is adjusting the system
DM interaction strength, and the other is regulating the energy gap of two-
quantum-dot system. This provides a possible new way to design a polarity-
controllable thermal diode. Compared with the spin-chains model studied
beforeY. Yan ; L. Zhang , this two-quantum-dot model can offer large thermal
rectification when the size is small (N=2). This two-quantum-dot system can be
used to build small size phonon devices, and has a potential application in
the phonon information.
## V acknowledgments
This work was 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.
## References
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* (9) R. Hanson and D. D. Awschalom, Nature (London) 453, 1043 (2008).
* (10) F. Kheirandisha, S. J. Akhtarshenasb, and H. Mohammadic, Eur. Phys. J. D. 57, 129 (2010).
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* (12) L. A. Wu, and D. A. Lidar, Phys. Rev. A 67, 050303(R) (2003).
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* (17) A. Pfund, I. Shorubalko, K. Ensslin, and R. Leturcq, Phys. Rev. B 76, 161308(R) (2007).
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* (19) L. A. Wu, C. X. Yu, and D. Segal, Phys. Rev. E 80, 041103 (2009).
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|
arxiv-papers
| 2012-11-13T10:11:52 |
2024-09-04T02:49:37.920547
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "T Chen and X.B. Wang",
"submitter": "Xiang-Bin Wang",
"url": "https://arxiv.org/abs/1211.2938"
}
|
1211.2955
|
# Nonlinear Transport through ultra-narrow Zigzag Graphene Nanoribbons: non-
equilibrium charge and bond currents
Hosein Cheraghchi School of Physics, Damghan University, P. O. Box: 36715-
364, Damghan, IRAN [email protected]
###### Abstract
The electronic nonlinear transport through ultra narrow graphene nanoribbons
(sub-$10nm$) is studied. A stable region of negative differential resistance
(NDR) appears in the I-V characteristic curve of odd zigzag graphene
nanoribbons (ZGNRs) in both positive and negative polarity. This NDR
originates from a transport gap inducing by a selection rule which blocks
electron transition between disconnecting energy bands of ZGNR. Based on this
transition rule, on/off ratio of the current increases exponentially with the
ribbon length up to $10^{5}$. In addition, charging effects and also spatial
distribution of bond currents was studied by using non-equilibrium Green’s
function formalism in the presence of e-e interaction at a mean field level.
On the other hand, we also performed an ab initio density functional theory
calculation of transmission through a passivated graphene nanoribbon to
demonstrate robustness of the transport gap against hydrogen termination of
the zigzag edges.
A. Graphene Nanoribbons; D. Negative Differential Resistance
###### pacs:
73.23.-b,73.63.-b
## I Introduction
Graphene is a two-dimensional carbon which has been recently explored
novoselov . Experiments in graphene-based devices experiment have shown the
possibility of controlling their electrical properties by the edge structure
and also application of an external gate voltage. Nowadays, by using a
chemical methodDai , it is possible to fabricate field effect transistor
graphene nanoribbons (GNRs) with ultra narrow widths and smooth edges which
are possibly well defined zigzag or armchair edge structures. A useful
transport gap is opened in sub-$10nm$ GNRs at room temperature which results
in a high on/off current switching up to $10^{6}$. The origin of the transport
gap can be understood by two factors: suppression of transport due to edge
disorder leading to Anderson localization, confinement along transverse
directionhan ; Molitor . Additional to these factors, in ultra narrow zigzag
graphene nanoribbons (ZGNR), flow of current is also blocked by the symmetric
selection rules cheraghchi1 ; grosso ; beenaker ; P-Ngraphene ; duan ;
kurihara ; Wakabayashi .
Negative differential resistance esaki in nanoelectronic devices has been
also observed in the metallic nanotube junctions farajian and in the case of
potential barriers in $2D$ graphene sheetsNDRgraphene .
Figure 1: Odd zigzag graphene nanoribbon considered as a central region
attached to two electrodes. Transverse zigzag carbon chains and unit cells
inside the central region are labeled by N and M.
In this paper, to shed light on the experimental work in Ref.[Dai, ], we study
non-linear transport in odd zigzag graphene nanoribbons by using non-
equilibrium Green’s function formalism (NEGF). It is shown that a stable NDR
against electrostatic interaction up to $10^{5}$ is appeared in ultra narrow
ZGNRs around $\pm 1t$ in both positive and negative polarity, where $t$ is
hopping integral between carbon atoms. On-off ratio of this NDR increases
exponentially with the ribbon length. The NDR phenomenon occurs in far-from-
equilibrium regime.
Although parity is not conserved in odd ZGNRs, current reduction is induced by
an other selection rule in which transition between disconnecting band groups
is forbidden (Cresti et al.grosso ). The edges of graphene ribbons can also
simply absorb some chemical compoundskobayashi . It is demonstrated that this
NDR is not much sensitive to the asymmetry of ribbons, while transport gap in
even ZGNRs which is based on the reflection symmetry, is completely destroyed
due to the edge disorder. The above mentioned selection rule is valid for both
even and odd ZGNRs, while parity selective tunneling just belongs to even
ZGNRs. This transport gap is robust against hydrogen termination. Additional
to the model calculation, by using TranSIESTA codesiesta , we performed an ab
initio density functional theory calculation of transmission through graphene
nanoribbon with hydrogen termination of the zigzag edges. Results of first
principle calculation confirms the transport gap originating from transition
rule between disconnected bands.
This NDR is stable against the presence of the electron-electron (e-e)
interaction at a mean-field approximation. It is concluded from self-
consistent charge and potential profiles so that external potential is
screened by charge redistribution around the contact junctions. So potential
profile deep inside the ribbon remains flat. Furthermore, the e-e interaction
increases on-off ratio of the current. Moreover, in low and high applied
biases, we study spatial profile of local currents in the presence of e-e
interaction which contrasts with non-equilibrium charge profile.
The paper is organized as the following sections. After an introduction, we
explain Hamiltonian and a short review of NEGF formalism in section II. In
section III, we present our results containing NDR in I-V curve and charging
effects in high applied bias and spatial bond currents. Before to end the
paper, we present the results of ab initio calculated by using Transiesta
code. Finally, we conclude our results.
## II Formalism
The single electron Hamiltonian of the molecule is defined as:
$\begin{array}[]{r}H\\{n\\}=\sum_{i=1}^{2N\times
M}[\varepsilon_{i}+u^{ext}_{i}+\sum_{j=1}^{2N\times M}V_{ij}\delta
n_{j}]c^{\dagger}_{i}c_{i}\\\
+\sum_{<ij>}t(c^{\dagger}_{i}c_{j}+c_{i}c^{\dagger}_{j})\end{array}$ (1)
where $c^{\dagger}_{i}$ and $c_{i}$ are the electron creation and annihilation
operators, respectively. $2N\times M$ and $t$ are number of atomic sites and
hopping energy between nearest neighbor atoms. One $\pi$ orbital is considered
per each site for graphene as a planar 2D system. Without losing any
generality, we set onsite energies ($\varepsilon_{i}$) of all sites to be
equal to zero. All energies and voltages are in unit $t=2.7eV$. The Fermi
energy of undoped graphene nanoribbon is at half-filling $E_{F}=0$ due to
perfect electron-hole symmetry. By applying a source-drain bias, the site
energies are shifted by a linear potential distribution along the molecule
$u^{ext}_{i}$ which is the solution of the Laplace equation. The applied bias
$V$ is divided symmetrically on the left and right electrodes as $-V/2$ and
$V/2$, respectively. $V_{ij}$ is the electrostatic Green’s function and
$\delta n_{i}=n_{i}-n_{i}^{0}$ is the change in the self-consistent charge
$n_{i}$ from its value in zero source-drain voltage $n_{i}^{0}$. This third
term is the direct Coulomb interaction created by the source-drain bias-
induced charges. It includes the Hartree term which is the solution of Poisson
equation and locates on the diagonal terms of Hamiltonian. For calculating the
electrostatic Green’s function, we have used the method explained in
Ref.[cheraghchi2, ] and its appendix.
Figure 2: Current-voltage characteristic curve of an odd zigzag graphene
nanoribbon with $N=5$ (zigzag chains). I-V curves are compared for two ribbon
lengths; $M=5$ and $10$. For the case of $(M,N)=(5,5)$ and for comparison
purpose, I-V curve is also plotted in the presence of electron-electron
interaction ($U$) and asymmetric factor($\varepsilon_{\alpha}$).
For self-containing, we present a very brief review of the NEGF formalism.
Charge density in non-equilibrium situation is calculated by [$-iG^{<}$] as
the occupation number in the presence of the two electrodes with an applied
source-drain biasTaylor .
$n_{i}=\frac{-1}{\pi}\int_{-\infty}^{E_{F}-\frac{V}{2}}Im[G^{r}(E)]_{ii}dE+n^{non-
eq}_{i}$ (2)
where non-equilibrium part of charge can be calculated by the following
integral,
$n^{non-
eq}_{i}=\frac{1}{2\pi}\int_{E_{F}-\frac{V}{2}}^{E_{F}+\frac{V}{2}}[-iG^{<}(E)]_{ii}dE$
(3)
where within a one-particle theory ,
$-iG^{<}=G^{r}(\Gamma_{L}f_{L}+\Gamma_{R}f_{R})G^{a}$ (4)
Here $f_{L/R}$ is the Fermi-Dirac distribution function of electrodes and
$G^{r}/a$ is the retarded/advanced Green’s function defining as the following:
$G^{r/a}=[(E\pm\eta)I-H\\{n\\}-\Sigma_{L}^{r/a}-\Sigma_{R}^{ra}]^{-1}$ (5)
and $\Gamma$ is the escaping rate of electrons to the electrodes which is
related to the self-energies as $\Gamma_{p}=i[\Sigma_{p}^{r}-\Sigma_{p}^{a}]$
with $p=L/R$Datta ; Taylor ; Munoz . Here $\eta\rightarrow 0^{+}$. Solving
equations 2 and 5 self-consistently results in a self consistent charge and
Green’s functions. Finally, the current passing through the molecule is
calculated by the Landauer formula for zero temperature Datta which is valid
for coherent transport.
$\displaystyle I(V)=\frac{2e}{h}\int_{E_{F}-V/2}^{E_{F}+V/2}\,dE\,T(E,V)\ $
(6)
where $T(E,V)$ are the bias dependent transmission coefficient.
$T={\rm Tr}[G^{r}\Gamma_{R}G^{a}\Gamma_{L}]$ (7)
Fig.(1) shows zigzag graphene nanoribbon with an odd number of zigzag chains
in width ($N$). Central interacting region, left and right electrodes divide
the ribbon into three regions. Odd ZGNRs has a bisection plane which its
intersection with the ribbon has been shown as the x-axis in Fig.(1). Although
this plane bisects the ribbon, it is not a reflection plane. If upper half of
the ribbon is displaced by $a/2$ in respect to the lower one, mirror symmetry
is achieved against x-axis. $a$ is the bond length of C-C. In other words,
part of the wave function which is a functional of $y$ variable, is an
eigenvector of the parity operator Polini .
## III Results
Figure 3: Energy spectrum of the left and right electrodes and transmission
through zigzag graphene nanoribbon with $(M,N)=(10,5)$ at voltage
($V_{SD}=1.4t>V_{T}$). The band structure is divided into three groups which
are called by upper , centeral and lower band groups. These groups are
classified based on the bands which are connected in terms of $k_{x}$. The
bold hollow arrows show the current integration window which based on Eq. 6,
is proportional to $V_{SD}$. The Fermi level is set to be as $E_{f}=0$. The
half-width of the central bands at $k_{x}=0$ is called as $\delta$ which is
equal to the threshold NDR voltage. $\Delta$ is energy separation of the upper
bands from the central bands at the Dirac points. Transport gaps $AB$ and $CD$
are equal to $\Delta-\delta+V_{SD}$.
Current-voltage characteristic curve of an odd ZGNR with $5$ zigzag chains
($N=5$) is shown in Fig.(2). Lower than the external bias $1.2t$, current
increases linearly with the applied bias as an Ohmic device. After a threshold
voltage ($1.2t$), NDR occurs at both positive and negative polarity.
The origin of NDR seen in odd ZGNRs is interpreted by analyzing their energy
spectrum accompanied to transmission curve. Fig.(3) shows energy spectrum
$E(n,k_{x})$ and transmission through 5-ZGNR at $V=1.4t$. $n$ counts bands
from the bottom ($n=1$) to the top of band structure ($n=2N$) and $k_{x}$ is
the longitudinal momentum.
In odd ZGNRs, parity has noncommutative relation with the Hamiltonian.
Therefore, parity has no conservation and consequently transmission is not
blocked by the parity selection rule, while parity conservation in even zigzag
nanoribbons opens transmission gap around Fermi levelcheraghchi1 . In the
range of $BC$ of Fig.(3), there exists one conducting channel which results in
the unity transmission around the Fermi level. In this range, electrons which
occupy $-k$ states of the lower band of the central band group (the dashed
blue bands) belonging to the right electrode are injected into the unoccupied
$-k$ states of the upper band of the central group belonging to the left
electrode. So at low biases, current increases linearly with the bias. This
single-channel transport around the Fermi level remains unchanged even for
high voltages. However, for voltages greater than the NDR threshold voltage
($V>V_{T}$), blocked regions marked by the ranges of $AB$ and $CD$ comes into
the current integration window. The current integration window, based on Eq. 6
and $E_{f}=0$, is proportional to $V_{SD}$ and is shown with the bold hollow
arrows in Fig.(3). Therefore, when the source-drain applied bias increases,
current begins to decrease.
Blocked regions ($AB$ and $CD$) arise from a selection rule which increases
back scattering in the lengthy ribbons. According to this rule, electron
transition between those bands which are disconnected from the view point of
longitudinal momentum, decreases exponentially with the length. Topology of
zigzag graphene ribbons divides the band structure into the three different
groupsgrosso . Three groups which are called by upper (n=1,2,3,4), centeral
(n=5,6) and lower (n=7,8,9,10) groups, are classified based on the bands which
are connected in terms of $k_{x}$. Variation of the momentum of electrons
passing through the system strongly depends on smoothness or sharpness of the
potential. The transition probability of electrons with the $k$ state injected
by the right electrode into the $q$ state as an empty state in the left
electrode is proportional to the Fourier transform of the longitudinal voltage
grosso as $<\psi(k)\mid V\mid\psi(q)>\propto\tilde{V}(k-q)$. So smooth
variation of the potential in longer ribbons results in a small momentum
variation of electron. Consequently, a smooth potential in the longitudinal
direction can just scatter electron into those states belonging to the energy
bands which are connected from the point of momentum viewgrosso ; cheraghchi1
.
Figure 4: Contour plot of transmission versus energy and applied bias for
graphene nanoribbon with 5 zigzag chains in width and 10 unit cells in length.
Dark solid lines show the current integration window and white dashed lines
show the forbidden region for electronic transition from the band groups of
upper or lower bands to the bands of central group.
Bias dependent of transmission leads us to plot a contour of transmission in
plane of energy and applied bias as shown in Fig.(4). In this figure, the
solid lines show the current integration window. Moreover, the dashed lines
clarify the regions $AB$ and $CD$ corresponding to transport gaps shown in
Fig.(3). The intersection of these blocked regions and also the current
integration window would be around $1.2t$ which is the threshold voltage
$V_{T}$ for the current reduction.
Band structure analyzing demonstrates that the threshold voltage is equal to
the half-width of the central bands at $k_{x}=0$ as
$V_{T}=\delta=[E(N+1,k_{x}=0)-E(N,k_{x}=0)]/2$. As shown in Fig.(3), $\Delta$
is energy separation of the upper bands from the central bands at the Dirac
points. There is a Log-Normal behavior of $\delta$ versus number of zigzag
chains ($N$) such that as $N\rightarrow\infty$, the NDR threshold voltage
asymptotically approaches to the value of $0.9738\pm 0.0002t$. So the NDR
threshold voltage slightly decreases with the ribbon width.
Analyzing transport gaps appeared in the band structure shows that they are
equal to $\Delta-\delta+V_{SD}$ where $\Delta\propto N^{-1}$. Since $\delta$
approaches to a constant values when $N\rightarrow\infty$, in a given voltage,
transport gap is disappeared for $N>30$ which is nearly equivalent to $10nm$.
The other effect which enhances performance of this electronic switch, is the
ribbon length. Fig.(2) shows an increase of on/off ratio with the ribbon
length. Moreover, NDR region ($V_{off}-V_{on}$) occurs in a more extended
range of the I-V curve. Exponential decay of transmission with the length in
the gap regions develops quality of switching. It is shown in Fig.(5) that
$I_{on}/I_{off}$ increases exponentially with the ribbon length.
Figure 5: Exponential increase of on/off current ratio as a function of ribbon
length ($M$).
Another parameter which affects I-V curve, is the e-e interaction. In this
case, interaction intensifies NDR effects such that off current reduces in
compared with non-interacting system and also the region of NDR becomes more
extended.
If one of the ribbon edges is doped by small impurity such as
$\varepsilon_{\alpha}=0.1t$, because of a band gap which is induced by edge
impurity at the band center, current at low biases decreases. Fig.(2)
represents that even with the presence of edge impurity, still the region
containing NDR exists. However, asymmetry decreases on/off ratio of the
current. Furthermore, asymmetric ZGNR behaves as a semiconductor while
symmetric ZGNRs behave as an Ohmic devicesRen . The effect of asymmetry on NDR
competes with the ribbon length. Since asymmetry can not be ignored in
experiment, longer ribbons are in favor of keeping NDR in the I-V curve.
On the other hand, additional to edge impurity, this asymmetry can be assigned
to a sublattice symmetry breaking induced by spontaneous ferromagnetic spin
ordering of the electrons localized at the zigzag edgesLouie . In fact, the
border atoms at the two opposite zigzag edges belong to different sublattices.
So spin orientation along the edges induces different magnetic potentials at
the edges. As a result, a small band gap is opened around Fermi level which
depending on the ribbon width, is about 0.15 eV. The asymmetry which we have
considered is about 0.3 eV which is stronger than the gap opened by spin-
polarization of the edges. We can conclude that spin-polarization along the
zigzag edges can not affect emerging of this NDR phenomenonRen .
Charging effects: To understand why details of electrostatic potential do not
affect the emergence of NDR, we compare transmission in the presence and
absence of e-e interaction. Figure (6) compares transmission curves for
voltages $1.2t$ at the NDR threshold $V_{T}$ and a larger value such as
$1.5t$. It is clear that for voltages lower than the NDR threshold,
transmission in the conducting region $BC$ is robust against e-e interaction.
However, for voltages $V>V_{T}$, transmission of conducting region $BC$
reduces with e-e interaction. This is the reason for reducing off-current in
the presence of e-e interaction. Furthermore, transmission curve in the
blocked regions $CD$ and $AB$ exhibits a little enhancement with the e-e
interaction for the whole range of voltages.
Figure 6: The effect of electron-electron interaction in the mean-field level
on transmission curves for two voltages a) NDR threshold voltage which is
equal to 1.2t, b) a voltage above the NDR threshold such as 1.5t.
To elucidate physics behind to the robustness of NDR phenomenon against e-e
interaction, we investigate self-consistent potential profiles which are shown
in Fig.(7.a,b). In fact, external potential is well screened by charge
redistribution so that potential mostly drops at the contact junctions. This
fact is represented in Fig.(7). So, electrostatic potential of atoms located
far from contact junctions (deep inside the ribbon) remains nearly flat. In
more details, depletion of charge close to the source electrode reduces the
source potential as $U\delta n$. Therefore, source potential can not penetrate
inside the central portion of the system. On the opposite side of system,
charge is accumulated close to the drain electrode such that dropping of the
potential is weakened.
Figure 7: a) Self-consistent charge profile at 1.5t. Charge compares with its
neutrality point ($n_{0}$) when there is no applied bias. Transferred charge
is equal to $(n-n_{0})$. Electrostatic potential profile is calculated
according this transferred charge. b) averaged electrostatic potential
variation on unit cell as a function of the ribbon axis. c) Total transferred
charge from/into the ribbon as a function of source-drain voltage.
It is useful to investigate bias dependence of total transferred charge
from/into the system. Fig.(7.c) shows that transferred charge in compared with
its neutrality point value ($n_{0}$ with zero source-drain applied bias)
decreases for voltages higher than the NDR threshold, while it remains
unchanged for voltages $V<V_{T}$. In other words, for $V<V_{T}$, in-flowing
and out-flowing charges are balanced with each other, while for $V>V_{T}$,
system becomes empty of charge. It can be seen that the threshold voltage for
depletion of charge is correlated with the NDR threshold voltage. This
behavior is closely similar to what is seen in graphene nano-
junctionscheraghchi2 . As a consequence, at high voltages, due to accumulation
of charge close to the drain electrode, external potential is always screened
in the middle of the graphene strip. Screening effect induces sharp variation
of electrostatic potential at the contacts which enhances transition
probability between disconnected band groups. Therefore, transmission in the
blocked regions ($CD$ and $AB$) increases in compared to non-interacting
system. However, blocked regions have not any contribution in the current
calculation.
Figure 8: Top: Spatial distribution of bonds current in three voltages: red
arrows) 1.0 t$<V_{T}$, black arrows) 1.2t $=V_{T}$ and blue arrows) 1.5
t$>V_{T}$. The length of arrows is proportional to the local current. The
arrows located on the edges are shorter than the arrows located in the bulk of
nanoribbon. By increasing the source-drain voltage, for voltages lower than
$V_{T}$, current increases and consequently arrows are longer, while higher
than $V_{T}$, current decreases and so arrows are shorter.
Bottom: Non-equilibrium charge profile at the threshold NDR voltage. Non-
equilibrium charge reaches in its maximum value at the ribbon’s edges.
Moreover, in agreement with the continuity equation, atomic sites with larger
local current corresponds to smaller non-equilibrium charge.
Spatial distribution of bond currents: Charge conservation based on continuity
equation yields non-equilibrium bond charge currentnikolic for a non-
interacting tight-binding Hamiltonian.
$J_{ij}=\frac{2et}{h}\int_{E_{F}-\frac{V}{2}}^{E_{F}+\frac{V}{2}}[G_{ij}^{<}(E)-G_{ji}^{<}(E)]dE$
(8)
where sites $i$ and $j$ are nearest-neighbor atoms which their hopping
integral is nonzero. Now, charge continuity equation is derived by using
Heisenberg equation.
$e\frac{d}{dt}n^{non-eq}_{i}+\sum_{j}[J_{ij}-J_{ji}]=0$ (9)
where $j$ are the nearest-neighbor atomic site around $i^{th}$ atomic site. It
can be simply proved that the integrands in Equations 3 and 8 are real. In
fact, transpose of matrix $-iG^{<}$ in Eq.(4) is equal to its conjugate. So,
diagonal terms of ($-iG^{<}_{ii}$) and also the terms of
($G^{<}_{ij}-G^{<}_{ji}$) are real.
Bond charge current formulated in Eq.8 has been derived for non-interacting
tight-binding Hamiltonian. However, by using Heisenberg equation, it can be
simply proved that this formula is still applicable for the Hamiltonian
described in Eq.1 in which Hartree interaction appears on diagonal elements of
the Hamiltonian. On the other hand, charging effects originating from
electrostatic interaction (Hartree term) adjust local currents by means of
lesser Green’s function.
Fig.(8.top) shows spatial profile of local current at each site. In the center
of the ribbon, magnitude of local current densities are larger than its value
in the zigzag edges of the ribbon. This feature is valid for both ranges of
low and high voltages not more than 1.5 t. The reason can be followed by
single-channel transport through ZGNR even for high voltagesnikolic . One
channel transport is an indirect result of transition rule in which transition
between disconnected bands is forbidden. For voltages higher than 1.5 t,
alignment of local currents along the ribbon axis is gradually disturbed.
Figure 9: Top: Zigzag graphene nanoribbon terminated by Hydrogen atoms.
Bottom: First principle calculation of transmission through the above system
as a function of energy for different values of external biases.
In contrast with the local current distribution, as shown in Fig.(8.bottom),
non-equilibrium charge reaches in its maximum value at the ribbon’s edges.
This result is compatible with the continuity equation (Eq.9) so that atomic
sites with larger local current corresponds to smaller non-equilibrium charge.
On the other hand, non-equilibrium charge increases for atomic sites with
higher onsite energy due to applied bias (close to the source electrode),
while its value tends to zero for sites close to the drain electrode.
Comparison of odd and even ZGNRs: There are some interesting differences
between results arising from odd ZGNRs with those results belonging to even
ZGNRscheraghchi1 which is kind of odd-even effect. Here, NDR appears in
voltages upper than $1t$ while in even ZGNR, NDR occurs for voltages lower
than $1t$. On/off ratio of the current in gated even ZGNRs increases as a
power law with the function of the ribbon length while here, on/off ratio
increases exponentially. Screening of the external bias by electrons of system
in even ZGNRs is so stronger than screening effects in odd ZGNRs. As a
consequence, the effect of electrostatic interaction on increase of the on/off
ratio in even ZGNRs is much effective than in odd ZGNRs. Furthermore,
transferred charge from/into the central portion of graphene nanoribbon
depends on odd or even zigzag chains in width.
## IV Hydrogen Terminated Nanoribbon
To show how passivation of zigzag edges of graphene nanoribbon affects
transport properties, we compare the results of the presented model with
transport properties calculated by using TranSIESTA codesiesta . This code is
based on density functional theory (DFT) approach. We have used the following
options to calculate SIESTA code: the generalized gradient approximation (GGA)
with the Perdew-Burke-Ernzerhof exchange-correlation functional (PBE)PBE ,
double-$\zeta$ plus polarization orbital (DZP) bases for all atoms, and
Troullier-Martins norm conserving pseudopotentials to represent the cores, 200
Ry real space mesh cutoff for charge density and a supercell within 20 $\AA$
of vacuum between the periodic graphene nanoribbons.
Fig.(9, Top) shows a schematic view of odd zigzag graphene nanoribbon which is
saturated by hydrogen atoms. Transmission around the Fermi energy is
represented in Fig. (9, bottom). The energy scale is shifted so that the Fermi
energy of the system, if there is no bias voltage, is zero ($E_{F}=0$). When
there is a finite bias, the Fermi energy of the left electrode is placed at
$V/2$ and the right electrode at $-V/2$. The occupation number on atoms in the
center part of the system is determined by $[-iG^{<}]_{ii}$.
As it is clear of Fig.(9, bottom), there exists one transmitting channel for
each spin around Fermi energy level. This result is in complete agreement with
the band structure analysis shown in section.III. There exists only one
transmitting channel in the region $BC$ presented in figures 3 and 4. The
transport gap in Fig. (9, bottom) begins to open at voltage around
$1.2eV\simeq 0.5t$ and for energy about $2.7eV\simeq t$. These points
correspond to the points marked by cross sign in Fig.4 in which the transport
gap begins to open as the applied bias increases. As a result, we demonstrate
that the transport gap which is responsible for emerging the NDR phenomena
could also be opened in graphene nanoribbons with passivated zigzag edges.
## V Conclusion
As a conclusion, based on a model calculation of non-equilibrium Green’s
function formalism, we found that there exists a region of negative
differential resistance in I-V curve of ultra narrow (lower than $10nm$)
zigzag graphene nanoribbons with odd number of zigzag chains in width. This
NDR is induced by a transport gap which originates from electronic transition
between disconnecting bands of energy from the view point of longitudinal
momentum. On/off ratio of the current exponentially increases up to $10^{5}$
as a function the ribbon length which proposes possibility of manipulation of
odd ZGNRs as high quality switch in nanoelectronic based on graphene
nanoribbons. In addition, e-e interaction enhances on-off ratio of the current
which originates from a flat electrostatic potential deep inside the ribbon
due to screening of the external bias by electrons close to the junctions. By
using continuity equation, spatial profile of local currents is calculated in
the presence of Hartree electron interactions. In both high and low biases,
local current reaches to its maximum values in the center of the ribbon while
in contrast with the local current profile, non-equilibrium charge has its
maximum values at the edges of the ribbon. Furthermore, this NDR is not much
sensitive to the edge asymmetry. So emerging of this NDR is robust against
spin orientation along the edges.
By using ab initio density functional theory, we also show that the transport
gap which is responsible for emerging NDR, exits in passivated graphene
nanoribbons by Hydrogen atoms.
## References
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|
arxiv-papers
| 2012-11-13T11:24:20 |
2024-09-04T02:49:37.927342
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Hosein Cheraghchi",
"submitter": "Hosein Cheraghchi",
"url": "https://arxiv.org/abs/1211.2955"
}
|
1211.2972
|
# Segregating event streams and noise
with a Markov renewal process model
Dan Stowell [email protected]
Mark D. Plumbley [email protected]
Centre for Digital Music
Queen Mary University of Londdon
###### Abstract
We describe an inference task in which a set of timestamped event observations
must be clustered into an unknown number of temporal sequences with
independent and varying rates of observations. Various existing approaches to
multi-object tracking assume a fixed number of sources and/or a fixed
observation rate; we develop an approach to inferring structure in timestamped
data produced by a mixture of an unknown and varying number of similar Markov
renewal processes, plus independent clutter noise. The inference
simultaneously distinguishes signal from noise as well as clustering signal
observations into separate source streams. We illustrate the technique via a
synthetic experiment as well as an experiment to track a mixture of singing
birds.
Keywords: Multi-target tracking, clustering, point processes, flow network,
sound
## 1 Introduction
Various approaches exist for the task of inferring the temporal evolution of
multiple sources based on joint observations (Mahler, 2007; Van Gael et al.,
2008). They are generally based on a model in which sources are continuously
observable, in the sense that they are expected to emit/return observations at
every time step (though there may be missed detections). Yet there are various
types of source for which observations are inherently intermittent, and for
which this intermittence exhibits temporal structure that can be characterised
as a point process. Examples include sound event sequences such as bird calls
or footsteps (Wang and Brown, 2006), internet access logs (Arlitt and
Williamson, 1997), pulsars in astronomy (Keane et al., 2010) and neural firing
patterns (Bobrowski et al., 2009). Intermittent observations are also often
output from sparse representation techniques, which transform signals into a
representation with activations distributed sparsely in time and state
(Plumbley et al., 2010).
In this paper we describe a generic problem setting that may be applied to
such data, along with an approach to estimation. We are given a set of
timestamped data, and we assume each datum is produced by one of a set of
similar but independent signal processes, or by a “clutter” noise process,
with known parameters. We do not know the true partitioning of the data into
sequences each generated by a single process, and wish to infer this. We do
not know how many processes are active, and we do not assume that each process
produces the same number of observations, or observations at the same time
points.
This specific type of clustering problem has applications in various domains.
For example, when sparse representation techniques are used for source
separation in time series, they often yield a set of atomic activations which
must be clustered according to their underlying source, and preferably to
discard any spurious noise activations (Plumbley et al., 2010). Temporal
dependence information may help to achieve this (cf. Mysore et al. (2010)).
Timestamped data such as internet access logs often contain no explicit user
association, yet it may be desirable to group such data by user for for
further analysis (Arlitt and Williamson, 1997). In computational audio scene
analysis, it is often the case that sound sources emit sound only
intermittently during their presence in the scene (e.g. bird calls,
footsteps), yet it is desirable to track their temporal evolution (Wang and
Brown, 2006).
### 1.1 Related Work
To our knowledge, this particular problem setting has not been directly
addressed in the literature. Temporal data is most commonly treated using a
model of sources which update continuously, or synchronously at an underlying
temporal sampling rate. Pertinent formulations for our purposes include the
infinite factorial hidden Markov model (infinite FHMM) of Van Gael et al.
(2008), or the probability hypothesis density filter (PHD filter) (Mahler,
2007), both of which infer an unknown number of independent Markov sources.
FHMMs assume that the underlying sources are not intermittent during their
lifetime, and also that they persist throughout the whole observation period.
Pragmatically, intermittent emissions may be handled by incorporating silence
states, though the duration of such states cannot take an arbitrary
distribution. The PHD filter allows for stochastic missed detections but not
for structured intermittency.
Among techniques which do not assume a synchronous update, graph clustering
approaches such as normalised cuts have similarities to our approach (Shi and
Malik, 2000). In particular, Lagrange et al. (2008) apply normalized cuts in
order to cluster temporally-ordered data. However, the normalised cuts method
is applied to undirected graphs, and Lagrange et al. (2008) use perceptually-
motivated similarity criteria rather than directed Markov dependencies as
considered herein. Further, the normalized cuts method does not include a
representation of clutter noise, and so Lagrange et al. (2008) perform
signal/noise cluster selection as a separate postprocessing step. In the
present work we include an explicit noise model.
Our problem setting also exhibits similarities with that of structure
discovery in Bayesian networks (Koivisto and Sood, 2004). However, in that
context the dependency structure is inferred from correlations present in
multiple observations from each vertex in the structure. In the present case
we have only one observation per vertex, plus the partial ordering implied by
temporality.
In the following we develop a model in which an unknown number of point-
process sources are assumed to be active as well as Poisson clutter, and
describe how to perform a maximum likelihood inference which clusters the
signal into individual identified tracks plus clutter noise. We then
demonstrate the performance of the approach in synthetic experiments, and in
an experiment analysing birdsong audio.
## 2 Preliminaries
Throughout we will consider sets of observations in the form $\\{(X,T)\\}$
where $X$ is state and $T$ is time. A Markov renewal process (MRP) generates a
sequence of such observations having the Markov property:
$\displaystyle P(\tau_{n+1}\leq
t,X_{n+1}=j|(X_{1},T_{1}),\ldots,(X_{n}=i,T_{n}))$
$\displaystyle=P(\tau_{n+1}\leq t,X_{n+1}=j|X_{n}=i)$
$\displaystyle\qquad\qquad\,\forall n\geq 1,\,t\geq 0,\,i,j\in\mathcal{S}$ (1)
where $\tau_{n+1}$ is the time difference $T_{n+1}-T_{n}$. Note that $\tau$ is
not explicitly given in observations $\\{(X,T)\\}$, but can be inferred if we
know that a particular pair of observations are adjacent members within a
sequence.
We will have cause to represent our data as a network flow problem (Bang-
Jensen and Gutin, 2007, Chapter 3). A network is a graph supplemented such
that each arc $A_{ij}$ has a lower capacity $l_{ij}$ and upper capacity
$u_{ij}$, and a cost $a_{ij}$. A flow is a function
$x:A\rightarrow\mathcal{R}_{0}$ that associates a value with each arc in the
network. We will be concerned with integer flows
$x:A\rightarrow\mathcal{Z}_{0}$. A flow is feasible if $l_{ij}\leq x_{ij}\leq
u_{ij}$ for all $A_{ij}$ in the graph, and for all vertices (except for any
source/sink vertices) the sum of the inward flow is equal to the sum of the
outward flow. For any flow we can calculate a total cost as the sum of
$a_{ij}x_{ij}$ over all $A_{ij}$. We define the value of a feasible flow to be
the sum of $x_{ij}$ over all arcs leading from source vertices.
The standard terminology of flow networks associates capacities, flows and
costs with arcs but not vertices. However, in the following we will have cause
to associate such attributes with vertices as well as with arcs. This can be
implemented transparently by the standard technique of vertex expansion, in
which each vertex is replaced by an in-vertex and an out-vertex, plus a single
arc between them which bears the associated attributes (Bang-Jensen and Gutin,
2007, Section 3.2.4).
## 3 Mixtures of Markov Renewal Processes with Clutter Noise
For the present task, we consider MRPs which are time-limited: each process
comes into being at a particular point in time (governed by an independent
Poisson process with intensity $\lambda_{b}(X)$), and after each observation
it may “die” with an independent death probability $p_{d}(X)$. Otherwise it
transitions to a new random state-and-time according to the transition
distribution $f_{x}(X,\tau)$. The overall system to be considered is not one
but a set of such time-limited MRPs, plus a separate Poisson process that
generates clutter noise with intensity $\lambda_{c}(X)$. The MRPs are
independent but share common parameters. We will refer to the overall system
(including the noise process) as a multiple Markov renewal process system or
MMRP, in order to clarify when we are referring to the whole system or to a
single MRP.
We receive a set of $N$ observations in the form $\\{(X,T)\\}$ and we assume
that they were generated by an MMRP for which the process parameters are
known, but the number $K$ of MRPs is unknown as well as the allocation of each
observation to its generating process. We assume that each observation is
generated either by one MRP or by the noise process. Given these observations
as well as model parameters $f_{x}(X,\tau)$, $\lambda_{b}$, $p_{d}$,
$\lambda_{c}$, there are many ways to cluster the observations into
$K\in[0,N]$ non-overlapping subsets to represent the assertion that each
cluster represents all the emissions from a single MRP, with $H$ of the
observations not included in any cluster and considered to be noise. The
overall likelihood under a chosen clustering is given by
$\displaystyle\textrm{likelihood}=\prod_{k=1}^{K}{p_{\textrm{MRP}}(k)}\prod_{\eta=1}^{H}{p_{\textrm{NOISE}}(\eta)}$
where $p_{\textrm{MRP}}(k)$ represents the likelihood of the observation
subsequence in cluster $k$ being generated by a single MRP, and
$p_{\textrm{NOISE}}(\eta)$ represents the likelihood of a single observation
datum under the noise model. (A set of clusters is arbitrarily indexed by
$k\in[1,K]$.)
In order to find the maximum likelihood solution, we may equivalently divide
the likelihood expression through by a constant factor, to give an alternative
expression to be maximised. We divide by the likelihood that all data were
generated by the noise process, to give the likelihood ratio:
$\displaystyle\textrm{L}=\prod_{k=1}^{K}{\frac{p_{\textrm{MRP}}(k)}{p_{\textrm{NOISE}}(k)}}$
(2)
where for notational simplicity we use $p_{\textrm{NOISE}}(k)$ as the joint
likelihood of all observations contained within cluster $k$ under the noise
model. This likelihood ratio $L$ will shortly be seen to be a convenient
expression to optimise.
The component likelihood ratio for a single cluster $k$ is given by
$\displaystyle\frac{p_{\textrm{MRP}}(k)}{p_{\textrm{NOISE}}(k)}=\frac{p_{b}(X_{k,1})\,\cdot
p_{d}(X_{k,n})\,\cdot\prod_{i=2}^{n_{k}}{f_{X_{k,i-1}}(X_{k,i},T_{k,i}-T_{k,i-1})}}{\prod_{i=1}^{n_{k}}{p_{c}(X_{k,i})}}$
(3)
where $(X_{k,i},T_{k,i})$ refers to the $i$th observation assigned to cluster
$k$, this cluster having $n_{k}$ observations indexed in ascending time order.
$p_{d}(\cdot)$ refers to the likelihood associated with a single observation
under the Poisson process parametrised by $\lambda_{d}$, and similarly for
$p_{c}(\cdot)$ for the clutter process parametrised by $\lambda_{c}$.
The overall likelihood ratio $L$ tells us the relative likelihood that the
observation set was generated by the selected clustering of signals and noise,
as opposed to the possibility that all observations were generated by clutter
noise. Our goal is to find the clustering that yields the highest likelihood
ratio, and therefore the set of MRP track identities that is most likely to
originate from signal rather than noise.
### 3.1 Network Flow Representation
For any observation set of non-trivial size, there is a combinatorial
explosion of possible clusterings available and enumerating them all is
intractable. In this subsection we propose to transform the problem into an
equivalent problem of network flow, which can be addressed using graph
theoretic techniques.
To maximise the likelihood ratio, we can equivalently minimise its negative
logarithm, which we will consider as a “cost” for any particular solution. We
define additive component costs for birth, death, transition and clutter
respectively as:
$\displaystyle a_{b}(X)$ $\displaystyle=-\log{p_{b}(X)}$ (4a) $\displaystyle
a_{d}(X)$ $\displaystyle=-\log{p_{d}(X)}$ (4b) $\displaystyle
a_{t}(X,X^{\prime},\tau)$ $\displaystyle=-\log{f_{X}(X^{\prime},\tau)}$ (4c)
$\displaystyle a_{c}(X)$ $\displaystyle=\log{p_{c}(X)}$ (4d)
which leads to the following expression for the overall cost under a
particular cluster assignment:
$\displaystyle-\log(\textrm{L})=\sum_{k=1}^{K}$
$\displaystyle\left(a_{b}(X_{k,1})+a_{d}(X_{k,n})\frac{}{}\right.$
$\displaystyle+\sum_{i=2}^{n_{k}}{a_{t}(X_{ik,i-1},X_{k,i},T_{k,i}-T_{k,i-1})}$
$\displaystyle\left.+\sum_{i=1}^{n_{k}}{a_{c}(X_{k,i})}\right).$ (5)
The Markov structure of transitions, as well as this representation as
additive costs, permit a natural representation as a problem defined on a
directed graph. If we construct a directed graph with observations as vertices
and possible transitions as arcs, then every possible path in the graph (from
any vertex to any other reachable vertex) corresponds to one potential MRP
cluster (Figure 1). A set of $K$ paths corresponds to a set of $K$ MRP
clusters. To reflect the assumption that each observation is generated by no
more than one MRP, we require that a vertex can be a member of no more than
one path in such a set. Vertices not included in any of the paths correspond
to noise observations.
Figure 1: Simple illustration of a path within a network that might correspond
to a single MRP sequence. Time increases along the horizontal axis. The bold
arrows indicate a path from the first to the third datum (the second datum
being left out of the corresponding cluster). The thin arrows indicate an
alternative possible path.
Given our restriction that a vertex can be included in no more than one path,
the problem of finding a mutually compatible set of MRP clusterings is
equivalent to solving a particular kind of network flow problem (Bang-Jensen
and Gutin, 2007, Chapter 3). In our case, the concept of a flow will be used
to pick out a set of arcs in the graph corresponding to a possible clustering,
by associating each arc with a value 1 or 0 indicating whether the arc is
included in the clustering. Therefore, in addition to the requirement that the
flow is integer-valued, all arcs will be defined to have unit capacity:
$l_{ij}=0,u_{ij}=1$ for all $A_{ij}$. To reflect our assumption that each
observation can be included in only one cluster, we will also specify unit
capacities for all vertices.
It remains to specify how we can associate the costs (4) with the network such
that we can solve for the minimum-cost solution to (5). Transition costs will
be associated with arcs, and clutter costs with vertices, but in order to
include birth and death costs we must modify the network by adding a single
“source” vertex with an outward arc to all other vertices, and a single “sink”
vertex with an inward arc from all other vertices, and by requiring that no
other vertices act as sources or sinks (i.e. in a feasible flow, their inward
and outward flows must balance). We then associate birth costs with arcs from
the source and death costs with arcs to the sink. This means that all feasible
flows in our network will be composed of paths which consist of one single
birth cost, plus a sequence of clutter and transition costs, and a single
death cost. The source and sink have infinite capacity, allowing for solutions
with unbounded $K$.
Figure 2: Constructing the weighted flow network for a set of three
observations. Figure 3: The network of Figure 2, with a single-path flow
indicated (s-2-3-t).
Putting these considerations together, constructing the directed graph
proceeds as follows:
* •
A unit-capacity vertex $V_{i}$ is created corresponding to each observation
$(X_{i},T_{i})$. The clutter noise cost $a_{c}(X_{i})$ is associated with this
vertex.
* •
A unit-capacity arc $A_{ij}$ is created corresponding to each possible
transition between two observations such that $T_{i}<T_{j}$. The transition
cost $a_{t}(X_{i},X_{j},T_{j}-T_{i})$ is associated with this arc.
* •
A “source” vertex $s$ is added, with one arc $A_{si}$ leading from $s$ to each
of the observation vertices. The birth cost $a_{b}(X_{i})$ is associated with
each arc $A_{si}$.
* •
A “sink” vertex $t$ is added, with one arc $A_{it}$ leading from each of the
observation vertices to $t$. The death cost $a_{d}(X_{i})$ is associated with
each arc $A_{it}$.
The temporal ordering of observations means that the graph will contain no
cycles.
An illustration of the network constructed for a set of three observations is
given in Figure 2. It is clear that any path from the source $s$ to a sink $t$
(we call this an ($s,t$)-path) visits a sequence of vertices representing a
temporal sequence of observations. In the case given in Figure 2, seven
different ($s,t$)-paths are possible, and various combinations of these can
form a feasible flow. For example the flow along the single path s-2-3-t
highlighted in Figure 3 represents the possibility that the observations
$X_{2}$ and $X_{3}$ were generated by a single MRP while $X_{1}$ is clutter:
the costs associated with flow along that path (the path flow) are related to
the birth of 2, the transition from 2 to 3, and the death of 3, plus the
clutter noise costs. The cost associated with any single-path flow corresponds
to one of the $K$ top-level summands in Equation (5). Since in our case each
(s,t)-path carries one unit of flow, the value of each feasible flow is the
number of paths it contains, and corresponds to the number of MRP processes
inferred in the data. The total cost of each feasible flow is the sum of the
path costs contained, and corresponds to the sum calculated in Equation (5).
### 3.2 Inference
The minimum cost flow in a network constructed according to our scheme
corresponds to the clustering with maximum likelihood ratio. So to perform
inference we can use existing algorithms that solve minimum-cost network flow
problems. The value of the minimum-cost flow, which gives the number of MRP
sources inferred, may be any integer between $0$ and $N$. We use the Edmonds-
Karp algorithm (Bang-Jensen and Gutin, 2007, Chapter 3), which iteratively
searches for single paths in a residual network representation and does not
get trapped in local optima. The Edmonds-Karp algorithm is often used to find
maximum-value flow but can be used to optimise cost in our case of binary
capacities.
We now consider the time complexity of our inference. The asymptotic time
complexity of the Edmonds-Karp search relates to the number of vertices and
arcs as $O(|V||A|^{2})$. The number of vertices is closely related to the
number of observations $N$; since we generate an arc for every possible
transition between a pair of observations, $|A|$ may be on the order of
$N^{2}$ in the worst case. Hence we add a constraint in constructing the arcs
which is reasonable in many applications: we assert that transitions have an
upper limit in the size of the time step, and so we do not create arcs for
time separations above some threshold $\tau_{\mathrm{max}}$. The cardinality
$|A|$ is then on the order of $NB$ where $B$ is the maximum number of
observations within a time window of size $\tau_{\mathrm{max}}$ (and often
$B<<N$).
If faster search is required at the cost of optimality, greedy search
strategies are available. One such strategy is to repeatedly apply a minimum-
cost path algorithm to the network, at each iteration taking the resulting
path as an identified cluster and removing its vertices from the network
before the next iteration. Since the graph is acyclic, finding a minimum-cost
path can be performed very efficiently with order $O(|A|+|V|)$ at each
iteration (Bang-Jensen and Gutin, 2007, Section 2.3.2); however there is no
guarantee of optimality since the overall minimum-cost flow is not guaranteed
to be composed of path flows of lowest individual cost. In our experiments we
will compare this greedy search empirically against the optimal search.
In the present work we primarily consider offline (batch) inference. However,
online inference is possible within the same framework, in which new
observations are received incrementally by updating the graph as observations
arrive. The Edmonds-Karp search cannot be used on such a dynamic network,
except by re-starting the search from scratch upon update. Alternative
strategies such as those based on cycle-cancelling can be used to provide an
updateable inference (Bang-Jensen and Gutin, 2007, Section 3.10.1). The speed
of cycle-cancelling relative to Edmonds-Karp may depend on the nature of the
data; we implemented both and found the cycle-cancelling relatively slow.
Thus far we have considered inference using a single set of MMRP model
parameters, encoded as the costs in (5). It may be of value to evaluate the
same data under different MMRP models, in situations where multiple types of
MRP process (having different parameters) may be active. Multiple
parametrisations cannot be represented together in a single flow network since
they would assign conflicting costs to arcs. To accommodate incompatible costs
is equivalent to the “multi-commodity” extension of the minimum-cost flow
problem, which is NP-complete (Even et al., 1975). However, if the clutter
noise model is held constant between two different MMRP inferences, then the
two likelihood ratios calculated by (2) can be divided through to give a
likelihood ratio between the two. This allows us to choose between possible
MMRP models although not to combine them in a single clustering.
To summarise the MMRP inference described in this section: given a set of
observations plus MRP process parameters and noise process parameters, one
first represents the data as a flow network, with added source and sink nodes,
and with costs representing component likelihoods (Section 3.1). One then
applies a minimum-cost flow algorithm to the network such as Edmonds-Karp.
Each ($s,t$)-path in the resulting minimum-cost flow represents a single
cluster (a single MRP sequence) in the maximum-likelihood result, while the
nodes which receive no flow represent data to be labelled as noise.
## 4 Experiments
We have described a multiple Markov renewal process (MMRP) inference technique
which takes an MRP model, an iid clutter noise model and a set of timestamped
data points, and finds a maximum-likelihood partition of the data into zero or
more MRP sequences plus clutter noise. In the following, we will illustrate
its properties with a synthetic experiment (Section 4.2), before applying it
to a specific task of tracking multiple singing birds in an audio mixture
(Section 4.3). We must first consider how to evaluate algorithm outputs.
### 4.1 Evaluation Measures
To judge the empirical performance of our inference procedure, we must
determine whether it can correctly separate signal from noise, and whether it
can correctly separate each individual MRP sequence into its own stream. MMRP
inference can be considered as a clustering task and could be evaluated
accordingly. However, the noise cluster is qualitatively different from the
MRP clusters, and the transitions within MRP sequences are the latent features
of primary interest, so we will focus our evaluation measures on signal/noise
separation and transitions.
In the following our statistics will be based on the standard F-measure
(Witten and Frank, 2005, Chapter 5), which summarises precision and recall as
follows:
$\displaystyle F$
$\displaystyle=2\cdot\frac{\text{precision}\cdot\text{recall}}{\text{precision}+\text{recall}}$
(6) $\displaystyle=\frac{2t_{+}}{(2t_{+}+f_{-}+f_{+})}$ (7)
where $t_{+}$ is the number of true positive detections, $f_{+}$ the number of
false positive detections (noise data labelled as signal), and $f_{-}$ the
number of false negative detections (signal data labelled as noise). However,
the task for which our MMRP inference is designed is not an ordinary
classification task: the signal/noise label for each ground-truth datum can be
treated as a class label to be inferred, but the individual signal streams to
be recovered do not have labels. To quantify performance we use the F-measure
in two ways. The first (which we denote $F_{\text{SN}}$) evaluates the
signal/noise classification performance without considering the clustering.
The second (which we denote $F_{\text{trans}}$) evaluates the performance at
recovering the pairwise transitions that are found in the ground-truth
signals, i.e. the arcs in the true dependency graph underlying the data.
Figure 4: Illustration of errors reflected in $F_{\text{trans}}$. The upper
diagram shows a hypothetical ground-truth transition through a sequence of
five observations (circles) accompanied by clutter noise (crosses). The lower
diagram shows what would happen if inference missed one of those observations
out of the chain, resulting in two false-negatives (dotted arrows) for ground-
truth transitions not recovered, plus one false-positive (dashed arrow) for a
transition that does not exist in the ground-truth. Considering these as well
as the two true-positives and applying (7), the $F_{\text{trans}}$ value here
is $\frac{4}{7}$.
To illustrate $F_{\text{trans}}$, consider a situation in which a ground-truth
sequence was perfectly recovered except that one datum in the middle was left
out (Figure 4). This would correspond to a number of true positives, but also
two false negatives (the omission of the transition into and out of the
missing datum) and one false positive (the mistaken inference of a transition
from the missing datum’s predecessor to its follower).
Correctly-classified noise observations do not affect $F_{\text{trans}}$ since
they are not associated with any signal transitions. Thus, $F_{\text{SN}}$ is
useful to measure signal/noise separation while $F_{\text{trans}}$ provides
complementary information about correctly recovering separate streams.
### 4.2 Synthetic Experiment
For our synthetic experiment we generated data in a one-dimensional state
space, with dependency structures inspired by the classic “audio streaming”
experiments used to explore human auditory grouping of sound sequences
(Winkler et al., 2012).
A strictly alternating sequence of the form ABABAB…, where A and B are
different tones (Figure 5, top row), can be interpreted either as a single
alternating sequence (the “coherent” interpretation) or as a simultaneous but
out-of-phase pair of constant sequences (the “segregated” interpretation).
Various factors can lead an observer to prefer one interpretation or the
other; here we focus on the case where drift in the timing of the events makes
one or the other model more likely (Cusack and Roberts, 2000, Experiment 2).
If the sequences drift such that the phase of the As and Bs remain in constant
relationship (Figure 5, second row), this is consistent with a “coherent”
alternating generator, though may by chance be generated by a “segregated”
pair of generators. If the sequences drift such that the phase relationship is
not maintained (third row), then this is inconsistent with the “coherent”
model but consistent with the “segregated” model. We can generate data with
these properties and observe how the MMRP inference behaves under the
assumptions of each model.
Figure 5: Examples of sequences generated by strict locked ABABAB repetition
(top), and by similar generators but with time offsets affected by process
noise reflecting either coherent (ABABAB, middle) or segregated (A_A_A_ and
_B_B_B, bottom) dependency structure.
For our synthetic experiment we defined two separate MRP transition models
(one “coherent” and one “segregated”) to emit values in a one-dimensional
state space $\mathcal{X}\in\mathbb{R}$. Each model was specified by a Gaussian
mixture probability distribution defined on state-delta and log-time-delta:
$\displaystyle P(\tau_{n+1}\leq t,X_{n+1}=j|X_{n}=i)$
$\displaystyle=f(X_{n+1}-X_{n},\log{\tau_{n+1}})$ (8)
Figure 6 illustrates the transition models. Time differences here are modelled
as log-Gaussian to reflect a simple yet perceptually plausible model for
lower-bounded time intervals. The variance of the Gaussian components leads to
process noise, and the two models tend to output different sequences in
general. We also define a “locked” model for generation only, which generates
a strict ABABAB sequence with no process noise. Its emissions could in
principle be explained by either of the two other models.
These models served two roles in our experiment, to synthesise data and to
analyse it. For synthesis, we generated one, two or four simultaneous
sequences each with a random offset in state space, and we also added iid
Poisson clutter noise in the same region of state space, whose intensity is
held constant within each run to create a given SNR. In the case of the
segregated model, each generator was a pair of such models, independent except
for the initial phase and offset, generating As and Bs as was done in Figure
6.
Figure 6: MRP transition probability densities for the two synthetic models:
coherent (upper) and segregated (lower).
The first column of Figure 7 shows the results of generating data under the
locked, coherent and segregated models, with two generated sequences present
in each case. The second column shows the sequences with added clutter noise
at an SNR of -12 dB. The final two columns show the maximum-likelihood signal
sequences inferred under the coherent and the segregated model. The MMRP
inference typically extracts clear traces corresponding to the ground-truth
signals, even in strongly adverse SNR. It is visually evident in the first
column that the generated sequences in the middle row have some drift in their
rate, but stay in order, while the As and Bs in the bottom row drift relative
to each other and do not maintain order. This leads to unlikely emission
sequences as judged by the coherent model, and so the coherent model finds the
maximum-likelihood solution to be that with no sequences (the blank plot in
the figure). Inference using the segregated model extracts traces in all three
cases, since the phase-locked drift of the coherent model is not unlikely
under the segregated model.
Figure 7: Results of generating observations under the locked, coherent or
segregated model (in each row), and then analysing them using the coherent
model or the segregated model (final two columns).
To evaluate our inference procedure, we ran this process multiple times,
varying the SNR level, the number of items present, and whether the true SNR
was known to the algorithm. When not known, the SNR estimate was arbitrarily
held fixed at 0 dB. We tested both the optimal and greedy inference algorithms
described in Section 3.2. For each setting we conducted 20 runs and recorded
the $F_{\text{SN}}$ and $F_{\text{trans}}$ statistics. Figures 8 and 9
illustrate the results, and show a consistent pattern according to both
statistics. Recovery performance is very strong in all but the most adverse
conditions, in most cases being well above 0.95. For these particular
scenarios, recovery is impaired under the strongest condition tested (4
simultaneous generators and SNR -24 dB). Under other conditions the recovery
is good, whether the true SNR is known to the algorithm or not. Knowing the
true SNR does not add a clear improvement to performance, showing that the
inference is robust to the SNR estimate parameter. Greedy inference has lower
time complexity than the full inference, but when there are multiple streams
to be recovered it yields poorer performance than the full algorithm even at
very favourable SNR.
Figure 8: F-measure for signal/noise separation ($F_{\text{SN}}$, upper) and
transitions ($F_{\text{trans}}$, lower). The ground truth in each case is a
single ABABAB stream, generated via the coherent (left) and segregated (right)
cases. Means and standard errors are shown; the vertical axis is reverse-log-
scaled so that the results very near 1.0 can be distinguished.
Figure 9: As Figure 8 but with four simultaneous generated streams rather than
one.
### 4.3 Birdsong Audio Experiment
Many natural sound sources produce signals with structured patterns of
emissions and silence, for example birdsong or footsteps. If the emissions due
to one such source can be modelled as an MRP, then our inference procedure
should be able to separate multiple simultaneous “streams” of emissions. In
the following experiment we studied the ability of our inference to perform
this separation in data derived from audio signals containing multiple
instances of a species of bird common in many European countries, the Common
Chiffchaff (Salomon and Hemim, 1992). Chiffchaff song consists of sequences of
typical length 8–20 “syllables”. Each syllable is a pitched note consisting of
a downward chirp to a briefly-held tone in the region of 5–8 kHz. Syllables
are separated by around 0.2–0.3 seconds. The exact note sequence has not to
our knowledge been studied in detail; it appears to exhibit only short-range
dependency, and is thus amenable to analysis under Markovian assumptions.
#### 4.3.1 Data Preparation
To aid reproducibility, we used recordings from the Xeno Canto database of
publicly-available bird recordings.111http://www.xeno-canto.org/europe We
located 25 recordings of song of the Chiffchaff (species Phylloscopus
collybita) recorded in Europe (excluding any recordings marked as having
“deviant” song or uncertain species identity; also excluding calls which are
different from song in sound and function). The recordings used are listed in
Table 1. We converted the recordings to 44.1 kHz mono wave files, high-pass
filtered them at 2 kHz, and normalised the amplitude of each file.
ID | Country
---|---
XC103404 | pl
XC25760 | dn
XC26762 | se
XC28027 | de
XC29706 | se
XC31881 | nl
XC32011 | nl
XC32094 | no
XC35097 | es
XC35974 | cz
XC36603 | cz
XC36902 | nl
XC46524 | nl
ID | Country
---|---
XC48263 | no
XC48383 | de
XC54052 | it
XC55168 | fr
XC56298 | de
XC56410 | ru
XC57168 | fr
XC65140 | es
XC77394 | dk
XC77442 | se
XC97737 | uk
XC99469 | pl
Table 1: Chiffchaff audio samples used in our dataset, giving the Xeno Canto
ID and the country code. Each recording can be accessed via a URL such as
http://www.xeno-canto.org/XC103404, and the dataset is also archived at
http://archive.org/details/chiffchaff25
Each audio file was analysed separately to create training data; during
testing, audio files were digitally mixed in groups of two to five files.
In order to convert an audio file into a sequence of events amenable to MMRP
inference, we used spectro-temporal cross-correlation to detect individual
syllables of song, as used by Osiejuk (2000). We designed a spectrotemporal
template using a Gaussian mixture (GM) to represent the main characteristics
of a single Chiffchaff syllable, a downward chirp to a briefly-held note
(Figure 10). The GM was modelled on a Chiffchaff recording from Xeno Canto
which was not included in our main dataset (ID number XC48101). Then to
analyse an audio file we converted the file into a spectrogram representation
(512 samples per frame, 50% overlap between frames, Hann window), and
converted the GM to a sampled grid template with the same time-frequency
granularity as the spectrogram, before sliding the grid template along the
time axis and along the frequency axis (between 3–8 kHz), evaluating the
correlation between the template and spectrogram at each location. Correlation
values were treated as detections if they were local peaks with value greater
than a threshold correlation of 0.8.
Figure 10: Template used for spectro-temporal cross-correlation detection. The
downward and horizontal bars have equal total weight; the latter appears
darker because shorter. The template is a manually-constructed Gaussian
mixture model having 40 components.
Figure 11: Example of cross-correlation detection: excerpt of spectrogram
shown (top), and the corresponding detections (bottom). In the lower image,
bold lines represent detections treated as “signal” in the filtering used for
training, while the fainter lines represent detections used to train the noise
model.
Such cross-correlation detection applied to an audio file produces a set of
observations, each having a time and frequency offset and a correlation
strength (Figure 11). It typically contains one detection for every Chiffchaff
syllable, with occasional doubled detections and spurious noise detections.
When applied to mixtures of audio, this produces data appropriate for MMRP
inference.
In order to derive a Gaussian mixture model (GMM) transition probability model
from monophonic Chiffchaff training data, for each audio file in a training
set we filtered the observations automatically to keep only the single
strongest detection within any 0.2 second window. This time limit corresponds
to the lower limit on the rate of song syllables; such filtering is only
appropriate for monophonic training sequences and was not applied to the audio
mixtures used for testing. The filtered sequences were then used to train a
10-component GMM with full covariance, defined on the vector space having the
following four dimensions:
* •
log(frequency) of syllable one
* •
log(frequency) of syllable two
* •
log(magnitude ratio between syllables)
* •
log(time separation between syllables)
We also trained a separate GMM to create a noise model, taking the set of
observations that had been discarded in the above filtering step and training
a 10-component GMM with full covariance to fit an iid distribution to the one-
dimensional log(frequency) data for the noise observations.
#### 4.3.2 Inference from Audio Mixtures
In order to test whether the MMRP approach could recover syllable sequences
from audio mixtures, we performed an experiment using five-fold cross-
validation. For each fold we used 20 audio files for training, and then with
the remaining five audio files we created audio mixtures of up to five
signals, testing recovery in each case.
The quality of signal/noise separation and of clustering the syllables
correctly could depend on various features of the experimental task, including
whether observations could be extracted from audio mixtures as reliably as
from single recordings, the generalisability of the fitted GMMs, noise levels,
and the MMRP inference procedure. In order to explore these factors we
compared various different analysis approaches:
Audio recovery:
The primary approach was to take a mixture audio file, apply spectro-temporal
cross-correlation as described above, then to apply MMRP inference using the
signal and noise GMMs.
Audio recovery (greedy):
This approach was as above, but using greedy recovery rather than the optimal
flow inference.
Ideal recovery:
There is no guarantee that the same observations will be recovered from the
mixture audio as were recovered from the individual recordings. To simulate
ideal-case recovery, instead of using the audio mixture we simply pooled the
signal and noise observations that had been derived from the test set’s
individual mono analysis, then performed MMRP inference as in the audio
recovery case.
Ideal recovery, synthetic noise:
To simulate ideal recovery but with more adverse noise conditions, we
proceeded as in the ideal case, but also added extra clutter noise at 0 dB. To
do this, we created a copy of every observation in the test set, but assigned
it an independent random time position, thus creating noise with the same
frequency distribution as the true signal.
Ideal recovery, tested on training set:
To measure an “upper limit” on performance and probe the generalisation
capability of the algorithm, we proceeded as in the ideal case, but used GMMs
trained on the actual test files to be analysed rather than on the separate
training data. If this resulted in stronger performance than the ideal-case,
it would indicate issues with generalising to signals outside the training
set.
Audio recovery, baseline:
In order to provide a low-complexity baseline showing the recovery quality
using only the marginal properties of the signal and noise, we created a
simple baseline system which treated both signal and noise as iid one-
dimensional log(frequency) data, using maximum likelihood to label each
observation as either signal or noise. The baseline system then clustered
together observations that were identified as signal and were separated by
less than 0.7 seconds.
We tested each of these approaches using mixtures of one, two, three, four or
five of the test recordings. As in the previous experiment, we measured the
$F_{\text{SN}}$ statistic to evaluate signal/noise separation, and the
$F_{\text{trans}}$ statistic to evaluate the performance at recovering
separate sequences.
Figure 12: The $F_{\text{SN}}$ and $F_{\text{trans}}$ evaluation measures for
the Chiffchaff audio analyses. Means and standard errors are shown taken over
the five folds of the cross-validation.
Results are shown in Figure 12. Although the two statistics we measure reflect
different aspects of performance, they both rank the different analysis
approaches in a very similar way. All the MMRP inference runs exhibit a
significant and very strong improvement over the baseline. Very strong
performance is achieved in the noiseless “ideal recovery” cases, achieving
results similar to those in the previous synthetic experiment. The small size
of the difference between training on the test data and on the training data
indicates that the algorithm can generalise across the data used in our
experiment.
When synthetic noise is added to the ideal-recovery case, performance is
reduced by a moderate but consistent amount. When we use recovery from audio
mixtures, performance reduces again. This shows that the practical task of
retrieving detections from audio mixtures has a significant effect on the
algorithm performance. However, even in this case our algorithm outperforms
the baseline system by a very wide margin, showing the value of MMRP inference
for separating signal from noise and clustering signals into MRP streams.
As we increase the number of recordings in the mixture, performance of all the
analysis approaches shows a mild decline. However even with five recordings
the performance of the MMRP remains relatively strong.
In this experiment, unlike the previous one, we see very little difference
between the performance of the full inference and the greedy inference. Thus
the faster greedy inference is appropriate in some but not all situations; in
this experiment it is not a limiting factor in performance.
## 5 Conclusions
In this paper we have introduced a specific clustering problem, that of
segregating timestamped data originating in multiple point processes plus
clutter noise. We developed an approach to inferring structure in data
produced by a mixture of an unknown number of similar Markov renewal processes
(MRPs) plus independent clutter noise. The inference simultaneously
distinguishes signal from noise as well as clustering signal observations into
separate source streams, by solving a network flow problem isomorphic to the
MMRP mixture problem.
In a synthetic experiment we have shown that inference can perform very well
even under high noise conditions (up to $-24$ dB SNR). In an experiment on
birdsong audio data we have also shown strong performance, albeit with a
dependence on the quality of the underlying representation to recover events
from audio data. Our method is general and has very few free parameters.
The inference in the present work is limited to models without hidden state
and with only single-order Markov dependencies. These limitations arise from
the combinatorial ambiguity in MMRP mixtures (unlike ordinary Markov models)
over which is the immediate predecessor for each observation. Future work will
aim to find techniques to broaden the class of models that can be treated in
this way.
## Acknowledgments
(Acknowledgments to be added in final version.)
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|
arxiv-papers
| 2012-11-13T12:43:45 |
2024-09-04T02:49:37.937646
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Dan Stowell and Mark D. Plumbley",
"submitter": "Dan Stowell",
"url": "https://arxiv.org/abs/1211.2972"
}
|
1211.3055
|
# The LHCb Trigger and its Performance in 2011
R. Aaijo J. Albrechtm F. Alessiom S. Amatoa E. Aslanidesd I. Belyaevi M. van
Beuzekomo E. Bonaccorsim R. Bonnefoyc L. Brardam O. Callote M. Cattaneom H.
Chanalc M. Chebbim X. Cid Vidall M. Clemencicm J. Closierm V. Cocoo J. Cogand
O. Deschampsc H. Dijkstram Corresponding author. C. Drancourtb R. Dzhelyadinj
M. Frankm M. Gandelmana C. Gasparm V.V. Gligorovm C. Göbelv L.A. Granado
Cardosom Yu. Guzj C. Haenm J. Hee E. van Herwijnenm W. Hulsbergeno R.
Jacobssonm B. Jostm T.M. Karbachm U. Kerzelm P. Koppenburgo G. Krockerf C.
Langenbruchm I. Laxg R. Le Gacd R. Lefèvrec J. Lefrançoise O. Leroyd L. Li
Gioic G. Lium F. Macheferte I.V. Machikhiliyanb,i M. Magnec G. Mancinellid U.
Marconig A. Martín Sáncheze M.-N. Minardb S. Monteilc N. Neufeldm V. Niessc S.
Oggeroo A. Pérez-Calero Yzquierdok P. Perretc M. Perrin-Terrind B. Pietrzykb
A. Puig Navarron G. Ravenp P. Robbee H. Ruizk M.-H. Schunee R. Schwemmerm J.
Serranod I. Shapovalq,m T. Skwarnickiu B. Souza De Paulaa P. Spradlinr S.
Stahlf V.K. Subbiahm S. T’Jampensb F. Teubertm C. Thomast M. Vesterinenm M.
Williamss,1 M. Witekh A. Zvyaginm
aUniversidade Federal do Rio de Janeiro (UFRJ)
Rio de Janeiro Brazil
bLAPP Université de Savoie CNRS/IN2P3
Annecy-Le-Vieux France
cClermont Université Université Blaise Pascal CNRS/IN2P3 LPC
Clermont-Ferrand France
dCPPM Aix-Marseille Université CNRS/IN2P3
Marseille France
eLAL Université Paris-Sud CNRS/IN2P3
Orsay France
fPhysikalisches Institut Ruprecht-Karls-Universität Heidelberg
Heidelberg Germany
gSezione INFN di Bologna
Bologna Italy
hHenryk Niewodniczanski Institute of Nuclear Physics Polish Academy of
Sciences
Kraków Poland
iInstitute of Theoretical and Experimental Physics (ITEP)
Moscow Russia
jInstitute for High Energy Physics (IHEP)
Protvino Russia
kUniversitat de Barcelona
Barcelona Spain
lUniversidad de Santiago de Compostela
Santiago de Compostela Spain
mEuropean Organization for Nuclear Research (CERN)
Geneva Switzerland
E-mail: [email protected]
nEcole Polytechnique Fédérale de Lausanne (EPFL)
Lausanne Switzerland
oNikhef National Institute for Subatomic Physics
Amsterdam The Netherlands
pNikhef National Institute for Subatomic Physics and VU University Amsterdam
Amsterdam The Netherlands
qNSC Kharkiv Institute of Physics and Technology (NSC KIPT)
Kharkiv Ukraine
rSchool of Physics and Astronomy University of Glasgow
Glasgow United Kingdom
sImperial College London
London United Kingdom
tDepartment of Physics University of Oxford
Oxford United Kingdom
uSyracuse University
Syracuse NY United States
vPontifícia Universidade Católica do Rio de Janeiro (PUC-Rio)
Rio de Janeiro Brazil associated to a
1Massachusetts Institute of Technology
Cambridge MA United States
###### Abstract
This paper presents the design of the LHCb trigger and its performance on data
taken at the LHC in 2011. A principal goal of LHCb is to perform flavour
physics measurements, and the trigger is designed to distinguish charm and
beauty decays from the light quark background. Using a combination of lepton
identification and measurements of the particles’ transverse momenta the
trigger selects particles originating from charm and beauty hadrons, which
typically fly a finite distance before decaying. The trigger reduces the
roughly $11$ MHz of bunch-bunch crossings that contain at least one inelastic
$pp$ interaction to 3 kHz. This reduction takes place in two stages; the first
stage is implemented in hardware and the second stage is a software
application that runs on a large computer farm. A data-driven method is used
to evaluate the performance of the trigger on several charm and beauty decay
modes.
###### keywords:
Trigger algorithms; Trigger concepts and systems (hardware and software)
## 1 Introduction
The LHCb detector [1] is a single-arm spectrometer that has been optimised to
perform flavour physics measurements at the LHC. LHCb has a pseudorapidity
acceptance of $2<\eta<5$. The detector layout is shown in Fig. 1.
Figure 1: Layout of the LHCb detector.
It consists of a silicon vertex detector surrounding the $pp$ interaction
region (VELO); a silicon strip detector (TT); a dipole magnet; two Ring
Imaging Cherenkov detectors (RICH 1$\&$2); tracking detectors (T1-T3), which
consist of silicon strip detectors (IT) near the beam and straw tubes (OT)
further out; a calorimeter system consisting of a Scintillating Pad detector
(SPD), an electromagnetic calorimeter with pre-shower (ECAL, PS) and a
hadronic calorimeter (HCAL); and muon chambers (M1-M5).
The LHCb trigger uses all of the above sub-systems. Its architecture consists
of two levels, the first level trigger (L0) and the High Level Trigger (HLT).
L0 is implemented in hardware and uses input from the calorimeter and muon
systems. L0 reduces the rate of crossings with at least one inelastic $pp$
interaction to below 1.1 MHz, at which the whole detector can be read out.
This maximum rate is imposed by the front-end (FE) electronics. The
implementation of L0 is only described briefly in Section 2; a fuller
treatment can be found in reference [1]. The HLT consists of a software
application that runs on a farm of Personal Computers (PCs). It has evolved
significantly compared to reference [1], in which it is assumed that the LHC
machine would operate with a 25 ns bunch separation [2], and that LHCb would
limit the number of visible $pp$ interactions111A visible interaction is
defined as one in which at least two tracks are reconstructed in the VELO,
that both point to the interaction envelope. such that the average number of
visible interactions per bunch crossing $\mu\simeq 0.4$. However, the smallest
bunch separation of the machine was 50 ns in the 2011 physics runs. To
compensate for the loss in number of bunches, and combined with the fact that
the LHCb detector performance did not degrade up to $\mu\approx 2.5$, LHCb
decided to run at $\mu\approx 1.4$. Therefore the HLT had to adapt to running
conditions rather different from those described in reference [1].
The HLT is described in detail in Section 3. The HLT reduces the rate of
accepted events to $\sim 3$ kHz, and all such events are written to storage.
The events written to storage are processed with a more accurate alignment and
calibration of the sub-detectors, and with reconstruction software that is
more elaborate and allows for more redundancy than is possible in the HLT.
This part of the reconstruction and subsequent event selection will henceforth
be referred to as the off-line reconstruction and selection.
The method used to obtain a data-driven determination of the trigger
performance is described in Section 4. Section 5 describes the performance of
the trigger in 2011 relative to off-line reconstruction and selection. Section
6 concludes with a summary of the trigger performance.
## 2 First Level Trigger
L0 is divided into three independent triggers; the L0-Calorimeter trigger, the
L0-Muon trigger and the L0-PileUp trigger. The last of these is not used to
select flavour physics events, but instead aids the determination of the
luminosity [5], and will not be further described in this paper.
The L0 system is fully synchronous with the 40 MHz bunch crossing signal of
the LHC. The latencies are fixed and depend neither on the occupancy nor on
the bunch crossing history. All of the L0 electronics are implemented in
custom-designed hardware that makes use of parallelism and pipelining to do
the necessary calculations within the maximum latency of $4\,\mu$s. The
trigger decisions are combined in a single L0 decision, which is transferred
to the Readout Supervisor board (RS). The RS generates in addition a small
rate of random (NoBias) triggers taking into account the bunch filling scheme
of the machine222Not all of the 3564 slots available for proton bunches around
the machine are filled with protons. Most of the luminosity in 2011 was
collected with 1296 bunches colliding in LHCb.. The RS emulates the state of
the FE buffers to protect against their overflow. It also has information on
the state of the buffers in the readout boards of all sub-detectors and the
availability of the PCs in the farm. Based on this information it can retain
or throttle a bunch crossing.
### 2.1 L0-Calorimeter Trigger Implementation
The L0-Calorimeter system uses information from the SPD, PS, ECAL and HCAL.
These four detectors are stacked along the beam axis ($z$-axis) and their
longitudinal segmentation offers the possibility to distinguish between
photon, electron and hadron showers. Transverse to the beam axis ($x$-$y$
plane) the detectors are segmented into square cells. SPD, PS and ECAL are
divided into three zones with ECAL cell rib sizes of 40.4 mm in the inner zone
close to the beam pipe, 60.6 mm and 121.2 mm further out. HCAL is divided into
two zones with rib size 131.3 mm and 262.6 mm. The SPD, PS and ECAL have the
same geometry and are projective, i.e. the sizes of the cells in the SPD and
PS are adjusted to take into account the different $z$ positions of the
detectors. The HCAL cells are larger but their boundaries always correspond to
the boundaries of the ECAL cells. The L0-Calorimeter system computes the
transverse energy deposited in clusters of $2\times 2$ cells, using only cells
located in the same zone. Hence the cluster energy of showers with energy
deposits in two adjacent zones will be too low. The transverse energy of a
cluster is defined as:
$\mbox{$E_{T}$}=\sum_{i=1}^{4}E_{i}\rm{sin}\theta_{\it i}\,,$ (1)
where $E_{i}$ is the energy deposited in cell $i$ and $\theta_{i}$ is the
angle between the $z$-axis and a neutral particle assumed to be coming from
the mean position of the interaction envelope hitting the centre of the cell.
The ECAL and HCAL signals are read out and processed in FE boards (FEB) that
cover an area of $(8+1)\times(4+1)$ cells, such that the (+1) cells are shared
between neighbouring FEB. Each FEB selects the highest $E_{T}$ cluster among
its 32 clusters. From these clusters, three types of candidates are built
combining information as follows:
1. 1.
Hadron candidate (L0Hadron): the highest $E_{T}$ HCAL cluster. If there is a
highest $E_{T}$ ECAL cluster located in front of the HCAL cluster, the $E_{T}$
of the hadron candidate is the sum of the $E_{T}$ of the HCAL and ECAL
clusters.
2. 2.
Photon candidate (L0Photon): the highest $E_{T}$ ECAL cluster with 1 or 2 PS
cells hit in front of the ECAL cluster and no hit in the SPD cells
corresponding to the PS cells. In the inner zone of the ECAL, an ECAL cluster
with 3 or 4 PS cells hit in front of it is also accepted as photon. The
$E_{T}$ of the candidate is the $E_{T}$ deposited in the ECAL alone.
3. 3.
Electron candidate (L0Electron): same requirements as for a photon candidate,
with in addition at least one SPD cell hit in front of the PS cells.
The $E_{T}$ of the candidates is compared to a fixed threshold and events
containing at least one candidate above threshold are retained by L0.
### 2.2 L0-Muon Trigger Implementation
The muon system contains five muon stations (M1-M5) consisting of pads in the
high occupancy regions and horizontal and vertical strips elsewhere. Strips
are combined to form logical pads for the muon trigger. The pad sizes are
chosen to obtain projectivity towards the interaction region in the $y$-$z$
plane. Each quadrant of the muon detector is connected to a L0 muon processor.
There is no exchange of information between quadrants, hence muons traversing
quadrant boundaries cannot be reconstructed in the trigger. Each of the four
L0 muon processors tries to identify the two muon tracks with the largest and
second largest momentum transverse to the $z$-axis ($p_{T}$) in their
quadrant. The processors search for hits that define a straight line through
the five muon stations and that points towards the interaction point in the
$y$-$z$ plane. In the $x$-$z$ plane the search is limited to muons with
$\mbox{$p_{T}$}\gtrsim 0.5$ ${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$. The
position of a track in the first two stations allows the determination of its
$p_{T}$ with a measured resolution of $\sim 25\,\%$ relative to off-line
reconstructed muon tracks. The trigger sets a single threshold on either the
largest $\mbox{$p_{T}$}^{\rm largest}$ of the eight candidates (L0Muon), or a
threshold on $\mbox{$p_{T}$}^{\rm largest}\times\mbox{$p_{T}$}^{\rm
2nd~{}largest}$ (L0DiMuon).
## 3 High Level Trigger
The HLT runs on the Event Filter Farm (EFF) that is a farm of multiprocessor
PCs. The HLT is a program written in C++, and 26110 copies of it run in the
EFF. An event that is accepted by L0 is transferred by the on-line system from
the FEB to the EFF and is assembled by one of the event builder programs that
run on one of the cores of each multicore node. The assembled events are
placed in a buffer that is accessed by the HLT programs that run on the cores
of the node. A detailed description of this process can be found in reference
[1] and references therein.
The HLT is based on the same software as used throughout LHCb data processing
and simulation [3]. The off-line event reconstruction and selection requires
about 2 s per event. During 2011 the L0 rate was about 870 kHz. Given the
available resources in the EFF this limits the time per event in the HLT to
$\sim$30 ms. The HLT is divided into two stages. The first stage (HLT1)
processes the full L0 rate and uses partial event reconstruction to reduce the
rate to 43 kHz. At this rate the second stage (HLT2) performs a more complete
event reconstruction.
A "trigger line" is composed of a sequence of reconstruction algorithms and
selections. The trigger line returns an accept or reject decision. An event
will be accepted by L0, HLT1 or HLT2 if it is accepted by at least one of its
trigger lines at the relevant stage. Combinations of trigger lines, together
with a L0 configuration, form a unique trigger with its associated Trigger
Configuration Key (TCK). The TCK is a 32 bit label pointing to a database that
contains the parameters that configure the trigger lines. The TCK is stored
for every event in the raw data, together with information on which trigger
lines accepted the event. During 2011 running, the HLT contained 38 HLT1 and
131 HLT2 lines. The trigger lines that cover the main physics goals of LHCb
[4], and accept the majority of events stored, are described below in addition
to the common reconstruction algorithms. The corresponding selection
parameters and their performance are given in Section 5.
The remaining trigger lines consist of lines for luminosity measurements, pre-
scaled physics trigger lines with looser cuts, lines that select very low
multiplicity events and lines that identify large transverse momentum jets.
The trigger also contains lines designed to accept NoBias events, lines that
monitor events with inconsistent raw data or other errors during the HLT
processing, lines that allow the VELO to monitor the position of the $pp$
interaction envelope and lines selecting calibration and monitoring data for
fast feedback on the quality of the data.
### 3.1 HLT1
The off-line VELO reconstruction software is fast enough to permit the full 3D
pattern recognition of all events that enter the HLT. In the off-line VELO
pattern recognition a second pass is made on unused hits to enhance the
efficiency for tracks that point far away from the beam-line, but in the HLT
this search is not executed. At the start of each LHC fill, the mean position
of the $pp$ interaction envelope in the $x$-$y$ plane,
$\rm{PV}_{\rm{xy}}^{\rm{mean}}$, is determined using VELO tracks. This
position is measured to be stable to within a few $\rm\mu m$ per fill. The
VELO tracks are used to construct vertices with at least 5 tracks originating
from them, and those vertices within a radius of $300~{}\mu\rm m$ of $\rm
PV_{xy}^{mean}$ are considered to be primary vertices (PV).
While in the off-line pattern recognition all VELO tracks are considered to
identify the corresponding hits in the tracking stations downstream of the
magnet, the pattern recognition in HLT1 limits the execution time by selecting
VELO tracks that have a larger probability to originate from signal decays.
HLT1 lines that do not require muons select VELO tracks based on their
smallest impact parameter (IP) to any PV. In addition, cuts are applied to the
quality of each VELO track based on the number of hits on a track and the
expected number of hits.
For events triggered by L0Muon or L0DiMuon, a fast muon identification is
performed in HLT1 to select VELO tracks that are muon candidates using the
following procedure. For every VELO track, a search window is defined in the
M3 station by extrapolating the VELO track in a straight line. The magnet does
not bend tracks in the vertical plane, and multiple scattering dominates the
vertical size of the search window. A muon candidate is required to have a
momentum of at least 6 ${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$, hence the
horizontal search window size corresponds to the deflection of a 6
${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ track. Hits found inside the search
window are combined with the VELO track to form candidate tracks that are used
in a search for additional muon hits in stations M2, M4 and M5. A candidate
track is provisionally accepted if it contains at least one hit in addition to
the M3 hit. In the final step of the algorithm, a linear $\chi^{2}$ fit of the
candidate track (containing both the VELO track and the muon hits) in the
horizontal plane is performed and the $\chi^{2}~{}\rm
divided~{}by~{}the~{}number~{}of~{}degrees~{}of~{}freedom~{}(ndf)$ is required
to be less than 25\. As soon as the first candidate is found, the algorithm
stops and the VELO track is tagged as a muon candidate.
For the VELO tracks that are selected by either their IP or by being tagged as
a muon candidate, the track-segments in the OT and IT-stations are
reconstructed to determine their momentum in a procedure known as forward
tracking. Imposing a minimum momentum and transverse momentum
($p,~{}\mbox{$p_{T}$}$) in the forward tracking significantly reduces the
search windows that have to be opened in the IT and OT tracking stations
thereby reducing the required processing time. Each reconstructed track is
fitted using a Kalman filter [6] based track fit to obtain its $\chi^{2}$ and
a full covariance matrix at the start of the track. Compared to off-line
reconstruction, this fit uses a simplified material geometry description, it
makes fewer iterations and consequently it performs a less sophisticated
removal of outlier hits. The invariant mass resolution of
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\rightarrow\mu^{+}\mu^{-}$
determined in the HLT is measured to be $3\,\%$ larger than the 14
${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ obtained off-line. This shows that
the resolution of the track parameters obtained in the HLT is sufficiently
close to off-line to allow selective cuts in IP, momentum and mass. For tracks
that are tagged as muon candidates, the off-line muon identification algorithm
[7] is applied to the tracks to improve the purity of the muon sample.
### 3.2 HLT2
As mentioned above, HLT1 reduces the rate from 870 kHz to 43 kHz. At this rate
forward tracking of all VELO tracks can be performed in HLT2. While the off-
line reconstruction uses two tracking algorithms, HLT2 only employs the
algorithm based on seeding the search with VELO tracks. This leads to a lower
efficiency compared to off-line of $1-2~{}\%$ per track. To further limit the
processing time only tracks with $p>5$ ${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$
and $\mbox{$p_{T}$}>0.5$ ${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ are
reconstructed by limiting the search windows.
Muon identification is performed using the off-line algorithm on all tracks
from the forward tracking. Tracks are also associated to ECAL clusters to
identify electrons.
A large share of the 3 kHz output rate of HLT2 is selected by "topological"
trigger lines, which are designed to trigger on partially reconstructed
$b$-hadron decays. The topological trigger lines in principle cover all
$b$-hadrons with at least two charged particles in the final state and a
displaced decay vertex. The efficiencies are less dependent on reconstruction
inefficiencies imposed by the minimum ($p,~{}\mbox{$p_{T}$}$) requirements and
loss due to the single, non-redundant, track reconstruction mentioned above.
In the following two sections the topological trigger lines are described in
more detail.
While the topological trigger lines target inclusive $b$-hadrons, a number of
dedicated "exclusive" trigger lines are also implemented in HLT2. These
require all decay particles to be reconstructed in HLT2 and use narrow mass
windows to reduce their rate. These exclusive trigger lines either target
prompt $c$-hadron production, or allow triggering on hadronic $b$-hadron
decays without the necessity to use lifetime-biasing selections to reduce the
rate. These lines are described in Section 3.2.3.
#### 3.2.1 Topological Trigger Lines
The decisions of the topological trigger lines are based on the properties of
combinations of 2, 3, or 4 “Topo-Tracks”. Topo-Tracks are a subset of HLT2
tracks selected with additional requirements on their track fit quality
($\chi^{2}/\rm ndf$), IP, and muon or electron identification. N-body (i.e. an
n track combination) candidates are built as follows: two Topo-Tracks are
combined into a 2-body object, requiring that their distance of closest
approach (DOCA) is less than 0.2 mm. A 3(4)-body object is made by combining a
2(3)-body object and another Topo-Track with the same DOCA <0.2 mm cut, where
the DOCA is calculated between the 2(3)-body object and the additional Topo-
Track. This sequence of DOCA selections enhances the efficiency of the
topological trigger lines on $B\rightarrow DX$ decays. Not all of the
$b$-hadron final state particles need to satisfy these criteria. The trigger
is designed to allow for the omission of one or more final state particles
when forming the trigger candidate.
If an n-body candidate only contains a subset of the daughter particles, its
invariant mass ($m$) will be less than the mass of a $b$-hadron. Thus, a mass
window would need to be very loose if the trigger is to be inclusive. Instead
a corrected mass ($m_{\rm corr}$) is used that is defined as:
$m_{\rm corr}=\sqrt{m^{2}+|p^{\prime}_{T\rm miss}|^{2}}+|p^{\prime}_{T\rm
miss}|,$ (2)
where $p^{\prime}_{T\rm miss}$ is the missing momentum transverse to the
direction of flight, as defined by the PV and the n-body vertex [8]. In case
of multiple PVs, the PV with respect to which the n-body combination has the
smallest IP is used. The quantity $m_{\rm corr}$ would be the minimal mass of
the parent if a massless particle was omitted from the trigger candidate.
Prompt $c$-hadrons that are erroneously combined with another track constitute
a significant fraction of the n-body candidates. These candidates are rejected
by requiring that all (n-1)-body objects used by a n-body trigger line either
have a mass greater than 2.5 ${\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$ or
that they have a significant IP to all PVs.
To select a n-body candidate, cuts are applied to the following variables:
$\sum|\mbox{$p_{T}$}|,~{}p^{min}_{T}$, $m$, $m_{\rm corr}$, DOCA, IP
significance (IP$\chi^{2}$) and flight distance significance (FD$\chi^{2}$).
Using NoBias events for background and Monte Carlo (MC) simulated signal
events, we find that a larger rejection power is achieved for the same signal
efficiency by combining the above variables in a multivariate selection.
#### 3.2.2 Topological Multivariate Lines
To combine the variables mentioned above a boosted decision tree (BDT) was
chosen; this classifier has already been successfully used elsewhere [9]. All
multivariate classifiers select n-dimensional regions of a multivariate space
by learning from the training samples provided to them. If selected regions
are small relative to the resolution of the detector, the signal could
oscillate between regions resulting in, at best, a less efficient trigger or,
at worst, a trigger that is very difficult to understand. To avoid this, all
of the variables are mapped onto discrete variables. The application of the
BDT to discrete variables is henceforth referred to as Bonsai BDT (BBDT).
The BBDT ensures that the smallest interval that can be used satisfies $\Delta
x_{\rm min}>\delta_{x}$ for all $x$ values, where $\delta_{x}={\rm
MIN}\\{|x_{i}-x_{j}|:~{}x_{i},~{}x_{j}\in x_{\rm discrete}\\}$. The
constraints governing the choice of $x_{\rm discrete}$ are then as follows:
firstly $\delta_{x}$ should be greater than the resolution on $x$ and be large
with respect to the expected variations in $x$, and secondly the
discretisation should reflect common $b$-hadron properties.
The discretisation scheme for each variable was determined by first training a
BBDT with a very large number of discretisation values and then gradually
decreasing this number while maintaining near optimal performance. The
training signal samples were MC simulated data that contained as signal
$B^{+},~{}B^{0},~{}B_{s}$ or $\Lambda_{b}$ decays333Charge conjugate hadrons
are always implied. with decay modes as given in Table 1, while the background
sample was NoBias data recorded in 2010.
Table 1: MC signal samples used to train the BBDT, where $K$ means $K^{\pm}$ and $\pi$ means $\pi^{\pm}$. Parent | Daughters
---|---
$B^{+}$ | $K\pi\pi,D_{[K\pi]}\pi,D_{[hhhh]}K,D_{[K_{S}\pi\pi]}K,D_{[K\pi\pi]}K\pi$
$B^{0}$ | $K^{*}_{[K\pi]}\mu\mu,K^{*}_{[K\pi]}ee,D_{[K\pi\pi]}\pi,K\pi,D_{[K\pi]}K\pi,D^{*}_{[D(K\pi)\pi]}\mu\nu,D_{[K\pi\pi]}K\pi\pi$
$B_{s}$ | $D_{s[KK\pi]}\pi,D_{s[KK\pi]}K\pi\pi,K^{*}_{[K\pi]}K^{*}_{[K\pi]}$
$\Lambda_{b}$ | $\Lambda_{c[pK\pi]}\pi,\Lambda_{c[pK\pi]}K\pi\pi$
Table 2 shows the discretisation scheme for each of the variables used in the
BBDT.
Table 2: Allowed mapping points in the BBDT. The variables are explained in the text. Variable | Cuts(2, 3, 4-body) | Intervals used in the BBDT
---|---|---
$\sum|p_{T}|$ [${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ ] | $>3$, 4, 4 | 3.5, 4, 4.5, 5, 6, 7, 8, 9, 10, 15, 20
$p_{T}^{\rm min}$ [${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ ] | $>0.5$ | 0.6, 0.7, 0.8, 0.9, 1, 1.25, 1.5, 1.75, 2, 2.5, 3, 4, 5, 10
$m$ [${\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$ ] | $<7$ | 2.5, 4.75
$m_{\rm corr}$ [${\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$ ] | | 2, 3, 4, 5, 6, 7, 8, 9, 10, 15
${\rm DOCA}$ [mm] | $<0.2$ | 0.05, 0.1, 0.15
${\rm IP}\chi^{2}$ | | 20
${\rm FD}\chi^{2}/100$ | $>1$ | 2, 3, 4, 5, 6, 7, 8, 9, 10, 25, 50, 100
#### 3.2.3 Exclusive Lines
In the topological trigger described in the previous section there is an
explicit veto on prompt charm. The selection of prompt charm decays is
achieved by HLT2 lines that require a reconstruction of all the decay
products, and have tight cuts on the invariant mass of the reconstructed
candidates. While HLT2 reconstruction and selection efficiencies for
$b$-hadrons are good, the $\mbox{$p_{T}$}>0.5$
${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ constraint reduces the efficiency for
the exclusive selection of charm decays with more than two final state
particles. To enhance the reconstruction efficiency for these exclusive
trigger lines, the trigger lines first try to identify a two-prong secondary
vertex. Selection cuts are imposed on the maximum invariant mass of the two
tracks, the quality of their vertex, the sum of the transverse momenta of the
tracks and $m_{\rm corr}$. These initial cuts reduce the rate sufficiently to
allow for the forward tracking of the remaining VELO tracks, but now with a
relaxed $\mbox{$p_{T}$}>0.25$ ${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$
constraint and only using the hits in the tracking system that have not been
used by the previous pass in forward tracking. Two-prong candidates are
subsequently combined with other tracks, which now include the low $p_{T}$
tracks, to form exclusively reconstructed candidates. The combinatorial
background is reduced by tight requirements on the mass and on the angle
between the momentum of the $D$ and the vector connecting the PV with the $D$
vertex. HLT2 contains 28 trigger lines dedicated to selecting prompt charm.
Another example of an exclusive trigger is a dedicated trigger line selecting
the decay $B_{s}\rightarrow\,K^{+}K^{-}$ while avoiding cuts that bias the
$B_{s}$ lifetime. This implies that a cut on IP, a powerful variable to reject
combinatorial background, cannot be used. In order to enrich the $B_{s}$
candidates two dedicated neural networks based on the NeuroBayes neural
network package [10] are used. In a first step, kinematic constraints such as
the transverse momentum of the final state particles and the helicity angle in
the rest frame of the $B_{s}$ candidate are used to reduce the rate. This
allows running the comparatively slow particle identification algorithm using
the RICH sub-detector on the events selected by the first neural network. This
information is then included in a second neural network that uses both
kinematic and particle ID information to make the final selection.
## 4 Data-driven Trigger Performance Determination
The trigger performance is evaluated relative to offline reconstruction and
selections, and thus contains only the additional inefficiency due to
simplifications used in the trigger, possible alignment inaccuracies, worse
resolution than the offline reconstruction or harder cuts imposed by rate
and/or processing time limitations. The following channels have been chosen to
show the performance of the trigger: $D^{0}\rightarrow K^{-}\pi^{+}$,
$D^{+}\rightarrow K^{-}\pi^{+}\pi^{+}$,
$B^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}(\mu^{+}\mu^{-})K^{+}$, $B^{0}\rightarrow
D^{-}(K^{+}\pi^{-}\pi^{-})\pi^{+}$, $B^{-}\rightarrow
D^{0}(K^{-}\pi^{+})\pi^{-}$,
$B^{0}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}(e^{+}e^{-})K^{\ast 0}(K^{+}\pi^{-})$ and $B^{0}\rightarrow
K^{*0}(K^{+}\pi^{-})\gamma$. These channels and their selections are
representative for those used in most analyses. In all off-line selected
signal samples the level of background is significantly lower than the signal.
Substantial differences in trigger efficiency, however, are observed for true
signal and background. The trigger performance on each channel is measured by
determining the signal component using fits to the invariant mass
distributions, hence avoiding any background contamination.
In what follows, the term “signal” refers to a combination of tracks that form
the off-line reconstructed and selected $b$ or $c$-hadron candidate. To
determine the trigger efficiency, trigger objects are associated to signal
tracks. The trigger records all the information needed for such an
association. All strips, straws, cells and pads of the sub-detectors have a
unique identifier, and these identifiers are written in a trigger report in
the data stream for every trigger line that accepts an event. The criteria
used to associate a trigger object with a signal track are as follows:
* •
L0-Calorimeter: the off-line track is extrapolated to the $z$-position of the
calorimeter (ECAL or HCAL), and the cells intersecting with the track and its
eight neighbours are considered signal cells. If any of the $2\times 2$ cells
of a L0-Calorimeter cluster above the threshold coincides with a signal cell,
this cluster is associated with the off-line track. If none of the cells
overlap, the cluster is not associated with the off-line track.
* •
L0-Muon: the trigger records the M1, M2 and M3 hits used to form the L0 muon
candidate. If at least two of the three hits are shared with an off-line
reconstructed muon the L0 muon is associated with the off-line track. Non-
associated L0 muons have no hits overlapping between the L0 muon and the muon
hits of the off-line track.
* •
HLT tracks: a HLT track has VELO hits and hits in the OT and/or the IT. In
addition it can have TT hits and hits in the muon chambers M2-M5. Associated
tracks require that the fraction of HLT track hits that overlaps with the off-
line track is at least $70~{}\%$ in the VELO, $70~{}\%$ in the TT if
applicable and $70~{}\%$ of the OT and IT combined. For muons the association
requirement is that at least $60~{}\%$ of the HLT muon hits overlap with the
off-line muon. Non-associated HLT tracks share no hits with the off-line
track.
An event is classified as TOS (Trigger on Signal) if the trigger objects that
are associated with the signal are sufficient to trigger the event. An event
is classified as TIS (Trigger Independent of Signal) if it could have been
triggered by those trigger objects that are not associated to the signal.
Global event variables, such as the number of primary vertices or the SPD
multiplicity, are not considered in this classification. A number of events
can be classified as TIS and TOS simultaneously ($N^{\rm TIS\&TOS}$), which
allows the extraction of the trigger efficiency relative to the off-line
reconstructed events from data alone. The efficiency to trigger an event
independently of the signal, $\epsilon\rm^{\rm TIS}$, is given by
$\epsilon^{\rm TIS}=N^{\rm TIS\&TOS}/N^{\rm TOS}$, where $N^{\rm TOS}$ is the
number of events classified as TOS. The efficiency to trigger an event on the
signal alone, $\epsilon\rm^{\rm TOS}$, is given by $\epsilon^{\rm TOS}=N^{\rm
TIS\&TOS}/N^{\rm TIS}$, where $N^{\rm TIS}$ is the number of events classified
as TIS. The total trigger efficiency for events containing the signal can then
be computed as $\epsilon^{\rm TIS}\times N^{\rm Trig}/N^{\rm TIS}$, where
$N^{\rm Trig}$ is the total number of triggered signal events.
The phase-space distribution of the signal is affected by the TIS requirement.
This is illustrated in Fig. 2, which shows the $p_{T}$ distribution of
$D^{+}\rightarrow K^{-}\pi^{+}\pi^{+}$ candidates selected from NoBias and TIS
events.
Figure 2: Comparison of the $p_{T}$ distributions of $D^{+}\rightarrow
K^{-}\pi^{+}\pi^{+}$ selected in NoBias and TIS events.
The $p_{T}$ of TIS events is harder, which would result in a too large signal
efficiency when integrated over all phase-space. This bias in phase-space can
be understood considering that $c\bar{c}$, or $b\bar{b}$, pairs are correlated
in phase-space at production, and TIS events are predominantly triggered by
the decay products of the hadron that contains the other heavy quark. Another
example of bias is that for charm decays TIS events could enhance the non
prompt charm component by triggering on the other decay products of a $B$ to
$D$ decay. These biases can only be evaluated individually for each analysis.
Therefore the trigger performance is presented as a function of the signal
$p_{T}$ and its lifetime ($\tau$).
## 5 Trigger Performance
For each channel $\varepsilon^{\rm TOS}$ is determined relative to the off-
line selection efficiency of a channel. $\varepsilon^{\rm TOS}$ for the
HLT1(2) performance is given for off-line selected events that have also been
classified as TOS in the previous trigger level(s), unless mentioned
otherwise.
At each trigger level the different trigger lines compete for their share of
the available resources. To determine the different selections for the trigger
lines, a "bandwidth division" procedure has been adopted, which is described
in the next section. The performance of the different trigger lines with the
thresholds as determined by the bandwidth division will be presented for L0,
HLT1 and HLT2 in the next sections for the channels listed in Section 4.
### 5.1 Bandwidth Division Procedure
The bandwidth division minimises the overall loss in efficiency by minimising
the following:
$\sum_{\rm signal}\left(\sum_{\rm lines}{\left(1-\frac{\varepsilon^{\rm
signal,~{}line}}{\varepsilon_{\rm max}^{\rm
signal,~{}line}}\right)}\right)^{2},$ (3)
where $\varepsilon^{\rm signal,~{}line}$ is the L0$\times$HLT trigger
efficiency obtained using a set of selections (corresponding to a single set
of cut values) for all signal channels and trigger lines simultaneously and
$\varepsilon_{\rm max}^{\rm signal,~{}line}$ is the maximum of
$\varepsilon^{\rm signal,~{}line}$ with the full computing resources dedicated
to that signal and specific trigger line alone. The score is evaluated for
each set of cuts by running an emulation of the L0 trigger and executing the
HLT application. This emulation includes FE-buffer overflow emulation, the
available processing power in the EFF and the maximal HLT2 output rate to disk
as boundary conditions. The configuration with the minimum score is found by
varying the cuts and running the trigger software for each variant.
For signal the following MC generated and off-line reconstructed and selected
channels have been chosen to represent both the main physics goals of LHCb and
to cover all the trigger components that need to be tuned: $B_{s}\rightarrow
J/\psi(\mu^{+}\mu^{-})\phi(K^{+}K^{-})$, $B_{s}\rightarrow\mu^{+}\mu^{-}$,
$B^{0}\rightarrow K^{*0}\mu^{+}\mu^{-}$, $B_{(s,d)}\rightarrow\mu^{+}X$,
$D^{*+}\rightarrow D(\mu^{+}\mu^{-})\pi^{+}$, $D^{+}\rightarrow
K^{-}\pi^{+}\pi^{+}$, $B^{0}\rightarrow K^{*0}\gamma$, $B^{0}\rightarrow
K^{+}\pi^{-}$, $B^{0}\rightarrow K^{*0}e^{-}e^{+}$, $B^{+}\rightarrow
K^{+}\pi^{-}\pi^{+}$, $B_{s}\rightarrow D_{s}^{-}(K^{+}K^{-}\pi^{-})K^{+}$,
$B^{+}\rightarrow\bar{D^{0}}(K_{S}(\pi^{+}\pi^{-})\pi^{+}\pi^{-})K^{+}$ and
$D^{0}\rightarrow K^{-}\pi^{+}$ with $K^{\ast 0}\rightarrow K^{+}\pi^{-}$.
NoBias events from 2010 with $\mu=1.4$ are used as background. Rather than
introducing weights favouring some channels, we have chosen to emphasize the
main physics goals by the number of channels included in the bandwidth
division procedure. For example, channels decaying with muons in the final
state are more abundant.
All performance results are given for 1296 colliding bunches in LHCb, which
corresponds to a bunch crossing rate with at least one visible $pp$
interaction of $\sim 11$ MHz. The bandwidth division yields the following
rates for NoBias events: 870 kHz for L0, 43 kHz for HLT1 and 3 kHz for HLT2.
### 5.2 L0 Performance
Events with a large occupancy in the OT and IT consume a disproportionately
large fraction of the available processing time in the HLT. The SPD
multiplicity measured at L0 is a good measure of this occupancy, permitting an
early rejection of events that require a relatively large processing time.
Using the bandwidth division the optimal SPD cut is determined to be $<~{}900$
for events triggered by L0DiMuon and $<~{}600$ for all other L0 triggers. On
average, events with a SPD multiplicity larger than 600 consume four times
more time in the HLT than events with less than 600 SPD hits. The fraction of
events rejected due to these cuts has been determined from real data for charm
hadron production to be $7.4\pm~{}0.3~{}(0.05\pm 0.01)\,\%$ for a cut on 600
(900) in SPD multiplicity. Similarly for $b$-hadron production the fraction of
events with a SPD multiplicity $>600~{}(900)$ is found to be $8.8\pm
0.6~{}(0.5\pm 0.2)\,\%$. All efficiencies quoted below are given relative to
the sample after the SPD multiplicity cut.
Table 3 lists the L0 cuts. About $20~{}\%$ of the events accepted by L0 are
selected by more than one trigger line, giving a total L0 rate of 870 kHz
prior to throttling.
Table 3: Cuts of L0 lines and their rates prior to throttling. The definition of the trigger lines is given in Section 2. | Threshold | SPD Multiplicity | Rate
---|---|---|---
L0Muon | $\mbox{$p_{T}$}>1.48$ ${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ | $<600$ | 340 kHz
L0DiMuon | $\sqrt{\mbox{$p_{T}$}^{\rm largest}\times\mbox{$p_{T}$}^{\rm 2nd~{}largest}}>1.296$ ${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ | $<900$ | 75 kHz
L0Hadron | $E_{T}>3.5$ $\mathrm{\,Ge\kern-1.00006ptV}$ | $<600$ | 405 kHz
L0Electron | $E_{T}>2.5$ $\mathrm{\,Ge\kern-1.00006ptV}$ | $<600$ | 160 kHz
L0Photon | $E_{T}>2.5$ $\mathrm{\,Ge\kern-1.00006ptV}$ | $<600$ | 80 kHz
L0Muon is the main trigger for particle decays with one or more muons in the
final state. L0DiMuon recovers part of the events with a SPD multiplicity
$>600$ for a small increase in rate. The performance of L0Muon and L0DiMuon
are shown in Fig. 4 for $B^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}(\mu^{+}\mu^{-})K^{+}$ as a function of $p_{T}$
(${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$). L0DiMuon increases the
number of signal events by $4.9~{}\%$, of which $87~{}\%$ have a SPD
multiplicity larger than 600 hits. The remaining $13~{}\%$ is due to the lower
$p_{T}$ cut in L0DiMuon. L0 requires a muon candidate to have a hit in all
five muon stations, while off-line as few as two stations are sufficient to
identify a muon. As a result L0DiMuon has a maximum efficiency of $\sim
80~{}\%$ even for a ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ with large
$p_{T}$. L0Muon recovers this loss for lower SPD multiplicities and decays
with more muons at large $p_{T}$.
Figure 3: Efficiency $\epsilon^{\rm TOS}$ of
$B^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}(\mu^{+}\mu^{-})K^{+}$ as a function of $p_{T}$
(${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$) for L0Muon and L0DiMuon
lines.
Figure 4: The efficiency $\epsilon^{\rm TOS}$ of L0Hadron is shown for
$B^{0}\\!\rightarrow D^{-}\pi^{+}$, $B^{-}\\!\rightarrow D^{0}\pi^{-}$,
$D^{0}\rightarrow K^{-}\pi^{+}$ and $D^{+}\rightarrow K^{-}\pi^{+}\pi^{+}$ as
a function of $p_{T}$ of the signal $B$ and $D$ mesons.
L0Hadron selects heavy flavour decays with hadrons in the final state. The
performance of L0Hadron is shown in Fig. 4 for $B^{0}\\!\rightarrow
D^{-}\pi^{+}$, $B^{-}\\!\rightarrow D^{0}\pi^{-}$, $D^{0}\rightarrow
K^{-}\pi^{+}$ and $D^{+}\rightarrow K^{-}\pi^{+}\pi^{+}$ as a function of
$p_{T}$ of the signal $B$ and $D$ mesons. At low $p_{T}$, L0Hadron has a
better efficiency for $b$-hadrons than for $c$-hadrons due to the larger
$b$-hadron mass. Once the $p_{T}$ of the hadron is above the $b$-hadron mass,
the decays with fewer final state tracks have a higher efficiency.
L0Electron selects decays with electrons in the final state. It also triggers
on radiative decays, with the photon being either converted, or with photon
clusters with SPD hits in front due to overlapping charged particles. The
performance of L0Electron is shown in Fig. 5 for ${B}^{0}\rightarrow
J/\psi{(\rm e^{+}e^{-})}K^{*0}$ as a function of
$\mbox{$p_{T}$}({J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu})$. Contrary to
L0Muon, L0Electron is not fully efficient for
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ with large $p_{T}$. This is due
to the hardware implementation (see Section 2.1) that prevents energy
deposited in different ECAL zones from being combined into one cluster.
Figure 5: The efficiency $\epsilon^{\rm TOS}$ of L0Electron is shown for
${B}^{0}\rightarrow J/\psi{(\rm e^{+}e^{-})}K^{*0}$ as a function of $p_{T}$
(${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$).
The performance of the trigger for high-energy photons from radiative penguin
decays is measured with the channel $B^{0}\rightarrow K^{*0}\gamma$. The
number of TIS events in this channel is insufficient to study the efficiency
as a function of $p_{T}$ of the $B^{0}$. The mean efficiency for the L0Photon
line integrated over $p_{T}$ is $50\pm 4~{}\%$. Selecting events with either
L0Photon or L0Electron gives an efficiency of $88\pm 5~{}\%$ .
### 5.3 HLT1 Performance
HLT1 muon lines are only executed for events that have been triggered by
L0Muon or L0DiMuon, and the lines require their tracks to be validated as a
muon candidate as described in Section 3.1. Table 4 gives the names of the
HLT1 muon lines and their cuts.
Table 4: HLT1 muon lines and their cuts. The rate is measured on events accepted by L0Muon or L0DiMuon. Hlt1line | TrackMuon | SingleMuon | DiMuon | DiMuon
---|---|---|---|---
| | HighPT | HighMass | LowMass
Track IP [mm] | $>0.1$ | - | - | -
Track IP$\chi^{2}$ | $>16$ | - | - | $>3$
Track $p_{T}$ [${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ ] | $>1$ | $>4.8$ | $>0.5$ | $>0.5$
Track $p$ [${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ ] | $>8$ | $>8$ | $>6$ | $>6$
Track $\chi^{2}/\rm ndf$ | $<2$ | $<4$ | $<4$ | $<4$
DOCA [mm] | - | - | $<0.2$ | $<0.2$
$\chi_{\rm vertex}^{2}$ | - | - | $<25$ | $<25$
Mass [${\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$ ] | - | - | $>2.7$ | $>1$
Rate [kHz] | 5 | 0.7 | 1.2 | 1.3
Hlt1TrackMuon accepts events with $B$, $D$ or $\tau$ decays with at least one
muon in its final state by identifying and accepting events with muon
candidates that have significant IP with respect to all PVs. To trigger on
muons originating from heavy particles with a negligible lifetime, like
$W^{\pm}$ or $Z^{0}$, an alternative line, Hlt1SingleMuonHighPT, is
implemented. It does not have any requirements on IP, but requires a hard
$p_{T}$ cut to reduce the rate. Hlt1DiMuonHighMass is complementary to
Hlt1TrackMuon in that it allows $b$-hadron decays to be selected without
imposing lifetime related cuts, and thus allows the lifetime acceptance bias
of the larger efficiency Hlt1TrackMuon line to be determined. Finally
Hlt1DiMuonLowMass allows triggering on final states with two muons with a
relatively small invariant mass. To reduce the rate the line requires that
both muons are not prompt.
The performance of the HLT1 muon lines is evaluated using
$B^{+}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+}$ decays.
Figure 6 shows the performance of Hlt1TrackMuon, Hlt1DiMuonHighMass and
Hlt1DiMuonLowMass as a function of the $p_{T}$ and $\tau$ of the $B^{+}$.
Hlt1TrackMuon gives the best performance overall, except at low lifetimes,
where Hlt1DiMuonHighMass recovers events. Hlt1DiMuonLowMass loses $\sim
10~{}\%$ in efficiency compared to Hlt1TrackMuon for
$B^{+}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+}$ due to
the requirement to have at least two muon candidates, but its cuts on IP and
$p_{T}$ are significantly relaxed to allow the selection of candidates with
the muon pair mass down to 1 ${\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$, which
is designed to select $b\rightarrow s\mu\mu$ decays like $B\rightarrow
K^{*}\mu^{+}\mu^{-}$.
The performance of Hlt1SingleMuonHighPT is not properly assessed using
$B^{+}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+}$ decays
because it is designed to accept events with decays of particles with a mass
larger than that of $b$-hadrons. Instead
$Z^{0}~{}\rightarrow~{}\mu^{+}~{}\mu^{-}$ events are used to measure the
efficiency, by requiring one of the two muons to be TIS. This yields an
efficiency of $77.1\pm 0.2~{}\%$ for the Hlt1SingleMuonHighPT line per single
muon, implying an efficiency for $Z^{0}~{}\rightarrow~{}\mu^{+}~{}\mu^{-}$ of
$95~{}\%$. The fast HLT1 muon reconstruction, as described in Section 3.1,
applies more stringent cuts than the off-line muon identification to keep
misidentification and as a consequence the rate and CPU time consumption under
control.
Figure 6: Efficiency $\epsilon^{\rm TOS}$ of Hlt1TrackMuon, Hlt1DiMuonHighMass
and Hlt1DiMuonLowMass for
$B^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}(\mu^{+}\mu^{-})K^{+}$ as a function of the $p_{T}$ and lifetime of the
$B^{+}$.
In addition to the muon lines mentioned above, HLT1 also contains a line that
is executed for all events accepted by L0, Hlt1TrackAllL0. It is designed to
select hadron decays which are significantly displaced from a PV. A trigger
line called Hlt1TrackPhoton is only executed for events that have a L0Photon
or a L0Electron with $E_{T}>4.2$ $\mathrm{\,Ge\kern-1.00006ptV}$. This trigger
line is designed to enhance the trigger efficiency for radiative $b$-hadron
decays with a high $p_{T}$ photon. The corresponding selection cuts are given
in Table 5. Both trigger lines require at least one track with sufficient IP
and $p_{T}$. Hlt1TrackPhoton is designed to select lower $p_{T}$ tracks, and
correspondingly also has relaxed track quality requirements compared to
Hlt1TrackAllL0.
Table 5: The cuts applied in Hlt1TrackAllL0 and Hlt1TrackPhoton lines. The rate is measured on events accepted by L0. Hlt1 line | Hlt1TrackAllL0 | Hlt1TrackPhoton
---|---|---
Track IP [mm] | $>0.1$ | $>$0.1
Number VELO hits/track | $>9$ | $>6$
Number missed VELO hits/track | $<3$ | $<3$
Number OT+IT$\times$2 hits/track | $>16$ | $>15$
Track IP$\chi^{2}$ | $>16$ | $>$16
Track $p_{T}$ [${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ ] | $>1.7$ | $>1.2$
Track $p$ [${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ ] | $>10$ | $>6$
Track $\chi^{2}/\rm ndf$ | $<2.5$ | $<2.5$
Rate [kHz] | 33 | 4.2
Figure 7 shows the performance of Hlt1TrackAllL0 as a function of
$\mbox{$p_{T}$}\rm~{}and~{}\tau$ for channels with hadronic decays.
Figure 7: Efficiency $\epsilon^{\rm TOS}$ of Hlt1TrackAllL0 is shown for
$B^{-}\\!\rightarrow D^{0}\pi^{-}$, $B^{0}\\!\rightarrow D^{-}\pi^{+}$,
$D^{0}\rightarrow K^{-}\pi^{+}$ and $D^{+}\rightarrow K^{-}\pi^{+}\pi^{+}$ as
a function of $p_{T}$ and $\tau$ of the $B$-meson and prompt $D$-meson
respectively.
Hlt1TrackAllL0 provides a very efficient trigger for all heavy flavour decays
with a significant flight distance from their PV, reducing the rate from 870
kHz to 33 kHz. At low $p_{T}$ the requirement of at least one decay particle
with $p_{T}$ $>1.7$ ${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ results in
selecting $b$-hadrons with a larger efficiency than $c$-hadrons, and low
multiplicity decays with a larger efficiency than higher multiplicity decays.
At large $p_{T}$ this condition favours the decays with larger multiplicities.
While in L0 special lines are used to select $b$-hadron decays with electrons
in the final state, in HLT1 these decays are covered by Hlt1TrackAllL0.
There are insufficient radiative $B$-decays to extract the performance of
Hlt1TrackPhoton as a function of $p_{T}$ in a data-driven way. Hlt1TrackPhoton
uses the same tracks as Hlt1TrackAllL0, but with a relaxed set of requirements
as shown in Table 5. The yield increase in $B^{0}\rightarrow K^{*0}\gamma$
events obtained by including Hlt1TrackPhoton in addition to Hlt1TrackAllL0 is
measured to be $12\pm 2~{}\%$.
### 5.4 HLT2 Performance
Similar to HLT1, HLT2 has lines that select events with one or two identified
muons in the final state. In HLT2 the muon identification is identical to the
off-line algorithm. The cuts corresponding to lines that are purely based on a
single identified muon are given in Table 6.
Table 6: HLT2 lines based on one identified muon. Hlt2Single | Muon | MuonHighPT
---|---|---
Hlt1TrackMuon | TOS | -
Track IP [mm] | $>0.5$ | -
Track IP$\chi^{2}$ | $>200$ | -
Track $p_{T}$ [${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ ] | $>1.3$ | $>$10
Track $\chi^{2}/\rm ndf$ | $<2$ | -
Pre-scale | 0.5 | 1.
Rate [Hz] | 480 | 45
Hlt2SingleMuon selects semileptonic $b$ and $c$-hadron decays. To minimise the
bias on the hadronic part of the decay the $p_{T}$ cut is set low, in
combination with scaling the rate down by a factor two, rather than tightening
the cut to reduce the rate. This trigger line also provides a large yield for
$J/\psi\rightarrow\mu\mu$ events that are selected by one of the two muons,
while the other muon is used for calibration of tracking and muon
identification efficiencies. Hlt2SingleMuonHighPT is designed to select heavy
particles decaying promptly to one or more muons, like $W^{\pm}$ or $Z^{0}$.
Contrary to Hlt2SingleMuon the rate is not a problem so there is no HLT1
requirement imposed. As for HLT1, $Z^{0}\rightarrow\mu^{+}\mu^{-}$ decays are
used to measure an efficiency of over $99\%$ for Hlt2SingleMuonHighPT per
muon. This small loss in efficiency is attributed to different alignment
constants and the non-redundant track reconstruction used in HLT2 as described
in Section 3.2.
The HLT2 lines that are based on two identified muons are grouped into two
categories. Those that are dedicated to prompt decays use the mass as the main
discriminant, while "detached" lines use the separation between the dimuon
vertex and the PV as the main discriminant. The names and corresponding cuts
of the prompt decay selections are given in Table 7.
Table 7: HLT2 lines based on two identified muon. Hlt2DiMuon | JPsi | Psi2S | B | JPsiHighPT | Psi2SHighPT
---|---|---|---|---|---
Track $\chi^{2}/\rm ndf$ | $<5$ | $<5$ | $<5$ | $<5$ | $<5$
Mass [${\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$ ] | M${}_{J/\psi}\pm 0.12$ | M${}_{\psi{(2S)}}\pm 0.12$ | $>4.7$ | M${}_{J/\psi}\pm 0.12$ | M${}_{\psi{(2S)}}\pm 0.12$
$\chi_{\rm vertex}^{2}$ | $<25$ | $<25$ | $<10$ | $<25$ | $<25$
$\mbox{$p_{T}$}^{\mu\mu}$ [${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ ] | - | - | - | $>2$ | $>3.5$
Pre-scale | 0.2 | 0.1 | 1. | 1. | 1.
Rate [Hz] | 50 | 5 | 80 | 115 | 15
Hlt2DiMuonJPsi(Psi2S) and Hlt2DiMuonJPsi(Psi2S)HighPT all select a mass region
around ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}~{}(\psi{(2S)})$.
Hlt2DiMuonJPsi(Psi2S) avoids explicit $p_{T}$ requirements but as a
consequence needs to be pre-scaled to reduce the rate.
Hlt2DiMuonJPsi(Psi2S)HighPT reduces the prompt
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}~{}(\psi{(2S)})$ rate by applying
a $p_{T}$ cut on the ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}~{}(\psi{(2S)})$ candidate. Hlt2DiMuonB has its mass cut set high enough
to have an acceptable rate.
The names and corresponding cuts of the detached decay selections are given in
Table 8.
Table 8: HLT2 lines based on two identified muons. Hlt2DiMuon | Detached | DetachedHeavy | DetachedJPsi
---|---|---|---
Track $\chi^{2}/\rm ndf$ | $<5$ | $<5$ | $<5$
Track IP$\chi^{2}$ | $>9$ | - | -
Mass [${\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$ ] | $>1$ | $>2.95$ | M${}_{J/\psi}\pm 0.12$
${\rm FD}\chi^{2}$ | $>$49 | $>$25 | $>$9
$\chi_{\rm vertex}^{2}$ | $<25$ | $<25$ | $<25$
$\mbox{$p_{T}$}^{\mu\mu}$ [${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ ] | $>1.5$ | - | -
Rate [Hz] | 70 | 75 | 35
Hlt2DiMuonDetached is the main trigger for low mass muon pairs.
Hlt2DiMuonDetachedHeavy is an analogous trigger line for
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ and higher mass muon pairs,
with relaxed lifetime selection criteria. Hlt2DiMuonDetachedJPsi enhances the
efficiency for ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ by reducing the
flight distance requirement for ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$
candidates even further. Figure 8 compares the performance of two
representative HLT2DiMuon lines: Hlt2DiMuonJPsiHighPT and
Hlt2DiMuonDetachedJPsi in
$B^{+}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+}$ decays.
Hlt2DiMuonJPsiHighPT avoids by design a bias in the proper lifetime, at the
price of losing efficiency at low
$\mbox{$p_{T}$}({J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu})$. The detached
lines allow the selection of decays with a significant flight distance with
high efficiency even at low
$\mbox{$p_{T}$}({J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu})$, but their
efficiency is reduced at small lifetimes.
Figure 8: Efficiencies $\epsilon^{\rm TOS}$ of Hlt2DiMuonJPsiHighPT and
Hlt2DiMuonDetachedJPsi for
$B^{+}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+}$ as a
function of $p_{T}$ and $\tau$ of the $B^{+}$.
There are nine BBDT topological lines: Hlt2Topo$n$Body, Hlt2TopoMu$n$Body and
Hlt2TopoE$n$Body, where $n$=2,3,4 for the multiplicities considered. TopoMu
(TopoE) require at least one of the decay particles to have been identified as
a muon (electron). Each line returns an output of the BBDT between 0 and 1.
Hlt2Topo$n$Body lines accept events with a combined rate of 930 Hz with a cut
on the BBDT output at 0.4, 0.4 and 0.3 for the 2, 3 and 4 body lines
respectively. While the TopoMu and TopoE lines are based on the same BBDT, the
extra requirement of either a muon or electron allows the cut on the BBDT
output to be reduced to 0.1 for all six lines, which results in rates of 290
and 260 Hz for TopoMu and TopoE respectively.
The performance of the topological lines is given in Fig. 9 for fully hadronic
$B$-decays and Fig. 10 for
$B^{+}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+}$ decays.
Figure 10 also shows the complementarity of Hlt2Topo$n$Body and
Hlt2TopoMu$n$Body; the efficiency increases if either of these lines has
selected the signal event. The inclusive performance of the topological lines
is demonstrated in Fig. 10 by giving the performance of Hlt2Topo2Body alone.
This line requires only two of the three decay tracks of
$B^{+}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+}$ to have
been reconstructed and selected. Adding Hlt2Topo3Body mainly recovers
efficiency at low $p_{T}$ compared to the Hlt2Topo2Body line alone.
Figure 9: Efficiency $\epsilon^{\rm TOS}$ if at least one of the lines
Hlt2Topo$n$Body, with $n=2,3$, selected the event for $B^{-}\\!\rightarrow
D^{0}\pi^{-}$ and one of the lines with $n=2,3,4$ for $B^{0}\\!\rightarrow
D^{-}\pi^{+}$ as a function of $p_{T}$ and $\tau$ of the $B$-meson. The
efficiency is measured relative to events that are TOS in Hlt1TrackAllL0.
Figure 10: Efficiency $\epsilon^{\rm TOS}$ if at least one of the lines
Hlt2Topo$n$Body or Hlt2TopoMu$n$Body, with $n=2,3$, selected events for
$B^{+}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+}$, as a
function of $p_{T}$ and $\tau$ of the $B$-meson. Also shown is $\epsilon^{\rm
TOS}$ if the line Hlt2Topo$n$Body, with $n=2,3$, selected the events.
Hlt2Topo2Body shows the inclusive performance of the topological lines. The
efficiency is measured relative to events that are TOS in either
Hlt1TrackAllL0 or Hlt1TrackMuon.
Table 9 lists the cuts applied in the two HLT2 exclusive lines
Hlt2CharmHadD02HH$\\_$D02KPi and Hlt2CharmHadD2HHH. The off-line selections of
$D^{0}\rightarrow K^{-}\pi^{+}$ are only slightly tighter than the cuts
applied in HLT2, resulting in an almost maximum efficiency of the
Hlt2CharmHadD02HH$\\_$D02KPi line for this channel, as shown in Fig. 11. This
figure also shows the performance of Hlt2CharmHadD2HHH for $D^{+}\rightarrow
K^{-}\pi^{+}\pi^{+}$. Here HLT2 loses efficiency due to the necessity of first
having to apply hard cuts to two of the three decay products before allowing
an extra reconstruction step for low $p_{T}$ tracks, as described in Section
3.2.3.
Table 9: HLT2 selection cuts applied for the exclusive lines Hlt2CharmHadD02HH$\\_$D02KPi and Hlt2CharmHadD2HHH. The 2-track cuts refer to a candidate constructed of two tracks, and $m_{\rm corr}$ is defined in equation 2. The angle $\alpha$ is the angle between the momentum of the $D$ and the vector connecting the PV with the $D$ vertex. Some selections require that at least one or two tracks pass a cut, indicated with "$\geqq$". Variable | Hlt2CharmHadD02HH$\\_$D02KPi | Hlt2CharmHadD2HHH
---|---|---
$\chi_{\rm track}^{2}/\rm ndf$ | $<3$ | $<$3
Track $p_{T}$ [${\mathrm{\,Me\kern-1.00006ptV\\!/}c}$ ] | $>$ 800 | $>$250
Track $p$ [${\mathrm{\,Me\kern-1.00006ptV\\!/}c}$ ] | $>5000$ | $>$2000
Track $\Sigma$$p_{T}$ [${\mathrm{\,Me\kern-1.00006ptV\\!/}c}$ ] | - | $>$ 2500
$\geqq 1$ track $p_{T}$ [${\mathrm{\,Me\kern-1.00006ptV\\!/}c}$ ] | $>1500$ | -
2-track mass [${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ ] | - | $<$ 2100
2-track $m_{\rm corr}$ [${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ ] | - | $<$3500
2-track IP$\chi^{2}$ | - | $>$40
$\geqq 2$ tracks $p_{T}$ [${\mathrm{\,Me\kern-1.00006ptV\\!/}c}$ ] | - | $>$500
$\geqq 2$ tracks $p$ [${\mathrm{\,Me\kern-1.00006ptV\\!/}c}$ ] | - | $>$5000
Track IP$\chi^{2}$ | $>9$ | $>5$
$\geqq 2$ tracks IP$\chi^{2}$ | - | $>10$
2-track DOCA [mm] | - | $<$0.1
$\chi_{\rm vertex}^{2}/\rm ndf$ | $<10$ | $<20$
FD$\chi^{2}$ | $>40$ | $>$ 150
D IP$\chi^{2}$ | - | $<$ 12
$D$ cos($\alpha$) | $>0.99985$ | -
$D$ $p_{T}$ [${\mathrm{\,Me\kern-1.00006ptV\\!/}c}$ ] | $>2000$ | $>$ 1000
$D$ mass interval [${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ ] | 1815-1915 | 1800-2040
Rate [Hz] | 260 | 390
Figure 11: Efficiency $\epsilon^{\rm TOS}$ of the lines Hlt2CharmHadD2HHH and
Hlt2CharmHadD02HH$\\_$D02KPi for $D^{+}\rightarrow K^{-}\pi^{+}\pi^{+}$ and
$D^{0}\rightarrow K^{-}\pi^{+}$ respectively as a function of $p_{T}$ and
$\tau$ of the $D$-meson. The efficiency is measured relative to events that
are TOS in Hlt1TrackAllL0.
## 6 Summary
The LHCb trigger is designed to select charm and beauty hadrons in a large
range of decay modes, and permits the measurement of its efficiency directly
from data. In 2011 the trigger has been tuned to cope with $pp$ interactions
at $\sqrt{s}=7$ TeV, with 1296 colliding bunches in LHCb and an average number
of visible $pp$ interactions per bunch crossing of 1.4. This corresponds to a
bunch crossing rate with at least one visible $pp$ interaction of $\sim 11$
MHz.
L0 reduces this rate to 870 kHz by applying $p_{T}$ cuts on muons and $E_{T}$
cuts on clusters in the calorimeters. HLT1 performs a partial reconstruction
of tracks and performs muon identification. It employs a combination of cuts
on $p_{T}$, invariant mass and IP to reduce the rate to around 43 kHz. HLT2
reconstructs all tracks in the event with $p_{T}$ $>$500
${\mathrm{\,Me\kern-1.00006ptV\\!/}c}$. It selects candidates based on lepton
identification, lifetime information and invariant mass. Its output rate is 3
kHz, consisting of 50 $\%$ inclusive hadronic triggers, 25 $\%$ triggers on
leptons and the remaining rate from exclusive triggers, mainly on charmed
hadrons. The efficiencies for the major trigger lines are presented for
representative decay modes as a function of $p_{T}$ and lifetime of $c$ and
$b$-hadrons.
The successful exploitation of the LHC as a beauty factory relies crucially on
the ability to trigger on heavy flavour decays in a hadronic environment. To
achieve this, the trigger is designed to be able to determine the impact
parameter of tracks at a high rate, and to measure the momentum of those
tracks with sufficiently large impact parameter, or to identify them as muon
candidates. The trigger managed to adapt to the larger pile-up conditions
imposed by the machine delivering only 1296 instead of the planned 2622
colliding bunches in the LHC. The trigger performance and the fact that its
efficiency can be evaluated in a data-driven way, in combination with the
excellent performance of the sub-detectors, allowed LHCb to publish more than
40 papers based on the data collected in 2011.
LHCb is preparing to upgrade the detector [11] in 2018. It will feature a
fully software based trigger that will allow it to explore its physics goals
at even larger luminosities.
###### 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); ANCS/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. A special acknowledgement goes to
all our LHCb collaborators who over the years have contributed to obtain the
results presented in this paper.
## References
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* [3] G. Corti et al., _Software for the LHCb experiment_ , IEEE Trans. Nucl. Sci. 53 (2006) 1323.
* [4] The LHCb Collaboration, _Roadmap for selected key measurements of LHCb_ , [0912.4179v3].
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* [6] R.E. Kalman, _A new approach to linear filtering and prediction problems_ , Trans. ASME J. Bas. Eng. D 82 (1960) 35.
R. Fr$\rm\ddot{u}$hwirth, _Application of Kalman Filtering to track and vertex
fitting_. Nucl. Instrum. Meth. A 262 (1987) 444.
* [7] M. Gandelman and E. Polycarpo, _The Performance of the LHCb Muon Identification Procedure_ , LHCb-2007-145. CERN-LHCb-2007-145.
* [8] SLD Collaboration, _Measurement of Rb using a Vertex Mass Tag_ , Phys. Rev. Lett., 80:660-665 (1998).
* [9] L. Brieman, J. H. Friedman, R. Olshen, and C. J. Stone, _Classication and Regression Trees_ , Wadsworth International Group, Belmont, California, 1984.
L. Breiman, _Bagging predictors_ , Machine Learning 24 (1996) 123.
H. Yang, B.P. Roe, J. Zhu., _Studies of Boosted Decision Trees for MiniBooNE
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|
arxiv-papers
| 2012-11-13T17:24:37 |
2024-09-04T02:49:37.954638
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "R.Aaij, J.Albrecht, F.Alessio, S.Amato, E.Aslanides, I.Belyaev,\n M.vanBeuzekom, E.Bonaccorsi, R.Bonnefoy, L.Brarda, O.Callot, M.Cattaneo,\n H.Chanal, M.Chebbi, X.CidVidal, M.Clemencic, J.Closier, V.Coco, J.Cogan,\n O.Deschamps, H.Dijkstra, C.Drancourt, R.Dzhelyadin, M.Frank, M.Gandelman,\n C.Gaspar, V.V.Gligorov, C.G\\\"obel, L.A.GranadoCardoso, Yu.Guz, C.Haen, J.He,\n E.vanHerwijnen, W.Hulsbergen, R.Jacobsson, B.Jost, T.M.Karbach, U.Kerzel,\n P.Koppenburg, G.Krocker, C.Langenbruch, I. Lax, R.LeGac, R.Lef\\`evre,\n J.Lefran\\c{c}ois, O.Leroy, L.LiGioi, G.Liu, F.Machefert, I.V.Machikhiliyan,\n M.Magne, G.Mancinelli, U.Marconi, A.Mart\\'inS\\'anchez, M.-N.Minard,\n S.Monteil, N.Neufeld, V.Niess, S.Oggero, A.P\\'erez-CaleroYzquierdo, P.Perret,\n M.Perrin-Terrin, B.Pietrzyk, A.PuigNavarro, G.Raven, P.Robbe, H.Ruiz,\n M.-H.Schune, R.Schwemmer, J.Serrano, I.Shapoval, T.Skwarnicki,\n B.SouzaDePaula, P.Spradlin, S.Stahl, V.K.Subbiah, S.T'Jampens, F.Teubert,\n C.Thomas, M.Vesterinen, M.Williams, M.Witek, A.Zvyagin",
"submitter": "Hans Dijkstra",
"url": "https://arxiv.org/abs/1211.3055"
}
|
1211.3072
|
# Computation of multi-region relaxed magnetohydrodynamic equilibria
S.R. Hudson [email protected] R.L. Dewar [email protected] G. Dennis
[email protected] M.J. Hole [email protected] M. McGann
[email protected] G. von Nessi [email protected] S. Lazerson
[email protected] Princeton Plasma Physics Laboratory, PO Box 451, Princeton
NJ 08543, USA Plasma Research Laboratory, Research School of Physics &
Engineering, The Australian National University, Canberra, ACT 0200, Australia
###### Abstract
We describe the construction of stepped-pressure equilibria as extrema of a
multi-region, relaxed magnetohydrodynamic (MHD) energy functional that
combines elements of ideal MHD and Taylor relaxation, and which we call
MRXMHD. The model is compatible with Hamiltonian chaos theory and allows the
three-dimensional MHD equilibrium problem to be formulated in a well-posed
manner suitable for computation. The energy-functional is discretized using a
mixed finite-element, Fourier representation for the magnetic vector potential
and the equilibrium geometry; and numerical solutions are constructed using
the stepped-pressure equilibrium code, SPEC. Convergence studies with respect
to radial and Fourier resolution are presented.
## I Introduction
Zero-Larmor-radius, single-fluid magnetohydrodynamics (MHD) is commonly used
for modeling the global, long-time-scale state of plasmas in the magnetic
confinement devices used for fusion power research. It is often reasonable to
approximate the plasma pressure tensor as isotropic and to ignore inertial
effects due to small mass flows. There is no minimum length scale in this
model, so spatial discontinuities are allowed McGann et al. (2010). To allow a
weak formulation, we write the equilibrium condition in conservation form,
$\displaystyle\nabla\cdot\left(p\,\textsf{{I}}+\frac{B^{2}}{2\mu_{0}}\textsf{{I}}-\frac{\mbox{$\bf
B$}\mbox{$\bf B$}}{\mu_{0}}\right)=0,$ (1)
where, using SI units, $\mu_{0}$ is the permeability of free space,
$p(\mbox{$\bf r$})\geq 0$ is the pressure as a function of position
$\mbox{$\bf r$}=x\mbox{$\bf i$}+y\mbox{$\bf j$}+z\mbox{$\bf k$}$, and
$\mbox{$\bf B$}(\mbox{$\bf r$})$ is the magnetic field, which must obey
$\nabla\cdot\mbox{$\bf B$}=0$. While MHD is a rather crude model for the
physics of a plasma, the Maxwell equations for the magnetic field and the
‘dynamics’ of field lines are exact.
The problem addressed in this paper is, treating both $p$ and $\bf B$ as
unknown fields within suitable function spaces, find general, weak solutions
of Eq. (1) in an arbitrary, three-dimensional (3D) toroidal domain, ${\cal
V}$, under the homogeneous boundary conditions,
$\displaystyle p=0,\;\mbox{$\bf n$}\cdot\mbox{$\bf B$}=0,\;\forall\>\mbox{$\bf
r$}\in{\partial\cal V},$ (2)
where $\bf n$ is the unit normal at the boundary, $\partial{\cal V}$. We take
the boundary to be fixed, being either the edge of a plasma confined by a
notional, tight-fitting shell, or the boundary of a surrounding vacuum region.
Our goal is to formulate the 3D equilibrium problem in a way that is as well-
posed mathematically as the two-dimensional (2D) problem, by which we mean
that a well-defined, unique solution exists. And, to develop an accurate,
robust and efficient numerical solution method, where the error between the
approximate numerical solution, e.g. $f_{h}$, and the exact solution, $f$, is
bounded and goes to zero as $f_{h}=f+{\cal O}(h^{n})$, where $h$ characterizes
the numerical resolution and $n$ depends on the numerical discretization.
If $p$ and ${\bf B}$ are assumed to be differentiable within a subregion of
$\cal V$, then Eq. (1) is locally equivalent to the force-balance condition
$\displaystyle\nabla p=\mbox{$\bf j$}\times\mbox{$\bf B$},$ (3)
where $\mbox{$\bf j$}=\nabla\times\mbox{$\bf B$}$ is the current density
(here, and hereafter, $\mu_{0}$ is ignored). We will not restrict attention to
differentiable solutions in the following, but we will work within the
approximation that ${\bf B}\cdot\nabla p=0$, which follows directly from Eq.
(3). Physically, this approximates the transport of heat and mass along the
magnetic field as infinite compared to that across the field. This immediately
implies that $p$ is invariant along the magnetic field: the spatial dependence
of the pressure and the phase space structure of the magnetic field are
intimately connected.
A specific equilibrium state is characterized by the pressure, i.e. $p$ is
considered to be a supplied, input function. The computational challenge is to
then determine the magnetic field that is consistent with the given pressure
and boundary. Generically, in 3D, there exist regions within ${\cal V}$ where
the magnetic field lines are chaotic. To admit numerically tractable solutions
for ${\bf B}$, it is necessary to restrict the class of admissible functions
for $p$; and to guarantee that ${\bf B}$ is consistent with a given $p$,
topological constraints on ${\bf B}$ must be enforced.
In Sec. II, we review the salient properties of 3D magnetic fields, which
generally have a fractal phase space, and we sketch the nature of continuous
solutions for $p$ and $\bf B$. This is based on the construction of an ergodic
partition; which, being fractal, is impractical from a standpoint of numerical
implementation. So, we describe a discrete partition which greatly simplifies
the equilibrium problem and leads naturally to stepped-pressure equilibria,
where the plasma is modeled as a set of nested volumes in each of which the
field satisfies the Beltrami equation, $\nabla\times{\bf B}=\mu{\bf B}$, and
across the interfaces that separate these volumes the total pressure is
continuous, $[[p+B^{2}/2]]=0$.
‘Sharp-boundary’ Berk et al. (1986) states and multi-volume Dewar et al.
(2008) sharp-boundary states have been considered previously, and Bruno &
Laurence Bruno and Laurence (1996) have presented theorems that insure the
existence of sharp boundary solutions, with an arbitrary number of pressure
jumps, for tori whose departure from axisymmetry is sufficiently small. In
Sec. III we introduce a variational approach to solving Eq. (1) based on the
notion of multi-region, relaxed MHD (MRXMHD), which is a generalization of
Taylor’s Taylor (1986) relaxed-MHD formulation: that a sufficiently
turbulent/chaotic, weakly non-ideal plasma will evolve so as to minimize the
energy subject to the constraint of conserved magnetic helicity, and in doing
so will break most of the constraints Yoshida and Dewar (2012) of ideal MHD,
thus allowing magnetic reconnection. In MRXMHD, a plasma with a non-trivial
pressure profile is constructed as a nested collection of relaxed states,
between which the ideal-MHD constraints apply. By deriving the Euler-Lagrange
equations, we see that the MRXMHD energy functional has stepped-pressure
equilibria as extremizing solutions. A close examination of the force-balance
condition, $[[p+B^{2}/2]]=0$, reveals that the rotational transform of the
interfaces must be strongly irrational.
In Sec. IV, the MRXMHD energy functional is discretized using a mixed Fourier,
finite-element representation for the vector potential and geometry. Setting
to zero the derivatives of the energy functional with respect to the vector-
potential in each volume gives a linear system for the magnetic field,
$\nabla\times{\bf B}=\mu{\bf B}$, where $\mu$ is a Lagrange multiplier
(sometimes called the Beltrami parameter). This can be adjusted in order to
preserve the helicity integral, or both $\mu$ and the enclosed poloidal flux
can be adjusted to satisfy the interface rotational transform constraints.
Assuming the Beltrami fields in each volume have been computed for a arbitrary
interface geometry, the problem of constructing an equilibrium solution is
standard: changes in the interface geometry are allowed to either minimize the
energy functional using conjugate gradient methods, or to find a zero of the
multi-dimensional gradient $\equiv$ force-balance vector using a Newton
method. To fully constrain the Fourier representation of the interface
geometry, we employ spectral-condensation Hirshman and Meier (1985); Hirshman
and Breslau (1998) methods to obtain a preferred poloidal angle coordinate.
Illustration of equilibrium states and convergence studies are then presented.
At appropriate points in the discourse, we contrast our approach to
constructing equilibrium solutions with others in the literature.
## II Hamiltonian chaos, partitioned
Magnetic-field-line flow is a Hamiltonian system Boozer (2005). The well-
developed theory of Hamiltonian dynamical systems (see, for example, the texts
by Wiggins Wiggins (1990) and Lichtenberg & Lieberman Lichtenberg and
Lieberman (1992), and the review by Meiss Meiss (1992)) provides a strong
foundation on which to build. We shall sometimes use general dynamical-systems
language rather than the more specialized plasma terminology; for instance,
using ‘orbit’ and ‘magnetic field line’ interchangeably. To facilitate the
following discussion, we use cylindrical coordinates, $(R,\phi,Z)$, which are
orthogonal and right handed, so that $x=R\cos(\phi)$, $y=R\sin(\phi)$, and
$z=Z$, and $(x,y,z)$ are Cartesian.
Devices of the tokamak and reversed-field-pinch (RFP) Boozer (2005) classes
use a large number of identical toroidal field coils arranged with a discrete
rotational symmetry about the $z$-axis. In the axisymmetric special case, it
is reasonable to seek solutions that are invariant under rotation.
Axisymmetric magnetic fields are representable as 1-degree-of-freedom (1-dof)
autonomous Hamiltonian systems Boozer (2005), with $\phi$, periodic, playing
the role of time. Such systems are integrable in the dynamical systems sense,
and action-angle coordinates may be constructed. The field lines lie on nested
invariant tori, $\psi=\mbox{const}$, that foliate the extended phase space,
$(\psi,\theta,\phi)$, where $\psi$ is a toroidal flux function and $\theta$ is
a poloidal angle that increases linearly against $\phi$. In the terminology of
magnetic confinement, the invariant tori are called magnetic flux surfaces,
and action-angle coordinates are called straight-field-line coordinates. In
the following, when we refer to an integrable system, we will assume that the
integrable system has shear.
The invariant tori $\equiv$ flux surfaces are characterized by their rotation
number $\equiv$ rotational transform, which is commonly denoted in magnetic
confinement plasma physics by $\iota\\!\\!$-. (Historically Kruskal and
Kulsrud (1958), the term rotational transform refers to $\iota$, the average
poloidal angle increase in each iteration of the return map, but in modern
usage Boozer (2005) it is used for $\mbox{$\iota\\!\\!$-}\equiv\iota/2\pi$,
and often the ‘bar’ is omitted.) If $\iota\\!\\!$- is a rational number,
$\mbox{$\iota\\!\\!$-}=n/m$ where $n$ and $m$ are integers, then the
corresponding surface is foliated by periodic orbits $\equiv$ closed field
lines, which close on themselves after $m$ toroidal transits, having undergone
$n$ poloidal transits. If $\iota\\!\\!$- is an irrational number, then the
flux surface is covered ergodically by a single quasi-periodic orbit, which
never closes on itself (but comes arbitrarily close), and each irrational
surface is the closure of an irrational field line.
In the axisymmetric case, the equilibrium problem can be reduced to the task
of solving a 2D partial differential equation, the Grad-Shafranov equation
Boozer (2005); Neches et al. (2008), which is well-posed (except for
bifurcations Solano (2004)). The equilibria are characterized by two free
profile functions, e.g. the pressure, $p(\psi)$, and the rotational transform,
$\mbox{$\iota\\!\\!$-}(\psi)$. Because space is foliated by flux surfaces,
equilibria with continuous, smooth profiles are admissible; in fact, the only
magnetic fields consistent with ${\bf B}\cdot\nabla p=0$ and globally smooth
profiles are integrable magnetic fields.
Axisymmetry is necessarily always broken to some extent by the modular nature
of the conductors and machine imperfection, or by intentionally applied
perturbation fields Evans et al. (2004), or by equilibrium bifurcations Cooper
et al. (2010). The stellarator family Spitzer (1958); Boozer (2005) of
confinement devices is intentionally nonaxisymmetric. This allows greater
freedom in the design of experiments and can provide enhanced plasma
stability. (The nonaxisymmetry of stellarators, however, generally leads to
degraded particle confinement; this can be ameliorated, somewhat, by the use
of ‘quasi-symmetric’ configurations Nuehrenberg and Zille (1988).)
The 3D magnetic field-line flow is still analogous to a Hamiltonian dynamical
system, but because there is no longer a symmetry coordinate, the 3D field-
line Hamiltonian is not autonomous. Such systems, still periodic in $\phi$,
are called $1\frac{1}{2}$-dof systems and are generically non-integrable,
meaning that the extended phase space is almost never foliated by invariant
tori.
The periodic orbits are fragile. Resonant magnetic fields associated with
geometric deformation destroy almost all of the periodic orbits; magnetic
islands form, and regions of chaotic magnetic field lines emerge. The
destruction of rational surfaces is related to the classical problem of small
denominators in the transformation to action-angle coordinates for the
perturbed system. The Poincar$\acute{\rm e}$-Birkhoff theorem Meiss (1992)
shows that, for every rational invariant torus present in the integrable case,
at least two of the periodic orbits survive. One orbit is hyperbolically
unstable, while the other is elliptically stable or has become hyperbolic
through a period-doubling bifurcation.
These orbits, known as Poincar$\acute{\rm e}$-Birkhoff orbits, form a robust
‘skeleton’ of invariant sets and provide crucial information about the
structure of phase space. The existence of a given KAM surface (described
below) can be inferred from the stability of nearby periodic orbits using
Greene’s residue criterion Greene (1979). Associated with each unstable
periodic orbit is an unstable manifold and a chaotic sea Wiggins (1990),
comprised of irregular trajectories without a well defined rotational
transform, i.e. the ratio $\Delta\theta/\Delta\phi$ does not converge as
$\Delta\phi\rightarrow\infty$, where $\Delta\theta$ and $\Delta\phi$ are the
increase in $\theta$ and $\phi$ along a field line. Although there is no
formal proof Meiss (1992), it is standard to assume, based on computational
evidence, that the closure of each chaotic sea is a three-dimensional subset
of $\mathbb{R}^{3}$, as each irregular trajectory seems to fill a volume.
Associated with the elliptic periodic orbits are local regions of regular
trajectories, the so-called magnetic islands.
The irrational field lines are quite robust to perturbation. Indeed, they are
guaranteed to survive by the Aubry-Mather theorem; however, a given irrational
field line may or may not come arbitrarily close to every point on a smooth
surface. The KAM theorem, named in honor of Kolmogorov, Arnold and Moser
Kolmogorov (1954); Moser (1962); Arnold (1963); Moser (1973); Arnold (1978),
shows that a finite measure of invariant tori do exist for sufficiently small,
smooth perturbations to an integrable system, provided that the rotational
transform, $\iota\\!\\!$-, is sufficiently irrational, i.e. $\iota\\!\\!$-
must satisfy a Diophantine condition: there exists an $r>0$ and $k\geq 2$ such
that, for all integers $n$ and $m$, $|\mbox{$\iota\\!\\!$-}-n/m|>r/m^{k}$.
About each rational, $n/m$, there is an excluded region of width $r/m^{k}$,
which is consistent with the emergence of a chaotic sea about every unstable
periodic orbit. KAM tori are two-dimensional subsets of $\mathbb{R}^{3}$ whose
union is of finite measure and forms a partition of phase space.
Typically, as the magnitude of the geometric deformation increases, the size
of the magnetic islands increases, the volume of the chaotic seas increases,
and each given KAM surface will become more geometrically deformed until a
critical point is reached, at which point the surface is continuous but no
longer smooth. These critical tori form fractal boundaries between the chaotic
seas associated with different island chains. By ‘fractal’ we simply mean
having a hierarchy of qualitatively self-similar structure on all scales, with
no minimum length scale, and is non-differentiable.
Some KAM tori are more robust than others. The most robust invariant tori are
those that have the ‘most irrational’ rotational transforms, where ‘most
irrational’ means most difficult to approximate with rationals. Such
irrationals are called noble, and their definition is made precise using the
continued fraction representation Niven (1956). The noble KAM tori are also
the smoothest, in that fewer Fourier harmonics are required for an accurate
description of their geometry.
After the destruction of a KAM surface, the closure of an irrational field
line has the structure of a Cantor set Aubry (1983); Mather (1982) and is
called a cantorus Percival (1979) (hint: Cantor + torus = cantorus). Cantori
are one-dimensional subsets of $\mathbb{R}^{3}$ Li and Bak (1986), and
constitute a set of zero measure that does not serve to partition phase space.
The cantori can, however, form effective partial barriers to field-line
transport MacKay et al. (1984a) and thus also to anisotropic diffusion Hudson
and Breslau (2008).
Ergodic invariant sets form a fractal hierarchy. The ‘primary’ chaotic seas
and KAM tori are infinitely intertwined, and each chaotic sea contains
‘secondary’ island chains, KAM tori, cantori, and chaotic seas in an ‘islands
around islands’ pattern repeated ad infinitum MacKay et al. (1984b); Meiss
(1986, 1992); Mezic and Wiggins. (1999); Levnajic and Mezic (2010). The
chaotic seas are infinitely multiply-connected, bounded externally by
critical, primary invariant tori, and internally by the infinite hierarchy of
islands.
### II.1 continuous solution on ergodic partition
To understand the general class of functions for $p$ and ${\bf B}$ that admit
solutions to the equilibrium problem we first consider the implications that
the non-integrability of the magnetic field has on the structure of the
pressure; and then, given a pressure that is consistent with a non-integrable
field, consider the implications this has on the field itself.
To understand the structure of the pressure function that satisfies ${\bf
B}\cdot\nabla p=0$, given a generic magnetic field, it is convenient to
represent phase space as a collection of pair-wise disjoint sets that are
invariant under the field-line flow map, $\bm{\varphi}_{\phi}:\mbox{$\bf
r$}_{0}\mapsto\mbox{$\bf r$}$. This map is constructed simply by following a
field line a distance $\phi$ in toroidal angle from a point, $\mbox{$\bf
r$}_{0}$, on a surface of section (for example the $\phi=0$ plane) to arrive
at point $\bf r$. The return map is generated by following field lines once
around the machine back to the initial surface of section. (In the case of the
RFP device, this discussion applies in a subdomain not containing points where
$B^{\phi}$ reverses sign – to treat the toroidal field-reversal region a
poloidal surface of section should be used instead.) An invariant set ${\cal
A}\subset\mathbb{R}^{2}$ within a surface of section is a set invariant under
the return map, $\bm{\varphi}_{2\pi}({\cal A})={\cal A}$. An invariant set
${\cal V}\subset\mathbb{R}^{3}$ within phase space may be constructed as the
continuous union of such sets, i.e. ${\cal
V}=\bigcup_{\phi\in[0,2\pi)}\bm{\varphi}_{\phi}({\cal A})$.
An invariant partition is a union of invariant tori, which are two-dimensional
magnetic surfaces, and three-dimensional invariant toroidal volumes or toroids
bounded by invariant tori. If an invariant volume contains an invariant
surface, e.g. a KAM surface, then the volume may be subdivided into two
distinct subvolumes, each of which is invariant under the return map. An
ergodic invariant set is a set with finite measure that allows no further
subdivision, and the ergodic partition of phase space is its decomposition
into ergodic invariant sets and a nonergodic (periodic) set of zero measure –
see definition 2.1 of Ref.Mezic and Wiggins. (1999).
For the purpose of constructing weak solutions to the equilibrium problem, we
are primarily interested in the sets of finite measure. We ignore the cantori
and periodic orbits and take our partition as having every chaotic sea, ${\cal
C}_{\alpha}$, each of which has finite volume, and the invariant surfaces,
${\cal C}_{\beta}$, the union of which has finite measure, as the only
elements with non-trivial measure, where $\alpha$ and $\beta$ are elements of
appropriate indexing sets (e.g. $\alpha$ is a rational and $\beta$ is
irrational). As any field line approaches every point arbitrarily closely in a
given ergodic set, the only solution for $p$ consistent with ${\bf
B}\cdot\nabla p=0$ is $p=p_{\alpha}=\mbox{const}$, $\forall\,\mbox{$\bf
r$}\in{\cal C}_{\alpha}$, and similarly for ${\cal C}_{\beta}$. The most
general solution for the pressure is
$\displaystyle p(\mbox{$\bf r$})=p_{\alpha}I_{\alpha}(\mbox{$\bf
r$})+p_{\beta}I_{\beta}(\mbox{$\bf r$}),$ (4)
where $I_{\alpha}$ is an indicator function on each ergodic component, i.e.
$I_{\alpha}(\mbox{$\bf r$})=1$ if $\mbox{$\bf r$}\in{\cal C}_{\alpha}$ and
$I_{\alpha}(\mbox{$\bf r$})=0$ otherwise, and similarly for $I_{\beta}$.
We now recognize that $\mbox{$\bf B$}(\mbox{$\bf r$})$ is not arbitrary and
seek a similarly general characterization of the constraints that Eq. (3)
places on this function. Each ${\cal C}_{\alpha}$ has finite volume, and we
assume that ${\bf B}$ is differentiable within ${\cal C}_{\alpha}$. That the
pressure is constant in ${\cal C}_{\alpha}$ implies that $\nabla p=0$. Then,
force balance, $\nabla p={\bf j}\times{\bf B}$, implies that
$\nabla\times\mbox{$\bf B$}=\mu(\mbox{$\bf r$})\mbox{$\bf B$}$, for some
scalar function $\mu({\bf r})$. Taking the divergence of this equation, we
find $\mbox{$\bf B$}\cdot\nabla\mu=0$. Thus, like $p$, $\mu$ must be constant
within each ergodic region Rusbridge (1977), $\mu=\mu_{\alpha}$ in ${\cal
C}_{\alpha}$, and $\bf B$ must be a linear force-free field, i.e. it satisfies
the Beltrami equation,
$\displaystyle\nabla\times\mbox{$\bf B$}=\mu_{\alpha}\mbox{$\bf B$}.$ (5)
This is a well-studied linear elliptic partial differential equation, about
which much is known Kress (1981, 1986); Yoshida and Giga (1990); Yoshida
(1992); Marsh (1996); Hudson et al. (2007). To construct a solution in a given
domain it is required to specify (i) the boundary of the domain; (ii)
appropriate boundary conditions, e.g. $\mbox{$\bf n$}\cdot\mbox{$\bf B$}=0$,
where $\bf n$ is the unit normal; and (iii) homological conditions, i.e. line
integrals (fluxes) around topologically inequivalent loops. To specify a
solution in a simple torus it is sufficient to specify $\mu$ and the toroidal
flux, while in a doubly-connected annulus the poloidal flux must also be
specified Hudson et al. (2007).
Solving the Beltrami equation in general ${\cal C}_{\alpha}$ is, however, an
intractable numerical problem. Because of the topological complexity resulting
from the infinity of islands embedded in the chaotic sea, there is an infinity
of inequivalent closed loops. The outer boundary of each chaotic sea is
presumably a critical KAM torus, which is not smooth, and the normal to the
fractal boundary is not defined.
Furthermore, a continuous, non-trivial pressure, that is consistent with a
generic non-integrable field, must be fractal. To see this, we may assume that
a finite pressure gradient is supported by the KAM tori. The Diophantine
condition serves as a simple, proxy indicator function describing the
existence of KAM tori in the fractal phase space of a generic non-integrable
field (though the more complicated Bruno function Locatelli et al. (2000) is
probably a better approximation). Let us consider a Diophantine pressure
profile, $p(\mbox{$\iota\\!\\!$-})$, defined
$p^{\prime}(\mbox{$\iota\\!\\!$-})=1$ if $|\mbox{$\iota\\!\\!$-}-n/m|>r/m^{k}$
for all integers $n$ and $m$, and $p^{\prime}(\mbox{$\iota\\!\\!$-})=0$
otherwise, supplemented with the condition $p(0)=0$. The function
$p(\mbox{$\iota\\!\\!$-})$ is continuous by construction (i.e. the derivative
is nowhere infinite) and we assume that $r$ and $k$ have been chosen so that
$p^{\prime}(\mbox{$\iota\\!\\!$-})$ is non-zero on a set of finite measure (so
that not all the excluded regions overlap) so that $p(\mbox{$\iota\\!\\!$-})$
is non-trivial. Even for this ‘toy’ model, numerically approximating the
function $p(\mbox{$\iota\\!\\!$-})$ given $p^{\prime}(\mbox{$\iota\\!\\!$-})$
is rather complicated.
An approximation to $p(\mbox{$\iota\\!\\!$-})$ may (in the case of continuous
$p^{\prime}$) be constructed using a tagged partition, i.e.
$p(\mbox{$\iota\\!\\!$-})\approx\sum_{i}p^{\prime}(x_{i})(\mbox{$\iota\\!\\!$-}_{i}-\mbox{$\iota\\!\\!$-}_{i-1})$,
where $x_{i}\in[\mbox{$\iota\\!\\!$-}_{i-1},\mbox{$\iota\\!\\!$-}_{i}]$ and
$0=\mbox{$\iota\\!\\!$-}_{0}<\mbox{$\iota\\!\\!$-}_{1}<\dots<\mbox{$\iota\\!\\!$-}_{N}=\mbox{$\iota\\!\\!$-}$.
However, because $p^{\prime}(\mbox{$\iota\\!\\!$-})$ is nowhere continuous
except where $p^{\prime}(\mbox{$\iota\\!\\!$-})=0$, the result depends on the
choice of $x_{i}$ even when
$|\mbox{$\iota\\!\\!$-}_{i}-\mbox{$\iota\\!\\!$-}_{i-1}|\rightarrow 0,\forall
i$. That is, the Riemann integral of $p^{\prime}(\mbox{$\iota\\!\\!$-})$ does
not exist. The error between this approximation and the exact solution is not
bounded.
More sophisticated numerical discretizations could be derived; for example, by
choosing the $x_{i}$ in the tagged partition to coincide with the locally most
irrational, i.e. by constructing an ‘irrational’ tagged partition Hudson
(2004). However, a precise treatment would involve numerically approximating
the Lebesgue integral and complicated measure theory. In 1967, Grad Grad
(1967) made a similar comment, describing the pressure as “pathological”.
Furthermore, the fractal structure of non-integrable fields in toroidal
confinement devices will be far more complicated than that described by the
Diophantine condition, and may not (will not!) be known apriori. There are
numerical diagnostics for determining the structure of phase space, such as
Greene’s residue criterion, but these diagnostics come at considerable
computational cost.
Considering (i) that a nonlinear equilibrium calculation will inevitably
require an iterative approach, in which the fractal phase space structure of
the field may need to be re-evaluated at each iteration, and (ii) that the
critical KAM tori are fragile, and an infinitesimal change in ${\bf B}$ can
cause an abrupt, finite change in the volume of any given chaotic sea, and
(iii) that the fractal structure of phase space will need to be resolved
sufficiently accurately in order to guarantee that an appropriately defined
error is below some bound, that can be made arbitrarily small as the numerical
resolution is increased; we may expect that this computational cost would be
excessive.
For our purpose of constructing a robust and efficient numerical solution of
well-defined, 3D MHD equilibria with non-integrable fields, with a bounded
error that can be made arbitrarily small, it is far better to work with smooth
functions, and to employ an algorithm that does not depend on resolving the
infinitely complicated structure of phase space. So, we extend our class of
functions for $p$ and ${\bf B}$ beyond globally continuous functions, as non-
trivial, continuous functions that satisfy force balance are necessarily
fractal, and consider instead functions that are continuous, and smooth,
almost everywhere; that is, we consider functions that are smooth except for a
finite set of discontinuities, which can be easily managed numerically.
### II.2 weak solution on discrete partition
Continuous pressure profiles are not the most general solutions of Eq. (1).
Discontinuous pressure profiles may seem unphysical, but they are a valid
solution class within the zero-Larmor-radius MHD model we have adopted. If
continuous, globally smooth solutions are required, then additional ‘non-
ideal’ physics should be included Park et al. (1986). For example, including a
small, but finite, diffusion of the pressure perpendicular to the magnetic
field will provide solutions with a globally smooth pressure; and including a
small resistivity will prevent the formation of singular currents. Appropriate
source terms are required to balance dissipative effects.
This is the approach adopted by various codes Cappello and Biskamp (1996);
Chacon (2008); Jardin et al. (2007); Sovinec et al. (2003) that can
approximate an MHD equilibrium as a resistive steady state, but which are best
described as initial-value, time-evolution codes and cannot, strictly, compute
an equilibrium that satisfies ${\bf B}\cdot\nabla p=0$, with the pressure
given. The algorithms these codes employ become increasingly ill-conditioned
as the non-ideal terms approach zero del Castillo-Negrete and Chacon (2011).
We now describe a restriction of the solution class that greatly simplifies
the equilibrium problem. A discrete invariant partition of phase space is
constructed. The disjoint, invariant sets that are surrounded by a given
primary chaotic sea, e.g. the hierarchy of island chains, are absorbed into
the chaotic sea itself, which then becomes either a simply or doubly connected
region; and the outer boundary of these regions is extended past the adjacent,
critical boundary surface to a smooth, noble surface, which also serves as the
boundary for the next ‘extended’ chaotic sea. That is, we choose a set of
smooth, noble KAM tori, ${\cal I}_{l}$, where $l=1,2,\ldots N_{V}$, that
partitions phase space into $N_{V}$ invariant toroidal or annular subvolumes,
${\cal V}_{l}$. Each ${\cal V}_{l}$ is an invariant set under the field line
map, but not necessarily an ergodic invariant set because the field may not be
totally chaotic.
In each region, ${\cal V}_{l}$, we equate all the $\mu_{\alpha^{\prime}}$ to a
single constant $\mu_{l}$, and all the $p_{\alpha^{\prime}}$ and
$p_{\beta^{\prime}}$ to a single constant $p_{l}$, where $\alpha^{\prime}$ and
$\beta^{\prime}$ label all the chaotic seas and invariant tori within ${\cal
V}_{l}$. Each ${\cal V}_{l}$ is simply or doubly connected with a smooth
boundary and it is a simple computational task to solve $\nabla\times{\bf
B}_{l}=\mu_{l}{\bf B}_{l}$ in each ${\cal V}_{l}$. We will enforce the
constraint that ${\bf n}\cdot{\bf B}=0$ on the ${\cal I}_{l}$, but otherwise
the topology of the field in each ${\cal V}_{l}$ is unconstrained.
For the pressure, rather than restricting attention to a globally continuous
pressure with finite pressure-gradient on the uncountably infinite ${\cal
C}_{\beta}$, we instead consider a piecewise continuous pressure with finite
pressure-jumps on the finite set ${\cal I}_{l}$. Intuitively, we imagine that
all of the pressure in the continuous-but-fractal model that is supported by
the ${\cal C}_{\beta}$ in the vicinity of a selected noble KAM torus is placed
on the noble torus itself: all of the pressure is placed on a finite selection
of the most irrational surfaces. The vanishing of the divergence of the stress
tensor, Eq. (1), in a neighborhood of a surface of discontinuity gives a
condition McGann et al. (2010) that must be satisfied at the interfaces,
namely that the total pressure must be continuous across the ${\cal I}_{l}$,
i.e. $[[p+B^{2}/2]]=0$.
We have described the interfaces where there exists a discontinuity in the
pressure and the tangential field as KAM surfaces, but this is rather loose
terminology. Such interfaces are perhaps ‘double-sided’ KAM surfaces, being
covered by an field line $\equiv$ integral curve with irrational frequency, of
the field, ${\bf B}_{-}$, immediately inside the torus, while also being
covered by an integral curve with the same irrational frequency of the perhaps
different field, ${\bf B}_{+}$, immediately outside the torus. In the next
section, where we describe the MRXMHD energy functional, we shall refer to the
${\cal I}_{l}$ as ideal interfaces and the ${\cal V}_{l}$ as relaxed volumes,
and describe why the ${\cal I}_{l}$ are required to have irrational rotational
transform.
In the above discussion, we have argued that stepped-pressure equilibria arise
naturally when one seeks a numerically tractable discretization of the
equilibrium problem that is consistent with the zero-Larmor-radius model of
MHD; satisfies ${\bf B}\cdot\nabla p=0$; and is consistent with what is known
about the fractal phase space structure of non-integrable fields. The
equilibria could also be described as multi-volume sharp-boundary states. A
major motivation for pursuing this model is that Bruno & Laurence Bruno and
Laurence (1996) have proven that such stepped-pressure equilibria exist
(provided the departure from axisymmetry is sufficiently small). The number of
volumes, $N_{V}$, and interfaces may be made arbitrarily large. We can,
depending on the numerical resources available, consider a sequence of
invariant partitions with increasing $N_{V}$, so the discontinuities in the
pressure are made arbitrarily small, in order to study the nature of a
continuous-but-fractal equilibrium via a sequence of well-defined, stepped-
pressure equilibria. Stepped-pressure profiles are sufficiently general to
represent observed profiles to within experimental error.
In order to explore the properties of these equilibria in arbitrary geometry,
i.e. to go beyond what may be proved analytically, this model has been
implemented numerically in the stepped-pressure equilibrium code, SPEC, as
will be described below in Sec. IV. We now show that there is an multi-region,
relaxed MHD energy functional, that we call MRXMHD, that has stepped-pressure
equilibria as extremizing solutions.
## III energy functional method
The classic MHD energy functional Kruskal and Kulsrud (1958) is given by the
integral
$\displaystyle W\equiv\int_{{\cal
V}}\left(\frac{p}{\gamma-1}+\frac{B^{2}}{2}\right)dv,$ (6)
where ${\cal V}$ is the plasma volume bounded by a toroidal surface,
$\partial{\cal V}$. Ideal equilibria are obtained when the plasma is in a
minimum energy state: more precisely, when the energy functional is extremized
allowing for a restricted class of variations, namely ideal variations. The
equation of state, $d_{t}(p/\rho^{\gamma})=0$, where
$d_{t}\equiv\partial_{t}+{\bf v}\cdot\nabla$ and ${\bf v}$ is the ‘velocity’
of an assumed plasma displacement, ${\bf
v}=\partial_{t}\mbox{\boldmath$\xi$\unboldmath}$, may be combined with mass
conservation, $\partial_{t}\rho+\nabla\cdot(\rho\,{\bf v})=0$, to obtain an
equation that constrains the variation in the pressure, $\delta
p=(\gamma-1)\mbox{\boldmath$\xi$\unboldmath}\cdot\nabla
p-\gamma\nabla\cdot(p\,\mbox{\boldmath$\xi$\unboldmath})$. Faraday’s law,
$\partial_{t}{\bf B}=\nabla\times{\bf E}$, may be combined with the ideal
Ohm’s law, ${\bf E}+{\bf v}\times{\bf B}=0$, where ${\bf E}$ is the electric
field, to obtain an equation that constrains the variation in the magnetic
field, $\delta{\bf B}=\nabla\times(\mbox{\boldmath$\xi$\unboldmath}\times{\bf
B})$. Note that this last constraint does not allow the topology of the field
to change.
The first variation in the energy due to an ideal displacement, $\xi$, that is
assumed to vanish on the boundary, is given by
$\displaystyle\delta W\equiv\int_{{\cal V}}(\nabla p-{\bf j}\times{\bf
B})\cdot\mbox{\boldmath$\xi$\unboldmath}\,dv.$ (7)
Extremizing solutions satisfy the ideal force-balance condition, $\nabla
p={\bf j}\times{\bf B}$. In order to uniquely define an equilibrium, in
addition to the shape of the plasma boundary, it is required to specify the
pressure, and either the rotational-transform or the parallel current density
Bauer et al. (1984); Boozer (2005). (Note that the constraints of ideal MHD
places a constraint on the differential toroidal and poloidal fluxes and the
rotational-transform, namely $d\psi_{p}/d\psi_{t}=\mbox{$\iota\\!\\!$-}$.)
The VMEC Hirshman and Whitson (1983) and BETAS/NSTAB codes Betancourt (1988);
Taylor (1994) are based on this approach. These codes assume that the magnetic
field is integrable and allow for smooth pressure and rotational-transform
profiles.
In general 3D geometry, there is a singularity in the resonant harmonic of the
parallel current in equilibria with nested flux surfaces Bhattacharjee et al.
(1995). Writing the current as ${\bf j}=\sigma{\bf B}+{\bf j}_{\perp}$, the
quasineutrality condition, $\nabla\cdot{\bf j}=0$, requires that the parallel
current must satisfy the magnetic differential equation, ${\bf
B}\cdot\nabla\sigma=-\nabla\cdot{\bf j}_{\perp}$, where we may consider the
perpendicular current to be driven by the pressure gradient, ${\bf
j}_{\perp}={\bf B}\times\nabla p/B^{2}$. Magnetic differential equations are
densely singular Newcomb (1959). The singularity may be exposed, in the
integrable case, by the use of straight field line coordinates, which allow
the directional derivative along the magnetic field to be written
$\sqrt{g}\,{\bf
B}\cdot\nabla\equiv\mbox{$\iota\\!\\!$-}\partial_{\theta}+\partial_{\phi}$.
Using a Fourier representation, e.g.
$\sigma=\sum_{m,n}\sigma_{m,n}\exp(im\theta-in\phi)$, we derive
$\sigma_{m,n}=-i(\sqrt{g}\,\nabla\cdot{\bf
j}_{\perp})_{m,n}/(m\mbox{$\iota\\!\\!$-}-n)$$+\hat{j}_{m,n}\,\delta(m\mbox{$\iota\\!\\!$-}-n)$.
The first term is called the Pfirsch-Schl$\ddot{\rm u}$ter current and has a
$1/x$ style singularity at the rational surface, where
$x\equiv\mbox{$\iota\\!\\!$-}-n/m$. The second term, the $\delta$-function
current, is generally required to ‘shield’ out resonant magnetic fields that
would otherwise destroy the nested family of flux surfaces (a more precise
discussion of the $\delta$-function current is provided in Boozer and Pomphrey
(2010)).
In general geometry, the only way to avoid the $1/x$ singular currents is
ensure that the pressure gradient is zero in the vicinity of the rational
surfaces (or to ensure that no rational surfaces are present). As the rational
surfaces are dense in space, to avoid the $1/x$ singularities the pressure
gradient must be zero everywhere.
(Despite these concerns near the rational surfaces, VMEC, in particular, does
an impressive job of robustly constructing global approximations to 3D
equilibria with arbitrary pressure profiles; presumably, this is because VMEC
seeks approximations to minima of the global energy functional, and does not
directly seek solutions to $\nabla p={\bf j}\times{\bf B}$ pointwise.)
### III.1 MRXMHD energy principle
The first step towards constructing the multi-region, relaxed MHD energy
functional is to partition space into discrete volumes. We introduce a set of
nested, toroidal surfaces, ${\cal I}_{l}$, for $l=1,2,\ldots N_{V}$ where
${\cal I}_{l}\equiv\partial{\cal V}$ for $l=N_{V}$. The energy local to each
volume is
$\displaystyle W_{l}\equiv\int_{{\cal
V}_{l}}\left(\frac{p}{\gamma-1}+\frac{B^{2}}{2}\right)dv,$ (8)
where ${\cal V}_{1}$ is the toroid enclosed by ${\cal I}_{1}$, and ${\cal
V}_{l}$ is the annular volume enclosed by ${\cal I}_{l-1}$ and ${\cal I}_{l}$
for $l=2,\ldots N_{V}$.
We again assume that the plasma is in a minimum energy state; however, we
allow for the effects of small resistivity: in each ${\cal V}_{l}$, the
magnetic field may relax and reconnect (and so topological constraints between
the toroidal and poloidal fluxes, the rotational-transform, and the helicity
are broken). But, in order to retain some control over the equilibria, we
consider the ${\cal I}_{l}$ to be preserved as ideal barriers that restrict
both pressure transport and field transport. Rather than continuously
constraining the topology, the topology is discretely constrained. This, or
something equivalent, is required in order to avoid trivial solutions.
In each ${\cal V}_{l}$, the mass and entropy constraints usually used in ideal
MHD do not apply to individual fluid elements, but apply instead to the entire
volume, giving the isentropic, ideal-gas constraint,
$\displaystyle p_{l}V_{l}^{\gamma}=a_{l},$ (9)
where $V_{l}$ is the volume of ${\cal V}_{l}$ and $a_{l}$ is a constant. The
internal energy in ${\cal V}_{l}$ is $\int_{{\cal V}}p_{l}/(\gamma-1)\,dv$ $=$
$a_{l}V_{l}^{(1-\gamma)}/(\gamma-1)$, and the first variation of this due to a
deformation, $\xi$, of the boundary is $-p\int_{\partial{\cal
V}}\mbox{\boldmath$\xi$\unboldmath}\cdot d{\bf s}$.
To constrain the relaxation of the magnetic field in each ${\cal V}_{l}$, we
follow Taylor Taylor (1974), who argued that the ‘most conserved’ invariant
for a weakly resistive plasma is the helicity Finn and Antonsen Jr. (1985);
Taylor (1986),
$\displaystyle K_{l}\equiv\int_{{\cal V}_{l}}\mbox{$\bf A$}\cdot\mbox{$\bf
B$}\,dv,$ (10)
where $\bf A$ is a vector potential, $\mbox{$\bf B$}=\nabla\times\mbox{$\bf
A$}$, which we consider to be differentiable and a single-valued function of
position. The helicity is related to the Gauss linking number: it reflects how
‘knotted’, or ‘twisted’, the magnetic field lines are Taylor (1986); Marsh
(1996); A.Berger (1999). The helicity in Eq. (10) is not gauge-invariant. A
gauge-invariant form is constructed by adding the loop integrals
$\Delta\psi_{p}\oint_{\cal S}\mathbf{A}\cdot d\mathbf{l}$ and
$\Delta\psi_{t}\oint_{\cal L}\mathbf{A}\cdot d\mathbf{l}$, where ${\cal S}$ is
a poloidal loop on ${\cal I}_{l-1}$ and ${\cal L}$ is a toroidal loop on
${\cal I}_{l}$.
In each ${\cal V}_{l}$, variations in the pressure and the field, and the
geometry of the interfaces, are allowed in order to extremize the energy
functional. These variations are arbitrary, except for (i) the mass-entropy
constraint, $p_{l}V_{l}^{\gamma}=const$; (ii) helicity conservation in each
${\cal V}_{l}$; (iii) the interfaces must remain tangential to the magnetic
field; and (iv) the magnetic fluxes are conserved.
The MRXMHD energy functional is
$\displaystyle
F=\sum_{l}\left[W_{l}-\frac{\mu_{l}}{2}\left(K_{l}-K_{l,o}\right)\right].$
(11)
The helicity constraint, $K_{l}=K_{l,o}$ where $K_{l,o}$ is a given constant,
is enforced explicitly by introducing a Lagrange multiplier, $\mu_{l}$, in
each ${\cal V}_{l}$. The flux constraints and the tangentiality condition at
the interfaces will be enforced implicitly by constraining the representation
of the magnetic field.
The most general function space for ${\bf B}$ in each volume is space of
vector-valued functions whose magnitude is square integrable, i.e. $\mbox{$\bf
B$}\in\mathcal{L}^{2}({\cal V}_{l})$, by which it is meant that
$B^{2}\in\mathcal{L}^{1}({\cal V}_{l})$, and $\mathcal{L}^{1}({\cal V}_{l})$
is the standard notation for the space of integrable scalar functions. More
precisely, we follow Yoshida et al. Yoshida and Giga (1990); Yoshida and Dewar
(2012) and restrict $\bf B$ to $\mathcal{L}_{\sigma}^{2}({\cal V}_{l})$, which
they define as the subspace of $\mathcal{L}^{2}({\cal V}_{l})$ occupied by
divergence-free fields that obey $\mbox{$\bf B$}\cdot{\bf n}=0$ on the
boundary. Similarly, the pressure is required to be integrable, i.e.
$p\in\mathcal{L}^{1}({\cal V}_{l})$. While these are the least restrictive
spaces required for a weak formulation, in order for the solutions to obey
tractable local differential equations almost everywhere we will, after
deriving the Euler Lagrange equations for states that extremize Eq. (11),
further restrict the allowed function spaces by assuming that $p$ is piecewise
constant and that $\bf B$ piecewise satisfies a simple elliptic partial
differential equation, which is solved numerically using a mixed Fourier and
finite element method.
The variation in the ‘local’ constrained energy functional, $F_{l}\equiv
W_{l}-\mu_{l}\left(K_{l}-K_{l,o}\right)/2$, due to arbitrary variations in the
field, $\delta{\bf B}=\nabla\times\delta{\bf A}$, and arbitrary variations,
$\xi$, in the interface geometry, is given by
$\displaystyle\delta F_{l}$ $\displaystyle=$ $\displaystyle\int_{{\cal
V}_{l}}\left(\nabla\times{\bf B}-\mu_{l}{\bf B}\right)\cdot\delta{\bf A}\,dv$
(12) $\displaystyle-$ $\displaystyle\int_{\partial{\cal
V}_{l}}\left(p_{l}+B^{2}/2\right)\mbox{\boldmath$\xi$\unboldmath}\cdot d{\bf
s}.$
The variation in the magnetic potential, $\delta\mbox{$\bf A$}$, is free
within ${\cal V}_{l}$, and so within each ${\cal V}_{l}$ the topology of the
field is arbitrary; but at the ${\cal I}_{l}$ it must obey
$\displaystyle\mbox{$\bf n$}\times\delta\mbox{$\bf A$}=-\mbox{$\bf
n$}\cdot\mbox{\boldmath$\xi$\unboldmath}\,\mbox{$\bf B$}+\mbox{$\bf
n$}\times\nabla\delta g,$ (13)
so that $\mbox{$\bf n$}\cdot\mbox{$\bf B$}=0$ remains satisfied; and where
$\delta g(\mbox{$\bf r$})$ is the variation in a single-valued gauge
potential, $g$, required for generality but physical quantities are invariant
with respect to gauge choice. The line integrals of $\bf A$ along arbitrary
loops $\mathcal{L}^{\rm pol}$ and $\mathcal{L}^{\rm tor}$ are related to the
poloidal and toroidal magnetic fluxes.
The enclosed toroidal fluxes in each volume and the poloidal fluxes in each
annular region constrain the magnetic field from being trivial. We use gauge
freedom to specify the loop integrals of $\oint\mathbf{A}\cdot d\mathbf{l}$ on
each interface. For gauges satisfying these conditions, the difference between
the gauge-invariant helicity and the gauge-dependent helicity is a constant.
The Euler-Lagrange equation for $F$ to be stationary with respect to
variations in the magnetic field in each ${\cal V}_{l}$ is the Beltrami
equation, $\nabla\times{\bf B}=\mu_{l}{\bf B}$. The Euler-Lagrange equation
for $F$ to be stationary with respect to variations in the interface geometry
is that the total pressure must be continuous across the interfaces,
$[[p+B^{2}/2]]=0$. States that extremize the MRXMHD energy functional are
stepped-pressure equilibria.
The pressure and tangential field are discontinuous across the interfaces, but
these comprise a finite set of measure zero and so $p$ and $B^{2}$ are both
integrable functions: the model is consistent with our goal of constructing
weak solutions via an energy-integral approach. The discontinuities are easily
accommodated for in the numerical discretization; within each volume a
continuous, smooth representation for the vector potential is allowed.
To avoid a problem with ‘small denominators’, as will be discussed below, we
will typically enforce the condition that the interfaces have irrational
transform. The problematic Pfirsch-Schl$\ddot{\rm u}$ter currents are
eliminated because the pressure gradient is identically zero across the
resonances. The $\delta$-function currents are also not present because the
topology of field is unrestricted at the rational surfaces, i.e. magnetic
islands are allowed to form.
There are, instead, a finite set of surface currents at the irrational, ideal
interfaces, given by $\mbox{\boldmath$\kappa$\unboldmath}=[[{\bf
B}]]\times{\bf n}$, where $[[{\bf B}]]$ is the tangential discontinuity in the
field. These are required to enforce the topological constraint, the
topological constraint in this case being that a noble irrational surface
exists; and the topological constraint is required so that the magnetic field
matches the given, stepped-pressure profile. Given the KAM theorem, this
topological constraint is presumably easier to enforce Ciraolo et al. (2004)
than forcing a rational flux surface. We expect these currents to be dominated
by the discontinuity in $p$, and may be thought of as a discrete approximation
to the pressure induced currents. These irrational surface currents may be
compared to (but are different from) the $\delta$-function currents shielding
at the rational surfaces, which are required in the linearly-perturbed, ideal-
equilibrium codes IPEC Park et al. (2007) and CAS3D N$\ddot{\rm u}$hrenberg
and Boozer (2003). The $\delta$-function currents at the rational surface
currents do not describe the pressure driven $1/x$ singularities.
We invoke multi-region energy minimization with helicity conservation
primarily as a mathematical device to achieve a variational formulation of the
restricted equilibrium class, namely stepped-pressure equilibria. MRXMHD is,
however, a generalization of the variational principle enunciated by Woltjer
Woltjer (1958) to generate linear force-free fields of interest in
astrophysics, and developed by Taylor Taylor (1974) to model fusion plasma
experiments.
The success of the Taylor relaxation theory in describing experimental data
suggests that the MRXMHD approach may likewise aid physical interpretation of
partial relaxation, reconnection and self-organization in toroidal plasmas
supporting a non-trivial pressure profile. Unlike Taylor’s globally relaxed
model, which gives a constant pressure across the plasma, MRXMHD is only
locally relaxed, i.e. it is partially constrained; arbitrarily many interfaces
may be included, each with an associated ideal $\equiv$ topological
constraint.
### III.2 transform constraint
A close examination of the interface force-balance condition,
$[[p+B^{2}/2]]=0$, reveals a Hamiltonian system, which we call the pressure-
jump Hamiltonian McGann et al. (2010). Let $p_{-}$ and ${\bf B}_{-}$ be the
pressure and field immediately inside a given interface and $p_{+}$ and ${\bf
B}_{+}$ be the pressure and field immediately outside. By combining (i) the
general, covariant representation for the field, ${\bf B}=B_{s}\nabla
s+B_{\theta}\nabla\theta+B_{\phi}\nabla\phi$, with (ii) $\nabla\times{\bf
B}=\mu{\bf B}$, and (iii) the tangentiality condition, ${\bf B}\cdot{\bf
n}=0$; we may write $B_{\theta}=\partial_{\theta}f_{\theta}$ and
$B_{\phi}=\partial_{\phi}f$, and
$B^{2}=$$(g_{\phi\phi}f_{\theta}f_{\theta}-2g_{\theta\phi}f_{\theta}f_{\phi}+g_{\theta\theta}f_{\phi}f_{\phi})$$/$$(g_{\theta\theta}g_{\phi\phi}-g_{\theta\phi}g_{\theta\phi})$,
where $f(\theta,\phi)$ is a surface potential and $g_{\theta\theta}$,
$g_{\theta\phi}$ and $g_{\phi\phi}$ are metric elements (local to the
interface). Now, consider the case where both ${\bf B}_{-}$ and the geometry
of the interface are known, and we seek a solution for ${\bf B}_{+}$ that
satisfies $H=const$, where
$\displaystyle H\equiv 2(p_{-}-p_{+})=B_{+}^{2}-B_{-}^{2}.$ (14)
We may write $H\equiv K(\theta,\phi,f_{\theta},f_{\phi})+V(\theta,\phi)$,
where $V\equiv-B_{-}^{2}$ is assumed known, and
$f_{\theta}\equiv\partial_{\theta}f$ and $f_{\phi}\equiv\partial_{\phi}f$,
where $f$ is an as-yet-unknown surface potential for ${\bf B}_{+}$.
To derive the solvability condition we treat $f_{\theta}$ and $f_{\phi}$ as
independent quantities (generalized momenta) and recognize $H$ as a 2-dof
Hamiltonian with a conserved energy, $2(p_{-}-p_{+})$. Then, $H=const$ along a
trajectory given by Hamilton’s equations: $d\theta/dt=\partial H/\partial
f_{\theta}$, $df_{\theta}/dt=-\partial H/\partial\theta$, $d\phi/dt=\partial
H/\partial f_{\phi}$, and $df_{\phi}/dt=-\partial H/\partial\phi$; where $t$
is an artificial ‘time’. This system may be reduced to a $1\frac{1}{2}$-dof
system by using $\phi$ as the time-like integration parameter (always possible
if $d_{t}\phi\neq 0$) and eliminating the integration of $f_{\phi}$ in favor
of inverting
$K(\theta,\phi,f_{\theta},f_{\phi})+V(\theta,\phi)=2(p_{-}-p_{+})$ for
$f_{\phi}$, so that $f_{\phi}$ is assumed to be a function of $\theta$, $\phi$
and $f_{\theta}$, i.e. $f_{\phi}=f_{\phi}(\theta,\phi,f_{\theta})$, where the
dependence on $2(p_{-}-p_{+})$ is implicit. The trajectory is then described
by $\dot{\theta}\equiv d_{t}\theta/d_{t}\phi$ and $\dot{f}_{\theta}\equiv
d_{t}f_{\theta}/d_{t}\phi$, which may be integrated in $\phi$ from an initial
starting point, $(\theta,f_{\theta})$, on a Poincar$\acute{\rm e}$ section,
e.g. $\phi=0$. If the trajectory lies on an invariant surface then it is
possible to construct $f_{\theta}=f_{\theta}(\theta,\phi)$, and $f_{\theta}$
and $f_{\phi}$ recover their interpretation as derivatives of a surface
function: there exists a well defined $f(\theta,\phi)$ such that
$f_{\theta}=\partial_{\theta}f$ and $f_{\phi}=\partial_{\phi}f$. That is, if
the trajectory lies on an invariant surface, then a solution for ${\bf B}_{+}$
that satisfies $2(p_{-}-p_{+})=B_{+}^{2}-B_{-}^{2}$ may be constructed.
An invariant surface can only exist if it avoids the problem of small
divisors. Note that $\dot{\theta}=B^{\theta}/B^{\phi}$, so there is a
fundamental relationship between the pressure-jump Hamiltonian and the field-
line Hamiltonian; and force-balance can only be satisfied if the rotational
transform of the interfaces is irrational McGann et al. (2010).
Many authors have considered sharp boundary equilibria either theoretically or
in simplified geometry Berk et al. (1986); Kaiser and Salat (1994); Lortz and
Spies (1994); Kaiser (1994); Spies and Li (1994); Spies et al. (2001); Spies
(2003); Kaiser and Uecker (2004); Hole et al. (2006, 2007); Hudson et al.
(2007); Mills et al. (2009); Hole et al. (2009). This paper, and our earlier
paper Hudson et al. (2012), represents the first numerical study of toroidal
3D equilibria with multiple Beltrami regions within the plasma. The only 3D
calculation of which we are aware is the early paper of Betancourt and
Garabedian Betancourt and Garabedian (1975), who consider a free-boundary
problem with both the vacuum region and the plasma being Beltrami regions with
$\mu=0$.
A number of 3D MHD equilibrium codes based on the assumption of continuity and
differentiability of $p$ and $\bf B$ have been written Betancourt and
Garabedian (1975); Bauer et al. (1978, 1982); Betancourt and Garabedian
(1982); Hirshman and Whitson (1983); Bhattacharjee et al. (1984); Bauer et al.
(1984); Hirshman et al. (1986); Park et al. (1986); Reiman and Greenside
(1986); Harafuji et al. (1989); Hirshman and Betancourt (1991); Taylor (1994);
Suzuki et al. (2006); Hirsman et al. (2011). These have either constrained the
magnetic field to be globally integrable; have not employed numerical
algorithms that explicitly accommodate the singularities in the parallel
current at the rational surfaces; do not constrain the profiles, and allow the
pressure, current and transform profiles to ‘evolve’ during the calculation in
a fashion more akin to initial-value, time-evolution codes rather than what is
suitable for an equilibrium code; have introduced small non-ideal terms, so
that the ${\bf B}\cdot\nabla p\neq 0$; have ignored the fractal hierarchy of
the ergodic invariant sets; or employ ill-posed numerical algorithms (e.g. the
so-called Spitzer Spitzer (1958) iterative approach Boozer (1984); Reiman and
Greenside (1986), which attempts to invert densely-singular magnetic
differential equations Hudson (2010)). While they have produced a variety of
results, their lack of formal foundations leads them to fall short of the
numerical rigor (e.g. demonstration of convergence, quantification of error,
estimate of stability Gardner and Blackwell (1992)) available in the
axisymmetric case.
We will now describe the stepped-pressure equilibrium code, SPEC, and
demonstrate that the solutions are well defined by presenting convergence
studies.
## IV Numerical Discretization
A Fourier representation is employed for all doubly-periodic, scalar
functions. Even functions, $f(-\theta,-\zeta)=f(\theta,\zeta)$, are written
$\displaystyle f$ $\displaystyle=$
$\displaystyle\sum_{n=0}^{N}f_{0,n}\cos(-nN_{P}\zeta)$ (15) $\displaystyle+$
$\displaystyle\sum_{m=1}^{M}\sum_{n=-N}^{N}f_{m,n}\cos(m\theta-nN_{P}\zeta),$
where $N_{P}$ is the field periodicity. The resolution of the Fourier
representation is determined by $M$ and $N$, and the total number of Fourier
harmonics is $N_{MN}\equiv(N+1)+M(2N+1)$. The poloidal, $\theta$, and
toroidal, $\zeta$, angles are, as yet, arbitrary. The Fourier summation will
be written concisely as $f=\sum_{j}f_{j}\cos(m_{j}\theta-n_{j}\zeta)$, where
$(m_{1},n_{1})=(0,0)$, etc. A similar description is used for odd (i.e. sine)
functions, $f(-\theta,-\zeta)=-f(\theta,\zeta)$.
An initial guess for the geometry of a set of $N_{V}$ nested, toroidal
surfaces, ${\cal I}_{l}$, is assumed given. For expedience, we assume
stellarator symmetry Dewar and Hudson (1998), so that the ${\cal I}_{l}$ may
be described by $R(\theta,\zeta)\,\hat{R}+Z(\theta,\zeta)\,\hat{k}$, with
$\displaystyle\begin{array}[]{ccc}R_{l}(\theta,\zeta)&=&\displaystyle\sum_{j}R_{l,j}\cos(m_{j}\theta-
n_{j}\zeta),\\\
Z_{l}(\theta,\zeta)&=&\displaystyle\sum_{j}Z_{l,j}\sin(m_{j}\theta-
n_{j}\zeta),\end{array}$ (18)
where $\hat{R}\equiv\cos\phi\,\,\hat{i}+\sin\phi\,\,\hat{j}$, and $\hat{i}$,
$\hat{j}$ and $\hat{k}$ are the Cartesian unit vectors.
To enforce various boundary conditions, it is convenient to use toroidal
coordinates, $(s,\theta,\zeta)$, that are adapted to the interfaces. These
coordinates are defined inversely via
$\displaystyle
R=R(s,\theta,\zeta)\,,\,\,\phi=-\zeta\,,\,\,Z=Z(s,\theta,\zeta).$ (19)
The Jacobian of the $(s,\theta,\zeta)$ coordinates is
$\sqrt{g}=R(R_{s}Z_{\theta}-R_{\theta}Z_{s})$. The ‘lower’ metric
coefficients, $g_{\alpha\beta}$, are given by
$g_{\alpha\beta}=R_{\alpha}R_{\beta}+Z_{\alpha}Z_{\beta}+\delta_{\alpha\beta}R^{2}$,
where $\delta_{\alpha\beta}=1$ if $\alpha=\beta=\zeta$ and
$\delta_{\alpha\beta}=0$ otherwise. The coordinate functions are given by
$\displaystyle\begin{array}[]{ccc}R(s,\theta,\zeta)&=&\displaystyle\sum_{j}R_{j}(s)\cos(m_{j}\theta-
n_{j}\zeta),\\\
Z(s,\theta,\zeta)&=&\displaystyle\sum_{j}Z_{j}(s)\sin(m_{j}\theta-
n_{j}\zeta),\end{array}$ (22)
where the $R_{j}(s)$, $Z_{j}(s)$ are a piecewise-linear interpolation of the
$R_{l,j}$ and $Z_{l,j}$. (A piecewise-cubic interpolation would give a
continuous Jacobian across the interfaces, but this is not required.) If the
toroidal flux is monotonic increasing, then $s\equiv\psi_{t}$, normalized to
its value at the outermost interface, is a suitable radial coordinate. In this
case, $s\sim r^{2}$, where $r$ is a polar-like radial coordinate. More
generally, we may use the interface label itself as the radial coordinate,
i.e. $s_{l}=l/N_{V}$.
In the innermost volume, regularization factors must be included to prevent
the interpolated coordinate surfaces from overlapping. These factors may be
derived by considering an arbitrary, regular (infinitely-differentiable)
function, $h(x,y)$, at the origin,
$h(x,y)=h+xh_{x}+yh_{y}+\frac{1}{2}\left(x^{2}h_{xx}+2xyh_{xy}+y^{2}h_{yy}\right)+\dots$.
By constructing a Fourier representation,
$h(r,\theta)=\sum_{m}\left[h_{m}^{c}(r)\cos(m\theta)+h_{m}^{s}(r)\sin(m\theta)\right]$
where $x=r\cos\theta$ and $y=r\sin\theta$, we obtain after repeated
applications of double-angle formulae,
$\displaystyle
h_{m}(r)=r^{m}(a_{0}r^{0}+a_{1}r^{2}+a_{2}r^{4}+a_{3}r^{6}+\dots).$ (23)
So, in the innermost volume, $s\leq s_{1}$, we write
$R_{j}(s)=R_{j,1}s^{m_{j}/2}/s^{m_{j}/2}_{1}$, and similarly for $Z_{j}(s)$,
where $s\sim r^{2}$.
In ${\cal V}_{l}$ that is bounded by ${\cal I}_{l-1}$ and ${\cal I}_{l}$, a
general covariant representation of the magnetic vector potential is
$\displaystyle{\bf\bar{A}}_{l}=\bar{A}_{s,l}\nabla
s+\bar{A}_{\theta,l}\nabla\theta+\bar{A}_{\zeta,l}\nabla\zeta.$ (24)
To this add a gauge term, $\nabla g_{l}(s,\theta,\zeta)$, where $g_{l}$
satisfies
$\displaystyle\begin{array}[]{ccl}\partial_{s}g_{l}(s,\theta,\zeta)&=&-\bar{A}_{s,l}(s,\theta,\zeta),\\\
\partial_{\theta}g_{l}(s_{l-1},\theta,\zeta)&=&-\bar{A}_{\theta,l}(s_{l-1},\theta,\zeta)+\psi_{t,l-1},\\\
\partial_{\zeta}g_{l}(s_{l-1},0,\zeta)&=&-\bar{A}_{\zeta,l}(s_{l-1},0,\zeta)+\psi_{p,l-1},\end{array}$
(28)
for arbitrary constants, $\psi_{t,l-1}$ and $\psi_{p,l-1}$. Then ${\bf
A}_{l}={\bf\bar{A}}_{l}+\nabla g_{l}$ is given by ${\bf
A}_{l}=A_{\theta,l}\nabla\theta+A_{\zeta,l}\nabla\zeta$ with
$\displaystyle\begin{array}[]{ccc}A_{\theta,l}(s_{l-1},\theta,\zeta)&=&\psi_{t,l-1},\\\
A_{\zeta,l}(s_{l-1},0,\zeta)&=&\psi_{p,l-1}.\end{array}$ (31)
For stellarator symmetric equilibria, $A_{\theta,l}$ and $A_{\zeta,l}$ may be
represented by cosine series,
$\displaystyle\begin{array}[]{ccc}A_{\theta,l}(s,\theta,\zeta)&=&\displaystyle\sum_{j}A_{\theta,l,j}(s)\cos(m_{j}\theta-
n_{j}\zeta),\\\
A_{\zeta,l}(s,\theta,\zeta)&=&\displaystyle\sum_{j}A_{\zeta,l,j}(s)\cos(m_{j}\theta-
n_{j}\zeta),\end{array}$ (34)
where $A_{\theta,l,j}(s)$ and $A_{\zeta,l,j}(s)$ are represented using finite-
elements, as will be described below.
The toroidal flux is given by
$\displaystyle\int_{\cal S}{\bf B}\cdot{\bf ds}$ $\displaystyle=$
$\displaystyle\oint_{\partial{\cal S}}{\bf A}\cdot{\bf
dl}=2\pi\;\psi_{t,l-1},$ (35)
where the surface ${\cal S}$ is that part of the $\zeta=0$ plane bounded by
$s=s_{l-1}$. The poloidal flux is given by
$\displaystyle\int_{\cal S}{\bf B}\cdot{\bf ds}$ $\displaystyle=$
$\displaystyle\oint_{\partial{\cal S}}{\bf A}\cdot{\bf
dl}=2\pi\;\psi_{p,l-1}.$ (36)
where the surface ${\cal S}$ is that part of the $\theta=0$ plane bounded by
$s=s_{l-1}$.
The boundary condition that the inner interface is a flux surface becomes
${\bf B}\cdot\nabla s=0$, which implies
$\displaystyle-m_{j}A_{\zeta,l,j}(s_{l-1})-n_{j}A_{\theta,l,j}(s_{l-1})=0.$
(37)
Combing Eq. (31) and Eq. (37) we have
$\displaystyle
A_{\theta,l,j}(s_{l-1})=\left\\{\begin{array}[]{c@{\quad,\quad}c}\psi_{t,l-1}&j=1,\\\
0&j>1,\end{array}\right.$ (40) $\displaystyle
A_{\zeta,l,j}(s_{l-1})=\left\\{\begin{array}[]{c@{\quad,\quad}c}\psi_{p,l-1}&j=1,\\\
0&j>1.\end{array}\right.$ (43)
The condition that the outer interface is a flux surface is satisfied by
writing
$\displaystyle
A_{\theta,l}(s_{l})=\partial_{\theta}f_{l}(\theta,\zeta)\,,\,\,\,A_{\zeta,l}(s_{l})=\partial_{\zeta}f_{l}(\theta,\zeta),$
(44)
for arbitrary $f$ of the form
$\displaystyle
f_{l}=\psi_{t,l}\theta+\psi_{p,l}\zeta+\sum_{j}f_{l,j}\sin(m_{j}\theta-
n_{j}\zeta),$ (45)
and $\psi_{t,l}\equiv A_{\theta,l,1}(s_{l})$ and $\psi_{p,l}\equiv
A_{\zeta,l,1}(s_{l})$. We have
$\displaystyle
A_{\theta,l,j}(s_{l})=\left\\{\begin{array}[]{c@{\quad,\quad}c}\psi_{t,l}&j=1,\\\
m_{j}f_{l,j}&j>1,\end{array}\right.$ (48) $\displaystyle
A_{\zeta,l,j}(s_{l})=\left\\{\begin{array}[]{c@{\quad,\quad}c}\psi_{p,l}&j=1,\\\
-n_{j}f_{l,j}&j>1.\end{array}\right.$ (51)
The radial dependence of the vector potential harmonics is described using
finite-elements. A continuous function, $f(x)$, with $x\in[0,1]$, may be
approximated using the linear basis functions, $\varphi_{L,0}(x)=1-x$ and
$\varphi_{R,0}(x)=x$, according to
$f(x)=f_{L,0}\varphi_{L,0}(x)+f_{R,0}\varphi_{R,0}(x)$, where $f_{L,0}\equiv
f(0)$ and $f_{R,0}\equiv f(1)$. A piecewise-linear interpolation of the vector
potential gives a discontinuous magnetic field in each ${\cal V}_{l}$. While
this is legitimate as far as the energy integral is concerned (the magnetic
field remains an integrable function), we prefer a smoother interpolation.
For piecewise-cubic interpolation, the basis functions are
$\varphi_{L,0}(x)=2x^{3}-3x^{2}+1$ and $\varphi_{L,1}(x)=x^{3}-2x^{2}+x$, and
their ‘reflections’, $\varphi_{R,p}(x)=(-1)^{p}\varphi_{L,p}(1-x)$. An
arbitrary smooth continuous function is approximated by
$f(x)=\sum_{p=0}^{N_{D}}[f_{L,p}\varphi_{L,p}(x)+f_{R,p}\varphi_{R,p}(x)]$,
where $f_{L,1}\equiv f^{\prime}(0)$ and $f_{R,1}\equiv f^{\prime}(1)$ and
$N_{D}=1$. For piecewise-quintic interpolation the same expression applies,
but with $N_{D}=2$ and $\varphi_{L,0}(x)=-6x^{5}+15x^{4}-10x^{3}+1$,
$\varphi_{L,1}(x)=-3x^{5}+8x^{4}-6x^{3}+1$, and
$\varphi_{L,2}(x)=-\frac{1}{2}x^{5}+\frac{3}{2}x^{4}-\frac{3}{2}x^{3}+\frac{1}{2}x^{2}$,
and their reflections.
In each ${\cal V}_{l}$, a regular, radial sub-grid is established,
$\displaystyle s_{l,i}=s_{l-1}+i\left(s_{i}-s_{l-1}\right)/N_{l},$ (52)
for $i=0,1,\ldots,N_{l}$. The resolution, $N_{l}$, of the radial sub-grid may
be different in each ${\cal V}_{l}$. The vector potential harmonics are
written
$\displaystyle
A_{\theta,l,j}(s)=\sum_{p=0}^{N_{D}}\left[A_{\theta,l,j,p,i-1}\;\varphi_{L,p}(x)+A_{\theta,l,j,p,i}\;\varphi_{R,p}(x)\right],$
where $x=(s-s_{l,i-1})/\Delta s_{l}$ with $\Delta s_{l}=s_{l,i}-s_{l,i-1}$.
The vector potential is completely specified by $A_{\theta,l,j,p,i}$ and
$A_{\zeta,l,j,p,i}$, which are the $p$-th derivatives of the $(m_{j},n_{j})$
harmonics of $A_{\theta}$ and $A_{\zeta}$ on the $i$-th grid-point in the
$l$-th annulus. Except for the subtlety required to ensure the field is
tangential to the outer interface, see Eq. (48) and Eq. (51), these are the
independent parameters that describe the vector potential, and thus the
magnetic field.
In the innermost volume, the condition that the field is tangential to the
inner interface is replaced (because there is no inner interface) by the
condition that the vector potential is analytic at the coordinate origin.
Assuming $s\sim r^{2}$, we may enforce regularity at the origin and restrict
the gauge by including $s^{m_{j}/2}$ radial factors with the
$A_{\theta,l,j,p,i}$ and $A_{\zeta,l,j,p,i}$, with the boundary conditions
$\displaystyle A_{\theta,1,j,0,0}$ $\displaystyle=$ $\displaystyle
0\;\;\;\mbox{\rm for all $j$,}$ (53) $\displaystyle A_{\zeta,1,j,0,0}$
$\displaystyle=$ $\displaystyle 0\;\;\;\mbox{\rm for $m_{j}=0$ and $n_{j}\neq
0$.}$ (54)
The mixed finite-element, Fourier representation of the magnetic vector
potential is inserted into the MRXMHD energy integral, $F=\sum_{l}F_{l}$,
where the ‘local’ energy functional is given by $F_{l}\equiv
W_{l}-\mu_{l}\left(K_{l}-K_{l,o}\right)/2$, where
$W_{l}\equiv\int\left[p/(\gamma-1)+B^{2}/2\right]dv$ and $K_{l}\equiv\int{\bf
A}\cdot{\bf B}\,dv$. With ${\bf
A}=A_{\theta}\nabla\theta+A_{\zeta}\nabla\zeta$, the magnetic field ${\bf
B}=B^{s}{\bf e_{s}}+B^{\theta}{\bf e_{\theta}}+B^{\zeta}{\bf e_{\zeta}}$ is
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\sqrt{g}\;{\bf B}$
$\displaystyle=$
$\displaystyle(\partial_{\theta}A_{\zeta}-\partial_{\zeta}A_{\theta})\;{\bf
e_{s}}-\partial_{s}A_{\zeta}\;{\bf e_{\theta}}+\partial_{s}A_{\theta}\;{\bf
e_{\zeta}},$ (55)
and
$\displaystyle B^{2}$ $\displaystyle=$ $\displaystyle
B^{s}B^{s}\,g_{ss}+2B^{s}B^{\theta}\,g_{s\theta}+2B^{s}B^{\zeta}\,g_{s\zeta}$
$\displaystyle+$ $\displaystyle
B^{\theta}B^{\theta}\,g_{\theta\theta}+2B^{\theta}B^{\zeta}\,g_{\theta\zeta}+B^{\zeta}B^{\zeta}\,g_{\zeta\zeta},$
$\displaystyle\sqrt{g}\;{\bf A}\cdot{\bf B}$ $\displaystyle=$ $\displaystyle-
A_{\theta}\partial_{s}A_{\zeta}+A_{\zeta}\partial_{s}A_{\theta}.$
After substituting these into the local energy and helicity integrals, the
$A_{\theta,l,j,p,i}$ and $A_{\zeta,l,j,p,i}$ are multiplied by terms such as
$\displaystyle\int_{s_{l,i-1}}^{s_{l,i}}\\!\\!\\!\\!\\!\\!\\!\\!ds\,\int_{0}^{2\pi}\\!\\!\\!\\!\\!\\!\\!d\theta\\!\int_{0}^{2\pi}\\!\\!\\!\\!\\!\\!\\!d\zeta\,\,\varphi_{L,p}(s)\,\,\varphi_{R,q}(s)\,f(s,\theta,\zeta),$
(56)
where, for example, $f\equiv\sin(m_{j}\theta-
n_{j}\zeta)\,g_{\alpha\beta}\,\cos(m_{k}\theta-n_{k}\zeta)$, and
$g_{\alpha\beta}$ are the metric elements (note that the $g_{\alpha\beta}$
depend on the $R_{l,j}$ and $Z_{l,j}$ that define the interface geometry).
These integrals are computed by constructing a fast Fourier transform of
$f(s,\theta,\zeta)$. The integrals over the angles then become trivial and we
obtain
$\bar{f}(s)\equiv\varphi_{L,p}(s)\varphi_{R,q}(s)\int\\!d\theta\\!\int\\!d\zeta\,f(s,\theta,\zeta)$.
The remaining radial quadrature is approximated using Gaussian integration,
$\displaystyle\int_{0}^{1}\\!\\!ds\,\bar{f}(s)\approx\sum_{i=1}^{N_{G}}\omega_{i}\,\bar{f}(s_{i}),$
(57)
where the ‘weights’, $\omega_{i}$ and the ‘abscissae’, $s_{i}$, are chosen to
optimize accuracy, and $N_{G}$ is a numerical resolution parameter that
depends on the order of the polynomial being integrated, i.e. $N_{G}$ depends
on the order of the finite-element basis expansion for the
$A_{\theta,l,j,p,i}$ and $A_{\zeta,l,j,p,i}$, and the order of the coordinate
interpolation of the $R_{l,j}$ and $Z_{l,j}$.
In each ${\cal V}_{l}$, the local energy functional, $F_{l}\equiv
W_{l}-\mu_{l}\left(K_{l}-K_{l,o}\right)/2$, depends on the interface geometry,
the vector potential, and various input parameters. Specifically, each $F_{l}$
depends on ${\bf x}\equiv\\{R_{l,j},Z_{l,j}\\}$; the enclosed toroidal flux,
$\Delta\psi_{t,l}\equiv\psi_{t,l}-\psi_{t,l-1}$; the enclosed poloidal flux,
$\Delta\psi_{p,l}\equiv\psi_{p,l}-\psi_{p,l-1}$ (except in the innermost
volume); the required helicity, $K_{l,o}$; the vector potential, ${\bf
a}_{l}\equiv\\{A_{\theta,l,j,p,i},A_{\zeta,l,j,p,i}\\}$; and the Lagrange
multiplier, $\mu_{l}$. We may indicate the dependence of $F$ as
$\displaystyle F_{l}=F_{l}[\Delta\psi_{t,l},\Delta\psi_{p,l},K_{l,o},{\bf
x};\mu_{l},{\bf a}_{l}].$ (58)
In the MRXMHD model, the enclosed fluxes and the helicity in each ${\cal
V}_{l}$ are assumed given, i.e. the $\Delta\psi_{t,l}$, $\Delta\psi_{p,l}$ and
$K_{l,o}$ are required input parameters. (Note: if the interface rotational-
transform constraint is to be given priority over the conservation of poloidal
flux and helicity, then $\Delta\psi_{p,l}$ and $K_{l,o}$ must generally be
allowed to vary.) The computational task is to then find extrema of
$F=\sum_{l}F_{l}$ with respect to the interface geometry, ${\bf
x}\equiv\\{R_{l,j},Z_{l,j}\\}$, and the Lagrange multipliers, $\mu_{l}$, and
the vector potentials, $\\{A_{\theta,l,j,p,i},A_{\zeta,l,j,p,i}\\}$.
The basic algorithm is to consider $\mu_{l}$ and ${\bf
a}_{l}\equiv\\{A_{\theta,l,j,p,i},A_{\zeta,l,j,p,i}\\}$ to be functions of the
interface geometry and the $\Delta\psi_{t,l}$, $\Delta\psi_{p,l}$, and
$K_{l,o}$. That is, first, the Beltrami field, $\nabla\times{\bf
B}=\mu_{l}{\bf B}$, in each ${\cal V}_{l}$ is constructed. We then may write
$F_{l}=F_{l}[\Delta\psi_{p,l},K_{l,o},{\bf x}]$, where the dependence on
$\Delta\psi_{t,l}$ is implicit. (Later, for computational expedience, we shall
modify this slightly by using $\mu_{l}$ to parametrize the solutions to the
Beltrami fields and remove $K_{l,o}$, so that
$F_{l}=F_{l}[\Delta\psi_{p,l},\mu_{l},{\bf x}]$.) Then, ‘global’ equilibrium
states are then constructed by extremizing $F=\sum_{l}F_{l}$ with respect to
the interface geometry, ${\bf x}\equiv\\{R_{l,j},Z_{l,j}\\}$.
### IV.1 solving $\nabla\times{\bf B}=\mu{\bf B}$ for ${\bf B}$, given
geometry
Assuming that the geometry of the interfaces, ${\bf x}$, is given, there are
various numerical methods that may be employed to construct the extremizing
fields. The first method is the standard Lagrange multiplier approach: a
multi-dimensional Newton method is used to find an extremum of the local
constrained energy functional. The solution satisfies
$\displaystyle\frac{\partial F_{l}}{\partial{\bf
a}_{l}}=0\;,\;\;\frac{\partial F_{l}}{\partial\mu_{l}}=0\;,$ (59)
where, in addition to ${\bf
a}_{l}\equiv\\{A_{\theta,l,j,p,i},A_{\zeta,l,j,p,i}\\}$, the Lagrange
multiplier is explicitly treated as an independent degree of freedom and must
be adjusted to satisfy the helicity constraint.
This approach cannot distinguish states that _minimize_ $W_{l}$ from states
which are saddle points or local maxima of $W_{l}$. When bifurcated solutions
exist, i.e. when there exist multiple stationary points of $W_{l}$ and hence
$F_{l}$, a gradient-descent algorithm such as sequential quadratic programming
Nocedal and Wright (1999) may instead be used to ensure that the constructed
solution is strictly a local minimum of $W_{l}$ subject to the constraint
$K_{l}=K_{l,o}$
Another method for constructing the Beltrami fields, on which we hereafter
concentrate, is to assume each ${\bf B}_{l}$ is parametrized by the enclosed
fluxes and $\mu_{l}$. The required value for the helicity, $K_{l,o}$, may be
dropped from the local energy functional, to obtain $F_{l}\equiv
W_{l}-\mu_{l}K_{l}/2$, and we write
$\displaystyle F_{l}=F_{l}[\Delta\psi_{p,l},\mu_{l},{\bf x};{\bf a}_{l}],$
(60)
where the dependence on $\Delta\psi_{t,l}$ is implicit.
The local energy functional, $F_{l}$, is quadratic in the $A_{\theta,l,j,p,i}$
and $A_{\zeta,l,j,p,i}$, and the ‘local’ equilibrium condition, $\partial
F_{l}/\partial{\bf a}_{l}=0$, gives a system of linear equations to be solved
for the vector potential. We call this the Beltrami linear system, as it is
analogous to $\nabla\times{\bf B}=\mu_{l}{\bf B}$ and can be represented as
$\displaystyle{\bf G}\cdot{\bf a}={\bf c},$ (61)
where the matrix ${\bf G}$ depends on the geometry, the fluxes and $\mu_{l}$,
i.e. ${\bf G}={\bf G}[\Delta\psi_{p,l},\mu_{l},{\bf x}]$; and similarly for
the right-hand-side vector, ${\bf c}$. This system is inhomogeneous (i.e.
${\bf c}$ is non-trivial) because of the ‘driving’ terms
$A_{\theta,1,l,0,0}=\psi_{t,l}$ and $A_{\zeta,1,l,0,0}=\psi_{p,l}$. Given
${\bf x}$, there is a two-dimensional family of solutions; each solution is
parametrized by $\Delta\psi_{p,l}$ and $\mu_{l}$.
There is an abundance of numerical methods and ‘canned’ numerical routines
available for solving linear equations, and any mathematical structure present
can be exploited. For example, usually the matrix ${\bf G}$ is positive
definite and it is typically very sparse, and an initial ‘guess’ for the
solution is often available (particularly so during an iterative calculation).
Employing numerical methods that exploit the sparsity and positive definite-
ness can significantly improve code performance.
It will be efficient to know how the Beltrami field in a given volume varies
with small variations in both the input parameters and the interface geometry.
In equilibria that are globally constrained by ideal MHD, variations in the
magnetic field are related to variations in geometry via $\delta{\bf
B}=\nabla\times(\mbox{\boldmath$\xi$\unboldmath}\times{\bf B})$. In our case,
we can compute the change in the vector potential resulting from an
infinitesimal change in $\Delta\psi_{p,l}$, or in $\mu_{l}$, or in the
interface geometry, ${\bf x}$, using matrix perturbation theory, $({\bf
G}+d{\bf G})\cdot({\bf a}+d{\bf a})=({\bf c}+d{\bf c})$, so that to lowest
order
$\displaystyle{\bf G}\cdot d{\bf a}=d{\bf c}-d{\bf G}\cdot{\bf a}.$ (62)
The infinitesimal variations, $d{\bf G}$ and $d{\bf c}$, resulting from
infinitesimal variations in $\Delta\psi_{l,p}$ and $\mu_{l}$ are rather simple
to write down because $\Delta\psi_{l,p}$ and $\mu_{l}$ just appear as factors
multiplying various geometric quantities in ${\bf G}$ and ${\bf c}$. The
derivatives with respect to the $R_{l,j}$ and $Z_{l,j}$ are more complicated
as these involve differentiating Eq. (56).
The Beltrami field in ${\cal V}_{l}$ depends on the geometry of both the
‘inner’ interface, i.e. the $R_{l-1,j}$ and $Z_{l-1,j}$, and the ‘outer’
interface, i.e. the $R_{l,j}$ and $Z_{l,j}$; giving a total of $4N_{MN}-2$
geometrical degrees of freedom, where $N_{MN}$ describes the Fourier
resolution. In preference over repeatedly inverting the Beltrami matrix
$4N_{MN}-2$ times, which would be the case if finite-differences for example
were used to compute the change in the Beltrami fields, we instead first
compute a Cholesky factorization of ${\bf G}$, i.e. ${\bf G}={\bf L}\,{\bf
L}^{T}$. Then, the solution to Eq. (62) is efficiently given by ${\bf
L}\cdot{\bf y}={\bf b}$, where ${\bf b}\equiv d{\bf c}-d{\bf G}\cdot{\bf a}$,
and ${\bf L}^{T}\cdot d{\bf a}={\bf y}$.
The helicity, $K_{l}\equiv\int{\bf A}\cdot{\bf B}\;dv$, depends on the
solution to Eq. (61), which in turn depends on $\mu_{l}$. The helicity
constraint, $K_{l}=K_{l,o}$, can be enforced by suitably adjusting $\mu_{l}$.
(This is not always possible; this is only for configurations in which
$\mu_{l}$ parametrizes states with different helicity.)
### IV.2 transform constraint, noble irrationals
The rotational-transform constraint can similarly be enforced. If only the
Beltrami field within a single annulus is to be constructed, then there is no
constraint on the allowed values of the transform on the interfaces. Recall
however, that if the Beltrami fields in multiple volumes are to be
consistently nested together in a fashion that satisfies force balance, an
analysis of the pressure-jump Hamiltonian derived from $[[p+B^{2}/2]]=0$ shows
that the interfaces should have irrational transform.
We restrict attention to noble irrationals Niven (1956), which play an
important role Meiss (1992) in the theory of chaos as invariant KAM surfaces
with noble transform are most likely to survive chaos-inducing perturbations
Greene (1979). A noble irrational is obtained as the limit of an infinite,
alternating path down a Farey tree, which is constructed as follows. Begin
with a pair of rationals, $p_{1}/q_{1}$ and $p_{2}/q_{2}$, which should be
neighboring, i.e. $|p_{1}q_{2}-p_{2}q_{1}|=1$, and without loss of generality
we assume that $p_{1}/q_{1}<p_{2}/q_{2}$. A Farey tree is constructed by
successively constructing the mediants, defined as
$p/q=(p_{1}+p_{2})/(q_{1}+q_{2})$. This is guaranteed to lie between the
original ‘parent’ rationals and so splits the original interval into left,
$[p_{1}/q_{1},p/q]$, and right, $[p/q,p_{2}/q_{2}]$, sub-intervals. The
mediant is neighboring to both parents, and the construction of the Farey tree
proceeds iteratively. An infinite, alternating path down the Farey tree is a
sequence of mediants for alternately the left and right subintervals.
Sequences of this type converge to noble irrationals, which have continued
fraction representations that terminate in an infinite sequence of $1$’s Meiss
(1992). It is easy to see that alternating paths give Fibonacci sequences for
the numerator and denominator of the successive rationals. For example,
beginning from $p_{1}/q_{1}=0/1$ and $p_{2}/q_{2}=1/1$ and constructing an
alternating path of mediants we obtain the sequence $\frac{0}{1}$,
$\frac{1}{1}$, $\frac{1}{2}$, $\frac{2}{3}$, $\frac{3}{5}$, $\frac{5}{8}$,
$\dots$ This allows noble irrationals to be written in the concise form
$\mbox{$\iota\\!\\!$-}(p_{1},q_{1},p_{2},q_{2})=(p_{1}+\gamma\,p_{2})/(q_{1}+\gamma\,q_{2})$,
where the golden-mean, $\gamma=(1+\sqrt{5})/2$, is the limiting ratio of
successive terms, $\gamma=F_{n+1}/F_{n}$ as $n\rightarrow\infty$, of the
Fibonacci sequence beginning from $F_{0}=0$ and $F_{0}=1$.
The poloidal angle parametrization of the interfaces is, at this stage,
arbitrary; it is not required, and will not be required, that the field lines
are ‘straight’. We may, however, construct the straight-field-line angle on
the interfaces, given the field, by calculating the angle transformation,
$\theta_{s}\equiv\theta+\lambda(\theta,\zeta)$, such that
$\displaystyle\frac{{\bf B}\cdot\nabla\theta_{s}}{{\bf
B}\cdot\nabla\zeta}=\mbox{$\iota\\!\\!$-},$ (63)
where $\iota\\!\\!$- is, as yet, an unknown constant to be determined. We
restrict attention to angle transformations of the form
$\lambda=\sum_{j}\lambda_{j}\sin(m_{j}\theta-n_{j}\zeta)$, which preserves
stellarator symmetry but is otherwise general. The Fourier resolution of the
angle transformation is independent of the Fourier resolution, $M$ and $N$,
used to represent the interfaces and Beltrami fields, and typically we use an
enhanced Fourier resolution for $\lambda$. With Eq. (55) and using $B^{s}=0$,
Eq. (63) becomes $A_{\theta}^{\prime}\,\partial_{\zeta}\lambda-
A_{\zeta}^{\prime}\,\partial_{\theta}\lambda-
A_{\theta}^{\prime}\,\mbox{$\iota\\!\\!$-}=A_{\zeta}^{\prime}$, where the
prime indicates radial derivative. By equating coefficients, we obtain a
system of linear equations for the unknowns,
${\mbox{\boldmath$\lambda$\unboldmath}}=(\mbox{$\iota\\!\\!$-},\lambda_{2},\lambda_{3},\dots)^{T}$,
which is represented as a matrix equation,
$\displaystyle{\bf\Lambda}\cdot\mbox{\boldmath$\lambda$\unboldmath}={\bf d},$
(64)
where ${\bf\Lambda}$ and ${\bf d}$ depend on the $A_{\theta}^{\prime}$ and
$A_{\zeta}^{\prime}$ harmonics at the interfaces. Solving this linear-system
determines $\lambda$, which gives the rotational-transform on the interface,
namely $\iota\\!\\!$-.
Figure 1: Poincar$\acute{\rm e}$ plot showing the Beltrami fields in multiple,
nested volumes and the ideal interfaces (thick lines) for the perturbed
equilibrium on the cross sections (A) $\zeta=0$; (B) $\zeta=\pi/2$; and (C)
$\zeta=\pi$.
Considering the geometry of the interfaces and the enclosed toroidal flux in
each ${\cal V}_{l}$ to be fixed, each Beltrami field depends only on $\mu_{l}$
and $\Delta\psi_{p,l}$. We thus have two degrees-of-freedom, and we must
satisfy two constraints; these constraints being that the field in ${\cal
V}_{l}$ provides the required rotational transform on the inner interface,
${\cal I}_{l-1}$, and on the outer interface, ${\cal I}_{l}$. (In the
innermost volume, ${\cal V}_{1}$, there is only one degree of freedom, namely
$\mu_{1}$, and only one constraint, namely that the field provides the
required rotational transform on ${\cal I}_{1}$.) Defining the function
$f(\mu_{l},\Delta\psi_{p,l})=(\mbox{$\iota\\!\\!$-}_{inn}-\mbox{$\iota\\!\\!$-}_{l-1},\mbox{$\iota\\!\\!$-}_{out}-\mbox{$\iota\\!\\!$-}_{l})$,
where $\mbox{$\iota\\!\\!$-}_{inn}$ and $\mbox{$\iota\\!\\!$-}_{out}$ are the
transforms determined from solving Eq. (64) on the inner and outer interface
for the magnetic field parametrized by $(\mu_{l},\Delta\psi_{p,l})$, and
$\mbox{$\iota\\!\\!$-}_{l-1}$ and $\mbox{$\iota\\!\\!$-}_{l}$ are prescribed
input values, we employ a simple Newton method to set
$f(\mu_{l},\Delta\psi_{p,l})=0$. Typically, if a reasonable guess is provided,
this converges in one or two iterations. Matrix perturbation methods are used
to compute the derivatives: the infinitesimal variation,
$d\mbox{\boldmath$\lambda$\unboldmath}$, resulting from an infinitesimal
variation in $\Delta\psi_{l}$ or $\mu_{l}$ is given by ${\bf\Lambda}\cdot
d{\mbox{\boldmath$\lambda$\unboldmath}}=d{\bf
d}-d{\bf\Lambda}\cdot\mbox{\boldmath$\lambda$\unboldmath}$, and
$d{\bf\Lambda}$ depends on ${\bf\Lambda}$ and $d{\bf G}$ via the chain rule;
and similarly for $d{\bf d}$. This search is computationally efficient: the
integral metric elements, Eq. (56), do not need to be recomputed if the
geometry does not change; and the matrix ${\bf\Lambda}$ is very sparse, and
sparse linear algorithms are employed.
In the MRXMHD model, the poloidal flux and helicity are assumed given. In the
stepped-pressure model, the interface rotational transforms are constrained.
(In both cases, the toroidal flux is constrained.) In either case, given the
interface geometry, the Beltrami field in each volume that satisfies these
constraints is unique – except for the possibility of bifurcations, which we
do not consider in this article. So, with this implicit, the dependence of
each $F_{l}$ on the degrees of freedom is reduced to $F_{l}=F_{l}[{\bf x}]$.
The task of constructing global equilibrium solutions is now the standard
mathematical problem of finding extrema of the global energy functional,
$F=F[{\bf x}]$. Before describing the two basic approaches we have adopted for
this, namely a preconditioned conjugate gradient algorithm for minimizing the
global energy functional and a Newton-style algorithm for finding a zero of
the multi-dimensional force-balance vector, we first present a convergence
study illustrating that the Beltrami field in each ${\cal V}_{l}$ may be
constructed to arbitrary accuracy.
### IV.3 illustration of Beltrami fields
For illustration, we show the Beltrami fields consistent with a multi-region
equilibrium. The equilibrium is defined by the toroidal flux enclosed by each
interface, the pressure in each ${\cal V}_{l}$, and the interface transforms,
and these are all given in Table. (1). The outer boundary is given by
$R=1+r\cos(\theta)$ and $Z=r\sin(\theta)$, with
$r=0.3+\delta\cos(2\theta-\zeta)+\delta\cos(3\theta-\zeta)$ and
$\delta=10^{-3}$. This choice of perturbation induces ‘primary’ islands at the
$\mbox{$\iota\\!\\!$-}=1/2$ and $\mbox{$\iota\\!\\!$-}=1/3$ rational surfaces.
The interior boundaries are consistent with force balance. The interface cross
sections and Poincar$\acute{\rm e}$ plots of the Beltrami fields are shown in
Fig. 1. Because of toroidal and poloidal coupling, magnetic islands (and
irregular field lines) will form at all rational surfaces within the
rotational transform range.
Table 1: Flux and transform constraints $l$ | $\psi_{t,l}$ | $p_{l}$ | $\mbox{$\iota\\!\\!$-}_{l}$ |
---|---|---|---|---
$1$ | $0.05950$ | $0.94168$ | $(5+\gamma\,6)/(6+\gamma\,7)$ | $=0.848898\dots$
$2$ | $0.35098$ | $0.63872$ | $(1+\gamma\,2)/(2+\gamma\,3)$ | $=0.618034\dots$
$3$ | $0.64902$ | $0.25740$ | $(1+\gamma\,1)/(2+\gamma\,3)$ | $=0.381966\dots$
$4$ | $1.00000$ | $0.04106$ | $(1+\gamma\,1)/(9+\gamma\,10)$ | $=0.103971\dots$
Figure 2: Scaling of components of error, $\delta{\bf j}\equiv{\bf j}-\mu{\bf
B}$, with respect to radial resolution. The diamonds are for the $n=3$ (cubic)
basis functions, the triangles are for the $n=5$ (quintic) basis functions.
The solid lines have gradient -3,-2 and -2, and the dotted lines have gradient
-5,-4 and -4.
With finite resolution, the equation $\nabla\times{\bf B}=\mu{\bf B}$ can of
course only approximately be solved. However, given a smooth boundary, the
solution to the Beltrami equation is well posed, and so the numerical error
can be made arbitrarily small. Assuming that the Fourier resolution is
sufficient to ensure the numerical error results from the finite-ness of the
radial discretization, an $n$-th order approximation to the vector potential
yields an error ${\cal O}(h^{n+1})$, where $h$ is the radial sub-grid size.
The Fourier harmonics of the contravariant components of ${\bf B}$ are
$\left(\sqrt{g}B^{s}\right)_{m,n}=-mA_{\theta,m,n}-nA_{\zeta,m,n}$,
$\left(\sqrt{g}B^{\theta}\right)_{m,n}=A_{\zeta,m,n}^{\prime}$, and
$\left(\sqrt{g}B^{\zeta}\right)_{m,n}=-A_{\theta,m,n}^{\prime}$, where the
prime denotes radial derivative. Radial derivatives reduce the order of the
error, and ${\bf B}=\nabla\times{\bf A}$ is generally an order less accurate
that ${\bf A}$ itself; with the exception of $B^{s}$, which remains accurate
to ${\cal O}(h^{n+1})$ as no radial derivatives are involved. Before computing
$\nabla\times{\bf B}$, the ‘contravariant’ components must be ‘lowered’,
$B_{\alpha}=\sum_{\beta}g_{\alpha\beta}B^{\beta}$, and the error in $B_{s}$,
$B_{\theta}$, $B_{\zeta}$ are each ${\cal O}(h^{n})$. The Fourier harmonics of
the contravariant components of ${\bf j}\equiv\nabla\times{\bf B}$ are
computed similarly, and the error in $j^{s}$, $j^{\theta}$ and $j^{\zeta}$ are
${\cal O}(h^{n})$, ${\cal O}(h^{n-1})$ and ${\cal O}(h^{n-1})$, respectively.
The components of the error, ${\bf j}-\mu{\bf B}$, are quantified by
$\displaystyle|\delta j^{\alpha}|$ $\displaystyle=$
$\displaystyle\left[\sum_{m,n}\left[(\sqrt{g}j^{\alpha})_{m,n}-\mu(\sqrt{g}B^{\alpha})_{m,n}\right]^{2}/N_{MN}\right]^{1/2},$
for $\alpha=s,\theta$, and $\zeta$. These quantities are shown as a function
of radial sub-grid resolution in Fig. 2, for the field in ${\cal V}_{3}$ of
the equilibrium shown in Fig. 1. The expected error scalings, $|\delta
j^{s}|\sim{\cal O}(h^{n})$, $|\delta j^{\theta}|\sim{\cal O}(h^{n-1})$ and
$|\delta j^{\zeta}|\sim{\cal O}(h^{n-1})$, are confirmed for both the cubic,
$n=3$, and quintic, $n=5$, finite-element representations.
Note that at no point does the algorithm depend of resolving the fractal
structure of phase space. The vector potential in each ${\cal V}_{l}$ is a
smooth function, both as a function of position within a given volume, and as
a function of interface geometry.
Before proceeding to the task of piecing together multiple, nested Beltrami
fields to obtain global, non-trivial equilibria, we must tie down a ‘loose-
end’ regarding the Fourier representation of the interfaces.
### IV.4 spectral condensation
To construct global equilibria, the $R_{l,j}$ and $Z_{l,j}$ describing the
interface geometry will be varied to extremize the energy functional and/or
satisfy force balance. Tangential geometric variations merely change the
angular parametrization of the interfaces and do not change the interface
geometry, and so do not affect the energy functional, but do alter the
$\\{R_{l,j},Z_{l,j}\\}$. This freedom may be exploited to obtain a preferred
angle parametrization.
A numerically insightful choice Hirshman and Meier (1985); Hirshman and
Breslau (1998) is to choose the angle that minimizes the ‘spectral width’, and
so obtain the most accurate representation of the interface geometry for a
given Fourier resolution. We define the spectral width as
$\displaystyle\frac{1}{2}\left.\sum_{j}\right.(m_{j}^{p}+n_{j}^{q})\left(R_{j}^{2}+Z_{j}^{2}\right),$
(65)
where $p$ and $q$ are arbitrary integers required as input.
The toroidal angle has already been constrained, $\zeta\equiv-\phi$, and the
geometry of the interfaces is to be constrained by force balance, so we are
left to minimize the spectral width with respect to poloidal variations,
$\delta R=\partial_{\theta}R\,\delta u$ and $\delta
Z=\partial_{\theta}Z\,\delta u$. To preserve stellarator symmetry we restrict
attention to odd functions, $\delta u=\sum_{j}u_{j}\sin(m_{j}\theta-
n_{j}\zeta)$. The variations in the Fourier harmonics, $R_{j}$ and $Z_{j}$,
are given by $\delta R_{j}=\oint\oint R_{\theta}\delta u\,\cos(m_{j}\theta-
n_{j}\zeta)d\theta d\zeta$ and $\delta Z_{j}=\oint\oint Z_{\theta}\delta
u\sin(m_{j}\theta-n_{j}\zeta)d\theta d\zeta$, where
$R_{\theta}\equiv\partial_{\theta}R$ and $Z_{\theta}\equiv\partial_{\theta}Z$.
The first variation in the spectral width is
$\displaystyle\oint\\!\\!\\!\oint d\theta
d\zeta\left(R_{\theta}X+Z_{\theta}Y\right)\delta u,$ (66)
where $X=\sum_{j}(m_{j}^{p}+n_{j}^{q})R_{j}\cos(m_{j}\theta-n_{j}\zeta)$ and
$Y=\sum_{j}(m_{j}^{p}+n_{j}^{q})Z_{j}\sin(m_{j}\theta-n_{j}\zeta)$. The
spectral width is decreased along $\delta u=-I$, where $I\equiv
R_{\theta}X+Z_{\theta}Y$, and is extremized when $I=0$.
### IV.5 illustrations of global equilibria
We have implemented two numerical methods for finding global equilibria. The
first is a minimization algorithm: we seek minima of $F[{\bf
x}]\equiv\sum_{l}\int_{\cal V}[p_{l}/(\gamma-1)+B^{2}/2]dv$ using a
preconditioned, conjugate gradient algorithm, where in each volume
$p_{l}V^{\gamma}=a_{l}$ is constant, and the field satisfies $\nabla\times{\bf
B}=\mu_{l}{\bf B}$. The gradient of $F$ with respect to the $R_{l,j}$ and
$Z_{l,j}$, with the fluxes and helicity constrained, can be derived from Eq.
(12) by recalling that the displacement is
$\mbox{\boldmath$\xi$\unboldmath}\equiv\delta R\,{\hat{R}}\,+\delta
Z\,{\hat{k}}$ and the surface element is $d{\bf s}\equiv-
RZ_{\theta}\,\hat{R}+(Z_{\theta}R_{\zeta}-R_{\theta}Z_{\zeta})\,\hat{\phi}+RR_{\theta}\,\hat{z}$.
The derivatives of $F$ are
$\displaystyle\\!\frac{\partial F}{\partial R_{l,j}}\\!$
$\displaystyle\\!=\\!$
$\displaystyle\\!-\left([[p+B^{2}/2]]_{l}\,R\,Z_{\theta}\right)_{j}-(R_{\theta}I_{l})_{j},$
(67) $\displaystyle\\!\frac{\partial F}{\partial Z_{l,j}}\\!$
$\displaystyle\\!=\\!$
$\displaystyle\\!+\left([[p+B^{2}/2]]_{l}\,R\,R_{\theta}\right)_{j}-(Z_{\theta}I_{l})_{j},$
(68)
where the $(R_{\theta}I_{l})_{j}$ and $(Z_{\theta}I_{l})_{j}$ terms are
included to reduce the spectral width.
The second approach for finding global equilibria is a globally-convergent,
multi-dimensional Newton method. We seek a zero of the ‘force-balance’ vector,
constructed as follows. Global force balance is satisfied when the total
pressure discontinuity across each interface is zero, $[[p+B^{2}/2]]=0$. The
Fourier representation of the interfaces minimizes the spectral width when $I$
is zero. Thus, we construct a ‘constraint’ vector, ${\bf f}({\bf x})$, by
collecting together the harmonics $[[p+B^{2}/2]]_{l,j}$ and $I_{l,j}$.
Figure 3: Comparison between the SPEC interfaces, with $N_{V}=6$, and the
corresponding VMEC surfaces (thick lines, upper half); and the SPEC radial
sub-grid (lower half). Figure 4: Location of magnetic axis as computed by
SPEC against resolution, $N_{V}$, of stepped-pressure approximation to smooth
pressure profile. The dotted line is the location of magnetic axis for a high
resolution VMEC calculation. Figure 5: Stepped pressure approximation, with
$N_{V}=64$, to smooth pressure profile.
Within the stellarator symmetric representation we have employed, $R$ is an
even function of $(\theta,\zeta)$, and $Z$ is odd. Force-balance,
$[[p+B^{2}/2]]$, and the spectral minimization condition, $I$, are similarly
even and odd functions. So, after suitably truncating $[[p+B^{2}/2]]$ and $I$
to match the truncated Fourier representation of the interface geometry, the
number of constraints equals the number of degrees-of-freedom.
Expanding about an arbitrary point, ${\bf f}({\bf x}+\delta{\bf x})\approx{\bf
f}({\bf x})+\nabla{\bf f}({\bf x})\cdot\delta{\bf x}$, the Newton correction
required to find the equilibrium point, ${\bf f}=0$, is given by $\delta{\bf
x}=-\nabla{\bf f}^{-1}\cdot{\bf f}.$ The Jacobian matrix, $\nabla{\bf f}$,
describes how the Beltrami fields change when the geometry is changed (more
precisely, how $B^{2}$ on the interfaces changes when the ${\cal I}_{l}$
change) and is computed in parallel using matrix perturbation methods as
described above.
In the following, we use this method for constructing global solutions. In all
calculations presented, either explicitly or implicitly, for any given Fourier
and radial sub-grid resolution, the error in force balance, $|f|$, and the
error in ‘position’, $|\delta x|$, are less than $10^{-12}$.
Before presenting illustrations of non-axisymmetric global equilibria, we
first present a comparison of stepped-pressure equilibria to axisymmetric MHD
equilibria with smooth profiles, the latter computed by VMEC Hirshman et al.
(1986). SPEC is intended for computing equilibria with partially chaotic
fields and only admits stepped-pressure profiles. In contrast, VMEC globally
constrains the field to be integrable, and so admits (and assumes) smooth
profiles. As it is the profiles that define an equilibrium, VMEC and SPEC will
differ. To obtain agreement, it is required to perform a convergence study in
the pressure profile: as the number of steps in the stepped profile increases,
the stepped profile will better approximate a smooth profile.
We consider an equilibrium with boundary described by
$R(\theta,\zeta)=1.0+0.3\cos\theta$ and $Z(\theta,\zeta)=0.3\sin\theta$. For
the (smooth) pressure profile we take $p(\psi)=p_{0}(1-2\psi+\psi^{2})$, where
$\psi$ is the normalized toroidal flux and $p_{0}$ is a scaling factor chosen
to give a Shafranov shift about one-third the minor radius. For the (smooth)
transform profile we take
$\mbox{$\iota\\!\\!$-}=\mbox{$\iota\\!\\!$-}_{0}-\mbox{$\iota\\!\\!$-}_{1}\psi$,
where $\mbox{$\iota\\!\\!$-}_{0}=$ $(8+9\gamma)/$ $(9+10\gamma)$ and
$\mbox{$\iota\\!\\!$-}_{1}=$ $\mbox{$\iota\\!\\!$-}_{0}-$ $(1+1\gamma)/$
$(9+10\gamma)$. We use high Fourier resolution, $M=16$, and high radial sub-
grid resolution; this will ensure that any discrepancy results from finite
$N_{V}$, and so the error will decrease as $N_{V}$ is increased.
As input to SPEC, we must ‘discretize’ the pressure and transform profiles.
Earlier we recalled that invariant surfaces of Hamiltonian systems with noble
rotational transform are the most likely to survive chaos-inducing
perturbations, and so the interface transforms should be strongly irrational;
however, in the axisymmetric and therefore integrable case, the interface
transforms may be chosen arbitrarily. For convergence studies, it is
preferable to have interfaces regularly spaced in radius and we choose the
interface rotational transforms
$\mbox{$\iota\\!\\!$-}_{l}=\mbox{$\iota\\!\\!$-}(\psi_{l})$, where
$\sqrt{\psi}_{l}=l/N_{V}$. We discretize the pressure profile in such a way as
to piecewise conserve the integrated pressure,
$\displaystyle
p_{l}\int_{\psi_{l-1}}^{\psi_{l}}\\!\\!\\!\\!\\!\\!\\!d\psi=\int_{\psi_{l-1}}^{\psi_{l}}\\!\\!\\!\\!\\!\\!p(\psi)\,d\psi.$
(69)
For illustration, the SPEC interfaces and the corresponding surfaces
(identified by toroidal flux) of a high radial resolution VMEC equilibrium are
shown for a low $N_{V}$ case in Fig. 3. Despite the fact that a smooth profile
was supplied to VMEC but a stepped-pressure profile was supplied to SPEC, the
agreement is reasonable. This may be expected: the Shafranov shift depends
White (2001) primarily on the integral of the pressure profile and not, to
lowest order, on the precise details of the pressure profile itself. To
quantify the discrepancy, we compare the location of the magnetic axes. As
shown in Fig. 4, as $N_{V}$ is increased the agreement improves. The stepped
pressure profile approximation and the smooth pressure profile are shown in
Fig. 5.
Indeed, it may be shown that the infinite interface limit of MRXMHD (i.e.
assuming continuously nested magnetic flux surfaces) reduces to $\nabla p={\bf
j}\times{\bf B}$ and ${\bf j}\cdot{\bf n}=0$, where ${\bf n}$ is normal to the
flux surfaces. These are the equations that define ideal MHD equilibria. The
details of this derivation will be presented in a forthcoming paper.
To illustrate a non-axisymmetric global equilibrium, we return to the
‘perturbed’ equilibrium shown in Fig. 1. Assuming that the radial sub-grid
resolution is sufficiently high so that the error is dominated by finite
Fourier resolution, we now confirm that the error in the interface geometry
decreases as $M$ and $N$ increase.
Let the exact solution for the $l$-th interface be described by
$R(\theta,\zeta)$ and $Z(\theta,\zeta)$. The error between this and an
approximation, $R_{M,N}(\alpha,\zeta)$ and $Z_{M,N}(\alpha,\zeta)$, with a
potentially different poloidal angle, $\alpha$, on the $\zeta_{0}$ plane is
quantified by $\Delta\equiv\int D(\theta)\,dl$, where $dl^{2}\equiv
dx^{2}+dy^{2}$ is the arc-length of the curve $x(\theta)\equiv
R(\theta,\zeta_{0})$ and $y(\theta)\equiv Z(\theta,\zeta_{0})$, and $D$ is the
distance between this reference curve and the curve described by
$x_{M,N}(\alpha)\equiv R_{M,N}(\alpha,\zeta_{0})$ and $y_{M,N}(\alpha)\equiv
Z_{M,N}(\alpha,\zeta_{0})$, i.e.
$D^{2}\equiv\left[x(\theta)-x_{M,N}(\alpha)\right]^{2}+\left[y(\theta)-y_{M,N}(\alpha)\right]^{2}$.
The comparison is made at the same polar angle, so that
$y(\theta)/[x(\theta)-x_{0}]=y_{M,N}(\alpha)/[x_{M,N}(\alpha)-x_{0}]$, where
$x_{0}$ is a reference point (e.g. the magnetic axis).
Figure 6: Difference between finite $M$, $N$ approximation to interface
geometry, and a high-resolution reference approximation (with $M=13$ and
$N=8$), plotted against Fourier resolution.
As the exact solution is not known apriori, we take as the reference
configuration the highest resolution approximation available. The error in the
interface geometry, for each of the internal interfaces, is shown as a
function of $(M,N)$ in Fig. 6, where we see that the error decreases as the
Fourier resolution is increased.
All properties of the equilibrium are defined by the interface geometries: if
the interface geometry has converged, then so too have the Beltrami fields,
and the location and size of the magnetic islands and chaotic seas – well, it
is difficult to resolve the infinitely complicated structure of phase space,
but ${\bf B}$ itself and all integral properties are converged.
The good convergence properties of the interface geometry with Fourier
resolution is because (i) the interfaces are chosen to have the most noble
transform, and so are ‘as far away as possible’, so to speak, from the lowest
order islands and associated chaos, and so are the smoothest surfaces (this is
in contrast to flux surfaces adjacent to a separatrix, or flux surfaces that
are nearly critical); and (ii) that the Fourier representation exploits a
preferred angle parametrization that minimizes the spectral width. More
importantly, perhaps, is that convergence is obtained because there is a well
defined, exact solution, and that the numerical discretization is capable of
resolving all the structure of the solution.
As a final illustration, we present a stepped-pressure equilibrium consistent
with the boundary and profiles obtained via a 3D STELLOPT reconstruction
Lazerson et al. (2012) of an up-down symmetric DIIID experimental shot with
applied resonant magnetic perturbation (RMP) fields. The reconstruction
process seeks to infer the experimental configuration by adjusting the MHD
equilibrium (presently, STELLOPT is built around VMEC) by varying the the
plasma boundary and the pressure and current profiles, so that derived
quantities (such as Thomson scattering, motional Stark effect polarimetry, and
magnetic diagnostics) match the experimental measurements. Because of the
applied error fields, and the plasma response to these error fields, the
reconstructed boundary is slightly, but significantly, perturbed from
axisymmetry. The pressure and $q$-profiles, where $q$ is the safety-factor
$q\equiv 1/\mbox{$\iota\\!\\!$-}$, derived from the reconstruction are shown
in Fig. 7. It is interesting to observe that the reconstructed pressure
profile appears quite flat across the lowest order rational surfaces.
Furthermore, the locations of locally maximum pressure gradient appear to
coincide with strongly-irrational surfaces.
We compute the stepped-pressure equilibrium using the reconstructed boundary,
a stepped, $N_{V}=32$ approximation (Fig. 7) to the reconstructed pressure
profile, and the reconstructed $q$-profile. The rotational transforms of the
interfaces are chosen by selecting the most noble irrationals that are within
range. The Fourier resolution is $M=10$ and $N=6$, and the total radial sub-
grid resolution is $279$. A Poincar$\acute{\rm e}$ plot is shown in Fig. 8;
most visible is a $q=2$ island at where VMEC has the $q=2$ rational surface.
In this ‘fixed-boundary’ calculation, the boundary is constrained to remain a
fixed, good flux surface. To determine to what extent the RMP fields and the
plasma response ergodize the field in the vicinity of the plasma edge, a
‘free-boundary’ calculation is required. This is left for future work.
Figure 7: Pressure profile (smooth) from a DIIID reconstruction using STELLOPT
and stepped-pressure approximation. Also shown is the inverse rotational
transform $\equiv$ safety factor. Figure 8: Poincar$\acute{\rm e}$ plot of a
DIIID equilibrium with perturbed boundary, calculated using SPEC.
## V comments, future Work
The fact that stepped-pressure equilibria can be derived as minima of an
energy functional is a great convenience numerically, as this allows
employment of minimization methods to construct the Beltrami fields and
interface geometries that satisfy force balance. Furthermore, the MRXMHD
energy functional provides a self-consistent approach for determining the
stability of partially chaotic equilibria Hole et al. (2007); Mills et al.
(2009); Hole et al. (2009). In future work, we hope to explore whether the
suppression of edge localized modes by resonant magnetic perturbations Evans
et al. (2004), that result in a stochastic plasma edge, may be understood
through an MRXMHD stability analysis.
In the above text we have distinguished MRXMHD equilibria, for which the
algorithm allows $\mu$ to vary to force the helicity constraint, from stepped-
pressure equilibria, for which the algorithm allows both $\mu$ and the
poloidal flux to vary to force the transform constraint and for which the
self-consistent helicity is computed aposteori. The distinction is, however,
superficial. The profiles could also be specified by keeping $\mu$ and the
poloidal flux as fixed input parameters. That the profiles can be prescribed
and constrained in a variety of ways illustrates the flexibility of the
theoretical and numerical method. (Note that because ideal MHD is not applied
globally, there is no explicit relationship between the toroidal and poloidal
fluxes and the rotational-transform profile, for example.) To decide how the
profiles should be described, one must depart from equilibrium theory and
develop a self-consistent model of transport, which may suggest how, for
example, the poloidal flux, the parallel current and the helicity should vary
in time to preserve the transform constraint.
Given that the MRXMHD equilibria do not have $1/x$ and $\delta$-function
currents at the rational surfaces, and that, in a pressure transport model
that includes a small perpendicular diffusion, maximum pressure gradients will
appear at the most irrational locations Hudson and Breslau (2008); Hudson
(2009) – and so too will, perhaps, the pressure driven currents – then it
would seem that MRXMHD equilibria are smoothly connected (e.g. as the
perpendicular-diffusion coefficient $\kappa_{\perp}\rightarrow 0$) to nearly-
ideal steady-state equilibria, and so it may be advantageous to initialize
resistive MHD initial value codes such as NIMROD Sovinec et al. (2003) or
M3DC1 Jardin et al. (2007) with SPEC equilibria. Local flattening of the
pressure gradient across the resonances can provide increased stability and
can allow access to higher plasma beta Ichiguchi et al. (2001); Ichiguchi and
Carreras (2011).
The equilibrium model considered in this article does not include plasma flow,
which can impact both the equilibrium and island healing phenomena Hegna
(2011, 2012). Extended MHD modeling of plasmas Ichiguchi et al. (2001);
Bonfiglio et al. (2010); Ferraro et al. (2010); Ichiguchi and Carreras (2011);
Schlutt et al. (2012) is crucial for understanding some key experimental
observations. Perhaps, by extending a stepped-pressure equilibrium, an
equilibrium with a continuous-pressure profile could be constructed by
replacing the pressure jump interfaces with finite-width ideal-MHD layers
Mills et al. (2009), each topologically constrained to avoid resonances and to
avoid ideal instabilities Mills et al. (2009), and, perhaps, capable of
supporting extended MHD behavior. A variational principle based on minimizing
the generalized enstrophy Yoshida and Mahajan (2002),
$F=\int\left|\nabla\times\left({\bf V}+{\bf A}\right)\right|^{2}dx$, may
better describe self-organization in two-fluid plasmas.
In addition to the incremental code improvements that are inevitably required,
the following in particular will be pursued. In this paper it was assumed that
the plasma boundary, $\partial\cal V$, is a given, fixed toroidal surface.
More generally, the magnetic field is part generated by external currents, and
the free-boundary problem could be solved with a little extra work, where
$\partial{\cal V}$ is to be determined as part of the solution. Also, most
modern tokamaks are not up-down symmetric, so it will be required to relax the
stellarator symmetry constraint.
It is worth exploring more efficient numerical methods for computing the
Beltrami fields. In a closed domain $\cal P$ in ${\mathbb{R}}^{3}$, in general
multiply connected, the solutions of $\nabla\times{\bf B}=\mu{\bf B}$ can be
represented Kress (1981, 1986) by
$\displaystyle\mbox{$\bf B$}=(\nabla\times\,+\mu)\\!\\!\int_{\partial\cal
P}\\!\\!G(\mbox{$\bf r$},\mbox{$\bf r$}^{\prime})\,\mbox{$\bf
B$}^{\prime}\times\mbox{$\bf n$}^{\prime}\,{d}S^{\prime}\;,$ (70)
where $\bf n$ is the outward unit normal on $\partial\cal P$ and $G(\mbox{$\bf
r$},\mbox{$\bf r$}^{\prime})$ satisfies $(\nabla^{2}+\mu^{2})G(\mbox{$\bf
r$},\mbox{$\bf r$}^{\prime})=-\delta(\mbox{$\bf r$}-\mbox{$\bf r$}^{\prime})$,
with $\delta(\cdot)$ being the 3D Dirac $\delta$ function.
Given the geometry of an interface, the maximum pressure jump that an
interface can support can be quickly determined by an analysis of the
pressure-jump Hamiltonian: the pressure discontinuity, $2(p_{+}-p_{-})$, is
increased until the appropriate irrational surface of the pressure-jump
Hamiltonian is critical. Looking beyond our present task of constructing an
equilibrium consistent with a given pressure that is not changed, this gives
an efficient method for distributing the pressure so that the most robust
interfaces support the most pressure. As the pressure across any interface is
altered, there will be a global response that requires re-computation of the
equilibrium.
We dedicate this article to Paul Garabedian, whom we consider to be a pioneer
in the field of 3D MHD calculations, and who also endorsed and employed weak
solutions Garabedian (1998). We are also indebted to Steve Hirshman, who has
provided VMEC. One of us (SRH) acknowledges stimulating discussions with Neil
Pomphrey, with Don Monticello regarding the construction of straight-field-
line coordinates, and with Allen Boozer regarding the choice of gauge Finn and
Chacon (2005) for the vector potential, and RLD acknowledges discussion with
Robert MacKay. The authors gratefully acknowledge support of the U.S.
Department of Energy and the Australian Research Council, through grants
DP0452728, FT0991899 and DP110102881.
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|
arxiv-papers
| 2012-11-13T18:46:05 |
2024-09-04T02:49:37.967243
|
{
"license": "Public Domain",
"authors": "S.R. Hudson, R.L. Dewar, G. Dennis, M.J. Hole, M. McGann, G. von\n Nessi, S. Lazerson",
"submitter": "Stuart Hudson",
"url": "https://arxiv.org/abs/1211.3072"
}
|
1211.3135
|
# Pointwise products of some Banach
function spaces and factorization
Paweł Kolwicz , Karol Leśnik∗ and Lech Maligranda Research partially supported
by the State Committee for Scientific Research, Poland, Grant N N201 362236
###### Abstract
The well-known factorization theorem of Lozanovskiĭ may be written in the form
$L^{1}\equiv E\odot E^{\prime}$, where $\odot$ means the pointwise product of
Banach ideal spaces. A natural generalization of this problem would be the
question when one can factorize $F$ through $E$, i.e., when $F\equiv E\odot
M(E,F)\,$, where $M(E,F)$ is the space of pointwise multipliers from $E$ to
$F$. Properties of $M(E,F)$ were investigated in our earlier paper [KLM12] and
here we collect and prove some properties of the construction $E\odot F$. The
formulas for pointwise product of Calderón-Lozanovskiĭ $E_{\varphi}$ spaces,
Lorentz spaces and Marcinkiewicz spaces are proved. These results are then
used to prove factorization theorems for these spaces. Finally, it is proved
in Theorem 11 that under some natural assumptions, a rearrangement invariant
Banach function space may be factorized through Marcinkiewicz space.
00footnotetext: 2010 Mathematics Subject Classification: 46E30, 46B20, 46B42,
46A4500footnotetext: Key words and phrases: Banach ideal spaces, Banach
function spaces, Calderón spaces, Calderón-Lozanovskiĭ spaces, symmetric
spaces, Orlicz spaces, sequence spaces, pointwise multipliers, pointwise
multiplication, factorization
1\. Introduction and preliminaries
The well-known factorization theorem of Lozanovskiĭ says that for any
$\varepsilon>0$ each $z\in L^{1}$ can be factorized by $x\in E$ and $y\in
E^{\prime}$ in such a way that
$z=xy~{}~{}{\rm
and}~{}~{}\|x\|_{E}\,\|y\|_{E^{\prime}}\leq(1+\varepsilon)\|z\|_{L^{1}}.$
Moreover, if $E$ has the Fatou property we may take $\varepsilon=0$ in the
above inequality. This theorem can be written in the form $L^{1}\equiv E\odot
E^{\prime}$, where
$E\odot F=\left\\{x\cdot y:x\in E\text{ and }y\in F\right\\}.$
Then natural question arises: when is it possible to factorize $F$ through
$E$, i.e., when
$F\equiv E\odot M\left(E,F\right)?$ (1)
Here $M\left(E,F\right)$ denotes the space of multipliers defined as
$M\left(E,F\right)=\left\\{x\in L^{0}:xy\in F\text{ for each }y\in E\right\\}$
with the operator norm
$\left\|x\right\|_{M\left(E,F\right)}=\sup_{\left\|y\right\|_{E}=1}\left\|xy\right\|_{F}.$
The space of multipliers between function spaces was investigated by many
authors, see for example [Ru79], [MP89] and [AZ90] (see also [Za66], [ZR67],
[Cr72], [Ma74], [AS76], [Ma89], [Ra92], [Na95], [DR00], [CN03], [CDS08],
[MN10], [Sc10] and [KLM12]). In this paper we are going to investigate general
properties of the product construction $E\odot F$ and calculate the product
space $E\odot F$ for Calderón-Lozanovskiĭ, Lorentz and Marcinkiewicz spaces.
This product space was of interest in [An60], [Wa63], [ON65], [ZR67], [Da74],
[Ru79], [Ma89], [RR91], [Ra92], [BL93], [DMM03], [AM09], [KM10] and [Sc10].
The results on product construction will be used to give answers to the
factorization question (1) in these special spaces.
Let $(\Omega,\Sigma,\mu)$ be a complete $\sigma$-finite measure space and
$L^{0}=L^{0}(\Omega)$ be the space of all classes of $\mu$-measurable real-
valued functions defined on $\Omega$. A (quasi-) Banach space
$E=\left(E,\|\cdot\|_{E}\right)$ is said to be a (quasi-) Banach ideal space
on $\Omega$ if $E$ is a linear subspace of $L^{0}(\Omega)$ and satisfies the
so-called ideal property, which means that if $y\in E,x\in L^{0}$ and
$|x(t)|\leq|y(t)|$ for $\mu$-almost all $t\in\Omega$, then $x\in E$ and
$\|x\|_{E}\leq\|y\|_{E}$. We will also assume that a (quasi-) Banach ideal
space on $\Omega$ is saturated, i.e. every $A\in\Sigma$ with $\mu(A)>0$ has a
subset $B\in\Sigma$ of finite positive measure for which $\chi_{B}\in E$. The
last statement is equivalent with the existence of a weak unit, i.e., an
element $x\in E$ such that $x(t)>0$ for each $t\in\Omega$ (see [KA77] and
[Ma89]). If the measure space $(\Omega,\Sigma,\mu)$ is non-atomic we should
say about (quasi-) Banach function space, if we replace the measure space
$\left(\Omega,\Sigma,\mu\right)$ by the counting measure space
$\left(\mathbb{N},2^{\mathbb{N}},m\right),$ then we say that $E$ is a (quasi-)
Banach sequence space (denoted by $e$).
A point $x\in E$ is said to have order continuous norm (or to be order
continuous element) if for each sequence $(x_{n})\subset E$ satisfying $0\leq
x_{n}\leq|x|$ and $x_{n}\rightarrow 0~{}\mu$-a.e. on $\Omega,$ one has
$\|x_{n}\|_{E}\rightarrow 0$. By $E_{a}$ we denote the subspace of all order
continuous elements of $E$. It is worth to notice that in case of Banach ideal
spaces on $\Omega$, $x\in E_{a}$ if and only if
$\|x\chi_{A_{n}}\|_{E}\downarrow 0$ for any sequence $\\{A_{n}\\}$ satisfying
$A_{n}\searrow\emptyset$ (that is $A_{n}\supset A_{n+1}$ and
$\mu(\bigcap_{n=1}^{\infty}A_{n})=0$). A Banach ideal space $E$ is called
order continuous if every element of $E$ is order continuous, i.e., $E=E_{a}$.
A space $E$ has the Fatou property if $0\leq x_{n}\uparrow x\in L^{0}$ with
$x_{n}\in E$ and $\sup_{n\in\mathbb{N}}\|x_{n}\|_{E}<\infty$ imply that $x\in
E$ and $\|x_{n}\|_{E}\uparrow\|x\|_{E}$.
We shall consider pointwise product of Calderón-Lozanovskiĭ spaces
$E_{\varphi}$ (which are generalizations of Orlicz spaces $L^{\varphi}$), but
very useful will be also general Calderón-Lozanovskiĭ construction
$\rho(E,F)$.
A function $\varphi:[0,\infty)\rightarrow[0,\infty]$ is called a Young
function (or an Orlicz function if it is finite-valued) if $\varphi$ is
convex, non-decreasing with $\varphi(0)=0$; we assume also that $\varphi$ is
neither identically zero nor identically infinity on $(0,\infty)$ and
$\lim_{u\rightarrow b_{\varphi}^{-}}\varphi(u)=\varphi(b_{\varphi})$ if
$b_{\varphi}<\infty$, where $b_{\varphi}=\sup\\{u>0:\varphi(u)<\infty\\}.$
Note that from the convexity of $\varphi$ and the equality $\varphi(0)=0$ it
follows that $\lim_{u\rightarrow 0+}\varphi(u)\newline =\varphi(0)=0$.
Furthermore, from the convexity and $\varphi\not\equiv 0$ we obtain that
$\lim_{u\rightarrow\infty}\varphi(u)=\infty$.
If we denote $a_{\varphi}=\sup\\{u\geq 0:\varphi(u)=0\\}$, then $0\leq
a_{\varphi}\leq b_{\varphi}\leq\infty$ and
$a_{\varphi}<\infty,~{}b_{\varphi}>0$, since a Young function is neither
identically zero nor identically infinity on $(0,\infty)$. The function
$\varphi$ is continuous and nondecreasing on $[0,b_{\varphi})$ and is strictly
increasing on $[a_{\varphi},b_{\varphi})$. If $a_{\varphi}=0$ then we write
$\varphi>0$, if $b_{\varphi}=\infty,$ then $\varphi<\infty$. Each Young
function $\varphi$ defines the function
$\rho_{\varphi}:[0,\infty)\times[0,\infty)\rightarrow[0,\infty)$ in the
following way
$\rho_{\varphi}(u,v)=\left\\{\begin{array}[]{cc}v\varphi^{-1}(\frac{u}{v})&\text{if
}u>0,\\\ 0&\text{if }u=0,\end{array}\right.$
where $\varphi^{-1}$ denotes the right-continuous inverse of $\varphi$ and is
defined by
$\varphi^{-1}(v)=\inf\\{u\geq
0:\varphi(u)>v\\}~{}\mathrm{for}~{}v\in[0,\infty)~{}~{}\mathrm{with}~{}~{}\varphi^{-1}(\infty)=\lim_{v\rightarrow\infty}\varphi^{-1}(v).$
If $\rho=\rho_{\varphi}$ and $E,F$ are Banach ideal spaces over the same
measure space $(\Omega,\Sigma,\mu)$, then the Calderón-Lozanovskiĭ space
$\rho(E,F)$ is defined as all $z\in L^{0}(\Omega)$ such that for some $x\in
E,y\in F$ with $\|x\|_{E}\leq 1,\|y\|_{F}\leq 1$ and for some $\lambda>0$ we
have
$|z|\leq\lambda\,\rho(|x|,|y|)~{}~{}\mu-{\rm a.e.~{}on}~{}~{}\Omega.$
The norm $\|z\|_{\rho}=\|z\|_{\rho(E,F)}$ of an element $z\in\rho(E,F)$ is
defined as the infimum values of $\lambda$ for which the above inequality
holds. It can be shown that
$\rho(E,F)=\left\\{z\in L^{0}(\Omega):|z|\leq\rho(x,y)~{}\mu-{\rm
a.e.~{}for~{}some}~{}x\in E_{+},~{}y\in F_{+}\right\\}$
with the norm
$\|z\|_{\rho\left(E,F\right)}=\inf\left\\{\max\left\\{\|x\|_{E},\|y\|_{F}\right\\}:\text{\
}|z|\leq\rho(x,y)\text{,}\ x\in E_{+},y\in F_{+}\right\\}\text{.}$ (2)
The Calderón-Lozanovskiĭ spaces, introduced by Calderón in [Ca64] and
developed by Lozanovskiĭ in [Lo71], [Lo73], [Lo78a] and [Lo78b], play crucial
role in the theory of interpolation since such construction is interpolation
functor for positive operators and under some additional assumptions on spaces
$E,F$, like Fatou property or separability, also for all linear operators (see
[Ov76], [Ov84], [KPS82], [Ma89]). If $\rho(u,v)=u^{\theta}v^{1-\theta}$ with
$0<\theta<1$ we write $E^{\theta}F^{1-\theta}$ instead of $\rho(E,F)$ and
these are Calderón spaces (cf. [Ca64], p. 122). Another important situation,
investigated by Calderón (cf. [Ca64], p. 121) and independently by Lozanovskiĭ
(cf. [Lo64, Theorem 2], [Lo65, Theorem 2]), appears when we put $F\equiv
L^{\infty}$. In this case, in the definition of the norm, it is enough to take
$y=\chi_{\Omega}$ and then
$\|z\|_{\rho_{\varphi}\left(E,L^{\infty}\right)}=\inf\left\\{\lambda>0:\left\|\varphi\left(|x|/\lambda\right)\right\|_{E}\leq
1\right\\}.$ (3)
Thus the Calderón-Lozanovskiĭ space $E_{\varphi}=\rho_{\varphi}(E,L^{\infty})$
for any Young function $\varphi$ is defined by
$E_{\varphi}=\\{x\in
L^{0}:I_{\varphi}(cx)<\infty~{}\mathrm{for~{}some}~{}c=c(x)>0\\},$
and it is a Banach ideal space on $\Omega$ with the so-called Luxemburg-Nakano
norm
$\|x\|_{E_{\varphi}}=\inf\left\\{\lambda>0:I_{\varphi}\left(x/\lambda\right)\leq
1\right\\},$
where the convex semimodular $I_{\varphi}$ is defined as
$I_{\varphi}(x):=\left\\{\begin{array}[]{cc}\|\varphi\left(|x|\right)\|_{E}&\text{if
}\varphi\left(|x|\right)\in E,\text{ }\\\
\infty&\text{otherwise.}\end{array}\right.$
If $E=L^{1}$ ($E=l^{1}$), then $E_{\varphi}$ is the classical Orlicz function
(sequence) space $L^{\varphi}$ ($l^{\varphi}$) equipped with the Luxemburg-
Nakano norm (cf. [KR61], [Ma89]). If $E$ is a Lorentz function (sequence)
space $\Lambda_{w}$ ($\lambda_{w}$), then $E_{\varphi}$ is the corresponding
Orlicz-Lorentz function (sequence) space $\Lambda_{\varphi,w}$
($\lambda_{\varphi,w}$), equipped with the Luxemburg-Nakano norm. On the other
hand, if $\varphi(u)=u^{p},1\leq p<\infty$, then $E_{\varphi}$ is the
$p$-convexification $E^{(p)}$ of $E$ with the norm
$\|x\|_{E^{(p)}}=\||x|^{p}\|_{E}^{1/p}$. In case $0<p<1,$ we will say about
$p$-concavification of $E$.
For two ideal (quasi-) Banach spaces $E$ and $F$ on $\Omega$ the symbol
$E\overset{C}{\hookrightarrow}F$ means that the embedding $E\subset F$ is
continuous with the norm which is not bigger than $C$, i.e., $\|x\|_{F}\leq
C\|x\|_{E}$ for all $x\in E$. In the case when the embedding
$E\overset{C}{\hookrightarrow}F$ holds with some (unknown) constant $C>0$ we
simply write $E\hookrightarrow F$. Moreover, $E=F$ (and $E\equiv F$) means
that the spaces are the same and the norms are equivalent (equal).
We will also need some facts from the theory of symmetric spaces. By a
symmetric function space (symmetric Banach function space or rearrangement
invariant Banach function space) on $I$, where $I=(0,1)$ or $I=(0,\infty)$
with the Lebesgue measure $m$, we mean a Banach ideal space
$E=(E,\|\cdot\|_{E})$ with the additional property that for any two
equimeasurable functions $x\sim y,x,y\in L^{0}(I)$ (that is, they have the
same distribution functions $d_{x}\equiv d_{y}$, where
$d_{x}(\lambda)=m(\\{t\in I:|x(t)|>\lambda\\}),\lambda\geq 0$) and $x\in E$ we
have that $y\in E$ and $\|x\|_{E}=\|y\|_{E}$. In particular,
$\|x\|_{E}=\|x^{\ast}\|_{E}$, where
$x^{\ast}(t)=\mathrm{\inf}\\{\lambda>0\colon\ d_{x}(\lambda)<t\\},\ t\geq 0$.
Similarly, if $e$ is a Banach sequence space with the above property, then we
say about symmetric sequence space. It is worth to point out that any Banach
ideal space with this property is equivalent to a symmetric space over one of
the above three measure spaces (cf. [LT79]).
The fundamental function $f_{E}$ of a symmetric function space $E$ on $I$ is
defined by the formula $f_{E}(t)=\|\chi_{[0,\,t]}\|_{E},t\in I$. It is well-
known that each fundamental function is quasi-concave on $I$, that is,
$f_{E}(0)=0,f_{E}(t)$ is positive, non-decreasing and $f_{E}(t)/t$ is non-
increasing for $t\in(0,m(I))$ or, equivalently,
$f_{E}(t)\leq\max(1,t/s)f_{E}(s)$ for all $s,t\in(0,m(I))$. Moreover, for each
fundamental function $f_{E}$, there is an equivalent, concave function ${\
\tilde{f}_{E},}$ defined by ${\
\tilde{f}_{E}}(t):=\inf_{s\in(0,m(I))}(1+\frac{t}{s})f_{E}(s)$. Then
$f_{E}(t)\leq{\tilde{f}_{E}}(t)\leq 2f_{E}(t)$ for all $t\in I$. For any
quasi-concave function $\phi$ on $I$ the Marcinkiewicz function space
$M_{\phi}$ is defined by the norm
$\|x\|_{M_{\phi}}=\sup_{t\in
I}\phi(t)\,x^{\ast\ast}(t),~{}~{}x^{\ast\ast}(t)=\frac{1}{t}\int_{0}^{t}x^{\ast}(s)ds.$
This is a symmetric Banach function space on $I$ with the fundamental function
$f_{M_{\phi}}(t)=\phi(t)$ and $E\overset{1}{\hookrightarrow}M_{f_{E}}$ since
$x^{\ast\ast}(t)\leq\frac{1}{t}\,\|x^{\ast}\|_{E}\|\chi_{[0,t]}\|_{E^{\prime}}=\|x\|_{E}\frac{1}{f_{E}(t)}~{}\mathrm{for~{}any}\,~{}t\in
I,$ (4)
(see, for example, [KPS82] or [BS88]). Although the fundamental function of a
symmetric function space $E$ need not be concave, there always exists
equivalent norm on $E$ for which new fundamental function is concave (cf.
Zippin [Zi71], Lemma 2.1). Then for a symmetric function space $E$ with the
concave fundamental function $f_{E}$ there is also the smallest symmetric
space with the same fundamental function. This space is the Lorentz function
space given by the norm
$\|x\|_{\Lambda_{f_{E}}}=\int_{I}x^{\ast}(t)df_{E}(t)=f_{E}(0^{+})\|x\|_{L^{\infty}(I)}+\int_{I}x^{\ast}(t)f_{E}^{\prime}(t)dt.$
Then the embeddings
$\Lambda_{f_{E}}\overset{1}{\hookrightarrow}E\overset{1}{\hookrightarrow}M_{f_{E}}$
(5)
are satisfied, where $f_{E}$ is the fundamental functions of $E$.
More information about Banach ideal spaces, quasi-Banach ideal spaces,
symmetric Banach and quasi-Banach spaces can be found, for example, in [KA77],
[LT79], [KPR84], [JMST], [KPS82] and [BS88].
The paper is organized as follows: In Section 1 some necessary definitions and
notations are collected including the Calderón-Lozanovskiĭ
$E_{\varphi}$-spaces. In Section 2 the product space $E\odot F$ is defined and
some general results are presented. We prove important representation of
$E\odot F$ as $\frac{1}{2}$-concavification of the Calderón space
$E^{1/2}F^{1/2}$, i.e., $E\odot F\equiv\left(E^{1/2}F^{1/2}\right)^{(1/2)}$.
Such an equality was used by Schep in [Sc10] but without any explanation,
which seems to be not so evident. Then we present some properties of $E\odot
F$ that follow from this representation. In particular, the symmetry is proved
and formula for the fundamental function of the product space is given
$f_{E\odot F}(t)=f_{E}(t)f_{F}(t)$. We finish Section 2 with some sufficient
conditions on $E$ and $F$ that $E\odot F$ is a Banach space (not only a quasi-
Banach space).
In Section 3 we collected properties connecting product spaces with the space
of multipliers. There is a proof of cancellation property of product operation
for multipliers $M(E\odot F,E\odot G)\equiv M(F,G)$.
Section 4 is devoted to products of the Calderón-Lozanovskiĭ spaces of the
type $E_{\varphi}$ as improvement of results on products known for Orlicz
spaces $L^{\varphi}$ proved by Ando [An60], Wang [Wa63] and O’Neil [ON65]. The
inclusion $E_{\varphi_{1}}\odot E_{\varphi_{2}}\hookrightarrow E_{\varphi}$
follows from the results proved in [KLM12]. The reverse inclusion
$E_{\varphi}\hookrightarrow E_{\varphi_{1}}\odot E_{\varphi_{2}}$ is
investigated here and we improve the sufficient and necessary conditions which
were given in the case of Orlicz spaces by Zabreĭko-Rutickiĭ [ZR], Dankert
[Da] and Maligranda [Ma89]. Combinig the above two inclusions we obtain
conditions on equality $E_{\varphi}=E_{\varphi_{1}}\odot E_{\varphi_{2}}$. For
example, for two Young functions $\varphi_{1},\varphi_{2}$ we always have
$E_{\varphi_{1}}\odot E_{\varphi_{2}}=E_{\varphi}$, where
$\varphi=\varphi_{1}\oplus\varphi_{2}$ is defined by
$(\varphi_{1}\oplus\varphi_{2})(u)=\inf_{u=vw}\,[\varphi_{1}(v)+\varphi_{2}(w)].$
In Section 5 we deal with the product space of Lorentz and Marcinkiewicz
spaces. The products of those spaces are calculated. One of the main tools in
the proof is commutativity of Calderón construction with the symmetrizations
(cf. Lemma 4).
Section 6 starts with some general discussion about factorization. We prove
that so-called $E$-perfectness of $F$ is necessary for the factorization
$F\equiv E\odot M(E,F)$. The rest of this section is divided into two parts.
The first is devoted to factorization of the Calderón-Lozanovskiĭ
$E_{\varphi}$-spaces. Using the results from Section 4 and the paper [KLM12]
we examine when $E_{\varphi}$ can be factorized through $E_{\varphi_{1}}$. In
the second part we investigate possibility of factorization for Lorentz and
Marcinkiewicz spaces. Finally, in Theorem 11 there is proved that under some
natural assumptions a rearrangement invariant Banach function $X$ space may be
factorized through Marcinkiewicz space and by duality, Lorentz space may be
factorized through a rearrangement invariant Banach function space $X$.
2\. On the product space $E\odot F$
Given two Banach ideal spaces (real or complex) $E$ and $F$ on
$(\Omega,\Sigma,\mu)$ define the pointwise product space $E\odot F$ as
$E\odot F=\left\\{x\cdot y:x\in E~{}{\rm and}~{}y\in F\right\\}.$
with a functional $\|\cdot\|_{E\odot F}$ defined by the formula
$\|z\|_{E\odot F}=\inf\left\\{\|x\|_{E}\,\|y\|_{F}:z=xy,x\in E,y\in
F\right\\}.$ (6)
We will show in the sequel that $E\odot F$ is, in general, a quasi-Banach
ideal space even if both $E$ and $F$ are Banach ideal spaces. Let us collect
some general properties of the product space and its norm.
Proposition 1. If $E$ and $F$ are Banach ideal spaces on
$(\Omega,\Sigma,\mu)$, then $E\odot F$ has an ideal property. Moreover,
$\displaystyle\|z\|_{E\odot F}$ $\displaystyle=$
$\displaystyle\|\,|z|\,\|_{E\odot F}$ $\displaystyle=$
$\displaystyle\inf\left\\{\|x\|_{E}\,\|y\|_{F}:|z|=xy,\,x\in E_{+},y\in
F_{+}\right\\}$ $\displaystyle=$
$\displaystyle\inf\left\\{\|x\|_{E}\,\|y\|_{F}:|z|\leq xy,\,x\in E_{+},y\in
F_{+}\right\\}.$
###### Proof.
We show first that $\|z\|_{E\odot F}=\|\,|z|\,\|_{E\odot F}$. If $z=xy$ with
$x\in E,y\in F$, then $|z|=ze^{i\theta}=xye^{i\theta}$, where
$\theta:\Omega\rightarrow\mathbb{R}$, and
$\|\,|z|\,\|_{E\odot F}\leq\|xe^{i\theta}\|_{E}\|y\|_{F}=\|x\|_{E}\|y\|_{F}.$
Hence, $\|\,|z|\,\|_{E\odot F}\leq\|z\|_{E\odot F}.$ Similarly, if $|z|=xy$
with $x\in E,y\in F$, then $z=|z|e^{-i\theta}=xye^{-i\theta}$ and
$\|z\|_{E\odot F}\leq\|xe^{-i\theta}\|_{E}\|y\|_{F}=\|x\|_{E}\|y\|_{F},$
from which we obtain the estimate $\|z\|_{E\odot F}\leq\|\,|z|\,\|_{E\odot
F}$. Combining these above estimates we obtain $\|z\|_{E\odot
F}=\|\,|z|\,\|_{E\odot F}$.
To show the ideal property of $E\odot F$ assume that $z\in E\odot F$ and
$|w|\leq|z|$. By definition for any $\varepsilon>0$ we can find $x\in E,y\in
F$ such that $z=xy$ and $\|x\|_{E}\|y\|_{F}\leq\|z\|_{E\odot F}+\varepsilon$.
We set $h(t)=\frac{w(t)}{z(t)}$ if $z(t)\neq 0$ and $h(t)=0$ if $z(t)=0$. Then
$w=hz=hxy$ and since $|hx|\leq|x|$ we have $w=hxy\in E\odot F$ with
$\|w\|_{E\odot
F}\leq\|hx\|_{E}\|y\|_{F}\leq\|x\|_{E}\|y\|_{F}\leq\|z\|_{E\odot
F}+\varepsilon.$
Since $\varepsilon>0$ was arbitrary, we have $\|w\|_{E\odot
F}\leq\|z\|_{E\odot F}$. Note that in the above proofs we needed only ideal
property of one of the spaces $E$ or $F$.
Next, if $|z(t)|=x(t)\,y(t),t\in\Omega$, then taking $x_{0}=|x|,y_{0}=|y|$ we
obtain $x_{0}\geq 0,y_{0}\geq 0,x_{0}\,y_{0}=|x|\,|y|=|xy|=|z|$, which gives
the proof of the second equality. The proof of the third equality follows from
the fact that if $0\leq x\in E,0\leq y\in F$ and $|z|\leq xy$, then
$|z|=u\,xy=x_{0}\,y_{0}$, where $x_{0}=ux,y_{0}=y$ and $u=\frac{|z|}{xy}$ on
the support of $xy$ and $u=0$ elsewhere. Since $0\leq u\leq 1$ it follows that
$x_{0}\leq x,y_{0}\leq y$ and this proves (the non-trivial part of) the last
equality. ∎
Proposition 1 shows that in investigation of product space it is enough to
consider real spaces, therefore from now we will consider only real Banach
ideal spaces.
The product space can be described with the help of the Calderón construction.
To come to this result we first prove some description of $E^{1/p}F^{1-1/p}$
spaces and $p$-convexification. Let us start in Theorem 1(i) below with a
reformulation of Lemma 31 from [KL10].
Theorem 1. Let $E$ and $F$ be a couple of Banach ideal spaces on
$(\Omega,\Sigma,\mu)$.
* $(i)$
If $1<p<\infty$ and $z\in E^{1/p}F^{1-1/p}$, then
$\|z\|_{E^{1/p}F^{1-1/p}}=\inf\left\\{\max\left\\{\|x\|_{E},\|y\|_{F}\right\\}:|z|=x^{1/p}y^{1-1/p},\,x\in
E_{+},y\in F_{+}\right\\}\\\ $
$=\inf\left\\{\max\left\\{\|x\|_{E},\|y\|_{F}\right\\}:|z|=x^{1/p}y^{1-1/p},\|x\|_{E}=\|y\|_{F},\,x\in
E_{+},y\in F_{+}\right\\}.$
* $(ii)$
If $1<p<\infty$, then
$E^{(p)}\odot F^{(p^{\prime})}\equiv E^{1/p}F^{1-1/p},~{}{\it
where}~{}1/p+1/p^{\prime}=1.$
* $(iii)$
For $0<p<\infty,~{}(E\odot F)^{(p)}\equiv E^{(p)}\odot F^{(p)}.$
* $(iv)$
We have
$E\odot F\equiv(E^{1/2}F^{1/2})^{(1/2)},$ (7)
that is,
$\|z\|_{E\odot
F}=\inf\left\\{\max\left\\{\|x\|_{E}^{2},\|y\|_{F}^{2}\right\\}:|z|=xy,\|x\|_{E}=\|y\|_{F},\,x\in
E_{+},y\in F_{+}\right\\}.$ (8)
###### Proof.
(i) Let $z\in E^{1/p}F^{1-1/p}$ and $z=x^{1/p}y^{1-1/p}$, where $0\leq x\in
E,0\leq y\in F$. Suppose $\frac{\|x\|_{E}}{\|y\|_{F}}=a>1$ (for $0<a<1$ proof
is similar). Put
$x_{1}=a^{-(1-1/p)}\,x,\,y_{1}=a^{\frac{1}{p}}\,y.$
Then $\|x_{1}\|_{E}=\|x\|_{E}^{1/p}\|y\|_{F}^{1-1/p}=\|y_{1}\|_{F}$ and
$z=x_{1}^{1/p}y_{1}^{1-1/p}$. Of course,
$\max\left\\{\|x_{1}\|_{E},\,\|y_{1}\|_{F}\right\\}\leq\max\left\\{\|x\|_{E},\,\|y\|_{F}\right\\},$
which ends the proof.
(ii) If $z\in E^{(p)}\odot F^{(p^{\prime})}$, then using Proposition 1 and
definition of $p$-convexification we obtain
$\displaystyle\|z\|_{E^{(p)}\odot F^{(p^{\prime})}}$ $\displaystyle=$
$\displaystyle\inf\left\\{\|g\|_{E^{(p)}}\|h\|_{F^{(p^{\prime})}}:|z|=gh,\,0\leq
g\in E^{(p)},0\leq h\in F^{(p^{\prime})}\right\\}$ $\displaystyle=$
$\displaystyle\inf\left\\{\|x\|_{E}^{1/p}\|y\|_{F}^{1-1/p}:|z|=x^{1/p}y^{1-1/p},\,x\in
E_{+},y\in F_{+}\right\\},$
and using Theorem 1(i) to the last expression we get
$\inf_{a>0}\left[\inf\left\\{\|x\|_{E}^{1/p}\|y\|_{F}^{1-1/p}:|z|=x^{1/p}y^{1-1/p},\,\frac{\|x\|_{E}}{\|y\|_{F}}=a,\,x\in
E_{+},y\in F_{+}\right\\}\right]$
$=\inf_{a>0}\left[\inf\left\\{a^{1/p}\|u\|_{E}^{1/p}\|y\|_{F}^{1-1/p}:|z|=a^{1/p}u^{1/p}y^{1-1/p},\|u\|_{E}=\|y\|_{F},\,u\in
E_{+},y\in F_{+}\right\\}\right]$
$=\inf_{a>0}\left[a^{1/p}\inf\left\\{\|u\|_{E}:\frac{|z|}{a^{1/p}}=u^{1/p}y^{1-1/p},\|u\|_{E}=\|y\|_{F},\,u\in
E_{+},y\in F_{+}\right\\}\right]$
$=\inf_{a>0}\left[a^{1/p}\|\frac{z}{a^{1/p}}\|_{E^{1/p}F^{1-1/p}}\right]=\|z\|_{E^{1/p}F^{1-1/p}}.$
(iii) One has
$\displaystyle\|z\|_{(E\odot F)^{(p)}}$ $\displaystyle=$
$\displaystyle\||z|^{p}\|_{E\odot F}^{1/p}$ $\displaystyle=$
$\displaystyle\inf\left\\{\|x\|_{E}^{1/p}\,\|y\|_{F}^{1/p}:|z|^{p}=xy,x\in
E_{+},y\in F_{+}\right\\}$ $\displaystyle=$
$\displaystyle\inf\left\\{\|u^{p}\|_{E}^{1/p}\,\|v^{p}\|_{F}^{1/p}:|z|^{p}=u^{p}v^{p},u\in
E^{(p)}_{+},v\in F^{(p)}_{+}\right\\}$ $\displaystyle=$
$\displaystyle\inf\left\\{\|u\|_{E^{(p)}}\,\|v\|_{F^{(p)}}:|z|=uv,\,u\in
E^{(p)}_{+},v\in F^{(p)}_{+}\right\\}$ $\displaystyle=$
$\displaystyle\|z\|_{E^{(p)}\odot F^{(p)}}.$
(iv) The proof is an immediate consequence of Theorem 1(ii) and (iii) since
$E\odot F\equiv\left(\left(E\odot
F\right)^{(2)}\right)^{(1/2)}\equiv\left(E^{1/2}F^{1/2}\right)^{(1/2)}.$
Moreover,
$\displaystyle\|z\|_{E\odot F}$ $\displaystyle=$
$\displaystyle\|z\|_{\left(E^{1/2}F^{1/2}\right)^{(1/2)}}=(\|\sqrt{|z|}\,\|_{E^{1/2}F^{1/2}})^{2}$
$\displaystyle=$
$\displaystyle\left[\inf\left\\{\max\left\\{\|x\|_{E},\|y\|_{F}\right\\}:\sqrt{|z|}=\sqrt{xy},\|x\|_{E}=\|y\|_{F},\,x\in
E_{+},y\in F_{+}\right\\}\right]^{2}$ $\displaystyle=$
$\displaystyle\inf\left\\{\max\left\\{\|x\|_{E}^{2},\|y\|_{F}^{2}\right\\}:|z|=xy,\|x\|_{E}=\|y\|_{F},\,x\in
E_{+},y\in F_{+}\right\\},$
and the proof is complete. ∎
As a consequence of the representation (7) we obtain the following results:
Corollary 1. Let $E$ and $F$ be a couple of Banach ideal spaces on
$(\Omega,\Sigma,\mu)$.
* $(i)$
Then $E\odot F$ is a quasi-Banach ideal space and the triangle inequality is
satisfied with constant $2$, i.e.,
$\|x+y\|_{E\odot F}\leq 2\left(\|x\|_{E\odot F}+\|y\|_{E\odot F}\right).$
* $(ii)$
If both $E$ and $F$ satisfy the Fatou property, then $E\odot F$ has the Fatou
property.
* $(iii)$
The space $E\odot F$ has order continuous norm if and only if the couple
$(E,F)$ is not jointly order discontinuous, i.e., $(E,F)\not\in(JOD)$.
Recall that $(E,F)\in\left(JOD\right)$ (see [KL10]) means that there exist
elements $x\in E\backslash E_{a},y\in F\backslash F_{a}$ and a sequence of
measurable sets $A_{n}\searrow\emptyset$ such that for any sequence $(B_{n})$
in $\Sigma$ with $B_{n}\subset A_{n}$ ($n\in\mathbb{N}$) there are a number
$a>0$ and a subsequence $(n_{k})$ in $\mathbb{N}$ such that either
$\|x\chi_{B_{n_{k}}}\|_{E}\geq a~{}{\rm
and}~{}~{}\|y\chi_{B_{n_{k}}}\|_{F}\geq a~{}~{}{\rm
for~{}all}~{}k\in\mathbb{N},$
or
$\|x\chi_{A_{n_{k}}\backslash B_{n_{k}}}\|_{E}\geq a~{}{\rm
and}~{}\|y\chi_{A_{n_{k}}\backslash B_{n_{k}}}\|_{F}\geq a~{}{\rm
for~{}all}~{}k\in\mathbb{N}.$
###### Proof.
(i) It is a consequence of the representation (7) $E\odot
F\equiv(E^{1/2}F^{1/2})^{(1/2)}$ and the fact that for $1/2$-concavification
of a Banach ideal space $G=E^{1/2}F^{1/2}$ we have
$\displaystyle\|\,|x+y|^{1/2}\,\|_{G}^{2}$ $\displaystyle\leq$
$\displaystyle\left(\|\,|x|^{1/2}\,\|_{G}+\|\,|y|^{1/2}\,\|_{G}\right)^{2}$
$\displaystyle\leq$ $\displaystyle
2\left(\|\,|x|^{1/2}\,\|_{G}^{2}+\|\,|y|^{1/2}\,\|_{G}^{2}\right).$
(ii) It is again a consequence of the representation (7) and the fact that
$E^{1/2}F^{1/2}$ has the Fatou property when $E$ and $F$ have the Fatou
property (see [Lo69], p. 595).
(iii) The representation (7) and Theorem 13 in [KL10] showing that
$E^{1/2}F^{1/2}$ has order continuous norm if and only if
$(E,F)\not\in\left(JOD\right)$ which gives the statement. ∎
Lozanovskiĭ [Lo65, Theorem 4] formulated result on the Köthe dual of
$p$-convexification $E^{(p)}$ with no proof. The proof can be found in paper
by Schep [Sc10, Theorem 2.9] and we present another proof which follows from
the Lozanovskiĭ duality result and our Theorem 1(ii).
Corollary 2. Let $E$ be a Banach ideal space and $1<p<\infty$. Then
$[E^{(p)}]^{\prime}\equiv(E^{\prime})^{(p)}\odot L^{p^{\prime}}.$
###### Proof.
Using Lozanovskiĭ theorem on duality of the Calderón spaces (see [Lo69],
Theorem 2) and our Theorem 1(ii) we obtain
$\displaystyle[E^{(p)}]^{\prime}$ $\displaystyle\equiv$
$\displaystyle[E^{1/p}(L^{\infty})^{1-1/p}]^{\prime}\equiv(E^{\prime})^{1/p}(L^{1})^{1-1/p}$
$\displaystyle\equiv$
$\displaystyle(E^{\prime})^{(p)}\odot(L^{1})^{(p^{\prime})}\equiv(E^{\prime})^{(p)}\odot
L^{p^{\prime}}.$
∎
Remark 1. In general, $[E^{(p)}]^{\prime}\neq(E^{\prime})^{(p)}$. In fact, for
the classical Lorentz space $E=L^{r,1}$ with $1<r<\infty$ we have
$[(L^{r,1})^{(p)}]^{\prime}=(L^{rp,p})^{\prime}=L^{q,p^{\prime}}$, where
$1/q+1/(pr)=1$ and
$[(L^{r,1})^{\prime}]^{(p)}=(L^{r^{\prime},\infty})^{(p)}=L^{r^{\prime}p,\infty}$.
Example 1. (a) If $1\leq p,q\leq\infty,1/p+1/q=1/r$, then $L^{p}\odot
L^{q}\equiv L^{r}$. In particular, $L^{p}\odot L^{p}\equiv L^{p/2}$. In fact,
by the Hölder-Rogers inequality $\|xy\|_{r}\leq\|x\|_{p}\|y\|_{q}$ for $x\in
L^{p},y\in L^{q}$ (see [Ma89], p. 69) we obtain $L^{p}\odot L^{q}\subset
L^{r}$ and $\|z\|_{r}\leq\|z\|_{L^{p}\odot L^{q}}$. On the other hand, if
$z\in L^{r}$, then $x=|z|^{r/p}{\rm sgn}z\in L^{p},y=|z|^{r/q}{\rm sgn}z\in
L^{q}$ and $xy=z$ with
$\|x\|_{p}\|y\|_{q}=\|z\|_{r}^{r/p}\|z\|_{r}^{r/q}=\|z\|_{r}^{r/p+r/q}=\|z\|_{r},$
which shows that $L^{r}\subset L^{p}\odot L^{q}$ and $\|z\|_{L^{p}\odot
L^{q}}\leq\|z\|_{r}$.
More general, if $1\leq p,q<\infty,1/p+1/q=1/r$ and $E$ is a Banach ideal
space, then $E^{(p)}\odot E^{(q)}\equiv E^{(r)}$ (cf. [MP89, Lemma 1] and
[ORS08, Lemma 2.21(i)]).
(b) We have $c_{0}\odot l^{1}\equiv l^{\infty}\odot l^{1}\equiv l^{1}$ and
$c_{0}\equiv c_{0}\odot l^{\infty}\neq l^{\infty}\odot l^{\infty}\equiv
l^{\infty}$.
This example shows that for the Fatou property of $E\odot F$ it is not
necessary that both $E$ and $F$ do have the Fatou property.
The next interesting question about product space is its symmetry.
Theorem 2. Let $E$ and $F$ be symmetric Banach spaces on $I=(0,1)$ or
$I=(0,\infty)$ with the fundamental functions $f_{E}$ and $f_{F}$,
respectively. Then $E\odot F$ is a symmetric quasi-Banach space on $I$ and its
fundamental function $f_{E\odot F}$ is given by the formula
$f_{E\odot F}(t)=f_{E}(t)f_{F}(t)~{}~{}{\rm for}~{}t\in I.$ (9)
###### Proof.
Using Lemma 4.3 from [KPS82, p. 93] we can easily show that $E^{1/2}F^{1/2}$
(even $\rho(E,F)$) is a Banach symmetric space and the representation (7)
gives the symmetry property of $E\odot F$.
The inequality $f_{E\odot F}(t)\leq f_{E}(t)f_{F}(t)$ for $t\in I$ is clear.
We prove the reverse inequality using the fact that each symmetric Banach
space $E$ satisfies $E\overset{1}{\hookrightarrow}M_{f_{E}},$ where
$M_{f_{E}}$ is the Marcinkiewicz space (see estimate (4)), some classical
inequality on rearrangement (see, for example, [HLP52], p. 277 or [BS88], p.
44 or [KPS82], p. 64) and the reverse Chebyshev inequality (see Lemma 1
below). For any $0\leq x\in E,0\leq y\in F$ such that $x\,y=\chi_{[0,t]}$ we
have
$\displaystyle\|x\|_{E}\,\|y\|_{F}$ $\displaystyle\geq$
$\displaystyle\|x\|_{M_{f_{E}}}\,\|y\|_{M_{f_{F}}}$ $\displaystyle\geq$
$\displaystyle\sup_{0<u\leq
t}\frac{f_{E}(u)}{u}\int_{0}^{u}x^{\ast}(s)\,ds\,\sup_{0<v\leq
t}\frac{f_{F}(v)}{v}\int_{0}^{v}y^{\ast}(s)\,ds$ $\displaystyle\geq$
$\displaystyle\frac{f_{E}(t)f_{F}(t)}{t}\,\frac{1}{t}\int_{0}^{t}x^{\ast}(s)\,ds\,\int_{0}^{t}y^{\ast}(s)\,ds$
$\displaystyle\geq$
$\displaystyle\frac{f_{E}(t)f_{F}(t)}{t}\frac{1}{t}\int_{0}^{t}x(s)\,ds\,\int_{0}^{t}y(s)\,ds$
$\displaystyle\geq$
$\displaystyle\frac{f_{E}(t)f_{F}(t)}{t}\int_{0}^{t}x(s)\,y(s)\,ds$
$\displaystyle=$
$\displaystyle\frac{f_{E}(t)f_{F}(t)}{t}\int_{0}^{t}\chi_{[0,t]}(s)\,ds=f_{E}(t)f_{F}(t),$
and so $\|\chi_{[0,t]}\|_{E\odot F}\geq f_{E}(t)f_{F}(t)$. ∎
In the fifth inequality above we used the reverse Chebyshev inequality which
we will prove in the lemma below. On the classical Chebyshev inequality for
decreasing functions (see, for example, [Mi70], p. 39 or [HM91-2], p. 213).
Lemma 1. Let $0<\mu(A)<\infty$ and $x(s)\,y(s)=a>0$ for all $s\in A$ with
$0\leq x,y\in L^{1}(A)$. Then
$\mu(A)\int_{A}xy\,d\mu\leq\int_{A}x\,d\mu\,\int_{A}y\,d\mu.$ (10)
###### Proof.
For any $s,t\in A$ we have
$\displaystyle x(s)y(s)$ $\displaystyle-$ $\displaystyle
x(s)y(t)-x(t)y(s)+x(t)y(t)=2a-x(s)y(t)-x(t)y(s)$ $\displaystyle=$
$\displaystyle
2a-\frac{x(s)a}{x(t)}-\frac{x(t)a}{x(s)}=a\,[2-\frac{x(s)}{x(t)}-\frac{x(t)}{x(s)}]$
$\displaystyle=$ $\displaystyle
a\,\frac{2x(s)x(t)-x(s)^{2}-x(t)^{2}}{x(s)x(t)}=-a\,\frac{[x(s)-x(t)]^{2}}{x(s)x(t)}\leq
0.$
Now integrating over $A$ with respect to $s$ and over $A$ with respect to $t$
we obtain the desired inequality (10). ∎
Remark 2. Formula (9) is a generalization of the well-known equality on
fundamental functions $f_{E}(t)f_{E^{\prime}}(t)=t=f_{L^{1}}(t)$ for $t\in I$
and it is also true for symmetric sequence spaces with the same proof.
Example 1(a) shows that $E\odot F$ is, in general, a quasi-Banach ideal space
even if both $E$ and $F$ are Banach ideal spaces. We can ask under which
additional conditions on $E$ and $F$ the product space $E\odot F$ is a Banach
ideal space. Before formulation the theorem we need notion of $p$-convexity. A
Banach lattice $E$ is said to be $p$-convex ($1\leq p<\infty$) with constant
$K\geq 1$ if
$\|(\sum_{k=1}^{n}|x_{k}|^{p})^{1/p}\|_{E}\leq
K\,(\sum_{k=1}^{n}\|x_{k}\|_{E}^{p})^{1/p}.$
for any sequence $\left(x_{k}\right)_{k=1}^{n}\subset X$ and any
$n\in\mathbb{N}$. If a Banach lattice $E$ is $p$-convex with constant $K\geq
1$ and $1\leq q<p$, then $E$ is also $q$-convex with constant at most $K$.
Moreover, $p$-convexification $E^{(p)}$ of a Banach lattice $E$ is $p$-convex
with constant $1$. More information on $p$-convexity we can find, for example,
in [LT79] and [Ma04].
Theorem 3. Suppose that $E,F$ are Banach ideal spaces such that $E$ is
$p_{0}$-convex with constant $1$, $F$ is $p_{1}$-convex with constant $1$ and
$\frac{1}{p_{0}}+\frac{1}{p_{1}}\leq 1$. Then $E\odot F$ is a Banach space
which is even $\frac{p}{2}$-convex, where
$\frac{1}{p}=\frac{1}{2}(\frac{1}{p_{0}}+\frac{1}{p_{1}})$.
Before the proof of Theorem 3 let us present the following lemma, which was
mentioned in [Re80] and proved in [TJ89], p. 219. For the sake of completeness
we give its proof.
Lemma 2. If $E$ is $p_{0}$-convex with constant $K_{0}$ and $F$ is
$p_{1}$-convex with constant $K_{1}$, then $E^{1-\theta}F^{\theta}$ is
$p$-convex with constant $K\leq K_{0}^{1-\theta}K_{1}^{\theta}$, where
$\frac{1}{p}=\frac{1-\theta}{p_{0}}+\frac{\theta}{p_{1}}$.
###### Proof.
Let $z_{1},\ldots,z_{n}\in E^{1-\theta}F^{\theta}$. There are $x_{k}\in
E,y_{k}\in F$ with $\|x_{k}\|_{E}\leq 1,\|y_{k}\|_{F}\leq 1$ and
$\lambda_{k}\geq 0$ be such that for all $1\leq k\leq n$
$|z_{k}|\leq\lambda_{k}|x_{k}|^{1-\theta}|y_{k}|^{\theta}~{}~{}{\rm
and}~{}~{}\lambda_{k}\leq\|z_{k}\|_{E^{1-\theta}F^{\theta}}(1+\varepsilon).$
By the Hölder-Rogers inequality
$\displaystyle(\sum_{k=1}^{n}|z_{k}|^{p})^{1/p}$ $\displaystyle\leq$
$\displaystyle(\sum_{k=1}^{n}\lambda_{k}^{p}\,|x_{k}|^{p(1-\theta)}|y_{k}|^{p\theta})^{1/p}$
$\displaystyle\leq$
$\displaystyle(\sum_{k=1}^{n}\lambda_{k}^{p}\,|x_{k}|^{p_{0}})^{(1-\theta)/p_{0}}(\sum_{k=1}^{n}\lambda_{k}^{p}\,|y_{k}|^{p_{1}})^{\theta/p_{1}}.$
Then, by the property
$\|u^{1-\theta}v^{\theta}\|_{E^{1-\theta}F^{\theta}}\leq\|u\|_{E}^{1-\theta}\,\|v\|_{F}^{\theta}$
and assumptions on convexity together with assumptions on elements, we obtain
$\displaystyle\|(\sum_{k=1}^{n}|z_{k}|^{p})^{1/p}\|_{E^{1-\theta}F^{\theta}}$
$\displaystyle\leq$
$\displaystyle\|(\sum_{k=1}^{n}\lambda_{k}^{p}\,|x_{k}|^{p_{0}})^{(1-\theta)/p_{0}}(\sum_{k=1}^{n}\lambda_{k}^{p}\,|y_{k}|^{p_{1}})^{\theta/p_{1}}\|_{E^{1-\theta}F^{\theta}}$
$\displaystyle\leq$
$\displaystyle\|(\sum_{k=1}^{n}\lambda_{k}^{p}\,|x_{k}|^{p_{0}})^{1/p_{0}}\|_{E}^{1-\theta}\,\|(\sum_{k=1}^{n}\lambda_{k}^{p}\,|y_{k}|^{p_{1}})^{1/p_{1}}\|_{F}^{\theta}$
$\displaystyle\leq$ $\displaystyle
K_{0}^{1-\theta}(\sum_{k=1}^{n}\|\lambda_{k}^{p/p_{0}}\,x_{k}\|_{E}^{p_{0}})^{(1-\theta)/{p_{0}}}\,K_{1}^{\theta}(\sum_{k=1}^{n}\|\lambda_{k}^{p/p_{1}}\,y_{k}\|_{F}^{p_{1}})^{\theta/{p_{1}}}$
$\displaystyle=$ $\displaystyle
K_{0}^{1-\theta}K_{1}^{\theta}\,(\sum_{k=1}^{n}\lambda_{k}^{p}\,\|x_{k}\|_{E}^{p_{0}})^{(1-\theta)/{p_{0}}}\,(\sum_{k=1}^{n}\lambda_{k}^{p}\,\|y_{k}\|_{F}^{p_{1}})^{\theta/{p_{1}}}$
$\displaystyle\leq$ $\displaystyle
K_{0}^{1-\theta}K_{1}^{\theta}\,(\sum_{k=1}^{n}\lambda_{k}^{p})^{(1-\theta)/{p_{0}}}\,(\sum_{k=1}^{n}\lambda_{k}^{p})^{\theta/{p_{1}}}=K_{0}^{1-\theta}K_{1}^{\theta}\,(\sum_{k=1}^{n}\lambda_{k}^{p})^{1/{p}}$
$\displaystyle\leq$
$\displaystyle(1+\varepsilon)\,K_{0}^{1-\theta}K_{1}^{\theta}\,(\sum_{k=1}^{n}\|z_{k}\|_{E^{1-\theta}F^{\theta}}^{p})^{1/{p}}.$
Since $\varepsilon>0$ is arbitrary, the $p$-convexity of
$E^{1-\theta}F^{\theta}$ is proved with the constant $K\leq
K_{0}^{1-\theta}K_{1}^{\theta}$. ∎
Proof of Theorem 3. By Lemma 2, we have that $Z=E^{1/2}F^{1/2}$ is $p$-convex
with constant $1$, where $\frac{1}{p}=\frac{1}{2p_{0}}+\frac{1}{2p_{1}}$. The
assumption on $p_{0},p_{1}$ gives that $p\geq 2$ and since
$1/2$-concavification of $Z$ is $p/2$-convex with constant $1$ ($p/2\geq 1$)
it follows that it is $1$-convex with constant $1$ which gives that the norm
of $E\odot F=Z^{(1/2)}$ satisfies the triangle inequality, and consequently is
a Banach space. This completes the proof. ∎
Remark 3. By duality arguments, Theorem 3 can be also formulated in the terms
of $q$-concavity of the Köthe dual spaces. A Banach lattice $E$ is $q$-concave
($1<q<\infty$) with constant $K\geq 1$ if
$(\sum_{k=1}^{n}\|x_{k}\|_{F}^{q})^{1/q}\leq
K\,\|(\sum_{k=1}^{n}|x_{k}|^{q})^{1/q}\|_{F}$ for any sequence
$\left(x_{k}\right)_{k=1}^{n}\subset X$ and any $n\in\mathbb{N}$.
Remark 4. Since imbedding $G\subset E\odot F$ means also factorization
$z=x\,y$, where $x\in E$ and $y\in F$, therefore sometimes these imbeddings or
identifications of product spaces $E\odot F=G$ are called factorizations of
concrete spaces as, for example, $l^{p}$ and Cesàro sequence spaces or $L^{p}$
and Cesàro function spaces (cf. [Be96], [AM09], [Sc10]), factorization of tent
spaces or other spaces (cf. [CV00], [CRW76], [Ho77]).
3\. The product spaces and multipliers
Let us collect properties connecting product space with the space of
multipliers. We start with the Cwikel and Nilsson result [CN03, Theorem 3.5].
They proved that if a Banach ideal space $E$ has the Fatou property and
$0<\theta<1$, then
$E\equiv M(F^{(1/\theta)},E^{1-\theta}F^{\theta})^{(1-\theta)}.$
We will prove a generalization of this equality, which in the case of
$G=L^{\infty}$ coincides with their result.
Proposition 2. Let $E,F,G$ be Banach ideal spaces. Suppose that $E$ has the
Fatou property and $0<\theta<1$. Then
$M(G,E)\equiv M(G^{1-\theta}F^{\theta},E^{1-\theta}F^{\theta})^{(1-\theta)}.$
###### Proof.
First, let us prove the imbedding $\overset{1}{\hookrightarrow}$. Let $x\in
M(G,E)$. We want to show that $x\in
M(G^{1-\theta}F^{\theta},E^{1-\theta}F^{\theta})^{(1-\theta)}$, that is,
$|x|^{1-\theta}\in M(G^{1-\theta}F^{\theta},E^{1-\theta}F^{\theta})$ or
equivalently $x^{1-\theta}|y|\in E^{1-\theta}F^{\theta}$ for any $y\in
G^{1-\theta}F^{\theta}$. Take arbitrary $y\in G^{1-\theta}F^{\theta}$ with the
norm $<1$. Then there are $w\in G,v\in F$ satisfying $\|w\|_{G}\leq
1,\|v\|_{F}\leq 1$ and $|y|\leq|w|^{1-\theta}|v|^{\theta}$. Clearly,
$|x|^{1-\theta}\,|y|\leq|xw|^{1-\theta}|v|^{\theta}\in E^{1-\theta}F^{\theta}$
since $x\in M(G,E),w\in G$ gives $xw\in E$. This proves the inclusion.
Moreover,
$\displaystyle\||x|^{1-\theta}\,|y|\|_{E^{1-\theta}F^{\theta}}$
$\displaystyle\leq$
$\displaystyle\||xw|^{1-\theta}|v|^{\theta}\|_{E^{1-\theta}F^{\theta}}$
$\displaystyle\leq$ $\displaystyle\|xw\|_{E}^{1-\theta}\|v\|_{F}^{\theta}$
$\displaystyle\leq$
$\displaystyle\|x\|_{M(G,E)}^{1-\theta}\|w\|_{G}^{1-\theta}\|v\|_{F}^{\theta}\leq\|x\|_{M(G,E)}^{1-\theta}.$
Thus,
$\||x|^{1-\theta}\|_{M(G^{1-\theta}F^{\theta},\,E^{1-\theta}F^{\theta})}^{1/(1-\theta)}\leq\|x\|_{M(G,E)}.$
The imbedding $\overset{1}{\hookleftarrow}$. Let
$\|x\|_{M(G^{1-\theta}F^{\theta},E^{1-\theta}F^{\theta})^{(1-\theta)}}=1$,
i.e.
$\||x|^{1-\theta}\|_{M(G^{1-\theta}F^{\theta},E^{1-\theta}F^{\theta})}=1$. We
need to show that for any $w\in G$ we have $xw\in
M(F^{(1/\theta)},E^{1-\theta}F^{\theta})^{(1-\theta)}$, that is,
$|xw|^{1-\theta}\in M(F^{(1/\theta)},E^{1-\theta}F^{\theta})$. Really, by the
Cwikel-Nilsson result, we obtain $xw\in E$ for any $w\in G$.
Let $w\in G$ and $v\in F$. Since the norm of $x$ is 1 it follows that
$\||x|^{1-\theta}\frac{z}{\|z\|_{G^{1-\theta}F^{\theta}}}\|_{E^{1-\theta}F^{\theta}}\leq
1$
for each $0\neq z\in G^{1-\theta}F^{\theta}$. Consequently, for
$z=|w|^{1-\theta}|v|^{\theta}$, we obtain
$\displaystyle\||xw|^{1-\theta}\,|v|^{\theta}\|_{E^{1-\theta}F^{\theta}}$
$\displaystyle=$
$\displaystyle\||x|^{1-\theta}|w|^{1-\theta}|v|^{\theta}\|_{E^{1-\theta}F^{\theta}}$
$\displaystyle\leq$
$\displaystyle\||w|^{1-\theta}|v|^{\theta}\|_{G^{1-\theta}F^{\theta}}\leq\|w\|_{G}^{1-\theta}\|v\|_{F}^{\theta}.$
This proves our inclusion part because from the assumption on $x$ and the fact
that $|w|^{1-\theta}\,|v|^{\theta}\in G^{1-\theta}F^{\theta}$ we have
$|xw|^{1-\theta}\,|v|^{\theta}=|x|^{1-\theta}|w|^{1-\theta}\,|v|^{\theta}\in
E^{1-\theta}F^{\theta}$. Moreover, by the Cwikel-Nilsson result and the last
estimate with $v=|m|^{1/\theta}$, we obtain
$\displaystyle\|x\|_{M(G,E)}$ $\displaystyle=$
$\displaystyle\sup_{\|w\|_{G}\leq 1}\|xw\|_{E}=\sup_{\|w\|_{G}\leq
1}\|xw\|_{M(F^{(1/\theta)},E^{1-\theta}F^{\theta})^{(1-\theta)}}$
$\displaystyle=$ $\displaystyle\sup_{\|w\|_{G}\leq
1}\sup_{\|m\|_{F^{(1/\theta)}}\leq
1}\||xw|^{1-\theta}m\|_{E^{1-\theta}F^{\theta}}^{1/(1-\theta)}$
$\displaystyle\leq$ $\displaystyle\sup_{\|w\|_{G}\leq
1}\sup_{\|m\|_{F^{(1/\theta)}}\leq
1}\|w\|_{G}\|m\|_{F^{(1/\theta)}}^{1/(1-\theta)}\leq 1,$
and the theorem is proved with the equality of the norms. ∎
Proposition 2 together with the representation of the product space as the
$1/2$-concavification of the Calderón space will give the “cancellation”
property for multipliers of products.
Theorem 4. Let $E,F,G$ be Banach ideal spaces. If $G$ has the Fatou property,
then
$M(E\odot F,E\odot G)\equiv M(F,G).$ (11)
###### Proof.
Applying Theorem 1(iv), property (g) from [MP89] and Proposition 2 we obtain
$\displaystyle M(E\odot F,E\odot G)$ $\displaystyle\equiv$ $\displaystyle
M[(E^{1/2}F^{1/2})^{(1/2)},(E^{1/2}G^{1/2})^{(1/2)}]$ $\displaystyle\equiv$
$\displaystyle M[E^{1/2}F^{1/2},(E^{1/2}G^{1/2})]^{(1/2)}\equiv M(F,G),$
and (11) is proved. ∎
Remark 5. Note that Proposition 2 and Theorem 4 are equivalent. Proposition 2
can be written in the following form: if $F$ has the Fatou property, then
$M(E^{\theta}G^{1-\theta},F^{\theta}G^{1-\theta})\equiv M(E,F)^{(1/\theta)},$
and it can be proved using Theorem 4. In fact, applying Theorem 1(ii),
cancellation property from Theorem 4 and property (g) from [MP89] we obtain
$\displaystyle M(E^{\theta}G^{1-\theta},F^{\theta}G^{1-\theta})$
$\displaystyle\equiv$ $\displaystyle M(E^{(1/\theta)}\odot
G^{(1/(1-\theta))},F^{(1/\theta)}\odot G^{(1/(1-\theta))})$
$\displaystyle\equiv$ $\displaystyle M(E^{(1/\theta)},F^{(1/\theta)})\equiv
M(E,F)^{(1/\theta)}.$
From Theorem 4 we can also get the equality mentioned by Raynaud [Ra92] which
can be proved also directly (cf. also [Sc10], Proposition 1.4).
Corollary 3. Let $E,F$ be Banach ideal spaces. If $E$ has the Fatou property,
then
$(E\odot F)^{\prime}\equiv M(F,E^{\prime})\equiv M(E,F^{\prime})$ (12)
###### Proof.
Using the Lozanovskiĭ factorization theorem (for more discussion see Part 6)
and the cancellation property (11) we obtain
$(E\odot F)^{\prime}\equiv M(E\odot F,L^{1})\equiv M(E\odot F,E\odot
E^{\prime})\equiv M(F,E^{\prime}).$
and
$(E\odot F)^{\prime}\equiv M(E\odot F,L^{1})\equiv M(E\odot F,F^{\prime}\odot
F)\equiv M(E,F^{\prime}).$
Note that the second identity in (12) follows also from the general properties
of multipliers (see [MP89, property (e)] or [KLM12, property (vii)]) because
we have $M(F,E^{\prime})\equiv M(E^{\prime\prime},F^{\prime})\equiv
M(E,F^{\prime})$. ∎
Corollary 4. Let $E,F$ be Banach ideal spaces. If $F$ has the Fatou property
and $\|xy\|_{E\odot F}\leq 1$ for all $x\in E$ with $\|x\|_{E}\leq 1$, then
$\|y\|_{F}\leq 1$.
###### Proof.
Since by assumption $\|y\|_{M(E,E\odot F)}\leq 1$, then using Theorem 4,
together with the facts that $M(L^{\infty},F)\equiv F,E\odot L^{\infty}\equiv
E$, we obtain
$\|y\|_{F}=\|y\|_{M(L^{\infty},F)}=\|y\|_{M(E\odot L^{\infty},E\odot
F)}=\|y\|_{M(E,E\odot F)}\leq 1.$
∎
Corollary 5. Let $E,F,G$ be Banach ideal spaces. If $F$ and $G$ have the Fatou
property, then
$M(E\odot F,G)\equiv M(E,M(F,G)).$
###### Proof.
Using Theorem 4, the Lozanovskiĭ factorization theorem, Corollary 3 with the
fact that the Fatou property of $F$ gives by Corollary 1(ii) that $F\odot
G^{\prime}$ has the Fatou property, again Corollary 3 and the Fatou property
of $G$ we obtain
$\displaystyle M(E\odot F,G)$ $\displaystyle\equiv$ $\displaystyle M(E\odot
F\odot G^{\prime},G\odot G^{\prime})$ $\displaystyle\equiv$ $\displaystyle
M(E\odot F\odot G^{\prime},L^{1})$ $\displaystyle\equiv$ $\displaystyle(E\odot
F\odot G^{\prime})^{\prime}\equiv(F\odot G^{\prime}\odot E)^{\prime}$
$\displaystyle\equiv$ $\displaystyle M(E,(F\odot G^{\prime})^{\prime})\equiv
M(E,M(F,G^{\prime\prime})\equiv M(E,M(F,G)),$
which establishes the formula. ∎
4\. The product of Calderón-Lozanovskiĭ $E_{\varphi}$-spaces
The pointwise product of Orlicz spaces was investigated already by
Krasnoselskiĭ and Rutickiĭ in their book, where sufficient conditions on
imbedding $L^{\varphi_{1}}\odot L^{\varphi_{2}}\subset L^{\varphi}$ are given
in the case when $\Omega$ is bounded closed subset of ${\mathbb{R}}^{n}$ (cf.
[KR61], Theorems 13.7 and 13.8). For the same set $\Omega$, Ando [An60] proved
that $L^{\varphi_{1}}\odot L^{\varphi_{2}}\subset L^{\varphi}$ if and only if
there exist $C>0$, $u_{0}>0$ such that
$\varphi(Cuv)\leq\varphi_{1}(u)+\varphi_{2}(v)$ for $u,v\geq u_{0}$.
O’Neil [ON65] presented necessary and sufficient conditions for the imbedding
$L^{\varphi_{1}}\odot L^{\varphi_{2}}\subset L^{\varphi}$ in the case when
measure space is either non-atomic and infinite or non-atomic and finite or
counting measure on $\mathbb{N}$. Moreover, he observed that condition
$\varphi(Cuv)\leq\varphi_{1}(u)+\varphi_{2}(v)$ for all [large, small] $u,v>0$
is equivalent to condition on inverse functions
$C_{1}\varphi_{1}^{-1}(u)\varphi_{2}^{-1}(u)\leq\varphi^{-1}(u)$ for all
[large, small] $u>0$. O’Neil’s results were also presented, with his proofs,
in the books [Ma89, pp. 71-75] and [RR91, pp. 179-184].
The reverse imbedding $L^{\varphi}\subset L^{\varphi_{1}}\odot
L^{\varphi_{2}}$ and the equality $L^{\varphi_{1}}\odot
L^{\varphi_{2}}=L^{\varphi}$ were considered by Zabreiko-Rutickiĭ [ZR67,
Theorem 8], Dankert [Da74, pp. 63-68] and Maligranda [Ma89, 69-71].
We will prove the above results for more general spaces, that is, for the
Calderón-Lozanovskiĭ $E_{\varphi}$-spaces. Results on the imbedding
$E_{\varphi_{1}}\odot E_{\varphi_{2}}\hookrightarrow E_{\varphi}$ need the
following relations between Young functions (cf. [ON65]): we say
$\varphi_{1}^{-1}\varphi_{2}^{-1}\prec\varphi^{-1}$ for all arguments [for
large arguments] (for small arguments) if that there is a constant $C>0$
[there are constants $C,u_{0}>0$] (there are constants $C,u_{0}>0$) such that
the inequality
$C\varphi_{1}^{-1}(u)\varphi_{2}^{-1}(u)\leq\varphi^{-1}(u)$ (13)
holds for all $u>0$ [for all $u\geq u_{0}$] (for all $u\leq u_{0}$),
respectively.
Remark 6. The inequality (13) implies a generalized Young inequality:
$\varphi(Cuv)\leq\varphi_{1}(u)+\varphi_{2}(v)~{}~{}{\rm
for~{}all}~{}~{}u,v>0~{}~{}{\rm
such~{}that}~{}\varphi_{1}(u),\varphi_{2}(v)<\infty.$ (14)
On the other hand, if $\varphi(Cuv)\leq\varphi_{1}(u)+\varphi_{2}(v)$ for all
$u,v>0$, then
$\varphi_{1}^{-1}(w)\varphi_{2}^{-1}(w)\leq\frac{2}{C}\varphi^{-1}(w)$ for
each $w>0$ (see [ON65] and [KLM12]). Similar equivalences hold for large and
small arguments.
In [KLM12] the question when the product $xy\in E_{\varphi}$ provided $x\in
E_{\varphi_{1}}$ and $y\in E_{\varphi_{2}}$ was investigated, as a
generalization of O’Neil’s theorems [ON65], and the following results were
proved (see [KLM12], Theorems 4.1, 4.2 and 4.5):
Theorem A. Let $\varphi_{1},\varphi_{2}$ and $\varphi$ be three Young
functions.
* $(a)$
If $E$ is a Banach ideal space with the Fatou property and one of the
following conditions holds:
$(a1)$ $\varphi_{1}^{-1}\varphi_{2}^{-1}\prec\varphi^{-1}$ for all arguments,
$(a2)$ $\varphi_{1}^{-1}\varphi_{2}^{-1}\prec\varphi^{-1}$ for large arguments
and $L^{\infty}\hookrightarrow E$,
$(a3)$ $\varphi_{1}^{-1}\varphi_{2}^{-1}\prec\varphi^{-1}$ for small arguments
and $E\hookrightarrow L^{\infty}$,
then, for every $x\in E_{\varphi_{1}}$ and $y\in E_{\varphi_{2}}$ the product
$xy\in E_{\varphi}$, which means that
$E_{\varphi_{1}}\odot E_{\varphi_{2}}\hookrightarrow E_{\varphi}.$
* $(b)$
If a Banach ideal space $E$ with the Fatou property is such that
$E_{a}\neq\left\\{0\right\\}$ and we have $E_{\varphi_{1}}\odot
E_{\varphi_{2}}\hookrightarrow E_{\varphi}$, then
$\varphi_{1}^{-1}\varphi_{2}^{-1}\prec\varphi^{-1}$ for large arguments.
* $(c)$
If a Banach ideal space $E$ has the Fatou property, ${\rm
supp}E_{a}=\Omega,L^{\infty}\not\hookrightarrow E$ and $E_{\varphi_{1}}\odot
E_{\varphi_{2}}\hookrightarrow E_{\varphi}$, then
$\varphi_{1}^{-1}\varphi_{2}^{-1}\prec\varphi^{-1}$ for small arguments.
* $(d)$
If $e$ is a Banach sequence space with the Fatou property,
$\sup_{k\in\mathbb{N}}\|e_{k}\|_{e}<\infty,l^{\infty}\not\hookrightarrow e$
and $e_{\varphi_{1}}\odot e_{\varphi_{2}}\hookrightarrow e_{\varphi}$, then
$\varphi_{1}^{-1}\varphi_{2}^{-1}\prec\varphi^{-1}$ for small arguments.
Note that in the case (c) we can even conclude the relation
$\varphi_{1}^{-1}\varphi_{2}^{-1}\prec\varphi^{-1}$ for all arguments, using
(b) and (c).
The sufficient and necessary conditions on the reverse inclusion
$E_{\varphi}\hookrightarrow E_{\varphi_{1}}\odot E_{\varphi_{2}}$ need also
the reverse relations between Young functions, the same as in [KLM12].
The symbol $\varphi^{-1}\prec\varphi_{1}^{-1}\varphi_{2}^{-1}$ for all
arguments [for large arguments] (for small arguments) means that there is a
constant $D>0$ [there are constants $D,u_{0}>0$] (there are constants
$D,u_{0}>0$) such that the inequality
$\varphi^{-1}(u)\leq D\,\varphi_{1}^{-1}(u)\varphi_{2}^{-1}(u)$ (15)
holds for all $u>0$ [for all $u\geq u_{0}$] (for all $0<u\leq u_{0}$),
respectively.
Theorem 5. Let $\varphi_{1},\varphi_{2}$ and $\varphi$ be three Young
functions.
* $(a)$
If $E$ is a Banach ideal space with the Fatou property and one of the
following conditions holds:
$(a1)$ $\varphi^{-1}\prec\varphi_{1}^{-1}\varphi_{2}^{-1}$ for all arguments,
$(a2)$ $\varphi^{-1}\prec\varphi_{1}^{-1}\varphi_{2}^{-1}$ for large arguments
and $L^{\infty}\hookrightarrow E$,
$(a3)$ $\varphi^{-1}\prec\varphi_{1}^{-1}\varphi_{2}^{-1}$ for small arguments
and $E\hookrightarrow L^{\infty}$,
then $E_{\varphi}\hookrightarrow E_{\varphi_{1}}\odot E_{\varphi_{2}}$.
* $(b)$
If $E$ is a symmetric Banach function space on $I$ with the Fatou property,
$E_{a}\neq\\{0\\}$ and $E_{\varphi}\hookrightarrow E_{\varphi_{1}}\odot
E_{\varphi_{2}}$, then $\varphi^{-1}\prec\varphi_{1}^{-1}\varphi_{2}^{-1}$ for
large arguments.
* $(c)$
If $E$ is a symmetric Banach function space on $I$ with the Fatou property,
${\it supp}E_{a}=\Omega,L^{\infty}\not\hookrightarrow E$ and
$E_{\varphi}\hookrightarrow E_{\varphi_{1}}\odot E_{\varphi_{2}}$, then
$\varphi^{-1}\prec\varphi_{1}^{-1}\varphi_{2}^{-1}$ for small arguments.
* $(d)$
Let $e$ be a symmetric Banach sequence space with the Fatou property and order
continuous norm. If $e_{\varphi}\hookrightarrow e_{\varphi_{1}}\odot
e_{\varphi_{2}}$, then $\varphi^{-1}\prec\varphi_{1}^{-1}\varphi_{2}^{-1}$ for
small arguments.
###### Proof.
$(a1)$ The idea of the proof is taken from [Ma89], Theorem 10.1(b). For $z\in
E_{\varphi}\backslash\left\\{0\right\\}$ let
$y=\varphi(\frac{|z|}{\|z\|_{E_{\varphi}}})$ and
$z_{i}(t)=\left\\{\begin{array}[]{ccc}\sqrt{\frac{|z(t)|}{\varphi_{1}^{-1}(y(t))\,\varphi_{2}^{-1}(y(t))}}\,\varphi_{i}^{-1}(y(t)),&\text{if}&t\in${\it
supp} z$,\\\ 0,&&\text{otherwise,}\end{array}\right.$
for $i=1,2$. The elements $z_{i}$ are well defined. Indeed, if
$a_{\varphi}=0$, then $y(t)>0$ for $\mu$-a.e. $t\in{\it supp}z$. If
$a_{\varphi}>0$, then assumption on functions implies that $a_{\varphi_{1}}>0$
and $a_{\varphi_{2}}>0$. Consequently, $\varphi_{1}^{-1}(0)=a_{\varphi_{1}}$
and $\varphi_{2}^{-1}(0)=a_{\varphi_{2}}$. Now we will prove the inequality
$\varphi_{i}(\frac{z_{i}}{\sqrt{D\|z\|_{E_{\varphi}}}})\leq y,~{}i=1,2.$ (16)
If $a_{\varphi}>0$, taking $u\rightarrow 0$ in inequality (15) we obtain
$a_{\varphi}\leq Da_{\varphi_{1}}a_{\varphi_{2}}.$ If $y(t)=0$, then
$z_{i}(t)=\sqrt{\frac{|z(t)|}{a_{\varphi_{1}}\,a_{\varphi_{2}}}}\,\varphi_{i}^{-1}(0)\leq\sqrt{\frac{\|z\|_{E_{\varphi}}a_{\varphi}}{a_{\varphi_{1}}a_{\varphi_{2}}}}\,\varphi_{i}^{-1}(0)\leq\sqrt{D\|z\|_{E_{\varphi}}}\,\varphi_{i}^{-1}(0)$
and consequently
$\varphi_{i}(\frac{z_{i}(t)}{\sqrt{D\|z\|_{E_{\varphi}}}})=0=y(t).$ If
$y\left(t\right)>0$, then
$z_{i}(t)=\sqrt{\frac{|z(t)|}{\varphi_{1}^{-1}(y(t))\varphi_{2}^{-1}(y(t))}}\,\varphi_{i}^{-1}(y(t))\leq\sqrt{\frac{D|z(t)|}{\varphi^{-1}(y(t))}}\,\varphi_{i}^{-1}(y(t))=\sqrt{D\|z\|_{\varphi}}\,\varphi_{i}^{-1}(y(t)).$
This proves (16) and consequently we obtain
$I_{\varphi_{1}}(\frac{z_{1}}{\sqrt{D\|z\|_{E_{\varphi}}}})\leq\|y\|_{E}=\|\varphi(\frac{|z|}{\|z\|_{E_{\varphi}}})\|_{E}\leq
1.$
Thus $\|z_{1}\|_{E_{\varphi_{1}}}\leq\sqrt{D\|z\|_{E_{\varphi}}}$ and
similarly $\|z_{2}\|_{E_{\varphi_{2}}}\leq\sqrt{D\|z\|_{E_{\varphi}}}$. Since
$|z|=z_{1}z_{2}$ it follows that $z\in E_{\varphi_{1}}\odot E_{\varphi_{2}}$
and $\|z\|_{E_{\varphi_{1}}\odot E_{\varphi_{2}}}\leq D\|z\|_{E_{\varphi}}$.
$(a2)$ If $b_{\varphi}<\infty$ and $L^{\infty}\hookrightarrow E$, then
$E_{\varphi}=L^{\infty}$ with equivalent norms and clearly
$E_{\varphi}=L^{\infty}\hookrightarrow E_{\varphi_{1}}\odot E_{\varphi_{2}}$.
Suppose $b_{\varphi}=\infty$. Set $v_{0}=\varphi^{-1}(u_{0})$, where $u_{0}$
is from (15) and let $v>0$ be such that
$\max[\varphi_{1}(v),\varphi_{2}(v)]\,\|\chi_{\Omega}\|_{E}\leq 1/2$. For
$\|z\|_{E_{\varphi}}=1$ let $y=\varphi(|z|)$ and
$A=\\{t\in{\it supp}z:|z(t)|\geq v_{0}\\},\,B={\it supp}z\backslash
A=\\{t\in{\it supp}z:|z(t)|<v_{0}\\},$
Define
$z_{i}(t)=\left\\{\begin{array}[]{ccc}\sqrt{\frac{|z(t)|}{\varphi_{1}^{-1}(y(t))\varphi_{2}^{-1}(y(t)}}\,\varphi_{i}^{-1}(y(t)),&\text{if}&t\in
A,\\\ \sqrt{|z(t)|},&&t\in B,\\\ 0,&&\text{otherwise},\end{array}\right.$
for $i=1,2$. Since $\varphi(v_{0})>0$ the functions $z_{i}$ are well defined.
If $t\in A$, then
$z_{i}(t)=\sqrt{\frac{|z(t)|}{\varphi_{1}^{-1}(y(t))\varphi_{2}^{-1}(y(t))}}\,\varphi_{i}^{-1}(y(t))\leq\sqrt{\frac{D|z(t)|}{\varphi^{-1}(y(t))}}\,\varphi_{i}^{-1}(y(t))\leq\sqrt{D}\,\varphi_{i}^{-1}(y(t)),$
whence
$I_{\varphi_{1}}(\frac{z_{1}}{2\sqrt{D}}\chi_{A})\leq\frac{1}{2}I_{\varphi_{1}}(\frac{z_{1}}{\sqrt{D}}\chi_{A})\leq\frac{1}{2}\|y\|_{E}\leq\frac{1}{2},$
and
$I_{\varphi_{1}}(\frac{vz_{1}}{\sqrt{v_{0}}}\chi_{B})=\|\varphi_{1}(\frac{vz_{1}}{\sqrt{v_{0}}}\chi_{B})\|_{E}\leq\varphi_{1}(v)\|\chi_{\Omega}\|_{E}\leq\frac{1}{2}.$
Then, for $\lambda=\max\\{\frac{\sqrt{v_{0}}}{v},2\sqrt{D}\\}$, we obtain
$I_{\varphi_{1}}(\frac{z_{1}}{\lambda})\leq
I_{\varphi_{1}}(\frac{z_{1}}{\lambda}\,\chi_{A})+I_{\varphi_{1}}(\frac{z_{1}}{\lambda}\,\chi_{B})\leq
I_{\varphi_{1}}(\frac{z_{1}}{2\sqrt{D}}\chi_{A})+I_{\varphi_{1}}(\frac{vz_{1}}{\sqrt{v_{0}}}\chi_{B})\leq
1.$
Thus $\|z_{1}\|_{E_{\varphi_{1}}}\leq\lambda$ and similarly
$\|z_{2}\|_{E_{\varphi_{2}}}\leq\lambda$. Since $|z|=z_{1}\,z_{2}$ it follows
that $z\in E_{\varphi_{1}}\odot E_{\varphi_{2}}$ and
$\|z\|_{E_{\varphi_{1}}\odot E_{\varphi_{2}}}\leq\lambda^{2}$. Consequently
$\|z\|_{E_{\varphi_{1}}\odot
E_{\varphi_{2}}}\leq\lambda^{2}\,\|z\|_{E_{\varphi}}$ for each $z\in
E_{\varphi}$.
$(a3)$ Since $\varphi^{-1}\prec\varphi_{1}^{-1}\varphi_{2}^{-1}$ for small
arguments it follows that for any $u_{1}>u_{0}$ there is a constant $D_{1}\geq
D$ such that
$\varphi^{-1}(u)\leq D_{1}\varphi_{1}^{-1}(u)\varphi_{2}^{-1}(u)$ (17)
for any $u\leq u_{1}$. We follow the same way as in the proof of $(a1)$
replacing $D$ by $D_{1}$ from (17) for $u_{1}=M,$ where $M$ is the constant of
the inclusion $E\overset{M}{\hookrightarrow}L^{\infty}$.
$(b)$ Suppose that condition
$\varphi^{-1}\prec\varphi_{1}^{-1}\varphi_{2}^{-1}$ for large arguments is not
satisfied. Then there is a sequence $(u_{n})$ with $u_{n}\nearrow\infty$ such
that
$2^{n}\varphi_{1}^{-1}(u_{n})\varphi_{2}^{-1}(u_{n})\leq\varphi^{-1}(u_{n})$
for all $n\in\mathbb{N}$.
We repeat a construction of the sequence $(z_{n})$, as it was given in [KLM12]
in the proof of Theorem 4.2(i), showing that
$\frac{\|z_{n}\|_{E_{\varphi_{1}}\odot
E_{\varphi_{2}}}}{\|z_{n}\|_{E_{\varphi}}}\rightarrow\infty$ as
$n\rightarrow\infty$.
Since $E_{a}\not=\\{0\\}$ it follows that there is a nonzero $0\leq x\in
E_{a}$ and so there is a set $A$ of positive measure such that $\chi_{A}\in
E_{a}$. Of course, for large enough $n$ one has $\|u_{n}\chi_{A}\|_{E}\geq 1$.
Applying Dobrakov’s result from [Do74] we conclude that the submeasure
$\omega(B)=\|u_{n}\chi_{B}\|_{E}$ has the Darboux property. Consequently, for
each $n\in\mathbb{N}$ there exists a set $A_{n}$ such that
$\|u_{n}\chi_{A_{n}}\|_{E}=1$. Define
$x_{n}=\varphi_{1}^{-1}(u_{n})\chi_{A_{n}},~{}y_{n}=\varphi_{2}^{-1}(u_{n})\chi_{A_{n}}\text{
and }z_{n}=x_{n}\,y_{n}.$
Let us consider two cases:
$1^{0}$. Let either $b_{\varphi_{1}}=\infty$ or $b_{\varphi_{1}}<\infty$ and
$\varphi_{1}(b_{\varphi_{1}})=\infty$. Then
$\varphi_{1}(\varphi_{1}^{-1}(u))=u$ for $u\geq 0$ and for $0<\lambda<1$, by
the convexity of $\varphi_{1}$, we obtain
$I_{\varphi_{1}}(\frac{x_{n}}{\lambda})=\|\varphi_{1}(\frac{\varphi_{1}^{-1}(u_{n})}{\lambda})\chi_{A_{n}}\|_{E}\geq\frac{1}{\lambda}\|\varphi_{1}(\varphi_{1}^{-1}(u_{n}))\chi_{A_{n}}\|_{E}=\frac{1}{\lambda}u_{n}\|\chi_{A_{n}}\|_{E}>1.$
$2^{0}$. Let $b_{\varphi_{1}}<\infty$ and
$\varphi_{1}(b_{\varphi_{1}})<\infty$. Then, for sufficiently large $n$ and
$0<\lambda<1$, we have $I_{\varphi_{1}}(\frac{x_{n}}{\lambda})=\infty$.
In both cases $\|x_{n}\|_{E_{\varphi_{1}}}\geq 1$ and similarly
$\|y_{n}\|_{E_{\varphi_{2}}}\geq 1$. Applying Theorem 2 we get
$\displaystyle\|z_{n}\|_{E_{\varphi_{1}}\odot E_{\varphi_{2}}}$
$\displaystyle=$
$\displaystyle\varphi_{1}^{-1}(u_{n})\varphi_{2}^{-1}(u_{n})\,f_{E_{\varphi_{1}}\odot
E_{\varphi_{2}}}(m(A_{n}))$ $\displaystyle=$
$\displaystyle\varphi_{1}^{-1}(u_{n})\varphi_{2}^{-1}(u_{n})\,f_{E_{\varphi_{1}}}(m(A_{n}))f_{E_{\varphi_{2}}}(m(A_{n}))$
$\displaystyle=$
$\displaystyle\|x_{n}\|_{E_{\varphi_{1}}}\|y_{n}\|_{E_{\varphi_{2}}}\geq 1.$
On the other hand, using the relation between functions on a sequence
$(u_{n})$ and the fact that $\varphi(\varphi^{-1}(u))\leq u$ for $u>0$ we
obtain
$I_{\varphi}(2^{n}z_{n})=\|\varphi(2^{n}\varphi_{1}^{-1}(u_{n})\varphi_{2}^{-1}(u_{n}))\chi_{A_{n}}\|_{E}\leq\|\varphi(\varphi^{-1}(u_{n}))\chi_{A_{n}}\|_{E}\leq\|u_{n}\,\chi_{A_{n}}\|_{E}=1,$
i.e., $\|z_{n}\|_{E_{\varphi}}\leq 1/2^{n}$ which gives
$\frac{\|z_{n}\|_{E_{\varphi_{1}}\odot
E_{\varphi_{2}}}}{\|z_{n}\|_{E_{\varphi}}}\geq 2^{n}\rightarrow\infty$ as
$n\rightarrow\infty$.
$(c)$ It can be done by combining methods from the proof of Theorem 5(b) and
Theorem A(c).
$(d)$ Suppose that condition
$\varphi^{-1}\prec\varphi_{1}^{-1}\varphi_{2}^{-1}$ for small arguments is not
satisfied. Then we can find a sequence $u_{n}\rightarrow 0$ such that
$2^{n}\varphi_{1}^{-1}(u_{n})\varphi_{2}^{-1}(u_{n})\leq\varphi^{-1}(u_{n})$
for all $n\in\mathbb{N}$.
Assumptions on a sequence space $e$ gives
$\lim_{m\rightarrow\infty}\|\sum_{k=0}^{m}e_{k}\|_{e}=\infty$. Thus, for each
$n\in\mathbb{N}$ there is a number $m_{n}$ such that
$u_{n}\,\|\sum_{k=0}^{m_{n}}e_{k}\|_{e}\leq
1<u_{n}\,\|\sum_{k=0}^{m_{n}+1}e_{k}\|_{e}.$
By symmetry of $e$, $\sup_{k\in\mathbb{N}}\|e_{k}\|_{e}=\|e_{1}\|_{e}=M$.
Therefore $u_{n}\,\|\sum_{k=1}^{m_{n}}e_{k}\|_{e}\rightarrow 1$ as
$n\rightarrow\infty$. Put
$x_{n}=\varphi_{1}^{-1}(u_{n})\,\sum_{k=1}^{m_{n}}e_{k},~{}y_{n}=\varphi_{2}^{-1}(u_{n})\,\sum_{k=1}^{m_{n}}e_{k}~{}{\rm
and}~{}z_{n}=x_{n}\,y_{n}.$
Then $I_{\varphi_{1}}(x_{n})\leq u_{n}\,\|\sum_{k=1}^{m_{n}}e_{k}\|_{e}\leq 1$
and
$I_{\varphi_{1}}(2x_{n})=\varphi_{1}(2\varphi_{1}^{-1}(u_{n}))\,\|\sum_{k=1}^{m_{n}}e_{k}\|_{e}\geq
2u_{n}\,\|\sum_{k=1}^{m_{n}}e_{k}\|_{e}\rightarrow 2\text{ as
}n\rightarrow\infty.$
Therefore, for $n$ large enough $1\geq\|x_{n}\|_{e_{\varphi_{1}}}\geq 1/2$ as
well as $1\geq\|y_{n}\|_{e_{\varphi_{2}}}\geq 1/2$. Consequently, explaining
like in (b) one has $\|z_{n}\|_{E_{\varphi_{1}}\odot E_{\varphi_{2}}}\geq 1/4$
and $I_{\varphi}(2^{n}z_{n})\leq 1$, which gives
$\frac{\|z_{n}\|_{E_{\varphi_{1}}\odot
E_{\varphi_{2}}}}{\|z_{n}\|_{E_{\varphi}}}\geq 2^{n-2}\rightarrow\infty$ as
$n\rightarrow\infty$ and the proof of Theorem 5 is complete. ∎
To formulate results on equality of product spaces we need to introduce
equivalences of inverses of Young functions $\varphi_{1},\varphi_{2}$ and
$\varphi$. The symbol $\varphi_{1}^{-1}\varphi_{2}^{-1}\approx\varphi^{-1}$
for all arguments [for large arguments] (for small arguments) means that
$\varphi_{1}^{-1}\varphi_{2}^{-1}\prec\varphi^{-1}$ and
$\varphi^{-1}\prec\varphi_{1}^{-1}\varphi_{2}^{-1},$ that is provided there
are constants $C,D>0$ [there are constants $C,D,u_{0}>0$] (there are constants
$C,D,u_{0}>0$) such that the inequalities
$C\varphi_{1}^{-1}(u)\varphi_{2}^{-1}(u)\leq\varphi^{-1}(u)\leq
D\varphi_{1}^{-1}(u)\varphi_{2}^{-1}(u)$ (18)
hold for all $u>0$ [for all $u\geq u_{0}$] (for all $0<u\leq u_{0}$),
respectively.
From the above Theorem A and Theorem 5 we obtain immediately results on the
product of Calderón-Lozanovskiĭ $E_{\varphi}$-spaces which are generalizations
of the results known for Orlicz spaces.
Corollary 6. Let $\varphi_{1},\varphi_{2}$ and $\varphi$ be three Young
functions.
* $(a)$
Suppose that $E$ is a symmetric Banach function space with the Fatou property,
$L^{\infty}\not\hookrightarrow E$ and ${\it supp}E_{a}=\Omega$. Then
$E_{\varphi_{1}}\odot E_{\varphi_{2}}=E_{\varphi}$ if and only if
$\varphi_{1}^{-1}\varphi_{2}^{-1}\approx\varphi^{-1}$ for all arguments.
* $(b)$
Suppose that $E$ is a symmetric Banach function space with the Fatou property,
$L^{\infty}\hookrightarrow E$ and $E_{a}\neq\\{0\\}$. Then
$E_{\varphi_{1}}\odot E_{\varphi_{2}}=E_{\varphi}$ if and only if
$\varphi_{1}^{-1}\varphi_{2}^{-1}\approx\varphi^{-1}$ for large arguments.
* $(c)$
Suppose that $e$ is a symmetric Banach sequence space with the Fatou property
and order continuous norm. Then $e_{\varphi_{1}}\odot
e_{\varphi_{2}}=e_{\varphi}$ if and only if
$\varphi_{1}^{-1}\varphi_{2}^{-1}\approx\varphi^{-1}$ for small arguments.
The following construction appeared in [ZR67] and in [DK67]: for two Young
functions $\varphi_{1},\varphi_{2}$ (or even for only the so-called
$\varphi$-functions) one can define a new function
$\varphi_{1}\oplus\varphi_{2}$ by the formula
$(\varphi_{1}\oplus\varphi_{2})(u)=\inf_{u=vw}[\varphi_{1}(v)+\varphi_{2}(w)]=\inf_{v>0}\,[\varphi_{1}(v)+\varphi_{2}(\frac{u}{v})],$
(19)
for $u\geq 0$. This operation was investigated in [ZR67], [BO78], [Ma89],
[MP89] and [St92]. Note that $\varphi_{1}\oplus\varphi_{2}$ is non-decreasing,
left-continous function and is 0 at $u=0$. Moreover, the proof of the
following estimates can be found in [ZR67, pp. 267, 271] and [St92, Theorem
1]:
$\varphi^{-1}(t)\leq\varphi_{1}^{-1}(t)\varphi_{2}^{-1}(t)\leq\varphi^{-1}(2t)~{}~{}{\rm
for~{}any}~{}t>0,$
where $\varphi_{1},\varphi_{2}$ are nondegenerate Orlicz functions (the proof
in general case is not difficult) $\varphi$-functions and
$\varphi=\varphi_{1}\oplus\varphi_{2}$. The function $\varphi$ need not be
convex even if both $\varphi_{1}$ and $\varphi_{2}$ are convex functions.
However, if $\varphi$ is a convex function, then
$\varphi^{-1}\leq\varphi_{1}^{-1}\varphi_{2}^{-1}\leq 2\varphi^{-1}$ and, by
Theorem A(a1) and 5(a1), we obtain $E_{\varphi_{1}}\odot
E_{\varphi_{2}}=E_{\varphi}$.
We will prove the last result without explicit assumption that $\varphi$ is
convex, but to do this we need to extend definition of the Calderón-
Lozanovskiĭ $E_{\varphi}$-space in this case (cf. [KMP03] for definition and
some results).
For a non-decreasing and left-continous function
$\varphi:[0,\infty)\rightarrow[0,\infty)$ with $\varphi(0)=0$ assume that
there exist $C,\alpha>0$ such that
$\varphi(st)\leq C\,t^{\alpha}\,\varphi(s)~{}~{}{\rm for~{}all}~{}s>0~{}{\rm
and}~{}0<t<1.$ (20)
Then the Calderón-Lozanovskiĭ $E_{\varphi}$-space is a quasi-Banach ideal
space (since $\varphi$ need not be convex) with the quasi-norm
$\|x\|_{E_{\varphi}}=\inf\\{\lambda>0:\|\varphi(|x|/\lambda)\|_{E}\leq 1\\}.$
Note that for a convex function $\varphi$ the condition (20) holds with
$C=\alpha=1$ and the space $E_{\varphi}$ is normable when the condition (20)
holds with $\alpha\geq 1$.
We know that if $\varphi_{1}$ and $\varphi_{2}$ are convex functions, then
$\varphi=\varphi_{1}\oplus\varphi_{2}$ is not necessary a convex function but
the condition (20) holds with $C=1$ and $\alpha=1/2$ and the space
$E_{\varphi}$ is a quasi-Banach ideal space. In fact, for any $s>0$ and
$0<t<1$ we have
$\displaystyle\varphi(st)$ $\displaystyle=$
$\displaystyle\inf_{st=vw}[\varphi_{1}(v)+\varphi_{2}(w)]=\inf_{s=ab}[\varphi_{1}(\sqrt{t}\,a)+\varphi_{2}(\sqrt{t}\,b)]$
$\displaystyle\leq$
$\displaystyle\sqrt{t}\,\inf_{s=ab}[\varphi_{1}(a)+\varphi_{2}(b)]=\sqrt{t}\,\varphi(s).$
Theorem 6. Let $\varphi_{1},\varphi_{2}$ be Young functions. If
$\varphi:=\varphi_{1}\oplus\varphi_{2}$, then $E_{\varphi_{1}}\odot
E_{\varphi_{2}}=E_{\varphi}$, where $E$ is a Banach ideal space with the Fatou
property.
###### Proof.
We prove that $E_{\varphi_{1}}\odot E_{\varphi_{2}}\hookrightarrow
E_{\varphi}$. By definition of $\varphi_{1}\oplus\varphi_{2}$ one has
$\varphi(uv)\leq\varphi_{1}(u)+\varphi_{2}(v)$
for each $u,v>0$ with $\varphi_{1}(u)<\infty,\varphi_{2}(v)<\infty$. Let $z\in
E_{\varphi_{1}}\odot E_{\varphi_{2}},z\neq 0$, and take arbitrary $0\leq x\in
E_{\varphi_{1}},0\leq y\in E_{\varphi_{2}}$ with $|z|=xy$. Since the condition
(20) holds with $C=1$ and $\alpha=1/2$ it follows that for $0<t<1/4$ we obtain
$\displaystyle
I_{\varphi}(\frac{t\,z}{\|x\|_{E_{\varphi_{1}}}\|y\|_{E_{\varphi_{2}}}})$
$\displaystyle\leq$
$\displaystyle\sqrt{t}\,\|\varphi(\frac{x}{\|x\|_{E_{\varphi_{1}}}}\frac{y}{\|y\|_{E_{\varphi_{2}}}})\|_{E}$
$\displaystyle\leq$
$\displaystyle\sqrt{t}\,\left[\|\varphi_{1}(\frac{x}{\|x\|_{E_{\varphi_{1}}}})\|_{E}+\|\varphi_{2}(\frac{y}{\|y\|_{E_{\varphi_{2}}}})\|_{E}\right]$
$\displaystyle\leq$ $\displaystyle 2\,\sqrt{t}<1.$
Thus
$\|z\|_{E_{\varphi}}\leq\frac{1}{t}\,\|x\|_{E_{\varphi_{1}}}\|y\|_{E_{\varphi_{2}}}$
and consequently
$\|z\|_{E_{\varphi}}\leq\frac{1}{t}\|z\|_{E_{\varphi_{1}}\odot
E_{\varphi_{2}}}$. Thus $E_{\varphi_{1}}\odot
E_{\varphi_{2}}\overset{1/t}{\hookrightarrow}E_{\varphi}$.
The proof of the imebedding $E_{\varphi}\hookrightarrow E_{\varphi_{1}}\odot
E_{\varphi_{2}}$ is exactly the same as the proof of Theorem 5(a1) since the
convexity of $\varphi$ has not been used there, which proves the theorem. ∎
5\. The product of Lorentz and Marcinkiewicz spaces
Before proving results on the product of Lorentz and Marcinkiewicz spaces on
$I=(0,1)$ or $I=(0,\infty)$ we need some auxiliary lemmas on the Calderón
construction and notion of the dilation operator.
The dilation operator $D_{s},s>0,$ defined by
$D_{s}x(t)=x(t/s)\chi_{I}(t/s),t\in I$ is bounded in any symmetric space $E$
on ${\rm I}$ and $\|D_{s}\|_{E\rightarrow E}\leq\max(1,s)$ (see [Sh68, Lemma
1] in the case $I=(0,1)$, [KPS82, pp. 96-98] for $I=(0,\infty)$ and [LT79, p.
130] for both cases). Moreover, the Boyd indices of $E$ are defined by
$\alpha_{E}=\lim_{s\rightarrow 0^{+}}\frac{\ln\|D_{s}\|_{E\rightarrow E}}{\ln
s},\beta_{E}=\lim_{s\rightarrow\infty}\frac{\ln\|D_{s}\|_{E\rightarrow E}}{\ln
s},$
and we have $0\leq\alpha_{E}\leq\beta_{E}\leq 1$.
Lemma 3. Let $E,F$ be symmetric function spaces on $I$ and $0<\theta<1$. Then
$\frac{1}{2}\,\|z^{*}\|_{E^{\theta}F^{1-\theta}}^{*}\leq\|z^{*}\|_{E^{\theta}F^{1-\theta}}\leq\|z^{*}\|_{E^{\theta}F^{1-\theta}}^{*},$
where
$\|z^{*}\|_{E^{\theta}F^{1-\theta}}^{*}:=\inf\\{\max(\|x^{*}\|_{E},\|y^{*}\|_{F}):z^{*}\leq(x^{*})^{\theta}\,(y^{*})^{1-\theta},x\in
E_{+},y\in F_{+}\\}.$
###### Proof.
Since
$\|x^{*}\|_{E}=\|D_{2}D_{\frac{1}{2}}x^{*}\|_{E}\leq\|D_{2}\|_{E\rightarrow
E}\,\|D_{\frac{1}{2}}x^{*}\|_{E}\leq 2\,\|D_{\frac{1}{2}}x^{*}\|_{E}.$
and $(|x|^{\theta}\,|y|^{1-\theta})^{*}(t)\leq
x^{*}(t/2)^{\theta}\,y^{*}(t/2)^{1-\theta}$ for any $t\in I$ (cf. [KPS82, p.
67]) it follows that
$\displaystyle\|z^{*}\|_{E^{\theta}F^{1-\theta}}$ $\displaystyle=$
$\displaystyle\inf\\{\max(\|x\|_{E},\|y\|_{F}):z^{*}\leq
x^{\theta}\,y^{1-\theta},x\in E_{+},y\in F_{+}\\}$ $\displaystyle\geq$
$\displaystyle\inf\\{\max(\|x\|_{E},\|y\|_{F}):z^{*}(t)\leq
x^{*}(t/2)^{\theta}\,y^{*}(t/2)^{1-\theta},x\in E_{+},y\in F_{+}\\}$
$\displaystyle=$
$\displaystyle\inf\\{\max(\|D_{\frac{1}{2}}x^{*}\|_{E},\|D_{\frac{1}{2}}y^{*}\|_{F}):z^{*}(t)\leq
x^{*}(t)^{\theta}\,y^{*}(t)^{1-\theta},x\in E_{+},y\in F_{+}\\}$
$\displaystyle\geq$
$\displaystyle\frac{1}{2}\,\inf\\{\max(\|x^{*}\|_{E},\|y^{*}\|_{F}):z^{*}(t)\leq
x^{*}(t)^{\theta}\,y^{*}(t)^{1-\theta},x\in E_{+},y\in F_{+}\\}$
$\displaystyle=$
$\displaystyle\frac{1}{2}\,\|z^{*}\|_{E^{\theta}F^{1-\theta}}^{*}.$
The other estimate is clear and the lemma follows. ∎
As a consequence of representation (8) and the above lemma with $\theta=1/2$
we obtain
Corollary 7. Let $E,F$ be symmetric function spaces on $I$. Then
$\|z^{*}\|_{E\odot F}\leq\inf\\{\|x\|_{E}\,\|y\|_{F}:z^{*}\leq
x^{*}\,y^{*},x\in E_{+},y\in F_{+}\\}\leq 2\,\|z^{*}\|_{E\odot F}.$
The idea of the proof of the next result is coming from Calderón [Ca64, Part
13.5]. For a Banach function space $E$ on $I=(0,1)$ or $(0,\infty)$ define new
spaces (symmetrizations of $E$) $E^{(*)}$ and $E^{(**)}$ as
$E^{(*)}=\\{x\in L^{0}(I):x^{*}\in E\\},~{}~{}E^{(**)}=\\{x\in
L^{0}(I):x^{**}\in E\\}$
with the functionals $\|x\|_{E^{(*)}}=\|x^{*}\|_{E}$ and
$\|x\|_{E^{(**)}}=\|x^{**}\|_{E}$. If $C_{E}$ denotes the smallest constant
$1\leq C<\infty$ such that
$\|D_{2}x^{*}\|_{E}\leq C\,\|x^{*}\|_{E}~{}~{}{\rm for~{}all}~{}x^{*}\in E,$
(21)
then $E^{(*)}$ is a quasi-Banach symmetric space. The space $E^{(**)}$ is
always a Banach symmetric space. Consider the Hardy operator $H$ and its dual
$H^{*}$ defined by
$Hx(t)=\frac{1}{t}\int_{0}^{t}x(s)\,ds,~{}H^{*}x(t)=\int_{t}^{l}\frac{x(s)}{s}\,ds~{}{\rm
with}~{}l=m(I),~{}t\in I.$ (22)
Remark 7. If $E$ is a Banach function space on $I$ and operator $H$ is bounded
in $E$, then (21) holds with $C_{E}\leq 2\,\|H\|_{E\rightarrow E}$. This
follows directly from the estimates
$\|Hx^{*}\|_{E}=\|\int_{0}^{1}x^{*}(st)\,ds\|_{E}\geq\|\int_{0}^{1/2}x^{*}(st)\,ds\|_{E}\geq\frac{1}{2}\,\|x^{*}(t/2)\|_{E}.$
As we already mentioned before the Calderón spaces $E^{\theta}F^{1-\theta}$
can be also defined for quasi-Banach spaces $E,F$ (cf. [Ov82], [Ni85],
[KMP03]).
Lemma 4. Let $E$ and $F$ be Banach function spaces on $I$ and $0<\theta<1$.
Suppose that both operators $H,H^{*}$ are bounded in $E$ and $F$. Then
$(E^{(*)})^{\theta}(F^{(*)})^{1-\theta}\overset{C_{1}}{\hookrightarrow}(E^{\theta}F^{1-\theta})^{(*)}\overset{C_{2}}{\hookrightarrow}(E^{(*)})^{\theta}(F^{(*)})^{1-\theta},$
(23)
where $C_{1}=C_{E}^{\theta}C_{F}^{1-\theta},C_{2}=\|HH^{*}\|_{E\rightarrow
E}^{\theta}\|HH^{*}\|_{F\rightarrow F}^{1-\theta}$ and
$(E^{(**)})^{\theta}(F^{(**)})^{1-\theta}\overset{1}{\hookrightarrow}(E^{\theta}F^{1-\theta})^{(**)}\overset{C_{3}}{\hookrightarrow}(E^{(**)})^{\theta}(F^{(**)})^{1-\theta},$
(24)
where $C_{3}=[\|H\|_{E\rightarrow E}\,\|HHH^{*}\|_{E\rightarrow
E}]^{\theta}\,[\|H\|_{F\rightarrow F}\,\|HHH^{*}\|_{F\rightarrow
F}]^{1-\theta}.$
###### Proof.
Embeddings (23). Let $z\in(E^{(*)})^{\theta}(F^{(*)})^{1-\theta}$. Then
$|z|\leq\lambda|x|^{\theta}|y|^{1-\theta}$ for some $\lambda>0$ and
$\|x^{*}\|_{E}\leq 1,\|y^{*}\|_{F}\leq 1$. Thus
$z^{*}(t)\leq\lambda(|x|^{\theta}|y|^{1-\theta})^{*}(t)\leq\lambda
x^{*}(t/2)^{\theta}y^{*}(t/2)^{1-\theta}=\lambda
C_{1}(\frac{x^{*}(t/2)}{C_{E}})^{\theta}(\frac{y^{*}(t/2)}{C_{F}})^{1-\theta}$
for any $t\in I$, which means that $z\in(E^{\theta}F^{1-\theta})^{(*)}$ with
the norm $\leq\lambda C_{1}$.
On the other hand, if $z\in(E^{\theta}F^{1-\theta})^{(*)}$, then $z^{*}\in
E^{\theta}F^{1-\theta}$ and so
$z^{*}\leq\lambda|x|^{\theta}|y|^{1-\theta}~{}{\rm
with~{}some}~{}\lambda>0,\|x\|_{E}\leq 1,\|y\|_{F}\leq 1.$
The following equality is true
$HH^{*}x(t)=Hx(t)+H^{*}x(t),~{}t\in I.$ (25)
In fact, using the Fubini theorem, we obtain for $x\geq 0$
$\displaystyle HH^{*}x(t)$ $\displaystyle=$
$\displaystyle\frac{1}{t}\int_{0}^{t}(\int_{s}^{l}\frac{x(r)}{r}\,dr)\,ds=\frac{1}{t}\int_{0}^{t}(\int_{0}^{r}ds)\,\frac{x(r)}{r}\,dr$
$\displaystyle+$
$\displaystyle\frac{1}{t}\int_{t}^{l}(\int_{0}^{t}ds)\,\frac{x(r)}{r}\,dr=Hx(t)+H^{*}x(t).$
Using then equality (25) and twice Hölder-Rogers inequality we obtain
$\displaystyle z^{*}$ $\displaystyle\leq$ $\displaystyle H(z^{*})\leq
H(z^{*})+H^{*}(z^{*})=HH^{*}(z^{*})$ $\displaystyle\leq$ $\displaystyle\lambda
HH^{*}(|x|^{\theta}|y|^{1-\theta})\leq\lambda
H[(H^{*}|x|)^{\theta}(H^{*}|y|)^{1-\theta}]$ $\displaystyle\leq$
$\displaystyle\lambda[HH^{*}(|x|)]^{\theta}\,[HH^{*}(|y|)]^{1-\theta}.$
By the Ryff theorem there exists a measure-preserving transformation
$\omega:I\rightarrow I$ such that $|z|=z^{*}(\omega)$ a.e. (cf. [BS88],
Theorem 7.5 for $I=(0,1)$ or Corollary 7.6 for $I=(0,\infty)$ under additional
assumption that $z^{*}(\infty)=0$). Thus
$|z|=z^{*}(\omega)\leq\lambda[HH^{*}(|x|)(\omega)]^{\theta}\,[HH^{*}(|y|)(\omega)]^{1-\theta}=\lambda
u^{\theta}v^{1-\theta}.$
Since $H^{*}|x|$ is non-increasing function it follows that $HH^{*}|x|$ is
also non-increasing function and
$HH^{*}|x|=[HH^{*}|x|]^{*}=[(HH^{*}|x|)(\omega)]^{*}$. Similarly with
$HH^{*}|y|$. Hence
$\displaystyle\|u\|_{E^{(*)}}$ $\displaystyle=$
$\displaystyle\|u^{*}\|_{E}=\|[(HH^{*}|x|)(\omega)]^{*}\|_{E}$
$\displaystyle=$ $\displaystyle\|HH^{*}|x|\|_{E}\leq\|HH^{*}\|_{E\rightarrow
E}\,\|x\|_{E}\leq\|HH^{*}\|_{E\rightarrow E}$
and
$\displaystyle\|v\|_{F^{(*)}}$ $\displaystyle=$
$\displaystyle\|v^{*}\|_{F}=\|[(HH^{*}|y|)(\omega)]^{*}\|_{F}$
$\displaystyle=$ $\displaystyle\|HH^{*}|y|\|_{F}\leq\|HH^{*}\|_{F\rightarrow
F}\,\|y\|_{F}\leq\|HH^{*}\|_{F\rightarrow F},$
which means that $z\in(E^{(*)})^{\theta}(F^{(*)})^{1-\theta}$ with the norm
$\leq\lambda C_{2}$.
To finish the proof in the case $I=(0,\infty)$ we need to show
$z^{*}(\infty)=0$. If we will have $z^{*}(\infty)=a>0$, then
$\lambda|x(t)|^{\theta}|y(t)|^{1-\theta}\geq a$ for almost all $t>0$ and
considering the sets $A=\\{t>0:|x(t)|\geq a/\lambda\\},~{}B=\\{t>0:|y(t)|\geq
a/\lambda\\}$ we obtain $A\cup B=(0,\infty)$ up to the set of measure zero.
Then
$H^{*}|x|(t)=\int_{t}^{\infty}\frac{|x(s)|}{s}\,ds\geq\int_{A\cap(t,\infty)}\frac{a}{\lambda
s}\,ds$
and
$H^{*}|y|(t)=\int_{t}^{\infty}\frac{|y(s)|}{s}\,ds\geq\int_{B\cap(t,\infty)}\frac{a}{\lambda
s}\,ds,$
which means $H^{*}|x|(t)+H^{*}|y|(t)=+\infty$ for all $t>0$. Since
$(0,\infty)=\\{t>0:H^{*}|x|(t)=\infty\\}\cup\\{t>0:H^{*}|y|(t)=\infty\\}$
(maybe except the set of measure zero) it follows that $H^{*}|x|\notin E$ or
$H^{*}|y|\notin F$, which is a contradiction.
Embeddings (24). Let $z\in(E^{(**)})^{\theta}(F^{(**)})^{1-\theta}$. Then
$|z|\leq\lambda|x|^{\theta}|y|^{1-\theta}$ for some $\lambda>0$ and
$\|x^{**}\|_{E}\leq 1,\|y^{**}\|_{F}\leq 1$. Thus
$z^{**}(t)\leq\lambda(|x|^{\theta}|y|^{1-\theta})^{**}(t)\leq\lambda
x^{**}(t)^{\theta}y^{**}(t)^{1-\theta}$
for any $t\in I$, which means that $z\in(E^{\theta}F^{1-\theta})^{(**)}$ with
the norm $\leq\lambda$.
On the other hand, if $z\in(E^{\theta}F^{1-\theta})^{(**)}$, then $z^{**}\in
E^{\theta}F^{1-\theta}$ and repeating the above arguments we obtain
$\displaystyle|z|$ $\displaystyle=$ $\displaystyle z^{*}(\omega)\leq
z^{**}(\omega)=Hz^{*}(\omega)$ $\displaystyle\leq$
$\displaystyle\lambda\,[HHH^{*}|x|(\omega)]^{\theta}\,[HHH^{*}(|y|)(\omega)]^{1-\theta}=\lambda\,u_{1}^{\theta}v_{1}^{1-\theta}.$
Since $HHH^{*}|x|=[HHH^{*}|x|]^{*}=[(HHH^{*}|x|)(\omega)]^{*}$ it follows that
$\displaystyle\|u_{1}\|_{E^{(**)}}$ $\displaystyle=$
$\displaystyle\|u_{1}^{**}\|_{E}=\|Hu_{1}^{*}\|_{E}\leq\|H\|_{E\rightarrow
E}\,\|u_{1}^{*}\|_{E}$ $\displaystyle=$ $\displaystyle\|H\|_{E\rightarrow
E}\,\|[(HHH^{*}|x|)(\omega)]^{*}\|_{E}=\|H\|_{E\rightarrow
E}\,\|HHH^{*}|x|\|_{E}$ $\displaystyle\leq$ $\displaystyle\|H\|_{E\rightarrow
E}\,\|HHH^{*}\|_{E\rightarrow E}\|x\|_{E}\leq\|H\|_{E\rightarrow
E}\,\|HHH^{*}\|_{E\rightarrow E}$
and
$\displaystyle\|v_{1}\|_{F^{(**)}}$ $\displaystyle=$
$\displaystyle\|v_{1}^{**}\|_{F}=\|Hv_{1}^{*}\|_{F}\leq\|H\|_{F\rightarrow
F}\,\|v_{1}^{*}\|_{F}$ $\displaystyle=$ $\displaystyle\|H\|_{F\rightarrow
F}\,\|[(HHH^{*}|y|)(\omega)]^{*}\|_{F}=\|H\|_{F\rightarrow
F}\,\|HHH^{*}|y|\|_{F}$ $\displaystyle\leq$ $\displaystyle\|H\|_{F\rightarrow
F}\,\|HHH^{*}\|_{F\rightarrow F}\|y\|_{F}\leq\|H\|_{F\rightarrow
F}\,\|HHH^{*}\|_{F\rightarrow F},$
which implies that $z\in(E^{(**)})^{\theta}(F^{(**)})^{1-\theta}$ with the
norm $\leq\lambda C_{3}$, and the lemma follows. ∎
Note that our proofs are working for both cases $I=(0,1)$ and $I=(0,\infty)$.
Our inclusions (23) were proved by Calderón but his result is true only in the
case when $I=(0,\infty)$ (cf. [Ca64], pp. 167-169). Since he was working with
the other composition $H^{*}H$, which in the case $I=(0,\infty)$ gives the
equality $H^{*}H=H+H^{*}$. For $I=(0,1)$ one gets another formula
$H^{*}Hx(t)=Hx(t)+H^{*}x(t)-\int_{0}^{1}x(s)\,ds$, which not allows then to
proof the same result in this case.
For the identification of product spaces we will need result on the Calderón
construction for weighted Lebesgue spaces $L^{p}(w)=\\{x\in L^{0}(\mu):xw\in
L^{p}(\mu)\\}$ with the norm $\|x\|_{L^{p}(w)}=\|xw\|_{L^{p}}$, where $1\leq
p\leq\infty$ and $w\geq 0$ . Result for $p_{0}=p_{1}$ was given in [Ov84, p.
459] and for general Banach ideal spaces in [KM03, Theorem 2] (for $1\leq
p_{0},p_{1}<\infty$ it also follows implicitely from [BL76, Theorem 5.5.3] and
results on relation between the complex method and the Calderón construction).
We present here a direct proof.
Lemma 5. Let $1\leq p_{0},p_{1}\leq\infty$ and $0<\theta<1$. Then
$L^{p_{0}}(w_{0})^{1-\theta}\,L^{p_{1}}(w_{1})^{\theta}\equiv L^{p}(w),$ (26)
where $\frac{1}{p}=\frac{1-\theta}{p_{0}}+\frac{\theta}{p_{1}}$ and
$w=w_{0}^{1-\theta}w_{1}^{\theta}$.
###### Proof.
Suppose $1\leq p_{0},p_{1}<\infty$. If $x\in
L^{p_{0}}(w_{0})^{1-\theta}\,L^{p_{1}}(w_{1})^{\theta}$, then
$|x|\leq\lambda|x_{0}|^{1-\theta}|x_{1}|^{\theta}$ with
$\|x_{0}\|_{L^{p_{0}}(w_{0})}\leq 1$ and $\|x_{1}\|_{L^{p_{1}}(w_{1})}\leq 1$.
Using the Hölder-Rogers inequality we obtain
$\displaystyle\int|x\,w|^{p}\,d\mu$ $\displaystyle\leq$
$\displaystyle\lambda^{p}\int|x_{0}w_{0}|^{(1-\theta)p}|x_{1}w_{1}|^{\theta
p}\,d\mu$ $\displaystyle\leq$
$\displaystyle\lambda^{p}(\int|x_{0}w_{0}|^{p_{0}}\,d\mu)^{(1-\theta)p/{p_{0}}}\,(\int|x_{1}w_{1}|^{p_{1}}\,d\mu)^{\theta
p/{p_{1}}}$ $\displaystyle=$
$\displaystyle\lambda^{p}\,\|x_{0}w_{0}\|_{L^{p_{0}}}^{(1-\theta)p}\,\|x_{1}w_{1}\|_{L^{p_{1}}}^{\theta
p},$
that is
$\|x\|_{L^{p}(w)}\leq\lambda\,\|x_{0}w_{0}\|_{L^{p_{0}}}^{1-\theta}\,\|x_{1}w_{1}\|_{L^{p_{1}}}^{\theta}\leq\lambda$
and so $x\in L^{p}(w)$ with the norm $\leq\lambda$.
On the other hand, if $0\neq x\in L^{p}(w)$ then, considering
$x_{i}(t)=\frac{|x(t)\,w(t)|^{p/{p_{i}}}}{\|x\|_{L^{p}(w)}^{p/{p_{i}}}}\frac{1}{w_{i}(t)}$
on the support of $w_{i}$ and $0$ otherwise ($i=0,1$), we obtain
$|x(t)|=\|x\|_{L^{p}(w)}\,x_{0}(t)^{1-\theta}\,x_{1}(t)^{\theta}$ and
$\|x_{i}\|_{L^{p_{i}}(w_{i})}^{p_{i}}=\int[x_{i}(t)\,w_{i}(t)]^{p_{i}}\,d\mu=\int\frac{|x(t)w(t)|^{p}}{\|x\|_{L^{p}(w)}^{p}}\,d\mu=1.$
Therefore, $x\in L^{p_{0}}(w_{0})^{1-\theta}\,L^{p_{1}}(w_{1})^{\theta}$ with
norm $\leq\|x\|_{L^{p}(w)}$ and equality (26) is proved. The proof for the
case when one or both $p_{0},p_{1}$ are $\infty$ is even simpler. ∎
We want to calculate product spaces of Lorentz space $\Lambda_{\phi}$ and
Marcinkiewicz space $M_{\phi}$ on $I$, where $\phi$ is a quasi-concave
function on $I$ with $\phi(0^{+})=0$. We will do this, in fact, for some other
closely connected spaces. Consider the Lorentz space $\Lambda_{\phi,1}$ and
more general Lorentz space $\Lambda_{\phi,p}$ with $0<p<\infty$ on $I$
defined, respectively, as
$\Lambda_{\phi,1}=\\{x\in
L^{0}(I):\|x\|_{\Lambda_{\phi,1}}=\int_{I}x^{*}(t)\frac{\phi(t)}{t}\,dt<\infty\\},$
$\Lambda_{\phi,p}=\\{x\in
L^{0}(I):\|x\|_{\Lambda_{\phi,p}}=\left(\int_{I}[\phi(t)\,x^{*}(t)]^{p}\frac{dt}{t}\right)^{1/p}<\infty\\},$
Space $\Lambda_{\phi,1}$ is a Banach space and if $\phi(t)\leq
at\phi^{\prime}(t)$ for all $t\in I$, then
$\Lambda_{\phi,1}\overset{1}{\hookrightarrow}\Lambda_{\phi}\overset{a}{\hookrightarrow}\Lambda_{\phi,1}$
(space $\Lambda_{\phi}$ was defined in Part 1). Consider also another
Marcinkiewicz space $M_{\phi}^{*}$ than the space $M_{\phi}$ defined in Part
1, as
$M_{\phi}^{*}=M_{\phi}^{*}(I)=\\{x\in L^{0}(I):\|x\|_{M_{\phi}^{*}}=\sup_{t\in
I}\phi(t)x^{*}(t)<\infty\\}.$
This Marcinkiewicz space need not be a Banach space and always we have
$M_{\phi}\overset{1}{\hookrightarrow}M_{\phi}^{*}$. Moreover,
$M_{\phi}^{*}\overset{C}{\hookrightarrow}M_{\phi}$ if and only if
$\int_{0}^{t}\frac{1}{\phi(s)}\,ds\leq C\frac{t}{\phi(t)}~{}{\rm
for~{}all}~{}~{}t\in I.$ (27)
In fact, since $\frac{1}{\phi}\in M_{\phi}^{*}$ then estimate (27) is
necessary for the imbedding. On the other hand, if (27) holds and $x\in
M_{\phi}^{*}$, then
$\displaystyle\|x\|_{M_{\phi}}$ $\displaystyle=$ $\displaystyle\sup_{t\in
I}\phi(t)x^{**}(t)=\sup_{t\in
I}\frac{\phi(t)}{t}\int_{0}^{t}\frac{1}{\phi(s)}\,\phi(s)x^{*}(s)\,ds$
$\displaystyle\leq$ $\displaystyle\sup_{s\in I}\phi(s)x^{*}(s)\,\sup_{t\in
I}\frac{\phi(t)}{t}\int_{0}^{t}\frac{1}{\phi(s)}\,ds\leq
C\|x\|_{M_{\phi}^{*}}.$
We can consider spaces $\Lambda_{w,1},\Lambda_{w,p}$ and $M_{w}^{*}$ for more
general weights $w\geq 0$, but then the problem of being quasi-Banach space or
Banach space will appear. Such investigations can be found in [CKMP] and
[KM04].
Since indices of the quasi-concave function on $I$ are useful in the
formulation of further results let us define them. The lower index
$p_{\phi,I}$ and upper index $q_{\phi,I}$ of a function $\phi$ on $I$ are
numbers defined as
$p_{\phi,I}=\lim_{t\rightarrow 0^{+}}\frac{\ln m_{\phi,I}(t)}{\ln
t},~{}q_{\phi,I}=\lim_{t\rightarrow\infty}\frac{\ln m_{\phi,I}(t)}{\ln
t},~{}{\rm where}~{}~{}m_{\phi,I}(t)=\sup_{s\in I,st\in
I}\frac{\phi(st)}{\phi(s)}.$
It is known (see, for example, [KPS82] and [Ma85], [Ma89]) that for a quasi-
concave function $\phi$ on $[0,\infty)$ we have $0\leq p_{\phi,[0,\infty)}\leq
p_{\phi,[0,1]}\leq q_{\phi,[0,1]}\leq q_{\phi,[0,\infty)}\leq 1$. Moreover,
estimate (27) is equivalent to $q_{\phi,I}<1$. We also need for a
differentiable increasing function $\phi$ on $I$ with $\phi(0^{+})=0$ the
Simonenko indices
$s_{\phi,I}=\inf_{t\in
I}\frac{t\phi^{\prime}(t)}{\phi(t)},~{}\sigma_{\phi,I}=\sup_{t\in
I}\frac{t\phi^{\prime}(t)}{\phi(t)}.$
They satisfy $0\leq s_{\phi,I}\leq p_{\phi,I}\leq
q_{\phi,I}\leq\sigma_{\phi,I}$ (cf. [Ma85, p. 22] and [Ma89, Theorem 11.11]).
Theorem 7.
* $(i)$
If $\phi,\psi$ are quasi-concave functions on $I$, then
$M_{\phi\psi}^{*}\overset{1}{\hookrightarrow}M_{\phi}^{*}\odot
M_{\psi}^{*}\overset{2}{\hookrightarrow}M_{\phi\psi}^{*}$.
* $(ii)$
Let $\phi,\psi$ and $\phi\psi$ be increasing concave functions on $I$ with
$\phi(0^{+})=\psi(0^{+})=0$. If $s_{\phi,I}\geq a>0$ and $s_{\phi\psi,I}\geq
b>0$, then $\Lambda_{\phi}\odot
M_{\psi}^{*}\overset{4+4/a}{\hookrightarrow}\Lambda_{\phi\psi}\overset{2/b}{\hookrightarrow}\Lambda_{\phi}\odot
M_{\psi}^{*}$.
* $(iii)$
Let $\phi,\psi$ be quasi-concave functions on $I$ such that $0<p_{\phi,I}\leq
q_{\phi,I}<1$ and $0<p_{\psi,I}\leq q_{\psi,I}<1$, then
$\Lambda_{\phi,1}\odot\Lambda_{\psi,1}=\Lambda_{\phi\psi,1/2},~{}\Lambda_{\phi,1}\odot
M_{\psi}^{*}=\Lambda_{\phi\psi,1},~{}M_{\phi}^{*}\odot
M_{\psi}^{*}=M_{\phi\psi}^{*}$
with equivalent quasi-norms.
###### Proof.
(i) For each $z\in M_{\phi\psi}^{\ast}$ one has
$z^{*}\leq\frac{\|z\|_{M_{\phi\psi}^{\ast}}}{\phi\psi}$, but since
$\frac{\|z\|_{M_{\phi\psi}^{\ast}}}{\phi\psi}\in M_{\phi}^{*}\odot
M_{\psi}^{*}$ it follows that $z\in M_{\phi}^{*}\odot M_{\psi}^{*}$ and
$\|z\|_{M_{\phi}^{*}\odot M_{\psi}^{*}}\leq\|z\|_{M_{\phi\psi}^{*}}$.
If $z\in M_{\phi}^{*}\odot M_{\psi}^{*}$, then, by Corollary 7, we have
$z^{*}\leq x^{*}y^{*}$ for some $x^{*}\in M_{\phi}^{*},y^{*}\in M_{\psi}^{*}$
and $\inf\\{\|x\|_{M_{\phi}^{*}}\|y\|_{M_{\psi}^{*}}:z^{*}\leq
x^{*}y^{*}\\}\leq 2\|z\|_{M_{\phi}^{*}\odot M_{\psi}^{*}}$. But
$x^{*}y^{*}\leq\frac{\|x\|_{M_{\phi}^{*}}}{\phi}\,\frac{\|y\|_{M_{\psi}^{*}}}{\psi}$
and so
$\|z\|_{M_{\phi\psi}^{*}}=\sup_{t\in I}\phi(t)\psi(t)\,z^{*}(t)\leq\sup_{t\in
I}\phi(t)\psi(t)\,x^{*}(t)y^{*}(t)\leq\|x\|_{M_{\phi}^{*}}\,\|y\|_{M_{\psi}^{*}}.$
Therefore,
$\|z\|_{M_{\phi\psi}^{*}}\leq\inf\\{\|x\|_{M_{\phi}^{*}}\|y\|_{M_{\psi}^{*}}:z^{*}\leq
x^{*}y^{*}\\}\leq 2\|z\|_{M_{\phi}^{*}\odot M_{\psi}^{*}}.$
(ii) Let $z\in\Lambda_{\phi}\odot M_{\psi}^{*}$. Then for any $\varepsilon>0$
we can find $x\in\Lambda_{\phi},y\in M_{\psi}^{*}$ such that $z=xy$ and
$\|x\|_{\Lambda_{\phi}}\,\|y\|_{M_{\psi}^{*}}\leq(1+\varepsilon)\|z\|_{\Lambda_{\phi}\odot
M_{\psi}^{*}}$. Since $\psi^{\prime}(t)\leq\psi(t)/t$ and
$\phi(t)\leq\frac{1}{a}\,t\phi^{\prime}(t)$ it follows that
$\displaystyle\int_{I}z^{*}(t)\,d(\phi\psi)(t)$ $\displaystyle\leq$
$\displaystyle\int_{I}x^{*}(t/2)\,y^{*}(t/2)\left[\phi^{\prime}(t)\psi(t)+\phi(t)\psi^{\prime}(t)\right]dt$
$\displaystyle\leq$ $\displaystyle
2\,\int_{I}x^{*}(t/2)\,y^{*}(t/2)\,\psi(t/2)\phi^{\prime}(t)\,dt$
$\displaystyle+$
$\displaystyle\frac{1}{a}\int_{I}x^{*}(t/2)\,y^{*}(t/2)\,t\phi^{\prime}(t)\frac{\psi(t)}{t}dt$
$\displaystyle\leq$ $\displaystyle 2\,\sup_{s\in
I}\psi(s)y^{*}(s)\,\int_{I}x^{*}(t/2)\,\phi^{\prime}(t)\,dt$ $\displaystyle+$
$\displaystyle\frac{2}{a}\,\sup_{s\in
I}\psi(s)y^{*}(s)\,\int_{I}x^{*}(t/2)\,\phi^{\prime}(t)\,dt$
$\displaystyle\leq$
$\displaystyle(2+2/a)\|y\|_{M_{\psi}^{*}}\,\|D_{2}x\|_{\Lambda_{\phi}}$
$\displaystyle\leq$
$\displaystyle(4+4/a)\|y\|_{M_{\psi}^{*}}\,\|x\|_{\Lambda_{\phi}}\leq(4+4/a)(1+\varepsilon)\,\|z\|_{\Lambda_{\phi}\odot
M_{\psi}^{*}}.$
Since $\varepsilon>0$ is arbitrary the first inclusion of (ii) is proved. To
prove the second inclusion assume that $z=z^{*}\in\Lambda_{\phi\psi}$. Then
$w(t)=z(t)\,\frac{\phi(t)\psi(t)}{t}\in L^{1}(I)~{}{\rm and}~{}w=w^{*}.$
Moreover,
$\|w\|_{L^{1}}=\int_{I}z^{*}(t)\frac{\phi(t)\psi(t)}{t}dt\leq\frac{1}{b}\,\int_{I}z^{*}(t)\,d(\phi\psi)(t)=\frac{1}{b}\|z\|_{\Lambda_{\phi\psi}}.$
Using the Lorentz result on the duality $(\Lambda_{\phi})^{\prime}\equiv
M_{t/\phi(t)}$ (see [Lo51, Theorem 6], [Lo53, Theorem 3.6.1]; see also [KPS82,
Theorem 5.2] for separable $\Lambda_{\phi}$, [HM92, Proposition 2.5(a)] with
$p=q=1$, [KM07, Theorem 2.2]) and Lozanovskiĭ’s factorization theorem
$L^{1}\equiv\Lambda_{\phi}\odot(\Lambda_{\phi})^{\prime}\equiv\Lambda_{\phi}\odot
M_{t/\phi(t)}$ we can find $u\in\Lambda_{\phi},~{}v\in M_{t/\phi(t)}$ such
that
$w^{*}=uv~{}~{}{\rm
and}~{}~{}\|u\|_{\Lambda_{\phi}}\,\|v\|_{M_{t/\phi(t)}}\leq\|w\|_{L^{1}}.$
By Corollary 7 we can find $u_{0}\in\Lambda_{\phi},~{}v_{0}\in
M_{t/\phi(t)}\overset{1}{\hookrightarrow}M_{t/\phi(t)}^{*}$ such that
$w^{*}\leq u_{0}^{*}v_{0}^{*}~{}~{}{\rm
and}~{}~{}\|u_{0}\|_{\Lambda_{\phi}}\,\|v_{0}\|_{M_{t/\phi(t)}}\leq
2\,\|u\|_{\Lambda_{\phi}}\,\|v\|_{M_{t/\phi(t)}}.$
Let
$x(t)=\frac{t}{\phi(t)}\frac{w^{*}(t)}{\|v_{0}\|_{M_{t/\phi(t)}}}~{}~{}{\rm
and}~{}~{}y=\frac{w^{*}}{u_{0}^{*}}.$
Then $x(t)\leq\frac{w^{*}(t)}{v_{0}^{*}(t)}\leq u_{0}^{*}(t)$ because
$M_{t/\phi(t)}\overset{1}{\hookrightarrow}M_{t/\phi(t)}^{*}$. Also $y(t)\leq
v_{0}^{*}(t)$ and
$z(t)\frac{\phi(t)\psi(t)}{t}=w(t)=x(t)\,\frac{\phi(t)}{t}\|v_{0}\|_{M_{t/\phi(t)}}$,
hence
$z(t)=x(t)\,\frac{1}{\psi(t)}\,\|v_{0}\|_{M_{t/\phi(t)}}$
with $x\in\Lambda_{\phi}$ and $\frac{\|v_{0}\|_{M_{t/\phi(t)}}}{\psi(t)}\in
M_{\psi}^{*}$. Moreover,
$\displaystyle\|z\|_{\Lambda_{\phi}\odot M_{\psi}^{*}}$ $\displaystyle\leq$
$\displaystyle\|x\|_{\Lambda_{\phi}}\,\|\frac{1}{\psi}\|_{M_{\psi}^{*}}\,\|v_{0}\|_{M_{t/\phi(t)}}\leq\|u_{0}\|_{\Lambda_{\phi}}\,\|v_{0}\|_{M_{t/\phi(t)}}$
$\displaystyle\leq$ $\displaystyle
2\,\|u\|_{\Lambda_{\phi}}\,\|v\|_{M_{t/\phi(t)}}\leq
2\,\|w\|_{L^{1}}\leq\frac{2}{b}\,\|z\|_{\Lambda_{\phi\psi}}$
and the proof of (ii) is complete.
(iii) If $0<p_{\phi,I}\leq q_{\phi,I}<1$, then both operators $H,H^{*}$ are
bounded on $L^{1}(\frac{\phi(t)}{t})$ (see [KMP07], Theorem 4) and using
Lemmas 4 and 5 we have
$\displaystyle\Lambda_{\phi,1}^{\theta}\,\Lambda_{\psi,1}^{1-\theta}$
$\displaystyle=$
$\displaystyle[L^{1}(\frac{\phi(t)}{t})^{(*)}]^{\theta}\,[L^{1}(\frac{\psi(t)}{t})^{(*)}]^{1-\theta}=[L^{1}(\frac{\phi(t)}{t})^{\theta}\,[L^{1}(\frac{\psi(t)}{t})^{1-\theta}]^{(*)}$
$\displaystyle=$ $\displaystyle
L^{1}(\frac{\phi(t)^{\theta}\psi(t)^{1-\theta}}{t})^{(*)}=\Lambda_{\phi^{\theta}\psi^{1-\theta},1}$
with equivalent quasi-norms. Thus by Theorem 1(iv) we obtain
$\Lambda_{\phi,1}\odot\Lambda_{\psi,1}=(\Lambda_{\phi,1}^{1/2}\Lambda_{\psi,1}^{1/2})^{(1/2)}=(\Lambda_{\phi^{1/2}\psi^{1/2},1})^{(1/2)}=\Lambda_{\phi\psi,1/2}$
with equivalent quasi-norms. The last space is not normable since it contains
isomorphic copy of $l^{1/2}$ (see [KM04], Theorem 1).
If $0<p_{\phi,I}\leq q_{\phi,I}<1$, then both operators $H,H^{*}$ are bounded
on $L^{\infty}(\phi)$ which can be proved directly. To show this we only need
here to see equivalence of the corresponding integral inequalities on $\phi$
with assumptions on indices of $\phi$ and this is proved, for example, in
[Ma85, Theorem 6.4] or [Ma89, Theorem 11.8] (see also [KPS], pp. 56-57). Then,
using Lemmas 4 and 5, we have
$\displaystyle\Lambda_{\phi,1}^{\theta}\,(M_{\psi}^{*})^{1-\theta}$
$\displaystyle=$
$\displaystyle[L^{1}(\frac{\phi(t)}{t})^{(*)}]^{\theta}\,[L^{\infty}(\psi)^{(*)}]^{1-\theta}=[L^{1}(\frac{\phi(t)}{t})^{\theta}\,[L^{\infty}(\psi)^{1-\theta}]^{(*)}$
$\displaystyle=$ $\displaystyle
L^{1/{\theta}}(\frac{\phi(t)^{\theta}\psi(t)^{1-\theta}}{t^{\theta}})^{(*)}=\Lambda_{\phi^{\theta}\psi^{1-\theta},1/{\theta}}$
with equivalent quasi-norms. Thus, by Theorem 1(iv), we obtain
$\Lambda_{\phi,1}\odot
M_{\psi}^{*}=[\Lambda_{\phi,1}^{1/2}(M_{\psi}^{*})^{1/2}]^{(1/2)}=(\Lambda_{\phi^{1/2}\psi^{1/2},2})^{(1/2)}=\Lambda_{\phi\psi,1}$
with equivalent quasi-norms. Similarly for Marcinkiewicz spaces
$\displaystyle(M_{\phi}^{*})^{\theta}\,(M_{\psi}^{*})^{1-\theta}$
$\displaystyle=$
$\displaystyle[L^{\infty}(\phi)^{(*)}]^{\theta}\,[L^{\infty}(\psi)^{(*)}]^{1-\theta}=[L^{\infty}(\phi)^{\theta}\,L^{\infty}(\psi)^{1-\theta}]^{(*)}$
$\displaystyle=$ $\displaystyle
L^{\infty}(\phi^{\theta}\psi^{1-\theta})^{(*)}=M_{\phi^{\theta}\psi^{1-\theta}}^{*}$
and, by Theorem 1(iv), we obtain
$M_{\phi}^{*}\odot
M_{\psi}^{*}=[(M_{\phi}^{*})^{1/2}(M_{\psi}^{*})^{1/2}]^{(1/2)}=(M_{\phi^{1/2}\psi^{1/2}}^{*})^{(1/2)}=M_{\phi\psi}^{*}$
with equivalent quasi-norms. This proves theorem completely. ∎
6\. Factorization of some Banach ideal spaces
The factorization theorem of Lozanovskiĭ states that for any Banach ideal
space $E$ the space $L^{1}$ has a factorization $L^{1}\equiv E\odot
E^{\prime}$. The natural generalization of the type
$F\equiv E\odot M(E,F)$ (28)
is not true without additional assumptions on the spaces, as we can see on the
example below.
Example 2. If $E=L^{p,1}$ with the norm
$\|x\|_{E}=\frac{1}{p}\int_{I}t^{1/p-1}x^{*}(t)\,dt$ for $1<p<\infty$, then
$M(L^{p,1},L^{p})\equiv L^{\infty}$ (cf. [MP89], Theorem 3) and
$L^{p,1}\odot M(L^{p,1},L^{p})\equiv L^{p,1}\odot L^{\infty}\equiv
L^{p,1}\mathchar 13608\relax L^{p}.$
Therefore, factorization (28) is not true and we even don’t have factorization
$L^{p}=E\odot M(E,L^{p})$ with equivalent norms. Similarly, if
$F=L^{p,\infty}$ with the norm $\|x\|_{F}=\sup_{t\in I}t^{1/p}x^{**}(t)$ for
$1<p<\infty$, then $M(L^{p},L^{p,\infty})\equiv
M(L^{p^{\prime},1},L^{p^{\prime}})\equiv L^{\infty}$ and
$L^{p}\odot M(L^{p},L^{p,\infty})\equiv L^{p}\odot L^{\infty}\equiv
L^{p}\mathchar 13608\relax L^{p,\infty}.$
Therefore, again factorization (28) is not true and we even don’t have
equality $F=L^{p}\odot M(L^{p},F)$ with equivalent norms.
Let us collect some factorization results of type (28). First of all the
Lozanovskiĭ factorization theorem was announced in 1967 (cf. [Lo67], Theorem
4) and published with detailed proof in 1969 (cf. [Lo69], Theorem 6). His
proof uses the Calderón space $F=E^{1/2}(E^{\prime})^{1/2}$ and result about
its dual $F^{\prime\prime}\equiv F^{\prime}\equiv L^{2}$ (cf. [Lo69], Theorem
5; see also [Ma89, p. 185] and [Re93]). Lozanovskiĭ factorization theorem was
new even for finite dimensional spaces. In 1976 Jamison and Ruckle [JR76]
proved that $l^{1}$ factors through every normal Banach sequence space and its
Köthe dual. Proof even in the finite dimensional case is indirect and it uses
the Brouwer fixed point theorem. Later on Lozanovskiĭ’s factorization result
was proved by Gillespie [Gi81], using different method, which inspiration was
coming from the theory of reflexive algebras of operators on Hilbert space.
If $E,F$ are finite dimensional ideal spaces and $B_{E},B_{F}$ denote their
unit balls, then Bollobás and Leader [BL95], with the help of Jamison-Ruckle
method, proved factorization $B_{E}\odot B_{M(E,F)}\equiv B_{F}$ under
assumptions that $B_{F}$ is a strictly unconditional body and $B_{M(E,F)}$ is
smooth.
Nilsson [Ni85, Lemma 2.5], using the Maurey factorization theorem (cf. [Ma74,
Theorem 8]; see also [Wo91, pp. 264-266]), proved the following result of type
(28): if $E$ is a Banach ideal space which is $p$-convex with constant $1$,
then
$E^{\prime}\equiv L^{p^{\prime}}\odot M(E,L^{p})\equiv L^{p^{\prime}}\odot
M(L^{p^{\prime}},E^{\prime}).$ (29)
By duality result and (29) we obtain that if $F$ is a Banach ideal space with
the Fatou property which is $q$-concave with constant $1$ for $1<q<\infty$,
then
$F=F^{\prime\prime}\equiv L^{q}\odot M(F^{\prime},L^{q^{\prime}})\equiv
L^{q}\odot M(L^{q},F).$ (30)
Factorization (30) was proved and used by Nilsson [Ni85, Theorem 2.4] in a new
proof of the Pisier theorem (cf. [Pi79a, Theorem 2.10], [Pi79b, Theorem 2.2];
see also [TJ89, Theorem 28.1]): if a Banach ideal space $E$ with the Fatou
property is $p$-convex and $q$-concave with constants $1,1<r<\infty$, and
$\frac{1}{p}=\frac{\theta}{r}+1-\theta,\frac{1}{q}=\frac{\theta}{r}$, then the
space $E_{0}\equiv M(L^{q},E)^{(1-\theta)}$ is a Banach ideal space and
$E\equiv E_{0}^{1-\theta}(L^{r})^{\theta}$. First part of the proof follows
from the facts that
$\||x|^{1-\theta}\|_{M(L^{q},E)}^{\frac{1}{1-\theta}}\equiv\||x|^{1/s^{\prime}}\|_{M(L^{s},E^{(1/p)})}^{s^{\prime}}\equiv\||x|^{1/s^{\prime}}\|_{M((E^{(1/p)})^{\prime},L^{s^{\prime}})}^{s^{\prime}},$
$E^{(1/p)}$ is a Banach space and $M((E^{(1/p)})^{\prime},L^{s^{\prime}})$ is
$s^{\prime}$-convex with constant $1$, where $s=\frac{r}{\theta p}$. Second
part uses (30) and by Theorem 1(ii) we obtain
$E_{0}^{1-\theta}(L^{r})^{\theta}\equiv E_{0}^{\frac{1}{1-\theta}}\odot
L^{\frac{r}{\theta}}\equiv E_{0}^{\frac{1}{1-\theta}}\odot L^{q}\equiv
M(L^{q},E)\odot L^{q}\equiv E.$
Schep proved factorization (30) and also the reverse implication, that is, if
(30) holds, then the space $F$ is $q$-concave with constant $1$ (cf. [Sc10],
Theorem 3.9). He has also proved another factorization result (even
equivalence - see [Sc10], Theorem 3.3): if Banach ideal space $E$ with the
Fatou property is $p$-convex with constant $1\,{\rm(}1<p<\infty{\rm)}$, then
$L^{p}\equiv E\odot M(E,L^{p}).$ (31)
His proof has misprints in Theorem 3.2. The proof should be as follows: using
property (g) from [MP89] we obtain
$M(E,L^{p})^{(1/p)}\equiv M(E^{(1/p)},(L^{p})^{(1/p)})\equiv
M(E^{(1/p)},L^{1})\equiv[E^{(1/p)}]^{\prime},$
and by the Lozanovskiĭ factorization theorem
$E^{(1/p)}\odot M(E,L^{p})^{(1/p)}\equiv
E^{(1/p)}\odot[E^{(1/p)}]^{\prime}\equiv L^{1}.$
Taking then $p$-convexification on both sides and using Theorem 1(iii) we get
$E\odot M(E,L^{p})\equiv[E^{(1/p)}\odot
M(E,L^{p})^{(1/p)}]^{(p)}\equiv(L^{1})^{(p)}\equiv L^{p}.$
Note that factorization theorem of the type (30): $F=l^{q}\odot M(l^{q},F)$
for any $q$-concave Banach space $F$ with a monotone unconditional basis was
proved already in 1980 (cf. [LT-J80], Corollary 3.2).
If a space $E$ has the Fatou property, then in the definition of the norm of
$E\odot E^{\prime}$ we may take “minimum” instead of “infimum”. It is known
that the Fatou property of $E$ is equivalent with the isometric equality
$E\equiv E^{\prime\prime}$. Then $E$ is called perfect. This notion can be
generalized to $F$-perfectness. We say that $E$ is $F$-perfect if
$M(M(E,F),F)\equiv E$ (see [MP89], [CDS08] and [Sc10] for more information
about $F$-perfectness). Is there any connection between factorization (28) and
to be $F$-perfect by $E$?
Theorem 8. Let $E,F$ be Banach ideal spaces with the Fatou property. Then
factorization $E\odot M(E,F)\equiv F$ implies $F$-perfectness of $E$, i.e.,
$M(M(E,F),F)\equiv E$.
###### Proof.
Schep [Sc10, Theorem 2.8] proved that if $E\odot F$ is a Banach ideal space,
then $M(E,E\odot F)\equiv F$ (see also Theorem 4 above). Since $E\odot
M(E,F)\equiv F$ is a Banach ideal space by assumption, therefore from the
above Schep result we obtain
$M(M(E,F),F)\equiv M(M(E,F),E\odot M(E,F))\equiv E,$
which is $F$-perfectness of $E$. ∎
The example of Bollobas and Brightwell [BB00], presented in [Sc10, Example
3.6], shows that the reverse implication is not true, even for three-
dimensional spaces.
Almost all proofs in factorization theorems are tricky or use powerful
theorems and, in fact, equality $E\odot M(E,F)\equiv F$ is proved without
calculating $M(E,F)$ directly. Except some special cases it seems to be the
only way to prove equality of the norms in (28). However, it seems to be also
useful to have equality (28) with just equivalence of the norms, that is,
$F=E\odot M(E,F).$ (32)
This can be done by finding $M(E,F)$ and $E\odot M(E,F)$ separately and we
will do so. Observe also that if a Banach ideal space $E$ is $p$-convex
($1<p<\infty$) with constant $K>1$, then $E^{(1/p)}$ is $1$-convex with
constant $K^{p}$ and
$\|x\|^{0}=\inf\\{\sum_{k=1}^{n}\|x_{k}\|_{E^{(1/p)}}:|x|\leq\sum_{k=1}^{n}|x_{k}|,x_{k}\in
E^{(1/p)},n\in{\mathbb{N}}\\}$
defines norm on $E^{(1/p)}$ with
$K^{-p}\|x\|_{E^{(1/p)}}\leq\|x\|^{0}\leq\|x\|_{E^{(1/p)}}$. Thus
$E_{0}=(E^{(1/p)},\|\cdot\|^{0})$ is a Banach ideal space and its
$p$-convexification $E_{0}^{(p)}=E$ with the norm
$\|x\|^{1}=[\||x|^{p}\|^{0})^{1/p}$ is $p$-convex with constant $1$ (cf.
[LT79, Lemma 1.f.11] and [ORS08, Proposition 2.23]), and we can use result
from (29) to obtain $E^{\prime}=L^{p^{\prime}}\odot
M(L^{p^{\prime}},E^{\prime})$.
As a straightforward conclusion from Corollary 6.1 in [KLM12] and Theorem A(a)
with Theorem 5(a) we get the following factorization theorem for the Calderón-
Lozanovskiĭ $E_{\varphi}$-spaces.
Theorem 9. Let $E$ be a Banach ideal space with the Fatou property and ${\it
supp}E=\Omega$. Suppose that for two Young functions $\varphi,\varphi_{1}$
there is a Young function $\varphi_{2}$ such that one of the following
conditions holds:
* $(i)$
$\varphi_{1}^{-1}\varphi_{2}^{-1}\approx\varphi^{-1}$ for all arguments,
* $(ii)$
$\varphi_{1}^{-1}\varphi_{2}^{-1}\approx\varphi^{-1}$ for large arguments and
$L^{\infty}\hookrightarrow E$,
* $(iii)$
$\varphi_{1}^{-1}\varphi_{2}^{-1}\approx\varphi^{-1}$ for small arguments and
$E\hookrightarrow L^{\infty}$.
Then the factorization $E_{\varphi_{1}}\odot
M\left(E_{\varphi_{1}},E_{\varphi}\right)=E_{\varphi}$ with equivalent norms
is valid and, in consequence, the space $E_{\varphi_{1}}$ is
$E_{\varphi}$–perfect up to equivalence of norms.
Moreover, applying Lemma 7.4 from [KLM12] to the Theorem 9(i) one has the
following special case.
Corollary 8. Let $\varphi,\varphi_{1}$ be two Orlicz functions, and let $E$
Banach an ideal space with the Fatou property and ${\it supp}E=\Omega$. If the
function $f_{v}(u)=\frac{\varphi(uv)}{\varphi_{1}(u)}$ is non-increasing on
$(0,\infty)$ for any $v>0$, then the factorization $E_{\varphi_{1}}\odot
M(E_{\varphi_{1}},E_{\varphi})=E_{\varphi}$ is valid with equivalent norms
and, in consequence, the space $E_{\varphi_{1}}$ is $E_{\varphi}$–perfect up
to equivalence of the norms.
###### Proof.
It is enough to take as $\varphi_{2}$ the function defined by
$\varphi_{2}(u)=(\varphi\ominus\varphi_{1})(u)=\sup_{v>0}[\varphi(uv)-\varphi_{1}(v)]$
and use the fact proved in [KLM12, Lemma 7.4] showing that
$\varphi_{1}^{-1}\varphi_{2}^{-1}\approx\varphi^{-1}$ for all arguments. ∎
Before we consider factorization of Lorentz and Marcinkiewicz let us calculate
“missing” multipliers spaces.
Proposition 3. Suppose $\phi,\psi$ are non-decreasing, concave functions on
$I$ with $\phi(0^{+})=\psi(0^{+})=0$. Let $E$ and $F$ be symmetric spaces on
$I$ with fundamental functions $f_{E}(t)=\phi(t)$ and $f_{F}(t)=\psi(t)$. If
$\omega(t)=\sup_{0<s\leq t}\frac{\psi(s)}{\phi(s)}$ is finite for any $t\in
I$, then
$M(\Lambda_{\phi},F)\overset{1}{\hookrightarrow}M_{\omega},~{}~{}M(E,M_{\psi})\overset{1}{\hookrightarrow}M_{\omega},~{}~{}{\rm
and}~{}~{}M_{\omega}^{*}\overset{1}{\hookrightarrow}M(\Lambda_{\phi,1},\Lambda_{\psi,1}).$
If, moreover, $s_{\phi,I}\geq a>0$, then
$M_{\omega}^{*}\overset{1/a}{\hookrightarrow}M(\Lambda_{\phi},\Lambda_{\psi})$.
###### Proof.
By Theorem 2.2(iv) in [KLM12] the function $\omega$ is a fundamental function
of $M(\Lambda_{\phi},F)$ and by the maximality of the space $M_{\omega}$ and
Theorem 2.2(i) in [KLM12] we obtain imbedding
$M(\Lambda_{\phi},F)\overset{1}{\hookrightarrow}M_{\omega}$. On the other
hand, using the property (e) from [MP89, p. 326] about duality of multipliers,
the duality $(M_{\psi})^{\prime}\equiv\Lambda_{t/{\psi(t)}}$ and the above
result we obtain
$M(E,M_{\psi})\equiv M((M_{\psi})^{\prime},E^{\prime})\equiv
M(\Lambda_{t/{\psi(t)}},E^{\prime})\overset{1}{\hookrightarrow}M_{\omega}.$
Two other imbeddings will be proved if we show that $\frac{1}{\omega}$ belongs
to the corresponding spaces. Since, by Theorem 2.2(ii) in [KLM12], we have
$\|y\|_{M(E,F)}=\sup_{\|x^{*}\|_{E}\leq 1}\|x^{*}y^{*}\|_{F}$ it follows that
$\displaystyle\|\frac{1}{\omega}\|_{M(\Lambda_{\phi,1},\Lambda_{\psi,1})}$
$\displaystyle=$ $\displaystyle\sup_{\|x\|_{\Lambda_{\phi,1}}\leq
1}\int_{I}\left(x^{*}\,\frac{1}{\omega}\right)^{*}(t)\frac{\psi(t)}{t}\,dt=\sup_{\|x\|_{\Lambda_{\phi,1}}\leq
1}\int_{I}x^{*}(t)\frac{1}{\omega(t)}\frac{\psi(t)}{t}\,dt$ $\displaystyle=$
$\displaystyle\sup_{\|x\|_{\Lambda_{\phi,1}}\leq
1}\int_{I}x^{*}(t)\inf_{0<s\leq
t}\frac{\phi(s)}{\psi(s)}\frac{\psi(t)}{t}\,dt\leq\sup_{\|x\|_{\Lambda_{\phi,1}}\leq
1}\int_{I}x^{*}(t)\frac{\phi(t)}{t}\,dt\leq 1,$
and, again by the above mentioned result in [KLM12],
$\displaystyle\|\frac{1}{\omega}\|_{M(\Lambda_{\phi},\Lambda_{\psi})}$
$\displaystyle=$ $\displaystyle\sup_{\|x\|_{\Lambda_{\phi}}\leq
1}\int_{I}x^{*}(t)\frac{1}{\omega(t)}\psi^{\prime}(t)\,dt=\sup_{\|x\|_{\Lambda_{\phi}}\leq
1}\int_{I}x^{*}(t)\inf_{0<s\leq
t}\frac{\phi(s)}{\psi(s)}\,\psi^{\prime}(t)\,dt$ $\displaystyle\leq$
$\displaystyle\sup_{\|x\|_{\Lambda_{\phi}}\leq
1}\int_{I}x^{*}(t)\frac{\phi(t)}{t}\,dt\leq\sup_{\|x\|_{\Lambda_{\phi}}\leq
1}\frac{1}{a}\,\int_{I}x^{*}(t)\,\phi^{\prime}(t)\,dt\leq 1/a,$
and all imbeddings are proved. ∎
Putting together previous results on products and multipliers of Lorentz and
Marcinkiewicz spaces we are ready to proof factorization of these spaces.
Theorem 10. Let $\phi,\psi$ be a non-decreasing, concave functions on $I$ with
$\phi(0^{+})=\psi(0^{+})=0$. Suppose $\frac{\psi(t)}{\phi(t)}$ is a non-
decreasing function on $I$.
* $(a)$
If $s_{\phi,I}>0$ and $s_{\psi,I}>0$, then $\Lambda_{\phi}\odot
M(\Lambda_{\phi},\Lambda_{\psi})=\Lambda_{\psi}$.
Moreover, for any symmetric space $F$ on $I$ with the fundamental function
$f_{F}(t)=\psi(t)$ and under the above assumptions on $\phi$ and $\psi$ we
have
$\Lambda_{\phi}\odot M(\Lambda_{\phi},F)=F~{}{\it
if~{}and~{}only~{}if}~{}~{}F=\Lambda_{\psi}.$ (33)
* $(b)$
If $\sigma_{\phi,I}<1$ and $\sigma_{\psi,I}<1$, then $M_{\phi}\odot
M(M_{\phi},M_{\psi})=M_{\psi}$.
Moreover, for any symmetric space $E$ on $I$ having Fatou property, with the
fundamental function $f_{E}(t)=\phi(t)$ and under the above assumptions on
$\phi$ and $\psi$ we have
$E\odot M(E,M_{\psi})=M_{\psi}~{}{\it if~{}and~{}only~{}if}~{}~{}E=M_{\phi}.$
(34)
* $(c)$
If $\sigma_{\phi,I}<1,s_{\psi,I}>0$ and $s_{\psi/\phi,I}>0$, then
$M_{\phi}\odot M(M_{\phi},\Lambda_{\psi})=\Lambda_{\psi}.$ (35)
###### Proof.
(a) Using Proposition 3 we have
$M(\Lambda_{\phi},\Lambda_{\psi})\overset{1}{\hookrightarrow}M_{\omega}\overset{1}{\hookrightarrow}M_{\omega}^{*}\overset{1/a}{\hookrightarrow}M(\Lambda_{\phi},\Lambda_{\psi}),~{}{\rm
where}~{}\omega(t)=\sup_{0<s\leq t}\frac{\psi(s)}{\phi(s)}.$
Since $\psi/\phi$ is a non-decreasing function on $I$ it follows that
$\phi\omega=\psi,s_{\phi\omega;I}=s_{\psi,I}>0$ and, by Theorem 7(ii),
$\Lambda_{\phi}\odot M(\Lambda_{\phi},\Lambda_{\psi})=\Lambda_{\phi}\odot
M_{\omega}^{*}=\Lambda_{\psi}$
with equivalent norms. Under the assumptions on $F$ we have from Proposition 3
the imbedding $M(\Lambda_{\phi},F)\overset{1}{\hookrightarrow}M_{\omega}$ and
then, by Theorem 7(ii),
$F=\Lambda_{\phi}\odot
M(\Lambda_{\phi},F)\overset{1}{\hookrightarrow}\Lambda_{\phi}\odot
M_{\omega}\overset{1}{\hookrightarrow}\Lambda_{\phi}\odot
M_{\omega}^{*}=\Lambda_{\psi}.$
Minimality of $\Lambda_{\psi}$ gives $F=\Lambda_{\psi}$.
(b) From the fact that $(M_{\phi})^{\prime}\equiv\Lambda_{t/\phi(t)}$, the
general duality property of multipliers (see [MP89], property (e)) and using
Proposition 3 we obtain
$M(M_{\phi},M_{\psi})\equiv M(M_{\psi}^{\prime},M_{\phi}^{\prime})\equiv
M(\Lambda_{t/\psi(t)},\Lambda_{t/\phi(t)})=M_{\omega}^{*}$
because $s_{t/\psi(t),I}=1-\sigma_{\psi,I}>0$. Since $\psi/\phi$ is a non-
decreasing function on $I$ it follows that
$\phi\omega=\psi,\sigma_{\phi\omega;I}=\sigma_{\psi,I}<1$ and by Theorem 7(i)
with the fact that $\sigma_{\phi,I}<1$ we have
$M_{\phi}\odot M(M_{\phi},M_{\psi})=M_{\phi}\odot M_{\omega}=M_{\phi}^{*}\odot
M_{\omega}^{*}=M_{\phi\omega}^{*}=M_{\psi}^{*}=M_{\psi}.$
Under the assumptions on $E$ we obtain from Proposition 3 that
$M(E,M_{\psi})\overset{1}{\hookrightarrow}M_{\omega}$ and
$M_{\psi}=E\odot M(E,M_{\psi})\overset{1}{\hookrightarrow}E\odot M_{\omega}.$
On the other hand, by Theorem 7(i) and assumption $q_{\psi;I}<1$
$M_{\phi}\odot M_{\omega}\overset{1}{\hookrightarrow}M_{\phi}^{*}\odot
M_{\omega}^{*}\overset{2}{\hookrightarrow}M_{\phi\omega}^{*}\equiv
M_{\psi}^{*}=M_{\psi}.$
Therefore, $M_{\phi}\odot M_{\omega}\overset{C}{\hookrightarrow}E\odot
M_{\omega}$. Using now Schep’s theorem, saying that if $E\odot
F\overset{C}{\hookrightarrow}E\odot G$, then $F\overset{C}{\hookrightarrow}G$
(see [Sc10], Theorem 2.5), we obtain $M_{\phi}\overset{C}{\hookrightarrow}E$.
Maximality of Marcinkiewicz space $M_{\phi}$ implies that $E=M_{\phi}$ since
fundamental function of $E$ is $f_{E}(t)=\phi(t)$ for all $t\in I$.
(c) Using Theorem 2.2(v) from [KLM12] we obtain
$M(M_{\phi},\Lambda_{\psi})\equiv\Lambda_{\eta}$, where
$\eta(t)=\int_{0}^{t}(\frac{s}{\phi(s)})^{\prime}\psi^{\prime}(s)\,ds<\infty$.
Since
$\displaystyle\eta(t)$ $\displaystyle=$
$\displaystyle\int_{0}^{t}\frac{\phi(s)\psi^{\prime}(s)-\phi^{\prime}(s)s\psi^{\prime}(s)}{\phi(s)^{2}}\,ds\leq\int_{0}^{t}\frac{\psi^{\prime}(s)}{\phi(s)}\,ds$
$\displaystyle\leq$
$\displaystyle\frac{1}{s_{\psi/\phi}}\,\int_{0}^{t}(\frac{\psi}{\phi})^{\prime}(s)\,ds=\frac{1}{s_{\psi/\phi}}\,\frac{\psi(t)}{\phi(t)},$
and
$\displaystyle\eta(t)$ $\displaystyle=$
$\displaystyle\int_{0}^{t}\frac{\phi(s)\psi^{\prime}(s)-\phi^{\prime}(s)s\psi^{\prime}(s)}{\phi(s)^{2}}\,ds\geq\int_{0}^{t}\frac{\phi(s)\psi^{\prime}(s)-\phi^{\prime}(s)\psi(s)}{\phi(s)^{2}}\,ds$
$\displaystyle=$
$\displaystyle\int_{0}^{t}(\frac{\psi}{\phi})^{\prime}(s)\,ds=\frac{\psi(t)}{\phi(t)},$
it follows that $M(M_{\phi},\Lambda_{\psi})=\Lambda_{\psi/\phi}$. Using to
this equality assumption $\sigma_{\phi,I}<1$ and result from Theorem 7(ii) we
obtain
$M_{\phi}\odot
M(M_{\phi},\Lambda_{\psi})=M_{\phi}\odot\Lambda_{\psi/\phi}=M_{\phi}^{*}\odot\Lambda_{\psi/\phi}=\Lambda_{\psi},$
and the theorem is proved. ∎
Applying the above theorem to classical Lorentz $L^{p,1}$ and Marcinkiewicz
$L^{p,\infty}$ spaces we obtain the following factorization results:
Example 3. (a) If $1\leq p\leq q<\infty$, then $L^{p,1}=L^{q,1}\odot
M(L^{q,1},L^{p,1})$.
(b) If $1<p\leq q\leq\infty$, then $L^{p,\infty}=L^{q,\infty}\odot
M(L^{q,\infty},L^{p,\infty})$.
(c) If $1<p<q\leq\infty$, then $L^{p,1}=L^{q,\infty}\odot
M(L^{q,\infty},L^{p,1})$.
What about factorization in classical Lorentz $L^{p,q}$-spaces?
Example 4. If either $1\leq r\leq p<q<\infty$ or $1<p<q\leq r\leq\infty$, then
$L^{p,r}=L^{q,r}\odot M(L^{q,r},L^{p,r}).$
In fact, if $1\leq r\leq p<q<\infty$ then using the commutativity of
$r$-convexification with multipliers (see property (g) in [MP89]) and
Proposition 3 we obtain
$M(L^{q,r},L^{p,r})\equiv M((L^{q/r,1})^{(r)},(L^{p/r,1})^{(r)})\equiv
M(L^{q/r,1},L^{p/r,1})^{(r)}=(L^{pq/[r(q-p)],\infty})^{(r)}.$
Finally, by Theorem 1(iii) and Theorem 7(ii) with $\phi(t)=t^{r/q}$ and
$\psi(t)=t^{r/p-r/q}$, we obtain
$\displaystyle L^{q,r}\odot M(L^{q,r},L^{p,r})$ $\displaystyle=$
$\displaystyle(L^{q/r,1})^{(r)}\odot(L^{pq/[r(q-p)],\infty})^{(r)}$
$\displaystyle=$ $\displaystyle(L^{q/r,1}\odot
L^{pq/[r(q-p)],\infty})^{(r)}=(L^{p/r,1})^{(r)}=L^{p,r}.$
The case $1<p<q\leq r\leq\infty$ can be proved by duality of multipliers and
the above calculations.
Theorem 11. Let $\phi$ be an increasing, concave function on $I$ with
$0<p_{\phi,I}\leq q_{\phi,I}<1$.
* $(a)$
Suppose that $F$ is a symmetric space on $I$ with the lower Boyd index
$\alpha_{F}>q_{\phi,I}$ and such that $M(M_{\phi}^{*},F)\neq\\{0\\}$. Then
$F=M_{\phi}^{*}\odot M(M_{\phi}^{*},F)=M_{\phi}\odot M(M_{\phi},F).$
* $(b)$
Suppose that $E$ is a symmetric space on $I$ with the Fatou property, which
Boyd indices satisfy $0<\alpha_{E}\leq\beta_{E}<p_{\phi,I}$ and such that
$M(E,\Lambda_{\phi})\neq\\{0\\}$. Then
$\Lambda_{\phi,1}=E\odot M(E,\Lambda_{\phi}).$
Let us start with the following identifications.
Lemma 6. Under assumptions on $\phi$ from Theorem 11 we have
$M(L^{\infty}(\phi)^{(*)},F)=M(L^{\infty}(\phi),F)^{(*)}\equiv
F(1/\phi)^{(*)}.$ (36)
###### Proof.
Since we have equivalences
$z\in M(L^{\infty}(\phi),F)^{(*)}\Leftrightarrow z^{*}\in
M(L^{\infty}(\phi),F)\Leftrightarrow\frac{z^{*}}{\phi}\in F\Leftrightarrow
z^{*}\in F(1/\phi)\Leftrightarrow z\in F(1/\phi)^{(*)}$
with equalities of the norms, the equality
$M(L^{\infty}(\phi),F)^{(\ast)}\equiv F(1/\phi)^{(\ast)}$ follows. Let
$I=(0,1)$. We prove the equality
$M(L^{\infty}(\phi)^{(\ast)},F)=F(1/\phi)^{(\ast)}.$ Clearly, it is enough to
show that the following conditions are equivalent:
$1^{0}$ $z\in M(L^{\infty}(\phi)^{(\ast)},F)$,
$2^{0}$ $\frac{z}{\phi\left(\omega\right)}\in F$ for every measure preserving
transformation $\left(mpt\right)$ $\omega:I\rightarrow I$,
$3^{0}$ $\frac{z^{\ast}}{\phi}\in F$.
Moreover, we prove that
$\|\frac{z^{*}}{\phi}\|_{F}\leq\sup_{\omega-{\rm
mpt}}\|\frac{z}{\phi(\omega)}\|_{F}=\|z\|_{M[L^{\infty}(\phi)^{(*)},F]}\leq\|D_{2}\|_{F\rightarrow
F}\,\|\frac{z^{*}}{\phi}\|_{F}.$ (37)
$1^{0}\Rightarrow 2^{0}$. Let $z\in M(L^{\infty}(\phi)^{*},F)$ and take
arbitrary mpt $\omega:I\rightarrow I.$ Since
$\left(\frac{1}{\phi(\omega)}\right)^{*}\phi=\frac{1}{\phi}\phi=1$ it follows
that $\frac{1}{\phi(\omega)}\in L^{\infty}(\phi)^{(\ast)}$. Whence
$\frac{z}{\phi(\omega)}\in F$ and
$\|\frac{z}{\phi(\omega)}\|_{F}\leq\|z\|_{M[L^{\infty}(\phi)^{(*)},F]}.$
Consequently,
$\sup_{\omega-{\rm
mpt}}\left\|\frac{z}{\phi(\omega)}\right\|_{F}\leq\|z\|_{M(L^{\infty}(\phi)^{(*)},F)}.$
$2^{0}\Rightarrow 1^{0}$. Let $x\in L^{\infty}(\phi)^{(*)}$ with the norm
$\leq 1$. Take a mpt $\omega_{0}$ such that $|x|=x^{*}(\omega_{0})$. Then
$|zx|=|z|\,x^{*}(\omega_{0})\leq\frac{|z|}{\phi(\omega_{0})}\in F,$
because
$1\geq\|x^{*}\phi\|_{L^{\infty}}=\|(x^{*}\phi)(\omega_{0})\|_{L^{\infty}}.$
Thus $z\in M(L^{\infty}(\phi)^{(*)},F)$ and
$\|z\|_{M(L^{\infty}(\phi)^{(*)},F)}\leq\sup\limits_{\omega-
mpt}\|\frac{z}{\phi(\omega)}\|_{F}.$
$2^{0}\Rightarrow 3^{0}.$ Take a mpt $\omega_{0}$ such that
$|z|=z^{*}(\omega_{0}).$ Then
$\frac{z^{*}}{\phi}\sim\frac{z^{*}}{\phi}(\omega_{0})=\frac{|z|}{\phi(\omega_{0})}\in
F\text{ and
}\|\frac{z^{*}}{\phi}\|_{F}=\|\frac{|z|}{\phi(\omega_{0})}\|_{F}\leq\sup\limits_{\omega-
mpt}\|\frac{z}{\phi(\omega)}\|_{F}.$
$3^{0}\Rightarrow 2^{0}.$ For each mpt $\omega:I\rightarrow I$ we have
$\frac{z}{\phi(\omega)}(t)\sim(\frac{z}{\phi(\omega)})^{*}(t)\leq
z^{*}(t/2)(\frac{1}{\phi(\omega)})^{*}(t/2)=\frac{z^{*}(t/2)}{\phi(t/2)}=D_{2}(\frac{z^{*}}{\phi})(t).$
By symmetry of $F$ we obtain $\frac{z}{\phi(\omega)}\in F$ and
$\sup\limits_{\omega-
mpt}\|\frac{z}{\phi(\omega)}\|_{F}\leq\|D_{2}\|_{F\rightarrow
F}\|\frac{z^{*}}{\phi}\|_{F}.$
The proof of (37) and also (36) is finished for $I=(0,1)$. If $I=(0,\infty)$,
then the existence of a measure preserving transformation
$\omega_{0}:I\rightarrow I$ requires additional assumption, in the first case
that $\phi(\infty)=\infty$, which we have because $p_{\phi,I}>0$. In the
second case, we need to have $z^{*}(\infty)=0$ when $z\in
M(M_{\phi}^{*},F)\neq\\{0\\}$. Suppose, on the contrary, $z^{*}(\infty)=a>0$.
Since $z^{*}\in M(M_{\phi}^{*},F)$ it follows that
$\frac{a}{\phi}\leq\frac{z^{*}}{\phi}\in F$ and $1/\phi\in F$ gives, by
maximality of the Marcinkiewicz space, that $1/\phi\in M_{\psi}^{*}$, where
fundamental function of $F$ is $f_{F}(t)=\psi(t)$. It means that
$\sup_{t>0}\frac{\psi(t)}{\phi(t)}<\infty$. On the other hand, since
$p_{\psi}\geq\alpha_{F}>q_{\phi}$ and $p_{\psi/\phi}\geq p_{\psi}-q_{\phi}>0$
it follows that for $0<\varepsilon<(p_{\psi}-q_{\phi})/2$ and for large $t$ we
obtain
$\frac{\psi(t)}{\phi(t)}\geq\frac{\psi(1)}{\phi(1)\,m_{\phi}(t)m_{\psi}(1/t)}\geq\frac{\psi(1)}{\phi(1)}\,t^{p_{\psi}-\varepsilon-(q_{\phi}+\varepsilon)}=\frac{\psi(1)}{\phi(1)}\,t^{p_{\psi}-q_{\phi}-2\varepsilon}\rightarrow\infty~{}{\rm
as}~{}t\rightarrow\infty,$
a contradiction. ∎
Proof of Theorem 11. (a) We have
$\displaystyle M_{\phi}^{*}\odot M(M_{\phi}^{*},F)$ $\displaystyle\equiv$
$\displaystyle L^{\infty}(\phi)^{(*)}\odot M[L^{\infty}(\phi)^{(*)},F]\hskip
31.29802pt({\rm using~{}Lemma~{}6})$ $\displaystyle=$ $\displaystyle
L^{\infty}(\phi)^{(*)}\odot F(1/\phi)^{(*)}\hskip 62.59605pt({\rm
by~{}Theorem~{}1(iv)})$ $\displaystyle\equiv$
$\displaystyle\left\\{[L^{\infty}(\phi)^{(*)}]^{1/2}[F(1/\phi)^{(*)}]^{1/2}\right\\}^{(1/2)}\hskip
5.69054pt({\rm using~{}Lemma~{}4})$ $\displaystyle=$
$\displaystyle\left\\{[L^{\infty}(\phi)^{1/2}F(1/\phi)^{1/2}]^{(*)}\right\\}^{(1/2)}$
$\displaystyle{\rm\hskip 2.84526pt(using~{}the~{}Krugljak-
Maligranda~{}result~{}[KM03],Thm~{}2})$ $\displaystyle=$
$\displaystyle\left\\{[(L^{\infty})^{1/2}F^{1/2}]^{(*)}\right\\}^{(1/2)}\equiv\left\\{[F^{(2)}]^{(*)}\right\\}^{(1/2)}\equiv
F^{(*)}\equiv F.$
Note that Lemma 4 can be used in the above equality since $0<p_{\phi,I}\leq
q_{\phi,I}<1$ implies that the operators $H,H^{*}:L^{\infty}(\phi)\rightarrow
L^{\infty}(\phi)$ are bounded. Moreover, $\alpha_{F}>q_{\phi,I}$ gives that
$H^{*}:F(1/\phi)\rightarrow F(1/\phi)$ is bounded and
$\beta_{F}<1+p_{\phi,I}$, which is satisfied, that $H:F(1/\phi)\rightarrow
F(1/\phi)$ is bounded (see [Ma80, Theorem 1] or [Ma83, Theorem 1).
(b) By the duality results
$\displaystyle E\odot M(E,\Lambda_{\phi})$ $\displaystyle\equiv$
$\displaystyle E\odot M((\Lambda_{\phi})^{\prime},E^{\prime})\equiv E\odot
M(M_{t/\phi(t)},E^{\prime})$ $\displaystyle\equiv$ $\displaystyle E\odot
M[L^{\infty}(\frac{t}{\phi(t)})^{(*)},E^{\prime}]\hskip 8.53581pt({\rm
using~{}Lemma~{}6~{}and~{}symmetry~{}of}~{}E)$ $\displaystyle=$ $\displaystyle
E\odot E^{\prime}(\frac{\phi(t)}{t})^{(*)}\equiv E^{(*)}\odot
E^{\prime}(\frac{\phi(t)}{t})^{(*)}\hskip 17.07164pt({\rm
by~{}Theorem~{}1(iv)})$ $\displaystyle\equiv$
$\displaystyle\left\\{[E^{(*)}]^{1/2}[E^{\prime}(\frac{\phi(t)}{t})^{(*)}]^{1/2}\right\\}^{(1/2)}\hskip
5.69054pt({\rm using~{}Lemma~{}4})$ $\displaystyle=$
$\displaystyle\left\\{[E^{1/2}E^{\prime}(\frac{\phi(t)}{t})^{1/2}]^{(*)}\right\\}^{(1/2)}$
$\displaystyle{\rm\hskip 14.22636pt(using~{}the~{}Krugljak-
Maligranda~{}result~{}[KM03],Thm~{}2})$ $\displaystyle=$
$\displaystyle\left\\{[E^{1/2}(E^{\prime})^{1/2}(t^{-1/2}\phi(t)^{1/2})]^{(*)}\right\\}^{(1/2)}\hskip
17.07164pt({\rm by~{}Theorem~{}1(iv)})$ $\displaystyle\equiv$
$\displaystyle\left[(E\odot
E^{\prime})^{(2)}(t^{-1/2}\phi(t)^{1/2})^{(1/2)}\right]^{(*)}$
$\displaystyle\hskip 14.22636pt({\rm
by~{}the~{}Lozanovskii~{}factorization~{}theorem)}$ $\displaystyle\equiv$
$\displaystyle\left[L^{2}(t^{-1/2}\phi(t)^{1/2})^{(1/2)}\right]^{(*)}\equiv
L^{1}(\frac{\phi(t)}{t})^{(*)}\equiv\Lambda_{\phi,1}.$
We must only control if the assumptions from Lemma 4 are satisfied in our
case, that is, if operators $H,H^{*}$ are bounded in $E$ and in
$E^{\prime}(\frac{\phi(t)}{t})$. Since
$0<\alpha_{E}\leq\beta_{E}<p_{\phi,I}<1$ it follows from Boyd’s result that
$H$ and $H^{*}$ are bounded in $E$ (cf. [KPS82], pp. 138-139, [BS88], Theorem
5.15 and [KMP07], pp. 126-129). The boundedness of $H$ in
$E^{\prime}(\frac{\phi(t)}{t})$ is equivalent to the estimate
$\|\frac{\phi(t)}{t^{2}}\int_{0}^{t}x(s)\frac{s}{\phi(s)}\,ds\|_{E^{\prime}}\leq
C_{1}\,\|x\|_{E^{\prime}}~{}{\rm for~{}all}~{}x\in E^{\prime},$
which is true if $\beta_{E^{\prime}}<p_{t^{2}/\phi(t),I}$ (cf. [Ma80, Theorem
1] or [Ma83, Theorem 1). The last strict inequality means that
$\beta_{E^{\prime}}=1-\alpha_{E}<p_{t^{2}/\phi(t),I}=2-q_{\phi,I}$ or
$\alpha_{E}>q_{\phi,I}-1$ which is true because $\alpha_{E}>0$ and
$q_{\phi,I}<1$. The boundedness of $H^{*}$ in $E^{\prime}(\frac{\phi(t)}{t})$
is equivalent to the estimate
$\|\frac{\phi(t)}{t}\int_{t}^{l}x(s)\frac{1}{\phi(s)}\,ds\|_{E^{\prime}}\leq
C_{2}\,\|x\|_{E^{\prime}}~{}{\rm for~{}all}~{}x\in E^{\prime},$
which is true if $\alpha_{E^{\prime}}>1-p_{\phi,I}$ (cf. [Ma80, Theorem 1] or
[Ma83, Theorem 1]). The last strict inequality means that
$\alpha_{E^{\prime}}=1-\beta_{E}>1-p_{\phi,I}$ or $\beta_{E}<p_{\phi,I}$, but
this is true by the assumption. ∎
Examples 5. (a) If $E=L^{q},F=L^{p}$ and $\phi(t)=t^{1/r}$, where $1\leq
p<r<q<\infty$, then from Theorem 11 we obtain
$L^{r,\infty}\odot M(L^{r,\infty},L^{p})=L^{p}~{}{\rm and}~{}~{}L^{q}\odot
M(L^{q},L^{r,1})=L^{r,1}.$ (38)
Equalities (38) we can also get from (31) and (30). In fact, space
$L^{r,\infty}$ satisfies upper $r$-estimate (cf. [Ma04], Theorem 5.4(a) and
[KK05], Theorem 3.1 and Corollary 3.9), therefore for $p<r$ is $p$-convex with
some constant $K\geq 1$ (cf. [LT79], Theorem 1.f.7). After renorming it is
$p$-convex with constant $1$ and we are getting from (31) the first equality
in (38) with equivalent norms. On the other hand, $L^{r,1}$ satisfies lower
$r$-estimate (cf. [Ma04], Theorem 5.1(a)), therefore for $q>r$ it is
$q$-concave with some constant $K\geq 1$ (cf. [LT79], Theorem 1.f.7). After
renorming is $q$-concave with constant $1$ and we are getting from (30) the
second equality in (38) with equivalent norms.
(b) If $E=L^{q,r},F=L^{p,r}$ and $\phi(t)=t^{1/s}$, where $1\leq p<s<q<\infty$
and $1\leq r\leq\infty$, then from Theorem 11 we obtain
$L^{s,\infty}\odot M(L^{s,\infty},L^{p,r})=L^{p,r}~{}{\rm
and}~{}~{}L^{q,r}\odot M(L^{q,r},L^{s,1})=L^{s,1}.$ (39)
(c) If $F=\Lambda_{\psi}$ and $\alpha_{F}=p_{\psi,I}>q_{\phi,I}$, then from
Theorem 11(a) we also obtain factorization (35) since $p_{\psi/\phi,I}\geq
p_{\psi,I}-q_{\phi,I}>0$.
Remark 8. In the case $I=(0,1)$ the assumption $\alpha_{F}>q_{\phi,I}$ implies
the imbedding $M_{\phi}^{*}\hookrightarrow F$, even the imbedding
$M_{\phi}^{*}\hookrightarrow\Lambda_{\psi}$, where $\psi$ is a fundamental
function of $F$ because $p_{\psi,1}\geq\alpha_{F}>q_{\phi,1}$ and
$p_{\psi/\phi,1}\geq p_{\psi,1}-q_{\phi,1}>0$ gives
$\int_{0}^{1}\frac{1}{\phi(t)}\psi^{\prime}(t)\,dt\leq\int_{0}^{1}\frac{\psi(t)}{\phi(t)}\,\frac{dt}{t}<\infty.$
Consequently, $M(M_{\phi}^{*},F)\neq\\{0\\}$.
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Paweł Kolwicz and Karol Leśnik, Institute of Mathematics of Electric Faculty
Poznań University of Technology, ul. Piotrowo 3a, 60-965 Poznań, Poland
E-mails: [email protected], [email protected]
Lech Maligranda, Department of Engineering Sciences and Mathematics
Luleå University of Technology, SE-971 87 Luleå, Sweden
E-mail: [email protected]
|
arxiv-papers
| 2012-11-13T21:18:11 |
2024-09-04T02:49:37.987099
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Pawe{\\l} Kolwicz, Karol Le\\'snik and Lech Maligranda",
"submitter": "Lech Maligranda",
"url": "https://arxiv.org/abs/1211.3135"
}
|
1211.3180
|
# Calculating $B$-meson decay constants using domain-wall light quarks and
nonperturbatively tuned relativistic $b$-quarks
Center for Computational Science, Boston University,
3 Cummington Mall, Boston, MA 02215, USA
E-mail
###### Abstract:
We calculate $B$-physics quantities using the RBC/UKQCD 2+1 flavor domain-wall
plus Iwasaki lattices and the relativistic heavy quark action developed by
Christ, Li and Lin. After tuning these parameters nonperturbatively, we
present our preliminary results for the calculation of the decay constants
$f_{B}$ and $f_{B_{s}}$ analyzing data at two lattice spacings of $a\approx
0.11$ fm and $a\approx 0.08$ fm.
## 1 Introduction
Using lattice QCD to compute the nonperturbative contributions to $B$-physics
quantities allows us to strengthen constraints on the CKM unitarity triangle
and determine CKM matrix elements. In the game of over-constraining the apex
of the CKM unitarity triangle, $B$-physics may also reveal signs of physics
beyond the Standard Model. The standard global fit of the CKM unitarity
triangle uses $B$-meson decay constants and mixing as an input [1, 2,
3].111For the latest results see: http://ckmfitter.in2p3.fr/,
http://utfit.roma1.infn.it/ and www.latticesaverages.org. Experiments such as
BaBar, Belle, and LHCb measure $B_{q}-\overline{B}_{q}$ mixing in terms of
mass differences (oscillation frequencies) $\Delta M_{q}$ to subpercent
accuracy [4]. In the Standard Model, these are parameterized by [5]
$\displaystyle\Delta
M_{q}=\frac{G_{F}^{2}m^{2}_{W}}{6\pi^{2}}\eta_{B}S_{0}M_{B_{q}}{f_{B_{q}}^{2}B_{B_{q}}}\lvert
V_{tq}^{*}V_{tb}\rvert^{2},$ (1)
where the index $q$ denotes a $d$\- or a $s$-quark and $M_{B_{q}}$ the mass of
the $B_{q}$-meson. Perturbatively computed inputs are the QCD coefficient
$\eta_{B}$ [5] and the Inami-Lim function $S_{0}$ [6]; contributions to be
computed nonperturbatively are the leptonic $B$-meson decay constant
$f_{B_{q}}$ and the bag parameter $B_{B_{q}}$. Precise theoretical
determinations of $f^{2}_{B_{q}}B_{B_{q}}$ are needed for precise extraction
of the product of CKM matrix elements $V^{*}_{tq}V_{tb}$. It is in particular
advantageous to compute the ratio [7]
$\displaystyle\frac{\Delta M_{s}}{\Delta
M_{d}}=\frac{M_{B_{s}}}{M_{B_{d}}}\,{\xi^{2}}\,\frac{\lvert
V_{ts}\rvert^{2}}{\lvert V_{td}\rvert^{2}},$ (2)
because statistical and systematic uncertainties largely cancel and the
nonperturbative contribution is contained in the $SU(3)$ breaking ratio
$\displaystyle\xi$
$\displaystyle=\frac{f_{B_{s}}\sqrt{B_{B_{s}}}}{f_{B_{d}}\sqrt{B_{B_{d}}}}.$
(3)
Further, constraints of the CKM triangle from $BR(B\to\tau\nu)$ require
precise knowledge of the decay constant $f_{B}$ [8, 9, 10]. Currently, only
HPQCD and Fermilab/MILC have published results for the decay constants $f_{B}$
and $f_{B_{s}}$ as well as $\xi$ using 2+1 dynamical flavor gauge field
configurations and ensembles with different lattice spacing. However, both
groups use MILC’s Asqtad lattices [11, 12, 13, 14] and hence an independent
cross-check is desirable.
Simulating $B$-physics on the lattice provides a special challenge because an
additional scale given by the large $b$-quark mass needs to be accommodated.
Here we report on our project to compute $B$-physics quantities using the
nonperturbatively tuned relativistic heavy quark action for $b$-quarks and
domain-wall fermions for the light $u,\,d,\,s$-quarks.
## 2 Computational setup
We base our $B$-physics project on gauge-field configurations generated by the
RBC/UKQCD collaborations with 2+1 flavors of domain-wall fermions and the
Iwasaki gluon action [15, 16]. The ensembles and their parameters are listed
in Tab. 1. Using the domain-wall action [17, 18] we generate on each
configuration six light valence-quark propagators with quark masses
$am_{\text{val}}^{24}$ = 0.005, 0.010, 0.020, 0.030, 0.0343 and $0.040$ on the
coarser $24^{3}$ ensembles and $am_{\text{val}}^{32}$ = 0.004, 0.006, 0.008,
0.025, 0.0272 and 0.030 on the finer $32^{3}$ ensembles. On the two “coarse”
$24^{3}$ ensembles ($a\approx 0.11$fm; $a^{-1}=1.729$ GeV) with light sea-
quark mass $am_{l}=0.005$ and $0.01$ we place one time source per
configuration, while on the “fine” $32^{3}$ ensembles ($a\approx 0.086$ fm;
$a^{-1}=2.281$ GeV) with light sea-quark masses $am_{l}=0.004,\,0.006$ and
$0.008$ we place two time sources per configuration separated by half the
temporal extent of the lattice. The mass of the light quark propagators with
$am^{24}_{\text{val}}=0.0343$ and $am^{32}_{\text{val}}=0.0272$ is extremely
close to the value of the physical strange quark mass on the corresponding
ensembles: $am_{s}^{24}=0.0348(11)$ and $am_{s}^{32}=0.0273(11)$.
For the $b$ quarks we use the anisotropic Sheikholeslami-Wohlert (clover)
action with the relativistic heavy-quark (RHQ) interpretation [19, 20]. We
require two experimental inputs for tuning the three parameters, $m_{0}a$,
$c_{P}$, $\zeta$, nonperturbatively. In our notation the RHQ action is defined
by
$\displaystyle
S=\sum_{n,n^{\prime}}\bar{\Psi}_{n}\left\\{\\!m_{0}+\gamma_{0}D_{0}-\\!\frac{aD_{0}^{2}}{2}+\zeta\left[\vec{\gamma}\cdot\vec{D}-\frac{a\left(\vec{D}\right)^{2}}{2}\right]\\!-a\sum_{\mu\nu}\frac{ic_{P}}{4}\sigma_{\mu\nu}F_{\mu\nu}\\!\right\\}_{\\!\\!n,n^{\prime}}\\!\\!\\!\\!\Psi_{n^{\prime}}.$
(4)
Table 1: Lattice simulation parameters used in our $B$-physics program. The columns list the lattice volume, approximate lattice spacing, light ($m_{l}$) and strange ($m_{h}$) sea-quark masses, unitary pion mass, and number of configurations and time sources analyzed. | | | | | | # time
---|---|---|---|---|---|---
$\left(L/a\right)^{3}\times\left(T/a\right)$ | $\approx a$(fm) | $am_{l}$ | $am_{h}$ | $M_{\pi}$(MeV) | # configs. | sources
$24^{3}\times 64$ | 0.11 | 0.005 | 0.040 | 329 | 1636 | 1
$24^{3}\times 64$ | 0.11 | 0.010 | 0.040 | 422 | 1419 | 1
$32^{3}\times 64$ | 0.086 | 0.004 | 0.030 | 289 | 628 | 2
$32^{3}\times 64$ | 0.086 | 0.006 | 0.030 | 345 | 889 | 2
$32^{3}\times 64$ | 0.086 | 0.008 | 0.030 | 394 | 544 | 2
We tune the parameters by probing seven points of the parameter space (see
Fig. 1) measuring each time the spin averaged $B_{s}$-meson mass
$\overline{M}$, the hyperfine splitting $\Delta_{M}$ and the ratio of the rest
mass over the kinetic mass $M_{1}/M_{2}$. Assuming linear dependence of these
measured quantities on the parameters for the seven points used, we obtain the
tuned parameters by matching $\overline{M}$ and $\Delta_{M}$ to experimental
results and demanding that $M_{1}/M_{2}$ equals one. Typically this requires a
few iterations to ensure that the tuned parameters are given by an
interpolation in a region where the measured parameter dependence is
sufficiently linear. Further details are presented in [21] where we also
predict bottomonium masses and mass splittings as test of our tuned
parameters.
Figure 1: The seven RHQ parameter sets are indicated by the six endpoints and the central point of the molecule-like structure in the $m_{0}a$, $c_{P}$ and $\zeta$ parameter space. Table 2: Tuned parameters $m_{0}a$, $c_{P}$, $\zeta$ of the RHQ action for simulating $b$-quarks on the ensembles listed in Table I. For details see [21]. $\approx a(fm)$ | $m_{sea}^{l}$ | $m_{0}a$ | $c_{P}$ | $\zeta$
---|---|---|---|---
0.11 | 0.005 | 8.43(7) | 5.7(2) | 3.11(9)
0.11 | 0.010 | 8.47(9) | 5.8(2) | 3.1(2)
0.086 | 0.004 | 4.07(6) | 3.7(1) | 1.86(8)
0.086 | 0.006 | 3.97(5) | 3.5(1) | 1.94(6)
0.086 | 0.008 | 3.95(6) | 3.6(1) | 1.99(8)
The computation of the $B$-meson decay constants presented here is performed
similarly to the tuning, i.e. we compute decay constants at the same seven
points in parameter space (see Fig. 1) and then interpolate these measurements
to the value of the tuned RHQ parameters given in Tab. 2. We measure the decay
constant $f_{B}$ by computing on the lattice the decay amplitude $\Phi_{B}$
which is proportional to the vacuum-to-meson matrix element of the heavy-light
axial vector current ${\cal A}_{\mu}=\bar{b}\gamma_{5}\gamma_{\mu}q$
$\displaystyle\langle 0|{\cal
A}_{\mu}|B(p)\rangle/\sqrt{M_{B}}=ip^{\mu}\Phi_{B}^{(0)}/M_{B}$ (5)
with $p_{\mu}$ the four-momentum of the $B$-meson. To reduce discretization
errors we use an $O(a)$-improved axial vector current on the lattice
$\displaystyle\Phi_{B}^{\text{imp}}=\Phi_{B}^{(0)}+c_{1}\Phi_{B}^{(1)},$ (6)
where the coefficient $c_{1}$ is computed at 1-loop with mean-field improved
lattice perturbation theory [22]. Multiplying the decay amplitude by the
renormalization factor $Z_{\Phi}$, the lattice spacing $a$ and the mass of the
$B_{q}$-meson $M_{B_{q}}$ we obtain the decay constant
$\displaystyle
f_{B_{q}}=Z_{\Phi}\Phi_{B}^{\text{imp}}a^{-3/2}/\sqrt{M_{B_{q}}}.$ (7)
In our final setup we intend to compute the renormalization factor $Z_{\Phi}$
following the mostly nonperturbative method outlined in [23], i.e. $Z_{\Phi}$
is split into a nonperturbative part containing the flavor-conserving factors
$Z_{V}^{ll}$ and $Z_{V}^{bb}$ and a perturbatively computed factor
$\varrho_{bl}$ which is expected to be close to one and to have a more
convergent series expansion in $\alpha_{s}$:
$\displaystyle Z_{\Phi}=\varrho_{bl}\sqrt{Z_{V}^{bb}Z_{V}^{ll}}.$ (8)
Here we use the nonperturbatively obtained $Z_{V}^{ll}$ [16], while we compute
$Z_{V}^{bb}$ and $\varrho_{bl}$ at 1-loop in mean-field improved lattice
perturbation theory [22]. Work is in progress to obtain $Z_{V}^{bb}$
nonperturbatively on all ensembles [24].
## 3 Preliminary results
In Fig. 2 we show our preliminary results for the renormalized,
$O(a)$-improved decay amplitude
$\Phi_{B}^{\text{ren}}=Z_{\Phi}\Phi_{B}^{\text{imp}}$. While a chiral
extrapolation to the physical $d$-quark mass is needed to obtain
$\Phi_{B_{d}}$, we may take advantage of the simulation data close to the
physical strange quark mass to read off a rough estimate of $\Phi_{B_{s}}$.
This indicates our result is in agreement with the existing literature and we
expect statistical uncertainties of a few MeV. Figure 3 shows our results for
the SU(3) breaking ratio of $\Phi_{B_{s}}/\Phi_{B_{q}}$. The data exhibit a
weak dependence on the light sea-quark mass as well as a mild dependence on
the lattice spacing.
## 4 Outlook
Currently work is in progress on the nonperturbative computation of
$Z_{V}^{bb}$ as well as checks on the perturbative computation of the
correction factor $\varrho_{bl}$ and the $O(a)$-improvement coefficient
$c_{1}$. We are also working on combined, correlated fits using expressions
based on SU(3) and SU(2) heavy-light meson chiral perturbation theory [26, 27,
28, 29]. With the individual parts being in good shape we hope to publish
results soon. We will move on to the computation of $B_{q}-\overline{B}_{q}$
mixing matrix elements and their ratio $\xi$.
Figure 2: Renormalized and $O(a)$-improved decay amplitude $\Phi_{B_{q}}$
computed on the full data set and shown in $a_{32}$ lattice units. For better
visibility data points for $m_{\text{sea}}^{l}=0.01$ and $0.008$ are plotted
with a small, positive, horizontal offset; data points for
$m_{\text{sea}}^{l}=0.004$ have a small, negative, horizontal offset. The
vertical gray lines with error-band indicate the physical values of the $d$\-
and $s$-quark masses [25, 15, 16]. Figure 3: Ratio of decay amplitudes
$\Phi_{B_{s}}/\Phi_{B_{q}}$ computed on the full data set and shown in
$a_{32}$ lattice units. For better visibility data points for
$m_{\text{sea}}^{l}=0.01$ and $0.008$ are plotted with a small, positive,
horizontal offset; data points for $m_{\text{sea}}^{l}=0.004$ have a small,
negative, horizontal offset. The vertical gray lines with error-band indicate
the physical values of the $d$\- and $s$-quark masses [25, 15, 16]. The open
black triangle (square) marks the value of the valence strange-quark mass in
the simulation at which the plotted ratio is one by construction.
## Acknowledgments
We thank our colleagues of the RBC and UKQCD collaborations for useful help
and discussions. Numerical computations for this work utilized USQCD resources
at Fermilab, in part funded by the Office of Science of the U.S. Department of
Energy, as well as computers at Brookhaven National Laboratroy and Columbia
University. O.W. acknowledges support at Boston University by the U.S. DOE
grant DE-FC02-06ER41440.
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|
arxiv-papers
| 2012-11-14T02:17:48 |
2024-09-04T02:49:38.006765
|
{
"license": "Public Domain",
"authors": "Oliver Witzel",
"submitter": "Oliver Witzel",
"url": "https://arxiv.org/abs/1211.3180"
}
|
1211.3204
|
# Seidel representations
and quantum cohomology of toric orbifolds
Hsian-Hua Tseng Department of Mathematics
Ohio State University
100 Math Tower
231 West 18th Avenue
Columbus, OH 43210-1174
USA [email protected] and Dongning Wang Department of Mathematics
University of Wisconsin-Madison
Van Vleck Hall
480 Lincoln Dr.
Madison, WI 53706
USA [email protected]
###### Abstract.
We use Seidel representation for symplectic orbifolds constructed in [TW] to
compute the quantum cohomology ring of a compact symplectic toric orbifold
$(\mathcal{X},\omega)$.
###### Contents
1. 1 Introduction
2. 2 Preliminary on Toric orbifolds
1. 2.1 Construction via Stacky Fans
2. 2.2 Symplectic Toric Orbifolds via Labelled Polytopes
3. 2.3 Chen-Ruan orbifold cohomology of toric orbifolds
4. 2.4 Hamiltonian Loops of Toric Orbifolds
5. 2.5 Hamiltonian Orbifiber Bundles over Sphere
3. 3 Seidel Elements
1. 3.1 Review of Seidel Representation
2. 3.2 Seidel Element for Toric Orbifolds
4. 4 Quantum Cohomology of Toric Orbifolds
1. 4.1 Main Result
2. 4.2 The Fano Case
## 1\. Introduction
In this paper we apply the Seidel representations for compact symplectic
orbifolds, constructed in [TW], to give a description of quantum cohomology
rings of symplectic toric orbifolds. The main idea in this work can be briefly
summarized as follows. Let $(\mathcal{X},\omega)$ be a compact symplectic
orbifold. Denote by $Ham(\mathcal{X},\omega)$ the ($2$-)group of Hamiltonian
diffeomorphisms of $(\mathcal{X},\omega)$. The Seidel representation of
$(\mathcal{X},\omega)$ is a group homomorphism
$\mathcal{S}:\pi_{1}(Ham(\mathcal{X},\omega))\to
QH_{orb}^{*}(\mathcal{X},\omega)^{\times}$
from the fundamental group of $Ham(\mathcal{X},\omega)$ to the group
$QH_{orb}^{*}(\mathcal{X},\omega)^{\times}$ of multiplicatively invertible
elements in the quantum cohomology ring of $(\mathcal{X},\omega)$. Suppose
there is a collection of loops $a_{1},a_{2},...,a_{k}$ in
$Ham(\mathcal{X},\omega)$ which compose to the identity loop $e$, namely
$a_{1}\cdot a_{2}\cdot...\cdot a_{k}=e,$
Since $\mathcal{S}$ is a homomorphism, we have
$\mathcal{S}(a_{1})*\mathcal{S}(a_{2})*...*\mathcal{S}(a_{k})=\mathcal{S}(e)=1.$
This gives a relation in $QH_{orb}^{*}(\mathcal{X},\omega)$.
Suppose $(\mathcal{X},\omega)$ is a compact symplectic toric orbifold. Then
$\mathcal{X}$ can be defined by a combinatorial object called the stacky fan
$\mathbf{\Sigma}=(\mathbf{N},\Sigma,\beta)$, see Section 2.1 for a more
detailed discussion. Every element $v\in\mathbf{N}$ in the lattice
$\mathbf{N}$ determines a $\mathbb{C}^{\times}$-action on $\mathcal{X}$ and
hence a loop in $Ham(\mathcal{X},\omega)$. Let $S_{v}\in
QH_{orb}^{*}(\mathcal{X},\omega)$ denote the Seidel element corresponding to
this loop. As discussed above, if $v_{1},v_{2},...,v_{k}\in\mathbf{N}$ are
such that $v_{1}+v_{2}+...+v_{k}=0$ in $\mathbf{N}$. Then we have
$S_{v_{1}}*S_{v_{2}}*...*S_{v_{k}}=S_{0}=1.$
This allows us to make use of additive relations in $\mathbf{N}$ to give a
presentation of $QH_{orb}^{*}(\mathcal{X},\omega)$.
We now describe our results in more details. Let $\mathcal{X}$ be a compact
symplectic toric orbifold associated with a labeled polytope111See Theorem 2.2
for the correspondence between symplectic toric orbifolds and labeled
polytopes. $\Delta$ and let $\mathbf{\Sigma}=(\mathbf{N},\Sigma,\beta)$ be the
stack fan associated to $\Delta$. Let $y_{1},...,y_{N}\in\mathbf{N}$ be
minimal generators of the rays in $\Sigma$. For each cone $\sigma$ in
$\Sigma$, define
$\text{SBox}(\sigma):=\left\\{b\in
N\,|\,b=\sum_{y_{i}\in\sigma}a_{i}y_{i},0\leq a_{i}<1\right\\},$
and let $\text{Gen}(\sigma)\subset\text{SBox}(\sigma)$ be the set of elements
which cannot be generated by other lattice points in $\sigma$. The union
$\text{Gen}(\Sigma):=\cup_{\sigma\in\Sigma}\text{Gen}(\sigma)$ is a finite set
and we write $\text{Gen}(\Sigma)=\\{y_{N+1},...,y_{M}\\}$. For each $y_{i}$ we
introduce a variable $X_{i}$.
###### Theorem 1.1 (See Theorem 4.4).
There is an isomorphism of graded rings
$\frac{\Lambda[X_{1},...,X_{M}]}{Clos_{\mathfrak{v}_{T}}(\left\langle\mathfrak{P}_{\xi}|\xi=1,...,n\right\rangle+SR_{\omega}+\mathcal{J}(\Sigma))}\simeq
QH_{orb}^{*}(\mathcal{X},\Lambda),$
where
1. (1)
$\Lambda$ is the Novikov ring in Definition 3.1;
2. (2)
The symbol $Clos_{\mathfrak{v}_{T}}(-)$ indicates that the closure with
respect to the valuation $\mathfrak{v}_{T}$ in (3.4).
3. (3)
$\mathfrak{P}_{\xi}\in\Lambda[X_{1},...,X_{M}],\xi=1,2,...,n$ are constructed
in Theorem 4.4;
4. (4)
$SR_{\omega}$ is the quantum Stanley-Reisner ideal in Definition 4.2.
5. (5)
$\mathcal{J}(\mathbf{\Sigma})$ is the cone ideal defined in (2.5).
Quantum cohomology rings of toric orbifolds have been studied in various
cases. The case of weighted projective lines is computed in [AGV, Section 9].
The quantum cohomology ring of an arbitrary weighted projective space is
computed as a consequence of a mirror theorem in [CCLT]. The case of orbifold
projective lines with at most two cyclic orbifold points is computed in
[MiTs]. The quantum cohomology ring of a weak Fano toric orbifold is computed
as a consequence of the mirror theorem in [CCIT]. For an arbitrary toric
orbifolds, the quantum cohomology ring is computed in [GW] using the quantum
Kirwan map [W]. These previous works use various methods algebraic in nature.
The method in this paper, based on Seidel representation, is symplectic.
The rest of this paper is organized as follows. Section 2 contains reviews of
preparatory materials including the basics of symplectic toric orbifolds,
Chen-Ruan cohomology, and Hamiltonian loops. In Section 3.1 we review the
results in [TW] on the construction of Seidel representation for symplectic
orbifolds. In Section 3.2 we calculate Seidel elements arising from circle
actions on a symplectic toric orbifold. This calculation is used in Section
4.1 to derive a presentation of quantum cohomology ring of a symplectic toric
orbifold. In Section 4.2 we discuss the case of Fano toric orbifolds.
Throughout this paper, $\mathcal{X}(\mathbf{\Sigma})$ is the $2n$-dimensional
compact symplectic toric orbifold associated with a labeled polytope $\Delta$
in the sense of Section 2.2. And $\mathbf{\Sigma}=(\mathbf{N},\Sigma,\beta)$
is the stack fan associated to $\Delta$.
## Acknowledgement
We thank Erkao Bao, Lev Borisov, Cheol-Hyun Cho, Conan Leung, and Yong-Geun Oh
for valuable discussions.
## 2\. Preliminary on Toric orbifolds
In this section we review some basic constructions and facts about symplectic
toric orbifolds. In this paper we only consider compact toric orbifolds whose
generic stabilizer group is trivial, and we limit our discussion to that case.
### 2.1. Construction via Stacky Fans
In algebraic geometry, toric orbifold are constructed using the combinatorial
object called stacky fans. In this subsection we review this construction
following [BCS].
By definition, a stacky fan consists of the following data
$\mathbf{\Sigma}=(\mathbf{N},\Sigma,\beta),$
where
1. (1)
$\mathbf{N}$ is a finitely generated free abelian group of rank $n$;
2. (2)
$\Sigma\subset\mathbf{N}_{\mathbb{Q}}=\mathbf{N}\otimes_{\mathbb{Z}}\mathbb{Q}$
is a complete simplicial fan, with $\rho_{1},...,\rho_{N}$ being its
$1$-dimensional cones;
3. (3)
$\beta:\mathbb{Z}^{N}\to\mathbf{N}$ is a map determined by the elements
$\\{b_{1},\cdots,b_{N}\\}$ in $\mathbf{N}$ satisfing that $b_{i}\in\rho_{i}$.
More precisely, let $e_{1},...,e_{N}\in\mathbb{Z}^{N}$ be the standard basis,
then $\beta(e_{i}):=b_{i}$.
We assume that $\beta$ has finite cokernel, and
$\\{b_{1},...,b_{N}\\}\subset\mathbf{N}\subset\mathbf{N}_{\mathbb{Q}}$
generate the simplicial fan $\Sigma$.
The toric orbifold (also known as toric Deligne-Mumford stack)
$\mathcal{X}(\mathbf{\Sigma})$ associated to $\mathbf{\Sigma}$ is defined to
be the following quotient stack
(2.1) $\mathcal{X}(\mathbf{\Sigma}):=[Z/G],$
whose definition may be explained as follows:
1. (1)
$Z$ is the open subvariety $\mathbb{C}^{N}\setminus\mathbb{V}(J_{\Sigma})$.
Here $J_{\Sigma}$ is the irrelevant ideal of the fan, defined as follows: let
$\mathbb{C}[z_{1},...,z_{N}]$ be the coordinate ring of $\mathbb{C}^{N}$, then
$J_{\Sigma}$ is the ideal of $\mathbb{C}[z_{1},...,z_{N}]$ generated by the
monomials $\prod_{\sigma_{i}\nsubseteq\sigma}z_{i}$ where $\sigma$ run through
all cones in $\Sigma$.
2. (2)
$G$ is an algebraic torus defined by
$G=\text{Hom}_{\mathbb{Z}}(\mathbf{N}^{\vee},\mathbb{C}^{*})$, where
$\mathbf{N}^{\vee}$ appears in the Gale dual
$\beta^{\vee}:\mathbb{Z}^{N}\to\mathbf{N}^{\vee}$ of $\beta$ (see [BCS]). The
$G$-action on $Z$ is given by a group homomorphism
$\alpha:G\to(\mathbb{C}^{*})^{N}$ obtained by applying the functor
$\text{Hom}_{\mathbb{Z}}(-,\mathbb{C}^{*})$ to the Gale dual
$\beta^{\vee}:\mathbb{Z}^{N}\to N^{\vee}$ of $\beta$.
The quotient (2.1) may be taken in different categories. In this paper we deal
with symplectic toric orbifolds. For this reason we consider the quotient as a
differentiable stack. Indeed the $G$-action on $Z$ defines a groupoid
$G\ltimes Z:=(G\times Z\rightrightarrows Z)$ where the source and target maps
are the projection to the second factor and the $G$-action respectively. It is
easy to see that $G\ltimes Z$ is a proper Lie groupoid and the quotient stack
(2.1) is the differentiable stack associated to this Lie groupoid.
The toric orbifold $\mathcal{X}(\mathbf{\Sigma})$ has a collection of
naturally defined étale charts which we describe. Let $\sigma\in\Sigma$ be a
$k$-dimensional cone generated by $\\{b_{i_{1}},...,b_{i_{k}}\\}$. Define the
open subset $U(\sigma)\subset\mathbb{C}^{N}$ as
$U(\sigma)=\\{(z_{1},...,z_{N})\in\mathbb{C}^{N}|z_{j}\neq 0\ \forall
j\notin\\{i_{1},...,i_{k}\\}\\}$. Then $U(\sigma)\subset U(\sigma^{\prime})$
if $\sigma$ is contained in $\sigma^{\prime}$, and
$\\{U(\sigma)|\sigma\in\Sigma,\ dim\ \sigma=n\\}$ is an open cover of $Z$.
Each $n$-dimensional cone $\sigma$ induces an orbifold chart of
$\mathcal{X}(\mathbf{\Sigma})$ which covers $U_{\sigma}:=U(\sigma)/G$. More
explicitly, let $\mathbf{N}_{\sigma}$ be the sublattice of $\mathbf{N}$
generated by $\\{b_{i_{1}},...,b_{i_{n}}\\}$, $N_{\sigma}^{*}$ be the dual
lattice of $\mathbf{N}_{\sigma}$, and $\\{u_{j}\\}_{j=1}^{n}$ be the dual
basis of $\mathbf{N}_{\sigma}^{*}$ so that $\langle
b_{i_{k}},u_{j}\rangle=\delta_{k,j}$, then we have a map from $U_{\sigma}$ to
$\mathbb{C}^{n}$ by
$w_{j}=z_{1}^{\langle b_{1},u_{j}\rangle}\cdot...\cdot z_{N}^{\langle
b_{N},u_{j}\rangle},\ \ \ \ j=1,...,n.$
The image of $U(\sigma)$ under this map, denoted as $V_{\sigma}$, carries a
group action by $G_{\sigma}:=\mathbf{N}/\mathbf{N}_{\sigma}$:
$g\cdot w_{j}=e^{2\pi i\langle\tilde{g},u_{j}\rangle}w_{j}\ \ \ \ \text{for}\
\tilde{g}\in\mathbf{N}\ \text{lifting}\ g\in\mathbf{N}/\mathbf{N}_{\sigma},\ \
\ j=1,...,n.$
Then $G_{\sigma}\ltimes V_{\sigma}$ defines an orbifold chart over
$U_{\sigma}$.
Now if $\tau$ is a $k$-dimensional cone contained in an $n$-dimensional cone
$\sigma$, then the orbit $O_{\tau}$ determined by $\tau$ has a neighborhood
$U_{\tau}$, the orbifold chart $G_{\sigma}\ltimes V_{\sigma}$ restricted to
$U_{\tau}$ defines an orbifold chart of $U_{\tau}$. Note that this chart is
not effective. After reduction, we get an orbifold chart $G_{\tau}\ltimes
V_{\tau}$, where
$G_{\tau}=(N_{\tau}\otimes_{\mathbb{Z}}\mathbb{Q})\cap\mathbf{N}\ \
/\mathbf{N}_{\tau}$, $V_{\tau}$ is an open set of $\mathbb{C}^{n}$.
Given an orbifold $\mathcal{X}$ one can consider its inertia orbifold
$I\mathcal{X}:=\mathcal{X}\times_{\mathcal{X}\times\mathcal{X}}\mathcal{X}$,
where the fiber product is taken over the diagonal map
$\mathcal{X}\to\mathcal{X}\times\mathcal{X}$. In the toric case we can give a
more combinatorial description of the inertia orbifolds, following [BCS]. Let
$\mathcal{X}(\mathbf{\Sigma})$ be the toric orbifold defined by the stacky fan
$\mathbf{\Sigma}:=(\mathbf{N},\Sigma,\beta)$. For a cone $\tau\in\Sigma$,
define
$\text{Box}(\tau):=\left\\{v\in\mathbf{N}\,|\,v=\sum_{b_{i}\in\tau}r_{i}b_{i},0\leq
r_{i}<1\right\\},$
and set
$\text{Box}(\mathbf{\Sigma}):=\bigcup_{\tau\in\Sigma}\text{Box}(\tau)$. Then
we have:
1. (1)
The set $\text{Box}(\mathbf{\Sigma})$ indexes the components of the inertia
orbifold of $\mathcal{X}(\mathbf{\Sigma})$:
$I\mathcal{X}(\mathbf{\Sigma})=\sqcup_{v\in\text{Box}(\mathbf{\Sigma})}\mathcal{X}_{(v)}.$
Here $\sigma(v)\in\Sigma$ is the minimal cone containing $v$ and
$\mathcal{X}_{(v)}=\mathcal{X}(\mathbf{\Sigma}/\sigma(v))$ is the toric
orbifold associated to the stacky fan $\mathbf{\Sigma}/\sigma(v)$ defined in
[BCS, Section 4]. Each component $\mathcal{X}_{(v)}$ is called a twisted
sector, and $\mathcal{X}_{(0)}\simeq\mathcal{X}$ is called the trivial twisted
sector or untwisted sector.
2. (2)
There is a natural involution $\mathcal{I}:I\mathcal{X}\to I\mathcal{X}$
defined by $\mathcal{I}((x,v)):=(x,v^{-1})$, where $v\in\text{Box}(\tau)$ and
$v^{-1}$ is the unique element in $\text{Box}(\tau)$ such that
$v^{-1}+v\in\mathbf{N}_{\tau}$. Note that $v^{-1}$ and $-v$ are different, the
later does not lie in $\text{Box}(\tau)$.
### 2.2. Symplectic Toric Orbifolds via Labelled Polytopes
Intrinsically a symplectic toric orbifold is a symplectic orbifold with a
Hamiltonian action by a half-dimensional compact torus. In [LT] symplectic
toric orbifolds are classified by the combinatorial objects called labelled
polytopes. This is a generalization of the classification of symplectic toric
manifolds by Delzant polytopes. In this subsection we review the relation
between symplectic toric orbifolds, labelled polytopes, and the stacky fan
construction in the previous subsection.
Let $(\mathcal{X},\omega)$ be a $2n$-dimensional (compact) symplectic
orbifold. There is a Hamiltonian $T^{n}$-action on $\mathcal{X}$ with moment
map
$\Phi:\mathcal{X}\to\textbf{t}^{*},$
where t is the Lie algebra of $T^{n}$ with a lattice l and $\textbf{t}^{*}$ is
the dual vector space of t. The image $\Phi(\mathcal{X})$ of the moment map is
a rational simple convex polytope which is defined as below:
###### Definition 2.1.
A convex polytope $\Delta\subset\textbf{t}^{*}$ is rational if
$\Delta=\bigcap_{i=1}^{N}\\{\alpha\in\textbf{t}^{*}|\left\langle\alpha,n_{i}\right\rangle\leq\lambda_{i}\\}$
for some $n_{i}\in\textbf{l}$ and $\lambda_{i}\in$.
A (closed) facet is a face of $\Delta$ of codimension one in $\Delta$. An open
facet is the relative interior of a facet. An n dimensional polytope is simple
if exactly $n$ facets meet at every vertex.
A convex rational simple polytope $\Delta$ such that $dim\Delta=dim\
\textbf{t}$, plus a positive integer attached to each open facet, is called a
labeled polytope. Two labeled polytopes are isomorphic if one can be mapped to
the other by a translation and the corresponding open facets have the same
integer labels.
The following result is due to Lerman and Tolman.
###### Theorem 2.2 ([LT], Theorem 1.5).
1. (1)
A compact symplectic toric orbifold $(\mathcal{X},\omega,T,\Phi)$ naturally
corresponds to a labeled polytope, namely the image of the moment map
$\Phi(\mathcal{X})$, which is a rational simple polytope. For every open facet
$\mathaccent 23{F}$ of $\Phi(\mathcal{X})$ there exists a positive integer
$n_{\mathaccent 23{F}}$ such that the structure group of every
$x\in\Phi^{-1}(\mathaccent 23{F})$ is $\mathbb{Z}/n_{\mathaccent
23{F}}\mathbb{Z}$.
2. (2)
Two compact symplectic toric orbifolds are isomorphic if and only if their
associated labeled polytopes are isomorphic.
3. (3)
Every labeled polytope can be realized as the image of the moment map for some
compact symplectic toric orbifold.
Now, we consider a symplectic toric orbifold $(\mathcal{X},\omega,T,\Phi)$
determined by a polytope $\Delta\subset\textbf{t}^{*}$ with labels $m_{i}$ on
facets $F_{i}$. Let $y_{i}$ be the primitive outward normal vectors of
$F_{i}$. For each face $F$ of $\Delta$, let $\sigma_{F}\in\Sigma$ be the cone
determined by the collection of vectors $\\{y_{i}|F\subset F_{i}\\}$. Define
* •
$\mathbf{N}=\textbf{l}=\\{\sum_{i}k_{i}y_{i}|k_{i}\in\mathbb{Z}\\}\subset\textbf{t}$,
* •
$\Sigma=\\{\sigma_{F}|F\ \text{a face of}\ \Delta\\}$,
* •
and $b_{i}=m_{i}y_{i}$, i.e. $\beta(e_{i})=m_{i}y_{i}$.
Then the data $(\mathbf{N},\Sigma,\\{b_{i}\\}_{i=1}^{N})$ give a stacky fan.
In this way every labelled polytope gives rise to a stacky fan.
Every corner (0-dimensional face) $C$ of $\Delta$ determines a point
$x\in\mathcal{X}$ which is fixed by the torus action. Let $\sigma_{C}$ be the
cone corresponding to $C$, then $\sigma_{C}$ determines an orbifold chart
$G_{\sigma(C)}\ltimes V_{\sigma(C)}$ covering $x$ as in Section 2.1. Let
$\\{u^{C}_{j}|j=1,...,n\\}\subset\textbf{t}^{*}$ be the dual basis of
$\\{y_{i}|y_{i}\in\sigma(C),\ 1\leq j\leq N\\}$. Then the following lemma is
an orbifold version of Theorem 3.1.2 in [dS].
###### Lemma 2.3.
The point $x$ is covered by an orbifold chart $G_{C}\ltimes V_{C}$ such that
* •
$G_{C}\ltimes V_{C}$ is isomorphic to $G_{\sigma(C)}\ltimes V_{\sigma(C)}$;
* •
the symplectic form can be written as
$\omega=\sum_{j=1}^{n}dp_{j}\wedge dq_{j}\ \ \ with\ w_{j}=p_{j}+iq_{j};$
* •
The moment map can be written as
$\Phi(p_{1},...,p_{n};q_{1},...,q_{n})=\Phi(x)-\sum_{j=1}^{n}u^{C}_{j}(p_{j}^{2}+q_{j}^{2}).$
### 2.3. Chen-Ruan orbifold cohomology of toric orbifolds
In this subsection we describe the calculation of the Chen-Ruan orbifold
cohomology ring of toric orbifolds, following [BCS]222Strictly speaking what
is computed in [BCS] is the orbifold Chow ring. However the computation for
Chen-Ruan orbifold cohomology ring is identical..
Let $\mathcal{X}(\mathbf{\Sigma})$ be a compact symplectic toric orbifold
given by a stacky fan $\mathbf{\Sigma}=(\mathbf{N},\Sigma,\beta)$. Recall that
the Chen-Ruan orbifold cohomology
$H_{CR}^{*}(\mathcal{X}(\mathbf{\Sigma}),\mathbb{Q})$ is defined as the direct
sum of the cohomology groups of its inertia orbifold with a shifted grading:
For $v=\sum_{b_{i}\in\sigma(v)}r_{i}b_{i}\in\text{Box}(\Sigma)$. The
corresponding twisted sector $\mathcal{X}_{(v)}$ is associated with a number
$\iota_{v}=\sum_{b_{i}\in\sigma(v)}r_{i}$ called age or degree shifting
number. We have
$H_{CR}^{*}(\mathcal{X}(\mathbf{\Sigma}),\mathbb{Q})=\oplus_{v\in\text{Box}(\mathbf{\Sigma})}H^{*-2\iota_{v}}(\mathcal{X}_{(v)},\mathbb{Q}).$
There is a product on $H_{CR}^{*}(\mathcal{X}(\mathbf{\Sigma}),\mathbb{Q})$,
called the Chen-Ruan cup product, which is defined using genus $0$ degree $0$
three-point Gromov-Witten invariants of $\mathcal{X}(\mathbf{\Sigma})$. We
refer to [CR1] for more details of the definition of this product. This
construction makes $H_{CR}^{*}(\mathcal{X}(\mathbf{\Sigma}),\mathbb{Q})$ into
a graded algebra.
Let $\mathbf{M}=\mathbf{N}^{*}=\text{Hom}_{\mathbb{Z}}(\mathbf{N},\mathbb{Z})$
be the dual of $\mathbf{N}$. Let $\mathbb{Q}[\mathbf{N}]^{\mathbf{\Sigma}}$ be
the group ring of $\mathbf{N}$, i.e.
$\mathbb{Q}[\mathbf{N}]^{\mathbf{\Sigma}}:=\bigoplus_{c\in\mathbf{N}}\mathbb{Q}\lambda^{v}$,
$\lambda$ is the formal variable. A $\mathbb{Q}$-grading on
$\mathbb{Q}[\mathbf{N}]^{\mathbf{\Sigma}}$ is defined as follows. For
$v\in\mathbf{N}$, if $v=\sum_{b_{i}\in\sigma(v)}r_{i}b_{i}$ where $\sigma(v)$
is the minimal cone in $\Sigma$ containing ${v}$ and $r_{i}$ are nonnegative
rational numbers, then we define
(2.2) $\text{deg}\,(\lambda^{v}):=\sum_{b_{i}\in\sigma(v)}r_{i}.$
Define the following multiplication on $\mathbb{Q}[N]^{\mathbf{\Sigma}}$:
(2.3)
$\lambda^{v_{1}}\cdot\lambda^{v_{2}}:=\begin{cases}\lambda^{v_{1}+v_{2}}&\text{if
there is a cone}~{}\sigma\in\Sigma~{}\text{such
that}~{}{v}_{1},{v}_{2}\in\sigma\,,\\\ 0&\text{otherwise}\,.\end{cases}$
Let $\mathcal{I}(\mathbf{\Sigma})$ be the ideal in
$\mathbb{Q}[\mathbf{N}]^{\mathbf{\Sigma}}$ generated by the elements
$\sum_{i=1}^{n}\theta(b_{i})\lambda^{b_{i}},\theta\in\mathbf{M}$. Then by
[BCS, Theorem 1.1], there is an isomorphism of $\mathbb{Q}$-graded algebras:
(2.4)
$H_{CR}^{*}\left(\mathcal{X}(\mathbf{\Sigma}),\mathbb{Q}\right)\cong\frac{\mathbb{Q}[\mathbf{N}]^{\mathbf{\Sigma}}}{\mathcal{I}(\mathbf{\Sigma})}.$
We will rewrite (2.4) in terms of a quotient of polynomial ring by some ideal.
Denote $X_{i}=\lambda^{y_{i}}$ for $i=1,...,N$.
Let
$\text{SBox}(\sigma):=\left\\{b\in
N\,|\,b=\sum_{y_{i}\in\sigma}a_{i}y_{i},0\leq a_{i}<1\right\\},$
and $\text{Gen}(\sigma)\subset\text{SBox}(\sigma)$ be the set of minimal
elements, which means these elements cannot be generated by other lattice
points in the cone $\sigma$. Obviously if $\sigma\subset\sigma^{\prime}$ then
$\text{Gen}(\sigma)\subset\text{Gen}(\sigma^{\prime})$.
Then $\text{Gen}(\Sigma):=\cup_{\sigma\in\Sigma}\text{Gen}(\sigma)$ is a
finite set. For convenience, we denote
$\text{Gen}(\Sigma)=\\{y_{N+1},...,y_{M}\\}$, and define
$X_{N+1}=\lambda^{y_{N+1}},...,X_{M}=\lambda^{y_{M}}$. We call
$I\subset\\{1,...,M\\}$ a generalized primitive collection if
* •
$\\{y_{i}|i\in I\\}$ is not contained in a cone,
* •
any proper subset of $\\{y_{i}|i\in I\\}$ is contained in some cone.
Note that when a generalized primitive collection $I$ is a subset of
$\\{1,...N\\}$, then it is a primitive collection in the sense of [Ba]. Denote
by $\mathcal{GP}$ the set of all generalized primitive collections.
For top-dimensional cones $\sigma_{j}$, $j=1,...,N$, define an ideal
$\mathcal{J}(\sigma_{j}):=\left\langle\prod_{t_{i}>0,y_{i}\in\sigma_{j}}X_{i}^{t_{i}}-\prod_{t_{i}<0,y_{i}\in\sigma_{j}}X_{i}^{-t_{i}}|\sum_{y_{i}\in\sigma_{j}}t_{i}y_{i}=0,\
t_{i}\in\mathbb{Z},\vec{t}\neq\vec{0}\right\rangle$
Set
(2.5) $\mathcal{J}(\mathbf{\Sigma})=\sum_{j=1}^{N}\mathcal{J}(\sigma_{j}).$
We call $\mathcal{J}(\mathbf{\Sigma})$ the cone ideal, and elements in
$\mathcal{J}(\mathbf{\Sigma})$ the cone relations.
Now the Chen-Ruan cohomology can be rewritten as:
$H^{*}_{CR}(\mathcal{X},\mathbb{Q})\cong\frac{\mathbb{Q}[[X_{1},...,X_{M}]]}{\mathcal{I}(\mathbf{\Sigma})+\left\langle\Pi_{i\in
I}X_{i}:I\in\mathcal{GP}\right\rangle+\mathcal{J}(\mathbf{\Sigma})}.$
Moreover, since we know $H^{*}_{CR}(\mathcal{X},\mathbb{Q})$ is finite
dimensional, a monomial in $X_{1},...,X_{M}$ in the right side vanishs if its
degree is large enough. Thus we have the following lemma:
###### Lemma 2.4.
$H^{*}_{CR}(\mathcal{X},\mathbb{Q})\cong\frac{\mathbb{Q}[X_{1},...,X_{M}]}{\mathcal{I}(\mathbf{\Sigma})+\left\langle\Pi_{i\in
I}X_{i}:I\in\mathcal{GP}\right\rangle+\mathcal{J}(\mathbf{\Sigma})}.$
###### Remark 2.5.
The grading of $\mathbb{Q}[X_{1},...X_{M}]$ is not the usual grading of
polynomial ring.
### 2.4. Hamiltonian Loops of Toric Orbifolds
Now, we consider a symplectic orbifold $(\mathcal{X},\omega,T,\Phi)$
determined by a polytope $\Delta\subset\textbf{t}^{*}$ with labels $m_{i}$ on
facets $F_{i}$. In this case, we give an explicit description of Hamiltonian
loops defined in [TW, Section 2.3].
Every integer vector $v\in\textbf{l}\subset\textbf{t}$ determines a
Hamiltonian function
$H_{v}:\mathcal{X}\to,\quad H_{v}(x):=\langle\Phi(x),v\rangle,$
whose flow determines a Hamiltonian loop
$\phi_{v}:I\times\mathcal{X}\to\mathcal{X}$ with some 2-morphism
$\phi_{v}|_{\\{0\\}\times\mathcal{X}}\Rightarrow\phi_{v}|_{\\{1\\}\times\mathcal{X}}$.
Let $\phi_{v_{1}}$, $\phi_{v_{2}}$ be the Hamiltonian loops determined by
$v,v^{\prime}\in\textbf{l}$, then the composition product of the two loop
$\phi_{v_{1}}\cdot_{cp}\phi_{v_{2}}$ is generated by
$H(x)=\langle\Phi(x),v_{1}\rangle+\langle\Phi(\phi_{v_{1}}(-t,x)),v_{2}\rangle=\langle\Phi(x),v_{1}\rangle+\langle\Phi(x),v_{2}\rangle,$
where the second equality holds because the moment map is invariant under the
torus action. So $H(x)=\langle\Phi(x),v_{1}+v_{2}\rangle$, and
$\phi_{v_{1}}\cdot_{cp}\phi_{v_{2}}$ is determined by the vector
$v_{1}+v_{2}\in\textbf{l}$.
Now we give a local description of Hamiltonian functions and Hamiltonian
loops. Let $C$ be a corner of $\Delta$, then it determines a chart
$G_{C}\ltimes V_{C}$ centered at the fixed point $x_{C}$ corresponding to $C$
as in Lemma 2.3. On this chart, the Hamiltonian loop generated by a vector
$v\in\textbf{l}$ can be represented by a groupoid morphism
$\displaystyle\gamma$ $\displaystyle:[0,1]\times G_{C}\ltimes V_{C}\to
G_{C}\ltimes V_{C},$ $\displaystyle\gamma_{0}(t,\vec{w})$
$\displaystyle=D(tv)\vec{w},$
$\displaystyle\gamma_{0}(t,\vec{w}\xrightarrow{h}h\cdot\vec{w})$
$\displaystyle=(D(tv)\vec{w}\xrightarrow{h}h\cdot D(tv)\vec{w}).$
where $\vec{w}=(w_{1},...,w_{n})$, and $D(tv)$ is the diagonal matrix
$diag(e^{-i2\pi\langle u_{1},tv\rangle},...,e^{-i2\pi\langle
u_{n},tv\rangle})$.
###### Lemma 2.6.
The natural transformation $\alpha_{v}=(\gamma|_{\\{1\\}\times G_{C}\ltimes
V_{C}}\Rightarrow Id_{G_{C}\ltimes V_{C}})$ is given by
$\displaystyle\alpha_{v}:V_{C}$ $\displaystyle\to\ \ G_{C},$ $\displaystyle x$
$\displaystyle\to\ \ D(-v):=diag(e^{-i2\pi\langle
u_{1},v\rangle},...,e^{-i2\pi\langle u_{n},v\rangle}).$
###### Example 2.7.
Consider the weighted projective line
$\mathbb{C}P(1,k):=\mathbb{C}^{2}\setminus\\{0\\}/\mathbb{C}^{*}$, where
$\mathbb{C}^{*}$ acts on $\mathbb{C}^{2}\setminus\\{0\\}$ by
$z\cdot(w_{1},w_{2})=(zw_{1},z^{k}w_{2})$. The corresponding moment polytope
is a line segment with one end labeled with $k$.
1k0Labelled Polytope0-1Stacky Fank$v$ Figure 1. Labelled Moment Polytope and
Stacky Fan of $\mathbb{C}P(1,k)$.
The point $[0,1]$ has non-trivial isotropy group $\mathbb{Z}_{k}$. There is a
groupoid chart $\mathbb{Z}_{k}\ltimes\mathbb{C}$ covering
$|\mathbb{C}(1,k)|\setminus\\{[1,0]\\}$, such that the Hamiltonian loop
determined by the vector $v$ in Figure 1 is given by
$\displaystyle\gamma$
$\displaystyle:[0,1]\times\mathbb{Z}_{k}\ltimes\mathbb{C}\to\mathbb{Z}_{k}\ltimes\mathbb{C},$
$\displaystyle\gamma_{0}(t,w)$ $\displaystyle=e^{\frac{2\pi}{k}it}w,$
$\displaystyle\gamma_{0}(t,w\xrightarrow{h}h\cdot w)$
$\displaystyle=(e^{\frac{2\pi}{k}it}w\xrightarrow{h}e^{\frac{2\pi}{k}it}w).$
### 2.5. Hamiltonian Orbifiber Bundles over Sphere
Given a Hamiltonian loop, one can construct a Hamiltonian orbifiber bundle
over $S^{2}$ as explained in [TW, Section 2.5]. In this section we will
consider the Hamiltonian orbifiber bundles determined by
$v\in\text{Gen}(\Sigma)$, because later we shall compute Seidel element for
these corresponding loops.
Denote by $H_{v}=\langle\Phi,v\rangle:\ \mathcal{X}\to$ the Hamiltonian
function, $\phi_{v}$ the Hamiltonian loop determined by $H_{v}$, and
$\mathcal{E}_{v}$ the corresponding orbifiber bundle. If $H_{v}$ attends the
maximum at $x\in\mathcal{X}$, then $x$ is fixed by the Hamiltonian loop, thus
$|x|\in|\mathcal{X}|$ determines a section $s_{x}$ of the topological fiber
bundle $|\mathcal{E}_{v}|\to S^{2}$ underlying $\mathcal{E}_{v}\to S^{2}$. The
main purpose of this section is to study the properties of sectional morphisms
in $\mathcal{E}_{v}$ which lift $s_{x}$.
It is easy to see that if $H_{v}$ attends the maximum at $x$, then $\Phi(x)\in
F_{v}$, where $F_{v}$ is the face of $\Delta$ determined by the minimal cone
containing $v$ according to the cone-face correspondence in Section 2.2.
Let $C$ be a corner of $\Delta$ such that $C\subset F_{v}$, and let $x_{C}$ be
the fixed point whose image under the moment map is $C$. Recall from Section
2.2 that there is a chart $G_{C}\ltimes V_{C}$ centered at $x_{C}$. The
orbifold $[G_{C}\ltimes V_{C}]$ can be regarded as an open suborbifold of
$\mathcal{X}$ containing $x$ and $x_{C}$. The Hamiltonian loop $\gamma_{v}$
restricts to a Hamiltonian loop of $[G_{C}\ltimes V_{C}]$. Then we have an
orbifiber bundle $\mathcal{E}_{v}(C)\to S^{2}$ determined by $\gamma_{v}$ with
fiber $[G_{C}\ltimes V_{C}]$. Then $\mathcal{E}_{v}(C)$ is an open suborbifold
of $\mathcal{E}_{v}$. We will study sectional morphisms in $\mathcal{E}_{v}$
lifting $s_{x}$ inside $\mathcal{E}_{v}(C)$ since it has a nice groupoid
chart.
Recall from the Section 2.4 that the Hamiltonian loop $\phi_{v}$ restricts to
a Hamiltonian loop of $[G_{C}\ltimes V_{C}]$ represented by:
$\displaystyle\gamma$ $\displaystyle:[0,1]\times G_{C}\ltimes V_{C}\to
G_{C}\ltimes V_{C},$ $\displaystyle\gamma_{0}(t,\vec{w})$
$\displaystyle=D(tv)\vec{w},$
$\displaystyle\gamma_{0}(t,\vec{w}\xrightarrow{h}h\cdot\vec{w})$
$\displaystyle=(D(tv)\vec{w}\xrightarrow{h}h\cdot D(tv)\vec{w}).$
Let
$U_{L}:=\\{e^{2\pi it}|t\in(0,1)\\}=\begin{aligned} \leavevmode\hbox
to31.51pt{\vbox to30.96pt{\pgfpicture\makeatletter\hbox{\hskip
15.37201pt\lower-17.21175pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } {}{{}}{}{{}{}{}{{}}{{{{}{}{}{}}}
{{{}{}{}{}}} {{{}{}{}{}}} {{}{}{}{}}}}{} {} {} {}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{}\pgfsys@moveto{14.22638pt}{0.0pt}\pgfsys@curveto{13.26886pt}{7.7985pt}{6.17064pt}{13.34416pt}{-1.62785pt}{12.38664pt}\pgfsys@curveto{-9.42635pt}{11.42912pt}{-14.97202pt}{4.3309pt}{-14.0145pt}{-3.46759pt}\pgfsys@curveto{-13.05698pt}{-11.26608pt}{-5.95876pt}{-16.81175pt}{1.83974pt}{-15.85423pt}\pgfsys@curveto{8.5066pt}{-15.03564pt}{13.69281pt}{-9.66492pt}{14.27805pt}{-2.97374pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@color@gray@fill{1}\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\pgfsys@fill@opacity{0}\pgfsys@invoke{
}{}\pgfsys@moveto{14.22638pt}{0.0pt}\pgfsys@moveto{15.93347pt}{0.0pt}\pgfsys@curveto{15.93347pt}{0.94281pt}{15.16919pt}{1.70709pt}{14.22638pt}{1.70709pt}\pgfsys@curveto{13.28357pt}{1.70709pt}{12.51929pt}{0.94281pt}{12.51929pt}{0.0pt}\pgfsys@curveto{12.51929pt}{-0.94281pt}{13.28357pt}{-1.70709pt}{14.22638pt}{-1.70709pt}\pgfsys@curveto{15.16919pt}{-1.70709pt}{15.93347pt}{-0.94281pt}{15.93347pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{14.22638pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\end{aligned},\ \ \ \
\ \ U_{R}:=\\{e^{2\pi it}|t\in(-\frac{1}{2},\frac{1}{2})\\}=\begin{aligned}
\leavevmode\hbox to31.34pt{\vbox
to30.51pt{\pgfpicture\makeatletter\hbox{\hskip
16.13347pt\lower-14.01682pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } {}{{}}{}{{}{}{}{{}}{{{{}{}{}{}}}
{{{}{}{}{}}} {{{}{}{}{}}} {{}{}{}{}}}}{} {} {} {}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{}\pgfsys@moveto{-14.22638pt}{0.0pt}\pgfsys@curveto{-13.54172pt}{-7.82693pt}{-6.64127pt}{-13.61682pt}{1.18567pt}{-12.93216pt}\pgfsys@curveto{9.0126pt}{-12.2475pt}{14.80249pt}{-5.34705pt}{14.11783pt}{2.47989pt}\pgfsys@curveto{13.43317pt}{10.30682pt}{6.53271pt}{16.09671pt}{-1.29422pt}{15.41205pt}\pgfsys@curveto{-8.17166pt}{14.81053pt}{-13.62486pt}{9.35733pt}{-14.22638pt}{2.47989pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@color@gray@fill{1}\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\pgfsys@fill@opacity{0}\pgfsys@invoke{
}{}\pgfsys@moveto{-14.22638pt}{0.0pt}\pgfsys@moveto{-12.51929pt}{0.0pt}\pgfsys@curveto{-12.51929pt}{0.94281pt}{-13.28357pt}{1.70709pt}{-14.22638pt}{1.70709pt}\pgfsys@curveto{-15.16919pt}{1.70709pt}{-15.93347pt}{0.94281pt}{-15.93347pt}{0.0pt}\pgfsys@curveto{-15.93347pt}{-0.94281pt}{-15.16919pt}{-1.70709pt}{-14.22638pt}{-1.70709pt}\pgfsys@curveto{-13.28357pt}{-1.70709pt}{-12.51929pt}{-0.94281pt}{-12.51929pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{-14.22638pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\end{aligned}.$
Define $U_{S^{1}}$ by
$\displaystyle Ob(U_{S^{1}})$ $\displaystyle=$ $\displaystyle U_{L}\sqcup
U_{R},$ $\displaystyle Mor(U_{S^{1}})$ $\displaystyle=$ $\displaystyle
U_{L}\times_{S^{1}}U_{L}\ \sqcup\ U_{R}\times_{S^{1}}U_{R}\ \sqcup\
U_{L}\times_{S^{1}}U_{R}\ \sqcup\ U_{R}\times_{S^{1}}U_{L}.$
Note that
$\displaystyle\begin{aligned}
U_{L}\times_{S^{1}}U_{L}\end{aligned}=\begin{aligned} \leavevmode\hbox
to31.51pt{\vbox to30.96pt{\pgfpicture\makeatletter\hbox{\hskip
15.37201pt\lower-17.21175pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } {}{{}}{}{{}{}{}{{}}{{{{}{}{}{}}}
{{{}{}{}{}}} {{{}{}{}{}}} {{}{}{}{}}}}{} {} {} {}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{}\pgfsys@moveto{14.22638pt}{0.0pt}\pgfsys@curveto{13.26886pt}{7.7985pt}{6.17064pt}{13.34416pt}{-1.62785pt}{12.38664pt}\pgfsys@curveto{-9.42635pt}{11.42912pt}{-14.97202pt}{4.3309pt}{-14.0145pt}{-3.46759pt}\pgfsys@curveto{-13.05698pt}{-11.26608pt}{-5.95876pt}{-16.81175pt}{1.83974pt}{-15.85423pt}\pgfsys@curveto{8.5066pt}{-15.03564pt}{13.69281pt}{-9.66492pt}{14.27805pt}{-2.97374pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@color@gray@fill{1}\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\pgfsys@fill@opacity{0}\pgfsys@invoke{
}{}\pgfsys@moveto{14.22638pt}{0.0pt}\pgfsys@moveto{15.93347pt}{0.0pt}\pgfsys@curveto{15.93347pt}{0.94281pt}{15.16919pt}{1.70709pt}{14.22638pt}{1.70709pt}\pgfsys@curveto{13.28357pt}{1.70709pt}{12.51929pt}{0.94281pt}{12.51929pt}{0.0pt}\pgfsys@curveto{12.51929pt}{-0.94281pt}{13.28357pt}{-1.70709pt}{14.22638pt}{-1.70709pt}\pgfsys@curveto{15.16919pt}{-1.70709pt}{15.93347pt}{-0.94281pt}{15.93347pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{14.22638pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\end{aligned}$
$\displaystyle,\ \ \ U_{R}\times_{S^{1}}U_{R}=\begin{aligned} \leavevmode\hbox
to31.34pt{\vbox to30.51pt{\pgfpicture\makeatletter\hbox{\hskip
16.13347pt\lower-14.01682pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } {}{{}}{}{{}{}{}{{}}{{{{}{}{}{}}}
{{{}{}{}{}}} {{{}{}{}{}}} {{}{}{}{}}}}{} {} {} {}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{}\pgfsys@moveto{-14.22638pt}{0.0pt}\pgfsys@curveto{-13.54172pt}{-7.82693pt}{-6.64127pt}{-13.61682pt}{1.18567pt}{-12.93216pt}\pgfsys@curveto{9.0126pt}{-12.2475pt}{14.80249pt}{-5.34705pt}{14.11783pt}{2.47989pt}\pgfsys@curveto{13.43317pt}{10.30682pt}{6.53271pt}{16.09671pt}{-1.29422pt}{15.41205pt}\pgfsys@curveto{-8.17166pt}{14.81053pt}{-13.62486pt}{9.35733pt}{-14.22638pt}{2.47989pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@color@gray@fill{1}\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\pgfsys@fill@opacity{0}\pgfsys@invoke{
}{}\pgfsys@moveto{-14.22638pt}{0.0pt}\pgfsys@moveto{-12.51929pt}{0.0pt}\pgfsys@curveto{-12.51929pt}{0.94281pt}{-13.28357pt}{1.70709pt}{-14.22638pt}{1.70709pt}\pgfsys@curveto{-15.16919pt}{1.70709pt}{-15.93347pt}{0.94281pt}{-15.93347pt}{0.0pt}\pgfsys@curveto{-15.93347pt}{-0.94281pt}{-15.16919pt}{-1.70709pt}{-14.22638pt}{-1.70709pt}\pgfsys@curveto{-13.28357pt}{-1.70709pt}{-12.51929pt}{-0.94281pt}{-12.51929pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{-14.22638pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\end{aligned},$
$\displaystyle U_{L}\times_{S^{1}}U_{R}=\begin{aligned} \leavevmode\hbox
to29.25pt{\vbox to15.03pt{\pgfpicture\makeatletter\hbox{\hskip
14.62637pt\lower-0.4pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } {}{{}}{}{{}{}{}{{}}{{{{}{}{}{}}}
{{}{}{}{}}}}{} {} {} {} {}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{}\pgfsys@moveto{14.22638pt}{0.0pt}\pgfsys@curveto{14.22638pt}{7.8571pt}{7.8571pt}{14.22638pt}{0.0pt}{14.22638pt}\pgfsys@curveto{-7.8571pt}{14.22638pt}{-14.22638pt}{7.8571pt}{-14.22638pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\end{aligned}\ \ \
\sqcup\ \ \ \ \begin{aligned} \leavevmode\hbox to29.25pt{\vbox
to15.03pt{\pgfpicture\makeatletter\hbox{\hskip
14.62637pt\lower-14.62637pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } {}{{}}{}{{}{}{}{{}}{{{{}{}{}{}}}
{{}{}{}{}}}}{} {} {} {} {}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{}\pgfsys@moveto{-14.22638pt}{0.0pt}\pgfsys@curveto{-14.22638pt}{-7.8571pt}{-7.8571pt}{-14.22638pt}{0.0pt}{-14.22638pt}\pgfsys@curveto{7.8571pt}{-14.22638pt}{14.22638pt}{-7.8571pt}{14.22638pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\end{aligned}$
$\displaystyle,\ \ \ \ \ \ U_{R}\times_{S^{1}}U_{L}=\begin{aligned}
\leavevmode\hbox to29.25pt{\vbox
to15.03pt{\pgfpicture\makeatletter\hbox{\hskip 14.62637pt\lower-0.4pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } {}{{}}{}{{}{}{}{{}}{{{{}{}{}{}}}
{{}{}{}{}}}}{} {} {} {} {}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{}\pgfsys@moveto{14.22638pt}{0.0pt}\pgfsys@curveto{14.22638pt}{7.8571pt}{7.8571pt}{14.22638pt}{0.0pt}{14.22638pt}\pgfsys@curveto{-7.8571pt}{14.22638pt}{-14.22638pt}{7.8571pt}{-14.22638pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\end{aligned}\ \ \
\sqcup\ \ \ \begin{aligned} \leavevmode\hbox to29.25pt{\vbox
to15.03pt{\pgfpicture\makeatletter\hbox{\hskip
14.62637pt\lower-14.62637pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } {}{{}}{}{{}{}{}{{}}{{{{}{}{}{}}}
{{}{}{}{}}}}{} {} {} {} {}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{}\pgfsys@moveto{-14.22638pt}{0.0pt}\pgfsys@curveto{-14.22638pt}{-7.8571pt}{-7.8571pt}{-14.22638pt}{0.0pt}{-14.22638pt}\pgfsys@curveto{7.8571pt}{-14.22638pt}{14.22638pt}{-7.8571pt}{14.22638pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\end{aligned}.$
We denote $(e^{2\pi it})_{*}\in U_{*}$ and $g=(e^{2\pi it})_{*}\to(e^{2\pi
it^{\prime}})_{\bullet}\in U_{*}\times_{S^{1}}U_{\bullet}$, for
$*,\bullet=R,L$, where
* •
$t^{\prime}=t-1$, if $g\in\begin{aligned} \leavevmode\hbox to29.25pt{\vbox
to15.03pt{\pgfpicture\makeatletter\hbox{\hskip
14.62637pt\lower-14.62637pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } {}{{}}{}{{}{}{}{{}}{{{{}{}{}{}}}
{{}{}{}{}}}}{} {} {} {} {}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{}\pgfsys@moveto{-14.22638pt}{0.0pt}\pgfsys@curveto{-14.22638pt}{-7.8571pt}{-7.8571pt}{-14.22638pt}{0.0pt}{-14.22638pt}\pgfsys@curveto{7.8571pt}{-14.22638pt}{14.22638pt}{-7.8571pt}{14.22638pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\end{aligned}\subset
U_{L}\times_{S^{1}}U_{R}$;
* •
$t^{\prime}=t+1$, if $g\in\begin{aligned} \leavevmode\hbox to29.25pt{\vbox
to15.03pt{\pgfpicture\makeatletter\hbox{\hskip
14.62637pt\lower-14.62637pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } {}{{}}{}{{}{}{}{{}}{{{{}{}{}{}}}
{{}{}{}{}}}}{} {} {} {} {}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{}\pgfsys@moveto{-14.22638pt}{0.0pt}\pgfsys@curveto{-14.22638pt}{-7.8571pt}{-7.8571pt}{-14.22638pt}{0.0pt}{-14.22638pt}\pgfsys@curveto{7.8571pt}{-14.22638pt}{14.22638pt}{-7.8571pt}{14.22638pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\end{aligned}\subset
U_{R}\times_{S^{1}}U_{L}$;
* •
$t^{\prime}=t$, otherwise.
Then define a groupoid morphism $\tilde{\gamma}:U_{S^{1}}\times G_{C}\ltimes
V_{C}\to G_{C}\ltimes V_{C}$ by:
$\displaystyle\tilde{\gamma}_{0}((e^{2\pi it})_{L},\vec{w})$
$\displaystyle:=\gamma_{0}(t,\vec{w})=D(tv)\vec{w},\ \ \ \ \ \ \ \ t\in(0,1),\
\ \ (e^{2\pi it})_{L}\in U_{L};$ $\displaystyle\tilde{\gamma}_{0}((e^{2\pi
it})_{R},\vec{w})$ $\displaystyle:=D(tv)\vec{w},\ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \ t\in(-\frac{1}{2},\frac{1}{2}),\ (e^{2\pi it})_{R}\in
U_{R};$ $\displaystyle\tilde{\gamma}_{1}((e^{2\pi it})_{L}\to(e^{2\pi
it})_{L},\vec{w}\xrightarrow{h}h\cdot\vec{w})$
$\displaystyle:=D(tv)\vec{w}\xrightarrow{h}h\cdot D(tv)\vec{w},\ \ \
t\in(0,1);$ $\displaystyle\tilde{\gamma}_{1}((e^{2\pi it})_{R}\to(e^{2\pi
it})_{R},\vec{w}\xrightarrow{h}h\cdot\vec{w})$
$\displaystyle:=D(tv)\vec{w}\xrightarrow{h}h\cdot D(tv)\vec{w},\ \ \ \
t\in(-\frac{1}{2},\frac{1}{2});$ $\displaystyle\tilde{\gamma}_{1}((e^{2\pi
it})_{L}\to(e^{2\pi it})_{R},\vec{w}\xrightarrow{h}h\cdot\vec{w})$
$\displaystyle:=D(tv)\vec{w}\xrightarrow{h}h\cdot D(tv)\vec{w},\ \ \
t\in(0,\frac{1}{2});$ $\displaystyle\tilde{\gamma}_{1}((e^{2\pi
it})_{R}\to(e^{2\pi it})_{L},\vec{w}\xrightarrow{h}h\cdot\vec{w})$
$\displaystyle:=D(tv)\vec{w}\xrightarrow{h}h\cdot D(tv)\vec{w},\ \ \ \
t\in(0,\frac{1}{2});$ $\displaystyle\tilde{\gamma}_{1}((e^{2\pi
it})_{L}\to(e^{2\pi i(t-1)}))_{R},\vec{w}\xrightarrow{h}h\cdot\vec{w})$
$\displaystyle:=D(tv)\vec{w}\xrightarrow{h\cdot D(-v)}h\cdot
D((t-1)v)\vec{w},\ \ \ t\in(\frac{1}{2},1);$
$\displaystyle\tilde{\gamma}_{1}((e^{2\pi it})_{R}\to(e^{2\pi
i(t+1)}))_{L},\vec{w}\xrightarrow{h}h\cdot\vec{w})$
$\displaystyle:=D(tv)\vec{w}\xrightarrow{h\cdot D(v)}h\cdot D((t+1)v)\vec{w},\
\ \ t\in(-\frac{1}{2},0);$
Consider the following groupoid chart of $S^{2}$:
$\displaystyle Ob(U_{S^{2}})$ $\displaystyle:=\begin{aligned} \leavevmode\hbox
to31.67pt{\vbox to31.23pt{\pgfpicture\makeatletter\hbox{\hskip
15.37498pt\lower-14.93193pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } {}{{}}{}{{}{}{}{{}}{{{{}{}{}{}}}
{{{}{}{}{}}} {{}{}{}{}}}}{} {} {} {}
{}{}{{}{}{}{{}}{{{{}{}{}{}}}{{}{}{}{}}}}{} {} {} {} {}{}
{}\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@invoke{
}\pgfsys@color@gray@fill{.5}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{.5,.5,.5}\pgfsys@stroke@opacity{0.05}\pgfsys@invoke{
}\pgfsys@fill@opacity{0.05}\pgfsys@invoke{
}{}\pgfsys@moveto{11.38092pt}{-8.5359pt}\pgfsys@curveto{16.09518pt}{-2.25021pt}{14.82138pt}{6.66687pt}{8.53569pt}{11.38113pt}\pgfsys@curveto{2.25pt}{16.0954pt}{-6.66708pt}{14.8216pt}{-11.38135pt}{8.5359pt}\pgfsys@curveto{-15.17499pt}{3.4778pt}{-15.17499pt}{-3.4778pt}{-11.38135pt}{-8.5359pt}\pgfsys@curveto{-11.38135pt}{-7.5931pt}{-6.28601pt}{-6.82881pt}{-0.00043pt}{-6.82881pt}\pgfsys@curveto{6.28516pt}{-6.82881pt}{11.3805pt}{-7.5931pt}{11.3805pt}{-8.5359pt}\pgfsys@closepath\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}
{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@invoke{
}\pgfsys@color@gray@fill{.5}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{.5,.5,.5}\pgfsys@stroke@opacity{0.5}\pgfsys@invoke{
}\pgfsys@fill@opacity{0.5}\pgfsys@invoke{
}\pgfsys@moveto{0.0pt}{-8.5359pt}\pgfsys@moveto{11.38092pt}{-8.5359pt}\pgfsys@curveto{11.38092pt}{-7.75015pt}{6.28558pt}{-7.11319pt}{0.0pt}{-7.11319pt}\pgfsys@curveto{-6.28558pt}{-7.11319pt}{-11.38092pt}{-7.75015pt}{-11.38092pt}{-8.5359pt}\pgfsys@curveto{-11.38092pt}{-9.32166pt}{-6.28558pt}{-9.95862pt}{0.0pt}{-9.95862pt}\pgfsys@curveto{6.28558pt}{-9.95862pt}{11.38092pt}{-9.32166pt}{11.38092pt}{-8.5359pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{-8.5359pt}\pgfsys@fill\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\definecolor[named]{tikz@ball}{rgb}{1,0,0}{}{{}}{}{{}{}{}{{}}{{{{}{}{}{}}}{{}{}{}{}}}}{}
{} {} {} {}{}{{}{}{}{{}}{{{{}{}{}{}}} {{}{}{}{}}}}{} {} {} {} {}{}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@moveto{11.38092pt}{-8.5359pt}\pgfsys@curveto{8.33142pt}{-12.60197pt}{3.34409pt}{-14.73193pt}{-1.70187pt}{-14.12413pt}\pgfsys@curveto{-5.55542pt}{-13.66003pt}{-9.05255pt}{-11.64098pt}{-11.38135pt}{-8.5359pt}\pgfsys@curveto{-11.38135pt}{-9.47871pt}{-6.28601pt}{-10.243pt}{-0.00043pt}{-10.243pt}\pgfsys@curveto{6.28516pt}{-10.243pt}{11.3805pt}{-9.47871pt}{11.3805pt}{-8.5359pt}\pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-0.00021pt}{-11.63391pt}\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{0.0pt}{0.0pt}\pgfsys@invoke{
}\pgfsys@transformcm{0.4536}{0.0}{0.0}{0.12347}{0.0pt}{0.0pt}\pgfsys@invoke{
}\definecolor{pgf@tempcol}{rgb}{1,0.85,0.85}\definecolor{pgf@tempcol}{rgb}{1,0.25,0.25}\definecolor{pgf@tempcol}{rgb}{0.7,0,0}\definecolor{pgf@tempcol}{rgb}{0.5,0,0}\definecolor[named]{pgf@tempcol}{rgb}{0,0,0}\definecolor{pgfshadetemp}{rgb}{1,0.85,0.85}\definecolor{pgfshadetemp}{rgb}{0,0,0}\definecolor{pgfshadetemp}{rgb}{1,0.25,0.25}\definecolor{pgfshadetemp}{rgb}{0.7,0,0}\definecolor{pgfshadetemp}{rgb}{0.5,0,0}\definecolor{pgfshadetemp}{rgb}{0,0,0}{}\lxSVG@sh@defs{}\lxSVG@sh@insert{-69.44pt}{-69.44pt}{\lxSVG@sh}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}\pgfsys@moveto{11.38092pt}{-8.5359pt}\pgfsys@curveto{8.33142pt}{-12.60197pt}{3.34409pt}{-14.73193pt}{-1.70187pt}{-14.12413pt}\pgfsys@curveto{-5.55542pt}{-13.66003pt}{-9.05255pt}{-11.64098pt}{-11.38135pt}{-8.5359pt}\pgfsys@curveto{-11.38135pt}{-9.47871pt}{-6.28601pt}{-10.243pt}{-0.00043pt}{-10.243pt}\pgfsys@curveto{6.28516pt}{-10.243pt}{11.3805pt}{-9.47871pt}{11.3805pt}{-8.5359pt}\pgfsys@closepath\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\end{aligned}\sqcup\begin{aligned}
\leavevmode\hbox to29.75pt{\vbox
to32.27pt{\pgfpicture\makeatletter\hbox{\hskip
14.42638pt\lower-16.13347pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }
\definecolor[named]{tikz@ball}{rgb}{1,0,0}{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{
}\pgfsys@color@gray@fill{1}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{14.22638pt}{0.0pt}\pgfsys@curveto{14.22638pt}{7.8571pt}{7.8571pt}{14.22638pt}{0.0pt}{14.22638pt}\pgfsys@curveto{-7.8571pt}{14.22638pt}{-14.22638pt}{7.8571pt}{-14.22638pt}{0.0pt}\pgfsys@curveto{-14.22638pt}{-7.8571pt}{-7.8571pt}{-14.22638pt}{0.0pt}{-14.22638pt}\pgfsys@curveto{7.8571pt}{-14.22638pt}{14.22638pt}{-7.8571pt}{14.22638pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{0.0pt}{0.0pt}\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{0.0pt}{0.0pt}\pgfsys@invoke{
}\pgfsys@transformcm{0.567}{0.0}{0.0}{0.567}{0.0pt}{0.0pt}\pgfsys@invoke{
}\lxSVG@sh@defs{}\lxSVG@sh@insert{-69.44pt}{-69.44pt}{\lxSVG@sh}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{14.22638pt}{0.0pt}\pgfsys@curveto{14.22638pt}{7.8571pt}{7.8571pt}{14.22638pt}{0.0pt}{14.22638pt}\pgfsys@curveto{-7.8571pt}{14.22638pt}{-14.22638pt}{7.8571pt}{-14.22638pt}{0.0pt}\pgfsys@curveto{-14.22638pt}{-7.8571pt}{-7.8571pt}{-14.22638pt}{0.0pt}{-14.22638pt}\pgfsys@curveto{7.8571pt}{-14.22638pt}{14.22638pt}{-7.8571pt}{14.22638pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@color@gray@fill{1}\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\pgfsys@fill@opacity{0}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{14.22638pt}\pgfsys@moveto{1.70709pt}{14.22638pt}\pgfsys@curveto{1.70709pt}{15.16919pt}{0.94281pt}{15.93347pt}{0.0pt}{15.93347pt}\pgfsys@curveto{-0.94281pt}{15.93347pt}{-1.70709pt}{15.16919pt}{-1.70709pt}{14.22638pt}\pgfsys@curveto{-1.70709pt}{13.28357pt}{-0.94281pt}{12.51929pt}{0.0pt}{12.51929pt}\pgfsys@curveto{0.94281pt}{12.51929pt}{1.70709pt}{13.28357pt}{1.70709pt}{14.22638pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@color@gray@fill{1}\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\pgfsys@fill@opacity{0}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{-14.22638pt}\pgfsys@moveto{1.70709pt}{-14.22638pt}\pgfsys@curveto{1.70709pt}{-13.28357pt}{0.94281pt}{-12.51929pt}{0.0pt}{-12.51929pt}\pgfsys@curveto{-0.94281pt}{-12.51929pt}{-1.70709pt}{-13.28357pt}{-1.70709pt}{-14.22638pt}\pgfsys@curveto{-1.70709pt}{-15.16919pt}{-0.94281pt}{-15.93347pt}{0.0pt}{-15.93347pt}\pgfsys@curveto{0.94281pt}{-15.93347pt}{1.70709pt}{-15.16919pt}{1.70709pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{{}}{}{{}{}{}{{}}{{{{}{}{}{}}}{{}{}{}{}}}}{} {} {} {}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{ }\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{2.20001pt}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{14.22638pt}\pgfsys@curveto{7.8571pt}{14.22638pt}{14.22638pt}{7.8571pt}{14.22638pt}{0.0pt}\pgfsys@curveto{14.22638pt}{-7.8571pt}{7.8571pt}{-14.22638pt}{0.0pt}{-14.22638pt}\pgfsys@stroke\pgfsys@invoke{
}\pgfsys@beginscope\pgfsys@invoke{
}{\pgfsys@setlinewidth{0.6pt}\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{
}\pgfsys@moveto{0.0pt}{14.22638pt}\pgfsys@curveto{7.8571pt}{14.22638pt}{14.22638pt}{7.8571pt}{14.22638pt}{0.0pt}\pgfsys@curveto{14.22638pt}{-7.8571pt}{7.8571pt}{-14.22638pt}{0.0pt}{-14.22638pt}\pgfsys@stroke\pgfsys@invoke{
}}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\end{aligned}\sqcup\begin{aligned}
\leavevmode\hbox to29.75pt{\vbox
to32.27pt{\pgfpicture\makeatletter\hbox{\hskip
15.32639pt\lower-16.13347pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }
\definecolor[named]{tikz@ball}{rgb}{1,0,0}{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{
}\pgfsys@color@gray@fill{1}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{14.22638pt}{0.0pt}\pgfsys@curveto{14.22638pt}{7.8571pt}{7.8571pt}{14.22638pt}{0.0pt}{14.22638pt}\pgfsys@curveto{-7.8571pt}{14.22638pt}{-14.22638pt}{7.8571pt}{-14.22638pt}{0.0pt}\pgfsys@curveto{-14.22638pt}{-7.8571pt}{-7.8571pt}{-14.22638pt}{0.0pt}{-14.22638pt}\pgfsys@curveto{7.8571pt}{-14.22638pt}{14.22638pt}{-7.8571pt}{14.22638pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{0.0pt}{0.0pt}\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{0.0pt}{0.0pt}\pgfsys@invoke{
}\pgfsys@transformcm{0.567}{0.0}{0.0}{0.567}{0.0pt}{0.0pt}\pgfsys@invoke{
}\lxSVG@sh@defs{}\lxSVG@sh@insert{-69.44pt}{-69.44pt}{\lxSVG@sh}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{14.22638pt}{0.0pt}\pgfsys@curveto{14.22638pt}{7.8571pt}{7.8571pt}{14.22638pt}{0.0pt}{14.22638pt}\pgfsys@curveto{-7.8571pt}{14.22638pt}{-14.22638pt}{7.8571pt}{-14.22638pt}{0.0pt}\pgfsys@curveto{-14.22638pt}{-7.8571pt}{-7.8571pt}{-14.22638pt}{0.0pt}{-14.22638pt}\pgfsys@curveto{7.8571pt}{-14.22638pt}{14.22638pt}{-7.8571pt}{14.22638pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@color@gray@fill{1}\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\pgfsys@fill@opacity{0}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{14.22638pt}\pgfsys@moveto{1.70709pt}{14.22638pt}\pgfsys@curveto{1.70709pt}{15.16919pt}{0.94281pt}{15.93347pt}{0.0pt}{15.93347pt}\pgfsys@curveto{-0.94281pt}{15.93347pt}{-1.70709pt}{15.16919pt}{-1.70709pt}{14.22638pt}\pgfsys@curveto{-1.70709pt}{13.28357pt}{-0.94281pt}{12.51929pt}{0.0pt}{12.51929pt}\pgfsys@curveto{0.94281pt}{12.51929pt}{1.70709pt}{13.28357pt}{1.70709pt}{14.22638pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@color@gray@fill{1}\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\pgfsys@fill@opacity{0}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{-14.22638pt}\pgfsys@moveto{1.70709pt}{-14.22638pt}\pgfsys@curveto{1.70709pt}{-13.28357pt}{0.94281pt}{-12.51929pt}{0.0pt}{-12.51929pt}\pgfsys@curveto{-0.94281pt}{-12.51929pt}{-1.70709pt}{-13.28357pt}{-1.70709pt}{-14.22638pt}\pgfsys@curveto{-1.70709pt}{-15.16919pt}{-0.94281pt}{-15.93347pt}{0.0pt}{-15.93347pt}\pgfsys@curveto{0.94281pt}{-15.93347pt}{1.70709pt}{-15.16919pt}{1.70709pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{{}}{}{{}{}{}{{}}{{{{}{}{}{}}}{{}{}{}{}}}}{} {} {} {}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{ }\pgfsys@beginscope\pgfsys@invoke{
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$\displaystyle Mor(U_{S^{2}})$ $\displaystyle:=\begin{aligned}
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}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\end{aligned}$
Define a Lie groupoid $\mathcal{G}_{\mathcal{E}_{v,C}}$ by
$\displaystyle Ob(\mathcal{G}_{\mathcal{E}_{v,C}}):=\begin{aligned}
\leavevmode\hbox to31.67pt{\vbox
to31.23pt{\pgfpicture\makeatletter\hbox{\hskip
15.37498pt\lower-14.93193pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }
\definecolor[named]{tikz@ball}{rgb}{1,0,0}{}{{}}{}{{}{}{}{{}}{{{{}{}{}{}}}{{}{}{}{}}}}{}
{} {} {} {}{}{{}{}{}{{}}{{{{}{}{}{}}} {{}{}{}{}}}}{} {} {} {} {}{}
{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@moveto{11.38092pt}{-8.5359pt}\pgfsys@curveto{8.33142pt}{-12.60197pt}{3.34409pt}{-14.73193pt}{-1.70187pt}{-14.12413pt}\pgfsys@curveto{-5.55542pt}{-13.66003pt}{-9.05255pt}{-11.64098pt}{-11.38135pt}{-8.5359pt}\pgfsys@curveto{-11.38135pt}{-9.47871pt}{-6.28601pt}{-10.243pt}{-0.00043pt}{-10.243pt}\pgfsys@curveto{6.28516pt}{-10.243pt}{11.3805pt}{-9.47871pt}{11.3805pt}{-8.5359pt}\pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-0.00021pt}{-11.63391pt}\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{0.0pt}{0.0pt}\pgfsys@invoke{
}\pgfsys@transformcm{0.4536}{0.0}{0.0}{0.12347}{0.0pt}{0.0pt}\pgfsys@invoke{
}\lxSVG@sh@defs{}\lxSVG@sh@insert{-69.44pt}{-69.44pt}{\lxSVG@sh}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}\pgfsys@moveto{11.38092pt}{-8.5359pt}\pgfsys@curveto{8.33142pt}{-12.60197pt}{3.34409pt}{-14.73193pt}{-1.70187pt}{-14.12413pt}\pgfsys@curveto{-5.55542pt}{-13.66003pt}{-9.05255pt}{-11.64098pt}{-11.38135pt}{-8.5359pt}\pgfsys@curveto{-11.38135pt}{-9.47871pt}{-6.28601pt}{-10.243pt}{-0.00043pt}{-10.243pt}\pgfsys@curveto{6.28516pt}{-10.243pt}{11.3805pt}{-9.47871pt}{11.3805pt}{-8.5359pt}\pgfsys@closepath\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}{{}{}{}{{}}{{{{}{}{}{}}} {{{}{}{}{}}} {{}{}{}{}}}}{} {} {} {}
{}{}{{}{}{}{{}}{{{{}{}{}{}}}{{}{}{}{}}}}{} {} {} {} {}{}
{}\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@invoke{
}\pgfsys@color@gray@fill{.5}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{.5,.5,.5}\pgfsys@stroke@opacity{0.05}\pgfsys@invoke{
}\pgfsys@fill@opacity{0.05}\pgfsys@invoke{
}{}\pgfsys@moveto{11.38092pt}{-8.5359pt}\pgfsys@curveto{16.09518pt}{-2.25021pt}{14.82138pt}{6.66687pt}{8.53569pt}{11.38113pt}\pgfsys@curveto{2.25pt}{16.0954pt}{-6.66708pt}{14.8216pt}{-11.38135pt}{8.5359pt}\pgfsys@curveto{-15.17499pt}{3.4778pt}{-15.17499pt}{-3.4778pt}{-11.38135pt}{-8.5359pt}\pgfsys@curveto{-11.38135pt}{-7.5931pt}{-6.28601pt}{-6.82881pt}{-0.00043pt}{-6.82881pt}\pgfsys@curveto{6.28516pt}{-6.82881pt}{11.3805pt}{-7.5931pt}{11.3805pt}{-8.5359pt}\pgfsys@closepath\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}
{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@invoke{
}\pgfsys@color@gray@fill{.5}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{.5,.5,.5}\pgfsys@stroke@opacity{0.5}\pgfsys@invoke{
}\pgfsys@fill@opacity{0.5}\pgfsys@invoke{
}\pgfsys@moveto{0.0pt}{-8.5359pt}\pgfsys@moveto{11.38092pt}{-8.5359pt}\pgfsys@curveto{11.38092pt}{-7.75015pt}{6.28558pt}{-7.11319pt}{0.0pt}{-7.11319pt}\pgfsys@curveto{-6.28558pt}{-7.11319pt}{-11.38092pt}{-7.75015pt}{-11.38092pt}{-8.5359pt}\pgfsys@curveto{-11.38092pt}{-9.32166pt}{-6.28558pt}{-9.95862pt}{0.0pt}{-9.95862pt}\pgfsys@curveto{6.28558pt}{-9.95862pt}{11.38092pt}{-9.32166pt}{11.38092pt}{-8.5359pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{-8.5359pt}\pgfsys@fill\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\end{aligned}\times
V_{C}\negthickspace$ $\displaystyle\sqcup(\begin{aligned} \leavevmode\hbox
to30.56pt{\vbox to30.36pt{\pgfpicture\makeatletter\hbox{\hskip
14.42638pt\lower-16.13347pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }
\definecolor[named]{tikz@ball}{rgb}{1,0,0}{}{{}}{}{{}{}{}{{}}{{{{}{}{}{}}}{{}{}{}{}}}}{}
{} {} {} {}{}{{}{}{}{{}}{{{{}{}{}{}}} {{}{}{}{}}}}{} {} {} {} {}{}
{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@moveto{14.22638pt}{0.0pt}\pgfsys@curveto{14.22638pt}{-7.8571pt}{7.8571pt}{-14.22638pt}{0.0pt}{-14.22638pt}\pgfsys@curveto{-7.8571pt}{-14.22638pt}{-14.22638pt}{-7.8571pt}{-14.22638pt}{0.0pt}\pgfsys@curveto{-14.22638pt}{-1.5715pt}{-7.8571pt}{-2.84544pt}{0.0pt}{-2.84544pt}\pgfsys@curveto{7.8571pt}{-2.84544pt}{14.22638pt}{-1.5715pt}{14.22638pt}{0.0pt}\pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{0.0pt}{-7.11319pt}\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{0.0pt}{0.0pt}\pgfsys@invoke{
}\pgfsys@transformcm{0.567}{0.0}{0.0}{0.2835}{0.0pt}{0.0pt}\pgfsys@invoke{
}\lxSVG@sh@defs{}\lxSVG@sh@insert{-69.44pt}{-69.44pt}{\lxSVG@sh}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}\pgfsys@moveto{14.22638pt}{0.0pt}\pgfsys@curveto{14.22638pt}{-7.8571pt}{7.8571pt}{-14.22638pt}{0.0pt}{-14.22638pt}\pgfsys@curveto{-7.8571pt}{-14.22638pt}{-14.22638pt}{-7.8571pt}{-14.22638pt}{0.0pt}\pgfsys@curveto{-14.22638pt}{-1.5715pt}{-7.8571pt}{-2.84544pt}{0.0pt}{-2.84544pt}\pgfsys@curveto{7.8571pt}{-2.84544pt}{14.22638pt}{-1.5715pt}{14.22638pt}{0.0pt}\pgfsys@closepath\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}{{}{}{}{{}}{{{{}{}{}{}}} {{}{}{}{}}}}{} {} {} {}
{}{}{{}{}{}{{}}{{{{}{}{}{}}}{{}{}{}{}}}}{} {} {} {} {}{}
{}\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@invoke{
}\pgfsys@color@gray@fill{.5}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{.5,.5,.5}\pgfsys@stroke@opacity{0.05}\pgfsys@invoke{
}\pgfsys@fill@opacity{0.05}\pgfsys@invoke{
}\pgfsys@moveto{14.22638pt}{0.0pt}\pgfsys@curveto{14.22638pt}{7.8571pt}{7.8571pt}{14.22638pt}{0.0pt}{14.22638pt}\pgfsys@curveto{-7.8571pt}{14.22638pt}{-14.22638pt}{7.8571pt}{-14.22638pt}{0.0pt}\pgfsys@curveto{-14.22638pt}{1.5715pt}{-7.8571pt}{2.84544pt}{0.0pt}{2.84544pt}\pgfsys@curveto{7.8571pt}{2.84544pt}{14.22638pt}{1.5715pt}{14.22638pt}{0.0pt}\pgfsys@closepath\pgfsys@fill\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}
{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@fill{.5}\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\pgfsys@stroke@opacity{0.5}\pgfsys@invoke{
}\pgfsys@fill@opacity{0.5}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{14.22638pt}{0.0pt}\pgfsys@curveto{14.22638pt}{1.5715pt}{7.8571pt}{2.84544pt}{0.0pt}{2.84544pt}\pgfsys@curveto{-7.8571pt}{2.84544pt}{-14.22638pt}{1.5715pt}{-14.22638pt}{0.0pt}\pgfsys@curveto{-14.22638pt}{-1.5715pt}{-7.8571pt}{-2.84544pt}{0.0pt}{-2.84544pt}\pgfsys@curveto{7.8571pt}{-2.84544pt}{14.22638pt}{-1.5715pt}{14.22638pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@color@gray@fill{1}\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\pgfsys@fill@opacity{0}\pgfsys@invoke{
}{}\pgfsys@moveto{14.22638pt}{0.0pt}\pgfsys@moveto{15.93347pt}{0.0pt}\pgfsys@curveto{15.93347pt}{0.94281pt}{15.16919pt}{1.70709pt}{14.22638pt}{1.70709pt}\pgfsys@curveto{13.28357pt}{1.70709pt}{12.51929pt}{0.94281pt}{12.51929pt}{0.0pt}\pgfsys@curveto{12.51929pt}{-0.94281pt}{13.28357pt}{-1.70709pt}{14.22638pt}{-1.70709pt}\pgfsys@curveto{15.16919pt}{-1.70709pt}{15.93347pt}{-0.94281pt}{15.93347pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{14.22638pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@color@gray@fill{1}\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\pgfsys@fill@opacity{0}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{-14.22638pt}\pgfsys@moveto{1.70709pt}{-14.22638pt}\pgfsys@curveto{1.70709pt}{-13.28357pt}{0.94281pt}{-12.51929pt}{0.0pt}{-12.51929pt}\pgfsys@curveto{-0.94281pt}{-12.51929pt}{-1.70709pt}{-13.28357pt}{-1.70709pt}{-14.22638pt}\pgfsys@curveto{-1.70709pt}{-15.16919pt}{-0.94281pt}{-15.93347pt}{0.0pt}{-15.93347pt}\pgfsys@curveto{0.94281pt}{-15.93347pt}{1.70709pt}{-15.16919pt}{1.70709pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{{}}{}{{}{}{}{{}}{{{}{}{}{}}}}{} {} {} {}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{ }\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{2.20001pt}\pgfsys@invoke{
}{}\pgfsys@moveto{14.22638pt}{0.0pt}\pgfsys@curveto{14.22638pt}{-7.8571pt}{7.8571pt}{-14.22638pt}{0.0pt}{-14.22638pt}\pgfsys@stroke\pgfsys@invoke{
}\pgfsys@beginscope\pgfsys@invoke{
}{\pgfsys@setlinewidth{0.6pt}\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{
}\pgfsys@moveto{14.22638pt}{0.0pt}\pgfsys@curveto{14.22638pt}{-7.8571pt}{7.8571pt}{-14.22638pt}{0.0pt}{-14.22638pt}\pgfsys@stroke\pgfsys@invoke{
}}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\end{aligned}\times
V_{C}\ \sqcup\begin{aligned} \leavevmode\hbox to30.56pt{\vbox
to30.36pt{\pgfpicture\makeatletter\hbox{\hskip
14.42638pt\lower-14.22638pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }
\definecolor[named]{tikz@ball}{rgb}{1,0,0}{}{{}}{}{{}{}{}{{}}{{{{}{}{}{}}}
{{}{}{}{}}}}{} {} {} {} {}{}{{}{}{}{{}}{{{{}{}{}{}}} {{}{}{}{}}}}{} {} {} {}
{}{} {}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@moveto{14.22638pt}{0.0pt}\pgfsys@curveto{14.22638pt}{7.8571pt}{7.8571pt}{14.22638pt}{0.0pt}{14.22638pt}\pgfsys@curveto{-7.8571pt}{14.22638pt}{-14.22638pt}{7.8571pt}{-14.22638pt}{0.0pt}\pgfsys@curveto{-14.22638pt}{-1.5715pt}{-7.8571pt}{-2.84544pt}{0.0pt}{-2.84544pt}\pgfsys@curveto{7.8571pt}{-2.84544pt}{14.22638pt}{-1.5715pt}{14.22638pt}{0.0pt}\pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{0.0pt}{5.69048pt}\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{0.0pt}{0.0pt}\pgfsys@invoke{
}\pgfsys@transformcm{0.567}{0.0}{0.0}{0.3402}{0.0pt}{0.0pt}\pgfsys@invoke{
}\lxSVG@sh@defs{}\lxSVG@sh@insert{-69.44pt}{-69.44pt}{\lxSVG@sh}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}\pgfsys@moveto{14.22638pt}{0.0pt}\pgfsys@curveto{14.22638pt}{7.8571pt}{7.8571pt}{14.22638pt}{0.0pt}{14.22638pt}\pgfsys@curveto{-7.8571pt}{14.22638pt}{-14.22638pt}{7.8571pt}{-14.22638pt}{0.0pt}\pgfsys@curveto{-14.22638pt}{-1.5715pt}{-7.8571pt}{-2.84544pt}{0.0pt}{-2.84544pt}\pgfsys@curveto{7.8571pt}{-2.84544pt}{14.22638pt}{-1.5715pt}{14.22638pt}{0.0pt}\pgfsys@closepath\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}{{}{}{}{{}}{{{{}{}{}{}}}{{}{}{}{}}}}{} {} {} {}
{}{}{{}{}{}{{}}{{{{}{}{}{}}} {{}{}{}{}}}}{} {} {} {} {}{}
{}\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@invoke{
}\pgfsys@color@gray@fill{.5}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{.5,.5,.5}\pgfsys@stroke@opacity{0.05}\pgfsys@invoke{
}\pgfsys@fill@opacity{0.05}\pgfsys@invoke{
}\pgfsys@moveto{14.22638pt}{0.0pt}\pgfsys@curveto{14.22638pt}{-7.8571pt}{7.8571pt}{-14.22638pt}{0.0pt}{-14.22638pt}\pgfsys@curveto{-7.8571pt}{-14.22638pt}{-14.22638pt}{-7.8571pt}{-14.22638pt}{0.0pt}\pgfsys@curveto{-14.22638pt}{-1.5715pt}{-7.8571pt}{-2.84544pt}{0.0pt}{-2.84544pt}\pgfsys@curveto{7.8571pt}{-2.84544pt}{14.22638pt}{-1.5715pt}{14.22638pt}{0.0pt}\pgfsys@closepath\pgfsys@fill\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}
{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@fill{.5}\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@invoke{
}\pgfsys@stroke@opacity{0.5}\pgfsys@invoke{
}\pgfsys@fill@opacity{0.5}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{14.22638pt}{0.0pt}\pgfsys@curveto{14.22638pt}{1.5715pt}{7.8571pt}{2.84544pt}{0.0pt}{2.84544pt}\pgfsys@curveto{-7.8571pt}{2.84544pt}{-14.22638pt}{1.5715pt}{-14.22638pt}{0.0pt}\pgfsys@curveto{-14.22638pt}{-1.5715pt}{-7.8571pt}{-2.84544pt}{0.0pt}{-2.84544pt}\pgfsys@curveto{7.8571pt}{-2.84544pt}{14.22638pt}{-1.5715pt}{14.22638pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{{}}{}{{}{}{}{{}}{{{{}{}{}{}}} {{}{}{}{}}}}{} {} {} {}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setdash{3.0pt,2.0pt}{0.0pt}\pgfsys@invoke{
}{}\pgfsys@moveto{14.22638pt}{0.0pt}\pgfsys@curveto{14.22638pt}{1.5715pt}{7.8571pt}{2.84544pt}{0.0pt}{2.84544pt}\pgfsys@curveto{-7.8571pt}{2.84544pt}{-14.22638pt}{1.5715pt}{-14.22638pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{{}}{}{{}{}{}{{}}{{{{}{}{}{}}} {{}{}{}{}}}}{} {} {} {}
{}{}{}\pgfsys@moveto{-14.22638pt}{0.0pt}\pgfsys@curveto{-14.22638pt}{-1.5715pt}{-7.8571pt}{-2.84544pt}{0.0pt}{-2.84544pt}\pgfsys@curveto{7.8571pt}{-2.84544pt}{14.22638pt}{-1.5715pt}{14.22638pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@color@gray@fill{1}\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\pgfsys@fill@opacity{0}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{14.22638pt}\pgfsys@moveto{1.70709pt}{14.22638pt}\pgfsys@curveto{1.70709pt}{15.16919pt}{0.94281pt}{15.93347pt}{0.0pt}{15.93347pt}\pgfsys@curveto{-0.94281pt}{15.93347pt}{-1.70709pt}{15.16919pt}{-1.70709pt}{14.22638pt}\pgfsys@curveto{-1.70709pt}{13.28357pt}{-0.94281pt}{12.51929pt}{0.0pt}{12.51929pt}\pgfsys@curveto{0.94281pt}{12.51929pt}{1.70709pt}{13.28357pt}{1.70709pt}{14.22638pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@color@gray@fill{1}\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\pgfsys@fill@opacity{0}\pgfsys@invoke{
}{}\pgfsys@moveto{14.22638pt}{0.0pt}\pgfsys@moveto{15.93347pt}{0.0pt}\pgfsys@curveto{15.93347pt}{0.94281pt}{15.16919pt}{1.70709pt}{14.22638pt}{1.70709pt}\pgfsys@curveto{13.28357pt}{1.70709pt}{12.51929pt}{0.94281pt}{12.51929pt}{0.0pt}\pgfsys@curveto{12.51929pt}{-0.94281pt}{13.28357pt}{-1.70709pt}{14.22638pt}{-1.70709pt}\pgfsys@curveto{15.16919pt}{-1.70709pt}{15.93347pt}{-0.94281pt}{15.93347pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{14.22638pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{{}}{}{{}{}{}{{}}{{{}{}{}{}}}}{} {} {} {}
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V_{C}\negthickspace\negthickspace\negthickspace$
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}\pgfsys@endscope{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@color@gray@fill{1}\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\pgfsys@fill@opacity{0}\pgfsys@invoke{
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} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{{}}{}{{}{}{}{{}}{{{}{}{}{}}}}{} {} {} {}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{ }\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{2.20001pt}\pgfsys@invoke{
}{}\pgfsys@moveto{-14.22638pt}{0.0pt}\pgfsys@curveto{-14.22638pt}{7.8571pt}{-7.8571pt}{14.22638pt}{0.0pt}{14.22638pt}\pgfsys@stroke\pgfsys@invoke{
}\pgfsys@beginscope\pgfsys@invoke{
}{\pgfsys@setlinewidth{0.6pt}\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{
}\pgfsys@moveto{-14.22638pt}{0.0pt}\pgfsys@curveto{-14.22638pt}{7.8571pt}{-7.8571pt}{14.22638pt}{0.0pt}{14.22638pt}\pgfsys@stroke\pgfsys@invoke{
}}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\end{aligned}\times
V_{C})/rel_{Ob,2}$ $\displaystyle
Mor(\mathcal{G}_{\mathcal{E}_{v,C}}):=\begin{aligned} \leavevmode\hbox
to31.67pt{\vbox to32.43pt{\pgfpicture\makeatletter\hbox{\hskip
15.37498pt\lower-16.13347pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
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to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }
\definecolor[named]{tikz@ball}{rgb}{1,0,0}{}{{}}{}{{}{}{}{{}}{{{{}{}{}{}}}{{}{}{}{}}}}{}
{} {} {} {}{}{{}{}{}{{}}{{{{}{}{}{}}} {{}{}{}{}}}}{} {} {} {} {}{}
{}\pgfsys@beginscope\pgfsys@invoke{
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}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{0.0pt}{0.0pt}\pgfsys@invoke{
}\pgfsys@transformcm{0.4536}{0.0}{0.0}{0.12347}{0.0pt}{0.0pt}\pgfsys@invoke{
}\lxSVG@sh@defs{}\lxSVG@sh@insert{-69.44pt}{-69.44pt}{\lxSVG@sh}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}\pgfsys@moveto{11.38092pt}{-8.5359pt}\pgfsys@curveto{8.33142pt}{-12.60197pt}{3.34409pt}{-14.73193pt}{-1.70187pt}{-14.12413pt}\pgfsys@curveto{-5.55542pt}{-13.66003pt}{-9.05255pt}{-11.64098pt}{-11.38135pt}{-8.5359pt}\pgfsys@curveto{-11.38135pt}{-9.47871pt}{-6.28601pt}{-10.243pt}{-0.00043pt}{-10.243pt}\pgfsys@curveto{6.28516pt}{-10.243pt}{11.3805pt}{-9.47871pt}{11.3805pt}{-8.5359pt}\pgfsys@closepath\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}{{}{}{}{{}}{{{{}{}{}{}}} {{{}{}{}{}}} {{}{}{}{}}}}{} {} {} {}
{}{}{{}{}{}{{}}{{{{}{}{}{}}}{{}{}{}{}}}}{} {} {} {} {}{}
{}\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@invoke{
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}\definecolor[named]{pgffillcolor}{rgb}{.5,.5,.5}\pgfsys@stroke@opacity{0.05}\pgfsys@invoke{
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{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@invoke{
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}\definecolor[named]{pgffillcolor}{rgb}{.5,.5,.5}\pgfsys@stroke@opacity{0.5}\pgfsys@invoke{
}\pgfsys@fill@opacity{0.5}\pgfsys@invoke{
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} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@color@gray@fill{1}\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
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} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@color@gray@fill{1}\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
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} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{{}}{}{{}{}{}{{}}{{{}{}{}{}}}}{} {} {} {}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{ }\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{2.20001pt}\pgfsys@invoke{
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}\pgfsys@beginscope\pgfsys@invoke{
}{\pgfsys@setlinewidth{0.6pt}\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{
}\pgfsys@moveto{11.38092pt}{-8.5359pt}\pgfsys@curveto{8.69438pt}{-12.1181pt}{4.47765pt}{-14.22638pt}{-0.00021pt}{-14.22638pt}\pgfsys@stroke\pgfsys@invoke{
}}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\end{aligned}\times
G_{C}\times V_{C}\ \sqcup\begin{aligned} \leavevmode\hbox to31.67pt{\vbox
to32.43pt{\pgfpicture\makeatletter\hbox{\hskip
15.37498pt\lower-16.13347pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
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}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }
\definecolor[named]{tikz@ball}{rgb}{1,0,0}{}{{}}{}{{}{}{}{{}}{{{{}{}{}{}}}{{}{}{}{}}}}{}
{} {} {} {}{}{{}{}{}{{}}{{{{}{}{}{}}} {{}{}{}{}}}}{} {} {} {} {}{}
{}\pgfsys@beginscope\pgfsys@invoke{
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}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-0.00021pt}{-11.63391pt}\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{0.0pt}{0.0pt}\pgfsys@invoke{
}\pgfsys@transformcm{0.4536}{0.0}{0.0}{0.12347}{0.0pt}{0.0pt}\pgfsys@invoke{
}\lxSVG@sh@defs{}\lxSVG@sh@insert{-69.44pt}{-69.44pt}{\lxSVG@sh}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}\pgfsys@moveto{11.38092pt}{-8.5359pt}\pgfsys@curveto{8.33142pt}{-12.60197pt}{3.34409pt}{-14.73193pt}{-1.70187pt}{-14.12413pt}\pgfsys@curveto{-5.55542pt}{-13.66003pt}{-9.05255pt}{-11.64098pt}{-11.38135pt}{-8.5359pt}\pgfsys@curveto{-11.38135pt}{-9.47871pt}{-6.28601pt}{-10.243pt}{-0.00043pt}{-10.243pt}\pgfsys@curveto{6.28516pt}{-10.243pt}{11.3805pt}{-9.47871pt}{11.3805pt}{-8.5359pt}\pgfsys@closepath\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}{{}{}{}{{}}{{{{}{}{}{}}} {{{}{}{}{}}} {{}{}{}{}}}}{} {} {} {}
{}{}{{}{}{}{{}}{{{{}{}{}{}}}{{}{}{}{}}}}{} {} {} {} {}{}
{}\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@invoke{
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}\definecolor[named]{pgffillcolor}{rgb}{.5,.5,.5}\pgfsys@stroke@opacity{0.05}\pgfsys@invoke{
}\pgfsys@fill@opacity{0.05}\pgfsys@invoke{
}{}\pgfsys@moveto{11.38092pt}{-8.5359pt}\pgfsys@curveto{16.09518pt}{-2.25021pt}{14.82138pt}{6.66687pt}{8.53569pt}{11.38113pt}\pgfsys@curveto{2.25pt}{16.0954pt}{-6.66708pt}{14.8216pt}{-11.38135pt}{8.5359pt}\pgfsys@curveto{-15.17499pt}{3.4778pt}{-15.17499pt}{-3.4778pt}{-11.38135pt}{-8.5359pt}\pgfsys@curveto{-11.38135pt}{-7.5931pt}{-6.28601pt}{-6.82881pt}{-0.00043pt}{-6.82881pt}\pgfsys@curveto{6.28516pt}{-6.82881pt}{11.3805pt}{-7.5931pt}{11.3805pt}{-8.5359pt}\pgfsys@closepath\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\par{}{{}}{}{{{}}
{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@invoke{
}\pgfsys@color@gray@fill{.5}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{.5,.5,.5}\pgfsys@stroke@opacity{0.5}\pgfsys@invoke{
}\pgfsys@fill@opacity{0.5}\pgfsys@invoke{
}\pgfsys@moveto{0.0pt}{-8.5359pt}\pgfsys@moveto{11.38092pt}{-8.5359pt}\pgfsys@curveto{11.38092pt}{-7.75015pt}{6.28558pt}{-7.11319pt}{0.0pt}{-7.11319pt}\pgfsys@curveto{-6.28558pt}{-7.11319pt}{-11.38092pt}{-7.75015pt}{-11.38092pt}{-8.5359pt}\pgfsys@curveto{-11.38092pt}{-9.32166pt}{-6.28558pt}{-9.95862pt}{0.0pt}{-9.95862pt}\pgfsys@curveto{6.28558pt}{-9.95862pt}{11.38092pt}{-9.32166pt}{11.38092pt}{-8.5359pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{-8.5359pt}\pgfsys@fill\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
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} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@color@gray@fill{1}\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\pgfsys@fill@opacity{0}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{-14.22638pt}\pgfsys@moveto{1.70709pt}{-14.22638pt}\pgfsys@curveto{1.70709pt}{-13.28357pt}{0.94281pt}{-12.51929pt}{0.0pt}{-12.51929pt}\pgfsys@curveto{-0.94281pt}{-12.51929pt}{-1.70709pt}{-13.28357pt}{-1.70709pt}{-14.22638pt}\pgfsys@curveto{-1.70709pt}{-15.16919pt}{-0.94281pt}{-15.93347pt}{0.0pt}{-15.93347pt}\pgfsys@curveto{0.94281pt}{-15.93347pt}{1.70709pt}{-15.16919pt}{1.70709pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{{}}{}{{}{}{}{{}}{{{}{}{}{}}}}{} {} {} {}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{ }\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{2.20001pt}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{-14.22638pt}\pgfsys@curveto{-4.47786pt}{-14.22638pt}{-8.6946pt}{-12.1181pt}{-11.38113pt}{-8.5359pt}\pgfsys@stroke\pgfsys@invoke{
}\pgfsys@beginscope\pgfsys@invoke{
}{\pgfsys@setlinewidth{0.6pt}\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{
}\pgfsys@moveto{0.0pt}{-14.22638pt}\pgfsys@curveto{-4.47786pt}{-14.22638pt}{-8.6946pt}{-12.1181pt}{-11.38113pt}{-8.5359pt}\pgfsys@stroke\pgfsys@invoke{
}}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\end{aligned}\times
G_{C}\times V_{C}\negthickspace\negthickspace\negthickspace$
$\displaystyle\sqcup$ $\displaystyle(\begin{aligned} \leavevmode\hbox
to32.27pt{\vbox to29.55pt{\pgfpicture\makeatletter\hbox{\hskip
16.13347pt\lower-15.32639pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
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}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }
\definecolor[named]{tikz@ball}{rgb}{1,0,0}{}{{}}{}{{}{}{}{{}}{{{{}{}{}{}}}{{}{}{}{}}}}{}
{} {} {} {}{}{{}{}{}{{}}{{{{}{}{}{}}} {{}{}{}{}}}}{} {} {} {} {}{}
{}\pgfsys@beginscope\pgfsys@invoke{
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G_{C}\times V_{C}\negthickspace\negthickspace\negthickspace$
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{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
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G_{C}\times V_{C}$ $\displaystyle\sqcup\begin{aligned} \leavevmode\hbox
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\pgfsys@invoke{\lxSVG@closescope
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{}{}{{}{}{}{{}}{{{{}{}{}{}}} {{}{}{}{}}}}{} {} {} {} {}{}
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{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
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G_{C}\times V_{C}\sqcup(\begin{aligned} \leavevmode\hbox to30.56pt{\vbox
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16.13347pt\lower-16.13347pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
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{}{}{{}{}{}{{}}{{{{}{}{}{}}}{{}{}{}{}}}}{} {} {} {} {}{}
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{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
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} \pgfsys@invoke{\lxSVG@closescope
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}\pgfsys@endscope{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
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} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{{}}{}{{}{}{}{{}}{{{}{}{}{}}}}{} {} {} {}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{ }\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{2.20001pt}\pgfsys@invoke{
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}{\pgfsys@setlinewidth{0.6pt}\pgfsys@invoke{
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}\pgfsys@endscope\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
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G_{C}\times V_{C}\negthickspace\negthickspace\negthickspace$
$\displaystyle\sqcup\begin{aligned} \leavevmode\hbox to30.56pt{\vbox
to30.36pt{\pgfpicture\makeatletter\hbox{\hskip
16.13347pt\lower-14.22638pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
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{}{} {}\pgfsys@beginscope\pgfsys@invoke{
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{}{}{{}{}{}{{}}{{{{}{}{}{}}} {{}{}{}{}}}}{} {} {} {} {}{}
{}\pgfsys@beginscope\pgfsys@invoke{
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{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
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{}{}\pgfsys@beginscope\pgfsys@invoke{
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}\pgfsys@endscope{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@color@gray@fill{1}\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\pgfsys@fill@opacity{0}\pgfsys@invoke{
}{}\pgfsys@moveto{-14.22638pt}{0.0pt}\pgfsys@moveto{-12.51929pt}{0.0pt}\pgfsys@curveto{-12.51929pt}{0.94281pt}{-13.28357pt}{1.70709pt}{-14.22638pt}{1.70709pt}\pgfsys@curveto{-15.16919pt}{1.70709pt}{-15.93347pt}{0.94281pt}{-15.93347pt}{0.0pt}\pgfsys@curveto{-15.93347pt}{-0.94281pt}{-15.16919pt}{-1.70709pt}{-14.22638pt}{-1.70709pt}\pgfsys@curveto{-13.28357pt}{-1.70709pt}{-12.51929pt}{-0.94281pt}{-12.51929pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{-14.22638pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{{}}{}{{}{}{}{{}}{{{}{}{}{}}}}{} {} {} {}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{ }\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{2.20001pt}\pgfsys@invoke{
}{}\pgfsys@moveto{-14.22638pt}{0.0pt}\pgfsys@curveto{-14.22638pt}{7.8571pt}{-7.8571pt}{14.22638pt}{0.0pt}{14.22638pt}\pgfsys@stroke\pgfsys@invoke{
}\pgfsys@beginscope\pgfsys@invoke{
}{\pgfsys@setlinewidth{0.6pt}\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{
}\pgfsys@moveto{-14.22638pt}{0.0pt}\pgfsys@curveto{-14.22638pt}{7.8571pt}{-7.8571pt}{14.22638pt}{0.0pt}{14.22638pt}\pgfsys@stroke\pgfsys@invoke{
}}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\end{aligned}\times
G_{C}\times V_{C})/rel_{Mor,4}.$
For notational convenience, we identify hemispheres with unit disks.
The glueing along the boundary is given by $\tilde{\gamma}_{v}:U_{S^{1}}\times
G_{C}\ltimes V_{C}\to G_{C}\ltimes V_{C}$. More explicitly,
* •
$rel_{Ob,1}$: for $(e^{i2\pi t},\vec{w})$ in the boundary of $\begin{aligned}
\leavevmode\hbox to30.56pt{\vbox
to30.36pt{\pgfpicture\makeatletter\hbox{\hskip
14.42638pt\lower-16.13347pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }
\definecolor[named]{tikz@ball}{rgb}{1,0,0}{}{{}}{}{{}{}{}{{}}{{{{}{}{}{}}}{{}{}{}{}}}}{}
{} {} {} {}{}{{}{}{}{{}}{{{{}{}{}{}}} {{}{}{}{}}}}{} {} {} {} {}{}
{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@moveto{14.22638pt}{0.0pt}\pgfsys@curveto{14.22638pt}{-7.8571pt}{7.8571pt}{-14.22638pt}{0.0pt}{-14.22638pt}\pgfsys@curveto{-7.8571pt}{-14.22638pt}{-14.22638pt}{-7.8571pt}{-14.22638pt}{0.0pt}\pgfsys@curveto{-14.22638pt}{-1.5715pt}{-7.8571pt}{-2.84544pt}{0.0pt}{-2.84544pt}\pgfsys@curveto{7.8571pt}{-2.84544pt}{14.22638pt}{-1.5715pt}{14.22638pt}{0.0pt}\pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{0.0pt}{-7.11319pt}\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{0.0pt}{0.0pt}\pgfsys@invoke{
}\pgfsys@transformcm{0.567}{0.0}{0.0}{0.2835}{0.0pt}{0.0pt}\pgfsys@invoke{
}\lxSVG@sh@defs{}\lxSVG@sh@insert{-69.44pt}{-69.44pt}{\lxSVG@sh}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}\pgfsys@moveto{14.22638pt}{0.0pt}\pgfsys@curveto{14.22638pt}{-7.8571pt}{7.8571pt}{-14.22638pt}{0.0pt}{-14.22638pt}\pgfsys@curveto{-7.8571pt}{-14.22638pt}{-14.22638pt}{-7.8571pt}{-14.22638pt}{0.0pt}\pgfsys@curveto{-14.22638pt}{-1.5715pt}{-7.8571pt}{-2.84544pt}{0.0pt}{-2.84544pt}\pgfsys@curveto{7.8571pt}{-2.84544pt}{14.22638pt}{-1.5715pt}{14.22638pt}{0.0pt}\pgfsys@closepath\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}{{}{}{}{{}}{{{{}{}{}{}}} {{}{}{}{}}}}{} {} {} {}
{}{}{{}{}{}{{}}{{{{}{}{}{}}}{{}{}{}{}}}}{} {} {} {} {}{}
{}\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@invoke{
}\pgfsys@color@gray@fill{.5}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{.5,.5,.5}\pgfsys@stroke@opacity{0.05}\pgfsys@invoke{
}\pgfsys@fill@opacity{0.05}\pgfsys@invoke{
}\pgfsys@moveto{14.22638pt}{0.0pt}\pgfsys@curveto{14.22638pt}{7.8571pt}{7.8571pt}{14.22638pt}{0.0pt}{14.22638pt}\pgfsys@curveto{-7.8571pt}{14.22638pt}{-14.22638pt}{7.8571pt}{-14.22638pt}{0.0pt}\pgfsys@curveto{-14.22638pt}{1.5715pt}{-7.8571pt}{2.84544pt}{0.0pt}{2.84544pt}\pgfsys@curveto{7.8571pt}{2.84544pt}{14.22638pt}{1.5715pt}{14.22638pt}{0.0pt}\pgfsys@closepath\pgfsys@fill\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}
{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@fill{.5}\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\pgfsys@stroke@opacity{0.5}\pgfsys@invoke{
}\pgfsys@fill@opacity{0.5}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{14.22638pt}{0.0pt}\pgfsys@curveto{14.22638pt}{1.5715pt}{7.8571pt}{2.84544pt}{0.0pt}{2.84544pt}\pgfsys@curveto{-7.8571pt}{2.84544pt}{-14.22638pt}{1.5715pt}{-14.22638pt}{0.0pt}\pgfsys@curveto{-14.22638pt}{-1.5715pt}{-7.8571pt}{-2.84544pt}{0.0pt}{-2.84544pt}\pgfsys@curveto{7.8571pt}{-2.84544pt}{14.22638pt}{-1.5715pt}{14.22638pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@color@gray@fill{1}\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\pgfsys@fill@opacity{0}\pgfsys@invoke{
}{}\pgfsys@moveto{14.22638pt}{0.0pt}\pgfsys@moveto{15.93347pt}{0.0pt}\pgfsys@curveto{15.93347pt}{0.94281pt}{15.16919pt}{1.70709pt}{14.22638pt}{1.70709pt}\pgfsys@curveto{13.28357pt}{1.70709pt}{12.51929pt}{0.94281pt}{12.51929pt}{0.0pt}\pgfsys@curveto{12.51929pt}{-0.94281pt}{13.28357pt}{-1.70709pt}{14.22638pt}{-1.70709pt}\pgfsys@curveto{15.16919pt}{-1.70709pt}{15.93347pt}{-0.94281pt}{15.93347pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{14.22638pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@color@gray@fill{1}\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\pgfsys@fill@opacity{0}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{-14.22638pt}\pgfsys@moveto{1.70709pt}{-14.22638pt}\pgfsys@curveto{1.70709pt}{-13.28357pt}{0.94281pt}{-12.51929pt}{0.0pt}{-12.51929pt}\pgfsys@curveto{-0.94281pt}{-12.51929pt}{-1.70709pt}{-13.28357pt}{-1.70709pt}{-14.22638pt}\pgfsys@curveto{-1.70709pt}{-15.16919pt}{-0.94281pt}{-15.93347pt}{0.0pt}{-15.93347pt}\pgfsys@curveto{0.94281pt}{-15.93347pt}{1.70709pt}{-15.16919pt}{1.70709pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{{}}{}{{}{}{}{{}}{{{}{}{}{}}}}{} {} {} {}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{ }\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{2.20001pt}\pgfsys@invoke{
}{}\pgfsys@moveto{14.22638pt}{0.0pt}\pgfsys@curveto{14.22638pt}{-7.8571pt}{7.8571pt}{-14.22638pt}{0.0pt}{-14.22638pt}\pgfsys@stroke\pgfsys@invoke{
}\pgfsys@beginscope\pgfsys@invoke{
}{\pgfsys@setlinewidth{0.6pt}\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{
}\pgfsys@moveto{14.22638pt}{0.0pt}\pgfsys@curveto{14.22638pt}{-7.8571pt}{7.8571pt}{-14.22638pt}{0.0pt}{-14.22638pt}\pgfsys@stroke\pgfsys@invoke{
}}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\end{aligned}\times
V_{C}$ , $(e^{i2\pi t^{\prime}},\vec{w}^{\prime})$ in the boundary of
$\begin{aligned} \leavevmode\hbox to30.56pt{\vbox
to30.36pt{\pgfpicture\makeatletter\hbox{\hskip
14.42638pt\lower-14.22638pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }
\definecolor[named]{tikz@ball}{rgb}{1,0,0}{}{{}}{}{{}{}{}{{}}{{{{}{}{}{}}}
{{}{}{}{}}}}{} {} {} {} {}{}{{}{}{}{{}}{{{{}{}{}{}}} {{}{}{}{}}}}{} {} {} {}
{}{} {}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@moveto{14.22638pt}{0.0pt}\pgfsys@curveto{14.22638pt}{7.8571pt}{7.8571pt}{14.22638pt}{0.0pt}{14.22638pt}\pgfsys@curveto{-7.8571pt}{14.22638pt}{-14.22638pt}{7.8571pt}{-14.22638pt}{0.0pt}\pgfsys@curveto{-14.22638pt}{-1.5715pt}{-7.8571pt}{-2.84544pt}{0.0pt}{-2.84544pt}\pgfsys@curveto{7.8571pt}{-2.84544pt}{14.22638pt}{-1.5715pt}{14.22638pt}{0.0pt}\pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{0.0pt}{5.69048pt}\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{0.0pt}{0.0pt}\pgfsys@invoke{
}\pgfsys@transformcm{0.567}{0.0}{0.0}{0.3402}{0.0pt}{0.0pt}\pgfsys@invoke{
}\lxSVG@sh@defs{}\lxSVG@sh@insert{-69.44pt}{-69.44pt}{\lxSVG@sh}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}\pgfsys@moveto{14.22638pt}{0.0pt}\pgfsys@curveto{14.22638pt}{7.8571pt}{7.8571pt}{14.22638pt}{0.0pt}{14.22638pt}\pgfsys@curveto{-7.8571pt}{14.22638pt}{-14.22638pt}{7.8571pt}{-14.22638pt}{0.0pt}\pgfsys@curveto{-14.22638pt}{-1.5715pt}{-7.8571pt}{-2.84544pt}{0.0pt}{-2.84544pt}\pgfsys@curveto{7.8571pt}{-2.84544pt}{14.22638pt}{-1.5715pt}{14.22638pt}{0.0pt}\pgfsys@closepath\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}{{}{}{}{{}}{{{{}{}{}{}}}{{}{}{}{}}}}{} {} {} {}
{}{}{{}{}{}{{}}{{{{}{}{}{}}} {{}{}{}{}}}}{} {} {} {} {}{}
{}\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@invoke{
}\pgfsys@color@gray@fill{.5}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{.5,.5,.5}\pgfsys@stroke@opacity{0.05}\pgfsys@invoke{
}\pgfsys@fill@opacity{0.05}\pgfsys@invoke{
}\pgfsys@moveto{14.22638pt}{0.0pt}\pgfsys@curveto{14.22638pt}{-7.8571pt}{7.8571pt}{-14.22638pt}{0.0pt}{-14.22638pt}\pgfsys@curveto{-7.8571pt}{-14.22638pt}{-14.22638pt}{-7.8571pt}{-14.22638pt}{0.0pt}\pgfsys@curveto{-14.22638pt}{-1.5715pt}{-7.8571pt}{-2.84544pt}{0.0pt}{-2.84544pt}\pgfsys@curveto{7.8571pt}{-2.84544pt}{14.22638pt}{-1.5715pt}{14.22638pt}{0.0pt}\pgfsys@closepath\pgfsys@fill\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}
{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@fill{.5}\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@invoke{
}\pgfsys@stroke@opacity{0.5}\pgfsys@invoke{
}\pgfsys@fill@opacity{0.5}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{14.22638pt}{0.0pt}\pgfsys@curveto{14.22638pt}{1.5715pt}{7.8571pt}{2.84544pt}{0.0pt}{2.84544pt}\pgfsys@curveto{-7.8571pt}{2.84544pt}{-14.22638pt}{1.5715pt}{-14.22638pt}{0.0pt}\pgfsys@curveto{-14.22638pt}{-1.5715pt}{-7.8571pt}{-2.84544pt}{0.0pt}{-2.84544pt}\pgfsys@curveto{7.8571pt}{-2.84544pt}{14.22638pt}{-1.5715pt}{14.22638pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{{}}{}{{}{}{}{{}}{{{{}{}{}{}}} {{}{}{}{}}}}{} {} {} {}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setdash{3.0pt,2.0pt}{0.0pt}\pgfsys@invoke{
}{}\pgfsys@moveto{14.22638pt}{0.0pt}\pgfsys@curveto{14.22638pt}{1.5715pt}{7.8571pt}{2.84544pt}{0.0pt}{2.84544pt}\pgfsys@curveto{-7.8571pt}{2.84544pt}{-14.22638pt}{1.5715pt}{-14.22638pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{{}}{}{{}{}{}{{}}{{{{}{}{}{}}} {{}{}{}{}}}}{} {} {} {}
{}{}{}\pgfsys@moveto{-14.22638pt}{0.0pt}\pgfsys@curveto{-14.22638pt}{-1.5715pt}{-7.8571pt}{-2.84544pt}{0.0pt}{-2.84544pt}\pgfsys@curveto{7.8571pt}{-2.84544pt}{14.22638pt}{-1.5715pt}{14.22638pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@color@gray@fill{1}\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\pgfsys@fill@opacity{0}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{14.22638pt}\pgfsys@moveto{1.70709pt}{14.22638pt}\pgfsys@curveto{1.70709pt}{15.16919pt}{0.94281pt}{15.93347pt}{0.0pt}{15.93347pt}\pgfsys@curveto{-0.94281pt}{15.93347pt}{-1.70709pt}{15.16919pt}{-1.70709pt}{14.22638pt}\pgfsys@curveto{-1.70709pt}{13.28357pt}{-0.94281pt}{12.51929pt}{0.0pt}{12.51929pt}\pgfsys@curveto{0.94281pt}{12.51929pt}{1.70709pt}{13.28357pt}{1.70709pt}{14.22638pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@color@gray@fill{1}\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\pgfsys@fill@opacity{0}\pgfsys@invoke{
}{}\pgfsys@moveto{14.22638pt}{0.0pt}\pgfsys@moveto{15.93347pt}{0.0pt}\pgfsys@curveto{15.93347pt}{0.94281pt}{15.16919pt}{1.70709pt}{14.22638pt}{1.70709pt}\pgfsys@curveto{13.28357pt}{1.70709pt}{12.51929pt}{0.94281pt}{12.51929pt}{0.0pt}\pgfsys@curveto{12.51929pt}{-0.94281pt}{13.28357pt}{-1.70709pt}{14.22638pt}{-1.70709pt}\pgfsys@curveto{15.16919pt}{-1.70709pt}{15.93347pt}{-0.94281pt}{15.93347pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{14.22638pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{{}}{}{{}{}{}{{}}{{{}{}{}{}}}}{} {} {} {}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{ }\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{2.20001pt}\pgfsys@invoke{
}{}\pgfsys@moveto{14.22638pt}{0.0pt}\pgfsys@curveto{14.22638pt}{7.8571pt}{7.8571pt}{14.22638pt}{0.0pt}{14.22638pt}\pgfsys@stroke\pgfsys@invoke{
}\pgfsys@beginscope\pgfsys@invoke{
}{\pgfsys@setlinewidth{0.6pt}\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{
}\pgfsys@moveto{14.22638pt}{0.0pt}\pgfsys@curveto{14.22638pt}{7.8571pt}{7.8571pt}{14.22638pt}{0.0pt}{14.22638pt}\pgfsys@stroke\pgfsys@invoke{
}}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\end{aligned}\times
V_{C}$, $(e^{i2\pi t},\vec{w})\sim(e^{i2\pi t^{\prime}},\vec{w}^{\prime})$ if
and only if $t^{\prime}=-t$, $\vec{w}^{\prime}=D(-tv)\vec{w}$. Note that here
$t\in(0,1)$ and $t^{\prime}\in(-1,0)$.
* •
$rel_{Ob,2}$: for $(e^{i2\pi t},\vec{w})$ in the boundary of $\begin{aligned}
\leavevmode\hbox to30.56pt{\vbox
to30.36pt{\pgfpicture\makeatletter\hbox{\hskip
16.13347pt\lower-16.13347pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }
\definecolor[named]{tikz@ball}{rgb}{1,0,0}{}{{}}{}{{}{}{}{{}}{{{{}{}{}{}}}{{}{}{}{}}}}{}
{} {} {} {}{}{{}{}{}{{}}{{{{}{}{}{}}} {{}{}{}{}}}}{} {} {} {} {}{}
{}\pgfsys@beginscope\pgfsys@invoke{
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} {}{{}}{}{{}{}{}{{}}{{{{}{}{}{}}} {{}{}{}{}}}}{} {} {} {}
{}{}{{}{}{}{{}}{{{{}{}{}{}}}{{}{}{}{}}}}{} {} {} {} {}{}
{}\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@invoke{
}\pgfsys@color@gray@fill{.5}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{.5,.5,.5}\pgfsys@stroke@opacity{0.05}\pgfsys@invoke{
}\pgfsys@fill@opacity{0.05}\pgfsys@invoke{
}\pgfsys@moveto{14.22638pt}{0.0pt}\pgfsys@curveto{14.22638pt}{7.8571pt}{7.8571pt}{14.22638pt}{0.0pt}{14.22638pt}\pgfsys@curveto{-7.8571pt}{14.22638pt}{-14.22638pt}{7.8571pt}{-14.22638pt}{0.0pt}\pgfsys@curveto{-14.22638pt}{1.5715pt}{-7.8571pt}{2.84544pt}{0.0pt}{2.84544pt}\pgfsys@curveto{7.8571pt}{2.84544pt}{14.22638pt}{1.5715pt}{14.22638pt}{0.0pt}\pgfsys@closepath\pgfsys@fill\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}
{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@fill{.5}\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\pgfsys@stroke@opacity{0.5}\pgfsys@invoke{
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} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@color@gray@fill{1}\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\pgfsys@fill@opacity{0}\pgfsys@invoke{
}{}\pgfsys@moveto{-14.22638pt}{0.0pt}\pgfsys@moveto{-12.51929pt}{0.0pt}\pgfsys@curveto{-12.51929pt}{0.94281pt}{-13.28357pt}{1.70709pt}{-14.22638pt}{1.70709pt}\pgfsys@curveto{-15.16919pt}{1.70709pt}{-15.93347pt}{0.94281pt}{-15.93347pt}{0.0pt}\pgfsys@curveto{-15.93347pt}{-0.94281pt}{-15.16919pt}{-1.70709pt}{-14.22638pt}{-1.70709pt}\pgfsys@curveto{-13.28357pt}{-1.70709pt}{-12.51929pt}{-0.94281pt}{-12.51929pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{-14.22638pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@color@gray@fill{1}\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\pgfsys@fill@opacity{0}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{-14.22638pt}\pgfsys@moveto{1.70709pt}{-14.22638pt}\pgfsys@curveto{1.70709pt}{-13.28357pt}{0.94281pt}{-12.51929pt}{0.0pt}{-12.51929pt}\pgfsys@curveto{-0.94281pt}{-12.51929pt}{-1.70709pt}{-13.28357pt}{-1.70709pt}{-14.22638pt}\pgfsys@curveto{-1.70709pt}{-15.16919pt}{-0.94281pt}{-15.93347pt}{0.0pt}{-15.93347pt}\pgfsys@curveto{0.94281pt}{-15.93347pt}{1.70709pt}{-15.16919pt}{1.70709pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{{}}{}{{}{}{}{{}}{{{}{}{}{}}}}{} {} {} {}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{ }\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{2.20001pt}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{-14.22638pt}\pgfsys@curveto{-7.8571pt}{-14.22638pt}{-14.22638pt}{-7.8571pt}{-14.22638pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
}\pgfsys@beginscope\pgfsys@invoke{
}{\pgfsys@setlinewidth{0.6pt}\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{
}\pgfsys@moveto{0.0pt}{-14.22638pt}\pgfsys@curveto{-7.8571pt}{-14.22638pt}{-14.22638pt}{-7.8571pt}{-14.22638pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
}}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\end{aligned}\times
V_{C}$, $(e^{i2\pi t^{\prime}},\vec{w}^{\prime})$ in the boundary of
$\begin{aligned} \leavevmode\hbox to30.56pt{\vbox
to30.36pt{\pgfpicture\makeatletter\hbox{\hskip
16.13347pt\lower-14.22638pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }
\definecolor[named]{tikz@ball}{rgb}{1,0,0}{}{{}}{}{{}{}{}{{}}{{{{}{}{}{}}}
{{}{}{}{}}}}{} {} {} {} {}{}{{}{}{}{{}}{{{{}{}{}{}}} {{}{}{}{}}}}{} {} {} {}
{}{} {}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@moveto{14.22638pt}{0.0pt}\pgfsys@curveto{14.22638pt}{7.8571pt}{7.8571pt}{14.22638pt}{0.0pt}{14.22638pt}\pgfsys@curveto{-7.8571pt}{14.22638pt}{-14.22638pt}{7.8571pt}{-14.22638pt}{0.0pt}\pgfsys@curveto{-14.22638pt}{-1.5715pt}{-7.8571pt}{-2.84544pt}{0.0pt}{-2.84544pt}\pgfsys@curveto{7.8571pt}{-2.84544pt}{14.22638pt}{-1.5715pt}{14.22638pt}{0.0pt}\pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{0.0pt}{5.69048pt}\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{0.0pt}{0.0pt}\pgfsys@invoke{
}\pgfsys@transformcm{0.567}{0.0}{0.0}{0.3402}{0.0pt}{0.0pt}\pgfsys@invoke{
}\lxSVG@sh@defs{}\lxSVG@sh@insert{-69.44pt}{-69.44pt}{\lxSVG@sh}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}\pgfsys@moveto{14.22638pt}{0.0pt}\pgfsys@curveto{14.22638pt}{7.8571pt}{7.8571pt}{14.22638pt}{0.0pt}{14.22638pt}\pgfsys@curveto{-7.8571pt}{14.22638pt}{-14.22638pt}{7.8571pt}{-14.22638pt}{0.0pt}\pgfsys@curveto{-14.22638pt}{-1.5715pt}{-7.8571pt}{-2.84544pt}{0.0pt}{-2.84544pt}\pgfsys@curveto{7.8571pt}{-2.84544pt}{14.22638pt}{-1.5715pt}{14.22638pt}{0.0pt}\pgfsys@closepath\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}{{}{}{}{{}}{{{{}{}{}{}}}{{}{}{}{}}}}{} {} {} {}
{}{}{{}{}{}{{}}{{{{}{}{}{}}} {{}{}{}{}}}}{} {} {} {} {}{}
{}\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@invoke{
}\pgfsys@color@gray@fill{.5}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{.5,.5,.5}\pgfsys@stroke@opacity{0.05}\pgfsys@invoke{
}\pgfsys@fill@opacity{0.05}\pgfsys@invoke{
}\pgfsys@moveto{14.22638pt}{0.0pt}\pgfsys@curveto{14.22638pt}{-7.8571pt}{7.8571pt}{-14.22638pt}{0.0pt}{-14.22638pt}\pgfsys@curveto{-7.8571pt}{-14.22638pt}{-14.22638pt}{-7.8571pt}{-14.22638pt}{0.0pt}\pgfsys@curveto{-14.22638pt}{-1.5715pt}{-7.8571pt}{-2.84544pt}{0.0pt}{-2.84544pt}\pgfsys@curveto{7.8571pt}{-2.84544pt}{14.22638pt}{-1.5715pt}{14.22638pt}{0.0pt}\pgfsys@closepath\pgfsys@fill\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}
{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@fill{.5}\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@invoke{
}\pgfsys@stroke@opacity{0.5}\pgfsys@invoke{
}\pgfsys@fill@opacity{0.5}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{14.22638pt}{0.0pt}\pgfsys@curveto{14.22638pt}{1.5715pt}{7.8571pt}{2.84544pt}{0.0pt}{2.84544pt}\pgfsys@curveto{-7.8571pt}{2.84544pt}{-14.22638pt}{1.5715pt}{-14.22638pt}{0.0pt}\pgfsys@curveto{-14.22638pt}{-1.5715pt}{-7.8571pt}{-2.84544pt}{0.0pt}{-2.84544pt}\pgfsys@curveto{7.8571pt}{-2.84544pt}{14.22638pt}{-1.5715pt}{14.22638pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{{}}{}{{}{}{}{{}}{{{{}{}{}{}}} {{}{}{}{}}}}{} {} {} {}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setdash{3.0pt,2.0pt}{0.0pt}\pgfsys@invoke{
}{}\pgfsys@moveto{14.22638pt}{0.0pt}\pgfsys@curveto{14.22638pt}{1.5715pt}{7.8571pt}{2.84544pt}{0.0pt}{2.84544pt}\pgfsys@curveto{-7.8571pt}{2.84544pt}{-14.22638pt}{1.5715pt}{-14.22638pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{{}}{}{{}{}{}{{}}{{{{}{}{}{}}} {{}{}{}{}}}}{} {} {} {}
{}{}{}\pgfsys@moveto{-14.22638pt}{0.0pt}\pgfsys@curveto{-14.22638pt}{-1.5715pt}{-7.8571pt}{-2.84544pt}{0.0pt}{-2.84544pt}\pgfsys@curveto{7.8571pt}{-2.84544pt}{14.22638pt}{-1.5715pt}{14.22638pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@color@gray@fill{1}\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\pgfsys@fill@opacity{0}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{14.22638pt}\pgfsys@moveto{1.70709pt}{14.22638pt}\pgfsys@curveto{1.70709pt}{15.16919pt}{0.94281pt}{15.93347pt}{0.0pt}{15.93347pt}\pgfsys@curveto{-0.94281pt}{15.93347pt}{-1.70709pt}{15.16919pt}{-1.70709pt}{14.22638pt}\pgfsys@curveto{-1.70709pt}{13.28357pt}{-0.94281pt}{12.51929pt}{0.0pt}{12.51929pt}\pgfsys@curveto{0.94281pt}{12.51929pt}{1.70709pt}{13.28357pt}{1.70709pt}{14.22638pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@color@gray@fill{1}\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\pgfsys@fill@opacity{0}\pgfsys@invoke{
}{}\pgfsys@moveto{-14.22638pt}{0.0pt}\pgfsys@moveto{-12.51929pt}{0.0pt}\pgfsys@curveto{-12.51929pt}{0.94281pt}{-13.28357pt}{1.70709pt}{-14.22638pt}{1.70709pt}\pgfsys@curveto{-15.16919pt}{1.70709pt}{-15.93347pt}{0.94281pt}{-15.93347pt}{0.0pt}\pgfsys@curveto{-15.93347pt}{-0.94281pt}{-15.16919pt}{-1.70709pt}{-14.22638pt}{-1.70709pt}\pgfsys@curveto{-13.28357pt}{-1.70709pt}{-12.51929pt}{-0.94281pt}{-12.51929pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{-14.22638pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{{}}{}{{}{}{}{{}}{{{}{}{}{}}}}{} {} {} {}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{ }\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{2.20001pt}\pgfsys@invoke{
}{}\pgfsys@moveto{-14.22638pt}{0.0pt}\pgfsys@curveto{-14.22638pt}{7.8571pt}{-7.8571pt}{14.22638pt}{0.0pt}{14.22638pt}\pgfsys@stroke\pgfsys@invoke{
}\pgfsys@beginscope\pgfsys@invoke{
}{\pgfsys@setlinewidth{0.6pt}\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{
}\pgfsys@moveto{-14.22638pt}{0.0pt}\pgfsys@curveto{-14.22638pt}{7.8571pt}{-7.8571pt}{14.22638pt}{0.0pt}{14.22638pt}\pgfsys@stroke\pgfsys@invoke{
}}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\end{aligned}\times
V_{C}$, $(e^{i2\pi t},\vec{w})\sim(e^{i2\pi t^{\prime}},\vec{w}^{\prime})$ if
and only if $t^{\prime}=-t$, $\vec{w}^{\prime}=D(-tv)\vec{w}$. Note that here
$t,t^{\prime}\in(-\frac{1}{2},\frac{1}{2})$.
* •
$rel_{Mor,i},\ i=1,2,3,4$: for boundary elements $(e^{i2\pi t_{1}}\to e^{i2\pi
t_{2}},\vec{w}\xrightarrow{g}g\cdot\vec{w})$ and $(e^{i2\pi t_{1}^{\prime}}\to
e^{i2\pi
t_{2}^{\prime}},\vec{w}^{\prime}\xrightarrow{g^{\prime}}g^{\prime}\cdot\vec{w}^{\prime})$,
$(e^{i2\pi t_{1}}\to e^{i2\pi
t_{2}},\vec{w}\xrightarrow{g}g\cdot\vec{w})\sim(e^{i2\pi t_{1}^{\prime}}\to
e^{i2\pi
t_{2}^{\prime}},\vec{w}^{\prime}\xrightarrow{g^{\prime}}g^{\prime}\cdot\vec{w}^{\prime})$
if and only if
$\displaystyle t_{1}=-t_{1}^{\prime}$ $\displaystyle,\ \
t_{2}=-t_{2}^{\prime},$ $\displaystyle\vec{w}^{\prime}=D(-t_{1}v)\vec{w}$
$\displaystyle,\ g^{\prime}\cdot\vec{w}^{\prime}=D(-t_{2}v)g\cdot\vec{w}.$
We denote $m_{v}:=ord(D(-v))=ord(D(v))$. Then $D(-v)$ generates a cyclic
subgroup of $G_{C}$ which is isomorphic to $\mathbb{Z}_{m_{v}}$. We denote
$\mathfrak{i}:\mathbb{Z}_{m_{v}}\to G_{C}$ the inclusion of the subgroup.
Note that when $v=y_{i}$, $i=1,...,N$, $m_{v}$ is the number labeling the
corresponding facet of the polytope $\Delta$.
###### Proposition 2.8.
If $\boldsymbol{s}_{x}:(S^{2}_{orb},p)\to\mathcal{E}_{v}$ is a sectional
orbifold morphism from an orbisphere with one orbipoint (possibly a trivial
one) lifting $s_{x}$, then the orbifold structure group at $p$ is
$\mathbb{Z}_{m_{v}}$.
###### Proof.
Given any orbifold morphism from a sphere with at most one orbifold point at
$p$, it determines an orbifold morphism $\check{\boldsymbol{s}}$ away from $p$
by restriction. Without loss of generality, we may assume $p$ to be the north
pole of the sphere. Since there is no other orbifold point on the sphere and
$\boldsymbol{s}$ lifts the zero section, $\check{\boldsymbol{s}}$ can be
represented by the groupoid morphism:
$\displaystyle\check{\boldsymbol{s}}_{0}=(id,0):\
Ob(U_{\check{S}^{2}})=\begin{aligned} \leavevmode\hbox to31.67pt{\vbox
to31.23pt{\pgfpicture\makeatletter\hbox{\hskip
15.37498pt\lower-14.93193pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
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}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }
\definecolor[named]{tikz@ball}{rgb}{1,0,0}{}{{}}{}{{}{}{}{{}}{{{{}{}{}{}}}{{}{}{}{}}}}{}
{} {} {} {}{}{{}{}{}{{}}{{{{}{}{}{}}} {{}{}{}{}}}}{} {} {} {} {}{}
{}\pgfsys@beginscope\pgfsys@invoke{
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}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-0.00021pt}{-11.63391pt}\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{0.0pt}{0.0pt}\pgfsys@invoke{
}\pgfsys@transformcm{0.4536}{0.0}{0.0}{0.12347}{0.0pt}{0.0pt}\pgfsys@invoke{
}\lxSVG@sh@defs{}\lxSVG@sh@insert{-69.44pt}{-69.44pt}{\lxSVG@sh}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}\pgfsys@moveto{11.38092pt}{-8.5359pt}\pgfsys@curveto{8.33142pt}{-12.60197pt}{3.34409pt}{-14.73193pt}{-1.70187pt}{-14.12413pt}\pgfsys@curveto{-5.55542pt}{-13.66003pt}{-9.05255pt}{-11.64098pt}{-11.38135pt}{-8.5359pt}\pgfsys@curveto{-11.38135pt}{-9.47871pt}{-6.28601pt}{-10.243pt}{-0.00043pt}{-10.243pt}\pgfsys@curveto{6.28516pt}{-10.243pt}{11.3805pt}{-9.47871pt}{11.3805pt}{-8.5359pt}\pgfsys@closepath\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}{{}{}{}{{}}{{{{}{}{}{}}} {{{}{}{}{}}} {{}{}{}{}}}}{} {} {} {}
{}{}{{}{}{}{{}}{{{{}{}{}{}}}{{}{}{}{}}}}{} {} {} {} {}{}
{}\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@invoke{
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{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
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}\definecolor[named]{pgffillcolor}{rgb}{.5,.5,.5}\pgfsys@stroke@opacity{0.5}\pgfsys@invoke{
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} \pgfsys@invoke{\lxSVG@closescope
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{}{}\pgfsys@beginscope\pgfsys@invoke{
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\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
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15.32639pt\lower-16.13347pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
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} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
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}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\pgfsys@fill@opacity{0}\pgfsys@invoke{
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} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{{}}{}{{}{}{}{{}}{{{{}{}{}{}}}{{}{}{}{}}}}{} {} {} {}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{ }\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{2.20001pt}\pgfsys@invoke{
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}\pgfsys@beginscope\pgfsys@invoke{
}{\pgfsys@setlinewidth{0.6pt}\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{
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}}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\end{aligned}\ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\longrightarrow\ \ \ \ \ \ \ \ \ Ob(\mathcal{G}_{\mathcal{E}})$
$\displaystyle\check{\boldsymbol{s}}_{1}=(id,0,\eta):\
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{}{}{{}{}{}{{}}{{{{}{}{}{}}}{{}{}{}{}}}}{} {} {} {} {}{}
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}\pgfsys@endscope{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
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} \pgfsys@invoke{\lxSVG@closescope
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{}{}\pgfsys@beginscope\pgfsys@invoke{
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\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
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{}{}{{}{}{}{{}}{{{{}{}{}{}}}{{}{}{}{}}}}{} {} {} {} {}{}
{}\pgfsys@beginscope\pgfsys@invoke{
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{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
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} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
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} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
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}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
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} \pgfsys@invoke{\lxSVG@closescope
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{}{}\pgfsys@beginscope\pgfsys@invoke{
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}\pgfsys@setlinewidth{2.20001pt}\pgfsys@invoke{
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}{\pgfsys@setlinewidth{0.6pt}\pgfsys@invoke{
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}}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\end{aligned}$
$\displaystyle\sqcup\begin{aligned} \leavevmode\hbox to30.65pt{\vbox
to30.65pt{\pgfpicture\makeatletter\hbox{\hskip
15.32639pt\lower-15.32639pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
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}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }
\definecolor[named]{tikz@ball}{rgb}{1,0,0}{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
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}\pgfsys@color@gray@fill{1}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@beginscope\pgfsys@invoke{
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}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{0.0pt}{0.0pt}\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{0.0pt}{0.0pt}\pgfsys@invoke{
}\pgfsys@transformcm{0.567}{0.0}{0.0}{0.567}{0.0pt}{0.0pt}\pgfsys@invoke{
}\lxSVG@sh@defs{}\lxSVG@sh@insert{-69.44pt}{-69.44pt}{\lxSVG@sh}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{14.22638pt}{0.0pt}\pgfsys@curveto{14.22638pt}{7.8571pt}{7.8571pt}{14.22638pt}{0.0pt}{14.22638pt}\pgfsys@curveto{-7.8571pt}{14.22638pt}{-14.22638pt}{7.8571pt}{-14.22638pt}{0.0pt}\pgfsys@curveto{-14.22638pt}{-7.8571pt}{-7.8571pt}{-14.22638pt}{0.0pt}{-14.22638pt}\pgfsys@curveto{7.8571pt}{-14.22638pt}{14.22638pt}{-7.8571pt}{14.22638pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{ }\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{2.20001pt}\pgfsys@invoke{
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}\pgfsys@beginscope\pgfsys@invoke{
}{\pgfsys@setlinewidth{0.6pt}\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{
}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{14.22638pt}{0.0pt}\pgfsys@curveto{14.22638pt}{7.8571pt}{7.8571pt}{14.22638pt}{0.0pt}{14.22638pt}\pgfsys@curveto{-7.8571pt}{14.22638pt}{-14.22638pt}{7.8571pt}{-14.22638pt}{0.0pt}\pgfsys@curveto{-14.22638pt}{-7.8571pt}{-7.8571pt}{-14.22638pt}{0.0pt}{-14.22638pt}\pgfsys@curveto{7.8571pt}{-14.22638pt}{14.22638pt}{-7.8571pt}{14.22638pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
}}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\end{aligned}\ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \longrightarrow\ \ \ \ \
Mor(\mathcal{G}_{\mathcal{E}})$ $\displaystyle\sqcup\begin{aligned}
\leavevmode\hbox to31.67pt{\vbox
to31.23pt{\pgfpicture\makeatletter\hbox{\hskip
15.37498pt\lower-14.93193pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
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}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }
\definecolor[named]{tikz@ball}{rgb}{1,0,0}{}{{}}{}{{}{}{}{{}}{{{{}{}{}{}}}{{}{}{}{}}}}{}
{} {} {} {}{}{{}{}{}{{}}{{{{}{}{}{}}} {{}{}{}{}}}}{} {} {} {} {}{}
{}\pgfsys@beginscope\pgfsys@invoke{
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}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-0.00021pt}{-11.63391pt}\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{0.0pt}{0.0pt}\pgfsys@invoke{
}\pgfsys@transformcm{0.4536}{0.0}{0.0}{0.12347}{0.0pt}{0.0pt}\pgfsys@invoke{
}\lxSVG@sh@defs{}\lxSVG@sh@insert{-69.44pt}{-69.44pt}{\lxSVG@sh}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}\pgfsys@moveto{11.38092pt}{-8.5359pt}\pgfsys@curveto{8.33142pt}{-12.60197pt}{3.34409pt}{-14.73193pt}{-1.70187pt}{-14.12413pt}\pgfsys@curveto{-5.55542pt}{-13.66003pt}{-9.05255pt}{-11.64098pt}{-11.38135pt}{-8.5359pt}\pgfsys@curveto{-11.38135pt}{-9.47871pt}{-6.28601pt}{-10.243pt}{-0.00043pt}{-10.243pt}\pgfsys@curveto{6.28516pt}{-10.243pt}{11.3805pt}{-9.47871pt}{11.3805pt}{-8.5359pt}\pgfsys@closepath\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}{{}{}{}{{}}{{{{}{}{}{}}} {{{}{}{}{}}} {{}{}{}{}}}}{} {} {} {}
{}{}{{}{}{}{{}}{{{{}{}{}{}}}{{}{}{}{}}}}{} {} {} {} {}{}
{}\pgfsys@beginscope\pgfsys@invoke{
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}\definecolor[named]{pgffillcolor}{rgb}{.5,.5,.5}\pgfsys@stroke@opacity{0.05}\pgfsys@invoke{
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}{}\pgfsys@moveto{11.38092pt}{-8.5359pt}\pgfsys@curveto{16.09518pt}{-2.25021pt}{14.82138pt}{6.66687pt}{8.53569pt}{11.38113pt}\pgfsys@curveto{2.25pt}{16.0954pt}{-6.66708pt}{14.8216pt}{-11.38135pt}{8.5359pt}\pgfsys@curveto{-15.17499pt}{3.4778pt}{-15.17499pt}{-3.4778pt}{-11.38135pt}{-8.5359pt}\pgfsys@curveto{-11.38135pt}{-7.5931pt}{-6.28601pt}{-6.82881pt}{-0.00043pt}{-6.82881pt}\pgfsys@curveto{6.28516pt}{-6.82881pt}{11.3805pt}{-7.5931pt}{11.3805pt}{-8.5359pt}\pgfsys@closepath\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\par{}{{}}{}{{{}}
{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@invoke{
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}\definecolor[named]{pgffillcolor}{rgb}{.5,.5,.5}\pgfsys@stroke@opacity{0.5}\pgfsys@invoke{
}\pgfsys@fill@opacity{0.5}\pgfsys@invoke{
}\pgfsys@moveto{0.0pt}{-8.5359pt}\pgfsys@moveto{11.38092pt}{-8.5359pt}\pgfsys@curveto{11.38092pt}{-7.75015pt}{6.28558pt}{-7.11319pt}{0.0pt}{-7.11319pt}\pgfsys@curveto{-6.28558pt}{-7.11319pt}{-11.38092pt}{-7.75015pt}{-11.38092pt}{-8.5359pt}\pgfsys@curveto{-11.38092pt}{-9.32166pt}{-6.28558pt}{-9.95862pt}{0.0pt}{-9.95862pt}\pgfsys@curveto{6.28558pt}{-9.95862pt}{11.38092pt}{-9.32166pt}{11.38092pt}{-8.5359pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{-8.5359pt}\pgfsys@fill\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\end{aligned}$
$\displaystyle\sqcup\begin{aligned} \leavevmode\hbox to29.75pt{\vbox
to32.27pt{\pgfpicture\makeatletter\hbox{\hskip
14.42638pt\lower-16.13347pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }
\definecolor[named]{tikz@ball}{rgb}{1,0,0}{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{
}\pgfsys@color@gray@fill{1}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{14.22638pt}{0.0pt}\pgfsys@curveto{14.22638pt}{7.8571pt}{7.8571pt}{14.22638pt}{0.0pt}{14.22638pt}\pgfsys@curveto{-7.8571pt}{14.22638pt}{-14.22638pt}{7.8571pt}{-14.22638pt}{0.0pt}\pgfsys@curveto{-14.22638pt}{-7.8571pt}{-7.8571pt}{-14.22638pt}{0.0pt}{-14.22638pt}\pgfsys@curveto{7.8571pt}{-14.22638pt}{14.22638pt}{-7.8571pt}{14.22638pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{0.0pt}{0.0pt}\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{0.0pt}{0.0pt}\pgfsys@invoke{
}\pgfsys@transformcm{0.567}{0.0}{0.0}{0.567}{0.0pt}{0.0pt}\pgfsys@invoke{
}\lxSVG@sh@defs{}\lxSVG@sh@insert{-69.44pt}{-69.44pt}{\lxSVG@sh}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{14.22638pt}{0.0pt}\pgfsys@curveto{14.22638pt}{7.8571pt}{7.8571pt}{14.22638pt}{0.0pt}{14.22638pt}\pgfsys@curveto{-7.8571pt}{14.22638pt}{-14.22638pt}{7.8571pt}{-14.22638pt}{0.0pt}\pgfsys@curveto{-14.22638pt}{-7.8571pt}{-7.8571pt}{-14.22638pt}{0.0pt}{-14.22638pt}\pgfsys@curveto{7.8571pt}{-14.22638pt}{14.22638pt}{-7.8571pt}{14.22638pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@color@gray@fill{1}\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\pgfsys@fill@opacity{0}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{14.22638pt}\pgfsys@moveto{1.70709pt}{14.22638pt}\pgfsys@curveto{1.70709pt}{15.16919pt}{0.94281pt}{15.93347pt}{0.0pt}{15.93347pt}\pgfsys@curveto{-0.94281pt}{15.93347pt}{-1.70709pt}{15.16919pt}{-1.70709pt}{14.22638pt}\pgfsys@curveto{-1.70709pt}{13.28357pt}{-0.94281pt}{12.51929pt}{0.0pt}{12.51929pt}\pgfsys@curveto{0.94281pt}{12.51929pt}{1.70709pt}{13.28357pt}{1.70709pt}{14.22638pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@color@gray@fill{1}\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\pgfsys@fill@opacity{0}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{-14.22638pt}\pgfsys@moveto{1.70709pt}{-14.22638pt}\pgfsys@curveto{1.70709pt}{-13.28357pt}{0.94281pt}{-12.51929pt}{0.0pt}{-12.51929pt}\pgfsys@curveto{-0.94281pt}{-12.51929pt}{-1.70709pt}{-13.28357pt}{-1.70709pt}{-14.22638pt}\pgfsys@curveto{-1.70709pt}{-15.16919pt}{-0.94281pt}{-15.93347pt}{0.0pt}{-15.93347pt}\pgfsys@curveto{0.94281pt}{-15.93347pt}{1.70709pt}{-15.16919pt}{1.70709pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{{}}{}{{}{}{}{{}}{{{{}{}{}{}}}{{}{}{}{}}}}{} {} {} {}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{ }\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{2.20001pt}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{14.22638pt}\pgfsys@curveto{7.8571pt}{14.22638pt}{14.22638pt}{7.8571pt}{14.22638pt}{0.0pt}\pgfsys@curveto{14.22638pt}{-7.8571pt}{7.8571pt}{-14.22638pt}{0.0pt}{-14.22638pt}\pgfsys@stroke\pgfsys@invoke{
}\pgfsys@beginscope\pgfsys@invoke{
}{\pgfsys@setlinewidth{0.6pt}\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{
}\pgfsys@moveto{0.0pt}{14.22638pt}\pgfsys@curveto{7.8571pt}{14.22638pt}{14.22638pt}{7.8571pt}{14.22638pt}{0.0pt}\pgfsys@curveto{14.22638pt}{-7.8571pt}{7.8571pt}{-14.22638pt}{0.0pt}{-14.22638pt}\pgfsys@stroke\pgfsys@invoke{
}}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\end{aligned}\sqcup\begin{aligned}
\leavevmode\hbox to29.75pt{\vbox
to32.27pt{\pgfpicture\makeatletter\hbox{\hskip
15.32639pt\lower-16.13347pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }
\definecolor[named]{tikz@ball}{rgb}{1,0,0}{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{
}\pgfsys@color@gray@fill{1}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{14.22638pt}{0.0pt}\pgfsys@curveto{14.22638pt}{7.8571pt}{7.8571pt}{14.22638pt}{0.0pt}{14.22638pt}\pgfsys@curveto{-7.8571pt}{14.22638pt}{-14.22638pt}{7.8571pt}{-14.22638pt}{0.0pt}\pgfsys@curveto{-14.22638pt}{-7.8571pt}{-7.8571pt}{-14.22638pt}{0.0pt}{-14.22638pt}\pgfsys@curveto{7.8571pt}{-14.22638pt}{14.22638pt}{-7.8571pt}{14.22638pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{0.0pt}{0.0pt}\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{0.0pt}{0.0pt}\pgfsys@invoke{
}\pgfsys@transformcm{0.567}{0.0}{0.0}{0.567}{0.0pt}{0.0pt}\pgfsys@invoke{
}\lxSVG@sh@defs{}\lxSVG@sh@insert{-69.44pt}{-69.44pt}{\lxSVG@sh}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{14.22638pt}{0.0pt}\pgfsys@curveto{14.22638pt}{7.8571pt}{7.8571pt}{14.22638pt}{0.0pt}{14.22638pt}\pgfsys@curveto{-7.8571pt}{14.22638pt}{-14.22638pt}{7.8571pt}{-14.22638pt}{0.0pt}\pgfsys@curveto{-14.22638pt}{-7.8571pt}{-7.8571pt}{-14.22638pt}{0.0pt}{-14.22638pt}\pgfsys@curveto{7.8571pt}{-14.22638pt}{14.22638pt}{-7.8571pt}{14.22638pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@color@gray@fill{1}\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\pgfsys@fill@opacity{0}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{14.22638pt}\pgfsys@moveto{1.70709pt}{14.22638pt}\pgfsys@curveto{1.70709pt}{15.16919pt}{0.94281pt}{15.93347pt}{0.0pt}{15.93347pt}\pgfsys@curveto{-0.94281pt}{15.93347pt}{-1.70709pt}{15.16919pt}{-1.70709pt}{14.22638pt}\pgfsys@curveto{-1.70709pt}{13.28357pt}{-0.94281pt}{12.51929pt}{0.0pt}{12.51929pt}\pgfsys@curveto{0.94281pt}{12.51929pt}{1.70709pt}{13.28357pt}{1.70709pt}{14.22638pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@color@gray@fill{1}\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\pgfsys@fill@opacity{0}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{-14.22638pt}\pgfsys@moveto{1.70709pt}{-14.22638pt}\pgfsys@curveto{1.70709pt}{-13.28357pt}{0.94281pt}{-12.51929pt}{0.0pt}{-12.51929pt}\pgfsys@curveto{-0.94281pt}{-12.51929pt}{-1.70709pt}{-13.28357pt}{-1.70709pt}{-14.22638pt}\pgfsys@curveto{-1.70709pt}{-15.16919pt}{-0.94281pt}{-15.93347pt}{0.0pt}{-15.93347pt}\pgfsys@curveto{0.94281pt}{-15.93347pt}{1.70709pt}{-15.16919pt}{1.70709pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{{}}{}{{}{}{}{{}}{{{{}{}{}{}}}{{}{}{}{}}}}{} {} {} {}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{ }\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{2.20001pt}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{-14.22638pt}\pgfsys@curveto{-7.8571pt}{-14.22638pt}{-14.22638pt}{-7.8571pt}{-14.22638pt}{0.0pt}\pgfsys@curveto{-14.22638pt}{7.8571pt}{-7.8571pt}{14.22638pt}{0.0pt}{14.22638pt}\pgfsys@stroke\pgfsys@invoke{
}\pgfsys@beginscope\pgfsys@invoke{
}{\pgfsys@setlinewidth{0.6pt}\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{
}\pgfsys@moveto{0.0pt}{-14.22638pt}\pgfsys@curveto{-7.8571pt}{-14.22638pt}{-14.22638pt}{-7.8571pt}{-14.22638pt}{0.0pt}\pgfsys@curveto{-14.22638pt}{7.8571pt}{-7.8571pt}{14.22638pt}{0.0pt}{14.22638pt}\pgfsys@stroke\pgfsys@invoke{
}}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\end{aligned}$
where
$\displaystyle\eta(re^{i\theta}\to re^{i\theta^{\prime}})=$ $\displaystyle
D(-v)\in G_{C}\ \ \ \ \ \ for\ \ re^{i\theta}\to
re^{i\theta^{\prime}}\in\begin{aligned} \leavevmode\hbox to32.27pt{\vbox
to29.55pt{\pgfpicture\makeatletter\hbox{\hskip
16.13347pt\lower-14.22638pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }
\definecolor[named]{tikz@ball}{rgb}{1,0,0}{}{{}}{}{{}{}{}{{}}{{{{}{}{}{}}}
{{}{}{}{}}}}{} {} {} {} {}{}{{}{}{}{{}}{{{{}{}{}{}}} {{}{}{}{}}}}{} {} {} {}
{}{} {}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@moveto{14.22638pt}{0.0pt}\pgfsys@curveto{14.22638pt}{7.8571pt}{7.8571pt}{14.22638pt}{0.0pt}{14.22638pt}\pgfsys@curveto{-7.8571pt}{14.22638pt}{-14.22638pt}{7.8571pt}{-14.22638pt}{0.0pt}\pgfsys@curveto{-14.22638pt}{-1.5715pt}{-7.8571pt}{-2.84544pt}{0.0pt}{-2.84544pt}\pgfsys@curveto{7.8571pt}{-2.84544pt}{14.22638pt}{-1.5715pt}{14.22638pt}{0.0pt}\pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{0.0pt}{5.69048pt}\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{0.0pt}{0.0pt}\pgfsys@invoke{
}\pgfsys@transformcm{0.567}{0.0}{0.0}{0.3402}{0.0pt}{0.0pt}\pgfsys@invoke{
}\lxSVG@sh@defs{}\lxSVG@sh@insert{-69.44pt}{-69.44pt}{\lxSVG@sh}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}\pgfsys@moveto{14.22638pt}{0.0pt}\pgfsys@curveto{14.22638pt}{7.8571pt}{7.8571pt}{14.22638pt}{0.0pt}{14.22638pt}\pgfsys@curveto{-7.8571pt}{14.22638pt}{-14.22638pt}{7.8571pt}{-14.22638pt}{0.0pt}\pgfsys@curveto{-14.22638pt}{-1.5715pt}{-7.8571pt}{-2.84544pt}{0.0pt}{-2.84544pt}\pgfsys@curveto{7.8571pt}{-2.84544pt}{14.22638pt}{-1.5715pt}{14.22638pt}{0.0pt}\pgfsys@closepath\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}{{{}} {}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@fill{.5}\pgfsys@invoke{
}\pgfsys@stroke@opacity{0.5}\pgfsys@invoke{
}\pgfsys@fill@opacity{0.5}\pgfsys@invoke{
}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{14.22638pt}{0.0pt}\pgfsys@curveto{14.22638pt}{1.5715pt}{7.8571pt}{2.84544pt}{0.0pt}{2.84544pt}\pgfsys@curveto{-7.8571pt}{2.84544pt}{-14.22638pt}{1.5715pt}{-14.22638pt}{0.0pt}\pgfsys@curveto{-14.22638pt}{-1.5715pt}{-7.8571pt}{-2.84544pt}{0.0pt}{-2.84544pt}\pgfsys@curveto{7.8571pt}{-2.84544pt}{14.22638pt}{-1.5715pt}{14.22638pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@fill\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{{}}{}{{}{}{}{{}}{{{{}{}{}{}}} {{}{}{}{}}}}{} {} {} {}
{}{}{{}{}{}{{}}{{{{}{}{}{}}}{{}{}{}{}}}}{} {} {} {} {}{}
{}\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@invoke{
}\pgfsys@color@gray@fill{.5}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{.5,.5,.5}\pgfsys@stroke@opacity{0.05}\pgfsys@invoke{
}\pgfsys@fill@opacity{0.05}\pgfsys@invoke{
}\pgfsys@moveto{-14.22638pt}{0.0pt}\pgfsys@curveto{-14.22638pt}{-7.8571pt}{-7.8571pt}{-14.22638pt}{0.0pt}{-14.22638pt}\pgfsys@curveto{7.8571pt}{-14.22638pt}{14.22638pt}{-7.8571pt}{14.22638pt}{0.0pt}\pgfsys@curveto{14.22638pt}{-1.5715pt}{7.8571pt}{-2.84544pt}{0.0pt}{-2.84544pt}\pgfsys@curveto{-7.8571pt}{-2.84544pt}{-14.22638pt}{-1.5715pt}{-14.22638pt}{0.0pt}\pgfsys@closepath\pgfsys@fill\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{{}}{}{{}{}{}{{}}{{{{}{}{}{}}} {{}{}{}{}}}}{} {} {} {}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@invoke{
}\pgfsys@color@gray@fill{.5}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{.5,.5,.5}\pgfsys@setdash{3.0pt,6.0pt}{0.0pt}\pgfsys@invoke{
}{}\pgfsys@moveto{14.22638pt}{0.0pt}\pgfsys@curveto{14.22638pt}{1.5715pt}{7.8571pt}{2.84544pt}{0.0pt}{2.84544pt}\pgfsys@curveto{-7.8571pt}{2.84544pt}{-14.22638pt}{1.5715pt}{-14.22638pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@color@gray@fill{1}\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\pgfsys@fill@opacity{0}\pgfsys@invoke{
}{}\pgfsys@moveto{14.22638pt}{0.0pt}\pgfsys@moveto{15.93347pt}{0.0pt}\pgfsys@curveto{15.93347pt}{0.94281pt}{15.16919pt}{1.70709pt}{14.22638pt}{1.70709pt}\pgfsys@curveto{13.28357pt}{1.70709pt}{12.51929pt}{0.94281pt}{12.51929pt}{0.0pt}\pgfsys@curveto{12.51929pt}{-0.94281pt}{13.28357pt}{-1.70709pt}{14.22638pt}{-1.70709pt}\pgfsys@curveto{15.16919pt}{-1.70709pt}{15.93347pt}{-0.94281pt}{15.93347pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{14.22638pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@color@gray@fill{1}\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\pgfsys@fill@opacity{0}\pgfsys@invoke{
}{}\pgfsys@moveto{-14.22638pt}{0.0pt}\pgfsys@moveto{-12.51929pt}{0.0pt}\pgfsys@curveto{-12.51929pt}{0.94281pt}{-13.28357pt}{1.70709pt}{-14.22638pt}{1.70709pt}\pgfsys@curveto{-15.16919pt}{1.70709pt}{-15.93347pt}{0.94281pt}{-15.93347pt}{0.0pt}\pgfsys@curveto{-15.93347pt}{-0.94281pt}{-15.16919pt}{-1.70709pt}{-14.22638pt}{-1.70709pt}\pgfsys@curveto{-13.28357pt}{-1.70709pt}{-12.51929pt}{-0.94281pt}{-12.51929pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{-14.22638pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{{}}{}{{}{}{}{{}}{{{{}{}{}{}}} {{}{}{}{}}}}{} {} {} {}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{ }\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{2.20001pt}\pgfsys@invoke{
}{}\pgfsys@moveto{14.22638pt}{0.0pt}\pgfsys@curveto{14.22638pt}{7.8571pt}{7.8571pt}{14.22638pt}{0.0pt}{14.22638pt}\pgfsys@curveto{-7.8571pt}{14.22638pt}{-14.22638pt}{7.8571pt}{-14.22638pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
}\pgfsys@beginscope\pgfsys@invoke{
}{\pgfsys@setlinewidth{0.6pt}\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{
}\pgfsys@moveto{14.22638pt}{0.0pt}\pgfsys@curveto{14.22638pt}{7.8571pt}{7.8571pt}{14.22638pt}{0.0pt}{14.22638pt}\pgfsys@curveto{-7.8571pt}{14.22638pt}{-14.22638pt}{7.8571pt}{-14.22638pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
}}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\end{aligned}\ \ \ \
(\text{where}\ \theta^{\prime}=\theta\pm 2\pi)$ $\displaystyle Id\ \ \ \in
G_{C}\ \ \ \ \ \ \ \ \ \text{everywhere else}.$
To extend $\check{\boldsymbol{s}}$ to $p$, we define $\boldsymbol{s}$ on
$\mathbb{Z}_{m_{v}}\ltimes\begin{aligned} \leavevmode\hbox to31.27pt{\vbox
to31.03pt{\pgfpicture\makeatletter\hbox{\hskip 15.17499pt\lower-16.0954pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
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}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }
\definecolor[named]{tikz@ball}{rgb}{1,0,0}{}{{}}{}{{}{}{}{{}}{{{{}{}{}{}}}
{{}{}{}{}}}}{} {} {} {} {}{}{{}{}{}{{}}{{{{}{}{}{}}} {{}{}{}{}}}}{} {} {} {}
{}{} {}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@moveto{11.38092pt}{8.5359pt}\pgfsys@curveto{8.33142pt}{12.60197pt}{3.34409pt}{14.73193pt}{-1.70187pt}{14.12413pt}\pgfsys@curveto{-5.55542pt}{13.66003pt}{-9.05255pt}{11.64098pt}{-11.38135pt}{8.5359pt}\pgfsys@curveto{-11.38135pt}{7.5931pt}{-6.28601pt}{6.82881pt}{-0.00043pt}{6.82881pt}\pgfsys@curveto{6.28516pt}{6.82881pt}{11.3805pt}{7.5931pt}{11.3805pt}{8.5359pt}\pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-0.00021pt}{10.78036pt}\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{0.0pt}{0.0pt}\pgfsys@invoke{
}\pgfsys@transformcm{0.4536}{0.0}{0.0}{0.15749}{0.0pt}{0.0pt}\pgfsys@invoke{
}\lxSVG@sh@defs{}\lxSVG@sh@insert{-69.44pt}{-69.44pt}{\lxSVG@sh}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}\pgfsys@moveto{11.38092pt}{8.5359pt}\pgfsys@curveto{8.33142pt}{12.60197pt}{3.34409pt}{14.73193pt}{-1.70187pt}{14.12413pt}\pgfsys@curveto{-5.55542pt}{13.66003pt}{-9.05255pt}{11.64098pt}{-11.38135pt}{8.5359pt}\pgfsys@curveto{-11.38135pt}{7.5931pt}{-6.28601pt}{6.82881pt}{-0.00043pt}{6.82881pt}\pgfsys@curveto{6.28516pt}{6.82881pt}{11.3805pt}{7.5931pt}{11.3805pt}{8.5359pt}\pgfsys@closepath\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}{{}{}{}{{}}{{{{}{}{}{}}}{{{}{}{}{}}}{{}{}{}{}}}}{} {} {} {}
{}{}{{}{}{}{{}}{{{{}{}{}{}}} {{}{}{}{}}}}{} {} {} {} {}{}
{}\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@invoke{
}\pgfsys@color@gray@fill{.5}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{.5,.5,.5}\pgfsys@stroke@opacity{0.05}\pgfsys@invoke{
}\pgfsys@fill@opacity{0.05}\pgfsys@invoke{
}\pgfsys@moveto{11.38092pt}{8.5359pt}\pgfsys@curveto{16.09518pt}{2.25021pt}{14.82138pt}{-6.66687pt}{8.53569pt}{-11.38113pt}\pgfsys@curveto{2.25pt}{-16.0954pt}{-6.66708pt}{-14.8216pt}{-11.38135pt}{-8.5359pt}\pgfsys@curveto{-15.17499pt}{-3.4778pt}{-15.17499pt}{3.4778pt}{-11.38135pt}{8.5359pt}\pgfsys@curveto{-11.38135pt}{7.5931pt}{-6.28601pt}{6.82881pt}{-0.00043pt}{6.82881pt}\pgfsys@curveto{6.28516pt}{6.82881pt}{11.3805pt}{7.5931pt}{11.3805pt}{8.5359pt}\pgfsys@closepath\pgfsys@fill\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}
{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@invoke{
}\pgfsys@color@gray@fill{.5}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{.5,.5,.5}\pgfsys@stroke@opacity{0.5}\pgfsys@invoke{
}\pgfsys@fill@opacity{0.5}\pgfsys@invoke{
}\pgfsys@moveto{0.0pt}{8.5359pt}\pgfsys@moveto{11.38092pt}{8.5359pt}\pgfsys@curveto{11.38092pt}{9.32166pt}{6.28558pt}{9.95862pt}{0.0pt}{9.95862pt}\pgfsys@curveto{-6.28558pt}{9.95862pt}{-11.38092pt}{9.32166pt}{-11.38092pt}{8.5359pt}\pgfsys@curveto{-11.38092pt}{7.75015pt}{-6.28558pt}{7.11319pt}{0.0pt}{7.11319pt}\pgfsys@curveto{6.28558pt}{7.11319pt}{11.38092pt}{7.75015pt}{11.38092pt}{8.5359pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{8.5359pt}\pgfsys@fill\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\end{aligned}\rightrightarrows
G_{C}\ltimes\begin{aligned} \leavevmode\hbox to31.27pt{\vbox
to31.03pt{\pgfpicture\makeatletter\hbox{\hskip 15.17499pt\lower-16.0954pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }
\definecolor[named]{tikz@ball}{rgb}{1,0,0}{}{{}}{}{{}{}{}{{}}{{{{}{}{}{}}}
{{}{}{}{}}}}{} {} {} {} {}{}{{}{}{}{{}}{{{{}{}{}{}}} {{}{}{}{}}}}{} {} {} {}
{}{} {}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@moveto{11.38092pt}{8.5359pt}\pgfsys@curveto{8.33142pt}{12.60197pt}{3.34409pt}{14.73193pt}{-1.70187pt}{14.12413pt}\pgfsys@curveto{-5.55542pt}{13.66003pt}{-9.05255pt}{11.64098pt}{-11.38135pt}{8.5359pt}\pgfsys@curveto{-11.38135pt}{7.5931pt}{-6.28601pt}{6.82881pt}{-0.00043pt}{6.82881pt}\pgfsys@curveto{6.28516pt}{6.82881pt}{11.3805pt}{7.5931pt}{11.3805pt}{8.5359pt}\pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-0.00021pt}{10.78036pt}\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{0.0pt}{0.0pt}\pgfsys@invoke{
}\pgfsys@transformcm{0.4536}{0.0}{0.0}{0.15749}{0.0pt}{0.0pt}\pgfsys@invoke{
}\lxSVG@sh@defs{}\lxSVG@sh@insert{-69.44pt}{-69.44pt}{\lxSVG@sh}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}\pgfsys@moveto{11.38092pt}{8.5359pt}\pgfsys@curveto{8.33142pt}{12.60197pt}{3.34409pt}{14.73193pt}{-1.70187pt}{14.12413pt}\pgfsys@curveto{-5.55542pt}{13.66003pt}{-9.05255pt}{11.64098pt}{-11.38135pt}{8.5359pt}\pgfsys@curveto{-11.38135pt}{7.5931pt}{-6.28601pt}{6.82881pt}{-0.00043pt}{6.82881pt}\pgfsys@curveto{6.28516pt}{6.82881pt}{11.3805pt}{7.5931pt}{11.3805pt}{8.5359pt}\pgfsys@closepath\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}{{}{}{}{{}}{{{{}{}{}{}}}{{{}{}{}{}}}{{}{}{}{}}}}{} {} {} {}
{}{}{{}{}{}{{}}{{{{}{}{}{}}} {{}{}{}{}}}}{} {} {} {} {}{}
{}\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@invoke{
}\pgfsys@color@gray@fill{.5}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{.5,.5,.5}\pgfsys@stroke@opacity{0.05}\pgfsys@invoke{
}\pgfsys@fill@opacity{0.05}\pgfsys@invoke{
}\pgfsys@moveto{11.38092pt}{8.5359pt}\pgfsys@curveto{16.09518pt}{2.25021pt}{14.82138pt}{-6.66687pt}{8.53569pt}{-11.38113pt}\pgfsys@curveto{2.25pt}{-16.0954pt}{-6.66708pt}{-14.8216pt}{-11.38135pt}{-8.5359pt}\pgfsys@curveto{-15.17499pt}{-3.4778pt}{-15.17499pt}{3.4778pt}{-11.38135pt}{8.5359pt}\pgfsys@curveto{-11.38135pt}{7.5931pt}{-6.28601pt}{6.82881pt}{-0.00043pt}{6.82881pt}\pgfsys@curveto{6.28516pt}{6.82881pt}{11.3805pt}{7.5931pt}{11.3805pt}{8.5359pt}\pgfsys@closepath\pgfsys@fill\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}
{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@invoke{
}\pgfsys@color@gray@fill{.5}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{.5,.5,.5}\pgfsys@stroke@opacity{0.5}\pgfsys@invoke{
}\pgfsys@fill@opacity{0.5}\pgfsys@invoke{
}\pgfsys@moveto{0.0pt}{8.5359pt}\pgfsys@moveto{11.38092pt}{8.5359pt}\pgfsys@curveto{11.38092pt}{9.32166pt}{6.28558pt}{9.95862pt}{0.0pt}{9.95862pt}\pgfsys@curveto{-6.28558pt}{9.95862pt}{-11.38092pt}{9.32166pt}{-11.38092pt}{8.5359pt}\pgfsys@curveto{-11.38092pt}{7.75015pt}{-6.28558pt}{7.11319pt}{0.0pt}{7.11319pt}\pgfsys@curveto{6.28558pt}{7.11319pt}{11.38092pt}{7.75015pt}{11.38092pt}{8.5359pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{8.5359pt}\pgfsys@fill\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\end{aligned}\times
V_{C}$
by
$\boldsymbol{s}_{0}(re^{i\theta})=(re^{i2\theta},0),$
$\boldsymbol{s}_{1}(re^{i\theta}\xrightarrow{g}g\cdot
re^{i\theta})=(re^{i2\theta},0)\xrightarrow{\mathfrak{i}(g)}(re^{i2\theta},0).$
It is straightforward to verify the morphism above together with
$\check{\boldsymbol{s}}$ define a morphism from an orbisphere with
$\mathbb{Z}_{m_{v}}$-orbipoint at $p$ to $\mathcal{G}_{\mathcal{E}}$.
This is the only representative orbifold morphism extends
$\check{\boldsymbol{s}}$. We refer the readers to [TW, Proposition 2.47] for a
detailed reason in an explicit example. ∎
Now we describe the pullback bundle $\boldsymbol{s}_{x}^{*}T\mathcal{E}_{v}$
which is the same as $\boldsymbol{s}_{x}^{*}T\mathcal{E}_{v,C}$ since the
image of $\boldsymbol{s}_{x}$ is contained in $\mathcal{E}_{v,C}$. The
pullback is obvious on $\check{S}^{2}$ since there is no orbipoint on the
domain. The orbipoint north pole is covered by:
$\mathbb{Z}_{m_{v}}\ltimes\ \ \ \ \ \begin{aligned} \leavevmode\hbox
to31.27pt{\vbox to31.03pt{\pgfpicture\makeatletter\hbox{\hskip
15.17499pt\lower-16.0954pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }
\definecolor[named]{tikz@ball}{rgb}{1,0,0}{}{{}}{}{{}{}{}{{}}{{{{}{}{}{}}}
{{}{}{}{}}}}{} {} {} {} {}{}{{}{}{}{{}}{{{{}{}{}{}}} {{}{}{}{}}}}{} {} {} {}
{}{} {}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@moveto{11.38092pt}{8.5359pt}\pgfsys@curveto{8.33142pt}{12.60197pt}{3.34409pt}{14.73193pt}{-1.70187pt}{14.12413pt}\pgfsys@curveto{-5.55542pt}{13.66003pt}{-9.05255pt}{11.64098pt}{-11.38135pt}{8.5359pt}\pgfsys@curveto{-11.38135pt}{7.5931pt}{-6.28601pt}{6.82881pt}{-0.00043pt}{6.82881pt}\pgfsys@curveto{6.28516pt}{6.82881pt}{11.3805pt}{7.5931pt}{11.3805pt}{8.5359pt}\pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-0.00021pt}{10.78036pt}\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{0.0pt}{0.0pt}\pgfsys@invoke{
}\pgfsys@transformcm{0.4536}{0.0}{0.0}{0.15749}{0.0pt}{0.0pt}\pgfsys@invoke{
}\lxSVG@sh@defs{}\lxSVG@sh@insert{-69.44pt}{-69.44pt}{\lxSVG@sh}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}\pgfsys@moveto{11.38092pt}{8.5359pt}\pgfsys@curveto{8.33142pt}{12.60197pt}{3.34409pt}{14.73193pt}{-1.70187pt}{14.12413pt}\pgfsys@curveto{-5.55542pt}{13.66003pt}{-9.05255pt}{11.64098pt}{-11.38135pt}{8.5359pt}\pgfsys@curveto{-11.38135pt}{7.5931pt}{-6.28601pt}{6.82881pt}{-0.00043pt}{6.82881pt}\pgfsys@curveto{6.28516pt}{6.82881pt}{11.3805pt}{7.5931pt}{11.3805pt}{8.5359pt}\pgfsys@closepath\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}{{}{}{}{{}}{{{{}{}{}{}}}{{{}{}{}{}}}{{}{}{}{}}}}{} {} {} {}
{}{}{{}{}{}{{}}{{{{}{}{}{}}} {{}{}{}{}}}}{} {} {} {} {}{}
{}\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@invoke{
}\pgfsys@color@gray@fill{.5}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{.5,.5,.5}\pgfsys@stroke@opacity{0.05}\pgfsys@invoke{
}\pgfsys@fill@opacity{0.05}\pgfsys@invoke{
}\pgfsys@moveto{11.38092pt}{8.5359pt}\pgfsys@curveto{16.09518pt}{2.25021pt}{14.82138pt}{-6.66687pt}{8.53569pt}{-11.38113pt}\pgfsys@curveto{2.25pt}{-16.0954pt}{-6.66708pt}{-14.8216pt}{-11.38135pt}{-8.5359pt}\pgfsys@curveto{-15.17499pt}{-3.4778pt}{-15.17499pt}{3.4778pt}{-11.38135pt}{8.5359pt}\pgfsys@curveto{-11.38135pt}{7.5931pt}{-6.28601pt}{6.82881pt}{-0.00043pt}{6.82881pt}\pgfsys@curveto{6.28516pt}{6.82881pt}{11.3805pt}{7.5931pt}{11.3805pt}{8.5359pt}\pgfsys@closepath\pgfsys@fill\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}
{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@invoke{
}\pgfsys@color@gray@fill{.5}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{.5,.5,.5}\pgfsys@stroke@opacity{0.5}\pgfsys@invoke{
}\pgfsys@fill@opacity{0.5}\pgfsys@invoke{
}\pgfsys@moveto{0.0pt}{8.5359pt}\pgfsys@moveto{11.38092pt}{8.5359pt}\pgfsys@curveto{11.38092pt}{9.32166pt}{6.28558pt}{9.95862pt}{0.0pt}{9.95862pt}\pgfsys@curveto{-6.28558pt}{9.95862pt}{-11.38092pt}{9.32166pt}{-11.38092pt}{8.5359pt}\pgfsys@curveto{-11.38092pt}{7.75015pt}{-6.28558pt}{7.11319pt}{0.0pt}{7.11319pt}\pgfsys@curveto{6.28558pt}{7.11319pt}{11.38092pt}{7.75015pt}{11.38092pt}{8.5359pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{8.5359pt}\pgfsys@fill\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\end{aligned}\times\mathbb{C}\times\mathbb{C}^{n}$
with $g\cdot(z,\xi,\eta)=(g\cdot z,\xi,g\eta)$.
Note that $g$ acts on the $\mathbb{C}$ component trivially since it acts on
the horizontal (base) direction of $\mathcal{E}_{v,C}$ trivially.
###### Remark 2.9.
We remark that the horizontal direction of the pullback bundle
$\boldsymbol{s}_{x}^{*}T\mathcal{E}_{v}$, is not the tangent bundle of
$\mathbb{C}P^{1}(1,m_{v})$.
The following Lemma follows from a direct computation.
###### Lemma 2.10.
The evaluation map of $\boldsymbol{s}_{x}$ at the north pole (possibly an
orbipoint) of $S^{2}$ lies in the twisted sector $\mathcal{E}_{v,(v^{-1})}$.
Now we describe $(\boldsymbol{s}_{x}^{*}T\mathcal{E}_{v})^{de}$, the
desingularized bundle of $\boldsymbol{s}_{x}^{*}T\mathcal{E}_{v}$. Let
$\\{i^{v}_{1},...,i^{v}_{n}\\}\subset\\{1,...,N\\}$ be the subset such that
the cone spanned by $\\{b_{i^{v}_{j}}|j=1,...,n\\}$ over $\mathbb{Q}$ is the
minimal cone $\sigma(v)$ containing $v$. Then
$(\boldsymbol{s}_{x}^{*}T\mathcal{E}_{v})^{de}$ is constructed by glueing two
copies of the trivial vector bundles over disk $D\times(\mathbb{C}\times
V_{c})\to D$ along the boundaries, where the glueing map $\rho:S^{1}\to
Sp(\mathbb{C}\times V_{c},\omega_{0})$ is given by
$\rho(e^{i2\pi t})=diag(e^{i2\pi\cdot(-2t)},e^{i2\pi\lceil r^{v}_{1}\rceil
t},...,e^{i2\pi\lceil r^{v}_{n}\rceil t}),$
where $\\{r^{v}_{j}\\}_{j=1}^{n}$ is defined by
$v=\sum_{j=1}^{n}r^{v}_{j}b_{i^{v}_{j}}$, $\lceil r^{v}_{j}\rceil$ is the
smallest integer no less than $r^{v}_{j}$.
###### Lemma 2.11.
1. (1)
The first Chern number of the desingularized bundle is
$c_{1}((\boldsymbol{s}_{x}^{*}T\mathcal{E}_{v})^{de})=2-dim\sigma(v).$
2. (2)
Each summand of $(\boldsymbol{s}_{x}^{*}T\mathcal{E}_{v})^{de}$ has Chern
number at least $-1$.
###### Proof.
To show (1), note that the chern number can be computed from the Maslov index
of the loop $\rho^{-1}$ in the group of symplectic matrices. The later is
$\mu(\rho^{-1})=2-\sum_{j=1}^{n}\lceil r^{v}_{j}\rceil=2-\sum_{r^{v}_{j}\neq
0}1=2-dim\sigma(v).$
The desingularized bundle splits into line bundles whose first Chern numbers
are 2 and $-\lceil r_{1}^{v}\rceil,...,-\lceil r_{n}^{v}\rceil$. Thus we have
(2). ∎
The desingularized bundle of the vertical subbundle
$\boldsymbol{s}_{x}^{*}T^{vert}\mathcal{E}_{v}$ is the same as the vertical
subbundle of the desingularized bundle
$(\boldsymbol{s}_{x}^{*}T\mathcal{E}_{v})^{de}$, which is constructed by
glueing two copies of the trivial vector bundles over disk
$D\times(\mathbb{C}\times V_{c})\to D$ along the boundaries, where the glueing
map $\rho:S^{1}\to Sp(\mathbb{C}\times V_{c},\omega_{0})$ is given by
$\rho(e^{i2\pi t})=diag(e^{i2\pi\lceil r^{v}_{1}\rceil t},...,e^{i2\pi\lceil
r^{v}_{n}\rceil t}).$
Now since the first Chern number of an orbibundle and the first Chern number
of its desingularized bundle differ by the degree shifting number, we have:
###### Lemma 2.12.
$c_{1}(\boldsymbol{s}_{x}^{*}T\mathcal{E}_{v})=2-dim\sigma(v)+\iota_{v^{-1}},$
and
$c_{1}(\boldsymbol{s}_{x}^{*}T^{vert}\mathcal{E}_{v})=-dim\sigma(v)+\iota_{v^{-1}}.$
Moreover when $m_{v}\neq 1$,
$c_{1}(\boldsymbol{s}_{x}^{*}T\mathcal{E}_{v})=2-\iota_{v}$ and
$c_{1}(\boldsymbol{s}_{x}^{*}T^{vert}\mathcal{E}_{v})=-\iota_{v}$.
In this paper, the Hamiltonian loops we consider is always Hamiltonian circle
action in the sense of [LM]. In this case, the orbifiber bundle has an
alternative description:
Let $S^{1}$ act on $S^{3}$ by
$e^{i\theta}\cdot(z_{1},z_{2}):=(e^{i\theta}z_{1},e^{i\theta}z_{2})$, where
$|z_{1}|^{2}+|z_{2}|^{2}=1$. Then
$\mathcal{E}_{v}=S^{3}\times\mathcal{X}/S^{1}$, where the $S^{1}$ action on
$\mathcal{X}$ corresponds to the lattice point $v$, i.e. generated by $H_{v}$.
To see that the quotient construction is equivalent to the glueing
construction, we need the following lemma:
###### Lemma 2.13.
For any atlas $\\{U_{\alpha}\\}$ of $S^{2}$, there exists an $S^{1}$-invariant
atlas $\\{\tilde{U}_{\alpha}\\}$ of $S^{3}$ such that
$U_{\alpha}=\tilde{U}_{\alpha}/S^{1}$.
###### Proof.
Let $\chi:S^{3}\to S^{2}$ be the Hopf map. An atlas of $S^{2}$ determines an
atlas of $S^{3}$ by pullback of $\chi$. This is the atlas with the required
property, since $S^{1}$ acts on $S^{3}$ preserving the fibers. ∎
Now the abstract construction of $\mathcal{X}_{v}$ by glueing stacks can be
carried out by glueing Lie groupoids333See [TW, Appendix] for definitions and
details as the following. Let $\mathcal{G}_{\mathcal{X}}$ be the translation
groupoid $G\ltimes Z$. The map $\tilde{\boldsymbol{\gamma}}_{v}$ associated to
the Hamiltonian loop is represented by
$\gamma:U_{S^{1}}\times\mathcal{G}_{\mathcal{X}}\to\mathcal{G}_{\mathcal{X}}$
where $U_{S^{1}}$ is the groupoid determined by an atlas of $S^{1}$. Take an
atlas of $S^{2}$ which gives arise the atlas of $S^{1}$ when cut along the
equator, denote the corresponding groupoid chart of $S^{2}$ to be $U_{S^{2}}$
and $U_{S^{2}}^{+}$, $U_{S^{2}}^{-}$ the groupoid charts of the two half
disks. Then $\mathcal{E}_{v}$ is represented by the groupoid glued from
$U_{S^{2}}^{+}\times\mathcal{G}_{\mathcal{X}}$ and
$U_{S^{2}}^{-}\times\mathcal{G}_{\mathcal{X}}$ using $\gamma$. A concrete
example of this construction can be found in [TW].
Note that Lemma 2.13 defines an atlas of $S^{3}$ for the atlas of $S^{2}$ used
above. Denote $U_{S^{3}}$ as the groupoid chart of $S^{3}$ determined by this
atlas. It is straightforward to construct an Lie groupoid isomorphism
$U_{S^{3}}\times\mathcal{G}_{\mathcal{X}}/S^{1}\to
U_{S^{2}}^{+}\times\mathcal{X}\sqcup
U_{S^{2}}^{-}\times\mathcal{X}/\sim_{\gamma}.$
Passing to stacks we have an diffeomorphism from
$S^{3}\times\mathcal{X}/S^{1}$ to $\mathcal{E}_{v}$. Consequently there is an
obvious orbifold morphism from $S^{3}\times\mathcal{X}$ to $\mathcal{E}_{v}$
determined by the quotient. We denote the morphism as
$pr:S^{3}\times\mathcal{X}\to\mathcal{E}_{v}$.
The coupling form $\textbf{u}_{v}$ has the following description: Let
$\alpha\in\Omega^{1}(S^{3})$ be the usual contact form on the unit sphere,
normalized so that $d\alpha=\chi^{*}(\tau)$ where $\tau$ is the standard area
form on $S^{2}$ with total area $1$. Then,
(2.6) $\textbf{u}_{v}=pr_{*}(\omega-dH_{v}\alpha).$
From the definition of the coupling class $\mathbf{u}_{v}$, it is easy to
check the following:
###### Lemma 2.14.
$\mathbf{u}_{v}([s_{x}])=-H_{v}(x).$
###### Lemma 2.15.
$\mathbf{u}_{v}(\sigma+\iota_{*}B)=\mathbf{u}_{v}(\sigma)+\omega(B)$ where
$\sigma\in H^{sec}_{2}(|\mathcal{E}_{v}|,\mathbb{Z}),\ B\in
H_{2}(|\mathcal{X}|,\mathbb{Z})$, $\iota:\mathcal{X}\to\mathcal{E}_{v}$ is the
inclusion of a fiber at the north pole and
$\iota_{*}:H_{2}(|\mathcal{X}|,\mathbb{Z})\to
H_{2}(|\mathcal{E}_{v}|,\mathbb{Z})$ is the induced pushforward map.
## 3\. Seidel Elements
### 3.1. Review of Seidel Representation
First we recall the notion of Novikov ring.
###### Definition 3.1.
Define a ring $\Lambda^{univ}$ as
$\Lambda^{univ}=\left\\{\sum_{k\in\mathbb{R}}r_{k}T^{k}|r_{k}\in\mathbb{Q},\\#\\{k<c|r_{k}\neq
0\\}<\infty\,\,\forall c\in\mathbb{R}\right\\}$
and equip it with a grading given by $deg(T)=0$.
Let $\mathcal{C}$ be the Mori cone of $\mathcal{X}$ which is a finitely
generated monoid. Then there is a maximal fraction $1/a$ with
$a\in\mathbb{Z}_{+}$ such that
$c_{1}(T\mathcal{X})(\mathcal{C})\subset\mathbb{Q}$ is contained in the monoid
generated by $1/a$. Define
$\Lambda:=\Lambda^{univ}[q^{\frac{1}{a}},q^{-\frac{1}{a}}]$ with the grading
given by $deg(q)=2/a$.
The Gromov-Witten theory of symplectic orbifolds is constructed by Chen-Ruan
in [CR2], to which we refer the readers for more details. Let
$\left\langle\alpha_{1},\alpha_{2},\alpha_{3}\right\rangle_{0,3,A}$
be the $3$-point genus $0$ degree $A$ Gromov-Witten invariants of
$\mathcal{X}$ with insertions $\alpha_{1},\alpha_{2},\alpha_{3}\in
H^{*}(I\mathcal{X},\mathbb{Q})$. We may assemble these genus zero orbifold
Gromov-Witten invariants with 3 marked points using the Novikov ring:
$\left\langle\alpha_{1},\alpha_{2},\alpha_{3}\right\rangle:=\sum_{A\in
H_{2}(|\mathcal{X}|,\mathbb{Z})}\left\langle\alpha_{1},\alpha_{2},\alpha_{3}\right\rangle_{0,3,A}q^{c_{1}[A]}t^{\omega[A]}.$
This is used to define the quantum product, an associative multiplication $*$
on $H^{*}(I\mathcal{X},\mathbb{Q})\otimes\Lambda$, as follows:
$\left\langle\alpha_{1}*\alpha_{2},\alpha_{3}\right\rangle_{orb}:=\left\langle\alpha_{1},\alpha_{2},\alpha_{3}\right\rangle,\
\ \text{for}\ \alpha_{i}\in H^{*}(I\mathcal{X},\mathbb{Q}).$
The resulting ring, denoted by $QH_{orb}^{*}(\mathcal{X},\Lambda)$, is called
the orbifold quantum cohomology ring of $(\mathcal{X},\Omega)$.
Let $QH^{*}_{orb}(\mathcal{X},\Lambda)^{\times}$ be the group of invertible
elements (with respect to the quantum product ”$*$”) in
$QH_{orb}^{*}(\mathcal{X},\Lambda)$. In [TW], the authors construct a group
homomorphism:
$\mathcal{S}:\pi_{1}(Ham(\mathcal{X},\omega))\to
QH^{*}_{orb}(\mathcal{X},\Lambda)^{\times}.$
Generalizing the manifold case, this is called the Seidel representation for
symplectic orbifold $(\mathcal{X},\omega)$. We briefly explain its
construction. Represent a homotopy class
$a\in\pi_{1}(Ham(\mathcal{X},\omega))$ by a Hamiltonian loop
$\boldsymbol{\gamma}$, then we can construct Hamiltonian orbifiber bundle
$\mathcal{E}_{\gamma}$ as in Section 2.5. Let $\\{f_{i}\\}$ be an additive
basis of $H^{*}(I\mathcal{X})$, $\\{f^{i}\\}$ another additive basis of
$H^{*}(I\mathcal{X})$ dual to $\\{f_{j}\\}$ with respect to the orbifold
Poincaré pairing. Denote $c_{1}(T^{vert}\mathcal{E})$ by $c_{1}^{vert}$. Let
$\iota$ be an inclusion of a fiber over a point in $S^{2}$ (we choose the
north pole throughout this paper). There is a Gysin map induced by this
inclusion: $\iota_{*}:H^{*}(I\mathcal{X},\mathbb{Q})\to
H^{*+2}(I\mathcal{E}_{\gamma},\mathbb{Q})$. One can think of this map as a
union of maps from $H^{*}(\mathcal{X}_{(g)},\mathbb{Q})$ to
$H^{*+2}(\mathcal{E}_{\gamma,(g)},\mathbb{Q})$, which makes sense because
there is not orbifoldness along the horizontal direction.
###### Definition 3.2.
Seidel representation for a symplectic orbifold $(\mathcal{X},\omega)$ is
defined as:
(3.1) $\mathcal{S}(a):=\sum_{\sigma\in
H_{2}^{sec}(|\mathcal{E}_{\gamma}|,\mathbb{Z})}\left(\sum_{i}\left\langle\iota_{*}f_{i}\right\rangle^{\mathcal{E}_{\gamma}}_{0,1,\sigma}f^{i}\right)\otimes
q^{c_{1}^{vert}(\sigma)}t^{\textbf{u}_{\gamma}(\sigma)}.$
The definition of $\mathcal{S}$ does not depend on the choice of Hamiltonian
loop $\gamma$ representing the homotopy class $a$, thus is a well-defined map.
Moreover it is a group homomorphism:
###### Theorem 3.3 ([TW], Theorem 1.2).
The map $\mathcal{S}$ has the following properties.
1. (1)
Triviality:
(3.2) $S(e)=1;$
2. (2)
Composition:
(3.3) $\mathcal{S}(a\cdot b)=\mathcal{S}(a)*\mathcal{S}(b).$
### 3.2. Seidel Element for Toric Orbifolds
In this section we consider a symplectic toric orbifold $\mathcal{X}$
associated to a labelled moment polytope:
$\Delta=\bigcap_{i=1}^{N}\\{\alpha\in\textbf{t}^{*}|\left\langle\alpha,b_{i}\right\rangle\leq\lambda_{i}\\},$
with $m_{i}$ the labeling number on the $i$-th facet, $y_{i}$ the primitive
outward normal vector and $b_{i}=m_{i}y_{i}$.
There is a naturally defined complex structure $\check{J}$ induced from the
complex structure on $\mathbb{C}^{N}$. For any orbifiber bundle
$\mathcal{E}\to S^{2}$ considered in Section 2.5, this complex structure
$\check{J}$ determines a complex structure $J$ on the total orbifold
$\mathcal{E}$, such that the projection $\pi:\mathcal{E}\to S^{2}$ is $j$-$J$
holomorphic, where $j$ is the complex structure on $S^{2}$ when identified
with $\mathbb{C}P^{1}$.
Recall from Section 2.3 that the Chen-Ruan cohomology of a toric orbifold can
be expressed as a quotient of a polynomial ring
$\mathbb{Q}[X_{1},X_{2},...,X_{M}]$. Define the following valuation
(3.4)
$\begin{split}&\mathfrak{v}_{T}:\mathbb{Q}[X_{1},X_{2},...,X_{M}]\otimes\Lambda\to\mathbb{Q};\\\
&\mathfrak{v}_{T}(\sum_{d,k}a_{d,k}\otimes q^{d}T^{k})=min\\{k|\exists
d:d_{d.k}\neq 0\\}.\end{split}$
This induces a valuation on the quantum cohomology
$QH_{orb}^{*}(\mathcal{X},\omega)$ which we still denote as $\mathfrak{v}_{T}$
when there is no ambiguity.
In this section we compute the Seidel elements for Hamiltonian loops
determined by lattice points $y_{k}\in\text{Gen}(\Sigma)$, $k=1,...,M$. Let
$H_{k}:\mathcal{X}\to\mathbb{R}$ be the Hamiltonian associated to $y_{k}$. Let
$\mathcal{E}^{k}$ be the Hamiltonian bundle associated to $y_{k}$. Let
$a_{k}\in\pi_{1}(Ham(\mathcal{X},\omega))$ be the homotopy class of the
Hamiltonian loop generated by $y_{k}$. Denote by $\sigma(y_{k})$ the minimal
cone in $\Sigma$ containing $y_{k}$, and
$y_{k}=\sum_{b_{i}\in\sigma(y_{k})}r_{ki}b_{i}$. Let $h.o.t.(T^{r})$ be terms
of order $>r$ with respect to the valuation $\mathfrak{v}_{T}$ on
$QH_{orb}^{*}(\mathcal{X},\omega)$. Then
###### Theorem 3.4.
$\mathcal{S}_{k}:=\mathcal{S}(\alpha_{k})=X_{k}\otimes
q^{-\sum_{b_{i}\in\sigma(y_{k})}r_{ki}}T^{-\sum_{b_{i}\in\sigma(y_{k})}r_{ki}\lambda_{i}}+h.o.t.(T^{-\sum_{b_{i}\in\sigma(y_{k})}r_{ki}\lambda_{i}}).$
###### Proof.
All we need is to compute
$\left\langle\iota_{*}f_{i}\right\rangle^{\mathcal{E}_{\gamma}}_{0,1,\sigma}$.
Denote $\textbf{x}_{k}=(\mathcal{E}^{k}_{(y_{k}^{-1})})$. Let
$\mathcal{F}_{max}$ be the (non-effective) suborbifold of $\mathcal{X}$ on
which $H_{k}$ is maximized. This suborbifold is fixed by the Hamiltonian loop.
Every $|x|\in|\mathcal{F}_{max}|$ defines a section class $\sigma_{x}\in
H_{2}(\mathcal{E}^{k},\mathbb{Z})$ of the topological bundle
$|\mathcal{E}^{k}|\to S^{2}$. It is easy to check that $\sigma_{x}$ does not
depend on the choice of $|x|\in|\mathcal{F}_{max}|$, so we can denote the
homology class as $\sigma_{max}$. Note that $\mathcal{F}_{max}$ swipes out a
suborbifold $S\mathcal{F}_{max}\subset\mathcal{E}_{k}$ which is also fibered
over $S^{2}$. It also determines a submanifold $S\mathcal{F}_{max,(g)}$ in
each stratum $\mathcal{E}^{k}_{(g)}$ of $I\mathcal{E}^{k}$, which is a
topological bundle over $S^{2}$ with fiber $|\mathcal{F}_{max,(g)}|$.
The computation of
$\left\langle\iota_{*}f_{i}\right\rangle^{\mathcal{E}_{\gamma}}_{0,1,\sigma}$
is divided into the following steps:
1. Step 1:
Every element in
$\overline{\mathcal{M}}_{0,1}(\mathcal{E}^{k},\sigma_{max},J,\textbf{x}_{k})$
is represented by a constant sectional morphism. This follows from the same
computation as [MT, Lemma 3.1].
2. Step 2:
The domain of the constant sectional morphism is $\mathbb{C}P(1,m_{k})$, with
$m_{k}=ord(D(-y_{k}))$. This is a direct conclusion from Proposition 2.8.
3. Step 3:
The constant sectional morphism is Fredholm regular.
Before prove Step 3, we run a quick check by dimension formula:
$\displaystyle
vdim\overline{\mathcal{M}}_{0,1}(\mathcal{E}^{k},\sigma_{max},J,\textbf{x}_{k})$
$\displaystyle=$ $\displaystyle
dim_{\mathbb{R}}\mathcal{E}^{k}+2c_{1}(T\mathcal{E}^{k})(\sigma_{max})+2-2\iota_{y_{k}^{-1}}-6$
$\displaystyle=$ $\displaystyle
2(n+1)+2(2-dim\sigma(y_{k})+\iota_{y_{k}^{-1}})+2-2\iota_{y_{k}^{-1}}-6$
$\displaystyle=$ $\displaystyle 2(n-dim\sigma(y_{k}))+2.$
On the other hand, by the above discussion
$\overline{\mathcal{M}}_{0,1}(\mathcal{E}^{k},\sigma_{x},J,\textbf{x}_{k})$
can be identified with $S\mathcal{F}_{max,(y_{k}^{-1})}$. So its dimension is
$2n-codim\Phi(\mathcal{F}_{max,(y_{k}^{-1})})+2=2n-2dim\sigma(y_{k})+2$.
Therefore the virtual dimension and the actual dimension match. This gives an
evidence for Fredholm regularity of constant sectional morphisms.
Let $p:\mathbb{C}P^{1}(1,m_{k}))\to\mathbb{C}P^{1}$ be the coarse moduli space
map. Then $p_{*}\boldsymbol{s}^{*}T\mathcal{E}^{k}$ is the desingularized
bundle $(\boldsymbol{s}^{*}T\mathcal{E}^{k})^{de}$ of
$\boldsymbol{s}^{*}T\mathcal{E}^{k}$. This vector bundle over
$\mathbb{C}P^{1}$ splits into a direct sum of line bundles, as discussed in
Lemma 2.11.
###### Lemma 3.5.
If each summand of $p_{*}\boldsymbol{s}^{*}T\mathcal{E}^{k}$ has Chern number
at least $-1$, then the linearized $\bar{\partial}_{J}$ operator
$D_{\boldsymbol{s}}\bar{\partial}_{J}$ is onto.
###### Proof.
To show that $D_{\boldsymbol{s}}$ is onto we need the cohomology group
$H^{1}(\boldsymbol{s}^{*}T\mathcal{E}^{k})$ to vanish. By a general property
of $p$, we have
$H^{1}(\boldsymbol{s}^{*}T\mathcal{E}^{k})=H^{1}(p_{*}\boldsymbol{s}^{*}T\mathcal{E}^{k})$.
Since $p_{*}\boldsymbol{s}^{*}T\mathcal{E}^{k}$ splits,
$H^{1}(p_{*}\boldsymbol{s}^{*}T\mathcal{E}^{k})$ splits into a direct sum of
$H^{1}$ of ths summands. Let L be any summand of
$p_{*}\boldsymbol{s}^{*}T\mathcal{E}^{k}$. We need to show that the cohomology
group $H^{1}(L)$ vanishes. By Serre duality this group is isomorphic to
$H^{0}(L^{*}\otimes K)^{*}$ where $K$ is the canonical line bundle of
$\mathbb{C}P^{1}$. We need $c_{1}(L^{*}\otimes K)<0$ in order for this group
to vanish. Now
$c_{1}(L^{*}\otimes K)=-c_{1}(L)+c_{1}(K)=-c_{1}(L)-1-1.$
So we need $-c_{1}(L)-1-1<0$, namely $c_{1}(L)>-2$, i.e. $c_{1}(L)\geq-1$. ∎
The above lemma together with Lemma 2.11 complete the proof of Step 3, i.e.,
the moduli space
$\overline{\mathcal{M}}_{0,1}(\mathcal{E}^{k},\sigma_{x},J,\textbf{x}_{k})$ is
regular.
4. Step 4:
The image of
$\overline{\mathcal{M}}_{0,1}(\mathcal{E}^{k},\sigma_{max},J,\textbf{x}_{k})$
under the evaluation lies in the twisted sector
$\mathcal{E}^{k}_{(y_{k}^{-1})}$. This follows from Lemma 2.10.
5. Step 5:
Show that
$\sum_{i}\left\langle\iota_{*}f_{i}\right\rangle^{\mathcal{E}^{k}}_{0,1,\sigma_{max}}f^{i}=X_{i}=\lambda^{y_{i}}\in
H^{*}_{CR}(\mathcal{X},\mathbb{Q})$.
From the previous steps, we have
$ev_{*}[\overline{\mathcal{M}}_{0,1}(\mathcal{E}^{k},\sigma_{x},J,\textbf{x}_{k})]=[S\mathcal{F}_{max,(y_{k}^{-1})}]$
as cycles in $\mathcal{I}\mathcal{E}^{k}$. Then
$\displaystyle\sum_{i}\left\langle\iota_{*}f_{i}\right\rangle^{\mathcal{E}^{k}}_{0,1,\sigma_{max}}f^{i}$
$\displaystyle=$
$\displaystyle\sum_{i}(\int_{ev_{*}[\overline{\mathcal{M}}_{0,1}(\mathcal{E}^{k},\sigma_{max},J,\textbf{x}_{k})]}\iota_{*}f_{i})\
f^{i}$ $\displaystyle=$ $\displaystyle\sum_{i:f_{i}\in
H^{*}(\mathcal{X}_{(y_{k}^{-1})},\mathbb{Q})}(\int_{ev_{*}[\overline{\mathcal{M}}_{0,1}(\mathcal{E}^{k},\sigma_{max},J,\textbf{x}_{k})]}\iota_{*}f_{i})\
f^{i}$ $\displaystyle=$ $\displaystyle\sum_{i:f_{i}\in
H^{*}(\mathcal{X}_{(y_{k}^{-1})},\mathbb{Q})}(\int_{[\mathcal{F}_{max,(y_{k}^{-1})}]}f_{i})\
f^{i}$ $\displaystyle=$ $\displaystyle X_{i}$
6. Step 6:
Any $J$-holomorphic non-constant sectional morphism $\boldsymbol{s}$ or a
constant sectional morphism constructed from a point not in
$\mathcal{F}_{max}$ satisfies
$\mathbf{u}_{y_{k}}([\boldsymbol{s}])>\mathbf{u}_{y_{k}}(\sigma_{max})=-max\
H_{k}=-\sum_{b_{i}\in\sigma(y_{k})}r_{ki}\lambda_{i}.$
If $\boldsymbol{s}$ is a constant sectional morphism determined by a point $x$
with $H(x)>max\ H$, by Lemma 2.14,
$\mathbf{u}_{y_{k}}([\boldsymbol{s}])=-H_{k}(x)>-max\
H_{k}=\mathbf{u}_{y_{k}}(\sigma_{max})$.
If $\boldsymbol{s}$ is a non-constant sectional morphism, compute
$\mathbf{u}_{y_{k}}([\boldsymbol{s}])$ by integrating the pullback of
$\mathbf{u}_{y_{k}}$. Recall from (2.6) that the coupling form is given by
$\textbf{u}_{y_{k}}=\omega-dH_{k}\alpha$. We choose the complex structure
$\check{J}$ on $\mathcal{X}$ which is induced from the complex structure on
$\mathbb{C}^{N}$. Consequently $\check{J}$ is invariant under the Hamiltonian
circle action generated by $y_{k}$. Let $\hat{j}$ be the standard complex
structure on $S^{3}$ induced from $\mathbb{C}^{2}$. Every non-zero tangent
vector $\xi\in T_{[z,x]}S^{3}\times\mathcal{X}/S^{1}$ can be uniquely
represented by a vector $\eta+v\in T_{(z,x)}S^{3}\times\mathcal{X}$ with
$\eta\in ker\alpha$ and $v\in T_{x}\mathcal{X}$. Then
$\textbf{u}_{y_{k}}(\xi,J\xi)=\omega(v,\check{J}v)-dH_{k}\alpha(\eta+v,\hat{j}\eta+\check{J}v)=\omega(v,\check{J}v)-H_{k}d\alpha(\eta,\hat{j}\eta)>-max\
H_{k}.$
Integrating over $S^{2}$ we get
$\mathbf{u}_{y_{k}}([\boldsymbol{s}])>\mathbf{u}_{y_{k}}(\sigma_{max})$.
Thus we complete the proof of Theorem 3.4. ∎
###### Definition 3.6.
For a toric orbifold $\mathcal{X}$, the reduced Seidel elements are defined
as:
$\tilde{\mathcal{S}}_{k}:=\mathcal{S}_{k}\otimes
q^{\sum_{b_{i}\in\sigma(y_{k})}r_{ki}}T^{\sum_{b_{i}\in\sigma(y_{k})}r_{ki}\lambda_{i}}$
###### Corollary 3.7.
$\tilde{\mathcal{S}}_{k}=X_{k}+h.o.t.(T^{0}).$
## 4\. Quantum Cohomology of Toric Orbifolds
In this section we give an explicit description of the quantum cohomology ring
of toric orbifolds using the Seidel elements computed in Section 3.2.
### 4.1. Main Result
Recall that for $k=1,...,N$, $y_{k}$ is the primitive vector of the $k$-th
ray, and for $k=N+1,...,M$, $y_{k}\in\text{Gen}(\Sigma)$. Let $\sigma(y_{k})$
be the minimal cone in $\Sigma$ containing $y_{k}$, we have
$y_{k}=\sum_{b_{i}\in\sigma(y_{k})}r_{ki}b_{i}$, where $r_{ki}$ are positive
rational numbers. Note that when $k=1,...,N$, $y_{k}=\frac{1}{m_{k}}b_{k}$.
For a generalized primitive collection $I\subset\\{1,...,M\\}$, let
$\sigma(I)$ be the minimal cone in $\Sigma$ containing $\sum_{k\in I}y_{k}$,
then there exists $\\{c_{j}\\}\subset\mathbb{Z}_{+}$ such that
$\sum_{k\in I}y_{k}=\sum_{y_{j}\in\sigma(I)}c_{j}y_{j}$
since the fan is complete.
Write both sides in terms of $b_{i}$’s:
$\sum_{k\in
I}\sum_{b_{i}\in\sigma(y_{k})}r_{ki}b_{i}=\sum_{y_{j}\in\sigma(I)}c_{j}\sum_{b_{i}\in\sigma(y_{j})}r_{ji}b_{i}=\sum_{y_{j}\in\sigma(I)}\sum_{b_{i}\in\sigma(y_{j})}c_{j}r_{ji}b_{i}.$
Let
(4.1) $\displaystyle C_{I}$ $\displaystyle=\sum_{k\in
I}\sum_{b_{i}\in\sigma(y_{k})}r_{ki}-\sum_{y_{j}\in\sigma(I)}\sum_{b_{i}\in\sigma(y_{j})}c_{j}r_{ji},$
(4.2) $\displaystyle\Omega_{I}$ $\displaystyle=\sum_{k\in
I}\sum_{b_{i}\in\sigma(y_{k})}r_{ki}\lambda_{i}-\sum_{y_{j}\in\sigma(I)}\sum_{b_{i}\in\sigma(y_{j})}c_{j}r_{ji}\lambda_{i}.$
###### Lemma 4.1.
For $\Omega_{I}$ defined as above, we have $\Omega_{I}>0$.
###### Proof.
Define a piecewise linear function $\phi_{\omega}:\textbf{t}\to$ as:
$\phi_{\omega}(u):=\sum_{b_{i}\in\sigma}-\lambda_{i}\langle
b_{i}^{\vee},u\rangle,\ \ \ if\ u\in\sigma,$
for $\sigma$ any full dimensional cone of the fan $\Sigma$ and
$\\{b_{i}^{\vee}\\}$ the dual basis of $\\{b_{i}|b_{i}\in\sigma\\}$.
Then
(4.3) $\Omega_{I}=-\sum_{k\in
I}\phi_{\omega}(y_{k})+\phi_{\omega}(\sum_{y_{j}\in\sigma(I)}c_{j}y_{j}).$
On the other hand, $\phi_{\omega}$ corresponds to the symplectic form $\omega$
of the toric orbifold $\mathcal{X}$ under the isomorphism between
$H^{2}(\mathcal{X},)\cong[b_{1}^{\vee},...,b_{N}^{\vee}]/\left\langle\sum_{i=1}^{n}\theta(b_{i})b_{i}^{\vee},\theta\in\mathbf{M}\right\rangle$.
Since $\omega$ lies in the Kähler cone of $\mathcal{X}$, $\phi_{\omega}$ is a
strictly convex function (in the sense that if $v_{1},v_{2}$ are not contained
in the same cone, then
$\phi_{\omega}(v_{1}+v_{2})>\phi_{\omega}(v_{1})+\phi_{\omega}(v_{2})$).
Together with (4.3), we have $\Omega_{I}>0$. ∎
###### Definition 4.2.
The quantum Stanley-Reisner relation associated with the generalized primitive
collection $I$ is defined to be
(4.4) $\prod_{k\in
I}Z_{k}-q^{C_{I}}T^{\Omega_{I}}\prod_{y_{j}\in\sigma(I)}Z_{j}^{c_{j}}=0.$
Let $SR_{\omega}$ be the ideal generated by expressions as the left-hand side
of the above equation. $SR_{\omega}$ is called the quantum Stanley-Reisner
ideal.
From the composition property of Seidel representation, we have
###### Theorem 4.3.
The reduced Seidel elements $\tilde{\mathcal{S}}_{k}$, $k=1,...,M$, satisfy
the quantum Stanley-Reisner relations and the cone relations.
Let
$\displaystyle\mathfrak{P}_{\xi}^{(0)}\negthickspace\negthickspace\negthickspace\negthickspace\negthickspace$
$\displaystyle:=\sum_{i=1}^{N}\langle e_{\xi},b_{i}\rangle X_{i}^{m_{i}}\ ,\ \
\ \ \ \xi=1,...,n,\ \text{where}\ \\{e_{\xi}\\}_{\xi=1}^{n}\ \text{is a basis
of}\ \mathbf{M}.$
###### Theorem 4.4.
For $\xi=1,...,n$, there exist $\mathfrak{P}_{\xi}\in\Lambda[X_{1},...,X_{M}]$
such that
1. (1)
$\mathfrak{P}_{\xi}(\tilde{\mathcal{S}}_{1},...,\tilde{\mathcal{S}}_{M})=0$ in
$H^{*}(I\mathcal{X},\Lambda)$;
2. (2)
$\mathfrak{P}_{\xi}=\mathfrak{P}^{(0)}_{\xi}+h.o.t.(T^{0})$;
3. (3)
The ring homomorphism
$\Psi:\frac{\Lambda[X_{1},...,X_{M}]}{Clos_{\mathfrak{v}_{T}}(\left\langle\mathfrak{P}_{\xi}|\xi=1,...,n\right\rangle+SR_{\omega}+\mathcal{J}(\Sigma))}\to
QH_{orb}^{*}(\mathcal{X},\Lambda),\quad X_{i}\mapsto\tilde{\mathcal{S}}_{k}$
is an isomorphism.
###### Proof.
Let $\hat{\Psi}:\Lambda[X_{1},...,X_{M}]\otimes\Lambda\to
QH_{orb}^{*}(\mathcal{X},\Lambda)$ be the map sending $X_{i}$ to
$\tilde{\mathcal{S}}_{i}$. Let $\delta_{\mathcal{X}}$ be a positive number
such that the symplectic area of any non-constant $J$-holomorphic curve is
bounded below by $\delta_{\mathcal{X}}$.
We first show that $\hat{\Psi}$ is surjective. For any $\alpha\in
QH_{orb}^{*}(\mathcal{X},\Lambda)$, Let $L(\alpha)$ be the leading term of
$\alpha$, namely $\mathfrak{v}_{T}(L(\alpha))=\mathfrak{v}_{T}(\alpha)$ and
$\mathfrak{v}_{T}(L(\alpha)-\alpha)>\mathfrak{v}_{T}(\alpha)$. Then
$L(\alpha)=\sum_{s}L^{\alpha}_{s}(X_{1},...,X_{M})q^{s}$, and
$\hat{\Psi}(L(\alpha))=\sum_{s}L^{\alpha}_{s}(\tilde{S}_{1},...,\tilde{S}_{M})q^{s}$.
Define $\alpha_{1}=\alpha-\hat{\Psi}(L(\alpha))$. The leading terms in
$\alpha$ and $\hat{\Psi}(L(\alpha))$ cancell, so
$\mathfrak{v}_{T}(\alpha_{1})\geq\mathfrak{v}_{T}(\alpha)+\delta_{\mathcal{X}}$.
We repeat the above procedure using $\alpha_{1}$, and continue the argument
inductively, then $\mathfrak{v}_{T}(\alpha_{k})\to\infty$, and
$\hat{\Psi}\left(\lim_{k\to\infty}\left(L(\alpha)+L(\alpha_{1})+L(\alpha_{2})+...+L(\alpha_{k})\right)\right)=\alpha.$
This proves surjectivity.
Next we show the kernel of $\hat{\Psi}$ is
$Clos_{\mathfrak{v}_{T}}(\left\langle\mathfrak{P}_{\xi}|\xi=1,...,n\right\rangle+SR_{\omega}+\left\langle\mathfrak{Q}_{\eta}|\eta=1,...,M-N\right\rangle).$
We need the following lemma which is analogous to [MT, Lemma 5.1]:
###### Lemma 4.5.
Let
$\displaystyle\hat{\phi}$ $\displaystyle:$
$\displaystyle\mathbb{Q}[X_{1},X_{2},...,X_{M}]\to
H_{CR}^{*}(\mathcal{X},\mathbb{Q})$ $\displaystyle\hat{\Phi}$ $\displaystyle:$
$\displaystyle\mathbb{Q}[X_{1},X_{2},...,X_{M}]\otimes\Lambda\to
QH_{orb}^{*}(\mathcal{X},\Lambda)$
be ring homomorphisms such that $\hat{\phi}(X_{i})=\hat{\Phi}(X_{i})=X_{i}$
for $i=1,...,M$. Let $w_{1},...w_{m}\in\mathbb{Q}[X_{1},X_{2},...,X_{M}]$
generate the kernel of $\hat{\phi}$, and suppose $v_{1},...v_{m}\in
Ker\hat{\Phi}$ and
(4.5) $\mathfrak{v}_{T}(w_{i}-v_{i})>0\ \text{for all}\ i.$
Then the kernel of $\hat{\Phi}$ is
$Ker\hat{\Phi}=Clos_{\mathfrak{v}_{T}}(\left\langle
v_{1},...,v_{m}\right\rangle).$
###### Proof.
Since in the orbifold case there is still a universal lower bound for
symplectic area of non-constant $J$-holomorphic curves, thus we have [MT,
Lemma 5.1] for orbifold quantum cohomology without any modification of the
proof. ∎
From Lemma 2.4, we know that $Ker\hat{\phi}$ is generated by the Stanley-
Reisner relations, cone relations, and $\mathfrak{P}_{\xi}^{(0)}$’s. We have
shown in Theorem 4.3 that the quantum Stanley-Reisner relations and cone
relations lie in the kernel of $\hat{\Phi}$. Moreover they satisfy (4.5).
Now we construct $\mathfrak{P}_{\xi}$’s which lie in the Kernel of
$\hat{\Phi}$ and satisfy the high order condition (4.5).
Let $\lambda_{0}=0$. By Corollary 3.7, there is a
$\lambda_{1}\geq\lambda_{0}+\delta_{\mathcal{X}}$, such that
(4.6)
$\mathfrak{P}_{\xi}^{(0)}(\tilde{\mathcal{S}}_{1},...,\tilde{\mathcal{S}}_{M})=\sum_{i=1}^{n}\langle
e_{\xi},b_{i}\rangle\tilde{\mathcal{S}}_{i}=\sum_{i=1}^{n}\langle
e_{\xi},b_{i}\rangle X_{i}^{m_{i}}+R^{1}\
T^{\lambda_{1}}+h.o.t.(T^{\lambda_{1}}),$
with $\mathfrak{v}_{T}(R^{1})=0$. Then define
$\mathfrak{P}_{\xi}^{(1)}=\mathfrak{P}_{\xi}^{(0)}-R^{1}\ T^{\lambda_{1}}$.
Since $R^{1}=\sum_{s}R^{1}_{s}(X_{1},...,X_{M})q^{s}$,
$\mathfrak{P}_{\xi}^{1}$ is again a polynomial in $X_{1},...,X_{M}$ with
coefficients in $\Lambda$. Replace $\mathfrak{P}_{\xi}^{(0)}$ with
$\mathfrak{P}_{\xi}^{(1)}$ in (4.6), we get
$\displaystyle\mathfrak{P}_{\xi}^{(1)}(\tilde{\mathcal{S}}_{1},...,\tilde{\mathcal{S}}_{M})$
$\displaystyle=\sum_{i=1}^{n}\langle e_{\xi},b_{i}\rangle
X_{i}^{m_{i}}+R^{1}(X_{1},...,X_{M})\
T^{\lambda_{1}}+h.o.t-(R^{1}(\tilde{S}_{1},...,\tilde{S}_{M})+h.o.t.(T^{\lambda_{2}}))$
$\displaystyle=\sum_{i=1}^{n}\langle e_{\xi},b_{i}\rangle X_{i}^{m_{i}}+R^{2}\
T^{\lambda_{2}}+h.o.t.(T^{\lambda_{2}}),$
for some $\lambda_{2}>\lambda_{1}+\delta_{\mathcal{X}}$.
Then we can construct $\mathfrak{P}_{\xi}^{(2)}$ as before. Continue the
procedure inductively, we construct $\mathfrak{P}_{\xi}^{(k)}$,
$k=0,1,2,....$. Then define
$\mathfrak{P}_{\xi}=\lim_{k\to\infty}\mathfrak{P}_{\xi}^{(k)}$.
By Lemma 4.5, we conclude that $Ker\hat{\Phi}$ is generated by quantum
Stanley-Reisner relations, cone relations, and
$\left\langle\mathfrak{P}_{\xi}|\xi=1,...,n\right\rangle$. This completes the
proof. ∎
###### Remark 4.6.
We remark that the quantum cohomology ring can also be described as some sort
of quantization of the group ring $\mathbb{Q}[\mathbf{N}]^{\mathbf{\Sigma}}$
by reversing the procedure in Section 2.3. Let
$\mathbb{Q}[\mathbf{N}]^{\mathbf{\Sigma}}_{\omega}$ be the ring with the same
elements as $\mathbb{Q}[\mathbf{N}]^{\mathbf{\Sigma}}\otimes\Lambda$ and the
product $*$ defined as the following:
$\lambda^{e_{1}}*\lambda^{e_{2}}:=q^{C(e_{1},e_{2})}T^{\Omega(e_{1},e_{2})}\lambda^{e},$
where $e_{1},e_{2}\in\mathbf{N}$, $e=e_{1}+e_{2}$, $C(e_{1},e_{2})$ and
$\Omega(e_{1},e_{2})$ are numbers defined in a similar fashion as (4.1). The
map $\mathbb{Q}[\mathbf{N}]^{\mathbf{\Sigma}}_{\omega}\to QH^{*}(\mathcal{X})$
defined by sending $\lambda^{e}$ to the reduced Seidel element $S_{e}$ is a
surjective ring morphism. The kernel is generated by
$\mathfrak{P}_{\xi}(\lambda^{y_{1}},...,\lambda^{y_{M}})$’s.
### 4.2. The Fano Case
When the toric orbifold $\mathcal{X}$ is Fano, namely every effective curve
has positive Chern number, we have the following lemma.
###### Lemma 4.7.
If $B\neq 0$, then
$\sum_{i}\left\langle\iota_{*}f_{i}\right\rangle^{\mathcal{E}^{k}}_{0,1,\sigma_{max}+\iota_{*}B}f^{i}=0$.
The following is an immediate consequence of Lemma 4.7.
###### Corollary 4.8.
$\tilde{\mathcal{S}}_{k}=X_{k}$.
###### Theorem 4.9.
The orbifold quantum cohomology ring $QH_{orb}^{*}(\mathcal{X},\Lambda)$ of a
Fano toric symplectic orbifold $(\mathcal{X},\omega)$ is isomorphic to
$\frac{\Lambda[X_{1},...,X_{M}]}{Clos_{\mathfrak{v}_{T}}(\left\langle\sum_{i=1}^{N}\langle
e_{\xi},b_{i}\rangle
X_{i}^{m_{i}}|\xi=1,...,n\right\rangle+\left\langle\prod_{k\in
I}X_{k}-q^{C_{I}}T^{\Omega_{I}}\prod_{y_{j}\in\sigma(I)}X_{j}^{c_{j}}|I\in\mathcal{GP}\right\rangle+\mathcal{J}(\Sigma))}.$
###### Proof of Lemma 4.7.
If $y_{k}\in\text{Gen}(\Sigma)$ such that $\iota_{y_{k}}\neq 0$, then
$\displaystyle
vdim\overline{\mathcal{M}}_{0,1}(\mathcal{E}^{k},\sigma_{max}+\iota_{*}B,J,(\mathcal{E}^{k}_{(v)}))$
$\displaystyle=vdim\overline{\mathcal{M}}_{0,1}(\mathcal{E}^{k},\sigma_{max},J,(\mathcal{X}_{(v)}))+2c_{1}^{\mathcal{X}}(B)$
$\displaystyle=2n+2+2(c_{1}^{\mathcal{X}}(B)-\iota_{v}-\iota_{y_{k}}).$
For $f\in H^{*}(\mathcal{X}_{(v)},\mathbb{Q})$, $deg\iota_{*}f\leq
2n+2-2dim\sigma(v)$. Then by Fanoness,
$vdim\overline{\mathcal{M}}_{0,1}(\mathcal{E}^{k},\sigma_{max}+\iota_{*}B,J,(\mathcal{E}^{k}_{(v)}))-deg\iota_{*}f\geq
2(c_{1}^{\mathcal{X}}(B)-\iota_{y_{k}}-\iota_{v})+2dim\sigma(v)>0.$
Consequently,
$\left\langle\iota_{*}f\right\rangle^{\mathcal{E}^{k}}_{0,1,\sigma_{max}+\iota_{*}B}=0$.
Now if $y_{k}\in\text{Gen}(\Sigma)$ such that $\iota_{y_{k}}=0$, then $y_{k}$
lies in a ray of the fan $\Sigma$ and $dim\sigma(y_{k})=1$. Therefore
$\displaystyle
vdim\overline{\mathcal{M}}_{0,1}(\mathcal{E}^{k},\sigma_{max}+\iota_{*}B,J,(\mathcal{E}^{k}_{(v)}))$
$\displaystyle=2n+2-2dim\sigma(y_{k})+2(c_{1}^{\mathcal{X}}(B)-\iota_{v})$
$\displaystyle=2n+2(c_{1}^{\mathcal{X}}(B)-\iota_{v}).$
Compute
$\displaystyle
vdim\overline{\mathcal{M}}_{0,1}(\mathcal{E}^{k},\sigma_{max}+\iota_{*}B,J,(\mathcal{E}^{k}_{(v)}))-deg\iota_{*}f$
$\displaystyle\geq
2n+2(c_{1}^{\mathcal{X}}(B)-\iota_{v})-(2n+2-2dim\sigma(v))$
$\displaystyle=2(c_{1}^{\mathcal{X}}(B)-\iota_{v})-2+2dim\sigma(v).$
By Fanoness, $2(c_{1}^{\mathcal{X}}(B)-\iota_{v})-2+2dim\sigma(v)=0$ only if
$dim\sigma(v)=0$, i.e. $v=0$. Thus
$vdim\overline{\mathcal{M}}_{0,1}(\mathcal{E}^{k},\sigma_{max}+\iota_{*}B,J,(\mathcal{E}^{k}_{(v)}))-deg\iota_{*}f=0$
only if $c_{1}^{\mathcal{X}}(B)=1$ and $deg\iota_{*}f=2n+2$.
In particular, we have shown
$\left\langle\iota_{*}f\right\rangle^{\mathcal{E}^{k}}_{0,1,\sigma_{max}+\iota_{*}B}\neq
0$ for $B\neq 0$ is possible only when the evaluation map lands in the trivial
twisted sector. Thus from now on the proof is similar to the manifold case as
in [MT]. We will sketch the idea below.
Because $deg\iota_{*}f=2n+2$, $f=\kappa PD([\mathcal{X}_{(0)}])$ for some
nonzero $\kappa\in\mathbb{Q}$. Then homological interpretation of
$\left\langle\iota_{*}f)\right\rangle^{\mathcal{E}^{k}}_{0,1,\sigma_{max}+\iota_{*}B}\neq
0$
is
$\left\langle\iota_{*}[pt])\right\rangle^{\mathcal{E}^{k}}_{0,1,\sigma_{max}+\iota_{*}B}[\mathcal{X}_{(0)}]\neq
0.$
In particular, its intersection product with a point class is non-zero. So
$\left\langle\iota_{*}[pt],\iota^{0}_{*}[pt]\right\rangle^{\mathcal{E}^{k}}_{0,2,\sigma_{max}+\iota_{*}B}\neq
0,$
where $\iota^{0}$ is the inclusion of a fiber at the south pole. We represent
the first point class by the point $x_{max}\in\mathcal{F}_{max,(0)}$ and the
second point class by $x_{min}\in\mathcal{F}_{min,(0)}$.
Let $\phi:S^{1}\times S^{2}\to S^{2}$ be the rotation of $S^{2}$ with respect
to the south pole and north pole. Consider a circle action on
$\mathcal{E}^{k}$ given by $\phi|_{D_{+}}\times\tilde{\boldsymbol{\gamma}}$
and $\phi|_{D_{-}}\times Id_{\mathcal{X}}$. This action induces a circle
action on the moduli space
$\overline{\mathcal{M}}_{0,2}(\mathcal{E}^{k},\sigma_{max}+\iota_{*}B)$. Then
by a version of localization proven in [MT],
$\left\langle\iota_{*}[x_{max}],\iota^{0}_{*}[x_{min}]\right\rangle^{\mathcal{E}^{k}}_{0,2,\sigma_{max}+\iota_{*}B}\neq
0$ only if there exist $S^{1}$-invariant stable orbifold morphisms which
consists of constant section determined by $x_{max}$ and a branch component
lying in the south pole. The branch component is an $S^{1}$-invariant
J-holomorphic curve in $\mathcal{X}$, representing $B$. It is the orbit of a
gradient flow of $H_{k}$ from $x_{min}$ to $x_{max}$. So
$B=p(\sigma_{max}-\sigma_{min})/q$ for $p\in\mathbb{Z}$ and
$q\in\mathbb{Z}_{>0}$. Thus $\omega(B)=p(minH_{k}-maxH_{k})/q$. Because
$\omega(B)>0$ and $minH_{k}-maxH_{k}<0$, so $p>0$. On the other side
$1=c_{1}^{\mathcal{X}}(B)=p(m_{min}-m_{max})$, where $m_{min}$ and $m_{max}$
is the weight of the circle action at $F_{min}$ and $F_{max}$ respectively.
Therefore $m_{min}=m_{max}+\frac{2}{p}\leq 0$, which contradicts the weight at
$\mathcal{F}_{min}$ is always positive. ∎
Now we look at some examples. As a convention we always put the barycenter of
the moment polytope at the original.
###### Example 4.10.
Weighted projective line $\mathbb{C}P(a,b)$ when $a$ and $b$ are coprime.
ab0$\lambda$0-ab Figure 2. Labelled Moment Polytope and Stacky Fan of
$\mathbb{C}P(a,b)$.
$QH^{*}_{orb}(\mathbb{C}P(a,b),\Lambda)\cong\frac{\Lambda[X_{1},X_{2}]}{Clos_{\mathfrak{v}_{T}}(\left\langle-
aX_{1}^{a}+bX_{2}^{b}\right\rangle+\left\langle
X_{1}X_{2}-q^{\frac{1}{a}+\frac{1}{b}}T^{\lambda}\right\rangle)}.$
###### Example 4.11.
Weighted projective space $\mathbb{C}P(1,1,2)$.
$(-\frac{2}{3}\lambda,-\frac{1}{3}\lambda)$$(\frac{4}{3}\lambda,-\frac{1}{3}\lambda)$$(-\frac{2}{3}\lambda,\frac{2}{3}\lambda)$$2\lambda$$\lambda$$\sqrt{5}\lambda$(0,0)$y_{1}$$y_{2}$$y_{3}$$y_{4}$
Figure 3. Moment Polytope and Fan of $F_{2}$.
Then the orbifold quantum cohomology ring
$QH^{*}_{orb}(\mathbb{C}P(1,1,2),\Lambda)$ is isomorphic to
$\frac{\Lambda[X_{1},X_{2},X_{3},X_{4}]}{Clos_{\mathfrak{v}_{T}}(\left\langle-
X_{1}+X_{3},-X_{2}+2X_{3}\right\rangle+\left\langle
X_{1}X_{2}X_{3}-q^{C}T^{\Omega}X_{4}\right\rangle+\left\langle
X_{1}X_{3}-X_{4}^{2}\right\rangle)},$
where
$C=1+1+1-\frac{1}{2}-\frac{1}{2}=2,$
$\Omega=\frac{2}{3}\lambda+\frac{1}{3}\lambda+\frac{2}{3}\lambda-\frac{1}{2}\cdot\frac{2}{3}\lambda-\frac{1}{2}\cdot\frac{2}{3}\lambda=\lambda.$
Note that the generalized primitive collections in this example are
$\\{y_{1},y_{2},y_{3}\\}$ and $\\{y_{2},y_{4}\\}$. But the quantum Stanley-
Reisner relation associated to $\\{y_{2},y_{4}\\}$ is already contained in
$\left\langle X_{1}X_{2}X_{3}-q^{C}T^{\Omega}X_{4}\right\rangle+\left\langle
X_{1}X_{3}-X_{4}^{2}\right\rangle.$
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|
arxiv-papers
| 2012-11-14T04:46:57 |
2024-09-04T02:49:38.020216
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Hsian-Hua Tseng and Dongning Wang",
"submitter": "Dongning Wang",
"url": "https://arxiv.org/abs/1211.3204"
}
|
1211.3376
|
arxiv-papers
| 2012-11-12T22:21:03 |
2024-09-04T02:49:38.042473
|
{
"license": "Public Domain",
"authors": "Erich L Foster and James R Overfelt",
"submitter": "Erich Foster",
"url": "https://arxiv.org/abs/1211.3376"
}
|
|
1211.3607
|
Classification by Boosting Differences in Input Vectors††thanks: By Ninan
Sajeeth Philip, email: [email protected]–LABEL:lastpage
# Classification by Boosting Differences in Input Vectors††thanks: By Ninan
Sajeeth Philip, email: [email protected]
N. S. Philip1, A. Mahabal2, S. Abraham.1, R. Williams2, S.G. Djorgovski2,3, A.
Drake2, C Donalek2 and M. Graham2
1St. Thomas College Kozhencheri Kerala India
2 Caltech 1200 E California Bl Pasadena CA 91125 USA
3 Distinguished Visiting Professor King Abdulaziz University Jeddah Saudi
Arabia
(2011; Received 27th August 2024)
###### Abstract
There are many occasions when one does not have complete information in order
to classify objects into different classes, and yet it is important to do the
best one can since other decisions depend on that. In astronomy, especially
time-domain astronomy, this situation is common when a transient is detected
and one wishes to determine what it is in order to decide if one must follow
it. We propose to use the Difference Boosting Neural Network (DBNN) which can
boost differences between feature vectors of different objects in order to
differentiate between them. We apply it to the publicly available data of the
Catalina Real-Time Transient Survey (CRTS) and present preliminary results. We
also describe another use with a stellar spectral library to identify spectra
based on a few features. The technique itself is more general and can be
applied to a varied class of problems.
###### keywords:
methods: data analysis, techniques: photometric, techniques: spectroscopic,
stars: general
††volume: 00
## 1 Introduction
It is common to have fragmented and fragmentary evidence when the sources of
information are varied as well as unequal in strength. In time-domain
astronomy, just like in forensic science, one tries to piece together the
information in order to obtain a verdict. An object has been observed that was
either not there before, or was much fainter. The only extra information
available is from sporadic past observations of the same area in archival
surveys that answer questions like: ‘was this seen at such and such radio
frequency?’, ‘was it seen in the Sloan Digital Sky Survey?’, ‘what is the
distance to the nearest galaxy?’ etc. Depending on the location of the object,
and its nature, the resulting information can be sparse and very different
from source to source. The Catalina Real-Time Transient Survey
(CRTS111http://crts.caltech.edu) publishes the transients it finds in real-
time. The events, broadcast as VOEvent packets, are ingested by
Skyalert222http://www.skyalert.org (Williams et al., 2009) where a portfolio
is gathered for each object by annotating the initial information using
programs that query individual surveys and archives to answer, as best as
possible, a pre-determined set of questions as mentioned earlier. If one
thinks of all such bits associated with an individual object as its feature,
the entire input data is just a vector of features (with many, often 50-80
percent, individual features for each vector missing). Making sense of this
dataset is not trivial, but classification is important (Mahabal et al.
(2012), Djorgovski et al. (2011a)). Classifying the features into classes that
are as unambiguous as possible, and with minimal number of false detections
needs a probabilistic approach that can make full use of all available prior
information.
The present work summarizes one possible method to handle such situations
where the sparseness of data is diverse and patchy whereby making it difficult
for existing machine learning techniques to handle them. It uses Bayesian
belief update rule that can be used to progressively update the belief in the
outcome based on plausible evidence. We apply the method to the CRTS
transients to generate quick classification probabilities which can be revised
as more data become available. We also show how it can be generalized to be
used with a stellar spectral library (Valdes et al., 2004) making it
especially useful in future when IFUs make available large number of spectra
simultaneously.
## 2 Catalina Real-Time Survey
The Catalina Real-Time Transient Survey (Drake et al. (2009), Djorgovski et
al. (2011b), Mahabal et al. (2011)) use data from the Catalina Sky Survey
(CSS333http://www.lpl.arizona.edu/css/) for near-Earth objects and potential
planetary hazard asteroids (NEO/PHA), conducted by Edward Beshore, Steve
Larson, and their collaborators at the University of Arizona. CRTS looks for
astrophysical transient and variable objects using real-time processing and
carries out characterization, and distribution of these events, as well as
follow-up observations of selected events.
Optical transients (OTs) are detected as sources displaying significant
changes in brightness, as compared to the baseline comparison images and
catalogs with significantly fainter limiting magnitudes. Data cover time
baselines from 10 min to several years. The detected transients are published
electronically in real time, using a variety of mechanisms. One of the methods
is a Skyalert stream where additional data on the transients is gathered by
harvesting archival datasets as well as information on the proximity of the
source in what is termed as passive follow-up.
It is this comprehensive dataset that we make use of with DBNN in order to try
to predict the nature of the transient. The training set is based upon
classification by human experts. Besides the extremely sparse matrix visible
to DBNN (described in the next section) the humans can use such aids as the
historic light curve of the transient which a human neural network can use to
trivially discriminate between SNe (single hump) versus a CV (multiple
brightenings over a few years) in many cases. Such features are being
incorporated into separate tools. Some derived characteristics from such aids
will be incorporated into DBNN in the future. The different techniques will
also be incorporated in to a fusion network.
## 3 Difference Boosting Neural Network (DBNN)
According to the Bayesian Theorem (von Toussaint, 2011), it is possible to
start with an initial belief about the probability of occurrence of an event
even when there is no compelling evidence and later update this belief
periodically as new evidence is found. In the context of this paper, the
initial belief is called prior and the likelihood for an observation to cause
an event is called evidence. Then according to Bayesian theorem, popularly
called the Bayes rule, the updated belief, or the posterior, is the product of
the prior and the evidence normalized over all possibilities. Mathematically,
this may be written as:
$P(A|B)=\frac{P(B|A)\times P(A)}{\sum_{i}P(B|A_{i})\times P(A_{i})}$ (1)
where $P(A|B)$ is the updated belief that the event A is caused by the
occurrence of event B and $P(B|A)$ is the likelihood that event B may cause A.
P(A) is the prior, the initial belief, which is independent of whether B is
observed or not. One may call it the probability that such an event may be
found even when none of the different types of evidence is seen. The index $i$
is used to normalize all the possible events including $A$ that could have
produced $B$.
The beauty of the Bayesian rule is that it allows sequential updating of the
belief or confidence in an outcome as more and more evidence arrives. Each
time an update is made, the computed new belief becomes the new prior for the
next update, thus essentially making the situation conducive for systems where
one has to deal with a diverse set of inputs and come up with plausible causes
that created them. The second advantage is that, all these estimates are based
on the statistical distribution of the likelihoods and hence the final
evidence is the probability and can be directly used as the confidence one may
have in the prediction of the Bayesian classifier.
We have used a Bayesian Classifier named Difference Boosting Neural Network
(Philip & Joseph, 2000) for this study. The DBNN distributes the evidence in a
feature space so that every feature is associated with its likelihood as
learned through a process called learning (Philip, 2009, 2010). It also
estimates the prior for each evidence in individual cases that maximizes the
prediction accuracy on the data used for training. This data is referred to as
training data. Assuming that we have a large set of training data, it is
possible to have a reasonable estimate of the prior and the likelihood even
for very complex cases. After the training process, the estimated likelihoods
and priors are saved for future use.
Training is usually followed by a testing cycle in which the classifier is
tested with a fresh set of data that are similar but never used in the
training process. This is to test for adequate learning of the classifier, in
which case, the accuracy obtained on the training data and that on the test
data will be comparable. If that is not the case, one has to add the failed
examples also into the training data and update the likelihoods taking them
into consideration. This results in a continued learning process that Bayesian
classifiers can very efficiently handle. The second advantage of this
progressive learning process is that the system asymptotically converges to a
global optimum as the learning progresses. This similarity of Bayesian rule to
human learning made Laplace comment that it is the mathematical equivalent of
common sense.
Preparing the data in a format required for use by the classifier is here
referred to as preprocessing. This is somewhat similar to preprocessing and
data reduction that are familiar to astronomers. The preprocessing ensures
that the input features are ordered in some fashion and an appropriate label
is used to represent the associated class of each example in the data. Usually
the number of input features may be fixed and the names of the classes all
known. But the new situation narrated above has no such restriction. One can
have a dynamic situation where new features are to be incorporated. For
managing this new situation, we have adopted a sequence coding method in which
the presence or absence of a feature is marked by a 1 or 0 in the form of a
chain of binary numbers. New entries are appended to the sequence when new
evidence becomes available, allowing dynamical growth of the feature vector.
Num. Code | Class
---|---
1 | Cataclysmic Variable
2 | Supernova
3 | other
5 | Blazar Outburst
6 | Active Galactic Nucleus Variability
7 | UVCeti Variable
8 | Asteroid
9 | Variable
10 | Mira Variable
11 | High Proper Motion Star
12 | Comet
16 | Nova
Table 1: The 12 different CRTS transient classes identified with numeric
labels and common names.
The preprocessed data may have many missing entries depending on the list of
evidence available per feature vector. The present scheme makes it possible
for us to use any additional information available about an entry to make more
meaningful prediction about its nature. The number of new entries and the
learning based on them dynamically vary as learning progress. However, because
all updating is done statistically, the subtle uncertainties and errors in the
features do not affect the decision making process. The DBNN code is able to
directly train and test on the preprocessed data and we describe the
application of the algorithm to CRTS and stellar spectral studies in the
following section.
Real $\rightarrow$ | 1 | 2 | 3 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 16 | Total
---|---|---|---|---|---|---|---|---|---|---|---|---|---
Predicted | | | | | | | | | | | | |
1 | 273 | 14 | 4 | 2 | 1 | 4 | 3 | 3 | 6 | 1 | 0 | 0 | 311
2 | 15 | 447 | 7 | 3 | 6 | 2 | 4 | 5 | 1 | 4 | 1 | 2 | 497
3 | 1 | 1 | 47 | 0 | 1 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 51
5 | 0 | 0 | 0 | 68 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 68
6 | 0 | 1 | 1 | 1 | 144 | 1 | 1 | 3 | 0 | 1 | 0 | 0 | 153
7 | 0 | 0 | 0 | 0 | 0 | 33 | 0 | 0 | 0 | 0 | 0 | 0 | 33
8 | 0 | 0 | 0 | 0 | 0 | 0 | 5 | 0 | 0 | 0 | 0 | 0 | 5
9 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 15 | 0 | 0 | 0 | 0 | 15
10 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 9 | 0 | 0 | 0 | 10
11 | 2 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 51 | 0 | 0 | 53
12 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 5 | 0 | 5
16 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0
Total | 291 | 463 | 59 | 75 | 152 | 40 | 13 | 26 | 17 | 57 | 6 | 2 | 1201
Table 2: The confusion matrix of the class labels: humans (Horizontal axis)
against predicted (Vertical).
## 4 Results
The CRTS data that was used for this study has 12 distinct classification
labels enumerated in Table 1 with their numerical class labels followed by the
common names. The input sequence used for this study had 39 input features
(including detection magnitudes, colors from archival photometry, distance to
nearest star/galaxy etc.) of which 50 – 80 percent were missing in some cases.
None of the cases had all 39 inputs. This forms only a subset of the data that
human experts usually use for discrimination. We have not yet translated all
the available information that human experts use into machine recognizable
format (for example, archival light curves). This is something we want to do
in the near future.
Despite these limitations, the classifier agrees with human experts in more
than 90% of the cases as the graduated training progresses and it learns about
the variety in the evidence. This is shown in the confusion matrix Table 2.
The confusion matrix is a convenient representation showing all relevant
information such as how many objects in each class were correctly identified
and into which classes they were incorrectly labeled etc. It may also be used
to have a rough estimate of the number density of the different classes by
normalizing the totals in each class (last row) by the total number of objects
(last column of the last row).
Num. Code | Object Type | Total Objects | Completeness (%) | Contamination (%)
---|---|---|---|---
1 | Cataclysmic Variable | 291 | 93.8 | 12.2
2 | Supernova | 463 | 96.5 | 10.1
3 | Other | 59 | 79.7 | 7.8
5 | Blazar Outburst | 75 | 90.7 | 0.0
6 | Active Galactic Nucleus Variability | 152 | 94.7 | 5.9
7 | UVCeti Variable | 40 | 82.5 | 0.0
8 | Asteroid | 13 | 38.5 | 0.0
9 | Variable | 26 | 57.7 | 0.0
10 | Mira Variable | 17 | 52.9 | 10
11 | High Proper Motion Star | 57 | 89.5 | 3.7
12 | Comet | 6 | 83.3 | 0.0
16 | Nova | 2 | 0.0 | 0.0
Table 3: Completeness and contamination details of predictions by the DBNN classifier based on Table 2 for the different type of objects in CRTS survey are shown. Num. Code | Object Type | Total Objects | Completeness (%) | Contamination (%)
---|---|---|---|---
1 | Cataclysmic Variable | 61 | 63.9 | 35.0
2 | Supernova | 74 | 74.3 | 40.9
3 | Other | 14 | 0.0 | 100
5 | Blazar Outburst | 9 | 55.6 | 16.7
6 | Active Galactic Nucleus Variability | 26 | 76.9 | 48.7
7 | UVCeti Variable | 4 | 25.0 | 0.0
8 | Asteroid | 1 | 0.0 | 0.0
9 | Variable | 3 | 0.0 | 0.0
10 | Mira Variable | 2 | 0.0 | 0.0
11 | High Proper Motion Star | 5 | 0.0 | 0.0
12 | Comet | 1 | 0.0 | 0.0
16 | Nova | 0 | 0.0 | 0.0
Table 4: Classification of transients from one of the test samples. Possible
reasons for relatively lower completeness are detailed in the text.
The performance of the classifier was quantified in terms of contamination and
completeness. The fraction of the total number of objects in a class that were
correctly recognized by the classifier is referred to as completeness and the
fraction that comes as contaminants into a class due to incorrect labelling by
the classifier is referred to as contamination. Both measures are indicative
of how reliable the classifier is. Table 3 shows the completeness and
contamination of the classifier predictions for the CRTS data taking the
verdict of the human expert as reference.
As explained earlier, the classifier evaluates the Bayesian posterior
probability that can be considered synonymous to the true confidence the
classifier predicts. A graphical representation of the completeness and
confidence measures gives a more intuitive picture of the classifier
performance (Fig. 1). It may be noted that for most of the failed cases, the
confidence was low indicating that they are not as reliable as the rest. It is
advisable to use this as a guide to determine how safe it is to rely on low
confidence predictions and to draw a cut-off line which states that anything
below is unreliable. Quantification of this is in progress as more data come
in.
We have also carried out tests with training the method on a smaller subset
and testing on feature vectors never seen before. Due to the sheer variety in
features, and the fact that a large majority of them are missing, the
performance worsens (Table 4). One other important reason for this is that a
few of the classes, like asteroid, comet and high proper motion stars (HPM)
are not static and hence the archives do not have any information about them
(at the discovery location). To incorporate that, additional features will
have to be introduced. Even for many of the other classes there are not enough
examples as yet. Including archival light curves will help greatly improve the
SNe/CV dichotomy. All that is part of on going research, and that should be
kept in mind when judging the current results.
Figure 1: The plots show the performance of the classifier in predicting the
nature of the objects in the CRTS survey. In the plot to the left, black
colour is used to represent the total count of objects in each class while
blue colour is used to represent the number of correctly recognized objects
and red colour is used to indicate the number of contaminants in each class.
The plot to the right shows the same with percent Bayesian confidence and the
counts (in log scale to improve clarity).
As another example, the classifier was used on stellar spectra to classify
them into 98 different classes as given in the Indo-US Stellar
library444http://www.noao.edu/cflib/ (Valdes et al., 2004). The input features
used were the major absorption lines in the spectra and the maximum flux
values from four regions of the continuum in a window of 200Å centered around
wavelengths 3700, 4500, 6300, 8500Å, respectively. These spectra have a
coverage from 3460Å to 9464Å with a few that have missing bands in between. It
was taken with a 0.9m Coud$e^{{}^{\prime}}$ telescope at Kitt Peak National
Observatory in five different grating settings.
The sequence length of the extracted features used by us for this study had 32
most distinctive features in the spectra quantified by their equivalent widths
and the said four flux values. We used 958 examples for training the
classifier and 1104 examples for testing the learning. It was found that 88%
of the classifications were in agreement with the classifications given in the
catalog. The purpose of this work was only to demonstrate that the method
might find some application in spectral analysis, especially for chemical
abundance measures in the stellar atmosphere at a larger scale than what is
possible otherwise.
## 5 Conclusions
We describe a new method to make use of diverse and sparse information from
various sources to classify astronomical data. One practical application we
found is in the case of transients where alerts need to be sent to astronomers
to carry out follow up observations whenever an object that is likely to be of
interest to them is detected. However, it is possible to extend the method to
other applications such as spectroscopic classification where it is difficult
to predefine important absorption/emission lines and we want to cluster them
in a high dimensional space.
## References
* Djorgovski et al. (2011a) Djorgovski, S. G., Donalek, C., Mahabal, A. A., et al. 2011a, in: Proc. CIDU 2011 Conf., eds. A. Srivasatva, et al., p. 174, NASA Ames Res. Ctr.
* Djorgovski et al. (2011b) Djorgovski, S. G., Drake, A. J., Mahabal, A. A., et al. 2011b, arXiv:1102.5004
* Drake et al. (2009) Drake, A. J., Djorgovski, S. G., Mahabal, A., et al. 2009, ApJ, 696, 870
* Mahabal et al. (2011) Mahabal, A. A., Djorgovski, S. G., Drake, A. J., et al. 2011, BASI, 39, 387
* Mahabal et al. (2012) Mahabal, A. A., Donalek, C., Djorgovski, S. G., et al. 2012, arXiv:1111.3699, in: Proc. IAU Symp. 285, New Horizons in Time Domain Astronomy, eds. E. Griffin et al., CUP, in press
* Philip & Joseph (2000) Philip, N. S., Joseph, K. B.,2000, Journal of Intelligent Data Analysis, 4, 463
* Philip (2009) Philip, N. S., World Conference on Nature and Biologically Inspired Computing,
(NaBIC-2009), IEEE, ISBN: 978-1-4244-5612-3.
* Philip (2010) Philip, N. S., A Learning Algorithm based on High School Teaching Wisdom,
Paladyn Journal of Behavioral Robotics, 2010, 1(3), 160
* Valdes et al. (2004) Valdes, F., Gupta, R., Rose, J. A., Singh, H. P., & Bell, D. J. 2004, ApJS, 152, 251
* von Toussaint (2011) von Toussaint, U. 2011, Reviews of Modern Physics, 83, 943
* Williams et al. (2009) Williams, R. D., Djorgovski, S. G., Drake, A. J., Graham, M. J., & Mahabal, A. 2009,
Astronomical Data Analysis Software and Systems XVIII, 411, 115
## Acknowledgements
The work at Caltech has been supported in part by the NSF grants AST-0407448,
CNS-0540369, AST-0834235, AST-0909182 and IIS-1118041; the NASA grant
08-AISR08-0085; and by the Ajax and Fishbein Family Foundations. The first
author wishes to thank Prof. Ranjan Gupta for useful discussions and
acknowledges the use of Indo-US Library of Coud$e^{{}^{\prime}}$ Feed Stellar
Spectra.
|
arxiv-papers
| 2012-11-15T13:55:02 |
2024-09-04T02:49:38.054035
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "N. S. Philip, A. Mahabal, S. Abraham.R. Williams, S.G. Djorgovski, A.\n Drake, C Donalek, and M. Graham",
"submitter": "Ninan Sajeeth Philip",
"url": "https://arxiv.org/abs/1211.3607"
}
|
1211.3625
|
# Stochastic Analysis on Path Space over Time-Inhomogeneous Manifolds with
Boundary
Li-Juan Cheng 111Correspondence should be addressed to Li-Juan Cheng (E-mail:
[email protected])
(School of Mathematical Sciences, Beijing Normal University,
Laboratory of Mathematics and Complex Systems, Ministry of Education,
Beijing 100875, The People’s Republic of China)
E-mail: [email protected](L.J. Cheng)
###### Abstract
Let $L_{t}:=\Delta_{t}+Z_{t}$ for a $C^{1,1}$-vector field $Z$ on a
differential manifold $M$ with possible boundary $\partial M$, where
$\Delta_{t}$ is the Laplacian induced by a time dependent metric $g_{t}$
differentiable in $t\in[0,T_{c})$. We first introduce the damp gradient
operator, defined on the path space with reference measure $\mathbb{P}$, the
law of the (reflecting) diffusion process generated by $L_{t}$ on the base
manifold; then establish the integration by parts formula for underlying
directional derivatives and prove the log-Sobolev inequality for the
associated Dirichlet form, which is further applied to the free path spaces;
and finally, establish numbers of transportation-cost inequalities associated
to the uniform distance, which are equivalent to the curvature lower bound and
the convexity of the boundary.
Keywords: Metric flow, log-Sobolev inequality, integration by parts formula,
path space over manifolds with boundary, reflecting $L_{t}$-diffusion process
MSC(2010): 60J60, 58J65, 53C44
## 1 Introduction
Let $M$ be a $d$-dimensional differential manifold with possible boundary
$\partial M$ equipped with a complete Riemannian metric
$(g_{t})_{t\in[0,T_{c})}$, which is $C^{1}$ in $t$. For simplicity, we take
the notations: for $X,Y\in TM$,
$\displaystyle{\rm Ric}_{t}^{Z}(X,Y):={\rm
Ric}_{t}(X,Y)-\left<\nabla^{t}_{X}Z_{t},Y\right>_{t},\ \
\mathcal{G}_{t}(X,X):=\partial_{t}g_{t}(X,X)$
$\displaystyle\mathcal{R}_{t}^{Z}(X,Y):={\rm
Ric}_{t}^{Z}(X,Y)-\frac{1}{2}\mathcal{G}_{t}(X,Y),$
where ${\rm Ric}_{t}$ is the Ricci curvature tensor with respect to $g_{t}$,
$(Z_{t})_{t\in[0,T_{c})}$ is a $C^{1}$-family of vector fields, and
$\left<\cdot,\cdot\right>_{t}:=g_{t}(\cdot,\cdot)$. Define the second
fundamental form of the boundary by
$\mathbb{I}_{t}(X,Y)=-\left<\nabla^{t}_{X}N_{t},Y\right>_{t},\ \ X,Y\in
T\partial M,$
where $N_{t}$ is the inward unit normal vector field of the boundary
associated with $g_{t}$; $T\partial M$ is the tangent space of $\partial M$.
Consider the elliptic operator $L_{t}:=\Delta_{t}+Z_{t}$. Let $X_{t}$ be the
reflecting inhomogeneous diffusion process generated by $L_{t}$ (called
reflecting $L_{t}$-diffusion process). Assume that $X_{t}$ is non-explosive
before $T_{c}$. In this case, for any $0\leq S<T<T_{c}$, the distribution
$\Pi^{S,T}$ of $X_{[S,T]}:=\\{X_{t}:t\in[S,T]\\}$ is a probability measure on
the path space
$W^{S,T}:=C([S,T];M),$
when $S=0$, we write $W^{T}:=W^{0,T}$ and $\Pi^{T}:=\Pi^{0,T}$ for simplicity.
For each point $x\in M$, let $X_{[0,T]}^{x}=\\{X_{t}^{x}:0\leq t\leq
T<T_{c}\\}$ and $W_{x}^{T}=\\{\gamma\in W^{T}:\gamma_{0}=x\\}$. It has been
well-known that there is a strong connection between the behavior of the
distribution of the $M$-valued Brownian motion associated with metric $g$ and
the geometry of their underlying space. This paper is devoted to further the
study of this relation over a inhomogeneous manifold. Note that although our
discussions base on the manifold with boundary, the results are also new for
the manifold without boundary.
When the metric is independent of $t$, the theory about stochastic analysis on
the path space over a complete Riemannian manifold has been well developed
since Driver [9] proved the quasi-invariance theorem for the Brownian motion.
Then, integration by parts formula for the associated gradient operator,
induced by the quasi-invariant flow, was established, which leads to the study
of the functional inequalities with respect to the corresponding Dirichlet
form (see e.g. [16, 12]). For the case with boundary, see [18, 25] for the
corresponding results on manifold with boundary; see [3, 5, 11] for an
intertwining formula for the differential of Itô development map.
A probabilistic approach to these problem was initiated by Abandon et al, who
constructed $g_{t}$-Brownian motion on time-inhomogeneous space in [2].
Recently, Chen [6] gives the interwining formula and log-Sobolev inequality
for usual Dirichlet form (without damp) on the path space over time-
inhomogeneous manifold. The main purpose of this paper is to prove the
integration by parts formula and further establish the log-Sobolev inequality
w.r.t. the associated Dirichlet form, defined by the damped gradient operator.
Note that from technical point of view, our method relies on [18, 25].
Our second purpose is to discuss the Talagrand type transportation-cost
inequalities on the path space with respect to the uniform distance on time-
inhomogeneous space. In 1996, Talagrand [20] found that the
$L^{2}$-Wasserstein distance to the standard Gaussian measure can be dominated
by the square root of twice relative entropy on $M=\mathbb{R}^{d}$ with
constant metric. This inequality has been extended to distributions on finite-
and infinite-dimensional spaces. In particular, this inequality was
established on the path space of diffusion processes with respect to several
different distances (i.e. cost functions). See [14] for the details on the
Wiener space with the Cameron-Martin distance; see [21, 8] on the path space
of diffusions with the $L^{2}$-distance; see [22] on the Riemannian path space
with intrinsic distance induced by the Malliavin gradient operator, and [13,
27, 24] on the path space of diffusions with the uniform distance.
The rest parts of the paper are organized as follows. In the following two
sections, we construct the Hsu’s multiplicative functional and then define the
corresponding damped gradient operator, which satisfies an integration by
parts formula induced by intrinsic quasi-invariant flows. In Section 4, the
log-Sobolev inequality for the associated Dirichlet form is established and
extends to the free path space. In Section 5, some transportation-cost
inequalities are presented to be equivalent to the curvature condition and the
convexity of the boundary, and parts of these are extended to non-convex case
in finial section.
## 2 (Reflecting) $L_{t}$-diffusion process and multiplicative functional
Let $\mathcal{F}(M)$ be the frame bundle over $M$ and $\mathcal{O}_{t}(M)$ be
the orthonormal frame bundle over $M$ with respect to $g_{t}$. Let
$\mathbf{p}:\mathcal{F}(M)\rightarrow M$ be the projection from
$\mathcal{F}(M)$ onto $M$. Let $\\{e_{\alpha}\\}_{\alpha=1}^{d}$ be the
canonical orthonormal basis of $\mathbb{R}^{d}$. For any $u\in\mathcal{F}(M)$,
let $H^{t}_{i}(u)$ be the $\nabla^{t}$ horizontal lift of $ue_{i}$ and
$\\{V_{\alpha,\beta}(u)\\}_{\alpha,\beta=1}^{d}$ be the canonical basis of
vertical fields over $\mathcal{F}(M)$, defined by
$V_{\alpha,\beta}(u)=Tl_{u}(\exp(E_{\alpha,\beta}))$, where $E_{\alpha,\beta}$
is the canonical basis of $\mathcal{M}_{d}(\mathbb{R})$, the $d\times d$
matric space over $\mathbb{R}$, and
$l_{u}:Gl_{d}(\mathbb{R})\rightarrow\mathcal{F}(M)$ is the left multiplication
from the general linear group to $\mathcal{F}(M)$, i.e.
$l_{u}\exp(E_{\alpha,\beta})=u\exp(E_{\alpha,\beta})$.
Let $B_{t}:=(B_{t}^{1},B_{t}^{2},\cdots,B_{t}^{d})$ be a
$\mathbb{R}^{d}$-valued Brownian motion on a complete filtered probability
space $(\Omega,\\{\mathscr{F}_{t}\\}_{t\geq 0},\mathbb{P})$ with the natural
filtration $\\{\mathscr{F}_{t}\\}_{t\geq 0}$. Assume the elliptic generator
$L_{t}$ is a $C^{1}$ functional of time with associated metric $g_{t}$:
$L_{t}=\Delta_{t}+Z_{t}$
where $Z_{t}$ is a $C^{1,1}$ vector field on $M$. As in the time-homogeneous
case, to construct the $L_{t}$-diffusion process, we first construct the
corresponding horizontal diffusion process generated by
$L_{\mathcal{O}_{t}(M)}:=\Delta_{\mathcal{O}_{t}(M)}+H_{Z_{t}}^{t}$ by solving
the Stratonovich stochastic diffusion equation (SDE):
$\begin{cases}\mathrm{d}u_{t}=\sqrt{2}\displaystyle\sum_{i=1}^{d}H_{i}^{t}(u_{t})\circ\mathrm{d}B_{t}^{i}+H_{Z_{t}}^{t}(u_{t})\mathrm{d}t-\frac{1}{2}\displaystyle\sum_{i,j}\partial_{t}g_{t}(u_{t}e_{i},u_{t}e_{j})V_{i,j}(u_{t})\mathrm{d}t+H^{t}_{N_{t}}(u_{t})\mathrm{d}l_{t},\vskip
6.0pt plus 2.0pt minus 2.0pt\\\ u_{0}\in\mathcal{O}_{0}(M),\end{cases}$
where $\Delta_{\mathcal{O}_{t}(M)}$ is the horizontal Laplace operator on
$\mathcal{O}_{t}(M)$; $H_{Z_{t}}^{t}$ and $H^{t}_{N_{t}}$ are the $\nabla^{t}$
horizontal lift of $Z_{t}$ and $N_{t}$ respectively; $l_{t}$ is an increasing
process supported on $\\{t\geq 0:X_{t}:={\bf p}u_{t}\in\partial M\\}$. By a
similar discussion as in [2, Proposition 1.2], we see that the last term
promises $u_{t}\in\mathcal{O}_{t}(M)$. Since $(H_{Z_{t}}^{t})_{t\in[0,T_{c})}$
is a $C^{1,1}$-family vector field, it is well-known that (see e.g. [17]) the
equation has a unique solution up to the life time
$\zeta:=\displaystyle\lim_{n\rightarrow\infty}\zeta_{n}$, and
$\zeta_{n}:=\inf\\{t\in[0,T_{c}):\rho_{t}({\bf p}u_{0},{\bf p}u_{t})\geq
n\\},\ n\geq 1,\ \ \inf\varnothing=T_{c},$
where $\rho_{t}(x,y)$ is the distance between $x$ and $y$ associated with
$g_{t}$. Let $X_{t}={\bf p}u_{t}$. It is easy to see that $X_{t}$ solves the
equation
$\mathrm{d}X_{t}=\sqrt{2}u_{t}\circ\mathrm{d}B_{t}+Z_{t}(X_{t})\mathrm{d}t+N_{t}(X_{t})\mathrm{d}l_{t},\
\ X_{0}=x:={\bf p}u$
up to the life time $\zeta$. By the Itô formula, for any $f\in
C_{0}^{1,2}([0,T_{c})\times M)$ with $N_{t}f_{t}:=N_{t}f_{t}|_{\partial M}=0$,
$f(t,X_{t})-f(0,x)-\int_{0}^{t}\left({\partial_{s}}+L_{s}\right)f(s,X_{s})\mathrm{d}s=\sqrt{2}\int_{0}^{t}\left<u_{s}^{-1}\nabla^{s}f(s,\cdot)(X_{s}),\mathrm{d}B_{s}\right>$
is a martingale up to the life time $\zeta$. Here and what follows, we denote
the inner product on $\mathbb{R}^{d}$ by $\left<\cdot,\cdot\right>$, and write
$f(t,\cdot)=f_{t}$ for simplicity. So, we call $X_{t}$ the reflecting
diffusion process generated by $L_{t}$. When $Z_{t}\equiv 0$, then
$\tilde{X}_{t}:=X_{t/2}$ is generated by $\frac{1}{2}\Delta_{t}$ and is called
the reflecting $g_{t}$-Brownian motion on $M$. In what follows, we assume the
process is non-explosive.
### 2.1 Multiplicative functional
To construct the desired continuous multiplicative functional, we need the
following assumption.
1. (A)
There exist two constants $K,\sigma\in C([0,T_{c}))$ such that
$\mathcal{R}^{Z}_{t}\geq K(t),\ \mathbb{I}_{t}\geq\sigma(t)\ \mbox{and}\ \
\mathbb{E}e^{\lambda\int_{0}^{t}\sigma^{-}(s)\mathrm{d}l_{s}^{x}}<\infty$
holds for $\lambda>0$, $t\in[0,T_{c})$, and $x\in M$, where
$\sigma^{-}=0\vee(-\sigma)$.
To introduce Hsu’s discontinuous multiplicative functional, we need the lift
operators of $\mathcal{R}^{Z}_{t}$, ${\mathcal{G}}_{t}$, ${\mathbb{I}}_{t}$.
For any $u\in\mathcal{O}_{t}(M)$, let
$\mathcal{R}_{u}^{Z}(t)(a,b)=\mathcal{R}^{Z}_{t}(ua,ub)$ and
$\mathcal{G}_{u}(t)(a,b)=\mathcal{G}_{t}(ua,ub),\ a,b\in\mathbb{R}^{d}$. Let
${\bf p}_{\partial}:TM\rightarrow T\partial M$ be the orthogonal projection at
points on $(\partial M,g_{t})$. For any $u\in\mathcal{O}_{t}(M)$ with ${\bf
p}u\in\partial M$, let
$\mathbb{I}_{u}(t)(a,b)=\mathbb{I}_{t}({\bf p}_{\partial}ua,{\bf
p}_{\partial}ub),a,b\in\mathbb{R}^{d}.$
For $u\in\partial\mathcal{O}_{t}(M)$, the boundary of $\mathcal{O}_{t}(M)$,
let
${P}_{u}(t)(a,b)=\left<ua,N_{t}\right>_{t}\left<ub,N_{t}\right>_{t},a,b\in\mathbb{R}^{d}.$
For any $\varepsilon>0$ and $r\geq 0$, let $Q^{x,\varepsilon}_{r,t}$ solve the
following SDE on $\mathbb{R}^{d}\otimes\mathbb{R}^{d}$:
$\displaystyle\mathrm{d}Q^{x,\varepsilon}_{r,t}=-Q^{x,\varepsilon}_{r,t}\left\\{\mathcal{R}_{u^{x}_{t}}^{Z}(t)\mathrm{d}t+(\varepsilon^{-1}P_{u_{t}^{x}}(t)+\mathbb{I}_{u_{t}^{x}}(t))\mathrm{d}l^{x}_{t}\right\\}.$
(2.1)
When the metric is independent of $t$ and $M$ is compact, letting
$\varepsilon\downarrow 0$, the process $Q^{x,\varepsilon}_{r,t}$ converges in
$L^{2}$ to an adapted right-continuous process $Q^{x}_{r,t}$ with left limit,
such that $Q^{x}_{r,t}P_{u^{x}_{t}}(t)=0$ if $X^{x}_{t}\in\partial M$ (see
[18, Theorem 3.4]). Here, we follow Wang [26, Theorem 4.1.1], and introduce a
slightly different but simpler construction of the multiplicative functional
by solving a random integral equation on
$\mathbb{R}^{d}\otimes\mathbb{R}^{d}$.
###### Theorem 2.1.
Assume ${\bf(A)}$, then
1. $(1)$
for any $x\in M$, $0\leq r\leq t<T_{c}$ and $u_{0}^{x}\in\mathcal{O}_{0}(M)$,
the equation
$Q^{x}_{r,t}=\left(I-\int_{r}^{t}Q_{r,s}^{x}\mathcal{R}^{Z}_{u_{s}^{x}}(s)\mathrm{d}s-\int_{s}^{t}Q_{r,s}^{x}\mathbb{I}_{u_{s}^{x}}(s)\mathrm{d}l^{x}_{s}\right)(I-{1}_{\\{X_{t}^{x}\in\partial
M\\}}P_{u_{t}^{x}}(t))$
has a unique solution;
2. $(2)$
for any $0\leq r\leq t<T_{c}$, $\|Q_{r,t}^{x}\|\leq
e^{-\int_{r}^{t}K(s)\mathrm{d}s-\int_{r}^{t}\sigma(s)\mathrm{d}l^{x}_{s}}$
a.s., where $\|\cdot\|$ is the operator norm for $d\times d$-matrices;
3. $(3)$
for any $0\leq r\leq s\leq t<T_{c}$, $Q^{x}_{r,t}=Q^{x}_{r,s}Q^{x}_{s,t}$
a.s..
###### Proof.
We only consider the existence of the solution up to a arbitrarily given time
$T\in(r,T_{c})$, since the uniqueness is obvious. In the following, we drop
the superscript $x$ for simplicity. By the assumption $\mathbf{(A)}$ and
(2.1), we have
$\displaystyle\|Q_{r,t}^{\varepsilon}\|^{2}\leq
1-2\int_{r}^{t}\|Q_{r,s}^{\varepsilon}\|^{2}K(s)\mathrm{d}s-2\int_{r}^{t}\|Q_{r,s}^{\varepsilon}\|^{2}\sigma(s)\mathrm{d}l_{s}-\frac{2}{\varepsilon}\int_{s}^{t}\|Q_{r,s}^{\varepsilon}P_{u_{s}}(s)\|^{2}\mathrm{d}l_{s},\
\ t>r.$
Therefore, we obtain $\|Q_{r,t}^{\varepsilon}\|^{2}\leq
e^{-2\int_{r}^{t}K(s)\mathrm{d}s-2\int_{r}^{t}\sigma(s)\mathrm{d}l_{s}},\ \
t\geq r$ and
$\displaystyle\int_{r}^{T}\|Q^{\varepsilon}_{r,s}P_{u_{s}}(s)\|^{2}\mathrm{d}l_{s}\leq\frac{\varepsilon}{2}\left[1+2\left(\int_{r}^{T}K^{-}(s)\mathrm{d}s+\int_{r}^{T}\sigma^{-}(s)\mathrm{d}l_{s}\right)e^{2\int_{r}^{T}K^{-}(s)\mathrm{d}s+\int_{r}^{T}\sigma^{-}(s)\mathrm{d}l_{s}}\right].$
(2.2)
Combining this with ${\bf(A)}$, we obtain
$\displaystyle\lim_{\varepsilon\rightarrow
0}\mathbb{E}\int_{r}^{T}\|Q_{r,s}^{\varepsilon}P_{u_{s}}(s)\|^{2}\mathrm{d}l_{s}=0$
(2.3)
and
$\sup_{\varepsilon\in(0,1)}\mathbb{E}\int_{r}^{T}\|Q_{r,t}^{\varepsilon}\|^{2}(\mathrm{d}t+\mathrm{d}l_{t})<\infty.$
Because of the latter, we may select a sequence $\varepsilon_{n}\downarrow 0$
and an adapted process $\overline{Q}_{r,\cdot}\in
L^{2}(\Omega\times[r,T])\rightarrow\mathbb{R}^{d}\otimes\mathbb{R}^{d};\mathbb{P}\times(\mathrm{d}t+\mathrm{d}l_{t})$,
such that for any $g\in
L^{2}(\Omega\times[r,T])\rightarrow\mathbb{R}^{d};\mathbb{P}\times(\mathrm{d}t+\mathrm{d}l_{t})$,
$\lim_{n\rightarrow\infty}\mathbb{E}\int_{r}^{T}Q^{\varepsilon_{n}}_{r,t}g_{t}(\mathrm{d}t+\mathrm{d}l_{t})=\mathbb{E}\int_{r}^{T}\overline{Q}_{r,t}g_{t}(\mathrm{d}t+\mathrm{d}l_{t}).$
The following discussion is almost the same as the case with constant metric,
see [25] for details. ∎
The following result is a direct conclusion of Theorem 2.1.
###### Proposition 2.2.
Assume (A). For any $\mathbb{R}^{d}$-valued continuous semi-martingale $h_{t}$
with
$1_{\\{X^{x}_{t}\in\partial M\\}}P_{u_{t}}(t)h_{t}=0,$
$\mathrm{d}Q^{x}_{r,t}h_{t}=\overline{Q}^{x}_{r,t}\mathrm{d}h_{t}-Q^{x}_{r,t}\mathcal{R}_{u_{t}^{x}}^{Z}(t)h_{t}\mathrm{d}t-Q^{x}_{r,t}\mathbb{I}_{u_{t}^{x}}(t)h_{t}\mathrm{d}l^{x}_{t},\
\ t\geq r,$
where
$\overline{Q}^{x}_{r,t}=\left(I-\int_{r}^{t}Q^{x}_{r,s}\mathcal{R}^{Z}_{u_{s}^{x}}(s)\mathrm{d}s-\int_{r}^{t}Q^{x}_{r,s}\mathbb{I}_{u_{s}^{x}}(s)\mathrm{d}l_{s}^{x}\right).$
###### Proof.
As $1_{\\{X^{x}_{t}\in\partial M\\}}P_{u_{t}}(t)h_{t}=0,$ and by Theorem 2.1,
we have
$Q^{x}_{r,t}h_{t}=\left(I-\int_{r}^{t}Q_{r,s}^{x}\mathcal{R}_{u_{s}^{x}}^{Z}(s)\mathrm{d}s+\frac{1}{2}\int_{r}^{t}Q_{r,s}^{x}\mathcal{G}_{u_{s}^{x}}(s)\mathrm{d}s-\int_{r}^{t}Q_{r,s}\mathbb{I}_{u_{s}^{x}}(s)\mathrm{d}l_{s}^{x}\right)h_{t}=\overline{Q}_{r,t}^{x}h_{t}.$
Then the proof is completed by using Itô formula. ∎
Recall that $\\{P_{r,t}\\}_{0\leq r\leq t<T_{c}}$ is the Neumann semigroup
generated by $L_{t}$. The following is a consequence of Proposition 2.2, which
is the derivative formula of the diffusion semigroup, known as Bismut-Elworthy
Li formula (see e.g. [3, 10]).
###### Corollary 2.3.
Assume ${\bf(A)}$. Let $f\in C^{\infty}_{b}(M)$. Then for any $0\leq
r<t<T_{c}$,
$[r,t]\ni s\rightarrow
Q^{x}_{r,s}(u^{x}_{s})^{-1}\nabla^{s}P_{s,t}f(X^{x}_{s})$
is a martingale. Consequently,
$\displaystyle(u_{r}^{x})^{-1}\nabla^{r}P_{r,t}f(X_{r}^{x})=\mathbb{E}(Q^{x}_{r,t}(u_{t}^{x})^{-1}\nabla^{t}f(X_{t}^{x})|\mathscr{F}_{r}),$
(2.4)
and for any adapted $\mathbb{R}_{+}$-valued precess $\xi$ satisfying
$\xi(r)=0,\xi(t)=1$, and
$\mathbb{E}(\int_{r}^{t}\xi^{\prime}(s)^{2}\mathrm{d}s)^{\alpha}<\infty$ for
$\alpha>1/2$, there holds
$(u_{r}^{x})^{-1}\nabla^{r}P_{r,t}f(X_{r}^{x})=\frac{1}{\sqrt{2}}\mathbb{E}\left(f(X_{t}^{x})\int_{r}^{t}\xi^{\prime}(s)(Q_{r,s}^{x})^{*}\mathrm{d}B_{s}\big{|}\mathscr{F}_{r}\right).$
###### Proof.
The proof is essentially due to [25, Corollary 3.4]. Without losing
generality, we assume $r=0$ and drop the superscript $x$ for simplicity.
We first prove that $Q_{s}h_{s}$ is a martingale. Let
$h_{s}=(u_{s})^{-1}\nabla^{s}f(X_{s})$. Since $\nabla^{s}P_{s,t}f$ is vertical
to $N_{s}$ on $\partial M$, $1_{\\{X_{s}\in\partial M\\}}P_{u}(s)h_{s}=0$.
Then we have
$\mathrm{d}Q_{s}h_{s}=\overline{Q}_{s}\mathrm{d}h_{s}-Q_{s}\mathcal{R}_{u}^{Z}(s)h_{s}\mathrm{d}s-Q_{s}\mathbb{I}_{u}(s)h_{s}\mathrm{d}l_{s}.$
Let $F(u,s):=u^{-1}\nabla^{s}P_{s,t}f(\mathbf{p}u),\ u\in\mathcal{O}_{s}(M)$,
then
$\frac{\mathrm{d}}{\mathrm{d}s}F(u,s)=-u^{-1}\nabla^{s}P_{s,t}f({\bf
p}u)=-L_{\mathcal{O}_{s}(M)}F(\cdot,s)(u)+(\mathcal{R}_{u}^{Z}(s)+\frac{1}{2}\mathcal{G}_{u}(s))F(u,s),s\in[0,t].$
On the other hand, noting that
$\mathrm{d}u_{t}=\sqrt{2}\sum_{i=1}^{d}H_{i}^{t}\circ\mathrm{d}B_{t}^{i}+H^{t}_{Z_{t}}(u_{t})\mathrm{d}t-\frac{1}{2}\sum_{\alpha,\beta=1}^{d}\mathcal{G}_{\alpha,\beta}(t,u_{t})V_{\alpha,\beta}(u_{t})\mathrm{d}t+H^{t}_{N_{t}}(u_{t})\mathrm{d}l_{t}.$
By the Itô formula, we have
$\displaystyle\mathrm{d}F(u_{s},t_{0})=$
$\displaystyle\mathrm{d}M_{s}+L_{\mathcal{O}_{s}{M}}F(\cdot,t_{0})(u_{s})\mathrm{d}s+H_{N_{s}}^{s}F(\cdot,t_{0})(u_{s})\mathrm{d}l_{s}$
$\displaystyle-\frac{1}{2}\sum_{\alpha,\beta}\mathcal{G}_{\alpha,\beta}(s,u_{s})V_{\alpha,\beta}(u_{s})F(\cdot,t_{0})(u_{s})\mathrm{d}s.$
(2.5)
where
$\mathrm{d}M_{s}:=\sqrt{2}\sum_{i=1}^{d}H^{s}_{i}F(\cdot,t_{0})(u_{s})\mathrm{d}B_{s}^{i}.$
Therefore,
$\mathrm{d}h_{s}=\mathrm{d}M_{s}+\mathcal{R}_{u}^{Z}(s)h_{s}\mathrm{d}s+H_{N_{s}}^{s}F(\cdot,s)(u_{s})\mathrm{d}l_{s}.$
Since $1_{\\{X_{s}\in\partial M\\}}Q_{s}P_{u_{s}}(s)=0$, combining this with
(2.1), we obtain
$\mathrm{d}Q_{s}h_{s}=Q_{s}\mathrm{d}M_{s}+Q_{s}(I-P_{u_{s}}(s))\left\\{H_{N_{s}}^{s}F(\cdot,s)(u_{s})-\mathbb{I}_{u}{(s)}F(u_{s},s)\right\\}\mathrm{d}l_{s}.$
Noting that for any $e\in\mathbb{R}^{d}$, it follows from that when
$X_{s}\in\partial M$,
$\displaystyle\left<(I-P_{u_{s}}(s))H^{s}_{N_{s}}F(\cdot,s)(u_{s}),e\right>$
$\displaystyle={\rm
Hess}^{s}_{P_{s,t}f}(N_{s},\mathbf{p}_{\partial}u_{s}e)=\mathbb{I}_{s}(\nabla^{s}P_{s,t}f(X_{s}),\mathbf{p}_{\partial}u_{s}e)$
$\displaystyle=\mathbb{I}_{u}(s)(F(u,s),e)=\left<\mathbb{I}_{u}(s)F(u,s),e\right>,$
we conclude that
$(1-P_{u}(s))\\{H^{s}_{N_{s}}F(\cdot,s)(u_{s})-\mathbb{I}_{u}(s)F(u_{s},s)\\}\mathrm{d}l_{s}=0.$
Therefore, $Q_{s}h_{s}$ is a local martingale. By (A), it is then a martingale
according to [7, Theorem 4.9].
The following step is similar as shown in step (b) in the proof of [25, Lemma
3.3]. We skip it here.
∎
## 3 Damped Gradient, quasi-invariant flows and integration by Parts
When the metric is independent of $t$, the Malliavin derivative can be
realized by quasi-invariant flows for diffusion on manifolds (see e.g. [17,
25]). In this section, by using the multiplicative functional constructed in
§2.1, we first introduce the damped gradient operator as in [11], then
introduce quasi-invariant flows induced by SDEs with refection, and finally
link them by establishing an integration by parts formula.
### 3.1 Damped Gradient operator and quasi-invariant flows
We shall use multiplicative functionals $\\{Q^{x}_{r,t}:0\leq r\leq
t<T_{c}\\}$ to define the damped gradient operator for functionals of $X^{x}$.
Let
$\mathscr{F}C_{0}^{\infty}=\\{W^{T}\ni\gamma\rightarrow
f(\gamma_{t_{1}},\gamma_{t_{2}},\cdots,\gamma_{t_{n}}):n\geq
1,0<t_{1}<t_{2}<\cdots<t_{n}\leq T,f\in C_{0}^{\infty}(M^{n})\\}$
be the class of smooth cylindrical functions on $W^{T}$. Let
${\bf H}_{0}:=\left\\{h\in C([0,T];\mathbb{R}^{d}):h(0)=0,\|h\|_{{\bf
H}_{0}}:=\int_{0}^{T}|\dot{h}(s)|^{2}\mathrm{d}s<\infty\right\\}$
be the Cameron-Martin space on the flat path space. For any
$F\in\mathscr{F}C^{\infty}_{0}$ with
$F(\gamma)=f(\gamma_{t_{1}},\gamma_{t_{2}},\cdots,\gamma_{t_{n}})$, define the
damped gradient $D^{0}F(X_{[0,T]}^{x})$ as an ${\bf H}_{0}$-valued random
variable by setting $(D^{0}F(X^{x}_{[0,T]}))(0)=0$ and
$\frac{\mathrm{d}}{\mathrm{d}t}(D^{0}F(X^{x}_{[0,T]}))(t)=\sum_{i=1}^{n}1_{\\{t<t_{i}\\}}Q^{x}_{t,t_{i}}(u^{x}_{t_{i}})^{-1}\nabla^{t_{i}}_{i}f(X_{t_{1}}^{x},X_{t_{2}}^{x},\cdots,X_{t_{n}}^{x}),\
t\in[0,T].$
where $\nabla^{t_{i}}_{i}$ denotes the gradient operator w.r.t. the $i$-th
component associated with $g_{t_{i}}$. Then, for any ${\bf H}_{0}$-valued
random variable $h$, let
$D^{0}_{h}F(X_{[0,T]}^{x})=\left<D^{0}F(X_{[0,T]}),h\right>_{{\bf
H_{0}}}=\sum_{i=1}^{n}\int_{0}^{t_{i}}\left<(u_{t_{i}}^{x})^{-1}\nabla^{t_{i}}_{i}f(X_{t_{1}}^{x},X_{t_{2}}^{x},\cdots,X_{t_{n}}^{x}),(Q_{t,t_{i}}^{x})^{*}h^{\prime}(t)\right>\mathrm{d}t,$
We would like to indicate that the formulation of $D^{0}_{h}F$ is consistent
with [11] for the case with constant metric. Note that compared with usual
gradient operator, it contains the multiplicative functional, which affects
the log-Sobolev constant. This operator links $D^{0}_{h}F$ to the directional
derivative induced by a quasi-invariant flow. We now turn to investigating
this relation. The main idea essentially due to [18], where quasi-invariant
flows are constructed for constant manifold $M$ with boundary $\partial M$.
Let ${\bf\tilde{H}_{0}}$ be the set of all adapted elements in
$L^{2}(\Omega\rightarrow\mathbf{H}_{0};\mathbb{P})$; i.e.
${\bf\tilde{H}_{0}}=\\{h\in
L^{2}(\Omega\rightarrow{\bf{H}_{0}};\mathbb{P}):h(t)\ \mbox{is\ a}\
\mathscr{F}_{t}\mbox{-measurable},t\in[0,T]\\}.$
Then, ${\bf\tilde{H}_{0}}$ is a Hilbert space with inner product
$\big{<}h,\tilde{h}\big{>}_{{\bf\tilde{H}_{0}}}:=\mathbb{E}\big{<}h,\tilde{h}\big{>}_{{\bf
H_{0}}}=\mathbb{E}\int_{0}^{T}\big{<}h^{\prime}(t),\tilde{h}^{\prime}(t)\big{>}\mathrm{d}t,h,\
\tilde{h}\in{{\bf\tilde{H}_{0}}}.$
For $h\in\mathbf{\tilde{H}_{0}}$ and $\varepsilon>0$, let
$X^{\varepsilon,h}_{t}$ solve the SDE
$\displaystyle\mathrm{d}X_{t}^{\varepsilon,h}=\sqrt{2}u_{t}^{\varepsilon,h}\circ\mathrm{d}B_{t}+Z_{t}(X_{t}^{\varepsilon,h})\mathrm{d}t++\varepsilon\sqrt{2}u_{t}^{\varepsilon,h}h^{\prime}(t)\mathrm{d}t+N_{t}(X_{t}^{\varepsilon,h})\mathrm{d}l_{t}^{\varepsilon,h},$
(3.1)
where $l_{t}^{\varepsilon,h}$ and $u_{t}^{\varepsilon,h}$ are, respectively,
the local time on $\partial M$ and the horizontal lift on $\mathcal{O}_{t}(M)$
for $X_{t}^{\varepsilon,h}$. The detailed construction of
$X^{\varepsilon,h}_{t}$ is similar as in Section 2. To see that
$\\{X_{[0,T]}^{\varepsilon,h}\\}_{\varepsilon\geq 0}$ has the flow property
also in our setting, let
$\Theta:W_{0}:=\\{\omega\in C([0,T];\mathbb{R}^{d}):\omega_{0}=0\\}\rightarrow
W^{T}$
be measurable such that $X=\Theta(B)$, $B\in W_{0}$. For any $\varepsilon>0$
and a function $\Phi:W_{0}\rightarrow W^{T}$, let
$(\theta^{h}_{\varepsilon}\Phi)(\omega)=\Phi(\omega+\varepsilon h)$. Then
$X_{[0,T]}^{\varepsilon,h}=(\theta^{h}_{\varepsilon}\Theta)(B),\
\varepsilon>0$. Hence,
$X^{\varepsilon_{1}+\varepsilon_{2},h}_{[0,T]}=\theta^{h}_{\varepsilon_{1}}X^{\varepsilon_{2},h}_{[0,T]},\
\varepsilon_{1},\varepsilon_{2}\geq 0.$
Moreover, let us explain that the flow is quasi-invariant, i.e. for each
$\varepsilon\geq 0$, the distribution of $X^{\varepsilon,h}_{[0,T]}$ is
absolutely continuous w.r.t. that of $X^{x}_{[0,T]}$. Let
$R^{\varepsilon,h}=\exp{\left[\varepsilon\int_{0}^{T}\left<h^{\prime}(t),\mathrm{d}B_{t}\right>-\frac{\varepsilon^{2}}{2}\int_{0}^{T}|h^{\prime}(t)|^{2}\mathrm{d}t\right]}.$
By the Girsanov theorem
$B_{t}^{\varepsilon,h}:=B_{t}-\varepsilon h(t)$
is the $d$-dimensional Brownian motion under the probability
$R^{\varepsilon,h}\mathbb{P}$. Thus, the distribution of $X^{x}_{[0,T]}$ under
$R^{\varepsilon,h}$ coincides with that of $X^{\varepsilon,h}_{[0,T]}$ under
$\mathbb{P}$. Therefore, the map $X^{x}_{[0,T]}\rightarrow
X^{\varepsilon,h}_{[0,T]}$ is quasi-invariant. The quasi-invarient property
leads us to prove the following property.
###### Proposition 3.1.
Let $x\in M$ and $F\in\mathscr{F}C^{\infty}$. Then
$\displaystyle\lim_{\varepsilon\downarrow
0}\mathbb{E}\frac{F(X^{\varepsilon,h}_{[0,T]})-F(X_{[0,T]}^{x})}{\varepsilon}=\mathbb{E}\left\\{F(X_{[0,T]}^{x})\int_{0}^{T}\left<h^{\prime}(t),\mathrm{d}B_{t}\right>\right\\}$
holds for $h\in{\bf\tilde{H}}_{0,b}$, the set of all elements in
${\bf\tilde{H}}_{0}$ with bounded $\|h\|_{\tilde{\mathbf{H}}_{0}}$.
###### Proof.
As we have explained that $B_{t}^{\varepsilon,h}=B_{t}-\varepsilon h(t)$ is a
$d$-dimensional Brownian motion under $R^{\varepsilon,h}\mathbb{P}$. By the
weak uniqueness of (3.1), we conclude that the distribution of $X^{x}$ under
$R^{\varepsilon,h}\mathbb{P}$ coincide with that of $X^{\varepsilon,h}$ under
$\mathbb{P}$. In particular,
$\mathbb{E}F(X_{[0,T]}^{\varepsilon,h})=\mathbb{E}[R^{\varepsilon,h}F(X_{[0,T]}^{x})]$.
Thus, the assertion follows from
$\frac{\mathrm{d}R^{\varepsilon,h}}{\mathrm{d}\varepsilon}|_{\varepsilon=0}$
and the dominated convergence theorem since
$\\{R^{\varepsilon,h}\\}_{\varepsilon\in[0,1]}$ is uniformly integrable for
$h\in\tilde{{\bf H}}_{0,b}$. ∎
### 3.2 Integration by parts formula
In this section, an integration by parts formula for $D^{0}_{h}F$ is
established and applied to clarifying the link between this formula and the
derivative induced by the flow
$\\{X^{\varepsilon,h}_{[0,T]}\\}_{\varepsilon\geq 0}.$ The main result of this
subsection is presented as follows.
###### Theorem 3.2.
Assume ${\bf(A)}$. For any $x\in M$ and $F\in\mathscr{F}C_{0}^{\infty}$,
$\displaystyle\lim_{\varepsilon\downarrow
0}\mathbb{E}\frac{F(X^{\varepsilon,h}_{[0,T]})-F(X^{x}_{[0,T]})}{\varepsilon}=\mathbb{E}\\{D^{0}_{h}F\\}(X^{x}_{[0,T]})=\mathbb{E}\left\\{F(X^{x}_{[0,T]})\int_{0}^{T}\left<h^{\prime}(t),\mathrm{d}B_{t}\right>\right\\}$
(3.2)
holds for all $h\in{\bf\tilde{H}}_{0,b}$.
###### Proof.
By the Proposition 3.1, it is sufficient for us to prove the second equality.
The proof is similar to the constant metric case (see [25, Theorem 2.1]) due
to the following Lemmas 3.3 and 3.5 and the Markov property, . Note that the
second equality holds for all $h\in\tilde{{\bf H}}_{0}$. ∎
The following lemma gives the gradient formula for special cylinder functions.
###### Lemma 3.3.
For any $n\geq 1$, $0<t_{1}<\cdots<t_{n}\leq T$, and $f\in C^{\infty}(M^{n})$,
$\displaystyle(u_{t_{1}}^{x})^{-1}\nabla^{t_{1}}_{1}\mathbb{E}\\{f(X_{t_{1}}^{x},X_{t_{2}}^{x},\cdots,X_{t_{n}}^{x})|\mathscr{F}_{t_{1}}\\}=\sum_{i=1}^{n}\mathbb{E}\\{Q_{t_{1},t_{i}}^{x}(u_{t_{i}}^{x})^{-1}\nabla^{t_{i}}_{i}f(X_{t_{1}}^{x},X_{t_{2}}^{x},\cdots,X_{t_{n}}^{x})|\mathscr{F}_{t_{1}}\\}$
holds for all $x\in M$ and $u_{0}^{x}\in\mathcal{O}_{0}(M)$, where
$\nabla^{t_{i}}_{i}$ denotes the $g_{t_{i}}$-gradient w.r.t. the $i$-th
component.
###### Proof.
It is obvious that the assertion is true for $n=1$. By (2.4), we have
$(u_{t_{1}}^{x})^{-1}\nabla^{t_{1}}\mathbb{E}(f(X_{t_{2}})|\mathscr{F}_{t_{1}})=\mathbb{E}(Q_{t_{1},t_{2}}^{x}(u_{t_{2}}^{x})^{-1}\nabla_{2}^{t_{2}}f(X_{t_{2}})|\mathscr{F}_{t_{1}}).$
which plus the case of $n=1$, we prove the result for $n=2$. Assume that it
holds for $n=k$, $k\geq 2$. It remains to prove the case for $n=k+1$. To this
end, by Markov property, set
$g(X_{t_{1}}^{x},X_{t_{2}}^{x})=\mathbb{E}f(X_{t_{1}}^{x},X_{t_{2}}^{x},\cdots,X_{t_{k+1}}^{x}|\mathscr{F}_{t_{2}})$
By the assumption for $n=k$, we have
$\displaystyle(u_{t_{1}}^{x})^{-1}\nabla^{t_{1}}_{1}\mathbb{E}\left\\{g(X_{t_{1}},X_{t_{2}})|\mathscr{F}_{t_{1}}\right\\}$
$\displaystyle=$
$\displaystyle\mathbb{E}\left\\{(u_{t_{1}}^{x})^{-1}\nabla_{1}^{t_{1}}g(X_{t_{1}}^{x},X_{t_{2}}^{x})|\mathscr{F}_{t_{1}}\right\\}+\mathbb{E}\\{Q_{t_{1},t_{2}}(u_{t_{2}}^{x})^{-1}\nabla^{t_{2}}_{2}g(X_{t_{1}},X_{t_{2}})|\mathscr{F}_{t_{1}}\\}$
(3.3)
for $x\in M$ and $u_{0}\in\mathcal{O}_{0}(M)$. Fix the value of
$X_{t_{1}}^{x}=x_{0}$, by the assumption for $n=k$, we have
$(u_{t_{2}}^{x})^{-1}\nabla^{t_{2}}_{2}\mathbb{E}(f(x_{0},X_{t_{2}}^{x},\cdots,X_{t_{k+1}}^{x})|\mathscr{F}_{t_{2}})=\sum_{i=2}^{k+1}\mathbb{E}\left\\{Q_{t_{2},t_{i}}^{x}(u_{t_{i}}^{x})^{-1}\nabla_{i}^{t_{i}}f(x_{0},X^{x}_{t_{2}},\cdots,X^{x}_{t_{k+1}})|\mathscr{F}_{t_{2}}\right\\}.$
Combining this with (3.2), we have
$\displaystyle(u_{t_{1}}^{x})^{-1}\nabla^{t_{1}}_{1}\mathbb{E}\left\\{f(X_{t_{1}}^{x},X_{t_{2}}^{x},\cdots,X^{x}_{t_{k+1}})\bigg{|}\mathscr{F}_{t_{1}}\right\\}=(u_{t_{1}}^{x})^{-1}\nabla^{t_{1}}_{1}\mathbb{E}\\{g(X^{x}_{t_{1}},X^{x}_{t_{2}})|\mathscr{F}_{t_{1}}\\}$
$\displaystyle=\sum_{i=1}^{k+1}\mathbb{E}\left\\{Q^{x}_{t,t_{i}}(u^{x}_{t_{i}})^{-1}\nabla^{t_{i}}_{i}f(X^{x}_{t_{1}},X^{x}_{t_{2}},\cdots,X^{x}_{t_{k+1}})|\mathscr{F}_{t_{1}}\right\\}.$
(3.4)
∎
###### Remark 3.4.
Especially, choosing $t_{1}=0$, we arrive at
$\displaystyle(u_{0}^{x})^{-1}\nabla^{0}\mathbb{E}\\{f(X_{t_{1}}^{x},X_{t_{2}}^{x},\cdots,X_{t_{n}}^{x})\\}=\sum_{i=1}^{n}\mathbb{E}\left\\{Q^{x}_{t_{i}}(u_{t_{i}}^{x})^{-1}\nabla_{i}^{t_{i}}f(X_{t_{1}}^{x},X_{t_{2}}^{x},\cdots,X_{t_{n}}^{x})\right\\}.$
(3.5)
The following result is a direct consequence of (2.4) and the Itô formula for
$f(X_{t}^{x})$, i.e.
$f(X_{t}^{x})=f(x)+\sqrt{2}\int_{0}^{t}\left<(u_{s}^{x})^{-1}\nabla^{s}P_{s,t}f(X_{s}^{x}),\mathrm{d}B_{s}\right>.$
###### Lemma 3.5.
For any $n\geq 1$, $0<t_{1}<t_{2}<\cdots<t_{n}\leq T$, and $f\in
C^{\infty}(M^{n})$,
$\displaystyle\mathbb{E}\left\\{f(X_{t}^{x})\int_{0}^{t}\left<h_{s}^{\prime},\mathrm{d}B_{s}\right>\right\\}=\mathbb{E}\int_{0}^{t}\left<(u_{t}^{x})^{-1}\nabla^{t}f(X_{t}^{x}),(Q^{x}_{s,t})^{*}h_{s}\right>\mathrm{d}s,h\in\mathbf{\tilde{H}}_{0},t\in[0,T]$
holds for all $x\in M$ and $u_{0}^{x}\in\mathcal{O}_{0}(M)$.
## 4 The Log-Sobolev Inequality
When the metric is independent of $t$, log-Sobolov inequalities on $W_{x}^{T}$
were established independently by Hsu [18] and by Aida and Elworthy [1]. In
this section, we first consider the path space with a fixed initiated point,
then move to the free path space following an idea of [11], where the (non-
damped) gradient operator is studied on the free path space over the constant
manifolds without boundary.
### 4.1 Log-Sobolev inequality on $W^{T}_{x}$
Let $\Pi^{T}_{x}$ be the distribution of $X^{x}_{[0,T]}$. Let
$\mathscr{E}^{x}(F,G)=\mathbb{E}\left\\{\left<D^{0}F,D^{0}G\right>_{\bf
H_{0}}(X^{x}_{[0,T]})\right\\},\ F,\ G\in\mathscr{F}C_{0}^{\infty}.$
Since both $D^{0}F$ and $D^{0}G$ are functionals of $X$,
$(\mathscr{E}^{x},\mathscr{F}C_{0}^{\infty})$ is a positive bilinear form on
$L^{2}(W^{T}_{x};\Pi^{T}_{x})$. It is standard that the integration by parts
formula (3.2) implies the closability of the form, see Lemma 4.1. We shall use
$(\mathscr{E}^{x},\mathscr{D}(\mathscr{E}^{x}))$ to denote the closure of
$(\mathscr{E}^{x},\mathscr{F}C^{\infty}_{0})$. Moreover, (3.2) also implies
the Clark-Ocône type martingale representation formula, see Lemma 4.2, which
leads to the standard Gross log-Sobolev inequality (see e.g. [15]).
###### Lemma 4.1.
Assume ${\bf(A)}$. $(\mathscr{E}^{x},\mathscr{F}C_{0}^{\infty})$ is closable
in $L^{2}(W^{T}_{x};\Pi^{T}_{x})$.
###### Proof.
By the integration by part formula, the discussion is standard (see [25, Lemma
4.1]), we omit it here. ∎
The following result gives us the Clark-Ocône type martingale representation
formula for $F(X_{[0,T]})$. Following the proof of [25, Lemma 4.2] for the
case with constant metric, we have
###### Lemma 4.2.
(Clark-Haussman-Ocône Formula) Assume ${\bf(A)}$. For any
$F\in\mathscr{F}C_{0}^{\infty}$, let $\tilde{D}^{0}F(X_{[0,T]}^{x})$ be the
projection of $D^{0}F(X_{[0,T]}^{x})$ on $\mathbf{\tilde{H}}_{0}$, i.e.
$\frac{\mathrm{d}}{\mathrm{d}t}(\tilde{D}^{0}F(X_{[0,T]}^{x}))(t)=\mathbb{E}\left(\frac{\mathrm{d}}{\mathrm{d}t}(D^{0}F(X_{[0,T]}^{x}))\big{|}\mathscr{F}_{t}\right),\
t\in[0,T],\ (\tilde{D}^{0}F(X_{[0,T]}^{x}))(0)=0.$
Then
$F(X_{[0,T]}^{x})=\mathbb{E}F(X^{x}_{[0,T]})+\int_{0}^{T}\left<\frac{\mathrm{d}}{\mathrm{d}t}(\tilde{D}^{0}F(X_{[0,T]}^{x})(t),\mathrm{d}B_{t}\right>.$
It is standard that the martingale representation in Lemma 4.2 implies the
following log-Sobolev inequality. Since the parameter $T$ and the information
of Ricci curvature and the second fundamental form have been properly
contained in the Dirichlet form $\mathscr{E}$, the resulting log-Sobolev
constant is independent of $T$, $K$ and $\sigma$. Moreover, it is well-known
that the constant 2 in the inequality is sharp constant for compact manifolds
with constant metric.
###### Theorem 4.3.
Assume $\bf{(A)}$. For any $T>0$ and $x\in M$,
$(\mathscr{E}^{x},\mathscr{D}(\mathscr{E}^{x}))$ satisfies the following log-
Sobolev inequality,
$\Pi^{T}_{x}(F^{2}\log F^{2})\leq 2\mathscr{E}^{x}(F,F),\
F\in\mathscr{D}(\mathscr{E}^{x}),\ \Pi_{x}^{T}(F^{2})=1.$
###### Proof.
Due to lemma 4.1, it sufficient to prove the inequality for
$F\in\mathscr{F}C_{0}^{\infty}$. Let
$m_{t}:=\mathbb{E}\left(F(X^{x}_{[0,T]})^{2}\big{|}\mathscr{F}_{t}\right),\ \
t\in[0,T].$
By the Itô formula,
$\displaystyle\mathrm{d}m_{t}\log m_{t}=(1+\log
m_{t})\mathrm{d}m_{t}+\frac{|\frac{\mathrm{d}}{\mathrm{d}t}(\tilde{D}^{0}F(X_{[0,T]}^{x}))(t)|^{2}}{2m_{t}}\mathrm{d}t.$
Therefore,
$\displaystyle\Pi_{x}^{T}(F^{2}\log F^{2})$
$\displaystyle=\mathbb{E}_{x}m_{T}\log
m_{T}=\int_{0}^{T}\frac{2\mathbb{E}\left(F(X_{[0,T]}^{x})\frac{\mathrm{d}}{\mathrm{d}t}(D^{0}F(X_{[0,T]}^{x})(t))\
\big{|}\mathscr{F}_{t}\right)^{2}}{\mathbb{E}\left(F(X_{[0,T]}^{x})^{2}|\mathscr{F}_{t}\right)}\mathrm{d}t$
$\displaystyle\leq
2\int_{0}^{T}\mathbb{E}\left|\frac{\mathrm{d}}{\mathrm{d}t}(D^{0}F(X_{[0,T]}^{x}))(t)\right|^{2}\mathrm{d}t$
$\displaystyle=2\mathbb{E}\|D^{0}F(X_{[0,T]}^{x})\|^{2}_{{\bf H}_{0}}$
$\displaystyle=2\mathscr{E}^{x}(F,F).$
∎
Note that on manifolds without boundary equipped with time-depending metric,
the log-Sobolov inequality with respect to the Dirichlet form induced by usual
gradient derivative is recently established in [6]. And the log-Sobolov
constant is $2\exp\\{\sup_{t\in[0,T]}|\mathcal{R}_{t}^{Z}|\\}$.
### 4.2 Application to free path spaces
Let $\Pi^{T}_{\mu}$ be the distribution of the (reflecting) diffusion process
generated by $L_{t}:=\nabla^{t}+Z_{t}$ with initial distribution $\mu$ and
time-interval $[0,T]$. Due to the freedom of the initial point, it is natural
for us to make use of the following Cameron-Martin space:
${\bf H}=\left\\{h\in
C([0,T];\mathbb{R}^{d}):\int_{0}^{T}|h^{\prime}(t)|^{2}\mathrm{d}t<\infty\right\\}.$
Then ${\bf H}$ is a Hilbert space under the inner product
$\left<h_{1},h_{2}\right>_{\bf
H}=\left<h_{1}(0),h_{2}(0)\right>+\int_{0}^{T}\left<h^{\prime}_{1}(t),h^{\prime}_{2}(t)\right>\mathrm{d}t.$
To defined the damped gradient operator on the free path space, let
$\overline{\Omega}=M\times\Omega,\
\overline{\mathscr{F}}_{t}=\mathscr{B}(M)\times\mathscr{F}_{t},\ \mbox{and}\
\overline{\mathbb{P}}=\mu\times\mathbb{P}.$
Let $X_{0}(x,\omega)=x$ for $(x,\omega)\in M\times\Omega$. Then, under the
filtered probability space $(\overline{\Omega},\ \overline{\mathscr{F}}_{t},\
\overline{\mathbb{P}})$, $X_{t}(x,\omega):=X^{x}_{t}(\omega)$ is the
(reflecting) diffusion process generated by $L_{t}$ stating from $x$, and
$u_{t}(x,\omega):=u^{x}_{t}(\omega)$ is its horizontal lift. Moreover, let
$Q_{r,t}(x,\omega)=Q^{x}_{r,t}(\omega)$ for $0\leq r\leq t$, and write
$Q_{t}^{x}(\omega):=Q_{0,t}(x,\omega)$ for simplicity. Now, for any $F\in
FC^{\infty}_{0}$ with $F(\gamma)=f(\gamma_{t_{1}},\cdots,\gamma_{t_{n}})$, let
$\displaystyle
DF(X)=D^{0}F(X)+\sum_{i=1}^{n}Q_{t_{i}}u_{t_{i}}^{-1}\nabla^{t_{i}}_{i}f(X_{t_{1}},\cdots,X_{t_{n}}),$
(4.1)
where
$D^{0}F(X):=\sum_{i=1}^{n}\int_{0}^{t_{i}}Q_{t,t_{i}}u_{t_{i}}^{-1}\nabla^{t_{i}}_{i}f(X_{t_{1}},X_{t_{2}},\cdots,X_{t_{n}})\mathrm{d}t$
is the damped gradient on the path space with fixed initial point. Obviously,
$DF(X)\in L^{2}(\overline{\Omega}\rightarrow{\bf H};\overline{\mathbb{P}})$.
Define the Dirichlet form by
$\mathscr{E}^{\mu}(F,G)=\mathbb{E}_{\overline{\mathbb{P}}}\left<DF,DG\right>_{{\bf
H}},F,G\in\mathscr{F}C_{0}^{\infty}.$
We aim to prove that $(\mathscr{E}^{\mu},FC^{\infty}_{0})$ is closable in
$L^{2}(W^{T};\Pi^{T}_{\mu})$ and then establish the log-Sobolev inequality for
its closure $(\mathscr{E}^{\mu},\mathscr{D}(\mathscr{E}^{\mu}))$. To prove the
closability, we need the following two lemmas modified from [11]. Let
$\mathscr{H}_{0}(M)$ be the class of all smooth vector fields on $M$ with
compact support.
Let div${}^{0}_{\mu}$ be the divergence operator w.r.t. $\mu$, which is the
minus adjoint of $\nabla^{0}$ in $L^{2}(\mu)$; that is, for any smooth vector
field $U$,
$\displaystyle\int_{M}(Uf)\mathrm{d}\mu=-\int_{M}f({\rm
div}^{0}_{\mu}U)\mathrm{d}\mu,\ f\in C^{1}_{0}(M).$ (4.2)
###### Lemma 4.4.
Assume ${\bf(A)}$. For any $F\in\mathscr{F}C^{\infty}_{0}$,
$U\in\mathscr{H}_{0}(M)$, and $\overline{\mathscr{F}}_{t}$-adapted $h\in
L^{2}(\Omega\rightarrow{\bf H};\overline{\mathbb{P}})$,
$\mathbb{E}_{\overline{\mathbb{P}}}\left<DF(X),h+u_{0}^{-1}U(X_{0})\right>_{{\bf
H}}=\mathbb{E}_{\overline{\mathbb{P}}}\left\\{F(X)\left(\int_{0}^{T}\left<h^{\prime}(t),\mathrm{d}B_{t}\right>-({\rm
div}^{0}_{\mu}U)(X_{0})\right)\right\\}$
One can mimic the proof of [12], we omit it here. Due to this lemma, we have
the following result, the main idea is standard.
###### Theorem 4.5.
Assume ${\bf(A)}$. $(\mathscr{E}^{\mu},\mathscr{F}C^{\infty}_{0})$ is closable
in $L^{2}(W^{T};\Pi^{T}_{\mu})$, and its closure is symmetric Dirichlet form.
###### Proof.
It suffices to prove the closability. Let $\\{F_{n}\\}_{g\geq
1}\subset\mathscr{F}C_{0}^{\infty}$ such that
$\lim\limits_{n\rightarrow\infty}F_{n}=0$ in $L^{2}(W^{T};\Pi^{T}_{\mu})$ and
$V:=\lim\limits_{n\rightarrow\infty}DF_{n}(X)$ exists in
$L^{2}(\overline{\Omega}\rightarrow H;\overline{\mathbb{P}})$. We intend to
prove that $V=0$. Under the condition ${\bf(A)}$, by the decomposition of
identity, there is a smooth ONB $\\{U_{i}\\}_{i=1}^{d}$ for the tangent space.
Then for any $f\in C_{0}^{\infty}(M^{n})$, define
$\xi_{j}=\sum_{i=1}^{d}\left<u_{0}Q_{t_{i}}u_{t_{i}}^{-1}\nabla_{i}^{t_{i}}f(X_{t_{1}},X_{t_{2}},\cdots,X_{t_{n}}),U_{j}(X_{0})\right>_{0},\
\ j=1,2,\cdots,d,$
and there holds
$\displaystyle\sum_{i=1}^{n}Q_{t_{i}}u_{t_{i}}^{-1}\nabla^{t_{i}}_{i}f(X_{t_{1}},X_{t_{2}},\cdots,X_{t_{n}})=\sum_{i=1}^{\infty}\xi_{j}u_{0}^{-1}U_{j}(X_{0}).$
(4.3)
Combining this with (4.1), it suffices to prove
$\mathbb{E}_{\overline{\mathbb{P}}}\left<V,h+\xi
u_{0}^{-1}U(X_{0})\right>_{\bf H}=0$ for $\overline{\mathscr{F}}_{t}$-adapted
$h\in L^{2}(\overline{\Omega}\rightarrow{\bf H}_{0};\overline{\mathbb{P}}),\
\xi\in L^{2}(\overline{\Omega};\overline{\mathbb{P}})$ and
$U\in\mathscr{H}_{0}(M)$. Since $\mathscr{F}C_{0}^{\infty}$ is dense in
$L^{2}(W^{T};\Pi^{T}_{\mu})$, we may assume that $\xi=G(X)$ for some
$G\in\mathscr{F}C_{0}^{\infty}$. In this case, it follows from Lemma 4.4 and
Eq. (4.2) that
$\displaystyle\mathbb{E}_{\overline{\mathbb{P}}}\left<V,h+\xi
u_{0}^{-1}U(X_{0})\right>_{\bf
H}=\mathbb{E}_{\overline{\mathbb{P}}}\left<V(0),\xi
u_{0}^{-1}U(X_{0})\right>_{\bf
H}=\lim_{n\rightarrow\infty}\mathbb{E}_{\overline{\mathbb{P}}}\left<DF_{n}(X),G(X)u_{0}^{-1}U(X_{0})\right>_{\bf
H}$
$\displaystyle=\lim_{n\rightarrow\infty}\mathbb{E}_{\overline{\mathbb{P}}}\left\\{\left<\\{D(F_{n}G)(X)\\}(0),u_{0}^{-1}U(X_{0})\right>_{\bf
H}-F_{n}(X)\left<((DG)(X))(0),u_{0}^{-1}U(X_{0})\right>_{\bf H}\right\\}$
$\displaystyle=-\lim_{n\rightarrow\infty}\mathbb{E}_{\overline{\mathbb{P}}}\left\\{F_{n}(X)\left(G(X)({\rm
div}_{\mu}^{0}U)(X_{0})+\left<((DG)(X))(0),u_{0}^{-1}U(X_{0})\right>_{\bf
H}\right)\right\\}$ $\displaystyle=0.$
∎
By Theorems 4.3 and 4.5, we obtain the main result of this subsection as
follows.
###### Theorem 4.6.
Assume ${\bf(A)}$. If the log-Sobolev inequality
$\displaystyle\mu(f^{2}\log f^{2})\leq C\mu(|\nabla^{0}f|_{0}^{2}),\ f\in
C_{b}^{1}(M),\ \mu(f^{2})=1$ (4.4)
holds for some constant $C>0$, then
$\Pi^{T}_{\mu}(F^{2}\log F^{2})\leq(2\vee C)\mathscr{E}^{\mu}(F,F),\
F\in\mathscr{D}(\mathscr{E}^{\mu}),\ \Pi^{T}_{\mu}(F^{2})=1.$
###### Proof.
From Theorem 4.5, it suffice to prove $F\in\mathscr{F}C_{0}^{\infty}$. By
Theorem 4.3 and the condition (4.4), we have
$\displaystyle\Pi^{T}_{\mu}(F^{2}\log F^{2})$
$\displaystyle=\int_{M}\Pi_{x}^{T}(F^{2}\log F^{2})\mu(\mathrm{d}x)$
$\displaystyle\leq
2\int_{M}\mathscr{E}^{x}(F,F)\mu(\mathrm{d}x)+C\int_{M}\Pi_{x}^{T}(F^{2})\log\Pi_{x}^{T}(F^{2})\mu(\mathrm{d}x)$
$\displaystyle\leq 2\mathbb{E}_{\mathbb{P}}\|D^{0}F(X)\|^{2}_{{\bf
H}_{0}}+C\int_{M}|\nabla^{0}\sqrt{\mathbb{E}^{\cdot}F^{2}(X)}|_{0}^{2}\mathrm{d}\mu.$
(4.5)
Moreover, letting $F(X)=f(X_{t_{1}},X_{t_{2}},\cdots,X_{t_{n}})$, it follows
from Lemma 3.3 that
$\displaystyle|\nabla^{0}\sqrt{\mathbb{E}^{\cdot}F^{2}(X)}|_{0}^{2}$
$\displaystyle=\frac{|\mathbb{E}^{\cdot}F(X)\sum_{i=1}^{n}Q_{t_{i}}u_{t_{i}}^{-1}\nabla^{t_{i}}_{i}f(X_{t_{1}},\cdots,X_{t_{n}})|^{2}}{\mathbb{E}^{\cdot}F^{2}(X)}$
$\displaystyle\leq\mathbb{E}\big{|}\sum_{i=1}^{n}Q_{t_{i}}u_{t_{i}}^{-1}\nabla_{i}^{t_{i}}f(X_{t_{1}},\cdots,X_{t_{n}})\big{|}^{2}.$
(4.6)
Combining (4.2) with (4.2), we complete the proof. ∎
## 5 Transportation-cost inequalities on path spaces over convex manifolds
The main purpose of this section is to investigate the Talagrand type
inequalities on the path space $W^{S,T}:=C([S,T];M),\ 0\leq S<T<T_{c}$ of the
(reflecting) diffusion processes on the manifold with convex boundary under
$g_{t}$, $t\in[0,T_{c})$. Let $X_{t}$ be the (reflecting if $\partial
M\neq\varnothing$) diffusion process generated by $L_{t}$ with initial
distribution $\mu\in\mathscr{P}(M)$. Assume that $X_{t}$ is non-explosive,
which is the case if
$\mathcal{R}_{t}^{Z}\geq K(t),\,\mbox{for\ some}\ K\in C([0,T_{c})),\
\mathbb{I}_{t}\geq 0.$ (5.1)
We call the metric flow is convex flow if $(\partial M,g_{t})$ keeps convex,
i.e. $\mathbb{I}_{t}\geq 0$. Let $\Pi^{S,T}_{\mu}$ be the distribution of
$X_{[S,T]}:=\\{X_{t}:t\in[S,T]\\}$, $0\leq S<T<T_{c}$, which is a probability
measure on the (free) path space $W^{S,T}$. When $\mu=\delta_{x}$, we denote
$\Pi_{\delta_{x}}^{S,T}=\Pi_{x}^{S,T}$. For any nonnegative measurable
function $F$ on $W^{S,T}$ such that $\Pi^{S,T}_{\mu}(F)=1$, one has
$\displaystyle\mu_{F}^{S,T}(\mathrm{d}x):=\Pi^{S,T}_{x}(F)\mu(\mathrm{d}x)\in\mathscr{P}(M).$
(5.2)
Consider the uniform distance on $W^{S,T}$:
$\rho_{\infty}(\gamma,\eta):=\sup_{t\in[S,T]}\rho(\gamma_{t},\eta_{t}),\gamma,\eta\in
W^{S,T}.$
Let $W^{\rho_{\infty}}_{2}$ be the $L^{2}$-Wasserstein distance (or
$L^{2}$-transportation cost) induced by $\rho_{\infty}$. In general, for any
$p\in[1,\infty)$ and for two probability measures $\Pi_{1},\Pi_{2}$ on
$W^{S,T}$,
$W^{\rho_{\infty}}_{p}(\Pi_{1},\Pi_{2}):=\inf_{\pi\in\mathscr{C}(\Pi_{1},\Pi_{2})}\left\\{\int_{W^{S,T}\times
W^{S,T}}\rho_{\infty}(\gamma,\eta)^{p}\pi(\mathrm{d}\gamma,\mathrm{d}\eta)\right\\}^{1/p}$
is the $L^{p}$-Wasserstein distance (or $L^{p}$-transportation cost) of
$\Pi_{1}$ and $\Pi_{2}$, induced by the uniform norm, where
$\mathscr{C}(\Pi_{1},\Pi_{2})$ is the set of all couplings for $\Pi_{1}$ and
$\Pi_{2}$. Similarly the $L^{p}$-Wasserstein distance of $\mu$ and $\nu$
induced by $g_{t}$-distance defined by
$W_{p,t}(\nu,\mu)=\inf_{\eta\in\mathscr{C}(\nu,\mu)}\left\\{\int_{M\times
M}\rho_{t}(x,y)^{p}\mathrm{d}\eta(x,y)\right\\}^{1/p}.$
The following Theorem 5.2 provides that there are some transportation-cost
inequalities to be equivalent to the lower bound of $\mathcal{R}^{Z}_{t}$ and
the convexity of $(\partial M,g_{t})$, $t\in[0,T_{c})$ (when $\partial
M\neq\varnothing$). To prove this result, we need the following lemma due to
[OW].
###### Lemma 5.1.
Let $\mu$ be a probability measure on $M$ and $f\in C^{2}_{b}(M)$ such that
$\mu(f)=0$. For small enough $\varepsilon>0$ such that
$f_{\varepsilon}:=1+\varepsilon f\geq 0$, there holds
$\mu(f^{2})\leq\frac{1}{\varepsilon}\sqrt{\mu{(|\nabla^{s}f|_{s}^{2})}}W_{2,s}(f_{\varepsilon}\mu,\mu)+\frac{\|{\rm
Hess}^{s}_{f}\|_{\infty}}{2\varepsilon}W_{2,s}(f_{\varepsilon}\mu,\mu)^{2},$
where $\|{\rm Hess}^{s}_{f}\|_{\infty}=\sup_{x\in M}\|{\rm
Hess}^{s}_{f}\|_{op}$ for $\|\cdot\|_{op}$ the operator norm in
$\mathbb{R}^{d}$.
The main result of the section is presented as follows.
###### Theorem 5.2.
Let $P_{S,T}(x,\cdot)$ be the distribution of $X_{T}$ with conditional
$X_{S}=x$. Denote the corresponding inhomogeneous semigroup by
$\\{P_{S,T}\\}_{0\leq S\leq T<T_{c}}$. For any $p\in[1,\infty)$, the following
statements are equivalent to each other:
1. $(1)$
(5.1) holds.
2. $(2)$
For any $0\leq S\leq T<T_{c}$, $\mu\in\mathscr{P}(M)$ and nonnegative $F$ with
$\Pi^{S,T}_{\mu}(F)=1$,
$W_{2}^{\rho_{\infty}}(F\Pi^{S,T}_{\mu},\Pi^{S,T}_{\mu^{S,T}_{F}})\leq
4C(S,T,K)\Pi^{S,T}_{\mu}(F\log F)$
holds, where $\mu^{S,T}_{F}\in\mathscr{P}(M)$ is fixed by $(\ref{def2})$ and
$C(S,T,K):=\sup\limits_{t\in[S,T]}\int_{S}^{t}e^{-2\int_{u}^{t}K(r)\mathrm{d}r}\mathrm{d}u$,
which will keep the same meaning in $(3),(7)$ and $(8)$.
3. $(3)$
For any $x\in M$ and $0\leq S\leq T<T_{c}$,
$W^{\rho_{\infty}}_{2}(F\Pi_{x}^{S,T},\Pi_{x}^{S,T})^{2}\leq
4C(S,T,K)\Pi_{x}^{S,T}(F\log F),\ F\geq 0,\ \Pi_{x}^{T}(F)=1.$
4. $(4)$
For any $x\in M$ and $0\leq S\leq T<T_{c}$,
$W_{2,T}(P_{S,T}(x,\cdot),fP_{S,T}(x,\cdot))^{2}\leq
4\left(\int_{S}^{T}e^{-2\int_{u}^{T}K(r)\mathrm{d}r}\mathrm{d}u\right)P_{S,T}(f\log
f)(x),f\geq 0,\ P_{S,T}f(x)=1.$
5. $(5)$
For any $x\in M$ and $0\leq S\leq T<T_{c}$,
$\displaystyle W_{2,T}(P_{S,T}(x,\cdot),fP_{S,T}(x,\cdot))^{2}\leq
4\left(\int_{S}^{T}e^{-2\int_{u}^{T}K(r)\mathrm{d}r}\mathrm{d}u\right)^{2}P_{S,T}$
$\displaystyle\frac{|\nabla^{T}f|_{T}^{2}}{f}(x),$
where $f\geq 1$ and $P_{S,T}f=1$.
6. $(6)$
For $0\leq S\leq T<T_{c}$ and $\mu,\nu\in\mathscr{P}(M)$,
$W^{\rho_{\infty}}_{p}(\Pi^{S,T}_{\mu},\Pi_{\nu}^{S,T})\leq
e^{-\int_{S}^{T}K(r)\mathrm{d}r}W_{p,S}(\mu,\nu).$
7. $(7)$
For any $0\leq S\leq T<T_{c}$, $\mu\in\mathscr{P}(M)$, and $F\geq 0$ with
$\Pi^{S,T}_{\mu}(F)=1$,
$W^{\rho_{\infty}}_{2}(F\Pi^{S,T}_{\mu},\Pi^{S,T}_{\mu})\leq
2\left\\{C(S,T,K)\Pi^{S,T}_{\mu}(F\log
F)\right\\}^{1/2}+e^{-\int_{S}^{T}K(r)\mathrm{d}r}W_{2,S}(\mu_{F}^{S,T},\mu).$
8. $(8)$
For any $\mu\in\mathscr{P}(M)$ and $C\geq 0$ such that
$W_{2,S}(f\mu,\mu)^{2}\leq C\mu(f\log f),\ f\geq 0,\ \mu(f)=1,$
there holds
$\displaystyle
W^{\rho_{\infty}}_{2}(F\Pi^{S,T}_{\mu},\Pi^{S,T}_{\mu})\leq\left(2\sqrt{C(S,T,K)}+\sqrt{C}e^{-\int_{S}^{T}K(r)\mathrm{d}r}\right)^{2}\Pi^{S,T}_{\mu}(F\log
F),$
for $F\geq 0,\ \Pi^{S,T}_{\mu}(F)=1$.
###### Proof.
By taking $\mu=\delta_{x}$, we have
$\mu_{F}^{T}=\Pi^{T}_{x}(F)\delta_{x}=\delta_{x}$. It is easy to see that (3)
follows from (2), (7) and (8). (4) follows from (3) by taking
$F(X_{[S,T]})=f(X_{T})$. (6) implies
$W_{p,S}(\delta_{x}P_{S,T},\delta_{y}P_{S,T})\leq
e^{-\int_{S}^{T}K(r)\mathrm{d}r}\rho_{S}(x,y)$
and thus implies (1) by [7, Theorem 5.3]. Moreover, it is clear that (8)
follows from (7) while (7) is implied by each of (2) and (6).
“(3)$\Rightarrow$(2)” is the same as explained in time-homogeneous case (see
[24, the proof of Theorem 1.1 (b)]). So it suffices to prove (1) $\Rightarrow$
(3), each of (4) and (5) $\Rightarrow$ (1), (1) $\Rightarrow$ (5), (1)
$\Rightarrow$ (6). Without loss generality, we assume $S=0$ for simplicity.
${(a)}\ \mathbf{(1)\ implies\ (3)}$ We shall only consider the case where
$\partial M$ is non-empty. For the case without boundary, the following
argument works well by taking $l_{t}=0$ and $N_{t}=0$. Simply denote
$X^{x}_{[0,T]}=X_{[0,T]}$. Let $F$ be a positive bounded measurable function
on $W^{T}$ such that $\inf F>0$ and $\Pi^{T}_{x}(F)=1$. Let
$\mathrm{d}\mathbb{Q}=F(X_{[0,T]})\mathrm{d}\mathbb{P}$. Since
$\mathbb{E}F(X_{[0,T]})=\Pi^{T}_{\mu}(F)=1$, $\mathbb{Q}$ is a probability
measure on $\Omega$. Then, with a similar discussion as in [25], we conclude
that there exists a unique $\mathscr{F}_{t}$-predict process $\beta_{t}$ on
$\mathbb{R}^{d}$ such that
$F(X_{[0,T]})=m_{T}=e^{\int_{0}^{T}\left<\beta_{s},\mathrm{d}B_{s}\right>-\frac{1}{2}\int_{0}^{T}\|\beta_{s}\|^{2}\mathrm{d}s}$
and
$\displaystyle\int_{0}^{T}\mathbb{E}_{\mathbb{Q}}\|\beta_{s}\|^{2}\mathrm{d}s=2\mathbb{E}F(X_{[0,T]}\log
X_{[0,T]}).$ (5.3)
Then by the Girsanov theorem,
$\tilde{B}_{t}:=B_{t}-\int_{0}^{t}\beta_{s}\mathrm{d}s,\ \ t\in[0,T]$ is a
$d$-dimensional Brownian motion under the probability measure $\mathbb{Q}$.
Let $Y_{t}$ solve the following SDE
$\displaystyle\mathrm{d}Y_{t}=\sqrt{2}P^{t}_{X_{t},Y_{t}}u_{t}\circ\mathrm{d}\tilde{B}_{t}+Z_{t}(Y_{t})\mathrm{d}t+N_{t}(Y_{t})\mathrm{d}\tilde{l}_{t},\
\ Y_{0}=x,$ (5.4)
where $P^{t}_{X_{t},Y_{t}}$ is the $g_{t}$-parallel displacement along the
minimal geodesic from $X_{t}$ to $Y_{t}$ and $\tilde{l}_{t}$ is the local time
of $Y_{t}$ on $\partial M$. As announced, under $\mathbb{Q}$, $\tilde{B}_{t}$
is a $d$-dimensional Brownian motion, the distribution of $Y_{[0,T]}$ is
$\Pi_{x}^{T}$.
On the other hand, since
$\tilde{B}_{t}=B_{t}-\int_{0}^{t}\beta_{s}\mathrm{d}s$, we have
$\displaystyle\mathrm{d}X_{t}=\sqrt{2}u_{t}\circ\mathrm{d}\tilde{B}_{t}+Z_{t}(X_{t})\mathrm{d}t+\sqrt{2}u_{t}\beta_{t}\mathrm{d}t+N_{t}(X_{t})\mathrm{d}l_{t}.$
(5.5)
Moreover, for any bounded measurable function $G$ on $W^{T}$,
$\mathbb{E}_{\mathbb{Q}}G(X_{[0,T]}):=\mathbb{E}(FG)(X_{[0,T]})=\Pi_{x}^{T}(FG).$
We conclude that the distribution of $X_{[0,T]}$ under $\mathbb{Q}$ coincides
with $F\Pi^{T}_{x}$. Therefore,
$\displaystyle W^{\rho_{\infty}}_{2}(F\Pi_{x}^{T},\Pi_{x}^{T})^{2}$
$\displaystyle\leq\mathbb{E}_{\mathbb{Q}}\rho_{\infty}(X_{[0,T]},Y_{[0,T]})^{2}=\mathbb{E}_{\mathbb{Q}}\max_{t\in[0,T]}\rho_{t}(X_{t},Y_{t})^{2}.$
(5.6)
By the convexity of $(\partial M,g_{t})$, we have
$\left<N_{t}(x),\nabla^{t}\rho_{t}(\cdot,y)(x)\right>_{t}=\left<N_{t}(x),\nabla^{t}\rho_{t}(y,\cdot)(x)\right>_{t}\leq
0.$
Combining this with (5.4) and (5.5), and by the Itô formula, we obtain from
$\mathcal{R}_{t}^{Z}\geq K(t)$ that
$\displaystyle\mathrm{d}\rho_{t}(X_{t},Y_{t})\leq$
$\displaystyle-K(t)\rho_{t}(X_{t},Y_{t})\mathrm{d}t+\sqrt{2}\left<u_{t}\beta_{t},\nabla^{t}\rho_{t}(\cdot,Y_{t})(X_{t})\right>_{t}\mathrm{d}t$
$\displaystyle\leq$
$\displaystyle(-K(t)\rho_{t}(X_{t},Y_{t})+\sqrt{2}\|\beta_{t}\|)\mathrm{d}t.$
(5.7)
Since $X_{0}=Y_{0}=x$, this implies
$\displaystyle\rho_{t}(X_{t},Y_{t})^{2}$ $\displaystyle\leq
e^{-2\int_{0}^{t}K(r)\mathrm{d}r}\left(\sqrt{2}\int_{0}^{t}e^{\int_{0}^{s}K(r)\mathrm{d}r}\|\beta_{s}\|\mathrm{d}s\right)^{2}$
$\displaystyle\leq
2e^{-2\int_{0}^{t}K(r)\mathrm{d}r}\int_{0}^{t}e^{2\int_{0}^{s}K(r)\mathrm{d}r}\mathrm{d}s\cdot\int_{0}^{t}\|\beta_{s}\|^{2}\mathrm{d}s,\
\ t\in[0,T].$ (5.8)
Therefore,
$\displaystyle\mathbb{E}_{\mathbb{Q}}\max_{t\in[0,T]}\rho_{t}(X_{t},Y_{t})^{2}\leq
2\max_{t\in[0,T]}\int_{0}^{t}e^{-2\int_{s}^{t}K(r)\mathrm{d}r}\mathrm{d}s\int_{0}^{t}\mathbb{E}_{\mathbb{Q}}\|\beta_{s}\|^{2}\mathrm{d}s.$
(5.9)
Therefore, (3) follows from (5.6) and (5.3).
${(b)}\ \mathbf{(4)\ implies\ (1)\ }$ Let $f\in C_{b}^{2}(M)$ such that
$P_{0,T}f(x)=0$. Then, for small $\varepsilon>0$ such that
$f_{\varepsilon}:=1+\varepsilon f\geq 0$, we have
$\displaystyle P_{0,T}(f_{\varepsilon}\log
f_{\varepsilon})=P_{0,T}\left\\{(1+\varepsilon f)\left(\varepsilon
f-\frac{1}{2}(\varepsilon
f)^{2}+o(\varepsilon^{2})\right)\right\\}(x)=\frac{\varepsilon^{2}}{2}P_{0,T}f^{2}(x)+o(\varepsilon^{2}).$
Combining with lemma 5.1 and (4), we obtain
$\displaystyle(P_{0,T}f^{2})^{2}(x)$ $\displaystyle\leq
4\int_{0}^{T}e^{-2\int_{u}^{T}K(r)\mathrm{d}r}\mathrm{d}u\cdot
P_{0,T}|\nabla^{T}f|^{2}_{T}(x)\cdot\lim_{\varepsilon\rightarrow
0}\frac{P_{0,T}f_{\varepsilon}\log f_{\varepsilon}(x)}{\varepsilon^{2}}$
$\displaystyle 4\int_{0}^{T}e^{-2\int_{u}^{T}K(r)\mathrm{d}r}\mathrm{d}u\cdot
P_{0,T}|\nabla^{T}f|^{2}_{T}(x)P_{0,T}f^{2}(x).$
This is equivalent to [7, Theorem 5.3] for $\sigma=0,p=2$ and continuous
function $K$. Therefore,by [7, Theorem 5.3] “(2)$\Leftrightarrow$ (1)”, (4)
implies (1).
${(c)}\mathbf{(5)\ is\ equivalent\ to\ (1)}$. Similarly to $\mathbf{(b)}$,
combining condition (5) with Lemma 5.1, we obtain
$\displaystyle P_{0,T}f^{2}(x)$ $\displaystyle\leq
2\int_{0}^{T}e^{-2\int_{s}^{T}K(r)\mathrm{d}r}\mathrm{d}s\sqrt{P_{0,T}|\nabla^{T}f|^{2}_{T}(x)}\lim_{\varepsilon\rightarrow
0}\sqrt{P_{0,T}\frac{|\nabla^{T}f_{\varepsilon}|^{2}_{T}}{f_{\varepsilon}\varepsilon^{2}}(x)}$
$\displaystyle=2\int_{0}^{T}e^{-2\int_{s}^{T}K(r)\mathrm{d}r}\mathrm{d}sP_{0,T}|\nabla^{T}f|^{2}_{T}.$
Hence, (1) holds.
On the other hand, due to $(1)\Rightarrow(4)$ and (5.1),
$\displaystyle P_{0,T}(f\log
f)(x)\leq\int_{0}^{T}e^{-2\int_{s}^{T}K(r)\mathrm{d}r}\mathrm{d}s\cdot
P_{0,T}\frac{|\nabla^{T}f|_{T}^{2}}{f}(x),\ \ f\geq 0,\ \ P_{0,T}f(x)=1,$
we conclude that (1) implies (5).
${(e)}\mathbf{(1)\ implies\ (2)}$. For any $x,y\in M$, there exists
$\Pi_{x,y}\in\mathscr{C}(\Pi_{x}^{T},\Pi_{y}^{T})$ such that
$\int_{W^{T}\times W^{T}}\rho_{\infty}^{p}\mathrm{d}\Pi_{x,y}\leq
e^{-p\int_{0}^{T}K(r)\mathrm{d}r}\rho_{0}(x,y)^{p}.$
The following discussion is similar with the constant matric case (see [24,
Theorem 1.1]). ∎
## 6 Extension to non-convex setting
In this section, we first consider $L_{t}=\psi^{2}_{t}(\nabla^{t}+Z_{t})$ with
diffusion coefficient $\psi_{t}$ on manifolds with convex flow; then extend
these results to non-convex case.
### 6.1 The case with a diffusion coefficient
Let $\psi_{t}(\cdot)=\psi(t,\cdot)>0$ be a smooth function on $(M,g_{t})$ and
$\Pi^{T}_{\mu,\psi}$ be the distribution of the (reflecting if $\partial
M\neq\varnothing$) diffusion process generated by
$L_{t}=\psi_{t}^{2}(\nabla^{t}+Z_{t})$ on time interval
$[0,T]\subset[0,T_{c})$ with initial distribution $\mu$. Set
$\Pi^{T}_{x,\psi}=\Pi^{T}_{\delta_{x},\psi}$ for $x\in M$. Moreover, for
$F\geq 0$ with $\Pi^{T}_{\mu,\psi}=0$, let
$\mu^{T}_{F,\psi}(\mathrm{d}x)=\Pi^{T}_{s,\psi}(F)\mu(\mathrm{d}x).$
###### Theorem 6.1.
Assume that $\mathbb{I}_{t}\geq 0$ for $t\in[0,T_{c})$ and
$\mathrm{Ric}_{t}^{Z}\geq K_{1}(t),\ \mathcal{G}_{t}\leq K_{2}(t)$ for some
continuous function on $[0,T_{c})$. Let $\psi\in
C^{1,\infty}_{b}([0,T_{c})\times M)$ be strictly positive. Let
$K_{\psi}(t)=(d-1)\|\nabla^{t}\psi_{t}\|_{\infty}^{2}+K_{1}^{-}(t)\|\psi_{t}\|_{\infty}^{2}+2\|Z_{t}\|_{\infty}\|\psi_{t}\|_{\infty}\|\nabla^{t}\psi_{t}\|_{\infty}+K_{2}(t).$
Then
$W^{\rho_{\infty}}_{2}(F\Pi^{T}_{\mu,\psi},\Pi^{T}_{\mu^{T}_{F,\psi},\psi})^{2}\leq
C(T,\psi)\Pi^{T}_{\mu,\psi}(F\log F),\ \mu\in\mathscr{P}(M),\ F\geq 0,\
\Pi^{T}_{\mu,\psi}(F)=1$
holds for
$C(T,\psi):=\inf_{R>0}\left\\{4(1+R^{-1})\int_{0}^{T}\|\psi_{s}\|_{\infty}^{2}e^{2\int_{s}^{T}K_{\psi}(r)\mathrm{d}r}\mathrm{d}s\cdot\exp\bigg{[}8(1+R)\sup_{s\in[0,T]}\|\nabla^{s}\psi_{s}\|_{\infty}\bigg{]}\right\\}.$
###### Proof.
We shall only consider the case that $\partial M$ is non-empty. As explained
in the proof of “(3)$\Rightarrow$(2)” in [24, Theorem 4.1], it suffices to
prove for $\mu=\delta_{x}$, $x\in M$. In this case, the desired inequality
reduces to
$W_{2}^{\rho_{\infty}}(F\Pi_{x,\psi}^{T},\Pi_{x,\psi}^{T})\leq
2C(T,\psi)\Pi_{x,\psi}^{T}(F\log F),\ \ F\geq 0,\ \Pi_{x,\psi}^{T}(F)=1.$
Since the diffusion coefficient is non-constant, it is convenient to adopt the
Itô differential $\mathrm{d}_{I}$ for the Girsanov transformation. So the
$L_{t}$-reflecting diffusion process can be constructed by solving the Itô SDE
$\mathrm{d}_{I}X_{t}=\sqrt{2}\psi_{t}(X_{t})u_{t}\mathrm{d}B_{t}+\psi_{t}^{2}(X_{t})Z_{t}(X_{t})\mathrm{d}t+N_{t}(X_{t})\mathrm{d}l_{t},\
\ \ X_{0}=x,$
where $B_{t}$ is the $d$-dimensional Brownian motion with natural filtration
$\mathscr{F}_{t}$. Let $\beta_{t}$, $\mathbb{Q}$ and $\tilde{B}_{t}$ be the
same as in the proof of Theorem 5.2. Then
$\displaystyle\mathrm{d}_{I}X_{t}=\sqrt{2}\psi_{t}(X_{t})u_{t}\mathrm{d}\tilde{B}_{t}+\\{\psi_{t}^{2}(X_{t})Z_{t}(X_{t})+\sqrt{2}\psi_{t}(X_{t})u_{t}\beta_{t}\\}\mathrm{d}t+N_{t}(X_{t})\mathrm{d}l_{t}.$
(6.1)
Let $Y_{t}$ solve
$\displaystyle\mathrm{d}_{I}Y_{t}=\sqrt{2}\psi_{t}(Y_{t})P^{t}_{X_{t},Y_{t}}u_{t}\mathrm{d}\tilde{B}_{t}+\psi_{t}^{2}(Y_{t})Z_{t}(Y_{t})\mathrm{d}t+N_{t}(Y_{t})\mathrm{d}\tilde{l}_{t},\
\ Y_{0}=y,$ (6.2)
where $\tilde{l}_{t}$ is the local time of $Y_{t}$ on $\partial M$. As
explained in the above theorem (see e.g. the proof of Theorem 5.2 (a)), under
$\mathbb{Q}$, the distribution of $Y_{[0,T]}$ and $X_{[0,T]}$ are
$\Pi_{x,\psi}^{T}$ and $F\Pi_{x,\psi}^{T}$, so
$\displaystyle
W_{2}^{\rho_{\infty}}(F\Pi_{x,\psi}^{T},\Pi_{x,\psi}^{T})\leq\mathbb{E}_{\mathbb{Q}}\max_{t\in[0,T]}\rho_{t}(X_{t},Y_{t})^{2}.$
(6.3)
Noting that due to the convexity of the boundary,
$\left<N_{t}(x),\nabla^{t}\rho(\cdot,y)(x)\right>_{t}=\left<N_{t}(y),\nabla^{t}\rho_{t}(y,\cdot)(x)\right>_{t}\leq
0,\ \ x\in\partial M,$
Combining this with (6.1), (6.2) and the comparison theorem [7, Theorem 4.1],
we obtain
$\displaystyle\mathrm{d}\rho_{t}(X_{t},Y_{t})\leq$
$\displaystyle\sqrt{2}(\psi_{t}(X_{t})-\psi_{t}(Y_{t}))\left<\nabla^{t}\rho_{t}(\cdot,Y_{t})(X_{t}),u_{t}\mathrm{d}B_{t}\right>_{t}$
$\displaystyle+K_{\psi}(t)\rho_{t}(X_{t},Y_{t})\mathrm{d}t+\sqrt{2}\|\psi_{t}\|_{\infty}\|\beta_{t}\|\mathrm{d}t,$
where
$K_{\psi}(t)=(d-1)\|\nabla^{t}\psi_{t}\|_{\infty}^{2}+K^{-}_{1}(t)\|\psi_{t}\|^{2}_{\infty}+2\|Z_{t}\|_{\infty}\|\nabla^{t}\psi_{t}\|_{\infty}\|\psi_{t}\|_{\infty}+K_{2}(t).$
Then
$M_{t}:=\sqrt{2}\int_{0}^{t}e^{-\int_{0}^{s}K_{\psi}(r)\mathrm{d}r}(\psi_{s}(X_{s})-\psi_{s}(Y_{s}))\left<\nabla^{s}\rho_{s}(\cdot,Y_{s})(X_{s}),u_{s}\mathrm{d}\tilde{B}_{s}\right>_{s}$
is a $\mathbb{Q}$-martingale such that
$\displaystyle\rho_{t}(X_{t},Y_{t})\leq
e^{\int_{0}^{t}K_{\psi}(r)\mathrm{d}r}\left(M_{t}+\sqrt{2}\int_{0}^{t}e^{-\int_{0}^{s}K_{\psi}(r)\mathrm{d}r}\|\psi_{s}\|_{\infty}\|\beta_{s}\|\mathrm{d}s\right),\
\ t\in[0,T].$
So by the Doob inequality, we obtain
$\displaystyle h_{t}:=$ $\displaystyle
e^{-2\int_{0}^{t}K_{\psi}(s)\mathrm{d}s}\mathbb{E}\max_{s\in[0,t]}\rho_{s}(X_{s},Y_{s})^{2}$
$\displaystyle\leq$
$\displaystyle(1+R)\mathbb{E}_{\mathbb{Q}}\max_{s\in[0,t]}M_{s}^{2}+2(1+R^{-1})\mathbb{E}_{\mathbb{Q}}\left(\int_{0}^{t}e^{-\int_{0}^{s}K_{\psi}(r)\mathrm{d}r}\|\psi_{s}\|_{\infty}\|\beta_{s}\|\mathrm{d}s\right)^{2}$
$\displaystyle\leq$ $\displaystyle
4(1+R)\mathbb{E}_{\mathbb{Q}}M_{t}^{2}+2(1+R^{-1})\int_{0}^{t}e^{-2\int_{0}^{s}K_{\psi}(r)\mathrm{d}r}\|\psi_{s}\|_{\infty}^{2}\mathrm{d}s\int_{0}^{t}\mathbb{E}_{\mathbb{Q}}\|\beta_{s}\|^{2}\mathrm{d}s$
$\displaystyle\leq$ $\displaystyle
8(1+R)\sup_{s\in[0,T]}\|\nabla^{s}\psi_{s}\|_{\infty}\int_{0}^{t}h_{s}\mathrm{d}s$
$\displaystyle+2(1+R^{-1})\int_{0}^{T}\|\psi_{s}\|_{\infty}^{2}e^{-2\int_{0}^{s}K_{\psi}(r)\mathrm{d}r}\mathrm{d}s\cdot\int_{0}^{T}\mathbb{E}\|\beta_{s}\|^{2}\mathrm{d}s,\
\ \ t\in[0,T].$
Since $h_{0}=0$, this inequality implies
$\displaystyle
e^{-2\int_{0}^{T}K_{\psi}(s)\mathrm{d}s}\mathbb{E}_{\mathbb{Q}}\max_{s\in[0,T]}\rho_{s}(X_{s},Y_{s})^{2}=h_{T}$
$\displaystyle\leq
2(1+R^{-1})\int_{0}^{T}\|\psi_{s}\|_{\infty}^{2}e^{-2\int_{0}^{s}K_{\psi}(r)\mathrm{d}r}\mathrm{d}s\cdot\exp\left[8(1+R)\sup_{s\in[0,T]}\|\nabla^{s}\psi_{s}\|_{\infty}\right]\cdot\int_{0}^{T}\mathbb{E}_{\mathbb{Q}}\|\beta_{s}\|^{2}\mathrm{d}s.$
As explained in (5.3), it holds
$\int_{0}^{T}\mathbb{E}_{\mathbb{Q}}\|\beta_{s}\|^{2}\mathrm{d}s=2\mathbb{E}F(X_{[0,T]})\log
F(X_{[0,T]}).$
Therefore
$\displaystyle\mathbb{E}\max_{s\in[0,T]}\rho_{s}(X_{s},Y_{s})^{2}$
$\displaystyle\leq
4(1+R^{-1})\int_{0}^{T}\|\psi_{s}\|_{\infty}^{2}e^{2\int_{s}^{T}K_{\psi}(r)\mathrm{d}r}\mathrm{d}s\cdot\exp{\big{[}8(1+R)\sup_{s\in[0,T]}\|\nabla^{s}\psi_{s}\|_{\infty}\big{]}\Pi_{o,\psi}^{T}(F\log
F)}.$
Combining with (6.3), we complete the proof. ∎
###### Theorem 6.2.
In the situation of Theorem 6.1,
$W^{\rho_{\infty}}_{2}(\Pi^{T}_{\nu,\psi},\Pi^{T}_{\mu,\psi})\leq
2e^{\int_{0}^{T}(K_{\psi}(t)+\|\nabla^{t}\psi_{t}\|_{\infty})\mathrm{d}t}W_{2,0}(\nu,\mu).$
###### Proof.
As explained in the proof of Theorem 5.2 $``(6)\Rightarrow(5)"$, we only
consider $\nu=\delta_{x}$, and $\nu=\delta_{y}$. Let $X_{t}$ and $Y_{t}$ solve
the following SDEs respectively.
$\displaystyle\mathrm{d}_{I}X_{t}=\sqrt{2}\psi_{t}(X_{t})u_{t}\mathrm{d}B_{t}+\psi_{t}^{2}(X_{t})Z_{t}(X_{t})\mathrm{d}t+N_{t}(X_{t})\mathrm{d}l_{t},\
\ X_{0}=x;$
$\displaystyle\mathrm{d}_{I}Y_{t}=\sqrt{2}\psi_{t}(Y_{t})P_{X_{t},Y_{t}}^{t}u_{t}\mathrm{d}B_{t}+\psi_{t}^{2}(Y_{t})Z_{t}(Y_{t})\mathrm{d}t+N_{t}(Y_{t})\mathrm{d}\tilde{l_{t}},\
\ Y_{0}=y.$
Then, as explained in Theorem 6.1, by the Itô formula,
$\displaystyle\mathrm{d}\rho_{t}(X_{t},Y_{t})$
$\displaystyle\leq\sqrt{2}(\psi_{t}(X_{t})-\psi_{t}(Y_{t}))\left<\nabla^{t}\rho_{t}(\cdot,Y_{t})(X_{t}),u_{t}\mathrm{d}B_{t}\right>_{t}+K_{\psi}(t)\rho_{t}(X_{t},Y_{t})\mathrm{d}t.$
(6.4)
Therefore,
$\displaystyle\rho_{t}(X_{t},Y_{t})\leq
e^{\int_{0}^{t}K_{\psi}(s)\mathrm{d}s}(M_{t}+\rho_{0}(x,y)),\ t\geq 0,$ (6.5)
for
$M_{t}:=\sqrt{2}\int_{0}^{t}e^{-\int_{0}^{s}K_{\psi}(u)\mathrm{d}u}(\psi_{s}(X_{s})-\psi_{s}(Y_{s}))\left<\nabla^{s}\rho_{s}(\cdot,Y_{s})(X_{s}),u_{s}\mathrm{d}B_{s}\right>_{s}.$
Again using the Itô formula,
$\displaystyle\mathrm{d}\rho_{t}^{2}(X_{t},Y_{t})\leq\mathrm{d}\tilde{M}_{t}+2\left[K_{\psi}(t)+\|\nabla^{t}\psi_{t}\|_{\infty}^{2}\right]\rho_{t}(X_{t},Y_{t})^{2}\mathrm{d}t,$
where
$\mathrm{d}\tilde{M}_{t}=2\rho_{t}(X_{t},Y_{t})(\psi_{t}(X_{t})-\psi_{t}(Y_{t}))\left<\nabla^{t}\rho_{t}(\cdot,Y_{t})(X_{t}),u_{t}\mathrm{d}B_{t}\right>.$
This implies
$\displaystyle\mathrm{d}\rho_{t}^{2}(X_{t},Y_{t})\leq
e^{2\int_{0}^{t}(K_{\psi}(s)+\|\nabla^{s}\psi_{s}\|_{\infty})\mathrm{d}s}\rho_{0}(x,y)^{2}.$
Combining this with (6.5), we arrive at
$\displaystyle W_{2}^{\rho_{\infty}}(\Pi^{T}_{x,\psi},\Pi^{T}_{y,\psi})^{2}$
$\displaystyle\leq\mathbb{E}\max_{t\in[0,T]}\rho_{t}(X_{t},Y_{t})^{2}\leq
e^{2\int_{0}^{T}K_{\psi}(t)\mathrm{d}t}\mathbb{E}\max_{t\in[0,T]}(M_{t}+\rho_{0}(x,y))^{2}$
$\displaystyle\leq
4e^{2\int_{0}^{T}K_{\psi}(t)\mathrm{d}t}\mathbb{E}(M_{T}+\rho_{0}(x,y))^{2}=4e^{2\int_{0}^{T}K_{\psi}(t)\mathrm{d}t}\mathbb{E}(M_{T}^{2}+\rho_{0}^{2}(x,y))$
$\displaystyle\leq
4e^{2\int_{0}^{T}K_{\psi}(t)\mathrm{d}t}\left(\rho_{0}(x,y)^{2}+2\int_{0}^{T}e^{-2\int_{0}^{t}K_{\psi}(s)\mathrm{d}s}\|\nabla^{t}\psi_{t}\|_{\infty}\mathbb{E}\rho_{t}(X_{t},Y_{t})^{2}\mathrm{d}t\right)$
$\displaystyle\leq
4e^{2\int_{0}^{T}(K_{\psi}(t)+\|\nabla^{t}\psi_{t}\|_{\infty})\mathrm{d}t}\rho_{0}(x,y)^{2}$
where the second inequality is due to the Doob inequality. This implies the
desired inequality for $\mu=\delta_{x}$ and $\nu=\delta_{y}$. ∎
### 6.2 Non-convex manifold
As discussed in Theorem 6.1 and with a proper conformal change of metric, we
are able to establish the following transportation-cost inequality on a class
of manifolds with non-convex boundary. Let
$\mathscr{D}=\\{\phi\in C_{b}^{2}([0,T_{c})\times M):\inf\phi_{t}=1,\
\mathbb{I}_{t}\geq-N_{t}\log\phi_{t}\\}.$
Assume that $\mathscr{D}\neq\varnothing$ and for some $K_{1},K_{2}\in
C{([0,T_{c}))}$ such that
$\displaystyle{\rm Ric}_{t}^{Z}\geq K_{1}(t),\ \ \mathcal{G}_{t}\leq K_{2}(t)$
(6.6)
holds. To make the boundary convex, let $\phi_{t}\in\mathscr{D}$. By Theorem
[23, Lemma 2.1], $\partial M$ become convex under
$\tilde{g}_{t}=\phi_{t}^{-2}g_{t}$. Let $\tilde{\Delta}_{t}$ and
$\tilde{\nabla}^{t}$ be the Laplacian and gradient induced by the new metric
$\tilde{g}_{t}$. Since $\phi_{t}\geq 1$, $\rho_{t}(x,y)$ is large than
$\tilde{\rho}_{t}(x,y)$, the Riemannian $\tilde{g}_{t}$-distance between $x$
and $y$.
###### Theorem 6.3.
Let $\partial M\neq\varnothing$ and $\mathbb{I}_{t}\geq-\sigma(t)$ for
positive $\sigma\in C([0,T_{c}))$. Assume (6.6) holds. For
$\phi\in\mathscr{D}$, let
$K_{\phi}(t):=(d-1)\|\nabla^{t}\phi_{t}\|_{\infty}^{2}+K_{\phi,1}^{-}(t)+2\|\phi_{t}Z_{t}+(d-2)\nabla^{t}\phi_{t}\|_{\infty}\|\nabla^{t}\phi_{t}\|_{\infty}+K_{\phi,2}(t),$
where
$\displaystyle
K_{\phi,1}(t):=\inf\\{\phi_{t}K_{1}(t)+\frac{1}{2}L_{t}\phi_{t}^{2}-|\nabla^{t}\phi_{t}^{2}|_{t}|Z_{t}|_{t}-(d-2)|\nabla^{t}\phi_{t}|_{t}^{2}\\},$
$\displaystyle K_{\phi,2}(t):=\sup\\{-2\partial_{t}\log\phi_{t}+K_{2}(t)\\}.$
Then for any $\mu\in\mathscr{P}(M)$,
$W^{\rho_{\infty}}_{2}(F\Pi_{\mu}^{T},\Pi_{\mu_{F}^{T}}^{T})\leq\sup_{s\in[0,T]}\|\phi_{t}\|_{\infty}^{2}C(T,\phi)\Pi_{\mu}^{T}(F\log
F),\ \ F\geq 0,\ \Pi_{\mu}^{T}(F)=1$
holds for
$C(T,\phi)=\inf_{R>0}\left\\{4(1+R^{-1})\int_{0}^{T}e^{2\int_{s}^{T}K_{\phi}(r)\mathrm{d}r}\mathrm{d}s\exp{\bigg{[}8(1+R)\sup_{s\in[0,T]}\|\nabla^{s}\phi_{s}\|_{\infty}\bigg{]}}\right\\}.$
###### Proof.
According to the proof of [7, Proposition 4.7]. We have
$L_{t}:=\phi_{t}^{-2}(\tilde{\Delta}_{t}+\tilde{Z}_{t})$, where
$\tilde{Z}_{t}=\phi_{t}^{2}Z_{t}+\frac{d-2}{2}\nabla^{t}\phi_{t}^{2}$, and
$\widetilde{{\rm Ric}}^{\tilde{Z}}_{t}\geq K_{\phi,1}(t),\ \ \
\tilde{\mathcal{G}}_{t}\leq K_{\phi,2}(t).$
Let $K_{\psi}$ be defined in Theorem 6.3 for the manifold equipped with
$\tilde{g}_{t}$. Then, $L_{t}=\psi_{t}^{2}(\tilde{\Delta}_{t}+\tilde{Z}_{t})$,
where $\psi_{t}=\phi_{t}^{-1}$, we see that $K_{\psi}(t)\leq K_{\phi}(t)$ and
thus $C(t,\psi)\leq C(t,\phi)$. Hence, it follows from Theorem 6.3 that
$W^{\tilde{\rho}_{\infty}}_{2}(F\Pi_{\mu}^{T},\Pi^{T}_{\mu^{T}_{F}})^{2}\leq
C(T,\phi)\Pi_{\mu}^{T}(F\log F),\ \ F\geq 0,\ \ \Pi_{\mu}^{T}(F)=1,$
where $\tilde{\rho}_{\infty}$ is the uniform distance on $W^{T}$ induced by
the metric $\tilde{g}_{t}$. The proof is completed by
$\rho_{\infty}\leq\sup_{t\in[0,T]}\|\phi_{t}\|_{\infty}\tilde{\rho}_{\infty}.$
∎
Since $K_{\psi}(t)\leq K_{\phi}(t)$ and
$\tilde{\rho}_{t}\leq\rho_{t}\leq\|\phi_{t}\|_{\infty}\tilde{\rho}_{t}$, the
following result follows from the proof of Theorem 3.2 by taking
$\psi=\phi^{-1}$.
###### Theorem 6.4.
In the situation of Theorem 6.3,
$W^{\rho_{\infty}}_{2}(\Pi^{T}_{\mu},\Pi_{\nu}^{T})\leq
2\sup_{t\in[0,T]}\|\phi_{t}\|_{\infty}e^{\int_{0}^{T}(K_{\phi}(t)+\|\nabla^{t}\phi_{t}\|_{\infty})\mathrm{d}t}W_{2,0}(\nu,\mu),\
\ \mu,\nu\in\mathscr{P},T>0.$
Acknowledgements The author would thank Professor Feng-Yu Wang for valuable
suggestions and this work is supported in part by 985 Project, 973 Project.
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|
arxiv-papers
| 2012-11-15T15:00:38 |
2024-09-04T02:49:38.061810
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Lijuan Cheng",
"submitter": "Lijuan Cheng",
"url": "https://arxiv.org/abs/1211.3625"
}
|
1211.3900
|
CORRELATED COHERENT STATES - QUANTUM ANALOGUE OF THERMAL STATES
(ON THE PROBLEM OF INCORPORATING THERMODYNAMICS INTO QUANTUM THEORY )
A.D.Sukhanov1, O.N. Golubjeva2, and V.G.Bar’yakhtar3
1Theoretical Physics Laboratory, Joint Institute for Nuclear Research, Dubna,
Russia. $<[email protected]$>$
2People’s Friendship University of Russia, Moscow, Russia. $<[email protected]$>$
3 Institute of Magnetism, National Academy of Sciences of Ukraine, Kiev,
Ukraine. $<[email protected]$>$
###### Abstract
In this paper, we show that correlated coherent states (CCSs) are the most
adequate candidates for the role of quantum analogues of the thermal states.
The main result of our study reduces to the fact that quantum thermal effects
under conditions of the equilibrium of an object with the stochastic
environment at any temperatures can be explained consistently only on the
basis of the set of CCSs.
## 1\. Wave functions of correlated coherent states of an arbitrary vacuum
In our paper [1], we proposed an original approach to the incorporation of
stochastic thermodynamics into quantum theory. It is based on the concept of
consistent inclusion of the holistic stochastic environmental action
consisting of the thermal environmental action in addition to the quantum one.
Bogoliubov arbitrary vacua were used as a model of the generalized thermostat.
In this case, it becomes possible to bring the set of quantum states described
by the complex wave function in correspondence with equilibrium thermal
states. In the next paper [2], it was shown that squeezed correlated states
can adequately describe only the cases of the contact with the cold vacuum,
which exists at the zero Kelvin temperature, which is not significant for
thermodynamics.
Now we consider a system (a quantum oscillator) under the conditions of the
contact with the thermostat modeled by an arbitrary vacuum. Under these
conditions, this system can be described by the complex wave function that is
dependent on the parameter $\alpha\neq\frac{\pi}{2}$. 111The states
corresponding to $\alpha=0$ were considered in our paper [2] and were
qualified as squeezed correlated states (SCSs), which are inadequate to the
thermal states. In the coordinate representation, in the general case, it has
the form
$\psi_{\alpha}(q)=[2\pi(\Delta
q_{0})^{2}\frac{1}{\cos\alpha}]^{-1/4}\exp\left\\{-\frac{q^{2}}{4(\Delta
q_{0})^{2}}e^{i\alpha}\right\\}.$ (1)
Here,
$(\Delta q_{0})^{2}=\frac{\hbar}{2\gamma},\;\;\;\;\;\;$ (2)
where the coefficient $\gamma>0$ and $\hbar$ is the Planck constant. The
coordinate variance $\overline{(\Delta q_{\alpha})^{2}}\equiv(\Delta
q_{\alpha})^{2}$ in the arbitrary-vacuum state $|\psi_{\alpha}\rangle$
calculated using wave function (1) has the form
$(\Delta
q_{\alpha})^{2}=\int_{-\infty}^{+\infty}\psi^{*}_{\alpha}q^{2}\psi_{\alpha}dq=\frac{\hbar}{2\gamma}\cdot\frac{1}{\cos\alpha}=(\Delta
q_{0})^{2}\cdot\frac{1}{\cos\alpha}.$ (3)
Similar calculations for the momentum variance $\overline{(\Delta
p_{\alpha})^{2}}\equiv(\Delta p_{\alpha})^{2}$ in this state
$|\psi_{\alpha}\rangle$ lead to the result
$(\Delta
p_{\alpha})^{2}=\frac{\hbar\gamma}{2}\cdot\frac{1}{\cos\alpha}=(\Delta
p_{0})^{2}\cdot\frac{1}{\cos\alpha},$ (4)
where $\dfrac{\hbar\gamma}{2}\equiv(\Delta p_{0})^{2}.$
We show that wave function (1) is completely identical to the state of the
contact with the arbitrary vacuum $\psi_{\tau,\phi}$, which was found in [3]
using the Bogoliubov $(u,v)$-transformations from the equations
$\hat{b}\psi_{\tau,\varphi}(q,\omega)=0,$ (5)
where the operator $\hat{b}$ is the quasiparticle annihilation operator for
the arbitrary vacuum, or
$\frac{d\psi_{\tau,\varphi}}{dq}+\frac{u-v}{u+v}\cdot\frac{\omega}{\hbar}\;q\;\psi_{\tau,\varphi}=0$
(6)
In the general case [4], the complex functions $u$ and $v$ containing in (6)
are determined in terms of the free parameters $(\tau,\theta,\varphi)$ as
follows:
$u=\mathop{\rm ch}\tau\cdot e^{i\varphi};\;\;\;\;v=\mathop{\rm sh}\tau\cdot
e^{-i\varphi}.$ (7)
In this case, $(\tau,\theta,\varphi)$ can be interpreted as Euler angles,
which are used to parametrize the group of rotations $O$(3). Without loss of
generality, we assume that $\theta=0$ in this case.
The solution of Eq. (6) for arbitrary $\tau$ and $\varphi$ has the form of the
complex Gaussoid
$\psi_{\tau,\varphi}(q,\omega)=C\exp\left\\{-\frac{q^{2}}{4(\Delta
q_{0})^{2}}\cdot\frac{u-v}{u+v}\right\\},$ (8)
where $(\delta q_{0})^{2}\equiv\dfrac{\hbar}{2\omega}$.
If formulas (7) and the normalization conditions are taken into account,
expression (8) becomes
$\psi_{\tau,\varphi}(q,\omega)=\left[2\pi(\ \Delta
q_{\tau,\varphi})^{2}\right]^{-1/4}\exp\left\\{-\frac{q^{2}}{4(\Delta
q_{\tau,\varphi})^{2}}(1-i\beta_{\tau,\varphi})\right\\}.$ (9)
Here,
$(\Delta q_{\tau,\varphi})^{2}=(\Delta q_{0})^{2}(\mathop{\rm
ch}2\tau-\mathop{\rm sh}2\tau\cdot\cos 2\varphi)$ (10)
$\beta_{\tau,\varphi}=\mathop{\rm sh}2\tau\cdot\sin 2\varphi$ (11)
For the convenience of comparison of different representations (1) and (9) of
the wave function, we endow them with the same forms. Writing the exponent
$e^{i\alpha}$ in (1) in the trigonometric form, we obtain the result
$\psi_{\alpha}(q)=[2\pi(\Delta
q_{0})^{2}\frac{1}{\cos\alpha}]^{-1/4}\exp\left\\{-\frac{q^{2}}{4(\Delta
q_{0})^{2}}\cos\alpha(1+i\tan\alpha)\right\\}.$ (12)
To make expressions (1) and (12) more similar, we assume
$\beta_{\alpha}\equiv\tan\alpha$ (13)
and use the value of $(\Delta q_{\alpha})^{2}$ in accordance with (3).
Finally, expression (1) can be written in the form
$\psi_{\alpha}=\left[2\pi(\ \Delta
q_{\alpha})^{2}\right]^{-1/4}\exp\left\\{-\frac{q^{2}}{4(\Delta
q_{\alpha})^{2}}(1-i\beta_{\alpha})\right\\}.$ (14)
It is natural to assume that formulas (1) and (14) obtained using different
initial preconditions and, accordingly, expressed in terms of different
parameters are nevertheless related to identical states. Then the condition
for the coincidence between $(\Delta q_{\alpha})^{2}$ (3) and $(\Delta
q_{\tau,\varphi})^{2}$ (10) can be obtained if it is required that the
following conditions be satisfied at the same time:
$(\mathop{\rm ch}2\tau-\mathop{\rm sh}2\tau\cdot\cos
2\varphi)\Leftrightarrow\frac{1}{\cos\alpha}$ (15)
$\mathop{\rm sh}2\tau\cdot\sin 2\varphi\Leftrightarrow\tan\alpha.$ (16)
This turns out to be possible if in (15) $\cos 2\varphi=0$ and in (16) $\sin
2\varphi=1$, which agrees with the well-known assertion that the correlated
coherent states (CCS) in the $(u,v)$-transformations are fixed by the
parameter $\varphi=\frac{\pi}{4}.$ Thus, we demonstrated that the states
$\psi_{\alpha}(q)$ (1) and
$\psi_{\tau,\varphi}(q)\Big{|}_{\varphi=\frac{\pi}{4}}$ (14) under condition
$\tan\alpha=\sinh 2\tau$ agree well with each other.
## 2\. Correlated coherent states as thermal ones
To study the possibility of endowing the states $|\psi_{\alpha}\rangle$ with
the meaning of thermal states, it is necessary to bring the parameter $\alpha$
in correspondence with the temperature that has no pre-image in quantum
mechanics. To do this, we consider the expression for the Planck energy
$\mathcal{E}_{Pl}=\frac{\hbar\omega}{2}\coth\frac{\hbar\omega}{k_{B}T}.$ (17)
We note that this formula is strictly equilibrium and corresponds to the
Kelvin temperature $T$. Therefore, based on formula (17), we can bring the
parameter $\alpha\neq\frac{\pi}{2},0$ in correspondence with the temperature
explicitly. In what follows, we use the fact that the average values of the
kinetic $\overline{K}$ and potential $\overline{U}$ energies are equal for a
quantum oscillator in the thermal-equilibrium state, so that from formula
(17), we obtain
$\overline{K}=\overline{U}=\frac{\mathcal{E}_{Pl}}{2}=\frac{\hbar\omega}{4}\coth\frac{\hbar\omega}{k_{B}T}.$
(18)
Taking into account that $\overline{K}=\cfrac{1}{2m}(\overline{\Delta p)^{2}}$
and $\overline{U}=\cfrac{m\omega^{2}}{2}(\overline{\Delta q)^{2}}$ and
assuming that $m=1$, from formula (18), we obtain the coordinate and momentum
variances, letting the subscript $T$ denote their relation with the Planck
distribution:
$(\Delta q_{\scriptscriptstyle T})^{2}=\frac{\hbar}{2\omega}\mathop{\rm
coth}\frac{\hbar\omega}{k_{B}T}=(\Delta q_{0})^{2}\mathop{\rm
coth}\frac{\hbar\omega}{k_{B}T}.$ (19)
$(\Delta p_{\scriptscriptstyle T})^{2}=\frac{\hbar\omega}{2}\mathop{\rm
coth}\frac{\hbar\omega}{k_{B}T}.$ (20)
As follows from formulas (19) and (20), as $T$ increases, the coordinate and
momentum variances increase synchronously for thermal states (unlike the
SCSs), so that their product $(\mathcal{U}\mathcal{P})_{pq}$ also increases in
this case.
The qualitative distinction between the thermal-like SCSs ($\alpha=0$) and the
thermal states (CCS) can be demonstrated obviously by giving them the
geometric interpretation in the phase plane. For convenience, we put $\hbar=1$
and choose the dimensionless variables $\hat{\mathcal{P}}=\dfrac{\scriptstyle
1}{i}\dfrac{d}{d\mathcal{Q}};\hat{\mathcal{Q}}$. Then, for cold-vacuum states
(SCSs of the $|\psi_{0}\rangle$ type), the quantities $\Delta\mathcal{P}_{0}$
and $\Delta\mathcal{Q}_{0}$ (in view of their equality in accordance with
formulas (3) and (4)) have the meaning of the sides of a single square with
area of $1/4$. At the same time, the thermal CCSs (of the
$|\psi_{\scriptscriptstyle T}\rangle$ type) in this plane correspond to
different squares whose sides and areas increase consistently with
temperature.
We now compare the coordinate $(\Delta q_{\alpha})^{2}$ (3) and momentum
$(\Delta p_{\alpha})^{2}$ (4) variances calculated using wave function (1)
with their corresponding values $(\Delta q_{\scriptscriptstyle T})^{2}$ (19)
and $(\Delta p_{\scriptscriptstyle T})^{2}$ (20) obtained using the Planck
energy. They turn out to be completely identical if we set
$\gamma=\omega$ (21)
and
$\frac{1}{\cos\alpha}=\mathop{\rm coth}\frac{\hbar\omega}{k_{B}T}.$ (22)
Thus, in accordance with relation (22), the parameter $\alpha$ in formula (1)
fixes the states corresponding to the equilibrium at the temperature $T$.
Expression (22) allows representing the exponent $e^{i\alpha}$ in formula (1)
in the trigonometric form, explicitly indicating its relation with the
temperature $T$ in this case. Calculating
$\sin\alpha=\sqrt{1-\tanh^{2}\frac{\hbar\omega}{k_{B}T}}=\dfrac{1}{\cosh\frac{\hbar\omega}{k_{B}T}}$
in advance, we obtain
$e^{i\alpha}=\cos\alpha+i\sin\alpha=\tanh(\frac{\hbar\omega}{k_{B}T})+i\frac{1}{\cosh(\frac{\hbar\omega}{k_{B}T})}=\tanh(\frac{\hbar\omega}{k_{B}T})\left[1+i\frac{1}{\sinh\frac{\hbar\omega}{k_{B}T}}\right]$
(23)
Returning to wave function (1), we can demonstrate the explicit temperature
dependence of its amplitude and phase by labeling it with the subscript $T$
$\psi_{\scriptscriptstyle T}(q)=[2\pi(\Delta q_{0})^{2}\mathop{\rm
coth}\frac{\hbar\omega}{k_{B}T}]^{-1/4}\exp\left\\{-\frac{q^{2}}{4(\Delta
q_{0})^{2}}\tanh(\frac{\hbar\omega}{k_{B}T})[1+i\frac{1}{\sinh\frac{\hbar\omega}{k_{B}T}}]\right\\}.$
(24)
## 3\. Saturation of the Schrödinger UR in the correlated coherent states
It is interesting to analyze certain CCS peculiarities. As is well known, the
most general _Schrödinger uncertainties relation_ (SUR) for the coordinate and
momentum has the form
$\Delta p\cdot\Delta
q\geqslant\big{|}\langle\psi|\delta{\hat{p}}\cdot\delta{\hat{q}}|\psi\rangle\big{|}.$
(25)
Here, the left-hand side of the inequality contains the product of the
momentum and coordinate uncertainties in the state $|\psi\rangle$ calculated
using the definition
$(\Delta
p)^{2}\equiv\langle\psi|(\delta{\hat{p}})^{2}|\psi\rangle;\;\;\;\;(\Delta
q)^{2}\equiv\langle\psi|(\delta{\hat{q}})^{2}|\psi\rangle.$ (26)
The expression in the right-hand side
$\Big{|}\langle\psi|\delta{\hat{p}}\cdot\delta{\hat{q}}|\psi\rangle\Big{|}=\Big{|}\langle\delta
p|\delta q\rangle\Big{|}$ (27)
has the meaning of the correlator of object momentum and coordinate
fluctuations in the same state, which is expressed in terms of the fluctuation
operators $\delta\hat{p}$ and $\delta\hat{q}$. We recall that the left- and
right-hand sides of relation (22) must be calculated independently.
In the arbitrary-vacuum state, the equality for the means
$(\Delta p_{\alpha})^{2}=\gamma^{2}(\Delta q_{\alpha})^{2}$ (28)
is valid [5] so that the _left-hand_ side of SUR (25) becomes equal to
$\Delta p_{\alpha}\cdot\Delta q_{\alpha}=\gamma\Delta q_{\alpha}^{2}.$ (29)
Calculating the correlator in the _right-hand_ side of SUR (25) with respect
to the state $|\psi_{\alpha}\rangle$, we obtain
$\Big{|}\langle\psi_{\alpha}|\hat{p}\cdot\hat{q}|\psi_{\alpha}\rangle\Big{|}=\sqrt{\frac{\hbar^{2}}{4}\tan^{2}\alpha+\frac{\hbar^{2}}{4}}=\frac{\hbar}{2}\cdot\frac{1}{\cos\alpha}.$
(30)
We note that the correlator here is expressed in terms of the $\alpha$ \-
phase of the wave function. Taking into account that in accordance with (3),
$\dfrac{1}{\cos\alpha}=\dfrac{2}{\hbar}\cdot\gamma\Delta q_{\alpha}^{2},$ we
reduce correlator (30) to the final form
$\Big{|}\langle\psi_{\alpha}|\hat{p}\cdot\hat{q}|\psi_{\alpha}\rangle\Big{|}=\gamma\Delta
q_{\alpha}^{2},$ (31)
which completely coincides with product (29) of uncertainties, i.e., with the
left-hand side of the SUR (25).
Thus, we see that the correlated state $|\psi_{\alpha}\rangle$ is outlined by
the fact that the SUR in it indeed acquires the form of the _equality_ ,
i.e., becomes saturated
$\Delta p_{\alpha}\cdot\Delta
q_{\alpha}=\Big{|}\langle\psi_{\alpha}|\hat{p}\cdot\hat{q}|\psi_{\alpha}\rangle\Big{|}.$
(32)
The interpretation of the radicand in (30) can be related to that of a similar
expression for the cold-vacuum state $\psi_{0}$. For $\alpha=0$, saturated SUR
(30) transforms into the saturated Heisenberg UR
$\Delta p_{0}\cdot\Delta
q_{0}=\big{|}\langle\psi_{0}|\frac{1}{2}[\hat{p},\hat{q}]|\psi_{0}\rangle\big{|}=\frac{\hbar}{2}.$
(33)
Here, as is known, $\dfrac{\hbar}{2}$ is the measure of a purely quantum
environmental action occurring in the cold vacuum. Thus, for $\alpha=0$, the
correlator
$\big{|}\langle\psi_{\alpha}|\hat{p}\cdot\hat{q}|\psi_{\alpha}\rangle\big{|}$
has a value that is as minimum as possible. As was expected, the state
$|\psi_{0}\rangle$ indeed has the meaning of the state of equilibrium with the
cold vacuum because it corresponds to the minimum value of the vacuum energy
$\dfrac{\hbar\omega}{2}$.
It is adopted to regard the Heisenberg relation as a fundamental equality
reflecting the presence of unavoidable purely quantum effects in the Nature.
We assume that in comparison with (33), the origin of the additional term in
the radicand of correlator (30) in the form
$\dfrac{\hbar^{2}}{4}\tan^{2}\alpha$ is related to precisely the inclusion of
the thermal influence of the arbitrary vacuum, which is manifested in the
complex character of the wave function. The fact that the macroparameter of
the effective quantum-thermostat action
$\mathbb{J}_{\alpha}\equiv\sqrt{\frac{\hbar^{2}}{4}\tan^{2}\alpha+\mathbb{J}_{0}^{2}}=\frac{\hbar}{2}\frac{1}{\cos\alpha},$
(34)
(here, $\mathbb{J}_{0}^{2}=\dfrac{\hbar}{2}$ relates to the purely quantum
influence) which was introduced previously in the framework of stochastic
thermodynamics [6], coincides with the right-hand side of formula (30) is
evidence in favor of this assertion. This fact gives grounds to assume from
now on that for the arbitrary vacuum, saturated SUR (32) also corresponds to
the more general equilibrium state $|\psi_{\alpha}\rangle$ occurring at the
simultaneous presence of the quantum and thermal actions. Thus, among the
functions providing the saturation of SUR (32), there exist the functions
$\psi_{\alpha}(q)\Big{|}_{\alpha\neq 0,\frac{\pi}{2}}$, which allow taking
into account extra thermal effects in addition to the quantum ones in a
certain temperature range. Thus, the CCSs can be regarded as quantum analogues
of thermal states in a sufficiently substantiated way.
References
[1] A.D. Sukhanov and O.N. Golubjeva. Toward a quantum generalization of
equilibrium statistical thermodynamics: $(\hbar,k)$\- dynamics. Theoretical
and Mathematical Physics, 160(2): 1177 (2009)
[2] A.D. Sukhanov, O.N. Golubjeva, V.G. Bar’yakhtar. arXiv:1211.3017v1 [quant-
ph] 13 Nov 2012
[3] A.D. Sukhanov and O.N. Golubjeva. Arbitrary vacuum as a model of
stochastic influence of environment: on the problem of incorporating
thermodynamics into quantum theory. Physics of Particles and Nuclei Letters,
Vol. 9, No. 3, p. 303. Pleiades Publishing, Ltd., 2012.
[4] V.V. Dodonov and V.I.Man’ko. Trudy Fiz. Inst. Lebedev, 1987. 183, 71
[5] J.von Neumann. Mathematical Foundations of Quantum Mechanics. Princeton
University Press,1996
[6] Thermodynamics. Ed. Tadashi Mizutani, 2011. InTech. P.P. 73-98.
|
arxiv-papers
| 2012-11-16T14:25:53 |
2024-09-04T02:49:38.081091
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "A. D. Sukhanov, O. N. Golubjeva, and V. G. Bar'yakhtar",
"submitter": "Aleksander Sukhanov D.",
"url": "https://arxiv.org/abs/1211.3900"
}
|
1211.3942
|
# Convergence of equilibria
for incompressible elastic plates
in the von Kármán regime
Marta Lewicka Marta Lewicka, University of Pittsburgh, Department of
Mathematics, 301 Thackeray Hall, Pittsburgh, PA 15260, USA
[email protected] and Hui Li Hui Li, Penn State University, 2030 Mary
Ellen Lane, State College, PA 16803, USA [email protected]
###### Abstract.
We prove convergence of critical points $u^{h}$ of the nonlinear elastic
energies $E^{h}$ of thin incompressible plates
$\Omega^{h}=\Omega\times(-h/2,h/2)$, which satisfy the von Kármán scaling:
$E^{h}(u^{h})\leq Ch^{4}$, to critical points of the appropriate limiting
(incompressible von Kármán) functional.
## 1\. Introduction and the main result
In this paper we prove convergence of critical points of the nonlinear elastic
energies on thin incompressible plates in the von Kármán scaling regime, to
critical points of the appropriate limiting (incompressible von Kármán)
functional.
### 1.1. Elastic energy of thin incompressible plates
Let $\Omega\subset\mathbb{R}^{2}$ be an open, bounded, simply connected
domain. For $h>0$, define $\Omega^{h}$ to be the $3$d plate with the midplate
$\Omega$ and thickness $h$:
$\Omega^{h}=\left\\{x=(x^{\prime},x_{3});~{}x^{\prime}\in\Omega,~{}x_{3}\in\left(-\frac{h}{2},\frac{h}{2}\right)\right\\}.$
The elastic energy of a deformation $u^{h}\in
W^{1.2}(\Omega^{h},\mathbb{R}^{3})$ of the homogeneous plate $\Omega^{h}$,
scaled by its unit thickness, is given by:
(1.1) $I^{h}(u^{h})=\frac{1}{h}\int_{\Omega^{h}}W_{in}(\nabla
u^{h})~{}\mbox{d}x,$
while the total energy, relative to the external force with the density
$f^{h}\in L^{2}(\Omega^{h},\mathbb{R}^{3})$, is:
(1.2) $J^{h}(u^{h})=\frac{1}{h}\int_{\Omega^{h}}W_{in}(\nabla
u^{h})~{}\mbox{d}x-\frac{1}{h}\int_{\Omega^{h}}f^{h}\cdot u^{h}~{}\mbox{d}x.$
The elastic energy density $W_{in}:\mathbb{R}^{3\times
3}\rightarrow[0,\infty]$ in (1.1) is assumed to be infinite at compressible
deformations:
$W_{in}(F)=\left\\{\begin{array}[]{ll}W(F)&\mbox{ if}~{}~{}\det F=1,\\\
+\infty&\mbox{ otherwise}.\end{array}\right.$
The effective density $W:\mathbb{R}^{3\times 3}\rightarrow[0,\infty)$ above,
which acts when $\det F=1$, is required to satisfy the following conditions:
* (i)
(frame invariance) $W(RF)=W(F)$, for each proper rotation $R\in SO(3)$, and
each $F\in\mathbb{R}^{3\times 3}$.
* (ii)
(normalisation) $W(F)=0$ for all $F\in SO(3)$.
* (iii)
(non-interpenetration) $W(F)=+\infty$ if $\det F\leq 0$, and $W(F)\to+\infty$
as $\det F\to 0+$.
* (iv)
(bound from below) $W(F)\geq c~{}\mbox{dist}^{2}(F,SO(3))$ with a constant
$c>0$ independent of $F$.
* (v)
(bound from above) There exists a constant $C>0$ such that for each $F$ with
$\det F>0$, i.e. for each $F\in\mathbb{R}_{+}^{3\times 3}$ there holds:
(1.3) $|DW(F)F^{T}|\leq C(W(F)+1).$
* (vi)
(regularity) $W$ is of class $\mathcal{C}^{1}$ on $\mathbb{R}_{+}^{3\times
3}$.
* (vii)
(local regularity) $W$ is of class $\mathcal{C}^{2}$ in a small neighborhood
of $SO(3)$.
The growth conditions in (iv) and (v) will be crucial in the present analysis.
Condition (iv) has been introduced in the context of [6] and it allows to use
the nonlinear version of Korn’s inequality [5], ultimately serving to control
the local deviations of the deformation $u^{h}$ from rigid motions, by the
elastic energy $I^{h}(u^{h})$. Condition (v) has been introduced in [1] (see
also [2]) in the context of inner variations, in order to control the related
strain in terms of the energy. Both conditions are compatible with other
requirements above. Indeed, examples of $W$ satisfying (i) – (vii) are:
$\begin{split}W_{1}(F)&=|(F^{T}F)^{1/2}-\mbox{Id}|^{2}+|\log\det F|^{q},\\\
W_{2}(F)&=|(F^{T}F)^{1/2}-\mbox{Id}|^{2}+\left|\frac{1}{\det
F}-1\right|^{q}\mbox{ for }\det F>0,\end{split}$
where $q>1$ and $W$ equals $+\infty$ if $\det F\leq 0$ [11].
### 1.2. Notation
Given a matrix $F\in\mathbb{R}^{n\times n}$, we denote its trace by
$\mbox{Tr}~{}F$ and its transpose by $F^{T}$. The symmetric part of $F$ is
given by $\mbox{sym}~{}F=\frac{1}{2}(F+F^{T})$. The cofactor of $F$ is the
matrix: $\mbox{cof}~{}F$, where $[\mbox{cof
}F]_{ij}=(-1)^{i+j}\det\hat{F}_{ij}$ and each
$\hat{F}_{ij}\in\mathbb{R}^{(n-1)\times(n-1)}$ is obtained from $F$ by
deleting its $i$th row and $j$th column. The identity matrix is denoted by
$\mbox{Id}_{n}$.
In what follows, we shall use the matrix norm $|F|=(\mbox{Tr}(F^{T}F))^{1/2}$,
which is induced by the inner product:
$F_{1}:F_{2}=\mbox{Tr}(F_{1}^{T}F_{2})$. To avoid notational confusion, we
will often write $\langle F_{1}:F_{2}\rangle$ instead of $F_{1}:F_{2}$. In
general, $3\times 3$ matrices will be denoted by $F$ and $2\times 2$ matrices
will be denoted by $F^{\prime\prime}$. Unless noted otherwise,
$F^{\prime\prime}$ is the principal $2\times 2$ minor of $F$.
Finally, by $\mathcal{C}^{k}_{b}(\mathbb{R}^{n},\mathbb{R}^{s})$ we denote the
space of continuous functions whose derivatives up to the order $k$ are
continuous and bounded in $\mathbb{R}^{n}$.
### 1.3. The limiting energy
The following $2$d energy functional has been rigorously derived in [10] as
the $\Gamma$-limit of the scaled incompressible energies $h^{-4}I^{h}$ in
(1.1), when $h\to 0$:
(1.4)
$\mathcal{I}(w,v)=\frac{1}{2}\int_{\Omega}\mathcal{Q}_{2}^{in}\left(\mbox{sym}\nabla
u+\frac{1}{2}\nabla v\otimes\nabla
v\right)~{}\mbox{d}x+\frac{1}{24}\int_{\Omega}\mathcal{Q}_{2}^{in}\left(\nabla^{2}v\right)~{}\mbox{d}x,$
acting on couples $w\in W^{1,2}(\Omega,\mathbb{R}^{2}),v\in
W^{2,2}(\Omega,\mathbb{R})$. The fields $(w,v)$ may be identified as the in-
plane and the out-of-plane displacements, respectively. Roughly speaking, any
minimizing sequence of $h^{-4}J^{h}$, where $f^{h}(x)\approx
h^{3}f(x^{\prime})e_{3}$ and $\int_{\Omega}f=0$, will have the structure:
$u^{h}_{|\Omega}\approx(\bar{R})^{T}\left(\mbox{id}+hve_{3}+h^{2}w\right)-c^{h}$
asymptotically as $h\to 0$, with $(w,v)$ as above and $\bar{R}\in SO(3)$
maximizing $\int_{\Omega}f(x^{\prime})e_{3}\cdot
Rx^{\prime}~{}\mbox{d}x^{\prime}$ among all rotations $R$, while
$c^{h}\in\mathbb{R}^{3}$ are constant translation vectors. Moreover,
$(w,v,\bar{R})$ minimize the following total limiting energy:
$\mathcal{J}(w,v,\bar{R})=\mathcal{I}(w,v)-\bar{R}_{33}\int_{\Omega}fv.$
A precise formulation of the statements above can be found in [9].
The energy in (1.4) is the incompressible version of the von Kármán
functional, which has been derived (for compressible case, i.e. without the
assumption that $\det\nabla u^{h}=1$) by means of $\Gamma$-convergence in [6].
The quadratic forms $\mathcal{Q}_{2}^{in}$ differ from the standard
$\mathcal{Q}_{2}$ in [6] in as much as minimization in (1.5) below is taken
over the out-of-plane stretches which preserve the incompressibility
constraint. Namely, $\mathcal{Q}_{2}^{in}$ in (1.4) are given as:
(1.5) $\begin{split}&\forall F^{\prime\prime}\in\mathbb{R}^{2\times
2}\quad\mathcal{Q}_{2}^{in}(F^{\prime\prime})=\min_{d\in\mathbb{R}^{3}}\Big{\\{}Q_{3}(F^{\prime\prime}+d\otimes
e_{3}+e_{3}\otimes d);~{}\mbox{Tr}(F^{\prime\prime}+d\otimes
e_{3}+e_{3}\otimes d)=0\Big{\\}},\\\ &\forall F\in\mathbb{R}^{3\times
3}\quad\mathcal{Q}_{3}(F)=D^{2}W(\mbox{Id})(F,F).\end{split}$
Both forms $\mathcal{Q}$ above are positive semidefinite, and strictly
positive definite on symmetric matrices. We also introduce the linear
operators $\mathcal{L}_{2}^{in}:\mathbb{R}^{2\times
2}\rightarrow\mathbb{R}^{2\times 2}$ and $\mathcal{L}_{3}:\mathbb{R}^{3\times
3}\rightarrow\mathbb{R}^{3\times 3}$ such that:
(1.6) $\begin{split}&\forall F^{\prime\prime}\in\mathbb{R}^{2\times
2}\quad\langle\mathcal{L}_{2}^{in}(F^{\prime\prime}):F^{\prime\prime}\rangle=\mathcal{Q}_{2}^{in}(F^{\prime\prime}),\\\
&\forall F\in\mathbb{R}^{3\times
3}\quad\langle\mathcal{L}_{3}(F):F\rangle=\mathcal{Q}_{3}(F).\end{split}$
Note that symmetric operators $\mathcal{L}$ are uniquely given by:
$\langle\mathcal{L}(F_{1}):F_{2}\rangle=\frac{1}{4}\left(\mathcal{Q}(F_{1}+F_{2})-\mathcal{Q}(F_{1}-F_{2})\right)$.
### 1.4. Critical points and the incompressible inner variations
Following [2], we now define the critical points $u^{h}$ of the functionals
$J^{h}$ in (1.2) with respect to inner variations, that is requesting that the
derivative of $J^{h}$ at an incompressible equilibrium $u^{h}$ be zero:
$\frac{\mathrm{d}}{\mathrm{d}\epsilon}_{|\epsilon=0}J^{h}(u^{h}_{\epsilon})=0,$
along all curves $\epsilon\mapsto u^{h}_{\epsilon}$ of incompressible
deformations of $\Omega^{h}$ having the form:
$u^{h}_{\epsilon}(x)=\Phi(\epsilon,u^{h}(x))$, with $u^{h}_{0}=u^{h}$ at
$\epsilon=0$. This requirement is translated into the following condition:
(1.7) $\int_{\Omega^{h}}\left\langle DW(\nabla u^{h})(\nabla
u^{h})^{T}:\nabla\phi(u^{h}(x))\right\rangle~{}\mbox{d}x=\int_{\Omega^{h}}f^{h}\cdot\phi(u^{h})~{}\mbox{d}x,\quad\forall\phi\in\mathcal{C}_{b}^{1}(\mathbb{R}^{3},\mathbb{R}^{3})\mbox{
with }\mbox{div}~{}\phi=0.$
We refer to section 2 for the derivation and discussion of (1.7). Let us only
note now that the incompressible inner variations:
$u^{h}_{\epsilon}(x)=\Phi(\epsilon,u^{h}(x))=u^{h}(x)+\epsilon\phi(u^{h}(x))+\mathcal{O}(\epsilon^{2}).$
replace the classical variations $u^{h}_{\epsilon}(x)=u^{h}(x)+\epsilon
w^{h}(x)$ used in definition of minimizers of $J^{h}$, and also they replace
the inner variations $u^{h}_{\epsilon}(x)=u^{h}(x)+\epsilon\phi(u^{h}(x))$
considered in [2] and [11] in the compressible case.
### 1.5. The main result
The following is our main result:
###### Theorem 1.1.
For each $h<<1$, let $u^{h}\in W^{1,2}(\Omega^{h},\mathbb{R}^{3})$ be a
critical point of $J^{h}$, i.e. it satisfies (1.7) subject to the external
forces $f^{h}(x)=h^{3}f(x^{\prime})e_{3}$. Assume that:
(1.8) $I^{h}(u^{h})\leq Ch^{4},$
for a constant $C>0$ independent of $h$. Then there exists a sequence of
proper rotations $\bar{R}^{h}\in SO(3)$, and translations
$c^{h}\in\mathbb{R}^{3}$, such that for the renormalized deformations:
(1.9)
$y^{h}(x^{\prime},x_{3})=(\bar{R}^{h})^{T}u^{h}(x^{\prime},hx_{3})-c^{h}\in
W^{1,2}(\Omega^{1},\mathbb{R}^{3}),$
the following convergences hold, up to a subsequence in $h$, as $h\to 0$:
* (i)
$\bar{R}^{h}\to\bar{R}=[\bar{R}_{ij}]_{i,j:1..3}\in SO(3)$.
* (ii)
$y^{h}\to x^{\prime}$ in $W^{1,2}(\Omega^{1})$.
* (iii)
For the scaled out-of-plane displacements:
(1.10)
$v^{h}(x^{\prime})=\frac{1}{h}\int_{-1/2}^{1/2}y_{3}^{h}(x^{\prime},x_{3})~{}\mathrm{d}x_{3},$
there exists $v\in W^{2,2}(\Omega,\mathbb{R})$ such that $v^{h}\to v$ strongly
in $W^{1,2}(\Omega)$.
* (iv)
For the scaled in-plane displacements:
(1.11)
$w^{h}(x^{\prime})=\frac{1}{h^{2}}\int_{-1/2}^{1/2}\left((y^{h})^{\prime}(x^{\prime},x_{3})-x^{\prime}\right)~{}\mathrm{d}x_{3}$
there exists $w\in W^{1,2}(\Omega,\mathbb{R}^{2})$ such that
$w^{h}\rightharpoonup w$ weakly in $W^{1,2}(\Omega,\mathbb{R}^{2})$.
* (v)
The limiting displacements $(w,v)$ solve the following Euler-Lagrange
equations of the functional (1.4), expressed in the variational form:
(1.12) $\int_{\Omega}\left\langle\mathcal{L}_{2}^{in}\left(\mathrm{sym}\nabla
w+\frac{1}{2}\nabla v\otimes\nabla
v\right):\nabla\tilde{w}\right\rangle~{}\mathrm{d}x^{\prime}=0$ (1.13)
$\begin{split}&\int_{\Omega}\left\langle\mathcal{L}_{2}^{in}\left(\mathrm{sym}\nabla
w+\frac{1}{2}\nabla v\otimes\nabla v\right):(\nabla
v\otimes\nabla\tilde{v})\right\rangle~{}\mathrm{d}x^{\prime}\\\
&\qquad\qquad\qquad\qquad\qquad\qquad+\frac{1}{12}\int_{\Omega}\left\langle\mathcal{L}_{2}^{in}(\nabla^{2}v):\nabla^{2}\tilde{v}\right\rangle~{}\mathrm{d}x^{\prime}=\bar{R}_{33}\int_{\Omega}f\tilde{v}~{}\mathrm{d}x^{\prime},\end{split}$
for every $\tilde{w}\in W^{1,2}(\Omega,\mathbb{R}^{2})$ and every
$\tilde{v}\in W^{2,2}(\Omega,\mathbb{R})$.
We note that (1.8) are automatically satisfied by any minimizing sequence of
$u^{h}$ of the total energy $J^{h}$, under the assumption that
$f^{h}(x)=h^{3}f(x^{\prime})e_{3}$ [6]. Also, (1.7) holds for every minimum of
$J^{h}$ (see Theorem 2.3), and the assertions (i) - (v) are then a direct
consequence [10] of the fact that $\frac{1}{h^{4}}J^{h}$ $\Gamma$-converges to
$\mathcal{J}$. In general, $\Gamma$-convergence does not assure that a limit
of a sequence of equlibria is an equilibrium of the $\Gamma$-limit. In the
present situation, this turns out to be the case.
### 1.6. Relation to other works
Our work is largely inspired by [11] and [10]. To put it in a larger
perspective, recall that one of the fundamental questions in the mathematical
theory of elasticity has been to rigorously justify various 2d plate models
present in the engineering literature, in relation to the three-dimensional
theory. This goal has been largely accomplished in [6], where a hierarchy of
limiting 2d energies has been derived; the distinct theories are
differentiated by their validity in the corresponding scaling regimes
$h^{\beta},$ $\beta\geq 2$, i.e. in presence of assumption (1.8) where $h^{4}$
is replaced by $h^{\beta}$.
Under the additional incompressibility constraint, the works [3, 4] proved
compactness properties and the $\Gamma$-convergence of the functionals
$\frac{1}{h^{\beta}}I^{h}$ as in (1.1), for the so-called Kirchhoff scaling
$\beta=2$, while [10] treated the case $\beta=4$ including as well a more
complex case of shells when the midsurface $\Omega$ is a generic 2d
hypersurface in $\mathbb{R}^{3}$. In view of the fundamental property of
$\Gamma$-convergence, it follows that the global almost-minimizers of the
energies (1.2) converge to the minimizers of the limiting energy (given by
(1.4) in the von Kármán regime).
Regarding convergence of stationary points for thin plates, the first result
has been obtained in [12] under the von Kármán scaling $\beta=4$ (see also [7]
for an extension to thin shells). These results relied on the crucial
assumption that the elastic energy density $W$ is differentiable everywhere
and its derivative satisfies a linear growth condition: $|DW(F)|\leq
C(|F|+1)$. This assumption is contradictory with the physically expected non-
interpenetration condition, and subsequently it has been removed in [11] and
exchanged with Ball’s condition (1.3), while the equilibrium equations have
been rephrased in terms of the inner variations. In the present paper we
follow the same approach; indeed the concept of inner variations comes up
naturally in the context of incompressible elasticity.
To conclude, we now comment on the isotropic case. For an isotropic energy
density $W$ with the Lamé constants $\lambda$ and $\mu$, the Euler-Lagrange
equations (1.12) – (1.13) of (1.4) are:
(1.14)
$\frac{\mu}{3}\Delta^{2}v=[v,\Phi],\qquad\Delta^{2}\Phi=-\frac{3\mu}{2}[v,v],$
where $v$ is the out-of-plane displacement, while the in-plane displacement
$w$ can be recovered through the Airy stress potential $\Phi$, by means of:
$\mbox{cof}\nabla^{2}\Phi=2\mu\Big{[}\mbox{sym}\nabla w+\frac{1}{2}\nabla
v\otimes\nabla v+\Big{(}\mbox{div}w+\frac{1}{2}|\nabla
v|^{2}\Big{)}\mbox{Id}\Big{]}.$
The Airy’s bracket $[\cdot,\cdot]$ is defined as:
$[v,\Phi]=\nabla^{2}v:(\mbox{cof}\nabla^{2}\Phi)$. As expected, the system
(1.14) can be now obtained as the incompressible limit, i.e. when passing with
the Poisson ratio $\nu\to\frac{1}{2}$, of the classical (compressible) von
Kármán system:
$B\Delta^{2}v=[v,\Phi],\qquad\Delta^{2}\Phi=-\frac{S}{2}[v,v],$
where $S=2\mu(1+\nu)$ is Young’s modulus, $\nu=\frac{\lambda}{2(\mu+\lambda)}$
is the Poisson ratio, and $B=\frac{S}{12(1-\nu^{2})}$ is bending stiffness. By
the change of variable $\Phi=2\mu\Phi_{1}$ one can eliminate the parameter
$\mu$ entirely and write (1.14) in its equivalent form:
$\Delta^{2}v=6[v,\Phi_{1}],\qquad\Delta^{2}\Phi_{1}=-\frac{3}{4}[v,v].$
Acknowledgments. M.L. was partially supported by the NSF Career grant
DMS-0846996 and by the Polish MN grant N N201 547438.
## 2\. Incompressible inner variations and critical points
Following [2], we want to define the critical points $u^{h}$ of the
functionals $J^{h}$ in (1.2) by taking inner variations. That is, we request
that the derivative of $J^{h}$ at an incompressible equilibrium $u^{h}$ be
zero along all curves $\epsilon\mapsto u^{h}_{\epsilon}$ of incompressible
deformations of $\Omega^{h}$ having the form:
$u^{h}_{\epsilon}(x)=\Phi(\epsilon,u^{h}(x))$, with $u^{h}_{0}=u^{h}$ at
$\epsilon=0$. This requirement imposes the following conditions on the flow
$\Phi:[0,\epsilon_{0})\times\mathbb{R}^{3}\rightarrow\mathbb{R}^{3}$:
(2.1) $\begin{split}&\forall\epsilon\quad\Phi(\epsilon,\cdot)\mbox{ is
incompressible, i.e }\quad\forall
y\in\mathbb{R}^{3}\quad\det\nabla\Phi(\epsilon,y)=1,\\\ &\forall
y\in\mathbb{R}^{3}\quad\Phi(0,y)=y.\end{split}$
Assuming sufficient smoothness of $\Phi$, the above immediately implies:
$\begin{split}0&=\frac{\mathrm{d}}{\mathrm{d}\epsilon}\det\nabla\Phi(0,y)=\left\langle\mbox{cof}\nabla\Phi(0,y):\frac{\mathrm{d}}{\mathrm{d}\epsilon}\nabla\Phi(0,y)\right\rangle=\left\langle\mbox{Id}:\frac{\mathrm{d}}{\mathrm{d}\epsilon}\nabla\Phi(0,y)\right\rangle\\\
&=\mbox{Tr}\left(\frac{\mathrm{d}}{\mathrm{d}\epsilon}\nabla\Phi(0,y)\right)=\mbox{div}\left(\frac{\mathrm{d}}{\mathrm{d}\epsilon}\Phi(0,y)\right)=:\mbox{div}~{}\phi(y).\end{split}$
On the other hand, any divergence-free vector field $\phi$ generates a path of
incompressible deformations. We recall this standard fact below, for the sake
of completeness.
###### Lemma 2.1.
Let $\phi\in\mathcal{C}^{1}_{b}(\mathbb{R}^{n},\mathbb{R}^{n})$ such that
$\mathrm{div}~{}\phi=0$. Consider the ODE:
(2.2) $\left\\{\begin{array}[]{ll}u^{\prime}(\epsilon)=\phi(u(\epsilon)),\\\
u(0)=y.\end{array}\right.$
and denote its flow by $\Phi(\epsilon,y)=u(\epsilon)$ solving (2.2). Then
$\Phi$ satisfies (2.1).
###### Proof.
Let $\epsilon,\delta>0$ and note that:
$\Phi(\epsilon+\delta,y)=\Phi(\delta,\Phi(\epsilon,y))=\Phi(\delta,y_{1})$
where we put $y_{1}=\Phi(\epsilon,y)$. Hence, denoting the spacial gradient by
$\nabla$, we obtain:
$\det\nabla\Phi(\epsilon+\delta,y)=\det\nabla\Phi(\delta,y_{1})\det\nabla\Phi(\epsilon,y),$
Consequently:
(2.3)
$\begin{split}\frac{\mbox{d}}{\mbox{d}\epsilon}\left(\det\nabla\Phi(\epsilon+\delta,y)\right)&=\frac{\mbox{d}}{\mbox{d}\delta}\left(\det\nabla\Phi(\epsilon+\delta,y)\right)=\frac{\mbox{d}}{\mbox{d}\delta}\left(\det\nabla\Phi(\delta,y_{1})\right)\left(\det\nabla\Phi(\epsilon,y)\right)\\\
&=\left\langle\mbox{cof}~{}\nabla\Phi(\delta,y_{1}):\frac{\mbox{d}}{\mbox{d}\delta}\nabla\Phi(\delta,y_{1})\right\rangle\det\nabla\Phi(\epsilon,y).\end{split}$
Above, we used the formula for the derivative of the determinant of a matrix
function $A(t)$, namely: $(\det A(t))^{\prime}=\mbox{cof}A(t):A(t)^{\prime}.$
For $\delta=0$, (2.3) implies:
$\frac{\mbox{d}}{\mbox{d}\epsilon}\left(\det\nabla\Phi(\epsilon,y)\right)=\langle\mbox{cof}\nabla\Phi(0,y_{1}):\nabla\phi(y_{1})\rangle=\langle\mbox{Id}:\nabla\phi(y_{1})\rangle=\mbox{Tr}\nabla\phi=\mbox{div}~{}\phi=0.$
But $\det\nabla\Phi(0,y)=\det\mbox{Id}_{n}=1$, which achieves the claim.
We are now ready to derive the equilibrium equations (1.7). The result is
essentially similar to Theorem 2.4 [2], which dealt with the compressible
inner variations $u^{h}_{\epsilon}=u^{h}(x)+\epsilon\phi\circ u^{h}$ of a
deformation $u^{h}$ with clamped boundary conditions. The growth condition
(1.3) will be crucial in passing to the limit in the nonlinear term in
$J^{h}$, to which end we are going to use the following Lemma from [2]:
###### Lemma 2.2.
(Lemma 2.5 (i) [2]) Assume that $W$ satisfies (1.3). Then there exists
$\gamma>0$ such that if $A\in\mathbb{R}^{3\times 3}_{+}$ and
$|A-\mathrm{Id}|<\gamma$, then:
$|DW(AF)F^{T}|\leq 3C(W(F)+1)\qquad\forall F\in\mathbb{R}^{3\times 3}_{+},$
where $C$ is the constant in condition (1.3).
###### Theorem 2.3.
Let $\phi\in\mathcal{C}^{1}_{b}(\mathbb{R}^{3},\mathbb{R}^{3})$ be such that
$\mathrm{div}~{}\phi=0$. Given a deformation $u^{h}\in
W^{1,2}(\Omega^{h},\mathbb{R}^{3})$ with $\det\nabla u^{h}=1$, and such that
$\int_{\Omega^{h}}W(\nabla u^{h})~{}\mathrm{d}x<+\infty$, define
$u_{\epsilon}^{h}(x)=\Phi(\epsilon,u^{h}(x))$. Then:
$\frac{\mathrm{d}}{\mathrm{d}\epsilon}_{|\epsilon=0}J^{h}(u^{h}_{\epsilon})=0$
is equivalent to:
$\int_{\Omega^{h}}\left\langle DW(\nabla u^{h})(\nabla
u^{h})^{T}:\nabla\phi(u^{h}(x))\right\rangle~{}\mathrm{d}x=\int_{\Omega^{h}}f^{h}\cdot\phi(u^{h})~{}\mathrm{d}x.$
###### Proof.
For the notational convenience, in what follows we drop the index $h$ and
write $U$ instead of $\Omega^{h}$, which stands now for a fixed open bounded
domain in $\mathbb{R}^{3}$. It is easy to notice that:
(2.4) $\lim_{\epsilon\to
0}\frac{1}{\epsilon}(\Phi(\epsilon,y)-y)=\phi(y)\quad\mbox{ uniformly in
}\mathbb{R}^{3}.$
It directly implies that:
$\lim_{\epsilon\to
0}\frac{1}{\epsilon}\int_{U}f\cdot(\Phi(\epsilon,u(x))-u(x))~{}\mbox{d}x=\int_{U}f\cdot\phi(u(x))~{}\mbox{d}x.$
To treat the nonlinear term, consider:
(2.5) $\begin{split}\frac{1}{\epsilon}&\int_{U}\big{(}W(\nabla
u_{\epsilon})-W(\nabla u)\big{)}~{}\mbox{d}x-\int_{U}\big{\langle}DW(\nabla
u)(\nabla u)^{T}:\nabla\phi(u)\big{\rangle}~{}\mbox{d}x\\\
&=\int_{U}\fint_{0}^{\epsilon}\Big{\langle}DW\big{(}\nabla\Phi(s,u)\nabla
u\big{)}(\nabla
u)^{T}:\nabla\phi(\Phi(s,u))\Big{\rangle}-\Big{\langle}DW(\nabla u)(\nabla
u)^{T}:\nabla\phi(u)\Big{\rangle}~{}\mbox{d}s~{}\mbox{d}x.\end{split}$
Since the integrand below converges to $0$ pointwise by (2.4), and it is
bounded by the function $2\|\nabla\phi\|_{L^{\infty}}|DW(\nabla u)(\nabla
u)^{T}|$ which is integrable in view of (1.3), we obtain:
$\lim_{\epsilon\to 0}\int_{U}\Big{\langle}DW(\nabla u)(\nabla
u)^{T}:\fint_{0}^{\epsilon}\nabla\phi(\Phi(s,u))-\nabla\phi(u)~{}\mbox{d}s\Big{\rangle}~{}\mbox{d}x=0,$
by the dominated convergence theorem. Similarly:
$\lim_{\epsilon\to
0}\int_{U}\fint_{0}^{\epsilon}\Big{\langle}\Big{(}DW(\nabla\Phi(s,u)\nabla
u)-DW(\nabla u)\Big{)}(\nabla
u)^{T}:\nabla\phi(\Phi(s,u))\Big{\rangle}~{}\mbox{d}s~{}\mbox{d}x=0,$
where the pointwise convergence follows by the formula (2.4), its counterpart
for $\nabla\Phi$, and the continuity of $DW$ on $\mathbb{R}^{3\times 3}_{+}$.
The integrands, for small $\epsilon$, are dominated by the $L^{1}(U)$ function
$4C\|\nabla\phi\|_{L^{\infty}}(W(\nabla u)+1)$ in view of Lemma 2.2 and the
growth condition (1.3).
Therefore, the left hand side in (2.5) converges to $0$ as well. This
completes the proof.
## 3\. The equilibrium equation (1.7)
In this section, we review several facts from [6] and [11], to set the stage
for a proof of Theorem 1.1 and to rewrite the equation (1.7) using the change
of variables (1.9).
The first crucial step in the dimension reduction argument of [6] is finding
the appropriate approximations of the deformations gradients $u^{h}$. Under
the sole assumption:
(3.1) $\frac{1}{h}\int_{\Omega^{h}}W(\nabla u^{h})~{}\mbox{d}x\leq Ch^{4},$
an application of a nonlinear verion of Korn’s inequality [5], yields
existence of rotation fields $R^{h}\in W^{1,2}(\Omega,\mathbb{R}^{3\times 3})$
with $R^{h}(x)\in SO(3)$ a.e. in $\Omega$, so that:
(3.2) $\|\nabla u^{h}(x^{\prime},hx_{3})-R^{h}\|_{L^{2}(\Omega^{1})}\leq
Ch^{2}\qquad\mbox{and}\qquad\|\nabla R^{h}\|_{L^{2}(\Omega)}\leq Ch.$
Recall that $\Omega^{1}=\Omega\times(-\frac{1}{2},\frac{1}{2})$ is the common
domain of the rescaled deformations
$y^{h}(x^{\prime},x_{3})=(\bar{R}^{h})^{T}u^{h}(x^{\prime},hx_{3})-c^{h}$, and
the typical point in $\Omega^{1}$ is denoted by $x=(x^{\prime},x_{3})$. Then,
the detailed analysis in [6] shows that convergences in (i) – (iv) of Theorem
1.1 hold, as a consequence of (1.8) implying (3.1). The constant rotations
$\bar{R}^{h}\in SO(3)$ are given by:
$\bar{R}^{h}=\mathbb{P}_{SO(3)}\left(\fint_{\Omega^{h}}\nabla
u^{h}~{}\mbox{d}x\right),$
where the orthogonal projection $\mathbb{P}_{SO(3)}$ onto $SO(3)$ above is
well defined; see also [8] for detailed calculations. Further, there holds:
(3.3) $\|R^{h}-\bar{R}^{h}\|_{L^{2}(\Omega)}\leq
Ch\qquad\mbox{and}\qquad\lim_{h\to
0}(\bar{R}^{h})^{T}R^{h}=\mbox{Id}\quad\mbox{in
}W^{1,2}(\Omega,\mathbb{R}^{3\times 3}),$
and upon defining the matrix fields $A^{h}\in
W^{1,2}(\Omega,\mathbb{R}^{3\times 3})$:
(3.4)
$A^{h}(x^{\prime})=\frac{1}{h}\left((\bar{R}^{h})^{T}R^{h}(x^{\prime})-\mbox{Id}\right),$
it also follows that:
(3.5) $A^{h}\rightharpoonup
A=\left[\begin{array}[]{c|c}0&\begin{minipage}{22.76228pt}$$\vspace{3mm}-\nabla
v$$\end{minipage}\\\ \hline\cr\nabla
v&\begin{minipage}{22.76228pt}$$\vspace{3mm}0$$\end{minipage}\end{array}\right]\qquad\mbox{weakly
in }W^{1,2}(\Omega,\mathbb{R}^{3\times 3}).$
The same convergence holds strongly in $L^{q}(\Omega,\mathbb{R}^{3\times 3})$
for each $q\geq 1$.
###### Lemma 3.1.
We have:
(3.6) $\lim_{h\to 0}y^{h}=(x^{\prime},0)\quad\mbox{ and }\quad\lim_{h\to
0}\frac{y^{h}_{3}}{h}=x_{3}+v(x^{\prime})\qquad\mbox{in
}~{}W^{1,2}(\Omega^{1}).$
Consequently, for every $\omega_{h}>0$ and $p\in[1,5]$:
(3.7)
$\left|\left\\{x\in\Omega^{1};~{}\frac{|y^{h}_{3}(x)|}{h}\geq\omega_{h}\right\\}\right|\leq\frac{C}{\omega_{h}^{2}}\quad\mbox{
and
}\quad\int_{\left\\{x\in\Omega^{1};~{}\frac{|y^{h}_{3}(x)|}{h}\geq\omega_{h}\right\\}}\left|\frac{y^{h}_{3}(x)}{h}\right|^{p}~{}\mathrm{d}x\leq\frac{C}{\omega_{h}^{\frac{2}{p+1}}}.$
###### Proof.
By (3.2), (3.3), and applying the Poincaré-Wirtinger inequality on segments
$\\{x^{\prime}\\}\times(-\frac{1}{2},\frac{1}{2})$, we see that:
$\begin{split}\left\|\frac{y_{3}^{h}}{h}-x_{3}-v^{h}(x^{\prime})\right\|_{L^{2}(\Omega^{1})}&\leq
C\Big{\|}\frac{\partial_{3}y_{3}^{h}}{h}-1\Big{\|}_{L^{2}(\Omega^{1})}=C\Big{\|}[(\bar{R}^{h})^{T}\nabla
u^{h}(x^{\prime},hx_{3})]_{33}-1\Big{\|}_{L^{2}(\Omega^{1})}\\\ &\leq
C\|(\bar{R}^{h})^{T}\nabla
u^{h}(x^{\prime},hx_{3})-\mbox{Id}\|_{L^{2}(\Omega^{1})}\\\ &\leq C\|\nabla
u^{h}(x^{\prime},hx_{3})-R^{h}\|_{L^{2}(\Omega^{1})}+C\|R^{h}-\bar{R}^{h}\|_{L^{2}(\Omega^{1})}\leq
Ch.\end{split}$
Together with (1.10), the above inequality implies the second assertion in
(3.6). The first assertion follows then directly in view of (1.11).
To prove (3.7), note that for every $p\in[1,5]$:
(3.8)
$\begin{split}\int_{\left\\{x\in\Omega^{1};~{}\frac{|y^{h}_{3}(x)|}{h}\geq\omega_{h}\right\\}}\left|\frac{y^{h}_{3}(x)}{h}\right|^{p}~{}\mathrm{d}x&\leq\left\|\frac{y^{h}_{3}}{h}\right\|^{p}_{L^{p+1}}\left|\left\\{x\in\Omega^{1};~{}\frac{|y^{h}_{3}(x)|}{h}\geq\omega_{h}\right\\}\right|^{\frac{1}{p+1}}\\\
&\leq
C\left|\left\\{x\in\Omega^{1};~{}\frac{|y^{h}_{3}(x)|}{h}\geq\omega_{h}\right\\}\right|^{\frac{1}{p+1}},\end{split}$
by the Hölder inequality and the Sobolev embedding
$W^{1,2}(\Omega^{1})\hookrightarrow L^{6}(\Omega^{1})$ combined with (3.6).
When $p=1$, it implies:
$\left|\left\\{x\in\Omega;~{}\frac{|y^{h}_{3}(x)|}{h}\geq\omega_{h}\right\\}\right|\leq\frac{1}{\omega_{h}}\int_{\left\\{x\in\Omega;~{}\frac{|y^{h}_{3}(x)|}{h}\geq\omega_{h}\right\\}}\frac{|y^{h}_{3}(x)|}{h}~{}\mathrm{d}x\leq\frac{C}{\omega_{h}}\left|\left\\{x\in\Omega;~{}\frac{|y^{h}_{3}(x)|}{h}\geq\omega_{h}\right\\}\right|^{1/2}$
Hence, the first assertion in (3.7) follows, as well as the second one, in
view of (3.8).
Define the strain $G^{h}\in L^{2}(\Omega^{1},\mathbb{R}^{3\times 3})$ and the
scaled stress $E^{h}\in L^{1}(\Omega^{1},\mathbb{R}^{3\times 3})$ as:
$\begin{split}&G^{h}(x^{\prime},x_{3})=\frac{1}{h^{2}}\left((R^{h})^{T}\nabla
u^{h}(x^{\prime},hx_{3})-\mbox{Id}\right),\\\
&E^{h}(x^{\prime},x_{3})=\frac{1}{h^{2}}DW(\mbox{Id}+h^{2}G^{h}(x^{\prime},x_{3}))(\mbox{Id}+h^{2}G^{h}(x^{\prime},x_{3}))^{T}.\end{split}$
We now gather the fundamental properties of $E^{h}$ and $G^{h}$ from [11],
that will be used in the sequel.
###### Lemma 3.2.
(Section 4, [11])
* (i)
Up to a subsequence, $G^{h}\rightharpoonup G$ weakly in
$L^{2}(\Omega^{1},\mathbb{R}^{3\times 3})$, where $G$ is the limiting strain
whose principal $2\times 2$ minor $G^{\prime\prime}$ satisfies:
(3.9)
$\begin{split}&G^{\prime\prime}(x^{\prime},x_{3})=G_{0}(x^{\prime})-x_{3}G_{1}(x^{\prime}),\quad\mbox{
with: }\\\ &\mathrm{sym}~{}G_{0}=\mathrm{sym}\nabla w+\frac{1}{2}\nabla
v\otimes\nabla v,\qquad G_{1}=\nabla^{2}v.\end{split}$
* (ii)
Each $E^{h}(x)$ is symmetric, and there holds:
(3.10) $|E^{h}|\leq
C\left(\frac{1}{h^{2}}W(\mathrm{Id}+h^{2}G^{h})+|G^{h}|\right).$
* (iii)
Up to a subsequence, $E^{h}\rightharpoonup E$ weakly in
$L^{1}(\Omega^{1},\mathbb{R}^{3\times 3})$, and $E=\mathcal{L}_{3}(G)\in
L^{2}(\Omega^{1},\mathbb{R}^{3\times 3})$.
* (iv)
For a given, fixed $\gamma\in(0,2)$, define
$B_{h}=\\{x\in\Omega^{1};~{}h^{2-\gamma}|G^{h}(x)|\leq 1\\}$. Then:
(3.11) $|\Omega^{1}\setminus B_{h}|\leq
Ch^{2(2-\gamma)}\qquad\mbox{and}\qquad\int_{\Omega^{1}\setminus
B_{h}}|E^{h}|~{}\mathrm{d}x\leq Ch^{2-\gamma}.$
Moreover, calling $\chi_{h}$ the characteristic function of $B_{h}$, we have:
(3.12) $\chi_{h}E^{h}\rightharpoonup E\quad\mbox{weakly in
}L^{2}(\Omega^{1},\mathbb{R}^{3\times 3}).$
The below more convenient form of the equilibrium condition will be repeatedly
used in the proof of Theorem 1.1.
###### Lemma 3.3.
Condition (1.7) is equivalent to:
(3.13)
$\begin{split}&\int_{\Omega^{1}}\left\langle(\bar{R}^{h})^{T}R^{h}E^{h}(x^{\prime},x_{3})(R^{h})^{T}\bar{R}^{h}:\nabla\phi(y^{h}(x^{\prime},x_{3}))\right\rangle~{}\mathrm{d}x_{3}~{}\mathrm{d}x^{\prime}\\\
&\qquad\qquad\qquad\qquad\qquad=h\int_{\Omega^{1}}\left\langle
f(x^{\prime})e_{3},\bar{R}^{h}\phi(y^{h}(x^{\prime},x_{3}))\right\rangle~{}\mathrm{d}x_{3}~{}\mathrm{d}x^{\prime},\end{split}$
for each $\phi\in\mathcal{C}_{b}^{1}(\mathbb{R}^{3},\mathbb{R}^{3})$ with
$\mathrm{div}\phi=0$.
###### Proof.
For a given divergence free
$\phi\in\mathcal{C}^{1}_{b}(\mathbb{R}^{3},\mathbb{R}^{3})$, consider:
$\psi(y)=\bar{R}^{h}\phi\left((\bar{R}^{h})^{T}y-c^{h}\right),$
which satisfies $\psi\in\mathcal{C}^{1}_{b}$ and $\mbox{div}\psi=0$, and
moreover:
$\nabla\psi\left(u^{h}(x^{\prime},hx_{3})\right)=\bar{R}^{h}\nabla\phi\left(y^{h}(x^{\prime},x_{3})\right)(\bar{R}^{h})^{T}.$
Use now (1.7) with the divergence-free test function $\psi$:
$\begin{split}&\int_{\Omega}\int_{-1/2}^{1/2}\left\langle DW\left(\nabla
u^{h}(x^{\prime},hx_{3})\right)\left(\nabla
u^{h}(x^{\prime},hx_{3})\right)^{T}:\bar{R}^{h}\nabla\phi(y^{h}(x^{\prime},x_{3}))(\bar{R}^{h})^{T}\right\rangle~{}\mbox{d}x_{3}~{}\mbox{d}x^{\prime}\\\
&\hskip
170.71652pt=h^{3}\int_{\Omega}\int_{-1/2}^{1/2}f(x^{\prime})e_{3}\cdot\bar{R}^{h}\phi(y^{h}(x^{\prime},x_{3}))~{}\mbox{d}x_{3}~{}\mbox{d}x^{\prime}.\end{split}$
The formula (3.13) now follows directly, in view of:
$\begin{split}DW(\nabla u^{h}(x^{\prime},hx_{3}))(\nabla
u^{h}(x^{\prime},hx_{3}))^{T}&=R^{h}DW(\mbox{Id}+h^{2}G^{h}(x))(\mbox{Id}+h^{2}G^{h}(x))^{T}(R^{h})^{T}\\\
&=h^{2}R^{h}E^{h}(x^{\prime},x_{3})(R^{h})^{T}.\end{split}$
## 4\. Identification of the operators in (1.12) – (1.13)
###### Lemma 4.1.
Let $G\in\mathbb{R}^{3\times 3}$ and a symmetric matrix
$E\in\mathbb{R}^{3\times 3}$ satisfy:
$\mathcal{L}_{3}(G)=E,\quad\mathrm{Tr}~{}G=0\quad\mbox{and}\quad
E_{13}=E_{23}=0.$
Then:
(4.1)
$\mathcal{L}_{2}^{in}(G^{\prime\prime})=E^{\prime\prime}-E_{33}\mathrm{Id}_{2}.$
###### Proof.
Since $\mathcal{L}$ and $\mathcal{Q}$ depend only on the symmetric parts of
their arguments, we may without loss of generality assume that $G$ is
symmetric.
Firstly, by definitions in (1.5), (1.6), it follows that for every
$F^{\prime\prime}\in\mathbb{R}^{2\times 2}$ there is a unique tangential
minimizer $d=d(F^{\prime\prime})\in\mathbb{R}^{2}$, in the sense that:
(4.2)
$\mathcal{Q}_{2}^{in}(F^{\prime\prime})=\mathcal{Q}_{3}(\left[\begin{array}[]{cc}F^{\prime\prime}&d\\\
d&-\mbox{Tr }F^{\prime\prime}\end{array}\right])\quad\mbox{ and
}\quad\Big{\langle}\mathcal{L}_{3}(\left[\begin{array}[]{cc}F^{\prime\prime}&d\\\
d&-\mbox{Tr
}F^{\prime\prime}\end{array}\right]):\left[\begin{array}[]{cc}0&c\\\
c&0\end{array}\right]\Big{\rangle}=0\quad\forall c\in\mathbb{R}^{2}.$
The second identity above is just the Euler-Lagrange equation for the
minimization in (1.5). By convexity of this minimization problem, it also
follows that $d$ is linear:
(4.3)
$d(F^{\prime\prime}+G^{\prime\prime})=d(F^{\prime\prime})+d(G^{\prime\prime})$
Observe now that:
$\begin{split}\mathcal{Q}_{2}(G^{\prime\prime})&=\mathcal{Q}_{3}(\left[\begin{array}[]{cc}G^{\prime\prime}&d(G^{\prime\prime})\\\
d(G^{\prime\prime})&G_{33}\end{array}\right])=\Big{\langle}\mathcal{L}_{3}(\left[\begin{array}[]{cc}G^{\prime\prime}&d(G^{\prime\prime})\\\
d(G^{\prime\prime})&G_{33}\end{array}\right]):\left[\begin{array}[]{cc}G^{\prime\prime}&d(G^{\prime\prime})\\\
d(G^{\prime\prime})&G_{33}\end{array}\right]\Big{\rangle}\\\
&=\Big{\langle}\Big{(}E+\mathcal{L}_{3}(\left[\begin{array}[]{cc}0&d(G^{\prime\prime})-G_{13,23}\\\
d(G^{\prime\prime})-G_{13,23}&0\end{array}\right])\Big{)}:\left[\begin{array}[]{cc}G^{\prime\prime}&d(G^{\prime\prime})\\\
d(G^{\prime\prime})&G_{33}\end{array}\right]\Big{\rangle}\\\ &=\langle
E^{\prime\prime}:G^{\prime\prime}\rangle+E_{33}G_{33}\\\
&\qquad\qquad\quad+\Big{\langle}\mathcal{L}_{3}(\left[\begin{array}[]{cc}G^{\prime\prime}&d(G^{\prime\prime})\\\
d(G^{\prime\prime})&G_{33}\end{array}\right]):\left[\begin{array}[]{cc}0&d(G^{\prime\prime})-G_{13,23}\\\
d(G^{\prime\prime})-G_{13,23}&0\end{array}\right]\Big{\rangle}\\\ &=\langle
E^{\prime\prime}:G^{\prime\prime}\rangle+E_{33}G_{33}=\langle
E:G\rangle=\mathcal{Q}_{3}(G),\end{split}$
where we repeatedly used the assumptions on $G$ and $E$, and (4.2).
Consequently, by uniqueness of the minimizer $d$, it follows that:
(4.4) $d(G^{\prime\prime})=G_{13,23}.$
Take any $F^{\prime\prime}\in\mathbb{R}^{2\times 2}$. By (4.2) and (4.3), we
see that:
$\mathcal{Q}_{2}(G^{\prime\prime}+F^{\prime\prime})=\mathcal{Q}_{3}(\left[\begin{array}[]{cc}G^{\prime\prime}+F^{\prime\prime}&d(G^{\prime\prime})+d(F^{\prime\prime})\\\
d(G^{\prime\prime})+d(F^{\prime\prime})&G_{33}-\mbox{Tr
}F^{\prime\prime}\end{array}\right]).$
Expanding the above and removing $\mathcal{Q}_{2}(G^{\prime\prime})$ and
$\mathcal{Q}_{2}(F^{\prime\prime})$ from both sides, we obtain:
$\begin{split}\langle\mathcal{L}_{2}(G^{\prime\prime}):F^{\prime\prime}\rangle&=\Big{\langle}\mathcal{L}_{3}(\left[\begin{array}[]{cc}G^{\prime\prime}&d(G^{\prime\prime})\\\
d(G^{\prime\prime})&-\mbox{Tr
}G^{\prime\prime}\end{array}\right]):\left[\begin{array}[]{cc}F^{\prime\prime}&d(F^{\prime\prime})\\\
d(F^{\prime\prime})&-\mbox{Tr
}F^{\prime\prime}\end{array}\right]\Big{\rangle}\\\
&=\Big{\langle}\mathcal{L}_{3}(\left[\begin{array}[]{cc}G^{\prime\prime}&d(G^{\prime\prime})\\\
d(G^{\prime\prime})&-\mbox{Tr
}G^{\prime\prime}\end{array}\right]):\left[\begin{array}[]{cc}F^{\prime\prime}&0\\\
0&-\mbox{Tr }F^{\prime\prime}\end{array}\right]\Big{\rangle}\\\
&=\Big{\langle}\mathcal{L}_{3}(G):\left[\begin{array}[]{cc}F^{\prime\prime}&d(F^{\prime\prime})\\\
d(F^{\prime\prime})&-\mbox{Tr
}F^{\prime\prime}\end{array}\right]\Big{\rangle}=\Big{\langle}E:\left[\begin{array}[]{cc}F^{\prime\prime}&d(F^{\prime\prime})\\\
d(F^{\prime\prime})&-\mbox{Tr
}F^{\prime\prime}\end{array}\right]\Big{\rangle}\\\ &=\langle
E^{\prime\prime}-E_{33}\mbox{Id}_{2}:F^{\prime\prime}\rangle,\end{split}$
by (4.4) and assumptions on $E$ and $G$. The expression (4.1) follows now
directly.
In section 5 below we shall prove that for almost every $x\in\Omega^{1}$ there
holds:
(4.5) $\mbox{Tr }G(x)=0\quad\mbox{and}\quad E_{13}(x)=E_{23}(x)=0.$
Therefore, recalling Lemma 3.2 (iii), we observe that the limiting stress and
strain satisfy the assumptions of Lemma 4.1 pointwise almost everywhere. We
now record the following simple conclusion which will be used in deriving the
Euler-Lagrange equations (1.12), (1.13).
###### Lemma 4.2.
Let $E,G\in L^{2}(\Omega^{1},\mathbb{R}^{3\times 3})$ be the limiting strain
and stress as in Lemma 3.2, which are related to $(w,u)$ by (3.9). Then, for
almost every $x^{\prime}\in\Omega$, there holds:
(4.6)
$\begin{split}&\int_{-1/2}^{1/2}(E^{\prime\prime}-E_{33}\mathrm{Id}_{2})~{}\mathrm{d}x_{3}=\mathcal{L}_{2}^{in}\left(\mathrm{sym}\nabla
w+\frac{1}{2}\nabla v\otimes\nabla v\right),\\\
&\int_{-1/2}^{1/2}x_{3}(E^{\prime\prime}-E_{33}\mathrm{Id}_{2})~{}\mathrm{d}x_{3}=-\frac{1}{12}\mathcal{L}_{2}^{in}\left(\nabla^{2}v\right).\end{split}$
###### Proof.
By Lemma 5.1, Lemma 5.3, Lemma 4.1 and (3.9) we see that:
$\begin{split}&\int_{-1/2}^{1/2}(E^{\prime\prime}-E_{33}\mathrm{Id}_{2})~{}\mathrm{d}x_{3}=\int_{-1/2}^{1/2}\mathcal{L}_{2}^{in}(G^{\prime\prime})~{}\mathrm{d}x_{3}\\\
&\qquad\qquad=\mathcal{L}_{2}^{in}\left(\int_{-1/2}^{1/2}G^{\prime\prime}(x^{\prime},x_{3})~{}\mathrm{d}x_{3}\right)=\mathcal{L}_{2}^{in}(G_{0}(x^{\prime}))=\mathcal{L}_{2}^{in}(\mbox{sym
}G_{0}(x^{\prime}))\\\
&\int_{-1/2}^{1/2}x_{3}(E^{\prime\prime}-E_{33}\mathrm{Id}_{2})~{}\mathrm{d}x_{3}=\int_{-1/2}^{1/2}x_{3}\mathcal{L}_{2}^{in}(G^{\prime\prime})~{}\mathrm{d}x_{3}\\\
&\qquad\qquad=\mathcal{L}_{2}^{in}\left(\int_{-1/2}^{1/2}x_{3}G^{\prime\prime}(x^{\prime},x_{3})~{}\mathrm{d}x_{3}\right)=-\mathcal{L}_{2}^{in}\left(\int_{-1/2}^{1/2}x_{3}^{2}G_{1}(x^{\prime})~{}\mathrm{d}x_{3}\right)=-\frac{1}{12}\mathcal{L}_{2}^{in}(G_{1}(x^{\prime})).\end{split}$
This concludes the proof, in view of (3.9).
## 5\. Two further properties of $G$ and $E$
In this section we derive the two fundamental properties of the incompressible
stress and strain, allowing for pointwise application of Lemma 4.1, and
ultimately leading to formulas in (4.6).
###### Lemma 5.1.
The limiting strain $G(x)$ is traceless, for almost every $x\in\Omega^{1}$.
###### Proof.
Recall that $\nabla
u^{h}(x^{\prime},hx_{3})=R^{h}(x^{\prime})\big{(}\mbox{Id}+h^{2}G^{h}(x^{\prime},x_{3})\big{)}$.
Therefore:
$1=\det\nabla u^{h}=\det(\mbox{Id}+h^{2}G^{h})=1+h^{2}\mbox{Tr
}G^{h}+h^{4}\mbox{Tr }\mbox{cof }G^{h}+h^{6}\det G^{h},$
and consequently:
(5.1) $\mbox{Tr }G^{h}+h^{2}\mbox{Tr }\mbox{cof }G^{h}+h^{4}\det G^{h}=0.$
Fix an exponent $\gamma\in(\frac{2}{3},2)$ and define
$B_{h}=\\{x\in\Omega^{1};~{}h^{2-\gamma}|G^{h}(x)\leq 1\\}$ as in Lemma 3.2
(iv). Then:
$\begin{split}\int_{\Omega^{1}\setminus B_{h}}|h^{4}\det
G^{h}|&=\int_{\Omega^{1}\setminus B_{h}}|\mbox{Tr }G^{h}+h^{2}\mbox{Tr
}\mbox{cof }G^{h}|\\\ &\leq|\Omega^{1}\setminus
B_{h}|^{1/2}\left(\int_{\Omega^{1}\setminus B_{h}}|\mbox{Tr
}G^{h}|^{2}\right)^{1/2}+h^{2}\int_{\Omega^{1}}|\mbox{Tr }\mbox{cof
}G^{h}|\leq C(h^{2-\gamma}+h^{2}),\end{split}$
where we used (3.11) and the boundedness of $G^{h}$ in $L^{2}(\Omega^{1})$. On
the other hand, we have:
$\int_{B_{h}}|h^{4}\det
G^{h}|=\frac{h^{4}}{h^{6-3\gamma}}\int_{B_{h}}|\det(h^{2-\gamma}G^{h})|\leq
Ch^{3\gamma-2}.$
Hence, by (5.1) and, again the boundedness of $\mbox{Tr }\mbox{cof }G^{h}$ in
$L^{1}(\Omega^{1})$, it follows that:
$\int_{\Omega^{1}}|\mbox{Tr }G^{h}|\leq\int_{\Omega^{1}}|h^{2}\mbox{Tr
}\mbox{cof }G^{h}|+\int_{\Omega^{1}}|h^{4}\det G^{h}|\to 0,\quad\mbox{as }h\to
0.$
Observing that $\mbox{Tr }G^{h}\rightharpoonup\mbox{Tr }G$ weakly in
$L^{2}(\Omega^{1})$, we conclude that $\mbox{Tr }G=0$.
We now prove the remaining property of the strain $E$ in (4.5). The strategy
of proof is the same as in the later proofs of the Euler-Lagrange equations;
we will apply the equilibrium equation (3.13) to appropriate test functions
$\phi^{h}$, such that after passing to the limit with $h\to 0$ only some
chosen terms will survive, yielding the week formulation of (4.5). One
difficulty with (3.13) is that it only allows for globaly bounded $\phi^{h}$.
For this reason, following [11], we introduce a family of truncation functions
$\theta^{h}$ which coincide with the identity on intervals
$(-\omega_{h},\omega_{h})$ with a suitable rate of convergence of
$\omega_{h}\to\infty$.
###### Lemma 5.2.
Let $\\{\omega_{h}\\}$ be a sequence of positive numbers, increasing to
$+\infty$ as $h\to 0$. There exists a sequence of nondecreasing functions
$\theta^{h}\in\mathcal{C}_{b}^{2}(\mathbb{R},\mathbb{R})$ with the following
properties:
(5.2)
$\begin{split}&\theta^{h}(t)=t\quad\forall|t|\leq\omega_{h}\qquad\mbox{and}\qquad\theta^{h}(t)=(\mathrm{sgn}~{}t)\frac{3}{2}\omega_{h}\quad\forall|t|\geq
2\omega_{h}\\\ &|\theta^{h}(t)|\leq t\quad\forall
t\qquad\mbox{and}\qquad\|\theta^{h}\|_{L^{\infty}}\leq\frac{3}{2}\omega_{h}\\\
&\|\frac{\mathrm{d}}{\mathrm{d}t}{\theta^{h}}\|_{L^{\infty}}\leq
1\qquad\mbox{and}\qquad\|\frac{\mathrm{d^{2}}}{\mathrm{d}t^{2}}{\theta^{h}}\|_{L^{\infty}}\leq\frac{C}{\omega_{h}}.\end{split}$
###### Proof.
One may take:
$\theta^{h}(t)=\left\\{\begin{array}[]{ll}t&\mbox{for }|t|\leq\omega_{h}\\\
\displaystyle{(\mbox{sgn
}t)\frac{1}{2}\left(|t|+\omega_{h}+\frac{\omega_{h}}{\pi}\sin\left(\frac{\pi|t|-\omega_{h}}{\omega_{h}}\right)\right)}&\mbox{for
}|t|\in[\omega_{h},2\omega_{h}]\\\ \displaystyle{(\mbox{sgn
}t)\frac{3}{2}\omega_{h}}&\mbox{for }|t|\geq\omega_{h}\end{array}\right.$
###### Lemma 5.3.
The limiting stress $E(x)$ satisfies: $E_{13}(x)=E_{23}(x)=0$ for almost every
$x\in\Omega^{1}$.
###### Proof.
1. Let $\eta=(\eta_{1},\eta_{2})\in\mathcal{C}_{b}^{2}(\mathbb{R}^{3},\mathbb{R}^{2})$ be a given test function, and define:
(5.3) $\eta_{3}(x^{\prime},x_{3})=-\int_{0}^{x_{3}}\mbox{div
}\eta(x^{\prime},s)~{}\mbox{d}s.$
Since $\partial_{3}\eta_{3}=-\mbox{div }\eta$, the following test functions
$\phi^{h}\in\mathcal{C}^{1}_{b}(\mathbb{R}^{3},\mathbb{R}^{3})$ are
divergence-free:
$\phi^{h}(x^{\prime},x_{3})=\left[\begin{array}[]{c}\displaystyle{h{\theta^{h}}^{\prime}\left(\frac{x_{3}}{h}\right)\eta\left(x^{\prime},\theta^{h}\left(\frac{x_{3}}{h}\right)\right)\vspace{2mm}}\\\
\displaystyle{h^{2}\eta_{3}\left(x^{\prime},\theta^{h}\left(\frac{x_{3}}{h}\right)\right)}\end{array}\right],$
and denoting $\nabla_{tan}$ the gradient in the tangential directions
$e_{1},e_{2}$, we have:
$\nabla\phi^{h}(x^{\prime},x_{3})=\left[\begin{array}[]{c|c}\displaystyle{h{\theta^{h}}^{\prime}\left(\frac{x_{3}}{h}\right)\nabla_{tan}\eta\left(x^{\prime},\theta^{h}\left(\frac{x_{3}}{h}\right)\right)}&\begin{minipage}{170.71652pt}$$\displaystyle{\left({\theta^{h}}^{\prime}\left(\frac{x_{3}}{h}\right)\right)^{2}\partial_{3}\eta\left(x^{\prime},\theta^{h}\left(\frac{x_{3}}{h}\right)\right)}$$
$$\qquad\qquad+~{}\displaystyle{{\theta^{h}}^{\prime\prime}\left(\frac{x_{3}}{h}\right)\eta\left(x^{\prime},\theta^{h}\left(\frac{x_{3}}{h}\right)\right)}$$\vspace{2mm}\end{minipage}\\\
\hline\cr\displaystyle{h^{2}\nabla_{tan}\eta_{3}\left(x^{\prime},\theta^{h}\left(\frac{x_{3}}{h}\right)\right)}&\begin{minipage}{170.71652pt}$$\vspace{2mm}\displaystyle{h{\theta^{h}}^{\prime}\left(\frac{x_{3}}{h}\right)\partial_{3}\eta_{3}\left(x^{\prime},\theta^{h}\left(\frac{x_{3}}{h}\right)\right)}$$\end{minipage}\end{array}\right].$
The truncations $\theta^{h}$ are chosen as in Lemma 5.2 and such that:
(5.4) $\lim_{h\to 0}\omega_{h}=+\infty\quad\mbox{ and }\quad
h^{2}\omega_{h}\leq C.$
2. Applying the equilibrium equation (3.13) with $\phi=\phi^{h}$, we obtain:
(5.5)
$\begin{split}h\int_{\Omega^{1}}\Big{\langle}&\Big{(}(\bar{R}^{h})^{T}R^{h}E^{h}(R^{h})^{T}\bar{R}^{h}\Big{)}^{\prime\prime}-\Big{(}(\bar{R}^{h})^{T}R^{h}E^{h}(R^{h})^{T}\bar{R}^{h}\Big{)}_{33}\mbox{Id}_{2}:{\theta^{h}}^{\prime}\left(\frac{y^{h}_{3}}{h}\right)\nabla_{tan}\eta({y^{h}}^{\prime},\theta^{h}\left(\frac{y_{3}^{h}}{h}\right))\Big{\rangle}\\\
&\qquad\qquad+\int_{\Omega^{1}}\Big{\langle}\Big{(}(\bar{R}^{h})^{T}R^{h}E^{h}(R^{h})^{T}\bar{R}^{h}\Big{)}_{13,23},({\theta^{h}}^{\prime}\left(\frac{y^{h}_{3}}{h}\right))^{2}\partial_{3}\eta({y^{h}}^{\prime},\theta^{h}\left(\frac{y_{3}^{h}}{h}\right))\Big{\rangle}\\\
&\qquad\qquad+\int_{\Omega^{1}}\Big{\langle}\Big{(}(\bar{R}^{h})^{T}R^{h}E^{h}(R^{h})^{T}\bar{R}^{h}\Big{)}_{13,23},{\theta^{h}}^{\prime\prime}\left(\frac{y^{h}_{3}}{h}\right)\eta({y^{h}}^{\prime},\theta^{h}\left(\frac{y_{3}^{h}}{h}\right))\Big{\rangle}\\\
&\qquad\qquad+h^{2}\int_{\Omega^{1}}\Big{\langle}\Big{(}(\bar{R}^{h})^{T}R^{h}E^{h}(R^{h})^{T}\bar{R}^{h}\Big{)}_{31,32},\nabla_{tan}\eta_{3}({y^{h}}^{\prime},\theta^{h}\left(\frac{y_{3}^{h}}{h}\right))\Big{\rangle}\\\
&=h^{2}\int_{\Omega^{1}}\Big{\langle}f(x^{\prime})(\bar{R}^{h})_{31,32},\eta({y^{h}}^{\prime},\theta^{h}\left(\frac{y_{3}^{h}}{h}\right))\Big{\rangle}+h^{3}\int_{\Omega^{1}}f(x^{\prime})(\bar{R}^{h})_{33}\eta_{3}({y^{h}}^{\prime},\theta^{h}\left(\frac{y_{3}^{h}}{h}\right)).\end{split}$
Now, we will discuss the convergence as $h\to 0$ of each term in (5.5). The
first term converges to $0$, because
$\Big{(}(\bar{R}^{h})^{T}R^{h}E^{h}(R^{h})^{T}\bar{R}^{h}\Big{)}^{\prime\prime}-\Big{(}(\bar{R}^{h})^{T}R^{h}E^{h}(R^{h})^{T}\bar{R}^{h}\Big{)}_{33}\mbox{Id}_{2}$
is bounded in $L^{1}(\Omega^{1})$ in view of Lemma 3.2 (iii), while
${\theta^{h}}^{\prime}\left(\frac{y^{h}_{3}}{h}\right)\nabla_{tan}\eta({y^{h}}^{\prime},\theta^{h}\left(\frac{y_{3}^{h}}{h}\right))$
is pointwise bounded by (5.2).
3. The second term in (5.5) when integrated over $\Omega^{1}\setminus B_{h}$, goes to $0$ in view of (3.11) and of the pointwise boundedness of $({\theta^{h}}^{\prime}\left(\frac{y^{h}_{3}}{h}\right))^{2}\partial_{3}\eta({y^{h}}^{\prime},\theta^{h}\left(\frac{y_{3}^{h}}{h}\right))$ by (5.2). On the other hand, the limit of this integral over $B_{h}$ is the same as the limit of:
(5.6)
$\int_{\Omega^{1}}\Big{\langle}\chi_{h}E^{h}_{13,23},({\theta^{h}}^{\prime}\left(\frac{y^{h}_{3}}{h}\right))^{2}\partial_{3}\eta({y^{h}}^{\prime},\theta^{h}\left(\frac{y_{3}^{h}}{h}\right))\Big{\rangle}~{}\mbox{d}x$
because of (3.3). We now conclude that the integrals in (5.6) converge to:
$\int_{\Omega^{1}}\Big{\langle}E_{13,23},\partial_{3}\eta(x^{\prime},x_{3}+v(x^{\prime}))\Big{\rangle}~{}\mbox{d}x.$
This follows by recalling (3.12) and observing that:
(5.7)
$({\theta^{h}}^{\prime}\left(\frac{y^{h}_{3}}{h}\right))^{2}\partial_{3}\eta({y^{h}}^{\prime},\theta^{h}\left(\frac{y_{3}^{h}}{h}\right))\rightarrow\partial_{3}\eta(x^{\prime},x_{3}+v(x^{\prime}))\quad\mbox{
in }L^{2}(\Omega^{1})$
Indeed:
$\begin{split}\int_{\Omega^{1}}&\left|({\theta^{h}}^{\prime}\left(\frac{y^{h}_{3}}{h}\right))^{2}\partial_{3}\eta({y^{h}}^{\prime},\theta^{h}\left(\frac{y_{3}^{h}}{h}\right))-\partial_{3}\eta(x^{\prime},x_{3}+v(x^{\prime}))\right|^{2}~{}\mbox{d}x\\\
&\leq
C\int_{\Omega^{1}}\left|{\theta^{h}}^{\prime}\left(\frac{y^{h}_{3}}{h}\right)\right|^{4}\left(|{y^{h}}^{\prime}-x^{\prime}|^{2}+\left|\theta^{h}\left(\frac{y_{3}^{h}}{h}\right)-(x_{3}+v(x^{\prime}))\right|^{2}\right)~{}\mbox{d}x\\\
&\qquad+C\int_{\Omega^{1}}\left|{\theta^{h}}^{\prime}\left(\frac{y^{h}_{3}}{h}\right)-1\right|^{2}~{}\mbox{d}x\\\
&\leq
C\int_{\Omega^{1}}|{y^{h}}^{\prime}-x^{\prime}|^{2}+\left|\frac{y_{3}^{h}}{h}-(x_{3}+v(x^{\prime}))\right|^{2}~{}\mbox{d}x+C\int_{\left\\{x\in\Omega^{1};~{}\frac{|y^{h}_{3}|}{h}\geq\omega_{h}\right\\}}1+\left|\frac{y_{3}^{h}}{h}\right|^{2}~{}\mbox{d}x\end{split}$
converges to $0$ as $h\to 0$, by (3.6), (3.7) and (5.4), proving hence (5.7).
4. The third term in (5.5) is bounded by: $\frac{C}{\omega_{h}}\int_{\Omega^{1}}|E^{h}|$ by (5.2). It therefore converges to $0$ in view of the boundedness of $E^{h}$ in $L^{1}(\Omega^{1})$ and (5.4).
The fourth term in (5.5) is bounded by:
$\begin{split}Ch^{2}\int_{\Omega^{1}}|E^{h}|\left|\theta^{h}\left(\frac{y^{h}_{3}}{h}\right)\right|~{}\mbox{d}x&\leq
Ch^{2}\omega_{h}\int_{\Omega^{1}\setminus
B_{h}}|E^{h}|+Ch^{2}\int_{\Omega^{1}}\chi_{h}|E^{h}|\frac{|y^{h}_{3}|}{h}\\\
&\leq
Ch^{2}\omega_{h}~{}{o}(1)+Ch^{2}\|\chi_{h}E^{h}\|_{L^{2}(\Omega^{1})}\left\|\frac{y^{h}_{3}}{h}\right\|_{L^{2}(\Omega^{1})},\end{split}$
and it converges to $0$ by (3.11), (3.12), (5.4) and the boundedness of
$\frac{y^{h}_{3}}{h}$ in $L^{2}(\Omega^{1})$.
Finally, both terms in the right hand side of (5.5) are bounded by:
$\begin{split}Ch^{2}\int_{\Omega^{1}}|f(x^{\prime})|\left(\left|{\theta^{h}}^{\prime}\left(\frac{y^{h}_{3}}{h}\right)\right|+h\left|\theta^{h}\left(\frac{y^{h}_{3}}{h}\right)\right|\right)~{}\mbox{d}x\leq
Ch^{2}\int_{\Omega^{1}}|f(x^{\prime})|(1+h\omega_{h})~{}\mbox{d}x\leq
Ch\|f\|_{L^{2}(\Omega)},\end{split}$
which clearly converges to $0$. Above, we used (5.2) and (5.4).
5. In conclusion, passing to the limit with $h\to 0$ in (5.5), results in:
(5.8)
$\int_{\Omega^{1}}\Big{\langle}E_{13,23},\partial_{3}\eta(x^{\prime},x_{3}+v(x^{\prime}))\Big{\rangle}~{}\mbox{d}x=0\qquad\forall\eta\in\mathcal{C}^{2}_{b}(\mathbb{R}^{3},\mathbb{R}^{2}).$
We now reproduce an argument from [11], in order to deduce that $E_{13,23}=0.$
Take an arbitrary $\phi\in\mathcal{C}^{2}_{c}(\Omega,\mathbb{R}^{2}).$ Let
$\mathcal{C}_{c}^{\infty}(\Omega,\mathbb{R})\ni{v}_{k}\rightarrow v$ in
$L^{2}(\Omega)$, and define:
$\phi_{k}(x^{\prime},x_{3})=\phi(x^{\prime},x_{3}-v_{k}(x^{\prime})),\qquad\eta(x^{\prime},x_{3})=\int_{0}^{x_{3}}\phi_{k}(x^{\prime},s)~{}\mbox{d}s$
Clearly $\phi_{k}\in\mathcal{C}^{2}_{c}(\mathbb{R}^{3},\mathbb{R}^{2})$,
$\eta\in\mathcal{C}^{2}_{b}(\mathbb{R}^{3},\mathbb{R}^{2})$, and thus by (5.8)
we obtain:
$0=\int_{\Omega^{1}}\Big{\langle}E_{13,23},\phi_{k}(x^{\prime},x_{3}+v(x^{\prime}))\Big{\rangle}~{}\mbox{d}x=\int_{\Omega^{1}}\Big{\langle}E_{13,23},\phi(x^{\prime},x_{3}+v(x^{\prime})-v_{k}(x^{\prime}))\Big{\rangle}~{}\mbox{d}x$
Passing to the limit with $k\to\infty$, it follows that:
$\int_{\Omega^{1}}E_{13,23}\phi(x^{\prime},x_{3})~{}\mbox{d}x=0\qquad\forall\phi\in\mathcal{C}^{2}_{c}(\Omega,\mathbb{R}^{2})$
which concludes the proof.
## 6\. Derivation of the first Euler-Lagrange equation (1.12)
1. Let $\eta=(\eta_{1},\eta_{2})\in\mathcal{C}_{b}^{2}(\mathbb{R}^{2},\mathbb{R}^{2})$ be a given test function, and let $\eta_{3}(x^{\prime})=-\mbox{div }\eta(x^{\prime})$. Given $\theta^{h}$ as in Lemma 5.2, with:
(6.1) $\lim_{h\to 0}\omega_{h}=\lim_{h\to 0}h\omega_{h}^{2}=+\infty\quad\mbox{
and }\quad h\omega_{h}\leq C,$
consider the following divergence-free test functions
$\phi^{h}\in\mathcal{C}^{1}_{b}(\mathbb{R}^{3},\mathbb{R}^{3})$:
$\phi^{h}(x^{\prime},x_{3})=\left[\begin{array}[]{c}\displaystyle{{\theta^{h}}^{\prime}\left(\frac{x_{3}}{h}\right)\eta(x^{\prime})\vspace{2mm}}\\\
\displaystyle{h{\theta^{h}}\left(\frac{x_{3}}{h}\right)\eta_{3}(x^{\prime})}\end{array}\right],$
Denoting $\nabla_{tan}$ the gradient in the tangential directions
$e_{1},e_{2}$, we have:
$\nabla\phi^{h}(x^{\prime},x_{3})=\left[\begin{array}[]{c|c}\displaystyle{{\theta^{h}}^{\prime}\left(\frac{x_{3}}{h}\right)\nabla_{tan}\eta(x^{\prime})}&\begin{minipage}{85.35826pt}$$\vspace{3mm}\displaystyle{\frac{1}{h}{\theta^{h}}^{\prime\prime}\left(\frac{x_{3}}{h}\right)\eta(x^{\prime})}$$\end{minipage}\\\
\hline\cr\displaystyle{h{\theta^{h}}\left(\frac{x_{3}}{h}\right)\nabla_{tan}\eta_{3}(x^{\prime})}&\begin{minipage}{85.35826pt}$$\displaystyle{{\theta^{h}}^{\prime}\left(\frac{x_{3}}{h}\right)\eta_{3}(x^{\prime})}$$\end{minipage}\end{array}\right].$
2. Applying the equilibrium equation (3.13) with $\phi=\phi^{h}$, we obtain:
(6.2)
$\begin{split}\int_{\Omega^{1}}\Big{\langle}&\Big{(}(\bar{R}^{h})^{T}R^{h}E^{h}(R^{h})^{T}\bar{R}^{h}\Big{)}^{\prime\prime}-\Big{(}(\bar{R}^{h})^{T}R^{h}E^{h}(R^{h})^{T}\bar{R}^{h}\Big{)}_{33}\mbox{Id}_{2}:{\theta^{h}}^{\prime}\left(\frac{y^{h}_{3}}{h}\right)\nabla_{tan}\eta({y^{h}}^{\prime})\Big{\rangle}\\\
&\qquad\qquad+h\int_{\Omega^{1}}\Big{\langle}\Big{(}(\bar{R}^{h})^{T}R^{h}E^{h}(R^{h})^{T}\bar{R}^{h}\Big{)}_{31,32},{\theta^{h}}\left(\frac{y^{h}_{3}}{h}\right)\nabla_{tan}\eta_{3}({y^{h}}^{\prime})\Big{\rangle}\\\
&\qquad\qquad+\frac{1}{h}\int_{\Omega^{1}}\Big{\langle}\Big{(}(\bar{R}^{h})^{T}R^{h}E^{h}(R^{h})^{T}\bar{R}^{h}\Big{)}_{13,23},{\theta^{h}}^{\prime\prime}\left(\frac{y^{h}_{3}}{h}\right)\eta({y^{h}}^{\prime})\Big{\rangle}\\\
&=h\int_{\Omega^{1}}\left\langle
f(x^{\prime})(\bar{R}^{h})_{31,32},{\theta^{h}}^{\prime}\left(\frac{y^{h}_{3}}{h}\right)\eta({y^{h}}^{\prime})\right\rangle~{}\mbox{d}x+h^{2}\int_{\Omega^{1}}f(x^{\prime})(\bar{R}^{h})_{33}{\theta^{h}}\left(\frac{y^{h}_{3}}{h}\right)\eta_{3}({y^{h}}^{\prime})~{}\mbox{d}x.\end{split}$
Now, we will check convergence as $h\to 0$ of each of the four terms in the
identity (6.2). Regarding the first term, it converges to $0$ when integrated
over $\Omega^{1}\setminus B_{h}$, by (3.11) and by the pointwise boundedness
of
${\theta^{h}}^{\prime}\left(\frac{y^{h}_{3}}{h}\right)\nabla_{tan}\eta({y^{h}}^{\prime})$
in view of (5.2). On the other hand, the limit of this integral over $B_{h}$
is the same as the limit of:
(6.3)
$\int_{\Omega^{1}}\Big{\langle}\chi_{h}\big{(}{E^{h}}^{\prime\prime}-E^{h}_{33}\mbox{Id}_{2}\big{)}:{\theta^{h}}^{\prime}\left(\frac{y^{h}_{3}}{h}\right)\nabla_{tan}\eta({y^{h}}^{\prime})\Big{\rangle}~{}\mbox{d}x,$
because of the convergence in (3.3). Now, the limit of integrals in (6.3)
equals:
$\int_{\Omega^{1}}\Big{\langle}{E}^{\prime\prime}-E_{33}\mbox{Id}_{2}:\nabla\eta(x^{\prime})\Big{\rangle}~{}\mbox{d}x,$
in view of (3.12) and:
$\begin{split}\int_{\Omega^{1}}&\left|{\theta^{h}}^{\prime}\left(\frac{y^{h}_{3}}{h}\right)\nabla_{tan}\eta({y^{h}}^{\prime})-\nabla\eta(x^{\prime})\right|^{2}~{}\mbox{d}x\\\
&\qquad\qquad\leq
C\int_{\Omega^{1}}\left|\nabla_{tan}\eta({y^{h}}^{\prime})-\nabla\eta(x^{\prime})\right|^{2}+C\int_{\Omega^{1}}\left|{\theta^{h}}^{\prime}\left(\frac{y^{h}_{3}}{h}\right)-1\right|^{2}\\\
&\qquad\qquad\leq
C\int_{\Omega^{1}}\left|{y^{h}}^{\prime}-x^{\prime}\right|^{2}~{}\mbox{d}x+C\left|\left\\{x\in\Omega^{1};~{}\frac{|y^{h}_{3}(x)|}{h}\geq\omega_{h}\right\\}\right|\\\
&\qquad\qquad\leq
C\int_{\Omega^{1}}\left|{y^{h}}^{\prime}-x^{\prime}\right|^{2}~{}\mbox{d}x+\frac{C}{\omega_{h}^{2}},\end{split}$
where we apply (3.7), and then (3.7) and (6.1) to conclude the convergence of
both terms in the right hand side of the above displayed expression to $0$.
3. The second term in (6.2) is bounded by:
$\begin{split}Ch\int_{\Omega^{1}\setminus
B_{h}}{\theta^{h}}&\left(\frac{|y^{h}_{3}|}{h}\right)|E^{h}|~{}\mbox{d}x+Ch\int_{\Omega^{1}}|\chi_{h}E^{h}|\frac{|y^{h}_{3}|}{h}~{}\mbox{d}x\\\
&\leq Ch\omega_{h}\int_{\Omega^{1}\setminus
B_{h}}|E^{h}|~{}\mbox{d}x+C\|y^{h}_{3}\|_{L^{2}(\Omega^{1})}\|\chi_{h}E^{h}\|_{L^{2}(\Omega^{1})}\end{split}$
and it clearly converges to $0$ by (3.11), (3.12), (3.6) and (6.1).
The third term in (6.2) is bounded by:
$\begin{split}&\frac{C}{h\omega_{h}}\int_{\left\\{x\in\Omega^{1};~{}\frac{|y^{h}_{3}(x)|}{h}\geq\omega_{h}\right\\}}|E^{h}|~{}\mbox{d}x\leq\frac{C}{h\omega_{h}}\int_{\left\\{x\in\Omega^{1};~{}\frac{|y^{h}_{3}(x)|}{h}\geq\omega_{h}\right\\}}\frac{1}{h^{2}}W(\mbox{Id}+h^{2}G^{h})+|G^{h}|~{}\mbox{d}x\\\
&\qquad\leq\frac{C}{h^{3}\omega_{h}}\int_{\Omega^{1}}W\big{(}\nabla
u^{h}(x^{\prime},hx_{3})\big{)}~{}\mbox{d}x+\frac{C}{h\omega_{h}}\|G^{h}\|_{L^{2}(\Omega^{1})}\left|\left\\{x\in\Omega^{1};~{}\frac{|y^{h}_{3}(x)|}{h}\geq\omega_{h}\right\\}\right|^{1/2}\\\
&\qquad\leq
C\left(\frac{h}{\omega_{h}}+\frac{1}{h\omega_{h}^{2}}\right),\end{split}$
by (3.10), (3.7), the boundedness of $G^{h}$ in $L^{2}(\Omega^{1})$ and (1.8).
Then, the right hand side above converges to $0$ by (6.1).
Finally, the right hand side of (6.2) converges to $0$ as well, as it is
bounded by:
$Ch\int_{\Omega^{1}}|f(x^{\prime})|(1+h\omega_{h})~{}\mbox{d}x\leq
Ch\|f\|_{L^{2}(\Omega)}.$
In conclusion, passing to the limit with $h\to 0$ in (6.2) we obtain:
(6.4)
$\int_{\Omega^{1}}\Big{\langle}{E}^{\prime\prime}-E_{33}\mbox{Id}_{2}:\nabla\eta(x^{\prime})\Big{\rangle}~{}\mbox{d}x=0\quad\forall\eta\in\mathcal{C}^{2}_{b}(\mathbb{R}^{2},\mathbb{R}^{2}).$
and thus the Euler-Lagrange equation (1.12) follows directly, in view of (4.6)
and the density of test functions $\eta$ as above in
$W^{1,2}(\Omega,\mathbb{R}^{2})$.
## 7\. Derivation of the second Euler-Lagrange equation (1.13)
###### Lemma 7.1.
For every $\eta_{3}\in\mathcal{C}_{b}^{3}(\mathbb{R}^{2},\mathbb{R})$, it
follows that:
(7.1)
$\begin{split}\int_{\Omega^{1}}\Big{\langle}(E^{\prime\prime}-E_{33}\mathrm{Id}_{2}):\nabla
v\otimes\nabla\eta_{3}\Big{\rangle}\mathrm{d}x~{}+~{}&\lim_{h\to
0}\frac{1}{h}\int_{\Omega^{1}}\left\langle
E^{h}_{31,32},\nabla\eta_{3}({y^{h}}^{\prime})\right\rangle\mathrm{d}x\\\
&\qquad=\bar{R}_{33}\int_{\Omega}f(x^{\prime})\eta_{3}(x^{\prime})~{}\mathrm{d}x^{\prime}.\end{split}$
###### Proof.
1. Given $\eta_{3}\in\mathcal{C}_{b}^{3}(\mathbb{R}^{2},\mathbb{R})$ consider the divergence-free test functions $\phi^{h}\in\mathcal{C}_{b}^{!}(\mathbb{R}^{3},\mathbb{R}^{3})$:
$\phi^{h}(x^{\prime},x_{3})=\left[\begin{array}[]{c}0\vspace{2mm}\\\
\displaystyle{\frac{1}{h}\eta_{3}(x^{\prime})}\end{array}\right],\quad\mbox{so
that}\quad\nabla\phi^{h}(x^{\prime},x_{3})=\left[\begin{array}[]{c|c}0&\begin{minipage}{28.45274pt}$$\vspace{3mm}0$$\end{minipage}\\\
\hline\cr\displaystyle{\frac{1}{h}\nabla_{tan}\eta_{3}(x^{\prime})}&\begin{minipage}{28.45274pt}$$\vspace{3mm}0$$\end{minipage}\end{array}\right].$
Applying the equilibrium equation (3.13) with $\phi=\phi^{h}$, we obtain:
(7.2)
$\frac{1}{h}\int_{\Omega^{1}}\Big{\langle}\Big{(}(\bar{R}^{h})^{T}R^{h}E^{h}(R^{h})^{T}\bar{R}^{h}\Big{)}_{31,32},\nabla_{tan}\eta_{3}({y^{h}}^{\prime})\Big{\rangle}~{}\mbox{d}x=\bar{R}_{33}^{h}\int_{\Omega^{1}}f(x^{\prime})\eta_{3}({y^{h}}^{\prime})~{}\mbox{d}x.$
Recall that the tensor field $A^{h}$ in (3.4) is defined as:
$A^{h}(x^{\prime})=\frac{1}{h}\left((\bar{R}^{h})^{T}R^{h}(x^{\prime})-\mbox{Id}\right)$.
Hence:
(7.3)
$\begin{split}\frac{1}{h}(\bar{R}^{h})^{T}R^{h}E^{h}(R^{h})^{T}\bar{R}^{h}=A^{h}E^{h}(R^{h})^{T}\bar{R}^{h}+E^{h}(A^{h})^{T}+\frac{1}{h}E^{h},\end{split}$
and therefore the left hand side of (7.2) can be written as:
(7.4)
$\begin{split}\int_{\Omega^{1}}&\left\langle(A^{h}E^{h}(R^{h})^{T}\bar{R}^{h})_{31,32},\nabla\eta_{3}({y^{h}}^{\prime})\right\rangle~{}\mbox{d}x\\\
&+\int_{\Omega^{1}}\left\langle(E^{h}(A^{h})^{T})_{31,32},\nabla\eta_{3}({y^{h}}^{\prime})\right\rangle~{}\mbox{d}x+\frac{1}{h}\int_{\Omega^{1}}\left\langle
E_{31,32}^{h},\nabla\eta_{3}({y^{h}}^{\prime})\right\rangle~{}\mbox{d}x.\end{split}$
2. Let the sets $B_{h}$ be defined as in Lemma 3.2 (iv), for some exponent $\gamma\in(0,1)$. The first two terms in (7.4), when considered on $\Omega^{1}\setminus B_{h}$, converge to $0$ because they are bounded by:
$C\int_{\Omega^{1}\setminus
B_{h}}|A^{h}||E^{h}|~{}\mbox{d}x\leq\frac{C}{h}\int_{\Omega^{1}\setminus
B_{h}}|E^{h}|~{}\mbox{d}x\leq\frac{C}{h}h^{2-\gamma},$
in view of (3.11) and $|A^{h}|\leq\frac{C}{h}$. On the other hand, the same
two terms while on $B_{h}$, converge to:
$\int_{\Omega^{1}}\left\langle(AE)_{31,32},\nabla\eta_{3}(x^{\prime})\right\rangle+\left\langle(EA^{T})_{31,32},\nabla\eta_{3}(x^{\prime})\right\rangle~{}\mbox{d}x,$
where we used the convergence (3.12) and the following strong convergences in
$L^{3}(\Omega^{1})$: of $A^{h}$ to $A$ by (3.5), of $(R^{h})^{T}\bar{R}^{h}$
to Id by (3.3), and of $\nabla\eta_{3}({y^{h}}^{\prime})$ to
$\nabla\eta_{3}(x^{\prime})$ in view of the Sobolev embedding and the strong
convergence in $W^{1,2}(\Omega^{1},\mathbb{R}^{2})$ in (3.6).
Concluding, the first two terms in (7.4) converge to:
$\int_{\Omega^{1}}\Big{\langle}E^{\prime\prime}\nabla
v,\nabla\eta_{3}(x^{\prime})\Big{\rangle}-\Big{\langle}E_{33}\nabla
v,\nabla\eta_{3}(x^{\prime})\Big{\rangle}~{}\mbox{d}x$
in view of the structure of the limiting tensor $A$ in (3.5). Since the right
hand side of (7.2) converges to
$\bar{R}_{33}\int_{\Omega}f(x^{\prime})\eta_{3}(x^{\prime})$ by (3.6), passing
to the limit in all terms of (7.2) yields the desired equality (7.1) and thus
proves the lemma.
###### Lemma 7.2.
For every $\eta\in\mathcal{C}_{b}^{2}(\mathbb{R}^{2},\mathbb{R}^{2})$, it
follows that:
(7.5)
$\begin{split}\int_{\Omega^{1}}&\Big{\langle}(E^{\prime\prime}-E_{33}\mathrm{Id}_{2}):(x_{3}+v(x^{\prime}))\nabla_{tan}\eta(x^{\prime})\Big{\rangle}~{}\mathrm{d}x\\\
&+\int_{\Omega^{1}}\Big{\langle}(E^{\prime\prime}-E_{33}\mathrm{Id}_{2}):\nabla
v(x^{\prime})\otimes\eta(x^{\prime})\Big{\rangle}~{}\mathrm{d}x~{}+~{}\lim_{h\to
0}\frac{1}{h}\int_{\Omega^{1}}\left\langle
E^{h}_{13,23},\nabla\eta_{3}({y^{h}}^{\prime})\right\rangle~{}\mathrm{d}x=0.\end{split}$
###### Proof.
1. Let $\eta\in\mathcal{C}_{b}^{2}(\mathbb{R}^{2},\mathbb{R}^{2})$ be a given test function, and define $\eta_{3}(x^{\prime})=-\mbox{div }\eta(x^{\prime})$. Given $\theta^{h}$ as in Lemma 5.2, with:
(7.6) $\lim_{h\to 0}\omega_{h}=\lim_{h\to 0}h\omega_{h}=+\infty\quad\mbox{ and
}\quad\lim_{h\to 0}h^{1+\frac{1-\gamma}{2}}\omega_{h}=0~{}~{}\mbox{ for some
fixed }\gamma\in(0,1),$
consider the divergence-free test functions
$\phi^{h}\in\mathcal{C}^{1}_{b}(\mathbb{R}^{3},\mathbb{R}^{3})$:
$\phi^{h}(x^{\prime},x_{3})=\left[\begin{array}[]{c}\displaystyle{{\theta^{h}}^{\prime}\left(\frac{x_{3}}{h}\right){\theta^{h}}\left(\frac{x_{3}}{h}\right)\eta(x^{\prime})\vspace{2mm}}\\\
\displaystyle{\frac{h}{2}({\theta^{h}}\left(\frac{x_{3}}{h}\right))^{2}\eta_{3}(x^{\prime})}\end{array}\right].$
Denoting $\nabla_{tan}$ the gradient in the tangential directions
$e_{1},e_{2}$, we have:
$\nabla\phi^{h}(x^{\prime},x_{3})=\left[\begin{array}[]{c|c}\displaystyle{{\theta^{h}}^{\prime}\left(\frac{x_{3}}{h}\right){\theta^{h}}\left(\frac{x_{3}}{h}\right)\nabla_{tan}\eta(x^{\prime})}&\begin{minipage}{199.16928pt}$$\vspace{3mm}\displaystyle{\frac{1}{h}\left({\theta^{h}}^{\prime\prime}\left(\frac{x_{3}}{h}\right){\theta^{h}}\left(\frac{x_{3}}{h}\right)+({\theta^{h}}\left(\frac{x_{3}}{h}\right))^{2}\right)\eta(x^{\prime})}$$\end{minipage}\\\
\hline\cr\displaystyle{\frac{h}{2}({\theta^{h}}\left(\frac{x_{3}}{h}\right))^{2}\nabla_{tan}\eta_{3}(x^{\prime})}&\begin{minipage}{199.16928pt}\vspace{3mm}$$\displaystyle{{\theta^{h}}^{\prime}\left(\frac{x_{3}}{h}\right){\theta^{h}}\left(\frac{x_{3}}{h}\right)\eta_{3}(x^{\prime})}$$\end{minipage}\end{array}\right].$
2. Applying now the equilibrium equation (3.13) with $\phi=\phi^{h}$, we obtain:
(7.7)
$\begin{split}\int_{\Omega^{1}}\Big{\langle}&\Big{(}(\bar{R}^{h})^{T}R^{h}E^{h}(R^{h})^{T}\bar{R}^{h}\Big{)}^{\prime\prime}-\Big{(}(\bar{R}^{h})^{T}R^{h}E^{h}(R^{h})^{T}\bar{R}^{h}\Big{)}_{33}\mbox{Id}_{2}:{\theta^{h}}^{\prime}\left(\frac{y^{h}_{3}}{h}\right){\theta^{h}}\left(\frac{y^{h}_{3}}{h}\right)\nabla_{tan}\eta({y^{h}}^{\prime})\Big{\rangle}\\\
&\qquad\quad+\frac{1}{h}\int_{\Omega^{1}}\Big{\langle}\Big{(}(\bar{R}^{h})^{T}R^{h}E^{h}(R^{h})^{T}\bar{R}^{h}\Big{)}_{13,23},\left({\theta^{h}}^{\prime\prime}\left(\frac{x_{3}}{h}\right){\theta^{h}}\left(\frac{x_{3}}{h}\right)+({\theta^{h}}\left(\frac{x_{3}}{h}\right))^{2}\right)\eta({y^{h}}^{\prime})\Big{\rangle}\\\
&\qquad\quad+\frac{h}{2}\int_{\Omega^{1}}\Big{\langle}\Big{(}(\bar{R}^{h})^{T}R^{h}E^{h}(R^{h})^{T}\bar{R}^{h}\Big{)}_{31,32},({\theta^{h}}\left(\frac{y^{h}_{3}}{h}\right))^{2}\nabla_{tan}\eta({y^{h}}^{\prime})\Big{\rangle}\\\
&=h\int_{\Omega^{1}}\Big{\langle}f(x^{\prime})(\bar{R}^{h})_{31,32},{\theta^{h}}^{\prime}\left(\frac{y_{3}^{h}}{h}\right)\theta^{h}\left(\frac{y^{h}_{3}}{h}\right)\eta({y^{h}}^{\prime})\Big{\rangle}~{}\mbox{d}x\\\
&\qquad\qquad+\frac{h^{2}}{2}\int_{\Omega^{1}}f(x^{\prime})(\bar{R}^{h})_{33}({\theta^{h}}\left(\frac{y^{h}_{3}}{h}\right))^{2}\eta_{3}({y^{h}}^{\prime})~{}\mbox{d}x.\end{split}$
In what follows, we will check convergence as $h\to 0$ of each of the five
terms in the identity (7.7). We first easily notice that the two terms in the
right hand side converge to $0$, as they are bounded by:
$\begin{split}C\int_{\Omega^{1}}|f(x^{\prime})|\Big{(}h\left|\theta^{h}\left(\frac{y_{3}^{h}}{h}\right)\right|+h^{2}\left|\theta^{h}\left(\frac{y_{3}^{h}}{h}\right)\right|^{2}\Big{)}~{}\mbox{d}x&\leq
C\int_{\Omega^{1}}|f(x^{\prime})|\left(|y_{3}^{h}|+|y_{3}^{h}|^{2}\right)~{}\mbox{d}x\\\
&\leq
C\|f\|_{L^{2}(\Omega^{1})}\left(\|y_{3}^{h}\|_{L^{2}(\Omega^{1})}+\|y_{3}^{h}\|^{2}_{L^{4}(\Omega^{1})}\right).\end{split}$
Since $\frac{y_{3}^{h}}{h}$ has a strong limit in $W^{1,2}(\Omega^{1})$ by
(3.6), it results that $\|y_{3}^{h}\|_{L^{2}}$ and $\|y_{3}^{h}\|_{L^{4}}$
converge to $0$.
3. The third term in (7.7) is bounded by the following expression, in view of (5.2), (3.12), (3.6) and (3.11):
$\begin{split}Ch\int_{\Omega^{1}}\chi_{h}|E^{h}|&(\theta^{h}\left(\frac{y_{3}^{h}}{h}\right))^{2}~{}\mbox{d}x+Ch\int_{\Omega^{1}}(1-\chi_{h})|E^{h}|(\theta^{h}\left(\frac{y_{3}^{h}}{h}\right))^{2}~{}\mbox{d}x\\\
&\leq
Ch\int_{\Omega^{1}}\chi_{h}|E^{h}|\left|\frac{y_{3}^{h}}{h}\right|^{2}~{}\mbox{d}x+Ch\omega^{2}_{h}\int_{\Omega^{1}\setminus
B_{h}}|E^{h}|~{}\mbox{d}x\\\ &\leq
Ch\|\chi_{h}E^{h}\|_{L^{2}}\left\|\frac{y^{h}_{3}}{h}\right\|^{2}_{L^{4}}+Ch\omega_{h}^{2}h^{2-\gamma}\leq
Ch+C\left(h^{1+\frac{1-\gamma}{2}}\omega_{h}\right)^{2}\end{split}$
which converges to $0$ by (7.6).
4. We will now investigate the first term in (7.7). Integrated on $\Omega^{1}\setminus B_{h}$, it is bounded by:
$C\omega_{h}\int_{\Omega^{1}}(1-\chi_{h})|E^{h}|~{}\mbox{d}x\leq
C\omega_{h}h^{2-\gamma}\leq Ch^{1+\frac{1-\gamma}{2}}\omega_{h},$
by (3.11) and hence it converges to $0$ through (7.6). The same term
integrated on $B_{h}$ equals now the following sum:
(7.8)
$\begin{split}&\int_{\Omega^{1}}\left({\theta^{h}}^{\prime}\left(\frac{y_{3}^{h}}{h}\right)-1\right)\theta^{h}\left(\frac{y_{3}^{h}}{h}\right)\cdot\\\
&\qquad\cdot\Big{\langle}\left((\bar{R}^{h})^{T}R^{h}\chi_{h}E^{h}(R^{h})^{T}\bar{R}^{h}\right)^{\prime\prime}-\left((\bar{R}^{h})^{T}R^{h}\chi_{h}E^{h}(R^{h})^{T}\bar{R}^{h}\right)_{33}\mbox{Id}_{2}:\nabla_{tan}\eta({y^{h}}^{\prime})\Big{\rangle}~{}\mbox{d}x\\\
&+\int_{\Omega^{1}}\theta^{h}\left(\frac{y_{3}^{h}}{h}\right)\cdot\\\
&\qquad\cdot\Big{\langle}\left((\bar{R}^{h})^{T}R^{h}\chi_{h}E^{h}(R^{h})^{T}\bar{R}^{h}\right)^{\prime\prime}-\left((\bar{R}^{h})^{T}R^{h}\chi_{h}E^{h}(R^{h})^{T}\bar{R}^{h}\right)_{33}\mbox{Id}_{2}:\nabla_{tan}\eta({y^{h}}^{\prime})\Big{\rangle}~{}\mbox{d}x.\end{split}$
The first term in (7.8) goes to $0$, as it is bounded by:
$C\int_{\left\\{\frac{|y_{3}^{h}|}{h}\geq\omega_{h}\right\\}}\left|\frac{y_{3}^{h}}{h}\right||\chi_{h}E^{h}|~{}\mbox{d}x\leq
C\left\|\frac{y_{3}^{h}}{h}\right\|_{L^{4}(\Omega^{1})}\left|\left\\{x\in\Omega^{1};~{}\frac{|y_{3}^{h}|}{h}\geq\omega_{h}\right\\}\right|^{1/4}\|\chi_{h}E^{h}\|_{L^{2}(\Omega^{1})}\leq\frac{C}{\omega_{h}^{1/2}},$
in view of (5.2), (3.7), (3.12) and recalling (7.6). The second term of (7.8)
converges to:
(7.9)
$\int_{\Omega^{1}}\Big{\langle}E^{\prime\prime}-E_{33}\mbox{Id}_{2}:(x_{3}+v(x^{\prime}))\nabla_{tan}\eta({x}^{\prime})\Big{\rangle}~{}\mbox{d}x$
because of (3.12) and through the following strong convergences: convergence
of $\nabla_{tan}\eta({y^{h}}^{\prime})$ to $\nabla_{tan}\eta(x^{\prime})$ in
$L^{5}(\Omega^{1})$ by (3.6), of $(\bar{R}^{h})^{T}R^{h}$ to Id in
$L^{20}(\Omega)$ by (3.3), and of $\theta^{h}\left(\frac{y^{h}_{3}}{h}\right)$
to $(x_{3}+v(x^{\prime}))$ in $L^{5}(\Omega^{1})$. The last convergence can be
seen from:
$\begin{split}\int_{\Omega^{1}}\left|\theta^{h}\left(\frac{y_{3}^{h}}{h}\right)-(x_{3}+v(x^{\prime}))\right|^{5}~{}\mbox{d}x&\leq
C\int_{\Omega^{1}}\left|\theta^{h}\left(\frac{y_{3}^{h}}{h}\right)-\frac{y_{3}^{h}}{h}\right|^{5}~{}\mbox{d}x+C\int_{\Omega^{1}}\left|\frac{y_{3}^{h}}{h}-(x_{3}+v(x^{\prime}))\right|^{5}~{}\mbox{d}x\\\
&\leq
C\int_{\left\\{\frac{|y_{3}^{h}|}{h}\geq\omega_{h}\right\\}}\left|\frac{y_{3}^{h}}{h}\right|^{5}~{}\mbox{d}x+o(1)\leq\frac{C}{\omega_{h}^{1/3}}+o(1)\leq
o(1)\end{split}$
by (3.6), (3.7) and (7.6). Concluding, we obtain that the first term in (7.7)
converges to the expression in (7.9).
5. Regarding the second term in (7.7), using (3.10), (5.2), (3.1) and (3.7) we note that:
$\begin{split}\Big{|}\int_{\Omega^{1}}&\left({\theta^{h}}^{\prime\prime}\left(\frac{y_{3}^{h}}{h}\right)\theta^{h}\left(\frac{y_{3}^{h}}{h}\right)+{\theta^{h}}^{\prime}\left(\frac{y_{3}^{h}}{h}\right)^{2}-1\right)\Big{\langle}\left((\bar{R}^{h})^{T}R^{h}E^{h}(R^{h})^{T}\bar{R}^{h}\right)_{13,23},\eta({y^{h}}^{\prime})\Big{\rangle}~{}\mbox{d}x\Big{|}\\\
&\qquad\leq\frac{C}{h}\int_{\left\\{x\in\Omega^{1};~{}\frac{|y_{3}^{h}(x)|}{h}\geq\omega_{h}\right\\}}\left(\frac{1}{\omega_{h}}\omega_{h}+1\right)|E^{h}|~{}\mbox{d}x\\\
&\qquad\leq\frac{C}{h}\int_{\left\\{\frac{|y_{3}^{h}|}{h}\geq\omega_{h}\right\\}}\frac{1}{h^{2}}W(\nabla
u^{h}(x^{\prime},hx_{3}))+|G^{h}|~{}\mbox{d}x\\\
&\qquad\leq\frac{C}{h}\left(h^{2}+\|G^{h}\|_{L^{2}(\Omega^{1}}\left|\left\\{x\in\Omega^{1};~{}\frac{|y^{h}(x)|}{h}\geq\omega_{h}\right\\}\right|^{1/2}\right)\leq\frac{C}{h}\left(h^{2}+\frac{1}{\omega_{h}}\right),\end{split}$
which converges to $0$ by (7.6). The remaining part of the second term in
(7.7) is:
(7.10)
$\begin{split}\frac{1}{h}\int_{\Omega^{1}}&\Big{\langle}\left((\bar{R}^{h})^{T}R^{h}E^{h}(R^{h})^{T}\bar{R}^{h}\right)_{13,23},\eta({y^{h}}^{\prime})\Big{\rangle}~{}\mbox{d}x\\\
&=\int_{\Omega^{1}}\Big{\langle}\left(A^{h}E^{h}(R^{h})^{T}\bar{R}^{h}\right)_{13,23},\eta({y^{h}}^{\prime})\Big{\rangle}~{}\mbox{d}x+\int_{\Omega^{1}}\Big{\langle}\left(E^{h}(A^{h})^{T}\right)_{13,23},\eta({y^{h}}^{\prime})\Big{\rangle}~{}\mbox{d}x\\\
&\qquad+\frac{1}{h}\int_{\Omega^{1}}\Big{\langle}(E^{h})_{13,23},\eta({y^{h}}^{\prime})\Big{\rangle}~{}\mbox{d}x,\end{split}$
where we used the decomposition (7.3). Now, exactly as in the proof of Lemma
7.1 and recalling the block structure of the limiting tensor $A$ in (3.5), we
see that (7.10) converges to:
$\begin{split}&\int_{\Omega^{1}}\Big{\langle}\left(AE\right)_{13,23},\eta(x^{\prime})\Big{\rangle}~{}\mbox{d}x+\int_{\Omega^{1}}\Big{\langle}\left(EA^{T}\right)_{13,23},\eta(x^{\prime})\Big{\rangle}~{}\mbox{d}x+\frac{1}{h}\int_{\Omega^{1}}\Big{\langle}(E^{h})_{13,23},\eta({y^{h}}^{\prime})\Big{\rangle}~{}\mbox{d}x\\\
&=\int_{\Omega^{1}}\Big{\langle}\left(E^{\prime\prime}-E_{33}\mbox{Id}_{2}\right)\nabla
v,\eta(x^{\prime})\Big{\rangle}~{}\mbox{d}x+\frac{1}{h}\int_{\Omega^{1}}\Big{\langle}(E^{h})_{13,23},\eta({y^{h}}^{\prime})\Big{\rangle}~{}\mbox{d}x.\end{split}$
In conclusion, passing to the limit in (7.7) clearly yields (7.5) and achieves
the lemma.
Proof of the second Euler-Lagrange equation (1.13).
Let now $\xi\in\mathcal{C}^{3}_{b}(\mathbb{R}^{2},\mathbb{R})$. Applying Lemma
7.1 with $\eta_{3}=\xi$, and Lemma 7.2 with $\eta=\nabla\xi$, it follows:
(7.11)
$-\int_{\Omega^{1}}\Big{\langle}E^{\prime\prime}-E_{33}\mbox{Id}_{2}:(x_{3}+v(x^{\prime})\nabla^{2}\xi\Big{\rangle}~{}\mbox{d}x=\bar{R}_{33}\int_{\Omega}f(x^{\prime})\xi(x^{\prime})~{}\mbox{d}x^{\prime}.$
By the first Euler-Lagrange equation in (6.4) applied with $\eta=v\nabla\xi\in
W^{2,2}(\Omega,\mathbb{R}^{2})$, we see that:
$\int_{\Omega^{1}}\Big{\langle}E^{\prime\prime}-E_{33}\mbox{Id}_{2}:\nabla
v\otimes\nabla\xi+v(x^{\prime})\nabla^{2}\xi\Big{\rangle}~{}\mbox{d}x=0.$
Thus, (7.11) becomes:
$\int_{\Omega^{1}}\Big{\langle}E^{\prime\prime}-E_{33}\mbox{Id}_{2}:\nabla
v\otimes\nabla\xi\Big{\rangle}~{}\mbox{d}x-\int_{\Omega^{1}}\Big{\langle}E^{\prime\prime}-E_{33}\mbox{Id}_{2}:x_{3}\nabla^{2}\xi\Big{\rangle}~{}\mbox{d}x=\bar{R}_{33}\int_{\Omega}f(x^{\prime})\xi(x^{\prime}).$
The equality in (1.13) follows now from the above in view of (4.6), and by the
density of test functions $\xi\in\mathcal{C}^{3}_{b}$ in
$W^{2,2}(\Omega,\mathbb{R})$.
## References
* [1] Ball, J.M. “Minimizers and the Euler-Lagrange equations.” In _Proc. ISIMM conference._ Paris: Springer, 1983.
* [2] Ball, J.M. “Some open problems in elasticity.” In _Geometry, Mechanics, and Dynamics,_ 3-59. New York: Springer, 2002.
* [3] Conti, S., and Dolzmann, G. “Derivation of a plate theory for incompressible materials. _C.R. Math. Acad. Sci. Paris_ 344 no. 8, (2007): 541 544.
* [4] Conti, S., and Dolzmann, G. “Gamma-convergence for incompressible elastic plates. _Calc. Var. PDE_ 34 (2009): 531–551.
* [5] Friesecke, G., James, R., and Müller, S. “A theorem on geometric rigidity and the derivation of nonlinear plate theory from three-dimensional elasticity.” _Comm. Pure Appl. Math._ 55 (2002), 1461–1506.
* [6] Friesecke, G., James, R., and Müller, S. “A hierarchy of plate models derived from nonlinear elasticity by gamma-convergence.” _Arch. Ration. Mech. Anal._ 180 no. 2, (2006): 183–236.
* [7] Lewicka, M. “A note on the convergence of low energy critical points of nonlinear elasticity functionals, for thin shells of arbitrary geometry. _ESAIM: Control, Optimisation and Calculus of Variations_ 17 (2011): 493–505.
* [8] Lewicka, M., Mahadevan, L., and Pakzad, M.R. “The Föppl-von Kármán equations for plates with incompatible strains.” _Proceedings of the Royal Society A_ 467 (2011): 402–426.
* [9] Lewicka, M., Mora, M.G., and Pakzad, M. R. “Shell theories arising as low energy $Gamma$-limit of 3d nonlinear elasticity.” _Ann. Scuola Norm. Sup. Pisa Cl. Sci._ 5 Vol. IX (2010), 1–43.
* [10] Li, H., and Chermisi, M. “The von Karman theory for incompressible elastic shells.” _Calc. Var. PDE_ (2012) DOI:10.1007/s00526-012-0549-5.
* [11] Mora, M. G. and Scardia, L. “Convergence of equilibria of thin elastic plates under physical growth conditions for the energy density.” _J. Differential Equations._ 252 (2012): 35–55.
* [12] Müller, S. and M.R. Pakzad, M. R. “Convergence of Equilibria of Thin Elastic Plates-The Von Kármán Case.” _Comm. Partial Differential Equations_ 33 (2008): 1018–1032.
|
arxiv-papers
| 2012-11-16T16:23:03 |
2024-09-04T02:49:38.091487
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Marta Lewicka and Hui Li",
"submitter": "Hui Li",
"url": "https://arxiv.org/abs/1211.3942"
}
|
1211.3966
|
# Lasso Screening Rules via Dual Polytope Projection
Jie Wang Computer Science and Engineering, Arizona State University, USA
Peter Wonka Computer Science and Engineering, Arizona State University, USA
Jieping Ye Computer Science and Engineering, Arizona State University, USA
###### Abstract
Lasso is a widely used regression technique to find sparse representations.
When the dimension of the feature space and the number of samples are
extremely large, solving the Lasso problem remains challenging. To improve the
efficiency of solving large-scale Lasso problems, El Ghaoui and his colleagues
have proposed the SAFE rules which are able to quickly identify the inactive
predictors, i.e., predictors that have $0$ components in the solution vector.
Then, the inactive predictors or features can be removed from the optimization
problem to reduce its scale. By transforming the standard Lasso to its dual
form, it can be shown that the inactive predictors include the set of inactive
constraints on the optimal dual solution. In this paper, we propose an
efficient and effective screening rule via Dual Polytope Projections (DPP),
which is mainly based on the uniqueness and nonexpansiveness of the optimal
dual solution due to the fact that the feasible set in the dual space is a
convex and closed polytope. Moreover, we show that our screening rule can be
extended to identify inactive groups in group Lasso. To the best of our
knowledge, there is currently no exact screening rule for group Lasso. We have
evaluated our screening rule using synthetic and real data sets. Results show
that our rule is more effective in identifying inactive predictors than
existing state-of-the-art screening rules for Lasso.
## 1 Introduction
Data with various structures and scales comes from almost every aspect of
daily life. To effectively extract patterns in the data and build
interpretable models with high prediction accuracy is always desirable. One
popular technique to identify important explanatory features is by sparse
regularization. For instance, consider the widely used $\ell_{1}$-regularized
least squares regression problem known as Lasso [31]. The most appealing
property of Lasso is the sparsity of the solutions, which is equivalent to
feature selection. Suppose we have $N$ observations and $p$ features. Let
${\bf y}$ denote the $N$ dimensional response vector and ${\bf X}=[{\bf
x}_{1},{\bf x}_{2},\ldots,{\bf x}_{p}]$ be the $N\times p$ feature matrix. Let
$\lambda\geq 0$ be the regularization parameter. The Lasso problem is
formulated as the following optimization problem:
$\inf_{\beta\in\mathbb{R}^{p}}\frac{1}{2}\left\|{\bf
y-X\beta}\right\|_{2}^{2}+\lambda\|{\beta}\|_{1}.$ (1)
Lasso has achieved great success in a wide range of applications [13, 12, 38,
9, 34] and in recent years many algorithms have been developed to efficiently
solve the Lasso problem [15, 20, 26, 14, 18, 4, 19]. However, when the
dimension of feature space and the number of samples are very large, solving
the Lasso problem remains challenging because we may not even be able to load
the data matrix into main memory. The idea of screening has been shown very
promising in solving Lasso for large-scale problems. Essentially, screening
aims to quickly identify the inactive features that have $0$ components in the
solution and then remove them from the optimization. Therefore, we can work on
a reduced feature matrix to solve the Lasso problem, which may lead to
substantial savings in computational cost and memory usage.
Existing screening methods for Lasso can be roughly divided into two
categories: the Heuristic Screening Methods and the Safe Screening Methods. As
the name indicated, the heuristic screening methods can not guarantee that the
discarded features have zero coefficients in the solution vector. In other
words, they may mistakenly discard the active features which have nonzero
coefficients in the sparse representations. Well-known heuristic screening
methods for Lasso include SIS [17] and strong rules [32]. SIS is based on the
associations between features and the prediction task, but not from an
optimization point of view. Strong rules rely on the assumption that the
absolute values of the inner products between features and the residue are
nonexpansive [3] with respect to the parameter values. Notice that, in real
applications, this assumption is not always true. In order to ensure the
correctness of the solutions, strong rules check the KKT conditions for
violations. In case of violations, they weaken the screened set and repeat
this process. In contrast to the heuristic screening methods, the safe
screening methods for Lasso can guarantee that the discarded features are
absent from the resulting sparse models. Existing safe screening methods for
Lasso includes SAFE [16] and DOME [36, 35], which are based on an estimation
of the dual optimal solution. The key challenge of searching for effective
safe screening rules is how to accurately estimate the dual optimal solution.
The more accurate the estimation is, the more effective the resulting
screening rule is in discarding the inactive features. Moreover, Xiang et al.
[36] have shown that the SAFE rule for Lasso can be read as a special case of
their testing rules.
In this paper, we develop novel efficient and effective screening rules for
the Lasso problem; our screening rules are safe in the sense that no active
features will be discarded. As the name indicated (DPP), the proposed
approaches heavily rely on the geometric properties of the Lasso problem.
Indeed, the dual problem of problem (1) can be formulated as a projection
problem. More specifically, the dual optimal solution of the Lasso problem is
the projection of the scaled response vector onto a nonempty closed and convex
polytope (the feasible set of the dual problem). This nice property provides
us many elegant approaches to accurately estimate the dual optimal solutions,
e.g., nonexpansiveness, firmly nonexpansiveness [3]. In fact, the estimation
of the dual optimal solution in DPP is a direct application of the
nonexpansiveness of the projection operators. Moreover, by further exploiting
the properties of the projection operators, we can significantly improve the
estimation of the dual optimal solution. Based on this estimation, we develop
the so called enhanced DPP (EDPP) rules which are able to detect far more
inactive features than DPP. Therefore, the speedup gained by EDPP is much
higher than the one by DPP.
In real applications, the optimal parameter value of $\lambda$ is generally
unknown and needs to be estimated. To determine an appropriate value of
$\lambda$, commonly used approaches such as cross validation and stability
selection involve solving the Lasso problems over a grid of tuning parameters
$\lambda_{1}>\lambda_{2}>\ldots>\lambda_{\mathcal{K}}$. Thus, the process can
be very time consuming. To address this challenge, we develop the sequential
version of the DPP families. Briefly speaking, for the Lasso problem, suppose
we are given the solution $\beta^{*}(\lambda_{k-1})$ at $\lambda_{k-1}$. We
then apply the screening rules to identify the inactive features of problem
(1) at $\lambda_{k}$ by making use of $\beta^{*}(\lambda_{k-1})$. The idea of
the sequential screening rules is proposed by [16] and [32] and has been shown
to be very effective for the aforementioned scenario. In [32], the authors
demonstrate that the sequential strong rules are very effective in discarding
inactive features especially for very small parameter values and achieve the
state-of-the-art performance. However, in contrast to the recursive SAFE (the
sequential version of SAFE rules) and the sequential version of DPP rules, it
is worthwhile to mention that the sequential strong rules may mistakenly
discard active features because they are heuristic methods. Moreover, it is
worthwhile to mention that, for the existing screening rules including SAFE
and strong rules, the basic versions are usually special cases of their
sequential versions, and the same applies to our DPP and EDPP rules. For the
DOME rule [36, 35], it is unclear whether its sequential version exists.
The rest of this paper is organized as follows. We present the family of DPP
screening rules, i.e., DPP and EDPP, in detail for the Lasso problem in
Section 2. Section 3 extends the idea of DPP screening rules to identify
inactive groups in group Lasso [37]. We evaluate our screening rules on
synthetic and real data sets from many different applications. In Section 4,
the experimental results demonstrate that our rules are more effective in
discarding inactive features than existing state-of-the-art screening rules.
We show that the efficiency of the solver can be improved by several orders of
magnitude with the enhanced DPP rules, especially for the high-dimensional
data sets (notice that, the screening methods can be integrated with any
existing solvers for the Lasso problem). Some concluding remarks are given in
Section 5.
## 2 Screening Rules for Lasso via Dual Polytope Projections
In this section, we present the details of the proposed DPP and EDPP screening
rules for the Lasso problem. We first review some basics of the dual problem
of Lasso including its geometric properties in Section 2.1; we also briefly
discuss some basic guidelines for developing safe screening rules for Lasso.
Based on the geometric properties discussed in Section 2.1, we then develop
the basic DPP screening rule in Section 2.2. As a straightforward extension in
dealing with the model selection problems, we also develop the sequential
version of DPP rules. In Section 2.3, by exploiting more geometric properties
of the dual problem of Lasso, we further improve the DPP rules by developing
the so called enhanced DPP (EDPP) rules. The EDPP screening rules
significantly outperform DPP rules in identifying the inactive features for
the Lasso problem.
### 2.1 Basics
Different from [36, 35], we do not assume ${\bf y}$ and all ${\bf x}_{i}$ have
unit length.The dual problem of problem (1) takes the form of (to make the
paper self-contained, we provide the detailed derivation of the dual form in
the appendix):
$\displaystyle\sup_{\theta}\quad\left\\{\frac{1}{2}\|{\bf
y}\|_{2}^{2}-\frac{\lambda^{2}}{2}\left\|\theta-\frac{{\bf
y}}{\lambda}\right\|_{2}^{2}:\,\,|{\bf x}_{i}^{T}\theta|\leq
1,\,i=1,2,\ldots,p\right\\},$ (2)
where $\theta$ is the dual variable. For notational convenience, let the
optimal solution of problem (2) be $\theta^{\ast}(\lambda)$ [recall that the
optimal solution of problem (1) with parameter $\lambda$ is denoted by
$\beta^{*}(\lambda)$]. Then, the KKT conditions are given by:
$\displaystyle{\bf y}$ $\displaystyle={\bf
X}{\beta}^{\ast}(\lambda)+\lambda\theta^{\ast}(\lambda),$ (3)
$\displaystyle{\bf x}_{i}^{T}\theta^{*}(\lambda)$
$\displaystyle\in\begin{cases}{\operatorname*{sign}([\beta^{\ast}(\lambda)]_{i})},\hskip
5.69054pt\rm{if}\hskip 2.84526pt[\beta^{\ast}(\lambda)]_{i}\neq 0,\\\
[-1,1],\hskip 39.83385pt\rm{if}\hskip
2.84526pt[\beta^{\ast}(\lambda)]_{i}=0,\\\ \end{cases}i=1,\ldots,p,$ (4)
where $[\cdot]_{k}$ denotes the $k^{th}$ component. In view of the KKT
condition in (4), we have
$\displaystyle|{\bf
x}_{i}^{T}(\theta^{*}(\lambda))^{T}|<1\Rightarrow[\beta^{*}(\lambda)]_{i}=0\Rightarrow{\bf
x}_{i}\,\,\mbox{\rm is an inactive feature.}$ (R1)
In other words, we can potentially make use of (R1) to identify the inactive
features for the Lasso problem. However, since $\theta^{*}(\lambda)$ is
generally unknown, we can not directly apply (R1) to identify the inactive
features. Inspired by the SAFE rules [16], we can first estimate a region
${\bf\Theta}$ which contains $\theta^{*}(\lambda^{\prime\prime})$. Then, (R1)
can be relaxed as follows:
$\displaystyle\sup_{\theta\in{\bf\Theta}}|{\bf
x}_{i}^{T}\theta|<1\Rightarrow[\beta^{*}(\lambda)]_{i}=0\Rightarrow{\bf
x}_{i}\,\,\mbox{\rm is an inactive feature.}$ (R1’)
Clearly, as long as we can find a region ${\bf\Theta}$ which contains
$\theta^{*}(\lambda)$, (R1’) will lead to a screening rule to detect the
inactive features for the Lasso problem. Moreover, in view of (R1) and (R1’),
we can see that the smaller the region ${\bf\Theta}$ is, the more accurate the
estimation of $\theta^{*}(\lambda)$ is. As a result, more inactive features
can be identified by the resulting screening rules.
Geometric Interpretations of the Dual Problem By a closer look at the dual
problem (2), we can observe that the dual optimal solution is the feasible
point which is closest to ${\bf y}/{\lambda}$. For notational convenience, let
the feasible set of problem (2) be $F$. Clearly, $F$ is the intersection of
$2p$ closed half-spaces, and thus a closed and convex polytope. (Notice that,
$F$ is also nonempty since $0\in F$.) In other words, $\theta^{\ast}(\lambda)$
is the projection of ${\bf y}/{\lambda}$ onto the polytope $F$.
Mathematically, for an arbitrary vector ${\bf w}$ and a convex set $C$ in a
Hilbert space $\mathcal{H}$, let us define the projection operator as
$P_{C}({\bf w})=\operatorname*{argmin}_{{\bf u}\in C}\|{\bf u}-{\bf w}\|_{2}.$
(5)
Then, the dual optimal solution $\theta^{*}(\lambda)$ can be expressed by
$\theta^{\ast}(\lambda)=P_{F}({\bf
y}/{\lambda})=\operatorname*{argmin}_{\theta\in F}\left\|\theta-\frac{\bf
y}{\lambda}\right\|_{2}.$ (6)
Indeed, the nice property of problem (2) illustrated by Eq. (6) leads to many
interesting results. For example, it is easy to see that ${\bf y}/\lambda$
would be an interior point of $F$ when $\lambda$ is large enough. If this is
the case, we immediately have the following assertions: 1) ${\bf y}/\lambda$
is an interior point of $F$ implies that none of the constraints of problem
(2) would be active on ${\bf y}/\lambda$, i.e., $|{\bf x}_{i}^{T}({\bf
y}/(\lambda)|)<1$ for all $i=1,\ldots,p$; 2) $\theta^{*}(\lambda)$ is an
interior point of $F$ as well since $\theta^{*}(\lambda)=P_{F}({\bf
y}/\lambda)={\bf y}/\lambda$ by Eq. (6) and the fact ${\bf y}/\lambda\in F$.
Combining the results in 1) and 2), it is easy to see that $|{\bf
x}_{i}^{T}\theta^{*}(\lambda)|<1$ for all $i=1,\ldots,p$. By (R1), we can
conclude that $\beta^{*}(\lambda)=0$, under the assumption that $\lambda$ is
large enough.
The above analysis may naturally lead to a question: does there exist a
specific parameter value $\lambda_{\rm max}$ such that the optimal solution of
problem (1) is $0$ whenever $\lambda>\lambda_{\rm max}$? The answer is
affirmative. Indeed, let us define
$\displaystyle\lambda_{\rm max}=\max_{i}|{\bf x}_{i}^{T}{\bf y}|.$ (7)
It is well known [32] that $\lambda_{\rm max}$ defined by Eq. (7) is the
smallest parameter such that problem (1) has a trivial solution, i.e.,
$\displaystyle\beta^{*}(\lambda)=0,\hskip 5.69054pt\forall\hskip
2.84526pt\lambda\in[\lambda_{\rm max},\infty).$ (8)
Combining the results in (8) and Eq. (3), we immediately have
$\displaystyle\theta^{*}(\lambda)=\frac{\bf y}{\lambda},\hskip
5.69054pt\forall\hskip 2.84526pt\lambda\in[\lambda_{\rm max},\infty).$ (9)
Therefore, through out the rest of this paper, we will focus on the cases with
$\lambda\in(0,\lambda_{\rm max})$.
In the subsequent sections, we will follow (R1’) to develop our screening
rules. More specifically, the derivation of the proposed screening rules can
be divided into the following three steps:
1. Step 1.
We first estimate a region ${\bf\Theta}$ which contains the dual optimal
solution $\theta^{*}(\lambda)$.
2. Step 2.
We solve the maximization problem in (R1’), i.e.,
$\sup_{\theta\in{\bf\Theta}}|{\bf x}_{i}^{T}\theta|$.
3. Step 3.
By plugging in the upper bound we find in Step ${\bf 2}$, it is
straightforward to develop the screening rule based on (R1’).
The geometric property of the dual problem illustrated by Eq. (6) serves as a
fundamentally important role in developing our DPP and EDPP screening rules.
### 2.2 Fundamental Screening Rules via Dual Polytope Projections (DPP)
In this Section, we propose the so called DPP screening rules for discarding
the inactive features for Lasso. As the name indicates, the idea of DPP
heavily relies on the properties of projection operators, e.g., the
nonexpansiveness [5]. We will follow the three steps stated in Section 2.1 to
develop the DPP screening rules.
First, we need to find a region ${\bf\Theta}$ which contains the dual optimal
solution $\theta^{*}(\lambda)$. Indeed, the result in (9) provides us an
important clue. That is, we may be able to estimate a possible region for
$\theta^{*}(\lambda)$ in terms of a known $\theta^{*}(\lambda_{0})$ with
$\lambda<\lambda_{0}$. Notice that, we can always set
$\lambda_{0}=\lambda_{\rm max}$ and make use of the fact that
$\theta^{*}(\lambda_{\rm max})={\bf y}/\lambda_{\rm max}$ implied by (9).
Another key ingredient comes from Eq. (6), i.e., the dual optimal solution
$\theta^{*}(\lambda)$ is the projection of ${\bf y}/\lambda$ onto the feasible
set $F$, which is nonempty closed and convex. A nice property of the
projection operators defined in a Hilbert space with respect to a nonempty
closed and convex set is the so called nonexpansiveness. For convenience, we
restate the definition of nonexpansiveness in the following theorem.
###### Theorem 1.
Let $C$ be a nonempty closed convex subset of a Hilbert space $\mathcal{H}$.
Then the projection operator defined in Eq. (5) is continuous and
nonexpansive, i.e.,
$\|P_{C}({\bf w}_{2})-P_{C}({\bf w}_{1})\|_{2}\leq\|{\bf w}_{2}-{\bf
w}_{1}\|_{2},\,\,\forall{\bf w}_{2},{\bf w}_{1}\in\mathcal{H}.$ (10)
In view of Eq. (6), a direct application of Theorem 1 leads to the following
result:
###### Theorem 2.
For the Lasso problem, let $\lambda,\lambda_{0}>0$ be two regularization
parameters. Then,
$\displaystyle\|\theta^{*}(\lambda)-\theta^{*}(\lambda_{0})\|_{2}\leq\left|\frac{1}{\lambda}-\frac{1}{\lambda_{0}}\right|\|{\bf
y}\|_{2}.$ (11)
For notational convenience, let a ball centered at ${\bf c}$ with radius
$\rho$ be denoted by $B({\bf c},\rho)$. Theorem 2 actually implies that the
dual optimal solution must be inside a ball centered at
$\theta^{*}(\lambda_{0})$ with radius
$\left|1/\lambda-1/\lambda_{0}\right|\|{\bf y}\|_{2}$, i.e.,
$\displaystyle\theta^{*}(\lambda)\in
B\left(\theta^{*}(\lambda_{0}),\left|\frac{1}{\lambda}-\frac{1}{\lambda_{0}}\right|\|{\bf
y}\|_{2}\right).$ (12)
We thus complete the first step for developing DPP. Because it is easy to find
the upper bound of a linear functional over a ball, we combine the remaining
two steps as follows.
###### Theorem 3.
For the Lasso problem, assume we are given the solution of its dual problem
$\theta^{\ast}(\lambda_{0})$ for a specific $\lambda_{0}$. Let $\lambda$ be a
positive value different from $\lambda_{0}$. Then
$[\beta^{\ast}(\lambda)]_{i}=0$ if
$\displaystyle\left|{\bf x}_{i}^{T}\theta^{\ast}(\lambda)\right|<1-\|{\bf
x}_{i}\|_{2}\|{\bf
y}\|_{2}\left|\frac{1}{\lambda}-\frac{1}{\lambda_{0}}\right|.$ (13)
###### Proof.
The dual optimal solution $\theta^{*}(\lambda)$ is estimated to be inside the
ball given by Eq. (12). To simplify notations, let ${\bf
c}=\theta^{\ast}(\lambda_{0})$ and
$\rho=\left|1/\lambda-1/\lambda_{0}\right|\|{\bf y}\|_{2}$. To develop a
screening rule based on (R1’), we need to solve the optimization problem:
$\sup_{\theta\in B({\bf c},\rho)}|{\bf x}_{i}^{T}\theta|$.
Indeed, for any $\theta\in B({\bf c},\rho)$, it can be expressed by:
$\displaystyle\theta=\theta^{*}(\lambda_{0})+{\bf v},\hskip 5.69054pt\|{\bf
v}\|_{2}\leq\rho.$
Therefore, the optimization problem can be easily solved as follows:
$\displaystyle\sup_{\theta\in B({\bf c},\rho)}\left|{\bf
x}_{i}^{T}\theta\right|=\sup_{\|{\bf v}\|_{2}\leq\rho}\left|{\bf
x}_{i}^{T}\left(\theta^{*}(\lambda_{0})+{\bf v}\right)\right|=\left|{\bf
x}_{i}^{T}\theta^{*}(\lambda_{0})\right|+\rho\|{\bf x}_{i}\|_{2}.$ (14)
By plugging the upper bound in Eq. (14) to (R1’), we obtain the statement in
Theorem 3, which completes the proof. ∎
Theorem 3 implies that we can develop applicable screening rules for Lasso as
long as the dual optimal solution $\theta^{\ast}(\cdot)$ is known for a
certain parameter value $\lambda_{0}$. By simply setting
$\lambda_{0}=\lambda_{\rm max}$ and noting that $\theta^{*}(\lambda_{\rm
max})={\bf y}/\lambda_{\rm max}$ [please refer to Eq. (9)], Theorem 3
immediately leads to the following result.
###### Corollary 4.
Basic DPP: For the Lasso problem (1), let $\lambda_{\rm max}=\max_{i}|{\bf
x}_{i}^{T}{\bf y}|$. If $\lambda\geq\lambda_{\rm max}$, then
$[\beta^{\ast}]_{i}=0,\forall i\in\mathcal{I}$. Otherwise,
$[\beta^{\ast}(\lambda)]_{i}=0$ if
$\left|{\bf x}_{i}^{T}\frac{\bf y}{\lambda_{\rm
max}}\right|<1-\left(\frac{1}{\lambda}-\frac{1}{\lambda_{\rm
max}}\right)\|{\bf x}_{i}\|_{2}\|{\bf y}\|_{2}.$
###### Remark 1.
Notice that, DPP is not the same as ST1 [36] and SAFE [16], which discards the
$i^{th}$ feature if
$\displaystyle|{\bf x}_{i}^{T}{\bf y}|<\lambda-\|{\bf x}_{i}\|_{2}\|{\bf
y}\|_{2}\frac{\lambda_{\rm max}-\lambda}{\lambda_{\rm max}}.$ (15)
From the perspective of the sphere test, the radius of ST1/SAFE and DPP are
the same. But the centers of ST1 and DPP are ${\bf y}/\lambda$ and ${\bf
y}/\lambda_{\rm max}$ respectively, which leads to different formulas, i.e.,
Eq. (15) and Corollary 4.
In real applications, the optimal parameter value of $\lambda$ is generally
unknown and needs to be estimated. To determine an appropriate value of
$\lambda$, commonly used approaches such as cross validation and stability
selection involve solving the Lasso problem over a grid of tuning parameters
$\lambda_{1}>\lambda_{2}>\ldots>\lambda_{\mathcal{K}}$, which is very time
consuming. Motivated by the ideas of [32] and [16], we develop a sequential
version of DPP rules. We first apply the DPP screening rule in Corollary 4 to
discard inactive features for the Lasso problem (1) with parameter being
$\lambda_{1}$. After solving the reduced optimization problem for
$\lambda_{1}$, we obtain the exact solution $\beta^{\ast}(\lambda_{1})$. Hence
by Eq. (3), we can find $\theta^{\ast}(\lambda_{1})$. According to Theorem 3,
once we know the optimal dual solution $\theta^{\ast}(\lambda_{1})$, we can
construct a new screening rule by setting $\lambda_{0}=\lambda_{1}$ to
identify inactive features for problem (1) with parameter being $\lambda_{2}$.
By repeating the above process, we obtain the sequential version of the DPP
rule as in the following corollary.
###### Corollary 5.
Sequential DPP: For the Lasso problem (1), suppose we are given a sequence of
parameter values $\lambda_{\rm
max}=\lambda_{0}>\lambda_{1}>\ldots>\lambda_{m}$. Then for any integer $0\leq
k<m$, we have $[\beta^{\ast}(\lambda_{k+1})]_{i}=0$ if
$\beta^{\ast}(\lambda_{k})$ is known and the following holds:
$\left|{\bf x}_{i}^{T}\frac{{\bf y}-{\bf
X}\beta^{\ast}(\lambda_{k})}{\lambda_{k}}\right|<1-\left(\frac{1}{\lambda_{k+1}}-\frac{1}{\lambda_{k}}\right)\|{\bf
x}_{i}\|_{2}\|{\bf y}\|_{2}.$
###### Remark 2.
From Corollaries 4 and 5, we can see that both of the DPP and sequential DPP
rules discard the inactive features for the Lasso problem with a smaller
parameter value by assuming a known dual optimal solution at a larger
parameter value. This is in fact a standard way to construct screening rules
for Lasso [32, 16, 36, 35].
###### Remark 3.
For illustration purpose, we present both the basic and sequential version of
the DPP screening rules. However, it is easy to see that the basic DPP rule
can be easily derived from its sequential version by simply setting
$\lambda_{k}=\lambda_{\rm max}$ and $\lambda_{k+1}=\lambda$. Therefore, in
this paper, we will focus on the development and evaluation of the sequential
version of the proposed screening rules. To avoid any confusions, DPP and EDPP
all refer to the corresponding sequential versions.
### 2.3 Enhanced DPP Rules for Lasso
In this section, we further improve the DPP rules presented in Section 2.2 by
a more careful analysis of the projection operators. Indeed, from the three
steps by which we develop the DPP rules, we can see that the first step is a
key. In other words, the estimation of the dual optimal solution serves as a
fundamentally important role in developing the DPP rules. Moreover, (R1’)
implies that the more accurate the estimation is, the more effective the
resulting screening rule is in discarding the inactive features. The
estimation of the dual optimal solution in DPP rules is in fact a direct
consequence of the nonexpansiveness of the projection operators. Therefore, in
order to improve the performance of the DPP rules in discarding the inactive
features, we propose two different approaches to find more accurate
estimations of the dual optimal solution. These two approaches are presented
in detail in Sections 2.3.1 and 2.3.2 respectively. By combining the ideas of
these two approaches, we can further improve the estimation of the dual
optimal solution. Based on this estimation, we develop the enhanced DPP rules
(EDPP) in Section 2.3.3. Again, we will follow the three steps in Section 2.1
to develop the proposed screening rules.
#### 2.3.1 Improving the DPP rules via Projections of Rays
In the DPP screening rules, the dual optimal solution $\theta^{*}(\lambda)$ is
estimated to be inside the ball
$B\left(\theta^{*}(\lambda_{0}),|1/\lambda-1/\lambda_{0}|\|{\bf
y}\|_{2}\right)$ with $\theta^{*}(\lambda_{0})$ given. In this section, we
show that $\theta^{*}(\lambda)$ lies inside a ball centered at
$\theta^{*}(\lambda_{0})$ with a smaller radius.
Indeed, it is well known that the projection of an arbitrary point onto a
nonempty closed convex set $C$ in a Hilbert space $\mathcal{H}$ always exists
and is unique [3]. However, the converse is not true, i.e., there may exist
${\bf w}_{1},{\bf w}_{2}\in\mathcal{H}$ such that ${\bf w}_{1}\neq{\bf w}_{2}$
and $P_{C}({\bf w}_{1})=P_{C}({\bf w}_{2})$. In fact, it is known that the
following result holds:
###### Lemma 6.
[3] Let $C$ be a nonempty closed convex subset of a Hilbert space
$\mathcal{H}$. For a point ${\bf w}\in\mathcal{H}$, let ${\bf w}(t)=P_{C}({\bf
w})+t({\bf w}-P_{C}({\bf w}))$. Then, the projection of the point ${\bf w}(t)$
is $P_{C}({\bf w})$ for all $t\geq 0$, i.e.,
$\displaystyle P_{C}({\bf w}(t))=P_{C}({\bf w}),\forall t\geq 0.$ (16)
Clearly, when ${\bf w}\neq P_{C}({\bf w})$, i.e., ${\bf w}\notin C$, ${\bf
w}(t)$ with $t\geq 0$ is the ray starting from $P_{C}({\bf w})$ and pointing
in the same direction as ${\bf w}-P_{C}({\bf w})$. By Lemma 6, we know that
the projection of the ray ${\bf w}(t)$ with $t\geq 0$ onto the set $C$ is a
single point $P_{C}({\bf w})$. [When ${\bf w}=P_{C}({\bf w})$, i.e., ${\bf
w}\in C$, ${\bf w}(t)$ with $t\geq 0$ becomes a single point and the statement
in Lemma 6 is trivial.]
By making use of Lemma 6 and the nonexpansiveness of the projection operators,
we can improve the estimation of the dual optimal solution in DPP [please
refer to Theorem 2 and Eq. (12)]. More specifically, we have the following
result:
###### Theorem 7.
For the Lasso problem, suppose the dual optimal solution $\theta^{*}(\cdot)$
at $\lambda_{0}\in(0,\lambda_{\rm max}]$ is known. For any
$\lambda\in(0,\lambda_{0}]$, let us define
$\displaystyle{\bf v}_{1}(\lambda_{0})=\begin{cases}\frac{\bf
y}{\lambda_{0}}-\theta^{*}(\lambda_{0}),\hskip 12.80373pt{\rm if}\hskip
5.69054pt\lambda_{0}\in(0,\lambda_{\rm max}),\\\ {\rm sign}({\bf
x}_{*}^{T}{\bf y}){\bf x}_{*},\hskip 8.53581pt{\rm if}\hskip
5.69054pt\lambda_{0}=\lambda_{\rm max},\end{cases}{\rm where}\hskip
5.69054pt{\bf x}_{*}={\rm argmax}_{{\bf x}_{i}}|{\bf x}_{i}^{T}{\bf y}|,$ (17)
$\displaystyle{\bf v}_{2}(\lambda,\lambda_{0})=\frac{\bf
y}{\lambda}-\theta^{*}(\lambda_{0}),$ (18) $\displaystyle{\bf
v}_{2}^{\perp}(\lambda,\lambda_{0})={\bf
v}_{2}(\lambda,\lambda_{0})-\frac{\langle{\bf v}_{1}(\lambda_{0}),{\bf
v}_{2}(\lambda,\lambda_{0})\rangle}{\|{\bf v}_{1}(\lambda_{0})\|_{2}^{2}}{\bf
v}_{1}(\lambda_{0}).$ (19)
Then, the dual optimal solution $\theta^{*}(\lambda)$ can be estimated as
follows:
$\displaystyle\theta^{*}(\lambda)\in B\left(\theta^{*}(\lambda_{0}),\|{\bf
v}_{2}^{\perp}(\lambda,\lambda_{0})\|_{2}\right)\subseteq
B\left(\theta^{*}(\lambda_{0}),\left|\frac{1}{\lambda}-\frac{1}{\lambda_{0}}\right|\|{\bf
y}\|_{2}\right).$ (20)
###### Proof.
By making use of Lemma 6, we present the proof of the statement for the cases
with $\lambda_{0}\in(0,\lambda_{\rm max})$. We postpone the proof of the
statement for the case with $\lambda_{0}=\lambda_{\rm max}$ after we introduce
more general technical results.
In view of the assumption $\lambda_{0}\in(0,\lambda_{\rm max})$, it is easy to
see that
$\displaystyle\frac{\bf y}{\lambda_{0}}\notin F\Rightarrow\frac{\bf
y}{\lambda_{0}}\neq P_{F}\left(\frac{\bf
y}{\lambda_{0}}\right)=\theta^{*}(\lambda_{0})\Rightarrow\frac{\bf
y}{\lambda_{0}}-\theta^{*}(\lambda_{0})\neq 0.$ (21)
For each $\lambda_{0}\in(0,\lambda_{\rm max})$, let us define
$\displaystyle\theta_{\lambda_{0}}(t)=\theta^{*}(\lambda_{0})+t{\bf
v}_{1}(\lambda_{0})=\theta^{*}(\lambda_{0})+t\left(\frac{\bf
y}{\lambda_{0}}-\theta^{*}(\lambda_{0})\right),\hskip 5.69054ptt\geq 0.$ (22)
By the result in (21), we can see that $\theta_{\lambda_{0}}(\cdot)$ defined
by Eq. (22) is a ray which starts at $\theta^{*}(\lambda_{0})$ and points in
the same direction as ${\bf y}/\lambda_{0}-\theta^{*}(\lambda_{0})$. In view
of Eq. (6), a direct application of Lemma 6 leads to that:
$\displaystyle P_{F}(\theta_{\lambda_{0}}(t))=\theta^{*}(\lambda_{0}),\hskip
5.69054pt\forall\hskip 2.84526ptt\geq 0.$ (23)
By applying Theorem 1 again, we have
$\displaystyle\|\theta^{*}(\lambda)-\theta^{*}(\lambda_{0})\|_{2}$
$\displaystyle=\left\|P_{F}\left(\frac{\bf
y}{\lambda}\right)-P_{F}(\theta_{\lambda_{0}}(t))\right\|_{2}$ (24)
$\displaystyle\leq\left\|\frac{\bf
y}{\lambda}-\theta_{\lambda_{0}}(t)\right\|_{2}=\left\|t\left(\frac{\bf
y}{\lambda_{0}}-\theta^{*}(\lambda_{0})\right)-\left(\frac{\bf
y}{\lambda}-\theta^{*}(\lambda_{0})\right)\right\|_{2}$ $\displaystyle=\|t{\bf
v}_{1}(\lambda_{0})-{\bf v}_{2}(\lambda,\lambda_{0})\|_{2},\hskip
5.69054pt\forall\hskip 2.84526ptt\geq 0.$
Because the inequality in (24) holds for all $t\geq 0$, it is easy to see that
$\displaystyle\|\theta^{*}(\lambda)-\theta^{*}(\lambda_{0})\|_{2}$
$\displaystyle\leq\min_{t\geq 0}\,\,\|t{\bf v}_{1}(\lambda_{0})-{\bf
v}_{2}(\lambda,\lambda_{0})\|_{2}$ (25) $\displaystyle=\begin{cases}\|{\bf
v}_{2}(\lambda,\lambda_{0})\|_{2},\hskip 14.22636pt\mbox{if }\langle{\bf
v}_{1}(\lambda_{0}),{\bf v}_{2}(\lambda,\lambda_{0})\rangle<0,\\\ \left\|{\bf
v}_{2}^{\perp}(\lambda,\lambda_{0})\right\|_{2},\hskip
8.53581pt\mbox{otherwise}.\end{cases}$
The inequality in (25) implies that, to prove the first half of the statement,
i.e.,
$\theta^{*}(\lambda)\in B(\theta^{*}(\lambda_{0}),\|{\bf
v}_{2}^{\perp}(\lambda,\lambda_{0})\|_{2}),$
we only need to show that $\langle{\bf v}_{1}(\lambda_{0}),{\bf
v}_{2}(\lambda,\lambda_{0})\rangle\geq 0$.
Indeed, it is easy to see that $0\in F$. Therefore, in view of Eq. (23), the
distance between $\theta_{\lambda_{0}}(t)$ and $\theta^{*}(\lambda_{0})$ must
be shorter than the one between $\theta_{\lambda_{0}}(t)$ and $0$ for all
$t\geq 0$, i.e.,
$\displaystyle\|\theta_{\lambda_{0}}(t)-\theta^{*}(\lambda_{0})\|_{2}^{2}\leq\|\theta_{\lambda_{0}}(t)-0\|_{2}^{2}$
(26) $\displaystyle\Rightarrow$ $\displaystyle\hskip
5.69054pt0\leq\|\theta^{*}(\lambda_{0})\|_{2}^{2}+2t\left(\left\langle\theta^{*}(\lambda_{0}),\frac{\bf
y}{\lambda_{0}}\right\rangle-\|\theta^{*}(\lambda_{0})\|_{2}^{2}\right),\hskip
5.69054pt\forall\hskip 2.84526ptt\geq 0.$
Since the inequality in (26) holds for all $t\geq 0$, we can conclude that:
$\displaystyle\left\langle\theta^{*}(\lambda_{0}),\frac{\bf
y}{\lambda_{0}}\right\rangle-\|\theta^{*}(\lambda_{0})\|_{2}^{2}\geq
0\Rightarrow\frac{\|{\bf
y}\|_{2}}{\lambda_{0}}\geq\|\theta^{*}(\lambda_{0})\|_{2}.$ (27)
Therefore, we can see that:
$\displaystyle\langle{\bf v}_{1}(\lambda_{0}),{\bf
v}_{2}(\lambda,\lambda_{0})\rangle$ $\displaystyle=\left\langle\frac{\bf
y}{\lambda_{0}}-\theta^{*}(\lambda_{0}),\frac{\bf y}{\lambda}-\frac{\bf
y}{\lambda_{0}}+\frac{\bf
y}{\lambda_{0}}-\theta^{*}(\lambda_{0})\right\rangle$ (28)
$\displaystyle\geq\left(\frac{1}{\lambda}-\frac{1}{\lambda_{0}}\right)\left\langle\frac{\bf
y}{\lambda_{0}}-\theta^{*}(\lambda_{0}),{\bf y}\right\rangle$
$\displaystyle=\left(\frac{1}{\lambda}-\frac{1}{\lambda_{0}}\right)\left(\frac{\|{\bf
y}\|_{2}^{2}}{\lambda_{0}}-\langle\theta^{*}(\lambda_{0}),{\bf
y}\rangle\right)$
$\displaystyle\geq\left(\frac{1}{\lambda}-\frac{1}{\lambda_{0}}\right)\left(\frac{\|{\bf
y}\|_{2}^{2}}{\lambda_{0}}-\|\theta^{*}(\lambda_{0})\|_{2}\|{\bf
y}\|_{2}\right)\geq 0.$
The last inequality in (28) is due to the result in (27).
Clearly, in view of (25) and (28), we can see that the first half of the
statement holds, i.e., $\theta^{*}(\lambda)\in
B(\theta^{*}(\lambda_{0}),\|{\bf v}_{2}^{\perp}(\lambda,\lambda_{0})\|_{2})$.
The second half of the statement, i.e., $B(\theta^{*}(\lambda_{0}),\|{\bf
v}_{2}^{\perp}(\lambda,\lambda_{0})\|_{2})\subseteq
B(\theta^{*}(\lambda_{0}),|1/\lambda-1/\lambda_{0}|\|{\bf y}\|_{2})$, can be
easily obtained by noting that the inequality in (24) reduces to the one in
(12) when $t=1$. This completes the proof of the statement with
$\lambda_{0}\in(0,\lambda_{\rm max})$. ∎
Before we present the proof of Theorem 7 for the case with
$\lambda_{0}=\lambda_{\rm max}$, let us briefly review some technical results
from convex analysis first.
###### Definition 8.
[28] Let $C$ be a nonempty closed convex subset of a Hilbert space
$\mathcal{H}$ and ${\bf w}\in C$. The set
$\displaystyle N_{C}({\bf w}):=\\{{\bf v}:\langle{\bf v},{\bf u}-{\bf
w}\rangle\leq 0,\forall{\bf u}\in C\\}$ (29)
is called the normal cone to $C$ at ${\bf w}$.
In terms of the normal cones, the following theorem provides an elegant and
useful characterization of the projections onto nonempty closed convex subsets
of a Hilbert space.
###### Theorem 9.
[3] Let $C$ be a nonempty closed convex subset of a Hilbert space
$\mathcal{H}$. Then, for every ${\bf w}\in\mathcal{H}$ and ${\bf w}_{0}\in C$,
${\bf w}_{0}$ is the projection of ${\bf w}$ onto $C$ if and only if ${\bf
w}-{\bf w}_{0}\in N_{C}({\bf w}_{0})$, i.e.,
$\displaystyle{\bf w}_{0}=P_{C}({\bf w})\Leftrightarrow\langle{\bf w}-{\bf
w}_{0},{\bf u}-{\bf w}_{0}\rangle\leq 0,\forall{\bf u}\in C.$ (30)
In view of the proof of Theorem 7, we can see that Eq. (23) is a key step.
When $\lambda_{0}=\lambda_{\rm max}$, similar to Eq. (22), let us define
$\displaystyle\theta_{\lambda_{\rm max}}(t)=\theta^{*}(\lambda_{\rm
max})+t{\bf v}_{1}(\lambda_{\rm max}),\hskip 5.69054pt\forall\hskip
2.84526ptt\geq 0.$ (31)
By Theorem 9, the following lemma shows that Eq. (23) also holds for
$\lambda_{0}=\lambda_{\rm max}$.
###### Lemma 10.
For the Lasso problem, let ${\bf v}_{1}(\cdot)$ and $\theta_{\lambda_{\rm
max}}(\cdot)$ be given by Eq. (17) and Eq. (31), then the following result
holds:
$\displaystyle P_{F}(\theta_{\lambda_{\rm max}}(t))=\theta^{*}(\lambda_{\rm
max}),\hskip 2.84526pt\forall\,\,t\geq 0.$ (32)
###### Proof.
To prove the statement, Theorem 9 implies that we only need to show:
$\displaystyle\langle{\bf v}_{1}(\lambda_{\rm
max}),\theta-\theta^{*}(\lambda_{\rm max})\rangle\leq 0,\hskip
5.69054pt\forall\hskip 2.84526pt\theta\in F.$ (33)
Recall that ${\bf v}_{1}(\lambda_{\rm max})={\rm sign}({\bf x}_{*}^{T}{\bf
y}){\bf x}_{*}$, ${\bf x}_{*}={\rm argmax}_{{\bf x}_{i}}|{\bf x}_{i}^{T}{\bf
y}|$ [Eq. (17)], and $\theta^{*}(\lambda_{\rm max})={\bf y}/\lambda_{\rm max}$
[Eq. (9)]. It is easy to see that
$\displaystyle\langle{\bf v}_{1}(\lambda_{\rm max}),\theta^{*}(\lambda_{\rm
max})\rangle=\left\langle{\rm sign}({\bf x}_{*}^{T}{\bf y}){\bf
x}_{*},\frac{\bf y}{\lambda_{\rm max}}\right\rangle=\frac{|{\bf x}_{*}^{T}{\bf
y}|}{\lambda_{\rm max}}=1.$ (34)
Moreover, assume $\theta$ is an arbitrary point of $F$. Then, we have
$|\langle{\bf x}_{*},\theta\rangle|\leq 1$, and thus
$\displaystyle\langle{\bf v}_{1}(\lambda_{\rm max}),\theta\rangle=\langle{\rm
sign}({\bf x}_{*}^{T}{\bf y}){\bf x}_{*},\theta\rangle\leq|\langle{\bf
x}_{*},\theta\rangle|\leq 1.$ (35)
Therefore, the inequality in (33) easily follows by combing the results in
(34) and (35), which completes the proof. ∎
We are now ready to give the proof of Theorem 7 for the case with
$\lambda_{0}=\lambda_{\rm max}$.
###### Proof.
In view of Theorem 1 and Lemma 10, we have
$\displaystyle\|\theta^{*}(\lambda)-\theta^{*}(\lambda_{\rm max})\|_{2}$
$\displaystyle=\left\|P_{F}\left(\frac{\bf
y}{\lambda}\right)-P_{F}(\theta_{\lambda_{\rm max}}(t))\right\|_{2}$ (36)
$\displaystyle\leq\left\|\frac{\bf y}{\lambda}-\theta_{\lambda_{\rm
max}}(t)\right\|_{2}=\left\|t{\bf v}_{1}(\lambda_{\rm max})-\left(\frac{\bf
y}{\lambda}-\theta^{*}(\lambda_{\rm max})\right)\right\|_{2}$
$\displaystyle=\|t{\bf v}_{1}(\lambda_{\rm max})-{\bf
v}_{2}(\lambda,\lambda_{\rm max})\|_{2},\hskip 5.69054pt\forall\hskip
2.84526ptt\geq 0.$
Because the inequality in (36) holds for all $t\geq 0$, we can see that
$\displaystyle\|\theta^{*}(\lambda)-\theta^{*}(\lambda_{\rm max})\|_{2}$
$\displaystyle\leq\min_{t\geq 0}\,\,\|t{\bf v}_{1}(\lambda_{\rm max})-{\bf
v}_{2}(\lambda,\lambda_{\rm max})\|_{2}$ (37)
$\displaystyle=\begin{cases}\|{\bf v}_{2}(\lambda,\lambda_{\rm
max})\|_{2},\hskip 14.22636pt\mbox{if }\langle{\bf v}_{1}(\lambda_{\rm
max}),{\bf v}_{2}(\lambda,\lambda_{\rm max})\rangle<0,\\\ \left\|{\bf
v}_{2}^{\perp}(\lambda,\lambda_{\rm max})\right\|_{2},\hskip
8.53581pt\mbox{otherwise}.\end{cases}$
Clearly, we only need to show that $\langle{\bf v}_{1}(\lambda_{\rm max}),{\bf
v}_{2}(\lambda,\lambda_{\rm max})\rangle\geq 0$.
Indeed, Lemma 10 implies that ${\bf v}_{1}(\lambda_{\rm max})\in
N_{F}(\theta^{*}(\lambda_{\rm max}))$ [please refer to the inequality in
(33)]. By noting that $0\in F$, we have
$\displaystyle\left\langle{\bf v}_{1}(\lambda_{\rm max}),0-\frac{\bf
y}{\lambda_{\rm max}}\right\rangle\leq 0\Rightarrow\langle{\bf
v}_{1}(\lambda_{\rm max}),{\bf y}\rangle\geq 0.$ (38)
Moreover, because ${\bf y}/\lambda_{\rm max}=\theta^{*}(\lambda_{\rm max})$,
it is easy to see that
$\displaystyle\langle{\bf v}_{1}(\lambda_{\rm max}),{\bf
v}_{2}(\lambda,\lambda_{\rm max})\rangle$ $\displaystyle=\left\langle{\bf
v}_{1}(\lambda_{\rm max}),\frac{\bf y}{\lambda}-\frac{\bf y}{\lambda_{\rm
max}}\right\rangle$ (39)
$\displaystyle=\left(\frac{1}{\lambda}-\frac{1}{\lambda_{\rm
max}}\right)\langle{\bf v}_{1}(\lambda_{\rm max}),{\bf y}\rangle\geq 0.$
Therefore, in view of (37) and (39), we can see that the first half of the
statement holds, i.e., $\theta^{*}(\lambda)\in B(\theta^{*}(\lambda_{\rm
max}),\|{\bf v}_{2}^{\perp}(\lambda,\lambda_{\rm max})\|_{2})$. The second
half of the statement, i.e.,
$B(\theta^{*}(\lambda_{\rm max}),\|{\bf v}_{2}^{\perp}(\lambda,\lambda_{\rm
max})\|_{2})\subseteq B(\theta^{*}(\lambda_{\rm
max}),|1/\lambda-1/\lambda_{\rm max}|\|{\bf y}\|_{2}),$
can be easily obtained by noting that the inequality in (37) reduces to the
one in (12) when $t=0$. This completes the proof of the statement with
$\lambda_{0}=\lambda_{\rm max}$. Thus, the proof of Theorem 7 is completed. ∎
Theorem 7 in fact provides a more accurate estimation of the dual optimal
solution than the one in DPP, i.e., $\theta^{*}(\lambda)$ lies inside a ball
centered at $\theta^{*}(\lambda_{0})$ with a radius $\|{\bf
v}_{2}^{\perp}(\lambda,\lambda_{0})\|_{2}$. Based on this improved estimation
and (R1’), we can develop the following screening rule to discard the inactive
features for Lasso.
###### Theorem 11.
For the Lasso problem, assume the dual optimal solution $\theta^{\ast}(\cdot)$
at $\lambda_{0}\in(0,\lambda_{\rm max}]$ is known. Then, for each
$\lambda\in(0,\lambda_{0})$, we have $[\beta^{\ast}(\lambda)]_{i}=0$ if
$\displaystyle|{\bf x}_{i}^{T}\theta^{\ast}(\lambda_{0})|<1-\|{\bf
v}_{2}^{\perp}(\lambda,\lambda_{0})\|_{2}\|{\bf x}_{i}\|_{2}.$
We omit the proof of Theorem 11 since it is very similar to the one of Theorem
3. By Theorem 11, we can easily develop the following sequential screening
rule.
Improvement 1: For the Lasso problem (1), suppose we are given a sequence of
parameter values $\lambda_{\rm
max}=\lambda_{0}>\lambda_{1}>\ldots>\lambda_{\mathcal{K}}$. Then for any
integer $0\leq k<\mathcal{K}$, we have $[\beta^{\ast}(\lambda_{k+1})]_{i}=0$
if $\beta^{\ast}(\lambda_{k})$ is known and the following holds:
$\left|{\bf x}_{i}^{T}\frac{{\bf y}-{\bf
X}\beta^{\ast}(\lambda_{k})}{\lambda_{k}}\right|<1-\|{\bf
v}_{2}^{\perp}(\lambda_{k+1},\lambda_{k})\|_{2}\|{\bf x}_{i}\|_{2}.$
The screening rule in Improvement 1 is developed based on (R1’) and the
estimation of the dual optimal solution in Theorem 7, which is more accurate
than the one in DPP. Therefore, in view of (R1’), the screening rule in
Improvement 1 are more effective in discarding the inactive features than the
DPP rule.
#### 2.3.2 Improving the DPP rules via Firmly Nonexpansiveness
In Section 2.3.1, we improve the estimation of the dual optimal solution in
DPP by making use of the projections of properly chosen rays. (R1’) implies
that the resulting screening rule stated in Improvement 1 is more effective in
discarding the inactive features than DPP. In this Section, we present another
approach to improve the estimation of the dual optimal solution in DPP by
making use of the so called firmly nonexpansiveness of the projections onto
nonempty closed convex subset of a Hilbert space.
###### Theorem 12.
[3] Let $C$ be a nonempty closed convex subset of a Hilbert space
$\mathcal{H}$. Then the projection operator defined in Eq. (5) is continuous
and firmly nonexpansive. In other words, for any ${\bf w}_{1},{\bf
w}_{2}\in\mathcal{H}$, we have
$\displaystyle\|P_{C}({\bf w}_{1})-P_{C}({\bf w}_{2})\|_{2}^{2}+\|({\rm
Id}-P_{C})({\bf w}_{1})-({\rm Id}-P_{C})({\bf w}_{2})\|_{2}^{2}\leq\|{\bf
w}_{1}-{\bf w}_{2}\|_{2}^{2},$ (40)
where ${\rm Id}$ is the identity operator.
In view of the inequalities in (40) and (10), it is easy to see that firmly
nonexpansiveness implies nonexpansiveness. But the converse is not true.
Therefore, firmly nonexpansiveness of the projection operators is a stronger
property than the nonexpansiveness. A direct application of Theorem 12 leads
to the following result.
###### Theorem 13.
For the Lasso problem, let $\lambda,\lambda_{0}>0$ be two parameter values.
Then
$\displaystyle\theta^{*}(\lambda)\in
B\left(\theta^{*}(\lambda_{0})+\frac{1}{2}\left(\frac{1}{\lambda}-\frac{1}{\lambda_{0}}\right){\bf
y},\frac{1}{2}\left|\frac{1}{\lambda}-\frac{1}{\lambda_{0}}\right|\|{\bf
y}\|_{2}\right)\subset
B\left(\theta^{*}(\lambda_{0}),\left|\frac{1}{\lambda}-\frac{1}{\lambda_{0}}\right|\|{\bf
y}\|_{2}\right).$ (41)
###### Proof.
In view of Eq. (6) and the firmly nonexpansiveness in (40), we have
$\displaystyle\|\theta^{*}(\lambda)-\theta^{*}(\lambda_{0})\|_{2}^{2}+\left\|\left(\frac{\bf
y}{\lambda}-\theta^{*}(\lambda)\right)-\left(\frac{\bf
y}{\lambda_{0}}-\theta^{*}(\lambda_{0})\right)\right\|_{2}^{2}\leq\left\|\frac{\bf
y}{\lambda}-\frac{\bf y}{\lambda_{0}}\right\|_{2}^{2}$ (42)
$\displaystyle\Leftrightarrow\hskip 5.69054pt$
$\displaystyle\|\theta^{*}(\lambda)-\theta^{*}(\lambda_{0})\|_{2}^{2}\leq\left\langle\theta^{*}(\lambda)-\theta^{*}(\lambda_{0}),\frac{\bf
y}{\lambda}-\frac{\bf y}{\lambda_{0}}\right\rangle$
$\displaystyle\Leftrightarrow\hskip 5.69054pt$
$\displaystyle\left\|\theta^{*}(\lambda)-\left(\theta^{*}(\lambda_{0})+\frac{1}{2}\left(\frac{1}{\lambda}-\frac{1}{\lambda_{0}}\right){\bf
y}\right)\right\|_{2}\leq\frac{1}{2}\left|\frac{1}{\lambda}-\frac{1}{\lambda_{0}}\right|\|{\bf
y}\|_{2},$
which completes the proof of the first half of the statement. The second half
of the statement is trivial by noting that the first inequality in (42)
(firmly nonexpansiveness) implies the inequality in (11) (nonexpansiveness)
but not vice versa. Indeed, it is easy to see that the ball in the middle of
(41) is inside the right one and has only a half radius. ∎
Clearly, Theorem 13 provides a more accurate estimation of the dual optimal
solution than the one in DPP, i.e., the dual optimal solution must be inside a
ball which is a subset of the one in DPP and has only a half radius. Again,
based on the estimation in Theorem 13 and (R1’), we have the following result.
###### Theorem 14.
For the Lasso problem, assume the dual optimal solution $\theta^{\ast}(\cdot)$
at $\lambda_{0}\in(0,\lambda_{\rm max}]$ is known. Then, for each
$\lambda\in(0,\lambda_{0})$, we have $[\beta^{\ast}(\lambda)]_{i}=0$ if
$\displaystyle\left|{\bf
x}_{i}^{T}\left(\theta^{*}(\lambda_{0})+\frac{1}{2}\left(\frac{1}{\lambda}-\frac{1}{\lambda_{0}}\right){\bf
y}\right)\right|<1-\frac{1}{2}\left(\frac{1}{\lambda}-\frac{1}{\lambda_{0}}\right)\|{\bf
y}\|_{2}\|{\bf x}_{i}\|_{2}.$
We omit the proof of Theorem 14 since it is very similar to the proof of
Theorem 3. A direct application of Theorem 14 leads to the following
sequential screening rule.
Improvement 2: For the Lasso problem (1), suppose we are given a sequence of
parameter values $\lambda_{\rm
max}=\lambda_{0}>\lambda_{1}>\ldots>\lambda_{\mathcal{K}}$. Then for any
integer $0\leq k<\mathcal{K}$, we have $[\beta^{\ast}(\lambda_{k+1})]_{i}=0$
if $\beta^{\ast}(\lambda_{k})$ is known and the following holds:
$\left|{\bf x}_{i}^{T}\left(\frac{{\bf y}-{\bf
X}\beta^{\ast}(\lambda_{k})}{\lambda_{k}}+\frac{1}{2}\left(\frac{1}{\lambda_{k+1}}-\frac{1}{\lambda_{k}}\right){\bf
y}\right)\right|<1-\frac{1}{2}\left(\frac{1}{\lambda_{k+1}}-\frac{1}{\lambda_{k}}\right)\|{\bf
y}\|_{2}\|{\bf x}_{i}\|_{2}.$
Because the screening rule in Improvement 2 is developed based on (R1’) and
the estimation in Theorem 13, it is easy to see that Improvement 2 is more
effective in discarding the inactive features than DPP.
#### 2.3.3 The Proposed Enhanced DPP Rules
In Sections 2.3.1 and 2.3.2, we present two different approaches to improve
the estimation of the dual optimal solution in DPP. In view of (R1’), we can
see that the resulting screening rules, i.e., Improvements 1 and 2, are more
effective in discarding the inactive features than DPP. In this section, we
give a more accurate estimation of the dual optimal solution than the ones in
Theorems 7 and 13 by combining the aforementioned two approaches together. The
resulting screening rule for Lasso is the so called enhanced DPP rule (EDPP).
Again, (R1’) implies that EDPP is more effective in discarding the inactive
features than the screening rules in Improvements 1 and 2. We also present
several experiments to demonstrate that EDPP is able to identify more inactive
features than the screening rules in Improvements 1 and 2. Therefore, in the
subsequent sections, we will focus on the generalizations and evaluations of
EDPP.
To develop the EDPP rules, we still follow the three steps in Section 2.1.
Indeed, by combining the two approaches proposed in Sections 2.3.1 and 2.3.2,
we can further improve the estimation of the dual optimal solution in the
following theorem.
###### Theorem 15.
For the Lasso problem, suppose the dual optimal solution $\theta^{*}(\cdot)$
at $\lambda_{0}\in(0,\lambda_{\rm max}]$ is known, and $\forall\hskip
2.84526pt\lambda\in(0,\lambda_{0}]$, let ${\bf
v}_{2}^{\perp}(\lambda,\lambda_{0})$ be given by Eq. (19). Then, we have
$\displaystyle\left\|\theta^{*}(\lambda)-\left(\theta^{*}(\lambda_{0})+\frac{1}{2}{\bf
v}_{2}^{\perp}(\lambda,\lambda_{0})\right)\right\|_{2}\leq\frac{1}{2}\|{\bf
v}_{2}^{\perp}(\lambda,\lambda_{0})\|_{2}.$ (43)
###### Proof.
Recall that $\theta_{\lambda_{0}}(t)$ is defined by Eq. (22) and Eq. (31). In
view of (40), we have
$\displaystyle\left\|P_{F}\left(\frac{\bf
y}{\lambda}\right)-P_{F}\left(\theta_{\lambda_{0}}(t)\right)\right\|_{2}^{2}+\left\|({\rm
Id}-P_{F})\left(\frac{\bf y}{\lambda}\right)-({\rm
Id}-P_{F})\left(\theta_{\lambda_{0}}(t)\right)\right\|_{2}^{2}\leq\left\|\frac{\bf
y}{\lambda}-\theta_{\lambda_{0}}(t)\right\|_{2}^{2}.$ (44)
By expanding the second term on the left hand side of (44) and rearranging the
terms, we obtain the following equivalent form:
$\displaystyle\left\|P_{F}\left(\frac{\bf
y}{\lambda}\right)-P_{F}\left(\theta_{\lambda_{0}}(t)\right)\right\|_{2}^{2}\leq\left\langle\frac{\bf
y}{\lambda}-\theta_{\lambda_{0}}(t),P_{F}\left(\frac{\bf
y}{\lambda}\right)-P_{F}\left(\theta_{\lambda_{0}}(t)\right)\right\rangle.$
(45)
In view of Eq. (6), Eq. (23) and Eq. (32), the inequality in (45) can be
rewritten as
$\displaystyle\|\theta^{*}(\lambda)-\theta^{*}(\lambda_{0})\|_{2}^{2}$
$\displaystyle\leq\left\langle\frac{\bf
y}{\lambda}-\theta_{\lambda_{0}}(t),\theta^{*}(\lambda)-\theta^{*}(\lambda_{0})\right\rangle$
(46) $\displaystyle=\left\langle\frac{\bf
y}{\lambda}-\theta^{*}(\lambda_{0})-t{\bf
v}_{1}(\lambda_{0}),\theta^{*}(\lambda)-\theta^{*}(\lambda_{0})\right\rangle$
$\displaystyle=\langle{\bf v}_{2}(\lambda,\lambda_{0})-t{\bf
v}_{1}(\lambda_{0}),\theta^{*}(\lambda)-\theta^{*}(\lambda_{0})\rangle,\hskip
5.69054pt\forall t\geq 0.$
[Recall that ${\bf v}_{1}(\lambda_{0})$ and ${\bf v}_{2}(\lambda,\lambda_{0})$
are defined by Eq. (17) and Eq. (18) respectively.] Clearly, the inequality in
(46) is equivalent to
$\displaystyle\left\|\theta^{*}(\lambda)-\left(\theta^{*}(\lambda_{0})+\frac{1}{2}({\bf
v}_{2}(\lambda,\lambda_{0})-t{\bf
v}_{1}(\lambda_{0}))\right)\right\|_{2}^{2}\leq\frac{1}{4}\|{\bf
v}_{2}(\lambda,\lambda_{0})-t{\bf v}_{1}(\lambda_{0})\|_{2}^{2},\hskip
5.69054pt\forall t\geq 0.$ (47)
The statement follows easily by minimizing the right hand side of the
inequality in (47), which has been done in the proof of Theorem 7. ∎
Indeed, Theorem 15 is equivalent to bounding $\theta^{*}(\lambda)$ in a ball
as follows:
$\displaystyle\theta^{*}(\lambda)\in
B\left(\theta^{*}(\lambda_{0})+\frac{1}{2}{\bf
v}_{2}^{\perp}(\lambda,\lambda_{0}),\frac{1}{2}\|{\bf
v}_{2}^{\perp}(\lambda,\lambda_{0})\|_{2}\right).$ (48)
Based on this estimation and (R1’), we immediately have the following result.
###### Theorem 16.
For the Lasso problem, assume the dual optimal problem $\theta^{*}(\cdot)$ at
$\lambda_{0}\in(0,\lambda_{\rm max}]$ is known, and
$\lambda\in(0,\lambda_{0}]$. Then $[\beta^{*}(\lambda)]_{i}=0$ if the
following holds:
$\displaystyle\left|{\bf
x}_{i}^{T}\left(\theta^{*}(\lambda_{0})+\frac{1}{2}{\bf
v}_{2}^{\perp}(\lambda,\lambda_{0})\right)\right|<1-\frac{1}{2}\|{\bf
v}_{2}^{\perp}(\lambda,\lambda_{0})\|_{2}\|{\bf x}_{i}\|_{2}.$
We omit the proof of Theorem 16 since it is very similar to the one of Theorem
3. Based on Theorem 16, we can develop the EDPP rules as follows.
###### Corollary 17.
EDPP: For the Lasso problem, suppose we are given a sequence of parameter
values $\lambda_{\rm
max}=\lambda_{0}>\lambda_{1}>\ldots>\lambda_{\mathcal{K}}$. Then for any
integer $0\leq k<\mathcal{K}$, we have $[\beta^{*}(\lambda_{k+1})]_{i}=0$ if
$\beta^{*}(\lambda_{k})$ is known and the following holds:
$\displaystyle\left|{\bf x}_{i}^{T}\left(\frac{{\bf y}-{\bf
X}\beta^{*}(\lambda_{k})}{\lambda_{k}}+\frac{1}{2}{\bf
v}_{2}^{\perp}(\lambda_{k+1},\lambda_{k})\right)\right|<1-\frac{1}{2}\|{\bf
v}_{2}^{\perp}(\lambda_{k+1},\lambda_{k})\|_{2}\|{\bf x}_{i}\|_{2}.$ (49)
It is easy to see that the ball in (48) has the smallest radius compared to
the ones in Theorems 7 and 13, and thus it provides the most accurate
estimation of the dual optimal solution. According to (R1’), EDPP is more
effective in discarding the inactive features than DPP, Improvements 1 and 2.
(a) Prostate Cancer, ${\bf X}\in\mathbb{R}^{132\times 15154}$
(b) PIE, ${\bf X}\in\mathbb{R}^{1024\times 11553}$
(c) MNIST, ${\bf X}\in\mathbb{R}^{784\times 50000}$
Figure 1: Comparison of the family of DPP rules on three real data sets:
Prostate Cancer digit data set (left), PIE data set (middle) and MNIST image
data set (right). The first row shows the rejection ratios of DPP, Improvement
1, Improvement 2 and EDPP. The second row presents the speedup gained by these
four methods.
Comparisons of the Family of DPP rules We evaluate the performance of the
family of DPP screening rules, i.e., DPP, Improvement 1, Improvement 2 and
EDPP, on three real data sets: a) the Prostate Cancer [27]; b) the PIE face
image data set [30]; c) the MNIST handwritten digit data set [21]. To measure
the performance of the screening rules, we compute the following two
quantities:
1. 1.
the rejection ratio, i.e., the ratio of the number of features discarded by
screening rules to the actual number of zero features in the ground truth;
2. 2.
the speedup, i.e., the ratio of the running time of the solver with screening
rules to the running time of the solver without screening.
For each data set, we run the solver with or without the screening rules to
solve the Lasso problem along a sequence of $100$ parameter values equally
spaced on the $\lambda/\lambda_{\rm max}$ scale from $0.05$ to $1.0$. Fig. 1
presents the rejection ratios and speedup by the family of DPP screening
rules. Table 1 reports the running time of the solver with or without the
screening rules for solving the $100$ Lasso problems, as well as the time for
running the screening rules.
Data | solver | DPP+solver | Imp.1+solver | Imp.2+solver | EDPP+solver | DPP | Imp.1 | Imp.2 | EDPP
---|---|---|---|---|---|---|---|---|---
Prostate Cancer | 121.41 | 23.36 | 6.39 | 17.00 | 3.70 | 0.30 | 0.27 | 0.28 | 0.23
PIE | 629.94 | 74.66 | 11.15 | 55.45 | 4.13 | 1.63 | 1.34 | 1.54 | 1.33
MNIST | 2566.26 | 332.87 | 37.80 | 226.02 | 11.12 | 5.28 | 4.36 | 4.94 | 4.19
Table 1: Running time (in seconds) for solving the Lasso problems along a
sequence of $100$ tuning parameter values equally spaced on the scale of
${\lambda}/{\lambda_{\rm max}}$ from $0.05$ to $1$ by (a): the solver [22]
(reported in the second column) without screening; (b): the solver combined
with different screening methods (reported in the $3^{rd}$ to the $6^{th}$
columns). The last four columns report the total running time (in seconds) for
the screening methods.
The Prostate Cancer Data Set The Prostate Cancer data set [27] is obtained by
protein mass spectrometry. The features are indexed by time-of-flight values,
which are related to the mass over charge ratios of the constituent proteins
in the blood. The data set has $15154$ measurements of $132$ patients. $69$ of
the patients have prostate cancer and the rest are healthy. Therefore, the
data matrix ${\bf X}$ is of size $132\times 15154$, and the response vector
${\bf y}\in\\{1,-1\\}^{132}$ contains the binary labels of the patients.
The PIE Face Image Data Set The PIE face image data set used in this
experiment111http://www.cad.zju.edu.cn/home/dengcai/Data/FaceData.html [10]
contains $11554$ gray face images of $68$ people, taken under different poses,
illumination conditions and expressions. Each of the images has $32\times 32$
pixels. Therefore, in each trial, we first randomly pick an image as the
response ${\bf y}\in\mathbb{R}^{1024}$, and then use the remaining images to
form the data matrix ${\bf X}\in\mathbb{R}^{1024\times 11553}$. We run $100$
trials and report the average performance of the screening rules.
The MNIST Handwritten Digit Data Set This data set contains grey images of
scanned handwritten digits, including $60,000$ for training and $10,000$ for
testing. The dimension of each image is $28\times 28$. We first randomly
select $5000$ images for each digit from the training set (and in total we
have $50000$ images) and get a data matrix ${\bf X}\in\mathbb{R}^{784\times
50000}$. Then in each trial, we randomly select an image from the testing set
as the response ${\bf y}\in\mathbb{R}^{784}$. We run $100$ trials and report
the average performance of the screening rules.
From Fig. 1, we can see that both Improvements 1 and 2 are able to discard
more inactive features than DPP, and thus lead to a higher speedup. Compared
to Improvement 2, we can also observe that Improvement 1 is more effective in
discarding the inactive features. For the three data sets, the second row of
Fig. 1 shows that Improvement 1 leads to about $20$, $60$, $70$ times speedup
respectively, which are much higher than the ones gained by Improvement 1
(roughly $10$ times for all the three cases).
Moreover, the EDPP rule, which combines the ideas of both Improvements 1 and
2, is even more effective in discarding the inactive features than Improvement
1. We can see that, for all of the three data sets and most of the $100$
parameter values, the rejection ratios of EDPP are very close to $100\%$. In
other words, EDPP is able to discard almost all of the inactive features.
Thus, the resulting speedup of EDPP is significantly better than the ones
gained by the other three DPP rules. For the PIE and MNIST data sets, we can
see that the speedup gained EDPP is about $150$ and $230$ times, which are two
orders of magnitude. In view of Table 1, for the MNIST data set, the solver
without screening needs about $2566.26$ seconds to solve the $100$ Lasso
problems. In contrast, the solver with EDPP only needs $11.12$ seconds,
leading to substantial savings in the computational cost. Moreover, from the
last four columns of Table 1, we can also observe that the computational cost
of the family of DPP rules are very low. Compared to that of the solver
without screening, the computational cost of the family of DPP rules is
negligible.
In Section 4, we will only compare the performance of EDPP against several
other state-of-the-art screening rules.
## 3 Extensions to Group Lasso
To demonstrate the flexibility of the family of DPP rules, we extend the idea
of EDPP to the group Lasso problem [37] in this section. Although the Lasso
and group Lasso problems are very different from each other, we will see that
their dual problems share a lot of similarities. For example, both of the dual
problems can be formulated as looking for projections onto nonempty closed
convex subsets of a Hilbert space. Recall that, the EDPP rule for the Lasso
problem is entirely based on the properties of the projection operators.
Therefore, the framework of the EDPP screening rule we developed for Lasso is
also applicable for the group Lasso problem. In Section 3.1, we briefly review
some basics of the group Lasso problem and explore the geometric properties of
its dual problem. In Section 3.2, we develop the EDPP rule for the group Lasso
problem.
### 3.1 Basics
With the group information available, the group Lasso problem takes the form
of:
$\inf_{\beta\in\mathbb{R}^{p}}\frac{1}{2}\left\|{\bf
y}-\sum\nolimits_{g=1}^{G}{\bf
X}_{g}\beta_{g}\right\|_{2}^{2}+\lambda\sum\nolimits_{g=1}^{G}\sqrt{n_{g}}\|\beta_{g}\|_{2},$
(50)
where ${\bf X}_{g}\in\mathbb{R}^{N\times n_{g}}$ is the data matrix for the
$g^{th}$ group and $p=\sum_{g=1}^{G}n_{g}$. The dual problem of (50) is (see
detailed derivation in the appendix):
$\displaystyle\sup_{\theta}\quad\left\\{\frac{1}{2}\|{\bf
y}\|_{2}^{2}-\frac{\lambda^{2}}{2}\left\|\theta-\frac{{\bf
y}}{\lambda}\right\|_{2}^{2}:\,\,\|{\bf
X}_{g}^{T}\theta\|_{2}\leq\sqrt{n_{g}},\,g=1,2,\ldots,G\right\\}$ (51)
The KKT conditions are given by
$\displaystyle{\bf y}$ $\displaystyle=\sum\nolimits_{g=1}^{G}{\bf
X}_{g}\beta_{g}^{\ast}(\lambda)+\lambda\theta^{\ast}(\lambda),$ (52)
$\displaystyle(\theta^{\ast}(\lambda))^{T}{\bf X}_{g}$
$\displaystyle\in\begin{cases}\sqrt{n_{g}}\frac{\beta_{g}^{\ast}(\lambda)}{\|\beta_{g}^{\ast}(\lambda)\|_{2}},\hskip
25.60747pt{\rm if}\beta_{g}^{\ast}(\lambda)\neq 0,\\\ \sqrt{n_{g}}{\bf
u},\,\|{\bf u}\|_{2}\leq 1,\hskip 5.69054pt{\rm
if}\beta_{g}^{\ast}(\lambda)=0.\\\ \end{cases}$ (53)
for $g=1,2,\ldots,G$. Clearly, in view of Eq. (53), we can see that
$\displaystyle\|(\theta^{\ast}(\lambda))^{T}{\bf
X}_{g}\|_{2}<\sqrt{n_{g}}\Rightarrow\beta_{g}^{\ast}(\lambda)=0$ (R2)
However, since $\theta^{*}(\lambda)$ is generally unknown, (R2) is not
applicable to identify the inactive groups, i.e., the groups which have $0$
coefficients in the solution vector, for the group Lasso problem. Therefore,
similar to the Lasso problem, we can first find a region
$\overline{\bf\Theta}$ which contains $\theta^{*}(\lambda)$, and then (R2) can
be relaxed as follows:
$\displaystyle\sup_{\theta\in\overline{\bf\Theta}}\|(\theta)^{T}{\bf
X}_{g}\|_{2}<\sqrt{n_{g}}\Rightarrow\beta_{g}^{\ast}(\lambda)=0.$ (R2′)
Therefore, to develop screening rules for the group Lasso problem, we only
need to estimate the region $\overline{\bf\Theta}$ which contains
$\theta^{*}(\lambda)$, solve the maximization problem in (R2′), and plug it
into (R2′). In other words, the three steps proposed in Section 2.1 can also
be applied to develop screening rules for the group Lasso problem. Moreover,
(R2′) also implies that the smaller the region $\overline{\bf\Theta}$ is, the
more accurate the estimation of the dual optimal solution is. As a result, the
more effective the resulting screening rule is in discarding the inactive
features.
Geometric Interpretations For notational convenience, let $\overline{F}$ be
the feasible set of problem (51). Similar to the case of Lasso, problem
(51)implies that the dual optimal $\theta^{\ast}(\lambda)$ is the projection
of ${\bf y}/{\lambda}$ onto the feasible set $\overline{F}$, i.e.,
$\displaystyle\theta^{*}(\lambda)=P_{\overline{F}}\left(\frac{\bf
y}{\lambda}\right),\hskip 5.69054pt\forall\hskip 2.84526pt\lambda>0.$ (54)
Compared to Eq. (6), the only difference in Eq. (54) is that the feasible set
$\overline{F}$ is the intersection of a set of ellipsoids, and thus not a
polytope. However, similar to $F$, $\overline{F}$ is also a nonempty closed
and convex (notice that $0$ is a feasible point). Therefore, we can make use
of all the aforementioned properties of the projection operators, e.g., Lemmas
6 and 10, Theorems 9 and 12, to develop screening rules for the group Lasso
problem.
Moreover, similar to the case of Lasso, we also have a specific parameter
value [32] for the group Lasso problem, i.e.,
$\displaystyle\overline{\lambda}_{\rm max}=\max_{g}\frac{\|{\bf X}_{g}^{T}{\bf
y}\|_{2}}{\sqrt{n_{g}}}.$ (55)
Indeed, $\overline{\lambda}_{\rm max}$ is the smallest parameter value such
that the optimal solution of problem (50) is 0. More specifically, we have:
$\displaystyle\beta^{*}(\lambda)=0,\hskip 5.69054pt\forall\hskip
2.84526pt\lambda\in[\overline{\lambda}_{\rm max},\infty).$ (56)
Combining the result in (56) and Eq. (52), we immediately have
$\displaystyle\theta^{*}(\lambda)=\frac{\bf y}{\lambda},\hskip
5.69054pt\forall\hskip 2.84526pt\lambda\in[\overline{\lambda}_{\rm
max},\infty).$ (57)
Therefore, all through the subsequent sections, we will focus on the cases
with $\lambda\in(0,\overline{\lambda}_{\rm max})$.
### 3.2 Enhanced DPP rule for Group Lasso
In view of (R2′), we can see that the estimation of the dual optimal solution
is the key step to develop a screening rule for the group Lasso problem.
Because $\theta^{*}(\lambda)$ is the projection of ${\bf y}/\lambda$ onto the
nonempty closed convex set $\overline{F}$ [please refer to Eq. (54)], we can
make use of all the properties of projection operators, e.g., Lemmas 6 and 10,
Theorems 9 and 12, to estimate the dual optimal solution. First, let us
develop a useful technical result as follows.
###### Lemma 18.
For the group Lasso problem, let $\overline{\lambda}_{\rm max}$ be given by
Eq. (55) and
$\displaystyle{\bf X}_{*}:={\rm argmax}_{{\bf X}_{g}}\frac{\|{\bf
X}_{g}^{T}{\bf y}\|_{2}}{\sqrt{n_{g}}}.$ (58)
Suppose the dual optimal solution $\theta^{*}(\cdot)$ is known at
$\lambda_{0}\in(0,\overline{\lambda}_{\rm max}]$, let us define
$\displaystyle\overline{\bf v}_{1}(\lambda_{0})$
$\displaystyle=\begin{cases}\frac{\bf
y}{\lambda_{0}}-\theta^{*}(\lambda_{0}),\hskip 5.69054pt{\rm if}\hskip
5.69054pt\lambda_{0}\in(0,\overline{\lambda}_{\rm max}),\\\ {\bf X}_{*}{\bf
X}_{*}^{T}{\bf y},\hskip 24.18483pt{\rm if}\hskip
5.69054pt\lambda_{0}=\overline{\lambda}_{\rm max}.\\\ \end{cases}$ (59)
$\displaystyle\overline{\theta}_{\lambda_{0}}(t)$
$\displaystyle=\theta^{*}(\lambda_{0})+t\overline{\bf
v}_{1}(\lambda_{0}),\hskip 5.69054ptt\geq 0.$ (60)
Then, we have the following result holds
$\displaystyle
P_{\overline{F}}(\overline{\theta}_{\lambda_{0}}(t))=\theta^{*}(\lambda_{0}),\hskip
2.84526pt\forall\,\,t\geq 0.$ (61)
###### Proof.
Let us first consider the cases with $\lambda_{0}\in(0,\overline{\lambda}_{\rm
max})$. In view of the definition of $\overline{\lambda}_{\rm max}$, it is
easy to see that ${\bf y}/\lambda_{0}\notin\overline{F}$. Therefore, in view
of Eq. (54) and Lemma 6, the statement in Eq. (61) follows immediately.
We next consider the case with $\lambda_{0}=\overline{\lambda}_{\rm max}$. By
Theorem 9, we only need to check if
$\displaystyle\overline{\bf v}_{1}(\overline{\lambda}_{\rm max})\in
N_{\overline{F}}(\theta^{*}(\overline{\lambda}_{\rm
max}))\Leftrightarrow\left\langle\overline{\bf v}_{1}(\overline{\lambda}_{\rm
max}),\theta-\theta^{*}(\overline{\lambda}_{\rm max})\right\rangle\leq
0,\hskip 5.69054pt\forall\hskip 2.84526pt\theta\in\overline{F}.$ (62)
Indeed, in view of Eq. (55) and Eq. (57), we can see that
$\displaystyle\langle\overline{\bf v}_{1}(\overline{\lambda}_{\rm
max}),\theta^{*}(\overline{\lambda}_{\rm max})\rangle=\left\langle{\bf
X}_{*}{\bf X}_{*}^{T}{\bf y},\frac{\bf y}{\overline{\lambda}_{\rm
max}}\right\rangle=\frac{\|{\bf X}_{*}^{T}{\bf
y}\|_{2}^{2}}{\overline{\lambda}_{\rm max}}.$ (63)
On the other hand, by Eq. (55) and Eq. (58), we can see that
$\displaystyle{\|{\bf X}_{*}^{T}{\bf y}\|_{2}}=\overline{\lambda}_{\rm
max}\sqrt{n_{*}},$ (64)
where $n_{*}$ is the number of columns of ${\bf X}_{*}$. By plugging Eq. (64)
into Eq. (63), we have
$\displaystyle\langle\overline{\bf v}_{1}(\overline{\lambda}_{\rm
max}),\theta^{*}(\overline{\lambda}_{\rm max})\rangle=\overline{\lambda}_{\rm
max}\cdot n_{*}.$ (65)
Moreover, for any feasible point $\theta\in\overline{F}$, we can see that
$\displaystyle\|{\bf X}_{*}^{T}\theta\|_{2}\leq\sqrt{n_{*}}.$ (66)
In view of the result in (66) and Eq. (64), it is easy to see that
$\displaystyle\left\langle\overline{\bf v}_{1}(\overline{\lambda}_{\rm
max}),\theta\right\rangle=\left\langle{\bf X}_{*}{\bf X}_{*}^{T}{\bf
y},\theta\right\rangle=\left\langle{\bf X}_{*}^{T}{\bf y},{\bf
X}_{*}^{T}\theta\right\rangle\leq\|{\bf X}_{*}^{T}{\bf y}\|_{2}\|{\bf
X}_{*}^{T}\theta\|_{2}=\overline{\lambda}_{\rm max}\cdot n_{*}.$ (67)
Combining the result in Eq. (63) and (67), it is easy to see that the
inequality in (62) holds for all $\theta\in\overline{F}$, which completes the
proof. ∎
By Lemma 18, we can accurately estimate the dual optimal solution of the group
Lasso problem in the following theorem. It is easy to see that the result in
Theorem 19 is very similar to the one in Theorem 15 for the Lasso problem.
###### Theorem 19.
For the group Lasso problem, suppose the dual optimal solution
$\theta^{*}(\cdot)$ at $\theta_{0}\in(0,\overline{\lambda}_{\rm max}]$ is
known, and $\overline{\bf v}_{1}(\lambda_{0})$ is given by Eq. (59). For any
$\lambda\in(0,\lambda_{0}]$, let us define
$\displaystyle\overline{\bf v}_{2}(\lambda,\lambda_{0})=\frac{\bf
y}{\lambda}-\theta^{*}(\lambda_{0}),$ (68) $\displaystyle\overline{\bf
v}_{2}^{\perp}(\lambda,\lambda_{0})=\overline{\bf
v}_{2}(\lambda,\lambda_{0})-\frac{\langle\overline{\bf
v}_{1}(\lambda_{0}),\overline{\bf
v}_{2}(\lambda,\lambda_{0})\rangle}{\|\overline{\bf
v}_{1}(\lambda_{0})\|_{2}^{2}}\overline{\bf v}_{1}(\lambda_{0}).$ (69)
Then, the dual optimal solution $\theta^{*}(\lambda)$ can be estimated as
follows:
$\displaystyle\left\|\theta^{*}(\lambda)-\left(\theta^{*}(\lambda_{0})+\frac{1}{2}\overline{\bf
v}_{2}^{\perp}(\lambda,\lambda_{0})\right)\right\|_{2}\leq\frac{1}{2}\|\overline{\bf
v}_{2}^{\perp}(\lambda,\lambda_{0})\|_{2}.$ (70)
We omit the proof of Theorem 19 since it is exactly the same as the one of
Theorem 15. Indeed, Theorem 19 is equivalent to estimating
$\theta^{*}(\lambda)$ in a ball as follows:
$\displaystyle\theta^{*}(\lambda)\in
B\left(\theta^{*}(\lambda_{0})+\frac{1}{2}\overline{\bf
v}_{2}^{\perp}(\lambda,\lambda_{0}),\frac{1}{2}\|\overline{\bf
v}_{2}^{\perp}(\lambda,\lambda_{0})\|_{2}\right).$ (71)
Based on this estimation and (R2′), we immediately have the following result.
###### Theorem 20.
For the group Lasso problem, assume the dual optimal solution
$\theta^{*}(\cdot)$ is known at $\lambda_{0}\in(0,\overline{\lambda}_{\rm
max}]$, and $\lambda\in(0,\lambda_{0}]$. Then $\beta_{g}^{\ast}(\lambda)=0$ if
the following holds
$\left\|{\bf X}_{g}^{T}\left(\theta^{*}(\lambda_{0})+\frac{1}{2}\overline{\bf
v}_{2}^{\perp}(\lambda,\lambda_{0})\right)\right\|_{2}<\sqrt{n_{g}}-\frac{1}{2}\|\overline{\bf
v}_{2}^{\perp}(\lambda,\lambda_{0})\|_{2}\|{\bf X}_{g}\|_{2}.$ (72)
###### Proof.
In view of (R2′), we only need to check if
$\displaystyle\left\|{\bf
X}_{g}^{T}\theta^{\ast}(\lambda)\right\|_{2}<\sqrt{n_{g}}.$
To simplify notations, let
${\bf o}=\theta^{*}(\lambda_{0})+\frac{1}{2}\overline{\bf
v}_{2}^{\perp}(\lambda,\lambda_{0}),\hskip
5.69054ptr=\frac{1}{2}\|\overline{\bf
v}_{2}^{\perp}(\lambda,\lambda_{0})\|_{2}.$
It is easy to see that
$\displaystyle\left\|{\bf X}_{g}^{T}\theta^{*}(\lambda)\right\|_{2}$
$\displaystyle\leq\|{\bf X}_{g}^{T}(\theta^{\ast}(\lambda)-{\bf
o})\|_{2}+\|{\bf X}_{g}^{T}{\bf o}\|_{2}$ (73) $\displaystyle<\|{\bf
X}_{g}\|_{2}\|\theta^{\ast}(\lambda)-{\bf o}\|_{2}+\sqrt{n_{g}}-r\|{\bf
X}_{g}\|_{2}$ $\displaystyle\leq r\|{\bf X}_{g}\|_{2}+\sqrt{n_{g}}-r\|{\bf
X}_{g}\|_{2}=\sqrt{n_{g}},$
which completes the proof. The second and third inequalities in (73) are due
to (72) and Theorem 19, respectively. ∎
In view of Eq. (52) and Theorem 20, we can derive the EDPP rule to discard the
inactive groups for the group Lasso problem as follows.
###### Corollary 21.
EDPP: For the group Lasso problem (50), suppose we are given a sequence of
parameter values $\overline{\lambda}_{\rm
max}=\lambda_{0}>\lambda_{1}>\ldots>\lambda_{\mathcal{K}}$. For any integer
$0\leq k<\mathcal{K}$, we have $\beta_{g}^{\ast}(\lambda_{k+1})=0$ if
$\beta^{\ast}(\lambda_{k})$ is known and the following holds:
$\displaystyle\left\|{\bf X}_{g}^{T}\left(\frac{{\bf y}-\sum_{g=1}^{G}{\bf
X}_{g}\beta_{g}^{\ast}(\lambda_{k})}{\lambda_{k}}+\frac{1}{2}\overline{\bf
v}_{2}^{\perp}(\lambda_{k+1},\lambda_{k})\right)\right\|_{2}<\sqrt{n_{g}}-\frac{1}{2}\|\overline{\bf
v}_{2}^{\perp}(\lambda_{k+1},\lambda_{k})\|_{2}\|{\bf X}_{g}\|_{2}.$
## 4 Experiments
In this section, we evaluate the proposed EDPP rules for Lasso and group Lasso
on both synthetic and real data sets. To measure the performance of our
screening rules, we compute the rejection ratio and speedup (please refer to
Section 2.3.3 for details). Because the EDPP rule is safe, i.e., no active
features/groups will be mistakenly discarded, the rejection ratio will be less
than one.
In Section 4.1, we conduct two sets of experiments to compare the performance
of EDPP against several state-of-the-art screening methods. We first compare
the performance of the basic versions of EDPP, DOME, SAFE, and strong rule.
Then, we focus on the sequential versions of EDPP, SAFE, and strong rule.
Notice that, SAFE and EDPP are safe. However, strong rule may mistakenly
discard features with nonzero coefficients in the solution. Although DOME is
also safe for the Lasso problem, it is unclear if there exists a sequential
version of DOME. Recall that, real applications usually favor the sequential
screening rules because we need to solve a sequence of of Lasso problems to
determine an appropriate parameter value [32]. Moreover, DOME assumes special
structure on the data, i.e., each feature and the response vector should be
normalized to have unit length.
In Section 4.2, we compare EDPP with strong rule for the group Lasso problem
on synthetic data sets. We are not aware of any safe screening rules for the
group Lasso problem at this point. For SAFE and Dome, it is not
straightforward to extend them to the group Lasso problem.
### 4.1 EDPP for the Lasso Problem
For the Lasso problem, we first compare the performance of the basic versions
of EDPP, DOME, SAFE and strong rule in Section 4.1.1. Then, we compare the
performance of the sequential versions of EDPP, SAFE and strong rule in
Section 4.1.2.
#### 4.1.1 Evaluation of the Basic EDPP Rule
In this section, we perform experiments on six real data sets to compare the
performance of the basic versions of SAFE, DOME, strong rule and EDPP. Briefly
speaking, suppose that we are given a parameter value $\lambda$. Basic
versions of the aforementioned screening rules always make use of
$\beta^{*}(\lambda_{\rm max})$ to identify the zero components of
$\beta^{*}(\lambda)$. Take EDPP for example. The basic version of EDPP can be
obtained by replacing $\beta^{*}(\lambda_{k})$ and ${\bf
v}_{2}^{\perp}(\lambda_{k+1},\lambda_{k})$ with $\beta^{*}(\lambda_{0})$ and
${\bf v}_{2}^{\perp}(\lambda_{k},\lambda_{0})$, respectively, in (49) for all
$k=1,\ldots,\mathcal{K}$.
In this experiment, we report the rejection ratios of the basic SAFE, DOME,
strong rule and EDPP along a sequence of $100$ parameter values equally spaced
on the $\lambda/\lambda_{\rm max}$ scale from $0.05$ to $1.0$. We note that
DOME requires that all features of the data sets have unit length. Therefore,
to compare the performance of DOME with SAFE, strong rule and EDPP, we
normalize the features of all the data sets used in this section. However, it
is worthwhile to mention that SAFE, strong rule and EDPP do not assume any
specific structures on the data set. The data sets used in this section are
listed as follows:
1. a)
Colon Cancer data set [1];
2. b)
Lung Cancer data set [6];
3. c)
Prostate Cancer data set [27];
4. d)
PIE face image data set [30, 10];
5. e)
MNIST handwritten digit data set [21];
6. f)
COIL-100 image data set [24, 11].
(a) Colon Cancer, ${\bf X}\in\mathbb{R}^{62\times 2000}$
(b) Lung Cancer, ${\bf X}\in\mathbb{R}^{203\times 12600}$
(c) Prostate Cancer, ${\bf X}\in\mathbb{R}^{132\times 15154}$
(d) PIE, ${\bf X}\in\mathbb{R}^{1024\times 11553}$
(e) MNIST, ${\bf X}\in\mathbb{R}^{784\times 50000}$
(f) COIL-100, ${\bf X}\in\mathbb{R}^{1024\times 7199}$
Figure 2: Comparison of basic versions of SAFE, DOME, Strong Rule and EDPP on
six real data sets.
The Colon Cancer Data Set This data set contains gene expression information
of 22 normal tissues and 40 colon cancer tissues, and each has 2000 gene
expression values.
The Lung Cancer Data Set This data set contains gene expression information of
186 lung tumors and 17 normal lung specimens. Each specimen has 12600
expression values.
The COIL-100 Image Data Set The data set consists of images of 100 objects.
The images of each object are taken every 5 degree by rotating the object,
yielding 72 images per object. The dimension of each image is $32\times 32$.
In each trial, we randomly select one image as the response vector and use the
remaining ones as the data matrix. We run 100 trials and report the average
performance of the screening rules.
The description and the experimental settings for the Prostate Cancer data
set, the PIE face image data set and the MNIST handwritten digit data set are
given in Section 2.3.3.
Fig. 2 reports the rejection ratios of the basic versions of SAFE, DOME,
strong rule and EDPP. We can see that EDPP significantly outperforms the other
three screening methods on five of the six data sets, i.e., the Colon Cancer,
Lung Cancer, Prostate Cancer, MNIST, and COIL-100 data sets. On the PIE face
image data set, EDPP and DOME provide similar performance and both
significantly outperform SAFE and strong rule.
However, as pointed out by Tibshirani et al. [32], the real strength of
screening methods stems from their sequential versions. The reason is because
the optimal parameter value is unknown in real applications. Typical
approaches for model selection usually involve solving the Lasso problems many
times along a sequence of parameter values. Thus, the sequential screening
methods are more suitable in facilitating the aforementioned scenario and more
useful than their basic-version counterparts in practice [32].
#### 4.1.2 Evaluation of the Sequential EDPP Rule
In this section, we compare the performance of the sequential versions of
SAFE, strong rule and EDPP by the rejection ratio and speedup. We first
perform experiments on two synthetic data sets. We then apply the three
screening rules to six real data sets.
Synthetic Data Sets
First, we perform experiments on several synthetic problems, which have been
commonly used in the sparse learning literature [7, 39, 31]. We simulate data
from the true model
$\displaystyle{\bf y}={\bf X}\beta^{*}+\sigma\epsilon,\hskip
5.69054pt\epsilon\sim N(0,1).$ (74)
We generate two data sets with $250\times 10000$ entries: Synthetic 1 and
Synthetic 2. For Synthetic 1, the entries of the data matrix ${\bf X}$ are
i.i.d. standard Gaussian with pairwise correlation zero, i.e., ${\rm
corr}({\bf x}_{i},{\bf x}_{i})=0$. For Synthetic 2, the entries of the data
matrix ${\bf X}$ are drawn from i.i.d. standard Gaussian with pairwise
correlation $0.5^{|i-j|}$, i.e., ${\rm corr}({\bf x}_{i},{\bf
x}_{j})=0.5^{|i-j|}$. To generate the response vector ${\bf
y}\in\mathbb{R}^{250}$ by the model in (74), we need to set the parameter
$\sigma$ and construct the ground truth $\beta^{*}\in\mathbb{R}^{10000}$.
Throughout this section, $\sigma$ is set to be $0.1$. To construct
$\beta^{*}$, we randomly select $\overline{p}$ components which are populated
from a uniform $[-1,1]$ distribution, and set the remaining ones as $0$. After
we generate the data matrix ${\bf X}$ and the response vector ${\bf y}$, we
run the solver with or without screening rules to solve the Lasso problems
along a sequence of $100$ parameter values equally spaced on the
$\lambda/\lambda_{\rm max}$ scale from $0.05$ to $1.0$. We then run $100$
trials and report the average performance.
(a) Synthetic 1, $\overline{p}=100$
(b) Synthetic 1, $\overline{p}=1000$
(c) Synthetic 1, $\overline{p}=5000$
(d) Synthetic 2, $\overline{p}=100$
(e) Synthetic 2, $\overline{p}=1000$
(f) Synthetic 2, $\overline{p}=5000$
Figure 3: Comparison of SAFE, Strong Rule and EDPP on two synthetic datasets
with different numbers of nonzero components of the groud truth.
We first apply the screening rules, i.e., SAFE, strong rule and EDPP to
Synthetic 1 with $\overline{p}=100,1000,5000$ respectively. Fig. 3(a), Fig.
3(b) and Fig. 3(c) present the corresponding rejection ratios and speedup of
SAFE, strong rule and EDPP. We can see that the rejection ratios of strong
rule and EDPP are comparable to each other, and both of them are more
effective in discarding inactive features than SAFE. In terms of the speedup,
EDPP provides better performance than strong rule. The reason is because
strong rule is a heuristic screening method, i.e., it may mistakenly discard
active features which have nonzero components in the solution. Thus, strong
rule needs to check the KKT conditions to ensure the correctness of the
screening result. In contrast, the EDPP rule does not need to check the KKT
conditions since the discarded features are guaranteed to be absent from the
resulting sparse representation. From the last two columns of Table 2, we can
observe that the running time of strong rule is about twice of that of EDPP.
Fig. 3(d), Fig. 3(e) and Fig. 3(f) present the rejection ratios and speedup of
SAFE, strong rule and EDPP on Synthetic 2 with $\overline{p}=100,1000,5000$
respectively. We can observe patterns similar to Synthetic 1. Clearly, our
method, EDPP, is very robust to the variations of the intrinsic structures of
the data sets and the sparsity of the ground truth.
Data | $\overline{p}$ | solver | SAFE+solver | Strong Rule+solver | EDPP+solver | SAFE | Strong Rule | EDPP
---|---|---|---|---|---|---|---|---
Synthetic 1 | 100 | 109.01 | 100.09 | 2.67 | 2.47 | 4.60 | 0.65 | 0.36
1000 | 123.60 | 111.32 | 2.97 | 2.71 | 4.59 | 0.66 | 0.37
5000 | 124.92 | 113.09 | 3.00 | 2.72 | 4.57 | 0.65 | 0.36
Synthetic 2 | 100 | 107.50 | 96.94 | 2.62 | 2.49 | 4.61 | 0.67 | 0.37
1000 | 113.59 | 104.29 | 2.84 | 2.67 | 4.57 | 0.63 | 0.35
5000 | 125.25 | 113.35 | 3.02 | 2.81 | 4.62 | 0.65 | 0.36
Table 2: Running time (in seconds) for solving the Lasso problems along a
sequence of $100$ tuning parameter values equally spaced on the scale of
${\lambda}/{\lambda_{\rm max}}$ from $0.05$ to $1$ by (a): the solver [22]
(reported in the third column) without screening; (b): the solver combined
with different screening methods (reported in the $4^{th}$ to the $6^{th}$
columns). The last four columns report the total running time (in seconds) for
the screening methods.
Real Data Sets
In this section, we compare the performance of the EDPP rule with SAFE and
strong rule on six real data sets along a sequence of $100$ parameter values
equally spaced on the $\lambda/\lambda_{\rm max}$ scale from $0.05$ to $1.0$.
The data sets are listed as follows:
1. a)
Breast Cancer data set [33, 29];
2. b)
Leukemia data set [2];
3. c)
Prostate Cancer data set [27];
4. d)
PIE face image data set [30, 10];
5. e)
MNIST handwritten digit data set [21];
6. f)
Street View House Number (SVHN) data set [25].
We present the rejection ratios and speedup of EDPP, SAFE and strong rule in
Fig. 4. Table 3 reports the running time of the solver with or without
screening for solving the $100$ Lasso problems, and that of the screening
rules.
(a) Breast Cancer, ${\bf X}\in\mathbb{R}^{44\times 7129}$
(b) Leukemia, ${\bf X}\in\mathbb{R}^{55\times 11225}$
(c) Prostate Cancer, ${\bf X}\in\mathbb{R}^{132\times 15154}$
(d) PIE, ${\bf X}\in\mathbb{R}^{1024\times 11553}$
(e) MNIST, ${\bf X}\in\mathbb{R}^{784\times 50000}$
(f) SVHN, ${\bf X}\in\mathbb{R}^{3072\times 99288}$
Figure 4: Comparison of SAFE, Strong Rule, and EDPP on six real data sets.
The Breast Cancer Data Set This data set contains $44$ tumor samples, each of
which is represented by $7129$ genes. Therefore, the data matrix ${\bf X}$ is
of $44\times 7129$. The response vector ${\bf y}\in\\{1,-1\\}^{44}$ contains
the binary label of each sample.
The Leukemia Data Set This data set is a DNA microarray data set, containing
$52$ samples and $11225$ genes. Therefore, the data matrix ${\bf X}$ is of
$55\times 11225$. The response vector ${\bf y}\in\\{1,-1\\}^{52}$ contains the
binary label of each sample.
The SVHN Data set The SVHN data set contains color images of street view house
numbers, including $73257$ images for training and $26032$ for testing. The
dimension of each image is $32\times 32$. In each trial, we first randomly
select an image as the response ${\bf y}\in\mathbb{R}^{3072}$, and then use
the remaining ones to form the data matrix ${\bf X}\in\mathbb{R}^{3072\times
99288}$. We run $100$ trials and report the average performance.
The description and the experiment settings for the Prostate Cancer data set,
the PIE face image data set and the MNIST handwritten digit data set are given
in Section 2.3.3.
Data | solver | SAFE+solver | Strong Rule+solver | EDPP+solver | SAFE | Strong Rule | EDPP
---|---|---|---|---|---|---|---
Breast Cancer | 12.70 | 7.20 | 1.31 | 1.24 | 0.44 | 0.06 | 0.05
Leukemia | 16.99 | 9.22 | 1.15 | 1.03 | 0.91 | 0.09 | 0.07
Prostate Cancer | 121.41 | 47.17 | 4.83 | 3.70 | 3.60 | 0.46 | 0.23
PIE | 629.94 | 138.33 | 4.84 | 4.13 | 19.93 | 2.54 | 1.33
MNIST | 2566.26 | 702.21 | 15.15 | 11.12 | 64.81 | 8.14 | 4.19
SVHN | 11023.30 | 5220.88 | 90.65 | 59.71 | 583.12 | 61.02 | 31.64
Table 3: Running time (in seconds) for solving the Lasso problems along a
sequence of $100$ tuning parameter values equally spaced on the scale of
${\lambda}/{\lambda_{\rm max}}$ from $0.05$ to $1$ by (a): the solver [22]
(reported in the second column) without screening; (b): the solver combined
with different screening methods (reported in the $3^{rd}$ to the $5^{th}$
columns). The last three columns report the total running time (in seconds)
for the screening methods.
From Fig. 4, we can see that the rejection ratios of strong rule and EDPP are
comparable to each other. Compared to SAFE, both of strong rule and EDPP are
able to identify far more inactive features, leading to a much higher speedup.
However, because strong rule needs to check the KKT conditions to ensure the
correctness of the screening results, the speedup gained by EDPP is higher
than that by strong rule. When the size of the data matrix is not very large,
e.g., the Breast Cancer and Leukemia data sets, the speedup gained by EDPP are
slightly higher than that by strong rule. However, when the size of the data
matrix is large, e.g., the MNIST and SVHN data sets, the speedup gained by
EDPP are significantly higher than that by strong rule. Moreover, we can also
observe from Fig. 4 that, the larger the data matrix is, the higher the
speedup can be gained by EDPP. More specifically, for the small data sets,
e.g., the Breast Cancer, Leukemia and Prostate Cancer data sets, the speedup
gained by EDPP is about $10$, $17$ and $30$ times. In contrast, for the large
data sets, e.g., the PIE, MNIST and SVHN data sets, the speedup gained by EDPP
is two orders of magnitude. Take the SVHN data set for example. The solver
without screening needs about $3$ hours to solve the $100$ Lasso problems.
Combined with the EDPP rule, the solver only needs less than $1$ minute to
complete the task.
Clearly, the proposed EDPP screening rule is very effective in accelerating
the computation of Lasso especially for large-scale problems, and outperforms
the state-of-the-art approaches like SAFE and strong rule. Notice that, the
EDPP method is safe in the sense that the discarded features are guaranteed to
have zero coefficients in the solution.
EDPP with Least-Angle Regression (LARS)
As we mentioned in the introduction, we can combine EDPP with any existing
solver. In this experiment, we integrate EDPP and strong rule with another
state-of-the-art solver for Lasso, i.e., Least-Angle Regression (LARS) [15].
We perform experiments on the same real data sets used in the last section
with the same experiment settings. Because the rejection ratios of screening
methods are irrelevant to the solvers, we only report the speedup. Table 4
reports the running time of LARS with or without screening for solving the 100
Lasso problems, and that of the screening methods. Fig. 5 shows the speedup of
these two methods. We can still observe a substantial speedup gained by EDPP.
The reason is that EDPP has a very low computational cost (see Table 4) and it
is very effective in discarding inactive features (see Fig. 4).
Data | LARS | Strong Rule+LARS | EDPP+LARS | Strong Rule | EDPP
---|---|---|---|---|---
Breast Cancer | 1.30 | 0.06 | 0.04 | 0.04 | 0.03
Leukemia | 1.46 | 0.09 | 0.05 | 0.07 | 0.04
Prostate Cancer | 5.76 | 1.04 | 0.37 | 0.42 | 0.24
PIE | 22.52 | 2.42 | 1.31 | 2.30 | 1.21
MNIST | 92.53 | 8.53 | 4.75 | 8.36 | 4.34
SVHN | 1017.20 | 65.83 | 35.73 | 62.53 | 32.00
Table 4: Running time (in seconds) for solving the Lasso problems along a
sequence of $100$ tuning parameter values equally spaced on the scale of
${\lambda}/{\lambda_{\rm max}}$ from $0.05$ to $1$ by (a): the solver [15, 23]
(reported in the second column) without screening; (b): the solver combined
with different screening methods (reported in the $3^{rd}$ and $4^{th}$
columns). The last two columns report the total running time (in seconds) for
the screening methods.
(a) Breast Cancer, ${\bf X}\in\mathbb{R}^{44\times 7129}$
(b) Leukemia, ${\bf X}\in\mathbb{R}^{55\times 11225}$
(c) Prostate Cancer, ${\bf X}\in\mathbb{R}^{132\times 15154}$
(d) PIE, ${\bf X}\in\mathbb{R}^{1024\times 11553}$
(e) MNIST, ${\bf X}\in\mathbb{R}^{784\times 50000}$
(f) SVHN, ${\bf X}\in\mathbb{R}^{3072\times 99288}$
Figure 5: The speedup gained by Strong Rule and EDPP with LARS on six real
data sets.
### 4.2 EDPP for the Group Lasso Problem
In this experiment, we evaluate the performance of EDPP and strong rule with
different numbers of groups. The data matrix ${\bf X}$ is fixed to be
$250\times 200000$. The entries of the response vector ${\bf y}$ and the data
matrix ${\bf X}$ are generated i.i.d. from a standard Gaussian distribution.
For each experiment, we repeat the computation $20$ times and report the
average results. Moreover, let $n_{g}$ denote the number of groups and $s_{g}$
be the average group size. For example, if $n_{g}$ is $10000$, then
$s_{g}=p/n_{g}=20$.
(a) $n_{g}=10000$
(b) $n_{g}=20000$
(c) $n_{g}=40000$
Figure 6: Comparison of EDPP and strong rules with different numbers of
groups.
From Figure 6, we can see that EDPP and strong rule are able to discard more
inactive groups when the number of groups $n_{g}$ increases. The intuition
behind this observation is that the estimation of the dual optimal solution is
more accurate with a smaller group size. Notice that, a large $n_{g}$ implies
a small average group size. Figure 6 also implies that compared to strong
rule, EDPP is able to discard more inactive groups and is more robust with
respect to different values of $n_{g}$.
$n_{g}$ | solver | Strong Rule+solver | EDPP+solver | Strong Rule | EDPP
---|---|---|---|---|---
$10000$ | 4535.54 | 296.60 | 53.81 | 13.99 | 8.32
$20000$ | 5536.18 | 179.48 | 46.13 | 14.16 | 8.61
$40000$ | 6144.48 | 104.50 | 37.78 | 13.13 | 8.37
Table 5: Running time (in seconds) for solving the group Lasso problems along
a sequence of $100$ tuning parameter values equally spaced on the scale of
${\lambda}/{\lambda_{\rm max}}$ from $0.05$ to $1.0$ by (a): the solver from
SLEP (reported in the second column) without screening; (b): the solver
combined with different screening methods (reported in the $3^{rd}$ and
$4^{th}$ columns). The last two columns report the total running time (in
seconds) for the screening methods. The data matrix ${\bf X}$ is of size
$250\times 200000$.
Table 5 further demonstrates the effectiveness of EDPP in improving the
efficiency of the solver. When $n_{g}=10000$, the efficiency of the solver is
improved by about $80$ times. When $n_{g}=20000$ and $40000$, the efficiency
of the solver is boosted by about $120$ and $160$ times with EDPP
respectively.
## 5 Conclusion
In this paper, we develop new screening rules for the Lasso problem by making
use of the properties of the projection operators with respect to a closed
convex set. Our proposed methods, i.e., DPP screening rules, are able to
effectively identify inactive predictors of the Lasso problem, thus greatly
reducing the size of the optimization problem. Moreover, we further improve
DPP rule and propose the enhanced DPP rule, which is more effective in
discarding inactive features than DPP rule. The idea of the family of DPP
rules can be easily generalized to identify the inactive groups of the group
Lasso problem. Extensive numerical experiments on both synthetic and real data
demonstrate the effectiveness of the proposed rules. It is worthwhile to
mention that the family of DPP rules can be combined with any Lasso solver as
a speedup tool. In the future, we plan to generalize our ideas to other sparse
formulations consisting of more general structured sparse penalties, e.g.,
tree/graph Lasso, fused Lasso.
## Appendix A.
In this appendix, we give the detailed derivation of the dual problem of
Lasso.
### A1. Dual Formulation
Assuming the data matrix is ${\bf X}\in\mathbb{R}^{N\times p}$, the standard
Lasso problem is given by:
$\inf_{\beta\in\mathbb{R}^{p}}\frac{1}{2}\|{\bf
y-X\beta}\|_{2}^{2}+\lambda\|{\beta}\|_{1}.$ (75)
For completeness, we give a detailed deviation of the dual formulation of (75)
in this section. Note that problem (75) has no constraints. Therefore the dual
problem is trivial and useless. A common trick [8] is to introduce a new set
of variables ${\bf z=y-X\beta}$ such that problem (75) becomes:
$\displaystyle\inf_{\beta}\qquad$ $\displaystyle\frac{1}{2}\|{\bf
z}\|_{2}^{2}+\lambda\|{\beta}\|_{1},$ (76) subject to $\displaystyle{\bf
z=y-X\beta}.$
By introducing the dual variables $\eta\in\mathbb{R}^{N}$, we get the
Lagrangian of problem (76):
$L({\beta,{\bf z},\eta})=\frac{1}{2}\|{\bf
z}\|_{2}^{2}+\lambda\|{\beta}\|_{1}+\eta^{T}\cdot({\bf y-X\beta-z}).$ (77)
For the Lagrangian, the primal variables are ${\beta}$ and ${\bf z}$. And the
dual function $g(\eta)$ is:
$\displaystyle g(\eta)=\inf_{\beta,{\bf z}}L({\beta,{\bf
z},\eta})=\eta^{T}{\bf y}+\inf_{\beta}(-\eta^{T}{\bf
X\beta}+\lambda\|{\beta}\|_{1})+\inf_{\bf z}\big{(}\frac{1}{2}\|{\bf
z}\|_{2}^{2}-\eta^{T}{\bf z}\big{)}.$ (78)
In order to get $g(\eta)$, we need to solve the following two optimization
problems.
$\inf_{\beta}-\eta^{T}{\bf X\beta}+\lambda\|{\beta}\|_{1},$ (79)
and
$\inf_{\bf z}\frac{1}{2}\|{\bf z}\|_{2}^{2}-\eta^{T}{\bf z}.$ (80)
Let us first consider problem (79). Denote the objective function of problem
(79) as
$f_{1}({\beta})=-\eta^{T}{\bf X\beta}+\lambda\|{\beta}\|_{1}.$ (81)
$f_{1}(\beta)$ is convex but not smooth. Therefore let us consider its
subgradient
$\partial f_{1}({\beta})=-{\bf X}^{T}\eta+\lambda{\bf v},$
in which $\|{\bf v}\|_{\infty}\leq 1$ and ${\bf v^{T}\beta}=\|{\beta}\|_{1}$,
i.e., ${\bf v}$ is the subgradient of $\|\beta\|_{1}$.
The necessary condition for $f_{1}$ to attain an optimum is
$\exists\,{\beta}^{\prime},\mbox{ such that }0\in\partial
f_{1}({\beta}^{\prime})=\\{-{\bf X}^{T}\eta+\lambda{\bf v}^{\prime}\\},$
where ${\bf v}^{\prime}\in\partial\|\beta^{\prime}\|_{1}$. In other words,
${\beta^{\prime},{\bf v}^{\prime}}$ should satisfy
${\bf v^{\prime}}=\frac{{\bf X}^{T}\eta}{\lambda},\|{\bf
v}^{\prime}\|_{\infty}\leq 1,{\bf
v^{\prime}}^{T}{\beta}^{\prime}=\|{\beta}^{\prime}\|_{1},$
which is equivalent to
$|{\bf x}_{i}^{T}\eta|\leq\lambda,i=1,2,\ldots,p.$ (82)
Then we plug ${\bf v^{\prime}}=\frac{{\bf X}^{T}\eta}{\lambda}$ and ${\bf
v^{\prime}}^{T}{\beta}^{\prime}=\|{\beta}^{\prime}\|_{1}$ into Eq. (81):
$f_{1}({\beta}^{\prime})=\inf_{\beta}f_{1}({\beta})=-\eta^{T}{\bf
X}{\beta}^{\prime}+\lambda\big{(}\frac{{\bf
X}^{T}\eta}{\lambda}\big{)}^{T}{\beta}^{\prime}=0.$ (83)
Therefore, the optimum value of problem (79) is $0$.
Next, let us consider problem (80). Denote the objective function of problem
(80) as $f_{2}({\bf z})$. Let us rewrite $f_{2}({\bf z})$ as:
$f_{2}({\bf z})=\frac{1}{2}(\|{\bf z}-\eta\|_{2}^{2}-\|\eta\|_{2}^{2}).$ (84)
Clearly,
${\bf z}^{\prime}=\mathop{\operatorname*{argmin}}_{\bf z}f_{2}({\bf z})=\eta,$
and
$\inf_{\bf z}f_{2}({\bf z})=-\frac{1}{2}\|\eta\|_{2}^{2}.$
Combining everything above, we get the dual problem:
$\displaystyle\sup_{\eta}\quad$ $\displaystyle g(\eta)=\eta^{T}{\bf
y}-\frac{1}{2}\|\eta\|_{2}^{2},$ (85) subject to $\displaystyle|{\bf
x}_{i}^{T}\eta|\leq\lambda,\,i=1,2,\ldots,p.$
which is equivalent to
$\displaystyle\sup_{\eta}\quad$ $\displaystyle g(\eta)=\frac{1}{2}\|{\bf
y}\|_{2}^{2}-\frac{1}{2}\|\eta-{\bf y}\|_{2}^{2},$ (86) subject to
$\displaystyle|{\bf x}_{i}^{T}\eta|\leq\lambda,\,i=1,2,\ldots,p.$
By a simple re-scaling of the dual variables $\eta$, i.e., let
$\theta=\frac{\eta}{\lambda}$, problem (86) transforms to:
$\displaystyle\sup_{\theta}\quad$ $\displaystyle g(\theta)=\frac{1}{2}\|{\bf
y}\|_{2}^{2}-\frac{\lambda^{2}}{2}\|\theta-\frac{{\bf y}}{\lambda}\|_{2}^{2},$
(87) subject to $\displaystyle|{\bf x}_{i}^{T}\theta|\leq 1,\,i=1,2,\ldots,p.$
### A2. The KKT Conditions
Problem (76) is clearly convex and its constraints are all affine. By Slater’s
condition, as long as problem (76) is feasible we will have strong duality.
Denote ${\beta}^{\ast}$, ${\bf z}^{\ast}$ and $\theta^{\ast}$ as optimal
primal and dual variables. The Lagrangian is
$L({\beta,{\bf z},\theta})=\frac{1}{2}\|{\bf
z}\|_{2}^{2}+\lambda\|{\beta}\|_{1}+\lambda\theta^{T}\cdot({\bf y-X\beta-z}).$
(88)
From the KKT condition, we have
$0\in\partial_{\beta}L({\beta}^{\ast},{\bf
z}^{\ast},\theta^{\ast})=-\lambda{\bf X}^{T}\theta^{\ast}+\lambda{\bf
v},\mbox{ in which }\|{\bf v}\|_{\infty}\leq 1\mbox{ and }{\bf
v}^{T}{\beta}^{\ast}=\|{\beta}^{\ast}\|_{1},$ (89)
$\nabla_{\bf z}L({\beta}^{\ast},{\bf z}^{\ast},\theta^{\ast})={\bf
z}^{\ast}-\lambda\theta^{\ast}=0,$ (90)
$\nabla_{\theta}L({\beta}^{\ast},{\bf z}^{\ast},\theta^{\ast})=\lambda({\bf
y}-{\bf X}{\beta}^{\ast}-{\bf z}^{\ast})=0.$ (91)
From Eq. (90) and (91), we have:
${\bf y}={\bf X}{\beta}^{\ast}+\lambda\theta^{\ast}.$ (92)
From Eq. (89), we know there exists ${\bf
v}^{\ast}\in\partial\|\beta^{\ast}\|_{1}$ such that
${\bf X}^{T}\theta^{\ast}={\bf v}^{\ast},\,\|{\bf v}^{\ast}\|_{\infty}\leq
1\mbox{ and }({\bf v^{\ast}})^{T}{\beta}^{\ast}=\|{\beta}^{\ast}\|_{1},$
which is equivalent to
$|{\bf x}_{i}^{T}\theta^{\ast}|\leq 1,i=1,2,\ldots,p,\mbox{ and
}(\theta^{\ast})^{T}{\bf X}{\beta}^{\ast}=\|{\beta}^{\ast}\|_{1}.$ (93)
From Eq. (93), it is easy to conclude:
$(\theta^{\ast})^{T}{\bf
x}_{i}\in\begin{cases}{\operatorname*{sign}(\beta^{*}_{i})}\mbox{ if
}\beta_{i}^{\ast}\neq 0,\\\ [-1,1]\hskip 9.95845pt\mbox{ if
}\beta_{i}^{\ast}=0.\\\ \end{cases}$ (94)
## Appendix B.
In this appendix, we present the detailed derivation of the dual problem of
group Lasso.
### B1. Dual Formulation
Assuming the data matrix is ${\bf X}_{g}\in\mathbb{R}^{N\times n_{g}}$ and
$p=\sum_{g=1}^{G}n_{g}$, the group Lasso problem is given by:
$\inf_{\beta\in\mathbb{R}^{p}}\frac{1}{2}\|{\bf y}-\sum_{g=1}^{G}{\bf
X}_{g}\beta_{g}\|_{2}^{2}+\lambda\sum_{g=1}^{G}\sqrt{n_{g}}\|\beta_{g}\|_{2}.$
(95)
Let ${\bf z}={\bf y}-\sum_{g=1}^{G}{\bf X}_{g}\beta_{g}$ and problem (95)
becomes:
$\displaystyle\inf_{\beta}\qquad$ $\displaystyle\frac{1}{2}\|{\bf
z}\|_{2}^{2}+\lambda\sum_{g=1}^{G}\sqrt{n_{g}}\|\beta_{g}\|_{2},$ (96) subject
to $\displaystyle{\bf z}={\bf y}-\sum_{g=1}^{G}{\bf X}_{g}\beta_{g}.$
By introducing the dual variables $\eta\in\mathbb{R}^{N}$, the Lagrangian of
problem (96) is:
$L({\beta,{\bf z},\eta})=\frac{1}{2}\|{\bf
z}\|_{2}^{2}+\lambda\sum_{g=1}^{G}\sqrt{n_{g}}\|\beta_{g}\|_{2}+\eta^{T}\cdot({\bf
y}-\sum_{g=1}^{G}{\bf X}_{g}\beta_{g}-{\bf z}).$ (97)
and the dual function $g(\eta)$ is:
$\displaystyle g(\eta)=\inf_{\beta,{\bf z}}L({\beta,{\bf
z},\eta})=\eta^{T}{\bf y}+\inf_{\beta}\bigg{(}-\eta^{T}\sum_{g=1}^{G}{\bf
X}_{g}\beta_{g}+\lambda\sum_{g=1}^{G}\sqrt{n_{g}}\|\beta_{g}\|_{2}\bigg{)}+\inf_{\bf
z}\big{(}\frac{1}{2}\|{\bf z}\|_{2}^{2}-\eta^{T}{\bf z}\big{)}.$ (98)
In order to get $g(\eta)$, let us solve the following two optimization
problems.
$\inf_{\beta}-\eta^{T}\sum_{g=1}^{G}{\bf
X}_{g}\beta_{g}+\lambda\sum_{g=1}^{G}\sqrt{n_{g}}\|\beta_{g}\|_{2},$ (99)
and
$\inf_{\bf z}\frac{1}{2}\|{\bf z}\|_{2}^{2}-\eta^{T}{\bf z}.$ (100)
Let us first consider problem (99). Denote the objective function of problem
(99) as
$\displaystyle\hat{f}({\beta})=-\eta^{T}\sum_{g=1}^{G}{\bf
X}_{g}\beta_{g}+\lambda\sum_{g=1}^{G}\sqrt{n_{g}}\|\beta_{g}\|_{2},$ (101)
Let
$\hat{f}_{g}(\beta_{g})=-\eta^{T}{\bf
X}_{g}\beta_{g}+\lambda\sqrt{n_{g}}\|\beta_{g}\|_{2},\qquad g=1,2,\ldots,G.$
then we can split problem (99) into a set of subproblems. Clearly
$\hat{f}_{g}(\beta_{g})$ is convex but not smooth because it has a singular
point at $0$. Consider the subgradient of $\hat{f}_{g}$,
$\partial\hat{f}_{g}({\beta}_{g})=-{\bf X}_{g}^{T}\eta+\lambda\sqrt{n_{g}}{\bf
v}_{g},\qquad g=1,2,\ldots,G,$
where ${\bf v}_{g}$ is the subgradient of $\|\beta_{g}\|_{2}$:
${\bf
v}_{g}\in\begin{cases}\frac{\beta_{g}}{\|\beta_{g}\|_{2}}\qquad\qquad\mbox{ if
}\beta_{g}\neq 0,\\\ {\bf u},\,\|{\bf u}\|_{2}\leq 1\quad\mbox{ if
}\beta_{g}=0.\\\ \end{cases}$ (102)
Let $\beta_{g}^{\prime}$ be the optimal solution of $\hat{f}_{g}$, then
$\beta_{g}^{\prime}$ satisfy
$\exists{\bf v}_{g}^{\prime}\in\partial\|\beta_{g}^{\prime}\|_{2},\quad-{\bf
X}_{g}^{T}\eta+\lambda\sqrt{n_{g}}{\bf v}_{g}^{\prime}=0.$
If $\beta_{g}^{\prime}=0$, clearly, $\hat{f}_{g}(\beta_{g}^{\prime})=0$.
Otherwise, since $\lambda\sqrt{n_{g}}{\bf v}_{g}^{\prime}={\bf X}_{g}^{T}\eta$
and ${\bf
v}_{g}^{\prime}=\frac{\beta_{g}^{\prime}}{\|\beta_{g}^{\prime}\|_{2}}$, we
have
$\hat{f}_{g}(\beta_{g}^{\prime})=-\lambda\sqrt{n_{g}}\frac{(\beta_{g}^{\prime})^{T}}{\|\beta_{g}^{\prime}\|_{2}}\beta_{g}^{\prime}+\lambda\sqrt{n_{g}}\|\beta_{g}^{\prime}\|_{2}=0.$
All together, we can conclude the
$\inf_{\beta_{g}}\hat{f}_{g}(\beta_{g})=0,\quad g=1,2,\ldots,G$
and thus
$\inf_{\beta}\hat{f}(\beta)=\inf_{\beta}\sum_{g=1}^{G}\hat{f}_{g}(\beta_{g})=\sum_{g=1}^{G}\inf_{\beta_{g}}\hat{f}_{g}(\beta_{g})=0.$
The second equality is due to the fact that $\beta_{g}$’s are independent.
Note, from Eq. (102), it is easy to see $\|{\bf v}_{g}\|_{2}\leq 1$. Since
$\lambda\sqrt{n_{g}}{\bf v}_{g}^{\prime}={\bf X}_{g}^{T}\eta$, we get a
constraint on $\eta$, i.e., $\eta$ should satisfy:
$\|{\bf X}_{g}^{T}\eta\|_{2}\leq\lambda\sqrt{n_{g}},\qquad g=1,2,\ldots,G.$
Next, let us consider problem (100). Since problem (100) is exactly the same
as problem (80), we conclude:
${\bf z}^{\prime}=\mathop{\operatorname*{argmin}}_{\bf z}\frac{1}{2}\|{\bf
z}\|_{2}^{2}-\eta^{T}{\bf z}=\eta,$
and
$\inf_{\bf z}\frac{1}{2}\|{\bf z}\|_{2}^{2}-\eta^{T}{\bf
z}=-\frac{1}{2}\|\eta\|_{2}^{2}.$
Therefore the dual function $g(\eta)$ is:
$g(\eta)=\eta^{T}{\bf y}-\frac{1}{2}\|\eta\|_{2}^{2}.$
Combining everything above, we get the dual formulation of the group Lasso:
$\displaystyle\sup_{\eta}\quad$ $\displaystyle g(\eta)=\eta^{T}{\bf
y}-\frac{1}{2}\|\eta\|_{2}^{2},$ (103) subject to $\displaystyle\|{\bf
X}_{g}^{T}\eta\|_{2}\leq\lambda\sqrt{n_{g}},\,g=1,2,\ldots,G.$
which is equivalent to
$\displaystyle\sup_{\eta}\quad$ $\displaystyle g(\eta)=\frac{1}{2}\|{\bf
y}\|_{2}^{2}-\frac{1}{2}\|\eta-{\bf y}\|_{2}^{2},$ (104) subject to
$\displaystyle\|{\bf
X}_{g}^{T}\eta\|_{2}\leq\lambda\sqrt{n_{g}},\,g=1,2,\ldots,G.$
By a simple re-scaling of the dual variables $\eta$, i.e., let
$\theta=\frac{\eta}{\lambda}$, problem (104) transforms to:
$\displaystyle\sup_{\theta}\quad$ $\displaystyle g(\theta)=\frac{1}{2}\|{\bf
y}\|_{2}^{2}-\frac{\lambda^{2}}{2}\|\theta-\frac{{\bf y}}{\lambda}\|_{2}^{2},$
(105) subject to $\displaystyle\|{\bf
X}_{g}^{T}\theta\|_{2}\leq\sqrt{n_{g}},\,g=1,2,\ldots,G.$
### B2. The KKT Conditions
Clearly, problem (96) is convex and its constraints are all affine. By
Slater’s condition, as long as problem (96) is feasible we will have strong
duality. Denote ${\beta}^{\ast}$, ${\bf z}^{\ast}$ and $\theta^{\ast}$ as
optimal primal and dual variables. The Lagrangian is
$L({\beta,{\bf z},\theta})=\frac{1}{2}\|{\bf
z}\|_{2}^{2}+\lambda\sum_{g=1}^{G}\sqrt{n_{g}}\|\beta_{g}\|_{2}+\lambda\theta^{T}\cdot({\bf
y}-\sum_{g=1}^{G}{\bf X}_{g}\beta_{g}-{\bf z}).$ (106)
From the KKT condition, we have
$0\in\partial_{\beta_{g}}L({\beta}^{\ast},{\bf
z}^{\ast},\theta^{\ast})=-\lambda{\bf
X}_{g}^{T}\theta^{\ast}+\lambda\sqrt{n_{g}}{\bf v}_{g},\mbox{ in which }{\bf
v}_{g}\in\partial\|\beta_{g}^{\ast}\|_{2},\quad g=1,2,\ldots,G,$ (107)
$\nabla_{\bf z}L({\beta}^{\ast},{\bf z}^{\ast},\theta^{\ast})={\bf
z}^{\ast}-\lambda\theta^{\ast}=0,$ (108)
$\nabla_{\theta}L({\beta}^{\ast},{\bf
z}^{\ast},\theta^{\ast})=\lambda\cdot({\bf y}-\sum_{g=1}^{G}{\bf
X}_{g}\beta_{g}^{\ast}-{\bf z}^{\ast})=0.$ (109)
From Eq. (108) and (109), we have:
${\bf y}=\sum_{g=1}^{G}{\bf X}_{g}\beta_{g}^{\ast}+\lambda\theta^{\ast}.$
(110)
From Eq. (107), we know there exists ${\bf
v}_{g}^{\prime}\in\partial\|\beta_{g}^{\ast}\|_{2}$ such that
${\bf X}_{g}^{T}\theta^{\ast}=\sqrt{n_{g}}{\bf v}_{g}^{\prime}$
and
${\bf
v}_{g}^{\prime}\in\begin{cases}\frac{\beta_{g}^{\ast}}{\|\beta_{g}^{\ast}\|_{2}}\qquad\qquad\mbox{
if }\beta_{g}^{\ast}\neq 0,\\\ {\bf u},\,\|{\bf u}\|_{2}\leq 1\quad\mbox{ if
}\beta_{g}^{\ast}=0,\\\ \end{cases}$
Then the following holds:
${\bf
X}_{g}^{T}\theta^{\ast}\in\begin{cases}\sqrt{n_{g}}\frac{\beta_{g}^{\ast}}{\|\beta_{g}^{\ast}\|_{2}}\hskip
29.87538pt\mbox{ if }\beta_{g}^{\ast}\neq 0,\\\ \sqrt{n_{g}}{\bf u},\,\|{\bf
u}\|_{2}\leq 1\mbox{ if }\beta_{g}^{\ast}=0,\\\ \end{cases}$ (111)
for $g=1,2,\ldots,G$. Clearly, if $\|{\bf
X}_{g}^{T}\theta^{\ast}\|_{2}<\sqrt{n_{g}}$, we can conclude
$\beta_{g}^{\ast}=0$.
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|
arxiv-papers
| 2012-11-16T17:48:42 |
2024-09-04T02:49:38.103768
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Jie Wang, Peter Wonka, Jieping Ye",
"submitter": "Jie Wang",
"url": "https://arxiv.org/abs/1211.3966"
}
|
1211.3997
|
# Chandra, Keck and VLA Observations of the Crab Nebula during the 2011-April
Gamma-ray Flare
Martin C. Weisskopf11affiliation: NASA Marshall Space Flight Center,
Astrophysics Office (ZP12), Huntsville, AL 35812, USA , Allyn F.
Tennant11affiliation: NASA Marshall Space Flight Center, Astrophysics Office
(ZP12), Huntsville, AL 35812, USA , Jonathan Arons22affiliation: Astronomy
Department and Theoretical Astrophysics Center, University of California,
Berkeley, 601 Campbell Hall, Berkeley, CA 94720, USA , Roger
Blandford33affiliation: W. W. Hansen Experimental Physics Laboratory, Kavli
Institute for Particle Astrophysics and Cosmology, Department of Physics and
SLAC National Accelerator Laboratory, Stanford University, Stanford, CA 94305,
USA ,
Rolf Buehler44affiliation: DESY, Platanenallee 6, 15738 Zeuthen, Germany ,
Patrizia Caraveo55affiliation: INAF-IASF Milano, via E. Bassini 15, 20133
Milano, Italy & INFN Pavia, via A. Bassi 6, 27100 Pavia, Italy , C. C. Teddy
Cheung66affiliation: National Research Council Research Associate, National
Academy of Sciences, Washington, DC 20001, resident at Naval Research
Laboratory, Washington, DC 20375, USA , Enrico Costa77affiliation: INFN Roma
Tor Vergata, via della Ricerca Scientifica 1, 00133 Roma, Italy , Andrea de
Luca55affiliation: INAF-IASF Milano, via E. Bassini 15, 20133 Milano, Italy &
INFN Pavia, via A. Bassi 6, 27100 Pavia, Italy , Carlo Ferrigno88affiliation:
ISDC, Data Center for Astrophysics of the University of Geneva, chemin
d’Écogia 16, 1290, Versoix, Switzerland , Hai Fu99affiliation: Department of
Physics & Astronomy, University of California, Irvine, CA 92697, USA , Stefan
Funk33affiliation: W. W. Hansen Experimental Physics Laboratory, Kavli
Institute for Particle Astrophysics and Cosmology, Department of Physics and
SLAC National Accelerator Laboratory, Stanford University, Stanford, CA 94305,
USA , Moritz Habermehl1010affiliation: Institut für Experimentalphysik,
Universität Hamburg, Luruper Chaussee 149, D-22761 Hamburg, Germany , Dieter
Horns1010affiliation: Institut für Experimentalphysik, Universität Hamburg,
Luruper Chaussee 149, D-22761 Hamburg, Germany ,
Justin D. Linford1111affiliation: Department of Physics and Astronomy,
University of New Mexico, MSC07 4220, Albuquerque, NM 87131-0001, USA. ,
Andrei Lobanov1212affiliation: Max-Planck-Institut für Radioastronomie, Auf
dem Hügel 69, 53121, Bonn, Germany , Claire Max1313affiliation: Department of
Astronomy & Astrophysics, University of California, Santa Cruz, CA 95064, USA
, Roberto Mignani1414affiliation: Mullard Space Science Laboratory, University
College London, Holmbury St. Mary Dorking, Surrey RH5 6NT, England & Kepler
Institute of Astronomy, University of Zielona Góra, Lubuska 2, 65-265, Zielona
Góra, Poland ,
Stephen L. O’Dell11affiliation: NASA Marshall Space Flight Center,
Astrophysics Office (ZP12), Huntsville, AL 35812, USA , Roger W.
Romani33affiliation: W. W. Hansen Experimental Physics Laboratory, Kavli
Institute for Particle Astrophysics and Cosmology, Department of Physics and
SLAC National Accelerator Laboratory, Stanford University, Stanford, CA 94305,
USA , Edoardo Striani77affiliation: INFN Roma Tor Vergata, via della Ricerca
Scientifica 1, 00133 Roma, Italy , Marco Tavani77affiliation: INFN Roma Tor
Vergata, via della Ricerca Scientifica 1, 00133 Roma, Italy ,
Gregory B. Taylor1111affiliation: Department of Physics and Astronomy,
University of New Mexico, MSC07 4220, Albuquerque, NM 87131-0001, USA. ,
Yasunobu Uchiyama33affiliation: W. W. Hansen Experimental Physics Laboratory,
Kavli Institute for Particle Astrophysics and Cosmology, Department of Physics
and SLAC National Accelerator Laboratory, Stanford University, Stanford, CA
94305, USA , Yajie Yuan33affiliation: W. W. Hansen Experimental Physics
Laboratory, Kavli Institute for Particle Astrophysics and Cosmology,
Department of Physics and SLAC National Accelerator Laboratory, Stanford
University, Stanford, CA 94305, USA
###### Abstract
We present results from our analysis of Chandra X-ray Observatory, W. M. Keck
Observatory, and Karl G. Jansky Very Large Array (VLA) images of the Crab
Nebula that were contemporaneous with the $\gamma$-ray flare of 2011 April.
Despite hints in the X-ray data, we find no evidence for statistically
significant variations that pinpoint the specific location of the flares
within the Nebula. The Keck observations extend this conclusion to the “inner
knot”, i.e., the feature within an arcsecond of the pulsar. The VLA
observations support this conclusion. We also discuss theoretical implications
of the $\gamma$-ray flares and suggest that the most dramatic $\gamma$-ray
flares are due to radiation-reaction-limited synchrotron emission associated
with sudden, dissipative changes in the current system sustained by the
central pulsar.
## 1 Introduction
The Crab Nebula, the relic of a stellar explosion recorded by Chinese
astronomers in 1054, has a special place in the history of astronomy. It is
our most frequently observed laboratory for high-energy astrophysics. Located
at a distance of $\approx 2$ kpc, the system is energized by a pulsar of
spindown luminosity $L_{plsr}\approx 5\times 10^{38}$ erg/s and current spin
period P $\approx 34$ ms. The history and general properties of the system are
nicely summarized in the review by Hester (2008). Optical and X-ray images
(Hester et al. 1995; Weisskopf et al. 2000; Hester et al. 2002) of the inner
nebula show features such as an inner ring, toroidal structure, knots, and two
opposing jets originating from the pulsar –– these latter presumably aligned
with its rotation axis and proper motion vector (Caraveo & Mignani 1999; Ng &
Romani 2007; Kaplan et al. 2008 and references therein). The “inner-ring”,
prominent in X-rays, is commonly accepted as being the termination shock
produced by the relativistic wind of particles accelerated by the pulsar. Many
of the optical and X-ray features brighten and fade and/or move over weeks or
months (e.g., Hester et al. 1995; Hester et al. 2002).
The quiescent or average spectral energy distribution (SED) of the Crab Nebula
has a characteristic two-humped form (see, e.g., Figure 7; Atoyan & Aharonian
1996; Bucciantini, Arons, & Amato 2011 and references therein). The
synchrotron spectrum extends from $\approx 30$ MHz to $\approx 1.2\times
10^{22}$ Hz (500 ${\rm Me\\!V}$). Most of the power is radiated by $\approx$
${\rm Te\\!V}$ electrons in the near UV $\approx 10$ ${\rm e\\!V}$ with an
associated luminosity of $\approx 1.3\times 10^{38}$ ergs/s (Hester 2008).
This roughly matches the loss of rotational energy by the pulsar, which
releases its energy electromagnetically, generating a current $\approx 200$ TA
and inducing an electro-motive force (EMF) $\approx 50$ PV. However, the
nebula is currently varying on a few-year timescale (Wilson-Hodge et al.
2011). At higher energies, Compton scattering has a luminosity of $\approx
10^{36}$ erg/s, peaking around 60 ${\rm Ge\\!V}$ (Albert et al. 2008) and
measured up to $\approx 80$ ${\rm Te\\!V}$ (e.g., Aharonian et al. 2004; Abdo
et al. 2010).
Since 2007, the AGILE and Fermi satellites have detected several $\gamma$-ray
flares from the Crab Nebula (Tavani et al. 2011; Abdo et al. 2011; Striani et
al. 2011a; Buehler et al. 2012) in the $0.1-1$ ${\rm Ge\\!V}$ range. The most
dramatic flares exhibit variability on timescales as short as a few hours,
although it is unclear whether they are distinct events or just the largest
variations from a stationary power spectrum of fluctuations. Prior to the
2011-April event, the only Crab $\gamma$-ray flare covered by a multi-
wavelength observing program was the 2010-September flare, which triggered
observations in radio, optical (using both ground-based telescopes and HST),
and X-ray bands. Despite the $\gamma$-ray brightness of the flares, there has
been no evidence for correlated variations in radio (Lobanov, Horns, & Muxlow
2011; this paper), near infrared (Kanbach, et al. 2010, this paper), optical
(Caraveo et al. 2010), or X-ray bands (Evangelista et al. 2010; Shaposhnikov
et al. 2010; Tennant et al. 2010; Ferrigno et al. 2010; Horns et al. 2010;
Cusumano et al. 2011; Tennant et al. 2011; Tavani et al. 2011; Striani et al.
2011b; this paper).
Here we focus on the Fermi-LAT results for the 2011-April flare (Buehler et
al. 2012), which allow us to assess the source behavior in detail. The source
doubled its $\gamma$-ray flux within eight hours and reached a peak flux
30-times its average. The isotropic luminosity increased to $\approx 2\times
10^{37}$ erg/s in $\approx 10$ hr and the spectrum peaked at $\approx 400$
${\rm Me\\!V}$. Table 1 gives the $\gamma$-ray powerlaw photon spectral index,
the integrated photon flux above 100 ${\rm Me\\!V}$, and the photon spectral
flux at 100 ${\rm Me\\!V}$, as measured during the 10-ks time intervals when
X-ray data (§2) were taken.
Notification as to the level of flaring prompted us to trigger pre-approved
Target of Opportunity observations with Chandra and with the NRAO111The
National Radio Astronomy Observatory is a facility of the National Science
Foundation operated under cooperative agreement by Associated Universities,
Inc. Karl G. Jansky Very Large Array (VLA). We were also fortunate to obtain a
Keck image in the near infrared, albeit not under ideal conditions. Figure 7
shows the Fermi-LAT $\gamma$-ray counting rate as a function of time and also
indicates the times of the Chandra, Keck (§3) and some of the VLA (§4)
observations.
## 2 X-ray Observations and Data Analysis
With the back-illuminated ACIS S3 CCD on the Chandra X-ray Observatory
approximately centered on the Crab pulsar, we obtained five observations
(Table 1) during and somewhat after the 2011-April $\gamma$-ray flare. For
these observations, the spacecraft dithered with an amplitude set to
$1\arcsec$. Although standard processing typically produces an aspect solution
better than $0.5\arcsec$, even this small uncertainty can introduce noticeable
shifts when comparing different data sets. Thus, we re-registered images for
our analysis using the read-out streak and the pulsar as guides. As each of
these 5 images was placed at approximately the same CCD location, spatial non-
uniformity in the ACIS response (e.g., due to contamination) does not
introduce spurious temporal variability.
Owing to the Crab’s high flux, the ACIS observations employed a special mode
with 0.2-s frame time, which limits the CCD read-out to a $300\times 300$
ACIS-pixel ($\approx 150\arcsec\times 150\arcsec$) subarray. Although each
observation lasted about 10 ks, telemetry saturation reduced the effective
integration time to approximately 1200 s per observation. Despite the short
frame time of the special ACIS mode, regions of high surface brightness suffer
somewhat from pile-up effects. We consider only data in the range 0.5–8.0
${\rm ke\\!V}$ because of severe interstellar absorption at low energies and
declining flux at high energies. Using these data, we then search for X-ray
variations approximately contemporaneous with the 2011-April $\gamma$-ray
flare.
### 2.1 X-ray Image Analysis
Figure 7 shows an image of the number of counts per ACIS pixel, summed over
the 5 observations. For each observation, we re-binned a $120\times 120$ ACIS-
pixel image centered on the pulsar into a $60\times 60$ array of $2\times 2$
ACIS pixels. Each of these $I=3600$ “analysis pixels” is sufficiently large
(about 1 square arcsec) to enclose most of the Chandra point spread function
anywhere in the field of view. Note that we also performed an analysis using a
circular bin of radius $1$ ACIS pixel on an oversampled grid of cadence
$0.1\times$ ACIS pixel — i.e., a spherical top hat smoothing of the events. As
each method gave similar results, we here report the results for the “analysis
pixel” binning, for which each pixel is statistically independent.
For each analysis pixel $i$, we calculate the mean count rate $r_{i}$ averaged
over the $J=5$ observations, weighted222
$r_{i}=\sum_{j=1}^{J}\\{r_{ij}/\sigma_{ij}^{2}\\}/\sum_{j=1}^{J}\\{1/\sigma_{ij}^{2}\\}$
by the respective (counting-rate) statistical error $\sigma_{ij}$ for each
analysis pixel and observation. For evaluating statistical significance of
temporal variations over the $J=5$ observations, we compute333
$\chi_{i}^{2}=\sum_{j=1}^{J}\\{(r_{ij}-r_{i})^{2}/\sigma_{ij}^{2}\\}$ and
$S_{i}\equiv(\chi_{i}^{2}-\nu_{i})/\sqrt{2\nu_{i}}$ where $\nu_{i}=(J-1)$.
$\chi_{i}^{2}$ along with a derived significance measure $S_{i}$.
For purposes of discussion, we also compute the appropriately weighted444
$\sigma_{i}^{2}=J/\sum_{j=1}^{J}\\{1/\sigma_{ij}^{2}\\}$ and
$s_{i}^{2}=\frac{J}{(J-1)}\sum_{j=1}^{J}\\{(r_{ij}-r_{i})^{2}/\sigma_{ij}^{2}\\}/\sum_{j=1}^{J}\\{1/\sigma_{ij}^{2}\\}=\frac{\chi_{i}^{2}}{(J-1)}\sigma_{i}^{2}$
statistical error $\sigma_{i}$ and sample standard deviation $s_{i}$ for each
pixel $i$. As properly weighted,
$\chi_{i}^{2}=(J-1)\,s_{i}^{2}/\sigma_{i}^{2}$. While we have rigorously
calculated $r_{i}$, $\chi_{i}^{2}$, $\sigma_{i}$, and $s_{i}$ for each pixel
$i$ using appropriate weightings, we note that the weightings are nearly
uniform as the effective duration of the each of the $J=5$ observations was
nearly the same—about 1200 s.
### 2.2 Variability of the X-ray images
The (counting) statistical error $\sigma_{i}$ is the primary noise term and
thus governs the sensitivity for detecting temporal variations at the
analysis-pixel (square-arcsec) scale. Figure 7 shows the image and the
corresponding histogram of the distribution of $\sigma_{i}$, which ranges from
0.0025 to 0.024 ct/s per analysis pixel. Based upon the $\chi^{2}$ probability
distribution and the number of “tries” ($I=3600$ independent analysis pixels),
a 99%-confidence detection would require a $\chi_{i,99\%}^{2}>31.2$ on
$(J-1)=4$ degrees of freedom. This corresponds to a sample standard deviation
$s_{i,99\%}>2.80\,\sigma_{i}$, which ranges from 0.0071 to 0.068 ct/s over the
field.
We do not here display the analogous image and histogram for the sample
standard deviation $s_{i}$, which ranges from 0.0014 to 0.048 ct/s per
analysis pixel. Instead, Figure 7 shows the image and histogram of the
distribution of a calculated (§2.1) significance measure $S_{i}$, related to
$\chi_{i}^{2}=(J-1)\,s_{i}^{2}/\sigma_{i}^{2}$. The statistically most
significant variation has $\chi_{i}^{2}=23.5$ on $\nu=(J-1)=4$ degrees of
freedom giving $S_{i}=6.9$. Such a fluctuation is expected statistically in at
least 1 of 3600 pixels in 31% of realizations. Table 2 gives the sample
standard deviation and 99%-confidence upper limit to the count-rate variation,
for the analysis pixel with the statistically most significant X-ray
variation. While we detect no variations statistically significant at
99%confidence, it is curious that the 3 most significant variations occur at
locations on the inner ring.
Note that if a feature, such as one of the knots, doesn’t change in intensity
but moves from one analysis pixel to another, then our $\chi_{i}^{2}$ test
would detect this as a variation. We know that features in the inner ring of
the Nebula do move and expect to detect some variability due to this motion.
However, as our 5 observations span only 14 days and $1\arcsec$ corresponds to
11.5 light days, only relativistic motion would be detectable. Other effects,
such as changes in the roll angle of the read-out streak, can also lead to
spurious variability. Indeed, this may play a role for the analysis pixel with
the most significant variation, which lies adjacent to the average read-out
streak (Figure 7).
### 2.3 Limits to the X-ray flux
Thus far, we have described the X-ray data for each analysis pixel in units of
ACIS count rate. Neglecting for the moment pile-up effects, the photon
spectral flux (or other related radiation quantity) is proportional to the
count rate for an assumed spectral shape. Consequently, any change in count
rate corresponds to a proportionate change in photon spectral flux (for an
assumed spectral shape). Using the Chandra
PIMMS555http://asc.harvard.edu/toolkit/pimms.jsp for the ACIS-S detector and
an absorption column $N_{\rm H}=3.1\times 10^{21}\ {\rm cm}^{-2}$, we
determine (ignoring pile-up) this constant of proportionality for an X-ray
power-law photon index $\Gamma_{x}=\frac{2}{3}$, 1, and 2: At $E_{x}=1$ ${\rm
ke\\!V}$, $N_{E}(E_{x})/r=$ 0.99, 1.26, and 2.46 $\times 10^{-3}$ ph/(cm2 s
${\rm ke\\!V}$) per ct/s, respectively. Correcting for pile-up has little
effect in low-count-rate regions, but would raise these flux upper limits by
$\approx 10$% or so for high-count-rate regions.
Table 2 calculates the photon spectral flux $N_{E}(E_{x})$, the energy
spectral flux $F_{E}(E_{x})$, and the indicative (isotropic) luminosity
$EL_{E}(E_{x})=4\pi D^{2}EF_{E}(E_{x})$ at $D=2$ kpc, corresponding to the
sample standard deviation and 99%-confidence upper limit for the count-rate
variation in the analysis pixel with the most significant X-ray variation.
Figure 7 displays an image of the energy spectral flux $F_{E}(E_{x})$ at
$E_{x}=1$ ${\rm ke\\!V}$ for $\Gamma_{x}=1$, based upon the sample standard
deviation $s_{i}$ of the count rate in each analysis pixel.
### 2.4 Constraints on the X-ray to $\gamma$-ray Spectral Index
We now compare the X-ray data with the $\gamma$-ray data to quantify the
implications of our lack of detection of time variations in the X-ray data.
Our approach compares a variability measure for the X-ray (1-${\rm ke\\!V}$)
photon spectral flux $\Delta N_{E}(E_{x})$ in each analysis pixel with the
analogous variability measure for the $\gamma$-ray (100-${\rm Me\\!V}$) photon
spectral flux $\Delta N_{E}(E_{\gamma})$. In particular, we calculate the
sample standard deviation of the $\gamma$-ray spectral flux at 100 ${\rm
Me\\!V}$, using power-law fits to the 5 Fermi-LAT measurements that were
simultaneous with the 5 Chandra observations. Table 1 lists the 5 Chandra
ObsIDs, their dates, along with the $\gamma$-ray photon index
$\Gamma_{\gamma}$, integrated photon flux $N(>$100 ${\rm Me\\!V}$), and photon
spectral flux $N_{E}$(100 ${\rm Me\\!V}$). For the 5 Fermi-LAT observations,
the mean and sample standard deviation of the photon spectral flux at 100
${\rm Me\\!V}$ are $1.21\times 10^{-10}$ and $5.77\times 10^{-11}$ ph/(cm2 s
${\rm ke\\!V}$), respectively.
Based upon the sample standard deviation ($s_{i}$) of photon spectral flux at
$E_{x}=1$ ${\rm ke\\!V}$ for each X-ray analysis pixel and the measured
standard deviation ($5.77\times 10^{-11}$ ph/(cm2 s ${\rm ke\\!V}$)) at
$E_{\gamma}=100$ ${\rm Me\\!V}$, we constrain the effective X-ray to
$\gamma$-ray photon index of the flaring component:
$\Gamma_{x\gamma}\equiv-\log[\Delta N_{E}(E_{\gamma})/\Delta
N_{E}(E_{x})]/\log[E_{\gamma}/E_{x}]$. Figure 7 shows the image and
corresponding histogram of the distribution of upper limits to
$\Gamma_{x\gamma}$ based upon the sample standard deviation of the X-ray
measurements and assuming $\Gamma_{x}=1$.
In that the $\gamma$-ray variations are statistically significant and the
X-ray variations are not, we compute 99%-confidence upper limits to
$\Gamma_{x\gamma}$ (Table 2 last row). Note that the upper limits to
$\Gamma_{x\gamma}$ are marginally consistent with the low-energy extrapolation
of the $\gamma$-ray spectrum ($\Gamma_{\gamma}=1.27\pm 0.12$) of the flaring
component (Buehler et al. 2012).
### 2.5 Variability within an X-ray Image
In sections 2.1–2.2, the search for variability focused on sensitivity to flux
changes amongst the five pointings with a minimum cadence of 0.6 days. Here,
we search for variability on shorter time-scales—namely within each pointing.
As described in Section 2, the ACIS S3 CCD was read at most roughly 6000 times
in each pointing due to telemetry saturation (deadtime). In this study, rather
than using the $2\times 2$ ACIS analysis pixel, we employed a circular search
bin with a radius of $1$ ACIS-pixel ($0.49\arcsec$) on a grid with $0.1$ ACIS
pixel spacing. Note that this oversampling implies that the results of the
test in adjacent pixels are not statistically independent. (We also analyzed
these data using the statistically independent analysis pixels of §2.1 with
similar results as below.)
Using the frame number of each detected photon, we derive the empirical
cumulative distribution function (ECDF) of the frames with a photon arriving
in the analysis pixels of the CCD. This ECDF is then compared with the
corresponding ECDF of the exposure given by the sequence of frames actually
read. Finally, we compare the two ECDFs using a Kolmogorov-Smirnov test,
resulting in a probability estimate $Q$ that the two ECDFs are derived from
the same parent distribution. A low value of $Q$ would indicate possible
variability. The results of the test for the five pointings were very similar.
The smallest value, $Q_{\mathrm{min}}=6.7\times 10^{-7}$, was obtained in
observation $13151$. Note that selecting this point represents a tuning bias,
as the noise in neighboring points is highly correlated due to the
oversampling. The probability of finding at least one pixel with
$Q_{\mathrm{min}}$ considering that there are $120\times 120/(\pi\times
1^{2})\times 5$ statistically independent trials is 0.015, which we regard as
a lower limit due to the tuning bias. A 0.015 probability is tantalizing but
not compellingly significant: Hence, we do not claim detection of short-time-
scale variability. The fact that the location of the point with minimum Q is
very close to the pulsar, a region in which pileup plays a strong role in
blotting out the image, bolsters our somewhat conservative conclusion.
## 3 Near-Infrared Image of the Inner Knot
The extreme saturation of the pulsar in the X-ray images means that we cannot
easily study the central 2′′ in X-rays. However, this region does contain a
nebular structure of particular interest: the “inner knot” whose peak is
$0.65^{\prime\prime}$ southeast of the pulsar at position angle $118^{\circ}$
East from North (Hester 2008). This structure, an oval shape extending
$\approx 0.75^{\prime\prime}$, is well measured in HST and ground-based near-
IR images. Given its relatively red spectrum (energy spectral index
$\alpha_{\nu}=-1.3\pm 0.1$ versus $\alpha_{\nu}=0.27\pm 0.03$ for the pulsar;
Sandberg and Sollerman 2009), it is one of the near-IR brightest structures in
the Nebula. Sandberg and Sollerman (2009) note that the knot varies by a
factor of 2; we confirm typical variability of $20-30\%$ in archival HST
images. Komissarov & Lyutikov (2011) have proposed that this structure
represents radiation from an oblique termination shock in the pulsar wind
nebula. In this picture, the Earth line-of-sight is tangent to the flow at the
inner knot position, and thus the intensity experiences substantial Doppler
boosting for synchrotron emission in the mildly relativistic post-shock flow.
Indeed, in relativistic MHD simulations they find that this bright spot is
highly variable and can dominate the $\gamma$-ray synchrotron emission.
Alternatively, the knot could be a time varying standing shock in the polar
jet flow itself, a flow known to be highly variable from HST imaging (Hester
1995, 2002, 2008).
It is thus of interest to check the status of the knot during the 2011-April
$\gamma$-ray flare. Unlike the sequence of multiwavelength observations
performed after the 2010 September flare, it was impossible to trigger an
allocated HST Target of Opportunity observation owing to solar constraints in
April. Happily we were able to obtain a Keck Near Infrared Camera (NIRC2)
$K^{\prime}$ exposure (Figure 7 left image) on MJD 55667.250, almost precisely
at the peak of the $\gamma$-ray flux and 2.5 h before the ACIS image ObsID
13152 (Figure 7). Unhappily, the observations occurred during twilight and
only one $20\times 4$-s integration without dithering was obtained. Under
these conditions the adaptic-optics (AO) loop did not close, leaving an
undithered image with native 0.46′′ full width at half maximum (FWHM) seeing.
This frame was dark subtracted and an approximate background was removed using
an immediately subsequent image. Despite the modest image quality, the inner
knot was well detected. After subtracting the pulsar with a scaled image of
the comparably bright companion star $4^{\prime\prime}$ northeast, we measured
the knot flux and position. We find a magnitude $K^{\prime}=15.60\pm 0.03$ and
an offset $0.64\pm 0.04^{\prime\prime}$ from the pulsar. For comparison we
measured a high-quality NIRC2 $K^{\prime}$ image (Figure 7 right image)
obtained 2005 Nov 10. Here the knot is $K^{\prime}=15.94\pm 0.02$ at offset
$0.58\pm 0.02^{\prime\prime}$. We also note that Sandberg and Sollerman (2009)
measured $K_{s}=15.80\pm 0.03$ on 2003 Oct 18. We conclude that the knot was
in a relatively bright state during the flare ($\approx 35\%$ brighter than in
2005), but well within the normal range of flux (and position) variation.
Thus, there is no dramatic change in the inner knot in the near-IR band. We
use the amplitude of the measured variation as an upper limit to any variation
in the inner knot associated with the $\gamma$-ray flare (Figure 7).
## 4 Radio Observations
On 2011 April 14, we triggered a prompt radio follow-up program with the VLA.
The VLA observations occurred in 8 epochs starting April 15 and ending July
10. These observations detected the pulsar at 2 epochs, but found no other
point sources in the field. Observations were predominantly in the range $4-8$
GHz, with additional observations at 1.4 GHz (not reported; see below) and at
22 GHz (for some later epochs). Unless otherwise noted, all observations used
two sub-bands, each with a 128-MHz bandwidth. In each run, observations of the
target were bracketed with scans of a phase calibrator (J0559+2353, except
where noted) and a flux calibrator (3C 147). The fields of view are limited to
the primary beam response of the antennas, with full width at half power of
9′/$\nu_{\rm 5}$ with $\nu_{\rm 5}\equiv\nu$/(5 GHz). Table 3 provides a
summary of the observational parameters and results.
Our initial observations were obtained through a Fermi guest-investigator
cycle-3 program (S3184) approved for four 1-hour runs in the L-band and C-band
($\approx\,1.4$ and $5$ GHz, respectively). The VLA was in its B-array
configuration during these observations, resulting in images with angular
resolution $\approx\,$1″/$\nu_{5}$. We found that the L-band data for the Crab
were highly confused due to the brightness and complexity of the steep-
spectrum nebular emission in the first (April 15) and second (April 19)
epochs. Consequently, we modified our strategy for subsequent observations.
After the first epoch, we split the C-band observations into two widely spaced
side-bands centered at 4.2 and 7.8 GHz, aiming better to constrain the
spectrum of any detected source. We also began scheduling observations only at
frequencies greater than 4 GHz after the second epoch. In these B-array data,
our point-source limits at the lower frequency are $\approx 3-4\times$ larger
than at the higher frequency, again due to the Crab Nebula’s steep-spectrum
radio emission.
After non-detection of any significant radio point-source emission down to
$\approx\,$1-7 mJy (3-sigma) sensitivities in the initial three VLA
observations 1–7 days after the $\gamma$-ray peak (Hays et al. 2011), we
purposely delayed the fourth observation until 9 days after the previous
observation to probe longer time scales. Only in this last observation (April
30) did we obtain a significant point-source detection, which was coincident
with the Crab pulsar position, but only in the upper side-band centered at 7.8
GHz. The detection is a factor of 5 greater than the 3-sigma limit from 9 days
prior. The source was not detected at the lower frequency side-band (4.2 GHz)
with a limit indicating a source with a flat radio spectrum.
Following the point-source detection on April 30, we became aware of VLA TEST
observations of the Crab obtained on April 22 (program TDEM0007, PI: D.
Frail). These data were obtained with wide bandwidth (16 $\times$ 128 MHz wide
sidebands), so were more sensitive than those from our observing runs. The
flux densities were scaled to 3C 147; J0534+1927 was utilized for phase
calibration. The flat-spectrum radio source coincident with the Crab pulsar
detected in our Apr-30 observation was confirmed in the Apr-22 data in two
bands, but with a much lower ($10\times$) flux. Also, the source spectrum was
rather steep, with an energy spectral index $\alpha=2.21\pm 0.34$
($S_{\nu}\propto\nu^{-\alpha}$) between 5 GHz and 8.6 GHz.
Following the radio detections of the pulsar, we requested further VLA
monitoring of the Crab through Director’s Discretionary Time (program 11A-268
= AC1052). In addition to the C-band observations, we obtained exposures in
the K band (centered at 22.396 and 22.254 GHz) aiming to constrain further the
spectrum of any detected radio source. Through this program, we obtained
2-hour runs on May 12/13 (while the VLA was in its hybrid BnA array) and on
July 10/11 (in A array), and an additional 1-hour run on May 28, using one of
the early (April 19, from program S3184) frequency setups. An angular
resolution of $\approx$(0.3″/$\nu_{\rm 5}$) is typically achieved in A-array
VLA observations. With the higher resolution, we obtained systematically
4$\times$ lower flux limits than in the lower resolution B-array data,
presumably due to lesser contribution from the extended nebular emission. In
none of these later epoch follow-up observations, did we detect a point
source, to typical limits of $1-2$ mJy at each of the three frequencies (see
Table 3).
### 4.1 Discussion of the Radio Data
Previously, it was argued that the $\gamma$-ray flaring possibly originates in
a knot 5.7″ east of the pulsar (Tavani et al. 2011). Indeed, this knot is the
site of the most significant X-ray variability we observe (§ 2.2) during the
2011-April flaring episode. Variable radio emission was detected around the
time of the previous Crab $\gamma$-ray flaring episode (Lobanov, Horns, &
Muxlow 2011) with fainter flux densities than achieved in our VLA
observations. However, we found no significant radio point-source counterpart
to this knot in any of our 8 epoch VLA observations following the 2011-April
$\gamma$-ray flare. Rather, we detected a variable continuum radio source with
the VLA, coincident within $0.2\arcsec$ of the Crab pulsar position in 2 of 8
epochs. However, the flux level and cadence of the radio detections is
consistent with previous observations by Moffett & Hankins (1996), who
detected the pulsar $20-40\%$ of the times they observed. Moreover, the dates
of the radio point-source detections coincident with the pulsar do not
coincide with any feature in the $\gamma$-ray lightcurve (Figure 7), having
occurred 8-16 days after the brightest $\gamma$-ray peak. Consequently, our
VLA follow-up observations provide no conclusive evidence for the site of the
$\gamma$-ray flares.
## 5 Discussion
Here we discuss possible explanations for the absence in non-$\gamma$-ray
bands of variability that is obviously correlated with the $\gamma$-ray flare.
In addition we present a conceptual model for the production of the
$\gamma$-ray flares.
### 5.1 $\gamma$-ray Emission
As was recognized immediately, the SED of $\gamma$-ray flares peak near a
characteristic energy about 5 times the energy $\alpha^{-1}m_{e}c^{2}\approx
70$ ${\rm Me\\!V}$, which is identified with radiation-reaction-limited
synchrotron (magneto-bremsstrahlung) emission (e.g., Landau & Lifshitz 1959).
Subjected to comparable parallel and perpendicular electromagnetic
acceleration, an electron radiates at this energy, independent of the strength
of the acceleration. An electron emitting synchrotron radiation in a
magnetostatic field with peak emission at several $100$ ${\rm Me\\!V}$ would
cool in turning through $\approx 0.2$ radian and would thus require a parallel
electric field $E\approx 5cB$ to compensate the radiative loss.
The Crab pulsar releases energy in an essentially electromagnetic form.
Poynting flux flows radially outward from the the pulsar through the light
cylinder at $cP/(2\pi)\approx 1500$ km and into an outflowing wind, where at
least some of the electromagnetic-energy flux may transform into a plasma-
energy flux. How, where, and to what extent this happens has long been a
matter of debate (e.g., Arons 2010; Kirk et al. 2009). Furthermore, the
electromagnetic component has a DC toroidal part with an associated
quadrupolar current distribution, and an AC, “striped” part containing current
sheets separated by $\frac{1}{2}cP\approx 5000$ km. The transformation from
electromagnetic to plasma energy might be non-dissipative—through the action
of a Lorentz force (e.g., Bogovalov 1997; Bogovalov 2001)—or
dissipative—through particle heating and acceleration (e.g., Coroniti 1990;
Lyubarsky & Kirk 2001; Sironi & Spitkovsky 2011). However, it must occur
somewhere as magnetic flux would otherwise accumulate in the nebula,
ultimately reacting back on the pulsar. Some of this transformation from
electromagnetic to plasma energy may occur at a shock (Pétri & Lyubarsky 2007;
Sironi & Spitkovsky 2011) with radius $\approx 10^{17}$ cm, where the wind
momentum flux balances the ambient nebular pressure (Rees & Gunn 1974; Kennel
& Coroniti 1984). It has also been proposed that the toroidal field loops
contract to form an axial pinch (identified with the X-ray jet) and reconnect
at an equatorial current sheet (the torus) (Komissarov & Lyubarsky 2003; Del
Zanna, Amato, & Bucciantini 2004; Camus et al. 2009).
In many respects the pulsar is a current generator. The supersonic wind
contains outflowing fluxes of electrons and positrons. (Any ions that are
present behave like positrons of similar rigidity but do not radiate.) The
difference in their fluxes determine the current density. This current may
concentrate into sheets and filaments, where strong dissipation can occur—as
happens in heliospheric and laboratory plasmas (Gosling et al. 2005; Sui &
Holman 2003; Sergeev et al. 1993). In particular, the inner wind, the shock,
the jet, and the torus are all natural sites of rapid dissipation and
$\gamma$-ray emission. If we consider this dissipation more generally under
electromagnetic conditions, a current $I$ may be associated with a potential
difference $V\approx IZ_{0}$ where $Z_{0}=\mu_{0}c=377\,\Omega$ is the
impedance of free space and we drop model-dependent constants of order unity.
(This result can be anticipated on the basis of dimensional analysis or
exhibited in particular simple cases.) The maximum energy to which an electron
or positron can be accelerated is $\gamma_{{\rm max}}m_{e}c^{2}\approx$ ${\rm
e\\!V}$ $\approx eIZ_{0}$ and the expected power is then $L\approx IV\approx
I^{2}Z_{0}$. On this basis, a spectrum of currents extending up to $\approx
30$ TA should suffice to account for the $\gamma$-ray variations.
However, currents do not automatically dissipate as just described. Large
electric fields are normally discharged in a few plasma periods. The best way
to create them here is transiently over a few Larmor periods and radii. This,
in turn, requires local charge separation of the plasma in the emission site.
To be more precise, the density $n=n_{-}+n_{+}$ of electrons plus
positrons/ions will combine to create a local current density $j\approx nec$.
For example, in the case of a pinch, the gradient, polarization and curvature
drifts automatically produce the axial current. However, a supporting local
electric field of strength $E\approx cB$ also requires that
$|n_{-}-n_{+}|\approx n$.
Put another way, the charge and current are mostly in an emission site that is
a few Larmor radii in size and survives for a few Larmor periods of the
$\gamma$-ray emitting particles. When this happens, the “Ohmic” dissipation is
radiative, not collisional as is normally the case. For this to occur, the
particles must be sufficiently energetic to radiate efficiently. This requires
that most of the particles are concentrated in an emission site that is as
small as $\approx\gamma_{{\rm
max}}^{3}r_{e}\approx(eIZ_{0}/m_{e}c^{2})^{3}r_{e}\approx 10^{16}$ cm. The key
point is that there should be extensive and sustained radiation-reaction-
limited emission at the peak $\gamma$-ray-flare energy, even though the
particle energy and magnetic field might be changing. Note that when this
condition is unsatisfied, efficient particle acceleration to lower energy
should still result: Most of the magnetic dissipation and particle
acceleration in the nebula might occur in this fashion. Detailed modeling is
necessary to determine whether or not such a scheme can reproduce the
powerful, narrow-band $\gamma$-ray variation that is observed (Uzdensky et al.
2011; Cerutti et al. 2012; Cerutti, Uzdensky, & Begelman 2012; Lyutikov,
Balsara, & Matthews 2012; Bykov et al. 2012; Sturrock & Aschwanden 2012;
Blandford & Yuan 2012) and to see if unstable magnetized plasmas, carrying
large currents, evolve to satisfy these conditions.
### 5.2 Associated Emission
Whether we interpret the $\gamma$-rays as coming from radiation-reaction-
limited synchrotron emission or simply extrapolate the observed $\gamma$-ray
spectra to lower energy, it should not be surprising that direct, associated
emission has not yet been observed in the X-ray, optical, or radio bands: In
these bands the contrast with the steady emission is too small to be easily
noticed. However, the indirect effects could be larger and detectable. For
example, the large 2011-April flare produced a radiant energy of $6\times
10^{40}$ ergs if isotropic, equivalent to the energy contained within a region
of size $\approx 2\times 10^{16}$ cm subtending an angle $\approx 0.3$ arcsec.
It seems unlikely that the dynamical aftermath of a major flare would not
alter the ambient emission—either through compression or rarefaction that
would cause the magnetic field strength and the electron distribution function
to change significantly. The associated surface brightness change should be
several percent, assuming a total emission region of size $\approx 0.3$
arcsec, consistent with our upper limits. Even if future observations fail to
exhibit associated emission, they may still rule out specific detailed
mechanisms in local sites.
Understanding the emission mechanism could have a significance beyond pulsar
wind nebulae. In particular, it could provide a clue to the surprisingly
rapidly variable emission seen in relativistic jets in the radio, optical,
X-ray, and ${\rm Te\\!V}$ bands. If so, the Crab Nebula would once again be
the source of fresh and important astrophysical insight.
## 6 Summary
Using the Chandra, Keck, and VLA Observatories, we acquired X-ray, near-IR and
radio images of the Crab Nebula, contemporaneous with the 2011-April
$\gamma$-ray flare. We searched for variability in the X-ray data over two
time-scale ranges: First we tested for pointing-to-pointing variations amongst
the 5 pointings, each with an effective exposure time $\approx 1200$ s and a
minimum separation of 0.6 days. Second we tested for variations within each of
the 5 observations. In neither case did we detect statistically significant
X-ray variations; thus we can set only upper limits to any X-ray variations
associated with the $\gamma$-ray flare. As the Chandra ACIS images suffer
severe pile-up near the Crab pulsar, our search for variability in the X-ray
images was not sensitive to variations within the central $\approx 1.5\arcsec$
or so.
Comparing the upper limits to X-ray variations with the Fermi-LAT-measured
$\gamma$-ray variations, we set upper limits at $99$%-confidence to the
effective X-ray–$\gamma$-ray photon power-law index $\Gamma_{x\gamma}\leq
1.20$ to $\leq 1.27$, dependent upon assumptions about the X-ray index
$\Gamma_{x}$. As Fermi-LAT measures a $\gamma$-ray index
$\Gamma_{\gamma}=1.27\pm 0.12$ for the flaring component, it is statistically
possible that the flaring component’s spectrum extends as a simple power-law
from $\gamma$-rays to X-rays. Further, we note that our upper limit to
$\Gamma_{x\gamma}$ is consistent with transparent synchrotron emission, whose
photon index must be $>\frac{2}{3}$.
Comparison of two Keck near-IR observations found that the inner knot
($\approx 0.65\arcsec$ from the pulsar) was somewhat brighter than average
during the $\gamma$-ray flare, but well within the normal range of brightness
fluctuations typically observed. We used the measured ($\approx 35\%$) change
in the near-IR flux from this knot as an upper limit to near-IR variations
associated with the $\gamma$-ray flare. We also performed a number of VLA
observations searching for a point source appearing either at an unusual
location and/or contemporaneous with the $\gamma$-ray flare. Other than the
pulsar itself, no such source was detected.
Figure 7 shows the spectral energy distribution (SED) of the Crab Nebula over
the observed electromagnetic spectrum. The plot also shows the SED of the
2011-April $\gamma$-ray flare and the various limits determined here on
variable radio, near-infrared, and X-ray emission possibly associated with the
$\gamma$-ray flare.
Finally, we reviewed and discussed potential implications of $\gamma$-ray
flares and theoretical issues to be addressed. We concluded that, apart from
lower-energy emission directly associated with the $\gamma$-ray flare itself,
the dynamical aftermath of a major flare could alter the ambient emission —
e.g., through compression of the magnetic field. The associated surface
brightness change would likely be only several percent, assuming a total
emission region of size $\approx 0.3\arcsec$, consistent with our upper
limits.
Although no “smoking gun” has been identified, one should be encouraged that
we have identified a number of regions in the X-ray images that are possible
candidates. We have also established further Target of Opportunity
observations with Chandra and HST that will be triggered at the onset of the
next $\gamma$-ray flare. The X-ray observations will also probe the region
very close to the pulsar using the Chandra High-Resolution Camera (HRC).
## 7 Acknowledgments
The work of MCW, SLO, and AFT is supported by the Chandra Program. The Chandra
data was obtained in response to a pre-approved target of opportunity request
granted under Chandra Director’s Discretionary Time. The Fermi LAT
Collaboration acknowledges generous ongoing support from a number of agencies
and institutes that have supported both the development and the operation of
the LAT as well as scientific data analysis. These include the National
Aeronautics and Space Administration and the Department of Energy in the
United States, the Commissariat à l’Energie Atomique and the Centre National
de la Recherche Scientifique / Institut National de Physique Nucléaire et de
Physique des Particules in France, the Agenzia Spaziale Italiana and the
Istituto Nazionale di Fisica Nucleare in Italy, the Ministry of Education,
Culture, Sports, Science and Technology (MEXT), High Energy Accelerator
Research Organization (KEK) and Japan Aerospace Exploration Agency (JAXA) in
Japan, and the K. A. Wallenberg Foundation, the Swedish Research Council and
the Swedish National Space Board in Sweden. Additional support for science
analysis during the operations phase is gratefully acknowledged from the
Istituto Nazionale di Astrofisica in Italy and the Centre National d’Études
Spatiales in France. CCC, GBT, and JBL thank Tim Hankins for useful
discussions, and the NRAO scheduling committee for alerting us to the VLA TEST
data and for their prompt consideration of our Director’s Discretionary Time
observations. CCC, GBT, and JBL were supported in part by NASA through a Fermi
cycle-3 guest investigator grant. In addition, work by CCC at NRL is supported
in part by NASA DPR S-15633-Y. Our analyses utilized software tools provided
by the Chandra X-ray Center (CXC) in the application package CIAO and from the
High-Energy Astrophysics Science Archive Research Center (HEASARC, operated by
the NASA Goddard Space Flight Center, Greenbelt MD, and by the Smithsonian
Astrophysical Observatory, Cambridge MA).
Facilities: CXO, Fermi, VLA
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Table 1: Time-ordered list of Chandra observations and Fermi-LAT $\gamma$-ray
measurementsa.
ObsID | Date b | $\Gamma_{\gamma}^{\ c}$ | $N(>$ 100 ${\rm Me\\!V}$)${}^{\ d}$ | $N_{E}$(100 ${\rm Me\\!V}$)${}^{\ e}$
---|---|---|---|---
13150 | 4851.039 | $2.42\pm 0.08$ | $(1.27\pm 0.08)\times 10^{-5}$ | ($1.80\pm 0.15)\times 10^{-10}$
13151 | 4851.667 | $2.25\pm 0.15$ | $(8.05\pm 0.15)\times 10^{-6}$ | $(1.01\pm 0.12)\times 10^{-10}$
13152 | 4853.423 | $2.20\pm 0.06$ | $(1.54\pm 0.06)\times 10^{-5}$ | $(1.84\pm 0.12)\times 10^{-10}$
13153 | 4859.032 | $2.27\pm 0.21$ | $(4.68\pm 0.21)\times 10^{-6}$ | $(6.0\pm 1.0)\times 10^{-11}$
13154 | 4865.335 | $2.76\pm 0.40$ | $(4.50\pm 0.40)\times 10^{-6}$ | $(7.9\pm 1.9)\times 10^{-11}$
a Analyzed following all the procedures in Buehler et al. (2012)
b Days after MJD 50814 to the middle of the observation, which is 10-ks long.
c $\gamma$-ray powerlaw number index. The error is the maximum of the two-
sided uncertainty
d Photon integrated flux [ph/(cm2 s)] above 100 ${\rm Me\\!V}$. The error is
the maximum of the two-sided uncertainty
e Photon spectral flux [ph/(cm2 s ${\rm ke\\!V}$)] at 100 ${\rm Me\\!V}$
Table 2: X-ray results at 1 ${\rm ke\\!V}$ for the analysis pixel with the most significant variation. Quantity | Unit | Sample stdev ($s$) | 99%-upper limit
---|---|---|---
Rate | ct/s | 0.0480 | 0.0554
$\Gamma_{x}$ | | $\frac{2}{3}$ | 1 | 2 | $\frac{2}{3}$ | 1 | 2
$N_{E}$ | $10^{-4}$ ph/(cm2 s ${\rm ke\\!V}$) | 0.48 | 0.61 | 1.18 | 0.55 | 0.70 | 1.36
$F_{E}$ | $10^{-13}$ erg/(cm2 s ${\rm ke\\!V}$) | 0.76 | 0.97 | 1.89 | 0.88 | 1.12 | 2.18
$EL_{E}$ | $10^{32}$ erg/s | 0.37 | 0.47 | 0.91 | 0.42 | 0.54 | 1.05
$\Gamma_{x\gamma}$ | | 1.18 | 1.20 | 1.26 | 1.20 | 1.22 | 1.27
Table 3: VLA point-source detections and limits in the Crab field
Epoch | Datea | Array | Frequency | Detection or Limitc
---|---|---|---|---
(in 2011) | | | (GHz) | (mJy)
Apr 15 | 4852.980 | B | 4.959 | $<$ 7.1
Apr 19 | 4856.006 | B | 4.195 | $<$ 5.7
Apr 19 | 4856.006 | B | 7.795 | $<$ 1.7
Apr 21 | 4858.011 | B | 4.195 | $<$ 4.3
Apr 21 | 4858.011 | B | 7.795 | $<$ 1.0
Apr 22 | 4859.919 | B | 4.910 | 1.78 $\pm$ 0.086
Apr 22 | 4859.919 | B | 8.566 | 0.52 $\pm$ 0.094
Apr 30 | 4867.901 | B | 4.195 | $<$ 7.9
Apr 30 | 4867.901 | B | 7.762b | 5.55 $\pm$ 0.86
Apr 30 | 4867.901 | B | 7.827b | 4.59 $\pm$ 0.96
May 12 | 4879.985 | BnA | 4.195 | $<$2.1
May 12 | 4879.985 | BnA | 7.795 | $<$0.8
May 12 | 4879.985 | BnA | 22.46 | $<$0.4
May 28 | 4895.925 | BnA | 4.195 | $<$2.4
May 28 | 4895.925 | BnA | 7.795 | $<$1.2
Jul 10 | 4938.791 | A | 4.195 | $<$0.7
Jul 10 | 4938.791 | A | 7.795 | $<$1.1
Jul 10 | 4938.791 | A | 22.46 | $<$1.7
Notes –
a Days after MJD 50814 to the middle of the on-target exposures.
b For the 7.8 GHz detection on Apr 30, the data were further split into two
sub-bands.
c Limits list 3-sigma uncertainties. Detections list 1-sigma uncertainties;
corresponding point-source limits during these 2 epochs should take 3 times
the 1-sigma values.
Fermi-LAT photon flux ($10^{-7}$ ph/(cm2 s)) above 100 ${\rm Me\\!V}$ during
the 2011-April flare as a function of time. Displayed data extend beyond the
time span shown in Buehler et al. (2012) but follow the same data processing
as described there: Data are adaptively binned with a 20-minute average bin
duration. The full-range vertical lines denote times of the 5 ($\approx$
1200-s) Chandra observations (black), of the Keck observation (blue, fourth
from left), and of the first 5 VLA observations (green).
Summed image for the 5 Chandra ACIS observations occurring near the 2011-April
$\gamma$-ray flare, at the native CCD resolution. The color bar gives summed
counts per ACIS pixel over a total effective exposure of about 6 ks. North is
up and the pulsar is at (0,0) in the displayed ACIS-pixel coordinates. The
nearly horizontal read-out streak through the pulsar’s location is the trailed
(out-of-time) image, resulting from exposure of each CCD pixel as the image is
read out at 40 $\mu$s per row. As the 5 observations occurred at slightly
different roll angles, the read-out streak is slightly blurred azimuthally.
The $\mathsf{X}$ symbols mark locations of the 3 statistically most
significant variations ($S_{i}>6$, §2.2), the most significant lying to the
east of the pulsar.
Left: Image of the statistical error $\sigma_{i}$ (ct/s) in the counting rate
per analysis pixel (about 1 square arcsec) for the 5 Chandra ACIS
observations. North is up and the pulsar is at (30,30) in the displayed
analysis-pixel coordinates. Right: Histogram of number of occurrences of each
value in the image to the left.
Left: Image of the significance measure
$S_{i}\equiv(\chi_{i}^{2}-\nu_{i})/\sqrt{2\nu_{i}}$ of sample counting-rate
variations amongst the 5 observations. North is up and the pulsar is at
(30,30) in the displayed analysis-pixel coordinates. Right: Histogram of
number of occurrences of each value in the image to the left.
Image of the energy spectral flux $F_{E}(E_{x})$, in erg/(cm2 s ${\rm
ke\\!V}$) at $E_{x}=1$ ${\rm ke\\!V}$, based upon the sample standard
deviation ($s_{i}$) of the counting rate and assuming $\Gamma_{x}=1$. Note
that the indicative energy flux $F(E)\equiv EF_{E}(E)$ in erg/(cm2 s) happens
to have the same numerical value as $F_{E}(E)$ at $E=1$ ${\rm ke\\!V}$. North
is up and the pulsar is at (30,30) in the displayed analysis-pixel
coordinates.
Left: Image of upper limits to the effective photon index $\Gamma_{x\gamma}$
between 1 ${\rm ke\\!V}$ and 100 ${\rm Me\\!V}$, based upon the sample
standard deviation ($s_{i}$) of the counting rate and assuming $\Gamma_{x}=1$.
North is up and the pulsar is at (30,30) in the displayed analysis-pixel
coordinates. Right: Histogram of number of occurrences of each value in the
image to the left.
Keck $K^{\prime}$ observations of the Crab, after subtraction of a field-star
image from the pulsar position (marked with $\mathsf{x}$). Left: (MJD-50814) =
4853.250 without adaptive optics (AO). Right: (MJD-50814) = 2870, with laser
guide star AO. Residuals from imperfect subtraction of the point spread
function are visible at the pulsar position; the ‘inner knot’ is the extended
structure to the southeast.
Spectral energy distribution (SED) of archival data (purple, orange, cyan,
light green, turquoise, light blue) compiled by Meyer, Horns, & Zechlin
(2010). Power is scaled from flux assuming isotropic emission at 2 kpc. The
Fermi-LAT data for the 2011-April flare component appear in dark blue (Buehler
et al. 2012). The solid black and red curves are fits to the flare spectrum
with a power-law extrapolation to lower energies of photon index
$\Gamma=1.27\pm 0.12$ (spectrum 7 in Buehler et al. 2012). The three downward
blue arrows at $\log_{10}\nu=17.4$ mark 99%-confidence upper limits to a
variable X-ray component, in increasing $EL_{E}=\nu L_{\nu}$ for
$\Gamma_{x}=\frac{2}{3}$, 1, and 2 respectively. (NB: Values for $EL_{E}$ from
Table 2 are multiplied by $2\sqrt{2}$ to scale from a standard deviation to a
peak-to-valley, for comparison with the plotted SED of the $\gamma$-ray
flare.) The red downward arrow at $\log_{10}\nu=14.1$ indicates an upper limit
to infrared variability of the inner knot, determined from the difference
between the two Keck images. Finally, the black downward arrow at
$\log_{10}\nu=9.7$ gives an upper limit to 5-GHz radio variability, based upon
the April-15 VLA measurement.
|
arxiv-papers
| 2012-11-16T19:22:13 |
2024-09-04T02:49:38.118873
|
{
"license": "Public Domain",
"authors": "Martin C. Weisskopf, Allyn F. Tennant, Roger Blandford, Rolf Buehler,\n Patrizia Caraveo, C. C. Teddy Cheung, Enrico Costa, Andrea de Luca, Carlo\n Ferrigno, Hai Fu, Stefan Funk, Moritz Habermehl, Dieter Horns, Justin D.\n Linford, Andrei Lobanov, Claire Max, Roberto Mignani, Stephen L. O'Dell,\n Roger W. Romani, Edoardo Striani, Marco Tavani, Gregory B. Taylor, Yasunobu\n Uchiyama, Yajie Yuan",
"submitter": "Martin C. Weisskopf",
"url": "https://arxiv.org/abs/1211.3997"
}
|
1211.4000
|
11institutetext: Old Dominion University, Department of Computer Science
Norfolk, VA 23529
# The Performance of Betting Lines for Predicting the Outcome of NFL Games
Greg Szalkowski Michael L. Nelson
###### Abstract
We investigated the performance of the collective intelligence of NFL fans
predicting the outcome of games as realized through the Vegas betting lines.
Using data from 2560 games (all post-expansion, regular- and post-season games
from 2002-2011), we investigated the opening and closing lines, and the margin
of victory. We found that the line difference (the difference between the
opening and closing line) could be used to retroactively predict divisional
winners with no less accuracy than 75% accuracy (i.e., “straight up”
predictions). We also found that although home teams only beat the spread 47%
of the time, a strategy of betting the home team underdogs (from 2002-2011)
would have produced a cumulative winning strategy of 53.5%, above the
threshold of 52.38% needed to break even.
## 1 Introduction
Week one of the 2007 National Football League (NFL) season had the New England
Patriots on the road against the New York Jets. The sportscasters had been
talking about how the Jets were looking good this year and were ready for
payback from the previous year when New England beat them in the playoffs.
While New England had performed well in the preseason games, they had two
starters out with injuries and Randy Moss was questionable for the week one
game. The future was looking grim for the Patriots and Vegas was favoring the
Jets at home by 6 points. When betting opened, many people placed bets on New
England to win even though the experts predicted that the Jets would triumph.
The lopsided nature of the betting forced the sportsbooks to move the line
repeatedly in order to keep the volume of bets even on both sides of the game.
Eventually the line moved a total of 13 points to New England being a 7 point
favorite by game day. New England went on to win this game 38 to 14, easily
covering the spread. This is one example where the collective intelligence of
the NFL fans was confident that New England would win even when the “experts”
thought otherwise.
Collective intelligence is a way of synthesizing information from a group of
people that no one person would have known on their own. Collating information
from many individuals and making new conclusions based on the assimilated
knowledge is the core of collective intelligence. Harnessing this collective
intelligence using network applications is one of the fundamental ideas
underlying Web 2.0 [10]. The archetypical example of collective intelligence
is the “guess how many jelly beans in a jar” prediction. Ask a group of people
to estimate the number of beans in a jar, and many of the answers will be far
from correct. But their average answer will be quite close to the real answer
and maybe even closer than the prediction of most of the people who guessed.
“Under the right circumstances, groups are remarkably intelligent, and are
often smarter than the smartest people in them”. Surowiecki [15] outlines four
conditions that must hold for collective intelligence:
> There are four key qualities that make a crowd smart. It needs to be
> diverse, so that people are bringing different pieces of information to the
> table. It needs to be decentralized, so that no one at the top is dictating
> the crowd’s answer. It needs a way of summarizing people’s opinions into one
> collective verdict. And the people in the crowd need to be independent, so
> that they pay attention mostly to their own information, and not worrying
> about what everyone around them thinks.
The NFL betting line implements each of the four qualities that Surowiecki
mentions as requirements for a crowd to be smart. NFL fans come from all walks
of life and while focused in the United States, they are a quite diverse group
of people. The line is initially set by sportsbooks but it then moves from its
initial point due to forces applied by betting volume and it neatly summarizes
the views of the betting pool. The betting line itself could be considered a
form of feedback so each of the individual bettors is not completely
independent but each person is acting in their own self interest. This
research is focused on leveraging the collective intelligence of the football
community realized in the NFL point spreads and how they change as a result of
the bets placed on the outcome of NFL games placed by many people [15].
Our question is: can the collective intelligence of the NFL fans consistently
be leveraged to predict the outcome of future games?
## 2 The Point Spread
Frequently teams have a talent discrepancy that makes predicting the outcome
easy; “straight up” is the term used for simply predicting the outcome of the
game and a “point spread” is used to handicap the prediction process. The
point spread by itself is not intended to predict a winner but to ensure that
the sportsbooks make money. The sportsbooks make money by charging a
commission, sometimes called the vigorish, on each bet placed. The
sportsbook’s intentions are to separate the betting population in half in
order to minimize risk and maximize profit. In order to split the population
of bettors in half, the point spread on a particular game may have to be
adjusted due to betting pressure on one side or the other. This way, the
bookmaker can guarantee 5% profit regardless of the outcome of the game (10%
from the losing half of the betting population).
Every week of the NFL season the initial point spreads are established at the
beginning of the week by a small group of sportsbooks and consultants. Las
Vegas Sports Consultants [3] bills itself as a provider of sports betting
lines to most of the Las Vegas casinos. The initial point spreads are
initially opened for bidding to a group of high level knowledgeable bettors
and the line is adjusted based on the bets placed by those bettors [7]. This
initial line is sometimes referred to as the “virgin” line and it moves to
equalize the early betting so that the amount of money bet on either side of
each game is roughly equal. After this initial adjustment the point spread is
released to the public and opened for betting as the opening line.
A bettor can place a bet on either team. The bettor wins if the bet is on the
favorite and the favorite wins by more than the point spread, or if the bet is
on the predicted loser and the predicted loser either wins or loses by fewer
points than the point spread. If the predicted winner wins by a margin equal
to the point spread, a tie is declared, and the bettor neither wins or loses.
This is usually referred to as a “push”. In all other cases, the bettor loses.
As bets are placed the point spread values will move to continue to keep an
equal volume of betting on both sides of each game.
As an example of the betting process, in the introduction New England was a 6
point underdog to the Jets in the opening line. The point spread was New
England +6 or New York -6. In order to place a bet, a potential bettor must be
willing to risk $11 to win $10. When betting opened, many people started
betting on New England to win, so the sportsbooks were forced to move the line
toward New England in order to entice people to bet on New York so that there
would be an even betting volume on both sides. By taking an equal volume of
bets on each side, the sportsbook is in effect charging the players betting
the losing side an extra dollar. When the game is over and the closing line
was New England favored by 7 points, a person who bet $10 on New England would
receive $21 (the $11 risked plus the $10 won) while the player who bet on New
York loses $11. The sportsbook then makes $1 on this transaction, the $11 lost
by the New York bettor less the $10 won by the New England bettor.
If a $100 bet is placed ($110 with the ante) and wins, the bettor is paid $100
and keeps the ante; if the bettor loses she pays $110. In order to break even
the bettors must win at least 52.38% of their bets. This can be found solving
equation (2). Let the win ratio (WR) be the proportion of winning games,
therefore 1-WR is the number of losing games. In order to determine the
proportion of winning bets, WR, necessary to break even on wagering, set the
expected winnings equal to the expected losses and solve for WR. Solving
yields WR=0.5238 or 52.38% to break even.
$\displaystyle 100*WR=110(1-WR)$ (1)
Every sportsbook that takes bets has the same incentive to maintain an equal
volume of bets on each side of a given game. The line offered by each
sportsbook is unique to that sportsbook and they can change independently of
other sportsbooks. There is an entire industry related to “line shopping”,
that seeks to exploit differences in lines offered by different sportsbooks.
If the line for a certain game is trending down, the “line shoppers” attempt
to place bets at sportsbooks that move a little slower and have not adjusted
down as quickly as other. Overall the entire “sportsbook system” has a form of
collective intelligence feedback loop that keeps most of the lines offered by
all of the sportsbooks very close if not equal to each other. In comparing
against the spread records, there may be some differences when using line
values from different sources. The majority of line values used in this study
were obtained from The GoldSheet [16].
A significant factor in determining the line is the home field advantage. Home
teams typically have an advantage because of familiarity, travel requirements,
and factors related to the crowd [13]. The magnitude of this advantage is
typically realized in the betting market by a three point benefit for the home
team. At first glance the betting line appears to be a good indicator of
performance, as home teams have won 58% of the NFL games from 1981 to 1996
[18] and 57% from 2002 to 2011. However winning against the spread was found
to be much less decisive with the home team beating the spread in only 48.9%
of the games from 2002 to 2011 and 49.9% of the games from 1981 to 1996 [18].
Home team performance against the spread for 2002 to 2011 is detailed in Table
1. Historically it appears that the home game factors may have been
overcompensated as the performance is below 50%. One data point that sticks
out is the performance of the home underdog. The home-underdog effect has been
the subject of some research and has been attributed to late season biases [2]
and weather conditions [1]. Betting on the home team underdog from 1973 to
1979 would have resulted in a 58.1% win ratio [5], a 52.5% win ratio from 1981
to 1996 [18] and 53.5% from 2002 to 2011. Results from a number of studies
indicate that the home-underdog bias has been diminishing over the years [5,
6]. This diminishing bias is quite evident when plotted over time as seen in
Figure 1.
Figure 1: Home Underdogs That Covered the Point Spread Table 1: Home Team Records Against the Spread | Favorites | Underdogs | Pick-ems | All Home Games
---|---|---|---|---
| Win | Lose | WR | Win | Lose | WR | Win | Lose | WR | Win | Lose | WR
2002 | 80 | 93 | 0.462 | 51 | 36 | 0.586 | 1 | 1 | 0.5 | 132 | 130 | 0.504
2003 | 90 | 86 | 0.511 | 36 | 35 | 0.507 | 2 | 2 | 0.5 | 128 | 123 | 0.510
2004 | 77 | 94 | 0.450 | 40 | 40 | 0.500 | 1 | 4 | 0.2 | 118 | 138 | 0.461
2005 | 91 | 76 | 0.545 | 31 | 45 | 0.408 | 1 | 0 | 1 | 123 | 121 | 0.504
2006 | 74 | 101 | 0.423 | 45 | 32 | 0.584 | 3 | 1 | 0.75 | 122 | 134 | 0.477
2007 | 82 | 79 | 0.509 | 46 | 44 | 0.511 | 0 | 1 | 0 | 128 | 124 | 0.508
2008 | 81 | 96 | 0.458 | 35 | 41 | 0.461 | 2 | 0 | 1 | 118 | 137 | 0.463
2009 | 78 | 84 | 0.481 | 39 | 46 | 0.459 | 3 | 2 | 0.6 | 120 | 132 | 0.476
2010 | 80 | 86 | 0.482 | 41 | 38 | 0.519 | 2 | 2 | 0.5 | 123 | 126 | 0.494
2011 | 83 | 93 | 0.472 | 45 | 39 | 0.536 | 0 | 0 | 0 | 128 | 132 | 0.492
Total | 816 | 888 | 0.479 | 409 | 396 | 0.508 | 15 | 13 | 0.535 | 1240 | 1297 | 0.489
0.500 | | z= | -1.858 | | z= | 0.404 | | z= | 0.044 | | z= | -1.907
0.524 | | z= | -4.002 | | z= | -0.955 | | z= | -0.165 | | z= | -5.977
Over the years a number of researchers have investigated betting strategies in
relation to NFL football [19, 17]. Numerous market efficiency metrics have
found the betting market to be efficient [18, 11]. A common theme in much of
the research was searching for a bias in the betting line. When investigating
the home field advantage for bias it would be useful to compare the line to
two win rates, 0.5 for a straight up win and 0.5238 to cover the spread [18].
The data in Table 1 shows that the favorite home team lost more often than it
won from 2002 to 2011 with a z of -1.85 compared to 0.5, which is similar to
the -1.36 for 1981 to 1996 [18]. The against the spread comparison was worse
with a z of -4.002 compared to 0.5238 for 2002 to 2011 and -3.66 for 1981 to
1996.
## 3 Comparing the Line with Reality
The NFL box scores and line values for all NFL games from 2002 to 2011 were
collected from the Internet and inserted into a MySQL database. The box scores
contain over 30 statistics about each game from the final score, to statistics
about individual players. A starting year of 2002 was chosen as this was the
first year after the last NFL expansion and included all of the current 32
teams. Data collection resulted in 2560 regular season games in the database
at 256 games per year over 10 years.
A histogram of the values of the point spreads for regular season games from
2002 to 2011 appears in Figure 2. The most popular point spread values 3, -3,
and 7 are evident and quite logical as they are common values for NFL scores
[9]. Visually there appears to be a bias towards positive values. A positive
point spread value in this system indicates that the home team is favored,
reinforcing the concept that the home team is favored more often than the
visitor. Principal component analysis was used to dimensionally reduce the box
score data and investigate the importance of each of the box score statistics.
The betting line value had a high coefficient in almost every analysis and was
ranked better than the other box score statistics with a high orthogonal
variance.
Figure 2: Most Popular Closing Line Values (2002-2011); positive values
indicate the home team is favored.
To compare the point spread with actual game results, the Margin of Victory
(MOV) will be compared to the point spread for each game. The MOV is
traditionally calculated as shown in equation (2).
$MOV=WinnerScore-LoserScore$ (2)
The MOV and the line values can be seen in Figure 3. The absolute value of the
closing line is on the x-axis and the MOV on the y-axis. If MOV was accurate
then the data in the scatter plot should be clustered about the diagonal. The
actual results are much different and show that the line is not a good match
for the actual MOV. Indeed the main purpose of the line is not to predict the
actual MOV but to split the betting population in half. The games above the
diagonal are games in which the favorite won. There were 1194 games in which
the favorite beat the spread. The games below the diagonal but above the
x-axis are games in which the favorite won but did not cover the spread. Games
below the x-axis are games in which the favorite lost. There were 412 games
where the favorite did not cover and 853 losses for a total of 1265 games in
which the favorite lost against the spread for a loss rate of 51.5%. There
were 101 games which resulted in a push and are not included in the numbers on
the graph. The line value on its own is not a good indicator of the actual
MOV.
Figure 3: Closing Line vs. MOV (2002-2011); above the diagonal represents
beating the spread, between the diagonal and above the x-axis is winning
straight up but not beating the spread, below the x-axis is losing.
In order to overcome the deficiencies of the MOV we defined a similar metric
called the Line Difference (LD) shown in equation (3). The LD for a given game
is the magnitude of how far off the point spread was from the actual outcome
of the game. A positive value will indicate that the favorite was undervalued
or the underdog overvalued, and a negative value indicates that the favorite
was overvalued or the underdog undervalued.
$LD=(FavoriteScore-UnderdogScore)-|ClosingLine|$ (3)
Figure 4: Histogram of Line Difference (LD) values (2002-2011)
A histogram of the Line Difference values appears in Figure 4. Visually the
data appears close to a normal distribution with a mean of -0.009 and standard
deviation on 13.588. This is similar to a mean of 0.07 and standard deviation
13.86 obtained from NFL seasons 1981 to 1984 [14], 1980 to 1985 [4], and data
for NFL seasons 1992 to 2001 [12]. A comparison of the mean and standard
deviation of the LD values from 1992 to 2011 is shown in Figure 5. The mean
Line Difference form 1992 to 2011 is rather close with a value of -0.009 but
the standard deviation of 13.588 which is close to two touchdowns demonstrates
quite a bit of volatility in the line values. Additionally it appears that the
line values have not improved over the years as the Line Difference values
going back to at least 1980 demonstrate a rather amazing consistency.
Figure 5: Mean and Deviation of Line Difference (1992-2011)
A chi-squared goodness of fit test comparing the LD curve to a Gaussian
distribution with a mean of 0 and a standard deviation of 13.588 indicates
that the distribution of LD is not statistically different. Given that our
data with a mean of 0 and standard deviation of 13.588 is an adequate
approximation of a normal distribution, we can use the cumulative distribution
function to approximate the probability of winning a game. Equation (4) was
used with p = point spread for an individual game to determine the probability
that a p-point favorite would win the game [14]. Going back to our New England
example, New England was a 7 point favorite at game time. Using equation (4)
with p=7 results in a New England win probability of 69.6%. The resulting
normal approximations for the probability of victory of a sample of point
spreads is shown in Table 2.
$Pr(F>U|P=p)=\phi\left(\frac{p}{13.588}\right)$ (4)
Table 2: The Normal Approximation and the Empirical Probability of Winning Point Spread | $Pr(F>U|P)$ | Actual
---|---|---
1 | 0.529 | 0.509
3 | 0.587 | 0.581
5 | 0.644 | 0.597
7 | 0.697 | 0.689
The probability of winning a given game could be calculated using Equation (4)
and the probability of a team winning multiple games could be envisioned as a
sequence of events and calculated as the product of each of the individual
events. For example, New England is favored by 7 points in one game and 4
points in another. The probability of New England winning both games is
$\phi\left.(\frac{7}{13.588})\right.\cdot\phi\left.(\frac{4}{13.588})\right.=0.429$.
The probability of winning k games in a season can be found by adding the the
probabilities for all $\binom{16}{k}$ sequences of outcomes. The probabilities
for all games in each season from 2002 to 2011 were calculated and each season
was simulated 1000 times and the results averaged. The results can be found in
Tables 4 to 13. The estimates from the normal approximation are consistent
with the results obtained from the actual games. The predicted division
winners for each season were calculated and compared to the actual outcome.
The rules for division winners become detailed in the event of ties. When
there is a tie in the predicted division winner, the predicted outcome was
decided in our favor. The results of the predictions are in Table 3.
Table 3: Predicting the Division Winners Year | Correct
---|---
2002 | 7/8
2003 | 7/8
2004 | 6/8
2005 | 8/8
2006 | 6/8
2007 | 7/8
2008 | 7/8
2009 | 7/8
2010 | 6/8
2011 | 6/8
Because we used the point spreads for the entire regular season, this is
essentially a retrospective analysis and does not in isolation lend itself to
being a good predictor of an entire season. In the 1970s a study of over 1,800
horse races and the accompanying betting odds was conducted. The winning
frequency of every horse in every race was compared to the odds in the betting
market. With only a few exceptions the betting odds precisely predicted the
actual order in which the horses finished [8]. Both the NFL and the horse
races were retrospective but they reinforce the concept that the line (or
odds) would be a valid input into a prediction algorithm.
## 4 Line Movement
In football, sports bettors have a whole week to bet on games. This is a week
in which new information becomes available daily. Sports Illustrated puts a
team on their cover111The Sports Illustrated cover jinx is an urban legend
that states that individuals or teams who appear on the cover of the Sports
Illustrated magazine will subsequently be jinxed (experience bad luck).,
injury reports are released, weather reports become more accurate, and paid
handicappers release their picks to the public. All of these things have an
influence on the volume and direction of betting, which directly impacts the
direction and magnitude of line movement.
Line movement is essential to sports betting. It is a game that begins
immediately after the line makers create the virgin line. During early betting
the sportsbooks will adjust the spread on these early lines until they get an
equivalent volume of betting on both sides. Sportsbooks do not hope for, but
expect, this early action to help firm the line. If the volume of betting is
equal on both sides, the line is considered good and will not move. If the
line is biased to one side or the other, heavier betting will occur one one
side and continue until the line has reached a point that the collective
intelligence indicates is a “correct” line for that particular game. Once the
line is firmed up, it will not move much until game time, but it still can.
Sportsbooks are typically reluctant to move the line too far due to the
possibility of losing money on both sides of the bet. If the opening line has
a team favored by 7 points and heavy betting forces the sportsbook to adjust
the line to 9 points and the team wins by 8 points the sportsbook could be
forced to pay out on bets on both sides and lose money. The most infamous
example of this is the Stardust during the 1978 Superbowl [19]. For Superbowl
XIII the line varied between 3.5 and 4.5 with Pittsburgh favored. Pittsburgh
won the game by 4 points resulting in a push for a majority of the bets. The
sportsbooks do not make money off a push and they lost millions of dollars
that year. Frequently it is observed that a spread may only move from 2 to 2.5
or from 3 to 3.5 before there is an equal amount of bets coming in on both
sides. The half point moves help the sportsbooks to prevent having to pay out
to both sides and to eliminate the probability of a push. If the games results
in a push then the sportsbook needs to refund many of the bets and the
overhead costs typically outweigh any profit from that game.
Figure 6: Line Movement in Points (2002-2012)
For the purposes of this study, the line movement is defined as the difference
between opening line and the closing line. The line movement for each of the
regular season games from 2002 to 2011 was calculated and a histogram of the
values is shown in Figure 6. Out of the 2560 regular season games, over 2000
games had a line movement of 1 point or less, 1548 games had a line movement
of 0.5 points or less. Therefore only 20% of the games had a line movement
greater than one point. In testing the opening line versus the closing line it
was found that the opening line had a smaller mean square error than the
closing line but the difference between the lines was generally small and
statistically insignificant [4]. Subsequently all of the calculations for this
study were then completed with both the opening line and the closing line. The
results were compared and resulted in no statistically significant difference
between the results.
There is anecdotal evidence of using the temporal movement of the line for
prediction of the game. Similar to the opening line, if sportsbooks receive
more money on just one side of the spread during the week, they will adjust
the line. This is a much slower process as bets come slowly during the week,
and sportsbooks do not like to move the line once it has stabilized. Unless a
team has a major injury or venue change, the spread should not move at this
point 95% of the time.
The last period is which the line can move is on game day. Big bettors that
did not hammer the opening line often place their bets a few hours up to 15
minutes before game time. These bettors come with large amounts of money and
desire the most accurate information available before the game starts. They
utilize accurate weather reports and have access to the most current injury
reports as they have delayed placing bets until the last day. This last minute
action can be large enough to move the line right up until game time. While
the temporal aspect of line movement may be interesting we did not have the
historical data to include in our study. The low frequency of late week line
changes would tend to lead to small sample sizes that would not have much
statistical significance.
Figure 7: Percentage of Games with Line Movement $\geq 1$ By Week (2002-2011)
Figure 8: Percentage of Games with Line Movement $\geq 2$ By Week (2002-2011)
One aspect of temporal line movement that we could analyze is to compare the
amount of line movement by week to see if there is a difference. Because the
number of games per week is not consistent throughout the year, we plotted the
number of games that had a line movement greater than or equal to one point in
Figure 7 and two points in Figure 8. The movement was calculated as the
difference between the opening and closing line. The values were plotted along
with the mean and standard deviation in Figure 7 and Figure 8. The peak in
week 1 could be attributed to indeterminate performance as it is the first
week of the regular season and the first time that actual game play strategies
are being revealed. The peak in week 17 is usually attributed to teams that
have locked in playoff positions who rest their starting lineup or teams that
have to win to make it into the playoffs. With so few games demonstrating line
movement of this magnitude we were unable to find a statistically significant
difference that could be leveraged as a predictive value.
## 5 Future Work and Conclusions
We investigated many aspects of the betting line and its relation to NFL
football. In comparing the opening and closing lines we found no statistically
significant difference in the predictive values. Line movement was
investigated and typically only 10% of the games experience a difference of 2
or more points between the opening and closing line and those games did not
demonstrate any consistent behavior. We showed that the line value using the
cumulative distribution function is a good predictor of the team that will win
the game straight up but historically less than 50% accurate at predicting the
winner against the spread.
Our investigation and the Principal Component Analysis of the box score data
demonstrate that the line value would be a valuable input to a machine
learning algorithm for predicting the outcome of NFL games.
We also found that from 2002-2011, a strategy of betting the home underdog
would produce a cumulative winning percentage of 53.5%, which is above the
threshold of 52.38% needed to break even. Pre-2002 data suggests that while
this strategy was historically effective, its effectivness has been reduced in
later years.
Future work will include incorporating the closing line as a feature in a
machine learning algorithm for predicting the outcome of games, and
investigation to determine if the results here hold in college football games
as well as professional (the former has more teams, but the latter has more
parity between the teams).
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Table 4: Results of 1,000 Simulations of the 2011 NFL Season 2011
---
Team | Division | Predicted Wins | Actual Wins | Outcome
New England | AFC East | 11 | 13 | Division Winner
NY Jets | AFC East | 9 | 8 |
Buffalo | AFC East | 7 | 6 |
Miami | AFC East | 7 | 6 |
Baltimore | AFC North | 11 | 12 | Division Winner
Pittsburgh | AFC North | 10 | 12 | Wild Card
Cincinnati | AFC North | 8 | 9 | Wild Card
Cleveland | AFC North | 6 | 4 |
Houston | AFC South | 9 | 10 | Division Winner
Tennessee | AFC South | 8 | 9 |
Jacksonville | AFC South | 6 | 5 |
Indianapolis | AFC South | 5 | 2 |
Denver | AFC West | 7 | 8 | Division Winner
San Diego | AFC West | 9 | 8 |
Oakland | AFC West | 8 | 8 |
Kansas City | AFC West | 6 | 7 |
NY Giants | NFC East | 8 | 9 | Division Winner
Dallas | NFC East | 9 | 8 |
Philadelphia | NFC East | 10 | 8 |
Washington | NFC East | 7 | 5 |
Green Bay | NFC North | 12 | 15 | Division Winner
Detroit | NFC North | 9 | 10 | Wild Card
Chicago | NFC North | 8 | 8 |
Minnesota | NFC North | 6 | 3 |
New Orleans | NFC South | 11 | 13 | Division Winner
Atlanta | NFC South | 9 | 10 | Wild Card
Carolina | NFC South | 7 | 6 |
Tampa Bay | NFC South | 7 | 4 |
San Francisco | NFC West | 9 | 13 | Division Winner
Arizona | NFC West | 7 | 8 |
Seattle | NFC West | 6 | 7 |
St. Louis | NFC West | 5 | 2 |
Table 5: Results of 1,000 Simulations of the 2010 NFL Season 2010
---
Team | Division | Predicted Wins | Actual Wins | Outcome
New England | AFC East | 10 | 14 | Division Winner
NY Jets | AFC East | 9 | 11 | Wild Card
Miami | AFC East | 8 | 7 |
Buffalo | AFC East | 6 | 4 |
Pittsburgh | AFC North | 10 | 12 | Division Winner
Baltimore | AFC North | 10 | 12 | Wild Card
Cleveland | AFC North | 6 | 5 |
Cincinnati | AFC North | 7 | 4 |
Indianapolis | AFC South | 10 | 10 | Division Winner
Jacksonville | AFC South | 7 | 8 |
Houston | AFC South | 8 | 6 |
Tennessee | AFC South | 8 | 6 |
Kansas City | AFC West | 8 | 10 | Division Winner
San Diego | AFC West | 11 | 9 |
Oakland | AFC West | 7 | 8 |
Denver | AFC West | 7 | 4 |
Philadelphia | NFC East | 10 | 10 | Division Winner
NY Giants | NFC East | 9 | 10 |
Dallas | NFC East | 8 | 6 |
Washington | NFC East | 6 | 6 |
Chicago | NFC North | 8 | 11 | Division Winner
Green Bay | NFC North | 10 | 10 | Wild Card
Detroit | NFC North | 6 | 6 |
Minnesota | NFC North | 8 | 6 |
Atlanta | NFC South | 10 | 13 | Division Winner
New Orleans | NFC South | 11 | 11 | Wild Card
Tampa Bay | NFC South | 7 | 10 |
Carolina | NFC South | 5 | 2 |
Seattle | NFC West | 8 | 7 | Division Winner
St. Louis | NFC West | 7 | 7 |
San Francisco | NFC West | 6 | 6 |
Arizona | NFC West | 7 | 5 |
Table 6: Results of 1,000 Simulations of the 2009 NFL Season 2009
---
Team | Division | Predicted Wins | Actual Wins | Outcome
New England | AFC East | 10 | 10 | Division Winner
NY Jets | AFC East | 9 | 9 | Wild Card
Miami | AFC East | 7 | 9 |
Buffalo | AFC East | 6 | 6 |
Cincinnati | AFC North | 8 | 10 | Division Winner
Baltimore | AFC North | 10 | 9 | Wild Card
Pittsburgh | AFC North | 11 | 9 |
Cleveland | AFC North | 5 | 5 |
Indianapolis | AFC South | 10 | 14 | Division Winner
Houston | AFC South | 9 | 9 |
Tennessee | AFC South | 8 | 8 |
Jacksonville | AFC South | 8 | 7 |
San Diego | AFC West | 10 | 13 | Division Winner
Denver | AFC West | 8 | 8 |
Oakland | AFC West | 4 | 5 |
Kansas City | AFC West | 5 | 4 |
Dallas | NFC East | 10 | 11 | Division Winner
Philadelphia | NFC East | 10 | 11 | Wild Card
NY Giants | NFC East | 10 | 8 |
Washington | NFC East | 7 | 4 |
Minnesota | NFC North | 11 | 12 | Division Winner
Green Bay | NFC North | 10 | 11 | Wild Card
Chicago | NFC North | 8 | 7 |
Detroit | NFC North | 4 | 2 |
New Orleans | NFC South | 11 | 13 | Division Winner
Atlanta | NFC South | 8 | 9 |
Carolina | NFC South | 7 | 8 |
Tampa Bay | NFC South | 4 | 3 |
Arizona | NFC West | 9 | 10 | Division Winner
Seattle | NFC West | 8 | 8 |
San Francisco | NFC West | 7 | 5 |
St. Louis | NFC West | 4 | 1 |
Table 7: Results of 1,000 Simulations of the 2008 NFL Season 2008
---
Team | Division | Predicted Wins | Actual Wins | Outcome
Miami | AFC East | 8 | 11 | Division Winner
New England | AFC East | 10 | 11 |
NY Jets | AFC East | 9 | 9 |
Buffalo | AFC East | 8 | 7 |
Pittsburgh | AFC North | 9 | 12 | Division Winner
Baltimore | AFC North | 8 | 11 | Wild Card
Cincinnati | AFC North | 5 | 4 |
Cleveland | AFC North | 6 | 4 |
Tennessee | AFC South | 10 | 13 | Division Winner
Indianapolis | AFC South | 10 | 12 | Wild Card
Houston | AFC South | 8 | 8 |
Jacksonville | AFC South | 8 | 5 |
San Diego | AFC West | 10 | 8 | Division Winner
Denver | AFC West | 8 | 8 |
Oakland | AFC West | 5 | 5 |
Kansas City | AFC West | 5 | 2 |
NY Giants | NFC East | 10 | 12 | Division Winner
Philadelphia | NFC East | 10 | 9 | Wild Card
Dallas | NFC East | 10 | 9 |
Washington | NFC East | 8 | 8 |
Minnesota | NFC North | 9 | 10 | Division Winner
Chicago | NFC North | 8 | 9 |
Green Bay | NFC North | 8 | 6 |
Detroit | NFC North | 4 | 0 |
Carolina | NFC South | 9 | 12 | Division Winner
Atlanta | NFC South | 8 | 11 | Wild Card
Tampa Bay | NFC South | 9 | 9 |
New Orleans | NFC South | 8 | 8 |
Arizona | NFC West | 9 | 9 | Division Winner
Seattle | NFC West | 7 | 7 |
San Francisco | NFC West | 6 | 4 |
St. Louis | NFC West | 4 | 2 |
Table 8: Results of 1,000 Simulations of the 2007 NFL Season 2007
---
Team | Division | Predicted Wins | Actual Wins | Outcome
New England | AFC East | 13 | 16 | Division Winner
Buffalo | AFC East | 6 | 7 |
NY Jets | AFC East | 6 | 4 |
Miami | AFC East | 5 | 1 |
Pittsburgh | AFC North | 11 | 10 | Division Winner
Cleveland | AFC North | 8 | 10 |
Cincinnati | AFC North | 8 | 7 |
Baltimore | AFC North | 8 | 5 |
Indianapolis | AFC South | 11 | 13 | Division Winner
Jacksonville | AFC South | 9 | 11 | Wild Card
Tennessee | AFC South | 9 | 10 | Wild Card
Houston | AFC South | 7 | 8 |
San Diego | AFC West | 10 | 11 | Division Winner
Denver | AFC West | 8 | 7 |
Kansas City | AFC West | 6 | 4 |
Oakland | AFC West | 6 | 4 |
Dallas | NFC East | 11 | 13 | Division Winner
NY Giants | NFC East | 9 | 10 | Wild Card
Washington | NFC East | 8 | 9 | Wild Card
Philadelphia | NFC East | 9 | 8 |
Green Bay | NFC North | 9 | 13 | Division Winner
Minnesota | NFC North | 8 | 8 |
Chicago | NFC North | 8 | 7 |
Detroit | NFC North | 7 | 7 |
Tampa Bay | NFC South | 8 | 9 | Division Winner
Carolina | NFC South | 7 | 7 |
New Orleans | NFC South | 9 | 7 |
Atlanta | NFC South | 6 | 4 |
Seattle | NFC West | 10 | 10 | Division Winner
Arizona | NFC West | 8 | 8 |
San Francisco | NFC West | 5 | 5 |
St. Louis | NFC West | 6 | 3 |
Table 9: Results of 1,000 Simulations of the 2006 NFL Season 2006
---
Team | Division | Predicted Wins | Actual Wins | Outcome
New England | AFC East | 10 | 12 | Division Winner
NY Jets | AFC East | 7 | 10 | Wild Card
Buffalo | AFC East | 6 | 7 |
Miami | AFC East | 8 | 6 |
Baltimore | AFC North | 9 | 13 | Division Winner
Cincinnati | AFC North | 9 | 8 |
Pittsburgh | AFC North | 9 | 8 |
Cleveland | AFC North | 6 | 4 |
Indianapolis | AFC South | 11 | 12 | Division Winner
Jacksonville | AFC South | 9 | 8 |
Tennessee | AFC South | 5 | 8 |
Houston | AFC South | 5 | 6 |
San Diego | AFC West | 11 | 14 | Division Winner
Denver | AFC West | 10 | 9 |
Kansas City | AFC West | 8 | 9 | Wild Card
Oakland | AFC West | 5 | 2 |
Philadelphia | NFC East | 9 | 10 | Division Winner
Dallas | NFC East | 10 | 9 | Wild Card
NY Giants | NFC East | 8 | 8 | Wild Card
Washington | NFC East | 7 | 5 |
Chicago | NFC North | 11 | 13 | Division Winner
Green Bay | NFC North | 6 | 8 |
Minnesota | NFC North | 8 | 6 |
Detroit | NFC North | 6 | 3 |
New Orleans | NFC South | 8 | 10 | Division Winner
Carolina | NFC South | 9 | 8 |
Atlanta | NFC South | 8 | 7 |
Tampa Bay | NFC South | 6 | 4 |
Seattle | NFC West | 9 | 9 | Division Winner
St. Louis | NFC West | 8 | 8 |
San Francisco | NFC West | 6 | 7 |
Arizona | NFC West | 6 | 5 |
Table 10: Results of 1,000 Simulations of the 2005 NFL Season 2005
---
Team | Division | Predicted Wins | Actual Wins | Outcome
New England | AFC East | 9 | 10 | Division Winner
Miami | AFC East | 7 | 9 |
Buffalo | AFC East | 7 | 5 |
NY Jets | AFC East | 6 | 4 |
Cincinnati | AFC North | 10 | 11 | Division Winner
Pittsburgh | AFC North | 10 | 11 | Wild Card
Baltimore | AFC North | 7 | 6 |
Cleveland | AFC North | 6 | 6 |
Indianapolis | AFC South | 12 | 14 | Division Winner
Jacksonville | AFC South | 9 | 12 | Wild Card
Tennessee | AFC South | 6 | 4 |
Houston | AFC South | 5 | 2 |
Denver | AFC West | 10 | 13 | Division Winner
Kansas City | AFC West | 8 | 10 |
San Diego | AFC West | 10 | 9 |
Oakland | AFC West | 7 | 4 |
NY Giants | NFC East | 9 | 11 | Division Winner
Washington | NFC East | 9 | 10 | Wild Card
Dallas | NFC East | 8 | 9 |
Philadelphia | NFC East | 8 | 6 |
Chicago | NFC North | 8 | 11 | Division Winner
Minnesota | NFC North | 8 | 9 |
Detroit | NFC North | 7 | 5 |
Green Bay | NFC North | 7 | 4 |
Carolina | NFC South | 10 | 11 | Wild Card
Tampa Bay | NFC South | 9 | 11 | Division Winner
Atlanta | NFC South | 9 | 8 |
New Orleans | NFC South | 6 | 3 |
Seattle | NFC West | 10 | 13 | Division Winner
St. Louis | NFC West | 8 | 6 |
Arizona | NFC West | 7 | 5 |
San Francisco | NFC West | 4 | 4 |
Table 11: Results of 1,000 Simulations of the 2004 NFL Season 2004
---
Team | Division | Predicted Wins | Actual Wins | Outcome
New England | AFC East | 11 | 14 | Division Winner
NY Jets | AFC East | 9 | 10 | Wild Card
Buffalo | AFC East | 8 | 9 |
Miami | AFC East | 6 | 4 |
Pittsburgh | AFC North | 9 | 15 | Division Winner
Baltimore | AFC North | 9 | 9 |
Cincinnati | AFC North | 7 | 8 |
Cleveland | AFC North | 6 | 4 |
Indianapolis | AFC South | 10 | 12 | Division Winner
Jacksonville | AFC South | 8 | 9 |
Houston | AFC South | 6 | 7 |
Tennessee | AFC South | 8 | 5 |
San Diego | AFC West | 8 | 12 | Division Winner
Denver | AFC West | 10 | 10 | Wild Card
Kansas City | AFC West | 9 | 7 |
Oakland | AFC West | 7 | 5 |
Philadelphia | NFC East | 11 | 13 | Division Winner
Washington | NFC East | 7 | 6 |
NY Giants | NFC East | 7 | 6 |
Dallas | NFC East | 7 | 6 |
Green Bay | NFC North | 9 | 10 | Division Winner
Minnesota | NFC North | 10 | 8 | Wild Card
Detroit | NFC North | 7 | 6 |
Chicago | NFC North | 6 | 5 |
Atlanta | NFC South | 9 | 11 | Division Winner
New Orleans | NFC South | 7 | 8 |
Carolina | NFC South | 8 | 7 |
Tampa Bay | NFC South | 8 | 5 |
Seattle | NFC West | 10 | 9 | Division Winner
St. Louis | NFC West | 9 | 8 | Wild Card
Arizona | NFC West | 6 | 6 |
San Francisco | NFC West | 5 | 2 |
Table 12: Results of 1,000 Simulations of the 2003 NFL Season 2003
---
Team | Division | Predicted Wins | Actual Wins | Outcome
New England | AFC East | 9 | 14 | Division Winner
Miami | AFC East | 9 | 10 |
Buffalo | AFC East | 8 | 6 |
NY Jets | AFC East | 7 | 6 |
Baltimore | AFC North | 9 | 10 | Division Winner
Cincinnati | AFC North | 7 | 8 |
Pittsburgh | AFC North | 8 | 6 |
Cleveland | AFC North | 7 | 5 |
Indianapolis | AFC South | 10 | 12 | Division Winner
Tennessee | AFC South | 10 | 12 | Wild Card
Houston | AFC South | 5 | 5 |
Jacksonville | AFC South | 7 | 5 |
Kansas City | AFC West | 11 | 13 | Division Winner
Denver | AFC West | 9 | 10 | Wild Card
Oakland | AFC West | 7 | 4 |
San Diego | AFC West | 6 | 4 |
Philadelphia | NFC East | 9 | 12 | Division Winner
Dallas | NFC East | 8 | 10 | Wild Card
Washington | NFC East | 7 | 5 |
NY Giants | NFC East | 8 | 4 |
Green Bay | NFC North | 10 | 10 | Division Winner
Minnesota | NFC North | 10 | 9 |
Chicago | NFC North | 6 | 7 |
Detroit | NFC North | 5 | 5 |
Carolina | NFC South | 9 | 11 | Division Winner
New Orleans | NFC South | 8 | 8 |
Tampa Bay | NFC South | 10 | 7 |
Atlanta | NFC South | 6 | 5 |
St. Louis | NFC West | 10 | 12 | Division Winner
Seattle | NFC West | 9 | 10 | Wild Card
San Francisco | NFC West | 8 | 7 |
Arizona | NFC West | 5 | 4 |
Table 13: Results of 1,000 Simulations of the 2002 NFL Season 2002
---
Team | Division | Predicted Wins | Actual Wins | Outcome
NY Jets | AFC East | 8 | 9 | Division Winner
Miami | AFC East | 9 | 9 |
New England | AFC East | 9 | 9 |
Buffalo | AFC East | 8 | 8 |
Pittsburgh | AFC North | 10 | 10 | Division Winner
Cleveland | AFC North | 8 | 9 | Wild Card
Baltimore | AFC North | 7 | 7 |
Cincinnati | AFC North | 6 | 2 |
Tennessee | AFC South | 9 | 11 | Division Winner
Indianapolis | AFC South | 9 | 10 | Wild Card
Jacksonville | AFC South | 8 | 6 |
Houston | AFC South | 4 | 4 |
Oakland | AFC West | 10 | 11 | Division Winner
Denver | AFC West | 9 | 9 |
Kansas City | AFC West | 8 | 8 |
San Diego | AFC West | 8 | 8 |
Philadelphia | NFC East | 10 | 12 | Division Winner
NY Giants | NFC East | 8 | 10 | Wild Card
Washington | NFC East | 7 | 7 |
Dallas | NFC East | 7 | 5 |
Green Bay | NFC North | 10 | 12 | Division Winner
Minnesota | NFC North | 7 | 6 |
Chicago | NFC North | 7 | 4 |
Detroit | NFC North | 6 | 3 |
Tampa Bay | NFC South | 10 | 12 | Division Winner
Atlanta | NFC South | 9 | 9 | Wild Card
New Orleans | NFC South | 9 | 9 |
Carolina | NFC South | 6 | 7 |
San Francisco | NFC West | 10 | 10 | Division Winner
St. Louis | NFC West | 9 | 7 |
Seattle | NFC West | 7 | 7 |
Arizona | NFC West | 6 | 5 |
|
arxiv-papers
| 2012-11-16T19:46:18 |
2024-09-04T02:49:38.128963
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Greg Szalkowski and Michael L. Nelson",
"submitter": "Greg Szalkowski",
"url": "https://arxiv.org/abs/1211.4000"
}
|
1211.4062
|
A NEW APPROACH TO MULTIVARIATE $q$-EULER POLYNOMIALS BY USING UMBRAL CALCULUS
†Serkan ARACI, ‡Xiangxing KONG, †Mehmet ACIKGOZ, and ♯Erdoğan ŞEN
†University of Gaziantep, Faculty of Arts and Science, Department of
Mathematics, 27310 Gaziantep, Turkey
‡Department of Mathematics and Statistics, Central South University, Changsha
410075, China
♯Department of Mathematics, Faculty of Science and Letters, Namik Kemal
University, 59030 Tekirdağ, TURKEY
[email protected]; [email protected]; [email protected];
[email protected]
Abstract
In the present paper, we derive numerous identities for multivariate $q$-Euler
polynomials by using umbral calculus.
2010 Mathematics Subject Classification 11S80, 11B68.
Key Words and Phrases Appell sequence, sheffer sequence, multivariate
$q$-Euler polynomials, formal power series.
## 1\. Preliminaries
Throughout this paper, we use the following notation, where $\mathbb{C}$
denotes the set of complex numbers, $\mathcal{F}$ denotes the set of all
formal power series in the variable $t$ over $\mathbb{C}$ with
$\mathcal{F}=\left\\{f\left(t\right)=\sum_{k=0}^{\infty}a_{k}\frac{t^{k}}{k!}\mid
a_{k}\in\mathbb{C}\right\\}$, $\mathcal{P}=\mathbb{C}\left[x\right]$ and
$\mathcal{P}^{\ast}$ denotes the vector space of all linear functional on
$\mathcal{P}$, $\left\langle L\mid p\left(x\right)\right\rangle$ denotes the
action of the linear functional $L$ on the polynomial $p\left(x\right)$, and
it is well-known that the vector space operation on $\mathcal{P}^{\ast}$ is
defined by
$\displaystyle\left\langle L+M\mid p\left(x\right)\right\rangle$
$\displaystyle=$ $\displaystyle\left\langle L\mid
p\left(x\right)\right\rangle+\left\langle M\mid p\left(x\right)\right\rangle,$
$\displaystyle\left\langle cL\mid p\left(x\right)\right\rangle$
$\displaystyle=$ $\displaystyle c\left\langle L\mid
p\left(x\right)\right\rangle,$
where $c$ is some constant in $\mathbb{C}$ (for details, see [10, 11, 14,
17]).
The formal power series are known by the rule
$f\left(t\right)=\sum_{k=0}^{\infty}a_{k}\frac{t^{k}}{k!}\in\mathcal{F}$
which defines a linear functional on $\mathcal{P}$ as $\left\langle
f\left(t\right)\mid x^{n}\right\rangle=a_{n}$ for all $n\geq 0$ (for details,
see [10, 11, 14, 17]). Additionally,
$\left\langle t^{k}\mid x^{n}\right\rangle=n!\delta_{n,k},$ (1.1)
where $\delta_{n,k}$ is the Kronecker symbol. When we take
$f_{L}\left(t\right)=\sum_{k=0}^{\infty}\left\langle L\mid
x^{k}\right\rangle\frac{t^{k}}{k!},$ then we obtain $\left\langle
f_{L}\left(t\right)\mid x^{n}\right\rangle=\left\langle L\mid
x^{n}\right\rangle$ and so as linear functionals $L=f_{L}\left(t\right)$ (see
[10, 11, 14, 17]). Additional, the map $L\rightarrow f_{L}\left(t\right)$ is a
vector space isomorphism from $\mathcal{P}^{\ast}$ onto $\mathcal{F}$.
Henceforth, $\mathcal{F}$ will denote both the algebra of the formal power
series in $t$ and the vector space of all linear functionals on $\mathcal{P}$,
and so an element $f\left(t\right)$ of $\mathcal{F}$ will be thought of as
both a formal power series and a linear functional. $\mathcal{F}$ will be
called as umbral algebra ( see [10, 11, 14, 17]).
Also, the evaluation functional for $y$ in $\mathbb{C}$ is defined to be power
series $e^{yt}$. We can write that $\left\langle e^{yt}\mid
x^{n}\right\rangle=y^{n}$ and so $\left\langle e^{yt}\mid
p\left(x\right)\right\rangle=p\left(y\right)$ ( see [10, 11, 12, 14, 17]). We
want to note that for all $f\left(t\right)$ in $\mathcal{F}$
$f\left(t\right)=\sum_{k=0}^{\infty}\left\langle f\left(t\right)\mid
x^{k}\right\rangle\frac{t^{k}}{k!}$ (1.2)
and for all polynomial $p\left(x\right)$,
$p\left(x\right)=\sum_{k=0}^{\infty}\left\langle t^{k}\mid
p\left(x\right)\right\rangle\frac{x^{k}}{k!},$ (1.3)
(for details, see [10, 11, 14, 17]). The order $o\left(f\left(t\right)\right)$
of the power series $f\left(t\right)\neq 0$ is the smallest integer $k$ for
which $a_{k}$ does not vanish. It is considered
$o\left(f\left(t\right)\right)=\infty$ if $f\left(t\right)=0$. We see that
$o\left(f\left(t\right)g\left(t\right)\right)=o\left(f\left(t\right)\right)+o\left(g\left(t\right)\right)$
and
$o\left(f\left(t\right)+g\left(t\right)\right)\geq\min\left\\{o\left(f\left(t\right)\right),o\left(g\left(t\right)\right)\right\\}$.
The series $f\left(t\right)$ has a multiplicative inverse, denoted by
$f\left(t\right)^{-1}$ or $\frac{1}{f\left(t\right)}$, if and only if
$o\left(f\left(t\right)\right)=0$. Such series is called an invertible series.
A series $f\left(t\right)$ for which $o\left(f\left(t\right)\right)=1$ is
called a delta series ( see [10, 11, 12, 14, 17]). For
$f\left(t\right),g\left(t\right)\in\mathcal{F}$, we have $\left\langle
f\left(t\right)g\left(t\right)\mid p\left(x\right)\right\rangle=\left\langle
f\left(t\right)\mid g\left(t\right)p\left(x\right)\right\rangle$.
A delta series $f\left(t\right)$ has a compositional inverse
$\overline{f}\left(t\right)$ such that
$f\left(\overline{f}\left(t\right)\right)=\overline{f}\left(f\left(t\right)\right)=t$.
For $f\left(t\right),g\left(t\right)\in\mathcal{F}$ , we have $\left\langle
f\left(t\right)g\left(t\right)\mid p\left(x\right)\right\rangle=\left\langle
f\left(t\right)\mid g\left(t\right)p\left(x\right)\right\rangle$. By (1.2), we
have
$p^{\left(k\right)}\left(x\right)=\frac{d^{k}p\left(x\right)}{dx^{k}}=\sum_{l=k}^{\infty}\frac{\left\langle
t^{l}\mid
p\left(x\right)\right\rangle}{l!}l\left(l-1\right)\cdots\left(l-k+1\right)x^{l-k}\text{.}$
(1.4)
Thus, we see that
$p^{\left(k\right)}\left(0\right)=\left\langle t^{k}\mid
p\left(x\right)\right\rangle=\left\langle 1\mid
p^{\left(k\right)}\left(x\right)\right\rangle\text{.}$ (1.5)
By (1.4), we get
$t^{k}p\left(x\right)=p^{\left(k\right)}\left(x\right)=\frac{d^{k}p\left(x\right)}{dx^{k}}\text{.}$
(1.6)
So, we have
$e^{yt}p\left(x\right)=p\left(x+y\right)\text{.}$ (1.7)
Let $S_{n}\left(x\right)$ be a polynomial with $\deg S_{n}\left(x\right)=n$.
Let $f\left(t\right)$ be a delta series and let $g\left(t\right)$ be an
invertible series. Then there exists a unique sequence $S_{n}\left(x\right)$
of polynomials such that $\left\langle g\left(t\right)f\left(t\right)^{k}\mid
S_{n}\left(x\right)\right\rangle=n!\delta_{n,k}$ for all $n,k\geq 0$. The
sequence $S_{n}\left(x\right)$ is called the sheffer sequence for
$\left(g\left(t\right),f\left(t\right)\right)$ or that $S_{n}\left(t\right)$
is sheffer for $\left(g\left(t\right),f\left(t\right)\right)$.
The sheffer sequence for $\left(1,f\left(t\right)\right)$ is called the
associated sequence for $f\left(t\right)$ or $S_{n}\left(x\right)$ is
associated with $f\left(t\right)$. The sheffer sequence for
$\left(g\left(t\right),t\right)$ is called the appell sequence for
$g\left(t\right)$ or $S_{n}\left(x\right)$ is Appell for $g\left(t\right)$.
Let $p\left(x\right)\in\mathcal{P}$. Then we have
$\displaystyle\left\langle\frac{e^{yt}-1}{t}\mid p\left(x\right)\right\rangle$
$\displaystyle=$ $\displaystyle\int_{0}^{y}p\left(u\right)du,$
$\displaystyle\left\langle f\left(t\right)\mid xp\left(x\right)\right\rangle$
$\displaystyle=$ $\displaystyle\left\langle\partial_{t}f\left(t\right)\mid
p\left(x\right)\right\rangle=\left\langle f{\acute{}}\left(t\right)\mid
p\left(x\right)\right\rangle,$ (1.8) $\displaystyle\left\langle e^{yt}-1\mid
p\left(x\right)\right\rangle$ $\displaystyle=$ $\displaystyle
p\left(y\right)-p\left(0\right),\text{ ( see [10, 11, 14, 17]).}$
Let $S_{n}\left(x\right)$ be sheffer for
$\left(g\left(t\right),f\left(t\right)\right)$. Then
$\displaystyle h\left(t\right)$ $\displaystyle=$
$\displaystyle\sum_{k=0}^{\infty}\frac{\left\langle h\left(t\right)\mid
S_{k}\left(x\right)\right\rangle}{k!}g\left(t\right)f\left(t\right)^{k},\text{
}h\left(t\right)\in\mathcal{F}$ $\displaystyle p\left(x\right)$
$\displaystyle=$ $\displaystyle\sum_{k=0}^{\infty}\frac{\left\langle
g\left(t\right)f\left(t\right)^{k}\mid
p\left(x\right)\right\rangle}{k!}S_{k}\left(x\right),\text{
}p\left(x\right)\in\mathcal{P},$
$\displaystyle\frac{1}{g\left(\overline{f}\left(t\right)\right)}e^{y\overline{f}\left(t\right)}$
$\displaystyle=$
$\displaystyle\sum_{k=0}^{\infty}S_{k}\left(y\right)\frac{t^{k}}{k!},\text{
for all }y\in\mathbb{C},$ (1.9) $\displaystyle
f\left(t\right)S_{n}\left(x\right)$ $\displaystyle=$ $\displaystyle
nS_{n-1}\left(x\right)\text{.}$
Let $a_{1},\cdots,a_{r},b_{1},\cdots,b_{r}$ be positive integers. Kim and Rim
[1] defined the generating function for multivariate $q$-Euler polynomials as
follows:
$\displaystyle F_{q}\left(t,x\mid
a_{1},\cdots,a_{r};b_{1},\cdots,b_{r}\right)=\sum_{n=0}^{\infty}E_{n,q}\left(x\mid
a_{1},\cdots,a_{r};b_{1},\cdots,b_{r}\right)\frac{t^{n}}{n!}$ (1.10)
$\displaystyle=\frac{2^{r}}{\left(q^{b_{1}}e^{a_{1}t}+1\right)\cdots\left(q^{b_{r}}e^{a_{r}t}+1\right)}e^{xt}\text{.}$
Note that
$E_{0,q}\left(x\mid
a_{1},\cdots,a_{r};b_{1},\cdots,b_{r}\right)=\frac{2^{r}}{\left[2\right]_{q^{b_{1}}}\left[2\right]_{q^{b_{2}}}\cdots\left[2\right]_{q^{b_{r}}}},$
where $\left[x\right]_{q}$ is $q$-extension of $x$ defined by
$\left[x\right]_{q}=\frac{q^{x}-1}{q-1}=1+q+q^{2}+\cdots+q^{x-1}.$
We assume that $q\in\mathbb{C}$ with $\left|q\right|<1$. Also, we want to note
that $\lim_{q\rightarrow 1}\left[x\right]_{q}=x$ (see [1-16]). In the special
case, $x=0$, $E_{n,q}\left(0\mid
a_{1},\cdots,a_{r};b_{1},\cdots,b_{r}\right):=E_{n,q}\left(a_{1},\cdots,a_{r};b_{1},\cdots,b_{r}\right)$
are called multivariate $q$-Euler numbers. By (1.10), we procure the
following:
$E_{n,q}\left(x\mid
a_{1},\cdots,a_{r};b_{1},\cdots,b_{r}\right)=\sum_{k=0}^{n}\binom{n}{k}x^{k}E_{n-k,q}\left(a_{1},\cdots,a_{r};b_{1},\cdots,b_{r}\right)\text{.}$
(1.11)
Kim et al [10] studied some interesting identities for Frobenius-Euler
polynomials arising from umbral calculus. They derived not only new but also
fascianting identities in modern classical umbral calculus.
By the same motivation, we also get numerous identities for multivariate
$q$-Euler polynomials by utilizing from the umbral calculus.
## 2\. On the multivariate $q$-Euler polynomials arising from umbral calculus
Assume that $S_{n}\left(x\right)$ is an appell sequence for $g\left(t\right)$,
by (1.9), we have
$\frac{1}{g\left(t\right)}x^{n}=S_{n}\left(x\right)\text{ if and only if
}x^{n}=g\left(t\right)S_{n}\left(x\right)\text{, }\left(n\geq 0\right).$ (2.1)
Let us take
$g\left(t\mid
a_{1},\cdots,a_{r};b_{1},\cdots,b_{r}\right)=\frac{\left(q^{b_{1}}e^{a_{1}t}+1\right)\cdots\left(q^{b_{r}}e^{a_{r}t}+1\right)}{2^{r}}\in\mathcal{F}\text{.}$
Then we readily see that $g\left(t\mid
a_{1},\cdots,a_{r};b_{1},\cdots,b_{r}\right)$ is an invertible series. By
(2.1), we have
$\sum_{n=0}^{\infty}E_{n,q}\left(x\mid
a_{1},\cdots,a_{r};b_{1},\cdots,b_{r}\right)\frac{t^{n}}{n!}=\frac{1}{g\left(t\mid
a_{1},\cdots,a_{r};b_{1},\cdots,b_{r}\right)}e^{xt}\text{.}$ (2.2)
By (2.2), we procure the following
$\frac{1}{g\left(t\mid
a_{1},\cdots,a_{r};b_{1},\cdots,b_{r}\right)}x^{n}=E_{n,q}\left(x\mid
a_{1},\cdots,a_{r};b_{1},\cdots,b_{r}\right)\text{.}$ (2.3)
Also, by (1.9), we have
$\displaystyle tE_{n,q}\left(x\mid
a_{1},\cdots,a_{r};b_{1},\cdots,b_{r}\right)=E{\acute{}}_{n,q}\left(x\mid
a_{1},\cdots,a_{r};b_{1},\cdots,b_{r}\right)$ (2.4)
$\displaystyle=nE_{n-1,q}\left(x\mid
a_{1},\cdots,a_{r};b_{1},\cdots,b_{r}\right)\text{.}$
By (2.3) and (2.4), we have the following proposition.
###### Proposition 1.
For $n\geq 0$, $E_{n,q}\left(x\mid
a_{1},\cdots,a_{r};b_{1},\cdots,b_{r}\right)$ is an Appell sequence for
$g\left(t\mid
a_{1},\cdots,a_{r};b_{1},\cdots,b_{r}\right)=\frac{\left(q^{b_{1}}e^{a_{1}t}+1\right)\cdots\left(q^{b_{r}}e^{a_{r}t}+1\right)}{2^{r}}\text{.}$
By (1.10), we see that
$\displaystyle\sum_{n=1}^{\infty}E_{n,q}\left(x\mid
a_{1},\cdots,a_{r};b_{1},\cdots,b_{r}\right)\frac{t^{n}}{n!}$
$\displaystyle=\frac{xge^{xt}-g{\acute{}}e^{xt}}{g^{2}}$ (2.5)
$\displaystyle=\sum_{n=0}^{\infty}\left(x\frac{1}{g}x^{n}-\frac{g{\acute{}}}{g}\frac{1}{g}x^{n}\right)\frac{t^{n}}{n!}$
where we used $g:=g\left(t\mid a_{1},\cdots,a_{r};b_{1},\cdots,b_{r}\right)$.
Because of (2.3) and (2.5), we discover the following:
$\displaystyle E_{n+1,q}\left(x\mid
a_{1},\cdots,a_{r};b_{1},\cdots,b_{r}\right)$ (2.6)
$\displaystyle=xE_{n,q}\left(x\mid
a_{1},\cdots,a_{r};b_{1},\cdots,b_{r}\right)-\frac{g{\acute{}}}{g}E_{n,q}\left(x\mid
a_{1},\cdots,a_{r};b_{1},\cdots,b_{r}\right)\text{.}$
Therefore, we deduce the following theorem.
###### Theorem 1.
Let $g:=g\left(t\mid
a_{1},\cdots,a_{r};b_{1},\cdots,b_{r}\right)=\frac{\left(q^{b_{1}}e^{a_{1}t}+1\right)\cdots\left(q^{b_{r}}e^{a_{r}t}+1\right)}{2^{r}}\in
F$. Then we have for $n\geq 0:$
$E_{n+1,q}\left(x\mid
a_{1},\cdots,a_{r};b_{1},\cdots,b_{r}\right)=\left(x-\frac{g{\acute{}}}{g}\right)E_{n,q}\left(x\mid
a_{1},\cdots,a_{r};b_{1},\cdots,b_{r}\right)\text{.}$ (2.7)
From (1.10), we derive that
$\displaystyle\sum_{n=0}^{\infty}\left(q^{b_{r}}E_{n,q}\left(x+a_{r}\mid
a_{1},\cdots,a_{r};b_{1},\cdots,b_{r}\right)+E_{n,q}\left(x\mid
a_{1},\cdots,a_{r};b_{1},\cdots,b_{r}\right)\right)\frac{t^{n}}{n!}$ (2.8)
$\displaystyle=2\sum_{n=0}^{\infty}E_{n,q}\left(x\mid
a_{1},\cdots,a_{r-1};b_{1},\cdots,b_{r-1}\right)\frac{t^{n}}{n!}\text{.}$
By comparing the coefficients in the both sides of $\frac{t^{n}}{n!}$ on the
above, we procure the following
$\displaystyle 2E_{n,q}\left(x\mid
a_{1},\cdots,a_{r-1};b_{1},\cdots,b_{r-1}\right)=q^{b_{r}}E_{n,q}\left(x+a_{r}\mid
a_{1},\cdots,a_{r};b_{1},\cdots,b_{r}\right)$ (2.9)
$\displaystyle+E_{n,q}\left(x\mid
a_{1},\cdots,a_{r};b_{1},\cdots,b_{r}\right)\text{.}$
From theorem 1, we get the following equation
$\displaystyle gE_{n+1,q}\left(x\mid
a_{1},\cdots,a_{r};b_{1},\cdots,b_{r}\right)$ (2.10)
$\displaystyle=gxE_{n,q}\left(x\mid
a_{1},\cdots,a_{r};b_{1},\cdots,b_{r}\right)-g{\acute{}}E_{n,q}\left(x\mid
a_{1},\cdots,a_{r};b_{1},\cdots,b_{r}\right).$
By using (2.9) and (2.10), we obtain the following theorem.
###### Theorem 2.
For $n\geq 0$, then we have
$\displaystyle 2E_{n,q}\left(x\mid
a_{1},\cdots,a_{r-1};b_{1},\cdots,b_{r-1}\right)=q^{b_{r}}E_{n,q}\left(x+a_{r}\mid
a_{1},\cdots,a_{r};b_{1},\cdots,b_{r}\right)$ (2.11)
$\displaystyle+E_{n,q}\left(x\mid
a_{1},\cdots,a_{r};b_{1},\cdots,b_{r}\right)\text{.}$
Now, we consider that
$\displaystyle\int_{x}^{x+y}E_{n,q}\left(u\mid
a_{1},\cdots,a_{r};b_{1},\cdots,b_{r}\right)du$ $\displaystyle=$
$\displaystyle\frac{1}{n+1}\left(E_{n,q}\left(x+y\mid
a_{1},\cdots,a_{r};b_{1},\cdots,b_{r}\right)-E_{n,q}\left(x\mid
a_{1},\cdots,a_{r};b_{1},\cdots,b_{r}\right)\right)$ $\displaystyle=$
$\displaystyle\frac{1}{n+1}\sum_{j=1}^{\infty}\binom{n+1}{j}E_{n+1-j,q}\left(x\mid
a_{1},\cdots,a_{r};b_{1},\cdots,b_{r}\right)y^{j}$ $\displaystyle=$
$\displaystyle\sum_{j=1}^{\infty}\frac{n\left(n-1\right)\left(n-2\right)\cdots\left(n-j+2\right)}{j!}E_{n+1-j,q}\left(x\mid
a_{1},\cdots,a_{r};b_{1},\cdots,b_{r}\right)y^{j}$ $\displaystyle=$
$\displaystyle\frac{1}{t}\left(\sum_{j=0}^{\infty}\frac{y^{j}t^{j}}{j!}-1\right)E_{n,q}\left(x\mid
a_{1},\cdots,a_{r};b_{1},\cdots,b_{r}\right)$ $\displaystyle=$
$\displaystyle\frac{e^{yt}-1}{t}E_{n,q}\left(x\mid
a_{1},\cdots,a_{r};b_{1},\cdots,b_{r}\right)\text{.}$
Therefore, we discover the following theorem:
###### Theorem 3.
For $n\geq 0$, then we have
$\int_{x}^{x+y}E_{n,q}\left(u\mid
a_{1},\cdots,a_{r};b_{1},\cdots,b_{r}\right)du=\frac{e^{yt}-1}{t}E_{n,q}\left(x\mid
a_{1},\cdots,a_{r};b_{1},\cdots,b_{r}\right)\text{.}$ (2.12)
By (2.4) and proposition 1, we have
$t\left\\{\frac{1}{n+1}E_{n+1,q}\left(x\mid
a_{1},\cdots,a_{r};b_{1},\cdots,b_{r}\right)\right\\}=E_{n,q}\left(x\mid
a_{1},\cdots,a_{r};b_{1},\cdots,b_{r}\right)\text{.}$ (2.13)
Thanks to (1.8), we readily derive the following:
$\displaystyle\left\langle e^{yt}-1\mid\frac{E_{n+1,q}\left(x\mid
a_{1},\cdots,a_{r};b_{1},\cdots,b_{r}\right)}{n+1}\right\rangle$
$\displaystyle=$ $\displaystyle\left\langle\frac{e^{yt}-1}{t}\mid
t\left\\{\frac{E_{n+1,q}\left(x\mid
a_{1},\cdots,a_{r};b_{1},\cdots,b_{r}\right)}{n+1}\right\\}\right\rangle$
$\displaystyle=$ $\displaystyle\left\langle\frac{e^{yt}-1}{t}\mid
E_{n,q}\left(x\mid a_{1},\cdots,a_{r};b_{1},\cdots,b_{r}\right)\right\rangle.$
On account of (2.13) and (2), we get
$\displaystyle\left\langle\frac{e^{yt}-1}{t}\mid E_{n,q}\left(x\mid
a_{1},\cdots,a_{r};b_{1},\cdots,b_{r}\right)\right\rangle=\left\langle
e^{yt}-1\mid\frac{E_{n+1,q}\left(x\mid
a_{1},\cdots,a_{r};b_{1},\cdots,b_{r}\right)}{n+1}\right\rangle$
$\displaystyle=\frac{1}{n+1}\left\\{E_{n+1,q}\left(y\mid
a_{1},\cdots,a_{r};b_{1},\cdots,b_{r}\right)-E_{n+1,q}\left(a_{1},\cdots,a_{r};b_{1},\cdots,b_{r}\right)\right\\}$
$\displaystyle=\int_{0}^{y}E_{n,q}\left(u\mid
a_{1},\cdots,a_{r};b_{1},\cdots,b_{r}\right)du.$
Consequently, we obtain the following theorem.
###### Theorem 4.
For $n\geq 0$, then we have
$\left\langle\frac{e^{yt}-1}{t}\mid E_{n,q}\left(x\mid
a_{1},\cdots,a_{r};b_{1},\cdots,b_{r}\right)\right\rangle=\int_{0}^{y}E_{n,q}\left(u\mid
a_{1},\cdots,a_{r};b_{1},\cdots,b_{r}\right)du\text{.}$ (2.15)
Assume that
$\mathcal{P}\left(q\mid
a_{1},\cdots,a_{r};b_{1},\cdots,b_{r}\right)=\left\\{p\left(x\right)\in
Q\left(q\mid
a_{1},\cdots,a_{r};b_{1},\cdots,b_{r}\right)\left[x\right]\mid\deg
p\left(x\right)\leq n\right\\}$
is a vector space over $Q\left(q\mid
a_{1},\cdots,a_{r};b_{1},\cdots,b_{r}\right)$.
For $p\left(x\right)\in\mathcal{P}\left(q\mid
a_{1},\cdots,a_{r};b_{1},\cdots,b_{r}\right)$, let us consider
$p\left(x\right)=\sum_{k=0}^{n}b_{k}E_{k,q}\left(x\mid
a_{1},\cdots,a_{r};b_{1},\cdots,b_{r}\right)\text{.}$ (2.16)
By proposition 1, $E_{n,q}\left(u\mid
a_{1},\cdots,a_{r};b_{1},\cdots,b_{r}\right)$ is an appell sequence for
$g:=g\left(t\mid
a_{1},\cdots,a_{r};b_{1},\cdots,b_{r}\right)=\frac{\left(q^{b_{1}}e^{a_{1}t}+1\right)\cdots\left(q^{b_{r}}e^{a_{r}t}+1\right)}{2^{r}}\text{.}$
Thus we have
$\left\langle g\left(t\mid
a_{1},\cdots,a_{r};b_{1},\cdots,b_{r}\right)t^{k}\mid E_{n,q}\left(x\mid
a_{1},\cdots,a_{r};b_{1},\cdots,b_{r}\right)\right\rangle=n!\delta_{n,k}\text{.}$
(2.17)
From (2.16) and (2.17), we compute
$\displaystyle\left\langle g\left(t\mid
a_{1},\cdots,a_{r};b_{1},\cdots,b_{r}\right)t^{k}\mid
p\left(x\right)\right\rangle=\sum_{l=0}^{n}b_{l}\left\langle gt^{k}\mid
E_{l,q}\left(x\mid a_{1},\cdots,a_{r};b_{1},\cdots,b_{r}\right)\right\rangle$
(2.18) $\displaystyle=\sum_{l=0}^{n}b_{l}l!\delta_{l,k}=k!b_{k}\text{.}$
Thus, by (2.18), we derive
$\displaystyle b_{k}$ $\displaystyle=$ $\displaystyle\frac{1}{k!}\left\langle
gt^{k}\mid p\left(x\right)\right\rangle$ $\displaystyle=$
$\displaystyle\frac{1}{2^{r}k!}\left\langle\left(q^{b_{1}}e^{a_{1}t}+1\right)\cdots\left(q^{b_{r}}e^{a_{r}t}+1\right)\mid
p^{\left(k\right)}\left(x\right)\right\rangle\text{.}$
It is not difficult to show the following
$\left(q^{b_{1}}e^{a_{1}t}+1\right)\cdots\left(q^{b_{r}}e^{a_{r}t}+1\right)=\sum_{\underset{k_{1}+k_{2}+\cdots+k_{r}=1}{k_{1},\cdots,k_{r}\geq
0}}q^{\sum_{l=1}^{r}b_{l}k_{l}}e^{t\sum_{j=1}^{r}a_{j}k_{j}}\text{.}$ (2.20)
Via the (2) and (2.20), we easily see that
$\displaystyle b_{k}$ $\displaystyle=$
$\displaystyle\frac{1}{2^{r}k!}\sum_{\underset{k_{1}+k_{2}+\cdots+k_{r}=1}{k_{1},\cdots,k_{r}\geq
0}}q^{\sum_{l=1}^{r}b_{l}k_{l}}\left\langle e^{t\sum_{j=1}^{r}a_{j}k_{j}}\mid
p^{\left(k\right)}\left(x\right)\right\rangle$ $\displaystyle=$
$\displaystyle\frac{1}{2^{r}k!}\sum_{\underset{k_{1}+k_{2}+\cdots+k_{r}=1}{k_{1},\cdots,k_{r}\geq
0}}q^{\sum_{l=1}^{r}b_{l}k_{l}}p^{\left(k\right)}\left(\sum_{j=1}^{r}a_{j}k_{j}\right)\text{.}$
As a result, we state the following theorem.
###### Theorem 5.
For $p\left(x\right)\in P\left(q\mid
a_{1},\cdots,a_{r};b_{1},\cdots,b_{r}\right)$, when we consider
$p\left(x\right)=\sum_{k=0}^{n}b_{k}E_{k,q}\left(x\mid
a_{1},\cdots,a_{r};b_{1},\cdots,b_{r}\right)\text{,}$
then we have
$b_{k}=\frac{1}{2^{r}k!}\sum_{\underset{k_{1}+k_{2}+\cdots+k_{r}=1}{k_{1},\cdots,k_{r}\geq
0}}q^{\sum_{l=1}^{r}b_{l}k_{l}}p^{\left(k\right)}\left(\sum_{j=1}^{r}a_{j}k_{j}\right)\text{,}$
where
$p^{\left(k\right)}\left(\sum_{j=1}^{r}a_{j}k_{j}\right)=\frac{d^{k}p\left(x\right)}{dx^{k}}\mid_{x=\sum_{j=1}^{r}a_{j}k_{j}}$.
## References
* [1] T. Kim and S. H. Rim, New Changhee $q$-Euler numbers and polynomials associated with $p$-adic $q$-integrals, Computers & Mathematics with Applications, 54 (2007) 484–489.
* [2] T. Kim, Note on Dedekind type DC sums, Adv. Stud. Contemp. Math. 18 (2009), 249–260.
* [3] T. Kim, The modified $q$-Euler numbers and polynomials, Adv. Stud. Contemp. Math. 16 (2008), 161–170.
* [4] T. Kim, $p$-adic $q$-integrals associated with the Changhee–Barnes’ $q$-Bernoulli polynomials, Integral Transforms Spec. Funct. 15 (2004) 415–420
* [5] T. Kim, On $p$-adic interpolating function for $q$-Euler numbers and its derivatives, J. Math. Anal. Appl. 339 (2008), 598–608.
* [6] T. Kim, On a $q$-analogue of the $p$-adic log gamma functions and related integrals, J. Number Theory 76 (1999), 320-329.
* [7] T. Kim, Symmetry of power sum polynomials and multivariate fermionic $p$-adic invariant integral on $\mathbb{Z}_{p}$, Russ. J. Math. Phys. 16, no. 1, 93-96 (2009).
* [8] T. Kim, On the $q$-extension of Euler and Genocchi numbers, J. Math. Anal. Appl. 326 (2007) 1458–1465.
* [9] T. Kim, Non-Archimedean $q$-integrals associated with multiple Changhee $q$-Bernoulli polynomials, Russ. J. Math. Phys. 10 (2003) 91–98.
* [10] D. S. Kim and T. Kim, Some identities of Frobenius-Euler polynomials arising from umbral calculus, Advances in Difference Equations (In press).
* [11] T. Kim, D. S. Kim, S-H. Lee and S-H. Rim, Umbral calculus and Euler polynomials, Ars Combinatoria (In press).
* [12] M. Maldonado, J. Prada and M. J. Senosiain, Appell bases on sequence spaces, Journal of Nonlinear Mathematical Physics, Vol. 18, Suppl. 1 (2011) 189–194.
* [13] M. Acikgoz and Y. Simsek, On multiple interpolation function of the Nörlund-type $q$-Euler polynomials, Abst. Appl. Anal. 2009 (2009), Article ID 382574, 14 pages.
* [14] R. Dere and Y. Simsek, Applications of umbral algebra to some special polynomials, Adv. Stud. Contemp. Math. 22 (2012), no. 3, 433-438.
* [15] S. Araci, D. Erdal and J. J. Seo, A study on the fermionic $p$-adic $q$-integral representation on $\mathbb{Z}_{p}$ associated with weighted $q$-Bernstein and $q$-Genocchi polynomials, Abstract and Applied Analysis, Volume 2011, Article ID 649248, 10 pages.
* [16] S. Araci, M. Acikgoz and J. J. Seo, Explicit formulas involving $q$-Euler numbers and polynomials, Abstract and Applied Analysis, Volume 2012, Article ID 298531, 11 pages.
* [17] S. Roman, The umbral calculus, Dover Publ. Inc. New York, 2005.
|
arxiv-papers
| 2012-11-16T23:16:42 |
2024-09-04T02:49:38.139439
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Serkan Araci, Xiangxing Kong, Mehmet Acikgoz, and Erdo\\u{g}an \\c{S}en",
"submitter": "Serkan Araci",
"url": "https://arxiv.org/abs/1211.4062"
}
|
1211.4132
|
# Helicity Condensation as the Origin of Coronal and Solar Wind Structure
S. K. Antiochos NASA Goddard Space Flight Center, Greenbelt, MD, 20771
[email protected]
###### Abstract
Three of the most important and most puzzling features of the Sun’s atmosphere
are the smoothness of the closed field corona, the accumulation of magnetic
shear at photospheric polarity inversion lines (PIL), and the complexity of
the slow wind. We propose that a single process, helicity condensation, is the
physical mechanism giving rise to all three features. A simplified model is
presented for how helicity is injected and transported in the closed corona by
magnetic reconnection. With this model we demonstrate that helicity must
condense onto PILs and coronal hole boundaries, and estimate the rate of
helicity accumulation at PILs and the loss to the wind. Our results can
account for many of the observed properties of the closed corona and wind.
Sun: magnetic field — Sun: corona
## 1 Introduction
A classic, but puzzling feature of the Sun’s high-temperature ($>1$MK)
atmosphere is its apparent lack of complexity. High-resolution XUV and X-ray
images of the closed-field corona, such as those from the Transition Region
and Coronal Explorer (TRACE) mission, invariably show a smooth collection of
loops (e.g. Schrijver et al., 1999). If the underlying photospheric flux
distribution is highly structured with several polarity regions, then the
topology of the loops in the corona will appear complex in XUV images, but
this is only due to seeing through multiple flux systems. The surprising
result is that the loops in any one flux system, such as a bipolar active
region, are generally not observed to be twisted or tangled. Since the coronal
field is line-tied to the photosphere, which is undergoing continuous chaotic
motions due to the convective flows, it would seem that geometrical complexity
should eventually appear in the coronal field. Instead, the magnetic field
appears to remain laminar and not far from a potential state over most of the
corona. Extrapolations of the field do indicate the presence of coronal
currents (Leka et al., 1996; Tian et al., 2005), but these are large-scale
volumetric currents that produce only a global shear or twist rather than
field line tangling (Schrijver, 2007).
The observation that loops are untangled, at least on present observable
scales, is especially surprising given that the corona is being heated
continuously. The standard theory for the heating is that the energy is due to
stressing of the coronal field by the random motions of the photospheric
footpoints. This is the basic idea of Parker’s nanoflare model and similar
theories (Parker, 1972, 1983, 1988; van Ballegooijen, 1986; Mikic et al.,
1989; Berger, 1991; Rappazzo et al., 2008). The photospheric motions are
postulated to tangle and braid the field lines, producing small scale current
sheets, which then release their energy via reconnection. A great deal of work
has been done applying this nanoflare scenario to coronal observations with
considerable success (Klimchuk, 2006).
The problem with such reconnection-heating models is that any helicity
injected into the corona as a result of the motions is expected to survive,
because reconnection in a high Lundquist-number system like the corona
conserves magnetic helicity (Taylor, 1974; Berger, 1984). Consequently, even
if it is injected on scales below present-day resolution, $<1$ arcsecond, the
helicity should build up and appear as twisting or tangling of the large-scale
coronal field. Note also that even if the degree of tangling required for the
heating is small – for example, Parker estimates a misalignment angle between
the reconnecting stressed field and the initial potential state of only
$20^{\circ}$ or so (Parker, 1983) – any net helicity injected by the stressing
should continue to accumulate and eventually produce large-scale observable
effects. We conclude, therefore, that both the basic observations of
photospheric motions and the reconnection theories for coronal heating imply
that the coronal field should have a complex geometry, in direct disagreement
with observations.
There are, at least, two seemingly likely explanations for this disagreement.
The first is that photospheric motions produce equal and opposite helicity
everywhere, so that no net helicity is injected into the corona. Nanoflare
reconnection then simply cancels out the positive and negative helicities.
This explanation, however, has both observational and theoretical
difficulties. Numerous observations imply that photospheric motions, including
flux emergence, do inject a net helicity into each hemisphere. For example,
observations of prominence structure (Martin et al., 1992; Rust, 1994; Zirker
et al., 1997; Pevtsov et al., 2003) and of active region vector fields
(Seehafer, 1990; Pevtsov et al., 1995) indicate a strongly preferred sign for
the helicity injected into each solar hemisphere, possibly related to the
differential rotation (DeVore, 2000). Furthermore, theory and numerical
simulations (Linton et al., 2001) have shown that, for magnetic flux tubes
with parallel axial fields, reconnection occurs only if the tubes have the
same sign of helicity, the so-called co-helicity case as discussed by Yamada
et al. (1990). Flux tubes with opposite helicity only bounce when they collide
(Linton et al., 2001); therefore, reconnection cannot cancel out positive and
negative injected helicity in interacting coronal loops. This point will be
clarified in Figure 2 below.
The other possible explanation for the lack of helicity buildup is that the
heating is due not to reconnection, but to true diffusion, in which case
helicity is not conserved. This hypothesis was proposed by Schrijver (2007) to
explain the TRACE images. He argued that continual reconnection induced by the
rapidly varying field of the magnetic carpet (Harvey, 1985; Schrijver et al.,
1997) causes the chromosphere and transition region to act like a high-
resistivity layer. Coronal loop field lines can slip along this layer and,
thereby, lose their tangles. This explanation, however, also has theoretical
and observational difficulties. Reconnection is not physically equivalent to
diffusion, no matter how frequent the reconnection. Diffusion does destroy
helicity and will relax a magnetic field back down to its minimum energy
potential state, but reconnection can relax the system down only to some state
compatible with total helicity conservation, such as a linear force-free state
(Taylor, 1974, 1986). In fact, for the line-tied corona, we have argued that
helicity imposes very stringent constraints on the possible end state of a
system undergoing reconnection relaxation (Antiochos et al., 2002).
Furthermore, observations imply that whenever helicity is, indeed, present in
the corona, it does not show evidence for diffusive decay. The largest
concentration of coronal helicity is in filaments and prominences, or more
precisely, in the strongly sheared field that defines a filament channel (e.g.
Tandberg-Hanssen, 1995; Mackay et al., 2010). Although there is still debate
over the exact topology of the filament channel magnetic field; in particular,
whether it is a sheared arcade (Antiochos et al,, 1994) or a twisted flux
rope, the models agree that for a physically realistic 3D topology all the
filament flux must be connected to the photosphere. Consequently, if the
chromosphere or transition region really did contain a high-resistivity layer,
the filament channel shear would simply disappear by field-line slippage. Such
slippage is never observed; if anything, filament shear seems to increase
continuously until it is ejected from the corona with a filament eruption/CME.
From the discussion above, we conclude that a net helicity is injected into
each coronal hemisphere by the photosphere, and that reconnection preserves
this helicity. But, in that case, where does the helicity go? In a sense, the
answer is obvious: The helicity injected into the closed-field corona must end
up as the magnetic shear in filament channels. These are the only locations in
the corona where the magnetic field is strongly non-potential and, hence, has
a strong helicity concentration.
Although this answer is intuitively appealing, it seems extremely unlikely. It
naturally raises the long-standing questions: What exactly are filament
channels and how do they form? Along with laminar coronal loops, filament
channels are also classic but puzzling features of the Sun’s atmosphere. These
structures consist of low-lying magnetic flux centered about photospheric
polarity inversion lines (PIL), in which the chromospheric and coronal
magnetic field lines run almost parallel to the inversion line rather than
perpendicular, as expected for a potential field (Rust, 1967; Leroy et al.,
1983; Martin, 1998). Direct measurements of the filament vector field both in
the photosphere (Kuckein et al., 2012) and corona (Casini et al., 2003) show
that the component parallel to the PIL is dominant. The channels have narrow
widths, of order 10 Mm, but their lengths can be greater than a solar diameter
for PILs that encircle the Sun. Filament channels are very common, invariably
appearing about any long-lived PIL, both in active regions and quiet Sun. It
should be emphasized that the channels are much more common than observable
filaments and prominences, which require the presence of substantial amounts
of cold plasma as well as the magnetic shear.
Two general mechanisms have been proposed for filament channel formation. One
mechanism is flux emergence, specifically the emergence of a sub-photospheric
twisted flux rope. Most simulations of flux rope emergence find that the
resulting structure in the corona is a sheared arcade localized near the PIL
(e.g. Mancester, 2001; Fan, 2001; Magara & Longcope, 2003; Archontis, 2004;
Leake et al., 2010; Fang et al., 2012). The basic process is straightforward;
the twist component of the sub-photospheric flux rope emerges to become the
overlying quasi-potential arcade in the corona, while the axial sub-surface
component emerges to become the shear field of the filament channel. It is
interesting to note that, in general, the resulting filament channel in the
corona is not a twisted flux rope, but a sheared arcade, because the concave-
up portion of the flux-rope field lines stays trapped below the surface even
in 3D (Fang et al., 2012).
Although flux emergence can yield a sheared arcade, there are major
theoretical and observational difficulties with this process as the general
mechanism for filament channel formation. First, the simulations have yet to
show that flux emergence agrees quantitatively with the amount of flux and
degree of shear measured in observed filament channels. In fact, Leake et al.
(2010) argue that only a small amount of axial flux emerges, at least in 2.5D
simulations, far too small to account for the magnetic free energy in observed
filament channels. Similar conclusions have been reached from recent 3D
simulations, as well (Fang et al., 2012). A much greater problem for the model
is that filament channels are frequently observed to form in regions where
there is no apparent flux emergence, such as at PILs between decaying regions
and high latitude PILs (e.g. Mackay et al., 2010). Moreover, the emergence of
a simple bipolar active region rarely produces a filament channel at its PIL.
The channel usually forms only well after the end of the flux emergence, when
the active region has decayed and dispersed to interact with surrounding flux
regions. Consequently, flux emergence cannot be the only mechanism for
filament channel formation.
The second, and perhaps, the most popular mechanism that has been proposed for
filament channel formation is flux cancellation (Martin, 1998). The basic
picture is that large-scale shear due to differential rotation or flux
emergence concentrates at PILs as opposite-polarity photospheric flux
converges and cancels there. Note that this mechanism inherently requires
reconnection at the photospheric PIL in order to form low-lying loops that can
sink and disappear and concave-up loops that can rise into the corona (van
Ballegooijen & Martens, 1989). A fundamental difficulty with such
reconnection, however, is that it produces a twisted flux rope in the corona
just like flare reconnection produces the highly twisted flux rope of a CME.
On the other hand, high resolution observations of filaments both from the
ground (Lin et al., 2005) and space (Vourlidas et al., 2010) show a field
geometry consisting of long, parallel strands, with no evidence of twist or
tangling. In fact, empirical models for filaments derived solely from
observations, have a laminar field geometry exactly like that of the TRACE
loops, except that the field lines are stretched out and primarily horizontal
rather than arched (Martin ref). It has been suggested that the large twist
component resulting from reconnection may diffuse away (van Ballegooijen,
2004), but as argued above, any diffusion would also decrease the shear
component, contrary to observations. Note also that the twist produced by
flare reconnection, which is physically identical to flux cancellation
reconnection, is never observed to diffuse away, but is measured to persist
out to 1 AU (Kumar & Rust, 1996; Qiu et al., 2007).
In addition to the lack of observed twist, the prevalence of filament channels
poses severe difficulties for the flux cancellation model and, indeed, for any
model. As stated above, filament channels are ubiquitous, appearing over all
types of PILs ranging from the most complex and strongest active regions to
very quiet high-latitude regions. In fact, it is not uncommon to observe a
filament channel that continues unbroken over a PIL that passes from an active
region into neighboring quiet region with the cold material, itself,
transitioning smoothly from a typical low-lying active region filament to a
high-lying quiet sun filament (e.g. Su & van Ballegooijen, 2012). Given these
observations, it seems improbable that filament channels are due to some
phenomenon in the plasma-dominated photosphere, because the dynamics there are
insensitive to the magnetic field structure and, in particular, to whether a
PIL is present or not. This is especially true in the weak-field regions where
quiescent prominences typically form (Klimchuk, 1987). It seems much more
likely that filament channel formation is due to some generic process
occurring in the magnetically-dominated corona and upper chromosphere.
We propose that the origin of filament channels is the reconnection-driven
evolution of helicity injected into the closed-field corona. This hypothesis
seems counterintuitive, because coronal-loop helicity is injected on small
scales more-or-less uniformly throughout the corona, whereas filament channels
are coherent structures, localized only around PILs and extending to very
large scale. In this paper, we describe a process, helicity condensation, that
performs exactly the required transformation of small-scale coronal loop
helicity into large-scale filament-channel shear. Helicity condensation keeps
coronal loops laminar while shearing filament channels.
Furthermore, helicity condensation may be responsible for much of the complex
structure and dynamics observed in the slow solar wind. We argued above that
the helicity injected into the corona must end up in filament channels, but in
regions containing coronal holes, another possibility is that some helicity is
ejected out into the wind by the opening of closed flux at the coronal hole
boundary. Such helicity transfer is implicitly present in the S-Web model for
the slow wind (Antiochos et al., 2011, 2012), which postulates that this wind
is due to continual dynamics of the open-closed flux boundary. If so, then
helicity condensation also will play a major role in the origin and properties
of the slow solar wind.
We describe below the basic process of helicity condensation and derive
estimates of its effectiveness in the Sun’s corona.
## 2 A Model for Helicity Injection and Transport
In order to understand how magnetic helicity is likely to evolve in the
corona, we must first consider the injection process. Assume, for the moment,
that the photospheric flux distribution consists of only two polarity regions
as shown in Figure 1: a negative northern hemisphere and a positive south, so
that all the flux closes across the equatorial PIL (dashed line). The yellow
arches in the figure denote two arbitrary small flux tubes corresponding to
coronal loops or to the strands inside an observable coronal loop. The quasi-
random photospheric motions will introduce small-scale structure and inject
helicity to this coronal field. Helicity will also be injected by large-scale
motions, such as differential rotation, and by flux emergence/cancellation,
such as the magnetic carpet, but for simplicity let us model the injection as
due to the continual small-scale photospheric motions, in particular, the
granular or supergranular flows. Note that if the magnetic carpet dynamics do
not change the net coronal flux, their effect on the coronal helicity can be
captured by effective photospheric motions. Furthermore, recent analysis of
high-resolution Solar Dynamics Observatory (SDO) data indicates that the bulk
of the helicity injected into active regions is due to photospheric motions
rather than flux emergence (Liu & Schuck, 2012).
Following the arguments of Sturrock & Uchida (1981), the energy and,
certainly, the helicity injected into coronal loops by stochastic horizontal
flows at the photosphere will be primarily in the form of twist. Therefore, we
model the motions as a set of randomly located and randomly occurring
rotations that have fixed spatial and temporal scales. The true photospheric
motions are more complex than a set of fixed-scale rotations, but we are
interested only in that part of the flow that injects helicity to the corona.
Note also that there is some evidence for exactly the pattern of photospheric
rotations of Fig. 1 in measurements of the vorticity of the supergranulation
(Duvall & Gizon, 2000; Gizon & Duvall, 2003; Komm et al., 2007).
It has been known since the time of Hale (1927) that sunspot whirls have a
clear hemispheric preference, counterclockwise in the north and clockwise in
the south (Pevtsov et al., 1995), indicating a preferred sense for the
helicity of the subsurface solar motions. The same helicity preference,
negative in the north and positive in the south, has been well documented to
occur in all types of coronal magnetic structures ranging from quiet Sun field
to active region complexes (Pevtsov & Balasubramaniam, 2003) and has been
observed out in the heliospheric magnetic field (Bieber et al., 1987). As
shown in Figure 1, this hemispheric “rule” is in the same sense as would be
expected from the surface differential rotation, but the actual mechanism is
still not clear. In any case, we expect there to be a preferred sense to the
helicity injecting rotations as illustrated in Figure 1. Note that this is
only a preference; a fraction of the rotation in each hemisphere could well
have the “unpreferred” sense.
The motions shown in Figure 1 have a number of interesting implications for
the coronal field. Assuming, for simplicity, that the rotations are solid
body, have size $d$, and have magnitude, $\Theta$, then each rotation of a
photospheric flux tube with axial flux,
$\Phi_{d}=\pi d^{2}B_{p}/4,$ (1)
produces in the corona a twist flux,
$\Phi_{t}=\Theta\Phi_{d}/\pi,$ (2)
where $B_{p}$ is the average normal field at the photosphere. In open field
regions (not shown in the Figure), this twist flux simply propagates outward,
resulting in a net helicity to the turbulence in the fast wind (e.g. Leamon et
al, 1998). We will discuss the implications for the slow wind below. In the
closed field regions, however, the coronal loops acquire a twist component to
their magnetic field, as shown in the Figure. If the loop is perfectly
symmetric about the equator, then on average, the twists imposed by the two
footpoints cancel out so that no net helicity is injected. Basically, the loop
is twisted at one end, but untwisted at the other. On the other hand, if the
loop has both footpoints in one hemisphere, as is usually the case when the
PIL is not exactly at the equator, then the twist from each footpoint will
add. Even if the loop is transequatorial, we do not expect any symmetry for a
real coronal loop, so a net twist will still be produced by the footpoint
motions. Note also, that the effect of any unpreferred-sense rotations
(clockwise in the north and counterclockwise in the south) is only to decrease
the rate of twisting. To first order, the unpreferred rotations simply untwist
the loops, but it should be emphasized that since the rotations are time
varying, they can create higher order topological structure in the field even
if the net injected helicity vanishes. All higher order topological features,
however, such as the braiding of three flux tubes, are not conserved by
reconnection (e.g. Pontin et al., 2011) and are not expected to build up in
the corona.
Consider now the interaction of the two twisted flux tubes of Fig. 1 due to
some random motion that causes them to collide. In fact, the twist itself will
cause the flux tubes to expand and interact. Since their main axial fields are
parallel, only the twist components of the flux tubes can reconnect. If the
tubes have the same sense of twist (helicity), then at the contact point
between the tubes, their twist components will be oppositely directed and,
hence, will reconnect. This is illustrated in the three sequences of Fig. 2,
where the red and blue circles correspond to field lines of the twist magnetic
component. In the top sequence the twist components are in the same sense, so
they are oppositely directed at their contact point; consequently, they will
reconnect there. Another way of understanding this result is to note that for
interacting tubes with the same helicity, the photospheric rotations have a
stagnation point between them. It is well known that such stagnation point
flows lead to exponentially growing separation of magnetic footpoints and,
hence, to exponentially growing currents in the corona (e.g. Antiochos &
Dahlburg, 1997), which can drive efficient reconnection. The effect of this
reconnection is to spread the twist component over the flux of the two tubes,
in other words, the two tubes merge into one globally twisted tube as
illustrated in the Figure and as found in simulations of flux-tube collisions
(Linton et al., 2001).
On the other hand, if the tubes have opposite twist (helicity), then at their
point of interaction the twist components are parallel. As illustrated in the
second sequence of Figure 2, there is no reconnection in this case. The tubes
simply “bounce” (Linton et al., 2001). This result emphasizes the point made
in Antiochos & Dahlburg (1997) that reconnection in the solar corona is highly
constrained by line-tying. As a result, the coronal magnetic field cannot
simply relax to a minimum energy Taylor (1974) state, which for the opposite
twist case corresponds to the initial potential field.
The third sequence in Figure 2 illustrates the effect of continued interaction
of same-helicity flux tubes. If a larger-scale merged flux tube reconnects
with another tube, the result is further merging of the flux and the spread of
the twist to even larger scale. This reflects the well-known result from
turbulence studies that magnetic helicity tends to cascade upward in scale
(e.g. Biskamp, 1993). A key point is that the scale referred to in our cascade
process is the amount of axial flux, which is closely related to, but not
identical to the spatial scale. As argued directly below, the helicity
cascades up to the largest possible flux scale, which corresponds to all the
axial flux inside a single polarity region.
Let us consider the end result of this reconnection-driven helicity cascade.
Assume a flux system, as in Figure 3, with simple topology given by a PIL
(heavy dashed curve in the Figure) and a separatrix curve somewhere on the
photosphere (light dashed curve) that defines all the flux that closes across
the PIL. There must be additional PILs on the photosphere, but these are not
shown. If the separatrix curve lies in the north, then the flux system is
entirely in the north and the twist injected by the photosphere will be
predominately counter-clockwise. Since only the relative footpoint motions are
significant, we can assume without loss of generality that all the twist is
imparted inside the PIL as shown in the Figure.
The expected evolution of this twist is seen in Figure 4, which shows a top
view of the flux system. As a result of reconnection, the helicity “condenses”
onto the largest scale in the flux system, the PIL, since this encompasses all
the flux in the system. The PIL defines the boundary of the polarity region.
We conclude, therefore, that the net effect of the many small-scale
photospheric twists and the coronal reconnection is to impart a coherent,
global twist of the whole flux system that concentrates at the PIL. This
global twist is not a true physical motion; the large-scale flux system does
not actually rotate as a coherent body, but the photospheric helicity
injection and subsequent transport by reconnection does result in an effective
global rotation of the magnetic field.
The key point is that such a rotation of the whole flux system corresponds to
a coherent localized shear all along the PIL, exactly what is needed to
explain the formation of filament channels. Such an effective motion produces
a channel consisting of field lines that are sheared but smooth and laminar,
with no twist or tangles in agreement with high-resolution observations of
prominence threads (Lin et al., 2005; Vourlidas et al., 2010). The coronal
reconnection in our helicity condensation model results in a structure that is
the direct opposite to that of flux cancellation reconnection, which
invariably produces a highly twisted flux rope at the PIL.
Another important point is that the helicity condensation mechanism is
unaffected by the shape of the PIL, in particular whether the PIL contains so-
called switchbacks where it forms a sharp zigzag. As long as the flux system
defined by the PIL is primarily in one hemisphere, helicity condensation will
form a filament channel with the same chirality all along that PIL and with
roughly the same amount of shear. This result is in contrast to the
predictions of some of the flux cancellation models (van Ballegooijen et al.,
1998), but is in good agreement with observations (Pevtsov et al., 2003).
Furthermore, since the photospheric convection in either quiet or active
regions is not observed to change significantly with phase of the solar cycle,
the model predicts that the hemispheric helicity rule should hold independent
of solar cycle. Again, this conclusion appears to be in good agreement with
observations (Pevtsov et al., 2003), unlike some flux cancellation models
(Mackay & van Ballegooijen, 2001).
### 2.1 Rate of Helicity Cascade
Prominences typically form on time scales of several days, which sets a
constraint that any model must satisfy; therefore, we calculate below the rate
of filament channel formation predicted by the helicity condensation model.
Let us consider a polarity region as in Figure 4 with scale $L$ that is large
compared to the helicity injection scale $d$. The rate of helicity injection
$\eta$ into a flux tube $\Phi_{d}$ by a photospheric twist of average angular
velocity $V_{d}/d$ is given by the product of the rate of change of the twist
flux of Eq. (2) and the axial flux $\Phi_{d}$, (which stays constant):
$\eta\approx\Phi_{d}(V_{d}/d)\Phi_{d}$ (3)
(e.g. Berger, 2000). The total helicity injection rate at the scale $d$ over
the whole flux system region with scale $L$ is therefore given by:
$h_{d}=\Phi_{d}^{2}(V_{d}/d)(L/d)^{2}=(\Phi_{L}/L)^{2}d^{2}\,V_{d}/d,$ (4)
where $\Phi_{L}$ is the flux of the whole system. Following Kolmogorov’s
classic theory for hydrodynamic turbulence (Kolmogorov, 1941), we assume a
constant helicity transfer rate at any scale $\lambda$. Therefore:
$h_{\lambda}=(\Phi_{L}/L)^{2}\lambda^{2}V_{\lambda}/\lambda=h_{d},$ (5)
which implies that:
$V_{\lambda}=V_{d}(d/\lambda)\ {\rm and,thus},\ V_{L}=V_{d}(d/L).$ (6)
It should be noted that unlike $V_{d}$, which is the actual velocity of the
photospheric flows, the quantity $V_{L}$ does not represent a true plasma
velocity. It is only an effective velocity for the transfer of twist to the
largest scale by coronal reconnection. The physical plasma velocities will
likely be dominated by reconnection jets and will have both larger magnitude
and smaller scale than $V_{L}$. The physical velocities and kinetic energy are
expected to cascade downward, not upward, in scale. Consequently, the velocity
spectrum $V_{\lambda}$ derived in Eq. 6 may not be directly observable, but
the effective velocity $V_{L}$ is indeed physically significant. It quantifies
the rate at which magnetic helicity “condenses” out of the corona at the
largest scale of the flux system and, hence, $V_{L}$ corresponds to the
effective shear velocity along the PIL.
Note also, that the helicity cascade process derived above is somewhat
different than the usual hydrodynamic turbulence in which velocity is injected
statistically uniformly at some scale and then cascades down to where it is
dissipated, usually at a kinetic scale. In such an energy cascade, the
injection of kinetic energy at a global scale results in the slow increase of
thermal energy (temperature) approximately uniformly throughout the system. In
our cascade, however, helicity is injected statistically uniformly at some
intermediate scale and then simply piles up at the largest global scale. Even
though the helicity injection (i.e., photospheric motions) are uniform, the
cascade produces a localized spatial structure in the corona. Helicity
condensation in the Sun’s atmosphere is a striking example of self-
organization in a complex system.
The time scale for filament channel formation can now be calculated directly
from Eq. (6). We note that for the photospheric helicity injection the
important parameter is the product of the velocity and the coherence scale of
that velocity. Granules typically have $V_{d}\sim 1$ km/s and $d\sim 700$ km,
while supergranules have: $V_{d}\sim.25$ km/sec and $d\sim 30,000$ km. Eq. (6)
implies that the product $V_{d}\,d$ is the important quantity; therefore,
supergranules are expected to dominate the helicity injection. Taking the
whole flux system to have scale $d/L\sim 10-100$ implies that a shear of order
1,000 – 10,000 km will build up in $\sim 10^{5}$ s, where we assume that equal
helicity is injected at both ends of a flux tube. These results indicate that
a high-latitude filament channel with typical shear scales of 100,000 km will
form in several days or so, which is consistent with observations (Tandberg-
Hanssen, 1995; Mackay et al., 2010). Note also that we expect that the width
of the helicity condensation region to be of order the width of the elemental
photospheric rotation, $\sim 15,000$ km for supergranules, which again is
consistent with the observed widths of filaments (Tandberg-Hanssen, 1995;
Mackay et al., 2010).
The supergranular rate of helicity injection estimated above is also
consistent with estimates of solar helicity loss to the wind. Taking the
system size $L$ to be of order the solar radius $\sim 10^{11}$ cm, and the
average field strength at the photosphere to be $~{}\sim 10$ G, we derive from
Eq. (4) and the numbers above, a helicity injection rate over the solar
surface of $\sim 10^{38}$ Mx/s. Over the course of a full solar cycle, this
yields a total helicity loss of $\sim 3\times 10^{46}$ Mx, which agrees well
with the inferred losses from observations of CMEs and the wind (DeVore,
2000).
### 2.2 Implications of the Model
We conclude from the derivation above that helicity condensation can account
for filament channel formation, at least, in regions that do not exhibit
strong flux emergence. The mechanism can also account for the observed
smoothness of coronal loops. Let $\tau$ be the time scale required for the
system to establish a steady state (except, of course, at the largest scale
$L$ where no steady-state is possible). We expect that $\tau$ is determined by
the twist required to produce intense current sheets, of order a full rotation
or so (Antiochos, 1998), and not by the rate of reconnection. The driving
velocity is only $\sim$ 0.1% of the coronal Alfven speed, so that the
reconnection need not be fast in order to keep pace with the driving. For a
given $\tau$, the twist angle produced by the effective velocities of Eq. (6)
scales as:
$\Theta_{\lambda}=\tau V_{\lambda}/\lambda\sim\lambda^{-2}$ (7)
This result implies that the corona will exhibit the most structure at the
scale at which the twist is injected (presumably the supergranular scale), and
at the largest scale where the helicity piles up, the whole length of the PIL.
The so-called coronal cells recently discovered by Sheeley & Warren (2012)
appear to be evidence for just this type of structure separation. These
authors observe that the large-scale corona breaks up into three distinct
structures: flux tubes twisted on a scale of 30,000 km or so, long filment
channels along PILs that typically span the whole Sun, and coronal holes. Our
helicity condensation model is in excellent agreement with these observations.
An important issue that is raised by the observations and that we have yet to
discuss is the effect on the model of a coronal hole or, more generally, of an
open field region. Note, also, that even if no coronal hole is present, the
simple picture of Figs. 3 and 4 is topologically incomplete. The PIL cannot be
the only boundary that defines the closed negative polarity region. At the
very least, there must be a point somewhere in the region where a magnetic
spine line connects up to a null point Lau & Finn (1990); Antiochos (1990);
Priest & Titov (1996); thereby, making this field line effectively open. Of
course, real solar polarity regions tend to have much more complexity often
containing intricate open field areas and corridors (Antiochos et al., 2011).
Assume that the polarity region of Figs. 3 contains a coronal hole, as
illustrated in Figure 5. The presence of the coronal hole introduces
subtleties to the calculation of helicity evolution, because the helicity of a
truly open field that extends to infinity is not physically meaningful. An
open field can have arbitrary helicity due to linkages at infinity where the
field vanishes, but the topology does not. Therefore, let us consider instead
a system where all the field lines remain closed so the helicity is well-
defined throughout the evolution, and let us model the coronal hole as a
region where no photospheric twists are imposed; hence, no helicity is
injected into this region. In an actual coronal hole twist is injected by
photospheric motions, exactly as in closed field regions, but the twist
propagates away at the Alfven speed and presumably has no effect on the
subsequent evolution in the low corona. Consequently, we can simply model this
region as being untwisted.
The analysis of the helicity cascade in the large annular flux region bounded
by the PIL and the coronal hole boundary proceeds exactly as above. The only
difference is that the total twisted area $L^{2}$ is replaced by $L^{2}-H^{2}$
where $H$ is the scale of the coronal hole. Therefore, the largest scale for
the helicity is given by:
$L^{\prime}=\sqrt{L^{2}-H^{2}}$ (8)
and the effective velocity for helicity condensation is given by Eq (6) above
except that $L$ is replaced by $L^{\prime}$. Also, the twist spectrum, Eq (7)
is unchanged. As long as $H<<L$, the presence of the coronal hole has minimal
effect on the filament channel formation process, and on the smoothness of
coronal loops.
It is evident from Fig. 5, however, that helicity condensation does have an
effect on the magnetic field near the coronal hole boundary. We note that
twist, or more accurately magnetic shear, also condenses at this boundary, but
curiously enough the shear has the sense opposite to that at the PIL. This
result may seem physically unlikely, but in fact it is mandated by helicity
conservation. The key point is that the shear flux $\Phi_{1}$ that condenses
onto the PIL encircles all the photospheric flux in the polarity region,
including that in the coronal hole region (“CH”). Therefore, the helicity
$H_{1}$ due to this shear flux is given by:
$H_{1}=\Phi_{1}\Phi_{L^{\prime}}+\Phi_{1}\Phi_{CH},$ (9)
where $\Phi_{L^{\prime}}$ is the amount of closed photospheric flux inside the
PIL and $\Phi_{CH}$ is the amount of photospheric flux in the coronal hole,
which can be arbitrary compared to $\Phi_{L^{\prime}}$. But the CH flux is not
twisted and never contributes to the helicity injection; hence, it should not
affect the helicity condensation at the PIL. The only way to ensure that the
CH flux has no effect is to have a shear flux $\Phi_{2}$ that condense at the
CH boundary and is exactly equal and oppositely directed to that at the PIL.
Such a shear flux encircles only the CH flux and, thereby, adds a helicity
contribution:
$H_{2}=-\Phi_{1}\Phi_{CH},$ (10)
which exactly cancels out that between the PIL shear flux and the CH. Fig. 5
shows that reconnection would produce just this required shear flux at the CH
boundary. In other words, helicity condensation predicts that a filament
channel should form at coronal hole boundaries, at the same rate as at the PIL
but with the opposite handedness.
These results clearly have major implications for observations. At PILs, the
magnetic shear builds up until eventually, it is ejected as a prominence
eruption/CME. We expect that a similar process of buildup and ejection occurs
at the CH boundaries, but with much less explosive dynamics. The closed field
lines near the CH boundary consist of very long, high-lying loops that form
the outer shell of the streamer belt. Consequently, it requires far less shear
and free energy to open up these loops than to eject the filament channel. We
expect that helicity condensation at CH boundaries results in continual small
bursts of flux opening and closing there, as is required by models for the
slow wind (Antiochos et al., 2011, 2012). An interesting prediction is that
the helicity of the closed flux opening at the CH boundary should be opposite
to that of the photospheric injection into the coronal hole open field lines,
i.e., into the fast wind. It may be possible to test this prediction with in
situ measurements.
In summary, we argue that a single deeply-profound process, helicity
condensation, can explain three long-standing observational challenges in
solar/heliospheric physics: the formation of filament channels, the smoothness
of coronal loops, and the origin of the slow wind. Furthermore, the model
implies major new predictions for solar structure and dynamics. We look
forward to many more theoretical and observational studies of helicity
condensation in the Sun’s corona and wind.
This work has been supported, in part, by the NASA TR&T Program. The work has
benefited greatly from the authors’ participation in the NASA TR&T focused
science teams on multiscale coupling and the slow solar wind. The author
thanks C. R. DeVore, J. T. Karpen, and J. A. Klimchuk for invaluable
scientific discussions and J. T. Karpen for help with the graphics.
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Figure 1: Model for helicity injection into the corona by photospheric
motions. The primary effect of the motions is to inject an effective twist of
scale d. The sense of the twist in each hemisphere is determined by the
differential rotation, large arrows. The yellow arches represent two
neighboring coronal loops (magnetic flux tubes).
Figure 2: Interaction of the twist component of interacting flux tube. Red
and blue circles correspond to oppositely-oriented twist components of the
magnetic field.
Figure 3: Model of negative polarity flux region fully in the northern
hemisphere. The dark dashed line correspond to the PIL of the flux region.
Figure 4: The polarity region of the previous figure as viewed from the north
pole.
Figure 5: The polarity region of the Figs 3 and 4, but now containing a
coronal hole region, indicated by “CH”.
|
arxiv-papers
| 2012-11-17T15:44:19 |
2024-09-04T02:49:38.147510
|
{
"license": "Public Domain",
"authors": "S. K. Antiochos",
"submitter": "Spiro K. Antiochos",
"url": "https://arxiv.org/abs/1211.4132"
}
|
1211.4204
|
# Upper bounds, cofiniteness, and artinianness of local cohomology modules
defined by a pair of ideals
M. Aghapournahr1 1 Department of Mathematics, Faculty of Science, Arak
University, Arak, 38156-8-8349, Iran. [email protected] , Kh. Ahmadi-
amoli2 2 Department of Mathematics, Payame Noor University, Tehran,
19395-3697, Iran. [email protected] and M. Y. Sadeghi3 3 Department of
Mathematics, Payame Noor University, Tehran, 19395-3697, Iran.
[email protected]
###### Abstract.
Let $R$ be a commutative noetherian ring, $I,J$ be two ideals of $R$, $M$ be
an $R$-module, and $\mathcal{S}$ be a Serre class of $R$-modules. A positive
answer to the Huneke,s conjecture is given for a noetherian ring $R$ and
minimax $R$-module $M$ of krull dimension less than 3, with respect to
$\mathcal{S}$. There are some results on cofiniteness and artinianness of
local cohomology modules with respect to a pair of ideals. For a ZD-module $M$
of finite krull dimension and an integer $n\in\mathbb{N}$, if
$\operatorname{H}^{i}_{I,J}(M)\in\mathcal{S}$ for all $i>n$, then
$\operatorname{H}^{i}_{I,J}(M)/\mathfrak{a}^{j}\operatorname{H}^{i}_{I,J}(M)\in\mathcal{S}$
for any $\mathfrak{a}\in\tilde{W}(I,J)$, all $i\geq n$, and all $j\geq 0$. By
introducing the concept of Seree cohomological dimension of $M$ with respect
to $(I,J)$, for an integer $r\in\mathbb{N}_{0}$,
$\operatorname{H}^{j}_{I,J}(R)\in\mathcal{S}$ for all $j>r$ iff
$\operatorname{H}^{j}_{I,J}(M)\in\mathcal{S}$ for all $j>r$ and any finite
$R$-module $M$.
###### Key words and phrases:
local cohomology modules defined by a pair of ideals, local cohomology, Serre
subcategory, associated primes, cofinite modules, ZD-modules, minimax modules,
Bass numbers.
###### 2010 Mathematics Subject Classification:
13D45, 13E05, 14B15.
## 1\. Introduction
As a generalization of the notion of local cohomology modules, R. Takahashi,
Y. Yoshino, and T. Yoshizawa [36], introduced local cohomology modules with
respect to a pair of ideals. This paper is concerned about this new notion of
local cohomology and Serre subcategories. For notations and terminologies not
given in this paper, if necessary, the reader is referred to [36] and [1].
Throughout this paper, $R$ is denoted a commutative noetherian ring with non-
zero identity, $I$ , $J$ are denoted two ideals of $R$, and $M$ is denoted an
arbitrary $R$-module. The $(I,J)$-torsion submodule $\Gamma_{I,J}(M)$ of $M$
is a submodule of $M$ consists of all elements $x$ of $M$ with
Supp$(Rx)\subseteq W(I,J)$, in which
$W(I,J)=\\{~{}\mathfrak{p}\in\textmd{Spec}(R)\mid
I^{n}\subseteq\mathfrak{p}+J\textmd{\ \ for an integer}\ n\geq 1\\}.$
For an integer $n$, the $n$-th local cohomology functor
$\operatorname{H}^{n}_{I,J}$ with respect to $(I,J)$ is the $n$-th right
derived functor of $\Gamma_{I,J}$. The $R$-module
$\operatorname{H}^{n}_{I,J}(M)$ is called the $n$-th local cohomology module
of $M$ with respect to $(I,J)$. In the case $J=0$,
$\operatorname{H}^{n}_{I,J}$ coincides with the ordinary functor
$\operatorname{H}^{n}_{I}$. Also, we are concerned with the following set of
ideals of $R$:
$\tilde{W}(I,J)=\\{~{}\mathfrak{a}\trianglelefteq R\mid
I^{n}\subseteq\mathfrak{a}+J\textmd{\ \ for an integer}\ n\geq 0\\}.$
A class of $R$-modules is called a Serre subcategory (or Serre class) of the
category of $R$-modules when it is closed under taking submodules, quotients
and extensions. Always, $\mathcal{S}$ stands for a Serre class.
According to the third Huneke,s problem on local cohomology [23, Conjecture
4.3], one of the main problem in commutative algebra is finiteness of the
socle of local cohomology modules on a local ring. Solving this problem, gives
an answer to the finiteness of the set of associated primes of local
cohomology modules.
On this area, some remarkable attempts have been done, e.g. see [24], [28],
[29], [31], and [7].
In section 2, we give a positive answer to the Huneke,s conjecture more
general for an arbitrary Serre subcategory $\mathcal{S}$, instead of the
category of finitely generated modules. Let $R$ be a noetherian (not necessary
local) ring. Let $R/\mathfrak{m}\in\mathcal{S}$ for all
$\mathfrak{m}\in\textmd{Max}(R)$. For any minimax $R$-module $M$ of krull
dimension less than 3, we show that
$\operatorname{Ext}^{j}_{R}\big{(}R/\mathfrak{m},\operatorname{H}^{i}_{I}(M)\big{)}\in\mathcal{S}$
for any $\mathfrak{m}\in\textmd{Max}(R)\cap\operatorname{V}(I)$ and all
$i,j\geq 0$. In particular
$\operatorname{Hom}_{R}\big{(}R/\mathfrak{m},\operatorname{H}^{i}_{I}(M)\big{)}\in\mathcal{S}$
for any $\mathfrak{m}\in\textmd{Max}(R)\cap\operatorname{V}(I)$ and all $i\geq
0$ (see Theorem 2.14).
We get the same result for local cohomology modules with respect to a pair of
ideals, but in local case (see Theorem 2.18).
In section 3, we obtain some results on cofiniteness and artinianness of local
cohomology with respect to a pair of ideals.
M. Aghapournahr and L. Melkersson in [1], obtained some conditions in which
the ordinary local cohomology $\operatorname{H}^{i}_{I}(M)$ belongs to
$\mathcal{S}$ for all $i<n$ (from below). In section 4, as a complement of
this work, we study some conditions in which the local cohomology
$\operatorname{H}^{i}_{I,J}(M)$ belongs to $\mathcal{S}$ for all $i>n$ (from
top). For a ZD-module $M$ of finite krull dimension, we show that if the
integer $n\in\mathbb{N}$ is such that
$\operatorname{H}^{i}_{I,J}(M)\in\mathcal{S}$ for all $i>n$, then the modules
$\operatorname{H}^{i}_{I,J}(M)/\mathfrak{a}^{j}\operatorname{H}^{i}_{I,J}(M)\in\mathcal{S}$
for any $\mathfrak{a}\in\tilde{W}(I,J)$, all $i\geq n$, and all $j\geq 0$ (see
Theorem 4.4). Replacing $\mathcal{S}$ with some familiar Serre subcategories
such as zero modules, finite modules, and artinian modules (resp.), we show
that a necessary and sufficient condition for $\operatorname{H}^{n}_{I,J}(M)$
to be zero, inite, and artinian (resp.), is the existence of an integer
$m\in\mathbb{N}_{0}$ such that $I^{m}\operatorname{H}^{n}_{I,J}(M)$ is zero,
finite, and artinian (resp.) (see Corollary 4.6).
More generally, for a finite $R$-module $M$ if $n\in\mathbb{N}_{0}$ is such
that $\operatorname{H}^{i}_{I,J}(M)$ belongs to $\mathcal{S}$ for all $i>n$
and $\mathfrak{b}$ is an ideal of $R$ such that
$\operatorname{H}^{n}_{I,J}(M/\mathfrak{b}M)$ belongs to $\mathcal{S}$, then
$\operatorname{H}^{n}_{I,J}(M)/{\mathfrak{b}}\operatorname{H}^{n}_{I,J}(M)$
belongs to $\mathcal{S}$ (see Theorem 4.7). As a consequence of this theorem,
we obtain a similar result as Corollary 4.6, for the finite module $M$ (see
Corollary 4.10). Other consequence of this theorem is concerned about
finiteness of $\operatorname{H}^{n}_{I,J}(M)$ where $n=\textmd{cd}(I,J,M)$ or
$n=\textmd{dim}M\geq 1$ (see Corollary 4.11).
Finally, these results motivate us to introduce the concept of Serre
cohomological dimension of $M$ with respect to $(I,J)$,
$\textmd{cd}_{\mathcal{S}}(I,J,M)$, as the supremum of non-negative integers
$i$ such that $\operatorname{H}^{i}_{I,J}(M)\not\in\mathcal{S}$. We give some
characterizations to $\textmd{cd}_{\mathcal{S}}(I,J,M)$ (Corollary 4.20) and
as a main result of its properties, we show that for an integer
$r\in\mathbb{N}_{0}$, $\operatorname{H}^{j}_{I,J}(R)\in\mathcal{S}$ for all
$j>r$ iff $\operatorname{H}^{j}_{I,J}(M)\in\mathcal{S}$ for all $j>r$ and any
finite $R$-module $M$ (see Corolary 4.23)
## 2\. The membership of the
$\textmd{Hom}_{R}\big{(}-,\operatorname{H}^{t}_{I,J}(M)\big{)}$ in Serre
classes
In this section, Proposition 2.5 plays a main role to obtain our results. For
this purpose we need the following Lemmas.
###### Lemma 2.1.
For a Serre class $\mathcal{S}$, we have $\mathcal{S}\neq{0}$ if and only if
$R/\mathfrak{m}\in\mathcal{S}$ for some
$\mathfrak{m}\in\operatorname{Max}(R)$.
###### Proof.
$(\Rightarrow)$ Let $L\in\mathcal{S}$ and $0\neq x\in L$. Then
$(0:_{R}x)\subseteq\mathfrak{m}$ for some $\mathfrak{m}\in$ Max$(R)$. Now,
since $Rx\in\mathcal{S}$, the assertion follows from the natural epimorphism
$Rx\cong R/(0:_{R}x)\rightarrow R/\mathfrak{m}$.
$(\Leftarrow)$ It is obvious. ∎
###### Lemma 2.2.
Let $\mathcal{FL}$ be the class of finite length $R$-modules. Then
$\mathcal{FL}\subseteq\mathcal{S}$ if and only if
$R/\mathfrak{m}\in\mathcal{S}$ for all $\mathfrak{m}\in\operatorname{Max}(R)$.
###### Proof.
$(\Rightarrow)$ It is obvious.
$(\Leftarrow)$ Let $N\in\mathcal{FL}$ and $\ell:=\ell_{R}(N)$. So, consider
the following chain of $R$-submodules of $N$:
$0=N_{0}\subseteq N_{1}\subseteq\cdots\subseteq N_{\ell}=N$
in which, for all $1\leq j\leq\ell$, $N_{j}/N_{j-1}\cong R/\mathfrak{m}$ for
some $\mathfrak{m}\in$ Max$(R)$. Now, the assertion can be followed by
induction on $\ell$. ∎
###### Corollary 2.3.
Let $(R,\mathfrak{m})$ be a local ring and $\mathcal{S}\neq 0$. Then
$\mathcal{FL}\subseteq\mathcal{S}$.
###### Example 2.4.
_In general case, it not true that $R/\mathfrak{m}\in\mathcal{S}$ for all
$\mathfrak{m}\in\operatorname{Max}(R)$. To see this, let $R$ be a non-local
ring and $\mathfrak{m}\in$ Max$(R)$. Let $I$ be an ideal of $R$ such that
$I\nsubseteq\mathfrak{m}$. Let $\mathcal{S}$ be the class of $I$-cofinite
minimax $R$-modules, (see [34, Corollary 4.4]). Then
Supp$(R/\mathfrak{m})\nsubseteq V(I)$ and so
$R/\mathfrak{m}\not\in\mathcal{S}$. For example, let $R:=\mathbb{Z}[x]$,
$\mathfrak{m}:=(x-1)R$, and $I:=xR$._
###### Proposition 2.5.
For a noetherian ring $R$, we have
1. (i)
If $R/\mathfrak{m}\in\mathcal{S}$, for any $\mathfrak{m}\in$ Max$(R)$, and $M$
is a finite or an artinian $R$-module, then
$\operatorname{Ext}^{j}_{R}(R/\mathfrak{m},M)\in\mathcal{S}$ for any
$\mathfrak{m}\in$ Max$(R)$ and all $j\geq 0$.
2. (ii)
If $R/\mathfrak{m}\in\mathcal{S}$, for any $\mathfrak{m}\in$ Max$(R)$, and $M$
is a minimax $R$-module, then
$\operatorname{Ext}^{j}_{R}(R/\mathfrak{m},M)\in\mathcal{S}$ for any
$\mathfrak{m}\in$ Max$(R)$ and all $j\geq 0$.
3. (iii)
If $(R,\mathfrak{m})$ be a local ring, $\mathcal{S}\neq 0$, and $M$ be a
minimax $R$-module, then
$\operatorname{Ext}^{j}_{R}(R/\mathfrak{m},M)\in\mathcal{S}$ for all $j\geq
0$.
###### Proof.
(i) Let $\mathfrak{m}\in$ Max$(R)$ and $j\geq 0$. Since
$\operatorname{Ext}^{j}_{R}(R/\mathfrak{m},M)$ is finite and is annihilated by
$\mathfrak{m}$, hence $\operatorname{Ext}^{j}_{R}(R/\mathfrak{m},M)$ has
finite length. Now, the result follows from Lemma 2.2.
(ii) Since $M$ is minimax, there exists a short exact sequence
$0\rightarrow N\rightarrow M\rightarrow A\rightarrow 0,$
where $N$ is a finite module and $A$ is an artinian module. This induces the
exact sequence
$\cdots\rightarrow\operatorname{Ext}^{j}_{R}(R/\mathfrak{m},N)\rightarrow\operatorname{Ext}^{j}_{R}(R/\mathfrak{m},M)\rightarrow\operatorname{Ext}^{j}_{R}(R/\mathfrak{m},A)\rightarrow\cdots.$
Now, the assertion follows from part (i).
(iii) Apply Lemma 2.1 and part (ii). ∎
###### Corollary 2.6.
Let $R/\mathfrak{m}\in\mathcal{S}$ for all $\mathfrak{m}\in$ Max$(R)$. Let
$t\in\mathbb{N}_{0}$ be such that $\operatorname{H}^{t}_{I,J}(M)$ is a minimax
$R$-module. Then
$\operatorname{Ext}^{j}_{R}\big{(}R/\mathfrak{m},\operatorname{H}^{t}_{I,J}(M)\big{)}\in\mathcal{S}$
for all $i\geq 0$.
###### Lemma 2.7.
Let $\mathfrak{a}\in\tilde{W}(I,J)$. Let $X$ be an $R$-module. Then
$\big{(}0:_{X}\mathfrak{a}\big{)}=\big{(}0:_{\Gamma_{\mathfrak{a}}(X)}\mathfrak{a}\big{)}=\big{(}0:_{\Gamma_{\mathfrak{a},J}(X)}\mathfrak{a}\big{)}=\big{(}0:_{\Gamma_{I,J}(X)}\mathfrak{a}\big{)},$
in particular, for any ideal $\mathfrak{b}$ of $R$ with
$\mathfrak{b}\supseteq\ I$, we have
1. (i)
$\big{(}0:_{X}\mathfrak{b}\big{)}=\big{(}0:_{\Gamma_{\mathfrak{b}}(X)}\mathfrak{b}\big{)}=\big{(}0:_{\Gamma_{\mathfrak{b},J}(X)}\mathfrak{b}\big{)}=\big{(}0:_{\Gamma_{I,J}(X)}\mathfrak{b}\big{)}=\big{(}0:_{\Gamma_{I}(X)}\mathfrak{b}\big{)}$.
2. (ii)
$\big{(}0:_{X}\mathfrak{b}\big{)}=\big{(}0:_{\Gamma_{\mathfrak{b},J}(X)}\mathfrak{b}\big{)}\subseteq\big{(}0:_{\Gamma_{\mathfrak{b},J}(X)}I\big{)}\subseteq\big{(}0:_{\Gamma_{I,J}(X)}I\big{)}=\big{(}0:_{X}I\big{)}$.
###### Proof.
All proofs are easy and we leave them to the reader. ∎
###### Proposition 2.8.
Let $\mathfrak{a}\in\tilde{W}(I,J)$ and $t\in\mathbb{N}_{0}$. Consider the
natural homomorphism
$\psi:\operatorname{Ext}^{t}_{R}\big{(}R/\mathfrak{a},M\big{)}\longrightarrow\operatorname{Hom}_{R}\big{(}R/\mathfrak{a},\operatorname{H}^{t}_{I,J}(M)\big{)}.$
1. (i)
If
$\operatorname{Ext}^{t-j}_{R}\big{(}R/\mathfrak{a},\operatorname{H}^{j}_{I,J}(M)\big{)}\in\mathcal{S}$
for all $j<t$, then _Ker_ $\psi\in\mathcal{S}$.
2. (ii)
If
$\operatorname{Ext}^{t+1-j}_{R}\big{(}R/\mathfrak{a},\operatorname{H}^{j}_{I,J}(M)\big{)}\in\mathcal{S}$
for all $j<t$, then _Coker_ $\psi\in\mathcal{S}$.
3. (iii)
If
$\operatorname{Ext}^{n-j}_{R}\big{(}R/\mathfrak{a},\operatorname{H}^{j}_{I,J}(M)\big{)}\in\mathcal{S}$
for $n=t,\ t+1$ and for all $j<t$, then _Ker_ $\psi$ and _Coker_ $\psi$ both
belong to $\mathcal{S}$. Thus
$\operatorname{Ext}^{t}_{R}\big{(}R/\mathfrak{a},M\big{)}\in\mathcal{S}$ iff
$\operatorname{Hom}_{R}\big{(}R/\mathfrak{a},\operatorname{H}^{t}_{I,J}(M)\big{)}\in\mathcal{S}$.
###### Proof.
Let $\textmd{F}(-)=\operatorname{Hom}_{R}\big{(}R/\mathfrak{a},-\big{)}$ and
$\textmd{G}(-)={\Gamma_{I,J}\big{(}-\big{)}}$. By Lemma 2.7,
$\textmd{FG}=\textmd{F}$. Now, the result can be followed by [2, Proposition
3.1]. ∎
The next result can be a generalization of main results of [7], [25], [11],
[18], [26], [8], and [6].
###### Theorem 2.9.
Let $\mathfrak{a}\in\tilde{W}(I,J)$ and $~{}t\in\mathbb{N}_{0}$ be such that
$\operatorname{Ext}^{t}_{R}\big{(}R/\mathfrak{a},M\big{)}\in\mathcal{S}$ and
$\operatorname{Ext}^{n-j}_{R}\big{(}R/\mathfrak{a},\operatorname{H}^{j}_{I,J}(M)\big{)}$
$\in\mathcal{S}$ for $n=t,\ t+1$ and for all $j<t$. Then for any submodule $N$
of $\operatorname{H}^{t}_{I,J}(M)$ such that
$\operatorname{Ext}^{1}_{R}\big{(}R/\mathfrak{a},N\big{)}\in\mathcal{S}$, we
have
1. (i)
$\operatorname{Hom}_{R}\big{(}R/\mathfrak{a},\operatorname{H}^{t}_{I,J}(M)/N\big{)}\in\mathcal{S}$.
2. (ii)
$\operatorname{Hom}_{R}\big{(}L,\operatorname{H}^{t}_{I,J}(M)/N\big{)}\in\mathcal{S}$
for any finite $R$-module $L$ with
$\operatorname{Supp}(L)\subseteq\textmd{V}(\mathfrak{a})$.
3. (iii)
$\operatorname{Hom}_{R}\big{(}R/\mathfrak{p},\operatorname{H}^{t}_{I,J}(M)/N\big{)}\in\mathcal{S}$
for any $\mathfrak{p}\in\textmd{V}(\mathfrak{a})$.
All the statements are hold for $\mathfrak{a}=I$.
###### Proof.
(i) Apply Proposition 2.8 and the exact sequence
$0\rightarrow
N\rightarrow\operatorname{H}^{t}_{I,J}(M)\rightarrow\operatorname{H}^{t}_{I,J}(M)/N\rightarrow
0.$
For (ii) and (iii) apply [5, Theorem 2.10]. ∎
###### Corollary 2.10.
Let $(R,\mathfrak{m})$ be a local ring and $\mathfrak{a}\in\tilde{W}(I,J)$.
Let $t\in\mathbb{N}_{0}$ be such that
$\operatorname{Ext}^{t}_{R}(R/\mathfrak{a},M)\in\mathcal{S}$ and
$\operatorname{Ext}^{n-j}_{R}\big{(}R/\mathfrak{a},\operatorname{H}^{j}_{I,J}(M)\big{)}\in\mathcal{S}$
for $n=t,\ t+1$ and all $j<t$. Then for any submodule $N$ of
$\operatorname{H}^{t}_{I,J}(M)$ such that
$\operatorname{Ext}^{1}_{R}\big{(}R/\mathfrak{a},N\big{)}\in\mathcal{S}$, we
have
$\operatorname{Hom}_{R}\big{(}R/\mathfrak{a},\operatorname{H}^{t}_{I,J}(M)/N\big{)}\in\mathcal{S}$,
specially
$\operatorname{Hom}_{R}\big{(}R/\mathfrak{m},\operatorname{H}^{t}_{I,J}(M)/N\big{)}\in\mathcal{S}$.
###### Example 2.11.
_In Theorem 2.9, the assumption
$\operatorname{Ext}^{t}_{R}\big{(}R/\mathfrak{a},M\big{)}\in\mathcal{S}$ is
necessary. To see this, let $(R,\mathfrak{m})$ be a local Gorenstein ring of
positive dimension $d$, and $\mathcal{S}=0$. Then
$\operatorname{H}^{i}_{\mathfrak{m}}(R)=0$ for $i<d$. But we have
$\operatorname{Ext}^{d}_{R}\big{(}R/\mathfrak{m},R\big{)}\cong\operatorname{Hom}_{R}\big{(}R/\mathfrak{m},\operatorname{H}^{d}_{\mathfrak{m}}(R)\big{)}\cong\operatorname{Hom}_{R}\big{(}R/\mathfrak{m},\emph{E}(R/\mathfrak{m})\big{)}\cong
R/\mathfrak{m}\neq 0$_.
One of the main result of this paper is the following theorem which is a
generalization of [7, Theorem 2.12]
###### Theorem 2.12.
Let $R/\mathfrak{m}\in\mathcal{S}$ for all $\mathfrak{m}\in$ $\emph{Max}(R)$.
Let $M$ be a minimax $R$-module and $t\in\mathbb{N}_{0}$ be such that
$\operatorname{Ext}^{n-j}_{R}\big{(}R/\mathfrak{m},\operatorname{H}^{j}_{I,J}(M)\big{)}$
$\in\mathcal{S}$ for $n=t,\ t+1$, all $j<t$, and all $\mathfrak{m}\in$
$\emph{Max}(R)$. Then for any submodule $N$ of $\operatorname{H}^{t}_{I,J}(M)$
such that
$\operatorname{Ext}^{1}_{R}\big{(}R/\mathfrak{m},N\big{)}\in\mathcal{S}$, we
have
$\operatorname{Hom}_{R}\big{(}R/\mathfrak{m},\operatorname{H}^{t}_{I,J}(M)/N\big{)}\in\mathcal{S}$,
for any
$\mathfrak{m}\in\emph{W}_{\emph{Max}}(I,J):=\emph{Max}(R)\cap\emph{W}(I,J)$.
###### Proof.
The assertion follows from Proposition 2.5 (ii) and Theorem 2.9. ∎
###### Corollary 2.13.
Let $(R,\mathfrak{m})$ be a local ring, $\mathcal{S}\neq 0$ and $M$ be a
minimax $R$-module. Let $~{}t\in\mathbb{N}_{0}$ be such that
$\operatorname{Ext}^{n-j}_{R}\big{(}R/\mathfrak{m},\operatorname{H}^{j}_{I,J}(M)\big{)}$
$\in\mathcal{S}$ for $n=t,\ t+1$ and for all $j<t$. Then for any minimax
submodule $N$ of $\operatorname{H}^{t}_{I,J}(M)$, we have
$\operatorname{Hom}_{R}\big{(}R/\mathfrak{m},\operatorname{H}^{t}_{I,J}(M)/N\big{)}\in\mathcal{S}.$
###### Proof.
Apply Proposition 2.5 and Theorem 2.12. ∎
The following familiar conjecture is due to Huneke [23].
Conjecture. Let $(R,\mathfrak{m},k)$ be a regular local ring and $I$ be an
ideal of $R$. For all $n$, soc$\big{(}\operatorname{H}^{n}_{I}(R)\big{)}$ is
finitely generated.
As we mentioned in the introduction, the following theorem which is one of the
main result of this section, can be a positive answer to Huneke,s conjecture.
In fact the following theorem proves a generalization of the conjecture for an
arbitrary noetherian ring $R$ (not necessary regular local one), a minimax
$R$-module $M$ of krull dimension less than 3, and a Serre class
$\mathcal{S}$.
###### Theorem 2.14.
Let $R$ be a noetherian ring and $M$ be a minimax $R$-module of krull
dimension less than _3_. Let $R/\mathfrak{m}\in\mathcal{S}$ for any
$\mathfrak{m}\in\emph{Max}(R)$. Then
$\operatorname{Ext}^{j}_{R}\big{(}R/\mathfrak{m},\operatorname{H}^{i}_{I}(M)\big{)}\in\mathcal{S}$
for any $\mathfrak{m}\in\emph{Max}(R)\cap\operatorname{V}(I)$ and all $i,j\geq
0$. In particular
$\operatorname{Hom}_{R}\big{(}R/\mathfrak{m},\operatorname{H}^{i}_{I}(M)\big{)}\in\mathcal{S}$
for any $\mathfrak{m}\in\emph{Max}(R)\cap\operatorname{V}(I)$ and all $i\geq
0$.
###### Proof.
Let $\mathfrak{m}\in\textmd{Max}(R)\cap\operatorname{V}(I)$. By Proposition
2.5 (ii) and the Grothendieck,s vanishing theorem there is nothing to prove
for cases $i=0$ and $i>2$. Now, assume that $0<i\leq 2$. If dim $M=2$ and
$i=2$, then the result follows from [19, Corollary 3.3] and Proposition 2.5
(i). Also, in the case $i=1$, the result is obtained from [4, Theorem 2.3 ],
by replacing $s:=j$, $t:=1$, and $N:=R/\mathfrak{m}$, and Proposition 2.5
(ii). Finally if dim$M\leq 1$, we can obtain the desired result in similar
way. ∎
###### Corollary 2.15.
Let $R$ be a noetherian ring of dimension less than _3_ and let $M$ be a
minimax $R$-module. Let $R/\mathfrak{m}\in\mathcal{S}$ for any
$\mathfrak{m}\in\emph{Max}(R)$. Then
$\operatorname{Ext}^{j}_{R}\big{(}R/\mathfrak{m},\operatorname{H}^{i}_{I}(M)\big{)}\in\mathcal{S}$
for any $\mathfrak{m}\in\emph{Max}(R)\cap\operatorname{V}(I)$ and all $i,j\geq
0$. In particular for the class of finite $R$-modules.
###### Corollary 2.16.
Let $R$ be a noetherian ring. Let $M$ be a minimax $R$-module of krull
dimension less than _3_. Then the Bass numbers of
$\operatorname{H}^{i}_{I}(M)$ are finite for all $i\geq 0$, in particular it
is true when $\dim R\leqslant 2$.
One can generalize Theorem 2.14 for local cohomology modules with respect to a
pair of ideals, but in local case. To do this, we need the following lemma.
###### Lemma 2.17.
Let $(R,\mathfrak{m})$ be a local ring and $M$ be a minimax $R$-module of
finite krull dimension d. Then $\operatorname{H}^{d}_{I,J}(M)$ is artinian.
###### Proof.
Let $N$ be a finite submodule of $M$ such that $A:=M/N$ is artinian. Since
$\textmd{dim}~{}N\leq d$, by [15, Theorem 2.1],
$\operatorname{H}^{d}_{I,J}(N)$ is artinian. Also, the exact sequence
$0\rightarrow N\rightarrow M\rightarrow A\rightarrow 0$ induces the following
exact sequence
$0\rightarrow\Gamma_{I,J}(N)\rightarrow\Gamma_{I,J}(M)\rightarrow\Gamma_{I,J}(A)\rightarrow\operatorname{H}^{1}_{I,J}(N)\rightarrow\operatorname{H}^{1}_{I,J}(M)\rightarrow
0,$
and $\operatorname{H}^{i}_{I,J}(N)\cong\operatorname{H}^{i}_{I,J}(M)$ for all
$i\geq 2$. By the exactness of the above sequence, it is easy to see that
$\operatorname{H}^{d}_{I,J}(M)$ is artinian. ∎
###### Theorem 2.18.
Let $(R,\mathfrak{m})$ be a local ring and $M$ be a minimax $R$-module of
krull dimension less than _3_. Let $\mathcal{S}\neq 0$. Then
$\operatorname{Ext}^{j}_{R}\big{(}R/\mathfrak{m},\operatorname{H}^{i}_{I,J}(M)\big{)}\in\mathcal{S}$
for all $i,j\geq 0$.
###### Proof.
Apply Lemma 2.17, Proposition 2.5, and the same method of the proof of Theorem
2.14. ∎
The next ressults can be useful for finiteness of Bass numbers of local
cohomology modules over a noetherian ring and modules of krull dimension 3.
Recall that an $R$-module $M$ is called locally minimax if $M_{\mathfrak{m}}$
is minimax for any $\mathfrak{m}\in$ Max$(R)$ (see [3]).
###### Theorem 2.19.
Let $R$ be a noetherian ring and $M$ be an $R$-module of krull dimension _3_.
Let $R/\mathfrak{m}\in\mathcal{S}$ for any $\mathfrak{m}\in\emph{Max}(R)$.
Then
$\operatorname{Ext}^{j}_{R}\big{(}R/\mathfrak{m},\operatorname{H}^{i}_{I}(M)\big{)}\in\mathcal{S}$
for any $\mathfrak{m}\in\emph{Max}(R)\cap V(I)$ and all $i,j\geq 0$, if one of
the following conditions holds:
1. (i)
$M$ and $\operatorname{H}^{1}_{I}(M)$ are minimax;
2. (ii)
$M$ is finite and $\operatorname{H}^{1}_{I}(M)$ is locally minimax;
3. (iii)
$M$ and $I^{m}\operatorname{H}^{2}_{I}(M)$ are minimax for some
$m\in\mathbb{N}_{0}$;
4. (iv)
$M$ is minimax, $I^{m}\operatorname{H}^{2}_{I}(M)$ is locally minimax, and
$\operatorname{Hom}_{R}\big{(}R/I,I^{m}\operatorname{H}^{2}_{I}(M)\big{)}$ is
finite for some $m\in\mathbb{N}_{0}$;
5. (v)
$(R,\mathfrak{m})$ is local, $M$ is minimax, and
$I^{m}\operatorname{H}^{2}_{I}(M)$ is locally minimax for some
$m\in\mathbb{N}_{0}$.
###### Proof.
Let $\mathfrak{m}\in\emph{Max}(R)\cap\operatorname{V}(I)$.
(i) By [19, Corollary 3.3], $\operatorname{H}^{3}_{I}(M)$ is artinian. Thus
the assertion is true for $i=0,1,3$, by Proposition 2.5. Also, for $i=2$, we
apply [4, Theorem 2.3], by replacing $s:=j$, $t:=2$, and $N:=R/\mathfrak{m}$.
(ii) By [8, Theorem 2.3] and [3, Theorem 2.6], $\operatorname{H}^{1}_{I}(M)$
is minimax. Now, the assertion follows from part (i).
(iii) By the short exact sequence
$0\rightarrow
I^{m}\operatorname{H}^{2}_{I}(M)\rightarrow\operatorname{H}^{2}_{I}(M)\rightarrow\operatorname{H}^{2}_{I}(M)/I^{m}\operatorname{H}^{2}_{I}(M)\rightarrow
0,$
and [7, Theorem 3.1], for the class of minimax $R$-modules, we get
$\operatorname{H}^{2}_{I}(M)$ is minimax. Now, for $i=0,2,3$, the claim
follows from [19, Corollary 3.3] and Proposition 2.5. In the case $i=2$, apply
[4, Theorem 2.3 ], by replacing $s:=j$, $t:=1$, $N:=R/\mathfrak{m}$ and
Proposition 2.5.
(iv) Apply [3, Theorem 2.6] and part (iii).
(v) By proof of Lemma 2.17, we may assume that $M$ is finite. Now, the
assertion follows from [3, Propositions 3.4 , 2.2] and part (iii). ∎
###### Corollary 2.20.
Let $R$ be a noetherian ring and $M$ be a minimax $R$-module of krull
dimension $d\leq\emph{3}$. Then the Bass numbers of
$\operatorname{H}^{i}_{I}(M)$ are finite for all $i\geq 0$, if one of the
following conditions holds:
1. (i)
$M$ and $\operatorname{H}^{1}_{I}(M)$ are minimax;
2. (ii)
$M$ is finite and $\operatorname{H}^{1}_{I}(M)$ is locally minimax;
3. (iii)
$M$ and $I^{m}\operatorname{H}^{2}_{I}(M)$ are minimax for some
$m\in\mathbb{N}_{0}$;
4. (iv)
$M$ is minimax, $I^{m}\operatorname{H}^{2}_{I}(M)$ is locally minimax, and
$\operatorname{Hom}_{R}\big{(}R/I,I^{m}\operatorname{H}^{2}_{I}(M)\big{)}$ is
finite for some $m\in\mathbb{N}_{0}$;
5. (v)
$M$ is minimax, and $I^{m}\operatorname{H}^{2}_{I}(M)$ is locally minimax, for
some $m\in\mathbb{N}_{0}$.
In particular the statments are true when $\dim R\leqslant 3$.
###### Proof.
For part (v), apply localization and Theorem 2.19 (v). ∎
## 3\. Cofiniteness and Artinianness of $\operatorname{H}^{i}_{I,J}(M)$
In this section, we need the concept of $(I,J)$-cofinite modules and
$\operatorname{cd}(I,J,M)$. An $R$-module $M$ is called $(I,J)$-cofinite if
Supp$(M)\subseteq W(I,J)$ and $\operatorname{Ext}^{i}_{R}(R/I,M)$ is a finite
$R$-module, for all $i\geq 0$. Also
$~{}\operatorname{cd}(I,J,M)={\sup\big{\\{}i\in\mathbb{N}_{0}}\mid
H^{i}_{I,J}(M)\neq 0\big{\\}}$ (see [37] and [15], resp.)
###### Theorem 3.1.
Let $(R,\mathfrak{m})$ be a local ring and $M$ be a finite $R$-module. Let
$t\in\mathbb{N}$ be an integer such that
$\operatorname{Supp}(\operatorname{H}^{i}_{I,J}(M))\subseteq\\{\mathfrak{m}\\}$
for all $i<t$. Then, for all $i<t$, the $R$-module
$\operatorname{H}^{i}_{I,J}(M)$ is artinian and $I$-cofinite.
###### Proof.
We do this by induction on $t$. When $t=1$, it is obvious that
$\operatorname{H}^{0}_{I,J}(M)$ is artinian $I$-cofinite $R$-module, since it
is a finite module with support in $\\{\mathfrak{m}\\}$. Now, suppose that
$t\geq 2$ and the case $t-1$ is settled. By [36, Corollary 1.13], we may
assume that $M$ is $(I,J)$-torsion free, and so $I$-torsion free $R$-module.
Therefore, by [12, Lemma 2.1.1], there exists $x\in
I\setminus\bigcup_{\mathfrak{p}\in\operatorname{Ass}(M)}\mathfrak{p}$ such
that $0\rightarrow M^{{.x}\atop{\longrightarrow}}M\rightarrow M/xM\rightarrow
0$ is exact. Now, by the exact sequence
$\operatorname{H}^{t-2}_{I,J}(M)\rightarrow\operatorname{H}^{t-2}_{I,J}(M/xM)\rightarrow\operatorname{H}^{t-1}_{I,J}(M)^{{.x}\atop{\longrightarrow}}\operatorname{H}^{t-1}_{I,J}(M),$
we get the following exact sequence
$\operatorname{H}^{t-2}_{I,J}(M)\rightarrow\operatorname{H}^{t-2}_{I,J}(M/xM)\rightarrow\big{(}0:_{\operatorname{H}^{t-1}_{I,J}(M)}x\big{)}\rightarrow
0.$
Thus by inductive hypothesis $\operatorname{H}^{t-2}_{I,J}(M/xM)$ is artinian
$I$-cofinite . So $\big{(}0:_{\operatorname{H}^{t-1}_{I,J}(M)}x\big{)}$ is
artinian $I$-cofinite. As
Supp$(\operatorname{H}^{t-1}_{I,J}(M))\subseteq\\{\mathfrak{m}\\}$, hence
$\operatorname{H}^{t-1}_{I,J}(M)$ is $I$-torsion. Now, the assertion follows
from [34, Proposition 4.1]. ∎
###### Theorem 3.2.
Let $M$ be an $R$-module such that $\operatorname{Ext}^{i}_{R}(R/I,M)$ is
finite for all $i\geq 0$. Let $~{}t\in\mathbb{N}_{0}$ be such that
$\operatorname{H}^{i}_{I,J}(M)$ is $(I,J)$-cofinite for all $i\neq t$. Then
$\operatorname{H}^{t}_{I,J}(M)$ is $(I,J)$-cofinite.
###### Proof.
Apply [5, Theorem 3.11]. ∎
###### Theorem 3.3.
Let $M$ be a finite $R$-module. If $~{}\emph{cd}(I,J,M)\leq 1$, then
$~{}\operatorname{H}^{i}_{I,J}(M)$ is $(I,J)$-cofinite for all $i\geqslant 0$.
###### Proof.
When $i=0$ and $i\geq 2$, the claim is true, since $\Gamma_{I,J}(M)$ is finite
and $\operatorname{cd}(I,J,M)\leq 1$. For $i=1$, apply Theorem 3.2. ∎
###### Theorem 3.4.
Let $(R,\mathfrak{m})$ be a local ring and $M$ be a finite $R$-module with
$\dim M=n$. Then $\operatorname{H}^{n}_{I,J}(M)$ is artinian and $I$-cofinite.
In fact,
$\operatorname{Ext}^{i}_{R}\big{(}R/I,\operatorname{H}^{n}_{I,J}(M)\big{)}$
has finite length for all $i$.
###### Proof.
The assertion follows from [15, Theorem 2.1], [14, Theorem 2.3] and [16,
Theorem 3]. ∎
###### Theorem 3.5.
Let $(R,\mathfrak{m})$ be a local ring and $M$ be a finite $R$-module. If
$\dim M\leqslant 2$, then $\operatorname{H}^{i}_{I,J}(M)$ is $(I,J)$-cofinite
for all $i\geqslant 0$, in particular it is true when $\dim R\leqslant 2$.
###### Proof.
Apply Theorems 3.4 , 3.2. ∎
## 4\. Upper bounds of $\operatorname{H}^{i}_{I,J}(M)$
In this section, we introduce the concept of Serre cohomological dimension of
$M$ with respect to a pair of ideals $(I,J)$, but first, we characterize the
membership of $\operatorname{H}^{i}_{I,J}(M)$ in a Serre class from upper
bound.
###### Theorem 4.1.
Let $n\in\mathbb{N}_{0}$ and $M$ be a finite $R$–module. Then the following
statements are equivalent:
1. (i)
$\operatorname{H}^{i}_{I,J}(M)$ is in $\mathcal{S}$ for all $i>n$.
2. (ii)
$\operatorname{H}^{i}_{I,J}(N)$ is in $\mathcal{S}$ for all $i>n$ and for any
finite $R$-module $N$ such that
$\operatorname{Supp}_{R}(N)\subseteq\operatorname{Supp}_{R}(M)$.
3. (iii)
$\operatorname{H}^{i}_{I,J}(R/\mathfrak{p})$ is in $\mathcal{S}$ for all
$\mathfrak{p}\in\operatorname{Supp}_{R}(M)$ and for all $i>n$.
4. (iv)
$\operatorname{H}^{i}_{I,J}(R/\mathfrak{p})$ is in $\mathcal{S}$ for all
$\mathfrak{p}\in\operatorname{Min}\operatorname{Ass}_{R}(M)$ and for all
$i>n$.
###### Proof.
Apply the method of the proof of Theorem 3.1 in [1] to
$\operatorname{H}^{i}_{I,J}$ . ∎
###### Corollary 4.2.
Let $M$ , $N$ be finite $R$-modules such that
$~{}\operatorname{Supp}(N)\subseteq\operatorname{Supp}(M)$ and
$n\in\mathbb{N}_{0}$. If $\operatorname{H}^{i}_{I,J}(M)\in\mathcal{S}$ for all
$i>n$, then $\operatorname{H}^{i}_{I,J}(N)\in\mathcal{S}$ for all $i>n$.
###### Lemma 4.3.
For any $R$-module $N$, the following statements are fulfilled.
1. (i)
$\Gamma_{\mathfrak{a}}(N)\subseteq\Gamma_{\mathfrak{a},J}(N)\subseteq\Gamma_{I,J}(N)$
for any $\mathfrak{a}\in\tilde{W}(I,J)$.
2. (ii)
$\Gamma_{I,J}(N)=0$ if and only if $\Gamma_{\mathfrak{a},J}(N)=0$ for any
$\mathfrak{a}\in\tilde{W}(I,J)$.
3. (iii)
$\Gamma_{I,J}(N)=N$ if and only if there exists
$\mathfrak{a}\in\tilde{W}(I,J)$ such that $\Gamma_{\mathfrak{a},J}(N)=N$.
4. (iv)
If there exists $\mathfrak{a}\in\tilde{W}(I,J)$ such that
$\Gamma_{\mathfrak{a}}(N)=N$, then $\Gamma_{I,J}(N)=N$.
5. (v)
$\Gamma_{I,J}(N)=\cup_{\mathfrak{a}\in\tilde{W}(I,J)}\Gamma_{\mathfrak{a}}(N)=\cup_{\mathfrak{a}\in\tilde{W}(I,J)}\Gamma_{\mathfrak{a},J}(N)$
###### Proof.
All these statements follow easily from the definitions. We will only prove
the statement (v). Since
$\Gamma_{\mathfrak{b}}(N)\subseteq\Gamma_{\mathfrak{b},J}(N)$, for any ideal
$\mathfrak{b}$ of $R$, so by [36, Theorem 3.2] and part(i), we get
$\Gamma_{I,J}(N)\subseteq\cup_{\mathfrak{a}\in\tilde{W}(I,J)}\Gamma_{\mathfrak{a}}(N)\subseteq\cup_{\mathfrak{a}\in\tilde{W}(I,J)}\Gamma_{\mathfrak{a},J}(N)\subseteq\Gamma_{I,J}(N).$
∎
In [19], authors introduced the concept of ZD-modules. An $R$-module $M$ is
said to be a ZD-module (zero-divisor module) if for every submodule $N$ of
$M$, the set of zero divisors of $M/N$ is a union of finitely many prime
ideals in Ass${}_{R}(M/N)$. By [19, Example 2.2], the class of ZD-modules
contains modules with finite support, finitely generated, Laskerian, weakly
Laskerian, linearly compact, Matlis reflexive and minimax $R$-modules.
Now, we are in position to prove two other main results of this paper (Theorem
4.4 and Theorem 4.7), which the first one can be considered as a
generalization of [7, Theorem 3.1].
###### Theorem 4.4.
Let $M$ be a ZD-module of finite Krull dimension. Let $n\in\mathbb{N}$ be such
that $\operatorname{H}^{i}_{I,J}(M)\in\mathcal{S}$ for all $i>n$. Then
$\operatorname{H}^{i}_{I,J}(M)/\mathfrak{a}^{j}\operatorname{H}^{i}_{I,J}(M)\in\mathcal{S}$
for any $\mathfrak{a}\in\tilde{W}(I,J)$, all $i\geq n$, and all $j\geq 0$.
###### Proof.
It is enough to verify the assertion for just $i=n$ and $j=1$. To do this, we
use induction on $d$:=dim$M$. When $d=0$, the result follows from [36, Theorem
3.2] and Grothendieck,s Vanishing theorem.
Next, we assume that $d>0$ and the claim is true for all $R$-modules of
dimension less than $d$. By [36, Theorem 1.3 (4)], we have
$\operatorname{H}^{j}_{I,J}(M)\cong\operatorname{H}^{j}_{I,J}\big{(}M/\Gamma_{I,J}(M)\big{)}$
for all $j>0$. Also, $M/\Gamma_{I,J}(M)$ has dimension not exceeding $d$, and
is an $(I,J)$-torsion-free $R$-module. Therefore we may assume that
$\Gamma_{I,J}(M)=0$ and so, by Lemma 4.3, $\Gamma_{\mathfrak{a}}(M)=0$. By
[19, Lemma 2.4],
$x\in\mathfrak{a}\setminus\bigcup_{\mathfrak{p}\in\textmd{Ass}(M)}\mathfrak{p}$.
Now, the $R$-module $M/xM$ is ZD-module of dimension $d-1$. Considering the
exact sequence
$0\rightarrow M^{{.x}\atop{\longrightarrow}}M\rightarrow M/xM\rightarrow 0$
induces a long exact sequence of local cohomology modules, which shows that
$\operatorname{H}^{i}_{I,J}(M/xM)\in\mathcal{S}$ for all $i>n$. Thus by
inductive hypothesis
$\operatorname{H}^{n}_{I,J}(M/xM)/\mathfrak{a}\operatorname{H}^{n}_{I,J}(M/xM)\in\mathcal{S}$.
Now, the exact sequence
$\operatorname{H}^{n}_{I,J}(M)^{{.x}\atop{\longrightarrow}}\operatorname{H}^{n}_{I,J}(M)^{{\alpha}\atop{\longrightarrow}}\operatorname{H}^{n}_{I,J}(M/xM)^{{\beta}\atop{\longrightarrow}}\operatorname{H}^{n+1}_{I,J}(M),$
induces the following exact sequences
$\operatorname{H}^{n}_{I,J}(M)^{{.x}\atop{\longrightarrow}}\operatorname{H}^{n}_{I,J}(M)\rightarrow
N:=\operatorname{Im}\alpha\rightarrow 0,$ $0\rightarrow
N\rightarrow\operatorname{H}^{n}_{I,J}(M/xM)\rightarrow
K:=\operatorname{Im}\beta\rightarrow 0.$
Therefore, the following two sequences
1. $(\ast)$ $~{}\operatorname{H}^{n}_{I,J}(M)/\mathfrak{a}\operatorname{H}^{n}_{I,J}(M)^{{.x}\atop{\longrightarrow}}\operatorname{H}^{n}_{I,J}(M)/\mathfrak{a}\operatorname{H}^{n}_{I,J}(M)\rightarrow N/\mathfrak{a}N\rightarrow 0,$
1. $(\ast\ast)$ ${\textmd{Tor}^{R}_{1}(R/\mathfrak{a},K)}\rightarrow N/\mathfrak{a}N\rightarrow\operatorname{H}^{n}_{I,J}(M/xM)/\mathfrak{a}\operatorname{H}^{n}_{I,J}(M/xM)\rightarrow K/\mathfrak{a}K\rightarrow 0$
are both exact. Since $x\in\mathfrak{a}$ and from the exact sequence $(\ast)$,
we get
$N/\mathfrak{a}N\cong\operatorname{H}^{n}_{I,J}(M)/\mathfrak{a}\operatorname{H}^{n}_{I,J}(M)$.
On the other hand, by [7, Lemma 2.1], we have
Tor${}^{R}_{1}(R/\mathfrak{a},K)\in\mathcal{S}$. Therefore
$N/\mathfrak{a}N\in\mathcal{S}$, by $(\ast\ast)$, as required. ∎
The following result is an immediate consequence of the above theorem.
###### Corollary 4.5.
Let $M$ be a ZD-module of finite Krull dimension. Let $n\in\mathbb{N}$ be such
that $\operatorname{H}^{i}_{I,J}(M)\in\mathcal{S}$ for all $i>n$. Then
$\operatorname{H}^{n}_{I,J}(M)\in\mathcal{S}$ if and only if there exist
$\mathfrak{a}\in\tilde{W}(I,J)$ and $m\in\mathbb{N}_{0}$ such that
$\mathfrak{a}^{m}\operatorname{H}^{n}_{I,J}(M)\in\mathcal{S}$.
###### Proof.
$(\Rightarrow)$ It is obvious.
$(\Leftarrow)$ Apply the short exact sequence
$0\rightarrow\mathfrak{a}^{m}\operatorname{H}^{n}_{I,J}(M)\rightarrow\operatorname{H}^{n}_{I,J}(M)\rightarrow\operatorname{H}^{n}_{I,J}(M)/\mathfrak{a}^{m}\operatorname{H}^{n}_{I,J}(M)\rightarrow
0$ and Theorem 4.3. ∎
Applying Corollary 4.5, for some familiar Serre classes of modules, we get
some results as follows.
###### Corollary 4.6.
Let $M$ be a ZD-module of finite Krull dimension and let $n\in\mathbb{N}$.
Then the following statements are fulfilled.
1. (i)
If $\operatorname{H}^{i}_{I,J}(M)$ is finite for all $i>n$, then
$\operatorname{H}^{n}_{I,J}(M)$ is finite if and only if there exist
$\mathfrak{a}\in\tilde{W}(I,J)$ and $m\in\mathbb{N}_{0}$ such that
$\mathfrak{a}^{m}\operatorname{H}^{n}_{I,J}(M)$ is finite if and only if there
exist $\mathfrak{a}\in\tilde{W}(I,J)$ and $m\in\mathbb{N}_{0}$ such that
$\operatorname{H}^{n}_{I,J}(M)/(0:_{\operatorname{H}^{n}_{I,J}(M)}\mathfrak{a}^{m})$
is finite .
2. (ii)
If $\operatorname{H}^{i}_{I,J}(M)$ is artinian for all $i>n$, then
$\operatorname{H}^{n}_{I,J}(M)$ is artinian if and only if there exist
$\mathfrak{a}\in\tilde{W}(I,J)$ and $m\in\mathbb{N}_{0}$ such that
$\mathfrak{a}^{m}\operatorname{H}^{n}_{I,J}(M)$ is artinian.
3. (iii)
If $\operatorname{H}^{i}_{I,J}(M)=0$ for all $i>n$, then
$\operatorname{H}^{n}_{I,J}(M)={\mathfrak{a}}^{j}\operatorname{H}^{n}_{I,J}(M)$
for any $\mathfrak{a}\in\tilde{W}(I,J)$ and all $j\geq 0$. Thus
$\operatorname{H}^{n}_{I,J}(M)=0$ if and only if there exist
$\mathfrak{a}\in\tilde{W}(I,J)$ and $m\in\mathbb{N}_{0}$ such that
${\mathfrak{a}}^{m}\operatorname{H}^{n}_{I,J}(M)=0$.
In particular for $n=\emph{dim}M$ and $\emph{cd}(I,J,M)$
###### Proof.
Apply Corollary 4.5 and [3, Theorem 3.1]. ∎
Although the following theorem is seemed to be similar to Theorem 4.4, but it
is more useful and general than 4.4 for finite $R$-modules.
###### Theorem 4.7.
Let $M$ be a finite $R$ module and $n\in\mathbb{N}_{0}$ be such that
$\operatorname{H}^{i}_{I,J}(M)$ belongs to $\mathcal{S}$ for all $i>n$. If
$\mathfrak{b}$ is an ideal of $R$ such that
$\operatorname{H}^{n}_{I,J}(M/{\mathfrak{b}}M)$ belongs to $\mathcal{S}$, then
the module
$\operatorname{H}^{n}_{I,J}(M)/{\mathfrak{b}}\operatorname{H}^{n}_{I,J}(M)$
belongs to $\mathcal{S}$.
###### Proof.
Let $\mathfrak{b}=(b_{1},\dots,b_{r})$ and consider the map $f:M^{r}\to M$,
defined by $f(x_{1},\dots,x_{r})=\sum_{1}^{r}b_{i}x_{i}$. Then
$\operatorname{Im}f=\mathfrak{b}M$ and
$\operatorname{Coker}f=M/\mathfrak{b}M$. Since $\operatorname{H}^{i}_{I,J}(M)$
is in $\mathcal{S}$ for all $i>n$ and
$\operatorname{Supp}(\operatorname{Ker}f)\subseteq\operatorname{Supp}(M)$, it
follows from Theorem 4.1 that
$\operatorname{H}^{n+1}_{I,J}(\operatorname{Ker}f)$ is also in $\mathcal{S}$.
By hypothesis $\operatorname{H}^{n}_{I,J}(\operatorname{Coker}f)$ belongs to
$\mathcal{S}$. Hence by [34, Corollary 3.2]
$\operatorname{Coker}{\operatorname{H}^{n}_{I,J}(f)}$, which equals to
$\operatorname{H}^{n}_{I,J}(M)/{{\mathfrak{b}}\operatorname{H}^{n}_{I,J}(M)}$,
is in $\mathcal{S}$. ∎
###### Corollary 4.8.
Let $M$ be a finite $R$-module and $n\in\mathbb{N}$ be such that
$\operatorname{H}^{i}_{I,J}(M)\in\mathcal{S}$ for all $i>n$. Then
$\operatorname{H}^{n}_{I,J}(M)/{\mathfrak{a}}\operatorname{H}^{n}_{I,J}(M)\in\mathcal{S}$
for any $\mathfrak{a}\in\tilde{W}(I,J)$, in particular for $\mathfrak{a}=I$.
###### Proof.
Let $\mathfrak{a}\in\tilde{W}(I,J)$. Since $M/{\mathfrak{a}}M$ is
$\mathfrak{a}$-torsion $R$-module, thus the assertion follows from Lemma 4.3
(iv), [36, Corollary 1.13] and Theorem 4.7. ∎
###### Corollary 4.9.
Let $M$ be a finite $R$-module. Let $n\in\mathbb{N}$ be such that
$\operatorname{H}^{i}_{I,J}(M)\in\mathcal{S}$ for all $i>n$. Then
$\operatorname{H}^{n}_{I,J}(M)\in\mathcal{S}$ if and only if there exists
$m\in\mathbb{N}_{0}$ such that
$\mathfrak{a}^{m}\operatorname{H}^{n}_{I,J}(M)\in\mathcal{S}$.
###### Corollary 4.10.
Let $M$ be a finite $R$-module. Let $\mathfrak{a}\in\tilde{W}(I,J)$ and
$n\in\mathbb{N}$.
1. (i)
If $\operatorname{H}^{i}_{I,J}(M)$ is finite for all $i>n$, then
$\operatorname{H}^{n}_{I,J}(M)$ is finite if and only if there exist
$\mathfrak{a}\in\tilde{W}(I,J)$ and $m\in\mathbb{N}_{0}$ such that
$\mathfrak{a}^{m}\operatorname{H}^{n}_{I,J}(M)$ is finite if and only if there
exist $\mathfrak{a}\in\tilde{W}(I,J)$ and $m\in\mathbb{N}_{0}$ such that
$\operatorname{H}^{n}_{I,J}(M)/(0:_{\operatorname{H}^{n}_{I,J}(M)}\mathfrak{a}^{m})$
is finite
2. (ii)
If $\operatorname{H}^{i}_{I,J}(M)$ is artinian for all $i>n$, then
$\operatorname{H}^{n}_{I,J}(M)$ is artinian if and only if there exists
$m\in\mathbb{N}_{0}$ such that
${\mathfrak{a}}^{m}\operatorname{H}^{n}_{I,J}(M)$ is artinian.
3. (iii)
If $\operatorname{H}^{i}_{I,J}(M)=0$ for all $i>n$, then
$\operatorname{H}^{n}_{I,J}(M)={\mathfrak{a}}^{j}\operatorname{H}^{n}_{I,J}(M)$
for all $j\in\mathbb{N}_{0}$. Thus $\operatorname{H}^{n}_{I,J}(M)=0$ if and
only if there exists $m\in\mathbb{N}_{0}$ such that
${\mathfrak{a}}^{m}\operatorname{H}^{n}_{I,J}(M)=0$, in particular for $n=\dim
M$ and $\operatorname{cd}(I,J,M)$.
4. (iv)
If $R$ is local and $\operatorname{H}^{i}_{I,J}(M)$ is finite for all $i>n$,
then
$\operatorname{H}^{n}_{I,J}(M)={\mathfrak{a}}^{j}\operatorname{H}^{n}_{I,J}(M)$
for all $j\geq 0$.
###### Proof.
Apply Corollary 4.9, [27, Proposition 1], and [3, Theorem 3.1]. ∎
The following result is more useful whenever $R$ is a local ring and $I$ is a
proper ideal. (see [10, Lemma 2.1])
###### Corollary 4.11.
Let $M$ be a non-zero ZD-module with $d:=\dim M$ and
$~{}t:=\operatorname{cd}(I,J,M)$. Let $\mathfrak{a}\in\tilde{W}(I,J)$ be such
that ${\mathfrak{a}}^{m}\subseteq\emph{Jac}(R)$ for some $m\in\mathbb{N}_{0}$.
1. (i)
If $t\geq 1$, then ${\mathfrak{a}}^{j}\operatorname{H}^{t}_{I,J}(M)$ is not
finite for all $j\geq 0$.
2. (ii)
If $d\geq 1$, then $\operatorname{H}^{d}_{I,J}(M)$ is finite if and only if
$~{}\operatorname{H}^{d}_{I,J}(M)=0$.
3. (iii)
If $d\geq 2$ and $\operatorname{H}^{d}_{I,J}(M)$ is finite, then
$\operatorname{H}^{d-1}_{I,J}(M)/{\mathfrak{a}}^{j}\operatorname{H}^{d-1}_{I,J}(M)$
has finite length for all $j\geq 0$.
In particular when $R$ is local ring and $I\neq R$.
###### Proof.
(i), (ii) Apply Corollary 4.6 (iii) and Nakayama,s Lemma.
(iii) Apply part (ii) and Theorem 4.4 for the class of finite length. ∎
###### Corollary 4.12.
Let $M$ be a non-zero finite $R$-module and set $t:=\emph{sup}\big{\\{}i\geq
1\mid\operatorname{H}^{i}_{I,J}(M)\emph{~{}is not finite}\big{\\}}$,
$n:=\emph{cd}(I,J,M)$, and $r:=\emph{dim}M/JM$.
1. (i)
If $\mathfrak{a}\in\tilde{W}(I,J)$ is such that
${\mathfrak{a}}^{m}\subseteq\emph{Jac}(R)$ for some $m\in\mathbb{N}_{0}$, then
$n\geq 1$ if and only if $n=t$.
2. (ii)
Let $(R,\mathfrak{m})$ be a local ring and $r\geq 1$ be an integer such that
$\operatorname{H}^{r}_{I,J}(M)$ is finite. Let $\mathfrak{a}\in\tilde{W}(I,J)$
be such that ${\mathfrak{a}}^{m}+J\subseteq\mathfrak{m}$ for some
$m\in\mathbb{N}_{0}$. Then $~{}\operatorname{H}^{r}_{I,J}(M)=0$.
3. (iii)
If $(R,\mathfrak{m})$ is a local ring and $I+J$ is an $\mathfrak{m}$-primary
ideal, then $r=n=t$.
###### Proof.
Apply Corollaries 4.9, 4.10, and [36, Theorems 4.3,4.5]. ∎
###### Corollary 4.13.
Let $M$ be a finite $R$-module and $n\in\mathbb{N}$. If
$\operatorname{H}^{i}_{I,J}(M)$ is artinian for all $i>n$. Then
$\operatorname{H}^{n}_{I,J}(M)/{\mathfrak{a}}\operatorname{H}^{n}_{I,J}(M)$ is
artinian for any $\mathfrak{a}\in\tilde{W}(I,J)$.
###### Proof.
Note that $\operatorname{H}^{n}_{I,J}(M/{\mathfrak{a}}M)=0$ for any
$\mathfrak{a}\in\tilde{W}(I,J)$ and all $n\geq 1$. Now, apply Theorem 4.7 for
the class of artinian modules. ∎
###### Remark 4.14.
_In 4.13 we have to assume that $n\geq 1$. Take an ideal $I$ in a ring $R$
such that $R/I$ is not artinian. Let $J=0$ and $M=R/I$. Then
$\operatorname{H}^{i}_{I,J}(M)=0$ for $i\geq 1$, and
$\operatorname{\Gamma}_{I}(M)=M$. On the other hand $M/{I}M\cong M$. Thus
$\operatorname{H}^{0}_{I,J}(M)/{I}\operatorname{H}^{0}_{I,J}(M)$ is not
artinian_.
Properties of Serre classes of modules and the previous results motivate us to
introduce the following definition as a generalization of the concept of
cohomological dimension. (see [15]).
###### Definition 4.15.
_Let $I$ , $J$ be two ideals of $R$ and let $M$ be an $R$-module. For a Serre
subcategory $\mathcal{S}$ of the category of $R$-modules, we define Serre
cohomological dimension of $M$ with respect to $(I,J)$, by_
${\textmd{cd}}_{\mathcal{S}}(I,J,M)={\textmd{sup}}\big{\\{}{i\in\mathbb{N}_{0}}\mid\operatorname{H}^{i}_{I,J}(M)\not\in\mathcal{S}\big{\\}},$
_if this supremum exists, and $\infty$ otherwise.
It is easy to see that
${\textmd{cd}}_{\mathcal{S}}(I,J,M)={\textmd{inf}}\big{\\{}{n\in\mathbb{N}_{0}}\mid\operatorname{H}^{i}_{I,J}(M)\in\mathcal{S}~{}\textmd{~{}for
~{}all}~{}~{}i>n\big{\\}}$_.
###### Remark 4.16.
_For an arbitrary Serre class $\mathcal{S}$ , we have
${\textmd{cd}}_{\mathcal{S}}(I,J,M)\leq{\emph{cd}}(I,J,M)$, and if in
Definition 4.16, we let $\mathcal{S}:=0$ then we have_
${\textmd{cd}}_{\mathcal{S}}(I,J,M)={\textmd{sup}}\big{\\{}{i\in\mathbb{N}_{0}}\mid\operatorname{H}^{i}_{I,J}(M)\neq
0\big{\\}}=\textmd{cd}(I,J,M).$
_Also, if $\mathcal{S}$ is the class of Artinian $R$-modules, we get_
${\textmd{cd}}_{\mathcal{S}}(I,J,M)={\textmd{sup}}\big{\\{}{i\in\mathbb{N}_{0}}\mid\operatorname{H}^{i}_{I,J}(M)~{}{\textmd{is
not Artinian $R$-module}}\big{\\}}.$
_We denote it by $\operatorname{q}(I,J,M)$_.
###### Proposition 4.17.
Let $M$ be a ZD-module of finite krull dimension or finite $R$-module. Let
$\mathcal{S}$ be a Serre class and $n:=\emph{cd}_{\mathcal{S}}(I,J,M)\geq 1$.
Then ${\mathfrak{a}}^{j}\operatorname{H}^{n}_{I,J}(M)\not\in\mathcal{S}$ for
any $\mathfrak{a}\in\tilde{W}(I,J)$ and all $j\geq 0$, in particular
${\mathfrak{a}}^{j}\operatorname{H}^{n}_{I,J}(M)\neq 0$ for any
$\mathfrak{a}\in\tilde{W}(I,J)$ and all $j\geq 0$.
###### Proof.
Apply Corollaries 4.5 , 4.9. ∎
It is well known that if $\mathfrak{a}$ is an ideal of $R$ and $M$ , $N$ are
finite $R$-modules with $\textmd{Supp}(N)\subseteq\textmd{Supp}(M)$, then
${\textmd{cd}}(\mathfrak{a},N)\leq{\textmd{cd}}(\mathfrak{a},M)$. The next
result is a generalization of this fact.
###### Proposition 4.18.
Let $M$ , $N$ be finite $R$-modules such that
$\emph{Supp}(N)\subseteq\emph{Supp}(M)$. Then
${\emph{cd}}_{\mathcal{S}}(I,J,N)\leq{\emph{cd}}_{\mathcal{S}}(I,J,M)$.
###### Proof.
Apply Corollary 4.2. ∎
###### Corollary 4.19.
Let $M$ , $N$ be finite $R$-modules such that
$\emph{Supp}(N)\subseteq\emph{Supp}(M)$. Then
${\emph{cd}}(I,J,N)\leq{\emph{cd}}(I,J,M)$.
###### Corollary 4.20.
For a finite $R$-module $M$, there exist the following equalities.
$\displaystyle\emph{cd}_{\mathcal{S}}(I,J,M)$
$\displaystyle={\emph{max}~{}}\big{\\{}\emph{cd}_{\mathcal{S}}(I,J,R/\mathfrak{p})\mid\mathfrak{p}\in\
{\emph{Ass}}(M)\big{\\}}$
$\displaystyle={\emph{max}~{}}\big{\\{}\emph{cd}_{\mathcal{S}}(I,J,R/\mathfrak{p})\mid\mathfrak{p}\in\
{\emph{Min Ass}}(M)\big{\\}}$
$\displaystyle={\emph{max}~{}}\big{\\{}\emph{cd}_{\mathcal{S}}(I,J,R/\mathfrak{p})\mid\mathfrak{p}\in\
{\emph{Supp}}(M)\big{\\}}$
$\displaystyle={\emph{max}~{}}\big{\\{}\emph{cd}_{\mathcal{S}}(I,J,R/\mathfrak{p})\mid\mathfrak{p}\in\
{\emph{Min Supp}}(M)\big{\\}}$
$\displaystyle={\emph{max}~{}}\big{\\{}\emph{cd}_{\mathcal{S}}(I,J,N)\mid
N~{}\emph{is a finite submodule }M\big{\\}}$
$\displaystyle={\emph{max}~{}}\big{\\{}i\geq 0\mid
H^{i}_{I,J}(R/\mathfrak{p})\not\in\mathcal{S},~{}\emph{for
some}~{}\mathfrak{p}\in{\emph{Ass}}(M)\big{\\}}$
$\displaystyle=\emph{min~{}}\big{\\{}{n\geq
0}\mid\operatorname{H}^{i}_{I,J}(R/\mathfrak{p})\in\mathcal{S}~{}\emph{~{}for
~{}all}~{}~{}i>n~{}\emph{and ~{}all}~{}~{}i>n\big{\\}}.$
###### Proof.
Apply Corollary 4.1 and Proposition 4.19. ∎
###### Corollary 4.21.
Let $M$ , $N$ be finite $R$-modules such that
$\emph{Supp}(N)\subseteq\emph{Supp}(M)$. Then
$\operatorname{q}(I,J,N)\leq\operatorname{q}(I,J,M)$.
###### Proposition 4.22.
Let $0\rightarrow L\rightarrow M\rightarrow N\rightarrow 0$ be an exact
sequence of finite $R$-modules. Then
$\emph{cd}_{\mathcal{S}}(I,J,M)={\emph{max}}~{}\big{\\{}\emph{cd}_{\mathcal{S}}(I,J,L)~{},~{}\emph{cd}_{\mathcal{S}}(I,J,N)\big{\\}}$
###### Proof.
Let $t:=\textmd{cd}_{\mathcal{S}}(I,J,M)$ and
$s:={\textmd{max}~{}}\big{\\{}\emph{cd}_{\mathcal{S}}(I,J,L)~{},~{}\emph{cd}_{\mathcal{S}}(I,J,N)\big{\\}}$.
By, Corollary 4.2 , we have $t\geq s$. Now, let $t>s$. Then by the following
exact sequence
$\cdots\rightarrow H^{t}_{I,J}(L)\rightarrow H^{t}_{I,J}(M)\rightarrow
H^{t}_{I,J}(N)\rightarrow\cdots,$
we get $H^{t}_{I,J}(M)\in\mathcal{S}$ which is a contradiction with
$\textmd{cd}_{\mathcal{S}}(I,J,M)=t$. ∎
###### Corollary 4.23.
For a noetherian ring $R$ there exists the following equality.
$\emph{cd}_{\mathcal{S}}(I,J,R)={\emph{sup}~{}}\big{\\{}\emph{cd}_{\mathcal{S}}(I,J,N)\mid~{}N~{}{\textmd{is
a finite $R$-module}}\big{\\}}.$
In particular, for $r\in\mathbb{N}_{0}$ the following statements are
equivalent:
1. (i)
$H^{j}_{I,J}(R)\in\mathcal{S}$ for all $j>r$.
2. (ii)
$H^{j}_{I,J}(M)\in\mathcal{S}$ for all $j>r$ and all finite $R$-module $M$.
## References
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* [2] M. Aghapournahr, L. Melkersson, A natural map in local cohomology, Ark. Mat. 48 (2010), 243–251.
* [3] M. Aghapournahr, L. Melkersson, Finiteness properties of minimax and coatomic local cohomology modules, Arch. Math., 94 (2010), 519–528.
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|
arxiv-papers
| 2012-11-18T09:06:13 |
2024-09-04T02:49:38.160782
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "M. Aghapournahr, KH. Ahmadi-amoli, and M. Y. Sadeghi",
"submitter": "Moharram Aghapournahr",
"url": "https://arxiv.org/abs/1211.4204"
}
|
1211.4330
|
# Sub-barrier capture reactions with 16,18O beams
V.V.Sargsyan1, G.G.Adamian1, N.V.Antonenko1, W. Scheid2, and H.Q.Zhang3 1Joint
Institute for Nuclear Research, 141980 Dubna, Russia
2Institut für Theoretische Physik der Justus–Liebig–Universität, D–35392
Giessen, Germany
3China Institute of Atomic Energy, Post Office Box 275, Beijing 102413, China
###### Abstract
Various sub-barrier capture reactions with beams 16,18O are treated within the
quantum diffusion approach. The role of neutron transfer in these capture
reactions is discussed.
###### pacs:
25.70.Jj, 24.10.-i, 24.60.-k
Key words: sub-barrier capture, neutron transfer, quantum diffusion approach
The purpose of this Brief Report is the theoretical explanation of the sub-
barrier capture reactions 16,18O+52,50Cr, 16,18O+76,74Ge, 16,18O+94,92Mo, and
16,18O+112,114,118,120,124,126Sn. Within the quantum diffusion approach EPJSub
; EPJSub1 we try to answer the question how strong the influence of neutron
transfer in these capture reactions. This study is important for future
experiments indicated in Ref. Jia .
In the quantum diffusion approach EPJSub ; EPJSub1 the collisions of nuclei
are described with a single relevant collective variable: the relative
distance between the colliding nuclei. This approach takes into consideration
the fluctuation and dissipation effects in collisions of heavy ions which
model the coupling with various channels (for example, coupling of the
relative motion with low-lying collective modes such as dynamical quadrupole
and octupole modes of the target and projectile Ayik333 ). We have to mention
that many quantum-mechanical and non-Markovian effects accompanying the
passage through the potential barrier are taken into consideration in our
formalism EPJSub ; EPJSub1 ; PRCPOP . The nuclear deformation effects are
taken into account through the dependence of the nucleus-nucleus potential on
the deformations and mutual orientations of the colliding nuclei. To calculate
the nucleus-nucleus interaction potential $V(R)$, we use the procedure
presented in Refs. EPJSub ; EPJSub1 . For the nuclear part of the nucleus-
nucleus potential, the double-folding formalism with the Skyrme-type density-
dependent effective nucleon-nucleon interaction is used. With this approach
many heavy-ion capture reactions at energies above and well below the Coulomb
barrier have been successfully described EPJSub ; EPJSub1 ; PRCPOP . One
should stress that the diffusion models, which include the quantum statistical
effects, were also treated in Refs. Hofman .
Following the hypothesis of Ref. Broglia , we assume that the sub-barrier
capture in the reactions under consideration mainly depends on the two-neutron
transfer with the positive $Q_{2n}$-value. Our assumption is that, just before
the projectile is captured by the target-nucleus (just before the crossing of
the Coulomb barrier) which is a slow process, the $2n$-transfer ($Q_{2n}>0$)
occurs that can lead to the population of the excited collective states in the
recipient nucleus SSzilner . So, the motion to the $N/Z$ equilibrium starts in
the system before the capture because it is energetically favorable in the
dinuclear system in the vicinity of the Coulomb barrier. For the reactions
considered, the average change of mass asymmetry is related to the two-neutron
transfer. In these reactions the $2n$-transfer channel is more favorable than
$1n$-transfer channel ($Q_{2n}>Q_{1n}$). Since after the $2n$-transfer the
mass numbers, the deformation parameters of the interacting nuclei, and,
correspondingly, the height $V_{b}=V(R_{b})$ and shape of the Coulomb barrier
are changed, one can expect an enhancement or suppression of the capture. If
after the neutron transfer the deformations of interacting nuclei increase
(decrease), the capture probability increases (decreases). If after the
transfer the deformations of interacting nuclei do not change, there is no
effect of the neutron transfer on the capture. This scenario was verified in
the description of many reactions EPJSub1 .
Figure 1: The calculated (solid line) capture cross sections versus $E_{\rm
c.m.}$ for the reactions 16O+76Ge and 18O+74Ge (the curves coincide). For the
18O+74Ge reaction, the calculated capture cross sections without the neutron
transfer are shown by dotted line. The experimental data for the reactions
16O+76Ge (open circles) and 18O+74Ge (open squares) are from Ref. Jia . The
experimental data for the 16O+76Ge reaction (solid circles) are from Ref.
16OAGe . Figure 2: The calculated capture cross sections versus $E_{\rm
c.m.}$ for the reactions 16O+92Mo (dashed line) and 18O+92Mo (solid line). For
the 18O+92Mo reaction, the calculated capture cross sections without the
neutron transfer are shown by dotted line. The experimental data for the
reactions 16O+92Mo (solid stars) and 18O+92Mo (solid squares) are from Ref.
AO92Mo .
All calculated results are obtained with the same set of parameters as in Ref.
EPJSub and are rather insensitive to the reasonable variation of them EPJSub
; EPJSub1 . Realistic friction coefficient in the momentum $\hbar\lambda$=2
MeV is used which is close to those calculated within the mean field
approaches obzor . The parameters of the nucleus-nucleus interaction potential
$V(R)$ are adjusted to describe the experimental data at energies above the
Coulomb barrier corresponding to spherical nuclei. The absolute values of the
quadrupole deformation parameters $\beta_{2}$ of even-even deformed nuclei are
taken from Ref. Ram . In Ref. Ram the quadrupole deformation parameters
$\beta_{2}$ are given for the first excited 2+ states of nuclei. For the
nuclei deformed in the ground state, the $\beta_{2}$ in 2+ state is similar to
the $\beta_{2}$ in the ground state and we use $\beta_{2}$ from Ref. Ram in
the calculations. For the double magic nucleus 16O, in the ground state we
take $\beta_{2}=0$. Since there are uncertainties in the definition of the
values of $\beta_{2}$ in light- and medium-mass nuclei, one can extract the
quadrupole deformation parameters of these nuclei from a comparison of the
calculated capture cross sections with the existing experimental data. By
describing the reactions 18O+208Pb, where there are no neutron transfer
channels with positive $Q$-values, we extract $\beta_{2}=0.1$ for the ground-
state of 18O EPJSub1 . This extracted value is used in our calculations.
Figure 3: The calculated capture cross sections versus $E_{\rm c.m.}$ for the
reactions 16O+114Sn and 18O+112Sn (solid line), 16O+120Sn and 18O+118Sn
(dashed line), 16O+126Sn and 18O+124Sn (dotted line). The calculated results
for the reactions 16O+114,120,126Sn and 18O+112,118,124Sn coincide,
respectively. The experimental data for the reactions 18O+112Sn (solid
squares), 18O+118Sn (open squares), and 18O+124Sn (open stars) are from Ref.
AOASn . Figure 4: The calculated capture cross sections versus $E_{\rm c.m.}$
for the reactions 16O+52Cr (dashed line) and 18O+50Cr (solid line).
Figures 1-4 show the capture excitation function for the reactions
16,18O+76,74Ge, 16,18O+94,92Mo, 16,18O+114,112,120,118,126,124Sn, and
16,18O+52,50Cr as a function of bombarding energy. One can see a relatively
good agreement between the calculated results and the experimental data Jia ;
16OAGe ; AO92Mo ; AOASn for the reactions 16O+76Ge, 16,18O+92Mo, and
18O+112,118,124Sn. The $Q_{2n}$-values for the $2n$-transfer processes are
positive (negative) for all reactions with 18O (16O). Thus, the neutron
transfer can be important for the reactions with the 16O beam. However, our
results show that cross sections for reactions 16O+76Ge
(16O+114,120,126Sn,52Cr) and 18O+74Ge (18O+114,118,124Sn,50Cr) are very
similar. The reason of such behavior is that after the $2n$-transfer in the
system 18O+A-2X$\to^{16}$O+AX the deformations remain to be similar. As a
result, the corresponding Coulomb barriers of the systems 18O+A-2X and 16O+AX
are almost the same and, correspondingly, their capture cross sections
coincide. The similar behaviour was observed in the recent experiments
16,18O+76,74Ge Jia .
One can see in Figs. 1-4 that at energies above and near the Coulomb barrier
the cross sections with and without two-neutron transfer are quite similar.
After the $2n$-transfer (before the capture) in the reactions
18O($\beta_{2}=0.1$) + 92Mo($\beta_{2}=0.05$)$\to^{16}$O($\beta_{2}=0$) +
94Mo($\beta_{2}=0.151$), 18O($\beta_{2}=0.1$) +
74Ge($\beta_{2}=0.283$)$\to^{16}$O($\beta_{2}=0$) + 76Ge($\beta_{2}=0.262$),
18O($\beta_{2}=0.1$)+112Sn($\beta_{2}=0.123$)$\to^{16}$O($\beta_{2}=0$)+114Sn($\beta_{2}=0.121$),
18O($\beta_{2}=0.1$)+118Sn($\beta_{2}=0.111$)$\to^{16}$O($\beta_{2}=0$)+120Sn($\beta_{2}=0.104$),
and
18O($\beta_{2}=0.1$)+124Sn($\beta_{2}=0.095$)$\to^{16}$O($\beta_{2}=0$)+126Sn($\beta_{2}=0.09$)
the deformations of the nuclei decrease and the values of the corresponding
Coulomb barriers increase. As a result, the transfer suppresses the capture
process at the sub-barrier energies. The suppression becomes stronger with
decreasing energy. As examples, in Fig. 1 and 2 we show this effect for the
reactions 18O+74Ge,92Mo.
The quantum diffusion approach was applied to study the role of the neutron
transfer with positive $Q$-value in the capture reactions 18O+50Cr, 18O+74Ge,
18O+92Mo, and 18O+112,118,124Sn at sub-barrier energies. We found that the
change of the magnitude of the capture cross section after the neutron
transfer occurs due to the change of the deformations of nuclei. The effect of
the neutron transfer is an indirect effect of the quadrupole deformation. If
in the reaction under consideration the deformations of nuclei decrease after
the neutron transfer, the neutron transfer suppresses the capture cross
section. As shown, the capture cross sections for the reactions
16O+52Cr,76Ge,94Mo,114,120,126Sn and 18O+50Cr,74Ge,92Mo,112,118,124Sn are
almost coincide, respectively.
This work was supported by DFG, NSFC, and RFBR. The IN2P3(France) -
JINR(Dubna) and Polish - JINR(Dubna) Cooperation Programmes are gratefully
acknowledged.
## References
* (1) V.V. Sargsyan, G.G. Adamian, N.V. Antonenko, and W. Scheid, Eur. Phys. J. A 45, 125 (2010); V.V. Sargsyan, G.G. Adamian, N.V. Antonenko, W. Scheid, and H.Q. Zhang, Eur. Phys. J. A 47, 38 (2011); J. of Phys.: Conf. Ser. 282, 012001 (2011); EPJ Web Conf. 17, 04003 (2011); V.V. Sargsyan, G.G. Adamian, N.V. Antonenko, W. Scheid, C.J. Lin, and H.Q. Zhang, Phys. Rev. C 85, 017603 (2012); Phys. Rev. C 85, 037602 (2012).
* (2) V.V. Sargsyan, G.G. Adamian, N.V. Antonenko, W. Scheid, and H.Q. Zhang, Phys. Rev. C 84, 064614 (2011); Phys. Rev. C 85, 024616 (2012); Phys. Rev. C 86, 014602 (2012).
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* (6) H. Hofmann, Phys. Rep. 284, 137 (1997); S. Ayik, B. Yilmaz, A. Gokalp, O. Yilmaz, and N. Takigawa, Phys. Rev. C 71, 054611 (2005); V.V. Sargsyan , Z. Kanokov, G.G. Adamian, and N.V. Antonenko, Part. Nucl. 41, 175 (2010); G. Hupin and D. Lacroix, Phys. Rev. C 81, 014609 (2010).
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|
arxiv-papers
| 2012-11-19T08:24:06 |
2024-09-04T02:49:38.171582
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "V. V. Sargsyan, G. G. Adamian, N. V. Antonenko, W. Scheid, and H. Q.\n Zhang",
"submitter": "Vazgen Sargsyan Dr.",
"url": "https://arxiv.org/abs/1211.4330"
}
|
1211.4341
|
# On the flat transverse momentum dependence of the single-spin asymmetry in
inclusive neutral pion production
S.M. Troshin, N.E. Tyurin
Institute for High Energy Physics,
Protvino, Moscow Region, 142281, Russia
We discuss recent experimental results from RHIC where the flat transverse
momentum dependence of a single-spin asymmetry has been found in the inclusive
production of neutral pions. This dependence takes place in a wide region of
the transverse momenta up to $p_{T}=10$ GeV/c. We emphasize that similar
dependence has been predicted in the nonperturbative spin filtering mechanism
for the single-spin asymmetries in hadron interactions and present some
implications for this mechanism from the new experimental data.
## Introduction
It should be noted that decrease with transverse momentum $p_{T}$ of
single–spin asymmetries (SSA) is a common feature of perturbative QCD
approaches including those based on account for the various modifications
implemented into the calculation scheme originally grounded on collinear
factorization. The most recent progress in this field is described in [1, 2].
The decreasing dependence has not been directly observed experimentally, but
the experimental data were not very conclusive due to large statistcal errors.
However, the existing data are consistent with the flat transverse dependence
of SSA in inclusive processes. This conclusion is valid for the old data on of
$\Lambda$–hyperon polarization [3], for example, and for the most recent data
obtained at RHIC [4]. It is essential that the new data cover the wide region
of the transverse momentum values up to $p_{T}\simeq 10$ GeV/c. In [6] it was
noted that experimental data at higher values of $p_{T}$ would be needed to
perform a more conclusive test of various pQCD theoretical approaches and
their predictions. The model proposed in [6] provided flat $p_{T}$ -dependence
of SSA. The new data which have appeared very recently[4, 5] are consistent
with the observed earlier trend, i.e. flat dependence on transverse momenta
can be extended to the region of higher values of $p_{T}$. Since such high
values of transverse momenta have been reached experimentally, one forced to
conclude that the mechanism of the SSA’s generation can have a nonperturbative
origin. Of course, presence of the significant statistical errors in the
current experimental data (cf. e.g [4, 5]) is the serious obstacle on the way
of derivation of a completely unambiguous final conclusion on the
impossibility of a decreasing dependence of the SSA with $p_{T}$.
## Highlights of the filtering mechanism of SSA generation and new
large-$p_{T}$ experimental data
A nonperturbative QCD dynamics is closely interrelated with the two well–known
phenomena, namely, color confinement and spontaneous breaking of chiral
symmetry ($\chi$SB)( cf. e.g. [7]). The $\chi$SB–mechanism resulting in
transition of current into constituent quarks is directly responsible for
generation of their masses and appearance of quark condensates. Constituent
quarks are colored objects, they appear to be quasiparticles and a hadron is
often represented as a loosely bounded system of the constituent quarks.
Simultaneously, the Goldstone bosons which are the excitations of the
condensates appear and mediate interactions of the constituent quarks.
This interaction is mainly due to a pseudoscalar pion field and has therefore
a spin–flip nature. Spin states filtering results from the unitarization
process in the $s$-channel and connects SSA with asymmetries in the position
(impact parameter) space [6].
The common features of SSA measurements at RHIC and Tevatron (linear increase
of asymmetry with $x_{F}$ and flat transverse momentum dependence at large
transverse momentum, $p_{T}>1$ GeV/c) are reproduced and described in the
framework of the semiclassical picture based on the further development of the
chiral quark model suggested in [8] and results of its use for the treatment
of the polarized and unpolarized inclusive cross-sections including those
obtained at RHIC [9].
We summarize now the essential features of the mechanism. Valence constituent
quarks are scattered simultaneously (due to strong coupling with Goldstone
bosons) and in a quasi-independent way by the effective strong field. In the
initial state of the reaction $pp_{\uparrow}\to\pi^{0}X$ the proton is
polarized and for the wave function of the proton we use the simple SU(6)
model. The constituent quark $Q_{\uparrow}$ with transverse up spin fluctuates
into Goldstone boson and another constituent quark $Q^{\prime}_{\downarrow}$
with down spin, performing spin-flip transition [10]:
$Q_{\uparrow}\to GB+Q^{\prime}_{\downarrow}.$ (1)
It should be noted that $\pi^{0}$-fluctuation of constituent quark does not
change its flavor and color. Assuming the equal probabilities of the processes
with $U$ and $D$ quarks:
$U_{\uparrow,\downarrow}\to\pi^{0}+U_{\downarrow,\uparrow}\quad\mbox{and}\quad
D_{\uparrow,\downarrow}\to\pi^{0}+D_{\downarrow,\uparrow},$ (2)
the production of $\pi^{0}$ by the polarized proton $p_{\uparrow}$ in this
simple $SU(6)$ picture can be treated as a result of the fluctuation of the
constituent quark $Q_{\uparrow}$ ($Q=U$ or $D$) in the effective field into
the system $\pi^{0}+Q_{\downarrow}$ (Fig. 1).
Figure 1: Chiral mechanism of $\pi^{0}$–production in polarized proton-proton
interaction.
Since the total angular momentum is conserved, the compensation of quark spin
flip should occur, i.e. to compensate quark spin flip $\delta{\bf S}$, an
orbital angular momentum $\delta{\bf L}=-\delta{\bf S}$ should be associated
with the final state of reaction (1). The introduction of $\delta{\bf L}$
implies a shift in the impact parameter value of the pion $\pi^{0}$:
$\delta{\bf S}\Rightarrow\delta{\bf L}\Rightarrow\delta\tilde{\bf b}.$
Note, that outside the hadron interior the Goldstone bosons are the usual
pions and kaons. Due to different strengths of interaction at the different
impact distances, i.e.
$\displaystyle p_{\uparrow}\Rightarrow Q_{\uparrow}$ $\displaystyle\to$
$\displaystyle\pi^{0}+Q_{\downarrow}\Rightarrow\;\;-\delta\tilde{\bf b},$
$\displaystyle p_{\downarrow}\Rightarrow Q_{\downarrow}$ $\displaystyle\to$
$\displaystyle\pi^{0}+Q_{\uparrow}\Rightarrow\;\;+\delta\tilde{\bf b}.$ (3)
the processes of transition $Q_{\uparrow}$ and $Q_{\downarrow}$ to $\pi^{0}$
will have different probabilities. It eventually leads to nonzero asymmetry
$A_{N}(\pi^{0})$. When the shift in impact parameter is negative,
$-\delta\tilde{\bf b}$, the interaction is stronger than that with the
positive shift, $+\delta\tilde{\bf b}$, and therefore the asymmetry
$A_{N}(\pi^{0})$ is positive too. The shift in $\tilde{\bf b}$ (the impact
parameter of final pion) is correlated with the shift of the impact parameter
of the collision according to the relation between impact parameters in the
multiparticle production process:
${\bf b}=\sum_{i}x_{i}{\tilde{\bf b}_{i}}.$ (4)
The production of $\pi^{0}$ with impact parameter $\tilde{\bf b}$ is
considered in the fragmentation region, i.e. at large $x_{F}$ the approximate
relation
${\bf b}\simeq x_{F}\tilde{\bf b},$ (5)
which results from Eq. (4) has been used with additional assumption on the
small values of Feynman $x_{F}$ for the other particles.
It should be noted that direction of spin of the proton is chosen to be along
or opposite to the impact parameter vector ${\bf b}$. The integration over the
asimuthal angle $\varphi$ selects therefore the finctions $I$ (see below)
corresponding to the respective directions of the proton spin. The mechanism
is illustrated in Fig. 2.
Figure 2: Schematic illustration of the SSA’s generation mechanism due to spin
flip of the constituent quark Q.
An essential feature of the mechanism is an account of unitarity in the direct
channel of reaction. The corresponding formulas for inclusive cross–sections
of the process
$p^{\uparrow,\downarrow}+p\to\pi^{0}+X,$
have been obtained in [11]:
${d\sigma^{\uparrow,\downarrow}}/{d\xi}=8\pi\int_{0}^{\infty}bdb{I^{\uparrow,\downarrow}(s,b,\xi)}/{|1-iU(s,b)|^{2}},$
(6)
$b$ is the collison impact parameter. The function $U(s,b)$ is the generalized
reaction matrix (averaged over initial spin states) which is determined by the
basic dynamics of the elastic scattering. The elastic scattering amplitude in
the impact parameter representation $F(s,b)$ is then given [8] by the
relation:
$F(s,b)=U(s,b)/[1-iU(s,b)].$ (7)
The equation (7) allows one to obey unitarity for the elastic scattering
amplitude provided the inequality $\mbox{Im}\,U(s,b)\geq 0\,$ takes place. The
model [8] has been used for construction of the functional dependence of
$U(s,b)$, namely, this function was chosen as a product of the factors
corresponding to the averaged amplitudes of the individual valence quarks. The
strong interaction radius of the quarks is determined by its mass,
$r_{Q}=\zeta/m_{Q}$
The parameter $\zeta$ was extracted from the experimental data for the
differential cross-section of the elastic $pp$-scattering. In the region of
medium values of $t$ this model provides[8] the familiar Orear-type behavior:
$\frac{d\sigma}{dt}\sim\exp\left(-\frac{2\pi\zeta}{M}\sqrt{-t}\right),$
where $M$ is equal to the total mass of the constituent quarks in the two
colliding protons, i.e. $M=6m_{Q}\simeq 2$ GeV/c, and the value of parameter
$\zeta\simeq 2$ since from the experimental data $m_{Q}/\zeta=150-200$ MeV and
to reproduce the standard constituent quark masses the value of $\zeta$ should
be around 2.
The functions $I^{\uparrow,\downarrow}$ in Eq. (6) can be expressed through
the functions $U_{n}^{\uparrow,\downarrow}$ – the multiparticle analogs of the
function $U$ [11] in the polarized case. The set of the kinematical variables
$\xi$ ($x_{F}$ and $p_{T}$ for example) describe the state of the produced
pion.
We assume that the shift $\delta\tilde{b}$ can be connected with the radius of
quark interaction $r_{Q}^{flip}$, which is responsible for the quark
transition flipping its spin, i.e.:
$\delta\tilde{b}\simeq r_{Q}^{flip}.$
Asymmetry $A_{N}$ can be written in terms of the functions $I_{-}$, $I_{0}$
and $U$:
$A_{N}(s,\xi)=\frac{\int_{0}^{\infty}bdbI_{-}(s,b,\xi)/|1-iU(s,b)|^{2}}{2\int_{0}^{\infty}bdbI_{0}(s,b,\xi)/|1-iU(s,b)|^{2}},$
(8)
where $I_{0}=1/2(I^{\uparrow}+I^{\downarrow})$ and
$I_{-}=(I^{\uparrow}-I^{\downarrow})$ and $I_{0}$ obey the sum rule
$\int I_{0}(s,b,\xi)d\xi=\bar{n}(s,b)ImU(s,b),$
here $\bar{n}(s,b)$ stands for the mean multiplicity in the impact parameter
representation.
With the above relation for the shift in the impact parameter due to the
transition flipping quark spin the following expression for asymmetry
$A_{N}^{\pi^{0}}$ can be written
$A_{N}^{\pi^{0}}(s,\xi)\simeq-
x_{F}r_{Q}^{flip}\frac{1}{3}\frac{\int_{0}^{\infty}bdbI^{\prime}_{0}(s,b,\xi)db/|1-iU(s,b)|^{2}}{\int_{0}^{\infty}bdbI_{0}(s,b,\xi)/|1-iU(s,b)|^{2}},$
(9)
where $I^{\prime}_{0}(s,b,\xi)={dI_{0}(s,b,\xi)}/{db}$. It is evident that
$A_{N}^{\pi^{0}}(s,\xi)$ should be positive because
$I^{\prime}_{0}(s,b,\xi)<0$.
In the model [8] the function $U(s,b)$ is chosen to be a product of the
averaged quark amplitudes under assumption of the the quasi-independence of
valence constituent quark scattering in the self-consistent mean field. The
function $U(s,b)$ in a pure imaginary case for the elastic scattering
amplitude, which we consider here for simplicity has been written in the
following form
$U(s,b)=i\tilde{U}(s,b)=ig(s)\exp(-Mb/\zeta),$ (10)
where the factor $g(s)$ increases at large values of $s$ like a power of
energy:
$g(s)=\left[1+\alpha\frac{\sqrt{s}}{m_{Q}}\right]^{N},$
$M$ is the total mass of $N$ constituent quarks with mass $m_{Q}$ in the
initial hadrons and parameter $\zeta$ determines a universal scale for the
quark interaction radius without flipping its spin, i.e. $r_{Q}=\zeta/m_{Q}$.
To evaluate asymmetry dependence on $x_{F}$ and $p_{T}$ the semiclassical
correspondence between transverse momentum and impact parameter has been used.
Integrating by parts, we can rewrite the expression for the asymmetry in the
form:
$A_{N}^{\pi^{0}}(s,\xi)\simeq
x_{F}r_{Q}^{flip}\frac{M}{3\zeta}\frac{\int_{0}^{\infty}bdbI_{0}(s,b,\xi)\tilde{U}(s,b)/[1+\tilde{U}(s,b)]^{3}}{\int_{0}^{\infty}bdbI_{0}(s,b,\xi)/[1+\tilde{U}(s,b)]^{2}},$
(11)
At small values of $b$ the values of $U$-matrix are large, and we can neglect
unity in the denominators of the integrands. Thus the ratio of the two
integrals (after integration by parts of nominator in Eq. (11)) is of order of
unity, i.e. the energy and $p_{T}$-independent behavior of asymmetry
$A_{N}^{\pi^{0}}$ takes place at the values of transverse momentum $p_{T}\gg
x_{F}/R(s)$:
$A_{N}^{\pi^{0}}(s,\xi)\simeq x_{F}r_{Q}^{flip}\frac{M}{3\zeta}.$ (12)
This flat transverse momentum dependence of asymmetry results from the
similarity of the rescattering effects for the different spin states, i.e.
spin-flip and spin-nonflip interactions undergo similar absorption at short
distances and the relative magnitude of this absorption does not depend on
energy. It can be considered as a one of the manifestations of the unitarity.
The numerical value of polarization $A_{N}^{\pi^{0}}$ can be significant.
Indeed, there is no small factor in (12). The function $R(s)$ is the hadron
interaction radius ($R(s)\sim\ln s$), the typical numerical falue of $R$ has
been taken to be equal to 1 fm. Thus, the typical value of $x_{F}/R(s)$ is 0.1
GeV/c and the Eq. (12) which is valid at $p_{T}\gg x_{F}/R(s)$, should be
applicable in the region $p_{T}>1$ GeV/c. The value of $r_{Q}^{flip}$ is of
order $\sim 10^{-1}$ fm on the basis of the model estimates [6, 8, 11]. The
radius of quark interaction $r_{Q}^{flip}$ responsible for the transition
$Q_{\uparrow}\to Q_{\downarrow}$ changing quark spin. The linear increase of
asymmetry with $x_{F}$ follows from the above considerations which, of course,
are approximate and valid at $x_{F}$ around unity. Therefore, at smaller
values of $x_{F}$ the linear dependence is distorted.
Thus, Eq. (12 is valid in the region of large $x_{F}$. The flat dependence of
asymmetry on $p_{T}$ provided by this relation is consistent with the new data
from RHIC (we have used data available at the largest values of $x_{F}$ due to
approximation made in the model and discussed above) [4, 5] (cf. Fig. 2,3).
Figure 3: $p_{T}$ -dependence of the asymmetry $A_{N}$ in the process
$p_{\uparrow}+p\to\pi^{0}+X$ at RHIC, preliminary data from [4, 5] correspond
to pion isolation of 70 mR, $\sqrt{s}=500$ GeV and $0.32<x_{F}<0.40$.
Figure 4: The same plot as in Fig.3, but with data corresponding to pion
isolation of 30 mR.
Comparison with the data allows one to estimate the value of $r_{Q}^{flip}$
more precisely , namely $r_{Q}^{flip}\simeq 0.05$ fm. Similar mechanism
generates SSA in the inclusive production of charged pions. It should be noted
that dependencies of SSA on $p_{T}$ consistent with the flat ones have also
been observed at lower values of $x_{F}$, namely, in the two regions
$0.16<x_{F}<0.24$ and $0.24<x_{F}<0.32$ [4, 5]. The data demonstrate increase
of SSA with $x_{F}$. Due to limitation of the model for the large $x_{F}$
region, we have not used those data, but, in principle, the model is in
agreement with them too. To be honest, one should not compare the predictions
of the model which are valid for $x_{F}$ in the vicinity of unity with the
experimental data wich have highest value of $x_{F}$ around 0.4. However, the
data demonstrate similar flat dependence on transverse momentum in the the
rather wide range $x_{F}$ from 0.16 to 0.4. We therefore assume that the
similar dependence will be valid at $x_{F}$ values around unity and perform
comparison with the experiment. It should be clarified, that the above
agreement with the data has a qualitative value only and for the quantitative
analysis the data at higher values of $x_{F}$ are required.
The mechanism of chiral quark fluctuation in the effective field with spin
flip is suppressed compared to the direct elastic scattering of quarks and,
therefore, it should not play a significant role e.g. in the reaction
$pp_{\uparrow}\to pX$ in the fragmentation region, but evidently it is not the
case for the reaction $pp_{\uparrow}\to nX$. The above features can be
observed experimentally: asymmetry $A_{N}$ is consistent with zero for proton
production and significantly deviates from zero for the neutron production in
the forward region.
In ref. [14] an important issue has been raised, namely a model when trying to
explain spin asymmetries should simultaneously describe the data for the
unpolarized inclusive cross-sections. In this approach with the effective
degrees of freedom – constituent quarks and Goldstone bosons – unpolarized
inclusive cross–section at high transverse momenta fllows a generic power-like
dependencies on $p_{T}$ . At high $p_{T}$ the power-like dependence
$p_{T}^{-n}$ with $n=6$ takes place. It originates from the singularity at
zero impact parameter $b=0$. The exponent $n$ does not depend on $x_{F}$. This
$p_{T}^{-6}$–dependence of the unpolarized inclusive cross–section is
consistent with the respective data dependence on the transverse momentum
(Fig. 5) [6].
Figure 5: Transverse momentum dependence of unpolarized inclusive
cross–section, experimental data from [9].
## Conclusion
It was shown that prediction of the spin filtering mechanism [6] on the flat
$p_{T}$-dependence of single-spin asymmetries is consistent with the new
experimental data from RHIC. As it often happens, this interpretation is not
unique. Such flat dependence can result from finite size of a constituent
quark and a presence of the orbital angular momentum of the current quarks
residing inside the constituent one [12, 13]. This mechanism is based on the
similar ideas as the spin filtering one but is, in principle, different and
has no predictive power for $x_{F}$ dependence of SSA. Thus, we would like to
treat the seemingly flat $p_{T}$-dependence of SSA in favor of the spin
filtering mechanism, while the presence of the internal orbital momentum in
the structure of constituent quarks still remains to be an interesting option
and cannot be excluded at the moment (cf. e.g. [15]).
Finally, we would like to stress again that the experimental data set [4, 5]
is the preliminary one and we hope that the final data will show smaller error
bars and allow one to provide a quantitative discrimination of the model
predictions for SSA.
## Acknowledgment
We are grateful to Yuri Kovchegov for the information on the new experimental
data from RHIC and the interesting discussions.
## References
* [1] Y.V. Kovchegov, M.D. Sievert, Phys. Rev. D 86 (2012) 034028.
* [2] A. Metz, D. Pitonyak, arXiv:1212.5037v1, 2012.
* [3] L. Pondrom, Phys. Rep. 122 (1985) 57.
* [4] S. Heppelmann (for STAR Collaboration), Talk at 2012 RHIC and AGS Annual Users’ Meeting, BNL, Upton, June 12-15, 2012.
* [5] G. Igo, Talk at the Interanational Workshop on Diffraction in High Energy Physics (Diffraction 2012), Puerto del Carmen, Lanzarote, Canary Islands, Spain, September 10-15, 2012.
* [6] S.M. Troshin, N.E. Tyurin, Phys. Part. Nucl. 41 (2010) 54.
* [7] H. Georgi, A. Manohar, Nucl. Phys. B 234 (1984) 189.
* [8] S.M. Troshin, N.E. Tyurin, Phys. Rev. D 49 (1994) 4427.
* [9] J. Adams et al.(STAR Collaboration), Phys. Rev. Lett. 92 (2004) 171801.
* [10] T.P. Cheng, L.F. Li, Phys. Rev. Lett. 80 (1998) 2789.
* [11] S.M. Troshin, N.E. Tyurin, Z. Phys. C 45 (1989) 171.
* [12] S.M. Troshin, N.E. Tyurin, Phys. Rev. D 52 (1995) 3862.
* [13] V.V. Mochalov, S.M. Troshin, A.N. Vasiliev, Phys. Rev. D69 (2004) 077503.
* [14] C. Bourrely, J. Soffer, Eur. Phys. J. C 36 (2004) 371.
* [15] F. Arash, Talk at the Interanational Workshop on Diffraction in High Energy Physics (Diffraction 2012), Puerto del Carmen, Lanzarote, Canary Islands, Spain, September 10-15, 2012.
|
arxiv-papers
| 2012-11-19T09:14:27 |
2024-09-04T02:49:38.177077
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "S. M. Troshin, N. E. Tyurin",
"submitter": "Sergey Troshin",
"url": "https://arxiv.org/abs/1211.4341"
}
|
1211.4567
|
# Superradiant Superconductivity
G. Baskaran The Institute of Mathematical Sciences, C.I.T. Campus, Chennai
600 113, India
Perimeter Institute for Theoretical Physics, Waterloo, Ontario, Canada N2L 2Y5
###### Abstract
We suggest possibility of Dicke superradiance in superconductors. The
necessary 2-level atoms are identified with Anderson pseudo spins in k-space,
seeing a k-dependent self consistent mean field. A way to couple these 2-level
bose atoms to a macroscopically excited coherent boson mode and create a novel
nonequilibrium superradiant superconductivity (SRSC) is suggested. Our
coherence transfer mechanism offers a hope to realize transient
superconductivity, even at room temperatures, in the pseudo gap phase of
certain underdoped cuprates. Recent experiments are briefly discussed in the
light of our theory. Quantum entanglement, QCP and superfluorescence
properties follow.
Introduction Superconductivity is a remarkable macroscopic manifestation of
quantum mechanics. A rich physics and phenomenology, including Meissner and
Josephson effects are parts of superconductivity BCS . Dicke’s Superradiance
Dicke1954 is another macroscopic manifestation, exhibited by a collection of
2-level atoms interacting with a single boson mode. The coupled system can
develop quantum coherence, enhanced emission properties and complex dynamics.
Certain phenomena in NMR, ESR, optics and cold atoms are related to
superradiance.
In the present work we suggest a way to combine superconductivity and
superradiance, We call the resultant non equilibrium state as superradiant
superconductivity (SRSC). In our proposal a macroscopically occupied long
wavelength single boson mode interacts with a collection of independent
2-level atoms located in k-space and creates a Dicke superradiant situation,
under certain conditions. In this state certain deformation of Cooper pair
wave function is entangled with a coherent external bosonic mode.
Interaction of coherent electromagnetic radiation and ultrasound with
superconductor is a well studied subjectMicrowaveExpt ; Eliashberg ;
OwenScalapino ; KumarSinha ; McIntoshLindesay . Our proposal of SRSC may have
relevance to some known results. An exciting recent development is
experimental observation of transient superconductivity well above Tc, induced
by certain femtosecond laser pulses, in the pseudo gap phase of cuprates
liscLBCO ; liscYBCO .
In a pioneering theoretical work Eliashberg Eliashberg in 1970 showed that
microwave induced quasi particle redistribution self consistently enhances gap
values and Jc. Works by Scalapino, Owen and Chang OwenScalapino , also focused
on quasi particle redistribution. In a later theory in 1994, McIntosh and
Lindesey McIntoshLindesay showed that stimulated emission and reabsorption of
photon by correlated electron pairs play a fundamental role in
superconductivity enhancement. This key insight is one of the triggers for our
proposal. Interestingly, in 1968, there was a theoretical suggestion
KumarSinha for photon induced room temperature superconductivity.
In what follows, we start with an ideal BCS superconductor and show how Dicke
superradiance emerges, when the wavelength of the macroscopically occupied
external single boson mode $\lambda\geq L$, the sample size $L$. Then we
discuss how our mechanism could gnerate transient superconductivity abouve Tc
and discuss recent experiments liscLBCO and (see note liscYBCO in the light
of our mechanism.
In our work we make the tacit assumption that there are suitable relaxation
processes involving quasiparticles and phonons that drains energy to the heat
bath efficiently to avoid heating. At the same time some energy gets pumped to
the electronic sub system to help reach a new non equilibrium coherent state
for a short time scale. It is the nature of non equilibrium coherent state
that we are after. To achieve this we assume that the coherent state of the
single boson mode is long lived and does not radiate away its energy. It
exchanges its quanta with the electron subsystem only and gets quantum
entangled. Ours is an equilibrium statistical mechanics approximation tailored
to get a glimpse of a remarkable non equilibrium situation.
Model. To develop our theory we follow Anderson’s pseudo spin formulation of
BCS theory PWApseudoSpin . It helps us to view BCS mean field eigen states as
a k-space lattice containing 2-level bose atoms and free fermions. Consider
time reversed single particle states $({\bf k}\uparrow,{\bf-k}\downarrow)$,
with empty state written as $|0\rangle_{\bf k}$. To generate complete Fock
space, we need only 4 states in each $({\bf k}\uparrow,{\bf-k}\downarrow)$ :
i) $|0\rangle_{\bf k}$, ii)
$c^{\dagger}_{k\uparrow}c^{\dagger}_{-k\downarrow}|0\rangle_{\bf k}$, iii)
$c^{\dagger}_{k\uparrow}|0\rangle_{\bf k}$ and iv)
$c^{\dagger}_{-k\downarrow}|0\rangle_{\bf k}$. BCS interaction mixes only the
0 and 2-fermion states. Resulting ground and excited paired states are two
orthogonal states:
$|g\rangle_{k}\equiv(u_{k}+v_{k}c^{\dagger}_{k\uparrow}c^{\dagger}_{-k\downarrow})|0\rangle_{k}$
and
$|e\rangle_{k}\equiv(u_{k}c^{\dagger}_{k\uparrow}c^{\dagger}_{-k\downarrow}-v_{k})|0\rangle_{k}$.
We call these 2-level bosonic states as Anderson atom or A-atom. A-atom
carries zero total momentum. Single fermion states
$c^{\dagger}_{k\uparrow}|0\rangle_{\bf k}$ and
$c^{\dagger}_{-k\downarrow}|0\rangle_{\bf k}$, in $({\bf
k}\uparrow,{\bf-k}\downarrow)$ remain unaffected by BCS interaction.
An A-atom close to fermi suface is special (see note note2 ). It is a coherent
superposition of 0 and 2-electron states. Consequent non zero value of the
product $u_{k}v_{k}$, around the fermi surface quantifies superconductivity.
BCS mean field Hamiltonian has the familiar form:
$H_{mf}=\sum\varepsilon_{k}\alpha^{\dagger}_{k\sigma}\alpha_{k\sigma},$ (1)
where, Bogoliubov quasi particle operators $\alpha^{\dagger}_{k\sigma}\equiv
u_{k}c^{\dagger}_{k\sigma}+\sigma v_{k}c_{-k-\sigma}$ and
$\alpha_{k\sigma}\equiv u_{k}^{*}c_{k\sigma}+\sigma
v_{k}^{*}c^{\dagger}_{-k-\sigma}$. The quasi particle energy
$\varepsilon_{k}\equiv\sqrt{(\frac{\hbar
k^{2}}{2m}-\mu)^{2}+\triangle_{k}^{2}}$.
Complete set of BCS mean field eigen states can be written as product over all
states, $({\bf k}\uparrow,{\bf-k}\downarrow)$, each containing either an
A-atom in the ground or excited state or a single upspin or down spin fermion
state. Bogoliubov quasi particle operators have very simple action on the BCS
eigen states. BCS vacuum,
$|BCS\rangle=\prod_{k}(u_{k}+v_{k}c^{\dagger}_{k\uparrow}c^{\dagger}_{-k\downarrow})|0\rangle$
is annihilated by the annihilation operator, $\alpha_{q\sigma}|BCS\rangle=0$.
Bogoliubov creation operator, while acting on the BCS ground state, removes an
A-atom and replaces it by a fermion:
$\alpha^{\dagger}_{q\uparrow}|BCS\rangle=c^{\dagger}_{q\uparrow}\prod_{k\neq
q}(u_{k}+v_{k}c^{\dagger}_{k\uparrow}c^{\dagger}_{-k\downarrow})|0\rangle$ and
$\alpha^{\dagger}_{-q\downarrow}|BCS\rangle=c^{\dagger}_{-q\downarrow}\prod_{k\neq
q}(u_{k}+v_{k}c^{\dagger}_{k\uparrow}c^{\dagger}_{-k\downarrow})|0\rangle$.
What is the operator that excites an A-atom ? Pair of Bogoliubov quasi
particle operators
$\alpha^{\dagger}_{q\uparrow}\alpha^{\dagger}_{-q\downarrow}$, with total
momentum zero and total spin projection zero, acting within $({\bf
q}\uparrow,{\bf-q}\downarrow)$ excites an A-atom:
$\alpha^{\dagger}_{q\uparrow}\alpha^{\dagger}_{-q\downarrow}|BCS\rangle=(u_{q}c^{\dagger}_{q\uparrow}c^{\dagger}_{-q\downarrow}-v_{q})\prod_{k\neq
q}(u_{k}+v_{k}c^{\dagger}_{k\uparrow}c^{\dagger}_{-k\downarrow})|0\rangle$.
The 2-level (bosonic) A-atom subspace can be studied using pseudo spin (Pauli)
operators. Pseudo spin operators (see note note3 ) are defined as,
$\sigma^{z}_{k}\equiv(1-\alpha^{\dagger}_{k\uparrow}\alpha_{k\uparrow}-\alpha^{\dagger}_{-k\downarrow}\alpha_{-k\downarrow}),\leavevmode\nobreak\
\sigma^{+}_{k}\equiv\alpha^{\dagger}_{k\uparrow}\alpha^{\dagger}_{-k\downarrow}$
and $\sigma^{-}_{k}\equiv\alpha_{-k\downarrow}\alpha_{k\uparrow}$.
The BCS mean field Hamiltonian (equation 1) in the boson subspace takes a
suggestive form :
$H_{mf}=-\sum\varepsilon_{k}\sigma_{k}^{z},$ (2)
It describes a collection of non-interacting pseudo spins in the prsence of a
k-dependent magnetic field of magnitude $\varepsilon_{k}$. Energy level
separation of a 2-level A-atom is 2$\varepsilon_{k}$. Notice that long range
interaction in k-space in the BCS Hamiltonian leads to free spins in the mean
field description, but in the presence of a self consistent mean field of
magnitude $\varepsilon_{k}$ in k-space. In our pseudo spin basis BCS ground
state is a fully aligned ferromagnet, while in Andersons basis pseudo spins
twist to form Bloch wall across the fermi surface in k-space (see note note2
).
Now we consider a simple way to couple A-atoms selectively to a single
external boson mode, with creation and annihilation operators
($b^{\dagger},b$). Interaction of electrons with this mode, in the long wave
length (zero momentum transfer) limit, $\lambda>>L$, where $L$ is the size of
the sample, has a simple form:
$H_{int}=\frac{1}{{\sqrt{N}}}\sum
B_{k}(c^{\dagger}_{k\sigma}c_{k\sigma}+H.c.)(b+b^{\dagger})$ (3)
Here Bk is a momentum dependent coupling constant and N $\sim$ number of
electrons in the interaction region. In terms of Bogoliubov quasiparticle
operators,
$\displaystyle H_{int}=\frac{1}{{\sqrt{N}}}\sum
B_{k}(u_{k}^{2}-B_{-k}v_{k}^{2})\alpha^{\dagger}_{k\sigma}\alpha_{k\sigma}(b+b^{\dagger})+\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ $ (4) $\displaystyle+$
$\displaystyle\frac{1}{{\sqrt{N}}}\sum(B_{k}+B_{-k})u_{k}v_{k}(\alpha^{\dagger}_{k\uparrow}\alpha^{\dagger}_{-k\downarrow}+H.c.)(b+b^{\dagger})$
We ignore non resonant terms using rotating wave approximation. Further
quasiparticle number operators can be also be taken care of using Hartree type
of approximations. We are left with the important pair annihilation and
creation terms:
$H_{int}\approx\frac{1}{{\sqrt{N}}}\sum
B_{k}u_{k}v_{k}(\alpha^{\dagger}_{k\uparrow}\alpha^{\dagger}_{-k\downarrow}b+\alpha_{-k\downarrow}\alpha_{k\uparrow}b^{\dagger})$
(5)
Interms of pseudo spin operators it takes the form
$H_{int}\approx\frac{1}{{\sqrt{N}}}\sum
B_{k}u_{k}v_{k}(\sigma^{+}_{k}b+\sigma^{-}_{k}b^{\dagger})$. Thus the final
form of the Hamiltonian of the superconductor interacting with a single boson
mode is:
$H=\hbar\omega_{0}(b^{\dagger}b+\frac{1}{2})-\sum\varepsilon_{k}\sigma_{k}^{z}+\frac{1}{{\sqrt{N}}}\sum\lambda_{k}(\sigma_{k}^{+}b+\sigma_{k}^{-}b^{\dagger})$
(6)
where $\lambda_{k}\equiv(B_{k}+B_{-k})u_{k}v_{k}$. Equation 5 is a generalized
Dicke Hamiltonian Dicke1954 , where 2-level atoms in k-space have a
k-dependent energy level separation, The sum, N${}_{t}\equiv$ N∗ \+ Nboson of
number of excited N∗ atoms and number of photons Nboson, commutes with the
Hamiltonian (equation 6)2$\varepsilon_{k}$.
Finding a Dicke like Hamiltonian is a key result of our paper, from which
several consequences follow.
Notice that A-atom-boson mode coupling
$\lambda_{k}\equiv(B_{k}+B_{-k})u_{k}v_{k}$ is appreciable only in regions
where the product $u_{k}v_{k}$ is appreciable. That is, possibility of
superrandiace is intimately connected with pairing phenomenon. The matrix
element Bk = - B-kfor electron-photon coupling. And Bk = + B-k for electron-
acoustic phonon coupling BCS . Thus in simple geometries, $\lambda_{k}=0$ for
electron-electromagnetic radiation coupling.
Our restriction to bosonic subspace and our effective Hamiltonian is a good
low temperature approximation because i) kBT << $\Delta_{0}$, the minimum
superconducting gap and density of thermal fermionic quasi particles is small
and ii) when $\lambda>>L$, the boson mode excites only the A-atoms.
More importantly, we have ignored back reaction, i.e., self consistent
modification of uk, vk or gap function $\Delta_{k}$, arising from interaction
with the boson mode. We will see later that selfconsistent modification
reinforces superradiant superconductivity.
To illustrate superradiance, consider a simple Dicke Hamiltonian, with
identical two level atoms in resonance with the boson mode,
$H_{D}=\hbar\omega_{0}(b^{\dagger}b+\frac{1}{2})-\frac{\hbar\omega_{0}}{2}\sum_{i}\sigma^{z}_{i}+\frac{g}{\sqrt{N}}\sum_{i}(b^{\dagger}\sigma^{-}_{i}+b\sigma^{+}_{i})$.
For every value of Nt there is an unique ground state, a nodeless in phase
superposition of degenerate states with real positive coefficients. The ground
state is a superradiant state capable of undergoing a spontaneous emission
with an emission strength that scales as N${}^{2}_{t}$.
For our purpose consider a superconductor at T = 0 in the presence of a
macroscopically occupied single boson mode $|N_{b}\rangle$ and allow the
coupled system to evolve in time. When $\hbar\omega_{0}$ start increasing
towards $\triangle_{0}$, minimum of the two quasi particle gap, a set of
k-points which are near resonance with energy of a boson quanta actively
participate in superradiance and modify the ground state wave function. Net
density of these active A-atoms depend on quasi particle density of states and
the coupling constant $\lambda_{k}$.
Dicke Hamiltonian, equation 6, admits Bethe Ansatz solutionBetheAnsatz for
k-independent $\lambda_{k}=\lambda_{0}$. Using the approximation,
$\lambda_{k}\approx\lambda_{0}$ for our set of near resonant A-atoms, our
ground state wave function has Bethe Ansatz form:
$\displaystyle|SRSC\rangle$ $\displaystyle\sim$
$\displaystyle(b^{\dagger}+\sum_{k}w_{k}\leavevmode\nobreak\
\alpha^{\dagger}_{k\uparrow}\alpha^{\dagger}_{-k\downarrow})^{N_{b}}|BCS\rangle\otimes|0_{b}\rangle$
(7) $\displaystyle\equiv$
$\displaystyle(b^{\dagger}+\sum_{k}w_{k}\leavevmode\nobreak\
\sigma^{+}_{k})^{N_{b}}|BCS\rangle\otimes|0_{b}\rangle$
Here $|0_{b}\rangle$ is the vacuum of the single boson mode. Superradiance
mixes (hybridizes or entangles) two nearly degenerate neutral modes. One is
the single mode external Bose oscillator. Second is a coherent sum of zero
momentum Bogoliubov pair excitations,
$\sum_{k}w_{k}\alpha^{\dagger}_{k\uparrow}\alpha^{\dagger}_{-k\downarrow}$; or
equivalently an Anderson pseudo spin wave packet mode in k-space. It is easy
to show that the second boson mode is a dynamic deformation of the Cooper pair
wave function (in the relative coordinate of the two electrons, characterized
by wk). The center of mass degree of freedom of the Cooper pairs, and hence
the phase of superconducting order parameter is not directly influenced by
superradiance phenomenon.
Superradiance effect in an s-wave superconductor is maximum, when the boson
frequency $\hbar\omega_{0}$ passes through minimum gap 2$\Delta_{o}$, where
quasi particle density of states has a maximum. If a superconductor supports
excited Cooper bound states below $2\Delta_{0}$, depending on the symmetry of
the excited states, there will be enhanced superradiance around these bound
state energies.
It follows from our work that one should be able to see i) a well known
quantum phase transition HeppLieb , as a function $\omega,\lambda$ ii)
enhanced quantum entanglement TobiasEntanglement around the transition point
and iii) superfluorescence Superfluorescence .
Application to Pseudogap Phase of Cuprates Having theoretically suggested
possibility of Dicke superradiance in a BCS superconductor, we will address
recent experimental observation liscLBCO ; liscYBCO of femtosecond laser
induced transient superconductivity in the pseudogap normal state of some
cuprates. In the two experiments two different Cu-O bond stretching modes are
resonantly excited by an 80 meV ( $\sim$ 20 THz) femtosecond laser. In view of
resonance, laser pumps its energy and coherence to the infrared phonon mode.
Electronic subsystem receives its energy and coherence from the infrared mode.
We have a phonon-photon polariton Hamiltonian:
$H=\hbar\omega_{0}a^{\dagger}a+\hbar\omega_{0}b^{\dagger}b+g(a^{\dagger}b+H.c.)$
(8)
Here $(a^{\dagger},a)$ and $(b^{\dagger},b)$ are the photon and phonon
operators respectively. As wavelengths of 20 THz infrared radiation and the
optic modes is $\sim 150$ microns, we will approximate the wavelengths by size
of the sample. The phonon optical mode coupling ‘g’ is of the order of 10 meV.
This coupling will lead to interesting Rabi oscillation between two modes,
after the femtosecond photon pulse impinges on the superconducting crystal.
It is safe to assume that Cu-O stretching lattice modes in both experiments
modulate i) site energy and ii) the hopping matrix element ‘t’ of the tight
binding electronic Hamiltonian for cuprates. To leading order in the normal
coordinate displacement u of this mode we have $t=t_{0}+\frac{\partial
t}{\partial u}|_{0}u\equiv
t_{0}+\alpha_{t}(b^{\dagger}+b){\rm\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ and}$ to is the value of hopping
integral in the absence of resonant excitation of Cu-O stretching infrared
mode.
As far as the pseudogap normal state of cuprates is concerned, there are
experimental evidences Ong ; RamanPseudoGap ; STM and theoretical support
BZAEmeryKivelson that this metallic state has substantial pairing amplitude
and a strong phase fluctuations. It is well described as a 2D vortex liquid
above a Kosterlitz-Thouless transition point. This is borne out by Nernst
effect Ong , Raman effect RamanPseudoGap , among other experiments. In what
follows we propose an effective Hamilotonian that is an expression of the fact
that pseudogap phase gap supports local superconductivity. We assumes presence
of equal density of positive and negative vortices that are quasi static and
spatially random. That is, the thermal vortices are slowly moving compared to
time scale of interest to us.
Our effective Hamiltonian for pseudogap normal state has the form:
$H_{\rm normal}=\sum\varepsilon_{m}\alpha^{\dagger}_{\sigma}\alpha_{m\sigma}$
(9)
The index m denotes eigen modes of Bogolibov quasiparticle operator
$\alpha^{\dagger}_{m\sigma}\equiv u_{m}c^{\dagger}_{m\sigma}+\sigma
v_{m}c_{-m-\sigma}$ . In view of presence of disordered vortices in the
background, single particle eigen modes are not Bloch states; some localized
and rest extended. Our conclusions hold good even for the d-wave symmetry
situation in cuprates.
In the absence of external magnetic field we have pairs of degenerate single
particle eigen states (m$\uparrow$, - m $\downarrow$), connected by time
reversal symmetry. As in the BCS case, a pair subspace (m$\uparrow$, - m
$\downarrow$) is occupied by A-atom in its ground or excited state or an
unpaired fermion. By using same arguments as in the BCS case, bosonic
excitation sector in the normal state pseudo gap phase, coupled to the single
phonon mode gives us a Dicke type pseudo spin Hamiltonian:
$H=\hbar\omega_{0}(b^{\dagger}b+\frac{1}{2})+\sum\varepsilon_{m}\sigma^{z}_{m}+\frac{1}{{\sqrt{N}}}\sum\lambda_{i}(\sigma^{+}_{m}b+\sigma^{-}_{m}b^{\dagger})$
(10)
Here the operator
$\alpha^{\dagger}_{m\uparrow}\alpha^{\dagger}_{{-m}\downarrow}\equiv\sigma^{+}_{m}$
excites an A-atom.
In terms of A-atom and ferminic quasi particle there is a key difference
between the BCS supercodnuctor and the cuprate superconductors above Tc. In a
standard BCS superconductor, the pair subspace $({\bf
k}\uparrow,{\bf-k}\downarrow)$, is dominated by fermionic quasi particles and
nearly vanishing density of A-atom. Whereas, in the pseudogap, which exists
over a wide temperature range above Tc, the pair subspace (m$\uparrow$, -m
$\downarrow$) is dominated by ground and excited A-atoms and nearly vanishing
density of fermions. This makes pseudogap phase special and susceptible for
transient superconductivity.
To understand how superradiance induces transient superconductivity in the
pseudo gap phase, we have to go beyond our model Hamiltonian (equation 10) and
consider selfconsistent modification of um and vm’s. We offer a feed back
mechanism. Qualitatively it is as follows. A subspace (m$\uparrow$, - m
$\downarrow$) contains A-atoms with high probability, in ground or excited
states; fermions with low probability. A fraction of excited A-atoms are in
resonance with the macroscopically occupied phonon mode. In view of
macroscopic occupancy, the boson mode stimulates the near resonant excited
A-atom to emit a boson and reach its ground state. In the process we create an
excess population of ground state A-atoms. Increase in density of ground state
A-atoms means increased superconducting correlation (increase in magnitude of
ukvk); consequently an increase in superradiance interaction. Thus there is a
positive feedback, which could establishes a transient long range
superconducting order. As pseudo gap phase extends to room temperatures in
some of the underdoped cupraets, our mechamism offers a possibility to observe
room temperature transient superconductivity. This is one more incentive for
authors of reference liscYBCO to confirm their exciting observations.
To establish superconductivity in the normal state of a Kosterlitz Thouless
superconductor, what we need is only a spatial reorganization of random
thermal vortices into either i) a fluid of bound vortex-antivortex pairs as in
Kosterlitz-Thouless phase or ii) an ordered lattice of positive and negative
vortices (see note note5 ). The increased pairing correlation from
superradiance increases the core energy of the thermal vortices and a
corresponding increase of vortex pair binding energy. Resulting increase in
population of paired vortices help create transient superconductivity.
In addition to superconductors, it will be interesting look for superradiant
superfluidity in pairing dominatated fermion systems: superfluid He3, cold
atoms, heavy nucleii and nuclear matter.
Acknowledgement I thank - N Kumar, K P Sinha, R K Shankar and R Nityananda for
early discussions on photoinduced superconductivity; P W Anderson and N P Ong
for an encouraging discussion; N P Ong for bringing to my attention reference
RamanPseudoGap ; B. Keimer for an encouraging information liscYBCO ; DAE,
India for a Raja Ramanna Fellowship. This research was supported by Perimeter
Institute for Theoretical Physics.
## References
* (1) J. Bardeen, J R Schreiffer and L. Cooper, Phys. Rev., 108, 1175 (1957); Introduction to Superconductivity, M. Tinkham (Dover, NY 2004)
* (2) R. H. Dicke, Phys. Rev., 93, 99 (1954); Super-radiance, M.G. Benedict et al., (IOP Publishing, Bristol 1996)
* (3) P. W. Anderson and A. H. Dayem, Phys. Rev. Lett., 13, 195 (1964); A. F. G. Wyatt et al., Phys. Rev. Lett., 16, 1166 (1966); A. H. Dayem, J. J. Wiegand, Phys. Rev., 155, 419 (1967); R. Escudero and H.J.T. Smith, Phys. Rev. B31, 2725 (1985); S. I. Vedeneev, D. K. Maude, and J. M. Byrne, Phys. Rev. B78, 052509 (2008)
* (4) G. M. Eliashberg, JETP Letters, 11, 114 (1970); B. I. Ivlev and G. M. Eliashberg, JETP Letters 13, 333 (1971)
* (5) C.S. Owen and D.J. Scalapino, Phys. Rev Lett. 25, 1559 (1972)
* (6) N Kumar and K P Sinha, Phys. Rev., 1̱74, 482 (1968)
* (7) D. R. McIntosh and J. Lindesay, Phys. Rev., B50, 15852 (1994)
* (8) D. Fausti et al., Science, 331, 6014 (2011)
* (9) S. Kaiser et al., arXiv:1205.466 v2. According to version 3, in view of a calibration error, part of the claim needs to be verified; experiments are being repeated. However, signal for transient superconductivitty, seen as an appearance of c-axis plasma edge, remains robust (B. Keimer, private communication)
* (10) P.W. Anderson, Phys. Rev.,112, 1900 (1958)
* (11) In circuit QED, a collective degree of freedom of a Josephson junction is called Josephson atom (see for example, M. Devoret, S. Girvin and R. Schoelkopf, Ann. Phys. (Leipzig), 16, 767 .(2007)). A-atom is different - it fills k-space and is a bulk property of the superconductor.
* (12) Our pseudo spin ${\vec{\sigma}}_{\bf k}$ is related to Anderson’s pseudo spin, $\tau^{z}_{k}\equiv(1-c^{\dagger}_{k\uparrow}c_{k\uparrow}-c^{\dagger}_{-k\downarrow}c_{-k\downarrow}),\leavevmode\nobreak\ \tau^{+}_{k}\equiv c^{\dagger}_{k\uparrow}c^{\dagger}_{-k\downarrow}$ and $\tau^{-}_{k}\equiv c_{-k\downarrow}c_{k\uparrow}$ by a ${\bf k}$-dependent rotation of quantization direction to $(\theta,\phi_{k})$, where $u_{k}\equiv\cos\frac{\phi_{k}}{2}$ and $v_{k}\equiv\sin\frac{\phi_{k}}{2}e^{i\theta}$.
* (13) M. Gaudin, J. Phys. (Paris), 37, 1087 (1976); A. Kundu, J. Phys. A: Math. Gen., 37, L281 (2004); J. Dukelsky et al., Phys. Rev. Lett., 93, 050403 (2004). Depending on wk, pseudo spin wave packet mode may have a overlap with the Higgs amplitude mode of the superconducting order parameter.
* (14) N. Lambert, C. Emary, and T. Brandes, Phys. Rev. Lett., 92, 073602 (2004)
* (15) K. Hepp and E. Lieb, Annals of Physics, 76, 360 (1973); Y. K. Wang and F. T. Hioe, Phys. Rev., A7, 831 (1973)
* (16) R. Bonifacio, L. A. Lugiato, Phys. Rev., A11 1507 (1975)
* (17) Z.A. Xu et al., Nature 406, 486 (2000); Y. Wang, Lu Li and N.P. Ong, Phys. Rev. B73, 024510 (2006)
* (18) I. Iguchi, T. Yamaguchi and A. Sugimoto NATURE, 412, 420 (2001); C.V. Parker et al., Nature, 468, 677 (2010) vanishing phase coherence J. Corson et al., Nature 398, 221-223 (18 March 1999)
* (19) A. Dubroka et al., Phys. Rev. Lett. 106, 047006 (2011)
* (20) G. Baskaran, Z. Zou and P. W. Anderson, Sol. St. Commn., 63, 973 (1987); V. Emery and S. Kivelson, Nature, 374, 434 (1995)
* (21) I thank P.W. Anderson for suggesting this possibility.
|
arxiv-papers
| 2012-11-19T20:54:23 |
2024-09-04T02:49:38.193217
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "G. Baskaran",
"submitter": "Ganapathy Baskaran",
"url": "https://arxiv.org/abs/1211.4567"
}
|
1211.4638
|
# Proof of Kac and Rudakov’s Conjecture on Generalized Verma Module over Lie
Superalgebra $\mbox{E}(5,10)$111Research supported by CSC
Yufeng Zhao
LMAM, School of Mathematical Sciences
Peking University, Beijing, 100871, P. R. China
###### Abstract
The exceptional infinite-dimensional linearly compact simple Lie superalgebra
$\mbox{E}(5,10)$, which Kac believes, is the algebra of symmetries of the
$\mbox{SU}_{5}$ Grand Unified Model. In this paper, we give a proof of Kac and
Rudakov’s conjecture about the classification of all the degenerate
generalized Verma module over $\mbox{E}(5,10)$. Also, we work out all the
nontrivial singular vectors degree by degree. It is a potential that the
representation theory of $\mbox{E}(5,10)$ will shed new light on various
features of the the $\mbox{SU}_{5}$ Grand unified model.
## 1 Introduction
A linearly compact infinite-dimensional Lie algebra is a topological Lie
algebra whose underlying space is a topological space isomorphic to the space
of formal power series over complex field in finite number of variables with
formal topology. Cartan’s list of linearly compact infinite-dimensional simple
Lie algebras consists of four series: the Lie algebra of all complex vector
fields and its subalgebras of divergence 0 vector fields, symplectic vector
fields and contact vector fields.
Kac proved the “super” version of this result. In other words, he classified
linearly compact infinite-dimensional Lie superalgebras [K1]. There turn out
to be 10 families and 5 exceptions, which are called
$\mbox{E}(1,6),\mbox{E}(3,6),\mbox{E}(3,8),\mbox{E}(4,4)$ and
$\mbox{E}(5,10)$. Many of the families are straightforward “super”
generalizations of the 4 families of linearly compact infinite-dimensional
simple Lie algebras. Some are stranger. Most important for us today are the 5
exceptions discovered by Irina Shchepochkina [Sh].
The representation theory of $E(3,6)$ and $E(3,8)$ was developed by Kac and
Rudakov [KR1-KR3], and some further observations were made on its connections
to the Standard Model [K2]. It was found quite remarkable that the
$\mbox{SU}_{5}$ Grand unified model of Georgi-Glashow combines the left
multiplets of fundamental fermions in precisely the negative part of the
consistent gradation of $\mbox{E}(5,10)$. This is perhaps an indication of the
possibility that an extension from $\mbox{su}_{5}$ to algebra of internal
symmetries may resolve the difficulties with the proton decay. It is a
potential that the representation theory of $\mbox{E}(5,10)$ will shed new
light on various features of the the $\mbox{SU}_{5}$ Grand unified model.
As to the representation theory of $E(5,10)$, Kac and Rudakov formulate an
conjecture [KR3], which can be stated as follows. The Lie superalgebra
$L=E(5,10)$ carries a unique consistent irreducible $\mathbb{Z}$-gradation
$L=\bigoplus\limits_{j\geq-2}L_{j}$, where $L_{0}$ is isomorphic to simple Lie
algebra $sl_{5}$. Given $L_{0}$\- module $V$, we extend it to a $L$ module by
letting $L_{+}$ acts trivially, and define the induced module
$M(V)=U(L)\otimes_{U(L_{0})}V\cong U(L_{-})V.$
If $V$ is finite-dimensional irreducible $L_{0}$ -module, the $L$ module
$M(V)$ is called a generalized Verma module associated to $V$, and it is
called degenerate if it is not irreducible.
We denote by $\mbox{V}(\lambda_{1},\lambda_{2},\lambda_{3},\lambda_{4})$ the
finite-dimensional irreducible $L_{0}$ -module with highest weight
$\sum\limits_{i=1}^{4}\lambda_{i}\omega_{i}$, where
$\omega_{1},\omega_{2},\omega_{3},\omega$ are the fundamental weights for
$sl_{5}$. Let
$M=M(\lambda_{1},\lambda_{2},\lambda_{3},\lambda_{4})=M(V(\lambda_{1},\lambda_{2},\lambda_{3},\lambda_{4}))$
denote the corresponding generalized Verma module over $\mbox{E}(5,10)$.
Denote by $\mathbb{N}$ the additive semigroup of nonnegative integers.
Kac and Rudakov’s Conjecture The following is a complete list of degenerate
Verma modules over $\mbox{E}(5,10)$:
$\mbox{M}(m,n,0,0);\mbox{M}(m,0,0,n);\mbox{M}(0,0,m,n)\ (m,n\in\mathbb{N}).$
In this paper, we give a proof of this conjecture and work out all the
nontrivial singular vectors for any generalized Verma module over
$\mbox{E}(5,10)$. The first key point of our proof is investigating that there
exists a grading on the generalized Verma module, through which we find that
any singular vector is controlled by its leading term via an exponential-like
differential operator, where the leading term lies in certain tensor product
module of $sl_{5}$; the second one is the observation of an irreducible tensor
operator of rank $\omega_{1}+\omega_{2}$ for simple Lie algebra $sl_{5}$,
which plays the center role in our calculation of all the singular vectors.
The paper is organized as follows: In section 2, we recall Kac’s geometric
construction of Lie superalgebra $\mbox{E}(5,10)$ and the KR conjecture. In
Section 3, we provide some techniques concerning the irreducible tensor
operators and tensor module decomposition theory of simple Lie algebra. In
Section 4, we prove that all the nontrivial singular vectors are of degree
less than or equal to four. Also, the leading term of any singular vector must
lie in one of the tensor decomposition of four tensor product module of
$sl_{5}$ (cf. Theorem 4.6). In Section 5, we work out all the nontrivial
singular vectors degree by degree (cf. Theorem 5.3, Theorem 5.4, Theorem 5.5,
Theorem 5.6).
## 2 Lie superalgebra $\mbox{E}(5,10)$ and KR conjecture
In this section, we recall Kac’s geometric construction of Lie superalgebra
$\mbox{E}(5,10)$ and KR Conjecture which are stated in [KR3].
For two integers $m<n$, we denote $\overline{m,n}=\\{m,m+1,\cdots,n\\}$. Let
$\mbox{W}_{n}=\\{\sum\limits_{i=1}^{n}p_{i}(x)\partial_{i}\ |\
p_{i}(x)\in\mathbb{C}[[x_{1},\cdots,x_{n}]],\partial_{i}=\partial_{x_{i}}\\}$
$None$
denote the Lie algebra of formal vector fields in $n$ indeterminates;
$\mbox{S}_{n}=\\{D=\sum\limits_{i=1}^{n}p_{i}\partial_{i}\ |\
\mbox{div}D=\sum\limits_{i=1}^{n}\partial_{i}(p_{i})=0\\}$ $None$
denote the Lie subalgebra of divergenceless formal vector fields;
$\Omega^{k}(n)$ denote the associative algebra of formal differential forms of
degree k in $n$ indeterminates, $\Omega^{k}_{\mbox{cl}}(n)$ denote the
subspace of closed forms.
The exceptional infinite-dimensional linearly compact Lie superalgebra
$\mbox{E}(5,10)=\mbox{E}(5,10)_{\underline{0}}+\mbox{E}(5,10)_{\underline{1}}$
is constructed as follows:
$\mbox{E}(5,10)_{\underline{0}}=S_{5},\
\mbox{E}(5,10)_{\underline{1}}=\Omega_{\mbox{cl}}^{2}(5),$ $None$
where $\mbox{E}(5,10)_{\underline{0}}$ acts on
$\mbox{E}(5,10)_{\underline{1}}$ via the Lie derivative,
$[\omega_{2},\omega_{2}^{\prime}]=\omega_{2}\wedge\omega_{2}^{\prime}\in\Omega_{\mbox{cl}}^{4}(5)=S_{5}$
$None$
for $\omega_{2},\omega_{2}^{\prime}\in\mbox{E}(5,10)_{\underline{1}}$.
We use for the odd elements of $\mbox{E}(5,10)$ the notation
$d_{ij}=dx_{i}\wedge dx_{j}(i,j\in\overline{1,5})$; recall that we have the
following commutation relation ($f,g\in C[[x_{1},\cdots,x_{5}]]$ ):
$[fd_{jk};gd_{lm}]=\varepsilon_{ijklm}fg\partial_{i};$ $None$
where
$\varepsilon_{ijklm}=\left\\{\begin{array}[]{ll}\mbox{the \ sign \ of \ the \
permutation}\ $(ijklm)$,&\mbox{ if \ all \ indices \ $ijklm$ \ are \
distinct},\\\ 0,&\mbox{otherwise}.\end{array}\right.$ $None$
And the Lie superalgebra $\mbox{L}=\mbox{E}(5,10)$ carries a unique consistent
irreducible $\mathbb{Z}$-gradation
$\mbox{L}=\bigoplus\limits_{j\geq-2}\mbox{L}_{j}$. It is defined by:
$degx_{i}=2=-\partial_{i},degd_{ij}=-1$ $None$
One has: $\mbox{L}_{0}\simeq\mbox{sl}_{5}$ and the $\mbox{L}_{0}$-modules
occurring in the negative part are:
$\mbox{L}_{-1}=\mbox{Span}_{\mathbb{C}}\\{d_{ij}\ |\
i,j\in\overline{1,5}\\}\simeq\Lambda^{2}\mathbb{C}^{5},$
$\mbox{L}_{-2}=\mbox{Span}_{\mathbb{C}}\\{\partial_{i}\ |\
i\in\overline{1,5}\\}\simeq\mathbb{C}^{5*}$ $None$
Recall also that $\mbox{L}_{1}$ consist of closed 2-forms with linear
coefficients, that $\mbox{L}_{1}$ is an irreducible $\mbox{L}_{0}$-module and
$\mbox{L}_{j}=[\mbox{L}_{1}[\cdots]]=\mbox{L}_{1}^{j}$ for $j\geq 1$. We take
for the Borel subalgebra of $\mbox{L}_{0}\simeq\mbox{sl}_{5}$the subalgebra of
the vector fields
$\mbox{Span}\\{x_{i}\partial_{j}(1\leq i\leq j\leq 5),\
x_{i}\partial_{i}-x_{i+1}\partial_{i+1}(i\in\overline{1,4})\\}.$ $None$
Given $\mbox{L}_{0}$ module $V$, we extend it to a L module by letting
$\mbox{L}_{+}$ acts trivially, and define the induced module
$M(V)=U(\mbox{L})\otimes_{U(\mbox{L}_{0})}V\cong U(\mbox{L}_{-})V.$ $None$
We denote by $\mbox{V}(\lambda_{1},\lambda_{2},\lambda_{3},\lambda_{4})$ the
finite-dimensional irreducible $\mbox{L}_{0}$ -module with highest weight
$\sum\limits_{i=1}^{4}\lambda_{i}\omega_{i}$, where $\omega_{i}$
($i\in\overline{1,4}$) are the fundamental weights for $sl_{5}$. Let
$M=M(\lambda_{1},\lambda_{2},\lambda_{3},\lambda_{4})=M(V(\lambda_{1},\lambda_{2},\lambda_{3},\lambda_{4}))$
$None$
denote the corresponding generalized Verma module over $\mbox{E}(5,10)$.
Definition 2.1 If $\xi\in M$ satisfies :
$(x_{i}\partial_{x_{i+1}}).\xi=0(i\in\overline{1,4}),$ $None$
$x_{5}d_{45}.\xi=0,$ $None$
then we call $\xi$ a singular vector for generalized verma module $M$ of
$E(5,10)$.
The aim of the following sections is to determine all the nontrivial singular
vectors for $\mbox{E}(5,10)$-module $M$.
## 3 Preliminary
In this section, we give some preparatory techniques about the irreducible
tensor operators and the decomposition of tensor product module of simple Lie
algebra.
Following the notations of Humphreys [H], let $H$ be the Cartan subalgebra of
simple Lie algebra $L$, and let $\Delta=\\{\alpha_{1},\cdots,\alpha_{l}\\}$ be
a base for the root system $\phi$ of $H^{*}$. The corresponding fundamental
dominant weights $\\{\omega_{1},\cdots,\omega_{l}\\}$ are defined from the
root system via the form $<\cdot,\cdot>$ given by:
$<\omega_{i},\alpha_{j}>\equiv\frac{2(\omega_{i},\alpha_{j})}{(\alpha_{j},\alpha_{j})}=\delta_{ij},$
$None$
where $(\cdot,\cdot)$ denotes the inner product induced on $H^{*}$ by the
Killing form on $H$. Consider a basis
$\\{h_{1},\cdots,h_{l},x_{\alpha},\alpha\in\phi\\}$ of $L$ where
$h_{1},\cdots,h_{l}$ is a basis for $H$ and $x_{\alpha}$ is a nonzero element
of the root space $L_{\alpha}$. The dual basis may therefore be written
$\\{h^{1},\cdots,h^{l},x^{\alpha},\alpha\in\phi\\}$ where $x^{\alpha}$ is the
unique element of $L_{-\alpha}$ which is dual to $x_{\alpha}$ under the
Killing form of $L$. Write the universal Casimir element in the form:
$c_{L}=\sum\limits_{i=1}^{l}h_{i}h^{i}+\sum\limits_{\alpha\in\phi}x_{\alpha}x^{\alpha}.$
$None$
Let $V(\mu)$ be an irreducible highest weight module over $L$ and let
$\pi_{\mu}$ be the representation afforded by $V(\mu)$. Choose an ordeded
basis $\\{e_{1},\cdots,e_{d}\\}$ of $V(\mu)$, let $\pi_{\mu}(x)$ denote the
matrix representing $x\in L$ on $V(\mu)$ with respect to this basis.
Definition 3.1 We call a collection of linear operators $\\{T_{i}:V\rightarrow
W\ |\ i\in\overline{1,d}\\}$ an irreducible tensor operator of rank $\mu$ if
these components transform according to the rule:
$[x,T_{i}]=\pi_{W}(x)T_{i}-T_{i}\pi_{V}(x)=\sum\limits_{j=1}^{d}\pi_{\mu}(x)_{ji}T_{j},\
x\in L,$ $None$
where $V$,$W$ are (possibly infinite dimensional) $L$-modules and $\pi_{V}$
(resp. $\pi_{W}$) is the representation afforded by $V$ (resp. $W$).
Then we can define the following intertwining operator between $L$-modules
$V(\mu)\otimes V$ and $W$:
$T:V(\mu)\otimes V\rightarrow W,\ T(e_{i}\otimes v)=T_{i}(v),\
i\in\overline{1,d},\ v\in V.$ $None$
In other words, $T\in\mbox{Hom}_{L}(V(\mu)\otimes V,W)$ is an element of the
set of all operators from $V(\mu)\otimes V$ to $W$ commuting with the action
of $L$.
Remark 3.2 In Section 5, we find an irreducible tensor operator of rank
$\omega_{1}+\omega_{2}$ for simple Lie algebra $sl_{5}$, which play the center
role in our determining all the singular vectors.
In the following two Lemmas, we record some well-known facts concerning the
decomposition of tensor modules:
Lemma 3.3 (1) (cf. [H]) The $\alpha$\- string through any weight $\nu$ of
$V(\mu)$ is of length $<\nu,\alpha>$, for $\alpha\in\phi$.
(2) (cf. [EG]) Denote $\mu_{1},\cdots,\mu_{m}$ the weights occurring in
$V(\mu)$ with multiplicities $n_{1},\cdots,n_{m}$ respectively. For each
$i\in\overline{1,m}$, let $V_{i}(\mu)$ denote the space of weight vectors of
weight $\mu_{i}$. The decomposition of the tensor product module
$V(\mu)\otimes V(\lambda)$ is written:
$V(\mu)\otimes
V(\lambda)=\sum\limits_{i=1}^{m}m(\lambda+\mu_{i}:\mu\otimes\lambda)V(\lambda+\mu_{i}),\
\lambda+\mu_{i}\in\Lambda^{+},$ $None$
where the multiplicities are given by
$m(\lambda+\mu_{i}:\mu\otimes\lambda)=dimV_{i,\lambda}(\mu),$ $None$
$V_{i,\lambda}(\mu)=\\{v\in V_{i}(\mu)\ |\
e_{j}^{<\lambda+\delta,\alpha_{j}>}v=0,j\in\overline{1,l}.\\}$ $None$
(3) (cf. [EG]) Assume $\\{e_{i,j}\ |\
j\in\overline{1,m(\lambda+\mu_{i}:\mu\otimes\lambda)}\\}$ is a basis for the
space $V_{i,\lambda}(\mu)$ and $v_{\lambda}$ is the maximal weight vector of
$V(\lambda)$. A full set of independent maximal weight states of weight
$\lambda+\mu_{i}$ is given by the vectors:
$\\{P_{i}(e_{i,j}\otimes v_{\lambda}),\
j\in\overline{1,m(\lambda+\mu_{i}:\mu\otimes\lambda)}\\},$
where
$P_{i}=\prod\limits_{\mu_{i}<\sigma\leq\mu}\frac{\tilde{c_{L}}-\chi_{\sigma+\lambda}(\tilde{c_{L}})}{\chi_{\mu_{i}+\lambda}(\tilde{c_{L}})-\chi_{\sigma+\lambda}(\tilde{c_{L}})},$
$\chi_{\sigma+\lambda}(\tilde{c_{L}})=\frac{(\sigma+\lambda,\sigma+\lambda+2\delta)-(\mu,\mu+2\delta)-(\lambda,\lambda+2\delta)}{2},$
$\tilde{c_{L}}=\sum\limits_{i=1}^{l}\pi_{\mu}(h_{i})\otimes\pi_{\lambda}(h^{i})+\sum\limits_{\alpha\in\phi}\pi_{\mu}(x_{\alpha})\otimes\pi_{\lambda}(x^{\alpha}).$
$None$
(4) (cf. [MS]) The tensor product module $V(\mu)\otimes V(\lambda)$ is a
cyclic module which is cyclically generated by the vector $v^{\mu}\otimes
v_{\lambda}$, where $v^{\mu}$ is the lowest weight vector for $V(\mu)$ and
$v_{\lambda}$ is the highest weight vector for $V(\lambda)$.
For $\overrightarrow{a}=(a_{1},\cdots,a_{n-1})\in\mathbb{N}^{n-1}$ and
$0<k\in\mathbb{N}$, we denote
$\overrightarrow{a}^{*}=(a_{n-1},a_{n-2},\cdots,a_{1}),$ $None$
$I(\overrightarrow{a},k)=\\{(a_{1}+c_{1}-c_{2},a_{2}+c_{2}-c_{3},\cdots,a_{n-1}+c_{n-1}-c_{n})\
|\ c_{i}\in\mathbb{N}\ $ $\mbox{such \ that}\ \sum\limits_{i=1}^{n}c_{i}=k\
\mbox{and}\ c_{s+1}\leq a_{s}\mbox{for}\ s\in\overline{1,n-1}\\}.$ $None$
Set
$\omega_{\overrightarrow{a}}=\sum\limits_{i=1}^{n-1}a_{i}\omega_{i},\ \
\mbox{for}\ \overrightarrow{a}\in\mathbb{N}^{n-1},$ $None$
Lemma 3.4 (Pieri’s formula cf. [FH]) (1) For any
$\overrightarrow{a}\in\mathbb{N}^{n-1}$, the tensor product of $sl_{n}$-module
$V(\omega_{\overrightarrow{a}})$ with $V(k\omega_{1})$ decomposes into a
direct sum:
$V(\omega_{\overrightarrow{a}})\otimes
V(k\omega_{1})=\bigoplus_{\overrightarrow{b}\in
I(\overrightarrow{a},k)}V(\omega_{\overrightarrow{b}}).$ $None$
(2) For $sl_{n}$, we have
$V(\omega_{\overrightarrow{a}})^{*}=V(\omega_{\overrightarrow{a}^{*}})$ and
$V(\omega_{\overrightarrow{a}})\otimes
V(k\omega_{n-1})=\bigoplus_{\overrightarrow{b}\in
I(\overrightarrow{a}^{*},k)}V(\omega_{\overrightarrow{b}^{*}}).$ $None$
In the rest of this section, we will concentrate on some special wedge and
tensor modules for $sl_{5}$. Take
$\\{h_{i}=E_{i,i}-E_{i+1,i+1}(i\in\overline{1,4}),E_{ij}(1\leq i\neq j\leq
5)\\}$ as a basis for Lie algebra $sl_{5}$. Then
$\\{\frac{h_{i}^{*}}{10}(i\in\overline{1,4}),\frac{E_{ji}}{10}(1\leq i\neq
j\leq 5)\\}$ is its dual basis via the Killing form, where
$h_{1}^{*}=\frac{4}{5}h_{1}+\frac{3}{5}h_{2}+\frac{2}{5}h_{3}+\frac{1}{5}h_{4},\
h_{2}^{*}=\frac{3}{5}h_{1}+\frac{6}{5}h_{2}+\frac{4}{5}h_{3}+\frac{2}{5}h_{4},$
$h_{3}^{*}=\frac{2}{5}h_{1}+\frac{4}{5}h_{2}+\frac{6}{5}h_{3}+\frac{3}{5}h_{4},\
h_{4}^{*}=\frac{1}{5}h_{1}+\frac{2}{5}h_{2}+\frac{3}{5}h_{3}+\frac{4}{5}h_{4}.$
$None$
And the Casimir operator $c$ of the universal enveloping algebra of $sl_{5}$
is
$c=\frac{1}{10}(\sum\limits_{i=1}^{4}h_{i}h_{i}^{*}+\sum\limits_{i\neq
j\in\overline{1,5}}E_{i,j}.E_{j,i}).$ $None$
Relative to the ordered basis $\omega_{1},\omega_{2},\omega_{3},\omega_{4}$,
the coordinates of the simple roots $\alpha_{i}(i\in\overline{1,4})$ are:
$\alpha_{1}=(2,-1,0,0),\ \alpha_{2}=(-1,2,-1,0),\ \alpha_{3}=(0,-1,2,-1),\
\alpha_{4}=(0,0,-1,2).$ $None$
And the killing form for the simple root $\alpha_{i}(i\in\overline{1,4})$ are:
$(\alpha_{i},\alpha_{j})=\left\\{\begin{array}[]{lll}0,&|i-j|>1,\\\
\frac{1}{5},&i=j,\\\ -\frac{1}{10},&|i-j|=1.\end{array}\right.$ $None$
Lemma 3.5 Assume $L=sl_{5}$, $C_{L}=c$ and
$\sigma=\mu-\sum\limits_{i=1}^{4}k_{i}\alpha_{i}$ in Lemma 3.4. Then
$\chi_{\sigma+\lambda}(\tilde{c})$ in (3.6) is explicitly given by:
$\displaystyle\chi_{\sigma+\lambda}(\tilde{c})=\frac{\lambda_{1}(4\mu_{1}+3\mu_{2}+2\mu_{3}+\mu_{4})}{50}+\frac{\lambda_{2}(3\mu_{1}+6\mu_{2}+4\mu_{3}+2\mu_{4})}{50}+\frac{\lambda_{3}(2\mu_{1}+4\mu_{2}+6\mu_{3}+3\mu_{4})}{50}$
$\displaystyle+\frac{\lambda_{4}(\mu_{1}+2\mu_{2}+3\mu_{3}+4\mu_{4})}{50}+\frac{\sum\limits_{i=1}^{4}k_{i}^{2}-k_{1}k_{2}-k_{2}k_{3}-k_{3}k_{4}-\sum\limits_{i=1}^{4}k_{i}-k_{i}(\lambda_{i}+\mu_{i})}{10}.\hskip
56.9055pt(3.18)$
From (2.7) and (2.9), we know that $L_{0}\simeq sl_{5}$. And $L_{0}$-module
$L_{-1}$ is isomorphic to fundamental module $V(\omega_{2})=W$. The set of its
weights and the basis for the corresponding weight space are tabulated in
Table 1. The $L_{0}$-module $L_{1}$ is isomorphic to highest weight module
$V(\omega_{1}+\omega_{2})$ with lowest weight vector $x_{5}d_{45}$ (cf. Table
9).
Lemma 3.6 The wedge module $\Lambda^{k}W$ ($k\in\overline{1,10}$) for
$sl_{5}$ are decomposed multiplicity freely into irreducible components, which
are listed in Table 2.
Proof By Weyl’s dimension formula, we get:
$\mbox{dim}V(\omega_{1}+\omega_{3})=45$, $\mbox{dim}V(2\omega_{3})=50$,
$\mbox{dim}V(2\omega_{1}+\omega_{4})=70$, $\mbox{dim}V(3\omega_{1})=35$,
$\mbox{dim}V(\omega_{1}+\omega_{3}+\omega_{4})=175$,
$\mbox{dim}V(2\omega_{1}+\omega_{3})=126$,
$\mbox{dim}V(\omega_{2}+2\omega_{4})=126$. Since $\mbox{dim}W=10$,
$\mbox{dim}\wedge^{k}W=C_{10}^{k}$. Thus the decomposition follows through
comparing the dimensions of both sides. $\Box$
Lemma 3.7 The tensor module $V(k\omega_{4})\otimes\Lambda^{n}W$
($k\in\mathbb{N},\ n\in\overline{1,10}$) for $sl_{5}$ are decomposed into
irreducible components, which are listed in Table 3.
For any highest weight module $V(\mu)$ of simple Lie algebra $sl_{5}$, denote
the set of its weights by $\Pi(\mu)$, which are listed by
$\\{\overrightarrow{w}_{j}^{\mu}\ |\ j\in\overline{1,|\Pi(\mu)|}\\}$. Let
$\\{v_{j,k}^{\mu}\ |\
j\in\overline{1,|\Pi(\mu)|},k\in\overline{1,\mbox{mult}(\overrightarrow{w}_{j}^{\mu})}\\}$
be the Verma basis for the weight space of weight
$\overrightarrow{w}_{j}^{\mu}$, where
$\mbox{mult}(\overrightarrow{w}_{j}^{\mu})$ denotes the multiplicity of the
weight $\overrightarrow{w}_{j}^{\mu}$.
Lemma 3.8 For
$\mu\in\\{\omega_{1}+\omega_{3},2\omega_{1}+\omega_{4},3\omega_{1}\\}$, the
set $\Pi(\mu)$ of weights for $V(\mu)$ and their corresponding Verma bases for
every weight space are listed in Table 5-Table 8 in the Appendix.
Proof Assume $\mu=\sum\limits_{i=1}^{4}m_{i}\omega_{i}$. The set $\Pi(\mu)$ is
obtained by the algorithm from [W]. The Verma bases for the weight space with
weight $\mu-\sum\limits_{i=1}^{4}k_{i}\alpha_{i}$ are (cf. [LMNP], [RS]):
$(f_{1}^{a_{10}}f_{2}^{a_{9}}f_{3}^{a_{8}}f_{4}^{a_{7}})(f_{1}^{a_{6}}f_{2}^{a_{5}}f_{3}^{a_{4}})(f_{1}^{a_{3}}f_{2}^{a_{2}})f_{1}^{a_{1}}v_{\mu},$
$None$
where
$a_{10}+a_{6}+a_{3}+a_{1}=k_{1},a_{9}+a_{5}+a_{2}=k_{2},a_{8}+a_{4}=k_{3},a_{7}=k_{4},$
$0\leq a_{1}\leq m_{1},0\leq a_{2}\leq m_{2}+a_{1},0\leq
a_{3}\leq\mbox{min}(m_{2},a_{2}),$ $0\leq a_{4}\leq m_{3}+a_{2},0\leq
a_{5}\leq\mbox{min}(m_{3}+a_{3},a_{4}),0\leq
a_{6}\leq\mbox{min}(m_{3},a_{5}),$ $0\leq a_{7}\leq m_{4}+a_{4},0\leq
a_{8}\leq\mbox{min}(m_{4}+a_{5},a_{7}),0\leq
a_{9}\leq\mbox{min}(a_{4}+a_{6},a_{8}),0\leq
a_{10}\leq\mbox{min}(m_{4},a_{9}).$ $None$
$\Box$
Remark 3.9 The coordinates of the weights appearing in Table1, Table5-Table9
are with respect to the ordered basis
$\omega_{1},\omega_{2},\omega_{3},\omega_{4}$. The basis of every weight space
appearing in these tables are Verma basis.
## 4 Singular vectors for GVM of $\mbox{E}(5,10)$
In Section 4.1, we analyze the detailed structure of the generalized Verma
module $M$ over $\mbox{E}(5,10)$. It turns out that there is a grading on $M$
and each graded subspace is a finite dimensional $sl_{5}$\- module (cf.
Equation (4.6) and (4.7)). Moreover, any singular vector for $M$ is controlled
by its leading term through an exponential-like differential operator (cf.
Equation (4.27) ). In section 4.2, we inductively prove that any leading term
must satisfy three equations, i.e. (4.28), (4.35) and (4.37). Based on the
Lemmas in Section 3, we simplify these three differential equations and prove
that any singular vector is of degree less than or equal to four. Also, the
leading term of any singular vector must lie in one of the tensor
decomposition of four tensor product module for $sl_{5}$ (cf. Theorem 4.6 ).
### 4.1 Gradation for GVM
Set
$T=\\{0,1\\},\ \
T^{\prime}=\\{(45),(35),(25),(15),(34),(24),(14),(23),(13),(12)\\}.$ $None$
Define order $``\prec^{\prime\prime}$ on the set $T^{\prime}$ by:
$(45)\prec(35)\prec(25)\prec(15)\prec(34)\prec(24)\prec(14)\prec(23)\prec(13)\prec(12).$
$None$
For
$\underline{n}=(n_{12},n_{13},n_{14},n_{24},n_{34},n_{15},n_{25},n_{35},n_{45})\in
T^{10}$ and $\underline{m}\in\mathbb{N}^{5}$, we take the following notations:
$\underline{n}\pm\varepsilon_{ij}=(n_{12},\cdots,n_{ij}\pm 1,\cdots,n_{45}),\
\underline{m}\pm\varepsilon_{i}=(m_{1},\cdots,m_{i}\pm 1,\cdots,m_{5}).$
$None$
Let
$d^{\underline{n}}=d_{12}^{n_{12}}d_{13}^{n_{13}}d_{23}^{n_{23}}d_{14}^{n_{14}}d_{24}^{n_{24}}d_{34}^{n_{34}}d_{15}^{n_{15}}d_{25}^{n_{25}}d_{35}^{n_{35}}d_{45}^{n_{45}},$
$None$
$\partial^{\underline{m}}=\partial_{1}^{m_{1}}\partial_{2}^{m_{2}}\partial_{3}^{m_{3}}\partial_{4}^{m_{4}}\partial_{5}^{m_{5}}.$
$None$
Then the induced module $M$ is spanned by
$\\{\partial^{\underline{m}}d^{\underline{n}}v_{\nu}\ |\ \underline{n}\in
T^{10},\underline{m}\in\mathbb{N}^{5},\nu\in\Pi(\lambda)\\}$. Define
$\partial^{m}\wedge^{n}V=\mbox{Span}\\{\partial^{\underline{m}}d^{\underline{n}}v\
|\ |\underline{m}|=m,|\underline{n}|=n\\},\
M_{k}=\mbox{Span}\\{\partial^{\underline{m}}d^{\underline{n}}v\ |\ 2m+n=k\\}.$
$None$
Then
$M=\bigoplus\limits_{k\in\mathbb{N}}M_{k}.$ $None$
Definition 4.1 We say any nonzero vector of $M_{k}$ is of degree $k$.
The equations
$[x_{i}\partial_{x_{j}},d_{kl}]=\delta_{j,k}d_{il}-\delta_{jl}d_{ik},[x_{5}d_{45},d_{12}]=x_{5}\partial_{x_{3}},\
[x_{5}d_{45},d_{13}]=-x_{5}\partial_{x_{2}},\
[x_{5}d_{45},d_{23}]=x_{5}\partial_{x_{1}},$
$[x_{5}d_{45},d_{i4}]=0(i\in\overline{1,3}),\
[x_{5}d_{45},d_{i5}]=0(i\in\overline{1,4}).$ $None$
yield
$L_{0}.\partial^{m}\wedge^{n}V\subseteq\partial^{m}\wedge^{n}V+\partial^{m+1}\wedge^{n-2}V,\
x_{5}d_{45}.M_{k}\subseteq M_{k-1}.$ $None$
That is to say, every graded vector subspace $M_{k}$ is an $sl_{5}$-module and
every singular vector for $\mbox{E}(5,10)$-module $M$ is in a certain graded
subspace $M_{k}$.
In the following of this section, we consider the maximal vectors for
$sl_{5}$-module $M_{k}$. On any linear vector space $\partial^{m}\wedge^{n}V$,
we define the following linear operators:
$(-1)^{|ij|}:\partial^{m}\wedge^{n}V\rightarrow\partial^{m}\wedge^{n}V;\partial^{\underline{m}}d^{\underline{n}}v\mapsto(-1)^{\sum\limits_{(kl)\prec(ij)}n_{kl}}\partial^{\underline{m}}d^{\underline{n}}v,$
$(-1)^{|ij,kl|}:\partial^{m}\wedge^{n}V\rightarrow\partial^{m}\wedge^{n}V;\partial^{\underline{m}}d^{\underline{n}}v\mapsto(-1)^{\sum\limits_{(kl)\prec(pq)\prec(ij)}n_{pq}}\partial^{\underline{m}}d^{\underline{n}}v,$
$y_{ij}\partial_{y_{kl}}:\partial^{m}\wedge^{n}V\rightarrow\partial^{m}\wedge^{n}V;\partial^{\underline{m}}d^{\underline{n}}v\mapsto
n_{kl}\partial^{\underline{m}}d^{\underline{n}+\varepsilon_{ij}-\varepsilon_{kl}}v,$
$z_{i}:\partial^{m}\wedge^{n}V\rightarrow\partial^{m+1}\wedge^{n}V;\partial^{\underline{m}}d^{\underline{n}}v\mapsto\partial^{\underline{m}+\varepsilon_{i}}d^{\underline{n}}v,$
$\partial_{z_{i}}:\partial^{m}\wedge^{n}V\rightarrow\partial^{m-1}\wedge^{n}V;\partial^{\underline{m}}d^{\underline{n}}v\mapsto
m_{i}\partial^{\underline{m}-\varepsilon_{i}}d^{\underline{n}}v,$
$E_{i,j}:\partial^{m}\wedge^{n}V\rightarrow\partial^{m}\wedge^{n}V;\partial^{\underline{m}}d^{\underline{n}}v\mapsto\partial^{\underline{m}}d^{\underline{n}}(E_{i,j}.v).$
$None$
Set
$(x_{i}\partial_{x_{j}})_{0}^{{}^{\prime}}=\sum\limits_{k\in\overline{1,5},k\neq
i,j}(-1)^{|ki,kj|}y_{ki}\partial_{y_{kj}},\
(x_{i}\partial_{x_{j}})_{0}=-z_{j}\partial_{z_{i}}+(x_{i}\partial_{x_{j}})_{0}^{{}^{\prime}}+E_{i,j}(i\neq
j);$ $None$
$(x_{3}\partial_{x_{4}})_{-2}=z_{5}\partial_{y_{14}}\partial_{y_{24}},$ $None$
$(x_{4}\partial_{x_{5}})_{-2}=z_{1}\partial_{y_{25}}\partial_{y_{35}}+(-1)^{1+|15,35|}z_{2}\partial_{y_{15}}\partial_{y_{35}}+z_{3}\partial_{y_{15}}\partial_{y_{25}}.$
$None$
Using these settings, we could formulate the equation (2.12) in the following
explicit form:
$x_{1}\partial_{x_{2}}=(x_{1}\partial_{x_{2}})_{0},\quad
x_{2}\partial_{x_{3}}=(x_{2}\partial_{x_{3}})_{0},$
$x_{3}\partial_{x_{4}}=(x_{3}\partial_{x_{4}})_{0}+(x_{3}\partial_{x_{4}})_{-2},\
x_{4}\partial_{x_{5}}=(x_{4}\partial_{x_{5}})_{0}+(x_{4}\partial_{x_{5}})_{-2}.$
$None$
According to the Cartan subalgebra of $L_{0}$, $M$ can be decomposed to the
following direct sum of subspaces:
$M=\bigoplus\limits_{\mu\in\Gamma}M^{\mu},\
M^{\mu}=\mbox{Span}\\{\partial^{\underline{m}}d^{\underline{n}}v_{\nu}\ |\
(x_{i}\partial_{x_{i}}-x_{i+1}\partial_{x_{i+1}}).\partial^{\underline{m}}d^{\underline{n}}v_{\nu}=\mu_{i}\partial^{\underline{m}}d^{\underline{n}}v_{\nu}\\},$
$None$
where
$\mu_{i}=m_{i+1}-m_{i}+t_{i}(\underline{n})+\nu_{i},$
$t_{1}(\underline{n})=n_{13}+n_{14}+n_{15}-n_{23}-n_{24}-n_{25},\
t_{2}(\underline{n})=n_{12}+n_{24}+n_{25}-n_{13}-n_{34}-n_{35},$
$t_{3}(\underline{n})=n_{13}+n_{23}+n_{35}-n_{14}-n_{24}-n_{45},\
t_{4}(\underline{n})=n_{14}+n_{24}+n_{34}-n_{15}-n_{25}-n_{35}.$ $None$
For any vectors $v\in M^{\mu}$, we say that it is of weight $\mu$ and denote
$\mbox{wt}(v)=\mu,\ |\mbox{wt}(v)|=|\mu|=\sum\limits_{i=1}^{4}\mu_{i}.$
Proposition 4.2 The differential operators $(x_{i}\partial_{x_{j}})_{0}(1\leq
i\neq j\leq 5)$ and
$x_{i}\partial_{x_{i}}-x_{i+1}\partial_{x_{i+1}}(i\in\overline{1,4})$ give
every vector space $\partial^{m}\wedge^{n}V$ an $sl_{5}$-module structure,
which is isomorphic to tensor module $V(m\omega_{4})\otimes\wedge^{n}W\otimes
V$ for $sl_{5}$.
Proof The module isomorphism is given by:
$\phi:V(m\omega_{4})\otimes\wedge^{n}W\otimes
V\rightarrow\partial^{m}\wedge^{n}V;\partial^{\underline{m}}\otimes(d_{i_{1}j_{1}}\wedge\cdots\wedge
d_{i_{n}j_{n}})\otimes v\mapsto\partial^{\underline{m}}d_{i_{1}j_{1}}\cdots
d_{i_{n}j_{n}}v.$ $None$
$\Box$
Denote
$\Gamma_{k}=\\{(m,n)\in\mathbb{N}^{2}\ |\ 2m+n=k\\}$ $None$
For any $(m,n)\in\Gamma_{k}$, let
$\Gamma_{k}^{(m,n)}=\\{(m^{\prime},n^{\prime})\in\Gamma_{k}\ |\ m^{\prime}\geq
m\\}$ $None$
Assume $\xi\in M_{k}$ is any $E(5,10)$ singular vector. Then there exists
$(m,n)\in\Gamma_{k}$ such that
$\xi\in\bigoplus\limits_{(m^{\prime}n^{\prime})\in\Gamma_{k}^{(m,n)}}\partial^{m^{\prime}}\wedge^{n^{\prime}}V$
$None$
For emphasis, we write
$\xi=\xi^{m,n}=\sum\limits_{(m^{\prime}n^{\prime})\in\Gamma_{k}^{(m,n)}}\xi_{m^{\prime},n^{\prime}}$
$None$
We say that $\xi_{m,n}$ is the leading term of $\xi^{m,n}$. It follows from
(4.14) that $\xi^{m,n}$ must satisfy the following equations inductively:
$(x_{i}\partial_{x_{i+1}})_{0}.\xi_{m,n}=0(i\in\overline{1,4}),$
$(x_{i}\partial_{x_{i+1}})_{-2}.\xi_{m^{\prime},n^{\prime}}+(x_{i}\partial_{x_{i+1}})_{0}.\xi_{m^{\prime}+1,n^{\prime}-2}=0,\
i\in\overline{1,4},\ (m^{\prime},n^{\prime})\in\Gamma_{k}^{(m,n)}.$ $None$
Remark 4.3 From Proposition 4.2 and (4.22), we derive that the leading term
$\xi_{m,n}$ of any singular vector $\xi=\xi^{m,n}$ is also a singular vector
of the tensor product module
$V(m\omega_{4})\otimes\wedge^{n}V(\omega_{2})\otimes V(\lambda)$ for simple
Lie algebra $sl_{5}$. In the following, we will point out that any singular
vector $\xi=\xi^{m,n}$ is completely controlled by its leading term
$\xi_{m,n}$ through certain exponential-like differential operator.
Set
$P=\sum\limits_{(kl)\prec(ij)\in
T^{\prime},m\in\overline{1,5}}\varepsilon_{mijkl}(-1)^{|ij,kl|}z_{m}\partial_{y_{ij}}\partial_{y_{kl}},$
$None$
where $\varepsilon_{mijkl}$ is defined in (2.6). The operator $P$ is checked
to satisfy the following equations:
$[(x_{3}\partial_{x_{4}})_{-2},P]=0,[(x_{4}\partial_{x_{5}})_{-2},P]=0,\
[(x_{1}\partial_{x_{2}})_{0},P]=0,[(x_{2}\partial_{x_{3}})_{0},P]=0,$
$[(x_{3}\partial_{x_{4}})_{0},P]=2(x_{3}\partial_{x_{4}})_{-2},\
[(x_{4}\partial_{x_{5}})_{0},P]=2(x_{4}\partial_{x_{5}})_{-2}.$ $None$
Inductively,
$[(x_{i}\partial_{x_{i+1}})_{0},P^{k}]=[x_{i}\partial_{x_{i+1}},P^{k}]=2kP^{k-1}(x_{i}\partial_{x_{i+1}})_{-2},k\in\mathbb{N}.$
$None$
It implies
$x_{i}\partial_{x_{i+1}}.e^{-\frac{1}{2}P}\xi_{m,n}=0,\quad(x_{i}\partial_{x_{i+1}})_{0}.e^{\frac{1}{2}P}\xi^{m,n}=0.$
$None$
Thus we prove the following formula:
Proposition 4.3 Assume
$\xi^{m,n}=\sum\limits_{(m^{\prime}n^{\prime})\in\Gamma_{k}^{(m,n)}}\xi_{m^{\prime},n^{\prime}}\in
M_{k}$ is any singular vector for $E(5,10)$-module $M$, then
$\xi^{m,n}=e^{-\frac{1}{2}P}\xi_{m,n}.$ $None$
### 4.2 Singular vectors for GVM
In this section, we continue the discussion concerning the equation (2.13) in
Definition 2.1. Recall the notations in (4.10), set
$(x_{5}d_{45})_{1}=(-1)^{1+|45|}\partial_{z_{5}}y_{45},$ $None$
$\displaystyle(x_{5}d_{45})_{-1}$ $\displaystyle=$ $\displaystyle-
z_{3}\partial_{z_{5}}\partial_{y_{12}}+(-1)^{|13|}z_{2}\partial_{z_{5}}\partial_{y_{13}}+(-1)^{1+|23|}z_{1}\partial_{z_{5}}\partial_{y_{23}}$
$\displaystyle+$
$\displaystyle\partial_{y_{12}}E_{53}+(-1)^{1+|13|}\partial_{y_{13}}E_{52}+(-1)^{|23|}\partial_{y_{23}}E_{51}$
$\displaystyle+$
$\displaystyle(-1)^{|13,15|}y_{15}\partial_{y_{12}}\partial_{y_{13}}+(-1)^{|23,25|}y_{25}\partial_{y_{12}}\partial_{y_{23}}+(-1)^{|13|+|23,35|}y_{35}\partial_{y_{13}}\partial_{y_{23}}$
$\displaystyle+$
$\displaystyle(-1)^{1+|34,45|}y_{45}\partial_{y_{12}}\partial_{y_{34}}+(-1)^{1+|23|+|14,45|}y_{45}\partial_{y_{23}}\partial_{y_{14}}+(-1)^{|13|+|24,45|}y_{45}\partial_{y_{13}}\partial_{y_{24}},$
$None$ $\displaystyle(x_{5}d_{45})_{-3}$ $\displaystyle=$
$\displaystyle(-1)^{|23,34|}z_{1}\partial_{y_{12}}\partial_{y_{23}}\partial_{y_{34}}+(-1)^{1+|23,24|+|13|}z_{1}\partial_{y_{13}}\partial_{y_{23}}\partial_{y_{24}}$
$\displaystyle+$
$\displaystyle(-1)^{|13|}z_{2}\partial_{y_{13}}\partial_{y_{23}}\partial_{y_{14}}+(-1)^{1+|13,34|}z_{2}\partial_{y_{12}}\partial_{y_{13}}\partial_{y_{34}}$
$\displaystyle+$
$\displaystyle(-1)^{|13,24|}z_{3}\partial_{y_{12}}\partial_{y_{13}}\partial_{y_{24}}-z_{3}\partial_{y_{12}}\partial_{y_{23}}\partial_{y_{14}}-z_{4}\partial_{y_{12}}\partial_{y_{13}}\partial_{y_{23}}.\hskip
108.12054pt(4.30)$
It follows from the equation (4.8) that
$x_{5}d_{45}.\partial^{m}\wedge^{n}V\subseteq\partial^{m-1}\wedge^{n+1}V+\partial^{m}\wedge^{n-1}V+\partial^{m+1}\wedge^{n-3}V,\
x_{5}d_{45}.M_{k}\subseteq M_{k-1},$
$x_{5}d_{45}=(x_{5}d_{45})_{1}+(x_{5}d_{45})_{-1}+(x_{5}d_{45})_{-3}.$ $None$
Furthermore, $\xi^{m,n}$ must satisfy the following equations inductively:
$(x_{5}d_{45})_{1}.\xi_{m,n}=0,\
(x_{5}d_{45})_{-1}.\xi_{m,n}+(x_{5}d_{45})_{1}.\xi_{m+1,n-2}=0,$
$(x_{5}d_{45})_{-3}.\xi_{m,n}+(x_{5}d_{45})_{-1}.\xi_{m+1,n-2}+(x_{5}d_{45})_{1}.\xi_{m+2,n-4}=0,$
$(x_{5}d_{45})_{-3}.\xi_{m^{\prime},n^{\prime}}+(x_{5}d_{45})_{-1}.\xi_{m^{\prime}+1,n^{\prime}-2}+(x_{5}d_{45})_{1}.\xi_{m^{\prime}+2,n^{\prime}-4}=0,\
\mbox{ for \ any}\ (m^{\prime},n^{\prime})\in\Gamma_{k}^{(m,n)}.$ $None$
Applying (4.27), the leading term $\xi_{m,n}$ should be killed by the
following three operators:
$(x_{5}d_{45})_{1},\ (x_{5}d_{45})_{-1}+(x_{5}d_{45})_{1}(-\frac{1}{2}P),\
(x_{5}d_{45})_{-3}+(x_{5}d_{45})_{-1}(-\frac{1}{2}P)+(x_{5}d_{45})_{1}(\frac{1}{8}P^{2}).$
$None$
We can reduce the last two differential operators to be of more explicit
forms. Indeed, the following relations are easily checked:
$\displaystyle[(x_{5}d_{45})_{1},P]$ $\displaystyle=$ $\displaystyle-
z_{3}\partial_{z_{5}}\partial_{y_{12}}+(-1)^{|13|}z_{2}\partial_{z_{5}}\partial_{y_{13}}+(-1)^{1+|23|}z_{1}\partial_{z_{5}}\partial_{y_{23}}$
$\displaystyle+$
$\displaystyle(-1)^{1+|34,45|}y_{45}\partial_{y_{12}}\partial_{y_{34}}+(-1)^{1+|23|+|14,45|}y_{45}\partial_{y_{23}}\partial_{y_{14}}+(-1)^{|13|+|24,45|}y_{45}\partial_{y_{13}}\partial_{y_{24}},$
$[[(x_{5}d_{45})_{1},P],P]=2(x_{5}d_{45})_{-3}+2z_{4}\partial_{y_{12}}\partial_{y_{13}}\partial_{y_{23}},\
[(x_{5}d_{45})_{-1},P]=3(x_{5}d_{45})_{-3}.$ $None$
Therefore,
$\displaystyle[(x_{5}d_{45})_{-1}+(x_{5}d_{45})_{1}(-\frac{1}{2}P)]\xi_{m,n}$
$\displaystyle=\\{(x_{5}d_{45})_{-1}-\frac{1}{2}P(x_{5}d_{45})_{1}-\frac{1}{2}[(x_{5}d_{45})_{1},P]\\}\xi_{m,n}$
$\displaystyle\stackrel{{\scriptstyle\mbox{by}(4.34)}}{{=}}[-\frac{1}{2}z_{3}\partial_{z_{5}}\partial_{y_{12}}+\frac{1}{2}(-1)^{|13|}z_{2}\partial_{z_{5}}\partial_{y_{13}}+(-1)^{1+|23|}\frac{1}{2}z_{1}\partial_{z_{5}}\partial_{y_{23}}$
$\displaystyle+\partial_{y_{12}}E_{53}+(-1)^{1+|13|}\partial_{y_{13}}E_{52}+(-1)^{|23|}\partial_{y_{23}}E_{51}$
$\displaystyle+(-1)^{|13,15|}y_{15}\partial_{y_{12}}\partial_{y_{13}}+(-1)^{|23,25|}y_{25}\partial_{y_{12}}\partial_{y_{23}}$
$\displaystyle+(-1)^{|13|+|23,35|}y_{35}\partial_{y_{13}}\partial_{y_{23}}+\frac{1}{2}(-1)^{1+|34,45|}y_{45}\partial_{y_{12}}\partial_{y_{34}}$
$\displaystyle+\frac{1}{2}(-1)^{1+|23|+|14,45|}y_{45}\partial_{y_{23}}\partial_{y_{14}}+\frac{1}{2}(-1)^{|13|+|24,45|}y_{45}\partial_{y_{13}}\partial_{y_{24}}]\xi_{m,n}.$
$\displaystyle\stackrel{{\scriptstyle\mbox{by}(4.11)}}{{=}}[(E_{53}+\frac{1}{2}(x_{5}\partial_{x_{3}})_{0}^{\prime}-\frac{1}{2}z_{3}\partial_{z_{5}})\partial_{y_{12}}+(E_{52}+\frac{1}{2}(x_{5}\partial_{x_{2}})_{0}^{\prime}-\frac{1}{2}z_{2}\partial_{z_{5}})(-1)^{1+|13|}\partial_{y_{13}}$
$\displaystyle+(E_{51}+\frac{1}{2}(x_{5}\partial_{x_{1}})_{0}^{\prime}-\frac{1}{2}z_{1}\partial_{z_{5}})(-1)^{|23|}\partial_{y_{23}}]\xi_{m,n}\hskip
204.85974pt(4.35)$
Hence,
$P(x_{5}d_{45})_{-1}\xi_{m,n}=\frac{1}{2}P(x_{5}d_{45})_{1}P\xi_{m,n}=\frac{1}{2}P[(x_{5}d_{45})_{1},P]\xi_{m,n}.$
$None$
Furthermore, (4.34) and (4.35) imply that
$\displaystyle[(x_{5}d_{45})_{-3}+(x_{5}d_{45})_{-1}(-\frac{1}{2}P)+(x_{5}d_{45})_{1}(\frac{1}{8}P^{2})]\xi_{m,n}$
$\displaystyle=\\{(x_{5}d_{45})_{-3}-\frac{3}{2}(x_{5}d_{45})_{-3}-\frac{1}{2}P(x_{5}d_{45})_{-1}$
$\displaystyle+\frac{1}{8}[[(x_{5}d_{45})_{1},P],P]+\frac{1}{8}P[(x_{5}d_{45})_{1},P]+\frac{1}{8}P(x_{5}d_{45})_{1}P\\}\xi_{m,n}$
$\displaystyle\stackrel{{\scriptstyle\mbox{by}(4.34),(4.35),(4.36)}}{{=}}[-\frac{1}{4}(x_{5}d_{45})_{-3}+\frac{1}{4}z_{4}\partial_{y_{12}}\partial_{y_{13}}\partial_{y_{23}}]\xi_{m,n}$
$\displaystyle=\frac{1}{4}[z_{1}(-1)^{|13|+|23,24|}\partial_{y_{13}}\partial_{y_{23}}\partial_{y_{24}}-z_{1}(-1)^{|23,34|}\partial_{y_{12}}\partial_{y_{23}}\partial_{y_{34}}$
$\displaystyle+z_{2}(-1)^{|13,34|}\partial_{y_{12}}\partial_{y_{13}}\partial_{y_{34}}-z_{2}(-1)^{|13|}\partial_{y_{13}}\partial_{y_{23}}\partial_{y_{14}}$
$\displaystyle+z_{3}\partial_{y_{12}}\partial_{y_{23}}\partial_{y_{14}}-z_{3}(-1)^{|13,24|}\partial_{y_{12}}\partial_{y_{13}}\partial_{y_{24}}+2z_{4}\partial_{y_{12}}\partial_{y_{13}}\partial_{y_{23}}]\xi_{m,n}.\hskip
119.50148pt(4.37)$
Denote the set of all the highest weight vectors for tensor modules
$V(m\omega_{4})\bigotimes\wedge^{n}W$ by
$S_{m,n}=\\{e_{m,n}^{1},\cdots,e_{m,n}^{\nu(m,n)}\\}$ $None$
By Lemma 3.3, any singular vector $\xi_{m,n}$ of the tensor product module
$V(m\omega_{4})\bigotimes\bigwedge^{n}V(\omega_{2})\bigotimes V(\lambda)$ for
$sl_{5}$ can be written by the following form:
$\xi_{m,n}=e_{m,n}^{i}\otimes v_{\vartheta}+\cdots.$ $None$
We consider the set
$S_{m,n}^{\prime}=\\{e_{m,n}^{i}\in S_{m,n}\ |\
(-1)^{1+|45|}\partial_{z_{5}}y_{45}.\phi(e_{m,n}^{i}\otimes v)=0,$
$[(x_{5}d_{45})_{-3}+(x_{5}d_{45})_{-1}(-\frac{1}{2}P)+(x_{5}d_{45})_{1}(\frac{1}{8}P^{2})].\phi(e_{m,n}^{i}\otimes
v)=0,\ \forall\ v\in V(\lambda)\\}.$ $None$
Proposition 4.4 All the non empty set of $S_{m,n}^{\prime}$ are listed in the
following:
$S_{0,0}^{\prime}=\\{1\\},S_{0,1}^{\prime}=\\{d_{12}\\},S_{0,2}^{\prime}=\\{d_{12}\wedge
d_{13}\\},S_{0,3}^{\prime}=\\{d_{12}\wedge d_{13}\wedge
d_{14}\\},S_{0,4}^{\prime}=\\{d_{12}\wedge d_{13}\wedge d_{14}\wedge
d_{15}\\}.$
Proof Let $V(\mu)$ be any highest weight module appearing in the decomposition
of the $sl_{5}$ wedge module $\wedge^{n}V(\omega_{2})$ (cf. Table 2). And the
highest weights appearing in the decomposition of $V(m\omega_{4})\otimes
V(\mu)$ are listed in Table 3. By Lemma 3.3, the maximal vector in the tensor
module $V(m\omega_{4})\otimes V(\mu)$ is written as:
$\partial_{5}^{m}\otimes
l_{\mu}+\sum\limits_{\underline{q}\in\mathbb{N}^{5}}\partial^{\underline{q}}\otimes
v_{\underline{q}},$ $None$
where $l_{\mu}$ satisfies
$E_{12}l_{\mu}=0,E_{23}l_{\mu}=0,E_{34}l_{\mu}=0,E_{45}^{m+1}l_{\mu}=0$. By
detailed calculation, we get all the $l_{\mu}$, which are listed in Table-4. A
straightforward but messy check case by case shows that the assertion holds.
$\Box$
Remark 4.5 For the 10-tuple $d_{12}\wedge d_{13}\wedge\cdots\wedge d_{45}$, we
use the notation
${\hat{d}_{i_{1}j_{1}}}\wedge{\hat{d}_{i_{2}j_{2}}}\wedge\cdots\wedge{\hat{d}_{i_{k}j_{k}}}$
to denote the (10-k)-tuple where $d_{i_{1}j_{1}},\cdots,d_{i_{k}j_{k}}$ have
been omitted in Table 4.
To summarize Proposition 4.3 and Proposition 4.4, we have proved the following
statement in this section:
Theorem 4.6 Any singular vector for $E(5,10)$-module $M$ is of the form:
$\xi^{0,n}=e^{-\frac{1}{2}P}\xi_{0,n},\ \ n\in\overline{1,4}$
where the leading term $\xi_{0,n}$ satisfies the equation:
$[(E_{53}+\frac{1}{2}(x_{5}\partial_{x_{3}})_{0}^{\prime})\partial_{y_{12}}+(E_{52}+\frac{1}{2}(x_{5}\partial_{x_{2}})_{0}^{\prime})(-1)^{1+|13|}\partial_{y_{13}}+(E_{51}+\frac{1}{2}(x_{5}\partial_{x_{1}})_{0}^{\prime})(-1)^{|23|}\partial_{y_{23}}].\xi_{0,n}=0.$
$None$
Moreover, $\xi_{0,n}$ is the maximal vector lying in one of the following
$sl_{5}$-tensor modules:
$\xi_{0,1}\in V(\omega_{2})\otimes V(\lambda),\xi_{0,2}\in
V(\omega_{1}+\omega_{3})\otimes V(\lambda),\xi_{0,3}\in
V(2\omega_{1}+\omega_{4})\otimes V(\lambda),\xi_{0,4}\in V(3\omega_{1})\otimes
V(\lambda).$ $None$
## 5 Singular vectors degree by degree
In this section, we work out all the singular vectors in Theorem 4.6
explicitly degree by degree. Before turning to the calculation, we introduce
some formula which we are going to use in the remainder of this section.
Recall that we could endow any vector space $\Lambda^{m}V$ an $sl_{5}$\-
module structure with the action
$(x_{i}\partial_{x_{i}})_{0}^{{}^{\prime}}-(x_{i+1}\partial_{x_{i+1}})_{0}^{{}^{\prime}}\
(i\in\overline{1,4}),\ (x_{i}\partial_{x_{j}})_{0}^{\prime}\ (i\neq j)$, which
is isomorphic to the tensor product module $\wedge^{m}V(\omega_{2})\otimes
V(\lambda)$ in Section 4.1. Now we define the following differential operator
on the $sl_{5}$\- module $\Lambda^{m}V$:
$\tilde{c}=\frac{1}{10}[\sum\limits_{i=1}^{4}((x_{i}\partial_{x_{i}})_{0}^{{}^{\prime}}-(x_{i+1}\partial_{x_{i+1}})_{0}^{{}^{\prime}})h_{i}^{*}+\sum\limits_{1\leq
i\neq j\leq 5}(x_{i}\partial_{x_{j}})_{0}^{{}^{\prime}}E_{ji}],$ $None$
$T_{i,jkl}=[E_{ij}+\frac{1}{2}(x_{i}\partial_{x_{j}})_{0}^{\prime}](-1)^{|kl|}\partial_{y_{kl}}+[E_{ik}+\frac{1}{2}(x_{i}\partial_{x_{k}})_{0}^{\prime}](-1)^{1+|jl|}\partial_{y_{jl}}+[E_{il}+\frac{1}{2}(x_{i}\partial_{x_{l}})_{0}^{\prime}](-1)^{|jk|}\partial_{y_{jk}}.$
$None$
Lemma 5.1 Assume
$Q_{ij}^{0}\in\mbox{Span}_{\mathbb{F}}\\{y_{ij}\partial_{y_{kl}}\ |\ 1\leq
i<j\leq 5,\ 1\leq k<l\leq 5\\}$. Then
$\sum\limits_{1\leq i<j\leq
5}Q_{ij}^{0}.(-1)^{|ij|}\partial_{y_{ij}}.\tilde{c}|_{\Lambda^{m}V}=\sum\limits_{1\leq
i<j\leq 5}Q_{ij}^{1}(-1)^{|ij|}\partial_{y_{ij}}|_{\wedge^{m}V},$
where
$Q_{ij}^{1}=Q_{ij}^{0}(\tilde{c}+\sum\limits_{k=1}^{4}\frac{s_{k}^{ij}}{10}h_{k}^{*})+\frac{1}{10}\sum\limits_{m\neq
i,j}(Q_{im}^{0}E_{jm}-Q_{jm}^{0}E_{im}),$
$s_{k}^{ij}\stackrel{{\scriptstyle\mbox{by}(4.16)}}{{=}}t_{k}(\underline{n})-t_{k}(\underline{n}-\epsilon_{ij})$
$None$
for any $k\in\overline{1,4},\underline{n}\in T^{10},(ij)\in S^{\prime}$.
Proof Indeed, the formula (5.3) follows from:
$[(-1)^{|ij|}\partial_{y_{ij}},\tilde{c}]|_{\wedge^{m}V}=\frac{1}{10}(\sum\limits_{k\neq
i,j}(-1)^{|ik|}\partial_{y_{ik}}E_{kj}-\sum\limits_{k\neq
i,j}(-1)^{|jk|}\partial_{y_{jk}}E_{ki}+\sum\limits_{k=1}^{4}s_{k}^{ij}(-1)^{|ij|}\partial_{y_{ij}}h_{k}^{*})|_{\wedge^{m}V}.$
$None$
$\Box$
Lemma 5.2 We could define the following intertwining operators between the
$sl_{5}$-module $V(\omega_{1}+\omega_{2})\otimes\Lambda^{m}V$ and
$\Lambda^{m-1}V$ by:
$T^{m}:V(\omega_{1}+\omega_{2})\otimes\Lambda^{m}V\rightarrow\Lambda^{m-1}V;v_{30,1}^{\omega_{1}+\omega_{2}}\otimes\xi\mapsto
T_{5,123}(\xi),$ $None$
where $v_{30,1}^{\omega_{1}+\omega_{2}}$ is the lowest weight vector for
$V(\omega_{1}+\omega_{2})$ (cf. Table 9) and $\xi$ is any maximal vector in
$sl_{5}$-module $\Lambda^{m}V$.
Proof Since the $sl_{5}$-module $V(\omega_{1}+\omega_{2})\otimes\Lambda^{m}V$
is generated by such vectors of $v_{30,1}^{\omega_{1}+\omega_{2}}\otimes\xi$
by part (4) of Lemma 3.3, the assertion follows from the following formula:
$[(x_{i+1}\partial_{x_{i}})_{0}^{\prime},T_{5,123}]|_{\wedge^{m}V}=0,\
i\in\overline{1,4},$
$[(x_{s}\partial_{x_{t}})_{0}^{\prime},T_{i,jkl}]|_{\wedge^{m}V}=\delta_{t,i}T_{s,jkl}-\delta_{s,j}T_{i,tkl}-\delta_{s,k}T_{i,jtl}-\delta_{s,l}T_{i,jkt}.$
$None$
$\Box$
### 5.1 Singular vectors of degree one
Theorem 5.3 All the possible degree one singular vectors are listed in the
following:
$d_{12}v_{\lambda},\ \mbox{where}\ \lambda=(m,n,0,0),\ m,n\in\mathbb{N};$
$\prod\limits_{\mbox{wt}(d_{15})<\sigma\leq\omega_{2}}\frac{\tilde{c}-\chi_{\sigma+\lambda}(\tilde{c})}{\chi_{\mbox{wt}(d_{15})+\lambda}(\tilde{c})-\chi_{\sigma+\lambda}(\tilde{c})}.d_{15}v_{\lambda},\
\mbox{where}\ \lambda=(m,0,0,n),\ m\in\mathbb{N},\ 1\leq n\in\mathbb{N};$
$\prod\limits_{\mbox{wt}(d_{45})<\sigma\leq\omega_{2}}\frac{\tilde{c}-\chi_{\sigma+\lambda}(\tilde{c})}{\chi_{\mbox{wt}(d_{45})+\lambda}(\tilde{c})-\chi_{\sigma+\lambda}(\tilde{c})}.d_{45}v_{\lambda},\
\mbox{where}\ \lambda=(0,0,m,n),\ 1\leq m\in\mathbb{N},\ n\in\mathbb{N}.$
Proof The leading term of any singular vector of degree one can be written as
$\xi_{0,1}=\sum\limits_{1\leq i<j\leq 5}d_{ij}v_{ij},\ v_{ij}\in V(\lambda),$
$None$
which should satisfy :
$T_{5,123}.\xi_{0,1}=[\partial_{y_{12}}E_{53}+(-1)^{1+|13|}\partial_{y_{13}}E_{52}+(-1)^{|23|}\partial_{y_{23}}E_{51}].\xi_{0,1}=0,$
$None$
i.e.
$E_{53}v_{12}-E_{52}v_{13}+E_{51}v_{23}=0.$ $None$
Note that $(x_{i}\partial_{x_{j}})_{0}.\xi_{0,1}=0$ $(1\leq i<j\leq 5)$ imply
that
$v_{13}=-E_{23}v_{12},\ v_{23}=E_{13}v_{12}=-E_{12}v_{13},\
v_{14}=-E_{34}v_{13},\ v_{15}=-E_{25}v_{12}=-E_{35}v_{13},\ $
$v_{25}=E_{15}v_{12}=-E_{35}v_{23},\
v_{35}=E_{15}v_{13}=-E_{23}v_{25}=E_{25}v_{23},\ v_{45}=-E_{34}v_{35}.$ $None$
Obviously, $v_{12}\neq 0$.
Case 1. wt($\xi_{0,1})=$ wt$(d_{12}v_{\lambda})$.
In this case, $v_{13}=v_{23}=0$, $v_{12}=v_{\lambda}$. And (5.9) implies that
$E_{5,3}.v_{12}=E_{5,3}.v_{\lambda}=0$. That is to say,
$\lambda=(m,n,0,0),(m,n)\in\mathbb{N}^{2}$.
Case 2.
wt($\xi_{0,1})\in\\{\mbox{wt}(d_{13}v_{\lambda}),\mbox{wt}(d_{14}v_{\lambda}),\mbox{wt}(d_{15}v_{\lambda})\\}$.
In these three cases, we have $v_{23}=0,v_{13}\neq 0$.
Case 2.1
$wt(\xi_{0,1})\in\\{\mbox{wt}(d_{13}v_{\lambda}),\mbox{wt}(d_{14}v_{\lambda})\\}$
In these two cases, $v_{15}=0,v_{13}\neq 0$,
$(h_{2}+h_{3}+h_{4}).v_{13}=(\lambda_{2}+\lambda_{3}+\lambda_{4})v_{13}.$
Hence,
$\displaystyle
0=E_{25}(E_{53}.v_{12}-E_{52}.v_{13})=(E_{23}+E_{53}E_{25})v_{12}-(h_{2}+h_{3}+h_{4}+E_{52}E_{25})v_{13}$
$\displaystyle=$
$\displaystyle-(1+h_{2}+h_{3}+h_{4})v_{13}-E_{53}v_{15}=-(1+h_{2}+h_{3}+h_{4})v_{13}=-(1+\lambda_{2}+\lambda_{3}+\lambda_{4})v_{13}\hskip
28.45274pt(5.11)$
provides a contradiction.
Case 2.2 $wt(\xi_{0,1})=\mbox{wt}(d_{15}v_{\lambda})$
In this case, $v_{15}=v_{\lambda}$ and
$\mbox{wt}(v_{13})=\lambda-\alpha_{3}-\alpha_{4}$. And
$0=E_{25}(E_{53}.v_{12}-E_{52}.v_{13})=-(\lambda_{2}+\lambda_{3}+\lambda_{4})v_{13}-E_{53}v_{15},$
$0=E_{35}E_{25}(E_{53}.v_{12}-E_{52}.v_{13})=-E_{35}(\lambda_{2}+\lambda_{3}+\lambda_{4})v_{13}-E_{35}E_{53}v_{15}=(\lambda_{2}+\lambda_{3}+\lambda_{4}-h_{3}-h_{4})v_{15}=\lambda_{2}v_{15}$
$None$
forces $\lambda_{2}=0$.
Case 2.2.1. $\lambda_{3}=0,\ \lambda_{4}>0$.
Suppose
$T_{5,123}.\tilde{c}^{k}=\sum\limits_{1\leq i<j\leq
5}Q_{ij}^{k-1}(-1)^{|ij|}\partial_{y_{ij}}.$ $None$
By (5.3),
$Q_{15}^{1}v_{\lambda}=(Q_{12}^{0}E_{52}+Q_{13}^{0}E_{53})v_{\lambda}=0$. Then
$T_{5,123}.\xi_{0,1}=[\partial_{y_{12}}E_{53}+(-1)^{1+|13|}\partial_{y_{13}}E_{52}+(-1)^{|23|}\partial_{y_{23}}E_{51}].\tilde{c}.d_{15}v_{\lambda}=Q_{15}^{1}.v_{\lambda}=0.$
$None$
Therefore, $\lambda=(m,0,0,n),m\in\mathbb{N},n>0$.
Case 2.2.2. $\lambda_{3}>0,\ \lambda_{4}>0$.
Note that $E_{54}v_{\lambda}\neq 0$. Since
$T_{5,123}.\tilde{c}.d_{15}v_{\lambda}=Q_{15}^{1}.v_{\lambda}=0$, the equation
(3.8) implies
$T_{5,123}.\xi_{0,1}=T_{5,123}.\tilde{c}^{2}.d_{15}v_{\lambda}=Q_{15}^{2}.v_{\lambda}=(E_{53}E_{42}E_{54}-E_{52}E_{43}E_{54})v_{\lambda}=0;$
$None$
which yields
$0=E_{34}E_{25}(E_{53}E_{42}E_{54}-E_{52}E_{43}E_{54})v_{\lambda}=-\lambda_{3}(1+\lambda_{3}+\lambda_{4})E_{54}v_{\lambda}.$
$None$
A contradiction arises.
Case 3.
wt($\xi_{0,1})\in\\{\mbox{wt}(d_{23}v_{\lambda}),\mbox{wt}(d_{24}v_{\lambda}),\mbox{wt}(d_{34}v_{\lambda}),\mbox{wt}(d_{25}v_{\lambda}),\mbox{wt}(d_{35}v_{\lambda}),\mbox{wt}(d_{45}v_{\lambda})\\}$.
In these cases, $v_{13}\neq 0,v_{23}\neq 0$. Set
$Q=E_{53}.v_{12}-E_{52}.v_{13}+E_{51}.v_{23},\ E_{15}Q=Q_{1},\
E_{25}Q_{1}=Q_{2},E_{35}Q_{1}=Q_{2}^{\prime}\ .$ $None$
Then
$0=Q_{1}=E_{13}v_{12}+E_{53}E_{15}v_{12}-E_{12}v_{13}-E_{52}E_{15}v_{13}+(h_{1}+h_{2}+h_{3}+h_{4})v_{23}.$
$None$
Case 3.1.
$wt(\xi_{0,1})\in\\{\mbox{wt}(d_{23}v_{\lambda}),\mbox{wt}(d_{24}v_{\lambda}),\mbox{wt}(d_{34}v_{\lambda})\\}$.
In these three cases, we have $v_{23}\neq 0$, $v_{25}=v_{35}=0$. So
$0=Q_{1}=(2+|\mbox{wt}(v_{23})|)v_{23}$ $None$
induces a contradiction.
Case 3.2. $wt(\xi_{0,1})=\mbox{wt}(d_{25}v_{\lambda})$.
In this case, $v_{35}=0$ and
$\mbox{wt}(v_{23})=\lambda-\alpha_{3}-\alpha_{4}$. So
$0=Q_{1}=(2+|\mbox{wt}(v_{23})|)v_{23}+E_{53}v_{25},0=Q_{2}^{\prime}=-(|\lambda|+2-h_{3}-h_{4})v_{25}=-(\lambda_{1}+\lambda_{2}+1)v_{25}$
$None$
force $\lambda_{1}+\lambda_{2}+1=0$. A contradiction arises.
Case 3.3.
$wt(\xi_{0,1})\in\\{\mbox{wt}(d_{35}v_{\lambda}),\mbox{wt}(d_{45}v_{\lambda})\\}$.
In these two cases, $v_{35}\neq 0$. And the equations
$0=Q_{1}=(2+|\mbox{wt}(v_{23})|)v_{23}+E_{53}v_{25}-E_{52}v_{35},0=Q_{2}=(1+|\mbox{wt}(v_{23})|-h_{2}-h_{3}-h_{4})v_{35}=\lambda_{1}v_{35}$
$None$
imply $\lambda_{1}=0$. Recall the intertwining operator defined in Lemma 5.2.
In these two cases, $T_{5,123}(\xi_{0,1})=0$ is equivalent to
$T^{1}|_{V(\omega_{1}+\omega_{2})\otimes V(\mbox{wt}(\xi_{0,1}))}=0$. Assume
$v_{(\omega_{1}+\omega_{2})\otimes(\mbox{wt}(\xi_{0,1}))}^{\lambda}$ is any
maximal vector of weight $\lambda$ appearing in the tensor decomposition
$V(\omega_{1}+\omega_{2})\otimes V(\mbox{wt}(\xi_{0,1}))$. Then
$T^{1}|_{V(\omega_{1}+\omega_{2})\otimes V(\mbox{wt}(\xi_{0,1}))}=0$ iff
$T^{1}(v_{(\omega_{1}+\omega_{2})\otimes(\mbox{wt}(\xi_{0,1}))}^{\lambda})=0$.
Case 3.3.1
$wt(\xi_{0,1})=\mbox{wt}(d_{35}v_{\lambda})=(0,\lambda_{2}-1,\lambda_{3}+1,\lambda_{4}-1)$.
Indeed,
$T^{1}(v_{(1,1,0,0)\otimes(0,\lambda_{2}-1,\lambda_{3}+1,\lambda_{4}-1)}^{\lambda})$
in this case could be written as:
$\displaystyle
T^{1}(v_{(1,1,0,0)\otimes(0,\lambda_{2}-1,\lambda_{3}+1,\lambda_{4}-1)}^{\lambda})$
$\displaystyle=$
$\displaystyle(T_{4,345}+T_{2,235})\xi_{0,1}+\frac{1}{2}(T_{1,135}-T_{2,235})\xi_{0,1}-\frac{1}{1+\lambda_{3}}(T_{3,345}-T_{2,245}).(x_{4}\partial_{x_{3}})_{0}.\xi_{0,1}$
$\displaystyle-$
$\displaystyle\frac{1}{2+2\lambda_{3}}(T_{2,245}-T_{1,145}).(x_{4}\partial_{x_{3}})_{0}.\xi_{0,1}-\frac{3}{1+\lambda_{2}+\lambda_{3}}T_{2,345}.x_{4}\partial_{x_{3}}.x_{3}\partial_{x_{2}}\xi_{0,1}$
$\displaystyle+$
$\displaystyle\frac{6+3\lambda_{3}}{(1+\lambda_{2}+\lambda_{3})(1+\lambda_{3})}T_{2,345}.(x_{3}\partial_{x_{2}})_{0}.(x_{4}\partial_{x_{3}})_{0}.\xi_{0,1}$
$\displaystyle-$
$\displaystyle\frac{6+3\lambda_{3}}{(1+\lambda_{2}+\lambda_{3})(1+\lambda_{3})}T_{1,345}.(x_{2}\partial_{x_{1}})_{0}.(x_{3}\partial_{x_{2}})_{0}.(x_{4}\partial_{x_{3}})_{0}.\xi_{0,1}$
$\displaystyle+$
$\displaystyle\frac{3}{1+\lambda_{2}+\lambda_{3}}T_{1,345}.(x_{4}\partial_{x_{3}})_{0}.(x_{2}\partial_{x_{1}})_{0}.(x_{3}\partial_{x_{2}})_{0}.\xi_{0,1}=\frac{(2+\lambda_{3})(\lambda_{2}+\lambda_{3}+7))}{(1+\lambda_{3})(\lambda_{2}+\lambda_{3}+1))}v_{35}\neq
0.\hskip 34.14322pt(5.22)$
Case 3.3.2
$wt(\xi_{0,1})=\mbox{wt}(d_{45}v_{\lambda})=(0,\lambda_{2},\lambda_{3}-1,\lambda_{4})$.
Suppose $\lambda_{2}\neq 0$. Then
$\displaystyle
T^{1}(v_{(1,1,0,0)\otimes(0,\lambda_{2},\lambda_{3}-1,\lambda_{4})}^{\lambda})=\frac{-2\lambda_{2}}{3}(T_{3,345}-T_{2,245})\xi_{0,1}-\frac{\lambda_{2}}{3}(T_{2,245}-T_{1,145})\xi_{0,1}$
$\displaystyle+$ $\displaystyle
T_{2,345}.(x_{3}\partial_{x_{2}})_{0}.\xi_{0,1}-T_{1,345}.(x_{2}\partial_{x_{1}})_{0}.(x_{3}\partial_{x_{2}})_{0}.\xi_{0,1}=\frac{2\lambda_{2}(\lambda_{2}+3)}{3}v_{45}\neq
0\hskip 79.6678pt(5.23)$
induces a contradiction. Assume $\lambda_{2}=0$. Then it is easily checked
that
$T^{1}(v_{(1,1,0,0)\otimes(0,\lambda_{2},\lambda_{3}-1,\lambda_{4})}^{\lambda})=[2(T_{3,345}-T_{2,245})+(T_{2,245}-T_{1,145})]\xi_{0,1}=0.$
$None$
Thus $\lambda=(0,0,m,n)$. The proof is complete by Lemma 3.3. $\Box$
### 5.2 Singular vectors of degree two
Theorem 5.4 All the possible degree two singular vectors are listed in the
following:
$\prod\limits_{\mbox{wt}(d_{12}d_{15})<\sigma\leq\omega_{1}+\omega_{3}}\frac{\tilde{c}-\chi_{\sigma+\lambda}(\tilde{c})}{\chi_{\mbox{wt}(d_{12}d_{15})+\lambda}(\tilde{c})-\chi_{\sigma+\lambda}(\tilde{c})}.d_{12}d_{15}v_{\lambda},\
\mbox{where}\ \ \lambda=(m,0,0,1),\ m\in\mathbb{N}.$
Proof The leading term of any singular vector of degree two could be written
as:
$\xi_{0,2}=\sum\limits_{j\in\overline{1,35},k\in\overline{1,\mbox{mult}(\overrightarrow{w}_{j}^{\omega_{1}+\omega_{3}})}}v_{j,k}^{\omega_{1}+\omega_{3}}v_{j,k}^{\lambda},\
v_{j,k}^{\lambda}\in V(\lambda),$ $None$
which should satisfy $T_{5,123}.\xi_{0,2}=0$. Assume
$T_{5,123}.\xi_{0,2}=\sum\limits_{1\leq i<j\leq 5}d_{ij}t_{ij},t_{ij}\in
V(\lambda).$ $None$
Then we could derive the following equations:
$t_{12}=E_{52}v_{1,1}^{\lambda}-E_{51}v_{2,1}^{\lambda}=0,t_{15}=E_{53}v_{5,1}^{\lambda}-E_{52}v_{10,1}^{\lambda}+E_{51}(v_{11,2}^{\lambda}+v_{11,3}^{\lambda})+v_{1,1}^{\lambda}=0,$
$t_{13}=E_{53}v_{1,1}^{\lambda}-E_{51}v_{6,1}^{\lambda}=0,t_{25}=E_{53}v_{9,1}^{\lambda}+E_{52}(v_{11,1}^{\lambda}+v_{11,3}^{\lambda})+E_{51}v_{19,1}^{\lambda}+v_{2,1}^{\lambda}=0,$
$t_{23}=E_{53}v_{2,1}^{\lambda}-E_{52}v_{6,1}^{\lambda}=0,t_{35}=E_{53}(v_{11,1}^{\lambda}+v_{11,2}^{\lambda})-E_{52}v_{17,1}^{\lambda}+E_{51}v_{22,1}^{\lambda}+v_{6,1}^{\lambda}=0.$
$None$
It follows from $v_{1,1}^{\lambda}\neq 0$ that one of $v_{5,1}^{\lambda},\
v_{10,1}^{\lambda},\ v_{11,2}^{\lambda}+v_{11,3}^{\lambda}$ should be nonzero.
Hence, the information of the weights in Table 5 implies that
$\mbox{wt}(\xi_{0,2})$ should be restricted to the following cases:
$\displaystyle\mbox{wt}(\xi_{0,2})\in\\{\lambda+\overrightarrow{w}_{5}^{\omega_{1}+\omega_{3}},\lambda+\overrightarrow{w}_{9}^{\omega_{1}+\omega_{3}},\lambda+\overrightarrow{w}_{10}^{\omega_{1}+\omega_{3}},\lambda+\overrightarrow{w}_{11}^{\omega_{1}+\omega_{3}},\lambda+\overrightarrow{w}_{15}^{\omega_{1}+\omega_{3}},\lambda+\overrightarrow{w}_{16}^{\omega_{1}+\omega_{3}},$
$\displaystyle\lambda+\overrightarrow{w}_{17}^{\omega_{1}+\omega_{3}},\lambda+\overrightarrow{w}_{19}^{\omega_{1}+\omega_{3}},\lambda+\overrightarrow{w}_{21}^{\omega_{1}+\omega_{3}},\lambda+\overrightarrow{w}_{22}^{\omega_{1}+\omega_{3}},\lambda+\overrightarrow{w}_{i}^{\omega_{1}+\omega_{3}}(i\in\overline{24,35})\\}\hskip
73.97733pt(5.30)$
Case 1
$\mbox{wt}(\xi_{0,2})\in\\{\lambda+\overrightarrow{w}_{5}^{\omega_{1}+\omega_{3}},\lambda+\overrightarrow{w}_{10}^{\omega_{1}+\omega_{3}},\lambda+\overrightarrow{w}_{15}^{\omega_{1}+\omega_{3}}\\}$
In these three cases, $v_{2,1}^{\lambda}=v_{6,1}^{\lambda}=0$.
Case 1.1
$wt(\xi_{0,2})=\lambda+\overrightarrow{w}_{5}^{\omega_{1}+\omega_{3}}$
We have $wt(v_{1,1}^{\lambda})=\lambda-\alpha_{3}-\alpha_{4}$ and
$\lambda_{4}>0$. So
$0=E_{25}.t_{12}=E_{25}.E_{52}v_{1,1}^{\lambda}=(h_{2}+h_{3}+h_{4})v_{1,1}^{\lambda}=(\lambda_{2}+\lambda_{3}+\lambda_{4}-1)v_{1,1}^{\lambda}$
yields $(\lambda_{2},\lambda_{3},\lambda_{4})=(0,0,1).$ Then
$\chi_{\overrightarrow{w}_{1}^{\omega_{1}+\omega_{3}}}(\tilde{c})=\frac{3\lambda_{1}+2}{25}$
by Lemma 3.5. And
$T_{5,123}.\xi_{0,2}=T_{5,123}.(\tilde{c}-\chi_{\overrightarrow{w}_{1}^{\omega_{1}+\omega_{3}}}(\tilde{c})).d_{12}d_{15}.v_{\lambda}=T_{5,123}.\tilde{c}.d_{12}d_{15}.v_{\lambda}-\chi_{\overrightarrow{w}_{1}^{\omega_{1}+\omega_{3}}}(\tilde{c})d_{15}E_{53}.v_{\lambda}=0,$
$None$
since
$T_{5,123}.\tilde{c}.d_{12}d_{15}.v_{\lambda}\stackrel{{\scriptstyle\mbox{by}\
(5.13)}}{{=}}Q_{12}^{1}d_{15}v_{\lambda}-Q_{15}^{1}d_{12}v_{\lambda};$
where
$Q_{12}^{1}\stackrel{{\scriptstyle\mbox{by}(5.3)}}{{=}}Q_{12}^{0}(\tilde{c}+\frac{h_{2}^{*}}{10})+\frac{1}{10}Q_{13}^{0}E_{23}-\frac{1}{10}Q_{23}^{0}E_{13},\
Q_{15}^{1}\stackrel{{\scriptstyle\mbox{by}(5.3)}}{{=}}\frac{1}{10}(Q_{12}^{0}E_{52}+Q_{13}^{0}E_{53}),$
$Q_{12}^{0}=E_{53}+\frac{(x_{5}\partial_{x_{3}})_{0}^{\prime}}{2},\
Q_{13}^{0}=-E_{52}-\frac{(x_{5}\partial_{x_{2}})_{0}^{\prime}}{2},Q_{23}^{0}=E_{51}+\frac{(x_{5}\partial_{x_{1}})_{0}^{\prime}}{2},$
$\tilde{c}.d_{15}v_{\lambda}\stackrel{{\scriptstyle\mbox{by}(5.1)}}{{=}}\frac{1}{10}[d_{15}(h_{1}^{*}-h_{4}^{*}).v_{\lambda}+\sum\limits_{i=2}^{4}d_{1i}E_{5i}v_{\lambda}]=\frac{1}{10}[\frac{3\lambda_{1}-3}{5}d_{15}v_{\lambda}+\sum\limits_{i=2}^{4}d_{1i}E_{5i}v_{\lambda}].$
$None$
That is, $\lambda=(m,0,0,1).$
Case 1.2
$wt(\xi_{0,2})=\lambda+\overrightarrow{w}_{10}^{\omega_{1}+\omega_{3}}$
We have
$\mbox{wt}(v_{1,1}^{\lambda})=\lambda-\alpha_{2}-\alpha_{3}-\alpha_{4}$ and
$\lambda_{2}>0,\lambda_{4}>0$, $E_{25}.v_{1,1}^{\lambda}=-v_{10,1}^{\lambda}$.
So
$0=E_{25}t_{12}=(h_{2}+h_{3}+h_{4})v_{1,1}^{\lambda}+E_{52}.E_{25}.v_{1,1}^{\lambda}=(\lambda_{2}+\lambda_{3}+\lambda_{4}-2)v_{1,1}^{\lambda}-E_{52}.v_{10,1}^{\lambda},$
$None$
$0=E_{25}^{2}t_{12}=E_{25}.[(\lambda_{2}+\lambda_{3}+\lambda_{4}-2)v_{1,1}^{\lambda}-E_{52}.v_{10,1}^{\lambda}]=-2(\lambda_{2}+\lambda_{3}+\lambda_{4}-1)v_{10,1}^{\lambda}$
$None$
yields a contradiction.
Case 1.3
$wt(\xi_{0,2})=\lambda+\overrightarrow{w}_{15}^{\omega_{1}+\omega_{3}}$
We have $\mbox{wt}(v_{10,1}^{\lambda})=\lambda-\alpha_{3}$ and
$\lambda_{3}>0$, $E_{23}.v_{1,5}^{\lambda}=-v_{10,1}^{\lambda}$. Then
$E_{53}v_{5,1}^{\lambda}-E_{52}v_{10,1}^{\lambda}+v_{1,1}^{\lambda}=0,$ $None$
$0=E_{25}(E_{53}v_{5,1}^{\lambda}-E_{52}v_{10,1}^{\lambda}+v_{1,1}^{\lambda})=E_{23}v_{5,1}^{\lambda}-(h_{2}+h_{3}+h_{4})v_{10,1}^{\lambda}+E_{25}v_{1,1}^{\lambda}$
$None$
imply $(\lambda_{2},\lambda_{3},\lambda_{4})=(0,0,0).$ A contradiction arises.
Case 2
$\mbox{wt}(\xi_{0,2})\in\\{\lambda+\overrightarrow{w}_{9}^{\omega_{1}+\omega_{3}},\lambda+\overrightarrow{w}_{11}^{\omega_{1}+\omega_{3}},\lambda+\overrightarrow{w}_{16}^{\omega_{1}+\omega_{3}},\lambda+\overrightarrow{w}_{17}^{\omega_{1}+\omega_{3}},\lambda+\overrightarrow{w}_{19}^{\omega_{1}+\omega_{3}},\lambda+\overrightarrow{w}_{21}^{\omega_{1}+\omega_{3}},\lambda+\overrightarrow{w}_{22}^{\omega_{1}+\omega_{3}},\lambda+\overrightarrow{w}_{i}^{\omega_{1}+\omega_{3}}(i\in\overline{24,35})\\}$
Case 2.1
$wt(\xi_{0,2})=\lambda+\overrightarrow{w}_{9}^{\omega_{1}+\omega_{3}}$
We have $v_{2,1}^{\lambda}\neq 0$, since
$E_{35}v_{2,1}^{\lambda}=-v_{9,1}^{\lambda}$. Note that $v_{6,1}^{\lambda}=0$,
$E_{15}v_{1,1}^{\lambda}=E_{15}v_{2,1}^{\lambda}=0$,
$E_{12}v_{1,1}^{\lambda}=-v_{2,1}^{\lambda}$,
$\mbox{wt}(v_{2,1}^{\lambda})=\lambda-\alpha_{3}-\alpha_{4}$ and
$\lambda_{1}>0,\lambda_{4}>0$. Then
$0=E_{15}.t_{12}=E_{15}.(E_{52}v_{1,1}^{\lambda}-E_{51}v_{2,1}^{\lambda})=(E_{12}+E_{52}E_{15})v_{1,1}^{\lambda}-(\sum\limits_{i=1}^{4}h_{i}+E_{51}E_{15})v_{2,1}^{\lambda}$
$None$
implies $|\lambda|=0$. A contradiction arises.
Case 2.2
$\mbox{wt}(\xi_{0,2})\in\\{\lambda+\overrightarrow{w}_{11}^{\omega_{1}+\omega_{3}},\lambda+\overrightarrow{w}_{16}^{\omega_{1}+\omega_{3}},\lambda+\overrightarrow{w}_{17}^{\omega_{1}+\omega_{3}},\lambda+\overrightarrow{w}_{19}^{\omega_{1}+\omega_{3}},\lambda+\overrightarrow{w}_{21}^{\omega_{1}+\omega_{3}},\lambda+\overrightarrow{w}_{22}^{\omega_{1}+\omega_{3}},\lambda+\overrightarrow{w}_{i}^{\omega_{1}+\omega_{3}}(i\in\overline{24,35})\\}$
Case 2.2.1
$\mbox{wt}(\xi_{0,2})\in\\{\lambda+\overrightarrow{w}_{11}^{\omega_{1}+\omega_{3}},\lambda+\overrightarrow{w}_{16}^{\omega_{1}+\omega_{3}},\lambda+\overrightarrow{w}_{17}^{\omega_{1}+\omega_{3}},\lambda+\overrightarrow{w}_{19}^{\omega_{1}+\omega_{3}},\lambda+\overrightarrow{w}_{21}^{\omega_{1}+\omega_{3}},\lambda+\overrightarrow{w}_{22}^{\omega_{1}+\omega_{3}},\lambda+\overrightarrow{w}_{24}^{\omega_{1}+\omega_{3}},\lambda+\overrightarrow{w}_{26}^{\omega_{1}+\omega_{3}},\lambda+\overrightarrow{w}_{27}^{\omega_{1}+\omega_{3}},\lambda+\overrightarrow{w}_{29}^{\omega_{1}+\omega_{3}},\lambda+\overrightarrow{w}_{30}^{\omega_{1}+\omega_{3}},\lambda+\overrightarrow{w}_{33}^{\omega_{1}+\omega_{3}}.\\}$
In these cases, $v_{25,1}^{\lambda}=v_{28,1}^{\lambda}=0$. Assume
$v_{11,2}^{\lambda}+v_{11,3}^{\lambda}\neq 0$. Then
$0=E_{15}t_{15}\stackrel{{\scriptstyle\mbox{by}(5.29)}}{{=}}(2+\sum\limits_{i=1}^{4}h_{i})(v_{11,2}^{\lambda}+v_{11,3}^{\lambda})=0$
$None$
yields a contradiction. Hence, $v_{11,2}^{\lambda}+v_{11,3}^{\lambda}=0$.
Furthermore, either the assertion $v_{10,1}^{\lambda}\neq 0,\
(h_{2}+h_{3}+h_{4})v_{10,1}^{\lambda}=0$ or the assertion
$v_{10,1}^{\lambda}=0,\ v_{5,1}^{\lambda}\neq 0,\
(h_{3}+h_{4}-1)v_{5,1}^{\lambda}=0$ holds. By detailed check case by case,
only the cases
$\mbox{wt}(\xi_{0,2})\in\\{\lambda+\overrightarrow{w}_{16}^{\omega_{1}+\omega_{3}},\lambda+\overrightarrow{w}_{24}^{\omega_{1}+\omega_{3}},\lambda+\overrightarrow{w}_{26}^{\omega_{1}+\omega_{3}}\\}$
satisfy this assertion.
For the case
$\mbox{wt}(\xi_{0,2})=\lambda+\overrightarrow{w}_{16}^{\omega_{1}+\omega_{3}}$,
we get $v_{10,1}^{\lambda}=0,\ v_{5,1}^{\lambda}\neq 0,\
(\lambda_{3},\lambda_{4})=(1,0)$. And we could write
$\xi_{0,2}=y_{12}\xi_{0,2}^{\prime}$, where
$\xi_{0,2}^{\prime}=\sum\limits_{1\leq i<j\leq 5}d_{ij}v_{ij}$. Hence,
$T_{5,123}.\xi_{0,2}=T_{5,123}.y_{12}\xi_{0,2}^{\prime}=([T_{5,123},y_{12}]+y_{12}T_{5,123}).\xi_{0,2}^{\prime}=(x_{5}\partial_{x_{3}}+\frac{1}{2}(-1)^{|34,45|}y_{45}\partial_{34}).\xi_{0,2}^{\prime}=\frac{1}{2}(-1)^{|34,45|}y_{45}\partial_{34}.\xi_{0,2}^{\prime}\neq
0$.
For the case
$\mbox{wt}(\xi_{0,2})=\lambda+\overrightarrow{w}_{24}^{\omega_{1}+\omega_{3}}$,
we get $v_{19,1}^{\lambda}\neq 0$. The equation $E_{15}t_{25}=0$ implies that
$0=E_{15}v_{2,1}^{\lambda}+(E_{13}+E_{53}E_{15})v_{9,1}^{\lambda}+(E_{12}+E_{52}E_{15})(v_{11,1}^{\lambda}+v_{11,3}^{\lambda})+\sum\limits_{i=1}^{4}h_{i}v_{19,1}^{\lambda}=(3+|\lambda|)v_{19,1}^{\lambda}.$
$None$
For the case
$\mbox{wt}(\xi_{0,2})=\lambda+\overrightarrow{w}_{26}^{\omega_{1}+\omega_{3}}$,
one of $v_{19,1}^{\lambda}$ and $v_{22,1}^{\lambda}$ should be nonzero,
otherwise $v_{26,i}^{\lambda}=0$. Then, the equation $E_{15}t_{35}=0$ implies
that
$0=E_{15}v_{6,1}^{\lambda}+(E_{13}+E_{53}E_{15})(v_{11,1}^{\lambda}+v_{11,2}^{\lambda})-(E_{12}+E_{52}E_{15})v_{17,1}^{\lambda})+\sum\limits_{i=1}^{4}h_{i}.v_{22,1}^{\lambda}=(1+|\lambda|)v_{22,1}^{\lambda}.$
$None$
Case 2.2.2
$\mbox{wt}(\xi_{0,2})=\lambda+\overrightarrow{w}_{25}^{\omega_{1}+\omega_{3}}$
In this case, $v_{25,1}^{\lambda}\neq 0$, $v_{28,1}^{\lambda}=0$ and
$wt(\xi_{0,2})=\lambda+\mbox{wt}(d_{15}d_{25})$. Then
$0=E_{15}t_{15}=(2+\sum\limits_{i=1}^{4}h_{i})(v_{11,2}^{\lambda}-v_{11,3}^{\lambda})-E_{53}v_{25,1}^{\lambda},$
$E_{35}.E_{15}t_{15}=(2+|\mbox{wt}(\overrightarrow{w}_{25}^{\omega_{1}+\omega_{3}})|-h_{3}-h_{4})v_{25,1}^{\lambda}=0.$
$None$
So $\lambda_{1}+\lambda_{2}+1=0$. A contradiction arises.
Case 2.2.3
$\mbox{wt}(\xi_{0,2})\in\\{\lambda+\overrightarrow{w}_{28}^{\omega_{1}+\omega_{3}},\lambda+\overrightarrow{w}_{31}^{\omega_{1}+\omega_{3}},\lambda+\overrightarrow{w}_{32}^{\omega_{1}+\omega_{3}},\lambda+\overrightarrow{w}_{34}^{\omega_{1}+\omega_{3}},\lambda+\overrightarrow{w}_{35}^{\omega_{1}+\omega_{3}},\\}$
In these cases, $v_{25,1}^{\lambda}\neq 0$ and $v_{28,1}^{\lambda}\neq 0$.
Case 2.2.3.1
$\mbox{wt}(\xi_{0,2})\in\\{\lambda+\overrightarrow{w}_{28}^{\omega_{1}+\omega_{3}},\lambda+\overrightarrow{w}_{31}^{\omega_{1}+\omega_{3}}\\}$
The equations
$0=E_{15}t_{12}=(E_{12}+E_{52}E_{15})v_{1,1}^{\lambda}-(\sum\limits_{i=1}^{4}h_{i}+E_{51}E_{15})v_{2,1}^{\lambda}=-(1+|\mbox{wt}(\overrightarrow{w}_{2}^{\omega_{1}+\omega_{3}})|)v_{1,1}^{\lambda}+E_{52}(v_{11,2}^{\lambda}+v_{11,3}^{\lambda}),$
$None$
$E_{25}^{2}E_{15}t_{12}=2(\lambda_{2}+\lambda_{3}+\lambda_{4}-1)v_{28,1}^{\lambda}=0$
$None$
induce a contradiction.
Case 2.2.3.2
$\mbox{wt}(\xi_{0,2})\in\\{\lambda+\overrightarrow{w}_{32}^{\omega_{1}+\omega_{3}},\lambda+\overrightarrow{w}_{34}^{\omega_{1}+\omega_{3}},\lambda+\overrightarrow{w}_{35}^{\omega_{1}+\omega_{3}}\\}$
First, we have $v_{32,1}^{\lambda}\neq 0$ in these cases, since
$E_{34}v_{32,1}^{\lambda}=-v_{34,1}^{\lambda},\
E_{24}v_{32,1}^{\lambda}=v_{35,1}^{\lambda}$. Then
$\displaystyle
0=E_{15}t_{35}=E_{15}v_{6,1}^{\lambda}+(E_{13}+E_{53}E_{15})(v_{11,1}^{\lambda}+v_{11,2}^{\lambda})-(E_{12}+E_{52}E_{15})v_{17,1}^{\lambda}$
$\displaystyle+(\sum\limits_{i=1}^{4}h_{i})v_{22,1}^{\lambda}=(1+|\mbox{wt}(\overrightarrow{w}_{22}^{\omega_{1}+\omega_{3}})|)v_{22,1}^{\lambda}+E_{53}v_{32,1}^{\lambda},$
$E_{35}E_{15}t_{35}=(-1-|\mbox{wt}(\overrightarrow{w}_{22}^{\omega_{1}+\omega_{3}})|+h_{3}+h_{4})v_{32,1}^{\lambda}=0.$
$None$
If
$\mbox{wt}(\xi_{0,2})=\lambda+\overrightarrow{w}_{32}^{\omega_{1}+\omega_{3}}$,
then $\mbox{wt}(\overrightarrow{w}_{32}^{\omega_{1}+\omega_{3}})=\lambda$ and
$\mbox{wt}(\overrightarrow{w}_{22}^{\omega_{1}+\omega_{3}})=\lambda-\alpha_{3}-\alpha_{4}$.
If
$\mbox{wt}(\xi_{0,2})=\lambda+\overrightarrow{w}_{34}^{\omega_{1}+\omega_{3}}$,
then
$\mbox{wt}(\overrightarrow{w}_{32}^{\omega_{1}+\omega_{3}})=\lambda-\alpha_{3}$
and
$\mbox{wt}(\overrightarrow{w}_{22}^{\omega_{1}+\omega_{3}})=\lambda-2\alpha_{3}-\alpha_{4}$.
If
$\mbox{wt}(\xi_{0,2})=\lambda+\overrightarrow{w}_{35}^{\omega_{1}+\omega_{3}}$,
then
$\mbox{wt}(\overrightarrow{w}_{32}^{\omega_{1}+\omega_{3}})=\lambda-\alpha_{2}-\alpha_{3}$
and
$\mbox{wt}(\overrightarrow{w}_{22}^{\omega_{1}+\omega_{3}})=\lambda-\alpha_{2}-2\alpha_{3}-\alpha_{4}$.
Thus, (5.44) yields $\lambda_{1}+\lambda_{2}=0$ or
$\lambda_{1}+\lambda_{2}+1=0$. A contradiction arises. $\Box$
### 5.3 Singular vectors of degree three
Theorem 5.5 All the possible degree three singular vectors are listed in the
following:
$e^{-\frac{P}{2}}.\prod\limits_{\mbox{wt}(d_{15}d_{25}d_{45})<\sigma\leq
2\omega_{1}+\omega_{4}}\frac{\tilde{c}-\chi_{\sigma+\lambda}(\tilde{c})}{\chi_{\mbox{wt}(d_{15}d_{25}d_{45})+\lambda}(\tilde{c})-\chi_{\sigma+\lambda}(\tilde{c})}.d_{15}d_{25}d_{45}v_{\lambda},$
$\ \mbox{where}\ \ \lambda=(0,0,m,n),\ 1\leq m\in\mathbb{N},\ 2\leq
n\in\mathbb{N}.$
Proof The leading term of any singular vector of degree three could be written
as:
$\xi_{0,3}=\sum\limits_{j\in\overline{1,55},k\in\overline{1,\mbox{mult}(\overrightarrow{w}_{j}^{2\omega_{1}+\omega_{4}})}}v_{j,k}^{2\omega_{1}+\omega_{4}}v_{j,k}^{\lambda},\
v_{j,k}^{\lambda}\in V(\lambda),$ $None$
which should satisfy $T_{5,123}.\xi_{0,3}=0$ and
$(x_{i}\partial_{x_{j}})_{0}.\xi_{0,3}=0$ ( $1\leq i<j\leq 5$). Assume
$T_{5,123}.\xi_{0,3}=\sum\limits_{i_{1}<j_{1},i_{2}<j_{2},(i_{1},j_{1})\neq(i_{2},j_{2})}d_{i_{1}j_{1}}d_{i_{2}j_{2}}t_{i_{1}j_{1},i_{2}j_{2}},\
t_{i_{1}j_{1},i_{2}j_{2}}\in V(\lambda).$ $None$
Since $v_{1,1}^{\lambda}\neq 0$, the equation
$0=t_{14,15}=-v_{1,1}^{\lambda}+E_{53}v_{7,1}^{\lambda}-E_{52}v_{13,1}^{\lambda}+E_{51}(v_{18,1}^{\lambda}-v_{18,4}^{\lambda})$
$None$
implies that one of the terms $v_{7,1}^{\lambda}$, $v_{13,1}^{\lambda}$ and
$v_{18,1}^{\lambda}-v_{18,4}^{\lambda}$ should be nonzero. Hence, the
information of the weights of Table 6-7 induces that $wt(\xi_{0,3})$ could be
restricted to the following cases:
$\displaystyle
wt(\xi_{0,3})\in\\{\lambda+\overrightarrow{w}_{7}^{2\omega_{1}+\omega_{4}},\lambda+\overrightarrow{w}_{12}^{2\omega_{1}+\omega_{4}},\lambda+\overrightarrow{w}_{13}^{2\omega_{1}+\omega_{4}},\lambda+\overrightarrow{w}_{17}^{2\omega_{1}+\omega_{4}},\lambda+\overrightarrow{w}_{18}^{2\omega_{1}+\omega_{4}},$
$\displaystyle\lambda+\overrightarrow{w}_{21}^{2\omega_{1}+\omega_{4}},\lambda+\overrightarrow{w}_{22}^{2\omega_{1}+\omega_{4}},\lambda+\overrightarrow{w}_{23}^{2\omega_{1}+\omega_{4}},\lambda+\overrightarrow{w}_{24}^{2\omega_{1}+\omega_{4}},\lambda+\overrightarrow{w}_{i}^{2\omega_{1}+\omega_{4}}(i\in\overline{26,55})\\}.\hskip
51.21504pt(5.48)$
Case 1 $wt(\xi_{0,3})=\lambda+\overrightarrow{w}_{7}^{2\omega_{1}+\omega_{4}}$
In this case, $v_{6,1}^{\lambda}=0$, $v_{3,1}^{\lambda}\neq 0$,
$wt(v_{3,1}^{\lambda})=\lambda-\alpha_{3}$,
$E_{34}v_{3,1}^{\lambda}=-v_{7,1}^{\lambda}$. Then
$E_{25}t_{12,15}=E_{25}(E_{52}v_{3,1}^{\lambda}-E_{51}v_{6,1}^{\lambda})=(h_{2}+h_{3}+h_{4})v_{3,1}^{\lambda}=(\lambda_{2}+\lambda_{3}+\lambda_{4})v_{3,1}^{\lambda}=0$
$None$
contradicts $\lambda_{3}>0$.
Case 2
$wt(\xi_{0,3})\in\\{\lambda+\overrightarrow{w}_{12}^{2\omega_{1}+\omega_{4}},\lambda+\overrightarrow{w}_{17}^{2\omega_{1}+\omega_{4}}\\}$
Note that
$E_{34}v_{6,1}^{\lambda}=-v_{12,1}^{\lambda},E_{12}v_{12,1}^{\lambda}=-v_{17,1}^{\lambda},E_{12}v_{3,1}^{\lambda}=-2v_{6,1}^{\lambda}.$
Hence, $v_{3,1}^{\lambda}\neq 0$ and $v_{6,1}^{\lambda},\neq 0$. Since
$wt(v_{6,1}^{\lambda})=\lambda-\alpha_{3}$ for
$wt(\xi_{0,3})=\lambda+\overrightarrow{w}_{12}^{2\omega_{1}+\omega_{4}}$;
$wt(v_{6,1}^{\lambda})=\lambda-\alpha_{1}-\alpha_{3}$ for
$wt(\xi_{0,3})=\lambda+\overrightarrow{w}_{17}^{2\omega_{1}+\omega_{4}}$. The
equation
$E_{15}t_{12,15}=-(2+|\mbox{wt}v_{6,1}^{\lambda}|)v_{6,1}^{\lambda}=0$ induces
that $|\lambda|<0$ in both cases.
Case 3
$wt(\xi_{0,3})=\lambda+\overrightarrow{w}_{13}^{2\omega_{1}+\omega_{4}}\vspace{0.1cm}$
We have $E_{25}v_{1,1}^{\lambda}=-v_{13,1}^{\lambda}$,
$E_{23}v_{7,1}^{\lambda}=-v_{13,1}^{\lambda}$. Then
$0=E_{25}t_{14,15}\stackrel{{\scriptstyle\mbox{by}(5.47)}}{{=}}-E_{25}v_{1,1}^{\lambda}+E_{23}v_{7,1}^{\lambda}-\sum\limits_{i=2}^{4}h_{i}v_{13,1}^{\lambda}=-(\lambda_{2}+\lambda_{3}+\lambda_{4})v_{13,1}^{\lambda}=0,$
$None$
contradicts $\lambda_{2}>0$.
Case 4
$wt(\xi_{0,3})=\lambda+\overrightarrow{w}_{22}^{2\omega_{1}+\omega_{4}}$
We have $v_{11,1}^{\lambda}\neq 0$, since
$E_{24}v_{11,1}^{\lambda}=v_{22,1}^{\lambda}$. Then
$0=E_{15}t_{13,15}=E_{15}(E_{53}v_{3,1}^{\lambda}-E_{51}v_{11,1}^{\lambda})=-(2+|\lambda|)v_{11,1}^{\lambda}$
$None$
yields a contradiction.
Case 5
$wt(\xi_{0,3})\in\\{\lambda+\overrightarrow{w}_{24}^{2\omega_{1}+\omega_{4}},\lambda+\overrightarrow{w}_{28}^{2\omega_{1}+\omega_{4}},\lambda+\overrightarrow{w}_{30}^{2\omega_{1}+\omega_{4}},\lambda+\overrightarrow{w}_{35}^{2\omega_{1}+\omega_{4}},\lambda+\overrightarrow{w}_{36}^{2\omega_{1}+\omega_{4}}\\}$
For
$wt(\xi_{0,3})\in\\{\lambda+\overrightarrow{w}_{24}^{2\omega_{1}+\omega_{4}},\lambda+\overrightarrow{w}_{30}^{2\omega_{1}+\omega_{4}},\lambda+\overrightarrow{w}_{36}^{2\omega_{1}+\omega_{4}}\\}$,
we have $v_{28,1}^{\lambda}=0$. Then
$0=E_{25}t_{15,25}=E_{25}(E_{52}v_{24,1}^{\lambda}-E_{51}v_{28,1}^{\lambda})=(h_{2}+h_{3}+h_{4})v_{24,1}^{\lambda}$
$None$
yields $\lambda_{2}+\lambda_{3}+\lambda_{4}=0$ or $1$. For
$wt(\xi_{0,3})\in\\{\lambda+\overrightarrow{w}_{28}^{2\omega_{1}+\omega_{4}},\lambda+\overrightarrow{w}_{35}^{2\omega_{1}+\omega_{4}}\\},$
the equation
$0=E_{15}t_{15,25}=-(h_{1}+h_{2}+h_{3}+h_{4})v_{28,1}^{\lambda}+(E_{12}+E_{52}E_{15})v_{24,1}^{\lambda}$
$None$
yields $1+|\lambda|=0$.
Case 6
$wt(\xi_{0,3})\in\\{\lambda+\overrightarrow{w}_{23}^{2\omega_{1}+\omega_{4}},\lambda+\overrightarrow{w}_{29}^{2\omega_{1}+\omega_{4}}\\}$
We have
$E_{35}v_{10,1}^{\lambda}=v_{23,1}^{\lambda},E_{25}v_{10,1}^{\lambda}=-v_{29,1}^{\lambda},E_{23}v_{23,1}^{\lambda}=-v_{29,1}^{\lambda}$.
Consider the equation
$t_{14,45}=-v_{10,1}^{\lambda}+E_{53}v_{23,1}^{\lambda}-E_{52}v_{29,1}^{\lambda}-E_{51}(v_{32,1}^{\lambda}-v_{32,4}^{\lambda}).$
$None$
For $wt(\xi_{0,3})=\lambda+\overrightarrow{w}_{23}^{2\omega_{1}+\omega_{4}}$,
$E_{35}t_{14,45}=(\lambda_{3}+\lambda_{4}-1)v_{23,1}^{\lambda}=0$ induces
$\lambda_{3}+\lambda_{4}=1$, which contradicts $\lambda_{3}>1$. For
$wt(\xi_{0,3})=\lambda+\overrightarrow{w}_{29}^{2\omega_{1}+\omega_{4}}$, we
have
$E_{25}t_{14,45}=-(\lambda_{2}+\lambda_{3}+\lambda_{4})v_{29,1}^{\lambda}=0$
induces $\lambda_{2}+\lambda_{3}+\lambda_{4}=0$, which contradicts
$\lambda_{2}>0,\lambda_{3}>0$.
Case 7
$wt(\xi_{0,3})\in\\{\lambda+\overrightarrow{w}_{27}^{2\omega_{1}+\omega_{4}},\lambda+\overrightarrow{w}_{39}^{2\omega_{1}+\omega_{4}},\lambda+\overrightarrow{w}_{42}^{2\omega_{1}+\omega_{4}}\\}$
Note that in these cases, the equations are derived:
$E_{35}v_{15,1}^{\lambda}=v_{27,1}^{\lambda},E_{15}v_{15,1}^{\lambda}=v_{39,1}^{\lambda},E_{13}v_{27,1}^{\lambda}=v_{39,1}^{\lambda},E_{12}v_{32,4}^{\lambda}=-v_{39,1}^{\lambda},E_{12}v_{32,1}^{\lambda}=0=E_{12}v_{32,3}^{\lambda},E_{25}v_{15,1}^{\lambda}=-\sum\limits_{i=1}^{4}v_{32,i}^{\lambda},E_{23}v_{27,1}^{\lambda}=-v_{32,1}^{\lambda}-2v_{32,3}^{\lambda}-v_{32,4}^{\lambda},E_{34}v_{32,1}^{\lambda}=-v_{42,1}^{\lambda},E_{34}v_{32,2}^{\lambda}=v_{42,1}^{\lambda},E_{34}v_{32,3}^{\lambda}=0=E_{34}v_{32,4}^{\lambda}.$
Consider the equation
$t_{24,45}=-v_{15,1}^{\lambda}+E_{53}v_{27,1}^{\lambda}-E_{52}(v_{32,1}^{\lambda}+v_{32,3}^{\lambda}+v_{32,4}^{\lambda})+E_{51}v_{39,1}^{\lambda}.$
$None$
For $wt(\xi_{0,3})=\lambda+\overrightarrow{w}_{27}^{2\omega_{1}+\omega_{4}}$,
$E_{35}t_{24,45}=(\lambda_{3}+\lambda_{4}-1)v_{27,1}^{\lambda}=0$ induces
$\lambda_{3}+\lambda_{4}=1$, which contradicts $\lambda_{3}>1$. For
$wt(\xi_{0,3})=\lambda+\overrightarrow{w}_{39}^{2\omega_{1}+\omega_{4}}$,
$E_{15}t_{24,45}=(1+|\lambda|)v_{39,1}^{\lambda}=0$ induces $1+|\lambda|=0$.
For $wt(\xi_{0,3})=\lambda+\overrightarrow{w}_{42}^{2\omega_{1}+\omega_{4}}$,
Then
$E_{15}t_{24,45}=v_{32,2}^{\lambda}-v_{32,3}^{\lambda}-(\lambda_{2}+\lambda_{3}+\lambda_{4})(v_{32,1}^{\lambda}+v_{32,3}^{\lambda}+v_{32,4}^{\lambda})=0$,
and
$E_{34}E_{15}t_{24,45}=(1+\lambda_{2}+\lambda_{3}+\lambda_{4})v_{42,1}^{\lambda}=0$
induces $1+\lambda_{2}+\lambda_{3}+\lambda_{4}=0$.
Case 8
$wt(\xi_{0,3})\in\\{\lambda+\overrightarrow{w}_{31}^{2\omega_{1}+\omega_{4}},\lambda+\overrightarrow{w}_{33}^{2\omega_{1}+\omega_{4}}\\}$
Observe these equations are derived:
$E_{15}v_{4,1}^{\lambda}=v_{31,1}^{\lambda},E_{13}v_{17,1}^{\lambda}=v_{31,1}^{\lambda},E_{12}v_{21,1}^{\lambda}=-v_{31,1}^{\lambda},E_{12}v_{21,2}^{\lambda}=v_{31,1}^{\lambda},E_{12}v_{21,3}^{\lambda}=0,E_{12}v_{21,4}^{\lambda}=0.$
Consider the equation
$t_{24,25}=-v_{4,1}^{\lambda}+E_{53}v_{17,1}^{\lambda}-E_{52}(2v_{21,1}^{\lambda}+v_{21,2}^{\lambda}+v_{21,3}^{\lambda})+E_{51}v_{31,1}^{\lambda}.$
$None$
It follows from $E_{15}t_{24,25}=0$ that $(1+|\lambda|)v_{31,1}^{\lambda}=0$.
Case 9
$wt(\xi_{0,3})\in\\{\lambda+\overrightarrow{w}_{37}^{2\omega_{1}+\omega_{4}},\lambda+\overrightarrow{w}_{38}^{2\omega_{1}+\omega_{4}}\\}$
Note
$E_{15}v_{14,1}^{\lambda}=v_{38,1}^{\lambda},E_{13}v_{26,2}^{\lambda}=v_{38,1}^{\lambda},E_{12}v_{37,1}^{\lambda}=-v_{38,1}^{\lambda},E_{25}v_{14,1}^{\lambda}=-v_{37,1}^{\lambda},E_{23}v_{26,2}^{\lambda}=-v_{37,1}^{\lambda}.$
Consider the equation
$t_{34,35}=-v_{14,1}^{\lambda}+E_{53}v_{26,2}^{\lambda}-E_{52}v_{37,1}^{\lambda}+E_{51}v_{38,1}^{\lambda}.$
$None$
For $wt(\xi_{0,3})=\lambda+\overrightarrow{w}_{37}^{2\omega_{1}+\omega_{4}}$,
$E_{25}t_{34,35}=-(\lambda_{2}+\lambda_{3}+\lambda_{4})v_{37,1}^{\lambda}=0$
implies $\lambda_{2}+\lambda_{3}+\lambda_{4}=0$, which which contradicts
$\lambda_{2}>2$. For
$wt(\xi_{0,3})=\lambda+\overrightarrow{w}_{38}^{2\omega_{1}+\omega_{4}}$,
$E_{15}t_{34,35}=(1+|\lambda|)v_{38,1}^{\lambda}=0$.
Case 10
$wt(\xi_{0,3})\in\\{\lambda+\overrightarrow{w}_{40}^{2\omega_{1}+\omega_{4}},\lambda+\overrightarrow{w}_{44}^{2\omega_{1}+\omega_{4}},\overrightarrow{w}_{48}^{2\omega_{1}+\omega_{4}},\overrightarrow{w}_{51}^{2\omega_{1}+\omega_{4}}\\}$
Note
$E_{25}v_{19,1}^{\lambda}=-v_{40,1}^{\lambda},E_{23}v_{32,3}^{\lambda}=-v_{40,1}^{\lambda},E_{15}v_{19,1}^{\lambda}=v_{44,1}^{\lambda},E_{13}v_{32,3}^{\lambda}=v_{44,1}^{\lambda},E_{12}v_{40,1}^{\lambda}=-v_{44,1}^{\lambda}.$
Consider the equation
$t_{34,45}=-v_{19,1}^{\lambda}+E_{53}v_{32,3}^{\lambda}-E_{52}v_{40,1}^{\lambda}+E_{51}v_{44,1}^{\lambda},$
$None$
For
$wt(\xi_{0,3})=\lambda+\overrightarrow{w}_{40}^{2\omega_{1}+\omega_{4}},\lambda+\overrightarrow{w}_{48}^{2\omega_{1}+\omega_{4}},$
the equation $E_{25}t_{34,45}=0$ implies
$-(\lambda_{2}+\lambda_{3}+\lambda_{4})v_{40,1}^{\lambda}=0$. For
$wt(\xi_{0,3})=\lambda+\overrightarrow{w}_{44}^{2\omega_{1}+\omega_{4}},\lambda+\overrightarrow{w}_{51}^{2\omega_{1}+\omega_{4}},$
the equation $E_{15}t_{34,45}=0$ implies $(1+|\lambda|)v_{44,1}^{\lambda}=0$.
Case 11
$wt(\xi_{0,3})\in\\{\lambda+\overrightarrow{w}_{45}^{2\omega_{1}+\omega_{4}},\lambda+\overrightarrow{w}_{52}^{2\omega_{1}+\omega_{4}}\\}$
Note
$E_{15}v_{21,3}^{\lambda}=-v_{45,1}^{\lambda},E_{15}v_{21,4}^{\lambda}=v_{45,1}^{\lambda},E_{13}v_{35,1}^{\lambda}=v_{45,1}^{\lambda},E_{12}v_{41,1}^{\lambda}=-v_{45,1}^{\lambda},E_{12}v_{41,i}^{\lambda}=0\
(i\in\overline{2,4}),E_{15}v_{18,3}^{\lambda}=v_{41,1}^{\lambda}-v_{41,4}^{\lambda},E_{15}v_{18,4}^{\lambda}=-v_{41,1}^{\lambda}-v_{41,3}^{\lambda},E_{13}v_{30,1}^{\lambda}=v_{41,1}^{\lambda}-v_{41,2}^{\lambda}-v_{41,3}^{\lambda},E_{12}v_{36,1}^{\lambda}=-2v_{41,1}^{\lambda}-v_{41,2}^{\lambda},E_{15}v_{30,1}^{\lambda}=E_{35}v_{41,3}^{\lambda}=-E_{35}v_{41,4}^{\lambda}=-v_{52,1}^{\lambda}.$
Consider the equation
$t_{25,45}=-v_{21,3}^{\lambda}-v_{21,4}^{\lambda}+E_{53}v_{35,1}^{\lambda}-E_{52}(v_{41,1}^{\lambda}+v_{41,2}^{\lambda}+v_{41,3}^{\lambda})+E_{51}v_{45,1}^{\lambda}=0.$
$None$
For $wt(\xi_{0,3})=\lambda+\overrightarrow{w}_{45}^{2\omega_{1}+\omega_{4}},$
the equation $E_{15}t_{25,45}=0$ implies $(2+|\lambda|)v_{45,1}^{\lambda}=0$.
Now suppose
$wt(\xi_{0,3})=\lambda+\overrightarrow{w}_{52}^{2\omega_{1}+\omega_{4}}.$ The
equation $t_{15,45}=0$ implies that
$E_{15}t_{15,45}=0=(|\lambda|+2)v_{41,1}^{\lambda}+(1-|\lambda|)v_{41,3}^{\lambda}+v_{41,4}^{\lambda}-E_{53}v_{52,1}^{\lambda},$
$0=E_{35}E_{15}t_{15,45}=(\lambda_{1}+\lambda_{2})v_{52,1}^{\lambda},$ $None$
i.e. $\lambda_{1}=\lambda_{2}=0$. Then
$wt(\xi_{0,3})=\lambda+\overrightarrow{w}_{52}^{2\omega_{1}+\omega_{4}}=(0,1,\lambda_{3}-1,\lambda_{4}-2).$
Observe that
$\Pi((\omega_{1}+\omega_{2})\otimes\mbox{wt}(\xi_{0,3}))\bigcap\Pi((\omega_{1}+\omega_{3})\otimes\lambda)=\\{\mbox{wt}(\xi_{0,3})+(0,0,1,0),\mbox{wt}(\xi_{0,3})+(1,-1,0,1)\\}.$
$None$
$\Pi((\omega_{1}+\omega_{2})\otimes(\mbox{wt}(\xi_{0,3})+(0,0,1,0)))\bigcap\\{\lambda\\}=\emptyset,\
\Pi((\omega_{1}+\omega_{2})\otimes(\mbox{wt}(\xi_{0,3})+(1,-1,0,1)))\bigcap\\{\lambda\\}=\emptyset.$
$None$
Then (5.62) implies that
$T^{2}|_{V(\omega_{1}+\omega_{2})\otimes
V(\mbox{wt}(\xi_{0,3})+(0,0,1,0))}=0,T^{2}|_{V(\omega_{1}+\omega_{2})\otimes
V(\mbox{wt}(\xi_{0,3})+(1,-1,0,1))}=0.$ $None$
Note that $T_{5,123}\xi_{0,3}=0$ iff $T^{3}|_{V(\omega_{1}+\omega_{2})\otimes
V(\mbox{wt}(\xi_{0,3}))}=0$ by Lemma 5.2. Assume
$v_{(\omega_{1}+\omega_{2})\otimes\mbox{wt}(\xi_{0,3})}^{(0,1,\lambda_{3},\lambda_{4}-2)}$
(resp.
$v_{(\omega_{1}+\omega_{2})\otimes\mbox{wt}(\xi_{0,3})}^{(1,0,\lambda_{3}-1,\lambda_{4}-1)}$
) is any maximal vector of weight $(0,1,\lambda_{3},\lambda_{4}-2)$ (resp.
$(1,0,\lambda_{3}-1,\lambda_{4}-1)$ ) appearing in the tensor decomposition
$V(\omega_{1}+\omega_{2})\otimes V(\mbox{wt}(\xi_{0,3}))$. Since
$T^{3}{V(\omega_{1}+\omega_{2})\otimes V(\mbox{wt}(\xi_{0,3}))}\subseteq
V(\omega_{1}+\omega_{3})\otimes V(\lambda)$,
$T^{3}|_{V(\omega_{1}+\omega_{2})\otimes V(\mbox{wt}(\xi_{0,3}))}=0$ iff
$T^{3}.(v_{(\omega_{1}+\omega_{2})\otimes\mbox{wt}(\xi_{0,3})}^{(0,1,\lambda_{3},\lambda_{4}-2)})=0,T^{3}(v_{(\omega_{1}+\omega_{2})\otimes\mbox{wt}(\xi_{0,3})}^{(1,0,\lambda_{3}-1,\lambda_{4}-1)})=0$.
Otherwise, we could get a singular vector of degree two with weights
$(0,1,\lambda_{3},\lambda_{4}-2)$ and $(1,0,\lambda_{3}-1,\lambda_{4}-1)$
respectively by (5.63), which contradicts proposition 5.3. Hence, we get
$\lambda=(0,0,m,n)$.
Case 12
$wt(\xi_{0,3})\in\\{\lambda+\overrightarrow{w}_{43}^{2\omega_{1}+\omega_{4}},\lambda+\overrightarrow{w}_{47}^{2\omega_{1}+\omega_{4}}\\}$
Consider the equation
$t_{15,45}=-v_{18,3}^{\lambda}-v_{18,4}^{\lambda}+E_{53}v_{30,1}^{\lambda}-E_{52}v_{36,1}^{\lambda}+E_{51}v_{41,1}^{\lambda}-E_{51}v_{41,3}^{\lambda},$
$None$
Note
$E_{15}v_{18,3}^{\lambda}=v_{41,1}^{\lambda}-v_{41,4}^{\lambda},E_{15}v_{18,4}^{\lambda}=-v_{41,1}^{\lambda}-v_{41,3}^{\lambda},E_{13}v_{30,1}^{\lambda}=v_{41,1}^{\lambda}-v_{41,2}^{\lambda}-v_{41,3}^{\lambda},E_{12}v_{36,1}^{\lambda}=-2v_{41,1}^{\lambda}-v_{41,2}^{\lambda}.$
Then $E_{15}t_{15,45}=0$ induces that
$3v_{41,1}^{\lambda}+v_{41,4}^{\lambda}+\sum\limits_{i=1}^{4}h_{i}(v_{41,1}^{\lambda}-v_{41,3}^{\lambda})=0$.
Since
$E_{45}v_{41,4}^{\lambda}=-v_{43,1}^{\lambda},E_{45}v_{41,i}^{\lambda}=0(i\in\overline{1,3})$
and
$E_{34}v_{41,3}^{\lambda}=-v_{47,1}^{\lambda},E_{34}v_{41,i}^{\lambda}=0(i\in\\{1,2,4\\})$.
Hence, $E_{45}E_{15}t_{15,45}=0$ and $E_{34}E_{15}t_{15,45}=0$ imply
$v_{41,3}^{\lambda}=0$ and $|\lambda|v_{47,1}^{\lambda}=0$ in these two cases,
respectively.
Case 13
$wt(\xi_{0,3})\in\\{\lambda+\overrightarrow{w}_{46}^{2\omega_{1}+\omega_{4}},\lambda+\overrightarrow{w}_{49}^{2\omega_{1}+\omega_{4}},\lambda+\overrightarrow{w}_{50}^{2\omega_{1}+\omega_{4}},\lambda+\overrightarrow{w}_{53}^{2\omega_{1}+\omega_{4}},\lambda+\overrightarrow{w}_{54}^{2\omega_{1}+\omega_{4}},\lambda+\overrightarrow{w}_{55}^{2\omega_{1}+\omega_{4}}\\}$
Observe
$E_{25}v_{26,3}^{\lambda}=2v_{46,1}^{\lambda},E_{25}v_{26,4}^{\lambda}=-v_{46,1}^{\lambda},E_{23}v_{41,2}^{\lambda}=-v_{46,1}^{\lambda},E_{15}v_{26,3}^{\lambda}=-2v_{49,1}^{\lambda},E_{15}v_{26,4}^{\lambda}=v_{49,1}^{\lambda},E_{13}v_{41,2}^{\lambda}=v_{49,1}^{\lambda},E_{12}v_{46,1}^{\lambda}=-v_{49,1}^{\lambda},E_{25}v_{41,2}^{\lambda}=-v_{54,1}^{\lambda},E_{35}v_{46,1}^{\lambda}=-v_{54,1}^{\lambda}.$
Consider the equation
$t_{35,45}=-v_{26,3}^{\lambda}-v_{26,4}^{\lambda}+E_{53}v_{41,2}^{\lambda}-E_{52}v_{46,1}^{\lambda}+E_{51}v_{49,1}^{\lambda},$
$None$
For
$wt(\xi_{0,3})=\lambda+\overrightarrow{w}_{46}^{2\omega_{1}+\omega_{4}},\overrightarrow{w}_{50}^{2\omega_{1}+\omega_{4}},$
the equation $E_{25}t_{35,45}=0$ implies
$2+\lambda_{2}+\lambda_{3}+\lambda_{4}=0$. For
$wt(\xi_{0,3})=\lambda+\overrightarrow{w}_{49}^{2\omega_{1}+\omega_{4}},\overrightarrow{w}_{53}^{2\omega_{1}+\omega_{4}},$
the equation $E_{15}t_{35,45}=0$ implies $3+|\lambda|=0$. For
$wt(\xi_{0,3})=\lambda+\overrightarrow{w}_{54}^{2\omega_{1}+\omega_{4}},$ the
equation $E_{35}E_{25}t_{35,45}=0$ implies
$(1+\lambda_{2})v_{54,1}^{\lambda}=0$. For
$wt(\xi_{0,3})=\lambda+\overrightarrow{w}_{55}^{2\omega_{1}+\omega_{4}},$ the
equation $E_{35}E_{15}t_{35,45}=0$ implies $\lambda_{1}+\lambda_{2}+2=0$.
Case 14
$wt(\xi_{0,3})=\lambda+\overrightarrow{w}_{21}^{2\omega_{1}+\omega_{4}}$
We have $v_{21,3}^{\lambda}+2v_{21,4}^{\lambda}+E_{45}v_{15,1}^{\lambda}=0$.
And the equations $t_{25,45}=0$ and $t_{24,45}=0$ induce that
$v_{21,3}^{\lambda}+v_{21,4}^{\lambda}=0$, $v_{15,1}^{\lambda}=0$. Thus,
$v_{21,3}^{\lambda}=v_{21,4}^{\lambda}=0$. Then
$0=E_{25}.t_{24,25}=E_{25}.[-v_{4,1}^{\lambda}+E_{53}v_{17,1}^{\lambda}-E_{52}(2v_{21,1}^{\lambda}+v_{21,2}^{\lambda})]=0,$
$None$
$0=E_{15}.t_{14,25}=E_{15}[-v_{2,1}^{\lambda}+E_{53}v_{12,1}^{\lambda}-E_{52}(v_{18,1}^{\lambda}+v_{18,2}^{\lambda}-v_{18,3}^{\lambda})+E_{51}v_{21,1}^{\lambda}]=0$
$None$
yield
$2(1+\lambda_{2}+\lambda_{3}+\lambda_{4})v_{21,1}^{\lambda}+(2+\lambda_{2}+\lambda_{3}+\lambda_{4})v_{21,2}^{\lambda}=0,\
(2+|\lambda|)v_{21,1}^{\lambda}=0$ $None$
i.e. $v_{21,i}^{\lambda}=0$ for $i\in\overline{1,4}$. A contradiction arises.
Case 15
$wt(\xi_{0,3})=\lambda+\overrightarrow{w}_{26}^{2\omega_{1}+\omega_{4}}$
Note that
$t_{24,35}=-v_{8,1}^{\lambda}+E_{53}\sum\limits_{i=1}^{4}v_{21,i}^{\lambda}-E_{52}(v_{26,2}^{\lambda}+\sum\limits_{i=1}^{4}v_{26,i}^{\lambda}).$
$None$
We have
$v_{26,2}^{\lambda}+v_{26,3}^{\lambda}+2v_{26,4}^{\lambda}+E_{45}v_{19,1}^{\lambda}=0$,
$v_{26,2}^{\lambda}+v_{26,3}^{\lambda}+E_{23}v_{21,3}^{\lambda}=0$ and
$v_{26,4}^{\lambda}+E_{23}v_{21,4}^{\lambda}=0$. And the equations
$t_{35,45}=0$ , $t_{34,45}=0$ and $t_{25,45}=0$ induce that
$v_{26,3}^{\lambda}+v_{26,4}^{\lambda}=0$, $v_{19,1}^{\lambda}=0$ and
$v_{26,2}^{\lambda}+v_{26,3}^{\lambda}+v_{26,4}^{\lambda}=0$. Thus,
$v_{26,i}^{\lambda}=0$ for $i\in\overline{2,4}$. The equation $t_{24,35}=0$
induce that
$E_{25}.[-v_{8,1}^{\lambda}+E_{53}(v_{21,1}^{\lambda}+v_{21,2}^{\lambda})-E_{52}v_{26,1}^{\lambda}]=0,$
$None$
which means $\lambda_{2}+\lambda_{3}+\lambda_{4}=0$. A contradiction arises.
Case 16
$wt(\xi_{0,3})=\lambda+\overrightarrow{w}_{32}^{2\omega_{1}+\omega_{4}}$
Note that
$v_{32,1}^{\lambda}+v_{32,2}^{\lambda}+E_{14}v_{18,3}^{\lambda}=0,v_{32,1}^{\lambda}+v_{32,4}^{\lambda}+E_{14}v_{18,4}^{\lambda}=0,v_{32,1}^{\lambda}-2v_{32,3}^{\lambda}-v_{32,4}^{\lambda}+E_{24}v_{21,3}^{\lambda}=0,v_{32,1}^{\lambda}+v_{32,2}^{\lambda}+E_{24}v_{21,4}^{\lambda}=0,2v_{32,1}^{\lambda}+E_{34}v_{26,3}^{\lambda}=0,v_{32,1}^{\lambda}+v_{32,2}^{\lambda}+E_{34}v_{26,4}^{\lambda}=0.$
And the equations follow:
$t_{25,45}=v_{21,3}^{\lambda}+v_{21,4}^{\lambda}=0,t_{35,45}=v_{26,3}^{\lambda}+v_{26,4}^{\lambda}=0,$
$t_{34,45}=-v_{19,1}^{\lambda}+E_{53}v_{32,3}^{\lambda}=0,t_{15,45}=-v_{18,3}^{\lambda}-v_{18,4}^{\lambda}+E_{53}v_{30,1}^{\lambda}=0.$
$None$
Thus,
$-2v_{32,1}^{\lambda}+2v_{32,3}^{\lambda}-v_{32,2}^{\lambda}+v_{32,4}^{\lambda}=0,\
3v_{32,1}^{\lambda}+v_{32,2}^{\lambda}=0,v_{32,2}^{\lambda}+(\lambda_{3}+\lambda_{4}-1)v_{32,3}^{\lambda}=0,2v_{32,1}^{\lambda}+v_{32,2}^{\lambda}+v_{32,4}^{\lambda}=0.$
$None$
yield the contradiction:$2+\lambda_{3}+\lambda_{4}=0$.
Case 17
$wt(\xi_{0,3})=\lambda+\overrightarrow{w}_{41}^{2\omega_{1}+\omega_{4}}$
Observe that
$-v_{41,1}^{\lambda}-v_{41,2}^{\lambda}+v_{41,3}^{\lambda}+v_{41,4}^{\lambda}+E_{24}v_{28,1}^{\lambda}=0,v_{41,2}^{\lambda}+2v_{41,3}^{\lambda}+v_{41,4}^{\lambda}+E_{34}v_{34,1}^{\lambda}=0,-v_{41,1}^{\lambda}-v_{41,2}^{\lambda}+v_{41,3}^{\lambda}+E_{25}v_{21,3}^{\lambda}=0,v_{41,1}^{\lambda}+v_{41,2}^{\lambda}+v_{41,4}^{\lambda}+E_{25}v_{21,4}^{\lambda}=0,v_{41,1}^{\lambda}+2v_{41,2}^{\lambda}+v_{41,3}^{\lambda}+E_{23}v_{35,1}^{\lambda}=0,2v_{41,3}^{\lambda}+E_{35}v_{26,3}^{\lambda}=0,-v_{41,2}^{\lambda}-v_{41,3}^{\lambda}+v_{41,4}^{\lambda}+E_{35}v_{26,4}^{\lambda}=0.$
Then
$0=E_{25}t_{25,45}=E_{25}[-v_{21,3}^{\lambda}-v_{21,4}^{\lambda}+E_{53}v_{35,1}^{\lambda}-E_{52}(v_{41,1}^{\lambda}+v_{41,2}^{\lambda}+v_{41,3}^{\lambda})]=0,$
$None$
$0=E_{35}t_{35,45}=E_{35}[-v_{26,3}^{\lambda}-v_{26,4}^{\lambda}+E_{53}v_{41,2}^{\lambda}]=0.$
$None$
which mean
$v_{41,4}^{\lambda}-v_{41,1}^{\lambda}-2v_{41,2}^{\lambda}=(\sum\limits_{i=2}^{4}\lambda_{i})(\sum\limits_{i=1}^{3}v_{41,i}^{\lambda}),\
v_{41,3}^{\lambda}-v_{41,2}^{\lambda}+v_{41,4}^{\lambda}+(\lambda_{3}+\lambda_{4})v_{41,2}^{\lambda}).$
$None$
We claim that one of $v_{28,1}^{\lambda}$ and $v_{34,1}^{\lambda}$ should be
nonzero. Otherwise, the equation implies:
$-v_{41,1}^{\lambda}-v_{41,2}^{\lambda}+v_{41,3}^{\lambda}+v_{41,4}^{\lambda}=0$
and $v_{41,2}^{\lambda}+2v_{41,3}^{\lambda}+v_{41,4}^{\lambda}=0$, which
provide a contradiction. Now,
$0=E_{15}t_{15,35}=E_{15}(E_{53}v_{24,1}^{\lambda}-E_{51}v_{34,1}^{\lambda})=E_{13}v_{24,1}^{\lambda}-\sum\limits_{i=1}^{4}h_{i}v_{34,1}^{\lambda}=0$
and
$0=E_{15}t_{15,25}=E_{15}(E_{52}v_{24,1}^{\lambda}-E_{51}v_{28,1}^{\lambda})=E_{12}v_{24,1}^{\lambda}-\sum\limits_{i=1}^{4}h_{i}v_{34,1}^{\lambda}=0$.
Thus $1+|\lambda|=0$. A contradiction arises.
Case 18
$wt(\xi_{0,3})=\lambda+\overrightarrow{w}_{18}^{2\omega_{1}+\omega_{4}}$
Note that
$E_{15}v_{1,1}^{\lambda}=v_{18,1}^{\lambda}-2v_{18,3}^{\lambda},E_{25}v_{2,1}^{\lambda}=-v_{18,1}^{\lambda}-v_{18,2}^{\lambda}-v_{18,3}^{\lambda},E_{13}v_{7,1}^{\lambda}=v_{18,1}^{\lambda}-2v_{18,2}^{\lambda}-2v_{18,4}^{\lambda},E_{13}v_{3,1}^{\lambda}=-2v_{11,1}^{\lambda},E_{23}v_{12,1}^{\lambda}=-v_{18,1}^{\lambda}-2v_{18,2}^{\lambda}-v_{18,4}^{\lambda},E_{34}v_{11,1}^{\lambda}=-v_{18,2}^{\lambda}-v_{18,3}^{\lambda}-2v_{18,4}^{\lambda}.$
We have $v_{11,1}^{\lambda}=0$. Otherwise,
$0=E_{15}t_{13,15}=-(2+|\lambda|)v_{11,1}^{\lambda}=0$. Hence,
$0=-v_{18,2}^{\lambda}-v_{18,3}^{\lambda}-2v_{18,4}^{\lambda}=E_{34}v_{11,1}^{\lambda}.$
$None$
Moreover,
$0=t_{15,45}=v_{18,3}^{\lambda}+v_{18,4}^{\lambda},0=E_{15}t_{14,15}=3v_{18,1}^{\lambda}+2v_{18,3}^{\lambda}-2v_{18,4}^{\lambda}+|\lambda|(v_{18,1}^{\lambda}-v_{18,4}^{\lambda}).$
$None$
Observe that
$t_{14,25}=-v_{2,1}^{\lambda}+E_{53}v_{12,1}^{\lambda}-E_{52}(v_{18,1}^{\lambda}+v_{18,2}^{\lambda}-v_{18,3}^{\lambda})+E_{51}(v_{21,1}^{\lambda}-v_{21,3}^{\lambda}-v_{21,4}^{\lambda})=0.$
$None$
The equation $E_{25}t_{14,25}=0$ induces
$-v_{18,2}^{\lambda}+v_{18,3}^{\lambda}-v_{18,4}^{\lambda}-(\lambda_{2}+\lambda_{3}+\lambda_{4})(v_{18,1}^{\lambda}+v_{18,2}^{\lambda}-v_{18,3}^{\lambda})=0.$
$None$
All these equations yield the contradiction:
$\frac{3+|\lambda|}{4+|\lambda|}=-(\lambda_{2}+\lambda_{3}+\lambda_{4}).$
$None$
$\Box$
### 5.4 Singular vectors of degree four
Theorem 5.6 All the possible degree four singular vectors are listed in the
following:
$d_{12}d_{13}d_{14}d_{15}v_{\lambda},\ \lambda=(m,0,0,0),\ m\in\mathbb{N}.$
Proof The leading term of any singular vector of degree four could be written
as:
$\xi_{0,4}=\sum\limits_{j\in\overline{1,35}}v_{j}^{3\omega_{1}}v_{j}^{\lambda},$
$None$
which should satisfy $T_{5,123}.\xi_{0,4}=0$. Since
$[T_{5,123}-(E_{53}\partial_{y_{12}}+E_{52}(-1)^{1+|13|}\partial_{y_{13}}+E_{51}(-1)^{|23|}\partial_{y_{23}}),(x_{i}\partial_{x_{j}})_{0}^{\prime}]=0$
$None$
for $1\leq j<i\leq 5$, we have
$[T_{5,123}-(E_{53}\partial_{y_{12}}+E_{52}(-1)^{1+|13|}\partial_{y_{13}}+E_{51}(-1)^{|23|}\partial_{y_{23}})].|_{V(3\omega_{1})\otimes
V(\lambda)}=0.$ $None$
Hence,
$T_{5,123}.\xi_{0,4}=[E_{53}\partial_{y_{12}}+E_{52}(-1)^{1+|13|}\partial_{y_{13}}+E_{51}(-1)^{|23|}\partial_{y_{23}}].\xi_{0,4}=0.$
$None$
Case 1
$wt(\xi_{0,4})=\overrightarrow{w}_{1}^{3\omega_{1}}=\mbox{wt}(d_{12}d_{13}d_{14}d_{15}v_{\lambda})$
The vector $d_{12}d_{13}d_{14}d_{15}v_{\lambda}$ is singular iff
$\lambda=(m,0,0,0)$.
Case 2
$wt(\xi_{0,4})\in\\{\lambda+\overrightarrow{w}_{2}^{3\omega_{1}},\lambda+\overrightarrow{w}_{3}^{3\omega_{1}},\lambda+\overrightarrow{w}_{5}^{3\omega_{1}},\lambda+\overrightarrow{w}_{6}^{3\omega_{1}},\lambda+\overrightarrow{w}_{8}^{3\omega_{1}},\lambda+\overrightarrow{w}_{10}^{3\omega_{1}},\lambda+\overrightarrow{w}_{12}^{3\omega_{1}},\lambda+\overrightarrow{w}_{13}^{3\omega_{1}},\lambda+\overrightarrow{w}_{15}^{3\omega_{1}},\lambda+\overrightarrow{w}_{16}^{3\omega_{1}},\lambda+\overrightarrow{w}_{17}^{3\omega_{1}},\lambda+\overrightarrow{w}_{22}^{3\omega_{1}},\lambda+\overrightarrow{w}_{23}^{3\omega_{1}},\lambda+\overrightarrow{w}_{27}^{3\omega_{1}},\lambda+\overrightarrow{w}_{30}^{3\omega_{1}}\\}$
In these cases, the equations are derived:
$E_{51}v_{i}^{\lambda}-E_{52}v_{j}^{\lambda}=0,\
E_{12}v_{j}^{\lambda}=-s_{ij}v_{i}^{\lambda},\
\mbox{wt}(v_{i}^{\lambda})=\lambda,(i,j,s_{ij})\in\\{(2,1,3),(3,2,2),(5,3,1),(8,6,1),(10,7,2),(12,9,1),(13,10,1),(15,11,2),(16,14,1),(17,15,1)\\}.$
Then, the equation
$E_{15}.(E_{51}v_{i}^{\lambda}-E_{52}v_{j}^{\lambda})=(|\lambda|+s_{ij})v_{i}^{\lambda}=0$
yields a contradiction.
Case 3
$wt(\xi_{0,4})\in\\{\lambda+\overrightarrow{w}_{4}^{3\omega_{1}},\lambda+\overrightarrow{w}_{9}^{3\omega_{1}},\lambda+\overrightarrow{w}_{14}^{3\omega_{1}},\lambda+\overrightarrow{w}_{19}^{3\omega_{1}},\lambda+\overrightarrow{w}_{20}^{3\omega_{1}},\lambda+\overrightarrow{w}_{21}^{3\omega_{1}},\lambda+\overrightarrow{w}_{25}^{3\omega_{1}},\lambda+\overrightarrow{w}_{26}^{3\omega_{1}},\lambda+\overrightarrow{w}_{29}^{3\omega_{1}},\lambda+\overrightarrow{w}_{33}^{3\omega_{1}}\\}$
In these cases, we derive that:
$E_{51}v_{i}^{\lambda}-E_{53}v_{j}^{\lambda}=0,\
E_{13}v_{j}^{\lambda}=-t_{ij}v_{i}^{\lambda},\
\mbox{wt}(v_{i}^{\lambda})=\lambda,(i,j,t_{ij})\in\\{(4,1,3),(9,4,2),(14,7,2),(19,11,2),(20,9,1),(21,14,1),(25,18,1),(26,19,1),(29,24,1),(33,28,1)\\}.$Then,
$E_{15}.(E_{51}v_{i}^{\lambda}-E_{53}v_{j}^{\lambda})=(|\lambda|+t_{ij})v_{i}^{\lambda}=0$
yields a contradiction.
Case 4
$wt(\xi_{0,4})\in\\{\lambda+\overrightarrow{w}_{7}^{3\omega_{1}},\lambda+\overrightarrow{w}_{11}^{3\omega_{1}},\lambda+\overrightarrow{w}_{18}^{3\omega_{1}},\lambda+\overrightarrow{w}_{24}^{3\omega_{1}},\lambda+\overrightarrow{w}_{28}^{3\omega_{1}}\\}$
In these cases, we have $\lambda_{3}+\lambda_{4}>0$. And
$E_{52}v_{\lambda}=E_{53}v_{\lambda}=0$, i.e. $\lambda=(m,0,0,0)$. A
contradiction arises.
Case 5
$wt(\xi_{0,4})\in\\{\lambda+\overrightarrow{w}_{31}^{3\omega_{1}},\lambda+\overrightarrow{w}_{32}^{3\omega_{1}},\lambda+\overrightarrow{w}_{34}^{3\omega_{1}},\lambda+\overrightarrow{w}_{35}^{3\omega_{1}}\\}$
Case 5.1
$wt(\xi_{0,4})\in\\{\lambda+\overrightarrow{w}_{31}^{3\omega_{1}},\lambda+\overrightarrow{w}_{32}^{3\omega_{1}},\lambda+\overrightarrow{w}_{34}^{3\omega_{1}}\\}$
In these cases, the following equations are derived:
$E_{51}v_{i}^{\lambda}-E_{53}v_{j}^{\lambda}=0,\
E_{13}v_{j}^{\lambda}=-q_{ij}v_{i}^{\lambda},\
\mbox{wt}(v_{i}^{\lambda})=\lambda-\alpha_{3},\mbox{wt}(v_{j}^{\lambda})=\lambda-\alpha_{1}-\alpha_{2}-\alpha_{3},(i,j,t_{ij})\in\\{(25,18,1),(30,24,1),(34,28,1).$
Then,
$E_{15}.(E_{51}v_{i}^{\lambda}-E_{53}v_{j}^{\lambda})=(|\lambda|+q_{ij})v_{i}^{\lambda}=0$
yields a contradiction.
Case 5.2 $wt(\xi_{0,4})=\lambda+\overrightarrow{w}_{35}^{3\omega_{1}}$
Note that $E_{51}v_{33}^{\lambda}-E_{53}v_{28}^{\lambda}=0,\
E_{51}v_{30}^{\lambda}-E_{52}v_{28}^{\lambda}=0,\
\mbox{wt}(v_{33}^{\lambda})=\lambda-\alpha_{3}-\alpha_{4},\
\mbox{wt}(v_{30}^{\lambda})=\lambda-\alpha_{2}-\alpha_{3}-2\alpha_{4},\
\mbox{wt}(v_{28}^{\lambda})=\lambda-\alpha_{1}-\alpha_{2}-\alpha_{3}-\alpha_{4},E_{13}v_{28}^{\lambda}=-v_{33}^{\lambda},E_{15}v_{28}^{\lambda}=-v_{35}^{\lambda},E_{35}v_{33}^{\lambda}=-v_{35}^{\lambda},E_{12}v_{28}^{\lambda}=-v_{30}^{\lambda},E_{25}v_{30}^{\lambda}=-v_{35}^{\lambda}.$
Then
$E_{35}E_{15}(E_{51}v_{33}^{\lambda}-E_{53}v_{28}^{\lambda})=0=E_{35}(|\lambda|v_{33}^{\lambda}+E_{53}v_{35}^{\lambda})=-(\lambda_{1}+\lambda_{2})v_{35}^{\lambda}$
$None$
induces $\lambda_{1}=\lambda_{2}=0$. And
$E_{35}E_{15}(E_{51}v_{30}^{\lambda}-E_{52}v_{28}^{\lambda})=0=E_{35}[(|\lambda|-1)v_{30}^{\lambda}+E_{52}v_{35}^{\lambda}]=(1-\lambda_{1})v_{35}^{\lambda}$
$None$
induces $\lambda_{1}=1$. A contradiction arises. $\Box$
Acknowledgement: Part of this work was completed while the author was visiting
the Massachusetts Institute of Technology in 2011. Se is very grateful to all
of the faculty and the staff members of the institution for their hospitality
and support during her visit. The author would like to thank Professor Victor
Kac for his introducing this interesting conjecture that motivates this work.
The author would also like to thank Professor Dihua Jiang for his
encouragements. Finally, the author acknowledges the financial support by CSC.
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Table 1: Weights and weight vectors for $\mbox{sl}_{5}$ module $V(\omega_{2})$ $i$ | $\overrightarrow{w}_{i}^{\omega_{2}}$ | $v_{i}^{\omega_{2}}$ | $i$ | $\overrightarrow{w}_{i}^{\omega_{2}}$ | $v_{i}^{\omega_{2}}$
---|---|---|---|---|---
1 | $(0,1,0,0)=\omega_{2}$ | $d_{12}$ | 6 | $(0,-1,0,1)=\omega_{2}-\alpha_{1}-2\alpha_{2}-\alpha_{3}$ | $d_{34}$
2 | $(1,-1,1,0)=\omega_{2}-\alpha_{2}$ | $d_{13}$ | 7 | $(1,0,0,-1)=\omega_{2}-\alpha_{2}-\alpha_{3}-\alpha_{4}$ | $d_{15}$
3 | $(-1,0,1,0)=\omega_{2}-\alpha_{1}-\alpha_{2}$ | $d_{23}$ | 8 | $(-1,1,0,-1)=\omega_{2}-\alpha_{1}-\alpha_{2}-\alpha_{3}-\alpha_{4}$ | $d_{25}$
4 | $(1,0,-1,1)=\omega_{2}-\alpha_{2}-\alpha_{3}$ | $d_{14}$ | 9 | $(0,-1,1,-1)=\omega_{2}-\alpha_{1}-2\alpha_{2}-\alpha_{3}-\alpha_{4}$ | $d_{35}$
5 | $(-1,1,-1,1)=\omega_{2}-\alpha_{1}-\alpha_{2}-\alpha_{3}$ | $d_{24}$ | 10 | $(0,0,-1,0)=\omega_{2}-\alpha_{1}-2\alpha_{2}-2\alpha_{3}-\alpha_{4}$ | $d_{45}$
Table 2: Decomposition for wedge module $\Lambda^{k}W$ ($k\in\overline{1,10}$) $\Lambda^{k}W$ | irreducible components for $\Lambda^{k}W$ | maximal vector for the irreducible components
---|---|---
$\wedge^{1}W$ | $V(\omega_{2})$ | $d_{12}$
$\wedge^{2}W$ | $V(\omega_{1}+\omega_{3})$ | ${d_{12}}\wedge{d_{13}}$
$\wedge^{3}W$ | $V(2\omega_{3})\oplus V(2\omega_{1}+\omega_{4})$ | ${d_{12}}\wedge{d_{13}}\wedge{d_{23}},{d_{12}}\wedge{d_{13}}\wedge{d_{14}}$
$\wedge^{4}W$ | $V(3\omega_{1})\oplus V(\omega_{1}+\omega_{3}+\omega_{4})$ | ${d_{12}}\wedge{d_{13}}\wedge{d_{14}}\wedge{d_{15}},{d_{12}}\wedge{d_{13}}\wedge{d_{23}}\wedge{d_{14}}$
$\wedge^{5}W$ | $V(2\omega_{1}+\omega_{3})\oplus V(\omega_{2}+2\omega_{4})$ | ${d_{12}}\wedge{d_{13}}\wedge{d_{23}}\wedge{d_{14}}\wedge{d_{15}},{d_{12}}\wedge{d_{13}}\wedge{d_{23}}\wedge{d_{14}}\wedge{d_{24}}$
$\wedge^{6}W$ | $V(3\omega_{4})\oplus V(\omega_{1}+\omega_{2}+\omega_{4})$ | ${d_{12}}\wedge{d_{13}}\wedge{d_{23}}\wedge{d_{14}}\wedge{d_{24}}\wedge{d_{34}},{d_{12}}\wedge{d_{13}}\wedge{d_{23}}\wedge{d_{14}}\wedge{d_{24}}\wedge{d_{15}}$
$\wedge^{7}W$ | $V(\omega_{1}+2\omega_{4})\oplus V(2\omega_{2})$ | ${d_{13}}\wedge{d_{23}}\wedge{d_{14}}\wedge{d_{24}}\wedge{d_{34}}\wedge{d_{15}},{d_{12}}\wedge{d_{13}}\wedge{d_{23}}\wedge{d_{14}}\wedge{d_{24}}\wedge{d_{15}}\wedge{d_{25}}$
$\wedge^{8}W$ | $V(\omega_{2}+\omega_{4})$ | ${d_{12}}\wedge{d_{13}}\wedge{d_{23}}\wedge{d_{14}}\wedge{d_{24}}\wedge{d_{34}}\wedge{d_{15}}\wedge{d_{25}}$
$\wedge^{9}W$ | $V(\omega_{3})$ | ${d_{12}}\wedge{d_{13}}\wedge{d_{23}}\wedge{d_{14}}\wedge{d_{24}}\wedge{d_{34}}\wedge{d_{15}}\wedge{d_{25}}\wedge{d_{35}}$
$\wedge^{10}W$ | $V(0)$ | ${d_{12}}\wedge{d_{13}}\wedge{d_{23}}\wedge{d_{14}}\wedge{d_{24}}\wedge{d_{34}}\wedge{d_{15}}\wedge{d_{25}}\wedge{d_{35}}\wedge{d_{45}}$
Table 3: Tensor decomposition for $V(k\omega_{4})\otimes V(\mu)$ $\mu$ | highest weight in the decomposition $V(k\omega_{4})\otimes V(\mu)$
---|---
$\omega_{2}$ | $\omega_{(0,1,0,k)},\omega_{(1,0,0,k-1)}$
$\omega_{1}+\omega_{3}$ | $\omega_{(1,0,1,k)},\omega_{(0,0,1,k-1)},\omega_{(1,1,0,k-1)},\omega_{(0,1,0,k-2)}$
$2\omega_{3}$ | $\omega_{(0,0,2,k)},\omega_{(0,1,1,k-1)},\omega_{(0,2,0,k-2)}$
$2\omega_{1}+\omega_{4}$ | $\omega_{(2,0,0,k+1)},\omega_{(1,0,0,k)},\omega_{(0,0,0,k-1)},\omega_{(2,0,1,k-1)},\omega_{(1,0,1,k-2)},\omega_{(0,0,1,k-3)}$
$\omega_{1}+\omega_{3}+\omega_{4}$ | $\omega_{(1,0,1,k+1)},\omega_{(0,0,1,k)},\omega_{(1,1,0,k)},\omega_{(0,1,0,k-1)},\omega_{(1,0,2,k-1)},\omega_{(0,0,2,k-2)},\omega_{(1,1,1,k-2)},\omega_{(0,1,1,k-3)}$
$3\omega_{1}$ | $\omega_{(3,0,0,k)},\omega_{(2,0,0,k-1)},\omega_{(1,0,0,k-2)},\omega_{(0,0,0,k-3)}$
$\omega_{2}+2\omega_{4}$ | $\omega_{(0,1,0,k+2)},\omega_{(1,0,0,k+1)},\omega_{(0,1,1,k)},\omega_{(1,0,1,k-1)},\omega_{(0,1,2,k-2)},\omega_{(1,0,2,k-3)}$
$2\omega_{1}+\omega_{3}$ | $\omega_{(2,0,1,k)},\omega_{(1,0,1,k-1)},\omega_{(0,0,1,k-2)},\omega_{(2,1,0,k-1)},\omega_{(1,1,0,k-2)},\omega_{(0,1,0,k-3)}$
$3\omega_{4}$ | $\omega_{(0,0,0,k+3)},\omega_{(0,0,1,k+1)},\omega_{(0,0,2,k-1)},\omega_{(0,0,3,k-3)}$
$\omega_{1}+\omega_{2}+\omega_{4}$ | $\omega_{(1,1,0,k+1)},\omega_{(0,1,0,k)},\omega_{(2,0,0,k)},\omega_{(1,0,0,k-1)},\omega_{(1,1,1,k-1)},\omega_{(0,1,1,k-2)},\omega_{(2,0,1,k-2)},\omega_{(1,0,1,k-3)}$
$\omega_{1}+2\omega_{4}$ | $\omega_{(1,0,0,k+2)},\omega_{(0,0,0,k+1)},\omega_{(1,0,1,k)},\omega_{(0,0,1,k-1)},\omega_{(1,0,2,k-2)},\omega_{(0,0,2,k-3)}$
$2\omega_{2}$ | $\omega_{(0,2,0,k)},\omega_{(1,1,0,k-1)},\omega_{(2,0,0,k-2)}$
$\omega_{2}+\omega_{4}$ | $\omega_{(0,1,0,k+1)},\omega_{(1,0,0,k)},\omega_{(0,1,1,k-1)},\omega_{(1,0,1,k-2)},$
$\omega_{3}$ | $\omega_{(0,0,1,k)},\omega_{(0,1,0,k-1)}$
Table 4: The leading term $l_{\mu}$ $\mu$ | $l_{\mu}$
---|---
$\omega_{2}$ | $d_{12},d_{15}$
$\omega_{1}+\omega_{3}$ | $d_{12}\wedge d_{13},2d_{12}\wedge d_{35}+d_{23}\wedge d_{15}-d_{13}\wedge d_{25},d_{12}\wedge d_{15},d_{15}\wedge d_{25}$
$2\omega_{3}$ | $d_{12}\wedge d_{13}\wedge d_{23},d_{12}\wedge d_{13}\wedge d_{25}-d_{12}\wedge d_{23}\wedge d_{15},d_{12}\wedge d_{15}\wedge d_{25}$
$2\omega_{1}+\omega_{4}$ | $d_{12}\wedge d_{13}\wedge d_{14},d_{15}\wedge d_{25}\wedge d_{35},2d_{12}\wedge d_{13}\wedge d_{45}-2d_{12}\wedge d_{14}\wedge d_{35}+2d_{13}\wedge d_{14}\wedge d_{25}-d_{13}\wedge d_{24}\wedge d_{15}-d_{23}\wedge d_{14}\wedge d_{15}+d_{12}\wedge d_{34}\wedge d_{15},d_{12}\wedge d_{35}\wedge d_{45}-d_{13}\wedge d_{25}\wedge d_{45}+d_{23}\wedge d_{15}\wedge d_{45}+d_{14}\wedge d_{25}\wedge d_{35}-d_{24}\wedge d_{15}\wedge d_{35}+d_{34}\wedge d_{15}\wedge d_{25},d_{12}\wedge d_{13}\wedge d_{15},d_{12}\wedge d_{15}\wedge d_{35}-d_{13}\wedge d_{15}\wedge d_{25}$
$\omega_{1}+\omega_{3}+\omega_{4}$ | $d_{12}\wedge d_{13}\wedge d_{23}\wedge d_{14},d_{12}\wedge d_{13}\wedge d_{23}\wedge d_{15},2d_{12}\wedge d_{13}\wedge d_{23}\wedge d_{45}-d_{12}\wedge d_{13}\wedge d_{24}\wedge d_{35}+d_{12}\wedge d_{13}\wedge d_{34}\wedge d_{25}+d_{12}\wedge d_{23}\wedge d_{14}\wedge d_{35}-d_{12}\wedge d_{23}\wedge d_{34}\wedge d_{15}-d_{13}\wedge d_{23}\wedge d_{14}\wedge d_{25}+d_{13}\wedge d_{23}\wedge d_{24}\wedge d_{15},d_{12}\wedge d_{23}\wedge d_{14}\wedge d_{15}-2d_{12}\wedge d_{13}\wedge d_{14}\wedge d_{25}+d_{12}\wedge d_{13}\wedge d_{24}\wedge d_{15},d_{12}\wedge d_{13}\wedge d_{25}\wedge d_{45}-d_{12}\wedge d_{23}\wedge d_{15}\wedge d_{45}-d_{12}\wedge d_{14}\wedge d_{25}\wedge d_{35}+d_{12}\wedge d_{24}\wedge d_{15}\wedge d_{35}-d_{12}\wedge d_{34}\wedge d_{15}\wedge d_{25},d_{12}\wedge d_{13}\wedge d_{25}\wedge d_{35}-d_{12}\wedge d_{23}\wedge d_{15}\wedge d_{35}+d_{13}\wedge d_{23}\wedge d_{15}\wedge d_{25},d_{12}\wedge d_{13}\wedge d_{15}\wedge d_{25},d_{12}\wedge d_{15}\wedge d_{25}\wedge d_{35}$
$3\omega_{1}$ | $d_{12}\wedge d_{13}\wedge d_{14}\wedge d_{15},d_{12}\wedge d_{13}\wedge d_{15}\wedge d_{45}-d_{12}\wedge d_{14}\wedge d_{15}\wedge d_{35}+d_{13}\wedge d_{14}\wedge d_{15}\wedge d_{25},d_{12}\wedge d_{15}\wedge d_{35}\wedge d_{45}-d_{13}\wedge d_{15}\wedge d_{25}\wedge d_{45}+d_{14}\wedge d_{15}\wedge d_{25}\wedge d_{35},d_{15}\wedge d_{25}\wedge d_{35}\wedge d_{45}$
$\omega_{2}+2\omega_{4}$ | $d_{12}\wedge d_{13}\wedge d_{23}\wedge d_{14}\wedge d_{24},2d_{12}\wedge d_{13}\wedge d_{23}\wedge d_{14}\wedge d_{45}+2d_{12}\wedge d_{13}\wedge d_{14}\wedge d_{24}\wedge d_{35}-2d_{12}\wedge d_{13}\wedge d_{14}\wedge d_{34}\wedge d_{25}+d_{12}\wedge d_{13}\wedge d_{24}\wedge d_{34}\wedge d_{15}+d_{12}\wedge d_{23}\wedge d_{14}\wedge d_{34}\wedge d_{15}-d_{13}\wedge d_{23}\wedge d_{14}\wedge d_{24}\wedge d_{15},d_{12}\wedge d_{13}\wedge d_{23}\wedge d_{14}\wedge d_{25}-d_{12}\wedge d_{13}\wedge d_{23}\wedge d_{24}\wedge d_{15},3d_{12}\wedge d_{23}\wedge d_{14}\wedge d_{15}\wedge d_{35}-3d_{13}\wedge d_{23}\wedge d_{14}\wedge d_{15}\wedge d_{25}+4d_{12}\wedge d_{13}\wedge d_{23}\wedge d_{15}\wedge d_{45}-2d_{12}\wedge d_{13}\wedge d_{14}\wedge d_{25}\wedge d_{35}-d_{12}\wedge d_{13}\wedge d_{24}\wedge d_{15}\wedge d_{35}+d_{12}\wedge d_{13}\wedge d_{34}\wedge d_{15}\wedge d_{25},d_{12}\wedge d_{13}\wedge d_{23}\wedge d_{15}\wedge d_{25},d_{12}\wedge d_{13}\wedge d_{15}\wedge d_{25}\wedge d_{35}$
$2\omega_{1}+\omega_{3}$ | $d_{12}\wedge d_{13}\wedge d_{23}\wedge d_{14}\wedge d_{15},4d_{12}\wedge d_{13}\wedge d_{23}\wedge d_{15}\wedge d_{45}-2d_{12}\wedge d_{13}\wedge d_{14}\wedge d_{25}\wedge d_{35}-d_{12}\wedge d_{13}\wedge d_{24}\wedge d_{15}\wedge d_{35}+3d_{12}\wedge d_{23}\wedge d_{14}\wedge d_{15}\wedge d_{35}-3d_{13}\wedge d_{23}\wedge d_{14}\wedge d_{15}\wedge d_{25}+d_{12}\wedge d_{13}\wedge d_{34}\wedge d_{15}\wedge d_{25},2d_{12}\wedge d_{13}\wedge d_{25}\wedge d_{35}\wedge d_{45}-2d_{12}\wedge d_{23}\wedge d_{15}\wedge d_{35}\wedge d_{45}+d_{12}\wedge d_{34}\wedge d_{15}\wedge d_{25}\wedge d_{35}+2d_{13}\wedge d_{23}\wedge d_{15}\wedge d_{25}\wedge d_{45}-d_{13}\wedge d_{24}\wedge d_{15}\wedge d_{25}\wedge d_{35}+d_{23}\wedge d_{14}\wedge d_{15}\wedge d_{25}\wedge d_{35},d_{12}\wedge d_{13}\wedge d_{14}\wedge d_{15}\wedge d_{25},d_{12}\wedge d_{15}\wedge d_{25}\wedge d_{35}\wedge d_{45}$
$3\omega_{4}$ | ${\hat{d}_{15}}\wedge{\hat{d}_{25}}\wedge{\hat{d}_{35}}\wedge{\hat{d}_{45}},{\hat{d}_{34}}\wedge{\hat{d}_{15}}\wedge{\hat{d}_{25}}\wedge{\hat{d}_{45}}-{\hat{d}_{24}}\wedge{\hat{d}_{15}}\wedge{\hat{d}_{35}}\wedge{\hat{d}_{45}}+{\hat{d}_{14}}\wedge{\hat{d}_{25}}\wedge{\hat{d}_{35}}\wedge{\hat{d}_{45}},{\hat{d}_{14}}\wedge{\hat{d}_{34}}\wedge{\hat{d}_{25}}\wedge{\hat{d}_{45}}-{\hat{d}_{24}}\wedge{\hat{d}_{34}}\wedge{\hat{d}_{15}}\wedge{\hat{d}_{45}}-{\hat{d}_{14}}\wedge{\hat{d}_{24}}\wedge{\hat{d}_{35}}\wedge{\hat{d}_{45}},{\hat{d}_{14}}\wedge{\hat{d}_{24}}\wedge{\hat{d}_{34}}\wedge{\hat{d}_{45}},$
$\omega_{1}+\omega_{2}+\omega_{4}$ | ${\hat{d}_{34}}\wedge{\hat{d}_{25}}\wedge{\hat{d}_{35}}\wedge{\hat{d}_{45}},{\hat{d}_{12}}\wedge{\hat{d}_{34}}\wedge{\hat{d}_{35}}\wedge{\hat{d}_{45}}-{\hat{d}_{13}}\wedge{\hat{d}_{34}}\wedge{\hat{d}_{25}}\wedge{\hat{d}_{45}}+{\hat{d}_{23}}\wedge{\hat{d}_{34}}\wedge{\hat{d}_{15}}\wedge{\hat{d}_{45}}+{\hat{d}_{24}}\wedge{\hat{d}_{34}}\wedge{\hat{d}_{15}}\wedge{\hat{d}_{35}}-{\hat{d}_{14}}\wedge{\hat{d}_{34}}\wedge{\hat{d}_{25}}\wedge{\hat{d}_{35}},{\hat{d}_{23}}\wedge{\hat{d}_{24}}\wedge{\hat{d}_{35}}\wedge{\hat{d}_{45}}-{\hat{d}_{23}}\wedge{\hat{d}_{34}}\wedge{\hat{d}_{25}}\wedge{\hat{d}_{45}}-{\hat{d}_{24}}\wedge{\hat{d}_{34}}\wedge{\hat{d}_{25}}\wedge{\hat{d}_{35}},{\hat{d}_{12}}\wedge{\hat{d}_{23}}\wedge{\hat{d}_{34}}\wedge{\hat{d}_{45}}+{\hat{d}_{12}}\wedge{\hat{d}_{24}}\wedge{\hat{d}_{34}}\wedge{\hat{d}_{35}}-{\hat{d}_{13}}\wedge{\hat{d}_{23}}\wedge{\hat{d}_{24}}\wedge{\hat{d}_{45}}-{\hat{d}_{13}}\wedge{\hat{d}_{24}}\wedge{\hat{d}_{34}}\wedge{\hat{d}_{25}}+{\hat{d}_{23}}\wedge{\hat{d}_{14}}\wedge{\hat{d}_{24}}\wedge{\hat{d}_{35}}-{\hat{d}_{23}}\wedge{\hat{d}_{14}}\wedge{\hat{d}_{34}}\wedge{\hat{d}_{25}}+2{\hat{d}_{23}}\wedge{\hat{d}_{24}}\wedge{\hat{d}_{34}}\wedge{\hat{d}_{15}},{\hat{d}_{24}}\wedge{\hat{d}_{34}}\wedge{\hat{d}_{35}}\wedge{\hat{d}_{45}},{\hat{d}_{13}}\wedge{\hat{d}_{24}}\wedge{\hat{d}_{34}}\wedge{\hat{d}_{45}}+2{\hat{d}_{14}}\wedge{\hat{d}_{24}}\wedge{\hat{d}_{34}}\wedge{\hat{d}_{35}}-{\hat{d}_{23}}\wedge{\hat{d}_{14}}\wedge{\hat{d}_{34}}\wedge{\hat{d}_{45}},{\hat{d}_{23}}\wedge{\hat{d}_{24}}\wedge{\hat{d}_{34}}\wedge{\hat{d}_{45}},{\hat{d}_{23}}\wedge{\hat{d}_{14}}\wedge{\hat{d}_{24}}\wedge{\hat{d}_{34}}$
$\omega_{1}+2\omega_{4}$ | ${\hat{d}_{25}}\wedge{\hat{d}_{35}}\wedge{\hat{d}_{45}},{\hat{d}_{34}}\wedge{\hat{d}_{15}}\wedge{\hat{d}_{25}}-{\hat{d}_{24}}\wedge{\hat{d}_{15}}\wedge{\hat{d}_{35}}+{\hat{d}_{14}}\wedge{\hat{d}_{25}}\wedge{\hat{d}_{35}}-{\hat{d}_{23}}\wedge{\hat{d}_{15}}\wedge{\hat{d}_{45}}+{\hat{d}_{13}}\wedge{\hat{d}_{25}}\wedge{\hat{d}_{45}}-{\hat{d}_{12}}\wedge{\hat{d}_{35}}\wedge{\hat{d}_{45}},{\hat{d}_{24}}\wedge{\hat{d}_{35}}\wedge{\hat{d}_{45}},2{\hat{d}_{24}}\wedge{\hat{d}_{34}}\wedge{\hat{d}_{15}}-2{\hat{d}_{14}}\wedge{\hat{d}_{34}}\wedge{\hat{d}_{25}}-{\hat{d}_{23}}\wedge{\hat{d}_{14}}\wedge{\hat{d}_{45}}+{\hat{d}_{13}}\wedge{\hat{d}_{24}}\wedge{\hat{d}_{45}}-{\hat{d}_{12}}\wedge{\hat{d}_{34}}\wedge{\hat{d}_{45}}+2{\hat{d}_{14}}\wedge{\hat{d}_{24}}\wedge{\hat{d}_{35}},{\hat{d}_{24}}\wedge{\hat{d}_{34}}\wedge{\hat{d}_{45}},{\hat{d}_{14}}\wedge{\hat{d}_{24}}\wedge{\hat{d}_{34}},$
$2\omega_{2}$ | ${\hat{d}_{34}}\wedge{\hat{d}_{35}}\wedge{\hat{d}_{45}},{\hat{d}_{24}}\wedge{\hat{d}_{34}}\wedge{\hat{d}_{35}}+{\hat{d}_{23}}\wedge{\hat{d}_{34}}\wedge{\hat{d}_{45}},{\hat{d}_{23}}\wedge{\hat{d}_{24}}\wedge{\hat{d}_{34}},$
$\omega_{2}+\omega_{4}$ | ${\hat{d}_{35}}\wedge{\hat{d}_{45}},{\hat{d}_{23}}\wedge{\hat{d}_{45}}+{\hat{d}_{24}}\wedge{\hat{d}_{35}}-{\hat{d}_{34}}\wedge{\hat{d}_{25}},{\hat{d}_{34}}\wedge{\hat{d}_{45}},{\hat{d}_{24}}\wedge{\hat{d}_{34}}$
$\omega_{3}$ | ${\hat{d}_{45}},{\hat{d}_{34}}$
Table 5: Weights and weight vectors for $\mbox{sl}_{5}$ module $V(\omega_{1}+\omega_{3})$ $(i,j)$ | $\overrightarrow{w}_{i}^{\omega_{1}+\omega_{3}}$ | $v_{i,j}^{\omega_{1}+\omega_{3}}$ | $(i,j)$ | $\overrightarrow{w}_{i}^{\omega_{1}+\omega_{3}}$ | $v_{i,j}^{\omega_{1}+\omega_{3}}$
---|---|---|---|---|---
(1,1) | $(1,0,1,0)=\omega_{1}+\omega_{3}$ | $d_{12}\wedge d_{13}$ | (18,1) | $(-1,-1,1,1)=\omega_{1}+\omega_{3}-2\alpha_{1}-2\alpha_{2}-\alpha_{3}$ | $d_{23}\wedge d_{34}$
(2,1) | $(-1,1,1,0)=\omega_{1}+\omega_{3}-\alpha_{1}$ | $d_{12}\wedge d_{23}$ | (19,1) | $(-2,1,1,-1)=\omega_{1}+\omega_{3}-2\alpha_{1}-\alpha_{2}-\alpha_{3}-\alpha_{4}$ | $d_{23}\wedge d_{25}$
(3,1) | $(1,1,-1,1)=\omega_{1}+\omega_{3}-\alpha_{3}$ | $d_{12}\wedge d_{14}$ | (20,1) | $(1,-1,-1,2)=\omega_{1}+\omega_{3}-\alpha_{1}-2\alpha_{2}-2\alpha_{3}$ | $d_{14}\wedge d_{34}$
(4,1) | $(-1,2,-1,1)=\omega_{1}+\omega_{3}-\alpha_{1}-\alpha_{3}$ | $d_{12}\wedge d_{24}$ | (21,1) | $(1,-1,0,0)==\omega_{1}+\omega_{3}-\alpha_{1}-2\alpha_{2}-2\alpha_{3}-\alpha_{4}$ | $d_{14}\wedge d_{35}-d_{13}\wedge d_{45}$
(5,1) | $(1,1,0,-1)=\omega_{1}+\omega_{3}-\alpha_{3}-\alpha_{4}$ | $d_{12}\wedge d_{15}$ | (21,2) | $(1,-1,0,0)=\omega_{1}+\omega_{3}-\alpha_{1}-2\alpha_{2}-2\alpha_{3}-\alpha_{4}$ | $d_{14}\wedge d_{35}-d_{34}\wedge d_{15}$
(6,1) | $(0,-1,2,0)=\omega_{1}+\omega_{3}-\alpha_{1}-\alpha_{2}$ | $d_{13}\wedge d_{23}$ | (21,3) | $(1,-1,0,0)=\omega_{1}+\omega_{3}-\alpha_{1}-2\alpha_{2}-2\alpha_{3}-\alpha_{4}$ | $d_{34}\wedge d_{15}+d_{13}\wedge d_{45}$
(7,1) | $(2,-1,0,1)=\omega_{1}+\omega_{3}-\alpha_{2}-\alpha_{3}$ | $d_{13}\wedge d_{14}$ | (22,1) | $(-1,-1,2,-1)=\omega_{1}+\omega_{3}-2\alpha_{1}-2\alpha_{2}-\alpha_{3}-\alpha_{4}$ | $d_{23}\wedge d_{35}$
(8,1) | $(0,0,0,1)=\omega_{1}+\omega_{3}-\alpha_{1}-\alpha_{2}-\alpha_{3}$ | $d_{12}\wedge d_{34}-d_{13}\wedge d_{24}$ | $(23,1)$ | $(-1,0,-1,2)=\omega_{1}+\omega_{3}-2\alpha_{1}-2\alpha_{2}-2\alpha_{3}$ | $d_{24}\wedge d_{34}$
(8,2) | $(0,0,0,1)=\omega_{1}+\omega_{3}-\alpha_{1}-\alpha_{2}-\alpha_{3}$ | $d_{12}\wedge d_{34}+d_{23}\wedge d_{14}$ | (24,1) | $(-2,2,-1,0)=\omega_{1}+\omega_{3}-2\alpha_{1}-\alpha_{2}-2\alpha_{3}-\alpha_{4}$ | $d_{24}\wedge d_{25}$
(8,3) | $(0,0,0,1)=\omega_{1}+\omega_{3}-\alpha_{1}-\alpha_{2}-\alpha_{3}$ | $d_{13}\wedge d_{24}-d_{23}\wedge d_{14}$ | (25,1) | $(0,1,0,-2)=\omega_{1}+\omega_{3}-\alpha_{1}-\alpha_{2}-2\alpha_{3}-2\alpha_{4}$ | $d_{15}\wedge d_{25}$
(9,1) | $(-1,2,0,-1)=\omega_{1}+\omega_{3}-\alpha_{1}-\alpha_{3}-\alpha_{4}$ | $d_{12}\wedge d_{25}$ | (26,1) | $(-1,0,0,0)=\omega_{1}+\omega_{3}-2\alpha_{1}-2\alpha_{2}-2\alpha_{3}-\alpha_{4}$ | $d_{34}\wedge d_{25}-d_{24}\wedge d_{35}$
(10,1) | $(2,-1,1,-1)=\omega_{1}+\omega_{3}-\alpha_{2}-\alpha_{3}-\alpha_{4}$ | $d_{13}\wedge d_{15}$ | (26,2) | $(-1,0,0,0)=\omega_{1}+\omega_{3}-2\alpha_{1}-2\alpha_{2}-2\alpha_{3}-\alpha_{4}$ | $d_{23}\wedge d_{45}-d_{24}\wedge d_{35}$
(11,1) | $(0,0,1,-1)=\omega_{1}+\omega_{3}-\alpha_{1}-\alpha_{2}-\alpha_{3}-\alpha_{4}$ | $d_{12}\wedge d_{35}-d_{13}\wedge d_{25}$ | (26,3) | $(-1,0,0,0)=\omega_{1}+\omega_{3}-2\alpha_{1}-2\alpha_{2}-2\alpha_{3}-\alpha_{4}$ | $d_{34}\wedge d_{25}+d_{23}\wedge d_{45}$
(11,2) | $(0,0,1,-1)=\omega_{1}+\omega_{3}-\alpha_{1}-\alpha_{2}-\alpha_{3}-\alpha_{4}$ | $d_{12}\wedge d_{35}+d_{23}\wedge d_{15}$ | (27,1) | $(1,0,-2,1)=\omega_{1}+\omega_{3}-\alpha_{1}-2\alpha_{2}-3\alpha_{3}-\alpha_{4}$ | $d_{14}\wedge d_{45}$
(11,3) | $(0,0,1,-1)=\omega_{1}+\omega_{3}-\alpha_{1}-\alpha_{2}-\alpha_{3}-\alpha_{4}$ | $d_{23}\wedge d_{15}-d_{13}\wedge d_{25}$ | (28,1) | $(1,-1,1,-2)=\omega_{1}+\omega_{3}-\alpha_{1}-2\alpha_{2}-2\alpha_{3}-2\alpha_{4}$ | $d_{15}\wedge d_{35}$
(12,1) | $(1,-2,1,1)=\omega_{1}+\omega_{3}-\alpha_{1}-2\alpha_{2}-\alpha_{3}$ | $d_{13}\wedge d_{34}$ | (29,1) | $(-1,1,-2,1)=\omega_{1}+\omega_{3}-2\alpha_{1}-2\alpha_{2}-3\alpha_{3}-\alpha_{4}$ | $d_{24}\wedge d_{45}$
(13,1) | $(-2,1,0,1)=\omega_{1}+\omega_{3}-2\alpha_{1}-\alpha_{2}-\alpha_{3}$ | $d_{23}\wedge d_{24}$ | (30,1) | $(0,-2,1,0)=\omega_{1}+\omega_{3}-2\alpha_{1}-3\alpha_{2}-2\alpha_{3}-\alpha_{4}$ | $d_{34}\wedge d_{35}$
(14,1) | $(0,1,-2,2)=\omega_{1}+\omega_{3}-\alpha_{1}-\alpha_{2}-2\alpha_{3}$ | $d_{14}\wedge d_{24}$ | (31,1) | $(1,0,-1,-1)=\omega_{1}+\omega_{3}-\alpha_{1}-2\alpha_{2}-3\alpha_{3}-2\alpha_{4}$ | $d_{15}\wedge d_{45}$
(15,1) | $(2,0,-1,0)=\omega_{1}+\omega_{3}-\alpha_{2}-2\alpha_{3}-\alpha_{4}$ | $d_{14}\wedge d_{15}$ | (32,1) | $(-1,0,1,-2)=\omega_{1}+\omega_{3}-2\alpha_{1}-2\alpha_{2}-2\alpha_{3}-2\alpha_{4}$ | $d_{25}\wedge d_{35}$
(16,1) | $(0,1,-1,0)=\omega_{1}+\omega_{3}-\alpha_{1}-\alpha_{2}-2\alpha_{3}-\alpha_{4}$ | $d_{12}\wedge d_{45}-d_{14}\wedge d_{25}$ | (33,1) | $(0,-1,-1,1)=\omega_{1}+\omega_{3}-2\alpha_{1}-3\alpha_{2}-3\alpha_{3}-\alpha_{4}$ | $d_{34}\wedge d_{45}$
(16,2) | $(0,1,-1,0)=\omega_{1}+\omega_{3}-\alpha_{1}-\alpha_{2}-2\alpha_{3}-\alpha_{4}$ | $d_{12}\wedge d_{45}+d_{24}\wedge d_{15}$ | (34,1) | $(-1,1,-1,-1)=\omega_{1}+\omega_{3}-2\alpha_{1}-2\alpha_{2}-3\alpha_{3}-2\alpha_{4}$ | $d_{25}\wedge d_{45}$
(16,3) | $(0,1,-1,0)=\omega_{1}+\omega_{3}-\alpha_{1}-\alpha_{2}-2\alpha_{3}-\alpha_{4}$ | $d_{14}\wedge d_{25}-d_{24}\wedge d_{15}$ | (35,1) | $(0,-1,0,-1)=\omega_{1}+\omega_{3}-2\alpha_{1}-3\alpha_{2}-3\alpha_{3}-2\alpha_{4}$ | $d_{35}\wedge d_{45}$
(17,1) | $(1,-2,2,-1)=\omega_{1}+\omega_{3}-\alpha_{1}-2\alpha_{2}-\alpha_{3}-\alpha_{4}$ | $d_{13}\wedge d_{35}$ | | |
Table 6: Weights and weight vectors for $\mbox{sl}_{5}$ module $V(2\omega_{1}+\omega_{4})$ $(i,j)$ | $\overrightarrow{w}_{i}^{2\omega_{1}+\omega_{4}}$ | $v_{i,j}^{2\omega_{1}+\omega_{4}}$
---|---|---
(1,1) | $(2,0,0,1)=2\omega_{1}+\omega_{4}$ | $d_{12}\wedge d_{13}\wedge d_{14}$
(2,1) | $(0,1,0,1)=2\omega_{1}+\omega_{4}-\alpha_{1}$ | $d_{12}\wedge d_{23}\wedge d_{14}+d_{12}\wedge d_{13}\wedge d_{24}$
(3,1) | $(2,0,1,-1)=2\omega_{1}+\omega_{4}-\alpha_{4}$ | $d_{12}\wedge d_{13}\wedge d_{15}$
(4,1) | $(-2,2,0,1)=2\omega_{1}+\omega_{4}-2\alpha_{1}$ | $d_{12}\wedge d_{23}\wedge d_{24}$
(5,1) | $(1,-1,1,1)=2\omega_{1}+\omega_{4}-\alpha_{1}-\alpha_{2}$ | $d_{13}\wedge d_{23}\wedge d_{14}+d_{12}\wedge d_{13}\wedge d_{34}$
(6,1) | $(0,1,1,-1)=2\omega_{1}+\omega_{4}-\alpha_{1}-\alpha_{4}$ | $d_{12}\wedge d_{23}\wedge d_{15}+d_{12}\wedge d_{13}\wedge d_{25}$
(7,1) | $(2,1,-1,0)=2\omega_{1}+\omega_{4}-\alpha_{3}-\alpha_{4}$ | $d_{12}\wedge d_{14}\wedge d_{15}$
(8,1) | $(-1,0,1,1)=2\omega_{1}+\omega_{4}-2\alpha_{1}-\alpha_{2}$ | $d_{13}\wedge d_{23}\wedge d_{24}+d_{12}\wedge d_{23}\wedge d_{34}$
(9,1) | $(-2,2,1,-1)=2\omega_{1}+\omega_{4}-2\alpha_{1}-\alpha_{4}$ | $d_{12}\wedge d_{23}\wedge d_{25}$
(10,1) | $(1,0,-1,2)=2\omega_{1}+\omega_{4}-\alpha_{1}-\alpha_{2}-\alpha_{3}$ | $-d_{13}\wedge d_{14}\wedge d_{24}+d_{12}\wedge d_{14}\wedge d_{34}$
(11,1) | $(1,-1,2,-1)=2\omega_{1}+\omega_{4}-\alpha_{1}-\alpha_{2}-\alpha_{4}$ | $d_{13}\wedge d_{23}\wedge d_{15}+d_{12}\wedge d_{13}\wedge d_{35}$
(12,1) | $(0,2,-1,0)=2\omega_{1}+\omega_{4}-\alpha_{1}-\alpha_{3}-\alpha_{4}$ | $d_{12}\wedge d_{24}\wedge d_{15}+d_{12}\wedge d_{14}\wedge d_{25}$
(13,1) | $(3,-1,0,0)=2\omega_{1}+\omega_{4}-\alpha_{2}-\alpha_{3}-\alpha_{4}$ | $d_{13}\wedge d_{14}\wedge d_{15}$
(14,1) | $(0,-2,2,1)=2\omega_{1}+\omega_{4}-2\alpha_{1}-2\alpha_{2}$ | $d_{13}\wedge d_{23}\wedge d_{34}$
(15,1) | $(-1,1,-1,2)=2\omega_{1}+\omega_{4}-2\alpha_{1}-\alpha_{2}-\alpha_{3}$ | $d_{12}\wedge d_{24}\wedge d_{34}-d_{23}\wedge d_{14}\wedge d_{24}$
(16,1) | $(-1,0,2,-1)=2\omega_{1}+\omega_{4}-2\alpha_{1}-\alpha_{2}-\alpha_{4}$ | $d_{12}\wedge d_{23}\wedge d_{35}+d_{13}\wedge d_{23}\wedge d_{25}$
(17,1) | $(-2,3,-1,0)=2\omega_{1}+\omega_{4}-2\alpha_{1}-\alpha_{3}-\alpha_{4}$ | $d_{12}\wedge d_{24}\wedge d_{25}$
(18,1) | $(1,0,0,0)=2\omega_{1}+\omega_{4}-\alpha_{1}-\alpha_{2}-\alpha_{3}-\alpha_{4}$ | $d_{13}\wedge d_{24}\wedge d_{15}+d_{13}\wedge d_{14}\wedge d_{25}+d_{23}\wedge d_{14}\wedge d_{15},$
(18,2) | $(1,0,0,0)=2\omega_{1}+\omega_{4}-\alpha_{1}-\alpha_{2}-\alpha_{3}-\alpha_{4}$ | $d_{13}\wedge d_{24}\wedge d_{15}+d_{13}\wedge d_{14}\wedge d_{25}+d_{12}\wedge d_{14}\wedge d_{35}+d_{12}\wedge d_{34}\wedge d_{15},$
(18,3) | $(1,0,0,0)=2\omega_{1}+\omega_{4}-\alpha_{1}-\alpha_{2}-\alpha_{3}-\alpha_{4}$ | $d_{13}\wedge d_{24}\wedge d_{15}-d_{13}\wedge d_{14}\wedge d_{25}-d_{12}\wedge d_{34}\wedge d_{15}+d_{12}\wedge d_{14}\wedge d_{35}$
(18,4) | $(1,0,0,0)=2\omega_{1}+\omega_{4}-\alpha_{1}-\alpha_{2}-\alpha_{3}-\alpha_{4}$ | $-d_{23}\wedge d_{14}\wedge d_{15}+d_{13}\wedge d_{24}\wedge d_{15}+d_{12}\wedge d_{14}\wedge d_{35}+d_{12}\wedge d_{13}\wedge d_{45}$
(19,1) | $(0,-1,0,2)=2\omega_{1}+\omega_{4}-2\alpha_{1}-2\alpha_{2}-\alpha_{3}$ | $d_{13}\wedge d_{24}\wedge d_{34}-d_{23}\wedge d_{14}\wedge d_{34}$
(20,1) | $(0,-2,3,-1)=2\omega_{1}+\omega_{4}-2\alpha_{1}-2\alpha_{2}-\alpha_{4}$ | $d_{13}\wedge d_{23}\wedge d_{35}$
(21,1) | $(-1,1,0,0)=2\omega_{1}+\omega_{4}-2\alpha_{1}-\alpha_{2}-\alpha_{3}-\alpha_{4}$ | $d_{12}\wedge d_{34}\wedge d_{25}+d_{12}\wedge d_{24}\wedge d_{35}+2d_{13}\wedge d_{24}\wedge d_{25}+d_{23}\wedge d_{14}\wedge d_{25}+d_{23}\wedge d_{24}\wedge d_{15}$
(21,2) | $(-1,1,0,0)=2\omega_{1}+\omega_{4}-2\alpha_{1}-\alpha_{2}-\alpha_{3}-\alpha_{4}$ | $d_{12}\wedge d_{24}\wedge d_{35}+d_{12}\wedge d_{34}\wedge d_{25}+d_{13}\wedge d_{24}\wedge d_{25}$
(21,3) | $(-1,1,0,0)=2\omega_{1}+\omega_{4}-2\alpha_{1}-\alpha_{2}-\alpha_{3}-\alpha_{4}$ | $-d_{23}\wedge d_{14}\wedge d_{25}+d_{13}\wedge d_{24}\wedge d_{25}+d_{12}\wedge d_{24}\wedge d_{35}+d_{12}\wedge d_{23}\wedge d_{45}$
(21,4) | $(-1,1,0,0)=2\omega_{1}+\omega_{4}-2\alpha_{1}-\alpha_{2}-\alpha_{3}-\alpha_{4}$ | $d_{12}\wedge d_{24}\wedge d_{35}-d_{12}\wedge d_{34}\wedge d_{25}+d_{23}\wedge d_{24}\wedge d_{15}-d_{23}\wedge d_{14}\wedge d_{25}$
(22,1) | $(2,-2,1,0)=2\omega_{1}+\omega_{4}-\alpha_{1}-2\alpha_{2}-\alpha_{3}-\alpha_{4}$ | $d_{13}\wedge d_{34}\wedge d_{15}+d_{13}\wedge d_{14}\wedge d_{35}$
(23,1) | $(1,1,-2,1)=2\omega_{1}+\omega_{4}-\alpha_{1}-\alpha_{2}-2\alpha_{3}-\alpha_{4}$ | $d_{12}\wedge d_{14}\wedge d_{45}+d_{14}\wedge d_{24}\wedge d_{15}$
(24,1) | $(1,0,1,-2)=2\omega_{1}+\omega_{4}-\alpha_{1}-\alpha_{2}-\alpha_{3}-2\alpha_{4}$ | $d_{12}\wedge d_{15}\wedge d_{35}-d_{13}\wedge d_{15}\wedge d_{25}$
(25,1) | $(0,0,-2,3)=2\omega_{1}+\omega_{4}-2\alpha_{1}-2\alpha_{2}-2\alpha_{3}$ | $d_{14}\wedge d_{24}\wedge d_{34}$
(26,1) | $(0,-1,1,0)=2\omega_{1}+\omega_{4}-2\alpha_{1}-2\alpha_{2}-\alpha_{3}-\alpha_{4}$ | $d_{13}\wedge d_{24}\wedge d_{35}+d_{13}\wedge d_{34}\wedge d_{25}+d_{23}\wedge d_{14}\wedge d_{35}+d_{23}\wedge d_{34}\wedge d_{15}$
(26,2) | $(0,-1,1,0)=2\omega_{1}+\omega_{4}-2\alpha_{1}-2\alpha_{2}-\alpha_{3}-\alpha_{4}$ | $-d_{23}\wedge d_{14}\wedge d_{35}+d_{13}\wedge d_{34}\wedge d_{25}+2d_{13}\wedge d_{24}\wedge d_{35}+d_{12}\wedge d_{34}\wedge d_{35}+d_{13}\wedge d_{23}\wedge d_{45}$
(26,3) | $(0,-1,1,0)=2\omega_{1}+\omega_{4}-2\alpha_{1}-2\alpha_{2}-\alpha_{3}-\alpha_{4}$ | $d_{13}\wedge d_{24}\wedge d_{35}+d_{13}\wedge d_{23}\wedge d_{45}-d_{23}\wedge d_{14}\wedge d_{35}$
(26,4) | $(0,-1,1,0)=2\omega_{1}+\omega_{4}-2\alpha_{1}-2\alpha_{2}-\alpha_{3}-\alpha_{4}$ | $d_{23}\wedge d_{34}\wedge d_{15}-d_{23}\wedge d_{14}\wedge d_{35}-d_{13}\wedge d_{34}\wedge d_{25}+d_{13}\wedge d_{24}\wedge d_{35}$
(27,1) | $(-1,2,-2,1)=2\omega_{1}+\omega_{4}-2\alpha_{1}-\alpha_{2}-2\alpha_{3}-\alpha_{4}$ | $d_{14}\wedge d_{24}\wedge d_{25}+d_{12}\wedge d_{24}\wedge d_{45}$
(28,1) | $(-1,1,1,-2)=2\omega_{1}+\omega_{4}-2\alpha_{1}-\alpha_{2}-\alpha_{3}-2\alpha_{4}$ | $d_{12}\wedge d_{25}\wedge d_{35}-d_{23}\wedge d_{15}\wedge d_{25}$
(29,1) | $(2,-1,-1,1)=2\omega_{1}+\omega_{4}-\alpha_{1}-2\alpha_{2}-2\alpha_{3}-\alpha_{4}$ | $d_{14}\wedge d_{34}\wedge d_{15}+d_{13}\wedge d_{14}\wedge d_{45}$
(30,1) | $(1,1,-1,-1)=2\omega_{1}+\omega_{4}-\alpha_{1}-\alpha_{2}-2\alpha_{3}-2\alpha_{4}$ | $d_{12}\wedge d_{15}\wedge d_{45}-d_{14}\wedge d_{15}\wedge d_{25}$
Table 7: Weights and weight vectors for $\mbox{sl}_{5}$ module $V(2\omega_{1}+\omega_{4})$ $(i,j)$ | $\overrightarrow{w}_{i}^{2\omega_{1}+\omega_{4}}$ | $v_{i,j}^{2\omega_{1}+\omega_{4}}$
---|---|---
(31,1) | $(-3,2,0,0)=2\omega_{1}+\omega_{4}-3\alpha_{1}-\alpha_{2}-\alpha_{3}-\alpha_{4}$ | $d_{23}\wedge d_{24}\wedge d_{25}$
(32,1) | $(0,0,-1,1)=2\omega_{1}+\omega_{4}-2\alpha_{1}-2\alpha_{2}-2\alpha_{3}-\alpha_{4}$ | $d_{13}\wedge d_{24}\wedge d_{45}+2d_{14}\wedge d_{24}\wedge d_{35}-d_{14}\wedge d_{34}\wedge d_{25}-d_{23}\wedge d_{14}\wedge d_{45}+d_{24}\wedge d_{34}\wedge d_{15}$
(32,2) | $(0,0,-1,1)=2\omega_{1}+\omega_{4}-2\alpha_{1}-2\alpha_{2}-2\alpha_{3}-\alpha_{4}$ | $d_{14}\wedge d_{24}\wedge d_{35}-d_{14}\wedge d_{34}\wedge d_{25}+d_{24}\wedge d_{34}\wedge d_{15}$
(32,3) | $(0,0,-1,1)=2\omega_{1}+\omega_{4}-2\alpha_{1}-2\alpha_{2}-2\alpha_{3}-\alpha_{4}$ | $d_{12}\wedge d_{34}\wedge d_{45}+d_{13}\wedge d_{24}\wedge d_{45}+d_{14}\wedge d_{34}\wedge d_{25}+d_{14}\wedge d_{24}\wedge d_{35}$
(32,4) | $(0,0,-1,1)=2\omega_{1}+\omega_{4}-2\alpha_{1}-2\alpha_{2}-2\alpha_{3}-\alpha_{4}$ | $d_{13}\wedge d_{24}\wedge d_{45}+d_{23}\wedge d_{14}\wedge d_{45}+d_{14}\wedge d_{34}\wedge d_{25}+d_{24}\wedge d_{34}\wedge d_{15}$
(33,1) | $(-2,0,1,0)=2\omega_{1}+\omega_{4}-3\alpha_{1}-2\alpha_{2}-\alpha_{3}-\alpha_{4}$ | $d_{23}\wedge d_{24}\wedge d_{35}+d_{23}\wedge d_{34}\wedge d_{25}$
(34,1) | $(0,-1,2,-2)=2\omega_{1}+\omega_{4}-2\alpha_{1}-2\alpha_{2}-\alpha_{3}-2\alpha_{4}$ | $d_{13}\wedge d_{25}\wedge d_{35}-d_{23}\wedge d_{15}\wedge d_{35}$
(35,1) | $(-1,2,-1,-1)=2\omega_{1}+\omega_{4}-2\alpha_{1}-\alpha_{2}-2\alpha_{3}-2\alpha_{4}$ | $d_{12}\wedge d_{25}\wedge d_{45}-d_{24}\wedge d_{15}\wedge d_{25}$
(36,1) | $(2,-1,0,-1)=2\omega_{1}+\omega_{4}-\alpha_{1}-2\alpha_{2}-2\alpha_{3}-2\alpha_{4}$ | $d_{13}\wedge d_{15}\wedge d_{45}-d_{14}\wedge d_{15}\wedge d_{35}$
(37,1) | $(1,-3,2,0)=2\omega_{1}+\omega_{4}-2\alpha_{1}-3\alpha_{2}-\alpha_{3}-\alpha_{4}$ | $d_{13}\wedge d_{34}\wedge d_{35}$
(38,1) | $(-1,-2,2,0)=2\omega_{1}+\omega_{4}-3\alpha_{1}-3\alpha_{2}-\alpha_{3}-\alpha_{4}$ | $d_{23}\wedge d_{34}\wedge d_{35}$
(39,1) | $(-2,1,-1,1)=2\omega_{1}+\omega_{4}-3\alpha_{1}-2\alpha_{2}-2\alpha_{3}-\alpha_{4}$ | $d_{23}\wedge d_{24}\wedge d_{45}+d_{24}\wedge d_{34}\wedge d_{25}$
(40,1) | $(1,-2,0,1)=2\omega_{1}+\omega_{4}-2\alpha_{1}-3\alpha_{2}-2\alpha_{3}-\alpha_{4}$ | $d_{13}\wedge d_{34}\wedge d_{45}+d_{14}\wedge d_{34}\wedge d_{35}$
(41,1) | $(0,0,0,-1)=2\omega_{1}+\omega_{4}-2\alpha_{1}-2\alpha_{2}-2\alpha_{3}-2\alpha_{4}$ | $d_{13}\wedge d_{25}\wedge d_{45}+d_{23}\wedge d_{15}\wedge d_{45}-d_{14}\wedge d_{25}\wedge d_{35}-d_{24}\wedge d_{15}\wedge d_{35}$
(41,2) | $(0,0,0,-1)=2\omega_{1}+\omega_{4}-2\alpha_{1}-2\alpha_{2}-2\alpha_{3}-2\alpha_{4}$ | $d_{12}\wedge d_{35}\wedge d_{45}+d_{13}\wedge d_{25}\wedge d_{45}-d_{24}\wedge d_{15}\wedge d_{35}-d_{34}\wedge d_{15}\wedge d_{25}$
(41,3) | $(0,0,0,-1)=2\omega_{1}+\omega_{4}-2\alpha_{1}-2\alpha_{2}-2\alpha_{3}-2\alpha_{4}$ | $d_{13}\wedge d_{25}\wedge d_{45}-d_{23}\wedge d_{15}\wedge d_{45}+d_{14}\wedge d_{25}\wedge d_{35}-d_{24}\wedge d_{15}\wedge d_{35}$
(41,4) | $(0,0,0,-1)=2\omega_{1}+\omega_{4}-2\alpha_{1}-2\alpha_{2}-2\alpha_{3}-2\alpha_{4}$ | $d_{14}\wedge d_{25}\wedge d_{35}-d_{24}\wedge d_{15}\wedge d_{35}+d_{34}\wedge d_{15}\wedge d_{25}$
(42,1) | $(0,1,-3,2)=2\omega_{1}+\omega_{4}-2\alpha_{1}-2\alpha_{2}-3\alpha_{3}-\alpha_{4}$ | $d_{14}\wedge d_{24}\wedge d_{45}$
(43,1) | $(0,0,1,-3)=2\omega_{1}+\omega_{4}-2\alpha_{1}-2\alpha_{2}-2\alpha_{3}-3\alpha_{4}$ | $d_{15}\wedge d_{25}\wedge d_{35}$
(44,1) | $(-1,-1,0,1)=2\omega_{1}+\omega_{4}-3\alpha_{1}-3\alpha_{2}-2\alpha_{3}-\alpha_{4}$ | $d_{23}\wedge d_{34}\wedge d_{45}+d_{24}\wedge d_{34}\wedge d_{35}$
(45,1) | $(-2,1,0,-1)=2\omega_{1}+\omega_{4}-3\alpha_{1}-2\alpha_{2}-2\alpha_{3}-2\alpha_{4}$ | $d_{23}\wedge d_{25}\wedge d_{45}-d_{24}\wedge d_{25}\wedge d_{35}$
(46,1) | $(1,-2,1,-1)=2\omega_{1}+\omega_{4}-2\alpha_{1}-3\alpha_{2}-2\alpha_{3}-2\alpha_{4}$ | $d_{13}\wedge d_{35}\wedge d_{45}-d_{34}\wedge d_{15}\wedge d_{35}$
(47,1) | $(0,1,-2,0)=2\omega_{1}+\omega_{4}-2\alpha_{1}-2\alpha_{2}-3\alpha_{3}-2\alpha_{4}$ | $d_{14}\wedge d_{25}\wedge d_{45}-d_{24}\wedge d_{15}\wedge d_{45}$
(48,1) | $(1,-1,-2,2)=2\omega_{1}+\omega_{4}-2\alpha_{1}-3\alpha_{2}-3\alpha_{3}-\alpha_{4}$ | $d_{14}\wedge d_{34}\wedge d_{45}$
(49,1) | $(-1,-1,1,-1)=2\omega_{1}+\omega_{4}-3\alpha_{1}-3\alpha_{2}-2\alpha_{3}-2\alpha_{4}$ | $d_{23}\wedge d_{35}\wedge d_{45}-d_{34}\wedge d_{25}\wedge d_{35}$
(50,1) | $(1,-1,-1,0)=2\omega_{1}+\omega_{4}-2\alpha_{1}-3\alpha_{2}-3\alpha_{3}-2\alpha_{4}$ | $d_{14}\wedge d_{35}\wedge d_{45}-d_{34}\wedge d_{15}\wedge d_{45}$
(51,1) | $(-1,0,-2,2)=2\omega_{1}+\omega_{4}-3\alpha_{1}-3\alpha_{2}-3\alpha_{3}-\alpha_{4}$ | $d_{24}\wedge d_{34}\wedge d_{45}$
(52,1) | $(0,1,-1,-2)=2\omega_{1}+\omega_{4}-2\alpha_{1}-2\alpha_{2}-3\alpha_{3}-3\alpha_{4}$ | $d_{15}\wedge d_{25}\wedge d_{45}$
(53,1) | $(-1,0,-1,0)=2\omega_{1}+\omega_{4}-3\alpha_{1}-3\alpha_{2}-3\alpha_{3}-2\alpha_{4}$ | $d_{24}\wedge d_{35}\wedge d_{45}-d_{34}\wedge d_{25}\wedge d_{45}$
(54,1) | $(1,-1,0,-2)=2\omega_{1}+\omega_{4}-2\alpha_{1}-3\alpha_{2}-3\alpha_{3}-3\alpha_{4}$ | $d_{15}\wedge d_{35}\wedge d_{45}$
(55,1) | $(-1,0,0,-2)=2\omega_{1}+\omega_{4}-3\alpha_{1}-3\alpha_{2}-3\alpha_{3}-3\alpha_{4}$ | $d_{25}\wedge d_{35}\wedge d_{45}$
Table 8: Weights and weight vectors for $\mbox{sl}_{5}$ module $V(3\omega_{1})$ $i$ | $\overrightarrow{w}_{i}^{3\omega_{1}}$ | $v_{i}^{3\omega_{1}}$
---|---|---
1 | $(3,0,0,0)=3\omega_{1}$ | $d_{12}\wedge d_{13}\wedge d_{14}\wedge d_{15}$
2 | $(1,1,0,0)=3\omega_{1}-\alpha_{1}$ | $d_{12}\wedge d_{23}\wedge d_{14}\wedge d_{15}+d_{12}\wedge d_{13}\wedge d_{24}\wedge d_{15}+d_{12}\wedge d_{13}\wedge d_{14}\wedge d_{25}$
3 | $(-1,2,0,0)=3\omega_{1}-2\alpha_{1}$ | $d_{12}\wedge d_{23}\wedge d_{24}\wedge d_{15}+d_{12}\wedge d_{23}\wedge d_{14}\wedge d_{25}+d_{12}\wedge d_{13}\wedge d_{24}\wedge d_{25}$
4 | $(2,-1,1,0)=3\omega_{1}-\alpha_{1}-\alpha_{2}$ | $d_{13}\wedge d_{23}\wedge d_{14}\wedge d_{15}+d_{12}\wedge d_{13}\wedge d_{34}\wedge d_{15}+d_{12}\wedge d_{13}\wedge d_{14}\wedge d_{35}$
5 | $(-3,3,0,0)=3\omega_{1}-3\alpha_{1}$ | $d_{12}\wedge d_{23}\wedge d_{24}\wedge d_{25}$
6 | $(0,0,1,0)=3\omega_{1}-2\alpha_{1}-\alpha_{2}$ | $d_{13}\wedge d_{23}\wedge d_{24}\wedge d_{15}+d_{13}\wedge d_{23}\wedge d_{14}\wedge d_{25}+d_{12}\wedge d_{23}\wedge d_{34}\wedge d_{15}+d_{12}\wedge d_{13}\wedge d_{34}\wedge d_{25}+d_{12}\wedge d_{23}\wedge d_{14}\wedge d_{35}+d_{12}\wedge d_{13}\wedge d_{24}\wedge d_{35}$
7 | $(2,0,-1,1)=3\omega_{1}-\alpha_{1}-\alpha_{2}-\alpha_{3}$ | $-d_{13}\wedge d_{14}\wedge d_{24}\wedge d_{15}+d_{12}\wedge d_{13}\wedge d_{14}\wedge d_{45}+d_{12}\wedge d_{14}\wedge d_{34}\wedge d_{15}$
8 | $(-2,1,1,0)=3\omega_{1}-3\alpha_{1}-\alpha_{2}$ | $d_{13}\wedge d_{23}\wedge d_{24}\wedge d_{25}+d_{12}\wedge d_{23}\wedge d_{34}\wedge d_{25}+d_{12}\wedge d_{23}\wedge d_{24}\wedge d_{35}$
9 | $(1,-2,2,0)=3\omega_{1}-2\alpha_{1}-2\alpha_{2}$ | $d_{13}\wedge d_{23}\wedge d_{34}\wedge d_{15}+d_{13}\wedge d_{23}\wedge d_{14}\wedge d_{35}+d_{12}\wedge d_{13}\wedge d_{34}\wedge d_{35}$
10 | $(0,1,-1,1)=3\omega_{1}-2\alpha_{1}-\alpha_{2}-\alpha_{3}$ | $-d_{23}\wedge d_{14}\wedge d_{24}\wedge d_{15}-d_{13}\wedge d_{14}\wedge d_{24}\wedge d_{25}+d_{12}\wedge d_{23}\wedge d_{14}\wedge d_{45}+d_{12}\wedge d_{13}\wedge d_{24}\wedge d_{45}+d_{12}\wedge d_{24}\wedge d_{34}\wedge d_{15}+d_{12}\wedge d_{14}\wedge d_{34}\wedge d_{25}$
11 | $(2,0,0,-1)=3\omega_{1}-\alpha_{1}-\alpha_{2}-\alpha_{3}-\alpha_{4}$ | $d_{13}\wedge d_{14}\wedge d_{15}\wedge d_{25}+d_{12}\wedge d_{13}\wedge d_{15}\wedge d_{45}-d_{12}\wedge d_{14}\wedge d_{15}\wedge d_{35}$
12 | $(-1,-1,2,0)=3\omega_{1}-3\alpha_{1}-2\alpha_{2}$ | $d_{13}\wedge d_{23}\wedge d_{34}\wedge d_{25}+d_{13}\wedge d_{23}\wedge d_{24}\wedge d_{35}+d_{12}\wedge d_{23}\wedge d_{34}\wedge d_{35}$
13 | $(-2,2,-1,1)=3\omega_{1}-3\alpha_{1}-\alpha_{2}-\alpha_{3}$ | $-d_{23}\wedge d_{14}\wedge d_{24}\wedge d_{25}+d_{12}\wedge d_{24}\wedge d_{34}\wedge d_{25}+d_{12}\wedge d_{23}\wedge d_{24}\wedge d_{45}$
14 | $(1,-1,0,1)=3\omega_{1}-2\alpha_{1}-2\alpha_{2}-\alpha_{3}$ | $-d_{23}\wedge d_{14}\wedge d_{34}\wedge d_{15}-d_{13}\wedge d_{14}\wedge d_{24}\wedge d_{35}+d_{13}\wedge d_{23}\wedge d_{14}\wedge d_{45}+d_{12}\wedge d_{13}\wedge d_{34}\wedge d_{45}+d_{13}\wedge d_{24}\wedge d_{34}\wedge d_{15}+d_{12}\wedge d_{14}\wedge d_{34}\wedge d_{35}$
15 | $(0,1,0,-1)=3\omega_{1}-2\alpha_{1}-\alpha_{2}-\alpha_{3}-\alpha_{4}$ | $d_{23}\wedge d_{14}\wedge d_{15}\wedge d_{25}+d_{13}\wedge d_{24}\wedge d_{15}\wedge d_{25}+d_{12}\wedge d_{23}\wedge d_{15}\wedge d_{45}+d_{12}\wedge d_{13}\wedge d_{25}\wedge d_{45}-d_{12}\wedge d_{24}\wedge d_{15}\wedge d_{35}-d_{12}\wedge d_{14}\wedge d_{25}\wedge d_{35}$
16 | $(-1,0,0,1)=3\omega_{1}-3\alpha_{1}-2\alpha_{2}-\alpha_{3}$ | $-d_{23}\wedge d_{14}\wedge d_{34}\wedge d_{25}-d_{23}\wedge d_{14}\wedge d_{24}\wedge d_{35}+d_{13}\wedge d_{24}\wedge d_{34}\wedge d_{25}+d_{13}\wedge d_{23}\wedge d_{24}\wedge d_{45}+d_{12}\wedge d_{24}\wedge d_{34}\wedge d_{35}+d_{12}\wedge d_{23}\wedge d_{34}\wedge d_{45}$
17 | $(-2,2,0,-1)=3\omega_{1}-3\alpha_{1}-\alpha_{2}-\alpha_{3}-\alpha_{4}$ | $d_{23}\wedge d_{24}\wedge d_{15}\wedge d_{25}-d_{12}\wedge d_{24}\wedge d_{25}\wedge d_{35}+d_{12}\wedge d_{23}\wedge d_{25}\wedge d_{45}$
18 | $(1,0,-2,2)=3\omega_{1}-2\alpha_{1}-2\alpha_{2}-2\alpha_{3}$ | $d_{14}\wedge d_{24}\wedge d_{34}\wedge d_{15}-d_{13}\wedge d_{14}\wedge d_{24}\wedge d_{45}+d_{12}\wedge d_{14}\wedge d_{34}\wedge d_{45}$
19 | $(1,-1,1,-1)=3\omega_{1}-2\alpha_{1}-2\alpha_{2}-\alpha_{3}-\alpha_{4}$ | $d_{23}\wedge d_{14}\wedge d_{15}\wedge d_{35}+d_{13}\wedge d_{34}\wedge d_{15}\wedge d_{25}+d_{13}\wedge d_{23}\wedge d_{15}\wedge d_{45}+d_{12}\wedge d_{13}\wedge d_{35}\wedge d_{45}-d_{12}\wedge d_{34}\wedge d_{15}\wedge d_{35}-d_{13}\wedge d_{14}\wedge d_{25}\wedge d_{35}$
20 | $(0,-3,3,0)=3\omega_{1}-3\alpha_{1}-3\alpha_{2}$ | $d_{13}\wedge d_{23}\wedge d_{34}\wedge d_{35}$
21 | $(0,-2,1,1)=3\omega_{1}-3\alpha_{1}-3\alpha_{2}-\alpha_{3}$ | $-d_{23}\wedge d_{14}\wedge d_{34}\wedge d_{35}+d_{13}\wedge d_{23}\wedge d_{34}\wedge d_{45}+d_{13}\wedge d_{24}\wedge d_{34}\wedge d_{35}$
22 | $(-1,1,-2,2)=3\omega_{1}-3\alpha_{1}-2\alpha_{2}-2\alpha_{3}$ | $d_{14}\wedge d_{24}\wedge d_{34}\wedge d_{25}-d_{23}\wedge d_{14}\wedge d_{24}\wedge d_{45}+d_{12}\wedge d_{24}\wedge d_{34}\wedge d_{45}$
23 | $(1,0,-1,0)=3\omega_{1}-3\alpha_{1}-2\alpha_{2}-\alpha_{3}-\alpha_{4}$ | $d_{23}\wedge d_{24}\wedge d_{15}\wedge d_{35}+d_{23}\wedge d_{34}\wedge d_{15}\wedge d_{25}+d_{13}\wedge d_{23}\wedge d_{25}\wedge d_{45}+d_{12}\wedge d_{23}\wedge d_{35}\wedge d_{45}-d_{12}\wedge d_{34}\wedge d_{25}\wedge d_{35}-d_{13}\wedge d_{24}\wedge d_{25}\wedge d_{35}$
24 | $(0,-1,-1,2)=3\omega_{1}-2\alpha_{1}-2\alpha_{2}-2\alpha_{3}-\alpha_{4}$ | $-d_{14}\wedge d_{24}\wedge d_{15}\wedge d_{35}+d_{14}\wedge d_{34}\wedge d_{15}\wedge d_{25}+d_{13}\wedge d_{24}\wedge d_{15}\wedge d_{45}+d_{12}\wedge d_{14}\wedge d_{35}\wedge d_{45}-d_{12}\wedge d_{34}\wedge d_{15}\wedge d_{45}-d_{13}\wedge d_{14}\wedge d_{25}\wedge d_{45}$
25 | $(0,-2,2,-1)=3\omega_{1}-3\alpha_{1}-3\alpha_{2}-2\alpha_{3}$ | $d_{14}\wedge d_{24}\wedge d_{34}\wedge d_{35}-d_{23}\wedge d_{14}\wedge d_{34}\wedge d_{45}+d_{13}\wedge d_{24}\wedge d_{34}\wedge d_{45}$
26 | $(0,-2,2,-1)=3\omega_{1}-3\alpha_{1}-3\alpha_{2}-\alpha_{3}-\alpha_{4}$ | $d_{23}\wedge d_{34}\wedge d_{15}\wedge d_{35}+d_{13}\wedge d_{23}\wedge d_{35}\wedge d_{45}-d_{13}\wedge d_{34}\wedge d_{25}\wedge d_{35}$
27 | $(-1,1,-1,0)=3\omega_{1}-3\alpha_{1}-2\alpha_{2}-2\alpha_{3}-\alpha_{4}$ | $d_{23}\wedge d_{24}\wedge d_{15}\wedge d_{45}+d_{24}\wedge d_{34}\wedge d_{15}\wedge d_{25}-d_{23}\wedge d_{14}\wedge d_{25}\wedge d_{45}+d_{12}\wedge d_{24}\wedge d_{35}\wedge d_{45}-d_{12}\wedge d_{34}\wedge d_{25}\wedge d_{45}-d_{14}\wedge d_{24}\wedge d_{25}\wedge d_{35}$
28 | $(1,0,0,-2)=3\omega_{1}-2\alpha_{1}-2\alpha_{2}-2\alpha_{3}-2\alpha_{4}$ | $d_{14}\wedge d_{15}\wedge d_{25}\wedge d_{35}-d_{13}\wedge d_{15}\wedge d_{25}\wedge d_{45}+d_{12}\wedge d_{15}\wedge d_{35}\wedge d_{45}$
29 | $(0,-1,0,0)=3\omega_{1}-3\alpha_{1}-3\alpha_{2}-2\alpha_{3}-\alpha_{4}$ | $d_{24}\wedge d_{34}\wedge d_{15}\wedge d_{35}-d_{14}\wedge d_{34}\wedge d_{25}\wedge d_{35}+d_{23}\wedge d_{34}\wedge d_{15}\wedge d_{45}-d_{23}\wedge d_{14}\wedge d_{35}\wedge d_{45}-d_{13}\wedge d_{34}\wedge d_{25}\wedge d_{45}+d_{13}\wedge d_{24}\wedge d_{35}\wedge d_{45}$
30 | $(-1,1,-1,0)=3\omega_{1}-3\alpha_{1}-2\alpha_{2}-2\alpha_{3}-\alpha_{4}$ | $-d_{23}\wedge d_{15}\wedge d_{25}\wedge d_{45}+d_{24}\wedge d_{15}\wedge d_{25}\wedge d_{35}+d_{12}\wedge d_{25}\wedge d_{35}\wedge d_{45}$
31 | $(0,0,-3,3)=3\omega_{1}-3\alpha_{1}-3\alpha_{2}-3\alpha_{3}$ | $d_{14}\wedge d_{24}\wedge d_{34}\wedge d_{45}$
32 | $(0,0,-2,1)=3\omega_{1}-3\alpha_{1}-3\alpha_{2}-3\alpha_{3}-\alpha_{4}$ | $d_{24}\wedge d_{34}\wedge d_{15}\wedge d_{45}-d_{14}\wedge d_{34}\wedge d_{25}\wedge d_{45}+d_{14}\wedge d_{24}\wedge d_{35}\wedge d_{45}$
33 | $(0,-1,1,-2)=3\omega_{1}-3\alpha_{1}-3\alpha_{2}-2\alpha_{3}-2\alpha_{4}$ | $-d_{23}\wedge d_{15}\wedge d_{35}\wedge d_{45}+d_{34}\wedge d_{15}\wedge d_{25}\wedge d_{35}+d_{13}\wedge d_{25}\wedge d_{35}\wedge d_{45}$
34 | $(0,0,-1,-1)=3\omega_{1}-3\alpha_{1}-3\alpha_{2}-3\alpha_{3}-2\alpha_{4}$ | $-d_{24}\wedge d_{15}\wedge d_{35}\wedge d_{45}+d_{14}\wedge d_{25}\wedge d_{35}\wedge d_{45}+d_{34}\wedge d_{15}\wedge d_{25}\wedge d_{45}$
35 | $(0,0,0,-3)=3\omega_{1}-3\alpha_{1}-3\alpha_{2}-3\alpha_{3}-3\alpha_{4}$ | $d_{15}\wedge d_{25}\wedge d_{35}\wedge d_{45}$
Table 9: Weights for $\mbox{sl}_{5}$ module $V(\omega_{1}+\omega_{2})$ $i$ | $\overrightarrow{w}_{i}^{\omega_{2}}$ | $i$ | $\overrightarrow{w}_{i}^{\omega_{2}}$
---|---|---|---
1 | $(1,1,0,0)=\omega_{1}+\omega_{2}$ | 16 | $(1,0,-2,2)=\omega_{1}+\omega_{2}-\alpha_{1}-2\alpha_{2}-2\alpha_{3}$
2 | $(-1,2,0,0)=\omega_{1}+\omega_{2}-\alpha_{1}$ | 17 | $(1,-1,1,-1)=\omega_{1}+\omega_{2}-\alpha_{1}-2\alpha_{2}-\alpha_{3}-\alpha_{4}$
3 | $(2,-1,1,0)=\omega_{1}+\omega_{2}-\alpha_{2}$ | 18 | $(0,-2,1,1)=\omega_{1}+\omega_{2}-2\alpha_{1}-3\alpha_{2}-\alpha_{3}$
4 | $(0,0,1,0)=\omega_{1}+\omega_{2}-\alpha_{1}-\alpha_{2}$ | 19 | $(-1,1,-2,2)=\omega_{1}+\omega_{2}-2\alpha_{1}-2\alpha_{2}-2\alpha_{3}$
5 | $(2,0,-1,1)=\omega_{1}+\omega_{2}-\alpha_{2}-\alpha_{3}$ | 20 | $(-1,0,1,-1)=\omega_{1}+\omega_{2}-2\alpha_{1}-2\alpha_{2}-\alpha_{3}-\alpha_{4}$
6 | $(-2,1,1,0)=\omega_{1}+\omega_{2}-2\alpha_{1}-\alpha_{2}$ | 21 | $(1,0,-1,0)=\omega_{1}+\omega_{2}-\alpha_{1}-2\alpha_{2}-2\alpha_{3}-\alpha_{4}$
7 | $(1,-2,2,0)=\omega_{1}+\omega_{2}-\alpha_{1}-2\alpha_{2}$ | 22 | $(0,-1,-1,2)=\omega_{1}+\omega_{2}-2\alpha_{1}-3\alpha_{2}-2\alpha_{3}$
8 | $(0,1,-1,1)=\omega_{1}+\omega_{2}-\alpha_{1}-\alpha_{2}-\alpha_{3}$ | 23 | $(0,-2,2,-1)=\omega_{1}+\omega_{2}-2\alpha_{1}-3\alpha_{2}-\alpha_{3}-\alpha_{4}$
9 | $(2,0,0,-1)=\omega_{1}+\omega_{2}-\alpha_{2}-\alpha_{3}-\alpha_{4}$ | 24 | $(-1,1,-1,0)=\omega_{1}+\omega_{2}-2\alpha_{1}-2\alpha_{2}-2\alpha_{3}-\alpha_{4}$
10 | $(-1,-1,2,0)=\omega_{1}+\omega_{2}-2\alpha_{1}-2\alpha_{2}$ | 25 | $(1,0,0,-2)=\omega_{1}+\omega_{2}-\alpha_{1}-2\alpha_{2}-2\alpha_{3}-2\alpha_{4}$
11 | $(-2,2,-1,1)=\omega_{1}+\omega_{2}-2\alpha_{1}-\alpha_{2}-\alpha_{3}$ | 26 | $(0,-1,0,0)=\omega_{1}+\omega_{2}-2\alpha_{1}-3\alpha_{2}-2\alpha_{3}-\alpha_{4}$
12 | $(1,-1,0,1)=\omega_{1}+\omega_{2}-\alpha_{1}-2\alpha_{2}-\alpha_{3}$ | 27 | $(-1,1,0,-2)=\omega_{1}+\omega_{2}-2\alpha_{1}-2\alpha_{2}-2\alpha_{3}-2\alpha_{4}$
13 | $(0,1,0,-1)=\omega_{1}+\omega_{2}-\alpha_{1}-\alpha_{2}-\alpha_{3}-\alpha_{4}$ | 28 | $(0,0,-2,1)=\omega_{1}+\omega_{2}-2\alpha_{1}-3\alpha_{2}-3\alpha_{3}-\alpha_{4}$
14 | $(-1,0,0,1)=\omega_{1}+\omega_{2}-2\alpha_{1}-2\alpha_{2}-\alpha_{3}$ | 29 | $(0,-1,1,-2)=\omega_{1}+\omega_{2}-2\alpha_{1}-3\alpha_{2}-2\alpha_{3}-2\alpha_{4}$
15 | $(-2,2,0,-1)=\omega_{1}+\omega_{2}-2\alpha_{1}-\alpha_{2}-\alpha_{3}-\alpha_{4}$ | 30 | $(0,0,-1,-1)=\omega_{1}+\omega_{2}-2\alpha_{1}-3\alpha_{2}-3\alpha_{3}-2\alpha_{4}$
|
arxiv-papers
| 2012-11-20T00:35:35 |
2024-09-04T02:49:38.204690
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Yufeng Zhao",
"submitter": "Zhao Yufeng",
"url": "https://arxiv.org/abs/1211.4638"
}
|
1211.4665
|
# A Decentralized Method for Joint Admission Control and Beamforming in
Coordinated Multicell Downlink
Hoi-To Wai, Wing-Kin Ma
This work was supported by a Direct Grant by the Chinese University of Hong
Kong (Project ID 2050506). Department of Electronic Engineering,
The Chinese University of Hong Kong, Shatin, N.T., Hong Kong
Email: [email protected],[email protected]
###### Abstract
In cellular networks, admission control and beamforming optimization are
intertwined problems. While beamforming optimization aims at satisfying users’
quality-of-service (QoS) requirements or improving the QoS levels, admission
control looks at how a subset of users should be selected so that the
beamforming optimization problem can yield a reasonable solution in terms of
the QoS levels provided. However, in order to simplify the design, the two
problems are usually seen as separate problems. This paper considers joint
admission control and beamforming (JACoB) under a coordinated multicell MISO
downlink scenario. We formulate JACoB as a user number maximization problem,
where selected users are guaranteed to receive the QoS levels they requested.
The formulated problem is combinatorial and hard, and we derive a convex
approximation to the problem. A merit of our convex approximation formulation
is that it can be easily decomposed for per-base-station decentralized
optimization, namely, via block coordinate decent. The efficacy of the
proposed decentralized method is demonstrated by simulation results.
###### Index Terms:
admission control, distributed optimization, downlink beamforming
## I Introduction
Coordinated beamforming (CoBF) [1, 2] is a recently studied technique to
mitigate intercell interference (ICI) in the downlink of multicell cooperative
systems. In CoBF, the neighboring BSs share the same frequency band and employ
beamforming for data transmission. The transmit beamformers at different BSs
are coordinately designed according to the channel conditions and certain
design formulation, e.g., maximum system throughput, minimum transmit power,
to name a few. Compared to fully multicell cooperative techniques such as
network MIMO [1], CoBF has an advantage that the BS cooperation overheads are
not as significant, and yet appealing performance may be achieved.
Meanwhile, admission control also plays an important role in cellular systems.
As cellular systems are usually congested, with lots of users awaiting
service, it is necessary for the BSs to decide which user is served or not.
Admission control refers to methods of selecting users. While admission
control and beamforming are commonly seen as two separate problems, they are
fundamentally dependent on each other. Recent work has demonstrated that by
considering admission control and beamforming jointly, promising system
performance can be achieved [3, 4]. However, joint admission control and
beamforming (JACoB) is a challenging problem. It has been shown that JACoB is
NP-hard even under a single-cell scenario [3]. Hence, as a compromise, one may
consider approximation approaches.
This paper describes a JACoB approach to CoBF in multicell MISO downlink. A
distinguishing part of the present work is that our proposed JACoB formulation
can be easily decomposed for decentralized implementation, and the
decentralized process is considered even more straightforward than those in
CoBF (without admission control), e.g. [2, 5]. A key idea of our approach is
to use the now popularized $\ell_{1}$ approximation method. We very recently
note that in a concurrent work [4], the authors have studied $\ell_{1}$
approximation for joint admission control and power control (i.e., no
beamforming). Nevertheless, the work [4] does not investigate the multicell
CoBF scenario and, more importantly, decentralized optimization considered
here. It is also worthwhile to mention [3], which considers JACoB under a
single cell scenario. The formulation used there is based on a mixed-integer
program formulation, and is processed by semidefinite relaxation (SDR). While
it is not difficult to see that the idea of [3] can be extended to the
multicell scenario, one needs to assume centralized optimization and presently
there is no reported work on how the method in [3] can be decentralized. In
our simulations, we will show that our decentralized method yields a
performance quite on a par with the centralized method in [3].
## II System model
Consider a cellular system with $M$ coordinating BSs. Each BS is equipped with
$N$ transmit antennas. In each cell, there are $K$ single-antenna user
terminals; thus the total number of users in the system is $KM$. The set of
users associated with the $i$th cell, or the $i$th BS, is denoted by
$\mathcal{K}_{i}\subset\mathcal{K}=\\{1,2,...,KM\\}$. We assume that
$\mathcal{K}=\mathcal{K}_{1}\cup\cdots\cup\mathcal{K}_{M}$ and
$\mathcal{K}_{i}\cap\mathcal{K}_{j}=\emptyset$ whenever $i\neq j$, i.e., each
user is served only by one BS. The scenario of interest is downlink, with an
emphasis on CoBF. Assuming that the BS-to-user channels are frequency-flat and
slow, and that the linear unicast transmit beamforming scheme is employed, we
can characterize the CoBF system performance by the received signal-to-
interference-and-noise ratios (SINRs) (for more complete system model
descriptions, see the literature, such as [2]):
${\sf
SINR}_{q}=\displaystyle\frac{|{\bm{h}}_{i(q),q}^{H}{\bm{w}}_{q}|^{2}}{\displaystyle\sigma_{q}^{2}+\sum_{m\in\mathcal{K}_{i(q)}\setminus\\{q\\}}|{\bm{h}}_{i(q),q}^{H}{\bm{w}}_{m}|^{2}+\sum_{j\neq
i(q)}\sum_{m\in\mathcal{K}_{j}}|{\bm{h}}_{j,q}^{H}{\bm{w}}_{m}|^{2}},$ (1)
where $q\in\\{1,2,\ldots,KM\\}$ is the user index, $i(q)\in\\{1,...,M\\}$
denotes the BS with which the $q$th user is associated (i.e., $i(q)$ is such
that $q\in\mathcal{K}_{i(q)}$), ${\bm{h}}_{j,q}\in\mathbb{C}^{N}$ is the
channel response from the $j$th BS to the $q$th user, $\sigma_{q}^{2}$ is the
noise variance, and ${\bm{w}}_{q}\in\mathbb{C}^{N}$ is the beamforming vector
for the $q$th user.
The difference between the SINR model in (1) and a single-cell-based SINR
model is that the former explicitly models the ICI, which is given by the
third term in the denominator of (1). On the contrary, in the single-cell
case, the ICI is usually treated as a constant, and is absorbed by
$\sigma_{q}^{2}$. The idea of CoBF is therefore to ask the BSs to coordinately
design $\\{{\bm{w}}_{q}\\}_{q=1}^{KM}$, so that ICI may be jointly mitigated.
To motivate the study of JACoB, let us quickly review a CoBF design problem,
specifically, the design proposed in [2]. In that design, the BSs jointly
design the beamforming vectors $\\{{\bm{w}}_{q}\\}_{q=1}^{KM}$ such that the
SINR of each user is no less than a user-requested threshold $\gamma_{q}$.
Moreover, the design objective aims at minimizing the total transmit power.
This amounts to the following design optimization problem:
$\begin{array}[]{rl}\displaystyle\min_{\\{{\bm{w}}_{q}\\}_{q=1}^{KM}}&\sum_{q=1}^{KM}\|{\bm{w}}_{q}\|_{2}^{2}\\\
{\rm s.t.}&{\sf SINR}_{q}\geq\gamma_{q},~{}q=1,2,...,KM,\\\
&\sum_{q\in\mathcal{K}_{i}}\|{\bm{w}}_{q}\|_{2}^{2}\leq
P_{max,i},~{}i=1,...,M,\end{array}$ (2)
where the last constraints in (2) are per-BS power budget constraints, with
$P_{max,i}$ specifying the maximum transmit power of the $i$th BS111As a minor
point to note, the previous study [2] does not incorporate the per-BS power
budget constraints., and $\|\cdot\|_{2}$ is the $\ell_{2}$ norm. At first
glance problem (2) seems to be nonconvex, since the SINR constraints are
nonconvex in $\\{{\bm{w}}_{q}\\}_{q=1}^{KM}$. Actually, problem (2) can be
solved in a convex and tractable fashion, using either the second-order cone
programming formulation or the semidefinite relaxation (SDR) formulation; see
[2] and [6] for more detail.
Cellular systems are usually congested, with lots of users awaiting service. A
subsequent issue relevant to the CoBF problem (2) is that we may be unable to
find a beamforming solution $\\{{\bm{w}}_{q}\\}_{q=1}^{KM}$ that satisfies all
the users’ SINR requests. In other words, problem (2) may be infeasible. To
illustrate this issue, we simulated the feasibility rate of problem (2)
against the total number of users $KM$. The simulation result is plotted in
Figure 1. The feasibility rate was evaluated by counting the number of
instances for which (2) is feasible, under randomly generated channels. We
observe that problem (2) has a low feasibility rate when the number of users
is large.
Figure 1: The feasibility rate of the CoBF problem (2). $M=3$, $N=8$,
$\gamma_{q}=6$ dB, and $P_{max,i}$ = 46 dBm.
## III Joint admission control and beamforming
This work considers joint admission control and beamforming (JACoB). The
problem is stated as follows:
Joint admission control and beamforming (JACoB): _Select a maximum number of
users, such that there exists a beamforming solution $\\{{\bm{w}}_{q}\\}$ that
satisfies all the selected users’ SINR requests._
In the following, we will first provide an optimization formulation for JACoB,
and derive a convex approximation to the formulated problem. Then, a
decentralized method based on the convex approximation will be developed.
### III-A Centralized method for JACoB
Our endeavor starts with formulating JACoB in a mathematically convenient
form. First of all, let
${\bm{W}}_{q}={\bm{w}}_{q}{\bm{w}}_{q}^{H},\quad q=1,\ldots,KM,$
and observe the following equivalence (cf. (1)):
${\sf SINR}_{q}\geq\gamma_{q}\Longleftrightarrow 0\geq
f_{q}(\\{{\bm{W}}_{m}\\}_{m=1}^{KM}),$ (3)
where we define
$\begin{array}[]{l}\displaystyle
f_{q}(\\{{\bm{W}}_{m}\\}_{m=1}^{KM})\triangleq 1+\sum_{j\neq i(q)}{\rm
Tr}\Big{(}{\bm{H}}_{j,q}\Big{(}\sum_{m\in\mathcal{K}_{j}}{\bm{W}}_{m}\Big{)}\Big{)}\\\
\displaystyle\hfill+{\rm
Tr}\Big{(}{\bm{H}}_{i(q),q}\Big{(}\sum_{m\in\mathcal{K}_{i(q)}\setminus\\{q\\}}{\bm{W}}_{m}-\frac{1}{\gamma_{q}}{\bm{W}}_{q}\Big{)}\Big{)},\end{array}$
and ${\bm{H}}_{j,q}\triangleq{\bm{h}}_{j,q}{\bm{h}}_{j,q}^{H}/\sigma_{q}^{2}$.
We claim that JACoB can be formulated as the following $\ell_{0}$ minimization
problem:
$\displaystyle\displaystyle\min_{{\bm{t}},\\{{\bm{W}}_{q}\\}_{q=1}^{KM}}$
$\displaystyle~{}~{}\|{\bm{t}}\|_{0}+\textstyle\epsilon\sum_{q=1}^{KM}{\rm
Tr}({\bm{W}}_{q})$ (4a) $\displaystyle{\rm s.t.}$
$\displaystyle~{}~{}\textstyle\sum_{q\in\mathcal{K}_{i}}{\rm
Tr}({\bm{W}}_{q})\leq P_{max,i},~{}i=1,...,M,$ (4b)
$\displaystyle~{}~{}{\bm{W}}_{q}\succeq{\bm{0}},~{}\forall~{}q,$ (4c)
$\displaystyle~{}~{}t_{q}=\max\\{0,f_{q}(\\{{\bm{W}}_{m}\\}_{m=1}^{KM})\\},~{}\forall~{}q,$
(4d) $\displaystyle~{}~{}{\rm rank}({\bm{W}}_{q})=1,~{}\hfill~{}\forall~{}q,$
(4e)
where $0<\epsilon<1/\sum_{i=1}^{M}P_{max,i}$ is a penalty parameter,
$\bm{W}_{q}\succeq\bm{0}$ means that $\bm{W}_{q}$ is positive semidefinite,
and $\|\bm{t}\|_{0}$ is the $\ell_{0}$ norm, which counts the number of
nonzero elements in $\bm{t}$.
Let us describe why problem (4) delivers the above defined JACoB goal.
Firstly, constraints (4c) and (4e) are equivalent to
${\bm{W}}_{q}={\bm{w}}_{q}{\bm{w}}_{q}^{H}$. Secondly, by substituting (4d)
into the first term of (4a), i.e., $\|{\bm{t}}\|_{0}$, and observing (3), we
can see that $\|{\bm{t}}\|_{0}$ is counting the number of unserved or
unadmitted users. Hence, if we ignore the second term of (4a), then problem
(4) minimizes the number of unadmitted users. Thirdly, the second term of
(4a), i.e, $\epsilon\sum_{q=1}^{KM}{\rm Tr}({\bm{W}}_{q})$, is a penalty term.
It is used to encourage more power-efficient beamforming solutions. It can be
shown that problem (4) with $\epsilon=0$ (i.e., direct unadmitted user
minimization) achieves the same number of unadmitted users as problem (4) with
$\epsilon<1/\sum_{i=1}^{M}P_{max,i}$.
Problem (4) is difficult to solve. As a remedy, we adopt a convex
approximation approach. Our approximation involves two steps. First, we
replace the hard $\ell_{0}$ norm function by the $\ell_{1}$ norm, which is now
a popularized trick in compressive sensing. Second, we remove the rank-one
constraints (4e), which is well known as SDR [6]. The above two approximations
lead us to the following $\ell_{1}$ approximate JACoB problem:
$\displaystyle\displaystyle\min_{{\bm{t}},\\{{\bm{W}}_{q}\\}_{q=1}^{KM}}$
$\displaystyle~{}~{}\textstyle\|{\bm{t}}\|_{1}+\epsilon\sum_{q=1}^{KM}{\rm
Tr}({\bm{W}}_{q})$ (5a) $\displaystyle{\rm s.t.}$
$\displaystyle~{}~{}\textstyle\sum_{q\in\mathcal{K}_{i}}{\rm
Tr}({\bm{W}}_{q})\leq P_{max,i},~{}i=1,...,M,$ (5b)
$\displaystyle\textstyle~{}~{}t_{q}\geq 0,~{}t_{q}\geq
f_{q}(\\{{\bm{W}}_{m}\\}_{m=1}^{KM}),$ (5c)
$\displaystyle~{}~{}{\bm{W}}_{q}\succeq{\bm{0}},~{}\forall~{}q,$ (5d)
where $\|\cdot\|_{1}$ is the $\ell_{1}$ norm. Note that in (5), we replace
(4d) by (5c), which can be easily verified to be equivalent.
The $\ell_{1}$ approximate JACoB problem (5) is convex. In fact, problem (5)
can be written as an SDP. Hence, for centralized implementation, we can solve
problem (5) by using a readily available SDP solver. Moreover, we show that
the second approximation, i.e., SDR, is a tight relaxation:
###### Proposition 1
For $\epsilon>0$, any optimal solution $\\{{\bm{W}}_{q}^{\star}\\}_{q=1}^{KM}$
of problem (5) must satisfy ${\rm rank}({\bm{W}}_{q}^{\star})\leq 1$ for all
$q$.
The proof of Proposition 1 is skipped here owing to the limit of space. The
idea behind the proof is to examine the KKT conditions and exploit the rank-
one structure of ${\bm{H}}_{i(q),q}$. We should also note that Proposition 1
is different from the SDR tightness results in [7], which may be seemingly
similar at first look. Simply speaking, [7] studies more general
${\bm{H}}_{j,q}$ (which may take any rank), but may not solve the problem in
Proposition 1. Proposition 1 means that solving problem (5) automatically
leads to a set of beamforming solutions (recall
$\bm{W}_{q}={\bm{w}}_{q}{\bm{w}}_{q}^{H}$ for rank-one positive semidefinite
$\bm{W}_{q}$), and there is no loss in applying SDR.
### III-B Decentralized method for JACoB
A significant advantage of the $\ell_{1}$ approximate JACoB formulation in (5)
is that it can be easily decomposed for decentralized optimization. To see
this, let
$\bm{\mathcal{W}}_{i}\triangleq\\{{\bm{W}}_{m}\\}_{m\in\mathcal{K}_{i}},~{}i=1,...,M.$
Notice that $\bm{\mathcal{W}}_{i}$ corresponds to the beamforming vectors
controlled by the $i$th BS. Now, by substituting (5c) into (5a), we can
reformulate (5) as
$\begin{array}[]{rl}\displaystyle\min_{\\{{\bm{W}}_{q}\\}_{q=1}^{KM}}&\displaystyle\sum_{q=1}^{KM}\Big{(}\max\\{0,f_{q}(\bm{\mathcal{W}}_{1},...,\bm{\mathcal{W}}_{M})\\}+\epsilon{\rm
Tr}({\bm{W}}_{q})\Big{)}\vspace{-.0cm}\\\ {\rm
s.t.}&\sum_{q\in\mathcal{K}_{i}}{\rm Tr}({\bm{W}}_{q})\leq
P_{max,i},~{}i=1,...,M,\\\
&{\bm{W}}_{q}\succeq{\bm{0}},~{}q=1,...,KM.\end{array}$ (6)
A unique feature with problem (6) is that the constraints are per-BS decoupled
(note that this is not the case with the CoBF problem in (2)). As a result, we
can directly apply per-BS alternating optimization. To be specific, we employ
block coordinate descent (BCD). In BCD, we update only one beamforming block
$\bm{\mathcal{W}}_{i}$, while holding the other blocks fixed. This BCD update
is done cyclically with respect to the BSs, until some stopping rule is
satisfied.
A curious question is whether the above-described BCD method would converge to
the optimum of problem (6). Unfortunately, this may not be guaranteed—BCD may
not converge to the optimum for problems whose objective functions are not
continuously differentiable, even if the problem is convex [8]. The function
$\max\\{0,x\\}$ seen in problem (6) exactly falls into this case. To remedy
this, we apply a smooth approximation to (6) using the one-sided Huber
function
$h(x)=\begin{cases}0&,~{}{\sf if}~{}x\leq 0,\\\ \displaystyle
0.5x^{2}&,~{}{\sf if}~{}0<x\leq 1,\\\ \displaystyle x-0.5&,~{}{\sf
if}~{}x>1.\end{cases}$
The Huber function $h(x)$ is continuously differentiable in $x$. Applying the
approximation $\max\\{0,x\\}\approx h(x)$, we obtain the Huber approximate
JACoB problem:
$\begin{array}[]{rl}\displaystyle\min_{\\{{\bm{W}}_{q}\\}_{q=1}^{KM}}&\displaystyle\sum_{q=1}^{KM}\Big{(}h(f_{q}(\bm{\mathcal{W}}_{1},...,\bm{\mathcal{W}}_{M}))+\epsilon{\rm
Tr}({\bm{W}}_{q})\Big{)}\\\ {\rm s.t.}&\sum_{q\in\mathcal{K}_{i}}{\rm
Tr}({\bm{W}}_{q})\leq P_{max,i},~{}i=1,...,M,\\\
&{\bm{W}}_{q}\succeq{\bm{0}},~{}q=1,...,KM.\end{array}$ (7)
From this point on, we will concentrate on the BCD of problem (7).
Let us consider the BCD update of problem (7) with respect to the $i$th block
$\bm{\mathcal{W}}_{i}$, holding the other blocks
$\\{\hat{\bm{\mathcal{W}}}_{j}\\}_{j\neq
i}\triangleq\\{\hat{\bm{W}}_{m}\\}_{m\notin\mathcal{K}_{i}}$ fixed. The
respective problem is
$\begin{array}[]{rl}\displaystyle\min_{\\{{\bm{W}}_{m}\\}_{m\in\mathcal{K}_{i}}}&\displaystyle\sum_{q=1}^{KM}h(f_{q}(\bm{\mathcal{W}}_{i},\\{\hat{\bm{\mathcal{W}}_{j}}\\}_{j\neq
i}))+\sum_{m\in\mathcal{K}_{i}}\epsilon{\rm Tr}({\bm{W}}_{m})\vspace{.2cm}\\\
{\rm s.t.}&\sum_{m\in\mathcal{K}_{i}}{\rm Tr}({\bm{W}}_{m})\leq
P_{max,i},~{}{\bm{W}}_{m}\succeq{\bm{0}},~{}m\in\mathcal{K}_{i},\\\
\end{array}$
which can be expressed as a convex problem (see, e.g., [9]):
$\displaystyle\displaystyle\min$
$\displaystyle~{}~{}\displaystyle\sum_{q=1}^{KM}\left(\frac{1}{2}u_{q}^{2}+v_{q}\right)+\sum_{m\in\mathcal{K}_{i}}\epsilon{\rm
Tr}({\bm{W}}_{m})\vspace{.2cm}$ (8a) $\displaystyle{\rm s.t.}$
$\displaystyle~{}~{}u_{q}+v_{q}\geq
f_{q}(\bm{\mathcal{W}}_{i},\\{\hat{\bm{\mathcal{W}}}_{j}\\}_{j\neq
i}),~{}\forall~{}q,$ (8b) $\displaystyle~{}~{}u_{q},~{}v_{q}\geq
0,~{}\forall~{}q,$ (8c) $\displaystyle\sum_{m\in\mathcal{K}_{i}}{\rm
Tr}({\bm{W}}_{m})\leq
P_{max,i},~{}{\bm{W}}_{m}\succeq{\bm{0}},~{}m\in\mathcal{K}_{i}.$ (8d)
Constraints (8b) seems to indicate that full knowledge of
$\\{\hat{\bm{\mathcal{W}}}_{j}\\}_{j\neq i}$ is required, in order to solve
the BCD update (8). Actually, this may be not necessary. Notice that for
$q\in\mathcal{K}_{i}$, constraint (8b) can be expressed as:
$u_{q}+v_{q}\geq 1+\sum_{j\neq i}\hat{\Omega}_{j,q}+{\rm
Tr}\Big{(}{\bm{H}}_{i,q}\Big{(}\sum_{m\in\mathcal{K}_{i}\setminus\\{q\\}}{\bm{W}}_{m}-\frac{{\bm{W}}_{q}}{\gamma_{q}}\Big{)}\Big{)},$
and for $q\notin\mathcal{K}_{i}$,
$u_{q}+v_{q}\geq 1+\sum_{j\neq i}\hat{\Omega}_{j,q}+{\rm
Tr}\Big{(}{\bm{H}}_{i,q}\Big{(}\sum_{m\in\mathcal{K}_{i}}{\bm{W}}_{m}\Big{)}\Big{)},$
where $\hat{\Omega}_{j,q}$ are scalar constants defined as:
$\hat{\Omega}_{j,q}=\begin{cases}\displaystyle{\rm
Tr}\Big{(}{\bm{H}}_{j,q}\Big{(}\sum_{m\in\mathcal{K}_{j}\setminus\\{q\\}}\hat{\bm{W}}_{m}-\frac{\hat{\bm{W}}_{q}}{\gamma_{q}}\Big{)}\Big{)}&,~{}q\in\mathcal{K}_{j},\\\
\displaystyle{\rm
Tr}\Big{(}{\bm{H}}_{j,q}\Big{(}\sum_{m\in\mathcal{K}_{j}}\hat{\bm{W}}_{m}\Big{)}\Big{)}&,~{}q\notin\mathcal{K}_{j}.\end{cases}$
Hence, if the $i$th BS knows i) the matrices
$\\{{\bm{H}}_{i,q}\\}_{q=1}^{KM}$, i.e., the channel response from the $i$th
BS to the users in the system; and ii) the scalar constants
$\\{\hat{\Omega}_{j,q}\\}_{j\neq i,q\in\mathcal{K}}$, then problem (8) can be
solved independently at the $i$th BS. In fact, the first premise can be
satisfied automatically as it can be assumed that each BS knows the channel
response from itself to the users in the system [2]. To satisfy the second
premise, we can utilize the backhaul link between the BSs. Specifically, the
scalars $\\{\hat{\Omega}_{i,q}\\}_{q\in\mathcal{K}}$ can be computed and
broadcast to the other BSs after the $i$th BCD update is solved. There are
$KM$ real numbers to be broadcast at each iteration. This justifies our claim
that the $i$th BS can solve (8) alone. The BCD method for (7) is summarized in
Algorithm 1.
Recall that the reason for employing the Huber function in (7) is to provide a
smooth approximation to JACoB, avoiding the original nondifferentiable
objective function which may result in BCD non-convergence problems. But can
the smooth approximation guarantee convergence to the optimum? By invoking an
available BCD convergence analysis result [10], we have the following claim:
###### Fact 1
[10] The sequence $\\{\\{{\bm{W}}_{q}^{(k)}\\}_{q=1}^{KM}\\}_{k}$ generated by
Algorithm 1 has limit points and every limit point of the sequence
$\\{\\{{\bm{W}}_{q}^{(k)}\\}_{q=1}^{KM}\\}_{k}$ is an optimal solution to (7).
Readers are referred to Proposition 6 in [10] for more detail. We should note
that the important premises for us to use this available result are that the
objective function of (7) is convex, continuously differentiable in
$\\{\bm{\mathcal{W}}_{i}\\}_{i=1}^{M}$ and the constraint set for each
$\bm{\mathcal{W}}_{i}$ is convex and compact. Furthermore, by extending
Proposition 1, we can prove that the intermediate solutions
$\\{{\bm{W}}_{q}^{(k)}\\}_{q\in\mathcal{K}_{i}}$ in Algorithm 1 at the $i$th
BCD update are always of rank-one:
###### Proposition 2
For $\epsilon>0$ and for each $i$, any optimal solution
$\\{{\bm{W}}_{q}^{(k)}\\}_{q\in\mathcal{K}_{i}}$ of the $i$th BCD update in
Algorithm 1 must satisfy ${\rm rank}({\bm{W}}_{q}^{(k)})\leq 1$ for all $q$ in
$\mathcal{K}_{i}$.
Algorithm 1 Block coordinate descent method for (7)
0: initialization - $\\{{\bm{W}}_{q}^{(0)}\\}_{q\in\mathcal{K}}$.
1: $k=1$;
2: For each $i=1,2,...,M$, the $i$th BS computes
$\\{\hat{\Omega}_{i,q}\\}_{q\in\mathcal{K}}$ and broadcasts them to the other
BSs;
3: repeat
4: for $i=1$ to $M$ do
5: The $i$th BS solves (8) given $\\{\hat{\Omega}_{j,q}\\}_{j\neq
i,q\in\mathcal{K}}$ to obtain
$\\{{\bm{W}}_{q}^{(k)}\\}_{q\in\mathcal{K}_{i}}$. The scalars
$\\{\hat{\Omega}_{i,q}\\}_{q\in\mathcal{K}}$ are computed and broadcast to the
other BSs;
6: end for
7: $k=k+1$;
8: until convergence.
9: return an optimal solution to (7) -
$\\{{\bm{W}}_{q}^{(k)}\\}_{q\in\mathcal{K}}$.
## IV Deflation heuristic
Both $\ell_{1}$ and Huber approximate JACoB problems (cf. problems (5) and
(7), respectively) can be seen as some kind of “soft decision” formulations
for handling admission control. In order to select more users for service, we
can apply a hard decision using the deflation heuristic. Similar to [3, 4],
the heuristic is initialized by considering all users in the system, then the
users are dropped one-by-one. At first the BSs solve (5) or (7) either
centrally or using the BCD method. Our user dropping rule is based on the
value of $t_{q}^{\star}\triangleq\max\\{0,f_{q}(\cdot)\\}$ which relates
directly to the satisfiability of the SINR threshold for user $q$. The user
with the largest $t_{q}^{\star}$ will be dropped.
When the number of users in the system is too large, we may encounter cases
where the optimal solution to (5) or (7) is trivial, i.e.,
${\bm{W}}_{q}={\bm{0}}$ for all $q$. Here we state an easy-to-check condition
for identifying such cases.
###### Fact 2
(Prescreening condition)222A similar condition has been discovered recently in
[4] for the joint power and admission control problem. Our results applies to
the case with CoBF. If
$\bm{\Phi}_{q}(\\{{\bm{H}}_{i(q),m}\\}_{m\in\mathcal{K}}))\succeq{\bm{0}},~{}\forall~{}q\in\mathcal{K},$
(9)
then solving (5) or (7) gives a trivial solution, i.e.,
${\bm{W}}_{q}={\bm{0}}$ for all $q$ in $\mathcal{K}$, where
$\bm{\Phi}_{q}(\\{{\bm{H}}_{i(q),m}\\}_{m\in\mathcal{K}}))\triangleq\epsilon{\bm{I}}+\sum_{m\neq
q}{\bm{H}}_{i(q),m}-\frac{1}{\gamma_{q}}{\bm{H}}_{i(q),q}.$
The proof is omitted due to space limitation. Inspired by fact 2, we now adopt
a _prescreening procedure_ where we drop the users gradually until condition
(9) gets violated. Specifically, at each time, we remove user $q$ with
$\bm{\Phi}_{q}(\cdot)$ that gives the largest minimum eigenvalue As condition
(9) can be checked in closed-form, the prescreening procedure can be run at a
low complexity. The deflation heuristic, together with prescreening, are
summarized as follows333Note that both the deflation heuristic and
prescreening procedure can be operated in a decentralized manner.:
Deflation heuristic: $\bullet$ Initialize: a set of users requesting service -
$\mathcal{K}=\\{1,2,...,KM\\}$. 1. (Prescreening) Check condition (9). If it
holds, then remove user $m$ from $\mathcal{K}$ according to
$m=\arg\max_{q}\lambda_{min}(\bm{\Phi}_{q}(\cdot))$ and repeat 1). Otherwise,
go to 2). 2. (Deflation) Solve (5) or (7) for
$\\{t_{q}^{\star},{\bm{W}}_{q}^{\star}\\}_{q\in\mathcal{K}}$. If
$t_{q}^{\star}=0$ for all $q\in\mathcal{K}$, terminate. Otherwise remove user
$m$ from $\mathcal{K}$ according to $m=\arg\max_{q}t_{q}^{\star}$, repeat 2).
$\bullet$ Return: a set of selected users $\mathcal{K}$ and rank-one matrices
$\\{{\bm{W}}_{q}^{\star}\\}_{q\in\mathcal{K}}$ that decomposes into
beamforming vectors satisfying the SINR requirements.
## V Numerical results
This section presents numerical results for the proposed JACoB methods. The
simulation environment is similar to [2]. We focus on a multicell scenario
with 3 coordinating BSs where each BS is separated from the others by 2.8 km.
For each simulation trial, the users’ positions and their respective channels
are randomly generated. The users are separated from their respective BS by at
least 0.7 km and are assigned to the nearest BS. The channel is assumed to
experience both small-scale and large-scale fading. The noise variance
$\sigma_{q}^{2}$ is -92 dBm and the receive antenna gain is 5 dBi. There are 8
transmit antennas and 15 users are assigned to each BS (i.e., 45 users
awaiting service in total).
Here, the centralized method and the decentralized method refer to the
deflation heuristic which uses a centralized solver for (5) and the
decentralized BCD method for (7), respectively. The scalar $\epsilon$ is
chosen as $10^{-5}$ and the BCD method terminates when the relative change in
objective value is less than $10^{-2}$.
Figure 2: Total no. of users admitted, $M=3$, $N=8$, $K=15$.
Figure 2 shows the performance of different JACoB methods in terms of the
total number of users selected for service. Here, two benchmarking methods are
compared. The “SDR deflation method” is adopted from [3] and modified to
operate in the multicell scenario. Note that it is a centralized method which
may not be decomposed straightforwardly. The “single cell deflation method” is
a modified algorithm from [11], which is originally proposed as a suboptimal
alternative to the CoBF problem (2). The main feature of [11] is that the ICI
levels are always constrained below a fixed threshold, and therefore the
beamforming design can be done independently at each BS. The subsequent
single-cell beamforming design is processed by our JACoB method. Note that the
resultant method can be implemented in a per-BS decentralized manner _without_
any BS coordination.
Turning back to Figure 2, we compare the performance of [3] with the proposed
centralized method. We observe that there are some performance gains with the
proposed method in the high SINR regime. Furthermore, the proposed
decentralized method achieves a performance on a par with its centralized
counterpart. The decentralized method should also be compared to the single
cell deflation method, where the numerical results has clearly demonstrated
the benefits of allowing BSs coordination.
Table I demonstrates the efficacies of the proposed decentralized BCD method
with prescreening procedure in terms of the total number of iterations. The
total number of iterations is defined as the total number of BCD iterations
consumed _throughout_ the deflation heuristic, where multiple instances of (7)
are solved. Note that the load on the backhaul link is directly proportional
to the iteration count. The iteration counts reported in Table I confirms that
significant reduction in the number of iterations can be achieved _with_ the
prescreening procedure.
## VI Conclusion
The contributions of this paper are twofold. First, we have developed a
formulation of joint admission control and beamforming (JACoB) for coordinated
multicell downlink, wherein an efficient convex approach is proposed. Second,
we have built a decentralized JACoB method via a simple BCD procedure.
Simulation results have shown that the decentralized method achieves a
performance on a par with the centralized method with fast convergence.
TABLE I: Total no. of iterations, $M=3,N=8,K=15$. Threshold ($\gamma$) | With prescreening | Without prescreening
---|---|---
12 dB | 53.080 | 61.790
20 dB | 31.030 | 59.290
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* [10] L. Grippo and M. Sciandrone, “On the convergence of the block nonlinear gauss-seidel method under convex constraints,” _Operations Research Letters_ , vol. 26, no. 3, pp. 127 – 136, 2000.
* [11] H. Huh, H. C. Papadopoulos, and G. Caire, “Multiuser MISO transmitter optimization for intercell interference mitigation,” _IEEE Trans. Signal Process._ , vol. 58, no. 8, pp. 4272–4285, Aug. 2010.
|
arxiv-papers
| 2012-11-20T05:17:44 |
2024-09-04T02:49:38.222212
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Hoi-To Wai and Wing-Kin Ma",
"submitter": "Hoi To Wai",
"url": "https://arxiv.org/abs/1211.4665"
}
|
1211.4670
|
# DOUBLY NONNEGATIVE RELAXATION METHOD FOR SOLVING MULTIPLE OBJECTIVE
QUADRATIC PROGRAMMING PROBLEMS
[email protected] [email protected]
###### Abstract.
Multicriterion optimization and Pareto optimality are fundamental tools in
economics. In this paper we propose a new relaxation method for solving
multiple objective quadratic programming problems. Exploiting the technique of
the linear weighted sum method, we reformulate the original multiple objective
quadratic programming problems into a single objective one. Since such single
objective quadratic programming problem is still nonconvex and NP-hard in
general. By using the techniques of lifting and doubly nonnegative relaxation,
respectively, this single objective quadratic programming problem is
transformed to a computable convex doubly nonnegative programming problem. The
optimal solutions of this computable convex problem are (weakly) Pareto
optimal solutions of the original problem under some mild conditions.
Moreover, the proposed method is tested with two examples and a practical
portfolio selection problem. The test problems are solved by CVX package which
is a solver for convex optimization. The numerical results show that the
proposed method is effective and promising.
###### Key words and phrases:
Multiple objective programming, quadratic programming, linear weighted sum
method, copositive programming, completely positive programming
###### 1991 Mathematics Subject Classification:
Primary: 90C29, 90C26; Secondary: 49M20
The research is supported by National Natural Science Foundation of China (No.
11071158) and the Key Disciplines of Shanghai Municipality (No. S30104).
yanqin bai
Department of Mathematics, Shanghai University
Shanghai 200444, China
chuanhao guo
Department of Mathematics, Shanghai University
Shanghai 200444, China
(Communicated by the associate editor name)
## 1\. Introduction
We consider multiple objective nonconvex quadratic programming problems as
follows
$\rm{(MOQP)}\begin{array}[]{llll}&\min&F(x)=(F_{1}(x),F_{2}(x),\ldots,F_{p}(x))^{T}&\\\
&{\rm s.t.}&Ax=b,\\\ &&x\geq 0,&\\\ \end{array}$
where $F_{i}(x)=x^{T}Q_{i}x+2c_{i}^{T}x$ and $x\in R^{n}$ is the decision
variable. $Q_{i}\in R^{n\times n},\ c_{i}\in R^{n},\ i=1,\ldots,p$,
$A=(a_{1},a_{2},\ldots,a_{m})^{T}\in R^{m\times n}$ and
$b=(b_{1},b_{2},\ldots,b_{m})\in R^{m}$ are given data. Without loss of
generality, $Q_{i}$ is symmetric and not positive semidefinite by assumption.
Multi-objective programming (MOP) also known as multi-criteria optimization,
is the process of simultaneously optimizing two or more conflicting objectives
subject to certain constraints. (MOP) problems are found in many fields, such
as facility location and optimal detector design [5], image processing [8].
Problem (MOQP) is a subclass of (MOP) problem and arises in portfolio
selection [16], reservoir optimal operation [18] and so on. Problem (MOQP)
also can be viewed as an extension of multiple objective quadratic-linear
programming (MOQLP) problem for which the objectives are a quadratic and
several linear functions and the constraints are linear functions which were
studied in [13, 16].
Problem (MOP) does not have a single solution that simultaneously minimizes
each objective function. A tentative solution is called Pareto optimal if it
impossible to make one objective function better off without necessarily
making the others worse off. And problem (MOP) may have many Pareto optimal
solutions. For solving problem (MOP), the linear weighted sum method is one of
the most widely used methods. The main idea is to choose the weighting
coefficients corresponding to objective functions. Then, problem (MOP) can be
transformed to a single objective one, and the Pareto optimal solutions for
problem (MOP) could be found by solving this single objective problem with the
appropriate weights. Ammar [1, 2] investigates problem (MOQP) with fuzzy
random coefficient matrices. Under the assumption that the coefficient
matrices in objectives are positive semidefinite, some results are discussed
to deduce Pareto optimal solutions for fuzzy problem (MOQP). However, many
practical problems of this class of problems are nonconvex in general. So,
these two methods have certain limitations in practical applications.
Burer [6] proves that a large class of NP-hard nonconvex quadratic program
with a mix of binary and continuous variables can be modeled as so called
completely positive programs (CPP), i.e., the minimization of a linear
function over the convex cone of completely positive matrices subject to
linear constraints (For more details and developments of this technique, one
may refer to [4, 6, 7, 14]). In order to solve such convex programs
efficiently, a computable relaxed problem is obtained by approximation the
completely positive matrices with doubly nonnegative matrices, resulting in a
doubly nonnegative programming [7], which can be efficiently solved by some
popular packages.
Motivated by the ideas of [6, 7], we propose a new relaxation method for
solving problem (MOQP) by combining with the linear weighted sum method. First
of all, in virtue of the linear weighted sum method, we first transform
problem (MOQP) into a single objective quadratic programming (SOQP) problem
over a linearly constrained subset of the cone of nonnegative orthant. Since
problem (SOQP) is a nonconvex in general, which is equivalently reformulated
as a completely positive programming problem, which is NP-hard. Furthermore, a
computable relaxed convex problem for this completely positive programming
problem is derived by using doubly nonnegative relaxation technique, and
resulting in a doubly nonnegative programming (DNNP) problem. Based on the
characteristics of optimal solutions of problem (DNNP), a sufficient condition
for (weakly) Pareto optimal solutions for problem (MOQP) is proposed.
Moreover, the proposed method is tested with two examples and a practical
portfolio selection problem. The test problems are solved by CVX package,
which is a solver for convex optimization. The numerical results show that the
proposed method is effective and promising.
The paper is organized as follows. In Section 2, we recall some basic
definitions and preliminaries for (weakly) Pareto optimal solution and the
linear weighted sum method, respectively. In Section 3, problem (MOQP) is
transformed into problem (SOQP) by using the linear weighted sum method. And
some optimality conditions for problem (MOQP) are established. In order to
solve problem (SOQP) effectively, problem (SOQP) is equivalently reformulated
as a convex problem (CP) which is further relaxed to a computable (DNNP)
problem in Section 4. In Section 5, numerical results are given to show the
performance of the proposed method. Some conclusions and remarks are given in
Section 6.
### 1.1. Notation and terminology
Let $\Omega=\\{x\in R^{n}:Ax=b,x\geq 0\\}$ be the feasible set of problem
(MOQP). Let $R^{n}_{+}$ (or $R^{n}_{++}$) denotes the cone of nonnegative (or
positive) vectors with dimension $n$, $S_{n}$ the cone of all $n\times n$
symmetric matrices, $S_{n}^{+}$ the cone of all $n\times n$ symmetric positive
semidefinite matrices and $(S_{n})^{+}$ the cone of all $n\times n$ symmetric
matrices with nonnegative elements. $C_{n}^{*}$ is the cone of all $n\times n$
completely positive matrices, i.e.,
$C_{n}^{*}=\\{X\in R^{{n}\times{n}}:\ X=\sum\limits_{k\in
K}z^{k}(z^{k})^{T}\\}\cup\\{0\\},$
where $\\{z^{k}\\}_{k\in K}\subset R^{n}_{+}\backslash\\{0\\}$. For two
vectors $x,\ y\in R^{n}$, $x\circ y$ is a vector in $R^{n}$ with $x_{i}y_{i}$
is its $i$-th component. For a matrix $M$, ${\rm{diag}(M)}$ is a column vector
whose elements are the diagonal elements of $M$. Given two conformal matrices
$A$ and $B$, $A\bullet B={\rm trace}(A^{T}B)$. For a given optimization
problem $(\ast)$, its optimal objective value is denoted by $\rm{Opt(\ast)}$.
## 2\. Preliminaries
### 2.1. Pareto optimal solutions
In multi-objective optimization with conflicting objectives, there is no
unique optimal solution. A simple optimal solution may exist here only when
the objectives are non-conflicting. For conflicting objectives one may at best
obtain what is called Pareto optimal solutions. For the sake of completeness,
we restate the definitions of some types of Pareto optimal solutions and ideal
point from [12].
###### Definition 2.1.
A solution $x^{*}$ is said to be Pareto optimal solution of problem (MOQP) if
and only if there does not exist another feasible solution $x\in\Omega$ such
that $F_{i}(x)\leq F_{i}(x^{*})$ for all $i=1,\ldots,p$ and
$F_{i}(x)<F_{i}(x^{*})$ for at least one index $i$.
All Pareto optimal points lie on the boundary of the feasible region $\Omega$.
Often, algorithms provide solutions that may not be Pareto optimal but may
satisfy other criteria, making them significant for practical applications.
For instance, weakly Pareto optimal is defined as follows.
###### Definition 2.2.
A solution $x^{*}$ is said to be weakly Pareto optimal solution of problem
(MOQP) if and only if there does not exist another feasible solution
$x\in\Omega$ such that $F_{i}(x)<F_{i}(x^{*})$ for all $i=1,\ldots,p$.
###### Remark 1.
A solution is weakly Pareto optimal if there is no other point that improves
all of the objective functions simultaneously. In contrast, a point is Pareto
optimal if there is no other point that improves at least one objective
function without detriment to another function. It is obvious that each Pareto
optimal point is weakly Pareto optimal, but weakly Pareto optimal point is not
Pareto optimal.
In order to illustrate (weakly) Pareto optimal solution intuitively, two
examples are given as follows.
###### Example 1.
Let $\Omega=[0,3]$, and
$F_{1}(x)=\frac{x^{2}}{2}+1,\
F_{2}(x)=\left\\{\begin{array}[]{llll}-x+2,&x\in[0,1],\\\ 1,&x\in(1,2],\\\
x-1,&x\in(2,3].\end{array}\right.$
Figure 1. Pareto optimal solution set [0, 1] and weakly Pareto optimal
solution set [0, 2]
The design space for this problem is shown in Figure 1. According to the above
two definitions about (weakly) Pareto optimal solution, we can easily get
Pareto optimal solution set for this problem is $[0,1]$, and weakly Pareto
optimal solution set is $[0,2]$. Note that each Pareto optimal solution is
weakly Pareto optimal solution for this problem, since $[0,1]\subseteq[0,2]$,
this also shows the conclusion holds in Remark 1.
###### Example 2.
Let $\Omega=[0,4]$, and
$F_{1}(x)=(x-1)^{2}+2,\ F_{2}(x)=(x-2)^{2}+1,\ F_{3}(x)=\|x\|.$
Figure 2. Pareto optimal solution set [0, 2] and weakly Pareto optimal
solution set [0, 2]
For this problem, the graphs for each objective function is plotted in Figure
2. By Definitions 2.1 and 2.2, it holds that Pareto optimal solution and
weakly Pareto optimal solution for this problem are the same, they all equal
to $[0,2]$.
### 2.2. The linear weighted sum method
One useful way of getting the efficiency of problem (MOQP) is to build a
utility function [17] according to the decision makers provided preference
information, such that each solution gained by this method is Pareto optimal
solution of problem (MOQP).
The linear weighted function
$u(F(\cdot)):=\sum\limits_{i=1}^{p}\lambda_{i}F_{i}(\cdot)$ (1)
is one of the most widely used utility function, where weight $\lambda_{i}$
corresponding to objective functions $F_{i}(\cdot),\ i=1,\ldots,p$ satisfy the
following conditions
$\sum\limits_{i=1}^{p}\lambda_{i}=1,\ \lambda_{i}\geq 0,\ i=1,\ldots,p,$ (2)
which is provided by the decision makers, and weights $\lambda_{i}$ imply that
the relative importance for $F_{i}(\cdot)$ in the heart of the decision
makers.
## 3\. Optimality conditions
In this section, we derive a single objective quadratic programming (SOQP)
problem corresponding to problem (MOQP) by the linear weighted sum method.
And, some optimality conditions for problem (MOQP) are proposed based on the
optimal solutions for problem (SOQP).
From (1) and (2), problem (MOQP) can be convert to a single objective
quadratic programming problem as follows
$\rm{(SOQP)}\begin{array}[]{llll}&\min&\sum\limits_{i=1}^{p}\lambda_{i}(x^{T}Q_{i}x+2c_{i}^{T}x)&\\\
&{\rm s.t.}&Ax=b,\\\ &&x\geq 0.&\\\ \end{array}$
Similar to Corollary 3.1 in [17], we can get the following theorem, its proof
can be found in [17].
###### Theorem 3.1.
Let $x^{*}$ be an optimal solution for problem (SOQP), it follows that $x^{*}$
is a Pareto optimal solution (weakly Pareto optimal solution) for problem
(MOQP) if weight $\lambda>0$ ($\lambda\geq 0$, $\lambda\neq 0$).
According to the above Theorem 3.1, varying weight $\lambda$ consistently and
continuously can result in a subset of Pareto optimal (weakly Pareto optimal)
set for problem (MOQP). The following theorem presents a sufficient optimality
condition for weakly Pareto solution of problem (MOQP).
Firstly, we quote the following definition which will be used in the sequel.
###### Definition 3.2.
Let $\bar{x}\in\Omega$ and $Y=F(\Omega)$, if there exists a convex set $H$
such that
$Y\subseteq H,\ {\rm{and}}\ H\cap(F(\bar{x})-R^{p}_{++})=\emptyset,$
then we say that $Y$ satisfies the convex inclusion condition at $F(\bar{x})$.
Based on the above convex inclusion condition, we have the following theorem.
The proof is omitted here for the reason that it is similar to the one of
Theorem 3.6 in [17].
###### Theorem 3.3.
Let $x^{*}$ be a weakly Pareto optimal solution for problem (MOQP), and $Y$
satisfies the convex inclusion condition at $F(x^{*})$. Then there exists a
weight $\bar{\lambda}\geq 0,\ \bar{\lambda}\neq 0$ such that $x^{*}$ is an
optimal solution for problem (SOQP) with $\bar{\lambda}$.
In particular, if problem (MOQP) is convex, Theorem 3.3 still holds without
the convex inclusion condition.
## 4\. Reformulation
Note that problem (SOQP) is a nonconvex quadratic programming problem in
general, and thus it is NP-hard. If problem (SOQP) is convex with appropriate
coefficient $\lambda$, we may use some popular convex packages to solve it
directly. In the following section, we will establish the computable convex
reformulation for problem (SOQP) when it is nonconvex, and the details are as
follows.
### 4.1. Completely positive reformulation
Motivated by the ideas in [6], problem (SOQP) can be reformulated as a
completely positive programming problem. First, the definition of completely
positive [3] is given as follows.
###### Definition 4.1.
A symmetric matrix $B$ of order $n$ is called completely positive if one can
find an integer $m$ and a matrix $V$ of size $n\times m$ with nonnegative
entries such that $B=VV^{T}$, where the smallest possible number $m$ is called
the CP-rank of $B$.
Based on above definition for completely positive, by using the techniques in
[6], problem (SOQP) can be reformulated as the following completely positive
programming problem
$\rm{(CP)}\begin{array}[]{llll}&\min&\sum\limits_{i=1}^{p}\lambda_{i}(Q_{i}\bullet
X+2c_{i}^{T}x)\\\ &{\rm s.t.}&Ax=b,\\\ &&{\rm{diag}}(AXA^{T})=b\circ b,\\\
&&\left(\begin{array}[]{lll}1&x^{T}\\\ x&X\end{array}\right)\in
C_{1+n}^{*},\end{array}$
which is a convex programming problem. Similar to Theorem 2.6 in [6], the
following theorem holds immediately, for more details can be seen in [6].
###### Theorem 4.2.
$\it{Opt}(SOQP)=\it{Opt}(CP)$, and if $(x^{*},X^{*})$ is an optimal solution
for problem (CP), then $x^{*}$ is in the convex hull of optimal solutions of
problem (SOQP).
According to Theorem 4.2, problem (SOQP) is equivalent to problem (CP).
However, problem (CP) is NP-hard, since there is a cone $C^{*}_{1+n}$
constraint, and check whether or not a given matrix belong to $C^{*}_{1+n}$ is
shown to be NP-hard [10], one must relax it in practice. Relaxing problem (CP)
in a natural way yields a doubly nonnegative programming (DNNP) problem.
### 4.2. Doubly nonnegative relaxation
As mentioned above, in order to establish the doubly nonnegative relaxation
for problem (CP), the definition of doubly nonnegative is given as follows.
###### Definition 4.3.
If matrix $Z\in S_{n}$ is not only nonnegative but also positive semidefinite,
then $Z$ is called doubly nonnegative.
Note that if $Z\in C^{*}_{n}$, it necessarily holds that $Z$ is doubly
nonnegative from the above Definitions 4.1 and 4.3. Moreover, the convex cone
$S_{n}^{+}$ is self-dual, and so is the convex cone $(S_{n})^{+}$. Hence,
Diananda’s decomposition theorem [9] can be reformulated as follows.
###### Theorem 4.4.
For all $n$, we have $C_{n}^{*}\subseteq S_{n}^{+}\cap(S_{n})^{+}$. The
relationship $"\supseteq"$ for two sets holds if and only if $n\leq 4$.
Regardless of the dimension $n$, one always has the inclusion
$C_{n}^{*}\subseteq S_{n}^{+}\cap(S_{n})^{+}$. Of course, in dimension $n\geq
5$ there are matrices which are doubly nonnegative but not completely
positive. The counterexample
$M=\left[\begin{array}[]{lllllll}1&1&0&0&1\\\ 1&2&1&0&0\\\ 0&1&2&1&0\\\
0&0&1&2&1\\\ 1&0&0&1&6\end{array}\right]$
proposed by Diananda [9] to illustrates this point.
Replacing $C^{*}_{1+n}$ by $S_{1+n}^{+}\cap(S_{1+n})^{+}$ according to Theorem
4.4, problem (CP) is relaxed to the following doubly nonnegative programming
problem
$\rm{(DNNP)}\begin{array}[]{llll}&\min&\sum\limits_{i=1}^{p}\lambda_{i}(Q_{i}\bullet
X+2c_{i}^{T}x)\\\ &{\rm s.t.}&Ax=b,\\\ &&{\rm{diag}}(AXA^{T})=b\circ b,\\\
&&\left(\begin{array}[]{lll}1&x^{T}\\\ x&X\end{array}\right)\in
S_{1+n}^{+}\cap(S_{1+n})^{+},\end{array}$
which is not only a convex problem but also can be solved in polynomial time
to any fixed precision from the theory of interior-point methods.
Up to now, problem (MOQP) is reformulated as above problem (DNNP), which can
be solved by some popular package CVX. It is obviously that problem (DNNP) is
a relaxation form for problem (MOQP).
In the last of this section, we will investigate the relationship between
optimal solutions for problems (MOQP) and (DNNP), i.e., a sufficient condition
for (weakly) Pareto optimal solutions of problem (MOQP) based on the
characteristics of optimal solutions for problem (DNNP) is established in the
following part.
###### Theorem 4.5.
Let $(x^{*},X^{*})$ be an optimal solution for problem (DNNP). If the
relationship $X^{*}=x^{*}(x^{*})^{T}$ holds, then
${\it{Opt(DNNP)}}={\it{Opt(CP)}}.$ Moreover, $(x^{*},X^{*})$ is an optimal
solution for problem (CP).
###### Proof.
On one hand, from Theorem 4.4, it is obviously holds that
${\rm{Opt}(CP)}\geq{\rm{Opt}(DNNP)}.$ (3)
On the other hand, from $X^{*}=x^{*}(x^{*})^{T}$, and constraints of problem
(CP), we have $(x^{*},X^{*})$ is also a feasible solution for problem (CP).
Since problems (DNNP) and (CP) have the same objective function, it follows
that
${\rm{Opt(CP)}}\leq\sum\limits_{i=1}^{p}\lambda_{i}(Q_{i}\bullet
X^{*}+2c_{i}^{T}x^{*})={\rm{Opt(DNNP)}}.$ (4)
Thus, combining (3) and (4), we have
${\rm{Opt(CP)}}={\rm{Opt(DNNP)}}.$
Again from problems (CP) and (DNNP) have the same objective function, it holds
that $(x^{*},X^{*})$ is an optimal solution for problem (CP). ∎
###### Remark 2.
It holds that ${\it{Opt(SOQP)}}={\it{Opt(CP)}}$ from Theorem 4.2. Let
$(x^{*},X^{*})$ be an optimal solution for problem (DNNP), if
$X^{*}=x^{*}(x^{*})^{T}$, by Theorem 4.5, we have
${\it{Opt(CP)}}={\it{Opt(DNNP)}}.$ Thus, we get
${\it{Opt(SOQP)}}={\it{Opt(DNNP)}}$ under the condition
$X^{*}=x^{*}(x^{*})^{T}$. Furthermore, since problems (SOQP) and (DNNP) have
the same objective function, again from Theorem 4.2, we can conclude that
$x^{*}$ is an optimal solution for problem (SOQP). From Theorem 3.1, we
further know that $x^{*}$ is a Pareto optimal solution (or weakly Pareto
optimal solution) for problem (MOQP) if $\lambda>0$ (or $\lambda\geq 0$).
## 5\. Numerical experiments
In this section, in order to show the effectiveness of our proposed method,
some examples are tested and corresponding numerical results are reported. To
solve test problems, we use CVX [11], a package for specifying and solving
convex programs. The software is implemented using MATLAB R2011b on Windows 7
platform, and on a PC with Intel(R) Core(TM) i3-2310M CPU 2.10 GHz.
In the following numerical experiments, three examples are solved by using the
proposed method, respectively. The first example is a given two-dimension
problem, which has four nonconvex objective functions. The second example is a
five-dimension nonconvex problem with five objective functions. Note that its
coefficients are generated by MATLAB function randn($\cdot$). The last example
is a practical portfolio selection problem, which is taken from [16]. The
weighted coefficient $\lambda$ is generated by the following procedure
lambda=zeros(p,1);
while lambda(p)==0
lambda(1:p-1)=rand(p-1,1); s=sum(lambda);
if s<1 lambda(p)=1-s; end
end
where $p$ is the number of objective functions.
###### Example 3.
First of all, a two-dimension problem with four objective functions is tested.
The corresponding coefficients $Q_{i},\ c_{i},\ i=1,\ldots,4$ and $A,\ b$ are
given in Table 1.
Table 1. Coefficients for Example 3 $Q_{1}=\left[\begin{array}[]{ll}1&3\\\ 3&8\end{array}\right]$ | $Q_{2}=\left[\begin{array}[]{lllll}2&0\\\ 0&-2\end{array}\right]$ | $Q_{3}=\left[\begin{array}[]{lllll}0&2\\\ 2&0\end{array}\right]$ | $Q_{4}=\left[\begin{array}[]{llllll}-1&0\\\ 0&0\end{array}\right]$
---|---|---|---
$c_{1}=[2.5\ \ -0.5]^{T}$ | $c_{2}=[-1\ \ -1.5]^{T}$ | $c_{3}=[1\ \ 0.5]^{T}$ | $c_{4}=[1\ \ 1.5]^{T}$
$A=[1\ \ 1]$ | $b=1$ | |
First, by relative simple computation, we obtain optimal solutions for each
objective function which is minimized independently. The corresponding optimal
numerical results of each objective function are given in Table 2.
Table 2. Optimal numerical results for each objective function of Example 3 | $F_{1}(x^{*})$ | | | | | $F_{2}(x^{*}$) | | | | | $F_{3}(x^{*}$) | | | | | $F_{4}(x^{*}$)
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
FV | $5.6667$ | | | | | $-5$ | | | | | $1$ | | | | | $1$
$x^{*}$ | $(0.6667\ 0.3333)^{T}$ | | | | | $(0\ 1)^{T}$ | | | | | $(0\ 1)^{T}$ | | | | | $(1\ 0)^{T}$
In Table 2, the labels FV and $x^{*}$ denotes optimal values and optimal
solutions corresponding to each objective function, respectively. The results
in Table 2 show that the objective functions $F_{2}(x)$ and $F_{3}(x)$ have
the same optimal solution, which is different from the other two functions
optimal solutions. These imply that we can not find a single solution that
simultaneously optimizes each objective function. Moreover, note that these
four optimal solutions corresponding to each objective function are all weakly
Pareto optimal solutions for Example 3.
Table 3. Optimal numerical results for Example 3 in (DNNP) $\lambda$ | FV | $x^{*}$ | $X^{*}$
---|---|---|---
$(0.5472\ 0.1386\ 0.1493\ 0.1649)$ | $\left[\begin{array}[]{ll}5.8345\\\ -2.8494\\\ 2.4106\\\ 2.3849\end{array}\right]$ | $\left[\begin{array}[]{ll}0.4301\\\ 0.5699\end{array}\right]$ | $\left[\begin{array}[]{ll}0.1850&0.2451\\\ 0.2451&0.3248\end{array}\right]$
$(0.3500\ 0.1966\ 0.2511\ 0.2023)$ | $\left[\begin{array}[]{ll}7.0000\\\ -5.0000\\\ 1.0000\\\ 3.0000\end{array}\right]$ | $\left[\begin{array}[]{ll}0.0000\\\ 1.0000\end{array}\right]$ | $\left[\begin{array}[]{ll}0.0000&0.0000\\\ 0.0000&1.0000\end{array}\right]$
$(0.0759\ 0.0540\ 0.5308\ 0.3394)$ | $\left[\begin{array}[]{ll}6.0000\\\ 0.0000\\\ 2.0000\\\ 1.0000\end{array}\right]$ | $\left[\begin{array}[]{ll}1.0000\\\ 0.0000\end{array}\right]$ | $\left[\begin{array}[]{ll}1.0000&0.0000\\\ 0.0000&0.0000\end{array}\right]$
$(0.2417\ 0.4039\ 0.0965\ 0.2579)$ | $\left[\begin{array}[]{ll}5.6670\\\ -1.6105\\\ 2.5513\\\ 1.8625\end{array}\right]$ | $\left[\begin{array}[]{ll}0.6779\\\ 0.3221\end{array}\right]$ | $\left[\begin{array}[]{ll}0.4596&0.2183\\\ 0.2183&0.1037\end{array}\right]$
$(0.5752\ 0.0598\ 0.2348\ 0.1302)$ | $\left[\begin{array}[]{ll}5.8570\\\ -2.9261\\\ 2.3857\\\ 2.4132\end{array}\right]$ | $\left[\begin{array}[]{ll}0.4148\\\ 0.5852\end{array}\right]$ | $\left[\begin{array}[]{ll}0.1720&0.2427\\\ 0.2427&0.3425\end{array}\right]$
$(0.0430\ 0.1690\ 0.6491\ 0.1389)$ | $\left[\begin{array}[]{ll}7.0000\\\ -5.0000\\\ 1.0000\\\ 3.0000\end{array}\right]$ | $\left[\begin{array}[]{ll}0.0000\\\ 1.0000\end{array}\right]$ | $\left[\begin{array}[]{ll}0.0000&0.0000\\\ 0.0000&1.0000\end{array}\right]$
$(0.2259\ 0.1707\ 0.2277\ 0.3757)$ | $\left[\begin{array}[]{ll}6.0000\\\ 0.0000\\\ 2.0000\\\ 1.0000\end{array}\right]$ | $\left[\begin{array}[]{ll}1.0000\\\ 0.0000\end{array}\right]$ | $\left[\begin{array}[]{ll}1.0000&0.0000\\\ 0.0000&0.0000\end{array}\right]$
$(0.2920\ 0.4317\ 0.0155\ 0.2609)$ | $\left[\begin{array}[]{ll}5.7480\\\ -2.4901\\\ 2.5020\\\ 2.2460\end{array}\right]$ | $\left[\begin{array}[]{ll}0.5020\\\ 0.4980\end{array}\right]$ | $\left[\begin{array}[]{ll}0.2520&0.2500\\\ 0.2500&0.2480\end{array}\right]$
$(0.2607\ 0.5944\ 0.0225\ 0.1224)$ | $\left[\begin{array}[]{ll}5.7480\\\ -2.4901\\\ 2.5020\\\ 2.2460\end{array}\right]$ | $\left[\begin{array}[]{ll}0.5020\\\ 0.4980\end{array}\right]$ | $\left[\begin{array}[]{ll}0.2520&0.2500\\\ 0.2500&0.2480\end{array}\right]$
It is very easy to verify that the given four objective functions are all
nonconvex by using MATLAB function eig. Furthermore, we obtain problem (SOQP)
is nonconvex with corresponding coefficients $\lambda$ which proposed in Table
3 by using eig. Hence, we use our method to solve this problem. The
corresponding optimal numerical results are reported in Table 3. In Table 3,
Example 3 is transformed into problem (DNNP), and then is solved with nine
different weighted coefficient $\lambda$. The results of $x^{*}$ and $X^{*}$
in Table 3 show that $X^{*}=x^{*}(x^{*})^{T}$ holds for nine different
weighted coefficients. Thus, from Theorem 4.5 and Remark 2, it holds that each
optimal solution $x^{*}$ in Table 3 also is Pareto optimal solution for
Example 3. Moreover, we obtain weakly Pareto optimal solutions for Example 3
when the weighted coefficient $\lambda$ is chosen appropriately. For instance,
if $\lambda$ is chosen as $(0.3500,0.1966,0.2511,0.2023)$ or
$(0.0430,0.1690,0.6491,0.1389)$, then weakly Pareto optimal solution
$(0.0000,1.0000)^{T}$ is obtained by using our method.
###### Example 4.
In this test problem, we set $n=5$, $m=4$ and $p=5$. The corresponding
coefficients $Q_{i},\ c_{i},\ i=1,\ldots,5$ and $A,\ b$ are generated by the
functions tril(randn(n,n),-1)+triu(randn(n,n)’,0), randn(n,1), randn(n,m) and
randn(m,1), respectively, and the details can be seen in Table 4.
Table 4. Coefficients for Example 4 $Q_{1}=\left[\begin{array}[]{lllllll}0.4056&-1.4193&-0.7294&1.1473&0.5979\\\ -1.4193&-2.2033&-0.5712&0.2140&0.9424\\\ -0.7294&-0.5712&0.3062&-1.1723&-0.9610\\\ 1.1473&0.2140&-1.1723&-0.9000&-0.2857\\\ 0.5979&0.9424&-0.9610&-0.2857&0.6103\end{array}\right]$ | $c_{1}=\left[\begin{array}[]{ll}0.7907\\\ 0.2877\\\ 0.0032\\\ 0.3656\\\ 3.5267\end{array}\right]$
---|---
$Q_{2}=\left[\begin{array}[]{lllll}0.0591&-1.4669&-1.6258&-1.9648&2.6052\\\ -1.4669&0.2570&-0.9742&-1.1464&0.5476\\\ -1.6258&-0.9742&-0.4494&-0.0843&-1.9920\\\ -1.9648&-1.1464&-0.0843&-0.3909&0.4092\\\ 2.6052&0.5476&-1.9920&0.4092&1.3018\end{array}\right]$ | $c_{2}=\left[\begin{array}[]{ll}-0.1124\\\ -1.5566\\\ 1.9151\\\ 0.6098\\\ -0.6479\end{array}\right]$
$Q_{3}=\left[\begin{array}[]{lllll}-0.5936&0.4364&-0.5044&0.1021&1.1963\\\ 0.4364&-1.0368&-0.8571&-0.1699&-0.1917\\\ -0.5044&-0.8571&1.2665&-0.2512&-0.2046\\\ 0.1021&-0.1699&-0.2512&-0.3862&0.5256\\\ 1.1963&-0.1917&-0.2046&0.5256&0.8175\end{array}\right]$ | $c_{3}=\left[\begin{array}[]{ll}2.6173\\\ 0.5510\\\ 0.2942\\\ -0.7778\\\ -1.0649\end{array}\right]$
$Q_{4}=\left[\begin{array}[]{llllll}0.4902&0.7653&0.7783&-1.4803&0.5404\\\ 0.7653&-0.7603&-0.6936&1.2815&-0.8097\\\ 0.7783&-0.6936&2.0108&0.0256&0.3083\\\ -1.4803&1.2815&0.0256&0.5301&-0.9521\\\ 0.5404&-0.8097&0.3083&-0.9521&-0.4506\end{array}\right]$ | $c_{4}=\left[\begin{array}[]{ll}-1.7684\\\ -0.4229\\\ -1.0531\\\ 0.6478\\\ -0.3176\end{array}\right]$
$Q_{5}=\left[\begin{array}[]{lllllll}0.1092&-0.2506&-0.1899&-1.0329&-0.3233\\\ -0.2506&1.7447&-1.1605&2.3774&1.5261\\\ -0.1899&-1.1605&-0.6987&0.8328&-0.6946\\\ -1.0329&2.3774&0.8328&0.8967&0.5047\\\ -0.3233&1.5261&-0.6946&0.5047&1.1867\end{array}\right]$ | $c_{5}=\left[\begin{array}[]{ll}1.7690\\\ 1.5106\\\ 0.1640\\\ -0.2828\\\ 1.1522\end{array}\right]$
$A=\left[\begin{array}[]{llllll}-1.1465&-0.6718&0.5530&0.2695&1.0393\\\ 0.6737&0.5756&-0.4234&-2.5644&0.9109\\\ -0.6691&-0.7781&0.3616&0.4659&-0.2397\\\ -0.4003&-1.0636&-0.3519&1.8536&0.1810\end{array}\right]$ | $b=\left[\begin{array}[]{ll}0.2442\\\ 0.0964\\\ -0.8305\\\ -0.3523\end{array}\right]$
In order to verify that whether the given objective functions in Table 4 are
nonconvex functions or not, the corresponding eigenvalues for $Q_{i},\
i=1,\ldots,5$ are given in Table 5. The results in Table 5 show that five
objective functions are all nonconvex. Thus, we will use the proposed method
to solve this problem. The results are given in Tables 6 and 7.
Table 5. Eigenvalues for $Q_{i},\ i=1,\ldots,5$ in Example 4 Quadratic matrices | Eigenvalues
---|---
$Q_{1}$ | $(-3.3892,-1.8503,-0.0780,1.1180,2.4184)$
$Q_{2}$ | $(-4.2248,-1.8386,0.7234,1.3339,4.7837)$
$Q_{3}$ | $(-1.6777,-1.1336,-0.3917,1.1973,2.0731)$
$Q_{4}$ | $(-2.5335,-1.0738,0.6848,1.6750,3.0677)$
$Q_{5}$ | $(-2.3512,-0.6382,0.1606,1.3837,4.6838)$
Table 6. Optimal numerical results for each objective function of Example 4 | | | | | FV | | | | | $x^{*}$
---|---|---|---|---|---|---|---|---|---|---
$F_{1}(x^{*})$ | | | | | $2.1900$ | | | | | $(0,\ 1.2165,\ 0.2253,\ 0.4747,\ 0.7783)^{T}$
$F_{2}(x^{*})$ | | | | | $-9.2542$ | | | | | $(2.7216,\ 0,\ 2.7773,\ 0.7727,\ 1.5592)^{T}$
$F_{3}(x^{*})$ | | | | | $-1.3755$ | | | | | $(0,\ 1.2165,\ 0.2253,\ 0.4747,\ 0.7783)^{T}$
$F_{4}(x^{*})$ | | | | | $-2.2276$ | | | | | $(0.4915,\ 0.9969,\ 0.6861,\ 0.5285,\ 0.9193)^{T}$
$F_{5}(x^{*})$ | | | | | $0.6759$ | | | | | $(2.7216,\ 0,,\ 2.7773,\ 0.7727,\ 1.5592)^{T}$
Table 6 shows that the optimal numerical results for each objective function
of Example 4. The results in Table 6 show that we can not find a single
solution that simultaneously optimizes these five objective functions. Note
that these five optimal solutions are also weakly Pareto optimal solutions for
Example 4.
Table 7. Optimal numerical results for Example 4 in (DNNP) $\lambda$ | FV | $x^{*}$ | $X^{*}$
---|---|---|---
$\left[\begin{array}[]{ll}0.3317\\\ 0.1522\\\ 0.3480\\\ 0.1217\\\ 0.0464\end{array}\right]$ | $\left[\begin{array}[]{llllll}2.1900\\\ -1.7656\\\ -1.3755\\\ -1.7914\\\ 7.0248\end{array}\right]$ | $\left[\begin{array}[]{ll}0.0000\\\ 1.2165\\\ 0.2253\\\ 0.4747\\\ 0.7783\end{array}\right]$ | $\left[\begin{array}[]{lllll}0.0000&0.0000&0.0000&0.0000&0.0000\\\ 0.0000&1.4799&0.2741&0.5775&0.9469\\\ 0.0000&0.2741&0.0508&0.1070&0.1753\\\ 0.0000&0.5775&0.1070&0.2254&0.3695\\\ 0.0000&0.9469&0.1753&0.3695&0.6058\end{array}\right]$
$\left[\begin{array}[]{ll}0.0712\\\ 0.1820\\\ 0.0930\\\ 0.4635\\\ 0.1903\end{array}\right]$ | $\left[\begin{array}[]{lll}2.2944\\\ -2.3834\\\ 0.1927\\\ -2.2196\\\ 6.2356\end{array}\right]$ | $\left[\begin{array}[]{ll}0.4250\\\ 1.0266\\\ 0.6238\\\ 0.5213\\\ 0.9003\end{array}\right]$ | $\left[\begin{array}[]{lllllll}0.1806&0.4363&0.2651&0.2215&0.3826\\\ 0.4363&1.0538&0.6403&0.5351&0.9242\\\ 0.2651&0.6403&0.3891&0.3252&0.5616\\\ 0.2215&0.5351&0.3252&0.2717&0.4693\\\ 0.3826&0.9242&0.5616&0.4693&0.8105\end{array}\right]$
$\left[\begin{array}[]{ll}0.0290\\\ 0.6371\\\ 0.0595\\\ 0.1692\\\ 0.1052\end{array}\right]$ | $\left[\begin{array}[]{ll}2.3045\\\ -2.4426\\\ 0.3234\\\ -2.2257\\\ 6.1681\end{array}\right]$ | $\left[\begin{array}[]{ll}0.4595\\\ 1.0111\\\ 0.6562\\\ 0.5251\\\ 0.9102\end{array}\right]$ | $\left[\begin{array}[]{llllll}0.2111&0.4646&0.3015&0.2413&0.4182\\\ 0.4646&1.0224&0.6635&0.5309&0.9203\\\ 0.3015&0.6635&0.4305&0.3445&0.5972\\\ 0.2413&0.5309&0.3445&0.2757&0.4779\\\ 0.4182&0.9203&0.5972&0.4779&0.8284\end{array}\right]$
$\left[\begin{array}[]{ll}0.0472\\\ 0.0559\\\ 0.0715\\\ 0.0157\\\ 0.8098\end{array}\right]$ | $\left[\begin{array}[]{ll}3.5136\\\ -9.2542\\\ 9.9301\\\ 6.7545\\\ 0.6759\end{array}\right]$ | $\left[\begin{array}[]{ll}2.7216\\\ 0.0000\\\ 2.7773\\\ 0.7727\\\ 1.5592\end{array}\right]$ | $\left[\begin{array}[]{llllll}7.4071&0.0000&7.5586&2.1030&4.2434\\\ 0.0000&0.0000&0.0000&0.0000&0.0000\\\ 7.5586&0.0000&7.7132&2.1460&4.3302\\\ 2.1030&0.0000&2.1460&0.5971&1.2048\\\ 4.2434&0.0000&4.3302&1.2048&2.4310\end{array}\right]$
$\left[\begin{array}[]{ll}0.1911\\\ 0.0504\\\ 0.5484\\\ 0.0095\\\ 0.2006\end{array}\right]$ | $\left[\begin{array}[]{ll}2.1900\\\ -1.7656\\\ -1.3755\\\ -1.7914\\\ 7.0248\end{array}\right]$ | $\left[\begin{array}[]{ll}0.0000\\\ 1.2165\\\ 0.2253\\\ 0.4747\\\ 0.7783\end{array}\right]$ | $\left[\begin{array}[]{lllll}0.0000&0.0000&0.0000&0.0000&0.0000\\\ 0.0000&1.4799&0.2741&0.5775&0.9469\\\ 0.0000&0.2741&0.0508&0.1070&0.1753\\\ 0.0000&0.5775&0.1070&0.2254&0.3695\\\ 0.0000&0.9469&0.1753&0.3695&0.6058\end{array}\right]$
$\left[\begin{array}[]{ll}0.2166\\\ 0.6291\\\ 0.0149\\\ 0.0433\\\ 0.0961\end{array}\right]$ | $\left[\begin{array}[]{ll}3.5136\\\ -9.2542\\\ 9.9301\\\ 6.7545\\\ 0.6759\end{array}\right]$ | $\left[\begin{array}[]{ll}2.7216\\\ 0.0000\\\ 2.7773\\\ 0.7727\\\ 1.5592\end{array}\right]$ | $\left[\begin{array}[]{lllll}7.4071&0.0000&7.5586&2.1030&4.2434\\\ 0.0000&0.0000&0.0000&0.0000&0.0000\\\ 7.5586&0.0000&7.7132&2.1460&4.3302\\\ 2.1030&0.0000&2.1460&0.5971&1.2048\\\ 4.2434&0.0000&4.3302&1.2048&2.4310\end{array}\right]$
$\left[\begin{array}[]{ll}0.2583\\\ 0.0041\\\ 0.0905\\\ 0.5958\\\ 0.0513\end{array}\right]$ | $\left[\begin{array}[]{ll}2.2080\\\ -1.8738\\\ -1.0651\\\ -1.9303\\\ 6.8716\end{array}\right]$ | $\left[\begin{array}[]{ll}0.0857\\\ 1.1782\\\ 0.3057\\\ 0.4841\\\ 0.8029\end{array}\right]$ | $\left[\begin{array}[]{lllll}0.0073&0.1010&0.0262&0.0415&0.0688\\\ 0.1010&1.3882&0.3601&0.5704&0.9460\\\ 0.0262&0.3601&0.0934&0.1480&0.2454\\\ 0.0415&0.5704&0.1480&0.2344&0.3887\\\ 0.0688&0.9460&0.2454&0.3887&0.6447\end{array}\right]$
Note that we can verify that problem (SOQP) is nonconvex with seven different
choices of weighted coefficient $\lambda$ which show in Table 7. So, Example 4
can be solved by using the proposed method. The results of $x^{*}$ and $X^{*}$
in Table 7 show that $X^{*}=x^{*}(x^{*})^{T}$ holds for these seven different
cases of weighted coefficient $\lambda$. Hence, we can conclude that each
$x^{*}$ in Table 7 also is Pareto optimal solution for Example 4. Furthermore,
compare the results of FV and $x^{*}$ in Tables 6 and 7, note that some weakly
Pareto optimal solutions of Example 4 can be obtained by using our method. For
example, if $\lambda=(0.3317,0.1522,0.3480,0.1217,0.0464)$, then we obtain
weakly Pareto optimal solution $(0.0000,1.2165,0.2253,0.4747,0.7783)^{T}$ for
Example 4.
###### Remark 3.
The results for Example 3 and Example 4 imply that we not only obtain Pareto
optimal solutions, but also obtain some weakly Pareto optimal solutions for
original problem with appropriate choices of weighted coefficient $\lambda$.
Summarizing these results, we can conclude that our method is effective for
solving some problems (MOQP).
###### Example 5.
(Portfolio Selection Problem) This problem is taken from [16]. It is a
practical portfolio selection problem in which objective function $F(x)$ has
the following expression
$F(x)=(\frac{1}{2}x^{T}Q_{1}x,\ -c_{2}^{T}x,\ -c_{3}^{T}x)^{T},$
where symmetric matrix $Q_{1}$ is called the risk matrix, $c_{2}$ denotes the
return rate vector, $c_{3}$ is a given weighting vector and its element is a
function of corresponding security liquidity. The corresponding coefficients
are given in Table 8.
Table 8. Coefficients for Example 5
$Q_{1}=\left[\begin{array}[]{llllllllllllllllllllll}12.35&-8.64&9.88&-8.43&9.42&-1.47&4.29&-0.96&15.2&0\\\
-8.64&6.42&-8.15&5.13&4.56&5.83&-6.72&0.78&3.42&0\\\
9.88&-8.15&18.27&-8.14&2.32&-4.12&-1.85&1.34&0.795&0\\\
-8.43&5.13&-8.14&12.6&-8.43&7.47&-4.67&-0.679&8.09&0\\\
9.42&4.56&2.32&-8.43&17.07&7.33&-3.18&-0.547&0.323&0\\\
-1.47&5.83&-4.12&7.47&7.33&13.38&-3.84&-0.478&0.616&0\\\
4.29&-6.27&-1.85&-4.67&-3.18&-3.84&3.41&0.107&0.464&0\\\
-0.96&-0.78&1.34&-0.679&-0.547&-0.478&0.107&1.33&11.23&0\\\
15.2&3.42&0.795&0.809&0.323&0.616&0.464&11.23&2.45&0\\\
0&0&0&0&0&0&0&0&0&0\end{array}\right]$
---
$c_{2}=[11.11\ \ 17.78\ \ 2.22\ \ 18.28\ \ 12.6\ \ 10.15\ \ 6.72\ \ 5.75\ \
6.1\ \ 2.25]^{T}$
$c_{3}=[0.069\ \ 0.092\ \ 0.046\ \ 0.096\ \ 0.086\ \ 0.068\ \ 0.06\ \ 0.145\ \
0.151\ \ 0.197]^{T}$
$A=[1\ 1\ 1\ 1\ 1\ 1\ 1\ 1\ 1\ 1]\ \ \ b=1$
Note that the objective functions contain only one quadratic function
$\frac{1}{2}x^{T}Q_{1}x$, by using eig function of MATLAB, it is easy to
verify that function $\frac{1}{2}x^{T}Q_{1}x$ is nonconvex. Thus, we use the
proposed method to solve this problem. The corresponding optimal results are
given in Table 9.
Table 9. Optimal numerical results for Example 5 $\lambda$ | FV | $x^{*}$
---|---|---
$\left[\begin{array}[]{ll}0.3482\\\ 0.1655\\\ 0.4863\end{array}\right]$ | $\left[\begin{array}[]{ll}0.1894\\\ -15.6800\\\ -0.0850\end{array}\right]$ | $(0.3240,0.5567,0.0000,0.1194,0.0000,0.0000,0.0000,0.0000,0.0000,0.0000)^{T}$
$\left[\begin{array}[]{ll}0.5141\\\ 0.1931\\\ 0.2927\end{array}\right]$ | $\left[\begin{array}[]{ll}0.1320\\\ -15.5462\\\ -0.0846\end{array}\right]$ | $(0.3433,0.5466,0.0000,0.1101,0.0000,0.0000,0.0000,0.0000,0.0000,0.0000)^{T}$
$\left[\begin{array}[]{ll}0.6167\\\ 0.3087\\\ 0.0746\end{array}\right]$ | $\left[\begin{array}[]{ll}0.2026\\\ -15.7070\\\ -0.0851\end{array}\right]$ | $(0.3200,0.5590,0.0000,0.1210,0.0000,0.0000,0.0000,0.0000,0.0000,0.0000)^{T}$
In Table 9, Example 5 is solved with three different values of weight
$\lambda$. The first element and the absolute value of the second element of
FV in Table 9 denote the expectation risk and return, respectively. The
results in Table 9 show that we obtain the lower risk and higher return when
weight $\lambda$ is chosen appropriately. Moreover, we also compare with the
results in [16], which are show in Table 10.
Table 10. Optimal numerical results for Example 5 in [16] FV | $x^{*}$
---|---
$\left[\begin{array}[]{ll}0.6823\\\ -7.5079\\\ -0.1343\end{array}\right]$ | $(0.0453,0.0706,0.0273,0.0703,0.078,0.045,0.0534,0.169,0.1665,0.2746)^{T}$
The results of FV in Tables 9 further imply that the risk and return are more
comparable with the results in Table 10. For example, when
$\lambda=(0.3482,0.1655,0.4863)$, we obtain lower risk $0.1894$ and higher
return $15.6800$, which are more comparable with the results of [16] $0.6823$
and $7.5079$, respectively. Furthermore, note that the optimal solutions
$x^{*}$ obtained by using our method are sparse, which imply that we mainly
focus on some kinds of important stocks. Hence, we can put together the
limited money, and invest these money in some of important stocks to obtain
more satisfied return with lower risk.
We also notice that there are infinite choices of weight $\lambda$, and not
all choices of weight $\lambda$ are reasonable for each investor. How to
select weight $\lambda$ to lower risk and higher return depends on the
investors’s preference. Therefore, it is reasonable that investors participate
in decision making and continuously revise their preferences according to
practical conditions. This also shows that our method is promising in solving
portfolio selection problems.
## 6\. Concluding remarks
In this paper, a class of (MOQP) problems is discussed. By using the linear
weighted sum method to deal with quadratic objective functions, problem (MOQP)
is transformed into problem (SOQP), which is nonconvex in general. Then,
taking advantage of lifting techniques, problem (SOQP) is equivalently
reformulated as problem (CP) which is a convex programming problem but NP-hard
in general. A computable relaxed convex problem (DNNP) for problem (CP) is
obtained by using doubly nonnegative relaxation method. Moreover, based on the
characteristics of optimal solutions for problem (DNNP), a sufficient
condition for (weakly) Pareto optimal solutions for problem (MOQP) is
proposed. Finally, the numerical results of two problems and a practical
portfolio selection problem show that the proposed method is effective and
promising.
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Received xxxx 20xx; revised xxxx 20xx.
|
arxiv-papers
| 2012-11-20T06:41:41 |
2024-09-04T02:49:38.229691
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Yan-Qin Bai, Chuan-Hao Guo",
"submitter": "Chuan-Hao Guo",
"url": "https://arxiv.org/abs/1211.4670"
}
|
1211.4680
|
# Novel Two-dimensional Carbon Allotrope with Strong Electronic Anisotropy
Cong Su International Center for Quantum Materials, Peking University,
Beijing 100871, China Yuanpei College, Peking University, Beijing 100871,
China Hua Jiang [email protected] International Center for Quantum
Materials, Peking University, Beijing 100871, China Department of Physics,
Soochow University, Suzhou 215006, China Ji Feng [email protected]
International Center for Quantum Materials, Peking University, Beijing 100871,
China
###### Abstract
Two novel two-dimensional carbon allotropes comprised of octagons and
pentagons are proposed based on the first-principle calculations. The two
carbon allotropes, named OPG-L and OPG-Z, are found to have distinct
properties. OPG-L is metallic, while OPG-Z is a gapless semimetal. Remarkably,
OPG-Z exhibits pronounced electronic anisotropy with highly anisotropic Dirac
points at the Fermi level. A tight-binding model is suggested to describe the
low-energy quasiparticles, which clarifies the origin of the anisotropic Dirac
points. The electronic anisotropy of OPG-Z is expected to have interesting
potential applications in electronic devices.
###### pacs:
61.48. Gh, 61.46.-w, 68.65.-k
## I introduction
There have been growing interests in exploring new structures of the two-
dimensional (2D) carbon in recent years. This is primarily stimulated by
extensive investigations on the intriguing properties of graphene Graphene ,
an atomically thin semimetal that harbors Dirac fermions at a pair of
inequivalent valleys in the k-space Graphene_eletronic . Among others,
graphyne, graphdiyne, graphane, the $sp^{2}$-like carbon layer with five-,
six- and seven-membered rings, the 2D amorphous carbon with four-membered
rings, the planar carbon pentaheptite, the 2D carbon semiconductor with
patterned defects, several carbon networks, octagraphene and T graphene
graphyne ; graphdiyne ; graphane ; 5-6-7-rings ; amorphous-2D-C ; pentaheptite
; caron-semiconductor ; Graphene_allotropes ; octagraphene ; T_graphene have
been studied theoretically. In a clever synthetic attempt, graphdiyne has been
successfully prepared experimentally graphdiyne . One-dimensional (1D)
topological defect containing octagonal and pentagonal $sp^{2}$-hybridized
carbon rings embedded in a perfect graphene has been studied by first-
principles approach ld_2 and produced experimentally line-defect . The result
is a line defect that mimics the 1D metallic wire, which has potential
application in all-carbon valleytronics valley_filter ; quantum_channel .
Consequently, one naturally suspects that the other 2D metastable carbon
allotropes with intriguing properties may be prepared, in particular,
comprised of five- and eight-membered carbon rings as inspired by the line
defect in graphene.
Figure 1: The structures of (a) OPG-L and (b) OPG-Z. The black dashed frames
are the orthogonal unit cells of OPG-L and OPG-Z, where OA and OB are lattice
vectors. These two structures can be tiled by the red-colored 558 structure by
copying it along the green arrows. The primitive cell of OPG-L is shown in
blue dashed frame while the primitive cell of OPG-Z is the same as the crystal
cell. Here we give the structural information of unit cells from the DFT
calculations. OPG-L: space group $Cmmm$, OA = 3.68 Å, OB = 9.12 Å, two atoms
in the asymmetric unit cell, (0, 0.42) and (0.69, 0.33); OPG-Z: space group
$Pmam$, OA = 6.90 Å, OB = 4.87 Å, four atoms in asymmetric unit cell,
(0.45,0.87), (0.56,0.62), (0.25,0.48), (0.25,0.78).
## II Structures of OP-graphenes
Apparently, viewing the line defects as structural motifs whose tiling covers
the 2D plane is a sensible pathway for discovering new stable 2D carbon
allotropes. Indeed, such tilings are geometrically viable. In this work, a 2D
carbon allotrope is suggested to have an intrinsic strong electronic
anisotropy, without the need for an external field anisotropic . By using the
first-principles calculations, we propose two novel energetically competitive,
kinetically stable 2D carbon allotropes. They can be viewed as 2D
tessellations of octagons and pentagons, called OP graphene-L (OPG-L) and
OPG-Z, as shown in Figs. 1(a) and (b), respectively. The structure of OPG-L
can be viewed as juxtaposing the five-five-eight-membered rings (558) ribbon
(indicated by the red atoms in Fig. 1) along a straight line path, while the
OPG-Z along a zigzag path, as indicated by the green arrows in Figs. 1 (a) and
(b). It is worth noting that this 558 ribbon occurs experimentally as a
topological line defect of graphene line-defect . We show computationally that
the OPG-L is a metal and OPG-Z is a gapless semimetal. Analysis of the
electronic structures reveal that OPG-Z displays a strong electronic
anisotropy, with anisotropic Dirac points near the Fermi level. These novel 2D
carbon structures, with the proposed electronic properties, may be useful for
novel electronic applications, in particular in all-carbon electronics dopants
.
## III Computational method
Our calculations are based on the density functional theory (DFT) within the
generalized gradient approximation (GGA), in the form of Perdew-Burke-
Ernzerhof’s exchange-correlation functional PBE . All the calculations are
performed using the Vienna Ab-initio Simulation Package (VASP) VASP . Periodic
boundary conditions were employed and vacuum slabs of 10 Å were used to
isolate the replicas of OPG layers. Geometrical optimizations are performed
until the Hellmann-Feynman forces on the ions are less than $1.0\times
10^{-4}$ eV/Å. The plane-wave basis is used, with a cut-off of 700 eV that
converges the total energy to 1 meV/atom. The Brillouin zone is sampled using
$9\times 9\times 1$ Monkhorst-Pack k-point scheme MP . The phonon spectra are
calculated using the finite-displacement method in a $3\times 3\times 1$
supercell phonopy ; PLK_method .
Figure 2: (a) The formation energies of OPG-L, OPG-Z, graphdiyne, T graphene
and pentaheptite as a function of area ratio in comparison to graphene, where
$A_{0}$ is the optimized lattice area. The formation energy of graphene is set
to 0. (b) The Helmholtz free energy as a function of temperature for the
above-mentioned 2D carbon allotropes. The phonon spectra of (c) OPG-L and (d)
OPG-Z. Inset: The first Brillouin zones and high symmetry points of OPG-L and
OPG-Z. The high symmetry points are: $\Gamma$(0,0), M(0,0.5), K(0.419,0.709),
H(0.581,0.291) in OPG-L, and $\Gamma$(0,0), X(0.5,0), M(0.5,0.5), Y(0,0.5) in
OPG-Z (fractional coordinates in reciprocal space).
## IV Stabilities of OP-Graphenes
To gauge the stability of the proposed carbon structures, we calculated the
formation energies, at $T=0$ K within the static lattice approximation, of
OPG-L, OPG-Z and four other typical 2D carbon allotropes, namely graphene,
graphdiyne graphdiyne , T graphene T_graphene and pentaheptite pentaheptite
for comparison [Fig. 2(a)]. The formation energy is defined with respect to
the free-standing graphene. Among all the structures, graphene is the most
stable energetically as expected T_carbon . It is found that OPG-L and OPG-Z
have fairly close formation energies, 0.31 eV/atom and 0.34 eV/atom
respectively. Therefore, OPG-L and OPG-Z are energetically metastable compared
to graphene and pentaheptite (whose formation energy is 0.21 eV/atom), though
much stabler than previously proposed T graphene T_graphene (0.52 eV/atom)
and graphdiyne graphdiyne (0.76 eV/atom). It should be noted that the
successful synthesis of graphdiyne in a previous work graphdiyne implies the
realization of OPG-L and OPG-Z is not unlikely.
We further estimated the Helmholtz free energy as a function of temperature
$T$. The Helmholtz free energy $A(T)$ is approximated as:
$A(T)\approx A_{ph}(T)+E(0)$ (1)
where $A_{ph}(T)$ is the vibrational free energy within the Born-Oppenheimer
and quasiharmonic approximation, and $E(0)$ is the total static-lattice energy
at 0 K. The finite-temperature Fermi-Dirac distribution of electronic level
occupation and the electron-phonon coupling are neglected. The vibrational
free energy is calculated by
$A_{ph}(T)=\frac{1}{2}\sum_{\textbf{q},s}\hbar\omega(\textbf{q},s)+k_{B}T\sum_{\textbf{q},s}\ln[1-exp(-\frac{\hbar\omega(\textbf{q},s)}{k_{B}T})]$
(2)
where q stands for the wavevector, $s$ the branch index, $\omega$ the
frequency at 0 K, $k_{B}$ and $\hbar$ the Boltzmann and Planck’s constants. As
a result, we find that the free energy of OPG-L and OPG-Z [Fig. 2(b)] falls
rapidly with increasing temperature. At low temperature, OPGs are
thermodynamically stabler than graphdiyne and pentaheptite. The phonon spectra
of OPG-L and OPG-Z are also calculated [see Fig. 2(c) and (d)] and no
imaginary phonon modes are found, confirming again the kinetic stability of
these two carbon sheets. In addition, the chemical stability of these two
structures are also examined. We put dioxygen and dihydrogen molecules close
to OPG-L and OPG-Z. After geometric relaxations, these molecules are all
repelled by the carbon sheet. Therefore, no spontaneous chemical reaction is
expected between these structures and oxygen or hydrogen molecules, suggesting
their redox stability in atmosphere.
Figure 3: Electronic band structures and density of states for (a) OPG-L and
(b) OPG-Z respectively. The inset figures are locations of high symmetry
points. The red dashed lines represent Fermi levels, which are set to 0 eV.
Blue lines are the results of DFT calculations, while orange dotted lines are
the results of tight-binding model. Figure 4: (a) The blow-up band structures
of Kohn-Sham quasiparticles of OPG-Z near the Fermi level around $\Gamma$
point. (b) The partial charge density (PCD) at $\kappa$ point near Fermi
level, calculated by DFT. (c) The wave function in a unit cell at the same
point, calculated by the tight-binding model [Eq.(1)]. (d) The component of
group velocity parallel to the k vector ($v_{k}$) measured from the $\kappa$
point in units of the Fermi velocity along $k_{y}$ ($v_{0}$) versus the angle
($\theta_{k}$) of the k vector. Inset: The isosurface of the band structure of
OPG-Z within the range of $k_{x}\in[0.139,0.141]$ and
$k_{y}\in[-0.0001,0.0001]$. The black dots in (b) and (c) are carbon atoms.
Figure 5: (a) OPG-Z as a distorted honeycomb lattice. Gray atoms are the atoms
with zero wavefunction amplitude around Fermi level in OPG-Z. The supercell of
graphene consists of eight atoms with its TB hopping matrix elements $t_{1}$
and $t_{2}$ shown. (b) The tight-binding band structure of graphene in the
supercell shown in (a), with $t_{2}=3$ eV. (c) The TB band structure of
graphene of the same primitive cell, with $t_{2}=0$ eV. Inset: high symmetry
points of the graphene supercell. (d) The different bond lengths of the zigzag
ribbon in OPG-Z.
## V electronic structures and possible realization
### V.1 Energy Bands
The Kohn-Sham electronic band structures and density of states (DOS) of OPG-L
and -Z are shown in Fig. 3. As we can see, OPG-L is a metal and OPG-Z is a
gapless semimetal. In Fig. 3(b), $\Gamma=$(0,0) point (fractional coordinates
with respect to the reciprocal unit cell; we follow the same notation
hereafter) near the Fermi level (FL) is dominated by linear dispersion,
resembling the Dirac fermions. A closer inspection reveals that OPG-Z has a
wedge-shaped conduction band and valence band near the FL where the Dirac
points emerge at $\kappa=(0.139,0)$ and $\kappa^{\prime}=(-0.139,0)$, as shown
in Fig. 4(a). The Fermi velocity along $k_{y}$ at these Dirac points are
estimated to be $2.2\times 10^{5}$ m/s, relatively high but an order of
magnitude smaller than in pristine graphene. The Kohn-Sham quasiparticle
energy isosurfaces of conduction band of OPG-Z near $\kappa$ point are very
elongated ellipses with an eccentricity near unity [inset of Fig. 4(d)]. In
addition, the Fermi velocities along different directions measured from
$\kappa$ point are also calculated [Fig. 4(d)], which shows that the group
velocity is redueced to $3.2\times 10^{3}$ m/s in the $k_{x}$ direction.
Therefore, OPG-Z is semimetallic with highly anisotropic Dirac fermions
anisotropic .
### V.2 Tight-Binding Approximation
To further understand the DFT calculated results, we construct a tight-binding
model of OPG-Z by allowing for the electrons hopping only to nearest
neighbors. The Hamiltonian can be written as
$H=-\sum_{\langle
i,j\rangle\sigma}t_{ij}(a_{i\sigma}^{\dagger}a_{j\sigma}+h.c.)+\varepsilon_{0}\sum_{i,\sigma}a_{i\sigma}^{\dagger}a_{i\sigma},$
(3)
where $a_{i\sigma}^{\dagger}$ and $a_{i\sigma}$ are, respectively, the
electronic creation and annihilation operators of the carbon valence $p_{z}$
orbital with spin $\sigma$ at site $i$; $\langle i,j\rangle$ stands for the
nearest-neighbor pairs of atoms at sites $i$ and $j$; $\varepsilon_{0}$ is the
on-site energy and $t_{ij}$ are the hopping matrix elements between site $i$
and site $j$, which are all taken to be 3.0 eV Review except for $t_{AB}$ [1
eV, site $A$ and site $B$ are shown in Fig. 4(c)]. Because all the atoms are
carbons, the on-site energy $\varepsilon_{0}$ can be set to zero. We
calculated the band structures (orange dotted lines in Fig. 3), which are
observed to match well the DFT calculations. The partial charge density at
$\kappa$ point in the conduction band is also calculated by DFT, shown in Fig.
4(b), which is in excellent agreement with the wavefunction calculated by the
TB [Fig. 4(c)]. Notice that the electron density is zero on atoms A and B in
Fig. 4(c), naturally splitting the whole plane into independent 1D electron
channels along OB direction [Fig. 4(b)]. The continuous electron density along
OB and the strong localization of electrons along OA bring about the
anisotropic electronic conductivity in OPG-Z. In this case, a perfect 1D
aligned electron transport is able to be realised in OPG-Z. Such stripe-like
electron transport channels are as wide as 3.38 Å [Fig. 4(b)]. The anisotropic
property of electronic transport in OPG-Z may give rise to some interesting
implications in nano-electronics. For example, the 1D electronic channels in
OPG-Z layer can be a replacement of the conducting wires in thin film
transistor.
It is also of interest to analyze how the Dirac points and flat band emerge
around FL in OPG-Z. We observe that the elimination of the nodal atoms [zero
wavefunction amplitude near the Dirac point; c.f., Fig. 4(b)] from OPG-Z leads
a distorted graphene lattice [Fig. 5(a)]. A TB model of graphene will be good
to describe some of the OPG-Z’s properties. The TB model has two hopping
parameters, conforming to the prescribed lattice distortion, as shown in Fig.
5(a). By expanding the primitive cell of graphene to an eight-atom supercell,
two linear dispersion bands and flat bands at $E=\pm t_{2}$ are folded to the
$\Gamma$ point in the first Brillouin zone [Fig. 5(b)]. Upon setting $t_{2}$
to zero, these two sets of band are shifted to FL, creating one band with
linear dispersion and one flat band at $\Gamma$ point [Fig. 5(c)]. As the
simple model demonstrates, the anisotropic property of OPG-Z is originated
from the separation of zigzag ribbons in graphene. The spliting of Dirac point
from $\Gamma$ point to $\kappa$ and $\kappa^{\prime}$ points in OPG-Z arises
from the further structural relaxation that leads to alternating bond lengths
as indicated by the red atom chain in Fig. 5(d).
### V.3 Possible Experimental Realization
As its formation energy and free energy is lower than the existing carbon
allotropes like graphdiynegraphdiyne , OPG-Z might be experimentally obtained.
Possible routes include nano-engineered synthesis Defect_exp ;
line_defect_TBMD ; nano_eng , epitaxial or chemical vapor deposition on a
suitable substrate epitaxial ; graphene ; graphene_2 and chemically,
acetylene scaffolding planar dicyclopenta pentalene scaffolding ; four_five .
## VI Summary
In summary, with first-principle calculations we have shown that two new 2D
carbon allotropes comprised of octagons and pentagons, named as OPG-L and
OPG-Z can be energetically and kinetically viable. The energetic and kinetic
stabilities of OPG-L and -Z are supported by calculating their zero
temperature energies, approximate finite temperature free energy, and phonon
band structure. Their electronic structures are calculated and analyzed in
detail. Our results show that OPG-L and OPG-Z are even more favorable in
energy than graphdiyne that was already synthesized experimentally. OPG-L is a
metal while OPG-Z is a gapless semimetal. It is found that the electronic
structure of OPG-Z is remarkably anisotropic, with a pair of anisotropic Dirac
points very close to Fermi level. The wavefunction and partial charge density
in the conduction band of OPG-Z at Dirac point, in combination with
transparent TB model, explain the origin of the electronic anisotropy of
OPG-Z. The computed stability and electronic structure argue for experimental
synthesis of these two-dimensional carbon structures, which are expected to
have potentially interesting applications.
###### Acknowledgements.
We are grateful to Ran Duan and Dr. Haiwen Liu for useful discussions. We
thank the financial support from National Science Foundation of China (Grant
No. 11174009, No. 91121004 and No. 11274032) and China 973 Project (Grant No.
2013CB921900).
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|
arxiv-papers
| 2012-11-20T08:12:02 |
2024-09-04T02:49:38.239176
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Cong Su, Hua Jiang and Ji Feng",
"submitter": "Cong Su",
"url": "https://arxiv.org/abs/1211.4680"
}
|
1211.4957
|
]}-<>()
# An Experiment on the Connection between the Description Logics’ Family
$\mathcal{DL}$ <$\mathcal{\forall}_{0}^{\pi}$> and the Real World
Domenico Cantone ([email protected])
Antonio Pisasale ([email protected])
###### Abstract
This paper describes the analysis of a selected testbed of Semantic Web
ontologies, by a SPARQL query, which determines those ontologies that can be
related to the description logic $\mathcal{DL}$
<$\mathcal{\forall}_{0}^{\pi}$>, introduced in Cantone _et al._ , [4] and
studied in Longo, [9]. We will see that a reasonable number of them is
expressible within such computationally efficient language. We expect that, in
a long-term view, a temporalization of description logics, and consequently,
of OWL(2), can open new perspectives for the inclusion in this language of a
greater number of ontologies of the testbed and, hopefully, of the “real
world”.
## INTRODUCTION
In the last years, Semantic Web has increasingly expanded its area of
influence. Being an innovative instrument for the retrieval of not expressly
stored information and a way of organizing concepts and relations by their
meaning, it broadens up plenty of horizons for knowledge representation.
Though, only a bunch of experts and researchers know that under a suprisingly
vast dimension of new features there lies a mathematical and logical structure
inside which they fight day by day for the balancing of expressiveness and
efficiency. The _Description Logics_ formalisms, which we will see in Section
1, are the formal bases for the so-called Web 3.0, that should allow one to
automatically infer (and retrieve) new information from _reasoning_ on
“sematicized” knowledge repositories. Some examples of families of logics are
described in the following, together with a short coverage of arguments such
as RDF graphs and OWL.
Section 2 introduces the description logic $\mathcal{DL}$
<$\mathcal{\forall}_{0}^{\pi}$>, showing notable characteristics of expressive
power and polynomial complexity. In the same section, a more detailed
description of the analysis, which may be seen as a conceptual experiment,
follows. Our purpose is to show that a good number of real-world ontologies
may be related to this family of logics. Results are promising enough to spur
us to stay on this path and complement it with studies on temporalization of
Description Logics and, consequently, of Semantic Web (briefly touched in this
report), which we will approach in the near future.
The SPARQL query which was specifically created to assess the membership of
some real-world ontologies to the aforementioned logic is reported in Appendix
A.
## 1 THEORETICAL FUNDAMENTALS
### 1.1 Description Logics
_Description Logics_ (_DL_ s) are a family of formalisms for knowledge
representation, built on logic-based semantics. They are founded on some
fundamental ideas:
* •
basic syntactic “blocks” are atomic concepts (1-ary predicates), atomic roles
(2-ary predicates) and individuals (costants);
* •
the expressiveness of a certain language depends on the use of a set of chosen
constructors, that give birth to complex concepts and roles starting from
existing ones;
* •
by means of classification of concepts, a _subsumption_ hierarchy is
established, which specifies what concept includes or is included by another;
* •
implied knowledge is automatically obtained through a logic procedure, called
_reasoning_ , essentially based on the application of inference rules to
subsumption between concepts, and an instance definitions between individuals
and the latter.
On the assumption that a Knowledge Representation System (_KR-System_) is to
give an answer to a user query, _reasoning_ algorithms for DLs should be
regarded as decisional procedures which return a positive or negative verdict.
This raises the decision problem for such languages. Furthermore, having an
answer does not always mean getting it in a reasonable lapse of time, and that
compels us to consider also the complexity of algorithms at stake.
Decidability and complexity directly depend on the expressive power of the
description logic we use: whereas very expressive DLs tend to have inference
problems and be computationally _hard_ (or even undecidable), the DLs that are
more efficient in _reasoning_ turn out to be not espressive enough to
represent all concepts and relations in the domain of interest. Research still
going on in the field of DLs just aims at the ability of balancing
expressiveness against efficiency, while not dropping semantic precision that
could make it applicable to real world situations.
A _Description Logics Knowledge Base_ (_DLKB_) is made of two components: TBox
(_Terminological Box_) and ABox (_Assertional Box_). The former contains the
vocabulary, i.e., the definitions of atomic and non-atomic concepts and roles,
called _axioms_ , whereas the latter describes individuals in terms of this
vocabulary, i.e. it includes the declarations, called _assertions_ , of their
instances. By means of the TBox we can name complex descriptions of concepts
and roles. The language for such a naming is what distinguishes one DL from
another and is based on a _model-theoretic_ semantics. Thus statements in TBox
and ABox can be regarded as first-order logic formulas. _Reasoning_ procedures
for the terminological part are used to verify the _satisfiability_ of a
description (i.e., its non-contradictoriness), or whether it may be _subsumed_
by another (i.e., whether the latter is more general than the former), while
those for the assertive part establish whether its set of assertion is
_consistent_ (i.e., it has a model or it entails that a certain individual is
an instance of a given concept). Satisfiability tests for descriptions and
consistency tests for a set of assertion allow one to establish whether the
knowledge base is meaningful or not, whereas subsumption tests allow one to
maintain a hierarchy of concepts _ab universali_ ; finally, instance tests
give one the ability of querying the system against single individuals.
More formally, a generic DL axiom is a formula of one of the following types:
* •
$C\equiv D$ (equivalence between concepts)
* •
$C\sqsubseteq D$ (subsumption between concepts)
* •
$R\equiv S$ (equivalence between roles)
* •
$R\sqsubseteq S$ (subsumption between roles),111The equivalence and
subsumption between roles may be indicated also by the symbols $=$ and
$\subseteq$, respectively.
where the symbols $C,D$ are names or expressions of complex concepts, which
are formed by 1-ary or 2-ary operations on/between atomic (indicated by $A$ in
the following) or complex concepts, while $R,S$ are names or expressions of
complex roles which are formed by 1-ary or 2-ary operations on/between atomic
(indicated by $P$ in the following) or complex roles.
A generic DL assertion is a formula of one of the following types:
* •
$C\left(a\right)$ (concept assertion)
* •
$R\left(a,b\right)$ (role assertion),
where the symbols $a,b$ are names of individuals, for which the concept $C$ or
the role $R$ holds.222A large number of constructs is listed in the table at
the end of this section.
From a semantical point of view, an _interpretation_ $\mathcal{I}$ is a pair
$(\Delta^{\mathcal{I}},\cdot^{\mathcal{I}})$, where the non-empty set
$\Delta^{\mathcal{I}}$ represents the domain of the interpretation and the
interpretation function $\cdot^{\mathcal{I}}$ associates a set
$A^{\mathcal{I}}\subseteq\Delta^{\mathcal{I}}$ to every atomic concept $A$ , a
relation
$P^{\mathcal{I}}\subseteq\Delta^{\mathcal{I}}\times\Delta^{\mathcal{I}}$ to
every atomic role $P$, and an element $a^{\mathcal{I}}\in\Delta^{\mathcal{I}}$
to every individual $a$. We write:333In the following,
$\mathcal{I}\models\phi$ means “$\mathcal{I}$ satisfies $\phi$, where $\phi$
can be an axiom or an assertion. A syntax/semantical reference for the main
axioms, assertions and property declarations is listed in the table at the end
of this section.
* •
$\mathcal{I}\models\left(C\equiv D\right)$ iff444“Iff” is short for “if and
only if”. $C^{\mathcal{I}}=D^{\mathcal{I}}$
* •
$\mathcal{I}\models\left(C\sqsubseteq D\right)$ iff $C^{\mathcal{I}}\subseteq
D^{\mathcal{I}}$
* •
$\mathcal{I}\models\left(R\equiv S\right)$ iff
$R^{\mathcal{I}}=S^{\mathcal{I}}$
* •
$\mathcal{I}\models\left(R\sqsubseteq S\right)$ iff $R^{\mathcal{I}}\subseteq
S^{\mathcal{I}}$
* •
$\mathcal{I}\models C\left(a\right)$ iff $a^{\mathcal{I}}\in C^{\mathcal{I}}$
* •
$\mathcal{I}\models R\left(a,b\right)$ iff $\left(a,b\right)^{\mathcal{I}}\in
R^{\mathcal{I}}$
Finally, we say that $\mathcal{I}$ is a _model_ for a DLKB $\mathcal{K}$ (and
write $\mathcal{I}\models\mathcal{K}$) if $\mathcal{I}$ satisfies all the
axioms and assertions of $\mathcal{K}$ . The latter is said to be _consistent_
if there exists at least an interpretation satisfying it, and the search of
this interpretation (_consistency problem_) is just the clue to _reasoning._
### 1.2 Families of logics
As already observed, distinct DLs are characterized by the constructs allowed
to form complex concepts and roles starting from atomic ones. The names are
usually specified by a series of alphabet letters and symbols. In the
following, we will not concentrate on formal semantics, but, for the sake of
clarity, we will only hint at the meaning of some constructs.
By way of an example, we briefly describe the $\mathcal{ALC}$ logic
(_Attributive Language with Complements_). Its syntax obeys the following
rules:
$C,D\rightarrow A\ |\ \top\ |\ \bot\ |\ \neg C\ |\ C\sqcap D\ |\ C\sqcup D\ |\
\forall R.C\ |\ \exists R.C\,\,,$
where $\top$ denotes the concept enclosing any other one (_top concept_),
$\bot$ denotes the concept enclosed in any other one (_bottom concept_),
$\neg$ negates a concept; $\sqcup$ represents the union of concepts (notice
the analogy with the corresponding set operators) and $\sqcap$ represents the
intersection of concepts. The last two constructs are called respectively
_universal_ and _existential restriction_ , and are pivotal in research
connected to this report. The concepts that can be built in $\mathcal{ALC}$
are called $\mathcal{ALC}$ -concepts.The axioms __ and assertions that may be
expressed in the $\mathcal{ALC}$ logic are
$C\equiv D,\ \,\ \,\ C\sqsubseteq D,\ \,\ \,\ C\left(a\right),$
which indicate, respectively, equivalence and subsumption (also called _GCI_ ,
_General Concept Inclusion_) between two concepts, and the membership of a
concept. However, _reasoning_ in $\mathcal{ALC}$ has a PSpace-complete
computational complexity.
In addition to those seen above, the most common constructs for concepts which
can be formed in DLs are $\leq 1.R$ (_functional restriction_ , denoted
$\mathcal{F}$ , that is equivalent to $\exists R.\top$), $\leq nR$ and $\geq
nR$ (_numerical restrictions_ , $\mathcal{N}$ , which enclose the functional
one), $\leq nR.C$ and $\geq nR.C$ (_qualified restrictions_ , $\mathcal{Q}$ ,
that include the numerical ones), $\left\\{a\right\\}$ and
$\left\\{a_{1},\ldots,a_{n}\right\\}$ (_nominals_ , $\mathcal{O}$ ), and
$\exists R.\mathit{Self}$ (_self-concept_). Contructs for roles are often
denoted by an operator symbol inside brackets after the name of the logic.
They are $R^{-}$ (inverse role, $\mathcal{I}$), $R\cup S,R\cap S$ and $\neg R$
(role union, intersection and complement, respectively), $R\circ S$ (role
composition, also used in chain), $R^{+}$ and $R^{*}$ (transitive and
reflexive-transitive closure of roles), $\mathit{id}\left(C\right)$(_concept
identity_), and $R_{C|},R_{|D}$ and $R_{C|D}$ (role restrictions). Sometimes,
the symbols $U$ (or _$\nabla$_ ,_universal role_ , defining the role which
encloses any other one) and _$N$ _(or __$\triangle$__ ,_empty role_ , defining
the role enclosed in any other one) are used. Other kinds of axioms which can
be introduced are $R\equiv S$ (equivalence between roles), $R\sqsubseteq S$
(hierarchy between roles, $\mathcal{H}$ , denoted also by $R\subseteq S$), and
the relative assertion $R\left(a,b\right)$ (role instance). The reflexive,
irreflexive, symmetric, antisymmetric, transitive, intransitive, disjunctive
(two roles having no pair of elements in common), functional and inverse
functional properties for roles are denoted by symbols that often differ in
literature, but are never ambiguous, e.g. respectively
$\mathit{Sym}\left(R\right),\mathit{Asym}\left(R\right),\mathit{Refl}\left(R\right),\mathit{Irrefl}\left(R\right),\mathit{Tr}\left(R\right),\mathit{Intr}\left(R\right),\mathit{Disj}\left(R\right),\mathit{Fn}\left(R\right),\mathit{InvFn}\left(R\right)$.
When transitive property is allowed, we use the symbol $\mathcal{S}$ , which
corresponds to $\mathcal{ALC}(^{+})$. Sometimes, even small differences among
logics (e.g., limitation on the use of atomic rather than complex roles on the
right or left part of an assertion) can make a big difference in
expressiveness and complexity. Thus, it is not always easy to concisely denote
DLs, that consequently constitute an ever-open research field.
One or more among the above descripted constructs, axioms and assertions are
present in the families of logics that are the base of languages used in
ontologies. Among these, an important example is the logic
$s\mathcal{ROIQ}_{(D)}$ (in literature always denoted by $\mathcal{SROIQ}(D)$
), at the bottom of the OWL 2 DL profile, which will be discussed in the
following (the letter in parenthesis indicates the use of _concrete domains_ ,
that will not be discussed here). Its syntax is easily inferred from the
symbols, while its complexity is N2ExpTime-hard.555By the symbol $\mathcal{R}$
, we mean that a DL allows complex inclusions of the kind $R\circ S\sqsubseteq
R$ and $R\circ S\sqsubseteq S$.
_Construct_ | _Syntax_ | _Semantics_
---|---|---
top concept | $\top$ | $\top^{\mathcal{I}}=\Delta^{\mathcal{I}}$
bottom concept | $\bot$ | $\bot^{\mathcal{I}}=\emptyset$
concept negation | $\neg C$ | $\left(\neg C\right)^{\mathcal{I}}=\Delta^{\mathcal{I}}\backslash C^{\mathcal{I}}$
concept intersection | $C\sqcap D$ | $\left(C\sqcap D\right)^{\mathcal{I}}=C^{\mathcal{I}}\cap D^{\mathcal{I}}$
concept union | $C\sqcup D$ | $\left(C\sqcup D\right)^{\mathcal{I}}=C^{\mathcal{I}}\cup D^{\mathcal{I}}$
universal restriction | $\forall R.C$ | $\left(\forall R.C\right)^{\mathcal{I}}=\left\\{a\in\Delta^{\mathcal{I}}|\left(\forall\left(a,b\right)\in R^{\mathcal{I}}\right)\left(b\in C^{\mathcal{I}}\right)\right\\}$
existential restriction | $\exists R.C$ | $\left(\exists R.C\right)^{\mathcal{I}}=\left\\{a\in\Delta^{\mathcal{I}}|\,\exists\left(a,b\right)\in R^{\mathcal{I}}\wedge b\in C^{\mathcal{I}}\right\\}$
numerical restriction | | $\leq nR$
---
$\geq nR$
| $\left(\leq
nR\right)^{\mathcal{I}}=\left\\{a\in\Delta^{\mathcal{I}}|\,\\#\left\\{b\in\Delta^{\mathcal{I}}|\left(a,b\right)\in
R^{\mathcal{I}}\right\\}\leq n\right\\}$
---
$\left(\geq
nR\right)^{\mathcal{I}}=\left\\{a\in\Delta^{\mathcal{I}}|\,\\#\left\\{b\in\Delta^{\mathcal{I}}|\left(a,b\right)\in
R^{\mathcal{I}}\\}\right\\}\geq n\right\\}$
qualified restriction | | $\leq nR.C$
---
$\geq nR$.C
| $\left(\leq
nR.C\right)^{\mathcal{I}}=\left\\{a\in\Delta^{\mathcal{I}}|\,\\#\left\\{b\in
C^{\mathcal{I}}|\left(a,b\right)\in R^{\mathcal{I}}\right\\}\leq n\right\\}$
---
$\left(\geq
nR.C\right)^{\mathcal{I}}=\left\\{a\in\Delta^{\mathcal{I}}|\,\\#\left\\{b\in
C^{\mathcal{I}}|\left(a,b\right)\in R^{\mathcal{I}}\right\\}\geq n\right\\}$
nominals | | $\left\\{a\right\\}$
---
$\left\\{a_{1},\ldots,a_{n}\right\\}$
| $\left\\{a\right\\}^{\mathcal{I}}=\left\\{a^{\mathcal{I}}\right\\}$
---
$\left\\{a_{1},\ldots,a_{n}\right\\}^{\mathcal{I}}=\left\\{a_{1}^{\mathcal{I}},\ldots,a_{n}^{\mathcal{I}}\right\\}$
self concept | $\exists R.\mathit{Self}$ | $\left(\exists R.Self\right)^{\mathcal{I}}=\left\\{a\in\Delta^{\mathcal{I}}|\left(a,a\right)\in R^{\mathcal{I}}\right\\}$
universal role | $U$, $\nabla$ | $U^{\mathcal{I}}=\Delta^{\mathcal{I}}\times\Delta^{\mathcal{I}}$
empty role | $N$, $\triangle$ | $N^{\mathcal{I}}=\emptyset\times\emptyset$
role inverse | $R^{-}$ | $\left(R^{-}\right)^{\mathcal{I}}=\left\\{\left(a,b\right)|\left(b,a\right)\in R^{\mathcal{I}}\right\\}$
role negation | $\neg R$ | $\left(\neg R\right)^{\mathcal{I}}=\left(\Delta^{\mathcal{I}}\times\Delta^{\mathcal{I}}\right)\backslash R^{\mathcal{I}}$
role intersection | $R\sqcap S$ | $\left(R\sqcap S\right)^{\mathcal{I}}=R^{\mathcal{I}}\cap S^{\mathcal{I}}$
role union | $R\sqcup S$ | $\left(R\sqcup S\right)^{\mathcal{I}}=R^{\mathcal{I}}\cup S^{\mathcal{I}}$
chaining | $R\circ S$ | $\left(R\circ S\right)^{\mathcal{I}}=R^{\mathcal{I}}\circ S^{\mathcal{I}}$
concept identity | $\mathit{id}\left(C\right)$ | $\left(id\left(C\right)\right)^{\mathcal{I}}=\left\\{\left(a,a\right)|\,a\in C^{\mathcal{I}}\right\\}$
transitive closure | $R^{+}$ | $\left(R^{+}\right)^{\mathcal{I}}=\left(R^{\mathcal{I}}\right)^{+}$
refl-trans closure | $R^{*}$ | $\left(R^{*}\right)^{\mathcal{I}}=\left(R^{+}\right)^{\mathcal{I}}\cup\left(id\left(\top\right)\right)^{\mathcal{I}}$
role restriction | | $R_{C|}$
---
$R_{|D}$
$R_{C|D}$
| $(R_{C|})^{\mathcal{I}}=\left\\{\left(a,b\right)\in R^{\mathcal{I}}|\,a\in
C^{\mathcal{I}}\right\\}$
---
$(R_{|D})^{\mathcal{I}}=\left\\{\left(a,b\right)\in R^{\mathcal{I}}|\,b\in
D^{\mathcal{I}}\right\\}$
$(R_{C|D})^{\mathcal{I}}=\left\\{\left(a,b\right)\in R^{\mathcal{I}}|\,a\in
C^{\mathcal{I}}\wedge b\in D^{\mathcal{I}}\right\\}$
_Table 1.1._ Main constructs for concepts and roles
_Syntax_ | _Semantics_
---|---
$C\equiv D$ | $C^{\mathcal{I}}=D^{\mathcal{I}}$
$C\sqsubseteq D$ | $C^{\mathcal{I}}\subseteq D^{\mathcal{I}}$
| $C\left(a\right)$
---
$\neg C\left(a\right)$
| $a^{\mathcal{I}}\in C^{\mathcal{I}}$
---
$a^{\mathcal{I}}\notin C^{\mathcal{I}}$
$R\equiv S$ | $R^{\mathcal{I}}=S^{\mathcal{I}}$
$R\sqsubseteq S$ | $R^{\mathcal{I}}\subseteq S^{\mathcal{I}}$
| $R\left(a,b\right)$
---
$\neg R\left(a,b\right)$
| $\left(a,b\right)^{\mathcal{I}}\in R^{\mathcal{I}}$
---
$\left(a,b\right)^{\mathcal{I}}\notin R^{\mathcal{I}}$
$R\sqcap S=N$, $\mathit{Disj}\left(R,S\right)$ | $R^{\mathcal{I}}\cap S^{\mathcal{I}}=\emptyset\times\emptyset$
| $\mathit{Refl}\left(R\right)$
---
$\neg\mathit{Refl}\left(R\right)$
| $\left(\forall a\in\Delta^{\mathcal{I}}\right)\left(\left(a,a\right)\in
R^{\mathcal{I}}\right)$
---
$\left(\forall a\in\Delta^{\mathcal{I}}\right)\left(\left(a,a\right)\notin
R^{\mathcal{I}}\right)$
| $\mathit{Sym}\left(R\right)$
---
$\neg\mathit{Sym}\left(R\right)$
| $\left(\forall\left(a,b\right)\in
R^{\mathcal{I}}\right)\left(\left(b,a\right)\in R^{\mathcal{I}}\right)$
---
$\left(\forall\left(a,b\right)\in
R^{\mathcal{I}}\right)\left(\left(b,a\right)\notin R^{\mathcal{I}}\right)$
| $\mathit{Trans}\left(R\right)$
---
$\neg\mathit{Trans}\left(R\right)$
| $\left(\forall\left(a,b\right)\in
R^{\mathcal{I}}\right)\left(\left(b,c\right)\in
R^{\mathcal{I}}\rightarrow\left(a,c\right)\in R^{\mathcal{I}}\right)$
---
$\left(\forall\left(a,b\right)\in
R^{\mathcal{I}}\right)\left(\left(b,c\right)\in
R^{\mathcal{I}}\rightarrow\left(a,c\right)\notin R^{\mathcal{I}}\right)$
_Table 1.2._ Main types of axioms and assertions on concepts and roles666The
use of $\mathit{Trans}\left(R\right)$, $\mathit{Refl}\left(R\right)$, and so
on, looks like adding a new type of axiom. Actually, they must be regarded as
abbreviations of more complex expressions involving equivalence, subsumption
and various constructs which we will not be treated here.
### 1.3 Semantic Web
The so-called Web 2.0 describes the current model of information search. The
difference with Web 1.0 is represented by the change in the origin of this
information (from below−users−rather than from above−webmasters). Yet, a
paradigm change in its fundamental structure has not occured, as the creator
of Web, Tim Berners-Lee, wished instead. Indeed, contents are still
exclusively made up of pages connected by links, whose nontrivial words,
together with metadata, are indexed by search engines, by means of which a
correspondence between those and the one inserted by users can be found. In
this way, notwithstanding the high efficiency and precision of the algorithms
involved, such engines constitute a “stupid” example of _information_
_retrieval_ , where the matching between terms, even if advanced, weighs more
than their meaning.
The change brought by the so-called Semantic Web (_SW_ or Web 3.0) addresses
the main issue that in the Web, as we know it, most of the contents are
structured in order to be read by humans, rather than investigated by programs
in an automatic way. A computer can jump from page to page by following links,
but it does not make any assumption on the semantics of their contents. SW is
an extension of “previous” Web that allows one to assign a meaning to
information and provides machines with the ability to elaborate and
“understand” what in the past they could only show. For all this to work,
computers should be able to access well-structured information schemas and
apply inference rules to permit automatic reasoning, in order to delocalize
knowledge representation and spread it over the Net. The languages these rules
apply should be expressive enough to make the Web capable of “reasoning” in a
versatile and widespread way. That is made possibile by tools such as
_eXstensible Markup Language_ (_XML_), _Resource Description Framework_
(_RDF_) and the ontologies described by the _Ontology Web Language_ (_OWL_).
### 1.4 RDF graphs
RDF is a model for representation of information on the Web. The basic idea is
that every resource (concept, class, property, object, value, etc.) can be
univocally described in terms of simple or identified-by-value properties. All
this permits to schematize an RDF model by means of an oriented graph, whose
nodes represent primitive objects and whose edges denote properties. The
limited vocabulary of RDF is extended by _RDFS_ (_RDF Schema_), which allows
one to define classes and properties in a more powerful way. E.g., it is
possibile to define a property as a relation by indicating its domain and/or
range, or say what class is a subclass of another.
From a data structure point of view, an RDF graph may be seen as a set of
triples of type (subject predicate object). For each triple, subject and
object are two nodes connected by an edge that represents a predicate; thus,
the subject of a triple may be the object of another and viceversa. Each of
these nodes may be identified by a URI/IRI, and so it can represent a
resource, or it may be simply a “connector” (in this case, it is called a
_blank node_ , _bnode_ , or _anonymous node_). If an object is not a subject
of another triple, it may also be a _literal_ , i.e., a datum representing a
value in a certain domain.
On the representation side, W3C suggests five types of _serialization_ s for
an RDF graph, the most used of which are RDF/XML (XML language is used) and
Turtle (descriptions through lists of triples).777The _serialization_ of an
object is the process which allows one to represent it in an accessible way,
in our case a text file. XML is used because of its precise representation of
data. Thus, they can be shared, by means of RDF, among various Web
applications while preserving their original meaning.
### 1.5 Ontologies
OWL ontologies extend RDFS vocabulary further.888Computer science ambitiously
draw on the philosophical term _ontology_ to show that the OWL language can
describe the world starting from its ultimate constituents and from relations
among them. If RDFS guarantees generality and precision in constructing
knowledge representation, with OWL we reach such an expressiveness that we can
“argue” about described information and “extract” more, through _reasoning_.
It provides powerful tools to define classes (that represent concepts),
properties (relations among classes) and individuals (their instances), and
gives the chance to combine them in logic constraints by means of which
necessity and/or sufficiency may be expressed.
Upon a W3C _recommendation_ , OWL is made up of three sublanguages with
increasing expressiveness: OWL Lite, OWL DL and OWL Full. The first two have
an almost total correspondence between two peculiar DLs (resp._
$\mathcal{SHIF}_{(D)}$_and _ $\mathcal{SHOIN}_{(D)}$_).999OWL-Lite and OWL-DL
provide the possibility of defining _annotation_ s, which correspondent DLs do
not do. In any case, annotations affect neither _reasoning_ , nor complexity,
nor decidability. For the sake of precision, OWL-Lite is considered a
“syntactic subset” of OWL-DL. The last one uses the same subset of OWL DL
constructs, but allows them to be unconstrained, according to RDF style. Due
to the lack of restrictions on transitive property and to the possibility of
handling concepts as individuals (_metamodeling_), OWL Full turns out to be
undecidable and thus, for the correspondence between that and any of the
aforesaid DLs, OWL DL is the most expressive decidable sublanguage of
OWL.101010Obviously, nothing excludes there is another decidable and more
expressive−but not studied yet−sublanguage.
OWL 2 extends OWL by enhancing its features and expressiveness. Apart from a
limited number of cases, OWL 2 is perfectly “backward compatible” with the old
OWL (which we call OWL 1 to differentiate it from the OWL 2), i.e., all the
ontologies of the latter keep the same semantics as that they had before, even
if “fed” to a _reasoner_ for OWL 2. Concisely, OWL 2 introduces a
simplification into the writing of the most common _statements_ , it permits
_metamodeling_ , together with some new constructs which increase its
expressiveness, and it extends support to data types.
As for sublanguages, in OWL 2 we can deal with OWL 2 Full and OWL 2 DL. As
mentioned before, the latter corresponds to the _$s\mathcal{ROIQ}_{(D)}$ _
logic. Though the former may be syntactically seen as the union of the latter
with RDFS, it is semantically compatible with OWL 2 DL (insofar as its
semantics allows one to draw all the inferences that can be drawn by using the
semantics of OWL 2 DL). Similarly to the analogue sublanguage OWL 1 Full, OWL
2 finds application in the modelling of concepts where _reasoning_ is not
required. A new feature of OWL 2 is the use of _profiles_ (i.e., fragments of
language) OWL 2 EL, OWL 2 QL and OWL 2 RL:111111In this way, OWL 1 Lite, OWL 1
DL and OWL 2 DL may be considered profiles of OWL 2. the first is useful for
applications involving ontologies that contain a great number of classes
and/or properties; the second aims at those applications with large volumes of
instances, where answering the queries is of primary importance (from which
the name QL, _Query Language_); the last profile is useful in applications not
requiring the sacrifice of too much expressiveness to efficiency. We will not
delve into the structural characteristics of these profiles. Here, we only say
that they are grounded on very different DLs, each of them fit for the
specific purpose they are designed for.
### 1.6 Syntactical correspondence of OWL(2) with DLs
As to its practical representation, an ontology is a RDF/XML-structured text
file, but, for our goals, it will be more useful to think to it in terms of an
RDF graph and, thus, of triples. Table 1.3 describes the correspondence
between the most common constructs of DLs and such triples (or a single
resource, where applicable), which use RDF(S) syntax together with that of
OWL(2). It also contains the names given by OWL to the fundamental concepts
and roles of DLs. It is interesting to notice how a DL construct often
corresponds to more triples, which sometimes makes interpretation quite hard.
The symbols _:x and _list in the Table 1.3 indicate respectively a _blank
node_ and a _list_ (a structure we will not deal with), whereas the prefixes
before the elements of triples represent standard W3C _namespaces_ , which
serve as abbreviations for the URIs they refer to.121212The correspondences
between namespaces and prefixes are: rdf: http://www.w3.org/1999/02/22-rdf-
syntax-ns# rdfs: http://www.w3.org/2000/01/rdf-schema# owl:
http://www.w3.org/2002/07/owl# xsd: http://www.w3.org/2001/XMLSchema#
### 1.7 SPARQL
One of the most widespread languages for querying against ontologies is
_SPARQL_ _(_ recursive acronym for _SPARQL Protocol And RDF Query Language_).
Although it has lots of analogies with SQL, most of which syntactical, it is
equipped with a fundamentally different semantics. As query languages for
databases essentially work on tables and handle logic conditions to select the
rows of these tables that satisfy them, the WHERE clause in SPARQL finds
matches between the triples of the query and those of the ontology indicated
in the FROM clause. The logic assessments are relegated to the FILTER
operator, which possesses a big expressive power, thanks to its several
functions, but is rarely used due to the loss of efficiency that its presence
may cause.131313Depending on implementation, the FROM clause may be implied
because software loads in memory the ontology model separately. The SELECT
clause is very similar to that of SQL. It accepts DISTINCT as a keyword and
the classical aggregation operators (COUNT, SUM, MIN, AVG and MAX), while the
GROUP BY and HAVING clauses are often used (another similarity) to refine the
selection. The involved variables are preceded by the symbol ’?’, whereas
constants do not have a particular syntax, even if they generally coincide
with URI/IRIs of resources present in the ontology one is handling.
Analyzing in finer detail the WHERE clause, one can say that the triples to
match __ are enclosed in a block between braces and separated by a dot, which
implies their intersection. Inside these braces, more sub-blocks may be found,
which are useful in making the union (UNION operator) and the difference
(MINUS operator) between sets. The matching is valid if the variables having
the same names in the clause have the same values in the ontology. There exist
some useful shortenings, such as the use of semicolon or colon in place of
dot, which act respectively in the following way:
?s ?p1 ?o1 ; ?p2 ?o2 . shortens ?s ?p1 ?o1 . ?s ?p2 ?o2 .
?s ?p ?o1 , ?o2 . shortens ?s ?p ?o1 . ?s ?p ?o2 .
In addition to selection, one can also have the DESCRIBE, ASK and CONSTRUCT
query types, which we will not review here. The employment of the PREFIX
clauses, that precede the real query and indicate the abbreviations for
namespaces used inside of it, is quite peculiar.
_DL construct_ | _RDF(S)/OWL(2) resource/triple(s)_
---|---
$\top$ | owl:Thing
$\bot$ | owl:Nothing
$C\sqsubseteq D$ | ($C$ rdfs:subClassOf $D$)
$C\equiv D$ | ($C$ owl:equivalentClass $D$)
$C\sqsubseteq\neg D$ opp. $C\sqcap D\sqsubseteq\bot$ | ($C$ owl:disjointWith $D$)
$C_{1}\sqcap C_{2}\sqcap\ldots\sqcap C_{n}$ | (_:x owl:intersectionOf _list$\left(C_{1},C_{2},\ldots,C_{n}\right)$)
$C_{1}\sqcup C_{2}\sqcup\ldots\sqcup C_{n}$ | (_:x owl:unionOf _list$\left(C_{1},C_{2},\ldots,C_{n}\right)$)
$\neg C$ | (_:x owl:complementOf $C$)
$\left\\{a_{1},a_{2},\ldots,a_{n}\right\\}$ | (_:x owl:oneOf _list$\left(a_{1},a_{2},\ldots,a_{n}\right)$)
$\exists R.C$ | | (_:x owl:someValuesFrom $C$) (_:x owl:onProperty $R$)
---
$\forall R.C$ | | (_:x owl:allValuesFrom $C$) (_:x owl:onProperty $R$)
---
$\exists R.\left\\{a\right\\}$ | | (_:x owl:hasValue $a$) (_:x owl:onProperty $R$)
---
$\leq nR$ | | (_:x owl:minCardinality $n$) (_:x owl:onProperty $R$)
---
$\geq nR$ | | (_:x owl:maxCardinality $n$) (_:x owl:onProperty $R$)
---
$\leq nR\,\sqcap\geq nR$ | | (_:x owl:cardinality $n$) (_:x owl:onProperty $R$)
---
$\leq nR.C$ | | (_:x owl:minQualifiedCardinality $n$)
---
(_:x owl:onClass $C$) (_:x owl:onProperty $R$)
$\geq nR.C$ | | (_:x owl:maxQualifiedCardinality $n$)
---
(_:x owl:onClass $C$) (_:x owl:onProperty $R$)
$\leq nR.C\,\sqcap\geq nR.C$ | | (_:x owl:qualifiedCardinality $n$)
---
(_:x owl:onClass $C$) (_:x owl:onProperty $R$)
$\exists R.Self$ | | (_:x owl:hasSelf _true_) (_:x owl:onProperty $R$)
---
$C\left(a\right)$ | ($a$ rdf:type $C$)
$\left\\{a_{i}\right\\}\equiv\left\\{a_{j}\right\\}$ | ($a_{i}$ owl:sameAs $a_{j}$)
$\left\\{a_{i}\right\\}\sqsubseteq\neg\left\\{a_{j}\right\\}$ | ($a_{i}$ owl:differentFrom $a_{j}$)
$U$ opp. ▽ | owl:topObjectProperty
$N$ opp. △ | owl:bottomObjectProperty
$R\sqsubseteq S$ | ($R$ owl:subPropertyOf $S$)
$R\equiv S$ | ($R$ owl:equivalentProperty $S$)
$R\sqcap S\sqsubseteq N$ | ($R$ owl:PropertyDisjointWith $S$)
$R^{-}$ | (_:x owl:inverseOf $R$)
$R_{1}\circ R_{2}\circ\ldots\circ R_{n}\sqsubseteq R$ | ($R$ owl:propertyChainAxiom _list$\left(R_{1},R_{2},\ldots,R_{n}\right)$)
$\mathit{Refl}\left(R\right)$ | ($R$ rdf:type owl:ReflexiveProperty)
$\mathit{Irrefl}\left(R\right)$ | ($R$ rdf:type owl:IrreflexiveProperty)
$\mathit{Sym}\left(R\right)$ | ($R$ rdf:type owl:SymmetricProperty)
$\mathit{Asym}\left(R\right)$ | ($R$ rdf:type owl:AsymmetricProperty)
$\mathit{Trans}\left(R\right)$ | ($R$ rdf:type owl:TransitiveProperty)
$\mathit{Fn}\left(R\right)$ | ($R$ rdf:type owl:FunctionalProperty)
$\mathit{InvFn}\left(R\right)$ | ($R$ rdf:type owl:InverseFunctionalProperty)
$R\left(a,b\right)$ | ($a$_property_name_ $b$)
$\neg R\left(a,b\right)$ | | (_:x rdf:type owl:NegativePropertyAssertion)
---
(_:x owl:sourceIndividual $a$)
(_:x owl:assertionProperty $R$)
(_:x owl:targetIndividual $b$)
_Table 1.3._ Correspondences between DLs and SW
## 2 ANALYSIS AND RESULTS
### 2.1 The family $\mathcal{DL}$ <$\mathcal{\forall}_{0}^{\pi}$>
Description Logics derived from decidable fragments of set theory, generally
denoted by the notation $\mathcal{DL}$ <language_name>, are having
considerable importance.
The family of logics underlying this class of ontologies, object of research
this project is based on, is focused on the fragment
$\mathcal{\forall}_{0}^{\pi}$ , which has a good expressiveness w.r.t.
knowledge representation in real-world applications. The interest aroused by
that is related to NP-completeness of its decision procedure in some cases of
practical relevance. $\mathcal{DL}$ <$\mathcal{\forall}_{0}^{\pi}$>-concepts
and $\mathcal{DL}$ <$\mathcal{\forall}_{0}^{\pi}$>-roles are formed according
to the syntax141414The symbol $\text{sym}\left(R\right)$ denotes symmetric
closure of $R$.
$\displaystyle C,D$ $\displaystyle\rightarrow$ $\displaystyle A\ |\ \top\ |\
\bot\ |\ \neg C\ |\ C\sqcap D\ |\ C\sqcup D\ |\ \left\\{a\right\\}\ |\ \exists
R.Self\ |\ \exists R.\left\\{a\right\\}$ $\displaystyle R,S$
$\displaystyle\rightarrow$ $\displaystyle P\ |\ U\ |\ R^{-}\ |\ \neg R\ |\
R\sqcap S\ |\ R\sqcup S\ |\ R_{C|}\ |\ R_{|D}\ |\ R_{C|D}\ |\
id\left(C\right)\ |\ \text{{\text{sym}}}\left(R\right)\,,$
while a $\mathcal{DL}$ <$\mathcal{\forall}_{0}^{\pi}$>-KB is made of axioms
and assertions of the following kind:
$\begin{array}[]{c}C\equiv D\ \,\ \,\ C\sqsubseteq D\ \,\ \,\
C\sqsubseteq\forall R.D\ \,\ \,\ \exists R.C\sqsubseteq D\ \,\ \,\
C\left(a\right)\ \,\ \,\ R\left(a,b\right)\\\ R\equiv S\ \,\ \,\ R\sqsubseteq
S\ \,\ \,\ R\circ R^{\prime}\sqsubseteq S\ \,\ \,\ Trans\left(R\right)\ \,\
\,\ Refl\left(R\right)\ \,\ \,\ Asym\left(R\right)\ \,\ \,\ Fn\left(R\right)\
\,\ \,\ InvFn\left(R\right)\,.\end{array}$
We may notice that universal restriction is allowed only in the right part of
a subsumption, whereas existential restriction can appear only in the left
part. In addition, neither numerical, nor qualified restrictions are allowed.
### 2.2 Description of the experiment
Our analysis is aimed at selecting ontologies corresponding to the
$\mathcal{DL}$ <$\mathcal{\forall}_{0}^{\pi}$> family, in order to use them as
a base of study for this family of logics in real-world applications. To do
that, it is necessary to query against as many as possible publicly available
ontologies in a quick and efficient way. The use of queries in SPARQL
conveniently lends itself to the purpose. The management is provided by
dOWLphin, a Java library specifically created in order to easily load the
ontologies and prepare the queries. The underlying library is Jena, one of the
most common collections of API for the Semantic Web, that is very handy
because it can directly deal with triples. As a _front-end_ GUI, the program
QuAny was used, which was born inside the experiment too and allows one to
query against local and remote ontologies, and save on disk the corresponding
results, for future verification.
### 2.3 Results and future objectives
The query was employed to test a significant number of ontologies of
BioPortal, the web portal of the National Center for Biomedical Ontology. This
choice was not random, but motivated mainly by two reasons: the first concerns
the large amount of ontologies present on the portal, coming from repositories
of biomedical resources spread all over the world; the second is relative to
the connection that these ontologies have with the real world in general, and
with human life and medicine in particular, which are fields offering several
matters for reflection on how widely knowledge of so important themes may be
schematized and represented, and, most of all, on the role _reasoning_ may
have in automatically inferring new information.
Around 30% of the ontologies resulted member of the language $\mathcal{DL}$
<$\mathcal{\forall}_{0}^{\pi}$>, which brings good hopes for reasoning on
them, given that−we remind−w.r.t. computational complexity we are in the NP-
completeness realm. Concerning the remaining 70%, efficient algorithms for
conversion will have to be considered and semantic tests will have to be done,
as previously happened for the $\mathcal{DL}$ <$\mathsf{MLSS}_{2,m}^{\times}$
> language (cfr. Cantone _et al._ , [3]).
In Appendix B, a table shows which ontologies were recognized as members of
the language $\mathcal{DL}$ <$\mathcal{\forall}_{0}^{\pi}$>.
> ## References
* Baader & Nutt, [2003] Franz Baader, Werner Nutt, _The Description Logic Handbook: Theory, Implementation, and Applications - 2: Basic Description Logics_ , Cambridge University Press, 2003
* Berners-Lee _et al._ , [2001] Tim Berners-Lee, James Hendler, Ora Lassila, _Il Web Semantico_ in _Le Scienze \- 393_ (Italian translation of _The Semantic Web_ in _Scientific American_), 2001
* Cantone _et al._ , [2010] Domenico Cantone, Cristiano Longo, Antonio Pisasale, _Comparing Description Logics with Multi-Level Syllogistics: the Description Logic $\mathcal{DL}$ <$\mathsf{MLSS}_{2,m}^{\times}$ >_ in _6th Workshop on Semantic Web Applications and Perspectives_ , University of Bozen/Bolzano, 2010
* Cantone _et al._ , [2011] Domenico Cantone, Cristiano Longo, Marianna Nicolosi Asmundo, _A Decidable Quantified Fragment of Set Theory Involving Ordered Pairs with Applications to Description Logics_ in _Computer Science Logic (CSL’11) - 25th International Workshop/20th Annual Conference of the EACSL_ , 2011
* DuCharme, [2011] Bob DuCharme, _Learning SPARQL_ , O’Reilly, 2011
* Hitzler _et al._ , [2009] Pascal Hitzler, Markus Krötzsch, Sebastian Rudolph, _Foundations of Semantic Web Technologies_ , Chapman & Hall, 2009
* Horrocks & Patel-Schneider, [2004] Ian Horrocks, Peter F. Patel-Schneider, _Reducing OWL entailment to description logic satisfiability_ in _Journal of Web Semantics_ , Elsevier, 2004
* Klyne & Carroll, [2004] Graham Klyne, Jeremy J. Carroll, _Resource Description Framework (RDF): Concepts and Abstract Syntax_ on _< http://www.w3.org/TR/rdf-concepts>_, W3C, 2004
* Longo, [2012] Cristiano Longo, _Set Theory for Knowledge Representation_ , Ph.D. Thesis, University of Catania, 2012
* Manola & Miller, [2004] Frank Manola, Eric Miller, _RDF Primer_ su _< http://www.w3.org/TR/2004/REC-rdf-primer-20040210_>, W3C, 2004
* Motik _et al._ , [2009] Boris Motik, Bernardo Cuenca Grau, Ian Horrocks, Zhe Wu, Achille Fokoue, Carsten Lutz, _OWL 2 Web Ontology Language Profiles_ on <_http://www.w3.org/TR/owl2-profiles_ >, W3C, 2009
* Nardi & Brachman, [2003] Daniele Nardi, Ronald J. Brachman, _The Description Logic Handbook: Theory, Implementation, and Applications - 1: An Introduction to Description Logics_ , Cambridge University Press, 2003
* Pan & Horrocks, [2007] Jeff Z. Pan, Ian Horrocks, _RDFS(FA): Connecting RDF(S) and OWL DL_ in _IEEE Transaction on Knowledge and Data Engineering - 19.2_ , 2007
* Pisasale, [2012] Antonio Pisasale, _Web Semantico e Rappresentazione della Conoscenza_ , University of Catania, 2012
* Zolin, [2010] Evgeny Zolin, _Complexity of reasoning in Description Logics_ on _< http://www.cs.man.ac.uk/~ezolin/dl/>_, 2010
## Appendix A ANALYSIS OF THE SPARQL QUERY
> As we are going to see in the implementation, to write declaratively what an
> algorithm could do through a list of well-constructed statements is not an
> immediate operation to undertake. The final query is the result of various
> logical lines of argument concerning the conversion from $\mathcal{DL}$
> <$\mathcal{\forall}_{0}^{\pi}$>-constructs into those of RDF(S)/OWL(2),
> opportunely applied to some test ontologies.
From Table 1.3, one infers that, for the ontology under examination to be
considered part of the family of logics seen before, the allowed elements of
RDF(S)/OWL(2) are the following:
* •
class or property definition (rdf:type)
* •
top concept (owl:Thing)
* •
bottom concept (owl:Nothing)
* •
concept negation (owl:complementOf)
* •
intersection and union of concepts (resp. owl:intersectionOf and owl:unionOf)
* •
singleton (owl:oneOf)
* •
self concept (owl:hasSelf)
* •
value restriction (owl:hasValue)
* •
universal role (owl:topObjectProperty)
* •
role inverse (owl:inverseOf)
* •
intersection and union of roles (not present in OWL 2)151515Obviously, we may
ignore the constructs not present in OWL 2, since they cannot be a cause of
rejection of the ontology.
* •
role restrictions (not present in OWL 2)
* •
identity concept (not present in OWL 2)
* •
symmetric closure (it can be emulated by means of union and inverse of roles)
* •
equivalence axiom between concepts (owl:equivalentClass)
* •
subsumption axiom between concepts (rdfs:subClassOf)
* •
exiistential restriction (owl:someValuesFrom) only in the left part of a
subsumption
* •
universal restriction (owl:allValuesFrom) only in the right part of a
subsumption
* •
minimal numerical restriction (owl:minCardinality) with cardinality not
greater than 1, only in the left part of a subsumption
* •
declaration of concept membership (rdf:type)
* •
declaration of role membership
* •
equivalece between roles (owl:equivalentProperty)
* •
inclusion between roles (owl:subPropertyOf)
* •
role chaining (owl:PropertyChainAxiom)
* •
transitive property (owl:TransitiveProperty)
* •
reflexive property (owl:ReflexiveProperty)
* •
asymmetric property (owl:AsymmetricProperty)
We have to add to these all the ones that can be deduced from a combination of
them, such as
* •
disjunction between concepts (owl:disjointWith, owl:members)
* •
union of disjoint concepts (owl:disjointUnionOf)
* •
equivalence and non-equivalence between individuals (resp. owl:sameAs and
owl:differentFrom)
* •
empty role (owl:bottomObjectProperty)
* •
role domain (rdfs:domain, some cases excluded)161616Role domain corresponds to
the DLs’ axiom $\exists R.\top\sqsubseteq C$, so it is necessary to verify
that $C$ is satisfactorily expressed by the language.
* •
role range (rdfs:range)
* •
disjunction of properties (owl:propertyDisjointWith, owl:members)
* •
irreflexive property (owl:IrreflexiveProperty)
* •
symmetric property (owl:SymmetricProperty)
* •
direct and inverse functional property (resp. owl:FunctionalProperty,
owl:InverseFunctionalProperty)
* •
declaration of role non-membership (owl:NegativePropertyAssertion)
Consequently, the constructs which cause the rejection of an ontology are:
* •
existential restriction not in the left part of a subsumption
* •
universal restriction not in the right part of a subsumption
* •
minimal numerical restriction with cardinality greater than 1 or not in the
left part of a subsumption
* •
qualified numerical restrictions (owl:[max|min]QualifiedCardinality,
owl:qualifiedCardinality)
* •
maximum and exact unqualified numerical restrictions (resp.
owl:maxCardinality, owl:cardinality)
* •
some cases of domain declaration171717Cf. footnote 16.
* •
datatype (rdfs:Datatype, owl:DatatypeProperty)
In order to create the query, we must consider that the language
$\mathcal{DL}$ <$\mathcal{\forall}_{0}^{\pi}$> is very expressive and provides
constructs and axioms which range over several elements of RDF(S)/OWL(2).
Thus, to abbreviate the task of enumerating them, the criterion of selecting
the triples not related to the language was adopted. Consequently, the result
of the query will contain at least a triple if the examined ontology is to be
rejected. There are as many clauses as the elements to consider, connected by
the UNION operator; thus, if at least a match is found, a non empty result
will be obtained, i.e., the analyzed ontology will not be a member of the
language.
Now, let us analyze the correctness of the query in detail (the relative row
number is indicated in brackets, when necessary).
* •
The first clause (lines 7-19) includes the triples corresponding to
existential restrictions (line 8). Yet, not all of them contribute to the
rejection of the ontology, but only those which are not in the left part of a
subsumption. Thus, in line 10 we force their exclusion, provided that any is
found. If this is not the case, we find a match, according to line 8;
otherwise, we must control that neither of the following cases occurs (lines
11-17):181818The second-level MINUS operator could be optimized by including
the various triples contained in its clauses after line 8; nonetheless, we
must not forget that the _matching_ of triples is a notoriusly not very much
efficient operation: the choice of double-nesting concurs to avoid that SPARQL
engine performs useless searches when no subsumption in line 10 is found.
* –
the anonymous node corresponding to a restriction is in the right part of any
triple (line 12): that allows one to include all the axioms with the
restriction and the various operators (e.g., union, intersection, complement,
etc.) which work on lists and single classes in the right part;
* –
the above-mentioned node is in the left part of a triple corresponding to an
axiom (lines 13-16): that allows one to include single and multiple
equivalences and disjunctions.
* •
The second clause (lines 19-33) includes the triples corresponding to
universal restrictions (line 20). When a triple of this kind is found, in
order for a _matching_ to be valid, it is also necessary to check that at
least either of the following cases occurs (lines 22-31):
* –
the anonymous node corresponding to a restriction is in the right part of a
triple (line 23), but this triple is not a subsumption (line 24), nor that
node is the domain of a property (line 25);
* –
the above-mentioned node is in the left part of a triple (line 27), but it is
not part of the definition of the restriction itself (lines 28-30) (otherwise,
an ever-false condition would be represented): that is to control that the
restriction is neither in the left part of an axiom, nor in the left part of
some class construct.
* •
The third clause (lines 33-46) includes the triples corresponding to an
operator of minimal cardinality (line 34). Here we proceed (line 36 and lines
39-43) as in the first clause, with the difference (line 37) that the
cardinality value must be not greater than 1 too.191919Footnote 18 holds also
for the nested MINUSes.
* •
The remaining clauses (lines 47-51) check for _datatype_ (lines 47 and 48),
qualified cardinality (line 49: the onClass property is peculiar to this kind
of restriction), exact and maximum cardinality (lines 50 and 51) triples.
* •
For the purpose of accelerating the execution, the use of the LIMIT operator
(line 52) reduces the size of results to only one element, which is enough to
reject the ontology.
⬇
1PREFIX rdf: <http://www.w3.org/1999/02/22-rdf-syntax-ns#>
2PREFIX rdfs: <http://www.w3.org/2000/01/rdf-schema#>
3PREFIX owl: <http://www.w3.org/2002/07/owl#>
4
5SELECT *
6WHERE {
7 {
8 ?left owl:someValuesFrom ?class .
9 MINUS {
10 ?left rdfs:subClassOf ?right .
11 MINUS {
12 { ?s ?p ?left . } UNION
13 { ?left owl:equivalentClass ?right . } UNION
14 { ?left owl:disjointWith ?right . } UNION
15 { ?left owl:members ?right . } UNION
16 { ?left owl:disjointUnionOf ?right . }
17 }
18 }
19 } UNION {
20 ?restr owl:allValuesFrom ?class .
21 {
22 {
23 ?x ?prop ?restr .
24 MINUS { ?x rdfs:subClassOf ?restr . }
25 MINUS { ?x rdfs:domain ?restr . }
26 } UNION {
27 ?restr ?prop ?x .
28 MINUS { ?restr rdf:type ?x . }
29 MINUS { ?restr owl:onProperty ?x . }
30 MINUS { ?restr owl:allValuesFrom ?x . }
31 }
32 }
33 } UNION {
34 ?left owl:minCardinality ?num .
35 MINUS {
36 ?left rdfs:subClassOf ?right .
37 FILTER (?num<=1) .
38 MINUS {
39 { ?s ?p ?left . } UNION
40 { ?left owl:equivalentClass ?right . } UNION
41 { ?left owl:disjointWith ?right . } UNION
42 { ?left owl:members ?right . } UNION
43 { ?left owl:disjointUnionOf ?right . }
44 }
45 }
46 } UNION
47 { ?s ?p owl:DatatypeProperty . } UNION
48 { ?s ?p rdfs:Datatype . } UNION
49 { ?s owl:onClass ?o . } UNION
50 { ?s owl:cardinality ?o . } UNION
51 { ?s owl:maxCardinality ?o . }
52} LIMIT 1
> ## Appendix B TEST AGAINST BIOPORTAL ONTOLOGIES
The SPARQL query was tested against the ontologies of BioPortal, available at
the link http://rest.bioontology.org/bioportal/virtual/download/ID?apikey=KEY.
The API KEY is an identifier for the registered users. The following table
associates the ID of the previous URL to the symbolic name of the
corrispondent ontology and highlights the ones which belong to the language.
_ID_ | _Name_ | _Res_ | _ID_ | _Name_ | _Res_ | _ID_ | _Name_ | _Res_
---|---|---|---|---|---|---|---|---
1033 | NMR Metabolomics Investig. | Y | 1362 | Hymenoptera Anatomy | N | 1552 | Reprod. Trait and Phenotype | Y
1039
& Proteomics Data N 1369 PhysicalFields Y 1560 Cognitive Paradigm N 1052
Proteins N 1381 NIF Dysfunction Y 1565 Medical Social Entities Y 1054 Amino-
acid N 1393 Information Artifacts N 1567 Pharmacovigilance Y 1056 Basic
Vertebrate Anatomy N 1394 Syndromic Surveillance N 1569 Host Pathogen
Interactions N 1058 SNP N 1398 Language Disorder in Autism Y 1570 Traditional
Med. Constitution Y 1059 Computer-based Patient Record N 1399 Pilot Ontology N
1571 Traditional Med. Other Factors Y 1060 Epoch Clinical Trial N 1401 Nursing
Practice N 1572 Trad. Med. Signs and Symptoms Y 1061 Pharmacogenomics N 1402
NIF Cell N 1573 Traditional Med. Meridian Y 1068 Subcellular Anatomy N 1406
LinkingKing2PEP N 1576 FDA Med. Devices N 1082 Gene Regulation (GRO) N 1407
Description of Dynamics N 1578 HOM-Helixhauser Scores N 1083 NanoParticles N
1409 PKO Re Y 1580 Adverse Event Reporting N 1084 NIFSTD Y 1410 Kinetic
Simulation Algorithm N 1581 Health Indicators N 1086 Disease Genetic
Investigation N 1411 Functioning, Disability and Health N 1582 CAO N 1087
Geographical Regions Y 1413 Software N 1588 General Purpose Datatypes N 1088
MaHCO N 1414 General Medical Science Y 1596 HOM_MDCs-DRGS N 1089 BIRNLex N
1415 CTCAE N 1613 Bone Dysplasia N 1092 Infectious Diseases N 1417 Influenza N
1615 Chemogenomics N 1100 Genetic Intervals N 1418 TOK N 1616 Phylogenetics Y
1104 Biomedical Resource N 1438 Breast tissue cell lines N 1625 HOM-ICD9PCS N
1106 Gene Regulation (BOOTStrep) N 1439 General Formal (GFO) N 1627 HOMERUN
Metadata N 1116 Bleeding History Phenotype N 1440 General Formal (GFO-Bio) N
1629 HOM-UCARE Demographics N 1122 Skin Physiology N 1444 Chemical Information
N 1631 HOM-Harvard N 1123 Biomedical Investigations N 1461 Translational
Medicine N 1633 Cognitive Alias N 1126 Family Health History N 1484 External
Causes of Injuries N 1638 Data Mining Y 1128 Comparative Data Analysis N 1487
Body System Y 1639 Epilepsy N 1130 Cancer Research and Mgmt N 1488 SysMO-JERM
N 1640 Pediatric Terminology N 1131 MGED N 1489 Adverse Events N 1641 HOM-
ICD9CM-ECODES N 1134 BioTop N 1494 Tissue Microarray N 1642 HOM-DXPROCS MDCDRG
N 1136 Experimental Factors Y 1497 PMA 2010 N 1643 HOM-ICD9_PROCS OSHPD N 1141
Physics for Biology N 1500 RNA N 1648 HOM-DATASOURCE OSHPD N 1142 Cardiac
Electrophysiology Y 1501 Neomark Oral Cancer-based N 1650 Units Y 1146
Electrocardiography N 1505 MicroRNA Target Prediction N 1652 HOM-OSHPD Use
cases N 1149 Dermatology Lexicon N 1515 Interaction Network Y 1653 HOM-PROCS2
OSHPD N 1172 Vaccines N 1521 Neural Motor Recovery Y 1658 Hewan Invertebrata N
1183 Lipids N 1522 BioPAX N 1665 Student Health Record N 1190 Parasite
LifeCycle N 1523 OBOE-SBC N 1666 Emotion N 1192 Proteomics Pipeline
Infrastructures N 1524 OBOE N 1667 HOM-DATASOURCE OSHPDSC N 1237 Situation-
based Access Control N 1530 Animal natural history N 1668 HOM-OSHPD-SC N 1247
GeoSpecies N 1532 SemanticScience Y 1671 Quantitative Imaging Biomarkers N
1249 Smoking Behavior Risk N 1533 BioAssay N 1672 DIKB-Evidence N 1290 ABA
Adult Mouse Brain Y 1534 Apollo-akesios Y 1676 Randomized Controlled Trials N
1304 Breast Cancer Grading N 1537 Brucellosis N 1686 Neomark Oral Cancer Y
1314 Cell Line (2) N 1538 Roles Y 1696 Synapses Y 1321 Neural ElectroMagnetic
Ontology N 1540 Drug Discovery Investigations N 3002 Mental Functioning Y 1332
Basic Formal Ontology Y 1541 Cell Line (MCCL 2) N 1335 Parasite Experiments N
1550 PHARE N
|
arxiv-papers
| 2012-11-21T08:18:54 |
2024-09-04T02:49:38.251119
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Antonio Pisasale, Domenico Cantone",
"submitter": "Antonio Pisasale",
"url": "https://arxiv.org/abs/1211.4957"
}
|
1211.5075
|
# Controlling cell-matrix traction forces by extracellular geometry
Shiladitya Banerjee1 and M. Cristina Marchetti1,2 1 Department of Physics,
Syracuse University, Syracuse New York, 13244-1130, USA 2 Syracuse
Biomaterials Institute, Syracuse University, Syracuse New York, 13244-1130,
USA [email protected]
###### Abstract
We present a minimal continuum model of strongly adhering cells as active
contractile isotropic media and use the model to study the effect of the
geometry of the adhesion patch in controlling the spatial distribution of
traction and cellular stresses. Activity is introduced as a contractile, hence
negative, spatially homogeneous contribution to the pressure. The model shows
that patterning of adhesion regions can be used to control traction stress
distribution and yields several results consistent with experimental
observations. Specifically, the cell spread area is found to increase with
substrate stiffness and an analytic expression for the dependence is obtained
for circular cells. The correlation between the magnitude of traction stresses
and cell boundary curvature is also demonstrated and analyzed.
## 1 Introduction
Living cells actively sense and respond to the physical geometry and stiffness
of their environment, which in turn affects a variety of cellular processes,
such as growth, differentiation, morphogenesis, spreading and motility [1].
Cell-matrix adhesion is mediated by integrin complexes, referred to as focal
adhesions, that bind to specific ligands on the underlying matrix. Focal
adhesions are mechanically linked to the actomyosin cytoskeleton inside the
cell, that in turn generates contractile forces on the extracellular matrix.
The interplay between substrate stiffness, intracellular contractility and the
extracellular adhesion forces controls the cell morphology and its mechanical
behavior. For instance, cells adhering to soft substrate are generally found
to spread less and have round morphology, while cells on stiff substrates have
greater spread area with more branched shapes [2].
Powerful techniques have been developed in recent years to measure the
traction forces exerted by adherent cells on compliant substrates [3].
Traction Force Microscopy is used to probe the traction stresses exerted by
cells on continuous elastic gels. The stresses are inferred from measurements
of the displacements of fiducial markers embedded in the gel before and after
cell detachment [4, 5]. In a second technique cells plated on microfabricated
pillar arrays induce bending of the elastic micropillars. The traction forces
are then obtained from by assuming a linear Hooke’s law relation between the
measured bending and the forces [6]. These experiments have demonstrated that
the mechanical response of adherent cells is controlled by a complex interplay
of substrate stiffness and geometry, myosin activity and extracellular matrix
proteins. Adhesive micro patterning has also been used as a tool for both
controlling cell shape and study the interplay between shape and cytoskeletal
organization and architecture [7]. These studies have shown that when strongly
adhesive patterns force the cell boundary to exhibit regions of high
curvature, traction stresses tend to be concentrated in these regions, while
stress fibers develop along cell boundaries linking non-adhesive zones,
confirming the crucial role of the cytoskeletal contractility and architecture
in controlling cellular stresses and morphology [8].
The role of adhesion geometry in controlling traction force distribution has
been addressed theoretically using network models and continuum mechanical
models. While models of continuum mechanical elements coupled to bio-chemical
agents have been used before to describe the traction force distribution by
adherent cells [9], continuum minimal models inspired by thermoelasticity [10]
or active gel theory [11] have recently provided new key analytical results.
Network models of the contractile cytoskeleton have also been used to describe
the relation between force distribution and shape of adherent cells [12, 13],
including networks of Hookean springs as well as cable networks that
incorporate the asymmetry of the elastic response of biopolymers such as
filamentary actin to compression version extension, with and without the
explicit inclusion of contractility. In particular, the active cable network
reproduces the arc morphology of cell boundaries pinned by strong local
adhesions that has been seen in experiments [14]. The relationship between
cell shape and adhesion geometry has also been studied by modeling cells as
contractile films bounded by the elastic cortex [15, 16, 17]. In this paper we
consider a continuum model of cells as linear, active elastic media and
demonstrate that the introduction of activity as a _spatially homogeneous_
contractile, hence negative, contribution to the pressure is sufficient to
reproduce the spatial inhomogeneous distribution of traction and cellular
stresses observed in experiments for a number of cell geometries. An
interesting extension of our work will be to introduce nonlinearity in the
continuum model to incorporate an asymmetric response to compression and
stretching. This asymmetry, arising from the nonlinear force-extension curve
of actin filaments, is known to be important in controlling the contractile
behavior of isotropic gels [18, 19] and may alter the stress distribution in
adhering cells.
In the next section we introduce our continuum model of adherent cells as
active contractile elastic media. We then use the model to study the effect of
the geometry of the adhesion region on controlling the spatial distribution of
stresses in the cell. The model can be solved analytically for a circular
cell, where we obtain an expression for the cell spread area as a function of
substrate stiffness and show that our results compares favorably to
experiments (inset of Fig. 2). The cases of elliptical, square and triangular
cells are solved numerically. We show that the geometry of the adhesive region
strongly affect the stress distribution, with traction stresses concentrated
in regions of high curvatures or at sharp corners (Fig. 1). In section 3.3 we
provide an analytical argument that quantifies the correlation between
traction stress magnitude and curvature of the cell boundary and discuss in
section 3.4 the relative roles of shear and compressional deformations in
controlling the stress distribution. We conclude with a brief discussion.
## 2 Adherent cell as a contractile gel
We consider a stationary cell adhering to an elastic substrate via stable
focal adhesion complexes. We further assume that the cell has attained its
optimum spread area on the substrate, with an average height $h$ much thinner
than its perimeter. In mechanical equilibrium, the condition of local force-
balance translates to $\partial_{\beta}\sigma_{\alpha\beta}=0$, where
${\bm{\sigma}}$ is the three-dimensional stress tensor of the cell with greek
indices taking values $x,y$ and $z$. For a thin cellular film we average the
cellular force-balance equation over the cell thickness $h$. In-plane force
balance is given by
$\partial_{j}\sigma_{ij}+\partial_{z}\sigma_{iz}=0\;,$ (1)
with $i,j$ denoting in-plane coordinates. We assume that the top surface of
the cell is stress free, $\sigma_{iz}({\bf r}_{\perp},z=h)=0$, whereas at the
cell-substrate interface $z=0$, the cell experiences lateral traction stresses
given by $\sigma_{iz}({\bf r}_{\perp},z=0)=Yu_{i}({\bf r}_{\perp},z=0)$. Here,
$Y$ denotes the substrate rigidity parameter, representing the cell-substrate
anchoring strength, and ${\bf u}({\bf r}_{\perp},z)$ is the in-plane
deformation field of the cellular medium. The thickness-averaged force balance
equation then reads [11, 10],
$h\partial_{j}\overline{\sigma}_{ij}=Yu_{i}\;,$ (2)
where $\overline{\sigma}_{ij}({\bf
r}_{\perp})=\int_{0}^{h}(dz/h)\sigma_{ij}({\bf r}_{\perp},z)$. It is
worthwhile to mention that the assumption of in-plane traction forces is a
good approximation for fully spread stationary cells making almost zero
contact angle with the substrate. During the early stages of spreading and
migration, cells can exert appreciable out-of-plane traction forces via
rotation of focal adhesions [20]. In the following we will drop the overbear
indicating the average and refer to thickness averaged quantities throughout.
The quantity $T_{i}=Yu_{i}$ is a stress in three dimensions, i.e., a force per
unit area. It describes the in-plane traction force per unit area that the
cell exerts on the substrate. The assumption of local elastic interactions
with the substrate strictly holds on elastic substrates that are much thinner
than the lateral size of the cell [21] or on micropillar substrates [10]. The
substrate rigidity parameter $Y$ depends on the stiffness of the underlying
substrate as well as on the density $\rho_{f}$ and stiffness $k_{f}$ of focal
adhesions. For an elastic substrate of shear modulus $\mu_{s}$ and thickness
$h_{s}$, $Y$ takes the simple form [21],
$Y^{-1}=\frac{1}{k_{f}\rho_{f}}+\frac{1}{\mu_{s}/h_{s}}$.
We model the cell as an isotropic and homogeneous elastic material with
additional internal active stresses due to actomyosin contractility. The
constitutive relation for the cellular stress tensor is then given by,
$\sigma_{ij}=\frac{E}{2(1+\nu)}\left(\frac{2\nu}{1-2\nu}\bm{\nabla}\cdot{\bf
u}\
\delta_{ij}+\partial_{j}u_{i}+\partial_{i}u_{j}\right)+\sigma_{a}\delta_{ij}\;,$
(3)
where $E$ and $\nu$ denote the Young modulus and Poisson ratio of the cellular
material, respectively. Actomyosin contractility is modeled as a negative
contribution to the local pressure, corresponding to $\sigma_{a}>0$. The
assumption of linear elasticity is valid on time scales shorter than
cytoskeletal turnovers, that are indeed slowed down by strong adhesion to the
substrate. Equations (2) and (3), subject to the boundary condition
$\sigma_{ij}n_{j}|_{\Omega}=0$, wtih $\Omega$ the cell boundary and ${\bf n}$
the outward unit normal on $\Omega$, completely describe the equilibrium of an
adherent cell. As a consequence of the stress free condition at the lateral
cell boundary, the net traction force transmitted by the cell to the substrate
vanishes, i.e., $\int_{A}d^{2}{\bf r}\ Yu_{i}=\oint_{\Omega}ds\
\sigma_{ij}n_{j}=0$. It is instructive to consider two limiting cases for the
anchoring strength. When the cell is rigidly anchored onto the substrate,
corresponding to $Y\rightarrow\infty$, we find ${\bf u}=0$, defining the
reference state for elastic deformations. In our model the reference cell
shape is then dictated by the geometry of the adhesion patch, which can be
controlled in experiments by micropatterning substrates by adhesion proteins.
In contrast, when $Y\rightarrow 0$, the cell does not adhere to the substrate
and the equilibrium state is uniformly contracted state, with a density
enhancement $\delta\rho=-\bm{\nabla}\cdot{\bf
u}=\sigma_{a}(1+\nu)(1-2\nu)/E(1-\nu)$. In the following, we investigate
analytically and numerically solutions of the cell elasticity equations (2)
and (3) subject to stress-free boundary conditions in various planar
geometries.
## 3 Results
### 3.1 Spatial distribution of traction stresses is sensitive to adhesion
geometry
Figure 1: Equilibrium cell shapes for various adhesion patterns : Circle (top
left), ellipse (top right), square (bottom left) and equilateral triangle
(bottom right). The color map indicates magnitude of the traction $|{\bf
T}|=Y|{\bf u}|$, and the arrows demote the direction of the traction vectors.
The reference shapes for all the four patterns have an equal area of $1000\
\mu m^{2}$. The other parameters are: $E=1~{}{\rm kPa}$, $\nu=0.4$,
$\sigma_{a}=1~{}{\rm kPa}$, $\mu_{s}=10~{}{\rm kPa}$, $h_{s}=30~{}\mu{\rm
m}m$, $h=0.2~{}\mu{\rm m}$.
The spatial distribution of traction stresses exerted by cells on substrate
and the corresponding organization of stress and deformation inside the cell
are affected by the geometry of adhesive patterns. Using micropatterning
techniques, cell shapes can be constrained to adhere to controlled geometrical
patterns [22, 23]. In our model the shape determined by the pattern in the
limit of infinite adhesion strength provides the reference shape for the cell.
Here we investigate four reference cell shapes: circle, ellipse, square and
equilateral triangle. These are chosen to have the same reference area but
different perimeters. The case of a circular cell can be treated analytically,
as described below. For the other shapes the elasticity equations (2) and (3)
are solved numerically using the MATLAB pde toolbox. We assume the
contractility $\sigma_{a}$ to be uniform and of order of the cellular Young’s
modulus. Heatmap of traction stresses are shown in Fig. 1. In all cases the
traction stresses are concentrated at the cell periphery, irrespective of the
reference shape. The magnitude of the local traction stress is, however,
higher in regions of high curvatures or at sharp corners.
For a circular cell, Eqs. (2) and (3) can be solved analytically [10, 24].
Assuming in-plane rotational symmetry, it is convenient to use polar
coordinates $r$ and $\theta$, denoting radial and angular coordinates, and
demand that no quantity depend on $\theta$. The equation for the radial
displacement $u_{r}$ about a circular reference state of radius $R_{0}$, is
then given by
$r^{2}\partial_{r}^{2}u_{r}+r\partial_{r}u_{r}-(1+r^{2}/\ell_{p}^{2})u_{r}=0\;,$
(4)
where the penetration length $\ell_{p}$ describes the localization of traction
stresses at the cell boundary. It is given by :
$\ell_{p}^{2}=\frac{Eh(1-\nu)}{Y(1+\nu)(1-2\nu)}\;,$ (5)
and is essentially controlled by the ratio of the cell stiffness $\sim E$ to
the substrate rigidity $\sim Y$. The penetration length is short on stiff
substrates and increases with decreasing substrate rigidity. The solution of
Eq. (4) with the boundary conditions $\sigma_{rr}(r=R_{0})=0$ and
$u_{r}(r=0)=0$ is given in terms of modified Bessel functions of the first
kind as,
$u_{r}(r)=-\sigma_{a}R_{0}\left[\frac{(1+\nu)(1-2\nu)}{E(1-\nu)}\right]I_{1}(r/\ell_{p})g(R_{0}/\ell_{p})\;,$
(6)
with $g(s)=\left[sI_{0}(s)-\frac{1-2\nu}{1-\nu}I_{1}(s)\right]^{-1}$. As
anticipated, the deformation $u_{r}$ vanishes for all $r$ when
$Y\rightarrow\infty$, when the adhering circular cell is maximally spread and
has its largest undeformed radius $R_{0}$.
### 3.2 Cell spread area is sensitive to substrate stiffness and
contractility
The optimal spread area of the cell is controlled by the interplay between
cell contractility, as described by the active pressure $\sigma_{a}$, and the
traction forces on the substrate. In the case of a circular cell, where the
deformation induced by adhesion is given by Eq. (6), the steady state cell
area is given by,
$A=\pi(R_{0}+u(R_{0}))^{2}\;,$ (7)
with $R_{0}$ the reference radius corresponding to the maximal spread area
$A_{\infty}=\pi R_{0}^{2}$ attained on an infinitely rigid substrate, where
$u_{r}(r)=0$. To make contact with experiments, we investigate the ratio
$A/A_{\infty}$, the relative cell spread area, as a function of substrate
stiffness and contractility.
Figure 2: Optimal shape of a triangular cell for different values of the
active pressure $\sigma_{a}$ and the substrate shear modulus $\mu_{s}$, with
$E=1~{}{\rm Pa}$. The color map represents the magnitude of the displacement
vector $|{\bf u}|$ (proportional to the traction force) about an equilateral
triangular reference shape of area $1000~{}\mu{\rm m}^{2}$. The cell spread
area increases with increasing substrate stiffness and decreases with
increasing $\sigma_{a}$. Inset (Left) : Least-square fit of the relative cell
spread area $A/A_{\infty}$ obtained from the model using Eq. (7) (solid) to
the experimental data reported in Ref. [25] (solid red circles). The fitting
parameters are $E=911~{}{\rm Pa}$ and $\sigma_{a}=1589~{}{\rm Pa}$. Inset
(Right) : Relationship between cellular Young’s modulus $E_{c}$ and
contractility $\sigma_{a}$. Here we tune $\sigma_{a}$ to desired values and
then determine the fitting parameter $E_{c}$ using data in Ref. [25]. Other
parameters : $\nu=0.4$, $h_{s}=30~{}\mu{\rm m}$, $h=0.2~{}\mu{\rm m}$.
On stiff substrates, where $R_{0}\gg\ell_{p}$, i.e., the traction stress
extends over a length much smaller than the reference cell radius,
$u_{r}(R_{0})\simeq-\sigma_{a}\ell_{p}/B$, where the compressional modulus $B$
is given by $B=E(1-\nu)/\left[(1+\nu)(1-2\nu)\right]$. The relative spread
area then takes the simple form
$A/A_{\infty}\simeq\left(1-\frac{\sigma_{a}}{R_{0}}\sqrt{h/BY}\right)^{2}$.
Letting $Y\simeq\mu_{s}/h_{s}$, we note that increasing substrate stiffness
increases relative spread area, with $A/A_{\infty}\rightarrow 1$ as
$\mu_{s}\rightarrow\infty$, in qualitative agreement with experiments [2, 26,
25]. In contrast, increasing the contractile pressure $\sigma_{a}$ reduces the
optimal cell spread area, consistent with the experimental observation that
myosin-II activity retards cell spreading [27]. To make a quantitative
comparison with experiments, we fit Eq. (7) to experimentally reported data on
the projected area of cardiac myocytes cultured on N-cadherin coated PA gels
of varying stiffnesses [25]. Here the maximal spread area $A_{\infty}$ is
taken to be equal to the cell projected area on a glass substrate (shear
modulus $\sim$ 30 GPa), which is $\simeq 690\ \mu m^{2}$. The fit, shown in
the left inset of Fig. 2, is obtained using the active contractility
$\sigma_{a}$ and the cellular Young’s modulus $E$ as the fitting parameters. A
least-square fit gives us $E=911~{}{\rm Pa}$ and $\sigma_{a}=1589~{}{\rm Pa}$.
Although the strength of contractility is likely to depend on cell type, it is
worth highlighting that the fit value for $\sigma_{a}$ is of the same order of
magnitude as previously used in Ref. [24] to fit the measured value of the
surface tension of a colony of epithelial cells. Next, we tune the
contractility $\sigma_{a}$, which can be artificially controlled through
pharmacological interventions, and determine the corresponding best fit value
of the cellular Young’s modulus $E_{c}$. Our result (Fig. 2, right inset)
indicates a linear relationship between the cellular Young’s modulus and the
contractile stress. There are indeed experimental data available [28] that
show that the cell stiffness increases linearly with contractility for
adherent cells 111We thank the anonymous referee for suggesting this fit and
pointing out to us Ref. [28].. This suggests that our model could be used to
infer contractility from measurements of cellular stiffness. Figure 2 also
demonstrates the competing roles of contractility and adhesion in controlling
optimal cell shapes for a chosen triangular reference state. On softer
substrates the triangular cell retains its topology and contracts by an amount
proportional to $\sigma_{a}$, whereas on stiffer substrates the corners tend
to form protrusions.
### 3.3 Traction forces increase with cell boundary curvature
Figure 3: (a) Force-balance on a thin slice of cellular material at the cell
boundary. (b) Force-balance at a generic sharp corner with opening angle
$\phi$. (c) Traction stress magnitude at the cell edge as a function of the
local curvature $\kappa$ for the elliptical cell of Fig. 1.
When the boundaries of the adhesion pattern exhibits non-uniform curvature,
the traction stresses are higher at regions of high curvatures. This is seen
for example in Fig. 1 for the case of an elliptical reference shape. To
justify this we propose a simple analytical argument based on local force
balance. Consider a thin slice of cellular material at the cell periphery of
width comparable to penetration length $\ell_{p}$ and arc length
$R\Delta\theta$ much less than the cell perimeter (Fig. 3(a)), with $1/R$ the
local curvature of the cell element. At the outer edge of this element, the
only force on the cell is the reaction to the traction by the cell on the
substrate traction, of areal density $-{\bf T}$, with ${\bf T}=Y{\bf u}$. This
yields an outward total force on the outer edge of the cell element of
magnitude $TR\Delta\theta\ell_{p}$, with $T>0$. At the interior edge, the
cellular element experiences a contractile force of magnitude
$\sigma_{n}(R-\ell_{p})\Delta\theta\ell_{p}$, where $\sigma_{n}$ is the normal
stress pulling the inner contour inwards and has contributions from active as
well as passive elastic stresses. The lateral stresses $\sigma_{t}$
contributes to an effective line tension $\sigma_{t}\ell_{p}R\Delta\theta$ of
the cell element. Due to the curvature of the boundary element, the line
tension generates an inward Laplace pressure of magnitude
$\sigma_{t}\ell_{p}/R$. Local balance of forces then yields,
$TR\Delta\theta\ell_{p}-\sigma_{n}(R-\ell_{p})\Delta\theta\ell_{p}=R\Delta\theta\ell_{p}\sigma_{t}\frac{\ell_{p}}{R}\;.$
(8)
The above law can be written down in a compact form as,
$T=\sigma_{n}+(\sigma_{t}-\sigma_{n})\ell_{p}\kappa\;,$ (9)
with $\kappa=1/R$, the local curvature of the boundary element. Equation (9)
then tells us that local magnitude of traction increases linearly with
increasing boundary curvature. The lateral and normal stresses $\sigma_{t}$
and $\sigma_{n}$ can be expressed in terms of the local cellular stresses in
polar coordinates as
$\sigma_{t}=\sigma_{\theta\theta}-\partial_{\theta}\sigma_{r\theta}$ and
$\sigma_{n}=\sigma_{rr}$. The linear dependence of $T$ on $\kappa$ strictly
holds in the limit $\ell_{p}\kappa\ll 1$. In addition, non-local elastic
interactions can also affect the dependence of traction magnitude on local
curvature. Figure. 3(c) shows the dependence of the magnitude of the traction
stress at the cell boundary on local curvature for an elliptical cell as shown
in Fig. 1. For low $\kappa$, the traction stress magnitude increases linearly
with $\kappa$ before reaching a plateau at higher values of $\kappa$.
When the cell boundary exhibits a sharp corner with opening angle $\phi$, as
shown in Fig. 3(b), the local force-balance is given by,
$T=\sigma_{n}+2\sigma_{t}\cos{(\phi/2)}\;,$ (10)
where $\sigma_{n}$ acts along the bisecting line of the corner. Hence smaller
the opening angle, the larger is the traction force.
### 3.4 Mechanical anisotropy induced by geometric anisotropy
Figure 4: Cell shape anisotropy correlates with internal stress anisotropy.
(a) Heatmap of internal compressive stress $\sigma$ (left) and maximum shear
stress $\sigma_{s}$ (right) corresponding to various reference shapes :
circle, ellipse, square and equilateral triangle. The reference shapes all
have an equal area of $1000\ \mu m^{2}$. (b) Average maximum shear
$\bar{\sigma}_{s}$ as a function of eccentricity $e$ for elliptical cells of
same reference area ($1000\ \mu m^{2}$). Equilibrium shapes with colorplot of
$\mu$ are given as plot markers. Parameters : $E=1$ kPa, $\nu=0.4$,
$\sigma_{a}=1$ kPa, $\mu_{s}=10$ kPa, $h_{s}=30\ \mu m$, $h=0.2\ \mu m$.
The spatial distribution of internal stresses $\sigma_{ij}$ within the cell
depends on cell shape, which is in turn controlled by the geometry of the
adhesive region. Experimentally ${\bm{\sigma}}(x,y)$ can be obtained from the
measured distribution of traction stresses ${\bf T}(x,y)$, inverting the local
force-balance condition $\partial_{j}\sigma_{ij}=T_{i}$ [29]. The elasticity
equations Eqs. (2) and (3) can be recast as a single partial differential
equation for the internal stress tensor $\sigma_{ij}$, given by
$\ell_{p}^{2}\left[\partial_{i}\partial_{k}\sigma_{kj}\right]^{S}+\delta_{ij}\sigma_{a}=\sigma_{ij}+\frac{1-2\nu}{\nu}\delta_{ij}\left(\sigma_{kk}-2\sigma_{a}\right)\;,$
(11)
where $[...]^{S}$ denotes symmetrization with respect to indices that are not
summed over, i.e.,
$\left[\partial_{i}\partial_{k}\overline{\sigma}_{kj}\right]^{S}=\frac{1}{2}\left[\partial_{i}\partial_{k}\overline{\sigma}_{kj}+\partial_{j}\partial_{k}\overline{\sigma}_{ki}\right]$.
We have investigated numerically the solution of Eq. (11) with stress free
boundary condition $\sigma_{ij}n_{j}=0$. To understand the role of shear and
compressional deformations in different geometries, it is instructive to
diagonalize the stress tensor and display the results in terms of linear
combinations of the eigenvalues $\sigma_{1}$ and $\sigma_{2}$. The sum
$\sigma=\frac{1}{2}(\sigma_{1}+\sigma_{2})$ is simply half the trace of the
stress tensor and describes compressional deformations. The difference
$\sigma_{s}=\frac{1}{2}|\sigma_{1}-\sigma_{2}|=\sqrt{[\sigma_{xx}-\sigma_{yy}]^{2}+4\sigma_{xy}^{2}}$,
is controlled by normal stress $\sigma_{xx}-\sigma_{yy}$ and shear stress
$\sigma_{xy}$. For an isotropic reference shape, such as the circle,
$\sigma_{1}=\sigma_{2}$ and $\sigma_{s}=0$, whereas for anisotropic shapes
such as the ellipse, one expects nonzero values for the local maximum shear
$\sigma_{s}$.
Fig. 4(a) shows heatmaps of the spatial distribution of $\sigma$ and
$\sigma_{s}$ for various reference shapes - circle, ellipse, square and
equilateral triangle. Irrespective of the shape of the adhesion geometry,
$\sigma$ is maximum at the cell center, indicating build-up of compressive
stresses. The compressional stress $\sigma$ always vanishes at the boundary,
and it does so more rapidly at regions of high curvature or at sharp corners.
In contrast, the shear stress $\sigma_{s}$ is identically zero for isotropic
shapes, defined as those that have a gyration tensor that is diagonal, with
equal eigenvalues. The circle, triangle and square are all in this class.
Local stress anisotropy as measured by $\sigma_{s}$ is nonzero for elliptical
shapes and shear stresses build up at the center of the ellipse. The shape
anisotropy of ellipses can be quantified by their eccentricity
$e=\sqrt{1-(b/a)^{2}}$, with $a$ and $b$ the semi-major and semi-minor axes.
Figure 4(b) shows the spatial average of $\sigma_{s}$ over the area $A$ of the
cell, defined as $\bar{\sigma}_{s}=\frac{1}{A}\int_{A}d^{2}{\bf r}\
\sigma_{s}$, as a function of the eccentricity $e$. The average shear stress
$\bar{\sigma}_{s}$ increases with $e$ with a sharp rise as $e\rightarrow 1$,
indicating a positive relationship between geometrical and mechanical
anisotropy in adherent cells. Our theoretical model thus confirms the
experimental result that cell mechanical anisotropy increases with increasing
aspect ratio, as previously reported for single endothelial cells with the
same spread area [30].
## 4 Discussion
We have used a continuum model of an adherent cell on a substrate as an active
contractile medium to study the role of adhesion geometry in controlling cell
shape, cell spreading and the spatial distribution of traction stresses. More
realistic future modeling should take into account that a cell is a highly
heterogeneous material with spatially varying stiffness [31]. It is however
intriguing to note that the simplified assumption of homogeneity and isotropy
in the underlying cytoskeletal network can reproduce several of the known
experimental results. The central input of the model is the cell contractility
or activity $\sigma_{a}$, a negative contribution to the pressure that enters
the constitutive equation for the cellular material. In general, $\sigma_{a}$
will be determined by the concentration and activity of myosin proteins cross
linking the actin cortex and controlling the formation of stress fiber. In our
model $\sigma_{a}$ is assumed to be a constant parameter, to be determined by
fitting experiments. We consider cells adhering to flat substrates that have
been patterned with adhesive patches, consisting for instance of fibronectin
coatings, of specific geometry and examine the role of the geometry of the
adhesive patch in controlling the spatial distribution of stresses in the
cellular material. The reference state for our cell is the limit of infinitely
strong adhesion, where the cell shape and lateral extent and determined
entirely by the shape and size of the adhesive patch. For finite adhesion
strength, cell elasticity and contractility yield deviations form this
reference state. We restrict ourselves to considering continuous or densely
spaced adhesion sites. For discrete or sparsely distributed adhesion sites,
non-adherent segments in the cell boundary could likely exhibit morphological
transitions induced by contractile activity and substrate stiffness [17]. In
agreement with experimental observations, we find that cells spread more on
stiff substrates and we provide an expression for the cell area versus
substrate stiffness for the case of a circular cell. We show that this
expression fit the data for spread areas of cardiac myocytes on substrates of
various sitffness values(see inset of Fig. 2). We demonstrate analytically and
numerically that strong traction stresses correlate with regions of high cell
boundary curvature, in agreement with experimental observations. Further, as
reported in experiments on single endothelial cells, our model demonstrates
that cell mechanical anisotropy is higher on elongated cells than on rounded
ones for fixed area [30].
Understanding the relation between cell morphology, the cell’s mechanical
response and cell fate is an important question in cellular biophysics. Our
simple model highlights the correlation between the geometry of adhesion sites
and cell morphology and demonstrates that traction forces by cells can be
tuned by controlling the geometry of adhesive regions. An important open
question not addressed by this simple model where the adhesive patch geometry
solely controls the cell shape is how cell morphology is determined by the
interplay of cell-substrate adhesion and dynamical reorganization of the
cytoskeletal architecture in response to the adhesion stimulus. To understand
this it will be necessary to incorporate the dynamical feedback between actin
reorganization and adhesion kinetics.
This work was supported by the National Science Foundation through award
DMR-1004789.
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arxiv-papers
| 2012-11-21T16:25:04 |
2024-09-04T02:49:38.263331
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Shiladitya Banerjee and M. Cristina Marchetti",
"submitter": "Shiladitya Banerjee",
"url": "https://arxiv.org/abs/1211.5075"
}
|
1211.5210
|
Polarized and transversity GPDs in kaon leptoproduction
S.V. Goloskokov
Bogoliubov Laboratory of Theoretical Physics, Joint Institute for Nuclear
Research, 141980 Dubna, Russia
Abstract
We study the kaon leptoproduction on the basis of the handbag approach. We
consider the leading-twist contribution together with the transversity twist-3
effects which were found to be important in the description of pseudo-scalar
meson production. We present our predictions for the cross section and spin
asymmetries in the kaon leptoproduction.
In this report, we analyze the process of kaons leptoproduction at large
photon virtualities within the handbag approach, where the amplitudes
factorize [1] into hard subprocesses and GPDs which keep the soft physics.
Different applications of GPDs were discussed at this conference [2]. At the
leading-twist accuracy the reactions of kaon production are sensitive only to
the GPDs $\widetilde{H}$ and $\widetilde{E}$ which contribute to the
amplitudes for longitudinally polarized virtual photons [3]. It was observed
that to be consistent with experimental data on the pion leptoproduction the
contributions of transversity GPDs $H_{T}$ and $\bar{E}_{T}$ are needed [4].
Within the handbag approach the transversity GPDs are accompanied by the
twist-3 meson wave function.
We consider here the transversity $H_{T}$ and $\bar{E}_{T}$ effects in the
leptoproduction of kaons. We present the model results for the cross section
of the $K^{+}\Lambda$ and $K^{+}\Sigma^{0}$ leptoproduction [4] and
predictions for the spin asymmetry in these reactions. It is shown that the
$H_{T}$ effects are essential in the $K^{+}\Lambda$ channel while in the
$K^{+}\Sigma^{0}$ leptoproduction the $\bar{E}_{T}$ contribution is mostly
important.
In what follows, we calculate the meson leptoproduction on the basis of the
handbag approach. The hard subprocess amplitudes are calculated within the
modified perturbative approach [5] in which the quark transverse degrees of
freedom as well as gluonic radiation, condensed in a Sudakov factor, are taken
into account.
The proton non- flip or helicity-flip amplitudes for longitudinally polarized
photons ${\cal M}^{K}_{0\pm,0+}$ can be written in the form:
${\cal
M}^{K}_{0+,0+}\propto[P^{K}_{0+,0+}+\langle\tilde{H}^{K}\rangle\rangle];\;{\cal
M}^{K}_{0-,0+}\propto\frac{\sqrt{-t^{\prime}}}{(m+M_{N^{f}})}\,[P^{K}_{0-,0+}+\xi\langle\widetilde{E}^{K}_{n.p.}\rangle].$
(1)
The amplitudes (1) dominate at large $Q^{2}$. The corresponding amplitudes
with transversally polarized photons are suppressed as $1/Q$.
The $P^{K}$ terms in (1) represent a kaon pole which appears in this reaction
for charged kaon production. We use the kaon-barion coupling constants [4]
$g_{K^{+}p\Lambda}\sim-13.3;\;g_{K^{+}p\Sigma^{0}}\sim-3.5,$ (2)
which are close to SU(3) predictions.
The second terms in (1) accumulate the handbag contribution to the kaons
production amplitude. The $<\tilde{F}>$ in (1) is a convolution of GPD
$\tilde{F}$ with the hard subprocess amplitude ${\cal
H}_{0\lambda,0\lambda}(\overline{x},...)$:
$<\tilde{F}>=\sum_{\lambda}\int_{-1}^{1}d\overline{x}{\cal
H}_{0\lambda,0\lambda}(\overline{x},...)\tilde{F}(\overline{x},\xi,t).$ (3)
The proton- hyperon transition GPDs in (3) can be related with the proton GPDs
by using the SU(3) flavor symmetry [6]
$F(p\to\Lambda)\sim[2F^{u}-F^{d}-F^{s}];\;F(p\to\Sigma^{0})\sim[F^{d}-F^{s}].$
(4)
It was found that the asymptotically dominant leading-twist contributions are
not sufficient to describe the experimental results on leptoproduction of
pseudoscalar mesons [4]. The data require also the contributions from the
transversity GPDs.
We estimate this contribution to the ${\cal M}_{0\pm,++}$ amplitudes by the
transversity GPDs $H_{T}$, $\bar{E}_{T}$, which are considered together with
the twist-3 meson wave function [4] in the hard subprocess amplitude
${\cal M}^{K,tw3}_{0-,\mu+}\propto\int_{-1}^{1}d\overline{x}{\cal
H}_{0-,\mu+}(\overline{x},...)\,H^{K}_{T},\,{\cal
M}^{K,tw3}_{0-,\mu+}\propto\frac{-t^{\prime}}{4\,m}\int_{-1}^{1}d\overline{x}{\cal
H}_{0-,\mu+}(\overline{x},...)\,\bar{E}^{K}_{T}.$ (5)
The $H_{T}$ GPD is connected with transversity PDFs as
$H^{a}_{T}(x,0,0)=\delta^{a}(x);\;\;\;\mbox{and}\;\;\;\delta^{a}(x)=C\,N^{a}_{T}\,x^{1/2}\,(1-x)\,[q_{a}(x)+\Delta
q_{a}(x)].$ (6)
We parameterize the PDF $\delta$ using the model [7]. The double distribution
representation [8] is used to calculate GPD $H_{T}$. Due to different signs of
$H_{T}^{u}$ and $H_{T}^{d}$ we find a quite large $H_{T}$ contribution
$K^{+}\Lambda$ and much smaller effect in the $K^{+}\Sigma^{0}$ production
(4).
The information on $\bar{E}_{T}$ is available only from the lattice QCD
estimations [9]. It was found that $\bar{E}_{T}^{u}$ and $\bar{E}_{T}^{d}$
should to be quite large, have the same sign and a similar size. From (4) we
can conclude that the $\bar{E}_{T}$ contributions to different kaon production
channels should be similar.
|
---|---
Figure 1: Left: the $K^{+}\Lambda$ production cross sections. Right: the $K^{+}\Sigma^{0}$ production cross sections at HERMES energies. Full line- unseparated cross section. dashed- $\sigma_{L}$, dashed-dotted line- $\sigma_{T}$. |
---|---
Figure 2: Predicted moments of $A_{UT}$ asymmetries at HERMES. Left: for the
$K^{+}\Lambda$ channel. Right: for the $K^{+}\Sigma^{0}$ production.
The large transversity $H_{T}$ effects in the $K^{+}\Lambda$ channel provide
to the large $\sigma_{T}$ cross section without a forward dip which dominated
with respect to $\sigma_{L}$, see Fig. 1 (Left). For the $K^{+}\Sigma^{0}$
production the $H_{T}$ contribution is much smaller and the $\bar{E}_{T}$
effects become essential. It provides the cross section with a forward dip,
Fig. 1 (Right). In both cases $\sigma_{T}$ determined by the transversity
$H_{T}$ and $\bar{E}_{T}$ contribution is large at low $Q^{2}$ with respect to
the leading twist $\sigma_{L}$ cross section. Note that the twist-3 effects
decrease rapidly with $Q^{2}$ growing and at sufficiently high $Q^{2}$ the
$\sigma_{L}$ will predominate.
In Fig. 2, we show our predictions for the moments of $A_{UT}$ asymmetry for
kaon production. The $\sin(\phi_{s})$ moment of asymmetry determined mainly by
the $H_{T}$ contribution is quite large in the $K^{+}\Lambda$ production, Fig.
2 (Left). In the $K^{+}\Sigma^{0}$ channel this moment of $A_{UT}$ asymmetry
is much smaller, Fig. 2 (Right). The $\sin(\phi-\phi_{s})$ moment of $A_{UT}$
asymmetry is predicted to be not small in this process, Fig. 2 (Right) with
respect to the $K^{+}\Lambda$ production.
To summarize, in this report we considered kaon leptoproduction within the
handbag approach. We calculated the leading twist and twist-3 transversity
contributions together. It was found that the $H_{T}$ and $\bar{E}_{T}$
contribution was quite large. They produce $\sigma_{T}$ which at low $Q^{2}$
exceeds substantially the leading twist $\sigma_{L}$ cross section. We observe
the same effect for most reactions of the pseudoscalar meson leptoproduction
[4]. The role of transversity effects can be investigated in future COMPASS
and JLAB12 experiments.
This work is supported in part by the Russian Foundation for Basic Research,
Grant 12-02-00613 and by the Heisenberg-Landau program.
## References
* [1] Ji X.// Phys. Rev. D. 1997. V.55. P.7114;
Radyushkin A.V.// Phys. Lett. B. 1996. V.380. P.417;
Collins J.C. et al.// Phys. Rev. D. 1997. V.56. P.2982.
* [2] Kroll P. // this proceedings.
* [3] Goloskokov S.V. and Kroll P.// Euro. Phys. J. C. 2010. V.65. P.137.
* [4] Goloskokov S.V. and Kroll P.// Euro. Phys. J. A. 2011. V.47. P.112.
* [5] Botts J. and Sterman G.// Nucl. Phys. B. 1989. V.325. P.62.
* [6] Frankfurt L.L., Pobylitsa P.V., Polyakov M.V and Strikman M.// Phys. Rev. D. 1999. V.60. P.014010.
* [7] Anselmino M., et al.// Nucl. Phys. Proc. Suppl. 2009\. V.191. P.98.
* [8] Musatov I.V. and Radyushkin A.V.// Phys. Rev. D. 2000. V.61. P.074027.
* [9] Gockeler M., et al. (QCDSF and UKQCD Collab.)// Phys. Rev. Lett. 2007\. V.98. P.222001.
|
arxiv-papers
| 2012-11-22T06:23:12 |
2024-09-04T02:49:38.273991
|
{
"license": "Public Domain",
"authors": "S. V. Goloskokov",
"submitter": "Sergey Goloskokov",
"url": "https://arxiv.org/abs/1211.5210"
}
|
1211.5307
|
$Id:espcrc1.tex,v1.22004/02/2411:22:11speppingExp$ On sum edge-coloring of
regular, bipartite and split graphsP.A. Petrosyan, R.R. Kamalian
# On sum edge-coloring of regular, bipartite and split graphs
P.A. Petrosyan, R.R. Kamalian[MCSD] email: pet_petros@{ipia.sci.am, ysu.am,
yahoo.com}email: [email protected]. Institute for Informatics and
Automation Problems,
National Academy of Sciences, 0014, Armenia Department of Informatics and
Applied Mathematics,
Yerevan State University, 0025, Armenia Department of Applied Mathematics and
Informatics,
Russian-Armenian State University, 0051, Armenia
###### Abstract
An edge-coloring of a graph $G$ with natural numbers is called a sum edge-
coloring if the colors of edges incident to any vertex of $G$ are distinct and
the sum of the colors of the edges of $G$ is minimum. The edge-chromatic sum
of a graph $G$ is the sum of the colors of edges in a sum edge-coloring of
$G$. It is known that the problem of finding the edge-chromatic sum of an
$r$-regular ($r\geq 3$) graph is $NP$-complete. In this paper we give a
polynomial time $\left(1+\frac{2r}{(r+1)^{2}}\right)$-approximation algorithm
for the edge-chromatic sum problem on $r$-regular graphs for $r\geq 3$. Also,
it is known that the problem of finding the edge-chromatic sum of bipartite
graphs with maximum degree $3$ is $NP$-complete. We show that the problem
remains $NP$-complete even for some restricted class of bipartite graphs with
maximum degree $3$. Finally, we give upper bounds for the edge-chromatic sum
of some split graphs.
Keywords: edge-coloring, sum edge-coloring, regular graph, bipartite graph,
split graph
## 1 Introduction
We consider finite undirected graphs that do not contain loops or multiple
edges. Let $V(G)$ and $E(G)$ denote sets of vertices and edges of $G$,
respectively. For $S\subseteq V(G)$, let $G[S]$ denote the subgraph of $G$
induced by $S$, that is, $V(G[S])=S$ and $E(G[S])$ consists of those edges of
$E(G)$ for which both ends are in $S$. The degree of a vertex $v\in V(G)$ is
denoted by $d_{G}(v)$, the maximum degree of $G$ by $\Delta(G)$, the chromatic
number of $G$ by $\chi(G)$, and the chromatic index of $G$ by
$\chi^{\prime}(G)$. The terms and concepts that we do not define can be found
in [4, 26].
A proper vertex-coloring of a graph $G$ is a mapping
$\alpha:V(G)\rightarrow\mathbf{N}$ such that $\alpha(u)\neq\alpha(v)$ for
every $uv\in E(G)$. If $\alpha$ is a proper vertex-coloring of a graph $G$,
then $\Sigma(G,\alpha)$ denotes the sum of the colors of the vertices of $G$.
For a graph $G$, define the vertex-chromatic sum $\Sigma(G)$ as follows:
$\Sigma(G)=\min_{\alpha}\Sigma(G,\alpha)$, where minimum is taken among all
possible proper vertex-colorings of $G$. If $\alpha$ is a proper vertex-
coloring of a graph $G$ and $\Sigma(G)=\Sigma(G,\alpha)$, then $\alpha$ is
called a sum vertex-coloring. The strength of a graph $G$ ($s(G)$) is the
minimum number of colors needed for a sum vertex-coloring of $G$. The concept
of sum vertex-coloring and vertex-chromatic sum was introduced by Kubicka [16]
and Supowit [22]. In [17], Kubicka and Schwenk showed that the problem of
finding the vertex-chromatic sum is $NP$-complete in general and polynomial
time solvable for trees. Jansen [12] gave a dynamic programming algorithm for
partial $k$-trees. In papers [5, 6, 10, 13, 18], some approximation algorithms
were given for various classes of graphs. For the strength of graphs,
Brook’s-type theorem was proved in [11]. On the other hand, there are graphs
with $s(G)>\chi(G)$ [8]. Some bounds for the vertex-chromatic sum of a graph
were given in [23].
Similar to the sum vertex-coloring and vertex-chromatic sum of graphs, in [5,
9, 11], sum edge-coloring and edge-chromatic sum of graphs was introduced. A
proper edge-coloring of a graph $G$ is a mapping
$\alpha:E(G)\rightarrow\mathbf{N}$ such that $\alpha(e)\neq\alpha(e^{\prime})$
for every pair of adjacent edges $e,e^{\prime}\in E(G)$. If $\alpha$ is a
proper edge-coloring of a graph $G$, then $\Sigma^{\prime}(G,\alpha)$ denotes
the sum of the colors of the edges of $G$. For a graph $G$, define the edge-
chromatic sum $\Sigma^{\prime}(G)$ as follows:
$\Sigma^{\prime}(G)=\min_{\alpha}\Sigma^{\prime}(G,\alpha)$, where minimum is
taken among all possible proper edge-colorings of $G$. If $\alpha$ is a proper
edge-coloring of a graph $G$ and
$\Sigma^{\prime}(G)=\Sigma^{\prime}(G,\alpha)$, then $\alpha$ is called a sum
edge-coloring. The edge-strength of a graph $G$ ($s^{\prime}(G)$) is the
minimum number of colors needed for a sum edge-coloring of $G$. For the edge-
strength of graphs, Vizing’s-type theorem was proved in [11]. In [5], Bar-Noy
et al. proved that the problem of finding the edge-chromatic sum is $NP$-hard
for multigraphs. Later, in [9], it was shown that the problem is $NP$-complete
for bipartite graphs with maximum degree $3$. Also, in [9], the authors proved
that the problem can be solved in polynomial time for trees and that
$s^{\prime}(G)=\chi^{\prime}(G)$ for bipartite graphs. In [20], Salavatipour
proved that determining the edge-chromatic sum and the edge-strength are
$NP$-complete for $r$-regular graphs with $r\geq 3$. Also he proved that
$s^{\prime}(G)=\chi^{\prime}(G)$ for regular graphs. On the other hand, there
are graphs with $\chi^{\prime}(G)=\Delta(G)$ and $s^{\prime}(G)=\Delta(G)+1$
[11]. Recently, Cardinal et al. [7] determined the edge-strength of the
multicycles.
In the present paper we give a polynomial time $\frac{11}{8}$-approximation
algorithm for the edge-chromatic sum problem of $r$-regular graphs for $r\geq
3$. Next, we show that the problem of finding the edge-chromatic sum remains
$NP$-complete even for some restricted class of bipartite graphs with maximum
degree $3$. Finally, we give upper bounds for the edge-chromatic sum of some
split graphs.
## 2 Definitions and necessary results
A proper $t$-coloring is a proper edge-coloring which makes use of $t$
different colors. If $\alpha$ is a proper $t$-coloring of $G$ and $v\in V(G)$,
then $S\left(v,\alpha\right)$ denotes set of colors appearing on edges
incident to $v$. Let $G$ be a graph and $R\subseteq V(G)$. A proper
$t$-coloring of a graph $G$ is called an $R$-sequential $t$-coloring [1, 2] if
the edges incident to each vertex $v\in R$ are colored by the colors
$1,\ldots,d_{G}(v)$. For positive integers $a$ and $b$, we denote by
$\left[a,b\right]$, the set of all positive integers $c$ with $a\leq c\leq b$.
For a positive integer $n$, let $K_{n}$ denote the complete graph on $n$
vertices.
We will use the following four results.
###### Theorem 1
[15]. If $G$ is a bipartite graph, then $\chi^{\prime}(G)=\Delta(G)$.
###### Theorem 2
[24]. For every graph $G$,
$\Delta(G)\leq\chi^{\prime}(G)\leq\Delta(G)+1$.
###### Theorem 3
[25]. For the complete graph $K_{n}$ with $n\geq 2$,
$\chi^{\prime}(K_{n})=\left\\{\begin{tabular}[]{ll}$n-1$,&if $n$ is even,\\\
$n$,&if $n$ is odd.\\\ \end{tabular}\right.$
###### Theorem 4
[9, 11]. If $G$ is a bipartite or a regular graph, then
$s^{\prime}(G)=\chi^{\prime}(G)$.
## 3 Edge-chromatic sums of regular graphs
In this section we consider the problem of finding the edge-chromatic sum of
regular graphs. It is easy to show that the edge-chromatic sum problem of
graphs $G$ with $\Delta(G)\leq 2$ can be solved in polynomial time. On the
other hand, in [19], it was proved that the problem of finding the edge-
chromatic sum of an $r$-regular ($r\geq 3$) graph is $NP$-complete. Clearly,
$\Sigma^{\prime}(G)\geq\frac{nr(r+1)}{4}$ for any $r$-regular graph $G$ with
$n$ vertices, since the sum of colors appearing on the edges incident to any
vertex is at least $\frac{r(r+1)}{2}$. Moreover, it is easy to see that
$\Sigma^{\prime}(G)=\frac{nr(r+1)}{4}$ if and only if $\chi^{\prime}(G)=r$ for
any $r$-regular graph $G$ with $n$ vertices.
First we give a result on $R$-sequential colorings of regular graphs and then
we use this result for constructing an approximation algorithm.
###### Theorem 5
If $G$ is an $r$-regular graph with $n$ vertices, then $G$ has an
$R$-sequential $(r+1)$-coloring with
$|R|\geq\left\lceil\frac{n}{r+1}\right\rceil$.
* Proof.
By Theorem 2, there exists a proper $(r+1)$-coloring $\alpha$ of the graph
$G$. For $i=1,2,\ldots,r+1$, define the set $V_{\alpha}(i)$ as follows:
$V_{\alpha}(i)=\left\\{v\in V(G):i\notin S(v,\alpha)\right\\}$.
Clearly, for any $i^{\prime},i^{\prime\prime},1\leq
i^{\prime}<i^{\prime\prime}\leq r+1$, we have
$V_{\alpha}(i^{\prime})\cap V_{\alpha}(i^{\prime\prime})=\emptyset$ and
$\underset{i=1}{\overset{r+1}{\bigcup}}V_{\alpha}(i)=V(G)$.
Hence,
$n=|V(G)|=\left|\underset{i=1}{\overset{r+1}{\bigcup}}V_{\alpha}(i)\right|=\underset{i=1}{\overset{r+1}{\sum}}|V_{\alpha}(i)|$.
This implies that there exists $i_{0}$, $1\leq i_{0}\leq r+1$, for which
$|V_{\alpha}(i_{0})|\geq\left\lceil\frac{n}{r+1}\right\rceil$. Let
$R=V_{\alpha}(i_{0})$.
If $i_{0}=r+1$, then $\alpha$ is an $R$-sequential $(r+1)$-coloring of $G$;
otherwise define an edge-coloring $\beta$ as follows: for any $e\in E(G)$, let
$\beta(e)=\left\\{\begin{tabular}[]{ll}$\alpha(e)$,&if $\alpha(e)\neq
i_{0},r+1$,\\\ $i_{0}$,&if $\alpha(e)=r+1$,\\\ $r+1$,&if $\alpha(e)=i_{0}$.\\\
\end{tabular}\right.$
It is easy to see that $\beta$ is an $R$-sequential $(r+1)$-coloring of $G$
with $|R|\geq\left\lceil\frac{n}{r+1}\right\rceil$. $\square$
###### Corollary 6
If $G$ is a cubic graph with $n$ vertices, then $G$ has an $R$-sequential
$4$-coloring with $|R|\geq\left\lceil\frac{n}{4}\right\rceil$.
Note that if $n$ is odd, then the lower bound in Theorem 5 cannot be improved,
since the complete graph $K_{n}$ has an $R$-sequential $n$-coloring with
$|R|=1$.
The theorem we are going to prove will be used in section 5.
###### Theorem 7
For any $n\in\mathbf{N}$, we have
$\Sigma^{\prime}(K_{n})=\left\\{\begin{tabular}[]{ll}$\frac{n(n^{2}-1)}{4}$,&if
$n$ is odd,\\\ $\frac{(n-1)n^{2}}{4}$,&if $n$ is even.\\\
\end{tabular}\right.$
* Proof.
Since for any $r$-regular graph $G$ with $n$ vertices,
$\Sigma^{\prime}(G)=\frac{nr(r+1)}{4}$ if and only if $\chi^{\prime}(G)=r$
and, by Theorems 3 and 4, we obtain
$\Sigma^{\prime}(K_{n})=\frac{(n-1)n^{2}}{4}$ if $n$ is even.
Now let $n$ be an odd number and $n\geq 3$. In this case by Theorems 3 and 4,
we have $s^{\prime}(K_{n})=\chi^{\prime}(K_{n})=n$. It is easy to see that in
any proper $n$-coloring of $K_{n}$ the missing colors at $n$ vertices are all
distinct. Hence,
$\displaystyle\Sigma^{\prime}(K_{n})=\frac{\frac{n^{2}(n+1)}{2}-\frac{n(n+1)}{2}}{2}=\frac{n(n^{2}-1)}{4}.$
$\square$
In [5], it was shown that there exists a $2$-approximation algorithm for the
edge-chromatic sum problem on general graphs. Now we show that there exists a
$\left(1+\frac{2r}{(r+1)^{2}}\right)$-approximation algorithm for the edge-
chromatic sum problem on $r$-regular graphs for $r\geq 3$. Note that
$1+\frac{2r}{(r+1)^{2}}$ decreases for increasing $r$ and $\frac{11}{8}$ is
its maximum value achieved for $r=3$. Thus, we show that there is a
$\frac{11}{8}$-approximation algorithm for the edge-chromatic sum problem on
regular graphs.
###### Theorem 8
For any $r\geq 3$, there is a polynomial time
$\left(1+\frac{2r}{(r+1)^{2}}\right)$-approximation algorithm for the edge-
chromatic sum problem on $r$-regular graphs.
* Proof.
Let $G$ be an $r$-regular graph with $n$ vertices and $m$ edges. Now we
describe a polynomial time algorithm $A$ for constructing a special proper
$(r+1)$-coloring of $G$. First we construct a proper $(r+1)$-coloring $\alpha$
of $G$ in $O(mn)$ time [21]. Next we recolor some edges as it is described in
the proof of Theorem 5 to obtain an $R$-sequential $(r+1)$-coloring $\beta$ of
$G$ with $|R|\geq\left\lceil\frac{n}{r+1}\right\rceil$. Clearly, we can do it
in $O(m)$ time. Now, taking into account that the sum of colors appearing on
the edges incident to any vertex is at most $\frac{r(r+3)}{2}$, we have
$\displaystyle\Sigma_{A}^{\prime}(G)=\Sigma^{\prime}\left(G,\beta\right)$
$\displaystyle\leq$
$\displaystyle\frac{\frac{r(r+1)}{2}\left\lceil\frac{n}{r+1}\right\rceil+\left(n-\left\lceil\frac{n}{r+1}\right\rceil\right)\frac{r(r+3)}{2}}{2}\leq\frac{\frac{r(r+1)}{2}\frac{n}{r+1}+\left(n-\frac{n}{r+1}\right)\frac{r(r+3)}{2}}{2}$
$\displaystyle=$
$\displaystyle\frac{\frac{r(r+1)}{2}\frac{n}{r+1}+\frac{nr}{r+1}\frac{r(r+3)}{2}}{2}=\frac{nr(r^{2}+4r+1)}{4(r+1)}.$
On the other hand, since $\Sigma^{\prime}(G)\geq\frac{nr(r+1)}{4}$, we get
$\displaystyle\frac{\Sigma_{A}^{\prime}(G)}{\Sigma^{\prime}(G)}\leq\frac{nr(r^{2}+4r+1)}{4(r+1)}\cdot\frac{4}{nr(r+1)}=\frac{r^{2}+4r+1}{(r+1)^{2}}=1+\frac{2r}{(r+1)^{2}}.$
This shows that there exists a
$\left(1+\frac{2r}{(r+1)^{2}}\right)$-approximation algorithm for the edge-
chromatic sum problem on $r$-regular graphs. Moreover, we can construct the
aforementioned coloring $\beta$ for a regular graph in $O(mn)$ time. $\square$
## 4 Edge-chromatic sums of bipartite graphs
In this section we consider the problem of finding the edge-chromatic sum of
bipartite graphs. Let $G=(U\cup W,E)$ be a bipartite graph with a bipartition
$(U,W)$. By $U_{i}\subseteq U$ and $W_{i}\subseteq W$, we denote sets of
vertices of degree $i$ in $U$ and $W$, respectively. Define sets $V_{\geq
i}\subseteq V(G)$ and $U_{\geq i}\subseteq U$ as follows: $V_{\geq
i}=\\{v:v\in V(G)\wedge d_{G}(v)\geq i\\}$ and $U_{\geq i}=\\{u\in V(G):u\in
U\wedge d_{G}(u)\geq i\\}$. It was proved the following:
###### Theorem 9
[1, 2, 3, 4] If $G=(U\cup W,E)$ is a bipartite graph with $d_{G}(u)\geq
d_{G}(w)$ for every $uw\in E(G)$, where $u\in U$ and $w\in W$, then $G$ has a
$U$-sequential $\Delta(G)$-coloring.
By this theorem, we obtain the following corollary:
###### Corollary 10
If $G=(U\cup W,E)$ is a bipartite graph with $d_{G}(u)\geq d_{G}(w)$ for every
$uw\in E(G)$, where $u\in U$ and $w\in W$, then a $U$-sequential
$\Delta(G)$-coloring of $G$ is a sum edge-coloring of $G$ and
$\Sigma^{\prime}(G)=\sum_{u\in U}\frac{d_{G}(u)(d_{G}(u)+1)}{2}$.
In [9], it was shown that the problem of finding the edge-chromatic sum of
bipartite graphs $G$ with $\Delta(G)=3$ is $NP$-complete. Now we give a short
proof of this fact. First we need the following
Problem 1. [2, 4, 14]
Instance: A bipartite graph $G=(U\cup W,E)$ with $\Delta(G)=3$.
Question: Is there a $U$-sequential $3$-coloring of $G$?
It was proved the following:
###### Theorem 11
[2, 14] Problem 1 is $NP$-complete.
Now let us consider the following
Problem 2.
Instance: A bipartite graph $G=(U\cup W,E)$ with $\Delta(G)=3$.
Question: Is
$\Sigma^{\prime}(G)=\underset{i=1}{\overset{3}{\sum}}i\cdot\left|U_{\geq
i}\right|$?
###### Theorem 12
Problem 2 is $NP$-complete.
* Proof.
Clearly, Problem 2 belongs to $NP$. For the proof of the $NP$-completeness, we
show a reduction from Problem 1 to Problem 2. We prove that a bipartite graph
$G=(U\cup W,E)$ with $\Delta(G)=3$ admits a $U$-sequential $3$-coloring if and
only if
$\Sigma^{\prime}(G)=\underset{i=1}{\overset{3}{\sum}}i\cdot\left|U_{\geq
i}\right|$. Let $G=(U\cup W,E)$ be a bipartite graph with $\Delta(G)=3$ and
$\alpha$ be a $U$-sequential $3$-coloring of $G$. In this case the colors
$1,2,3$ appear on the edges incident to each vertex $u\in U_{3}$, the colors
$1,2$ appear on the edges incident to each vertex $u\in U_{2}$ and the color
$1$ appears on the pendant edges incident to each vertex $u\in U_{1}$. Hence,
$\Sigma^{\prime}(G,\alpha)=\underset{i=1}{\overset{3}{\sum}}i\cdot\left|U_{\geq
i}\right|$. On the other hand, clearly,
$\Sigma^{\prime}(G)\geq\underset{i=1}{\overset{3}{\sum}}i\cdot\left|U_{\geq
i}\right|$, thus
$\Sigma^{\prime}(G)=\underset{i=1}{\overset{3}{\sum}}i\cdot\left|U_{\geq
i}\right|$.
Now suppose that
$\Sigma^{\prime}(G)=\underset{i=1}{\overset{3}{\sum}}i\cdot\left|U_{\geq
i}\right|$. By Theorems 1 and 4, there exists a proper $3$-coloring $\beta$ of
a bipartite graph $G$ with $\Delta(G)=3$. This implies that the colors $1,2,3$
appear on the edges incident to each vertex $u\in U_{3}$. If the color $3$
appears on the edges incident to some vertices $u\in U_{2}$ or the colors $2$
or $3$ appear on the pendant edges incident to some vertices $u\in U_{1}$,
then it is easy to see that
$\Sigma^{\prime}(G,\beta)>\underset{i=1}{\overset{3}{\sum}}i\cdot\left|U_{\geq
i}\right|$. Hence, $\beta$ is a $U$-sequential $3$-coloring of $G$. $\square$
Now we prove that the problem of finding the edge-chromatic sum of bipartite
graphs $G$ with $\Delta(G)=3$ and with additional conditions is $NP$-complete,
too. We need the following
Problem 3. [2, 14]
Instance: A bipartite graph $G=(U\cup W,E)$ with $\Delta(G)=3$ and
$|U_{i}|=|W_{i}|$ for $i=1,2,3$.
Question: Is there a $V(G)$-sequential $3$-coloring of $G$?
It was proved the following:
###### Theorem 13
[2, 14] Problem 3 is $NP$-complete.
Now let us consider the following
Problem 4.
Instance: A bipartite graph $G=(U\cup W,E)$ with $\Delta(G)=3$ and
$|U_{i}|=|W_{i}|$ for $i=1,2,3$.
Question: Is
$\Sigma^{\prime}(G)=\frac{1}{2}\underset{i=1}{\overset{3}{\sum}}i\cdot\left|V_{\geq
i}\right|$?
###### Theorem 14
Problem 4 is $NP$-complete.
* Proof.
Clearly, Problem 4 belongs to $NP$. For the proof of the $NP$-completeness, we
show a reduction from Problem 3 to Problem 4. We prove that a bipartite graph
$G=(U\cup W,E)$ with $\Delta(G)=3$ and $|U_{i}|=|W_{i}|$ for $i=1,2,3$, admits
a $V(G)$-sequential $3$-coloring if and only if
$\Sigma^{\prime}(G)=\frac{1}{2}\underset{i=1}{\overset{3}{\sum}}i\cdot\left|V_{\geq
i}\right|$. Let $\alpha$ be a $V(G)$-sequential $3$-coloring of $G$. In this
case the colors $1,2,3$ appear on the edges incident to each vertex $v\in
V(G)$ with $d_{G}(v)=3$, the colors $1,2$ appear on the edges incident to each
vertex $v\in V(G)$ with $d_{G}(v)=2$ and the color $1$ appears on the pendant
edges incident to each vertex $v\in V(G)$ with $d_{G}(v)=1$. Hence,
$\Sigma^{\prime}(G,\alpha)=\frac{1}{2}\underset{i=1}{\overset{3}{\sum}}i\cdot\left|V_{\geq
i}\right|$. On the other hand, clearly,
$\Sigma^{\prime}(G)\geq\frac{1}{2}\underset{i=1}{\overset{3}{\sum}}i\cdot\left|V_{\geq
i}\right|$, thus
$\Sigma^{\prime}(G)=\frac{1}{2}\underset{i=1}{\overset{3}{\sum}}i\cdot\left|V_{\geq
i}\right|$.
Now suppose that
$\Sigma^{\prime}(G)=\frac{1}{2}\underset{i=1}{\overset{3}{\sum}}i\cdot\left|V_{\geq
i}\right|$. By Theorems 1 and 4, there exists a proper $3$-coloring $\beta$ of
a bipartite graph $G$ with $\Delta(G)=3$ and $|U_{i}|=|W_{i}|$ for $i=1,2,3$.
This implies that the colors $1,2,3$ appear on the edges incident to each
vertex $v\in V(G)$ with $d_{G}(v)=3$. If the color $3$ appears on the edges
incident to some vertices $v\in V(G)$ with $d_{G}(v)=2$ or the colors $2$ or
$3$ appear on the pendant edges incident to some vertices $v\in V(G)$ with
$d_{G}(v)=1$, then it is easy to see that
$\Sigma^{\prime}(G,\beta)>\frac{1}{2}\underset{i=1}{\overset{3}{\sum}}i\cdot\left|V_{\geq
i}\right|$. Hence, $\beta$ is a $V(G)$-sequential $3$-coloring of $G$.
$\square$
In [19], it was proved that the problem of finding the edge-chromatic sum of
bipartite graphs $G$ with $\Delta(G)=3$ remains $NP$-hard even for planar
bipartite graphs.
## 5 Edge-chromatic sums of split graphs
In this section we consider the problem of finding the edge-chromatic sum of
split graphs. A split graph is a graph whose vertices can be partitioned into
a clique $C$ and an independent set $I$. Let $G=(C\cup I,E)$ be a split graph,
where $C=\\{u_{1},u_{2},\ldots,u_{n}\\}$ is clique and
$I=\\{v_{1},v_{2},\ldots,v_{m}\\}$ is independent set. Define a number
$\Delta_{I}$ as follows: $\Delta_{I}=\max_{1\leq j\leq m}d_{G}(v_{j})$. Define
subgraphs $H$ and $H^{\prime}$ of a graph $G$ as follows:
$H=(C\cup I,E(G)\setminus E(G[C]))$ and $H^{\prime}=G[C]$.
Clearly, $H$ is a bipartite graph with a bipartition $(C,I)$, and
$d_{H}(u_{i})=d_{G}(u_{i})-n+1$ for $i=1,2,\ldots,n$,
$d_{H}(v_{j})=d_{G}(v_{j})$ for $j=1,2,\ldots,m$.
###### Theorem 15
Let $G=(C\cup I,E)$ be a split graph, where $C=\\{u_{1},u_{2},\ldots,u_{n}\\}$
is clique and $I=\\{v_{1},v_{2},\ldots,v_{m}\\}$ is independent set. If
$d_{G}(u_{i})-d_{G}(v_{j})\geq n-1$ for every $u_{i}v_{j}\in E(G)$, then:
(1)
if $n$ is even, then
$\Sigma^{\prime}(G)\leq{\min}\left\\{\sum_{i=1}^{n}\frac{\left(d_{G}(u_{i})-n+1\right)\left(d_{G}(u_{i})-n+2\right)}{2}+\frac{\left(2\Delta(G)-n+2\right)n(n-1)}{4},\Sigma^{\prime}(K_{n})+\sum_{i=1}^{n}\frac{\left(d_{G}(u_{i})-n+1\right)\left(d_{G}(u_{i})+n\right)}{2}\right\\}$;
(2)
if $n$ is odd, then
$\Sigma^{\prime}(G)\leq{\min}\left\\{\sum_{i=1}^{n}\frac{\left(d_{G}(u_{i})-n+1\right)\left(d_{G}(u_{i})-n+2\right)}{2}+\frac{\left(2\Delta(G)-n+3\right)n(n-1)}{4},\Sigma^{\prime}(K_{n})+\sum_{i=1}^{n}\frac{\left(d_{G}(u_{i})-n+1\right)\left(d_{G}(u_{i})+n+2\right)}{2}\right\\}$.
* Proof.
For the proof, we are going to construct edge-colorings that satisfies the
specified conditions.
Since $d_{G}(u_{i})-d_{G}(v_{j})\geq n-1$ for every $u_{i}v_{j}\in E(G)$, we
have $d_{H}(u_{i})\geq d_{H}(v_{j})$ for each $u_{i}v_{j}\in E(H)$. By Theorem
9, there exists a $C$-sequential $\Delta(H)$-coloring $\alpha$ of the graph
$H$ and, by Corollary 10, we obtain
$\Sigma^{\prime}(H)=\Sigma^{\prime}(H,\alpha)=\sum_{i=1}^{n}\frac{d_{H}(u_{i})\left(d_{H}(u_{i})+1\right)}{2}$.
Now we consider two cases.
Case 1: $n$ is even.
In this case, by Theorem 3, we have $\chi^{\prime}(H^{\prime})=n-1$. Let
$\beta$ be a proper edge-coloring of a graph $H^{\prime}$ with colors
$\Delta(G)-n+2,\ldots,\Delta(G)$. Clearly, for each vertex $u_{i}$,
$i=1,2,\ldots,n$, the set of colors appearing on edges incident to $u_{i}$ in
$H^{\prime}$ is $[\Delta(G)-n+2,\Delta(G)]$. Thus, we obtain
$\Sigma^{\prime}(G)\leq\Sigma^{\prime}(H)+\frac{(2\Delta(G)-n+2)n(n-1)}{4}$.
On the other hand, let $\beta^{\prime}$ be a proper edge-coloring of a graph
$H^{\prime}$ with colors $1,2,\ldots,n-1$. Clearly, for each vertex $u_{i}$,
$i=1,2,\ldots,n$, the set of colors appearing on edges incident to $u_{i}$ in
$H^{\prime}$ is $[1,n-1]$. Next, we define an edge-coloring $\gamma$ of the
graph $H$ as follows: for every $e\in E(H)$, let $\gamma(e)=\alpha(e)+n-1$.
Thus, we obtain
$\Sigma^{\prime}(G)\leq\Sigma^{\prime}(K_{n})+\sum_{i=1}^{n}\frac{\left(d_{G}(u_{i})-n+1\right)\left(d_{G}(u_{i})+n\right)}{2}$.
Case 2: $n$ is odd.
In this case, by Theorem 3, we have $\chi^{\prime}(H^{\prime})=n$. Let $\beta$
be a proper edge-coloring of a graph $H^{\prime}$ with colors
$\Delta(G)-n+2,\ldots,\Delta(G)+1$. Without loss of generality, we may assume
that for each vertex $u_{i}$, $i=1,2,\ldots,n$, the set of colors appearing on
edges incident to $u_{i}$ in $H^{\prime}$ is
$[\Delta(G)-n+2,\Delta(G)+1]\setminus\\{\Delta(G)-n+1+i\\}$. Thus, we obtain
$\Sigma^{\prime}(G)\leq\Sigma^{\prime}(H)+\frac{(2\Delta(G)-n+3)n(n-1)}{4}$.
On the other hand, let $\beta^{\prime}$ be a proper edge-coloring of a graph
$H^{\prime}$ with colors $1,2,\ldots,n$. Without loss of generality, we may
assume that for each vertex $u_{i}$, $i=1,2,\ldots,n$, the set of colors
appearing on edges incident to $u_{i}$ in $H^{\prime}$ is
$[1,n]\setminus\\{i\\}$. Next, we define an edge-coloring $\gamma$ of the
graph $H$ as follows: for every $e\in E(H)$, let $\gamma(e)=\alpha(e)+n$.
Thus, we obtain
$\Sigma^{\prime}(G)\leq\Sigma^{\prime}(K_{n})+\sum_{i=1}^{n}\frac{\left(d_{G}(u_{i})-n+1\right)\left(d_{G}(u_{i})+n+2\right)}{2}$.
$\square$
###### Theorem 16
Let $G=(C\cup I,E)$ be a split graph, where $C=\\{u_{1},u_{2},\ldots,u_{n}\\}$
is clique and $I=\\{v_{1},v_{2},\ldots,v_{m}\\}$ is independent set. If
$d_{G}(u_{i})-d_{G}(v_{j})\leq n-1$ for every $u_{i}v_{j}\in E(G)$, then:
(1)
if $n$ is even, then
$\Sigma^{\prime}(G)\leq{\min}\left\\{\sum_{j=1}^{m}\frac{d_{G}(v_{j})\left(d_{G}(v_{j})+1\right)}{2}+\frac{\left(2\Delta_{I}+n\right)n(n-1)}{4},\Sigma^{\prime}(K_{n})+\sum_{j=1}^{m}\frac{d_{G}(v_{j})\left(d_{G}(v_{j})+2n-1\right)}{2}\right\\}$;
(2)
if $n$ is odd, then
$\Sigma^{\prime}(G)\leq{\min}\left\\{\sum_{j=1}^{m}\frac{d_{G}(v_{j})\left(d_{G}(v_{j})+1\right)}{2}+\frac{\left(2\Delta_{I}+n+1\right)n(n-1)}{4},\Sigma^{\prime}(K_{n})+\sum_{j=1}^{m}\frac{d_{G}(v_{j})\left(d_{G}(v_{j})+2n+1\right)}{2}\right\\}$.
* Proof.
For the proof, we are going to construct edge-colorings that satisfies the
specified conditions.
Since $d_{G}(u_{i})-d_{G}(v_{j})\leq n-1$ for every $u_{i}v_{j}\in E(G)$, we
have $d_{H}(u_{i})\leq d_{H}(v_{j})$ for each $u_{i}v_{j}\in E(H)$. By Theorem
9, there exists an $I$-sequential $\Delta_{I}$-coloring $\alpha$ of the graph
$H$ and, by Corollary 10, we obtain
$\Sigma^{\prime}(H)=\Sigma^{\prime}(H,\alpha)=\sum_{j=1}^{m}\frac{d_{H}(v_{j})\left(d_{H}(v_{j})+1\right)}{2}=\sum_{j=1}^{m}\frac{d_{G}(v_{j})\left(d_{G}(v_{j})+1\right)}{2}$.
Now we consider two cases.
Case 1: $n$ is even.
In this case, by Theorem 3, we have $\chi^{\prime}(H^{\prime})=n-1$. Let
$\beta$ be a proper edge-coloring of a graph $H^{\prime}$ with colors
$\Delta_{I}+1,\ldots,\Delta_{I}+n-1$. Clearly, for each vertex $u_{i}$,
$i=1,2,\ldots,n$, the set of colors appearing on edges incident to $u_{i}$ in
$H^{\prime}$ is $[\Delta_{I}+1,\Delta_{I}+n-1]$. Thus, we obtain
$\Sigma^{\prime}(G)\leq\Sigma^{\prime}(H)+\frac{\left(2\Delta_{I}+n\right)n(n-1)}{4}$.
On the other hand, let $\beta^{\prime}$ be a proper edge-coloring of a graph
$H^{\prime}$ with colors $1,2,\ldots,n-1$. Clearly, for each vertex $u_{i}$,
$i=1,2,\ldots,n$, the set of colors appearing on edges incident to $u_{i}$ in
$H^{\prime}$ is $[1,n-1]$. Next, we define an edge-coloring $\gamma$ of the
graph $H$ as follows: for every $e\in E(H)$, let $\gamma(e)=\alpha(e)+n-1$.
Thus, we obtain
$\Sigma^{\prime}(G)\leq\Sigma^{\prime}(K_{n})+\sum_{j=1}^{m}\frac{d_{G}(v_{j})\left(d_{G}(v_{j})+2n-1\right)}{2}$.
Case 2: $n$ is odd.
In this case, by Theorem 3, we have $\chi^{\prime}(H^{\prime})=n$. Let $\beta$
be a proper edge-coloring of a graph $H^{\prime}$ with colors
$\Delta_{I}+1,\ldots,\Delta_{I}+n$. Without loss of generality, we may assume
that for each vertex $u_{i}$, $i=1,2,\ldots,n$, the set of colors appearing on
edges incident to $u_{i}$ in $H^{\prime}$ is
$[\Delta_{I}+1,\Delta_{I}+n]\setminus\\{\Delta_{I}+i\\}$. Thus, we obtain
$\Sigma^{\prime}(G)\leq\Sigma^{\prime}(H)+\frac{\left(2\Delta_{I}+n+1\right)n(n-1)}{4}$.
On the other hand, let $\beta^{\prime}$ be a proper edge-coloring of a graph
$H^{\prime}$ with colors $1,2,\ldots,n$. Without loss of generality, we may
assume that for each vertex $u_{i}$, $i=1,2,\ldots,n$, the set of colors
appearing on edges incident to $u_{i}$ in $H^{\prime}$ is
$[1,n]\setminus\\{i\\}$. Next, we define an edge-coloring $\gamma$ of the
graph $H$ as follows: for every $e\in E(H)$, let $\gamma(e)=\alpha(e)+n$.
Thus, we obtain
$\Sigma^{\prime}(G)\leq\Sigma^{\prime}(K_{n})+\sum_{j=1}^{m}\frac{d_{G}(v_{j})\left(d_{G}(v_{j})+2n+1\right)}{2}$.
$\square$
* Acknowledgement
We would like to thank both referees for many useful suggestions.
## References
* [1] A.S. Asratian, Investigation of some mathematical model of scheduling theory, Doctoral Thesis, Moscow, 1980.
* [2] A.S. Asratian, R.R. Kamalian, Interval colorings of edges of a multigraph, Appl. Math. 5 (1987) 25-34 (in Russian).
* [3] A.S. Asratian, R.R. Kamalian, Investigation on interval edge-colorings of graphs, J. Combin. Theory Ser. B 62 (1994) 34-43.
* [4] A.S. Asratian, T.M.J. Denley, R. Haggkvist, Bipartite Graphs and their Applications, Cambridge University Press, Cambridge, 1998.
* [5] A. Bar-Noy, M. Bellare, M.M. Halldorsson, H. Shachnai, T. Tamir, On chromatic sums and distributed resource allocation, Inform. and Comput. 140 (1998) 183-202.
* [6] A. Bar-Noy, G. Kortsarz, The minimum color sum of bipartite graphs, J. Algorithms 28 (1998) 339-365.
* [7] J. Cardinal, V. Ravelomanana, M. Valencia-Pabon, Minimum sum edge colorings of multicycles, Discrete Appl. Mathematics 158 (2010) 1216-1223.
* [8] P. Erdős, E. Kubicka, A. Schwenk, Graphs that require many colors to achieve their chromatic sum, Congr. Numer. 71 (1990) 17-28.
* [9] K. Giaro, M. Kubale, Edge-chromatic sum of trees and bounded cyclicity graphs, Inform. Proc. Letters 75 (2000) 65-69.
* [10] K. Giaro, R, Janczewski, M. Kubale, M. Malafiejski, A $27/26$-approximation algorithm for the chromatic sum coloring of bipartite graphs, Lecture Notes in Computer Science 2462 (2002) 135-145.
* [11] H. Hajiabolhassan, M.L. Mehrabadi, R. Tusserkani, Minimal coloring and strength of graphs, Discrete Math. 215 (2000) 265-270.
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* [13] K. Jansen, Approximation results for the optimum cost chromatic partition problem, J. Algorithms 34 (2000) 54-89.
* [14] R.R. Kamalian, Interval edge-colorings of graphs, Doctoral Thesis, Novosibirsk, 1990.
* [15] D. König, Über Graphen und ihre Anwendung auf Determinantentheorie und Mengenlehre. Math. Ann. 77 (1916) 453-465.
* [16] E. Kubicka, The chromatic sum of a graph, PhD Thesis, Western Michigan University, 1989.
* [17] E. Kubicka, A.J. Schwenk, An introduction to chromatic sums, in: Proc. of the 17th Annual ACM Computer Science Conference, ACM Press, New York, 1989, pp. 39-45.
* [18] E. Kubicka, G. Kubicki, D. Kountanis, Approximation algorithms for the chromatic sum, in: Proc. of the First Great Lakes Computer Science Conference, Lecture Notes in Computer Science 507, 1989, pp. 15-21.
* [19] D. Marx, Complexity results for minimum sum edge coloring, Discrete Appl. Mathematics 157 (2009) 1034-1045.
* [20] M.R. Salavatipour, On sum coloring of graphs, Discrete Appl. Mathematics 127 (2003) 477-488.
* [21] A. Schrijver, Combinatorial Optimization, Polyhedra and efficiency, Springer-Verlag Heidelberg, 2004.
* [22] K.J. Supowit, Finding a maximum planar subset of nets in a channel, IEEE Trans. Comput. Aided Design CAD 6(1) (1987) 93-94.
* [23] C. Thomassen, P. Erdős, Y. Alavi, P.J. Malde, A.J. Schwenk, Tight bounds on the chromatic sum of a connected graph, J. Graph Theory 13 (1989) 353-357.
* [24] V.G. Vizing, On an estimate of the chromatic class of a $p$-graph, Disc. Analiz 3 (1964) 25-30 (in Russian).
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|
arxiv-papers
| 2012-11-22T15:16:03 |
2024-09-04T02:49:38.284103
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "P. A. Petrosyan, R. R. Kamalian",
"submitter": "Petros Petrosyan",
"url": "https://arxiv.org/abs/1211.5307"
}
|
1211.5311
|
$Id:espcrc1.tex,v1.22004/02/2411:22:11speppingExp$ Interval colorings of
complete balanced multipartite graphsPetros A. Petrosyan
# Interval colorings of complete balanced multipartite graphs
Petros A. Petrosyan email: pet_petros@{ipia.sci.am, ysu.am, yahoo.com}
Institute for Informatics and Automation Problems,
National Academy of Sciences, 0014, Armenia Department of Informatics and
Applied Mathematics,
Yerevan State University, 0025, Armenia
###### Abstract
A graph $G$ is called a complete $k$-partite ($k\geq 2$) graph if its vertices
can be partitioned into $k$ independent sets $V_{1},\ldots,V_{k}$ such that
each vertex in $V_{i}$ is adjacent to all the other vertices in $V_{j}$ for
$1\leq i<j\leq k$. A complete $k$-partite graph $G$ is a complete balanced
$k$-partite graph if $|V_{1}|=|V_{2}|=\cdots=|V_{k}|$. An edge-coloring of a
graph $G$ with colors $1,\ldots,t$ is an interval $t$-coloring if all colors
are used, and the colors of edges incident to each vertex of $G$ are distinct
and form an interval of integers. A graph $G$ is interval colorable if $G$ has
an interval $t$-coloring for some positive integer $t$. In this paper we show
that a complete balanced $k$-partite graph $G$ with $n$ vertices in each part
is interval colorable if and only if $nk$ is even. We also prove that if $nk$
is even and $(k-1)n\leq t\leq\left(\frac{3}{2}k-1\right)n-1$, then a complete
balanced $k$-partite graph $G$ admits an interval $t$-coloring. Moreover, if
$k=p2^{q}$, where $p$ is odd and $q\in\mathbb{N}$, then a complete balanced
$k$-partite graph $G$ has an interval $t$-coloring for each positive integer
$t$ satisfying $(k-1)n\leq t\leq\left(2k-p-q\right)n-1$.
Keywords: edge-coloring, interval coloring, complete multipartite graph,
complete graph, complete bipartite graph
## 1 Introduction
Throughout this paper all graphs are finite, undirected, and have no loops or
multiple edges. Let $V(G)$ and $E(G)$ denote the sets of vertices and edges of
$G$, respectively. For $F\subseteq E(G)$, the subgraph obtained by deleting
the edges of $F$ from $G$ is denoted by $G-F$. The maximum degree of $G$ is
denoted by $\Delta(G)$. The terms and concepts that we do not define can be
found in [19].
An edge-coloring of a graph $G$ is a mapping
$\alpha:E(G)\rightarrow\mathbb{N}$. A proper edge-coloring of a graph $G$ is
an edge-coloring $\alpha$ of $G$ such that $\alpha(e)\neq\alpha(e^{\prime})$
for any pair of adjacent edges $e,e^{\prime}\in E(G)$. The edge-chromatic
number $\chi^{\prime}(G)$ of $G$ is the least number of colors needed for a
proper edge-coloring of $G$. Clearly, $\chi^{\prime}(G)\geq\Delta(G)$ for
every graph $G$. On the other hand, the well-known theorem of Vizing [16]
states that the edge-chromatic number of any graph $G$ is either $\Delta(G)$
or $\Delta(G)+1$. One of the most important, interesting and long-standing
problem in this field is the problem of determining the exact value of the
edge-chromatic number of graphs. There are many results in this direction, in
particular, the exact value of the edge-chromatic number is known for
bipartite graphs [13], complete graphs [4, 18], complete multipartite graphs
[9, 14], split graphs with odd maximum degree [5], outerplanar graphs [7],
planar graphs $G$ with $\Delta(G)\geq 7$ [17, 20].
A graph $G$ is a complete $k$-partite ($k\geq 2$) graph if its vertices can be
partitioned into $k$ independent sets $V_{1},\ldots,V_{k}$ such that each
vertex in $V_{i}$ is adjacent to all the other vertices in $V_{j}$ for $1\leq
i<j\leq k$. A complete $k$-partite graph $G$ is a complete balanced
$k$-partite graph if $|V_{1}|=|V_{2}|=\cdots=|V_{k}|$. Clearly, if $G$ is a
complete balanced $k$-partite graph with $n$ vertices in each part, then
$\Delta(G)=(k-1)n$. Note that the complete graph $K_{n}$ and the complete
balanced bipartite graph $K_{n,n}$ are special cases of the complete balanced
$k$-partite graph. In [14], Laskar and Hare proved the following:
###### Theorem 1
If $G$ is a complete balanced $k$-partite graph with $n$ vertices in each
part, then
$\chi^{\prime}(G)=\left\\{\begin{tabular}[]{ll}$(k-1)n$,&if $nk$ is even,\\\
$(k-1)n+1$,&if $nk$ is odd.\\\ \end{tabular}\right.$
A more general result was obtained by Hoffman and Rodger [9]. Before we
formulate this result we need a definition of the overfull graph. A graph $G$
is overfull if $|E(G)|>\left\lfloor\frac{|V(G)|}{2}\right\rfloor\Delta(G)$.
Clearly, if $G$ is overfull, then $\chi^{\prime}(G)=\Delta(G)+1$.
###### Theorem 2
If $G$ is a complete $k$-partite graph, then
$\chi^{\prime}(G)=\left\\{\begin{tabular}[]{ll}$\Delta(G)$,&if $G$ is not
overfull,\\\ $\Delta(G)+1$,&if $G$ is overfull.\\\ \end{tabular}\right.$
An edge-coloring of a graph $G$ with colors $1,\ldots,t$ is an interval
$t$-coloring if all colors are used, and the colors of edges incident to each
vertex of $G$ are distinct and form an interval of integers. A graph $G$ is
interval colorable if $G$ has an interval $t$-coloring for some positive
integer $t$. For an interval colorable graph $G$, the least and the greatest
values of $t$ for which $G$ has an interval $t$-coloring are denoted by $w(G)$
and $W(G)$, respectively. The concept of interval edge-coloring was introduced
by Asratian and Kamalian [1]. In [1, 2], they proved the following:
###### Theorem 3
If $G$ is a regular graph, then
(1)
$G$ is interval colorable if and only if $\chi^{\prime}(G)=\Delta(G)$.
(2)
If $G$ is interval colorable and $w(G)\leq t\leq W(G)$, then $G$ has an
interval $t$-coloring.
In [10], Kamalian investigated interval colorings of complete bipartite graphs
and trees. In particular, he proved the following:
###### Theorem 4
For any $r,s\in\mathbb{N}$, the complete bipartite graph $K_{r,s}$ is interval
colorable, and
(1)
$w\left(K_{r,s}\right)=r+s-\gcd(r,s)$,
(2)
$W\left(K_{r,s}\right)=r+s-1$,
(3)
if $w\left(K_{r,s}\right)\leq t\leq W\left(K_{r,s}\right)$, then $K_{r,s}$ has
an interval $t$-coloring.
Later, Kamalian [11] obtained an upper bound on $W(G)$ for an interval
colorable graph $G$ depending on the number of vertices of $G$.
###### Theorem 5
If $G$ is a connected interval colorable graph, then $W(G)\leq 2|V(G)|-3$.
Clearly, this bound is sharp for the complete graph $K_{2}$, but if $G\neq
K_{2}$, then this upper bound can be improved to $2\left|V(G)\right|-4$ [8].
For an $r$-regular graph $G$, Kamalian and Petrosyan [12] showed that if $G$
with at least $2r+2$ vertices admits an interval $t$-coloring, then $t\leq
2\left|V(G)\right|-5$. For a planar graph $G$, Axenovich [3] showed that if
$G$ has an interval $t$-coloring, then $t\leq\frac{11}{6}\left|V(G)\right|$.
In [15], Petrosyan investigated interval colorings of complete graphs and
$n$-dimensional cubes. First note that $K_{2n+1}$ is not interval colorable,
but $K_{2n}$ is interval colorable and $w(K_{2n})=2n-1$ for any
$n\in\mathbb{N}$. For $W(K_{2n})$, Petrosyan [15] proved the following:
###### Theorem 6
If $n=p2^{q}$, where $p$ is odd and $q$ is nonnegative, then
$W\left(K_{2n}\right)\geq 4n-2-p-q$.
In this paper we investigate interval colorings of complete balanced
$k$-partite graphs. In particular, we generalize Theorem 6 for complete
balanced $k$-partite graphs. Also, we discuss some other corollaries of our
result.
## 2 Results
If $\alpha$ is a proper edge-coloring of $G$ and $v\in V(G)$, then
$S\left(v,\alpha\right)$ denotes the set of colors appearing on edges incident
to $v$.
Let $[t]$ denote the set of the first $t$ natural numbers. Let $\left\lfloor
a\right\rfloor$ denote the largest integer less than or equal to $a$. For two
positive integers $a$ and $b$ with $a\leq b$, the set
$\left\\{a,\ldots,b\right\\}$ is denoted by $[a,b]$ and called an interval.
For an interval $[a,b]$ and a nonnegative number $p$, the notation
$[a,b]\oplus p$ means: $[a+p,b+p]$.
We also need the following lemma.
###### Lemma 7
If $K_{n,n}$ is a complete balanced bipartite graph with a bipartition
$(U,V)$, where $U=\\{u_{1},\ldots,u_{n}\\}$ and $V=\\{v_{1},\ldots,v_{n}\\}$,
then $K_{n,n}$ has an interval $(2n-1)$-coloring $\alpha$ such that
$S(u_{i},\alpha)=S(v_{i},\alpha)=[i,i+n-1]$ for $1\leq i\leq n$
* Proof.
Let $(U,V)$ be a bipartition of $K_{n,n}$, where $U=\\{u_{1},\ldots,u_{n}\\}$
and $V=\\{v_{1},\ldots,v_{n}\\}$. Define a coloring $\alpha$ of the edges of
$K_{n,n}$ as follows: for each edge $u_{i}v_{j}\in E(K_{n,n})$, let
$\alpha(u_{i}v_{j})=i+j-1$, where $1\leq i\leq n,1\leq j\leq n$. Clearly
$\alpha$ is an interval $(2n-1)$-coloring of $K_{n,n}$ and
$S(u_{i},\alpha)=S(v_{i},\alpha)=[i,i+n-1]$ for $1\leq i\leq n$. $\square$
Let $G$ be a complete balanced $k$-partite graph with $n$ vertices in each
part. By Theorems 1 and 3, we have that $G$ is interval colorable if and only
if $nk$ is even. Moreover, if $nk$ is even, then $w(G)=\Delta(G)=(k-1)n$. On
the other hand, by Theorem 5, we obtain $W(G)\leq 2nk-3$ whenever $nk$ is
even. Now we derive a lower bound for $W(G)$.
###### Theorem 8
If $G$ is a complete balanced $k$-partite graph with $n$ vertices in each part
and $nk$ is even, then $W(G)\geq\left(\frac{3}{2}k-1\right)n-1$.
* Proof.
We distinguish our proof into two cases.
Case 1: $k$ is even.
Let $V(G)=\left\\{v^{(i)}_{j}\colon\,1\leq i\leq k,1\leq j\leq n\right\\}$ and
$E(G)=\left\\{v^{(i)}_{p}v^{(j)}_{q}\colon\,1\leq i<j\leq k,1\leq p\leq
n,1\leq q\leq n\right\\}$.
Define an edge-coloring $\alpha$ of the graph $G$.
For each edge $v^{(i)}_{p}v^{(j)}_{q}\in E(G)$ with $1\leq i<j\leq k$ and
$p=1,\ldots,n$, $q=1,\ldots,n$, define a color
$\alpha\left(v^{(i)}_{p}v^{(j)}_{q}\right)$ as follows:
for $i=1,\ldots,\left\lfloor\frac{k}{4}\right\rfloor$,
$j=2,\ldots,\frac{k}{2}$, $i+j\leq\frac{k}{2}+1$, let
$\alpha\left(v_{p}^{(i)}v_{q}^{(j)}\right)=\left(i+j-3\right)n+p+q-1$;
for $i=2,\ldots,\frac{k}{2}-1$,
$j=\left\lfloor\frac{k}{4}\right\rfloor+2,\ldots,\frac{k}{2}$,
$i+j\geq\frac{k}{2}+2$, let
$\alpha\left(v_{p}^{(i)}v_{q}^{(j)}\right)=\left(i+j+\frac{k}{2}-4\right)n+p+q-1$;
for $i=3,\ldots,\frac{k}{2}$, $j=\frac{k}{2}+1,\dots,k-2$,
$j-i\leq\frac{k}{2}-2$, let
$\alpha\left(v_{p}^{(i)}v_{q}^{(j)}\right)=\left(\frac{k}{2}+j-i-1\right)n+p+q-1$;
for $i=1,\ldots,\frac{k}{2}$, $j=\frac{k}{2}+1,\ldots,k$,
$j-i\geq\frac{k}{2}$, let
$\alpha\left(v_{p}^{(i)}v_{q}^{(j)}\right)=\left(j-i-1\right)n+p+q-1$;
for $i=2,\ldots,1+\left\lfloor\frac{k-2}{4}\right\rfloor$,
$j=\frac{k}{2}+1,\dots,\frac{k}{2}+\left\lfloor\frac{k-2}{4}\right\rfloor$,
$j-i=\frac{k}{2}-1$, let
$\alpha\left(v_{p}^{(i)}v_{q}^{(j)}\right)=\left(2i-3\right)n+p+q-1$;
for $i=\left\lfloor\frac{k-2}{4}\right\rfloor+2,\ldots,\frac{k}{2}$,
$j=\frac{k}{2}+1+\left\lfloor\frac{k-2}{4}\right\rfloor,\ldots,k-1$,
$j-i=\frac{k}{2}-1$, let
$\alpha\left(v_{p}^{(i)}v_{q}^{(j)}\right)=\left(i+j-3\right)n+p+q-1$;
for
$i=\frac{k}{2}+1,\dots,\frac{k}{2}+\left\lfloor\frac{k}{4}\right\rfloor-1$,
$j=\frac{k}{2}+2,\ldots,k-2$, $i+j\leq\frac{3}{2}k-1$, let
$\alpha\left(v_{p}^{(i)}v_{q}^{(j)}\right)=\left(i+j-k-1\right)n+p+q-1$;
for $i=\frac{k}{2}+1,\ldots,k-1$,
$j=\frac{k}{2}+\left\lfloor\frac{k}{4}\right\rfloor+1,\dots,k$,
$i+j\geq\frac{3}{2}k$, let
$\alpha\left(v_{p}^{(i)}v_{q}^{(j)}\right)=\left(i+j-\frac{k}{2}-2\right)n+p+q-1$.
Let us prove that $\alpha$ is an interval
$\left(\left(\frac{3}{2}k-1\right)n-1\right)$-coloring of the graph $G$.
First we show that for each $t\in\left[\left(\frac{3}{2}k-1\right)n-1\right]$,
there is an edge $e\in E(G)$ with $\alpha(e)=t$.
Consider the vertices
$v_{1}^{(1)},\ldots,v_{n}^{(1)},v_{1}^{(k)},\ldots,v_{n}^{(k)}$. Now, by Lemma
7 and the definition of $\alpha$, for $1\leq j\leq n$,
$S\left(v_{j}^{(1)},\alpha\right)=\bigcup_{l=1}^{k-1}\left([j,j+n-1]\oplus(l-1)n\right)$
and
$S\left(v_{j}^{(k)},\alpha\right)=\bigcup_{l=\frac{k}{2}}^{\frac{3}{2}k-2}\left([j,j+n-1]\oplus(l-1)n\right)$.
Let $\overline{C}$ and $\overline{\overline{C}}$ be the subsets of colors
appear on edges incident to the vertices $v_{1}^{(1)},\ldots,v_{n}^{(1)}$ and
$v_{1}^{(k)},\ldots,v_{n}^{(k)}$ in the coloring $\alpha$, respectively, that
is:
$\overline{C}=\bigcup_{j=1}^{n}S\left(v_{j}^{(1)},\alpha\right)$ and
$\overline{\overline{C}}=\bigcup_{j=1}^{n}S\left(v_{j}^{(k)},\alpha\right)$.
It is straightforward to check that
$\overline{C}\cup\overline{\overline{C}}=\left[\left(\frac{3}{2}k-1\right)n-1\right]$,
so for each $t\in\left[\left(\frac{3}{2}k-1\right)n-1\right]$, there is an
edge $e\in E(G)$ with $\alpha(e)=t$.
Next we show that the edges incident to any vertex of $G$ are colored by
$(k-1)n$ consecutive colors.
Let $v_{j}^{(i)}\in V(G)$, where $1\leq i\leq k$, $1\leq j\leq n$.
Subcase 1.1. $1\leq i\leq 2$, $1\leq j\leq n$.
By Lemma 7 and the definition of $\alpha$, we have
$S\left(v_{j}^{(1)},\alpha\right)=S\left(v_{j}^{(2)},\alpha\right)=\bigcup_{l=1}^{k-1}\left([j,j+n-1]\oplus(l-1)n\right)=\left[j,j+(k-1)n-1\right]$.
Subcase 1.2. $3\leq i\leq\frac{k}{2}$, $1\leq j\leq n$.
By Lemma 7 and the definition of $\alpha$, we have
$S\left(v_{j}^{(i)},\alpha\right)=\bigcup_{l=i-1}^{k-3+i}\left([j,j+n-1]\oplus(l-1)n\right)=\left[j+(i-2)n,j+(k-3+i)n-1\right]$.
Subcase 1.3. $\frac{k}{2}+1\leq i\leq k-2$, $1\leq j\leq n$.
By Lemma 7 and the definition of $\alpha$, we have
$S\left(v_{j}^{(i)},\alpha\right)=\bigcup_{l=i-\frac{k}{2}+1}^{\frac{k}{2}-1+i}\left([j,j+n-1]\oplus(l-1)n\right)=\left[j+\left(i-\frac{k}{2}\right)n,j+\left(\frac{k}{2}-1+i\right)n-1\right]$.
Subcase 1.4. $k-1\leq i\leq k,1\leq j\leq n$.
By Lemma 7 and the definition of $\alpha$, we have
$S\left(v_{j}^{(k-1)},\alpha\right)=S\left(v_{j}^{(k)},\alpha\right)=\bigcup_{l=\frac{k}{2}}^{\frac{3}{2}k-2}\left([j,j+n-1]\oplus(l-1)n\right)=\left[j+\left(\frac{k}{2}-1\right)n,j+\left(\frac{3k}{2}-2\right)n-1\right]$.
This shows that $\alpha$ is an interval
$\left(\left(\frac{3}{2}k-1\right)n-1\right)$-coloring of $G$; thus
$W(G)\geq\left(\frac{3}{2}k-1\right)n-1$ for even $k\geq 2$.
Case 2: $n$ is even.
Let $n=2m$, $m\in\mathbb{N}$. Let
$U_{i}=\left\\{v_{1}^{(i)},\ldots,v_{m}^{(i)},v_{1}^{(k+i)},\ldots,v_{m}^{(k+i)}\right\\}$
($1\leq i\leq k$) be the $k$ independent sets of vertices of $G$. For
$i=1,\ldots,2k$, define the set $V_{i}$ as follows:
$V_{i}=\left\\{v_{1}^{(i)},\ldots,v_{m}^{(i)}\right\\}$. Clearly,
$V(G)=\bigcup_{i=1}^{2k}V_{i}$. For $1\leq i<j\leq 2k$, define $(V_{i},V_{j})$
as the set of all edges between $V_{i}$ and $V_{j}$. It is easy to see that
for $1\leq i<j\leq 2k$, $\left|(V_{i},V_{j})\right|=m^{2}$ except for
$\left|(V_{i},V_{k+i})\right|=0$ whenever $i=1,\ldots,k$. If we consider the
sets $V_{i}$ as the vertices and the sets $(V_{i},V_{j})$ as the edges, then
we obtain that $G$ is isomorphic to the graph $K_{2k}-F$, where $F$ is a
perfect matching. Now we define an edge-coloring $\beta$ of the graph $G$.
For each edge $v^{(i)}_{p}v^{(j)}_{q}\in E(G)$ with $1\leq i<j\leq 2k$ and
$p=1,\ldots,m$, $q=1,\ldots,m$, define a color
$\beta\left(v^{(i)}_{p}v^{(j)}_{q}\right)$ as follows:
for $i=1,\ldots,\left\lfloor\frac{k}{2}\right\rfloor$, $j=2,\ldots,k$,
$i+j\leq k+1$, let
$\beta\left(v_{p}^{(i)}v_{q}^{(j)}\right)=\left(i+j-3\right)m+p+q-1$;
for $i=2,\ldots,k-1$, $j=\left\lfloor\frac{k}{2}\right\rfloor+2,\ldots,k$,
$i+j\geq k+2$, let
$\beta\left(v_{p}^{(i)}v_{q}^{(j)}\right)=\left(i+j+k-5\right)m+p+q-1$;
for $i=3,\ldots,k$, $j=k+1,\dots,2k-2$, $j-i\leq k-2$, let
$\beta\left(v_{p}^{(i)}v_{q}^{(j)}\right)=\left(k+j-i-2\right)m+p+q-1$;
for $i=1,\ldots,k-1$, $j=k+2,\ldots,2k$, $j-i\geq k+1$, let
$\beta\left(v_{p}^{(i)}v_{q}^{(j)}\right)=\left(j-i-2\right)m+p+q-1$;
for $i=2,\ldots,1+\left\lfloor\frac{k-1}{2}\right\rfloor$,
$j=k+1,\dots,k+\left\lfloor\frac{k-1}{2}\right\rfloor$, $j-i=k-1$, let
$\beta\left(v_{p}^{(i)}v_{q}^{(j)}\right)=\left(2i-3\right)m+p+q-1$;
for $i=\left\lfloor\frac{k-1}{2}\right\rfloor+2,\ldots,k$,
$j=k+1+\left\lfloor\frac{k-1}{2}\right\rfloor,\ldots,2k-1$, $j-i=k-1$, let
$\beta\left(v_{p}^{(i)}v_{q}^{(j)}\right)=\left(i+j-4\right)m+p+q-1$;
for $i=k+1,\dots,k+\left\lfloor\frac{k}{2}\right\rfloor-1$,
$j=k+2,\ldots,2k-2$, $i+j\leq 3k-1$, let
$\beta\left(v_{p}^{(i)}v_{q}^{(j)}\right)=\left(i+j-2k-1\right)m+p+q-1$;
for $i=k+1,\ldots,2k-1$,
$j=k+\left\lfloor\frac{k}{2}\right\rfloor+1,\dots,2k$, $i+j\geq 3k$, let
$\beta\left(v_{p}^{(i)}v_{q}^{(j)}\right)=\left(i+j-k-3\right)m+p+q-1$.
Let us prove that $\beta$ is an interval
$\left(\left(\frac{3}{2}k-1\right)n-1\right)$-coloring of the graph $G$.
First we show that for each $t\in\left[\left(\frac{3}{2}k-1\right)n-1\right]$,
there is an edge $e\in E(G)$ with $\beta(e)=t$.
Consider the vertices
$v_{1}^{(1)},\ldots,v_{m}^{(1)},v_{1}^{(2k)},\ldots,v_{m}^{(2k)}$. Now, by
Lemma 7 and the definition of $\beta$, for $1\leq j\leq m$,
$S\left(v_{j}^{(1)},\beta\right)=\bigcup_{l=1}^{2k-2}\left([j,j+m-1]\oplus(l-1)m\right)$
and
$S\left(v_{j}^{(2k)},\beta\right)=\bigcup_{l=k}^{3k-3}\left([j,j+m-1]\oplus(l-1)m\right)$.
Let $\tilde{C}$ and $\tilde{\tilde{C}}$ be the subsets of colors appear on
edges incident to the vertices $v_{1}^{(1)},\ldots,v_{m}^{(1)}$ and
$v_{1}^{(2k)},\ldots,v_{m}^{(2k)}$ in the coloring $\beta$, respectively, that
is:
$\tilde{C}=\bigcup_{j=1}^{m}S\left(v_{j}^{(1)},\beta\right)$ and
$\tilde{\tilde{C}}=\bigcup_{j=1}^{m}S\left(v_{j}^{(2k)},\beta\right)$.
It is straightforward to check that
$\tilde{C}\cup\tilde{\tilde{C}}=\left[\left(\frac{3}{2}k-1\right)n-1\right]$,
so for each $t\in\left[\left(\frac{3}{2}k-1\right)n-1\right]$, there is an
edge $e\in E(G)$ with $\beta(e)=t$.
Next we show that the edges incident to any vertex of $G$ are colored by
$(k-1)n$ consecutive colors.
Let $v_{j}^{(i)}\in V(G)$, where $1\leq i\leq 2k$, $1\leq j\leq m$.
Subcase 2.1. $1\leq i\leq 2$, $1\leq j\leq m$.
By Lemma 7 and the definition of $\beta$, we have
$S\left(v_{j}^{(1)},\beta\right)=S\left(v_{j}^{(2)},\beta\right)=\bigcup_{l=1}^{2k-2}\left([j,j+m-1]\oplus(l-1)m\right)=\left[j,j+(2k-2)m-1\right]$.
Subcase 2.2. $3\leq i\leq k$, $1\leq j\leq m$.
By Lemma 7 and the definition of $\beta$, we have
$S\left(v_{j}^{(i)},\beta\right)=\bigcup_{l=i-1}^{2k-4+i}\left([j,j+m-1]\oplus(l-1)m\right)=\left[j+(i-2)m,j+(2k-4+i)m-1\right]$.
Subcase 2.3. $k+1\leq i\leq 2k-2$, $1\leq j\leq m$.
By Lemma 7 and the definition of $\beta$, we have
$S\left(v_{j}^{(i)},\beta\right)=\bigcup_{l=i-k+1}^{k-2+i}\left([j,j+m-1]\oplus(l-1)m\right)=\left[j+(i-k)m,j+(k-2+i)m-1\right]$.
Subcase 2.4. $2k-1\leq i\leq 2k,1\leq j\leq m$.
By Lemma 7 and the definition of $\beta$, we have
$S\left(v_{j}^{(2k-1)},\beta\right)=S\left(v_{j}^{(2k)},\beta\right)=\bigcup_{l=k}^{3k-3}\left([j,j+m-1]\oplus(l-1)m\right)=\left[j+(k-1)m,j+(3k-3)m-1\right]$.
This shows that $\beta$ is an interval
$\left(\left(\frac{3}{2}k-1\right)n-1\right)$-coloring of $G$; thus
$W(G)\geq\left(\frac{3}{2}k-1\right)n-1$ for even $n\geq 2$. $\square$
From Theorems 3(1) and 8, and taking into account that a complete balanced
$k$-partite graph $G$ with $n$ vertices in each part is overfull when $nk$ is
odd, we have:
###### Corollary 9
If $G$ is a complete balanced $k$-partite graph with $n$ vertices in each
part, then $\chi^{\prime}(G)=(k-1)n$ if and only if $nk$ is even.
From Theorems 3(2) and 8, we have:
###### Corollary 10
Let $G$ be a complete balanced $k$-partite graph with $n$ vertices in each
part and $nk$ is even. If $(k-1)n\leq t\leq\left(\frac{3}{2}k-1\right)n-1$,
then $G$ has an interval $t$-coloring.
Also, note that the proof of the case 2 implies that if a graph $G$ with $n$
vertices is $(n-2)$-regular, then $\chi^{\prime}(G)=n-2$.
The next theorem improves the lower bound in Theorem 8 for complete balanced
$k$-partite graphs with even $k$.
###### Theorem 11
Let $G$ be a complete balanced $k$-partite graph with $n$ vertices in each
part. If $k=p2^{q}$, where $p$ is odd and $q\in\mathbb{N}$, then
$W(G)\geq(2k-p-q)n-1$.
* Proof.
Let $V(G)=\left\\{v^{(i)}_{j}\colon\,1\leq i\leq k,1\leq j\leq n\right\\}$ and
$V(K_{k})=\\{u_{1},\ldots,u_{k}\\}$. Also, let
$E(G)=\left\\{v^{(i)}_{r}v^{(j)}_{s}\colon\,1\leq i<j\leq k,1\leq r\leq
n,1\leq s\leq n\right\\}$ and $E(K_{k})=\\{u_{i}u_{j}\colon\,1\leq i<j\leq
k\\}$.
Since $k=p2^{q}$, where $p$ is odd and $q\in\mathbb{N}$, by Theorem 6, there
exists an interval $(2k-1-p-q)$-coloring $\alpha$ of $K_{k}$. Now we define an
edge-coloring $\beta$ of the graph $G$.
For each edge $v^{(i)}_{r}v^{(j)}_{s}\in E(G)$ with $1\leq i<j\leq k$ and
$r=1,\ldots,n$, $s=1,\ldots,n$, define a color
$\beta\left(v^{(i)}_{r}v^{(j)}_{s}\right)$ as follows:
$\beta\left(v^{(i)}_{r}v^{(j)}_{s}\right)=\left(\alpha\left(u_{i}u_{j}\right)-1\right)n+r+s-1$.
By Lemma 7 and the definition of $\beta$, and taking into account that $\max
S\left(u_{i},\alpha\right)-\min S\left(u_{i},\alpha\right)=k-2$ for
$i=1,\ldots,k$, we have
$\displaystyle S\left(v_{j}^{(i)},\beta\right)$ $\displaystyle=$
$\displaystyle\bigcup_{l=\min S(u_{i},\alpha)}^{\max
S(u_{i},\alpha)}\left([j,j+n-1]\oplus(l-1)n\right)=$ $\displaystyle=$
$\displaystyle[j+\left(\min S(u_{i},\alpha)-1\right)n,j+\max
S(u_{i},\alpha)n-1]$
for $i=1,\ldots,k$ and $j=1,\ldots,n$, and
$\displaystyle\bigcup_{i=1}^{k}\bigcup_{j=1}^{n}S\left(v_{j}^{(i)},\beta\right)$
$\displaystyle=$ $\displaystyle[(2k-p-q)n-1].$
This shows that $\beta$ is an interval $((2k-p-q)n-1)$-coloring of the graph
$G$; thus $W(G)\geq(2k-p-q)n-1$. $\square$
From Theorems 3(2) and 11, we have:
###### Corollary 12
Let $G$ be a complete balanced $k$-partite graph with $n$ vertices in each
part and $k=p2^{q}$, where $p$ is odd and $q\in\mathbb{N}$. If $(k-1)n\leq
t\leq(2k-p-q)n-1$, then $G$ has an interval $t$-coloring.
## 3 Problems
In the previous section we obtained some results on interval colorings of
complete balanced multipartite graphs, but very small is known about interval
colorings of complete unbalanced multipartite graphs. In fact, there are only
two results on interval colorings of complete unbalanced multipartite graphs.
Let $n_{1}\leq\cdots\leq n_{k}$ be positive integers. The complete
multipartite graph $K_{n_{1},\ldots,n_{k}}$ is a complete $k$-partite graph
for which $|V_{i}|=n_{i}$, $i=1,\ldots,k$. The first result is Theorem 4 which
gives all possible values of the number of colors in interval colorings of
$K_{n_{1},n_{2}}$. The second result was obtained by Feng and Huang [6]. In
[6], they proved that the complete $3$-partite graph $K_{1,1,n}$ is interval
colorable if and only if $n$ is even. Now we would like to formulate some
problems on interval colorings of complete multipartite graphs:
###### Problem 1
Characterize all interval colorable complete multipartite graphs.
###### Problem 2
Find the exact values of $w(G)$ and $W(G)$ for interval colorable complete
multipartite graphs G.
###### Problem 3
Find the exact value of $W\left(K_{n,\ldots,n}\right)$ for interval colorable
complete balanced $k$-partite graphs $K_{n,\ldots,n}$.
Note that even a special case of Problem 3 is open: the problem of determining
the exact value of $W(K_{2n})$ for complete graph $K_{2n}$.
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arxiv-papers
| 2012-11-22T15:33:36 |
2024-09-04T02:49:38.291191
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Petros A. Petrosyan",
"submitter": "Petros Petrosyan",
"url": "https://arxiv.org/abs/1211.5311"
}
|
1211.5406
|
# The equivalence between doubly nonnegative relaxation and semidefinite
relaxation for binary quadratic programming problems111This work is supported
by National Natural Science Foundation of China (Grant No. 11071158)
Chuan-Hao Guo [email protected] Yan-Qin Bai [email protected] Li-Ping Tang
[email protected] Department of Mathematics, Shanghai University, Shanghai
200444, China
###### Abstract
It has recently been shown (Burer, Math. Program Ser. A 120:479-495, 2009)
that a large class of NP-hard nonconvex quadratic programming problems can be
modeled as so called completely positive programming problems, which are
convex but still NP-hard in general. A basic tractable relaxation is gotten by
doubly nonnegative relaxation, resulting in a doubly nonnegative programming.
In this paper, we prove that doubly nonnegative relaxation for binary
quadratic programming (BQP) problem is equivalent to a tighter semidifinite
relaxation for it. When problem (BQP) reduces to max-cut (MC) problem, doubly
nonnegative relaxation for it is equivalent to the standard semidifinite
relaxation. Furthermore, some compared numerical results are reported.
###### keywords:
binary quadratic programming, semidefinite relaxation, completely positive
programming, doubly nonnegative relaxation, max-cut problem
###### MSC:
90C10, 90C26, 49M20
## 1 Introduction
In this paper, we consider the following binary quadratic programming problem
$\rm{(BQP)}\ \ \ \ \ \begin{array}[]{lll}&\min&x^{T}Qx+2c^{T}x\\\ &{\rm
s.t.}&a_{i}^{T}x=b_{i},\ i=1,2,\ldots,m,\\\ &&x\in\\{-1,1\\}^{n},\\\
\end{array}$
where $x\in R^{n}$ is the variable, $Q\in R^{n\times n}$, $c\in R^{n}$,
$a_{i}\in R^{n}$ and $b_{i}\in R$ for all $i\in I:=\\{1,2,\dots,m\\}$ are the
data. Without loss of generality, $Q$ is symmetric, and we assume $Q$ is not
positive semidefinite, which implies generally that problem (BQP) is nonconvex
and NP-hard [1].
Problem (BQP) arises in many applications, such as financial analysis [2],
molecular conformation problem [3] and cellular radio channel assignment [4].
Many combinatorial optimization problems are special cases of problem (BQP),
such as max-cut problem [5]. For solving this type of problem, a systematic
survey of the solution methods can be found in Chapter 10 in [6] and the
references therein.
It is well-known that semidefinite relaxation (SDR) is a powerful,
computationally efficient approximation technique for a host of very difficult
optimization problems, for instance, max-cut problem [5], Boolean quadratic
program [7]. It also has been at the center of some of the very exciting
developments in the area of signal processing and communications [8, 9]. The
standard SDR for problem (BQP) is as follows:
$\rm{(SDR)}\ \ \ \ \ \begin{array}[]{lll}&\min&X\bullet Q+2c^{T}x\\\ &{\rm
s.t.}&a_{i}^{T}x=b_{i},\ \forall i\in I,\\\ &&a_{i}^{T}Xa_{i}=b_{i}^{2},\
\forall i\in I,\\\ &&X_{ii}=1,\ \forall i=1,2,\ldots,n,\\\ &&X\succeq
0,\end{array}$
where symbol $\bullet$ denotes the trace for any two conformal matrices. It is
obviously that problem (SDR) is convex and gives a lower bound for problem
(BQP) if the feasible set of problem (BQP) is nonempty. Moreover, if the
optimal solution $(x^{*},\ X^{*})$ for problem (SDR) satisfy
$X^{*}=x^{*}(x^{*})^{T}$, then we can conclude that $x^{*}$ is an optimal
solution for problem (BQP).
Recently, Burer [10] proves that a large class of NP-hard nonconvex quadratic
programs with a mix of binary and continuous variables can be modeled as so
called completely positive programs, which are convex but still NP-hard in
general. In order to solve such convex programs efficiently, a computable
relaxed problem is obtained by approximation the completely positive matrices
with doubly nonnegative matrices, resulting in a doubly nonnegative
programming [11], which can be efficiently solved by some popular packages.
For more details and developments of this technique, one may refer to [10, 11,
12, 13] and the references therein.
In this paper, a tighter SDR problem and a doubly nonnegative relaxation
(DNNR) problem for problem (BQP) are established, respectively, according to
the features of the constraints in problem (BQP) and the techniques of DNNP.
And, we prove that doubly nonnegative relaxation for problem (BQP) is
equivalent to the tighter semidefinite relaxation for it. Applying this result
to max-cut (MC) problem, it is shown that doubly nonnegative relaxation for
problem (MC) is equivalent to the standard semidefinite relaxation for it.
Moreover, some compared numerical results are reported to illustrate the
features of doubly nonnegative relaxation and semidefinite relaxation,
respectively.
The paper is organized as follows. In Section 2, a new tighter semidefinite
relaxation for problem (BQP) is proposed in Section 2.1. Problem (BQP) is
relaxed to a doubly nonnegative programming problem in Section 2.2. Section 3
and Section 4 are devoted to show the equivalence of two relaxation problems
for problem (BQP) and problem (MC), respectively. Some conclusions are given
in Section 5.
## 2 New relaxation for problem (BQP)
### 2.1 New tighter SDR for problem (BQP)
First, note that problem (BQP) also can be relaxed to the following problem by
SDR
$\widetilde{\rm{(SDR)}}\begin{array}[]{lll}&\min&X\bullet Q+2c^{T}x\\\ &{\rm
s.t.}&a_{i}^{T}x=b_{i},\ \forall i\in I,\\\ &&a_{i}^{T}Xa_{i}=b_{i}^{2},\
\forall i\in I,\\\ &&X_{ii}=1,\ \forall i=1,2,\ldots,n,\\\ &&X-xx^{T}\succeq
0.\end{array}$
If the optimal solution $(x^{*},\ X^{*})$ for problem $\widetilde{\rm{(SDR)}}$
satisfy $X^{*}=x^{*}(x^{*})^{T}$, it holds that $x^{*}$ also is an optimal
solution for problem (BQP). On one hand, it is worth noting that
$X-xx^{T}\succeq 0\Longrightarrow X\succeq 0$ (1)
holds always, which further implies that any feasible solution of problem
$\widetilde{\rm{(SDR)}}$ is also feasible for problem (SDR). It follows that
$\rm{Opt(SDR)}\leq\rm{Opt(\widetilde{\rm{SDR)}}}$ since the two problems have
the same objective functions, where $\rm{Opt(\ast)}$ denotes the optimal value
for problem $(\ast)$. Therefore, we can conclude that problem
$\widetilde{\rm{(SDR)}}$ is a tighter SDR problem for problem (BQP) than
problem (SDR).
On the other hand, we can easily verify that the constraint $X-xx^{T}\succeq
0$ is nonconvex, since the quadratic term $-xx^{T}$ is nonconvex. Thus,
problem $\widetilde{\rm{(SDR)}}$ is nonconvex and not solved by some popular
packages for solving convex programs. In order to establish the convex
representation for problem $\widetilde{\rm{(SDR)}}$, a crucial theorem is
given below and the details of its proof can be seen in Appendix A.5.5 Schur
complement in [14].
###### Theorem 2.1.
Let matrix $M\in S^{n}$ is partitioned as
$M=\left[\begin{array}[]{ll}A&B\\\ B^{T}&C\end{array}\right].$
If $\rm{det}A\neq 0$, the matrix $H=C-B^{T}A^{-1}B$ is called the Schur
complement of $A$ in $M$. Then, we have the following relations:
(i) $M\succ 0$ if and only if $A\succ 0$ and $H\succ 0$.
(ii) If $A\succ 0$, then $M\succeq 0$ if and only if $H\succeq 0$.
According to Theorem 2.1(ii) and (1), it holds immediately that
$\left[\begin{array}[]{ll}1&x^{T}\\\ x&X\end{array}\right]\succeq
0\Longleftrightarrow X-xx^{T}\succeq 0\Longrightarrow X\succeq 0,$ (2)
i.e., the constraint $X-xx^{T}\succeq 0$ can be equivalently reformulated as
$\left[\begin{array}[]{ll}1&x^{T}\\\ x&X\end{array}\right]\succeq 0$, which is
not only convex, but also computable. So, problem $\widetilde{\rm{(SDR)}}$ is
equivalently reformulated as follows:
${\rm{(SDR1)}}\begin{array}[]{lll}&\min&X\bullet Q+2c^{T}x\\\ &{\rm
s.t.}&a_{i}^{T}x=b_{i},\ \forall i\in I,\\\ &&a_{i}^{T}Xa_{i}=b_{i}^{2},\
\forall i\in I,\\\ &&X_{ii}=1,\ \forall i=1,2,\ldots,n,\\\
&&\left[\begin{array}[]{ll}1&x^{T}\\\ x&X\end{array}\right]\succeq
0,\end{array}$
which is not only convex in form, also can be efficiently solved by some
popular packages for solving convex programs.
Here, some examples are given to show that problem (SDR1) is a tighter
relaxation problem compared to problem (SDR), and the corresponding numerical
results further show that problem (SDR1) is more efficient than problem (SDR).
These examples are solved by CVX, a package for specifying and solving convex
programs [15].
###### Example 2.1.
This is a two dimensional nonconvex problem with one linear equality
constraint, the corresponding coefficients are selected as follows:
$Q=\left[\begin{array}[]{cc}0&-3\\\ -3&-20\end{array}\right],\
c=\left[\begin{array}[]{cc}-8\\\ 9\end{array}\right],\ \ A=[10,\ -10],\ \
b=0,$
where $A=[a_{1},a_{2},\ldots,a_{m}]^{T}$, $b=[b_{1},b_{2},\ldots,b_{m}]^{T}$.
On one hand, we use CVX to solve problem (SDR), then we obtain
$\rm{Opt(SDR)}=-\infty$, since problem (SDR) is unbounded below. On the other
hand, when problem (SDR1) is solved, it follows that $\rm{Opt(SDR1)}=-28$ with
$X=\left[\begin{array}[]{cc}1&1\\\ 1&1\end{array}\right]$ and $x=[-1,\
-1]^{T}$. Note that the relationship $X=xx^{T}$ holds, thus we can conclude
that $x=[-1,\ -1]^{T}$ also is an optimal solution for problem (BQP). The
results show that problem (SDR1) is more tighter and efficient than problem
(SDR) for this problem.
###### Example 2.2.
This problem is five dimensions with three linear equality constraints, the
corresponding coefficients are chosen as follows:
$Q=\left[\begin{array}[]{cccccc}-52&31&49&-7&4\\\ 31&-16&-50&-13&-49\\\
49&-50&8&44&-30\\\ -7&-13&44&36&12\\\ 4&-49&-30&12&56\end{array}\right],\ \ \
\ \ c=\left[\begin{array}[]{cc}-20\\\ 37\\\ 43\\\ 25\\\ -6\end{array}\right],$
$A=\left[\begin{array}[]{ccccc}4&10&29&14&-36\\\ 38&9&1&-17&23\\\
48&39&5&-17&-13\end{array}\right],\ \ \ \ \ \
b=\left[\begin{array}[]{ccc}11\\\ -50\\\ -36\end{array}\right].$
If this problem is solved by CVX with problem (SDR), then it returns
$\rm{Opt(SDR)}=-\infty$ since problem (SDR) is unbounded below. When we use
problem (SDR1) to solve this problem, we have $\rm{Opt(SDR1)}=-307.548$,
however, the relationship $X=xx^{T}$ is not holds for this problem. Therefore,
we obtain a tighter lower bound $-307.548$ for original problem. These results
also show that problem (SDR1) is more effective than problem (SDR).
In fact, the constraint $x\in\\{-1,+1\\}^{n}$ in problem (BQP) further imply
that the following relationship
$(1-x_{i})(1-x_{j})\geq 0\Rightarrow 1-x_{i}-x_{j}+X_{ij}\geq 0,\ \forall
1\leq i\leq j\leq n$ (3)
always hold. Combing with (3) in problem (SDR1), we get the following new
semidefinite relaxation problem
$\rm{(SDR2)}\begin{array}[]{lll}&\min&X\bullet Q+2c^{T}x\\\ &{\rm
s.t.}&a_{i}^{T}x=b_{i},\ \forall i\in I,\\\ &&a_{i}^{T}Xa_{i}=b_{i}^{2},\
\forall i\in I,\\\ &&X_{ii}=1,\ \forall i=1,2,\ldots,n,\\\
&&1-x_{i}-x_{j}+X_{ij}\geq 0,\ \forall 1\leq i\leq j\leq n,\\\
&&\left[\begin{array}[]{ll}1&x^{T}\\\ x&X\end{array}\right]\succeq
0.\end{array}$
The above semidefinite relaxation problem $\rm{(SDR2)}$ is more tighter than
problem $\rm{(SDR1)}$ in form, since $\frac{n(n+1)}{2}$ inequality constraints
are added into corresponding problem $\rm{(SDR2)}$. Furthermore, we will prove
that problem (SDR2) is equivalent to another convex relaxation problem for
problem (BQP) in Section 3.
Now, we test some problems to show that problem (SDR2) is tighter than problem
(SDR1) from the computational point of view. These problems are of one of two
types:
Table 1: Statistics of the test problems Type | | Instances | | $n$ | | $m$ | | Function
---|---|---|---|---|---|---|---|---
RdnBQP | | $50$ | | $50$ | | $20$ | | randn($\cdot$)
RdiBQP | | $50$ | | $50$ | | $20$ | | randi($[-10,10],\cdot$)
$\bullet$ RdnBQP. We generate $50$ instances of problem (BQP) by MATLAB
function randn($\cdot$). The symmetric matrix $Q$ is generated by
tril(randn($\cdot$),
-1)+triu(randn($\cdot$)’,0), and all instances are nonconvex.
$\bullet$ RdiBQP. $50$ instances of problem (BQP) are generated by MATLAB
function randi([-10, 10], $\cdot$). The symmetric matrix $Q$ is generated by
randn($\cdot$)+randn($\cdot$)’. Each element in the data coefficients is a
random integer number in the range $[-10,10]$. All instances are nonconvex
binary quadratic programming problems.
To compare the performance of two relaxation problems for problem (BQP), by
using problem (SDR1) and problem (SDR2), respectively, we use performance
profiles as described in Dolan and Moré’s paper [16]. Our profiles are based
on optimal values for problems (SDR1) and (SDR2). These problems are solved by
CVX, and the results of performance are shown in Figure 1. From Figure 1, it
is obviously that the lower bound which got from problem (SDR2) is much
greater than that one of from problem (SDR1), for test problems RdnBQP and
RdiBQP, respectively. Moreover, we find that optimal value of problem (SDR2)
is strictly greater than that of problem (SDR1) for test problems in the
experiment. Thus, the performance of problem (SDR2) is much better than
problem (SDR1) for solving problem (BQP) in some sense.
Figure 1: Left figure is based on optimal values of problems RdnBQP, right
figure is based on optimal values of problems RdiBQP.
### 2.2 Doubly nonnegative relaxation for problem (BQP)
Recently, Burer [10] has shown that a large class of NP-hard nonconvex
quadratic problems with binary constraints can be modeled as so-called
completely positive programs (CPP), i.e., the minimization of a linear
function over the convex cone of completely positive matrices subject to
linear constraints. Motivated by the ideas, we first establish the CPP
representation for problem (BQP), and then give its doubly nonnegative
relaxation (DNNR) formulation. Subsequently, some compared numerical results
are presented in this section.
Let $z=\frac{1}{2}(e-x)$ in problem (BQP), it follows that
$z\in\\{0,1\\}^{n}$, and then problem (BQP) can be equivalently reformulated
as follows:
$\widehat{\rm{(BQP)}}\begin{array}[]{lll}&\min&4z^{T}Qz-4z^{T}(Qe+c)+e^{T}Qe+2c^{T}e\\\
&{\rm s.t.}&2a_{i}^{T}z=a_{i}^{T}e-b_{i},\ \forall i\in I,\\\ &&z\in\\{0,\
1\\}^{n},\end{array}$
where $e$ denote the vector of ones with appropriate dimension. According to
Theorem 2.6 in [10] and similar to the analysis in it, problem
$\widehat{\rm{(BQP)}}$ can be further equivalently transformed into the
following CPP problem
${\rm{(CPP)}}\begin{array}[]{lll}&\min&4Q\bullet
Z-4z^{T}(Qe+c)+e^{T}Qe+2c^{T}e\\\ &{\rm s.t.}&2a_{i}^{T}z=a_{i}^{T}e-b_{i},\
\forall i\in I,\\\ &&4a_{i}^{T}Za_{i}=(a_{i}^{T}e-b_{i})^{2},\ \forall i\in
I,\\\ &&Z_{ii}=z_{i},\ \forall i=1,2,\ldots,n,\\\
&&\left[\begin{array}[]{ll}1&z^{T}\\\ z&Z\end{array}\right]\in
C_{1+n},\end{array}$
where $C_{1+n}$ is defined as follows:
$C_{1+n}:=\left\\{X\in S^{1+n}:\ X=\sum\limits_{k\in
K}z^{k}(z^{k})^{T}\right\\}\cup\\{0\\},$
and for some finite $\\{z^{k}\\}_{k\in K}\subset
R^{1+n}_{+}\backslash\\{0\\}$.
In view of the definition of convex cone in [14], $C_{1+n}$ is a closed convex
cone, and is called the completely positive matrices cone. Thus, problem (CPP)
is a convex problem. However, problem (CPP) is NP-hard, since checking whether
or not a given matrix belongs to $C_{1+n}$ is NP-hard, which has been shown by
Dickinson and Gijen in [17]. Thus, it has to be replaced by some computable
cones, which can efficiently approximate cone $C_{1+n}$. Note that the convex
cone $(S_{n})^{+}$ is self-dual, and so is the convex cone $S_{n}^{+}$, where
$(S^{n})^{+}$ and $S^{n}_{+}$ denotes the cone of $n\times n$ nonnegative
symmetric matrices and the cone of $n\times n$ positive semidefinite matrices,
respectively. Hence, Diananda’s decomposition theorem [18] can be reformulated
as follows.
###### Theorem 2.2.
$C_{n}\subseteq S_{n}^{+}\cap(S_{n})^{+}$ holds for all $n$. If $n\leq 4$,
then $C_{n}=S_{n}^{+}\cap(S_{n})^{+}$.
By the way, the matrices in $S_{n}^{+}\cap(S_{n})^{+}$ sometimes are called
“doubly nonnegative”. Of course, in dimension $n\geq 5$ there are matrices
which are doubly nonnegative but not completely positive, the counterexample
can be seen in [19].
According to Theorem 2.2, problem (CPP) can be relaxed to the following DNNP
problem
${\rm{(DNNP)}}\begin{array}[]{lll}&\min&4Q\bullet
Z-4z^{T}(Qe+c)+e^{T}Qe+2c^{T}e\\\ &{\rm s.t.}&2a_{i}^{T}z=a_{i}^{T}e-b_{i},\
\forall i\in I,\\\ &&4a_{i}^{T}Za_{i}=(a_{i}^{T}e-b_{i})^{2},\ \forall i\in
I,\\\ &&Z_{ii}=z_{i},\ \forall i=1,2,\ldots,n,\\\
&&\left[\begin{array}[]{ll}1&z^{T}\\\ z&Z\end{array}\right]\in
S_{1+n}^{+}\cap(S_{1+n})^{+}.\end{array}$
Up to now, the other convex relaxation problem for problem (BQP) is
established, i.e., problem (DNNP), which is computable by some popular
packages for solving convex programs, such as CVX, etc.
Note that problem (DNNP) has $n+\frac{n(n+1)}{2}$ equality constraints more
than standard semidefinite relaxation problem (SDR1), and $n$ equality
constraints more than problem (SDR2), respectively. Thus, the lower bound
which get from problem (DNNP) is much greater than that one of by problem
(SDR1) and problem (SDR2), respectively.
In the following, two types of problems are tested to show the performance of
problem (SDR1), problem (SDR2) and problem (DNNP), respectively. The
statistics of the test problems are chosen as follows:
Table 2: Statistics of the test problems Type | | Instances | | $n$ | | $m$ | | Function
---|---|---|---|---|---|---|---|---
RdBQP | | $50$ | | $50$ | | $25$ | | rand($\cdot$)
RdsBQP | | $50$ | | $50$ | | $25$ | | rands($\cdot$)
$\bullet$ RdBQP. For this type of problems, we generate $50$ instances of
problem (BQP) by using MATLAB function rand($\cdot$). The symmetric matrix $Q$
is generated by rand($\cdot$)+rand($\cdot$)’, and all problems are nonconvex.
$\bullet$ RdsBQP. The coefficients of $50$ instances of problem (BQP) are
generated by using MATLAB function rands($\cdot$), and the symmetric matrix
$Q$ is generated by rands($\cdot$)+rands($\cdot$)’. All instances are
nonconvex.
We use performance profiles [16] to compare the performance of problem (SDR1),
problem (SDR2) and problem (DNNP), for problem (BQP), respectively. The
corresponding results of performance are shown in Figure 2. The profiles for
Figure 2 are based on optimal values of problem (SDR1), problem (SDR2) and
problem (DNNP), respectively, and these problems are solved by CVX. From
Figure 2, it is obviously that the performances of problem (SDR2) and problem
(DNNP) are almost the same, which are better than that one of problem (SDR1),
for problems RdBQP and RdsBQP, respectively. Thus, we can conclude that it is
more efficient to use problem (SDR2) and problem (DNNP) than problem (SDR1) to
solve problem (BQP), from the point of view of optimal values.
Furthermore, we will show the equivalence of the problems (DNNP) and (SDR2) in
Section 3.
Figure 2: Left figure is based on optimal values of problems RdBQP, right
figure is based on optimal values of problems RdsBQP.
## 3 Relationship between relaxation problems
In this section, we will investigate the relationship between two relaxation
problems (SDR2) and (DNNP). First of all, the definition of the equivalence of
two optimization problems is defined as follows.
###### Definition 3.1.
We call two problems are equivalent if they satisfy the following two
conditions:
(i) If from a solution of one problem, a solution of the other problem is
readily found, and vice versa.
(ii) The two problems have the same optimal value.
Now, based on the above Definition 3.1, the main theorem is given below.
###### Theorem 3.1.
Suppose that the feasible sets $\rm{Feas(SDR2)}$ and $\rm{Feas(DNNP)}$ are all
nonempty. Then, two problems (SDR2) and (DNNP) are equivalent.
###### Proof.
The proof can be divided into two parts. First of all, we will prove that
$\rm{Opt(SDR2)\geq Opt(DNNP)}$.
Suppose that $(x^{*},X^{*})$ is an optimal solution of problem $\rm(SDR2)$,
let $Z_{ij}=\frac{1}{4}(1-x_{i}^{*}-x_{j}^{*}+X_{ij}^{*})$ and
$z_{i}=\frac{1}{2}(1-x_{i}^{*}),\ \forall 1\leq i\leq j\leq n$, i.e.,
$Z=\frac{1}{4}(ee^{T}-e(x^{*})^{T}-x^{*}e^{T}+X^{*}),\
z=\frac{1}{2}(e-x^{*}).$ (4)
By $a_{i}^{T}x^{*}=b_{i}$ for all $i\in I$ and (4), we have
$a_{i}^{T}x^{*}=a_{i}^{T}(e-2z)=b_{i}\Rightarrow
2a_{i}^{T}z=a_{i}^{T}e-b_{i},\ \forall i\in I.$ (5)
From (4) and $a_{i}^{T}X^{*}a_{i}=b_{i}^{2}$ for all $i\in I$, it follows that
$\begin{array}[]{ll}a_{i}^{T}X^{*}a_{i}&=a_{i}^{T}(4Z-ee^{T}+e(x^{*})^{T}+x^{*}e^{T})a_{i}=b_{i}^{2}\\\
&\Rightarrow 4a_{i}^{T}Za_{i}=(a_{i}^{T}e-b_{i})^{2},\ \forall i\in
I.\end{array}$ (6)
Again from (4), which imply that
$Z_{ii}=\frac{1}{4}(1-2x_{i}^{*}+X_{ii}^{*})=\frac{1}{2}(1-x_{i}^{*})=z_{i},\
\forall i\in I,$ (7)
since $X_{ii}^{*}=1$.
From $1-x_{i}^{*}-x_{j}^{*}+X_{ij}^{*}\geq 0,\forall 1\leq i\leq j\leq n$, it
holds that
$Z_{ij}\geq 0,\forall 1\leq i\leq j\leq n,$ (8)
which combining with (7), further imply that
$z_{i}\geq 0,\forall 1\leq i\leq n.$ (9)
By Theorem 2.1(ii) and (4), it follows that
$\begin{array}[]{ll}Z-zz^{T}&=\frac{1}{4}(ee^{T}-e(x^{*})^{T}-x^{*}e^{T}+X^{*})-\frac{1}{4}(e-x^{*})(e-x^{*})^{T}\\\
&=\frac{1}{4}(X^{*}-x^{*}(x^{*})^{T})\succeq 0.\end{array}$ (10)
Combining (10) with (5), (6), (8) and (9), it follows that $(z,Z)$ defined by
(4) is a feasible solution for problem $\rm{(DNP)}$.
Moreover, again from (4), we have
$\begin{array}[]{ll}4Q\bullet Z-4z^{T}(Qe+c)+e^{T}Qe+2c^{T}e\\\
=Q\bullet(ee^{T}-e(x^{*})^{T}-x^{*}e^{T}+X^{*})-2(e-x^{*})^{T}(Qe+c)+e^{T}Qe+2c^{T}e\\\
=Q\bullet X^{*}+2c^{T}x^{*}=\rm{Opt(SDR2)},\end{array}$
which further imply that $\rm{Opt}(DNNP)\leq\rm{Opt}(SDR2)$.
On the other hand, given an optimal solution $(z^{*},Z^{*})$ to problem
(DNNP), and let
$X_{ij}=1-2z_{i}^{*}-2z_{j}^{*}+4Z_{ij}^{*},\ x_{i}=1-2z^{*}_{i},\ \forall
1\leq i\leq j\leq n,$ (11)
which imply that
$X_{ii}=1-4z^{*}_{i}+4Z^{*}_{ii}=1$ (12)
since $Z^{*}_{ii}=z^{*}_{i},\ \forall i=1,2,\ldots,n$. Moreover,
$\begin{array}[]{ll}1-x_{i}-x_{j}+X_{ij}&=1-(1-2z_{i}^{*})-(1-2z_{j}^{*})+1-2z_{i}^{*}-2z_{j}^{*}+4Z_{ij}^{*}\\\
&=4Z_{ij}^{*}\geq 0,\ \forall 1\leq i\leq j\leq n.\end{array}$ (13)
From (11) and $2a_{i}^{T}z^{*}=a_{i}^{T}e-b_{i},\ \forall i\in I$, it follows
that
$a_{i}^{T}x=a_{i}^{T}(e-2z^{*})=b_{i},\ \forall i\in I.$ (14)
Again from (11) and $4a_{i}^{T}Z^{*}a_{i}=(a_{i}^{T}e-b_{i})^{2},\ \forall
i\in I$, we have
$\begin{array}[]{ll}a_{i}^{T}Xa_{i}&=a_{i}^{T}(ee^{T}-2e(z^{*})^{T}-2z^{*}e^{T}+4Z^{*})a_{i}\\\
&=b_{i}^{2},\ \forall i\in I.\end{array}$ (15)
From (11) and Theorem 2.1(ii), it holds that
$\begin{array}[]{lll}X-xx^{T}&=ee^{T}-2z^{*}e^{T}-2e(z^{*})^{T}+4Z^{*}-(e-2z^{*})(e-2z^{*})^{T}\\\
&=4(Z^{*}-z^{*}(z^{*})^{T})\succeq 0.\end{array}$ (16)
By (13), (14), (15) and (16), we can conclude that $(x,X)$ defined by (11) is
a feasible solution for problem $\rm{(SDR2)}$. Furthermore, we have
$\begin{array}[]{ll}X\bullet
Q+2c^{T}x&=(ee^{T}-2e(z^{*})^{T}-2z^{*}e^{T}+4Z^{*})\bullet
Q+2c^{T}(e-2z^{*})\\\ &=4Z^{*}\bullet Q-4(z^{*})^{T}(Qe+c)+e^{T}Qe+2c^{T}e\\\
&=\rm{Opt(DNNP)},\end{array}$
which imply that $\rm{Opt}(SDR2)\leq\rm{Opt}(DNNP)$. Summarizing the analysis
above and according to Definition 3.1, we can conclude that problem (DNNP) is
equivalent to problem (SDR2). ∎
Although $\rm{Opt}(SDR2)=\rm{Opt}(DNNP)$ in view of Theorem 3.1 and Definition
3.1, problem (DNNP) has $n$ equality constraints more than problem (SDR2) in
form. So, the amount of computation for solving problem (DNNP) may be much
greater than that one of solving problem (SDR2). In order to illustrate this
point of view, the compared performance results are shown in Figure 3 and
Figure 4, respectively, which are based on the number of iterations and CPU
time for solving problems RdBQP and RdsBQP. The results in Figure 3 show that
the performance of problem (SDR2) is better than that one of problem (DNNP)
for problems RdBQP, but the performance of problem (DNNP) is better than that
one of problem (SDR2) for problems RdsBQP, in view of the points of the number
of iterations. From the results of the performance of CPU time, it is
obviously that problem (SDR2) is more efficient than problem (DNNP) for
solving problems RdBQP and RdsBQP, respectively. Summarizing the analysis
above, we can efficiently solving problem (BQP) by soling problem (SDR2) or
problem (DNNP) in practice.
Figure 3: Left figure is based on the number of iterations of problems RdBQP,
right figure is based on the number of iterations of problems RdsBQP.
Figure 4: Left figure is based on CPU time of problems RdBQP, right figure is
based on CPU time of problems RdsBQP.
## 4 An application to max-cut problem
The max-cut (MC) problem is a kind of important combinatorial optimization
problem on undirected graphs with weights on the edges, and also is NP-hard
[20]. Given such a graph, (MC) problem consists in finding a partition of the
set of nodes into two parts so as to maximize the total weight of edges cut by
the partition.
Let $G$ be an $n$-node graph, vertex set $V:=\\{1,2,\ldots,n\\}$, $A(G)$ the
adjacency matrix of graph $G$, $L$ the Laplacian matrix associated with the
graph, i.e., $L:={\rm{Diag}}(A(G)e)-A(G)$. Let the vector
$u\in\\{+1,-1\\}^{n}$ represent any cut in the graph $G$ via the
interpretation that the sets $\\{i:u_{i}=+1\\}$ and $\\{i:u_{i}=-1\\}$ form a
partition of the node set of $G$, we can get the following formulation for
(MC) problem
$\rm{(MC)}\begin{array}[]{lll}&\max&\frac{1}{4}u^{T}Lu\\\ &{\rm
s.t.}&u\in\\{+1,-1\\}^{n}.\end{array}$
On one hand, by using the standard semidefinite relaxation technique to (MC)
problem, we can get the following problem
$\rm{\widehat{(SDR)}}\begin{array}[]{lll}&\max&\frac{1}{4}L\bullet U\\\ &{\rm
s.t.}&U_{ii}=1,\ \forall i=1,2,\ldots,n,\\\ &&U\succeq 0.\end{array}$
Goemans and Williamson [5] have provided estimates for the quality of problem
$(\rm\widehat{SDR})$ bound for (MC) problem. By a randomly rounding a solution
to problem $(\rm\widehat{SDR})$, they propose a $0.878$-approximation
algorithm for solving problem (MC) based on problem $(\rm\widehat{SDR})$,
which is known to be the best approximation ration of polynomial-time
algorithm for solving problem (MC).
On the other hand, according to the technique introduced in Section 2.2,
problem (MC) can also be relaxed to the following doubly nonnegative
programming problem
$\rm{\widehat{(DNNP)}}\begin{array}[]{lll}&\max&L\bullet
X-x^{T}Le+\frac{1}{4}e^{T}Le\\\ &{\rm s.t.}&X_{ii}=x_{i},\ \forall
i=1,2,\ldots,n,\\\ &&\left[\begin{array}[]{ccc}1&x^{T}\\\
x&X\end{array}\right]\in S_{1+n}^{+}\cap(S_{1+n})^{+}.\end{array}$
###### Remark 4.1.
(i) Note that for two relaxation problems $\rm{\widehat{(SDR)}}$ and
$\rm{\widehat{(DNNP)}}$, the feasible sets are all nonempty. It is obviously
that the identity matrix $E$ is a feasible solution for problem
$\rm{\widehat{(SDR)}}$, and $(x,X)=(0,\mathbf{0})$ feasible for problem
$\rm{\widehat{(DNNP)}}$.
(ii) Compared with problem $\rm{\widehat{(SDR)}}$, problem
$\rm{\widehat{(DNNP)}}$ has not only $n+\frac{n(n+1)}{2}$ new inequality
constraints, but also $n$ variables.
Thus, according to Theorem 3.1 and Remark 4.1(i), we have the following
theorem.
###### Theorem 4.1.
Problem $\rm\widehat{(SDR)}$ is equivalent to problem $\rm\widehat{(DNNP)}$.
###### Proof.
On one hand, suppose that $U^{*}$ is an optimal solution for problem
$\rm\widehat{(SDR)}$, and let $X_{ij}=\frac{1}{4}(U^{*}_{ij}+1)$ and
$x_{i}=\frac{1}{2}$, $\forall 1\leq i\leq j\leq n$, i.e.,
$X=\frac{1}{4}(U^{*}+ee^{T}),\ x=\frac{1}{2}e,$ (17)
which imply that
$X_{ii}=\frac{1}{4}(U^{*}_{ii}+1)=x_{i}=\frac{1}{2}>0,\ \forall 1\leq i\leq
n,$ (18)
since $U_{ii}^{*}=1$. Then, from $U^{*}\succeq 0$, it follows that
$0\leq U^{*}_{ii}U^{*}_{jj}-(U^{*}_{ij})^{2}=1-(U^{*}_{ij})^{2},$
i.e.,
$-1\leq U^{*}_{ij}\leq 1,\ \forall 1\leq i<j\leq n,$
combining with (17), we have
$X_{ij}=\frac{1}{4}(U^{*}_{ij}+1)\geq 0,\ 1\leq i<j\leq n.$ (19)
Moreover, from (17), it follows that
$X-xx^{T}=\frac{1}{4}(U^{*}+ee^{T})-\frac{1}{4}ee^{T}=\frac{1}{4}U^{*}\succeq
0,$ (20)
it followed by $U^{*}\succeq 0$. Combining (18), (19) and (20) as well as
Theorem 2.1(ii), it holds that $(x,X)$ is a feasible solution for problem
$\rm\widehat{(DNNP)}$. Again from (17), we have $L\bullet
X-x^{T}Le+\frac{1}{4}e^{T}Le=\frac{1}{4}L\bullet
U^{*}=\rm{Opt}\widehat{(SDR)}$, which further imply that
$\rm{Opt}\widehat{(DNNP)}\geq\rm{Opt}\widehat{(SDR)}$.
On the other hand, suppose that $(x^{*},X^{*})$ is an optimal solution for
problem $\rm\widehat{(DNNP)}$, and let
$U=4X^{*}-2x^{*}e^{T}-2e(x^{*})^{T}+ee^{T},$ (21)
which imply that
$U_{ii}=4X_{ii}^{*}-2x^{*}_{i}-2x^{*}_{i}+1=1,\ \forall 1\leq i\leq n,$ (22)
since $X_{ii}^{*}=x_{i}^{*},\ \forall 1\leq i\leq n$. From (21), it follows
that
$\begin{array}[]{ll}U&=4X^{*}-2e(x^{*})^{T}-2x^{*}e^{T}+ee^{T}\\\
&=4(X^{*}-x^{*}(x^{*})^{T})+(2x^{*}-e)(2x^{*}-e)^{T}\succeq 0.\end{array}$
(23)
From (22) and (23), we can conclude that $U$ defined by (21) is a feasible
solution for problem $\rm\widehat{(SDP)}$. Furthermore, again from (21), we
have
$\frac{1}{4}L\bullet U=L\bullet
X^{*}-(x^{*})^{T}Le+\frac{1}{4}e^{T}Le=\rm{Opt}\widehat{(DNNP)},$
which imply that $\rm{Opt}\widehat{(SDR)}\geq\rm{Opt}\widehat{(DNNP)}$. The
proof is completed. ∎
By Theorem 4.1, we can obtain doubly nonnegative relaxation for problem (MC)
exactly equal to the standard semidefinite relaxation, i.e. problem
$\rm{\widehat{(DNNP)}}$ and problem $\rm{\widehat{(SDR)}}$ are equivalent
according to Definition 3.1, without the boundedness assumption of two
feasible sets of two problems.
## 5 Conclusions
In this paper, a class of nonconvex binary quadratic programming problem is
considered, which is NP-hard in general. In order to solve this problem
efficiently by some popular packages for solving convex programs, two convex
representation methods are proposed. One of the methods is semidefinite
relaxation, by the structure of the binary constraints of original problem,
which results in a new tighter semidefinite relaxation problem (SDR2). The
other method is doubly nonnegative relaxation. The original problem is
equivalently transformed into a convex problem (CPP), which is also NP-hard in
general. Then, by virtue of the features of constraints in this problem, a
computable convex problem (DNNP) is obtained through doubly nonnegative
relaxation. Moreover, the two convex relaxation problems are equivalent. These
results are applied to (MC) problem, we can conclude that doubly nonnegative
relaxation for problem (MC) is equivalent to the standard semidefinite
relaxation for it. Furthermore, some compared numerical results are reported
to show the performance of two relaxed problems.
## References
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|
arxiv-papers
| 2012-11-23T02:33:20 |
2024-09-04T02:49:38.299528
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Chuan-Hao Guo, Yan-Qin Bai and Li-Ping Tang",
"submitter": "Chuan-Hao Guo",
"url": "https://arxiv.org/abs/1211.5406"
}
|
1211.5416
|
11institutetext: Bogoliubov Laboratory of Theoretical Physics, Joint Institute
for Nuclear Research, Dubna, Russia
# Polarized GPDs in pions and kaons electroproduction. Tranversity effects.
S.V.Goloskokov [email protected]
###### Abstract
We analyze the electroproduction of pseudoscalar mesons within the handbag
approach. To investigate these reactions, we consider the leading-twist
contribution together with the transversity twist-3 effects that are crucial
in the description of experimental data. Our results on the cross section are
in agreement with experiment. We present our predictions for spin observables.
## 1 Introduction
We investigate the process of pseudoscalar meson leptoproduction (PML) at
large $Q^{2}$ within the handbag approach, where the amplitudes factorize into
a hard subprocess and soft part –Generalized Parton Distributions GPDs fact .
The hard subprocess amplitudes are calculated by using the modified
perturbative approach sterman that takes into account quark transverse
degrees of freedom as well as gluonic radiation condensed in a Sudakov factor.
The PML was analyzed in gk09 ; gk11 . It was shown that the leading-twist
contribution determined by the polarized GPDs is not sufficient to describe
processes of PML. The essential contributions from the transversity GPDs are
needed to be consistent with experiment. Within the handbag approach, these
twist-3 effects can be modeled by the transversity GPDs $H_{T}$,
$\bar{E}_{T}$, in conjunction with the twist-3 meson wave function.
In this report we study the cross sections of the pion leptoproduction in the
HERMES and CLAS energy range on the basis of the model gk09 ; gk11 . Our
results are in good agreement with experiment. We show that the transversity
GPDs lead to a large transverse cross section for most reactions of the
pseudoscalar meson production. Predictions for spin asymmetries in the pion
leptoproduction are presented as well.
At the end, we present the model results for the cross section of the
$K^{+}\Lambda$ leptoproduction, which is large due to the transversity
contribution gk11 , and predictions for the spin asymmetry in this reaction.
## 2 Leptoproduction of pseudoscalar mesons
Hard exclusive PML amplitudes were studied on the basis of the handbag
approach. The typical contributions are shown in Fig.1. In the left part of
the graph we present the meson pole contribution which appears for the charge
meson production. In Fig. 1 (right) the example of the handbag diagram is
shown. In the leading twist the last contribution is expressed in terms of the
polarized GPDs $\tilde{H}$ and $\tilde{E}$ whose parameterization can be found
in gk07q .
Figure 1: Examples of the graphs essential in PML. Left–pion pole and
right–handbag contributions to the $\pi^{+}$ production.
The proton non- flip and helicity-flip amplitudes for longitudinally polarized
photons ${\cal M}^{M}_{0\nu^{\prime},0\nu}$, which dominates at large $Q^{2}$,
can be written in the form:
$\displaystyle{\cal M}^{M}_{0+,0+}$ $\displaystyle\propto$
$\displaystyle\sqrt{1-\xi^{2}}\,\,[-\frac{\xi(m_{N^{i}}+M_{N^{f}})Q^{2}}{1-\xi^{2}}\frac{\rho_{M}}{t-m_{M}^{2}}+\langle\tilde{H}^{M}\rangle-\frac{\xi^{2}}{1-\xi^{2}}\langle\widetilde{E}^{M}_{n.p.}\rangle];\;$
$\displaystyle{\cal M}^{M}_{0-,0+}$ $\displaystyle\propto$
$\displaystyle\frac{\sqrt{-t^{\prime}}}{(m_{N^{i}}+M_{N^{f}})}\,\Big{[}(m_{N^{i}}+M_{N^{f}})Q^{2}\frac{\rho_{M}}{t-m_{M}^{2}}+\xi\langle\widetilde{E}^{M}_{n.p.}\rangle\Big{]}\,.$
(1)
Here $M$\- produced pseudoscalar meson, $N^{i}$-initial nucleon (proton),
$N^{f}$-final barion (neutron, $\Lambda$, $\Sigma$). The corresponding
amplitudes with transversally polarized photons are suppressed as $1/Q$.
The first terms in (2) appear for the charged meson production and are
connected with the M meson pole. The fully experimentally measured
electromagnetic form factor of M meson is included into $\rho_{M}$.
The second terms in (2) represent the handbag contribution to the PML
amplitude. The $<\tilde{F}>$ in (2) is a convolution of GPD $\tilde{F}$ with
the hard subprocess amplitude ${\cal
H}_{0\lambda,0\lambda}(\overline{x},...)$:
$<\tilde{F}>=\sum_{\lambda}\int_{-1}^{1}d\overline{x}{\cal
H}_{0\lambda,0\lambda}(\overline{x},...)\tilde{F}(\overline{x},\xi,t),\;$ (2)
The subprocess amplitude is calculated within the MPA sterman . The amplitude
${\cal H}^{a}$ is a contraction of the hard part ${\cal F}^{a}$, which is
calculated perturbatively and includes the transverse quark momentum ${\bf
k}_{\perp}$, and the nonperturbative ${\bf k}_{\perp}$-dependent meson wave
function $\Psi$ koerner . The gluonic corrections are treated in the form of
the Sudakov factors. The resummation and exponentiation of the Sudakov
corrections $S$ can be done in the impact parameter space ${\bf b}$ sterman .
The Fourier transformed subprocess amplitude from the ${\bf k}_{\perp}$ to
${\bf b}$ space reads as
$\displaystyle{\cal H}^{a}_{0\lambda,0\lambda}\propto\int d\tau
d^{2}b\,{\Psi}(\tau,-{\bf b})\,{\cal
F}^{a}_{0\lambda,0\lambda}(\overline{x},\xi,\tau,Q^{2},{\bf
b},)\,\alpha_{s}\,{\rm exp}{[-S(\tau,{\bf b},Q^{2})]}.$
Here $\tau$ is the momentum fraction of the quark that enters into the meson.
The GPDs are estimated using the double distribution representation mus99
which connects GPDs with PDFs through the double distribution function $f$.
For the valence quark contribution it looks like
$\displaystyle
f_{i}(\beta,\alpha,t)=h_{i}(\beta,t)\,\frac{3}{4}\,\frac{[(1-|\beta|)^{2}-\alpha^{2}]}{(1-|\beta|)^{3}}.$
(3)
The functions $h$ are determined in the terms of PDFs and are parameterized in
the form
$h(\beta,t)=N\,e^{b_{0}t}\beta^{-\alpha(t)}\,(1-\beta)^{n}.$ (4)
Here the $t$\- dependence is considered in a Regge form and $\alpha(t)$ is the
corresponding Regge trajectory. The parameters in (4) are obtained from the
known information about PDFs CTEQ6 e.g, or from the nucleon form factor
analysis pauli .
We calculate the leading-twist amplitudes together with the meson pole
contribution on the basis of (2). Unfortunately, these terms are insufficient
to describe experimental data at low $Q^{2}$. We can demonstrate this using
the $A_{UT}$ asymmetry in the $\pi^{+}$ leptoproduction as an example.
Figure 2: $A_{UT}^{\sin(\phi_{s})}$ asymmetry of the $\pi^{+}$ production.
Dashed line- leading twist contribution. Solid line- model results, including
twist-3 effects. Data are from HERMES airap
This asymmetry is expressed in terms of interference of $M_{0-,++}$ and proton
non-flip amplitude gk09 :
$A_{UT}^{\sin(\phi_{s})}\propto\mbox{Im}[M^{*}_{0-,++}M_{0+,0+}].$ (5)
The leading twist contributions cannot explain this asymmetry –see Fig. 2. A
new twist-3 contribution to the $M_{0-,++}$ amplitude, which is not small at
$t^{\prime}\sim 0$, is needed. We estimate this contribution to ${\cal
M}_{0-,++}$ by the transversity GPD $H_{T}$ in conjugation with the twist-3
pion wave function in the hard subprocess amplitude ${\cal H}$ gk11 . We have
${\cal M}^{M,twist-3}_{0-,\mu+}\propto\,\int_{-1}^{1}d\overline{x}{\cal
H}_{0-,\mu+}(\overline{x},...)\,[H^{M}_{T}+...O(\xi^{2}\,E^{M}_{T})].$ (6)
The $H_{T}$ GPD is connected with transversity PDFs as
$H^{a}_{T}(x,0,0)=\delta^{a}(x);\;\;\;\mbox{and}\;\;\;\delta^{a}(x)=C\,N^{a}_{T}\,x^{1/2}\,(1-x)\,[q_{a}(x)+\Delta
q_{a}(x)].$ (7)
Here $a$ is a quark flavor. We parameterize the PDF $\delta$ by using the
model ans . The double distribution (3) is used to calculate GPD $H_{T}$.
The amplitude $M_{0+,++}$ is extremely important in analyzes of PML as well.
The transversity twist-3 contribution to this amplitude is determined by
$\bar{E}_{T}$ GPDs and has the form gk11 similar to (6)
${\cal
M}^{M,twist-3}_{0+,\mu+}\propto\,\frac{\sqrt{-t^{\prime}}}{4m}\,\int_{-1}^{1}d\overline{x}{\cal
H}_{0-,\mu+}(\overline{x},...)\;\bar{E}^{M}_{T}.$ (8)
The hard scattering subprocess amplitude ${\cal
H}_{0-,\mu+}(\overline{x},...)$ in (8) is the same as in (6).
|
---|---
Figure 3: Left: Cross section of $\pi^{+}$ production at HERMES. Right: Cross
section of $\pi^{0}$ production at HERMES. Full line- unseparated cross
section. dashed- $d\sigma_{L}/dt$, dashed-dotted line- $d\sigma_{T}/dt$.
At the moment, the information on $\bar{E}_{T}$ is very poor. Some results
were obtained only in the lattice QCD lat . The lower moments of
$\bar{E}_{T}^{u}$ and $\bar{E}_{T}^{d}$ were found to be quite large, have the
same sign and a similar size. At the same time, $H_{T}^{u}$ and $H_{T}^{d}$
are different in sign. For the pion production we have the following
contribution to GPDs frankfurt99
$\displaystyle F(\pi^{+})$ $\displaystyle=$ $\displaystyle
F^{(3)}=F^{u}-F^{d},$ $\displaystyle F(\pi^{0})$ $\displaystyle=$
$\displaystyle 2/3\,F^{u}+1/3\,F^{d}.$ (9)
From these equations we find an essential compensation of the $\bar{E}_{T}$
contribution to the $\pi^{+}$ amplitude but $H_{T}$ effects are not small
there. For the $\pi^{0}$ production we have the opposite case – $\bar{E}_{T}$
contributions are large but $H_{T}$ effects are smaller.
In Fig. 3 (left), we show our results gk09 for the unseparated cross section
of the $\pi^{+}$ production which describes fine HERMES data airapsig . The
$\sigma_{L}$ and $\sigma_{T}$ are shown as well. The longitudinal cross
section determined by leading-twist contribution dominates at small momentum
transfer $-t<0.2\mbox{GeV}^{2}$. At larger $-t$ we find a not small transverse
cross section where the $H_{T}$ contribution is visible. In Fig. 3 (right),
our results for the cross section of the $\pi_{0}$ production are presented
which are very different from the $\pi^{+}$ process. The transverse cross
section
$\sigma_{T}\propto|{\cal M}^{M,twist-3}_{0+,++}|^{2}+|{\cal
M}^{M,twist-3}_{0-,++}|^{2},$ (10)
where the $\bar{E}_{T}$ and $H_{T}$ contributions are important gk09
dominates. At small momentum transfer the $H_{T}$ contribution is visible and
provides a nonzero cross section. At larger $-t^{\prime}\sim
0.2\mbox{GeV}^{2}$ the $E_{T}$ contribution is essential and gives a maximum
in the cross section.
The longitudinal cross section which is expected to play an important role is
much smaller with respect to the transverse cross section $\sigma_{T}$. The
essential contributions to the $\sigma_{T}$ cross section are determined by
the twist-3 $H_{T}$ and $\bar{E}_{T}$ effects and decreases quickly with
$Q^{2}$. At quite large $Q^{2}$ the leading-twist effects will dominate.
|
---|---
Figure 4: Left: $A_{UT}$ asymmetry of the $\pi^{+}$ production at HERMES
energies. Right: $A_{UL}$ asymmetry of the $\pi^{+}$ production at HERMES.
HERMES data are shown airap . Dashed line- results without transversity
$H_{T}$ effects.
In Fig.4, we demonstrate that the transversity $H_{T}$ effects are essential
in asymmetries of the $\pi^{+}$ production. When we omit the $H_{T}$
contributions, asymmetries change drastically.
In Fig 5 (left), our prediction for the $\pi^{0}$ production in the CLAS
energy range sgspin11 is shown together with experimental data bedl . Our
results are close to the experimental data and definitely show the same dip in
the unseparated cross section at low momentum transfer, as was observed for
HERMES –see Fig.3 (right). We present in this plot the interference
$\sigma_{LT}$ and $\sigma_{TT}$ cross sections too. The value of $\sigma_{LT}$
is quite small, compatible with zero. The $\sigma_{TT}$ cross section is
negative and large. Note that the $E_{T}$ contribution to $\sigma_{T}$ and
$\sigma_{TT}$ cross sections is strongly correlated. The fact that we describe
the CLAS data for both cross sections quite well can be an indication of
observation of large transversity effects at CLAS. However, the definite
conclusion on the importance of transversity effects in the $\pi^{0}$ cross
section can be made only if the data on the separated $\sigma_{L}$ and
$\sigma_{T}$ cross section will be available experimentally and $\sigma_{T}$
will be much larger than $\sigma_{L}$. Probably, such a study can be performed
at JLAB12.
|
---|---
Figure 5: Left: $\pi^{0}$ production in the CLAS energy range together with
the data. Dashed-dot-dotted line- $\sigma_{T}+\epsilon\sigma_{L}$, dashed
line-$\sigma_{LT}$, dashed-dotted- $\sigma_{TT}$. Right: $\eta/\pi^{0}$
production ratio in the CLAS energy range together with preliminary data.
In Fig. 5 (right), we analyze the transversity effects in the ratio of the
$\eta/\pi^{0}$ cross section at CLAS energies. The two parameterizations of
$H_{T}$ GPDs gk11 are presented there. Different combinations of the quark
contributions to these processes lead to the essential role of $H_{T}$ effects
in this ratio at small $-t<0.2\mbox{GeV}^{2}$. At larger momentum transfer
large $E_{T}$ effects in the $\pi^{0}$ production found in the model lead to a
rapid decrease of the $\eta/\pi^{0}$ cross section ratio with $t$\- growing.
At $-t>0.2\mbox{GeV}^{2}$ this ratio becomes close to $\sim 0.3$, which was
confirmed by CLAS vkubar .
|
---|---
Figure 6: Left: Moments of the $A_{UT}$ asymmetries at HERMES for the
$\pi^{0}$ production. Right: The predicted $A_{LU}$ asymmetry in the $\pi^{+}$
and $\pi^{0}$ production at HERMES.
In Fig.6 (left), we present our results for the moments of the $A_{UT}$
asymmetry in the $\pi^{0}$ production at HERMES. The predicted asymmetries are
large and can give additional information on transversity effects in this
reaction. In Fig.6 (right), we show the $A_{LU}$ asymmetry in the pion
production at HERMES. $A_{LU}(\pi^{+})$ is large because of the pion-pole
contribution in this channel. The predicted $A_{LU}$ asymmetry in the
$\pi^{0}$ production is small. Measurement of this asymmetry at HERMES can
give information on the nonpole term of $\widetilde{E}^{M}_{n.p.}$ in (2).
|
---|---
Figure 7: Left: The $K^{+}\Lambda$ production cross sections at HERMES
energies. Right: Predicted moments of $A_{UT}$ asymmetries for $K^{+}\Lambda$
channel at HERMES.
Using the same model we calculate the cross section and spin asymmetry for the
$K^{+}\Lambda$ production. To estimate proton- hyperon transition GPDs we use
the SU(3) flavor symmetry model frankfurt99
$H_{T}(p\to\Lambda)\sim[2H_{T}^{u}-H_{T}^{d}-H_{T}^{s}].$ (11)
Due to different signs of $H_{T}^{u}$ and $H_{T}^{d}$ we find a quite large
$H_{T}$ effect here. In this reaction, the kaon pole contribution should be
much smaller with respect to the $\pi^{+}$ case. The details of calculations
can be found in gk11 . The large transversity $H_{T}$ effects in the
$K^{+}\Lambda$ channel provide the large $\sigma_{T}$ cross section without a
forward dip which dominated with respect to $\sigma_{L}$, see Fig. 7 (left).
In Fig. 7 (right), we show our predictions for moments of the $A_{UT}$
asymmetry in this channel. The $\sin(\phi_{s})$ moment of asymmetry determined
by the $H_{T}$ transversity contribution (5) is quite large.
## 3 Conclusion
We calculate the PML amplitude within the handbag approach, in which the
amplitudes factorize into hard subprocesses and GPDs fact . The hard
subprocess amplitudes were calculated within the modified perturbative
approach sterman where quark transverse degrees of freedom and the gluonic
radiation, condensed in a Sudakov factor were taken into account.
At leading-twist accuracy the PML reactions are sensitive to the GPDs
$\widetilde{H}$ and $\widetilde{E}$ which contribute to the amplitudes for
longitudinally polarized virtual photons. This contribution should be
predominated at large $Q^{2}$. Unfortunately, now experimental data on these
reactions are available at small photon virtualities.
We observed that the experimental data on pseudoscalar meson leptoproduction
at low $Q^{2}$ also require contributions from the transversity GPDs, in
particular, from $H_{T}$ and $\bar{E}_{T}$. Within the handbag approach the
transversity GPDs are accompanied by a twist-3 meson wave function. At HERMES
and COMPASS energies the twist-3 $\bar{E}_{T}$ effects produce a large
transverse cross section $\sigma_{T}$ gk11 which exceeds substantially the
leading twist longitudinal cross section for most reactions with the exception
of the $\pi^{+}$ and $\eta^{\prime}$ channels.
The indication of large transversity effects are available now at CLASS. They
observe a large unseparated and large negative $\sigma_{TT}$ cross section
which can be described in our model by large transversity $\bar{E}_{T}$
effects. Essential $H_{T}$ and $\bar{E}_{T}$ effects are predicted at the
ratio $\eta/\pi^{0}$ cross section. Large $E_{T}$ effects in the $\pi^{0}$
production predict that for $-t>0.2\mbox{GeV}^{2}$ this ratio should be close
to $\sim 0.3$, which was confirmed by CLAS vkubar .
Nevertheless the experimental separation of the $\sigma_{L}$ and $\sigma_{T}$
cross section in the $\pi^{0}$ electroproduction is important. If it is found
that $\sigma_{T}$ is much larger than the $\sigma_{L}$ cross section, this
will be a definite demonstration of observation of transversity effects in
this reaction. We hope that it can be done at JLAB12. Essential $H_{T}$
effects in the $K^{+}\Lambda$ channel were predicted.
We describe well the cross section and spin observables for various PML. Thus,
we can conclude that the information on GPDs discussed above should be not far
from reality. Future experimental results at COMPASS, JLAB12 can give
important information on the role of transversity effects in these reactions.
This work is supported in part by the Russian Foundation for Basic Research,
Grant 12-02-00613 and by the Heisenberg-Landau program.
## References
* (1) X. Ji, Phys. Rev. D55, (1997) 7114;
A.V. Radyushkin, Phys. Lett. B380, (1996) 417;
J.C. Collins, et al., Phys. Rev. D56, (1997) 2982.
* (2) J. Botts and G. Sterman, Nucl. Phys. B325, (1989) 62.
* (3) S.V.Goloskokov, P.Kroll, Euro. Phys. J. C65, (2010) 137\.
* (4) S.V.Goloskokov, P.Kroll, Euro. Phys. J. A47, (2011) 112.
* (5) S.V. Goloskokov, P. Kroll, Euro. Phys. J. C53, (2008) 367.
* (6) R. Jakob, P. Kroll, Phys. Lett. B315, (1993) 463;
J. Bolz, J.G. Körner, P. Kroll, Z. Phys. A350, (1994) 145.
* (7) I.V. Musatov and A.V. Radyushkin, Phys. Rev. D61, (2000) 074027.
* (8) J. Pumplin, et al., JHEP 0207, (2002) 012.
* (9) M. Diehl, T. Feldmann, R. Jakob and P. Kroll, Eur0. Phys. J. C39, (2005) 1.
* (10) A. Airapetian et al. (HERMES Collab.), Phys. Lett. B682, (2010) 345;
A. Airapetian et al. (HERMES Collab.), Phys. Lett. B535, (2002) 85.
* (11) M. Anselmino, M. Boglione, U. D’Alesio, A. Kotzinian, F. Murgia, A. Prokudin and S. Melis, Nucl. Phys. Proc. Suppl. 191, (2009) 98.
* (12) M. Gockeler et al. [QCDSF Collaboration and UKQCD Collaboration], Phys. Rev. Lett. 98, (2007) 222001.
* (13) L. L. Frankfurt, P. V. Pobylitsa, M. V. Polyakov and M. Strikman, Phys. Rev. D60, (1999) 014010.
* (14) A. Airapetian et al. (HERMES Collab.) , Phys. Lett. B659, (2008) 486. .
* (15) S.V. Goloskokov. Proc. of DSPIN-11, Dubna, September 2011, p. 71.
* (16) I. Bedlinskiy, et al. (CLAS Collaboration) Phys. Rev. Lett. 109, (2012) 112001.
* (17) V.Kubarovsky et al., Proc. of DSPIN-11, Dubna, September 2011, p. 258.
|
arxiv-papers
| 2012-11-23T06:24:47 |
2024-09-04T02:49:38.307176
|
{
"license": "Public Domain",
"authors": "S. V. Goloskokov",
"submitter": "Sergey Goloskokov",
"url": "https://arxiv.org/abs/1211.5416"
}
|
1211.5610
|
# Precise asymptotics for large deviations of integral forms
Xiangfeng Yang
Grupo de Física Matemática
Universidade de Lisboa
Av. Prof. Gama Pinto 2, 1649-003 Lisboa, Portugal [email protected]
(September 13, 2012)
###### Abstract
For suitable families of locally infinitely divisible Markov processes
$\\{\xi^{{\epsilon}}_{t}\\}_{0\leq t\leq T}$ with frequent small jumps
depending on a small parameter $\epsilon>0,$ precise asymptotics for large
deviations of integral forms
$\mathbb{E}^{\epsilon}\left[\exp\\{{\epsilon}^{-1}F(\xi^{\epsilon})\\}\right]$
are proved for smooth functionals $F.$ The main ingredient of the proof in
this paper is a recent result regarding the asymptotic expansions of the
expectations $\mathbb{E}^{\epsilon}\left[G(\xi^{\epsilon})\\}\right]$ for
smooth $G.$ Several connections between these large deviation asymptotics and
partial integro-differential equations are included as well.
Keywords and phrases: Crámer’s transformation, large deviations, normal
deviations, locally infinitely divisible, compensating operators
AMS 2010 subject classifications: Primary 60F10, 60F17; secondary 60J75, 35C20
## 1 Introduction
The study of large deviations in limit theorems can be formulated as follows.
Let $\mathbb{X}$ be a metric space with metric $\rho$, and $\mu^{\epsilon}$ be
a family of probability measures on $\mathbb{X}$ depending on a parameter
$\epsilon>0.$ Suppose there is a point $x_{*}\in\mathbb{X}$ such that for any
$\delta>0$ and small $\epsilon,$ $\mu^{\epsilon}\\{y:\rho(x_{*},y)<\delta\\}$
have overwhelming probabilities: $\lim_{\epsilon\rightarrow
0}\mu^{\epsilon}\\{y:\rho(x_{*},y)<\delta\\}=1.$ Problems on large deviations
are concerned with the limiting behavior as $\epsilon\rightarrow 0$ of the
infinitesimal probabilities $\mu^{\epsilon}(A)$ for measurable sets
$A\subseteq\mathbb{X}$ that are situated at a positive distance from point
$x_{*}.$ Problems concerning asymptotics as $\epsilon\rightarrow 0$ of
integrals in the form $\int_{\mathbb{X}}f_{\epsilon}(x)\mu^{\epsilon}(dx)$
also belong to large deviations if the main part of such integrals for small
$\epsilon$ is due to the values of $x$ far away from point $x_{*}.$ This paper
deals with the later (asymptotics of integrals).
Classical large deviation problems are about empirical means
$\bar{S}_{n}=\sum_{i=1}^{n}\xi_{i}/n$ of random variables $\xi_{i}.$ In
general, results obtained deal with asymtotics up to logarithmic equivalence
$\ln\mathbb{P}\\{\bar{S}_{n}\in A\\}$ or
$\ln\mathbb{E}\exp\\{nf(\bar{S}_{n})\\},$ and we call these results _rough
large deviations_ , see [44], [11], [14] and [46]. If we assume that the
random variables $\xi_{i}$ are real-valued and independent identically
distributed (i.i.d.), Cramér in [10] made use of limit theorems on normal
deviations (asymptotic expansions in limit theorem for i.i.d. random
variables) and proved a precise large deviation result:
$\mathbb{P}\\{\bar{S}_{n}>a\\}\sim\frac{c}{\sqrt{n}}\exp\\{-nI(a)\\}$ for
$a>0,$ some constant $c$ and a rate function $I(x)$ provided $\xi_{i}$ are
non-lattice having zero mean and finite moment generating function. He used
what we will call Cramér’s transformation to define a new distribution
$\tilde{\mu}(dx)=e^{z_{0}x}\mu(dx)/\int e^{z_{0}x}\mu(dx)$ for some $z_{0}$ so
that new random variables $\tilde{\xi}_{i}$ corresponding to $\tilde{\mu}$
have mean $a.$ If more conditions are assumed on $\xi_{i},$ then Cramér
derived precise asymptotics for large deviation probabilities
$\mathbb{P}\\{\bar{S}_{n}>a\\}=\exp\\{-nI(a)\\}(\sum_{1\leq i\leq
N}l_{i}n^{-i/2}+o(n^{-N/2}))$ for an integer $N$ depending on the moments of
$\xi_{i}$ (see also [6], [26] and the references therein for related works).
If we use $\mu^{n}$ to denote the distributions of $\bar{S}_{n},$ then results
concerning integrals $\int_{\mathbb{R}}f_{n}(x)\mu^{n}(dx)$ with
$f_{n}(x)=\exp\\{nf(x)\\}$ can be obtained similarly in the form
$\int_{\mathbb{R}}\exp\\{nf(x)\\}\mu^{n}(dx)=\exp\\{n[f(x_{0})-I(x_{0})]\\}(\sum_{0\leq
i\leq M}k_{i}n^{-i}+o(n^{-M}))$ provided $\max[f(x)-I(x)]$ is reached uniquely
at $x_{0}$ for some integer $M$ depending on the smoothness of $f(x).$ If
$\xi_{i}$ are not independent or the moment generating function doesn’t exist,
similar precise large deviations can be also obtained (see for instance [31],
[32], [28] and the references therein). For related treatments on other types
of sequences of random variables (such as randomly indexed sums), we refer to
[29] and [34]. Precise large deviations are also called in the literature as
sharp (or exact) large deviations.
When we study large deviations for stochastic processes $\xi_{t}^{\epsilon}$
defined on probability spaces $(\Omega,\mathcal{F},\mathbb{P}^{\epsilon}),$
one usually investigates the asymtotics up to logarithmic equivalence
$\ln\mathbb{P}^{\epsilon}\\{\xi^{\epsilon}\in A\\}\sim g(\epsilon,A)$ or
$\ln\mathbb{E}^{\epsilon}f_{\epsilon}(\xi^{\epsilon})\sim
g(\epsilon,f_{\epsilon})$ (we use $\mathbb{E}^{\epsilon}$ to denote the
expectation with respect to the probability measure $\mathbb{P}^{\epsilon}$).
More precisely, if $(\mathbb{X},\mathcal{B})$ denotes a function space with a
metric and the Borel $\sigma$-algebra $\mathcal{B},$ then the family
$\\{\xi^{\epsilon}\\}$ is said to satisfy the large deviation principle with a
normalized action functional $S(x)$ on $(\mathbb{X},\mathcal{B})$ if for every
Borel measurable set $\Gamma\in\mathcal{B},$
$-\inf_{x\in\Gamma^{o}}S(x)\leq\liminf_{\epsilon\rightarrow
0}\epsilon\ln\mathbb{P}^{\epsilon}\\{\xi^{\epsilon}\in\Gamma\\}\leq\limsup_{\epsilon\rightarrow
0}\epsilon\ln\mathbb{P}^{\epsilon}\\{\xi^{\epsilon}\in\Gamma\\}\leq-\inf_{x\in\bar{\Gamma}}S(x)$
(1.1)
where $S(x)$ takes values in $[0,+\infty]$ such that each level set
$\Phi(s):=\\{x\in\mathbb{X}:S(x)\leq s\\}$ is compact ($s\geq 0$). The
normalized action functional $S(x)$ is also called a rate function in the
literature. Here we also consider the large deviation principle as rough large
deviations. We refer to [11], [44], [41], [30], [24], [1] and [27] for the
large deviation principles for various classes of stochastic processes. Of
note, references [1] and [27] study processes with jumps, which will be
included in this paper.
The following identity, to be called as Varadhan’s integral lemma according to
[11], was derived in [41] from (1.1)
$\lim_{\epsilon\rightarrow
0}\epsilon\ln\mathbb{E}^{\epsilon}\exp\left\\{\epsilon^{-1}F(\xi^{\epsilon})\right\\}=\max_{x\in\mathbb{X}}[F(x)-S(x)]$
(1.2)
for every bounded and continuous functional $F(x)$ on $\mathbb{X}.$ If the
metric space $\mathbb{X}$ is regular enough, then (1.2) and (1.1) are
equivalent, see Section 3.3 in [25], [7] and [11]. Related works were
considered in [12]. Two questions arise here. First, it is natural to expect
precise large deviation probabilities from (1.1) for suitable stochastic
processes. This direction has been extensively studied, such as for random
walks, actual aggregate loss processes, prospective-loss processes,
(fractional) Ornstein-Uhlenbeck processes, Gaussian quadratic forms, Markov
chains and so on (see [37], [33], [35], [2], [3], [4], [5], [21] and [22]).
Second, it is natural to expect precise large deviations of integral forms
from (1.2) for more regular $F$ such as what we had for sums of i.i.d. random
variables. Namely, we want to specify the conditions on $F$ and
$\xi^{\epsilon}$ under which
$\mathbb{E}^{\epsilon}\exp\left\\{\epsilon^{-1}F(\xi^{\epsilon})\right\\}$ has
precise asymptotics. Not many references can be found along this direction,
and below is a summary.
Indeed, for the family of stochastic processes
$\\{\sqrt{\epsilon}W_{t}\\}_{t\in[0,T]},$ where $\\{W_{t}\\}$ is the standard
Wiener process, it was proved by Schilder in [38] that the following precise
asymptotics hold
$\mathbb{E}^{\epsilon}\left[\exp\left\\{{\epsilon}^{-1}F(\sqrt{\epsilon}W)\right\\}\right]=\exp\\{{\epsilon}^{-1}[F(\phi_{0})-S(\phi_{0})]\\}\left(\sum_{0\leq
i\leq s/2}K_{i}\cdot{\epsilon}^{i}+o({\epsilon}^{s/2})\right)$ (1.3)
for a positive integer $s$ depending on the smoothness of $F,$ where the
normalized action functional
$S(\phi)=\frac{1}{2}\int_{0}^{T}\phi^{\prime}(t)^{2}dt$ for absolutely
continuous $\phi$ and $S(\phi)=\infty$ for other $\phi.$ We note that the
trajectory metric space $\mathbb{X}$ here is the continuous function space
$C[0,T],$ and $F(\phi_{0})-S(\phi_{0})$ indicates the maximum of
$F(\phi)-S(\phi)$ is reached uniquely at $\phi_{0}.$ The proof of (1.3) made
use of many particular properties of Wiener processes such as
$d\mu_{\phi+\sqrt{\epsilon}W}/d\mu_{W}$ and the distribution of
$\max_{t\in[0,T]}|W_{t}|.$ In 1970s, Dubrovskii and Wentzell showed precise
large deviation of the first order, i.e. $s=0$ in (1.3), for suitable Markov
processes by a transformation similar to Cramér’s. The general precise
asymptotics can’t be obtained due to the lack of tools which will be explained
below, see [13] and [44]. Ellis and Rosen in 1980s derived precise asymptotics
in the form (1.3) for Gaussian probability measures by suitable technical
arguments based on Hilbert spaces, see [16], [17], [18], [36] and [23]. The
purpose of this paper is to study precise asymptotics for large deviations in
form (1.3) for a wide class of families of stochastic processes including
diffusion processes, pure jump processes, deterministic processes and the
mixture processes of them - locally infinitely divisible processes.
Most of the proofs regarding the precise large deviations mentioned above made
use of a transformation
$\tilde{\mathbb{P}}^{\epsilon}(A)=\int_{A}\pi_{\phi}^{\epsilon}d\mathbb{P}$
($\pi_{\phi}^{\epsilon}$ is chosen such that
$\tilde{\mathbb{P}}^{\epsilon}(\Omega)=1$) in order that the main part of
$\mathbb{E}^{\epsilon}\left[\exp\\{{\epsilon}^{-1}F(\xi^{\epsilon})\\}\right]$
for small $\epsilon$ is due to the set of paths in a neighborhood of $\phi$
which has large $\tilde{\mathbb{P}}^{\epsilon}$ probability. We call such a
transformation the _generalized_ Cramér’s transformation which will be used
also in this paper.
Back to the classical precise asymptotics for large deviations
$\int_{\mathbb{R}}\exp\\{nf(x)\\}\mu^{n}(dx)$ on i.i.d. random variables,
Cramér’s main tools are the asymptotic expansions on normal deviations for
$\sqrt{n}\bar{S}_{n}$ in the form
$F_{\sqrt{n}\bar{S}_{n}}(x)=F_{\infty}(x)+\sum_{i=1}^{k}P_{i}(x)n^{-i/2}+o(n^{-k/2})$
where $F_{\infty}(x)$ is the limiting distribution of
$F_{\sqrt{n}\bar{S}_{n}}(x)$ (the distribution function of the random variable
$\sqrt{n}\bar{S}_{n}$). Equivalently, for smooth function $g(x),$ the normal
deviations take the following form
$\mathbb{E}g(\sqrt{n}\bar{S}_{n})=\mathbb{E}g(Y)+\sum_{i=1}^{k}p_{i}n^{-i/2}+o(n^{-k/2})$
(1.4)
where $Y$ is the random variable corresponding to the distribution function
$F_{\infty}(x).$ It is well-known that a family of stochastic processes
$\eta^{\epsilon}$ converges weakly to a process $\eta$ as probability measures
on the trajectory function space $\mathbb{X}$ if for any continuous and
bounded functional $G(x)$ on $\mathbb{X}$
$\mathbb{E}^{\epsilon}G(\eta^{\epsilon})=\mathbb{E}G(\eta)+o(1).$
The exact order for $o(1)$ is generally unknown. Thus, if one wants to follow
the idea of Cramér on random variables to derive precise asymptotics of large
deviations for stochastic processes by using the asymptotic expansions on
normal deviations for stochastic processes, then the first step would be to
obtain normal deviations for stochastic processes, namely,
$\mathbb{E}^{\epsilon}G(\eta^{\epsilon})=\mathbb{E}G(\eta)+\sum_{i=1}^{k}P_{i}\epsilon^{i/2}+o(\epsilon^{k/2}).$
(1.5)
But normal deviations (1.5) are far from clear until a recent result [47], and
we refer to [45] and the references therein for closely related works. The
method of deriving precise asymptotics of large deviations from precise normal
deviations for stochastic processes seems to appear for the first time in this
paper.
In Section 1.1 we give the definition of a locally infinitely divisible
process and list several related concepts. The main result of this paper is
contained in Section 2, where some examples are also included. After
appropriate recall from [47] on normal deviations for stochastic processes in
Section 3.1, we present the proof of our main theorem in the rest of Section
3.
As related problems, in Section 4 we study the connections between precise
asymptotics for large (or normal) deviations and for the solutions to partial
integro-differential equations
$\begin{cases}\begin{aligned} \frac{\partial}{\partial
t}u^{\epsilon}(t,x)=&\frac{\epsilon}{2}a(t,x)\Delta
u^{\epsilon}(t,x)+b(t,x)\nabla
u^{\epsilon}(t,x)+\epsilon^{-1}c(x)u^{\epsilon}(t,x)\\\
&+\epsilon^{-1}\int_{\mathbb{R}}\left[u^{\epsilon}(t,x+\epsilon
u)-u^{\epsilon}(t,x)-\epsilon u\nabla
u^{\epsilon}(t,x)\right]\nu_{t,x}(du)\end{aligned}\\\
u^{\epsilon}(0,x)=g(x)\end{cases}$
under suitable smooth and growth conditions on $a,b,c$ and $g.$ For instance,
if $c=1,$ $a(t,x)=a(x),$ $b(t,x)=b(x),$
$0<\inf_{x}a(x)\leq\sup_{x}a(x)<\infty,$ $\nu_{t,x}(du)=u^{2}1_{\\{|u|\leq
1\\}}(du),$ the smooth functions $a(x),b(x)$ and $g(x)$ are bounded together
with their derivatives $d^{j}a/dx^{j},$ $d^{j}b/dx^{j}$ and $d^{j}g/dx^{j},$
then the precise asymptotics for the solution $u^{\epsilon}(t,x)$ for fixed
$(t,x)$ is (with $n$ being an arbitrary integer)
$u^{\epsilon}(t,x)=e^{t/{\epsilon}}\cdot\left[\sum_{k=0}^{n}k_{i}(x)\epsilon^{k/2}+o(\epsilon^{n/2})\right],\quad\text{
for constants }k_{i}\text{ depending on }x.$
### 1.1 Locally infinitely divisible processes
If $(\xi_{t},\mathbb{P}_{s,x}),t\in[s,T],$ is a real-valued Markov process
(the subscript s,x means the process starts from $x$ at time $s$), we use
$P^{s,t},0\leq s\leq t\leq T,$ to denote the corresponding multiplicative
family of linear operators acting on functions according to the formula
$P^{s,t}f(x)=\mathbb{E}_{s,x}f(\xi_{t}),$
where $\mathbb{E}_{s,x}$ is the expectation with respect to probability
measure $\mathbb{P}_{s,x}.$ The _compensating operator_ $\mathfrak{A}$ of this
Markov process, taking functions $f(t,x)$ to functions of the same two
arguments, is defined by
$\displaystyle
P^{s,t}f(t,\cdot)(x)=f(s,x)+\int_{s}^{t}P^{s,u}\mathfrak{A}f(u,\cdot)(x)du$
(1.6)
under suitable assumptions on the measurability in $(t,x)$ of
$\mathfrak{A}f(t,x),$ where $P^{s,t}f(t,\cdot)(x)$ means that $P^{s,t}$ is
applied to the function $f(t,x)$ in its second argument $x,$ and
$P^{s,u}\mathfrak{A}f(u,\cdot)(x)$ means that $P^{s,u}$ is applied to function
$g(u,x):=\mathfrak{A}f(u,x)$ in its second argument $x.$ If some measurability
conditions are imposed on the process $\xi_{t}(\omega),$ then (1.6) is
equivalent to that
$f(t,\xi_{t})-\int_{s}^{t}\mathfrak{A}f(u,\xi_{u})du$
is a martingale with respect to the natural family of $\sigma$-algebras and
every probability measure $P_{s,x}.$ Of course, compensating operator
$\mathfrak{A}$ is not defined uniquely. Different versions are such that
$\mathfrak{A}f(u,\xi_{u})$ coincide almost surely except on a set of time
argument $u$ of zero Lebesgue measure.
We say $A_{t}$ is the generating operator of our process
$(\xi_{t},\mathbb{P}_{s,x})$ if for $s\leq t,$
$P^{s,t}f(x)=f(x)+\int_{s}^{t}P^{s,u}A_{u}f(x)du$
for suitable $f.$ Also a generating operator has different versions. For a
wide class of Markov processes, a version of the compensating operator
$\mathfrak{A}$ of process $\xi_{t}$ for smooth functions $f(t,x)$ is given by
$\mathfrak{A}f(t,x)=\frac{\partial f}{\partial t}(t,x)+A_{t}f(t,\cdot)(x),$
where generating operator $A_{t}$ acts on functions of the spatial argument
$x$ only.
For each fixed ${\epsilon}>0,$ let
$(\xi_{t}^{\epsilon},\mathbb{P}_{0,x}^{\epsilon}),t\in[0,T],$ be a one-
dimensional process with jumps whose trajectories are right continuous with
left limits. We assume that the generating operator of $\xi_{t}^{\epsilon}$ is
$\displaystyle
A_{t}^{\epsilon}f(x)={\epsilon}^{-1}\int_{\mathbb{R}}\left[f(x+{\epsilon}u)-f(x)-{\epsilon}uf^{\prime}(x)\right]\nu_{t,x}(du)+\alpha(t,x)f^{\prime}(x)+\frac{\epsilon}{2}a(t,x)f^{\prime\prime}(x)$
(1.7)
for functions $f$ that are bounded and continuous together with their first
and second derivatives, and that a version of its compensating operator is
given by $\mathfrak{A}f(t,x)=\frac{\partial f}{\partial
t}(t,x)+A_{t}f(t,\cdot)(x)$ for bounded functions $f(t,x)$ that are absolutely
continuous in $t,$ twice continuously differentiable in $x$ for fixed $t$ with
bounded derivatives $\partial f/\partial t,\partial f/\partial x$ and
$\partial^{2}f/\partial x^{2}.$ In order to make sense of the integral in
$A_{t}^{\epsilon}f(x)$ and also for the purpose of the proof, throughout this
paper we impose two conditions on measures $\nu_{t,x}:\,\int
u^{2}\nu_{t,x}(du)<\infty$ for every $(t,x),$ and there is a bounded support
$K$ for all $\nu_{t,x}(\cdot),$ i.e., $\nu_{t,x}(K^{c})\equiv 0.$ The family
$\\{\xi_{t}^{\epsilon}\\}$ is the underlying family of stochastic processes in
this paper, and we call them locally infinitely divisible processes with
bounded support, see also [45] and [44]. We note that this family contains
diffusion processes, pure jump processes, deterministic processes and the
processes coming from the mixture of them.
For each $\epsilon,$ the process $\xi^{\epsilon}$ makes jumps of size
$\epsilon\cdot u,$ according to the rate measure
${\epsilon}^{-1}\nu_{t,x}(du),$ and moves with velocity $\alpha(t,x)-\int
u\nu_{t,x}(du)$ between the jumps. We define the cumulant
$G^{\epsilon}(t,x;z)$ of $(\xi_{t}^{\epsilon},\mathbb{P}_{0,x}^{\epsilon})$
by, for $t\in[0,T],x,z\in\mathbb{R},$
$G^{\epsilon}(t,x;z)=z\alpha(t,x)+\frac{\epsilon}{2}a(t,x)z^{2}+{\epsilon}^{-1}\int_{\mathbb{R}}\left(e^{z{\epsilon}u}-1-z{\epsilon}u\right)\nu_{t,x}(du).$
Here $G^{\epsilon}(t,x;z)$ is well defined because of two conditions we
imposed on $\nu_{t,x},$ and it satisfies
$G^{\epsilon}(t,x;z)={\epsilon}^{-1}G_{0}(t,x;\epsilon z),$ where
$G_{0}(t,x;z)=z\alpha(t,x)+\frac{1}{2}a(t,x)z^{2}+\int_{\mathbb{R}}\left(e^{zu}-1-zu\right)\nu_{t,x}(du).$
Let $H_{0}(t,x;u),G_{0}(t,x;z)$ be coupled by the Legendre transformation in
the third argument,
$H_{0}(t,x;u)=\sup_{z\in\mathbb{R}}\left[zu-G_{0}(t,x;z)\right].$
For an absolutely continuous function $\phi_{0}$ (which will be specified
later as a maximizer) we define
$z^{\epsilon}(t)={\epsilon}^{-1}z_{0}(t),z_{0}(t)=\frac{\partial
H_{0}}{\partial u}(t,\phi_{0}(t);\phi^{\prime}_{0}(t))$ and generalized
Cramér’s transformation:
$\mathbb{P}_{0,x}^{z^{\epsilon}}(A):=\int_{A}\pi^{\epsilon}(0,T)d\mathbb{P}_{0,x}^{\epsilon},$
with
$\pi^{\epsilon}(0,T)=\exp\left\\{{\epsilon}^{-1}\int_{0}^{T}z_{0}(t)d\xi_{t}^{\epsilon}-{\epsilon}^{-1}\int_{0}^{T}G_{0}(t,\xi_{t}^{\epsilon};z_{0}(t))dt\right\\}.$
For each ${\epsilon}>0,$ this transformation gives us a new probability
measure $\mathbb{P}_{0,x}^{z^{\epsilon}}$ if we assume $\pi^{\epsilon}(0,T)$
is a martingale as a process in $T$ with respect to
$\mathbb{P}_{0,x}^{\epsilon}.$ For each ${\epsilon}>0,$ under
$\mathbb{P}_{0,x}^{z^{\epsilon}}$ it turns out $\xi^{\epsilon}$ is again a
jump process with compensating operator (see [44], Section 2.2.2),
$\displaystyle\mathfrak{A}^{z^{\epsilon}}f(t,x)=$ $\displaystyle\frac{\partial
f}{\partial t}(t,x)+\frac{\partial G_{0}}{\partial
z}(t,x;z_{0}(t))\frac{\partial f}{\partial
x}(t,x)+\frac{\epsilon}{2}a(t,x)\frac{\partial^{2}f}{\partial x^{2}}(t,x$
(1.8)
$\displaystyle+{\epsilon}^{-1}\int_{\mathbb{R}}\left[f(t,x+{\epsilon}u)-f(t,x)-\epsilon
u\frac{\partial f}{\partial x}(t,x)\right]e^{z_{0}(t)u}\nu_{t,x}(du).$
Let us define the normalized action functional as follows
$S(\phi)=S_{0,T}(\phi)=\int_{0}^{T}H_{0}(t,\phi(t);\phi^{\prime}(t))dt$
for absolutely continuous function $\phi,$ otherwise $S(\phi)=+\infty.$ At the
end of this section, we introduce several notations which are needed for our
formulation of the main theorem. Let $\phi_{0}$ be continuously
differentiable,
$\displaystyle G_{0}^{*}(t,x;z)=$ $\displaystyle
z\left[\alpha(t,\phi_{0}(t)+x)-\int_{\mathbb{R}}u\nu_{t,\phi_{0}(t)+x}(du)-\phi^{\prime}_{0}(t)\right]$
$\displaystyle+\frac{1}{2}a(t,\phi_{0}(t)+x)z^{2}+\int_{\mathbb{R}}\left(e^{zu}-1\right)e^{z_{0}(t)u}\nu_{t,\phi_{0}(t)+x}(du),$
and $H_{0}^{*}(t,x;u)$ be the Legendre transformation of $G_{0}^{*}(t,x;z)$ in
the third argument. For simplicity, throughout this paper we will only
consider $\mathbb{P}_{0,0}^{\epsilon},\mathbb{P}_{0,0}^{z^{\epsilon}},$ and
use symbols $\mathbb{P}^{\epsilon},\mathbb{P}^{z^{\epsilon}}$ for short. The
following symbols are also used,
$\displaystyle\alpha^{1}(t,x)=\frac{\partial G_{0}}{\partial
z}(t,x;z_{0}(t)),$ $\displaystyle\alpha^{2}(t,x)=a(t,x)+\int
u^{2}e^{z_{0}(t)u}\nu_{t,x}(du);$ (1.9) $\displaystyle\alpha^{j}(t,x)=\int
u^{j}e^{z_{0}(t)u}\nu_{t,x}(du),$
$\displaystyle\beta^{j}(t,x)=\int|u|^{j}e^{z_{0}(t)u}\nu_{t,x}(du),j\geq 3.$
### 1.2 Functional derivatives
Let us include in this section the function spaces related to the trajectory
spaces of our stochastic processes and the corresponding functional
derivatives. We use $D_{0}[0,T]$ to denote the space of all functions defined
on $[0,T]$ vanishing at $0$ which are right continuous with left limits;
$C_{0}^{1}[0,T]$ the space of all continuously differentiable functions on
$[0,T]$ vanishing at $0;$ and $W_{0}^{1,2}[0,T]$ the space of absolutely
continuous functions vanishing at $0$ having square integrable derivatives. In
the space $D_{0}[0,T],$ the uniform norm $||\phi||=\sup_{0\leq t\leq
T}|\phi(t)|$ will be used. Throughout this paper, we understand the
differentiability of a functional $F(\phi)$ on $D_{0}[0,T]$ as Fréchet
differentiability. Moreover, we assume that the derivatives
$F^{(j)}(\phi)(\delta_{1},\cdots,\delta_{j})$ can be represented as integrals
of the product $\delta_{1}(s_{1})\cdots\delta_{j}(s_{j})$ with respect to some
signed measures, denoted by $F^{(j)}(\phi;\bullet)$:
$\displaystyle
F^{(j)}(\phi)(\delta_{1},\cdots,\delta_{j})=\int_{[0,T]^{j}}\delta_{1}(s_{1})\cdots\delta_{j}(s_{j})F^{(j)}(\phi;ds_{1}\cdots
ds_{j}).$ (1.10)
The norm of the signed measure is defined by
$||F^{(j)}||:=\sup_{x[0,T]\in
D_{0}[0,T]}\left|F^{(j)}(x[0,T];\bullet)\right|([0,T]^{j}).$
The notation $F^{(j)}(\phi)(y[0,T]^{\otimes^{j}})$ stands for the $j$-th
derivative $F^{(j)}(\phi)(y[0,T],\cdots,y[0,T])$ of the functional $F$ at
point $\phi[0,T]$ in directions $y[0,T].$
## 2 The main theorem and examples
From now on, an integer $s\geq 2$ will be used. To precisely state our main
result, we make a list of assumptions on two pairs $G_{0},H_{0}$ and
$G_{0}^{*},H_{0}^{*}$ introduced in Section 1.1. The first five general
assumptions (A)-(E) can be found in [44]. Let $p(t,x;z)$ and $q(t,x;u)$ be
coupled by the Legendre transformation in the third arguments.
(A). $p(t,x;z)\leq\overline{p}_{0}(z)$ for all $t,x,z,$ where
$\overline{p}_{0}$ is a downward convex non-negative function, finite for all
$z,$ and such that $\overline{p}_{0}(0)=0.$
(B). $q(t,x;u)<\infty$ for the same $u$ for which $\underline{q}_{0}(u)$ is
finite, where $\underline{q}_{0}$ is the Legendre transformation of
$\overline{p}_{0}.$
(C).
$\mathop{\text{sup}}_{|t-s|<h,|x-y|<\delta,q(t,x;u)<\infty}\frac{q(s,y;u)-q(t,x;u)}{1+q(t,x;u)}\rightarrow
0\quad\text{ for }{h}\downarrow 0,\delta\downarrow 0.$
(D). The set $\\{u:\underline{q}_{0}(u)<\infty\\}$ is open, and
$\sup_{t,x}q(t,x;u_{0})<\infty$ for some point $u_{0}$ of it.
(E). For any compactum $U_{K}\subseteq\\{u:\underline{q}_{0}(u)<\infty\\},$
the partial derivative $\frac{\partial q}{\partial u}(t,x;u)$ is bounded and
continuous in $u\in U_{K}$ uniformly with respect to all $t,x.$
Besides, one more technical assumption is imposed directly on $G_{0}$ and
$\alpha^{j}.$
(F). Let $G_{0}(t,x;z)$ and $H_{0}(t,x;u)$ be twice differentiable with
respect to $(x;z)$ and $(x;u)$ respectively, and their first and second
derivatives be continuous with respect to $(t,x;z)$ and $(t,x;u).$
Furthermore, assume $G_{0}(t,x;z)$ is differentiable in $x$ up to $s+1$ times,
$\frac{\partial^{s+1}G_{0}}{\partial x^{s+1}}(t,x;z)$ is bounded for all
$x\in\mathbb{R},t\in[0,T]$ and bounded $z.$ Suppose
$\sup_{t,x,z}|\frac{\partial^{2}G_{0}}{\partial z\partial x}(t,x;z)|<\infty,$
$\inf_{t,x,z}|\frac{\partial^{2}G_{0}}{\partial z^{2}}(t,x;z)|>0,$
$\sup_{t,x}|a(t,x)|<\infty$, and
$||\alpha^{1}_{(i+1)}||+||\alpha^{j+1}_{(i)}||<\infty$ for all integers
$i,j\geq 1$ where the subscript (i) in
$\alpha^{j}_{(i)}(t,x):=\alpha^{j}_{\underbrace{22\cdots 2}_{i}}(t,x)$ means
the $i$-th partial derivative of $\alpha^{j}$ in its second argument $x.$
### 2.1 The main theorem
###### Theorem 2.1.
Let $(\xi_{t}^{\epsilon},\mathbb{P}^{\epsilon}),t\in[0,T],$ be a family of
one-dimensional locally infinitely divisible processes with bounded support
introduced in Section 1.1, For a continuous functional $F$ on $D_{0}[0,T]$
which is bounded above, let the maximum of functional $F-S$ be attained at a
unique function $\phi_{0}\in C_{0}^{1}[0,T],$ and the assumption (F) be
fulfilled. Furthermore, assume that the assumptions (A)-(E) are satisfied for
$p(t,x;z)=G_{0}(t,x;z)$ and $p(t,x;z)=G_{0}^{*}(t,x;z).$
Suppose $F$ is $s+1$ times differentiable at all points $\phi$ in a
neighborhood of $\phi_{0}$ and $F^{(2)}(\phi_{0})(x,x)<S^{(2)}(\phi_{0})(x,x)$
for any non-zero function $x\in W_{0}^{1,2}[0,T].$ For all $\phi$ in this
neighborhood of $\phi_{0},$ any $x[0,T],x_{i}[0,T]\in D_{0}[0,T],$ we assume
$\displaystyle
F^{(2)}(\phi)(x[0,T],x[0,T])+\int_{0}^{T}(x(t))^{2}\frac{\partial^{2}G_{0}}{\partial
x^{2}}(t,\phi_{0}(t);z_{0}(t))dt\leq 0,$ (2.1)
$\displaystyle\left|F^{(i)}(\phi)(x_{1}[0,T],\cdots,x_{i}[0,T])\right|\leq
p\left[\left|x_{1}(T)\cdots
x_{i}(T)\right|^{m}+\prod_{j=1}^{i}\left(1+\int_{0}^{T}\left|x_{j}(t)\right|^{n}dt\right)\right]$
with $2\leq i\leq s+1$ and some constants $m,n,p\geq 1.$ Another continuous
and bounded functional $H$ is assumed to be $s-1$ times differentiable at all
points $\phi$ in a neighborhood of $\phi_{0},$ and
$\displaystyle\left|H^{(i)}(\phi)(x_{1}[0,T],\cdots,x_{i}[0,T])\right|\leq
p^{\prime}\left[\left|x_{1}(T)\cdots
x_{i}(T)\right|^{m^{\prime}}+\prod_{j=1}^{i}\left(1+\int_{0}^{T}\left|x_{j}(t)\right|^{n^{\prime}}dt\right)\right]$
(2.2)
with $1\leq i\leq s-1$ and some positive constants
$m^{\prime},n^{\prime},p^{\prime}\geq 1,$ for all $x_{i}[0,T]\in D_{0}[0,T].$
Then as $\epsilon\rightarrow 0,$ the following precise asymptotics hold
$\displaystyle\mathbb{E}^{\epsilon}\left[H(\xi^{\epsilon})\exp\\{{\epsilon}^{-1}F(\xi^{\epsilon})\\}\right]=\exp\left\\{{\epsilon}^{-1}\left[F(\phi_{0})-S(\phi_{0})\right]\right\\}\left[\sum_{0\leq
i\leq(s-2)}K_{i}\cdot{\epsilon}^{i/2}+o\left({\epsilon}^{(s-2)/2}\right)\right]$
(2.3)
where the coefficients $K_{i}$ are determined by $F,H$ and their derivatives
at $\phi_{0};$ in particular,
$\displaystyle
K_{0}=H(\phi_{0})\cdot\mathbb{E}\left(\exp\\{Q(2,\eta)\\}\right),$
$\displaystyle
K_{1}=C_{01}H(\phi_{0})+C_{11}+\mathbb{E}\left[\exp\\{Q(2,\eta)\\}\left(Q(3,\eta)H(\phi_{0})+H^{(1)}(\phi_{0})(\eta)\right)\right]$
with the constants $C_{ij}$ depending on $F,H$ and $\phi_{0},$ the process
$\eta$ being a Gaussian diffusion having diffusion coefficient
$A(t)=\frac{\partial^{2}G_{0}}{\partial z^{2}}(t,\phi_{0}(t);z_{0}(t))$ and
drift coefficient $B(t,x)=x\cdot\frac{\partial^{2}G_{0}}{\partial z\partial
x}\left(t,\phi_{0}(t);z_{0}(t)\right),$ and
$Q(n,x[0,T])=\frac{1}{n!}F^{(n)}(\phi_{0})(x[0,T],\cdots,x[0,T])+\int_{0}^{T}\frac{1}{n!}(x(t))^{n}\frac{\partial^{n}G_{0}}{\partial
x^{n}}(t,\phi_{0}(t);z_{0}(t))dt.$
Remark: (1). The constants $C_{ij}$ are determined from the asymptotic
expansions for normal deviations formulated in (III) of Section 3.3. The
finiteness of the coefficients $K_{i}$ is proved in Section 3.5.
(2). In the spacial case when $\xi^{\epsilon}=\sqrt{\epsilon}W$, each
coefficient $K_{i}$ in (2.3) with an odd index $i$ is equal to zero because of
the symmetry of the distribution of $W.$
(3). If the initial position of $\xi^{\epsilon}$ is $x$ instead of $0,$ then
we think of $\xi^{\epsilon}-x$ as a new process with zero initial position and
$\mathbb{E}^{\epsilon}_{0,x}\Phi(\xi^{\epsilon})=\mathbb{E}^{\epsilon}_{0,0}\Phi(\xi^{\epsilon}-x).$
### 2.2 Examples
Example 1. Let us consider a family of one-dimensional pure jump processes
$\xi_{t}^{\epsilon},t\in[0,1],$ with generating operator given by
$A_{t}^{\epsilon}f(x)={\epsilon}^{-1}\int_{\mathbb{R}}\left(f(x+{\epsilon}u)-f(x)\right)\nu_{t,x}(du),$
where $\nu_{t,x}(du)=\frac{1}{2}\left(\delta_{1}(du)+\delta_{-1}(du)\right)$
with $\delta_{1}(\cdot)$ (resp. $\delta_{-1}(\cdot)$) denoting the probability
measure concentrating at point $1$ (resp. $-1$). For each $\epsilon,$ the
trajectories of $\xi^{\epsilon}$ are step functions with finitely many steps
on $[0,1].$ This process makes jumps of size $\pm\epsilon$ according to the
rate $\frac{1}{2}{\epsilon}^{-1}.$
The most probable trajectory for $\xi^{\epsilon}$ as $\epsilon\rightarrow 0$
is identically zero. To see this, we first note that
$\displaystyle G_{0}(t,x;z)$
$\displaystyle=\int_{\mathbb{R}}\left(e^{zu}-1\right)\nu_{t,x}(du)=\frac{1}{2}\left(e^{z}+e^{-z}-2\right);$
$\displaystyle H_{0}(t,x;u)$ $\displaystyle=\sup_{z\in\mathbb{R}}\left[zu-
G_{0}(t,x;z)\right]=u\ln\left(u+\sqrt{u^{2}+1}\right)+1-\sqrt{u^{2}+1};$
$\displaystyle S(\phi)$
$\displaystyle=\int_{0}^{1}H_{0}(t,\phi(t);\phi^{\prime}(t))dt,\text{ for
absolutely continuous }\phi\text{ in }D_{0}[0,1].$
As a function of $u,$ $H_{0}(t,x;u)$ is positive except at $u=0,$ strictly
increasing on $(0,\infty),$ and strictly decreasing on $(-\infty,0).$ Thus, in
order to make $S(\phi)=0,$ it is required $\phi^{\prime}(t)=0$ almost
everywhere with respect to Lebesgue measure. But $\phi(t)$ is absolutely
continuous and $\phi(0)=0,$ it follows that $\phi(t)\equiv 0.$ This proves
that the most probable trajectory is zero.
Let the functional $F$ on $D_{0}[0,1]$ be
$F(\phi)=\int_{0}^{1}\left(\phi(t)-\phi^{2}(t)\right)dt,$
and $H(\phi)\equiv 1.$ We need to show that $\max(F-S)$ is attained at a
unique function $\phi_{0}\in C_{0}^{1}[0,1],$ that is, the following
variational problem
$\displaystyle\max_{\phi\in
C_{0}^{1}[0,1]}\int_{0}^{1}\left[\phi(t)-\phi(t)^{2}-\left(\phi^{\prime}(t)\ln\left(\phi^{\prime}(t)+\sqrt{\phi^{\prime}(t)^{2}+1}\right)+1-\sqrt{\phi^{\prime}(t)^{2}+1}\right)\right]dt,$
(2.4)
has an unique (nonzero) solution. The existence and uniqueness for a nonzero
solution of (2.4) are shown in the Appendix. All other conditions of Theorem
2.1 can be easily checked.
We could have considered some wider families of processes and more general
functionals (e.g.
$F(\phi[0,T])=h\left(\int_{0}^{T}g\left(\phi\left(s\right)\right)ds\right)$),
and each time we will have to verify the existence and uniqueness of the
solution for the corresponding variational problem.
Example 2. Suppose $\xi_{t}^{\epsilon},t\in[0,1]$ is a family of one-
dimensional pure jump processes with generating operator
$A_{t}^{\epsilon}f(x)={\epsilon}^{-1}\int_{\mathbb{R}}\left(f(x+{\epsilon}u)-f(x)\right)\nu_{t,x}(du),$
where $\nu_{t,x}(du)=r(x)\delta_{1}(du)+l(x)\delta_{-1}(du)$ and
$r(x)=l(x)=\sin(x)+2.$ It is easy to get
$\displaystyle G_{0}(t,x;z)$
$\displaystyle=\int_{\mathbb{R}}\left(e^{zu}-1\right)\nu_{t,x}(du)=r(x)(e^{z}-1)+l(x)(e^{-z}-1).$
Again we consider functional
$F(\phi)=\int_{0}^{1}\left(\phi(t)-\phi^{2}(t)\right)dt.$ The existence and
uniqueness of the variational problem $\max_{\phi\in
C_{0}^{1}[0,1]}[F(\phi)-S(\phi)]$ can be similarly obtained as Example 1. Now
we check $G_{0}^{*}(t,x;z)$ satisfies conditions (A)-(E). Let us recall that
in this example
$\displaystyle G_{0}^{*}(t,x;z)=$
$\displaystyle-z[\phi_{0}^{\prime}(t)+r(\phi_{0}(t)+x)-l(\phi_{0}(t)+x)]$
$\displaystyle+r(\phi_{0}(t)+x)(e^{z}-1)e^{z_{0}(t)}+l(\phi_{0}(t)+x)(e^{-z}-1)e^{-z_{0}(t)}.$
For conditions (A) and (B): we choose $\bar{G}_{0}(z)=C(e^{|z|}-1)$ with a
positive constant $C,$ then $\underline{H}_{0}(u)=Ch(|u|/C),$ where function
$h(y)=y\ln y-y+1$ for $y\geq 1,$ and $=0$ for $0\leq y<1.$ For condition (C):
for any fixed $h>0,$ if $(t,x)$ and $(s,y)$ are close enough, then
$\displaystyle G_{0}^{*}(t,x;(1-h)z)-(1-h)G_{0}^{*}(s,y;z)\leq h.$ (2.5)
To see (2.5), note that for large $z\rightarrow\infty$ or
$z\rightarrow-\infty,$ the left hand side of (2.5) goes to $-\infty$ uniformly
in $t$ and $x.$ This means that we just need to consider bounded $z,$ which
proves (2.5). Conditions (D) and (E) are easy to be checked.
## 3 Proof of Theorem 2.1
As mentioned in the introduction, we will use precise normal deviations for
stochastic processes in our proof. The normal deviations needed here are not
for the processes $\xi^{\epsilon},$ but for another family of processes
$\eta^{\epsilon}$ related to $\xi^{\epsilon}.$
Because of assumption (F) in Section 2, the ordinary differential equation
$x^{\prime}(t)=\alpha^{1}(t,x(t))$ with an initial condition $x(0)=0$ has a
unique solution with $\alpha^{1}$ defined in (1.9). Furthermore, this unique
solution can be proved to be $\phi_{0}$ from Legendre transformation. Let us
set
$\eta^{\epsilon}_{t}=\epsilon^{-1/2}(\xi^{\epsilon}_{t}-\phi_{0}(t)).$
Note that here the initial point $\eta^{\epsilon}_{0}=0.$ More generally, we
consider an initial point $\eta^{\epsilon}_{0}=x$ in this section in order to
fully exhibit the normal deviations. It was proved in [45] that as
$\epsilon\rightarrow 0$ the family $\eta^{\epsilon}$ under
$\mathbb{P}^{z^{\epsilon}}_{0,x}$ converges weakly to a process $\eta$, which
is a Gaussian diffusion process on the real line with generating operator
$\displaystyle A^{\eta}_{t}f(x)=\alpha_{2}^{1}(t,\phi_{0}(t))\cdot x\cdot
f^{\prime}(x)+\frac{1}{2}\alpha^{2}(t,\phi_{0}(t))\cdot f^{\prime\prime}(x),$
(3.1)
where the subscript 2 means differentiation in the second spatial argument.
### 3.1 Results on normal deviations
Before the statement of normal deviations, let us recall a differential
operator $A_{1}$ which was defined in [45] for functionals $G$ on $D[0,T]$
(consisting of right continuous functions with left limits):
$\displaystyle A_{1}G(x[0,T])$
$\displaystyle=\sum_{k=1}^{3}\int_{[0,T]^{k}}\Gamma_{1}^{k}(x[0,T];s_{1},\cdots,s_{k})G^{(k)}(x[0,T];ds_{1}\cdots
ds_{k})$
where
$\displaystyle\Gamma_{1}^{1}(x[0,T];s_{1})$
$\displaystyle=\frac{1}{2}\int_{0}^{s_{1}}\alpha_{22}^{1}(t,\phi_{0}(t))x(t)^{2}\exp\left\\{\int_{0}^{s_{1}}\alpha_{2}^{1}(v,\phi_{0}(v))dv\right\\}dt,$
$\displaystyle\Gamma_{1}^{2}(x[0,T];s_{1},s_{2})$
$\displaystyle=\frac{1}{2}\int_{0}^{\min\\{s_{1},s_{2}\\}}\alpha_{2}^{2}(t,\phi_{0}(t))x(t)\exp\left\\{\sum_{i=1}^{2}\int_{0}^{s_{i}}\alpha_{2}^{1}(v,\phi_{0}(v))dv\right\\}dt,$
$\displaystyle\Gamma_{1}^{3}(x[0,T];s_{1},s_{2},s_{3})$
$\displaystyle=\frac{1}{6}\int_{0}^{\min\\{s_{1},s_{2},s_{3}\\}}\alpha^{3}(t,\phi_{0}(t))\exp\left\\{\sum_{i=1}^{3}\int_{0}^{s_{i}}\alpha_{2}^{1}(v,\phi_{0}(v))dv\right\\}dt.$
A crucial functional in [45] and [47] is defined through a conditional
expectation on the past path
$\displaystyle f(t,x[0,t])=\mathbb{E}_{t,x[0,t]}G(\eta),\quad t\in[0,T].$
###### Theorem 3.1 (Theorem 5.2 in [47]).
Let a functional $G(x[0,T])$ on $D[0,T]$ be $3(s-2)$ times differentiable with
the following conditions:
(I). there is a constant $C>0$ such that for all $x[0,T],y[0,T]\in D[0,T]$
$\displaystyle\left|G(x[0,T])\right|$ $\displaystyle\leq
C\left(1+|x(T)|^{s}+\int_{0}^{T}|x(t)|^{s}dt\right),$
$\displaystyle\left|G^{(i)}(x[0,T])(y[0,T]^{\otimes^{i}})\right|$
$\displaystyle\leq(1+||y||^{i})C\left(1+|x(T)|^{s-2}+\int_{0}^{T}|x(t)|^{s-2}dt\right),1\leq
i\leq 3(s-2),$
(II). $G^{(i)}(x[0,T])(I_{[t,T]}\delta,\cdots,I_{[t,T]}\delta),1\leq i\leq
3(s-2),$ are continuous with respect to $x[0,T]$ uniformly as $x[0,T]$ changes
over an arbitrary compact subset of $D[0,T],$ $t$ over $[0,T],$ and
$\delta[0,T]$ over the set of Lipschitz continuous functions with constant
$1,$ $||\delta||\leq 1.$
Then as $\epsilon\rightarrow 0,$ under the assumptions of Theorem 2.1 the
precise normal deviations hold
$\displaystyle\mathbb{E}^{z^{\epsilon}}_{0,x}G(\eta^{\epsilon})=\mathbb{E}_{0,x}G(\eta)+\sum_{i=1}^{s-2}\epsilon^{i/2}\mathbb{E}_{0,x}A_{i}G(\eta)+o(\epsilon^{(s-2)/2})$
(3.2)
where $A_{1}$ is a third-order differential operator defined before, $A_{2}$
is a sixth-order differential operator given by
$\displaystyle A_{2}$ $\displaystyle
G(x[0,T])=\int_{0}^{T}A_{1}\widetilde{G}(x[0,t])dt+\int_{0}^{T}\Big{[}\frac{1}{3!}\alpha^{1}_{222}(t,\phi_{0}(t))x(t)^{3}f^{(1)}(t,x[0,t])(I_{\\{t\\}})$
$\displaystyle+\frac{1}{4}\alpha^{2}_{22}(t,\phi_{0}(t))x(t)^{2}f^{(2)}(t,x[0,t])(I_{\\{t\\}}^{\otimes^{2}})+\frac{1}{3!}\alpha^{3}_{2}(t,\phi_{0}(t))x(t)f^{(3)}(t,x[0,t])(I_{\\{t\\}}^{\otimes^{3}})$
$\displaystyle+\frac{1}{4!}\alpha^{4}(t,\phi_{0}(t))f^{(4)}(t,x[0,t])(I_{\\{t\\}}^{\otimes^{4}})\Big{]}dt$
with
$\displaystyle\widetilde{G}(x[0,t])=$
$\displaystyle\frac{1}{2}\alpha^{1}_{22}(t,\phi_{0}(t))x^{2}(t)f^{(1)}(t,x[0,t])(I_{\\{t\\}})+\frac{1}{2}\alpha^{2}_{2}(t,\phi_{0}(t))x(t)f^{(2)}(t,x[0,t])(I_{\\{t\\}}^{\otimes^{2}})$
$\displaystyle+\frac{1}{6}\alpha^{3}(t,\phi_{0}(t))f^{(3)}(t,x[0,t])(I_{\\{t\\}}^{\otimes^{3}}),$
and $A_{3},\cdots,A_{s-2}$ are suitable differential operators defined through
derivatives of $f(t,x[0,t]).$
### 3.2 Large deviations for $\epsilon^{1/2}\eta^{\epsilon}$
It can be easily seen that the process
$\eta^{\epsilon}=\epsilon^{-1/2}(\xi^{\epsilon}-\phi_{0})$ under the measure
$\mathbb{P}^{z^{\epsilon}},$ has compensating operator
$\displaystyle\mathfrak{A}^{\eta^{\epsilon}}f(t,x)$
$\displaystyle=\frac{\partial f}{\partial
t}(t,x)+\epsilon^{-1/2}\left[\frac{\partial G_{0}}{\partial
z}(t,\phi_{0}(t)+\epsilon^{1/2}x;z_{0}(t))-\phi_{0}^{\prime}(t)\right]\frac{\partial
f}{\partial x}(t,x)$ (3.3)
$\displaystyle\qquad+\frac{1}{2}a(t,\phi_{0}(t)+\epsilon^{1/2}x)\frac{\partial^{2}f}{\partial
x^{2}}(t,x)$
$\displaystyle\qquad+{\epsilon}^{-1}\int_{\mathbb{R}}\left[f(t,x+\epsilon^{1/2}u)-f(t,x)-\epsilon^{1/2}\frac{\partial
f}{\partial x}(t,x)\cdot
u\right]e^{z_{0}(t)u}\nu_{t,\phi_{0}(t)+\epsilon^{1/2}x}(du),$
and cumulant
$\displaystyle G^{\eta^{\epsilon}}(t,x;z)=$ $\displaystyle
z\cdot\epsilon^{-1/2}\left[\frac{\partial G_{0}}{\partial
z}(t,\phi_{0}(t)+\epsilon^{1/2}x;z_{0}(t))-\phi_{0}^{\prime}(t)\right]+\frac{1}{2}a(t,\phi_{0}(t)+\epsilon^{1/2}x)z^{2}$
$\displaystyle+{\epsilon}^{-1}\int_{\mathbb{R}}\left[e^{z\sqrt{\epsilon}u}-1-z\sqrt{\epsilon}u\right]e^{z_{0}(t)u}\nu_{t,\phi_{0}(t)+\epsilon^{1/2}x}(du).$
The limiting process $\eta$ has compensating operator given by
$\displaystyle\mathfrak{A}^{\eta}f(t,x)=\frac{\partial f}{\partial
t}(t,x)+x\cdot\frac{\partial^{2}G_{0}}{\partial z\partial x}(t,\phi_{0}(t);$
$\displaystyle z_{0}(t))\cdot\frac{\partial f}{\partial
x}(t,x)+\frac{1}{2}\frac{\partial^{2}G_{0}}{\partial
z^{2}}(t,\phi_{0}(t);z_{0}(t))\cdot\frac{\partial^{2}f}{\partial x^{2}}(t,x),$
and cumulant
$\displaystyle G^{\eta}(t,x;z)$
$\displaystyle=zx\cdot\frac{\partial^{2}G_{0}}{\partial z\partial
x}(t,\phi_{0}(t);z_{0}(t))+\frac{1}{2}z^{2}\cdot\frac{\partial^{2}G_{0}}{\partial
z^{2}}(t,\phi_{0}(t);z_{0}(t)).$
In this case, the Legendre transformation of $G^{\eta}(t,x;z)$ in $z$ becomes
$\displaystyle H^{\eta}(t,x;u)$
$\displaystyle=\frac{1}{2}\frac{\partial^{2}H_{0}}{\partial
u^{2}}(t,\phi_{0}(t);\phi_{0}^{\prime}(t))\left(u-x\cdot\frac{\partial^{2}G_{0}}{\partial
z\partial x}(t,\phi_{0}(t);z_{0}(t))\right)^{2}.$
It then follows from Section 5.2.6 of [44] that the normalized action
functional for the family of processes $\epsilon^{1/2}\eta$ is
$\displaystyle
I^{\eta}(f(\cdot))=\int_{0}^{T}H^{\eta}(t,f(t);f^{\prime}(t))dt.$
An important tool we need for the proof of Theorem 2.1 is the following
convergence
$\displaystyle\mathbb{P}^{z^{\epsilon}}\left\\{||\eta^{\epsilon}||\geq\epsilon^{-1/2}\right\\}\rightarrow
0\text{ exponentially fast as }\epsilon\downarrow 0.$ (3.4)
#### 3.2.1 Proof of (3.4)
Let us consider the family of processes
$\zeta^{\epsilon}=\epsilon^{1/2}\eta^{\epsilon}=\xi^{\epsilon}-\phi_{0}$ with
respect to probabilities $\mathbb{P}^{z^{\epsilon}}.$ Since $\eta^{\epsilon}$
converge weakly to $\eta,$ it is expected that the the most probable
trajectory of $\zeta^{\epsilon}$ as ${\epsilon}\downarrow 0$ is path zero.
Then the exponential convergence to zero of
$\mathbb{P}^{z^{\epsilon}}\left\\{||\eta^{\epsilon}||\geq\epsilon^{-1/2}\right\\}=\mathbb{P}^{z^{\epsilon}}\left\\{||\zeta^{\epsilon}||\geq
1\right\\}$ follows provided a large deviation result for $\zeta^{\epsilon}$
is proved. To be precise, we now prove that a large principle holds for
$\zeta^{\epsilon}$ under the assumptions of Theorem 2.1
First it is straightforward to compute the cumulant of $\zeta^{\epsilon}:$
$\displaystyle G^{\zeta^{\epsilon}}(t,x;z)=$ $\displaystyle
z\left[\frac{\partial G_{0}}{\partial
z}(t,\phi_{0}(t)+x;z_{0}(t))-\phi^{\prime}_{0}(t)\right]+\frac{\epsilon}{2}a(t,\phi_{0}(t)+x)z^{2}$
$\displaystyle+{\epsilon}^{-1}\int_{\mathbb{R}}\left[e^{z{\epsilon}u}-1-z{\epsilon}u\right]e^{z_{0}(t)u}\nu_{t,\phi_{0}(t)+x}(du),$
which satisfies $\epsilon
G^{\zeta^{\epsilon}}(t,x;{\epsilon}^{-1}z)=G_{0}^{*}(t,x;z).$ Then by taking
into account the assumptions (A)-(F), we can deduce a large deviation
principle with the following normalized action functional
$S_{0,T}^{*}(\phi)=\int_{0}^{T}H_{0}^{*}(t,\phi(t);\phi^{\prime}(t))dt,$
for absolutely continuous function $\phi$ (see Theorem 3.2.1 in [44] for
details).
Now let us consider a closed set $A$ in $D_{0}[0,T]$ given by
$A=\\{x[0,T]:||x[0,T]||\geq 1\\}.$ The large deviation principle for
$\zeta^{\epsilon}$ gives that, for any small $\gamma>0,$ there is
$\epsilon_{0}$ such that for all $\epsilon\in(0,\epsilon_{0}),$
$\displaystyle\mathbb{P}^{z^{\epsilon}}\left\\{||\eta^{\epsilon}||\geq\epsilon^{-1/2}\right\\}$
$\displaystyle=\mathbb{P}^{z^{\epsilon}}\left\\{||\zeta^{\epsilon}||\geq
1\right\\}=\mathbb{P}^{z^{\epsilon}}\left\\{\zeta^{\epsilon}\in
A\right\\}\leq\exp\left\\{-{\epsilon}^{-1}[\inf_{\phi\in
A}S_{0,T}^{*}(\phi)-\gamma]\right\\}.$
So (3.4) is proved if $C_{A}:=\inf_{\phi\in A}S_{0,T}^{*}(\phi)>0.$
Note that $C_{A}$ is reached at some point $\phi_{A}.$ This is because $A$ is
closed and any level set $\left\\{\phi\in D_{0}[0,T]:S_{0,T}^{*}(\phi)\leq
s\right\\}$ is compact for any $s>0.$ Now we set
$C_{A}=S_{0,T}^{*}(\phi_{A}).$ If $S_{0,T}^{*}(\phi_{A})=0,$ then
$H_{0}^{*}(t,\phi_{A}(t);\phi_{A}^{\prime}(t))=0$ almost everywhere with
respect to Lebesgue measure, i.e., for all $z\in\mathbb{R},$
$\displaystyle z\cdot\phi_{A}^{\prime}(t)-G_{0}^{*}(t,\phi_{A}(t);z)\leq
0,\text{ almost all }t.$
Thus for almost all $t,$
$\displaystyle\phi_{A}^{\prime}(t)\leq\lim_{z\downarrow
0}z^{-1}G_{0}^{*}(t,\phi_{A}(t);z)=\left[\frac{\partial G_{0}}{\partial
z}(t,\phi_{0}(t)+\phi_{A}(t);z_{0}(t))-\phi^{\prime}_{0}(t)\right]$
$\displaystyle+\lim_{z\downarrow
0}z^{-1}\int_{\mathbb{R}}\left(e^{zu}-1-zu\right)e^{z_{0}(t)u}\nu_{t,\phi_{0}(t)+\phi_{A}(t)}(du)=\frac{\partial
G_{0}}{\partial z}(t,\phi_{0}(t)+\phi_{A}(t);z_{0}(t))-\phi^{\prime}_{0}(t),$
where the fact that the second limit is equal to zero is from the assumption
that $\nu_{t,x}$ have a bounded support $K.$ Similarly,
$\displaystyle\phi_{A}^{\prime}(t)\geq\lim_{z\uparrow
0}z^{-1}G_{0}^{*}(t,\phi_{A}(t);z)=\frac{\partial G_{0}}{\partial
z}(t,\phi_{0}(t)+\phi_{A}(t);z_{0}(t))-\phi^{\prime}_{0}(t),$
thus $\phi_{A}^{\prime}(t)=\frac{\partial G_{0}}{\partial
z}(t,\phi_{0}(t)+\phi_{A}(t);z_{0}(t))-\phi^{\prime}_{0}(t).$ Taking the
initial condition $\phi_{A}(0)=0$ into account, we deduce that $\phi_{A}\equiv
0.$ But this is a contradiction with $||\phi_{A}||\geq 1,$ so
$S_{0,T}^{*}(\phi_{A})\neq 0,$ i.e. $C_{A}>0.$
### 3.3 Taylor’s expansions and estimates
We start this section with a technical lemma which suggests that when we
consider the precise asymptotics for large deviations, the part away from the
most probable path can be simply dropped.
###### Lemma 3.1.
Let the family $\xi^{\epsilon}$ satisfy a large deviation principle with a
normalized action functional $S,$ and $F$ be a continuous measurable
functional on $D_{0}[0,T].$ Suppose $F$ is bounded above and the difference
$F-S$ attains its maximum at a unique function $\phi_{0}\in D_{0}[0,T].$ Then
for any $h>0,$ there is a $\gamma>0$ such that as ${\epsilon}\rightarrow 0,$
$\mathbb{E}^{\epsilon}\left[1_{\\{||\xi^{\epsilon}-\phi_{0}||\geq
h\\}}\exp\\{{\epsilon}^{-1}F(\xi^{\epsilon})\\}\right]=o\left(\exp\left\\{{\epsilon}^{-1}[F(\phi_{0})-S(\phi_{0})-\gamma]\right\\}\right).$
The proof of this lemma is contained in the Appendix. It follows from this
lemma that for any $h>0,$ there exists some $\gamma>0$ such that as
${\epsilon}\rightarrow 0,$
$\displaystyle\mathbb{E}^{\epsilon}\left[H(\xi^{\epsilon})\exp\\{{\epsilon}^{-1}F(\xi^{\epsilon})\\}\right]=$
$\displaystyle\mathbb{E}^{\epsilon}\left[1_{\left\\{||\xi^{\epsilon}-\phi_{0}||<h\right\\}}H(\xi^{\epsilon})\exp\\{{\epsilon}^{-1}F(\xi^{\epsilon})\\}\right]$
$\displaystyle+o\left(\exp\left\\{{\epsilon}^{-1}\left[F(\phi_{0})-S(\phi_{0})-\gamma\right]\right\\}\right).$
Noticing that $\exp\\{-{\epsilon}^{-1}\gamma\\}$ tends to zero exponentially
fast, we thus only focus on the first part over the set
$\left\\{||\xi^{\epsilon}-\phi_{0}||<h\right\\}.$ Simple calculations yield
$\displaystyle\mathbb{E}^{\epsilon}\left[1_{\left\\{||\xi^{\epsilon}-\phi_{0}||<h\right\\}}H(\xi^{\epsilon})\exp\\{{\epsilon}^{-1}F(\xi^{\epsilon})\\}\right]$
(3.5)
$\displaystyle=\mathbb{E}^{z^{\epsilon}}\left[1_{\left\\{||\xi^{\epsilon}-\phi_{0}||<h\right\\}}H(\xi^{\epsilon})\exp\left\\{{\epsilon}^{-1}F(\xi^{\epsilon})-{\epsilon}^{-1}\int_{0}^{T}z_{0}(t)d\xi_{t}^{\epsilon}+{\epsilon}^{-1}\int_{0}^{T}G_{0}(t,\xi_{t}^{\epsilon};z_{0}(t))dt\right\\}\right]$
$\displaystyle=\mathbb{E}^{z^{\epsilon}}\Big{[}1_{\left\\{||\eta^{\epsilon}||<\epsilon^{-1/2}h\right\\}}H(\phi_{0}+{\epsilon}^{1/2}\eta^{\epsilon})\exp\Big{\\{}{\epsilon}^{-1}F(\phi_{0}+{\epsilon}^{1/2}\eta^{\epsilon})-{\epsilon}^{-1/2}\int_{0}^{T}z_{0}(t)d\eta_{t}^{\epsilon}$
$\displaystyle\qquad\qquad-{\epsilon}^{-1}\int_{0}^{T}\Big{(}z_{0}(t)\phi^{\prime}_{0}(t)-G_{0}(t,\phi_{0}(t)+{\epsilon}^{1/2}\eta_{t}^{\epsilon};z_{0}(t))\Big{)}dt\Big{\\}}\Big{]}.$
Now we apply Taylor’s expansion for $F$ at $\phi_{0}$ up to order $s$ with an
integral’s remainder ($IR$) (see [8] for details),
$\displaystyle
F(\phi_{0}+{\epsilon}^{1/2}\eta^{\epsilon})=F(\phi_{0})+{\epsilon}^{1/2}F^{(1)}(\phi_{0})(\eta^{\epsilon})+\cdots+\frac{{\epsilon}^{s/2}}{s!}F^{(s)}(\phi_{0})(\eta^{\epsilon},\cdots,\eta^{\epsilon})+IR_{1},$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad
IR_{1}={\epsilon}^{\frac{s+1}{2}}\int_{0}^{1}\frac{(1-u)^{s}}{s!}F^{(s+1)}(\phi_{0}+u{\epsilon}^{1/2}\eta^{\epsilon})(\eta^{\epsilon},\cdots,\eta^{\epsilon})du.$
For $G_{0},$ Taylor’s expansion at $\phi_{0}(t)$ in the second argument up to
order $s$ with an integral’s remainders yields
$\displaystyle\int_{0}^{T}G_{0}(t,\phi_{0}(t)+{\epsilon}^{1/2}\eta_{t}^{\epsilon};z_{0}(t))dt=$
$\displaystyle\int_{0}^{T}\left(G_{0}(t,\phi_{0}(t);z_{0}(t))+{\epsilon}^{1/2}\eta_{t}^{\epsilon}\frac{\partial
G_{0}}{\partial
x}(t,\phi_{0}(t);z_{0}(t))+\cdots+\frac{\left({\epsilon}^{1/2}\eta_{t}^{\epsilon}\right)^{s}}{s!}\frac{\partial^{s}G_{0}}{\partial
x^{s}}(t,\phi_{0}(t);z_{0}(t))\right)dt$ $\displaystyle\qquad\qquad+IR_{2}$
where
$IR_{2}=\int_{0}^{T}\left({\epsilon}^{1/2}\eta_{t}^{\epsilon}\right)^{s+1}\int_{0}^{1}\frac{(1-u)^{s}}{s!}\frac{\partial^{s+1}G_{0}}{\partial
x^{s+1}}(t,\phi_{0}(t)+u{\epsilon}^{1/2}\eta_{t}^{\epsilon};z_{0}(t))dudt.$
We use these expansions to replace the exponential in (3.5) to get
$\displaystyle\exp\left\\{{\epsilon}^{-1}F(\phi_{0}+\epsilon^{1/2}\eta^{\epsilon})-\epsilon^{-1/2}\int_{0}^{T}z_{0}(t)d\eta_{t}^{\epsilon}\right.$
(3.6)
$\displaystyle\qquad\qquad\qquad\qquad\qquad\quad\left.-{\epsilon}^{-1}\int_{0}^{T}\Big{(}z_{0}(t)\phi^{\prime}_{0}(t)-G_{0}(t,\phi_{0}(t)+\epsilon^{1/2}\eta_{t}^{\epsilon};z_{0}(t))\Big{)}dt\right\\}$
$\displaystyle=\exp\Big{\\{}{\epsilon}^{-1}\left[F(\phi_{0})-\int_{0}^{T}\Big{(}z_{0}(t)\phi^{\prime}_{0}(t)-G_{0}(t,\phi_{0}(t);z_{0}(t))\Big{)}dt\right]$
$\displaystyle\qquad\qquad+{\epsilon}^{-1/2}\left[F^{(1)}(\phi_{0})(\eta^{\epsilon})-\int_{0}^{T}z_{0}(t)d\eta_{t}^{\epsilon}+\int_{0}^{T}\eta_{t}^{\epsilon}\frac{\partial
G_{0}}{\partial x}(t,\phi_{0}(t);z_{0}(t))dt\right]$
$\displaystyle\qquad\qquad+\left[\frac{1}{2!}F^{(2)}(\phi_{0})(\eta^{\epsilon},\eta^{\epsilon})+\int_{0}^{T}\frac{1}{2!}(\eta_{t}^{\epsilon})^{2}\frac{\partial^{2}G_{0}}{\partial
x^{2}}(t,\phi_{0}(t);z_{0}(t))dt\right]$
$\displaystyle\qquad\qquad+{\epsilon}^{1/2}\left[\frac{1}{3!}F^{(3)}(\phi_{0})(\eta^{\epsilon},\eta^{\epsilon},\eta^{\epsilon})+\int_{0}^{T}\frac{1}{3!}(\eta_{t}^{\epsilon})^{3}\frac{\partial^{3}G_{0}}{\partial
x^{3}}(t,\phi_{0}(t);z_{0}(t))dt\right]+\cdots$
$\displaystyle\qquad\qquad+{\epsilon}^{\frac{s-2}{2}}\left[\frac{1}{s!}F^{(s)}(\phi_{0})(\eta^{\epsilon},\cdots,\eta^{\epsilon})+\int_{0}^{T}\frac{1}{s!}(\eta_{t}^{\epsilon})^{s}\frac{\partial^{s}G_{0}}{\partial
x^{s}}(t,\phi_{0}(t);z_{0}(t))dt\right]$
$\displaystyle\qquad\qquad+{\epsilon}^{-1}(IR_{1}+IR_{2})\Big{\\}}$
$\displaystyle=\exp\Big{\\{}{\epsilon}^{-1}\left[F(\phi_{0})-S(\phi_{0})\right]+\left[\frac{1}{2!}F^{(2)}(\phi_{0})(\eta^{\epsilon},\eta^{\epsilon})+\int_{0}^{T}\frac{1}{2!}(\eta_{t}^{\epsilon})^{2}\frac{\partial^{2}G_{0}}{\partial
x^{2}}(t,\phi_{0}(t);z_{0}(t))dt\right]$
$\displaystyle\qquad\qquad+{\epsilon}^{1/2}\left[\frac{1}{3!}F^{(3)}(\phi_{0})(\eta^{\epsilon},\eta^{\epsilon},\eta^{\epsilon})+\int_{0}^{T}\frac{1}{3!}(\eta_{t}^{\epsilon})^{3}\frac{\partial^{3}G_{0}}{\partial
x^{3}}(t,\phi_{0}(t);z_{0}(t))dt\right]+\cdots$
$\displaystyle\qquad\qquad+{\epsilon}^{\frac{s-2}{2}}\left[\frac{1}{s!}F^{(s)}(\phi_{0})(\eta^{\epsilon},\cdots,\eta^{\epsilon})+\int_{0}^{T}\frac{1}{s!}(\eta_{t}^{\epsilon})^{s}\frac{\partial^{s}G_{0}}{\partial
x^{s}}(t,\phi_{0}(t);z_{0}(t))dt\right]$
$\displaystyle\qquad\qquad+{\epsilon}^{-1}(IR_{1}+IR_{2})\Big{\\}}$
$\displaystyle=\exp\left\\{{\epsilon}^{-1}\left[F(\phi_{0})-S(\phi_{0})\right]+Q(2,\eta^{\epsilon})\right\\}\times$
$\displaystyle\qquad\exp\Big{\\{}{\epsilon}^{1/2}Q(3,\eta^{\epsilon})+\cdots+{\epsilon}^{\frac{s-2}{2}}Q(s,\eta^{\epsilon})+{\epsilon}^{-1}(IR_{1}+IR_{2})\Big{\\}}.$
For the second exponential function in the last equality of (3.6), we apply
$e^{a}=1+a+\cdots+a^{s-2}/(s-2)!+\frac{e^{\theta(a)\cdot a}}{(s-1)!}a^{s-1}$
with $0\leq\theta(a)\leq 1,$ then formula (3.6) is equal to
$\displaystyle(\ref{first-step-2})=$
$\displaystyle\exp\left\\{{\epsilon}^{-1}\left[F(\phi_{0})-S(\phi_{0})\right]+Q(2,\eta^{\epsilon})\right\\}\times$
$\displaystyle\left\\{1+\epsilon^{1/2}Q(3,\eta^{\epsilon})+\epsilon\left(Q(4,\eta^{\epsilon})+\frac{1}{2}Q(3,\eta^{\epsilon})^{2}\right)+\cdots+{\epsilon}^{\frac{s-2}{2}}\ell(\epsilon,\eta^{\epsilon})+\Re(\epsilon,\eta^{\epsilon})\right\\}$
where the coefficient $\ell(\epsilon,\eta^{\epsilon})$ of
${\epsilon}^{\frac{s-2}{2}}$ is a functional of $\eta^{\epsilon}$ and depends
on $\epsilon.$ Special attention needs to be paid to the structure of the
remainder term $\Re(\epsilon,\eta^{\epsilon}).$ There are two different
aspects
$\Re(\epsilon,\eta^{\epsilon})=\Re_{1}(\epsilon,\eta^{\epsilon})+\Re_{2}(\epsilon,\eta^{\epsilon}),$
where the first $\Re_{1}(\epsilon,\eta^{\epsilon})$ can be bounded by powers
of $\eta^{\epsilon}$, while the second $\Re_{2}(\epsilon,\eta^{\epsilon})$
involves a part $e^{\theta(a)\cdot a}.$ By taking conditions (2.1)-(2.2) into
account,
$|\ell(\epsilon,\eta^{\epsilon})|+|\Re_{1}(\epsilon,\eta^{\epsilon})|\leq\epsilon^{\frac{s-1}{2}}\cdot
c\cdot\left(1+\left|\eta_{T}^{\epsilon}\right|^{k}+\int_{0}^{T}\left|\eta_{t}^{\epsilon}\right|^{k}dt\right),$
for some nonnegative constants $c$ and $k.$ The second one
$\Re_{2}(\epsilon,\eta^{\epsilon})$ on the set
$\\{||\eta^{\epsilon}||<\epsilon^{-1/2}h\\}$) can be estimated as
$|\Re_{2}(\epsilon,\eta^{\epsilon})|\leq\epsilon^{\frac{s-1}{2}}\cdot
c\cdot\left(1+\left|\eta_{T}^{\epsilon}\right|^{k}+\int_{0}^{T}\left|\eta_{t}^{\epsilon}\right|^{k}dt\right)e^{h\cdot||\eta^{\epsilon}||^{2}}.$
Hölder’s inequality and the fact
$\mathbb{E}^{z^{\epsilon}}1_{\\{||\eta^{\epsilon}||<\epsilon^{-1/2}h\\}}e^{h\cdot||\eta^{\epsilon}||^{2}}<C<\infty$
uniformly in $\epsilon$ (see Section 5.2.6 in [44]) suggest that we only need
to take care of $\Re_{1}(\epsilon,\eta^{\epsilon}).$
Taylor’s expansion for $H(\phi_{0}+{\epsilon}^{1/2}\eta^{\epsilon})$ at
$\phi_{0}$ up to $(s-2)$-derivative gives
$\displaystyle H(\phi_{0}+{\epsilon}^{1/2}\eta^{\epsilon})$
$\displaystyle=H(\phi_{0})+{\epsilon}^{1/2}H^{(1)}(\phi_{0})(\eta^{\epsilon})+\cdots+\frac{{\epsilon}^{\frac{s-2}{2}}}{(s-2)!}H^{(s-2)}(\phi_{0})(\eta^{\epsilon},\cdots,\eta^{\epsilon})+IR_{3},$
(3.7) $\displaystyle\qquad\qquad
IR_{3}=\epsilon^{\frac{s-1}{2}}\int_{0}^{1}\frac{(1-u)^{s-2}}{(s-2)!}H^{(s-1)}(\phi_{0}+u{\epsilon}^{1/2}\eta^{\epsilon})(\eta^{\epsilon},\cdots,\eta^{\epsilon})du,$
Now we combine (3.6) and (3.7) to rewrite (3.5) on the set
$\\{||\eta^{\epsilon}||<\epsilon^{-1/2}h\\}$ as follows,
$\displaystyle\mathbb{E}^{\epsilon}\left[1_{\left\\{||\eta^{\epsilon}||<\epsilon^{-1/2}h\right\\}}H(\xi^{\epsilon})\exp\\{{\epsilon}^{-1}F(\xi^{\epsilon})\\}\right]$
(3.8)
$\displaystyle=\exp\left\\{{\epsilon}^{-1}\left[F(\phi_{0})-S(\phi_{0})\right]\right\\}\times$
$\displaystyle\quad\mathbb{E}^{z^{\epsilon}}\Big{\\{}\exp\\{Q(2,\eta^{\epsilon})\\}\cdot\Big{(}H(\phi_{0})+{\epsilon}^{1/2}\left[Q(3,\eta^{\epsilon})H(\phi_{0})+H^{(1)}(\phi_{0})(\eta^{\epsilon})\right]$
$\displaystyle\qquad+\epsilon\left[\left(Q(4,\eta^{\epsilon})+\frac{1}{2}[Q(3,\eta^{\epsilon})]^{2}\right)H(\phi_{0})+H^{(2)}(\phi_{0})(\eta^{\epsilon},\eta^{\epsilon})+Q(3,\eta^{\epsilon})H^{(1)}(\phi_{0})(\eta^{\epsilon})\right]$
$\displaystyle\qquad+\cdots+{\epsilon}^{\frac{s-2}{2}}\tilde{\ell}(\epsilon,\eta^{\epsilon})\Big{)}\Big{\\}}$
$\displaystyle\,\,\,\,\,\,-\exp\left\\{{\epsilon}^{-1}\left[F(\phi_{0})-S(\phi_{0})\right]\right\\}\times$
$\displaystyle\quad\quad\mathbb{E}^{z^{\epsilon}}\Big{\\{}1_{\left\\{||\eta^{\epsilon}||\geq\epsilon^{-1/2}h\right\\}}\exp\\{Q(2,\eta^{\epsilon})\\}\cdot\Big{(}H(\phi_{0})+{\epsilon}^{1/2}\left[Q(3,\eta^{\epsilon})H(\phi_{0})+H^{(1)}(\phi_{0})(\eta^{\epsilon})\right]$
$\displaystyle\qquad+\epsilon\left[\left(Q(4,\eta^{\epsilon})+\frac{1}{2}[Q(3,\eta^{\epsilon})]^{2}\right)H(\phi_{0})+H^{(2)}(\phi_{0})(\eta^{\epsilon},\eta^{\epsilon})+Q(3,\eta^{\epsilon})H^{(1)}(\phi_{0})(\eta^{\epsilon})\right]$
$\displaystyle\qquad+\cdots+{\epsilon}^{\frac{s-2}{2}}\bar{\ell}(\epsilon,\eta^{\epsilon})\Big{)}\Big{\\}}$
$\displaystyle\,\,\,\,\,\,+\exp\left\\{{\epsilon}^{-1}\left[F(\phi_{0})-S(\phi_{0})\right]\right\\}\cdot
E^{z^{\epsilon}}\left\\{1_{\left\\{||\eta^{\epsilon}||<\epsilon^{-1/2}h\right\\}}\exp\\{Q(2,\eta^{\epsilon})\\}\times\Re_{1}(\epsilon,\eta^{\epsilon})\right\\}$
for two functionals $\tilde{\ell}(\epsilon,\eta^{\epsilon})$ and
$\bar{\ell}(\epsilon,\eta^{\epsilon}).$ Now it becomes quite clear that
Theorem 2.1 is proved if the following (I)-(III) are proved:
(I).
$\displaystyle\mathbb{E}^{z^{\epsilon}}\left\\{1_{\left\\{||\eta^{\epsilon}||<\epsilon^{-1/2}h\right\\}}\exp\\{Q(2,\eta^{\epsilon})\\}\times\Re_{1}(\epsilon,\eta^{\epsilon})\right\\}=o\left({\epsilon}^{\frac{s-2}{2}}\right).$
(3.9)
(II). Each term over the set
$\left\\{||\eta^{\epsilon}||\geq\epsilon^{-1/2}h\right\\}$ in (3.8) tends to
zero exponentially fast.
(III). The expectations $\mathbb{E}^{z^{\epsilon}}$ in (3.8) without
$\left\\{||\eta^{\epsilon}||\geq\epsilon^{-1/2}h\right\\}$ have precise
asymptotic expansions. Obviously, this is the place where normal deviations
for stochastic processes are used. To make (III) more precise, we need to show
the following asymptotic expansions for normal deviations
$\displaystyle\mathbb{E}^{z^{\epsilon}}\left\\{\exp\\{Q(2,\eta^{\epsilon})\\}\right\\}=\mathbb{E}\left\\{\exp\\{Q(2,\eta)\\}\right\\}+{\epsilon}^{1/2}C_{01}+\epsilon
C_{02}+\cdots+o({\epsilon}^{\frac{s-2}{2}}),$
$\displaystyle\mathbb{E}^{z^{\epsilon}}\left\\{\exp\\{Q(2,\eta^{\epsilon})\\}\left[Q(3,\eta^{\epsilon})H(\phi_{0})+H^{(1)}(\phi_{0})(\eta^{\epsilon})\right]\right\\}$
$\displaystyle\quad=\mathbb{E}\left\\{\exp\\{Q(2,\eta)\\}\left[Q(3,\eta)H(\phi_{0})+H^{(1)}(\phi_{0})(\eta)\right]\right\\}+{\epsilon}^{1/2}C_{11}+\epsilon
C_{12}+\cdots+o(\epsilon^{\frac{s-3}{2}}),$
$\displaystyle\mathbb{E}^{z^{\epsilon}}\left\\{\exp\\{Q(2,\eta^{\epsilon})\\}\left[\left(Q(4,\eta^{\epsilon})+\frac{1}{2}[Q(3,\eta^{\epsilon})]^{2}\right)H(\phi_{0})+H^{(2)}(\phi_{0})(\eta^{\epsilon},\eta^{\epsilon})+Q(3,\eta^{\epsilon})H^{(1)}(\phi_{0})(\eta^{\epsilon})\right]\right\\}$
$\displaystyle\quad=\mathbb{E}\left\\{\exp\\{Q(2,\eta)\\}\left[\left(Q(4,\eta)+\frac{1}{2}[Q(3,\eta)]^{2}\right)H(\phi_{0})\right.\right.$
$\displaystyle\qquad\left.\left.+H^{(2)}(\phi_{0})(\eta,\eta)+Q(3,\eta)H^{(1)}(\phi_{0})(\eta)\right]\right\\}+{\epsilon}^{1/2}C_{21}+\epsilon
C_{22}+\cdots+o(\epsilon^{\frac{s-4}{2}})$
and so on, where $C_{ij}$ are constants which can be determined by Theorem 3.1
in Section 3.1. If we replace all terms in (3.8) by these asymptotic
expansions for normal deviations, then we get
$\displaystyle\mathbb{E}^{\epsilon}\left[H(\xi^{\epsilon})\exp\\{{\epsilon}^{-1}F(\xi^{\epsilon})\\}\right]=\mathbb{E}^{\epsilon}\left[||\xi^{\epsilon}-\phi_{0}||<h;H(\xi^{\epsilon})\exp\\{{\epsilon}^{-1}F(\xi^{\epsilon})\\}\right]$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad+o(\exp\\{{\epsilon}^{-1}[F(\phi_{0})-S(\phi_{0})-\gamma]\\})$
$\displaystyle=\exp\left\\{{\epsilon}^{-1}\left[F(\phi_{0})-S(\phi_{0})\right]\right\\}\cdot\left[\sum_{0\leq
i\leq(s-2)}K_{i}{\epsilon}^{i/2}+o\left({\epsilon}^{(s-2)/2}\right)\right],$
which is exactly (2.3). That is, Theorem 2.1 is proved if (I), (II), (III) and
the finiteness of $K_{i}$ are proved.
### 3.4 Proofs of (I)-(III)
#### 3.4.1 Proof of (I)
It is clear that (I) will be proved if for any positive integer $k,$
$\displaystyle\mathbb{E}^{z^{\epsilon}}\left\\{1_{\left\\{||\eta^{\epsilon}||<\epsilon^{-1/2}h\right\\}}\exp\\{Q(2,\eta^{\epsilon})\\}\epsilon^{\frac{s-1}{2}}\left(1+\left|\eta_{T}^{\epsilon}\right|^{k}+\int_{0}^{T}\left|\eta_{t}^{\epsilon}\right|^{k}dt\right)\right\\}=o\left({\epsilon}^{\frac{s-2}{2}}\right).$
(3.10)
In order to prove (3.10), we establish a lemma first.
###### Lemma 3.2.
Under the assumption (F), for any positive $k,$ there are constants $c_{1}$
and $c_{2}$ (depending only on $k$) such that for all $t\in[0,T],$
$1>{\epsilon}>0,$
$\displaystyle\mathbb{E}^{z^{\epsilon}}\left[\left(\eta^{\epsilon}_{t}\right)^{k}\right]\leq
t\cdot c_{1}+c_{1}\cdot c_{2}\cdot\int_{0}^{t}s\cdot e^{c_{2}\cdot(t-s)}ds.$
###### Proof.
We consider a sequence of functions
$f_{n}(x)=\frac{x^{k}}{1+\left(x/n\right)^{k}}$ with an even positive integer
$k.$ Then, according to (3.3), the generating operator
$A_{t}^{\eta^{\epsilon}}$ applying to $f_{n}$ gives
$\displaystyle A_{t}^{\eta^{\epsilon}}f_{n}(x)$
$\displaystyle=\epsilon^{-1/2}\left[\frac{\partial G_{0}}{\partial
z}(t,\phi_{0}(t)+\epsilon^{1/2}x;z_{0}(t))-\phi_{0}^{\prime}(t)\right]f_{n}^{\prime}(x)+\frac{1}{2}a(t,\phi_{0}(t)+\epsilon^{1/2}x)f^{\prime\prime}_{n}(x)$
$\displaystyle\qquad+{\epsilon}^{-1}\int_{\mathbb{R}}\left[f_{n}(x+\epsilon^{1/2}u)-f_{n}(x)-\epsilon^{1/2}f_{n}^{\prime}(x)\cdot
u\right]e^{z_{0}(t)u}\nu_{t,\phi_{0}(t)+\epsilon^{1/2}x}(du)$
$\displaystyle=\frac{\partial^{2}G_{0}}{\partial z\partial
x}(t,\phi_{0}(t)+\theta_{1}\epsilon^{1/2}x;z_{0}(t))\cdot x\cdot
f_{n}^{\prime}(x)$
$\displaystyle\qquad+\frac{f_{n}^{\prime\prime}(x)}{2}\left(\int_{\mathbb{R}}u^{2}e^{z_{0}(t)u}\nu_{t,\phi_{0}(t)+\epsilon^{1/2}x}(du)+a(t,\phi_{0}(t)+\epsilon^{1/2}x)\right)$
$\displaystyle\qquad+\cdots+\frac{1}{k!}\int_{\mathbb{R}}\epsilon^{\frac{k-2}{2}}u^{k}f_{n}^{(k)}(x+\theta_{2}\epsilon^{1/2}u)e^{z_{0}(t)u}\nu_{t,\phi_{0}(t)+\epsilon^{1/2}x}(du),$
which is less than or equal to $c_{3}+c_{4}\cdot f_{n}(x),$ since
$f_{n}^{(k)}$ is bounded, and $x\cdot
f^{\prime}_{n}(x),f_{n}^{\prime\prime},f_{n}^{\prime\prime\prime},\cdots,f_{n}^{(k-1)}$
are bounded by $c_{5}+c_{6}\cdot f_{n}(x).$ So
$\displaystyle\mathbb{E}^{z^{\epsilon}}\left[f_{n}\left(\eta^{\epsilon}_{t}\right)\right]\leq
f_{n}(0)+\int_{0}^{t}\left(c_{1}+c_{2}\cdot\mathbb{E}^{z^{\epsilon}}\left[f_{n}\left(\eta^{\epsilon}_{s}\right)\right]\right)ds.$
Applying Gronwall’s lemma with such nonnegative
$\mathbb{E}^{z^{\epsilon}}\left[f_{n}\left(\eta^{\epsilon}_{t}\right)\right],$
we obtain
$\displaystyle\mathbb{E}^{z^{\epsilon}}\left[f_{n}\left(\eta^{\epsilon}_{t}\right)\right]\leq
t\cdot c_{1}+c_{1}\cdot c_{2}\cdot\int_{0}^{t}s\cdot e^{c_{2}\cdot(t-s)}ds,$
then the proof is done by sending $n$ to infinity. ∎
From this lemma, we have
$\sup_{\epsilon\in(0,1)}\mathbb{E}^{z^{\epsilon}}\left[(\eta^{\epsilon}_{T})^{k}\right]<\infty$
and
$\sup_{\epsilon\in(0,1)}\mathbb{E}^{z^{\epsilon}}\left[\int_{0}^{T}(\eta^{\epsilon}_{t})^{k}dt\right]<\infty$
for any integer $k.$ These together with the fact
$\exp\\{Q(2,\eta^{\epsilon})\\}\leq 1$ yield (3.10).
#### 3.4.2 Proof of (II)
It is immediate that the first term goes to zero exponentially fast,
$\displaystyle\mathbb{E}^{z^{\epsilon}}\left\\{1_{\left\\{||\eta^{\epsilon}||\geq\epsilon^{-1/2}h\right\\}}\exp\\{Q(2,\eta^{\epsilon})\\}\cdot
H(\phi_{0})\right\\}\rightarrow 0\text{ exponentially fast},$
which is from (3.4). Every other term has an upper bound by using estimates
(2.1)-(2.2):
$c\cdot\mathbb{E}^{z^{\epsilon}}\left\\{1_{\left\\{||\eta^{\epsilon}||\geq\epsilon^{-1/2}h\right\\}}\epsilon^{\frac{j-2}{2}}\left(1+\left|\eta^{\epsilon}_{T}\right|^{k}+\int_{0}^{T}\left|\eta^{\epsilon}_{t}\right|^{k}dt\right)\right\\},$
(3.11)
for some nonnegative constants $c$ and $k.$ By applying Hölder’s inequality to
(3.11) and using Lemma 3.2, it follows each term over the set
$\left\\{||\eta^{\epsilon}||\geq\epsilon^{-1/2}h\right\\}$ in (3.8) tends to
zero exponentially fast.
#### 3.4.3 Proof of (III)
(a) Proof of the first expansion
The task is to prove
$\mathbb{E}^{z^{\epsilon}}\left\\{\exp\\{Q(2,\eta^{\epsilon})\\}\right\\}=\mathbb{E}\left\\{\exp\\{Q(2,\eta)\\}\right\\}+{\epsilon}^{1/2}C_{01}+\epsilon
C_{02}+\cdots+{\epsilon}^{(s-2)/2}C_{0(s-s)}+o({\epsilon}^{(s-2/2}).$
Denoting
$\displaystyle\widetilde{F}(x[0,T])=\exp\left\\{Q(2,x[0,T])\right\\}=$
$\displaystyle\exp\left\\{\frac{1}{2}F^{(2)}(\phi_{0})(x[0,T],x[0,T])\right.$
$\displaystyle\left.+\int_{0}^{T}\frac{1}{2}(x(t))^{2}\frac{\partial^{2}G_{0}}{\partial
x^{2}}(t,\phi_{0}(t);z_{0}(t))dt\right\\},$
we need to check such $\widetilde{F}(x[0,T])$ satisfies all conditions of
Theorem 3.1 in Section 3.1.
Claim 1. $\widetilde{F}(x[0,T])$ is $3(s-2)$ times differentiable. It is easy
to see that $\widetilde{F}$ is infinitely differentiable. Furthermore, the
derivatives can be computed as follows
$\displaystyle\widetilde{F}^{(1)}$
$\displaystyle(x[0,T])(\delta)=\lim_{h\rightarrow
0}{h}^{-1}\left[\widetilde{F}(x[0,T]+h\delta)-\widetilde{F}(x[0,T])\right]$
$\displaystyle=\widetilde{F}(x[0,T])\left(F^{(2)}(\phi_{0})(x[0,T],\delta)+\int_{0}^{T}x(t)\delta(t)\frac{\partial^{2}G_{0}}{\partial
x^{2}}(t,\phi_{0}(t);z_{0}(t))dt\right),$ $\displaystyle\widetilde{F}^{(2)}$
$\displaystyle(x[0,T])(\delta_{1},\delta_{2})=\lim_{h\rightarrow
0}h^{-1}\left[\widetilde{F}^{(1)}(x[0,T]+h\delta_{2})(\delta_{1})-\widetilde{F}^{(1)}(x[0,T])(\delta_{1})\right]$
$\displaystyle=\widetilde{F}(x[0,T])\left(F^{(2)}(\phi_{0})(x[0,T],\delta_{2})+\int_{0}^{T}x(t)\delta_{2}(t)\frac{\partial^{2}G_{0}}{\partial
x^{2}}(t,\phi_{0}(t);z_{0}(t))dt\right)$
$\displaystyle\qquad\qquad\qquad\cdot\left(F^{(2)}(\phi_{0})(x[0,T],\delta_{1})+\int_{0}^{T}x(t)\delta_{1}(t)\frac{\partial^{2}G_{0}}{\partial
x^{2}}(t,\phi_{0}(t);z_{0}(t))dt\right)$
$\displaystyle\,\,+\widetilde{F}(x[0,T])\left(F^{(2)}(\phi_{0})(\delta_{1},\delta_{2})+\int_{0}^{T}\delta_{1}(t)\delta_{2}(t)\frac{\partial^{2}G_{0}}{\partial
x^{2}}(t,\phi_{0}(t);z_{0}(t))dt\right)$
and so on.
Claim 2. $\widetilde{F}$ satisfies condition (I) of Theorem 3.1. First we have
$\left|\widetilde{F}(x[0,T])\right|\leq 1.$ The derivatives of $\widetilde{F}$
satisfy (I). For instance, here we check for $\widetilde{F}^{(2)}.$
$\displaystyle\left|\widetilde{F}^{(2)}(x[0,T])(\delta,\delta)\right|\leq\left(F^{(2)}(\phi_{0})(x[0,T],\delta)+\int_{0}^{T}x(t)\delta(t)\frac{\partial^{2}G_{0}}{\partial
x^{2}}(t,\phi_{0}(t);z_{0}(t))dt\right)^{2}+c(||\delta||),$
where $c(||\delta||)$ is a constant depending on the uniform norm of $\delta.$
Taking into account the assumptions on $F^{(2)},$ the above is less than or
equal to
$\displaystyle
p^{2}\cdot\left(|x(T)\delta(T)|^{m}+(1+T||\delta||^{n})(1+\int_{0}^{T}|x(t)|^{n}dt)+||\delta\frac{\partial^{2}G_{0}}{\partial
x^{2}}||\int_{0}^{T}x(t)^{2}dt\right)^{2}+c(||\delta||).$
We apply Hölder inequality several times to get an upper bound
$c_{1}(||\delta||)\left(1+|x(T)|^{2m}+\int_{0}^{T}|x(t)|^{2n}dt\right).$
Claim 3.
$\widetilde{F}^{(i)}(x[0,T])(I_{[t,T]}\delta,\cdots,I_{[t,T]}\delta),3\leq
i\leq 3(s-2),$ are continuous with respect to $x[0,T]$ uniformly as $x[0,T]$
changes over an arbitrary compact subset of $D_{0}[0,T],$ $t$ over $[0,T],$
and $\delta[0,T]$ over the set of Lipschitz continuous functions with constant
$1,$ $||\delta||\leq 1.$
It follows from [39] that any compact subset of $D_{0}[0,T]$ is a bounded set
in uniform topology. Taking $|I_{[t,T]}\delta|\leq 1$ into account, Claim 3 is
done easily. For instance, the uniform continuity of
$F^{(2)}(\phi_{0})(x[0,T],I_{[t,T]}\delta)$ in $x[0,T]$ can be achieved as
follows:
$\displaystyle\left|F^{(2)}(\phi_{0})(y[0,T],I_{[t,T]}\delta)-F^{(2)}(\phi_{0})(x[0,T],I_{[t,T]}\delta)\right|$
$\displaystyle=\left|F^{(2)}(\phi_{0})(y[0,T]-x[0,T],I_{[t,T]}\delta)\right|\leq
C||y-x||,$
where $C$ is independent of $x,y,t,\delta.$ Uniform continuity of
$F^{(2)}(\phi_{0})(x[0,T],x[0,T])$ in $x[0,T]$ can be also obtained by
$\displaystyle\left|F^{(2)}(\phi_{0})(y[0,T],y[0,T])-F^{(2)}(\phi_{0})(x[0,T],x[0,T])\right|$
$\displaystyle\leq\left|F^{(2)}(\phi_{0})(y[0,T]-x[0,T],y[0,T])\right|+\left|F^{(2)}(\phi_{0})(y[0,T]-x[0,T],x[0,T])\right|$
$\displaystyle\leq C\cdot\left(||y||+||x||\right)\cdot||x-y||\leq C\cdot
C_{1}\cdot||x-y||,$
where $C$ is independent of $x,y,$ and $C_{1}$ can be chosen independently of
$x,y,$ because a compact set is a bounded set in uniform topology.
(b) Proofs of the second expansion and the other expansions
The second expansion is
$\displaystyle\mathbb{E}^{z^{\epsilon}}\left\\{\exp\\{Q(2,\eta^{\epsilon})\\}\left[Q(3,\eta^{\epsilon})H(\phi_{0})+H^{(1)}(\phi_{0})(\eta^{\epsilon})\right]\right\\}$
$\displaystyle\quad=\mathbb{E}\left\\{\exp\\{Q(2,\eta)\\}\left[Q(3,\eta)H(\phi_{0})+H^{(1)}(\phi_{0})(\eta)\right]\right\\}+{\epsilon}^{1/2}C_{11}+\epsilon
C_{12}+\cdots+o(\epsilon^{(s-3)/2}).$
We use $\widehat{F}$ to denote
$\displaystyle\widehat{F}(x[0,T])$
$\displaystyle=\exp\left\\{Q(2,x[0,T])\right\\}[Q(3,x[0,T])+H^{(1)}(\phi_{0})(x[0,T])]$
$\displaystyle=\widetilde{F}(x[0,T])\cdot[Q(3,x[0,T])+H^{(1)}(\phi_{0})(x[0,T])].$
Claim 1. $\widehat{F}(x[0,T])$ is infinitely differentiable. The first two
derivatives are
$\displaystyle\widehat{F}^{(1)}(x[0,T])(\delta)=\lim_{{h}\rightarrow
0}{h}^{-1}\left[\widehat{F}(x[0,T]+h\delta)-\widehat{F}(x[0,T])\right]$
$\displaystyle=\widetilde{F}^{(1)}(x[0,T])(\delta)\cdot[Q(3,x[0,T])+H^{(1)}(\phi_{0})(x[0,T])]+\frac{3}{3!}\widetilde{F}(x[0,T])$
$\displaystyle\quad\times\left(F^{(3)}(\phi_{0})(x[0,T],x[0,T],\delta)+\int_{0}^{T}x^{2}(t)\delta(t)\frac{\partial^{3}G_{0}}{\partial
x^{3}}(t,\phi_{0}(t);z_{0}(t))dt+H^{(1)}(\phi_{0})(\delta)\right),$
$\displaystyle\widehat{F}^{(2)}(x[0,T])(\delta_{1},\delta_{2})=\lim_{{h}\rightarrow
0}{h}^{-1}\left[\widehat{F}^{(1)}(x[0,T]+h\delta_{2})(\delta_{1})-\widehat{F}^{(1)}(x[0,T])(\delta_{1})\right]$
$\displaystyle=\widetilde{F}^{(2)}(x[0,T])(\delta_{1},\delta_{2})\cdot[Q(3,x[0,T])+H^{(1)}(\phi_{0})(x[0,T])]$
$\displaystyle+\frac{3}{3!}\sum_{1\leq i\neq j\leq
2}\widetilde{F}^{(1)}(x[0,T])(\delta_{j})\Big{(}F^{(3)}(\phi_{0})(x[0,T],x[0,T],\delta_{i})$
$\displaystyle\qquad\qquad\qquad\qquad\qquad+\int_{0}^{T}x^{2}(t)\delta_{i}(t)\frac{\partial^{3}G_{0}}{\partial
x^{3}}(t,\phi_{0}(t);z_{0}(t))dt+H^{\prime}(\phi_{0})(\delta_{i})\Big{)}$
$\displaystyle+\frac{6}{3!}\widetilde{F}(x[0,T])\left(F^{(3)}(\phi_{0})(x[0,T],\delta_{1},\delta_{2})+\int_{0}^{T}x(t)\delta_{1}(t)\delta_{2}(t)\frac{\partial^{3}G_{0}}{\partial
x^{3}}(t,\phi_{0}(t);z_{0}(t))dt\right).$
Claim 2. $\widehat{F}$ satisfies condition (I) of Theorem 3.1. We notice that
$Q(3,x[0,T])$ satisfies
$\displaystyle|Q(3,x[0,T])|=\left|\frac{1}{3!}F^{(3)}(\phi_{0})(x[0,T],x[0,T],x[0,T])+\int_{0}^{T}\frac{1}{3!}x^{3}(t)\frac{\partial^{3}G_{0}}{\partial
x^{3}}(t,\phi_{0}(t);z_{0}(t))dt\right|$
$\displaystyle\leq\frac{p}{3!}\left[|x(T)|^{3m}+\left(1+\int_{0}^{T}|x(t)|^{n}dt\right)^{3}\right]+\frac{1}{3!}\left|\left|\frac{\partial^{3}G_{0}}{\partial
x^{3}}\right|\right|\cdot\int_{0}^{T}|x(t)|^{3}dt.$
This together with (2.2) imply that the first part of (I) is fulfilled. For
the second part of (I) on derivatives of $\widehat{F},$ similar arguments can
be applied. We can also prove the following
Claim 3.
$\widehat{F}^{(i)}(x[0,T])(I_{[t,T]}\delta,\cdots,I_{[t,T]}\delta),3\leq i\leq
3(s-3),$ are continuous with respect to $x[0,T]$ uniformly as $x[0,T]$ changes
over an arbitrary compact subset of $D_{0}[0,T],$ $t$ over $[0,T],$ and
$\delta[0,T]$ over the set of Lipschitz continuous functions with constant $1$
and $||\delta||\leq 1.$
The other expansions are proved in the same way.
### 3.5 Finiteness of the coefficients
First it is obvious that $K_{0}$ is finite since $H$ is bounded and
$\exp\\{Q(2,\eta)\\}\leq 1.$ For the rest, we will use Theorem 2.3.1 in [44]
to prove each finiteness. The very first requirement of Theorem 2.3.1 is that
$G^{\eta}(t,x;z)$ satisfies condition (A) in Section 2. Let us recall
$G^{\eta}(t,x;z):$
$\displaystyle G^{\eta}(t,x;z)$
$\displaystyle=zx\cdot\frac{\partial^{2}G_{0}}{\partial z\partial
x}(t,\phi_{0}(t);z_{0}(t))+\frac{1}{2}z^{2}\cdot\frac{\partial^{2}G_{0}}{\partial
z^{2}}(t,\phi_{0}(t);z_{0}(t)),$
which doesn’t satisfies condition (A) obviously because of the linear term in
$x.$ So we turn to consider the following transformation
$\widetilde{\eta}_{t}=\exp\left\\{-\int_{0}^{t}g(s)ds\right\\}\eta_{t},$
where $g(t)=\frac{\partial^{2}G_{0}}{\partial z\partial
x}(t,\phi_{0}(t);z_{0}(t)).$ Straightforward computation will give us
$\displaystyle\mathfrak{A}^{\widetilde{\eta}}f(t,x)$
$\displaystyle=\frac{\partial f}{\partial
t}(t,x)+\frac{1}{2}\frac{\partial^{2}G_{0}}{\partial
z^{2}}(t,\phi_{0}(t);z_{0}(t))\cdot\frac{\partial^{2}f}{\partial
x^{2}}(t,x)\exp\left\\{-2\int_{0}^{t}g(s)ds\right\\},$ $\displaystyle
G^{\widetilde{\eta}}(t,x;z)$
$\displaystyle=\frac{1}{2}z^{2}\cdot\frac{\partial^{2}G_{0}}{\partial
z^{2}}(t,\phi_{0}(t);z_{0}(t))\exp\left\\{-2\int_{0}^{t}g(s)ds\right\\},$
$\displaystyle H^{\widetilde{\eta}}(t,x;u)$
$\displaystyle=\frac{1}{2}\frac{\partial^{2}H_{0}}{\partial
u^{2}}(t,\phi_{0}(t);\phi_{0}^{\prime}(t))\exp\left\\{2\int_{0}^{t}g(s)ds\right\\}u^{2}.$
Now $G^{\widetilde{\eta}}(t,x;z)$ satisfies condition (A). We will apply
Theorem 2.3.1 restricting ourself to $G^{\widetilde{\eta}}(t,x;z).$ It is then
shown that each finiteness can be deduced from this.
#### 3.5.1 Finiteness of $K_{1}$
We now show an auxiliary result.
###### Lemma 3.3.
For any positive integer $j,$
$\displaystyle\mathbb{E}(||\widetilde{\eta}||^{j})<\infty.$ (3.12)
###### Proof.
The normalized action functional for the family of processes
$\sqrt{\epsilon}\cdot\widetilde{\eta}$ is
$\displaystyle
I^{\widetilde{\eta}}(f(\cdot))=\int_{0}^{T}H^{\widetilde{\eta}}(t,f(t);f^{\prime}(t))dt.$
Let us consider, for some positive $\alpha,$ positive integer $m,$
$\displaystyle\Phi_{0}^{\widetilde{\eta}}(m)=\left\\{f\in
D_{0}[0,T]:I^{\widetilde{\eta}}(f(\cdot))\leq m\right\\};$
$\displaystyle\Phi_{0}^{\widetilde{\eta}}(m)_{+\alpha\sqrt{m}}:\alpha\sqrt{m}-\text{neighbourhood
of }\Phi_{0}^{\widetilde{\eta}}(m).$
The space $D_{0}[0,T]$ decomposes into the union
$\Phi_{0}^{\widetilde{\eta}}(1)_{+\alpha}\cup\bigcup_{m=1}^{\infty}\Phi_{0}^{\widetilde{\eta}}(m+1)_{+\alpha\sqrt{m+1}}\backslash\Phi_{0}^{\widetilde{\eta}}(m)_{+\alpha\sqrt{m}},$
thus we have
$\displaystyle\mathbb{E}(||\widetilde{\eta}||^{j})$
$\displaystyle\leq\sum_{m=0}^{\infty}\mathbb{E}\left\\{\widetilde{\eta}\in\Phi_{0}^{\widetilde{\eta}}(m+1)_{+\alpha\sqrt{m+1}}\backslash\Phi_{0}^{\widetilde{\eta}}(m)_{+\alpha\sqrt{m}};||\widetilde{\eta}||^{j}\right\\}$
(3.13)
$\displaystyle\leq\sum_{m=0}^{\infty}\mathbb{P}\left\\{\widetilde{\eta}\notin\Phi_{0}^{\widetilde{\eta}}(m)_{+\alpha\sqrt{m}}\right\\}\cdot\sup\left\\{||f||^{j}:f\in\Phi_{0}^{\widetilde{\eta}}(m+1)_{+\alpha\sqrt{m+1}}\right\\}.$
We first analyze the supremum term in (3.13). Let us recall a fact that, for
any fixed $a>0,$ any integer $l>0,$ there exists a constant $A>0$ (only
depends on $a$ and $l,$ independent of $x$) such that
$\displaystyle|x|^{l}\leq Ae^{ax^{2}},\quad\text{ for all }x\in\mathbb{R}.$
(3.14)
Thus, for any $a>0,$ any positive integer $j,$ there is some $A=A(a,j)>0$ such
that
$\displaystyle||f||^{j}\leq A\exp\left\\{a||f||^{2}\right\\}.$
And for any $f\in\Phi_{0}^{\widetilde{\eta}}(m+1)_{+\alpha\sqrt{m+1}},$ we can
choose a small $a$ such that (such $a$ can be chosen independent of $m$ by
using Lemma 5.2.5 on in [44])
$a||f||^{2}\leq\frac{1}{3}(m+1).$
So the supremum term can be estimated as follows
$\displaystyle\sup\left\\{||f||^{j}:f\in\Phi_{0}^{\widetilde{\eta}}(m+1)_{+\alpha\sqrt{m+1}}\right\\}\leq
A\exp\left\\{\frac{1}{3}\cdot(m+1)\right\\}.$ (3.15)
Now we analyze the probabilities in (3.13) by using Theorem 2.3.1 in [44]. We
check all the conditions of Theorem 2.3.1 as follows. We suppose
$\sup_{t}|\frac{\partial^{2}G_{0}}{\partial
z^{2}}(t,\phi_{0}(t);z_{0}(t))\exp\left\\{-2\int_{0}^{t}g(s)ds\right\\}|\leq
c$ for some constant $c.$ Let us consider a constant $Z,$ an integer $n,$
whose values will be determined a little later. We set
$\epsilon_{2}=\frac{mZ^{2}c^{2}\kappa}{6T},t_{i}=\frac{iT}{n},\triangle
t_{\min}=\triangle t_{\max}=\frac{T}{n},k=2,$
$z(1)=\sqrt{m}Z,z(2)=-\sqrt{m}Z,$ $d(1)=d(2)=mZ^{2}c,$
$\delta^{\prime}=\frac{\alpha\sqrt{m}}{3},A=m,$
$U_{0}=\left\\{u:z(j)\cdot
u<d(j),j=1,2\right\\}=\left(-\sqrt{m}Zc,\sqrt{m}Zc\right).$
Now we define a small $\epsilon_{1}$ such that
$\epsilon_{1}(2-\epsilon_{1})+T\epsilon_{1}(3-\epsilon_{1})+\frac{Z^{2}c^{2}\kappa}{6}(1-\epsilon_{1})\leq\frac{Z^{2}c^{2}\kappa}{3}.$
Firstly we know $G^{\widetilde{\eta}}$ satisfies condition $A$ with
$\overline{G}(z)=\frac{1}{2}z^{2}c.$ Secondly $G^{\widetilde{\eta}}$ has the
property:
$\displaystyle
G^{\widetilde{\eta}}(t,y;(1-\epsilon_{1})z)\leq(1-\epsilon_{1})G^{\widetilde{\eta}}(s,x;z),$
for $t,s$ which are close enough (this can be guaranteed by choosing a large
$n$). Finally we can approximate $H^{\widetilde{\eta}}$ on $U_{0}$ by tangent
lines from below with any accuracy. More precisely, we can obviously find some
$z_{0}\\{1\\},\cdots,z_{0}\\{N\\}$ such that
$\displaystyle\sup_{u\in[-1,1]}\left(H^{\widetilde{\eta}}(t,x;u)-\max_{1\leq
j\leq
N}\left[z_{0}\\{j\\}u-G^{\widetilde{\eta}}(t,x;z_{0}\\{j\\})\right]\right)\leq\frac{\kappa}{6T}.$
For general $u\in U_{0},$ we set $z\\{j\\}=\sqrt{m}Zcz_{0}\\{j\\},1\leq j\leq
N$ (here $N$ can be chosen independent of $m$). For short, we will use
$h_{1}(t)=\frac{\partial^{2}H_{0}}{\partial
u^{2}}(t,\phi_{0}(t);\phi_{0}^{\prime}(t))\exp\\{2\int_{0}^{t}g(s)ds\\}$ and
$h_{2}(t)=\frac{\partial^{2}G_{0}}{\partial
z^{2}}(t,\phi_{0}(t);z_{0}(t))\exp\\{-2\int_{0}^{t}g(s)ds\\}.$ Then
$\displaystyle\sup_{u\in U_{0}}\left(H^{\widetilde{\eta}}(t,x;u)-\max_{1\leq
j\leq N}\left[z\\{j\\}u-G^{\widetilde{\eta}}(t,x;z\\{j\\})\right]\right)$
$\displaystyle=\sup_{u\in\left(-\sqrt{m}Zc,\sqrt{m}Zc\right)}\left(\frac{1}{2}h_{1}(t)u^{2}-\max_{1\leq
j\leq N}\left[z\\{j\\}u-\frac{1}{2}z\\{j\\}^{2}h_{2}(t)\right]\right)$
$\displaystyle=mZ^{2}c^{2}\cdot\sup_{u\in\left(-\sqrt{m}Zc,\sqrt{m}Zc\right)}\Big{(}\frac{1}{2}h_{1}(t)\left(\frac{u}{\sqrt{m}Zc}\right)^{2}-$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad-\max_{1\leq j\leq
N}\left[\frac{z\\{j\\}}{\sqrt{m}Zc}\frac{u}{\sqrt{m}Zc}-\frac{1}{2}\left(\frac{z\\{j\\}}{\sqrt{m}Zc}\right)^{2}h_{2}(t)\right]\Big{)}$
$\displaystyle=mZ^{2}c^{2}\cdot\sup_{u\in(-1,1)}\left(\frac{1}{2}h_{1}(t)u^{2}-\max_{1\leq
j\leq N}\left[z_{0}\\{j\\}u-\frac{1}{2}z_{0}\\{j\\}^{2}h_{2}(t)\right]\right)$
$\displaystyle\leq\frac{mZ^{2}c^{2}\cdot\kappa}{6T}=\epsilon_{2}.$
All conditions of Theorem 2.3.1 are thus checked. Applying this theorem with
$\delta^{\prime}\geq\frac{T}{n}\sqrt{m}Zc,$ i.e. $n\geq\frac{3ZTc}{\alpha},$
we get
$\displaystyle\mathbb{P}\left\\{\widetilde{\eta}\notin\Phi_{0}^{\widetilde{\eta}}(m)_{+\alpha\sqrt{m}}\right\\}=\mathbb{P}\left\\{\text{dist}\left(\eta^{\epsilon},\Phi_{0}^{\widetilde{\eta}}(m)\right)\geq\alpha\sqrt{m}\right\\}$
$\displaystyle\leq
4n\exp\left\\{\frac{T}{n}\left[\frac{Z^{2}cm}{2}-Z^{2}cm\right]\right\\}$
$\displaystyle\qquad\qquad+N^{n}\exp\left\\{-m+m\epsilon_{1}(2-\epsilon_{1})+T\left(m\epsilon_{1}(3-\epsilon_{1})+\frac{mZ^{2}c^{2}\kappa}{6T}(1-\epsilon_{1})\right)\right\\}$
$\displaystyle=4n\exp\left\\{-m\left[\frac{TZ^{2}c}{2n}\right]\right\\}$
$\displaystyle\qquad\qquad+N^{n}\exp\left\\{-m\left[1-\epsilon_{1}(2-\epsilon_{1})-T\epsilon_{1}(3-\epsilon_{1})-\frac{Z^{2}c^{2}\kappa}{6}(1-\epsilon_{1})\right]\right\\}$
$\displaystyle\leq
4n\exp\left\\{-m\left[\frac{TZ^{2}c}{2n}\right]\right\\}+N^{n}\exp\left\\{-m\left(1-\frac{Z^{2}c^{2}\kappa}{3}\right)\right\\},\text{
definition of }\epsilon_{1}$
$\displaystyle=4n\exp\left\\{-m\right\\}+N^{n}\exp\left\\{-m\left(1-\frac{Z^{2}c^{2}\kappa}{3}\right)\right\\},\text{
choose }Z=\sqrt{\frac{2n}{Tc}},n\geq\frac{18T}{\alpha^{2}c}.$
Noticing that $\frac{Z^{2}c^{2}\kappa}{3}$ can be as small as possible by
choosing a small $\kappa,$ we thus assume $1-\frac{Z^{2}c^{2}\kappa}{3}>1/2.$
Then it follows
$\mathbb{P}\left\\{\widetilde{\eta}\notin\Phi_{0}^{\widetilde{\eta}}(m)_{+\alpha\sqrt{m}}\right\\}\leq
4n\exp\left\\{-m\right\\}+N^{n}\exp\left\\{-\frac{m}{2}\right\\}.$
Let us now go back to (3.13) combining above estimate and (3.15)
$\displaystyle\mathbb{E}(||\widetilde{\eta}||^{j})$
$\displaystyle\leq\sum_{m=0}^{\infty}\left(4n\exp\left\\{-m\right\\}+N^{n}\exp\left\\{-\frac{m}{2}\right\\}\right)A\exp\left\\{\frac{1}{3}\cdot(m+1)\right\\}$
$\displaystyle\leq
Ae^{1/3}(4n+N^{n})\sum_{m=0}^{\infty}\exp\left\\{-\frac{m}{6}\right\\}<\infty.$
∎
By observing the transformation
$\widetilde{\eta}_{t}=\exp\left\\{-\int_{0}^{t}g(s)ds\right\\}\eta_{t},$ we
immediately derive
$\displaystyle\mathbb{E}(||\eta||^{j})\leq
Be^{1/3}(4n+N^{n})\sum_{m=0}^{\infty}\exp\left\\{-\frac{m}{6}\right\\}<\infty.$
It is clear that $K_{1}$ can be bounded by expectation of
$c_{1}+c_{2}\cdot||\eta||^{j}$ for some $j,c_{1}$ and $c_{2},$ from which
finiteness of $K_{1}$ follows according to Lemma 3.3.
#### 3.5.2 Finiteness of the rest of the coefficients
Since all derivatives of $F$ and $H$ are bounded symmetric linear functionals,
we can use Lemma 3.3 to prove the finiteness of the rest of the coefficients.
## 4 Connections with partial integro-differential equations
The connections are between large (or normal) deviations and solutions to
$\begin{cases}\begin{aligned} \frac{\partial}{\partial
t}u^{\epsilon}(t,x)=&\frac{\epsilon}{2}a(t,x)\Delta
u^{\epsilon}(t,x)+b(t,x)\nabla
u^{\epsilon}(t,x)+\epsilon^{-1}c(x)u^{\epsilon}(t,x)\\\
&+\epsilon^{-1}\int_{\mathbb{R}}\left[u^{\epsilon}(t,x+\epsilon
u)-u^{\epsilon}(t,x)-\epsilon u\nabla
u^{\epsilon}(t,x)\right]\nu_{t,x}(du)\end{aligned}\\\
u^{\epsilon}(0,x)=g(x)\end{cases}$ (4.1)
over $(t,x)\in\mathbb{R}_{+}\times\mathbb{R}.$ More precisely, it is expected
that
$u^{\epsilon}(t,x)=\mathbb{E}^{\epsilon}_{0,x}\left[g(\xi^{\epsilon}_{t})\exp\left\\{\epsilon^{-1}\int_{0}^{t}c(\xi^{\epsilon}_{s})ds\right\\}\right],$
(4.2)
then the precise asymptotics for large deviations (or normal deviations)
developed in Theorem 2.1 (or Theorem 3.1) can be applied. Of course, formula
(4.2) is not always true unless suitable conditions are imposed. In the first
part of this section, we prove (4.2) for a special case when $c(x)$ is a
constant. Then from Theorem 2.1, it follows
$u^{\epsilon}(t,x)=e^{t/{\epsilon}}\cdot\left[\sum_{k=0}^{n}k_{i}(x)\epsilon^{k/2}+o(\epsilon^{n/2})\right].$
The second part of this section is on the study of precise asymptotics of
$u^{\epsilon}(t,x)$ in more general settings.
### 4.1 The specific case
In (4.1), we set $c=1$ and $\nu_{t,x}(du)=u^{2}1_{\\{|u|\leq 1\\}}(du).$ What
is more, we assume $a(t,x)=a(x),$ $b(t,x)=b(x),$ and
$0<\inf_{x}a(x)\leq\sup_{x}a(x)<\infty,$ the smooth functions $a(x),b(x)$ and
$g(x)$ are bounded together with their derivatives $d^{j}a/dx^{j},$
$d^{j}b/dx^{j}$ and $d^{j}g/dx^{j}.$ In this case, we consider a family of
jump processes $\xi^{\epsilon}$ with generating operators
$A^{\epsilon}f(x)=\frac{\epsilon}{2}a(x)f^{\prime\prime}(x)+b(x)f^{\prime}(x)+\epsilon^{-1}\int_{-1}^{1}\left[f(x+\epsilon
u)-f(x)\right]u^{2}du$
for continuous bounded $f$ together with its first and second derivatives.
From the theory of semigroups, the function
$v^{\epsilon}(t,x):=\mathbb{E}^{\epsilon}_{0,x}f(\xi^{\epsilon}_{t})$
is the unique solution to the problem, for $f$ in the domain of
$A^{\epsilon},$
$\begin{cases}\begin{aligned} \frac{\partial}{\partial
t}v^{\epsilon}(t,x)=A^{\epsilon}v^{\epsilon}(t,x),\end{aligned}\\\
v^{\epsilon}(0,x)=f(x).\end{cases}$
Now it is easy to see that
$u^{\epsilon}(t,x):=e^{t/{\epsilon}}\cdot\mathbb{E}^{\epsilon}_{0,x}f(\xi^{\epsilon}_{t})$
is the unique solution of
$\begin{cases}\begin{aligned} \frac{\partial}{\partial
t}u^{\epsilon}(t,x)=A^{\epsilon}u^{\epsilon}(t,x)+\epsilon^{-1}u^{\epsilon}(t,x),\end{aligned}\\\
u^{\epsilon}(0,x)=f(x).\end{cases}$
The conditions imposed on $a,b$ and $g$ are mainly for the smooth and growth
assumptions in Theorem 2.1, such as (F) and (2.2). The condition $c(x)=1$ is
crucial in this special case since it forces the $\max[F-S]$ is reached
uniquely at $\phi_{0}\equiv 0.$ This example should be considered as the
asymptotics for normal (not large) deviations since the main part of the
integral (4.2) is due to the most probable sample path (which is identically
zero). Asymptotics for large (not normal) deviations can be seen below.
### 4.2 In more general settings
Let $\xi^{\epsilon}$ now be the locally infinitely divisible family of
processes considered in Theorem 2.1 satisfying all the assumptions. Then the
corresponding partial integro-differential equation is (4.1) with $b(t,x)$
replaced by $\alpha(t,x).$ In order to show (4.2), it is natural to impose
suitable conditions on two new functions $c(x)$ and $g(x).$ What is more, more
conditions on the processes are also expected. This leads to a theorem
borrowed from [42].
###### Theorem 4.1 (Section 10.3 in [42]).
Let $\xi^{\epsilon}$ be uniformly stochastically continuous, the function
$g(x)$ be in the domain of the generating operator $A^{\epsilon}_{t},$ and the
function $c(x)$ be bounded uniformly continuous. Then the function given by
(4.2) is the unique solution of (4.1).
For (4.2), we need further assumptions in order to apply Theorem 2.1. For
instance, it is assumed that $\max\left[\int_{0}^{t}c(\phi(s))-H_{0}ds\right]$
is reached uniquely at non-zero $\phi_{0}$.
## 5 Appendix
### 5.1 Compensating operators after transformations
In preceding sections, we presented many compensating operators after
transformations without any proofs. In order to show the method, we give the
details for deriving the compensating operator appeared in Section 3.5. There,
Gaussian process $\eta$ was considered with compensating operator
$\displaystyle\mathfrak{A}^{\eta}f(t,x)=\frac{\partial f}{\partial
t}(t,x)+x\cdot\frac{\partial^{2}G_{0}}{\partial z\partial x}(t,\phi_{0}(t);$
$\displaystyle z_{0}(t))\cdot\frac{\partial f}{\partial
x}(t,x)+\frac{1}{2}\frac{\partial^{2}G_{0}}{\partial
z^{2}}(t,\phi_{0}(t);z_{0}(t))\cdot\frac{\partial^{2}f}{\partial x^{2}}(t,x).$
for $f(t,x)$ which is bounded and continuous together with its first
derivatives in $t$ and $x$ and its second derivative in $x.$ The following
transformation was used
$\widetilde{\eta}_{t}=\exp\left\\{-\int_{0}^{t}g(s)ds\right\\}\eta_{t},\quad
t\in[0,T].$
After such a transformation, we give the details in this section that
$\widetilde{\eta}$ has compensating operator given by
$\displaystyle\mathfrak{A}^{\widetilde{\eta}}f(t,x)$
$\displaystyle=\frac{\partial f}{\partial
t}(t,x)+\frac{1}{2}\frac{\partial^{2}G_{0}}{\partial
z^{2}}(t,\phi_{0}(t);z_{0}(t))\cdot\frac{\partial^{2}f}{\partial
x^{2}}(t,x)\exp\left\\{-2\int_{0}^{t}g(s)ds\right\\}$
for the same class of functions $f.$
The generating operator $A^{\eta}_{t}$ of process $\eta$ is
$A^{\eta}_{t}f(x)=\mathfrak{A}^{\eta}f(t,x)$ for $f(t,x)=f(x).$ Let us assume
the starting position of process $\eta$ is $\eta_{s}=x.$ From definitions of
compensating operator and generating operator, we have the following two
equalities:
$\displaystyle P^{s,t}_{\eta}f(x)-f(x)$
$\displaystyle=\int_{s}^{t}P^{s,v}_{\eta}A^{\eta}_{v}f(x)dv,$ $\displaystyle
P^{s,t}_{\eta}f(t,\cdot)(x)-f(s,x)$
$\displaystyle=\int_{s}^{t}P^{s,v}_{\eta}\mathfrak{A}^{\eta}f(v,\cdot)(x)dv,$
for suitable $f(x)$ and $f(t,x),$ where $P^{s,t}_{\eta}$ is the multiplicative
family of operators of Markov process $\eta$ given by
$P^{s,t}_{\eta}f(x)=\mathbb{E}^{\eta}_{s,x}f(\eta_{t}).$ The following two
connections between multiplicative families of $\eta$ and $\widetilde{\eta}$
are easily derived:
$\displaystyle P^{s,t}_{\eta}f(x)$
$\displaystyle=\mathbb{E}^{\eta}_{s,x}f(\eta_{t})=\mathbb{E}^{\widetilde{\eta}}_{s,x\exp\\{-\int_{0}^{s}g(u)du\\}}f\left(\widetilde{\eta}_{t}\exp\\{\int_{0}^{t}g(u)du\\}\right)$
$\displaystyle=P^{s,t}_{\widetilde{\eta}}G(t,\cdot)\left(x\exp\\{-\int_{0}^{s}g(u)du\\}\right),\quad
G(t,x)=f\left(x\exp\\{\int_{0}^{t}g(u)du\\}\right),$ $\displaystyle
P^{s,t}_{\widetilde{\eta}}f(x)$
$\displaystyle=\mathbb{E}^{\widetilde{\eta}}_{s,x}f(\widetilde{\eta}_{t})=\mathbb{E}^{\eta}_{s,x\exp\\{\int_{0}^{s}g(u)du\\}}f\left(\eta_{t}\exp\\{-\int_{0}^{t}g(u)du\\}\right)$
$\displaystyle=P^{s,t}_{\eta}F(t,\cdot)\left(x\exp\\{\int_{0}^{s}g(u)du\\}\right),\quad
F(t,x)=f\left(x\exp\\{-\int_{0}^{t}g(u)du\\}\right).$
Now we look for $A_{t}^{\widetilde{\eta}}$ in the following way.
$\displaystyle P^{s,t}_{\widetilde{\eta}}f(x)-f(x)$
$\displaystyle=P^{s,t}_{\eta}F(t,\cdot)\left(x\exp\\{\int_{0}^{s}g(u)du\\}\right)-F\left(s,x\exp\\{\int_{0}^{s}g(u)du\\}\right)$
$\displaystyle=\int_{s}^{t}P^{s,v}_{\eta}\mathfrak{A}^{\eta}F(v,\cdot)\left(x\exp\\{\int_{0}^{s}g(u)du\\}\right)dv$
$\displaystyle=\int_{s}^{t}P^{s,v}_{\eta}g(v,\cdot)\left(x\exp\\{\int_{0}^{s}g(u)du\\}\right)dv,\text{
set }h(v,x)=\mathfrak{A}^{\eta}F(v,\cdot)(x)$
$\displaystyle=\int_{s}^{t}P^{s,v}_{\widetilde{\eta}}\widetilde{h}(v,\cdot)(x)dv,\text{
where }\widetilde{h}(v,x)=h\left(v,x\exp\\{\int_{0}^{v}g(u)du\\}\right).$
It is straightforward to compute
$h(v,x)=\frac{1}{2}\frac{\partial^{2}G_{0}}{\partial
z^{2}}(v,\phi_{0}(v);z_{0}(v))\cdot
f^{\prime\prime}\left(x\exp\\{-\int_{0}^{v}g(u)du\\}\right)\cdot\exp\left\\{-2\int_{0}^{v}g(u)du\right\\},$
thus
$\displaystyle A^{\widetilde{\eta}}_{t}g(x)$
$\displaystyle=\widetilde{h}(t,x)=h\left(t,x\exp\\{\int_{0}^{t}g(u)du\\}\right)$
$\displaystyle=\frac{1}{2}\frac{\partial^{2}G_{0}}{\partial
z^{2}}(t,\phi_{0}(t);z_{0}(t))\cdot
f^{\prime\prime}\left(x\right)\cdot\exp\left\\{-2\int_{0}^{v}g(u)du\right\\}.$
### 5.2 Proof of Lemma 3.1
###### Proof.
The same conclusion with a continuous and bounded functional $F$ was given in
[44] without a proof. For completeness, we first present the proof for
continuous and bounded $F$ and then extend the argument to include the
continuous functional $F$ which is only bounded above.
We take
$\gamma=\left[F(\phi_{0})-S(\phi_{0})-\max_{\|(\phi_{0}-\phi\|\geq\delta}[F(\phi)-S(\phi)]\right]/2.$
Here the $\max_{\|(\phi_{0}-\phi\|\geq\delta}[F(\phi)-S(\phi)]$ is reached at
some point $\phi_{1}\in X.$ To see this, we note that
$\sup_{\|(\phi_{0}-\phi\|\geq\delta}[F(\phi)-S(\phi)]=\sup_{\phi\in
A}[F(\phi)-S(\phi)]$
where
$A=\left\\{\phi\in D_{0}[0,T]:\|(\phi_{0}-\phi\|\geq\delta\text{ and
}S(\phi)\leq\sup_{x\in X}F(x)-[F(\tilde{\phi})-S(\tilde{\phi})]\right\\}$
given a fixed $\tilde{\phi}$ such that $\|\phi_{0}-\tilde{\phi}\|\geq\delta$
and $\left|F(\tilde{\phi})-S(\tilde{\phi})\right|<\infty.$ The compactness of
$A$ implies that $\sup_{\phi\in A}[F(\phi)-S(\phi)]$ attains its maximum at
some $\phi_{1}.$
We first assume that $F$ is bounded, thus $F$ can be split into finitely many
parts as
$F(x)\in\bigcup_{i=-k}^{k}[i\gamma/4,(i+1)\gamma/4].$
It then follows
$\displaystyle\int_{\left\\{\|\xi^{\epsilon}-\phi_{0}\|\geq\delta\right\\}}\exp\left\\{F(\xi^{\epsilon})/\epsilon\right\\}d\,\mathbb{P}^{\epsilon}$
$\displaystyle\leq\sum_{i=-k}^{k}\int_{\left\\{\|\xi^{\epsilon}-\phi_{0}\|\geq\delta,F(\xi^{\epsilon})\in[i\gamma/4,(i+1)\gamma/4]\right\\}}\exp\left\\{F(\xi^{\epsilon})/\epsilon\right\\}d\,\mathbb{P}^{\epsilon}$
$\displaystyle\leq\sum_{i=-k}^{k}\exp\left\\{(i+1)\gamma/(4\epsilon)\right\\}\cdot\mathbb{P}^{\epsilon}\left\\{\|\xi^{\epsilon}-\phi_{0}\|\geq\delta,F(\xi^{\epsilon})\in[i\gamma/4,(i+1)\gamma/4]\right\\}.$
On each set
$\left\\{\|\xi^{\epsilon}-\phi_{0}\|\geq\delta,F(\xi^{\epsilon})\in[i\gamma/4,(i+1)\gamma/4]\right\\},$
$S(\xi^{\epsilon})\geq F(\xi^{\epsilon})-[F(\phi_{1})-S(\phi_{1})]\geq
i\gamma/4-[F(\phi_{1})-S(\phi_{1})],$
then according to large deviation principle, for small enough $\epsilon,$
$\displaystyle\mathbb{P}^{\epsilon}\left\\{\|\xi^{\epsilon}-\phi_{0}\|\geq\delta,F(\xi^{\epsilon})\in[i\gamma/4,(i+1)\gamma/4]\right\\}$
$\displaystyle\leq\exp\left\\{-\inf_{\left\\{\phi:\left\\{\|\xi^{\epsilon}-\phi_{0}\|\geq\delta,F(\phi)\in[i\gamma/4,(i+1)\gamma/4]\right\\}\right\\}}S(\phi)/\epsilon+\gamma/(4\epsilon)\right\\}$
$\displaystyle\leq\exp\left\\{[F(\phi_{1})-S(\phi_{1})]/\epsilon-i\gamma/(4\epsilon)+\gamma/(4\epsilon)\right\\}.$
Therefore,
$\displaystyle\int_{\left\\{\|\xi^{\epsilon}-\phi_{0}\|\geq\delta\right\\}}\exp\left\\{F(\xi^{\epsilon})/\epsilon\right\\}d\,\mathbb{P}^{\epsilon}$
$\displaystyle\leq\sum_{i=-k}^{k}\exp\left\\{(i+1)\gamma/(4\epsilon)\right\\}\cdot\exp\left\\{[F(\phi_{1})-S(\phi_{1})]/\epsilon-i\gamma/(4\epsilon)+\gamma/(4\epsilon)\right\\}$
$\displaystyle=\sum_{i=-k}^{k}\exp\left\\{\frac{1}{\epsilon}\left[F(\phi_{0})-S(\phi_{0})-\frac{3\gamma}{2}\right]\right\\}=o\left(\exp\left\\{[F(\phi_{0})-S(\phi_{0})-\gamma]/\epsilon\right\\}\right).$
Now we assume $F$ to be bounded above, i.e., $M:=\sup_{x\in
D_{0}[0,T]}F(x)<\infty.$ Let us define a sequence of truncated functionals
$G^{N}$ as follows
$\displaystyle G^{N}(x)=\begin{cases}F(x)&\text{ if }F(x)\geq-N,\\\ -N&\text{
if }F(x)<-N.\end{cases}$
Taking into account the fact that $F\leq G,$ it is clear that the proof is
complete if we can prove $G^{N}-S$ attains its maximum uniquely at $\phi_{0}$
for large $N$ given the condition that $F-S$ attains its maximum uniquely at
$\phi_{0}.$ We will argue this by contradiction. Suppose for every large $N,$
there is a point $\phi_{N}\in X$ different from $\phi_{0}$ such that
$\displaystyle G^{N}(\phi_{N})-S(\phi_{N})\geq
G^{N}(\phi_{0})-S(\phi_{0})=F(\phi_{0})-S(\phi_{0})\text{ for large }N.$ (5.1)
Noticing that
$\sup_{\phi\in X}[G^{N}(\phi)-S(\phi)]=\sup_{\phi\in B}[G^{N}(\phi)-S(\phi)]$
for a compact set $B=\left\\{\phi\in X:S(\phi)\leq
M-[F(\phi_{0})-S(\phi_{0})]\right\\},$ we have $\\{\phi_{N}\\}\subseteq B.$
Then there must be a limiting point $\widehat{\phi}$ of a subsequence of
$\\{\phi_{N}\\}$ (we still denote the subsequence as $\phi_{N}$),
$\lim_{N\rightarrow\infty}\phi_{N}=\widehat{\phi}.$
From (5.1), it follows
$F(\phi_{N})\leq-N$
because of the uniqueness of the maximizer of $F-S.$ Now we take the limit
$F(\widehat{\phi})=\lim_{N\rightarrow\infty}F(\phi_{N})\leq\lim_{N\rightarrow\infty}-N=-\infty,$
which is impossible. ∎
### 5.3 On a variational problem
In this section, we show the existence and uniqueness of the variational
problem of Example 1 in Section 2.2 (note that $T=1$ in Example 1)
$\max_{\phi\in
C_{0}^{1}[0,T]}\int_{0}^{T}\left[\phi(t)-\phi(t)^{2}-\left(\phi^{\prime}(t)\ln\left(\phi^{\prime}(t)+\sqrt{\phi^{\prime}(t)^{2}+1}\right)+1-\sqrt{\phi^{\prime}(t)^{2}+1}\right)\right]dt.$
The proof of uniqueness of our problem is standard and is included in Section
5.3.1. For existence, many references deal with problems having two fixed
boundaries and satisfying coercivity assumption (see (5.7) in Section 5.3.2),
our problem fails to meet these two requirements. A proof for the existence is
given in Section 5.3.2 mainly based on nice properties of the functional
$F(\phi)-S(\phi)$ and the analysis on absolutely continuous function space.
#### 5.3.1 Uniqueness
For short, let us define,
$\displaystyle H(u)=u\ln\left(u+\sqrt{u^{2}+1}\right)+1-\sqrt{u^{2}+1},$ (5.2)
$\displaystyle
v(\phi)=\int_{0}^{T}\left[\phi(t)-\phi(t)^{2}-H(\phi^{\prime}(t))\right]dt.$
(5.3)
Let $f(x,y)=H(y)+x^{2}-x,$ then the variational problem becomes
$\displaystyle\alpha=\max_{\phi\in
C_{0}^{1}[0,T]}\int_{0}^{T}\left[\phi(t)-\phi(t)^{2}-H(\phi^{\prime}(t))\right]dt=-\min_{\phi\in
C_{0}^{1}[0,T]}\int_{0}^{T}f(\phi(t),\phi^{\prime}(t))dt.$ (5.4)
Now suppose $\phi_{1}$ and $\phi_{2}$ are two minimizers of problem (5.4). Let
$w(t)=[\phi_{1}(t)+\phi_{2}(t)]/2,$ then on one hand,
$\int_{0}^{T}f(w(t),w^{\prime}(t))dt\geq-\alpha;$ on the other hand, convexity
of $f$ yields
$\displaystyle\int_{0}^{T}f(w(t),w^{\prime}(t))dt$
$\displaystyle=\int_{0}^{T}f\left(\frac{1}{2}(\phi_{1}(t),\phi_{1}^{\prime}(t))+\frac{1}{2}(\phi_{2}(t),\phi_{2}^{\prime}(t))\right)dt$
$\displaystyle\leq\frac{1}{2}\int_{0}^{T}f(\phi_{1}(t),\phi_{1}^{\prime}(t))dt+\frac{1}{2}\int_{0}^{T}f(\phi_{2}(t),\phi_{2}^{\prime}(t))dt=-\alpha,$
which indicates that $w(t)$ is also a minimizer of (5.4). From equality
$\displaystyle\int_{0}^{T}\left[\frac{1}{2}f(\phi_{1}(t),\phi_{1}^{\prime}(t))+\frac{1}{2}f(\phi_{2}(t),\phi_{2}^{\prime}(t))-f(w(t),w^{\prime}(t))\right]dt=-\frac{1}{2}\alpha-\frac{1}{2}\alpha+\alpha=0$
(5.5)
where the integrand of (5.5) is always nonpositive (from convexity of $f$), we
have
$\frac{1}{2}f(\phi_{1}(t),\phi_{1}^{\prime}(t))+\frac{1}{2}f(\phi_{2}(t),\phi_{2}^{\prime}(t))=f(w(t),w^{\prime}(t)),\quad\text{
for all }t\in[0,T].$
Rewrite above identity as follows
$\displaystyle\frac{1}{2}\phi_{1}^{2}(t)+\frac{1}{2}\phi_{2}^{2}(t)-\left(\frac{\phi_{1}(t)+\phi_{2}(t)}{2}\right)^{2}$
$\displaystyle=H(\frac{\phi_{1}^{\prime}(t)+\phi_{2}^{\prime}(t)}{2})-\left(\frac{1}{2}H(\phi_{1}^{\prime}(t))+\frac{1}{2}H(\phi_{2}^{\prime}(t))\right).$
(5.6)
If there were a point $t_{0}\in[0,T]$ such that
$\phi_{1}(t_{0})\neq\phi_{2}(t_{0}),$ then left hand side of (5.6) would be
strictly $<0$ (which is from strict convexity of function $x^{2}$), while the
right hand side is always $\geq 0$ from convexity of $H.$ This contradiction
proves that (5.4) has at most one minimizer.
#### 5.3.2 Existence
When one deals with the existence of variational problems, the following
coercivity condition is in general assumed: for all $p,z\in R,$
$\displaystyle
H(p)-(z-z^{2})\geq\alpha|p|^{q}-\beta,\qquad\exists\,\,\alpha>0,\beta\geq
0,q>1.$ (5.7)
(see Section 8.2 in [20], or see [40] for the case $q=2$). But this condition
is not satisfied for our problem since
$\lim_{|p|\rightarrow\infty}\frac{H(p)-(z-z^{2})}{|p|^{q}}=0$ for any fixed
$z.$ What is more, calculus of variations in references were given in general
with two fixed boundaries: $\phi(0)=A$ and $\phi(T)=B.$ But our problem has
one movable boundary $\phi(T).$
We define a space $\text{AC}_{0}[0,T]$ consisting of absolutely continuous
functions on $[0,T]$ vanishing at zero:
$\text{AC}_{0}[0,T]=\left\\{f:[0,T]\rightarrow\mathbb{R}\text{ being
absolutely continuous with }f(0)=0\right\\}.$
The existence of our variational problem is solved in the following way. We
first prove the existence for $\max_{\phi\in\text{AC}_{0}[0,T]}v(\phi),$ which
implies that this variational problem coincides with a two fixed boundary
problem
$\mathop{\max_{\phi\in\text{AC}_{0}[0,T]}}_{\phi(T)=c}v(\phi),\text{ for some
}c.$
Then we show $C^{1}$ regularity of the maximizer by means of two fixed
boundary variational results, which immediately implies the existence of
$\max_{\phi\in C_{0}^{1}[0,T]}v(\phi).$
#### Existence of $\max_{\phi\in\text{AC}_{0}[0,T]}v(\phi)$
Obviously, we can find some $\phi_{*}\in\text{AC}_{0}[0,T]$ with
$|v(\phi_{*})|<T/4$ (for instance $\phi_{*}\equiv 0$). We define a subset
$\mathcal{A}$ of $\text{AC}_{0}[0,T],$
$\mathcal{A}=\left\\{\phi\in\text{AC}_{0}[0,T]:\int_{0}^{T}H(\phi^{\prime}(t))dt\leq\frac{T}{4}-v(\phi_{*})\right\\}.$
Then
$\displaystyle\sup_{\phi\in\text{AC}_{0}[0,T]}v(\phi)=\sup_{\phi\in\mathcal{A}}v(\phi).$
(5.8)
To see (5.8), we notice that for any $\phi\notin\mathcal{A},$
$\int_{0}^{T}H(\phi^{\prime}(t))dt>\frac{T}{4}-v(\phi_{*}),\text{ then
}v(\phi_{*})>\frac{T}{4}-\int_{0}^{T}H(\phi^{\prime}(t))dt\geq v(\phi).$
Let us write
$\alpha=\sup_{\phi\in\mathcal{A}}v(\phi),$
and let $\\{\phi_{n}(t)\\}_{n\geq 1}\subseteq\mathcal{A}$ be chosen such that
$\displaystyle\lim_{n\rightarrow\infty}v(\phi_{n})=\alpha,\text{ and
}\lim_{n\rightarrow\infty}\max_{0\leq t\leq
T}|\phi_{n}(t)-\phi_{0}(t)|=0\text{ for some }\phi_{0}\in\text{AC}_{0}[0,T].$
The reason why we can choose such a sequence $\phi_{n}$ is from the fact that
$\mathcal{A}$ is compact in $\text{AC}_{0}[0,T]$ according to the following
Lemma 5.1 (after passing to a subsequence). We now show $v(\phi_{0})=\alpha.$
In fact, $v(\phi_{0})\leq\alpha$ is trivial. Lower semi-continuity of
$-v(\cdot)$ in Lemma 5.1 gives
$v(\phi_{0})\geq\limsup_{n\rightarrow\infty}v(\phi_{n})=\alpha.$
So (5.8) can be rewritten as
$\displaystyle\max_{\phi\in\text{AC}_{0}[0,T]}v(\phi)=\max_{\phi\in\mathcal{A}}v(\phi),$
which proves the existence of $\max_{\phi\in\text{AC}_{0}[0,T]}v(\phi).$
###### Lemma 5.1.
$\mathcal{A}$ defined above is compact in $\text{AC}_{0}[0,T]$ and $-v(\phi)$
is lower semi-continuous in $\text{AC}_{0}[0,T]$ in uniform topology.
###### Proof.
We will finish the proof in several steps. First we show $\mathcal{A}$ is an
absolutely euqicontinuous family of functions: for any $\epsilon>0,$ there is
$\delta(\epsilon)>0$ such that whenever finitely many non-overlapping
intervals $\sum_{i}(t_{i}-s_{i})\leq\delta,$ then
$\displaystyle\sum_{i}|\phi(t_{i})-\phi(s_{i})|<\epsilon,\qquad\forall\,\,\phi\in\mathcal{A}.$
(5.9)
To see (5.9), first we have a nice property for $H:$
$\displaystyle
H(p)\geq|p|\cdot\ln\left(|p|+\sqrt{p^{2}+1}\right)+1-\sqrt{2}|p|-\sqrt{2},\qquad\text{
for all }p\in\mathbb{R}.$ (5.10)
Then $\lim_{|p|\rightarrow\infty}H(p)/|p|=\infty,$ so there is some
$P(\epsilon)>0,$ such that when $|p|>P,$
$H(p)/|p|\geq 2\left[\frac{1}{4}-v(\phi_{*})\right]/{\epsilon}.$
Let $\delta(\epsilon)=\epsilon/(2P),$ then for all $\phi\in\mathcal{A},$
$\displaystyle\frac{1}{4}-v(\phi_{*})$
$\displaystyle\geq\int_{0}^{T}H(\phi^{\prime}(t))dt\geq\sum_{i}\int_{s_{i}}^{t_{i}}H(\phi^{\prime}(t))dt\geq\sum_{i}\int_{s_{i}}^{t_{i}}\frac{H(\phi^{\prime}(t))}{|\phi^{\prime}(t)|}|\phi^{\prime}(t)|1_{\\{|\phi^{\prime}(t)|>P\\}}(t)dt$
$\displaystyle\geq\sum_{i}\int_{s_{i}}^{t_{i}}|\phi^{\prime}(t)|1_{\\{|\phi^{\prime}(t)|>P\\}}(t)dt\cdot
2\left[\frac{1}{4}-v(\phi_{*})\right]/{\epsilon},$
so
$\displaystyle\epsilon/2\geq\sum_{i}\int_{s_{i}}^{t_{i}}|\phi^{\prime}(t)|1_{\\{|\phi^{\prime}(t)|>P\\}}(t)dt=\sum_{i}\int_{s_{i}}^{t_{i}}|\phi^{\prime}(t)|dt-\sum_{i}\int_{s_{i}}^{t_{i}}|\phi^{\prime}(t)|1_{\\{|\phi^{\prime}(t)|\leq
P\\}}(t)dt,$ $\displaystyle\Rightarrow\,\,\,$
$\displaystyle\sum_{i}|\phi(t_{i})-\phi(s_{i})|\leq\sum_{i}\int_{s_{i}}^{t_{i}}|\phi^{\prime}(t)|dt\leq\epsilon/2+\sum_{i}\int_{s_{i}}^{t_{i}}|\phi^{\prime}(t)|1_{\\{|\phi^{\prime}(t)|\leq
P\\}}(t)dt$ $\displaystyle\leq\epsilon/2+\epsilon/2=\epsilon.$
The second step is to prove lower semi-continuity of $-v(\cdot).$ Let
$\phi_{n}\in\text{AC}_{0}[0,T]$ be a family of absolutely continuous functions
such that $\max_{0\leq t\leq T}|\phi_{n}(t)-\phi_{\infty}(t)|\rightarrow 0$ as
$n\rightarrow\infty.$ It turns out that $\phi_{\infty}$ is also absolutely
continuous. More precisely, according to absolute equicontinuity (5.9), for
any $\epsilon>0,$ there is $\delta(\epsilon)>0$ such that if
$\sum_{i}(t_{i}-s_{i})<\delta,$ then
$\sup_{n}\sum_{i}|\phi_{n}(t_{i})-\phi_{n}(s_{i})|<\epsilon.$ Sending
$n\rightarrow\infty$ we get
$\sum_{i}|\phi_{\infty}(t_{i})-\phi_{\infty}(s_{i})|<\epsilon,$ which proves
the absolute continuity of $\phi_{\infty}.$ Now we show the lower semi-
continuity of $\int_{0}^{T}H(\phi(t))dt.$ Let $0=t_{0}<t_{1}\cdots<t_{k}=T,$
Jensen’s inequality implies
$\displaystyle\liminf_{n\rightarrow\infty}\int_{0}^{T}H(\phi_{n}^{\prime}(t))dt=\liminf_{n\rightarrow\infty}\sum_{i=0}^{k-1}\int_{t_{i}}^{t_{i+1}}H(\phi_{n}^{\prime}(t))dt$
$\displaystyle\geq\liminf_{n\rightarrow\infty}\sum_{i=0}^{k-1}(t_{i+1}-t_{i})H\left(\frac{\phi_{n}(t_{i+1})-\phi_{n}(t_{i})}{t_{i+1}-t_{i}}\right)=\sum_{i=0}^{k-1}(t_{i+1}-t_{i})H\left(\frac{\phi_{\infty}(t_{i+1})-\phi_{\infty}(t_{i})}{t_{i+1}-t_{i}}\right)$
$\displaystyle=\sum_{i=0}^{k-1}\int_{t_{i}}^{t_{i+1}}H\left(\Psi(t)\right)dt,\quad\text{where
}\Psi(t)=\frac{\phi_{\infty}(t_{i+1})-\phi_{\infty}(t_{i})}{t_{i+1}-t_{i}}\text{
for }t_{i}\leq t<t_{i+1}$ $\displaystyle=\int_{0}^{T}H\left(\Psi(t)\right)dt.$
Now let a sequence $\triangle_{m}$ of partitions be infinitely small, then the
corresponding functions $\Psi_{m}(t)$ converge to $\phi_{\infty}^{\prime}(t)$
almost everywhere (because of absolute continuity of $\phi_{\infty}$). Using
continuity of $H$ and Fatou’s lemma we get
$\int_{0}^{T}H\left(\phi_{\infty}^{\prime}(t)\right)dt\leq\liminf_{n\rightarrow\infty}\int_{0}^{T}H(\phi_{n}^{\prime}(t))dt,$
which gives us the lower semi-continuity of $\int_{0}^{T}H(\phi(t))dt.$ Then
the lower semi-continuity of $-v(\cdot)$ is from lower semi-continuity of
$\int_{0}^{T}H(\phi(t))dt.$
The last step will present the compactness of $\mathcal{A}$ in
$\text{AC}_{0}[0,T].$ Lower semi-continuity of $\int_{0}^{T}H(\phi(t))dt$
shows $\mathcal{A}$ is closed in $\text{AC}_{0}[0,T].$ What’s more, the
equicontinuity in step one and the fact all functions in $\mathcal{A}$ have
zero initial value imply that $\mathcal{A}$ is pre-compact in $C_{0}[0,T],$
thus $\mathcal{A}$ is compact in $\text{AC}_{0}[0,T].$ ∎
#### $C^{1}$ regularity of a maximizer of
$\max_{\phi\in\text{AC}_{0}[0,T]}v(\phi)$
Let us consider two fixed boundaries problem as follows
$\displaystyle
g(c):=\mathop{\max_{\phi\in\text{AC}_{0}[0,T]}}_{\phi(T)=c}v(\phi).$
First we note that $g(c)$ is well defined because of the existence of a
maximizer of $v(\phi)$ under restrictions $\phi\in\text{AC}_{0}[0,T]$ and
$\phi(T)=c.$ Clarke and Vinter in their paper [9] showed several powerful
regularity theorems under pretty mild hypotheses by using nonsmooth analysis.
More precisely, Clarke and Vinter in [9] considered the basic problem in the
calculus of variation, which is to minimize
$J(\phi):=\int_{0}^{T}-L(\phi(t),\phi^{\prime}(t))dt$
over the class of absolutely continuous functions $\phi$ having two fixed
boundaries $\phi(0)=A$ and $\phi(T)=B.$ Assuming $L$ satisfies suitable
conditions, they proved that if there is a $\phi(t)$ solving the variational
problem, then at every point $t\in[0,T],$ the function $\phi$ is $C^{\infty}$
in a neighborhood of $t,$ see Theorem 2.1 and Corollary 3.1 in [9]. Our
functional $\phi(t)-\phi(t)^{2}-H(\phi^{\prime}(t))$ has nice properties which
make it satisfy all the hypotheses in [9], so we immediately deduce the
$C^{1}$ regularity (actually $C^{\infty}$ regularity) of the maximizer of
$g(c).$
Now, according to Section 5.3,
$\displaystyle\max_{\phi\in\text{AC}_{0}[0,T]}v(\phi)=\mathop{\max_{\phi\in\text{AC}_{0}[0,T]}}_{\phi(T)=c}v(\phi)\,\,\text{
for some (possibly not unique) }c\in\mathbb{R}.$
because of the existence of $\max_{\phi\in\text{AC}_{0}[0,T]}v(\phi).$
## Acknowledgment
The author wishes to thank Professor Alexander Wentzell for his guidance and
suggestions on this work, and Portuguese Science Foundation project
(PTDC/MAT/120354/2010) for the support.
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|
arxiv-papers
| 2012-11-23T21:17:47 |
2024-09-04T02:49:38.320945
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Xiangfeng Yang",
"submitter": "Xiangfeng Yang",
"url": "https://arxiv.org/abs/1211.5610"
}
|
1211.5622
|
# $\tau$-rigid modules for algebras with radical square zero ††thanks: 2000
Mathematics Subject Classification: 16G10, 16G70, 16E10. ††thanks: Keywords:
$\tau$-rigid modules, simple modules, almost split sequences, algebras with
radical square zero
Xiaojin Zhang
School of Mathematics and Statistics, Nanjing University of Information
Science and Technology,
Nanjing 210044, P. R. China E-mail address: [email protected],
[email protected]
###### Abstract
In this paper, we show that for an algebra $\Lambda$ with radical square zero
and an indecomposable $\Lambda$-module $M$ such that $\Lambda$ is Gorenstein
of finite type or $\tau M$ is $\tau$-rigid, $M$ is $\tau$-rigid if and only if
the first two projective terms of a minimal projective resolution of $M$ have
no on-zero direct summands in common. We also determined all $\tau$-tilting
modules for Nakayama algebras with radical square zero. Moreover, by giving a
construction theorem we show that a basic connected radical square zero
algebra admitting a unique $\tau$-tilting module is local.
1 Introduction
In October of 2012, Adachi, Iyama and Reiten introduced the notion of
$\tau$-tilting modules which is a generalization of the classical tilting
modules [APR, BB, HR]. $\tau$-tilting modules which admit very similar
properties to the classical tilting modules are very close to silting objects
in [AiI] and the cluster tilting objects in 2-Calabi-Yau triangulated
categories [IY]. So it is interesting to find $\tau$-tilting modules for a
given algebra. It is showed in [AIR] that all $\tau$-tilting modules can be
written as finite copies of direct sums of $\tau$-rigid modules which was
firstly introduced in [AuS]. To find the $\tau$-tilting modules for given
algebras, what we need to do is just to find the (indecomposable) $\tau$-rigid
modules for them.
Notice that Adachi, Iyama and Reiten showed that every $\tau$-rigid module $M$
has no common non-zero direct summands in the first and second projective
terms of its minimal projective resolution. It is interesting to consider
whether the $\tau$-rigid modules can be determined by the non-existence of
common direct summands in the first and second projective terms of their
minimal projective resolutions. A positive answer to this question would make
us be able to judge $\tau$-rigid modules straightly. Unfortunately, it is far
from being true. So we have to ask: (1) When can $\tau$-rigid modules be
determined by the non-existence of common direct summands in their minimal
projective resolution?
In addition, what we also want to know is to determine the structures of
algebras from the properties of their $\tau$-rigid modules. It is well-known
that a local algebra admits a unique $\tau$-tilting module, that is, all
indecomposable $\tau$-rigid modules are projective. So it is natural to ask:
(2) Is an algebra $\Lambda$ local if it admits a unique $\tau$-tilting module?
We should remark that a similar question for the classical tilting modules is
not true in general since every non-local self-injective algebra admits a
unique classical tilting module.
On the other hand, algebras with radical square zero have been studied by
Auslander, Reiten and Smal$\phi$ in [AuRS], which play an important role in
classifying Nakayama algebras and stable equivalence. For the recent
development of this class of algebras, we refer to [C] and [RX]. We should
note that this kind of algebras make us be able to give more examples for
algebras with best properties of $\tau$-rigid modules and non-trivial CM-free
algebras (all finitely generated indecomposable Gorenstein projective modules
are projective).
In this paper, we try to answer the two questions above over algebras with
radical square zero. The paper is organized as follows:
In Section 2, we will recall some preliminaries on algebras with radical
square zero. In Section 3, we give an answer to the first question above and
prove the following:
Theorem 1 Let $\Lambda$ be a basic and connected Nakayama algebra with
$r^{2}=0$ which is not self-injective local and let $n$ be the number of non-
isomorphic simple modules. Then
(1) Every indecomposable module $M$ is $\tau$-rigid.
(2) Every $\tau$-tilting module $T$ is of the form $S_{1}\bigoplus
S_{2}\bigoplus\cdots
S_{t}\bigoplus(\Lambda/P_{0}(\tau(S_{1}\bigoplus\cdots\bigoplus S_{t}))$,
where $S_{j}$ is simple for $1\leq j\leq t$, $t$ is an integer such that
$0\leq t\leq{\rm int}(n/2)$ and ${\rm int}(m)$ denotes the largest integer
less than or equal to $m$ for any real number $m$.
Theorem 2 Let $\Lambda$ be a basic and connected algebra with $r^{2}=0$.
(1) If $\Lambda$ is self-injective local, then every indecomposable
$\tau$-rigid module is projective.
(2) If $\Lambda$ is self-injective but not local, then every indecomposable
module is $\tau$-rigid.
(3) Let $M$ be an indecomposable $\Lambda$-module. If $\Lambda$ is
representation finite of finite global dimension or $\tau M$ is $\tau$-rigid,
then $M$ is $\tau$-rigid if and only if there is no non-zero direct summand of
$P_{0}(M)$ and $P_{1}(M)$ in common, where $P_{0}(M)$ and $P_{1}(M)$ are the
first and second projective terms of a minimal projective resolution of $M$,
respectively.
In Section 4, we will give a construction theorem to get indecomposable
$\tau$-rigid modules from simple modules. This is very different from the
mutation theorem in [AIR]. As a result, we can give an answer to the second
question and prove the following:
Theorem 3 Let $\Lambda$ be a basic and connected algebra with $r^{2}=0$. If
$\Lambda$ admits a unique $\tau$-tilting module, then it is local.
In Section 5, we will give examples to show our results.
Throughout this paper, all algebras are basic connected non-semi-simple Artin
algebras over a commutative Artin ring R. $\mathbb{D}={\rm
Hom}_{R}(-,I^{0}(R/r))$ is the ordinary dual, where $r$ is the Jacobson
radical of $R$ and $I^{0}(R/r)$ is the injective envelope of $R/r$. All
modules are finitely generated left $\Lambda$-modules if not claimed.
2 Properties for algebras with radical square zero
In this section we will recall some properties for algebras with radical
square zero. Denote by $r$ the Jacobson radical of an algebra $\Lambda$.
$\Lambda$ is called radical square zero if $r^{2}=0$. Let $\Gamma$ be another
algebra. We say that $\Lambda$ is stable equivalent to $\Gamma$ if there is an
equivalence functor $F:\underline{\mathop{\rm
mod}\nolimits}\Lambda\rightarrow\underline{\mathop{\rm mod}\nolimits}\Gamma$,
where $\underline{\mathop{\rm mod}\nolimits}\Lambda$ and
$\underline{\mathop{\rm mod}\nolimits}\Gamma$ denote the associate module
categories modulo the projective modules, respectively.
Now we can recall the following result for algebras with radical square zero
from [AuRS, X, Theorem 2.4, Lemma 2.1].
Lemma 2.1 Let $\Lambda$ be an algebra with $r^{2}=0$. Denote by $\Gamma$ the
triangular matrix algebra
$\left(\begin{array}[]{cc}\Lambda/r&0\\\ r&\Lambda/r\par\end{array}\right)$
(1)
and denote by $F:\underline{\mathop{\rm
mod}\nolimits}\Lambda\rightarrow\underline{\mathop{\rm mod}\nolimits}\Gamma$
the functor via $F(M)=(M/rM,rM,f)$ and $F(g)=(g_{1},g_{2})$ for any
$M,N,L\in\mathop{\rm mod}\nolimits\Lambda$ and $g:N\rightarrow L$, where
$f:r\bigotimes_{\Lambda/r}M/rM\rightarrow rM$ is an epimorphism,
$g_{1}:N/rN\rightarrow L/rL$ and $g_{2}:rN\rightarrow rL$ are induced by $g$.
Then
(1) $F$ is an equivalence and hence $\Lambda$ is stable equivalent to
$\Gamma$.
(2) $F(M)$ is indecomposable if and only if $M$ is indecomposable.
(3) $F(M)$ is projective if and only if $M$ is projective.
Recall that a morphism $h:E\rightarrow M$ is called right minimal if for any
$l:E\rightarrow E$ $h=hl$ implies that $l$ is an isomorphism. $h$ is right
almost split if $h$ is not a spit epimorphism and for any $m:N\rightarrow E$
which is not a split epimorphism there exists a $t:N\rightarrow E$ such that
$m=ht$. Dually, one can define left minimal morphisms and left almost split
sequences. An exact sequence $0\rightarrow A\stackrel{{\scriptstyle
g}}{{\rightarrow}}B\stackrel{{\scriptstyle h}}{{\rightarrow}}C\rightarrow 0$
is called almost split if $g$ is left almost split and $h$ is right almost
split. Now we are ready to recall the following properties of almost split
sequences for algebras with $r^{2}=0$ from [AuRS, V, Proposition 3.5, X,
Proposition 2.5].
Lemma 2.2 Let $\Lambda$ be an algebra with $r^{2}$=0 and let $0\rightarrow
A\stackrel{{\scriptstyle g}}{{\rightarrow}}B\stackrel{{\scriptstyle
h}}{{\rightarrow}}C\rightarrow 0$ be an almost split sequence. Then
(1) $B$ is projective if and only if $A$ is non-injective simple. If $A$ is
simple non-injective, then $h:B\rightarrow C$ is a projective cover.
(2) $B$ is injective if and only if $C$ is non-projective simple. If $C$ is
simple non-projective, then $g:A\rightarrow B$ is an injective envelope.
Lemma 2.3 Let $\Lambda$, $F$ and $\Gamma$ be as in Lemma 2.1 and let
$0\rightarrow A\stackrel{{\scriptstyle
g}}{{\rightarrow}}B\stackrel{{\scriptstyle h}}{{\rightarrow}}C\rightarrow 0\ \
(*)$ be an exact sequence such that $A$ and $C$ are indecomposable and $A$ is
not simple. Then
(1) The sequence $(*)$ is almost split in $\mathop{\rm mod}\nolimits\Lambda$
if and only if $0\rightarrow F(A)\stackrel{{\scriptstyle
F(g)}}{{\rightarrow}}F(B)\stackrel{{\scriptstyle
F(h)}}{{\rightarrow}}F(C)\rightarrow 0$ is almost split in $\mathop{\rm
mod}\nolimits\Gamma$.
(2) If $(*)$ is almost split, then
$F(A)=F(\tau_{\Lambda}C)=\tau_{\Gamma}F(C)$.
Proof. (2) follows from (1). $\hfill{\square}$
The following result which gives a connection between morphisms in
$\mathop{\rm mod}\nolimits\Lambda$ and $\mathop{\rm mod}\nolimits\Gamma$ is
very important to the proof of the main results.
Lemma 2.4 Let $\Lambda$, $F$ and $\Gamma$ be as in Lemma 2.1. Then
(1) For any $M,N\in\mathop{\rm mod}\nolimits\Lambda$ we have the following
exact sequence of Abelian groups:
$0\rightarrow{\rm Hom}_{\Lambda}(M,rN)\rightarrow{\rm
Hom}_{\Lambda}(M,N)\rightarrow{\rm Hom}_{\Gamma}(F(M),F(N))\rightarrow 0$
(2) $\underline{\rm Hom}_{\Lambda}(M,N)\simeq{\rm Hom}_{\Gamma}(F(M),F(N))$ if
both $M$ and $N$ have no projective direct summands.
Proof. (1) follows from [AuRS, X, Lemma 2.1] and (2) follows from [AuRS, X,
Lemma 2.3] and (1). $\hfill{\square}$
In order to show the main result on Nakayama algebra with $r^{2}=0$, we need
the following:
Lemma 2.5 Let $\Lambda$ be a Nakayama algebra with $r^{2}=0$. Then every
indecomposable module $M\in\mathop{\rm mod}\nolimits\Lambda$ is either simple
or projective.
Proof. By using $r^{2}=0$ and [AsSS, V, Theorem 4.1]. $\hfill{\square}$
3 $\tau$-rigid modules and minimal projective resolution
In this section, we will determine the $\tau$-rigid modules in terms of
minimal projective resolution and try to answer the first question (see
Theorem 3.4, Theorem 3.12 and Theorem 3.15). Firstly, we recall the notions of
$\tau$-tilting modules and $\tau$-rigid modules in [AIR] and [AuS],
respectively.
Definition 3.1 For an algebra $\Lambda$, a $\Lambda$-module $M$ is called
$\tau$-rigid if ${\rm Hom}(M,\tau M)=0$, where $\tau$ denotes the Auslander-
Reiten translation. A module $N$ is $\tau$-tilting if it is $\tau$-rigid and
$|N|=|\Lambda|$, where $|N|$ denotes the number of non-isomorphic direct
summands of $N$. Any $\tau$-rigid module is a direct summand of a
$\tau$-tilting module. We also note that if $\Lambda$ is hereditary then
$\tau$-tilting modules and $\tau$-rigid modules coincide with tilting modules
and rigid modules, respectively.
For any indecomposable $M$ in $\mathop{\rm mod}\nolimits\Lambda$, if $M$ is
projective, then it is $\tau$-rigid. So we can assume that $M$ is not
projective. Denote by $\cdots\rightarrow P_{t}(M)\rightarrow\cdots\rightarrow
P_{1}(M)\rightarrow P_{0}(M)\rightarrow M\rightarrow 0$ be a minimal
projective resolution of $M$, where $t$ is a non-negative integer. And denote
by $\Omega^{i}M$ the $i$-th syzygy of $M$ for any $i\geq 0$. Considering the
almost split sequence $0\rightarrow\tau M\rightarrow E\rightarrow M\rightarrow
0$, we have the following:
Proposition 3.2 Let $\Lambda$ be an algebra with $r^{2}=0$. Then $\Lambda$ is
self-injective local if and only if there is an almost split sequence
$0\rightarrow S\rightarrow P\rightarrow S\rightarrow 0$, where $S$ is a simple
$\Lambda$-module.
Proof. $\Rightarrow$ Since $\Lambda$ is a basic connected local algebra, one
can get the following exact sequence: $0\rightarrow
r\rightarrow\Lambda\rightarrow S\rightarrow 0,$ where $S$ is simple and $r$ is
the radical of $\Lambda$. Notice that $r^{2}=0$ then $r$ is semi-simple.
Because $\Lambda$ is self-injective, we get $r$ is simple by [HuZ, Lemma 2.6],
and hence $r\simeq S$. Then the sequence is almost split by [AsSS, IV,
Proposition 3.11].
$\Leftarrow$ By Lemma 2.2, one gets that $P$ is projective and injective. It
is enough to prove that $\Lambda$ has a unique simple module $S$ up to
isomorphism. On the contrary, Suppose that there is another simple
$S^{\prime}\not\simeq S$. We claim that ${\rm
Hom}_{\Lambda}(P_{0}(S^{\prime}),P_{0}(S))={\rm
Hom}_{\Lambda}(P_{0}(S),P_{0}(S^{\prime}))=0$, where $P_{0}(M)$ is the
projective cover of $M$.
(1) ${\rm Hom}_{\Lambda}(P_{0}(S^{\prime}),P_{0}(S))=0$.
Suppose that there is an $f\in{\rm Hom}_{\Lambda}(P_{0}(S^{\prime}),P_{0}(S))$
such that $f\not=0$, then $f$ is not epic since $P_{0}(S)$ is projective and
$S\not\simeq S^{\prime}$. Denote by ${\rm Im}f$ the image of $f$, then ${\rm
Im}f\subseteq rP_{0}(S)$. Notice that $0\rightarrow S\rightarrow P\rightarrow
S\rightarrow 0$ is almost split, then $P_{0}(S)\simeq P$ and $rP_{0}(S)\simeq
S$ by Lemma 2.2, and hence ${\rm Im}f=S$, then $f:P_{0}(S^{\prime})\rightarrow
S$ is epic, and hence $P_{0}(S^{\prime})\simeq P_{0}(S)$. One gets a
contradiction since $S\not\simeq S^{\prime}$.
(2) ${\rm Hom}_{\Lambda}(P_{0}(S),P_{0}(S^{\prime}))=0$.
Suppose that there is a $g\in{\rm Hom}_{\Lambda}(P_{0}(S),P_{0}(S^{\prime}))$
such that $g\not=0$, then $g$ is not epic and ${\rm Im}g\subseteq
rP_{0}(S^{\prime})$ by a similar argument in (1). Notice that $r^{2}=0$, then
$rP_{0}(S^{\prime})$ is semi-simple. So we get ${\rm Im}g=S$, and hence
$j:S\hookrightarrow P_{0}(S^{\prime})$. Then we have the following commutative
diagram:
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By Lemma 2.2 we get that $P_{0}(S)\simeq P,$ $P$ is injective and
$i:S\rightarrow P$ is an injective envelope. Then $i$ is an essential
monomorphism implies that $h:P\rightarrow P_{0}(S^{\prime})$ is monic, and
hence $P\simeq P_{0}(S^{\prime})$, a contradiction.
Since $\Lambda$ is connected, one gets the assertion by the claim and [AsSS,
II, Lemma 1.6]. $\hfill\square$
Now we can give a class of $\tau$-rigid modules over algebras with $r^{2}=0$
which are not local-self-injective.
Proposition 3.3 Let $\Lambda$ be an algebra with $r^{2}=0$ which is not self-
injective local and let $M$ be an indecomposable $\Lambda$-module. If $M$
satisfies (1) $\tau M$ is simple projective, or (2) both $\tau M$ and $M$ are
simple, then $M$ is $\tau$-rigid.
Proof. Suppose that there is a non-zero $f\in$ ${\rm Hom}_{\Lambda}(M,\tau
M)$. For both cases, we have $f$ is epic, and hence $M\simeq\tau M$. For the
first case, we get that $M$ is projective, a contradiction. For the second
one, by Lemma 2.2 there is an almost split sequence $0\rightarrow S\rightarrow
P\rightarrow S\rightarrow 0$ with $S$ a simple module. By Proposition 3.2
$\Lambda$ is self-injective local, a contradiction. $\hfill\square$
Remark For any algebra $\Sigma$ and an indecomposable $\Sigma$-module $N$ with
$\tau N$ simple projective, one can show that $N$ is $\tau$-rigid by
formulating the proof of Proposition 3.3 (1).
Now we are in a position to state the $\tau$-tilting and $\tau$-rigid modules
for Nakayama algebras with $r^{2}=0$.
Theorem 3.4 Let $\Lambda$ be a Nakayama algebra with $r^{2}=0$ which is not
self-injective local and let $n$ be the number of non-isomorphic simple
modules. Then
(1) Every indecomposable module $M$ is $\tau$-rigid.
(2) Every $\tau$-tilting module $T$ is of the form $S_{1}\bigoplus
S_{2}\bigoplus\cdots
S_{t}\bigoplus(\Lambda/P_{0}(\tau(S_{1}\bigoplus\cdots\bigoplus S_{t}))$,
where $S_{j}$ is simple for $1\leq j\leq t$, $t$ is an integer such that
$0\leq t\leq{\rm int}(n/2)$ and ${\rm int}(m)$ denotes the largest integer
less than or equal to $m$ for any real number $m$.
Proof. (1) By Lemma 2.5, $M$ is simple or projective for any indecomposable
$M\in\mathop{\rm mod}\nolimits\Lambda$. If $M$ is projective, there is nothing
to prove. If $M$ is simple non-projective, then $\tau M$ is simple, by
Proposition 3.3, $M$ is $\tau$-rigid.
(2) For any $\tau$-tilting module $T$, we claim that there is at least one
indecomposable $P$ as a direct summand of $T$.
On the contrary, suppose that $T=S_{1}\bigoplus S_{2}\bigoplus\cdots\bigoplus
S_{n}$ with all $S_{j}$ simple non-projective for $1\leq j\leq n$. Without
loss of generality, we can assume that $\tau S_{1}=S_{2}$ by Lemma 2.5 or (1).
Then $0\not={\rm Hom}_{\Lambda}(S_{2},S_{2})\subseteq{\rm
Hom}_{\Lambda}(T,\tau T)=0$, a contradiction.
Next we will show if $S$ is a direct summand of $T$, then $P_{0}(S)$ is a
direct summand of $T$.
If $S$ is projective, then the assertion holds true. We can assume that $S$ is
not projective. Then we get the following almost split sequence: $0\rightarrow
S^{\prime}\rightarrow P\rightarrow S\rightarrow 0$ with $S^{\prime}$ simple by
Lemma 2.2 and Lemma 2.5. Again by Lemma 2.2, $P$ is indecomposable projective
and $P\simeq P_{0}(S)$. Because $T$ is $\tau$-tilting and $S$ is direct
summand of $T$, it is not difficult to show that $S^{\prime}$ is not a direct
summand of $T$. Similarly, if $S$ is not injective, then a simple module
$S^{*}$ with $\tau S^{*}\simeq S$ is not a direct summand of $T$. Since $\tau
T$ is semi-simple, then ${\rm Hom}_{\Lambda}(P_{0}(S),\tau T)=0$, that is,
$P_{0}(S)$ is in $Fac(T)$ by [AIR, Theorem 2.10], where ${\rm Fac(T)}$ denotes
the category consisting of factor modules of finite copies of direct sums of
$T$. So $P_{0}(S)$ is a direct summand of $T$. Notice that $|T|=|\Lambda|=n$,
so the number of simple direct summands of $T$ has to be at most ${\rm
int(n/2)}$. $\hfill\square$
The conditions in Proposition 3.3 are not easy to be satisfied. In the
following we will generalize it into a general framework. Denote by
$\mathop{\rm gl.dim}\nolimits\Lambda$ the global dimension of $\Lambda$ and
denote by $\mathop{\rm pd}\nolimits_{\Lambda}M$ the projective dimension of
$M$. We have:
Lemma 3.5 Let $\Lambda$ be an algebra with $r^{2}=0$. If $S$ is a simple
module with $\mathop{\rm pd}\nolimits_{\Lambda}S=m<\infty$. Then $S$ is
$\tau$-rigid. Moreover, if $\mathop{\rm gl.dim}\nolimits\Lambda=m<\infty$,
then every simple module $S$ is $\tau$-rigid.
Proof. We only have to the first one since the last follows from the first. By
[AIR, Proposition 1.2] a simple module $S$ is $\tau$-rigid if and only if it
is rigid, that is, ${\rm Ext}_{\Lambda}^{1}(S,S)=0$. If $S$ is projective,
there is nothing to prove. So we can assume that $m\geq 1$. Take the following
part of a minimal projective resolution of $S$:
$0\rightarrow\Omega^{1}S\rightarrow P_{0}(S)\rightarrow S\rightarrow 0$, where
$\Omega^{1}S$ denotes the first syzygy of $S$. One gets that $\mathop{\rm
pd}\nolimits_{\Lambda}\Omega^{1}S=m-1$ since $\mathop{\rm
pd}\nolimits_{\Lambda}S=m<\infty$. Since $r^{2}=0$, we have $\Omega^{1}S\simeq
rP_{0}(S)$ is semi-simple and any direct summand of it is of projective
dimension at most $m-1$. So it is not difficult to show ${\rm
Ext}_{\Lambda}^{1}(S,S)\simeq{\rm Hom}_{\Lambda}(\Omega^{1}S,S)=0.$
$\hfill{\square}$
Denote by $\overline{\mathop{\rm mod}\nolimits}\Lambda$ the associate modules
category modulo injective modules and denote by $\overline{{\rm
Hom}}_{\Lambda}(L,N)$ and $\underline{{\rm Hom}}_{\Lambda}(L,N)$ classes of
morphisms from $L$ to $N$ in $\overline{\mathop{\rm mod}\nolimits}\Lambda$ and
$\underline{\mathop{\rm mod}\nolimits}\Lambda$, respectively. Now we are in a
position to state another main result on judging the $\tau$-rigid properties
by simple modules.
Theorem 3.6 Let $\Lambda$ be an algebra with $r^{2}=0$ and let $M$ be
indecomposable with $\tau M$ simple. We have (1) $M$ is $\tau$-rigid if and
only if $\tau M$ is $\tau$-rigid. (2) If $\mathop{\rm
pd}\nolimits_{\Lambda}M<\infty$, then $M$ is $\tau$-rigid. Moreover, if
$\mathop{\rm gl.dim}\nolimits\Lambda<\infty$, then $M$ is $\tau$-rigid.
Proof. Since (2) is a straight result of (1), Lemma 2.2 and Lemma 3.5, we only
show (1).
$\Leftarrow$ By the remark of Proposition 3.3, it is enough to show the case
of $\tau M$ is not projective.
On the contrary, suppose that $M$ is not $\tau$-rigid, that is, ${\rm
Hom}(M,\tau M)\not=0$. We get that $f$ is epic for any $0\not=f\in{\rm
Hom}(M,\tau M)$ since $\tau M$ is simple. By Lemma 2.2, one gets the following
almost split sequence: $0\rightarrow\tau M\stackrel{{\scriptstyle
i}}{{\rightarrow}}P_{0}(M)\rightarrow M\rightarrow 0$. So $P_{0}(\tau M)$ is a
direct summand of $P_{0}(M)$. Notice that $i$ is left minimal, then by [AuRS,
I, Theorem 2.4] $\tau M$ can be embedded into $P_{0}(\tau M)$, and hence a
direct summand of $rP_{0}(\tau M)$ since $r^{2}=0$. Then we have the following
commutative diagram:
$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{rP_{0}(\tau
M)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha}$$\textstyle{P_{0}(\tau
M)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi}$$\scriptstyle{\beta}$$\textstyle{\tau
M\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\tau
M\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{E\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\tau
M\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$
with $\alpha$ an epimorphism. By the snake lemma, $\beta$ is also an
epimorphism. Since $\mathop{\rm pd}\nolimits_{\Lambda}M<\infty$, one can show
that $\mathop{\rm pd}\nolimits_{\Lambda}\tau M$ is of finite projective
dimension by Lemma 2.2. Then by the assumption $\tau M$ is $\tau$-rigid and
hence the bottom row in the commutative diagram is split. So one gets an
epimorphism $P_{0}(\tau M)\rightarrow\tau M\bigoplus\tau M$, a contradiction.
$\Rightarrow$ Since $M$ is $\tau$-rigid, one gets ${\rm Hom}_{\Lambda}(M,\tau
M)=0$ which implies that $\underline{{\rm Hom}}_{\Lambda}(M,\tau M)=0$. Notice
that $\tau:\underline{\mathop{\rm
mod}\nolimits}\Lambda\rightarrow\overline{\mathop{\rm mod}}\nolimits\Lambda$
is an equivalence, one can get $\overline{{\rm Hom}}_{\Lambda}(\tau
M,\tau^{2}M)=\underline{{\rm Hom}}_{\Lambda}(M,\tau M)=0$. By AR-formula one
gets ${\rm Ext}_{\Lambda}^{1}(\tau M,\tau M)\simeq\mathbb{D}\overline{{\rm
Hom}}_{\Lambda}(\tau M,\tau^{2}M)=0$. Then by [AIR, Proposition 1.2] $\tau M$
is $\tau$-rigid since $\tau M$ is simple. $\hfill\square$
To answer the first question, we need the following properties for
$\tau$-rigid modules over hereditary algebras.
Lemma 3.7 Let $\Lambda$ be a hereditary algebra and let $M$ be an
indecomposable non-projective module. If $\tau M$ is projective, then $M$ is
$\tau$-rigid.
Proof. Suppose that ${\rm Hom}_{\Lambda}(M,\tau M)\not=0$. Then there is a
non-zero morphism $f:M\rightarrow\tau M$. Since $\Lambda$ is hereditary and
$\tau M$ is projective, one gets that ${\rm Im}f$ is projective and hence $M$
is projective, a contradiction. $\hfill\square$
Lemma 3.8 Let $\Lambda$ be a hereditary algebra and let $M$ be an
indecomposable non-projective module. If $\tau M$ is $\tau$-rigid, then $M$ is
$\tau$-rigid.
Proof. If $\tau M$ is projective, the assertion holds from Lemma 3.7. We only
show the case $\tau M$ is not projective. By [AsSS, IV, Corollary 2.15 (b)],
one gets ${\rm Hom}_{\Lambda}(M,\tau M)\simeq{\rm Hom}_{\Lambda}(\tau
M,\tau^{2}M)=0$. $\hfill\square$
Recall that an indecomposable module $M$ over a hereditary algebra is
preprojective if there is a non-negative integer $j$ such that $\tau^{j}M$ is
a non-zero projective module. Then we have:
Proposition 3.9 [AuRS, VIII, Propositions 1.7, 1.13] Let $\Lambda$ be a
hereditary algebra. Then (1) Every preprojective module $M$ is $\tau$-rigid.
(2) If $\Lambda$ is of finite type, then every indecomposable module is
$\tau$-rigid, and hence rigid.
Proof. (1) We can assume that $\tau^{j}M$ is projective for some non-negative
integer $j$. By induction on $j$ and Lemma 3.8, one gets the assertion. Then
by [AuRS, VIII, Proposition 1.13] and (1), one can show (2). $\hfill\square$
Denote by $\mathop{\rm id}\nolimits_{\Lambda}M$ $(resp.\ \mathop{\rm
id}\nolimits_{\Lambda^{o}}M)$ the injective dimension of $M$ for an $M$ in
$\mathop{\rm mod}\nolimits\Lambda$ $(resp.\ \mathop{\rm
mod}\nolimits\Lambda^{o})$. Recall that an algebra $\Lambda$ is called
Gorenstein if $\mathop{\rm id}\nolimits_{\Lambda}\Lambda=\mathop{\rm
id}\nolimits_{\Lambda^{o}}\Lambda=n$ for some integer $n\geq 0$. We have the
following:
Lemma 3.10 Let $\Lambda$ be a Gorenstein algebra with $r^{2}=0$. Then
$\Lambda$ is either self-injective or of finite global dimension.
Proof. By [C], we can get that every algebra with $r^{2}=0$ is either self-
injective or CM-free. Recall that an algebra is called CM-free if every
finitely generated Gorenstein projective module is projective. We will show
$\mathop{\rm gl.dim}\nolimits\Lambda=n$ if $\mathop{\rm
id}\nolimits_{\Lambda}\Lambda=\mathop{\rm id}\nolimits_{\Lambda^{o}}\Lambda=n$
for some $n>0$. By [AuR, Proposition 3.1] one can show that $\Omega^{n}M$ is
Gorenstein projective and hence projective for any $M$ in $\mathop{\rm
mod}\nolimits\Lambda$ since $\Lambda$ is CM-free. The assertion holds true.
$\hfill\square$
Notice that a self-injective algebra with $r^{2}=0$ is Nakayama by [AuRS, IV,
Proposition 2.16]. Since the Nakayama case is completely classified in
Proposition 3.2 and Theorem 3.4, we only have to find the $\tau$-rigid modules
for algebras of finite global dimension with $r^{2}=0$.
Denote by ${\rm Soc}M$ the socle of $M$. We have the following easy
observation:
Lemma 3.11 Let $\Lambda$ be an algebra and let $M$ be an indecomposable and
non-projective $\Lambda$-module. Then (1) ${\rm Soc}\tau
M\simeq\Omega^{1}M/r\Omega^{1}M$. (2) ${\rm Soc}\tau M\simeq\Omega^{1}M$ if
$r^{2}=0$.
Proof. (1) Taking the following part of a minimal projective resolution of
$M$: $P_{1}(M)\rightarrow P_{0}(M)\rightarrow M\rightarrow 0$ and then
applying the functor ${\rm Hom}_{\Lambda}(-,\Lambda)=(-)^{*}$, one gets the
following part of minimal projective resolution of ${\rm Tr}M$:
$P_{0}(M)^{*}\rightarrow P_{1}(M)^{*}\rightarrow{\rm Tr}M\rightarrow 0$, where
${\rm Tr}$ is the Auslander-Bridger transpose. Then applying the functor
$\mathbb{D}$, one has a minimal injective resolution of $\tau M:$
$0\rightarrow\tau
M\rightarrow\mathbb{D}P_{1}(M)^{*}\rightarrow\mathbb{D}P_{0}(M)^{*}$. Then
${\rm Soc}\tau M\simeq{\rm
Soc}\mathbb{D}P_{1}(M)^{*}\simeq\Omega^{1}M/r\Omega^{1}M.$
(2) Taking the following part of a minimal projective resolution of $M$:
$0\rightarrow\Omega^{1}M\rightarrow P_{0}(M)\rightarrow M\rightarrow 0$, one
can get the following commutative diagram:
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where $f$ is epic. Then by the snake lemma, one gets $g$ is a monomorphism.
Since $r^{2}=0$, we have that $rP_{0}(M)$ is semi-simple and hence
$\Omega^{1}M$ is semi-simple. We are done. $\hfill\square$
Now we are ready to determine all the $\tau$-rigid modules for a Gorenstein
algebra of finite type with $r^{2}=0$. Combined with Proposition 3.2, Lemma
3.11 and Theorem 3.4, one can show:
Theorem 3.12 Let $\Lambda$ be a Gorenstein algebra of finite type with
$r^{2}=0$.
(1) If $\Lambda$ is self-injective local, then every indecomposable
$\tau$-rigid module is projective.
(2) If $\Lambda$ is self-injective but not local, then every indecomposable
module $M$ is $\tau$-rigid
(3) If $\Lambda$ is of finite global dimension, then an indecomposable module
$M$ is $\tau$-rigid if and only if there is no non-zero direct summand of
$P_{0}(M)$ and $P_{1}(M)$ in common.
Proof. (1) is clear and (2) is showed in Theorem 3.4.
(3) $\Rightarrow$ It is a straight result of [AIR, Proposition 2.5].
$\Leftarrow$ Without loss of generality, we can assume that $\tau M$ is not
zero. If $\tau M$ is simple, then the assertion holds true by Theorem 3.6. Now
we can assume that $\tau M$ is not simple. Let $\Gamma$ and $F$ be as in Lemma
2.1. By Lemma 2.1 and [AuRS, X, Proposition 1.1], we get that $\Gamma$ is
hereditary of finite type. Then by Proposition 3.9(2), $F(M)\in\mathop{\rm
mod}\nolimits\Gamma$ is $\tau$-rigid. So ${\rm Hom}_{\Gamma}(F(M),F(\tau
M))\simeq{\rm Hom}_{\Gamma}(F(M),\tau F(M))=0$ by Lemma 2.3. Then by Lemma 2.4
$M$ is $\tau$-rigid if and only if ${\rm Hom}_{\Lambda}(M,r\tau M)=0$.
We show that ${\rm Hom}_{\Lambda}(M,r\tau M)=0$. Since $r^{2}=0$, we get that
$r\tau M$ is semi-simple and hence a direct summand of ${\rm Soc}\tau
M\simeq\Omega^{1}M$ by Lemma 3.11. Notice that there is no common direct
summand of $P_{0}(M)$ and $P_{1}(M)$, one can show ${\rm Hom}_{\Lambda}(M,{\rm
Soc}\tau M)=0$ which implies that ${\rm Hom}_{\Lambda}(M,r\tau M)=0$. Then $M$
is $\tau$-rigid by Lemma 2.4. $\hfill\square$
In general, for algebras mentioned in Theorem 3.12 (3) we don’t know whether
there is a common direct summand in $P_{0}(M)$ and $P_{1}(M)$ for an
indecomposable $M$ (see Example 5.3). However, we get the following:
Proposition 3.13 Let $\Lambda$ be an algebra of finite global dimension with
$r^{2}=0$ and let $M$ be an indecomposable module. If $M/rM$ is simple, then
$P_{0}(M)$ and $P_{1}(M)$ have no non-zero direct summands in common.
Proof. Denote by $S=M/rM$. It is enough to show that ${\rm
Hom}_{\Lambda}(\Omega^{1}M,S)=0$. On the contrary, suppose that ${\rm
Hom}_{\Lambda}(\Omega^{1}M,S)\not=0$. Then $S$ is a direct summand of
$\Omega^{1}M$ since $r^{2}=0$ and $S$ is simple. Moreover, we have the
following commutative diagram with exact rows:
$\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
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29.5pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
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80.31387pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
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180.28403pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
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0.0pt\hbox{$\textstyle{0}$}}}}}}}{\hbox{\kern-5.5pt\raise-41.13776pt\hbox{\hbox{\kern
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Since $g$ is epic, one can show that $f:\Omega^{1}M\rightarrow\Omega^{1}S$ is
a monomorphism, and hence $\Omega^{1}M$ is a direct summand of $\Omega^{1}S$
because $r^{2}=0$. So $S$ is a direct summand of $\Omega^{1}S$, that is
$\mathop{\rm pd}\nolimits_{\Lambda}S\leq\mathop{\rm
pd}\nolimits_{\Lambda}S-1<\infty$, a contradiction. $\hfill\square$
Denote by $l(M)$ the length of $M$. As a result of Proposition 3.13, we can
get:
Corollary 3.14 Let $\Lambda$ be an algebra of finite global dimension with
$r^{2}=0$ and $M$ be an indecomposable module. if $l(M)\leq 2$, then
$P_{0}(M)$ and $P_{1}(M)$ have no non-zero direct summands in common.
Proof. By Proposition 3.13, it suffice to prove that $M/rM$ is simple. If
$l(M)=1$ then $M$ is simple, the assertion holds true. If $l(M)=2$ then
$rM={\rm Soc}M$ is simple, and hence $M/rM$ is simple. $\hfill\square$
At the end of this section we will give a method to find more $\tau$-rigid
modules for any algebra with $r^{2}=0$. As we know all the projective
$\Lambda$-modules are $\tau$-rigid. In case that the algebra $\Lambda$ is not
self-injective, there must be some indecomposable $M$ such that $\tau M$ is
projective. It is interesting to know whether $M$ is $\tau$-rigid. A more
general question is: Whether is $M$ $\tau$-rigid if $\tau M$ is $\tau$-rigid?
To answer this question, we have
Theorem 3.15 Let $\Lambda$ be an algebra with $r^{2}=0$ and let $M$ be an
indecomposable $\Lambda$-module such that $\tau M$ is $\tau$-rigid. Then $M$
is $\tau$-rigid if and only if there is no non-zero direct summand of
$P_{0}(M)$ and $P_{1}(M)$ in common.
Proof. $\Rightarrow$ By [AIR, Proposition 2.5].
$\Leftarrow$ If $\tau M$ is simple, then by Theorem 3.6 (1) $M$ is
$\tau$-rigid if and only if $\tau M$ is $\tau$-rigid. Then the assertion holds
by the assumption $\tau M$ is $\tau$-rigid.
Now we can assume that $\tau M$ is not simple. Let $\Gamma$ and $F$ be as in
Lemma 2.1. We claim that ${\rm Hom}_{\Gamma}(F(M),F(\tau M))=0$.
Since $\tau M$ is $\tau$-rigid, we get ${\rm Hom}_{\Lambda}(\tau
M,\tau^{2}M)=0$ and hence $\overline{{\rm Hom}}_{\Lambda}(\tau
M,\tau^{2}M)=0$. Notice that $\tau:\underline{\mathop{\rm
mod}\nolimits\Lambda}\rightarrow\overline{\mathop{\rm mod}\nolimits\Lambda}$
is an equivalence, we get that $\underline{{\rm Hom}}_{\Lambda}(M,\tau M)=0$.
If $\tau M$ is not projective, then we get ${\rm Hom}_{\Gamma}(F(M),F(\tau
M))=0$ by Lemma 2.4. If $\tau M$ is projective, then ${\rm
Hom}_{\Gamma}(F(M),F(\tau M))=0$ since $F(\tau M)$ is projective and $\Gamma$
is hereditary. Otherwise, one can get a non-zero $g:F(M)\rightarrow F(\tau
M)$. So ${\rm Im}g$ is projective, and hence $F(M)$ is projective, that is,
$M$ is projective by Lemma 2.1, a contradiction.
By Lemma 2.4, we only have to show ${\rm Hom}_{\Lambda}(M,r\tau M)=0$. One can
get the assertion by a similar argument in Theorem 3.12. $\hfill\square$
For a non-Nakayama algebra $\Lambda$ with $r^{2}=0$, by Theorem 3.15 one can
find the $\tau$-rigid modules one by one from the projective vertices of the
$AR$-quiver of $\Lambda$ since here $\Lambda$ is not self-injective. For the
Nakayama case with $r^{2}=0$, we refer to Theorem 3.4 or Theorem 3.12. On the
other hand, it is interesting to study the structure of algebras in terms of
indecomposable $\tau$-rigid modules. Compared with Theorem 3.10 and Lemma 2.1,
we end this section with an open question which is closed to algebras of
finite type.
Question Let $\Lambda$ be an algebra with radical square zero. If all the
indecomposable modules are $\tau$-rigid, then $\Lambda$ is of finite type.
4 $\tau$-rigid modules and local algebras
In this section we firstly introduce a theorem to get a class of
indecomposable $\tau$-rigid modules from simple modules (here we don’t need
$\Lambda$ to be radical square zero). This method is very different from the
mutation theorem in [AIR]. As a result, we give a partial answer to the second
question.
Theorem 4.1 Let $\Lambda$ be an algebra and let $S$ be a simple
$\Lambda$-module such that the first syzygy $\Omega^{1}S$ is non-zero semi-
simple.
(1) Suppose that $S$ is not a direct summand of $\Omega^{1}S$. Let $S_{1}$ be
a simple submodule of $\Omega^{1}S$ and let $m$ be the maximal integer such
that ${S_{1}}^{m}$ is a direct summand of $\Omega^{1}S$. Then there is an
exact sequence $0\rightarrow\Omega^{1}S/S_{1}^{m}\rightarrow
P_{0}(S)\rightarrow M\rightarrow 0$ with $M$ indecomposable $\tau$-rigid.
(2) Assume that $S$ is a direct summand of $\Omega^{1}S$. Let $n$ be the
maximal integer such that ${S}^{n}$ is a direct summand of $\Omega^{1}S$.
(a) If $\Omega^{1}S\simeq S^{n}$, then there is no non-projective
indecomposable $\tau$-rigid module $N$ with the projective cover
$P_{0}(N)\simeq P_{0}(S)$.
(b) If $\Omega^{1}S\not\simeq S^{n}$, then we can get the following exact
sequence $0\rightarrow\Omega^{1}S/S^{n}\rightarrow P_{0}(S)\rightarrow
N\rightarrow 0$ such that $N$ is indecomposable non-projective $\tau$-rigid.
Proof. (1) By the assumption of (1), one can get that the simple module $S$ is
non-projective $\tau$-rigid since ${\rm Ext}_{\Lambda}^{1}(S,S)\simeq{\rm
Hom(\Omega^{1}S,S)}=0$.
If $\Omega^{1}S\simeq{S_{1}}^{m}$, there is nothing to prove.
Now we can assume that $\Omega^{1}S\not\simeq{S_{1}}^{m}$. Since $\Omega^{1}S$
is semi-simple, we get that a monomorphism
$\Omega^{1}S/{S_{1}}^{m}\hookrightarrow\Omega^{1}S\hookrightarrow P_{0}(S)$,
and hence we have the desired exact sequence
$0\rightarrow\Omega^{1}S/S_{1}^{m}\rightarrow P_{0}(S)\rightarrow M\rightarrow
0$. It remains to prove that $M$ is indecomposable $\tau$-rigid.
Since $P_{0}(S)$ is indecomposable and projective, one can show that $M$ is
indecomposable and $P_{0}(M)\simeq P_{0}(S)$ by the exact sequence above. In
the following we show that $M$ is $\tau$-rigid. By [AIR, Proposition 1.2(a)],
it is enough to show that ${\rm Ext}_{\Lambda}^{1}(M,N)=0$ for any $N\in{\rm
Fac}M$, where ${\rm Fac}M$ is the full subcategory consisting of factor
modules of finite copies of direct sums of $M$.
By the construction of $M$, we have the following commutative diagram with
exact rows
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Here $\Omega^{1}M\simeq\Omega^{1}S/S_{1}^{m}$ and by snake lemma one gets an
exact sequence
$0\rightarrow\Omega^{1}M\rightarrow\Omega^{1}S\rightarrow S_{1}^{m}\rightarrow
0\ \ \ \ \ (*1)$
Since $N$ is in ${\rm Fac}M$, then there is a minimal positive integer $t\geq
1$ such that $g:M^{t}\rightarrow N$ is an epimorphism. By [AuRS, I, Theorem
2.2] it is not difficult to show that $P_{0}(N)\simeq P_{0}(M)^{t}\simeq
P_{0}(S)^{t}$. Hence we get an epimorphism $h:N\rightarrow S^{t}$. Then we
have the following commutative diagram with exact rows
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78.64719pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
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0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
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3.0pt}}}}}}\ignorespaces{\hbox{\kern
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220.73465pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
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0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
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29.6972pt\raise-42.2911pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{{\Omega^{1}S}^{t}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
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101.56975pt\raise-42.2911pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{P_{0}(S)^{t}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
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181.8069pt\raise-42.2911pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{S^{t}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
220.73465pt\raise-42.2911pt\hbox{\hbox{\kern 0.0pt\raise
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220.73465pt\raise-42.2911pt\hbox{\hbox{\kern 0.0pt\raise
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Notice that $h$ is an epimorphism and $\Omega^{1}S$ is semi-simple, by snake
lemma we have two exact sequences
$0\rightarrow\Omega^{1}N\rightarrow\Omega^{1}S^{t}\rightarrow L\rightarrow 0\
\ \ \ (*2)$
$0\rightarrow L\rightarrow N\rightarrow S^{t}\rightarrow 0\ \ \ (*3)$
On the other hand, we have the following commutative diagram
$\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
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3.0pt}}}}}}\ignorespaces{\hbox{\kern
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82.33612pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise 0.0pt\hbox{$\textstyle{P_{0}(M)^{t}(\simeq
P_{0}(S)^{t})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
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0.0pt\hbox{\lx@xy@droprule}}\hbox{\kern-1.0pt\raise
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0.0pt\hbox{\lx@xy@droprule}}\hbox{\kern-1.0pt\raise
0.0pt\hbox{\lx@xy@droprule}}}}{\hbox{\kern 188.98755pt\raise
0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
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3.0pt}}}}}}\ignorespaces{\hbox{\kern
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231.80142pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
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0.0pt\hbox{$\textstyle{0}$}}}}}}}{\hbox{\kern-5.5pt\raise-41.72223pt\hbox{\hbox{\kern
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84.18059pt\raise-41.72223pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{P_{0}(N)(\simeq
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0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{0}$}}}}}}}\ignorespaces}}}}\ignorespaces.$
Since $g$ is an epimorphism, we get a monomorphism
$l:\Omega^{1}M^{t}\rightarrow\Omega^{1}N$. Combining the exact sequence $(*1)$
and $(*2)$, we have the following commutative diagram:
$\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
5.5pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry@@#!@\cr&&&&\\\&&&&\crcr}}}\ignorespaces{\hbox{\kern-5.5pt\raise
0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
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0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
29.5pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{\Omega^{1}M^{t}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
82.33612pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
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0.0pt\raise-2.43056pt\hbox{$\scriptstyle{l}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern
43.91806pt\raise-27.90222pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
82.33612pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
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0.0pt\hbox{$\textstyle{\Omega^{1}S^{t}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
131.0889pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\hbox{\kern
1.0pt\raise 0.0pt\hbox{\lx@xy@droprule}}\hbox{\kern-1.0pt\raise
0.0pt\hbox{\lx@xy@droprule}}}}\ignorespaces{}{\hbox{\hbox{\kern 1.0pt\raise
0.0pt\hbox{\lx@xy@droprule}}\hbox{\kern-1.0pt\raise
0.0pt\hbox{\lx@xy@droprule}}}}{\hbox{\hbox{\kern 1.0pt\raise
0.0pt\hbox{\lx@xy@droprule}}\hbox{\kern-1.0pt\raise
0.0pt\hbox{\lx@xy@droprule}}}}{\hbox{\kern 131.0889pt\raise
0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
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0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-1.75pt\hbox{$\scriptstyle{\exists f}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern
138.84305pt\raise-29.70667pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
170.5972pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
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0.0pt\hbox{$\textstyle{0}$}}}}}}}{\hbox{\kern-5.5pt\raise-39.53998pt\hbox{\hbox{\kern
0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
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31.34447pt\raise-39.53998pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
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82.33612pt\raise-39.53998pt\hbox{\hbox{\kern 0.0pt\raise
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132.44026pt\raise-39.53998pt\hbox{\hbox{\kern 0.0pt\raise
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By snake lemma again, we get that $f$ is an epimorphism. Notice that
$S_{1}^{mt}$ is semi-simple, then $L$ is a direct summand of $S_{1}^{mt}$.
Applying the functor ${\rm Hom}_{\Lambda}(M,-)$ to the exact sequence $(*3)$,
one can get that ${\rm Ext}^{1}_{\Lambda}(M,S)\simeq{\rm
Hom}_{\Lambda}(\Omega^{1}M,S)=0$. Similarly, one can get ${\rm
Ext}^{1}_{\Lambda}(M,S_{1})=0$ and hence ${\rm Ext}^{1}_{\Lambda}(M,L)=0$.
Then one gets ${\rm Ext}^{1}_{\Lambda}(M,N)=0$. We are done.
(2) We only prove (a) since the proof of (b) is very similar to the proof of
(1). It is easy to show that $S$ is not $\tau$-rigid. Suppose that there is an
indecomposable $\tau$-rigid module $N$ such that $P_{0}(S)\simeq P_{0}(N)$.
Then $N\not\simeq S$ and we have the following commutative diagram
$\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
5.5pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
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30.70833pt\raise-41.13776pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
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By snake lemma, one get that $\Omega^{1}N$ is a direct summand of
$\Omega^{1}S$, and hence has $S$ as one of its direct summand. That means
$P_{0}(N)$ and $P_{1}(N)$ have a non-zero direct summand $P_{0}(S)$. But $N$
is $\tau$-rigid, by using [AIR, Proposition 2.5], one gets a contradiction.
$\hfill{\square}$
Remark One can easily show that algebras with radical square zero satisfy the
condition of Theorem 4.1. For a non-local algebra $\Gamma$ with radical square
zero, there is at least $2n-m$ indecomposable $\tau$-rigid modules, where $n$
and $m$ is the number of non-isomorphic simple modules and the number of non-
isomorphic simple projective modules, respectively.
In the following we will focus on the structure of algebras and the
homological properties of algebras for which all $\tau$-rigid modules are
projective. To prove the main result of this section, we need the following
lemmas.
Lemma 4.2 Let $\Lambda$ be an algebra such that all $\tau$-rigid modules are
projective, then $\Lambda$ has no simple projective module.
Proof. Suppose that there is a simple projective module $S$. Then one can get
an AR-sequence $0\rightarrow S\rightarrow E\rightarrow M\rightarrow 0$. By
Proposition 3.3, $M$ is $\tau$-rigid. But $M$ is not projective since $\tau
M\simeq S\not=0$. $\hfill{\square}$
Lemma 4.3 Let $\Lambda$ be an algebra such that all $\tau$-rigid modules are
projective and let $S$ be a simple $\Lambda$-module.
(1)Then there is a non-zero direct summand of $P_{0}(S)$ and $P_{1}(S)$ in
common.
(2) If in addition $\Lambda$ is radical square zero, then $S$ is a direct
summand of $\Omega^{1}S$.
Proof. (1) By Lemma 4.2, we get that there is no projective simple module. By
the assumption, $S$ is not $\tau$-rigid. By [AIR, Proposition 1.2 (a)],
$0\not={\rm Ext}_{\Lambda}^{1}(S,S)\simeq{\rm Hom}_{\Lambda}(\Omega^{1}S,S)$.
Then one gets the assertion.
(2) is a straight result of (1). $\hfill{\square}$
Now we can state the main theorem of this section.
Theorem 4.4 Let $\Lambda$ be an algebra with radical square zero. If $\Lambda$
admits a unique $\tau$-tilting module, then $\Lambda$ is local.
Proof. Firstly, we claim that for any simple module $S$, $\Omega^{1}S\simeq
S^{t}$ for some positive integer $t$. By Lemma 4.3, we get that $S$ is a
direct summand of $\Omega^{1}S$. By Theorem 4.1 (2)(b), $\Omega^{1}S/S^{t}$
must be zero (otherwise, there will be an indecomposable non-projective
$\tau$-rigid module). The assertion holds.
Next we will show that there is a unique simple module $S$ in $\mathop{\rm
mod}\nolimits\Lambda$. Suppose there is another simple module $S^{\prime}$.
Then by the claim above we get that there is a positive integer $m$ such that
$\Omega^{1}S^{\prime}\simeq{S^{\prime}}^{m}$. So one can get ${\rm
Hom}_{\Lambda}(P_{0}(S),P_{0}(S^{\prime}))={\rm
Hom}_{\Lambda}(P_{0}(S^{\prime}),P_{0}(S))=0$. Notice that $\Lambda$ is basic
and connected, this is a contradiction. $\hfill{\square}$
In Theorem 4.4, if $\Lambda$ is a finite dimensional algebra over an
algebraically closed field $K$, one can get that the quiver of $\Lambda$ is
just one vertex with several cycles. Then one determines the structure of the
algebras completely. After finishing Theorem 4.4, the author was told by
Professor Iyama that he can prove that a basic connected algebra with a unique
$\tau$-tilting module is a local algebra by mutation.
5 Examples
In this section we give examples to show our results. Let Q be a quiver.
Denote by $P(i)$, $I(i)$ and $S(i)$ the indecomposable projective module,
indecomposable injective module and the simple module according to the vertex
$i\in\textbf{Q}$, respectively. The following example is a Nakayama algebra
with $r^{2}=0$.
Example 5.1 Let $\Lambda$ be given by the quiver:
$\textstyle{2\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{a}$$\textstyle{3\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{a}$$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{a}$$\textstyle{4\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{a}$
with relations $a^{2}=0$. By Theorem 3.4, every indecomposable module is
$\tau$-rigid. The $\tau$-tilting modules are of the following forms:
(1) $0$-simple module. $P(1)\bigoplus P(2)\bigoplus P(3)\bigoplus P(4)$
(2) $1$-simple module. $P(1)\bigoplus S(1)\bigoplus P(3)\bigoplus
P(4),P(1)\bigoplus P(2)\bigoplus S(2)\bigoplus P(4)$
$P(1)\bigoplus P(2)\bigoplus P(3)\bigoplus S(3),S(4)\bigoplus P(2)\bigoplus
P(3)\bigoplus P(4)$
(3) $2$-simple modules. $P(1)\bigoplus S(1)\bigoplus P(3)\bigoplus
S(3),S(4)\bigoplus P(2)\bigoplus S(2)\bigoplus P(4)$
In the following we give an example to show that there does exist an algebra
of finite global dimension with $r^{2}=0$ which is of finite type but not
Nakayama.
Example 5.2 Let $\Lambda$ is given by the quiver:
$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{a}$
$\textstyle{3\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{a}$$\textstyle{4}$$\textstyle{2\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{a}$
with relations $a^{2}=0$. Then
(1) $\Lambda$ is a representation finite algebra of global dimension $2$ with
$r^{2}=0$.
(2) $\tau S(3)\simeq S(4)$ and $\tau I(3)\simeq S(3)$. So $S(3)$ and $I(3)$
are $\tau$-rigid by Theorem 3.6.
(3) By Theorem 3.12 or Theorem 3.15, Corollary 3.14 and (2), every
indecomposable $\Lambda$-module is $\tau$-rigid.
To show Theorem 3.12 and Theorem 3.15, in the following we will construct an
algebra $\Lambda$ and an indecomposable $\Lambda$-module $M$ such that
$\Lambda$ is of finite type and finite global dimension with $r^{2}=0$ and
there is no non-zero direct summand of $P_{0}(M)$ and $P_{1}(M)$ in common.
Example 5.3 Let $\Lambda$ is given by the quiver:
$\textstyle{2\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{a}$$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{a}$$\textstyle{3}$$\textstyle{4\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{a}$
with relations $a^{2}=0$. Then
(1) $\Lambda$ is a representation finite algebra of global dimension $2$ with
$r^{2}=0$.
(2) The injective module $I(3)$ has a minimal projective resolution:
$P(2)\bigoplus P(3)\rightarrow P(2)\bigoplus P(4)\rightarrow I(3)\rightarrow
0.$ By Theorem 3.12 (3), $I(3)$ is not $\tau$-rigid.
(3) $\tau^{2}I(3)\simeq S(2)$, then by Theorem 3.6 (2) $\tau I(3)$ is
$\tau$-rigid. So $\tau M$ is $\tau$-rigid can not imply that $M$ is
$\tau$-rigid in general.
Acknowledgement Part of the paper was written when the author was visiting
Bielefeld University in October and November of 2012 with the support of CRC
701. The author would like to thank Prof. Henning Krause for invitation and
hospitality. He also wants to thank Prof. Claus Michael Ringel, Dr. Zhe Han
and other people in Bielefeld for useful discussion and kind help. The author
is also indebted to Prof. Osamu Iyama for kind help and Tiwei Zhao for careful
reading. The research of the author is carried out with the support of NSFC
(Nos.11101217, 11171142), NSF of Jiangsu Province (No. BK20130983) and NSF for
Colleges and Universities in Jiangsu Province of China (No.11KJB110007).
## References
* [A] T. Adachi, $\tau$-tilting modules over Nakayama algebras, arXiv:1309.2216.
* [AIR] T. Adachi, O. Iyama and I. Reiten, $\tau$-TILTING THEORY, arxiv:1210.1036.
* [AiI] T. Aihara and O. Iyama, Silting mutation in triangulated categories, J. Lond. Math. Soc. 85 (2012), no. 3, 633-668.
* [AsSS] I. Assem, D. Simson and A. Skowroński, Elements of the Representation Theory of Associative Algebras. Vol. 1. Techniques of Reperesentation Theory. London Math. Soc. Student Texts 65, Cambridge Univ. Press, Cambridge, 2006.
* [AuPR] M. Auslander, M. I. Platzeck and I. Reiten, Coxeter functions without diagrams, Trans. Amer. Math. Soc. 250 (1979), 1-12.
* [AuR] M. Auslander and I. Reiten, Cohen-Macaulay and Gorenstein algebras. In: Representation theory of finite groups and finite-dimensional algebras, Bielefeld, 1991, edited by G. O. Michler and C. M. Ringel, Progr. Math., Vol. 95, Birkh auser, Basel, 1991, 221-245
* [AuRS] M. Auslander, I. Reiten and S.O. Smol$\phi$, Representation Theory of Artin Algebras. Corrected reprint of the 1995 original. Cambridge Studies in Adv. Math. 36, Cambridge Univ. Press, Cambridge, 1997.
* [AuS] M. Auslander and S.O. Smol$\phi$, Almost split sequences in subcategories, J. Algebra 69(1981), 426-454, Addendum; J. Algebra 71(1981), 592-594.
* [BB] S. Brenner and M. C. R. Bulter, Generalization of Bernstein-Gelfand-Ponomarev reflection functors, Lecture Notes in Math. 839, Springer-Verlag (1980), 103-169.
* [C] X. W. Chen, Algebras with radical square zero are either self-injective or CM-free, Proc. Amer. Math. Soc. 140 (2012), 93-98.
* [HR] D. Happel and C. M. Ringel, Tilted algebras, Trans. Amer. Math. Soc. 274 (1982), 399-443.
* [HuZ] Z. Y. Huang and X. J. Zhang, Higher Auslander algebras admitting trivial maximal orthogonal subcategories, J. Algebra 330 (2011), 375-387.
* [IY] O. Iyama and Y. Yoshino, Mutations in triangulated categories and rigid Cohen-Macaulay modules, Invent. Math. 172 (2008), 117-168.
* [RX] C. M. Ringel and B. L. Xiong, On radical square zero rings, arxiv: 1112.1422.
|
arxiv-papers
| 2012-11-23T22:48:47 |
2024-09-04T02:49:38.336785
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Xiaojin Zhang",
"submitter": "Xiaojin Zhang",
"url": "https://arxiv.org/abs/1211.5622"
}
|
1211.5624
|
# A Note On Gorenstein Projective Conjecture II
Xiaojin Zhang
School of Mathematics and Statistics, Nanjing University of Information
Science and Technology,
Nanjing 210044, P. R. China E-mail address: [email protected]
Abstract In this paper, we prove that Gorenstein projective conjecture is left
and right symmetric and the co-homology vanishing condition can not be reduced
in general. Moreover, Gorenstein projective conjecture is proved to be true
for CM-finite algebras.
Keywords Gorenstein projective, CM-finite algebras, Gorenstein projective
conjecture
AMS(2000) Subject Classification 16G10, 16E05.
1 Introduction
For the representation theory of Artinian algebras, the Auslander-Reiten
conjecture (ARC) which is related to generalized Nakayama conjecture (GNC) is
everything. It was proposed by Auslander and Reiten, which says that $M$ is
projective if ${\rm Ext}_{\Lambda}^{i}(M\bigoplus\Lambda,M\bigoplus\Lambda)=0$
for any $i\geq 1$(See [2,3]). Achievements for special cases have been got by
K. R. Fuller, B. Zimmermann-Huisgen, A. Mar$\dot{o}$ti and G. Wilson…(See
[10,15,16]). In general it is still open now. As a special case of Auslander-
Reiten conjecture, Luo and Huang proposed the following Gorenstein projective
conjecture (GPC) in 2008:
Let $\Lambda$ be an Artinian algebra and let $M$ be a Gorenstein projective
module. Then $M$ is projective if and only if ${\rm Ext}_{\Lambda}^{i}(M,M)=0$
for any $i\geq 1$(See [14]).
It is still unknown whether the Auslander-Reiten conjecture is left and right
symmetric. But as we stated Gorenstein projective conjecture is a special
case. So what about the left and right symmetric property of Gorenstein
projective conjecture? In this paper, we will give a positive answer to this
question.
By the definition of Gorenstein projective conjecture, for an algebra
$\Lambda$ the truth of Auslander-Reiten conjecture implies the truth of
Gorenstein projective conjecture. So we can get a large class of algebras
satisfying Gorenstein projective conjecture. It is interesting to ask: Is
there an algebra satisfying Gorenstein projective conjecture while for which
the Auslander-Reiten conjecture is unknown?
Recall that an algebra is called CM-finite (of finite Cohen-Macaulay type) if
there are only finitely many isomorphism classes of indecomposable finitely
generated Gorenstein projective modules. CM-finite algebras are studied by
several authors recently (see [5,6,7,12,13]). Although the Auslander-Reiten
conjecture for this class of algebras is unknown, we will give a positive
answer to the second question above.
The paper is organized as follows:
In Section 2, based on some facts of Gorenstein projective modules, we will
show the symmetric property of Gorenstein projective conjecture. Moreover, an
example is given to show that the condition ’${\rm Ext}_{\Lambda}^{i}(M,M)=0$
for any $i\geq 1$’ in Gorenstein projective conjecture can not be reduced to
’${\rm Ext}_{\Lambda}^{i}(M,M)=0$ for some positive integer $t$ and any $1\leq
i\leq t$ .’
In Section 3, we will prove that CM-finite algebras satisfy the Gorenstein
projective conjecture by showing the Gorenstein projective conjecture holds
for algebras with finite self-orthogonal indecomposable Gorenstein projective
modules (up to isomorphisms).
Throughout the paper, $\Lambda$ is an Artinian algebra and all modules are
finitely generated left $\Lambda$-modules.
2 Symmetric property of Gorenstein projective conjecture
In this section we will show the symmetric property of Gorenstein projective
conjecture. First, we need to recall some notions and lemmas. The following
definition is due to Auslander, Briger, Enochs and Jenda (see [1,8,9]).
Definition 2.1 A module $M$ is called Gorenstein projective if for any $i\geq
1$
(1) ${\rm Ext}_{\Lambda}^{i}(M,\Lambda)=0$
(2) ${\rm Ext}_{\Lambda}^{i}(\mathop{\rm Tr}\nolimits M,\Lambda)=0$
Where $\mathop{\rm Tr}\nolimits M$ denotes the Auslander transpose of $M$.
Let $\dots\rightarrow P_{2}(M)\rightarrow P_{1}(M)\rightarrow M\rightarrow 0$
be a minimal projective resolution of $M$. Denoted by $\Omega^{i}M$ the $i$-th
syzygy of $M$. Dually, one can define $\Omega^{-i}M$. We remark that for any
$i\geq 0$ $\Omega^{i}M$ is a Gorenstein projective if so is $M$. Let
$\mathscr{C}$ be the subcategory of $\mathop{\rm mod}\nolimits\Lambda$
consisting of modules $M$ such that ${\rm Ext}_{\Lambda}^{j}(M,\Lambda)=0$ for
any $j\geq 1$ and $\mathscr{D}$ a subcategory of $\mathop{\rm
mod}\nolimits\Lambda$ consisting of Gorenstein projective modules. We use
$\underline{\mathscr{C}}$ and $\underline{\mathscr{D}}$ to denote the stable
subcategory of $\mathscr{C}$ and $\mathscr{D}$ modulo projectives,
respectively. We recall the following proposition from [1].
Proposition 2.2 (1)
$\Omega:\underline{\mathscr{C}}\rightarrow\underline{\mathscr{C}}$ is a fully
faithful functor .
(2) $\Omega:\underline{\mathscr{D}}\rightarrow\underline{\mathscr{D}}$ is an
equivalence.
(3) $(-)^{*}={\rm Hom}(-,\Lambda):\mathscr{D}\rightarrow\mathscr{D}^{o}$ is a
duality, where $\mathscr{D}^{o}$ denotes the subcategory of Gorenstein
projective right $\Lambda$-modules.
Proof. (1) is a result of Auslander and Bridger. One can get (2) by (1) and
the remark above. (3) is well-known. $\hfill{\square}$
Recall that a module $M$ is called self-orthogonal if ${\rm
Ext}_{\Lambda}^{j}(M,M)=0$ for any $j\geq 1$. The following self-orthogonal
property is essential to the main result in this section.
Lemma 2.3 Let $\Lambda$ be an algebra. Then for any $M\in\mathscr{D}$ and
$i\geq 1$, $M$ is self-orthogonal if and only if $M^{*}$ is self-orthogonal.
Proof. Since $(-)^{*}$ is a duality between $\mathscr{D}$ and
$\mathscr{D}^{o}$, it is enough to show that ${\rm Ext}_{\Lambda}^{i}(M,M)=0$
implies ${\rm Ext}_{\Lambda}^{i}(M^{*},M^{*})=0$.
One can take the following minimal projective resolution of $M$:
$\dots\rightarrow P_{1}\rightarrow P_{0}\rightarrow M\rightarrow 0\ \ \ \ (1)$
Applying the functor ${\rm Hom}(-,M)$ to sequence $(1)$ above, since ${\rm
Ext}_{\Lambda}^{i}(M,M)=0$ we get the following exact sequence
$0\rightarrow{\rm Hom}(M,M)\rightarrow{\rm Hom}(P_{0},M)\rightarrow{\rm
Hom}(P_{1},M)\rightarrow\cdots\ \ \ (2)$
On the other hand, applying the functor $(-)^{*}$ to the sequence $(1)$, since
$M\in\mathscr{D}\subseteq\mathscr{C}$ one can show the following exact
sequence
$0\rightarrow
M^{*}\rightarrow{P_{0}}^{*}\rightarrow{P_{1}}^{*}\rightarrow\cdots\ \ \ (3)$
Then by using the functor ${\rm Hom}(M^{*},-)$ on the exact sequences (3), one
has the following exact sequence
$0\rightarrow{\rm Hom}(M^{*},M^{*})\rightarrow{\rm
Hom}(M^{*},{P_{0}}^{*})\rightarrow{\rm
Hom}(M^{*},{P_{1}}^{*})\rightarrow\cdots\ \ \ (4)$
Using Proposition 2.2(3), we get ${\rm
Ext}_{\Lambda}^{i}(M^{*},M^{*})\simeq{\rm Ext}_{\Lambda}^{i}(M,M)=0$ by
comparing sequences (2) with (4). $\hfill{\square}$
Although the symmetric property of Auslander-Reiten conjecture is still
unknown, we are able to show the symmetric properties of Gorenstein projective
conjecture.
Theorem 2.4 Let $\Lambda$ be an algebra and let $\Lambda^{o}$ be the opposite
ring of $\Lambda$. Then $\Lambda$ satisfies the Gorenstein projective
conjecture if and only if $\Lambda^{o}$ satisfies the Gorenstein projective
conjecture.
Proof. $\Rightarrow$ Assume that $N\in\mathscr{D}^{o}$ and ${\rm
Ext}_{\Lambda}^{i}(N,N)=0$ for any $i\geq 1$. By Proposition 2.2, there is a
$M\in\mathscr{D}$ such that $M^{*}\simeq N$. By Lemma 2.3 one gets that ${\rm
Ext}_{\Lambda}^{i}(M,M)=0$. Note that $\Lambda$ satisfies the Gorenstein
projective conjecture, we have $M$ is projective, and hence $N\simeq M^{*}$ is
projective. Conversely, one can formula the proof above. $\hfill{\square}$
Notice that Gorenstein projective conjecture is a special case of Auslander-
Reiten conjecture. It is natural to consider whether the assumption of
Gorenstein projective conjecture can be reduced. In particular, whether can
the condition ’${\rm Ext}_{\Lambda}^{i}(M,M)=0$ for any $i\geq 1$’ in GPC be
reduced to ’${\rm Ext}_{\Lambda}^{i}(M,M)=0$ for some positive integer $t$ and
any $1\leq i\leq t$ ? At the end of this section, we construct an example to
give a negative answer to the question.
Example 2.5 Let $n>t+1$ be a positive integer and let $\Lambda$ be the algebra
generated by the following quiver
$\textstyle{n\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{a_{n}}$$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{a_{1}}$$\textstyle{n-1\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{a_{n-1}}$$\textstyle{2\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{a_{2}}$$\textstyle{n-2}$$\textstyle{3\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{5\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{4\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$
modulo the ideal $\\{a_{n}a_{1}=0,a_{i}a_{i+1}=0|1\leq i\leq n-1\\}$. Denoted
by $S(j)$ the simple module according to the dot $j$. Then
(1) $\Lambda$ is a Nakayama self-injective algebra.
(2) $S(j)$ is Gorenstein projective such that ${\rm
Ext}_{\Lambda}^{i}(S(j),S(j))=0$ for $t\geq i\geq 1$ and $1\leq j\leq n$, but
it is not projective.
3 Gorenstein projective conjecture for CM-finite algebras
In this section we try to find a class of algebras which satisfy Gorenstein
projective conjecture and for which the Auslander-Reiten conjecture is
unknown. They are CM-finite algebras. We begin with the following definition
due to Beligiannis
Definition 3.1 An algebra is called CM-finite (of finite Cohen-Macaulay type )
if there are only finite number of indecomposable Gorenstein projective
modules (up to isomorphisms).
Remark 3.2 (1) Algebras of finite representation type or finite global
dimension are CM-finite.
(2) There does exist a CM-finite algebra $\Lambda$ such that $\Lambda$ is of
infinite type and the global dimension of $\Lambda$ is infinite [13].
(3) There does exist a CM-finite algebra which is not Gorenstein [5].
(4)An algebra with a trivial maximal n-orthogonal subcategory for some
positive integer $n$ is CM-finite [11].
Let $\mathscr{C}$, $\mathscr{D}$ and $\mathscr{D}^{o}$ be as in Section 2. The
following lemma partly from [1] plays an important role in the proof of the
main results.
Lemma 3.3 For any $M\in\mathscr{C}$ and $N\in\mathop{\rm
mod}\nolimits\Lambda$, then ${\rm Ext}_{\Lambda}^{1}(M,N)\simeq\underline{{\rm
Hom}}_{\Lambda}(\Omega^{1}M,N)$ and hence ${\rm
Ext}_{\Lambda}^{i}(M,N)\simeq\underline{{\rm Hom}}_{\Lambda}(\Omega^{i}M,N)$
for any $i\geq 1$.
Proof. The first assertion is a result of Auslander and Bridger. For the
second one, the case $i=1$ is clear. We only need to show the case $i\geq 2$.
Taking a minimal projective resolution of $M$, one gets ${\rm
Ext}_{\Lambda}^{i}(M,N)\simeq{\rm Ext}_{\Lambda}^{1}(\Omega^{i-1}M,N)$ for any
$i\geq 2$. Notice that $M\in\mathscr{C}$, by Proposition 2.2 one can show
$\Omega^{i-1}M\in\mathscr{C}$. Using the first assertion, we are done.
$\hfill{\square}$.
The following proposition gives a connection between the self-orthogonal
property of $M$ and that of $\Omega^{i}M$ for any $i\geq 0$.
Proposition 3.4 Let $M\in\mathscr{C}$ $(\mathscr{D})$. Then
(1) $\Omega^{i}M$ is self-orthogonal in $\mathscr{C}$ $(\mathscr{D})$ for any
$i\geq 0$ if $M$ is self-orthogonal.
(2) If $M\in\mathscr{D}$ is self-orthogonal, then ${\rm Tr}M$ is self-
orthogonal in $\mathscr{D}^{o}.$
Proof. (1) For the case $i=0$, there is nothing to prove. By Proposition 2.2,
we only need to prove the case $i=1$, that is, ${\rm Ext}_{\Lambda}^{j}(\Omega
M,\Omega M)=0$ for any $j\geq 1$. One gets ${\rm Ext}_{\Lambda}^{j}(\Omega
M,\Omega M)\simeq\underline{{\rm Hom}}_{\Lambda}(\Omega^{j+1}M,\Omega
M)\simeq\underline{{\rm Hom}}_{\Lambda}(\Omega^{j}M,M)$ by Proposition 2.2 and
Lemma 3.3. Using the second equation of Lemma 3.3, one can show
$\underline{{\rm Hom}}_{\Lambda}(\Omega^{j}M,M)\simeq{\rm
Ext}_{\Lambda}^{j}(M,M)=0$ since M is self-orthogonal.
(2) Taking a minimal projective resolution of $M$, it is not difficult to show
that ${\rm Tr}M\simeq(\Omega^{2}M)^{*}$ since $M\in\mathscr{D}$. By
Propositions 2.2 and 3.4(1), $\Omega^{2}M$ is also self-orthogonal in
$\mathscr{D}$. Then by Lemma 2.3 and Proposition 2.2 one gets the assertion.
$\hfill{\square}$
The following proposition is crucial to the main results.
Proposition 3.5 Let $\Lambda$ be an algebra with only finite (up to
isomorphism) self-orthogonal indecomposable modules in $\mathscr{D}$
($\mathscr{C}$) and let $M$ be a self-orthogonal indecomposable module in
$\mathscr{D}$ ($\mathscr{C}$). Then $M$ is projective.
Proof. Denoted by $\\{M_{1},M_{2},...,M_{t}\\}$ the complete set of non-
isomorphic self-orthogonal indecomposable modules in $\mathscr{D}$
($\mathscr{C}$). Then $M\simeq M_{i}$ for some $1\leq i\leq t$.
Suppose that M is not projective. Then by Proposition 2.2 and Proposition 3.4,
we have the following set of self-orthogonal indecomposable modules
$\mathcal{S}=\\{\Omega^{i}M|1\leq i\\}$ in $\mathscr{D}$ ($\mathscr{C}$).
We claim that there are two modules $\Omega^{i}M,\Omega^{j}M$ in $\mathcal{S}$
such that $\Omega^{i}M\simeq\Omega^{j}M$ for some $i<j$. Otherwise, one gets
infinite number of non-isomorphic self-orthogonal indecomposable modules in
$\mathscr{D}$ ($\mathscr{C}$), a contradiction. Again by Proposition 2.2, one
gets $M\simeq\Omega^{j-i}M$.
Considering the following exact sequence $0\rightarrow\Omega^{j-i}M\rightarrow
P\rightarrow\Omega^{j-i-1}M\rightarrow 0$, we will show ${\rm
Ext}_{\Lambda}^{1}(\Omega^{j-i-1}M,\Omega^{j-i}M)=0$. Since
$\Omega^{j-i-1}M\in\mathscr{D}$ ($\mathscr{C}$) and $M\simeq\Omega^{j-i}M$, we
get ${\rm
Ext}_{\Lambda}^{1}(\Omega^{j-i-1}M,\Omega^{j-i}M)\simeq\underline{{\rm
Hom}}_{\Lambda}(\Omega^{j-i}M,M)\simeq{\rm Ext}_{\Lambda}^{j-i}(M,M)=0$ by
Proposition 2.2 and Lemma 3.3, and hence $M$ is projective, a contradiction.
The assertion holds true.$\hfill\square$
Now we are in the position to show the main result of this section.
Theorem 3.6 Let $\Lambda$ be CM-finite. Then $\Lambda$ satisfies Gorenstein
projective conjecture.
Proof. Since $\Lambda$ is CM-finite, then there are only finite (up to
isomorphisms) indecomposable modules in $\mathscr{D}$. One can show the result
by Proposition 3.5. $\hfill{\square}$
Although the Auslander-Reiten conjecture for CM-finite algebras is unknown
now, we have the following
Proposition 3.7 Let $\Lambda$ be a CM-finite algebra and let $M$ be a
$\Lambda$-module satisfying ${\rm Ext}_{\Lambda}^{i}(M,M\bigoplus\Lambda)=0$
for any $i\geq 1$. Then the following are equivalent.
(1) $M$ is projective.
(2) $M$ is Gorenstein projective.
Proof. $(1)\Rightarrow(2)$ is trivial. The converse follows from Theorem 3.6.
$\hfill{\square}$
We end this section with two open questions related to this paper.
Question 1 Does the Gorenstein projective conjecture hold for virtually
Gorenstein algebras (see [5])?
Question 2 Does the Auslander-Reiten conjecture hold for CM-finite algebras?
Acknowledgements Part of the paper was finished when the author stayed in
University of Bielefeld with the support of DAAD Fellowship. The author would
like to thank people in Bielefeld for their help. He also wants to thank Prof.
Xiaowu Chen, Prof. Zhaoyong Huang, Prof. Shengyong Pan and Prof. Changchang Xi
for useful suggestions. The research of the author is supported by NSFC(Grant
No.11101217) and NSF for Colleges and Universities in Jiangsu Province (Grant
No.11KJB110007).
## References
* [1] Auslander M and Bridger M. Stable module theory, Memoirs Amer. Math. Soc. 94, Amer. Math. Soc., Providence, RI, 1969\.
* [2] Auslander M and Reiten I. On a generalized version of Nakayama conjecture, Proc. Amer. Math. Soc. 52(1975), 69–74.
* [3] Auslander M and Reiten I. Applications of contravariantly finite subcategories. Adv. Math. 86(1991), 111–152.
* [4] Bass H. Injective dimension in Noetherian Ring, Trans. Amer. Math. Soc. 102 (1962), 18-29.
* [5] Beligiannis A. On Algebras of Finite Cohen-Macaulay Type. Adv. Math 226, (2011), 1973–2019.
* [6] Chen X W. An Auslander-type result for gorenstein-projective modules. Adv. Math. 218 (2008), 2043–2050.
* [7] Christensen L W, Piepmeyer G, Striuli J and Takahashi R. Finite Gorenstein representation type implies simple singularity. Adv. Math. 218 (2008) 1012–1026.
* [8] Enochs E E and Jenda O M G. Gorenstein injective and projective modules. Math. Zeit. 220 (1995), 611–633.
* [9] Enochs E E and Jenda O M G. Relative Homological Algebra. de Gruyter Exp. Math., vol. 30, Walter de Gruyter Co., 2000.
* [10] Fuller K R and Zimmermann-Huisgen B. On the generalied Nakayama conjecture and the Cartan determinant problem. Tran. Amer. Math. Soc 294(2) (1986), 679–691.
* [11] Huang Z Y and Zhang X J. The exsistence of maximal n-orthogonal subcategories. J. Algebra 321(10) (2009), 2829–2842.
* [12] Li Z W and Zhang P. Gorenstein algebras of finite Cohen-Macaulay type. Adv. Math. 218 (2010), 728–734.
* [13] Li Z W and Zhang P. A construction of Gorenstein-projective modules. J. Algebra 323 (2010) 1802–1812.
* [14] Luo R and Huang Z Y. When are torsionless modules projective? J. Algebra, 320(5)(2008), 2156–2164.
* [15] Mar$\dot{o}$ti A. A proof of a generalized Nakayama conjecture. Bull. Lond. Math. Soc 38(5) (2006), 777–785.
* [16] Wilson G. The Cartan map on categories of graded modules. J. Algebra 85 (1983), 390–398.
|
arxiv-papers
| 2012-11-23T23:11:08 |
2024-09-04T02:49:38.346620
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Xiaojin Zhang",
"submitter": "Xiaojin Zhang",
"url": "https://arxiv.org/abs/1211.5624"
}
|
1211.5669
|
# Analysis-suitable T-splines: characterization, refineability, and
approximation
Xin Li
School of mathematical science, USTC,
Hefei, Anhui Province 230026, P. R. China
[email protected]
M. A. Scott
Department of Civil and Environmental Engineering,
Brigham Young University, Provo, UT 84602, USA
[email protected]
###### Abstract
We establish several fundamental properties of analysis-suitable T-splines
which are important for design and analysis. First, we characterize T-spline
spaces and prove that the space of smooth bicubic polynomials, defined over
the extended T-mesh of an analysis-suitable T-spline, is contained in the
corresponding analysis-suitable T-spline space. This is accomplished through
the theory of perturbed analysis-suitable T-spline spaces and a simple
topological dimension formula. Second, we establish the theory of analysis-
suitable local refinement and describe the conditions under which two
analysis-suitable T-spline spaces are nested. Last, we demonstrate that these
results can be used to establish basic approximation results which are
critical for analysis.
Keywords: T-splines; isogeometric analysis; local refinement; analysis-
suitable; approximation.
## 1 Introduction
T-splines were originally introduced in Computer Aided Design (CAD) as a
superior alternative to NURBS [1] and have since emerged as an important
technology across several disciplines including industrial, architectural, and
engineering design, manufacturing, and engineering analysis. T-splines can
model complicated designs as a single, watertight geometry and can be locally
refined [2, 3]. These basic properties make it possible to merge multiple
NURBS patches into a single T-spline [4, 1] and any trimmed NURBS model can be
represented as a watertight T-spline [5].
The use of T-splines as a basis for isogeometric analysis has gained
widespread attention [6, 7, 3, 8, 9, 10, 11, 12, 13]. Isogeometric analysis
was introduced in [14] and described in detail in [15]. The isogeometric
paradigm is simple: use the smooth spline basis that defines the geometry as
the basis for analysis. Traditional design-through-analysis procedures such as
geometry clean-up, defeaturing, and mesh generation are simplified or
eliminated entirely. Additionally, the higher-order smoothness provides
substantial gains to analysis in terms of accuracy and robustness of finite
element solutions [16, 17, 18].
An important development in the evolution of isogeometric analysis was the
advent of Analysis-suitable T-splines (ASTS). ASTS are a mildly restricted
subset of T-splines which are optimized to simultaneously meet the needs of
design and analysis [19, 3]. Linear independence of analysis-suitable T-spline
blending functions was established in [19]. An efficient local refinement
algorithm for ASTS was developed in [3]. Later, it was shown that a dual
basis, constructed as in the tensor product settting, could be generalized to
ASTS [20]. This characteristic of ASTS is called dual compatibility. These
results were then generalized to ASTS surfaces of arbitrary degree in [21].
In this paper we continue to develop the theory of ASTS spaces. Specifically,
we provide a rigorous characterization of ASTS and show that the space of
smooth parametric bicubic polynomials, defined over the extended T-mesh of an
ASTS, is contained in the corresponding ASTS space. To accomplish this, the
theory of perturbed ASTS spaces is developed and an ASTS dimension formula is
established in terms of the topology of the extended T-mesh. We note that,
unlike existing approaches, our dimension formula does not require that the
T-mesh have any particular nesting structure. We then show that this
characterization, coupled with the dual compatibility of ASTS, can be used to
prove that ASTS spaces possess the same optimal approximation properties as
tensor product B-spline spaces [22]. Next, we prove under what conditions two
ASTS spaces are nested. This provides the theoretical justification for the
analysis-suitable local refinement algorithm in [3] and provides a foundation
upon which adaptive isogeometric analysis procedures may be developed in the
future.
This paper is organized as follows. Section 2 describes the T-mesh in index
space, T-junctions, and the extended T-mesh. T-splines in the parametric
domain, blending functions, and T-spline spaces are defined in Section 3.
Section 4 describes the conditions under which a T-spline is analysis-
suitable. The theory of smoothly perturbed ASTS is developed in Section 5.
Section 6 proves the conditions under which two ASTS spaces are nested. Using
the characterization of ASTS spaces and dual compatibility several basic
approximation results are proven in Section 7. Finally, Section 8 proves the
dimension of ASTS spaces.
## 2 The T-mesh
An important object underlying T-spline spaces is the T-mesh. A T-mesh is used
to determine T-spline basis functions and how they are arranged with respect
to one another. In other words, the mesh topology of the T-mesh determines the
functional properties of the resulting space. In an attempt to adhere to a
single notation and to reduce confusion, we define a T-mesh following much of
the notation given in [19, 20]. For quick reference, A lists the most
important notational conventions used throughout the text and where they are
defined.
### 2.1 Definition
A T-mesh $\mathsf{T}$ is a rectangular partition of the index domain
$[\underline{m},\overline{m}]\times[\underline{n},\overline{n}]$,
$\underline{m},\overline{m},\underline{n},\overline{n}\in\mathbb{Z}$, where
all rectangle corners (or vertices) have integer coordinates and all
rectangles are open sets. Each vertex in $\mathsf{T}$ is a singleton subset of
$\mathbb{Z}^{2}$. We denote all vertices of $\mathsf{T}$ by $\mathsf{V}$. An
edge of $\mathsf{T}$ is a segment between vertices of $\mathsf{T}$ that does
not intersect any rectangle of $\mathsf{T}$. We note that edges do not contain
vertices and they are open at their endpoints. We denote all edges of
$\mathsf{T}$ by $\mathsf{E}$. Figure 1 shows an example of a T-mesh. The
notation $\mathsf{T}^{1}\subseteq\mathsf{T}^{2}$ will indicate that
$\mathsf{T}^{2}$ can be created by adding vertices and edges to
$\mathsf{T}^{1}$.
Figure 1: A T-mesh.
The valence of a vertex $V\in\mathsf{V}$ is the number of edges such that $V$
is an endpoint. We only allow valence three (called T-junctions) or four
vertices. Note that valence two vertices, other than the four corners, are
eliminated from the definition.
The horizontal (resp., vertical) skeleton of a T-mesh is denoted by
$h\mathsf{S}$ (resp., $v\mathsf{S}$), and is the union of all horizontal
(resp., vertical) edges and all vertices. Finally, we denote the skeleton to
be the union $\mathsf{S}=h\mathsf{S}\cup v\mathsf{S}$. For a given vertex
$a=\\{(i,j)\\}$ we define $h\mathsf{J}(a):=\\{k\in\mathbb{Z}:\\{k\\}\times
a\subset v\mathsf{S}\\}$ and
$v\mathsf{J}(a):=\\{k\in\mathbb{Z}:a\times\\{k\\}\subset h\mathsf{S}\\}$. We
assume that these two sets are ordered.
We split the index domain
$\mathsf{R}=[\underline{m},\overline{m}]\times[\underline{n},\overline{n}]$
into an active region $\mathsf{AR}$ and a frame region $\mathsf{FR}$ such that
$\mathsf{R}=\mathsf{FR}\cup\mathsf{AR}$ and
$\mathsf{AR}=[\underline{m}+2,\overline{m}-2]\times[\underline{n}+2,\overline{n}-2]$,
and
$\mathsf{FR}=([\underline{m},\underline{m}+2]\cup[\overline{m}-2,\overline{m}])\times[\underline{n},\overline{n}]\cup[\underline{m},\overline{m}]\times([\underline{n},\underline{n}+2]\cup[\overline{n}-2,\overline{n}])$.
Note that both $\mathsf{FR}$ and $\mathsf{AR}$ are closed.
A symbolic T-mesh [19] is created from a T-mesh $\mathsf{T}$ by assigning a
symbol in Table 1 to each vertex in a tensor product mesh formed from the
index coordinates,
$\\{\underline{m},\ldots,\overline{m}\\}\times\\{\underline{n},\ldots,\overline{n}\\}\subset\mathbb{Z}^{2}$.
The symbol is chosen to match the mesh topology of $\mathsf{T}$. The symbolic
T-mesh corresponding to the T-mesh in Figure 1 is shown in Figure 2.
Table 1: Definition of possible symbols in a symbolic T-mesh Symbol | Correspondence with $\mathsf{T}$
---|---
$+$ | Valence 4 vertex, corner vertex, or valence 3 boundary vertex in $\mathsf{T}$
$\vdash$, $\dashv$, $\bot$, $\top$ | Oriented valence three vertex in $\mathsf{T}$
$|$ | Vertical edge in $\mathsf{T}$
$-$ | Horizontal edge in $\mathsf{T}$
$\cdot$ | No corresponding vertex or edge in $\mathsf{T}$
Figure 2: The symbolic representation of the T-mesh in Figure 1.
### 2.2 Admissible T-meshes
We say that a T-mesh is admissible if it satisfies three basic conditions.
First, we require that $\mathsf{S}\cap\mathsf{FR}$ contains the vertical
segments $\\{i\\}\times[\underline{n},\overline{n}]$ for
$i=\underline{m},\underline{m}+1,\underline{m}+2,\overline{m}-2,\overline{m}-1,\overline{m}$
and the horizontal segments $[\underline{m},\overline{m}]\times\\{j\\}$ for
$j=\underline{n},\underline{n}+1,\underline{n}+2,\overline{n}-2,\overline{n}-1,\overline{n}$.
These horizontal and vertical lines are for basis function definition near the
boundary. Second, we require that $\mathsf{S}\cap\mathsf{AR}$ contains the
vertical segments $\\{i\\}\times[\underline{n},\overline{n}]$ for
$i=\underline{m}+2,\underline{m}+3,\overline{m}-3,\overline{m}-2$ and the
horizontal segments $[\underline{m},\overline{m}]\times\\{j\\}$ for
$j=\underline{n}+2,\underline{n}+3,\overline{n}-3,\overline{n}-2$. Third, we
require that for any two vertices
$V_{1}=\\{(i_{1},j_{1})\\},V_{2}=\\{(i_{2},j_{2})\\}$ in $\mathsf{V}$, such
that $V_{1},V_{2}\subset\partial Q$ for some $Q\in\mathsf{T}$, if
$i_{1}=i_{2}$ (resp., $j_{1}=j_{2}$), then
$\\{i_{1}\\}\times]j_{1},j_{2}[\subset\mathsf{S}$ (resp.,
$]i_{1},i_{2}[\times\\{j_{1}\\}\subset\mathsf{S}$). From a practical point of
view these are minor restrictions. The T-mesh in Figure 1 is admissible.
We note that for convenience and simplicity, we often refer to only the active
region of an admissible T-mesh when speaking of a T-mesh. In all cases, we
assume that the frame region has an admissible topology.
### 2.3 Anchors and T-junctions
We define the anchors
$\mathsf{A}(\mathsf{T})=\\{A\in\mathsf{V}\cap\mathsf{AR}\\}$. We denote the
total number of anchors in $\mathsf{T}$ by $n^{A}$. We define
$\mathsf{J}\subset\mathsf{A}(\mathsf{T})$ to be the set of all valence three
vertices. These are called T-junctions. The symbols $\vdash$, $\dashv$,
$\bot$, $\top$ indicate the four possible orientations of a T-junction in a
symbolic T-mesh. A T-junction $T_{h}\in\mathsf{J}$ (resp.,
$T_{v}\in\mathsf{J}$) of type $\vdash$ and $\dashv$ (resp., $\bot$, $\top$)
and their extensions are called horizontal (resp., vertical). The solid white
and red circles in Figure 1 are anchors and the red circles are T-junctions.
### 2.4 Segments
We define a segment to be a closed line segment of contiguous vertices and
edges whose beginning and ending vertices are T-junctions (interior or
boundary). Given two horizontal (resp., vertical) segments
$G_{1}^{h},G_{2}^{h}$ defined over the intervals $[i_{1},j_{1}]\times a$ and
$[i_{2},j_{2}]\times b$ we say that $G_{1}^{h}\leq G_{2}^{h}$ if $i_{1}\leq
i_{2}$. We denote by $h\mathsf{G}$ (resp., $v\mathsf{G}$) the collection of
all horizontal (resp., vertical) segments, and by $\mathsf{G}=h\mathsf{G}\cup
v\mathsf{G}$ the collection of all segments. We define
$h\mathsf{G}(a)=h\mathsf{G}\cap([\underline{m},\overline{m}]\times a)$ and
$v\mathsf{G}(a)=v\mathsf{G}\cap(a\times[\underline{n},\overline{n}])$. We
assume these two sets are ordered. We denote the total number of segments in
$\mathsf{T}$ by $n^{G}$. We denote the total number of horizontal (resp.,
vertical) segments in $\mathsf{T}$ by $n^{G}_{h}$ (resp., $n^{G}_{v}$). We
denote the number of line segments in $h\mathsf{G}(a)$ (resp.,
$v\mathsf{G}(a)$) by $n^{G}_{h}(a)$ (resp., $n^{G}_{v}(a)$).
### 2.5 The extended T-mesh
T-junction extensions can be associated with each T-junction. For example,
given a T-junction
$T=\\{(\overline{\imath},\overline{\jmath})\\}\in\mathsf{J}$ of type $\vdash$
we extract from $h\mathsf{J}(\\{\overline{\jmath}\\})$ four consecutive
indices $i_{1},\ldots,i_{4}$ such that $\overline{\imath}=i_{3}$. We call
$ext^{e}(T)=[i_{1},\overline{\imath}]\times\\{\overline{\jmath}\\}$ the face
extension, $ext^{f}(T)=]\overline{\imath},i_{4}]\times\\{\overline{\jmath}\\}$
the edge extension for such kind of T-junction. Similarly, we can define the
face and edge extensions for the other kinds of T-junctions $\dashv$, $\bot$,
$\top$ which are illustrated in Figure 3.
Figure 3: The face extension (red) and edge extension (yellow) for four
different kinds of T-junctions.
We denote $ext(T)=ext^{e}(\mathsf{T})\cup ext^{f}(T)$ the extension of
T-junction $T$ and the union of all horizontal (resp., vertical) face
extensions by $hext^{f}(\mathsf{T})$ (resp., $vext^{f}(\mathsf{T})$), the
union of all face extensions by $ext^{f}(\mathsf{T})$, and the union of all
extensions (face and edge) by $ext(\mathsf{T})$. We define the extended
T-mesh, $\mathsf{T}_{ext}$, as the T-mesh created by adding to $\mathsf{T}$
all the T-junction extensions. In other words,
$\mathsf{T}_{ext}=\mathsf{T}\cup ext(\mathsf{T})$. We denote the total number
of vertices in $\mathsf{T}_{ext}$ by $n^{ext}$. The extended T-mesh
corresponding to the T-mesh in Figure 1 is shown in Figure 4.
Figure 4: The extended T-mesh corresponding to the T-mesh in Figure 1. The
crossing vertices are denoted by red stars, the overlap vertices are denoted
by green triangles, and the extended vertices are denoted by black squares.
Notice that the inactive vertices (grey circles) are also regarded as extended
vertices. The active vertices are denoted by hollow circles and the
T-junctions are denoted by red circles.
Adding T-junction extensions to $\mathsf{T}$ may introduce three additional
collections of vertices. The first, called crossing vertices and denoted by
$\mathsf{CV}$, is created from the intersection of crossing face extensions.
In other words,
$\mathsf{CV}=hext(\mathsf{T})\cap vext(\mathsf{T}).$
We denote the number of crossing vertices in $\mathsf{T}_{ext}$ by $n^{+}$. In
Figure 4 the crossing vertices are denoted by red stars.
The second, called overlap vertices and denoted by $\mathsf{OV}$, is created
from the intersection of overlapping face extensions with $\mathsf{S}$. In
other words,
$\mathsf{OV}=((\bigcap_{T_{h}\in\mathsf{J}}ext^{f}(T_{h}))\cap
v\mathsf{S})\cup((\bigcap_{T_{v}\in\mathsf{J}}ext^{f}(T_{v}))\cap
h\mathsf{S}).$
We denote the number of overlap vertices in $\mathsf{T}_{ext}$ by $n^{-}$. In
Figure 4 the overlap vertices are denoted by green triangles.
The third, called extended vertices and denoted by $\mathsf{EV}$, is created
from the intersection of face extensions and $\mathsf{S}$ while removing those
vertices which already correspond to overlap vertices. Additionally, all non-
anchor vertices are classified as extended vertices. In other words,
$\mathsf{EV}=((ext^{f}(\mathsf{T})\cap\mathsf{S})\setminus\mathsf{OV})\cup(\mathsf{V}\setminus\mathsf{A}(\mathsf{T})).$
We denote the number of extended vertices in $\mathsf{T}_{ext}$ by $n^{*}$. In
Figure 4 the extended vertices are denoted by black squares.
## 3 The parametric domain and T-spline spaces
Let $\Xi=(\xi_{\underline{m}},\ldots,\xi_{\overline{m}})$ and
$\Pi=(\eta_{\underline{n}},\ldots,\eta_{\overline{n}})$ be two global knot
vectors defined on the interval $\mathbb{R}$. Interior knots may have a
multiplicity of three while end knots may have a multiplicity of four. The
global knot vectors define a full parametric domain,
$\tilde{\Omega}\subset\mathbb{R}^{2}$, where
$\tilde{\Omega}=[\xi_{\underline{m}},\xi_{\overline{m}}]\otimes[\eta_{\underline{n}},\eta_{\overline{n}}]$
and a reduced parametric domain, $\hat{\Omega}\subset\tilde{\Omega}$, where
$\hat{\Omega}=[\xi_{\underline{m}+3},\xi_{\overline{m}-3}]\otimes[\eta_{\underline{n}+3},\eta_{\overline{n}-3}]$.
The T-mesh in the parametric domain is defined as the collection of non-empty
elements of the form
$\tilde{Q}=]\xi_{i_{1}},\xi_{i_{2}}[\times]\eta_{j_{1}},\eta_{j_{2}}[$ where
$Q=]i_{1},i_{2}[\times]j_{1},j_{2}[\in\mathsf{T}$. We denote those elements
where $\tilde{Q}\cap\hat{\Omega}\neq\emptyset$ by $\hat{Q}$. The extended
T-mesh in the parametric domain as well as all element related concepts are
defined similarly. Throughout this paper we use the index and parametric
representation of a T-mesh interchangeably with the context making the use
clear.
For each anchor $A=a\times b\in\mathsf{A}(\mathsf{T})$ we define its
horizontal (vertical) index vector $hv(A)$ ($vv(A)$, respectively) as a subset
of $h\mathsf{J}(b)$ ($v\mathsf{J}(a)$, respectively) where
$hv(A)=(i_{1},\ldots,i_{5})\in\mathbb{Z}^{5}$ contains five unique consecutive
indices in $h\mathsf{J}(b)$ with $\\{i_{3}\\}=a$. The vertical index vector,
denoted by $vv(A)$, is constructed in an analogous manner. We then associate a
T-spline blending function $N_{A}(\xi,\eta)$ with anchor $A$. The T-spline
blending functions are given by
$N_{A}(\xi,\eta):=B[\Xi_{A}](\xi)B[\Pi_{A}](\eta)\quad\forall(\xi,\eta)\in\hat{\Omega}$
(1)
where $B[\Xi_{A}](\xi)$ and $B[\Pi_{A}](\eta)$ are the cubic B-spline basis
functions associated with the local knot vectors
$\displaystyle\Xi_{A}$
$\displaystyle=[\xi_{i_{1}},\ldots,\xi_{i_{5}}]\subset\Xi$ (2)
$\displaystyle\Pi_{A}$
$\displaystyle=[\eta_{j_{1}},\ldots,\eta_{j_{5}}]\subset\Pi$ (3)
and $hv(A)=(i_{1},\ldots,i_{5})$ and $vv(A)=(j_{1},\ldots,j_{5})$.
Figure 5 illustrates the construction of a T-spline blending function
corresponding to anchor $A=\\{(3,3)\\}$. In this case, the local knot vectors
are $\Xi_{A}=[\xi_{1},\xi_{2},\xi_{3},\xi_{4},\xi_{6}]$ and
$\Pi_{A}=[\eta_{1},\eta_{2},\eta_{3},\eta_{4},\eta_{5}]$.
Figure 5: Inferring a T-spline blending function from a T-mesh.
A T-spline space $\mathcal{T}=\mathcal{T}(\mathsf{T},\Xi,\Pi)$ is simply the
span of the blending functions, $N_{A}$, $A\in\mathsf{A}(\mathsf{T})$.
## 4 Analysis-suitable T-splines
Analysis-suitable T-splines form a practically useful subset of T-splines.
ASTS maintain the important mathematical properties of the NURBS basis while
providing an efficient and highly localized refinement capability. Several
important properties of ASTS have been proven:
* •
The blending functions are linearly independent for any choice of knots [19].
* •
The basis constitutes a partition of unity (see Corollary 6.7).
* •
Each basis function is non-negative.
* •
They can be generalized to arbitrary degree [21].
* •
An affine transformation of an analysis-suitable T-spline is obtained by
applying the transformation to the control points. We refer to this as affine
covariance. This implies that all “patch tests” (see [23]) are satisfied a
priori.
* •
They obey the convex hull property.
* •
They can be locally refined [1, 3].
* •
A dual basis can be constructed [20, 21].
###### Definition 4.1.
An analysis-suitable T-spline is a T-spline whose T-mesh is analysis-suitable
[19]. A T-mesh is said to be analysis-suitable if it is admissible and no
horizontal T-junction extension intersects a vertical T-junction extension.
An analysis-suitable T-mesh is shown in Figure 6a. The corresponding extended
T-mesh is shown in Figure 6b. Notice that no horizontal extension intersects a
vertical extension. The dual basis for an ASTS equips these spaces with a rich
mathematical structure which we leverage in this paper [20].
###### Lemma 4.2.
For a bicubic ASTS, each dual basis function, corresponding to a T-spline
basis function $N_{A}(\xi,\eta)$ with local knot vectors
$\Xi_{A}=[\xi_{i_{1}},\ldots,\xi_{i_{5}}]$ and
$\Pi_{A}=[\eta_{j_{1}},\ldots,\eta_{j_{5}}]$, is
$\lambda_{A}=\lambda[\xi_{i_{1}},\ldots,\xi_{i_{5}}]\otimes\lambda[\eta_{j_{1}},\ldots,\eta_{j_{5}}]$
where $\lambda[\xi_{i_{1}},\ldots,\xi_{i_{5}}]$ and
$\lambda[\eta_{j_{1}},\ldots,\eta_{j_{5}}]$ are dual basis functions
corresponding to univariate cubic B-splines [24] whose knot vectors are
$[\xi_{i_{1}},\ldots,\xi_{i_{5}}]$ and $[\eta_{j_{1}},\ldots,\eta_{j_{5}}]$,
respectively.
Figure 6: An analysis-suitable T-mesh (a) and the extended T-mesh (b).
## 5 Perturbed T-splines
From a theoretical point of view, developing a complete and rigorous
characterization of T-spline spaces is complicated by the presence of zero
knot intervals (especially near T-junctions) and overlap vertices. However,
allowing both is important when T-splines are used as a tool in design and
analysis.
To overcome this difficulty, we develop the theory of the perturbed T-mesh
(and resulting perturbed T-spline space). A perturbed T-spline can be used to
prove properties about the original T-spline. In other words, we will generate
a perturbed T-mesh, establish the result in the perturbed setting, and then
show that the result holds as the perturbation converges to the original
T-spline.
### 5.1 Perturbed T-meshes
A perturbed T-mesh is created by first generating perturbed global knot
vectors, $\Xi[\boldsymbol{\delta}]$, $\Pi[\boldsymbol{\delta}]$, where
$\boldsymbol{\delta}=(\delta_{1},\ldots,\delta_{n})^{T}$ is a vector of
perturbation parameters. A perturbed global knot vector is written as
$\Xi[\boldsymbol{\delta}]=(\xi[\boldsymbol{\delta}]_{\underline{\imath}},\ldots,\xi[\boldsymbol{\delta}]_{\imath},\ldots,\xi[\boldsymbol{\delta}]_{\overline{\imath}})$
where $\imath=\imath(i,g)$ takes the index of the $i^{th}$ knot in $\Xi$ and
the $g^{th}$ segment in $v\mathsf{G}(\\{i\\})$ and returns a unique index in
the perturbed global knot vector. The knot values are initialized as
$\xi[\boldsymbol{\delta}]_{\imath(i,g)}=\xi_{i}$. In other words, a knot index
which corresponds to a $v\mathsf{G}(\\{i\\})$ which contains multiple segments
in the T-mesh is repeated $n_{v}^{G}(\\{i\\})$ times. Notice that this
operation induces an index map $h\pi(\imath(i,g))=i$ (resp., $v\pi$) from the
indices in the perturbed global knot vector onto the original global knot
vector. The knot values are then perturbed using a small parameter
$\delta\in\mathbb{R}$ as
$\xi[\boldsymbol{\delta}]_{\imath}=\xi[\boldsymbol{\delta}]_{\underline{\imath}}+\sum_{\jmath=\underline{\imath}+1}^{\imath}\Delta\xi[\boldsymbol{\delta}]_{\jmath},\quad\imath=\underline{\imath},\ldots,\overline{\imath}$
where
$\Delta\xi[\boldsymbol{\delta}]_{\jmath}=c_{\alpha(\jmath)}\delta=\delta_{\alpha}$,
if $\xi_{h\pi(\jmath)}-\xi_{h\pi({\jmath}-1)}=0$, and is equal to
$\xi_{h\pi(\jmath)}-\xi_{h\pi({\jmath}-1)}$, otherwise. The constant,
$c_{\alpha}\in[0,\infty)$. This same procedure is applied to $\Pi$ to form
$\Pi[\boldsymbol{\delta}]$. The T-mesh, $\mathsf{T}$, is then modified to form
the perturbed T-mesh, $\mathsf{T}[\boldsymbol{\delta}]$, by associating the
vertices and edges contained in the $g^{th}$ segment of $v\mathsf{G}(\\{i\\})$
with knot $\xi[\boldsymbol{\delta}]_{\imath(i,g)}$. Notice that the number of
anchors does not change when forming $\mathsf{T}[\boldsymbol{\delta}]$. A
perturbed T-spline space $\mathcal{T}[\boldsymbol{\delta}]$ is a T-spline
space formed from perturbed global knot vectors and T-mesh. A strictly
perturbed T-mesh or T-spline space is one where $c_{\alpha}\in(0,\infty)$.
A perturbation of an analysis-suitable T-mesh is shown in Figure 7. The
analysis-suitable T-mesh is shown in Figure 7a and the perturbed T-mesh is
shown in Figure 7b. Knot intervals are shown instead of knots for simplicity.
Recall that a knot interval is simply the difference between adjacent knots in
a global knot vector. Notice that the horizontal and vertical zero knot
intervals have been replaced by non-zero knot intervals $\sigma_{2}$ and
$\sigma_{3}$. The vertical segments with T-junctions are perturbed resulting
in a new knot interval $\sigma_{1}$.
Figure 7: A perturbation of an analysis-suitable T-mesh $\mathsf{T}$ shown in
(a) results in the perturbed T-mesh $\mathsf{T}[\boldsymbol{\delta}]$ shown in
(b).
###### Proposition 5.1.
If $\mathsf{T}$ is analysis-suitable then $\mathsf{T}[\boldsymbol{\delta}]$ is
analysis-suitable.
###### Proof.
Suppose the extensions of two T-junctions
$ext(V_{1})=[i_{1}^{1},i_{4}^{1}]\times\\{j^{1}\\}$ ($\vdash$ or $\dashv$) and
$ext(V_{2})=\\{i^{2}\\}\times[j_{1}^{2},j_{4}^{2}]$ ($\bot$ or $\top$) in
$\mathsf{T}[\boldsymbol{\delta}]$ intersect. This implies that $j_{1}^{2}\leq
j^{1}\leq j_{4}^{2}$ and $i_{1}^{1}\leq i^{2}\leq i_{4}^{1}$. According to the
construction of $\mathsf{T}[\boldsymbol{\delta}]$, we don’t change the order
of the indices, so $h\pi(j_{1}^{2})\leq h\pi(j^{1})\leq h\pi(j_{4}^{2})$ and
$v\pi(i_{1}^{1})\leq v\pi(i^{2})\leq v\pi(i_{4}^{1})$, i.e., there are
intersecting extensions in $\mathsf{T}$. ∎
###### Lemma 5.1.
Let $\mathsf{T}$ be an analysis-suitable T-mesh. For every anchor,
$\\{(\imath,\jmath)\\}$, and horizontal index vector,
$hv(\\{(\imath,\jmath)\\})[\boldsymbol{\delta}]$, in the perturbed T-mesh,
$\mathsf{T}[\boldsymbol{\delta}]$,
$hv(\\{(h\pi(\imath),v\pi(\jmath))\\})=h\pi(hv(\\{(\imath,\jmath)\\})[\boldsymbol{\delta}])$
where
$h\pi(hv(\\{(\imath,\jmath)\\})[\boldsymbol{\delta}])=(h\pi(\imath_{1}),\ldots,h\pi(\imath_{5}))$.
This also holds for the vertical index vectors.
###### Proof.
If the topological symbols corresponding to the indices
$\\{(i_{1},\jmath),\ldots,(i_{5},\jmath)\\}$ where $i_{\ell}\in
hv(\\{(\imath,\jmath)\\})[\boldsymbol{\delta}]$, $\ell=1,\ldots,5$, are not
$\vdash$ or $\dashv$ then the result immediately follows since the indices are
contained in a single horizontal segment. Thus, we only need to prove that the
result holds when the topological symbol corresponding to
$\\{(\imath,\jmath)\\}$ is $\vdash$ or $\dashv$. Without loss of generality we
assume it to be $\vdash$. Since, according to Lemma 5.1, the T-mesh,
$\mathsf{T}[\boldsymbol{\delta}]$, is analysis-suitable the symbols for the
first two indices in $hv(\\{(\imath,\jmath)\\})[\boldsymbol{\delta}]$ can only
be $|$ or $\dashv$. Thus,
$hv(\\{(h\pi(\imath),v\pi(\jmath))\\})=h\pi(hv(\\{(\imath,\jmath)\\})[\boldsymbol{\delta}])$.
∎
###### Theorem 5.2.
If $\mathcal{T}$ is analysis-suitable and $\mathcal{T}[\boldsymbol{\delta}]$
is an offset perturbation then $N_{A}[\boldsymbol{\delta}]\rightarrow
N_{\pi(A)}$ as $\delta\rightarrow 0$,
$A\in\mathsf{A}(\mathsf{T}[\boldsymbol{\delta}])$ where
$\pi(A)=\\{(h\pi(\imath),v\pi(\jmath))\\}$.
###### Proof.
Notice that if $\delta\rightarrow 0$,
$\xi[\boldsymbol{\delta}]_{\imath}\rightarrow\xi_{h\pi(\imath)}$ for any
$\imath$ and $\eta[\delta]_{\jmath}\rightarrow\eta_{v\pi(\jmath)}$ for any
$\jmath$. According to Lemma 5.1, the local knot vectors for each basis
function $N_{A}[\boldsymbol{\delta}]$ converge to those for $N_{\pi(A)}$,
$A\in\mathsf{A}(\mathsf{T}[\boldsymbol{\delta}])$. Thus, according to ([24],
Theorem 4.36) we have the result. ∎
## 6 Refineability and nestedness
We now explore the refineability and nesting behavior of analysis-suitable
T-spline spaces. In other words, given two analysis-suitable T-splines spaces,
$\mathcal{T}^{1}$ and $\mathcal{T}^{2}$, we establish the conditions under
which $\mathcal{T}^{1}\subseteq\mathcal{T}^{2}$. We first establish basic
refineability properties when the analysis-suitable T-mesh does not have any
knot multiplicities or overlap vertices. Using the theory of perturbed
T-splines, we then extend those results to encompass T-meshes which do have
zero knot intervals and overlap vertices.
###### Definition 6.1.
The notation $\mathsf{T}^{1}[\boldsymbol{\delta},\mathsf{T}^{2}]$ denotes a
perturbed T-mesh where $\mathsf{T}^{1}\subseteq\mathsf{T}^{2}$ and
$\mathsf{T}^{1}[\boldsymbol{\delta},\mathsf{T}^{2}]$ is created by removing
those edges and vertices from the strictly perturbed T-mesh
$\mathsf{T}^{2}[\boldsymbol{\delta}]$ which correspond to unperturbed edges
and vertices in $\mathsf{T}^{2}\setminus\mathsf{T}^{1}$. By inspection, it is
clear that $\mathsf{T}^{1}[\boldsymbol{\delta},\mathsf{T}^{2}]$, constructed
in this way, is a perturbed T-mesh which satisfies Proposition 5.1, Lemma 5.1,
and Theorem 5.2 and that
$\mathsf{T}^{1}[\boldsymbol{\delta},\mathsf{T}^{2}]\subseteq\mathsf{T}^{2}[\boldsymbol{\delta}]$.
The construction of $\mathsf{T}^{1}[\boldsymbol{\delta},\mathsf{T}^{2}]$ is
depicted in Figure 8. Two analysis-suitable T-meshes are shown in Figure 8a
and Figure 8b. Notice that $\mathsf{T}^{1}\subseteq\mathsf{T}^{2}$. The
perturbed T-mesh $\mathsf{T}^{1}[\boldsymbol{\delta},\mathsf{T}^{2}]$ (shown
in Figure 8c) is formed by removing the dotted lines (shown in Figure 8c) from
$\mathsf{T}^{2}[\boldsymbol{\delta}]$ (shown in Figure 8d).
Figure 8: The construction of
$\mathsf{T}^{1}[\boldsymbol{\delta},\mathsf{T}^{2}]$. Two analysis-suitable
T-meshes are shown in (a) and (b) such that
$\mathsf{T}^{1}\subseteq\mathsf{T}^{2}$. The perturbed T-mesh
$\mathsf{T}^{1}[\boldsymbol{\delta},\mathsf{T}^{2}]$, shown in (c), is formed
by removing the dotted lines from $\mathsf{T}^{2}[\boldsymbol{\delta}]$, shown
in (d).
###### Definition 6.2.
Given a T-mesh, $\mathsf{T}$, with no knot multiplicities, and the
corresponding extended T-mesh, $\mathsf{T}_{ext}$, the homogeneous extended
spline space is defined as
$\mathcal{S}_{ext}=\left\\{f\in
C^{2,2}(\mathbb{R}^{2})\,|\,f|_{\tilde{Q}_{ext}}\in\mathbb{P}_{33},\forall\tilde{Q}_{ext}\subseteq\tilde{\Omega},\,\mathrm{and}\,f|_{\mathbb{R}^{2}\setminus\tilde{\Omega}}\equiv
0\right\\}$ (4)
where $C^{2,2}(\mathbb{R}^{2})$ is the space of bivariate functions which are
$C^{2}$-continuous in $\xi$ and $\eta$ over all of $\mathbb{R}^{2}$.
$\mathbb{P}_{33}$ is the space of bicubic polynomials. The extended spline
space is defined to be $\mathcal{T}_{ext}=\mathcal{S}_{ext}|_{\hat{\Omega}}$.
###### Proposition 6.1.
$\dim\mathcal{T}_{ext}=\dim\mathcal{S}_{ext}$
###### Proof.
We first prove that the dimension of $\mathcal{S}_{ext}$ is not less than the
dimension of $\mathcal{T}_{ext}$. Notice that for any function
$f\in\mathcal{S}_{ext}$, $f|_{\hat{\Omega}}\in\mathcal{T}_{ext}$. We now show
that the dimension of $\mathcal{T}_{ext}$ is not less than the dimension of
$\mathcal{S}_{ext}$. This is equivalent to showing that there is only one
function in $\mathcal{S}_{ext}$ which is zero over $\hat{\Omega}$. It is easy
to see that the only function which is zero over $\hat{\Omega}$ must be zero
over all of $\mathbb{R}^{2}$ since the minimum support of a cubic $C^{2}$
spline function is four intervals. ∎
###### Lemma 6.3.
If the extended T-mesh, $\mathsf{T}_{ext}$, of an analysis-suitable T-mesh,
$\mathsf{T}$, has no knot multiplicities or overlap vertices, then
$\mathcal{T}=\mathcal{T}_{ext}$. In other words, the analysis-suitable
T-spline space, $\mathcal{T}$, and the extended spline space,
$\mathcal{T}_{ext}$, are the same space.
###### Proof.
We have that $\mathcal{T}\subseteq\mathcal{T}_{ext}$ (see [20], Lemma 4.3), so
the dimension of $\mathcal{T}$ is less than that of $\mathcal{T}_{ext}$,
which, according to Theorem 8.5, Proposition 6.1, and Theorem 8.9 is the
number of active vertices. Since the blending functions for anaysis-suitable
T-splines are linearly independent the dimension of $\mathcal{T}$ is also the
number of active vertices. Thus, the two spline spaces are identical. ∎
###### Lemma 6.4.
Given two analysis-suitable T-meshes, $\mathsf{T}^{1}$ and $\mathsf{T}^{2}$,
neither of which has knot multiplicities or overlap vertices, if
$\mathsf{T}_{ext}^{1}\subseteq\mathsf{T}_{ext}^{2}$, then
$\mathcal{T}^{1}\subseteq\mathcal{T}^{2}$.
###### Proof.
Obviously, $\mathcal{T}_{ext}^{1}\subseteq\mathcal{T}_{ext}^{2}$. Since
$\mathsf{T}^{1}$ and $\mathsf{T}^{2}$ are analysis-suitable, according to
Lemma 6.3, $\mathcal{T}^{1}\subseteq\mathcal{T}^{2}$. ∎
###### Lemma 6.5.
Given two analysis-suitable T-spline spaces, $\mathcal{T}^{1}$ and
$\mathcal{T}^{2}$, if
$\mathcal{T}^{1}[\boldsymbol{\delta}]\subseteq\mathcal{T}^{2}[\boldsymbol{\delta}]$,
then $\mathcal{T}^{1}\subseteq\mathcal{T}^{2}$.
###### Proof.
Suppose the perturbed T-spline space, $\mathcal{T}^{1}[\boldsymbol{\delta}]$,
is spanned by the basis functions, $N_{A}^{1}[\boldsymbol{\delta}]$, and the
perturbed T-spline space, $\mathcal{T}^{2}[\boldsymbol{\delta}]$, is spanned
by the basis functions, $N_{B}^{2}[\boldsymbol{\delta}]$. We have that
$N_{A}^{1}[\boldsymbol{\delta}]=\sum\lambda_{B}(N_{A}[\boldsymbol{\delta}])N_{B}^{2}[\boldsymbol{\delta}].$
where $\lambda_{B}$, $B\in\mathsf{A}(\mathsf{T}^{2}[\boldsymbol{\delta}])$ are
the dual functionals for the analysis-suitable T-spline basis
$N_{B}[\boldsymbol{\delta}]$ as described in [20]. According to Theorem 5.2,
$\lim_{\delta\rightarrow
0}N_{B}^{2}[\boldsymbol{\delta}]=N_{\pi(B)}^{2},\quad\lim_{\delta\rightarrow
0}N_{A}^{1}[\boldsymbol{\delta}]=N_{\pi(A)}^{1}.$
According to Theorem 4.41 in [24], $\lambda_{B}(N_{A}[\boldsymbol{\delta}])$
is bounded, so
$\lim_{\delta\rightarrow 0}\lambda_{B}(N_{A}[\boldsymbol{\delta}])=c_{B}^{A},$
i.e.,
$N_{\pi(A)}^{1}=\sum c_{B}^{A}N_{\pi(B)}^{2}.$
∎
###### Theorem 6.6.
Given two analysis-suitable T-meshes, $\mathsf{T}^{1}$ and $\mathsf{T}^{2}$,
if
$\mathsf{T}^{1}_{ext}[\boldsymbol{\delta},\mathsf{T}^{2}]\subseteq\mathsf{T}^{2}_{ext}[\boldsymbol{\delta}]$,
then $\mathcal{T}^{1}\subseteq\mathcal{T}^{2}$.
###### Proof.
Obviously,
$\mathcal{T}_{ext}^{1}[\boldsymbol{\delta},\mathsf{T}^{2}]\subseteq\mathcal{T}_{ext}^{2}[\boldsymbol{\delta}]$.
Since $\mathsf{T}_{ext}^{2}[\boldsymbol{\delta}]$ has no knot multiplicities
or overlap vertices we have that
$\mathcal{T}^{2}[\boldsymbol{\delta}]=\mathcal{T}^{2}_{ext}[\boldsymbol{\delta}]$
according to Lemma 6.3. Since the extended T-mesh,
$\mathsf{T}^{1}_{ext}[\boldsymbol{\delta},\mathsf{T}^{2}]$, may have overlap
vertices but no knot multiplicities we have that
$\mathcal{T}^{1}[\boldsymbol{\delta},\mathsf{T}^{2}]\subseteq\mathcal{T}^{1}_{ext}[\boldsymbol{\delta},\mathsf{T}^{2}]$
according to Theorem 8.5 and Proposition 6.1. This immediately implies that
$\mathcal{T}^{1}\subseteq\mathcal{T}^{2}$. ∎
###### Proposition 6.2.
Every analysis-suitable T-spline space contains the space of bicubic
polynomials.
###### Proof.
We have that any bicubic polynomial
$f[\boldsymbol{\delta}]\in\mathcal{T}[\boldsymbol{\delta}]$ according to Lemma
6.3. Using the dual basis for $\mathcal{T}[\boldsymbol{\delta}]$ we have that
$f[\boldsymbol{\delta}]=\sum\lambda_{A}(f)N_{A}[\boldsymbol{\delta}].$
Thus, according to Theorem 5.2 and ([24], Theorem 4.41), as $\delta\rightarrow
0$,
$f=\sum c_{A}N_{A}.$
∎
###### Corollary 6.7.
Every analysis-suitable T-spline space forms a partition of unity. In other
words, $\sum_{A}N_{A}(\xi,\eta)=1$, $\forall(\xi,\eta)\in\hat{\Omega}$.
###### Proof.
This immediately follows from Proposition 6.2 and the fact that
$\lambda_{A}(1)=1$. ∎
## 7 Approximation
As described in [20, 21] approximation properties of analysis-suitable
T-splines are directly linked to Proposition 6.2. In other words, having the
bicubic polynomials in the T-spline space is the minimal requirement to obtain
an $O(h^{4})$ convergence rate in the mesh size.
Following the approach in [22, 20, 21], the dual basis for an analysis-
suitable T-spline space, $\mathcal{T}$, can be used to construct a projection
operator, $\mathbb{P}:L^{2}(\hat{\Omega})\rightarrow\mathcal{T}$, where
$\mathbb{P}(f)(\xi,\eta)=\sum_{A\in\mathsf{A}(\mathsf{T})}\lambda_{A}(f)N_{A}(\xi,\eta)\quad\forall
f\in L^{2}(\hat{\Omega}),\forall(\xi,\eta)\in\hat{\Omega}.$
We denote the open support of a T-spline basis function by
$Q_{A}\subset\tilde{\Omega}$, and the extended support of an element $\hat{Q}$
by $\Omega_{\hat{Q}}\subset\tilde{\Omega}$, where
$\Omega_{\hat{Q}}=\bigcup_{A\in\mathsf{A}(\hat{Q})}Q_{A},\quad\mathsf{A}(\hat{Q})=\\{A\in\mathsf{A}(\mathsf{T}):Q_{A}\cap\hat{Q}\neq\emptyset\\}.$
We will denote by $R(\Omega_{\hat{Q}})$ the smallest rectangle in
$\tilde{\Omega}$ containing $\Omega_{\hat{Q}}$ and .
###### Proposition 7.1.
Given an analysis-suitable T-spline space, $\mathcal{T}$, the projection
operator $\mathbb{P}$ is (locally) h–uniformly continuous in the $L^{2}$ norm.
In other words, there exists a constant $C$ independent of
$\mathsf{T},\Xi,\Pi$ such that
$||\mathbb{P}(f)||_{L^{2}(\hat{Q})}\leq
C||f||_{L^{2}(\Omega_{\hat{Q}})}\quad\forall\hat{Q}\in\mathsf{T}_{ext},\forall
f\in L^{2}(\hat{\Omega})$
Note that the constant $C$ may depend on the polynomial degree.
###### Proof.
The result follows immediately from ([20], Proposition 5.4) and Proposition
6.2. ∎
###### Proposition 7.2.
Given an analysis-suitable T-spline space, $\mathcal{T}$, there exists a
constant $C^{\prime}$ independent of $\mathsf{T},\Xi,\Pi$ such that for
$r\in[0,4]$
$||f-\mathbb{P}(f)||_{L^{2}(\hat{Q})}\leq
C^{\prime}(h_{R(\Omega_{\hat{Q}})})^{r}|f|_{H^{r}(R(\Omega_{\hat{Q}}))}\quad\forall\hat{Q}\in\mathsf{T}_{ext},\forall
f\in L^{2}(\hat{\Omega})$
where $h_{R(\Omega_{\hat{Q}})}$ denotes the diameter of $R(\Omega_{\hat{Q}})$.
Note that the constant $C^{\prime}$ may depend on the polynomial degree.
###### Proof.
The result follows immediately from ([20], Proposition 5.4) and Proposition
6.2. ∎
## 8 Dimension
In this section, we develop a dimension formula for polynomial spline spaces
defined over the extended T-mesh in the parametric domain of a T-spline and
establish the connection between this dimension formula and analysis-suitable
T-spline spaces. The dimension formula, written only in terms of topological
quantities of the original T-mesh, is an essential ingredient in establishing
the refineability properties in Section 6 and the approximation results in
Section 7 for analysis-suitable T-splines. The essential results are proven in
Theorems 8.5 and 8.9.
Unlike existing approaches, our dimension formula does not require that the
T-mesh have any nesting structure. Of critical importance is how this
dimension formula can be directly related to analysis-suitable T-spline spaces
which can then be used to construct a simple set of basis functions for the
spline space which are compatible with commercial CAD and analysis frameworks.
### 8.1 Smoothing cofactor-conformality method
We use the smoothing cofactor-conformality method [25, 24] to transform the
smoothness properties of $\mathcal{S}_{ext}$ into a linear constraint matrix,
$\mathbf{M}$. This constraint matrix is then analyzed to determine the
dimension of $\mathcal{S}_{ext}$. We recall that the spline space,
$\mathcal{S}_{ext}$, is defined using the extended T-mesh, $\mathsf{T}_{ext}$,
corresponding to a T-mesh, $\mathsf{T}$, which does not have any knot
multiplicities.
#### 8.1.1 Vertex and edge cofactors
As shown in Figure 9, for any vertex,
$V_{i,j}=(\xi_{i},\eta_{j})\in\mathsf{T}_{ext}$, the surrounding bicubic
polynomial patches are labeled, $p_{i,j}^{k}(\xi,\eta)$, $k=0,1,2,3$. If the
vertex, $V_{i,j}$, is a T-junction, then
$p_{i,j}^{k}(\xi,\eta)=p_{i,j}^{k+1}(\xi,\eta)$ for some $k$. Since
$p_{i,j}^{0}(\xi,\eta)$ and $p_{i,j}^{1}(\xi,\eta)$ are $C^{2}$-continuous
there exists a cubic polynomial $\gamma_{i,j}^{2}(\eta)$, called the edge
cofactor, such that
$p_{i,j}^{1}(\xi,\eta)-p_{i,j}^{0}(\xi,\eta)=\gamma_{i,j}^{2}(\eta)(\xi-\xi_{i})^{3}.$
(5)
Figure 9: The smoothing cofactors around a vertex.
Similarly, there exists cubic polynomials, $\gamma_{i,j}^{1}(\eta)$,
$\mu_{i,j}^{1}(\xi)$, and $\mu_{i,j}^{2}(\xi)$, such that
$\displaystyle p_{i,j}^{2}(\xi,\eta)-p_{i,j}^{1}(\xi,\eta)$
$\displaystyle=\mu_{i,j}^{1}(\xi)(\eta-\eta_{j})^{3},$ (6) $\displaystyle
p_{i,j}^{3}(\xi,\eta)-p_{i,j}^{2}(\xi,\eta)$
$\displaystyle=-\gamma_{i,j}^{1}(\eta)(\xi-\xi_{i})^{3},$ (7) $\displaystyle
p_{i,j}^{0}(\xi,\eta)-p_{i,j}^{3}(\xi,\eta)$
$\displaystyle=-\mu_{i,j}^{2}(\xi)(\eta-\eta_{j})^{3}.$ (8)
We note that if two patches are identical the edge cofactor is zero. Combining
(5) - (8) gives
$(\gamma_{i,j}^{1}(\eta)-\gamma_{i,j}^{2}(\eta))(\xi-\xi_{i})^{3}=(\mu_{i,j}^{1}(\xi)-\mu_{i,j}^{2}(\xi))(\eta-\eta_{j})^{3}.$
(9)
Since $(\xi-\xi_{i})^{3}$ and $(\eta-\eta_{j})^{3}$ are prime to each other
there exists a constant, $d_{i,j}$, called the vertex cofactor, such that
$\gamma_{i,j}^{1}(\eta)-\gamma_{i,j}^{2}(\eta)=d_{i,j}(\eta-\eta_{j})^{3},\quad\mu_{i,j}^{1}(\xi)-\mu_{i,j}^{2}(\xi)=d_{i,j}(\xi-\xi_{i})^{3}.$
(10)
#### 8.1.2 Assembling the constraint matrix, $\mathbf{M}$
Figure 10: The smoothing cofactors along a horizonal edge segment.
Referring to Figure 10, consider a horizontal segment $G_{j}^{h}$ with $k+1$
vertices and $k$ edge cofactors. Using (10) we have that
$\displaystyle\mu_{i_{0},j}^{1}-0$
$\displaystyle=d_{i_{0},j}(\xi-\xi_{i_{0}})^{3},$ (11)
$\displaystyle\mu_{i_{1},j}^{2}-\mu_{i_{1},j}^{1}$
$\displaystyle=d_{i_{1},j}(\xi-\xi_{i_{1}})^{3},$ (12)
$\displaystyle\,\,\,\vdots$ $\displaystyle 0-\mu_{i_{k},j}^{2}$
$\displaystyle=d_{i_{k},j}(\xi-\xi_{i_{k}})^{3},$ (13)
and
$\mu_{i_{\ell+1},j}^{2}=\mu_{i_{\ell},j}^{1},\,\,\ell=1,\ldots,k.$ (14)
Summing (11) - (13) and using (14) results in the linear system
$L_{j}^{h}:=\sum_{\ell=0}^{k}d_{i_{\ell},j}(\xi-\xi_{i_{\ell}})^{3}=0.$ (15)
We call the solution space, denoted by $W[G_{j}^{h}]$, for this linear system
the _edge conformality space_. Similarly, for a vertical segment $G_{i}^{v}$
we have that
$L_{i}^{v}:=\sum_{\ell=0}^{l}d_{i,j_{\ell}}(\eta-\eta_{j_{\ell}})^{3}=0$ (16)
where the solution space is denoted by $W[G_{i}^{v}]$. By (15) and (16), one
immediately has that
###### Lemma 8.1.
If each $\xi_{i_{\ell}}$ and $\eta_{j_{\ell}}$ are different, then the
dimension of $W[G_{j}^{h}]$ and $W[G_{i}^{v}]$ are $k-3$ and $l-3$
respectively.
The linear systems (15) and (16), associated with the horizontal and vertical
segments in $\mathsf{T}_{ext}$, can be assembled into the global system
$\mathbf{M}\mathbf{D}=\mathbf{0}$ (17)
where $\mathbf{D}=[d_{1},d_{2},\ldots,d_{n^{ext}}]^{T}$ is a column vector of
all vertex cofactors in $\mathsf{T}_{ext}$ and $\mathbf{M}$ is a
$4n_{seg}\times n_{ext}$ real matrix. Each edge conformality condition
corresponds to a submatrix consisting of $4$ rows of $\mathbf{M}$ and each
vertex cofactor corresponds to a column of $\mathbf{M}$.
###### Lemma 8.2.
The dimension of $\mathcal{S}_{ext}$ is the nullity of $\mathbf{M}$, i.e., the
dimension is $n_{ext}$ minus the rank of $\mathbf{M}$.
###### Proof.
Since the continuity constraints in $\mathcal{S}_{ext}$ have been converted
into the linear system in (17) using the smoothing cofactor-conformality
method (see [25]), the dimension of $\mathcal{S}_{ext}$ is the dimension of
the null space of $\mathbf{M}$, i.e., the dimension is $n_{ext}$ minus the
rank of $\mathbf{M}$. ∎
### 8.2 Simplifying the constraint matrix, $\mathbf{M}$, and
$\mathsf{T}_{ext}$
It is possible to simplify the constraint matrix, $\mathbf{M}$, and the
topology of the extended T-mesh in the parametric domain, $\mathsf{T}_{ext}$,
such that the null space of $\mathbf{M}$ is undisturbed. To remove a vertex
from $\mathsf{T}_{ext}$ means we delete the corresponding column from
$\mathbf{M}$ and to remove a segment from $\mathsf{T}_{ext}$ means we delete
the appropriate submatrix from $\mathbf{M}$. We form the reduced constraint
matrix $\overline{\mathbf{M}}$ by removing the eight segments and contained
vertices
$[\xi_{\underline{m}},\xi_{\overline{m}}]\times\\{\eta_{\underline{n}}\\}$,
$[\xi_{\underline{m}},\xi_{\overline{m}}]\times\\{\eta_{\underline{n}+1}\\}$,
$[\xi_{\underline{m}},\xi_{\overline{m}}]\times\\{\eta_{\overline{n}-1}\\}$,
$[\xi_{\underline{m}},\xi_{\overline{m}}]\times\\{\eta_{\overline{n}}\\}$,
$\\{\xi_{\underline{m}}\\}\times[\eta_{\underline{n}},\eta_{\overline{n}}]$,
$\\{\xi_{\underline{m}+1}\\}\times[\eta_{\underline{n}},\eta_{\overline{n}}]$,
$\\{\xi_{\overline{m}-1}\\}\times[\eta_{\underline{n}},\eta_{\overline{n}}]$,
and
$\\{\xi_{\overline{m}}\\}\times[\eta_{\underline{n}},\eta_{\overline{n}}]$. We
denote the T-mesh after the removals by $\overline{\mathsf{T}}_{ext}$ and the
number of vertices and segments in $\overline{\mathsf{T}}_{ext}$ by
$\overline{n}^{ext}$ and $\overline{n}^{G}$, respectively. Figure 11 shows the
simplified extended T-mesh $\overline{\mathsf{T}}_{ext}$ for the extended
T-mesh in Figure 4. The vertices and segments which remain after the removal
process have corresponding entries in $\overline{\mathbf{M}}$.
Figure 11: The simplified extended T-mesh $\overline{\mathsf{T}}_{ext}$ for
the extended T-mesh in Figure 4.
###### Lemma 8.3.
The dimension of the null space of $\mathbf{M}$ is the same as that for
$\overline{\mathbf{M}}$.
###### Proof.
The vertex cofactors which correspond to the removed corner vertices can be
uniquely determined by applying (15) to the four horizontal removed segments
or by applying (16) to the four vertical removed segments. To establish the
result we need to show that the constraints corresponding to the four vertical
removed segments can be derived from the constraints corresponding to the four
horizontal removed segments. We have that
$\displaystyle 0$
$\displaystyle=\sum_{m=0}^{k}\left[\sum_{\ell=0}^{l}d_{i_{m},j_{\ell}}(\eta-\eta_{j_{\ell}})^{3}\right](\xi-\xi_{i_{m}})^{3}$
(18)
$\displaystyle=\sum_{\ell=0}^{l}\left[\sum_{m=0}^{k}d_{i_{m},j_{\ell}}(\xi-\xi_{i_{m}})^{3}\right](\eta-\eta_{j_{\ell}})^{3}$
(19)
$\displaystyle=\sum_{\ell=0,1,l-1,l}\left[\sum_{m=0}^{k}d_{i_{m},j_{\ell}}(\xi-\xi_{i_{m}})^{3}\right](\eta-\eta_{j_{\ell}})^{3}.$
(20)
Equation (18) involves the sum of all edge conformality conditions for the
horizonal edge segments. Equation (20) holds because the linear systems for
the other vertical segments are satisfied. Since $(\eta-\eta_{j_{\ell}})^{3}$,
$\ell=0,1,l-1,l$, form a basis for a linear space of polynomials with degree
less than four, $\sum_{m=0}^{k}d_{i_{m},j_{\ell}}(\xi-\xi_{i_{m}})^{3}=0$, for
$m=0,\ldots,k$. In other words, the constraints for the four vertical removed
segments can be derived from the other constraints. ∎
###### Remark 8.4.
There are many possible simplification techniques which could have been used.
This simplication technique was chosen because it leaves the null space of
$\mathbf{M}$ undisturbed and each resulting segment in $\overline{\mathbf{M}}$
contains exaclty four extended vertices.
###### Theorem 8.5.
If $\overline{\mathbf{M}}$ has full column rank, then the dimension of
$\mathcal{S}_{ext}$ is
$\begin{array}[]{r@{\;}l}\dim\mathcal{S}_{ext}=n^{a}+n^{+}+n^{-}\end{array}$
(21)
where $n^{a}$ is the number of active vertices in $\mathsf{T}$ and $n^{+}$ and
$n^{-}$ are the number of crossing and overlap vertices, respectively, in
$\mathsf{T}_{ext}$.
###### Proof.
Since there are $\overline{n}^{ext}$ and $\overline{n}^{G}$ vertices and
segments, respectively, in $\overline{\mathsf{T}}_{ext}$,
$\overline{\mathbf{M}}$ is a $4\overline{n}^{G}\times\overline{n}^{ext}$
matrix. Since $\overline{\mathbf{M}}$ has full column rank the dimension of
$\mathcal{S}_{ext}$ is $\overline{n}^{ext}-4\overline{n}^{G}$. As every
segment in $\overline{\mathsf{T}}_{ext}$ has exactly four extended vertices
and these four extended vertices are not extended vertices for any other
segment, the number of extended vertices in $\overline{\mathsf{T}}_{ext}$ is
$4\overline{n}^{G}$. Thus,
$\begin{array}[]{r@{\;}l}\dim\mathcal{S}_{ext}=\overline{n}^{ext}-4\overline{n}^{G}=n^{a}+n^{+}+n^{-}.\end{array}$
(22)
∎
### 8.3 Rank of the constraint matrix $\overline{\mathbf{M}}$
Since every extended vertex in $\overline{\mathsf{T}}_{ext}$ is an extended
vertex in exactly one segment, the matrix $\overline{\mathbf{M}}$ has more
columns than rows, i.e., $\overline{n}^{ext}>4\overline{n}^{G}$. After
arranging the order of edge conformality conditions and the order of vertex
cofactors, an appropriate partition of the linear system of constraints,
$\overline{\mathbf{M}}\,\,\overline{\mathbf{D}}=\mathbf{0}$, is
$\left[\begin{array}[]{c|c}\overline{\mathbf{M}}_{1}&\overline{\mathbf{M}}_{2}\end{array}\right]\left[\begin{array}[]{c}\overline{\mathbf{D}}_{1}\vspace{2pt}\\\
\hline\cr\vspace{-10pt}\hfil\\\
\overline{\mathbf{D}}_{2}\end{array}\right]=\mathbf{0}$ (23)
where $\overline{\mathbf{M}}_{1}$ is a $4\overline{n}^{G}\times
4\overline{n}^{G}$ matrix and $\overline{\mathbf{M}}_{1}$ is a
$4\overline{n}^{G}\times(\overline{n}^{ext}-4\overline{n}^{G})$ matrix,
$\overline{\mathbf{D}}_{1}$ is a vector of the first $4\overline{n}^{G}$
vertex cofactors, and $\overline{\mathbf{D}}_{2}$ is a vector of the remaining
vertex cofactors.
###### Definition 8.6.
A simplified extended T-mesh is called diagonalizable if we can arrange all
the segments $G_{i},i=1,\dots,\overline{n}^{G}$ such that the number of
vertices on segment $G_{j}$ but not on segment $G_{i},i<j$ is at least 4.
###### Lemma 8.7.
If a simplified extended T-mesh is diagonalizable, then the matrix
$\overline{\mathbf{M}}$ has full column rank.
###### Proof.
Since the simplified extended T-mesh is diagonalizable, the segments can be
ordered as described in Definition 8.6. Given this ordering of segments, for
$i=1,\ldots,\overline{n}^{G}$, we place the edge conformality conditions
corresponding to segment $G_{i}$ in rows $4(\overline{n}^{G}-i)+1$ through
$4(\overline{n}^{G}-i)+4$ of $\overline{\mathbf{M}}$, and place the first four
vertex cofactors which appear in $G_{i}$ but not in $G_{j},j<i$ in columns
$4(\overline{n}^{G}-i)+1$ through $4(\overline{n}^{G}-i)+4$ of
$\overline{\mathbf{D}}_{1}$. Then the matrix $\overline{\mathbf{M}}_{1}$ is in
upper block triangular form and according to Lemma 8.1 each diagonal block
$4\times 4$ matrix is full rank, thus matrix $\overline{\mathbf{M}}_{1}$ is
obviously of full rank. ∎
###### Lemma 8.8.
$\overline{\mathsf{T}}_{ext}$ is diagonalizable if and only if for any set of
segments there exists at least one segment in the set that has at least four
vertices which are not in the other segments in the set.
###### Proof.
Assume the T-mesh is diagonalizable but there exists a set of segments
$\\{G_{i_{j}},j=1,\dots,s\\}$ such that any segment in the set has at most
three vertices which are not on the other segments in the set. Without loss of
generality, assume the diagonalizable segment ordering for
$\overline{\mathsf{T}}_{ext}$ is $G_{i},i=1,\dots,\overline{n}^{G}$. Let $k$
be the maximal index for all $i_{j},j=1,\dots,s$ and consider the set
$\\{G_{i},i=1,\dots,k\\}$. Since $G_{k}$ has at most three vertices which are
not in the segments $G_{i_{j}},j=1,\dots,s$, there exists an index $i,i<k$
such that the number of vertices on segment $G_{k}$ but not on segment $G_{i}$
is at most three. This violates the assumption that
$\overline{\mathsf{T}}_{ext}$ is diagonalizable.
Suppose for any set of segments in $\overline{\mathsf{T}}_{ext}$ there exists
at least one segment which has at least four vertices which are not on the
other segments in the set. For the set containing every segment, according to
the assumption, there exists one segment, $G_{\overline{n}^{G}}$, which has at
least four vertices which are not on the other segments. Now, removing
$G_{\overline{n}^{G}}$ from the set, we have that in the set of remaining
segments there exists one segment, $G_{\overline{n}^{G}-1}$, which has at
least four vertices which are not on the other segments. Continuing this
process, we can arrange all the segments $G_{i},i=1,2,\dots,\overline{n}^{G}$
such that the number of vertices on segment $G_{j}$ but not on segment
$G_{i},i<j$ is at least 4\. Thus $\overline{\mathsf{T}}_{ext}$ is
diagonalizable. ∎
###### Theorem 8.9.
For an analysis-suitable T-mesh, matrix $\overline{\mathbf{M}}$ has full
column rank.
###### Proof.
For an analysis-suitable T-mesh we will prove that the corresponding
simplified extended T-mesh is diagonalizable. Otherwise, according to Lemma
8.8, there exists a set of segments such that each segment in the set has at
most three vertices which are not on the other segments in the set. It is
evident that the set must contain horizonal segments. Otherwise, any vertical
segment violates the assumption because it must have at least four vertices
which are not in the other segments in the set. Let $G_{i}$ be the bottommost
horizonal segment in the set (if there is more than one such segment choose
one of them). Since $G_{i}$ has at most three vertices which are not on the
other segments in the set, one of the four extended vertices of $G_{i}$ must
lie on a vertical segment, $G_{j}$. Now, referring to Figure 12, since the
T-mesh is analysis-suitable, $G_{j}$ must have two anchor vertices whose
vertical index coordinate is less than $a$. Otherwise, the T-mesh is not
analysis-suitable due to intersecting T-juction extensions. Additionally,
$G_{j}$ must have two extended vertices whose vertical index coordinate is
less than $a$. Thus, there are four vertices in $G_{j}$ which do not belong to
any other segment in the set which contradicts the assumption. ∎
Figure 12: A schematic for Theorem 8.9.
## 9 Conclusion
We have established several important properties of analysis-suitable
T-splines. We developed a characterization of ASTS spaces and proved that the
space of bicubic polynomials, defined over the extended T-mesh of an ASTS, is
contained in the corresponding ASTS space. We then proved the conditions under
which two ASTS spaces are nested. This provides the theoretical foundation for
the analysis-suitable local refinement algorithm in [3]. Using the
characterization of ASTS we then proved several basic approximation results.
Additionally, we developed the theory of perturbed ASTS and a simple mesh-
based dimension formula which is written in terms of the vertices in the
extended T-mesh of a T-spline. Both of these developments were critical for
the proofs in this paper and may have important applications in other
contexts. While the developments in this paper are restricted to bicubic
surfaces the extension to arbitrary degree should be straightforward.
## Acknowledgements
The authors thank Prof. A. Buffa, G. Sangalli, R. Vazquez, University of
Pavia, and T. W. Sederberg, Brigham Young University, for several fruitful and
insightful discussions on the technical aspects of this work. This work was
supported by grants from the NSF of China (Nos.11031007, 60903148), the
Chinese Universities Scientific Fund, SRF for ROCS SE, and the Chinese Academy
of Science (Startup Scientific Research Foundation). M.A. Scott was partially
supported by an ICES CAM Graduate Fellowship. This support is gratefully
acknowledged.
## Appendix A Notational conventions
In Tables 2 and 3 the notational conventions used throughout the text are
listed as well as the section where they are defined.
Table 2: Notational conventions used throughout the text and where they are defined. Symbols | Description | Section
---|---|---
$\mathsf{T}$ | A T-mesh | 2.1
$\mathsf{V}$ | All vertices in a T-mesh | 2.1
$\mathsf{E}$ | All edges in a T-mesh | 2.1
$\mathsf{S}$ | The skeleton of a T-mesh | 2.1
$h\mathsf{S}$ $(v\mathsf{S})$ | The horizontal (vertical) skeleton of a T-mesh | 2.1
$Q$ | An element of a T-mesh | 2.1
$\mathsf{G}$ | All segments in a T-mesh | 2.4
$n^{G}$ | The number of segments in a T-mesh | 2.4
$h\mathsf{G}$ ($v\mathsf{G}$) | All horizontal (vertical) segments in a T-mesh | 2.4
$h\mathsf{G}(a)$ $(v\mathsf{G}(a))$ | The horizontal (vertical) segments at vertical (horizontal) index $a$ | 2.4
$\mathsf{R}$ | The index domain of a T-mesh | 2.1
$\mathsf{AR}$ | The active region of an index domain | 2.1
$\mathsf{FR}$ | The frame region of an index domain | 2.1
$\mathsf{T}_{ext}$ | An extended T-mesh | 2.5
$ext^{f}(\mathsf{T})$ | All face extensions in $\mathsf{T}$ | 2.5
$ext^{f}(T)$ | The face extension of T-junction $T$ | 2.5
$ext^{e}(\mathsf{T})$ | All edge extensions in $\mathsf{T}$ | 2.5
$ext^{e}(T)$ | The edge extension of T-junction $T$ | 2.5
$ext(\mathsf{T})$ | All extensions in $\mathsf{T}$ | 2.5
$ext(T)$ | The extension of T-junction $T$ | 2.5
$hext^{f}(\mathsf{T})$ $vext^{f}{\mathsf{T}}$ | All horizontal (vertical) faces extensions in a T-mesh | 2.5
$\mathsf{CV}$ | All crossing vertices in $\mathsf{T}_{ext}$ | 2.5
$n^{+}$ | The number of crossing vertices in $\mathsf{T}_{ext}$ | 2.5
$\mathsf{OV}$ | All overlap vertices in $\mathsf{T}_{ext}$ | 2.5
$n^{-}$ | The number of overlap vertices in $\mathsf{T}_{ext}$ | 2.5
$\mathsf{EV}$ | All extended vertices in $\mathsf{T}_{ext}$ | 2.5
$n^{*}$ | The number of overlap vertices in $\mathsf{T}_{ext}$ | 2.5
$n^{ext}$ | The number of vertices in $\mathsf{T}_{ext}$ | 2.5
Table 3: Notational conventions used throughout the text and where they are defined. Symbols | Description | Section
---|---|---
$\mathsf{A}(\mathsf{T})$ | All anchors in a T-mesh | 2.3
$n^{A}$ | The number of anchors in a T-mesh | 2.3
$\mathsf{J}$ | The set of T-junctions in a T-mesh | 2.3
$hv(A)$ ($vv(A)$) | The horizontal (vertical) local index vector for $N_{A}(\xi,\eta)$ | 3
$\Xi_{A}$ ($\Pi_{A}$) | The horizontal (vertical) local knot vector for $N_{A}(\xi,\eta)$ | 3
$\tilde{\Omega}$ | The full parametric domain of $\mathsf{T}$ | 3
$\tilde{Q}$ | A T-spline element in the full parametric domain | 3
$\hat{\Omega}$ | The reduced parametric domain of $\mathsf{T}$ | 3
$\hat{Q}$ | A T-spline element in the reduced parametric domain | 3
$\Omega_{\hat{Q}}$ | The extended support of $\hat{Q}$ | 7
$R(\Omega_{\hat{Q}})$ | The smallest rectangle containg $\Omega_{hat{Q}}$ | 7
$N_{A}(\xi,\eta)$ | The T-spline blending function for anchor $A$ | 3
$Q_{A}$ | The support of $N_{A}$ | 7
$\lambda_{A}$ | The dual basis function associated with $N_{A}$ | 3
$\mathcal{T}$ | A T-spline space | 3
$\mathcal{S}_{ext}$ | A homogeneous extended spline space | 6
$\mathcal{T}_{ext}$ | An extended spline space | 6
$\Xi[\boldsymbol{\delta}]$($\Pi[\boldsymbol{\delta}]$) | A horizontal (vertical) perturbed global knot vector | 5
$\imath=\imath(i,g)$ ($\jmath=\imath(j,g)$) | the index of the $g^{th}$ segment in $h\mathsf{G}(\\{i\\})$ ($v\mathsf{G}(\\{j\\})$) | 5
$\pi$ | the map from perturbed index $\imath(i,g)$ to index $i$ | 5
$\mathsf{T}(\boldsymbol{\delta})$ | A perturbed T-mesh | 5
$\mathcal{T}(\boldsymbol{\delta})$ | A perturbed T-spline space | 5
$\mathsf{T}(\boldsymbol{\delta},\mathsf{T}^{i})$ | A perturbed T-mesh generated from $\mathsf{T}^{i}(\boldsymbol{\delta})$ | 5
## References
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|
arxiv-papers
| 2012-11-24T12:56:58 |
2024-09-04T02:49:38.354041
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Xin Li and M. A. Scott",
"submitter": "Xin Li",
"url": "https://arxiv.org/abs/1211.5669"
}
|
1211.5673
|
Vol.0 (200x) No.0, 000–000
11institutetext: National Astronomical Observatories, Chinese Academy of
Sciences, Beijing 100012, China; [email protected]
22institutetext: Department of Astronomy, Peking University, Beijing 100871,
China
33institutetext: Kavli Institute for Astronomy and Astrophysics, Peking
University, Beijing 100871, China
# Delayed Onset and Fast Rise of Prompt Optical-UV Emission from Gamma-Ray
Bursts in Molecular Clouds
Xiao-Hong Cui Zhuo Li 112233 Li-Ping Xin 11
(Received 2009 month day; accepted 2009 month day)
###### Abstract
Observations imply that long $\gamma$-ray bursts (GRBs) are originated from
explosions of massive stars, therefore they may occur in the molecular clouds
where their progenitors were born. We show here that the prompt optical-UV
emission from GRBs may be delayed due to the dust extinction, which can well
explain the observed optical delayed onset and fast rise in GRB 080319B. The
density and the size of the molecular cloud around GRB 080319B are roughly
constrained to be $\sim 10^{3}$cm-3 and $\sim 8$pc, respectively. We also
investigate the other GRBs with prompt optical-UV data, and find similar
values of the densities and sizes of the local molecular clouds. The future
observations of prompt optical-UV emission from GRBs in subsecond timescale,
e.g., by UFFO-Pathfinder and SVOM-GWAC, will provide more evidence and probes
of the local GRB environments.
###### keywords:
radiation mechanisms: non-thermal – gamma-rays: bursts – dust: extinction
## 1 Introduction
The properties of $\gamma$-ray burst (GRB) circumburst and host-galaxy
environment are important for the studies of GRB progenitors and the
fundamental conditions required within a galaxy to form a GRB. The multi-
wavelength observations about the emission from GRBs and that from their host
galaxies would provide a unique tool to understand the nature of GRBs and the
properties of interstellar medium (ISM) around the bursts.
Observations imply that long GRBs are originated from massive star explosions.
First, they are observed to lie in star-forming galaxies, or even within the
active star-forming regions of the host galaxies (e.g., Paczyński 1998; Bloom,
Djorgovski, & Kulkarni 2002). More precise HST images of afterglows reveals
that they occur within a few kiloparsecs of the flux-weighted centroid of
their host galaxies (Fruchter et al. 2006). Second, X-ray observations show
evidence for high column densities of gas around long GRBs, implying giant
molecular clouds around them (e.g., Galama & Wijers 2001). Finally, at least
some long GRBs are associated with core-collapse supernovae (SNe). The
discovery of four clear associations between long, soft GRBs and Type Ib/c SNe
and many SN-like bumps in the late optical afterglow light curves (see, e.g.,
review by Woosley & Bloom 2006) directly tell that their progenitors are
massive stars.
Now that the progenitors of long GRBs are massive stars, they may occur in the
birth place of the progenitors since massive stars are short-lived, i.e., the
long GRBs may lie in the molecular clouds that the massive stars are born. The
optical-UV and X-ray emission from GRBs can be affected significantly by the
extinction of dust and absorption of gas in the local environment,
$\gamma$-ray emission is almost unaffected. Therefore, one may naturally
expect that the behavior of prompt optical-UV emission is different in light
curves from that of prompt $\gamma$-ray emission. The difference may hint the
properties of the dust environments around the GRBs. The interaction of a GRB
with the environment can yield powerful clues on the properties of the medium
in which the burst occur. The behavior of the X-ray and optical opacities in
the nearby of a GRB have been studied (Perna, Raymond, & Loeb 2000; Perna&
Raymond 2000). A time-dependent photoionization code has been developed to
study the modificaitons in the dust distribution and the graphite in the
medium around the GRB was found to be more resistent than silicates (Perna &
Lazzati 2002; Lazzati & Perna 2002).
The varieties of observed GRB prompt optical behaviors are rich. The prompt
optical emission was first observed in GRB 990123 and was found to be
uncorrelated with the ongoing $\gamma$-ray emission (Akerlof et al. 1999; but
see Liang et al. 1999). Then the prompt optical emission from GRB 050820A
(Vestrand et al. 2006) was reported and a strong correlation between
$\gamma$-energy and optical emission in the prompt phase was discovered.
Similar cases of some degree of correlation are observed in GRB 041219A
(Vestrand et al. 2005; Blake et al. 2005), GRB 060526 (Thöne et al. 2010), and
“naked eye” burst GRB 080319B (Racusin et al. 2008; Beskin et al. 2010). GRB
080319B with the richest prompt optical observation attracted much attention
about it’s nature. The detailed observation of this burst presented by Racusin
et al. (2008) showed not only a correlation between $\gamma$-ray and optical
emission in the prompt phase but also an obvious delayed onset $\sim$ 15s
between them.
In this work, we show that if a GRB is located in a molecular cloud, its
prompt optical-UV emission may be absorbed by the dust in the molecular cloud,
and only emerges after the dusts on the line of sight are all destroyed. This
can well explain the observed delayed onset of the prompt optical-UV emission
in GRB 080319B, and the density and the size of the molecular cloud around
this burst can be roughly constrained. For other bursts with prompt optical
observations, the properties of local environment can also be constrained. We
find similar properties of the clouds, with density and size being $n_{\rm
H}\sim 10^{3}-10^{4}$ cm-3 and $\Delta R\sim 6$ pc. The paper is arranged as
the following: a simple model of the radiation-dust interaction and the
resulted prompt optical-UV light curve are presented in $\S$ 2; in $\S$ 3, we
apply the model to GRB 080319B and other GRBs with prompt optical-UV
observations; in $\S$ 4 discussions and conclusion are presented.
## 2 Radiation-dust interaction and emergent optical-UV emission
Consider a GRB that is located in a molecular cloud. The prompt optical-UV
emission from this GRB may be absorbed by the dust in the cloud, but in the
same time the dusts may also be destroyed by the emission. If the optical-UV
emission is strong and lasts long enough, it may emerge from the cloud after
the dusts on the way are all destroyed. The dust destruction by the optical-UV
radiation has been discussed by Waxman & Draine (2000). Here we will follow
their model in the radiation-dust interaction, and focus on the back effect of
the dust on the optical-UV emission, i.e., how the dusts in a cloud of finite
size affects the apparent light curve of the prompt optical-UV emission. On
the other hand, from the observed light curve profile of the prompt optical-UV
emission, we can also give some constraints on the properties of the molecular
cloud. We will only consider dust destruction due to thermal sublimation and
neglect the effect of grain charging, since, as argued by Waxman & Draine
(2000) and Draine & Hao (2002), the thermal sublimation is likely to be more
effective (see, but, Fruchter et al. 2001).
Considering a simple picture as shown in Fig.1, the molecular cloud is assumed
to be uniform in density, and the distance of the GRB from the edge of the
cloud on the side to the observer is $\Delta R$. The cloud contains dust
grains of characteristic radius $a$ and dust number density $n_{d}$. Assuming
a standard dust-to-gas mass ratio, $n_{d}$ is related to the cloud density
$n_{\rm H}$ as $n_{d}=0.01n_{\rm H}m_{\rm H}/(4\pi/3)a^{3}\rho$, where $\rho$
is mass density of the grain material. A characteristic value of $\rho=3.5$g
cm3 (Guhathakurta & Draine 1989) will be taken in the following calculations.
Considering that the source radiates a $1-7.5$ eV radiation with luminosity
$L_{1-7.5}$, a grain at a distance $r$ can be heated up leading to thermal
sublimation and thermal emission. The temperature $T$ of the grain at distance
$r$ from the source is governed by
$\frac{L_{1-7.5}}{4\pi r^{2}}Q_{\rm UV}\pi a^{2}=\langle Q\rangle_{T}4\pi
a^{2}\sigma T^{4}-4\pi a^{2}{da\over dt}{\rho\over m}B,$ (1)
where $m$ is the mean atomic mass, $B$ is the chemical binding energy per
atom, $Q_{\rm UV}$ is the absorption efficiency factor averaged over the
$1-7.5$ eV spectrum of the source emission, and $\langle Q\rangle_{T}$ is the
usual Planck-averaged absorption efficiency. We will assume $Q_{\rm UV}\approx
1$ for $a\ga 10^{-5}$cm and approximate $\langle Q\rangle_{T}$ by
$\langle Q\rangle_{T}\approx\frac{0.1a_{-5}(T/2300~{}\rm
K)}{1+0.1a_{-5}(T/2003~{}\rm K)}$ (2)
with $a_{-5}=a/10^{-5}\rm cm$. The thermal sublimation rate can be
approximated by (Guhathakurta & Draine 1989)
${da\over dt}=-(\frac{m}{\rho})^{1/3}\nu_{0}e^{-B/kT}.$ (3)
We adopt the frequency $\nu_{0}=1\times 10^{15}\rm s^{-1}$, $B/k=7\times
10^{4}$K, and $\rho/m=10^{23}\rm cm^{-3}$ as representative values
(Guhathakurta & Draine 1989; Waxman & Draine 2000). If we assume $T$ is
approximately constant during the illumination, then the grain survival time
at $T$ is $t_{\rm surv}(T)=a/|da/dt|$. The grain will be completely destructed
by thermal sublimation if it is illuminated over a time longer than $t_{\rm
surv}(T)$.
Figure 1: The sketch of an observed GRB located inside a molecular cloud.
”obs” denotes the direction to the Earth (observer). The cloud is assumed to
be with uniform density and a clear boundary. The distance of the GRB to the
edge of the cloud is $\Delta R$, as marked.
As the dust is destructed by the radiation, the radiation is also extinguished
by the dust. We consider below the effects of dust extinction on the observed
flash light curves. Following Waxman & Draine (2000) let us approximate the
$1-7.5$ eV emission from the GRB as a rectangular pulse of duration $\Delta t$
and luminosity $L_{1-7.5}$. The problem can be simplified by assuming that the
effects of extinction can be approximated as a narrowing of the optical pulse,
retaining a rectangular profile. The leading edge of the radiation is just at
the dust destruction front. We assume a sharp disruption front within which
the dust grains are all destructed whereas the grains further are not
affected. Define $f(r)$ the fraction of the flash energy that is absorbed by
dust interior to radius $r$. If $t_{\rm surv}<(1-f)\Delta t$, the grains are
destructed and $f(r)$ satisfies
$\frac{df}{dr}=Q_{\rm UV}n_{d}\pi a^{2}\frac{t_{\rm surv}}{\Delta t}.$ (4)
The relation between the radius of dust destruction front $R_{f}$ and observer
time $t_{\rm obs}$ can be given by
$t_{\rm obs}=f(R_{f})\Delta t(1+z),$ (5)
with $z$ the redshift of the GRB source. A (maximum) dust destruction radius
$R_{d}$ is determined by the condition $t_{\rm
surv}[T(R_{d})]=[1-f(R_{d})]\Delta t$. At $r>R_{d}$, the dust grain survives
the illumination and the destruction front does not move any more, therefore
we can simply assume $R_{f}=R_{d}$ and $f=1$ at $t_{\rm obs}>f(R_{d})\Delta
t(1+z)$.
The above discussion on $R_{f}$ propagation omits the existence of the edge of
the cloud at $\Delta R$. If $R_{f}<\Delta R$, the optical depth due to dust
extinction is
$\tau=Q_{\rm UV}n_{d}\pi a^{2}(\Delta R-R_{f}),$ (6)
and the attenuated luminosity observed outside is
$L_{\rm obs}(t_{\rm obs})=L_{1-7.5}\exp\\{-\tau[R_{f}(t_{\rm obs})]\\}.$ (7)
If $R_{d}>\Delta R$ then the destruction front can reach the edge
($R_{f}=\Delta R$) at time $t_{\rm obs}=f(\Delta R)\Delta t(1+z)$, which means
all the dust in the beam of the radiation is cleared and the radiation is not
attenuated, $\tau=0$. However, if $R_{d}<\Delta R$ then the dust is not
destructed completely, and the disruption front stays at $R_{d}$ at $t_{\rm
obs}>f(R_{d})\Delta t(1+z)$ while the dust optical depth is fixed at
$\tau=Q_{\rm UV}n_{d}\pi a^{2}(\Delta R-R_{d})$.
Note, in the former case $R_{d}>\Delta R$, the $1-7.5$ eV emission is firstly
totally attenuated, since $R_{f}<\Delta R$ and $\tau\gg 1$; when the
destruction front propagates to be close to the edge of the cloud,
$R_{f}\la\Delta R$ and $\tau\sim 1$, it starts to emerge by some fraction; and
after the destruction front reaches the edge, $R_{f}=\Delta R$, the emission
emerge completely without any extinction. Thus, the end time of the light
curve rising up depends on the edge of cloud $\Delta R$, while the slope of
the light curve rising up depends on the propagation speed of the destruction
front which is sensitive to the cloud density, $n_{\rm H}$.
## 3 Applications
As discussed above, in the case of $R_{d}>\Delta R$, the radiation-dust
interaction leads to that only the later part of the prompt optical-UV photons
emerges, but the $\gamma$-ray photons from the GRB are without any
attenuation. Thus, if the prompt $\gamma$-ray and optical-UV radiation is
emitted together from the GRB source, there should be a time delay between the
onset of the apparent prompt optical-UV and $\gamma$-ray emission. So far
there are quite a few GRBs that are detected with prompt optical-UV emission.
We will apply the simple radiation-dust interaction model to all these
detected GRBs, with the goal to explain the time delays of the prompt optical-
UV emission relative to $\gamma$-ray emission, and roughly give some
implications to the properties, e.g., the densities and the sizes of the
molecular clouds around them.
The observed luminosity is usually given in a single band for a single filter,
e.g., U, B, V, R bands etc. A spectrum with the form $f_{\nu}\propto\nu^{-1}$
is assumed for the prompt optical-UV flash in 1–7.5 eV, which is consistent
with the fast-cooling electrons expected in the standard internal shock model.
Thus, a cosmological $\kappa$–correction factor can be defined to account for
the transformation of the single passband of filter to the band of 1–7.5 eV in
the proper GRB frame,
$\kappa=\frac{\int^{7.5{\rm eV}/h(1+z)}_{1{\rm
eV}/h(1+z)}f_{\nu}d\nu}{\int^{b_{2}}_{b_{1}}f_{\nu}d\nu}$ (8)
where $b_{2}$ and $b_{1}$ are the frequency boundaries of the passband for the
observed filter, $z$ is the GRB redshift.
### 3.1 GRB 080319B
So far the so-called “naked-eye” GRB 080319B is the only one that happens to
occur in the field of view of an optical telescope, without the trigger by
high-energy detector, thus it is by luck monitored in optical band from before
the beginning of the GRB. The broadband observations of it has been presented
by Racusin et al. (2008) and Beskin et al. (2010). The $\gamma$-ray emission
was found to begin at about 4 s before the BAT trigger and last $\sim$ 57 s.
The bright optical transient begins at $\sim$ 10 s after the BAT trigger,
peaks at $\sim$18 s and then fades below the threshold to magnitude $\sim$12
after 5 min. That is to say, there is a time delay $\sim$ 14 s between the
onsets of $\gamma$-ray and optical emission. It should be noticed that the
optical rising is too fast to be accounted for by the afterglow model, either
forward shock emission (Sari et al. 1998) or reverse shock emission (Kobayashi
et al. 2000). The optical light curve during the plateau phase shows
fluctuation, similar to the $\gamma$-ray one. Moreover, the optical and
$\gamma$-ray emission is found to be correlated. All these features suggest
that the prompt optical emission from this burst is not produced by afterglow
shock. Thus the delayed up-rising optical emission needs other explanation. We
show below that the delay can be well explained by the radiation-dust
interaction.
We, again, approximate the intrinsic optical-UV emission as a rectangle pulse
of duration $\Delta t$ (in the rest frame of the GRB). Since the optical
emission is observed to decay at $\approx 50$s (similar to the $\gamma$-ray
duration), the duration is $\Delta t\approx 50/(1+z)\approx 25$s, where the
GRB redshift is $z=0.937$ (Vreeswijk et al. 2008). Apparently in observations,
the optical flux rises from zero to a plateau phase at a time $t_{b}\approx
15$s after trigger, and the plateau phase ends at $\sim$ 50 s. The mean
luminosity after $t_{b}$ (i.e.,in the range of 15–50 s) and in 1–7.5 eV energy
band can be given by $L_{1-7.5}=4\pi D_{L}(z)^{2}\kappa f_{p}\approx 2.6\times
10^{50}$ erg s-1. Here $D_{L}(z)$ is the luminosity distance calculated
(adopting a Universe model with $\Omega_{\mathrm{M}}=0.3$,
$\Omega_{\Lambda}=0.7$, and $H_{0}=71$ km $\rm{s}^{-1}$ $\rm{Mpc}^{-1}$), and
$f_{p}=9.39\times 10^{-9}$ erg cm${}^{-}2$ s-1 is the mean flux observed
during the time range 15–50 s in V band observed by TORTORA (Pagani et al.
2008; Racusin et al. 2008). The correction factor $\kappa=6.17$ for this burst
is calculated by assuming a power law spectrum $f_{\nu}\sim\nu^{-1}$.
If the duration of the optical-UV emission $\Delta t=25$ s is taken and
assuming the radius of dust grain as $a=1\times 10^{-5}$ cm, we calculate the
absorbed energy fraction $f(r)$ of the flash up to the destruction radius
$R_{d}$ for a cloud density range of $n_{\rm H}=10^{2}-10^{5}$cm-3. The result
is shown in Fig.2. We can see the denser the cloud the faster the flash energy
is absorbed. However, after the destruction front reaches the destruction
radius $r=R_{d}$, the absorbed fraction rapidly reaches unity, $f(R_{d})=1$.
Figure 2: The fraction $f(r)$ of flash energy absorbed by dust interior to
radius $r$ up to dust destruction radius $r=R_{d}$ in the case of GRB 080319B.
The duration and luminosity of prompt emission in 1–7.5 eV are $\Delta t=25$ s
and $L_{1-7.5}=2.6\times 10^{50}$ erg s-1, respectively, and the dust grain
size is assumed to be $a=10^{-5}$ cm. Different lines correspond to different
values of cloud density $n_{\rm H}$, as marked in the plot. The dash line
shows $f(r=\Delta R)=0.3$. The dotted line presents the case of
$f(r=R_{d})=0.3$, i.e., $R_{d}=\Delta R$.
Apparently in observations, the optical emission rises to the mean flux level
at about 15 s, and then keeps this level until $\sim$ 50 s. This implies that
the absorbed fraction of the flash energy, when the dust destruction front
reaches the edge of the cloud, is $f(R_{f}=\Delta R)\approx 15/50=0.3$. Thus,
given the cloud density $n_{\rm H}$, the cloud size, roughly implicated by
$\Delta R$, can be determined for this burst, i.e. $n_{\rm H}$ and $\Delta R$
are one by one related for fixed $f(r=\Delta R)$ value. For example, if
$n_{\rm H}=(10^{3},10^{4},10^{5})$cm-3, we have $\Delta R=(2.4,1.7,1.1)\times
10^{19}$cm, respectively with fixed $f(r=\Delta R)=0.3$. Then the value of
$\Delta R$ can be found to decrease with larger value of $n_{\rm H}$. However,
for too small $n_{\rm H}$, the destruction front reaches the maximum
destruction radius $R_{d}$ before reaching the edge of the cloud, i.e., the
absorbed fraction $f(r=R_{d})<0.3$, as the case of $n_{\rm H}=10^{2}$cm-3 in
Fig 2.
In order to decouple $n_{\rm H}$ and $\Delta R$, we need to further consider
the temporal profile of the observed optical-UV emission. For different values
of $n_{\rm H}$ and $t_{b}$, we have calculated the optical-UV light curve
using equation (7). The resulted light curves are shown in Fig.3, also plotted
are the $\gamma$-ray and optical-UV data that are adopted from Racusin et al.
(2008). Note, as Beskin et al. (2010) found that the optical emission is 2 s
delayed relative to the $\gamma$-ray emission in the plateau phase, we also
assume a time delay of 2 s for the intrinsic onset of optical-UV emission
compared to the $\gamma$-ray one. The plotted light curves in Fig. 3 take this
into account. We see that compared with the observed optical data of GRB
080319B, the case with $n_{\rm H}=10^{3}$cm-3 and $t_{b}=16$ s fits the light
curve profile better. Therefore, it can be concluded that the cloud that hosts
GRB 080319B has a density of $n_{\rm H}\approx 10^{3}$cm-3 and a size of
$R\sim\Delta R\approx 8$ pc.
Figure 3: The light curves of GRB 080319B in $\gamma$-ray and optical bands.
The black triangles are optical data from TORTORA. For comparison, the Konus-
Wind background subtracted $\gamma$-ray light curve (18–1,160 keV), with
respect with the trigger time by Swift-BAT, is shown with dashed line. The
solid lines are the calculated optical light curves in the simple radiation-
dust interaction model. Left panel: The cases with the cloud density $n_{\rm
H}=10^{2},10^{3},10^{4}$cm-3 (corresponding to three light curves from left to
right respectively) and the fixed end time of the rising part $t_{b}=16$ s.
Right panel: The cases with $t_{b}=12,16,20$ s (from left to right) and the
fixed $n_{\rm H}=10^{3}$cm-3. The other parameters are the same as Fig.2. The
case with $t_{b}=16$ s and $n_{\rm H}=10^{3}$cm-3 gives the best fit to the
rising part of the optical flash of GRB 080319B.
It should be noted that in the above calculations we have taken $\rho=3.5$g
cm3 (Guhathakurta & Draine 1989), $a=1\times 10^{-5}$cm and a standard dust-
to-gas mass ratio of 0.01. The resulted values of $n_{\rm H}$, $\Delta R$ and
$R_{d}$ are not sensitive to the values of them, i.e., the resulted $n_{\rm
H}$, $\Delta R$ and $R_{d}$ values vary within a factor of a few if changing
$\rho$, $a$, $\kappa$, and the gas-to-dust ratio by one order of magnitude.
This is good enough for order of magnitude estimate with the simple model
here.
### 3.2 Other GRBs with prompt optical detections
Besides GRB 080319B, there are quite a few other GRBs with prompt optical
detections during the $\gamma$-ray bursting phase. They are all detected by
rapid slew of optical telescopes to the GRB location after trigger by
$\gamma$-ray detectors. So usually there is a gap between the trigger time and
the start time of optical observation. Nevertheless, we can still try to make
some constraints on the local GRB environments based on the simple radiation-
dust interaction model.
All the GRBs detected after December of 2004 and with optical detections
during the prompt $\gamma$-ray emission are collected and analyzed with the
simple radiation-dust interaction model here. We separate these GRBs into two
samples. In Sample I, the GRBs satisfie the following three criterions: (1)
There are optical detections before the end of the GRB, specifically, the
optical detection is within the duration of $T_{90}$; (2) The optical light
curve within $T_{90}$ shows a rising of the flux, i.e., if the optical light
curve shows a decay or flat plateau then the GRB is not included; (3) The
number of optical data points, excluding upper limits, in the rising part is
not less than three. All the other GRBs only satisfy criterion (1) are grouped
into Sample II.
We find besides GRB 080319B, there are 7 other GRBs satisfy the three
criterions: GRB 041219A (Vestrand et al. 2005; Blake et al. 2005); GRB050820A
(Vestrand et al. 2006); GRB 060218 (Mirabal et al. 2006; Ferrero et al. 2006;
Sollerman et al. 2006; Kocevski et al. 2007); GRB 060418 (Molinari1 et al.
2007; Dupree et al. 2006; Vreeswijk & Jaunsen 2006), GRB 060607A (Molinari1 et
al. 2007; Ledoux et al. 2006), GRB 080810 (Page et al. 2009; Burenin et al.
2008), and GRB 100906A (Gorbovskoy et al. 2011; Barthelmy et al. 2010;
Markwardt et al. 2010; Tanvir et al. 2010). However, we exclude GRB 041219A
and GRB 060218 from Sample I for reasons as follows. GRB 041219A shows
correlation between $\gamma$-ray and optical emission, thus the observed
initial rising in the optical band is likely intrinsic (Vestrand et al. 2005)
other than due to radiation-dust interaction. This GRB is included in Sample
II instead. As for GRB 060218, its early optical-UV emission is likely
associated with the supernova shock breakout (Campana et al. 2006; Waxman et
al 2007), thus not due to radiation-dust interaction either. Sample I GRBs are
listed in Table 1. All the other GRBs only satisfy criterion (1) are grouped
into Sample II. For example, GRB 110205A (Klotz et al. 2011a, b; Cucchiara et
al. 2011) was started to be detected in optical band 166 s after trigger but
within the duration of $T_{90}=257$ s. The observed optical light curve
already appears to be a plateau, without a rising part, which may occur before
the start of optical detections. We exclude this burst in Sample I but include
in Sample II. As shown in Table 2, there are 13 GRBs in Sample II.
Table 1: The observational results of GRBs in Sample I and the constraints of
their local molecular clouds
GRB | $z$ | $T_{90}$ | $t_{\rm op}$ | $\kappa^{*}$ | $L_{1-7.5}$ | $R_{\rm d}$ | $\Delta R$ | $n_{\rm H}$ | $\Delta t_{\rm obs}$ | $t_{b}$ | ref
---|---|---|---|---|---|---|---|---|---|---|---
| | (s) | (s) | | ($10^{48}$ erg s-1) | (pc) | (pc) | ($10^{3}$ cm-3) | (s) | (s) |
050820A | 2.6 | 750 ∗∗ | 84 | 3.10 (R) | 0.6 | 3.35 | 3.23 | 9 | 646 | 305 | 1, 2, 3
060418 | 1.49 | 103.1 | 40 | 4.07 (H) | 2.4 | 6.91 | 6.87 | 4 | 140 | 107 | 4, 5, 6
060607A | 3.082 | 102 | 73 | 4.07 (H) | 2.3 | 5.90 | 5.87 | 5 | 200 | 150 | 4, 7
080319B | 0.937 | 57 | 8.9 † | 6.17 (V) | 260 | 7.78 | 7.67 | 1 | 50 | 16 | 8, 9, 10
080810 | 3.35 | 106 | 38 | 3.10 (W) | 7.1 | 11.7 | 11.4 | 3 | 150 | 67 | 11, 12
100906A | 1.727 | 114.4 | 48.5 | 3.10 (W) | 1.2 | 4.60 | 4.48 | 15 | 190 | 83 | 13, 14, 15, 16
* *
* In the bracket is the passband of filter. Letters “V”, “H” and “W” denote V, H and white bands, respectively.
* *
** From the work of Vestrand et al. (2006), rather than Swift data.
* *
${\dagger}$ The time corresponds to the first optical data by TORTORA. In
fact, the optical observations start before the trigger of this GRB.
* *
References: (1) Prochaska et al. 2005; (2) Ledoux et al. 2005; (3) Vestrand et
al. 2006; (4) Molinari1 et al. 2007; (5) Dupree et al. 2006; (6) Vreeswijk &
Jaunsen 2006; (7) Ledoux et al. 2006; (8) Vreeswijk et al. 2008; (9) Racusin
et al. 2008; (10) Beskin et al. 2010; (11) Page et al. 2009; (12) Burenin et
al. 2008; (13) Gorbovskoy et al. 2011; (14) Barthelmy et al., 2010; (15)
Markwardt et al., 2010; (16) Tanvir et al. 2010
Table 2: The observational results of GRBs in sample II and the constraints of
their local molecular clouds
GRB | $z$ | $T_{90}$ | $t_{\rm op}$ | $L_{1-7.5}$ | ref
---|---|---|---|---|---
| | (s) | (s) | (1047erg s-1) |
041219A | 0.31 | 520 | 460 | 3.7$\times 10^{-5}$ | 1,2
050319 | 3.24 | 160.5 | 30.4 | 3.9 | 3,4
050904 | 6.29 | 174.2 | 150.3 | 1.1 | 5
060526 | 3.21 | 298.2 | 16.1 | 3.4 | 6
060904B | 0.703 | 171.5 | 21 | $<$6.8$\times 10^{-2}$ | 7
061126 | 1.16 | 70.8 | 42 | 1.9$\times 10^{-2}$ | 8,9
071003 | 1.1 | 150 | 44.5 | 3.0 | 10
071031 | 2.69 | 180 | 59.6 | 2.3 | 11,12
080603A | 1.69 | 150 | 105 | 5.8$\times 10^{-3}$ | 13
080607 | 3.036 | 79 | 24.5 | 2.7 | 14
100901A | 1.408 | 439 | 113.4 | 9.6$\times 10^{-2}$ | 15,16,17
100902A | 4.5 | 428.8 | 104 | $<$1.4 | 17
110205A | 1.98 | 257 | 166 | 0.2 | 18
* *
References: (1) Vestrand et al. 2005; (2) Blake et al. 2005 (3) Quimby et al.
(2006); (4) Wozniak et al. 2005; (5) Boër et al. 2006; (6) Thöne et al. 2010:
(7) Klotz et al. 2008; (8) Gomboc et al. 2008; (9) Perley et al. 2008a; (10)
Perley et al. 2008b; (11) Kruehler et al. 2007; (12) Antonelli et al. 2007;
(13) Guidorzi et al. 2011; (14) Perley et al. 2011; (15) Chornock et al. 2010;
(16) Immler et al. 2010; (17) Gorbovskoy et al. 2011; (18) Cucchiara et al.
2011
#### 3.2.1 Sample I
For GRBs in Sample I, we can follow the same approach we carry for GRB
080319B, and the optical rising can be accounted for by the simple radiation-
dust interaction model, in the same time the local environments of these
bursts are constrained by fitting the observed promptly optical-UV light
curves.
We assume there is intrinsic optical-UV emission associated with the
$\gamma$-ray emission, with approximated rectangle light curve profile. The
optical-UV duration $\Delta t_{\rm obs}=\Delta t(1+z)$ is obtained from
observations (which is usually comparable or somewhat larger than the
$\gamma$-ray duration). The luminosity $L_{1-7.5}$ is calculated from observed
optical emission, by correction with $\kappa$ factor assuming a
$f_{\nu}\propto\nu^{-1}$ spectrum (For white band the same $\kappa$ factor as
R band is assumed). There are usually fluctuations of optical flux in the
plateau phase, thus we use the average of optical flux during the plateau
phase (i.e., after the rising part and before the decay phase) to calculate
the $L_{1-7.5}$ values: We average the optical data of GRB 050820A during the
period of $t_{\rm obs}=230-722$s; GRB 060418 of $107-137$s; GRB 060607A of
$159-205$s; GRB 080810 of $67-261$s; and for GRB 100906A we use the peak flux
at 115s.
Given $L_{1-7.5}$ and $\Delta t$, the maximum dust destruction radius $R_{d}$
can be determined to be as function as density $n_{\rm H}$. Furthermore, the
time $t_{b}$ that the optical flux rise to the top value can be estimated from
the observed optical light curve. Once given $f(r=\Delta R)=t_{b}/\Delta
t_{\rm obs}$ and combined with the condition of $R_{d}=\Delta R$, one obtains
a minimum $n_{\rm H}$ value, $n_{\rm H}>n_{\rm H,0}$, otherwise, the
destruction front cannot reach the edge of the cloud and no optical-UV
emission escapes from the cloud. Finally, we apply eq. (7) to fit the rising
part of the optical light curve by taking $n_{\rm H}$ (in the range of $n_{\rm
H}>n_{\rm H,0}$) and $t_{b}$ as free parameters. The best fit gives us the
resulted values of $n_{\rm H}$ and $\Delta R$.
The resulted values of $n_{\rm H}$ and $\Delta R$ is also listed in Table 1.
Illustrations of our fitting results for the four bursts included in Sample I
are shown in Fig. 4. From the fitting results, we find that the density of the
surrounding molecular clouds are in the range of $10^{3}-10^{4}$cm-3, while
the size, as implicated by $\Delta R$, is in the order of $\sim 10$ pc.
However, due to small number of GRBs with optical rising part detected in
prompt emission, it is impossible to give the statistic discussion of the
properties of local molecular clouds. Furthermore, the observed data points in
the optical rising part are usually sparse for individual GRB, which may
induce large errors in light curve fitting. The future precise observations
are needed to test the model and constrain the properties of local environment
more precisely.
Figure 4: The light curves of prompt optical emission from four other GRBs in
Sample I besides GRB 080319B: GRBs 050820A, 060418, 060607A, 080810 and
100906A. The black triangles are the optical data. The solid lines are the
predictions from radiation-dust interaction model, with the parameters $n_{\rm
H}$ and $t_{b}$ marked.
#### 3.2.2 Sample II
For GRBs in Sample II, because there is no optical rising detected due to the
delay of optical observations, we cannot well constrain the properties of the
surrounding environments. However we still try to make some constraints,
although rough. In these bursts, the time $t_{b}$ when the optical-UV flux
reaches the plateau, i.e., when the dust destruction front reaches the edge of
the cloud, $R_{f}=\Delta R$, can be considered to be smaller than the start
time of the optical observations, $t_{\rm op}$. Thus we have $t_{b}<t_{\rm
op}$. Moreover, we take the flux of the first optical data point to calculate
the mean luminosity in optical-UV band. In these bursts, the maximum dust
destruction radius must be within the boundary of the cloud, $R_{d}<\Delta R$,
otherwise the optical emission from these GRBs cannot emerge.
Thus we constrain the properties of the molecular clouds of sample-II GRBs as
follows. Considering an optical-UV flash with luminosity $L_{1-7.5}$ and
duration of $\Delta t=t_{\rm op}/(1+z)$, we can calculate the maximal dust
destruction radius as function of surrounding density. This puts an upper
limit to the value of $\Delta R$ of the relevant GRB. The results for all GRBs
in sample II are shown in Fig5. We see that although there is no good
constraints on density $n_{\rm H}$, the value of $\Delta R$ is quite well
constrained since $\Delta R$ does not vary much with $n_{\rm H}$. All except
GRB 041219A have upper limits of $\Delta R<0.1-2$ pc, somewhat less than those
of sample-I GRBs. This might be reasonable since the luminosity of sample-II
GRBs are generally smaller than that of sample-I GRBs. GRB 041219A has
exceptionally small luminosity then its value of $\Delta R$ is smaller than
$\sim 0.01$ pc.
Figure 5: The constraints on the sizes and densities of molecular clouds
around GRBs in Sample II. The allowed parameter region for each GRB is that
below the line corresponding to it.
## 4 Discussion and conclusions
Long GRBs are believed to be the explosions of massive stars, therefore the
GRBs may occur in the molecular clouds where their progenitors were born. We
show in this work that the prompt optical-UV emission from GRBs, if originally
emitted simultaneously with $\gamma$-ray emission, may appear with relative
time delay in observations, due to the dust extinction. This can well explain
the optical delayed onset observed in GRB 080319B, and the number density and
the size of the molecular cloud are roughly constrained to be $n_{\rm H}\sim
10^{3}$cm-3 and $\Delta R\sim 8$pc, respectively. We also investigate the
other GRBs with good optical-UV data, and find the densities and sizes of the
molecular clouds in the range of $n_{\rm H}\sim 10^{3}-10^{4}$cm-3 and $\Delta
R\sim 10$pc.
We use a simple picture that the effects of extinction is approximated as a
narrowing of the optical pulse, retaining a rectangular profile. This neglects
that there may be fluctuation of the original flux with time, and that the
dust destruction front is not a zero-thickness one. Thus the constraints on
the molecular clouds only make sense by order of magnitude.
The resulted $n_{\rm H}$ and $\Delta R$ constraints suggest high column
densities of gas around GRBs, $\sim 10^{22}-10^{23}$cm-3. It is interesting to
note that Galama & Wijers (2001) obtain similar range of column densities by
observations of X-ray afterglow spectra. Moreover, our constraints are also
consistent with those giant molecular clouds found in Milky Way, which are
observed to have sizes of $10-30$ pc and average gas densities of
$10^{2}-10^{3}$cm-3 (Winnewisser et al. 1979; Goldsmith 1987). Although the
GRB local molecular clouds from our constraints seem to be slightly denser,
the low statistics of GRBs with prompt optical detection and the sparse data
points for individual GRBs prevent us from giving clear conclusions.
One may expect that the initial fast rise of the prompt optical flux can be
produced by the afterglow forward shock due to sweep-up the circumburst medium
before deceleration. However, the multi-band observations of two sample-I GRBs
060418 and 060607A show spectral index in the optical band of
$F_{\nu}\propto\nu^{-0.9}$ and $F_{\nu}\propto\nu^{-0.8}$, respectively. This
implies the injection frequency below the optical band, $\nu_{m}<\nu_{\rm
opt}$, and require extremely unusual afterglow model parameters, e.g.,
postshock electron energy far smaller than typical value, $\epsilon_{e}\la
10^{-3}$. Moreover, most GRBs in sample I show up-rising even faster than
$F_{\nu}\propto t^{3}$ (e.g., GRBs 060418, 060607A, 080319B and 100906A),
which is faster than the model prediction for the pre-deceleration forward
shock emission at $\nu>\nu_{m}$. One may also expect the up-rising part can be
accounted for by the reverse shock emission due to shock swept-up of outflow
material but the predicted temporal slope is not faster than $F_{\nu}\propto
t^{2}$ (Kobayashi 2000). Thus the prompt optical emission is more likely to be
generated within the outflow.
The density of the surrounding from our constraints is higher in general than
the medium density implicated by afterglow modeling. The X-ray absorber must
lie within $1-5$ pc from the GRB thus probing the innermost region in the
close vicinity of the GRB explosion. Comparing the HI column densities from
Ly$\alpha$ absorption to the metal column densities from X-ray absorption in
GRB afterglows, Watson et al. (2007) found there no correlation between the
column density values, and the X-ray absorptions often far exceed the HI
column densities. Based on a detailed study of the absorption pattern, Campana
et al. (2011) found a high-metallicity absorbing medium for GRB 090618 and a
best-fitting column densities $6\times 10^{17}$ cm2 in Ne and Si. However
there is no contradiction here because the size of observed afterglow is
usually sub-pc scale, but the region concerned here is in much larger scale,
$\sim 10$pc, as resulted from the constraints. Thus, it may be that in the
place very close to the GRB location the medium density is low while the
further-out region has much denser gas. This is reasonable that the vicinity
of the GRB source may be affected by the progenitor before the GRB explosion.
In our simple model, for given luminosity and duration of the prompt optical-
UV emission, the maximum dust destruction radius can be determined. Once it is
within the boundary of the cloud, $R_{d}<\Delta R$, there will be no prompt
optical emission observed, neither the optical afterglow emission. The GRB
will appear as being optically dark in this case. It is interesting to note
that only 60% GRBs observed by BAT/Swift are detected by UVOT/Swift in the
optical afterglows. The “dark burst” (van der Horst et al. 2009) are still
mystery now. If dust extinction is the reason, then by our simple model, this
suggests that the maximum dust destruction radii and the sizes of the
molecular cloud are statistically comparable, i.e., $R_{d}\sim\Delta R$, thus
the bright and dark bursts are comparable in numbers. Indeed, as shown in
Table 1, those bright GRBs in sample I with better observations and hence
better constraints, $R_{d}$ and $\Delta R$ values are similar.
There are quite a few small robotic telescopes that have been built and
installed around the world in order to detect the optical counterparts in the
early phase of $\gamma$-ray bursts, such as Super-LOTIS (Park et al. 1997),
TAROT (Klotz et al. 2009), PROMPT (Reichart et al. 2005), ROTSE-III (RykoR et
al. 2009), SkyNet111http://skynet.unc.edu/, WIDGET (Urata et al. 2011),
MASTER222http://observ.pereplet.ru/, Pi of the sky (Burd et al. 2005), and
TORTORA333http://www.eso.org/public/images/eso0808a/ etc. With their large
Field of View (FOV) and fast slewing abilities, these telescopes are able to
detect the prompt optical emission in minute timescale after the trigger of
GRBs by $\gamma$-ray detectors. In the case of the “naked-eye GR” 080319B
(Racusin et al. 2008), the prompt optical emission was caught by the TORTORA
and Pi of the sky even with zero time delay. Besides, the UFFO-Pathfinder
(Chen et al. 2011), which aims at prompt optical detection in subsecond
timescale, will be launched soon; the Ground-based Wide-Angle Camera array
(GWAC), with a larger field of view ($\sim$ 8000 square degrees), as a part of
ground system of the Chinese-French SVOM mission (Paul et al. 2011), aiming at
search for the optical emission in zero delay, will be constructed in the near
future. All these robotic telescopes and the planed projects will make a
larger and better sample of prompt optical emission from GRBs in the future,
leading to more precise constraints on the GRB local environments.
###### Acknowledgements.
We would like to thank the useful discussions at pulsar group of PKU and at
SVOM group of NAOC. This work is supported by National Basic Research Program
of China-973 Program 2009CB824800, China Postdoctoral Science Foundation
funded project (No. 20110490590), the National Natural Science Foundation of
China (Grant Nos. 111030262) and the Foundation for the Authors of National
Excellent Doctoral Dissertations of China.
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|
arxiv-papers
| 2012-11-24T14:04:26 |
2024-09-04T02:49:38.365092
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Xiao-Hong Cui, Zhuo Li and Li-Ping Xin",
"submitter": "Xiaohong Cui",
"url": "https://arxiv.org/abs/1211.5673"
}
|
1211.5748
|
# Cofiniteness of weakly Laskerian local cohomology modules
Moharram Aghapournahr∗ and Kamal Bahmanpour
Dedicated to Professor Leif Melkersson Department of Mathematic, Faculty of
Science, Arak University, Arak, 38156-8-8349, Iran. [email protected]
Department of Mathematics, Islamic Azad University-Ardabil branch, P.O. Box
5614633167, Ardabil, Iran. [email protected]
###### Abstract.
Let $I$ be an ideal of a Noetherian ring R and M be a finitely generated
R-module. We introduce the class of extension modules of finitely generated
modules by the class of all modules $T$ with $\dim T\leq n$ and we show it by
${\rm FD_{\leq n}}$ where $n\geq-1$ is an integer. We prove that for any ${\rm
FD_{\leq 0}}$(or minimax) submodule N of $H^{t}_{I}(M)$ the R-modules ${\rm
Hom}_{R}(R/I,H^{t}_{I}(M)/N)\,\,\,{\rm and}\,\,\,{\rm
Ext}^{1}_{R}(R/I,H^{t}_{I}(M)/N)$ are finitely generated, whenever the modules
$H^{0}_{I}(M)$, $H^{1}_{I}(M)$, …, $H^{t-1}_{I}(M)$ are ${\rm FD_{\leq 1}}$ (
or weakly Laskerian). As a consequence, it follows that the associated primes
of $H^{t}_{I}(M)/N$ are finite. This generalizes the main results of
Bahmanpour and Naghipour [4] and [5], Brodmann and Lashgari [7],
Khashyarmanesh and Salarian [21] and Hong Quy [18]. We also show that the
category $\mathscr{FD}^{1}(R,I)_{cof}$ of $I$-cofinite ${\rm FD_{\leq 1}}$
$R$-modules forms an Abelian subcategory of the category of all $R$-modules.
###### Key words and phrases:
Local cohomology module, cofinite module, Weakly Laskerian modules.
2000 Mathematics Subject Classification: 13D45, 14B15, 13E05.
∗Corresponding author: e-mail: [email protected] (Moharram Aghapournahr)
## 1\. Introduction
The following conjecture was made by Grothendieck in [15]:
Conjecture: _For any ideal $I$ of a Noetherian ring $R$ and any finite
$R$–module $M$, the module $\operatorname{Hom}_{R}(R/I,H^{j}_{I}(M))$ is
finitely generated for all $j\geq 0$_.
Here, $H^{j}_{I}(M)$ denotes the $j^{th}$ local cohomology module of $M$ with
support in $I$. Although the conjecture is not true in general as was shown by
Hartshorne in [16], there are some attempts to show that under some
conditions, for some number $t$, the module
$\operatorname{Hom}_{R}(R/I,H^{t}_{I}(M))$ is finite, see [2, Theorem 3.3],
[11, Theorem 6.3.9], [13, Theorem 2.1], [4, Theorem 2.6] and [5, Theorem 2.3].
In [16], Hartshorne defined an $R$-module $L$ to be $I$-cofinite, if ${\rm
Supp}(L)\subseteq V(I)$ and ${\rm Ext}^{i}_{R}(R/I,L)$ is finitely generated
module for all $i$. He asked:
_If $I$ is an ideal of $R$ and $M$ is a finitely generated $R$-module, when is
$H^{t}_{I}(M)$ $I$–cofinite for all i ?_
In this direction in section 3 we generalize [2, Theorem 3.3], [4, Theorem
2.6] and [5, Theorem 2.3] to the class of extension modules of finitely
generated modules by the class of all modules $T$ with $\dim T\leq 1$ (${\rm
FD_{\leq 1}}$). Note that the class of weakly Laskerian modules is contained
in the class of ${\rm FD_{\leq 1}}$ modules. More precisely, we shall show
that:
###### Theorem 1.1.
Let $R$ be a Noetherian ring and $I$ an ideal of $R$. Let $M$ be a finitely
generated $R$-module and $t\geq 1$ be a positive integer such that the
$R$-modules $H^{i}_{I}(M)$ are ${\rm FD_{\leq 1}}$ $R$-modules (or weakly
Laskerian) for all $i<t$. Then, the following conditions hold:
(i) The $R$-modules $H^{i}_{I}(M)$ are $I$-cofinite for all $i<t$.
(ii) For all ${\rm FD_{\leq 0}}$ (or minimax) submodule $N$ of $H^{t}_{I}(M)$,
the $R$-modules
${\rm Hom}_{R}(R/I,H^{t}_{I}(M)/N)\,\,\,{\rm and}\,\,\,{\rm
Ext}^{1}_{R}(R/I,H^{t}_{I}(M)/N)$
are finitely generated.
As an immediate consequence we prove the following corollary that is a
generalization of Bahmanpour-Naghipour’s results in [4] and also the Delfno-
Marley’s result in [10] and Yoshida’s result in [27] for an arbitrary
Noetherian ring.
###### Corollary 1.2.
Let $R$ be a Noetherian ring and $I$ an ideal of $R$. Let $M$ be a finitely
generated $R$-module such that the $R$-modules $H^{i}_{I}(M)$ are ${\rm
FD_{\leq 1}}$ (or weakly Laskerian) $R$-modules for all $i$. Then,
(i) the $R$-modules $H^{i}_{I}(M)$ are $I$-cofinite for all $i$.
(ii) For any $i\geq 0$ and for any ${\rm FD_{\leq 0}}$ (or minimax) submodule
$N$ of $H^{i}_{I}(M)$, the $R$-module $H^{i}_{I}(M)/N$ is $I$-cofinite.
Abazari and Bahmanpour in [1] studied cofiniteness of extension functors of
cofinite modules as a generalization of Huneke-Koh’s results in [17]. In
Corollary 3.8 we generalis the results of Abazari and Bahmanpour.
Hartshorn also posed the following question:
Whether the category $\mathscr{M}(R,I)_{cof}$ of $I$-cofinite modules forms an
Abelian subcategory of the category of all $R$-modules? That is, if
$f:M\longrightarrow N$ is an $R$-module homomorphism of $I$-cofinite modules,
are $\ker f$ and ${\rm coker}f$ $I$-cofinite?
Hartshorne proved that if $I$ is a prime ideal of dimension one in a complete
regular local ring $R$, then the answer to his question is yes. On the other
hand, in [10], Delfino and Marley extended this result to arbitrary complete
local rings. Recently, Kawasaki [20] generalized the Delfino and Marley’s
result for an arbitrary ideal $I$ of dimension one in a local ring $R$.
Finally, more recently, Sedghi, Bahmanpour and Naghipour in [6] completely
have removed local assumption on $R$. One of the main results of this section
is to prove that the class of $I$-cofinite ${\rm FD_{\leq 1}}$ modules compose
an Abelian category (see Theorem 3.7).
Let $R$ denote a commutative Noetherian ring, and let $I$ be an ideal of $R$.
Throughout this paper, $R$ will always be a commutative Noetherian ring with
non-zero identity and $I$ will be an ideal of $R$. We denote
$\\{\mathfrak{p}\in{\rm Spec}\,R:\,\mathfrak{p}\supseteq\mathfrak{a}\\}$ by
$V(\mathfrak{a})$. For any unexplained notation and terminology we refer the
reader to [9] and [24].
## 2\. Preliminaries
Yoshizawa in [28, Definition 2.1] defined classes of extension modules of
Serre subcategory by another one as below.
###### Definition 2.1.
Let $\mathcal{S}_{1}$ and $\mathcal{S}_{2}$ be Serre subcategories of the
category of all R-modules. We denote by $(\mathcal{S}_{1},\mathcal{S}_{2})$
the class of all $R$-modules M with some $R$-modules $S_{1}\in\mathcal{S}_{1}$
and $S_{2}\in\mathcal{S}_{2}$ such that a sequence $0\longrightarrow
S_{1}\longrightarrow M\longrightarrow S_{2}\longrightarrow 0$ is exact.
We will denote the class of all modules $M$ with $\dim M\leq n$ by ${\rm
D_{\leq n}}$ and the class of extension modules of finitely generated modules
by the class of ${\rm D_{\leq n}}$ modules by ${\rm FD_{\leq n}}$ where
$n\geq-1$ is an integer. Note that the class of ${\rm FD_{\leq-1}}$ is the
same as finitely generated $R$-modules. Recall that a module $M$ is a
_minimax_ module if there is a finitely generated submodule $N$ of $M$ such
that the quotient module $M/N$ is artinian. Thus the class of minimax modules
is the class of extension modules of finitely generated modules by the class
of Artinian modules. Minimax modules have been studied by Zink in [29] and
Zöschinger in [30, 31]. See also [26]. Recall too that an $R$-module $M$ is
called _weakly Laskerian_ if ${\rm Ass}(M/N)$ is a finite set for each
submodule $N$ of $M$. The class of weakly Laskerian modules introduced in
[14], by Divaani-Aazar and Mafi. Recently, Hung Quy [18], introduced the class
of extension modules of finitely generated modules by the class of all modules
of finite support and named it ${\rm FSF}$ modules. By the following theorem
over a Noetherian ring $R$ an $R$-module $M$ is weakly Laskerian if and only
if is ${\rm FSF}$.
###### Theorem 2.2.
Let $R$ be a Noetherian ring and $M$ a nonzero $R$-module. The following
statements are equivalent:
1. (1)
$M$ is a weakly Laskerian module;
2. (2)
M is an ${\rm FSF}$ module.
###### Proof.
See [3, Theorem 3.3].∎
###### Lemma 2.3.
Let $R$ be a Noetherian ring. Then the following conditions hold:
(i) Any finitely generated $R$-module and any ${\rm D_{\leq n}}$ $R$-module
are ${\rm FD_{\leq n}}$.
(ii) The class of ${\rm FD_{\leq n-1}}$ modules is contained in the class of
${\rm FD_{\leq n}}$ modules for all $n\geq 0$.
(iii) The class of minimax modules is contained in the class of ${\rm FD_{\leq
0}}$ that is the class of extension modules of finitely generated modules by
semiartinian modules.
(iv) The class of weakly Laskerian modules is contained in the class of ${\rm
FD_{\leq 1}}$.
(v) The class of ${\rm FD_{\leq n}}$ $R$-modules forms a Serre subcategory of
the category of all $R$-modules.
###### Proof.
(i), (ii), (iii) are trivial.
(iv) Use Theorem 2.2.
(v) See [28, Corollary 4.3 or 4.5].∎
###### Example 2.4.
(i) Let $R$ be a Notherian ring with $\dim R\geq 2$ and let
$\mathfrak{p}\in\operatorname{Spec}(R)$ such that $\dim R/\mathfrak{p}=1$. Let
$M=R\oplus E(R/\mathfrak{p})$. It is easy to see that $M$ is an ${\rm FD_{\leq
1}}$ $R$-module that is neither finitely generated nor ${\rm D_{\leq 1}}$.
(ii) Suppose the set $\Omega$ of maximal ideals of $R$ is infinite. Then the
module $\oplus_{\mathfrak{m}\in\Omega}R/\mathfrak{m}$ is ${\rm FD_{\leq 0}}$
module and thus ${\rm FD_{\leq 1}}$ but it is not a weakly Laskerian module.
###### Proposition 2.5.
Let $I$ be an ideal of a Noetherian ring $R$ and $M$ be a ${\rm D_{\leq 1}}$
module such that $\operatorname{Supp}M\subseteq V(I)$. Then the following
statements are equivalent:
(i) $M$ is $I$-cofinite,
(ii) The $R$-modules $\operatorname{Hom}_{R}(R/I,M)$ and
$\operatorname{Ext}^{1}_{R}(R/I,M)$ are finitely generated.
###### Proof.
See [6, Proposition 2.6].∎
## 3\. Cofinitness of local cohomology
In what follows the next theorem plays an important role.
###### Theorem 3.1.
Let $I$ be an ideal of a Noetherian ring $R$ and $M$ be an ${\rm FD_{\leq 1}}$
$R$-module such that $\operatorname{Supp}M\subseteq V(I)$. Then the following
statements are equivalent:
(i) $M$ is $I$-cofinite,
(ii) The $R$-modules $\operatorname{Hom}_{R}(R/I,M)$ and
$\operatorname{Ext}^{1}_{R}(R/I,M)$ are finitely generated.
###### Proof.
$(i)\Rightarrow(ii)$ is clear. In order to prove $(ii)\Rightarrow(i)$, by
Definition there is a finitely generated submodule $N$ of $M$ such that the
$R$-module ${\rm dim}(M/N)\leq 1$ and $\operatorname{Supp}M/N\subseteq V(I)$.
Also, the exact sequence
$0\rightarrow N\rightarrow M\rightarrow M/N\rightarrow 0,\,\,\,\,\,\,(*)$
induces the following exact sequence
$0\longrightarrow{\rm Hom}_{R}(R/I,N)\longrightarrow{\rm
Hom}_{R}(R/I,M)\longrightarrow{\rm Hom}_{R}(R/I,M/N)$ $\longrightarrow{\rm
Ext}^{1}_{R}(R/I,N)\longrightarrow{\rm Ext}^{1}_{R}(R/I,M)\longrightarrow{\rm
Ext}^{1}_{R}(R/I,M/N)\longrightarrow{\rm Ext}^{2}_{R}(R/I,N).$
Whence, it follows that the $R$-modules $\operatorname{Hom}_{R}(R/I,M/N)$ and
$\operatorname{Ext}^{1}_{R}(R/I,M/N)$ are finitely generated. Therefore, in
view of Proposition 2.5, the $R$-module $M/N$ is $I$-cofinite. Now it follows
from the exact sequence $(*)$ that $M$ is $I$-cofinite.∎
###### Lemma 3.2.
Let $I$ be an ideal of Noetherian ring $R$, $M$ a non-zero $R$-module and
$t\in\mathbb{N}_{0}$. Suppose that the $R$-module $H^{i}_{I}(M)$ is
$I$-cofinite for all $i=0,...,t-1$, and the $R$-modules ${\rm
Ext}^{t}_{R}(R/I,M)$ and ${\rm Ext}^{t+1}_{R}(R/I,M)$ are finitely generated.
Then the $R$-modules ${\rm Hom}_{R}(R/I,H^{t}_{I}(M))$ and ${\rm
Ext}^{1}_{R}(R/I,H^{t}_{I}(M))$ are finitely generated.
###### Proof.
See [13, Theorem 2.1] and [12, Theorem A].∎
###### Lemma 3.3.
Let $I$ be an ideal of a Noetherian ring $R$ and $M$ be an ${\rm FD_{\leq 0}}$
$R$-module such that $\operatorname{Supp}M\subseteq V(I)$. Then the following
statements are equivalent:
(i) $M$ is $I$-cofinite,
(ii) The $R$-module $\operatorname{Hom}_{R}(R/I,M)$ is finitely generated.
###### Proof.
The proof is similar to the proof of [25, Proposition 4.3].∎
We are now ready to state and prove the following main results (Theorem 3.4
and the Corollaries 3.5 and 3.6) which are extension of Bahmanpour-Naghipour’s
results in [4] and [5], Brodmann-Lashgari’s result in [7], Khashyarmanesh-
Salarian’s result in [21], Hong Quy’s result in [18], and also the Delfno-
Marley’s result in [10] and Yoshida’s result in [27] for an arbitrary
Noetherian ring.
###### Theorem 3.4.
Let $R$ be a Noetherian ring and $I$ an ideal of $R$. Let $M$ be a finitely
generated $R$-module and $t\geq 1$ be a positive integer such that the
$R$-modules $H^{i}_{I}(M)$ are ${\rm FD_{\leq 1}}$ $R$-modules for all $i<t$.
Then, the following conditions hold:
(i) The $R$-modules $H^{i}_{I}(M)$ are $I$-cofinite for all $i<t$.
(ii) For all ${\rm FD_{\leq 0}}$ (or minimax) submodule $N$ of $H^{t}_{I}(M)$,
the $R$-modules
${\rm Hom}_{R}(R/I,H^{t}_{I}(M)/N)\,\,\,{\rm and}\,\,\,{\rm
Ext}^{1}_{R}(R/I,H^{t}_{I}(M)/N)$
are finitely generated. In particular the set
$\operatorname{Ass}_{R}(H^{t}_{I}(M)/N)$ is a finite set.
###### Proof.
(i) We proceed by induction on $t$. By Lemma 3.2 the case $t=1$ is obvious
since $H^{0}_{I}(M)$ is finitely generated. So, let $t>1$ and the result has
been proved for smaller values of $t$. By the inductive assumption,
$H^{i}_{I}(M)$ is I-cofinite for $i=0,1,...,t-2$. Hence by Lemma 3.2 and
assumption, ${\rm Hom}_{R}(R/I,H^{t-1}_{I}(M))\,\,\,{\rm and}\,\,\,{\rm
Ext}^{1}_{R}(R/I,H^{t-1}_{I}(M))$ are finitely generated. Therefore by
Corollary 3.1 , $H^{i}_{I}(M)$ is I-cofinite for all $i<t$. This completes the
inductive step.
(ii) In view of (i) and lemma 3.2, ${\rm Hom}_{R}(R/I,H^{t}_{I}(M))\,\,\,{\rm
and}\,\,\,{\rm Ext}^{1}_{R}(R/I,H^{t}_{I}(M))$ are finitely generated. On the
other hand, according to Lemma 3.3 or Melkersson’s result [25, Proposition
4.3], N is $I$-cofinite. Now, the exact sequence
$0\longrightarrow N\longrightarrow H^{t}_{I}(M)\longrightarrow
H^{t}_{I}(M)/N\longrightarrow 0$
induces the following exact sequence,
${\rm Hom}_{R}(R/I,H^{t}_{I}(M))\longrightarrow{\rm
Hom}_{R}(R/I,H^{t}_{I}(M)/N)\longrightarrow{\rm
Ext}^{1}_{R}(R/I,N)\longrightarrow$ ${\rm
Ext}^{1}_{R}(R/I,H^{t}_{I}(M))\longrightarrow{\rm
Ext}^{1}_{R}(R/I,H^{t}_{I}(M)/N)\longrightarrow{\rm Ext}^{2}_{R}(R/I,N).$
Consequently
${\rm Hom}_{R}(R/I,H^{t}_{I}(M)/N)\,\,\,{\rm and}\,\,\,{\rm
Ext}^{1}_{R}(R/I,H^{t}_{I}(M)/N)$
are finitely generated, as required.∎
###### Corollary 3.5.
Let $R$ be a Noetherian ring and $I$ an ideal of $R$. Let $M$ be a finitely
generated $R$-module such that the $R$-modules $H^{i}_{I}(M)$ are ${\rm
FD_{\leq 1}}$ (or weakly Laskerian) $R$-modules for all $i$. Then, the
(i) The $R$-modules $H^{i}_{I}(M)$ are $I$-cofinite for all $i$.
(ii) For any $i\geq 0$ and for any ${\rm FD_{\leq 0}}$ (or minimax) submodule
$N$ of $H^{i}_{I}(M)$, the $R$-module $H^{i}_{I}(M)/N$ is $I$-cofinite.
###### Proof.
(i) Clear.
(ii) In view of (i) the R-module $H^{i}_{I}(M)$ is $I$-cofinite for all $i$.
Hence the R-module ${\rm Hom}_{R}(R/I,N)$ is finitely generated, and so it
follows from Lemma 3.3 or [25, Proposition 4.3] that N is $I$-cofinite . Now,
the exact sequence
$0\longrightarrow N\longrightarrow H^{t}_{I}(M)\longrightarrow
H^{t}_{I}(M)/N\longrightarrow 0$
,
implies that the R-module $H^{i}_{I}(M)/N$ is $I$-cofinite.∎
###### Corollary 3.6.
Let $R$ be a Noetherian ring and $I$ an ideal of $R$. Let $M$ be a finitely
generated $R$-module and $t\geq 1$ be a positive integer such that the
$R$-modules $H^{i}_{I}(M)$ are weakly Laskerian for all $i<t$. Then, the
following conditions hold:
(i) The $R$-modules $H^{i}_{I}(M)$ are $I$-cofinite for all $i<t$.
(ii) For all ${\rm FD_{\leq 0}}$ (or minimax) submodule $N$ of $H^{t}_{I}(M)$,
the $R$-modules
${\rm Hom}_{R}(R/I,H^{t}_{I}(M)/N)\,\,\,{\rm and}\,\,\,{\rm
Ext}^{1}_{R}(R/I,H^{t}_{I}(M)/N)$
are finitely generated. In particular the set
$\operatorname{Ass}_{R}(H^{t}_{I}(M)/N)$ is a finite set.
###### Proof.
Use Theorem 2.2 and note that the category of weakly Laskerian modules is
contained in the category of ${\rm FD_{\leq 1}}$ modules.∎
One of the main result of this section is to prove that for an arbitrary ideal
$I$ of a Noetherian ring $R$, the Category of $I$-cofinite ${\rm FD_{\leq 1}}$
modules compose an Abelian category.
###### Theorem 3.7.
Let $I$ be an ideal of a Noetherian ring $R$. Let
$\mathscr{FD}^{1}(R,I)_{cof}$ denote the category of $I$-cofinite ${\rm
FD_{\leq 1}}$ $R$-modules. Then $\mathscr{FD}^{1}(R,I)_{cof}$ is an Abelian
category.
###### Proof.
Let $M,N\in\mathscr{FD}^{1}(R,I)_{cof}$ and let $f:M\longrightarrow N$ be an
$R$-homomorphism. It is enough that to show that the $R$-modules $\ker f$ and
${\rm coker}f$ are $I$-cofinite.
To this end, the exact sequence
$0\longrightarrow\ker f\longrightarrow M\longrightarrow{\rm
im}f\longrightarrow 0,$
induces an exact sequence
$0\longrightarrow{\rm Hom}_{R}(R/I,\ker f)\longrightarrow{\rm
Hom}_{R}(R/I,M)\longrightarrow{\rm Hom}_{R}(R/I,{\rm im}f)$
$\longrightarrow{\rm Ext}^{1}_{R}(R/I,\ker
f)\longrightarrow\operatorname{Ext}^{1}_{R}(R/I,M),$
that implies the $R$-modules $\operatorname{Hom}_{R}(R/I,\ker f)$ and
$\operatorname{Ext}^{1}_{R}(R/I,\ker f)$ are finitely generated. Therefore it
follows from Theorem 3.1 that $\ker f$ is $I$-cofinite. Now, the assertion
follows from the following exact sequences
$0\longrightarrow\ker f\longrightarrow M\longrightarrow{\rm
im}f\longrightarrow 0,$
and
$0\longrightarrow{\rm im}f\longrightarrow N\longrightarrow{\rm
coker}f\longrightarrow 0.$
∎
The following corrollary is a generalization of [1, Theorem 2.7].
###### Corollary 3.8.
Let $I$ be an ideal of a Noetherian ring $R$. Let $M$ be an ${\rm FD_{\leq
1}}$ $I$-cofinite $R$-module. Then, the R-modules ${\rm Ext}^{i}_{R}(N,M)$ and
${\rm Tor}^{R}_{i}(N,M)$ are $I$-cofinite and ${\rm FD_{\leq 1}}$ modules, for
all finitely generated R-modules $N$ and all integers $i\geq 0$.
###### Proof.
Since $N$ is finitely generated it follows that $N$ has a free resolution of
finitely generated free modules. Now the assertion follows using Theorem 3.7
and computing the modules ${\rm Tor}_{i}^{R}(N,M)$ and ${\rm
Ext}^{i}_{R}(N,M)$, by this free resolution. ∎
###### Corollary 3.9.
Let $I$ be an ideal of a Noetherian ring R, M a non-zero finite $R$-module
such that $\dim M/IM\leq 1$(e.g., $\dim R/I\leq 1)$. Then for each finite
$R$-module $N$, the $R$-modules ${\rm Ext}^{j}_{R}(N,H^{i}_{I}(M))$ and ${\rm
Tor}^{R}_{j}(N,H^{i}_{I}(M))$ are $I$-cofinite for all $i\geq 0$ and $j\geq
0$.
###### Proof.
Note that $\dim\operatorname{Supp}H^{i}_{I}(M)\leq\dim M/IM\leq 1$ thus it is
an ${\rm FD_{\leq 1}}$ module for all $i\geq 0$, now use Corollary 3.8. ∎
###### Lemma 3.10.
Let $R$ be a Noetherian ring, $I$ a proper ideal of $R$ and $M$ be a non-zero
${\rm D_{\leq 1}}$ and $I$-cofinite $R$-module. Then for each non-zero
finitely generated $R$-module $N$ with support in $V(I)$, the $R$-modules
$\operatorname{Ext}^{i}_{R}(M,N)$ are finitely generated, for all integers
$i\geq 0$.
###### Proof.
See [19, Theorem 2.8].∎
###### Corollary 3.11.
Let $R$ be a Noetherian ring and $I$ be an ideal of $R$. Let $M$ be an ${\rm
FD_{\leq 1}}$ and $I$-cofinite $R$-module. Then, the R-modules ${\rm
Ext}^{i}_{R}(M,N)$ and ${\rm Tor}^{R}_{i}(M,N)$ are finitely generated, for
all finitely generated R-modules $N$ with ${\rm Supp}(N)\subseteq V(I)$ and
all integers $i\geq 0$.
###### Proof.
The assertion follows from the definition using Lemma 3.9 and [25, Theorem
2.1].∎
## References
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|
arxiv-papers
| 2012-11-25T09:37:09 |
2024-09-04T02:49:38.382461
|
{
"license": "Public Domain",
"authors": "Moharram Aghapournahr and Kamal Bahmanpour",
"submitter": "Moharram Aghapournahr",
"url": "https://arxiv.org/abs/1211.5748"
}
|
1211.5765
|
Holography principle for twistor spaces
Misha Verbitsky111Partially supported by RFBR grants 12-01-00944- ,
10-01-93113-NCNIL-a, and AG Laboratory NRI-HSE, RF government grant, ag.
11.G34.31.0023.
[email protected]
Abstract
Let $S$ be a smooth rational curve on a complex manifold $M$. It is called
ample if its normal bundle is positive: $NS=\bigoplus{\cal O}(i_{k})$,
$i_{k}>0$. We assume that $M$ is covered by smooth holomorphic deformations of
$S$. The basic example of such a manifold is a twistor space of a hyperkähler
or a 4-dimensional anti-selfdual Riemannian manifold $X$ (not necessarily
compact). We prove “a holography principle” for such a manifold: any
meromorphic function defined in a neighbourhood $U$ of $S$ can be extended to
$M$, and any section of a holomorphic line bundle can be extended from $U$ to
$M$. This is used to define the notion of a Moishezon twistor space: this is a
twistor space admitting a holomorphic embedding to a Moishezon variety
$M^{\prime}$. We show that this property is local on $X$, and the variety
$M^{\prime}$ is unique up to birational transform. We prove that the twistor
spaces of hyperkähler manifolds obtained by hyperkähler reduction of flat
quaternionic-Hermitian spaces by the action of reductive Lie groups (such as
Nakajima’s quiver varieties) are always Moishezon.
###### Contents
1. 1 Introduction
1. 1.1 Quasilines on complex manifolds
2. 1.2 Holography principle
3. 1.3 Moishezon twistor spaces
2. 2 Twistor spaces and ample rational curves
1. 2.1 Hyperkähler manifolds
2. 2.2 Twistor spaces
3. 2.3 Geometry of twistor spaces
4. 2.4 Quasilines and ample curves in the twistor spaces
3. 3 Holography principle for manifolds with ample rational curves
1. 3.1 Holography principle for line bundles
2. 3.2 The local holography principle
3. 3.3 Holography principle for meromorphic functions
4. 4 Moishezon twistor spaces
1. 4.1 Vector bundles in a neighbourhood of an ample curve
2. 4.2 Algebraic dimension of the field of meromorphic functions
3. 4.3 Moishezon manifolds and ample rational curves
4. 4.4 Moishezon twistor spaces
5. 4.5 Twistor spaces and hyperkähler reduction
5. 5 Appendix: Formal geometry and holography principle (by Dmitry Kaledin)
## 1 Introduction
### 1.1 Quasilines on complex manifolds
The present paper was written as an attempt to answer the following question.
Let $S\subset M$ be a smooth rational curve in a complex manifold, with normal
bundle $NS$ isomorphic to ${\cal O}(1)^{n}$.111Such rational curves are called
quasilines, see e.g. [BBI]. Is there a notion of a normal form for a tubular
neighbourhood of such a curve?
When the normal bundle is ${\cal O}(-1)^{n}$ instead of ${\cal O}(1)^{n}$, a
tubular neighbourhood of the curve has a normal form, obtained by blowing down
this curve to a point, and taking a sufficiently small Stein neighbourhood of
this point in the corresponding singular variety. When $NS={\cal O}(1)^{n}$,
no such normal form can be obtained. In fact, the birational type of the
manifold can be reconstructed from a complex analytic (and even formal)
neighbourhood of $S$. This was known already to Hartshorne ([Har, Theorem
6.7]).
However, one can associate with a quasiline an infinite-dimensional bundle
$\hat{\cal O}_{S}(M)$ over $S\cong{\mathbb{C}}P^{1}$, with
$\hat{\cal O}_{S}(M)=\lim_{\leftarrow}{\cal O}_{M}/I^{n}_{S}{\cal O}_{M},$
where $I_{S}$ is an ideal sheaf of $S$. This bundle is called a formal
completion of $M$ at $S$.
It is not hard to observe that the space of sections $H^{0}(S,\hat{\cal
O}_{S}(M))$ is finite-dimensional, and, moreover, $H^{0}(S,\hat{\cal
O}_{S}(M)\otimes_{{\cal O}_{S}}{\cal O}(i))$ is finite-dimensional for each
$i$ (4.1). For a long time, we expected that the algebra
$A_{S}:=\bigoplus_{i}H^{0}(S,\hat{\cal O}_{S}(M)\otimes_{{\cal O}_{S}}{\cal
O}(i))$ would provide a sort of an algebraic “normal form” of $S$ in $M$, in
such a way that the algebraic structure on $\hat{\cal O}_{S}(M)$ can be
reconstructed from this ring.
In this paper, we show that this approach works when $M$ is a Moishezon
manifold, and $M$ can be reconstructed from the ring $A_{S}$, up to birational
isomorphism (Subsection 4.3; this is not very surprising due to the above-
mentioned theorem of Hartshorne, [Har, Theorem 6.7]).
For non-Moishezon $M$, such as a twistor space of a simply connected, compact
hyperkähler manifold, this conjecture is spectacularly wrong. In this case,
$A_{S}=\bigoplus_{i}H^{0}({\mathbb{C}}P^{1},{\cal O}(i))$, and this ring has
no information about $M$ whatsoever (2.3).
### 1.2 Holography principle
The main technical tool of the present paper is the following theorem, called
the holography principle. Recall that an ample rational curve on a complex
variety $M$ is a smooth curve $S\cong{\mathbb{C}}P^{1}\subset M$ such that the
normal bundle $NS$ is decomposed as $NS=\bigoplus_{k}{\cal O}(i_{k})$, with
all $i_{k}>0$.
Theorem 1.1: Let $S\subset M$ be an ample curve in a simply connected complex
manifold, and $U$ its connected neighbourhood. Suppose that $M$ is covered by
smooth complex-analytic deformations of $S$. Then
(i)
For any holomorphic vector bundle $B$ on $M$, the restriction map
$H^{0}(M,B){\>\longrightarrow\>}H^{0}(U,B)$
is an isomorphism.
(ii)
Let $\operatorname{Mer}(M)$, $\operatorname{Mer}(U)$ be the fields of
meromorphic functions on $M$ and $U$. Then the restriction map
$\operatorname{Mer}(M){\>\longrightarrow\>}\operatorname{Mer}(U)$ is an
isomorphism.
Proof: For line bundles, 1.2 (i) is implied by 3.1. For general vector
bundles, an elegant argument is given by D. Kaledin in the appendix to this
paper (Section 5). Holography principle for meromorphic functions is proven in
3.3.
The holography principle is not very surprising of one looks at the
neighbourhood of $S$ from the point of view of complex analysis. The normal
bundle to $S$ is obviously positive. Choose a Hermitian metric $h$ on a
neighbourhood of $S$ such that the Chern connection on $NS$ induced by $h$ has
positive curvature. Let $d_{S}:\;M{\>\longrightarrow\>}{\mathbb{R}}$ be the
Riemannian distance to $S$ in this Hermitian metric. Since $d_{S}$ around $S$
is close to the distance in $NS$, the form $dd^{c}d_{S}$ has $n-1$ positive
and 1 negative eigenvalue in a sufficiently small neighbourhood of $S$ ([D]).
This means that $S$ has a neighbourhood $U$ with a smooth boundary $\partial
U$ such that the Levi form on $\partial U$ has one negative and $\dim M-2$
positive eigenvalues. Then, a holomorphic function defined on an open subset
of $U$ and continuous on $\partial U$ can be extended outside of a boundary;
at least, this is the expectation one has from the solution of the Levi
problem.
### 1.3 Moishezon twistor spaces
Definition 1.2: Let $M$ be a compact complex manifold. Define the algebraic
dimension as $a(M):=\deg_{tr}\operatorname{Mer}(M)$, where
$\deg_{tr}\operatorname{Mer}(M)$ denotes the transcendence degree of the field
of global meromorphic functions on $M$.
Definition 1.3: A Moishezon variety is a compact complex variety satisfying
$a(M)=\dim M$.
The notion of a Moishezon manifold, as it is usually stated, makes no sense
for non-compact varieties. Indeed, $\deg_{tr}\operatorname{Mer}(M)=\infty$
even if $M$ is an open disk.
However, when $M$ contains an ample curve, the situation changes drastically.
Theorem 1.4: Let $M$ be a complex manifold containing an ample rational
curve. Then $a(M)\leqslant\dim M$. Moreover, if $a(M)=\dim M$, there exists an
open embedding of $M$ to a Moishezon variety $M^{\prime}$ which satisfies
$\operatorname{Mer}(M^{\prime})=\operatorname{Mer}(M)$.
Proof: See 4.3.
Let now $M$ be a twistor space of $X$. Here, $X$ can be either a hypercomplex
(hyperkähler) manifold, a quaternionic, or quaternionic-Kähler manifold, or
Riemannian anti-selfdual manifold. We are not very specific, because the only
thing about $M$ which is used is existence of a large number of quasilines.
The twistor spaces are complex manifold covered by quasilines, usually non-
Kähler and non-quasiprojective (see 2.3 and 4.5).
Since $M=\operatorname{Tw}(X)$ is covered by quasilines which are by
definition ample, we can apply 4.2, and obtain that
$\deg_{tr}\operatorname{Mer}(M)\leqslant\dim M$. We call $M$ a Moishezon
twistor space if $\deg_{tr}\operatorname{Mer}(M)=\dim M$. This is equivalent
to an existence of an open embedding $M{\>\longrightarrow\>}M^{\prime}$ of $M$
to a Moishezon manifold (1.3).
Moishezon twistor spaces for compact 4-dimensional anti-selfdual manifolds
were discovered by Y.-S. Poon in [P], and much studied since then. Structure
theorems about such manifolds were obtained by F. Campana, [C], and partial
classification results by N. Honda (see e.g. [Ho]).
The definition given above extends the class of “Moishezon twistor manifolds”
significantly. Let, for instance, $M:=V{/\\!\\!/\\!\\!/}G$ be a hyperkähler
manifold which can be obtained using the hyperkähler reduction, where $V$ is a
flat hyperkähler manifold, and $G$ a compact Lie group (such as the Nakajima
quiver variety). Then the twistor space $\operatorname{Tw}(M)$ is always
Moishezon (4.5).
Acknowledgements: Many thanks to Hans-Joachim Hein, Claude LeBrun, Nobuhiro
Honda and Dima Kaledin for interesting discussions on the subject of this
article.
## 2 Twistor spaces and ample rational curves
### 2.1 Hyperkähler manifolds
Definition 2.1: Let $M$ be a manifold, and $I,J,K\in\operatorname{End}(TM)$
endomorphisms of the tangent bundle satisfying the quaternionic relation
$I^{2}=J^{2}=K^{2}=IJK=-\operatorname{Id}_{TM}.$
The manifold $(M,I,J,K)$ is called hypercomplex if the almost complex
structures $I$, $J$, $K$ are integrable. If, in addition, $M$ is equipped with
a Riemannian metric $g$ which is Kähler with respect to $I,J,K$, the manifold
$(M,I,J,K,g)$ is called hyperkähler.
Consider the Kähler forms $\omega_{I},\omega_{J},\omega_{K}$ on $M$:
$\omega_{I}(\cdot,\cdot):=g(\cdot,I\cdot),\ \
\omega_{J}(\cdot,\cdot):=g(\cdot,J\cdot),\ \
\omega_{K}(\cdot,\cdot):=g(\cdot,K\cdot).$
An elementary linear-algebraic calculation implies that the 2-form
$\Omega:=\omega_{J}+\sqrt{-1}\>\omega_{K}$ is of Hodge type $(2,0)$ on
$(M,I)$. This form is clearly closed and non-degenerate, hence it is a
holomorphic symplectic form.
In algebraic geometry, the word “hyperkähler” is essentially synonymous with
“holomorphically symplectic”, due to the following theorem, which is implied
by Yau’s solution of Calabi conjecture.
Theorem 2.2: Let $(M,I)$ be a compact, Kähler, holomorphically symplectic
manifold. Then there exists a unique hyperkähler metric on $(M,I)$ with the
same Kähler class.
Proof: See [Y], [Bes].
Remark 2.3: The hyperkähler metric is unique, but there could be several
hyperkähler structures compatible with a given hyperkähler metric on $(M,I)$,
if the holonomy of its Levi-Civita connection is strictly less than
$\operatorname{Sp}(n)$.
Definition 2.4: Let $M$ be a hypercomplex manifold, and $L$ a quaternion
satisfying $L^{2}=-1$. Then $L=aI+bJ+cK$, $a^{2}+b^{2}+c^{2}=1$. The
corresponding complex structure on $M$ is called an induced complex structure.
The space $M$, considered as a complex manifold, is denoted by $(M,L)$. The
set of induced complex structures is naturally identified with $S^{2}$, which
we often consider as ${\mathbb{C}}P^{1}$ with the standard complex structure.
Definition 2.5: ([V1]) Let $X\subset M$ be a closed subset of a hyperkähler
manifold $M$. Then $X$ is called trianalytic if $X$ is a complex analytic
subset of $(M,L)$ for every induced complex structure $L$.
Trianalytic subvarieties were a subject of a long study. Most importantly,
consider a generic induced complex structure $L$ on $M$. Then all closed
complex subvarieties of $(M,L)$ are trianalytic. Moreover, a trianalytic
subvariety can be canonically desingularized ([V2]), and this
desingularization is hyperkähler.
Theorem 2.6: ([V1], [V3]) Let $(M,I,J,K)$ be a hyperkähler manifold (not
necessarily compact). Then there exists a countable subset
$R\subset{\mathbb{C}}P^{1}$, such that for any induced complex structure
$L\notin R$, all compact complex subvarieties of $(M,L)$ are trianalytic.
Remark 2.7: For hypercomplex manifolds, 2.1 is (generally speaking) false,
though for manifolds with trivial canonical bundle a weaker form of this
result was obtained ([SV]).
### 2.2 Twistor spaces
Definition 2.8: Let $M$ be a Riemannian 4-manifold. Consider the action of the
Hodge $*$-operator: $*:\;\Lambda^{2}M{\>\longrightarrow\>}\Lambda^{2}M$. Since
$*^{2}=1$, the eigenvalues are $\pm 1$, and one has a decomposition
$\Lambda^{2}M=\Lambda^{+}M\oplus\Lambda^{-}M$ onto selfdual ($*\eta=\eta$) and
anti-selfdual ($*\eta=-\eta$) forms.
Remark 2.9: If one changes the orientation of $M$, leaving metric the same,
$\Lambda^{+}M$ and $\Lambda^{-}M$ are exchanged. Therefore, their dimensions
are equal, and $\dim\Lambda^{2}M=6$ implies $\dim\Lambda^{\pm}(M)=3$.
Remark 2.10: Using the isomorphism $\Lambda^{2}M=\mathfrak{so}(TM)$, we
interpret $\eta\in\Lambda^{2}_{m}M$ as an endomorphisms of $T_{m}M$. Then the
unit vectors $\eta\in\Lambda^{+}_{m}M$ correspond to oriented, orthogonal
complex structures on $T_{m}M$.
Definition 2.11: Let $\operatorname{Tw}(M):=S\Lambda^{+}M$ be the set of unit
vectors in $\Lambda^{+}M$. At each point $(m,s)\in\operatorname{Tw}(M)$,
consider the decomposition $T_{m,s}\operatorname{Tw}(M)=T_{m}M\oplus
T_{s}S\Lambda^{+}_{m}M$, induced by the Levi-Civita connection. Let $I_{s}$ be
the complex structure on $T_{m}M$ induced by $s$, $I_{S\Lambda^{+}_{m}M}$ the
complex structure on $S\Lambda^{+}_{m}M=S^{2}$ induced by the metric and
orientation, and
${\cal
I}:\;T_{m,s}\operatorname{Tw}(M){\>\longrightarrow\>}T_{m,s}\operatorname{Tw}(M)$
be equal to ${\cal I}_{s}\oplus I_{S\Lambda^{+}_{m}M}$. An almost complex
manifold $(\operatorname{Tw}(M),{\cal I})$ is called the twistor space of $M$.
Given a hyperkähler or hypercomplex manifold (2.1), one defines its twistor
space in a similar manner.
Definition 2.12: A twistor space $\operatorname{Tw}(M)$ of a hypercomplex
manifold $M$ is $S^{2}\times M$ equipped with a complex structure which is
defined as follows. Consider the complex structure $I_{m}:T_{m}M\to T_{m}M$ on
$M$ induced by $J\in S^{2}\subset{\mathbb{H}}$. Let $I_{J}$ denote the complex
structure on $S^{2}={\mathbb{C}}P^{1}$. The operator
$I_{\operatorname{Tw}}=I_{m}\oplus I_{J}:T_{x}\operatorname{Tw}(M)\to
T_{x}\operatorname{Tw}(M)$ satisfies
$I_{\operatorname{Tw}}^{2}=-\operatorname{Id}$. It defines an almost complex
structure on $\operatorname{Tw}(M)$.
The almost complex structure on the twistor space of a Riemannian 4-manifold
$X$ is integrable whenever $X$ is anti-selfdual ([AHS]) For a hypercomplex
manifold it is integrable as well ([K]). Twistor spaces are the main example
of the geometries we are working with.
### 2.3 Geometry of twistor spaces
Proposition 2.13: Let $\operatorname{Tw}(M)$ be a twistor space of a compact
hyperkähler manifold. Then
(i)
$\operatorname{Tw}(M)$ is non-Kähler.
(ii)
The algebraic dimension of $\operatorname{Tw}(M)$ is 1.
Proof of (i): Let $\omega$ be the standard Hermitian form of
$\operatorname{Tw}(M)$. Then $dd^{c}\omega$ is a positive (2,2)-form ([KV,
(8.2)]). For any Kähler form $\omega_{0}$ on $\operatorname{Tw}(M)$, this
would imply
$\int_{\operatorname{Tw}(M)}d\left(\omega_{0}^{\dim_{\mathbb{C}}M-1}\wedge
d^{c}\omega\right)=\int_{\operatorname{Tw}(M)}\omega_{0}^{\dim_{\mathbb{C}}M-1}\wedge
dd^{c}\omega>0,$
which is impossible by Stokes’ theorem.
Proof of (ii): See 4.4.
### 2.4 Quasilines and ample curves in the twistor spaces
Definition 2.14: An ample rational curve on a complex manifold $M$ is a smooth
curve $S\cong{\mathbb{C}}P^{1}\subset M$ such that its normal bundle $N\\!S$
satisfies $N\\!S=\bigoplus_{k=1}^{n-1}{\cal O}(i_{k})$, with all $i_{k}>0$
(see [Ko]). It is called a quasiline if all $i_{k}=1$.
Claim 2.15: Let $M$ be a twistor space of a hyperkähler or 4-dimensional ASD
manifold, $m\in M$ a point, and $S_{m}$ the corresponding $S^{2}$ in
$\operatorname{Tw}(M)={\mathbb{C}}P^{1}\times M$. Then $S_{m}$ is a quasiline.
Proof: Since the claim is essentially infinitesimal, it suffices to check it
when $M$ is flat. Then $\operatorname{Tw}(M)=\operatorname{Tot}({\cal
O}(1)^{\oplus 2n})\cong{\mathbb{C}}P^{2n+1}\backslash{\mathbb{C}}P^{2n-1}$,
and $S_{m}$ is a section of ${\cal O}(1)^{\oplus 2n}$.
Existence of quasilines in twistor spaces is a very strong condition, and can
be used to obtain all kinds of geometric information; for example, see [C] and
[V4].
## 3 Holography principle for manifolds
with ample rational curves
### 3.1 Holography principle for line bundles
Throughout this paper, all neighbourhoods and manifolds are silently assumed
to be connected. One of the main results of the present paper is the following
theorem.
Theorem 3.1: (holography principle for line bundles) Let $S\subset M$ be an
ample rational curve in a simply connected complex manifold, which is covered
by smooth, ample deformations of $S$, and $L$ a holomorphic line bundle on
$M$. Consider an open neighbourhood $U\supset S$. Then the restriction map
$H^{0}(M,L){\>\longrightarrow\>}H^{0}(U,L)$ is an isomorphism.
We deduce 3.1 from the following local result (3.1).
Remark 3.2: Since $S$ is an ample curve, $S$ can be deformed in any normal
direction. Therefore, there exists an open neighbourhood $U\supset S$ which is
contained in a union of the set ${\mathfrak{S}}$ of all smooth, ample
deformations of $S$ intersecting $S$. Further on, we choose this neighbourhood
in such a manner that any $S_{1}\in{\mathfrak{S}}$ can be connected to $S$ by
a continuous family of deformations intersecting $S$.
Proposition 3.3: Let $S\subset M$ be an ample rational curve in a connected
complex manifold, which is covered by deformations of $S$, and $L$ a
holomorphic line bundle on $M$. Consider a neighbourhood $U\supset S$ which is
is contained in a union of the set of all deformations of $S$ intersecting $S$
(3.1). Then for any smaller open neighbourhood $V\subset U$ of $S$, the
restriction map $H^{0}(U,L){\>\longrightarrow\>}H^{0}(V,L)$ is an isomorphism.
Proof of an implication “3.1 $\Rightarrow$ 3.1”.
Step 1: Choose a continuous, connected family $S_{b}$ of ample curves
parametrized by $B$ such that $\bigcup_{b\in B}S_{b}=M$, and choose a tubular
neighbourhood $U_{b}$ for each $S_{b}$, continuously depending on $b$. Then
the intersection $U_{b}\cap U_{b^{\prime}}$ for sufficiently close
$b,b^{\prime}$ always contains $S_{b}$ and $S_{b^{\prime}}$. By 3.1, $U_{b}$
can be chosen in such a way that $H^{0}(U_{b}\cap
U_{b^{\prime}},L)=H^{0}(U_{b},L)=H^{0}(U_{b^{\prime}},L)$.
Step 2: Since $B$ is connected, all the spaces $H^{0}(U_{b},L)$ are
isomorphic, and these isomorphisms are compatible with the restrictions to the
intersections $U_{b}\cap U_{b^{\prime}}$. Let now $f\in H^{0}(U_{b},L)$, and
let $\tilde{M}_{f}$ be the domain of holomorphy for $f$, that is, a maximal
domain (non-ramified over $M$) such that $f$ admits a holomorphic extension to
$\tilde{M}_{f}$. Since $\cup U_{b}=M$, and $f$ can be holomorphically extended
to any $U_{b}$, the domain $\tilde{M}_{f}$ is a covering of $M$. Now, 3.1
follows, because $M$ is simply connected.
### 3.2 The local holography principle
To prove 3.1 it remains to prove 3.1. If $\deg
L{\left|{}_{{\phantom{|}\\!\\!}_{S}}\right.}<0$, the statement of 3.1 is
vacuous. Therefore, we may always assume that $\deg
L{\left|{}_{{\phantom{|}\\!\\!}_{S}}\right.}\geqslant 0$, hence
$l{\left|{}_{{\phantom{|}\\!\\!}_{S}}\right.}$ is generated by global
sections. Let $S(M)$ be the space of deformations of the ample curve $S$ which
remain smooth, and $SS(M)$ the space of pairs $\\{(x,S_{1}):\ \ S_{1}\in
S(M),x\in S_{1}\\}$. Consider the natural forgetful maps
$\tau_{1}:\;SS(M){\>\longrightarrow\>}M$,
$\tau_{2}:\;SS(M){\>\longrightarrow\>}S(M)$, and let $E$ be the bundle
${\tau_{2}}_{*}\tau_{1}^{*}L$ on $S(M)$. Denote by $\deg_{S}L$ the degree of
the restriction of $L$ to $S$. Since $\dim H^{0}(S_{1},L)=\deg L+1$, $E$ is a
$(\deg L+1)$-dimensional vector bundle. Given a section $f$ of $L$ on $M$,
denote the corresponding section of $E$ by
$\tilde{f}:={\tau_{2}}_{*}\tau_{1}^{*}f$. When $\deg_{S}L=d$, the value of
$\tilde{f}$ at $S_{1}$ is uniquely determined by the restriction of $f$ to any
$d+1$ distinct points of $S_{1}$. Indeed,
$E{\left|{}_{{\phantom{|}\\!\\!}_{S_{1}}}\right.}$ is $d+1$-dimensional, and
any section $h\in H^{0}(S_{1},L)$ is uniquely determined by its values in
$d+1$ points. This gives a map
$L{\left|{}_{{\phantom{|}\\!\\!}_{z_{1}}}\right.}\times
L{\left|{}_{{\phantom{|}\\!\\!}_{z_{2}}}\right.}\times...\times
L{\left|{}_{{\phantom{|}\\!\\!}_{z_{d+1}}}\right.}\tilde{\>\longrightarrow\>}H^{0}(S_{1},L).$
(3.1)
Now, let $f$ be a section of $L$ on $V$. For any $S_{1}\in S(M)$ intersecting
$V$, we choose $d_{1}$ distinct points $z_{1},...,z_{d+1}\in S_{1}\cap V$, and
consider the section
$\varphi\left(f{\left|{}_{{\phantom{|}\\!\\!}_{z_{1}}}\right.},f{\left|{}_{{\phantom{|}\\!\\!}_{z_{2}}}\right.},...,f{\left|{}_{{\phantom{|}\\!\\!}_{z_{d+1}}}\right.}\right)\in
H^{0}(S_{1},L)$ defined using (3.1). When $S_{1}\subset V$, this section is
independent from the choice of $z_{1},...,z_{d+1}\in S_{1}$. Let $R_{V}$ be a
connected component of the set of all $S_{1}\in S(M)$ intersecting $V$ and
containing $S$. Since the map
$(z_{1},...,z_{d+1}){\>\longrightarrow\>}\varphi\left(f{\left|{}_{{\phantom{|}\\!\\!}_{z_{1}}}\right.},f{\left|{}_{{\phantom{|}\\!\\!}_{z_{2}}}\right.},...,f{\left|{}_{{\phantom{|}\\!\\!}_{z_{d+1}}}\right.}\right)$
is holomorphic and independent from the choice of $z_{1},...,z_{d+1}$ on an
open subset of $R_{V}$, it is independent of $z_{1},...,z_{d+1}$ everywhere on
$R_{V}$. This gives a section $\tilde{f}\in H^{0}(R_{V},E)$ extending the
section $\tilde{f}:={\tau_{2}}_{*}\tau_{1}^{*}f\in H^{0}(S(V),E)$.
By construction, $U$ is contained in a connected part $U_{1}$ of the union of
all deformations of $S$ intersecting $V$. To extend $f$ from $V$ to $U$, we
use $\tilde{f}$ to obtain an extension of $f$ to $U_{1}$, as follows.
Any section $g\in H^{0}(R_{V},E)$ gives a function $\psi_{g}$ mapping a pair
$(x,S_{1})$, $x\in S_{1}\in R_{V}$ to
$g(S_{1}){\left|{}_{{\phantom{|}\\!\\!}_{x}}\right.}\in
L{\left|{}_{{\phantom{|}\\!\\!}_{x}}\right.}$. For the section $\tilde{f}$
constructed above, $\psi_{\tilde{f}}(x,S_{1})$ is independent from the choice
of $S_{1}$ whenever $S_{1}$ lies in $V$. The same analytic continuation
argument as above implies that $\psi_{\tilde{f}}(x,S_{1})$ is independent of
$S_{1}$ everywhere. For any $x\in U_{1}$, the set $f\in
L{\left|{}_{{\phantom{|}\\!\\!}_{x}}\right.}$ equal to
$\psi_{\tilde{f}}(x,S_{1})$, where $S_{1}\in R_{V}$ is an arbitrary curve
passing through $x$. This gives an extension of $f$ to $U_{1}$. 3.1 is proven.
We finished the proof of 3.1.
### 3.3 Holography principle for meromorphic functions
The following theorem is proven in the same way as 3.1. Given a complex
variety $M$, we denote the field of meromorphic functions on $M$ by
$\operatorname{Mer}(M)$.
Theorem 3.4: Let $S\subset M$ be an ample curve in a simply connected complex
manifold, and $U\supset S$ a connected neighbourhood of $S$. Suppose that $M$
is covered by the union of all smooth, ample deformations of $S$. Then the
restriction map
$\operatorname{Mer}(M){\>\longrightarrow\>}\operatorname{Mer}(U)$ is an
isomorphism.
Proof: The same argument as used to deduce 3.1 from 3.1 can be used to reduce
3.3 to the following statement.
Proposition 3.5: Let $S\subset M$ be an ample rational curve in a connected
complex manifold, which is covered by smooth, ample deformations of $S$.
Consider a neighbourhood $U\supset S$ which is contained in a union
${\mathfrak{S}}$ of all deformations of $S$ intersecting $S$ (3.1). Then for
any smaller open neighbourhood $V\subset U$ of $S$, the restriction map
$\operatorname{Mer}(U){\>\longrightarrow\>}\operatorname{Mer}(V)$ is an
isomorphism.
Proof: Define the degree $\deg_{S}(f)$ of a meromorphic function
$f\in\operatorname{Mer}(V)$ as the degree of the pole divisor of
$f{\left|{}_{{\phantom{|}\\!\\!}_{S_{1}}}\right.}$ for any deformation $S_{1}$
of $S$ transversal to the pole divisor of $f$. Denote by
$\operatorname{Mer}_{d}(V)$ the space meromorphic functions of degree
$\leqslant d$. To prove 3.3 it would suffice to show that the restriction map
$\operatorname{Mer}_{d}(U){\>\longrightarrow\>}\operatorname{Mer}_{d}(V)$ is
an isomorphism, for all $d$.
For each rational curve $S_{1}$, a degree $\leqslant d$ meromorphic function
is uniquely determined by its values in any $d+1$ distinct points on $S_{1}$.
Given a meromorphic function $f\in\operatorname{Mer}_{d}(V)$, and a
deformation $S_{1}$ of $S$ intersecting $V$, we can extend
$f{\left|{}_{{\phantom{|}\\!\\!}_{S_{1}\cap V}}\right.}$ to a degree $d$
meromorphic function $f_{1}$ on $S_{1}$ by computing its values at $d+1$
distinct points $z_{1},...,z_{d+1}$ of $S_{1}\cap V$. Whenever $S_{1}$ is in
$V$, this procedure gives $f{\left|{}_{{\phantom{|}\\!\\!}_{S_{1}}}\right.}$.
By analytic continuation, the values of $f_{1}(z)$ at any $z\in S_{1}$ are
independent from the choice of $z_{i}$ and $S_{1}$.
We have shown that $f_{1}$ is a well-defined meromorphic function on the union
$U_{1}$ of all deformations of $S$ intersecting $V$. By construction, on $V$
we have $f_{1}=f$. Since $U_{1}$ contains $U$, this implies that $f$ can be
extended from $V$ to $U$.
## 4 Moishezon twistor spaces
### 4.1 Vector bundles in a neighbourhood of an ample curve
Proposition 4.1: Let $S\subset M$ be an ample rational curve, $U\supset S$
its neighbourhood, and $B$ a holomorphic bundle on $U$. Then $H^{0}(U,B)$ is
finite-dimensional.
Proof. Step 1: Let $I_{S}\subset{\cal O}_{U}$ be the ideal sheaf of $S$, and
$J^{r}_{S}(B):=B/I^{r+1}_{S}B$ the sheaf of $r$-jets of the sections of $B$.
Since $I^{r}_{S}/I^{r+1}_{S}=\operatorname{Sym}^{r}(N^{*}S)$, the bundle
$I^{r}_{S}/I^{r+1}_{S}$ is a direct sum of ${\cal O}(k_{i})$ with $k_{i}<-r$.
Therefore, $H^{0}(I^{r}_{S}B/I^{r+1}_{S}B)=0$ for $r$ sufficiently big.
Step 2: The sheaf $I^{r}_{S}B$ admits a filtration $I^{r}_{S}B\supset
I^{r+1}_{S}B\supset I^{r+2}_{S}B\supset...$ with associated graded sheaves
$I^{r}_{S}B/I^{r+1}_{S}B$ having no sections for $r\gg 0$. Therefore,
$H^{0}(I^{r}_{S}B)=0$ for sufficiently byg $r$.
Step 3: If $H^{0}(U,B)$ is infinite-dimensional, the map
$H^{0}(U,B){\>\longrightarrow\>}H^{0}(B/I^{r}_{S}(B))$
cannot be injective. Then, for each $r$, there exists a non-zero section with
vanishing $r$-jet: $f_{r}\in H^{0}(U,I^{r+1}_{S}B)$. This is impossible, as
shown in Step 2.
This proof is effective, and gives the following bound on the dimension of the
space of sections of $B$.
Corollary 4.2: Let $S\subset M$ be an ample rational curve, $U\supset S$ its
neighbourhood, and $B$ a holomorphic bundle on $U$. Then
$\dim H^{0}(U,B)\leqslant\dim
H^{0}\left(S,\bigoplus_{d}\operatorname{Sym}^{d}(N^{*}S)\otimes_{{\cal
O}_{S}}B{\left|{}_{{\phantom{|}\\!\\!}_{S}}\right.}\right).$
The same argument can be applied to degree $d$ meromorphic functions. Recall
that the degree $\deg_{S}(f)$ of a meromorphic function
$f\in\operatorname{Mer}(V)$ is the degree of the pole divisor of
$f{\left|{}_{{\phantom{|}\\!\\!}_{S_{1}}}\right.}$ for any deformation of $S$
transversal to the pole divisor of $f$. We denote the space of meromorphic
functions of degree $\leqslant d$ on $U$ by $\operatorname{Mer}_{d}(U)$.
Corollary 4.3: Let $S\subset M$ be an ample rational curve, and $U\supset S$
its neighbourhood. Then
$\dim\operatorname{Mer}_{d}(U)\leqslant\dim
H^{0}\left(S,\operatorname{Sym}^{\leqslant d}(N^{*}S)\otimes_{{\cal
O}_{S}}{\cal O}(d)\right).$ (4.1)
### 4.2 Algebraic dimension of the field of meromorphic functions
The conormal bundle $N^{*}\\!S$ is negative; clearly, the dimension (4.1) is
maximal when $S$ is a quasiline, and $N^{*}\\!S=\oplus{\cal O}(1)$. In this
case, the bound (4.1) is realized for a rational line in ${\mathbb{C}}P^{n}$.
Indeed, for a rational line $S$ in ${\mathbb{C}}P^{n}$, the sheaf of algebraic
functions in a neighbourhood of $S$ is isomorphic to of
$\bigoplus_{i}\operatorname{Sym}^{i}(N^{*}\\!S)$. This implies the following
simple numerical result.
Claim 4.4: Let $S\subset M$ be an ample rational curve, and $U\supset S$ its
neighbourhood. Then
$\dim\operatorname{Mer}_{d}(U)\leqslant\dim H^{0}({\mathbb{C}}P^{n},{\cal
O}(d)).$ (4.2)
where $n=\dim M$.
Corollary 4.5: Let $M$ be a complex variety containing an ample rational
curve. Then the transcendence degree of $\operatorname{Mer}(M)$ satisfies
$\deg_{tr}\operatorname{Mer}(M)\leqslant\dim M$.
Proof: Consider the graded ring $\bigoplus_{d}\operatorname{Mer}_{d}(M)$.
Since $\dim\operatorname{Mer}_{d}(U)\leqslant\dim
H^{0}({\mathbb{C}}P^{n},{\cal O}(d))$, the Krull dimension of this ring is
$\leqslant n$. Therefore, the transcendence degree of its ring of fractions is
also bounded by $n$.
This observation is not new: it was known already to Hartshorne (in the
context of formal neighbourhoods). Applied to complex analytic spaces,
Hartshorne’s theorem can be stated as follows.
Theorem 4.6: Let $S\subset M$ be a connected, positive-dimensional, smooth
subvariety in a complex manifold. Assume that the normal bundle of $S$ is
ample. Then the transcendence degree of the field $\operatorname{Mer}(M)$ of
meromorphic functions is no bigger than the dimension of $M$:
$\deg_{tr}\operatorname{Mer}(M)\leqslant\dim(M).$
Moreover, if equality is reached, $\operatorname{Mer}(M)$ is a finitely
generated extension of ${\mathbb{C}}$.
Proof: [Har, Theorem 6.7]; see also [KST].
### 4.3 Moishezon manifolds and ample rational curves
Hartshorne’s methods are already sufficient to prove the following general
result.
Proposition 4.7: Let $S\subset M$ be an ample rational curve in a simply
connected complex manifold. Assume that
$\deg_{tr}\operatorname{Mer}(M)=\dim(M).$ Then there exists a meromorphic map
$\varphi:\;M{\>\longrightarrow\>}Z_{0}$ to an open subset of a projective
variety $M$, which is bijective onto its image outside of a complex analytic
subset of positive codimension.
Proof: By Hartshorne’s theorem (4.2), $\operatorname{Mer}(M)$ is a finitely
generated extension of ${\mathbb{C}}$. Let $\xi_{1},...,\xi_{N}$ be generators
of $\operatorname{Mer}(M)$, $D_{i}$ their pole divisors, and $L_{i}:={\cal
O}(D_{i})$ the corresponding line bundles. Then $\xi_{i}$ can be considered as
sections of $L_{i}$, and $\xi_{1},...,\xi_{N}$ – as sections of
$L:=\bigotimes_{i}L_{i}$. Consider now the subring of
$\bigoplus_{d}H^{0}(M,L^{d})$ generated by $\xi_{i}$, and let $Z$ be its
spectre. Clearly, $\dim M=\dim Z$, $Z$ is projective, and the natural rational
map $M\stackrel{{\scriptstyle\varphi}}{{{\>\longrightarrow\>}}}Z$ induces an
isomorphism
$\operatorname{Mer}(Z)\tilde{\>\longrightarrow\>}\operatorname{Mer}(M)$.
Let $\tilde{M}$ be a resolution of the base set of $\varphi$, such that
$\tilde{M}\stackrel{{\scriptstyle\tilde{\varphi}}}{{{\>\longrightarrow\>}}}Z$
is holomorphic. If $\tilde{\varphi}$ is ramified at some divisor $D$ in
$\varphi(\tilde{M})$, this divisor can be extended to $Z$ using 3.3 applied to
$U=\varphi(\tilde{M})\subset Z$. Taking the corresponding ramified covering
$\tilde{Z}$ of $Z$, we obtain a holomorphic map from $\tilde{M}$ to
$\tilde{Z}$, which is impossible, because $\operatorname{Mer}(\tilde{Z})$ is
strictly bigger than $\operatorname{Mer}(Z)$, and
$\operatorname{Mer}(\tilde{M})=\operatorname{Mer}(Z)$. Therefore,
$\tilde{\varphi}$ is bijective to its impage at its general point. Openness of
its image is a general property of bimeromorphic maps.
Theorem 4.8: Let $S\subset M$ be an ample rational curve in a simply
connected complex manifold. Assume that
$\deg_{tr}\operatorname{Mer}(M)=\dim(M).$ Then there exists an open embedding
of $M$ to a Moishezon variety.
Proof: From 4.3 we obtain a line bundle $L$ on $M$ inducing a bimeromorphic
map $\varphi:\;M{\>\longrightarrow\>}Z_{0}$ to an open subset of a projective
variety $Z$. Resolving the base points of the inverse map if necessary, we may
assume that the inverse map $\psi:\;Z_{0}{\>\longrightarrow\>}M$ is
holomorphic. Then, $M$ is obtained from $Z_{0}$ by blowing down a certain
number of exceptional subvarieties $E_{i}$, obtained as common zero sets of a
certain number of meromorphic functions. Applying 3.3 to $Z_{0}\subset Z$, we
extend the meromorphic functions and the corresponding subvarieties $E_{i}$
and obtain closed exceptional subvarieties $E_{i}^{\prime}\subset Z$. Blowing
these down, we obtain a Moishezon variety which contains $M$ as an open
subset.
### 4.4 Moishezon twistor spaces
Definition 4.9: Let $M=\operatorname{Tw}(X)$ be a twistor space of a simply
connected hyperkähler, hypercomplex, quaternionic or or 4-dimensional anti-
selfdual manifold, not necessarily compact. We say that $M$ is a Moishezon
twistor space if $\deg_{tr}\operatorname{Mer}(M)=\dim(M)$ (see 4.2).
From 4.3, we immediately obtain the following corollary.
Corollary 4.10: Let $M$ be a Moishezon twistor space. Then $M$ admits an open
embedding to a Moishezon variety $M_{1}$. Moreover, $M_{1}$ is unique up to a
bimeromorphic equivalence.
Proof: The open embedding to a Moishezon variety follows from 4.3, and its
uniqueness is implied by an isomorphism
$\operatorname{Mer}(M)=\operatorname{Mer}(M_{1})$ (3.3).
It is easy to construct an example of a twistor manifold which does not belong
to this class. The twistor space of a K3 surface, and, more generally, any
compact hyperkähler manifold is never Moishezon.
Theorem 4.11: Let $M$ be a compact hyperkähler manifold, and
$\operatorname{Tw}(M)$ its twistor space. Then
$\deg_{tr}\operatorname{Mer}(\operatorname{Tw}(M))=1$.
Proof: Let $Z\subset\operatorname{Tw}(M)$ be any divisor, and
$R\subset{\mathbb{C}}P^{1}$ a countable subset constructed in 2.1. For any
induced complex structure $L\notin R$, all complex subvarieties of $(M,L)$ are
even-dimensional. Therefore, $Z$ intersects the twistor fiber
$(M,L)=\pi^{-1}(L)\subset\operatorname{Tw}(M)$ non-transversally, or not at
all. By Thom’s transversality theorem, the intersection $Z$ with all fibers of
$\pi$ except a finite number is transversal. This means that $Z$ can intersect
only finitely many of the fibers of $\pi$. However, all these fibers are
irreducible divisors. Therefore, $D$ is a union of several fibers of $\pi$.
Since a meromorphic function is uniquely determined by its pole or zero
divisor, all meromorphic functions on $\operatorname{Tw}(M)$ are pull-backs of
meromorphic functions on ${\mathbb{C}}P^{1}$.
Remark 4.12: When $M$ is hypercomplex, no effective bounds on the
transcendence degree $\deg_{tr}\operatorname{Mer}(\operatorname{Tw}(M))$ are
known. We conjecture (based on empirical evidence) that the twistor space of a
compact hypercomplex manifold is not Moishezon, but this conjecture seems to
be difficult.
### 4.5 Twistor spaces and hyperkähler reduction
We recall the definition of hyperkähler reduction, following [HKLR] and [Nak].
This material is fairly standard.
We denote the Lie derivative along a vector field as
$\operatorname{Lie}_{x}:\;\Lambda^{i}M{\>\longrightarrow\>}\Lambda^{i}M$, and
contraction with a vector field by
$i_{x}:\;\Lambda^{i}M{\>\longrightarrow\>}\Lambda^{i-1}M$. Recall the Cartan’s
formula:
$d\circ i_{x}+i_{x}\circ d=\operatorname{Lie}_{x}$
Let $(M,\omega)$ be a symplectic manifold, $G$ a Lie group acting on $M$ by
symplectomorphisms, and $\mathfrak{g}$ its Lie algebra. For any
$g\in{\mathfrak{g}}$, denote by $\rho_{g}$ the corresponding vector field.
Cartan’s formula gives $\operatorname{Lie}_{\rho_{g}}\omega=0$, hence
$d(i_{\rho_{g}}(\omega))=0$. We obtain that $i_{\rho_{g}}(\omega)$ is closed,
for any $g\in{\mathfrak{g}}$.
Definition 4.13: A Hamiltonian of $g\in{\mathfrak{g}}$ is a function $h$ on
$M$ such that $dh=i_{\rho_{g}}(\omega)$.
Definition 4.14: $(M,\omega)$ be a symplectic manifold, $G$ a Lie group acting
on $M$ by symplectomorphisms. A moment map $\mu$ of this action is a linear
map ${\mathfrak{g}}{\>\longrightarrow\>}C^{\infty}M$ associating to each
$g\in{\mathfrak{g}}$ its Hamiltonian.
Remark 4.15: It is more convenient to consider $\mu$ as an element of
${\mathfrak{g}}^{*}\otimes_{\mathbb{R}}C^{\infty}M$, or, as it is usually
done, a function on $M$ with values in ${\mathfrak{g}}^{*}$.
Remark 4.16: Note that the moment map always exists, if $M$ is simply
connected.
Definition 4.17: A moment map $M{\>\longrightarrow\>}{\mathfrak{g}}^{*}$ is
called equivariant if it is equivariant with respect to the coadjoint action
of $G$ on ${\mathfrak{g}}^{*}$.
Remark 4.18:
$M\stackrel{{\scriptstyle\mu}}{{{\>\longrightarrow\>}}}{\mathfrak{g}}^{*}$ is
a moment map if and only if for all $g\in{\mathfrak{g}}$, $\langle
d\mu,g\rangle=i_{\rho_{g}}(\omega)$. Therefore, a moment map is defined up to
a constant ${\mathfrak{g}}^{*}$-valued function. An equivariant moment map is
is defined up to a constant ${\mathfrak{g}}^{*}$-valued function which is
$G$-invariant.
Definition 4.19: A $G$-invariant $c\in\mathfrak{g}^{*}$ is called central.
Claim 4.20: An equivariant moment map exists whenever
$H^{1}(G,{\mathfrak{g}}^{*})=0$. In particular, if $G$ is reductive and $M$ is
simply connected, an equivariant moment map is always possible to define.
Definition 4.21: Let $(M,\omega)$ be a symplectic manifold, $G$ a compact Lie
group acting on $M$ by symplectomorphisms,
$M\stackrel{{\scriptstyle\mu}}{{{\>\longrightarrow\>}}}{\mathfrak{g}}^{*}$ an
equivariant moment map, and $c\in{\mathfrak{g}}^{*}$ a central element. The
quotient $\mu^{-1}(c)/G$ is called the symplectic reduction of $M$, denoted by
$M{/\\!\\!/}G$.
Claim 4.22: The symplectic quotient $M{/\\!\\!/}G$ is a symplectic manifold of
dimension $\dim M-2\dim G$.
Theorem 4.23: Let $(M,I,\omega)$ be a Kähler manifold, $G_{\mathbb{C}}$ a
complex reductive Lie group acting on $M$ by holomorphic automorphisms, and
$G$ is a compact form of $G_{\mathbb{C}}$ acting isometrically. Then
$M{/\\!\\!/}G$ is a Kähler orbifold.
Remark 4.24: In such a situation, $M{/\\!\\!/}G$ is called the Kähler
quotient, or GIT quotient.
Remark 4.25: The points of $M{/\\!\\!/}G$ are in bijective correspondence with
the orbits of $G_{\mathbb{C}}$ which intersect $\mu^{-1}(c)$. Such orbits are
called polystable, and the intersection of a $G_{\mathbb{C}}$-orbit with
$\mu^{-1}(c)$ is a $G$-orbit.
Definition 4.26: Let $G$ be a compact Lie group, $\rho$ its action on a
hyperkähler manifold $M$ by hyperkähler isometries, and ${\mathfrak{g}}^{*}$ a
dual space to its Lie algebra. A hyperkähler moment map is a $G$-equivariant
smooth map $\mu:M\to{\mathfrak{g}}^{*}\otimes{\mathbb{R}}^{3}$ such that
$\langle\mu_{i}(v),g\rangle=\omega_{i}(v,d\rho(g))$, for every $v\in TM$,
$g\in{\mathfrak{g}}$ and $i=1,2,3$, where $\omega_{i}$ are three Kähler forms
associated with the hyperkähler structure.
Definition 4.27: Let $\xi_{1},\xi_{2},\xi_{3}$ be three $G$-invariant vectors
in ${\mathfrak{g}}^{*}$. The quotient manifold
$M{/\\!\\!/\\!\\!/}G:=\mu^{-1}(\xi_{1},\xi_{2},\xi_{3})/G$ is called the
hyperkähler quotient of $M$.
Theorem 4.28: ([HKLR]) The quotient $M{/\\!\\!/\\!\\!/}G$ is hyperkaehler.
Proof: We sketch the proof of Hitchin-Karlhede-Lindström-Roček theorem,
because we make use of it further on.
Let $\Omega:=\omega_{J}+\sqrt{-1}\>\omega_{K}$. This is a holomorphic
symplectic (2,0)-form on $(M,I)$. Let $\mu_{J},\mu_{K}$ be the moment map
associated with $\omega_{J},\omega_{K}$, and
$\mu_{\mathbb{C}}:=\mu_{J}+\sqrt{-1}\>\mu_{K}$. Then $\langle
d\mu_{\mathbb{C}},g\rangle=i_{\rho_{g}}(\Omega)$ Therefore,
$d\mu_{\mathbb{C}}\in\Lambda^{1,0}(M,I)\otimes{\mathfrak{g}}^{*}$. This
implies that the map $\mu_{\mathbb{C}}$ is holomorphic. It is called a
holomorphic moment map.
By definition, $M{/\\!\\!/\\!\\!/}G=\mu_{{\mathbb{C}}}^{-1}(c){/\\!\\!/}G$,
where $c\in{\mathfrak{g}}^{*}\otimes_{\mathbb{R}}{\mathbb{C}}$ is a central
element. This is a Kähler manifold, because it is a Kähler quotient of a
Kähler manifold.
We obtain 3 complex structures $I,J,K$ on the hyperkähler quotient
$M{/\\!\\!/\\!\\!/}G$. They are compatible in the usual way, as seen from a
simple local computation.
Theorem 4.29: Let $V$ be a quaterionic Hermitian vector space, and
$G\subset\operatorname{Sp}(V)$ a compact Lie group acting on $V$ by
quaternionic isometries. Denote by $M$ the hyperkähler reduction of $V$. Then
$\operatorname{Tw}(M)$ is a Moishezon twistor space, in the sense of 4.4.
Proof: The holomorphic symplectic form on $(V,I)$ depends on $I$
holomorphically, giving a section
$\Omega_{tw}\in\Omega^{2}_{\pi}(\operatorname{Tw}(V))\otimes_{{\cal
O}_{\operatorname{Tw}(V)}}\pi^{*}{\cal O}(2).$
Here, $\pi:\;\operatorname{Tw}(V){\>\longrightarrow\>}{\mathbb{C}}P^{1}$ is
the twistor projection, and $\Omega^{2}_{\pi}(\operatorname{Tw}(V))$ the sheaf
of fiberwise holomorphic 2-forms. Consider the fiberwise holomorphic moment
map given by this form,
$\mu_{tw}:\;\operatorname{Tw}(V){\>\longrightarrow\>}\operatorname{Tot}({\mathfrak{g}}^{*}\otimes_{\mathbb{C}}{\cal
O}(2))$. Replacing the moment map by its translate, we can always assume that
$M{/\\!\\!/\\!\\!/}G=\mu_{{\mathbb{C}}}^{-1}(0){/\\!\\!/}G$ (that is, we
assume that the central vector $c$ used to define $M{/\\!\\!/\\!\\!/}G$
vanishes). Then $\operatorname{Tw}(M)$ is obtained as the space of polystable
$G_{\mathbb{C}}$-orbits in $\mu_{tw}^{-1}(0)\subset\operatorname{Tw}(V)$.
Here, “polystability” of an orbit is understood as the non-emptiness of the
intersection of this orbit with $\mu^{-1}(0)$; the set of such orbits is open.
The space
$\operatorname{Tw}(V)={\mathbb{C}}P^{2n+1}\backslash{\mathbb{C}}P^{2n-1}$ is
quasiprojective. Averaging the ring of rational functions with $G$, we obtain
that the field $G_{\mathbb{C}}$-invariant rational functions on
$\operatorname{Tw}(V)$ has dimension $\dim\mu_{tw}^{-1}(0)-\dim
G_{\mathbb{C}}=\dim\operatorname{Tw}(M)$, and hence $\operatorname{Tw}(M)$ is
Moishezon.
Corollary 4.30: Let $U$ be an open subset of a compact, simply connected
hyperkähler manifold, and $U^{\prime}$ an open subset of a hyperkähler
manifold obtained as $V{/\\!\\!/\\!\\!/}G$, where $V$ is flat and $G$
reductive. Then $U$ is not isomorphic to $U^{\prime}$ as hyperkähler manifold.
Proof:
$\dim_{tr}\operatorname{Tw}(U^{\prime})=\dim\operatorname{Tw}(U^{\prime})$ by
4.5, and $\dim_{tr}\operatorname{Tw}(U)=1$ by 4.4.
A twistor space of a manifold obtained by hyperkähler reduction is Moishezon,
and from the above argument it is easy to see that it is Zariski open in a
compact Moishezon variety. However, it is (almost) never quasiprojective.
Theorem 4.31: Let $M$ be a hyperkähler manifold such that its twistor space
$\operatorname{Tw}(M)$ can be embedded to a projective manifold. Then, for
each induced complex structure $L$, the complex manifold $(M,L)$ has no
compact subvarieties of positive dimension.
Proof: Consider the anticomplex involution $\iota$ on $\operatorname{Tw}(M)$
mapping $(m,L)$ to $(m,-L)$. Suppose that $(M,L)\subset\operatorname{Tw}(M)$
has a compact subvariety. Since the $(M,L)$ is quasiprojective, this would
imply that $(M,L)$ contains a compact curve $S$. The curve $\iota(S)$ is also
holomorphic in $(M,-L)$.
Consider a Kähler form $\omega$ on $\operatorname{Tw}(M)$. Since $\iota$ is
antiholomorphic, $-\iota(\omega)$ is a closed, positive $(1,1)$-form.
Replacing $\omega$ by $\omega-\iota(\omega)$, we may assume that $\omega$
satisfies $\iota(\omega)=-\omega$.
Now, since the cohomology classes of $S$ and $\iota(S)$ are equal, we have
$\int_{S}\omega=\int_{\iota(S)}\omega=-\int_{\iota(S)}\iota(\omega)$
On the other hand, $\int_{S}\omega=\int_{\iota(S)}\iota(\omega)$ by functorial
properties of integral. This implies $\int_{S}\omega=0$, giving a
contradiction.
Remark 4.32: As shown in [V3], a general fiber of the map
$\operatorname{Tw}(M){\>\longrightarrow\>}{\mathbb{C}}P^{1}$ has only even-
dimensional complex subvarieties.
## 5 Appendix: Formal geometry and holography principle (by Dmitry Kaledin)
In this Appendix, we will try to explain the constructions of the paper in a
slightly more general context of $\mathcal{D}$-modules and “formal geometry”
of Gelfand and Kazhdan [GK]. To save space, we only sketch the proofs, and we
work in the algebraic setting (generalization to complex-analytic varieties is
immediate, exactly the same arguments work).
Assume given a smooth algebraic variety $X$ over a field $k$ of characteristic
$0$. We will work with coherent $\mathcal{D}$-modules over $X$, that is, with
sheaves of left modules over the algebra $\mathcal{D}_{X}$ of differential
operators on $X$ which are finitely generated over $\mathcal{D}$. Any coherent
$\mathcal{D}$-module $\mathcal{E}$ is also a quasicoherent sheaf of
$\mathcal{O}_{X}$-modules; recall that if $\mathcal{E}$ is coherent over
$\mathcal{O}_{X}$, then it comes from a vector bundle on $X$ equipped with a
flat connection. In particular, the structure sheaf $\mathcal{O}_{X}$ is a
$\mathcal{D}$-module; it corresponds to the trivial line bundle. For any
coherent sheaf $\mathcal{E}$ on $X$, we can consider the induced
$\mathcal{D}$-module $\mathcal{D}_{X}\otimes_{\mathcal{O}_{X}}\mathcal{E}$;
this is coherent over $\mathcal{D}_{X}$ but not over $\mathcal{O}_{X}$.
Coherent $\mathcal{D}$-modules form an abelian category, and we can consider
its derived category.
Assume given an open subvariety $U\subset X$ with the embedding map $j:U\to
X$, and let $Z=X\setminus U\subset X$ be the complement $U$. Then the complex
of quascoherent sheaves
$R^{\>\raisebox{3.0pt}{\text{\circle*{1.5}}}}j_{*}\mathcal{O}_{U}$ on $X$ has
a natural structure of a complex of $\mathcal{D}$-modules, and we have an
exact triangle
$\begin{CD}\delta_{Z}@>{}>{}>\mathcal{O}_{X}@>{}>{}>R^{\>\raisebox{3.0pt}{\text{\circle*{1.5}}}}j_{*}\mathcal{O}_{X}@>{}>{}>\end{CD}$
(5.1)
of complexes of $\mathcal{D}$-modules on $X$, where $\delta_{Z}$ is supported
at $Z$. We will need the following standard result.
Lemma 5.1: In the notation above, for any coherent sheaf $\mathcal{E}$ on
$X$, we have a natural identification
$\operatorname{Hom}_{\mathcal{O}_{X}}(\delta_{Z},\mathcal{E})\cong\Gamma(X,\mathcal{E}\otimes\widehat{\mathcal{O}}_{X,Z}),$
where $\widehat{\mathcal{O}}_{X,Z}$ is the formal completion of structure
sheaf $\mathcal{O}_{X}$ at the closed subscheme $Z\subset X$.
Sketch of a proof. Since $\delta_{Z}$ is supported at $Z$, the natural map
$\operatorname{Hom}_{\mathcal{O}_{X}}(\delta_{Z},\mathcal{E})\to\operatorname{Hom}_{\mathcal{O}_{X}}(\delta_{Z},\mathcal{E}\otimes\widehat{\mathcal{O}}_{X,Z})$
is an isomorphism. On the other hand, we obviously have
$\operatorname{Hom}_{\mathcal{O}_{X}}(R^{\>\raisebox{3.0pt}{\text{\circle*{1.5}}}}j_{*}\mathcal{O}_{U},E\otimes\widehat{\mathcal{O}}_{X,Z})=0,$
and the claim then immediately follows from the long exact sequence associated
to the exact triangle (5.1). $\square$
For any map $f:X\to Y$ of smooth algebraic varieties, the pullback functor
$f^{*}$ extends to a functor between $\mathcal{D}$-modules, and its derived
functor extends to a functor between the derived categories of
$\mathcal{D}$-modules; we will denote this last functor by $f^{?}$. We have
$f^{?}=f^{*}[\dim X-\dim Y],$
where $f^{*}$ is the standard pullback functor for $\mathcal{D}$-modules, and
$[-]$ stands for cohomological shift. If $f$ is a smooth map, or a closed
embedding, or a composition of the two, then $f^{?}$ has a left-adjoint
functor $f_{?}$ given by
$f_{?}=f_{!}[\dim Y-\dim X],$
where $f_{!}$ is the standard functor of direct image with compact supports.
Then 5 has the following corollary.
Lemma 5.2: Assume given a map $f:Y\to X$ of smooth algebraic varieties, and
assume that $f$ factors as
$\begin{CD}Y@>{\pi}>{}>Z@>{\iota}>{}>X,\end{CD}$
where $Z$ is smooth, $\iota$ is a closed embedding, and $\pi$ is a smooth map
with contractible fibers. Then for any vector bundle $\mathcal{E}$ on $X$, we
have a natural identification
$\operatorname{Hom}_{\mathcal{D}_{Y}}(f^{?}(\mathcal{D}_{X}\otimes_{\mathcal{O}_{X}}\mathcal{E}^{*}),\mathcal{O}_{Y})\cong\Gamma(X,\mathcal{E}\otimes\widehat{\mathcal{O}}_{X,Z}),$
(5.2)
where $\mathcal{E}^{*}$ is the dual vector bundle, and
$\mathcal{D}_{X}\otimes_{\mathcal{O}_{X}}\mathcal{E}^{*}$ is the corresponding
induced $\mathcal{D}$-module.
Proof. For any $\mathcal{D}$-module $\mathcal{F}$ and $\mathcal{O}_{X}$-module
$\mathcal{E}$, we have
$\operatorname{Hom}_{\mathcal{D}_{X}}(\mathcal{D}_{X}\otimes_{\mathcal{O}_{X}}\mathcal{E},\mathcal{F})\cong\operatorname{Hom}_{\mathcal{O}_{X}}(\mathcal{E},F),$
so that 5 provides an identification
$\operatorname{Hom}_{\mathcal{D}_{X}}(\mathcal{D}_{X}\otimes_{\mathcal{O}_{X}}\mathcal{E}^{*}\otimes_{\mathcal{O}_{X}}\delta_{Z},\mathcal{O}_{X})\cong\Gamma(X,\mathcal{E}\otimes\widehat{\mathcal{O}}_{X,Z}),$
and by adjunction, it suffices to construct an isomorphism
$f_{?}f^{?}(\mathcal{D}_{X}\otimes_{\mathcal{O}_{X}}\mathcal{E}^{*})\cong\mathcal{D}_{X}\otimes_{\mathcal{O}_{X}}\mathcal{E}^{*}\otimes_{\mathcal{O}_{X}}\delta_{Z}.$
But under our assumptions on $f$, we have
$f_{?}\mathcal{O}_{X}\cong\delta_{Z}$, and we are done by the projection
formula. $\square$
We note that the left-hand side of (5.2) admits a slightly different
interpretation. Recall that for any coherent sheaf $\mathcal{E}$ on $X$, the
jet bundle $J^{\infty}\mathcal{E}$ of $\mathcal{E}$ is a (pro)coherent sheaf
on $X$ given by
$J^{\infty}\mathcal{E}=\pi_{2*}\pi_{1}^{*}\mathcal{E},$
where $\pi_{1},\pi_{2}:\widehat{X}\to X$ are the two natural projections of
the completion $\widehat{X}$ of the product $X\times X$ near the diagonal
$X\subset X\times X$. In terms of $\mathcal{D}$-modules, we have
$J^{\infty}\mathcal{E}\cong\operatorname{\mathcal{H}om}_{\mathcal{O}_{X}}(\mathcal{D}_{X},\mathcal{E}).$
The jet bundle $J^{\infty}\mathcal{E}$ carries a canonical flat connection,
and in the assumptions of 5, we have
$\displaystyle\operatorname{Hom}_{\mathcal{D}_{Y}}(f^{?}(\mathcal{D}_{X}\otimes_{\mathcal{O}_{X}}\mathcal{E}^{*}),\mathcal{O}_{Y})$
$\displaystyle\cong\operatorname{Hom}_{\mathcal{D}_{Y}}(\mathcal{O}_{Y},\operatorname{\mathcal{H}om}_{\mathcal{O}_{Y}}(f^{?}(\mathcal{D}_{X}\otimes_{\mathcal{O}_{X}}\mathcal{E}^{*}),\mathcal{O}_{Y}))$
$\displaystyle\cong\operatorname{Hom}_{\mathcal{D}_{Y}}(\mathcal{O}_{Y},f^{?}(\operatorname{\mathcal{H}om}_{\mathcal{O}_{X}}(\mathcal{D}_{X}\otimes_{\mathcal{O}_{X}}\mathcal{E}^{*},\mathcal{O}_{X})))$
$\displaystyle\cong\Gamma^{\nabla}(Y,f^{*}J^{\infty}\mathcal{E}),$
where $\Gamma^{\nabla}(-)$ stands for the space of flat global sections. Then
(5.2) reads as
$\Gamma^{\nabla}(Y,f^{*}J^{\infty}\mathcal{E})\cong\Gamma(X,\mathcal{E}\otimes\widehat{\mathcal{O}}_{X,Z}).$
(5.3)
Remark 5.3: The assumptions of 5 are in fact too strong. Firstly, it is
clearly enough to require that the fibers of $\pi$ are non-empty and
connected, so that their top degree cohomology with compact supports is one-
dimensional – and under the assumptions as stated, we not only obtain an
isomorphism of $\operatorname{Hom}$’s but also of the $\operatorname{RHom}$’s,
so that (5.3) extends to an isomorphism
$H^{\>\raisebox{3.0pt}{\text{\circle*{1.5}}}}_{DR}(Y,f^{*}J^{\infty}\mathcal{E})\cong
H^{\>\raisebox{3.0pt}{\text{\circle*{1.5}}}}(X,\mathcal{E}\otimes\widehat{\mathcal{O}}_{X,Z})$
of cohomology groups. Secondly, one probably does not have to require that
$\pi$ and $Z$ are smooth – some assumptions are needed, but they can
considerably relaxed. However, since even the stronger assumptions work for
us, we did not pursue this.
Assume now that we are given smooth proper algebraic varieties $Y$, $X$, and a
family of closed embedding from $Y$ to $X$ parametrized by a smooth algebraic
variety $T$ – that is, we have a map
$f:Y\times T\to X$
such that for any $t\in T$, the corresponding map $f_{t}:Y=Y\times t\to X$ is
a closed embedding. Moreover, assume given a coherent sheaf $\mathcal{E}$ on
$X$, denote by $\rho:Y\times T\to T$ the projection, and denote
$\Phi_{f}\mathcal{E}=\rho_{*}^{\nabla}f^{*}J^{\infty}\mathcal{E},$
where $f^{*}J^{\infty}\mathcal{E}$ is the pullback of the jet bundle
$J^{\infty}\mathcal{E}$ equipped with its natural flat connection $\nabla$,
and $\rho_{*}^{\nabla}$ stands for sheaf of relative flat sections. Then
$\Phi_{a}\mathcal{E}$ is a sheaf on $T$, the base of the family, and by (5.3),
the fiber $(\Phi_{f}\mathcal{E})_{t}$ at a point $t\in T$ is given by
$(\Phi_{f}\mathcal{E})_{t}=\Gamma^{\nabla}(Y,f_{t}^{*}J^{\infty}\mathcal{E})\cong\Gamma(X,\mathcal{E}\otimes\widehat{\mathcal{O}}_{X,Y_{t}}),$
where $Y_{t}\subset X$ is the image of the closed embedding $f_{t}:Y\to X$. On
the other hand, $\Phi_{f}\mathcal{E}$ carries a natural flat connection, and
if we assume that $f$ is smooth with contractible fibers, we have
$\Gamma^{\nabla}(T,\Phi_{f}\mathcal{E})=\Gamma^{\nabla}(T\times
Y,f^{*}J^{\infty}\mathcal{E})\cong\Gamma(X,\mathcal{E}),$
again by (5.3).
We now note that this is exactly the situation that we have in the paper.
Namely, we take $X$ to be the twistor space of a hyperkähler manifold $M$, we
take $T=M_{\mathbb{C}}$ to be the complexification of the real-analytic
manifold underlying $M$, we take $Y=\mathbb{C}P^{1}$, and we let
$f:T\times Y\to X$
be the standard family of twistor lines (real points in this family correspond
to horizontal sections of the twistor fibration $X\to\mathbb{C}P^{1}$
parametrized by points of $M$). Then since the normal bundle to any line in
our family is a sum of several copies of $\mathcal{O}(1)$, the family is
unobstructed, and moreover, it remains unobstructed even if we fix a point at
a twistor line, so that the map $f$ is smooth. Its fibers are small polydiscs,
thus contractible, and all the assumptions of 5 are therefore satisfied.
## References
* [BBI] Bǎdescu, Lucian; Beltrametti, Mauro C.; Ionescu, Paltin, Almost-lines and quasi-lines on projective manifolds, Complex analysis and algebraic geometry, 1-27, de Gruyter, Berlin, 2000.
* [AHS] Atiyah, M. F., Hitchin, N. J., and Singer, I. M. Self-duality in four-dimensional Riemannian geometry, Proc. Roy. Soc. London Ser. A 362 (1978), 425-461.
* [Bes] Besse, A., Einstein Manifolds, Springer-Verlag, New York (1987).
* [C] F. Campana, On twistor spaces of the class ${\cal C}$, J. Differential Geom. 33 (1991) 541-549.
* [D] Demailly, Jean-Pierre, Pseudoconvex-concave duality and regularization of currents, Several complex variables (Berkeley, CA, 1995-1996), 233-271, Math. Sci. Res. Inst. Publ., 37, Cambridge Univ. Press, Cambridge, 1999.
* [GK] I.M. Gelfand and D.A. Kazhdan, Some problems of differential geometry and the calculation of cohomologies of Lie algebras of vector fields, Soviet Math. Dokl. 12 (1971), 1367-1370.
* [Har] Hartshorne, Robin, Cohomological dimension of algebraic varieties, Ann. of Math. (2) 88 1968 403-450.
* [HKLR] N. J. Hitchin, A. Karlhede, U. Lindström, M. Roček, Hyperkähler metrics and supersymmetry, Comm. Math. Phys. 108, (1987) 535–589.
* [Ho] Nobuhiro Honda Moishezon twistor spaces on $4\mathbb{C}P^{2}$, arXiv:1112.3109, 51 pages.
* [K] D. Kaledin, Integrability of the twistor space for a hypercomplex manifold, Sel. math., New ser. 4 (1998) 271-278.
* [KV] Kaledin, D., Verbitsky, M., Non-Hermitian Yang-Mills connections, Selecta Math. (N.S.) 4 (1998), no. 2, 279–320.
* [KST] Stefan Kebekus, Luis Sola Conde, Matei Toma, Rationally connected foliations after Bogomolov and McQuillan, J. Algebraic Geom. 16 (2007), no. 1, 65-81.
* [Ko] Kollár, J., Rational curves on algebraic varieties, Springer, 1996, viii+320 pp..
* [Nak] H. Nakajima, Lectures on Hilbert schemes of points on surfaces, Providence: American Mathematical Society, 1999.
* [P] Y. S. Poon, On the algebraic structure of twistor spaces, J. Diff. Geom. 36 (1992), 451-491.
* [SV] Andrey Soldatenkov, Misha Verbitsky, Subvarieties of hypercomplex manifolds with holonomy in $SL(n,{\mathbb{H}})$, Journal of Geometry and Physics, Volume 62, Issue 11 (2012), Pages 2234-2240, arXiv:1202.0222
* [V1] Verbitsky M., Tri-analytic subvarieties of hyper-Kaehler manifolds, also known as Hyperkähler embeddings and holomorphic symplectic geometry II, GAFA 5 no. 1 (1995), 92-104, alg-geom/9403006.
* [V2] Verbitsky, M., Hypercomplex Varieties, alg-geom/9703016, Comm. Anal. Geom. 7 (1999), no. 2, 355–396.
* [V3] Verbitsky, M., Subvarieties in non-compact hyperkähler manifolds, Math. Res. Lett., vol. 11 (2004), no. 4, pp. 413–418.
* [V4] Verbitsky, M., Rational curves and special metrics on twistor spaces, arXiv:1210.6725, 12 pages.
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Misha Verbitsky
Laboratory of Algebraic Geometry,
Faculty of Mathematics, National Research University HSE,
7 Vavilova Str. Moscow, Russia
|
arxiv-papers
| 2012-11-25T14:07:24 |
2024-09-04T02:49:38.389537
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Misha Verbitsky",
"submitter": "Misha Verbitsky",
"url": "https://arxiv.org/abs/1211.5765"
}
|
1211.5768
|
# Accelerator Control System at KEKB and the Linac
Kazuro Furukawa, Atsuyoshi Akiyama, Eiichi Kadokura,
Miho Kurashina, Katsuhiko Mikawa, Tatsuro Nakamura,
Jun-ichi Odagiri, Masanori Satoh, Tsuyoshi Suwada
KEK, Tsukuba, Ibaraki, 305-0801, Japan < kazuro.furukawa @ kek.jp >
###### Abstract
KEKB has completed all of the technical milestones and has offered important
insights into the flavor structure of elementary particles, especially CP
violation. The accelerator control system at KEKB and the injector linac was
initiated by a combination of scripting languages at the operation layer and
EPICS (experimental physics and industrial control system) at the equipment
layer. During the project, many features were implemented to achieve extreme
performance from the machine. In particular, the online linkage to the
accelerator simulation played an essential role. In order to further improve
the reliability and flexibility, two major concepts were additionally
introduced later in the project, namely, channel access everywhere and dual-
tier controls. Based on the improved control system, a virtual accelerator
concept was realized, allowing the single injector linac to serve as three
separate injectors to KEKB’s high-energy ring, low-energy ring, and Photon
Factory, respectively. These control technologies are indispensable for future
particle accelerators.
## 1 Introduction
The KEKB B-Factory was designed as an asymmetric electron/positron collider in
order to study the violation of CP symmetry in the B-meson system. It
consisted of double storage rings of an 8 GeV electron high-energy ring (HER)
and a 3.5 GeV positron low-energy ring (LER) with a diameter of 1 km, and a
full energy injector linac of 600 m [1]. It achieved a collision luminosity of
$2.11\times 10^{34}\,{\rm cm}^{-2}\,{\rm s}^{-1}$, twice as much as the
designed luminosity, and led to Kobayashi and Maskawa winning the Nobel Prize
for the theory of quarks and CP symmetry violation.
During a decade of successful operation, KEKB completed all of the technical
milestones and offered important insights into the flavor structure of
elementary particles.
KEKB collider and Belle detector were constructed and operated at almost the
same time as PEP-II and Babar at SLAC with the same scientific goal. This
provided a friendly competition between these two projects and led to many
collaborative efforts [2].
## 2 KEKB accelerator control system
The success of the high performance operation of KEKB owed much to the control
system. It was designed more than 15 years ago and started beam operation in
1998. While it inherited some of its resources from the previous project,
TRISTAN, it restructured most of the software and hardware components. It
employed the EPICS (experimental physics and industrial control system)
toolkit for the low-level control mechanism and scripting languages for high-
level operational applications. This combination provided a flexible and
robust operational environment. The details of the hardware and software
architecture at the commissioning of KEKB are described elsewhere [3].
### 2.1 Lower-level controls with EPICS
Before KEKB, projects in the institute repeated the development of their own
control systems. As technologies such as computers, system software, and field
networks became de facto standards at the beginning of the 1990s, the sharing
of control system architectures between projects was considered. After SSC
(superconducting super collider) chose EPICS as its main control toolkit,
EPICS became a candidate for future controls at KEK [4]. EPICS was studied at
the injector linac, and EPICS gateways were constructed based on the existing
control system [5]. In contrast, for the KEKB ring it was decided to employ
EPICS controls and the previous software resources were not employed. The main
reason for this was that the linac already had a network-based control system
and it had to continue operation for light sources even during the upgrade
construction towards KEKB, but the ring could have shut down the accelerator
completely for 4 years.
The KEKB ring employed several fieldbuses such as VME, VXI, CAMAC, GPIB, and
ARCNET depending on the intended purpose. Approximately 100 VME systems with
the VxWorks operating system served as EPICS input/output controllers (IOCs)
for all the hardware devices, including 200 VXI mainframes, 50 CAMAC crates,
and 200 ARCNET segments.
For the networks and computers in the global control system, a quite standard
environment was applied, taking account of reliability, such as redundant GbE
network systems, redundant network file servers, blade CPU servers, and
commodity console machines with X11 server software at KEKB and the linac.
At the linac most of the devices employed controllers with Ethernet and TCP/IP
instead of other kinds of fieldbuses. Before the KEKB project, network-based
PLCs (programmable logic controllers), CAMAC crate controllers, and VME
computers were managed by middle-layer software on Unix servers. During the
upgrade construction for KEKB, these network-based controllers were shared
between the old control software and the EPICS gateways. Gateways were
implemented with a portable channel access server (PCAS) at the beginning, and
were eventually replaced by soft IOCs as EPICS started to support Unix-based
IOCs [6].
The number of EPICS process variables in the system was approximately 300,000,
and that of hard-disk-archived ones was 150,000. They were distributed over
150 VME-based and Linux-based EPICS IOCs.
### 2.2 High-level application with scripting languages
At the linac, the Tcl/Tk scripting language was effectively employed for its
commissioning [7] after the language had been utilized for testing tools for
many years. Later, for both the ring and the linac, Python was employed as it
had more strong points [8]. Much of the device manipulation software was
written in these two languages, as well as MEDM.
For beam operation, SAD (strategic accelerator design program) was extended to
have an interpreter, SADscript, which emulated most of the Mathematica
language [9, 10]. This provided most of the functionalities required for the
accelerator operation, such as linear beam optics, symplectic beam tracking,
non-linear beam dynamics, optimization, list processing, numerical
manipulation, EPICS channel access, and a graphical user interface. Thus,
online linkage between the beam simulation and the machine operation was
provided through the SAD and EPICS environments.
In fact, many slow closed feedback loops were implemented using these script
languages in order to stabilize the beam characteristics and to maintain the
ultimate collision condition. Script languages are very suitable for rapid
prototyping. If a program needed higher performance, the algorithm was
eventually transported into a faster IOC process. Such feedback loops were
often effective in suppressing interim instability of hardware devices before
repair. They were also important during the beam studies because the beam
conditions were very different from the normal ones. While a certain parameter
was scanned in such studies, other parameters often had to be maintained
stable.
During normal operation it is necessary to measure the beam response on
certain parameter changes, and then to optimize those parameters. Such a
process can be interactively carried by SADscript, and then turned into a
graphical user interface that is performed routinely. New ideas for luminosity
optimization were often proposed in the morning meeting, and corresponding
operational tools were realized within a day or two. Some of the ideas turned
out to be favorable, and the tool became utilized routinely. As many equipment
and beam-physics ideas were proposed day by day, rapid tool development was
crucial and SADscript played a significant role. Actually, it is difficult to
name a single mechanism that enabled the KEKB’s high performance; however, the
accumulation of a hundred 1% improvements provided twice the performance.
Besides script-based application programs, many software packages were
developed, including a data archiver (KEKBlog), an alarm handler (KEKBalarm),
an electric operational logbook (Zlog), an archive browser, etc. The
combination of them refined the everyday KEKB operation.
## 3 Two additional concepts towards higher luminosity
As an upgrade to a control system needs considerable effort, it is preferable
to maintain the same environment during a project. However, an accelerator
project can span more than ten years, and related software, hardware, persons,
companies, and their policies may change substantially during that period.
Thus, it is necessary to introduce advanced technologies to improve the
machine even during the project. On the other hand, it is often difficult to
accommodate new technologies with existing components. If a component is
modified to accept them, others may have to be modified.
Actually, during the KEKB project, new operation schemes were introduced
almost every year. As most components in the control system were kept the
same, it was rather difficult to catch up with the requirements at a later
period. One modification might trigger another, and several modifications had
to be performed at the same time. Because the shutdown period in a year was
limited, if the extent of modifications exceeded the limit, it became very
hard.
Thus, we needed to be prepared to accommodate small upgrades each year not
only for application programs but also for control system infrastructure,
including base software and hardware components, in order to manage a project
for a long period.
Without changing the architecture of the control system, featured with EPICS
and scripting languages, new concepts were incorporated at later years in the
project. These included channel access everywhere and the dual-tier control
system, described below.
### 3.1 Channel access everywhere
The accelerator control architecture in KEK evolved in several steps. Some
time ago some control systems were standardized with a combination of several
fieldbuses, VME field computers, and Unix computers. In order to consolidate
the efforts on the development and maintenance, some of the fieldbuses were
gradually removed and many controllers were directly attached to IP networks.
At the same time, the EPICS control software framework was employed at control
systems at KEK. Eventually, many controllers evolved to embed the same EPICS
IOC software as on VME field computers, as illustrated in Fig. 1. Common
pieces of IOC software communicated with others using a common protocol called
channel access (CA), which realized a unified application development
environment from top to bottom. We call this embedded EPICS framework “CA
everywhere”, and it enabled both rapid development and smooth maintenance [6,
11].
Figure 1: Evolution of device controllers (from left to right) from fieldbus
devices towards CA everywhere with embedded IOC.
Different kinds of controllers have been developed in the framework of CA
everywhere, and they greatly reduced the control efforts and improved the
reliability. They include the following controllers. Many of them were
realized employing PLC [12], embedded computers [13], or FPGA (field
programmable gate array) [14].
* •
Temperature monitor with Linux
* •
Yokogawa FAM3 PLC with realtime Linux
* •
Tektronix oscilloscope for 50 Hz measurement with Windows
* •
Time-to-digital converter (TDC) with FPGA and Linux
* •
MicroTCA LLRF controller with FPGA and Linux
* •
Microwave power modulator with FPGA and Linux
* •
Libera BPM readout at 50 Hz with FPGA and realtime Linux
* •
NI Compact-RIO with CAS and FPGA
### 3.2 Dual-tier control system
For higher experimental performance at KEKB and the light sources that share
the same injector, it was favorable to inject beams in top-up mode into all
the storage rings. In the Photon Factory (PF), a stable beam current would
provide precise experimental results. In KEKB, stability was desired for
sensitive beam collision tuning to increase the luminosity.
To that end, simultaneous top-up injection had been established for three
storage rings at the KEKB HER, LER, and PF since 2009. Electron and positron
beams with very different characteristics, charged from 0.1 nC to 8 nC and
with energies between 2.5 GeV and 8 GeV, were exchanged at a rate of 50 Hz (20
ms). As a result, stored beam stabilities of 0.05% (1 mA) at the KEKB HER and
LER, and 0.01% (0.05 mA) at PF were achieved, improving the quality of
experiments.
While it initially took 30 seconds to 2 minutes to switch the beam injection
modes between these storage rings, it was preferable to change many parameters
synchronously within 20 ms. Thus, global and fast controls were established
for such pulse-to-pulse beam modulation. This kind of beam modulation was
realized a long time ago but the speed was much slower, around one to several
seconds [15]. As the existing control system, based on ten-year-old hardware
and conventional EPICS software, was inadequate for controlling the beam
within 20 ms, a new control system with an event notification mechanism,
capable of regulating $\sim$150 parameters at 50 Hz, was installed. This
event-based control system covered the controls of the low-level RF, high-
power RF, pulsed magnets, an electron gun, injection systems, and beam
instrumentation, whose devices were spread over a distance of 1 km. While the
event-based control system was supervised by the EPICS control software, it
had a dedicated communication link for fast, global, and robust controls [16].
The event generator sent timing signals and control data to 17 event receiver
stations arranged in a star-like topology as in Fig. 2. Each link between the
event generator and a receiver was carried over a single optical fiber. It
provided both synchronized timing signals, with approximately 10 ps precision,
and synchronized controls through a realtime software mechanism, at about 10
$\mu$s precision. Recent technological advances in FPGA and SFP (small form-
factor pluggable) enabled reliable controls in this configuration.
Figure 2: Overall configuration of the event-based control system at the
injector linac. 17 event receiver stations cover the 1 km facility.
VME-based event control modules of the generator (EVG230) and the receiver
(EVR230RF) from MRF were utilized [17]. The event generator provided several
events corresponding to beam and device controls synchronized to one of the
linac RF clock (114.24 MHz).
The same dual-tier control system with a conventional EPICS control and an
event-based control depicted in Fig. 3 will also be essential in the future
SuperKEKB. Simultaneous injection will be maintained, as the beam lifetime
will be more limited at the SuperKEKB HER and LER. Many more parameters will
have to be managed precisely in order to realize lower-emittance beams for
higher luminosity.
Figure 3: Dual-tier controls with EPICS channel access at the top and fast
event synchronized control at the bottom.
The event-based control tier manages global and fast controls in the pico- to
microsecond range. The EPICS control tier covers slower parameter controls for
the event-based controls as well as existing conventional controls. The EPICS
tier arbitrates operation requests of average beam repetition rates from the
three rings, and schedules different beams pulse by pulse.111Under typical
operating conditions, the average injection rates were 25 Hz for LER, 12.5 Hz
for HER and 0.5 Hz for PF. However, they were frequently changed to maintain
the stored beam current. Such requests occurred every several seconds, and
upon each request the beam mode schedule was reprogrammed at the event-control
tier through EPICS CA. Such a dual-tier control system is also optimal for the
next generation of accelerator systems.
## 4 Virtual accelerators
Under the simultaneous injection configuration, the event-based control system
provided beam-mode dependent control parameters. Moreover, these parameters in
different beam modes were organized to be independent both for controls and
measurements. Thus, we can see these independent parameter sets as independent
virtual accelerators. For each 20 ms time slot, the event system associated
one of the virtual accelerators with the real accelerator.
Because these control parameters for each virtual accelerator continue to
exist, human operators and operational software can act on one of these
virtual accelerators without any interference from the others.
### 4.1 Event-based beam feedback loops
BPM information and RF control parameters were also handled independently in
each virtual accelerator. At first, energy feedback loops at the 180 degree
arc and at the end of the linac were installed using event control parameters
on each virtual accelerator as in Fig. 4. As the parameters are independently
managed, no modifications to the existing software were necessary.
Figure 4: Independent closed feedback loops (F.B.) on three virtual
accelerators for KEKB-HER, KEKB-LER, and PF.
The performance of these closed loops was observed with small feedback gains
during normal operation. In these feedback operations no beam stability
improvements were achieved. In other words, no signs of instabilities were
observed other than white noise, since the hardware stability was much
improved in the later years of the project [18].
For energy stabilization at the PF beam-transport line, it turned out that the
separation of the betatron and dispersion functions was not optimal and the
resolution of BPMs was insufficient because of the low beam charges. The
procedures of betatron oscillation compensation and the weighed average of
beam positions were applied later. Because the processing speed with a
scripting language was not sufficient in a certain case, the EPICS EPID
(enhanced proportional, integral, and derivative controller) record mechanism
was employed as well.
The orbit and energy spread stabilizations can be implemented in the same way.
These beam feedback signals provided valuable information for the accelerator
operation.
We could insert a test beam pulse between injection beam pulses and associate
it with one of the virtual accelerators. Using such a virtual accelerator we
could perform a kind of beam study during normal operation. The same could be
applied to one of the stored beam bunches in the ring as well. This virtual
accelerator concept might play an important role in the future SuperKEKB
operation.
## 5 Conclusion
The accelerator control system in the KEKB project achieved successful
operation of the project, leading satisfactory physics results with the two
basic facilities of the EPICS software toolkit and scripting languages. Later,
two additional concepts of “channel access everywhere” and “dual-tier
controls” were introduced for further improvements. Based on such control
foundations the virtual accelerator mechanism was successfully tested; this
could be the basis of next-generation accelerators, including SuperKEKB.
## Acknowledgments
The authors are grateful to all the staff and operators at KEKB and the linac
for their suggestions and continuous encouragement during the project. They
also thank the developers from Mitsubishi Electric System & Service Co. Ltd.
and East Japan Institute of Technology Co. Ltd. for their system development.
## References
* [1] Y. Funakoshi et al., “Achievements of KEKB”, to be published in Prog. Theor. Exp. Phys.
* [2] K. Oide, “KEKB B-Factory, The Luminosity Frontier”, Prog. Theor. Phys. 122 (2009) p. 69.
* [3] N. Akasaka et al., “KEKB Accelerator Control System”, Nucl. Instrum. Meth. A 499 (2003) p. 138.
* [4] EPICS (Experimental physics and industrial control system, available at http://www.aps.anl.gov/epics/, last accessed December 2012).
* [5] K. Furukawa et al., “Integration Feasibility of the Existing Linac Control System and the Ring EPICS System at KEKB”, Proc. ICALEPCS1995, Chicago, USA. (1995) p. 863.
* [6] K. Furukawa et al., “Control System of the KEKB Accelerator Complex”, Proc. ICALEPCS2007, Knoxville, USA. (2007) p. 268.
* [7] K. Furukawa, N. Kamikubota, T. Suwada, and T. Urano, “Accelerator Controls in KEKB Linac Commissioning”, Proc. ICALEPCS1999, Trieste, Italy (1999) p. 98.
* [8] T.T. Nakamura, K. Furukawa, J. Odagiri, and N. Yamamoto, “Development of the Software Tools Using Python for EPICS-Based Control System”, Proc. ICALEPCS2007, Knoxville, USA. (2007) p. 120.
* [9] N. Akasaka, H. Koiso, and K. Oide, “Operation Software for Commissioning of KEKB Linac Programmed with SAD”, Proc. APAC1998, Tsukuba, Japan (1998) p. 495.
* [10] SAD: <http://acc-physics.kek.jp/SAD/>.
* [11] K. Furukawa, “Modern Accelerator Control Systems”, Proc. PAC2007, Albuquerque, USA. (2007) p. 873.
* [12] J. Odagiri et al., “Application of EPICS on F3RP61 to Accelerator Control”, Proc. ICALEPCS2009, Kobe, Japan (2009) p. 916.
* [13] M. Satoh et al., “EPICS IOC of WindowsXP-based Oscilloscope for Fast BPM Data Acquisition System”, ibid., p. 567.
* [14] K. Furukawa et al., “Embedded LLRF Controller with Channel Access on MicroTCA Backplane Interconnect”, Proc. ICALEPCS2011, Grenoble, France (2011) p. 1274.
* [15] J.P. Potier et al., “Pulse to Pulse Modulation of the CERN PS Complex”, Proc. PAC1977, Chicago, USA (1977) p.1518.
* [16] K. Furukawa et al., “New Event-based Control System for Simultaneous Top-up Operation at KEKB and PF”, Proc. ICALEPCS2009, Kobe, Japan (2009) p. 765.
* [17] Micro-Research Finland Oy: <http://www.mrf.fi/>.
* [18] K. Furukawa, M. Satoh, T. Suwada, and T.T. Nakamura, “Pulse-to-pulse Beam Modulation and Event-based Beam Feedback Systems at KEKB Linac”, Proc. IPAC2010, Kyoto, Japan (2010) p. 1271.
|
arxiv-papers
| 2012-11-25T15:19:22 |
2024-09-04T02:49:38.399332
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Kazuro Furukawa, Atsuyoshi Akiyama, Eiichi Kadokura, Miho Kurashina,\n Katsuhiko Mikawa, Tatsuro Nakamura, Jun-ichi Odagiri, Masanori Satoh,\n Tsuyoshi Suwada",
"submitter": "Kazuro Furukawa",
"url": "https://arxiv.org/abs/1211.5768"
}
|
1211.5994
|
EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN)
CERN-PH-EP-2012-340 LHCb-PAPER-2012-030 26 November 2012
First observation of the decay $B_{s2}^{*}(5840)^{0}\rightarrow B^{*+}K^{-}$
and studies of excited $B^{0}_{s}$ mesons
The LHCb collaboration†††Authors are listed on the following pages.
Properties of the orbitally excited ($L=1$) $B^{0}_{s}$ states are studied
using 1.0$\mbox{\,fb}^{-1}$ of $pp$ collisions at
$\sqrt{s}=7$$\mathrm{\,Te\kern-1.00006ptV}$ collected with the LHCb detector.
The first observation of the $B_{s2}^{*}(5840)^{0}$ meson decaying to
$B^{*+}K^{-}$ is reported, and the corresponding branching fraction measured
relative to the $B^{+}K^{-}$ decay mode. The $B_{s1}(5830)^{0}\rightarrow
B^{*+}K^{-}$ decay is observed as well. The width of the
$B_{s2}^{*}(5840)^{0}$ state is measured for the first time and the masses of
the two states are determined with the highest precision to date. The
observation of the $B_{s2}^{*}(5840)^{0}\rightarrow B^{*+}K^{-}$ decay favours
the spin-parity assignment $J^{P}=2^{+}$ for the $B_{s2}^{*}(5840)^{0}$ meson.
In addition, the most precise measurement of the mass difference
$m(B^{*+})-m(B^{+})=45.01\pm 0.30\,({\rm stat})\pm 0.23\,({\rm
syst}){\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ is obtained.
Submitted to Phys. Rev. Lett.
LHCb collaboration
R. Aaij38, C. Abellan Beteta33,n, A. Adametz11, B. Adeva34, M. Adinolfi43, C.
Adrover6, A. Affolder49, Z. Ajaltouni5, J. Albrecht35, F. Alessio35, M.
Alexander48, S. Ali38, G. Alkhazov27, P. Alvarez Cartelle34, A.A. Alves
Jr22,35, S. Amato2, Y. Amhis36, L. Anderlini17,f, J. Anderson37, R.B.
Appleby51, O. Aquines Gutierrez10, F. Archilli18, A. Artamonov32, M. Artuso53,
E. Aslanides6, G. Auriemma22,m, S. Bachmann11, J.J. Back45, C. Baesso54, V.
Balagura28, W. Baldini16, R.J. Barlow51, C. Barschel35, S. Barsuk7, W.
Barter44, A. Bates48, Th. Bauer38, A. Bay36, J. Beddow48, I. Bediaga1, S.
Belogurov28, K. Belous32, I. Belyaev28, E. Ben-Haim8, M. Benayoun8, G.
Bencivenni18, S. Benson47, J. Benton43, A. Berezhnoy29, R. Bernet37, M.-O.
Bettler44, M. van Beuzekom38, A. Bien11, S. Bifani12, T. Bird51, A.
Bizzeti17,h, P.M. Bjørnstad51, T. Blake35, F. Blanc36, C. Blanks50, J.
Blouw11, S. Blusk53, A. Bobrov31, V. Bocci22, A. Bondar31, N. Bondar27, W.
Bonivento15, S. Borghi51, A. Borgia53, T.J.V. Bowcock49, C. Bozzi16, T.
Brambach9, J. van den Brand39, J. Bressieux36, D. Brett51, M. Britsch10, T.
Britton53, N.H. Brook43, H. Brown49, A. Büchler-Germann37, I. Burducea26, A.
Bursche37, J. Buytaert35, S. Cadeddu15, O. Callot7, M. Calvi20,j, M. Calvo
Gomez33,n, A. Camboni33, P. Campana18,35, A. Carbone14,c, G. Carboni21,k, R.
Cardinale19,i, A. Cardini15, H. Carranza-Mejia47, L. Carson50, K. Carvalho
Akiba2, G. Casse49, M. Cattaneo35, Ch. Cauet9, M. Charles52, Ph.
Charpentier35, P. Chen3,36, N. Chiapolini37, M. Chrzaszcz23, K. Ciba35, X. Cid
Vidal34, G. Ciezarek50, P.E.L. Clarke47, M. Clemencic35, H.V. Cliff44, J.
Closier35, C. Coca26, V. Coco38, J. Cogan6, E. Cogneras5, P. Collins35, A.
Comerma-Montells33, A. Contu15, A. Cook43, M. Coombes43, G. Corti35, B.
Couturier35, G.A. Cowan36, D.C. Craik45, S. Cunliffe50, R. Currie47, C.
D’Ambrosio35, P. David8, P.N.Y. David38, I. De Bonis4, K. De Bruyn38, S. De
Capua51, M. De Cian37, J.M. De Miranda1, L. De Paula2, P. De Simone18, D.
Decamp4, M. Deckenhoff9, H. Degaudenzi36,35, L. Del Buono8, C. Deplano15, D.
Derkach14, O. Deschamps5, F. Dettori39, A. Di Canto11, J. Dickens44, H.
Dijkstra35, P. Diniz Batista1, M. Dogaru26, F. Domingo Bonal33,n, S.
Donleavy49, F. Dordei11, A. Dosil Suárez34, D. Dossett45, A. Dovbnya40, F.
Dupertuis36, R. Dzhelyadin32, A. Dziurda23, A. Dzyuba27, S. Easo46,35, U.
Egede50, V. Egorychev28, S. Eidelman31, D. van Eijk38, S. Eisenhardt47, U.
Eitschberger9, R. Ekelhof9, L. Eklund48,35, I. El Rifai5, Ch. Elsasser37, D.
Elsby42, A. Falabella14,e, C. Färber11, G. Fardell47, C. Farinelli38, S.
Farry12, V. Fave36, D. Ferguson47, V. Fernandez Albor34, F. Ferreira
Rodrigues1, M. Ferro-Luzzi35, S. Filippov30, M. Fiore16, C. Fitzpatrick35, M.
Fontana10, F. Fontanelli19,i, R. Forty35, O. Francisco2, M. Frank35, C.
Frei35, M. Frosini17,f, S. Furcas20, A. Gallas Torreira34, D. Galli14,c, M.
Gandelman2, P. Gandini52, Y. Gao3, J-C. Garnier35, J. Garofoli53, P. Garosi51,
J. Garra Tico44, L. Garrido33, C. Gaspar35, R. Gauld52, E. Gersabeck11, M.
Gersabeck35, T. Gershon45,35, Ph. Ghez4, V. Gibson44, V.V. Gligorov35, C.
Göbel54, D. Golubkov28, A. Golutvin50,28,35, A. Gomes2, H. Gordon52, M.
Grabalosa Gándara5, R. Graciani Diaz33, L.A. Granado Cardoso35, E. Graugés33,
G. Graziani17, A. Grecu26, E. Greening52, S. Gregson44, O. Grünberg55, B.
Gui53, E. Gushchin30, Yu. Guz32,35, T. Gys35, C. Hadjivasiliou53, G.
Haefeli36, C. Haen35, S.C. Haines44, S. Hall50, T. Hampson43, S. Hansmann-
Menzemer11, N. Harnew52, S.T. Harnew43, J. Harrison51, P.F. Harrison45, T.
Hartmann55, J. He7, V. Heijne38, K. Hennessy49, P. Henrard5, J.A. Hernando
Morata34, E. van Herwijnen35, E. Hicks49, D. Hill52, M. Hoballah5, P.
Hopchev4, W. Hulsbergen38, P. Hunt52, T. Huse49, N. Hussain52, D.
Hutchcroft49, D. Hynds48, V. Iakovenko41, P. Ilten12, J. Imong43, R.
Jacobsson35, A. Jaeger11, M. Jahjah Hussein5, E. Jans38, F. Jansen38, P.
Jaton36, B. Jean-Marie7, F. Jing3, M. John52, D. Johnson52, C.R. Jones44, B.
Jost35, M. Kaballo9, S. Kandybei40, M. Karacson35, T.M. Karbach35, I.R.
Kenyon42, U. Kerzel35, T. Ketel39, A. Keune36, B. Khanji20, Y.M. Kim47, O.
Kochebina7, I. Komarov36, R.F. Koopman39, P. Koppenburg38, M. Korolev29, A.
Kozlinskiy38, L. Kravchuk30, K. Kreplin11, M. Kreps45, G. Krocker11, P.
Krokovny31, F. Kruse9, M. Kucharczyk20,23,j, V. Kudryavtsev31, T.
Kvaratskheliya28,35, V.N. La Thi36, D. Lacarrere35, G. Lafferty51, A. Lai15,
D. Lambert47, R.W. Lambert39, E. Lanciotti35, G. Lanfranchi18,35, C.
Langenbruch35, T. Latham45, C. Lazzeroni42, R. Le Gac6, J. van Leerdam38,
J.-P. Lees4, R. Lefèvre5, A. Leflat29, J. Lefrançois7, O. Leroy6, Y. Li3, L.
Li Gioi5, M. Liles49, R. Lindner35, C. Linn11, B. Liu3, G. Liu35, J. von
Loeben20, J.H. Lopes2, E. Lopez Asamar33, N. Lopez-March36, H. Lu3, J.
Luisier36, H. Luo47, A. Mac Raighne48, F. Machefert7, I.V. Machikhiliyan4,28,
F. Maciuc26, O. Maev27,35, S. Malde52, G. Manca15,d, G. Mancinelli6, N.
Mangiafave44, U. Marconi14, R. Märki36, J. Marks11, G. Martellotti22, A.
Martens8, L. Martin52, A. Martín Sánchez7, M. Martinelli38, D. Martinez
Santos39, D. Martins Tostes2, A. Massafferri1, R. Matev35, Z. Mathe35, C.
Matteuzzi20, M. Matveev27, E. Maurice6, A. Mazurov16,30,35,e, J. McCarthy42,
G. McGregor51, R. McNulty12, F. Meier9, M. Meissner11, M. Merk38, J. Merkel9,
D.A. Milanes13, M.-N. Minard4, J. Molina Rodriguez54, S. Monteil5, D. Moran51,
P. Morawski23, R. Mountain53, I. Mous38, F. Muheim47, K. Müller37, R.
Muresan26, B. Muryn24, B. Muster36, J. Mylroie-Smith49, P. Naik43, T.
Nakada36, R. Nandakumar46, I. Nasteva1, M. Needham47, N. Neufeld35, A.D.
Nguyen36, T.D. Nguyen36, C. Nguyen-Mau36,o, M. Nicol7, V. Niess5, R. Niet9, N.
Nikitin29, T. Nikodem11, A. Nomerotski52, A. Novoselov32, A. Oblakowska-
Mucha24, V. Obraztsov32, S. Oggero38, S. Ogilvy48, O. Okhrimenko41, R.
Oldeman15,d, M. Orlandea26, J.M. Otalora Goicochea2, P. Owen50, B.K. Pal53, A.
Palano13,b, M. Palutan18, J. Panman35, A. Papanestis46, M. Pappagallo48, C.
Parkes51, C.J. Parkinson50, G. Passaleva17, G.D. Patel49, M. Patel50, G.N.
Patrick46, C. Patrignani19,i, C. Pavel-Nicorescu26, A. Pazos Alvarez34, A.
Pellegrino38, G. Penso22,l, M. Pepe Altarelli35, S. Perazzini14,c, D.L.
Perego20,j, E. Perez Trigo34, A. Pérez-Calero Yzquierdo33, P. Perret5, M.
Perrin-Terrin6, G. Pessina20, K. Petridis50, A. Petrolini19,i, A. Phan53, E.
Picatoste Olloqui33, B. Pie Valls33, B. Pietrzyk4, T. Pilař45, D. Pinci22, S.
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Raven39, S. Redford52, M.M. Reid45, A.C. dos Reis1, S. Ricciardi46, A.
Richards50, K. Rinnert49, V. Rives Molina33, D.A. Roa Romero5, P. Robbe7, E.
Rodrigues51, P. Rodriguez Perez34, G.J. Rogers44, S. Roiser35, V.
Romanovsky32, A. Romero Vidal34, J. Rouvinet36, T. Ruf35, H. Ruiz33, G.
Sabatino22,k, J.J. Saborido Silva34, N. Sagidova27, P. Sail48, B. Saitta15,d,
C. Salzmann37, B. Sanmartin Sedes34, M. Sannino19,i, R. Santacesaria22, C.
Santamarina Rios34, R. Santinelli35, E. Santovetti21,k, M. Sapunov6, A.
Sarti18,l, C. Satriano22,m, A. Satta21, M. Savrie16,e, D. Savrina28,29, P.
Schaack50, M. Schiller39, H. Schindler35, S. Schleich9, M. Schlupp9, M.
Schmelling10, B. Schmidt35, O. Schneider36, A. Schopper35, M.-H. Schune7, R.
Schwemmer35, B. Sciascia18, A. Sciubba18,l, M. Seco34, A. Semennikov28, K.
Senderowska24, I. Sepp50, N. Serra37, J. Serrano6, P. Seyfert11, M. Shapkin32,
I. Shapoval35,40, P. Shatalov28, Y. Shcheglov27, T. Shears49,35, L.
Shekhtman31, O. Shevchenko40, V. Shevchenko28, A. Shires50, R. Silva
Coutinho45, T. Skwarnicki53, N.A. Smith49, E. Smith52,46, M. Smith51, K.
Sobczak5, F.J.P. Soler48, F. Soomro18, D. Souza43, B. Souza De Paula2, B.
Spaan9, A. Sparkes47, P. Spradlin48, F. Stagni35, S. Stahl11, O. Steinkamp37,
S. Stoica26, S. Stone53, B. Storaci37, M. Straticiuc26, U. Straumann37, V.K.
Subbiah35, S. Swientek9, V. Syropoulos39, M. Szczekowski25, P. Szczypka36,35,
T. Szumlak24, S. T’Jampens4, M. Teklishyn7, E. Teodorescu26, F. Teubert35, C.
Thomas52, E. Thomas35, J. van Tilburg11, V. Tisserand4, M. Tobin37, S. Tolk39,
D. Tonelli35, S. Topp-Joergensen52, N. Torr52, E. Tournefier4,50, S.
Tourneur36, M.T. Tran36, M. Tresch37, A. Tsaregorodtsev6, P. Tsopelas38, N.
Tuning38, M. Ubeda Garcia35, A. Ukleja25, D. Urner51, U. Uwer11, V. Vagnoni14,
G. Valenti14, R. Vazquez Gomez33, P. Vazquez Regueiro34, S. Vecchi16, J.J.
Velthuis43, M. Veltri17,g, G. Veneziano36, M. Vesterinen35, B. Viaud7, I.
Videau7, D. Vieira2, X. Vilasis-Cardona33,n, J. Visniakov34, A. Vollhardt37,
D. Volyanskyy10, D. Voong43, A. Vorobyev27, V. Vorobyev31, C. Voß55, H.
Voss10, R. Waldi55, R. Wallace12, S. Wandernoth11, J. Wang53, D.R. Ward44,
N.K. Watson42, A.D. Webber51, D. Websdale50, M. Whitehead45, J. Wicht35, D.
Wiedner11, L. Wiggers38, G. Wilkinson52, M.P. Williams45,46, M. Williams50,p,
F.F. Wilson46, J. Wishahi9, M. Witek23, W. Witzeling35, S.A. Wotton44, S.
Wright44, S. Wu3, K. Wyllie35, Y. Xie47,35, F. Xing52, Z. Xing53, Z. Yang3, R.
Young47, X. Yuan3, O. Yushchenko32, M. Zangoli14, M. Zavertyaev10,a, F.
Zhang3, L. Zhang53, W.C. Zhang12, Y. Zhang3, A. Zhelezov11, A. Zhokhov28, L.
Zhong3, A. Zvyagin35.
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 Roma Tor Vergata, Roma, Italy
22Sezione INFN di Roma La Sapienza, Roma, Italy
23Henryk Niewodniczanski Institute of Nuclear Physics Polish Academy of
Sciences, Kraków, Poland
24AGH - University of Science and Technology, Faculty of Physics and Applied
Computer Science, Kraków, Poland
25National Center for Nuclear Research (NCBJ), Warsaw, Poland
26Horia Hulubei National Institute of Physics and Nuclear Engineering,
Bucharest-Magurele, Romania
27Petersburg Nuclear Physics Institute (PNPI), Gatchina, Russia
28Institute of Theoretical and Experimental Physics (ITEP), Moscow, Russia
29Institute of Nuclear Physics, Moscow State University (SINP MSU), Moscow,
Russia
30Institute for Nuclear Research of the Russian Academy of Sciences (INR RAN),
Moscow, Russia
31Budker Institute of Nuclear Physics (SB RAS) and Novosibirsk State
University, Novosibirsk, Russia
32Institute for High Energy Physics (IHEP), Protvino, Russia
33Universitat de Barcelona, Barcelona, Spain
34Universidad de Santiago de Compostela, Santiago de Compostela, Spain
35European Organization for Nuclear Research (CERN), Geneva, Switzerland
36Ecole Polytechnique Fédérale de Lausanne (EPFL), Lausanne, Switzerland
37Physik-Institut, Universität Zürich, Zürich, Switzerland
38Nikhef National Institute for Subatomic Physics, Amsterdam, The Netherlands
39Nikhef National Institute for Subatomic Physics and VU University Amsterdam,
Amsterdam, The Netherlands
40NSC Kharkiv Institute of Physics and Technology (NSC KIPT), Kharkiv, Ukraine
41Institute for Nuclear Research of the National Academy of Sciences (KINR),
Kyiv, Ukraine
42University of Birmingham, Birmingham, United Kingdom
43H.H. Wills Physics Laboratory, University of Bristol, Bristol, United
Kingdom
44Cavendish Laboratory, University of Cambridge, Cambridge, United Kingdom
45Department of Physics, University of Warwick, Coventry, United Kingdom
46STFC Rutherford Appleton Laboratory, Didcot, United Kingdom
47School of Physics and Astronomy, University of Edinburgh, Edinburgh, United
Kingdom
48School of Physics and Astronomy, University of Glasgow, Glasgow, United
Kingdom
49Oliver Lodge Laboratory, University of Liverpool, Liverpool, United Kingdom
50Imperial College London, London, United Kingdom
51School of Physics and Astronomy, University of Manchester, Manchester,
United Kingdom
52Department of Physics, University of Oxford, Oxford, United Kingdom
53Syracuse University, Syracuse, NY, United States
54Pontifícia Universidade Católica do Rio de Janeiro (PUC-Rio), Rio de
Janeiro, Brazil, associated to 2
55Institut für Physik, Universität Rostock, Rostock, Germany, associated to 11
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
pMassachusetts Institute of Technology, Cambridge, MA, United States
Heavy quark effective theory (HQET) describes mesons with one heavy and one
light quark where the heavy quark is assumed to have infinite mass [1]. It is
an important tool for calculating meson properties which may be modified by
physics beyond the Standard Model, such as $C\\!P$ violation in charm meson
decays [2] or the mixing and lifetimes of $B$ mesons [3]. It also predicts the
properties of excited $B$ and $B^{0}_{s}$ mesons [4, 5, 6, 7], and precise
measurements of these properties are a sensitive test of the validity of the
theory. Within HQET the $B^{0}_{s}$ mesons are characterised by three quantum
numbers: the relative orbital angular momentum $L$ of the two quarks, the
total angular momentum of the light quark $j_{q}=|L\pm\frac{1}{2}|$, and the
total angular momentum of the $B^{0}_{s}$ meson $J=|j_{q}\pm\frac{1}{2}|$. For
$L=1$ there are four different possible ($J$, $j_{q}$) combinations, all with
even parity. These are collectively termed the orbitally excited states. Such
states can decay to $B^{+}K^{-}$ and/or $B^{*+}K^{-}$ (the inclusion of
charge-conjugate states is implied throughout this Letter), depending on their
quantum numbers and mass values. The two states with $j_{q}=1/2$, named
$B_{s0}^{*}$ and $B_{s1}^{\prime}$, are expected to decay through an S-wave
transition and to have a large
$\mathcal{O}(100{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}})$ decay width. In
contrast, the two states with $j_{q}=3/2$, named $B_{s1}(5830)^{0}$ and
$B_{s2}^{*}(5840)^{0}$ (henceforth $B_{s1}$ and $B^{*}_{s2}$ for brevity), are
expected to decay through a D-wave transition and to have a narrow
$\mathcal{O}(1{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}})$ decay width. Table 1
gives an overview of these states.
Table 1: Summary of the orbitally excited ($L=1$) $B^{0}_{s}$ states. | $j_{q}$ | $J^{P}$ | Allowed decay mode | Mass (${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$) [8]
---|---|---|---|---
| | | $B^{+}K^{-}$ | $B^{*+}K^{-}$ |
$B_{s0}^{*}$ | 1/2 | $0^{+}$ | yes | no | Unobserved
$B_{s1}^{\prime}$ | 1/2 | $1^{+}$ | no | yes | Unobserved
$B_{s1}$ | 3/2 | $1^{+}$ | no | yes | $5829.4\pm 0.7$
$B^{*}_{s2}$ | 3/2 | $2^{+}$ | yes | yes | $5839.7\pm 0.6$
In this Letter a $1.0$ fb-1 sample of data collected by the LHCb detector is
used to search for the orbitally excited $B^{0}_{s}$ mesons in the mass
distribution of $B^{+}K^{-}$ pairs, where the $B^{+}$ mesons are selected in
the four decay modes: $B^{+}\rightarrow J/\psi(\mu^{+}\mu^{-})K^{+}$,
$B^{+}\rightarrow\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}(K^{+}\pi^{-})\pi^{+}$,
$B^{+}\rightarrow\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}(K^{+}\pi^{-}\pi^{+}\pi^{-})\pi^{+}$,
and $B^{+}\rightarrow\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}(K^{+}\pi^{-})\pi^{+}\pi^{-}\pi^{+}$.
Two narrow peaks were observed in the $B^{+}K^{-}$ mass distribution by the
CDF collaboration [9]. Putatively, they are identified with the states of the
$j_{q}=3/2$ doublet expected in HQET[4] and are named $B_{s1}$ and
$B^{*}_{s2}$. As the $B_{s1}\rightarrow B^{+}K^{-}$ decay is forbidden, one of
the mass peaks observed is interpreted as the $B_{s1}\rightarrow B^{*+}K^{-}$
decay followed by $B^{*+}\rightarrow B^{+}\gamma$, where the photon is not
observed. This peak is shifted by the $B^{*+}-B^{+}$ mass difference due to
the missing momentum of the photon in the $B^{*+}\rightarrow B^{+}\gamma$
decay. While the $B^{*}_{s2}\rightarrow B^{+}K^{-}$ decay has been observed by
the D0 collaboration as well [10], a confirmation of the $B_{s1}$ meson is
still missing. The identification of the $B_{s1}$ and $B^{*}_{s2}$ mesons in
the $B^{+}K^{-}$ mass spectrum is based on the expected mass splitting between
the $j_{q}=3/2$ states. The $B_{s1}$ and $B^{*}_{s2}$ widths are very
sensitive to their masses, due to their proximity to the $BK$ and $B^{*}K$
thresholds. Measurements of the widths thus provide fundamental information
concerning the nature of these states. In addition the $B_{s1}$ and
$B^{*}_{s2}$ quantum numbers have not yet been directly determined and the
observation of other decay modes can constrain the spin-parity combinations of
the states. In particular the $B^{*}_{s2}\rightarrow B^{*+}K^{-}$ decay has
not yet been observed but could manifest itself in the $B^{+}K^{-}$ mass
spectrum in a similar fashion to the corresponding $B_{s1}$ meson decay. The
$B^{*}_{s2}\rightarrow B^{*+}K^{-}$ branching fraction relative to
$B^{*}_{s2}\rightarrow B^{+}K^{-}$ is predicted to be between 2$\%$ and
10$\%$, depending on the $B^{*}_{s2}$ mass [11, 12, 13, 14].
Recently the Belle collaboration has reported observation of charged
bottomonium-like $Z_{b}(10610)^{+}$ and $Z_{b}(10650)^{+}$ states [15, 16],
that could be interpreted as $B\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{*}$ and $B^{*}\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{*}$ molecules respectively [17]. To
test this interpretation, improved measurements of the $B^{*+}$ mass are
necessary, and can be obtained from the difference in peak positions between
$B^{*}_{s2}\rightarrow B^{*+}K^{-}$ and $B^{*}_{s2}\rightarrow B^{+}K^{-}$
decays in the $B^{+}K^{-}$ mass spectrum.
The LHCb detector [18] is a single-arm forward spectrometer covering the
pseudo-rapidity range $2<\eta<5$, designed for studying 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 has a momentum resolution ($\Delta p/p$), 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 a decay time resolution of
50$\rm\,fs$. The resolution of the impact parameter, the transverse distance
of closest approach between the track and a primary interaction, is about
20$\,\upmu\rm m$ for tracks with large transverse momentum. The transverse
component is measured in the plane normal to the beam axis. Charged hadrons
are identified using two ring-imaging Cherenkov 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 trigger system [19] consists of a hardware stage, based on information
from the calorimeter and muon systems, followed by a software stage that
applies a full event reconstruction. Events likely to contain a $B$ meson are
selected by searching for a dimuon vertex detached from the primary
interaction or two-, three-, and four-track vertices detached from the primary
interaction which have high total transverse momentum. These are respectively
referred to as dimuon and topological triggers.
The samples of simulated events used in this analysis are based on the Pythia
6.4 generator [20], with a choice of parameters specifically configured for
LHCb [21]. The EvtGen package [22] describes the decay of the $B$ mesons, and
the Geant4 toolkit [23, *Agostinelli:2002hh, 25] is used to simulate the
detector response. QED radiative corrections are generated with the Photos
package [26].
Figure 1: Invariant mass spectra of the final $B^{+}$ candidates. The signal
lineshape is fitted with a double Gaussian distribution, while the background
is modelled with a second order polynomial . (a) $B^{+}\rightarrow J/\psi
K^{+}$, (b) $B^{+}\rightarrow\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}(K^{+}\pi^{-})\pi^{+}$, (c)
$B^{+}\rightarrow\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}(K^{+}\pi^{-}\pi^{+}\pi^{-})\pi^{+}$,
and (d) $B^{+}\rightarrow\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}(K^{+}\pi^{-})\pi^{+}\pi^{-}\pi^{+}$
decays. The ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ and $D^{0}$ masses
are constrained to their world average values.
In the offline analysis the $B$ mesons are reconstructed using a set of loose
selection criteria to suppress the majority of the combinatorial backgrounds.
The $B^{+}\rightarrow J/\psi K^{+}$ selection requires a $B^{+}$ candidate
with a transverse momentum of at least 2${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$
and a decay time of at least 0.3${\rm\,ps}$. For the other decay modes, the
selection explicitly requires that the topological trigger, which selected the
event, is based exclusively on tracks from which the $B$ meson candidate is
formed. Additional loose selection requirements are placed on variables
related to the $B$ meson production and decay such as transverse momentum and
quality of the track fits for the decay products, detachment of the $B^{+}$
candidate from the primary interaction, whether the momentum of the $B^{+}$
candidate points back to the primary interaction, and the impact parameter
$\chi^{2}$. The impact parameter $\chi^{2}$ is defined as the difference
between the $\chi^{2}$ of the primary vertex reconstructed with and without
the considered track.
Following these selections, $B^{+}$ signals are visible above backgrounds in
all four decay modes. In order to improve their purity, four boosted decision
tree classifiers [27, *Roe] are trained on variables common to all four decay
modes: the transverse momenta and impact parameters of the final state tracks,
the transverse momentum and impact parameter of the $B^{+}$ candidate, the
detachment of the $B^{+}$ candidate from the primary interaction, the cosine
of the angle between the $B^{+}$ candidate momentum and the direction of
flight from the primary vertex to the decay vertex, the fit $\chi^{2}$ of the
tracks, and particle identification information. The classifier is trained on
data using the sWeights technique [29], with the $B^{+}$ candidate mass as a
discriminating variable, to unfold the signal and background distributions.
The cut on the classifier response is chosen by optimizing the significance of
each $B^{+}$ signal. The final mass distributions for the $B^{+}$ candidates
are shown in Fig. 1.
The $B^{+}$ candidate mass spectra are fitted using a double Gaussian function
for the signal and a second order polynomial for the background. The average
mass resolution, $\sigma_{B^{+}}$, is defined as the weighted average of the
Gaussian widths. The purities of the samples, defined as the fraction of the
signal events in a $\pm 2\sigma_{B^{+}}$ mass region, are $96\%$, $91\%$,
$90\%$, and $85\%$ for the $B^{+}\rightarrow J/\psi K^{+}$,
$B^{+}\rightarrow\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}(K^{+}\pi^{-})\pi^{+}$,
$B^{+}\rightarrow\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}(K^{+}\pi^{-}\pi^{+}\pi^{-})\pi^{+}$,
and $B^{+}\rightarrow\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}(K^{+}\pi^{-})\pi^{+}\pi^{-}\pi^{+}$
decays respectively. The $B^{+}$ candidates, within a $\pm 2\sigma_{B^{+}}$
mass region, are selected for each decay mode. A sample of about $1\,000\,000$
$B^{+}$ candidates is obtained and combined with any track of opposite charge
that is identified as a kaon.
Multiple $pp$ interactions can occur in LHC bunch crossings. In order to
reduce combinatorial backgrounds, the $B^{+}$ and kaon candidates are required
to be consistent with coming from the same interaction point. The signal
purity is improved by a boosted decision tree classifier, whose inputs are the
$B^{+}$ and the kaon transverse momenta, the log-likelihood difference between
the kaon and pion hypotheses, and the vertex fit and impact parameter
$\chi^{2}$. The training is performed using simulated events for the signal
and the like-charge $B^{+}K^{+}$ candidates in the data for the background.
The same selection is subsequently applied to all $B^{+}$ decay modes. The cut
on the classifier response is chosen by optimizing the significance of the
$B^{*}_{s2}\rightarrow B^{+}K^{-}$ signal. It retains 57% of the signal events
and rejects 92% of the background events. In order to improve the mass
resolution, the $B^{+}K^{-}$ mass fits are performed constraining the $J/\psi$
(or $D^{0}$) and $B^{+}$ particles to their respective world average masses
[8] and constraining the $B^{+}$ and $K^{-}$ momenta to point to the
associated primary vertex.
Figure 2 shows the mass difference for the selected candidates, summed over
all $B^{+}$ decay modes. The mass difference is defined as $Q\equiv
m(B^{+}K^{-})-m(B^{+})-m(K^{-})$ where $m(B^{+})$ and $m(K^{-})$ are the known
masses of the $B^{+}$ and $K^{-}$ mesons [8], respectively. The two narrow
peaks at 10 and $67{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ are identified as
the $B_{s1}\rightarrow B^{*+}K^{-}$ and $B^{*}_{s2}\rightarrow B^{+}K^{-}$
signals, respectively, as previously observed. In addition, a smaller
structure is seen around $20{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$,
identified as the previously unobserved $B^{*}_{s2}\rightarrow B^{*+}K^{-}$
decay mode.
Simulated events are used to compute the detector resolutions corresponding to
the three signals. The values obtained are increased by 20$\%$ to account for
differences between the $B^{+}$ resolutions in data and simulated events. The
corrected resolutions are 0.4 ${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$, 0.6
${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ and 1.0
${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ for the $B_{s1}\rightarrow
B^{*+}K^{-}$, $B^{*}_{s2}\rightarrow B^{*+}K^{-}$, and $B^{*}_{s2}\rightarrow
B^{+}K^{-}$ signals respectively. A discrepancy of 40% between the mass
resolutions in data and simulated events is observed for decays with small Q
values, such as $D^{*+}\rightarrow D^{0}\pi^{+}$. Therefore we assign an
uncertainty of $\pm 20\%$ to the resolution in the systematic studies.
Figure 2: Mass difference distribution $m(B^{+}K^{-})-m(B^{+})-m(K^{-})$. The
three peaks are identified as (left) $B_{s1}\rightarrow B^{*+}K^{-}$, (middle)
$B^{*}_{s2}\rightarrow B^{*+}K^{-}$, and (right) $B^{*}_{s2}\rightarrow
B^{+}K^{-}$. The total fit function is shown as a solid blue line, while the
shaded red region is the spectrum of like-charge $B^{+}K^{+}$ combinations.
The inset shows an expanded view of the $B_{s1}/B^{*}_{s2}\rightarrow
B^{*+}K^{-}$ signals. The bottom plot shows the fit pulls.
An unbinned fit of the mass difference distribution is performed to extract
the $Q$ values and event yields of the three peaks. The $B^{*}_{s2}\rightarrow
B^{+}K^{-}$ signal is parameterized by a relativistic Breit-Wigner function
with natural width $\Gamma$ convolved with a Gaussian function that accounts
for the detector resolution. Its width is fixed to the value obtained from
simulated events. The lineshapes of the $B_{s1}/B^{*}_{s2}\rightarrow
B^{*+}K^{-}$ signals, expected to be Breit-Wigner functions in the
$B^{*+}K^{-}$ mass spectrum, are affected by the phase space and the angular
distribution of the decays as the photon is not reconstructed. The resulting
shapes can not be properly simulated due to the lack of knowledge of the
$B_{s1}/B^{*}_{s2}$ properties. Therefore a Gaussian function is used for each
$B_{s1}/B^{*}_{s2}\rightarrow B^{*+}K^{-}$ signals as effective
parameterization. The background is modelled by a threshold function,
$f(Q)=Q^{\alpha}e^{\beta Q+\delta}$, where $\alpha$, $\beta$ and $\delta$ are
free parameters in the fit. Its analytical form is verified by fitting the
like charge $B^{+}K^{+}$ combinations where no signal is expected.
The parameters allowed to vary in the fit are: the yield
$N_{B^{*}_{s2}\rightarrow B^{+}K^{-}}$, the yield ratios $N_{B_{s1}\rightarrow
B^{*+}K^{-}}/N_{B^{*}_{s2}\rightarrow B^{+}K^{-}}$ and
$N_{B^{*}_{s2}\rightarrow B^{*+}K^{-}}/N_{B^{*}_{s2}\rightarrow B^{+}K^{-}}$,
the $Q$ values of the $B_{s1}\rightarrow B^{*+}K^{-}$ and
$B^{*}_{s2}\rightarrow B^{+}K^{-}$ signals, the mass difference between the
$B^{*}_{s2}\rightarrow B^{+}K^{-}$ and $B^{*}_{s2}\rightarrow B^{*+}K^{-}$
peaks, the natural width of the $B^{*}_{s2}$ state, the Gaussian widths of
$B_{s1}/B^{*}_{s2}\rightarrow B^{*+}K^{-}$ signals and the parameters of the
threshold function. From the yield ratios, the relative branching fraction
$\frac{\mathcal{B}(B^{*}_{s2}\rightarrow
B^{*+}K^{-})}{\mathcal{B}(B^{*}_{s2}\rightarrow
B^{+}K^{-})}=\frac{N_{B^{*}_{s2}\rightarrow
B^{*+}K^{-}}}{N_{B^{*}_{s2}\rightarrow B^{+}K^{-}}}\times\epsilon^{\rm
rel}_{2,2}=R^{B^{*}_{s2}}$ (1)
is measured. The $B_{s1}$ to $B^{*}_{s2}$ ratio of production cross-sections
times the ratio of branching fractions of $B_{s1}\rightarrow B^{*+}K^{-}$
relative to that of $B^{*}_{s2}\rightarrow B^{+}K^{-}$ is also determined from
$\frac{\sigma(pp\rightarrow B_{s1}X)\mathcal{B}(B_{s1}\rightarrow
B^{*+}K^{-})}{\sigma(pp\rightarrow
B^{*}_{s2}X)\mathcal{B}(B^{*}_{s2}\rightarrow
B^{+}K^{-})}=\frac{N_{B_{s1}\rightarrow B^{*+}K^{-}}}{N_{B^{*}_{s2}\rightarrow
B^{+}K^{-}}}\times\epsilon^{\rm
rel}_{1,2}=\sigma^{B_{s1}/B^{*}_{s2}}R^{B_{s1}/B^{*}_{s2}}$ (2)
These ratios are corrected by the relative selection efficiencies,
$\epsilon^{\rm rel}_{2,2}=1.05\pm 0.02$ and $\epsilon^{\rm rel}_{1,2}=1.03\pm
0.01$, using simulated decays. The fit results are given in Table 2. The
widths of the two Gaussian functions are $0.73\pm
0.04{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ and $1.9\pm
0.3{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ for the $B_{s1}\rightarrow
B^{*+}K^{-}$ and $B^{*}_{s2}\rightarrow B^{*+}K^{-}$ signals respectively. A
binned $\chi^{2}$ test gives a confidence level of $43\%$ for the fit.
Table 2: Results of the fit to the mass difference distributions $m(B^{+}K^{-})-m(B^{+})-m(K^{-})$. The first uncertainties are statistical and the second are systematic. Parameter | Fit result | Best previous measurement
---|---|---
$m(B_{s1})$ | $-\,m(B^{*+})-m(K^{-})$ | $10.46\,\pm\,$ | $0.04$ | $\,\pm\,0.04$ | ${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ | $10.73\,\pm\,$ | $0.21$ | $\,\pm\,0.14$ | ${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ [9]
$m(B^{*}_{s2})$ | $-\,\,m(B^{+})\,-m(K^{-})$ | $67.06\,\pm\,$ | $0.05$ | $\,\pm\,0.11$ | ${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ | $66.96\,\pm\,$ | $0.39$ | $\,\pm\,0.14$ | ${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ [9]
$m(B^{*+})$ | $-\,\,m(B^{+})$ | $45.01\,\pm\,$ | $0.30$ | $\,\pm\,0.23$ | ${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ | $45.6\pm 0.8$ | ${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ [30]
$\Gamma(B^{*}_{s2})$ | $1.56\,\pm\,$ | $0.13$ | $\,\pm\,0.47$ | ${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ | | | |
$\frac{\mathcal{B}(B^{*}_{s2}\rightarrow B^{*+}K^{-})}{\mathcal{B}(B^{*}_{s2}\rightarrow B^{+}K^{-})}$ | $(9.3\phantom{0}\,\pm\,$ | $1.3\phantom{0}$ | $\,\pm\,1.2)$ | % | | | |
$\frac{\sigma(pp\rightarrow B_{s1}X)\mathcal{B}(B_{s1}\rightarrow B^{*+}K^{-})}{\sigma(pp\rightarrow B^{*}_{s2}X)\mathcal{B}(B^{*}_{s2}\rightarrow B^{+}K^{-})}$ | $(23.2\phantom{0}\,\pm\,$ | $1.4\phantom{0}$ | $\,\pm\,1.3)$ | % | | | |
$N_{B_{s1}\rightarrow B^{*+}K^{-}}$ | $750\pm 36$ | | | | |
$N_{B^{*}_{s2}\rightarrow B^{*+}K^{-}}$ | $307\pm 46$ | | | | |
$N_{B^{*}_{s2}\rightarrow B^{+}K^{-}}$ | $3140\pm 100$ | | | | |
To determine the significance of the $B^{*}_{s2}\rightarrow B^{*+}K^{-}$
signal, a similar maximum likelihood fit is performed, where all parameters of
the signal are fixed according to expectation, except its yield. The
likelihood of this fit is compared to the result of a fit where the yield of
the signal is fixed to zero. The statistical significance of the
$B^{*}_{s2}\rightarrow B^{*+}K^{-}$ signal is $8\sigma$.
A number of systematic uncertainties are considered. For the signal model, the
signal shape is changed to a double Gaussian function and an alternative
threshold function is used for the background. The changes in the fit results
are assigned as the associated uncertainties. The $B^{+}$ decay modes are
fitted independently to test for effects that may be related to differences in
their selection requirements. For each observable quoted in Table 2, the
difference between the weighted average of these independent fits and the
global fit is taken as a systematic uncertainty. Additional systematic
uncertainties are assigned based on the change in the results when varying the
selection criteria and the $B^{+}$ signal region. The detector resolution of
$B^{*}_{s2}\rightarrow B^{+}K^{-}$ signal is varied by $\pm 20\%$. In
addition, the momentum scale in the processing of the data used in this
analysis is varied within the estimated uncertainty of $0.15\%$. The
corresponding uncertainty on the measured masses is assigned as a systematic
uncertainty. The uncertainty on the determination of the selection efficiency
ratios caused by finite samples of simulated events is taken as a systematic
uncertainty for the branching fractions. Finally simulated events are used to
estimate the mass shifts of the $B_{s1}/B^{*}_{s2}\rightarrow B^{*+}K^{-}$
signals from the nominal values when the radiated photon is excluded from
their reconstructed decays. The absolute systematic uncertainties are given in
Table 3. The $B^{*}_{s2}\rightarrow B^{*+}K^{-}$ signal is observed with the
expected frequency in each of the four resconstructed decay modes and the
systematic error for the $\frac{\mathcal{B}(B^{*}_{s2}\rightarrow
B^{*+}K^{-})}{\mathcal{B}(B^{*}_{s2}\rightarrow B^{+}K^{-})}$ branching
fraction ratio, related to the different $B^{+}$ decay modes, is small. The
final results are shown in Table 2. The measured mass differences are more
precise than the previous best measurements of a factor two at least. The
measured $\frac{\mathcal{B}(B^{*}_{s2}\rightarrow
B^{*+}K^{-})}{\mathcal{B}(B^{*}_{s2}\rightarrow B^{+}K^{-})}$ branching
fraction ratio and $B^{*}_{s2}$ width are in good agreement with theoretical
predictions [12, 13, 14].
The mass differences given in Table 2 are translated into absolute masses by
adding the masses of the $B^{+}$ and kaon [8] and, in the case of the $B_{s1}$
meson, the $B^{*+}-B^{+}$ mass difference measured in this Letter. The results
are
$\begin{array}[]{rccccccccl}m(B^{*+})&=&5324.26&\pm&0.30&\pm&0.23&\pm&0.17&{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}},\\\
m(B_{s1})&=&5828.40&\pm&0.04&\pm&0.04&\pm&0.41&{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}},\\\
m(B^{*}_{s2})&=&5839.99&\pm&0.05&\pm&0.11&\pm&0.17&{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}},\end{array}$
where the first uncertainty is statistical and the second is systematic. The
third uncertainty corresponds to the uncertainty on the $B^{+}$ mass [8] and,
in the case of the $B_{s1}$ mass measurement, the uncertainty on the
$B^{*+}-B^{+}$ mass difference measured in this analysis.
Table 3: Absolute systematic uncertainties for each measurement, which are
assumed to be independent and are added in quadrature.
Source | $Q(B_{s1})$ | $Q(B^{*}_{s2})$ | $m(B^{*+})-m(B^{+})$ | $\Gamma(B^{*}_{s2})$ | $R^{B^{*}_{s2}}$ | $\sigma^{B_{s1}/B^{*}_{s2}}R^{B_{s1}/B^{*}_{s2}}$
---|---|---|---|---|---|---
| $({\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}})$ | $({\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}})$ | $({\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}})$ | $({\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}})$ | (%) | (%)
Fit model | $0.00$ | $0.02$ | $0.03$ | $0.01$ | $0.2$ | $0.5$
$B^{+}$ decay mode | $0.01$ | $0.01$ | $0.02$ | $0.01$ | $0.1$ | $0.1$
Selection | $0.03$ | $0.02$ | $0.19$ | $0.05$ | $1.1$ | $0.6$
$B^{+}$ signal region | $0.01$ | $0.03$ | $0.11$ | $0.07$ | $0.2$ | $0.4$
Mass resolution | $0.00$ | $0.01$ | $0.02$ | $0.46$ | $0.2$ | $0.9$
Momentum scale | $0.02$ | $0.10$ | $0.03$ | - | - | -
Efficiency ratios | - | - | - | - | $0.2$ | $0.2$
Missing photon | $0.01$ | - | $0.01$ | - | - | -
Total | $0.04$ | $0.11$ | $0.23$ | $0.47$ | $1.2$ | $1.3$
The significance of the non-zero $B^{*}_{s2}$ width is determined by comparing
the likelihood for the nominal fit with a fit in which the width is fixed to
zero. To account for systematic effects, the minimum
$\sqrt{2\Delta\rm{log}\mathcal{L}}$ among all systematic variations is taken;
the significance including systematic uncertainties is $9\sigma$.
In conclusion, using 1.0$\mbox{\,fb}^{-1}$ of data collected with the LHCb
detector at $\sqrt{s}=7$$\mathrm{\,Te\kern-1.00006ptV}$, the decay mode
$B^{*}_{s2}\rightarrow B^{*+}K^{-}$ is observed for the first time and its
branching fraction measured relative to that of $B^{*}_{s2}\rightarrow
B^{+}K^{-}$. The observation of the $B^{*}_{s2}$ meson decaying to two
pseudoscalars ($B^{*}_{s2}\rightarrow B^{+}K^{-}$) and to a vector and a
pseudoscalar ($B^{*}_{s2}\rightarrow B^{*+}K^{-}$) favours the assignment of
$J^{P}=2^{+}$ for this state. The $B^{*}_{s2}$ width is measured for the first
time, while the masses of the $B_{s1}$ and $B^{*}_{s2}$ states are measured
with the highest precision to date and are consistent with previous
measurements [9, 10]. Finally, the observed $B^{*}_{s2}\rightarrow
B^{*+}K^{-}$ decay is used to make the most precise measurement to date of the
$B^{*+}-B^{+}$ mass difference. This measurement, unlike others reported in
the literature, does not require the reconstruction of the soft photon from
$B^{*+}$ decays and therefore has significantly smaller systematic
uncertainty. High precision measurements of the $B^{*+}$ mass are important
for the understanding of the exotic $Z_{b}^{+}$ states recently observed[15].
Using the $B^{*+}$ mass measured in this analysis, we compute that the
$Z_{b}(10610)^{+}$ and $Z_{b}(10650)^{+}$ masses are $3.69\pm
2.05{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ and $3.68\pm
1.71{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ above the $B\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{*}$ and $B^{*}\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{*}$ thresholds respectively.
## 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); ANCS/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
| 2012-11-26T15:44:48 |
2024-09-04T02:49:38.410755
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "LHCb collaboration: R. Aaij, C. Abellan Beteta, A. Adametz, B. Adeva,\n M. Adinolfi, C. Adrover, A. Affolder, Z. Ajaltouni, J. Albrecht, F. Alessio,\n M. Alexander, S. Ali, G. Alkhazov, P. Alvarez Cartelle, A. A. Alves Jr, S.\n Amato, Y. Amhis, L. Anderlini, J. Anderson, R. B. Appleby, O. Aquines\n Gutierrez, F. Archilli, A. Artamonov, M. Artuso, E. Aslanides, G. Auriemma,\n S. Bachmann, J. J. Back, C. Baesso, W. Baldini, R. J. Barlow, C. Barschel, S.\n Barsuk, W. Barter, A. Bates, Th. Bauer, A. Bay, J. Beddow, I. Bediaga, S.\n Belogurov, K. Belous, I. Belyaev, E. Ben-Haim, M. Benayoun, G. Bencivenni, S.\n Benson, J. Benton, A. Berezhnoy, R. Bernet, M.-O. Bettler, M. van Beuzekom,\n A. Bien, S. Bifani, T. Bird, A. Bizzeti, P. M. Bj{\\o}rnstad, T. Blake, F.\n Blanc, C. Blanks, J. Blouw, S. Blusk, A. Bobrov, V. Bocci, A. Bondar, N.\n Bondar, W. Bonivento, S. Borghi, A. Borgia, T. J. V. Bowcock, C. Bozzi, T.\n Brambach, J. van den Brand, J. Bressieux, D. Brett, M. Britsch, T. Britton,\n N. H. Brook, H. Brown, A. B\\\"uchler-Germann, I. Burducea, A. Bursche, J.\n Buytaert, S. Cadeddu, O. Callot, M. Calvi, M. Calvo Gomez, A. Camboni, P.\n Campana, A. Carbone, G. Carboni, R. Cardinale, A. Cardini, H. Carranza-Mejia,\n L. Carson, K. Carvalho Akiba, G. Casse, M. Cattaneo, Ch. Cauet, M. Charles,\n Ph. Charpentier, P. Chen, N. Chiapolini, M. Chrzaszcz, K. Ciba, X. Cid Vidal,\n G. Ciezarek, P. E. L. Clarke, M. Clemencic, H. V. Cliff, J. Closier, C. Coca,\n V. Coco, J. Cogan, E. Cogneras, P. Collins, A. Comerma-Montells, A. Contu, A.\n Cook, M. Coombes, G. Corti, B. Couturier, G. A. Cowan, D. Craik, S. Cunliffe,\n R. Currie, C. D'Ambrosio, P. David, P. N. Y. David, I. De Bonis, K. De Bruyn,\n S. De Capua, M. De Cian, J. M. De Miranda, L. De Paula, P. De Simone, D.\n Decamp, M. Deckenhoff, H. Degaudenzi, L. Del Buono, C. Deplano, D. Derkach,\n O. Deschamps, F. Dettori, A. Di Canto, J. Dickens, H. Dijkstra, P. Diniz\n Batista, M. Dogaru, F. Domingo Bonal, S. Donleavy, F. Dordei, A. Dosil\n Su\\'arez, D. Dossett, A. Dovbnya, F. Dupertuis, R. Dzhelyadin, A. Dziurda, A.\n Dzyuba, S. Easo, U. Egede, V. Egorychev, S. Eidelman, D. van Eijk, S.\n Eisenhardt, R. Ekelhof, L. Eklund, I. El Rifai, Ch. Elsasser, D. Elsby, A.\n Falabella, C. F\\\"arber, G. Fardell, C. Farinelli, S. Farry, V. Fave, V.\n Fernandez Albor, F. Ferreira Rodrigues, M. Ferro-Luzzi, S. Filippov, C.\n Fitzpatrick, M. Fontana, F. Fontanelli, R. Forty, O. Francisco, M. Frank, C.\n Frei, M. Frosini, S. Furcas, A. Gallas Torreira, D. Galli, M. Gandelman, P.\n Gandini, Y. Gao, J-C. Garnier, J. Garofoli, P. Garosi, J. Garra Tico, L.\n Garrido, C. Gaspar, R. Gauld, E. Gersabeck, M. Gersabeck, T. Gershon, Ph.\n Ghez, V. Gibson, V. V. Gligorov, C. G\\\"obel, D. Golubkov, A. Golutvin, A.\n Gomes, 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, O.\n Gr\\\"unberg, B. Gui, E. Gushchin, Yu. Guz, T. Gys, C. Hadjivasiliou, G.\n Haefeli, C. Haen, S. C. Haines, S. Hall, T. Hampson, S. Hansmann-Menzemer, N.\n Harnew, S. T. Harnew, J. Harrison, P. F. Harrison, T. Hartmann, J. He, V.\n Heijne, K. Hennessy, P. Henrard, J. A. Hernando Morata, E. van Herwijnen, E.\n Hicks, D. Hill, M. Hoballah, P. Hopchev, W. Hulsbergen, P. Hunt, T. Huse, N.\n Hussain, D. Hutchcroft, D. Hynds, V. Iakovenko, P. Ilten, J. Imong, R.\n Jacobsson, A. Jaeger, M. Jahjah Hussein, E. Jans, F. Jansen, P. Jaton, B.\n Jean-Marie, F. Jing, M. John, D. Johnson, C. R. Jones, B. Jost, M. Kaballo,\n S. Kandybei, M. Karacson, T. M. Karbach, I. R. Kenyon, U. Kerzel, T. Ketel,\n A. Keune, B. Khanji, Y. M. Kim, O. Kochebina, V. Komarov, R. F. Koopman, P.\n Koppenburg, M. Korolev, A. Kozlinskiy, L. Kravchuk, K. Kreplin, M. Kreps, G.\n Krocker, P. Krokovny, F. Kruse, M. Kucharczyk, V. Kudryavtsev, T.\n Kvaratskheliya, V. N. La Thi, D. Lacarrere, G. Lafferty, A. Lai, D. Lambert,\n R. W. Lambert, E. Lanciotti, G. Lanfranchi, C. Langenbruch, T. Latham, C.\n Lazzeroni, R. Le Gac, J. van Leerdam, J.-P. Lees, R. Lef\\`evre, A. Leflat, J.\n Lefran\\c{c}ois, O. Leroy, T. Lesiak, Y. Li, L. Li Gioi, M. Liles, R. Lindner,\n C. Linn, B. Liu, G. Liu, J. von Loeben, J. H. Lopes, E. Lopez Asamar, N.\n Lopez-March, H. Lu, J. Luisier, H. Luo, A. Mac Raighne, F. Machefert, I. V.\n Machikhiliyan, F. Maciuc, O. Maev, J. Magnin, M. Maino, S. Malde, G. Manca,\n G. Mancinelli, N. Mangiafave, U. Marconi, R. M\\\"arki, J. Marks, G.\n Martellotti, A. Martens, L. Martin, A. Mart\\'in S\\'anchez, M. Martinelli, D.\n Martinez Santos, D. Martins Tostes, A. Massafferri, R. Matev, Z. Mathe, C.\n Matteuzzi, M. Matveev, E. Maurice, A. Mazurov, J. McCarthy, G. McGregor, R.\n McNulty, M. Meissner, M. Merk, J. Merkel, D. A. Milanes, M.-N. Minard, J.\n Molina Rodriguez, S. Monteil, D. Moran, P. Morawski, R. Mountain, I. Mous, F.\n Muheim, K. M\\\"uller, R. Muresan, B. Muryn, B. Muster, J. Mylroie-Smith, P.\n Naik, T. Nakada, R. Nandakumar, I. Nasteva, M. Needham, N. Neufeld, A. D.\n Nguyen, T. D. Nguyen, C. Nguyen-Mau, M. Nicol, V. Niess, N. Nikitin, T.\n Nikodem, A. Nomerotski, A. Novoselov, A. Oblakowska-Mucha, V. Obraztsov, S.\n Oggero, S. Ogilvy, O. Okhrimenko, R. Oldeman, M. Orlandea, J. M. Otalora\n Goicochea, P. Owen, B. K. Pal, A. Palano, M. Palutan, J. Panman, A.\n 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. Pellegrino, G. Penso, M. Pepe Altarelli, S. Perazzini, D. L.\n Perego, E. Perez Trigo, A. P\\'erez-Calero Yzquierdo, P. Perret, M.\n Perrin-Terrin, G. Pessina, K. Petridis, A. Petrolini, A. Phan, E. Picatoste\n Olloqui, B. Pie Valls, B. Pietrzyk, T. Pila\\v{r}, D. Pinci, S. Playfer, M.\n Plo Casasus, F. Polci, G. Polok, A. Poluektov, E. Polycarpo, D. Popov, B.\n Popovici, C. Potterat, A. Powell, J. Prisciandaro, V. Pugatch, A. Puig\n Navarro, W. Qian, J. H. Rademacker, B. Rakotomiaramanana, M. S. Rangel, I.\n Raniuk, N. Rauschmayr, G. Raven, S. Redford, M. M. Reid, A. C. dos Reis, S.\n Ricciardi, A. Richards, K. Rinnert, V. Rives Molina, D. A. Roa Romero, P.\n Robbe, E. Rodrigues, P. Rodriguez Perez, G. J. Rogers, S. Roiser, V.\n Romanovsky, A. Romero Vidal, J. Rouvinet, T. Ruf, H. Ruiz, G. Sabatino, J. J.\n Saborido Silva, N. Sagidova, P. Sail, B. Saitta, C. Salzmann, B. Sanmartin\n Sedes, M. Sannino, R. Santacesaria, C. Santamarina Rios, R. Santinelli, E.\n Santovetti, M. Sapunov, A. Sarti, C. Satriano, A. Satta, M. Savrie, P.\n Schaack, M. Schiller, H. Schindler, S. Schleich, 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, P. Shatalov, Y.\n Shcheglov, T. Shears, L. Shekhtman, O. Shevchenko, V. Shevchenko, A. Shires,\n R. Silva Coutinho, T. Skwarnicki, N. A. Smith, E. Smith, M. Smith, K.\n Sobczak, 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. Stoica, S. Stone,\n B. Storaci, M. Straticiuc, U. Straumann, V. K. Subbiah, S. Swientek, M.\n Szczekowski, P. Szczypka, T. Szumlak, S. T'Jampens, M. Teklishyn, E.\n Teodorescu, F. Teubert, C. Thomas, E. Thomas, J. van Tilburg, V. Tisserand,\n M. Tobin, S. Tolk, D. Tonelli, S. Topp-Joergensen, N. Torr, E. Tournefier, S.\n Tourneur, M. T. Tran, A. Tsaregorodtsev, P. Tsopelas, N. Tuning, M. Ubeda\n Garcia, A. Ukleja, D. Urner, U. Uwer, V. Vagnoni, G. Valenti, R. Vazquez\n Gomez, P. Vazquez Regueiro, S. Vecchi, J. J. Velthuis, M. Veltri, G.\n Veneziano, M. Vesterinen, B. Viaud, I. Videau, D. Vieira, X. Vilasis-Cardona,\n J. Visniakov, A. Vollhardt, D. Volyanskyy, D. Voong, A. Vorobyev, V.\n Vorobyev, C. Vo{\\ss}, H. Voss, R. Waldi, R. Wallace, S. Wandernoth, J. Wang,\n D. R. Ward, N. K. Watson, A. D. Webber, D. Websdale, M. Whitehead, J. Wicht,\n D. Wiedner, L. Wiggers, G. Wilkinson, M. P. Williams, M. Williams, F. F.\n Wilson, J. Wishahi, M. Witek, W. Witzeling, S. A. Wotton, S. Wright, S. Wu,\n K. Wyllie, Y. Xie, F. Xing, Z. Xing, Z. Yang, R. Young, X. Yuan, O.\n Yushchenko, M. Zangoli, M. Zavertyaev, F. Zhang, L. Zhang, W. C. Zhang, Y.\n Zhang, A. Zhelezov, L. Zhong, A. Zvyagin",
"submitter": "Marco Pappagallo",
"url": "https://arxiv.org/abs/1211.5994"
}
|
1211.6004
|
# SPIN SQUEEZING AND ENTANGLEMENT VIA FINITE-DIMENSIONAL DISCRETE PHASE-SPACE
DESCRIPTION
Marcelo A. Marchiolli Avenida General Osório 414, centro, Jaboticabal, SP
14870-100, Brazil
[email protected] Diógenes Galetti Instituto de Física Teórica,
Universidade Estadual Paulista,
Rua Dr Bento Teobaldo Ferraz 271, Bloco II, Barra Funda, São Paulo, SP
01140-070, Brazil
[email protected] Tiago Debarba Instituto de Ciências Exatas,
Universidade Federal de Minas Gerais, Departamento de Física,
Avenida Antônio Carlos 6627, Belo Horizonte, MG 31270-901, Brazil
[email protected]
###### Abstract
We show how mapping techniques inherent to ${\rm N}^{2}$-dimensional discrete
phase spaces can be used to treat a wide family of spin systems which exhibits
squeezing and entanglement effects. This algebraic framework is then applied
to the modified Lipkin-Meshkov-Glick (LMG) model in order to obtain the time
evolution of certain special parameters related to the Robertson-Schrödinger
(RS) uncertainty principle and some particular proposals of entanglement
measure based on collective angular-momentum generators. Our results reinforce
the connection between both the squeezing and entanglement effects, as well as
allow to investigate the basic role of spin correlations through the discrete
representatives of quasiprobability distribution functions. Entropy
functionals are also discussed in this context. The main sequence correlations
$\mapsto$ entanglement $\mapsto$ squeezing of quantum effects embraces a new
set of insights and interpretations in this framework, which represents an
effective gain for future researches in different spin systems.
###### keywords:
Spin squeezing; entanglement; finite-dimensional discrete phase spaces.
## 1 Introduction
Originally introduced by E. Schrödinger in his seminal work on probability
relations between separated systems,[1] “entanglement” indeed corresponds to a
fundamental concept in physics that lies at the heart of many conceptual
problems in quantum mechanics.[2, 3] The mere comprehension of this abstract
concept and its respective changing of status, in the recent past, for
experimental measure, certainly represents a concatenation of efforts with
significant progress in theoretical and experimental physics. It is important
to stress that ‘quantum entanglement’ plays an essential role in multipartite
quantum systems,[4] since its underlying physical properties can be used as a
specific resource for determined quantum-information tasks[5] — recently,
different proposals of quantumness correlations have appeared in current
literature,[6] emphasizing the possible connections with quantum
entanglement.[7] Therefore, understanding, identifying, measuring and,
consequently, exploiting genuine quantum effects (or “quantumness”[8]) in
multipartite systems constitute a set of obligatory prerequisites for grasping
the fundamental implications of any quantum theory.[9]
Recent theoretical proposals[10] and experiments[11, 12] involving
measurements upon collective angular-momentum generators in different physical
systems fulfil, in part, the aforementioned prerequisites, as well as
corroborate the subtle match between spin-squeezing[13] and entanglement
effects. However, some necessary questions concerning the correlations among
different spin components as chief agents responsible for spin-squeezing
effects deserve be properly answered. Then, it seems reasonable to adopt a
particular algebraic framework that, within all the inherent mathematical
virtues, allows to: (i) map the kinematical and dynamical contents of a given
spin system with a finite space of states into a ${\rm N}^{2}$-dimensional
discrete phase space; (ii) comprehend the role of correlations through
discrete Wigner functions and their connections with spin-squeezing effects;
and finally, (iii) gain new insights on the effects exhibited in the Venn
diagram below which lead us to take advantage operationally of their
quantumness in many-particle experiments. Notwithstanding the appreciable
number of papers in current literature proposing similar theoretical
frameworks with different intrinsic mathematical properties,[14] let us focus
our attention upon the formalism developed in Refs. GP1–MR which complies such
demands.
CorrelationsEntanglementSpin Squeezing Figure 1: This particular Venn diagram
depicts, in a general form, the interplay among correlations, entanglement,
and spin-squeezing effects in finite-size quantum spin systems.
Based on the technique of constructing unitary operator bases initially
formulated by Schwinger,[25] this particular discrete quantum phase-space
approach embraces a well-defined algebraic structure where the ${\rm
N}$-dimensional pre-Hilbert spaces equipped with a Hilbert-Schmidt inner
product[26] endorse the finite space of states. Moreover, it leads us (i) to
exhibit and handle the pair of complementary variables related to a specific
degree of freedom we are dealing with (as well as to recognize the quantum
correlations between them), and also (ii) to obtain additional quantum
information about the physical system through the analysis of the
corresponding discrete quasiprobability distribution functions. Henceforth,
the basic idea consists in exploring this mathematical tool in order to study
the role of those correlations in connection with spin-squeezing and
entanglement processes.
Initially focused on some mathematically appealing features inherent to the
collective angular-momentum generators, the first part of this paper basically
discuss the spin coherent states and their relations with unitary
transformations through a constructive point of view. Next, we establish a
$\mbox{mod}({\rm N})$-invariant operator basis which can be immediately
employed in the mapping of quantum operators (acting into that particular
${\rm N}$-dimensional space of states) onto well-behaved functions of discrete
variables by means of a trace operation.[27] These functions correspond to the
representatives of the operators in a ${\rm N}^{2}$-dimensional phase space
labeled by a pair of discrete variables for each degree of freedom of the
physical system under investigation. Thus, all the necessary quantities for
describing its kinematical and dynamical contents can now be promptly mapped
one-to-one on such phase spaces. For completeness sake, we also present two
different prescriptions that essentially determine the time evolution for both
the discrete Wigner and Weyl functions.
These results are then applied to an extended version of the LMG model,[28]
which was initially introduced by Vidal and coworkers[29] with the aim of
investigating the statistical-mechanical properties and entanglement effects
of a particular interacting spin system. Summarizing, the modified LMG model
describes a finite set of spins $\frac{1}{2}$ with a mutual anisotropic $(XY)$
ferromagnetic interaction, and also subjected to a transverse magnetic field.
The second part of this paper shows how the quantum correlations (here
controlled by parameters related to the transverse magnetic field and
anisotropy) affect the connection between the spin-squeezing and entanglement
effects. To develop this specific task, we first employ the RS uncertainty
principle[30] in order to determine a spin-squeezing measure which
incorporates, in its definition, the covariance function — being such a
function responsible for introducing, within the Heisenberg uncertainty
principle, important contributions associated with the anticommutation
relations of the angular-momentum generators. The subsequent comparison with
the entanglement measure proposed in Ref. Hald establishes, in this way, the
aforementioned link and reinforces the fundamental role of correlations in our
initial analysis. Moreover, we also study the behaviour of the discrete Wigner
function for certain particular values of time where occur an ‘almost perfect
match’ between both the spin-squeezing and entanglement measurements (such
procedure leads us to comprehend the intricate role of correlations in a
finite-dimensional phase space). We have finalized this work with the analysis
of some entropy functionals.[31] For completeness sake, it is worth stressing
that we have worked with a number of spins which preserves basic quantum
effects, contrary to the thermodynamic limit.
This paper is structured as follows. In Section 2, we present an important set
of essential mathematical tools related to the angular-momentum generators and
spin coherent states, as well as an interesting discussion on unitary
transformations and their corresponding geometric interpretations. In Section
3, we introduce a mapping kernel for finite-dimensional discrete phase spaces
whose inherent properties allow, within other important virtues, to describe
the kinematical and dynamical contents of a given physical system with a
finite space of states. In Section 4, we apply the quantum-algebraic approach
developed in the previous section to the modified LMG model, with the aim of
analysing the influence of quantum correlations on the spin-squeezing and
entanglement effects. Section 5 summarizes the main results obtained in this
paper and discuss some possible perspectives for future research. We have
added three appendixes related to the calculational details of certain topics
and expressions used in the previous sections: Appendix A shows how the
unitary transformations associated with angular-momentum generators affect the
expressions for variances, covariances, and RS uncertainty relation; Appendix
B discuss the Kitagawa-Ueda model[13] and the inherent spin-squeezing and
entanglement effects; while Appendix C exhibits an important set of specific
numerical computations which leads us to establish a validity domain for the
entanglement criteria studied in this work.
## 2 Definitions and background for spin coherent states
In this section, we will briefly survey some mathematically appealing features
inherent to the $\mathfrak{su}(2)$ Lie algebra with emphasis on a specific
group of unitary transformations involving the generators of this algebra
which leads us to properly define the spin coherent states. For this task, let
us initially consider the standard angular-momentum generators $\left\\{{\bf
J}_{x},{\bf J}_{y},{\bf J}_{z}\right\\}$ which act on a finite-dimensional
Hilbert space $\mathcal{H}_{\mbox{\tiny${\rm N}$}}$. The familiar commutation
relation $\left[{\bf J}_{a},{\bf J}_{b}\right]=\mathrm{i}\epsilon_{abc}{\bf
J}_{c}$ for $a,b,c=x,y,z$ (letting $\hbar=1$ for convenience), where
$\epsilon_{abc}$ denotes the Levi-Civita symbol associated with the three
orthogonal directions, permits us to define the raising and lowering operators
through the particular decomposition ${\bf J}_{\pm}\mathrel{\mathop{:}}={\bf
J}_{x}\pm\mathrm{i}{\bf J}_{y}$ with $\left[{\bf J}_{+},{\bf
J}_{-}\right]=2{\bf J}_{z}$ and $\left[{\bf J}_{z},{\bf
J}_{\pm}\right]=\pm{\bf J}_{\pm}$. Moreover, $\vec{{\bf
J}}^{2}\mathrel{\mathop{:}}={\bf J}_{x}^{2}+{\bf J}_{y}^{2}+{\bf
J}_{z}^{2}={\bf J}_{z}^{2}+\frac{1}{2}\left\\{{\bf J}_{+},{\bf
J}_{-}\right\\}$ characterizes the total spin operator and constitutes an
important element of this algebra since $\vec{{\bf J}}^{2}$ (also known as a
Casimir operator) commutes with all the generators ${\bf J}_{a}$. In
particular, these results allow to construct, for example, a set of
simultaneous eigenstates of $\vec{{\bf J}}^{2}$ and ${\bf J}_{z}$, that is,
$\left\\{|j,m\rangle:|m|\leq
j\;,\;j=0,\frac{1}{2},1,\frac{3}{2},\ldots\right\\}$, with well-known
mathematical properties.[32] [(ii)]
It is interesting to note that each particular member of this representation
(here characterized by a specific eigenvalue $j$) is analogous to the Fock
state[26] of the electromagnetic field, since it can be created by the
repeated application of the raising operator (remembering that ${\bf
J}_{\pm}^{2j+1}=0$) on the ‘vaccum state’,
$|j,m\rangle\mathrel{\mathop{:}}=\mathscr{C}_{2j,j+m}^{-\frac{1}{2}}\frac{{\bf
J}_{+}^{j+m}}{(j+m)!}|j,-j\rangle\quad\mbox{for}\quad\mathscr{C}_{r,s}\equiv\frac{r!}{s!(r-s)!}.$
Thus, the set $\left\\{|j,m\rangle\right\\}$ corresponds to a discrete,
orthonormal and complete basis for the $(2j+1)$-dimensional vector space
$\mathbb{C}^{2j+1}$ of angular-momentum states,[33] whose completeness and
orthonormality relations are expressed as follow:
$\sum_{m=-j}^{j}|j,m\rangle\langle j,m|={\bf 1}\quad\mbox{and}\quad\langle
j^{\prime},m^{\prime}|j,m\rangle=\delta_{j^{\prime},j}\,\delta_{m^{\prime},m}.$
Consequently, the expansion of any quantum state $|\Psi\rangle$ belonging to
this finite-dimensional space can now be prompty obtained,
$|\Psi\rangle=\sum_{m=-j}^{j}c_{j,m}|j,m\rangle$
where $c_{j,m}\equiv\langle j,m|\Psi\rangle$ denotes the coefficients of such
a discrete expansion.
The nondiagonal matrix elements $\langle j^{\prime},m^{\prime}|{\bf
J}_{\pm}^{k}|j,m\rangle$ associated with the moments ${\bf J}_{\pm}^{k}$ show
a dependence on the binomial coefficients $\mathscr{C}_{r,s}$, that is
$\langle j^{\prime},m^{\prime}|{\bf J}_{\pm}^{k}|j,m\rangle=\frac{(j\pm
m^{\prime})!}{(j\pm
m)!}\left(\frac{\mathscr{C}_{2j,j+m^{\prime}}}{\mathscr{C}_{2j,j+m}}\right)^{\frac{1}{2}}\delta_{j^{\prime},j}\,\delta_{m^{\prime},m\pm
k}\qquad(0\leq k\leq 2j),$
while $\langle j^{\prime},m^{\prime}|{\bf
J}_{z}^{k}|j,m\rangle=m^{k}\delta_{j^{\prime},j}\,\delta_{m^{\prime},m}$
demonstrates an explicit connection with the discrete component $m$. These
results are extremely useful when applied to the evaluation of nondiagonal
matrix elements related to well-known families of unitary transformations
involving the linear combination of spin operators,[32, 33] and also in the
study of spin coherent states.[34, 35, 36]
### 2.1 Unitary transformations
Let ${\bf
T}(\Omega_{\pm},\Omega_{z})\mathrel{\mathop{:}}=\exp\left(\Omega_{+}{\bf
J}_{+}+\Omega_{z}{\bf J}_{z}+\Omega_{-}{\bf J}_{-}\right)$ denote a kind of
general abstract operator expressed in terms of generators of the
$\mathfrak{su}(2)$ Lie algebra and arbitrary c-number parameters
$\Omega_{\pm}$ and $\Omega_{z}$. For $\Omega_{+}=\xi$,
$\Omega_{-}=-\xi^{\ast}$ and $\Omega_{z}=\mathrm{i}\omega$, with
$\xi\in\mathbb{C}$ and $\omega\in\mathbb{R}$, such an abstract operator
represents a generator of unitary transformations whose respective generalized
‘normal’- and ‘antinormal’-order decomposition formulae display the following
expressions:[37, 38]
$\displaystyle\\!\\!\\!\\!\\!{\bf T}(\xi,\omega)$ $\displaystyle=$
$\displaystyle\exp\left(\Lambda_{+}{\bf
J}_{+}\right)\exp\left[\ln\left(\Lambda_{z}\right){\bf
J}_{z}\right]\exp\left(\Lambda_{-}{\bf J}_{-}\right)$ (1) $\displaystyle=$
$\displaystyle\exp\left(-\Lambda_{+}^{\ast}{\bf
J}_{-}\right)\exp\left[-\ln\left(\Lambda_{z}^{\ast}\right){\bf
J}_{z}\right]\exp\left(-\Lambda_{-}^{\ast}{\bf J}_{+}\right)$
with
$\displaystyle\Lambda_{+}$ $\displaystyle=$
$\displaystyle\frac{\left(\xi/\phi\right)\sin(\phi)}{\cos(\phi)-\mathrm{i}\left(\omega/2\phi\right)\sin(\phi)},\qquad\Lambda_{-}=-\frac{\left(\xi^{\ast}/\phi\right)\sin(\phi)}{\cos(\phi)-\mathrm{i}\left(\omega/2\phi\right)\sin(\phi)},$
$\displaystyle\Lambda_{z}$ $\displaystyle=$
$\displaystyle\left[\cos(\phi)-\mathrm{i}\left(\omega/2\phi\right)\sin(\phi)\right]^{-2}\quad\mbox{and}\quad\phi=\left[|\xi|^{2}+\left(\omega/2\right)^{2}\right]^{\frac{1}{2}}.$
So, the action of ${\bf T}(\xi,\omega)$ on the generators $\\{{\bf J}_{a}\\}$
— here defined through the relation $\overline{{\bf
J}}_{a}\mathrel{\mathop{:}}={\bf T}^{\dagger}(\xi,\omega){\bf J}_{a}{\bf
T}(\xi,\omega)$ for $a=x,y,z$ — produces effectively a new set of angular-
momentum operators $\\{\overline{{\bf J}}_{a}\\}$ expressed in terms of the
old ones that, by their turn, are multiplied by determined coefficients which
depend on the parameters $\xi$ and $\omega$, namely,
$\displaystyle\overline{{\bf J}}_{x}$ $\displaystyle=$
$\displaystyle\left[\cos(2\phi)+2\mbox{${\rm
Im}$}^{2}(\xi)\frac{\sin^{2}(\phi)}{\phi^{2}}\right]{\bf
J}_{x}+\left[\omega\,\frac{\sin(2\phi)}{2\phi}+2\mbox{${\rm
Re}$}(\xi)\mbox{${\rm Im}$}(\xi)\frac{\sin^{2}(\phi)}{\phi^{2}}\right]{\bf
J}_{y}$ (2) $\displaystyle+\left[\omega\mbox{${\rm
Im}$}(\xi)\frac{\sin^{2}(\phi)}{\phi^{2}}-\mbox{${\rm
Re}$}(\xi)\frac{\sin(2\phi)}{\phi}\right]{\bf J}_{z},$
$\displaystyle\overline{{\bf J}}_{y}$ $\displaystyle=$
$\displaystyle\left[-\omega\,\frac{\sin(2\phi)}{2\phi}+2\mbox{${\rm
Re}$}(\xi)\mbox{${\rm Im}$}(\xi)\frac{\sin^{2}(\phi)}{\phi^{2}}\right]{\bf
J}_{x}+\left[\cos(2\phi)+2\mbox{${\rm
Re}$}^{2}(\xi)\frac{\sin^{2}(\phi)}{\phi^{2}}\right]{\bf J}_{y}$ (3)
$\displaystyle+\left[\omega\mbox{${\rm
Re}$}(\xi)\frac{\sin^{2}(\phi)}{\phi^{2}}+\mbox{${\rm
Im}$}(\xi)\frac{\sin(2\phi)}{\phi}\right]{\bf J}_{z},$
$\displaystyle\overline{{\bf J}}_{z}$ $\displaystyle=$
$\displaystyle\left[\omega\mbox{${\rm
Im}$}(\xi)\frac{\sin^{2}(\phi)}{\phi^{2}}+\mbox{${\rm
Re}$}(\xi)\frac{\sin(2\phi)}{\phi}\right]{\bf J}_{x}+\left[\omega\mbox{${\rm
Re}$}(\xi)\frac{\sin^{2}(\phi)}{\phi^{2}}-\mbox{${\rm
Im}$}(\xi)\frac{\sin(2\phi)}{\phi}\right]{\bf J}_{y}$ (4)
$\displaystyle+\left[\cos(2\phi)+\frac{\omega^{2}}{2}\frac{\sin^{2}(\phi)}{\phi^{2}}\right]{\bf
J}_{z}.$
Note that $\vec{{\bf J}}^{2}$ remains invariant under the unitary
transformation (1), which implies in the identity $\vec{\overline{{\bf
J}}}^{2}\equiv\vec{{\bf J}}^{2}$; besides, the geometric counterpart of this
particular result is directly associated with arbitrary rotations on the
surface of a sphere of radius $j(j+1)$. Indeed, let ${\bf T}(\theta,\varphi)$
denote a particular case of ${\bf T}(\xi,\omega)$ when
$\xi=\frac{\theta}{2}\exp(-\mathrm{i}\varphi)$ and $\omega=0$, which implies
in specific decomposition formulae characterized by
$\Lambda_{\pm}=\pm\tan(\theta/2)\exp(\mp\mathrm{i}\varphi)$ and
$\Lambda_{z}=\sec^{2}(\theta/2)$. The connection between $\xi$ and
$\Lambda_{+}$ reflects the stereographic projection of a two-dimensional
sphere $S^{2}$ on the complex plane $\mathbb{C}$ with one-point
compactification (in this situation, the infinite point corresponds to the
north pole of $S^{2}=\mathbb{C}\cup\\{\infty\\}$).
As a consequence, the action of ${\bf T}(\theta,\varphi)$ on the vacuum state
$|j,-j\rangle$ will define our next object of study: the spin coherent states
— also known in current literature as $\mathrm{SU}(2)$ coherent states or spin
coherent states.[36] In fact, we will establish some few important
mathematical results connected with the spin coherent states which constitute
the first basic tools for discussing the subtle link between spin squeezing
and entanglement.
### 2.2 Spin coherent states
In general, the spin coherent states can be defined by means of two different
mathematical procedures: the first one[39] basically follows the Schwinger’s
prescription of angular momentum (note that LMG model[40] represents a typical
example of this prescription) in order to produce such states, while the
second one[41] pursues an algebraic framework analogous to that adopted for
the field coherent states which allows us to describe physical systems
consisting of $N$ two-level atoms111See Refs. N1–Ag2 for a detailed discussion
on the atomic coherent-state representation and its practical applications in
multitime-correlation functions and phase-space quasidistributions. (for
instance, see the Dicke model[48]). Here, we follow closely the last way that
consists in defining the spin coherent states through the mathematical
relation $|\theta,\varphi\rangle\mathrel{\mathop{:}}={\bf
T}(\theta,\varphi)|j,-j\rangle$, with the vacuum state $|j,-j\rangle$ written
in terms of the previously mentioned basis $\\{|j,m\rangle\\}_{-j\leq m\leq
j}$. Thus, after some algebra, it is easy to show that
$\\!\\!\\!\\!\\!|\theta,\varphi\rangle=\sum_{k=0}^{2j}\left[\mathscr{C}_{2j,k}\sin^{2k}(\theta/2)\cos^{2(2j-k)}(\theta/2)\right]^{\frac{1}{2}}\exp(-\mathrm{i}k\varphi)|j,k-j\rangle$
(5)
presents a bijective mapping with the $N$-photon generalized binomial states
related to the electromagnetic field, being this correspondence properly
explored in Ref. Messina.
Next, we focus our attention on an important set of mathematical properties
inherent to the spin coherent states (5) that leads us to discuss the
squeezing effects in determined physical systems under different
circumstances. Note that a complete list of such properties can be promptly
found in Refs. Thomas and Zhang.
Non-orthogonality.
The inner product of two distinct spin coherent states can be evaluated
through the completeness relation associated with $\\{|j,m\rangle\\}$,
yielding the well-known result
$\langle\theta^{\prime},\varphi^{\prime}|\theta,\varphi\rangle=\left\\{\cos(\theta^{\prime})\cos(\theta)+\sin(\theta^{\prime})\sin(\theta)\exp[\mathrm{i}(\varphi^{\prime}-\varphi)]\right\\}^{2j}.$
For $\theta^{\prime}=\theta$ and $\varphi^{\prime}=\varphi$, we achieve
$\langle\theta,\varphi|\theta,\varphi\rangle=1$ (normalization condition).
Now, excepting for the antipodal points, the spin coherent states are in
general not orthogonal. Hence, the Majorana-Bloch sphere[51] of unit radius
represents the ideal geometric element to describe such states, since its
respective north $(\theta=\pi)$ and south $(\theta=0)$ poles correspond to the
highest/lowest states $|j,\pm j\rangle$. For completeness sake, let us briefly
mention that
$|\langle\theta^{\prime},\varphi^{\prime}|\theta,\varphi\rangle|^{2}=\cos^{4j}(\Theta/2)$
(overlap probability) with
$\cos(\Theta)\equiv\cos(\theta^{\prime})\cos(\theta)+\sin(\theta^{\prime})\sin(\theta)\cos(\varphi^{\prime}-\varphi)$
is limited to the closed interval $[0,1]$, which reinforces the overcomplete
character of the states under consideration (they do not form an orthonormal
set).
Completeness relation.
Let ${\bf P}(\theta,\varphi)=|\theta,\varphi\rangle\langle\theta,\varphi|$
denote the diagonal projector operator related to the spin coherent states
which satisfies the property[34]
$\int\mathrm{d}\Omega(\theta,\varphi){\bf P}(\theta,\varphi)={\bf
1}\quad\mbox{with}\quad\mathrm{d}\Omega(\theta,\varphi)=\frac{2j+1}{4\pi}\sin(\theta)\mathrm{d}\theta\mathrm{d}\varphi.$
In particular, such a completeness relation asserts that any quantum state
$|\Psi\rangle$ belonging to $\mathbb{C}^{2j+1}$ can be properly expanded in
this overcomplete basis, namely
$|\Psi\rangle=\int\mathrm{d}\Omega(\theta,\varphi)\Psi(\theta,\varphi)|\theta,\varphi\rangle.$
In this case, $\Psi(\theta,\varphi)$ represents a polynomial function of
degree $2j$, whose analytical expression is given by
$\Psi(\theta,\varphi)=\sum_{k=0}^{2j}\left[\mathscr{C}_{2j,k}\sin^{2k}(\theta/2)\cos^{2(2j-k)}(\theta/2)\right]^{\frac{1}{2}}\exp(\mathrm{i}k\varphi)\,\langle
j,k-j|\Psi\rangle.$
It is worth mentioning that operators acting on this Hilbert space also admit
expansions in both the nondiagonal and diagonal forms, the diagonal
representation being of particular interest to deal with statistical operators
for atoms.[41]
Minimum uncertainty states.
Note that spin coherent states constitute an important class of minimum
uncertainty states. To demonstrate this assertion, let us initially consider
the RS uncertainty principle[30] for ${\bf J}_{x}$ and ${\bf J}_{y}$, i.e.,
$\mathscr{V}_{\mathrm{J}_{x}}\mathscr{V}_{\mathrm{J}_{y}}-\left(\mathscr{V}_{\mathrm{J}_{x}\mathrm{J}_{y}}\right)^{2}\geq\frac{1}{4}\left|\langle\left[{\bf
J}_{x},{\bf J}_{y}\right]\rangle\right|^{2}=\frac{1}{4}|\langle{\bf
J}_{z}\rangle|^{2}$ (6)
where
$\mathscr{V}_{\mathrm{J}_{x}\mathrm{J}_{y}}\equiv\langle\frac{1}{2}\left\\{{\bf
J}_{x},{\bf J}_{y}\right\\}\rangle-\langle{\bf J}_{x}\rangle\langle{\bf
J}_{y}\rangle$ represents the covariance function, and
$\mathscr{V}_{\mathrm{J}_{a}}\equiv\langle{\bf J}_{a}^{2}\rangle-\langle{\bf
J}_{a}\rangle^{2}$ stands for usual variance when $a=x,y$. So, if one
considers the previous mean values evaluated for the spin coherent states, it
is immediate to verify the equality sign in this equation since both the
expressions are equal to $\frac{1}{4}j^{2}\cos^{2}(\theta)$. In fact, this
particular result can be extended in order to include any non-parametrized
complex variable — see Appendix A for further results.
## 3 Mappings via finite-dimensional discrete phase spaces
In this section, we will introduce certain basic mathematical tools
constituents of the pioneering formalism initially developed in Refs. GP1 and
GP2 for physical systems with finite-dimensional space of states
$\mathcal{H}_{\mbox{\tiny${\rm N}$}}$. It is worth stressing that such tools
will represent our guidelines for the fundamentals of the formal description
of finite-dimensional phase spaces by means of discrete variables. For this
specific task, let us first establish the mod(N)-invariant operator basis[17]
${\bf G}(\mu,\nu)\mathrel{\mathop{:}}=\frac{1}{\sqrt{{\rm
N}}}\sum_{\eta,\xi=-\ell}^{\ell}\exp\left[-\frac{2\pi\mathrm{i}}{{\rm
N}}\left(\eta\mu+\xi\nu\right)\right]{\bf S}(\eta,\xi)$ (7)
expressed in terms of a discrete double Fourier transform of the symmetrized
unitary operator basis[15]
${\bf S}(\eta,\xi)\mathrel{\mathop{:}}=\frac{1}{\sqrt{{\rm
N}}}\exp\left(\frac{\pi\mathrm{i}}{{\rm N}}\eta\xi\right){\bf U}^{\eta}{\bf
V}^{\xi},$
where the labels $\eta$ and $\xi$ are associated with the dual momentum and
coordinatelike variables of an ${\rm N}^{2}$-dimensional discrete phase space
here endorsed by an underlying presymplectic structure of geometric
origin.[16] Note that these discrete labels obey the arithmetic modulo N and
assume integer values in the symmetric interval $[-\ell,\ell]$ for
$\ell=\frac{{\rm N}-1}{2}$ fixed.222Henceforth, for convenience, we assume
${\rm N}$ odd throughout this paper. However, it is important to stress that
even dimensionalities can also be dealt with simply by working on non-
symmetrized intervals. A comprehensive and useful compilation of results and
properties of the unitary operators ${\bf U}$ and ${\bf V}$ can be found in
Ref. DM1, since the primary focus of our attention is the essential features
exhibited by $\\{{\bf G}(\mu,\nu)\\}_{\mu,\nu=-\ell,\ldots,\ell}$.
The set of ${\rm N}^{2}$ operators $\\{{\bf G}(\mu,\nu)\\}$ allows, for
instance, to decompose any linear operator ${\bf O}$ acting on
$\mathcal{H}_{\mbox{\tiny${\rm N}$}}$ by means of the expansion
${\bf O}=\frac{1}{{\rm N}}\sum_{\mu,\nu=-\ell}^{\ell}\mathscr{O}(\mu,\nu){\bf
G}(\mu,\nu),$ (8)
where the coefficients $\mathscr{O}(\mu,\nu)\equiv\mbox{${\rm Tr}$}\left[{\bf
G}^{\dagger}(\mu,\nu){\bf O}\right]$ are evaluated through trace operation and
correspond, in this context, to a one-to-one mapping between operators and
functions embedded in a finite phase-space characterized by the discrete
variables $\mu$ and $\nu$.[27] So, if one considers the density operator
$\rho$ in such a case, we verify that
$\mbox{\boldmath$\rho$}=\frac{1}{{\rm
N}}\sum_{\mu,\nu=-\ell}^{\ell}\mathscr{W}_{\rho}(\mu,\nu){\bf G}(\mu,\nu)$ (9)
admits a plausible expansion with intrinsic mathematical properties since the
coefficients result in the discrete Wigner function
$\mathscr{W}_{\rho}(\mu,\nu)\mathrel{\mathop{:}}=\mbox{${\rm Tr}$}\left[{\bf
G}^{\dagger}(\mu,\nu)\mbox{\boldmath$\rho$}\right]$. The first practical
consequence of this decomposition yields the mean value
$\langle{\bf O}\rangle\mathrel{\mathop{:}}=\mbox{${\rm Tr}$}[{\bf
O}\mbox{\boldmath$\rho$}]=\frac{1}{{\rm
N}}\sum_{\mu,\nu=-\ell}^{\ell}\mathscr{O}(\mu,\nu)\mathscr{W}_{\rho}(\mu,\nu),$
(10)
whose formal expression depends explicitly on the product of both mapped forms
related to ${\bf O}$ and $\rho$. Consequently, the moments $\langle{\bf
J}_{a}^{k}\rangle$ for $a=x,y,z$ and $0\leq k\leq 2j$ can now be promptly
obtained from this formalism for any finite quantum state belonging to
$\mathcal{H}_{2j+1}$. Next, we apply such a mapping technique for the spin
coherent states in order to establish an important link between angular-
momentum operators and $(2j+1)^{2}$-dimensional discrete phase spaces.
### 3.1 Applications
According to a prescription adopted in Ref. GP3, let
$\\{|j,m\rangle\\}_{-j\leq m\leq j}$ denote the eigenstates of ${\bf U}$ with
eigenvalues $\\{\omega^{m}\\}$ for
$\omega=\exp\left(\frac{2\pi\mathrm{i}}{2j+1}\right)$ fixed. This assumption
leads us to establish the general properties ${\bf
U}^{\eta}|j,m\rangle=\omega^{m\eta}|j,m\rangle$ and ${\bf
V}^{\xi}|j,m\rangle=|j,m-\xi\rangle$ (together with the relations ${\bf
U}^{2j+1}={\bf 1}$, ${\bf V}^{2j+1}={\bf 1}$, and ${\bf V}^{\xi}{\bf
U}^{\eta}=\omega^{\eta\xi}{\bf U}^{\eta}{\bf V}^{\xi}$), which represent a
satisfactory mathematical connection for the present purpose. Next, we
determine some useful results related to the angular-momentum operators and
spin coherent states, remembering that ${\bf G}^{\dagger}(\mu,\nu)={\bf
G}(\mu,\nu)$ for $-\ell\leq\mu,\nu\leq\ell$.
As a first and pertinent application, let us consider the mapped expressions
$\displaystyle\left({\bf J}_{z}^{k}\right)(\mu,\nu)$ $\displaystyle=$
$\displaystyle\sum_{m=-j}^{j}m^{k}\underbrace{\langle j,m|{\bf
G}(\mu,\nu)|j,m\rangle}_{\mathscr{G}_{j,m|j,m}(\mu,\nu)}$
$\displaystyle\left({\bf J}_{+}^{k}\right)(\mu,\nu)$ $\displaystyle=$
$\displaystyle\sum_{m=-j}^{j-k}\frac{(j+m+k)!}{(j+m)!}\left(\frac{\mathscr{C}_{2j,j+m+k}}{\mathscr{C}_{2j,j+m}}\right)^{\frac{1}{2}}\underbrace{\langle
j,m|{\bf G}(\mu,\nu)|j,m+k\rangle}_{\mathscr{G}_{j,m|j,m+k}(\mu,\nu)}$
$\displaystyle\left({\bf J}_{-}^{k}\right)(\mu,\nu)$ $\displaystyle=$
$\displaystyle\sum_{m=-j+k}^{j}\frac{(j-m+k)!}{(j-m)!}\left(\frac{\mathscr{C}_{2j,j+m-k}}{\mathscr{C}_{2j,j+m}}\right)^{\frac{1}{2}}\underbrace{\langle
j,m|{\bf G}(\mu,\nu)|j,m-k\rangle}_{\mathscr{G}_{j,m|j,m-k}(\mu,\nu)},$
which depend on the nondiagonal matrix elements
$\mathscr{G}_{j,m|j,m^{\prime}}(\mu,\nu)=\frac{1}{2j+1}\sum_{\beta=-j}^{j}\exp\left\\{-\frac{2\pi\mathrm{i}}{2j+1}\left[\beta(\mu-m^{\prime})+(m^{\prime}-m)\left(\nu+\frac{\beta}{2}\right)\right]\right\\}$
associated with the discrete mapping kernel ${\bf G}(\mu,\nu)$ for
$m^{\prime}=m\pm k$ and $0\leq k\leq 2j$. Since the diagonal matrix element
$\mathscr{G}_{j,m|j,m}(\mu,\nu)=\delta_{m,\mu}^{[2j+1]}$ gives the Kronecker
delta function (in such a case, the superscript $[2j+1]$ denotes that this
function is different from zero when its labels are congruent modulo $2j+1$),
it seems reasonable to obtain a simple expression in the first example, that
is $\left({\bf J}_{z}^{k}\right)(\mu,\nu)=\mu^{k}$; however, this fact is not
verified for $\left({\bf J}_{\pm}^{k}\right)(\mu,\nu)$. From the kinematical
and dynamical point of view, these mapped expressions represent an
advantageous set of mathematical tools which allows us to comprehend certain
intriguing problems related to discrete symmetries of spin systems.[18]
Basically, the second application consists in evaluating the discrete Wigner
function $\mathscr{W}_{\theta,\varphi}(\mu,\nu)$ present in the expansion of
the projector ${\bf P}(\theta,\varphi)$ via Eq. (9), namely,
${\bf
P}(\theta,\varphi)=\frac{1}{2j+1}\sum_{\mu,\nu=-j}^{j}\mathscr{W}_{\theta,\varphi}(\mu,\nu){\bf
G}(\mu,\nu).$
In general, this particular quasidistribution can be written in terms of its
respective discrete Weyl function
$\widetilde{\mathscr{W}}_{\theta,\varphi}(\eta,\xi)\mathrel{\mathop{:}}=\mbox{${\rm
Tr}$}[{\bf S}(\eta,\xi){\bf P}(\theta,\varphi)]$ as follows:
$\\!\\!\\!\\!\\!\mathscr{W}_{\theta,\varphi}(\mu,\nu)=\frac{1}{\sqrt{2j+1}}\sum_{\mu,\nu=-j}^{j}\exp\left[-\frac{2\pi\mathrm{i}}{2j+1}(\eta\mu+\xi\nu)\right]\widetilde{\mathscr{W}}_{\theta,\varphi}(\eta,\xi).$
(11)
So, after some lengthy calculations, the analytical expression for
$\widetilde{\mathscr{W}}_{\theta,\varphi}(\eta,\xi)$ assumes the exact form
$\widetilde{\mathscr{W}}_{\theta,\varphi}(\eta,\xi)=\frac{1}{\sqrt{2j+1}}\sum_{m=-j}^{j}\exp\left[\frac{2\pi\mathrm{i}}{2j+1}\eta\left(m-\frac{\xi}{2}\right)\right]\langle\theta,\varphi|j,m-\xi\rangle\langle
j,m|\theta,\varphi\rangle,$
where $\langle j,m|\theta,\varphi\rangle$ can be promptly obtained from Eq.
(5). It is interesting to stress that both the north $(\theta=\pi)$ and south
$(\theta=0)$ poles of the Majorana-Bloch sphere correspond to the limit cases
$\mathscr{W}_{\pi,\varphi}(\mu,\nu)=\delta_{\mu,j}^{[2j+1]}$ and
$\mathscr{W}_{0,\varphi}(\mu,\nu)=\delta_{\mu,-j}^{[2j+1]}$, which reflect
directly the highest/lowest states $|j,\pm j\rangle$ for any
$\varphi\in[0,2\pi)$. Furthermore, numerical calculations related to Eq. (11)
suggest that some relevant contributions are connected with discrete values of
the angles $\theta$ and $\varphi$ — in particular, integer multiples of
$\frac{\pi}{2j+1}$. This evidence supports the ideas of Buniy and
coworkers[52] about a ‘possible discretization’ of the Majorana-Bloch sphere,
as well as reveals some important features — and not yet properly explored —
on the discrete nature of quantum states in finite-dimensional Hilbert
spaces.333In the quantum-gravity scope,[53] it is worth mentioning that
certain theoretical approaches to the generalized uncertainty principle (GUP)
also suggest the breakdown of the spacetime continuum picture near to Planck
scale.[54] From an experimental point of view, it seems reasonable to argue
that ‘discreteness is actually less speculative than absolute continuity’.
### 3.2 Time evolution
Here, we establish a mathematical recipe that permits to investigate the
dynamics of a particular quantum system characterized by a finite space of
(discrete) states. For this task, let $\mbox{\boldmath$\rho$}(t)$ describe the
state of this physical system whose interaction with any dissipative
environment is, in principle, automatically discarded. Besides, let us
consider (for convenience) only time-independent Hamiltonians; consequently,
the time-evolution of that density operator will be governed, in such a case,
by the well-known von Neumann-Liouville equation
$\mathrm{i}\hbar\partial_{t}\mbox{\boldmath$\rho$}(t)=[{\bf
H},\mbox{\boldmath$\rho$}(t)]$. So, those initial premises represent the
constituting blocks for the aforementioned recipe.
As a first proposal in our prescription, we determine a mapped expression for
the von Neumann-Liouville equation which describes the time evolution of the
discrete Wigner function $\mathscr{W}_{\rho}(\mu,\nu;t)$. In this sense, the
mapping technique sketched in this section yields the differential equation
$\mathrm{i}\hbar\partial_{t}\mathscr{W}_{\rho}(\mu,\nu;t)=\sum_{\mu^{\prime},\nu^{\prime}=-\ell}^{\ell}\mathscr{L}_{\mbox{\tiny${\rm
H}$}}(\mu,\nu,\mu^{\prime},\nu^{\prime})\mathscr{W}_{\rho}(\mu^{\prime},\nu^{\prime};t),$
(12)
where $\mathscr{L}_{\mbox{\tiny${\rm H}$}}(\mu,\nu,\mu^{\prime},\nu^{\prime})$
represents the mapped form of the Liouville operator written in terms of
$\mathit{H}(\mu^{\prime\prime},\nu^{\prime\prime})=\mbox{${\rm Tr}$}\left[{\bf
G}(\mu^{\prime\prime},\nu^{\prime\prime}){\bf H}\right]$, that is,
$\displaystyle\mathscr{L}_{\mbox{\tiny${\rm
H}$}}(\mu,\nu,\mu^{\prime},\nu^{\prime})$ $\displaystyle=$
$\displaystyle\frac{2\mathrm{i}}{{\rm
N}^{4}}\sum_{\Delta}\sin\left[\frac{\pi}{{\rm
N}}(\alpha\beta^{\prime}-\alpha^{\prime}\beta)\right]\exp\left\\{\frac{2\pi\mathrm{i}}{{\rm
N}}\left[\alpha(\mu-\mu^{\prime})+\beta(\nu-\nu^{\prime})\right]\right\\}$
$\displaystyle\times\exp\left\\{\frac{2\pi\mathrm{i}}{{\rm
N}}\left[\alpha^{\prime}(\mu-\mu^{\prime\prime})+\beta^{\prime}(\nu-\nu^{\prime\prime})\right]\right\\}\mathit{H}(\mu^{\prime\prime},\nu^{\prime\prime}).$
Note that $\Delta$ denotes the set
$\\{\alpha,\beta,\alpha^{\prime},\beta^{\prime},\mu^{\prime\prime},\nu^{\prime\prime}\\}\in[-\ell,\ell]$
in this expression. In the following, let us mention some few words about the
formal solution of Eq. (12): it can be expressed analytically in terms of the
series[20]
$\mathscr{W}_{\rho}(\mu,\nu;t)=\sum_{\kappa,\tau=-\ell}^{\ell}\mathscr{P}(\mu,\nu;t|\kappa,\tau;t_{0})\mathscr{W}_{\rho}(\kappa,\tau;t_{0}),$
(13)
whose ${\rm N}^{2}$-dimensional discrete phase-space propagator admits the
expansion
$\displaystyle\mathscr{P}(\mu,\nu;t|\kappa,\tau;t_{0})$ $\displaystyle=$
$\displaystyle\delta_{\kappa,\mu}^{[\mbox{\tiny${\rm
N}$}]}\delta_{\tau,\nu}^{[\mbox{\tiny${\rm
N}$}]}+\frac{\mathrm{i}}{1!\hbar}(t-t_{0})\mathscr{L}_{\mbox{\tiny${\rm
H}$}}(\mu,\nu,\kappa,\tau)$
$\displaystyle+\frac{\mathrm{i}^{2}}{2!\hbar^{2}}(t-t_{0})^{2}\sum_{\kappa^{\prime},\tau^{\prime}=-\ell}^{\ell}\mathscr{L}_{\mbox{\tiny${\rm
H}$}}(\mu,\nu,\kappa^{\prime},\tau^{\prime})\mathscr{L}_{\mbox{\tiny${\rm
H}$}}(\kappa^{\prime},\tau^{\prime},\kappa,\tau)+\cdots$
which allows to evaluate directly the time evolution of
$\mathscr{W}_{\rho}(\mu,\nu;t)$ by using the series related to the iterated
application of the mapped Liouville operator. The advantages and/or
disadvantages from this particular formal solution were adequately discussed
in Ref. GR1, and subsequently applied with great success in the discrete
Husimi-function context for the LMG model.[23]
The alternative proposal establishes a differential equation for the discrete
Weyl function analogous to Eq. (12). Then, after some calculations related to
the product of three symmetrized unitary operator bases and its respective
trace operation,[19] it is immediate to show that such an equation can be
written as
$\mathrm{i}\hbar\partial_{t}\widetilde{\mathscr{W}}_{\rho}(\eta,\xi;t)=\sum_{\eta^{\prime},\xi^{\prime}=-\ell}^{\ell}\mathcal{L}_{\mbox{\tiny${\rm
H}$}}(\eta,\xi,\eta^{\prime},\xi^{\prime})\widetilde{\mathscr{W}}_{\rho}(\eta^{\prime},\xi^{\prime};t),$
(14)
where
$\widetilde{\mathscr{W}}_{\rho}(\eta,\xi;t)\mathrel{\mathop{:}}=\mbox{${\rm
Tr}$}\left[{\bf S}(\eta,\xi)\mbox{\boldmath$\rho$}(t)\right]$ corresponds to
our object of study and
$\mathcal{L}_{\mbox{\tiny${\rm
H}$}}(\eta,\xi,\eta^{\prime},\xi^{\prime})=\frac{2\mathrm{i}}{\sqrt{N}}\sin\left[\frac{\pi}{N}\left(\eta^{\prime}\xi-\xi^{\prime}\eta\right)\right]\underbrace{\mathcal{H}(\eta-\eta^{\prime},\xi-\xi^{\prime})}_{\mathrm{Tr}\left[{\bf
S}(\eta-\eta^{\prime},\xi-\xi^{\prime}){\bf H}\right]}$
yields the mapped forms of the Liouville and Hamiltonian operators. As
expected, Eq. (14) describes the time evolution of the discrete Weyl function
whose formal solution is given by
$\widetilde{\mathscr{W}}_{\rho}(\eta,\xi;t)=\sum_{\eta^{\prime},\xi^{\prime}=-\ell}^{\ell}\mathcal{P}(\eta,\xi;t|\eta^{\prime},\xi^{\prime};t_{0})\widetilde{\mathscr{W}}_{\rho}(\eta^{\prime},\xi^{\prime};t_{0}),$
(15)
with $\mathcal{P}(\eta,\xi;t|\eta^{\prime},\xi^{\prime};t_{0})$ being the
discrete dual phase-space propagator which admits a time expansion similar to
$\mathscr{P}(\mu,\nu;t|\kappa,\tau;t_{0})$, that is
$\displaystyle\mathcal{P}(\eta,\xi;t|\eta^{\prime},\xi^{\prime};t_{0})$
$\displaystyle=$ $\displaystyle\delta_{\eta^{\prime},\eta}^{[\mbox{\tiny${\rm
N}$}]}\delta_{\xi^{\prime},\xi}^{[\mbox{\tiny${\rm
N}$}]}+\frac{\mathrm{i}}{1!\hbar}(t-t_{0})\mathcal{L}_{\mbox{\tiny${\rm
H}$}}(\eta,\xi,\eta^{\prime},\xi^{\prime})$
$\displaystyle+\frac{\mathrm{i}^{2}}{2!\hbar^{2}}(t-t_{0})^{2}\sum_{\eta^{\prime\prime},\xi^{\prime\prime}=-\ell}^{\ell}\mathcal{L}_{\mbox{\tiny${\rm
H}$}}(\eta,\xi,\eta^{\prime\prime},\xi^{\prime\prime})\mathcal{L}_{\mbox{\tiny${\rm
H}$}}(\eta^{\prime\prime},\xi^{\prime\prime},\eta^{\prime},\xi^{\prime})+\cdots.$
Since the discrete Wigner and Weyl functions are connected by means of a
double Fourier transform, the low operational costs involved in this
mathematical recipe are advantageous — from a computational point of view — if
compared with the previous one. Notwithstanding the apparent advantage, it is
worth remembering that $\widetilde{\mathscr{W}}_{\rho}(\eta,\xi;t)$ is a
complex function and hence it represents an intermediate step for the main
goal, namely, the time evolution of the discrete Wigner function — and
consequently, the time evolution of certain mean values via Eq. (10). The
schematic diagram shown below
(24)
represents a summary of the previous proposals for describing the dynamics of
a finite quantum system represented in a ${\rm N}^{2}$-dimensional discrete
phase space.[20]
## 4 Spin squeezing and entanglement
This section will illustrate how the mapping techniques leading to finite-
dimensional discrete phase spaces can be effectively used in the study of
important quantum effects, such as spin squeezing and entanglement. Basically,
we will invoke the LMG model already discussed in current literature (e.g.,
see Ref. Ring) which exhibits not only strong mathematical and physical
appeals,[23] but also a fundamental physical property: quantum correlations
due to different types of interactions and mediated by a transverse magnetic
field. Since quantum correlations are responsible for the aforementioned
effects, it is natural to investigate how the interactions involved in this
model affect the standard quantum noise related to the spin coherent
states.444In Appendix B, we investigate a soluble spin model with lowest-order
nonlinear interaction in ${\bf J}_{z}$,[13] whose algebraic features permit to
attain a particular set of analytical results very useful in the study of spin
squeezing and entanglement effects via quantum correlations.[11]
### 4.1 The modified Lipkin-Meshkov-Glick model
Here, we adopt the prescription established by Vidal and coworkers[29] for the
modified LMG model through the Hamiltonian operator (written in the spin
language)
${\bf H}=-2h{\bf J}_{z}-2\wp_{+}\left[\vec{{\bf J}}^{2}-{\bf
J}_{z}^{2}-(N/2){\bf I}\right]-\wp_{-}\left({\bf J}_{+}^{2}+{\bf
J}_{-}^{2}\right),$ (25)
which describes a set of $N$ spins half mutually interacting in the $xy$ plane
subjected to a transverse magnetic field $h$. The coefficients
$\wp_{\pm}=\frac{\lambda}{2N}(1\pm\gamma)$ have an important role in this
description: they allow to (i) investigate the different anti-ferromagnetic
$(\lambda<0)$ and ferromagnetic $(\lambda>0)$ cases inherent to the model for
any anisotropy parameter $|\gamma|\leq 1$ ($\gamma=1$ refers to the isotropic
case), and also (ii) ensure that the free energy per spin is finite in the
thermodynamical limit. In particular, such a model presents a second-order
quantum phase transition at $\lambda=|h|$ for $\lambda>0$ fixed, whose
symmetric $(\lambda<|h|)$ and broken $(\lambda>|h|)$ phases are well-defined
within the mean-field approach; furthermore, its ground-state entanglement
properties exhibit a rich structure which reflects the internal symmetries of
the Hamiltonian operator ${\bf H}$.[29, 55] Indeed, such an operator preserves
the magnitude of the total spin operator since $[{\bf H},\vec{{\bf J}}^{2}]=0$
for all $\gamma$, and does not couple states having a different number of
spins pointing in the field direction, namely,
$\biggl{[}{\bf
H},\prod_{j=1}^{N}\mbox{\boldmath$\sigma$}_{j,z}\biggr{]}=0\qquad(\mbox{spin-
flip symmetry}).$
Consequently, it is immediate to verify that Eq. (25) can be diagonalized
within each $(2j+1)$-dimensional multiplet labelled by the eigenvalues of
$\vec{{\bf J}}^{2}$ and ${\bf J}_{z}$, this fact being responsible for the
soluble character of the associated spin model.555It is important to stress
that the prescritpion here adopted does not coincide with the original Lipkin-
Meshkov-Glick model[28], which was initially introduced over 40 years ago in
nuclear physics[40] for treating certain fermionic systems. In fact, this new
modified version brings to scene the modern statistical mechanics point of
view, where the collective properties of spin systems can be worked out with
great success.[29, 55] The extended original version of the LMG model, which
explores the parity symmetry via finite-dimensional discrete phase spaces, can
be found in Ref. GP3.
Next, let us mention some few words about two discrete conserved quantities
inherent to the spin model under investigation which reflect certain
additional symmetry properties for $\gamma=1$ fixed. In this case, the
simplest operator commuting with ${\bf H}$, therefore giving a constant of
motion, is the parity operator $\Pi$ — here defined as
$\mbox{\boldmath$\Pi$}\mathrel{\mathop{:}}={\bf
R}_{z}(\pi)\equiv\exp(\mathrm{i}\pi{\bf J}_{z})$. This result tells us that
the Hamiltonian matrix, in the ${\bf J}_{z}$ representation, breaks into two
disjoint blocks involving only even and odd eigenvalues of ${\bf J}_{z}$,
respectively. The second interesting quantity comes from the anticommutation
relation $\\{{\bf H},{\bf R}\\}=0$ for ${\bf R}(\pi,0,\pi/2)\equiv{\bf
R}_{x}(\pi){\bf R}_{z}(\pi/2)$: it corresponds to a particular rotation of the
angular-momentum quantization frame by the Euler angles $(\pi,0,\pi/2)$,
transforming, in this way, ${\bf H}\rightarrow-{\bf H}$ for the isotropic
case. Thus, if $|E_{j}\rangle$ is an energy eigenstate with eigenvalue
$E_{j}$, then ${\bf R}|E_{j}\rangle$ is also an eigenstate of ${\bf H}$ with
eigenvalue $-E_{j}$. This specific symmetry property gives rise to an energy
spectrum that is symmetric about zero.[18, 23, 29]
After this condensed review, we establish below a sequence of steps that
allows us to calculate the time evolution of several quantities necessary for
investigating both the squeezing and entanglement effects associated with the
aforementioned spin model in terms of the parameters $h$ and $\wp_{\pm}$. The
first one consists in adopting the theoretical framework described in the
previous section for the time-dependent discrete Wigner function
$\mathscr{W}_{\rho}(\mu,\nu;t)$ defined upon a $(2j+1)^{2}$-dimensional
discrete phase space labeled by the angular-momentum and angle pair
$(\mu,\nu)\in[-j,j]$ — in particular, we adopt the theoretical prescription
established in Ref. GR1 for the angle variable, that is
$\theta_{\nu}=\frac{2\pi}{2j+1}\nu$. Since this approach depends on the Wigner
function evaluated at time $t_{0}=0$, the second step consists in fixing the
spin coherent states as initial state — see Eq. (11). The last step refers to
the numerical calculation of the moments
$\langle{\bf
J}_{a}^{k}\rangle_{\theta,\varphi}(t)=\frac{1}{2j+1}\sum_{\mu,\nu=-j}^{j}\left({\bf
J}_{a}^{k}\right)(\mu,\nu)\mathscr{W}_{\theta,\varphi}(\mu,\nu;t)$ (26)
and covariance functions in the Schrödinger picture fixing the initial state
in $\theta=\frac{\pi}{2}$ and $\varphi=0$ (in this way, the uncertainties are
redistributed between the orthogonal components in the $yz$-plane), and also
considering (for convenience, not necessity) only the ferromagnetic case
$\lambda=1$. Henceforth, Eq. (25) will assume the simplified form ${\bf
H}^{\prime}=-h{\bf J}_{z}-\frac{1}{N}\left({\bf J}_{x}^{2}+\gamma{\bf
J}_{y}^{2}\right)$, where the constant term $\frac{1}{4}(1+\gamma){\bf I}$ was
suppressed at this initial stage since it represents only a phase for the time
evolution operator with null contribution (as expected) within our
computational approach.
### 4.2 The match between squeezing and entanglement effects
In this subsection, we adopt the theoretical framework established for the
Kitagawa-Ueda model (see Appendix B) concerning the match between squeezing
and entanglement effects. In particular, let us initially consider the results
obtained in Table 4.2 for the RS uncertainty principle related to the angular-
momentum generators and its connections with the $\mathcal{S}$-inequalities.
Since the covariance function $\mathscr{V}_{\mathrm{J}_{a}\mathrm{J}_{b}}(t)$
has a central role in this algebraic approach and requires only numerical
computations of $\langle{\bf J}_{a}^{k}\rangle_{\theta,\varphi}(t)$ and
$\langle\\{{\bf J}_{a},{\bf J}_{b}\\}\rangle_{\theta,\varphi}(t)$, it is
convenient to clarify certain subtle steps that are inherent to the exact
analytical calculation of the last quantity. According to the mapping
technique discussed in Section 3, the expression for
$\\!\\!\\!\\!\\!\langle\\{{\bf J}_{a},{\bf
J}_{b}\\}\rangle_{\theta,\varphi}(t)=\frac{1}{2j+1}\sum_{\mu,\nu=-j}^{j}\left(\\{{\bf
J}_{a},{\bf J}_{b}\\}\right)(\mu,\nu)\mathscr{W}_{\theta,\varphi}(\mu,\nu;t)$
(27)
basically depends on the mapped form of the anticommutation relation, that is
$\left(\\{{\bf J}_{a},{\bf
J}_{b}\\}\right)(\mu,\nu)=\frac{1}{(2j+1)^{2}}\sum_{\Omega}\Gamma_{{\rm
A}}(\mu,\nu|\mu^{\prime},\nu^{\prime},\mu^{\prime\prime},\nu^{\prime\prime})\left({\bf
J}_{a}\right)(\mu^{\prime},\nu^{\prime})\left({\bf
J}_{b}\right)(\mu^{\prime\prime},\nu^{\prime\prime})$
where $\Omega$ stands for
$\\{\mu^{\prime},\nu^{\prime},\mu^{\prime\prime},\nu^{\prime\prime}\\}\in[-j,j]$.
It is worth stressing that[19]
$\displaystyle\Gamma_{{\rm
A}}(\mu,\nu|\mu^{\prime},\nu^{\prime},\mu^{\prime\prime},\nu^{\prime\prime})=\frac{2}{(2j+1)^{2}}\sum_{\Omega^{\prime}}\exp\left\\{\frac{2\pi\mathrm{i}}{2j+1}\left[\eta^{\prime}(\mu-\mu^{\prime})+\xi^{\prime}(\nu-\nu^{\prime})\right]\right\\}$
$\displaystyle\qquad\times\exp\left\\{\frac{2\pi\mathrm{i}}{2j+1}\left[\eta^{\prime\prime}(\mu-\mu^{\prime\prime})+\xi^{\prime\prime}(\nu-\nu^{\prime\prime})\right]\right\\}\cos\left[\frac{\pi}{2j+1}(\eta^{\prime}\xi^{\prime\prime}-\xi^{\prime}\eta^{\prime\prime})\right]$
shows explicitly the embryonic structure of the continuous cosine function
presents in the well-known Weyl-Wigner-Moyal phase space approach[56] — here,
$\Omega^{\prime}$ denotes the set
$\\{\eta^{\prime},\xi^{\prime},\eta^{\prime\prime},\xi^{\prime\prime}\\}\in[-j,j]$.
Thus, the $\mathcal{S}$-inequalities can be properly estimated for the
modified LMG model, which lead us to investigate the squeezing effects. For
completeness sake, let us briefly mention that $\langle[{\bf J}_{a},{\bf
J}_{b}]\rangle_{\theta,\varphi}(t)$ yields an expression analogous to Eq.
(27), but with two minor modifications: the function $\cos(z)$ in
$\Gamma_{{\rm A}}$ should be replaced by $\mathrm{i}\sin(z)$ (keeping constant
the argument of the trigonometric functions), which implies in the change
$\Gamma_{{\rm A}}\rightarrow\Gamma_{{\rm C}}$.
The $\mathcal{S}$-inequalities exhibit unique mathematical virtues since they
yield a direct connection with the RS uncertainty principle related to the
non-commuting pair $\\{{\bf J}_{a},{\bf J}_{b}\\}_{a,b=x,y,z}$ of angular-
momentum operators, where the covariance functions have an important role.
Indeed, such inequalities lead us to investigate the squeezing effects through
a well-established criterion in literature: for $\mathcal{S}_{a}^{(c)}<1$
(squeezing condition) and $\mathcal{S}_{b}^{(c)}>1$, the inequality
$\mathcal{S}_{a}^{(c)}\mathcal{S}_{b}^{(c)}>1$ is always preserved; moreover,
the saturation $\mathcal{S}_{a}^{(c)}\mathcal{S}_{b}^{(c)}=1$ describes
minimum uncertainty states. In this table, we show all the possible links
among RS uncertainty principles and $\mathcal{S}$-inequalities, with
$\mathscr{V}_{\mathrm{J}_{a}\mathrm{J}_{b}}$ restricted to the closed interval
$\left[-\sqrt{\mathscr{V}_{\mathrm{J}_{a}}\mathscr{V}_{\mathrm{J}_{b}}},\sqrt{\mathscr{V}_{\mathrm{J}_{a}}\mathscr{V}_{\mathrm{J}_{b}}}\,\right]$;
moreover, the superscript $(c)$ of the product
$\mathcal{S}_{a}^{(c)}\mathcal{S}_{b}^{(c)}$ denotes the angular-momentum
component resulting from the commutation relation between ${\bf J}_{a}$ and
${\bf J}_{b}$. RS uncertainty principles $\mathcal{R}$-denominators
$\mathcal{S}$-inequalities $\\{{\bf J}_{x},{\bf J}_{y},{\bf
J}_{z}\\}\in\mathcal{H}_{\mbox{\tiny${\rm N}$}}$ $\mathcal{R}_{abc}\neq
0\;\forall\mbox{\boldmath$\rho$}\in\mathcal{H}_{\mbox{\tiny${\rm N}$}}$
$\mathcal{S}_{a}^{(c)}\mathcal{S}_{b}^{(c)}\mathrel{\mathop{:}}=\mathscr{V}_{\mathrm{J}_{a}}\mathscr{V}_{\mathrm{J}_{b}}/\mathcal{R}_{abc}^{2}$
$\mathscr{V}_{\mathrm{J}_{x}}\mathscr{V}_{\mathrm{J}_{y}}-\left(\mathscr{V}_{\mathrm{J}_{x}\mathrm{J}_{y}}\right)^{2}\geq\frac{1}{4}|\langle{\bf
J}_{z}\rangle|^{2}$
$\mathcal{R}_{xyz}\mathrel{\mathop{:}}=\left[(\mathscr{V}_{\mathrm{J}_{x}\mathrm{J}_{y}})^{2}+\frac{1}{4}|\langle{\bf
J}_{z}\rangle|^{2}\right]^{\frac{1}{2}}$
$\mathcal{S}_{x}^{(z)}\mathcal{S}_{y}^{(z)}\geq 1$
$\mathscr{V}_{\mathrm{J}_{x}}\mathscr{V}_{\mathrm{J}_{z}}-\left(\mathscr{V}_{\mathrm{J}_{x}\mathrm{J}_{z}}\right)^{2}\geq\frac{1}{4}|\langle{\bf
J}_{y}\rangle|^{2}$
$\mathcal{R}_{xzy}\mathrel{\mathop{:}}=\left[(\mathscr{V}_{\mathrm{J}_{x}\mathrm{J}_{z}})^{2}+\frac{1}{4}|\langle{\bf
J}_{y}\rangle|^{2}\right]^{\frac{1}{2}}$
$\mathcal{S}_{x}^{(y)}\mathcal{S}_{z}^{(y)}\geq 1$
$\mathscr{V}_{\mathrm{J}_{y}}\mathscr{V}_{\mathrm{J}_{z}}-\left(\mathscr{V}_{\mathrm{J}_{y}\mathrm{J}_{z}}\right)^{2}\geq\frac{1}{4}|\langle{\bf
J}_{x}\rangle|^{2}$
$\mathcal{R}_{yzx}\mathrel{\mathop{:}}=\left[(\mathscr{V}_{\mathrm{J}_{y}\mathrm{J}_{z}})^{2}+\frac{1}{4}|\langle{\bf
J}_{x}\rangle|^{2}\right]^{\frac{1}{2}}$
$\mathcal{S}_{y}^{(x)}\mathcal{S}_{z}^{(x)}\geq 1$
In order to illustrate the theoretical framework developed in this work, let
us now consider the $\mathcal{S}$-inequality
$\mathcal{S}_{y}^{(x)}\mathcal{S}_{z}^{(x)}\geq 1$ and the
$\mathcal{E}$-inequalities $\\{\mathcal{E}_{y},\mathcal{E}_{z}\\}$ described
in Appendix B by means of Eq. (42). In particular, this procedure allows not
only to establish a direct link between squeezing and entanglement effects for
the modified LMG model, but also to investigate how these effects are affected
by the transverse magnetic field $h$ and anisotropy parameter $\gamma$ with
$N=20$ fixed. For instance, Fig. 2(a,c) represents the plots of
$\mathcal{S}_{y}^{(x)}(t)$ (dot-dashed line) and $\mathcal{S}_{z}^{(x)}(t)$
(solid line) versus $t\in[0,50]$ for different values of $(h,\gamma)$: (a)
$(-0.1,0.2)$ and (c) $(-0.13,0.1)$. It is interesting to observe at a first
glance how the squeezing effect is sensitive to small variations in the
parameters $|h|$ and $\gamma$. Further numerical computations corroborate this
sensitivity and allow us to produce the following proper description about
such an evidence as time goes on: for $\gamma>|h|$, both the parameters
$\mathcal{S}_{y}^{(x)}(t)$ and $\mathcal{S}_{z}^{(x)}(t)$ exhibit squeezing
effect and oscillatory behaviour due to the prevalence of quantum correlation
effects (here introduced by the anisotropy parameter $\gamma$) on the
transverse magnetic field $h$; however, if one considers $\gamma<|h|$, the
squeezing effect is strongly reduced (in fact, it always survives for small
values of time) in both the parameters, which reveals the relative strong
influence of $h$ on the quantum correlation effects. With this in mind, let us
now consider Fig. 2(b,d) where the plots of $\mathcal{E}_{y}(t)$ (dot-dashed
line) and $\mathcal{E}_{z}(t)$ (solid line) are exhibited for the same values
of $(h,\gamma)$ used, respectively, in 2(a,c). The ‘almost perfect match’
between squeezing and entanglement effects demonstrate the previous assertions
and indeed reinforce the fundamental sequence correlation $\mapsto$
entanglement $\mapsto$ squeezing of quantum effects.
Figure 2: The ‘almost perfect match’ between squeezing and entanglement
effects is clearly visible in this selection of pictures, where the parameters
(a,c) $\mathcal{S}_{y}^{(x)}(t)$ (dot-dashed line) and
$\mathcal{S}_{z}^{(x)}(t)$ (solid line) as well as (b,d) $\mathcal{E}_{y}(t)$
(dot-dashed line) and $\mathcal{E}_{z}(t)$ (solid line) are plotted as
functions of time and placed side by side for immediate comparision.
Furthermore, this set of plots corresponds to two different values of
$(h,\gamma)$, namely, (a,b) $(-0.1,0.2)$ and (c,d) $(-0.13,0.1)$, which leads
us to investigate how the quantum correlations are affected by the transverse
magnetic field $h$ and anisotropy parameter $\gamma$ via
$(2j+1)^{2}$-dimensional discrete phase space inherent to the modified LMG
model.
At this moment, we pay special attention to the results obtained by Tóth and
co-workers[10] where a full set of generalized spin-squeezing inequalities for
entanglement detection was derived and studied in detail. For such task, let
us formally establish an entanglement criterion for separable states through a
set of inequalities involving the mean values $\left\\{\langle{\bf
J}_{a}\rangle,\langle{\bf J}_{a}^{2}\rangle\right\\}_{a=x,y,z}$ that
summarizes those aforementioned results.
Entanglement criterion.
Let $\rho$ describe an $N$-particle density operator, as well as
$\left\\{\langle{\bf J}_{x}\rangle,\langle{\bf J}_{y}\rangle,\langle{\bf
J}_{z}\rangle,\langle{\bf J}_{x}^{2}\rangle,\langle{\bf
J}_{y}^{2}\rangle,\langle{\bf J}_{z}^{2}\rangle\right\\}$ characterize the
mean values of collective angular-momentum operators associated with the
physical system of interest which, by hypothesis, are previously known. For
separable states represented by the convex sum
$\mbox{\boldmath$\rho$}=\sum_{k}p_{k}\mbox{\boldmath$\rho$}_{k}^{(1)}\otimes\mbox{\boldmath$\rho$}_{k}^{(2)}\otimes\ldots\otimes\mbox{\boldmath$\rho$}_{k}^{(N-1)}\otimes\mbox{\boldmath$\rho$}_{k}^{(N)}$,
where $p_{k}$ features the probability distribution for a given $k$, one
verifies that
$\displaystyle\langle{\bf J}_{x}^{2}\rangle+\langle{\bf
J}_{y}^{2}\rangle+\langle{\bf J}_{z}^{2}\rangle$ $\displaystyle\leq$
$\displaystyle\frac{1}{4}N(N+2)$ (28)
$\displaystyle\frac{1}{2}(N+2)\left(\mathscr{V}_{\mathrm{J}_{x}}+\mathscr{V}_{\mathrm{J}_{y}}+\mathscr{V}_{\mathrm{J}_{z}}\right)$
$\displaystyle\geq$ $\displaystyle\frac{1}{4}N(N+2)$ (29)
$\displaystyle\frac{1}{2}(N+2)\left[\langle{\bf J}_{a}^{2}\rangle+\langle{\bf
J}_{b}^{2}\rangle-(N-1)\mathscr{V}_{\mathrm{J}_{c}}\right]$
$\displaystyle\leq$ $\displaystyle\frac{1}{4}N(N+2)$ (30)
$\displaystyle(N-1)\left(\mathscr{V}_{\mathrm{J}_{a}}+\mathscr{V}_{\mathrm{J}_{b}}\right)-\langle{\bf
J}_{c}^{2}\rangle+N$ $\displaystyle\geq$ $\displaystyle\frac{1}{4}N(N+2)$ (31)
are always fulfilled, $\frac{1}{4}N(N+2)$ being considered a lower/upper bound
in all cases, with $\\{a,b,c\\}$ labelling all the possible permutations of
$\\{x,y,z\\}$. Hence, the violation of any inequality (29)-(31) leads, in this
framework, to entangled states since Eq. (28) is valid for all quantum states
— indeed, such inequalities define a polytope in the three-dimensional
$(\langle{\bf J}_{x}^{2}\rangle,\langle{\bf J}_{y}^{2}\rangle,\langle{\bf
J}_{z}^{2}\rangle)$-space, where the separable states lie inside this
geometric representation. It is important to stress that physical systems with
particle-number fluctuations (i.e., the particle number $N$ is not constant,
such as in BEC experiments) are discarded in this criterion.[57]
###### Remark 4.1.
Let us now apply these inequalities to the modified LMG model with the main
aim of corroborating the previous results and presenting alternative forms for
detection of spin-squeezing and entanglement effects. Numerical computations
show that Eqs. (30) and (31) are not satisfied for determined values of time,
which leads us to choose one of them whose particular time evolution combines
with those exhibited by $\mathcal{S}$ and $\mathcal{E}$-parameters — see Fig.
2(a)-(d). So, let us define the parameter
$E_{a}\mathrel{\mathop{:}}=\frac{(N-1)\mathscr{V}_{\mathrm{J}_{a}}}{\langle{\bf
J}_{b}^{2}\rangle+\langle{\bf J}_{c}^{2}\rangle-\frac{N}{2}}\geq
1\qquad(a,b,c=x,y,z)$ (32)
based on Eq. (30), where the violation $E_{a}<1$ directly implies in the
quantum effects under investigation. Note that, compared with Eq. (42),
$E_{a}$ has subtle differences: it requires the sum of mean values related to
the angular-momentum-squared operators (see denominator of both expressions),
while the previous one involves only squared mean values of the angular-
momentum operators. Figure 3 shows the plots of $E_{y}(t)$ (dot-dashed line)
and $E_{z}(t)$ (solid line) versus $t\in[0,50]$ for the same set of values
associated with the transverse magnetic field and anisotropy parameters used
in the previous figure, i.e., (a) $(-0.1,0.2)$ and (b) $(-0.13,0.1)$. It is
important to stress that the functional characteristics of the involved
parameters in the study of spin-squeezing and entanglement effects, as well as
the associated entanglement dynamics to the modified LMG model, represent two
essential factors that could explain (since the time evolution also depends on
the initial state chosen for the physical system), in part, the similarities
of behaviour related to the $\mathcal{S}$, $\mathcal{E}$, and $E$-parameters
viewed in Figs. 2 and 3. In summary, the entanglement criterion conceived by
Tóth et al.[10] reinforces the ‘almost perfect match’ between spin-squeezing
and entanglement effects, which leads us to investigate, in this particular
case, the quantum correlation rules via discrete Wigner and Husimi
functions.666For completeness sake, we present a case study for $h=0$ (absence
of transverse magnetic field) in appendix C, whose numerical computations lead
us to discuss the real necessity of demanding the respective validity domains
of the entanglement criteria used in this work.
Figure 3: Plots of $E_{y}(t)$ (dot-dashed line) and $E_{z}(t)$ (solid line) as
a function of time for the same values of $(h,\gamma)$ used in the previous
figure, that is, (a) $(-0.1,0.2)$ and (b) $(-0.13,0.1)$. Here, two factors
immediately emerge from our considerations on these pictures: (i) their
similarities with the previous cases (emphasizing, mainly, the strong
influence of $h$ on the entanglement effects), and consequently, (ii) the
important role of quantum correlations in such measurements. It is worth
mentioning that the particle number $N$ used in the numerical calculations
remains the same (i.e., $N=20$), which preserves important basic quantum
effects (contrary to the thermodynamic limit).
Henceforth, let us mention some few words on the entanglement dynamics of the
Hamiltonian operator ${\bf H}^{\prime}$ (Schrödinger picture): it basically
involves the action of collective spin operators responsible for introducing
the quantum effects attributed to two-body correlations — here mediated by the
anisotropy parameter $\gamma$ — upon a nonseparable initial state, that is,
the collective spin-coherent state (5) with $\theta=\frac{\pi}{2}$ and
$\varphi=0$ fixed. In this sense, since the discrete Wigner function reflects
the action of the time-evolution operator ${\bf
U}(t)\mathrel{\mathop{:}}=\exp\left(-\frac{\mathrm{i}}{\hbar}{\bf
H}^{\prime}t\right)$ upon the initial state $|\frac{\pi}{2},0\rangle$ in a
$(2j+1)^{2}$-dimensional discrete phase space, it seems natural to investigate
its behaviour for those specific values of time where effectively occur the
spin-squeezing and entanglement effects.
### 4.3 Finite-dimensional discrete phase spaces: a case study of quantum
correlations via Wigner and Husimi functions
Initially, let us consider the time evolution of
$\mathscr{W}_{\frac{\pi}{2},0}(\mu,\nu;t)$ upon a $21^{2}$-dimensional phase
space labeled by the pair $(\mu,\nu)$ of dimensionless discrete angular-
momentum and angle variables,[20] where the particular operator ${\bf
H}^{\prime}=-h{\bf J}_{z}-\frac{1}{N}\left({\bf J}_{x}^{2}+\gamma{\bf
J}_{y}^{2}\right)$ will take place at this description. As usual, the
parameters employed in the numerical computations are the same as those
mentioned in figure 2(b), i.e., $N=20$ and $(h,\gamma)=(-0.1,0.2)$.
Furthermore, we choose six representative time values of $\mathcal{E}_{z}(t)$
which reflect the first local minimum and maximum points exhibited in figure
2(b). In this way, figure 4 shows the contour plots of
$\mathscr{W}_{\frac{\pi}{2},0}(\mu,\nu;t)$ versus $(\mu,\nu)\in[-10,10]$
where, in particular, (a) $t=0$ (initial Wigner function), (b) $t=2.15$ (first
local minimum point), (c) $t=4.75$ (first local maximum point), (d) $t=7.10$
(second local minimum point), (e) $t=9.05$ (second local maximum point), and
(f) $t=9.95$ (corresponds to $\mathcal{E}_{z}(9.95)\approx 1$). In all these
pictures, it is interesting to observe how quantum correlations associated
with two-body interaction term modify the initial correlations present in the
collective spin-coherent state $|\frac{\pi}{2},0\rangle$; besides, note that
both $\mu$ and $\nu$ were conveniently shifted in the snapshots. Next, let us
describe certain specific points inherent to the time evolution of the
discrete Wigner function.
Figure 4: (Color online) Time evolution of
$\mathscr{W}_{\frac{\pi}{2},0}(\mu,\nu;t)$ for the modified LMG model with
$N=20$ and $(h,\gamma)=(-0.1,0.2)$ fixed, where the labels $\mu$ and $\nu$
characterize, respectively, the dimensionless angular-momentum and angle pair.
In particular, these pictures show how the two-body interaction term present
in the Hamiltonian operator ${\bf H}^{\prime}$ affects the initial Wigner
distribution $\mathscr{W}_{\frac{\pi}{2},0}(\mu,\nu;0)$ (via contour plots)
for determined values of time. We have adopted in our numerical computations
the values (a) $t=0$, (b) $t=2.15$, (c) $t=4.75$, (d) $t=7.10$, (e) $t=9.05$,
and (f) $t=9.95$, which illustrate a representative but not complete evolution
of the discrete Wigner function. Moreover, these values correspond to the
first local minimum and maximum points of $\mathcal{E}_{z}(t)$ — see solid
curve in figure 2(b) — while (f) represents the situation
$\mathcal{E}_{z}(9.95)\approx 1$. Although this phase space appears to the
eyes as continuous (due to the large number of points associated with $N=20$),
it is important to stress that such a particular phase space is (by
construction) genuinely discrete.
* •
4(a). The function $\mathscr{W}_{\frac{\pi}{2},0}(\mu,\nu;0)$ exhibits a
rotational symmetry with negative values located at the proximities of
$(-10,0)$ and $(10,0)$ (see small orange zones); as well as different widths
in the respective $\mu$ and $\nu$-directions. This important fact suggests, if
one considers both the directions, that marginal distributions related to the
angular momentum and angle variables can be somehow relevant and necessary in
the analysis of quantum correlations. The approximate format of an ellipse —
white zone in the middle part of the contour plot — represents a region where
there exists raised probability peaks which feature, by their turn, the
initial correlations associated with $|\frac{\pi}{2},0\rangle$.
* •
4(b). This first case of time evolution corresponds to
$\mathscr{W}_{\frac{\pi}{2},0}(\mu,\nu;t=2.15)$ and exemplifies the first
local minimum point of $\mathcal{E}_{z}(t)$ (see solid curve in figure 2(b)
when $t=2.15$). The entanglement effects (associated with the two-body
interaction, and synonymous of quantum correlation) introduce additional
symmetries that modify the statistical weights of the initial quantum state
$|\frac{\pi}{2},0\rangle$, originating, in this way, new interference patterns
which should explain the changed widths of the discrete Wigner distribution
function in $t=2.15$, as well as the appearance of new zones where negative
probabilities occur. Note that $\mathscr{W}_{\frac{\pi}{2},0}(\mu,\nu;t)$
exhibits a motion towards the frontier of the discrete phase space labeled by
$\mu=-10$, the transverse magnetic field $h$ being the agent responsible for
such behaviour. Numerical computations indeed corroborate this assertion and
allow us to show that, for $h=0$, the Wigner function remains frozen in the
discrete phase space, having its widths modified as time goes by.
* •
4(c). This particular case depicts
$\mathscr{W}_{\frac{\pi}{2},0}(\mu,\nu;t=4.75)$ and represents the first local
maximum point of $\mathcal{E}_{z}(t)$. Here,
$\mathscr{W}_{\frac{\pi}{2},0}(\mu,\nu;t)$ begins its motion towards the
centre of such a discrete phase space, where the interference patterns
associated with the components of ${\bf U}(t=4.75)|\frac{\pi}{2},0\rangle$
have specifically yielded the portrait verified in this situation. The small
white ‘island’ located in the right hand side of this picture and quite near
to the raised probability peaks (main white zone) depicts a small region with
negative probabilities.
* •
4(d). The second minimum point of $\mathcal{E}_{z}(t)$ has as discrete phase-
space representative the contour plot of
$\mathscr{W}_{\frac{\pi}{2},0}(\mu,\nu;t=7.10)$. These unique interference
patterns (once ${\bf H}^{\prime}$ describes for $h\neq 0$ a nonperiodic
system) with small ‘islands’ of positive and negative probabilities represent
the accumulated effects attributed to the transverse magnetic field and
anisotropy parameter. Moreover, let us briefly mention that ${\bf
U}(t=7.10)|\frac{\pi}{2},0\rangle$ has a coincidence probability (also known
as fidelity and/or time correlation) with the original state close to $89\%$,
while 4(b) results in $55\%$.
* •
4(e). Note that $\mathscr{W}_{\frac{\pi}{2},0}(\mu,\nu;t=9.05)$ corresponds to
the second local maximum point of $\mathcal{E}_{z}(t)$, i.e., we are
describing a situation where the entanglement does not occur. For the sake of
comparison, the coincidence probabilities associated with 4(c) and 4(e) assume
the respective percentages of $48\%$ and $78\%$. This difference can be
justified, in principle, by means of a simple inspection between both the
pictures: while 4(e) is located near the centre of discrete phase space, 4(c)
stays in the proximity of $\mu=-10$. However, if one considers the previous
case 4(d) for subsequent comparison, different contributions should be taken
into account in the analysis.
* •
4(f). This last case depicts $\mathscr{W}_{\frac{\pi}{2},0}(\mu,\nu;t=9.95)$
for $\mathcal{E}_{z}(9.95)\approx 1$. It is interesting to observe how the
nonperiodic dynamics related to the Hamiltonian operator ${\bf H}^{\prime}$
affects any reconstruction process of
$\mathscr{W}_{\frac{\pi}{2},0}(\mu,\nu;0)$: some numerical estimates of
$\mathfrak{F}(t)\mathrel{\mathop{:}}=\mbox{${\rm
Tr}$}[\mbox{\boldmath$\rho$}(0)\mbox{\boldmath$\rho$}(t)]$ for $t=9.95$
results in $51\%$ of time correlation.
Now, let us show an essential formal result that explores the connection
between discrete Husimi and Wigner distribution functions,[58]
$\mathscr{H}_{\rho}(\mu,\nu;t)=\frac{1}{2j+1}\sum_{\mu^{\prime},\nu^{\prime}=-j}^{j}E(\mu,\nu|\mu^{\prime},\nu^{\prime})\,\mathscr{W}_{\rho}(\mu^{\prime},\nu^{\prime};t).$
(33)
Here, $E(\mu,\nu|\mu^{\prime},\nu^{\prime})$ defines a smoothing process
characterized by the discrete phase-space function[21]
$E(\mu,\nu|\mu^{\prime},\nu^{\prime})=\frac{1}{2j+1}\sum_{\eta,\xi=-j}^{j}\exp\left\\{\frac{2\pi\mathrm{i}}{2j+1}\left[\eta(\mu^{\prime}-\mu)+\xi(\nu^{\prime}-\nu)\right]\right\\}\mathscr{K}(\eta,\xi)$
that closely resembles the role of a usual Gaussian function in the continuous
phase space, since
$\mathscr{K}(\eta,\xi)\mathrel{\mathop{:}}=\mathscr{M}(\eta,\xi)/\mathscr{M}(0,0)$
is here responsible for the sum of products of Jacobi $\vartheta$-functions
evaluated at integer arguments, namely,
$\displaystyle\mathscr{M}(\eta,\xi)$ $\displaystyle=$
$\displaystyle\frac{\sqrt{\mathfrak{a}}}{2}\bigl{\\{}\vartheta_{3}(\mathfrak{a}\eta|\mathrm{i}\mathfrak{a})\left[\vartheta_{3}(\mathfrak{a}\xi|\mathrm{i}\mathfrak{a})+\exp(\mathrm{i}\pi\eta)\vartheta_{4}(\mathfrak{a}\xi|\mathrm{i}\mathfrak{a})\right]$
$\displaystyle+\exp(\mathrm{i}\pi\xi)\vartheta_{4}(\mathfrak{a}\eta|\mathrm{i}\mathfrak{a})\left[\vartheta_{3}(\mathfrak{a}\xi|\mathrm{i}\mathfrak{a})+\exp[\mathrm{i}\pi(\eta+2j+1)]\vartheta_{4}(\mathfrak{a}\xi|\mathrm{i}\mathfrak{a})\right]\bigr{\\}}$
where $\mathfrak{a}=(4j+2)^{-1}$ — in particular, see Appendix A of Ref. MR
for definitions and technical details on Jacobi $\vartheta$-functions. So, in
order to illustrate how the smoothing process acts on the summand of Eq. (33),
let us consider $\mathscr{W}_{\frac{\pi}{2},0}(\mu,\nu;t)$ and also the same
set of parameters used in the previous figure. In addition, it should be
noticed that $\mathscr{H}_{\frac{\pi}{2},0}(\mu,\nu;t)$ is strictly positive
as well as limited to the closed interval $[0,1]$ for any $t\geq 0$;
consequently, all the negative values and non-regular patterns described in
figure 4 will be washed out by $E(\mu,\nu|\mu^{\prime},\nu^{\prime})$. Figure
5 exhibits the contour plots of $\mathscr{H}_{\frac{\pi}{2},0}(\mu,\nu;t)$,
where such effects are effectively checked along the pictures 5(a-f). In these
cases, the width changes observed for the $(\mu,\nu)$-directions are related
to the two-body interaction term (here mediated by means of $N$ and $\gamma$),
while the motion towards frontier located at $\mu=-10$ is associated with the
transverse magnetic field $h$. Since the widths associated with the angular-
momentum and angle distributions are also affected by those effects, it seems
convenient at this moment to employ the Wehrl-type entropy functionals[31] for
understanding how the correlations between the discrete variables $\mu$ and
$\nu$ of a finite-dimensional phase space change for $t\geq 0$.
Figure 5: (Color online) Time evolution of
$\mathscr{H}_{\frac{\pi}{2},0}(\mu,\nu;t)$ for the same set of parameters
employed in the previous figure. In what concerns the discrete Husimi
function, it must be stressed that such a function is particularly obtained
from $\mathscr{W}_{\frac{\pi}{2},0}(\mu,\nu;t)$ through a smoothing process
characterized by a discrete phase-space function that closely resembles the
role of the Gaussian function in the continuous phase space. This process
essentially explains the disappearance of all irregular regions and negative
values exhibited by the discrete Wigner distribution, which implies in a well-
behaved function along its time evolution and limited to the closed interval
$[0,1]$. Similarly to the previous case, it is worth mentioning that such a
particular phase space is genuinely discrete (by ab initio construction), even
though it appears to the eyes as contionuous.
To this end, let us now consider the mutual correlation functional
$\mathrm{I}[\mathrm{H};t]$ defined as follows:[22]
$\mathrm{I}[\mathrm{H};t]\mathrel{\mathop{:}}=\mathrm{E}[\mathrm{Q};t]+\mathrm{E}[\mathrm{R};t]-\mathrm{E}[\mathrm{H};t]\geq
0$. In this particular definition,
$\mathrm{E}[\mathrm{H};t]\equiv-\frac{1}{2j+1}\sum_{\mu,\nu=-j}^{j}\mathscr{H}_{\rho}(\mu,\nu;t)\ln\left[\mathscr{H}_{\rho}(\mu,\nu;t)\right]$
(34)
corresponds to the time-dependent joint entropy functional here expressed in
terms of the discrete Husimi function (33), while
$\mathrm{E}[\mathrm{Q};t]\equiv-\frac{1}{\sqrt{2j+1}}\sum_{\mu=-j}^{j}\mathscr{Q}_{\rho}(\mu;t)\ln\left[\mathscr{Q}_{\rho}(\mu;t)\right]$
(35)
and
$\mathrm{E}[\mathrm{R};t]\equiv-\frac{1}{\sqrt{2j+1}}\sum_{\nu=-j}^{j}\mathscr{R}_{\rho}(\nu;t)\ln\left[\mathscr{R}_{\rho}(\nu;t)\right]$
(36)
represent the marginal entropies — which are directly related, by their turn,
to the respective marginal distribution functions (for technical details, see
Ref. Ruzzi)
$\mathscr{Q}_{\rho}(\mu;t)=\frac{1}{\sqrt{2j+1}}\sum_{\nu=-j}^{j}\mathscr{H}_{\rho}(\mu,\nu;t)\quad\mbox{and}\quad\mathscr{R}_{\rho}(\nu;t)=\frac{1}{\sqrt{2j+1}}\sum_{\mu=-j}^{j}\mathscr{H}_{\rho}(\mu,\nu;t).$
Note that $\mathrm{E}[\mathrm{H};t]$, $\mathrm{E}[\mathrm{Q};t]$ and
$\mathrm{E}[\mathrm{R};t]$ constitute some basic mathematical elements for
describing functional correlations between $\mu$ and $\nu$, and consequently,
the width changes associated with the discrete Husimi function
$\mathscr{H}_{\rho}(\mu,\nu;t)$. Within several important properties inherent
to the entropy functionals, the Araki-Lieb inequality[59]
$\left|\mathrm{E}[\mathrm{Q};t]-\mathrm{E}[\mathrm{R};t]\right|\leq\mathrm{E}[\mathrm{H};t]\leq\mathrm{E}[\mathrm{Q};t]+\mathrm{E}[\mathrm{R};t]$
has a central role in such a description: for instance, it leads us to
investigate how the dynamic correlations (introduced by means of the
Hamiltonian operator ${\bf H}^{\prime}$) affect the underlying correlations of
the initial state $\mbox{\boldmath$\rho$}(t_{0})$ — here mapped onto
$\mathscr{H}_{\rho}(\mu,\nu;t_{0})$. So, if one computes the entropy
functionals from $\mathscr{H}_{\frac{\pi}{2},0}(\mu,\nu;t)$, some interesting
results can be promptly achieved.
Figure 6: Plots of (a) $\mathrm{E}[\mathrm{H};t]$ and (b)
$\mathrm{I}[\mathrm{H};t]$ versus $t\in[0,50]$ for two different values of
transverse magnetic field $h$ and anisotropy parameter $\gamma$ with $N=20$
fixed, namely, the solid curves in both pictures correspond to $(-0.1,0.2)$,
while the dot-dashed curves are related to $(-0.13,0.1)$. In each case, the
orange region describes particular situations of
$\mathscr{H}_{\frac{\pi}{2},0}(\mu,\nu;t)$ where the entropy functionals are
restricted to (a) $\mathrm{E}[\mathrm{H};t]\leq\mathrm{E}[\mathrm{H};0]\approx
0.1994$ and (b) $\mathrm{I}[\mathrm{H};t]\leq\mathrm{I}[\mathrm{H};0]\approx
0.7144$. It is important to stress that $\mathrm{E}[\mathrm{H};t]$ and
$\mathrm{I}[\mathrm{H};t]$ are sensitive to small variations of $h$.
Figure 6 illustrates the time evolution of (a) $\mathrm{E}[\mathrm{H};t]$ and
(b) $\mathrm{I}[\mathrm{H};t]$ as a function of $t\in[0,50]$ for the same
values of $(h,\gamma)$ used in figure 2. Then, let us initially focus on
$(-0.1,0.2)$ (see solid curves in both the pictures) since this case coincides
with that used in the contour plots of
$\mathscr{H}_{\frac{\pi}{2},0}(\mu,\nu;t)$ for specific times. Those
concentric ellipses observed in figure 5(a) can be considered as a phase-space
signature of the initial state $|\frac{\pi}{2},0\rangle$ at $t=0$, here
endorsed by $\mathrm{E}[\mathrm{H};0]\approx 0.1994$ and
$\mathrm{I}[\mathrm{H};0]\approx 0.7144$, which means that such a state
presents an initial correlation between $\mu$ and $\nu$ featured by a spread
angular-momentum distribution and ‘localized’ angle distribution. Indeed, the
sum $\mathrm{E}[\mathrm{Q};0]+\mathrm{E}[\mathrm{R};0]\approx 0.5150$
justifies such an assertion and (as expected) also corroborates the right-hand
side of the Araki-Lieb inequality. Already at $t=2.15$, one observes the
displacement of the discrete Husimi function towards the frontier $\mu=-10$
accompanied, in this situation, by a decrease of asymmetry between its
intrinsic widths in the $(\mu,\nu)$-directions. Since
$\mathrm{E}[\mathrm{H};t=2.15]\approx 0.1958$ implies in effective information
gain upon the discrete angular-momentum and angle collective variables, the
extra correlations introduced through the two-body interaction term explain,
in principle, the effects depicted in pictures 2(a,b). However, the deformed
widths viewed in 5(c-f) after the first rebound from that frontier and
subsequent return towards the centre of the finite-dimensional discrete phase
space are responsible for the inscreased values of both entropy functionals
when $4.75\leq t\leq 9.95$. Let us now briefly discuss the case $(-0.13,0.1)$
here characterized by a significant reduction of the effects related to
$\gamma$. Although the information gain happens more frequently in such a
case, the spin squeezing and entanglement effects exhibited in pictures 2(c,d)
are associated with frequent motions of
$\mathscr{H}_{\frac{\pi}{2},0}(\mu,\nu;t)$ towards the frontier $\mu=-10$,
followed by small variations of its widths.
A last pertinent question then emerges from our considerations about discrete
Husimi function and Wehrl-type entropy functionals: “How the smoothing process
characterized by Eq. (33) affects the description of correlation strength
between the discrete labels $\mu$ and $\nu$ here used to describe the finite-
dimensional phase space”? To answer this question, let us initially comprehend
the role of $\mathscr{K}(\eta,\xi)$ present in the function
$E(\mu,\nu|\mu^{\prime},\nu^{\prime})$: it was basically written as a sum of
products of Jacobi $\vartheta$-functions evaluated at integer arguments, which
plays, in such an aforementioned discrete phase space, the role reserved to
the Gaussian functions in the continuous case ($N\rightarrow\infty$); besides,
$\sqrt{4j+2}$ represents its respective width and it assumes a constant value,
in this situation, for a given $j$. Hence, any sub-Planck structures[60] or
even relevant correlations within this range are smeared in the smoothing
process, which directly affects the description of correlation strength via
Wehrl-type entropy functionals. Another important restriction associated with
$\mathrm{E}[\mathrm{H};t]$ emerges from the mathematical property
$\mathrm{S}[\mbox{\boldmath$\rho$}(t)]=-\mbox{${\rm
Tr}$}[\mbox{\boldmath$\rho$}(t)\ln\mbox{\boldmath$\rho$}(t)]\leq\mathrm{E}[\mathrm{H};t]$,
i.e., Eq. (34) consists of an upper bound for the von Neumann entropy;[31]
consequently, “if there are small distance fluctuations or if
$\mathscr{H}_{\rho}(\mu,\nu;t)$ is concentrated on small regions of discrete
phase space”, then the percent error estimated $\delta=\mathrm{S}/\mathrm{E}$
will be very bad. However, these apparent limitations can be circumvented by:
(i) adopting the prescription of Manfredi and Feix[61] for quantum entropy
based on Wigner functions in continuous phase-space, (ii) reformulating the
function $\mathscr{K}(\eta,\xi)$ in order to modify its respective width,[22]
and finally (iii) including a parallel study on von Neumann entropy[62] and
quantum discord[7] which permits us to increase our knowledge base on the
intricate mechanisms of correlation strength related to the spin-squeezing and
entanglement effects for any spin systems. In many ways the analysis presented
in this work of the modified LMG model is complementary to that provided in
Ref. Dusuel; moreover, in what concerns the discrete phase-space approach and
its implications for different physical systems, our results sound promising
at a first glance.
## 5 Conclusions
In this non-trivial quantum mechanical scenario of correlations, entanglement
and spin-squeezing effects, as well as their connections, any associated
theoretical and/or experimental proposals for measures of the aforementioned
effects shall necessarily be accomplished by exhaustive tests of confidence
within a wide class of analogous physical systems. The mere comprehension of
these effects by means of a ‘specific theoretical/experimental measure applied
to a particular physical system’ does not change its respective status of
proposal per se for standard measure: such transition demands ‘time, patience,
and efforts’ to understand the subtle role of correlations and their intrinsic
mechanisms in quantum mechanics. In this paper, we have used a theoretical
framework of finite-dimensional discrete phase space as an alternative
approach for the study of the spin-squeezing and entanglement effects. The
emphasis on covariances and its fundamental role in the investigation of spin-
squeezing effects is not accidental: additional correlations related to the
anticommutation relations of the angular-momentum generators are now included
in the analysis of spin systems. This first effective gain does not imply in
the match between the previous effects, since both the measures here adopted
for describing entanglement exhibit ‘slightly different functional relations’
and also present ‘speculative features’, as expected.
In what concerns the connection between the spin-squeezing and entanglement
effects, both time-evolution operator and initial state of a given spin system
play a fundamental role in such process: they describe the short-range and/or
long-range correlations of the multipartite system under investigation, which
could explain the similarity degree of the spin-squeezing and entanglement
measures studied in this paper, as well as its ‘almost perfect match’. In this
sense, the modified LMG model fulfils a reasonable set of mathematical and
physical prerequisites that lead us to corroborate, by means of numerical
computations, the aforementioned link, for then establishing, subsequently,
its inherent limitations. Besides, these results open new possibilities for
future investigations in different physical systems which encompass underlying
$\mathfrak{su}(2)$ structures with distinct physical properties — in
particular, those multipartite systems where contributions related to the
long-range correlations are indeed necessary in the effective description of
entanglement.
Our particular phase-space approach, nevertheless, also reveals its drawbacks.
To begin with, Eq. (12) and its constituting blocks coming from the numerical
recipe described in subsection 3.2 do not satisfy the basic criterion ‘easy-
to-compute’. The large number of sums that appeared in the time evolution of
discrete Wigner function and mean values obligatorily implies in high
computational and operational costs, whose complexities grow as $N$ increases.
Moreover, the modulo $N$ extraction phase here adopted for the discrete
variables still follows that mathematical prescription discussed in Ref. DM1,
which represents a necessary operational cost inherent to the
mod($N$)-invariant operator basis ${\bf G}(\mu,\nu)$. However, such apparent
limitations can be circumvented in this context by adopting the theoretical
framework exposed in Ref. MR for the mod($N$)-invariant unitary operator basis
$\mbox{\boldmath$\Delta$}(\mu,\nu)$, and reformulating the content associated
with the time evolution of the discrete Wigner function. This procedure will
allow a real computational gain in the numerical calculations.
Now, let us briefly mention some possibilities for future research that stem
from the present paper. As a first example, we recall from Ref. Modi those
considerations on the concept of quantum discord and its important link with
the subtle boundary between entanglement and classical correlations. Given
that the Wehrl-type entropy functionals failed in the description of
correlations related to the entanglement and spin-squeezing effects, it seems
reasonable to introduce such a measure in this context since quantum discord
can lead us to a more efficient analysis on quantum and classical correlations
in multipartite physical systems. Another possible example of research
consists in employing that finite-dimensional discrete phase-space framework
in order to corroborate the Lieb’s conjecture for the spin coherent states[63]
via quantum dynamics of spin systems.
## Acknowledgements
The authors thank Maurizio Ruzzi and anonymous referee for providing valuable
suggestions on an earlier version of this manuscript. Tiago Debarba is
supported by Conselho Nacional de Desenvolvimento Científico e Tecnológico
(CNPq), Brazil.
## Appendix A Variances, covariances, and uncertainty relation
There are interesting results relating covariance functions and variances
associated with certain summations of two non-commuting observables ${\bf X}$
and ${\bf Y}$, namely,
$\mathscr{V}_{\mbox{\scriptsize${\rm X}$}\pm\mbox{\scriptsize${\rm
Y}$}}=\mathscr{V}_{\mbox{\scriptsize${\rm
X}$}}+\mathscr{V}_{\mbox{\scriptsize${\rm Y}$}}\pm
2\mathscr{V}_{\mbox{\scriptsize${\rm X}$}\mbox{\scriptsize${\rm
Y}$}}\quad\mbox{and}\quad\mathscr{V}_{\mbox{\scriptsize${\rm
X}$}\pm\mathrm{i}\mbox{\scriptsize${\rm
Y}$}}=\mathscr{V}_{\mbox{\scriptsize${\rm
X}$}}-\mathscr{V}_{\mbox{\scriptsize${\rm Y}$}}\pm
2\mathrm{i}\mathscr{V}_{\mbox{\scriptsize${\rm X}$}\mbox{\scriptsize${\rm
Y}$}}.$
Note that such connections can be generalized for a large number of non-
commuting observables. For instance, if one considers the set $\\{{\bf X},{\bf
Y},{\bf Z}\\}$, it is easy to show that
$\displaystyle\mathscr{V}_{\mbox{\scriptsize${\rm X}$}+\mbox{\scriptsize${\rm
Y}$}+\mbox{\scriptsize${\rm Z}$}}$ $\displaystyle=$
$\displaystyle\mathscr{V}_{\mbox{\scriptsize${\rm
X}$}}+\mathscr{V}_{\mbox{\scriptsize${\rm
Y}$}}+\mathscr{V}_{\mbox{\scriptsize${\rm
Z}$}}+2\mathscr{V}_{\mbox{\scriptsize${\rm X}$}\mbox{\scriptsize${\rm
Y}$}}+2\mathscr{V}_{\mbox{\scriptsize${\rm X}$}\mbox{\scriptsize${\rm
Z}$}}+2\mathscr{V}_{\mbox{\scriptsize${\rm Y}$}\mbox{\scriptsize${\rm Z}$}}$
$\displaystyle=$ $\displaystyle\mathscr{V}_{\mbox{\scriptsize${\rm
X}$}+\mbox{\scriptsize${\rm Y}$}}+\mathscr{V}_{\mbox{\scriptsize${\rm
X}$}+\mbox{\scriptsize${\rm Z}$}}+\mathscr{V}_{\mbox{\scriptsize${\rm
Y}$}+\mbox{\scriptsize${\rm Z}$}}-\mathscr{V}_{\mbox{\scriptsize${\rm
X}$}}-\mathscr{V}_{\mbox{\scriptsize${\rm
Y}$}}-\mathscr{V}_{\mbox{\scriptsize${\rm Z}$}}$
yields the statistical balance equation
$\mathscr{V}_{\mbox{\scriptsize${\rm X}$}+\mbox{\scriptsize${\rm
Y}$}+\mbox{\scriptsize${\rm Z}$}}+\mathscr{V}_{\mbox{\scriptsize${\rm
X}$}}+\mathscr{V}_{\mbox{\scriptsize${\rm
Y}$}}+\mathscr{V}_{\mbox{\scriptsize${\rm
Z}$}}=\mathscr{V}_{\mbox{\scriptsize${\rm X}$}+\mbox{\scriptsize${\rm
Y}$}}+\mathscr{V}_{\mbox{\scriptsize${\rm X}$}+\mbox{\scriptsize${\rm
Z}$}}+\mathscr{V}_{\mbox{\scriptsize${\rm Y}$}+\mbox{\scriptsize${\rm Z}$}}.$
(37)
This particular ‘conservation law’ describes, if applied to the angular-
momentum generators, how the statistical fluctuations are distributed among
the observables $\\{{\bf J}_{x},{\bf J}_{y},{\bf J}_{z}\\}$ in the measurement
process for a given initial quantum state. Table A shows some analytical
expressions obtained for the variances and covariances related to the angular-
momentum generators and spin coherent states, which exemplify Eq. (37).
Following, let us discuss how the unitary transformations modify the signal-
to-noise ratio for an ideal experimental situation where the deleterious
effects of a low-efficiency detection process are absent.[64]
The generator of unitary transformations ${\bf T}(\xi,\omega)$ maintains, by
definition, the linearity of the angular-momentum operators. Indeed, the new
set
$\left\\{\overline{{\bf J}}_{a}\mathrel{\mathop{:}}={\bf
T}^{\dagger}(\xi,\omega){\bf J}_{a}{\bf
T}(\xi,\omega)\;:\;\xi\in\mathbb{C}\;\,\mbox{and}\;\,\omega\in\mathbb{R}\right\\}_{a=x,y,z}$
shows explicitly such a mathematical property through the relations
$\displaystyle\overline{{\bf J}}_{x}$ $\displaystyle=$ $\displaystyle
A_{11}{\bf J}_{x}+A_{12}{\bf J}_{y}+A_{13}{\bf J}_{z}$
$\displaystyle\overline{{\bf J}}_{y}$ $\displaystyle=$ $\displaystyle
A_{21}{\bf J}_{x}+A_{22}{\bf J}_{y}+A_{23}{\bf J}_{z}$
$\displaystyle\overline{{\bf J}}_{z}$ $\displaystyle=$ $\displaystyle
A_{31}{\bf J}_{x}+A_{32}{\bf J}_{y}+A_{33}{\bf J}_{z},$ (38)
whose coefficients $\\{A_{ij}\\}$ can be immediately determined from the
results obtained in subsection 2.1. Since the commutation relation
$[\overline{{\bf J}}_{a},\overline{{\bf
J}}_{b}]=\mathrm{i}\epsilon_{abc}\overline{{\bf J}}_{c}$ keeps its invariable
form, some restrictions on the aforementioned coefficients should be
established in this context, that is,
$\displaystyle A_{22}A_{33}-A_{23}A_{32}$ $\displaystyle=$ $\displaystyle
A_{11},\quad A_{13}A_{32}-A_{12}A_{33}=A_{21},\quad
A_{12}A_{23}-A_{13}A_{22}=A_{31},$ $\displaystyle A_{23}A_{31}-A_{21}A_{33}$
$\displaystyle=$ $\displaystyle A_{12},\quad
A_{11}A_{33}-A_{13}A_{31}=A_{22},\quad A_{13}A_{21}-A_{11}A_{23}=A_{32},$
$\displaystyle A_{21}A_{32}-A_{22}A_{31}$ $\displaystyle=$ $\displaystyle
A_{13},\quad A_{12}A_{31}-A_{11}A_{32}=A_{23},\quad
A_{11}A_{22}-A_{12}A_{21}=A_{33}.$
It is worth stressing that these general equations are valid for any unitary
transformations which preserve Eq. (A); moreover, after certain proper
manipulations of such equations, this set can be reduced to
$\displaystyle A_{11}A_{12}+A_{21}A_{22}+A_{31}A_{32}$ $\displaystyle=$
$\displaystyle 0$ $\displaystyle A_{11}A_{13}+A_{21}A_{23}+A_{31}A_{33}$
$\displaystyle=$ $\displaystyle 0$ $\displaystyle
A_{12}A_{13}+A_{22}A_{23}+A_{32}A_{33}$ $\displaystyle=$ $\displaystyle 0.$
As mentioned in the main part of the text, the identity $\vec{\overline{{\bf
J}}}^{2}\equiv\vec{{\bf J}}^{2}$ also preserves the original form of the total
spin operator, which implies in the additional relations
$A_{11}^{2}+A_{21}^{2}+A_{31}^{2}=1,\quad
A_{12}^{2}+A_{22}^{2}+A_{32}^{2}=1,\quad A_{13}^{2}+A_{23}^{2}+A_{33}^{2}=1.$
Note that
$\mathscr{V}_{\overline{\mathrm{J}}_{x}}+\mathscr{V}_{\overline{\mathrm{J}}_{y}}+\mathscr{V}_{\overline{\mathrm{J}}_{z}}\equiv\mathscr{V}_{\mathrm{J}_{x}}+\mathscr{V}_{\mathrm{J}_{y}}+\mathscr{V}_{\mathrm{J}_{z}}=j$
represents an important by-product in this process since the sum of such
variances remains invariant under the unitary transformation described by Eq.
(A).
The explicit results for the variances and covariances — as shown on the table
below — allow us, in principle, to illustrate both the Eqs. (6) and (37).
Besides, they also lead us to verify the uncertainty relation
$\mathscr{V}_{\mathrm{J}_{a}}\mathscr{V}_{\mathrm{J}_{b}}-\left(\mathscr{V}_{\mathrm{J}_{a}\mathrm{J}_{b}}\right)^{2}\geq\frac{1}{4}\left|\langle\left[{\bf
J}_{a},{\bf J}_{b}\right]\rangle\right|^{2}$ for any $a,b=x,y,z$. Note that
the saturation is reached in this case for $\theta=0$ and $\pi$, which
correspond to the fiducial states $|j,-j\rangle$ and $|j,j\rangle$. Variances
and covariances associated with $\\{{\bf J}_{x},{\bf J}_{y},{\bf J}_{z}\\}$
for the spin coherent states
$\mathscr{V}_{\mathrm{J}_{x}}=(j/2)\left[1-\cos^{2}(\varphi)\sin^{2}(\theta)\right]$
$\mathscr{V}_{\mathrm{J}_{y}}=(j/2)\left[1-\sin^{2}(\varphi)\sin^{2}(\theta)\right]$
$\mathscr{V}_{\mathrm{J}_{z}}=(j/2)\sin^{2}(\theta)$
$\mathscr{V}_{\mathrm{J}_{x}\mathrm{J}_{y}}=-(j/4)\sin(2\varphi)\sin^{2}(\theta)$
$\mathscr{V}_{\mathrm{J}_{x}\mathrm{J}_{z}}=(j/4)\cos(\varphi)\sin(2\theta)$
$\mathscr{V}_{\mathrm{J}_{y}\mathrm{J}_{z}}=(j/4)\sin(\varphi)\sin(2\theta)$
$\mathscr{V}_{\mathrm{J}_{x}+\mathrm{J}_{y}}=(j/2)\left\\{2-\sin^{2}(\theta)\left[1+\sin(2\varphi)\right]\right\\}$
$\mathscr{V}_{\mathrm{J}_{x}+\mathrm{J}_{z}}=(j/2)\left[1+\sin^{2}(\varphi)\sin^{2}(\theta)+\cos(\varphi)\sin(2\theta)\right]$
$\mathscr{V}_{\mathrm{J}_{y}+\mathrm{J}_{z}}=(j/2)\left[1+\cos^{2}(\varphi)\sin^{2}(\theta)+\sin(\varphi)\sin(2\theta)\right]$
$\mathscr{V}_{\mathrm{J}_{x}+\mathrm{J}_{y}+\mathrm{J}_{z}}=(j/2)\left\\{2-\sin(2\varphi)\sin^{2}(\theta)+[\cos(\varphi)+\sin(\varphi)]\sin(2\theta)\right\\}$
See Ref. Inomata for certain mean values involving some quadratic forms of
${\bf J}_{\pm}$ and ${\bf J}_{z}$.
The evaluation of the signal-to-noise ratio (SNR) associated with collective
and intrinsinc degrees of freedom is crucial in any experimental data
analysis: indeed, it provides quantitative information on the measured signal
and the noise inherent to the experiment under investigation. Thus, let us
define SNR through the expression
$\mathfrak{R}_{a}\mathrel{\mathop{:}}=|\langle{\bf
J}_{a}\rangle|/\sqrt{\mathscr{V}_{\mathrm{J}_{a}}}$, where each component of
$\vec{{\bf J}}$ presents an important role in the measurement process.
However, if one considers experimental situations — rotations and/or time
evolutions — that transform ${\bf J}_{a}\mapsto\overline{{\bf J}}_{a}$, it
seems natural to analyse the case
$\mathfrak{R}_{a}\mapsto\overline{\mathfrak{R}}_{a}\mathrel{\mathop{:}}=|\langle\overline{{\bf
J}}_{a}\rangle|/\sqrt{\mathscr{V}_{\overline{\mathrm{J}}_{a}}}$. In this
sense, the variances
$\displaystyle\mathscr{V}_{\overline{\mathrm{J}}_{x}}$ $\displaystyle=$
$\displaystyle
A_{11}^{2}\mathscr{V}_{\mathrm{J}_{x}}+A_{12}^{2}\mathscr{V}_{\mathrm{J}_{y}}+A_{13}^{2}\mathscr{V}_{\mathrm{J}_{z}}$
$\displaystyle+2\left(A_{11}A_{12}\mathscr{V}_{\mathrm{J}_{x}\mathrm{J}_{y}}+A_{11}A_{13}\mathscr{V}_{\mathrm{J}_{x}\mathrm{J}_{z}}+A_{12}A_{13}\mathscr{V}_{\mathrm{J}_{y}\mathrm{J}_{z}}\right)$
$\displaystyle\mathscr{V}_{\overline{\mathrm{J}}_{y}}$ $\displaystyle=$
$\displaystyle
A_{21}^{2}\mathscr{V}_{\mathrm{J}_{x}}+A_{22}^{2}\mathscr{V}_{\mathrm{J}_{y}}+A_{23}^{2}\mathscr{V}_{\mathrm{J}_{z}}$
$\displaystyle+2\left(A_{21}A_{22}\mathscr{V}_{\mathrm{J}_{x}\mathrm{J}_{y}}+A_{21}A_{23}\mathscr{V}_{\mathrm{J}_{x}\mathrm{J}_{z}}+A_{22}A_{23}\mathscr{V}_{\mathrm{J}_{y}\mathrm{J}_{z}}\right)$
$\displaystyle\mathscr{V}_{\overline{\mathrm{J}}_{z}}$ $\displaystyle=$
$\displaystyle
A_{31}^{2}\mathscr{V}_{\mathrm{J}_{x}}+A_{32}^{2}\mathscr{V}_{\mathrm{J}_{y}}+A_{33}^{2}\mathscr{V}_{\mathrm{J}_{z}}$
$\displaystyle+2\left(A_{31}A_{32}\mathscr{V}_{\mathrm{J}_{x}\mathrm{J}_{y}}+A_{31}A_{33}\mathscr{V}_{\mathrm{J}_{x}\mathrm{J}_{z}}+A_{32}A_{33}\mathscr{V}_{\mathrm{J}_{y}\mathrm{J}_{z}}\right)$
exhibit an explicit dependence on all the previous variance and covariance
functions, which modifies substantially the estimate of
$\\{\mathfrak{R}_{a}\\}$. Besides, the uncertainty relation
$\mathscr{V}_{\overline{\mathrm{J}}_{a}}\mathscr{V}_{\overline{\mathrm{J}}_{b}}-\left(\mathscr{V}_{\overline{\mathrm{J}}_{a}\overline{\mathrm{J}}_{b}}\right)^{2}\geq\frac{1}{4}\left|\langle[\overline{{\bf
J}}_{a},\overline{{\bf J}}_{b}]\rangle\right|^{2}$ (39)
is also modified in this context, since the covariances
$\displaystyle\mathscr{V}_{\overline{\mathrm{J}}_{x}\overline{\mathrm{J}}_{y}}$
$\displaystyle=$ $\displaystyle
A_{11}A_{21}\mathscr{V}_{\mathrm{J}_{x}}+A_{12}A_{22}\mathscr{V}_{\mathrm{J}_{y}}+A_{13}A_{23}\mathscr{V}_{\mathrm{J}_{z}}+(A_{11}A_{22}+A_{12}A_{21})\mathscr{V}_{\mathrm{J}_{x}\mathrm{J}_{y}}$
$\displaystyle+(A_{11}A_{23}+A_{13}A_{21})\mathscr{V}_{\mathrm{J}_{x}\mathrm{J}_{z}}+(A_{12}A_{23}+A_{13}A_{22})\mathscr{V}_{\mathrm{J}_{y}\mathrm{J}_{z}}$
$\displaystyle\mathscr{V}_{\overline{\mathrm{J}}_{x}\overline{\mathrm{J}}_{z}}$
$\displaystyle=$ $\displaystyle
A_{11}A_{31}\mathscr{V}_{\mathrm{J}_{x}}+A_{12}A_{32}\mathscr{V}_{\mathrm{J}_{y}}+A_{13}A_{33}\mathscr{V}_{\mathrm{J}_{z}}+(A_{11}A_{32}+A_{12}A_{31})\mathscr{V}_{\mathrm{J}_{x}\mathrm{J}_{y}}$
$\displaystyle+(A_{11}A_{33}+A_{13}A_{31})\mathscr{V}_{\mathrm{J}_{x}\mathrm{J}_{z}}+(A_{12}A_{33}+A_{13}A_{32})\mathscr{V}_{\mathrm{J}_{y}\mathrm{J}_{z}}$
$\displaystyle\mathscr{V}_{\overline{\mathrm{J}}_{y}\overline{\mathrm{J}}_{z}}$
$\displaystyle=$ $\displaystyle
A_{21}A_{31}\mathscr{V}_{\mathrm{J}_{x}}+A_{22}A_{32}\mathscr{V}_{\mathrm{J}_{y}}+A_{23}A_{33}\mathscr{V}_{\mathrm{J}_{z}}+(A_{21}A_{32}+A_{22}A_{31})\mathscr{V}_{\mathrm{J}_{x}\mathrm{J}_{y}}$
$\displaystyle+(A_{21}A_{33}+A_{23}A_{31})\mathscr{V}_{\mathrm{J}_{x}\mathrm{J}_{z}}+(A_{22}A_{33}+A_{23}A_{32})\mathscr{V}_{\mathrm{J}_{y}\mathrm{J}_{z}}$
possess a similar dependence if one compares them with the previous case.
Finally, it is worth stressing that such results are extremely relevant in the
investigative process of squeezing and entanglement effects for spin
systems.[10, 11, 12, 13]
For instance, Muñoz and Klimov[65] have recently introduced a particular set
of discrete displacement generators upon a $2^{n}\otimes 2^{n}$ discrete phase
space, which allows us to establish a specific family of discrete spin
coherent states for $n$-qubit systems with interesting mathematical and
physical properties. Generated by application of the discrete displacement
generators to a symmetric fiducial state, such states have isotropic
fluctuations in a tangent plane whose geometric features are well-defined: in
general, its direction does not coincide with that chosen for the mean value
$\langle\vec{{\bf J}}\rangle$. Besides, these reference states constitute the
essential basic elements necessary for investigating the squeezing effects
associated with the non-symmetric $n$-qubit states (resulting from the
application of XOR gates) through the covariances
$\mathscr{V}_{\mathrm{J}_{a}\mathrm{J}_{b}}$. It is important to emphasize
that the number of XOR gates applied to the discrete spin coherent states in
order to minimize fluctuations in the original homogeneous plane strongly
depends on the number of qubits. Since
$\\{\mathscr{V}_{\mathrm{J}_{a}\mathrm{J}_{b}}\\}_{a,b=x,y,z}$ is also
modified by the action of XOR gates, we believe that our results can be
somehow useful for the detailed examination of such squeezing effects.
## Appendix B The Kitagawa-Ueda model
Let us initiate our investigation on spin squeezing and entanglement through
the Kitagawa-Ueda model[13] — here described by the Hamiltonian ${\bf
H}=\chi{\bf J}_{z}^{2}$ — which describes a nonlinear interaction proportional
to ${\bf J}_{z}^{2}$. The unitary transformations generated by the time-
evolution operator ${\bf U}(\tau)=\exp\left(-\mathrm{i}\tau{\bf
J}_{z}^{2}\right)$ for $\tau=\chi t$ represents a basic set of mathematical
tools that allows us, in principle, to calculate certain important quantities
necessary to the investigative process. Indeed, if one considers the unitary
transformations
$\displaystyle{\bf J}_{x}(\tau)$ $\displaystyle\equiv$ $\displaystyle{\bf
U}^{\dagger}(\tau){\bf J}_{x}{\bf U}(\tau)=\exp(\mathrm{i}\tau)\left[{\bf
J}_{x}\cos(2\tau{\bf J}_{z})-{\bf J}_{y}\sin(2\tau{\bf J}_{z})\right]$
$\displaystyle{\bf J}_{y}(\tau)$ $\displaystyle\equiv$ $\displaystyle{\bf
U}^{\dagger}(\tau){\bf J}_{y}{\bf U}(\tau)=\exp(\mathrm{i}\tau)\left[{\bf
J}_{x}\sin(2\tau{\bf J}_{z})+{\bf J}_{y}\cos(2\tau{\bf J}_{z})\right]$
$\displaystyle{\bf J}_{z}(\tau)$ $\displaystyle\equiv$ $\displaystyle{\bf
U}^{\dagger}(\tau){\bf J}_{z}{\bf U}(\tau)={\bf
J}_{z}\;\rightleftharpoons\;\left[{\bf J}_{z},{\bf U}(\tau)\right]=0,$ (40)
it is not so hard to yield the mean values (Heisenberg picture)
$\displaystyle\langle{\bf J}_{x}(\tau)\rangle_{\theta,\varphi}$
$\displaystyle\equiv$ $\displaystyle\langle\theta,\varphi|{\bf
J}_{x}(\tau)|\theta,\varphi\rangle=\mathscr{A}_{\tau}^{2j-1}j\cos\left[\varphi-(2j-1)\delta_{\tau}\right]\sin(\theta)$
$\displaystyle\langle{\bf J}_{y}(\tau)\rangle_{\theta,\varphi}$
$\displaystyle\equiv$ $\displaystyle\langle\theta,\varphi|{\bf
J}_{y}(\tau)|\theta,\varphi\rangle=\mathscr{A}_{\tau}^{2j-1}j\sin\left[\varphi-(2j-1)\delta_{\tau}\right]\sin(\theta)$
$\displaystyle\langle{\bf J}_{z}(\tau)\rangle_{\theta,\varphi}$
$\displaystyle\equiv$ $\displaystyle\langle\theta,\varphi|{\bf
J}_{z}(\tau)|\theta,\varphi\rangle=-j\cos(\theta)$ (41)
for the spin coherent states, where
$\mathscr{A}_{\tau}=\left[\cos^{2}(\tau)+\sin^{2}(\tau)\cos^{2}(\theta)\right]^{\frac{1}{2}}$
denotes the amplitude function, and
$\delta_{\tau}=\arctan\left[\tan(\tau)\cos(\theta)\right]$ the phase function.
Note that such results lead us to conclude that unitary transformations
involving nonlinear forms of the angular-momentum generators do not preserve,
in general, the linear feature of such operators — for instance, see Eqs. (B)
and (A). This particularity is directly associated with the generators of the
$\mathfrak{su}(2)$ Lie algebra and does not apply to the squeeze operator
${\bf
S}(\zeta)\mathrel{\mathop{:}}=\exp\left[\frac{1}{2}\left(\zeta^{\ast}{\bf
a}^{2}-\zeta{\bf a}^{\dagger 2}\right)\right]$, since ${\bf S}(\zeta)$ is
responsible for unitary transformations in the Heisenberg-Weyl algebra which
preserve the linearity of the boson annihilation and creation operators.[37,
38]
Another interesting point inherent to the time-evolution operator ${\bf
U}(\tau)$ establishes the relation $\langle\vec{{\bf
J}}^{2}(\tau)\rangle_{\theta,\varphi}=\langle\vec{{\bf
J}}^{2}(0)\rangle_{\theta,\varphi}=j(j+1)$, which preserves the mean value of
the total spin operator in $\tau=0$. In fact, this result can be interpreted
as a direct consequence of the unitary transformations originated from the
action of ${\bf U}(\tau)$ on the angular-momentum operators. Following, let us
discuss some important points related to uncertainty relation, squeezing
effects and their links with entanglement, considering the spin coherent
states as initial states of the physical system.
The time-dependent variance and covariance functions, as shown on the table
below for the spin coherent states, constitute an important group of formal
mathematical results related to the Kitagawa-Ueda model that permits us to
investigate not only the Robertson-Schrödinger uncertainty principle, but also
both the squeezing and entanglement effects associated with the collective
angular-momentum operators. Time-dependent mean values, variances and
covariance function for the Kitagawa-Ueda model $\langle{\bf
J}_{x}^{2}(\tau)\rangle_{\theta,\varphi}=j/2+(j/4)(2j-1)\left[1+\mathscr{A}_{2\tau}^{2j-2}\cos[2\varphi-(2j-2)\delta_{2\tau}]\right]\sin^{2}(\theta)$
$\langle{\bf
J}_{y}^{2}(\tau)\rangle_{\theta,\varphi}=j/2+(j/4)(2j-1)\left[1-\mathscr{A}_{2\tau}^{2j-2}\cos[2\varphi-(2j-2)\delta_{2\tau}]\right]\sin^{2}(\theta)$
$\langle{\bf J}_{z}^{2}(\tau)\rangle_{\theta,\varphi}\equiv\langle{\bf
J}_{z}^{2}(0)\rangle_{\theta,\varphi}=j^{2}\cos^{2}(\theta)+(j/2)\sin^{2}(\theta)$
$\mathscr{V}_{\mathrm{J}_{x}}(\tau)=j/2+(j/4)(2j-1)\left[1+\mathscr{A}_{2\tau}^{2j-2}\cos[2\varphi-(2j-2)\delta_{2\tau}]\right]\sin^{2}(\theta)$
$\qquad\qquad\quad-j^{2}\mathscr{A}_{\tau}^{4j-2}\cos^{2}\left[\varphi-(2j-1)\delta_{\tau}\right]\sin^{2}(\theta)$
$\mathscr{V}_{\mathrm{J}_{y}}(\tau)=j/2+(j/4)(2j-1)\left[1-\mathscr{A}_{2\tau}^{2j-2}\cos[2\varphi-(2j-2)\delta_{2\tau}]\right]\sin^{2}(\theta)$
$\qquad\qquad\quad-j^{2}\mathscr{A}_{\tau}^{4j-2}\sin^{2}\left[\varphi-(2j-1)\delta_{\tau}\right]\sin^{2}(\theta)$
$\mathscr{V}_{\mathrm{J}_{z}}(\tau)\equiv\mathscr{V}_{\mathrm{J}_{z}}(0)=(j/2)\sin^{2}(\theta)$
$\mathscr{V}_{\mathrm{J}_{x}\mathrm{J}_{y}}(\tau)=(j/4)(2j-1)\mathscr{A}_{2\tau}^{2j-2}\sin[2\varphi-(2j-2)\delta_{2\tau}]\sin^{2}(\theta)$
$\qquad\qquad\quad-(j^{2}/2)\mathscr{A}_{\tau}^{4j-2}\sin[2\varphi-(4j-2)\delta_{\tau}]\sin^{2}(\theta)$
See Refs. SS1 and Hald for some measurement criteria involving such physical
quantities.
* •
Initially, let us introduce the parameters
$\mathcal{S}_{x}\mathrel{\mathop{:}}=\mathscr{V}_{\mathrm{J}_{x}}/\mathcal{R}_{xyz}$
and
$\mathcal{S}_{y}\mathrel{\mathop{:}}=\mathscr{V}_{\mathrm{J}_{y}}/\mathcal{R}_{xyz}$
with
$\mathcal{R}_{xyz}\mathrel{\mathop{:}}=\left[(\mathscr{V}_{\mathrm{J}_{x}\mathrm{J}_{y}})^{2}+\frac{1}{4}|\langle{\bf
J}_{z}\rangle|^{2}\right]^{\frac{1}{2}}$ (in this definition,
$\mathcal{R}_{xyz}\neq 0$ represents a condition sine qua non), which lead us
to rewrite Eq. (6) as follows: $\mathcal{S}_{x}\mathcal{S}_{y}\geq 1$. In
principle, this simplified form allows us to investigate the squeezing effects
related to the aforementioned model through exact expressions obtained in
Table B for the time-dependent variance and covariance functions. Figure 7(a)
shows the plots of $\mathcal{S}_{x}$ (dot-dashed line) and $\mathcal{S}_{y}$
(dashed line) versus $\tau\in[0,2\pi]$ for $j=2$ and
$\theta=\varphi=\frac{\pi}{4}$ fixed. The hachured area exhibited in the
picture describes, in this case, that region where the squeezing effects occur
for $\mathcal{S}_{x(y)}(\tau)<1$ (but not both simultaneously). Such a
particular evidence of squeezing effect generalizes, through an effective way,
those results obtained by Kitagawa and Ueda[13] in the absence of one-axis
twisting mechanism.777In fact, this mathematical procedure introduces
artificially additional quantum correlations (by means of time-dependent
unitary transformations that consist of rotations around the $x$-axis) for the
angular-momentum operators, which allow to reduce the standard quantum noise
related to the spin coherent states down to
$\frac{1}{2}\sqrt[3]{\frac{j}{3}}$. It is important to emphasize that such an
estimate does not consider the contributions originated from the covariance
function.
* •
How the squeezing and entanglement effects can be connected? Sørensen et
al[12] have proposed an interesting experiment for the Bose-Einstein
condensates (and attainable with present technology) which allows us, in
principle, to answer this intriguing question through a fundamental effect in
quantum mechanics: the many-particle entanglement. Since correlations among
spins yield the squeezing effect, it is natural to build a bridge connecting
both the entanglement and spin squeezing effects by means of a solid
mathematical framework which allows us to establish certain reliable
measurement criteria.[10] In this sense, let us comment some few words on the
entanglement criterion adopted in such an experiment: it is basically focussed
on the separability criterion for the $N$-particle density operator involving
the variances and squared mean values associated with each orthogonal
component of the collective angular-momentum operators, that is,
$\mathcal{E}_{a}\equiv\frac{N\mathscr{V}_{\mathrm{J}_{a}}}{\langle{\bf
J}_{b}\rangle^{2}+\langle{\bf J}_{c}\rangle^{2}}<1\qquad(a,b,c=x,y,z).$ (42)
Thus, the parameters $\\{\mathcal{E}_{a}\\}$ characterize the atomic
entanglement in this context, and quantum states with $\mathcal{E}_{a}<1$ are
referred to as spin squeezed states.[12] For instance, Figure 7(b) shows the
plots of $\mathcal{E}_{x}$ (dot-dashed line), $\mathcal{E}_{y}$ (dashed line),
as well as $\mathcal{E}_{z}$ (solid line) versus $\tau\in[0,2\pi]$ for the
same values of $\\{j,\theta,\varphi\\}$ adopted in the squeezing criterion —
see Fig. 7(a). Note that $\mathcal{E}_{x}$ and $\mathcal{E}_{y}$ exhibit
values less than one, except $\mathcal{E}_{z}$, which implies in entanglement
effect (see hachured region) for the orthogonal components ${\bf J}_{x}(\tau)$
and ${\bf J}_{y}(\tau)$, as expected, since $[{\bf H},{\bf J}_{z}]=0$. The
similarities between both the figures and the evidence of the squeezing and
entanglement effects for the same values of $\tau$ justify, in such a case,
the above connection per se.
Figure 7: (a) represents the plots of $\mathcal{S}_{x}$ (dot-dashed line) and
$\mathcal{S}_{y}$ (dashed line) versus $\tau\in[0,2\pi]$ with $j=2$ ($N=4$
spins) and $\theta=\varphi=\frac{\pi}{4}$ fixed. Note that both the parameters
$\mathcal{S}_{x}(\tau)$ and $\mathcal{S}_{y}(\tau)$ assume values less than
one in different intervals of $\tau$ (see hachured region), which implies in
the squeezing effect for each case obeying the relation
$\mathcal{S}_{x}(\tau)\mathcal{S}_{y}(\tau)\geq 1$ (the saturation is reached
for $\tau=m\pi$ with $m\in\mathbb{N}$). (b) corresponds to the plots of
$\mathcal{E}_{x}$ (dot-dashed line), $\mathcal{E}_{y}$ (dashed line), and
$\mathcal{E}_{z}$ (solid line) versus $\tau\in[0,2\pi]$, for the same values
of $\\{j,\theta,\varphi\\}$ adopted in the previous picture. The hachured area
denotes, in this case, that region where the entanglement effect occurs.
## Appendix C A case study for $h=0$
To what extent the anisotropy parameter $\gamma$ can be effectively employed
in order to validate the entanglement criteria here adopted in the modified
LMG model? Since the two-body term of the Hamiltonian ${\bf H}^{\prime}=-h{\bf
J}_{z}-\frac{1}{N}\left({\bf J}_{x}^{2}+\gamma{\bf J}_{y}^{2}\right)$ plays an
essential role in such a case, we cannot refrain from studying it per se in a
deeper perspective — in this context, we will assume $h=0$ hereafter. Note
that the plethora of results exposed and discussed along the text for
assigning the existence of entanglement in a multi-spin system can then be
directly compared in the particular case of the two-body correlations
associated with the model at hand. This important procedure will produce an
effective range for $\gamma$ with $N=20$ fixed, via extensive numerical
computations of Eqs. (42) and (32), whose physical implications permit, within
other features, to answer (in part, at least) the aforementioned question.
Figure 8: Plots of (a,c) $\mathcal{E}_{z}(t)$ (solid line), $E_{z}(t)$ (dot-
dashed line), $\mathcal{S}_{z}^{(x)}(t)$ (dashed line) and (b,d)
$\mathscr{V}_{\mathrm{J}_{y}\mathrm{J}_{z}}(t)$ versus $t\in[0,50]$ for (a,b)
$\gamma=0.5$ and (c,d) $\gamma=0.948$ (critical value), with $N=20$ fixed and
$h=0$ (absence of transverse magnetic field). The transition between the
different values of $\gamma$, here depicted by pictures (a) and (c),
illustrates how the squeezing and entanglement criteria (as well as the
covariance function) are affected by the anysotropy parameter $\gamma$. Such a
numerical evidence suggests the existence of a validity domain for the
aforementioned criteria, where the ‘almost perfect match’ between squeezing
and entanglement effects should be preserved for any $t\geq 0$.
Now, we discuss some essential points inherent to the numerical results
obtained through the discrete Wigner function (13) evaluated at equally spaced
time intervals (i.e., $0.05$ time units) and $|\gamma|\leq 1$. The first one
confirms the link between squeezing and entanglement effects verified in
section 4 for values up to $\gamma=0.5$; however, if one considers the
interval $0.5<\gamma<0.9$, small differences concerning the parameters
$\\{\mathcal{S}_{a}^{(x)}(t),\mathcal{E}_{a}(t),E_{a}(t)\\}$ for $a=y,z$ begin
to appear, this fact being considered by us as a first numerical evidence
about the ‘limitation’ of the particular entanglement measures
$\mathcal{E}_{a}(t)$ and $E_{a}(t)$ here studied. The second essential point
establishes that, for $\gamma\approx 0.948$ (critical value),
$\mathcal{E}_{a}(t)$ does not predict entanglement effects anymore, while
$E_{a}(t)$ still does — in this case, it is worth stressing that
$\mathcal{S}_{a}^{(x)}(t)$ also indicates the presence of the squeezing effect
for determined values of time.
With respect to the proposals of squeezing and entanglement measures discussed
in this work, let us draw attention to the presence of the covariance function
$\mathscr{V}_{\mathrm{J}_{y}\mathrm{J}_{z}}(t)$ in the expression for
$\mathcal{S}_{a}^{(x)}(t)$ (see table 4.2 and appendix A) which depends
explicitly on the mean value (27). Our numerical calculations allow to show,
in particular, that $\mathscr{V}_{\mathrm{J}_{y}\mathrm{J}_{z}}(t)$ is
responsible for the occurrence of squeezing effect in the entire domain of
$\gamma$, since it carries essential additional information, when we take into
account the RS inequality (instead of the Heisenberg one), associated with the
anticommutation relation of the angular-momentum operators $\mathbf{J}_{y}$
and $\mathbf{J}_{z}$.
Therefore, our results reinforce the real necessity of a deeper discussion on
the validity domain related to the entanglement criteria here used to
characterize the important link between squeezing and entanglement
effects.[13] Thus, it seems that we cannot assure beforehand the viability of
such criteria without also especifying their respective validity domains in
order to be employed with a consistent degree of confidence.
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|
arxiv-papers
| 2012-11-26T16:00:46 |
2024-09-04T02:49:38.422119
|
{
"license": "Public Domain",
"authors": "Marcelo A. Marchiolli, Di\\'ogenes Galetti and Tiago Debarba",
"submitter": "Tiago Debarba",
"url": "https://arxiv.org/abs/1211.6004"
}
|
1211.6013
|
In this paper, we are interested in the development of efficient algorithms for convex
optimization problems in the simultaneous presence of multiple objectives and stochasticity in the first-order information. We cast the stochastic multiple objective optimization problem into a constrained optimization problem by choosing one function as the objective and try to bound other objectives by appropriate thresholds. We first examine a two stages exploration-exploitation based algorithm which first approximates the stochastic objectives by sampling and then solves a constrained stochastic optimization problem by projected gradient method. This method attains a suboptimal convergence rate even under strong assumption on the objectives. Our second approach is an efficient primal-dual stochastic algorithm. It leverages on the theory of Lagrangian method in constrained optimization and attains the optimal convergence rate of $O(1/ \sqrt{T})$ in high probability for general Lipschitz continuous objectives.
§ INTRODUCTION
Although both stochastic optimization [17, 4, 18, 10, 26, 20, 22] and multiple objective optimization [9] are well studied subjects in Operational Research and Machine Learning [11, 12, 24], much less is developed for stochastic multiple objective optimization, which is the focus of this work. Unlike multiple objective optimization where we have access to the complete objective functions, in stochastic multiple objective optimization, only stochastic samples of objective functions are available for optimization. Compared to the standard setup of stochastic optimization, the fundamental challenge of stochastic multiple objective optimization is how to make appropriate tradeoff between different objectives given that we only have access to stochastic oracles for different objectives. In particular, an algorithm for this setting has to ponder conflicting objective functions and accommodate the uncertainty in the objectives.
A simple approach toward stochastic multiple objective optimization is to linearly combine multiple objectives with a fixed weight assigned to each objective. It converts stochastic multiple objective optimization into a standard stochastic optimization problem, and is guaranteed to produce Pareto efficient solutions. The main difficulty with this approach is how to decide an appropriate weight for each objective, which is particularly challenging when the complete objective functions are unavailable. In this work, we consider an alternative formulation that casts multiple objective optimization into a constrained optimization problem. More specifically, we choose one of the objectives as the target to be optimized, and use the rest of the objectives as constraints in order to ensure that each of these objectives is below a specified level. Our assumption is that although full objective functions are unknown, their desirable levels can be provied due to the prior knowledge of the domain. Below, we provide a few examples that demonstrate the application of stochastic multiple objective optimization in the form of stochastic constrained optimization.
Robust Investment. Let $\r \in \R^n$ denote random returns of the $n$ risky assets, and $\w \in \K = \{ \w \in \R_{+}^d : \sum_{i}^{d}{w_i} = 1 \}$ denote the distribution of an investor's wealth over all $n$ risky assets. The return for an investment distribution is defined as $\dd{\w}{\r}$. The investor needs to consider conflicting objectives such as rate of return, liquidity and risk in maximizing his wealth [2]. Suppose that $\mathbf{r}$ has a unknown probability distribution with mean vector $\boldsymbol{\mu}$ and covariance matrix $\Sigma$. Then the target of the investor is to choose an optimal portfolio $\w$ that lies on the mean-risk efficient frontier. In mean-variance theory [15], which trades off between the expected return (mean) and risk (variance) of a portfolio, one is interested in minimizing the variance subject to budget constraints which leads to a formulation like:
\[ \min_{\w \in \R_{+}^d, \sum_{i}^{d}{w_i} = 1} \left[ \dd{\w}{\E[{\r}{\r}^{\top}] \w} \right] \quad \text{subject to} \quad \E[\dd{\mathbf{r}}{\w}] \geq \gamma. \]
Neyman-Pearson Classification. In the Neyman-Pearson (NP) classification paradigm (see e.g, [19]), the goal is to learn a classifier from
labeled training data such that the probability of a false negative is minimized while the probability
of a false positive is below a user-specified level $\gamma \in (0, 1)$. Let hypothesis class be a parametrized convex set $\W = \{ \x \mapsto \langle \w, \mathbf{x} \rangle: \w \in \R^d, \|\w\| \leq R\}$ and for all $(\mathbf{x}, y) \in \R^d \times \{-1, +1\}$, the loss function $\ell(\cdot, (\mathbf{x}, y))$ be a non-negative convex function. While the goal of classical binary classification problem is to minimize the risk as $\min_{\w \in \W} \left[ \L(\w) = \E\left[\ell(\w; (\mathbf{x},y))\right]\right]$, the Neyman-Pearson targets on
\[ \min_{\w \in \W} \L^{+}(\w) \quad \text{subject to} \quad \L^{-}(\w) \leq \gamma, \]
where $\L^{+}(\w) = \E[\ell(\w; (\mathbf{x},y))|y = +1]$ and $\L^{-}(\w) = \E[\ell(\w; (\mathbf{x},y))|y = -1]$.
Linear Optimization with Stochastic Constraints. In many applications in economics, most notably in welfare and utility theory, and management parameters are known only stochastically and it is unreasonable to assume that the objective functions and the solution domain are deterministically fixed. These situations involve the challenging task of pondering both conflicting goals and random data concerning the uncertain parameters of the problem. Mathematically, the goal in multi-objective linear programming with stochastic information is to solve:
\[ \min_{\w} \left[\dd{\c_1(\xi)}{\w}, \cdots, \dd{\c_K(\xi)}{\w}\right]\quad \text{subject to} \quad \w \in \W = \{\w \in \R_{+}^d: A(\xi)\w \leq \mathbf{b}(\xi)\}, \]
where $\xi$ is the randomness in the parameters, $c_i, i \in [K]$ are the objective functions, and $A$ and $\mathbf{b}$ formulate the constraints on the solution.
In this paper, we first examine two methods that try to eliminate the multi-objective aspect or the stochastic nature of stochastic multiple objective optimization and reduce the problem to a standard convex optimization problem. We show that both methods fail to tackle the problem of stochastic multiple objective optimization in general and require strong assumptions on the stochastic objectives, which limits their applications to real world problems. Having discussed these negative results, we propose an algorithm that can solve the problem optimally and efficiently. We achieve this by an efficient primal-dual stochastic gradient descent method that is able to attain $O(1/\sqrt{T})$ convergence rate for all the objectives under the standard assumption of the Lipschitz continuity of objectives which is known to be optimal (see for instance [3]). We note that there is a flurry of research on heuristics-based methods to address the multi-objective stochastic optimization problem (see e.g., [8] and [1] for a recent survey on existing methods). However, in contrast to this study, most of these approaches do not have theoretical guarantees.
Finally, we would like to distinguish our work from robust optimization [5] and online learning with long term constraint [13]. Robust optimization was designed to deal with uncertainty within the optimization systems. Although it provides a principled framework for dealing with stochastic constraints, it often ends up with non-convex optimization problems that are not computationally tractable. Online learning with long term constraint generalizes online learning. Instead of requiring the constraints to be satisfied by every solution generated by online learning, it allows the constraints to be satisfied by the entire sequence of solutions. However, unlike stochastic multiple objective optimization, in online learning with long term constraints, constraint functions are fixed and known before the start of online learning.
Outline The remainder of the paper is organized as follows. In Section <ref> we establish the necessary notation and introduce the problem under consideration. Section <ref> introduces the problem reduction methods and elaborates their disadvantages. Section <ref> presents our efficient primal-dual stochastic optimization algorithm. Finally, we conclude the paper with open questions in Section <ref>.
§ PRELIMINARIES
NotationThroughout this paper, we use the following notation. We use bold-face letters to denote vectors. We denote the inner product between two vectors $\w,\w' \in \W$ by $\dd{\w}{\w'}$ where $\W \subseteq \R^d$ is a compact closed domain. For $m \in \mathbb{N}$, we denote
by $[m]$ the set $\{1, 2, \cdots, m\}$. We only consider the $\ell_2$ norm throughout the paper. The ball with radius $R$ is denoted by $\B = \left\{\x \in \R^d: \|\x\| \leq R \right\}$.
Statement of the Problem In this work, we generalize online convex optimization (OCO) to the case of multiple objectives. In particular, at each iteration, the learner is asked to present a solution $\x_t$, which will be evaluated by multiple loss functions $f^0_t(\x), f^1_t(\x), \ldots, f^m_t(\x)$. A fundamental difference between single- and multi-objective optimization is that for the latter it is not obvious how to evaluate the optimization quality. Since it is impossible to simultaneously minimize multiple loss functions and in order to avoid complications caused by handling more than one objective, we choose one function as the objective and try to bound other objectives by appropriate thresholds. Specifically, the goal of OCO with multiple objectives becomes to minimize $\sum_{t=1}^T f^0_t(\x_t)$ and at the same time keep the other objective functions below a given threshold, i.e.
\begin{eqnarray*}
\frac{1}{T}\sum_{t=1}^T f^i_t(\x_t) \leq \gamma_i,\; i \in [m],
\end{eqnarray*}
where $\x_1, \ldots, \x_T$ are the solutions generated by the online learner and $\gamma_i$ specifies the level of loss that is acceptable to the $i$th objective function.
Since the general setup (i.e., full adversarial setup) is challenging for online convex optimization even with two objectives [14], in this work, we consider a simple scenario where all the loss functions $\{f_t^i(\w)\}_{i=1}^m$ are i.i.d samples from unknown distribution [21]. We also note that our goal is NOT to find a Pareto efficient solution (a solution is Pareto efficient if it is not dominated by any solution in the decision
space). Instead, we aim to find a solution that (i) optimizes one selected objective, and (ii) satisfies all the other objectives with respect to the specified thresholds.
We denote by $\fb_i(\w) = \E_t[f_t^i(\w)], i =0, 1, \ldots, m$ the expected loss function of sampled function $f_t^i(\w)$. In stochastic multiple objective optimization, we assume that we do not have direct access to the expected loss functions and the only information available to the solver is through a stochastic oracle that returns a stochastic realization of the expected loss function at each call. We assume that there exists a solution $\x$ strictly satisfying all the constraints, i.e. $\fb_i(\x) < \gamma_i, i \in [m]$. We denote by $\x_*$ the optimal solution to multiple objective optimization, i.e.,
\begin{eqnarray}
\x_* = \mathop{\arg\min}\left\{\fb_0(\x): \fb_i(\x) \leq \gamma_i, i \in [m]\right\}. \label{eqn:opt-1}
\end{eqnarray}
Our goal is to efficiently compute a solution $\xh_T$ after $T$ trials that (i) obeys all the constraints, i.e. $\fb_i(\xh_T) \leq \gamma_i, i \in [m]$ and (ii) minimizes the objective $\fb_0$ with respect to the optimal solution $\x_*$, i.e. $\fb_0(\xh_T) - \fb_0(\x_*)$. For the convenience of discussion, we refer to $f^0_t(\w)$ and $\fb_0(\w)$ as the objective function, and to $f_t^i(\w)$ and $\fb_i(\w)$ as the constraint functions.
Before discussing the algorithms, we first mention a few assumptions made in our analysis. We assume that the optimal solution $\x_*$ belongs to $\B$. We also make the standard assumption that all the loss functions, including both the objective function and constraint functions, are Lipschitz continuous, i.e., $|f_t^i(\x) - f_t^i(\x')| \leq L\|\x - \x'\|$ for any $\x, \x' \in \B$.
§ PROBLEM REDUCTION AND ITS LIMITATIONS
Here we examine two algorithms to cope with the complexity of stochastic optimization with multiple objectives and discuss some negative results which motivate the primal-dual algorithm presented in Section <ref>. The first method transforms a stochastic multi-objective problem into a stochastic single-objective optimization problem and then solves the latter problem by any stochastic programming approach. Alternatively, one can eliminate the randomness of the problem by estimating the stochastic objectives and transform the problem into a deterministic multi-objective problem.
§.§ Linear Scalarization with Stochastic Optimization
A simple approach to solve stochastic optimization problem with multiple objectives is to eliminate the multi-objective aspect of the problem by aggregating the $m+1$ objectives into a single objective $\sum_{i=0}^{m}{\alpha_i f^i_t(\x_t)}$, where $\alpha_i, i \in \{0,1, \cdots, m\}$ is the weight of $i$th objective, and then solving the resulting single objective stochastic problem by stochastic optimization methods. This approach is in general known as the weighted-sum or scalarization method [1]. Although this naive idea considerably facilitates the computational challenge of the problem, unfortunately, it is difficult to decide the weight for each objective, such that the specified levels for different objectives are obeyed. Beyond the hardness of optimally determining the weight of individual functions, it is also unclear how to bound the sub-optimality of final solution for individual objective functions.
§.§ Projected Gradient Descent with Estimated Objective Functions
The main challenge of the proposed problem is that the expected constraint functions $\fb_i(\w)$ are not given. Instead, only a sampled function is provided at each trial $t$. Our naive approach is to replace the expected constraint function $\fb_i(\w)$ with its empirical estimation based on sampled objective functions.
This approach circumvents the problem of stochastically optimizing multiple objective into the original online convex optimization with complex projections, and therefore can be solved by projected gradient descent. More specifically, at trial $t$, given the current solution $\x_t$ and received loss functions $f^i_t(\x), i=0,1,\ldots, m$, we first estimate the constraint functions as
\[
\fh_t^i(\x) = \frac{1}{t}\sum_{k=1}^t f_k^i(\x), i \in [m],
\]
and then update the solution by $\x_{t+1} = \Pi_{\K_t}\left(\x_{t} - \eta \nabla f^0_t(\x_t)\right\}$ where $\eta > 0$ is the step size, $\Pi_{\K}(\x) = \min_{\z \in \K} \|\z-\x \|$ projects a solution $\w$ into domain $\W$, and $\K_t$ is an approximate domain given by $\K_t = \{\x: \fh_t^i(\x) \leq \gamma_i, i \in [m] \}$.
One problem with the above approach is that although it is feasible to satisfy all the constraints based on the true expected constraint functions, there is no guarantee that the approximate domain $\K_t$ is not empty. One way to address this issue is to estimate the expected constraint functions by burning the first $b T$ trials, where $b \in (0, 1)$ is a constant that needs to be adjusted to obtain the optimal performance, and keep the estimated constraint functions unchanged afterwards. Given the sampled functions $f_1^i, \ldots, f_{bT}^i$ received in the first $bT$ trials, we compute the approximate domain $\K'$ as
\[
\fh^i(\x) = \frac{1}{bT}\sum_{t=1}^{bT} f^i_t(\x), i \in [m], \; \K' = \left\{\x: \fh_i(\x) \leq \gamma_i + \hat{\gamma}_i, i=1, \ldots, m \right\}
\]
where $\hat{\gamma}_i > 0 $ is a relaxed constant introduced to ensure that with a high probability, the approximate domain $\W_t$ is not empty provided that the original domain $\W$ is not empty.
To ensure the correctness of the above approach, we need to establish some kind of uniform (strong) convergence assumption to make sure that the solutions obtained by projection onto the estimated domain $\W'$ will be close to the true domain $\W$ with high probability. It turns out that the following assumption ensures the desired property.
[Uniform Convergence] Let $\fh^i(\x), i = 0,1,\cdots, m$ be the estimated functions obtained by averaging over $bT$ i.i.d samples for $\fb_i(\w), i \in [m]$. We assume that, with a high probability,
\[\sup_{\w \in \W} \left|\fh^i(\x)-\fb_i(\w)\right| \leq O([bT]^{-q}), i = 0,1,\cdots, m.\]
where $q > 0$ decides the convergence rate.
It is straightforward to show that under Assumption <ref>, with a high probability, for any $\x \in \K$, we have $\x \in \K'$, with appropriately chosen relaxation constant $\hat{\gamma}_i, i \in [m]$. Using the estimated domain $\K'$, for trial $t \in [b T + 1, T]$, we update the solution by
\[
\x_{t+1} = \Pi_{\K'}(\x_{t} - \eta \nabla f^0_t(\x_t)).
\]
There are however several drawbacks with this naive approach. Since the first $bT$ trials are used for estimating the constraint functions, only the last $(1 - b)T$ trials are used for searching for the optimal solution. The total amount of violation of individual constraint functions for the last $(1 - b)T$ trials, given by $\sum_{t=bT+1}^T \fb_i(\x_t)$, is $O((1 - b) b^{-q} T^{1 - q})$, where each of the $(1 -b)T$ trials receives a violation of $O([bT]^{-q})$. Similarly, following the conventional analysis of online learning [26], we have $\sum_{t=bT+1}^T (f_t^0(\w_t) - f_t^0(\w_*)) \leq O(\sqrt{(1 - b)T})$. Using the same trick as in [13], to obtain a solution with zero violation of constraints, we will have a regret bound $O((1 - b) b^{-q} T^{1 - q} + \sqrt{(1 - b)T})$, which yields a convergence rate of $O(T^{-1/2} + T^{-q})$ which could be worse than the optimal rate $O(T^{-1/2})$ when $q < 1/2$. Additionally, this approach requires memorizing the constraint functions of the first $bT$ trials. This is in contrast to the typical assumption of online learning where only the solution is memorized.
We finally remark on the uniform convergence assumption, which holds when the constraint functions are linear [25], but unfortunately does not hold for general convex Lipschitz functions. In particular, one can simply show examples where there is no uniform convergence for stochastic convex Lipchitz functions in infinite dimensional spaces [21]. Without uniform convergence assumption, the approximate domain $\W'$ may depart from the true $\W$ significantly at some unknown point, which makes the above approach to fail for general convex objectives.
Primal-Dual Stochastic Optimization with Multiple Objectives
[1]
INPUT: step size $\eta$, $\lambda^{0}_i > 0, i \in [m]$ and total iterations $T$
$\x_1 = \bl_{1} = \bz$
$t = 1, \ldots, T$
Submit the solution $\x_t$
Receive loss functions $f^i_t, i=0, 1, \ldots, m$
Compute the gradients $\nabla f^i_t(\x_t), i=0, 1, \ldots, m$
Update the solution $\x$ and $\blambda$ by
\begin{eqnarray*}
\x_{t+1} & = & \Pi_{\B}\left(\x_t - \eta \nabla_\x \L_t(\x_t, \lambda_t)\right) = \Pi_{\B}\left(\x_t - \eta \left[ \nabla f_t^0(\x_t) + \sum_{i=1}^m \lambda_t^i \nabla f_t^i(\x_t) \right]\right), \\
\lambda^i_{t+1} & = & \Pi_{[0, \lambda_i^0]}\left(\lambda^i_t + \eta \nabla_{\lambda_i} \L_t(\x_t, \lambda_t)\right) = \Pi_{[0, \lambda_i^0]}\left(\lambda^i_t + \eta \left[f_t^i(\x_t) - \gamma_i\right]\right).
\end{eqnarray*}
Return $\hat{\x}_T = \sum_{t=1}^T \x_t/T$
To address these limitations and in particular the dependence on uniform convergence assumption, we present an algorithm that does not require projection when updating the solution and does not require to impose any additional assumption on the stochastic functions except for the standard Lipschitz continuity assumption. We show that with a high probability, the solution found by the proposed algorithm will exactly satisfy the expected constraints and achieves a regret bound of $O(\sqrt{T})$. We note that our result is closely related to the recent studies of learning from the viewpoint of optimization [23], which state that solutions found by stochastic gradient descent can be statistically consistent even when uniform convergence theorem does not hold.
§ A PRIMAL-DUAL ALGORITHM FOR STOCHASTIC MULTI-OBJECTIVE OPTIMIZATION
We now turn to devise a tractable formulation of the problem, followed by an efficient primal-dual optimization algorithm and the statements of our main results. The main idea of the proposed algorithm is to design an appropriate objective that combines the loss function $\fb_0(\w)$ with $\{\fb_i(\w)\}_{i=1}^m$. As mentioned before, owing to the presence of conflicting goals and the randomness nature of the objective functions, we resort to seek for a solution that satisfies all the objectives instead of an optimal one. To this end, we define the following objective function
\[
\Lb(\x, \blambda) = \fb_0(\x) + \sum_{i=1}^m \lambda_i (\fb_i(\x) - \gamma_i).
\]
Note that the objective function consists of both the primal variables $\x$ and dual variables $\bl = (\lambda_1, \ldots, \lambda_m)^{\top} \in \Lambda$, where $\Lambda \subseteq \R_{+}^m$ is a compact convex set that bounds the set of dual variables and will be discussed later. In the proposed algorithm, we will simultaneously update solutions for both $\x$ and $\blambda$. By exploring convex-concave optimization theory [16], we will show that with a high probability, the solution of regret $O(\sqrt{T})$ exactly obeyes the constraints.
As the first step, we consider a simple scenario where the obtained solution is allowed to violate the constraints. The detailed steps of our primal-dual algorithm is presented in Algorithm <ref> . It follows the same procedure as convex-concave optimization. Since at each iteration, we only observed a randomly sampled loss functions $f^i_t(\w), i=0,1,\ldots, m$, the objective function given by
\[
\L_t(\x, \bl) = f^0_t(\x) + \sum_{i=1}^m \lambda_i (f_t^i(\x) - \gamma_i)
\]
provides an unbiased estimate of $\Lb(\x, \blambda)$. Given the approximate objective $\L_t(\x, \blambda)$, the proposed algorithm tries to minimize the objective $\L_t(\w, \blambda)$ with respect to the primal variable $\x$ and maximize the objective with respect to the dual variable $\blambda$.
To facilitate the analysis, we first rewrite the the constrained optimization problem
\begin{eqnarray*}
\min\limits_{\x \in \B \cap \K} \fb_0(\x)
\end{eqnarray*}
where $\K$ is defined as $\K = \left\{\x: \fb_i(\x) \leq \gamma_i, i=1, \ldots m \right\}$ in the following equivalent form:
\begin{eqnarray}
\min\limits_{\x \in \B} \max\limits_{\blambda \in \R_+^m} \fb_0(\x) + \sum_{i=1}^m \lambda_i(\fb_i(\x) - \gamma_i). \label{eqn:dual}
\end{eqnarray}
We denote by $\x_*$ and $\blambda_* = (\lambda_1^*, \ldots, \lambda_m^*)^{\top}$ as the optimal primal and dual solutions to the above convex-concave optimization problem, i.e.,
\begin{eqnarray}
\x_* & = & \mathop{\arg\min}\limits_{\x \in \B} \fb_0(\x) + \sum_{i=1}^m \lambda_i^*(\fb_i(\x) - \gamma_i), \label{eqn:x*} \\
\blambda_* & = & \mathop{\arg\max}\limits_{\blambda \in \R_+^m} \fb_0(\x_*) + \sum_{i=1}^m \lambda_i(\fb_i(\x_*) - \gamma_i). \label{eqn:lambda*}
\end{eqnarray}
The following assumption establishes upper bound on the gradients of $\L(\w, \bl)$ with respect to $\w$ and $\bl$. We later show that this assumption holds under a mild condition on the objective functions.
[Gradient Boundedness] The gradients $\nabla_\x \L(\x, \blambda)$ and $\nabla_{\blambda} \L(\x, \blambda)$ are uniformly bounded, i.e., there exist a constant $G > 0$ such that
\[ \max\left(\nabla_\x \L(\x, \blambda), \nabla_{\blambda} \L(\x, \blambda) \right) \leq G, \;\; \text{for any} \;\; \w \in \B \;\;\text{and} \;\; \blambda \in \Lambda.\]
Under the preceding assumption, in the following theorem, we show that under appropriate conditions, the average solution $\xh_T$ generated by of Algorithm <ref> attains a convergence rate of $O(1/\sqrt{T})$ for both the regret and the violation of the constraints.
Set $\lambda_0^i \geq \lambda_i^* + \theta, i \in [m]$, where $\theta > 0$ is a constant. Let $\xh_T$ be the solution obtained by Algorithm <ref> after $T$ iterations. Then, with a probability $1 - (2m+1)\delta$, we have
\[
\fb_0(\xh_T) - \fb_0(\x_*) \leq \frac{\mu(\delta)}{\sqrt{T}}\;\; \text{and} \;\; \fb_i(\xh_T) - \gamma_i \leq \frac{\mu(\delta)}{\theta\sqrt{T}}, i \in [m]
\]
where $D^2 = \sum_{i=1}^m [\lambda^0_i]^2$, $\eta = [\sqrt{ (R^2+D^2)/2T}]/G$, and
\begin{eqnarray}
\mu(\delta) = \sqrt{2} G\sqrt{R^2 + D^2} + 2G(R+D)\sqrt{2\ln\frac{1}{\delta}}. \label{eqn:mu}
\end{eqnarray}
The parameter $\theta \in \R_{+}$ is a quantity that may be set to obtain sharper
upper bound on the violation of constraints and may be chosen arbitrarily. In particular, a larger value for $\theta$ imposes larger penalty on the violation of the constraints and results in a smaller violation for the objectives.
We also can develop an algorithm that allows the solution to exactly satisfy all the constraints. To this end, we define $ \gammah_i = \gamma_i - \frac{\mu(\delta)}{\theta\sqrt{T}} $. We will run Algorithm <ref> but with $\gamma_i$ replaced by $\gammah_i$. Let $G'$ denote the upper bound in Assumption <ref> for $\nabla_{\bl}\L(\w,\bl)$ with $\gammah_i$ is replaced by $\gamma_i, i \in [m]$. The following theorem shows the property of the obtained average solution $\xh_T$.
Let $\xh_T$ be the solution obtained by Algorithm <ref> with $\gamma_i$ replaced by $\gammah_i$ and $\lambda_0^i = \lambda_i^* + \theta, i \in [m]$. Then, with a probability $1 - (2m+1)\delta$, we have
\[
\fb_0(\xh_T) - \fb_0(\x_*) \leq \frac{(1 + \sum_{i=1}^m \lambda_i^0)\mu'(\delta)}{\sqrt{T}} \;\; \text{and} \;\; \fb_i(\xh_T) \leq \gamma_i, i \in [m],
\]
where $\mu'(\delta)$ is same as (<ref>) with $G$ is replaced by $G'$ and $\eta = [\sqrt{ (R^2+D^2)/2T}]/G'$.
§.§ Convergence Analysis
Here we provide the proofs of main theorems stated above. We start by proving Theorem <ref> and then extend it to prove Theorem <ref>.
(of Theorem <ref>)
Using the standard analysis of convex-concave optimization, from the convexity of $\Lb(\x, \cdot)$ with respect to $\w$ and concavity of $\Lb(\cdot, \blambda)$ with respect to $\blambda$, for any $\x \in \B$ and $\lambda_i \in [0, \lambda^0_i], i \in [m]$, we have
\begin{eqnarray*}
\lefteqn{\Lb(\x_t, \blambda) - \Lb(\x, \blambda_t) }\\
& \leq & \dd{\x_t - \x}{\nabla_\x\Lb(\x_t, \blambda_t)} - \dd{\blambda_t - \blambda}{\nabla_{\blambda}\Lb(\x_t, \blambda_t)} \\
& = & \dd{\x_t - \x}{\nabla_\x\L_t(\x_t, \blambda_t)} - \dd{\blambda_t - \blambda}{\nabla_{\blambda}\L_t(\x_t, \blambda_t)} \\
& & + \dd{\x_t - \x}{\nabla_\x\Lb(\x_t, \blambda_t) - \nabla_\x\L_t(\x_t, \blambda_t)} - \dd{\blambda_t - \blambda}{\nabla_{\blambda}\Lb(\x_t, \blambda_t) - \nabla_{\blambda}\L_t(\x_t, \blambda_t)} \\
& \leq & \frac{\|\x_t - \x\|^2 - \|\x_{t+1} - \x\|^2}{2\eta} + \frac{\|\blambda_t - \blambda\|^2 - \|\blambda_{t+1} - \blambda\|^2}{2\eta} + \frac{\eta}{2}\left(\|\nabla_\x \L_t(\x_t, \blambda_t)\|^2 + \|\nabla_{\blambda} \L_t(\x_t, \blambda_t)\|^2\right) \\
& & + \dd{\x_t - \x}{\nabla_\x\Lb(\x_t, \blambda_t) - \nabla_\x\L_t(\x_t, \blambda_t)} - \dd{\blambda_t - \blambda}{\nabla_{\blambda}\Lb(\x_t, \blambda_t) - \nabla_{\blambda}\L_t(\x_t, \blambda_t)},
\end{eqnarray*}
where the last inequality follows from the updating rules for $\w_{t+1}$ and $\blambda_{t+1}$ and non-expensiveness property of the projection operation. By adding all the inequalities together, we have
\begin{eqnarray*}
\lefteqn{\sum_{t=1}^T \Lb(\x_t, \blambda) - \Lb(\x, \blambda_t)} \\
& \leq & \frac{\|\x - \x_1\|^2 + \|\blambda - \blambda_1\|^2}{2\eta} + \frac{\eta}{2}\sum_{t=1}^T \|\nabla_\x \L_t(\x_t, \blambda_t)\|^2 + \|\nabla_{\blambda} \L_t(\x_t, \blambda_t)\|^2 \\
& & + \sum_{t=1}^T \dd{\x_t - \x}{\nabla_\x\Lb(\x_t, \blambda_t) - \nabla_\x\L_t(\x_t, \blambda_t)} - \dd{\blambda_t - \blambda}{\nabla_{\blambda}\Lb(\x_t, \blambda_t) - \nabla_{\blambda}\L_t(\x_t, \blambda_t)} \\
& \leq & \frac{R^2 + D^2}{2\eta} + {\eta G^2 T} \\
& \hspace{0.5cm}&+\sum_{t=1}^T \dd{\x_t - \x}{\nabla_\x\Lb(\x_t, \blambda_t) - \nabla_\x\L_t(\x_t, \blambda_t)} - \dd{\blambda_t - \blambda}{\nabla_{\blambda}\Lb(\x_t, \blambda_t) - \nabla_{\blambda}\L_t(\x_t, \blambda_t)} \\
& \leq & \frac{R^2 + D^2}{2\eta} + {\eta G^2 T} + 2G(R+D)\sqrt{2 T \ln\frac{1}{\delta}} \mbox{\quad(w.p. $1 - \delta$)},
\end{eqnarray*}
where the last inequality follows from the Hoeffiding inequality for Martingales [6]. By expanding the left hand side, substituting the stated value of $\eta$, and applying the Jensen's inequality for the average solutions $\hat{\x}_T = \sum_{t=1}^T \x_t/T$ and $\blh_T = \sum_{t=1}^T \bl_t/T$, for any fixed $\lambda_i \in [0, \lambda_i^0], i\in [m]$ and $\x \in \B$, with a probability $1 - \delta$, we have
\begin{eqnarray}
\fb_0(\xh_T) + \sum_{i=1}^m \lambda_i (\fb_i(\xh_T) - \gamma_i) - \fb_0(\x) - \sum_{i=1}^m \lambdah^i_T (\fb_i(\x) - \gamma_i) \label{eqn:temp-bound-1} \\ \leq \sqrt{2}G\sqrt{\frac{R^2 + D^2}{T}} + 2G(R+D)\sqrt{\frac{2}{T}\ln\frac{1}{\delta}}. \nonumber
\end{eqnarray}
By fixing $\x = \x_*$ and $\blambda = \mathbf{0}$ in (<ref>), we have $\fb_i(\x_*) \leq \gamma_i, i \in [m]$, and therefore, with a probability $1 - \delta$, have
\[
\fb_0(\xh_T) \leq \fb_0(\x_*) + \sqrt{2}G\sqrt{\frac{R^2 + D^2}{T}} + 2G(R+D)\sqrt{\frac{2}{T}\ln\frac{1}{\delta}}.
\]
To bound the violation of constraints, for each $i \in [m]$, we set $\x = \x_*$, $\lambda_i = \lambda_i^0$, and $\lambda_j = \lambda_j^*, j \neq i$ in (<ref>). We have
\begin{eqnarray*}
& & \fb_0(\xh_T) + \lambda^0_i(\fb_i(\xh_T) - \gamma_i) + \sum_{j \neq i} \lambda^*_j(\fb_j(\xh_T) - \gamma_j) - \fb_0(\x_*) - \sum_{i=1}^m \lambdah_T^i(\fb_i(\x_*) - \gamma_i) \\
& \geq & \fb_0(\xh_T) + \lambda^0_i(\fb_i(\xh_T) - \gamma_i) + \sum_{j \neq i} \lambda^*_j(\fb_j(\xh_T) - \gamma_j) - \fb_0(\x_*) - \sum_{i=1}^m \lambda_*^i(\fb_i(\x_*) - \gamma_i) \\
& \geq & \theta (\fb_i(\xh_T) - \gamma_i),
\end{eqnarray*}
where the first inequality utilizes (<ref>) and the second inequality utilizes (<ref>).
We thus have, with a probability $1 - \delta$,
\begin{eqnarray*}
\fb_i(\xh_T) - \gamma_i \leq \frac{\sqrt{2}G}{\theta}\sqrt{\frac{R^2 + D^2}{T}} + \frac{2G(R+D)}{\theta}\sqrt{\frac{2}{T}\ln\frac{1}{\delta}}.
\end{eqnarray*}
We complete the proof by taking the union bound over all the random events.
We now turn to the proof of Theorem <ref> that gives high probability bound on the convergence of the modified algorithm which obeys all the constraints.
(of Theorem <ref>)
Following the proof of Theorem <ref>, with a probability $1 - \delta$, we have
\begin{eqnarray}
\fb_0(\xh_T) + \sum_{i=1}^m \lambda_i (\fb_i(\xh_T) - \gammah_i) - \fb_0(\x) - \sum_{i=1}^m \lambdah^i_T (\fb_i(\x) - \gammah_i) \nonumber \\ \leq \sqrt{2}G'\sqrt{\frac{R^2 + D^2}{T}} + 2G'(R+D)\sqrt{\frac{2}{T}\ln\frac{1}{\delta}} \nonumber
\end{eqnarray}
Define $\widetilde{\w}_*$ and $\{\widetilde{\lambda}_j^*\}_{j=1}^m$ be the saddle point for the following minimax optimization problem
\[
\min\limits_{\w \in \B} \max\limits_{\blambda \in \R_+^m} \fb_0(\w) + \sum_{i=1}^m \lambda_i (\fb_i(\w) - \widehat{\gamma}_i)
\]
Following the same analysis as Theorem <ref>, for each $i \in [m]$, by setting $\w = \widetilde{\w}_*$, $\lambda_i = \lambda_i^0$, and $\lambda_j = \widetilde{\lambda}_j^*$, using the fact that $\widetilde{\lambda}_j^* \leq \lambda_j^*$, we have, with a probability $1 - \delta$
\[
\theta(\fb_i(\wh_T) - \gamma_i) \leq \sqrt{2}G'\sqrt{\frac{R^2 + D^2}{T}} + 2G'(R+D)\sqrt{\frac{2}{T}\ln\frac{1}{\delta}} - \frac{\mu(\delta)}{\sqrt{T}} \leq 0,
\]
which completes the proof.
§.§ Implementation Issues
In order to run Algorithm <ref>, we need to estimate the parameter $\lambda_i^0, i \in [m]$, which requires to decide the set $\Lambda$ by estimating an upper bound for the optimal dual variables $\lambda_i^*, i \in [m]$. To this end, we consider an alternative problem to the convex-concave optimization problem in (<ref>), i.e.
\begin{eqnarray}
\min\limits_{\x \in \B} \max\limits_{\lambda \geq 0} \fb_0(\x) + \lambda \max\limits_{1 \leq i \leq m} (\fb_i(\x) - \gamma_i). \label{eqn:dual-max}
\end{eqnarray}
Evidently $\x_*$ is the optimal primal solution to (<ref>). Let $\lambda_a$ be the optimal dual solution to the problem in (<ref>). We have the following proposition that links $\lambda_i^*, i\in [m]$, the optimal dual solution to (<ref>), with $\lambda_a$, the optimal dual solution to (<ref>).
Let $\lambda_a$ be the optimal dual solution to (<ref>) and $\lambda_*^i, i \in [m]$ be the optimal solution to (<ref>). We have $ \lambda_a = \sum_{i=1}^m \lambda_*^i$.
We can rewrite (<ref>) as $ \min\limits_{\x \in \B} \max\limits_{\lambda \geq 0, \mathbf{p} \in \Delta_m} \fb_0(\x) + \sum_{i=1}^m p_i \lambda (\fb_i(\x) - \gamma_i)$,
where domain $\Delta_m$ is defined as $\Delta_m =\{\alpha \in \R_+^m: \sum_{i=1}^m \alpha_i = 1 \}$. By redefining $\lambda_i = p_i\lambda$, we have the problem in (<ref>) equivalent to (<ref>) and consequently $\lambda = \sum_{i=1}^m \lambda_i$ as claimed.
Given the result from Proposition <ref>, it is sufficient to bound $\lambda_a$. In order to bound $\lambda_a$, we need to make certain assumption about $\fb_i(\w), i \in [m]$. The purpose of introducing this assumption is to ensure that the optimal dual variable for the constrained optimization problem is well bounded from the above.
We assume $\min\limits_{\alpha \in \Delta_m} \left\|\sum_{i=1}^m \alpha_i \nabla \fb_i(\x) \right\| \geq \tau$, where $\tau > 0$ is a constant.
Equipped with Assumption <ref>, we are able to bound $\lambda_a$ by $\tau$. To this end, using the first order optimality condition of (<ref>) [7], we have $\lambda_a = \frac{\|\nabla \fb(\x_*)\|}{\|\partial g(\x)\|}$, where $g(\x) = \max_{1 \leq i \leq m} \fb_i(\x)$. Since $\partial g(\x) \in \left\{\sum_{i=1}^{m} \alpha_i \nabla \fb_i(\x): \alpha \in \Delta_m \right\}$, under Assumption <ref>, we have $\lambda_a \leq \frac{L}{\tau}$. By combining Proposition <ref> with the upper bound on $\lambda_a$, we obtain $\lambda_i^* \leq \frac{L}{\tau}, i \in [m]$ as desired.
Finally, we note that by having $\blambda_*$ bounded, Assumption <ref> is guaranteed by setting $G^2 = \max( L^2\left(1 +\sum_{i=1}^m \lambda_i^0\right)^2, \; \max\limits_{\x \in \B} \sum_{i=1}^m (\fb_i(\x)-\gamma_i)^2)$ which follows from Lipschitz continuity of the objective functions. In a similar way we can set $G'$ in Theorem <ref> by replacing $\gamma_i$ with $\gammah_i$.
§ CONCLUSION AND OPEN QUESTIONS
In this paper we have addressed the problem of stochastic convex optimization with multiple objectives underlying many applications in machine learning. We first examined a simple problem reduction technique that eliminates the stochastic aspect of constraint functions by approximating them using the sampled functions from each iteration. We showed that this simple idea fails to attain the optimal convergence rate and requires to impose a strong assumption, i.e., uniform convergence, on the objective functions. Then, we presented a novel efficient primal-dual algorithm which attains the optimal convergence rate $O(1/\sqrt{T})$ for all the objectives relying only on the Lipschitz continuity of the objective functions. This work leaves few direction for further elaboration. In particular, it would be interesting to see whether or not making stronger assumptions on the analytical properties of objective functions such as smoothness or strong convexity may yield improved convergence rate.
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|
arxiv-papers
| 2012-11-26T16:27:18 |
2024-09-04T02:49:38.437581
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Mehrdad Mahdavi, Tianbao Yang, Rong Jin",
"submitter": "Mehrdad Mahdavi",
"url": "https://arxiv.org/abs/1211.6013"
}
|
1211.6036
|
# The Adiabatic Phase Mixing and Heating of Electrons in Buneman Turbulence
H. Che1, J. F. Drake2, M. Swisdak2, M. L. Goldstein1 1:Goddard Space Flight
Center, NASA, Greenbelt, MD, 20771, USA 2: IREAP, University of Maryland,
College Park, MD, 20742, USA
###### Abstract
The nonlinear development of the strong Buneman instability and the associated
fast electron heating in thin current layers with $\Omega_{e}/\omega_{pe}<1$
are explored. Phase mixing of the electrons in wave potential troughs and a
rapid increase in temperature are observed during the saturation of the
instability. We show that the motion of trapped electrons can be described
using a Hamiltonian formalism in the adiabatic approximation. The process of
separatrix crossing as electrons are trapped and de-trapped is irreversible
and guarantees that the resulting electron energy gain is a true heating
process.
††preprint: APS/123-QED
The exploration of how waves and particles interact in strong turbulence has
been an important challenge in plasma physics. Kadomtsev (1965); O’Neil
(1965); Dupree (1966); Sagdeev and Galeev (1969); Galeev _et al._ (1975);
Goldman (1984); Krommes (2002); Yampolsky and Fisch (2009); Bénisti, Morice,
and Gremillet (2012) Using particle-in-cell simulations we explore the
nonlinear development and nonlinear wave-particle interactions of the Buneman
instability to reveal how particle acceleration and heating take place. The
Buneman instabilityBuneman (1958) is driven by the relative drift between ions
and electrons. Its quasi-linear theory is well understood, but strong Buneman
turbulence is still a subject with open questions though it has been widely
discussedDavidson _et al._ (1970); Ishihara, Hirose, and Langdon (1981);
Hirose, Ishihara, and Langdon (1982); Cargill and Papadopoulos (1988). The
previous work either did not consider the trapping regime (where the wave
electric field is large enough to trap thermal particles) or treated it under
the assumption that the particle heating growth rate was slow compared to the
instability. We investigate the regime in which rapid electron heating takes
place near the saturation of the Buneman instability when the particle’s
bounce rate in the wave potential is far larger than the growth rate of the
instability. As a consequence, the trapped particle’s motion is approximately
adiabatic. Heating is thus a consequence of coherent trapping, phase mixing
and de-trapping of the particles. Our simulations also demonstrate the
difference between the nonlinear development of the Buneman instability and an
idealized adiabatically-growing single sine wave, which supports that the
heating can be achieved by adiabatic motion and de-trapping .
Electron heating as a result of the Buneman instability is associated with the
intense electron current layers formed during magnetic reconnectionDrake _et
al._ (2003); Che _et al._ (2010); Khotyaintsev _et al._ (2010),
shocksCargill and Papadopoulos (1988); Riquelme and Spitkovsky (2009);
Matsumoto, Amano, and Hoshino (2012) and turbulent energy cascades to sub-
proton scalesAlexandrova _et al._ (2009); Sahraoui _et al._ (2010). In
particular, understanding how kinetic turbulence transfers momentum and energy
is important for revealing the role of anomalous resistivity in magnetic
reconnection, which has been a long-standing puzzleKulsrud _et al._ (2005);
Che, Drake, and Swisdak (2011).
We propose a new mechanism that is responsible for extremely fast electron
heating, in a few tens of electron plasma periods, during the nonlinear
evolution of the Buneman instability. The dynamics is dominated by the
coherent trapping and de-trapping of streaming electrons (with drift $v_{de}$)
in the nearly non-propagating electric field from the instability. The wave
amplitude grows until nearly all of the streaming electrons have been trapped.
Thus the electrostatic potential at saturation is approximately given by
$e\phi\sim m_{e}v_{de}^{2}$. The bounce frequency
$\omega_{b}=k_{0}\sqrt{e\phi/m_{e}}\sim\omega_{pe}$ of electrons trapped in
the potential $\phi$ greatly exceeds the characteristic growth rate of the
wave $\gamma\sim(m_{e}/m_{i})^{1/3}\omega_{pe}$. As a result, the electrons
trapped in the growing potential behave adiabatically, preserving their phase
space area as the wave amplitude slowly changes in time. Phase mixing of the
electrons in the wave potential troughs guarantees that, as the wave amplitude
decreases following saturation, the de-trapping of electrons leaves a
distribution of particles that forms a velocity-space plateau over the
interval ($-v_{de},v_{de}$). The process of separatrix crossing as electrons
are trapped and de-trapped is irreversible and guarantees that the resulting
electron energy gain $\sim m_{e}v_{de}^{2}/2$ is a true heating process.
We carry out 3D PIC simulations with strong electron drifts in an
inhomogeneous current-carrying plasma with a guide field. We apply no external
perturbations to initiate reconnection, and consequently reconnection does not
develop during relatively short duration of the simulation. We specify the
initial magnetic field to be $B_{x}/B_{0}=\tanh[(y-L_{y}/2)/w_{0}]$, where
$B_{0}$ is the asymptotic amplitude of $B_{x}$, and $w_{0}$ and $L_{y}$ are
the half-width of the initial current sheet and the box size in the $y$
direction, respectively. The guide field $B_{z}^{2}=B^{2}-B_{x}^{2}$ is chosen
so that the total field $B$ is constant. In our simulation, we take
$w_{0}=0.1d_{i}$ and $B=\sqrt{26}B_{0}$, where $d_{i}=c/\omega_{pi}$ and
$\omega_{pi}$ is the ion plasma frequency. The initial temperature is
$T_{e0}=T_{i0}=0.04m_{i}c_{A0}^{2}$, the mass ratio is $m_{i}/m_{e}=100$, and
$\Omega_{e}\sim 0.625\omega_{pe}$, where $c_{A0}=B_{0}/(4\pi
n_{0}m_{i})^{1/2}$ is the asymptotic ion Alfvén wave speed. The simulation
domain has dimensions $L_{x}\times L_{y}\times L_{z}=0.5\times 1\times 4$
$d_{i}$ with periodic boundaries in $x$ and $z$ and a conducting boundary in
$y$. The initial electron drift along $z$, $v_{de}\sim 10c_{A0}$, is above the
threshold for triggering the Buneman instability.
Figure 1: a: The time evolution of $d\langle T_{ezz}\rangle/dt$ (black solid
line), $|dW/dt|=-\dfrac{m_{e}}{e^{2}}d\langle j_{ez}^{2}/n_{e}^{2}\rangle/dt$
(black dot-dot-dashed line), and the parallel electric field variance
$Var(E_{z})=\langle\sqrt{(E_{z}-\langle E_{z}\rangle)^{2}}\rangle$ (blue dot-
dashed line), where $n_{e}$ is the electron density and $j_{ez}$ is the
$z$-component of the electron current density. b: Electron velocity
distribution functions in the current sheet $f(v_{ez})$ at $\Omega_{e}t=25.5$,
38.2, 51, 63.7 and 102 plotted as black dot-dashed, red short-dashed, orange
long-dashed, blue dot-dot-dashed and black solid lines respectively. The
narrow ion velocity distribution function $f(v_{iz})$ at $\Omega_{e}t=102$ is
reduced by 5 times. c: Electric field $E_{z}$ parallel to $B$ at
$\Omega_{e}t=$25.5 (blue dot-dashed), 38.2 (black solid), 63.7 (red dashed).
During the simulation the Buneman instability onsets around $\Omega_{e}t\simeq
25.5$ with a wave vector that is aligned along the magnetic field. The
instability reaches its peak around $\Omega_{e}t\simeq 40$ and ceases around
$\Omega_{e}t\simeq 102$, as indicated by the turbulence strength $Var(E_{z})$
in Fig.1a (blue dot-dashed line). The electric field $E_{z}$ parallel to
$B_{z}$ abruptly increases from a few $E_{0}=c_{A0}B_{0}/c$ to $E_{z}\sim
40-60E_{0}$ at $\Omega_{e}t\sim 40$ and then falls to a value $20E_{0}$ at
$\Omega_{e}t\sim 64$ (Fig.1c). At the same time, the average parallel
component of the electron temperature, $\langle T_{ezz}\rangle$ sharply
increases, from 0.04 to 0.5, by more than a factor of 10 while the ion
temperature increases only slightly. $\langle\rangle$ denotes an average over
the mid-plane of the current sheet at $y=L_{y}/2$. The electron drift velocity
decreases from $9c_{A0}$ to $\sim 6c_{A0}$. It is noteworthy in Fig. 1a that
the increase of $\langle T_{ezz}\rangle$ nearly matches the damping rate of
the electron parallel kinetic energy $W$, which implies that the streaming
kinetic energy is nearly fully converted into thermal energy, i.e.
$\dfrac{m_{e}}{2}\triangle\langle
v_{de}^{2}\rangle\approx\dfrac{1}{2}\triangle\langle T_{ezz}\rangle$, where
the Boltzmann constant has been absorbed into T. Panel b in Fig. 1 shows the
electron velocity distribution function in the current sheet at
$\Omega_{e}t=25.5$, 38.3, 51, 63.7 and 102. The narrow ion distribution
function at $\Omega_{e}t=102$ is shown with a solid line. We can see that the
electron velocity distribution functions $f(v_{ez})$ become flatter and
broader at late times, but the significant change takes place during
$\Omega_{e}t\sim 38-51$. The electron distribution functions reveal that a few
electrons are accelerated to very high velocity, which is a consequence of the
inductive electric field $E_{z}$ that maintains the integrated current. The
ion velocity distribution function is slightly affected by the turbulence.
Figure 2: Electron distribution functions in phase space $(z,v_{ez})$ at
$\Omega_{e}t=$25.5, 38.2, 51 and 63.7.
To fully understand the fast electron thermalization, we plot the electron
distribution function in phase space $(z,v_{ez})$ at times $\Omega_{e}t=$
25.5, 38.2, 51, and 63.7 in Fig. 2. We see that at $\Omega_{e}t=38.2$,
electrons begin their trapped orbits in the potential wells and by
$\Omega_{e}t=51$ they have phase mixed along their trapped orbits. The
localized electric field structure remains intact. Phase mixing does not
change significantly after $\Omega_{e}t=63.7$. The period of fast phase mixing
is coincident with that of the flattening of electron distribution functions
shown in Fig. 1. The phase mixing occurs near the time of saturation when the
change in the electric field is small. We now explore the physical mechanism
that produces the fast phase mixing and electron heating.
To reveal the physics behind this phase mixing and electron heating, we need
to identify the properties of the instability driving the turbulence. We use a
double drifting-Maxwellian kinetic model to trace the evolution of the
instabilities in the simulation in the following way Che (2009); Che _et al._
(2009, 2010). We fit the ion and electron distribution functions averaged over
$z$ at $x=0.025$ in the mid-plane $x-z$ of the current sheet at $\Omega_{e}t=$
25.5, 38.2, 51, and 63.7, and then substitute the fits into the local
dispersion relation derived from a double drifting-Maxwellian kinetic model
for waves with $\Omega_{i}\ll\omega\ll\Omega_{e}$:
$1+\frac{2\omega_{pi}^{2}}{k^{2}v^{2}_{ti}}[1+\zeta_{i}Z(\zeta_{i})]+\frac{2\delta\omega^{2}_{pe}}{k^{2}v^{2}_{zte1}}[1+I_{0}(\lambda)e^{-\lambda}\zeta_{e1}Z(\zeta_{e1})]\\\
+\frac{2(1-\delta)\omega^{2}_{pe}}{k^{2}v^{2}_{zte2}}[1+I_{0}(\lambda)e^{-\lambda}\zeta_{e2}Z(\zeta_{e2})]=0,$
(1)
where $\zeta_{i}=(\omega-k_{z}v_{di})/kv_{ti}$, $\zeta_{e1}=(\omega-
k_{z}v_{de1})/k_{z}v_{zte1}$, $\zeta_{e2}=(\omega-
k_{z}v_{de2})/k_{z}v_{zte2}$, $\lambda=k^{2}_{x}v^{2}_{xte}/2\Omega^{2}_{e}$,
$Z$ is the plasma dispersion function and $I_{0}$ is the modified Bessel
function of the first kind with order zero. The thermal velocity of species
$j$ is defined by $v^{2}_{tj}=2T_{tj}/m_{j}$ and the drift speed by $v_{dj}$
which is parallel to the $z$ direction. The electron temperature takes on
different values parallel and across the magnetic field while the ions are
assumed to be isotropic. $\delta$ is the weight of the high velocity drifting
Maxwellian.
We numerically solve the dispersion relation in Eq. 1 and obtain the unstable
modes at $\Omega_{e}t=$ 25.5, 38.2, 51, 63.7 and 102. We find that the Buneman
instability dominates. The growth rate of the fastest growing Buneman mode
decreases with time from $\gamma\sim 0.12\omega_{pe}$ (close to the linear
value given by Ishihara et al Ishihara, Hirose, and Langdon (1981)
$\gamma\sim\sqrt{3}/2(m_{e}/2m_{i})^{1/3}(1-(m_{e}/2m_{i})^{1/3}/2)\omega_{pe}\sim
0.13\omega_{pe}$) at $\Omega_{e}t=25.5$ to $\gamma\sim 0.06\omega_{pe}$ at
$\Omega_{e}t=38.2$ and $\gamma\sim 0.006\omega_{pe}$ at $\Omega_{e}t=102$. The
frequency of the fastest growing mode is about $\omega_{0}\sim
0.013\omega_{pe}$ and its wavenumber $k_{0}d_{i}$ decreases from 90 to 75. The
phase speed $v_{p}$ increases slightly with time and has a value of $v_{p}\sim
0.05c_{A0}$. A transient two stream instability with growth rate $\gamma\sim
0.006\omega_{pe}$ appears at $\Omega_{e}t=51$ and is stable by
$\Omega_{e}t=63.7$. An oblique lower hybrid instability develops with growth
rate $\gamma\sim 0.02\omega_{pe}$ after $\Omega_{e}t=51$.
Figure 3: Panel (a) and (b) are electron distributions in phase space
$z,v_{ez}$ at the middle and the end time of the test particle simulation with
$E_{max}\sim 80$. The black dashed lines in Panel (c) and (d) are the initial
test electron velocity distribution functions. Panel (c) is at the middle time
of the test particle simulations and panel (d) is at the end time of the
simulations. The black solid lines are for the test particle simulation with
$E_{max}\sim 80$ and the red dot-dashed line are for the test particle
simulation with $E_{max}\sim 40$.
It’s interesting to notice that during $\Omega_{e}t\sim 38-64$, the typical
parallel electric field is about $40E_{0}$ and the wavenumber of the fastest
mode is $k_{0}d_{i}\sim 90$. The corresponding bounce frequency is
$\omega_{b}=k_{0}v_{b}/\sqrt{2}\sim\omega_{pe}$, where
$v_{b}\sim\sqrt{2e\phi/m_{e}}\sim 10c_{A0}$. The bounce rate is more than ten
times larger than the growth rate. Thus the amplitude of the electric field
$E_{z}$ evolves slowly compared with $\omega_{b}^{-1}$ during this interval.
By assuming the slowly evolving and slowly propagating (i.e. $z-v_{p}t\approx
z$) wave potential is $\phi(z,t)$, the electron Hamiltonian can be
approximated as:
$H\approx\frac{m_{e}}{2}v_{ez}^{2}-e\phi(z,t).$ (2)
Eq. (2) shows that during $\Omega_{e}t=38-64$, it is possible to choose $z$
and $v_{ez}$ as two approximate Hamiltonian canonical coordinates so that the
area $S=\frac{1}{2\pi}\oint v_{ez}dz$ enclosed by the electron’s trajectory in
phase space $(z,v_{ez})$ is an adiabatic invariant for trapped electrons,
where $v_{ez}=\sqrt{2(W(t,z)+e\phi)/m_{e}}$ and $W(t,z)$ is the electron’s
total energy. With the slow variation of the electric field, the electron’s
trajectory in phase space $(z,v_{ez})$ becomes narrower in $z$ and longer in
$v_{ez}$ as $E_{z}$ increases and becomes wider in $z$ and shorter in $v_{ez}$
as $E_{z}$ decreases. The electrons are trapped when the electric field grows
and are de-trapped when the electric field decays. The trapping and de-
trapping are non-adiabatic due to the change of the phase area inside and
outside of the wave potential Cary, Escande, and Tennyson (1986). The final
electron velocity depends on whether it crosses the upper or lower separatrix
as it is de-trapped. The upper (lower) separatrix crossing results in a
positive (negative) velocity in the wave frame.
To investigate how the adiabatic process converts kinetic energy into thermal
energy through non-adiabatic separatrix crossings of the wave potential, we
perform two test particle simulations with 5000 electrons in one single
standing wave $E=E_{z0}e^{\gamma t}\sin kz$. We take $kd_{i}\sim 80\sim
k_{0}$; $E_{z0}$ and $|\gamma|$ are constant and small. They satisfy
$\omega_{b}/|\gamma|\sim 80$ at the peak value of $E_{z}$. $\gamma>0$ for the
first half of the total duration and $\gamma<0$ for the second half so that
$E$ grows and decays sufficiently slowly to guarantee that the motion of the
trapped particles is adiabatic during the entire duration. The duration is
determined by the peak value of $E_{max}$. We investigate the cases with
$E_{max}\sim 40$ and $E_{max}\sim 80$. The initial electron velocity
distribution is a Maxwellian with a drift $v_{de}\sim 9c_{A0}$ and
$T_{e}=0.04m_{i}c_{A0}^{2}$ and the electron density is uniform in space. The
value $E_{max}=40$ is similar to the peak value of $E_{z}$ observed when the
PIC simulations can trap electrons with velocity below $v_{de}$. $E_{max}=80$
is higher than the peak value of $E_{z}$ observed in the PIC simulations. The
test single wave with $E_{max}=80$ can trap almost all of the electrons. The
results are shown in Fig.3.
More and more electrons are trapped as the electric field slowly increases and
the most are trapped at the peak value of $E$. The slight energy difference
between two trapped electrons leads to a large separation in their phase angle
around their trapped orbit since their angular velocity depends on energy.
Thus, at the time of the maximum trapping the trapped electrons are nearly
uniformly distributed along their trajectories as shown in panel a and the
velocity distribution of trapped electrons become flat as shown in panel c. As
$E_{z}$ decreases, the electron energy gain during trapping reverses and the
electrons are eventually de-trapped with the same value of $W$. The total
energy $W$ is symmetric with respect to positive and negative velocity.
Therefore, at the end of the simulation, due to the same probability for de-
trapping at the positive and negative velocity (Fig.3 panel b), a dip appears
near zero velocity in the velocity distribution function shown in panel d. The
red lines in panel c and d are for $E_{max}\sim 40$ where $E_{max}$ is not
strong enough to trap all of the electrons. As a result, the distribution
functions are not completely symmetric around $v_{ez}=0$.
Figure 4: The contours of total electron energy in phase space $(z,v_{ez})$ at
$\Omega_{e}t=$ 40, 102 from the PIC simulations.
The red line in panel c is similar to the electron velocity distribution
function at the saturation stage of the Buneman instability displayed in
Fig.1b.
However, even at $\Omega_{e}t=102$ in Fig. 1b, the electron velocity
distribution function keeps a similar shape to that at the saturation stage
rather than show a dip near zero as seen in the test particle simulation.
There are two reasons for the missing dip in the PIC simulations. First, the
wave amplitude is not spatially uniform. When the Buneman instability enters
the nonlinear stage, strong wave-wave interactions cause the collapse of the
uniformly distributed waves in space, and form localized solitary waves. As a
result, the trapping/de-trapping is more complex than in the sample model. In
Fig. 4, we plot the constant energy contours of electrons in phase space at
$\Omega_{e}t=$ 40 and 102 which correspond, respectively, to the peak of the
Buneman instability and late time of turbulence. We can see that the islands
between $z\in[0,0.2]$ and $z\in[0.3,0.4]$ are longer than those between
$z\in[0.2,0.3]$ and $z\in[0.4,0.5]$ at $\Omega_{i}t=40$ due to the
corresponding variation of the electric field $E_{z}$ in $z$ as shown in Fig.
1c. The long islands in phase space correspond to weak electric field and weak
trapping. Second, after $\Omega_{e}t=51$, the electron two-stream and Buneman
instabilities remain unstable, albeit weaker, and trapping continues. At the
late stage of the turbulence $\Omega_{e}t=102$, the islands in phase space
(Fig. 4) still exist even though the islands become longer.
In the PIC simulation, the electron trapping velocity covers the range
$[-10,10]c_{A0}\sim[-v_{de},v_{de}]$. The heating stops when the source of
kinetic energy is completely drained, i.e. the electron distribution with
velocity below $v_{de}$ becomes flat, as shown in Fig. 1 b. Nearly half of the
kinetic energy is dissipated, and the final electron temperature can be
estimated by $\triangle T_{ezz}\sim\triangle W\sim m_{e}v_{de}^{2}/2\sim 0.4$
which is consistent with what is observed in the simulation (Fig. 1a).
In reconnection, current sheets shrink as reconnection evolves and the Buneman
instability might occur in a wider current sheet with a reduced drift but a
similar growth rate $\sim(m_{e}/2m_{i})^{1/3}\omega_{pe}$. The implications of
these results in reconnection are being explored.
This research was supported by the NASA Postdoctoral Program at NASA/GSFC
administered by Oak Ridge Associated Universities through a contract with
NASA. The simulations and analysis were partially carried out at NASA/Ames
High-End Computing Capacity, at the National Energy Research Scientific
Computing Center, and at the National Institute for Computation Sciences.
## References
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|
arxiv-papers
| 2012-11-26T17:47:46 |
2024-09-04T02:49:38.446843
|
{
"license": "Public Domain",
"authors": "H. Che, J. F. Drake, M. Swisdak, and M. Goldstein",
"submitter": "Haihong Che",
"url": "https://arxiv.org/abs/1211.6036"
}
|
1211.6059
|
# On the spectrum of bounded immersions
G. Pacelli Bessa Luquésio P. Jorge Luciano Mari
###### Abstract
In this paper, we investigate the relationship between the discreteness of the
spectrum of a non-compact, extrinsically bounded submanifold $\varphi\colon
M^{m}\rightarrow N^{n}$ and the Hausdorff dimension of its limit set
$\lim\varphi$. In particular, we prove that if
$\varphi\colon\\!M^{2}\rightarrow D\subseteq\mathbb{R}^{3}$ is a minimal
immersion into an open, bounded, strictly convex subset $D$ with
$C^{2}$-boundary, then $M$ has discrete spectrum provided that
$\mathcal{H}_{\Psi}(\lim\varphi\cap D)=0$, where $\mathcal{H}_{\Psi}$ is the
generalized Hausdorff measure of order $\Psi(t)=t^{2}|\log t|$. Our theorem
applies to a number of examples recently constructed by various authors in the
light of N. Nadirashvili’s discovery of complete, bounded minimal disks in
$\mathbb{R}^{3}$, as well as to solutions of Plateau’s problems, giving a
fairly complete answer to a question posed by S.T. Yau in his Millenium
Lectures. Suitable counter-examples show the sharpness of our results: in
particular, we develop a simple criterion for the existence of essential
spectrum which is suited for the techniques developed after Jorge-Xavier and
Nadirashvili’s examples.
Departamento de Matemática, Universidade Federal do Ceará
Campus do PICI, 60455-760 Fortaleza-Ce, (Brazil)
E-mail address: [email protected], [email protected], [email protected]
## 1 Introduction
The Calabi-Yau conjectures have their origin in a set of two problems proposed
by E. Calabi in the 1960’s, about the non-existence of complete minimal
hypersurfaces of $\mathbb{R}^{n}$ subjected to certain extrinsic bounds ([7],
see also [8, p. 212]).
* •
Calabi proposed the first conjecture as an exercise. He wrote: “Prove that any
complete minimal hypersurface in $\mathbb{R}^{n}$ must be unbounded.”
* •
The second problem, on the other hand, was proposed almost as an unlikely
conjecture. He wrote “A more ambitious conjecture is: a complete minimal
hypersurface in $\mathbb{R}^{n}$ has an unbounded projection in every
$(n-2)$-dimensional flat subspace.”
It is known by works of L. Jorge-F. Xavier [18] and N. Nadirashvili [25] that
both conjectures turned out to be false. More precisely, Jorge-Xavier
constructed a non-flat, complete minimal surface lying between two parallel
planes in $\mathbb{R}^{3}$, showing that the second conjecture was false in
general, whereas N. Nadirashvili constructed a bounded, complete minimal
immersion of the unit disk $\mathbb{D}$ into $\mathbb{R}^{3}$, contradicting
the statement of the first conjecture. Both counter-examples lie on the
construction of suitable labyrinths of compact subsets and clever use of
Runge’s Theorem in order to find appropriate Weierstrass data (the interested
reader could consult [10] for a detailed proof of the main theorems in [25]).
In view of these results, in his Millennium Lectures [33], [34], S. T. Yau
“calibrated” the Calabi conjectures, proposing a new set of questions about
bounded minimal hypersurfaces.
* He wrote: “It is known [25] that there are complete minimal surfaces properly immersed into the $[$open$]$ ball. What is the geometry of these surfaces? Can they be embedded? Since the curvature must tend to minus infinity, it is important to find the precise asymptotic behavior of these surfaces near their ends. Are their $[$Laplacian$]$ spectra discrete?”.
These questions became known in the literature as the Calabi-Yau conjectures
on minimal hypersurfaces. The problem about the existence of a bounded,
complete, _embedded_ minimal surfaces in $\mathbb{R}^{3}$ was negatively
answered by T. Colding-W. Minicozzi in the finite topology case, see [9].
Although Yau’s question suggests that Nadirashvili’s example is _properly_
immersed into an open ball $D\subset\mathbb{R}^{3}$ (that is, the pre-image of
compact subsets of $D$ are compacts in the surface), this is not clear and one
may consider, as the first problem in the Calabi-Yau conjectures, the
question: “Does there exist a complete minimal surface properly immersed into
a ball of $\mathbb{R}^{3}$? ”.
This question, and more generally the search for understanding the shape of
the limit set of bounded minimal surfaces, has stimulated intense research in
the last fifteen years111Given a map $\varphi:M\rightarrow N$ between
topological manifolds, the limit set of $\varphi$, $\lim\varphi$, is defined
as $\displaystyle\lim\varphi=\\{x\in N;\;\exists\,\\{p_{n}\\}\subseteq
M\text{ divergent in }M,\text{ such that }\varphi(p_{n})\rightarrow x\text{ in
}N\big{\\}}.$ (1.1) By this definition, a map $\varphi:M\rightarrow N$ whose
image lie in a bounded, open subset $D\subseteq N$ is proper in $D$ if and
only if $\lim\varphi\subseteq\partial D$.. Regarding properness, we briefly
recall the main achievements:
* $(i)$
with a highly nontrivial refinement of Nadirashvili’s technique, F. Martin and
S. Morales in [20] proved that, for each convex domain
$D\subseteq\mathbb{R}^{3}$ not necessarily bounded or smooth, there exists a
complete minimal disk $B\subseteq\mathbb{C}$ properly immersed into $D$. Later
on, M. Tokuomaru in [30] constructed a complete minimal annulus which is
proper in the unit ball of $\mathbb{R}^{3}$;
* $(ii)$
in [21], Martin-Morales improved the results in [20] showing that, if $D$ is a
bounded strictly convex domain of $\mathbb{R}^{3}$ with $\partial D$ is of
class $C^{2,\alpha}$, then there exists a complete, minimal disk properly
immersed into $D$ whose limit set is close to a prescribed Jordan curve222We
recall that a Jordan curve is defined as a continuous embedding of
$I\subseteq\mathbb{R}$ (or $I\subseteq\mathbb{S}^{1}$) in $\mathbb{R}^{3}$. on
$\partial D$;
* $(iii)$
Alarcon, Ferrer and Martin in [1] extended the results in [20] and [30] from
disks and annuli to open surfaces $M$ with any finite topology;
* $(iv)$
improving on [21], Ferrer, Martin and Meeks in [15] showed that if
$D\subset\mathbb{R}^{3}$ is bounded, convex and smooth, then every open
surface $M$ with finite topology can be properly and minimally immersed into
$D$ in such a way that the limit sets of the ends are disjoint compact
connected subsets of $\partial D$. Furthermore (see Proposition 1 in [15]),
they proved that, for every convex, open set $D$ and every non-compact,
orientable surface $M$, there exists a complete, proper minimal immersion
$\varphi:M\rightarrow D$ such that $\lim\varphi\equiv\partial D$.
A parallel line of research focused on the problem of controlling the size of
$\lim\varphi$ from the measure-theoretic point of view. In this respect,
* $(v)$
Martin and Nadirashvili, [23], constructed complete minimal immersions
$\varphi\colon B\rightarrow\mathbb{R}^{3}$ of the unit disk
$B\subseteq\mathbb{C}$ admitting continuous extensions to the closed disk
$\overline{\varphi}\colon\overline{B}\rightarrow\mathbb{R}^{3}$ in such a way
that $\overline{\varphi}_{|\partial B}\colon\partial
B=\mathbb{S}^{1}\to\overline{\varphi}(\mathbb{S}^{1})$ is an homeomorphism and
$\overline{\varphi}(\mathbb{S}^{1})$ is a non-rectifiable Jordan curve of
Hausdorff dimension
$\dim_{\mathcal{H}}(\overline{\varphi}(\mathbb{S}^{1}))=1$. Moreover, they
showed that the set of Jordan curves $\overline{\varphi}(\mathbb{S}^{1})$
constructed via this procedure is dense in the space of Jordan curves of
$\mathbb{R}^{3}$ with respect to the Hausdorff metric.
In this paper, we address Yau’s question of deciding whether the spectrum of
bounded, minimal surfaces is discrete or not, and we provide a sharp, general
criterion that applies to each of the examples in $(i),\ldots,(v)$, as well as
to many other minimal surfaces. The first answer to this question was given in
Bessa-Jorge-Montenegro in [5], where they proved that the spectrum of a
complete minimal surface properly immersed into a ball of $\mathbb{R}^{3}$ is
always discrete. Despite the generality of this result, there is the technical
unpleasant drawback that the possible convex ambient domains
$D\subset\mathbb{R}^{3}$ are restricted to balls. Much more important, their
approach uses in a crucial way the properness condition, and cannot be
generalized to deal with non-proper immersions. On the other hand, it is
reasonable to guess that a well-behaved limit set, close (loosely speaking) to
a smooth curve as in $(v)$, could imply that $\Delta$ has discrete spectrum on
$M$. In some sense, we could say that the minimal surface is not too far from
a compact surface with boundary. Indeed, it is not necessary that
$\lim\varphi$ resembles a curve from the measure-theoretic point of view, but
a much more general condition is sufficient:
###### Theorem 1.
Let $\varphi\colon M^{2}\to D\subset N$ be a minimal immersion into an open,
bounded, $2$-convex subset of a Cartan-Hadamard manifold $N$. Set
$\Psi(t)=t^{2}|\log t|$. If the $\Psi$-Hausdorff measure of $\lim\varphi\cap
D$ satisfies $\mathcal{H}_{\Psi}(\lim\varphi\cap D)=0$, then the spectrum of
$-\Delta$ is discrete.
###### Remark 1.
_We stress that no geodesic completeness of $M$ is required in Theorem 1. When
$M$ is incomplete, $-\Delta$ may fail to be essentially self-adjoint on
$C^{\infty}_{c}(M)$, in sharp contrast with the complete case where the self-
adjointness of $-\Delta$ is automatic. For this reason, we agree on always
considering the Friedrichs extension of $\Delta$, that is, the unique self-
adjoint extension of $(-\Delta,C^{\infty}_{c}(M))$ whose domain lies in that
of the closure of the quadratic form $Q(u,v)=(-\Delta u,v)_{L^{2}}$, for
$u,v\in C^{\infty}_{c}(M)$. _
###### Remark 2.
_We also underline that there is no codimension restriction, that is, $N$ may
be of any dimension._
Later, we will recall the notion of $2$-convexity for subsets with $C^{2}$
boundary, and the definition of Hausdorff measure $\mathcal{H}_{\Psi}$. Here,
we just observe that $2$-convexity relaxes standard convexity, so $D$ can be
any bounded strictly convex subset of $M$ with $C^{2}$-boundary. As for
$\mathcal{H}_{\Psi}$, condition $\mathcal{H}_{\Psi}(\lim\varphi\cap D)=0$ is
satisfied, for instance, whenever the Hausdorff dimension of $\lim\varphi\cap
D$ is strictly less than $2$. It is worth to underline that we are considering
only the part of $\lim\varphi$ _inside_ $D$: there is no requirement on the
portion of $\lim\varphi$ contained in $\partial D$. As an immediate corollary,
we have the following
###### Corollary 1.
All the examples of complete, bounded minimal surfaces constructed in
$(i),\ldots,(v)$ $($if the convex sets are bounded and $C^{2})$ have discrete
spectrum.
The fact that the completeness of $M$ is irrelevant enables us to apply our
result to solutions of the Plateau problem for arbitrary Jordan curves, whose
existence is granted by R. Douglas in [14] (see also the treatises [11] and
[12, 13], as well as [19], [31]):
###### Corollary 2.
Let $\Gamma\colon\mathbb{S}^{1}\rightarrow\mathbb{R}^{3}$ be a Jordan curve.
If $\mathcal{H}_{\Psi}(\Gamma(\mathbb{S}^{1}))=0$, where $\Psi(t)=t^{2}|\log
t|$, then every minimal surface spanning $\Gamma$ has discrete spectrum.
###### Remark 3.
_It is known (see[26]) that there exist Jordan curves whose image have non-
zero $2$-dimensional Hausdorff measure, whence the measure condition in
Corollary 2 is not automatically satisfied. _
There is a trivial reason why the estimate in Theorem 1 is sharp, and it is
the presence of infinite-sheet coverings.
###### Example 1.
_Consider a bounded, complete minimal annulus $f\colon
A\rightarrow\mathbb{R}^{3}$ (either the proper one constructed in [30], or the
one in [22]). Then, taking the universal covering $\pi\colon M\rightarrow A$,
$\varphi=f\circ\pi$ is a bounded, complete minimal surface with non-empty
essential spectrum, as it is for every infinite-sheet covering (for instance,
one can note that $M$ has the ball property, see Definition 1 and Corollary 3
below). Clearly, $\dim_{\mathcal{H}}(\lim\varphi)\geq 2$ since, by
construction, $\lim\varphi\supseteq f(A)$. _
As we shall see, Theorem 1 is a particular case of a more general result,
Theorem 2 below, where we deal with complete submanifolds $\varphi\colon
M^{m}\to N^{n}$ immersed with sufficiently small mean curvature in an
$m$-convex and regular ball $B_{R}\Subset N$. Thus, the property described in
Theorem 1 does not, indeed, exclusively pertain to the realm of minimal
surfaces. For technical reason, we postpone the general statement of our main
theorem to Section 3.
###### Remark 4.
_As a consequence of Theorem 3, an analogous of Corollary 2 holds when the
mean curvature is non-zero and sufficiently small. In this case, the existence
of disks solving Plateau’s problem is granted by H. Werner in [32]. We leave
the correspondent statement to the interested reader._
To inspect more closely the sharpness of our results, Section 4 is devoted to
find sufficient conditions for a manifold to have non-empty essential
spectrum. These conditions will be used to investigate minimal submanifolds
subjected to some general extrinsic constrain, such as, for instance, Jorge-
Xavier minimal surface in a slab. In particular, we shall concentrate on
criteria that do not involve the curvature of the submanifold, as this is
often a hardly available information: for example, it seems extremely
difficult to control the curvature when exploiting the techniques used for the
examples in $(i),\ldots,(v)$. We are led to the following
###### Definition 1.
A Riemannian manifold $(M,\langle\,,\,\rangle)$ is said to have the _ball
property_ if there exists $R>0$ and a collection of disjoint balls
$\\{B_{R}(x_{j})\\}_{j=1}^{+\infty}$ of radius $R$ for which, for some
constants $C>0$, $\delta\in(0,1)$ possibly depending on $R$,
$\mathrm{vol}\big{(}B_{\delta R}(x_{j})\big{)}\geq
C^{-1}\mathrm{vol}\big{(}B_{R}(x_{j})\big{)}\qquad\forall\,j\in\mathbb{N}$
(1.2)
###### Remark 5.
_Note that ( 1.2) is _not_ a doubling condition, as $C$ may depend on $R$. _
###### Proposition 1.
If a Riemannian manifold has the ball property (with parameters $R,\delta,C$),
then
$\inf\sigma_{\mathrm{ess}}(-\Delta)\leq\frac{C}{R^{2}(1-\delta)^{2}}.$ (1.3)
The ball property is reasonably easy to check, since it only requires a
$C^{0}$ control on the metric in some region of $M$, and is very suited to
deal with surfaces constructed via developments of Jorge-Xavier original
technique in [18]. For example, we easily prove the next
###### Proposition 2.
For a suitable choice of the parameters, the Jorge-Xavier complete minimal
surface $(M^{2},\mathrm{d}s^{2})$ in a slab of $\mathbb{R}^{3}$, constructed
in [18], satisfies $\inf\sigma_{\mathrm{ess}}(-\Delta)=0$.
For more results and comments, we refer to Section 4. There, the reader can
also find some open problems that we believe to be worth investigating.
## 2 Setting, notations and preliminaries
Let $M$, $N$ be connected (smooth) Riemannian manifolds of dimensions $m$ and
$n$ respectively and let $\varphi\colon M\rightarrow N$ be an isometric
immersion. In what follows, $\nabla\mathrm{d}\varphi$ will stand for the
second fundamental form of $\varphi$ and
$H=\frac{1}{m}\mathrm{tr}(\nabla\mathrm{d}\varphi)$ for its mean curvature.
All the elements describing the Riemannian structure of $N$ will be marked
with a bar, so that, for instance, $\overline{\nabla}$ and $\overline{K}$ will
denote the Riemannian connection and the sectional curvature of $N$. Having
fixed $o\in N$, we will write $\overline{K}_{\mathrm{rad}}(x)$ for the radial
sectional curvature of $N$, that is, the restriction of $\overline{K}(x)$ to
the subset of $2$-planes at $x$ containing tangent vectors of minimizing
geodesics issuing from $o$. For $x,y\in N$, we define
$\rho(x,y)=\mathrm{dist}(x,y)$, and $\rho(x)=\mathrm{dist}(x,o)$ whenever the
second point is considered as fixed. The symbol $B_{r}(x)$ indicates the ball
of radius $r$ centered at $x$, and we simply write $B_{r}$ when $x=o$.
Similarly, for $A\subseteq N$ the symbol $B_{r}(A)$ denotes the open set of
points whose distance from $A$ is less than $r$. We use $\mathbb{R}^{+}$ and
$\mathbb{R}^{+}_{0}$ to denote, respectively, $(0,+\infty)$ and $[0,+\infty)$.
Hereafter, we will consider a relaxed notion of convexity, called
$j$-convexity, which is widely used in the literature (for instance, see
[17]).
###### Definition 2.
For an integer $j\leq n$, We say that an open subset $D\subseteq N^{n}$ is
(strictly) $j$-convex if there exists $F\in C^{2}(N)$, satisfying the
following properties:
* $(i)$
$F<0$ on $D$, $F=0$ on $\partial D$;
* $(ii)$
for every $p\in D$, denoting with
$\lambda_{1}(p)\leq\lambda_{2}(p)\leq\ldots\leq\lambda_{n}(p)$ the eigenvalues
of the Hessian $\nabla\mathrm{d}F$ at $p$ written in increasing order, it
holds $\lambda_{1}(p)+\ldots+\lambda_{j}(p)>0$.
The set $D$ will be called uniformly convex with constant $c>0$ if at every
point $p\in D$ it holds $\lambda_{1}(p)+\ldots+\lambda_{j}(p)\geq c$. For a
uniformly $j$-convex subset $D$, the pair $(F,c)$ will be called the data of
uniform convexity of $D$.
In particular, if $j=1$, the definition of $j$-convexity coincides with the
standard definition of (strict) convexity, and for this reason we simply say
that $D$ is convex. Moreover, we observe that $n$-convexity means that $D$ is
$F^{-1}((-\infty,0))$ for a strictly subharmonic function $F$. Since each
$\lambda_{i}$ is a Lipschitz function of $p\in N$, by a compactness argument
the properties of $j$-convexity and uniform $j$-convexity coincide for
relatively compact $D$.
###### Example 2.
_Let $N$ be a complete Riemannian manifold, let $o\in N$ be a reference origin
and suppose that $\overline{K}_{\mathrm{rad}}(x)\leq-G(\rho(x))$, for some
$G\in C^{0}(\mathbb{R}^{+}_{0})$. Consider a solution $h(t)\in
C^{2}(\mathbb{R}^{+}_{0})$ of_
$\left\\{\begin{array}[]{l}h^{\prime\prime}(t)-G(t)h(t)=0,\\\\[2.84544pt]
h(0)=0,\quad h^{\prime}(0)=1,\end{array}\right.$ (2.1)
_and let $R$ be such that $h^{\prime}>0$ on $[0,R]$. Suppose that
$B_{R}\subseteq N$ does not intersect the cut-locus $\mathrm{cut}(o)$, that
is, that $B_{R}$ is a regular ball of $N$. Then, by the Hessian comparison
theorem (see for instance [28], Theorem 2.3), for $x\in B_{R}$_
$\nabla\mathrm{d}\rho\geq\frac{h^{\prime}(\rho)}{h(\rho)}\Big{(}(\,,\,)-\mathrm{d}\rho\otimes\mathrm{d}\rho\Big{)},\qquad\rho=\rho(x).$
_Having set_
$f(t)=\int_{0}^{t}h(s)\mathrm{d}s,\qquad F(x)=f(\rho(x))\quad\text{on }B_{R},$
_then by the chain rule_
$\nabla\mathrm{d}F=f^{\prime\prime}(\rho)\mathrm{d}\rho\otimes\mathrm{d}\rho+f^{\prime}(\rho)\nabla\mathrm{d}\rho\geq
h^{\prime}(\rho)(\,,\,)\geq\left[\inf_{[0,R]}h^{\prime}\right](\,,\,),$
_whence the ball $B_{R}\subseteq N$ is convex, with constant
$c=\inf_{[0,R]}h^{\prime}$. For instance, if $G(t)=k\leq 0$ is constant, then
every ball centered at a pole $o$ of $N$ is convex (we recall that a pole is a
point $o\in N$ such that $\mathrm{cut}(o)=\emptyset$, or equivalently such
that $\exp_{o}:T_{o}N\rightarrow N$ is a diffeomorphism). The same happens
when $k>0$ whenever $B_{R}\subseteq N$ is a regular ball of radius
$R<\pi/2\sqrt{k}$. If $o$ is a pole for $N$, there are some sufficient
conditions ensuring that each $B_{R}$ is convex. For a general case, we
suggest the reader to consult Remark 8 below. _
Before stating our result in its stronger form, we also recall some general
notions on Hausdorff measures. We follow the exposition in [24], Chapter 4,
although with a different notation. According to Carathéodory construction, we
consider a function $\Psi\geq 0$ defined and continuous on some right
neighborhood $[0,2\delta_{0})$ of zero, and such that $\Psi(0)=0$, together
with a family $\mathcal{F}$ of Borel subsets of $M$ satisfying the following
property:
* For every $\delta\in(0,\delta_{0})$, there exist $\\{E_{i}\\}\subseteq\mathcal{F}$ such that $M=\bigcup_{i=1}^{+\infty}E_{i}$ and $\mathrm{diam}(E_{i})\leq\delta$.
For each $\delta\in(0,\delta_{0})$ and for every subset $A\subseteq M$, we set
$\begin{array}[]{l}\displaystyle\mathcal{H}_{\Psi,\delta}(A)=\displaystyle\inf\left\\{\sum_{i=1}^{+\infty}\Psi\big{(}\mathrm{diam}(E_{i})\big{)}\
:\ \\{E_{i}\\}\subseteq\mathcal{F},\ A\subseteq\bigcup_{i=1}^{+\infty}E_{i},\
\mathrm{diam}(E_{i})\leq\delta\right\\}\\\\[14.22636pt]
\displaystyle\mathcal{H}_{\Psi}(A)=\lim_{\delta\downarrow
0}\mathcal{H}_{\Psi,\delta}(A)=\sup_{\delta\in(0,\delta_{0})}\mathcal{H}_{\Psi,\delta}(A).\end{array}$
(2.2)
If $\mathcal{F}\equiv\\{\text{borel subsets of }M\\}$, then the resulting
measure $\mathcal{H}_{\Psi}$ is a borel regular measure ([24], Theorem 4.2),
and we call it the Hausdorff measure related to $\Psi$. By Theorem 4.4 in
[24], the same $\mathcal{H}_{\Psi}$ can be obtained if we restrict to the
subfamily $\mathcal{F}=\\{\text{open subsets of }M\\}$. The particular choice
$\Psi(t)=t^{\beta}$, for some fixed $\beta>0$, gives the standard Hausdorff
measure $\mathcal{H}^{\beta}$ of order $\beta$, up to an unessential constant.
###### Remark 6.
_If we restrict $\mathcal{F}$ to the subfamily of geodesic balls of $M$, the
resulting measure $\overline{\mathcal{H}}_{\Psi}$ does not coincide, in
general, with $\mathcal{H}_{\Psi}$ (see [24], Chapter 5). However, if for some
constant $C>0$ it holds_
$\Psi(2t)\leq C\Psi(t)\qquad\text{for }t\in(0,\delta_{0}),$ (2.3)
_then $\mathcal{H}_{\Psi}\leq\overline{\mathcal{H}}_{\Psi}\leq
C\mathcal{H}_{\Psi}$. The first inequality is obvious from definitions. To
prove the second one, since every open set $E_{j}$ is contained in a ball
$B_{j}$ of diameter $2\mathrm{diam}(E_{j})$, for every covering $\\{E_{j}\\}$
of $A\subseteq M$ with $\mathrm{diam}(E_{j})<\delta$ it holds_
$\sum_{i=1}^{+\infty}\Psi\big{(}\mathrm{diam}(E_{j})\big{)}\geq\frac{1}{C}\sum_{i=1}^{+\infty}\Psi\big{(}2\mathrm{diam}(E_{j})\big{)}=\frac{1}{C}\sum_{i=1}^{+\infty}\Psi\big{(}\mathrm{diam}(B_{j})\big{)}.$
_Now, taking the infimum, in the right hand-side, with respect to all covering
$\\{B_{j}\\}$ with balls of diameter less than $2\delta$, and then doing the
same on the left hand side, letting $\delta\downarrow 0$ we deduce the desired
$\overline{\mathcal{H}}_{\Psi}\leq C\mathcal{H}_{\Psi}$. _
## 3 The main theorem and its proof
We are ready to state our main result in its general form.
###### Theorem 2.
Let $\varphi\colon M^{m}\rightarrow D\subset N^{n}$ be an isometric immersion
into a bounded, $m$-convex open subset $D$ of a Cartan-Hadamard manifold $N$
satisfying $\overline{K}_{\mathrm{rad}}\leq-B^{2}$, for some $B\geq 0$. Let
$(F,c)$ be the uniform convexity data of $D$ as in Definition 2, and let
$R_{0}>\frac{1}{2}\mathrm{diam}(D)$. Set
$\mu(t)=\left\\{\begin{array}[]{ll}t&\quad\text{if }\,B=0,\\\\[2.84544pt]
B^{-1}\tanh(Bt)&\quad\text{if }B>0\end{array}\right.$ (3.1)
Suppose that the mean curvature $H$ of $\varphi$ satisfies
$\|H\|=\|H\|_{L^{\infty}(M)}<\left\\{\begin{array}[]{ll}\displaystyle\min\left\\{\frac{m-1}{m\mu(2R_{0})}\,,\,\frac{c}{m\|\overline{\nabla}F\|_{L^{\infty}(D)}}\right\\}&\quad\text{if
}\lim\varphi\cap\partial D\neq\emptyset,\\\\[14.22636pt]
\displaystyle\frac{m-1}{m\mu(2R_{0})}&\quad\text{if }\lim\varphi\cap\partial
D=\emptyset,\end{array}\right.$ (3.2)
and set
$\theta=m-1-m\mu(2R_{0})\|H\|.$
If $\mathcal{H}_{\Psi}(\lim\varphi\cap D)=0$, where
$\begin{array}[]{ll}\Psi(t)=t^{2}&\quad\text{if }\theta>1\\\\[2.84544pt]
\Psi(t)=t^{2}|\log t|&\quad\text{if }\theta=1\\\\[2.84544pt]
\Psi(t)=t^{\theta+1}&\quad\text{if }\theta\in(0,1),\end{array}$ (3.3)
then the spectrum of $-\Delta$ is discrete.
In the minimal case, the result is particularly transparent and extends
Theorem 1 to the higher-dimensional case.
###### Theorem 3.
Let $\varphi\colon M^{m}\hookrightarrow D\subset N^{n}$ be a minimal immersion
into a bounded, $m$-convex subset of a Cartan-Hadamard manifold $N$. If
$\mathcal{H}_{\Psi}(\lim\varphi\cap D)=0$, where
$\Psi(t)=t^{2}\quad\text{if }m>2,\qquad\text{or}\qquad\Psi(t)=t^{2}|\log
t|\quad\text{if }m=2,$ (3.4)
then the spectrum of $-\Delta$ is discrete.
###### Remark 7.
_We underline that, in Theorem 3, raising the dimension of the manifold $M$
does not yield an improvement of the allowed Hausdorff dimension of
$\lim\varphi\cap D$. Indeed, the exponent $2$ is essential for the arguments
of the proof of Theorem 2 to work. We will come back to this observation
later. _
We now come to a brief description of the strategy of the proof. To show that
$-\Delta$ has discrete spectrum, as usual we rely on a combination of Persson
formula, see [27], and Barta’s inequality ([3], and its generalized version in
[4]). Persson formula relates the infimum $\inf\sigma_{\mathrm{ess}}(-\Delta)$
of the essential spectrum of $-\Delta$ to the fundamental tone of the
complementary of compact sets:
$\inf\sigma_{\mathrm{ess}}(-\Delta)=\sup_{\footnotesize K\subset M\text{
compact}}\lambda^{*}(M\backslash K)$ (3.5)
where $\lambda^{*}(M\backslash K)$ is the bottom of the spectrum of the
Friedrichs extension of $(-\Delta,C^{\infty}_{c}(M\backslash K))$. On the
other hand, Barta inequality gives a lower bound for $\lambda^{*}(M\backslash
K)$ via positive functions:
$\lambda^{*}(M\backslash K)\geq\inf_{M\backslash K}\frac{-\Delta
w}{w}\qquad\text{for every }\,0<w\in C^{2}(M\backslash K).$ (3.6)
Combining (3.5) and (3.6), to prove that $-\Delta$ has discrete spectrum or
equivalently, by the min-max theorem, that
$\inf\sigma_{\mathrm{ess}}(-\Delta)=+\infty$ it is therefore enough to find an
increasing sequence of compact subsets $\\{K_{l}\\}$ of $M$, and functions
$0<w_{l}\in C^{2}(M\backslash K_{l})$, such that
$\frac{-\Delta w_{l}}{w_{l}}\geq c_{l}\,\,\,\text{on }M\backslash
K_{l},\,\,\,\,\,\text{with}\,\,\,c_{l}\rightarrow+\infty\,\,\,\text{as
}\,l\rightarrow+\infty.$ (3.7)
Each $w_{l}$ will be constructed as a sum of suitable positive strictly
superharmonic functions, depending on the balls of a good covering of
$\lim\varphi$. The key point to construct them is the next lemma. We state it
in a rather general form in order to put in evidence the flexibility of the
procedure, and to underline some subtle phenomena.
### The fundamental lemma
Given a bounded immersion $\varphi\colon M^{m}\rightarrow N^{n}$, the next key
lemma enables us to construct bounded, strictly subharmonic functions on $M$
with a very precise control both on their Laplacian and on their
$L^{\infty}$-norm. We remark that this is possible since, in our assumptions
on $\|H\|$ and on $N$, $M$ turns out to be non-parabolic.
Throughout this section, we assume the following:
* $(\mathcal{H}_{1})$
$N^{n}$ has a pole $x_{0}$, and the radial sectional curvatures
$\overline{K}_{\mathrm{rad}}$ of radii issuing from $x_{0}$ in $N$ satisfy
$\overline{K}_{\mathrm{rad}}(y)\leq-G(\rho(y)),\qquad\rho(y)=\mathrm{dist}(y,x_{0}),$
for some $G\in C^{0}(\mathbb{R}^{+}_{0})$.
* $(\mathcal{H}_{2})$
The solution $h(t)$ of (2.1) satisfies
$h,h^{\prime}>0\quad\text{on }\mathbb{R}^{+},\qquad
h(t)\uparrow+\infty\quad\text{as }t\rightarrow+\infty.$ (3.8)
###### Remark 8.
_By Proposition 1.21 in[6], $h$ satisfies $(\mathcal{H}_{2})$ whenever the
negative part of $G$ is small in the following sense:_
$G_{-}(s)\leq\frac{1}{4s^{2}}\qquad\text{on }\mathbb{R}^{+}.$ (3.9)
_Furthermore, ( 3.9) ensures the validity of the lower bound $h(t)\geq
C\sqrt{t}\log t$ for $t\geq t_{0}\geq 2$, for some positive constant $C$. _
We set for convenience $\mu(t)=\|h/h^{\prime}\|_{L^{\infty}([0,t])}$. Since
$h^{\prime}(0)=1$, fix $\bar{a}\in(0,1)$ small enough that
$h^{\prime}(t)\geq\frac{1}{2}\qquad\text{for every }t\in[0,\bar{a}].$ (3.10)
###### Lemma 1.
Suppose that the conditions $(\mathcal{H}_{1})$, $(\mathcal{H}_{2})$ are met,
and let $\varphi\colon M^{m}\rightarrow N^{n}$ be an isometric immersion into
a ball $B_{R}(x_{0})\subset N$, with mean curvature satisfying
$\|H\|=\|H\|_{L^{\infty}(M)}<\frac{m-1}{m\mu(R)}.$ (3.11)
Set
$\theta=m-1-m\|H\|\mu(R)>0,$
and choose a positive, non-increasing function $S\in
C^{0}(\mathbb{R}^{+}_{0})$ satisfying
$S(0)=1,\qquad\int_{0}^{+\infty}t^{\theta}S(t)\mathrm{d}t=\hat{S}<+\infty.$
(3.12)
Then, there exists a positive constant $C$, depending on $m,R,\theta,S$ and
$h_{|_{[0,R]}}$ such that the following holds: for each $a\in(0,\bar{a}]$,
there is a smooth function $u_{x_{0}}\colon M\rightarrow\mathbb{R}$ such that
$\displaystyle(i)$ $\displaystyle u_{x_{0}}\geq 0,\quad u_{x_{0}}(p)=0\ \text{
if and only if }\varphi(p)=x_{0},$ (3.13) $\displaystyle(ii)$
$\displaystyle\|u_{x_{0}}\|_{L^{\infty}}\leq\left\\{\begin{array}[]{ll}Ca^{2}&\quad\text{if
}\theta>1\\\\[2.84544pt] Ca^{2}|\log a|&\quad\text{if }\theta=1\\\\[2.84544pt]
Ca^{\theta+1}&\quad\text{if }\theta\in(0,1)\end{array}\right.$ (3.17)
$\displaystyle(iii)$ $\displaystyle\Delta
u_{x_{0}}\geq\left\\{\begin{array}[]{ll}\displaystyle\frac{(\theta+1)}{2}&\quad\text{on
}\varphi^{-1}\big{\\{}B_{a}(x_{0})\big{\\}},\\\\[8.5359pt] \displaystyle\theta
h^{\prime}(\rho\circ\varphi)S\left(\frac{h(\rho\circ\varphi)-h(a)}{h(a)}\right)&\quad\text{on
}\varphi^{-1}\big{\\{}N\backslash B_{a}(x_{0})\big{\\}}.\end{array}\right.$
(3.20)
###### Remark 9.
_Note that $\mu(R)\|H\|$ is scale-invariant, thus $\theta$ defines a genuine
geometrical object associated to the immersion. _
###### Proof.
First of all we recall that, for a function $f\in C^{2}(N)$, by the chain
rule, the composition $f\circ\varphi\in C^{2}(M)$ satisfies
$\nabla\mathrm{d}(f\circ\varphi)=\overline{\nabla}\mathrm{d}f(\mathrm{d}\varphi,\mathrm{d}\varphi)+\mathrm{d}f(\nabla\mathrm{d}\varphi),$
Now, let $\\{e_{i},e_{\alpha}\\}$ be a local Darboux frame along $\varphi$,
with $\\{e_{i}\\}$ tangent to $M$. Then, tracing the above equality, it yields
$\Delta(f\circ\varphi)=\displaystyle\sum_{j=1}^{m}\overline{\nabla}\mathrm{d}f(e_{j},e_{j})+m\mathrm{d}f(H).$
(3.21)
Define $f(y)=g(\rho(y))$, for some suitable $g\in C^{2}(\mathbb{R}^{+}_{0})$,
$g^{\prime}\geq 0$, that will be chosen in a moment. By the Hessian comparison
theorem (see [6], Theorem 1.15)
$\overline{\nabla}\mathrm{d}\rho\geq\frac{h^{\prime}(\rho)}{h(\rho)}\Big{(}\langle\,,\,\rangle-\mathrm{d}\rho\otimes\mathrm{d}\rho\Big{)}.$
Hence, if $g$ is increasing,
$\overline{\nabla}\mathrm{d}f\geq\frac{g^{\prime}(\rho)h^{\prime}(\rho)}{h(\rho)}\Big{(}\langle\,,\,\rangle-\mathrm{d}\rho\otimes\mathrm{d}\rho\Big{)}+g^{\prime\prime}(\rho)\mathrm{d}\rho\otimes\mathrm{d}\rho.$
(3.22)
By (3.22), and using that $|\mathrm{d}\rho|=1$,
$\begin{array}[]{l}\displaystyle\sum_{j=1}^{m}\overline{\nabla}\mathrm{d}f(e_{j},e_{j})+m\mathrm{d}f(H)=\frac{g^{\prime}h^{\prime}}{h}\Big{(}m-\sum_{j=1}^{m}\mathrm{d}\rho(e_{j})^{2}\Big{)}+g^{\prime\prime}\sum_{j=1}^{m}\mathrm{d}\rho(e_{j})^{2}+mg^{\prime}\mathrm{d}\rho(H)\\\\[11.38092pt]
\displaystyle\geq\frac{g^{\prime}h^{\prime}}{h}\Big{(}m-\sum_{j=1}^{m}\mathrm{d}\rho(e_{j})^{2}-m\frac{h}{h^{\prime}}\|H\|\Big{)}+g^{\prime\prime}\sum_{j=1}^{m}\mathrm{d}\rho(e_{j})^{2}\\\\[11.38092pt]
\displaystyle\geq\frac{g^{\prime}h^{\prime}}{h}\Big{(}m-\sum_{j=1}^{m}\mathrm{d}\rho(e_{j})^{2}-m\mu(R)\|H\|\Big{)}+g^{\prime\prime}\sum_{j=1}^{m}\mathrm{d}\rho(e_{j})^{2}.\end{array}$
(3.23)
Define
$w(t)=\left\\{\begin{array}[]{ll}\displaystyle(\theta+1)h^{\prime}(t)&\quad\text{if
}t\leq a\\\\[5.69046pt]
\displaystyle(\theta+1)h^{\prime}(t)S\left(\frac{h(t)-h(a)}{h(a)}\right)&\quad\text{if
}t\geq a,\end{array}\right.$
and set
$g(t)=\int_{0}^{t}\frac{1}{h(s)^{\theta}}\left[\int_{0}^{s}h(\sigma)^{\theta}w(\sigma)\mathrm{d}\sigma\right]\mathrm{d}s.$
(3.24)
Note that
$g(t)=\int_{0}^{t}h(s)\mathrm{d}s\qquad\text{for }t\leq a,$ (3.25)
so $g\in C^{2}(\mathbb{R}^{+}_{0})$, and that
$g^{\prime}(t)=\frac{1}{h(t)^{\theta}}\int_{0}^{t}h(s)^{\theta}w(s)\mathrm{d}s>0,\qquad\big{(}h(t)^{\theta}g^{\prime}(t)\big{)}^{\prime}=h(t)^{\theta}w(t)\qquad\text{on
}\mathbb{R}^{+}.$ (3.26)
Since, on $[0,a]$, by (3.25) it holds
$g^{\prime\prime}(t)/g^{\prime}(t)=h^{\prime}(t)/h(t)$, the estimate (3.23)
implies
$\sum_{j=1}^{m}\overline{\nabla}\mathrm{d}f(e_{j},e_{j})+m\mathrm{d}f(H)\geq\left(m-m\mu(R)\|H\|\right)g^{\prime\prime}(\rho)=(\theta+1)h^{\prime}(\rho)$
(3.27)
at every point $p\in M$ such that $\varphi(p)\in B_{a}(x_{0})$. Consider now a
point $p\in M$ such that $\rho(\varphi(p))\geq a$ and
$g^{\prime\prime}(\rho(\varphi(p)))\geq 0$. Using $|\mathrm{d}\rho|=1$, (3.26)
and the fact that $g,h$ are increasing we deduce that, at $p$,
$\begin{array}[]{lcl}\displaystyle\sum_{j=1}^{m}\overline{\nabla}\mathrm{d}f(e_{j},e_{j})+m\mathrm{d}f(H)&\geq&\displaystyle\left(m-1-m\mu(R)\|H\|\right)\frac{g^{\prime}(\rho)h^{\prime}(\rho)}{h(\rho)}\\\\[11.38092pt]
\displaystyle&=&\displaystyle\theta\frac{h^{\prime}(\rho)}{h(\rho)^{\theta+1}}\int_{0}^{\rho}h(s)^{\theta}w(s)\mathrm{d}s.\end{array}$
(3.28)
Now, by the definition of $w(s)$ and since $h$ is increasing and $S$ is non-
increasing,
$\displaystyle\displaystyle\frac{h^{\prime}(\rho)}{h(\rho)^{\theta+1}}\int_{0}^{\rho}h(s)^{\theta}w(s)\mathrm{d}s$
$\displaystyle=$
$\displaystyle\displaystyle\frac{h^{\prime}(\rho)h(a)^{\theta+1}}{h(\rho)^{\theta+1}}$
$\displaystyle\displaystyle+\frac{h^{\prime}(\rho)}{h(\rho)^{\theta+1}}\int_{a}^{\rho}h(s)^{\theta}(\theta+1)h^{\prime}(s)S\left(\frac{h(s)-h(a)}{h(a)}\right)\mathrm{d}s$
$\displaystyle\geq$
$\displaystyle\\!\\!\\!\displaystyle\frac{h^{\prime}(\rho)h(a)^{\theta+1}}{h(\rho)^{\theta+1}}+\frac{h^{\prime}(\rho)}{h(\rho)^{\theta+1}}S\left(\frac{h(\rho)-h(a)}{h(a)}\right)\left[h(\rho)^{\theta+1}-h(a)^{\theta+1}\right]$
$\displaystyle\geq$ $\displaystyle\displaystyle
h^{\prime}(\rho)S\left(\frac{h(\rho)-h(a)}{h(a)}\right).$
Combining with (3.28) we get
$\displaystyle\sum_{j=1}^{m}\overline{\nabla}\mathrm{d}f(e_{j},e_{j})+m\mathrm{d}f(H)\geq\theta
h^{\prime}(\rho)S\left(\frac{h(\rho)-h(a)}{h(a)}\right)$ (3.30)
at those points for which $\rho(\varphi(p))\geq a$ and
$g^{\prime\prime}(\rho(\varphi(p)))\geq 0$. On the other hand, at those points
for which $g^{\prime\prime}(\rho(\varphi(p)))\leq 0$ we can bound as follows:
$\begin{array}[]{lcl}\displaystyle\sum_{j=1}^{m}\overline{\nabla}\mathrm{d}f(e_{j},e_{j})+m\mathrm{d}f(H)&\geq&\displaystyle\frac{g^{\prime}(\rho)h^{\prime}(\rho)}{h(\rho)}\Big{(}m-1-m\mu(R)\|H\|\Big{)}-|g^{\prime\prime}(\rho)|\\\\[11.38092pt]
&=&\displaystyle\theta\frac{g^{\prime}(\rho)h^{\prime}(\rho)}{h(\rho)}+g^{\prime\prime}(\rho)\\\\[11.38092pt]
&=&\displaystyle
h(\rho)^{-\theta}\big{(}h(t)^{\theta}g^{\prime}(t)\big{)}^{\prime}_{|t=\rho}=w(\rho)\\\\[11.38092pt]
&=&\displaystyle(\theta+1)h^{\prime}(\rho)S\left(\frac{h(\rho)-h(a)}{h(a)}\right)\end{array}$
(3.31)
the second equality following from (3.26), and the third from the definition
of $w(s)$. Putting together (3.30) and (3.31) we get
$\displaystyle\sum_{j=1}^{m}\overline{\nabla}\mathrm{d}f(e_{j},e_{j})+m\mathrm{d}f(H)\geq\theta
h^{\prime}(\rho)S\left(\frac{h(\rho)-h(a)}{h(a)}\right)$ (3.32)
for every $p$ with $\rho(\varphi(p))\geq a$. Concluding, from (3.21), (3.27),
(3.30) and (3.31) we get
$\left\\{\begin{array}[]{ll}\Delta(f\circ\varphi)\geq(\theta+1)h^{\prime}(\rho\circ\varphi)&\quad\text{on
}\varphi^{-1}\big{\\{}B_{a}(x_{0})\big{\\}},\\\\[8.5359pt]
\displaystyle\Delta(f\circ\varphi)\geq\theta
h^{\prime}(\rho\circ\varphi)S\left(\frac{h(\rho\circ\varphi)-h(a)}{h(a)}\right)&\quad\text{on
}\varphi^{-1}\big{\\{}N\backslash B_{a}(x_{0})\big{\\}}.\end{array}\right.$
(3.33)
Define $u_{x_{0}}=f\circ\varphi$. Property (3.13) is immediate, and (3.20)
follows from (3.10) and (3.33). We are left to prove the $L^{\infty}$-bounds
(3.17). By (3.12), there exists $S^{*}=S^{*}(\theta,S)$ such that
$(\theta+1)\sum_{k=1}^{+\infty}S(k)(k+1)^{\theta}=S^{*}<\infty.$ (3.34)
Define a sequence $\\{a_{k}\\}_{k=0}^{+\infty}$ in such a way that
$h(a_{k})=kh(a).$ (3.35)
In our assumption $(\mathcal{H}_{2})$, $h$ is increasing and
$h(t)\rightarrow+\infty$ if $t\rightarrow+\infty$, thus each $a_{k}$ is well
defined, $\\{a_{k}\\}$ is increasing and $a_{k}\rightarrow+\infty$ as
$k\rightarrow+\infty$. For every $s\in\mathbb{R}^{+}$, define $N(s)$ to be the
greatest $k$ such that $a_{k}<s$. Then, for $s\geq a$,
$\begin{array}[]{l}\displaystyle\int_{0}^{s}h(\sigma)^{\theta}w(\sigma)\mathrm{d}\sigma=\int_{0}^{a}\ldots+\int_{a}^{s}\ldots=h(a)^{\theta+1}+\int_{a}^{s}h(\sigma)^{\theta}w(\sigma)\mathrm{d}\sigma\\\\[11.38092pt]
\displaystyle\leq
h(a)^{\theta+1}+\sum_{k=1}^{N(s)}\int_{a_{k}}^{a_{k+1}}(\theta+1)h(\sigma)^{\theta}h^{\prime}(\sigma)S\left(\frac{h(\sigma)-h(a)}{h(a)}\right)\mathrm{d}\sigma.\end{array}$
(3.36)
Since $h$ is increasing, $h\geq h(a_{k})$ on $[a_{k},a_{k+1}]$. Property
(3.35) and the fact that $S$ is non-increasing thus imply
$\begin{array}[]{l}\displaystyle\int_{a}^{s}h(\sigma)^{\theta}w(\sigma)\mathrm{d}\sigma\leq\sum_{k=1}^{N(s)}S(k)\int_{a_{k}}^{a_{k+1}}(\theta+1)h(\sigma)^{\theta}h^{\prime}(\sigma)\mathrm{d}\sigma\\\\[11.38092pt]
\displaystyle\leq\sum_{k=1}^{N(s)}S(k)\big{[}h(a_{k+1})^{\theta+1}-h(a_{k})^{\theta+1}\big{]}\\\\[11.38092pt]
\displaystyle\leq
h(a)^{\theta+1}\sum_{k=1}^{N(s)}S(k)\big{[}(k+1)^{\theta+1}-k^{\theta+1}\big{]}\leq
h(a)^{\theta+1}\sum_{k=1}^{N(s)}(\theta+1)S(k)(k+1)^{\theta}\end{array}$
(3.37)
the last inequality following by the mean value theorem. Therefore, putting
together (3.24), (3.36) and (3.37) we argue that, for $t\geq a$,
$\begin{array}[]{lcl}\displaystyle g(t)&=&\displaystyle
g(a)+\int_{a}^{t}\frac{1}{h(s)^{\theta}}\left[\int_{0}^{s}h(\sigma)^{\theta}w(\sigma)\mathrm{d}\sigma\right]\mathrm{d}s\\\\[11.38092pt]
&\leq&\displaystyle
g(a)+\int_{a}^{t}\frac{1}{h(s)^{\theta}}\left[h(a)^{\theta+1}+h(a)^{\theta+1}\sum_{k=1}^{N(s)}(\theta+1)S(k)(k+1)^{\theta}\right]\mathrm{d}s\\\\[11.38092pt]
&\leq&\displaystyle
g(a)+(1+S^{*})h(a)^{\theta+1}\int_{a}^{t}\frac{\mathrm{d}s}{h(s)^{\theta}},\end{array}$
(3.38)
where the last inequality comes from the definition of $S^{*}$ in (3.34).
Combining with (3.27) we conclude that
$\begin{array}[]{lcl}\displaystyle\|g\|_{L^{\infty}([0,R])}&=&\displaystyle
g(R)\leq
g(a)+(1+S^{*})h(a)^{\theta+1}\int_{a}^{R}\frac{\mathrm{d}s}{h(s)^{\theta}}\\\\[11.38092pt]
&=&\displaystyle\int_{0}^{a}h(s)\mathrm{d}s+(1+S^{*})h(a)^{\theta+1}\int_{a}^{R}\frac{\mathrm{d}s}{h(s)^{\theta}}\end{array}$
(3.39)
From $h(s)=s+O(s^{2})$ as $s\rightarrow 0$ it is easy to deduce that
$\int_{0}^{a}h(s)\mathrm{d}s=\frac{a^{2}}{2}+O(a^{3}),\qquad\int_{a}^{R}\frac{\mathrm{d}s}{h(s)^{\theta}}\sim\left\\{\begin{array}[]{ll}\displaystyle\frac{a^{1-\theta}}{\theta-1}&\quad\text{if
}\theta>1\\\\[8.5359pt] \displaystyle|\log a|&\quad\text{if
}\theta=1,\\\\[8.5359pt] \displaystyle C&\quad\text{if
}\theta\in(0,1),\\\\[8.5359pt] \end{array}\right.$ (3.40)
For some $C$ depending on $m,R,\theta$ and $h$ on $[0,R]$. From (3.39) and
(3.40), there exists a positive constant $C$ depending only on $m,R,\theta,S$
and on $h_{|_{[0,R]}}$ such that, if if $a\in(0,\bar{a}]$,
$\|g\|_{L^{\infty}([0,R])}\leq\left\\{\begin{array}[]{ll}Ca^{2}&\quad\text{if
}\theta>1\\\\[5.69046pt] a^{2}+Ca^{2}|\log a|\leq Ca^{2}|\log a|&\quad\text{if
}\theta=1\\\\[5.69046pt] a^{2}+Ca^{\theta+1}\leq Ca^{\theta+1}&\quad\text{if
}\theta\in(0,1).\end{array}\right.$
Noting that
$\|u_{x_{0}}\|_{L^{\infty}(M)}\leq\|f\|_{L^{\infty}(B_{R}(x))}=\|g\|_{L^{\infty}([0,R])},$
the desired bounds in (3.17) are proved. ∎
###### Remark 10.
_If $M$ is minimal then, the constant $C$ in (3.17) does _not_ depend on $R$
whenever the non-parabolicity condition_
$\frac{1}{h(s)^{m-1}}\in L^{1}(+\infty)$ (3.41)
_holds. Indeed, if $M$ is minimal then $\theta=m-1$. Once $S$ is chosen, the
value $S^{*}$ in (3.34) is thus independent of $R$. By (3.41), in the bound
(3.39) we can let $R\rightarrow+\infty$ to obtain_
$\displaystyle\|g\|_{L^{\infty}(\mathbb{R}^{+}_{0})}\leq\int_{0}^{a}h(s)\mathrm{d}s+(1+S^{*})h(a)^{m}\int_{a}^{+\infty}\frac{\mathrm{d}s}{h(s)^{m-1}}.$
(3.42)
_The asymptotics $h(a)\sim a$ and_
$h(a)^{m}\int_{a}^{+\infty}\frac{\mathrm{d}s}{h(s)^{m-1}}\sim\left\\{\begin{array}[]{ll}\frac{a^{2}}{m-2}&\quad\text{if
}m\geq 3,\\\\[5.69046pt] a^{2}|\log a|&\quad\text{if }m=2\end{array}\right.$
_prove that $\|g\|_{L^{\infty}(\mathbb{R}^{+}_{0})}\leq Ca^{2}$ if $m\geq 3$
(respectively, $\|g\|_{L^{\infty}(\mathbb{R}^{+}_{0})}\leq Ca^{2}|\log a|$ if
$m=2$), for some constant $C$ only depending on $m,S^{*}$, proving the claim._
_By Remark 8, condition (3.41) is always satisfied whenever (3.9) holds and
$m\geq 3$. We further observe that, via Proposition 3.1 in [16], requirement
(3.41) is necessary and sufficient for the non-parabolicity of the radially
symmetric model $M^{m}_{h}$, which is, roughly speaking, compared to $M$ by
means of formulas (3.22). We recall that $M^{m}_{h}$ is defined as the
manifold $\mathbb{R}^{m}$, with a fixed origin $o$ and metric given, in polar
geodesic coordinates $(r,\theta)$ centered at $o$, by
$\mathrm{d}s_{h}^{2}=\mathrm{d}r^{2}+h(r)^{2}g_{\mathbb{S}^{m-1}}(\theta)$,
where $g_{\mathbb{S}^{m-1}}(\theta)$ is the standard metric on the unit
sphere. Clearly, since we are constructing bounded, non-constant and strictly
subharmonic functions on $M$, some non-parabolicity condition must necessarily
appear. _
### Proof of Theorem 2
Property $R_{0}>\frac{1}{2}\mathrm{diam}(D)$ and the completeness of $N$
enable us to choose $x\in N$ such that $D\Subset B_{R_{0}}=B_{R_{0}}(x)$. For
notational convenience, define $\lim_{0}\varphi=\lim\varphi\cap D$. Choose a
small $r_{1}<<\min\\{R_{0},1\\}$ in such a way that
$B_{2r_{1}}(\mathrm{lim}_{0}\,\varphi)\subseteq B_{R_{0}}.$
Since the function $\Psi$ defined in (3.3) satisfies (2.3), by Remark 6 the
measure $\overline{\mathcal{H}}_{\Psi}(\lim_{0}\varphi)$ computed by only
using balls is zero. Therefore, we can find a countable covering $\\{B_{j}\\}$
of balls $B_{j}=B_{\varepsilon_{j}}(x_{j})\subseteq M$ of radius
$\varepsilon_{j}\leq r_{1}$ such that
$\mathrm{lim}_{0}\,\varphi\subseteq\bigcup_{j}B_{j}\qquad\text{and}\qquad\left|\sum_{j}\Psi(\varepsilon_{j})\right|\leq
r_{1}.$ (3.43)
For notational convenience, set $b_{1}=8\sqrt{r_{1}}$. Consider
$\lim_{1}\varphi=\lim\varphi\cap\big{(}D\backslash B_{b_{1}/2}(\partial
D)\big{)}$. This set is compact and contained in $\lim_{0}\varphi$. By
compactness, we can select a finite subcovering $\\{B_{j}\\}_{j=1}^{k_{1}}$ of
balls touching $\lim_{1}\varphi$, in such a way that the covering is contained
in $B_{2r_{1}}(\lim_{0}\varphi)\subseteq B_{R_{0}}$. We can suppose $k_{1}\geq
2$. Clearly, the second property in (3.43) still holds for the subcovering.
Define the compact set
$K_{r_{1}}=M\backslash\left(\varphi^{-1}\Big{(}\bigcup_{j=1}^{k_{1}}B_{j}\Big{)}\cup\varphi^{-1}\big{(}B_{b_{1}}(\partial
D)\big{)}\right)$
For each $j$, we choose $a_{j}=\varepsilon_{j}\leq r_{1}$. Since $x_{j}\in
B_{R_{0}}$, we can apply Lemma 1 with the choices $G=B^{2}$ (so that $h(s)=s$
if $B=0$, $h(s)=B^{-1}\sinh(Bs)$ if $B>0$), $x=x_{j}$, $R=2R_{0}$ and
$S(t)=\max\\{t,1\\}^{-\theta-2}$ to deduce the existence of constants
$\bar{a},C>0$ ($C$ only depending on $m,R,\theta$ and on $h_{|_{[0,2R_{0}]}}$)
such that, if $r_{1}\leq\bar{a}$, there exists $u_{j}=u_{x_{j}}$ with the
properties
$\left\\{\begin{array}[]{l}u_{j}\geq 0,\quad u_{j}(p)=0\ \text{ if and only if
}\varphi(p)=x_{j},\\\\[5.69046pt] \|u_{j}\|_{L^{\infty}}\leq
C\Psi(\varepsilon_{j})\\\\[8.5359pt] \Delta u_{j}>0\ \text{ on }M,\quad\Delta
u_{j}\geq(\theta+1)/2\ \text{ on }\varphi^{-1}(B_{j}).\end{array}\right.$
(3.44)
We consider the case $\lim\varphi\cap\partial D\neq\emptyset$ in (3.11), the
other case being easier. Define $u_{\infty}=b_{1}F(\varphi(x))$. Then,
$u_{\infty}\leq 0$ on $M$. By formula (3.21) with $f=F$ and the $m$-convexity
of $D$ with constant $c$ we deduce that, on the whole $M$,
$\begin{array}[]{lcl}\Delta u_{\infty}&=&\displaystyle
b_{1}\Delta(F\circ\varphi)\geq
b_{1}\left(\sum_{j=1}^{m}\overline{\nabla}\mathrm{d}F(e_{j},e_{j})-m\|\overline{\nabla}F\|_{L^{\infty}(D)}\|H\|\right)\\\\[11.38092pt]
&\geq&b_{1}\Big{(}\displaystyle
c-m\|\overline{\nabla}F\|_{L^{\infty}(D)}\|H\|\Big{)}\geq
b_{1}\overline{C},\end{array}$ (3.45)
for some positive constant $\overline{C}$. Set
$w_{1}=\sum_{j=1}^{k_{1}}(2\|u_{j}\|_{L^{\infty}}-u_{j})-u_{\infty}.$
Note that, since $k_{1}\geq 2$, $w_{1}$ is strictly positive on $M$ by
construction. For every $p\in M\backslash K_{r_{1}}$, either
$(1)\ \ \ \ \varphi(p)\in B_{b_{1}}(\partial D)\qquad\text{or}\qquad(2)\ \ \ \
\varphi(p)\in\bigcup_{j=1}^{k_{1}}B_{j},$
and the two cases are not mutually excluding. We first examine $(1)$. In this
case, since $F=0$ on $\partial D$ and is Lipschitz on $D$,
$|u_{\infty}(p)|\leq b_{1}^{2}\|\overline{\nabla}F\|_{L^{\infty}(D)},$
whence by (3.43), (3.44) and the fact that $b_{1}^{2}=64r_{1}$ we obtain
$\begin{array}[]{lcl}\displaystyle-\frac{\Delta
w_{1}}{w_{1}}(p)&\geq&\displaystyle\frac{\Delta
u_{\infty}(p)}{2\left(\sum_{j}\|u_{j}\|_{L^{\infty}}\right)+|u_{\infty}(p)|}\geq\frac{b_{1}\overline{C}}{C\left(\sum_{j}\Psi(\varepsilon_{j})\right)+b_{1}^{2}\|\overline{\nabla}F\|_{L^{\infty}(D)}}\\\\[17.07182pt]
&\geq&\displaystyle
C\frac{b_{1}}{r_{1}+b_{1}^{2}}\geq\frac{C}{b_{1}}=\frac{C}{\sqrt{r_{1}}}\end{array}$
(3.46)
for some constant $C>0$ depending on $m,R_{0},\theta,F,h_{|_{[0,2R_{0}]}}$
(hereafter, we will say that it depends on the data of the immersion) but not
on $r_{1}$ or on the covering $\\{B_{j}\\}$, and that may vary from line to
line in (3.46).
We now examine case $(2)$. We split $\\{1,\ldots,k_{1}\\}$ in two subsets
$J_{1}^{p}=\big{\\{}j\in\\{1,\ldots,k_{1}\\}:\varphi(p)\in
B_{j}\big{\\}},\qquad
J_{2}^{p}=\big{\\{}j\in\\{1,\ldots,k_{1}\\}:\varphi(p)\in N\backslash
B_{j}\big{\\}}.$
By construction and by $(2)$, $J_{1}^{p}\neq\emptyset$. Then, again by (3.43),
(3.44), and since $b_{1}=8\sqrt{r_{1}}\geq 8r_{1}$ by our choice of $r_{1}$,
it holds
$\begin{array}[]{lcl}\displaystyle-\frac{\Delta
w_{1}}{w_{1}}(p)&\geq&\displaystyle\frac{\sum_{J^{p}_{1}\cup J^{p}_{2}}\Delta
u_{j}(p)}{2\left(\sum_{j}\|u_{j}\|_{L^{\infty}}\right)+\|u_{\infty}\|_{L^{\infty}}}\geq\frac{\sum_{J^{p}_{1}}\Delta
u_{j}(p)}{C\left(\sum_{j}\Psi(\varepsilon_{j})\right)+b_{1}\|F\|_{L^{\infty}(D)}}\\\\[17.07182pt]
&\geq&\displaystyle\frac{(\theta+1)|J^{p}_{1}|}{C(r_{1}+b_{1})}\geq\frac{C}{b_{1}}=\frac{C}{\sqrt{r_{1}}}\end{array}$
(3.47)
for some $C>0$ depending, as before, on the data of the immersion but not on
$r_{1}$ or on the covering $\\{B_{j}\\}$. Summarizing, there exists $C>0$ only
depending on the data of the immersion such that
$-\frac{\Delta w_{1}}{w_{1}}\geq\frac{C}{\sqrt{r_{1}}}\qquad\text{on
}K_{r_{1}}.$ (3.48)
Now, choose a positive
$r_{2}<\min\\{\frac{r_{1}}{2},\frac{b_{1}}{16},\frac{1}{2}\\}$ and set
$b_{2}=8\sqrt{r_{2}}$.
Again, we can find a countable covering $\\{B_{j}\\}$ of balls
$B_{j}=B_{\varepsilon_{j}}(x_{j})\subseteq M$ of radius $\varepsilon_{j}\leq
r_{2}$ such that
$\mathrm{lim}_{0}\,\varphi\subseteq\bigcup_{j}B_{j}\qquad\text{and}\qquad\left|\sum_{j}\Psi(\varepsilon_{j})\right|\leq
r_{2}.$ (3.49)
Consider the compact set $\lim_{2}\varphi=\lim\varphi\cap\big{(}D\backslash
B_{b_{2}/2}(\partial D)\big{)}$, contained in $\lim_{0}\varphi$ and containing
$\lim_{1}\varphi$. By compactness, we can select a finite subcovering
$\\{B_{j}\\}_{j=1}^{k_{2}}$ of balls touching $\lim_{2}\varphi$, so that the
subcovering is contained in $B_{2r_{2}}(\lim_{2}\varphi)$. Since $b_{2}\geq
8\sqrt{r_{2}}\geq 8r_{2}$, $B_{2r_{2}}(\lim_{2}\varphi)\subseteq D$. Moreover,
by our choice of $r_{2}$,
$B_{2r_{2}}(\mathrm{lim}_{2}\varphi)\Subset
B_{2r_{1}}(\mathrm{lim}_{1}\varphi)\cup B_{b_{1}}(\partial D).$
Consequently, the compact set
$K_{r_{2}}=M\backslash\left(\varphi^{-1}\Big{(}\bigcup_{j=1}^{k_{2}}B_{j}\Big{)}\cup\varphi^{-1}\big{(}B_{b_{2}}(\partial
D)\big{)}\right)$
satisfies $K_{r_{1}}\subseteq\mathrm{int}(K_{r_{2}})$. The same construction
procedure as above can be repeated verbatim, yielding a superharmonic function
$w_{2}$ on $M$ such that
$-\frac{\Delta w_{2}}{w_{2}}\geq\frac{C}{\sqrt{r_{2}}}\qquad\text{on
}M\backslash K_{r_{2}},$
for the same constant $C$ as in (3.48), only depending on the data of the
immersion. Inductively, if at each step we select a positive
$r_{l+1}<\min\\{\frac{r_{l}}{2},\frac{b_{l}}{16},2^{-l}\\}$ and proceed to
find $K_{r_{l+1}}$ satisfying $\mathrm{int}(K_{r_{l+1}})\supseteq K_{r_{l}}$,
and a positive $w_{l+1}$ solving $-\Delta w_{l+1}/w_{l+1}\geq
C/\sqrt{r_{l+1}}$ on $M\backslash K_{r_{l+1}}$. Note that, although
$r_{l}\downarrow 0^{+}$, we cannot infer that $K_{r_{l}}$ is an exhaustion of
$M$ because $\lim\varphi$ could actually contain points of $\varphi(M)$.
However, by Persson formula and Barta inequality, for each $l$
$\inf\sigma_{\mathrm{ess}}(-\Delta)\geq\lambda^{*}(M\backslash
K_{r_{l}})\geq\inf_{M\backslash K_{r_{l}}}\left(-\frac{\Delta
w_{l}}{w_{l}}\right)\geq\frac{C}{\sqrt{r_{l}}}\rightarrow+\infty$
as $l\rightarrow+\infty$, which concludes the proof.
## 4 Remarks on $\sigma_{\mathrm{ess}}(-\Delta)$ and open questions
We conclude this paper with some observations on the links between
$\sigma_{\mathrm{ess}}(-\Delta)$ and the topology of the limit set of an
immersion, and with some open problems. First, we prove Proposition 1 of the
Introduction, that is, that the ball property in Definition 1 implies the
existence of essential spectrum.
###### Proof of Proposition 1.
For each $j$, define the compactly supported, Lipschitz function
$\varphi_{j}(x)=\psi(\rho_{j}(x))$, where $\rho_{j}(x)=\mathrm{dist}(x,x_{j})$
and
$\psi(t)=\left\\{\begin{array}[]{ll}1,&t\leq\delta R\\\\[2.84544pt]
\frac{R-t}{R(1-\delta)}&t\in\left[\delta R,R\right]\\\\[2.84544pt] 0&t\geq
R\end{array}\right.\quad\Longrightarrow\quad|\psi^{\prime}|\leq\frac{1}{R(1-\delta)}.$
(4.1)
Then, by the ball property (1.2),
$\displaystyle I_{\lambda}(\phi_{j},\phi_{j})$ $\displaystyle=$
$\displaystyle\int_{B_{R}(x_{j})}|\nabla\phi_{j}|^{2}-\lambda\int_{B_{R}(x_{j})}\phi_{j}^{2}\leq\frac{\mathrm{vol}(B_{j})}{R^{2}(1-\delta)^{2}}-\lambda\text{vol}\big{(}B_{\delta
R}(x_{j})\big{)}$ (4.2) $\displaystyle\leq$
$\displaystyle\mathrm{vol}(B_{R})\left(\frac{1}{R^{2}(1-\delta)^{2}}-\lambda
C^{-1}\right)<0$ (4.3)
provided that $\lambda>\lambda^{*}=\frac{C}{R^{2}(1-\delta)^{2}}$. Since
$\\{\phi_{j}\\}$ span an infinite-dimensional subspace of the domain of
$-\Delta$, the (Friedrichs extension of) the operator $-\Delta-\lambda$ has
infinite index, or equivalently $-\Delta$ has infinite eigenvalues below
$\lambda$, for each $\lambda>\lambda^{*}$. By the min-max theorem (see for
instance [29], or Section 3 in [28] for a concise account), the inequality
$\inf\sigma_{\mathrm{ess}}(-\Delta)\leq\lambda^{*}$ follows at once. ∎
###### Remark 11.
_In virtue of Bishop-Gromov volume comparison theorem, the ball property in
Definition 1 is met, for instance, when the Ricci curvature of $B_{R}(x_{j})$
is uniformly bounded from below by a constant, say $-(m-1)H^{2}<0$, $m=\dim
M$. Indeed, denoting with $\mathrm{vol}_{H}(r)$ the volume of a geodesic ball
of radius $r$ in the hyperbolic space $M^{m}_{H}$ of sectional curvature
$-H^{2}$, by Bishop-Gromov theorem
$\mathrm{vol}(B_{r}(x_{j}))/\mathrm{vol}_{H}(r)$ is non-increasing on $[0,R]$.
Hence, for each $\delta>0$_
$\mathrm{vol}\big{(}B_{\delta
R}(x_{j})\big{)}\geq\frac{\mathrm{vol}_{H}(\delta
R)}{\mathrm{vol}_{H}(R)}\mathrm{vol}\big{(}B_{R}(x_{j})\big{)}=C(\delta,R)^{-1}\mathrm{vol}\big{(}B_{R}(x_{j})\big{)}.$
As already underlined, the independence of Proposition 1 from curvature
requirements makes it suited to investigate minimal surfaces, particularly
those described in the Introduction. The reason is that the approach via
Runge’s approximation theorem, shared by all these constructions, guarantees a
$C^{0}$-control of the metric (hence, of lengths and volumes) in some known
regions, while it seems more difficult to control curvatures in the same
region. We now come to the proof of Proposition 2.
###### Proof of Proposition 2.
Let $K_{n}$ the compact sets used in the construction of [18], let $r_{n}$
denote the Euclidean distance between the inner and the outer circle of
$K_{n}$, and let $p_{n}\in K_{n}$ be the point in the middle of the segment of
the real axis crossed by $K_{n}$. By construction, $r_{n}\rightarrow 0^{+}$ as
$n\rightarrow+\infty$, and $\mathrm{d}s^{2}$ is $C^{0}$-close to a multiple of
the Euclidean metric $\mathrm{d}s^{2}_{E}$ on $K_{n}$, more precisely
$\mathrm{d}s^{2}=\lambda^{2}\mathrm{d}s^{2}_{E}$ with
$\lambda=\frac{1}{2}\left(|e^{h}|+|e^{-h}|\right),\quad\text{for some
holomorphic $h$ on $K_{n}$ with }|h-c_{n}|<1\ \text{ on }K_{n},$
$c_{n}$ being some positive constant chosen in such a way that
$\sum_{n=1,\,n\text{
even}}^{+\infty}r_{n}e^{c_{n}-1}=+\infty,\qquad\sum_{n=1,\,n\text{
odd}}^{+\infty}r_{n}e^{c_{n}-1}=+\infty.$ (4.4)
The line elements thus satisfy $C_{n}\mathrm{d}s_{E}\leq\mathrm{d}s\leq
e^{2}C_{n}\mathrm{d}s_{E}$, where $C_{n}=(e^{c_{n}-1}+e^{-c_{n}-1})/2$.
Consequently, every curve passing $K_{n}$ from the inner to the outer circle
has length at least $r_{n}e^{c_{n}-1}/2$. The choice $c_{n}=r_{n}$ guarantees
both (4.4) and the following property that, for each fixed $R>0$, one can find
$n_{R}$ large enough such that $B_{R}(p_{n})\subseteq K_{n}$ for each $n\geq
n_{R}$. By the relation of the line elements, denoting with
$\mathbb{B}_{\rho}(z_{j})$ the Euclidean ball of radius $\rho$ centered at
$z_{j}$, $\mathbb{B}_{Re^{-2}/C_{n}}(z_{j})\subseteq
B_{R}(z_{j})\subseteq\mathbb{B}_{R/C_{n}}(z_{j})$, thus
$\mathrm{vol}\big{(}B_{\frac{R}{2}}(p_{n})\big{)}\geq
C_{n}^{2}\mathrm{vol}_{E}\big{(}\mathbb{B}_{\frac{R}{e^{2}2C_{n}}}(p_{n})\big{)},\quad\mathrm{vol}\big{(}B_{R}(p_{n})\big{)}\leq
e^{4}C_{n}^{2}\mathrm{vol}_{E}\big{(}\mathbb{B}_{\frac{R}{C_{n}}}(p_{n})\big{)}.$
(4.5)
By the doubling property of Euclidean space $\mathbb{R}^{3}$, it is immediate
from (4.5) that (1.2) holds for $\mathrm{d}s^{2}$ with $\delta=1/2$ and a
suitable $C$ independent of $n$, but even of $R$. Therefore, by Proposition 1,
there exists an absolute constant $C>0$ such that
$\inf\sigma_{\mathrm{ess}}(-\Delta)\leq C/R^{2}$. Since $R$ can be chosen to
be arbitrarily large, the thesis follows. ∎
Now, let $\varphi:M\rightarrow N$ be an isometric immersion. The ball property
(1.2) can also be deduced from suitable convergences of pieces of $\varphi(M)$
in $N$. We formalize this fact in the next
###### Definition 3.
Let $\varphi:M^{m}\rightarrow N^{n}$ be an isometric immersion with
$\lim\varphi\neq\emptyset$. We will say that $\varphi$ has the _extrinsic ball
property_ if there is a point $p\in\lim\varphi$, a bounded subset with compact
closure $\Sigma_{p}\subset\lim\varphi$ containing $p$ and diffeomorphic to the
unit ball $B_{1}(0)\subseteq\mathbb{R}^{m}$, and a sequence of disjoint balls
$B_{j}=B_{R}(x_{j})\subseteq M^{m}$ such that $\varphi(B_{j})\to\Sigma_{p}$ in
the $C^{0}$-norm.
###### Corollary 3.
If $\varphi:M^{m}\rightarrow N^{n}$ has the extrinsic ball property, then $M$
has the ball property. In particular, $\sigma_{\mathrm{ess}}(-\Delta)$ is non-
empty.
###### Proof.
Let $c>0$ be such that the curvature tensor $R_{\Sigma}$ of $\Sigma_{p}$
satisfies $R_{\Sigma}\geq-c$. Then, by Bishop-Gromov theorem, there exists
$C=C(R)$ such that $\mathrm{vol}(B_{R/2}(x))\geq C\mathrm{vol}(B_{R}(x))$. By
$C^{0}$-convergence, there exists $j_{0}$ such that
$\mathrm{vol}\big{(}B_{R/2}(x_{j})\big{)}\geq\frac{C}{2}\mathrm{vol}\big{(}B_{R}(x_{j})\big{)}\qquad\text{for
each }j\geq j_{0},$
thus $M$ has the ball property, and non-empty essential spectrum thanks to
Proposition 1. ∎
We underline that, if $\varphi:M^{m}\rightarrow N^{n}$ has the extrinsic ball
property, then $\dim_{\mathcal{H}}(\lim\varphi)\geq m$. The extrinsic ball
property is closely related to the existence of immersions into $\lim\varphi$
that are generated by $\varphi$ itself. Indeed, we report here the following
well-known convergence result:
###### Lemma 2.
Let $M^{m}_{j}$ a sequence of complete smooth manifolds, all with injectivity
radius $\mathrm{inj}(M_{j})\geq\epsilon_{0}>0$ and isometrically immersed in a
complete manifold $N^{n}$ via $\varphi_{j}:M_{j}\rightarrow N$. Let
$\Omega_{1}\Subset\Omega_{2}\Subset N$ be two open, relatively compact sets of
$N$. Assume that $\Omega_{1}$ intersects all $\varphi_{j}(M_{j})$ and that the
second fundamental forms of all $\varphi_{j}(M_{j})\cap\Omega_{2}$ are
uniformly bounded. Then, there exists a sufficiently small $R>0$ such that
1. 1.
If $\\{x_{j}\\}$, $x_{j}\in M_{j}$ is such that
$p_{j}=\varphi_{j}(x_{j})\in\Omega_{1},\;p_{j}\to p$ and
$\mathrm{d}\varphi_{j}(T_{x_{j}}M_{j})\to\Pi\subseteq T_{p}N$, then
$\varphi_{j}$ restricted to the ball $\Sigma_{j}=B_{R}(x_{j})$ is an embedding
such that $\varphi_{j}(\Sigma_{j})$ converges in the $C^{\infty}$ topology to
an embedded $m$-ball $\Sigma\ni p$, and $T_{p}\Sigma=\Pi$.
2. 2.
If $M_{j}=M$ and $\varphi_{j}=\varphi$ for every $j$, and if
$p\in\lim\varphi$, $M$ has the extrinsic ball property and we can take
$\Sigma_{p}\subseteq\lim\varphi$.
3. 3.
In addition, if each $M_{j}$ is a complete, simply connected manifold and the
norms of the second fundamental forms $II_{j}$ of $\varphi_{j}$ satisfy the
following property:
$\forall\,R>0,\ \exists\,C_{R}>0\,\text{ such that
}\,\left|(II_{j})_{\varphi_{j}^{-1}(B_{R}(p))}\right|^{2}\leq C_{R}\,\text{
for each }j,$
then there is an isometric immersion of a complete, simply connected manifold
$M_{p}$ into $N$ build by convergence over compact sets. In the case when
$M_{j}=M$, $\varphi_{j}=\varphi$ for each $j$, and $p\in\lim\varphi$, we have
$M_{p}\subseteq\lim\varphi$.
###### Remark 12.
_One idea to prove this result is to find $\varepsilon^{\prime}>0$ such that
each $\varphi_{j}(M_{j})$ can be written as the image of a section of the
normal bundle over the tangent plane at some point
$p_{j}\in\varphi_{j}(M_{j})\cap\Omega_{2}$ via the normal exponential map.
Then, use the convergence and regularity theory for the system of elliptic
PDEs describing the mean curvature vector. Simply connectedness is important
to avoid period problems. _
A particular case when the extrinsic ball property holds is when $\varphi$ has
locally bounded geometry, in the sense of the following
###### Definition 4.
An isometric immersion $\varphi:M^{m}\to N^{n}$ has locally bounded geometry
if, for each compact set $K\subseteq N$, then $\varphi|_{\varphi^{-1}(K)}$ has
bounded second fundamental form.
###### Corollary 4.
Let $\varphi:M^{m}\to N^{n}$ be an isometric immersion between complete
manifolds. If $\lim\varphi\neq\emptyset$, $\varphi$ has locally bounded
geometry and $\mathrm{inj}(M)>0$, then $\varphi$ has the extrinsic ball
property. In particular, $M$ has non-empty essential spectrum.
We conclude by exhibiting an example of a bounded, minimal surface whose
essential spectrum is non-empty, whose limit set has big Hausdorff dimension
and which is not a covering. This, again, shows the sharpness of Theorem 1,
and introduces us to some open questions.
###### Example 3.
_We consider a portion of Andrade minimal surface,[2]. We recall its
construction and some of its properties, referring the reader to [2] for full
proofs. Choose $r_{1},r_{2}>0$ such that $r_{1}/r_{2}$ is irrational and
strictly less than $1$, and set $d=r_{2}-r_{1}$. Define the map
$\chi:\mathbb{C}\rightarrow\mathbb{R}^{3}=\mathbb{C}\times\mathbb{R}$,
$\chi(z)=(L(z)-\overline{H(z)},h(z))$, for the following choice of holomorphic
functions $L,H$ and harmonic function $h$:_
$L(z)=(r_{1}-r_{2})e^{z},\quad
H(z)=-de^{\left(\frac{r_{1}}{r_{2}}-1\right)z},\quad
h(z)=4\left(\frac{d}{r_{2}}\right)^{1/2}\left|\frac{r_{2}}{r_{1}}\right||r_{2}-r_{1}|\Re\left(ie^{\frac{r_{1}}{2r_{2}}z}\right),$
_where $\Re$ means the real part. Then, a computation gives that_
$|L^{\prime}(z)|+|H^{\prime}(z)|>0,\qquad
L^{\prime}H^{\prime}=\left(\frac{\partial h}{\partial
z}\right)^{2}\qquad\text{on }\mathbb{C},$
_which is a necessary and sufficient set of condition on $\chi$ to be a
conformal minimal immersion of $\mathbb{C}$ in $\mathbb{R}^{3}$. Restricting
$\chi$ to the region $U=\\{z=u+iv\in\mathbb{C}:|u|<1\\}$, we get a bounded,
simply-connected minimal immersion $\varphi=\chi_{|U}$. For each fixed
$u\in(-1,1)$, $\varphi(u+iv)$ is a dense immersed trochoid in the cylinder
$\Gamma_{u}=\left[B_{s_{1}(u)}\backslash
B_{s_{2}(u)}\right]\times(-l(u),l(u))$, where $s_{1},s_{2},l$ are explicit
functions of $u$ depending on $r_{1}$ and $r_{2}$. Therefore, $\lim\varphi$ is
dense in the open subset $\bigcup_{u\in(-1,1)}\Gamma_{u}$ of $\mathbb{R}^{3}$,
which gives $\dim_{\mathcal{H}}(\lim\varphi)=3$. Moreover, the induced metric
$\mathrm{d}s^{2}$ satisfies_
$\mathrm{d}s^{2}=\left(|L^{\prime}|+|H^{\prime}|\right)^{2}|\mathrm{d}z|^{2}=\left(|r_{2}-r_{1}|e^{u}+de^{\left(\frac{r_{1}}{r_{2}}-1\right)u}\right)^{2}|\mathrm{d}z|^{2}\geq
4(r_{2}-r_{1})^{2}|\mathrm{d}z|^{2}.$ (4.6)
_Considering $z_{k}=2ik\in U$, each of the unit balls
$\mathbb{B}_{1}(z_{k})\subseteq U$ in the metric $|\mathrm{d}z|^{2}$ contains
a ball $B_{R}(z_{k})$ in the metric $\mathrm{d}s^{2}$ of radius at least
$R=2|r_{2}-r_{1}|$. Since the sectional curvature of $\chi$ satisfies_
$K=-c_{1}\left(e^{\left(1-\frac{r_{1}}{4r_{2}}\right)u}+c_{2}e^{\left(\frac{3r_{1}}{4r_{2}}-1\right)u}\right)^{-4},$
_for some positive constants $c_{1},c_{2}$, and $1-\frac{r_{1}}{4r_{2}}$ and
$\frac{3r_{1}}{4r_{2}}-1$ have opposite signs, $\chi$ has globally bounded
curvature. In particular, $\\{B_{R}(z_{k})\\}$ is a collection of disjoint
balls in $(U,\mathrm{d}s^{2})$ with uniformly bounded sectional curvature,
thus $\sigma_{\mathrm{ess}}(-\Delta)$ on $(U,\mathrm{d}s^{2})$ is non-empty by
Proposition 1 and Remark 11 ._
### Open problems
* $(1)$
As Theorem 3 shows, raising the dimension of $M$ does not yield an improvement
of the allowed Hausdorff dimension of the limit set. However, from some point
of view this fact seems of technical nature. It seems to us reasonable to
state the following
###### Conjecture.
Let $\varphi:M^{m}\rightarrow N^{n}$ a minimally immersed submanifold of
dimension $m\geq 3$ into an open, $m$-convex subset $D$ of a Cartan-Hadamard
manifold $N$. If $\mathcal{H}^{m}(\lim\varphi\cap D)=0$, then $-\Delta$ has
discrete spectrum on $M$.
* $(2)$
Although Theorem 1 is suited for each of the examples $(i),\ldots,(v)$ in the
Introduction, as well as to deal with solutions of general Plateau problems,
it is still unapplicable for the original example of Nadirashvili in [25]. The
reason is that it seems hard to deduce the behaviour of the limit set from the
original construction. Could it be possible that, for some choice of the
parameters in Nadirashvili’s construction, the spectrum of the resulting
minimal surface is discrete?
* $(3)$
As we have seen in the Introduction, infinite sheet coverings of complete
bounded minimal surfaces always have non-empty essential spectrum. On the
other hand, Example 3 establishes the existence of incomplete minimal surfaces
with $\sigma_{\mathrm{ess}}(-\Delta)\neq\emptyset$ and whose immersion map
$\varphi$ cannot factorize via some Riemannian covering. One could naturally
ask the following
###### Question.
Is it possible to find a complete, bounded minimal surface
$\varphi:M\rightarrow\mathbb{R}^{3}$ with non-empty essential spectrum and
such that $\varphi$ cannot factorize via a Riemannian covering map?
Acknowledgements: this paper has been completed while the third author was
enjoying the hospitality of the Universidade Federal do Ceará-Brazil. The
authors are partially supported by PRONEX/FUNCAP/CNPq. The third author is
indebted to Andrea Carlo Mennucci and Carlo Mantegazza for useful e-mail
discussions about the regularity of distance functions. He also wants to thank
Matteo Novaga for his kind willingness.
## References
* [1] A. Alarcón, L. Ferrer and F. Martin, Density theorems for complete minimal surfaces in $\mathbb{R}^{3}$. Geom. Funct. Anal. 18 (2008), no. 1, 1–49.
* [2] P. Andrade, _A wild minimal plane in $\mathbb{R}^{3}$_. Proc. Amer. Math. Soc. 128 (2000), no. 5, 1451–1457.
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|
arxiv-papers
| 2012-11-26T18:42:26 |
2024-09-04T02:49:38.454604
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Gregorio Pacelli Bessa, Luquesio P. Jorge, Luciano Mari",
"submitter": "Luciano Mari",
"url": "https://arxiv.org/abs/1211.6059"
}
|
1211.6074
|
# Singular Quadrature Rules and Fast Convolutions for Fourier Spectral Methods
Jae-Seok Huh Computational Mathematics Group, Computer Science and
Mathematics Division, Oak Ridge National Laboratory, Oak Ridge, TN 37831.
[email protected] and George Fann Computational Mathematics Group, Computer
Science and Mathematics Division, Oak Ridge National Laboratory, Oak Ridge, TN
37831. [email protected]
(Date: June 28, 2012)
###### Abstract.
We present a generic scheme to construct corrected trapezoidal rules with
spectral accuracy for integral operators with weakly singular kernels in
arbitrary dimensions. We assume that the kernel factorization of the form,
$K=\alpha\,\phi+\widetilde{K}$ with smooth $\alpha$ and $\widetilde{K}$, is
available so that the operations on the smooth factors can be performed
accurately on the basis of standard Fourier spectral methods. To achieve high
precision results, our approach utilizes the exact evaluation of the Fourier
coefficients of the radial singularity $\phi$, which can be obtained in
arbitrary dimensions by the singularity isolation/truncation described in this
article. We provide a complete set of formulae for singularities of the type:
$\log(r)$ and $r^{-\nu}$. Convergence analysis shows that the constructed
quadrature rules exhibit almost identical rate of convergence to the
trapezoidal rule applied for non-singular integrands. Especially, for smooth
data, the corrected trapezoidal rules converge super-algebraically.
###### Key words and phrases:
corrected trapezoidal rule, singular kernel, fast convolution
###### 2010 Mathematics Subject Classification:
Primary 65R20, 65T99
## 1\. Introduction
We consider the problem of evaluating the integral,
(1.1) $(If)(x)=\int_{D}K(x,y)\,f(y)\,dy,$
where the kernel $K$ may have a point singularity at $x=y$. We assume that $f$
is compactly supported on $D$ and can be represented satisfactorily by a
Fourier series. When the kernel is also smooth (or sufficiently regular for
the purpose), the usual trapezoidal rule with the integrand $(K\cdot f)$
evaluated on a uniform grid is a classical rule of thumb for the construction
of a spectral scheme involving the integral operator (1.1). Additionally, many
kernels of interest are given by $K(x,y)=K(x-y)$. Then, the Fourier
representation of the kernel and the data enables the fast evaluation of the
multiplier operator $I$ via the FFT. However, the singularities in most
kernels of interest render the trapezoidal rule unapplicable, and we need a
corrected one.
We assume that the kernel can be factored to the form,
(1.2) $K(x,y)=\alpha(x,y)\,\phi(r)+\widetilde{K}(x,y)$
where $\alpha$ and $\widetilde{K}$ are smooth functions and $r=\|{y-x}\|$ is
the Euclidean distance in ${\mathbb{R}}^{m}$. We assume that the singularity
is known and carried entirely by the radial function $\phi$. For the
derivation of the quadrature weights, $\alpha$ and $\widetilde{K}$ may depend
explicitly on the target point $x$, in which case the weights should be
constructed for each $x$. The main subject of this paper is the construction
of corrected trapezoidal rules which exhibit the same spectral accuracy as the
usual trapezoidal rule applied for smooth kernels.
The constructed quadrature rules can be written as
(1.3) $(If)(x)\sim h\sum_{y_{\ell}\in
B_{R}}w_{\ell}f(y_{\ell})+h\sum_{y_{\ell}\neq x}K(x,y_{\ell})f(y_{\ell}),$
where the contribution of the singularity is reproduced by the correction
weights $w_{\ell}$ in a neighborhood $B_{R}$ of $x$. The presented scheme can
also be viewed as a regularization of the kernel such that the replacement of
$\phi$ with the regularized $\widetilde{\phi}$ does not impair the accuracy of
the regular trapezoidal rule with a smooth kernel at the given sampling
frequency. In what follows are a few comments regarding the key features of
our method and the relationship to other quadrature rules and applications
available in the literature.
1. (a)
One of the main advantages of our approach is the high precision of the
resulting quadrature rules. The construction does not involve any other non-
trivial quadrature rule or a solve of a linear system both of which typically
limit the achievable order of accuracy. In our approach, the FFT is the only
numerical algorithm used and the accumulation of the rounding error of the FFT
is only $O(\epsilon_{mach}\log N)$. To achieve the high precision, the Fourier
coefficients of the truncated singularity $\phi$ are evaluated up to the
machine precision. One of the major contributions of this paper is the
complete recipe we present for the logarithmic and the power-law
singularities.
2. (b)
The second advantage is its applicability for any dimension. The key functions
required for the construction can obtained by recurrence relations presented
in main body of the paper. Except for the $r^{-\nu}$ singularities with a non-
integer $\nu$ (which is less significant in applications than the integer
cases, we think), the construction does not require an implementation of a new
special function.
3. (c)
We present the factored forms of the Helmholtz kernels in arbitrary dimensions
so that the corrected quadrature rules can be readily obtained from them. The
Helmholtz kernels are of great importance in applications. We present
numerical examples which demonstrate the advantage of the presented quadrature
rules in the construction of high order integral operators especially for
oscillating kernels with high wavenumbers.
4. (d)
We do not pursue in this study the end-point corrected trapezoidal rules for
non-periodic data, which can be found in [2, 14, 19]. In principle, an end-
point correction is equivalent to an accurate estimation of derivatives at and
near the end-points, which can be done by an over-sampling or by a certain
type of data extension. For the latter, a simple extrapolation results in a
terrible oscillation, which limits the achievable order of accuracy. Hence,
most successful _high-order_ methods involves a solve of a least-squares
problem. An accurate and stable extension is possible also for the
trigonometric basis functions (cf. [13]). We can represent a smooth non-
periodic function by a smooth compactly supported interior and a near boundary
part. In this paper, we focus on the former case and the correction for the
near boundary part will be considered in a separate study. The quadrature
rules in [2, 14, 19] are for one-dimensional singular integrals and the
multidimensional extension is not so straightforward since a weak singularity
in a higher dimension can not always be represented by a product of one-
dimensional weak singularities. Multi-dimensional singular quadrature rules of
relatively low orders can be found in [21], which are applied for non-
oscillatory kernels.
5. (e)
When $f$ is periodic and the domain of the convolution (1.1) is unbounded, the
kernel can be factored as $K=\varphi\,K+(1-\varphi)K$ by a smooth cut-off
function $\varphi$. The convolution can be recast to
(1.4) $If=\int K\cdot(\varphi f)+\int\big{(}(1-\varphi)K\big{)}\cdot f,$
where the first integral involves the singular kernel and the compactified
$(\varphi f)$, and the second integral the convolution of the periodic $f$
with the regularized kernel $(1-\varphi)K$ where the trapezoidal rule can be
applied without any correction. The Ewald summation (cf. [10]) focuses on the
design of the smooth kernel so that the Fourier coefficients of the
regularized kernel decays rapidly. Classical applications of the Ewald
summation are for point sources, hence, the first integral is merely a
summation. Our focus is more on the accurate evaluation of the first integral
for continuous $f$.
6. (f)
The data $f$ or their Fourier coefficients do not necessarily have to be given
on a uniform grid as long as equivalently fast and accurate algorithms are
provided. For the extension of the quadrature rules to non-uniformly sampled
data, one can refer to [11, 12]. For sparse data as well as rapidly decreasing
kernels, the use of the partial FFT (cf. [3]) can also be utilized.
7. (g)
The kernel factorization of the form (1.2) can be found in numerous articles.
Probably, the most well-known examples are the Nyström methods for boundary
integral equations such as [9, §3.5]. In [7, 22], combined with partition-of-
unity boundary decomposition, Nyström-type schemes are presented for boundary
integral equations in ${\mathbb{R}}^{3}$. Spectral schemes in [7, 22] commonly
remove the $1/r$-singularity by a polar change of variables, and evaluate each
of the resulting one-dimensional integrals by the trapezoidal rule using
resampled data without building quadrature weights explicitly.
8. (h)
In order to achieve a proper spectral convergence, the kernel should be
correctly factored so that $\alpha$ and $\widetilde{K}$ are both smooth.
Regarding this issue, we should keep the fact in mind that $r^{2}$ is smooth
but $r$ is not even in $C^{1}$. The following factorization might look
attractive since the singularity is isolated independently of the wavenumber.
(1.5) $\frac{e^{ikr}}{4\pi r}=\frac{1}{4\pi r}+\left(\frac{\cos(kr)-1}{4\pi
r}+i\frac{\sin(kr)}{4\pi r}\right).$
However, the above form should be avoided since $(\cos(kr)-1)/r$ is not
smooth. A correct form of factorization is
(1.6) $\frac{e^{ikr}}{4\pi r}=\frac{\cos(kr)}{4\pi r}+i\frac{\sin(kr)}{4\pi
r}$
where $\alpha(r)=\cos(kr)/(4\pi)$, $\phi(r)=1/r$, and
$\widetilde{K}(r)=i\sin(kr)/(4\pi r)$. Readers will soon notice that all the
well-behaved functions appearing in this paper have power series expansions
containing only even powers of $r$.
This paper is organized as follows: in §2, we begin with the detailed
derivation of the quadrature rules, which will be followed by a complete set
of formulae for the evaluation of the Fourier coefficients of the radially-
truncated singularities of the type: $\log(r)$ and $r^{-\nu}$. Then, we
present the factored forms for the Helmholtz kernels in arbitrary dimensions.
In §3, we discuss the order of accuracy of the quadrature rules related to the
regularity of the data. Finally, the results of numerical experiments will be
presented in §4.
## 2\. Construction of Quadrature Weights
### 2.1. Notations, grids, and DFTs
Let $U=[-1,1]^{m}$ be the computational domain in ${\mathbb{R}}^{m}$. We
consider a uniform grid on ${\mathbb{R}}^{m}$ with the spacing $1/N_{j}$ in
the $j$th dimension. Thus, for any multi-index
$\ell=(\ell_{1},\ldots,\ell_{m})\in\mathbb{Z}^{m}$, the corresponding grid
point $y_{\ell}=(y_{\ell_{1}},\ldots,y_{\ell_{m}})$ is given by
$y_{\ell_{j}}=\ell_{j}/N_{j}$. We assume that the physical domain is given as
an affine image of $U$ (without translation for the sake of simplicity),that
is, a parallelotope in ${\mathbb{R}}^{m}$. We denote by $\chi$ this non-
degenerate linear mapping $U\mapsto\chi U$, then the related metric tensor is
constant and is given by $\chi^{T}\chi$. We use the symbol $r$ for the
Euclidean distance in the physical domain. That is,
(2.1) $r(x,y)\equiv\|{\chi(y-x)}\|.$
With the target $x\in U$ fixed and $r$ viewed as a function of $y\in U$, we
view (1.1) as an integral on $U$ with the Jacobian determinant
$|\chi|\equiv\sqrt{\det(\chi^{T}\chi)}$ multiplied afterward. The function $r$
is not smooth at the point of singularity due to the square root but $r^{2}$
is a smooth quadratic polynomial. Note also that the singularity $\phi$ is not
radial in $U$ (but is radial in $\chi U$).
Let $\overline{N}\equiv\prod_{j=1}^{m}N_{j}$ and define
$\mathbb{I}_{N}\subset\mathbb{Z}^{m}$ by
(2.2)
$\mathbb{I}_{N}\equiv\big{\\{}(\ell_{1},\ldots,\ell_{m})\,|\,-N_{j}\leq\ell_{j}\leq
N_{j}-1\big{\\}}.$
Let $F$ be a periodic function on $U$. Application of the trapezoidal rule to
the (periodic) Fourier transform
(2.3) $\widehat{F}(k)=\frac{1}{2^{m}}\int_{{\mathbb{R}}^{m}}F(y)\,e^{-i\pi
k\cdot y}\,dy$
results in the formula for the DFT
(2.4)
$\widehat{F}_{k}=\frac{1}{2^{m}\overline{N}}\sum_{\ell\in\mathbb{I}_{N}}F(y_{\ell})\,e^{-i\pi
k\cdot y_{\ell}}.$
The truncation applied to the inverse Fourier transform (of delta functions on
$\mathbb{Z}^{m}$)
(2.5) $F(y)=\sum_{k\in\mathbb{Z}^{m}}\widehat{F}(k)\,e^{i\pi k\cdot y}$
defines the inverse DFT by
(2.6) $F_{\ell}=\sum_{k\in\mathbb{I}_{N}}\widehat{F}_{k}\,e^{i\pi k\cdot
y_{\ell}}.$
Note that the Fourier coefficients $\widehat{F}_{k}$ approximated by the DFT
contains error due to the aliasing. Suppose we are given exact samples, i.e.
$F_{\ell}=F(y_{\ell})$. The interpolation $\widetilde{F}$ of the samples
defined by
(2.7) $\widetilde{F}(y)=\sum_{k\in\mathbb{I}_{N}}\widehat{F}_{k}\,e^{i\pi
k\cdot y}\quad\text{where $\widehat{F}_{k}$ is the DFT of $F_{\ell}$}$
involves two sources of error: (1) the aliasing contained in $\widehat{F}_{k}$
and (2) the truncation error of the inverse DFT. The decay characteristics of
the exact Fourier coefficients $\widehat{F}(k)$ takes a central role in the
estimation of the error.
### 2.2. The support of the data
We assume that the data (or source) $f$ in (1.1) viewed as a function on the
computational domain is supported on $[0,1]^{m}$. Then, $f$ can be extended on
$U$ by padding zeroes. The rationale behind this is quite obvious; since we
take $x+U$ as the support of the kernel $K(x,\cdot)$ (or equivalently, the
domain of integral) for the target $x$, if $f$ is not extended, the
convolution will include the effect of the _fictitious_ portion of the
periodized source. Similarly, if $x$ is not in $[0,1]^{m}$, the result
$(If)(x)$ will include the effect of the fictitious source, hence, the portion
on $U\setminus[0,1]^{m}$ of $(If)(x)$ obtained on $U$ should be discarded.
For $x\not\in[0,1]^{m}$, the integrand becomes smooth, hence, the usual
trapezoidal rule serves our purpose well without any correction. The resulting
summation can be accelerated by any fast multipole (or an equivalent) method
designed for the kernel. In this paper, we focus ourselves on the near-field
solution for $x\in[0,1]^{m}$.
On this setting, the translation $f(y-x)$ by $x\in[0,1]^{m}$ results in the
translated source vanishing on the boundary of $U$ and satisfying all the
assumptions. Therefore, in this section, without loss of generality, we assume
that the target point $x$ is at the origin. Every function is viewed as a
function of the source point $y$ only and the symbol $x$ will be omitted.
Thus, we recast (1.1) to the integral,
(2.8) $I(f)=|\chi|\int_{U}K(y)\,f(y)\,dy.$
### 2.3. Localization of the singularity
Let $r(y)\equiv\|{\chi y}\|$. With a slight abuse of notation, we denote by
$B_{R}$ an ellipsoid in $U$ given by $B_{R}\equiv\\{\,r(y)\leq R\,\\}$ for
some $R>0$, where $R$ is chosen such that the image $\chi\,B_{R}$ (a ball of
radius $R$ in the physical domain) is contained in $\chi\,U$. For the
accuracy, the best choice of $R$ is $\max_{y\in\partial U}r(y)$ so that
$B_{R}$ contains as many grid points as possible.
Let $\varphi$ be any smooth even function on ${\mathbb{R}}$ such that (1)
$\varphi(0)=1$, (2) $\varphi=0$ on ${\mathbb{R}}\setminus(-R,R)$, and (3)
$\varphi^{c}=(1-\varphi)$ vanishes smoothly at the origin. Then,
$(\phi(r)\,\varphi^{c}(r))$ is also a smooth function and vanishes at the
origin. For all the numerical experiments in this paper, we utilized
$\varphi(r)=\varphi_{1}(r/R)$ where $\varphi_{1}$ is the sigmoidal function,
(2.9) $\varphi_{1}(t)=\begin{cases}e^{-e^{-2/|t|}/(1-|t|)^{2}}&|t|<1\\\
0&|t|\geq 1\end{cases}.$
Then, we isolate the singularity by using the identity
(2.10) $\phi(r)=\phi(r)\,\varphi(r)+\phi(r)\,\varphi^{c}(r)$
and rewrite (2.8) as
(2.11) $I(f)=|\chi|\int_{B_{R}}\phi(r)\,F(y)\,dy+|\chi|\int_{U}G(y)\,dy$
where $F$ and $G$ are regular periodic functions given by
(2.12) $\displaystyle F(y)$ $\displaystyle=\varphi(r)\,\alpha(y)\,f(y)$ (2.13)
$\displaystyle G(y)$
$\displaystyle=\varphi^{c}(r)\,\phi(r)\,\alpha(y)\,f(y)+\widetilde{K}(y)\,f(y).$
The second integral without the singularity can be treated well by the usual
trapezoidal rule; that is,
(2.14)
$\int_{U}G(y)\,dy\sim\frac{|\chi|}{\overline{N}}\sum_{\ell\in\mathbb{I}_{N}}\left(\varphi^{c}(r_{\ell})\,\phi(r_{\ell})\,\alpha(y_{\ell})+\widetilde{K}(y_{\ell})\right)f(y_{\ell}).$
For the first integral with the singularity, we utilize the interpolation
(2.7) to obtain
(2.15) $\displaystyle|\chi|\int_{B_{R}}\phi(r)\,F(y)\,dy$
$\displaystyle\sim|\chi|\sum_{k\in\mathbb{I}_{N}}\widehat{F}_{k}\int_{B_{R}}\phi(r)\,e^{i\pi
k\cdot y}\,dy$ (2.16)
$\displaystyle=\frac{|\chi|}{\overline{N}}\sum_{\ell\in\mathbb{I}_{N}}F(y_{\ell})\sum_{k\in\mathbb{I}_{N}}\left(\frac{1}{2^{m}}\int_{B_{R}}\phi(r)\,e^{i\pi
k\cdot y}\,dy\right)$ (2.17)
$\displaystyle\equiv\frac{|\chi|}{\overline{N}}\sum_{\ell\in\mathbb{I}_{N}}F(y_{\ell})\sum_{k\in\mathbb{I}_{N}}\widehat{\phi}(k)$
(2.18)
$\displaystyle\equiv\frac{|\chi|}{\overline{N}}\sum_{\ell\in\mathbb{I}_{N}}F(y_{\ell})\,\widetilde{\phi}_{\ell},$
where we defined
(2.19) $\widehat{\phi}(k)\equiv\frac{1}{2^{m}}\int_{B_{R}}\phi(r)\,e^{-i\pi
k\cdot
y}\,dy\quad\text{and}\quad\widetilde{\phi}_{\ell}\equiv\sum_{k\in\mathbb{I}_{N}}\widehat{\phi}(k).$
We can change the sign in the exponential function arbitrarily since
$r(-y)=r(y)$. Notice that $\widehat{\phi}(k)$ are the _exact_ Fourier
coefficients of $\phi(r)$ which is truncated to zero on the exterior of
$B_{R}$ and viewed as a periodic function on $U$. Or equivalently, we can
consider $\widehat{\phi}$ as the (non-periodic) Fourier transform (scaled by
$2^{-m}$) of $\phi$ truncated on ${\mathbb{R}}^{m}\setminus B_{R}$. By
definition, $\widetilde{\phi}_{\ell}$ are the inverse DFT of the finite
samples $\widehat{\phi}(k)$ in the frequency domain. Consider the
interpolation
(2.20)
$\widetilde{\phi}(y)=\sum_{k\in\mathbb{I}_{N}}\widehat{\phi}(k)\,e^{i\pi
k\cdot y}.$
Then, $\widetilde{\phi}_{\ell}=\widetilde{\phi}(y_{\ell})$ and (2.18) is
simply the trapezoidal rule applied to the product of the two functions,
$\widetilde{\phi}$ and $F$. Thus, the procedure is equivalent to the
regularization of the singular $\phi$ such that the regularized kernel
$\widetilde{\phi}$ results in the exact integral by the trapezoidal rule if
$F$ can be exactly represented on the given grid.
Care should be taken not to confuse $\widehat{\phi}$ with the Fourier
transform of the original _non-truncated_ $\phi$. To be more precise, we
should use a notation like $\widehat{\phi}_{R}$, but we choose the notational
simplicity. Typically, $\phi$ is a slowly decaying function with the point
singularity at the origin, hence, without the truncation, its Fourier
transform possesses the same nature in the frequency domain. By truncating in
the space, we regularize the Fourier transform to a smooth (but still slowly
decaying) $\widehat{\phi}_{R}$. By truncating in the frequency domain, that
is, by sampling only up to the given sampling frequency, we obtain
$\widetilde{\phi}$ regularized in the space.
As one can notice from the above derivation, our construction requires the
exact evaluation of $\widehat{\phi}$. In §2.5, we present a detailed
discussion on the nature of $\widehat{\phi}$ as well as the explicit formulae
for the logarithmic and the power-law singularities.
### 2.4. The corrected trapezoidal rule
Merging (2.14) and (2.18), the quadrature rule can be written as
(2.21)
$I(f)\sim\frac{|\chi|}{\overline{N}}\sum_{\ell\in\mathbb{I}_{N}}\left(\alpha(y_{\ell})\left(\varphi(r_{\ell})\,\widetilde{\phi}_{\ell}+\varphi^{c}(r_{\ell})\,\phi(r_{\ell})\right)+\widetilde{K}(y_{\ell})\right)f(y_{\ell}).$
For $\ell=0$, since $\varphi(0)=1$ and $\varphi^{c}(0)=0$, the weight becomes
(2.22) $\alpha(0)\,\widetilde{\phi}_{0}+\widetilde{K}(0).$
For $\ell\neq 0$, we can rewrite
(2.23)
$\varphi(r_{\ell})\,\widetilde{\phi}_{\ell}+\varphi^{c}(r_{\ell})\,\phi(r_{\ell})=\left(\widetilde{\phi}_{\ell}-\phi(r_{\ell})\right)\varphi(r_{\ell})+\phi(r_{\ell}),$
Thus, we present the quadrature rule by the sum of the trapezoidal rule and
the correction rule; that is,
(2.24)
$I(f)\sim\frac{|\chi|}{\overline{N}}\sum_{\begin{subarray}{c}\ell\in\mathbb{I}_{N}\\\
\ell\neq
0\end{subarray}}K(y_{\ell})\,f(y_{\ell})+\frac{|\chi|}{\overline{N}}\sum_{\begin{subarray}{c}\ell\in\mathbb{I}_{N}\\\
r_{\ell}<R\end{subarray}}w_{\ell}\,f(y_{\ell})$
where the correction weights $w_{\ell}$ are given by
(2.25)
$w_{\ell}=\begin{cases}\alpha(0)\,\widetilde{\phi}_{0}+\widetilde{K}(0)&\ell=0\\\
\alpha(y_{\ell})\left(\widetilde{\phi}_{\ell}-\phi(r_{\ell})\right)\varphi(r_{\ell})&\ell\neq
0\end{cases}.$
The separated representation (2.24,2.25) is the final form of the quadrature
rule we present in this paper. The quadrature weights (2.25) is quite self-
explanatory; at the point of singularity, $\phi$ has been represented by the
equivalent finite weight $\widetilde{\phi}_{0}$ which accompanies the
balancing neighbors
$(\widetilde{\phi}_{\ell}-\phi(r_{\ell}))\varphi(r_{\ell})$ to achieve the
desired spectral accuracy. The advantage of using the separated form
(2.24,2.25) over the primitive one (2.21) is obvious; oftentimes, the smooth
remainder term $\widetilde{K}$ has a quite complicated form. Since the
separated form requires only the limiting value of $\widetilde{K}$ at the
origin, the implementation can be simpler and more readable.
There must be a few careful readers concerning the numerical soundness of the
expression $(\widetilde{\phi}_{\ell}-\phi(r_{\ell}))$. With finite precision
arithmetics, it sometimes comes to a catastrophic end to take the difference
of two potentially large values; the cancellation error can corrupt the
result. However, on uniform grids where grid points are not clustered, the
issue is not so significant. Moreover, for spectral methods like ours, when
applied to quite smooth data, the solution often converges to the desired
precision before the grid spacing becomes small enough to bring such an issue
to the surface. In all the numerical experiments we conducted including all
the examples in this paper, we utilized the separated representation but have
never experience a trouble. Actually, the classical Nyström methods frequently
use the expression $\widetilde{K}=K-\alpha\,\phi$ to avoid the explicit
evaluation of a complicated $\widetilde{K}$ (see, for example, [9, §3.5]). For
rare cases when the issue becomes significant, one can utilize the original
form (2.21) for grid points close to the singularity.
We can also bring the convolution (2.24) to the frequency domain. In this
sense, our construction is equivalent to the representation of the Fourier
transform of $K$ truncated on ${\mathbb{R}}^{m}\setminus U$ by
(2.26)
$\widehat{K}=\widehat{\alpha}*\widehat{\phi}*\widehat{\varphi}+\left(\widehat{\alpha}*\widehat{\phi}*\widehat{\varphi^{c}}+\widehat{\widetilde{K}}\right)$
where the terms in the parentheses decays rapidly. The key idea is that the
application of the radial cut-off function $\varphi$ enables us to truncate
$\phi$ outside of $B_{R}$ and we can evaluate the Fourier transform (which is
now the one-dimensional Hankel transform) of the truncated $\phi$ exactly.
Numerically, we evaluate $\widehat{K}$ by the DFT of
$\\{w_{0},w_{\ell}+K(y_{\ell})\\}$, where $w_{\ell}$ are obtained by the
inverse DFT of $\widehat{\phi}(k)$.
Now, the true meaning of the _grid_ we used so far has become clearer. It is
not necessary for the data $f$ to be given on that (or on any) grid. Regarding
$f$, the only information we need is the required minimum sampling frequency.
The grid we have used so far is in principle to perform the DFT of
$\widetilde{K}$ and the convolutions (by multiplications in the spatial
domain) in (2.26), hence, we call it more specifically the construction grid.
1. (A)
A recommended sampling frequency for the construction (which is greater or
equal to the given sampling frequency of the data) is such that the
interpolation errors of the smooth functions $\alpha$, $\varphi$, and
$\widetilde{K}$ are comparable to that of the data. Otherwise, the quadrature
weights will still work but the error in the weights will be the dominating
factor (cf. §4.2 and §4.2).
2. (B)
For the inverse DFT to obtain $\widetilde{\phi}$, we do not need to use the
entire construction grid. Only a subset of the grid containing $B_{R}$ is
sufficient. The use of the subset can be very useful if the kernel exhibits
exponential decay like the Helmholtz kernel with a complex wavenumber $k$ with
$\operatorname{Im}(k)>1$. Such an exponentially decaying kernel requires a
very high sampling frequency for the construction (cf. §4.2). However, due to
the rapid decay, the kernel outside of a certain small $B_{R}$ is practically
zero. Hence, we can compute $(w_{\ell}+\widetilde{K}(y_{\ell}))$ only within
the small subset of the conceptually huge construction grid. The DFT on the
entire grid can be performed by padding zeros and the unnecessary high
frequency terms can be discarded, both of which can be done in a single
efficient procedure without using huge temporary memory by (so called) the
partial FFT.
### 2.5. The Fourier transform $\bm{\widehat{\phi}}$
The construction of the correction weights requires the exact evaluation of
the Fourier transform $\widehat{\phi}$ of the truncated singularity. We begin
with the definition of $\widehat{\phi}$,
(2.27) $\widehat{\phi}(k)=\frac{1}{2^{m}}\int_{B_{R}}\phi(r)\,e^{-i\pi k\cdot
y}\,dy.$
Note that $\phi$ is not a radial function in the computational domain $U$ and
$B_{R}$ is not a ball in $U$. First, we perform the change of variables
$\eta=R^{-1}\chi y$ back to a scaled physical domain. Utilizing $k\cdot
y=k^{\prime}\cdot\eta$ with $k^{\prime}=R\chi^{-T}k$, we obtain
(2.28) $\widehat{\phi}(k)=\frac{R^{m}}{2^{m}|\chi|}\int_{\|{\eta}\|\leq
1}\phi\left(R\|{\eta}\|\right)\,e^{-i\pi k^{\prime}\cdot\eta}\,d\eta.$
The function $\phi(R\|{\cdot}\|)$ is radial and supported on the unit ball.
Hence, the integral on the right side is a radial function of $k^{\prime}$ and
is given by the following Hankel transform on one-dimension.
(2.29)
$\widehat{\phi}(k)=\frac{R^{m}}{2^{m}|\chi|}\frac{(2\pi)^{m/2}}{\rho_{k}^{m}}\int_{0}^{\rho_{k}}\phi\left(\frac{R}{\rho_{k}}t\right)\,t^{m/2}\,J_{(m-2)/2}(t)\,dt$
where $J_{\mu}$ is the Bessel function of the first kind and
(2.30) $\rho_{k}=\pi R\|{\chi^{-T}k}\|.$
Then, we can rewrite (2.29) as
(2.31)
$\widehat{\phi}(\rho)=\frac{\sqrt{\pi}^{m}}{2^{m}|\chi|\,\Gamma(m/2+1)}\frac{R^{m}}{\rho^{m}}\int_{0}^{\rho}\phi\left(\frac{R}{\rho}t\right)\,m\,t^{m-1}\,A_{m}(t)\,dt,$
where the Bessel function of degree $(m-2)/2$ has been replaced with a better
behaved function $A_{m}$ defined by
(2.32) $A_{m}(t)\equiv\Gamma(m/2)\frac{J_{(m-2)/2}(t)}{(t/2)^{(m-2)/2}}.$
Then, $A_{m}(0)=1$ and $|A_{m}|\leq 1$. The series representation of $A_{m}$
can be obtained from that of $J_{(m-2)/2}$.
(2.33)
$A_{m}(t)=\sum_{\ell=0}^{\infty}\frac{(-1)^{\ell}\Gamma(m/2)}{\ell!\,\Gamma(\ell+m/2)}\left(\frac{t}{2}\right)^{2\ell}.$
Thus, $A_{m}$ is an entire function on ${\mathbb{C}}$ and is even on
${\mathbb{R}}$. The asymptotic behavior of $A_{m}$, which can be obtained from
that of $J_{(m-2)/2}$, governs the decay characteristics of $\widehat{\phi}$.
For $t\gg m^{2}$,
(2.34)
$A_{m}(t)\sim\frac{\Gamma(m/2)}{\sqrt{\pi}}\frac{\cos(t-(m-1)\pi/4)}{(t/2)^{(m-1)/2}}.$
Note that $A_{m}$ includes a Bessel function with an integer index for an even
$m$ and with a half-integer index for an odd $m$, the latter of which can be
expressed by a finite series of trigonometric functions. Although the series
representation may look complicated, $A_{m}$ actually consists of familiar
functions. For the first 4 indices,
(2.35) $\displaystyle A_{1}(t)$ $\displaystyle=\cos(t)$ (2.36) $\displaystyle
A_{2}(t)$ $\displaystyle=J_{0}(t)$ (2.37) $\displaystyle A_{3}(t)$
$\displaystyle=\sin(t)/t\equiv\operatorname{sinc}(t)$ (2.38) $\displaystyle
A_{4}(t)$ $\displaystyle=2J_{1}(t)/t.$
For indices greater than 4, closed-forms can be obtained by utilizing the
three-term recurrence relation,
(2.39) $A_{m+4}(t)=\frac{m(m+2)}{t^{2}}\left(A_{m+2}(t)-A_{m}(t)\right),$
which can be easily verified from the series form. Thus, $A_{m}$ involves only
$J_{0}$ and $J_{1}$ for an even $m$, and only cosine and sine for an odd $m$,
which does not introduce any implementation issue.
Another useful identity is
(2.40) $\left(t^{m}A_{m+2}(t)\right)^{\prime}=m\,t^{m-1}A_{m}(t),$
which results in another expression for (2.31); suppose $\phi$ is $C^{1}$ on
$(0,R\,]$ and $(\phi(t)\,t^{m-\epsilon})\rightarrow 0$ as $t\rightarrow 0$ for
some $\epsilon>0$. Applying the integration-by-part, (2.31) becomes
(2.41)
$\widehat{\phi}(\rho)=\frac{\sqrt{\pi}^{m}R^{m}}{2^{m}|\chi|\,\Gamma(m/2+1)}\bigg{\\{}\phi(R)A_{m+2}(\rho)-\int_{0}^{1}\phi_{1}(R\,t)\,t^{m-1}\,A_{m+2}(\rho\,t)\,dt\bigg{\\}},$
where $\phi_{1}(t)=t\,\phi^{\prime}(t)$ and the change of variables
$t/\rho\mapsto t$ is applied for the second integral. Note that
$\phi_{1}(t)=1$ for $\phi(t)=\log(t)$ and $\phi_{1}(t)=-\nu\,\phi(t)$ for
$\phi(t)=t^{-\nu}$, which provides us with useful recurrence relations for the
integral term. The formula (2.41) also reveals the fact that the asymptotic
behavior of $\widehat{\phi}$ is determined by that of $A_{m+2}$ (not $A_{m}$),
which is summarized in the following lemma.
###### Lemma 2.1.
Let the grid parameters, $R$ and $\chi$, be fixed. Suppose $\phi$ is $C^{1}$
on $(0,R\,]$ and $(\phi(t)\,t^{m-\epsilon})\rightarrow 0$ as $t\rightarrow 0$
for some $\epsilon>0$. Then, for $k\in\mathbb{Z}^{m}$,
(2.42) $|\widehat{\phi}(k)|=O\left(\|{k}\|^{-(m+1)/2}\right).$
###### Proof.
First, consider the integral in (2.41).
(2.43)
$\left|\int_{0}^{1}\phi_{1}(R\,t)\,t^{m-1}\,A_{m+2}(\rho\,t)\,dt\right|\leq\frac{\|{\phi_{1}(t)\,t^{m}}\|_{\infty}}{R^{m}}\int_{0}^{1}\frac{\left|A_{m+2}(\rho\,t)\right|}{t^{1-\epsilon}}\,dt$
where we denote by $\|{\cdot}\|_{\infty}$ the $L^{\infty}$-norm on
$[\,0,R\,]$. From (2.34), there is a constant $C>0$ such that
$|A_{m+1}(\rho\,t)|\leq C(\rho\,t)^{-(m+1)/2}$. Define $a$ by
$C=(\rho\,a)^{(m+1)/2}$. Then, $a<1$ if $\rho>C^{2/(m+1)}$. Since
$|A_{m+2}|\leq 1$,
(2.44)
$\begin{split}\int_{0}^{1}\frac{\left|A_{m+2}(\rho\,t)\right|}{t^{1-\epsilon}}\,dt&\leq\int_{0}^{a}t^{\epsilon-1}\,dt+C\,\rho^{-(m+1)/2}\int_{a}^{1}t^{\epsilon-(m+1)/2-1}\,dt\\\
&\leq\delta\frac{a^{\epsilon}}{\epsilon}+\frac{C\,\rho^{-(m+1)/2}}{\epsilon-(m+1)/2}\left(1-a^{\epsilon-(m+1)/2}\right)\\\
&=\frac{C\,\rho^{-(m+1)/2}}{\epsilon-(m+1)/2}+a^{\epsilon}\left(\frac{\delta}{\epsilon}-\frac{1}{\epsilon-(m+1)/2}\right),\end{split}$
where $\delta$ is an arbitrary constant $\geq 1$. When $\epsilon\geq(m+1)/2$,
$a^{\epsilon}=O\left(\rho^{-(m+1)/2}\right)$ since
$a=O\left(\rho^{-1}\right)$. When $\epsilon<(m+1)/2$, we can choose
$\delta=\epsilon/(\epsilon-(m+1)/2)>1$ so that the second term vanishes. Thus,
$\int_{0}^{1}t^{\epsilon-1}|A_{m+2}(\rho\,t)|\,dt=O\left(\rho^{-(m+1)/2}\right)$.
Combined (2.34) for the first term in (2.41), the above result shows that
$|\widehat{\phi}(\rho)|=O\left(\rho^{-(m+1)/2}\right)$. Since $\rho_{k}\leq\pi
R\|{\chi^{-T}}\|\|{k}\|$ where $\|{\chi^{-T}}\|$ is the operator norm of
$\chi^{-T}$, the same relation holds for $\|{k}\|$ also. ∎
In what follows, we present the formulae of $\widehat{\phi}$ for
$\phi(r)=\log(r)$ and for $\phi(r)=r^{-\nu}$ with $\nu<m$, which enables us to
evaluate them up to the machine precision. We begin with the logarithmic
singularity.
#### 2.5.1. $\bm{\phi(r)=\log(r)}$
From the integration-by-parts formula (2.41), we obtain
(2.45)
$\widehat{\phi}^{L}_{m}(\rho)=\frac{\sqrt{\pi}^{m}R^{m}}{2^{m}|\chi|\Gamma(m/2+1)}\bigg{\\{}\log(R)A_{m+2}(\rho)-L_{m}(\rho)\bigg{\\}}$
where
(2.46) $L_{m}(\rho)\equiv\int_{0}^{1}t^{m-1}\,A_{m+2}(\rho\,t)\,dt.$
For $m=1$ and 2,
(2.47) $\displaystyle L_{1}(\rho)$
$\displaystyle=\int_{0}^{1}\frac{\sin(\rho\,t)}{\rho\,t}\,dt=\frac{\operatorname{Si}(\rho)}{\rho}$
(2.48) $\displaystyle L_{2}(\rho)$
$\displaystyle=\int_{0}^{1}\frac{2J_{1}(\rho\,t)}{\rho}\,dt=\frac{2(1-J_{0}(\rho))}{\rho^{2}}.$
For $m\geq 3$, utilize the recurrence relations (2.39) to obtain
(2.49)
$L_{m+2}(\rho)=\frac{m(m+2)}{\rho^{2}}\int_{0}^{1}t^{m-1}\left(A_{m+2}(\rho\,t)-A_{m}(\rho\,t)\right)dt$
where the second term can be integrated by using (2.40), which results in the
recurrence relation for $L_{m}$,
(2.50)
$L_{m+2}(\rho)=\frac{(m+2)}{\rho^{2}}\bigg{\\{}m\,L_{m}(\rho)-A_{m+2}(\rho)\bigg{\\}}.$
With the formulae for $L_{1}$ and $L_{2}$, the above recurrence relation
enables us to construct an explicit formula of $\widehat{\phi}^{L}_{m}$ for
any $m$. In order to avoid possible cancellation error of the recurrence
formula for small $\rho$, we need to evaluate the series representation,
(2.51)
$L_{m}(\rho)=\sum_{\ell=0}^{\infty}\frac{(-1)^{\ell}\Gamma(m/2+1)}{\ell!\,\Gamma(\ell+m/2+1)(m+2\ell)}\bigg{(}\frac{\rho}{2}\bigg{)}^{2\ell},$
which is also more efficient for small $\rho$.
#### 2.5.2. $\bm{\phi(r)=r^{-\nu}\,\,(\nu<m)}$
First, notice that $r^{-\nu}$ with any $\nu<m$ can be factored as
$r^{-\nu}=r^{2n}\cdot r^{-\epsilon}$ with a non-negative integer $n$ and
$\epsilon\in[m-2,m)$. Since $r^{2}$ is smooth, the smooth factor $r^{2n}$ can
be included in $\alpha$. Interestingly, due to the factor $r^{2n}$ in
$\alpha$, the correction weight $w_{0}$ at origin vanishes for $r^{-\nu}$ with
$\nu<m-2$. For such _weaker_ singularities, the correction is made by the
correction weights (with relatively smaller magnitude) near the location of
the singularity.
Therefore, from now on, we presume $\nu\in[m-2,m)$. It is convenient for the
presentation to use a new notation, $\mu\equiv m-\nu$, where $\mu\in(0,2\,]$.
Then, (2.31) is written in terms of $\mu$ as,
(2.52)
$\widehat{\phi}^{(\mu)}_{m}(\rho)=\frac{\sqrt{\pi}^{m}R^{\mu}}{2^{m}|\chi|\,\Gamma(m/2+1)}\,M^{(\mu)}_{m}(\rho)$
where
(2.53)
$M^{(\mu)}_{m}(\rho)\equiv\int_{0}^{1}m\,t^{\mu-1}\,A_{m}(\rho\,t)\,dt.$
The series representation is given by
(2.54)
$M^{(\mu)}_{m}(\rho)=\sum_{\ell=0}^{\infty}\frac{(-1)^{\ell}\Gamma(m/2)\,m}{\ell!\,\Gamma(\ell+m/2)(\mu+2\ell)}\bigg{(}\frac{\rho}{2}\bigg{)}^{2\ell},$
which can be evaluated efficiently for small $\rho$ avoiding the cancellation
error of the recurrence relation.
We can derive a recurrence relation for $M_{m}$ like the one for $L_{m}$ in
the previous section. By integrating by parts and utilizing
$t\,A^{\prime}_{m+2}(t)=m(A_{m}(t)-A_{m+2}(t))$,
(2.55)
$\begin{split}\mu\,M^{(\mu)}_{m+2}(\rho)&=(m+2)\,A_{m+2}(\rho)-(m+2)\int_{0}^{1}\rho\,t^{\mu}\,A^{\prime}_{m+2}(\rho\,t)\,dt\\\
&=(m+2)\,A_{m+2}(\rho)-(m+2)\,M^{(\mu)}_{m}(\rho)+m\,M^{(\mu)}_{m+2}(\rho).\end{split}$
Hence,
(2.56)
$(m-\mu)\,M^{(\mu)}_{m+2}(\rho)=(m+2)\big{(}A_{m+2}(\rho)-\,M^{(\mu)}_{m}(\rho)\big{)}.$
And, the above equation implies that $\,M^{(m)}_{m}(\rho)=A_{m+2}(\rho)$.
Since $\mu\in(0,2\,]$, there are only two cases with $\mu=m$: $\mu=m=1$ and
$\mu=m=2$. Except those two cases, we can utilize the recurrence relation,
(2.57)
$M^{(\mu)}_{m+2}(\rho)=\frac{m+2}{m-\mu}\bigg{(}A_{m+2}(\rho)-\,M^{(\mu)}_{m}(\rho)\bigg{)}.$
Among $\mu\in(0,2\,]$, the two integer cases are of prime interest; (1)
$r^{2-m}$ ($\mu=2$) is the principal singularity of the Helmholtz kernel in
${\mathbb{R}}^{m}$ (in even dimensions with $m\geq 4$, the Helmholtz kernel
contains an additional logarithmic singularity, which can be treated by
$\widehat{\phi}^{L}_{m}$ in the previous section). Interestingly, this
(probably) most important class of singularities has the simplest description;
$M^{(2)}_{m}$ can be written explicitly without using the recurrence relation.
(2) Singularities with $\mu=1$ arise when the Helmholtz kernel in
${\mathbb{R}}^{m+1}$ is acting on $m$-dimensional flat boundary. For $m=1$,
the domain of integral need not be a flat manifold (see an example in §4.4 for
the application of the quadrature rule on curves). We are studying the
extension of our method for higher dimensional general (non-flat) manifolds.
1. (I)
$\bm{\mu=2\quad(\nu=m-2)}.$
(2.58) $M^{(2)}_{m}(\rho)=\int_{0}^{1}m\,t\,A_{m}(\rho\,t)\,dt$
The evaluation of $M^{(2)}_{1}$ is straightforward. The formula for
$M^{(2)}_{2}$ is the result of (2.56) with $\mu=m$. For $m>2$, it is not
difficult to obtain the formula from the series representation of $A_{m}$.
(2.59) $\displaystyle M^{(2)}_{1}$
$\displaystyle=\frac{\cos(\rho)-1}{\rho^{2}}-\operatorname{sinc}(\rho)$ (2.60)
$\displaystyle M^{(2)}_{2}$ $\displaystyle=A_{4}(\rho)$ (2.61) $\displaystyle
M^{(2)}_{m}$
$\displaystyle=\frac{m(m-2)}{\rho^{2}}\bigg{(}1-A_{m-2}(\rho)\bigg{)}\quad\text{for
$m>2$}$
2. (II)
$\bm{\mu=1\quad(\nu=m-1)}.$
(2.62) $M^{(1)}_{m}(\rho)=\int_{0}^{1}m\,A_{m}(\rho\,t)\,dt$
Formulae for $M^{(1)}_{1}$ and $M^{(1)}_{3}$ requires elementary calculus
only. To the best of our knowledge, there is not a simple representation of
$M^{(1)}_{2}$ by well-known functions. Hence, we treat the following integral
form of $M^{(1)}_{2}$ as the definition of a special function (see [1, p. 480]
for the properties of the integral).
(2.63) $\displaystyle M^{(1)}_{1}(\rho)$ $\displaystyle=A_{3}(\rho)$ (2.64)
$\displaystyle M^{(1)}_{2}(\rho)$
$\displaystyle=2\int_{0}^{1}J_{0}(\rho\,t)\,dt=\frac{2}{\rho}\int_{0}^{\rho}J_{0}(t)\,dt$
(2.65) $\displaystyle M^{(1)}_{3}(\rho)$
$\displaystyle=\frac{3\operatorname{Si}(\rho)}{\rho}$
One may implement his/her own version of $M^{(1)}_{2}$ from the series
representation $J_{0}$ and the asymptotic expansion of the integral, or can
simply use an implementation of $\int_{0}^{\rho}J_{0}(t)\,dt$ in Algorithm 757
(MISCFUN) of ACM Transactions on Mathematical Software (TOMS) (cf. [17]). For
$m>1$, $M^{(1)}_{m+2}$ can be obtained from the recurrence relation,
(2.66)
$M^{(1)}_{m+2}(\rho)=\frac{m+2}{m-1}\bigg{(}M^{(1)}_{m}(\rho)-A_{m+2}(\rho)\bigg{)}.$
3. (III)
$\bm{\mu\in(0,1)\cup(1,2)}.$
The recurrence relation (2.57) can be applied to obtain $M^{(\mu)}_{m+2}$ for
any $m>1$. Hence, we only need to consider two initial cases $m=1,2$.
1. (a)
$m=1.$
(2.67)
$\begin{split}M^{(\mu)}_{1}(\rho)&=\int_{0}^{1}t^{\mu-1}\cos(\rho\,t)\,dt=\frac{\operatorname{Ci}(\mu,\rho)}{\rho^{\mu}}\\\
&=\sum_{\ell=0}^{\infty}\frac{(-1)^{\ell}\rho^{2\ell}}{(2\ell+1)!(2\ell+\mu)}.\end{split}$
The function
$\operatorname{Ci}(\mu,\rho)\equiv\int_{0}^{\rho}t^{\mu-1}\cos(t)\,dt$ is
known as the generalized cosine integral, which is related to the lower
incomplete gamma function with pure imaginary argument,
(2.68)
$\operatorname{Ci}(\mu,\rho)=\operatorname{Re}\big{(}(-i)^{\mu}\,\gamma(\mu,i\rho)\big{)}.$
2. (b)
$m=2.$
(2.69)
$M^{(\mu)}_{2}(\rho)=\int_{0}^{1}t^{\mu-1}J_{0}(\rho\,t)\,dt=\sum_{\ell=0}^{\infty}\frac{(-1)^{\ell}(\rho/2)^{2\ell}}{(\ell!)^{2}(2\ell+\mu)}.$
To the best our knowledge, there is no available/reliable implementation of
either $M^{(\mu)}_{1}$ or $M^{(\mu)}_{2}$ (or any special function which can
be used to compute them) in the public domain. We can follow the standard
implementation procedure of special functions – the partial sums of the above
power series’ for small $\rho$ and the asymptotic expansions for large $\rho$.
For $M^{(\mu)}_{1}$, the error of the 18-term partial sum is less than
$10^{-16}$ on $0\leq\rho\leq 2\pi$. For $M^{(\mu)}_{2}$, 20 terms are enough
for the error less than $10^{-16}$ on $0\leq\rho\leq 7.01558666981561875$ (the
upper limit is the second positive zero of $J_{1}$). For $\rho>14\pi$ for
$M^{(\mu)}_{1}$ and $\rho>44.7593189976528217$ (the 14th positive zero of
$J_{1}$), asymptotic expansions described in Appendix A and B produce 16-digit
accurate results. For $\rho$ in the intermediate range, we divide the domain
as $\cup_{n=1}^{6}[a_{n},a_{n+1}]$ where $a_{n}=2\pi n$ for $M^{(\mu)}_{1}$
and $a_{n}$ is the (2n)th positive zero of $J_{1}$. When
$\rho\in[a_{n},a_{n+1}]$ for some $n=1,\ldots,6$,
(2.70)
$M^{(\mu)}_{m}(\rho)=M^{(\mu)}_{m}(a_{n})+\frac{1}{\rho^{\mu}}\int_{a_{n}}^{\rho}t^{\mu-1}\begin{cases}\cos(t)\,dt&m=1\\\
J_{0}(t)\,dt&m=2\end{cases}$
where $M^{(\mu)}_{m}(a_{n})$ can be precomputed and reused, and the integrals
on $[a_{n},\rho\,]$ are evaluated at each time by a quadrature rule. Since the
integrands are very smooth and oscillate less than one cycle in the interval,
any high order numerical quadrature (such as Clenshaw-Curtis) with a small
number of samples can compute the result up to the machine precision.
Thus, we have presented our scheme for the construction of corrected
quadrature weights and the required formulae for the Fourier transforms of
logarithmic and power-law singularities. We conclude this section by
presenting the factored form of the Helmholtz kernel in arbitrary dimension,
for which the results we have developed so far turn out to be well-suited.
### 2.6. Helmholtz kernels
Denote by $K^{k}_{n}(r)$ the Helmholtz kernel in ${\mathbb{R}}^{n}$ with
complex wavenumber $k$ such that $\operatorname{Im}(k)\geq 0$. The domain of
the integral (i.e. the domain of the convolution) is not necessarily
$n$-dimensional. If the convolution is performed on ${\mathbb{R}}^{m}$, $n$
needs only to satisfy $n\leq m+1$. Let $z\equiv kr$ and $\nu\equiv(n-2)/2$,
then the Helmholtz kernels are given by
(2.71) $K^{k}_{n}(r)=\frac{i}{4}\left(\frac{k}{2\pi
r}\right)^{\nu}H^{(1)}_{\nu}(z)=\frac{1}{4}\left(\frac{k}{2\sqrt{\pi}}\right)^{n-2}\left(-\frac{Y_{\nu}(z)}{(z/2)^{\nu}}+i\frac{J_{\nu}(z)}{(z/2)^{\nu}}\right)$
where $H^{(1)}_{\nu}$ is the Hankel function of the first kind. More familiar
forms in the first three dimensions are
(2.72) $K^{k}_{1}(r)=\frac{ie^{ikr}}{2k},\quad
K^{k}_{2}(r)=\frac{i}{4}H^{(1)}_{0}(kr),\quad K^{k}_{3}(r)=\frac{e^{ikr}}{4\pi
r}.$
In the limiting case with $k=0$, $K^{0}_{n}$ is simply a constant multiple of
the logarithmic or the power-law singularity;
(2.73) $\displaystyle K^{0}_{1}(r)$ $\displaystyle=-\frac{1}{2}r,$
$\displaystyle K^{0}_{2}(r)$ $\displaystyle=-\frac{1}{2\pi}\log(r),$
$\displaystyle K^{0}_{n}(r)$
$\displaystyle=\frac{\Gamma\left(n/2-1\right)}{4\sqrt{\pi}^{n}}\frac{1}{r^{n-2}}\quad\text{for
$n\geq 3$}.$
Recall that $J_{\nu}(z)/(z/2)^{\nu}=A_{n}(z)/\Gamma(n/2)$ with $A_{n}$ defined
and extensively used in previous sections. Like $A_{n}$, the imaginary part of
$K^{k}_{n}$ is analytic, and has the limiting value
(2.74) $\lim_{r\rightarrow
0}\operatorname{Im}(K^{k}_{n}(r))=\frac{i}{4\Gamma(n/2)}\left(\frac{k}{2\sqrt{\pi}}\right)^{n-2}.$
Thus, the singularity is carried entirely by the real part. In principle, for
any $n$, $\operatorname{Re}(K^{k}_{n})$ contains the same type of singularity
as $K^{0}_{n}$ given above. However, for even $n>3$, the kernel contains an
additional logarithmic singularity.
#### 2.6.1. Odd $\bm{n}$
Since $\nu=(n-2)/2$ is an half-integer, we can utilize the identity,
$Y_{\nu}(z)=(-1)^{\lceil\nu\rceil}\,J_{-\nu}(z)$, to obtain
(2.75) $\displaystyle\frac{Y_{\nu}(z)}{(z/2)^{\nu}}$
$\displaystyle=(-1)^{\lceil\nu\rceil}\,\frac{J_{-\nu}(z)}{(z/2)^{-\nu}}\,\frac{1}{(z/2)^{2\nu}}$
(2.76) $\displaystyle=-(-1)^{\lceil
n/2\rceil}\left(\frac{2}{k}\right)^{n-2}\frac{A_{4-n}(kr)}{\Gamma(2-n/2)}\frac{1}{r^{n-2}}.$
Therefore, the factored form is given by
(2.77)
$K^{k}_{n}(r)=\frac{\alpha^{k}_{n}(r)}{r^{n-2}}+\widetilde{K}^{k}_{n}(r)$
with smooth functions,
(2.78) $\displaystyle\alpha^{k}_{n}(r)$
$\displaystyle=\alpha^{k}_{n}(0)\,A_{4-n}(kr)\quad\text{where}\quad\alpha^{k}_{n}(0)=\frac{(-1)^{\lceil
n/2\rceil}}{4\,\Gamma(2-n/2)\,\sqrt{\pi}^{n-2}}$ (2.79)
$\displaystyle\widetilde{K}^{k}_{n}(r)$
$\displaystyle=\widetilde{K}^{k}_{n}(0)\,A_{n}(kr)\quad\;\;\;\text{where}\quad\widetilde{K}^{k}_{n}(0)=\frac{i}{4\,\Gamma(n/2)}\left(\frac{k}{2\sqrt{\pi}}\right)^{n-2}.$
It is convenient to use the following recurrence relations.
(2.80)
$\alpha^{k}_{n+2}(0)=\frac{n-2}{2\pi}\,\alpha^{k}_{n}(0)\quad\text{and}\quad\widetilde{K}^{k}_{n+2}(0)=\frac{k^{2}}{2n\pi}\,\widetilde{K}^{k}_{n}(0).$
Recall that, in the separated form of the quadrature rule (2.24), we do not
need to evaluate $\widetilde{K}^{k}_{n}$ explicitly; only the limiting value
at the origin is required. However, the values of $\alpha^{k}_{n}$ at grid
points are still needed. The following table shows explicit formulae and
values for the first few dimensions.
$\displaystyle n$ | $\displaystyle\phi(r)$ | $\displaystyle\alpha^{k}_{n}(r)$ | $\displaystyle\alpha^{k}_{n}(0)$ | $\displaystyle\widetilde{K}^{k}_{n}(0)$
---|---|---|---|---
1 | $\displaystyle r$ | $\displaystyle-\operatorname{sinc}(kr)/2$ | $\displaystyle-1/2$ | $\displaystyle i/(2k)$
3 | $\displaystyle r^{-1}$ | $\displaystyle\cos(kr)/(4\pi)$ | $\displaystyle 1/(4\pi)$ | $\displaystyle ik/(4\pi)$
5 | $\displaystyle r^{-3}$ | $\displaystyle\big{(}\cos(kr)+kr\sin(kr)\big{)}/(8\pi^{2})$ | $\displaystyle 1/(8\pi^{2})$ | $\displaystyle ik^{3}/(24\pi^{2})$
7 | $\displaystyle r^{-5}$ | $\displaystyle\big{(}(3-(kr)^{2})\cos(kr)+3kr\sin(kr)\big{)}/(16\pi^{3})$ | $\displaystyle 3/(16\pi^{3})$ | $\displaystyle ik^{5}/(240\pi^{3})$
#### 2.6.2. Even $\bm{n}$
The Bessel function of the second kind with an integer index $\nu=(n-2)/2$ can
written (cf. [1, p. 358]) as
(2.81)
$Y_{\nu}(z)=-\frac{1}{\pi}\left(\frac{z}{2}\right)^{-\nu}P_{\nu}(z)+\frac{2}{\pi}\log\left(\frac{z}{2}\right)\,J_{\nu}(z)-\frac{1}{\pi\nu!}\left(\frac{z}{2}\right)^{\nu}Q_{\nu}(z)$
where
(2.82) $\displaystyle P_{\nu}(z)$
$\displaystyle\equiv\sum_{\ell=0}^{\nu-1}\frac{(\nu-1-\ell)!}{\ell!}\left(\frac{z}{2}\right)^{2\ell}\qquad\qquad\qquad\bigg{(}P_{0}(z)\equiv
0\bigg{)}$ (2.83) $\displaystyle Q_{\nu}(z)$
$\displaystyle\equiv\nu!\sum_{\ell=0}^{\infty}(-1)^{\ell}\,\frac{h_{\ell}+h_{\nu+\ell}-2\gamma_{e}}{\ell!\,(\nu+\ell)!}\left(\frac{z}{2}\right)^{2\ell}$
where $h_{\ell}\equiv\sum_{k=1}^{\ell}1/k$ ($h_{0}\equiv 0$) and $\gamma_{e}$
is the Euler constant. Thus, the kernel involves two types of singularities
for $n\geq 4$ and can be factored as
(2.84)
$K^{k}_{n}(r)=\frac{\alpha^{k}_{n}(r)}{r^{n-2}}+\beta^{k}_{n}(r)\log(r)+\widetilde{K}^{k}_{n}(r)$
where
(2.85) $\displaystyle\alpha^{k}_{n}(r)$
$\displaystyle=\frac{P_{\nu}(kr)}{4\sqrt{\pi}^{n}},$ (2.86)
$\displaystyle\beta^{k}_{n}(r)$
$\displaystyle=\beta^{k}_{n}(0)\,A_{n}(kr)\quad\text{where}\quad\beta^{k}_{n}(0)=-\frac{1}{2\pi\Gamma(n/2)}\left(\frac{k}{2\sqrt{\pi}}\right)^{n-2},$
and
(2.87)
$\widetilde{K}^{k}_{n}(r)=\frac{1}{4\Gamma(n/2)}\left(\frac{k}{2\sqrt{\pi}}\right)^{n-2}\bigg{\\{}\frac{Q_{\nu}(kr)}{\pi}-\frac{2}{\pi}\log\left(\frac{k}{2}\right)A_{n}(kr)+i\,A_{n}(kr)\bigg{\\}}.$
The limiting values of $\alpha^{k}_{n}$ and $\beta^{k}_{n}$ are given by
(2.88) $\displaystyle\alpha^{k}_{n}(0)$
$\displaystyle=\frac{(n-4)!}{4\sqrt{\pi}^{n}},\quad\alpha^{k}_{2}(0)=0$ (2.89)
$\displaystyle\beta^{k}_{n+2}(0)$
$\displaystyle=\frac{k^{2}}{2n\pi}\beta^{k}_{n}(0),\quad\beta^{k}_{2}(0)=-\frac{1}{2\pi}$
The following table summarizes formulae and values of $\alpha^{k}_{n}$ and
$\beta^{k}_{n}$ for the first few even dimensions.
$\displaystyle n$ | $\displaystyle\alpha^{k}_{n}(r)$ | $\displaystyle\alpha^{k}_{n}(0)$ | $\displaystyle\beta^{k}_{n}(r)$ | $\displaystyle\beta^{k}_{n}(0)$
---|---|---|---|---
2 | 0 | 0 | $\displaystyle-J_{0}(kr)/(2\pi)$ | $\displaystyle-1/(2\pi)$
4 | $\displaystyle 1/(4\pi^{2})$ | $\displaystyle 1/(4\pi^{2})$ | $\displaystyle-kJ_{1}(kr)/(4\pi^{2}r)$ | $\displaystyle-k^{2}/(8\pi^{2})$
6 | $\displaystyle\big{(}1+(kr/2)^{2}\big{)}/(4\pi^{3})$ | $\displaystyle 1/(4\pi^{3})$ | $\displaystyle-k^{2}J_{2}(kr)/(8\pi^{3}r^{2})$ | $\displaystyle-k^{4}/(64\pi^{3})$
8 | $\displaystyle\big{(}2+(kr/2)^{2}+(kr/2)^{4}/2\big{)}/(4\pi^{4})$ | $\displaystyle 1/(2\pi^{4})$ | $\displaystyle-k^{3}J_{3}(kr)/(16\pi^{4}r^{3})$ | $\displaystyle-k^{6}/(768\pi^{4})$
The value of $\widetilde{K}^{k}_{n}$ at the origin is given by
(2.90)
$\widetilde{K}^{k}_{n}(0)=\frac{1}{4\Gamma(n/2)}\left(\frac{k}{2\sqrt{\pi}}\right)^{n-2}\bigg{\\{}\frac{h_{(n-2)/2}-2\gamma_{e}}{\pi}-\frac{2}{\pi}\log\left(\frac{k}{2}\right)+i\bigg{\\}}.$
Hence, $\widetilde{K}^{k}_{n}(0)$ for any even $n>2$ can be generated by the
recurrence relation
(2.91)
$\widetilde{K}^{k}_{n+2}(0)=\frac{k^{2}}{2n\pi}\widetilde{K}^{k}_{n}(0)+\frac{1}{n^{2}\Gamma(n/2)\pi}\left(\frac{k}{2\sqrt{\pi}}\right)^{n}$
from the initial value
(2.92)
$\widetilde{K}^{k}_{2}(0)=\frac{i}{4}-\frac{\gamma_{e}}{2\pi}-\frac{1}{2\pi}\log\left(\frac{k}{2}\right)$
The existence of the additional logarithmic singularity does not add any
difficulty to the construction of the quadrature weights; now, the formula
(2.24) simply contains one more term.
(2.93)
$w_{\ell}=\begin{cases}\alpha^{k}_{n}(0)\,\widetilde{\phi}^{\alpha}_{0}+\beta^{k}_{n}(0)\,\widetilde{\phi}^{\beta}_{0}+\widetilde{K}^{k}_{n}(0)&\ell=0\\\
\alpha^{k}_{n}(r_{\ell})\left(\widetilde{\phi}^{\alpha}_{\ell}-\phi^{\alpha}(r_{\ell})\right)\varphi(r_{\ell})+\beta^{k}_{n}(r_{\ell})\left(\widetilde{\phi}^{\beta}_{\ell}-\phi^{\beta}(r_{\ell})\right)\varphi(r_{\ell})&\ell\neq
0\end{cases}$
where $\phi^{\alpha}(r)=1/r^{n-2}$ and $\phi^{\beta}(r)=\log(r)$. The
regularized singularities $\widetilde{\phi}^{\alpha}_{\ell}$ and
$\widetilde{\phi}^{\beta}_{\ell}$ are obtained by the same procedure,
independently of each other.
## 3\. Convergence Analysis
In this section, we present the rate of convergence of the presented
quadrature rules depending on the regularity of the data. The main result is
that our corrected trapezoidal rules applied to smooth data converge faster
than any algebraic order of accuracy. We begin with defining useful regularity
classes. The Fourier coefficients of functions in each class shares a common
form of upper bounds. Then, we present the accuracy of the (uncorrected)
trapezoidal rule for the regular integral (2.14). Finally, the accuracy of the
corrected trapezoidal rule for the singular integral (2.18) is presented.
### 3.1. Decay characteristics of Fourier coefficients
In spectral contexts, the classification of functions by the decay
characteristics of their spectral coefficients will always be the best one
since the error of a numerical scheme is in principle determined by the rate
of decay of the coefficients. However, in common situations, an _a priori_
estimate is not likely to be available. On the contrary, the smoothness of a
function is more accessible, and upper bounds for the Fourier coefficients
$|\widehat{F}(k)|$ can be obtained by repeated applications of integration-by-
parts. One of the earliest application of this technique (in a somewhat
diffrerent direction) can be found in [16]. We present the extension of one-
dimensional results (cf. [5, 6]) to higher dimensions. Although this approach
has been widely exercised, the employed regularity conditions vary depending
on authors resulting, sometimes, a less tight error bound for the trapezoidal
rule. We begin with the following definition.
###### Definition 3.1.
For any non-negative integer $P$, we denote by $C^{P}_{per}$ the class of
periodic (or periodized) $C^{P}$ functions with $U$ as a period, which satisfy
the following conditions. Let $F\in C^{P}_{per}$.
1. (1)
For $P>0$ and $P$ is even, $\Delta^{(P/2)}F\in C^{0}_{per}$.
2. (2)
For $P>0$ and $P$ is odd, $\partial_{j}\Delta^{\lfloor P/2\rfloor}F\in
C^{0}_{per}$ for all $j=1,\ldots,m$.
3. (3)
For $P=0$, (a) $U$ consists of a finite number of disjoint sub-domains on each
of which $F$ is $C^{1}$ up to the sub-domain boundary and $C^{2}$ with $\Delta
F$ in $L^{1}$, and (b) the boundary of each sub-domain is of $C^{1}$ and each
connected component of the boundary has two adjacent sub-domains.
Then, an $F\in C^{P}_{per}$ is of the Hölder class $C^{P,1}$. With an
arbitrary assignment of the orientation, unit normal vectors are well-defined
on the sub-domain boundaries, and so is the trace of $\partial(\Delta^{\lfloor
P/2\rfloor}F)/\partial n$. The conditions on $C^{0}_{per}$ regarding the
piecewise smoothness are to enable the application of the Green’s identities.
Those piecewise smoothness enable a more precise estimate by considering two
types of manageable singularities in the (P+2)nd derivatives of a function –
$L^{1}$ and $H^{-1}$. Take the function $F\equiv\max(0,1-4t^{2})$ on $[-1,1]$
for example. $F$ is in $C^{0}_{per}$ and its Fourier coefficients are given by
$\widehat{F}(k)=(\operatorname{sinc}(\pi k/2)-\cos(\pi k/2))/(\pi k/2)^{2}$
($O(|k|^{-2})$). Less rigorously, we can expect the result without the precise
evaluation; the second derivative $F^{(2)}$ consists of two delta functions at
$\pm 1/2$, hence, the Fourier coefficients of $F^{(2)}$ are $O(1)$, which
results in the $O(|k|^{-2})$-decay of $\widehat{F}(k)$. On the contrary, the
function $\sqrt{\max(0,1-4t^{2})}$ is $C^{0}$ but does not satisfy the
piecewise regularity conditions (hence, is not in $C^{0}_{per}$). Its Fourier
coefficients are $J_{1}(\pi k/2)/(2k)$, which is of $O(|k|^{-3/2})$. We
consider the first example can be observed more frequently than the second.
The second example illustrates that we can enrich the classification by
introducing (weaker) classes with half-integer indices ($L^{1}$ first
derivatives). However, in this paper, we follow the virtue of simplicity.
###### Lemma 3.2.
Let $F\in C^{P}_{per}$. Then,
(3.1) $\left|\widehat{F}(k)\right|=O\left(\|{k}\|^{-(P+2)}\right)$
###### Proof.
(1) For $P=0$, we apply Green’s second identity on each sub-domain. Let
$\Gamma$ be the union of the sub-domain boundaries with arbitrarily assigned
orientation. Utilizing $\Delta(e^{-i\pi k\cdot
y})=-\pi^{2}\|{k}\|^{2}\,e^{-i\pi k\cdot y}$ and the continuity of
$\partial(e^{-i\pi k\cdot y})/\partial n$, we obtain
(3.2)
$2^{m}\widehat{F}(k)=\frac{1}{\pi^{2}\|{k}\|^{2}}\Bigg{\\{}\int_{\Gamma}\bigg{[}\\!\\!\bigg{[}\frac{\partial
F}{\partial n}\bigg{]}\\!\\!\bigg{]}(y)\,e^{-i\pi k\cdot
y}\,dS_{y}-\int_{U}\Delta F(y)\,e^{-i\pi k\cdot y}\,dy\Bigg{\\}},$
where $[\\![\cdot]\\!]$ is the jump discontinuity across $\Gamma$. Therefore,
$|\widehat{F}(k)|=O\left(\|{k}\|^{-2}\right)$. (2) For $P=1$, we can apply
Green’s first identity. By the continuity of the first derivatives, the
boundary integral disappears.
(3.3)
$\widehat{F}(k)=-\frac{i}{\pi\|{k}\|^{2}}\sum_{j=1}^{m}k_{j}\int_{U}\partial_{j}F(y)\,e^{-i\pi
k\cdot y}\,dy.$
Since $\partial_{j}F\in C^{0}_{per}$ by definition, the integral for each $j$
is $O\left(\|{k}\|^{-2}\right)$. By the Hölder inequality, $|\sum
k_{j}|\leq\sqrt{m}\|{k}\|$. (3) For $P>1$, apply Green’s second identity
$\lfloor P/2\rfloor$ times with the boundary integrals vanishing. Then,
(3.4) $\widehat{F}(k)=\left(\frac{-1}{\pi^{2}\|{k}\|^{2}}\right)^{\lfloor
P/2\rfloor}\int_{U}\Delta^{\lfloor P/2\rfloor}F(y)\,e^{-i\pi k\cdot y}\,dy.$
Apply the result of (1) if $P$ is even, and apply (2) if odd. ∎
The upper bound given by Lemma 3.2 is somewhat conservative; for $m>1$, the
integrals in (3.2) contribute to an additional decay (of possibly fractional
order). However, in one-dimension, the integral on $\Gamma$ becomes point-wise
evaluations of the jump discontinuity, hence, we cannot expect a faster decay.
Also, one should note that the smoothness is not the only factor which governs
the decay characteristics. Consider the function $\max(0,1-4r^{2})^{2}$ which
is in $C^{1}_{per}$. Its Fourier coefficients are $((12-4(\pi
k/2)^{2})\operatorname{sinc}(\pi k/2)-12\cos(\pi k/2))/(\pi k/2)^{4}$, hence,
are of $O(|k|^{-4})$, which is faster than the estimated $O(|k|^{-3})$.
Although they are in different regularity classes, the function
$\max(0,1-4r^{2})^{2}\in C^{1}_{per}$ exhibits the same rate of convergence by
the trapezoidal rule as a more smooth function $\max(0,1-4r^{2})^{3}\in
C^{2}_{per}$.
###### Lemma 3.3.
Let $F\in C^{P}_{per}$. Then,
$\sum_{k\in\mathbb{Z}^{m}}|\widehat{F}(k)|<\infty$ if $P>m-2$.
###### Proof.
Rewrite the infinite series by
$|\widehat{F}(0)|+\sum_{n=1}^{\infty}\sum_{\|{k}\|^{2}=n}|\widehat{F}(k)|$.
Let $r(n)$ be the number of lattice points on the sphere
$\big{\\{}\|{k}\|^{2}=n\big{\\}}$, which is also known as the sum-of-squares
function. Then, $R(n)\equiv\sum_{n=0}^{N}r(n)$ is the number of lattice points
in the ball $\big{\\{}\|{k}\|^{2}\leq n\big{\\}}$, and the Gauss’ circle
problem in $\mathbb{Z}^{m}$ states that $R(n)=O(n^{m/2})$ (cf. [18]). Since
$|\widehat{F}(k)|=O(n^{-(P+2)/2})$ for $\|{k}\|^{2}=n$ by Lemma 3.2, the
series is bounded above by
$\widehat{F}(0)+C\sum_{n=1}^{\infty}r(n)/n^{(P+2)/2}$ for some $C$. Since
$r(n)/n^{(P+2)/2}=(R(n)-R(n-1))/n^{(P+2)/2}=O((n^{m/2}-(n-1)^{m/2})/n^{(P+2)/2})$
and $n^{m/2}-(n-1)^{m/2}=O(n^{m/2-1})$, $r(n)/n^{(P+2)/2}=O(n^{-(P-m+4)/2})$,
which implies that the series converges if $P>m-2$. ∎
###### Corollary 3.4.
Let $F\in C^{P}_{per}$. The multiple Fourier series
(3.5) $\sum_{k\in\mathbb{Z}^{m}}\widehat{F}(k)e^{i\pi k\cdot y}$
converges uniformly to $F(y)$ if $P>m-2$.
The criterion $P>m-2$ is somewhat excessive in higher dimensions. As mentioned
earlier, if we consider the additional decay, the convergence criterion can be
relaxed.
### 3.2. Accuracy of the trapezoidal rule
By substituting the Fourier series (3.5) into the definition of DFT (2.4), we
obtain
(3.6)
$\widehat{F}_{k}=\frac{1}{2^{m}\overline{N}}\sum_{k^{\prime}\in\mathbb{Z}^{m}}\widehat{F}(k^{\prime})\sum_{\ell\in\mathbb{I}_{N}}e^{i\pi(k^{\prime}-k)\cdot
y_{\ell}}.$
The inner sum is $2^{m}\overline{N}$ if $k^{\prime}_{j}=k_{j}+2q_{j}N_{j}$ for
all $j$ and for some $q\in\mathbb{Z}^{m}$, and vanishes otherwise. Therefore,
$\displaystyle\widehat{F}_{k}$
$\displaystyle=\sum_{q\in\mathbb{Z}^{m}}\widehat{F}(k_{1}+2q_{1}N_{1},\ldots,k_{m}+2q_{m}N_{m})$
(3.7)
$\displaystyle=\widehat{F}(k)+\sum_{q\in\mathbb{Z}^{m}\setminus\\{0\\}}\widehat{F}(k_{1}+2q_{1}N_{1},\ldots,k_{m}+2q_{m}N_{m}).$
The above equation describes that the coefficients $\widehat{F}_{k}$ obtained
from the DFT contain the aliasing error from the harmonics ($\sum_{q\neq 0}$).
Notice that $\widehat{F}_{0}$ is the integral of $F$ on $U$ obtained by the
trapezoidal rule (for periodic functions, hence, considering only one side of
the boundary), and $\widehat{F}(0)$ is the exact integral. Hence, an error
bound of the trapezoidal rule is given by
(3.8)
$|\widehat{F}(0)-\widehat{F}_{0}|\leq\sum_{q\in\mathbb{Z}^{m}\setminus\\{0\\}}|\widehat{F}(2q_{1}N_{1},\ldots,2q_{m}N_{m})|.$
Let $N_{min}=\min(N_{1},\ldots,N_{m})$. The following theorem states the order
of accuracy of the trapezoidal rule applied to a $C^{P}_{per}$ function.
###### Theorem 3.5 (The trapezoidal rule).
For any $F\in C^{P}_{per}$ with $P>m-2$. The error of the trapezoidal rule
applied to $F$ is $O(N_{min}^{-(P+2)})$.
###### Proof.
Let $p=(2q_{1}N_{1},\ldots,2q_{m}N_{m})$. Then, $\|{p}\|\geq 2N_{min}\|{q}\|$
and
$|\widehat{F}(p)|\leq C/(N_{min}\|{q}\|)^{(P+2)}$
for some $C$ by Lemma 3.2. Rewriting
$\sum_{q\in\mathbb{Z}^{m}\setminus\\{0\\}}$ by
$\sum_{n=1}^{\infty}\sum_{\|{q}\|^{2}=n}$,
$|\widehat{F}(0)-\widehat{F}_{0}|\leq\frac{C}{N_{min}^{P+2}}\sum_{n=1}^{\infty}\frac{r(n)}{n^{(P+2)/2}}$
where $r(n)$ is defined in the proof of Lemma 3.3 and is $O(n^{m/2-1})$. The
series on the right side converges if $P>m-2$. ∎
### 3.3. Accuracy of the singular trapezoidal rule
From (3.5) and (3.2), the error of the interpolation $\widetilde{F}$ defined
by (2.7) can be written as
(3.9)
$\begin{split}F(y)-\widetilde{F}(y)&=\sum_{k\not\in\mathbb{I}_{N}}\widehat{F}(k)\,e^{i\pi
k\cdot y}\\\
&-\sum_{k\in\mathbb{I}_{N}}\sum_{q\in\mathbb{Z}^{m}\setminus\\{0\\}}\widehat{F}(k_{1}+2q_{1}N_{1},\ldots,k_{m}+2q_{m}N_{m})\,e^{i\pi
k\cdot y}.\end{split}$
Hence, the error of the singular quadrature rule (2.18) is given by
(3.10)
$\begin{split}&\left|\int_{U}(F(y)-\widetilde{F}(y))\,\phi(r)\,dy\right|\leq\sum_{k\not\in\mathbb{I}_{N}}|\widehat{F}(k)|\,|\widehat{\phi}(k)|\\\
&+\sum_{k\in\mathbb{I}_{N}}|\widehat{\phi}(k)|\sum_{q\in\mathbb{Z}^{m}\setminus\\{0\\}}|\widehat{F}(k_{1}+2q_{1}N_{1},\ldots,k_{m}+2q_{m}N_{m})|\end{split}$
###### Theorem 3.6 (The singular quadrature rule).
Let $F\in C^{P}_{per}$ with $P>m-2$. The error of the quadrature rule (2.18)
is $O(N_{min}^{-(P+2-(m-1)/2)})$.
###### Proof.
(1) First, consider the first summation in (3.10). Since
$\widehat{F}(k)=O(\|{k}\|^{-(P+2)})$ by Lemma 3.2 and
$\widehat{\phi}(k)=O(\|{k}\|^{-(m+1)/2})$ by Lemma 2.1,
$\sum_{k\not\in\mathbb{I}_{N}}|\widehat{F}(k)||\widehat{\phi}(k)|\leq\sum_{n=N_{min}^{2}}^{\infty}\sum_{\|{k}\|^{2}=n}\frac{C}{n^{(P+m/2+5/2)/2}}\\\
=\sum_{n=N_{min}^{2}}^{\infty}\frac{C\,r(n)}{n^{(P+m/2+5/2)/2}}$
where $r(n)=O(n^{m/2-1})$ as in the proof of Lemma 3.3. Thus, the right side
is bounded by $C\sum_{n=N_{min}^{2}}^{\infty}n^{-(P-m/2+9/2)/2}$, which
converges if $P>(m-5)/2$ and is $O(N_{min}^{-(P+2-(m-1)/2)})$. (2) Consider
the second term in (3.10). Since $|k_{j}|\leq N_{j}$ for $j=1,\ldots,m$, the
minimum of the parabola $(k_{j}+2q_{j}N_{j})^{2}$ is $\min((2q_{j}\pm
1)^{2}N_{j}^{2})$ if $q_{j}\neq 0$. Hence, for $q_{j}\neq 0$,
$(k_{j}+2q_{j}N_{j})^{2}\geq(2|q_{j}|-1)^{2}N_{j}^{2}\geq q_{j}^{2}N_{j}^{2}$,
and the last inequality holds for $q_{j}=0$ also. Therefore,
$\|{(k_{1}+2q_{1}N_{1},\ldots,k_{m}+2q_{m}N_{m})}\|\geq\|{q}\|N_{min}$ and
$|\widehat{F}(k_{1}+2q_{1}N_{1},\ldots,k_{m}+2q_{m}N_{m})|=O((N_{min}\|{q}\|)^{-(P+2)})$
by Lemma 3.2. Then, the second term is bounded by
(3.11)
$C\bigg{\\{}|\widehat{\phi}(0)|+\sum_{n=1}^{N_{min}^{2}}\sum_{\|{k}\|^{2}=n}n^{-(m+1)/4}\bigg{\\}}\bigg{\\{}N_{min}^{-(P+2)}\sum_{p=1}^{\infty}\sum_{\|{q}\|^{2}=p}p^{-(P+2)/2}\bigg{\\}},$
where we utilized Lemma 2.1 to get the upper bound for $|\widehat{\phi}(k)|$.
The infinite series in the second pair of braces converges if $P>m-2$. Since
(3.12) $\sum_{n=1}^{N_{min}^{2}}\sum_{\|{k}\|^{2}=n}n^{-(m+1)/4}\leq
C^{\prime}\sum_{n=1}^{N_{min}^{2}}n^{(m-5)/4}=O(N_{min}^{(m-1)/2}),$
the second term in (3.10) is $O(N_{min}^{-(P+2-(m-1)/2)})$. Thus, (1) and (2)
complete the proof. When the kernel is less singular and $|\widehat{\phi}(k)|$
decays faster than $O(\|{k}\|^{-(m+1)/2})$, (3.11) becomes dominant and
determines the rate of convergence. If $|\widehat{\phi}(k)|$ decays faster
than $O(\|{k}\|^{-m})$, the first term in (3.11) is $O(1)$ and the rate of
convergence is the same as the regular trapezoidal rule, i.e. $(P+2)$. ∎
Combined with Theorem 3.5, Theorem 3.6 indicates that the order of accuracy of
the corrected trapezoidal rule presented in this paper is at least
$(P+2-(m-1)/2)$ for the integral (1.1) with $f\in C^{P}_{per}$. If $f$ is
smooth, the corrected trapezoidal rules converges faster than any algebraic
order. The dimension-dependent degradation from the intended $(P+2)$ (as the
regular trapezoidal rule) by the amount of $-(m-1)/2$ is more like a technical
outcome; in actual experiments with $m\leq 3$, we have not experienced any
obvious degradation in the rate of convergence and results strongly imply that
the corrected trapezoidal rule exhibits the same rate of convergence as the
usual trapezoidal rule without the singular kernel. The same is true for the
condition $P>m-2$. Our cautious conjecture is that, as briefly mentioned
previously, there is an additional decay for $m>1$ originating from the
integrals in (3.2), which cancels (at least, a part of) the degradation.
The intention of the somewhat complicated proof of Theorem 3.6 can be well
illustrated by considering simple one-dimensional cases. For $C^{P}_{per}$
functions, both the trapezoidal rule and the corrected trapezoidal rule
exhibit $(P+2)$nd order convergence. The interpolation error, however, is only
of $(P+1)$st order. The mutual cancellation of overshooting and undershooting,
which has been well explained for the regular trapezoidal rule, holds
identically for our singular quadrature rules. In the singular cases, the
additional one in the rate of convergence originates from the decay
characteristics of $|\widehat{\phi}(k)|$.
## 4\. Numerical Examples
### 4.1. Test of convergence: Helmholtz kernels with $\bm{k=0}$
We evaluate the convolution with the non-oscillatory kernels on
${\mathbb{R}}^{m}$ ($m=1,2,3$):
(4.1) $u=K^{0}_{n}*f$
for $n=m$ and $m+1$. For $n=m$, $u$ corresponds to the volume potential
induced by the surce $f$. For $n=m+1$, the convolution corresponds to the
application of the single layer operator as a boundary to boundary integral
operator on $m$-dimensional flat boundary in ${\mathbb{R}}^{m+1}$.
Convolutions were performed on uniform grids with the spacing $6/N$ on the
physical domain $[-3,3]^{m}$. The errors are reported in $L_{\infty}$ norm
from the values of the numerical and the exact solutions obtained at grid
points.
Table 1 and 2 summarize the results of one-dimensional convolution with the
logarithmic kernel: $K^{0}_{2}=-\log(r)/(2\pi)$. Reference solutions are
computed up to the machine precision using adaptive quadratures. For the
demonstration, we selected three different sources,
(4.2) $\displaystyle f_{G}(r)$ $\displaystyle=\exp\left(-(r/a)^{2}\right)$
$\displaystyle(a=1/2)$ $\displaystyle f_{B}(r)$
$\displaystyle=\exp\left(12-12/(1-(r/a)^{2})\right)$ $\displaystyle(a=2)$
$\displaystyle f_{P}(r)$ $\displaystyle=\max\left(0,1-(r/a)^{2}\right)^{7}$
$\displaystyle(a=2),$
based on their regularity characteristics. The Gaussian ($f_{B}$) is analytic
but is not compactly supported. However, on the boundary of the domain
employed, $|f_{G}|\sim 10^{-16}$ and any significant error of the domain
truncation has not been observed. Actually, the spectrum of the Gaussian (in
any dimension) exhibits the most rapid decay, which makes it a perfect
specimen for the test of the spectral accuracy. Table 1 shows a clear super-
algebraic convergence of our quadrature rule; each refinement doubles the
number of correct digits. The bump function $f_{B}$ is smooth and compactly
supported, hence, satisfies our assumptions faithfully. However, its Fourier
coefficients decays quite slowly compared with the Gaussian. The left column
of Table 2 shows that the estimated order keeps increasing but the increase is
slower than that of the Gaussian cases in Table 1. The function $f_{P}$ is not
smooth but in $C^{6}_{per}$. Hence, the estimated order is fixed to the
algebraic order of 8 (right column in Table 2), which is expected in the
convergence analysis.
Recall §2.4 that we can use arbitrarily refined grid during the construction
of the quadrature weights. Such a refined construction increases only the
construction time but does not affect the convolution time since the obtained
weights will be truncated in the frequency domain. The objective of the use of
a refined construction grid is to suppress errors originating from
insufficient samplings of $\alpha$, $\varphi$, and $K$. Since $K^{0}_{n}$ are
non-oscillatory ($\alpha$ is a constant and $\widetilde{K}=0$), the error
during the construction originates mostly from $\varphi$. In Table 1, the
results on the left column are obtained using the weights constructed on the
same grid as the data. For the results on the right column, we doubled the
sampling frequency of the construction grid. For small $N$, the difference is
negligible since the error from the data is dominant. However, the
interpolation error of the Gaussian decreases more rapidly than that of
$\varphi$. Hence, if we use the same grid, the construction error stand out
eventually. The remedy is simple; we can use a slightly higher sampling
frequency for the construction. As for $f_{B}$ and $f_{P}$, their
interpolation errors decrease more slowly than $\varphi$, and the results are
almost indistinguishable with or without using a refined construction grid
(see Table 2).
Table 3 shows the results for $(K^{0}_{n}*f_{G})$ in higher dimensions
($m=2,3$ and $n=m,m+1$). For the Gaussian, we know the exact solutions, which
are given by
(4.3) $\displaystyle(m=2,n=2)\quad u(r)$
$\displaystyle=\frac{a^{2}}{4}\log\left(\frac{\exp(-E_{1}(\rho^{2}))}{\rho^{2}}\right)-\frac{a^{2}}{2}\log(a)$
$\displaystyle(m=2,n=3)\quad u(r)$
$\displaystyle=\frac{a\sqrt{\pi}}{4}\,\exp\left(-\frac{\rho^{2}}{2}\right)I_{0}\left(\frac{\rho^{2}}{2}\right)$
$\displaystyle(m=3,n=3)\quad u(r)$
$\displaystyle=\frac{a^{2}\sqrt{\pi}}{4}\frac{\operatorname{erf}(\rho)}{\rho}$
$\displaystyle(m=3,n=4)\quad u(r)$
$\displaystyle=\frac{a}{2\sqrt{\pi}}\exp\left(-\frac{\rho^{2}}{2}\right)\int_{0}^{1}\exp\left(-\frac{\rho^{2}}{2}t^{2}\right)I_{0}\left(\frac{\rho^{2}}{2}(1-t^{2})\right)\,dt$
where $\rho=r/a$ and $a=1/2$. As in the one-dimensional case, we can observe
clear spectral rates of convergences.
| $f_{G}$ (without ref.) | $f_{G}$ (with ref.)
---|---|---
$N$ | $E_{N}$ | $\log_{2}\frac{E_{N/2}}{E_{N}}$ | $E_{N}$ | $\log_{2}\frac{E_{N/2}}{E_{N}}$
5 | $5$. | $58\times 10^{-2}$ | — | $5$. | $59\times 10^{-2}$ | —
10 | $3$. | $26\times 10^{-3}$ | $\;\>4.1$ | $3$. | $26\times 10^{-3}$ | $\;\>4.1$
20 | $1$. | $30\times 10^{-6}$ | $11.3$ | $1$. | $30\times 10^{-6}$ | $11.3$
40 | $3$. | $32\times 10^{-13}$ | $21.9$ | $3$. | $89\times 10^{-16}$ | $31.6$
Table 1. $L_{\infty}$ errors of $(K^{0}_{2}*f_{G})$ on ${\mathbb{R}}$: (left) without and (right) with using the doubly-refined grid for the construction of the weights. | $f_{B}$ | $f_{P}$
---|---|---
$N$ | $E_{N}$ | $\log_{2}\frac{E_{N/2}}{E_{N}}$ | $E_{N}$ | $\log_{2}\frac{E_{N/2}}{E_{N}}$
5 | $4$. | $17\times 10^{-2}$ | — | $1$. | $46\times 10^{-2}$ | —
10 | $7$. | $21\times 10^{-4}$ | $\;\>5.8$ | $5$. | $65\times 10^{-5}$ | $8.0$
20 | $1$. | $45\times 10^{-6}$ | $\;\>9.0$ | $2$. | $36\times 10^{-7}$ | $7.9$
40 | $9$. | $25\times 10^{-10}$ | $10.6$ | $7$. | $31\times 10^{-10}$ | $8.3$
80 | $2$. | $36\times 10^{-14}$ | $15.3$ | $4$. | $33\times 10^{-12}$ | $7.4$
Table 2. $L_{\infty}$ errors of $(K^{0}_{2}*f_{B})$ and $(K^{0}_{2}*f_{P})$ on
${\mathbb{R}}$.
| $(m=2,n=2)$ | $(m=2,n=3)$
---|---|---
$N$ | $E_{N}$ | $\log_{2}\frac{E_{N/2}}{E_{N}}$ | $E_{N}$ | $\log_{2}\frac{E_{N/2}}{E_{N}}$
5 | $1$. | $06\times 10^{-1}$ | — | $4$. | $88\times 10^{-2}$ | —
10 | $3$. | $96\times 10^{-3}$ | $\;\>4.7$ | $4$. | $70\times 10^{-3}$ | $\;\>3.4$
20 | $8$. | $99\times 10^{-7}$ | $12.1$ | $2$. | $35\times 10^{-6}$ | $11.0$
40 | $5$. | $55\times 10^{-16}$ | $30.6$ | $3$. | $33\times 10^{-16}$ | $32.7$
| $(m=3,n=3)$ | $(m=3,n=4)$
---|---|---
$N$ | $E_{N}$ | $\log_{2}\frac{E_{N/2}}{E_{N}}$ | $E_{N}$ | $\log_{2}\frac{E_{N/2}}{E_{N}}$
5 | $4$. | $04\times 10^{-2}$ | — | $1$. | $41\times 10^{-2}$ | —
10 | $4$. | $10\times 10^{-3}$ | $\;\>3.3$ | $5$. | $03\times 10^{-3}$ | $\;\>1.5$
20 | $1$. | $19\times 10^{-6}$ | $11.8$ | $3$. | $22\times 10^{-6}$ | $10.6$
40 | $1$. | $05\times 10^{-15}$ | $30.1$ | $3$. | $05\times 10^{-16}$ | $33.3$
Table 3. $L_{\infty}$ errors of $(K^{0}_{n}*f_{G})$ on ${\mathbb{R}}^{m}$.
### 4.2. Test of convergence: Helmholtz kernels with $\bm{k\neq 0}$
Convergence tests were performed for the Helmholtz kernels $K^{k}_{n}$ with
nonzero wavenumber. First, we present results with real wavenumber $k=2\pi$ on
$[-3,3]^{m}$. The Gaussian $f_{G}$ was selected as the source. Since the
Fourier coefficients of $f_{G}$ decay rapidly, the quadrature rule, if is is
properly constructed, should exhibit a similar fast convergence. Unlike non-
oscillatory kernels of the previous section, we do not know the exact solution
even for the Gaussian. Moreover, it is very difficult and expensive with
adaptive quadratures to obtain the reference solutions accurate up to the
machine-precision. Hence, the reference solutions were evaluated only at the
origin, where the error is likely to be largest. The results are summarized in
Table 4 and 5, from both of which we can observe the expected spectral rates
of convergences.
| $(m=1,n=2,\text{without ref.})$ | $(m=1,n=2,\text{with ref.})$
---|---|---
$N$ | $E_{N}$ | $\log_{2}\frac{E_{N/2}}{E_{N}}$ | $E_{N}$ | $\log_{2}\frac{E_{N/2}}{E_{N}}$
5 | $4$. | $11\times 10^{-2}$ | — | $7$. | $95\times 10^{-2}$ | —
10 | $4$. | $66\times 10^{-2}$ | $\\!\\!-0.2$ | $6$. | $47\times 10^{-3}$ | $\;\>3.6$
20 | $2$. | $89\times 10^{-4}$ | $\;\>7.3$ | $2$. | $82\times 10^{-6}$ | $11.2$
40 | $2$. | $61\times 10^{-11}$ | $23.4$ | $3$. | $93\times 10^{-17}$ | $36.1$
Table 4. $L_{\infty}$ errors of $(K^{k}_{2}*f_{G})$ on ${\mathbb{R}}$ with
$k=2\pi$: (left) without and (right) with using the doubly-refined grid for
the construction of the weights.
| $(m=2,n=2)$ | $(m=2,n=3)$
---|---|---
$N$ | $E_{N}$ | $\log_{2}\frac{E_{N/2}}{E_{N}}$ | $E_{N}$ | $\log_{2}\frac{E_{N/2}}{E_{N}}$
5 | $4$. | $01\times 10^{-2}$ | — | $1$. | $02\times 10^{-1}$ | —
10 | $1$. | $14\times 10^{-2}$ | $\;\>1.8$ | $1$. | $26\times 10^{-2}$ | $\;\>3.0$
20 | $2$. | $46\times 10^{-6}$ | $12.2$ | $4$. | $77\times 10^{-6}$ | $11.4$
40 | $2$. | $08\times 10^{-17}$ | $36.8$ | $2$. | $55\times 10^{-16}$ | $34.1$
| $(m=3,n=3)$ | $(m=3,n=4)$
---|---|---
$N$ | $E_{N}$ | $\log_{2}\frac{E_{N/2}}{E_{N}}$ | $E_{N}$ | $\log_{2}\frac{E_{N/2}}{E_{N}}$
5 | $4$. | $61\times 10^{-2}$ | — | $1$. | $13\times 10^{-1}$ | —
10 | $1$. | $52\times 10^{-2}$ | $\;\>1.6$ | $1$. | $81\times 10^{-2}$ | $\;\>2.6$
20 | $2$. | $95\times 10^{-6}$ | $12.3$ | $6$. | $17\times 10^{-6}$ | $11.5$
40 | $2$. | $96\times 10^{-17}$ | $36.5$ | $4$. | $13\times 10^{-16}$ | $33.8$
Table 5. $L_{\infty}$ errors of $(K^{k}_{n}*f_{G})$ on ${\mathbb{R}}^{m}$ with
$k=2\pi$.
Similarly to Table 1, Table 4 illustrates the enhancement in the accuracy by
using a higher sampling frequency for the construction grid. However, the
results with the oscillating kernels exhibit quite different aspects: first,
the level of the construction error is significantly larger than the previous
non-oscillatory cases. Second, the corruption occurs also for small $N$.
Recall, for non-oscillatory kernels in the previous section, the main source
of the construction error was $\varphi$, whose Fourier coefficients decay
quite rapidly. Although the interpolation error of the Gaussian exhibits
faster decay than that of $\varphi$, the Gaussian is a somewhat special case.
For many other functions (such as $f_{B}$ and $f_{P}$), $\varphi$ is good
enough not to cause such a issue. However, for oscillatory kernels, the major
source of the construction error is $\alpha$ and $\widetilde{K}$. Simply, we
need a construction grid fine enough to represent the oscillating $\alpha$
with similar accuracy as the data ($\widetilde{K}$ oscillates similarly to
$\alpha$). Hence, the sampling frequency of the construction grid should be
increased proportionally to the wavenumber.
On the other hand, if the sampling frequency of the data is not high enough
for the representation of the oscillating kernel, the sampling frequency is
not high enough for the representation of the solution either. Hence,
ironically, in most of practical applications such as examples in §4.3 and
§4.4, we cannot expect the enhanced accuracy by merely increasing the sampling
frequency of the construction grid. The first convolution from the exactly
given data may be accurate at each grid point. However, the interpolation
error of the first solution will not be as accurate as that of the data, due
to the insufficient grid resolution. This first solution with insufficient
accuracy will be (a part of ) the source of the next convolution. Hence, if an
appropriate grid is chosen for the data and the solution, we can use the same
grid for the construction.
Figure 1. The Helmholtz kernel $K^{k}_{2}$ on ${\mathbb{R}}$: (left) the
weights for $k=4i$ in the frequency domain and (right) errors up to $k=50i$.
The situation becomes more complicated if $\operatorname{Im}(k)\neq 0$. Take
$K^{k}_{2}$ with a pure imaginary wavenumber $k=i\lambda$ for example.
(4.4) $K^{k}_{2}(r)=\frac{i}{4}H^{(1)}_{0}(i\lambda r)=\frac{K_{0}(\lambda
r)}{2\pi}\sim\sqrt{\frac{1}{8\pi\lambda r}}e^{-\lambda r}$
As described in §2.6, the kernel can be factored as
$K^{k}_{2}(r)=\beta^{k}_{2}(r)\log(r)+\widetilde{K^{k}_{2}}(r)$ with
(4.5) $\beta^{k}_{2}(r)=-\frac{J_{0}(i\lambda r)}{2\pi}=-\frac{I_{0}(\lambda
r)}{2\pi}\sim\sqrt{\frac{1}{8\pi^{3}\lambda r}}\left(e^{\lambda
r}-ie^{-\lambda r}\right).$
$I_{0}$ and $K_{0}$ are the modified Bessel functions of the first and the
second kinds. Notice that the exponentially decaying kernel is decomposed by
two exponentially increasing $\beta^{k}_{2}$ and $\widetilde{K}^{k}_{2}$. With
finite precision arithmetic, the meaningful signal ($\sim e^{-\lambda
r}/\sqrt{r}$) in $\beta^{k}_{2}$ is completely lost for $\lambda r\gg 1$.
Thus, we should choose sufficiently small $B_{R}$ so that the $\beta^{k}_{2}$
evaluated within $B_{R}$ remains small $\sim 1$. The left figure in Figure 1
shows the spectrums of two sets of the weights generated for $\lambda=4$. For
the accurate weights, the radius of $B_{R}$ is set to $1$. The inaccurate
weights obtained with $B_{R}$ of the radius 3 deviate wildly after the mid-
frequency. As we increase $\lambda$ with the fixed radius of 3, the points of
deviation move to lower frequencies, and soon only the noise remains. Since
the construction grid should be fine enough to resolve $\varphi$ with $B_{R}$,
its sampling frequency should be increased accordingly. In order to prevent
the generation of a huge construction grid due to the high sampling frequency,
we can take only a small subset of grid containing $B_{R}$. Outside of the
subset, the kernel is practically zero, hence, the spectrum of the weights can
be obtained by zero-padded DFT followed by the truncation of the unnecessary
high frequency results.
In the right figure of Figure 1, the errors with the appropriately constructed
weights are less than $10^{-14}$. Without the $\lambda$-dependent control on
$B_{R}$, the error rapidly increases after $\lambda\geq 5$.
### 4.3. Application: Lippmann-Schwinger equation
Consider the acoustic scattering problem in an inhomogeneous medium. Let the
constant $k$ be the wavenumber of the medium at infinity (or we may call it
the ambient wavenumber) and let $n$ the non-constant refractive index such
that $(n-1)$ is compactly supported. Then, for time-harmonic problems, the
acoustic pressure $u$ satisfies the equation,
(4.6) $\Delta u+k^{2}n(x)\,u=0.$
Let $u^{i}$ be the given incident wave which satisfies $\Delta
u^{i}+k^{2}\,u^{i}=0$. We assume the scattered field $u^{s}=u-u^{i}$ satisfies
the radiation condition at infinity. The explicit form of the radiation
condition depends on the dimension. In this paper, we consider an example in
${\mathbb{R}}^{2}$, where the radiating scattered field satisfies
(4.7) $\lim_{r\rightarrow\infty}\sqrt{r}\left(\frac{\partial u^{s}}{\partial
r}-i\,k\,u\right)=0.$
The above scattering problem is equivalent to solving the integral equation,
(4.8) $u(x)-k^{2}\int_{D}K_{2}^{k}(r)\left(n(y)-1\right)u(y)\,dy=u^{i}(x),$
which is also known as the Lippmann-Schwinger equation. The linear operator on
the left side of the equation (denoted by $\mathcal{L}$) can be written as
(4.9) $\mathcal{L}=I-k^{2}\,\mathcal{K}\,\mathcal{N}$
in terms of the convolution $\mathcal{K}u=K_{2}^{k}*u$ and the multiplication
$\mathcal{N}u=(n-1)u$. Since $\mathcal{K}$ is compact and $\mathcal{N}$ is
bounded, $\mathcal{L}$ is a Fredholm operator of the second kind and the
problem is well-posed (cf. [9, §8]).
In this example, we consider a medium with three _bumps_ ,
(4.10)
$n(y)=1-0.9\sum_{i=1}^{3}e^{2\left(1-\left(1-(y_{1}-a_{i})^{2}-(y_{2}-b_{i})^{2}\right)^{-1}\right)},$
with the centers at $(a_{i},b_{i})=(1,0)$, $(-1,3)$, and $(-1,-3)$. Inside of
the bumps, the wavespeed slows down. The equation $\mathcal{L}u=u^{i}$ is
solved using GMRES for incident planewave with the wavenumber $k=5\pi$ and the
direction $(1,0)$. The domain of the computation is $[-6,6]^{2}$. In order to
measure the error, the reference solution is computed on the uniform
$1280\times 1280$ grid. For the convergence test presented in Table 6, we
measured $L_{\infty}$ error on the grid points. Note that, unlike the
convergence tests in previous sections, the reported errors are not just for
the convolution itself but includes the combined effect of the whole solution
procedure. Figure 2 depicts the total field the scattering problem.
$N$ | $E_{N}$ | $\log_{2}\frac{E_{N/2}}{E_{N}}$
---|---|---
80 | $1$. | $42\times 10^{-1}$ | —
160 | $2$. | $08\times 10^{-4}$ | $\;\>9.4$
320 | $2$. | $07\times 10^{-7}$ | $10.0$
640 | $7$. | $42\times 10^{-11}$ | $11.4$
Table 6. $L_{\infty}$ error of the solution of (4.8): $k=5\pi$ on
$[-6,6]^{2}$. Figure 2. Solution of the Lippmann-Schwinger equation (4.8):
$k=5\pi$ on $[-6,6]^{2}$.
### 4.4. Application: boundary integral equation
The presented quadrature rules can be applied for integral operators on curves
in ${\mathbb{R}}^{m}$. Let each curve be represented by a smooth periodic
parameterization: $s\in[0,L]\mapsto y(s)\in{\mathbb{R}}^{m}$. Assume
$y^{\prime}(s)\neq 0$ for every $s\in[0,L]$. For $s,t\in[0,L]$, define the
ratio,
(4.11) $\gamma(s,t)=\frac{r(s,t)}{|t-s|}\quad\text{where}\quad
r(s,t)=\|{y(t)-y(s)}\|.$
Then, $\gamma(t,t)=\|{y^{\prime}(t)}\|$ and $\gamma(s,\cdot)$ is smooth,
positive, and periodic on $[0,L]$. The logarithmic and the power-law
singularities can be rewritten as
(4.12) $\displaystyle\log(r)$ $\displaystyle=\log(|s-t|)+\log(\gamma)$ (4.13)
$\displaystyle r^{-\nu}$ $\displaystyle=\gamma^{-\nu}|s-t|^{-\nu}.$
Thus, the above radial singularities in ${\mathbb{R}}^{m}$ can be recast to
singularities of the same kind on $[0,L]$.
In this example, we solve the exterior scattering problem,
(4.14)
$\frac{1}{2}\,\psi(s)+\int_{\Gamma}D(s,t)\,\psi(t)\,\|{y^{\prime}(t)}\|\,dt-
ik\int_{\Gamma}S(s,t)\,\psi(t)\,\|{y^{\prime}(t)}\|\,dt=-u^{i}(s),$
where the combine integral equation is employed to avoid the interior
resonance issue. The single layer kernel $S$ is identical to the Helmholtz
kernel $K^{k}_{2}$ except the additional $\log(\gamma)$ in the smooth
remainder;
(4.15) $\displaystyle S(s,t)$
$\displaystyle\equiv\frac{i}{4}H^{(1)}_{0}(kr)\equiv\alpha_{s}(s,t)\log(|s-t|)+\widetilde{S}(s,t)$
(4.16) $\displaystyle\alpha_{s}(s,t)$ $\displaystyle=-\frac{J_{0}(kr)}{2\pi}$
(4.17) $\displaystyle\alpha_{s}(t,t)$ $\displaystyle=-\frac{1}{2\pi}$ (4.18)
$\displaystyle\widetilde{S}(t,t)$
$\displaystyle=\frac{i}{4}-\frac{\gamma_{euler}}{2\pi}-\frac{1}{2\pi}\log\left(\frac{k\|{y^{\prime}(t)}\|}{2}\right)$
where $\gamma_{euler}$ is the Euler constant. The double layer kernel $D$ is
given by
$\displaystyle D(s,t)$ $\displaystyle\equiv n(t)\cdot\nabla
K^{k}_{2}(r)=\frac{d}{dr}\left(\frac{i}{4}H^{(1)}_{0}(kr)\right)\left\\{\frac{n(t)\cdot(y(t)-y(s))}{r}\right\\}$
(4.19)
$\displaystyle=\frac{kr}{4}\bigg{\\{}Y_{1}(kr)-iJ_{1}(kr)\bigg{\\}}\bigg{\\{}\frac{n(t)\cdot(y(t)-y(s))}{r^{2}}\bigg{\\}},$
where $n(t)$ is the outward normal vector at $t$. Since (cf. §2.6.2)
(4.20) $z\,Y_{1}(z)=\frac{2zJ_{1}(z)}{\pi}\log(z)-\frac{2}{\pi}+O(z^{2}),$
the kernel can be factored by
(4.21) $\displaystyle D(s,t)$
$\displaystyle\equiv\alpha_{d}(s,t)\log(|s-t|)+\widetilde{D}(s,t)$ (4.22)
$\displaystyle\alpha_{d}(s,t)$
$\displaystyle=\frac{krJ_{1}(kr)}{2\pi}\bigg{\\{}\frac{n(t)\cdot(y(t)-y(s))}{r^{2}}\bigg{\\}}$
(4.23) $\displaystyle\alpha_{d}(t,t)$ $\displaystyle=0$ (4.24)
$\displaystyle\widetilde{D}(t,t)$ $\displaystyle=-\frac{1}{2\pi}\lim_{s\to
t}\bigg{\\{}\frac{n(t)\cdot(y(t)-y(s))}{r^{2}}\bigg{\\}}=\frac{c(t)}{2\pi\|{y^{\prime}(t)}\|^{2}}$
where $c(t)$ is the curvature at $t$. The above information is all we need to
construct the quadrature weights (at each target point). Beware that a fast
convolution cannot be applied for this case since the kernels are not
functions of $(t-s)$, hence, the resulting discrete operator is not a circular
matrix.
Figure 3 shows the total field constructed from the obtained solution $\psi$.
Each scatterer is a translation of the popular _kite_ shape,
(4.25) $y(t)=\left(\cos t+0.65\cos(2t)-0.65,1.5\sin t\right)\quad
t\in[0,2\pi].$
The incident planewave with $k=5\pi$ comes from the upper-left corner of the
domain $[-8,8]^{2}$ to the lower-right corner. For the interaction between
disconnected curves, the kernels become smooth periodic, hence, the usual
trapezoidal rule can be used.
Table 7 shows the error of the solution $\psi$ on the boundary and the error
of the far-field evaluate from the obtained $\psi$. The far-field signature
can be computed by
(4.26) $u_{\infty}(\widehat{x})=\frac{e^{-i\pi/4}}{\sqrt{8\pi
k}}\int_{\Gamma}\left(k\,n(t)\cdot\widehat{x}+k\right)e^{-ik\,\widehat{x}\cdot
y(t)}\,\psi(t)\,\|{y^{\prime}(t)}\|\,dt$
where $\widehat{x}=x/|x|$, $x\in{\mathbb{R}}^{2}$. The reference solution is
obtained with $N=640$.
| solution error | far-field error
---|---|---
$N$ | $E_{N}$ | $\log_{2}\frac{E_{N/2}}{E_{N}}$ | $E_{N}$ | $\log_{2}\frac{E_{N/2}}{E_{N}}$
80 | $1$. | $16\times 10^{-1}$ | — | $1$. | $45\times 10^{-1}$ | —
160 | $1$. | $61\times 10^{-5}$ | $12.8$ | $2$. | $06\times 10^{-7}$ | $19.4$
320 | $4$. | $17\times 10^{-14}$ | $28.5$ | $1$. | $34\times 10^{-14}$ | $23.9$
Table 7. $L_{\infty}$ errors of the exterior scattering problem (4.14):
$k=5\pi$. Figure 3. Total field of the exterior scattering problem (4.14) on
$[-8,8]^{2}$: $k=5\pi$. The obstacles are centered at $\pm 2$ on $x$ axis.
### 4.5. Conclusions
We have presented the construction scheme for corrected trapezoidal rules for
integral operators with weakly singular kernels. Numerical results show that
the quadrature rules converge as fast as the trapezoidal rule applied for non-
singular periodic integrands as predicted by the presented convergence
analysis. Especially, for smooth data, the quadrature rules exhibit super-
algebraic convergence. The examples in §4.3 and §4.4 demonstrate the
effectiveness of the proposed quadratures in the high precision solve of
integral equations.
## Appendix A Asymptotic Expansion of $M^{(\mu)}_{1}$
Dividing the domain of integral, we rewrite $M^{(\mu)}_{1}(\rho)$ as
(A.1)
$M^{(\mu)}_{1}(\rho)=M^{(\mu)}_{1}(a)+I_{a}(\mu,\rho)\quad\text{where}\quad
I_{a}(\mu,\rho)\equiv\frac{1}{\rho^{\mu}}\int_{a}^{\rho}t^{\mu-1}\cos(t)\,dt.$
Applying integration-by-parts $(2N)$ times, we obtain
(A.2)
$I_{a}(\mu,\rho)=I_{a}^{N}(\mu,\rho)+\frac{(-1)^{N}}{\rho^{\mu}}\prod_{\ell=1}^{2N}(\ell-\mu)\int_{0}^{\rho}t^{\mu-1-2N}\cos(t)\,dt,$
where
(A.3)
$\begin{split}I_{a}^{N}(\mu,\rho)&=\rho^{-1}\sin(\rho)\,P^{N}(\mu,\rho)-a^{-1}(a/\rho)^{\mu}\sin(a)\,P^{N}(\mu,a)\\\
&+\rho^{-2}\cos(\rho)\,Q^{N}(\mu,\rho)-a^{-2}(a/\rho)^{\mu}\cos(a)\,Q^{N}(\mu,a).\end{split}$
Polynomials $P^{N}$ and $Q^{N}$ are defined by
(A.4) $\displaystyle P^{N}(\mu,t)$
$\displaystyle=\sum_{\ell=0}^{N-1}C_{\ell}\,t^{-2\ell}\quad\text{where}\quad
C_{\ell}=(-1)^{\ell}\prod_{j=1}^{2\ell}(j-\mu)$ (A.5) $\displaystyle
Q^{N}(\mu,t)$ $\displaystyle=(\mu-1)\,P^{N}(\mu-1,t).$
Then, the relative error is given by
(A.6)
$\left|I_{a}(\mu,\rho)-I_{a}^{N}(\mu,\rho)\right|<\frac{(2N)!}{a^{2N}}\left|I_{a}(\rho)\right|.$
By choosing $a=14\pi$ and $N=13$, the relative error is less than $10^{-16}$.
Moreover, with $a$ being an integer multiple of $\pi$, $I^{N}_{a}$ has a
slightly simpler form.
## Appendix B Asymptotic Expansion of $M^{(\mu)}_{2}$
Similarly to $M^{(\mu)}_{1}$, we divide the domain of integral;
(B.1)
$M^{(\mu)}_{2}(\rho)=M^{(\mu)}_{2}(a)+I_{a}(\mu,\rho)\quad\text{where}\quad
I_{a}(\mu,\rho)\equiv\frac{1}{\rho^{\mu}}\int_{a}^{\rho}t^{\mu-1}J_{0}(t)\,dt$
Utilizing $t\,J_{0}(t)=(t\,J_{1}(t))^{\prime}$ and
$J_{1}(t)=-J^{\prime}_{0}(t)$, $(2N)$ applications of the integration-by-parts
result in
(B.2)
$I_{a}(\mu,\rho)=I_{a}^{N}(\mu,\rho)+\frac{(-1)^{N}}{\rho^{\mu}}\prod_{\ell=1}^{N}(2\ell-\mu)^{2}\int_{0}^{\rho}t^{\mu-1-2N}J_{0}(t)\,dt$
where
(B.3)
$\begin{split}I_{a}^{N}(\mu,\rho)&=\rho^{-1}J_{1}(\rho)\,P^{N}(\mu,\rho)-a^{-1}(a/\rho)^{\mu}J_{1}(a)\,P^{N}(\mu,a)\\\
&+\rho^{-2}J_{0}(\rho)\,Q^{N}(\mu,\rho)-a^{-2}(a/\rho)^{\mu}J_{0}(a)\,Q^{N}(\mu,a)\end{split}$
with
(B.4) $\displaystyle P^{N}(\mu,t)$
$\displaystyle=\sum_{\ell=0}^{N-1}C_{\ell}\,t^{-2\ell}\quad\text{where}\quad
C_{\ell}=(-1)^{\ell}\prod_{j=1}^{\ell}(2j-\mu)^{2}$ (B.5) $\displaystyle
Q^{N}(\mu,t)$
$\displaystyle=\sum_{\ell=0}^{N-1}(\mu-2\ell-2)\,C_{\ell}\,t^{-2\ell}.$
The relative error of the asymptotic expansion is given by
(B.6)
$\left|I_{a}(\mu,\rho)-I_{a}^{N}(\mu,\rho)\right|<\frac{4^{N}(N!)^{2}}{a^{2N}}\left|I_{a}(\rho)\right|.$
With $a=44.7593189976528217$ (which is the 14th zero of $J_{1}$) and $N=15$,
the relative error is less than $10^{-16}$.
## References
* [1] M. Abramowitz and I. A. Stegun (Eds.), Handbook of mathematical functions with formulas, graphs, and mathematical tables, 10th printing, Dover, New York, 1972.
* [2] B. Alpert, High-order quadratures for integral operators with singular kernels, J. Comput. Appl. Math., 60 (1995), pp. 367–378.
* [3] D. Bailey and P. Swarztrauber, The fractional Fourier transform and applications, SIAM Rev., 33 (1991), pp. 389–404.
* [4] J. Boyd, Sum-accelerated pseudospectral methods: the Euler-accelerated sinc algorithm, Appl. Numer. Math., 7 (1991), pp. 287–296.
* [5] J. Boyd, Chebyshev and Fourier spectral methods, Dover, New York, 2000.
* [6] J. Boyd, Large-degree asymptotics and exponential asymptotics for Fourier, Chebyshev and Hermite coefficients and Fourier transforms, Eng. Math., 63 (2009), pp. 355–399.
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* [8] E. Candés, L. Demanet, and L. Ying, Fast computation of Fourier integral operators, SIAM J. Sci. Comput., 29 (2007), pp. 2464 2493
* [9] D. Colton and R. Kress, Inverse acoustic and electromagnetic scattering theory, Springer-Verlag, Berlin, 1998.
* [10] T. Darden, D. York, and L. Pedersen, Particle mesh Ewald: An $N\cdot\log(N)$ method for Ewald sums in large systems, J. Chem. Phys., 98 (1993), pp. 10089–10092.
* [11] A. Dutt and V. Rokhlin, Fast Fourier transforms for nonequispaced data, SIAM J. Sci. Comput., 14 (1993), pp. 1368–1393.
* [12] L. Greengard and J.-Y. Lee, Accelerating the nonuniform fast Fourier transform, SIAM Rev., 46 (2004), pp. 443–454.
* [13] D. Huybrechs, On the Fourier extension of nonperiodic functions, SIAM J. Numer. Anal., 47 (2009), pp. 4326–4355.
* [14] S. Kapur and V. Rokhlin, High-order corrected trapezoidal quadrature rules for singular functions, SIAM J. Num. Anal., 34 (1997), pp. 1331–1356.
* [15] R. Kress, Linear integral equations, Springer-Verlag, New York, 1999.
* [16] J. Lyness, Adjusted forms of Fourier coefficient asymptotic expansion and applications in numerical quadrature, Math. Comput., 25 (1971), pp. 87–104.
* [17] A. MacLeod, Algorithm 757: MISCFUN, a software package to compute uncommon special functions, ACM Trans. Math. Softw., 22 (1996), pp. 288–301.
* [18] S. Minakshisundaram and O. Szasz, On the absolute convergence of multiple Fourier series, Trans. Amer. Math. Soc., 61 (1947), pp. 36–53.
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|
arxiv-papers
| 2012-11-26T19:46:45 |
2024-09-04T02:49:38.467766
|
{
"license": "Public Domain",
"authors": "Jae-Seok Huh, George Fann",
"submitter": "Jae-Seok Huh",
"url": "https://arxiv.org/abs/1211.6074"
}
|
1211.6076
|
11institutetext: Jae-Seok Huh 22institutetext: Computational Mathematics
Group, Computer Science and Mathematics Division, Oak Ridge National
Laboratory, Oak Ridge, TN 37831
Tel.: 1-865-574-3133
Fax: 1-865-241-4811
22email: [email protected]
# An Accurate and Quadrature-Free Evaluation of Multipole Expansion of
Functions Represented by Multiwavelets
Jae-Seok Huh
(Received: date / Accepted: date)
###### Abstract
We present formulae for accurate numerical conversion between functions
represented by multiwavelets and their multipole/local expansions with respect
to the kernel of the form, $e^{-\lambda r}/r$ (cf. GH02 ). The conversion is
essential for the application of fast multipole methods for functions
represented by multiwavelets. The corresponding separated kernels exhibit
near-singular behaviors at large $\lambda$. Moreover, a multiwavelet basis
function oscillates more wildly as its degree increases. These characteristics
in combination render any brute-force approach based on numerical quadratures
impractical. Our approach utilizes the series expansions of the modified
spherical Bessel functions and the Cartesian expansions of solid harmonics so
that the multipole–multiwavelet conversion matrix can be evaluated like a
special function. The result is a quadrature-free, fast, reliable, and machine
precision accurate scheme to compute the conversion matrix with predictable
sparsity patterns.
###### Keywords:
Multipole expansion Multiwavelet Screened Coulomb Potential
## 1 Introduction
Multiwavelets A90 ; ABGV02 developed originally by Alpert generalize Haar
wavelets with piecewise polynomial scale functions up to any given degree,
hence, enjoy higher order of accuracy in the representation of sufficiently
smooth functions. A detailed discussion on the sparse representation of
differential operators and exponential operators for evolution equations can
be found in ABGV02 . Recent advances in multi-resolution algorithms for
integral operators can be found in various articles including BM04 ; HFYGB04 ;
YFGHB04 ; YFGHB04b , which are based on the dimensional kernel separation via
representation of kernels by weighted sum of Gaussians, thus, applicable to a
variety of kernels in arbitrary dimensions.
In this paper, we focus ourselves on a more traditional, but very popular and
well-studied fast convolution algorithm – the fast multipole method. Our goal
is to establish the connection between multiwavelet representation and the
fast multipole method. The problem can be reduced to finding the multipole
expansion of multiwavelet basis functions.
Following one of the most recent version of the fast multipole algorithm
presented in GH02 , we begin with the kernel,
$G({\bm{x}},{\bm{y}})=\frac{e^{-\lambda\|{\bm{x}}-{\bm{y}}\|}}{\|{\bm{x}}-{\bm{y}}\|}$
(1)
where $\lambda$ is a non-negative real number. This kernel is the fundamental
solution of the linear differential operator,
$\Delta-\lambda^{2},$ (2)
which appears in various applications involving damped Coulomb forces. The
resulting potential is also known as Yukawa potential in nuclear physics. In
the derivation of the scheme, we rely on the multipole expansion formula for
strictly positive $\lambda$. However, it turns out that the first term of the
series representation of multipole expansion (with $\lambda>0$) corresponds to
the case of $\lambda=0$. Hence, readers can assume that our scheme can be
applied to _the_ Laplacian kernel also.
## 2 The Multipole Expansion of Multiwavelet Basis Functions
### 2.1 The Multipole Expansion
Denote by ${\langle{\cdot,\cdot}\rangle}$ the $L^{2}$ inner product of complex
functions on a bounded domain in ${\mathbb{R}}^{3}$. Let $\phi$ be a scalar
_source_ function supported on the domain. The potential generated by $\phi$
(outside of its support) is given by the following multipole series,
$\displaystyle\Phi({\bm{x}})$
$\displaystyle={\langle{G({\bm{x}},{\bm{y}}),\phi({\bm{y}})}\rangle}$ (3)
$\displaystyle=\sum_{p=0}^{\infty}\sum_{q=-p}^{p}M_{p}^{q}\,k_{p}(\lambda\|{\bm{x}}\|)\,Y_{p}^{q}({\widehat{\bm{x}}})$
(4)
where ${\widehat{\bm{x}}}={\bm{x}}/\|{\bm{x}}\|$ is identified to a point on
$S^{2}$. The multipole coefficients $M_{p}^{q}$ are given by
$M_{p}^{q}=8\,\lambda\,{\langle{i_{p}(\lambda\|{\bm{y}}\|)Y_{p}^{q}({\widehat{\bm{y}}}),\phi({\bm{y}})}\rangle}.$
(5)
The functions $i_{p}$ and $k_{p}$ are the modified spherical Bessel and Hankel
functions,
$\displaystyle i_{p}(r)$ $\displaystyle=\sqrt{\frac{\pi}{2r}}\,I_{p+1/2}(r)$
(6) $\displaystyle k_{p}(r)$
$\displaystyle=\sqrt{\frac{\pi}{2r}}\,K_{p+1/2}(r).$ (7)
Since $i_{p}(r)$ and $k_{p}(r)$ exhibit exponential growth and decay
respectively, an algorithm based on the above formula is likely to experience
an overflow/underflow. To avoid the issue, as suggested in GH02 , we replace
$i_{p}$ and $k_{p}$ in the formula by their scaled forms,
$\displaystyle{\widehat{i}^{\lambda_{0}}}_{p}(\lambda\,r)$
$\displaystyle=i(\lambda\,r)/\lambda_{0}^{p}$ (8)
$\displaystyle{\widehat{k}^{\lambda_{0}}}_{p}(\lambda\,r)$
$\displaystyle=k(\lambda\,r)\cdot\lambda_{0}^{p}.$ (9)
Assuming $O(r)=1$ by appropriate geometric scaling, the appropriate choice of
$\lambda_{0}$ is $\lambda$ itself, however, in order to maintain the
generality, we keep clear notational distinction between them.
###### Remark 1
To avoid any confusion, the normalization of spherical harmonics $Y_{p}^{q}$
used in this paper needs to be clearly stated before we begin any formulation.
We utilize exactly the same form presented in GH02 , that is,
$Y_{p}^{q}({\widehat{\bm{y}}})=\sqrt{\frac{2p+1}{4\pi}}\sqrt{\frac{(p-|q|)!}{(p+|q|)!}}\,P_{p}^{|q|}(\cos\theta)\,e^{i\,q\,\phi}.$
(10)
An obvious advantage of using this form is that
$Y_{p}^{-q}=\overline{Y_{p}^{q}}$, hence, in (5), $Y_{p}^{q}$ appears without
minus sign in front of $q$. We can observe, later in this paper, that this
property provides us with a better symmetry/sparsity pattern of the multipole
expansion matrix.
### 2.2 Symmetries
The key equation in the above formula is the multipole expansion (5). For
notational simplicity, in this paper, we omit the constant $8\,\lambda$, which
can be multiplied afterward. We denote the product of
${\widehat{i}^{\lambda_{0}}}_{p}$ and $Y_{p}^{q}$ in (5) by $Q_{p}^{q}$;
$Q_{p}^{q}(\lambda,{\bm{y}})\equiv{\widehat{i}^{\lambda_{0}}}_{p}(\lambda\|{\bm{y}}\|)\,Y_{p}^{q}({\widehat{\bm{y}}})$
(11)
The function $Q_{p}^{q}$ should be replaced with the regular solid harmonics
when $\lambda=0$. In the following section, we can observe that the first term
of the series representation of $Q_{p}^{q}$ is the regular solid harmonics. It
is obvious that
$Q_{p}^{q}(\lambda,\alpha\,{\bm{y}})=Q_{p}^{q}(\alpha\,\lambda,{\bm{y}}).$
(12)
The function $Q_{p}^{q}$ enjoys two useful symmetries: firstly, from the
normalization of $Y_{p}^{q}$ employed in this paper, it follows that
$Q_{p}^{-q}=\overline{Q_{p}^{q}}.$ (13)
Secondly, by change of variables $\phi\rightarrow\pi/2-\phi$ in $Y_{p}^{q}$,
we can obtain
$Q_{p}^{q}(\lambda,y_{2},y_{1},y_{3})=i^{q}\overline{Q_{p}^{q}}(\lambda,y_{1},y_{2},y_{3}).$
(14)
As a result, a multipole expansion matrix presented in this paper possesses
similar symmetries, which we utilize to reduce the number of elements we have
to compute.
### 2.3 Multiwavelet Basis Functions
In this section, we briefly introduce multiwavelet representation of
functions. A detailed discussion on the subject can be found in ABGV02 .
Denote by ${\bm{k}}$ non-negative multi-indices and by $\phi^{\bm{k}}$ multi-
dimensional orthonormal polynomials of degree $k_{i}$ in $i$th dimension. We
further assume that the generating functions $\phi^{\bm{k}}$ are constructed
by the Cartesian product of 1-d orthonormal polynomials on $[-1,1]$,
$\phi^{\bm{k}}({\bm{y}})=\sum_{i=1}^{d}\,\phi^{k_{i}}(y_{i}).$ (15)
The above $\phi^{\bm{k}}$ generate the orthonormal multiwavelet basis
functions at arbitrary level $n=0,1,\ldots$, and translation characterized by
multi-indices ${\bm{l}}=(l_{1},\ldots,\l_{d})$ with $l_{i}=0,\ldots,2^{n}-1$
by formula,
$\phi^{\bm{k}}_{n,{\bm{l}}}({\bm{x}})=\begin{cases}\sqrt{2}^{d(n+1)}\phi^{\bm{k}}(2\,(2^{n}{\bm{x}}-{\bm{l}})-1)&\text{on
$b_{(n,{\bm{l}})}$}\\\ 0&\text{elsewhere}\end{cases}$ (16)
where $b_{(n,{\bm{l}})}=\prod_{i=1}^{d}[2^{-n}l_{i},2^{-n}(l_{i}+1)]$. In this
paper, we take $[0,1]^{d}$ ($=b_{0}$ by definition) as the computational
domain. Beware that we assume that the 1-d generating functions $\phi^{k_{i}}$
are orthogonal polynomials defined on $[-1,1]$ (not on $[0,1]$). This choice
of _unshifted_ orthogonal polynomials as the generating functions is to
simplify the notations in multipole related formulae; we have to evaluate
multipole expansions with respect to the center of each $b_{(n,{\bm{l}})}$.
###### Remark 2
The term _orthogonal polynomials_ can be a source of confusion, which we need
to clarify before we present any related formula. By the term, we mean a
sequence of polynomials $\phi^{k}$ of degree $k$ orthogonal to each other with
respect to an underlying weighting function (as in “orthogonal polynomials and
quadratures”). Since a non-trivial weighting function loses its meaning under
scaling, readers may think $\phi^{k}$ a synonym of (normalized) Legendre
polynomial of degree $k$. This limitation of generating functions to
orthogonal polynomials greatly simplifies the resulting formulae and makes the
conversion matrix more sparse.
###### Remark 3
There can be different choices of polynomial basis which are mutually
orthogonal such as the _interpolating basis_ presented in ABGV02 . Conversion
between them and Legendre-generated basis is not complicated. The advantage
(by symmetry and sparsity) of using orthogonal polynomials exceeds the
additional cost of basis conversion.
### 2.4 The Multipole Expansion of $\phi^{\bm{k}}_{(n,{\bm{l}})}$
For any $p=0,1,\ldots$ and $q=-p,\ldots,p$, define
$E^{(p,q)}_{\bm{k}}(n,\lambda)$ by
$E^{(p,q)}_{\bm{k}}(n,\lambda)={\langle{Q_{p}^{q}(\lambda,{\bm{y}}-{\bm{c}}_{(n,{\bm{l}})}),\phi^{\bm{k}}_{(n,{\bm{l}})}({\bm{y}})}\rangle}_{b_{(n,{\bm{l}})}}$
(17)
where ${\bm{c}}_{(n,{\bm{l}})}$ is the center of $b_{(n,{\bm{l}})}$. The above
equation gives the multipole coefficient $M_{p}^{q}$ (without $8\,\lambda$) of
(5) with respect to the center, ${\bm{c}}_{(n,{\bm{l}})}$. The inner product
can be scaled and translated to the standard domain $[-1,1]^{3}$,
$\displaystyle E^{(p,q)}_{\bm{k}}(n,\lambda)$
$\displaystyle=\frac{1}{\sqrt{2}^{3(n+1)}}\,{\langle{Q_{p}^{q}(\lambda,2^{-(n+1)}{\bm{y}}),\phi^{\bm{k}}({\bm{y}})}\rangle}$
(18)
$\displaystyle=\frac{1}{\sqrt{2}^{3n}}\,E^{(p,q)}_{\bm{k}}(0,\lambda/2^{n})$
(19)
where
$E^{(p,q)}_{\bm{k}}(0,\lambda_{n})=\frac{1}{\sqrt{2}^{3}}\,{\langle{Q_{p}^{q}(\lambda_{n}/2,\cdot),\phi^{\bm{k}}}\rangle}.$
(20)
Thus, we are required to evaluate (20) for arbitrary $\lambda_{n}$ which
depends on $\lambda$ and the level, $n$. Viewing $(p,q)$ as a row multi-index
and ${\bm{k}}$ as a column multi-index, $E^{(p,q)}_{\bm{k}}$ acts as the
conversion (multipole expansion) matrix for multiwavelet represented
functions; let $s^{\bm{k}}_{(n,{\bm{l}})}$ be multiwavelet coefficients for a
fixed $(n,{\bm{l}})$. The multipole expansion centered at
${\bm{c}}_{(n,{\bm{l}})}$ of the function
$\sum_{\bm{k}}s^{\bm{k}}_{(n,l)}\,\phi^{\bm{k}}_{(n,{\bm{l}})}$ is given by
the matrix-vector multiplication,
$\sum_{\bm{k}}E^{(p,q)}_{\bm{k}}(n,\lambda)\,s^{\bm{k}}_{(n,{\bm{l}})}.$
The same matrix can be used for the conversion from a local expansion to its
multiwavelet representation. Consider a local expansion with the coefficients
$L_{p}^{q}$,
$\Phi({\bm{x}})=\sum_{p=0}^{\infty}\sum_{q=-p}^{p}\,L_{p}^{q}\,Q_{p}^{q}(\lambda,{\bm{x}}).$
(21)
The projection of $\Phi$ on to the span of the multiwavelet basis is, from the
orthonormality, given by
$s^{\bm{k}}_{(n,{\bm{l}})}={\langle{\overline{\Phi},\phi^{\bm{k}}_{(n,{\bm{l}})}}\rangle}=\sum_{(p,q)}\,\overline{E^{(p,q)}_{\bm{k}}}(n,\lambda)\,L_{p}^{q},$
(22)
i.e., by the multiplication with the conjugate transpose of
$E^{(p,q)}_{\bm{k}}$.
### 2.5 Symmetries
Recall the symmetries of $Q_{p}^{q}$. The following two conditions are the
immediate consequences of (13) and (14).
$E^{(p,-q)}_{\bm{k}}(n,\lambda)=\overline{E^{(p,q)}_{\bm{k}}}(n,\lambda)$ (23)
and
$E^{(p,q)}_{(k_{2},k_{1},k_{3})}(n,\lambda)=(-i)^{q}\,\overline{E^{(p,q)}_{(k_{1},k_{2},k_{3})}}(n,\lambda).$
(24)
In a later section, we will show that, depending on $(p,q,{\bm{k}})$, (1)
$E^{(p,q)}_{\bm{k}}$ is either real or pure imaginary and (2) has pre-
determined sparsity patterns. Combined with the above symmetries, we recommend
the following storage for the multipole expansion matrices. For each level
$n$, we compute $E^{(p,q)}_{\bm{k}}$ for $q\geq 0$ and $k_{2}>k_{1}$, and
store non-negative $q$ portion of the matrices in two sparse matrices, one for
real and the other for imaginary. Separated storage is simple and advantageous
in the implementation; (i) since each element is either real or imaginary, the
two sparse matrices have disjoint index sets; it does not require any
additional storage or computation cost due to duplicated indices. (ii) For
rows with $q<0$, the multiplication can be omitted; for a complex vector
$s^{\bm{k}}_{(n,{\bm{l}})}$,
$\sum_{{\bm{k}}}E^{(p,-q)}_{{\bm{k}},\Re/\Im}\,s^{\bm{k}}_{(n,{\bm{l}})}=\pm\sum_{{\bm{k}}}E^{(p,q)}_{{\bm{k}},\Re/\Im}\,s^{\bm{k}}_{(n,{\bm{l}})}$
with the negative sign for imaginary matrix.
### 2.6 Numerical Issues
There are three major numerical issues which make the evaluation of
$E^{(p,q)}_{\bm{k}}(n,\lambda)$ non-trivial.
1. (1)
Non-homogeneity of $\bm{Q_{p}^{q}(\lambda,\cdot)}$: Unlike regular solid
harmonics, $Q_{p}^{q}$ are not homogeneous. Since we cannot extract $\lambda$
out of the integral, we have to build different
$E^{(p,q)}_{\bm{k}}(n,\lambda)$ depending on $\lambda$ and $n$, which rules
out the possibility of utilizing a precomputed table. Since they are not even
polynomials, there is no simple quadrature which produces the exact integral.
Any naive approach using adaptive quadrature becomes impractical for the
following reasons.
2. (2)
Rapid growth of $\bm{Q_{p}^{q}(\lambda,\cdot)}$: The function
$i_{p}(\lambda\,r)$ grows exponentially. Scaling by using
${\widehat{i}^{\lambda_{0}}}_{p}(\lambda\,r)$ helps preventing overflow.
However, the function still exhibits near singularity for large $\lambda$.
3. (3)
Oscillating behavior of $\bm{\phi^{\bm{k}}}$: Although they are polynomials,
$\phi^{k}$ have all their zeros on $(-1,1)$. Hence, for large $k$, an adaptive
integrator will encounter with highly oscillating integrands.
From the above characteristics, any adaptive integration requires a large
amount of computation, or simply it fails to converge especially for highly
oscillating cases. Beware that, to build $N$ conversion matrices (up to depth
level $N-1$) for the multipole expansion (up to degree $P$) of functions
represented by multiwavelets (of degree up to $K$), we have to compute
$O(N\times P^{2}\times K^{3})$ elements!
Our approach begins with rewriting the $Q_{p}^{q}$ in a series form. Each term
in the resulting power series involves a regular solid harmonics weighted by
an even power of $\|{\bm{x}}\|$, hence, a homogeneous polynomial in
${\mathbb{R}}^{3}$. The Cartesian expansion of this polynomial can be
explicitly written and its projection on multiwavelet basis can be obtained
exactly without using any numerical quadrature.
## 3 The Series Form
In this section, we present a series representation of the multipole expansion
matrix, $E^{(p,q)}_{\bm{k}}(n,\lambda)$. We begin with the identity,
$I_{\alpha}(r)=\sum_{m=0}^{\infty}\frac{1}{m!\,\Gamma(m+\alpha+1)}\left(\frac{r}{2}\right)^{2m+\alpha}.$
(25)
Utilizing $\Gamma(m+1/2)=\sqrt{\pi}\,(2m)!/(4^{m}\,m!)$, we obtain
${\widehat{i}^{\lambda_{0}}}_{p}(\lambda\,r)=r^{p}\left(\frac{\lambda}{\lambda_{0}}\right)^{p}\sum_{m=0}^{\infty}\frac{1}{m!\,(2m+2p+1)!!}\,\left(\frac{\lambda\,r}{\sqrt{2}}\right)^{2m}.$
(26)
Therefore $Q_{p}^{q}$ can be written in power series given by
$Q_{p}^{q}(\lambda,{\bm{x}})=\left(\frac{\lambda}{\lambda_{0}}\right)^{p}\sum_{m=0}^{\infty}\frac{1}{m!\,(2m+2p+1)!!}\,\left(\frac{\lambda^{2}}{2}\right)^{m}\|{\bm{x}}\|^{2m+p}\,Y_{p}^{q}({\widehat{\bm{x}}}).$
(27)
Define $R_{p,m}^{q}$ by
$R_{p,m}^{q}({\bm{x}})={C_{Y}}(p,q)^{-1}\,\|{\bm{x}}\|^{2m+p}\,Y_{p}^{q}({\widehat{\bm{x}}})$
(28)
where
${C_{Y}}(p,q)=\sqrt{\frac{2p+1}{4\pi}}\,\sqrt{\frac{(p-|q|)!}{(p+|q|)!}},$
(29)
Notice that the function $R_{p,m}^{q}$ is a regular solid harmonics multiplied
by $\|{\bm{x}}\|^{2m}$, hence, a homogeneous polynomial of degree $(2m+p)$.
The factor ${C_{Y}}(p,q)^{-1}$ simplifies the Cartesian expansion of
$R_{p,m}^{q}$, which we introduced in the following section.
From the above series representation, $E^{(p,q)}_{\bm{k}}(0,\lambda_{n})$ is
given by
$\displaystyle E^{(p,q)}_{\bm{k}}(0,\lambda_{n})$
$\displaystyle=\frac{1}{\sqrt{2}^{3}}{\langle{Q_{p}^{q}(\lambda_{n}/2,\cdot),\phi^{\bm{k}}}\rangle}$
$\displaystyle={C_{E}}(\lambda_{n}/\lambda_{0},p,q)\sum_{m=0}^{\infty}A_{m}(p)\,\left(\frac{\lambda_{n}^{2}}{8}\right)^{m}I_{m}(p,q,{\bm{k}})$
(30)
where
${C_{E}}(\lambda_{n}/\lambda_{0},p,q)=\frac{{C_{Y}}(p,q)}{\sqrt{2}^{3}\,(2p+1)!!}\left(\frac{\lambda_{n}}{\lambda_{0}}\right)^{p},$
(31)
$A_{m}(p)=\frac{(2p+1)!!}{m!\,(2m+2p+1)!!},$ (32)
$I_{m}(p,q,{\bm{k}})=\int_{[-1,1]^{3}}\overline{R_{p,m}^{q}}({\bm{x}})\,\phi^{\bm{k}}({\bm{x}})\,d{\bm{x}}.$
(33)
The factor $(2p+1)!!$ in $A_{m}(p)$ is added for the normalization,
$A_{0}(p)=1$.
In (30), $\lambda_{n}$ is now taken out of the integral. We will observe that
the $\lambda$-independent term, $I_{m}(p,q,{\bm{k}})$, can be further reduced
to a finite sum of products of 1-d integrals with two integer parameters,
namely, ${\widehat{I}}_{k}^{l}$. We can construct ${\widehat{I}}_{k}^{l}$
exactly without using any numerical quadrature via the recurrence relation of
orthogonal polynomials. Our strategy is to tabulate ${\widehat{I}}_{k}^{l}$
and use the table to evaluate the series (30) for various $\lambda$, $n$, $p$,
$q$, and ${\bm{k}}$.
### 3.1 Properties of $I_{m}$ and the Convergence Criterion
Most of the properties of $I_{m}$ presented in this section will be explained
in detail in §4. For a more comprehensive presentation, we think it would be
more appropriate to discuss the behavior of (30) prior to the presentation of
detailed formulae for $I_{m}$. Followings are the summary of the relevant
properties:
1. 1.
$I_{m}(p,q,{\bm{k}})$ is either real or pure imaginary depending only on
${\bm{k}}$.
2. 2.
$I_{m}(p,q,{\bm{k}})=0$ if $2m<k_{x}+k_{y}+k_{z}-p$.
3. 3.
Sign of $I_{m}(p,q,{\bm{k}})$ is determined by $q$ only and is independent of
$m$.
4. 4.
$|I_{m}(p,q,{\bm{k}})|$ is monotonically increasing as $m$ increases.
5. 5.
$I_{m+1}(p,q,{\bm{k}})/I_{m}(p,q,{\bm{k}})\rightarrow 3$ as
$m\rightarrow\infty$.
###### Remark 4
Property (5) can be supported by the following estimate: Since
$|{C_{Y}}^{-1}\overline{Y_{p}^{q}}|=|P_{p}^{|q|}(\cos\theta)\,e^{i\,q\,\phi}|\leq
1$,
$\displaystyle|I_{m}(p,q,{\bm{k}})|$
$\displaystyle\leq\|\phi^{\bm{k}}\|_{\infty}\,\int_{[-1,1]^{3}}\|{\bm{x}}\|^{2m+p}d{\bm{x}}$
$\displaystyle<4\pi\,\|\phi^{\bm{k}}\|_{\infty}\,\int_{0}^{\sqrt{3}}r^{2m+p+2}dr=\frac{12\pi\,\sqrt{3}^{p}\,3^{m}}{(2m+p+3)}\,\|\phi^{\bm{k}}\|_{\infty}$
Thus, the series consists of two parts: $A_{m}$ decreasing factorially and
$(\lambda^{2}/8)^{m}\,I_{m}$ which behaves asymptotically
$\sim(3\lambda^{2}/8)^{m}$. Their product
$C_{m}=A_{m}(\lambda^{2}/8)^{m}I_{m}$ has a fixed sign for a fixed
$(p,q,{\bm{k}})$ independently of $m$. Hence, the partial sum of the series
increases (decreases) monotonically to the upper (lower) bound which is
potentially huge in the absolute sense. The non-alternating feature of the
series suppresses any necessity of considerations of cancellation errors, and
suggest the following simple convergence criterion: for given absolute and
relative tolerances $\epsilon_{a}$ and $\epsilon_{r}$, stop the summation if
$|C_{M}|<\epsilon_{a}\quad\text{or}\quad|C_{M}|<\epsilon_{r}\,\left|\sum_{m=0}^{M}C_{m}\right|.$
(34)
We can numerically observe that the number of terms to convergence $M$ is
$O(\lambda)$ in a conservative estimation. For example, for
$\epsilon_{r}=10^{-16}$, $M\sim\lambda$ and slightly smaller if $\lambda$ is
large; e.g., when $\lambda\sim 300$, $M\sim 200$. The condition (2) combined
with the convergence criterion provides us with additional sparsity of
$E^{(p,q)}_{\bm{k}}$; if $2M<k_{x}+k_{y}+k_{z}-p$, the corresponding
$E^{(p,q)}_{\bm{k}}$ can be considered to be zero.
###### Remark 5
We represent $E^{(p,q)}_{\bm{k}}$ like a special function of $\lambda$ with
exponential growth. The number of terms $M$ can grow indefinitely as $\lambda$
increases. Although, in many practical applications, $\lambda$ are quite
limited and $\lambda_{n}=\lambda/2^{n}$ decreases as the depth of the
multiwavelet representation increases, a more complete algorithm requires an
asymptotic expansion of $E^{(p,q)}_{\bm{k}}$ with respect to $\lambda$. Yet,
we haven’t found a closed formula for the asymptotic expansion, which is an
on-going work.
## 4 The Formula for $I_{m}$ and the Sparsity Pattern
In this section, we present an explicit Cartesian expansion form of
$\overline{R_{p,m}^{q}}$ in $I_{m}$. Each term can be written as a product of
1-d integrals which can be evaluated exactly by the recurrence relations of
the orthogonal polynomials $\phi^{k}$. We begin with the series form of the
spherical harmonics. With the Rodrigues’ formula, the associated Legendre
functions $P_{p}^{|q|}$ in $Y_{p}^{q}$ can be written as
$\displaystyle P_{p}^{|q|}(z)$
$\displaystyle=\frac{(-1)^{|q|}}{2^{p}\,p!}\,(1-z^{2})^{{|q|}/2}\,\frac{d^{p+{|q|}}}{dz^{p+{|q|}}}\,(z^{2}-1)^{p}$
$\displaystyle=\frac{(-1)^{|q|}}{2^{p}}\,(1-z^{2})^{{|q|}/2}\,\sum_{\nu=0}^{\lfloor\frac{p-{|q|}}{2}\rfloor}\,\frac{(-1)^{\nu}(2p-2\nu)!}{\nu!\,(p-\nu)!\,(p-{|q|}-2\nu)!}\,z^{(p-{|q|}-2\nu)}.$
(35)
Hence, using notations ${\bm{x}}=(x,y,z)$, $r=\|{\bm{x}}\|$, and
$s={\operatorname{sign}}(q)$,
$\overline{R_{p,m}^{q}}({\bm{x}})=r^{2m+p}P_{p}^{|q|}(z/r)\left(\frac{x-s\,i\,y}{\sqrt{r^{2}-z^{2}}}\right)^{|q|}\\\
=\frac{(-1)^{|q|}}{2^{p}}\,(x-s\,i\,y)^{|q|}\sum_{\nu=0}^{\lfloor\frac{p-{|q|}}{2}\rfloor}\frac{(-1)^{\nu}(2p-2\nu)!}{\nu!\,(p-\nu)!\,(p-{|q|}-2\nu)!}\,r^{2(m+\nu)}\,z^{(p-{|q|}-2\nu)}$
(36)
By expanding $(x-s\,i\,y)^{|q|}$ and $r^{2(m+\nu)}$, we obtain
$=(s\,i)^{|q|}\\\
\sum_{\mu=0}^{|q|}(s\,i)^{\mu}a_{\mu}\sum_{\nu=0}^{\lfloor\frac{p-{|q|}}{2}\rfloor}b_{\nu}\sum_{\alpha=0}^{m+\nu}c_{\nu\alpha}\,z^{(p-{|q|}+2m-2\alpha)}\sum_{\beta=0}^{\alpha}d_{\alpha\beta}\,y^{({|q|}+2\beta-\mu)}\,x^{(2\alpha-2\beta+\mu)}$
(37)
where the coefficients are given by
$\displaystyle a_{\mu}$ $\displaystyle=\begin{pmatrix}q\\\ \mu\end{pmatrix}$
(38) $\displaystyle b_{\nu}$
$\displaystyle=\frac{(-1)^{\nu}}{2^{\nu}}\frac{(2p-2\nu-1)!!}{\nu!\,(p-q-2\nu)!}$
(39) $\displaystyle c_{\nu\alpha}$ $\displaystyle=\begin{pmatrix}m+\nu\\\
\alpha\end{pmatrix}$ (40) $\displaystyle d_{\alpha\beta}$
$\displaystyle=\begin{pmatrix}\alpha\\\ \beta\end{pmatrix}$ (41)
where we use the definition, $(-1)!!=0!!=1$.
### 4.1 The Formula
From (37), we obtain our final formula for $I_{m}$:
$I_{m}(p,q,{\bm{k}})=(s\,i)^{|q|}\,I_{m}^{(1)}(p,q,{\bm{k}})+(s\,i)^{{|q|}+1}\,I_{m}^{(2)}(p,q,{\bm{k}})$
(42)
where
$I_{m}^{(1)}(p,q,{\bm{k}})=\sum_{\mu=0}^{\lfloor\frac{{|q|}}{2}\rfloor}(-1)^{\mu}a_{2\mu}\sum_{\nu=0}^{\lfloor\frac{p-{|q|}}{2}\rfloor}b_{\nu}\\\
\cdot\sum_{\alpha=0}^{m+\nu}c_{\nu\alpha}\,{\widehat{I}}_{k_{z}}^{(p-{|q|}+2m-2\alpha)}\sum_{\beta=0}^{\alpha}d_{\alpha\beta}\,{\widehat{I}}_{k_{y}}^{({|q|}+2\beta-2\mu)}\,{\widehat{I}}_{k_{x}}^{(2\alpha-2\beta+2\mu)},$
(43)
$I_{m}^{(2)}(p,q,{\bm{k}})=\sum_{\mu=0}^{\lfloor\frac{{|q|}-1}{2}\rfloor}(-1)^{\mu}a_{2\mu+1}\sum_{\nu=0}^{\lfloor\frac{p-{|q|}}{2}\rfloor}b_{\nu}\\\
\cdot\sum_{\alpha=0}^{m+\nu}c_{\nu\alpha}\,{\widehat{I}}_{k_{z}}^{(p-{|q|}+2m-2\alpha)}\sum_{\beta=0}^{\alpha}d_{\alpha\beta}\,{\widehat{I}}_{k_{y}}^{({|q|}+2\beta-2\mu-1)}\,{\widehat{I}}_{k_{x}}^{(2\alpha-2\beta+2\mu+1)},$
(44)
and
${\widehat{I}}_{k}^{l}=\int_{-1}^{1}\zeta^{l}\phi^{k}(\zeta)d\zeta.$ (45)
Note that $I_{m}^{(1)}$ and $I_{m}^{(2)}$ are real functions and, with factors
$(s\,i)^{|q|}$ and $(s\,i)^{{|q|}+1}$ respectively, they determine real and
imaginary parts of $I_{m}$ separately. The above representation of $I_{m}$ by
two separate parts $I_{m}^{(1)}$ and $I_{m}^{(2)}$ is to signify the following
very useful fact: at least, one of $I_{m}^{(1)}$ and $I_{m}^{(2)}$ vanishes
for any $(p,q,{\bm{k}})$ independently of $m$, which implies that
$E^{(p,q)}_{\bm{k}}$ is either a real or a pure imaginary.
Moreover, depending on the parameter $(p,q,{\bm{k}})$, many of $I_{m}$ vanish,
which results in the nice sparsity of the multipole expansion matrix. These
properties are the immediate consequence of the following properties of
orthogonal polynomials.
#### (1) Oddity
Any orthogonal polynomial $\phi^{k}$ with symmetric domain and weight is even
(odd) if the degree $k$ is even (odd). Hence,
${\widehat{I}}_{k}^{l}=0\quad\text{if $(l+k)$ is odd.}$
Since $m$ always appears in the equation with the factor of 2, any consequence
of the oddity condition is $m$-independent; the resulting sparsity of
$E^{(p,q)}_{\bm{k}}$ is pre-determined by $(p,q,{\bm{k}})$ only (independently
of the level $n$ and $\lambda$). We can observe that
1. (a)
$I_{m}^{(2)}=0$ if $q=0$.
2. (b)
$I_{m}^{(1)}=0$ if $k_{x}$ is odd or $({|q|}+k_{y})$ is odd or
$(p+{|q|}+k_{z})$ is odd.
3. (c)
$I_{m}^{(2)}=0$ if $k_{x}$ is even or $({|q|}+k_{y})$ is even or
$(p+{|q|}+k_{z})$ is odd.
Therefore,
$E^{(p,q)}_{\bm{k}}=0\quad\text{if}\quad\begin{cases}\text{$(k_{z}+p+{|q|})$
is odd}\\\ \text{$(k_{x}+k_{y}+{|q|})$ is odd}\\\ \text{$q=0$ and at least one
of $k_{x}$ and $k_{y}$ is odd}\end{cases}$ (46)
Suppose $E^{(p,q)}_{\bm{k}}\neq 0$ from the above test. Then, the oddity of
$k_{x}$ must be the same as the oddity of $({|q|}+k_{y})$, which results in
$I_{m}=c\cdot\begin{cases}I_{m}^{(2)}&\text{if $k_{x}$ is odd}\\\
I_{m}^{(1)}&\text{if $k_{x}$ is even}\end{cases}\quad
c=\begin{cases}(-1)^{\lfloor\frac{{|q|}}{2}\rfloor}\,{\operatorname{sign}}(q)\,i&\text{if
$k_{y}$ is odd}\\\ (-1)^{\lceil\frac{{|q|}}{2}\rceil}&\text{if $k_{y}$ is
even}\end{cases}$ (47)
Notice, $E^{(p,q)}_{\bm{k}}$ is real (imaginary) if $k_{y}$ is even (odd).
The following table illustrates the sparsity of the multipole expansion matrix
for parameters: $0\leq p\leq 10$, $0\leq q\leq p$, and $0\leq k_{x,y,z}\leq
10$. We can observe that about a quarter of elements are non-zeroes. (See
Table 1.)
total elements | real non-zeroes | imaginary non-zeroes
---|---|---
87846 | 12186 (13.9%) | 8450 (9.6%)
Table 1: $\lambda$-independent sparsity estimated by the oddity condition.
#### (2) Moment condition
Recall the moment conditions satisfied by orthogonal polynomials.
${\widehat{I}}_{k}^{l}=0\quad\text{if $l<k$.}$
Consider a term with a fixed set of indices
$(p,q,{\bm{k}},\mu,\nu,\alpha,\beta)$ in (42). The term vanishes if
$k_{z}>p-{|q|}+2m-2\alpha\quad\text{or}\quad
k_{y}>{|q|}+2\beta-\mu\quad\text{or}\quad k_{x}>2\alpha-2\beta+\mu$
which is true if
$k_{x}+k_{y}+k_{z}>p+2m.$
Thus,
$I_{m}(p,q,{\bm{k}})=0\quad\text{if}\quad 2m<k_{x}+k_{y}+k_{z}-p$ (48)
This condition is $m$-dependent, hence, cannot be used to pre-determine the
sparsity pattern of $E^{(p,q)}_{\bm{k}}$. However, it still can affect the
sparsity for a given $\lambda$; suppose that the convergence criterion (34) is
satisfied at $M$ for $2M<k_{x}+k_{y}+k_{z}-p$. Then, the corresponding
$E^{(p,q)}_{\bm{k}}(n,\lambda)$ is effectively zero. Since $M\sim\lambda$ and
$E^{(p,q)}_{\bm{k}}(n,\lambda)=constant\cdot
E^{(p,q)}_{\bm{k}}(0,\lambda/2^{n})$, the multipole expansion matrix becomes
more sparse as $\lambda$ decreases and as $n$ increases. Table 2 shows the
number of vanishing elements (for $n=0$) among those predicted to be non-
zeroes by the oddity condition. With parameters, $0\leq p\leq 10$, $0\leq
q\leq p$, and $0\leq k_{x,y,z}\leq 10$, the number of elements is 87846 and
the numbers of non-zero real and imaginary elements (predicted by the oddity
condition) are 12186 and 8450 respectively (same as the above example).
Tolerances are $\epsilon_{a}=\epsilon_{r}=10^{-16}$.
$\lambda$ | additional real zeroes | additional imaginary zeroes
---|---|---
1 | 9567 | 6679
2 | 8813 | 6154
4 | 7478 | 5235
6 | 6340 | 4439
8 | 5439 | 3775
10 | 4630 | 3203
50 | 1309 | 851
100 | 1301 | 848
200 | 1273 | 835
300 | 1251 | 824
Table 2: $\lambda$-dependent sparsity estimated from the moment condition.
The additional sparsity decreases as $\lambda$, hence $M$, increases. There
are two factors which controls $M$ and, hence, the the additional sparsity –
the absolute tolerance $\epsilon_{a}$ and the relative tolerance
$\epsilon_{r}$. Among them, the contribution of $\epsilon_{r}$ decreases
rapidly and becomes quite negligible when $\lambda\gg\max\,k_{i}$. However,
the contribution of $\epsilon_{a}$ is persistent. From the table, we can
observe that the additional sparsities by $\epsilon_{a}$ are $\sim 1200$ for
the real matrix and $\sim 800$ for the imaginary matrix.
Also, the moment condition enhances the computational efficiency slightly. We
can simply skip the evaluation of $I_{m}$ if
$m<\lceil(k_{x}+k_{y}+k_{z}-p)/2\rceil$. Like the sparsity by $\epsilon_{r}$,
the effect decreases quite rapidly as $\lambda$ increases. However, in
practical applications of fast multipole and multiwavelet representation, the
levels of terminal boxes where we need to perform the expansion is likely to
be high. Hence, the additional sparsity by $\epsilon_{r}$ should not be
considered insignificant.
### 4.2 The Laplacian Kernel ($\bm{\lambda=0}$)
In this case, $Q_{p}^{q}$ becomes simply the regular solid harmonics. The
corresponding $E^{(p,q)}_{\bm{k}}(n,0)$ is just the first term ($m=0$) of the
series form (30) with an appropriate adjustment of the constant factor. In
this case, the moment condition becomes
$E^{(p,q)}_{\bm{k}}=0\quad\text{if}\quad p<k_{x}+k_{y}+k_{z}$
which results in a more sparse multipole expansion matrices. With the same
condition as the previous examples, $\max p=\max k_{i}=10$, the number of non-
zero elements are given in Table 3.
total elements | real non-zeroes | imaginary non-zeroes
---|---|---
87846 | 1512 (1.72%) | 1001 (1.14%)
Table 3: Sparsity of the case with $\lambda=0$.
We can observe that the resulting matrices are very sparse – only less than 3%
of total elements are non-zeroes. This example illustrates the efficiency of
the multipole expansion on multiwavelet representations based on _orthogonal_
(almost synonymously in this paper, Legendre) polynomials.
### 4.3 Recurrence Relations for ${\widehat{I}}_{k}^{l}$
Finally, we present the algorithm to build the table of
${\widehat{I}}_{k}^{l}$ required for the evaluation of $I_{m}$. Let $p\leq
p_{\max}$, $k_{i}\leq k_{\max}$, and $M\leq M_{\max}$ (for a given $\lambda$).
Then, the required size of table is $(2M_{\max}+p_{\max})\times k_{\max}$, and
the half of the elements are zero by the oddity condition.
Each element ${\widehat{I}}_{k}^{l}$ can be calculated by the identical
recurrence relations to those of the orthogonal polynomials $\phi^{k}$. Recall
any sequence orthogonal polynomials satisfy a three term recurrence relation
of the form,
$\phi^{k+1}=(\alpha_{k}\,\zeta+\beta_{k})\,\phi^{k}-\gamma_{k}\,\phi^{k-1}.$
(49)
It immediately follows that
${\widehat{I}}_{k+1}^{l}=\alpha_{k}\,{\widehat{I}}_{k}^{l+1}+\beta_{k}\,{\widehat{I}}_{k}^{l}-\gamma_{k}\,{\widehat{I}}_{k-1}^{l}.$
(50)
From the oddity condition, ${\widehat{I}}_{k+1}^{l}\neq 0$ if and only if
${\widehat{I}}_{k}^{l}=0$, and ${\widehat{I}}^{l}_{k}=0$ for $l<k$. Hence, for
$k<l$,
${\widehat{I}}_{k}^{l+1}=\begin{cases}0&\text{if $(l+k)$ is even}\\\
\alpha_{k}^{-1}\,{\widehat{I}}_{k+1}^{l}+\alpha_{k}^{-1}\gamma_{k}\,{\widehat{I}}_{k-1}^{l}&\text{if
$(l+k)$ is odd}\end{cases}$ (51)
and
${\widehat{I}}_{k}^{k}=\alpha_{k}^{-1}\,\gamma_{k}\,{\widehat{I}}_{k-1}^{k-1}\quad(\because{\widehat{I}}_{k+1}^{k-1}=0).$
(52)
The recurrence relation can be evaluated from the initial data,
${\widehat{I}}_{0}^{l}=a_{0}\,\frac{1+(-1)^{l}}{l+1}\quad\text{and}\quad{\widehat{I}}_{1}^{l}=a_{1}\,\frac{1-(-1)^{l}}{l+2}$
(53)
where $\phi^{0}=a_{0}$ and $\phi^{1}=a_{1}\zeta$.
For _normalized_ Legendre polynomials, the coefficients are given by
$\alpha_{k}^{-1}=\frac{k+1}{2k+1}\,\sqrt{\frac{2k+1}{2k+3}}\quad\text{and}\quad\alpha_{k}^{-1}\gamma_{k}=\frac{k}{2k+1}\,\sqrt{\frac{2k+1}{2k-1}}.$
(54)
## 5 Results and Conclusions
Most of the implementation is done very faithfully with the formulae presented
in this paper. The only special treatment is that, in order to suppress the
accumulation of round-off errors affecting the result, we used higher
precision floating point arithmetics for internal calculations including the
table for ${\widehat{I}}^{l}_{k}$; for example, to generate matrices with
64bit double precision, we employed 80bit long(extended)-double arithmetics.
The computational cost is governed by the number of terms to be added, $m$,
and is not significantly affected by the augmented internal precision. By
comparing with values obtained by applying adaptive numerical integrator to
(15), we could validate the presented formula. When $\lambda$ or ${\bm{k}}$ is
only moderately large, an adaptive integrator typically fails to converge
since the integrand of (15) becomes near-singular or highly oscillating. Thus,
the presented formula can be viewed as a reliable way to evaluate (15) (or a
similar form of integral) when a typical numerical quadrature is not
applicable due to the near-singularity and/or the oscillation of the
integrand. It is also observed that the computing time of building
${\widehat{I}}_{k}^{l}$ table is negligible compared to the computing time of
$E^{(p,q)}_{\bm{k}}$. We summarize the contributions of this paper as follows.
1. 1.
We presented a method to build the multipole expansion matrices for functions
represented by multiwavelets.
2. 2.
The presented method does not involve any numerical quadrature and based
entirely on a series representation like a special function of $\lambda$.
3. 3.
The proposed scheme generates highly accurate multipole conversion matrices
stably and reliably for a wide range of parameters $(p,q,{\bm{k}})$ and
$\lambda$.
## References
* (1) B. Alpert, _Sparse Representation of Smooth Linear Operators_ , Ph.D. Thesis, Yale University (1990).
* (2) B. Alpert, G. Beylkin, D. Gines, and L. Vozovoi, _Adaptive Solution of Partial Differential Equations in Multiwavelet Bases_ , J. Comput. Phys., 182(1), 149-190 (2002).
* (3) G. Beylkin and M. Mohlenkamp, _Algorithms for Numerical Analysis in High Diemensions_ , SIAM J. Sci. Comput., 26 (6), 2133 2159 (2005).
* (4) R. Harrison, G. Fann, T. Yanai, Z. Gan, and G. Beylkin, _Multiresolution quantum chemistry: Basic theory and initial applications_ , J. Chem. Phys. 121 (23), 11587 11598 (2004).
* (5) T. Yanai, G. Fann, Z. Gan, R. Harrison, and G. Beylkin, _Multiresolution quantum chemistry: Hartree Fock exchange_ , J. Chem. Phys. 121 (14), 6680 6688 (2004).
* (6) T. Yanai, G. Fann, Z. Gan, R. Harrison, and G. Beylkin, _Multiresolution quantum chemistry: Analytic derivatives for Hartree Fock and density functional theory_ , J. Chem. Phys. 121 (7), 2866 2876 (2004).
* (7) L. Greengard, and J. Huang, _A New Version of the Fast Adaptive Multipole Method for Screened Coulomb Interactions in Three Dimensions_ , Journal of Computational Physics, 180, 642-658 (2002).
|
arxiv-papers
| 2012-11-26T19:57:43 |
2024-09-04T02:49:38.481601
|
{
"license": "Public Domain",
"authors": "Jae-Seok Huh",
"submitter": "Jae-Seok Huh",
"url": "https://arxiv.org/abs/1211.6076"
}
|
1211.6093
|
EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN)
CERN-PH-EP-2012-331 LHCb-PAPER-2012-035 November 26, 2012
Measurement of the
time-dependent $C\\!P$ asymmetry
in $B^{0}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{0}_{\rm\scriptscriptstyle S}$ decays
The LHCb collaboration†††Authors are listed on the following pages.
This Letter reports a measurement of the $C\\!P$ violation observables
$S_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{0}_{\rm\scriptscriptstyle
S}}$ and $C_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{0}_{\rm\scriptscriptstyle S}}$ in the decay channel
$B^{0}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{0}_{\rm\scriptscriptstyle S}$ performed with 1.0$\mbox{\,fb}^{-1}$ of
$pp$ collisions at $\sqrt{s}=7\mathrm{\,Te\kern-1.00006ptV}$ collected by the
LHCb experiment. The fit to the data yields
$S_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{0}_{\rm\scriptscriptstyle
S}}=0.73\pm 0.07\,\text{(stat)}\pm 0.04\,\text{(syst)}$ and
$C_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{0}_{\rm\scriptscriptstyle
S}}=0.03\pm 0.09\,\text{(stat)}\pm 0.01\,\text{(syst)}$. Both values are
consistent with the current world averages and within expectations from the
Standard Model.
Published as Phys. Lett. B721 (2013) 24–31
LHCb collaboration
R. Aaij38, C. Abellan Beteta33,n, A. Adametz11, B. Adeva34, M. Adinolfi43, C.
Adrover6, A. Affolder49, Z. Ajaltouni5, J. Albrecht35, F. Alessio35, M.
Alexander48, S. Ali38, G. Alkhazov27, P. Alvarez Cartelle34, A.A. Alves Jr22,
S. Amato2, Y. Amhis36, L. Anderlini17,f, J. Anderson37, R.B. Appleby51, O.
Aquines Gutierrez10, F. Archilli18,35, A. Artamonov 32, M. Artuso53, E.
Aslanides6, G. Auriemma22,m, S. Bachmann11, J.J. Back45, C. Baesso54, W.
Baldini16, R.J. Barlow51, C. Barschel35, S. Barsuk7, W. Barter44, A. Bates48,
Th. Bauer38, A. Bay36, J. Beddow48, I. Bediaga1, S. Belogurov28, K. Belous32,
I. Belyaev28, E. Ben-Haim8, M. Benayoun8, G. Bencivenni18, S. Benson47, J.
Benton43, A. Berezhnoy29, R. Bernet37, M.-O. Bettler44, M. van Beuzekom38, A.
Bien11, S. Bifani12, T. Bird51, A. Bizzeti17,h, P.M. Bjørnstad51, T. Blake35,
F. Blanc36, C. Blanks50, J. Blouw11, S. Blusk53, A. Bobrov31, V. Bocci22, A.
Bondar31, N. Bondar27, W. Bonivento15, S. Borghi48,51, A. Borgia53, T.J.V.
Bowcock49, C. Bozzi16, T. Brambach9, J. van den Brand39, J. Bressieux36, D.
Brett51, M. Britsch10, T. Britton53, N.H. Brook43, H. Brown49, A. Büchler-
Germann37, I. Burducea26, A. Bursche37, J. Buytaert35, S. Cadeddu15, O.
Callot7, M. Calvi20,j, M. Calvo Gomez33,n, A. Camboni33, P. Campana18,35, A.
Carbone14,c, G. Carboni21,k, R. Cardinale19,i, A. Cardini15, H. Carranza-
Mejia47, L. Carson50, K. Carvalho Akiba2, G. Casse49, M. Cattaneo35, Ch.
Cauet9, M. Charles52, Ph. Charpentier35, P. Chen3,36, N. Chiapolini37, M.
Chrzaszcz 23, K. Ciba35, X. Cid Vidal34, G. Ciezarek50, P.E.L. Clarke47, M.
Clemencic35, H.V. Cliff44, J. Closier35, C. Coca26, V. Coco38, J. Cogan6, E.
Cogneras5, P. Collins35, A. Comerma-Montells33, A. Contu52,15, A. Cook43, M.
Coombes43, G. Corti35, B. Couturier35, G.A. Cowan36, D. Craik45, S.
Cunliffe50, R. Currie47, C. D’Ambrosio35, P. David8, P.N.Y. David38, I. De
Bonis4, K. De Bruyn38, S. De Capua51, M. De Cian37, J.M. De Miranda1, L. De
Paula2, P. De Simone18, D. Decamp4, M. Deckenhoff9, H. Degaudenzi36,35, L. Del
Buono8, C. Deplano15, D. Derkach14, O. Deschamps5, F. Dettori39, A. Di
Canto11, J. Dickens44, H. Dijkstra35, P. Diniz Batista1, M. Dogaru26, F.
Domingo Bonal33,n, S. Donleavy49, F. Dordei11, A. Dosil Suárez34, D.
Dossett45, A. Dovbnya40, F. Dupertuis36, R. Dzhelyadin32, A. Dziurda23, A.
Dzyuba27, S. Easo46,35, U. Egede50, V. Egorychev28, S. Eidelman31, D. van
Eijk38, S. Eisenhardt47, R. Ekelhof9, L. Eklund48, I. El Rifai5, Ch.
Elsasser37, D. Elsby42, A. Falabella14,e, C. Färber11, G. Fardell47, C.
Farinelli38, S. Farry12, V. Fave36, V. Fernandez Albor34, F. Ferreira
Rodrigues1, M. Ferro-Luzzi35, S. Filippov30, C. Fitzpatrick35, M. Fontana10,
F. Fontanelli19,i, R. Forty35, O. Francisco2, M. Frank35, C. Frei35, M.
Frosini17,f, S. Furcas20, A. Gallas Torreira34, D. Galli14,c, M. Gandelman2,
P. Gandini52, Y. Gao3, J-C. Garnier35, J. Garofoli53, P. Garosi51, J. Garra
Tico44, L. Garrido33, C. Gaspar35, R. Gauld52, E. Gersabeck11, M. Gersabeck35,
T. Gershon45,35, Ph. Ghez4, V. Gibson44, V.V. Gligorov35, C. Göbel54, D.
Golubkov28, A. Golutvin50,28,35, A. Gomes2, H. Gordon52, M. Grabalosa
Gándara33, R. Graciani Diaz33, L.A. Granado Cardoso35, E. Graugés33, G.
Graziani17, A. Grecu26, E. Greening52, S. Gregson44, O. Grünberg55, B. Gui53,
E. Gushchin30, Yu. Guz32, T. Gys35, C. Hadjivasiliou53, G. Haefeli36, C.
Haen35, S.C. Haines44, S. Hall50, T. Hampson43, S. Hansmann-Menzemer11, N.
Harnew52, S.T. Harnew43, J. Harrison51, P.F. Harrison45, T. Hartmann55, J.
He7, V. Heijne38, K. Hennessy49, P. Henrard5, J.A. Hernando Morata34, E. van
Herwijnen35, E. Hicks49, D. Hill52, M. Hoballah5, P. Hopchev4, W.
Hulsbergen38, P. Hunt52, T. Huse49, N. Hussain52, D. Hutchcroft49, D. Hynds48,
V. Iakovenko41, P. Ilten12, J. Imong43, R. Jacobsson35, A. Jaeger11, M. Jahjah
Hussein5, E. Jans38, F. Jansen38, P. Jaton36, B. Jean-Marie7, F. Jing3, M.
John52, D. Johnson52, C.R. Jones44, B. Jost35, M. Kaballo9, S. Kandybei40, M.
Karacson35, T.M. Karbach35, I.R. Kenyon42, U. Kerzel35, T. Ketel39, A.
Keune36, B. Khanji20, Y.M. Kim47, O. Kochebina7, V. Komarov36,29, R.F.
Koopman39, P. Koppenburg38, M. Korolev29, A. Kozlinskiy38, L. Kravchuk30, K.
Kreplin11, M. Kreps45, G. Krocker11, P. Krokovny31, F. Kruse9, M.
Kucharczyk20,23,j, V. Kudryavtsev31, T. Kvaratskheliya28,35, V.N. La Thi36, D.
Lacarrere35, G. Lafferty51, A. Lai15, D. Lambert47, R.W. Lambert39, E.
Lanciotti35, G. Lanfranchi18,35, C. Langenbruch35, T. Latham45, C.
Lazzeroni42, R. Le Gac6, J. van Leerdam38, J.-P. Lees4, R. Lefèvre5, A.
Leflat29,35, J. Lefrançois7, O. Leroy6, T. Lesiak23, Y. Li3, L. Li Gioi5, M.
Liles49, R. Lindner35, C. Linn11, B. Liu3, G. Liu35, J. von Loeben20, J.H.
Lopes2, E. Lopez Asamar33, N. Lopez-March36, H. Lu3, J. Luisier36, H. Luo47,
A. Mac Raighne48, F. Machefert7, I.V. Machikhiliyan4,28, F. Maciuc26, O.
Maev27,35, J. Magnin1, M. Maino20, S. Malde52, G. Manca15,d, G. Mancinelli6,
N. Mangiafave44, U. Marconi14, R. Märki36, J. Marks11, G. Martellotti22, A.
Martens8, L. Martin52, A. Martín Sánchez7, M. Martinelli38, D. Martinez
Santos35, D. Martins Tostes2, A. Massafferri1, R. Matev35, Z. Mathe35, C.
Matteuzzi20, M. Matveev27, E. Maurice6, A. Mazurov16,30,35,e, J. McCarthy42,
G. McGregor51, R. McNulty12, M. Meissner11, M. Merk38, J. Merkel9, D.A.
Milanes13, M.-N. Minard4, J. Molina Rodriguez54, S. Monteil5, D. Moran51, P.
Morawski23, R. Mountain53, I. Mous38, F. Muheim47, K. Müller37, R. Muresan26,
B. Muryn24, B. Muster36, J. Mylroie-Smith49, P. Naik43, T. Nakada36, R.
Nandakumar46, I. Nasteva1, M. Needham47, N. Neufeld35, A.D. Nguyen36, T.D.
Nguyen36, C. Nguyen-Mau36,o, M. Nicol7, V. Niess5, N. Nikitin29, T. Nikodem11,
A. Nomerotski52,35, A. Novoselov32, A. Oblakowska-Mucha24, V. Obraztsov32, S.
Oggero38, S. Ogilvy48, O. Okhrimenko41, R. Oldeman15,d,35, M. Orlandea26, J.M.
Otalora Goicochea2, P. Owen50, B.K. Pal53, A. Palano13,b, M. Palutan18, J.
Panman35, A. Papanestis46, M. Pappagallo48, C. Parkes51, C.J. Parkinson50, G.
Passaleva17, G.D. Patel49, M. Patel50, G.N. Patrick46, C. Patrignani19,i, C.
Pavel-Nicorescu26, A. Pazos Alvarez34, A. Pellegrino38, G. Penso22,l, M. Pepe
Altarelli35, S. Perazzini14,c, D.L. Perego20,j, E. Perez Trigo34, A. Pérez-
Calero Yzquierdo33, P. Perret5, M. Perrin-Terrin6, G. Pessina20, K.
Petridis50, A. Petrolini19,i, A. Phan53, E. Picatoste Olloqui33, B. Pie
Valls33, B. Pietrzyk4, T. Pilař45, D. Pinci22, S. Playfer47, M. Plo Casasus34,
F. Polci8, G. Polok23, A. Poluektov45,31, E. Polycarpo2, D. Popov10, B.
Popovici26, C. Potterat33, A. Powell52, J. Prisciandaro36, V. Pugatch41, A.
Puig Navarro36, W. Qian4, J.H. Rademacker43, B. Rakotomiaramanana36, M.S.
Rangel2, I. Raniuk40, N. Rauschmayr35, G. Raven39, S. Redford52, M.M. Reid45,
A.C. dos Reis1, S. Ricciardi46, A. Richards50, K. Rinnert49, V. Rives
Molina33, D.A. Roa Romero5, P. Robbe7, E. Rodrigues48,51, P. Rodriguez
Perez34, G.J. Rogers44, S. Roiser35, V. Romanovsky32, A. Romero Vidal34, J.
Rouvinet36, T. Ruf35, H. Ruiz33, G. Sabatino22,k, J.J. Saborido Silva34, N.
Sagidova27, P. Sail48, B. Saitta15,d, C. Salzmann37, B. Sanmartin Sedes34, M.
Sannino19,i, R. Santacesaria22, C. Santamarina Rios34, R. Santinelli35, E.
Santovetti21,k, M. Sapunov6, A. Sarti18,l, C. Satriano22,m, A. Satta21, M.
Savrie16,e, P. Schaack50, M. Schiller39, H. Schindler35, S. Schleich9, M.
Schlupp9, M. Schmelling10, B. Schmidt35, O. Schneider36, A. Schopper35, M.-H.
Schune7, R. Schwemmer35, B. Sciascia18, A. Sciubba18,l, M. Seco34, A.
Semennikov28, K. Senderowska24, I. Sepp50, N. Serra37, J. Serrano6, P.
Seyfert11, M. Shapkin32, I. Shapoval40,35, P. Shatalov28, Y. Shcheglov27, T.
Shears49,35, L. Shekhtman31, O. Shevchenko40, V. Shevchenko28, A. Shires50, R.
Silva Coutinho45, T. Skwarnicki53, N.A. Smith49, E. Smith52,46, M. Smith51, K.
Sobczak5, F.J.P. Soler48, F. Soomro18,35, D. Souza43, B. Souza De Paula2, B.
Spaan9, A. Sparkes47, P. Spradlin48, F. Stagni35, S. Stahl11, O. Steinkamp37,
S. Stoica26, S. Stone53, B. Storaci38, M. Straticiuc26, U. Straumann37, V.K.
Subbiah35, S. Swientek9, M. Szczekowski25, P. Szczypka36,35, D. Szilard2, T.
Szumlak24, S. T’Jampens4, M. Teklishyn7, E. Teodorescu26, F. Teubert35, C.
Thomas52, E. Thomas35, J. van Tilburg11, V. Tisserand4, M. Tobin37, S. Tolk39,
D. Tonelli35, S. Topp-Joergensen52, N. Torr52, E. Tournefier4,50, S.
Tourneur36, M.T. Tran36, A. Tsaregorodtsev6, P. Tsopelas38, N. Tuning38, M.
Ubeda Garcia35, A. Ukleja25, D. Urner51, U. Uwer11, V. Vagnoni14, G.
Valenti14, R. Vazquez Gomez33, P. Vazquez Regueiro34, S. Vecchi16, J.J.
Velthuis43, M. Veltri17,g, G. Veneziano36, M. Vesterinen35, B. Viaud7, I.
Videau7, D. Vieira2, X. Vilasis-Cardona33,n, J. Visniakov34, A. Vollhardt37,
D. Volyanskyy10, D. Voong43, A. Vorobyev27, V. Vorobyev31, C. Voß55, H.
Voss10, R. Waldi55, R. Wallace12, S. Wandernoth11, J. Wang53, D.R. Ward44,
N.K. Watson42, A.D. Webber51, D. Websdale50, M. Whitehead45, J. Wicht35, D.
Wiedner11, L. Wiggers38, G. Wilkinson52, M.P. Williams45,46, M. Williams50,p,
F.F. Wilson46, J. Wishahi9, M. Witek23, W. Witzeling35, S.A. Wotton44, S.
Wright44, S. Wu3, K. Wyllie35, Y. Xie47,35, F. Xing52, Z. Xing53, Z. Yang3, R.
Young47, X. Yuan3, O. Yushchenko32, M. Zangoli14, M. Zavertyaev10,a, F.
Zhang3, L. Zhang53, W.C. Zhang12, Y. Zhang3, A. Zhelezov11, L. Zhong3, A.
Zvyagin35.
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 Roma Tor Vergata, Roma, Italy
22Sezione INFN di Roma La Sapienza, Roma, Italy
23Henryk Niewodniczanski Institute of Nuclear Physics Polish Academy of
Sciences, Kraków, Poland
24AGH University of Science and Technology, Kraków, Poland
25National Center for Nuclear Research (NCBJ), Warsaw, Poland
26Horia Hulubei National Institute of Physics and Nuclear Engineering,
Bucharest-Magurele, Romania
27Petersburg Nuclear Physics Institute (PNPI), Gatchina, Russia
28Institute of Theoretical and Experimental Physics (ITEP), Moscow, Russia
29Institute of Nuclear Physics, Moscow State University (SINP MSU), Moscow,
Russia
30Institute for Nuclear Research of the Russian Academy of Sciences (INR RAN),
Moscow, Russia
31Budker Institute of Nuclear Physics (SB RAS) and Novosibirsk State
University, Novosibirsk, Russia
32Institute for High Energy Physics (IHEP), Protvino, Russia
33Universitat de Barcelona, Barcelona, Spain
34Universidad de Santiago de Compostela, Santiago de Compostela, Spain
35European Organization for Nuclear Research (CERN), Geneva, Switzerland
36Ecole Polytechnique Fédérale de Lausanne (EPFL), Lausanne, Switzerland
37Physik-Institut, Universität Zürich, Zürich, Switzerland
38Nikhef National Institute for Subatomic Physics, Amsterdam, The Netherlands
39Nikhef National Institute for Subatomic Physics and VU University Amsterdam,
Amsterdam, The Netherlands
40NSC Kharkiv Institute of Physics and Technology (NSC KIPT), Kharkiv, Ukraine
41Institute for Nuclear Research of the National Academy of Sciences (KINR),
Kyiv, Ukraine
42University of Birmingham, Birmingham, United Kingdom
43H.H. Wills Physics Laboratory, University of Bristol, Bristol, United
Kingdom
44Cavendish Laboratory, University of Cambridge, Cambridge, United Kingdom
45Department of Physics, University of Warwick, Coventry, United Kingdom
46STFC Rutherford Appleton Laboratory, Didcot, United Kingdom
47School of Physics and Astronomy, University of Edinburgh, Edinburgh, United
Kingdom
48School of Physics and Astronomy, University of Glasgow, Glasgow, United
Kingdom
49Oliver Lodge Laboratory, University of Liverpool, Liverpool, United Kingdom
50Imperial College London, London, United Kingdom
51School of Physics and Astronomy, University of Manchester, Manchester,
United Kingdom
52Department of Physics, University of Oxford, Oxford, United Kingdom
53Syracuse University, Syracuse, NY, United States
54Pontifícia Universidade Católica do Rio de Janeiro (PUC-Rio), Rio de
Janeiro, Brazil, associated to 2
55Institut für Physik, Universität Rostock, Rostock, Germany, associated to 11
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
pMassachusetts Institute of Technology, Cambridge, MA, United States
## 1 Introduction
The source of $C\\!P$ violation in the electroweak sector of the Standard
Model (SM) is the single irreducible complex phase of the Cabibbo-Kobayashi-
Maskawa (CKM) quark mixing matrix [1, 2]. The decay
$B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{0}_{\rm\scriptscriptstyle S}$ is one of the theoretically cleanest
modes for the study of $C\\!P$ violation in the $B^{0}$ meson system. Here,
the $B^{0}$ and $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ mesons
decay to a common $C\\!P$-odd eigenstate allowing for interference through
$B^{0}$–$\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ mixing.
In the $B^{0}$ system the decay width difference $\Delta\Gamma_{d}$ between
the heavy and light mass eigenstates is negligible. Therefore, the time-
dependent decay rate asymmetry can be written as [3, 4]
$\displaystyle{\cal A}_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{0}_{\rm\scriptscriptstyle S}}(t)$
$\displaystyle\equiv\frac{\Gamma(\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}(t)\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{0}_{\rm\scriptscriptstyle
S})-\Gamma(B^{0}(t)\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{0}_{\rm\scriptscriptstyle S})}{\Gamma(\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}(t)\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{0}_{\rm\scriptscriptstyle
S})+\Gamma(B^{0}(t)\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{0}_{\rm\scriptscriptstyle S})}$
$\displaystyle=S_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{0}_{\rm\scriptscriptstyle S}}\sin(\Delta
m_{d}t)-C_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{0}_{\rm\scriptscriptstyle S}}\cos(\Delta m_{d}t).$ (1)
Here $B^{0}(t)$ and $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}(t)$ are
the states into which particles produced at $t=0$ as $B^{0}$ and $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ respectively have evolved, when
decaying at time $t$. The parameter $\Delta m_{d}$ is the mass difference
between the two $B^{0}$ mass eigenstates. The sine term results from the
interference between direct decay and decay after $B^{0}$–$\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ mixing. The cosine term arises
either from the interference between decay amplitudes with different weak and
strong phases (direct $C\\!P$ violation) or from $C\\!P$ violation in
$B^{0}$–$\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ mixing.
In the SM, $C\\!P$ violation in mixing and direct $C\\!P$ violation are both
negligible in $B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{0}_{\rm\scriptscriptstyle S}$ decays, hence
$C_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{0}_{\rm\scriptscriptstyle
S}}\approx 0$, while $S_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{0}_{\rm\scriptscriptstyle S}}\approx\sin 2\beta$, where the CKM angle
$\beta$ can be expressed in terms of the CKM matrix elements as
$\arg\left|-V_{cd}^{\phantom{\ast}}V_{cb}^{\ast}/V_{td}^{\phantom{\ast}}V_{tb}^{\ast}\right|$.
It can also be measured in other $B^{0}$ decays to final states including
charmonium such as ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{0}_{\rm\scriptscriptstyle L}$, ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{*0}$, $\psi{(2S)}K^{(*)0}$, which have been used in measurements by
the BaBar and Belle collaborations [5, 6]. Currently, the world averages are
$S_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{0}_{\rm\scriptscriptstyle
S}}=0.679\pm 0.020$ and $C_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{0}_{\rm\scriptscriptstyle S}}=0.005\pm 0.017$ [7].
The time-dependent measurement of the $C\\!P$ parameters
$S_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{0}_{\rm\scriptscriptstyle
S}}$ and $C_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{0}_{\rm\scriptscriptstyle S}}$ requires flavour tagging, i.e. the
knowledge whether the decaying particle was produced as a $B^{0}$ or a $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ meson. If a fraction $\omega$ of
candidates is tagged incorrectly, the accessible time-dependent asymmetry
${\cal A}_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{0}_{\rm\scriptscriptstyle S}}(t)$ is diluted by a factor
$(1-2\omega)$. Hence, a measurement of the $C\\!P$ parameters requires precise
knowledge of the wrong tag fraction. Additionally, the asymmetry between the
production rates of $B^{0}$ and $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ has to be determined as it affects
the observed asymmetries.
In this Letter, the most precise measurement of
$S_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{0}_{\rm\scriptscriptstyle
S}}$ and $C_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{0}_{\rm\scriptscriptstyle S}}$ to date at a hadron collider is
presented using approximately 8200 flavour-tagged
$B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{0}_{\rm\scriptscriptstyle S}$ decays.
## 2 Data samples and selection requirements
The data sample consists of $1.0$$\mbox{\,fb}^{-1}$ of $pp$ collisions
recorded in 2011 at a centre-of-mass energy of
$\sqrt{s}=7$$\mathrm{\,Te\kern-1.00006ptV}$ with the LHCb experiment at CERN.
The detector [8] is a single-arm forward spectrometer covering the
pseudorapidity range $2$ to $5$, designed for the study of particles
containing $b$ or $c$ quarks. It 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 $4$ Tm, and three stations of silicon-
strip detectors and straw drift-tubes placed downstream. The combined tracking
system has a momentum resolution $\Delta p/p$ that varies from $0.4$ % at
$5$${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ to $0.6$ % at
100${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$, and an impact parameter resolution
of $20$$\,\upmu\rm m$ for tracks with high transverse momentum. Charged
hadrons are identified using two ring-imaging Cherenkov detectors. Photon,
electron and hadron candidates are identified by a calorimeter system
consisting of scintillating-pad and preshower detectors, an electromagnetic
and a hadronic calorimeter. Muons are identified by a system composed of
alternating layers of iron and multiwire proportional chambers.
The analysis is performed on events with reconstructed
$B^{0}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{0}_{\rm\scriptscriptstyle S}$ candidates with subsequent
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\rightarrow\mu^{+}\mu^{-}$ and
$K^{0}_{\rm\scriptscriptstyle S}\rightarrow\pi^{+}\pi^{-}$ decays. Events are
selected by the trigger consisting of hardware and software stages. The
hardware stage accepts events if muon or hadron candidates with high
transverse momentum ($p_{\rm T}$) with respect to the beam axis are detected.
In the software stage, events are required to contain two oppositely-charged
particles, both compatible with a muon hypothesis, that form an invariant mass
greater than $2.7{\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$. The resulting
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ candidate has to be clearly
separated (decay length significance greater than $3$) from the production
vertex (PV) with which it is associated on the basis of the impact parameter.
The overall signal efficiency of these triggers is found to be $64\%$.
Further selection criteria are applied offline to decrease the number of
background candidates. The ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$
candidates are reconstructed from two oppositely-charged, well identified
muons with $\mbox{$p_{\rm T}$}>500{\mathrm{\,Me\kern-1.00006ptV\\!/}c}$ that
form a common vertex with a fit $\chi^{2}/\text{ndf}$ of less than $11$, where
ndf is the number of degrees of freedom, and with an invariant mass in the
range $3035$–$3160$${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$. It is required
that the ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ candidate fulfils the
trigger requirements described above. The $K^{0}_{\rm\scriptscriptstyle S}$
candidates are formed from two oppositely-charged pions, both with (long
$K^{0}_{\rm\scriptscriptstyle S}$ candidate) or without (downstream
$K^{0}_{\rm\scriptscriptstyle S}$ candidate) hits in the vertex detector. Any
$K^{0}_{\rm\scriptscriptstyle S}$ candidates where both pion tracks have hits
in the tracking stations but only one has additional hits in the vertex
detector are ignored, as they would only contribute to $<2\%$ of the events.
Each pion must have $p>2{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ and a clear
separation from any PV. Furthermore, they must form a common vertex with a fit
$\chi^{2}/\text{ndf}$ of less than $20$ and an invariant mass within the range
$485.6$–$509.6$${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ (long
$K^{0}_{\rm\scriptscriptstyle S}$ candidates) or
$476.6$–$518.6$${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ (downstream
$K^{0}_{\rm\scriptscriptstyle S}$ candidates). Different mass windows are
chosen to account for different mass resolutions for long and downstream
$K^{0}_{\rm\scriptscriptstyle S}$ candidates. The
$K^{0}_{\rm\scriptscriptstyle S}$ candidate’s decay vertex is required to be
significantly displaced with respect to the associated PV.
The $B^{0}$ candidates are constructed from combinations of
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ and
$K^{0}_{\rm\scriptscriptstyle S}$ candidates that form a vertex with a
reconstructed mass $m_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{0}_{\rm\scriptscriptstyle S}}$ in the range
$5230$–$5330$${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$. The value of
$m_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{0}_{\rm\scriptscriptstyle
S}}$ is computed constraining the invariant masses of the $\mu^{+}\mu^{-}$ and
$\pi^{+}\pi^{-}$ to the known ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$
and $K^{0}_{\rm\scriptscriptstyle S}$ masses [9], respectively. As most events
involve more than one reconstructed PV, $B^{0}$ candidates are required to be
associated to one PV only and are therefore omitted if their impact parameter
significance with respect to other PVs in the event is too small.
Additionally, the $K^{0}_{\rm\scriptscriptstyle S}$ candidate’s decay vertex
is required to be separated from the $B^{0}$ decay vertex by a decay time
significance of the $K^{0}_{\rm\scriptscriptstyle S}$ greater than 5.
The decay time $t$ of the $B^{0}$ candidates is determined from a vertex fit
to the whole decay chain under the constraint that the $B^{0}$ candidate
originates from the associated PV [10]. Only candidates with a good quality
vertex fit and with $0.3<t<18.3$${\rm\,ps}$ are retained. In case more than
one candidate is selected in an event, that with the best vertex fit quality
is chosen. The fit uncertainty on $t$ is used as an estimate of the decay time
resolution $\sigma_{t}$, which is required to be less than $0.2$${\rm\,ps}$.
Finally, candidates are only retained if the flavour tagging algorithms
provide a prediction for the production flavour of the candidate, as discussed
in Section 3.
Simulated samples are used for cross-checks and studies of decay time
distributions. For the simulation, $pp$ collisions are generated using Pythia
6.4 [11] with a specific LHCb configuration [12]. Decays of hadronic particles
are described by EvtGen [13] in which final state radiation is generated using
Photos [14]. The interaction of the generated particles with the detector is
implemented using the Geant4 toolkit [15, *Agostinelli:2002hh] as described in
Ref. [17].
## 3 Flavour tagging
A mandatory step for the study of $C\\!P$ violating quantities is to tag the
initial, i.e. production, flavour of the decaying $B^{0}$ meson. Since $b$
quarks are predominantly produced in $b\overline{}b$ pairs in LHCb, the
flavour tagging algorithms used in this analysis [18] reconstruct the flavour
of the non-signal $b$ hadron. The flavour of the non-signal $b$ hadron is
determined by identifying the charge of its decay products, such as that of an
electron or a muon from a semileptonic $b$ decay, a kaon from a $b\rightarrow
c\rightarrow s$ decay chain, or the charge of its inclusively reconstructed
decay vertex. The algorithms use this information to provide a tag $d$ that
takes the value $+1$ ($-1$) in the case where the signal candidate is tagged
as an initial $B^{0}$ ($\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}$)
meson.
A careful study of the fraction of candidates that are wrongly tagged (mistag
fraction) is necessary as the measured asymmetry is diluted due to the
imperfect tagging performance. The mistag fraction ($\omega$) is extracted on
an event-by-event basis from the combined per-event mistag probability
prediction $\eta$ of the tagging algorithms. On average, the mistag fraction
is found to depend linearly on $\eta$ and is parameterised as
$\displaystyle\omega(\eta)=p_{1}\cdot\left(\eta-\langle\eta\rangle\right)+p_{0}\
.$ (2)
Using events from the self-tagging control channel
$B^{+}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+}$, the
parameters are determined to be $p_{1}=1.035\pm 0.021\,\text{(stat)}\pm
0.012\,\text{(syst)}$, $p_{0}=0.392\pm 0.002\,\text{(stat)}\pm
0.009\,\text{(syst)}$ and $\langle\eta\rangle=0.391$ [19]. The systematic
uncertainties on the tagging calibration parameters are estimated by comparing
the tagging performance obtained in different decay channels such as
$B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{*0}$, in
$B^{+}$ and $B^{-}$ subsamples separately, and in different data taking
periods.
The difference in tagging response between $B^{0}$ and $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ is parameterised by using
$\displaystyle\omega=\omega(\eta)\pm\frac{\Delta p_{0}}{2}\ ,$ (3)
where the $+$ ($-$) is used for a $B^{0}$ ($\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}$) meson at production and $\Delta
p_{0}$ is the mistag fraction asymmetry parameter, which is the difference of
$p_{0}$ for $B^{0}$ and $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}$
mesons. It is measured as $\Delta p_{0}=0.011\pm 0.003$ using events from the
control channel $B^{+}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{+}$. By using $\Delta p_{0}$ in the analysis, the systematic
uncertainty on the $p_{0}$ parameter is reduced to $0.008$. The difference of
tagging efficiency for $B^{0}$ and $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ mesons is measured in the same
control channel as $\Delta\varepsilon_{\text{tag}}=0.000\pm 0.001$ and is
therefore negligible. Thus, it is only used to estimate possible systematic
uncertainties in the analysis.
The effect of imperfect tagging is the reduction of the statistical power by a
factor $\varepsilon_{\text{tag}}\mathcal{D}^{2}$, where
$\varepsilon_{\text{tag}}$ is the tagging efficiency and
$\mathcal{D}=1-2\omega$ is the dilution factor. The effective
$\varepsilon_{\text{tag}}$ and $\mathcal{D}$ values are measured as
$\varepsilon_{\text{tag}}=(32.65\pm 0.31)\%$ and $\mathcal{D}=0.270\pm 0.015$,
resulting in $\varepsilon_{\text{tag}}\mathcal{D}^{2}=(2.38\pm 0.27)\%$ ,
where combined systematic and statistical uncertainties are quoted. The
measured dilution corresponds to a mistag fraction of $\omega=0.365\pm 0.008$.
## 4 Decay time acceptance and resolution
The bias on the decay time distribution due to the trigger is estimated by
comparing candidates selected using different trigger requirements. In the
selection, the reconstructed decay times of the
$B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{0}_{\rm\scriptscriptstyle S}$ candidates are required to be greater
than $0.3$${\rm\,ps}$. This requirement makes the acceptance effects of the
trigger nearly negligible. However, some small efficiency loss remains for
small decay times. Neglecting this efficiency loss is treated as a source of
systematic uncertainty.
A decrease of efficiency is also observed at large decay times, mostly
affecting the candidates in the long $K^{0}_{\rm\scriptscriptstyle S}$
subsample. This can be described with a linear efficiency function with
parameters determined from simulated data for the downstream and long
$K^{0}_{\rm\scriptscriptstyle S}$ subsamples separately. The efficiency
function is then used to correct the description of the decay time
distribution.
The finite decay time resolution of the detector leads to an additional
dilution of the experimentally accessible asymmetry. It is modelled event-by-
event with a triple Gaussian function,
$\displaystyle\mathcal{R}(t-t^{\prime}|\sigma_{t})=\sum_{i=1}^{3}f_{i}\frac{1}{\sqrt{2\pi}s_{i}\sigma_{t}}\exp{\left(-\frac{(t-t^{\prime}-b\sigma_{t})^{2}}{2(s_{i}\sigma_{t})^{2}}\right)}\
,$ (4)
where $t$ is the reconstructed decay time, $t^{\prime}$ is the true decay
time, and $\sigma_{t}$ is the per-event decay time resolution estimate. The
parameters are: the three fractions $f_{i}$, which sum to unity, the three
scale factors $s_{i}$, and a relative bias $b$, which is found to be small.
They are determined from a fit to the $t$ and $\sigma_{t}$ distributions of
prompt ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ events that pass the
selection and trigger criteria for
$B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{0}_{\rm\scriptscriptstyle S}$, except for decay time biasing
requirements. The parameters are determined separately for the subsamples
formed from downstream and long $K^{0}_{\rm\scriptscriptstyle S}$ candidates.
This results in an average effective decay time resolution of
$55.6\,\mathrm{fs}$ ($65.6\,\mathrm{fs}$) for candidates with long
(downstream) $K^{0}_{\rm\scriptscriptstyle S}$.
## 5 Measurement of $S_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{0}_{\rm\scriptscriptstyle S}}$ and
$C_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{0}_{\rm\scriptscriptstyle
S}}$
The analysis is performed using the following set of observables: the
reconstructed mass $m_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{0}_{\rm\scriptscriptstyle S}}$, the decay time $t$, the estimated
decay time resolution $\sigma_{t}$, the flavour tag $d$, and the per-event
mistag probability $\eta$. The $C\\!P$ observables
$S_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{0}_{\rm\scriptscriptstyle
S}}$ and $C_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{0}_{\rm\scriptscriptstyle S}}$ are determined as parameters in an
unbinned extended maximum likelihood fit to the data.
Due to different resolution and acceptance effects for the downstream and long
$K^{0}_{\rm\scriptscriptstyle S}$ subsamples, a simultaneous fit to both
subsamples is performed. In each subsample, the probability density function
(PDF) is defined as the sum of two individual PDFs, one for each of the
components of the fit: the $B^{0}$ signal and the background. The latter
component contains both combinatorial background and mis-reconstructed
$b$-hadron decays.
The reconstructed mass distribution of the signal is described by the sum of
two Gaussian PDFs with common mean but different widths. Only the mean is
shared between the two subsamples. The background component is parameterised
as an exponential function, different for each subsample.
The signal and background distributions of the per-event mistag probability
$\eta$ are modelled with PDFs formed from histograms obtained with the sPlot
technique [20] on the reconstructed mass distribution. In both subsamples the
same signal and background models are used.
The distributions of the estimated decay time resolution $\sigma_{t}$ are
different in each component and each subsample. Hence, no parameters are
shared between subsamples or components. All $\sigma_{t}$ PDFs are modelled
with lognormal functions
$\displaystyle\text{Ln}(\sigma_{t};M_{\sigma_{t}},k)=\frac{1}{\sqrt{2\pi}\sigma_{t}\ln
k}\exp\left(-\frac{\ln^{2}(\sigma_{t}/M_{\sigma_{t}})}{2\ln^{2}(k)}\right),$
(5)
where $M_{\sigma_{t}}$ is the median and $k$ the tail parameter. The
background components in both subsamples are parameterised by single lognormal
functions. For the signal a sum of two lognormals with common (different)
median parameter(s) is chosen for the long $K^{0}_{\rm\scriptscriptstyle S}$
(downstream $K^{0}_{\rm\scriptscriptstyle S}$) subsample.
The background PDFs of the decay time are modelled in each subsample by the
sum of two exponential functions. These are convolved with the corresponding
resolution function $\mathcal{R}(t-t^{\prime}|\sigma_{t})$. The parameters are
not shared between the two subsamples. The background distribution of tags $d$
is described as a uniform distribution.
The signal PDF for the decay time simultaneously describes the distribution of
tags $d$, and is given by
$\displaystyle\mathcal{P}(t,d|\sigma_{t},\eta)=\epsilon(t)\cdot\mathcal{P}_{C\\!P}(t^{\prime},d|\sigma_{t},\eta)\otimes\mathcal{R}(t-t^{\prime}|\sigma_{t})\
,$ (6)
with
$\displaystyle\mathcal{P}_{C\\!P}(t^{\prime},d|\sigma_{t},\eta)\propto
e^{-t^{\prime}/\tau}\Big{(}1$ $\displaystyle-d\Delta
p_{0}-dA_{\text{P}}(1-2\omega(\eta))$
$\displaystyle-(d(1-2\omega(\eta))-A_{\text{P}}(1-d\Delta
p_{0}))S_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{0}_{\rm\scriptscriptstyle S}}\sin\Delta m_{d}t^{\prime}$
$\displaystyle+(d(1-2\omega(\eta))-A_{\text{P}}(1-d\Delta
p_{0}))C_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{0}_{\rm\scriptscriptstyle S}}\cos\Delta m_{d}t^{\prime}\Big{)}\ .$
(7)
This PDF description exploits time-dependent asymmetries, while its
normalisation adds sensitivity by accessing time-integrated asymmetries. The
lifetime $\tau$, the mass difference $\Delta m_{d}$, and the $C\\!P$
parameters $S_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{0}_{\rm\scriptscriptstyle S}}$ and
$C_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{0}_{\rm\scriptscriptstyle
S}}$ are shared in the PDFs of the downstream and long
$K^{0}_{\rm\scriptscriptstyle S}$ subsamples, as well as the asymmetry
$A_{\text{P}}=(R_{\kern
1.25995pt\overline{\kern-1.25995ptB}{}^{0}}-R_{B^{0}})/(R_{\kern
1.25995pt\overline{\kern-1.25995ptB}{}^{0}}+R_{B^{0}})$ of the production
rates $R$ for $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ and $B^{0}$
mesons in $pp$ collisions at LHCb. The latter value has been measured in Refs.
[21, 22] to be $A_{\text{P}}=-0.015\pm 0.013$.
In the fit all parameters related to decay time resolution and acceptance are
fixed. The tagging parameters and the production asymmetry parameter are
constrained within their statistical uncertainties by Gaussian constraints in
the likelihood. The fit yields
$\displaystyle S_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{0}_{\rm\scriptscriptstyle S}}=0.73\pm 0.07\ ,\quad
C_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{0}_{\rm\scriptscriptstyle
S}}=0.03\pm 0.09,$
with a correlation coefficient $\rho(S_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{0}_{\rm\scriptscriptstyle
S}},C_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{0}_{\rm\scriptscriptstyle S}})=0.42$. Both of the uncertainties and
the correlation are statistical only. The lifetime is fitted as $\tau=1.496\pm
0.018$${\rm\,ps}$ and the oscillation frequency as $\Delta m_{d}=0.53\pm
0.05$${\rm\,ps^{-1}}$, both in good agreement with the world averages [7, 23].
The mass and decay time distributions are shown in Fig. 1. The measured signal
asymmetry and the projection of the signal PDF are shown in Fig. 2.
Figure 1: Invariant mass (left) and decay time (right) distributions of the
$B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{0}_{\rm\scriptscriptstyle S}$ candidates. The solid line shows the
projection of the full PDF and the shaded area the projection of the
background component. Figure 2: Time-dependent asymmetry $(N_{\kern
1.25995pt\overline{\kern-1.25995ptB}{}^{0}}-N_{B^{0}})/(N_{\kern
1.25995pt\overline{\kern-1.25995ptB}{}^{0}}+N_{B^{0}})$. Here, $N_{B^{0}}$
($N_{\kern 1.25995pt\overline{\kern-1.25995ptB}{}^{0}}$) is the number of
$B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{0}_{\rm\scriptscriptstyle S}$ decays with a $B^{0}$ ($\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}$) flavour tag. The data points are
obtained with the sPlot technique, assigning signal weights to the events
based on a fit to the reconstructed mass distributions. The solid curve is the
signal projection of the PDF. The green shaded band corresponds to the one
standard deviation statistical error.
## 6 Systematic uncertainties
Most systematic uncertainties are estimated by generating a large number of
pseudo-experiments from a modified PDF and fitting each sample with the
nominal PDF. The PDF used in the generation is chosen according to the source
of systematic uncertainty that is being investigated. The variation of the
fitted values of the $C\\!P$ parameters is used to estimate systematic effects
on the measurement.
The largest systematic uncertainty arises from the limited knowledge of the
accuracy of the tagging calibration. It is estimated by varying the
calibration parameters within their systematic uncertainties in the pseudo-
experiments. Another minor systematic uncertainty related to tagging emerges
from ignoring a possible difference of tagging efficiencies of $B^{0}$ and
$\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}$.
The effect of an incorrect description of the decay time resolution model is
derived from pseudo-experiments in which the scale factors of the resolution
model are multiplied by a factor of either $0.5$ or $2$ in the generation. As
the mean decay time resolution of LHCb is much smaller than the oscillation
period of the $B^{0}$ system this variation leads only to a small systematic
uncertainty. The omission of acceptance effects for low decay times is
estimated from pseudo-experiments where the time-dependent efficiencies
measured from data are used in the generation but omitted in the fits.
Additionally, a possible inaccuracy in the description of the efficiency
decrease at large decay times is checked by varying the parameters within
their errors, but is found to be negligible.
The uncertainty induced by the limited knowledge of the background
distributions is evaluated from a fit method based on the sPlot technique. A
fit with the PDFs for the reconstructed mass is performed to extract signal
weights for the distributions in the other observable dimensions. These
weights are then used to perform a fit with the PDF of the signal component
only. The difference in fit results is treated as an estimate of the
systematic uncertainty.
To estimate the influence of possible biases in the $C\\!P$ parameters
emerging from the fit method itself, the method is probed with a large set of
pseudo-experiments. Systematic uncertainties of $0.004$ for
$S_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{0}_{\rm\scriptscriptstyle
S}}$ and $0.005$ for $C_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{0}_{\rm\scriptscriptstyle S}}$ are assigned based on the biases
observed in different fit settings.
The uncertainty on the scale of the longitudinal axis and on the scale of the
momentum [24] sum to a total uncertainty of $<0.1\%$ on the decay time. This
has a negligible effect on the $C\\!P$ parameters. Likewise, potential biases
from a non-random choice of the $B^{0}$ candidate in events with multiple
candidates are found to be negligible.
The sources of systematic effects and the resulting systematic uncertainties
on the $C\\!P$ parameters are quoted in Table 1 where the total systematic
uncertainty is calculated by summing the individual uncertainties in
quadrature.
Table 1: Summary of systematic uncertainties on the $C\\!P$ parameters. Origin | $\sigma(S_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{0}_{\rm\scriptscriptstyle S}})$ | $\sigma(C_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{0}_{\rm\scriptscriptstyle S}})$
---|---|---
Tagging calibration | $0.034$ | $0.001$
Tagging efficiency difference | $0.002$ | $0.002$
Decay time resolution | $0.001$ | $0.002$
Decay time acceptance | $0.002$ | $0.006$
Background model | $0.012$ | $0.009$
Fit bias | $0.004$ | $0.005$
Total | $0.036$ | $0.012$
The analysis strategy makes use of the time-integrated and time-dependent
decay rates of $B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{0}_{\rm\scriptscriptstyle S}$ decays that are tagged as
$B^{0}$/$\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ meson. Cross-check
analyses exploiting only the time-integrated or only the time-dependent
information show that both give results that are in good agreement and
contribute to the full analysis with comparable statistical power.
## 7 Conclusion
In a dataset of $1.0$$\mbox{\,fb}^{-1}$ collected with the LHCb detector,
approximately $8200$ flavour tagged decays of
$B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{0}_{\rm\scriptscriptstyle S}$ are selected to measure the $C\\!P$
observables $S_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{0}_{\rm\scriptscriptstyle S}}$ and
$C_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{0}_{\rm\scriptscriptstyle
S}}$, which are related to the CKM angle $\beta$. A fit to the time-dependent
decay rates of $B^{0}$ and $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}$
decays yields
$\displaystyle S_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{0}_{\rm\scriptscriptstyle S}}$ $\displaystyle=0.73\pm
0.07\text{\,(stat)}\pm 0.04\text{\,(syst)},$ $\displaystyle
C_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{0}_{\rm\scriptscriptstyle
S}}$ $\displaystyle=0.03\pm 0.09\text{\,(stat)}\pm 0.01\text{\,(syst)},$
with a statistical correlation coefficient of
$\rho(S_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{0}_{\rm\scriptscriptstyle
S}},C_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{0}_{\rm\scriptscriptstyle S}})=0.42$. This is the first significant
measurement of $C\\!P$ violation in
$B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{0}_{\rm\scriptscriptstyle S}$ decays at a hadron collider [25]. The
measured values are in agreement with previous measurements performed at the
$B$ factories [5, 6] and with the world averages [7].
## 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); ANCS/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
| 2012-11-26T20:55:55 |
2024-09-04T02:49:38.490422
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "LHCb collaboration: R. Aaij, C. Abellan Beteta, A. Adametz, B. Adeva,\n M. Adinolfi, C. Adrover, A. Affolder, Z. Ajaltouni, J. Albrecht, F. Alessio,\n M. Alexander, S. Ali, G. Alkhazov, P. Alvarez Cartelle, A. A. Alves Jr, S.\n Amato, Y. Amhis, L. Anderlini, J. Anderson, R. B. Appleby, O. Aquines\n Gutierrez, F. Archilli, A. Artamonov, M. Artuso, E. Aslanides, G. Auriemma,\n S. Bachmann, J. J. Back, C. Baesso, W. Baldini, R. J. Barlow, C. Barschel, S.\n Barsuk, W. Barter, A. Bates, Th. Bauer, A. Bay, J. Beddow, I. Bediaga, S.\n Belogurov, K. Belous, I. Belyaev, E. Ben-Haim, M. Benayoun, G. Bencivenni, S.\n Benson, J. Benton, A. Berezhnoy, R. Bernet, M.-O. Bettler, M. van Beuzekom,\n A. Bien, S. Bifani, T. Bird, A. Bizzeti, P. M. Bj{\\o}rnstad, T. Blake, F.\n Blanc, C. Blanks, J. Blouw, S. Blusk, A. Bobrov, V. Bocci, A. Bondar, N.\n Bondar, W. Bonivento, S. Borghi, A. Borgia, T. J. V. Bowcock, C. Bozzi, T.\n Brambach, J. van den Brand, J. Bressieux, D. Brett, M. Britsch, T. Britton,\n N. H. Brook, H. Brown, A. B\\\"uchler-Germann, I. Burducea, A. Bursche, J.\n Buytaert, S. Cadeddu, O. Callot, M. Calvi, M. Calvo Gomez, A. Camboni, P.\n Campana, A. Carbone, G. Carboni, R. Cardinale, A. Cardini, H. Carranza-Mejia,\n L. Carson, K. Carvalho Akiba, G. Casse, M. Cattaneo, Ch. Cauet, M. Charles,\n Ph. Charpentier, P. Chen, N. Chiapolini, M. Chrzaszcz, K. Ciba, X. Cid Vidal,\n G. Ciezarek, P. E. L. Clarke, M. Clemencic, H. V. Cliff, J. Closier, C. Coca,\n V. Coco, J. Cogan, E. Cogneras, P. Collins, A. Comerma-Montells, A. Contu, A.\n Cook, M. Coombes, G. Corti, B. Couturier, G. A. Cowan, D. Craik, S. Cunliffe,\n R. Currie, C. D'Ambrosio, P. David, P. N. Y. David, I. De Bonis, K. De Bruyn,\n S. De Capua, M. De Cian, J. M. De Miranda, L. De Paula, P. De Simone, D.\n Decamp, M. Deckenhoff, H. Degaudenzi, L. Del Buono, C. Deplano, D. Derkach,\n O. Deschamps, F. Dettori, A. Di Canto, J. Dickens, H. Dijkstra, P. Diniz\n Batista, M. Dogaru, F. Domingo Bonal, S. Donleavy, F. Dordei, A. Dosil\n Su\\'arez, D. Dossett, A. Dovbnya, F. Dupertuis, R. Dzhelyadin, A. Dziurda, A.\n Dzyuba, S. Easo, U. Egede, V. Egorychev, S. Eidelman, D. van Eijk, S.\n Eisenhardt, R. Ekelhof, L. Eklund, I. El Rifai, Ch. Elsasser, D. Elsby, A.\n Falabella, C. F\\\"arber, G. Fardell, C. Farinelli, S. Farry, V. Fave, V.\n Fernandez Albor, F. Ferreira Rodrigues, M. Ferro-Luzzi, S. Filippov, C.\n Fitzpatrick, M. Fontana, F. Fontanelli, R. Forty, O. Francisco, M. Frank, C.\n Frei, M. Frosini, S. Furcas, A. Gallas Torreira, D. Galli, M. Gandelman, P.\n Gandini, Y. Gao, J-C. Garnier, J. Garofoli, P. Garosi, J. Garra Tico, L.\n Garrido, C. Gaspar, R. Gauld, E. Gersabeck, M. Gersabeck, T. Gershon, Ph.\n Ghez, V. Gibson, V. V. Gligorov, C. G\\\"obel, D. Golubkov, A. Golutvin, A.\n Gomes, 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, O.\n Gr\\\"unberg, B. Gui, E. Gushchin, Yu. Guz, T. Gys, C. Hadjivasiliou, G.\n Haefeli, C. Haen, S. C. Haines, S. Hall, T. Hampson, S. Hansmann-Menzemer, N.\n Harnew, S. T. Harnew, J. Harrison, P. F. Harrison, T. Hartmann, J. He, V.\n Heijne, K. Hennessy, P. Henrard, J. A. Hernando Morata, E. van Herwijnen, E.\n Hicks, D. Hill, M. Hoballah, P. Hopchev, W. Hulsbergen, P. Hunt, T. Huse, N.\n Hussain, D. Hutchcroft, D. Hynds, V. Iakovenko, P. Ilten, J. Imong, R.\n Jacobsson, A. Jaeger, M. Jahjah Hussein, E. Jans, F. Jansen, P. Jaton, B.\n Jean-Marie, F. Jing, M. John, D. Johnson, C. R. Jones, B. Jost, M. Kaballo,\n S. Kandybei, M. Karacson, T. M. Karbach, I. R. Kenyon, U. Kerzel, T. Ketel,\n A. Keune, B. Khanji, Y. M. Kim, O. Kochebina, V. Komarov, R. F. Koopman, P.\n Koppenburg, M. Korolev, A. Kozlinskiy, L. Kravchuk, K. Kreplin, M. Kreps, G.\n Krocker, P. Krokovny, F. Kruse, M. Kucharczyk, V. Kudryavtsev, T.\n Kvaratskheliya, V. N. La Thi, D. Lacarrere, G. Lafferty, A. Lai, D. Lambert,\n R. W. Lambert, E. Lanciotti, G. Lanfranchi, C. Langenbruch, T. Latham, C.\n Lazzeroni, R. Le Gac, J. van Leerdam, J.-P. Lees, R. Lef\\`evre, A. Leflat, J.\n Lefran\\c{c}ois, O. Leroy, T. Lesiak, Y. Li, L. Li Gioi, M. Liles, R. Lindner,\n C. Linn, B. Liu, G. Liu, J. von Loeben, J. H. Lopes, E. Lopez Asamar, N.\n Lopez-March, H. Lu, J. Luisier, H. Luo, A. Mac Raighne, F. Machefert, I. V.\n Machikhiliyan, F. Maciuc, O. Maev, J. Magnin, M. Maino, S. Malde, G. Manca,\n G. Mancinelli, N. Mangiafave, U. Marconi, R. M\\\"arki, J. Marks, G.\n Martellotti, A. Martens, L. Martin, A. Mart\\'in S\\'anchez, M. Martinelli, D.\n Martinez Santos, D. Martins Tostes, A. Massafferri, R. Matev, Z. Mathe, C.\n Matteuzzi, M. Matveev, E. Maurice, A. Mazurov, J. McCarthy, G. McGregor, R.\n McNulty, M. Meissner, M. Merk, J. Merkel, D. A. Milanes, M.-N. Minard, J.\n Molina Rodriguez, S. Monteil, D. Moran, P. Morawski, R. Mountain, I. Mous, F.\n Muheim, K. M\\\"uller, R. Muresan, B. Muryn, B. Muster, J. Mylroie-Smith, P.\n Naik, T. Nakada, R. Nandakumar, I. Nasteva, M. Needham, N. Neufeld, A. D.\n Nguyen, T. D. Nguyen, C. Nguyen-Mau, M. Nicol, V. Niess, N. Nikitin, T.\n Nikodem, A. Nomerotski, A. Novoselov, A. Oblakowska-Mucha, V. Obraztsov, S.\n Oggero, S. Ogilvy, O. Okhrimenko, R. Oldeman, M. Orlandea, J. M. Otalora\n Goicochea, P. Owen, B. K. Pal, A. Palano, M. Palutan, J. Panman, A.\n 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. Pellegrino, G. Penso, M. Pepe Altarelli, S. Perazzini, D. L.\n Perego, E. Perez Trigo, A. P\\'erez-Calero Yzquierdo, P. Perret, M.\n Perrin-Terrin, G. Pessina, K. Petridis, A. Petrolini, A. Phan, E. Picatoste\n Olloqui, B. Pie Valls, B. Pietrzyk, T. Pila\\v{r}, D. Pinci, S. Playfer, M.\n Plo Casasus, F. Polci, G. Polok, A. Poluektov, E. Polycarpo, D. Popov, B.\n Popovici, C. Potterat, A. Powell, J. Prisciandaro, V. Pugatch, A. Puig\n Navarro, W. Qian, J. H. Rademacker, B. Rakotomiaramanana, M. S. Rangel, I.\n Raniuk, N. Rauschmayr, G. Raven, S. Redford, M. M. Reid, A. C. dos Reis, S.\n Ricciardi, A. Richards, K. Rinnert, V. Rives Molina, D. A. Roa Romero, P.\n Robbe, E. Rodrigues, P. Rodriguez Perez, G. J. Rogers, S. Roiser, V.\n Romanovsky, A. Romero Vidal, J. Rouvinet, T. Ruf, H. Ruiz, G. Sabatino, J. J.\n Saborido Silva, N. Sagidova, P. Sail, B. Saitta, C. Salzmann, B. Sanmartin\n Sedes, M. Sannino, R. Santacesaria, C. Santamarina Rios, R. Santinelli, E.\n Santovetti, M. Sapunov, A. Sarti, C. Satriano, A. Satta, M. Savrie, P.\n Schaack, M. Schiller, H. Schindler, S. Schleich, 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, P. Shatalov, Y.\n Shcheglov, T. Shears, L. Shekhtman, O. Shevchenko, V. Shevchenko, A. Shires,\n R. Silva Coutinho, T. Skwarnicki, N. A. Smith, E. Smith, M. Smith, K.\n Sobczak, 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. Stoica, S. Stone,\n B. Storaci, M. Straticiuc, U. Straumann, V. K. Subbiah, S. Swientek, 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, A. Tsaregorodtsev, P. Tsopelas, N.\n Tuning, M. Ubeda Garcia, A. Ukleja, D. Urner, U. Uwer, V. Vagnoni, G.\n Valenti, R. Vazquez Gomez, P. Vazquez Regueiro, S. Vecchi, J. J. Velthuis, M.\n Veltri, G. Veneziano, M. Vesterinen, B. Viaud, I. Videau, D. Vieira, X.\n Vilasis-Cardona, J. Visniakov, A. Vollhardt, D. Volyanskyy, D. Voong, A.\n Vorobyev, V. Vorobyev, C. Vo{\\ss}, H. Voss, R. Waldi, R. Wallace, S.\n Wandernoth, J. Wang, D. R. Ward, N. K. Watson, A. D. Webber, D. Websdale, M.\n Whitehead, J. Wicht, D. Wiedner, L. Wiggers, G. Wilkinson, M. P. Williams, M.\n Williams, F. F. Wilson, J. Wishahi, M. Witek, W. Witzeling, S. A. Wotton, S.\n Wright, S. Wu, K. Wyllie, Y. Xie, F. Xing, Z. Xing, Z. Yang, R. Young, X.\n Yuan, O. Yushchenko, M. Zangoli, M. Zavertyaev, F. Zhang, L. Zhang, W. C.\n Zhang, Y. Zhang, A. Zhelezov, L. Zhong, A. Zvyagin",
"submitter": "Julian Wishahi",
"url": "https://arxiv.org/abs/1211.6093"
}
|
1211.6123
|
We present the results of fitting simulations of an unbiased
of SDSS galaxies utilizing the fitting routine GALFIT and analysis pipeline
PyMorph. These simulations are used to test the two-dimensional decompositions
of SDSS galaxies. The simulations show that single
Sérsic models of SDSS data are recovered with $\sigma_{\mathrm{mag}}
\approx 0.025$ mag and $\sigma_{\mathrm{radius}} \approx 5\%$. The global
values (half-light radius and magnitude) are equally well constrained when a
two-component model is used. Sub-components of two-component models present
more scatter. SDSS resolution is the primary
source of error in the recovery of models. We use a
simple statistical correction of the biases in fitted parameters,
providing an example using the Sérsic index.
Fitting a two-component Sérsic + Exponential model to a
single Sérsic galaxy results in a noisier, but unbiased, recovery of the
input parameters ($\sigma_{\mathrm{total mag}}
\approx 0.075$ mag and $\sigma_{\mathrm{radius}} \approx 10\%$); fitting a
single Sérsic profile to a two-component system
results in biases of total magnitude and halflight radius of
$\approx 0.05-0.10$ mag and 5%-10% in radius. Using an
F-test to select the best fit model from among the single- and two-component
models is sufficient to remove this bias. We recommend fitting a
two-component model to all galaxies when attempting to
measure global parameters such as total magnitude and halflight radius.
galaxies: structural parameters – galaxies: fundamental parameters –
galaxies: catalogs – methods: numerical – galaxies: evolution
§ INTRODUCTION
Measurement of fundamental galaxy properties is an essential step
in analyzing galaxy structure, formation, and evolution. At
the most basic level, luminosity, size, and morphology are important
properties, useful in evaluating dynamical and evolutionary models
<cit.>. Non-parametric methods exist to estimate luminosity,
size, and structure without imposing a functional form on a galaxy
<cit.>. However,
non-parametric methods are sensitive to the depth of the image and to the PSF.
This can result in underestimating the luminosity and size of an object due to
missing flux in faint regions of the galaxy or when the true size of the galaxy
becomes small relative to the size of the angular PSF [Blanton et al, 2001, Blanton et al, 2003].
Parametric methods offer a reasonable way to extrapolate galaxy light profiles
into fainter regions at the expense of introducing a potentially incorrect
functional form for the galaxy.
Common functional forms used in parametric fitting include the $r^{1/4}$ and the
$r^{1/n}$ models originally
presented by [de Vaucouleurs, 1948] and [Sérsic, 1963].
Empirical study suggests that bulges and elliptical galaxies are better
described by de Vacouleurs profiles or Sérsic profiles with Sérsic
index $n\approx 4$. Disks and late-type spirals are best described by
exponential profiles or Sérsic profiles with Sérsic index $n\approx
1$ [Freeman, 1970].
More recent work has shown that the relationship between Sérsic index and
the photometric or kinematic components of a galaxy is more complicated.
Following [Kent, 1985], many
studies simultaneously fit a second component in order to better
accommodate the qualitative differences of bulges and disks in galaxies.
Additionally, [Caon, Capaccioli & D'Onofrio, 1993] showed that the Sérsic profile is a better
fit to many early-type galaxies than the traditional de Vacouleurs profile.
There have been several catalogs of photometric galaxy decompositions
presented recently [Simard et al, 2011, Kelvin et al, 2012, Lackner & Gunn, 2012] with particular
interest on the applicability of large sets of image decompositions to
evolutionary models. However, systematic effects continue to be of concern, and
the reliability of two-component decompositions in cases of low to moderate
signal-to-noise are often viewed with some skepticism. In order to quantify the
systematics and robustness of the $\sim 7\times10^5$ fits of g, r, and i band
SDSS spectroscopic galaxies to be presented in [Meert, Vikram & Bernardi, 2013], hereafter referred to
as , we generate simulations of single and two-component
galaxies, referred to as “mocks,” and fit them using the same PyMorph pipeline
[Vikram et al, 2010] used for
the photometric decompositions presented in . The catalog has
already been used in [Bernardi et al, 2013] to study systematics in the
size-luminosity relation, in [Shankar et al, 2013] to study size-evolution of
spheroids, and in [Huertas-Company et al, 2012] to study the environmental dependence of the
mass-size relation of early-type galaxies.
Following several detailed studies which have used simulations to test the
robustness of different fitting algorithms <cit.>,
the main goal of this paper is to test the robustness of PyMorph pipeline
software on SDSS galaxies.
We use these simulations to test the effects of
increased signal-to-noise as
well as increased resolution, PSF errors, and sky determination.
Our simulations are specifically applicable to SDSS galaxies and are
useful for evaluating the decompositions presented in . We use unbiased
samples to estimate and correct the systematic error on recovered parameters
as well as estimate reasonable uncertainties on fit parameters.
A description of the simulation process is presented in Section <ref>.
This includes constructing a catalog of realistic galaxy parameters
(Section <ref>); generating galaxy surface brightness
profiles based
on these parameters (Section <ref>); generating sky and noise
(Sections <ref> and <ref>); and including seeing
effects in the final image. The completed simulations are run through the fitting
pipeline, and the fits are analyzed in Section <ref>. We examine the
recovery of structural parameters in noise-free images
(Section <ref>) and parameter recovery in realistic observing
conditions including both neighboring sources and the effects of incorrect PSF
(Section <ref>). Recovery of mock galaxies is unbiased
for single Sérsic models. However, two-component mocks are biased when fitted with
single Sérsic profiles. This bias consists of an overestimate of
size and luminosity of the galaxy. PyMorph is further tested by inserting
mocks into real SDSS images to test the dependence on local density
(Section <ref>). We examine dependence of the fits on resolution and
signal-to-noise (Section <ref>). The effect of changing the fitted
cutout size (Section <ref>) and the effect of incorrect background
estimation (Section <ref>) are also examined.
In Section <ref> we discuss the overall scatter in our fits and the
implications of the simulations.
Finally, in Section <ref> we provide concluding remarks.
We generate single-component Sérsic galaxy models (hereafter
referred to as ) and two forms of two-component galaxy models:
one is a linear combination of de Vacouleurs and an exponential
profile () and the other replaces the de Vacouleurs with a Sérsic
profile ().
A good overview of the Sérsic profile
used throughout this paper is presented in [Graham & Driver, 2005].
Throughout the paper, a $\Lambda$CDM cosmology is assumed with
($h$,$\Omega_m$,$\Omega_{\Lambda}$) = (0.7,0.28,0.72) when necessary.
§ CREATING THE SIMULATIONS
§.§ Selecting the simulation catalog
We create a set of mocks using fits from the
photometric decompositions presented in . These galaxy parameters
represent the r-band image decompositions of a complete sample of
the SDSS spectroscopic catalog containing all galaxies with spectroscopic
information in SDSS DR7 [Abazajian et al, 2009].
The sample contains galaxies
with extinction-corrected
r-band Petrosian magnitudes between 14 and 17.77. The lower limit of 17.77 mag
in the r-band is the lower limit for completeness of the SDSS Spectroscopic
Survey [Strauss et al, 2002]. The galaxies are also
required to be identified by the SDSS pipeline <cit.>
as a galaxy
(), and the spectrum must also be identified as a galaxy
(). Additional cuts on the data following
[Shen et al, 2003] and [Simard et al, 2011] are applied. Any galaxies with redshift $<$
0.005 are removed to prevent redshift contamination by peculiar velocity.
Galaxies with saturation, deblended as a PSF as indicated by
the flags, or not included in the Legacy
survey[A list of fields in the Legacy survey
is provided at <http://www.sdss.org/dr7/coverage/allrunsdr7db.par>] are
also removed from the sample. In addition, following [Strauss et al, 2002] and
[Simard et al, 2011], we apply a surface-brightness cut of $\mu_{\textrm{50, r}} <
23.0$ mag/arcsec$^2$ because there is incomplete spectroscopic target selection
beyond this
threshold. After applying all data cuts, a sample of 670,722 galaxies
We select an unbiased sample of galaxies from the DR7 sample and use the fitted
models from PyMorph to generate the mocks used in this paper.
For each model (, , and ), we select a representative sub-sample physically
meaningful photometric decompositions. In order to ensure
that the galaxies are representative of the full catalog, we examined
the distributions of basic observational parameters of SDSS galaxies
(surface brightness, redshift, apparent Petrosian magnitude,
Petrosian half-light radius, and absolute magnitude).
Some restrictions on fit parameters are necessary to ensure that
outliers are removed from the parameter space used to generate the
simulations. Galaxies that do not satisfy these basic cuts
are removed to ensure that the parameters used to generate the images
are physically motivated. The cuts do not significantly bias our galaxy
distribution as is shown in Figure <ref>. The cuts are:
* Any Sérsic components must have Sérsic index less than 8.
* Half-light radius of any Sérsic component must be
less than 40 kpc.
* In the two-component fits, the ratio of the bulge halflight radius to
disk scale radius should be less than 1, or the galaxy should be bulge
dominated (B/T $>$ 0.5).
Conditions (i) and (ii) are used to prevent selection of models with
extended profiles that are likely the result of incorrect sky estimation during
the fitting process. Condition (iii) ensures that any disk dominated
galaxies have a bulge component that is smaller than the disk.
After enforcing the cuts on the sample, 10,000 fitted galaxy profiles for each
of the , , and models are selected at random
without regard to the morphological classification of the original galaxy.
The fitted parameters of these sample galaxies
are used to generate the mocks used in testing the pipeline.
Selecting galaxy samples independent of galaxy
morphology allows the and samples to
contain some galaxies that do not truly possess a second component.
Additionally, there will be some truly two-component galaxies (both bulge
and disk components are present) that are misrepresented by a single Sérsic
fit. However, this sampling method will not invalidate the results of our tests.
Since we seek to test the ability to recover simulated galaxy parameters, we only
require a realistic sample of galaxy profiles. Our samples satisfy this requirement.
Single Sérsic galaxies in the original sample, simulated as mock galaxies and fit with models, test the ability to recover Sérsic
parameters. Similarly, mocks with models, show bias resulting
from over-fitting a galaxy. Fitting the mocks with a model
shows the bias due to under-fitting.
Fitting a single-component model regardless of galaxy
structure or morphology is a common practice <cit.>. In Figure <ref> we show
that bias of 0.05 mags and 5% of the halflight radius result from fitting a
two-component galaxy with a single component and that this bias increases to 0.1 mags
and 10% of the halflight radius for brighter galaxies.
These biases are important in analyzing the results of a
single-component fitting catalog. For example, [Bernardi et al, 2013] shows that
intermediate B/T galaxies can often be fit by Sérsic models with large
Sérsic indicies, which can lead to misclassification if cuts similar to
[Shen et al, 2003] are used.
subfig:orig_dist:a The surface brightness
subfig:orig_dist:b redshift distribution,
subfig:orig_dist:c extinction-corrected r-band Petrosian
subfig:orig_dist:d r-band Petrosian halflight radius,
subfig:orig_dist:e V$_{\mathrm{max}}$-weighted
luminosity function, and subfig:orig_dist:f signal-to-noise
distribution of the samples used in this paper drawn from the DR7 SDSS
spectroscopic galaxy sample. The distribution of all SDSS spectroscopic
galaxies is shown in black. Distributions of the , , and
mocks are shown in red, green, and blue, respectively.
Bin counts are normalized to integrate to 1. The distributions of the
mocks are representative of the full sample fitted in
and are appropriate to compare to the SDSS spectroscopic
sample as verified by a Kolmogorov-Smirnov 2-sample test. The
signal-to-noise (S/N) will be discussed further in Section <ref>.
In calculating this S/N, we use the measurement of sky provided by
the PyMorph pipeline rather than SDSS to identify and separate target
counts from sky counts. PyMorph sky estimation is shown to be more
accurate than the SDSS estimation provided in the DR7 catalog.
Figure <ref> shows the distributions of surface brightness,
redshift, extinction-corrected r-band Petrosian magnitude,
r-band Petrosian halflight radius, and absolute magnitude of
all SDSS spectroscopic galaxies (in black) and our simulation samples:
(red), (green), and (blue). The distribution
of mock galaxies reproduces the observed distribution for all three samples
for each observational parameter as verified by a KS 2-sample
Figure <ref> also presents the signal-to-noise (S/N) of the
mock samples as compared to the parent distribution. The S/N of the
images is a limiting factor in the fitting process, so care must be taken to
ensure that the S/N is not artificially increased in the simulations when
compared to true SDSS galaxies. This S/N is calculated
using the r-band Petrosian magnitude and r-band Petrosian halflight radius.
Petrosian quantities are used to make a fairer comparison among all the
samples. Because the Petrosian quantities are non-parametric, they avoid the
complications that arise in assessing the possible biases introduced during
fitting. Any differences in S/N are not large enough to significantly bias the
distributions as verified by the KS 2-sample test. Therefore, we conclude that
our samples are fair representations of the underlying distribution of SDSS
spectroscopic galaxies. The S/N is discussed further in
Section <ref>.
Testing the accuracy of the PyMorph fitting routine does not necessarily require
an unbiased parameter distribution. In reality, all that is required is a sample
with sufficient coverage of the parameter space represented by the data. The
simulations use smooth profiles, simplifications of the true galaxies that are
observed in SDSS. Examination of the results of fitting these simplified models
comparison to fits of true observed galaxies can potentially yield useful
information regarding galaxy structure. In [Bernardi et al, 2013],
the simulations are used together with the decompositions of the SDSS
spectroscopic sample to characterize the scatter in the size-luminosity
relation as well as examine possible biases. In order to make these comparisons,
an unbiased sample is required. The distributions shown in
Figure <ref> show that the simulations are appropriate
to use for this purpose.
§.§ Generating the images
We generate the two-dimensional normalized photon
distributions from the one-dimensional Sérsic profiles and the
one-dimensional exponential profiles of each bulge and disk component.
Disk components are only simulated where required, as is the case for
two-component fits. When multiple components are to be simulated, each
component's normalized photon distribution is generated separately and
combined prior to generating the simulated exposure.
Two-dimensional galaxy profiles are treated as azimuthally symmetric
galaxy light profiles that are deformed according to an observed ellipticity.
The galaxy profiles are generated
using the structural parameters generated from photometric decompositions
as described in the previous section. Single-component galaxy
profiles and the bulges of two-component galaxies are generated according
to the Sérsic profile
\begin{equation} \label{eq:sersic}
\begin{aligned}
& I(r) = I_{e} \exp\left(-b_n\left[\left(\dfrac{r}{R_{e}}\right)^{\frac{1}{n}} - 1 \right]\right) \\
& b_n = 1.9992n - 0.3271 \\
\end{aligned}
\end{equation}
where Sérsic index (n), half-light radius ($R_{e}$), and surface brightness
$R_e$($I_{e}$) are selected simultaneously from the catalog described in the
previous section.
For the and cases, an exponential disk
(Equation <ref> with $n=1$) is added to the Sérsic component to
model the
disk component of the galaxies.
The disk is modeled using a slightly modified version of
Equation <ref>.
This model requires input parameters scale radius
($R_{d}$) and
central surface brightness ($I_{d}$).
\begin{equation} \label{eq:expdisk}
\begin{aligned}
& I_{Exp}(r) = I_{d} \exp\left(\dfrac{-r}{ R_{d}}\right). \\
\end{aligned}
\end{equation}
After generating the two-dimensional profile, the image is pixelated by
integrating over each pixel area. The details of this integration are
largely unimportant. However, the simulation must take careful account of the
integration in the central pixels, where the profile can vary greatly over a
single pixel. Various oversampling methods have been devised to properly
correct this common problem <cit.>.
The simulations in this paper have been tested to ensure that the
pixel-by-pixel integration is accurate to $\approx3\%$ of the corresponding
Poisson noise in a given pixel. Therefore, we treat the simulations as exact
calculations of the galaxy photon distributions since any noise from the
integration contributes only a small amount to the total noise budget.
The pixelated galaxy is numerically convolved with a
PSF extracted from SDSS DR7 data using program
distributed by SDSS[ is part of the
package available at
<http://www.sdss.org/dr7/products/images/read_psf.html>]. The choice of
this PSF is discussed in Section <ref>.
§.§ Creating the background
Two hundred background images, each equal in size to an SDSS fpC image, are
also simulated for testing purposes. These images contain constant background and a
randomly selected field of galaxies taken from an SDSS fpC image. The
SDSS catalog provides rudimentary photometric decompositions of each star and
galaxy. Galaxies are fit with an exponential disk and a de Vacouleurs
($n=4$) bulge independently. The best fit is reported as a linear combination
of the two fits using the parameter to express the ratio of
the de Vacouleurs model to the total light in the galaxy.
For the simulated
background used in this paper, each
galaxy is generated using the combined profile of the two fits. The
de Vacouleurs
bulge and exponential disk component are separately simulated according to the
magnitude, radius, ellipticity, and position angle reported in SDSS.
Each component is simulated using the method
described in Section <ref>. The background galaxy is
constructed by adding the two components using the parameter.
The galaxy is then inserted into the fpC image. Any foreground stars are also
simulated as point sources and inserted into the image.
The distribution of sky values for data in the SDSS CASJOBS catalog.
These data are drawn from the CASJOBS parameter and are
converted into counts (DN) per pixel per standard SDSS image exposure of
54 seconds.
We use this distribution to determine the sky value used in our simulations.
As an approximation, we use the mean value of 130 counts/pixel/exposure.
For the background sky counts in the image, we use
the mean sky of all SDSS observations as given in the SDSS
table by the parameter. The distribution of the sky flux is
plotted in Figure <ref> in units of counts (or DN) per pixel per
exposure. The median and mean values for a 54 second SDSS exposure are
$\approx$125 and $\approx$130 counts per pixel, respectively. We use
the mean value of 130 counts per pixel as the background in our simulations.
This sky background is applied to the entire chip as a constant background; no
gradient is simulated across the image. Background gradients should be
approximately constant across a single galaxy.
This assumption is verified by inserting the simulated galaxies into real SDSS
fpC images near known clusters, where the sky contribution should be higher and
gradients are more likely. In Section <ref> we show that there is
little change in the behavior of the fits in these types of environments.
Previous work has improved the measurements of sky background
<cit.>. However, these corrections tend to focus on
areas of large, bright galaxies or on making the sky subtraction stable
for purposes of tiling fpC images together. Since we
are only focused on maintaining the proper S/N for our simulations, the sky
levels provided in the SDSS database are sufficient, provided that they
maintain the correct S/N. We discuss the S/N distribution of our simulations
and the original SDSS galaxy sample in Section <ref> below.
Diffraction spikes and other image artifacts are not directly simulated.
However, the SDSS pipeline often misidentifies additional
phantom sources along an observed diffraction spike. These phantom sources are
modeled in our background, and so these effects are approximately modeled. It is
reasonable to expect that the diffraction effects should
not have a large effect on the fitting process, as their elongated straight
structure does not mimic galaxy structure. The dominant effect produced by the
bright stars in the field is bias in the background estimation
in the nearby neighborhood of a star.
After simulation of the background images, and prior to adding noise,
each background image is convolved with a random SDSS PSF selected from
original fpC image upon which the individual image is based.
Selecting PSFs from original SDSS images introduces a variation in PSF size
between mock galaxies inserted into images and the background galaxies.
However, this variation is not of concern for us in the fitting process because
the vast majority of galaxies (over 90% of all galaxies) do not have neighbors
near enough to the target galaxy to require simultaneous fitting. For these
galaxies, the PSF applied to neighboring galaxies is of no interest in the
fitting process because the sources are masked out. The details of this masking
are not discussed in the remainder of the paper. Modifying the masking conditions produce no
noticable difference in the fitted values for our simulations. For the remaining 10% of
galaxies, there may be some variation in the fit due to differing PSFs. PSF
sizes can differ between target and neighboring galaxies by up to a factor of
2. However in practice, this happens for less than 1% of galaxies of the
galaxies with neighbors. Furthermore, incorrect
PSF tends to only cause effects at the centers of galaxies. So although using
a PSF that is different from the background PSF will affect the recovered
parameters of the neighbor, it will not affect the target galaxy.
§.§ Noise
After generating a target galaxy and inserting it into a background,
Poisson noise is added using the average inverse gain of an SDSS image
(4.7 e$^-$/DN) and the
average contribution of the dark current and read noise, referred to as the
“dark variance,” (1.17 DN$^2$), to determine the standard deviation for each
pixel. Specifically,
\begin{equation}
\label{eq:Fij}
F_{i,j} \equiv I_{i,j} + \textrm{bkrd}_{i,j}
\end{equation}
is the total flux in pixel $(i,j)$ (the sum of the source and
background fluxes in the pixel), and
\begin{equation}
\label{eq:weight}
\sigma_{i,j} = \sqrt{\dfrac{F_{i,j}}{\textrm{gain}} + \textrm{dark variance}}
\end{equation}
\begin{equation}
\left(\dfrac{\textrm{S}}{\textrm{N}}\right)_{i,j} \equiv \dfrac{I_{i,j}}
\end{equation}
for a single pixel.
Since the fitting pipeline is dependent on the S/N, it is essential that the
simulated S/N is comparable to SDSS. The distribution of the average S/N per
pixel within the halflight radius for the simulations and the original galaxies
is plotted in Figure <ref>. The S/N
distribution of simulations and the SDSS
spectroscopic galaxies agree as verified by a KS 2-sample
test, therefore the simulations appropriately approximate the S/N of SDSS
galaxies contained in .
An unbiased selection in the previously mentioned parameters is not
sufficient to guarantee fair sampling of the S/N with respect to magnitude, nor
does it prevent fictitious correlations among multiple fit parameters. In fact,
correlations among fit parameters are to be expected if the PyMorph pipeline is
robustly measuring properties of the target galaxies (many correlations exist
among physical parameters). It is difficult, and largely
unnecessary, to examine every possible relationship for correlations introduced by
biases in the sample selection process.
Examining the S/N and the halflight radius versus apparent magnitude help
to ensure the appropriateness of the simulation. Systematic differences
in radius will lead to systematic variation in the S/N of the sample.
We also examine the scatter in recovered fitting parameters
as a function of magnitude. Therefore, the S/N as a
function of apparent magnitude should appropriately reflect that of the parent
sample from SDSS.
The distribution of galaxy radii as a function of apparent magnitude
for the parent SDSS sample in black, the model in red and the
model in blue. The median in each bin is shown with error bars
representing the 95% CI on the median.
Corresponding dashed lines show the extent of the middle 68% of data. The
model is in close agreement across the entire magnitude range while
the model begins to diverge at brighter magnitudes.
The distribution of galaxy S/N as a function of apparent magnitude is
presented in the same format as Figure <ref>. The
models are in close agreement with the full sample across the entire magnitude
Figure <ref> presents the halflight radius versus apparent
magnitude, and Figure <ref>
presents the S/N versus apparent magnitude. The points
shown in red and blue correspond to the and mocks, respectively. The underlying SDSS parent distribution is shown in black.
Figure <ref> shows that the and models
are in close agreement with the full SDSS sample. The and model
radii agree across the magnitude range. The S/N agrees with the
full SDSS sample or is slightly below that of SDSS. The lower signal-to-noise,
although not exactly that of SDSS, will not bias the tests toward better
results, so we deem these samples acceptable for testing. The sample,
which is not shown here,
tends to have smaller radii and higher S/N at brighter magnitude. The results
of tests using the model are not discussed in the remainder
of this paper. They can be found in .
Examples of mock galaxies and background shown before and
after adding Poisson noise. Top, middle, and bottom rows show randomly
selected sample , , and profiles, respectively.
From left to right, the columns show the mock galaxy, simulated
background, background+galaxy, and final image with noise.
§.§ Final processing for fitting
For each mock galaxy, we also generate a weight image of the $\sigma_{i,j}$
values according to Equation <ref>. This image is supplied along
with the input image to the pipeline in order to calculate the $\chi^2$
value for the fit.
Figure <ref> shows some examples of mock galaxies
throughout the simulation process. This includes the noiseless mock
galaxy, the noiseless simulated background, the composite image of galaxy
and background, and the composite image after adding Poisson noise with
$\sigma_{i,j}$ defined in Equation <ref>. The final image size
used for fitting is 20 times the Petrosian r-band halflight radius.
A discussion of this choice of stamp size is presented in Section <ref>.
§ TESTING PYMORPH IMAGE DECOMPOSITIONS
In order to test the parameter recovery of the PyMorph
pipeline on SDSS spectroscopic galaxies, we apply the PyMorph pipeline to the
mocks described in
Section <ref>. The PyMorph pipeline uses GALFIT to fit smooth profiles
to the the mock galaxies. We apply the pipeline to several different
realizations of our mock galaxies. These realizations increase in
complexity from a noiseless image to an image with real noise and
(possibly clustered) neighboring sources. We show that the ability of PyMorph
to reliably recover model parameters is limited by both the S/N and the
resolution of the mock galaxy. Understanding the systematic effects of S/N
and resolution is useful in interpreting the data presented
in . It may also be used to correct biases in the data as described
later in Section <ref>.
§.§ Noiseless images
As an initial test, the pipeline is applied to simulations prior to adding
noise, background counts, or neighboring sources. This produces
the minimum scatter in the data, serves to verify that our simulations are
correct, and shows that PyMorph is properly functioning.
The total apparent magnitude, halflight radius, and additional fit
parameters recovered by fitting the noiseless images of the and models are presented in Figures <ref>,
<ref>, <ref>,
<ref>, and <ref>. The
plots show the difference
in simulated and fitted values (fitted value - input value). The difference is
shown versus the input magnitude as well as the input value of the respective
fit parameter. The gray-scale shows the density of points in each
plane with red points showing the median value. Error bars on the median value
are the 95% confidence interval on the median obtained from bootstrapping.
Blue dashed lines show the regions which contain
68% of the objects.
Figures <ref> and <ref> show the
corresponding fit is well constrained ( fit with ,
and with ). The total magnitude and halflight
radius are both constrained well within 1% error on the flux or radius
($\sigma_{\mathrm{total\ mag}} \approx 0.01$ mag and
$\sigma_{\mathrm{radius}} \approx 1\%$). However, the scatter increases
somewhat for the sub-components of the fit (see Figure
<ref>). As the
components of the model become dim (bulge/disk magnitude approaches
18.5), the component contribution to the total light becomes small. The
origin of the magnitude limit is merely an artifact of our selection criteria
requiring that all galaxies have total magnitude brighter than 17.77. This
implies that components with magnitude of $\approx$18.5 or dimmer are
necessarily sub-dominant
components and contribute at most $\approx$50% of the light to the total profile. On
average, components dimmer than 18.5 magnitudes contribute about 25% of the
total light to a typical galaxy in this sample, and this contribution drops
rapidly to about 10% by 19 magnitudes. In these
cases, the sub-dominant component will be much less apparent in the image and,
therefore, less important to the overall $\chi^2$ of the fit, allowing for
greater error in the parameters of that component. In addition, once
Poisson noise is considered, these dimmer components suffer from much
lower S/N. Later tests (Section <ref>) show substantial error on
these components due to the low flux and resulting low S/N.
Additionally, sub-dominant components (in particular, bulges) may be much
smaller than the overall size of the galaxy. This makes bulge parameter
recovery susceptible to resolution effects. These effects are also explored in
Section <ref>.
The magnitude and halflight radius are also well constrained when a galaxy
is fit with a profile (Figure <ref>). However, a
galaxy fit with a profile produces large biases in the magnitude
and halflight radius (Figure <ref>).
As already mentioned, the total magnitude and halflight radius are well
($\sigma_{\mathrm{total\ mag}} \approx 0.01$ mag and
$\sigma_{\mathrm{radius}} \approx 1\%$) in cases where the correct model
is applied to the mocks ( mock fit with a model). This is not always the case when the wrong model is applied ( mock fit with a model). When attempting to fit the
simulated mocks with a model, we find measurable bias of
order .01 magnitudes in total magnitude. We also find the scatter of both the
size and magnitude to be increased by an order of magnitude. This bias and
increased scatter becomes even larger in later tests. It is obvious that a
single-component galaxy cannot properly model a two-component
galaxy in general, and therefore, we would expect significant problems in
attempting to fit a single-component profile to a two-component galaxy.
Nevertheless, this type of fit is often performed on real data at low to
moderate resolution and S/N where it is unlikely to
recover a robust two-component fit. An important observation is that the
fit provides the most stable estimate of the halflight radius and total
magnitude regardless of the true simulated galaxy model (, , or
). The additional freedom in the model and the
fact that the and models are special cases of the
model would lead us to expect this result. Therefore, it is
advisable to always use a fit in the course of fitting SDSS galaxies
unless there is specific evidence to the contrary.
One systematic effect in the pipeline that has been noted by other groups
<cit.>, is the underestimate of Sérsic index at
larger Sérsic indexes. At Sérsic indexes of $n\approx$ 4, we
underestimate the Sérsic index by less
than $1\%$. However, this underestimate increases in the later
tests. The data suggest that a substantial component of this error is due to the
resolution limits of the SDSS sample. At larger Sérsic index, a high
sampling rate at the center of the galaxy is useful in distinguishing the
preferred value of the Sérsic index. We further explore the effect of image
resolution in Section <ref>.
Since no Poisson noise is added to these images, the scatter
apparent in these fits is a combination of the limitations of the SDSS data (in
particular resolution), systematics inherent in the PyMorph routine (as well as
the GALFIT routine used by PyMorph), and any parameter degeneracies inherent in
the models.
GALFIT uses the Levenberg-Marquardt minimization method [Press et al, 1992]
to find the minimum of the $\chi^2$ distribution of the fit.
The Levenberg-Marquardt method is not a global search algorithm
but rather follows the steepest decent to a local minimum.
As the parameter space becomes more complicated,
GALFIT has more trouble accurately recovering parameters.
Adding components to the fit (going from a one-component to two-component
fit or going from a fixed Sérsic index component to one with a free Sérsic
index) will not only complicate the $\chi^2$ surface, making convergence less
likely, but may introduce true degeneracies in the parameter space.
For instance, the fit of a galaxy of very late type often suffers from
over-fitting. The bulge component will tend to fit the disk of the galaxy as a
second disk component with $n_{\mathrm{bulge}}\approx1$. This is obviously
an unintended solution to the fitting but one that is equally valid
from an $\chi^2$ perspective. In practice, it is difficult to prevent
this type of convergence without artificially constraining the fitting routine.
Such constraints are generally discouraged and can lead to other negative effects
including convergence to a non-optimal solution. The best
solution to the parameter degeneracy is close examination of any two-component
fits in cases where $n_{\mathrm{bulge}}\approx 1$, or B/T $\approx 0$ or 1.
In addition, PyMorph reports statistical error estimates on the
fitted parameters as returned from GALFIT. These errors are found to be an
underestimate of the true error in the fits by as much as an order of magnitude.
This gross underestimation of the error is also reported by [Häussler et al, 2007] as
well as being discussed in the GALFIT user notes[See the technical
FAQs at < http://users.obs.carnegiescience.edu/peng/work/galfit/TFAQ.html>].
Following [Häussler et al, 2007], we examine the ratio of the
uncertainty reported by GALFIT to the deviation of the measured parameters
(referred to as $\sigma/\Delta$). $\sigma/\Delta$ should be greater than 1
for approximately 68% of the data if the estimated uncertainty is appropriate.
However, this is not the case for any of the parameters in the fits.
We discuss a simple method for correcting the
systematic bias and estimating the uncertainty in Section <ref>.
§.§ The effects of background, neighbor sources, and incorrect PSF
When analyzing real data, it is not possible to extract the PSF at the
target galaxy to arbitrary accuracy. Interpolation is required and generally
performed on a network of the nearest stars to the target galaxy.
We test this effect through extraction of a neighboring PSF to
be used during fitting in place of the PSF used to generate the image.
The neighbor PSF used in fitting is randomly selected from a location within a
200 pixel box surrounding the source. This provides approximately even
sampling of distances from nearly 0 to about 170 pixels in separation from the
source which corresponds to a separation of $\approx0$ to $\approx67.32$
between the target galaxy and the location used for PSF extraction.
This inserts some PSF error into the process of fitting as would be expected
in the case of real data. However, it also retains the similarity between the PSF
used for simulation and the PSF used for fitting. A strong similarity between
the two would be expected since the PSF generally will not vary greatly over
the area of a single fpC image.
Target galaxies are randomly inserted into the
simulated fpC images described in Section <ref>.
The simulated fpC images contain sky as well as neighboring sources.
The PSF of the neighboring sources will
have a different PSF than the target galaxy. This effect is not of concern in
this work.
Prior to fitting, a new cutout is extracted from the
total image (containing the target galaxy and background) ensuring that the
target galaxy is
at the center of the stamp image. By constructing new postage stamp images in
this manner, we ensure that there is sufficient variation in the background
while preventing us from fitting the incorrect galaxy.
These fits (containing error in PSF reconstruction, neighboring sources, and
noise) are the closest simulation to actual observing conditions that
we have analyzed. Therefore the fits and the resulting measures of scatter and
bias are adopted as our fiducial estimates of scatter and bias when using the
Figures <ref>, <ref>,
<ref>, <ref>, and
show that we recover the input values with marginal scatter.
The total magnitude and halflight radius remain well constrained
($\sigma_{\mathrm{total mag}} \approx 0.05$ mag and
$\sigma_{\mathrm{radius}} \approx 5\%$) in cases where the
correct model is fit to the mock galaxy. However, this scatter
becomes larger when the wrong model is fit. The underestimate of
the Sérsic index, particularly at large values, persists.
Further examination of the two-component fits show that the pipeline has
difficulty extracting dim components (bulge or disk magnitude dimmer than
$\approx 18.5$). In these ranges, the components are observed at lower S/N
and the pipeline looses sensitivity to the model parameters.
The fit shows an underestimate of Sérsic index, which is even
stronger than in the single-component case, and an underestimate of
bulge radius. However, the disk parameters remain unbiased with an
increase in scatter of the model parameters. The increased stability of
the disk parameters relative to the bulge parameters was also noted in
[Simard et al, 2011]. In their paper, the authors comment that this may be due to the
fixed profile shape (due to the fixed Sérsic index, $n=1$) or to the fact that
on average bulges are more compact than disks leading to a resolution effect.
This stability is the result of the increased resolution as disk sizes
in our
sample are roughly 3 times the FWHM of the PSF while bulges are smaller, on average
approximately equal to the FWHM of the PSF in size. We discuss this further in
Section <ref>.
In general, the fits are problematic and require much care when
analyzing individual components. However, as we have already shown,
total magnitude and halflight radius are still tightly constrained.
The correlation matrix for a mock galaxy fit with a
Table <ref> summarizes the bias and scatter in the fits;
they exhibit trends with both the input value of the parameter
and the input magnitude of the galaxy. This behavior is not properly
encapsulated in the overall measure of bias, so these values are useful only
as an example of the relative scale of bias and scatter for each parameter.
Errors can be correlated across many fit parameters, so we
also calculate a correlation matrix for the parameter errors. Figure
<ref> shows an example of the
correlation matrix for the mocks fit with a model. We see the expected strong correlations between
bulge-to-light ratio and the bulge and disk magnitudes as well as the
correlation among the radii of the bulge component with the Sérsic index.
While the correlation matrix suggests that there is little correlation
between sky estimation error and the fitted parameters, we will show later that
there is indeed a strong correlation in model errors with sky estimation error.
The apparent lack of correlation of sky error with the other
fitting parameters is somewhat surprising. However,
Figures <ref> and <ref> suggest a
possible explanation for the apparent lack of correlation. Correlation of
parameter errors with sky errors is non-linear and asymmetric with respect to
over- or underestimating the sky. The fits discussed in this section are shown on
Figures <ref> and <ref> in red. These
points lie in a region where sky error does not significantly bias most
parameters. In addition, the scatter of the sky values is quite small. This
small scatter prevents us from sampling the broader covariance of the sky. If,
for example, the recovered sky value
was an underestimate of 0.5%, then there would be a measurable covariance of
fitting parameters with sky due to the steepness of the parameter bias with respect
to sky level. We discuss the sky estimation further in
Section <ref>.
The bias and scatter of the fitted parameters of the simulated
images with background and PSF effects. These values are provided for
illustrative purposes
only. There is much underlying structure in the errors when compared to their
respective input values or the magnitude of the component.
§.§ Testing with real images
To verify the validity of the simulated background and to test the fitting
in clustered environments, we insert the mock galaxies
into real SDSS fpC images. The fpC images are selected from SDSS DR7 images
containing spectroscopic galaxy targets.
We omit plots of the fitted values here, because the scatter and
the bias in the fits remain unchanged, suggesting that we have properly modeled
the sky background and neighboring sources common to an SDSS spectroscopic
Dense environments provide an additional test for our pipeline.
To select fpC images that contain dense environments, we use the GMBCG
catalog [Hao et al, 2011]. We match brightest cluster galaxies (BCGs) with
galaxies in our original
catalog to select fpC images with cluster members including the BCG.
Our mock galaxies are then inserted into the image which is run through the
In our previous simulations, intracluster light and gradients in the
sky were not modeled. These tests allow us to see what the effects may be.
Once again, the errors remain unchanged, showing that no environmental
correction is necessary when using the fits from the pipeline.
Placing mock galaxies near cluster members allows us to test for
systematic effects in crowded fields. However, further examination
of BCG galaxies is necessary before we are able to properly model them
for this purpose. For example, the curvature at the bright end observed
in the size-luminosity relation of early-type galaxies <cit.> appears to be due to
an increasing incidence of BCGs, which define steeper
relations than the bulk of the early-type population
<cit.>. However, the curvature could
also be due to intracluster light <cit.>. Our
ability to test the systematic effects associated with BCGs using the method
above is severely limited due to the existence of a BCG at the location we would
prefer to place our test galaxy (the center of the cluster).
Therefore, the stability of recovered fit parameters with respect to
environment cannot be assumed to extend to BCGs based on the tests presented
here alone. Further tests for the largest, brightest galaxies are needed to
explore this possibility. We have not presented these tests in this text.
§.§ Varying the S/N and pixel size
In addition to the previous tests, we isolate the effects of the S/N and image
resolution to quantify the contributions to the bias and scatter in
fits. Figures <ref>, <ref>,
<ref>, <ref>, and
show the effect of increasing the S/N by a factor of 4 while holding all other
parameters fixed. Similarly, Figures <ref>,
<ref>, <ref>, and
<ref> show the effect of increasing
resolution by a factor of 2 while holding S/N constant. Corresponding
decrements in these parameters were performed, although they are not presented
in this paper.
Improving the resolution by a factor of two substantially improves the
ability to recover the radius and Sérsic index with reduced bias. For
instance, the Sérsic index bias is reduced to $\approx0.1$ at the larger
values. Additionally, the bulge parameters of the fit
improve substantially with improved resolution.
Corresponding changes in the S/N reduce the scatter, but by a small amount
relative to the effect of the resolution
change. In addition, changing the S/N does not remove the observed bias in
Sérsic index or bulge size. This leads us to conclude that the limitations
of the resolution of SDSS are the leading factor
in causing systematic offsets in the halflight radius, Sérsic index, and
other fitting parameters (including the bulges of the fits). While
increasing the S/N will reduce the scatter in the fits, increased resolution is
necessary to properly recover unbiased values.
[Lackner & Gunn, 2012] also examined the effects of changing S/N and
resolution on SDSS galaxies (see Figures 5-11 of their paper). The authors
found that decreased resolution and S/N increases the relative error in the
Sérsic index and radius. They recommended that galaxies (and
the bulge and disk sub-components of two-component galaxies) have radii,
$R_{hl}\gtrsim0.5\times$FWHM. This cut removes $\approx1\%$ of the
mocks and $\approx 22\%$ of the mocks from our simulated samples
with a preference toward galaxies above $z=0.05$.
While this condition is sufficient to keep the relative error in the halflight
radius and Sérsic index comparable to the error in the magnitude, we find
that this condition fails to remove the bias in our galaxy samples.
Figure <ref> shows that
the underestimate of Sérsic index occurs at larger values. These galaxies
tend to exhibit radii larger than the PSF. Given that the average FWHM of PSFs in our sample
is $\approx1.3''$, if we apply the suggested cut in radius, we are unable to
remove the bias in Sérsic index. Clearly, reliable measurements
are dependent on both the Sérsic index of the object and its radius
relative to the resolution. Both parameters must be accounted for when
deciding on an appropriate resolution cut.
If we extend the [Lackner & Gunn, 2012] recommendation to include a
Sérsic index dependent term, this is sufficient to provide for recovery of
Sérsic index $>$ 4 with bias $\approx 0.1$ or $\approx1\%$. Galaxies should have
circularized halflight radii $R_{hl}\gtrsim$0.5*FWHM$\times n$. This
removes nearly 75% of the sample. While such large
cuts are sufficient to remove the bias in radius and Sérsic index for the
fits, the data are certainly biased relative to our original catalog after
the cuts. Rather than remove
these galaxies, we correct for the bias following a simple statistical argument
presented in Section <ref>.
§.§ Effect of cutout size
We select postage stamp cutouts for use in fitting. It is important to select a
cutout size that does not significantly bias the fits produced by PyMorph.
The most important consideration is to provide enough sky pixels to allow
the PyMorph program to properly determine the sky level in the images.
Reducing cutout size may cause overestimation of background and corresponding
errors in the other fit parameters.
However, we use the PyMorph pipeline and GALFIT to fit a constant background
to the galaxy image. Since a larger image could make sky gradients more
significant, this could bias the fits when a larger cutout is used.
We seek to minimize error when estimating the sky level without introducing a
gradient term and further complicating the fitting process.
To test for optimal cutout size, we fit mocks with cutout sizes
between 10 and 25 Petrosian half-light radii. We plot the average difference
between simulated and measured fit parameters below.
In Figure <ref> we present the error and 1-$\sigma$
scatter in the error on the total magnitude, halflight radius and sky
(showing SExtractor sky in blue and our estimates in red) as a function
of cutout size.
Smaller sizes clearly bias sky estimates made by SExtractor, but only minor
improvement in the scatter of any parameters is achieved by using cutout sizes
above 20 halflight radii. Since we use SExtractor sky as a starting point for
our fitting, we choose a size of 20 halflight radii for our images. The sky
estimates of SExtractor improve substantially. However, GALFIT sky
estimation is stable over these sizes. Because GALFIT sky estimation is
largely independent of the initial starting SExtractor value (which we
would expect if we are truly finding the best fit to the galaxy),
it is likely the case that cutout sizes smaller than even 10 halflight
radii could be used for analysis.
Additional plots of other parameters are omitted in this section.
The other fitted parameters show little or no sensitivity to cutout size
in the range of cutout sizes used. However, as previously discussed, the
bias and scatter may not be equally affected across all model parameters.
The effects may be concentrated in a small part of the parameter space.
The mean difference of the total magnitude
(left column), PSF-corrected halflight radius (center column), and sky estimation
(right column) as a function of cutout size for mocks
fitted with a model. Other simulated models behave similarly.
For sky estimation, the sky measured by GALFIT is plotted
in red. SExtractor sky measurements are plotted in blue for reference.
One-$\sigma$ scatter in the fits is plotted as a dashed line.
Improvement in scatter when fitting for cutout sizes above 20 Petrosian radii is
limited, so we use a 20 halflight radii cutout size for all images.
Fit parameters seem to have no sensitivity to cutout size in this range,
suggesting that it may even be possible to use smaller cutouts.
§.§ The effect of incorrect sky estimation
Estimation of the sky in the vicinity of the target galaxy has a high level of
uncertainty. Indeed, accurate sky determination is likely not even a solvable
problem as discussed briefly in [Blanton et al, 2011].
To determine the bias introduced by our sky estimation, we have
tested our fitting pipeline in cases of both underestimation and
overestimation of the sky. We fix the sky at the simulated sky level,
as well as at simulated sky level $\pm0.5\%$ and $\pm1.0\%$. These
ranges were chosen to represent the range of differences between our sky
estimations and those provided in the CASJOBS database for the SDSS
spectroscopic sample.
Figure <ref> shows a comparison of sky estimates using PyMorph to
those provided from the SDSS photometric data pipeline.
This comparison is performed on data from the catalog presented in .
The Figure shows the normalized distribution of differences in sky estimation
in bins of 0.1%. A negative difference indicates that the sky measured
by PyMorph is lower than that reported by SDSS.
The vertical red solid line indicates the median of the distribution.
The red dashed, dot-dashed, and dotted lines indicate the 68-95-99% ranges
of the data, respectively. The 95% range of sky values is approximately between
For the test, we adopt this range as the range to test for sky variation.
The results of incorrectly estimating the sky are shown in Figures
<ref> and <ref>.
In red, we show the results of fitting galaxies using
the standard PyMorph pipeline, treating sky level as a free parameter in the fit.
PyMorph systematically underestimates the sky at the 0.1% level. However, the
scatter is very tight as indicated by the vertical dashed red lines.
In black we have plotted the fitting results at fixed sky levels
of the correct value and $\pm 0.5\%$ and $\pm 1.0\%$. Errors approaching
0.5% clearly introduce a large bias in the fits. The
0.5% level is an important level because it is the
approximate level of overestimation shown in the preceding section
(Section <ref>) found by SExtractor.
The percent difference between the sky estimate of PyMorph for SDSS
galaxies and the sky estimated by the SDSS photometric pipeline for those same
The normalized distribution of differences is shown in bins of 0.1%. A negative
difference indicates that the sky measured by PyMorph is lower than that
reported by SDSS. The vertical red solid line indicates the median of the
The red dashed, dot-dashed, and dotted lines indicate the 68-95-99% ranges of the
data, respectively.
The 95% range of sky values is approximately between $\pm1\%$ difference, so we
adopt this as the range used to test the effects of improper sky estimation.
Note the asymmetry of the effects of incorrect sky estimation on fitting
parameters. In particular, an underestimate of sky is much more
detrimental to the fit than the corresponding overestimate.
The reason for this asymmetry is due to changes in the perceived
“flatness” of the profile at large radii. When the sky is overestimated, the
galaxy profile tends to 0 flux too early. This causes a decrease in the Sérsic
index and a decrease in the radius. However, when the sky is underestimated,
there will be an extended, approximately constant brightness profile at larger
radii. The only way to model such a profile is for Sérsic index to diverge to
larger values which produce flat, extended profiles at large radii.
[Guo et al, 2009] examined the effects of sky uncertainties in regards to the
covariance between magnitude and both Sérsic index and halflight radius.
They randomly sampled sky estimates from a distribution contained mostly within
They found similar variation of Sérsic index (varying by 2 or more in some
cases of underestimating the sky and varying by less than 1 in the case of
The asymmetry in bias due to incorrect sky estimation is apparent in Figure 5
of [Guo et al, 2009], but not explicitly commented upon.
Figure <ref> shows that PyMorph consistently
estimates the sky $\approx$0.25% lower than that of the SDSS pipeline.
Figures <ref>, <ref>, and
<ref> show that PyMorph has a systematic
underestimate of the true sky at the $\approx$0.1% level.
This bias is much smaller than the bias associated with using the SExtractor sky
estimate as shown in Figure <ref> (especially for smaller cutout
size), which suggests that the sky
values in SDSS are slightly overestimated.
disk components are remarkably robust to the errors in sky estimation,
while bulge parameters suffer greatly, especially when the sky is
underestimated. Upon further examination of Figure
the bulge parameters of the model are more accurately estimated when the sky
is treated as a free parameter in the fit rather than when the sky is fixed at the
correct value. However, this improvement does not suggest that underestimate of the
sky is the
preferred fitting outcome. It merely reflects the fact that the systematic
effects due to underestimation of the sky are opposite to the underlying biases
in halflight
radius and Sérsic index. If we were to apply the PyMorph pipeline to an
with higher S/N and increased resolution, we would prefer the correct estimate of
the sky to prevent systematic overestimate of these parameters.
§ DISCUSSION
In the preceding sections we have shown the covariance, bias, and
scatter in our parameter estimation for the and models. In reality, the effects above will combine to yield a total
scatter, covariance, and bias that should approach those shown in
<ref>. Our simulations give us an idea of the behavior
of the PyMorph pipeline when fitting SDSS galaxies as presented in .
The simulations show that the recovery of global fitting parameters
(total magnitude and halflight radius) in the case of SDSS galaxies is
remarkably robust, even in the case of the fits. Two-component fits
present a more difficult test for the pipeline. Both the bulge and disk
components exhibit increased scatter relative to the scatter of the global
parameters. In addition, the bulge component exhibits a systematic
underestimation of the radius, Sérsic index, and magnitude, particularly for
bulges with larger radii or higher Sérsic index.
The galaxies fit in have a median size roughly equivalent
to the average PSF of SDSS. For most galaxies, the
resolution necessary to accurately resolve bulge substructure is not present. As
shown in Section <ref>, the ability to recover small bulges is
improved by a factor of 2 increase in resolution. Finer resolution in central
regions of the galaxy is also necessary to fully recover larger Sérsic
indexes without bias. Even with these systematics, the two-component fits
are still necessary to recover unbiased global parameters and can
provide insight into the structure of galaxies.
The use of two-component models is potentially ill-advised for many
SDSS galaxies as the respective sub-components may be too small to be well-resolved.
This is suggested by [Simard et al, 2011] as well as [Lackner & Gunn, 2012] (if we use the
suggested resolution cut based on the PSF FWHM). However, our data show that
this recommendation should be conditional on the galaxy
parameters of interest. While it may be true that bulge parameters of the
fit become unreliable at small radii, we show that using only the fit
radius will bias a sample of SDSS galaxies containing both single and
two-component profiles (see Figure <ref>). However, there
are no cases where the fit introduces bias. It is advisable to
use the halflight radius and magnitude as the total magnitude
of the galaxy when examining a sample such as this.
The F-test offers a potentially powerful way to
distinguish when it is necessary to use a more complicated two-component model. The
F-test can compare the $\chi^2$ values
among nested linear models with Gaussian errors <cit.>. Although our
models are not linear and our error distribution is not strictly Gaussian, we
apply the F-test to our fits. Following [Simard et al, 2011], we adopt an F-test
probability of 0.32 as the cutoff indicating a more complicated model is required.
When we find a low F-test probability, P$_{correct}<0.32$, the more
complicated model (going from a one-component to two-component fit, or allowing
the Sérsic index of the bulge to vary) provides a better fit to the
observed profile. In cases where a fit is used rather than a fit, the improvement in fitting is large enough to justify using a model
with more free parameters. The improved fit is not merely the result of
using a more flexible model. A similar test was performed by [Lackner & Gunn, 2012]
to select among a pure disk or disk+bulge model.
If the selection based on the F-test is correct, then the
resulting measurements of total magnitude and halflight radius will be unbiased.
Using the mocks fit with each of the and models, we
select the
fitted model by performing the F-test comparing the and fits.
The preferred fit (either or ) of the mocks is then used to assess the bias in the halflight radius and magnitude.
Total magnitude and halflight radius of fits of mocks
shown by F-test to be sufficiently well fit by models. The fits are unbiased,
but the scatter in the recovered values are approximately twice as wide in halflight
radius and magnitude as compared to the fits in
Figure <ref>.
By examining the subset of mocks for which the F-test
determines the model to be the appropriate fit, we test the ability of the
F-test to select galaxies that are correctly represented by models. In
Figure <ref> we show the resulting distribution of total magnitude and
halflight radius of this subset of mocks fit with models. The bias
originally observed in Figure <ref> is not
evident. However, the scatter in the recovered values are approximately twice as
wide as in Figure <ref>, indicating that while the fits
are unbiased, some sensitivity is lost by using the simpler (and ultimately
incorrect) model. The remaining mocks, for which the fit is
determined by F-test to be most appropriate, are also unbiased in total
magnitude and halflight radius. From this test, we conclude that using the F-test
to determine the most appropriate fitted model allows for unbiased
measurement of the halflight radius and total magnitude.
Using the mocks, the false positive rate
( mocks classified as needing a fit according to the F-test)
for the F-test with a significance level of 0.32 is 5%,
suggesting that there is a low level of contamination in a two-component sample selected using the F-test.
Using mocks with $0.2<B/T<0.8$ and $n_{bulge}>2$, which we consider true two-component galaxies,
the false negative rate ( mocks classified as needing only a fit according to the F-test) is 34%, missing a substantial fraction of the
galaxies with two components. While selection using the F-test is sufficient
to remove the measured bias in global fitting parameters and is able to select a
relatively pure sample of two-component galaxies, it does not select a complete
sample of two-component galaxies. Clearly caution is necessary when using the
F-test to select two-component galaxies from fitting
routines. However, the F-test can indicate when the global parameters of a
model are likely unbiased regardless of the underlying galaxy type.
Following [Simard et al, 2011], we can also select the fitted model based
on a tiered approach, first performing the F-test on the and fits.
Galaxies for which the fit gives a statistically significant improvement
are then tested again to determine whether the fit is preferable to the
fit. The preferred fit (either , , or ) of the mocks is then used to assess the bias in the halflight radius and magnitude. We
tested this approach and found that it did not significantly alter the results.
Many galaxies exhibit more complex structure than a single- or two-component
structure. Even the case of a two-component model often
oversimplifies galaxy structure. Bars,
rings, central sources, clumpyness, or asymmetry cannot be effectively
modeled in our simulations. Because of this, we can only determine a
lower-bound on the uncertainty in our parameter estimates. However,
correcting fits using this lower bound improves the fit of the observed galaxy.
We can apply a simple example of bias correction following the
procedure outlined in [Simard et al, 2002]. Given the simulated and fitted values
of the Sérsic index for the model, we plot the bias as a function of
the fitted value output by PyMorph. In this case, the output value represents
the measured value in real data. The simulated value represents the true
underlying value of the galaxy Sérsic index. We can determine an average
bias and uncertainty in the bias, labeled as $Bias$ and $\Delta Bias$, as a
function of output Sérsic index. Additionally, we can measure the random
error in the fits from the width of the bias distribution as a function of
Sérsic index, labeled as $\Delta Random$. Then the corrected Sérsic
index and uncertainty on the corrected index is
\begin{equation}\label{eq:corrected}
\begin{aligned}
& n_{corrected} = n_{fitted} - Bias(n_{fitted}) \\
& \Delta n = \sqrt{\Delta_{galfit}^2 + \Delta Bias^2 + \Delta Random^2} \\
\end{aligned}
\end{equation}
Applying this correction allows us to correct bias as a function of
both simulated and fitted Sérsic index for the sample of galaxies used in
. We show the results of this process
in Figure <ref>.
An example of the bias correction of the Sérsic index of the model. The error in the Sérsic index ($n_{output}$-$n_{input}$) versus
output value is presented
before (top left panel) and after (top right panel) correction. The same
correction is shown in the bottom row versus the simulated value of the
Sérsic index. We apply the correction in the $n_{output}$ basis. This
appropriately corrects the bias as a function of $n_{input}$ except at high
$n$ where the correction fails. The reason for this failure is
due to the boundaries of the allowed $n$ parameter space. Galaxies in the
highest bins of output Sérsic index are a combination of poorly fit, low
Sérsic index galaxies that are artificially constrained to fall in the high
bins, and correctly fit, high Sérsic index galaxies. The result of this
mixture is a net negative correction on galaxies with high Sérsic index.
We are able to statistically correct for the bias in our sample in
both the simulated and fitted bases for most values of the Sérsic index.
However, there is an under-correction at high simulated Sérsic value. This
effect appears to be due to the boundaries of the parameter space that PyMorph
is allowed to search for the best fit model. By restricting PyMorph to values of
$n<8$, galaxies simulated with Sérsic index of 8 will be preferentially
underestimated. However, the highest bins of fitted Sérsic index
contain many more galaxies with over-estimated Sérsic index. Therefore the
net correction will be negative and not appropriate for the highest
bins. We could improve the error correction at higher bins by allowing GALFIT to
explore larger values of the Sérsic index. However, this is beyond the scope of
this paper.
Additional corrections may also be considered (divide in both magnitude and
Sérsic index prior to computing the bias correction) depending on the
specifics of a given study. For properties of the global population,
the corrections measured in this paper are applicable to the sample presented in
Our tests were performed on r-band data from SDSS. The performance of the
pipeline can change when observing in different bands. This change is primarily
dependent on the change in the S/N and resolution between bands (due to the
changing brightness of the sky, color of the galaxy, and size
relative to the PSF)
and on the different galactic structures to which neighboring SDSS filter bands
are sensitive. In principle, these effects could be measured from
the simulations presented in this paper by adjusting the S/N and background
level. Additionally, one may have to adjust the size of the galaxies or redraw
the sample to match the size distributions in the different band.
In , we fit the SDSS g, r, and i band data.
It is unlikely that the images change drastically enough over the wavelength
and redshift range observed to require additional testing in the i
band. However, these simulations become an increasingly poor estimate
of error in bluer bands where the photometry becomes more sensitive to
star forming regions. These regions tend to be clumpier and, therefore,
less well represented by a smooth profile. Therefore, g band fits may
present more scatter than the r or i band data. These clumpy regions are
difficult to model with the smooth models presented here. One could
attempt a hybrid approach to generating simulated data whereby one
isolates clumpyness in nearby galaxies and use this as a template to add
clumpyness to smaller SDSS galaxies. However, the details of this process are
beyond the scope of this paper.
It is also potentially useful to use information about the r-band to inform the
fits of neighboring bands. Indeed [Simard et al, 2011] attempted this by requiring
many parameters (Sérsic index, radius, ellipticity) of the fitting
model to be identical across the g and r bands, essentially using the two bands
as a form of coadded data to increase the S/N. This increase of S/N comes at
the expense of dis-allowing variation in the matched parameters, which may or
may not be an appropriate assumption (in a two-component fit, we might
expect the bulge size to change across bands, which is dis-allowed).
Additionally, [Häußler et al, 2013] enforced simple polynomial relationships in
parameters across bands, using the neighboring bands to further constrain the
acceptable parameter space to be searched by the fitting algorithm. The
most flexible method is to fit each band independently and examine the
systematic effects of each band as necessary, making additional
cross-band comparisons including color <cit.>.
This is our preferred method for the data presented here and in .
§ CONCLUSION
We presented the simulations used to test fitting of SDSS galaxies
using PyMorph. Simulations of the and models were
presented and examined in many different cases. The simulations were
generated using the results of the fits presented in . We showed
that our simulations are recoverable in the case of no noise, which
demonstrates that our simulations are correct. We then
showed that we can
recover the parameters in the case of a simulated background and noise
representative of the average SDSS image (see Figures <ref>,
<ref>,<ref>, <ref>, and
Several individual effects on the fitting were examined. We showed that our
choice of 20 halflight radii for cutout size does not significantly bias our
fitting results (see Figure <ref>). In addition, we examined the
effect of incorrect background
estimation, which can significantly affect fitting results
(Figure <ref> and <ref>). Effects of
increasing the S/N are somewhat limited for this sample. However, an
increase in the resolution of the sample would greatly improve parameter
measurements, removing many biases in the two-component fits and improving the
estimation radius and Sérsic index for galaxies as shown in
Figures <ref> and <ref>.
We also examined the bias created when fitting incorrect models to galaxies.
Fitting a two-component Sérsic + Exponential model to what is really
just a single Sérsic results in a noisier recovery of the input parameters,
but these are not biased (see Figure <ref>); fitting a single
Sérsic to what is truly a
two-component system results in an overestimate of 0.05 magnitudes in total magnitude
and 5% halflight radius for dim galaxies, increasing to 0.1 magnitudes and 10% for
galaxies at the brighter end of the apparent magnitude distribution (see
Figure <ref>). These biases are used to correct the
systematics of our fitted SDSS sample
and suggest that magnitude and radius values of a fit are
the least likely to be biased across many galaxy types. Therefore it is
advisable to use values when examining global parameters for galaxies.
These simulations can be analyzed together with the fits presented in
to give a more detailed understanding of galaxy structure and
formation as presented in [Bernardi et al, 2013].
§ ACKNOWLEDGMENTS
The authors would like to thank the anonymous referee for many useful comments that
helped to greatly improve the paper. AM and VV would also like to
thank Mike Jarvis and Joseph Clampitt for many helpful discussions.
This work was supported in part by NASA grant ADP/NNX09AD02G
and NSF/0908242.
Funding for the SDSS and SDSS-II has been provided by the Alfred P. Sloan
Foundation, the Participating Institutions, the National Science Foundation, the U.S.
Department of Energy, the National Aeronautics and Space Administration, the Japanese
Monbukagakusho, the Max Planck Society, and the Higher Education Funding Council for
England. The SDSS Web Site is http://www.sdss.org/.
The SDSS is managed by the Astrophysical Research Consortium for the
Participating Institutions. The Participating Institutions are the American Museum of
Natural History, Astrophysical Institute Potsdam, University of Basel, University of
Cambridge, Case Western Reserve University, University of Chicago, Drexel University,
Fermilab, the Institute for Advanced Study, the Japan Participation Group, Johns
Hopkins University, the Joint Institute for Nuclear Astrophysics, the Kavli Institute
for Particle Astrophysics and Cosmology, the Korean Scientist Group, the Chinese
Academy of Sciences (LAMOST), Los Alamos National Laboratory, the
Max-Planck-Institute for Astronomy (MPIA), the Max-Planck-Institute for Astrophysics
(MPA), New Mexico State University, Ohio State University, University of Pittsburgh,
University of Portsmouth, Princeton University, the United States Naval Observatory,
and the University of Washington.
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|
arxiv-papers
| 2012-11-26T21:00:47 |
2024-09-04T02:49:38.501345
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Alan Meert, Vinu Vikram, and Mariangela Bernardi",
"submitter": "Alan Meert",
"url": "https://arxiv.org/abs/1211.6123"
}
|
1211.6139
|
# The RHIC Beam Energy Scan Program: Results from the PHENIX Experiment
Jeffery T. Mitchell (for the PHENIX111A list of members of the PHENIX
Collaboration and acknowledgements can be found at the end of this
issue.Collaboration) Brookhaven National Laboratory, Building 510C, P.O. Box
5000, Upton, NY 11973-5000 USA
###### Abstract
The PHENIX Experiment at RHIC has conducted a beam energy scan at several
collision energies in order to search for signatures of the QCD critical point
and the onset of deconfinement. PHENIX has conducted measurements of
transverse energy production, muliplicity fluctuations, and the skewness and
kurtosis of net charge distributions. The data analyzed to date show no
significant indications of the presence of the critical point.
††journal: Nuclear Physics A
## 1 Introduction
Recent lattice QCD calculations predict that there is a first order phase
transition from hadronic matter to a Quark-Gluon Plasma that ends in a
critical point, with a continuous phase transition on the other side of the
critical point. The Relativistic Heavy Ion Collider (RHIC) has conducted a
program to probe different regions of the QCD phase diagram in the vicinity of
the possible critical point with a beam energy scan. During 2010 and 2011,
RHIC provided Au+Au collisions to PHENIX at $\sqrt{s_{NN}}=$ 200 GeV, 62.4
GeV, 39 GeV, 27 GeV, 19.6 GeV, and 7.7 GeV. Analysis of the data concentrates
on two strategies: looking for signs of the onset of deconfinement by
comparing to results at the top RHIC energy, and searching for direct
signatures of a critical point. Results from PHENIX covering transverse energy
production, multipicity fluctuations, and the skewness and kurtosis of net
charge distributions will be discussed in this article.
## 2 Transverse Energy Production
PHENIX has measured transverse energy ($E_{T}$) production in Au+Au collisions
at the following collision energies: 200, 62.4, 39, 27, 19.6, and 7.7 GeV.
These observables are closely related to the geometry of the system and are
fundamental measurements necessary to understand the global properties of the
collision. This work extends the previous PHENIX measurements in 200, 130, and
19.6 GeV Au+Au collisions [1]. Total $E_{T}$ production results are summarized
in Figure 1, which shows the excitation function of the estimated value of the
Bjorken energy density [2] expressed as
$\epsilon_{BJ}=\frac{1}{A_{\perp}\tau}\frac{dE_{T}}{dy},$ (1)
where $\tau$ is the formation time and $A_{\perp}$ is the transverse overlap
area of the nuclei. The Bjorken energy density increases monotonically over
the range of the RHIC beam energy scan. Also shown is the estimate for 200 GeV
U+U collisions taken during the 2012 running period. Although $E_{T}$
production dramatically increases at LHC energies compared to RHIC energies,
the shape of the distributions as a function of the number of participants,
$N_{part}$, is independent of the collision energy. This is illustrated in
Figure 2, which shows an overlay of the distributions of $dE_{T}/d\eta$
normalized by the number of participant pairs for 7.7 GeV, 200 GeV, and 2.76
TeV Au+Au collisions. The 200 GeV and 7.7 GeV distributions have been scaled
up by a factors of 2.6 and 9.7, respectively. The shape of the distribution as
a function of $N_{part}$ appears to be driven by the collision geometry.
Figure 1: The estimated value of the Bjorken energy density, $\epsilon_{BJ}$,
multiplied by the formation time in central Au+Au collisions at mid-rapidity
as a function of $\sqrt{s_{NN}}$. The open circle represents the estimate for
200 GeV U+U collisions. Figure 2: $dE_{T}/d\eta$ normalized by the number of
participant pairs as a function $N_{part}$. Overlayed are the distributions
from 7.7 GeV, 200 GeV, and 2.76 TeV Au+Au collisions. The PHENIX data has been
scaled up to overlay the ALICE data [3].
## 3 Charged Particle Multiplicity Fluctuations
Near the QCD critical point, it is expected that fluctuations in the charged
particle multiplicity will increase [4]. PHENIX has extended the previous
analysis of multiplicity fluctuations in 200 and 62.4 GeV Au+Au collisions [5]
to 39 and 7.7 GeV Au+Au collisions. Charged particle multiplicity fluctuations
are measured using the scaled variance, $\omega_{ch}=\sigma_{ch}/\mu_{ch}$,
which is the standard deviation scaled by the mean of the distribution. The
scaled variance is corrected for contributions due to non-dynamic impact
parameter fluctuations using the method described in [5]. Figure 3 shows the
PHENIX results for central collisions compared to results from the NA49
Collaboration [6] as a function of $\sqrt{s_{NN}}$. There is no indication of
the presence of a critical point from the PHENIX results alone.
Figure 3: Charged particle multiplicity fluctuations in central Au+Au (PHENIX)
and Pb+Pb (NA49) collisions expressed in terms of the scaled variance as a
function of $\sqrt{s_{NN}}$. Circles represent the results from PHENIX.
## 4 Higher Moments of Net Charge Distributions
The shapes of the distributions of the event-by-event net charge are expected
to be sensitive to the presence of the critical point [7]. PHENIX has measured
the skewness ($S=\langle(N-\langle N\rangle)^{3}\rangle/\sigma^{3}$) and the
kurtosis ($\kappa=\langle(N-\langle N\rangle)^{4}\rangle/\sigma^{4}-3$) of net
charge distributions in Au+Au collisions at 200, 62.4, 39, and 7.7 GeV. These
values are expressed in terms that can be associated with the quark number
susceptibilities, $\chi$: $S\sigma\approx\chi^{(3)}/\chi^{(2)}$ and
$\kappa\sigma^{2}\approx\chi^{(4)}/\chi^{(2)}$ [8]. The skewness and kurtosis
for central collisions are shown in Figure 4 as a function of $\sqrt{s_{NN}}$.
The data are compared to URQMD and HIJING simulation results processed through
the PHENIX acceptance and detector response. There is no excess above the
simulation results observed in the data at these four collision energies.
Figure 4: The skewness multiplied by the standard deviation and the kurtosis
multiplied by the variance from net charge distributions from central Au+Au
collisions. The circles represent the data. The grey error bars represent the
systematic errors. Also shown are URQMD and HIJING simulation results
processed through the PHENIX acceptance. The increase in the kurtosis from
URQMD and HIJING may be due to an increase in resonance production at 200 GeV.
The blue curve is the result from the Hadron Resonance Gas model [8].
## 5 Summary
Presented here are some of the PHENIX results from the RHIC beam energy scan
program. From the analyses completed to date, there is no significant
indication of the presence of the QCD critical point. However, many analyses
from PHENIX, particularly at $\sqrt{s_{NN}}=$ 27 GeV and 19.6 GeV, will be
available soon.
## References
* [1] S.S. Adler et al., Phys. Rev. C 71, 034908 (2005).
* [2] J. D. Bjorken, Phys. Rev. D 27, 140 (1983).
* [3] C. Loizides et al., arXiv:1106.6324v1 (2011).
* [4] M. Stephanov et al, Phys. Rev. D 60, 114028 (1999).
* [5] A. Adare et al, Phys. Rev. C 78, 044902 (2008).
* [6] C. Alt et al, Phys. Rev. C 78, 034914 (2008).
* [7] R. V. Gavai and S. Gupta, Phys. Lett. B 696, 459 (2011).
* [8] F. Karsch and K. Redlich, Phys. Lett. B 695, 136 (2011).
|
arxiv-papers
| 2012-11-26T21:35:33 |
2024-09-04T02:49:38.514289
|
{
"license": "Public Domain",
"authors": "J.T. Mitchell (for the PHENIX Collaboration)",
"submitter": "Jeffery T. Mitchell",
"url": "https://arxiv.org/abs/1211.6139"
}
|
1211.6141
|
# On Mannheim partner curves in three dimensional Lie groups
İsmaİl Gök Ankara University, Faculty of Science, Department of Mathematics,
06100, Tandog̃an, Ankara, Turkey [email protected] , O. Zekİ
Okuyucu Bilecik Şeyh Edebali University, Faculty of Science and Arts,
Department of Mathematics, Bilecik, Turkey [email protected] ,
Nejat Ekmekcİ Ankara University, Faculty of Science, Department of
Mathematics, 06100, Tandog̃an, Ankara, Turkey [email protected]
and Yusuf Yaylı Ankara University, Faculty of Science, Department of
Mathematics, 06100, Tandog̃an, Ankara, Turkey [email protected]
(Date: August 14, 2012)
###### Abstract.
In this paper, we define Mannheim partner curves in a three dimensional Lie
group $G$ with a bi-invariant metric. And then the main result in this paper
is given as (Theorem 3.3): A curve $\alpha:I\subset\mathbb{R\rightarrow}G$
with the Frenet apparatus $\left\\{T,N,B,\varkappa,\tau\right\\}$ is a
Mannheim partner curve if and only if
$\lambda\varkappa\left(1+H^{2}\right)=1$
where $\lambda$ is constants and $H$ is the harmonic curvature function of the
curve $\alpha.$
###### Key words and phrases:
Mannheim curves, Lie groups.
###### 2000 Mathematics Subject Classification:
Primary 53A04; Secondary 22E15
This paper is in final form and no version of it will be submitted for
publication elsewhere.
## 1\. Introduction
In the classical diferential geometry of curves, J. Bertrand studied curves in
Euclidean 3-space whose principal normals are the principal normals of another
curve. In (see [1]) he showed that a necessary and sufficient condition for
the existence of such a second curve is that a linear relationship with
constant coefficients shall exist between the first and second curvatures of
the given original curve. In other word, if we denote first and second
curvatures of a given curve by $k_{1}$ and $k_{2}$ respectively, then for
$\lambda,\mu$ $\in\mathbb{R}$ we have $\lambda k_{1}+\mu k_{2}=1$. Since the
time of Bertrand’s paper, pairs of curves of this kind have been called
Conjugate Bertrand Curves, or more commonly Bertrand Curves (see [2]).
Another kind of associated curve whose principal normal vector field is the
binormal vector field of another curve.is called Mannheim curve. Mannheim
partner curves was studied by Liu and Wang (see [3]) in Euclidean $3-$ space
and in the Minkowski $3-$space. After these papers lots of papers were
published about Mannheim curves in Euclidean $3-$space, Minkowski $3-$space,
dual $3-$space and Galilean spaces (see [4, 5, 6, 7, 8]). Matsuda and Yorozu
[9] gave a definition of generalized Mannheim curve in Euclidean $4-$space.
They show some characterizations and examples of generalized Mannheim curves.
Ersoy _et.al._ gave a definition of generalized Mannheim curve in Minkowski
$4-$space.
The degenarete semi-Riemannian geometry of Lie group is studied by Çöken and
Çiftçi [10]. Moreover, they obtanied a naturally reductive homogeneous semi-
Riemannian space using the Lie group. Then Çiftçi [11] defined general helices
in three dimensional Lie groups with a bi-invariant metric and obtained a
generalization of Lancret’s theorem and gave a relation between the geodesics
of the so-called cylinders and general helices.
Recently, Izumiya and Takeuchi [12] have introduced the concept of slant helix
in Euclidean $3$-space. A slant helix in Euclidean space $\mathbb{E}^{3}$ was
defined by the property that its principal normal vector field makes a
constant angle with a fixed direction. Also, Izumiya and Takeuchi showed that
$\alpha$ is a slant helix if and only if the geodesic curvature of spherical
image of principal normal indicatrix $\left(N\right)$ of a space curve
$\alpha$
$\sigma_{N}\left(s\right)=\left(\frac{\varkappa^{2}}{\left(\varkappa^{2}+\tau^{2}\right)^{3/2}}\left(\frac{\tau}{\varkappa}\right)^{\prime}\right)\left(s\right)$
is a constant function .
Harmonic curvature functions were defined earlier by Özdamar and Hacısalihoğlu
[13]. Recently, many studies have been reported on generalized helices and
slant helices using the harmonic curvatures in Euclidean spaces and Minkowski
spaces [14, 15, 16]. Then, Okuyucu et al. [17] defined slant helices in three
dimensional Lie groups with a bi-invariant metric and obtained some
characterizations using their harmonic curvature function.
In this paper, first of all, we define Mannheim partner curves in a three
dimensional Lie group $G$ with a bi-invariant metric and we obtain the
necessary and sufficient conditions for the Mannheim partner curves in a three
dimensional Lie group $G.$
## 2\. Preliminaries
Let $G$ be a Lie group with a bi-invariant metric $\left\langle\text{
},\right\rangle$ and $D$ be the Levi-Civita connection of Lie group $G.$ If
$\mathfrak{g}$ denotes the Lie algebra of $G$ then we know that $\mathfrak{g}$
is issomorphic to $T_{e}G$ where $e$ is neutral element of $G.$ If
$\left\langle\text{ },\right\rangle$ is a bi-invariant metric on $G$ then we
have
$\left\langle
X,\left[Y,Z\right]\right\rangle=\left\langle\left[X,Y\right],Z\right\rangle$
(2.1)
and
$D_{X}Y=\frac{1}{2}\left[X,Y\right]$ (2.2)
for all $X,Y$ and $Z\in\mathfrak{g}.$
Let $\alpha:I\subset\mathbb{R\rightarrow}G$ be an arc-lenghted regular curve
and $\left\\{X_{1},X_{2,}...,X_{n}\right\\}$ be an orthonormal basis of
$\mathfrak{g}.$ In this case, we write that any two vector fields $W$ and $Z$
along the curve $\alpha\ $as $W=\sum_{i=1}^{n}w_{i}X_{i}$ and
$Z=\sum_{i=1}^{n}z_{i}X_{i}$ where $w_{i}:I\rightarrow\mathbb{R}$ and
$z_{i}:I\rightarrow\mathbb{R}$ are smooth functions. Also the Lie bracket of
two vector fields $W$ and $Z$ is given
$\left[W,Z\right]=\sum_{i=1}^{n}w_{i}z_{i}\left[X_{i},X_{j}\right]$
and the covariant derivative of $W$ along the curve $\alpha$ with the notation
$D_{\alpha^{\shortmid}}W$ is given as follows
$D_{\alpha^{\shortmid}}W=\overset{\cdot}{W}+\frac{1}{2}\left[T,W\right]$ (2.3)
where $T=\alpha^{\prime}$ and
$\overset{\cdot}{W}=\sum_{i=1}^{n}\overset{\cdot}{w_{i}}X_{i}$ or
$\overset{\cdot}{W}=\sum_{i=1}^{n}\frac{dw}{dt}X_{i}.$ Note that if $W$ is the
left-invariant vector field to the curve $\alpha$ then $\overset{\cdot}{W}=0$
(see for details [18]).
Let $G$ be a three dimensional Lie group and
$\left(T,N,B,\varkappa,\tau\right)$ denote the Frenet apparatus of the curve
$\alpha$. Then the Serret-Frenet formulas of the curve $\alpha$ satisfies:
$D_{T}T=\varkappa N\text{, \ \ \ }D_{T}N=-\varkappa T+\tau B\text{, \ \ \
}D_{T}B=-\tau N$
where $D$ is Levi-Civita connection of Lie group $G$ and
$\varkappa=\overset{\cdot}{\left\|T\right\|}.$
###### Definition 2.1.
Let $\alpha:I\subset\mathbb{R\rightarrow}G$ be a parametrized curve. Then
$\alpha$ is called a _general helix_ if it makes a constant angle with a left-
invariant vector field $X$. That is,
$\left\langle T(s),X\right\rangle=\cos\theta\text{ for all }s\in I,$
for the left-invariant vector field $X\in g$ is unit length and $\theta$ is a
constant angle between $X$ and $T$, which is the tangent vector field of the
curve $\alpha$ (see [11]).
###### Definition 2.2.
Let $\alpha:I\subset\mathbb{R\rightarrow}G$ be a parametrized curve with the
Frenet apparatus $\left(T,N,B,\varkappa,\tau\right)$ then
$\tau_{G}=\frac{1}{2}\left\langle\left[T,N\right],B\right\rangle$ (2.4)
or
$\tau_{G}=\frac{1}{2\varkappa^{2}\tau}\overset{\cdot\cdot\text{ \ \ \ \ \ \ \
\ }\cdot}{\left\langle
T,\left[T,T\right]\right\rangle}+\frac{1}{4\varkappa^{2}\tau}\overset{\text{ \
\ }\cdot}{\left\|\left[T,T\right]\right\|^{2}}$
(see [11]).
###### Proposition 2.3.
Let $\alpha:I\subset\mathbb{R\rightarrow}G$ be an arc length parametrized
curve with the Frenet apparatus $\left\\{T,N,B\right\\}$. Then the following
equalities
$\displaystyle\left[T,N\right]$
$\displaystyle=\left\langle\left[T,N\right],B\right\rangle B=2\tau_{G}B$
$\displaystyle\left[T,B\right]$
$\displaystyle=\left\langle\left[T,B\right],N\right\rangle N=-2\tau_{G}N$
hold [17].
###### Definition 2.4.
Let $\alpha:I\subset\mathbb{R\rightarrow}G$ be an arc length parametrized
curve. Then $\alpha$ is called a _slant helix_ if its principal normal vector
field makes a constant angle with a left-invariant vector field $X$ which is
unit length. That is,
$\left\langle N(s),X\right\rangle=\cos\theta\text{ for all }s\in I,$
where $\theta\neq\frac{\pi}{2}$ is a constant angle between $X$ and $N$ which
is the principal normal vector field of the curve $\alpha$ (see [17]).
###### Definition 2.5.
Let $\alpha:I\subset\mathbb{R\rightarrow}G$ be an arc length parametrized
curve with the Frenet apparatus $\left\\{T,N,B,\varkappa,\tau\right\\}.$ Then
the _harmonic curvature function_ of the curve $\alpha$ is defined by
$H=\dfrac{\tau-\tau_{G}}{\varkappa}$
where $\tau_{G}=\frac{1}{2}\left\langle\left[T,N\right],B\right\rangle$ (see
[17]).
###### Theorem 2.6.
Let $\alpha:I\subset\mathbb{R\rightarrow}G$ be a parametrized curve with the
Frenet apparatus $\left(T,N,B,\varkappa,\tau\right)$. If the curve $\alpha$ is
a general helix, if and only if,
$\tau=c\varkappa+\tau_{G}$
where c is a constant (see [11]) or using the definition of the harmonic
curvature function of the curve $\alpha$ (see [17]) is constant function.
###### Theorem 2.7.
Let $\alpha:I\subset\mathbb{R\rightarrow}G$ be a unit speed curve with the
Frenet apparatus $\left(T,N,B,\varkappa,\tau\right)$. Then $\alpha$ is a slant
helix if and only if
$\sigma_{N}=\frac{\varkappa(1+H^{2})^{\frac{3}{2}}}{H^{\shortmid}}=\tan\theta$
is a constant where $H$ is a harmonic curvature function of the curve $\alpha$
and $\theta\neq\frac{\pi}{2}$ is a constant (see [17]).
## 3\. Mannheim partner curves in a three dimensional Lie group
In this section, we define Mannheim partner curves and their characterizations
are given in a three dimensional Lie group $G$ with a bi-invariant metric
$\left\langle\text{ },\right\rangle$. Also we give some characterizations of
Mannheim partner curves using the special cases of $G$.
###### Definition 3.1.
A curve $\alpha$ in $3$-dimensional Lie group $G$ is a _Mannheim curve_ if
there exists a special curve __ $\beta$ in $3$-dimensional Lie group $G$ such
that principal normal vector field of $\alpha$ is linearly dependent binormal
vector field of $\beta$ at corresponding point under $\psi$ which is bijection
from $\alpha$ to $\beta.$ In this case $\beta$ is called the _Mannheim
partner_ curve of $\alpha$ and $\left(\alpha,\beta\right)$ is called
_Mannheim_ curve _couple._
The curve $\alpha:I\subset\mathbb{R\rightarrow}G$ in $3$-dimensional Lie group
$G$ is parametrized by the arc-length parameter $s$ and from the Definition
3.1 Bertrand mate curve of $\alpha$ is given
$\beta:\overline{I}\subset\mathbb{R\rightarrow}G$ in $3$-dimensional Lie group
$G$ with the help of Figure 1 such that
$\text{{Figure1}}\mathtt{:}\text{{}Mannheim curve\ couple
}\left(\alpha,\beta\right)$
$\beta\left(s\right)=\alpha\left(s\right)+\lambda\left(s\right)N\left(s\right),\text{
}s\in I$
where $\lambda$ is a smooth function on $I$ and $N$ is the principal normal
vector field of $\alpha$.
###### Theorem 3.2.
Let $\alpha:I\subset\mathbb{R\rightarrow}G$ and
$\beta:\overline{I}\subset\mathbb{R\rightarrow}G$ be a Mannheim curve couple
with arc-length parameter $s$ and $\overline{s},$ respectively. Then
corresponding points are a fixed distance apart for all $s\in I$, that is,
$d\left(\alpha\left(s\right),\beta\left(s\right)\right)=\text{constant, \ \
for all }s\in I$
###### Proof.
From Definition 3.1, we can simply write
$\beta\left(s\right)=\alpha\left(s\right)+\lambda\left(s\right)N\left(s\right)$
(3.1)
Differentiating the Eq. (3.1) with respect to $s$ and using the Eq. (2.3), we
get
$\displaystyle\frac{d\beta\left(\overline{s}\right)}{d\overline{s}}\frac{d\overline{s}}{ds}$
$\displaystyle=\frac{d\alpha\left(s\right)}{ds}+\lambda^{\prime}\left(s\right)N\left(s\right)+\lambda(s)\overset{\cdot}{N}(s)$
$\displaystyle=T\left(s\right)+\lambda^{\prime}\left(s\right)N\left(s\right)+\lambda(s)\left[D_{T}N-\frac{1}{2}\left[T,N\right]\right]$
and with the help of Proposition 3.6 and Frenet equations, we obtain
$\frac{d\beta\left(\overline{s}\right)}{d\overline{s}}\frac{d\overline{s}}{ds}=\left(1-\lambda\left(s\right)\varkappa\left(s\right)\right)T(s)+\lambda^{\prime}\left(s\right)N\left(s\right)+\lambda\left(s\right)\left(\left(\tau-\tau_{G}\right)\left(s\right)\right)B\left(s\right)$
or
$T_{\beta}\left(\overline{s}\right)=\frac{ds}{d\overline{s}}\left[\left(1-\lambda(s)\varkappa\left(s\right)\right)T\left(s\right)+\lambda^{\prime}\left(s\right)N\left(s\right)+\lambda(s)\left(\tau-\tau_{G}\right)(s)B\left(s\right)\right].$
(3.2)
And then, we know that
$\left\\{N\left(s\right),B_{\beta}\left(\overline{s}\right)\right\\}$ is a
linearly dependent set, so we have
$\left\langle
T_{\beta}\left(\overline{s}\right),B_{\beta}\left(\overline{s}\right)\right\rangle=\frac{ds}{d\overline{s}}\left[\begin{array}[c]{c}\left(1-\lambda\left(s\right)\varkappa\left(s\right)\right)\left\langle
T(s),B_{\beta}\left(\overline{s}\right)\right\rangle+\lambda^{\prime}\left(s\right)\left\langle
N(s),B_{\beta}\left(\overline{s}\right)\right\rangle\\\
+\lambda(s)\left(\tau-\tau_{G}\right)(s)\left\langle
B(s),B_{\beta}(\left(\overline{s}\right))\right\rangle\end{array}\right]$
or
$\lambda^{\prime}\left(s\right)=0$
that is, $\lambda\left(s\right)$ is constant function on $I.$ This completes
the proof. ∎
###### Theorem 3.3.
Let $\alpha:I\subset\mathbb{R\rightarrow}G$ be a parametrized curve with arc
length parameter $s$ and the Frenet apparatus
$\left(T,N,B,\varkappa,\tau\right)$. Then, $\alpha$ is Mannheim curve if and
only if
$\lambda\varkappa\left(1+H^{2}\right)=1,\text{ for all }s\in I$ (3.3)
where $\lambda$ is constant and $H$ is the harmonic curvature function of the
curve $\alpha.$
###### Proof.
Let $\alpha:I\subset\mathbb{R\rightarrow}G$ be a parametrized Mannheim curve
with arc length parameter $s$ then we can write
$\beta\left(s\right)=\alpha\left(s\right)+\lambda N\left(s\right)$
Differentiating the above equality with respect to $s$ and by using the Frenet
equations, we get
$\frac{d\beta\left(s\right)}{ds}=\left(1-\lambda\varkappa\left(s\right)\right)T\left(s\right)+\lambda\left(\tau-\tau_{G}\right)(s)B\left(s\right).$
On the other hand, we have
$T_{\beta}=\frac{d\beta}{ds}\frac{ds}{d\overline{s}}=\left[\left(1-\lambda\varkappa\left(s\right)\right)T\left(s\right)+\lambda\left(\tau-\tau_{G}\right)(s)B\left(s\right)\right]\frac{ds}{d\overline{s}}.$
By taking the derivative of this equation with respect to $\overline{s}$ and
using the Frenet equations we obtain
$\displaystyle\frac{dT_{\beta}}{d\overline{s}}$
$\displaystyle=\left[-\lambda\frac{d\varkappa}{ds}T(s)+\left(\varkappa-\lambda\varkappa^{2}-\lambda\left(\tau-\tau_{G}\right)^{2}\right)N(s)+\lambda\left(\tau-\tau_{G}\right)^{\shortmid}B(s)\right]\left(\frac{ds}{d\overline{s}}\right)^{2}$
$\displaystyle+\left[\left(1-\lambda\varkappa\left(s\right)\right)T\left(s\right)+\lambda\left(\tau-\tau_{G}\right)(s)B\left(s\right)\right]\frac{d^{2}s}{d\overline{s}^{2}}$
From this equation we get
$\left(\varkappa-\lambda\varkappa^{2}-\lambda\left(\tau-\tau_{G}\right)^{2}\right)=0,$
$\lambda\varkappa\left(1+H^{2}\right)=1.$
Conversely, if $\lambda\varkappa\left(1+H^{2}\right)=1$ then we can easily see
that $\alpha$ is a Mannheim curve.
This completes proof. ∎
###### Corollary 3.4.
If $G$ is Abellian Lie group then $\tau_{G}=0.$ And so, if
$\alpha:I\subset\mathbb{R\rightarrow}G$ is a parametrized Mannheim curve with
arc length parameter $s$ and the Frenet apparatus
$\left(T,N,B,\varkappa,\tau\right)$ in Abellian Lie group $G$. Then, $\alpha$
satisfy the following equality
$\lambda\left(\varkappa^{2}+\tau^{2}\right)=\varkappa$
###### Proof.
If $G$ is Abellian Lie group then using the $\tau_{G}=0$ and Theorem 3.3 we
have the result. ∎
So, the above Corollary shows that the study is a generalization of Mannheim
curves defined by Liu and Wang [3] in Euclidean 3-space.
###### Theorem 3.5.
Let $\alpha:I\subset\mathbb{R\rightarrow}G$ be a parametrized curve with arc
length parameter $s$. Then $\beta$ is the Mannheim partner curve of $\alpha$
if and only if the curvature $\varkappa_{\beta}$ and the torsion
$\tau_{\beta}$ of $\beta$ satisfy the following equation
$\frac{d\varkappa_{\beta}H_{\beta}}{d\overline{s}}=\frac{\varkappa_{\beta}}{\mu}(1+\mu^{2}\varkappa_{\beta}^{2}H_{\beta}^{2})$
where $\mu$ is constant and $H_{\beta}$ is the harmonic curvature function of
the curve $\beta.$
###### Proof.
Let $\alpha:I\subset\mathbb{R\rightarrow}G$ be a parametrized Mannheim curve
with arc length parameter $s$ then we can write
$\alpha\left(\overline{s}\right)=\beta\left(\overline{s}\right)+\mu\left(\overline{s}\right)B_{\beta}\left(\overline{s}\right)$
for some function $\mu\left(\overline{s}\right)$. By taking the derivative of
this equation with respect to $\overline{s}$ and using the Frenet equations we
obtain
$T\frac{ds}{d\overline{s}}=T_{\beta}+\mu^{\shortmid}\left(\overline{s}\right)B_{\beta}\left(\overline{s}\right)-\mu\left(\overline{s}\right)\left(\tau_{\beta}-\tau_{G_{\beta}}\right)\left(\overline{s}\right)N_{\beta}\left(\overline{s}\right)$
or
$T\frac{ds}{d\overline{s}}=T_{\beta}+\frac{d\mu\left(\overline{s}\right)}{d\overline{s}}B_{\beta}\left(\overline{s}\right)-\mu\left(\overline{s}\right)\varkappa_{\beta}H_{\beta}N_{\beta}\left(\overline{s}\right)$
where $H_{\beta}$ is the harmonic curvature function of the curve $\beta.$ And
then, we know that
$\left\\{N\left(s\right),B_{\beta}\left(\overline{s}\right)\right\\}$ is a
linearly dependent set, so we have
$\frac{d\mu\left(\overline{s}\right)}{d\overline{s}}=0.$
This means that $\mu\left(\overline{s}\right)$ is a constant function. Thus we
have
$T\frac{ds}{d\overline{s}}=T_{\beta}-\mu\left(\overline{s}\right)\varkappa_{\beta}H_{\beta}N_{\beta}\left(\overline{s}\right).$
(3.4)
On the other hand, we have
$T=T_{\beta}\cos\theta+N_{\beta}\sin\theta$ (3.5)
where $\theta$ is the angle between $T$ and $T_{\beta}$ at the corresponding
points of $\alpha$ and $\beta.$ By taking the derivative of this equation with
respect to $\overline{s}$ and using the Frenet equations we obtain
$\varkappa
N\frac{ds}{d\overline{s}}=-\left(\varkappa_{\beta}+\frac{d\theta}{d\overline{s}}\right)\sin\theta
T_{\beta}+\left(\varkappa_{\beta}+\frac{d\theta}{d\overline{s}}\right)\cos\theta
N_{\beta}+\varkappa_{\beta}H_{\beta}\sin\theta B_{\beta}.$
From this equation and the fact that the
$\left\\{N\left(s\right),B_{\beta}\left(\overline{s}\right)\right\\}$ is a
linearly dependent set, we get
$\left\\{\begin{array}[c]{c}\left(\varkappa_{\beta}+\frac{d\theta}{d\overline{s}}\right)\sin\theta=0\\\
\left(\varkappa_{\beta}+\frac{d\theta}{d\overline{s}}\right)\cos\theta=0.\end{array}\right.$
For this reason we have
$\frac{d\theta}{d\overline{s}}=-\varkappa_{\beta}.$ (3.6)
From the Eq. (3.4) and Eq. (3.5) and notice that $T_{\beta}$ is orthogonal to
$B_{\beta}$, we find that
$\frac{ds}{d\overline{s}}=\frac{1}{\cos\theta}=-\frac{\mu\varkappa_{\beta}H_{\beta}}{\sin\theta}.$
Then we have
$\mu\varkappa_{\beta}H_{\beta}=-\tan\theta.$
By taking the derivative of this equation and applying Eq. (3.6), we get
$\mu\frac{d\varkappa_{\beta}H_{\beta}}{d\overline{s}}=\varkappa_{\beta}\left(1+\mu^{2}\varkappa_{\beta}^{2}H_{\beta}^{2}\right)$
that is
$\frac{d\varkappa_{\beta}H_{\beta}}{d\overline{s}}=\frac{\varkappa_{\beta}}{\mu}\left(1+\mu^{2}\varkappa_{\beta}^{2}H_{\beta}^{2}\right).$
Conversely, if the curvature $\varkappa_{\beta}$ and torsion $\tau_{\beta}$ of
the curve $\beta$ in three dimensional Lie group $G$ satisfy
$\frac{d\varkappa_{\beta}H_{\beta}}{d\overline{s}}=\frac{\varkappa_{\beta}}{\mu}\left(1+\mu^{2}\varkappa_{\beta}^{2}H_{\beta}^{2}\right)$
for constant $\mu\left(\overline{s}\right),$ then we define a curve a curve by
$\alpha\left(\overline{s}\right)=\beta\left(\overline{s}\right)+\mu
B_{\beta}\left(\overline{s}\right)$ (3.7)
and we will show that $\alpha$ is a Mannheim curve and $\beta$ is the partner
curve of $\alpha$ in three dimensional Lie group $G$. By taking the derivative
of Eq. (3.7) with respect to $\overline{s}$ twice, we get
$T\frac{ds}{d\overline{s}}=T_{\beta}-\mu\varkappa_{\beta}H_{\beta}N_{\beta},$
(3.8) $\varkappa
N\left(\frac{ds}{d\overline{s}}\right)^{2}+T\frac{d^{2}s}{d\overline{s}^{2}}=\mu\varkappa_{\beta}^{2}H_{\beta}T_{\beta}+\left(\varkappa_{\beta}-\mu\frac{d\varkappa_{\beta}H_{\beta}}{d\overline{s}}\right)N_{\beta}-\mu\varkappa_{\beta}^{2}H_{\beta}^{2}B_{\beta},$
(3.9)
respectively. Taking the cross product of Eq. (3.8) with Eq. (3.9) and
noticing that
$\varkappa_{\beta}-\mu\frac{d\varkappa_{\beta}H_{\beta}}{d\overline{s}}+\mu^{2}\varkappa_{\beta}^{3}H_{\beta}^{2}$
we have
$\varkappa
B\left(\frac{ds}{d\overline{s}}\right)^{3}=\mu^{2}\varkappa_{\beta}^{3}H_{\beta}^{3}T_{\beta}+\mu\varkappa_{\beta}^{2}H_{\beta}^{2}N_{\beta}.$
(3.10)
By taking the cross product of Eq. (3.8) with Eq. (3.10), we get
$\varkappa
N\left(\frac{ds}{d\overline{s}}\right)^{4}=-\mu\varkappa_{\beta}^{2}H_{\beta}^{2}\left(1+\mu^{2}\varkappa_{\beta}^{2}H_{\beta}^{2}\right)B_{\beta}.$
This means that the principal normal vector field of the curve $\alpha$ and
binormal vector field of the curve $\beta$ are linearly dependent set. And so
$\alpha$ is a Mannheim curve and $\beta$ is Mannheim partner curve of the
curve $\alpha$ in three dimensional Lie group $G.$ ∎
###### Proposition 3.6.
Let $\alpha:I\subset\mathbb{R\rightarrow}G$ be an arc-lenghted Mannheim curve
with the Frenet vector fields $\left\\{T,N,B\right\\}$ and
$\beta:\overline{I}\subset\mathbb{R\rightarrow}G$ be a Mannheim mate of
$\alpha$ with the Frenet vector fields
$\left\\{T_{\beta},N_{\beta},B_{\beta}\right\\}.$ Then
$\tau_{G}=\tau_{G_{\beta}}$ for the curves $\alpha$ and $\beta$ where
$\tau_{G}=\frac{1}{2}\left\langle\left[T,N\right],B\right\rangle$ and
$\tau_{G_{\beta}}=\frac{1}{2}\left\langle\left[T_{\beta},N_{\beta}\right],B_{\beta}\right\rangle.$
###### Proof.
Let $\left(\alpha,\beta\right)$ be a Mannheim curve couple. From the Eq. (3.2)
we have
$T_{\beta}\left(\overline{s}\right)=\left[\left(1-\lambda\varkappa\left(s\right)\right)T\left(s\right)+\lambda\left(\tau-\tau_{G}\right)(s)B\left(s\right)\right]\frac{ds}{d\overline{s}}.$
We take the norm this equation and by using Eq. (3.3), we obtain
$\frac{d\overline{s}}{ds}=\lambda\varkappa H\sqrt{1+H^{2}}.$
If we think together the last two equations, we get
$T_{\beta}\left(\overline{s}\right)=\frac{H}{\sqrt{1+H^{2}}}T(s)+\frac{1}{\sqrt{1+H^{2}}}B(s).$
(3.11)
Since $\left(\alpha,\beta\right)$ is a Mannheim curve couple we know
$B_{\beta}(\overline{s})=N(s).$ Then,
$N_{\beta}(\overline{s})=B_{\beta}(\overline{s})\times
T_{\beta}(\overline{s})$
$N_{\beta}(\overline{s})=\frac{1}{\sqrt{1+H^{2}}}T(s)-\frac{H}{\sqrt{1+H^{2}}}B(s).$
We know from Definition 2.2
$\left\langle\left[T_{\beta},N_{\beta}\right],B_{\beta}\right\rangle=2\tau_{G_{\beta}}$
for the curve $\beta.$ Then with the help of above equations for
$T_{\beta}\left(\overline{s}\right),N_{\beta}(\overline{s})$ and
$B_{\beta}(\overline{s})$, we obtain
$\left\langle\left[\frac{H}{\sqrt{1+H^{2}}}T+\frac{1}{\sqrt{1+H^{2}}}B,\frac{1}{\sqrt{1+H^{2}}}T(s)-\frac{H}{\sqrt{1+H^{2}}}B(s)\right],N\right\rangle=2\tau_{G_{\beta}},$
$\frac{H^{2}}{\sqrt{1+H^{2}}}\left\langle\left[T,N\right],B\right\rangle+\frac{1}{\sqrt{1+H^{2}}}\left\langle\left[T,N\right],B\right\rangle=2\tau_{G_{\beta}}.$
By using the Proposition 2.3 in last equation, we get
$\tau_{G}=\tau_{G_{\beta}.}$
This completes proof. ∎
###### Theorem 3.7.
Let $\alpha:I\subset\mathbb{R\rightarrow}G$ be a parametrized Mannheim curve
with curvature functions $\varkappa$, $\tau$ and
$\beta:\overline{I}\subset\mathbb{R\rightarrow}G$ be a Mannheim mate of
$\alpha$ with curvatures functions $\varkappa_{\beta}$, $\tau_{\beta}.$ Then
the relations between these curvature functions are
$\displaystyle\varkappa_{\beta}\left(\overline{s}\right)$
$\displaystyle=\frac{H^{\shortmid}\left(s\right)}{\lambda\varkappa(s)H(s)\left(1+H^{2}(s)\right)^{3/2}},$
$\displaystyle\tau_{\beta}\left(\overline{s}\right)$
$\displaystyle=\frac{1}{\lambda H\left(s\right)}+\tau_{G_{\beta}.}$
###### Proof.
If we differentiating the Eq. (3.11) and using the Frenet formulas, we have
$\varkappa_{\beta}N_{\beta}\lambda\varkappa
H\sqrt{1+H^{2}}=\frac{H^{\shortmid}}{\left(1+H^{2}(s)\right)^{3/2}}\left(T-HB\right).$
If we take the norm of last equation, we get
$\varkappa_{\beta}=\frac{H^{\shortmid}}{\lambda\varkappa
H\left(1+H^{2}(s)\right)^{3/2}}.$
Since $\left(\alpha,\beta\right)$ is a Mannheim curve couple, we know
$B_{\beta}=N.$ If we differentiating this equation and using the Frenet
formulas, we have
$-\left(\tau_{\beta}-\tau_{G_{\beta}}\right)N_{\beta}\lambda
H\sqrt{1+H^{2}}=-T+HB.$
If we take the norm of last equation, we get
$\tau_{\beta}=\frac{1}{\lambda H}+\tau_{G_{\beta}}.$
This completes proof. ∎
###### Theorem 3.8.
Let $\alpha:I\subset\mathbb{R\rightarrow}G$ be an arc-lenghted Mannheim curve
and $\beta:\overline{I}\subset\mathbb{R\rightarrow}G$ be a Mannheim mate of
$\alpha.$ The Mannheim curve $\alpha$ is a slant helix if and only if its
Mannheim mate $\beta$ is a general helix.
###### Proof.
If Mannheim curve $\alpha$ is a slant helix, than we have Theorem 2.7
$\sigma_{N}$ is a constant function. From Theorem 3.7 for the curve $\beta,$
we have
$\displaystyle\frac{\tau_{\beta}-\tau_{G_{\beta}}}{\varkappa_{\beta}}$
$\displaystyle=\frac{\frac{1}{\lambda
H}}{\frac{H^{\shortmid}}{\lambda\varkappa H\left(1+H^{2}\right)^{3/2}}}$
$\displaystyle\frac{\tau_{\beta}-\tau_{G_{\beta}}}{\varkappa_{\beta}}$
$\displaystyle=\frac{\varkappa\left(1+H^{2}\right)^{3/2}}{H^{\shortmid}}$
$\displaystyle\frac{\tau_{\beta}-\tau_{G_{\beta}}}{\varkappa_{\beta}}$
$\displaystyle=\sigma_{N}=\text{constant.}$
So $\beta$, which is Mannheim mate of $\alpha,$is a general helix.
Conversely, we assume that $\beta$, which is Mannheim mate of $\alpha,$ be a
general helix. So we have
$\frac{\tau_{\beta}-\tau_{G_{\beta}}}{\varkappa_{\beta}}=\text{constant.}$
From last equation $\sigma_{N}$ is a constant function. This completes the
proof. ∎
###### Theorem 3.9.
Let $\alpha:I\subset\mathbb{R\rightarrow}G$ be an arc-lenghted Mannheim curve
and $\beta:\overline{I}\subset\mathbb{R\rightarrow}G$ be a Mannheim mate of
$\alpha$. If $\alpha$ is a slant helix then the harmonic curvature function of
the curve $\alpha$: $H(s)$ is
$H(s)=\frac{1}{2}\left(ae^{bs}-\frac{1}{a}e^{-bs}\right)$ (3.12)
for some nonzero constant $a$ and $b$ and $s$ is the arc length parameter of
the curve $\alpha$. If we consider $a=b=1,$ we have the harmonic curvature
funtion of the curve $\alpha$ is hyperbolic sine function in arc length $s$,
that is., $H(s)=\sinh s$.
###### Proof.
Let $\alpha:I\subset\mathbb{R\rightarrow}G$ be an arc-lenghted Mannheim curve
with Frenet apparatus $\left\\{T,N,B,\varkappa,\tau\right\\}$ in three
dimensional Lie group. Assume that $\alpha$ be slant helix, we have
$\left\langle N,X\right\rangle=\cos\theta,\text{ }\theta\neq\frac{\pi}{2}$
(3.13)
for left invariant vector field $X.$ Differentiating the Eq. (3.13) twice, we
have
$-\varkappa\left\langle
T,X\right\rangle+\left(\tau-\tau_{G}\right)\left\langle B,X\right\rangle=0$
(3.14)
and
$-\varkappa^{\shortmid}\left\langle
T,X\right\rangle+\left(\tau-\tau_{G}\right)^{\shortmid}\left\langle
B,X\right\rangle=\left\\{\varkappa^{2}+\left(\tau-\tau_{G}\right)^{2}\right\\}\left\langle
N,X\right\rangle.$
Since $\alpha$ is a Mannheim curve using the Theorem 3.3, we rewrite the last
equation
$-\varkappa^{\shortmid}\left\langle
T,X\right\rangle+\left(\tau-\tau_{G}\right)^{\shortmid}\left\langle
B,X\right\rangle=\frac{\varkappa}{\lambda}\cos\theta$ (3.15)
where $\lambda$ is a non-zero constant. By a direct calculation using the Eq.
(3.14) and the Eq. (3.15), we obtain
$\left\langle T,X\right\rangle=\frac{H}{\lambda H^{\shortmid}}\cos\theta$
(3.16)
and
$\left\langle B,X\right\rangle=\frac{1}{\lambda H^{\shortmid}}\cos\theta.$
(3.17)
Differentiating the Eq. (3.16) and the Eq. (3.17), we have
$\varkappa=\frac{1}{\lambda}\left(1-\frac{HH^{\shortmid\shortmid}}{\left(H^{\shortmid}\right)^{2}}\right),$
$\tau-\tau_{G}=\frac{H^{\shortmid\shortmid}}{\lambda\left(H^{\shortmid}\right)^{2}},$
respectively. These equations give that
$H=\frac{\tau-\tau_{G}}{\varkappa}=\frac{H^{\shortmid\shortmid}}{\left(H^{\shortmid}\right)^{2}-HH^{\shortmid\shortmid}}.$
Then we have the following differential equation
$\left(1+H^{2}\right)H^{\shortmid\shortmid}-\left(H^{\shortmid}\right)^{2}=0.$
Solving the last equation, we obtain the Eq. (3.12). This completes the proof.
∎
###### Theorem 3.10.
Let $\left(\alpha,\beta\right)$ be a Mannheim curve couple in three
dimensional Lie group with bi-invariant metric. Then $\alpha$ is general helix
if and only if $\beta$ is a geodesic.
###### Proof.
If Mannheim curve $\alpha$ is a general helix, then its harmonic curvature $H$
is constant function. And so from Theorem 3.7,
$\varkappa_{\beta}=0.$
So, $\beta$ is a geodesic.
Conversely we assume that $\beta$ be a geodesic curve. From Theorem 3.7 we can
easily see that
$H^{\shortmid}\left(s\right)=0$
and so
$H(s)=\text{constant.}$
This comletes the proof. ∎
## References
* [1] J. M. Bertrand, Mémoire sur la théorie des courbes á double courbure, Comptes Rendus, vol.36, 1850.
* [2] W. Kuhnel, Differential geometry: curves-surfaces-manifolds, Braunschweig, Wiesbaden, 1999.
* [3] H., Liu, and F., Wang, Mannheim partner curves in 3-space, J. Geom., 88 (2008) 120 – 126.
* [4] K., Orbay and E., Kasap, On Mannheim partner curves in $E^{3}$, Int. J. Phys. Sci., Vol. 4 (5), (2009) 261–264.
* [5] M., K., Karacan, Weakened Mannheim curves, Int. J. Phys. Sci., Vol. 6 (20), (2011). 4700-4705.
* [6] H., B., Öztekin and Ma., Ergüt, Null mannheim curves in the minkowski 3-space $E_{1}^{3},$ Turk. J. Math., 35 (2011), 107–114.
* [7] S., Özkaldi, K. İlarslan and Y. Yayli, On Mannheim Partner Curve in Dual Space, An. Şt. Univ. Ovidius Constant¸a, Vol. 17 (2), (2009), 131–142.
* [8] M., A., Güngör and M., Tosun, A Study on Dual Mannheim Partner Curves, Int.. Math. Forum, 5, (2010), no. 47, 2319 - 2330.
* [9] H., Matsuda and S. Yorozu, On Generalized Mannheim Curves in Euclidean 4-Space, Vol.20, (2009), 33–56.
* [10] A. C. Çöken, Ü. Çiftçi, A note on the geometry of Lie groups, Nonlinear Analysis TMA 68 (2008) 2013-2016.
* [11] Ü. Çiftçi, A generalization of Lancert’s theorem, J. Geom. Phys. 59 (2009) 1597-1603.
* [12] S. Izumiya and N. Tkeuchi, New special curves and developable surfaces, Turk. J. Math 28 (2004), 153-163.
* [13] E. Özdamar, H.H. Hacısalihoğlu, A characterization of inclined curves in Euclidean n-space, Commun. Fac. Sci. Univ., Ser. 24A (1975) 15–22.
* [14] İ. Gök, Ç. Camcı, H.H. Hacısalihoğlu, Vn-slant helices in Euclidean n-space $E^{n}$, Math. Commun. 14 (2) (2009) 317–329.
* [15] Ç. Camcı, K. İlarslan, L. Kula, H.H. Hacısalihoğlu, Harmonic curvatures and generalized helices in $E^{n}$, Chaos Solitons Fract. 40 (2007) 1–7.
* [16] M., Külahcı, M., Bektaş, M. Ergüt, On Harmonic curvatures of Frenet curve in Lorentzian space., Chaos Solitons Fract. 41 (2009) 1668-1675.
* [17] O., Z., Okuyucu, İ., Gök, Y., Yaylı, and N., Ekmekci, Slant Helices in three Dimensional Lie Groups, arXiv:1203.1146v2 [math.DG].
* [18] P. Crouch, F. Silva Leite, The dynamic interpolation problem: on Riemannian manifoldsi Lie groups and symmetric spaces, J. Dyn. Control Syst. 1 (2) (1995) 177-202.
|
arxiv-papers
| 2012-11-26T21:38:52 |
2024-09-04T02:49:38.519781
|
{
"license": "Public Domain",
"authors": "\\.Ismail G\\\"ok, O. Zeki Okuyucu, Nejat Ekmekci and Yusuf Yayl{\\i}",
"submitter": "Osman Zeki Okuyucu",
"url": "https://arxiv.org/abs/1211.6141"
}
|
1211.6178
|
Topological Basis Associated with BWMA.....] Topological Basis Associated with BWMA, Extremes of $L_{1}$-norm in Quantum Information and Applications in Physics
Qing. Zhao^1, Ruo-Yang Zhang^2, Kang Xue^3 and Mo-Lin Ge^1,2
$^{1}$ College of Physics, Beijing Institute of Technology, Beijing, 100081, China
$^{2}$ Theoretical Physics Section, Chern Institute of Mathematics, Nankai University, Tianjin, 300071, China
$^{3}$ Dept. of Physics, Northeast Normal University, Changchun, 130024, China
[email protected], [email protected]
The topological basis associated with Birman-Wenzl-Murakami algebra (BWMA) is constructed and the three dimensional forms of braiding matrices S have been found for both $S^+=S$ and $S^+=S^{-1}$. A familiar spin-$1$ model related to braiding matrix associated with BWMA is discussed. The extreme points $(\theta=\pm\pi/2$ and $\pm\pi)$ of $L_{1}$-norm and von Neumann entropy are shown to be connected to each other. Through the general discussion and examples we then point out that the $L_{1}$-norm describes quantum entanglement.
In the ref.[1] two types of braiding matrices with
two distinct eigenvalues, i,e. those associated with Temperely-Lieb
algebra (TLA)[2] and their corresponding solutions of
Yang-Baxter equation (YBE)[3, 4, 5, 6, 7] had been discussed. Based on the
topological basis [8, 9, 10] the $4$-dimensional braiding
operators were mapped onto 2-dimensional ones [1, 9]. We had shown
that the two types of braiding matrices are related to the extremes
of $L_{1}$-norm of Wigner's D-function[1]. Especially the $2$-d
braiding matrix corresponding the Bell basis (the type-II) is
connected to the maximum of the $L_{1}$-norm, whereas the
permutation and extensions (the type-I) to the minimum. It hints
that the $L_{1}$-norm should relate to the quantum information. It
is natural to extend the discussions in the Ref.[1] to the
solution of YBE with three distinct eigenvalues. Among them, the
most important ones belong to Birman-Wenzl-Murakami algebra
[11, 12, 13](BWMA). As is well-known that the forms of braiding
matrices with three distinct eigenvalues were given by references,
say in [14, 15], and the connection with BWMA was shown in
Ref.[15] for both standard and non-standard solutions. In
parallel to the ref.[1], in this paper we shall firstly set up
the topological base $|e_{1}\rangle$, $|e_{2}\rangle$, and
$|e_{3}\rangle$ associated with BWMA, then map the $9$-d braiding
matrices to $3$-d forms. The physical application of BWMA is raised
through a familiar model which is different from the model
discussed in Ref. [16]. We shall point out that in general the
extremes of $L_{1}$-norm of the D-functions ($\theta=\pi/2$ and $\pi$)
are related to those of von Neumann entropy. We also take the
spin-$1$ models as examples to favor the statement.
§ TOPOLOGICAL BASIS FOR BWMA
For self-contain the BWMA relations have been given in the Appendix A through the graphs for three states. Denoting the eigenvalues of a braiding matrix $S$ with three eigenvalues by $\lambda_1$, $\lambda_2$, and $\lambda_3$, where $S$ satisfies braid relation
\begin{equation}
S_{12}S_{23}S_{12}=S_{23}S_{12}S_{23}\quad (S_1\equiv S_{12}=S\otimes I,\; S_2=S_{23}=I\otimes S)
\end{equation}
and without loss of generality by setting the eigenvalues of $S$ to be $\lambda_1$, $\lambda_2$ and $\lambda_3$ with
\begin{equation}
\lambda_1\lambda_2=-1,\quad W=\lambda_1+\lambda_2,\quad \lambda_3=\sigma
\end{equation}
we have for $S$ with three distinct eigenvalues.
\begin{equation}
\end{equation}
\begin{equation}
E=\frac{1}{\sigma W}(S^2-WS-I)
\end{equation}
(<ref>) becomes
\begin{equation}
\end{equation}
Where $S$ and $E$ occupy $k-th$ and $(k+1)-th$ sites and satisfy the relations shown in Appendix A, i. e. they form BWM algebra. Noting that a loop takes the value
\begin{equation}
\end{equation}
Following the philosophy for T-L algebra to set up the topological basis $|e_1\rangle$ and $|e_2\rangle$ [8, 9], we shall find the uni-orthogonal basis $|e_1\rangle$, $|e_2\rangle$ and $|e_3\rangle$ for $S$ and $E$ such that
\begin{equation}
S_{12}|e_{\mu}\rangle=\lambda_{\mu}|e_{\mu}\rangle\quad(\mu=1,2,3) \label{Eq1.7}
\end{equation}
\begin{equation}
S_{k,k+1}=\cross\quad\mathrm{and}\quad E_{k,k+1}=\Uniil,\quad d=\Nu=\myloop\quad \label{Eq1.8}
\end{equation}
where the eigenvalues $\lambda_{\mu}$ may be complex. The graphic expressions [17] of BWMA are shown in Appendix A. To satisfy all the relations for BWMA the base takes the forms:
\begin{eqnarray}
|e_3\rangle&=&d^{-1}\tsep \label{Eq1.9}\\
|e_i\rangle&=&f_i\left\{\tcross+\alpha_i\Uu+\beta_i\tsep\right\}\quad (i=1,2)\label{Eq1.10} .
\end{eqnarray}
In terms of the graphic calculations [17] it can be proved that the (<ref>) together with
\begin{equation}
\langle e_3|e_i\rangle=0,\quad \langle e_i|e_j\rangle=\delta_{ij}\quad (i,j=1,2),
\end{equation}
lead to the constraints to the parameters $\alpha_i$, $\beta_i$ and normalization constant $f_i$:
\begin{equation}
\alpha_i=\lambda_i\quad (\lambda_1\lambda_2=-1), \quad \alpha_i+\beta_i d=-\sigma^{-1}\quad (i=1,2)\label{Eq1.12}
\end{equation}
\begin{equation}
f_i=\{d(\lambda_i^2+1)[-\lambda_i^{-1}d^{-1}(\sigma^{-1}+\lambda_i)+\lambda_i^{-1}\sigma+d]\}^{1/2} \label{Eq1.13}
\end{equation}
for $\lambda_{\mu}^{*}=\lambda_{\mu}(\mu=1,2,3)$, i.e. $S^{\dag}=S$ (hermitian), whereas
\begin{equation}
f_i=\{(d-1)(\lambda_i+\lambda_i^{-1})(\sigma+\lambda_i d+\lambda_i^{-1})\}^{1/2}\label{Eq1.14}
\end{equation}
for $\lambda_{\mu}^{*}=\lambda_{\mu}^{-1}(\mu=1,2,3)$, i.e. $S^{\dag}=S^{-1}$ (unitary).
The (<ref>) takes the same form for $S$ being hermitian or unitary. The only difference between hermitian and unitary consists in the different normalized constants $f_i$ and the parameters being complex for $S^+=S^{-1}$. The proof can be seen in the Appendix B.
§ THREE-DIMENSIONAL MATRIX FORMS OF $E_{12}$, $E_{23}$, $S_{12}$ AND $S_{23}$ FOR $S^{\DAG}=S$
In terms of the uni-orthogonally topological basis the direct calculation gives the $3$-D matrix forms of $E$ and $S$ acting on 1-st and 2-nd sites, 2-nd and 3-rd sites, respectively:
\begin{equation}
\eqalign{
(E_{A})_{\mu\nu}=\langle e_{\mu}|E_{12}|e_{\nu}\rangle \\
(E_{B})_{\mu\nu}=\langle e_{\mu}|E_{23}|e_{\nu}\rangle\quad(\mu,\nu=1,2,3)\\
A_{\mu\nu}=\langle e_{\mu}|S_{12}|e_{\nu}\rangle\\
B_{\mu\nu}=\langle e_{\mu}|S_{23}|e_{\nu}\rangle
\end{equation}
where A represents the braiding between the 1-st and 2-nd sites, whereas B for the braiding between the 2-nd and 3-rd sties.
The explicit 3-D matrix forms are shown to be$(\lambda_1\lambda_2=-1)$:
\begin{equation}\label{Eq2.2}\fl
\end{array}\right], \qquad
\lambda_1&&\\
\end{array}\right]
\lambda_1&&\\
\end{array}\right]
\end{equation}
\begin{equation}\fl
\begin{array}{lcr}
\end{array}\!\right]
\end{equation}
\begin{equation}\label{Eq2.4}\fl
\lambda_1^{-1}(\lambda_3(d-1)-\lambda_1^{-1})&(1-\lambda_1^{-1}\lambda_3-\lambda_1\lambda_3d)f_1^{-1}f_2&\lambda_1^{-1}f_1^{-1}\\
(1-\lambda_1^{-1}\lambda_3-\lambda_1\lambda_3d)f_1^{-1}f_2& \lambda_1(\lambda_3(d-1)+\lambda_1)&\lambda_1f_2^{-1}\\
\lambda_1^{-1}f_1^{-1}&\lambda_1f_2^{-1}&(\lambda_1+\lambda_1^{-1})\lambda_3^{-1}
\end{array}\right]
\end{equation}
When $\lambda_1=q$,$\lambda_2=-q^{-1}$,$\lambda_3=q^{-2}$, i.e. the standard braiding matrix given in <cit.>, we have
\begin{equation}\fl
\end{array}\right]
\end{equation}
\begin{equation}\fl
\end{array}\right]\label{Eq2.6}
\end{equation}
The other relations can be sees in Appendix A.
§ $3$-D TOPOLOGICAL BASIS OF BWMA FOR UNITARY S
When $S$ is unitary with $q$ at root of unity, namely
\begin{equation} \fl
\lambda_{\mu}^{*}=\lambda_{\mu}^{-1}\quad (\mathrm{f.g.}\quad \lambda=q^m,\quad q \textrm{ at root of unity, where m may be $\pm$ inegers}),
\end{equation}
the basis reads ($i=1,2$)
\begin{equation}
\end{equation}
\begin{equation}
\langle e_i|=f_i^{*}\left(\ntcros+\alpha_i^{*}\Nn+\beta_i^{*}\tncup\right).
\end{equation}
The matrix forms of $E$ and $S$ read
\begin{equation} \fl
\begin{array}{ccc}
-(\lambda_1+\lambda_1^{-1})f_1^{*-1}f_2^{-1}& (\lambda_1+\lambda_1^{-1})^{-2}|f_2|^{-2}&-(\lambda_1+\lambda_1^{-1})^{-1}f_2^{-1}\\
\end{array}\right],
\end{equation}
\begin{equation}
\end{array}\right],
\end{equation}
\begin{equation} \fl
\eqalign{
\end{equation}
The 3-D matrix form is found:
\begin{equation}\label{Eq3.7}
\begin{array}{ccc}
\lambda_1&&\\
\end{array}\right],
\end{equation}
\begin{equation}\label{Eq3.8}\fl
\begin{array}{ccc}
\lambda_1^{-1}[\lambda_3(d-1)-\lambda_1^{-1}]&(1-\lambda_1^{-1}\lambda_3-\lambda_1\lambda_3d)f_1^{-1}f_2&\lambda_1^{-1}f_1^{-1}\\
\lambda_1^{-1}f_1^{*-1}&-\lambda_1f_2^{*-1}&(\lambda_1+\lambda_1^{-1})\lambda_3^{-1}\\
\end{array}\right],
\end{equation}
\begin{equation}
\begin{array}{rcl}
\because S-S^{-1}&=&W(I-E)=W(I-E^{+})\\
\end{array}
\end{equation}
i.e. $E$ must be real.
\begin{equation} \fl
W=\lambda_1+\lambda_2=\lambda_1-\lambda_1^{-1}\qquad W^*=\lambda_1^*-\lambda_1^{*-1}=-\lambda_1+\lambda_1^{-1}=-W.
\end{equation}
Following Ref. [15] for $q$ at root of unity it allows 'non-standard' braiding matrices, say, for $B_n$ some of $\lambda_k=q$, and for the others $\lambda_i=-q^{-1}$, then the general form of $\sigma$ is
\begin{equation}
\sigma=\lambda_3=q^{m} \quad(|m|\le 2n).
\end{equation}
It leads to
\begin{equation}
\begin{array}{rcl}
&=&1-\frac{\sin{m\alpha}}{\sin{\alpha}} \quad(\frac{\pi}{N}=\alpha, q=e^{i\frac{\pi}{N}})\\
&=& 1-[m]_q\\
\end{array}
\end{equation}
where $|m|$ is the difference between the positive power number and negative ones in the power of $q$ in the third eigenvalues of $S$ for the fundamental representations of $B_n$, $D_n$ and $C_n$.
§ SPIN-1 MODEL ASSOCIATED WITH BWM ALGEBRA
As was pointed out in [14, 15] that for $B_n$ algebra the corresponding braiding matrix $S$ has three distinct eigenvalues $\lambda_1=q$,$\lambda_2 =-q^{-1}$ whereas as the third one is given by
\begin{equation}
\lambda_{3}=\sigma=\prod_{k=1} u_{k}^{-2}
\end{equation}
where $u_{k}$ can be either $q$ (standard solution) or $-q^{-1}$ for nonstandard solution, so in general, we are able to take
\begin{equation}
\sigma=\lambda_{3}=q^{m}
\end{equation}
where $m$ can be arbitrary integers. To satisfy the spectral parameter dependent Yang-Baxter equation, the corresponding $\check{R}(x)$-matrix takes the form [15]
\begin{equation} \fl
\check{R}_{\alpha}(x)=(x-1)(x-x_{\alpha})S+Wx(x-1)E-Wx(x-x_{\alpha})I, \quad \alpha=a,b
\end{equation}
where $W=q-q^{-1}$, $x_a=-q\sigma^{-1}=-q^{(1-m)}$, $x_b=q^{-1}\sigma^{-1}=q^{-(m+1)}$.
In order to obtain the rational limit of the $\check{R}(x)$ the type-I solution is given by
\begin{equation}
\check{R}_b(x)=\mathrm{const}\,\left\{I-\left(\frac{x-1}{Wx}\right)S-\frac{x-1}{x-q^{-(m+1)}}E\right\}
\end{equation}
Under the rational limit and $q\rightarrow1$ we set $h\rightarrow0$ for
\begin{equation}
x=e^{hu} \quad\mathrm{and}\quad q=e^{-\frac{h\gamma}{2}}
\end{equation}
it leads to
\begin{equation}
\check{R}_b(x)|_{h\rightarrow0}=\mathrm{const}\cdot\left\{I+\frac{u}{\gamma}S-\frac{u}{u-\gamma(\frac{m+1}{2})}E\right\}|_{q=1}
\end{equation}
that under the rescaling $\frac{u}{\gamma}\rightarrow u$ becomes
\begin{equation}
\check{R}_b(u)|_{h\rightarrow0}\equiv\check{R}(u)=\mathrm{const}\cdot\left\{I+uT-\left(\frac{u}{u-\beta}\right)M\right\}
\end{equation}
where $T=S|_{q=1}$, $M=E|_{q=1}$, and $\beta=\frac{m+1}{2}$.
Following the standard way of Baxter [4] the Hamiltonian can be given by
\begin{equation}
H_{k,k+1}\propto\frac{\partial\check{R}_{k,k+1}(u)}{\partial u}|_{u=0}=T_{k,k+1}+\frac{1}{\beta}M_{k,k+1}
\end{equation}
Here the $k-th$ site has been indicated explicitly because $T$ and $M$ occupy the $k$-th and $(k+1)$-th sites. The $M_{k,k+1}$ is a new term added to the permutation-like operator $T_{k,k+1}$ due to BWMA.
In particular when $m=-3$, i.e. $\sigma=q^{-3}$ we have
\begin{equation}
\end{equation}
It is worthy noting that $m=-3$ means the solution of YBE being "nonstandard"[15]
The M works in the block for ${S_{z}(k)+S_{z}(k+1)=0}$,
where $S_{z}(k)$ means the third component of spin-$1$ at $k$-th site, or in terms of the basis for spin-$1$:
\begin{equation} \fl
\eqalign{
k,l}\left\langle1,1\right|+\left|-1,-1\right\rangle_{k.l\ k,l}\left\langle-1,-1\right|+\left|0,0\right\rangle_{k.l\ k,l}\left\langle0,0\right| \\
&+\left|1,0\right\rangle_{k,l\ k,l}\left\langle0,1\right|+\left|0,1\right\rangle_{k.l\ k,l}\left\langle1,0\right|+\left|0,-1\right\rangle_{k.l\ k,l}\left\langle-1,0\right| \\
&+\left|-1,0\right\rangle_{k.l\ k,l}\left\langle0,-1\right|+\left|1,-1\right\rangle_{k.l\ k,l}\left\langle-1,1\right|+\left|-1,1\right\rangle_{k.l\ k,l}\left\langle1,-1\right| \label{Eq5.11}
\end{equation}
where the $|S_z,S_z\rangle_{k,l}$ represents the state which occupies $k$-th and $l$-th sites. In general, $l$ is not necessary to be $k+1$.
\begin{equation} \fl
\eqalign{
M_{k,l}=&\left|0,0\right\rangle_{k.l\ k,l}\left\langle0,0\right|+\big(\left|1,-1\right\rangle_{k,l}+\left|-1,1\right\rangle_{k,l}\big)\left(_{k,l}\left\langle1,-1\right|+_{k,l}\left\langle-1,1\right|\right)
\nonumber\\
\nonumber\\
&+e^{-i\varphi}\left(\left|1,-1\right\rangle_{k,l}+\left|-1,1\right\rangle_{k,l}\right)_{k,l}\left\langle0,0\right| \label{Eq5.12}
\end{equation}
where $\varphi$ is real and $T^2_{k,l}=I$, $M^2_{k,l}=3M_{k,l}$ (i.e. $d=3$, here the braiding matrix is not unitary), then $T_{k,k+1}$ and $M_{k,k+1}$ satisfy BWM algebra.
It is easy to check that for spin-$1$ and $l=k+1$, in terms of
\begin{array}{ccc}
\end{array}
\right]$,
\begin{array}{ccc}
\end{array}
\right]$
\begin{array}{ccc}
\end{array}
\right]$,
we have folowing Baxter [4],
\begin{equation}\label{Eq5.13}
\frac{\partial\check{R}_{k,k+1}(u)}{\partial u}|_{u=0}=H_{k,k+1}=T_{k,k+1}-M_{k,k+1}=\overrightarrow{S}_k\cdot\overrightarrow{S}_{k+1}
\end{equation}
for $\varphi=\pi$. Therefore, for spin-$1$
\begin{equation}\label{Eq5.14}
\end{equation}
It is interesting to note that the Hamiltonian (<ref>) is known well for long time. Especially, it is not permutation operator, but plays role, say, in the Haldane conjuncture [17]. Here we have obtained $(\overrightarrow{S}_k.\overrightarrow{S}_{k+1})$ for spin-1 whose Hamiltonian is associated with BWM algebra.
Furthermore, as a demonstration example we show how to solve the model with $N=4$ in terms of the topological basis given by (<ref>), (<ref>).
Graphically the Hamiltonian can be expressed by the operators for $N=4$
\begin{equation}\label{Eq5.15}
\hat{\mathscr{H}}=J\sum_{k=1}^{4}\left(\Crosss-\Uniil\right)=J\sum_{k=1}^{4}\left(\hat{T}_{k,k+1}-\hat{M}_{k,k+1}\right)
\end{equation}
Its 9-d representation is given by acting the operator $\mathscr{H}$ on 9-d basis. Whereas acting (<ref>) on the 3-d basis (<ref>),(<ref>) for $m=-3$, we find
\begin{eqnarray}
\mathscr{H}\left|e_1\right\rangle&=&\frac{J}{3}\left(-3\left|e_1\right\rangle+\sqrt{15}\left|e_2\right\rangle\right)\nonumber\\
\mathscr{H}\left|e_2\right\rangle&=&\frac{J}{3}\left(\sqrt{15}\left|e_1\right\rangle-9\left|e_2\right\rangle+4\sqrt{3}\left|e_3\right\rangle\right)\\
\mathscr{H}\left|e_3\right\rangle&=&\frac{J}{3}\left(4\sqrt{3}\left|e_2\right\rangle-12\left|e_3\right\rangle\right)\nonumber
\end{eqnarray}
The $\mathscr{H}$ can be diagonalized in terms of the eigenstates $\left|g_{\mu}\right\rangle$: ($\mu=1,2,3$)
\begin{equation} \label{Eq5.17}
\mathscr{H}\left|g_{\mu}\right\rangle=E_{\mu}\left|g_{\mu}\right\rangle
\end{equation}
\begin{equation*}
E_1=-6J, \quad E_2=-2J, \quad E_3=0\nonumber\\
\end{equation*}
\begin{eqnarray}
\left|g_1\right\rangle&=&\frac{1}{\sqrt{6}}\left(\left|e_1\right\rangle-\sqrt{15}\left|e_2\right\rangle+2\sqrt{15}\left|e_3\right\rangle\right)\nonumber\\
\left|g_2\right\rangle&=&\frac{1}{2\sqrt{3}}\left(\sqrt{5}\left|e_1\right\rangle-\sqrt{3}\left|e_2\right\rangle-2\left|e_3\right\rangle\right)\nonumber\\
\left|g_3\right\rangle&=&\frac{1}{3}\left(\sqrt{5}\left|e_1\right\rangle+\sqrt{3}\left|e_2\right\rangle+\left|e_3\right\rangle\right) \label{Eq5.18}
\end{eqnarray}
How to extend the approach to any $N$ by using the topological basis more than four sites to solve the eigenvalues problem with the help of topological basis is far beyond the current discussion. Here we only discuss a four spin$-1$ model which may be a hint to look for how to solve the $N$-site chain problem based on the topological basis.
§ FOUR SPIN MODEL
The relations (<ref>) and (<ref>) are defined for any $k$ and $l$. To obtain the Hamiltonian (<ref>), the nearest neighborhood has been imposed through putting $l=k+1$. However for any $i$ and $j$, the operator $S_{ij}$ can be recast to
\begin{eqnarray}
kl}\left\langle1,1\right|+\left|-1,-1\right\rangle_{kl\ kl}\left\langle-1,-1\right|+\left|0,0\right\rangle_{kl\ kl}\left\langle0,0\right|\nonumber\\
&+\left|1,0\right\rangle_{kl\ kl}\left\langle0,1\right|+\left|0,1\right\rangle_{kl\ kl}\left\langle1,0\right|+\left|0,-1\right\rangle_{kl\ kl}\left\langle-1,0\right|\nonumber\\
&+\left|-1,0\right\rangle_{kl\ kl}\left\langle0,-1\right|. \label{Eq6.1}
\end{eqnarray}
It can be checked in terms of (<ref>), (<ref>)and (<ref>), it holds for any $i$ and $k$:
\begin{equation}\label{Eq6.2}
\overrightarrow{S}_{i}\cdot\overrightarrow{S}_{k}=T_{i,k}-M_{i,k}
\end{equation}
where at any $i$-th site,
\begin{eqnarray}
S_{+}=\sqrt{2}\left(\left|1\right\rangle\left\langle0\right|+\left|0\right\rangle\left\langle-1\right|\right), \nonumber\\ S_{-}=\sqrt{2}\left(\left|0\right\rangle\left\langle1\right|+\left|-1\right\rangle\left\langle0\right|\right), \nonumber\\
\end{eqnarray}
Noting that (<ref>) is valid for any i and k. Because ${S_{i}}^2=S_{i}(S_{i}+1)=2$ and total spin
\begin{equation}
{\overrightarrow{S}}^2=\left(\sum_{i=1}^{N}\overrightarrow{S}_{i}\right)^2=\sum_{i=1}^{N}{\overrightarrow{S}_{i}}^2+\sum_{i\ne j}^{N}(\overrightarrow{S}_i.\overrightarrow{S}_{j})=2N+2\sum_{i<j}^{N}(\overrightarrow{S}_i.\overrightarrow{S}_{j}).
\end{equation}
for $N=4$ we have
\begin{equation}
\overrightarrow{S}^2=2N+2\sum_{i<j}^{4}(T_{i,j}-M_{i,j}).
\end{equation}
There appear the additional terms other than $(T_{i,i+1}-M_{i,i+1})$, i.e. the term $(T_{13}-M_{13})$ and $(T_{24}-M_{24})$. Taking into account of
\begin{equation}
\end{equation}
we find
\begin{equation}\label{Eq6.5}
\sum_{i=1}^{2}(T_{i,i+2}-M_{i,i+2})\left|g_{k}\right\rangle=\mu_{k}\left|g_{k}\right\rangle
\end{equation}
where $\mu_{1}=1$,$\mu_{2}=-1$, $\mu_{3}=-3$, $(k=1,2,3)$
\begin{equation} \fl
\eqalign{
\overrightarrow{S}^{2}\left|g_{k}\right\rangle &=\{\sum_{i=1}^{4}\overrightarrow{S}_i^{2}+2\sum_{i=1}^{4}(T_{i,i+1}-M_{i,i+1})+2\sum_{i=1}^{2}(T_{i,i+2}-M_{i,i+2})\}\left|g_{k}\right\rangle \nonumber\\
4+E_{k}J^{-1}+2\mu_{k})\left|g_{k}\right\rangle \label{Eq6.6}
\end{equation}
Substituting (<ref>) and (<ref>) into (<ref>),we obtain for $N=4$
\begin{equation}
\overrightarrow{S}^{2}\left|g_{k}\right\rangle=0\left|g_{k}\right\rangle
\end{equation}
It is interesting to note that the topological eigenstates $\left|g_{k}\right\rangle (k=1,2,3)$ are spin singlet. From the point of view of Lie algebra, the direct product of four spin 1 can be decomposed to 5 subspaces; however, only the singlet with multiplicities three is the eigenstates of $\mathscr{H}$ for $N=4$.
§ $D_{MM'}^{J}(\THETA,\VARPHI)$ FUNCTION AS THE N-DIMENSIONAL SOLUTIONS OF YBE
The Yang-Baxterization (parametrization) of $N^2\times N^2$ braiding matrices can be made in the standard way, say, following Jimbo, Jones, and others $\cdots$ [14, 19, 20]. For $N\times N$ YBE there is another way to introduce spectral parameter to a given braiding matrix. The basic idea comes from the Wigner D-function [21].
\begin{equation} \fl
D(n)=e^{i\theta\overrightarrow{m}\cdot\overrightarrow{J}}=e^{\xi J_{+}-\xi^{*}J_{-}}, \quad \xi=-\frac{\theta}{2}e^{-i\varphi},\quad J_{\pm}=J_{x}\pm iJ_{y}, \quad J_0=J_3
\end{equation}
\begin{equation} \fl
\overrightarrow{m}=(\sin\varphi,-\cos\varphi,0),\quad \overrightarrow{n}=(\sin\theta\cos\varphi,\sin\theta\sin\varphi,\cos\theta),\quad n=-\tan\frac{\theta}{2}e^{-i\varphi}
\end{equation}
It can be proved that
\begin{equation}
\end{equation}
provided it holds
\begin{equation}
\cos\varphi=\frac{1}{2}\left[\frac{\tan\frac{\theta_1}{2}+\tan\frac{\theta_3}{2}-\tan\frac{\theta_2}{2}} {\tan\frac{\theta_1}{2}\tan\frac{\theta_2}{2}\tan\frac{\theta_3}{2}}-1\right] \label{Eq:6.4}
\end{equation}
When $\theta_1=\theta_2=\theta_3=\theta$, (<ref>) reduces to [22]
\begin{equation}\label{Eq63}
\cos\varphi=\frac{\cos\theta}{1-\cos\theta}
\end{equation}
which is the condition that (<ref>) reduces to braid relation.
Denoting by
\begin{equation}
A(\theta)=D^{J}(\theta,0), \quad B(\theta,\varphi)=D^{J}(\theta,\varphi)
\end{equation}
the ($N\times N$) YBE reads [23]
\begin{equation}
\end{equation}
\begin{equation}
A(x)B(xy)A(y)=B(y)A(xy)B(x), cos\theta=\frac{(1-x)}{\sqrt{2(1+x^{2})}}
\end{equation}
is satisfied under (<ref>) .
The examples for $N=2$ had been given in the ref.[1]. Here it holds for any ($2J+1$)-dimensional representations.
Let us condiser spin-1 example. The $D^{1}(\theta,\varphi)$ takes the form [21]
\begin{equation}\label{Eq:6.7}
\frac{1+\cos\theta}{2} & -\frac{\sin\theta}{\sqrt{2}}e^{-i\varphi} & \frac{1-\cos\theta}{2}e^{-2i\varphi} \\
\frac{\sin\theta}{2}e^{i\varphi} & \cos\theta & -\frac{\sin\theta}{\sqrt{2}} \\
\frac{1-\cos\theta}{2}e^{2i\varphi} & \frac{\sin\theta}{\sqrt{2}}e^{i\varphi} & \frac{1+\cos\theta}{2} \\
\end{array}\right)
\end{equation}
The 3-d braiding relation is given by
\begin{equation}
\end{equation}
\begin{equation}
\cos\varphi=\frac{\cos\theta}{1-\cos\theta}
\end{equation}
\begin{equation}\label{Eq:6.10}
A=D^{1}(\theta,\varphi=0),\quad B=D^{1}(\theta,\varphi)
\end{equation}
For the type-I, i.e. for $\varphi=2\pi/3$, $\theta=\pm\pi$ in (<ref>), after the unitary transformation, (<ref>) becomes
\begin{equation}\label{Eq:6.11} \fl
\eqalign{
1 & 0 & 0 \\
0 & -1 & 0 \\
0 & 0 & 1 \\
\end{array}\right),\quad
\frac{1}{2} & \frac{1}{\sqrt{2}} & \frac{1}{2} \\
\frac{i}{\sqrt{2}} & 0 & -\frac{i}{\sqrt{2}} \\
-\frac{1}{2} & \frac{1}{\sqrt{2}} & -\frac{1}{2} \\
\end{array}\right) \\
1 & i\sqrt{6} & -3 \\
-i\sqrt{6} & 2 & -i\sqrt{6} \\
-3 & i\sqrt{6} & 1
\end{array}\right)
\end{equation}
On the other hand, for $S^+=S$ on substituting
\begin{equation}
\lambda_1=q,\quad \lambda_2=-q^{-1}, \quad \lambda_3=q^s\quad (q\rightarrow1)
\end{equation}
into (<ref>) and (<ref>) the ($3\times 3$) matrices $A$ and $B$ given by the topological basis and on account of
\begin{equation}\fl
d=1-\left.\frac{q^s-q^{-s}}{q-q^{-1}}\right|_{q\rightarrow1}=1-s,\quad f_1=[2S(S-3)]^{\frac{1}{2}},\quad f_2=[2S(S-1)]^{\frac{1}{2}}
\end{equation}
we obtain for $s=-3$
\begin{equation} \fl
q & & \\
& -q^{-1} & \\
& & q^s\\
\end{array}\right)_{q\rightarrow1},\quad
1 & -\sqrt{6} & 3\\
-\sqrt{6} & 2 & \sqrt{6} \\
3 & \sqrt{6} & 1 \\
\end{array}\right)
\end{equation}
that under similar transformation becomes
\begin{equation}\label{Eq:6.15} \fl
\quad T^{\dag}B_{\rm{I}}'T=-\frac{1}{4}\left(\begin{array}{ccc}
1 & i\sqrt{6} & -3\\
-i\sqrt{6} & 2 & -i\sqrt{6} \\
-3 & i\sqrt{6} & 1\\
\end{array}\right),
\quad T^{\dag}=\left(\begin{array}{ccc}
1 & 0 & 0\\
0 & i & 0\\
0 & 0 & -1\\
\end{array}\right)
\end{equation}
(<ref>) is identified with (<ref>).
Namely, as was pointed out for $S^{\dag}=S$ that the type-I ($3\times 3$) braiding matrices based on the topological basis are the same as those given by $D^{1}(\theta,\varphi)$ with $\varphi=\frac{2\pi}{3}, \theta=\pm\pi$.
For the type-II, i.e. $\varphi=\pm\pi/2$, $\theta=\pm\pi/2$ with the same transformation $V$ as given by (<ref>) the $D^{1}$-function gives
\begin{equation}\label{Eq6.16}
-i & 0 & 0\\
0 & 1 & 0\\
0 & 0 & i\\
\end{array}\right),
\quad B=\frac{1}{2}\left(\begin{array}{ccc}
1 & -\sqrt{2} & 1\\
\sqrt{2} & 0 & -\sqrt{2} \\
1 & \sqrt{2} & 1\\
\end{array}\right)
\end{equation}
On the other hand, for $S^{\dag}=S^{-1}$ the ($3\times 3$) braiding matrices (<ref>) and (<ref>) based on the topological basis and on account of
\begin{equation}
\lambda_1=e^{i\pi/4},\quad \lambda_2=-\lambda_1^{-1}=-e^{-i\pi/4},\quad \lambda_3=\sigma=-e^{i\pi/4}
\end{equation}
are equal to $(f_1=\frac{1}{\sqrt{2}}, f_2=\frac{1}{2})$
\begin{equation}
-i & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & i \\
\end{array}\right),
-\frac{1}{2} & \frac{i}{\sqrt{2}} & \frac{1}{2} \\
\frac{i}{\sqrt{2}} & 0 & \frac{i}{\sqrt{2}} \\
\frac{1}{2} & \frac{i}{\sqrt{2}} & -\frac{1}{2} \\
\end{array}\right)
\end{equation}
that after the transformation becomes
\begin{equation}\label{Eq6.18}
-i & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & i \\
\end{array}\right), \quad
\frac{1}{2} & -\frac{1}{\sqrt{2}} & \frac{1}{2} \\
\frac{1}{\sqrt{2}} & 0 & -\frac{1}{\sqrt{2}} \\
\frac{1}{2} & \frac{1}{\sqrt{2}} & \frac{1}{2} \\
\end{array}\right)
\end{equation}
(<ref>) are the same as those given by $D^{1}$-function, subjecting to the unitary transformation V for the type-II as shown in (<ref>).
In short conclusion it turns out that the Yang-Baxterization for ($N\times N$) YBE takes the different way from the ($N^2\times N^2$) representations. The resultant solutions of YBE are simply the Wigner's $D^{J}$-functions. When $\theta_1=\theta_2=\theta_3=\theta$ it reduces to the braiding matrix. Two examples for spin-$\frac{1}{2}$ and spin-$1$ have been checked in Ref.[1] and in this section, respectively. However, the explicit correspondences between $(N\times N)$ and ($N^2\times N^2$) braiding matrices for any $J$ for type-II are still a challenge problem.
It is emphasized that the Yang-Baxterization for ($N\times N$) solutions of YBE through the D-function is a quite new parametrization way based on the topological basis in different from Refs.[14].
§ RELATIONSHIP BETWEEN EXTREMES OF $L_{1}$-NORM OF D-FUNCTION AND VON NEUMANN ENTROPY
As was pointed out in Ref[1] that the extreme of $D^{j}$-functions may take multiple values for the different $j$ with $2j+1$ components. However, for any half-integer $j$ (also for $j$ integer, but $J_z=0$ should be excluded, obviously) there exist the common maximum $\theta=\pm\pi/2$ and minimum $\theta=\pm\pi$.
The Bell basis can be regarded as a linear combination of the natural basis $|\psi_0\rangle=(|\uparrow\uparrow\rangle, |\downarrow\uparrow\rangle, |\uparrow\downarrow\rangle, |\downarrow\downarrow\rangle)^T$, i.e. subject to the rotational transformation with $\theta=\pi/2$ [24, 25]. In general
\begin{equation}
\end{equation}
where $|\Psi_0\rangle$ is an $(2j+1)$ vector serving as natural basis. For $j=1/2$, $|\Psi_0\rangle=|\Psi\rangle$ and $|\Psi(\pm\pi/2, \pm\pi/2)\rangle$ stands for the Bell basis [24, 25] which possesses the maximum of entanglement. For any $\theta$ other than $\pm\pi/2$ (the maximum of $L_1$-norm of $D^{1/2}(\theta)$), it decreases the entanglement. We naturally think the extreme of D-function should indicate the entangling degree. One of the descriptions of entangling degree is von Neumann entropy. We should show that the extreme points of the entropy and $L_1$-norm of D-function shares the common $\theta$-values.
The von Neumann entropy is defined by
\begin{equation}
S(\rho)=-Tr(\rho \textrm{log}_2\rho)
\end{equation}
where $\rho$ is the reduced density matrix of a quantum state. Following the YBE in $(2J+1)$ dimensions the Schmidt decomposition of a entangled state in $(2J+1)$ dimensions, (i.e. acting on topological basis) $|\Psi(\theta,\varphi)\rangle$ can be written in the form for a fixed m:
\begin{equation}
\end{equation}
where m and m' take over $N=2j+1$, and $|a\rangle$, $|b\rangle$ are "natural states".
Since the reduced density operator of the subsystem a is
\begin{equation}
\rho^{m}_{a}=\sum_{m'}|D^j_{m'm}(\theta,\varphi)|^2|a_{m'}\rangle \langle a_{m'}|
=\sum_{m'}|d^j_{m'm}(\theta)|^2|a_{m'}\rangle \langle a_{m'}|
\end{equation}
and for a fixed m
\begin{equation}\label{Eq6.19}
S(\rho_a)\leq2 \textrm{log}_2(\sum_{m'}|d^j_{m'm}(\theta)|)
\end{equation}
the extremes of $S(\rho_a)$ are shown in the following examples in comparison to the $L_1$-norm, i.e. $f_m=\sum_{m'}|d^j_{m'm}(\theta)|$.
When $\theta=\pm\pi$, the bipartite state is a direct product sate which is separable, therefore $S(\rho_a)$ and $f$ arrive at their minimum value simultaneously and the equality of (<ref>) holds. When $\theta=\pm\pi/2$, $|d^{1/2}_{-1/2\,1/2}|=|d^{1/2}_{1/2\,1/2}|=1/\sqrt{2}$, the bipartite state reaches the maximum of entanglement. We have $S(\rho_a)=2\log_2f=\log_22=1$ shown in Fig.1 and Fig.2.
Example 2: $j=1,\ m=\pm1$
When $\theta=\pm\pi$, the state is a separable, $S(\rho_a)$ and $f$ arrive at their minimum value simultaneously and the equality of (<ref>) holds. When $\theta=\pm\pi/2$, $S(\rho_a)$ and $f$ reach their maximum value simultaneously. Of course, the maximum value of $\sum_m'|d^j_{m'm}|$ is not the same as of $S(\rho_a)$, since $S(\rho_a)<2\log_2f<\log_23$, but both of them occur at $\theta=\pm\pi/2$, see Fig.3 and Fig.4.
The explicit forms of the common minimum $\theta=\pm\pi$ and maximum $\theta=\pm\pi/2$ for both $L_1$-norm of $|d^j_{m'm}|$ and $S(\varrho_a)$ for $j=1, 3/2$ can be seen in Appendix E.
Next, the Fig.$5$ shows the derivatives of $S(\rho_a)$ and $f$ with respect to $\theta$. The zero points correspond to the extreme points of $S(\rho_a)$ and $f$. Except the two common zero points $\theta=\pi/2$ and $\pi$, other four zero points of $S(\rho_a)$ do not coincide with the zero points of $f$.
In general, we can prove that $S(\rho)$ and $f$ always share the same common extreme points $\theta=\pi/2$ and $\pi$ in the period $[0,\pi]$ for arbitrary $j$ and $m$ (m=0 is excluded). Firstly, let's consider the case of $\theta=\pi$. Because $d^j_{m'm}(\pi)=(-1)^{j+m'}\delta_{m'\,-m}$,
we have $d^j_{-m\,m}(\pi)=(-1)^{j-m},\ d^j_{m'm}=0\ (m'\neq-m)$ for arbitrary $m$, and it exactly denotes a separable state. Therefore $S(\rho_a)$ and $f$ both take the minimum value at $\theta=\pi$, and
\begin{equation}
\left.S(\rho_a)\right|_{\theta=\pi}=2\log_2f(\pi)=0.
\end{equation}
When $\theta=\pi/2$, we have [21]
\begin{equation}\label{eq2}
\end{equation}
\begin{equation}
\left.\frac{d}{d\theta}d_{m'm}^{j}\right|_{\theta=\frac{\pi}{2}}=\mp(-1)^{2m'}\left.\frac{d}{d\theta}d_{-m'm}^{j\ \prime}\right|_{\theta=\frac{\pi}{2}}.
\end{equation}
In Ref.[1], it had proved that the $L_1$-norm of D-functions reaches the extreme value at $\theta=\pi/2$. We just need to prove $S(\rho_a)$ also to have extreme at $\theta=\pi/2$. Considering
\begin{equation}
\eqalign{
\frac{d}{d\theta}S(\rho_a)|_{\theta=\frac{\pi}{2}}&= -2\sum_{m'=-j}^{j}d_{m'm}^{j}\frac{d}{d\theta}d_{m'm}^{j}
\textrm{log}_2\left(|d_{m'm}^{j}|^2\right) \\
&=\delta_{0,(2j~\textrm{mod}~2)} d_{0m}^{j} \frac{d}{d\theta}\textrm{log}_2\left(|d_{m'm}^{j}|^2\right)}
\end{equation}
hence when $2j$ is odd, $\delta_{0,(2j\;\mathrm{mod}\;2)}=0$, thus $\left.\frac{d}{d\theta}S(\rho_a)\right|_{\theta=\frac{\pi}{2}}=0$. When $2j$ is even, according to Eq. (<ref>), we have
\[
\]
i.e. $d^j_{0m}\frac{d}{d\theta}d^j_{0m}=0$, therefore $\left.\frac{d}{d\theta}S(\rho_a)\right|_{\theta=\frac{\pi}{2}}=0$. To sum up, $\theta=\pi/2$ is the common extreme points of $S(\rho_a)$ and $f$.
§ CONCLUSION REMARKS
Similar to the standard strategy of the construction of the 2-D topological basis for TQFT associated with Temperley-Lieb algebra [1, 2, 3, 8, 9], the extension has been made to construct the 3-D basis for BWM algebra.
The point is to introduce the basis (<ref>) and (<ref>), then all of the 3-D representations of $S_{12}$, $S_{23}$, $E_{12}$ and $E_{23}$ can be shown in terms of the graphic technique [17] that yield the $3$-d representations of the corresponding braiding $A, B$ and $E$-operations.
For both $S^{\dag}=S$ and $S^{\dag}=S^{-1}$ we have given the explicit matrix forms of them in the Sec. <ref> and Sec. <ref>. In comparison with the case of TLA now the $E$-involved relations appear. Matching for the $S_{12}$, $S_{23}$, $E_{12}$ and $E_{23}$ in $9\times9$ matrix form the corresponding $A$, $B$, $E_A$ and $E_B$ have been found in $3\times3$ matrix forms that are nothing but the natural extension of $2\times2$ matrix forms for TLA.
The physical meaning of the TQFT associated with TLA has been well established [8, 9, 10]. However, the physical meaning of the counterpart for BWMA deserves to be explored in the future. In ref [16] a physical model was proposed, but here we present a different approach. As for the interaction model arisen from BWMA it deserves more discussions. The connection of (<ref>) with BWMA has been shown for spin-$1$. How to solve the model for any N based on the 3-D representation in terms of the topological basis is an open problem.
More interesting point should be emphasized. We have shown that the $L_1$-norm of $D^j$-function and von Neumann entropy share the common maximum and minimum points of $\theta$, i.e $\pm\pi/2$ and $\pm\pi$, respectively, as least for any half-integer $j$. As was proved that $D^j(\theta,\varphi)$ are $(2j+1)-d$ solutions of YBE. The connection of the extreme points for both the $L_1$-norm of $D^j$ and von Neumann entropy explores the deep meaning of $L_1$-norm in Quantum Mechanics. It also happened in other model [26]. It describes quantum entanglement in the quantum information. The reason is that braiding operation is very natural to describe entanglement for two particles. Because the entanglement only exists in the interaction between "particles" A and B themselves. An intuitive explanation of $(N\times N)$-dimensional braiding can easily be made in terms of $D(\theta,\varphi)$-function. Suppose two parallel lines along $Z$ axis (time pointed) represent A and B at the site $1$ and $2$, and form $X-Z$ plane. In the spherical coordinates an entanglement between A and B occurs by over crossing the two lines with angle $\theta$. The (<ref>) is to fix $\varphi$ for a given $\theta$. Different $\theta$ also gives rise the rotation of $X-Z$ plane about $Z$ axis by $\varphi$. The $\theta=\pi/2$ means that the two lines are perpendicular to each other locally at different $\varphi$. Whereas $\theta=\pi$ means parallel to each other. The former corresponds with the maximum entangled state and the later with decomposable one, i.e. disentangled. Since YBE is the factorization condition of multi-body S-matrix to 2-body scatterings. We look over the facts such as factorization of S-matrix, topological basis, D-function, $L_1$-norm and von Neumann entropy. Now all of them have been connected within the frame work of YBE for quantum information, especially for the entanglement.
We appreciate the interesting discussion with Prof. Z. H. Wang and Prof. J. Birman, and X. B. Peng. This work is in part supported by NSF of China with the Grant No.11075077 and 11275024. Additional support was provided by the Ministry of Science and Technology of China, No.2011AA120101.
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For the collection of articles, see M.Jimbo(ed), Yang-Baxter Eq. in Integrable Systems, World Scientific, Singapore, 1990. Also, C.N.Yang and M.L.Ge(eds), Braid Group, Knot Theory and Statistical Mechanics, World Scientific Pub. Singapore, 1990. J. Baxter, Exactly Solved Models in Statistical Mechanics, Dover Pub., Inc Mineola, New York, 1982.
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Z. Wang, Nankai Lectures on TQFT, June 5-7, 2006.
M.H.Freedman, M. Larsen, Z. Wang, Commu. Math. Phys, 227, 605(2002). S. Das Sarma, M. Freedman, C. Nayak, Phys.Rev.Lett.94,166802(2005)
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Y. Zhang, L. Kauffman, and M. Ge, Quantum Information Processing 4, 159 (2005).
J. L. Chen, K. Xue, andM. L. Ge, Phys. Rev. A 76, 042324 (Oct
S.W. Hu, K. Xue, andM. L. Ge, Phys. Rev. A 78, 022319
J. L. Chen, K. Xue, and M. L. Ge, Annals of Physics 323, 2614
[10]
J.Preskill Lecture Notes for Physics 219: Quantum Computation, 2004 http://www.theory.caltech.edu/preskill/ph229/.
C.Nayak, F. Wilczek, Nucl.Phys.B 479, 529(1996);
C. Nayak et al, Rev. Mod. Phys. 80, 1083(2008); J. K. Slingerland,
F. A. Bais, Nucl. Phys. B612[FS]229-290, 2001;
E. H.Rezayi, N.Read, Nucl.Phys.B56,16864(1996)
[11]
J. Birman and H. Wenzl, Trans. A.M.S. 313, 249 (1989).
[12] J. Murakami, Osaka J. Math. 24, 745 (1987).
[13] H. Wenzl, Ann. Math. 128, 179(1988).
[14] M. Jimbo, Commun. Math. Phys. 102, 537(1986); M. Jimbo, Lett. Math. Phys. 10, 63-69 (1985).
[15] Y. Cheng, M. L. Ge, K. Xue, Commun. Math. Phys. 136, 195(1991). M. L. Ge and A.C.T. Wu, J. Phys. A24,L725(1991)
[16] P. Fendley and E. Fradkin, Phys. Rev. B 72, 024412 (2005).
[17]
L. Kauffman, Knots in physics, World Scientific Pub. Singapore, 1991.
[18] F. D. M. Haldane, Phys. Rev. Lett. 50, 1153(1983). I. J. Affleck, Phys. Condens. Matter, 1, 3049(1989).
[19] V.F.R. Jones, Commun. Math. Phys. 125, 459, 1991
[20] M. L. Ge, Y. S. Wu, K. Xue, Inter. J. Mod. Phys. 6A, 3735(1991).
[21] D. A. Varshalovich, A. N. Moskalev, V. K. Khersonskii, Quantum Theory of Angular Momentum, World Scientific Pub. Singapore, 1988, (ref. p 53); A.M. Pelelomov, Sov. Phys. Usp. 20,703 (1997).
[22] A. Benvegnu, M. Spera, Rev. Math. Phys. 18, 1075 (2006).
[23] S.W. Hu, K. Xue and M.-L. Ge, Phys. Rev. A76, 042324, (2007).
[24] L. Kauffman and S. Lomonaco, Jr, New Journal of Physics 6, 134 (2004).
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[26] E. Baake, M. Baak and H. Wager, Phys. Rev. E57, 1191-1192 (1998).
§ GRAPHIC EXPLANATION OF BWA
For the self-contain we list the graphic expressions for the later use. The BWA reads
\begin{equation}
S=\cross,\quad S^{-1}=\cros,\quad E=\Un \quad (\mathrm{operators})
\end{equation}
\begin{equation}
\end{equation}
\begin{equation}
E_{i}E_{i\pm1}E_{i}=E_{i}, \quad E_{i}^{2}=dE_{i}
\end{equation}
\begin{equation}
E_{i}S_{i}=S_{i}E_{i}=\sigma E_{i}
\end{equation}
\begin{equation}
S_i\equiv S_{i,i+1},\quad E_i\equiv E_{i,i+1}
\end{equation}
\begin{equation}
S_{12}|e_{\mu}\rangle=\lambda_{\mu}|e_{\mu}\rangle \quad(\mu=1,2,3)
\end{equation}
The (<ref>) reads graphically as
\begin{equation}
\SSSSijk=\sssijk
\end{equation}
and the $E$ takes the simple graph and satisfies T-L algebra
\begin{equation}
E_{i}=\Unii,\quad \complexijk=d\,\Unll,\quad d=\myloop,\quad (\mathrm{T-L})
\end{equation}
All the other relations can then be expressed in terms of the graphs. Say,
\begin{equation}
E_{i}E_{i+1}E_{i}=E_{i}, \quad E_{i}^{2}=dE_{i}
\end{equation}
\begin{equation}
S_iE_i=E_iS_i=\sigma E_i
\end{equation}
\begin{equation}
\SicrossEi=\EiSicross=\sigma\Un
\end{equation}
\begin{equation}
\Ncross=\sigma\Ncup,\qquad \Ucros=\sigma\Ucup,\qquad \Ncros=\sigma^{-1}\Ncup
\end{equation}
\begin{equation}
\end{equation}
\begin{equation}
\EiSiSj=\SiSjEj=\EiEj
\end{equation}
\begin{equation}
\end{equation}
\begin{equation}
\EiSjEj=\SihSjhEj\Rightarrow\UcrossNn=\tcrossH
\end{equation}
\begin{equation}
\end{equation}
\begin{equation}
\EiSjEjb=\SihEjb\Rightarrow\UcrossNn=\lUn
\end{equation}
\begin{equation}
\EiEjSj=\SihEj
\end{equation}
\begin{equation}
\end{equation}
The dependent relations $E_iS_{i\pm1}E_i=\sigma^{-1}E_iE_{i\pm1}E_i=\sigma^{-1}E_i$ can also easily be expressed in terms of the similar graphs.
It can be checked that all of $E_{A}$,$E_{B}$,$A$ and $B$ obey BWM algebra in 3-D representation:
\begin{equation}
\end{equation}
\begin{equation}
\begin{array}{rcl}
\end{array}
\end{equation}
\begin{equation}
E_{A}^2=dE_{A},\quad E_{B}^2=dE_{B}
\end{equation}
\begin{equation}
\begin{array}{rcl}
\end{array}
\end{equation}
\begin{equation}
\begin{array}{ccccc}
AE_{A}&=&E_{A}A&=&\sigma E_{A}\\
BE_{B}&=&E_{B}B&=&\sigma E_{B}\\
\end{array}
\end{equation}
and other dependent relations:
\begin{equation}
\eqalign{
\end{equation}
\begin{equation}
\eqalign{
AE_{B}A=B^{-1}E_{A}B^{-1},\quad BE_{A}B=A^{-1}E_{B}A^{-1}\\
AE_{B}E_{A}=B^{-1}E_{A},\quad BE_{A}E_{B}=A^{-1}E_{B}\\
E_{A}E_{B}A=E_{A}B^{-1},\quad E_{B}E_{A}B=E_{B}A^{-1}\\
\end{equation}
§ PROOF OF (<REF>)
The basis $|e_{\mu}\rangle$ is defined by (<ref>), so
\begin{equation}
S_{12}E_{12}|e_3\rangle=\sigma E_{12}|e_3\rangle=d \cdot S_{12}|e_3\rangle \nonumber
\end{equation}
i. e.
\begin{equation}
S_{12}|e_3\rangle=\sigma|e_3\rangle\quad (\sigma=\lambda_3)
\end{equation}
For $i=1,2$ (no sum over repeating indices)
\begin{equation} \fl
\eqalign{
S_{12}|e_i\rangle&=f_i\left\{\Uu+\alpha_i [\crosUu+w\Uu-w\tsep+\sigma\beta_i\tsep]\right\} \nonumber\\
\end{equation}
In order that $|e_i\rangle$ are eigenstates for $S_{12}$ it should hold
\begin{eqnarray}
\alpha_{i}^{-1}(1+\alpha_{i}w)=\alpha_i \nonumber\\
\alpha_{i}^{-1}(\sigma\beta_i-\alpha_i w)=\beta_i. \nonumber
\end{eqnarray}
We then have
\begin{equation}
\alpha_1+\beta_1 d=d_2 +\beta_2 d= -\sigma^{-1} \nonumber
\end{equation}
i. e.
\begin{equation}
\beta_i =-d^{-1}(\sigma^{-1}+\lambda_i) \label{Eq.B2}
\end{equation}
Because of $\Ncross=\sigma\Ncup$ ($\sigma=\lambda_3$), $\Ncros=\sigma^{-1}\Ncup$, it holds ($i=1,2$)
\begin{eqnarray}
\langle e_3|e_i\rangle&=&d^{-1}f_i\left(\CcrossC+\alpha_i \myloop +\beta_i\tsquare\right) \qquad (\myloop=d) \nonumber \\
&=&d^{-1}f_i\left(\sigma^{-1}d+\alpha_i d+\beta_i d^2\right)=0 \nonumber
\end{eqnarray}
where $d=1+\frac{\sigma^{-1}-\sigma}{w}$, $\alpha_i=\lambda_i$ and (<ref>) have been used. Substituting (<ref>) into (<ref>) the basis $|e_{\mu}\rangle$ take the forms
\begin{eqnarray}
|e_i\rangle&=&f_i\left(\Utcross+\lambda_i \Uu+d^{-1}(\sigma^{-1}+\lambda_i)\tsep\right)
\end{eqnarray}
The relations for BWA in the Appendix A have been used.
To verify $\langle e_i|e_j\rangle=\delta_{ij}$ we have to distinguish two types of the braiding matrices from each other for $S^{\dag}=S$ (Hermitian) and $S^{\dag}=S^{-1}$ (unitary), respectively.
(a) For $S^{\dag}=S$ there are
\begin{eqnarray}
\langle\tcross|\Utcross\rangle=d[(\sigma^{-1}-\sigma)W+d]
\end{eqnarray}
\begin{eqnarray}
\langle\tcross|\Uu\rangle=\sigma d \qquad\
\end{eqnarray}
After calculation we find
\begin{eqnarray}
\langle e_1|e_2\rangle&=&d[(\sigma^{-1}-\sigma)W+d]+W\sigma d+(\beta_1 +\beta_2)\sigma^{-1}d\\
&&+(\lambda_1 \beta_2-\beta_1 \lambda_2)d+(\lambda_1 \lambda_2+\beta_1 \beta_2)d^2\\
&=&d\{\sigma^{-1}(W+\beta_1)+\lambda_2 \beta_1\}=0
\end{eqnarray}
The $\langle e_1|e_1\rangle=\langle e_2|e_2\rangle=1$ leads to (<ref>).
(b) For $S^{\dag}=S^{-1}$ we have ($i=1,2$)
\begin{equation}
\eqalign{
\langle e_i|=f_{i}^{*}\left\{\ntcros+\alpha_{i}^{*}\Nn+\beta_{i}^{*}\tncup\right\},\\
\crossUu=S_{12}\Uu, \quad \mathrm{and}\quad \langle e_1|e_2\rangle=\tloop=d^2
\end{equation}
Now $\alpha_i=\lambda_i$, $\beta_i$ are complex and $\lambda_i^{*}=\lambda_i^{-1}$ that gives
\begin{equation} \fl
\eqalign{
\langle e_i|e_j\rangle&=f_i^*f_j\left\{d^2+\alpha_j\sigma^{-1}d+\beta_j\sigma d
+\alpha_i^*\left(\sigma d+\alpha_j d^2+\beta_j d\right)
+\beta_i^*\left(\sigma^{-1}d+\alpha_j d+\beta_j d^2\right)\right\}\\
&=f_i^*f_j d\left\{d+\alpha_j \sigma^{-1}+\beta_j (\alpha_i^{*} +\sigma)+\alpha_j \alpha_i^{*} d+\alpha_i^{*}\sigma\right\}
\end{equation}
\begin{eqnarray}
\langle e_1|e_2\rangle =f_1^{*} f_2 d\left\{-\lambda_2W d+\lambda_2W+\lambda_2(\sigma^{-1}-\sigma)\right\}=0\\
f_i=\left[(d-1)(\lambda_i+\lambda_i^{-1})(\sigma+\lambda_i d+\lambda_i^{-1})\right]^{-1/2}
\end{eqnarray}
§ DERIVATION OF $3$-D MATRIX FORMS OF BRAIDING MATRICES
For $S^{\dag}=S$ ($i=1,2$)
\begin{eqnarray}
E_{12}|e_i\rangle&=&f_i \Unot \left\{\Utcross+\lambda_i \Uu+\beta_i \tsep\right\}\\
&=& d f_i (\sigma^{-1}+\beta_i d+\lambda_i) |e_3\rangle
\end{eqnarray}
\begin{equation}
\langle e_3|E_{12}|e_3\rangle=d
\end{equation}
\begin{equation} \fl
E_{23}|e_i\rangle=f_i \left\{\sigma+\lambda_i d-d^{-1}(\sigma^{-1}+\lambda_i)\right\}(\lambda_1-\lambda_2)^{-1}\left\{f_1^{-1}|e_1\rangle-f_2^{-1}|e_2\rangle+|e_3\rangle\right\}
\end{equation}
\begin{equation}
E_{23}|e_3\rangle=d^{-1} \left\{(\lambda_1 -\lambda_2^{-1})^{-1}(f_1^{-1}|e_1\rangle-f_2^{-1}|e_2\rangle)+|e_3\rangle\right\}
\end{equation}
\begin{equation}
\eqalign{
\end{equation}
\begin{equation}
S_{23}|e_3\rangle=d^{-1}\left\{(\lambda_2-\lambda_1)^{-1}[\lambda_2 f_1^{-1}|e_1\rangle-\lambda_1 f_2^{-1}|e_2\rangle]+\lambda_3^{-1}|e_3\rangle\right\}
\end{equation}
Obviously the base $|e_1\rangle$, $|e_2\rangle$ and $|e_3\rangle$ form a closed set for the operations $S$ and $E$. For $S^{\dag}=S^{-1}$ ($i=1,2$)
\begin{equation}\fl
E_{23}|e_i\rangle=f_i(\sigma+\beta_i+\lambda_i d)(\lambda_1+\lambda_1^{-1})^{-1}
\left\{f_1^{-1}|e_1\rangle-f_2^{-1}|e_2\rangle+(\lambda_1+\lambda_1^{-1})|e_3\rangle\right\}
\end{equation}
\begin{equation} \fl
\left\{f_1^{-1}|e_1\rangle-f_2^{-1}|e_2\rangle+(\lambda_1+\lambda_1^{-1})|e_3\rangle\right\}\\
\end{equation}
§ ACTING BRAIDING OPERATIONS ON THE TOPOLOGICAL BASIS
From (<ref>)-(<ref>) it follows that $d=3$ and for $m=-3$ as well as $q=1$ (for $S=S^\dagger$) we get:
\begin{eqnarray}
f_{1}=\frac{1}{2\sqrt{3}}, f_{2}=\frac{1}{2\sqrt{3}}, \alpha_{1}=1, \alpha_{2}=-1, \beta_{2}=0,
\end{eqnarray}
\begin{eqnarray}
\beta_{1}=\frac{q-q^{-1}}{q^{-3}-q}|_{q\rightarrow1}=-\frac{2}{3}
\end{eqnarray}
that leads to:
\begin{equation} \fl
\end{equation}
\begin{equation} \fl
\end{equation}
\begin{eqnarray}
\end{eqnarray}
\begin{eqnarray}
\end{eqnarray}
\begin{eqnarray}
\end{eqnarray}
\begin{eqnarray}
\end{eqnarray}
\begin{eqnarray}
\end{eqnarray}
\begin{eqnarray}
\end{eqnarray}
§ MORE EXAMPLES FOR THE COINCIDENCE OF EXTREMES OF $D^J$ AND VON NEUMANN ENTROPY
Example 3: $j=1,\ m=0$
When $\theta=\pm\pi$, the state is separable, and $S(\rho_a)=2\log_2f$ arrive at their minimum value simultaneously. The point $\theta=\pm\pi/2$ is both the local minimum point of $S(\rho_a)$ and $f$, however, $S(\rho_a)<2\log_2f$. In addition, $S(\rho_a)$ and $f$ shares another two common local maximum points in the period $(0,\pi]$, and the two common maximum points both correspond to the maximally entangled state, therefore $S(\rho_a)=2\log_2f=2\log_23$ at the two points. See Fig.E1 and Fig.E2.
Example 4: $j=3/2,\ m=\pm3/2$
When $\theta=\pm\pi$, the state is a separable sate and $S(\rho_a)=2\log_2f$ reach their minimum value simultaneously. When $\theta=\pm\pi/2$, $S(\rho_a)$ and $f$ reach their maximum values simultaneously, however both the values are not the same because $S(\rho_a)<2\log_2f<\log_24=2$. See Fig.E3 and Fig.E4.
Example 5: $j=3/2,\ m=\pm1/2$.
When $\theta=\pm\pi$, the state is separable, and $S(\rho_a)=2\log_2f$ arrives at the minimum value. The point $\theta=\pm\pi/2$ is both the local minimum point of $S(\rho_a)$ and $f$, with $S(\rho_a)<2\log_2f$. It is worth noting that $S(\rho_a)$ and $\ell_1$ norm $f$ both have other four local extreme points, in the period $(0, \pi)$. It is shown in Fig. $E5$ and Fig.$E6$.
|
arxiv-papers
| 2012-11-27T01:48:09 |
2024-09-04T02:49:38.528350
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Qing Zhao, Ruo-Yang Zhang, Kang Xue and Mo-Lin Ge",
"submitter": "Ruo-Yang Zhang",
"url": "https://arxiv.org/abs/1211.6178"
}
|
1211.6313
|
# On the relativistic heat equation in one space dimension
J.A. Carrillo , V. Caselles , S. Moll Department of Mathematics, Imperial
College London, London SW7 2AZ, UK. Email: [email protected]
de Tecnologia, Universitat Pompeu-Fabra, Barcelona, Spain. E-mail:
[email protected] d’Anàlisi Matmeàtica, Universitat de
València, Valencia, Spain. E-mail: [email protected]
###### Abstract
We study the relativistic heat equation in one space dimension. We prove a
local regularity result when the initial datum is locally Lipschitz in its
support. We propose a numerical scheme that captures the known features of the
solutions and allows for analysing further properties of their qualitative
behavior.
Key words: entropy solutions, flux limited diffusion equations, pseudo-inverse
distribution
AMS (MOS) subject classification: 35K55 (35K20 35K65)
## 1 Introduction
In this work, we explore both analytically and numerically the implications of
a new strategy to study flux-dominated nonlinear diffusions in one dimension.
To be more precise, we consider the so-called relativistic heat equation (RHE)
$u_{t}=\nu\,\left(\frac{uu_{x}}{\sqrt{u^{2}+\frac{\nu^{2}}{c^{2}}(u_{x})^{2}}}\right)_{x},\qquad
x\in I\\!\\!R,\,t>0.$ (1.1)
introduced by Rosenau in [37] and, later on, by Brenier in [14] based on
optimal transportation ideas. The name of RHE comes from the fact that (1.1)
converges as $c\to\infty$ to the heat equation both formally and rigorously
[20], while the flux in (1.1), understood as a conservation law, is bounded by
the speed of light $c$ whenever the solution is positive.
Many other models of nonlinear degenerate parabolic equations with flux
saturation as the gradient becomes unbounded have been proposed by Rosenau and
his coworkers [19, 37], and Bertsch and Dal Passo [12, 26]. Notice also [36]
for the presence of flux limited diffusion equations in the context of
radiation hydrodynamics.
The general class of flux limited diffusion equations and the properties of
the relativistic heat equation have been studied in a series of papers [5, 4,
6, 21]. An existence and uniqueness theory of entropy solutions for the Cauchy
problem associated to the quasi-linear parabolic equation
$\displaystyle\frac{\partial u}{\partial t}={\rm div}\ {\bf b}(u,Du),$ (1.2)
was developed in [5, 4]. Here, the flux function is given by ${\bf
b}(z,\xi)=\nabla_{\xi}f(z,\xi)$ and $f:I\\!\\!R\times I\\!\\!R^{N}\to
I\\!\\!R^{+}$ is a convex function with linear growth as $\|\xi\|\to\infty$,
such that $\nabla_{\xi}f(z,\xi)\in C(I\\!\\!R\times I\\!\\!R^{N})$ satisfying
other additional technical assumptions. In particular, the relativistic heat
equation (1.1) satisfies these assumptions, and other models considered in
[37]. To avoid the difficulty of the lack of a-priori estimates that ensure
the compactness in time of solutions of (1.2), the existence problem was
approached using Crandall-Liggett’s theorem [24]. For that, we first
considered the associated elliptic problem and we defined a notion of entropy
solution for which we developed a well-posedness theory. The notion of entropy
solution permits to prove a uniqueness result using Kruzhkov’s doubling
variables technique [30, 15]. This technique was suitably adapted to work with
functions whose truncatures are of bounded variation [5, 4], which is the
natural functional setting for (1.2) and its associated elliptic equation.
The evolution of the support of solutions of the relativistic heat equation
(1.1) was studied in [6]. By constructing sub- and super-solutions which are
fronts evolving at speed $c$ and using a comparison principle between entropy
solutions and sub- and super-solutions, it was proved in [6] that the support
of solutions evolves at speed $c$. Moreover, the existence of solutions which
have discontinuity fronts moving at the speed $c$ was again shown using the
comparison principle with sub-solutions. This implies, in particular, that the
maximal regularity in time that one can expect for general solutions of (1.1)
is that $u\in BV([\tau,T]\times I\\!\\!R^{N})$ for any $0<\tau<T$. That this
happens for a general class of initial conditions was proved in [8] and later
extended in [21]. This lack of regularity is at the origin of the notion of
entropy solutions for this type of equations. The only regularity result for
smooth initial conditions was proved in [20] and it guarantees that $\nabla\ln
u$ is bounded whenever initially is. But the study of the local regularity of
solutions of (1.1) is still an open question. One of the purposes of this
paper is to address this problem for the Cauchy problem associated to (1.1) in
one space dimension with compactly supported bounded probability densities as
initial data.
Assuming that the initial data in non-negative, we can easily change variables
to observe that $\tilde{u}(t,x)$ is a solution of (1.1) if and only if
$u(t,x)=\tilde{u}(\frac{\nu}{c^{2}}t,\frac{\nu}{c}x)$ is a solution of
$u_{t}=\,\left(\frac{uu_{x}}{\sqrt{u^{2}+(u_{x})^{2}}}\right)_{x}.$ (1.3)
Thus, without loss of generality we may assume that $\nu=c=1$, and for
simplicity we shall assume it in the rest unless explicitly stated. Notice
also that if $u(t)$ is a solution corresponding to $u_{0}$, then $\lambda
u(t)$ is a solution corresponding to $\lambda u_{0}$, $\lambda>0$. Thus,
without loss of generality we assume that $\|u_{0}\|_{1}=\|u(t)\|_{1}=1$ for
any $t>0$, and reduce our evolution to probability densities. In this
paragraph, the term solution refers to entropy solution for which the well-
posedness theory was developed and for which a summary of its concept is
reminded to the reader in the Appendix.
The local regularity of entropy solutions to (1.3) will be done by a change
variables, writing (1.3) in terms of its inverse distribution function. This
change of variables has its origin in using mass transport techniques to study
diffusion equations [18, 13]. It is known [14] that equation (1.1) has the
structure of a gradient flow of a certain functional (the physical entropy)
with respect to some transport distance. This structure was already used to
give well-posedness results to (1.1) in [35]. Nonlinear diffusions have
received lots of attention from optimal transport theory viewpoint starting
from the seminal works [29, 33].
Transport distances between probability measures in one dimension are much
easier to compute since they can be written in terms of distribution functions
and their generalized inverses (pseudo-inverse), the so-called Hoeffding-
Fréchet Lemma [39, Section 2.2]. This result led to the following change of
variables based on the distribution function $F$ associated to the probability
measure $u$, defined as
$F(t,x)=\int_{-\infty}^{x}u(t,y)\,dy\,.$
We formally consider its inverse $\varphi$ defined on the mass variable
$\eta\in(0,1)$ that verifies
$F(t,\varphi(t,\eta))=\eta,\qquad\eta\in(0,1).$
After straightforward computations assuming that all involved functions are
well-defined and smooth, one obtains the equation
$\varphi_{t}=\frac{\varphi_{\eta\eta}}{\sqrt{(\varphi_{\eta})^{4}+(\varphi_{\eta\eta})^{2}}}$
(1.4)
for the inverse distribution function $\varphi$. This change of variables has
first been used for nonlinear diffusions in [18] to show contractivity
properties of transport distances for porous-medium like equations. It is
worthy to remark that an implicit Euler discretization of (1.4) is equivalent
to the variational JKO scheme whose convergence is proved in [35] for (1.1)
under certain assumptions. Numerical schemes to solve the equation for the
pseudo-inverse function in the case of the porous medium equation were
analysed in [28]. This Lagrangian approach was generalized to several
dimensions in [16] in order to propose numerical schemes for equations with
gradient flow structure in optimal transport theory and general quasilinear
problems in divergence form.
In Section 2, we will first take advantage of this change of variables to
prove the following regularity result:
###### Theorem 1.1.
Let $u_{0}\in L^{\infty}(I\\!\\!R)$ with $u_{0}(x)\geq\kappa>0$ for
$x\in[a,b]$, and $u_{0}(x)=0$ for $x\not\in[a,b]$. Assume that $u_{0}\in
W^{1,\infty}([a,b])$. Let $u(t,x)$ be the entropy solution of (1.1) with
$u(0)=u_{0}$, $\|u_{0}\|_{1}=1$ . Then $u\in C([0,T],L^{1}(I\\!\\!R^{N}))$
satisfies:
* (i)
$u(t,x)\geq\kappa(t)>0$ for any $x\in(a-ct,b+ct)$ and any $t>0$, $u(t,x)=0$,
$x\not\in[a-ct,b+ct]$, $t\in(0,T)$,
* (ii)
$u(t)\in BV(I\\!\\!R)$, $u(t)\in W^{1,1}(a-ct,b+ct)$ for almost any
$t\in(0,T)$, and $u(t)$ is smooth inside its support,
* (iii)
if $u_{0}\in W^{2,1}(a,b)$, then $u_{t}$ is a Radon measure in $(0,T)\times
I\\!\\!R$.
We emphasize that the new parts of this result with respect to the literature
discussed above refer to the regularity stated on points (ii) and (iii). This
result implies that sharp corners on the support of the initial data are
immediately smoothed out by the evolution of the RHE. This result will be
extended in Section 3, in particular, we cover the case where the initial
condition $u_{0}$ vanishes at the boundary of its support.
In Section 4, we will propose an adaption of the numerical scheme in [16]
based on equation (1.4) with suitable boundary conditions that fully captures
the demonstrated behavior of the solutions of the RHE. Moreover, we will show
different numerical tests in situations where the theory has not been
developed yet. For instance, we numerically study the conditions for the
formation or not of discontinuities on the bulk of the solutions for RHE and
its porous medium counterparts
$u_{t}=\,\left(\frac{u^{m}u_{x}}{\sqrt{u^{2}+(u_{x})^{2}}}\right)_{x}$
with $m>1$ and their long-time asymptotic behaviour. Finally, we include in
Appendix A some basic material to describe the notion of entropy solutions for
(1.3) for the sake of completeness.
## 2 Regularity of Solutions
As proved in [5], there exists a unique entropy solution of the Cauchy problem
for (1.3) for any $u_{0}\in L^{1}(I\\!\\!R)\cap L^{\infty}(I\\!\\!R)$,
$u_{0}\geq 0$, see Appendix for the full notion of solution. Moreover if
$u_{0}$ has compact support in $I\\!\\!R$ and is locally bounded away from
zero in any interior point of its support, then
$\mathrm{supp}(u(t))=\mathrm{supp}(u_{0})\oplus B(0,t)$ [6]. The rest of this
Section is devoted to the proof of the regularity statements (ii) and (iii).
Let us recall that the entropy condition on the jump set of $u$ can be
expressed by saying that the profile of $u$ is vertical at those points. Since
the support of $u(t)$ is $(a-t,b+t)$, and $u(t)\geq\kappa(t)>0$ in $(a-t,b+t)$
for any $t>0$ [6], there is a jump at the points $x=a-t,b+t$ and we have [21]
$\frac{u_{x}}{\sqrt{u^{2}+(u_{x})^{2}}}(t,a-t)=1,\qquad\frac{u_{x}}{\sqrt{u^{2}+(u_{x})^{2}}}(t,b+t)=-1.$
(2.1)
Let us consider the change of variables discussed in the introduction and
define the function $\varphi(t,\eta)$ by the relation
$\int_{a-t}^{\varphi(t,\eta)}u(t,x)dx=\eta,\qquad\eta\in(0,1).$ (2.2)
Proceeding formally, assuming that the function is smooth inside its support
and differentiating with respect to $\eta$ we obtain
$u(t,x)\varphi_{\eta}=1,\qquad\hbox{\rm for $x=\varphi(t,\eta)$.}$
Differentiating with respect to $t$ we have
$u(t,x)\varphi_{t}+u(t,a-t)+\int_{a-t}^{\varphi(t,\eta)}u_{t}(t,x)dx=0.$
Taking into account the boundary conditions (2.1) [21], one has
$\displaystyle\int_{a-t}^{\varphi(t,\eta)}u_{t}(t,r)dr$ $\displaystyle=$
$\displaystyle\int_{a-t}^{\varphi(t,\eta)}\,\left(\frac{uu_{x}}{\sqrt{u^{2}+(u_{x})^{2}}}\right)_{x}dx=\frac{uu_{x}}{\sqrt{u^{2}+(u_{x})^{2}}}(t,x)-u(t,a-t),$
hence
$u(t,x)\varphi_{t}=-\frac{uu_{x}}{\sqrt{u^{2}+(u_{x})^{2}}}(t,x)\qquad\hbox{\rm
for $x=\varphi(t,\eta)$\,.}$
Then the equation satisfied by $\varphi$ is
$\varphi_{t}=\frac{\varphi_{\eta\eta}}{\sqrt{(\varphi_{\eta})^{4}+(\varphi_{\eta\eta})^{2}}}\,.$
### 2.1 Regularity result in mass variables
Now, let us consider the change of variables $v=\varphi_{\eta}$. The equation
satisfied by $v$ is
$v_{t}=\,\left(\frac{v_{x}}{\sqrt{v^{4}+(v_{x})^{2}}}\right)_{x}\qquad
t>0,\,\,x\in(0,1).$ (2.3)
where we have written $x$ instead of $\eta$. This will done through this
subsection for convenience.
The initial condition $v_{0}$ is determined from the initial condition
$u_{0}$. We assume that $u_{0}\in L^{\infty}(I\\!\\!R)$, $u_{0}\geq\kappa$,
and $u_{0}\in W^{1,\infty}([a,b])$. Since the relation between $u_{0}$ and $v$
is determined by $v_{0}(\eta)=\frac{1}{u_{0}(x)}$, then
$\alpha_{1}:=\frac{1}{\|u_{0}\|_{\infty}}\leq
v_{0}\leq\frac{1}{\kappa}:=\alpha_{2}$. We also have $v_{0}\in
W^{1,\infty}(0,1)$. Note that
$\int_{0}^{1}v_{0}(\eta)\,d\eta=\int_{a}^{b}\,dx=b-a.$
If we denote by $\nu$ the outer unit normal to $(0,1)$, that is $\nu(0)=-1$
and $\nu(1)=1$, the natural boundary conditions for (2.3) are
$\frac{v_{x}}{\sqrt{v^{4}+(v_{x})^{2}}}\nu=1\qquad\mbox{at
}x\in\partial(0,1)\,,$ (2.4)
with $\partial(0,1)=\\{0,1\\}$. The first step toward Theorem 1.1 is to show a
regularity result for the Cauchy problem (2.3)-(2.4).
###### Theorem 2.1.
Assume that $v_{0}\in W^{1,\infty}(0,1)$, $v_{0}\geq\alpha_{1}>0$. Then there
exists a smooth solution of (2.3) in $(0,T)\times(0,1)$ with $v(0,x)=v_{0}(x)$
and satisfying the boundary conditions (2.4) (in a weak sense).
###### Proof.
To prove this claim, we consider the following approximated Cauchy problem
$\displaystyle
v_{t}=\,\left(\frac{v_{x}}{\sqrt{v^{4}+(v_{x})^{2}}}\right)_{x}+\epsilon
v_{xx}\qquad t\in(0,T),\,\,x\in(0,1)\,,$ (2.5)
$\displaystyle\left(\frac{v_{x}}{\sqrt{v^{4}+(v_{x})^{2}}}+\epsilon
v_{x}\right)\nu=1-\epsilon^{1/3},\qquad t\in(0,T),\,\,x\in\partial(0,1),$
(2.6)
where $\epsilon>0$. The proof is divided in several Steps. In Steps 1 to 3 we
prove some formal estimates that are also useful to state the existence of
solutions of (2.5)-(2.6) in Step 4. For simplicity we write
${\bf a}(z,\xi)=\frac{\xi}{\sqrt{z^{4}+(\xi)^{2}}},\qquad z\geq 0,\,\xi\in
I\\!\\!R.$
Let us observe that
${\bf a}(z,\xi)\xi\geq|\xi|-z^{2}.$ (2.7)
Step 1. $L^{p}$ bounds on $v$ for $p\in[1,\infty)$. Let us first consider the
evolution of the $L^{1}$ norm. For that we integrate (2.5) on $(0,1)$. We have
$\frac{d}{dt}\int_{0}^{1}v(t,x)\,dx=({\bf a}(v,v_{x})+\epsilon v_{x})(1)-({\bf
a}(v,v_{x})+\epsilon v_{x})(0)=2(1-\epsilon^{1/3})\,,$
and thus,
$\int_{0}^{1}v(t,x)\,dx=\int_{0}^{1}v_{0}(x)\,dx+2(1-\epsilon^{1/3})t\,.$
(2.8)
Given $1\leq p<\infty$, we have
$\displaystyle\frac{1}{p+1}\frac{d}{dt}\int_{0}^{1}v^{p+1}(t,x)\,dx+\int_{0}^{1}{\bf
a}(v,v_{x})(v^{p})_{x}\,dx+\epsilon p\int_{0}^{1}v^{p-1}(v_{x})^{2}\,dx$
$\displaystyle={\color[rgb]{0,0,1}(}1-\epsilon^{1/3}{\color[rgb]{0,0,1})}\int_{\partial(0,1)}v^{p}$
$\displaystyle\leq\int_{0}^{1}v^{p}\,dx+\int_{0}^{1}|(v^{p})_{x}|\,dx\,,$
where the inequality
$v^{p}(0)+v^{p}(1)=\int_{\partial(0,1)}v^{p}\leq\int_{0}^{1}v^{p}\,dx+\int_{0}^{1}|(v^{p})_{x}|\,dx$
holds in one dimension. Using (2.7) we have
$\int_{0}^{1}{\bf
a}(v,v_{x})(v^{p})_{x}\geq\int_{0}^{1}|(v^{p})_{x}|-p\int_{0}^{1}v^{p+1}\,,$
hence
$\frac{1}{p+1}\frac{d}{dt}\int_{0}^{1}v^{p+1}(t,x)\,dx+\epsilon
p\int_{0}^{1}v^{p-1}(v_{x})^{2}\,dx\leq\int_{0}^{1}v^{p}\,dx+p\int_{0}^{1}v^{p+1}\,dx.$
Using this recurrence relation, by Gronwall’s inequality, we obtain that
$\int_{0}^{1}v(t,x)^{p}\,dx\leq C(T,p)\qquad\forall t\in(0,T),\forall
p\in[1,\infty),$
and that
$\epsilon\int_{0}^{T}\int_{0}^{1}v^{p-1}(v_{x})^{2}\,dxdt\leq
C(T,p),\qquad\forall p\in[1,\infty)\,,$ (2.9)
where the constant $C(T,p)$ does not depend on $\epsilon$.
Step 2. $L^{\infty}$ bounds above and below on $v$ independent of $\epsilon$.
Let us construct a supersolution to the Cauchy problem (2.5)-(2.6). Let
$V(t,x)=B(t)-\sqrt{\epsilon^{2/3}+x(1-x)}$ with $B$ smooth and increasing.
Take $B(0)$ such that
$V(0,x)=B(0)-\sqrt{\epsilon^{2/3}+x(1-x)}\geq v_{0}(x).$
We compute
$V_{t}=B^{\prime}(t),$
$V_{x}=\frac{(x-1/2)}{\sqrt{\epsilon^{2/3}+x(1-x)}},\qquad
V_{xx}=\frac{\epsilon^{2/3}+1/4}{(\epsilon^{2/3}+x(1-x))^{3/2}},$ ${\bf
a}(V,V_{x})=\frac{V_{x}}{(V^{4}+V_{x}^{2})^{1/2}}=\frac{(x-1/2)}{D(t,x)},$
where
$D(t,x)=\left(V(t,x)^{4}(\epsilon^{2/3}+x(1-x))+(x-1/2)^{2}\right)^{1/2}$.
Note that $D(t,x)$ is a smooth and strictly positive function in $[0,1]$.
Moreover, since $B$ is increasing,
$D\geq(V(0,x)^{4}(\varepsilon^{\frac{2}{3}}+x(1-x))+(x-\frac{1}{2})^{2})^{\frac{1}{2}}$.
Thus $|{\bf a}(V,V_{x})_{x}|\leq C$ for a constant $C$ that can be taken
independent of $\epsilon$ and $t\in[0,T]$. Thus, a direct computation shows
that
${\bf a}(V,V_{x})_{x}+\epsilon V_{xx}\leq
C+\epsilon\frac{\epsilon^{2/3}+1/4}{\epsilon}\leq
C+\epsilon^{2/3}+\frac{1}{4}\leq C+\frac{5}{4}=\tilde{C},$
where $\tilde{C}$ does not depend on $\epsilon\in(0,1]$. Take
$B^{\prime}(t)\geq\tilde{C}$, for instance $B(t)=B(0)+\tilde{C}t$. Let us
prove that given $T>0$, for $\epsilon>0$ small enough $V(t,x)$ satisfies
$({\bf a}(V,V_{x})+\epsilon V_{x})\nu\geq 1-\epsilon^{1/3},$
for $t\in[0,T]$, hence $V(t,x)$ is a supersolution of the Cauchy problem
(2.5)-(2.6) in $[0,T]$. Indeed, since
$D(t,0)=\left((B(t)-\epsilon^{1/3})^{4}\epsilon^{2/3}+1/4\right)^{1/2}\,,$
we have at $x=0$
$\displaystyle({\bf a}(V,V_{x})+\epsilon V_{x})\nu|_{x=0}$
$\displaystyle=\frac{1/2}{D(t,0)}+\epsilon\frac{1/2}{\epsilon^{1/3}}=\frac{1}{\left(1+4(B(t)-\epsilon^{1/3})^{4}\epsilon^{2/3}\right)^{1/2}}+\frac{1}{2}\epsilon^{2/3}\geq
1-\epsilon^{1/3}$
for $\epsilon>0$ small enough, and analogously at $x=1$. Since $V(t,x)$ is a
supersolution for the Cauchy problem (2.5)-(2.6), by the classical comparison
principle we get $v\leq V$ in $[0,T]\times[0,1]$, and thus there exists $M>0$
depending only on $u_{0}$ and $T$ such that $v(t,x)\leq M$ in
$(t,x)\in[0,T]\times[0,1]$.
Let us finally observe that $v\geq\alpha_{1}$. Indeed,
$\overline{v}=\alpha_{1}$ is a subsolution for the Cauchy problem (2.5)-(2.6)
and by the comparison principle in its weak version, we deduce that
$v\geq\alpha_{1}$.
Step 3. $L^{p}$ bounds on $v_{x}$ independent of $\epsilon$. Putting together
the estimates in Step 2 and (2.9), we deduce that
$\int_{0}^{T}\int_{0}^{1}|(v^{p})_{x}|\,dxdt\leq C(T,p),$
for any $p\in[1,\infty)$.
Step 4. Existence of smooth solutions for the Cauchy problem (2.5)-(2.6). The
existence of solutions of (2.5)-(2.6) follows from classical results in [31]
and [32, Theorem 13.24]. We note that thanks to the a priori bounds stated
above we could use the flux
${\bf a}_{M}(v,v_{x})=\frac{v_{x}}{\sqrt{\inf(|v|,M)^{4}+v_{x}^{2}}}\,,$
so that the assumptions of the existence theorems in [31] and [32, Theorem
13.24] hold. Finally, observe that we need to assume a compatibility condition
on $v_{0}$ so that $v_{0}$ satisfies (2.6). If $v_{0}$ does not satisfy (2.6),
we modify it to define a function $v_{0,\epsilon}\in W^{1,\infty}(0,1)$
satisfying (2.6). This modification is only done in a neighborhood of
$x\in\partial(0,1)$ which vanishes as $\epsilon\to 0+$, so that
$v_{0,\epsilon}$ is locally Lipschitz inside $(0,1)$ with bounds independent
of $\epsilon$. Finally, we observe that this modification can be done in such
a way that
$\sup_{\epsilon\in(0,1]}\epsilon\|v_{0\epsilon x}\|_{\infty}<\infty.$ (2.10)
Although we omit the details of the construction, let us check that (2.10) is
compatible with (2.6). For that, notice that we can take $v_{0\epsilon
x}=A(\epsilon)\epsilon^{-a}$ with $a=\frac{1}{6}$ and
$A(\epsilon)=\frac{1}{\sqrt{2}}v_{0\epsilon}(0)^{2}+O(\epsilon^{1/3})$. Indeed
substituting this expression in (2.6), we have
$\frac{A(\epsilon)/\epsilon^{a}}{\sqrt{v_{0\epsilon}(0)^{4}+A(\epsilon)^{2}/\epsilon^{2a}}}+\epsilon\frac{A(\epsilon)}{\epsilon^{a}}=1-\epsilon^{1/3}.$
An asymptotic expansion shows
$A(\epsilon)=\frac{1}{\sqrt{2}}v_{0\epsilon}(0)^{2}+O(\epsilon^{1/3})$, and
thus (2.10) is compatible with (2.6).
Let $v_{\epsilon}$ be the solution of the Cauchy problem (2.5)-(2.6). Then
$v_{\epsilon}$ has first derivatives Holder continuous up to the boundary and
for $g=v_{\epsilon xx},v_{\epsilon t}$, we have
$\sup_{x\neq
y}\left\\{\min(d((x,t),\mathcal{P}),d((y,s),\mathcal{P}))^{1-\delta}\frac{|g(x)-g(y)|}{(|x-y|^{2}+|s-t|)^{\alpha/2}}\right\\}$
for some $\alpha,\delta>0$, where $\mathcal{P}$ is the parabolic boundary of
$(0,1)\times(0,T)$, that is $[0,1]\times\\{0\\}\cup\\{0,1\\}\times(0,T)$, and
$d(\cdot,\mathcal{P})$ denotes the distance to $\mathcal{P}$. On the other
hand, by the interior regularity theorem [31, Chapter V, Theorem 3.1], the
solution is infinitely smooth in the interior of the domain. At this point the
smoothness bounds depend on $\epsilon$.
Step 5. A local Lipschitz bound on $v_{\epsilon}$ uniform on $\epsilon$. For
simplicity of notation, let us write $v$ instead of $v_{\epsilon}$. Let
$w=|v_{x}|^{2}\phi^{2}$ where $\phi\geq 0$ is smooth with compact support
$[a,b]\subset(0,1)$. This Step is a consequence of the following inequality
$w_{t}\leq A(t,x)w_{xx}+B(t,x)w_{x}+Cw+f(t,x),$ (2.11)
where $A,B$ are smooth functions, $C=(12+\frac{\epsilon}{2})$, and $0\leq
f=P(v,\phi,\phi_{x})|\phi_{x}|+\frac{7}{2}\epsilon\phi_{x}^{2}v_{x}^{2}$,
where $P$ is a polynomial in $v$ of degree $3$. Assume for the moment that the
last term $\epsilon\|v_{x}^{2}(t)\|_{\infty}\in L^{\infty}(0,T)$. Using Step
2, this implies that $f\in L^{\infty}([0,T]\times[0,1])$. Thus we may replace
$f$ by $\|f(t)\|_{\infty}$. The change of variables
$\bar{w}(t,x)=e^{-Ct}w(t,x)-\int_{0}^{t}f(s)\,ds$
permits to write (2.11) as $\bar{w}_{t}\leq
A(t,x)\bar{w}_{xx}+B(t,x)\bar{w}_{x}$. Then, using the maximum principle, this
implies
$\sup_{t\in[0,T]}\|\bar{w}(t)\|_{\infty}\leq\|\bar{w}(0)\|_{\infty}\,$
hence we get
$\sup_{t\in[0,T]}\|w(t)\|_{\infty}\leq C(T,\phi,\|w(0)\|_{\infty}).$
Let us now prove the claim (2.11). We first compute
${\bf a}_{z}(z,\xi)=\frac{-2z^{3}\xi}{(z^{4}+\xi^{2})^{3/2}},\qquad{\bf
a}_{zz}(z,\xi)=\frac{-6z^{2}\xi}{(z^{4}+\xi^{2})^{3/2}}+\frac{12z^{6}\xi}{(z^{4}+\xi^{2})^{5/2}},$
${\bf a}_{\xi}(z,\xi)=\frac{z^{4}}{(z^{4}+\xi^{2})^{3/2}},\qquad{\bf a}_{\xi
z}(z,\xi)=\frac{-2z^{7}+4z^{3}\xi^{2}}{(z^{4}+\xi^{2})^{5/2}},\qquad\mbox{and}\qquad{\bf
a}_{\xi\xi}(z,\xi)=\frac{-3z^{4}\xi}{(z^{4}+\xi^{2})^{5/2}}.$
We also compute $w_{x}=2\phi\phi_{x}v_{x}^{2}+2\phi^{2}v_{x}v_{xx}$ and
$w_{xx}=(2\phi_{x}^{2}+2\phi\phi_{xx})v_{x}^{2}+8\phi\phi_{x}v_{x}v_{xx}+2\phi^{2}v_{xx}^{2}+2\phi^{2}v_{x}v_{xxx}$.
Differentiating (2.5) with respect to $x$ and multiplying by $\phi^{2}$ we
obtain
$\frac{1}{2}w_{t}={\bf a}_{zz}v_{x}^{3}\phi^{2}+2{\bf a}_{\xi
z}v_{x}^{2}v_{xx}\phi^{2}+{\bf a}_{\xi\xi}v_{x}v_{xx}^{2}\phi^{2}+{\bf
a}_{z}v_{x}v_{xx}\phi^{2}+{\bf a}_{\xi}v_{x}v_{xxx}\phi^{2}+\epsilon
v_{x}v_{xxx}\phi^{2}.$
Now, we get
${\bf
a}_{zz}v_{x}^{3}\phi^{2}=-\frac{6v^{2}v_{x}^{4}\phi^{2}}{(v^{4}+v_{x}^{2})^{3/2}}+\frac{12v^{6}v_{x}^{4}\phi^{2}}{(v^{4}+v_{x}^{2})^{5/2}}\leq
12w\,,$ $2{\bf a}_{\xi z}v_{x}^{2}v_{xx}\phi^{2}={\bf a}_{\xi
z}v_{x}w_{x}-2{\bf a}_{\xi z}v_{x}^{3}\phi\phi_{x}\leq{\bf a}_{\xi
z}v_{x}w_{x}+12v^{3}\phi|\phi_{x}|\,,$
and
${\bf a}_{\xi\xi}v_{x}v_{xx}^{2}\phi^{2}=\frac{1}{2}{\bf
a}_{\xi\xi}v_{xx}w_{x}-{\bf
a}_{\xi\xi}v_{xx}v_{x}^{2}\phi\phi_{x}=\frac{1}{2}{\bf
a}_{\xi\xi}v_{xx}w_{x}-X,$
where $X={\bf a}_{\xi\xi}v_{xx}v_{x}^{2}\phi\phi_{x}$. Similarly, we obtain
${\bf a}_{z}v_{x}v_{xx}\phi^{2}=\frac{1}{2}{\bf a}_{z}w_{x}-{\bf
a}_{z}v_{x}^{2}\phi\phi_{x}\leq\frac{1}{2}{\bf
a}_{z}w_{x}+2v^{3}\phi|\phi_{x}|,$
and
$\displaystyle{\bf a}_{\xi}v_{x}v_{xxx}\phi^{2}$
$\displaystyle=\frac{1}{2}{\bf a}_{\xi}w_{xx}-{\bf
a}_{\xi}(\phi_{x}^{2}+\phi\phi_{xx})v_{x}^{2}-4{\bf
a}_{\xi}v_{xx}v_{x}\phi\phi_{x}-{\bf a}_{\xi}v_{xx}^{2}\phi^{2}$
$\displaystyle\leq\frac{1}{2}{\bf
a}_{\xi}w_{xx}+v^{2}(\phi_{x}^{2}+\phi|\phi_{xx}|)-Y-{\bf
a}_{\xi}v_{xx}^{2}\phi^{2},$
where $Y=4{\bf a}_{\xi}v_{xx}v_{x}\phi\phi_{x}$. Direct estimates show that
$|Y|\leq\frac{1}{2}{\bf a}_{\xi}v_{xx}^{2}\phi^{2}+8{\bf
a}_{\xi}v_{x}^{2}\phi_{x}^{2}\leq\frac{1}{2}{\bf
a}_{\xi}v_{xx}^{2}\phi^{2}+8v^{2}\phi_{x}^{2}$
and
$|X|\leq\frac{1}{2}{\bf a}_{\xi}v_{xx}^{2}\phi^{2}+\frac{{\bf
a}_{\xi\xi}^{2}}{2{\bf a}_{\xi}}v_{x}^{4}\phi_{x}^{2}\leq\frac{1}{2}{\bf
a}_{\xi}v_{xx}^{2}\phi^{2}+\frac{9}{2}v^{2}\phi_{x}^{2}.$
Finally, let us compute the term
$\displaystyle v_{x}v_{xxx}\phi^{2}$
$\displaystyle=\frac{1}{2}w_{xx}-(\phi_{x}^{2}+\phi\phi_{x})v_{x}^{2}-4\phi\phi_{x}v_{x}v_{xx}-\phi^{2}v_{xx}^{2}$
$\displaystyle\leq\frac{1}{2}w_{xx}-\phi_{x}^{2}v_{x}^{2}+\frac{1}{2}\phi^{2}v_{x}^{2}+\frac{1}{2}\phi_{x}^{2}v_{x}^{2}+4\phi_{x}^{2}v_{x}^{2}+\phi^{2}v_{xx}^{2}-\phi^{2}v_{xx}^{2}$
$\displaystyle=\frac{1}{2}w_{xx}+\frac{1}{2}w+\frac{7}{2}\phi_{x}^{2}v_{x}^{2}.$
Putting all together, we get the desired claim (2.11)
$\frac{1}{2}w_{t}\leq\frac{1}{2}({\bf a}_{\xi}+\epsilon)w_{xx}+\left({\bf
a}_{\xi z}v_{x}+\frac{1}{2}{\bf a}_{\xi\xi}v_{xx}+\frac{1}{2}{\bf
a}_{z}\right)w_{x}+\left(12+\frac{\epsilon}{2}\right)w+P(v,\phi,\phi_{x})|\phi_{x}|+\frac{7}{2}\epsilon\phi_{x}^{2}v_{x}^{2},$
(2.12)
where $P$ is a polynomial of degree $3$ in $v$.
Now, we have to show that $\epsilon\|v_{x}^{2}(t)\|_{\infty}\in
L^{\infty}(0,T)$. Let us first exploit the boundary condition in (2.6).
Multiplying it by $v_{x}$ and using (2.7), we get
$|v_{x}|-v^{2}\leq{\bf
a}(v,v_{x})v_{x}=\frac{|v_{x}|^{2}}{(v^{4}+v_{x}^{2})^{1/2}}+\epsilon
v_{x}^{2}=(1-\epsilon^{1/3})v_{x}\,,$
and thus we get that $\epsilon v_{x}^{2}\leq v^{2}$ on $\partial(0,1)$.
Moreover, using Step 2 we finally deduce that
$\epsilon v_{x}^{2}(t)\leq\sup_{t\in[0,T]}(|v((t,0)|,|v(t,1)|)\leq
M\,,\qquad\mbox{on }\partial(0,1)\,.$ (2.13)
Taking $\phi=1$ in (2.12), we obtain
$\frac{1}{2}w_{t}\leq\frac{1}{2}({\bf a}_{\xi}+\epsilon)w_{xx}+\left({\bf
a}_{\xi z}v_{x}+\frac{1}{2}{\bf a}_{\xi\xi}v_{xx}+\frac{1}{2}{\bf
a}_{z}\right)w_{x}+(12+\frac{\epsilon}{2})w,$
that together with (2.13) and the maximum principle, imply that
$\epsilon\|v_{x}(t)^{2}\|_{\infty}\leq C,$ (2.14)
for some constant $C$ that depends on the bound (2.10), and thus independent
of $\epsilon$.
Summarizing, now the term
$\frac{7}{2}\epsilon\|\phi_{x}^{2}v_{x}^{2}(t)\|_{\infty}\in L^{\infty}(0,T)$
with bounds independent of $\epsilon$. Again, Step 2 implies that
$\|f(t)\|_{\infty}\leq\|P(v(t),\phi,\phi_{x})|\phi_{x}|\|_{\infty}+\|\frac{7}{2}\epsilon\phi_{x}^{2}v_{x}^{2}(t)\|_{\infty}\in
L^{\infty}(0,T)$ with bounds independent of $\epsilon$. Then the argument
given above shows that there are local Lipschitz bounds on $v_{\epsilon}$
uniform in $\epsilon$.
Step 6. Interior regularity of higher order derivatives uniform in $\epsilon$.
Thanks to the smoothness results stated in Step 4 and the local uniform bounds
on the gradient in Step 5, the classical interior regularity results in [31,
Chapter V, Theorem 3.1] shows uniform (in $\epsilon$) interior bounds for any
space and time derivative of $v_{\epsilon}$.
Step 7. Passing to the limit as $\epsilon\to 0^{+}$. Letting $\epsilon\to
0^{+}$ is not completely obvious due to the boundary condition (2.4). Another
difficulty stems from the fact that we do not know if $v_{\epsilon t}$ are
Radon measures with uniform bounds in $\epsilon$. This means that the notion
of normal boundary trace has to be considered in a weak sense as considered in
[2] (see also [3, Section 5.6] or [9]). Thus, we only sketch the proof of this
result. Let us first prove that the interior regularity bounds on
$v_{\epsilon}$ permit to pass to the limit and obtain a solution $v$ of
$v_{t}=\,\left(\frac{v_{x}}{\sqrt{v^{4}+(v_{x})^{2}}}\right)_{x}\qquad\hbox{\rm
in $\mathcal{D}^{\prime}((0,T)\times(0,1))$}\,.$
Let
$\xi^{\epsilon}:=v_{\epsilon t}=\left(\frac{v_{\epsilon
x}}{\sqrt{v_{\epsilon}^{4}+(v_{\epsilon x})^{2}}}+\epsilon v_{\epsilon
x}\right)_{x}\qquad\mbox{and}\qquad{\bf a}_{\epsilon}=\frac{v_{\epsilon
x}}{\sqrt{v_{\epsilon}^{4}+(v_{\epsilon x})^{2}}}+\epsilon v_{\epsilon x}\,.$
Estimate (2.14) implies that ${\bf a}_{\epsilon}$ are uniformly bounded
independently of $\epsilon$. Then by extracting a subsequence, we may assume
that ${\bf a}_{\epsilon}\rightharpoonup{\bf a}\in
L^{\infty}((0,T)\times(0,1))$ weakly∗. On the other hand, the interior
regularity bounds on $v_{\epsilon}$ ensure that ${\bf
a}=\frac{v_{\eta}}{\sqrt{v^{4}+(v_{x})^{2}}}$. By passing to the limit as
$\epsilon\to 0$, we have $v_{t}={\bf a}_{x}$ in ${\cal
D}^{\prime}((0,T)\times(0,1))$. Finally, if we take $\varphi\in
C^{1}([0,T]\times[0,1])$ with $\varphi(0)=\varphi(T)=0$, multiply (2.5) by
$\varphi$ and integrate by parts, we obtain
$\int_{0}^{T}\int_{0}^{1}v_{\epsilon}\varphi_{t}\,dxdt=\int_{0}^{T}\int_{0}^{1}{\bf
a}_{\epsilon}\varphi_{x}\,dxdt-2(1-\epsilon^{1/3})T.$
Letting $\epsilon\to 0^{+}$, we obtain
$\int_{0}^{T}\int_{0}^{1}v\varphi_{t}\,dxdt=\int_{0}^{T}\int_{0}^{1}{\bf
a}\varphi_{x}\,dxdt-2T.$
This is a weak form of the boundary condition (2.4). The correct notion of
weak trace is much more technical and is described in [3]. Using Lemma 5.7 in
[9] one can directly obtain that $v$ satisfies (2.4) in this generalized
sense. Since we do not need this result here, we skip the details that would
need several technical definitions to be fully explained. ∎
###### Remark 2.2.
Note that we can apply Step 5 to the smooth solution obtained in Theorem 2.1
to the Cauchy problem (2.3)-(2.4). In this case
$\|f\|_{\infty}\leq\|P(v,\phi,\phi_{x})|\phi_{x}|\|_{\infty}$ and we obtain a
local Lipschitz bound for $v(t,x)$ which only depends on local uniform bounds
of $v(t,x)$ and on the local Lipschitz bound of $v_{0}(x)$.
###### Remark 2.3.
In Section 2.2 we will give sufficient conditions on $u_{0}$ that imply that
$v_{t}$ is a Radon measure. In that case, the notion of weak trace ${\bf
a}\cdot\nu$ can be found in [21, 23].
###### Remark 2.4.
We could define the notion on entropy solutions of equation (2.3) with
boundary condition (2.4) and prove that the solution constructed is indeed an
entropy solution of it. We will not pursue this here.
### 2.2 Getting an entropy solution of (1.3) from (2.3)
Here, we use several notations and definitions that are introduced in the
Appendix to which we refer for details. In this Section, we come back to the
notation $v(t,\eta)$ instead of $v(t,x)$, $\eta\in(0,1)$. Recall that by
passing to the limit as $\epsilon\to 0^{+}$ we have found a solution $v$ of
$v_{t}=\,\left(\frac{v_{\eta}}{\sqrt{v^{4}+(v_{\eta})^{2}}}\right)_{\eta}\qquad\hbox{\rm
in $\mathcal{D}^{\prime}((0,T)\times(0,1))$},$ (2.15)
for any $T>0$. Thus, let $v(t,\eta)$ be the solution of (2.15) constructed in
Theorem 2.1 which satisfies $[{\bf a}(t,\eta)\cdot\nu]=1$ for $\eta=0,1$ and
a.e. for $t\in(0,T)$ in a weak sense. As we shall see, we do not need this
here, we only need a weaker form of the boundary condition as expressed in
(2.17) below.
In the next Lemma we construct an entropy solution of (1.1) from a solution
$v(t,\eta)$ of (2.15). To prepare its statement, let $u_{0}\in
L^{\infty}(I\\!\\!R)$ with $u_{0}(x)\geq\kappa>0$ for $x\in[a,b]$, and
$u_{0}(x)=0$ for $x\not\in[a,b]$. Assume that $u_{0}\in W^{1,\infty}([a,b])$.
Let $v_{0}(\eta)=\frac{1}{u_{0}(x)}$, $\eta\in(0,1)$, where
$x=\varphi(0,\eta)$ is such that
$\int_{a}^{\varphi(0,\eta)}u_{0}(x)\,dx=\eta\,.$
Let $u(t,x)$ be defined in $[a-t,b+t]$ by
$u(t,x)=\frac{1}{v(t,\eta)}\,,\,\,\mbox{ where
}x=\varphi(t,\eta)=a-t+\int_{0}^{\eta}v(t,\bar{\eta})d\bar{\eta}\,.$ (2.16)
By (2.8), we have
$\int_{0}^{1}v(t,\eta)\,d\eta=b-a+2t\,,$ (2.17)
and $x=\varphi(t,\eta)\in[a-t,b+t]$ when $\eta$ varies in $[0,1]$. Note that
$\int_{a-t}^{\varphi(t,\eta)}u(t,x)dx=\eta,\qquad\eta\in(0,1).$
We define $u(t,x)=0$, $x\not\in[a-t,b+t]$, $t\in(0,T)$. Notice that
$u(t,x)\geq\kappa(t)>0$ for any $x\in(a-t,b+t)$ and any $t>0$.
The statement $(ii)$ in Theorem 1.1 follows from next Proposition.
###### Proposition 2.5.
Given $u$ defined by (2.16) where $v$ is a solution given by Theorem 2.1. Then
$u\in C([0,T],L^{1}(I\\!\\!R))$, $u(0)=u_{0}$, and satisfies
* (i)
$u(t)\in BV(I\\!\\!R)$, $u(t)\in W^{1,1}(a-t,b+t)$ for almost any $t\in(0,T)$,
and $u(t)$ is smooth inside its support,
* (ii)
$u_{t}={\bf z}_{x}$ in $\mathcal{D}^{\prime}((0,T)\times I\\!\\!R)$, where
${\bf z}(t)=\frac{u(t)u_{x}(t)}{\sqrt{u(t)^{2}+u_{x}(t)^{2}}}$ ,
* (iii)
$u(t,x)$ is the entropy solution of (1.3) with initial data $u_{0}$ in
$(0,T)$.
###### Proof.
$(i)$ Since $v$ is bounded and bounded away from zero from Step 2 in Theorem
2.1, then $u$ is bounded and bounded away from zero in its support. The
smoothness properties of $v$ prove that $u\in C([0,T],L^{1}(I\\!\\!R))$,
$u(0)=u_{0}$, and $u(t)$ is smooth inside its support. By Step 3 from Theorem
2.1, we have that $u(t)\in W^{1,1}(a-t,b+t)$ for almost any $t\in(0,T)$. This
implies that $u(t)\in BV(I\\!\\!R)$ for almost any $t\in(0,T)$. From the
change of variables (2.16) we have that
$\frac{u_{x}}{\sqrt{u^{2}+u_{x}^{2}}}=-\frac{v_{\eta}}{\sqrt{v^{4}+v_{\eta}^{2}}}.$
(2.18)
$(ii)$ For simplicity, let us write $Q_{T}=(0,T)\times I\\!\\!R$, and
$\Omega(t)=(a-t,b+t)$. Since
$Du(t)=u_{x}\raisebox{2.0pt}{\rm{$\chi$}}_{\Omega(t)}-u^{i}(t)\nu^{t}\mathcal{H}^{0}\mathbin{\vrule
height=9.0pt,width=0.1pt\vrule height=0.1pt,width=9.0pt}{\partial\Omega(t)},$
we have that $u\in L^{1}_{loc,w}(0,T;BV(I\\!\\!R))$. We have denoted by
$u^{i}(t)$ the trace of $u|_{\Omega(t)}$ on $\partial\Omega(t)$. Note that it
coincides with $u^{+}(t)$. Let us prove that
$u_{t}={\bf z}_{x}\quad\hbox{in}\ \ {\mathcal{D}}^{\prime}((0,T)\times
I\\!\\!R).$ (2.19)
Let $\phi\in{\mathcal{D}}(Q_{T})$. Let
$\overline{\phi}(t,\eta)=\phi(t,\varphi(t,\eta))$, $\eta\in[0,1]$. Then
$\overline{\phi}_{t}=\phi_{t}(t,\varphi(t,\eta))+\phi_{x}(t,\varphi(t,\eta))\varphi_{t}$
and
$\displaystyle-\int_{0}^{T}\int_{I\\!\\!R}u\phi_{t}\,dxdt$
$\displaystyle=-\int_{0}^{T}\int_{\Omega(t)}u\phi_{t}\,dxdt=-\int_{0}^{T}\int_{0}^{1}\frac{1}{v}(\overline{\phi}_{t}-\phi_{x}(t,\varphi(t,\eta))\varphi_{t})v\,d\eta
dt$
$\displaystyle=-\int_{0}^{T}\int_{0}^{1}(\overline{\phi}_{t}-\phi_{x}(t,\varphi(t,\eta))\varphi_{t})\,d\eta
dt=\int_{0}^{T}\int_{0}^{1}\phi_{x}(t,\varphi(t,\eta))\varphi_{t}\,d\eta dt$
$\displaystyle=\int_{0}^{T}\int_{0}^{1}\phi_{x}(t,\varphi(t,\eta))\frac{v_{\eta}}{\sqrt{v^{4}+v_{\eta}^{2}}}\,d\eta
dt=-\int_{0}^{T}\int_{\Omega(t)}\frac{uu_{x}}{\sqrt{u^{2}+u_{x}^{2}}}\phi_{x}(t,x)\,dxdt$
$\displaystyle=-\int_{0}^{T}\int_{I\\!\\!R}{\bf z}\phi_{x}\,dxdt\,,$
where (2.18) was used. Thus (2.19) holds.
$(iii)$ To prove that $u$ is an entropy solution of (1.3), we have to prove
that
$\displaystyle\int_{Q_{T}}h_{S}(u,DT(u))\phi\,dxdt$
$\displaystyle+\int_{Q_{T}}h_{T}(u,DS(u))\phi\,dxdt$
$\displaystyle\leq\int_{Q_{T}}J_{TS}(u)\phi_{t}\,dxdt-\int_{0}^{T}\int_{I\\!\\!R}{\bf
z}(t,x)\cdot\nabla\phi(t)T(u(t))S(u(t))\,dxdt,$ (2.20)
holds for any any $T,S\in{\mathcal{T}}^{+}$ and any
$\phi\in\mathcal{D}((0,T)\times I\\!\\!R)$, $\phi(t,x)=\eta(t)\rho(x)$. As in
[6, Proposition 1], we have
$(h_{S}(u(t),DT(u(t))))^{s}=\left|D^{j}J_{SRT^{\prime}}(u(t))\right|=J_{SRT^{\prime}}(u^{i}(t)){\mathcal{H}}^{0}\mathbin{\vrule
height=9.0pt,width=0.1pt\vrule height=0.1pt,width=9.0pt}{\partial\Omega(t)}$
(2.21)
and
$(h_{T}(u(t),DS(u(t))))^{s}=\left|D^{j}J_{TRS^{\prime}}(u(t))\right|=J_{TRS^{\prime}}(u^{i}(t)){\mathcal{H}}^{0}\mathbin{\vrule
height=9.0pt,width=0.1pt\vrule height=0.1pt,width=9.0pt}{\partial\Omega(t)},$
(2.22)
where $R(r)=r$, $r\in I\\!\\!R$. Thus, by (2.21) and (2.22), we get
$\displaystyle(h_{S}(u(t),DT(u(t))))^{s}+(h_{T}(u(t),DS(u(t))))^{s}$
$\displaystyle=\left(J_{SRT^{\prime}}(u^{i}(t))+J_{TRS^{\prime}}(u^{i}(t))\right){\mathcal{H}}^{0}\mathbin{\vrule
height=9.0pt,width=0.1pt\vrule height=0.1pt,width=9.0pt}{\partial\Omega(t)}$
$\displaystyle=\left(TSR(u^{i}(t))-J_{TS}(u^{i}(t))\right){\mathcal{H}}^{0}\mathbin{\vrule
height=9.0pt,width=0.1pt\vrule height=0.1pt,width=9.0pt}{\partial\Omega(t)}.$
(2.23)
On the other hand, it is easy to prove that
$\displaystyle\int_{Q_{T}}(h_{S}(u,DT(u)))^{ac}\phi\,dxdt\,+$
$\displaystyle\int_{Q_{T}}(h_{T}(u,DS(u)))^{ac}\phi\,dxdt$
$\displaystyle=\displaystyle\int_{0}^{T}\int_{\Omega(t)}{\bf
z}(t,x)\cdot[T(u(t,x))S(u(t,x))]_{x}\phi(t)\,dxdt.$ (2.24)
Adding (2.23) and (2.24), we obtain
$\displaystyle\int_{Q_{T}}\phi h_{S}(u(t),DT(u(t)))\,dxdt\,+$
$\displaystyle\,\int_{Q_{T}}\phi h_{T}(u(t),DS(u(t)))\,dxdt$ $\displaystyle=$
$\displaystyle\,\int_{0}^{T}\int_{\partial\Omega(t)}\left(TSR(u^{i}(t))-J_{TS}(u^{i}(t))\right)\phi(t)\,d{\mathcal{H}}^{0}\,dt$
$\displaystyle\,+\displaystyle\int_{0}^{T}\int_{\Omega(t)}{\bf
z}(t,x)\cdot[T(u(t,x))S(u(t,x))]_{x}\phi(t)\,dxdt.$ (2.25)
To simplify the subsequent notation let us denote
$p(u)=T(u)S(u)=J^{\prime}(u)$ and $J(u)=J_{TS}(u)$. Let us now prove that
$\displaystyle\int_{0}^{T}\int_{\partial\Omega(t)}\left(p(u^{i}(t))u^{i}(t)-J(u^{i}(t))\right)\phi(t)\,d{\mathcal{H}}^{0}\,dt\,+$
$\displaystyle\,\int_{0}^{T}\int_{\Omega(t)}{\bf z}\cdot[p(u)]_{x}\phi\,dxdt$
$\displaystyle\leq\displaystyle\int_{Q_{T}}J(u)\phi_{t}\,dxdt-\int_{0}^{T}\int_{\Omega(t)}\phi_{x}{\bf
z}p(u)\,dxdt.$ (2.26)
The main technical difficulty comes from the fact that we do not know that
$u_{t}={\bf z}_{x}$ is a Radon measure. We circumvent this difficulty by using
instead discrete derivatives. Let us denote
$\Delta_{\tau}^{+}w(t)=\frac{1}{\tau}(w(t+\tau)-w(t)),\qquad\Delta_{\tau}^{-}w(t)=\frac{1}{\tau}(w(t)-w(t-\tau)).$
Then, we can obtain
$\displaystyle-\int_{0}^{T}\int_{I\\!\\!R}up(u)\phi\,$
$\displaystyle\Delta_{\tau}^{-}\raisebox{2.0pt}{\rm{$\chi$}}_{\Omega(t)}\,dxdt=\int_{0}^{T}\int_{\Omega(t)}\Delta_{\tau}^{+}(up(u)\phi)\,dxdt$
$\displaystyle=$
$\displaystyle\int_{0}^{T}\int_{\Omega(t)}\Delta_{\tau}^{+}u(t)p(u(t+\tau))\phi(t+\tau)\,dxdt+\int_{0}^{T}\int_{\Omega(t)}u(t)\Delta_{\tau}^{+}(p(u)\phi)(t)\,dxdt$
$\displaystyle\geq$
$\displaystyle\int_{0}^{T}\int_{\Omega(t)}\Delta_{\tau}^{+}J(u)(t)\phi(t+\tau)\,dxdt+\int_{0}^{T}\int_{\Omega(t)}u(t)\Delta_{\tau}^{+}(p(u)\phi)(t)\,dxdt$
$\displaystyle=$
$\displaystyle-\int_{0}^{T}\int_{\Omega(t)}J(u)(t)\Delta_{\tau}^{-}[\phi(t+\tau)]\,dxdt-\int_{0}^{T}\int_{I\\!\\!R}J(u)(t)\phi(t)\Delta_{\tau}^{-}\raisebox{2.0pt}{\rm{$\chi$}}_{\Omega(t)}\,dxdt$
$\displaystyle+\int_{0}^{T}\int_{\Omega(t)}u(t)\Delta_{\tau}^{+}(p(u)\phi)(t)\,dxdt$
which is a discrete version of (2.26). Note that we have used the inequality
$\Delta_{\tau}^{+}u(t)p(u(t+\tau))\geq\Delta_{\tau}^{+}J(u)$ which is a
consequence of the convexity of $J$. By letting $\tau\to 0^{+}$, we need to
show that
$\int_{Q_{T}}(u(t)p(u(t))-J(u(t)))\phi(t)\Delta_{\tau}^{-}\raisebox{2.0pt}{\rm{$\chi$}}_{\Omega(t)}\,dxdt\to\int_{0}^{T}\int_{\partial\Omega(t)}\left(p(u^{i}(t))u^{i}(t)-J(u^{i}(t))\right)\phi(t)\,d{\mathcal{H}}^{0}\,dt,$
(2.27)
$\int_{0}^{T}\int_{\Omega(t)}J(u)(t)\Delta_{\tau}^{-}[\phi(t+\tau)]\,dxdt\to\int_{0}^{T}\int_{\Omega(t)}J(u)(t)\phi_{t}(t)\,dxdt,$
(2.28)
and
$\int_{0}^{T}\int_{\Omega(t)}u(t)\Delta_{\tau}^{+}(p(u)\phi)(t)\,dxdt\to\int_{0}^{T}\int_{\Omega(t)}(p(u)\phi)_{x}{\bf
z}\,dxdt.$ (2.29)
This will result in (2.26). The limit (2.27) follows since $u(t)\in
BV(I\\!\\!R)$ a.e. in $t$, $u\in L^{1}_{w}(0,T;BV(I\\!\\!R))$ (hence
$\|u_{x}(t)\|\in L^{1}(0,T)$) and the trace functions $u(t,a-t)$, $u(t,b+t)$
are integrable in $[0,T]$. The second limit (2.28) follows easily. To prove
(2.29), for any $\tau>0$ let
$\psi^{\tau}(t,x):=\frac{1}{\tau}\int_{t}^{t+\tau}\phi(s,x)p(u(s,x))\,ds,$
and observe that
$\Delta_{\tau}^{+}(p(u)\phi)(t,x)=\frac{\partial}{\partial
t}\psi^{\tau}(t,x).$
Observe also that
$\frac{d}{dt}[\psi^{\tau}(t,\varphi(t,\eta))]=\frac{\partial}{\partial
t}\psi^{\tau}(t,\varphi(t,\eta))+\psi^{\tau}_{x}(t,\varphi(t,\eta))\varphi_{t}(t,\eta).$
Then, as $\tau\to 0^{+}$
$\displaystyle\int_{0}^{T}\int_{\Omega(t)}u(t)\Delta_{\tau}^{+}(p(u)\phi)(t)\,dxdt=$
$\displaystyle\int_{0}^{T}\int_{\Omega(t)}u(t)\frac{\partial}{\partial
t}\psi^{\tau}(t,x)\,dxdt$ $\displaystyle=$
$\displaystyle\int_{0}^{T}\int_{0}^{1}\left(\frac{d}{dt}[\psi^{\tau}(t,\varphi(t,\eta))]-\psi^{\tau}_{x}(t,\varphi(t,\eta))\varphi_{t}(t,\eta)\right)\,d\eta
dt$ $\displaystyle=$
$\displaystyle-\int_{0}^{T}\int_{0}^{1}\psi^{\tau}_{x}(t,\varphi(t,\eta))\frac{v_{\eta}}{\sqrt{v^{4}+v_{\eta}^{2}}}\,d\eta
dt$ $\displaystyle=$
$\displaystyle\int_{0}^{T}\int_{0}^{1}\psi^{\tau}_{x}(t,x)\frac{uu_{x}}{\sqrt{u^{2}+u_{x}^{2}}}\,dxdt\to\int_{0}^{T}\int_{\Omega(t)}(p(u)\phi)_{x}{\bf
z}\,dxdt.$
We have proved (2.29). Finally we observe that from (2.25) and (2.26) we
obtain (2.20). ∎
###### Remark 2.6.
In a similar way, using this time
$\Delta_{\tau}^{+}(up(u)\phi)(t)=\Delta_{\tau}^{+}(p(u)\phi)(t)u(t+\tau)+p(u(t))\phi(t)\Delta_{\tau}^{+}(u)(t)$
and $\Delta_{\tau}^{+}(J(u))(t)\geq p(u(t))\Delta_{\tau}^{+}(u)(t)$ one can
prove that the opposite inequality in (2.26) holds, and we have equality. Note
also that equality holds also in the entropy conditions (2.20).
With some additional regularity on the initial condition, one has that $u_{t}$
is a Radon measure in $(0,T)\times I\\!\\!R$. Indeed, the following
proposition follows immediately from the results in [10, 21].
###### Proposition 2.7.
Let $u_{0}\in L^{\infty}(I\\!\\!R)$, $u_{0}(x)\geq\kappa>0$ for $x\in[a,b]$
and $u_{0}=0$ outside $[a,b]$. Assume that $u_{0}\in W^{2,1}(a,b)$. If $u$ is
the entropy solution of (1.3) with initial data $u_{0}$, then $u_{t}$ is a
Radon measure in $(0,T)\times I\\!\\!R$.
From Proposition 2.7 and the results in [21], it follows that $[{\bf
z}\cdot\nu^{\Omega(t)}]=-u^{i}(t)$ on $\partial\Omega(t)$ for almost any
$t\in(0,T)$. This permits also to define the notion of normal trace of ${\bf
a}(v,v_{\eta})$ in the sense of [21, 23].
## 3 Regularity for touching-down initial data
Let us start by getting local estimates.
###### Proposition 3.1.
Let $u_{0}\in L^{\infty}(I\\!\\!R)$ with $u_{0}(x)\geq\kappa>0$ for
$x\in[a,b]$, and $u_{0}(x)=0$ for $x\not\in[a,b]$. Assume that $u_{0}\in
W^{1,\infty}_{\rm loc}(a,b)$. The entropy solution $u(t,x)$ of (1.3) with
$u(0)=u_{0}$ satisfies $(i)$ and $(ii)$ in Theorem 1.1.
###### Proof.
Let $u_{0\delta}\in L^{\infty}(I\\!\\!R)$ with $u_{0\delta}(x)\geq\kappa>0$
for $x\in[a,b]$, $u_{0\delta}(x)=0$ for $x\not\in[a,b]$, $u_{0\delta}\to
u_{0}$ locally uniformly in $(a,b)$ as $\delta\to 0^{+}$, and $u_{0\delta}\in
W^{1,\infty}([a,b])$ with uniform local Lipschitz bounds in $(a,b)$. Let
$v_{0\delta}(\eta)$ be the functions obtained by the change of variables (2.2)
(with $t=0$). Let $u_{\delta}(t,x)$ be the entropy solution of (1.1) with
$u_{\delta}(0)=u_{0\delta}$. By Theorem 1.1 we know that each
$u_{\delta}(t,x)$ is smooth inside $(a,b)$. Let us note that the local bounds
on $u_{\delta}$ and its derivatives do not depend on $\delta$. It suffices to
observe that this is true for the associated functions $v_{\delta}(t,\eta)$
which are solutions of (2.3), (2.4), with initial data
$v_{\delta}(0,\eta)=v_{0\delta}(\eta)$. Note that the bounds in Steps 1, 2, 3
in the proof of Theorem 2.1 are independent of $\delta$. By Remark 2.2, the
Lipschitz bound in Step 5 depends only on the local Lipschitz bounds of
$v_{0\delta}(\eta)$ and are, thus, uniform in $\delta$. Step 6 proves uniform
(in $\delta$) interior bounds for any space and time derivative of
$v_{\delta}(t,\eta)$. By passing to the limit as $\delta\to 0+$ we conclude
that $u(t,x)$ is smooth inside its support and $(i)$ and $(ii)$ in Theorem 1.1
hold. ∎
We now generalize our main results to initial data vanishing at the boundary
of the support.
###### Proposition 3.2.
Let $u_{0}\in L^{\infty}(I\\!\\!R)$ with $u_{0}(x)>0$ for $x\in(a,b)$, and
$u_{0}(x)=0$ for $x\not\in(a,b)$. Assume that $u_{0}\in W^{1,\infty}_{\rm
loc}(a,b)$ and $u_{0}(x)\to 0$ as $x\to a,b$. The entropy solution $u(t,x)$ of
(1.3) with $u(0)=u_{0}$ satisfies $(i)$ and $(ii)$ in Theorem 1.1. Moreover,
if $u_{0}(x)\leq A(b-x)^{\alpha}(x-a)^{\alpha}$ for some $A,\alpha>0$, then
$u(t,x)\leq A(t)(b+t-x)^{\alpha}(x-a+t)^{\alpha}$ for any $x\in(a-t,b+t)$,
$t>0$ and some $A(t)$. In that case, $u(t,x)$ is a continuous function that
tends to $0$ as $x\to a-t,b+t$.
###### Proof.
Let $u_{0\delta}\in L^{\infty}(I\\!\\!R)$ with
$u_{0\delta}(x)=u_{0}(x)+\delta$ for $x\in[a,b]$, $u_{0\delta}(x)=0$ for
$x\not\in[a,b]$, and $u_{0\delta}\in W^{1,\infty}([a,b])$ with uniform local
Lipschitz bounds in $(a,b)$. Let $v_{0\delta}(\eta)$ be the functions obtained
by the change of variables (2.2) (with $t=0$). Let $u_{\delta}(t,x)$ be the
entropy solution of (1.1) with $u_{\delta}(0)=u_{0\delta}$. By Theorem 1.1 we
know that each $u_{\delta}(t,x)$ is smooth inside $(a,b)$. Let us note that
the local bounds on $u_{\delta}$ and its derivatives do not depend on
$\delta$. Again, it suffices to observe that this is true for the associated
functions $v_{\delta}(t,\eta)$ which are solutions of (2.3), (2.4), with
initial data $v_{\delta}(0,\eta)=v_{0\delta}(\eta)$.
The $L^{p}$ bounds follow from Step 1 in the proof of Theorem 2.1 for
$p\in[1,\infty)$ and they only depend on the $L^{p}$ bound of $v_{0\delta}$.
Actually, we have
$\int_{0}^{1}v_{0\delta}(\eta)^{p}\,d\eta=\int_{a}^{b}\frac{1}{u_{0\delta}(x)^{p-1}}\,dx\,,$
that depends on the integrability of $\frac{1}{u_{0\delta}(x)}$ at the
boundary points. But multiplying (2.3) by $v_{\delta}^{p}\phi$ where
$p\in[1,\infty)$ and $\phi$ is a positive smooth test function with compact
support in $(0,1)$ we obtain
$\frac{1}{p+1}\frac{d}{dt}\int_{0}^{1}v_{\delta}^{p+1}(t,\eta)\phi\,d\eta+\int_{0}^{1}|(v_{\delta}^{p})_{\eta}|\phi\leq
p\int_{0}^{1}v_{\delta}^{p+1}\phi\,d\eta+\int_{0}^{1}v_{\delta}^{p}|\phi_{\eta}|\,d\eta.$
Thus we derive local $L^{p}$ bounds for $v_{\delta}$ which are independent of
$\delta$. We also obtain local bounds on the total variation of
$v_{\delta}^{p}$ which are independent of $\delta$. To obtain a local
$L^{\infty}$ bound independent of $\delta$ we observe that this follows from
the identity $v_{\delta}(t,\eta)=\frac{1}{u_{\delta}(t,x)}$, where
$x=\varphi_{\delta}(t,\eta)$ is given by (2.2), since we know that
$u_{\delta}(t,x)$ is locally bounded away from zero in its support [6]. Thus
Steps 1, 2, 3 hold in their local versions. By Remark 2.2, the Lipschitz bound
in Step 5 depends only on the uniform local bounds on $v_{\delta}(t,\eta)$ and
on the local Lipschitz bounds of $v_{0\delta}(\eta)$ and are, thus, uniform in
$\delta$. Step 6 proves uniform (in $\delta$) interior bounds for any space
and time derivative of $v_{\delta}(t,\eta)$. By passing to the limit as
$\delta\to 0+$ we conclude that $u(t,x)$ is smooth inside its support and
$(i)$ and $(ii)$ in Theorem 1.1 hold. The last assertion is a consequence of
the comparison principle using Lemma 3.4 below. ∎
###### Remark 3.3.
Note that the last assertion implies that if the initial profile is not
vertical at the boundary at $t=0$ it remains non-vertical for any $t>0$.
Moreover, during the proof we have observed that if $u_{0}$ has a vertical
profile with $\frac{1}{u_{0}}\in L^{p}(a,b)$, then $\frac{1}{u(t,x)}\in
L^{p}(a-t,b+t)$ for any $t>0$. Thus in that case $u(t,x)$ has a vertical
profile at the boundary of its support.
Due to translational invariance of (1.3), we state our next Lemma in an
interval symmetric around zero.
###### Lemma 3.4.
Let $U(t,x)=A(t)(R(t)^{2}-x^{2})^{\alpha}$ where $R(t)=R_{0}+t$, $\alpha>0$.
If $A^{\prime}(t)\geq 0$, then $U(t,x)$ is a supersolution of (1.3).
###### Proof.
Computing the derivatives, we get
$U_{t}=A^{\prime}(R(t)^{2}-x^{2})^{\alpha}+2A\alpha
R(R^{2}-x^{2})^{\alpha-1},$ $U_{x}=-2A\alpha x(R^{2}-x^{2})^{\alpha-1}\,,$
and
$(U^{2}+U_{x}^{2})^{1/2}=A(R^{2}-x^{2})^{\alpha-1}Q(x)\,,$
where $Q(x)=\left((R^{2}-x^{2})^{2}+4\alpha^{2}x^{2}\right)^{1/2}$, and then
$\frac{UU_{x}}{(U^{2}+U_{x}^{2})^{1/2}}=-\frac{2A\alpha
x(R^{2}-x^{2})^{\alpha}}{Q}.$
Thus, the claim
$U_{t}\geq\,\left(\frac{UU_{x}}{\sqrt{U^{2}+(U_{x})^{2}}}\right)_{x}$
holds if and only if
$\displaystyle A^{\prime}(R^{2}-x^{2})^{\alpha}+2A\alpha
R(R^{2}-x^{2})^{\alpha-1}\geq$
$\displaystyle\,-\frac{2A\alpha(R^{2}-x^{2})^{\alpha}}{Q}+\frac{2A\alpha
x(R^{2}-x^{2})^{\alpha}Q_{x}}{Q^{2}}$
$\displaystyle\,+\frac{4A\alpha^{2}x^{2}(R^{2}-x^{2})^{\alpha-1}}{Q}\,.$
Let us prove that
$2A\alpha
R(R^{2}-x^{2})^{\alpha-1}\geq\frac{4A\alpha^{2}x^{2}(R^{2}-x^{2})^{\alpha-1}}{Q}.$
Indeed, the above inequality is implied by $2R\geq 4\alpha x^{2}/Q$ and
$4\alpha x^{2}\leq(2R)2\alpha|x|\leq 2RQ$. Now, we choose $A$ such that
$A^{\prime}(R^{2}-x^{2})^{\alpha}\geq-\frac{2A\alpha(R^{2}-x^{2})^{\alpha}}{Q}+\frac{2A\alpha
x(R^{2}-x^{2})^{\alpha}Q_{x}}{Q^{2}}\,,$
that is,
$A^{\prime}\geq-\frac{2A\alpha}{Q}+\frac{2A\alpha
xQ_{x}}{Q^{2}}=\frac{2A\alpha}{Q}\left(-1+\frac{xQ_{x}}{Q}\right)\,.$ (3.1)
Noticing that
$\frac{xQ_{x}}{Q}=\frac{4\alpha^{2}x^{2}-2x^{2}(R^{2}-x^{2})}{Q^{2}}\leq\frac{4\alpha^{2}x^{2}}{Q^{2}}\leq
1\,,$
hence (3.1) holds if $A^{\prime}\geq 0$. We have proved that if
$A^{\prime}\geq 0$, then $U(t,x)$ is a supersolution of (1.3). ∎
## 4 Numerical experiments and heuristics
In this section, we will propose a numerical scheme for more general equations
than the RHE (1.1). We deal with the Cauchy problem for the generic porous
media relativistic heat equation (RHEm) [22] given by
$u_{t}=\,\left(\frac{u^{m}u_{x}}{\sqrt{u^{2}+(u_{x})^{2}}}\right)_{x}$ (4.1)
with initial data $u_{0}$ a probability density with compact support. In order
to propose the numerical scheme, we make use of the change of variables to
Lagrangian coordinates. As in the introduction, let us denote by $F$ the
distribution function associated to the probability density $u$ and
$\varphi(t,\eta)$ its inverse or generalized inverse, defined by
$\varphi(t,\eta):=\left\\{\begin{array}[]{lc}-\infty&\eta=-\frac{1}{2}\\\\[5.69054pt]
\inf\\{x:\
F(t,x)>\eta+\tfrac{1}{2}\\}\,,&\qquad\eta\in(-\tfrac{1}{2},\tfrac{1}{2})\\\\[5.69054pt]
+\infty&\eta=\frac{1}{2}.\end{array}\right.$ (4.2)
Here, we have preferred to shift the mass variable to the interval
$(-\frac{1}{2},\frac{1}{2})$ to simplify the notations about boundary
conditions. In this way, we simply have the relation
$F(t,\varphi(t,\eta))=\eta,\qquad\eta\in(-\tfrac{1}{2},\tfrac{1}{2}).$ (4.3)
For simplicity, most of the numerical tests have been chosen for even initial
data. Observe that this change of variables is a weak diffeomorphism in case
of connected compactly supported smooth $u$, say on the interval
$(-A(t),A(t))$ in which case
$\lim_{\eta\to\pm\frac{1}{2}^{\mp}}\varphi(t,\eta)=\pm A(t).$ (4.4)
Straightforward computations show that the equation satisfied by $\varphi$ in
$(-\frac{1}{2},\frac{1}{2})$ is
$\varphi_{t}=-\frac{\left(\frac{1}{\varphi_{\eta}}\right)^{m-1}\left(\frac{1}{\varphi_{\eta}}\right)_{\eta}}{\sqrt{1+\left(\frac{1}{\varphi_{\eta}}\right)_{\eta}^{2}}}\,,$
(4.5)
while at the boundary, formally, by (4.2) and (4.4), we have to impose
$\varphi_{\eta}\left(t,\tfrac{1}{2}\right)=+\infty.$ (4.6)
Moreover, thanks to the vertical contact angle property (see (2.1) for the RHE
and [22] for the RHEm), we have that
$\lim_{\eta\to\pm\frac{1}{2}^{\mp}}\left(\frac{1}{\varphi_{\eta}}\right)_{\eta}(t,\eta)=\mp\infty.$
(4.7)
The purpose of this section is two-fold. On one hand, we heuristically observe
some qualitative properties from the Lagrangian viewpoint. On the other hand,
these properties are confirmed by numerical experiments with the use of an
adaptation of the algorithm proposed in [16] for general equations in
continuity form for the 2-dimensional case.
### 4.1 Numerical Method
Equations (1.3) and (4.1) have been numerically treated in [34, 38] using the
connection between nonlinear diffusions and Hamilton-Jacobi equations and
numerical methods for conservation laws and in [10] using an appropiate WENO
scheme. Here, we propose a completely different approach based on the optimal
transportation viewpoint. As we already mentioned in the introduction an
explicit Euler discretization of the equation satisfied by the generalized
inverse (4.5) coincides with the variational scheme introduced in [29, 33].
Moreover, the theoretical result proven in [35] shows that this scheme applied
to (1.3) is convergent for initial data compactly supported smooth in their
support and bounded below and above. Therefore, we plan to use a similar
algorithm for Eq. (4.1). This Lagrangian formulation in 1D for nonlocal and
nonlinear diffusion problems was numerically analysed in [28, 13]. These
Lagrangian coordinates ideas were generalized to several dimensions in [16].
The advantages of this method are the adaptation of the mesh to the mass
distribution of the solution in an automatic way, the immediate positivity of
the solutions, and the decay of the natural Liapunov functional of the
equations. We refer to [16] for more details and discussions on these issues.
Here, we propose an adaptation of the algorithm in [16]. First of all, the
discretization in the mass variable has been treated by finite difference
approximations of the derivatives of the unknown $\varphi$. We consider a
partition $\\{\eta_{i}\\}_{i=1:N}$ of the spatial interval
$[-\frac{1}{2},\frac{1}{2}]$ and we let $\Delta_{i}:=\eta_{i+1}-\eta_{i}$.
Note that, due to (4.2), first derivatives at the points corresponding to the
nodes $\eta_{2}$ and $\eta_{N-1}$ have to be taken from the inside of the
domain. In order to avoid higher errors in the approximation of the derivative
at the boundaries, we decide to approximate $\varphi_{\eta}$ as
$\varphi_{\eta}(\eta_{i}):=\left\\{\begin{array}[]{cc}\displaystyle\frac{\varphi(\eta_{i+1})-\varphi(\eta_{i})}{\Delta_{i}}&{\rm
if\ }\eta_{i}\leq\eta(t)\\\\[11.38109pt]
\displaystyle\frac{\varphi(\eta_{i})-\varphi(\eta_{i-1})}{\Delta_{i-1}}&{\rm
if\ }\eta_{i}>\eta(t)\end{array}\right.$
with $\eta(t)$ to be specified. The derivative of the term
$\frac{1}{\varphi_{\eta}}$ is computed in the other direction for better
stability properties of the approximation of
$\left(({\varphi_{\eta}})^{-1}\right)_{\eta}$. At the boundary we just impose
(4.6).
As explained in [16], the point $\eta(t)$ has to be taken as the global
maximum for $u$, which can be tracked at any time step. In all examples
computed, initial data are taken to be radially symmetric and decreasing from
the point $x=\eta=0$. In all of them, the global maximum stays at $x=\eta=0$.
Therefore, we choose to take an even number of points $N$ in the
discretization and to take a symmetric partition $\\{\eta_{i}\\}_{i=1:N}$ of
the spatial interval $[-\frac{1}{2},\frac{1}{2}]$. Let us point out that the
spatial partition is never uniform since the change to Lagrangian coordinates
produces the accumulation of nodes near the global maximum. We instead want to
follow some particular features of these type of equations such as propagation
of fronts with a vertical contact angle or formation of singularities.
Therefore, the partitions will be chosen accordingly in order to accumulate
more points around the points $\pm\frac{1}{2}$ and other points of interest.
The time derivative is evaluated through a simple explicit Euler scheme with
the CFL condition proposed in [16]; i.e:
$\left\|\left(\frac{1}{\varphi_{\eta}}\right)^{m}\right\|_{\infty}\frac{\Delta
t}{(\Delta\eta)^{2}}\leq\frac{1}{\alpha_{CFL}}\,,$
with $\alpha_{CFL}>2$, for the porous-medium equation which is the large-time
limit behaviour of (4.1), see [22] and subsection 5.3. All our simulations are
done with $\alpha_{CFL}=8$. Although the CFL analysis in [16] applies only to
equations written in variational form that includes (4.1) only for $m=1$, all
numerical tests seem not to be affected by the chosen CFL condition. Finally,
we point out that $u(t,\varphi(t,\eta_{1}))=u(t,\varphi(t,\eta_{N}))=0$.
Because of this fact, in all the plots which follow, the first and last nodes
are never plotted.
### 4.2 Formation of discontinuities
#### 4.2.1 Propagation of the support of solutions and waiting time
phenomenon
Observe that Eq. (4.5) and (4.7) imply that the speed of propagation of the
support is exactly
$\varphi_{t}(\pm\tfrac{1}{2}^{\mp})=\pm\left(\frac{1}{\varphi_{\eta}(\pm\tfrac{1}{2}^{\mp})}\right)^{m-1}=\pm
u^{m-1}(\pm A(t))\,.$ (4.8)
(here and from now on $f(a^{\pm}):=\lim_{x\to a^{\pm}}f(x)$ for a generic
function $f$ and point $a$). This coincides with well-known results in [21].
If we let $0\leq\psi(\eta)=\frac{1}{\varphi_{\eta}}(\eta)=u(\varphi(\eta))$,
then (4.5) transforms into:
$\psi_{t}=\psi^{2}\left(\frac{\psi^{m-1}\psi_{\eta}}{\sqrt{1+\psi_{\eta}^{2}}}\right)_{\eta}.$
(4.9)
Note that
$\psi_{\eta}(t,\eta)=\left(\frac{1}{\varphi_{\eta}}\right)_{\eta}(t,\eta)=\left(\frac{u_{x}}{u}\right)(t,\varphi(t,\eta))\,.$
(4.10)
In case $u(\pm A(t))\neq 0$ or $u_{x}(\pm A(t))\neq 0$ if $u(\pm A(t))=0$,
then the boundary condition for $\psi$ is just a vertical contact angle using
(4.7)-(4.10):
$\psi_{\eta}(t,\pm\tfrac{1}{2}^{\mp})=\mp\infty\,.$ (4.11)
Figure 1: Left: initial datum. Right: Evolution of $u_{0}$ in case $m=1$ at
different times.
Consider now $m=1$. By (1.4), $|\varphi_{t}|\leq 1$ and
$\varphi_{t}(\pm\frac{1}{2}^{\mp})=\pm 1$, it follows that
$(\varphi_{\eta})_{t}(\pm\frac{1}{2})\geq 0$. This implies that
$\psi_{t}(\pm\frac{1}{2}^{\mp})\leq 0$ by definition of $\psi(t,\eta)$. In
particular, this shows that in case $\psi(t_{0},\pm\frac{1}{2}^{\mp})=0$, this
condition remains true for all time as shown in Proposition 3.2.
We define next
$w(t,\eta):=\psi(t,\eta)\psi_{\eta}(t,\eta)=u_{x}(t,\varphi(t,\eta))$. The
analysis above also shows that, in case $\psi(t_{0},\pm\frac{1}{2}^{\mp})=0$,
then $|w(t,\pm\frac{1}{2})|\leq|w(0,\pm\frac{1}{2})|$. On the other hand, in
the bulk, $w$ verifies the following equation
$w_{t}=\frac{\psi^{5}w_{\eta\eta}}{(\psi^{2}+w^{2})^{\frac{3}{2}}}+\frac{3\psi
w}{(\psi^{2}+w^{2})^{\frac{5}{2}}}(2w^{2}w_{\eta}\psi^{2}-w_{\eta}^{2}\psi^{4}-w^{4}).$
Thus, if $w_{0}$ is initially bounded, $w$ remains bounded in
$[-\frac{1}{2},\frac{1}{2}]$ as proved in Section 3. Observe that at a point
$\eta_{0}$ of maximum of $w$, we have
$w_{t}(\eta_{0})\leq-\frac{3\psi
w^{5}}{(\psi^{2}+w^{2})^{\frac{5}{2}}}(\eta_{0})\leq 0,$
implying the claim.
We show a numerical experiment with $u_{0}(x)=(1-|x|)_{+}$ as initial datum
which does not satisfy the conditions of Theorem 1.1. We take $N=1000$ for the
simulations. We point out that since the initial datum is $0$ at the extremes
of the support, we need a lot of nodes in the discretization near them since
due to the change of variables (4.3), then
$\varphi_{\eta}(\pm\frac{1}{2}^{\mp})=\mp\infty$ and we want the numerical
scheme to be able to capture this feature. We report in Fig. 1 the precise
evolution of the support showing the smoothing effect at $x=0$, the
boundedness of the derivative all over the support including the boundaries,
and the expansion of the boundary at precise unit speed as expected by the
theory in Theorem 1.1 and the heuristic arguments above.
Figure 2: Evolution of $u_{0}$ in case $m=1.5$ at different times. Left:
Before the discontinuity at the tip of the support appears. Right: Evolution
of the discontinuity front after.
Let us now take $m>1$. In case $u(\pm A(0)^{\mp})=0$ (i.e.
$\psi(0,\pm\frac{1}{2}^{\mp})=0$), then (4.8) implies that the support of the
solution does not move at all whenever $u(\pm A(t))=0$. The solution will
become positive at the tip of the support $u(\pm A(t))>0$ with $t>t_{0}>0$ if
and only if $\psi_{t}(t_{0},\pm\frac{1}{2}^{\mp})\in(0,+\infty]$ with $u(\pm
A(t_{0})^{\mp})=\psi(t_{0},\pm\frac{1}{2}^{\mp})=0$. In case $u_{x}(\pm
A(t_{0}))\neq 0$, we can use (4.11) to approximate terms
$(1+\psi_{\eta}^{2})^{1/2}\simeq\psi_{\eta}$ around $\pm\frac{1}{2}$ in the
expression of (4.9) to get
$\psi_{t}(t_{0},\pm\tfrac{1}{2})=\lim_{\eta\to\pm\frac{1}{2}}\psi_{t}(t_{0},\eta)=\lim_{\eta\to\pm\frac{1}{2}}(m-1)(\psi^{m}\psi_{\eta})(t_{0},\eta).$
As a consequence, $\psi_{t}(t_{0},\pm\tfrac{1}{2}^{\mp})>0$ if and only if
$\lim_{\eta\to\pm\frac{1}{2}}(\psi^{m+1})_{\eta}(t_{0},\eta)>0\,.$ (4.12)
Observe that this condition in (4.12) is implied by
$\lim_{x\to\pm A(t)}(u^{m+1}(x))_{x}(t_{0},\pm A(t))>0\,.$
In such case, the solution becomes positive at $\pm A(t)$ and then, according
to (4.8), its support starts to increase. We note that this waiting time
phenomenon is similar to that of the classical porous medium equation but the
condition for the support to start moving is completely different to the one
obtained in [11]. Supposing a potential growth of $\psi$, i.e.
$\psi(t_{0},\eta)\simeq C\left(t_{0},\frac{1}{2}-|\eta|\right)^{p}$, $p>0$,
for $\eta\to\pm\frac{1}{2}^{\mp}$, then we obtain that
$\psi_{t}(t_{0}\pm\frac{1}{2}^{\mp})=+\infty$ if and only if
$p<\frac{1}{m+1}$.
Figure 3: Evolution of $u_{0}$ in case $m=3$ at different times. Top left:
Before a discontinuity on the bulk appears. Top right: After the discontinuity
front forms till it reaches the tip of the support. Bottom: After the
discontinuity front starts to move.
We point out that this behavior has already been numerically obtained in [10].
In Fig. 2, we show this waiting time phenomenon for $m=1.5$. One can observe
that initially the support does not move since the behavior near the boundary
is $\psi(0,\eta)\simeq C\left(0,\frac{1}{2}-|\eta|\right)^{\frac{1}{2}}$, then
the derivative at the boundary builds up until the behavior at the boundary
reaches the critical value producing the lift-off of the boundary point. More
interesting is the case $m=3$ which we show in Fig. 3. There, a discontinuity
in the bulk appears before the support starts to move.
#### 4.2.2 Formation of discontinuities in the bulk
In view of the first example in the last section, one may think that
discontinuities may appear only as a consequence of the waiting time
phenomenon; i.e. particles tend to dissipate but their support does not move,
which may create the discontinuities. In this section we heuristically study
that it is possible to create discontinuities inside the bulk even if the
solution are far away from zero as seen in Fig. 3.
First we treat the case $m=1$. In case of an upwards jump discontinuity or a
vertical angle at a point $\eta_{0}\in]-\frac{1}{2},\frac{1}{2}[$ such that
$\psi_{\eta}(\eta_{0})^{\pm}=+\infty$ , then we also have
$\varphi_{t}(\eta_{0})=-1$. Since $|\varphi_{t}|\leq 1$, then
$\varphi_{t}(\eta_{0})=-1$ implies that $\varphi_{t}$ is nonincreasing to the
left and nondecreasing to the right of $\eta_{0}$, i.e.,
$((\varphi_{\eta})^{-})_{t}\leq 0$ and $((\varphi_{\eta})^{+})_{t}\geq 0$.
This shows that $(\psi(\eta_{0})^{-})_{t}\geq 0$ while
$(\psi(\eta_{0})^{+})_{t}\leq 0$, which implies that the size of the
discontinuity reduces in for an upwards jump discontinuity or that no
discontinuity is created if initially there is a vertical angle.
This last phenomenon is not true if $m>1$ in the case of a vertical angle at a
point $\eta_{0}\in]-\frac{1}{2},\frac{1}{2}[$ such that
$\psi_{\eta}(\eta_{0})^{\pm}=+\infty$. From the equation (4.9) for $\psi$ as
in previous subsection, we deduce that
$\psi_{t}(\eta_{0})=(m-1)\psi^{m}\psi_{\eta}(\eta_{0})$, and thus, a
discontinuity is created. Once we have a discontinuity at $\eta_{0}$ the
evolution is theoretically unknown.
Figure 4: Evolution of solutions corresponding to $u_{0}$. Top left: Initial
datum $u_{0}$. Top right: Evolution for $m=1$ at small times. Bottom left:
Evolution for $m=1$ for larger times. Bottom right: Evolution for $m=4$.
In order to show this behavior we have taken two types of initial datum with
$N=1000$:
$u_{0}(x):=\frac{1}{4}\chi_{[-1,1]}+\frac{3}{2\sqrt{2}}\sqrt{\frac{1}{2}-|x|}\chi_{[-\frac{1}{2},\frac{1}{2}]}\qquad\mbox{
and
}\qquad\tilde{u}_{0}(x):=\frac{1}{4}\chi_{[-1,-\frac{1}{2}]\cup[\frac{1}{2},1]}+\frac{3}{4}\chi_{]-\frac{1}{2},\frac{1}{2}[}\,.$
We imposed a high concentration of nodes around the vertical angles or
discontinuities (i.e. $x=\pm\frac{1}{2}$).
Figure 5: Evolution of solutions for $\tilde{u}_{0}$. Top: Evolution for $m=1$
at different times. Bottom: Evolution for $m=2$ at different times.
In Fig. 4 we observe the evolution of the solutions corresponding to the
initial datum $u_{0}$, demonstrating the above heuristics. In Fig. 5, we see
how an initially discontinuous initial datum $\tilde{u}_{0}$ is smoothed
during the evolution both for $m=1$ (as heuristically deduced before) and for
$m=2$. We observe that the smoothing of the discontinuity is slower with $m>1$
than that of $m=1$.
### 4.3 Asymptotic behavior
In this Section, guided by heuristics, we numerically observe the asymptotic
behavior of solutions to (4.1) and the rate of convergence towards their
asymptotic steady state, for which no result is available in the literature.
Performing the classical self-similar change of variables [17] that translates
porous medium equation onto nonlinear Fokker-Planck equations given by
$v(x,t)=e^{t}u(e^{t}x,k(e^{\frac{t}{k}}-1))\,,$ (4.13)
with $k=\frac{1}{m+1}$, then equation (4.1) transforms into
$v_{t}={\rm div}\left(xv+\frac{v^{m}\nabla v}{\sqrt{v^{2}+e^{-2t}|\nabla
v|^{2}}}\right)\,.$ (4.14)
Therefore, formally, when $t\to\infty$ solutions of (4.14) should converge to
a stationary solution of $v_{t}={\rm div}\left(xv+v^{m-1}\nabla v\right)$,
i.e., to a Gaussian $V(x)$ for $m=1$ or to the corresponding Barenblatt
solution $V_{m}(x)$ when $m>1$ given by
$V(x)=\frac{1}{\sqrt{2\pi}}e^{-\frac{x^{2}}{2}}\qquad\mbox{and}\qquad
V_{m}(x)=\left(C_{m}-\frac{m-1}{2}x^{2}\right)_{+}^{\frac{1}{m-1}}\,,$
where $C_{m}$ is uniquely determined by the conservation of mass. In the
original variables, then solutions should converge to the corresponding self-
similar profiles obtained from $V$ and $V_{m}$ via the change of variables
(4.13) except time translations. To be precise, the self-similar solutions are
given by
$U(|x|,t)=\frac{e^{-\frac{x^{2}}{4t}}}{\sqrt{4\pi t}}\,\mbox{ for }m=1\mbox{
and
}\,\,U_{m}(|x|,t)=t^{\frac{-1}{m+1}}\left(\tilde{C}_{m}-\frac{m-1}{2m(m+1)}|x|^{2}t^{\frac{-2}{m+1}}\right)_{+}^{\frac{1}{m-1}}\mbox{
for }m>1,$
where $\tilde{C}_{m}$ is determined as above.
Figure 6: Top and Bottom left: Evolution of $u_{0}$ in case $m=1$ at different
times with $N=100$. Bottom right: log-log plot of the estimate
$\|u(t)-U(t)\|_{1}$ with $N=1000$.
In the following computations we have taken
$u_{0}:=\chi_{[-\frac{1}{2},\frac{1}{2}]}$, $N=100$. We plot the evolution of
the initial datum for different values of $m$ and an estimate of the
difference of $u-U_{m}$ in the $L^{1}$-norm. More precisely, we took
$\|u(t)-U_{m}(t)\|_{1}:=\frac{1}{N}\sum_{i=1}^{N}|u(x_{i},t)-U_{m}(x_{i},t)|$.
Some comments are in order. First of all, in Figure 6, we note that for $m=1$,
while time is small, the numerical solution satisfies both the linear
propagation of the support property, as well as the vertical contact angle
property. However, for larger times, these two conditions are lost during the
computation. This is due to the fact that we took a fixed number of nodes
($N=100$), and as time increases, this number of nodes is clearly
insufficient. We have observed that by increasing the number of nodes (for
instance to $N=1000$) the time in which the numerical solution is more
accurate increases. We can also see in Figure 6, that, in spite of this, the
numerical solution tends to a Gaussian with an algebraic rate of convergence
that seems to be $\tfrac{1}{2}$ the one of the heat equation. However, it is
exactly by the same reason as before that when time increases, the rate of
convergence degenerates. For this reason, we have included in Figure 6 the
$L^{1}$-convergence rate with $N=1000$.
Figure 7: Top and Bottom left: Evolution of $u_{0}$ in case $m=2$ at different
times. Bottom right: log-log plot of the estimate $\|u(t)-U_{2}(t)\|_{1}$.
Instead, when $m>1$, the support of the solution does not propagate so fast
and we can observe in Figures 7 and 8 how the vertical contact angle property
is preserved even for large times. Moreover, in Figures 7 and 8 we can see how
the numerical solution tends to $U_{m}$ for $m=2$ and $m=10$. In both cases,
the rate of convergence is algebraic and, numerically, it is surprisingly seen
that it might correspond to $\frac{1}{3}$ in the first case and to
$\frac{1}{11}$ in the second one; i.e.: the same convergence rate as for the
porous medium equation, see [17, 33].
Figure 8: Top: Evolution of $u_{0}$ in case $m=10$ at different times. Bottom
left: log-log plot of the estimate $\|u(t)-U_{10}(t)\|_{1}$. Bottom right:
zoom of the final time interval
### 4.4 Convergence toward the homogeneous relativistic heat equation
We finally show numerically how solutions to (1.1), converge to solutions of
the homogeneous relativistic heat equation
$\left\\{\begin{array}[]{cc}\displaystyle
u_{t}=\left(u\frac{u_{x}}{|u_{x}|}\right)_{x}&{\rm in\
}I\\!\\!R^{N}\times[0,T]\\\ u_{0}(x)=u_{0}&{\rm in\
}I\\!\\!R^{N}\end{array}\right.$
when the kinematic viscosity $\nu\to+\infty$ as already proved in [7]. In Fig.
9 we estimate the evolution in time of the difference in the $L^{1}$-norm for
solutions corresponding to the initial data
$u_{0}=\chi_{[-\frac{1}{2},\frac{1}{2}]}$ for different values of $\nu$ with
respect to the explicit solution $u_{hom}$, given by
$u_{hom}(x,t)=\frac{1}{1+2t}\chi_{[-\frac{1}{2}-t,\frac{1}{2}+t]}$
when $\nu\to\infty$.
Figure 9: Top left: Numerical solution at $t=1$ for different values of $\nu$.
Top Right: Numerical solution at $t=100$ for different values of $\nu$.
Bottom: Evolution of the $L^{1}$-difference with respect to $u_{hom}$.
## Appendix: A primer on Entropy Solutions
We collect in this Appendix some definitions that are needed to work with
entropy solutions of flux limited diffusion equations.
Note that the equation (1.3) can be written as
$u_{t}={\bf b}(u,u_{x})_{x},\qquad\hbox{in \hskip 5.69046pt $Q_{T}=(0,T)\times
I\\!\\!R$}$ (A.1)
where ${\bf b}(z,\xi)=\nabla_{\xi}f(z,\xi)$ and
$f(z,\xi)=z\sqrt{z^{2}+|\xi|^{2}}.$ (A.2)
As usual, we define
$h(z,\xi)={\bf b}(z,\xi)\cdot\xi=\frac{z|\xi|^{2}}{\sqrt{z^{2}+|\xi|^{2}}}.$
(A.3)
Note that $f$ is convex in $\xi$ and both $f,h$ have linear growth as
$|\xi|\to\infty$.
### A.1 Functions of bounded variation and some generalizations
Denote by ${\mathcal{L}}^{N}$ and ${\mathcal{H}}^{N-1}$ the $N$-dimensional
Lebesgue measure and the $(N-1)$-dimensional Hausdorff measure in
$I\\!\\!R^{N}$, respectively. Given an open set $\Omega$ in $I\\!\\!R^{N}$ we
denote by ${\mathcal{D}}(\Omega)$ the space of infinitely differentiable
functions with compact support in $\Omega$. The space of continuous functions
with compact support in $I\\!\\!R^{N}$ will be denoted by
$C_{c}(I\\!\\!R^{N})$.
Recall that if $\Omega$ is an open subset of $I\\!\\!R^{N}$, a function $u\in
L^{1}(\Omega)$ whose gradient $Du$ in the sense of distributions is a vector
valued Radon measure with finite total variation in $\Omega$ is called a
function of bounded variation. The class of such functions will be denoted by
$BV(\Omega)$. For $u\in BV(\Omega)$, the vector measure $Du$ decomposes into
its absolutely continuous and singular parts $Du=D^{ac}u+D^{s}u$. Then
$D^{ac}u=\nabla u\ \mathcal{L}^{N}$, where $\nabla u$ is the Radon–Nikodym
derivative of the measure $Du$ with respect to the Lebesgue measure
$\mathcal{L}^{N}$. We also split $D^{s}u$ in two parts: the jump part $D^{j}u$
and the Cantor part $D^{c}u$. It is well known (see for instance [1]) that
$D^{j}u=(u^{+}-u^{-})\nu_{u}\mathcal{H}^{N-1}\mathbin{\vrule
height=9.0pt,width=0.1pt\vrule height=0.1pt,width=9.0pt}J_{u},$
where $u^{+}(x),u^{-}(x)$ denote the upper and lower approximate limits of $u$
at $x$, $J_{u}$ denotes the set of approximate jump points of $u$ (i.e. points
$x\in\Omega$ for which $u^{+}(x)>u^{-}(x)$), and
$\nu_{u}(x)=\frac{Du}{|Du|}(x)$, being $\frac{Du}{|Du|}$ the Radon–Nikodym
derivative of $Du$ with respect to its total variation $|Du|$. For further
information concerning functions of bounded variation we refer to [1].
We need to consider the following truncation functions. For $a<b$, let
$T_{a,b}(r):=\max(\min(b,r),a)$, $T_{a,b}^{l}=T_{a,b}-l$. We denote
$\mathcal{T}_{r}:=\\{T_{a,b}\ :\ 0<a<b\\},\ \ \ $
$\mathcal{T}^{+}:=\\{T_{a,b}^{l}\ :\ 0<a<b,\,l\in I\\!\\!R,\,T_{a,b}^{l}\geq
0\\}.\ \ \ $
Given any function $w$ and $a,b\in I\\!\\!R$ we shall use the notation
$\\{w\geq a\\}=\\{x\in I\\!\\!R^{N}:w(x)\geq a\\}$, $\\{a\leq w\leq
b\\}=\\{x\in I\\!\\!R^{N}:a\leq w(x)\leq b\\}$, and similarly for the sets
$\\{w>a\\}$, $\\{w\leq a\\}$, $\\{w<a\\}$, etc.
We need to consider the following function space
$TBV_{\rm r}^{+}(I\\!\\!R^{N}):=\left\\{w\in L^{1}(I\\!\\!R^{N})^{+}\ :\ \
T_{a,b}(w)-a\in BV(I\\!\\!R^{N}),\ \ \forall\
T_{a,b}\in\mathcal{T}_{r}\right\\}.$
Notice that $TBV_{\rm r}^{+}(I\\!\\!R^{N})$ is closely related to the space
$GBV(I\\!\\!R^{N})$ of generalized functions of bounded variation introduced
by E. Di Giorgi and L. Ambrosio in [1]. Using the chain rule for BV-functions
(see for instance [1]), one can give a sense to $\nabla u$ for a function
$u\in TBV^{+}(I\\!\\!R^{N})$ as the unique function $v$ which satisfies
$\nabla T_{a,b}(u)=v\raisebox{2.0pt}{\rm{$\chi$}}_{\\{a<u<b\\}}\ \ \ \ \
{\mathcal{L}}^{N}-{\rm a.e.},\ \ \forall\ T_{a,b}\in\mathcal{T}_{r}.$
We refer to [1] for details.
### A.2 Functionals defined on BV
In order to define the notion of entropy solutions of (A.1) and give a
characterization of them, we need a functional calculus defined on functions
whose truncations are in $BV$.
Let $\Omega$ be an open subset of $I\\!\\!R^{N}$. Let $g:\Omega\times
I\\!\\!R\times I\\!\\!R^{N}\rightarrow[0,\infty[$ be a Borel function such
that
$C(x)|\zeta|-D(x)\leq g(x,z,\zeta)\leq M^{\prime}(x)+M|\zeta|$
for any $(x,z,\zeta)\in\Omega\times I\\!\\!R\times I\\!\\!R^{N}$, $|z|\leq R$,
and any $R>0$, where $M$ is a positive constant and $C,D,M^{\prime}\geq 0$ are
bounded Borel functions which may depend on $R$. Assume that
$C,D,M^{\prime}\in L^{1}(\Omega)$.
Following Dal Maso [25] we consider the functional:
$\displaystyle{\mathcal{R}}_{g}(u)$ $\displaystyle:=$
$\displaystyle\displaystyle\int_{\Omega}g(x,u(x),\nabla
u(x))\,dx+\int_{\Omega}g^{0}\left(x,\tilde{u}(x),\frac{Du}{|Du|}(x)\right)\,d|D^{c}u|$
$\displaystyle+\displaystyle\int_{J_{u}}\left(\int_{u_{-}(x)}^{u_{+}(x)}g^{0}(x,s,\nu_{u}(x))\,ds\right)\,d\mathcal{H}^{N-1}(x),$
for $u\in BV(\Omega)\cap L^{\infty}(\Omega)$, being $\tilde{u}$ is the
approximated limit of $u$ [1]. The recession function $g^{0}$ of $g$ is
defined by
$g^{0}(x,z,\zeta)=\lim_{t\to 0^{+}}tg\left(x,z,\frac{\zeta}{t}\right).$
It is convex and homogeneous of degree $1$ in $\zeta$.
In case that $\Omega$ is a bounded set, and under standard continuity and
coercivity assumptions, Dal Maso proved in [25] that ${\mathcal{R}}_{g}(u)$ is
$L^{1}$-lower semi-continuous for $u\in BV(\Omega)$. More recently, De Cicco,
Fusco, and Verde [27] have obtained a very general result about the
$L^{1}$-lower semi-continuity of ${\mathcal{R}}_{g}$ in $BV(I\\!\\!R^{N})$.
Assume that $g:I\\!\\!R\times I\\!\\!R^{N}\to[0,\infty[$ is a Borel function
such that
$C|\zeta|-D\leq g(z,\zeta)\leq M(1+|\zeta|)\qquad\forall(z,\zeta)\in
I\\!\\!R^{N},\,|z|\leq R,$ (A.4)
for any $R>0$ and for some constants $C,D,M\geq 0$ which may depend on $R$.
Observe that both functions $f,h$ defined in (A.2), (A.3) satisfy (A.4).
Assume that
$\raisebox{2.0pt}{\rm{$\chi$}}_{\\{u\leq
a\\}}\left(g(u(x),0)-g(a,0)\right),\raisebox{2.0pt}{\rm{$\chi$}}_{\\{u\geq
b\\}}\left(g(u(x),0)-g(b,0)\right)\in L^{1}(I\\!\\!R^{N}),$
for any $u\in L^{1}(I\\!\\!R^{N})^{+}$. Let $u\in TBV_{\rm
r}^{+}(I\\!\\!R^{N})\cap L^{\infty}(I\\!\\!R^{N})$ and
$T=T_{a,b}-l\in{\mathcal{T}}^{+}$. For each $\phi\in C_{c}(I\\!\\!R^{N})$,
$\phi\geq 0$, we define the Radon measure $g(u,DT(u))$ by
$\displaystyle\langle g(u,DT(u)),\phi\rangle$ $\displaystyle:=$
$\displaystyle{\mathcal{R}}_{\phi g}(T_{a,b}(u))+\displaystyle\int_{\\{u\leq
a\\}}\phi(x)\left(g(u(x),0)-g(a,0)\right)\,dx$ (A.5)
$\displaystyle\displaystyle+\int_{\\{u\geq
b\\}}\phi(x)\left(g(u(x),0)-g(b,0)\right)\,dx.$
If $\phi\in C_{c}(I\\!\\!R^{N})$, we write $\phi=\phi^{+}-\phi^{-}$ with
$\phi^{+}=\max(\phi,0)$, $\phi^{-}=-\min(\phi,0)$, and we define $\langle
g(u,DT(u)),\phi\rangle:=\langle g(u,DT(u)),\phi^{+}\rangle-\langle
g(u,DT(u)),\phi^{-}\rangle$.
Recall that, if $g(z,\zeta)$ is continuous in $(z,\zeta)$, convex in $\zeta$
for any $z\in I\\!\\!R$, and $\phi\in C^{1}(I\\!\\!R^{N})^{+}$ has compact
support, then $\langle g(u,DT(u)),\phi\rangle$ is lower semi-continuous in
$TBV^{+}(I\\!\\!R^{N})$ with respect to $L^{1}(I\\!\\!R^{N})$-convergence
[27]. This property is used to prove existence of solutions of (A.1).
We can now define the required functional calculus (see [4, 5, 21]).
Let us denote by ${\mathcal{P}}$ the set of Lipschitz continuous functions
$p:[0,+\infty[\rightarrow I\\!\\!R$ satisfying $p^{\prime}(s)=0$ for $s$ large
enough. We write ${\mathcal{P}}^{+}:=\\{p\in{\mathcal{P}}\ :\ p\geq 0\\}$.
Let $S\in\mathcal{P}^{+}$, $T\in\mathcal{T}^{+}$. We assume that $u\in
TBV_{\rm r}^{+}(I\\!\\!R^{N})\cap L^{\infty}(I\\!\\!R^{N})$ and note that
$\raisebox{2.0pt}{\rm{$\chi$}}_{\\{u\leq
a\\}}S(u)\left(f(u(x),0)-f(a,0)\right),\raisebox{2.0pt}{\rm{$\chi$}}_{\\{u\geq
b\\}}S(u)\left(f(u(x),0)-f(b,0)\right)\in L^{1}(I\\!\\!R^{N}).$
Since $h(z,0)=0$, the last assumption clearly holds also for $h$. We define by
$f_{S}(u,DT(u))$, $h_{S}(u,DT(u))$ as the Radon measures given by (A.5) with
$f_{S}(z,\zeta)=S(z)f(z,\zeta)$. and $h_{S}(z,\zeta)=S(z)h(z,\zeta)$,
respectively.
### A.3 The notion of of entropy solution
Let $L^{1}_{w}(0,T,BV(I\\!\\!R^{N}))$ be the space of weakly∗ measurable
functions $w:[0,T]\to BV(I\\!\\!R^{N})$ (i.e., $t\in[0,T]\to\langle
w(t),\phi\rangle$ is measurable for every $\phi$ in the predual of
$BV(I\\!\\!R^{N})$) such that $\int_{0}^{T}\|w(t)\|_{BV}\,dt<\infty$. Observe
that, since $BV(I\\!\\!R^{N})$ has a separable predual (see [1]), it follows
easily that the map $t\in[0,T]\to\|w(t)\|_{BV}$ is measurable. By
$L^{1}_{loc,w}(0,T,BV(I\\!\\!R^{N}))$ we denote the space of weakly∗
measurable functions $w:[0,T]\to BV(I\\!\\!R^{N})$ such that the map
$t\in[0,T]\to\|w(t)\|_{BV}$ is in $L^{1}_{loc}(]0,T[)$.
###### Definition 4.1.
Assume that $u_{0}\in(L^{1}(I\\!\\!R^{N})\cap L^{\infty}(I\\!\\!R^{N}))^{+}$.
A measurable function $u:]0,T[\times I\\!\\!R^{N}\rightarrow I\\!\\!R$ is an
entropy solution of (A.1) in $Q_{T}=]0,T[\times I\\!\\!R^{N}$ if $u\in
C([0,T];L^{1}(I\\!\\!R^{N}))$, $T_{a,b}(u(\cdot))-a\in
L^{1}_{loc,w}(0,T,BV(I\\!\\!R^{N}))$ for all $0<a<b$, and
* (i)
$u(0)=u_{0}$, and
* (ii)
the following inequality is satisfied
$\displaystyle\displaystyle\int_{0}^{T}\int_{I\\!\\!R^{N}}\phi
h_{S}(u,DT(u))\,dt+\int_{0}^{T}\int_{I\\!\\!R^{N}}\phi h_{T}(u,DS(u))\,dt$
$\displaystyle\leq\displaystyle\int_{0}^{T}\int_{I\\!\\!R^{N}}\Big{\\{}J_{TS}(u(t))\phi^{\prime}(t)-{\bf
b}(u(t),\nabla u(t))\cdot\nabla\phi\ T(u(t))S(u(t))\Big{\\}}dxdt,$
for truncation functions $S,T\in\mathcal{T}^{+}$, and any smooth function
$\phi$ of compact support, in particular those of the form
$\phi(t,x)=\phi_{1}(t)\rho(x)$, $\phi_{1}\in{\mathcal{D}}(]0,T[)$,
$\rho\in{\mathcal{D}}(I\\!\\!R^{N})$, where $J_{q}(r)$ denotes the primitive
of $q$ for any function $q$; i.e. $\displaystyle
J_{q}(r):=\int_{0}^{r}q(s)\,ds$
Acknowledgements. JAC acknowledges partial support by MICINN project,
reference MICINN MTM2011-27739-C04-02, by GRC 2009 SGR 345 by the Generalitat
de Catalunya, and by the Engineering and Physical Sciences Research Council
grant number EP/K008404/1. JAC also acknowledges support from the Royal
Society through a Wolfson Research Merit Award. VC acknowledges partial
support by MICINN project, reference MTM2009-08171, by GRC reference 2009 SGR
773 and by ”ICREA Acadèmia” prize for excellence in research funded both by
the Generalitat de Catalunya. S. Moll acknowledges partial support by MICINN
project, reference MTM2012-31103.
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|
arxiv-papers
| 2012-11-27T14:24:38 |
2024-09-04T02:49:38.541416
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "J.A. Carrillo, V. Caselles, S. Moll",
"submitter": "Jose A. Carrillo",
"url": "https://arxiv.org/abs/1211.6313"
}
|
1211.6317
|
# Group delay of electromagnetic pulses through multilayer dielectric mirrors
beyond special relativity
Jiang-Tao Liu1,2∗0001Department of Physics, Nanchang University, Nanchang
330031, China. 2Institute for Advanced Study, Nanchang University, Nanchang
330031, China. 3Department of Physics, Semiconductor Photonics Research
Center, Xiamen University, Xiamen 361005, China. 4Key Laboratory of Materials
Physics, Institute of Solid State Physics, Chinese Academy of Sciences Hefei
230031, China. ∗[email protected]. †[email protected]., Wu Xin1, Nian-Hua Liu1,2,
Jun Li3 and Fu-Hai Su4†
A number of unexpected phenomena historically broke through corresponding the
traditional physics framework, thus motivating scientists to develop more
advanced and accurate theories. A well-known example is quantum tunnelling in
$\alpha$-decay. The quantum tunnelling rate can be described accurately by the
quantum theory but cannot be explained by classical mechanics. In quantum
tunnelling, the length of time during which a particle tunnels through a
barrier remains undetermined[1, 2, 3, 4, 5, 6]. Based on classical quantum and
classical electrodynamics theories, the group delay during quantum tunnelling
is independent of barrier thickness. However, in more accurate theories, such
as the general theory of relativity[7], is this condition still valid? In this
Letter, we investigate the group delay of optical pulses through multilayer
dielectric mirrors (MDM) combined with gravitational wave (GW)[7, 8, 9, 10].
We find that the delay increases linearly with MDM length for the transmitted
wave packet, whereas the Hartman effect disappears. Our study provides insight
into the nature of both quantum tunnelling and GW.
The length of time during which quantum particles tunnel through a barrier has
attracted considerable attention for both fundamental and technological
reasons since the 1930s[1, 2, 3, 4, 5, 6]. Hartman calculated the tunneling of
a wavepacket through a rectangular potential barrier[3] and found that group
delay becomes constant as barrier length increases. This phenomenon, known as
the Hartman effect, implies that for sufficiently large barriers, the
effective group velocity of a particle may be superluminal. Although a number
of experiments have reported observations of electromagnetic waves propagating
with ”superluminal tunneling velocities,” the definition of tunneling time and
its exact physical meaning based on experimental results remain under heated
debate[4, 5, 6]. A large number of tunneling time definitions have been
proposed, including group delay or phase time[3], dwell time[6], Larmor
times[11, 12], and Büttiker-Landauer time[13]. Winful recently proposed that
the group delay in tunneling represents a lifetime of stored energy escaping
through both sides of the barrier and does not represent a transit time[6,
14]. Thus, the issues of superluminality, causality, or the speed of
information transfer do not even arise.
Beyond Schrödinger’s nonrelativistic quantum mechanics, the group delay for
Dirac particles traveling through a potential well was also studied by using
Dirac’s fully relativistic quantum theory[15]. The behavior of Dirac particles
is found to be the same as that in nonrelativistic quantum mechanics. Liu et
al. recently studied the effect of the electromagnetic fields of the group
delay of electrons and found that the group delay of the transmitted wave
packet increases linearly with barrier length for the transmitted wave
packet[16]. This peculiar tunneling effect is attributed to current leakage in
a time-dependent barrier generated via the electromagnetic fields[13, 16]. If
the quantum fluctuation or the zero-point field is considered, all potential
barriers are combined with electromagnetic fields. Thus, in the framework of
quantum field theory, the Hartman effect of electron tunneling may disappear
because of the existence of electromagnetic vacuum fields.
However, in photon tunnelling, a number of theories based on the special
relativistic covariant Maxwell equations have proven that the group delay of
photons becomes constant as the length of optical structures increases[4, 5,
6, 17, 18, 19, 20]. However, whether the Hartman effect still exists when a
more accurate theory (e.g. general relativity theory) remains unclear. One of
the unique predictions in general relativity theory is the existence of
gravitational wave (GW)[7, 8, 9, 10]. If optical structures such as a
multilayer dielectric mirror (MDM) (i.e. a one-dimensional photonic crystal)
is irradiated by GW, the center frequency and the width of the photonic
bandgap of the MDM will vary with the GW. Similar to the electron tunnelling
in a time-dependent barrier, variations of the photonic bandgap will result in
an additional leak photon current. Such current may propagate at the speed of
light. Thus, the Hartman effect may also be absent in photon tunnelling within
the framework of general relativity theory.
In this paper, we investigate the effect of GW on the group delay of photon
passing through MDM. Our simulation shows that with a thick MDM, group delay
increases linearly with increasing barrier width. The group velocity is
approximately $2.95\times 10^{8}$ m/s, slightly less than the speed of light
in in vacuum. Superluminality or causality no longer occurs. We also find that
the group delay of tunneling photons is sensitive to GW. Our study may
facilitate further understanding of both quantum tunnelling and GW as well as
provide a different method for the detection of GW. In particularly, the MDM
comprises alternating dielectric layers and vacuum layers [see Fig. 1(a)]. All
layers are nonmagnetic ($\mu=1$), and the thicknesses of the dielectric layers
(vacuum layers) satisfy $D_{1}=\lambda_{0}/4\sqrt{\varepsilon_{1}}$
($D_{2}=\lambda_{0}/4+\zeta_{2}\lambda_{0}/2$), where $\varepsilon_{1}$ is the
permittivity of dielectric layers, $\lambda_{0}$ is the center frequency of
the input electromagnetic pulse, and $\zeta_{2}$ is a positive integer. The
group delay of tunneling photons is generally more sensitive to GW at large
$\zeta_{2}$. We set $\zeta_{2}=5$ in this paper, unless otherwise specified.
The electromagnetic pulse is incident along the normal of the surface of MDM,
and the propagation direction of the plane-polarized GW is parallel to the
surface of MDM.
When the polarization GW occurs, the layer spacing of dielectric layers will
vary with the GW. Similar to the case of two masses separated by a distance D
along the Z direction that are coupled by a lossy spring, the equation for the
differential motion of the masses in the GW becomes[7, 21, 22]
$\frac{d^{2}z_{R}}{dt^{2}}+\frac{\omega_{m0}}{Q}\frac{dz_{R}}{dt}+\omega_{m0}^{2}z_{R}=\frac{1}{2}\frac{d^{2}h_{22}}{dt^{2}}D,$
(1)
where $z_{R}$ is the proper relative displacement of the two masses, $h_{22}$
is the the perturbation matrix (tensor) element resulting from the GW; and
$\omega_{m0}$ and $Q$ are the natural frequency and the q-factor of the lossy
spring, respectively. If the dielectric layers in MDM are free and under zero-
velocity initial conditions, the equation of differential motion can be
described by[7, 21, 22]
$z_{R}/{D}=h_{22}/2=A_{GW}\cos(\omega_{GW}t)/2,$ (2)
where $A_{GW}$ is the GW amplitude. Thus, in the propagation of
electromagnetic waves, the permittivity distribution will also change with
time. To study such a time-dependent photon scattering process, we employ the
finite-difference time-domain (FDTD) method to solve the time-dependent
Maxwell equations numerically[23, 24]. FDTD is a time-domain technique that
can provide animated displays of electromagnetic field movement through
various models, e.g., the photonic crystals, radar, etc. In the FDTD method,
the one-dimensional Maxwell equations are replaced by a finite set of finite
differential equations[23, 24]
$\left\\{\begin{array}[]{c}E_{x}^{n+1}(k)=E_{x}^{n}(k)-\frac{\Delta
t}{\varepsilon\Delta
z\Upsilon_{GW}}\left[H_{y}^{n+1/2}(k+1/2)-H_{y}^{n+1/2}(k-1/2)\right],\\\
H_{y}^{n+1/2}(k+1/2)=H_{y}^{n-1/2}(k+1/2)-\frac{\Delta t}{\mu\Delta
z\Upsilon_{GW}}\left[E_{x}^{n+1}(k+1)-E_{x}^{n}(k)\right],\end{array}\right.$
(3)
where $E_{x}$ ($H_{y}$) is the electric field (magnetic field) of the
electromagnetic wave, $(i,k)=(i\triangle x,k\triangle t)$ denote a grid point
of the space and time; and for any function of space and time $F(i\triangle
x,k\triangle t)=F^{k}(i)$, $\Upsilon_{GW}$ depends on the GW wave-induced
displacement. According to Eq. 2, in the vacuum layers
$\Upsilon_{GW}=1+A_{GW}\cos(\omega_{GW}t)/2$. The thickness changes of the
dielectric layers are minimal because the natural frequency of dielectric
layer is nonresonant with the frequency of GW, and the thickness the
dielectric layers is considerably less than that of the vacuum layers with
large $\mathcal{L}_{2}$. Thus, we set $\Upsilon_{GW}=1$ in dielectric layers.
At the input boundary, a Gaussian electromagnetic wave packet is injected. The
wave function at the input boundary is set as
$E_{x}=H_{y}=\frac{1}{\sqrt{2}}\exp\left[-4\pi\left(t-\tau_{0}\right)^{2}/\tau_{0}^{2}\right]e^{i\omega_{0}t}$.
To reduce distortion and numerical errors, a relatively long pulse is used:
$\tau_{0}=200T_{0}$, where $T_{0}$ is the period of the electromagnetic wave.
By numerically solving Eq. (3) directly in the time domain, the propagation of
a wave packet through a barrier can be demonstrated in real time. For
computational stability, the space increment $\Delta x$ and the time increment
$\Delta t$ need to satisfy the relation $\Delta x>c\Delta t$. Furthermore, the
space increment $\Delta x$ must be significantly smaller than the wavelength
of electromagnetic wave $\Delta x<\lambda_{0}/12$, and the time increment
$\Delta t$ must be considerably smaller than the period of the GW. To ensure
high precision, the space increment $\Delta x=\lambda_{0}/1.5\times 10^{3}$
and the time increment $\Delta t=2\times 10^{-4}T_{0}$ are used. When the
space and time increments are increased or reduced 10 times, the error is less
than 3%.
Figure 1: (a) Schematic diagram of the tunneling process in an MDM structure.
(b) Group delay for the reflected wave packet and (c) that for the transmitted
wave packet as a function of the number of MDM periods with different GW
amplitudes. Incident (black lines), tunneled (red lines) and reflected (green
lines) pulses with GW amplitude $A_{GW}=1\times 10^{-4}$ for the following
number of MDM periods: (d)$n_{MDM}=3$, (e)$n_{MDM}=19$, and (f) $n_{MDM}=30$.
The inset shows the normalized tunneled pulse overlaid with the incident
pulse.
Numerical results of the group delay (i.e., the delays of the peaks of the
reflected and transmitted pulses) are shown in Figs. 1(b)-(f). Fig. 1(b) shows
the group delay for the reflected wave packet as a function of the number of
MDM periods $n_{MDM}$. Similar to the traditional quantum tunneling, the group
delay is saturated by increasing $n_{MDM}$, and the saturated group delay is
identical to the dwell time. Meanwhile, the group delay for the reflected wave
packet is unaffected by the extrinsic GW.
However, the influence of the GW on the group delay of a transmitted wave
packet differs from the given case. As shown in Fig. 1(c), for the GW
amplitude $A_{GW}=1\times 10^{-4}$, the group delay for the transmitted wave
packet increases linearly when $n_{MDM}>24$. This result can be explained by
the variations of the photonic bandgap attributed to the GW. Similar to the
electron tunnelling in a time-dependent barrier[13, 16], the variations of the
photonic bandgap will result in an additional leak photon current. Given that
the amplitude of the additional leakage current attributed GW is quite small,
for the case of a small number of MDM periods, e.g., $n_{MDM}<15$, the
tunneling current is significantly larger than the additional leakage current
and is unaffected by the extrinsic GW [see Fig. 1(c)]. However, for the case
of large number of MDM periods, e.g., $n_{MDM}>24$, the tunneling current is
significantly weaker than the additional leakage current. The additional
leakage current determines the group delay, and the group delay increases with
the amplitude of GW. Under a weaker GW, the additional leakage current will
determine the group delay for the transmitted wave packet with a relatively
larger $n_{MDM}$. Specifically, for $A_{GW}=1\times 10^{-4}$ ($A_{GW}=2\times
10^{-5}$), the group delay increases linearly when $n_{MDM}>24$
($n_{MDM}>26$). Thus, if $n_{MDM}$ is sufficiently large, even under a
relatively weak GW (e.g., the GW background radiation[25]), no Hartman effect
occurs. On the other hand, for a non-strictly periodic GW emitted by various
sources(e.g., the chaos compact binary system[26]), the time-dependent
variations attributed to the GW will also result in an additional leak photon
current, thus modifying the group delay. From Fig. 1(c), we can also find that
the group velocity of the additional leakage current is independent of the
amplitude of GW. The group velocity is approximately $2.95\times 10^{8}$ m/s,
which is the same as the speed of light in vacuum. No superluminal appears.
The transmitted wave packet includes two components: the tunneling current and
the additional leakage current induced by the time-dependent modulation, which
may distort the transmitted wave packet. However, the relative weight of the
contribution of the tunneling current and that of the additional leakage
current vary with the number of MDM periods $n_{MDM}$. For a small number of
MDM periods, e.g., $n_{MDM}=3$, the amplitude of the tunneling current is
significantly larger than that of the additional leakage current. Similar to
in traditional quantum tunneling, the distortion is minimal [see Fig. 1(d)].
As the number of MDM periods increases, the tunneling rates decrease rapidly.
Meanwhile, the number of MDM periods only has a slight effect on the
additional leakage current. For a large number of MDM periods, e.g,
$n_{MDM}=19$, the amplitude of the additional leakage current and that of the
tunneling current are comparable. Considering that the group delay of the
additional leakage current is larger than that of the tunneling current, a
serious distortion of transmitted wave packet occurs [see Fig. 3(e)]. However,
if the number of MDM periods is sufficient, e.g., $n_{MDM}=30$, the tunneling
rate is very small, and the additional leakage current is the main contributor
to the transmitted wave packet. No distortion occurs at this scale [see Fig.
3(f)].
However, we have to determine whether the group delay of the undistorted
transmitted wave packet in a thick MDM with GW equates to tunneling time. In
traditional quantum tunneling, the consensus is that the group delay does not
equate to a tunneling time for the following reasons: the group delay is equal
to the dwell time, which denotes a lifetime of stored energy escaping through
the barrier, not a pulse traveling through the barrier[6, 14]; to reduce the
distortion, the length of the wave packet is often significantly larger than
the gap length of the barrier[6, 27], and the peak of the input pulse does not
even propagate into the barrier. However, the group delay of photons with GW
is different. The group delay with GW cannot be explained by the dwell time
because such delay is considerably larger than the dwell time. The full width
at half maximum (FWHM) of the injected wave packet is approximately
$66\lambda_{0}$ (or $66T_{0}$), which is smaller than the total optical path
length of MDM at $n_{MDM}=30$. Meanwhile, the group delay for $n_{MDM}=30$ is
approximately $80T_{0}$, which is larger than the FWHM of the injected wave
packet. The peaks of the injected wave packet and that of the transmitted wave
packet are distinguishable. Thus, the group delay of the transmitted wave
packet in a thick MDM with GW may be regarded as the tunneling time. However,
although the group delay equates to the tunneling time, the tunneling time is
increased with the number of MDM periods. Thus, the superluminality in the
tunneling process no longer occurs.
Figure 2: (a) Group delay of the transmitted wave packet as a function of the
number of MDM periods for different GW frequencies with $A_{GW}=1\times
10^{-4}$. (b) Group delay of the transmitted wave packet as a function of the
number of MDM periods for different vacuum layer thicknesses with
$A_{GW}=1\times 10^{-4}$, $\omega_{GW}=0.1\omega_{0}$, and
$\tau_{0}=2000T_{0}$. The inset shows the relative group delay as a function
of the number of MDM periods.
Notably, the group delay in the tunneling process also shows good sensitivity
to the GW. For instance, for $n_{MDM}=25$ and $A_{GW}=1\times 10^{-4}$, the
group delay without (with) GW is approximately $5.3T_{0}$ ($65T_{0}$). The
group delay is increased by approximately 12 times. In the GW detection by
using a Michelson interferometer, the interferometer measures the intensity
rather than the time delay of the interference light. We can still make a
comparison with Michelson interferometer. For a Michelson interferometer with
an arm length of $L_{o}=75\lambda_{0}$ (same as the total optical path length
of MDM with $n_{MDM}=25$), only when the GW amplitude $A_{GW}$ is
approximately $1.6\times 10^{-3}$ (i.e., the GW amplitude satisfies $4\pi
A_{GW}L_{o}/\lambda_{0}\approx\pi/2$), the intensity of the interference light
with and without GW can vary by approximately 12 times.
The sensitivity of the group delay to the GW depends on the frequency of the
GW [see Fig. 2(a)]. For high-frequency GW, e.g., $\omega_{GW}=\omega_{0}$, the
variations of the photonic bandgap occur rapidly, the stored electromagnetic
energy in MDM changes slightly, and the additional leak photon current is
relatively weak. The group delay increases remarkably only when $n_{MDM}>22$.
However, for a relatively low-frequency GW, the additional leak photon current
is enhanced. The group delay increases remarkably when $n_{MDM}>21$
($n_{MDM}>19$) for $\omega_{GW}=0.25\omega_{0}$ ($\omega_{GW}=0.1\omega_{0}$).
For an extremely low-frequency GW, e.g., $\omega_{GW}=0.01\omega_{0}$, the
period of the GW becomes larger than the FWHM of electromagnetic waves, and
the effect of the GW on the group delay becomes small. For $n_{MDM}<25$, the
group delay does not change significantly.
The sensitivity also depends on the thickness of the vacuum layers. For MDM
with thicker vacuum layers (i.e., large $\zeta_{2}$), the GW-induced variation
of layer spacing is enhanced, and a larger additional leak photon current can
be achieved. As shown in Fig. 2(b), group delay increases remarkably when
$n_{MDM}>12$ for $\zeta_{2}=25$. However, for $\zeta_{2}=5$, group delay
increases remarkably only when $n_{MDM}>19$. On the other hand, the relative
group delay $\tau_{DT}/L_{1o}$ of the additional leakage current is
independent of $\zeta_{2}$, where $L_{1o}$ is the optical path length of each
MDM period [see the inset of Fig. 2(b)], which indicates that the pulses
propagate with the same group velocity for different $\zeta_{2}$.
Finally, we discuss the experimental realization of our theoretical
predication. Although a strong GW is used in the numerical calculation, our
results show that the group delay of the tunneling photons is sensitive to GW
and that sensitivity can be increased for a relatively low-frequency GW or for
thick vacuum layers. Owing to ultrafast laser technology, an extremely high
time resolution can be achieved. Thus, the detection of the effect of the GW
on group delay may be feasible. Notably, a pulsar emits a beam of
electromagnetic radiation with ultra-high accuracy and stability. Thus, when
the pulsar electromagnetic radiation tunnels through a strong GW radiation
source such as the black hole binary, the variation of the group delay should
be detectable through astronomical observation.
In conclusion, we have calculated the group delay of optical pulses through
MDM combined with GW. We found that the group delay increases linearly with
MDM length for the transmitted wave packet. The Hartman effect disappears.
This peculiar tunneling effect is attributed to the additional current leakage
attributed to the GW-induced variations of the photonic bandgap. We also show
that the group delay of the tunneling photons is sensitive to GW. For a
relatively low-frequency GW or thick vacuum layers, the sensitivity can be
enhanced remarkably. Our study provides insight into the nature both of the
quantum tunnelling and GW as well as a novel process by which to detect the
GW.
This work was supported by the NSFC Grant Nos. 10904059, 11004199, 11104232,
11173012, and 11264030, the NSF from the Jiangxi Province Nos. 20122BAB212003.
## References
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* [26] Zhong, S. Y., Wu, X., Liu S. Q., & Deng X. F., Global symplectic structure-preserving integrators for spinning compact binaries. Phys. Rev. D 82, 124040 (2010).
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|
arxiv-papers
| 2012-11-27T14:39:11 |
2024-09-04T02:49:38.554415
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Jiang-Tao Liu, Wu Xin, Nian-Hua Liu, Jun Li, and Fu-Hai Su",
"submitter": "Jiangtao Liu",
"url": "https://arxiv.org/abs/1211.6317"
}
|
1211.6364
|
# Spontaneous creation and persistence of ground-state coherence in a
resonantly driven intra-cavity atomic ensemble
D. G. Norris Joint Quantum Institute, Department of Physics, University of
Maryland and National Institute of Standards and Technology, College Park, MD
20742 USA A. D. Cimmarusti Joint Quantum Institute, Department of Physics,
University of Maryland and National Institute of Standards and Technology,
College Park, MD 20742 USA L. A. Orozco Joint Quantum Institute, Department
of Physics, University of Maryland and National Institute of Standards and
Technology, College Park, MD 20742 USA P. Barberis-Blostein Instituto de
Investigación en Matemáticas Aplicadas y en Sistemas, Universidad Nacional
Autónoma de México, México, DF 01000, México H. J. Carmichael Department of
Physics, University of Auckland, Private Bag 92019, Auckland, New Zealand
###### Abstract
The spontaneous creation and persistence of ground-state coherence in an
ensemble of intracavity Rb atoms has been observed as a quantum beat. Our
system realizes a quantum eraser, where the detection of a first photon
prepares a superposition of ground-state Zeeman sublevels, while detection of
a second erases the stored information. Beats appear in the time-delayed
photon-photon coincidence rate (intensity correlation function). We study the
beats theoretically and experimentally as a function of system parameters, and
find them remarkably robust against perturbations such as spontaneous
emission. Although beats arise most simply through single-atom-mediated
quantum interference, scattering pathways involving pairs of atoms interfere
also in our intracavity experiment. We present a detailed model which
identifies all sources of interference and accounts for experimental realities
such as imperfect pre-pumping of the atomic beam, cavity birefringence, and
the transit of atoms across the cavity mode.
###### pacs:
42.50.Md,42.50.Pq,37.30.+i
## I Introduction
The interference of scattering amplitudes in quantum mechanics arises from the
indistinguishability of alternative scattering paths, as in the interference
of the paths of a photon passing through the celebrated double-slits of Young.
For example, a modern variation on the Young experiment Eichmann et al. (1993)
shows spatial fringes in the intensity of light scattered from two trapped
ions; but only so long as the scattering cannot be traced to one ion or the
other Itano et al. (1998). Even in the latter situation, other measurements
might be found which cannot distinguish between the paths, hence recovering
the interference. Typically they involve higher-order moments of the field
Agarwal et al. (2002), or post-selection, _i.e._ , partitioning of scattering
events into subensembles Scully and Drühl (1982). Such a measurement strategy
is termed a “quantum eraser”, since it recovers interference by “erasing” the
information that identifies the path.
A time-domain analogue of double-slit interference can occur inside multi-
level atoms, where photon emission via parallel transitions can result in a
modulation of the emission intensity at the frequency of a level splitting—the
phenomenon of “quantum beats” (see, _e.g._ , Aleksandrov (1964); Dodd et al.
(1967)). A distinction has traditionally been drawn between “Type-I” or “V”
atomic systems, where decay of a superposition of upper levels yields beats at
the transition difference frequency; and the inverted “Type-II” or “$\Lambda$”
systems, where decay to a superposition of lower levels _does not_ yield beats
Breit (1933); Ficek and Swain (2005). The typical argument for the latter
outcome is that a measurement of the ground-state population could always, in
principle, determine in which of the two available states the electron landed;
as there is no sum over alternative paths to one and the same final state,
there are no “ground-state quantum beats” Chow et al. (1975); Herman et al.
(1975).
Nevertheless, as with spatial fringes, a quantum eraser-type strategy can
recover time-domain interference in the ground state. Zajonc Zajonc (1983);
Ficek and Swain (2005) proposed one such implementation in a “Type-II” atomic
system, basing his proposal on two-photon scattering. In this case the second
scattered photon erases the path information written by the first—amplitudes
for the scattering of two photons in sequence interfere.
We recently published experimental results showing quantum beats in
spontaneous emission at the frequency of the ground-state Zeeman splitting in
Rb, i.e. ground-state quantum beats seen in spontaneous emission Norris et al.
(2010). Oscillations appear in the second-order intensity autocorrelation
function only, not in the average intensity, as follows from the
indistinguishability requirement above. The presence of a similar oscillation
hidden within the noise of spontaneous emission was demonstrated in 1955 by
Forrester _et al._ Forrester et al. (1955). They mixed two incoherent light
sources—a Zeeman doublet—on a photocathode, and used a resonant microwave
cavity to enhance the beat signal extracted from the photocurrent noise. The
interference in this case is classical, though the oscillation is recovered
from noise through intensity correlation. More than 50 years later, using
coherent excitation and single-photon detectors, we have realized a time-
resolved measurement of the ground-state quantum beat recovered from
spontaneous emission noise.
The oscillation in our system arises as a complicated mixture of quantum
eraser-type interferences within single atoms, pair-wise interference between
emission from different atoms, and a homodyne contribution due to the
superposition of a weak coherent background (similar to Gerber et al. (2009))
generated by birefringence in the cavity mirrors. By coupling spontaneous
emission into an optical cavity at moderate dipole coupling strength, we
overcome the signal-to-noise limitations set by a small coherence area in free
space Forrester et al. (1955), and enforce indistinguishability among
different atoms emitting into a common spatial mode. Moreover, we show below
that the complicated level structure of 85Rb actually aids in the survival of
ground-state coherence, counter to the conventional strategy of protecting
coherence by limiting the state space through which population can diffuse.
We distinguish here between ground-state coherences imposed by an external
drive and those arising spontaneously, selected through the detection process,
as in our experiment. In the former case, an external magnetic or optical
drive couples two ground states directly, with the resulting coherence read
out optically in forward scattering (see Alexandrov et al. (1993) for many
examples.) In the latter, levels couple only through the vacuum, with no
external drive to enforce coherence. The fact that spontaneous decay can
generate coherence is evident from the observation of quantum beats at the
intermediate level splitting in cascade decay Aspect et al. (1984); Willis et
al. (2010); that the same process occurs in transitions to ground or meta-
stable states is not therefore surprising. Schubert et al. Schubert et al.
(1995) measured such a coherence in the bichromatic cross-correlation of
fluorescence from a single ion, where detection of a first photon left the ion
in a superposition of meta-stable states. When considering isotropic emission,
however, spontaneously created coherences tend to vanish on the average, and
for this reason are often left out of density matrix calculations Aspect et
al. (1989). The recent interest follows a 1992 paper Javanainen (1992) in
which specific measurable consequences were claimed; various arrangements have
been explored theoretically Patnaik and Agarwal (1999); Kiffner et al. (2006).
A publication as recent as 2005 Dutt et al. (2005) claims evidence of the
first serious experimental consequences of spontaneously-generated coherence,
this in a quantum dot system. Other experiments are surprisingly few. We
direct the interested reader to Ref. Economou et al. (2005) for an overview.
In this article we expand upon the results presented in Norris et al. (2010).
In particular, we seek to explain the origin of the various individual
components of the beat signal, show which experimental conditions are
necessary for the robust survival and detection of beats, and explore their
sensitivity to various experimental controls. The paper is organized as
follows. Section II introduces the theoretical model, starting with a single
atom fixed in space and moving to a full atomic beam with realistic
fluctuations. Section III presents the details of our experimental method,
from the atomic source and optics to the detection apparatus. Section III.2
summarizes the evolution of the beat signal as we explore parameter space and
compares experiment with theory. The paper concludes in Section IV.
## II Theoretical model
We consider first the idealized system of one fixed 85Rb atom, then turn to a
realistic atomic ensemble, as realized in our experiment with a cold atomic
beam. The atom has Zeeman structure in its ground and excited states [Fig.
1(a)] and interacts through the $D_{2}$ line with degenerate, orthogonally
polarized cavity modes, designated $H$ (horizontal) and $V$ (vertical); a weak
magnetic field sets the quantization axis in the vertical direction, and mode
$V$ is weakly and continuously driven [Fig. 1(b)]. The atom is prepared in
state $|g_{0}\rangle$, from which it is excited to state $|e_{0}\rangle$ by
the $V$ mode [Fig. 2(a)]. It may return to the ground state by emitting a
$\pi,\sigma^{+}$ or $\sigma^{-}$ photon, or any linear combination conserving
angular momentum. In the assigned geometry, only $\sigma^{+}$ or $\sigma^{-}$
light couples to the $H$ mode, with the helicity undetermined. We assume that
the probability of reabsorption of an emitted photon is negligible; it escapes
the cavity and its detection places the atom in the superposition [Fig. 2(b)]:
$|\psi_{0}\rangle=(|g_{-1}\rangle+|g_{+1}\rangle)/\sqrt{2}\,.$ (1)
The atom is now in the ground state with its angular momentum perpendicular to
the magnetic field, and it performs Larmor precession. With subsequent
reexcitation by the $V$ mode, the state
$|\psi_{0}^{\prime}(t)\rangle=(e^{-i\phi(t)}|e_{-1}\rangle+e^{i\phi(t)}|e_{+1}\rangle)/\sqrt{2}$
(2)
is reached, with phase $\pm\phi(t)$ gained through precession in the ground
state [Fig. 2(c)]. From here the atom can decay back to $|g_{0}\rangle$ by
emitting a second $H$-mode photon [Fig. 2(d)]. The probability for this
emission depends on the phase difference, $2\phi(t)$, between the two parts of
the superposition. It oscillates and thus gives rise to beats in the rate of
detection of a second $H$ photon subsequent to the detection of a first.
Figure 1: (Color online) Experimental setup: (a) $\pi$-excitation of the
$F=3$ to $F^{\prime}=4$ transition in Rb showing scattering of a first (red)
and second (blue) photon into the $H$ mode; (b) schematic of the apparatus,
HWP: half-wave plate, PBS: polarizing beam-splitter, BS: beam-splitter, APD:
avalanche photodiode. Figure 2: Simplified model of the two-photon quantum
eraser process. (a) $\pi$-excitation from $g_{0}$ to $e_{0}$; (b) spontaneous
decay through $\sigma$ transitions to a superposition of $g_{-1}$ and
$g_{+1}$; (c) $\pi$-excitation to a superposition of $e_{-1}$ and $e_{+1}$;
(d) spontaneous decay through $\sigma$ transitions back to $g_{0}$.
As depicted in Fig. 2, there are two paths for scattering a pair of photons
into the $H$ mode:
$|g_{0}\rangle\rightarrow|e_{0}\rangle\rightarrow|g_{+1}\rangle\rightarrow|e_{+1}\rangle\rightarrow|g_{0}\rangle$
and
$|g_{0}\rangle\rightarrow|e_{0}\rangle\rightarrow|g_{-1}\rangle\rightarrow|e_{-1}\rangle\rightarrow|g_{0}\rangle$.
The phase gained from the ground-state Zeeman shift (Larmor precession) is
different along the two paths, which interfere to produce oscillations in the
rate of delayed coincidences—i.e., in the correlation function
$g^{(2)}(\tau)$. Note that after the first photon is detected, “which path”
information is available, since $|g_{+1}\rangle$ and $|g_{-1}\rangle$ are
distinguishable in principle, and their orthogonality precludes observation of
interference effects in the average intensity arising from cross-terms when
taking the expectation value over Eq. 1. This information is largely erased by
the second photon detection, where the amplitude for returning to the common
final state $|g_{0}\rangle$ allows survival of cross-terms in the two-photon
intensity expectation value, corresponding to interference between the two
scattering paths. We note that, as in Ref. Forrester et al. (1955), the
spectrum of scattered light still exhibits a doublet separated by the beat
frequency, but the random phase relation between the two fields destroys any
first-order coherence in the average intensity.
In the following sections we study the dependence of the quantum beats on the
different features and parameters of our experimental system: magnetic field
strength, number of atoms, initial state preparation, atomic beam
fluctuations, cavity birefringence, and mixing of the undriven $H$ field with
drive light ($V$ field) outside the cavity before detection. We first present
a detailed theory for the case of one fixed atom before we move on to the
treatment of many atoms and atomic motion.
### II.1 One fixed atom
The relevant 16-level structure of the atom is depicted in Fig. 1. Bold black
arrows indicate its interaction with the driven $V$ mode of the optical
cavity, and red and blue wavy lines with the undriven $H$ mode. We calculate
the second-order correlation function of the $H$ mode. We use the quantum
trajectory formalism Carmichael (1993), which provides insight into the
physical processes involved and facilitates efficient numerical calculations,
something of importance when many atoms are considered. Working in a frame
rotating at the frequency of the drive, the non-Hermitian Hamiltonian
governing coherent evolution between spontaneous emission events (modes other
than $H$ and $V$) or photon loss through the cavity mirrors is:
$H_{S}=H_{0}+H_{I}+H_{D}+H_{L}\,,$ (3)
with free Hamiltonian
$\displaystyle H_{0}$ $\displaystyle=$
$\displaystyle\hbar\delta_{v}a_{v}^{\dagger}a_{v}+\hbar\delta_{h}a_{h}^{\dagger}a_{h}$
(4a) $\displaystyle+\sum_{i=-3}^{3}\hbar\delta_{g_{i}}|g_{i}\rangle\langle
g_{i}|+\sum_{i=-4}^{4}\hbar\delta_{e_{i}}|e_{i}\rangle\langle e_{i}|\,,$
interaction and drive $\displaystyle H_{I}$ $\displaystyle=$
$\displaystyle\hbar
g[a_{v}^{\dagger}\Sigma_{\pi}+a_{h}^{\dagger}(\Sigma_{\sigma_{+}}+\Sigma_{\sigma_{-}})]+{\rm
h.c.}\,,$ (4b) $\displaystyle H_{D}$ $\displaystyle=$ $\displaystyle
i\hbar\mathcal{E}(a_{v}^{\dagger}-a_{v})\,,$ (4c) and non-Hermitian loss term
$\displaystyle H_{L}$ $\displaystyle=$
$\displaystyle-i\hbar\kappa(a_{v}^{\dagger}a_{v}+a_{h}^{\dagger}a_{h})$ (4d)
$\displaystyle-i\hbar\frac{\gamma}{2}(\Sigma^{\dagger}_{\pi}\Sigma_{\pi}+\Sigma^{\dagger}_{\sigma_{+}}\Sigma_{\sigma_{+}}+\Sigma^{\dagger}_{\sigma_{-}}\Sigma_{\sigma_{-}})\,,$
where $a_{v}$ and $a_{h}$ annihilate photons in the $V$ and $H$ modes,
respectively, detunings from the drive $\delta_{v}$ and $\delta_{h}$,
$\mathcal{E}$ is the drive amplitude for the $V$ mode, $g$ is the dipole
coupling constant, $2\kappa$ is the photon loss rate from each cavity mode,
$\gamma$ is the spontaneous emission rate, and $\Sigma_{\pi}$,
$\Sigma_{\sigma_{+}}$, and $\Sigma_{\sigma_{-}}$ are dipole lowering operators
for $\pi$, $\sigma_{+}$, and $\sigma_{-}$ transitions; atomic energy shifts in
the magnetic field and relative to the drive are $\hbar\delta_{e_{i}}$,
$i=-4,-3,\ldots,4$ (excited states) and $\hbar\delta_{g_{i}}$,
$i=-3,-2,\ldots,3$ (ground states). Explicit forms for the dipole operators
depend on Clebsh-Gordon coefficients and are relegated to the Appendix.
The evolution of the system under the non-Hermitian Hamiltonian $H_{S}$ is
calculated numerically, with photon-number truncation, using the fourth-order
Runge-Kutta method, including step-size correction. At regular intervals of
length $\Delta t$ the atom and the cavity modes are checked for quantum
jumps—spontaneous emission or photon leakage. If a scattering event occurs,
the system state is collapsed accordingly—jump operators $\Sigma_{\pi}$,
$\Sigma_{\sigma_{+}}$, $\Sigma_{\sigma_{-}}$, $a_{v}$, or $a_{h}$—before
continuing the coherent evolution. After a time $t$, we assume a photon is
scattered through the $H$ mode. The system state is collapsed (jump operator
$a_{h}$) and its conditional evolution, with the unnormalized correlation
function, $G^{(2)}(t,t+\tau)$, given by the $H$-mode-photon-number-weighted
mean of the $H$-mode photon number expectation in the conditional states.
Normalization by the photon number averages yields
$g^{(2)}(t,t+\tau)=G^{(2)}(t,t+\tau)/\langle(a_{h}^{\dagger}a_{h})(t)\rangle\langle(a_{h}^{\dagger}a_{h})(t+\tau)\rangle$.
We consider now a weak drive, such that $\langle
a_{v}^{\dagger}a_{v}\rangle\ll 1$, and $t$ long enough for the system to reach
a quasi-steady-state (overlooking the very slow process of optical pumping).
If the initial state is chosen to be any one of the ground states
$|g_{i}\rangle$, $i=-2,-1,0,+1,+2$, the correlation function, $g^{(2)}(\tau)$,
shows beats, with beat frequency twice the Larmor precession frequency of
$|g_{\pm 1}\rangle$; the frequency does not depend on the chosen initial
state, or the number of atoms when many are included (Sec. II.2).
The visibility of the beat depends, however, on a number of issues. In the
ideal case, the atom is in state $|g_{0}\rangle$ when the first $H$-photon is
detected; it is projected to $|\psi_{0}\rangle$, and the second photon is
detected as it returns to $|g_{0}\rangle$ via $|\psi^{\prime}_{0}(t)\rangle$
(Fig. 2). We note first that, for the level scheme of Fig. 1, the return is
not to $|g_{0}\rangle$ but (see the Appendix for the explicit form of the
lowering operator, $\Sigma_{\sigma_{+}}+\Sigma_{\sigma_{-}}$, for $H$-mode
scattering)
$2\cos[\phi(t)]\sqrt{\frac{5}{14}}|g_{0}\rangle+\sqrt{\frac{3}{28}}(e^{-i\phi(t)}|g_{-2}\rangle+e^{i\phi(t)}|g_{+2}\rangle).$
(5)
This brings a reduction of the visibility to a little more than 75%. Beyond
this, several trajectories deviate from the ideal and further reduce the
visibility. We divide them into two groups: (i) those that deviate prior to
the detection of the first photon, and (ii) those that deviate between the
detection of the first photon and the second.
Consider first group (i). If the atom is initially in state $|g_{i}\rangle$,
$i\neq 0$, then when the first $H$-photon is detected, it is projected into an
unequal superposition of $|g_{i\pm 1}\rangle$, with the different weights
given by Clebsch-Gordan coefficients; for example, if the initial state is
$|g_{+1}\rangle$, after detection of a scattered photon, the atom is projected
into $|\psi\rangle=(\sqrt{10}|g_{0}\rangle+\sqrt{3}|g_{2}\rangle)/\sqrt{13}$.
Table 1 lists the realized visibilities for all potential initial states. A
maximum visibility of $0.75$ is achieved when the initial state is
$|g_{0}\rangle$. The visibility decays rather quickly away from this maximum.
Experimentally, we optically pump the atoms before they enter the cavity;
however, the efficiency of the pumping is not perfect, and the result is
generally a distribution over ground states, peaked around $|g_{0}\rangle$.
Some reduction of the beat visibility must follow from the imperfect optical
pumping prior to an atom entering the cavity.
Initial state | $g_{0}$ | $g_{+1}$ | $g_{+2}$ | $g_{+3}$
---|---|---|---|---
Visibility | 0.75 | 0.5 | 0.15 | 0.03
Table 1: Quantum beat visibility for one fixed atom and different initial
ground states. The parameters are: $g/\gamma=0.25$, $\kappa/\gamma=0.5$,
$\mathcal{E}/\gamma=0.025$, and a Larmor frequency of $2.2\gamma/3$. The time
at which the first $H$-mode photon is detected is $t=25\gamma^{-1}$. Results
for $g_{-1}$, $g_{-2}$, $g_{-3}$ follow by symmetry. Note that a visibility of
zero (no quantum beat) is predicted for $g\pm 3$ in the absence of optical
pumping prior to $t$.
Other processes can redistribute population amongst the atomic levels and
contribute to the loss of visibility. The atom can spontaneously decay from an
excited state $|e_{i}\rangle$ to $|g_{i+1}\rangle$ or $|g_{i-1}\rangle$ with
the emission of a photon to the side rather than into the $H$ mode of the
cavity. At higher values of the drive, this process might be repeated many
times, redistributing population before a first $H$-photon is detected. The
distribution reached through such optical pumping by the drive depends on the
Clebsch-Gordan coefficients, the drive strength, and the time when the first
photon is detected. Table 2 displays the distribution reached in the long-time
limit. It shows that, even in the limit, states with $i=\pm 3$ have a very
small probability to be populated, and those with $i=\pm 2$ are populated at a
level of only about 10%.
State | $g_{0}$ | $g_{+1}$ | $g_{+}2$ | $g_{+}3$ | $e_{0}$ | $e_{+1}$ | $e_{+2}$ | $e_{+3}$
---|---|---|---|---|---|---|---|---
Probability | 0.23 | 0.15 | 0.04 | 0.003 | 0.16 | 0.09 | 0.02 | 0.001
Table 2: Energy level occupation probabilities for one fixed atom. The
parameters are: $g/\gamma=0.25$, $\kappa/\gamma=0.5$, $\mathcal{E}/\gamma=2$,
and a Larmor frequency of $2.2\gamma/3$. The number of $V$-mode photons inside
the cavity is $\langle a^{\dagger}_{v}a_{v}\rangle\approx 15$. Results for
$g_{-1}$, $g_{-2}$, $g_{-3}$ and $e_{-1}$, $e_{-2}$, $e_{-3}$ follow by
symmetry.
The distribution over atomic ground states prior to the detection of a first
$H$-photon strongly effects the state the atom is projected into on average.
If the time of the first detection is close to zero the atom is projected into
$|\psi_{0}\rangle$ [Eq. (1]. As this time increases, superpositions of, first,
$|g_{0}\rangle$ and $|g_{+2}\rangle$ (or $|g_{-2}\rangle$), and then $|g_{\pm
1}\rangle$ and $|g_{\pm 3}\rangle$ appear. Numerically we have checked that
the state immediately after the first $H$-photon detection may be written
approximately as
$\displaystyle\rho_{c}(t)$ $\displaystyle=$ $\displaystyle
p_{0}|\psi_{0}\rangle\langle\psi_{0}|+p_{1}(|\psi_{1}\rangle\langle\psi_{1}|+|\psi_{-1}\rangle\langle\psi_{-1}|)$
(6) $\displaystyle+p_{2}\Delta\rho_{c},$
with
$\displaystyle|\psi_{\pm 1}\rangle$ $\displaystyle=$
$\displaystyle(\sqrt{10}|g_{0}\rangle+\sqrt{3}|g_{\pm 2}\rangle)/\sqrt{13}\,,$
(7)
where $p_{i}$, $i=0,1$, is the probability distribution over
$|\psi_{0}\rangle$ and $|\psi_{\pm 1}\rangle$, and $p_{2}=1-p_{0}-2p_{1}$ is
the probability that the first $H$-photon is scattered out of one of the
ground states with $|i|=2,3$; $\Delta\rho_{c}$ is the state reached from such
scattering events. As can be seen from Fig. 3, as the wait for the first
$H$-photon detection becomes longer, the probability to realize
$|\psi_{s}\rangle$ decreases while that to realize $|\psi_{\pm 1}\rangle$
increases. For the parameters considered, the sum $p_{0}+2p_{1}$ is close to
0.97, which tells us that $|\psi_{s}\rangle$ and $|\psi_{\pm 1}\rangle$ cover
all relevant prepared superpositions. An increase in drive strength changes
only the time evolution of the probabilities, not their stationary values.
Figure 3: Probability for the preparation of superposition state
$|\psi_{s}\rangle$ (solid line) and either of the superposition states
$|\psi_{\pm 1}\rangle$ (dashed line), for one fixed atom, as a function of the
time of the first $H$-photon detection. The parameters are: $g/\gamma=0.25$,
$\kappa/\gamma=0.5$, $\mathcal{E}/\gamma=0.3$, and a Larmor frequency of
$2.2\gamma/3$.
We move now to trajectories that deviate from the ideal after the first
$H$-photon is detected [group (ii)]. We assume the prepared superposition is
$|\psi_{0}\rangle$. Interaction of the atom with the driven $V$ cavity mode
moves population to the superposition $|\psi_{0}^{\prime}(t)\rangle$ [Eq.
(2)]. It is then possible that spontaneous emission (to the side) moves
$|\psi_{0}^{\prime}(t)\rangle$ to a superposition of the ground states
$|g_{0}\rangle$ and $|g_{+2}\rangle$ (or $|g_{-2}\rangle$). As the Clebsch-
Gordan coefficient connecting $|e_{\mp 1}\rangle$ to $|g_{0}\rangle$ differs
from that connecting $|e_{\pm 1}\rangle$ to $|g_{\pm 2}\rangle$), such an
event yields unequal weights in the ground-state superposition. Nevertheless,
this aside, the described process recovers the initial setup—a ground-state
superposition—but in a manifold of states shifted to the right or left.
Continuing then with the standard story, the $\pi$-polarized drive transfers
the superposition of $|g_{0}\rangle$ and $|g_{\pm 2}\rangle$ to a
superposition of $|e_{0}\rangle$ and $|e_{\pm 2}\rangle$ in the excited state.
A second $H$-photon can then be emitted via the cavity, projecting the atom
into a superposition of $|g_{\pm 1}\rangle$, $|g_{\pm 3}\rangle$ and $|g_{\mp
1}\rangle$ [compare Eq. (5)]. Apart for the changed weight factors, the
quantum eraser process still takes place, only within a different manifold of
atomic states. Of course, the unequal weights yield a beat with diminished
visibility.
Such spontaneous emission events can happen several times in the interval
separating the $H$-photon detections; nevertheless, so long as there is no
$\sigma_{+}$ ($\sigma_{-}$) emission at a time when $|e_{-3}\rangle$
($|e_{+3}\rangle$) is part of the superposition in the excited state, the
superposition, with modified weights, survives. For an atom with six levels,
as in Fig. (2), a single $\sigma_{+}$ or $\sigma_{-}$ spontaneous emission
will destroy the prepared superposition, reducing the observed visibility far
more than in the 16-level case. Our use of the 16-level configuration produces
a particularly robust (against spontaneous emission) quantum beat.
Figure 4 shows how the spontaneously created coherence moves between different
ground states after the first $H$-photon is detected. We quantify the
coherence by off-diagonal matrix elements $|\langle
g_{i+1}|\rho_{c}(t+\tau)|g_{i-1}\rangle|$, $i=0$ and 1, which fall between a
maximum of 0.5 (equally weighted superposition) and zero. The solid line
follows $|\langle g_{+1}|\rho(t+\tau)|g_{-1}\rangle|$ as a function of $\tau$.
It begins at $\tau=0$ from approximately $0.46$, which shows that immediately
after the photon is detected the atom is to a good approximation in the
superposition $|\psi_{0}\rangle$; $p_{0}\approx 1$ in Eq. (6). As the time to
the second photon detection progresses, $p_{0}$ decreases as $p_{\pm 1}$ grows
and part of the coherence is transferred to a superposition of $|g_{0}\rangle$
and $|g_{\pm 2}\rangle$. Eventually phase diffusion (decoherence) which
accompanies repeated cycles of excitation and spontaneous emission sets in,
causing both displayed coherences to decay to zero Norris et al. (2012).
Coherences between ground states other than those shown in the figure are
negligible.
Figure 4: Evolution of spontaneously created coherences for one fixed atom;
off-diagonal matrix elements $|\langle|g_{+1}|\rho(t+\tau)|g_{-1}\rangle|$
(solid line) and $|\langle g_{\pm 2}|\rho(t+\tau)|g_{0}\rangle)|$ (dashed
line) are plotted as a function of the time after a first $H$-photon is
detected. The parameters are: $g/\gamma=0.25$, $\kappa/\gamma=0.5$,
$\mathcal{E}/\gamma=0.3$, and a Larmor frequency of $2.2\gamma/3$. The number
of $V$-photons inside the cavity is approximately $0.3$.
Before turning to the many atom case relevant to our experiment, one final
effect might usefully be introduce at the one-atom level. The mirrors in the
experiment show a small birefringence and mix a little of the $V$-polarized
light with the $H$-polarized cavity mode. We attempt to null this mixing with
a half-wave plate placed in the cavity output [Fig. 1(b)]; alternatively, in
some measurements we deliberately enhance it (see Fig. 11). The mixing
effectively performs a homodyne measurement with weak (at the one-photon
level) local oscillator field. Let us make the substitution $a_{h}\to
a_{h}+\epsilon$, where $\epsilon$ is the amplitude of the mixed drive light,
taken for simplicity to be classical, real, and constant. The (unnormalized)
intensity correlation function is now
$\displaystyle G^{(2)}(t,t+\tau)$ $\displaystyle=$ $\displaystyle
G^{(2)}_{h}(t,t+\tau)$ (8)
$\displaystyle+\epsilon^{2}\big{\\{}G^{(1)}_{h}(t,t)+G^{(1)}_{h}(t+\tau,t+\tau)$
$\displaystyle+2{\rm
Re}[G_{h}^{(1)}(t,t+\tau)+G_{h}^{(a)}(t,t+\tau)]\big{\\}}$
$\displaystyle+\epsilon^{4}\,,$
where $G^{(1)}_{h}(t,t+\tau)=\langle a_{h}^{\dagger}(t)a_{h}(t+\tau)\rangle$
is the first-order correlation function of the $H$ mode, and
$G_{h}^{(a)}(t,t+\tau)=\langle
a_{h}^{\dagger}(t)a_{h}^{\dagger}(t+\tau)\rangle$ is the $H$-mode anomalous
correlation. Third-order correlations vanish for weak drive because scattering
a second photon [blue wavy lines in Fig. 1(a)] leaves the atom in a manifold
orthogonal to that reached after scattering one photon [red wavy lines in Fig.
1(a)], i.e., one- and two-photon states entangle with orthogonal atomic
states. The anomalous correlation survives because the manifold reached by
scattering two photons contains the initial state—$|g_{0}\rangle$ in Fig.
1(a).
With increasing $\epsilon$, the frequency of the quantum beat changes from
twice the Larmor frequency to the Larmor frequency as the third term on the
righ-hand side of Eq. (8) comes to dominate the first. Figure 5 illustrates
the transition.
Figure 5: Size of the beat at the Larmor frequency (solid line) relative to
that at twice the Larmor frequency (dashed line), as a function of the
fraction of drive light mixed with the $H$-mode. The parameters are:
$g/\gamma=0.25$, $\kappa/\gamma=0.5$, $\mathcal{E}/\gamma=0.3$, and a Larmor
frequency of $2.2\gamma/3$.
### II.2 Many atoms
Our experiment is performed with a cold atomic beam and therefore the many-
atom case must be considered. We generalize the non-Hermitian Hamiltonian of
Eq. (3) by writing the free, interaction, and non-unitary loss terms for $N$
arbitrarily located atoms:
$\displaystyle H_{0}$ $\displaystyle=$
$\displaystyle\hbar\omega_{v}a_{v}^{\dagger}a_{v}+\hbar\omega_{h}a_{h}^{\dagger}a_{h}$
$\displaystyle+\sum_{j=1}^{N}\left(\sum_{i=-3}^{3}\hbar\omega_{g_{i}}|g_{i}\rangle^{j}\langle
g_{i}|^{j}+\sum_{i=-4}^{4}\hbar\omega_{e_{i}}|e_{i}\rangle^{j}\langle
e_{i}|^{j}\right),$ $\displaystyle H_{I}$ $\displaystyle=$
$\displaystyle\sum_{j=1}^{N}\hbar
g_{j}[a_{v}^{\dagger}\Sigma_{\pi}^{j}+a_{h}^{\dagger}(\Sigma_{\sigma_{+}}^{j}+\Sigma_{\sigma_{-}}^{j})]+{\rm
h.c.}\,,$ (9b) $\displaystyle H_{L}$ $\displaystyle=$
$\displaystyle-i\hbar\kappa(a_{v}^{\dagger}a_{v}+a_{h}^{\dagger}a_{h})$ (9c)
$\displaystyle-\sum_{j=1}^{N}i\hbar\frac{\gamma}{2}(\Sigma^{j\dagger}_{\pi}\Sigma^{j}_{\pi}+\Sigma^{j\dagger}_{\sigma_{+}}\Sigma^{j}_{\sigma_{+}}+\Sigma^{j\dagger}_{\sigma_{-}}\Sigma^{j}_{\sigma_{-}}),$
where the dipole coupling constants, $g_{j}$, $j=1,\ldots,N$, vary with the
location of the atoms within the cavity mode function.
We aim to write the measured correlation function as a sum of terms applying
to emission pathways for different atoms and their interference. To this end,
we formally integrate the Heisenberg equation of motion for the $H$-mode
annihilation operator, including its coupling to the reservoir. This yields
(see, e.g., Ref. Carmichael (2008) page 206)
$\displaystyle a_{h}(t)=a_{h}(0)e^{-\kappa
t}+\displaystyle{\sum_{j=1}^{N}}\mkern
3.0mug_{j}\displaystyle{\int_{0}^{t}}\Sigma^{j}_{h}(t-t^{\prime})e^{-\kappa
t^{\prime}}dt^{\prime}+{\rm v.f.},$ (10)
$\displaystyle\Sigma^{i}_{h}(t)=\Sigma^{i}_{\sigma_{-}}(t)+\Sigma^{i}_{\sigma_{+}}(t),$
(11)
where v.f. indicates the presence of a vacuum field (reservoir) noise
operator, which may be dropped from the calculation of a normal- and time-
ordered average. We also drop the first term on the right-hand side of Eq.
(II.2), as we are interested in times much longer than the cavity decay time.
Adding the mixed drive amplitude $\epsilon$, as above Eq. (8), the $H$-mode
cavity output is treated with the substitution
$\displaystyle a_{h}(t)\to\displaystyle{\sum_{j=1}^{N}}\mkern
2.0muA_{j}(t)+\epsilon,$ (12) $\displaystyle
A_{j}(t)=g_{j}\displaystyle{\int_{0}^{t}}\Sigma^{j}_{h}(t-t^{\prime})e^{-\kappa
t^{\prime}}dt^{\prime}.$ (13)
We further assume that (i) the probability that an emitted $H$-photon be re-
absorbed before leaving the cavity is negligible and (ii) no additional
$H$-photon is emitted in between the detection of photons at $t$ and $t+\tau$.
The assumptions are justified, respectively, for moderate-to-weak dipole
coupling and weak drive. They allow us to treat the atoms as independent and
write the intensity correlation function as a generalization of Eq. (8):
$\displaystyle G^{(2)}(t,t+\tau)$ $\displaystyle=$
$\displaystyle\sum_{j=1}^{N}\left(G^{(2)}_{j}(t,t+\tau)+\sum_{k\neq
j=1}^{N}G^{(a)}_{j}(t,t+\tau)\Big{(}G^{(a)}_{k}(t,t+\tau)\Big{)}^{\mkern-2.0mu*}\right)$
(14) $\displaystyle+\sum_{j=1}^{N}\sum_{k\neq
j=1}^{N}\left[G^{(1)}_{j}(t,t)G^{(1)}_{k}(t+\tau,t+\tau)+G^{(1)}_{j}(t,t+\tau)\Big{(}G^{(1)}_{k}(t,t+\tau)\Big{)}^{\mkern-2.0mu*}\mkern
2.0mu\right]$
$\displaystyle+\epsilon^{2}\sum_{j=1}^{N}\mkern-2.0mu\Big{\\{}G_{j}^{(1)}(t,t)+G_{j}^{(1)}(t+\tau,t+\tau)+2{\rm
Re}\Big{[}G_{j}^{(1)}(t,t+\tau)+G_{j}^{(a)}(t,t+\tau)\Big{]}\Big{\\}}+\epsilon^{4},$
where we introduce individual atom correlation functions:
$\displaystyle G^{(2)}_{j}(t,t+\tau)=\langle
A_{j}^{\dagger}(t)A_{j}^{\dagger}(t+\tau)A_{j}(t+\tau)A_{j}(t)\rangle,$ (15a)
$\displaystyle G^{(1)}_{j}(t,t+\tau)=\langle
A_{j}^{\dagger}(t)A_{j}(t+\tau)\rangle,\qquad G^{(a)}_{j}(t,t+\tau)=\langle
A_{j}^{\dagger}(t)A_{j}^{\dagger}(t+\tau)\rangle.$ (15b)
In order to help with the interpretation of this expression, let us assume
stationarity (correlation functions independent of $t$) and identical atoms.
Equation (14) then reduces to
$\displaystyle G^{(2)}(\tau)$ $\displaystyle=$ $\displaystyle
NG^{(2)}_{A}(\tau)+N(N-1)|G^{(a)}_{A}(\tau)|^{2}$ (16)
$\displaystyle+N(N-1)[I_{A}^{2}+|G^{(1)}_{A}(\tau)|^{2}]$
$\displaystyle+2\epsilon^{2}N\\{I_{A}+{\rm
Re}[G^{(1)}_{A}(\tau)+G^{(a)}_{A}(\tau)]\\}$ $\displaystyle+\epsilon^{4},$
where $I_{A}=G^{(1)}_{A}(0)$ and the subscript denotes any atom. The result is
similar to Eq. (4) in Carmichael et al. (1978). The only change is the
anomalous correlation $G^{(a)}_{A}(\tau)$, which is retained here because we
consider scattering into a single cavity mode, not free-space scattering as in
Carmichael et al. (1978); in the latter case, the random phases accompanying
propagation from randomly located atoms to the detector cause the anomalous
correlation to vanish. It is also important to recall that $G^{(a)}(\tau)$ is
only nonzero because scattering two $H$-photons places the atom in a ground-
state manifold that is not orthogonal to the initial state. $G^{(a)}(\tau)$ is
therefore intimately related to the erasure of which-path information by the
scattering of a second photon. Of course it approaches zero as
$\tau\rightarrow\infty$, due to dephasing induced by spontaneous emission
Norris et al. (2012). The time scale for this is long compared with the
transit time through the cavity if the drive is weak.
Leaving aside the homodyne terms, three correlation functions contribute to
the quantum beat in Eq. (16). There is first and foremost $G^{(2)}_{A}(\tau)$.
It records the beat due to the interference of indistinguishable pathways for
scattering two photons by one atom; there are $N$ such one-atom terms. There
are then $N(N-1)$ copies of the first-order correlation function
$|G^{(1)}_{A}(\tau)|^{2}$. These arise from the sum over cross-terms
$G^{(1)}_{j}(\tau)[G^{(1)}_{k}(\tau)]^{*}$, $k\neq j$, in Eq. (14). They also
exhibit a beat at twice the Larmor frequency. It records the interference of
indistinguishable pathways for scattering a first $H$-photon from atom $j$
($k$) and a second from atom $k$ ($j$)—i.e., the interference of scattering
events with reversed time-order, when both orders leave the same two atoms in
the same final state. Finally, there are $N(N-1)$ copies of
$|G_{A}^{(a)}(\tau)|^{2}$. These anomalous correlation functions record the
many-atom extension of the $G^{(2)}_{A}(\tau)$ quantum beat. They are present
because when one atom scatters two photons, our cavity setup is unable to tell
from which of the $N$ atoms the two photons come.
### II.3 Atomic motion
Our experiment is performed with a slow atomic beam. The atoms move through
the cavity mode function and the dipole coupling coefficients, $g_{j}(t)$,
$j=1,\ldots,N$, are randomly determined functions of time. If we neglect the
effect of photon scattering on the atomic motion, $g_{j}(t)$, for a particular
atom, is defined by a (constant) velocity ${\bm{v}}_{j}$, a time, $t_{j}$, at
which the atom crosses the plane perpendicular to the beam containing the
cavity axis, and a position ${\bm{r}}_{j}$ on that plane; the velocity may be
further specified by a speed $v_{j}$, and polar and azimuthal angles,
$\theta_{j}$ and $\phi_{j}$, defined with respect to the cavity axis. The
quantity relevant for our experiment is the time average
$G^{(2)}(\tau)=\frac{1}{T}\int_{0}^{T}G^{(2)}(t,t+\tau)dt\,$ (17)
of Eq. (14); we need also, for normalization, the mean intensity
$I=\frac{1}{T}\int_{0}^{T}\sum_{j=1}^{N}G^{(1)}_{j}(t,t)dt+\epsilon^{2}.$ (18)
The sums in Eq. (14) may be considered to range over every atom that enters
the cavity during the course of the experiment. The atoms are uniformly
distributed in a beam of rectangular cross section, height $d$ and extension
$l$ along the cavity axis, at flux density $F$ (number of atoms per second
through unit area). Correlation functions for atom $j$ depend on
${\bm{r}}_{j}$, ${\bm{v}}_{j}$, and the time difference $t-t_{j}$. They fall
to zero for $|t-t_{j}|$ much larger than the mean transit time across the
cavity, $\tau_{0}=w_{0}/\langle v\rangle$, where $w_{0}$ is the mode function
waist and $\langle v\rangle$ is the mean speed of an atom along the axis of
the atomic beam.
We consider the mean intensity as an illustration of the way to proceed and
then pass directly to the result for $G^{(2)}(\tau)$. For the time integral on
the right-hand side of Eq. (18), we may write (with $t^{\prime}=t-t_{j}$)
$\int_{0}^{T}\sum_{j=1}^{N}G^{(1)}_{j}(t,t)dt=FT\ell
d\left\langle\int_{-\infty}^{\infty}G^{(1)}_{{\bm{r}}_{j},{\bm{v}}_{j},t_{j}}(t^{\prime},t^{\prime})dt^{\prime}\right\rangle,$
(19)
where it is clear that the number of nonzero contributions arising from the
sum is just the mean number of atoms, $FT\ell d$, crossing the plane
containing the cavity axis during time $T$, while the angle bracket denotes an
ensemble average, over ${\bm{r}}_{j}$ and ${\bm{v}}_{j}$, for atoms
distributed within the beam cross-section; the ensemble average can be taken
numerically. We then define a mean intensity per (effective) atom normalized
to the profile of the mode function,
$I_{A}=\frac{\ell d}{\pi\ell
w_{0}/4}\frac{1}{\tau_{0}}\left\langle\int_{-\infty}^{\infty}G^{(1)}_{{\bm{r}}_{j},{\bm{v}}_{j},t_{j}}(t^{\prime},t^{\prime})dt^{\prime}\right\rangle,$
(20)
and arrive at
$I=\bar{N}_{\rm eff}I_{A}+\epsilon^{2},$ (21)
where $\bar{N}_{\rm eff}=\rho\pi w_{0}^{2}\ell/4$ is the effective number of
atoms (see Ref. Carmichael and Sanders (1999), for example), with
$\rho=F/\langle v\rangle$ the atomic density.
The time average of each sum in Eq. (14) is treated in a similar way. This
yields a straightforward generalization of Eq. (16):
$\displaystyle G^{(2)}(\tau)$ $\displaystyle=$ $\displaystyle\bar{N}_{\rm
eff}G^{(2)}_{A}(\tau)+\bar{N}_{\rm eff}^{2}|G^{(a)}_{A}(\tau)|^{2}$ (22)
$\displaystyle+\bar{N}_{\rm eff}^{2}[I_{A}^{2}+|G^{(1)}_{A}(\tau)|^{2}]$
$\displaystyle+2\epsilon^{2}\bar{N}_{\rm eff}\\{I_{A}+{\rm
Re}[G^{(1)}_{A}(\tau)+G^{(a)}_{A}(\tau)]\\}$ $\displaystyle+\epsilon^{4},$
with ($\xi=1,2,a$)
$G_{A}^{(\xi)}(\tau)=\frac{\ell d}{\pi\ell
w_{0}/4}\frac{1}{\tau_{0}}\left\langle\int_{-\infty}^{\infty}G^{(\xi)}_{{\bm{r}}_{j},{\bm{v}}_{j},t_{j}}(t^{\prime},t^{\prime}+\tau)dt^{\prime}\right\rangle.$
(23)
We compare experimental results with a numerical evaluation of Eq. (22) in
Section III.3. In the next subsection we introduce simplifications that lead
to a closed expression for $G^{(2)}(\tau)$.
### II.4 Bad-cavity and adiabatic limit
We make two simplifying assumptions and focus on the weak-field limit. First,
we assume that the cavity decay rate $\kappa$ is sufficiently large compared
with all other rates that the dipole operator may be taken outside the
integral in Eq. (13) (bad-cavity limit Carmichael (2008)), allowing us to
write $A_{j}(t)=g_{j}(t)\Sigma^{j}_{h}(t)/\kappa$. We assume also that the
atomic motion is slow and atomic states follow the changing coupling
constants, $g_{j}(t)$, $j=1,\ldots,N$, adiabatically. Then noting that the
dipole operator is expected to carry an atomic excitation proportional to
$g_{j}(t)$ in the weak-field limit, which we verify numerically, we may factor
out the $g_{j}(t)$-dependence and write each individual atom correlation
function in the form
$G^{(\xi)}_{{\bm{r}}_{j},{\bm{v}}_{j},t_{j}}(t,t+\tau)=\left[\frac{g_{\rm
max}}{\kappa}\bar{g}_{j}(t)\bar{g}_{j}(t+\tau)\right]^{i_{\xi}}G^{(\xi)}_{\rm
max}(\tau),$ (24)
with $i_{\xi}=2,4,2$ for $\xi=1,2,a$, where $G^{(\xi)}_{\rm max}(\tau)$ is the
correlation function, in the long-time limit, for a fixed atom at maximum
coupling strength $g_{\rm max}$. The scaled coupling functions,
$\bar{g}_{j}(t)$, $j=1,\ldots,N$, follow from the cavity mode function:
$\bar{g}_{j}(t)=\cos[kz_{i}(t)]e^{-[x_{j}(t)^{2}+y_{j}(t)^{2}]/w_{0}^{2}}\,,$
(25)
where $k=2\pi/\lambda$ is the wavenumber, and $x_{j}(t)$, $y_{j}(t)$,
$z_{j}(t)$ define the trajectory of atom $j$. We assign the $z$-axis parallel
to the cavity axis and the $x$-axis as the axis of the atomic beam; at time
$t^{\prime}=t-t_{j}=0$, we identify
${\bm{r}}_{j}=(y_{j}(t_{j}),z_{j}(t_{j}))$, and
${\bm{v}}_{j}=(\dot{x}_{j}(t),\dot{y}_{j}(t),\dot{z}_{j}(t))$.
With these simplifications, the time integration and average of Eq. (23) are
defined entirely by the prescribed coupling functions $\bar{g}_{j}(t)$,
$j=1,\ldots,N$, and the velocity distribution of the atomic beam. In some
simple cases analytical results can be obtained. The first is when the atoms
move along parallel trajectories perpendicular to the cavity and at a common
speed $v$. Equations (23) and (24) then yield
$G^{(2)}_{A}(\tau)=\frac{35}{256}\frac{g_{\rm
max}^{4}}{\kappa^{4}}e^{-2\tau^{2}/\tau_{0}^{2}}G_{\rm max}^{(2)}(\tau),$
(26a) and $G_{A}^{(1,a)}(\tau)=\frac{3}{8}\frac{g_{\rm
max}^{2}}{\kappa^{2}}e^{-\tau^{2}/\tau_{0}^{2}}G^{(1,a)}_{\rm max}(\tau),$
(26b)
and setting $\epsilon=0$, for simplicity, and introducing the normalized
correlation functions
$g^{(2)}(\tau)=\frac{G^{(2)}(\tau)}{(\bar{N}_{\rm eff}I_{A})^{2}},\qquad
g^{(2)}_{\rm max}(\tau)=\frac{G^{(2)}_{\rm max}(\tau)}{[G^{(1)}_{\rm
max}(0)]^{2}},$ (27a) and ($\xi=1,a$) $g^{(\xi)}_{\rm
max}(\tau)=\frac{G^{(\xi)}_{\rm max}(\tau)}{G^{(1)}_{\rm max}(0)},$ (27b)
we arrive at a correlation function with Gaussian transit time decay:
$\displaystyle g^{(2)}(\tau)$ $\displaystyle=$ $\displaystyle
1+e^{-2\tau^{2}/\tau_{0}^{2}}\left[\vphantom{\frac{35}{36}}|g^{(1)}_{\rm
max}(\tau)|^{2}+|g^{(a)}_{\rm max}(\tau)|^{2}\right.$ (28)
$\displaystyle\left.+\frac{35}{36}\bar{N}_{\rm eff}^{-1}g^{(2)}_{\rm
max}(\tau)\right].$
Note how the dominant term is $g^{(2)}_{\rm max}(\tau)$ (interfering pathways
for two photons emitted by one atom) at small values of $\bar{N}_{\rm eff}$,
less than one, whereas for $\bar{N}_{\rm eff}\gg 1$, $|g^{(1)}_{\rm
max}(\tau)|^{2}$ (interfering time orders for two photons emitted by different
atoms) and $|g^{(a)}_{\rm max}(\tau)|^{2}$ (interfering pathways for two
photons emitted by one atom or by another) are the dominant terms.
To illustrate the quantum beat we recover the case of fixed atoms (averaged
over locations) by taking $\tau_{0}\to\infty$. Two examples are shown in Fig.
6(a): the first with all three terms making equal contributions to the beat,
and the second where $|g^{(1)}_{\rm max}(\tau)|^{2}$ and $|g^{(a)}_{\rm
max}(\tau)|^{2}$ dominate. Figure 6(b) illustrates the behavior of each term
separately for the former case. All terms oscillate with the same
frequency—twice the ground-state Larmor frequency—in the weak-drive limit.
There is a phase difference between the oscillations of $g^{(2)}_{\rm
max}(\tau)$ and $|g^{(a)}_{\rm max}(\tau)|^{2}$ (two-photon amplitudes) and
those of $|g^{(1)}_{\rm max}(\tau)|^{2}$ (one-photon amplitudes). This arises
from different gyromagnetic ratios in the ground and excited states.
Figure 6: (a) Quantum beat plotted from Eq. (28) in the fixed atom limit
($\tau_{0}\to\infty$): $\bar{N}_{\rm eff}=35/36$ (solid) and $\bar{N}_{\rm
eff}\gg 1$ (dashed). (b) Intensity $I(\tau)$ of the three terms that
contribute to make up the oscillatory part of the solid curve in (a):
$g^{(2)}_{\rm max}(\tau)$ (solid), $|g^{(1)}_{\rm max}(\tau)|^{2}$ (dashed),
and $|g^{(a)}_{\rm max}(\tau)|^{2}$ (dot-dashed). The parameters are: $g_{\rm
max}/\gamma=0.005$, $\kappa/\gamma=0.5$, $\mathcal{E}/\gamma=0.005$, and a
Larmor frequency of $\gamma/3$
.
A more realistic modeling of our experiment takes the speed distribution,
$D(v)$, of the atoms to correspond to a thermal effusive source:
$D(v)=2\alpha^{-4}v^{3}e^{-v^{2}/\alpha^{2}}$, where $\alpha^{2}=2k_{B}T/m$
Ramsey (1985). The additional average over speed yields
$\displaystyle g^{(2)}(\tau)$ $\displaystyle=$ $\displaystyle
1+\left(1+\frac{\pi}{4}\frac{\tau^{2}}{\tau_{0}^{2}}\right)^{\mkern-3.0mu-3}\mkern-3.0mu\left(|g^{(1)}_{\rm
max}(\tau)|^{2}+|g^{(a)}_{\rm max}(\tau)|^{2}\right)$ (29)
$\displaystyle+\frac{35}{36}\bar{N}_{\rm
eff}^{-1}\mkern-3.0mu\left(1+\frac{\pi}{2}\frac{\tau^{2}}{\tau_{0}^{2}}\right)^{\mkern-3.0mu-3/2}\mkern-3.0mug^{(2)}_{\rm
max}(\tau),$
where $\tau_{0}=w_{0}/\langle v\rangle$ is defined with $\langle
v\rangle=\sqrt{8k_{B}T/\pi m}$—the mean speed of an atom in the source.
Figure 7: Effect of a mean tilt of the atomic beam relative to the cavity
axis. The envelope of $g^{(2)}(\tau)-1$ is plotted for no tilt of the beam
(solid) and a mean tilt $\langle\theta\rangle=1.3\mkern 2.0mu{\rm deg}$
(dashed), with $\langle v\rangle=22\mkern 2.0mu{\rm m/s}$, $\Delta v=2\mkern
2.0mu{\rm m/s}$, and $\Delta\theta=0.025\mkern 3.0mu{\rm deg}$.
In practice the atoms do not move perfectly perpendicular to the cavity axis.
We do not have an analytical expressions for this most general case. Figure 7
shows the numerically calculated envelope of the correlation function for the
case $\bar{N}_{\rm eff}\gg 1$, assuming a triangular distribution for the
polar angle relative to the cavity axis $\theta$ and a Maxwell-Boltzmann speed
distribution. The figure shows how the standing-wave structure of the cavity
mode function becomes imprinted on the envelope of the quantum beats. The
local minimum of the dotted curve corresponds to the delay time when a
majority of atoms pass from an anti-node to a node between the detection of
the first and second photons. The spread in angle and speed explains why this
structure does not recur at longer delays.
## III Experiment and Results
### III.1 Apparatus
We perform measurements using a slightly modified version of the apparatus
described in Ref. Norris et al. (2009). A sketch of the experimental setup
appears in Fig. 1(b). We probe a small ensemble of 85Rb atoms coupled to a
Fabry-Perot resonator in vacuum. The $2.2\mkern 2.0mu{\rm mm}$ cavity has a
$56\mkern 2.0mu\mu{\rm m}$ mode waist and a finesse of $11,000$, with losses
shared approximately equally between mirror transmission and
scattering/absorption. The decay rates of the field and atomic
dipole—$(\kappa,\gamma/2)/2\pi=(2.8,3.0)\times 10^{6}\mkern 2.0mu{\rm
s}^{-1}$—are approximately matched, and twice as large as the dipole coupling
strength, $g_{\rm max}/2\pi=1.5\mkern 2.0mu{\rm MHz}$, on the $D_{2}$-line
$F=3$, $m=0$ to $F^{\prime}=4$, $m^{\prime}=0$ transition; this places our
system in the intermediate coupling regime of cavity QED—single-atom
cooperativity $C_{1}=g^{2}/\gamma\kappa=0.12$ and saturation photon number
$n_{0}=\gamma^{2}/3g^{2}=5.3$—with only a small probability of reabsorption
after a photon is emitted into the cavity mode.
A crossed-polarizer configuration separates the weak $H$-mode fluorescence
from the much stronger $V$-polarized drive, necessitating careful selection
and alignment of polarization elements. We drive the cavity with a laser
sideband generated by a polarization-maintaining single-mode $780\mkern
2.0mu{\rm nm}$ fiber modulator (EO Space) operated at $230\mkern 2.0mu{\rm
MHz}$. Before entering the mode-matching lens and vacuum chamber, the drive is
linearly polarized—extinction ratio less than $5\times 10^{-5}$—after passing
through a Glan-Thompson polarizer and zero-order half-wave plate (HWP). A
second zero-order HWP placed after the cavity aligns the polarization to a
calcite Wollaston prism for separation of the $H$\- and $V$-mode light at the
output. The extinction ratio after this splitter is limited by birefringence
in the cavity mirrors, vacuum chamber windows and lenses. Its exact value is a
function of drive intensity, likely due to thermoelastic stress-induced
birefringence in the components, but is generally of order $5\times 10^{-4}$.
The splitting of $H$\- and $V$-mode resonance frequencies due to cavity mirror
birefringence is less than $200\mkern 2.0mu{\rm kHz}$.
The separated beams go to two avalanche photodiodes (APDs, Perkin-Elmer) for
photon counting, except when measuring autocorrelation functions, in which
case the $V$-mode beam is blocked and the $H$-mode beam is split equally
between the two detectors by means of a separate HWP and polarizing beam
splitter. The TTL output channel of each detector is electronically split
between a counter unit, for measuring rates, and a PC time-stamp card (Becker
and Hickl DPC-230) for recording detection times with $164\mkern 2.0mu{\rm
ps}$ resolution. A typical time series measurement takes approximately
$300\mkern 2.0mu{\rm s}$. The output of the time-stamp card is written to a
text file and parsed by a C++ program to calculate the cross-correlation
between detection events (the autocorrelation of the $H$-mode). The correlated
events are recorded in a histogram, bin width 16.4 ns, extending out to
$\pm$16.4 $\mu$s. Division by the average (uncorrelated) bin count yields
$g^{(2)}(\tau)$. The power spectrum is calculated from a discrete fast fourier
transform (FFT) of this function.
Figure 8: (Color online) (a) Example of the $V$-mode absorption fraction as a
function of common laser and cavity detuning from the central $m=0$ to
$m^{\prime}=0$ atomic transition; for a magnetic field of $\sim 5\mkern
2.0mu{\rm G}$ and approximately one photon on average in the cavity. The
optical pumping beam (red/open circles “on”, blue/filled dots “off”) shifts
the population toward the center. Solid lines are least-square fits of the
data to a Lorentzian line shape. (b) $H$-mode count rates measured
concurrently with the data in (a).
The ${}^{85}{\rm Rb}$ atoms are extracted continuously from a magneto-optical
trap (MOT) operating as a low-velocity intense source (LVIS) Lu et al. (1996)
in a chamber directly above the cavity. The atoms have a mean speed of $\sim
22\mkern 2.0mu{\rm m/s}$, which yields an interaction time of a few
microseconds. Atoms leaving the MOT are primarily pumped to the $m=3$ ground
state, with quantization axis provided by the residual vertical magnetic field
from the MOT “anti-Helmholtz”coil pair ($\sim 7\mkern 2.0mu{\rm G}$ at the
location of the cavity). In order to change the magnetic field in the cavity,
a third coil is added directly below the “anti-Helmholtz” pair, with separate
currents applied to each of the three coils. We are thus able to vary the
vertical magnetic field in the cavity between $\pm 12\mkern 2.0mu{\rm G}$
while maintaining the required gradient for the MOT. An additional pair of
coils is oriented with axis parallel to the cavity axis to cancel any residual
field in that direction.
Before atoms enter the cavity mode they are optically pumped to the $m=0$
ground state using a beam resonant with the $F=3$ to $F^{\prime}=3$ transition
and polarization parallel to the vertical magnetic field. The optical pumping
beam is combined with light from the MOT re-pumper laser in a 50/50
polarization-maintaining single-mode fiber splitter, and collimated to a waist
diameter of $0.5\mkern 2.0mu{\rm mm}$ in order to pass between the top of the
cavity mirrors and the upper edge of the vacuum window. Due to strong
scattering from multiple reflections into the APDs, we are unable to use the
beam in a retro-reflected configuration. It therefore imparts a net momentum
kick to the atoms. The intensity of the optical pumping beam is chosen
optimally as a compromise between moving most atoms to $m=0$, while not
ejecting too many from the beam, and scattering too strongly into the APDs.
Figure 8 shows a typical measurement sequence used for optimizing the optical
pumping configuration. Frame (a) shows the absorption fraction (output
intensity over input intensity, denoted $X/Y$) as measured from the $V$-mode
count rates with the cavity and drive frequencies simultaneously swept across
the atomic resonance. The effect of the optical pumping is to shift and narrow
the absorption peak, ideally to yield a symmetric lineshape centered around
the resonance frequency of the $F=3$, $m=0$ to $F^{\prime}=4$, $m^{\prime}=0$
transition ($0\mkern 2.0mu{\rm MHz}$ in the plot). The departure of the center
frequency from zero results from a combination of incomplete optical pumping
and a small drift in the frequency setpoint of the Rb saturated absorption
spectroscopy reference used for the laser. In frame (b) count rates for the
$H$-mode light show a similar effect. We use the center frequency obtained
from these scans as the reference (zero-detuning) point for our measurements.
The extensive Optical Bistability literature is useful for understanding and
interpreting the effects of absorption and detuning in our experiment. When
making these connections (see for example Lugiato (1984)) it should be noted
that we operate in the low intensity limit, and simultaneously scan the laser
and cavity in order to address the atoms directly.
### III.2 Results
Figure 9: (a) Measured intensity correlation function $g^{(2)}(\tau)$ and (b)
its FFT power spectrum; for a $5\mkern 2.0mu{\rm G}$ magnetic field,
$\bar{N}_{\rm eff}=2.9$, and approximately $6.5$ photons in the $V$ mode with
no atoms present. The peak in the spectrum is located at $\approx 4.8\mkern
2.0mu{\rm MHz}$, twice the ground state Larmor frequency for ${}^{85}{\rm Rb}$
in a $5\mkern 2.0mu{\rm G}$ field.
Frame (a) of Fig. 9 displays a measured correlation function for a magnetic
field of $5\mkern 2.0mu{\rm G}$; its calculated power spectrum is displayed in
frame (b). The main peak near 4.8 MHz corresponds to the quantum beat
resonance. A smaller peak at half this frequency is also present, though
largely obscured by noise. It is the result of homodyne interference with
drive light mixed in by cavity birefringence [see the paragraph surrounding
Eq. (8)]. The small sidebands on the main peak correspond to a slight
modulation of the beat envelope. The modulation is visible in frame (a) and
results from the small ($1-2$ degree) deviation of the atomic beam from normal
incidence with the axis of the cavity, which introduces sinusoidally varying
coupling strengths, $g_{j}(t)$, and amplitude modulation of the spontaneous
emission rate (see Fig. 7).
Figure 10 illustrates the changing frequency of the quantum beat with
increasing magnetic field, where the expected linear dependence is observed.
We note that the beat frequencies also depend on the intensity of the drive
through an anomalous light-shift, which we report elsewhere Norris et al.
(2012). Those presented here are extracted as the zero-intensity (i.e.
unshifted) limit of the measured frequencies for each magnetic field. Our
zeroing of the magnetic field in all three directions is not better than
$10\mkern 2.0mu{\rm mG}$. This error in independent calibration makes the
small offset in the figure consistent with zero.
Figure 10: Measured linear dependence of the frequency of the quantum beat on
magnetic field.
Figure 11 illustrates the change in the observed beat when the polarization
presented to the detector is not taken orthogonal to that of the drive but
allowed to rotate by a few degrees. The rotation is controlled by changing the
angle of the HWP placed between the cavity and the PBS (Fig. 1b). This mixes a
small amount of drive light with the scattered light. With increasing mixed-in
fraction the beat is eventually dominated by a homodyne term, which arises
from the correlation of a photon scattered into the $H$ mode with a photon
from the drive [see Fig. 5 and terms proportional to $\epsilon^{2}$ in Eqs.
(8), (14), (16), and (22)]; thus, as in the two-atom case, interfering time
orders also yield a quantum beat. This beat oscillates at half the frequency
and allows the correlation function to dip below one. Generally, some drive
light is coupled into the $H$ mode through a small birefringence of the cavity
mirrors. Frames (c) and (d) of Fig. 11 are recorded at the HWP angle that
gives maximum visibility of the half-frequency beat. Frames (b) and (d) show
the dramatic increase in visibility gained when the driving laser is slightly
detuned from resonance. This is related to a decreased decoherence rate from
quantum jumps (see Ref. Norris et al. (2012)).
Figure 11: Evolution of the measured $g^{(2)}(\tau)$ with homodyne mixing and
detuning of the drive: (a) on resonance, maximum extinction; (b) $-6\mkern
2.0mu{\rm MHz}$ detuning, maximum extinction; (c) on resonance, HWP rotated by
$2.8$ degrees; (d) $-6\mkern 2.0mu{\rm MHz}$ detuning, HWP rotated by $1.2$
degrees. Data taken for a $5\mkern 2.0mu{\rm G}$ magnetic field, $\bar{N}_{\rm
eff}=2.9$, and approximately $2.0$ photons in the $V$ mode with no atoms
present.
In Fig. 12 we show how $g^{(2)}(\tau)$ evolves as the number of effective
atoms increases from less than one to nearly three. For the fewest atoms
[frame (a)], fluctuations in the number interacting with the cavity show up as
a broad Gaussian background peak, reflecting the increase in the scattering
rate when an atom is present (similar to Ref. Norris et al. (2009).) The beats
sit on top of this background, with small visibility, a consequence of optical
pumping and spontaneous emission (to modes other than the cavity mode). The
correlation function is dominated by the contribution from $g^{(2)}_{\rm
max}(\tau)$ in Eqs. (28) and (29). As the density of the atomic beam grows
[frames (b) and (c)] the background peak disappears as contributions from
multiple atoms become more prominent; contributions from $|g^{(1)}_{\rm
max}(\tau)|^{2}$, $|g^{(a)}_{\rm max}(\tau)|^{2}$, and $g^{(2)}_{\rm
max}(\tau)$ contribute with more-or-less weight in frame (c).
Figure 12: Evolution of the measured $g^{(2)}(\tau)$ with increasing atomic
beam density. Data taken for a $5\mkern 2.0mu{\rm G}$ magnetic field,
$\bar{N}_{\rm eff}=0.3,0,9,2.9$, (a,b,c) and approximately $3.1$ photons in
the $V$ mode with no atoms present.
Finally, Fig. 13 shows the change in the spectrum when the optical pumping
beam is added. The effect on the beat frequency is minimal, revealing the
robustness of the quantum interference to the initial distribution of the
atoms amongst the ground-state Zeeman levels. The biggest change is in the
low-frequency components of the FFT, which correspond to atomic motion through
the standing-wave mode. This is caused by the momentum kick imparted by the
optical pumping beam.
Figure 13: (Color online) Sample power spectrum [FFT of the measured
$g^{(2)}(\tau)$] with (red, dashed) and without (blue, solid) optical pumping
of the atoms prior to entering the cavity.
### III.3 Comparison of theory and experiment
Outside the bad-cavity and adiabatic limit, the normalized correlation
function is given by
$g^{(2)}(\tau)=G^{(2)}(\tau)/I^{2},$ (30)
with $I$ and $G^{(2)}(\tau)$ defined in Eqs. (21) and (22). For comparison
with experiment, we introduce a global scale parameter, $s$, writing
$\displaystyle g^{(2)}(\tau)$ $\displaystyle=$ $\displaystyle
1+s\left\\{\vphantom{\frac{\epsilon^{2}}{\bar{N}_{\rm
eff}I+\epsilon^{2}}}|g^{(1)}_{A}(\tau)|^{2}+|g_{A}^{(a)}(\tau)|^{2}\right.$
(31) $\displaystyle+\bar{N}_{\rm eff}^{-1}g_{A}^{(2)}(\tau)$
$\displaystyle\left.+\frac{\epsilon^{2}}{\bar{N}_{\rm eff}I+\epsilon^{2}}2{\rm
Re}[g_{A}^{(1)}(\tau)+g_{A}^{(a)}(\tau)]\right\\},\mkern 10.0mu$
where ($\xi=1,2,a$)
$g_{A}^{(\xi)}(\tau)=\frac{G_{A}^{(\xi)}(\tau)}{\bar{N}_{\rm
eff}I+\epsilon^{2}}.$ (32)
This expression is evaluated numerically by calculating the one-atom
correlation functions from the model of the atomic beam developed in Section
II.3. The speeds $v_{j}$ are selected from a Gaussian distribution, and the
angles $\theta_{j}$ and $\phi_{j}$ from a triangular distribution.
Figure 14(a) and (b) show a fit to the experimental data and its FFT power
spectrum. Experimental error bars are computed as the square root of the
number of photon counts in each bin. In order to improve the fit, we adjust
the mean speed and angles according to the following considerations. The
experimental data shows a low frequency modulation at short delays, the
signature of a small mean inclination of the atomic trajectories away from
normal to the cavity axis (see Fig. 7). Comparing the FFT of the measured
autocorrelation function with simulations at different mean beam angles
$\langle\theta\rangle$, with $\langle\phi\rangle=0$, we find that
$\langle\theta\rangle=1.4$ degrees optimizes the fit. For small angles like
this, the main parameter affecting the width of the peak at twice the Larmor
frequency, around $5\mkern 2.0mu{\rm MHz}$ in the figure, is the mean atomic
speed. A value of $\langle v\rangle=17\mkern 2.0mu{\rm m/s}$ optimizes the
fit, consistent with values expected from an LVIS Lu et al. (1996). The
amplitude of the peak around $2.5\mkern 2.0mu{\rm MHz}$ is determined by the
value of $\epsilon$, which is adjusted as another free parameter.
Frame (a) of Fig. 14 compares theory and experiment in the time domain. The
reduced chi-square value for this fit is $1.3$. The major differences between
the calculation and the experiment comes for delays close to zero; here the
measured correlation function is substantially larger in value compared to the
calculated one. This is at least partially due to the presence of uncorrelated
background light scattered into the detection path (primarily from the MOT
cooling beams). Frame (b) of Fig. 14 shows the fit in the frequency domain.
The calculated spectrum in fact goes to zero at high frequencies (larger than
7 MHz), while the experimental one remains flat due to the presence of
background light in the detectors (up to about 200 MHz). With a flat frequency
background added to the calculation to account for the residual background and
shot noise (as shown in the figure), the reduced chi-square value of the fit
is $0.99$. Frame (c) shows a similar correlation function after addition of a
small amount of coherent drive in order to enhance the homodyne signal. The
simulation accurately captures the pronounced change in frequency and shape of
the signal, with a reduced chi-square value of 1.6.
Figure 14: (Color online) Sample comparison of the measured (points) and
calculated (solid line) correlation function, (a), and its FFT power spectrum,
(b), under conditions where atomic spontaneous emission dominates the measured
photon counts. A flat background is added to the calculated spectrum in order
to account for residual noise in the detectors. The atomic beam parameters
are: $\bar{N}_{\rm eff}=3$, $s=0.53$, $\langle v\rangle=17\mkern 2.0mu{\rm
m/s}$, $\Delta v=2\mkern 2.0mu{\rm m/s}$, $\langle\theta\rangle=1.4\mkern
4.0mu{\rm deg}$, $\Delta\theta=0.9\mkern 4.0mu{\rm deg}$,
$\langle\phi\rangle=0\mkern 4.0mu{\rm deg}$, $\Delta\phi=0.7\mkern 4.0mu{\rm
deg}$. The mean number of scattered photons in the $V$ mode is $0.63$, and the
birefringence background $\epsilon^{2}$ is 1.2% of the driven $V$-mode photon
number. (c) Sample measured (dashed line) and calculated (points) correlation
functions with a small amount of coherent drive admixed in order to enhance
the homodyne signal. Parameters are: $\bar{N}_{\rm eff}=0.55$, $s=1$, $\langle
v\rangle=13.5\mkern 2.0mu{\rm m/s}$, $\Delta v=2\mkern 2.0mu{\rm m/s}$,
$\langle\theta\rangle=0.97\mkern 4.0mu{\rm deg}$, $\Delta\theta=0.9\mkern
4.0mu{\rm deg}$, $\langle\phi\rangle=0\mkern 4.0mu{\rm deg}$,
$\Delta\phi=0.7\mkern 4.0mu{\rm deg}$. The mean number of scattered photons in
the $V$ mode is $1.2$, and the mixed coherent field $\epsilon^{2}$ is 0.5% of
the driven $V$-mode photon number, which due to the smaller number of atoms is
strong enough to substantially change the signal.
## IV Summary and Conclusions
We have studied theoretically and experimentally how the ground-state quantum
beats reported in Norris et al. (2010) depend on different parameters. The
fundamental beat frequency occurs at twice the Larmor frequency and is found
to increase linearly with magnetic field as expected. Mixing of the driving
and scattered fields produces a beat at the Larmor frequency itself.
Increasing the number of atoms brings the minimum of the oscillation to the
shot noise level; this is because the many pairs of two-atom beats come to
dominate the one-atom signal. The stochastic evolution of coherence within the
atomic level structure shows that the many levels of the $F=3$ to
$F^{\prime}=4$ transition help make the observed quantum beats robust against
optical pumping.
## Appendix A Dipole operators
The one atom dipole operators are given by
$\displaystyle\Sigma_{\pi}$ $\displaystyle=$
$\displaystyle\sqrt{\frac{1}{4}}|g_{-3}\rangle\langle
e_{-3}|+\sqrt{\frac{3}{7}}|g_{-2}\rangle\langle
e_{-2}|+\sqrt{\frac{15}{28}}|g_{-1}\rangle\langle
e_{-1}|+\sqrt{\frac{4}{7}}|g_{0}\rangle\langle
e_{0}|+\sqrt{\frac{15}{28}}|g_{+1}\rangle\langle e_{+1}|$ (33)
$\displaystyle+\sqrt{\frac{3}{7}}|g_{+2}\rangle\langle
e_{+2}|+\sqrt{\frac{1}{4}}|g_{+3}\rangle\langle e_{+3}|\,,$
$\displaystyle\Sigma_{\sigma_{+}}$ $\displaystyle=$
$\displaystyle\sqrt{\frac{1}{28}}|g_{-3}\rangle\langle
e_{-2}|+\sqrt{\frac{3}{28}}|g_{-2}\rangle\langle
e_{-1}|+\sqrt{\frac{3}{14}}|g_{-1}\rangle\langle
e_{0}|+\sqrt{\frac{5}{14}}|g_{0}\rangle\langle
e_{+1}|+\sqrt{\frac{15}{28}}|g_{+1}\rangle\langle e_{+2}|$ (34)
$\displaystyle+\sqrt{\frac{3}{4}}|g_{+2}\rangle\langle
e_{+3}|+|g_{+3}\rangle\langle e_{+4}|\,,$ $\displaystyle\Sigma_{\sigma_{-}}$
$\displaystyle=$ $\displaystyle|g_{-3}\rangle\langle
e_{-4}|+\sqrt{\frac{3}{4}}|g_{-2}\rangle\langle
e_{-3}|+\sqrt{\frac{15}{28}}|g_{-1}\rangle\langle
e_{-2}|+\sqrt{\frac{5}{14}}|g_{0}\rangle\langle
e_{-1}|+\sqrt{\frac{3}{14}}|g_{+1}\rangle\langle e_{0}|$ (35)
$\displaystyle+\sqrt{\frac{3}{28}}|g_{+2}\rangle\langle
e_{+1}|+\sqrt{\frac{1}{28}}|g_{+3}\rangle\langle e_{+2}|\,.$
###### Acknowledgements.
Work supported by NSF, CONACYT, México, and the Marsden Fund of the Royal
Society of New Zealand.
## References
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|
arxiv-papers
| 2012-11-27T17:15:16 |
2024-09-04T02:49:38.566347
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "D. G. Norris (1), A. D. Cimmarusti (1), L. A. Orozco (1), P.\n Barberis-Blostein (2) and H. J. Carmichael (3) ((1) Joint Quantum Institute,\n Department of Physics, University of Maryland and National Institute of\n Standards and Technology, College Park, MD USA, (2) Instituto de\n Investigaci\\'on en Matem\\'aticas Aplicadas y en Sistemas, Universidad\n Nacional Aut\\'onoma de M\\'exico, M\\'exico, (3) Department of Physics,\n University of Auckland, New Zealand)",
"submitter": "Andres Cimmarusti",
"url": "https://arxiv.org/abs/1211.6364"
}
|
1211.6399
|
# weighted composition operators on weak vector-valued weighted bergman spaces
and Hardy spaces
Mostafa Hassanlou and Hamid Vaezi
###### Abstract.
In this paper we investigate weighted composition operators between weak and
strong vector-valued weighted Bergman spaces and Hardy spaces.
###### Key words and phrases:
Weighted composition operators, vector-valued Bergman spaces, vector-valued
Hardy spaces .
###### 2000 Mathematics Subject Classification:
47B35.
## 1\. Introduction and Preliminaries
Weighted composition operators have been studied on different spaces of
analytic functions. In [5], Contreras and Hernandez-Diaz have made a study of
weighted composition operators on Hardy spaces whereas Mirzakarimi and
Siddighi [12] have studied these operators on Bergman and Dirichlet spaces. On
Bloch-type spaces, these operators are explored by MacCluer and Zhao [11],
Ohno [13], Ohno and Zhao [14] and Ohno, Stroethoff and Zhao [15]. In [8] Kumar
studied weighted composition operators between spaces of Dirichlet type by
using Carleson measures.
Recently these studies are about spaces of vector-valued analytic functions.
For example, in [17], Wang presented some necessary and sufficient conditions
for weighted composition operators to be bounded on vector-valued Dirichlet
spaces and Laitila, Tylli and Wang [10] studied composition operators from
weak to strong vector-valued Bergman spaces Hardy spaces. For some information
about vector-valued Bergman spaces see [1, 3].
Let $X$ be a complex Banach space and $\mathbb{D}$ be the open unit ball of
$\mathbb{C}$. We consider weight as a strictly positive bounded continuous
function $v:\mathbb{D}\rightarrow\mathbb{\mathbb{R}}^{+}$. Let $p\geq 1$ and
$v$ be a weight. The vector-valued weighted Bergman space $A_{v}^{p}(X)$
consists of all analytic functions $f:\mathbb{D}\rightarrow X$ such that
$||f||_{A_{v}^{p}(X)}=\left(\int_{\mathbb{D}}||f(z)||_{X}^{p}v(z)dA(z)\right)^{\frac{1}{p}}<\infty.$
where $dA$ is the normalized area measure on $\mathbb{D}$. Also, the vector-
valued weighted Hardy space $H_{v}^{p}(X)$ consists of all analytic functions
$f:\mathbb{D}\rightarrow X$ for which
$||f||_{H_{v}^{p}(X)}=\sup_{0<r<1}\left(\int_{\mathbb{T}}||f(r\zeta)||_{X}^{p}v(r\zeta)\
dm(\zeta)\right)^{\frac{1}{p}}<\infty,$
where $dm(\zeta)$ is the normalized Lebesgue measure on the unit circle
$\mathbb{T}=\partial\mathbb{D}$. In the case $X=\mathbb{C}$, we write
$A_{v}^{p}(X)=A_{v}^{p}$ and $H_{v}^{p}(X)=H_{v}^{p}$. Also, if $v\equiv 1$,
then we have $A_{v}^{p}(X)=A^{p}(X)$ and $H_{v}^{p}(X)=H^{p}(X)$. The
following weak versions of these spaces were considered by e.g. Blasco [2] and
Bonet, Domanski and Lindstrom [4]: the weak spaces $wA_{v}^{p}(X)$ and
$wH_{v}^{p}(X)$ consist of all analytic functions $f:\mathbb{D}\rightarrow X$
for which
$||f||_{wA_{v}^{p}(X)}=\sup_{||x^{*}||\leq 1}||x^{*}\circ f||_{A_{v}^{p}},\ \
\ \ ||f||_{wH_{v}^{p}(X)}=\sup_{||x^{*}||\leq 1}||x^{*}\circ f||_{H_{v}^{p}},$
are finite, respectively. Here $x^{*}\in X^{*}$, the dual space of $X$.
Let $\varphi$ be an analytic self-map of $\mathbb{D}$; that is
$\varphi(\mathbb{D})\subset\mathbb{D}$, and $u$ a scaler-valued analytic
function on $\mathbb{D}$. We can define the weighted composition operator
$uC_{\varphi}$ on the space of analytic functions as follows:
$uC_{\varphi}(f)(z)=u(z)f(\varphi(z)).$
When $u(z)\equiv 1$, we just have the composition operator $C_{\varphi}$,
defined by $C_{\varphi}(f)=f\circ\varphi$. Also if $\varphi=I$, the identity
function, then we get the multiplication operator $M_{u}$ defined by
$M_{u}(f)(z)=u(z)f(z)$. It is well known that for every analytic map
$\varphi:\mathbb{D}\rightarrow\mathbb{D}$, $C_{\varphi}:A^{p}(X)\rightarrow
A^{p}(X)$ and $C_{\varphi}:H^{p}(X)\rightarrow H^{p}(X)$ are bounded, and and
also on $wA^{p}(X),wH^{p}(X)$. For complete discussion about composition
operators we refer to [6, 16]. We consider the infinite dimensional complex
Banach space $X$, since $wA^{p}(X)=A^{p}(X)$ and $wH^{p}(X)=H^{p}(X)$, for
$\alpha>-1$ and any finite dimensional Banach space $X$.
But for the infinite dimensional complex Banach space $X$,
$A^{p}(X)\not=wA^{p}(X)$ ($H^{p}(X)\not=wH^{p}(X)$) and $||.||_{wA^{p}(X)}$ is
not equivalent to $||.||_{A^{p}(X)}$ on $A^{p}(X)$ ($||.||_{wH^{p}(X)}$ is not
equivalent to $||.||_{H^{p}(X)}$ on $H^{p}(X)$), see [10] Proposition 3.1 ([9]
Example 15).
Our aim in this paper is to compute the norm of weighted composition operators
between $wA_{v}^{p}(X)$ and $A_{v}^{p}(X)$, for $p\geq 2$ and also between
$wH_{v}^{p}(X)$ and $H_{v}^{p}(X)$, for $p\geq 2$, where $v$ and $v^{\prime}$
are weights.
Throughout the remainder of this paper, c will denote a positive constant, the
exact value of which will vary from one appearance to the next. The notation A
$\approx$ B means that there are positive constants $c_{1}$ and $c_{2}$ such
that $c_{1}A\leq B\leq c_{1}A$.
## 2\. Main Results
###### Proposition 2.1.
Let $X$ be any complex Banach spaces, $v$ be a weight of the form $v=|\mu|$,
where $\mu$ is an analytic function without any zeros on $\mathbb{D}$,
$v^{\prime}$ be a weight and $1\leq p<\infty$. Then
(2.1) $||uC_{\varphi}:wA_{v}^{p}(X)\longrightarrow
A_{v^{\prime}}^{p}(X)||\leq\left(\int_{\mathbb{D}}\frac{|u(z)|^{p}v^{\prime}(z)}{(1-|\varphi(z)|^{2})^{2}v(\varphi(z))}\,dA(z)\right)^{1/p},and$
(2.2) $||uC_{\varphi}:wH_{v}^{p}(X)\longrightarrow
H_{v^{\prime}}^{p}(X)||\leq\sup_{0<r<1}\left(\int_{\mathbb{T}}\frac{|u(r\zeta)|^{p}v^{\prime}(r\zeta)}{(1-|\varphi(r\zeta)|^{2})v(\varphi(r\zeta))}\,dm(\zeta)\right)^{1/p}.$
Proof. By Lemma 2.1 of [18] we have
$|f(z)|\leq\frac{||f||_{A_{v}^{p}}}{(1-|z|^{2})^{\frac{2}{p}}v(z)^{\frac{1}{p}}},$
for any $f\in A_{v}^{p}$ and $z\in\mathbb{D}$. Thus, for $f\in wA_{v}^{p}(X)$,
we have
$||f(z)||_{X}^{p}=\sup_{||x^{*}||\leq 1}|(x^{*}\circ
f)(z)|^{p}\leq\frac{1}{(1-|z|^{2})^{2}v(z)}\sup_{||x^{*}||\leq 1}||x^{*}\circ
f||_{A_{v}^{p}}^{p}$
$=\frac{1}{(1-|z|^{2})^{2}v(z)}||f||_{wA_{v}^{p}(X)}^{p}.$
Hence
$\displaystyle||uC_{\varphi}f||_{A_{v^{\prime}}^{p}(X)}^{p}=$
$\displaystyle\int_{\mathbb{D}}|u(z)|^{p}||f(\varphi(z))||_{X}^{p}v^{\prime}(z)\,dA(z)$
$\displaystyle\leq$
$\displaystyle||f||_{wA_{v}^{p}(X)}^{p}\int_{\mathbb{D}}\frac{|u(z)|^{p}v^{\prime}(z)}{(1-|z|^{2})^{2}v(\varphi(z))}\,dA(z).$
The proof of the theorem is complete. $\square$
For the next results we need the following Dvoretzky,s well-known theorem.
###### Lemma 2.2.
[7] Suppose that $X$ is an infinite dimensional complex Banach space. Then for
any $\epsilon>0$ and $n\in\mathbb{N}$, there is a linear embedding
$T_{n}:l_{n}^{2}\rightarrow X$ such that
(2.3)
$(1+\epsilon)^{-1}\bigg{(}\sum_{j=1}^{n}|a_{j}|^{2}\bigg{)}^{1/2}\leq\bigg{\|}\sum_{j=1}^{n}a_{j}T_{n}e_{j}\bigg{\|}_{X}\leq\bigg{(}\sum_{j=1}^{n}|a_{j}|^{2}\bigg{)}^{1/2}$
for any scalars $a_{1},a_{2},\cdots,a_{n}$ and some orthonormal basis
$\\{e_{1},\cdots,e_{n}\\}$ of $l_{n}^{2}$.
Now, we prove a lower bound for the operator
$uC_{\varphi}:wA_{v}^{p}(X)\longrightarrow A_{v^{\prime}}^{p}(X)$, in the case
$2\leq p<\infty$.
###### Theorem 2.3.
Let $X$ be any complex infinite-dimensional Banach space, $v$ be a weight of
the form $v=|\mu|$, where $\mu$ is an analytic function without any zeros on
$\mathbb{D}$, $v^{\prime}$ be a weight and $2\leq p<\infty$. Then
(2.4) $||uC_{\varphi}:wA_{v}^{p}(X)\longrightarrow
A_{v^{\prime}}^{p}(X)||\approx\left(\int_{\mathbb{D}}\frac{|u(z)|^{p}v^{\prime}(z)}{(1-|\varphi(z)|^{2})^{2}v(\varphi(z))}\,dA(z)\right)^{1/p}.$
Proof. We only prove there exists a positive constant $c$ such that
$||uC_{\varphi}:wA_{v}^{p}(X)\longrightarrow A_{v^{\prime}}^{p}(X)||\geq
c\left(\int_{\mathbb{D}}\frac{|u(z)|^{p}v^{\prime}(z)}{(1-|\varphi(z)|^{2})^{2}v(\varphi(z))}\,dA(z)\right)^{1/p}.$
Suppose that $x\in X$ with $||x||=1$ and define $g:\mathbb{D}\rightarrow X$ by
$g(z)=\frac{1}{\mu(z)^{\frac{1}{p}}}x$. Then $g$ is an analytic function on
$\mathbb{D}$, and $||g||_{wA_{v}^{p}(X)}=1$, so that
$||uC_{\varphi}||^{p}\geq||ug\circ\varphi||_{A_{v^{\prime}}^{p}}^{p}=\int_{\mathbb{D}}\frac{|u(z)|^{p}v^{\prime}(z)}{v(\varphi(z))}\
dA(z).$
Hence
$\int_{\\{z\in\mathbb{D}:|\varphi(z)|^{2}<1/2\\}}\frac{|u(z)|^{p}v^{\prime}(z)}{(1-|\varphi(z)|^{2})^{2}v(\varphi(z))}\,dA(z)\leq
4\int_{\mathbb{D}}\frac{|u(z)|^{p}v^{\prime}(z)}{v(\varphi(z))}\ dA(z)\leq
4||uC_{\varphi}||^{p}.$
So, it will be sufficient to prove that there exists a positive constant $c$
such that
$||uC_{\varphi}||^{p}\geq c\int_{\\{z\in\mathbb{D}:|\varphi(z)|^{2}\geq
1/2\\}}\frac{|u(z)|^{p}v^{\prime}(z)}{(1-|\varphi(z)|^{2})^{2}v(\varphi(z))}\,dA(z).$
Let $\lambda_{k}=k^{2/p-1/2}$, for any $n\in\mathbb{N}$, we define functions
$f_{n}$ as follows
$f_{n}(z)=\frac{1}{\mu(z)^{\frac{1}{p}}}\sum_{k=1}^{n}\lambda_{k}z^{k}T_{n}e_{k},$
where the linear embedding $T_{n}$ is the same as in Lemma 2.2, $||T_{n}||=1$
and $||T_{n}^{-1}||\leq(1+\epsilon)$ and $(e_{1},\cdots,e_{n})$ is an
orthonormal basis of $\ell_{2}^{n}$. As in the proof of Theorem 3.2 [10],
there exists $c>0$ such that for $X^{*}$ with $||x^{*}||\leq 1$, we have
$\displaystyle\|x^{*}\circ f_{n}\|_{A_{v}^{p}}=$
$\displaystyle\|\frac{1}{\mu(z)^{\frac{1}{p}}}\sum_{k=1}^{n}\lambda_{k}z^{k}x^{*}T_{n}e_{k}\|_{A_{v}^{p}}$
$\displaystyle=$
$\displaystyle||\sum_{k=1}^{n}\lambda_{k}x^{*}(T_{n}e_{k})z^{k}\|_{A^{p}}$
$\displaystyle\leq$ $\displaystyle
c\left(\sum_{k=1}^{n}|x^{*}(T_{n}e_{k})|^{2}\right)^{1/2}\leq c.$
It follows that $\|f_{n}\|_{wA_{v}^{p}(X)}\leq c$. Thus, Fatou’s lemma implies
that
$\displaystyle\|uC_{\varphi}\|^{p}\geq$ $\displaystyle
c^{-p}\limsup_{n\rightarrow\infty}\|uC_{\varphi}f_{n}\|_{A_{v^{\prime}}^{p}(X)}^{p}$
$\displaystyle=$ $\displaystyle
c^{-p}\limsup_{n\rightarrow\infty}\int_{\mathbb{D}}|u(z)|^{p}\|\frac{1}{\mu(\varphi(z))^{\frac{1}{p}}}\sum_{k=1}^{n}\lambda_{k}\varphi(z)^{k}T_{n}e_{k}\|_{X}^{p}v^{\prime}(z)\,dA(z)$
$\displaystyle=$ $\displaystyle
c^{-p}\limsup_{n\rightarrow\infty}\int_{\mathbb{D}}\|\sum_{k=1}^{n}\lambda_{k}\varphi(z)^{k}T_{n}e_{k}\|_{X}^{p}\frac{|u(z)|^{p}v^{\prime}(z)}{v(\varphi(z))}\,dA(z)$
$\displaystyle\geq$
$\displaystyle\frac{c^{-p}}{(1+\epsilon)^{p}}\limsup_{n\rightarrow\infty}\int_{\mathbb{D}}\left(\sum_{k=1}^{n}k^{4/p-1}|\varphi(z)|^{2k}\right)^{p/2}\frac{|u(z)|^{p}v^{\prime}(z)}{v(\varphi(z))}\
dA(z)$ $\displaystyle=$
$\displaystyle\frac{c^{-p}}{(1+\epsilon)^{p}}\int_{\mathbb{D}}\left(\sum_{k=1}^{\infty}k^{4/p-1}|\varphi(z)|^{2k}\right)^{p/2}\frac{|u(z)|^{p}v^{\prime}(z)}{v(\varphi(z))}\
dA(z)$ $\displaystyle\geq$
$\displaystyle\frac{c_{1}c^{-p}}{(1+\epsilon)^{p}}\int_{\\{z\in\mathbb{D}:|\varphi(z)|^{2}\geq
1/2\\}}\frac{|u(z)|^{p}v^{\prime}(z)}{(1-|\varphi(z)|^{2})^{2}v(\varphi(z))}\
dA(z)$
and the last inequality is derived by Lemma 2.3 [10]. As $\epsilon>0$ was
arbitrary, we obtain the desired lower bound estimate.
$\square$
###### Theorem 2.4.
Let $X$ be any complex infinite-dimensional Banach space, $v$ be a weight of
the form $v=|\mu|$, where $\mu$ is an analytic function without any zeros on
$\mathbb{D}$, $v^{\prime}$ be a weight and $2\leq p<\infty$. Then
(2.5) $||uC_{\varphi}:wH_{v}^{p}(X)\longrightarrow
H_{v^{\prime}}^{p}(X)||\approx\left(\int_{\mathbb{T}}\frac{|u(\zeta)|^{p}v^{\prime}(\zeta)}{(1-|\varphi(\zeta)|^{2})v(\varphi(\zeta))}\,dm(\zeta)\right)^{1/p}.$
Proof. Similar to the proof of previous theorem, we only prove that there
exists $c>0$ such that
$||uC_{\varphi}||^{p}\geq
c\int_{\\{\zeta\in\mathbb{T}:|\varphi(r\zeta)|^{2}\geq
1/2\\}}\frac{|u(r\zeta)|^{p}v^{\prime}(r\zeta)}{(1-|\varphi(r\zeta)|^{2})v(\varphi(r\zeta))}\,dm(\zeta).$
Let $\lambda_{k}=k^{1/p-1/2}$ and define
$f_{n}(z):=\frac{1}{\mu(z)^{\frac{1}{p}}}\sum_{k=1}^{n}\lambda_{k}z^{k}T_{n}e_{k},$
where the linear embedding $T_{n}$ is the same as in Lemma 2.2, $||T_{n}||=1$
and $||T_{n}^{-1}||\leq(1+\epsilon)$ and $(e_{1},\cdots,e_{n})$ is an
orthonormal basis of $\ell_{2}^{n}$. As in the proof of Theorem 2.2 [10],
there exists $c>0$ such that for $X^{*}$ with $||x^{*}||\leq 1$, we have
$\displaystyle\|x^{*}\circ f_{n}\|_{H_{v}^{p}}=$
$\displaystyle\|\frac{1}{\mu(z)^{\frac{1}{p}}}\sum_{k=1}^{n}\lambda_{k}z^{k}x^{*}T_{n}e_{k}\|_{H_{v}^{p}}$
$\displaystyle=$
$\displaystyle||\sum_{k=1}^{n}\lambda_{k}x^{*}(T_{n}e_{k})z^{k}\|_{H^{p}}$
$\displaystyle\leq$ $\displaystyle
c\left(\sum_{k=1}^{n}|x^{*}(T_{n}e_{k})|^{2}\right)^{1/2}\leq c.$
Thus $\|f_{n}\|_{wH_{v}^{p}(X)}\leq c$ and by suing Fatou’s lemma and Lemma
2.3 [10], we have
$\displaystyle\|uC_{\varphi}\|^{p}\geq$ $\displaystyle
c^{-p}\limsup_{n\rightarrow\infty}\|uC_{\varphi}f_{n}\|_{H_{v^{\prime}}^{p}(X)}^{p}$
$\displaystyle=$ $\displaystyle
c^{-p}\limsup_{n\rightarrow\infty}\int_{\mathbb{T}}|u(r\zeta)|^{p}\|\frac{1}{\mu(\varphi(r\zeta))^{\frac{1}{p}}}\sum_{k=1}^{n}\lambda_{k}\varphi(r\zeta)^{k}T_{n}e_{k}\|_{X}^{p}v^{\prime}(r\zeta)\,dm(\zeta)$
$\displaystyle=$ $\displaystyle
c^{-p}\limsup_{n\rightarrow\infty}\int_{\mathbb{T}}\|\sum_{k=1}^{n}\lambda_{k}\varphi(r\zeta)^{k}T_{n}e_{k}\|_{X}^{p}\frac{|u(r\zeta)|^{p}v^{\prime}(r\zeta)}{v(\varphi(r\zeta))}\,dm(\zeta)$
$\displaystyle\geq$
$\displaystyle\frac{c^{-p}}{(1+\epsilon)^{p}}\limsup_{n\rightarrow\infty}\int_{\mathbb{T}}\left(\sum_{k=1}^{n}k^{2/p-1}|\varphi(r\zeta)|^{2k}\right)^{p/2}\frac{|u(r\zeta)|^{p}v^{\prime}(r\zeta)}{v(\varphi(r\zeta))}\
dm(\zeta)$ $\displaystyle=$
$\displaystyle\frac{c^{-p}}{(1+\epsilon)^{p}}\int_{\mathbb{D}}\left(\sum_{k=1}^{\infty}k^{2/p-1}|\varphi(r\zeta)|^{2k}\right)^{p/2}\frac{|u(r\zeta)|^{p}v^{\prime}(r\zeta)}{v(\varphi(r\zeta))}\
dm(\zeta)$ $\displaystyle\geq$
$\displaystyle\frac{c_{1}c^{-p}}{(1+\epsilon)^{p}}\int_{\\{z\in\mathbb{T}:|\varphi(r\zeta)|^{2}\geq
1/2\\}}\frac{|u(r\zeta)|^{p}v^{\prime}(r\zeta)}{(1-|\varphi(r\zeta)|^{2})v(\varphi(r\zeta))}\
dm(\zeta).$
Take $\epsilon=1$, then
$\|uC_{\varphi}\|^{p}\geq
c\int_{\mathbb{T}}\frac{|u(r\zeta)|^{p}v^{\prime}(r\zeta)}{(1-|\varphi(r\zeta)|^{2})v(\varphi(r\zeta))}\
dm(\zeta).$
As $r\rightarrow 1$,
$\displaystyle\|uC_{\varphi}\|^{p}\geq$ $\displaystyle c\limsup_{r\rightarrow
1}\int_{\mathbb{T}}\frac{|u(r\zeta)|^{p}v^{\prime}(r\zeta)}{(1-|\varphi(r\zeta)|^{2})v(\varphi(r\zeta))}\
dm(\zeta)$ $\displaystyle\geq$ $\displaystyle
c\int_{\mathbb{T}}\frac{|u(\zeta)|^{p}v^{\prime}(\zeta)}{(1-|\varphi(\zeta)|^{2})v(\varphi(\zeta))}\
dm(\zeta).$
### Acknowledgment
## References
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* [4] J. Bonet, P. Domanski, M. Lindstrom, Weakly compact composition operators on analytic vector-valued function spaces, Ann. Acad. Sci. Fenn. Math. 26 (2001), 233-248.
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* [8] S. Kumar, Weighted composition operators between spaces of Dirichlet type, Rev Mat Complut, 2009, 22: 469-488.
* [9] J. Laitila, Weakly compact composition operators on vector-valued BMOA. J. Math. Anal. Appl., 308 (2005), 730-745.
* [10] J. Laitila , H.-O. Tylli, and M. Wang , Composition operators from weak to strong spaces of vector-valued analytic functions, J Operator Theory, 2009, 62: 281-295
* [11] B. D. Maccluer and R. Zhao, Essential norms of weighted composition operators between Bloch-type spaces, Rocky Mountain J. Math. 33 (2003), no. 4, 1437-1458.
* [12] G. Mirzakarimi and K. Siddighi, Weighted composition operators on Bergman and Dirichlet spaces, Georgian Math. J. 4 (1997), no. 4, 373-383.
* [13] S. Ohno, Weighted composition operators between $H^{\infty}$ and the Bloch space, Taiwanese J. Math. 5 (2001), no. 3, 555-563.
* [14] S. Ohno and R. Zhao, Weighted composition operators on the Bloch space, Bull. Austral. Math. Soc. 63 (2001), no. 2, 177-185.
* [15] S. Ohno, K. Stroethoff, and R. Zhao, Weighted composition operators between Bloch-type spaces, Rocky Mountain J. Math. 33 (2003), no. 1, 191-215.
* [16] J. H. Shapiro, Composition operator and classical function theory, Springer-Verlag, New York, 1993.
* [17] M. Wang, Weighted composition operators between Dirichlet spaces, Acta Math. Sci. 2011,31B(2):641-651.
* [18] E. Wolf, Weighted composition operators between weighted Bergman spaces and weighted Banach spaces of holomorphic functions, Rev. Mat. Complut. 21 (2008), 4267-4273.
* [19] K. Zhu, Spaces of holomorphic functions in the unit ball, Springer, New York, 2005.
|
arxiv-papers
| 2012-11-27T19:41:28 |
2024-09-04T02:49:38.578760
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Mostafa Hassanlou and Hamid Vaezi",
"submitter": "Mostafa Hassanlou",
"url": "https://arxiv.org/abs/1211.6399"
}
|
1211.6403
|
Practical Explicitly Invertible Approximation to 4 Decimals of Normal
Cumulative Distribution Function Modifying Winitzki’s Approximation of erf
Alessandro Soranzo∗, Emanuela Epure⋆
∗ Dipartimento di Matematica e Geoscienze – University of Trieste – Trieste –
Italy
email: [email protected]
⋆ Esteco S.p.A. (Optimization Software) – Area Science Park – Trieste – Italy
email: [email protected]
Abstract. We give a new explicitly invertible approximation of the normal
cumulative distribution function:
$\Phi(x)\simeq\frac{1}{2}+\frac{1}{2}\sqrt{1-{e}^{-x^{2}\frac{17+{x}^{2}}{26.694+2x^{2}}}}$,
$\forall x\geq 0$, with absolute error $<4.00\cdot 10^{-5}$, absolute value of
the relative error $<4.53\cdot 10^{-5}$, which, beeing designed essentially
for practical use, is much simpler than a previously published formula and,
though less precise, still reaches 4 decimals of precision, and has a
complexity essentially comparable with that of the approximation of the normal
cumulative distribution function $\Phi(x)$ immediatly derived from Winitzki’s
approximation of erf$(x)$, reducing about $36\%$ the absolute error and about
$28\%$ the relative error with respect to that, overcoming the threshold of 4
decimals of precision.
2010 Mathematics Subject Classification: 33B20 , 33F05 , 65D20 , 97N50.
Keywords: normal cdf, $\Phi$, error function, erf, Winitzki, normal quantile,
probit, erf-1, approximation, non-linear fitting.
This paper is devoted to approximate some special functions, in particular the
normal cumulative distribution function.
Though computers now allow to compute them with arbitrary precision, such
approximations are still worth for several reasons, including to catch the
soul of the considered functions, allowing to understand at a glance their
behaviour. Let’s add that, despite technologic progress, those functions – of
wide practical use – not always are available on pocket calculators. In
practice, the ancient numerical tables are still widely used; but they give
approximations only for some values and, if linearly interpolating to
approximate intermediate values, both precision and simplicity of use reduce.
And surely by computers you may obtain graphs of those functions, but for a
mathematician the meaning content of formuls is greater.
Furthermore, here we produce an explicitly invertible (and, in fact, simply)
approximation, which allow to keep coherence working contemporarily with the
considered function and its inverse.
For the [6] [8] special functions
$\Phi(x):=\int_{-\infty}^{x}\frac{1}{\sqrt{2\pi}}e^{\frac{-t^{2}}{2}}dt,$ (1)
and the related error function $erf(x)=2\Phi(x\sqrt{2})-1$ $\big{(}\forall
x\in I\\!\\!R\big{)}$, there are several approximations; in particular see
classical [1] [3] [4] and recent [2] [5] [7] [9]; approximations for $x\geq 0$
are sufficient because of the symmetry formula $\Phi(-x)=1-\Phi(x)$
$\big{(}\forall x\in I\\!\\!R\big{)}$.
Restricting now our attention only to those approximations which are simply
explicitly invertible – in the sense, explicitly invertible without requiring
to solve cubic or quartic equations – the most precise appears to be [5]
$\Phi(x)\cong\frac{1}{2}+\frac{1}{2}\sqrt{1-{e}^{\frac{-1.2735457x^{2}-0.0743968x^{4}}{2+0.1480931x^{2}+0.0002580x^{4}}}}\quad\left\\{\begin{array}[]{ll}|\varepsilon(x)|<1.14\cdot
10^{-5}\\\ |\varepsilon_{r}(x)|<1.78\cdot 10^{-5}\end{array}\forall x\geq
0\right.$ (2)
which is an improvement preserving (despite the adding of the quartic
monomial) the simple explicit invertibility (essential solving a biquadratic
equation after obvious substitutions) of this approximation of $\Phi$
$\Phi(x)\simeq\frac{1}{2}+\frac{1}{2}\sqrt{1-{e}^{-\frac{{x}^{2}\,\left(\frac{4}{\pi}+0.0735\,{x}^{2}\right)}{2\,\left(1+0.0735\,{x}^{2}\right)}}}\quad\left\\{\begin{array}[]{ll}|\varepsilon(x)|<6.21\cdot
10^{-5}\\\ |\varepsilon_{r}(x)|<6.30\cdot 10^{-5}\end{array}\forall x\geq
0\right.$ (3)
immediately derived by
$\Phi(x)=\frac{1}{2}+\frac{1}{2}erf\Big{(}\frac{x}{\sqrt{2}}\Big{)}$
$\big{(}\forall x\geq 0\big{)}$ from this [7] Winitzki’s Approximation of erf
$erf(x)\cong\sqrt{1-e^{-x^{2}\frac{\frac{4}{\pi}+0.147x^{2}}{1+0.147x^{2}}}}\quad\left\\{\begin{array}[]{ll}|\varepsilon(x)|<1.25\cdot
10^{-4}\\\ |\varepsilon_{r}(x)|<1.28\cdot 10^{-4}\end{array}\forall x\geq
0\right..$ (4)
In this note we give this new (simply) explicitly invertible approximation of
the normal cumulative distribution function
$\Phi(x)\simeq\frac{1}{2}+\frac{1}{2}\sqrt{1-{e}^{-x^{2}\frac{17+{x}^{2}}{26.694+2x^{2}}}}\quad\left\\{\begin{array}[]{ll}|\varepsilon(x)|<4.00\cdot
10^{-5}\\\ |\varepsilon_{r}(x)|<4.53\cdot 10^{-5}\end{array}\forall x\geq
0\right.$
---
(5)
which, beeing designed essentially for practical use,
$\bullet$ is much simpler than (2) and, though less precise, still reaches 4
decimals of precision;
$\bullet$ has a complexity essentially comparable with that of (3) reducing
about $36\%$ the absolute error and about $28\%$ the relative error with
respect to that, overcoming the threshold of 4 decimals of precision.
Instead, the corresponding approximation of erf is not so worth, because,
though reduces about $36\%$ the absolute error of (4), it remains with the
precision of 3 decimals, and furthermore the absolute value of the relative
error $|\varepsilon_{r}(x)|<1.79\cdot 10^{-4}$ is quite greater than in (4).
See below the graphs (made by Mathematica ®) of the approximation, of the
absolute error and of the absolute value of the relative error for $0\leq
x\leq 7$. For $x\geq 7$ the trivial approximation $\Phi(x)\simeq 1$ has
absolute error and absolute value of the relative error highly less than
$4\cdot 10^{-5}$ and $4.53\cdot 10^{-5}$ respectively. (Nevertheless, if
interested in a formal proof of the majorization $|\varepsilon(x)|<4\cdot
10^{-5}$ for $x\geq 7$, you may follow [5]).
$\begin{array}[]{cc}\includegraphics[width=158.99377pt,height=108.405pt]{Wini26x694xGraph.pdf}&\includegraphics[width=158.99377pt,height=108.405pt]{Wini26x694x0x00004-AbsErrZoom2.pdf}\\\
\rm{Fig.\>1.\>The\>new\>approximation\>(\ref{eq:ourPhiSimple})}&\rm{Fig.\>4.\>Second\>zoom\>of\>Fig.\>2.}\\\
\includegraphics[width=158.99377pt,height=108.405pt]{WiniArxivSecondAbsErr.pdf}&\includegraphics[width=158.99377pt,height=108.405pt]{Wini26x694xRelErr.pdf}\\\
\rm{Fig.\>2.\>Absolute\>error}&\rm{Fig.\>5.\>Absolute\>value\>of\>relative\>error}\\\
\includegraphics[width=158.99377pt,height=108.405pt]{Wini26x694x0x00004-AbsErrZoom1.pdf}&\includegraphics[width=158.99377pt,height=108.405pt]{Wini26x694x0x0000453-RelErrZoom.pdf}\\\
\rm{Fig.\>3.\>First\>zoom\>of\>Fig.\>2}&\rm{Fig.\>6.\>Zoom\>of\>Fig.\>5.\par}\end{array}$
## References
* [1] Abramowitz, M. Stegun, I.A. (Eds.), (1972). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing New York: Dover, pagg. 932 and 299.
* [2] Dyer, S.A. Dyer, J.S. (2007). Approximations to error function, Instrumentation $\&$ Measurement Magazine, IEEE 10, no.6:45 – 48.
* [3] Hart, J.F. et al. (1968). Computer Approximations, SIAM series in applied mathematics, John Wiley & Sons, Inc., New York - London - Sydney, 140, pagg. 288 – 289.
* [4] Johnson, N. Kotz, S. Balakrishnan, N. (1994). Continuous Univariate Distributions, Vol. 1, 2nd ed. Boston, MA: Houghton Mifflin.
* [5] Soranzo, A. Epure, E. (2012). Simply Explicitly Invertible Approximations to 4 Decimals of Error Function and Normal Cumulative Distribution Function, www.intellectualarchive.com (selecting Mathematics) and http://arxiv.org/abs/1201.1320v1.
* [6] Wikipedia. http://en.wikipedia.org/wiki/Error$\\_$function (read 2011, September)
* [7] Winitzki, S. A handy approximation for the error function and its inverse, http://docs.google.com/viewer?a=v$\&$pid=sites$\&$srcid=ZGVmYXVsdGRv bWFpbnx3aW5pdHpraXxneDoxYTUzZTEzNWQwZjZlOWY2 http://sites.google.com/site/winitzki http://sites.google.com/site/winitzki/sergei-winitzkis-files (2008) (read 2011, December)
* [8] Wolfram Research, Inc. http://functions.wolfram.com/GammaBetaErf/Erf/10/01/ (read 2011, September)
* [9] Zogheib, B. Hlynka, M. (2009). Approximations of the Standard Normal Distribution, University of Windsor, Dept. of Mathematics and Statistics. (Legally) available at http://web2.uwindsor.ca/math/hlynka/zogheibhlynka.pdf (read 2012, March).
Remark. This text has been explicitly placed by the Authors in the Creative
Commons Public Domain.
|
arxiv-papers
| 2012-11-27T19:53:21 |
2024-09-04T02:49:38.584300
|
{
"license": "Public Domain",
"authors": "Alessandro Soranzo, Emanuela Epure",
"submitter": "Alessandro Soranzo",
"url": "https://arxiv.org/abs/1211.6403"
}
|
1211.6424
|
# Bertrand Curves in three Dimensional Lie Groups
O. Zeki Okuyucu${}^{{}^{(1)}}$ ${}^{{}^{(1)}}$Bilecik Şeyh Edebali University,
Faculty of Science and Arts, Department of Mathematics, 11210, Bilecik, Turkey
[email protected] , İsmail Gök${}^{{}^{(2)}}$
${}^{{}^{(2)}}$Ankara University, Faculty of Science, Department of
Mathematics, 06100, Tandog̃an, Ankara, Turkey [email protected] ,
Yusuf Yaylı${}^{{}^{(3)}}$ ${}^{{}^{(3)}}$Ankara University, Faculty of
Science, Department of Mathematics, 06100, Tandog̃an, Ankara, Turkey
[email protected] and Nejat Ekmekci${}^{{}^{(4)}}$
${}^{{}^{(4)}}$Ankara University, Faculty of Science, Department of
Mathematics, 06100, Tandog̃an, Ankara, Turkey [email protected]
(Date: September 08, 2012)
###### Abstract.
In this paper, we give the definition of harmonic curvature function some
special curves such as helix, slant curves, Mannheim curves and Bertrand
curves. Then, we recall the characterizations of helices [8], slant curves
(see [19]) and Mannheim curves (see [12]) in three dimensional Lie groups
using their harmonic curvature function.
Moreover, we define Bertrand curves in a three dimensional Lie group $G$ with
a bi-invariant metric and the main result in this paper is given as (Theorem
3.4): A curve $\alpha:I\subset\mathbb{R\rightarrow}G$ with the Frenet
apparatus $\left\\{T,N,B,\varkappa,\tau\right\\}$ is a Bertrand curve if and
only if
$\lambda\varkappa+\mu\varkappa H=1$
where $\lambda$, $\mu$ are constants and $H$ is the harmonic curvature
function of the curve $\alpha.$
###### Key words and phrases:
Bertrand curves, Lie groups.
###### 2000 Mathematics Subject Classification:
Primary 53A04; Secondary 22E15
This paper is in final form and no version of it will be submitted for
publication elsewhere.
## 1\. Introduction
The general theory of curves in a Euclidean space (or more generally in a
Riemannian manifolds) have been developed a long time ago and we have a deep
knowledge of its local geometry as well as its global geometry. In the theory
of curves in Euclidean space, one of the important and interesting problem is
characterizations of a regular curve. In the solution of the problem, the
curvature functions $k_{1}\left(\text{or }\varkappa\right)$ and
$k_{2}\left(\text{or }\tau\right)$ of a regular curve have an effective role.
For example: if $k_{1}=0=$ $k_{2}$, then the curve is a geodesic or if
$k_{1}=$constant$\neq 0$ and $k_{2}=0,$ then the curve is a circle with radius
$\left(1/k_{1}\right)$, etc. Thus we can determine the shape and size of a
regular curve by using its curvatures. Another way in the solution of the
problem is the relationship between the Frenet vectors of the curves (see
[15]). For instance Bertrand curves:
In the classical diferential geometry of curves, J. Bertrand studied curves in
Euclidean 3-space whose principal normals are the principal normals of another
curve. In (see [3]) he showed that a necessary and sufficient condition for
the existence of such a second curve is that a linear relationship with
constant coefficients shall exist between the first and second curvatures of
the given original curve. In other word, if we denote first and second
curvatures of a given curve by $k_{1}$ and $k_{2}$ respectively, then for
$\lambda,\mu$ $\in\mathbb{R}$ we have $\lambda k_{1}+\mu k_{2}=1$. Since the
time of Bertrand’s paper, pairs of curves of this kind have been called
Conjugate Bertrand Curves, or more commonly Bertrand Curves (see [15]).
In 1888, C. Bioche [4] give a new theorem to obtaining Bertrand curves by
using the given two curves $C_{1}$ and $C_{2}$ in Euclidean $3-$space. Later,
in 1960, J. F. Burke [5] give a theorem related with Bioche’s thorem on
Bertrand curves.
The following properties of Bertrand curves are well known: If two curves have
the same principal normals, (i) corresponding points are a fixed distance
apart; (ii) the tangents at corresponding points are at a fixed angle. These
well known properties of Bertrand curves in Euclidean 3-space was extended by
L. R. Pears in [21] to Riemannian $n-$space and found general results for
Bertrand curves. When we applying these general result to Euclidean $n$-space,
it is easily find that either $k_{2}$ or $k_{3}$ is zero; in other words,
Bertrand curves in ,$\mathbb{E}^{n}(n>3)$ are degenerate curves. This result
is restated by Matsuda and Yorozu [18]. They proved that there is no special
Bertrand curves in $E^{n}(n>3)$ and they define new kind, which is called
$\left(1,3\right)-$type Bertrand curves in $4-$dimensional Euclidean space.
Bertrand curves and their characterizations were studied by many authours in
Euclidean space as well as in Riemann–Otsuki space, in Minkowski 3- space and
Minkowski spacetime (for instance see [1, 2, 10, 14, 16, 22, 23].)
The degenarete semi-Riemannian geometry of Lie group is studied by Çöken and
Çiftçi [9]. Moreover, they obtanied a naturally reductive homogeneous semi-
Riemannian space using the Lie group. Then Çiftçi [8] defined general helices
in three dimensional Lie groups with a bi-invariant metric and obtained a
generalization of Lancret’s theorem. Also he gave a relation between the
geodesics of the so-called cylinders and general helices. Then, Okuyucu et al.
[19] defined slant helices in three dimensional Lie groups with a bi-invariant
metric and obtained some characterizations using their harmonic curvature
function.
Recently, Izumiya and Takeuchi [13] have introduced the concept of slant helix
in Euclidean $3$-space. A slant helix in Euclidean space $\mathbb{E}^{3}$ was
defined by the property that its principal normal vector field makes a
constant angle with a fixed direction. Also, Izumiya and Takeuchi showed that
$\alpha$ is a slant helix if and only if the geodesic curvature of spherical
image of principal normal indicatrix $\left(N\right)$ of a space curve
$\alpha$
$\sigma_{N}\left(s\right)=\left(\frac{\varkappa^{2}}{\left(\varkappa^{2}+\tau^{2}\right)^{3/2}}\left(\frac{\tau}{\varkappa}\right)^{\prime}\right)\left(s\right)$
is a constant function .
Harmonic curvature functions were defined earlier by Özdamar and Hacısalihoğlu
[20]. Recently, many studies have been reported on generalized helices and
slant helices using the harmonic curvatures in Euclidean spaces and Minkowski
spaces [6, 11, 17]. Then, Okuyucu et al. [19] defined slant helices in three
dimensional Lie groups with a bi-invariant metric and obtained some
characterizations using their harmonic curvature function.
In this paper, first of all, we give the definition of harmonic curvature
function some special curves such as helix, slant curves. Then, we recall the
characterizations of helices [8], slant curves (see [19]) and Mannheim curves
(see [12]) in three dimensional Lie groups using their harmonic curvature
function. Moreover, we define Bertrand curves in a three dimensional Lie group
$G$ with a bi-invariant metric and then the main result to this paper is given
as (Theorem 3.4): A curve $\alpha:I\subset\mathbb{R\rightarrow}G$ with the
Frenet apparatus $\left\\{T,N,B,\varkappa,\tau\right\\}$ is a Bertrand curve
if and only if
$\lambda\varkappa+\mu\varkappa H=1$
where $\lambda$, $\mu$ are constants and $H$ is the harmonic curvature
function of the curve $\alpha.$
Note that three dimensional Lie groups admitting bi-invariant metrics are
$SO\left(3\right),SU^{2}$ and Abellian Lie groups. So we believe that our
characterizations about Bertrand curves will be useful for curves theory in
Lie groups.
## 2\. Preliminaries
Let $G$ be a Lie group with a bi-invariant metric $\left\langle\text{
},\right\rangle$ and $D$ be the Levi-Civita connection of Lie group $G.$ If
$\mathfrak{g}$ denotes the Lie algebra of $G$ then we know that $\mathfrak{g}$
is issomorphic to $T_{e}G$ where $e$ is neutral element of $G.$ If
$\left\langle\text{ },\right\rangle$ is a bi-invariant metric on $G$ then we
have
$\left\langle
X,\left[Y,Z\right]\right\rangle=\left\langle\left[X,Y\right],Z\right\rangle$
(2.1)
and
$D_{X}Y=\frac{1}{2}\left[X,Y\right]$ (2.2)
for all $X,Y$ and $Z\in\mathfrak{g}.$
Let $\alpha:I\subset\mathbb{R\rightarrow}G$ be an arc-lenghted regular curve
and $\left\\{X_{1},X_{2,}...,X_{n}\right\\}$ be an orthonormal basis of
$\mathfrak{g}.$ In this case, we write that any two vector fields $W$ and $Z$
along the curve $\alpha\ $as $W=\sum_{i=1}^{n}w_{i}X_{i}$ and
$Z=\sum_{i=1}^{n}z_{i}X_{i}$ where $w_{i}:I\rightarrow\mathbb{R}$ and
$z_{i}:I\rightarrow\mathbb{R}$ are smooth functions. Also the Lie bracket of
two vector fields $W$ and $Z$ is given
$\left[W,Z\right]=\sum_{i=1}^{n}w_{i}z_{i}\left[X_{i},X_{j}\right]$
and the covariant derivative of $W$ along the curve $\alpha$ with the notation
$D_{\alpha^{\shortmid}}W$ is given as follows
$D_{\alpha^{\shortmid}}W=\overset{\cdot}{W}+\frac{1}{2}\left[T,W\right]$ (2.3)
where $T=\alpha^{\prime}$ and
$\overset{\cdot}{W}=\sum_{i=1}^{n}\overset{\cdot}{w_{i}}X_{i}$ or
$\overset{\cdot}{W}=\sum_{i=1}^{n}\frac{dw}{dt}X_{i}.$ Note that if $W$ is the
left-invariant vector field to the curve $\alpha$ then $\overset{\cdot}{W}=0$
(see for details [7]).
Let $G$ be a three dimensional Lie group and
$\left(T,N,B,\varkappa,\tau\right)$ denote the Frenet apparatus of the curve
$\alpha$. Then the Serret-Frenet formulas of the curve $\alpha$ satisfies:
$D_{T}T=\varkappa N\text{, \ \ \ }D_{T}N=-\varkappa T+\tau B\text{, \ \ \
}D_{T}B=-\tau N$
where $D$ is Levi-Civita connection of Lie group $G$ and
$\varkappa=\overset{\cdot}{\left\|T\right\|}.$
###### Definition 2.1.
Let $\alpha:I\subset\mathbb{R\rightarrow}G$ be a parametrized curve. Then
$\alpha$ is called a _general helix_ if it makes a constant angle with a left-
invariant vector field $X$. That is,
$\left\langle T(s),X\right\rangle=\cos\theta\text{ for all }s\in I,$
for the left-invariant vector field $X\in g$ is unit length and $\theta$ is a
constant angle between $X$ and $T$, which is the tangent vector field of the
curve $\alpha$ (see [8]).
###### Definition 2.2.
Let $\alpha:I\subset\mathbb{R\rightarrow}G$ be a parametrized curve with the
Frenet apparatus $\left(T,N,B,\varkappa,\tau\right)$ then
$\tau_{G}=\frac{1}{2}\left\langle\left[T,N\right],B\right\rangle$ (2.4)
or
$\tau_{G}=\frac{1}{2\varkappa^{2}\tau}\overset{\cdot\cdot\text{ \ \ \ \ \ \ \
\ }\cdot}{\left\langle
T,\left[T,T\right]\right\rangle}+\frac{1}{4\varkappa^{2}\tau}\overset{\text{ \
\ }\cdot}{\left\|\left[T,T\right]\right\|^{2}}$
(see [8]).
###### Definition 2.3.
Let $\alpha:I\subset\mathbb{R\rightarrow}G$ be an arc length parametrized
curve. Then $\alpha$ is called a _slant helix_ if its principal normal vector
field makes a constant angle with a left-invariant vector field $X$ which is
unit length. That is,
$\left\langle N(s),X\right\rangle=\cos\theta\text{ for all }s\in I,$
where $\theta\neq\frac{\pi}{2}$ is a constant angle between $X$ and $N$ which
is the principal normal vector field of the curve $\alpha$ (see [19]).
###### Definition 2.4.
Let $\alpha:I\subset\mathbb{R\rightarrow}G$ be an arc length parametrized
curve with the Frenet apparatus $\left\\{T,N,B,\varkappa,\tau\right\\}.$ Then
the _harmonic curvature function_ of the curve $\alpha$ is defined by
$H=\dfrac{\tau-\tau_{G}}{\varkappa}$
where $\tau_{G}=\frac{1}{2}\left\langle\left[T,N\right],B\right\rangle$ (see
[19]).
###### Theorem 2.5.
Let $\alpha:I\subset\mathbb{R\rightarrow}G$ be a parametrized curve with the
Frenet apparatus $\left(T,N,B,\varkappa,\tau\right)$. If the curve $\alpha$ is
a general helix, if and only if
$\tau=c\varkappa+\tau_{G}$
where c is a constant (see [8]) or using the definition of the harmonic
curvature function of the curve $\alpha$ (see [19]) is constant function.
###### Theorem 2.6.
Let $\alpha:I\subset\mathbb{R\rightarrow}G$ be a parametrized curve with the
Frenet apparatus $\left(T,N,B,\varkappa,\tau\right)$. If the curve $\alpha$ is
a general helix, if and only if the harmonic curvature function of the curve
$\alpha$ is a constant function.
###### Proof.
It is obvious using the Definition 2.4 and Theorem 2.1. ∎
###### Theorem 2.7.
Let $\alpha:I\subset\mathbb{R\rightarrow}G$ be a unit speed curve with the
Frenet apparatus $\left(T,N,B,\varkappa,\tau\right)$. Then $\alpha$ is a slant
helix if and only if
$\sigma_{N}=\frac{\varkappa(1+H^{2})^{\frac{3}{2}}}{H^{\shortmid}}=\tan\theta$
is a constant where $H$ is a harmonic curvature function of the curve $\alpha$
and $\theta\neq\frac{\pi}{2}$ is a constant (see [19]).
###### Theorem 2.8.
Let $\alpha:I\subset\mathbb{R\rightarrow}G$ be a parametrized curve with arc
length parameter $s$ and the Frenet apparatus
$\left(T,N,B,\varkappa,\tau\right)$. Then, $\alpha$ is Mannheim curve if and
only if
$\lambda\varkappa\left(1+H^{2}\right)=1,\text{ for all }s\in I$ (2.5)
where $\lambda$ is constant and $H$ is the harmonic curvature function of the
curve $\alpha$ (see [12]).
###### Theorem 2.9.
Let $\alpha:I\subset\mathbb{R\rightarrow}G$ be a parametrized curve with arc
length parameter $s$. Then $\beta$ is the Mannheim partner curve of $\alpha$
if and only if the curvature $\varkappa_{\beta}$ and the torsion
$\tau_{\beta}$ of $\beta$ satisfy the following equation
$\frac{d\varkappa_{\beta}H_{\beta}}{d\overline{s}}=\frac{\varkappa_{\beta}}{\mu}(1+\mu^{2}\varkappa_{\beta}^{2}H_{\beta}^{2})$
where $\mu$ is constant and $H_{\beta}$ is the harmonic curvature function of
the curve $\beta.$
## 3\. Bertrand curves in a three dimensional Lie group
In this section, we define Bertrand curves and their characterizations are
given in a three dimensional Lie group $G$ with a bi-invariant metric
$\left\langle\text{ },\right\rangle$. Also we give some characterizations of
Bertrand curves using the special cases of $G$.
###### Definition 3.1.
A curve $\alpha$ in $3$-dimensional Lie group $G$ is a _Bertrand curve_ if
there exists a special curve __ $\beta$ in $3$-dimensional Lie group $G$ such
that principal normal vector field of $\alpha$ is linearly dependent principal
normal vector field of $\beta$ at corresponding point under $\psi$ which is
bijection from $\alpha$ to $\beta.$ In this case $\beta$ is called the
Bertrand mate curve of $\alpha$ and $\left(\alpha,\beta\right)$ is called
Bertrand curve _couple._
The curve $\alpha:I\subset\mathbb{R\rightarrow}G$ in $3$-dimensional Lie group
$G$ is parametrized by the arc-length parameter $s$ and from the Definition
3.1 Bertrand mate curve of $\alpha$ is given
$\beta:\overline{I}\subset\mathbb{R\rightarrow}G$ in $3$-dimensional Lie group
$G$ with the help of Figure 1 such that
$\text{{Figure1}}\mathtt{:}\text{{}Bertrand curve\ couple
}\left(\alpha,\beta\right)$
$\beta\left(s\right)=\alpha\left(s\right)+\lambda\left(s\right)N\left(s\right),\text{
}s\in I$
where $\lambda$ is a smooth function on $I$ and $N$ is the principal normal
vector field of $\alpha$. We should remark that the parameter $s$ generally is
not an arc-length parameter of $\beta.$ So, we define the arc-length parameter
of the curve $\beta$ by
$\overline{s}=\psi\left(s\right)=\int\limits_{0}^{s}\left\|\frac{d\beta\left(s\right)}{ds}\right\|ds$
where $\psi:I\longrightarrow\overline{I}$ is a smooth function and holds the
following equality
$\psi^{\prime}\left(s\right)=\varkappa H\sqrt{\lambda^{2}+\mu^{2}}$ (3.1)
for $s\in I.$
###### Proposition 3.2.
Let $\alpha:I\subset\mathbb{R\rightarrow}G$ be an arc length parametrized
curve with the Frenet apparatus $\left\\{T,N,B\right\\}$. Then the following
equalities
$\displaystyle\left[T,N\right]$
$\displaystyle=\left\langle\left[T,N\right],B\right\rangle B=2\tau_{G}B$
$\displaystyle\left[T,B\right]$
$\displaystyle=\left\langle\left[T,B\right],N\right\rangle N=-2\tau_{G}N$
hold [19].
###### Theorem 3.3.
Let $\alpha:I\subset\mathbb{R\rightarrow}G$ and
$\beta:\overline{I}\subset\mathbb{R\rightarrow}G$ be a Bertrand curve couple
with arc-length parameter $s$ and $\overline{s},$ respectively. Then
corresponding points are a fixed distance apart for all $s\in I$, that is,
$d\left(\alpha\left(s\right),\beta\left(s\right)\right)=\text{constant, \ \
for all }s\in I$
###### Proof.
From Definition 3.1, we can simply write
$\beta\left(s\right)=\alpha\left(s\right)+\lambda\left(s\right)N\left(s\right)$
(3.2)
Differentiating the Eq. (3.2) with respect to $s$ and using the Eq. (2.3), we
get
$\displaystyle\frac{d\beta\left(\overline{s}\right)}{d\overline{s}}\psi^{\prime}\left(s\right)$
$\displaystyle=\frac{d\alpha\left(s\right)}{ds}+\lambda^{\prime}\left(s\right)N\left(s\right)+\lambda\left(s\right)\overset{\cdot}{N}$
$\displaystyle=\left(1-\lambda\left(s\right)\varkappa\left(s\right)\right)T(s)+\lambda^{\prime}\left(s\right)N\left(s\right)+\lambda\left(s\right)\tau\left(s\right)B\left(s\right)-\dfrac{1}{2}\left[T,N\right]$
and with the help of Proposition 3.2, we obtain
$\frac{d\beta\left(\overline{s}\right)}{d\overline{s}}\psi^{\prime}\left(s\right)=\left(1-\lambda\left(s\right)\varkappa\left(s\right)\right)T(s)+\lambda^{\prime}\left(s\right)N\left(s\right)+\lambda\left(s\right)\left(\left(\tau-\tau_{G}\right)\left(s\right)\right)B\left(s\right)$
or
$T_{\beta}\left(\overline{s}\right)=\frac{1}{\psi^{\prime}\left(s\right)}\left[\left(1-\lambda\left(s\right)\varkappa\left(s\right)\right)T(s)+\lambda^{\prime}\left(s\right)N\left(s\right)+\lambda\left(s\right)\left(\left(\tau-\tau_{G}\right)\left(s\right)\right)B\left(s\right)\right].$
And then, we know that
$\left\\{N_{\beta}(\left(\overline{s}\right)),N\left(s\right)\right\\}$ is a
linearly dependent set, so we have
$\left\langle
T_{\beta}\left(\overline{s}\right),N_{\beta}\left(\overline{s}\right)\right\rangle=\frac{1}{\psi^{\prime}\left(s\right)}\left[\begin{array}[c]{c}\left(1-\lambda\left(s\right)\varkappa\left(s\right)\right)\left\langle
T(s),N_{\beta}\left(\overline{s}\right)\right\rangle+\lambda^{\prime}\left(s\right)\left\langle
N(s),N_{\beta}\left(\overline{s}\right)\right\rangle\\\
+\lambda\left(s\right)\tau\left(s\right)\left\langle
B(s),N_{\beta}\left(\overline{s}\right)\right\rangle\end{array}\right]$
$\lambda^{\prime}\left(s\right)=0$
that is, $\lambda\left(s\right)$ is constant function on $I.$ This completes
the proof. ∎
###### Theorem 3.4.
If $\alpha:I\subset\mathbb{R\rightarrow}G$ is a parametrized Bertrand curve
with arc length parameter $s$ and the Frenet apparatus
$\left(T,N,B,\varkappa,\tau\right)$. Then, $\alpha$ satisfy the following
equality
$\lambda\varkappa\left(s\right)+\mu\varkappa\left(s\right)H\left(s\right)=1,\text{
for all }s\in I$ (3.3)
where $\lambda$, $\mu$ are constants and $H$ is the harmonic curvature
function of the curve $\alpha.$
###### Proof.
Let $\alpha:I\subset\mathbb{R\rightarrow}G$ be a parametrized Bertrand curve
with arc length parameter $s$ then we can write
$\beta\left(s\right)=\alpha\left(s\right)+\lambda N\left(s\right)$
Differentiating the above equality with respect to $s$ and by using the Frenet
equations, we get
$\displaystyle\frac{d\beta\left(\overline{s}\right)}{d\overline{s}}\psi^{\prime}\left(s\right)$
$\displaystyle=\frac{d\alpha\left(s\right)}{ds}+\lambda\left(s\right)\overset{\cdot}{N}$
$\displaystyle=\left(1-\lambda\left(s\right)\varkappa\left(s\right)\right)T(s)+\lambda\left(s\right)\tau\left(s\right)B\left(s\right)-\dfrac{1}{2}\left[T,N\right]$
and with the help of Proposition 3.2, we obtain
$T_{\beta}\left(\overline{s}\right)=\frac{\left(1-\lambda\varkappa\left(s\right)\right)}{\psi^{\prime}\left(s\right)}T(s)+\frac{\lambda\left(\left(\tau-\tau_{G}\right)\left(s\right)\right)}{\psi^{\prime}\left(s\right)}B\left(s\right).$
As $\left\\{N_{\beta}(\left(\overline{s}\right)),N\left(s\right)\right\\}$ is
a linearly dependent set, we can write
$T_{\beta}\left(\overline{s}\right)=\cos\theta\left(s\right)T(s)+\sin\theta\left(s\right)B(s)$
(3.4)
where
$\cos\theta\left(s\right)=\frac{\left(1-\lambda\varkappa\left(s\right)\right)}{\psi^{\prime}\left(s\right)},$
$\sin\theta\left(s\right)=\frac{\lambda\left(\left(\tau-\tau_{G}\right)\left(s\right)\right)}{\psi^{\prime}\left(s\right)}.$
If we differentiate the Eq. (3.4) and consider
$\left\\{N_{\beta}\left(\overline{s}\right),N\left(s\right)\right\\}$ is a
linearly dependent set we can easily see that $\theta$ is a constant function.
So, we obtain
$\frac{\cos\theta}{\sin\theta}=\frac{1-\lambda\varkappa\left(s\right)}{\lambda\left(\left(\tau-\tau_{G}\right)\left(s\right)\right)}$
or taking $c=\dfrac{\cos\theta}{\sin\theta},$ we get
$\lambda\varkappa\left(s\right)+c\lambda\left(\left(\tau-\tau_{G}\right)\left(s\right)\right)=1.$
Then denoting $\mu=c\lambda=$costant and using the Definition 2.4, we have
$\lambda\varkappa\left(s\right)+\mu\varkappa\left(s\right)H\left(s\right)=1,\text{
for all }s\in I$
which completes the proof. ∎
###### Corollary 3.5.
The measure of the angle between the tangent vector fields of the Bertrand
curve couple $\left(\alpha,\beta\right)$ is constant.
###### Proof.
It is obvious from the proof of above Theorem. ∎
###### Remark 3.6.
It is unknown whether the reverse of the above Theorem. Because, for the proof
of the reverse we must consider a special Frenet curve
$\beta\left(s\right)=\alpha\left(s\right)+\lambda N\left(s\right)$ in its
proof. So, we give the following Theorem.
###### Theorem 3.7.
Let $\alpha:I\subset\mathbb{R\rightarrow}G$ be a parametrized Bertrand curve
whose curvature functions $\varkappa$ and harmonic curvature function $H$ of
the curve $\alpha$ satisfy
$\lambda\varkappa\left(s\right)+\mu\varkappa\left(s\right)H\left(s\right)=1,$
for all $s\in I$. If the curve $\beta$ given by
$\beta\left(s\right)=\alpha\left(s\right)+\lambda N\left(s\right)$ for all
$s\in I$ is a special Frenet curve, then $\left(\alpha,\beta\right)$ is the
Bertrand curve couple.
###### Proof.
Let $\alpha:I\subset\mathbb{R\rightarrow}G$ be a parametrized Bertrand curve
whose curvature function $\varkappa$ and harmonic curvature function $H$ of
the curve $\alpha$ satisfy
$\lambda\varkappa\left(s\right)+\mu\varkappa\left(s\right)H\left(s\right)=1$
for all $s\in I$. If the curve $\beta$ given by
$\beta\left(s\right)=\alpha\left(s\right)+\lambda N\left(s\right)$ for all
$s\in I$ is a special Frenet curve, then differentiating this equality with
respect to $s$ and by using the Eq. (3.1) with the equation
$\lambda\varkappa\left(s\right)+\mu\varkappa\left(s\right)H\left(s\right)=1$,
we have
$T_{\beta}\left(\overline{s}\right)=\frac{\mu}{\sqrt{\lambda^{2}+\mu^{2}}}T(s)+\frac{\lambda}{\sqrt{\lambda^{2}+\mu^{2}}}B\left(s\right).$
(3.5)
Then, if we differentiate the last equation with respect to $s$ and by using
the Frenet formulas we obtain
$\varkappa_{\beta}\left(\overline{s}\right)N_{\beta}\left(\overline{s}\right)\psi^{\prime}\left(s\right)=\frac{\varkappa\left(s\right)}{\sqrt{\lambda^{2}+\mu^{2}}}\left(\mu-\lambda
H\left(s\right)\right)N\left(s\right).$ (3.6)
Thus, for each $s\in I,$ the vector field $N_{\beta}\left(\overline{s}\right)$
of $\beta$ is linearly dependent the vector field $N\left(s\right)$ of
$\alpha$ at corresponding point under the bijection from $\alpha$ to $\beta.$
This completes the proof. ∎
###### Proposition 3.8.
Let $\alpha:I\subset\mathbb{R\rightarrow}G$ be an arc-lenghted Bertrand curve
with the Frenet vector fields $\left\\{T,N,B\right\\}$ and
$\beta:\overline{I}\subset\mathbb{R\rightarrow}G$ be a Bertrand mate of
$\alpha$ with the Frenet vector fields
$\left\\{T_{\beta},N_{\beta},B_{\beta}\right\\}.$ Then
$\tau_{G_{\beta}}=\tau_{G}$ for the curves $\alpha$ and $\beta$ where
$\tau_{G}=\frac{1}{2}\left\langle\left[T,N\right],B\right\rangle$ and
$\tau_{G\beta}=\frac{1}{2}\left\langle\left[T_{\beta},N_{\beta}\right],B_{\beta}\right\rangle.$
###### Proof.
Let $\alpha:I\subset\mathbb{R\rightarrow}G$ be an arc-lenghted Bertrand curve
with the Frenet vector fields $\left\\{T,N,B\right\\}$ and
$\beta:\overline{I}\subset\mathbb{R\rightarrow}G$ be a Bertrand mate of
$\alpha$ with with the Frenet vector fields
$\left\\{T_{\beta},N_{\beta},B_{\beta}\right\\}.$ From the Eq. (3.5) and
considering $N_{\beta}=\mp N$ we have
$B_{\beta}\left(\overline{s}\right)=-\frac{\lambda}{\sqrt{\lambda^{2}+\mu^{2}}}T(s)+\frac{\mu}{\sqrt{\lambda^{2}+\mu^{2}}}B\left(s\right).$
(3.7)
Since
$\tau_{G\beta}=\frac{1}{2}\left\langle\left[T_{\beta},N_{\beta}\right],B_{\beta}\right\rangle$,
using the equalities of the Frenet vector fields $T_{\beta},N_{\beta}$ and
$B_{\beta}$ we obtain $\tau_{G\beta}=\tau_{G}.$ Which completes the proof. ∎
###### Theorem 3.9.
Let $\alpha:I\subset\mathbb{R\rightarrow}G$ be a parametrized Bertrand curve
with curvature functions $\varkappa$, $\tau$ and
$\beta:\overline{I}\subset\mathbb{R\rightarrow}G$ be a Bertrand mate of
$\alpha$ with curvatures functions $\varkappa_{\beta}$, $\tau_{\beta}.$ Then
the relations between these curvature functions are
$\displaystyle\varkappa_{\beta}\left(\overline{s}\right)$
$\displaystyle=\frac{\mu\varkappa\left(s\right)-\lambda\varkappa\left(s\right)H\left(s\right)}{\left(\lambda^{2}+\mu^{2}\right)H\left(s\right)},$
(3.8) $\displaystyle\tau_{\beta}\left(\overline{s}\right)$
$\displaystyle=\frac{\lambda\varkappa\left(s\right)+\mu\varkappa\left(s\right)H\left(s\right)}{\left(\lambda^{2}+\mu^{2}\right)H\left(s\right)}+\tau_{G}$
(3.9)
###### Proof.
If we take the norm of the Eq. (3.6) and use the Eq. (3.1), we get the Eq.
(3.8). Then differentiating the Eq. (3.7) and using the Frenet formulas, we
have
$\displaystyle\overset{\cdot}{B_{\beta}}\left(\overline{s}\right)\psi^{\prime}\left(s\right)$
$\displaystyle=-\frac{\lambda}{\sqrt{\lambda^{2}+\mu^{2}}}\overset{\cdot}{T}(s)+\frac{\mu}{\sqrt{\lambda^{2}+\mu^{2}}}\overset{\cdot}{B}\left(s\right),$
$\displaystyle=-\frac{\lambda}{\sqrt{\lambda^{2}+\mu^{2}}}\varkappa(s)N(s)+\frac{\mu}{\sqrt{\lambda^{2}+\mu^{2}}}\left(-\tau(s)N(s)-\frac{1}{2}\left[T,B\right]\right)$
In the above equality, using the Eq. (3.1) and the Proposition 3.2, we get
$\left(\tau_{\beta}-\tau_{G\beta}\right)N_{\beta}\left(\overline{s}\right)=\frac{1}{\varkappa
H\left(\lambda^{2}+\mu^{2}\right)}\left(\lambda\varkappa+\mu\varkappa
H\right)N(s).$
If we take the norm of the last equation and use the Proposition 3.8, we get
the Eq. (3.9). Which completes the proof. ∎
###### Theorem 3.10.
Let $\alpha:I\subset\mathbb{R\rightarrow}G$ be a parametrized curve with
Frenet apparatus $\left\\{T,N,B,\varkappa,\tau\right\\}$ and
$\beta:\overline{I}\subset\mathbb{R\rightarrow}G$ be a curve with Frenet
apparatus
$\left\\{T_{\beta},N_{\beta},B_{\beta},\varkappa_{\beta},\tau_{\beta}\right\\}.$
If $\left(\alpha,\beta\right)$ is a Bertrand curve couple then
$\varkappa\varkappa_{\beta}HH_{\beta}$ is a constant function.
###### Proof.
We assume that $\left(\alpha,\beta\right)$ is a Bertrand curve couple. Then we
can write
$\alpha\left(s\right)=\beta\left(s\right)-\lambda\left(s\right)N_{\beta}\left(\overline{s}\right).$
(3.10)
If we use the similar method in the proof of Theorem 3.4 and consider the Eq.
(3.10), then we can easily see that $\varkappa\varkappa_{\beta}HH_{\beta}$ is
a constant function. ∎
###### Theorem 3.11.
Let $\alpha:I\subset\mathbb{R\rightarrow}G$ be a parametrized Bertrand curve
with Frenet apparatus $\left\\{T,N,B,\varkappa,\tau\right\\}$ and
$\beta:\overline{I}\subset\mathbb{R\rightarrow}G$ be a Bertrand mate of the
curve $\alpha$ with Frenet apparatus
$\left\\{T_{\beta},N_{\beta},B_{\beta},\varkappa_{\beta},\tau_{\beta}\right\\}.$
Then $\alpha$ is a slant helix if and only if $\beta$ is a slant helix.
###### Proof.
Let $\sigma_{N}$ and $\sigma_{N\beta}$ be the geodesic curvatures of the
principal normal curves of $\alpha$ and $\beta,$ respectively. Then using the
Theorem 3.9 we can easily see that
$\sigma_{N\beta}=-\frac{\varkappa(1+H^{2})^{\frac{3}{2}}}{H^{\shortmid}}=-\sigma_{N}.$
So, with the help of the Theorem 2.7 we complete the proof. ∎
###### Theorem 3.12.
Let $\alpha:I\subset\mathbb{R\rightarrow}G$ be a parametrized Bertrand curve
with curvature functios $\varkappa$, $\tau$ and
$\beta:\overline{I}\subset\mathbb{R\rightarrow}G$ be a Bertrand mate of the
curve $\alpha$ with curvature functions $\varkappa_{\beta},\tau_{\beta}.$ Then
$\alpha$ is a general helix if and only if $\beta$ is a general helix.
###### Proof.
Let $\alpha$ be a helix. From Theorem 2.5, we have $H$ is a constant function.
Then using Theorem 3.9, we get
$\frac{\tau_{\beta}-\tau_{G\beta}}{\varkappa_{\beta}}=\frac{\lambda+\mu
H}{\mu-\lambda H}.$ (3.11)
Since $H$ is constant function, the Eq. (3.11) is constant. So, $\beta$ is a
general helix.
Conversly, assume that $\beta$ be a general helix. So,
$\frac{\tau_{\beta}-\tau_{G\beta}}{\varkappa_{\beta}}=$constant. From the Eq.
(3.11) $c=\frac{\lambda+\mu H}{\mu-\lambda H}=$constant and then
$H=\frac{c\mu-\lambda}{\mu+\lambda c}=$constant. Consequently $\alpha$ is a
general helix. Which completes the proof. ∎
## References
* [1] H. Balgetir, M. Bektaş and J. Inoguchi, Null Bertrand curves in Minkowski 3-space and their characterizations. Note Mat. 23 (2004/05), no. 1, 7-13.
* [2] H. Balgetir, M. Bektaş and M. Ergüt, Bertrand curves for non null curves in 3-dimensional Lorentzian space. Hadronic J. 27 (2004), no. 2, 229-236.
* [3] J. M. Bertrand, Mémoire sur la théorie des courbes á double courbure, Comptes Rendus, vol.36, 1850.
* [4] Ch. Bioche, Sur les courbes de M. Bertrand, Bull. Soc. Math. France 17 (1889), 109–112.
* [5] J. F. Burke, Bertrand Curves Associated with a Pair of Curves Mathematics Magazine, Vol. 34, No. 1 (Sep.- Oct.,1960), pp. 60-62.
* [6] Ç. Camcı, K. İlarslan, L. Kula, H.H. Hacısalihoğlu, Harmonic curvatures and generalized helices in $E^{n}$, Chaos Solitons Fract. 40 (2007) 1–7.
* [7] P. Crouch, F. Silva Leite, The dynamic interpolation problem: on Riemannian manifoldsi Lie groups and symmetric spaces, J. Dyn. Control Syst. 1 (2) (1995) 177-202.
* [8] Ü. Çiftçi, A generalization of Lancert’s theorem, J. Geom. Phys. 59 (2009) 1597-1603.
* [9] A. C. Çöken, Ü. Çiftçi, A note on the geometry of Lie groups, Nonlinear Analysis TMA 68 (2008) 2013-2016.
* [10] N. Ekmekci and K. İlarslan, On Bertrand curves and their characterization. Differ. Geom. Dyn. Syst. 3 (2001), no. 2, 17-24.
* [11] İ. Gök, Ç. Camcı, H.H. Hacısalihoğlu, Vn-slant helices in Euclidean n-space $E^{n}$, Math. Commun. 14 (2) (2009) 317–329.
* [12] İ. Gök, O. Z. Okuyucu, N. Ekmekci and Y. Yaylı, On Mannheim partner curves in three dimensional Lie groups, submitted, 2012.
* [13] S. Izumiya and N. Tkeuchi, New special curves and developable surfaces, Turk. J. Math 28 (2004), 153-163.
* [14] D. H. Jin, Null Bertrand curves in a Lorentz manifold. J. Korea Soc. Math. Educ. Ser. B Pure Appl. Math. 15 (2008), no. 3, 209–215.
* [15] W. Kuhnel, Differential geometry: curves-surfaces-manifolds, Braunschweig, Wiesbaden, 1999.
* [16] M. Külahcı and M. Ergüt, Bertrand curves of AW(k)-type in Lorentzian space, Nonlinear Analysis: Theory, Methods & Applications, Volume 70, Issue 4, 15 February 2009, Pages 1725-1731.
* [17] M. Külahcı, M. Bektaş, M. Ergüt, On Harmonic curvatures of Frenet curve in Lorentzian space., Chaos Solitons Fract. 41 (2009) 1668-1675.
* [18] H. Matsuda and S. Yorozu, Notes on Bertrand curves. Yokohama Math. J. 50 (2003), no. 1-2, 41-58.
* [19] O. Z. Okuyucu, İ. Gök, Y. Yaylı and N. Ekmekci, Slant Helices in three Dimensional Lie Groups, preprint 2012: arXiv:1203.1146v2 [math.DG].
* [20] E. Özdamar, H.H. Hacısalihoğlu, A characterization of inclined curves in Euclidean n-space, Commun. Fac. Sci. Univ., Ser. 24A (1975) 15–22.
* [21] L. R. Pears, Bertrand curves in Riemannian space, J. London Math. Soc. Volume s1-10, Number 2, 180-183 , July 1935.
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|
arxiv-papers
| 2012-11-26T22:13:15 |
2024-09-04T02:49:38.589730
|
{
"license": "Public Domain",
"authors": "O. Zeki Okuyucu, \\.Ismail G\\\"ok, Yusuf Yayl{\\i} and Nejat Ekmekci",
"submitter": "Osman Zeki Okuyucu",
"url": "https://arxiv.org/abs/1211.6424"
}
|
1211.6617
|
# A surgery formula for the second Yamabe invariant
###### Abstract.
Let $(M,g)$ be a compact Riemannian manifold of dimension $n\geq 3$. For a
metric $g$ on $M$, we let ${\lambda}_{2}(g)$ be the second eigenvalue of the
Yamabe operator $L_{g}:=\frac{4(n-1)}{n-2}\Delta_{g}+{\mathop{\rm Scal}}_{g}$.
Then, the second Yamabe invariant is defined as
${\sigma}_{2}(M)\mathrel{\raise
0.40903pt\hbox{\rm:}\mkern-5.2mu=}\sup\inf_{h\in[g]}{\lambda}_{2}(h){\mathop{\rm
Vol}}(M,h)^{2/n}.$
where the supremum is taken over all metrics $g$ and the infimum is taken over
the metrics in the conformal class $[g]$. Assume that ${\sigma}_{2}(M)>0$. In
the spirit of [4], we prove that if $N$ is obtained from $M$ by a
$k$-dimensional surgery ($0\leq k\leq n-3$), there exists a positive constant
$\Lambda_{n}$ depending only on $n$ such that
${\sigma}_{2}(N)\geq\min(\sigma_{2}(M),\Lambda_{n})$. We then give some
topological conclusions of this result.
S. El [email protected]
Mathematics Subject Classification. 35J60 (Primary), 35P30, 57R65, 58J50,
58C40 (Secondary). Key words and phrases. Yamabe operator, second Yamabe
invariant, surgery, $\alpha$-genus.
###### Contents
1. 1 Introduction
2. 2 Joining manifolds along a submanifold
1. 2.1 Surgery on manifolds
3. 3 The constants $\Lambda_{n,k}$
1. 3.1 Definition of $\Lambda_{n,k}$
4. 4 Limit spaces and limit solutions
5. 5 $L^{2}$-estimates on $WS$-bundles
6. 6 Main Theorem
1. 6.1 Construction of the metric $g_{\theta}$
1. 6.1.1 Modification of the metric $g$
2. 6.1.2 Definition of the metric $g_{\theta}$
2. 6.2 A preliminary result
1. 6.2.1 Proof of Theorem 6.3 Part 1
2. 6.2.2 Proof of Theorem 6.3 Part 2
3. 6.3 Proof of Theorem 6.1
7. 7 Some applications
1. 7.1 A preliminary result
## 1\. Introduction
Definition of the Yamabe operator $L_{g}$, eigenvalues of $L_{g}$, smooth
Yamabe invariant $\sigma(M)$
Let $(M,g)$ be a compact Riemannian manifold of dimension $n\geq 3$. We denote
the scalar curvature by ${\mathop{\rm Scal}}_{g}$. Let us define
$\mu(M,g):=\inf_{\tilde{g}\in[g]}\int_{M}{\mathop{\rm
Scal}}_{\tilde{g}}dv_{\tilde{g}}\left({\mathop{\rm
Vol}}_{\tilde{g}}(M)\right)^{-(n-2)/n}$
and
$\sigma(M):=\sup_{g}\mu(M,g)$
where, in the definition of $\mu(M,g)$, the infimum runs over all the metrics
$g^{\prime}$ in the conformal class $[g]$ of $g$ and where, in the definition
of $\sigma(M)$, the supremum is taken over all the Riemannian metrics $g$ on
$M$. The number $\mu(M,g)$, also denoted by $\mu(g)$ if no ambiguity, is
called the Yamabe constant while $\sigma(M)$ is called the Yamabe invariant.
The Yamabe constant played a crucial role in the solution of the Yamabe
problem solved between 1960 and 1984 by Yamabe, Trüdinger, Aubin and Schoen.
This problem consists in finding a metric $\widetilde{g}$ conformal to $g$
such that the scalar curvature ${\mathop{\rm Scal}}_{\widetilde{g}}$ of
$\widetilde{g}$ is constant. For more information, the reader may refer to
[17, 13, 7]. An important geometric meaning of $\mu(M,g)$ and $\sigma(M)$ is
contained in the following well known result:
###### Proposition 1.1.
Let $M$ be a compact differentiable manifold of dimension $n\geq 3$. Then,
* •
if $g$ is a Riemannian metric on $M$, the conformal class $[g]$ of $g$
contains a metric of positive scalar curvature if and only if $\mu(M,g)>0$.
* •
$M$ carries a metric $g$ with positive scalar curvature if and only if
$\sigma(M)>0$.
Classifying compact manifolds admitting a positive scalar curvature metric is
a hard open problem which was studied by many mathematicians. Significant
progresses were made thanks to surgery techniques. We recall briefly that a
surgery on $M$ is the procedure of constructing from $M$ a new manifold
$N:=M\setminus{S^{k}\times B^{n-k}}\cup_{S^{k}\times
S^{n-k-1}}\bar{B}^{k+1}\times S^{n-k-1},$
by removing the interior of $S^{k}\times B^{n-k}$ and gluing it with
$\bar{B}^{k+1}\times S^{n-k-1}$ along the boundaries. Gromov-Lawson and
Schoen-Yau proved in [12] and [19] the following
###### Theorem 1.2.
Let $M$ be a compact manifold of dimension $n\geq 3$ such that
${\sigma}(M)>0$. Assume that $N$ is obtained from $M$ by a surgery of
dimension $k$ ($0\leq k\leq n-3$). Then, ${\sigma}(N)>0$.
Using cobordism techniques, one deduces:
###### Corollary 1.3.
Every manifold $M$ of dimension $n\geq 5$ simply connected and non-spin,
carries a metric of positive scalar curvature.
Later, Kobayashi [15] and Petean-Yun [18] obtained new surgery formulas for
$\sigma(M)$. These works were generalized by B. Ammann, M. Dahl and E. Humbert
in [4] where they proved in particular
###### Theorem 1.4.
If $N$ is obtained from $M$ by a surgery of dimension $0\leq k\leq n-3$, then
$\sigma(N)\geq\min(\sigma(M),\Lambda_{n}),$
where $\Lambda_{n}$ is a positive constant depending only on $n$.
As a corollary, they obtained the following
###### Corollary 1.5.
Let $M$ be a simply connected compact manifold of dimension $n\geq 5$, then
one of this assumptions is satisfied
1. (1)
$\sigma(M)=0$ (which implies that $M$ is spin);
2. (2)
$\sigma(M)\geq\alpha_{n}$, where $\alpha_{n}$ is a positive constant depending
only on $n$.
Now, let us define the _Yamabe operator_ or _conformal Laplacian_
$L_{g}:=a\Delta_{g}+{\mathop{\rm Scal}}_{g},$
where $a=\frac{4(n-1)}{n-2}$ and where $\Delta_{g}$ is the Laplace-Beltrami
operator. The operator $L_{g}$ is an elliptic differential operator of second
order whose spectrum is discrete:
${\mathop{\rm Spec}}(L_{g})=\\{{\lambda}_{1}(g),{\lambda}_{2}(g),\cdots\\},$
where ${\lambda}_{1}(g)<{\lambda}_{2}(g)\leq\cdots$ are the eigenvalues of
$L_{g}$. The variational characterization of ${\lambda}_{i}(g)$ is given by
${\lambda}_{i}(g)=\inf_{V\in Gr_{i}(H_{1}^{2}(M))}\sup_{v\in
V\setminus\\{0\\}}\frac{\int_{M}vL_{g}v\,dv_{g}}{\int_{M}v^{2}\,dv_{g}},$
where $Gr_{i}(H_{1}^{2}(M))$ stands for the $i$-th dimensional Grassmannian in
$H_{1}^{2}(M).$ One important property of the eigenvalues of $L_{g}$ is that
their sign is a conformal invariant equal to the sign of the Yamabe constant
(see [10]). Consequently, a compact manifold $M$ possesses a metric with
positive ${\lambda}_{1}$ if and only if it admits a positive scalar curvature
metric.
Now, if $\mu(M,g)\geq 0,$ it is easy to check that
$\displaystyle\mu(M,g)=\inf_{\widetilde{g}\in\left[g\right]}{\lambda}_{1}(\widetilde{g}){\mathop{\rm
Vol}}(M,\widetilde{g})^{\frac{2}{n}},$ (1)
where $\left[g\right]$ is the conformal class of $g$ and ${\lambda}_{1}$ is
the first eigenvalue of the Yamabe operator $L_{g}$. Inspired by these
definitions, one can define the _second Yamabe constant_ and the _second
Yamabe invariant_ by
$\mu_{2}(M,g)=\inf_{\widetilde{g}\in\left[g\right]}{\lambda}_{2}(\widetilde{g}){\mathop{\rm
Vol}}(M,\widetilde{g})^{\frac{2}{n}},$
and
${\sigma}_{2}(M)=\sup_{g}\mu_{2}(M,g).$
The second Yamabe constant $\mu_{2}(M,g)$ or $\mu_{2}(g)$ if no ambiguity was
introduced and studied in [6] when $\mu(M,g)\geq 0$. This study was enlarged
in [10] where we started to investigate the relationships between the sign of
the second eigenvalue of the Yamabe operator $L_{g}$ and the existence of
nodal solutions of the equation $L_{g}u={\epsilon}|u|^{N-2}u,$ where
${\epsilon}=-1,0,+1$. The present paper establishes a surgery formula for
${\sigma}_{2}(M)$ in the spirit of Theorem 1.4. More precisely, our main
result is the following
###### Theorem 1.6.
Let $M$ be a compact manifold of dimension $n\geq 3$ such that
$\sigma_{2}(M)>0$. Assume that $N$ is obtained from $M$ by a surgery of
dimension $0\leq k\leq n-3$, then we have
$\sigma_{2}(N)\geq\min(\sigma_{2}(M),\Lambda_{n}),$
where $\Lambda_{n}$ is a positive constant depending only on $n$.
Note that Bär and Dahl in [8] proved a surgery formula for the spectrum of the
Yamabe operator with interesting topological consequences.
The proof of Theorem 1.6 is inspired by the one of Theorem 1.4 but some new
difficulties arise here. Let us recall the strategy: first, we fix a metric
$g$ on $M$ such that $\mu_{2}(M,g)$ is close to $\sigma_{2}(M)$. Then the goal
is to construct on $N$ a sequence of metrics $g_{\theta}$ such that
$\liminf_{{\theta}\to
0}\mu_{2}(N,g_{\theta})\geq\min(\mu_{2}(M,g),\Lambda_{n})$
where $\Lambda_{n}>0$ depends only on $n$ (see Theorem 6.1). Surprisingly, if
$\mu(M,g)=0$, we are not able to prove Theorem 6.1 directly. So the first step
is to show that one can assume that $\mu(M,g)\not=0$ (see Paragraph 6.1.1).
Here, we use exactly the same metrics than in [4] and use many of their
properties established in [4]. The proof consists in studying the first and
second eigenvalues ${\lambda}_{1}(u_{\theta}^{N-2}g_{\theta})$ and
${\lambda}_{2}(u_{\theta}^{N-2}g_{\theta})$ of
$L_{u_{\theta}^{N-2}g_{\theta}}$ where $u_{\theta}$ is such that
$\mu_{2}(g_{\theta})={\lambda}_{2}(u_{\theta}^{N-2}g_{\theta}){\mathop{\rm
Vol}}_{u_{\theta}^{N-2}g_{\theta}}(M)^{2/n},$
or in other words, $u_{\theta}$ is such that the metric
$u_{\theta}^{N-2}g_{\theta}$ achieves the infimum in the definition of
$\mu_{2}(N,g_{\theta})$. Two main difficulties arise in this situation:
* •
Contrary to what happened in [4], we could not show that
${\lambda}_{1}(u_{\theta}^{N-2}g_{\theta})$ and
${\lambda}_{2}(u_{\theta}^{N-2}g_{\theta})$ are bounded.
* •
The proof of Theorem 1.4 was consisting in obtaining some good “limit
equations“. The difficulty here is to ensure that
$\lim_{\theta}{\lambda}_{1}(u_{\theta}^{N-2}g_{\theta})\not=\lim_{\theta}{\lambda}_{2}(u_{\theta}^{N-2}g_{\theta}).$
The way to overcome these difficulties is to proceed in two steps: the first
one is to show that ${\lambda}_{2}(u_{\theta}^{N-2}g_{\theta})>0$. In a second
step, we are able to get the desired inequality.
Let us now come back to Theorem 1.6. Standard cobordism techniques allow to
deduce the following corollary
###### Corollary 1.7.
Let $M$ be a compact, spin, connected and simply connected manifold of
dimension $n\geq 5$ with $n\equiv 0,1,2,4$ mod $8$. If $|\alpha(M)|\leq 1$,
then
$\sigma_{2}(M)\geq\alpha_{n},$
where $\alpha_{n}$ is a positive constant depending only on $n$ and
$\alpha(M)$ is the $\alpha$-genus of $M$ (see Section 7).
When $M$ is not spin, the conclusion of the corollary still holds but is a
direct application of Corollary 1.5 and the fact that
$\sigma_{2}(M)\geq\sigma(M)$. Note that:
$\bullet$ In dimensions $1,2$ mod $8$, $\alpha(M)\in\mathbb{Z}/{2\mathbb{Z}}$
and hence the condition on the $\alpha$-genus $|\alpha(M)|\leq 1$ is always
satisfied. We then obtain that on any connected, simply connected manifold
(not necessarily spin) of dimension $n\equiv 1,2$ mod $8$
$\sigma_{2}(M)\geq\alpha_{n},$
for some $\alpha_{n}>0$ depending only on $n$.
$\bullet$ In dimensions $0$ mod $8$, when $M$ is spin, $\alpha(M)=\hat{A}(M),$
where $\hat{A}$ is the $\hat{A}$-genus. Hence if $M$ is simply connected (not
necessarily spin) connected of dimension $n\equiv 0$ mod $8$, $|\hat{A}|\leq
1$ then
$\sigma_{2}(M)\geq\alpha_{n},$
where $\alpha_{n}$ is a positive constant depending only on $n$.
$\bullet$ In dimensions $4$ mod $8$, when $M$ is spin, we have
$\alpha(M)=\frac{1}{2}\hat{A}(M)$. When $M$ is spin and $\hat{A}(M)\leq 2$, we
get that $|\alpha(M)|\leq 1$ and consequently, for any simply connected (not
necessarily spin) connected $M$ of dimension $n\geq 5$, $n\equiv 4$ mod $8$
with $|\hat{A}|\leq 2$, we obtain that
$\sigma_{2}(M)\geq\alpha_{n},$
where $\alpha_{n}$ is a positive constant depending only on $n$.
Acknowledgements: I would like to thank Emmanuel Humbert for his
encouragements, support and remarks along this work. I am also very grateful
to Bernd Ammann, Mattias Dahl, Romain Gicquaud and Andreas Hermann for their
remarks and their suggestions.
## 2\. Joining manifolds along a submanifold
### 2.1. Surgery on manifolds
###### Definition 2.1.
A surgery on a $n$-dimensional manifold $M$ is the procedure of constructing a
new $n$-dimensional manifold
$N=(M\setminus f(S^{k}\times B^{n-k}))\cup(\overline{B}^{k+1}\times
S^{n-k-1})/\sim,$
by cutting out $f(S^{k}\times B^{n-k})\subset M$ and replacing it by
$\overline{B}^{k+1}\times S^{n-k-1},$ where
$f:S^{k}\times\overline{B^{n-k}}\rightarrow M$ is a smooth embedding which
preserve the orientation and $\sim$ means that we paste along the boundary.
Then, we construct on the topological space $N$ a differential structure and
an orientation that makes a differentiable manifold such that the following
inclusions
$M\setminus f(S^{k}\times B^{n-k})\subset N,$
and
$\overline{B^{k+1}}\times S^{n-k-1}\subset N$
preserve the orientation. We say that $N$ is obtained from $M$ by a surgery of
dimension $k$ and we will denote $M\stackrel{{\scriptstyle
k}}{{\rightarrow}}N.$
Surgery can be considered from another point of view. In fact, it is a special
case of the connected sum: We paste $M$ and $S^{n}$ along a $k$-sphere. In
this section we describe how two manifolds are joined along a common
submanifold with trivialized normal bundle. Strictly speaking this is a
differential topological construction, but since we work with Riemannian
manifolds we will make the construction adapted to the Riemannian metrics and
use distance neighborhoods defined by the metrics etc. Let $(M_{1},g_{1})$ and
$(M_{2},g_{2})$ be complete Riemannian manifolds of dimension $n$. Let $W$ be
a compact manifold of dimension $k$, where $0\leq k\leq n$. Let
$\bar{w}_{i}:W\times\doba{R}^{n-k}\to TM_{i}$, $i=1,2$, be smooth embeddings.
We assume that $\bar{w}_{i}$ restricted to $W\times\\{0\\}$ maps to the zero
section of $TM_{i}$ (which we identify with $M_{i}$) and thus gives an
embedding $W\to M_{i}$. The image of this embedding is denoted by
$W_{i}^{\prime}$. Further we assume that $\bar{w}_{i}$ restrict to linear
isomorphisms $\\{p\\}\times\doba{R}^{n-k}\to
N_{\bar{w}_{i}(p,0)}W_{i}^{\prime}$ for all $p\in W_{i}$, where
$NW_{i}^{\prime}$ denotes the normal bundle of $W_{i}^{\prime}$ defined using
$g_{i}$. We set $w_{i}\mathrel{\raise
0.40903pt\hbox{\rm:}\mkern-5.2mu=}\exp^{g_{i}}\circ\bar{w}_{i}$. This gives
embeddings $w_{i}:W\times B^{n-k}(R_{\textrm{max}})\to M_{i}$ for some
$R_{\textrm{max}}>0$ and $i=1,2$. We have
$W_{i}^{\prime}=w_{i}(W\times\\{0\\})$ and we define the disjoint union
$(M,g)\mathrel{\raise 0.40903pt\hbox{\rm:}\mkern-5.2mu=}(M_{1}\amalg
M_{2},g_{1}\amalg g_{2}),$
and
$W^{\prime}\mathrel{\raise
0.40903pt\hbox{\rm:}\mkern-5.2mu=}W_{1}^{\prime}\amalg W_{2}^{\prime}.$
Let $r_{i}$ be the function on $M_{i}$ giving the distance to
$W_{i}^{\prime}$. Then $r_{1}\circ w_{1}(p,x)=r_{2}\circ w_{2}(p,x)=|x|$ for
$p\in W$, $x\in B^{n-k}(R_{\textrm{max}})$. Let $r$ be the function on $M$
defined by $r(x)\mathrel{\raise 0.40903pt\hbox{\rm:}\mkern-5.2mu=}r_{i}(x)$
for $x\in M_{i}$, $i=1,2$. For $0<{\epsilon}$ we set
$U_{i}({\epsilon})\mathrel{\raise 0.40903pt\hbox{\rm:}\mkern-5.2mu=}\\{x\in
M_{i}\,:\,r_{i}(x)<{\epsilon}\\}$ and $U({\epsilon})\mathrel{\raise
0.40903pt\hbox{\rm:}\mkern-5.2mu=}U_{1}({\epsilon})\cup U_{2}({\epsilon})$.
For $0<{\epsilon}<{\theta}$ we define
$N_{{\epsilon}}\mathrel{\raise
0.40903pt\hbox{\rm:}\mkern-5.2mu=}(M_{1}\setminus
U_{1}({\epsilon}))\cup(M_{2}\setminus U_{2}({\epsilon}))/{\sim},$
and
$U^{N}_{\epsilon}({\theta})\mathrel{\raise
0.40903pt\hbox{\rm:}\mkern-5.2mu=}(U({\theta})\setminus U({\epsilon}))/{\sim}$
where ${\sim}$ indicates that we identify $x\in\partial U_{1}({\epsilon})$
with $w_{2}\circ w_{1}^{-1}(x)\in\partial U_{2}({\epsilon})$. Hence
$N_{{\epsilon}}=(M\setminus U({\theta}))\cup U^{N}_{\epsilon}({\theta}).$
We say that $N_{\epsilon}$ is obtained from $M_{1}$, $M_{2}$ (and
$\bar{w}_{1}$, $\bar{w}_{2}$) by a connected sum along $W$ with parameter
${\epsilon}$.
The diffeomorphism type of $N_{\epsilon}$ is independent of ${\epsilon}$,
hence we will usually write $N=N_{\epsilon}$. However, in situations when
dropping the index causes ambiguities, we will keep the notation
$N_{\epsilon}$. For example the function $r:M\to[0,\infty)$ gives a continuous
function $r_{\epsilon}:N_{\epsilon}\to[{\epsilon},\infty)$ whose domain
depends on ${\epsilon}$. It is also going to be important to keep track of the
subscript ${\epsilon}$ on $U^{N}_{\epsilon}({\theta})$ since crucial estimates
on solutions of the Yamabe equation will be carried out on this set.
The surgery operation on a manifold is a special case of taking connected sum
along a submanifold. Indeed, let $M$ be a compact manifold of dimension $n$
and let $M_{1}=M$, $M_{2}=S^{n}$, $W=S^{k}$. Let $w_{1}:S^{k}\times B^{n-k}\to
M$ be an embedding defining a surgery and let $w_{2}:S^{k}\times B^{n-k}\to
S^{n}$ be the canonical embedding. Since $S^{n}\setminus w_{2}(S^{k}\times
B^{n-k})$ is diffeomorphic to $\overline{B^{k+1}}\times S^{n-k-1}$ we have in
this situation that $N$ is obtained from $M$ using surgery on $w_{1}$, see
[16, Section VI, 9].
## 3\. The constants $\Lambda_{n,k}$
### 3.1. Definition of $\Lambda_{n,k}$
In this paragraph, we define some constants $\Lambda_{n,k}$ in the same way
than in [4]. The only difference is that the functions we considered are not
necessarily positive. More precisely, let $(M,h)$ be a Riemannian manifold of
dimension $n\geq 3$. For $i=1,2$ we denote by $\Omega^{(i)}$ the set of
$C^{2}$ functions $v$ (not necessarily positive) solution of the equation
$L_{h}v=\mu|v|^{N-2}v,$
where $\mu\in\doba{R}$ . We assume that $v$ satisfies
$\displaystyle\bullet$ $\displaystyle v\not\equiv 0,$ $\displaystyle\bullet$
$\displaystyle\|v\|_{L^{N}(M)}\leq 1,$ $\displaystyle\bullet$ $\displaystyle
v\in L^{\infty}(M),$
together with
$\displaystyle\bullet$ $\displaystyle v\in L^{2}(M),\text{ for }i=1,$
$\displaystyle or$ $\displaystyle\bullet$
$\displaystyle\mu\|v\|_{L^{\infty}(M)}^{N-2}\geq\frac{(n-k-2)^{2}(n-1)}{8(n-2)},\text{
for }i=2.$
For $i=1,2$, we set
$\mu^{(i)}(M,h):=\inf_{v\in\Omega^{(i)}(M,h)}\mu(v).$
If $\Omega^{(i)}(M,h)$ is empty, we set $\mu^{(i)}=\infty.$
###### Definition 3.1.
For $n\geq 3$ and $0\leq k\leq n-3$, we define
$\Lambda_{n,k}^{(i)}:=\inf_{c\in[-1,1]}\mu^{(i)}({\mathbb{H}}_{c}^{k+1}\times{\mathbb{S}}^{n-k-1}),$
and
$\Lambda_{n,k}:=\min(\Lambda_{n,k}^{(1)},\Lambda_{n,k}^{(2)}),$
where
${\mathbb{H}}_{c}^{k+1}:=({\mathbb{R}}^{k}\times\mathbb{R},\eta_{c}^{k+1}=e^{2ct}\xi^{k}+dt^{2})$
When considering only positive functions $v$, B. Ammann, M. Dahl and E.
Humbert proved in [4] that these constants are positive. It is straightforward
to see that the positivity of $v$ has no role in their proof and hence it
remains true that $\Lambda_{n,k}>0$. They gave also explicit positive lower
bounds of these constants and many of their techniques still hold in this
context but we will not discuss this fact here. For more informations, the
reader may refer to [2], [3] and [5] .
## 4\. Limit spaces and limit solutions
###### Lemma 4.1.
Let $M$ be an $n$-dimensional manifold. let $(g_{\theta})$ be a sequence of
metrics which converges toward a metric $g$ in $C^{2}$ on all compact
$K\subset M$ when ${\theta}\to 0$. Assume that $v_{\theta}$ is a sequence of
functions such that $\|v_{\theta}\|_{L^{\infty}(M)}$ is bounded and
$\|L_{g_{\theta}}v_{\theta}\|_{L^{\infty}(M)}$ tends to $0$. Then, there
exists a smooth function $v$ solution of the equation
$L_{g}v=0$
such that $v_{\theta}$ tends to $v$ in $C^{1}$ on each compact set
$K\subset\subset V$.
Proof: Let $K,K^{\prime}$ be compact sets of $M$ such that $K^{\prime}\subset
K$, we have
$-g_{\theta}^{ij}\left(\partial_{i}\partial_{j}v_{\theta}-\Gamma_{ij}^{k}\partial_{k}v_{\theta}\right)+\frac{n-2}{4(n-1)}{\mathop{\rm
Scal}}_{g_{\theta}}v_{\theta}=f_{\theta}\to 0.$
Using Theorem 9.11 in [11], one easily checks that
$\|v_{\theta}\|_{H^{2,p}(K^{\prime},g)}\leq
C(\|L_{g_{\theta}}v_{\theta}\|_{L^{p}(K,g_{\theta})}+\|v_{\theta}\|_{L^{p}(K,g_{\theta})}).$
It follows that $v_{\theta}$ is bounded in $H^{2,p}(K^{\prime},g)$ for all
$p\geq 1$. Using Kondrakov’s theorem, there exists $v_{K^{\prime}}$ such that
$v_{\theta}$ tends to $v_{K^{\prime}}$ in $C^{1}(K^{\prime}).$ Taking an
increasing sequence of compact sets $K_{m}$ such that $\cup_{m}K_{m}=M$,
$(v_{\theta})$ converges to $v_{m}$ on $C^{1}(K_{m}),$ we define $v:=v_{m}$ on
$K_{m}$. Using the diagonal extraction process, we deduce that $v_{\theta}$
tends to $v$ in $C^{1}$ on any compact set and that $v$ verifies the same
Yamabe equation as $v_{\theta}$. Since for each compactly supported smooth
function ${\varphi}$, we have
$\int_{M}L_{g_{\theta}}{\varphi}v_{\theta}dv_{g_{\theta}}\to\int_{M}L_{g}{\varphi}vdv_{g},$
and
$\|L_{g_{\theta}}v_{\theta}\|_{L^{\infty}(M)}\to 0,$
we obtain that $L_{g}v=0$ in the sense of distributions. Using standard
regularity theorems, $v$ is smooth.
## 5\. $L^{2}$-estimates on $WS$-bundles
We suppose that the product $P\mathrel{\raise
0.40903pt\hbox{\rm:}\mkern-5.2mu=}I\times W\times S^{n-k-1}$ is equipped with
a metric $g_{\rm WS}$ of the form
$g_{\rm WS}=dt^{2}+e^{2{\varphi}(t)}h_{t}+\sigma^{n-k-1}$
and we mean by $WS$-bundle this product, where $h_{t}$ is a smooth family of
metrics on $W$ and depending on $t$ and ${\varphi}$ is a function on $I$. Let
$\pi:P\to I$ be the projection onto the first factor and
$F_{t}=\pi^{-1}(t)=\\{t\\}\times W\times S^{n-k-1}$, and the metric induced on
$F_{t}$ is defined by
$g_{t}:=dt^{2}+e^{2{\varphi}(t)}h_{t}+\sigma^{n-k-1}.$
Let $H_{t}$ be the mean curvature of $F_{t}$ in $P$, it is given by the
following
$H_{t}=-\frac{k}{n-1}{\varphi}^{\prime}(t)+e(h_{t}),$
with $e(h_{t}):=\frac{1}{2}tr_{h_{t}}(\partial_{t}h_{t}).$ The derivative of
the element of volume of $F_{t}$ is
$\partial_{t}dv_{g_{t}}=-(n-1)H_{t}dv_{g_{t}}.$
From the definition of $H_{t}$, when $t\to h_{t}$ is constant, we obtain that
$H_{t}=-\frac{k}{n-1}{\varphi}^{\prime}(t).$
###### Definition 5.1.
We say that the condition $(A_{t})$ is verified if the following assumptions
are satisfied:
$\ \ \begin{matrix}1.)&t\mapsto h_{t}\mbox{ is constant},\hfill\\\
2.)&e^{-2{\varphi}(t)}\inf_{x\in W}{\mathop{\rm
Scal}}^{h_{t}}(x)\geq-\frac{n-k-2}{32}a,\hfill\\\
3.)&|{\varphi}^{\prime}(t)|\leq 1,\hfill\\\
4.)&0\leq-2k{\varphi}^{\prime\prime}(t)\leq\frac{1}{2}(n-1)(n-k-2)^{2}.\end{matrix}\hbox{\kern
276.50574pt$(A_{t})$\hss}$
Similarly, for the condition $B_{t}$, we should have another assumptions to
verify
$\ \ \begin{matrix}1.)&t\mapsto{\varphi}(t)\mbox{ is constant,}\hfill\\\
2.)&\inf_{x\in F_{t}}{\mathop{\rm Scal}}^{g_{\rm
WS}}(x)\geq\frac{1}{2}{\mathop{\rm
Scal}}^{{\sigma}^{n-k-1}}=\frac{1}{2}(n-k-1)(n-k-2),\hfill\\\
3.)&\frac{(n-1)^{2}}{2}e(h_{t})^{2}+\frac{n-1}{2}\partial_{t}e(h_{t})\geq-\frac{3}{64}(n-k-2).\hfill\end{matrix}\hbox{\kern
211.99918pt$(B_{t})$\hss}$
###### Theorem 5.2.
Let $\alpha,$ $\beta\in\doba{R}$ such that $\left[\alpha,\beta\right]\subset
I.$ We suppose also that one of the conditions $(A_{t})$ and $(B_{t})$ is
satisfied. We assume that we have a solution $v$ of the equation
$L^{g_{\rm WS}}v=a\Delta^{g_{\rm WS}}v+{\mathop{\rm Scal}}^{g_{\rm WS}}v=\mu
u^{N-2}v+d^{*}A(dv)+Xv+{\epsilon}\partial_{t}v-sv$ (2)
where $s,{\epsilon}\in C^{\infty}(P)$, $A\in{\mathop{\rm End}}(T^{*}P)$, and
$X\in\Gamma(TP)$ are perturbation terms coming from the difference between $G$
and $g_{\rm WS}$. We assume that the endomorphism $A$ is symmetric and that
$X$ and $A$ are vertical, that is $dt(X)=0$ and $A(dt)=0$. Such that
$\mu\|u\|_{L^{\infty}(P)}^{N-2}\leq\frac{(n-k-2)^{2}(n-1)}{8(n-2)}.$ (3)
Then there exists $c_{0}>0$ independent of ${\alpha}$, $\beta$, and
${\varphi}$, such that if
$\|A\|_{L^{\infty}(P)},\|X\|_{L^{\infty}(P)},\|s\|_{L^{\infty}(P)},\|{\epsilon}\|_{L^{\infty}(P)},\|e(h_{t})\|_{L^{\infty}(P)}\leq
c_{0}$
then
$\int_{\pi^{-1}\left(({\alpha}+{\gamma},{\beta}-{\gamma})\right)}v^{2}\,dv_{g_{\rm
WS}}\leq\frac{4\|v\|_{L^{\infty}}^{2}}{n-k-2}\left({\mathop{\rm
Vol}}^{g_{\alpha}}(F_{{\alpha}})+{\mathop{\rm
Vol}}^{g_{\beta}}(F_{{\beta}})\right),$
where ${\gamma}\mathrel{\raise
0.40903pt\hbox{\rm:}\mkern-5.2mu=}\frac{\sqrt{32}}{n-k-2}$.
Remark that we should have $\beta-\alpha>2\gamma$ to obtain our result and
note that this theorem gives us an estimate of $\left\|v\right\|_{L^{2}}.$
For the proof of this Theorem, we mimic exactly the proof of Theorem 6.2 in
[4]. The only difference is that we consider here a nodal solution (and not a
positive solution) of the equation
$L^{g_{\rm WS}}v=\mu u^{N-2}v+d^{*}A(dv)+Xv+{\epsilon}\partial_{t}v-sv.$
Other details are exactly the same.
## 6\. Main Theorem
Theorem 1.6 is a direct corollary of
###### Theorem 6.1.
Let $(M,g)$ be a compact Riemannian manifold of dimension $n\geq 3$ such that
$\mu_{2}(M,g)>0$ and let $N$ be obtained from $M$ by a surgery of dimension
$0\leq k\leq n-3$. Then there exists a sequence of metrics $g_{\theta}$ such
that
$\liminf_{{\theta}\to
0}\mu_{2}(N,g_{\theta})\geq\min(\mu_{2}(M,g),\Lambda_{n}),$
where $\Lambda_{n}>0$ depends only on $n$.
Indeed, to get Theorem 1.6, it suffices to apply Theorem 6.1 with a metric $g$
such that $\mu_{2}(M,g)$ is arbitrary closed to $\sigma_{2}(M)$. The
conclusion easily follows since $\mu_{2}(N,g_{\theta})\leq\sigma_{2}(M)$. This
section is devoted to the proof of Theorem 6.1.
### 6.1. Construction of the metric $g_{\theta}$
#### 6.1.1. Modification of the metric $g$
For a technical reason, we will need in the proof of Theorem 6.1 that
$\mu(g)\not=0$. To get rid of this difficulty, we need the following
proposition:
###### Proposition 6.2.
There exists on $M$ a metric $g^{\prime}$ arbitrary close to $g$ in $C^{2}$
such that $\mu(g^{\prime})\not=0$.
Indeed, let us assume for a while that Theorem 6.1 is true if $\mu(g)\not=0$
and let us see that the result remains true if $\mu(g)=0$. A first observation
is that if $g^{\prime}$ is close enough to $g$ in $C^{2}$, then as one can
check, $\mu_{2}(g^{\prime})$ is close to $\mu_{2}(g)$. Let us consider a
metric $g^{\prime}$ given by Proposition 6.2 close enough to $g$ so that
$\mu_{2}(g^{\prime})>\mu_{2}(g)-{\epsilon}>0$ for an arbitrary small
${\epsilon}$. From Theorem 6.1 applied to $g^{\prime}$, we obtain a sequence
of metrics $g_{\theta}$ on $N$ such that
$\liminf_{{\theta}\to
0}\mu_{2}(N,g_{\theta})\geq\min(\mu_{2}(M,g^{\prime}),\Lambda_{n})\geq\min(\mu_{2}(M,g)-{\epsilon},\Lambda_{n}).$
Letting ${\epsilon}$ tend to $0$, we obtain Theorem 6.1. It remains to prove
Proposition 6.2.
Proof of Proposition 6.2: At first, in order to simplify notations, we will
consider $g$ as a metric on $M\amalg S^{n}$ and equal to the standard metric
$g={\sigma}^{n}$ on $S^{n}$. Since $\mu(g)=0$, we can assume that
${\mathop{\rm Scal}}_{g}=0$, possibly making a conformal change of metrics.
Let us consider a metric $h$ for which ${\mathop{\rm Scal}}_{h}$ is negative
and constant and whose existence is given in [7]. Consider the analytic family
of metrics $g_{t}:=th+(1-t)g$. Since the first eigenvalue ${\lambda}_{t}$ of
$L_{g_{t}}$ is simple, the function $t\to{\lambda}_{t}$ is analytic (see for
instance Theorem VII.3.9 in [14]). Since ${\lambda}_{0}=0$ and
${\lambda}_{1}<0$, it follows that for $t$ arbitrary close to $0$,
${\lambda}_{t}\not=0$. Proposition 6.2 follows since $\mu(g_{t})$ has the same
sign than ${\lambda}_{t}$.
#### 6.1.2. Definition of the metric $g_{\theta}$
As explained above, we will use the same construction as in [4]. Consequently,
we give the definition of $g_{\theta}$ without additional explanations. The
reader may refer to [4] for more details. We keep the same notations than in
Section 2. Let $h_{1}$ be the restriction of $g$ to the surgery sphere
$S_{1}^{\prime}\subset M$ and $h_{2}$ be the restriction of the standard
metric ${\sigma}^{n}=g$ on $S^{n}$ to $S_{2}^{\prime}\subset S^{n}$. Define
$S^{\prime}:=S_{1}^{\prime}\amalg S_{2}^{\prime}$ and $h\mathrel{\raise
0.40903pt\hbox{\rm:}\mkern-5.2mu=}h_{1}\amalg h_{2}$ on $S^{\prime}$. In the
following, $r$ denotes the distance function to $S^{\prime}$ in $(M\amalg
S^{n},g\amalg{\sigma}^{n})$. In polar coordinates, the metric $g$ has the form
$g=h+\xi^{n-k}+T=h+dr^{2}+r^{2}\sigma^{n-k-1}+T$ (4)
on $U(R_{\textrm{max}})\setminus S^{\prime}\cong
S^{\prime}\times(0,R_{\textrm{max}})\times S^{n-k-1}$. Here $T$ is a symmetric
$(2,0)$-tensor vanishing on $S^{\prime}$ which is the error term measuring the
fact that $g$ is not in general a product metric (at least near
$S_{1}^{\prime}$). We also define the product metric
$g^{\prime}\mathrel{\raise
0.40903pt\hbox{\rm:}\mkern-5.2mu=}h+\xi^{n-k}=h+dr^{2}+r^{2}{\sigma}^{n-k-1},$
(5)
on $U(R_{\textrm{max}})\setminus S^{\prime}$ so that $g=g^{\prime}+T$. As in
[4], we have
$\left\\{\begin{array}[]{ccc}|T(X,Y)|&\leq&Cr|X|_{g^{\prime}}|Y|_{g^{\prime}},\\\
|(\nabla_{U}T)(X,Y)|&\leq&C|X|_{g^{\prime}}|Y|_{g^{\prime}}|U|_{g^{\prime}},\\\
|(\nabla^{2}_{U,V})T(X,Y)|&\leq&C|X|_{g^{\prime}}|Y|_{g^{\prime}}|U|_{g^{\prime}}|V|_{g^{\prime}},\end{array}\right.$
for $X,Y,U,V\in T_{x}M$ and $x\in U(R_{\textrm{max}})$. We define
$T_{1}\mathrel{\raise 0.40903pt\hbox{\rm:}\mkern-5.2mu=}T|_{M}$ and
$T_{2}\mathrel{\raise 0.40903pt\hbox{\rm:}\mkern-5.2mu=}T|_{S^{n}}$. We fix
$R_{0}\in(0,R_{\textrm{max}})$, $R_{0}<1$ and choose a smooth positive
function $F:M\setminus S^{\prime}\to\doba{R}$ such that
$F(x)=\begin{cases}1,&\text{if $x\in M\setminus U_{1}(R_{\textrm{max}})\amalg
S^{n}\setminus U_{2}(R_{\textrm{max}})$;}\\\ r(x)^{-1},&\text{if $x\in
U_{i}(R_{0})\setminus S^{\prime}$.}\end{cases}$
Next we choose a sequence $\theta=\theta_{j}$ of positive numbers tending to
$0$. For any $\theta$ we then choose a number
${\delta}_{0}={\delta}_{0}({\theta})\in(0,{\theta})$ small enough to suit with
the arguments below. For any ${\theta}>0$ and sufficiently small
${\delta}_{0}$ there is $A_{\theta}\in[{\theta}^{-1},({\delta}_{0})^{-1})$ and
a smooth function $f:U(R_{\textrm{max}})\to\doba{R}$ depending only on the
coordinate $r$ such that
$f(x)=\begin{cases}-\ln r(x),&\text{if $x\in U(R_{\textrm{max}})\setminus
U({\theta})$;}\\\ \phantom{-}\ln A_{\theta},&\text{if $x\in
U({\delta}_{0})$,}\end{cases}$
and such that
$\left|r\frac{df}{dr}\right|=\left|\frac{df}{d(\ln r)}\right|\leq
1,\quad\text{and}\quad\left\|r\frac{d}{dr}\left(r\frac{df}{dr}\right)\right\|_{L^{\infty}}=\left\|\frac{d^{2}f}{d^{2}(\ln
r)}\right\|_{L^{\infty}}\to 0$ (6)
as ${\theta}\to 0$. Set ${\epsilon}=e^{-A_{\theta}}{\delta}_{0}$ that we
assume smaller than $1$ and use this ${\epsilon}$ to construct $M$ as in
Section 2. On
$U^{N}_{\epsilon}(R_{\textrm{max}})=\left(U(R_{\textrm{max}})\setminus
U({\epsilon})\right)/{\sim}$ we define $t$ by
$t\mathrel{\raise 0.40903pt\hbox{\rm:}\mkern-5.2mu=}\begin{cases}-\ln
r_{1}+\ln{\epsilon},&\text{on $U_{1}(R_{\textrm{max}})\setminus
U_{1}({\epsilon})$;}\\\ \phantom{-}\ln r_{2}-\ln{\epsilon},&\text{on
$U_{2}(R_{\textrm{max}})\setminus U_{2}({\epsilon})$.}\end{cases}$
One checks that
* •
$r_{i}=e^{|t|+\ln{\epsilon}}={\epsilon}e^{|t|};$
* •
$F(x)={\epsilon}^{-1}e^{-|t|}$ for $x\in U(R_{0})\setminus U^{N}({\theta})$,
or equivalently if $|t|+\ln{\epsilon}\leq\ln R_{0}$ and hence
$F^{2}g={\epsilon}^{-2}e^{-2|t|}(h+T)+dt^{2}+\sigma^{n-k-1}$
on $U(R_{0})\setminus U^{N}({\theta})$;
* •
and
$f(t)=\begin{cases}-|t|-\ln{\epsilon},&\text{if
$\ln{\theta}-\ln{\epsilon}\leq|t|\leq\ln R_{\textrm{max}}-\ln{\epsilon}$;}\\\
\ln A_{\theta},&\text{if $|t|\leq\ln{\delta}_{0}-\ln{\epsilon}$.}\end{cases}$
We have $|df/dt|\leq 1$, $\|d^{2}f/dt^{2}\|_{L^{\infty}}\to 0$. Now, we choose
a cut-off function $\chi:\doba{R}\to[0,1]$ such that $\chi=0$ on
$(-\infty,-1]$, $|d\chi|\leq 1$, and $\chi=1$ on $[1,\infty)$. Finally, we
define
$g_{{\theta}}\mathrel{\raise
0.40903pt\hbox{\rm:}\mkern-5.2mu=}\begin{cases}F^{2}g_{i},&\text{on
$M_{i}\setminus U_{i}({\theta})$;}\\\
e^{2f(t)}(h_{i}+T_{i})+dt^{2}+\sigma^{n-k-1},&\text{on
$U_{i}({\theta})\setminus U_{i}({\delta}_{0})$;}\\\ \begin{aligned}
&A_{{\theta}}^{2}\chi(t/A_{{\theta}})(h_{2}+T_{2})+A_{{\theta}}^{2}(1-\chi(t/A_{{\theta}}))(h_{1}+T_{1})\\\
&\quad+dt^{2}+\sigma^{n-k-1},\end{aligned}&\text{on
$U^{N}_{\epsilon}({\delta}_{0})$.}\end{cases}$
Moreover, the metric $g_{\theta}$ can be written as
$g_{\theta}:=g^{\prime}_{\theta}+\widetilde{T_{t}}\text{ on }U^{N}(R_{0}),$
where $g^{\prime}_{\theta}$ is the metric without error term and it is equal
to
$g^{\prime}_{\theta}=e^{2f(t)}\widetilde{h_{t}}+dt^{2}+\sigma^{n-k-1},$
where the metric $\widetilde{h_{t}}$ is given by
$\widetilde{h_{t}}:=\chi(\frac{t}{A_{\theta}})h_{2}+(1-\chi(\frac{t}{A_{\theta}}))h_{1},$
and $\widetilde{T_{t}}$ is the error term and his expression is given by the
following
$\widetilde{T_{t}}:=e^{2f(t)}(\chi(\frac{t}{A_{\theta}})T_{2}+(1-\chi(\frac{t}{A_{\theta}}))T_{1}).$
We further have the following properties of the error term $\widetilde{T_{t}}$
$\left\\{\begin{array}[]{ccc}|\widetilde{T}(X,Y)|&\leq&Cr|X|_{g^{\prime}_{\theta}}|Y|_{g^{\prime}_{\theta}},\\\
|\nabla{{\widetilde{T}}_{t}}|_{g^{\prime}_{\theta}}&\leq&Ce^{-f(t)},\\\
|\nabla^{2}{{\widetilde{T}}_{t}}|_{g^{\prime}_{\theta}}&\leq&Ce^{-f(t)},\end{array}\right.$
where $\nabla$ is the Levi-Civita connection with respect to the metric
$g^{\prime}_{\theta}$, for all $X$, $Y\in T_{x}N$ and $x\in U^{N}(R_{0}).$
### 6.2. A preliminary result
In order to prove Theorem 6.1, we will start by proving the following results.
###### Theorem 6.3.
Part 1: let $(u_{\theta})$ be a sequence of functions which satisfy
$L_{g_{\theta}}u_{\theta}={\lambda}_{\theta}|u_{\theta}|^{N-2}u_{\theta},$
such that $\int_{N}|u_{\theta}|^{N}dv_{g_{\theta}}=1$ and
${\lambda}_{\theta}\to_{{\theta}\to 0}{\lambda}_{\infty}$, where
${\lambda}_{\infty}\in\doba{R}$. Then, at least one of the two following
assertions is true
1. (1)
${\lambda}_{\infty}\geq\Lambda_{n}$, where $\Lambda_{n}>0$ depends only on
$n$;
2. (2)
there exists a function $u\in C^{\infty}(M\amalg S^{n})$, $u\equiv 0$ on
$S^{n}$, $u\not\equiv 0$ on $M$ solution of
$L_{g}u=\lambda_{\infty}|u|^{N-2}u,$
with
$\int_{M}|u|^{N}dv_{g}=1$
such that for all compact sets $K\subset M\amalg S^{n}\setminus S^{\prime}$
(note that $K$ can also be considered as a subset of $N$),
$F^{\frac{n-2}{2}}u_{\theta}$ tends to $u$ in $C^{2}(K)$, where $F$ is defined
in Section 6.1. Moreover, we have
1. (a)
the norm $L^{2}$ of $u_{\theta}$ is bounded uniformly in ${\theta}$;
2. (b)
$\lim_{b\to 0}\limsup_{{\theta}\to 0}\sup_{U^{N}(b)}u_{\theta}=0$;
3. (c)
$\lim_{b\to 0}\limsup_{{\theta}\to
0}\int_{U^{N}(b)}u_{\theta}^{N}\,dv_{g_{\theta}}=0.$
Part 2: let $u_{\theta}$ be as in Part 1 above and assume that Assertion 2) is
true. Let $v_{\theta}$ be a sequence of functions which satisfy
$L_{g_{\theta}}v_{\theta}=\mu_{\theta}|u_{\theta}|^{N-2}v_{\theta},$
such that $\int_{N}v_{\theta}^{N}dv_{g_{\theta}}=1,$
$\mu_{\theta}\to\mu_{\infty}$ where $\mu_{\infty}<\mu(\doba{S}^{n})$. Then,
there exists a function $v\in C^{\infty}(M\amalg S^{n})$, $v\equiv 0$ on
$S^{n}$, $v\not\equiv 0$ on $M$ solution of
$L_{g}v=\mu_{\infty}|u|^{N-2}v$
with
$\int_{M}|v|^{N}dv_{g}=1$
and such that for all compact sets $K\subset M\amalg S^{n}\setminus
S^{\prime}$, $F^{\frac{n-2}{2}}v_{\theta}$ tends to $v$ in $C^{2}(K).$
Moreover,
1. (1)
the norm $L^{2}$ of $v_{\theta}$ is bounded uniformly in ${\theta}$;
2. (2)
$\lim_{b\to 0}\limsup_{{\theta}\to 0}\sup_{U^{N}(b)}v_{\theta}=0;$
3. (3)
$\lim_{b\to 0}\limsup_{{\theta}\to
0}\int_{U^{N}(b)}v_{\theta}^{N}\,dv_{g_{\theta}}=0.$
#### 6.2.1. Proof of Theorem 6.3 Part 1
Let $(u_{\theta})$ be a sequence of functions which satisfy
$L_{g_{\theta}}u_{\theta}={\lambda}_{\theta}|u_{\theta}|^{N-2}u_{\theta},$
such that $\int_{N}|u_{\theta}|^{N}\,dv_{g_{\theta}}=1$ and
${\lambda}_{\theta}\to_{{\theta}\to 0}{\lambda}_{\infty}$, where
${\lambda}_{\infty}\in\doba{R}$. We proceed exactly as in [4] where here, the
manifold $M_{2}$ is $S^{n}$ equiped with the standard metric $\sigma^{n}$, and
where $W$ is the sphere $S^{k}$. The only difference will be that $u_{\theta}$
may now have a changing sign.
###### Remark 6.4.
In the proof of the main theorem in [4], it was proven that
${\lambda}_{\infty}>-\infty.$
Here, we made the assumption that ${\lambda}_{\infty}$ has a limit. Without
this assumption, one could again prove that ${\lambda}_{\infty}>-\infty$ but
the point here is that there is no reason why ${\lambda}_{\infty}$ should be
bounded from above contrary to what happened in [4].
The argument of Corollary 7.7 in [4] still holds here and shows that
$\displaystyle\liminf_{\theta}\|u_{\theta}\|_{L^{\infty}(N)}>0.$ (7)
Several cases are studied:
###### Case I.
$\limsup_{{\theta}\to 0}\|u_{\theta}\|_{L^{\infty}(N)}=\infty$.
Set $m_{\theta}:=\|u_{\theta}\|_{L^{\infty}(N)}$ and choose $x_{\theta}\in N$
such that $u_{\theta}(x_{\theta})=m_{\theta}$. After taking a subsequence, we
can assume that $\lim_{{\theta}\to 0}m_{\theta}=\infty$. We have to study the
following two subcases.
###### Subcase I.1.
There exists $b>0$ such that $x_{\theta}\in N\setminus U^{N}(b)$ for an
infinite number of ${\theta}$.
###### Subcase I.2.
For all $b>0$ it holds that $x_{\theta}\in U^{N}(b)$ for ${\theta}$
sufficiently small.
###### Case II.
There exists a constant $C_{0}$ such that $\|u_{\theta}\|_{L^{\infty}(N)}\leq
C_{0}$ for all ${\theta}$.
###### Subcase II.1.
There exists $b>0$ such that
$\liminf_{{\theta}\to
0}\left({\lambda}_{\theta}\sup_{U^{N}(b)}{u_{\theta}}^{N-2}\right)<\frac{(n-k-2)^{2}(n-1)}{8(n-2)}.$
###### Subsubcase II.1.1.
$\limsup_{b\to 0}\limsup_{{\theta}\to 0}\sup_{U^{N}(b)}u_{\theta}>0$.
###### Subsubcase II.1.2.
$\lim_{b\to 0}\limsup_{{\theta}\to 0}\sup_{U^{N}(b)}u_{\theta}=0$.
###### Subcase II.2.
${\lambda}_{\theta}\sup_{U^{N}(b)}{u_{\theta}}^{N-2}\geq\frac{(n-k-2)^{2}(n-1)}{8(n-2)}$
In Subcases I.1 and I.2, it is shown in [4] that
$\lambda_{\infty}\geq\mu(\doba{S}^{n})$. The proof still holds when
$u_{\theta}$ has a changing sign. In Subsubcase II.1.1 and Subcase II.2, we
obtain that $\lambda_{\infty}\geq\Lambda_{n,k}$ where $\Lambda_{n,k}$ is a
positive number depending only on $n$ and $k$. The definition of
$\Lambda_{n,k}$ in [4] is the infimum of energies of positive solutions of the
Yamabe equation on model spaces (see Section 3). This definition has to be
slightly modified to allow nodal solutions. As explained in Section 3 the
proof that $\Lambda_{n,k}>0$ remains the same.
In Subcases I.1, I.2, II.1.1 and II.2, we then get that
$\lambda_{\infty}\geq\Lambda_{n}$, where
$\Lambda_{n}:=\min_{k\in\\{0,\cdots,n-3\\}}\\{\Lambda_{n,k},\mu\\}.$
In particular, Assertion 1) of part 1 in Theorem 6.3 is true. So let us
examine Subsubcase II.1.2. The assumption of Subcase II.1 allows to obtain as
in [4] that
$\displaystyle\int_{N}u_{\theta}^{2}dv_{g_{\theta}}\leq C.$ (8)
for some $C>0$. The assumptions of Subcase II.1.2 are that
$\displaystyle\sup_{N}(u_{\theta})\leq C$ (9)
and that
$\displaystyle\limsup_{b\to 0}\limsup_{{\theta}\to
0}\sup_{U^{N}(b)}u_{\theta}=0.$ (10)
###### Step 1.
We prove that $\lim_{b\to 0}\limsup_{{\theta}\to
0}\int_{U^{N}(b)}{|u_{\theta}|}^{N}\,dv_{g_{\theta}}=0$.
Let $b>0$. We have, by Relation (8)
$\int_{U^{N}(b)}|u_{\theta}|^{N}\,dv_{g_{\theta}}\leq
A_{0}\sup_{U^{N}(b)}|u_{\theta}|^{N-2},$
where $A_{0}$ is a positive number which does not depend on $b$ and
${\theta}.$ The claim then follows from (10).
###### Step 2.
$C^{2}$ convergence on all compact sets of $M\amalg S^{n}\setminus
S^{\prime}.$
Let $(\Omega_{j})_{j}$ be an increasing sequence of subdomains of $(M\amalg
S^{n}\setminus S^{\prime})$ with smooth boundary such that
$\bigcup_{j}\Omega_{j}=M\amalg S^{n}\setminus S^{\prime},$
$\Omega_{j}\subset\Omega_{j+1}$. The norm
$\left\|u_{\theta}\right\|_{L^{\infty}(N)}$ is bounded, then so is
$\left\|u_{\theta}\right\|_{L^{\infty}(\Omega_{j+1})}$. Using standard results
on elliptic regularity (for more details, see for example [11]), we see that
the sequence $(u_{\theta})$ is bounded in the Sobolev space
$H^{2,p}(\Omega^{\prime}_{j})$ $\forall p\in(1,\infty)$ where
$\Omega^{\prime}_{j}$ is any domain such that
$\overline{{\Omega}}_{j}\subset{\Omega}^{\prime}_{j}\subset\overline{{\Omega}^{\prime}_{j}}\subset{\Omega}_{j+1}$.
The Sobolev embedding Theorem implies that $(u_{\theta})$ is bounded in
$C^{1,\alpha}(\overline{\Omega_{j}})$ for any $\alpha\in(0,1)$. (See Theorem
4.12 in [1] for more informations on Sobolev embedding Theorems).
Now we use a diagonal extraction process, by taking successive subsequences,
it follows that $(u_{\theta})$ converges to functions $\widetilde{u_{j}}\in
C^{1}(\overline{\Omega_{j}})$ and such that
$\widetilde{u_{j}}|_{\overline{\Omega}_{j-1}}=\widetilde{u}_{j-1}.$
We define
$\widetilde{u}=\widetilde{u_{j}}\text{ on }\overline{\Omega_{j}}.$
By taking a diagonal subsequence of $u_{\theta}$, we get that $u_{\theta}$
tends to $\widetilde{u}$ in $C^{1}$ on any compact subset of $M\amalg
S^{n}\setminus S^{\prime}$ and by $C^{1}$-convergence of the functions
$u_{\theta}$, the function $\widetilde{u}$ satisfies the equation
$L_{g_{\theta}}\widetilde{u}={\lambda}_{\infty}|\widetilde{u}|^{N-2}{\widetilde{u}}\text{
on }M\amalg S^{n}\setminus S^{\prime}.$ (11)
We recall that $g_{\theta}=F^{2}g=(F^{\frac{n-2}{2}})^{\frac{4}{n-2}}g$ on
$U^{N}(b)$. By conformal invariance of the Yamabe operator we obtain for all
$v$
$L_{F^{2}g}v=F^{-\frac{n+2}{2}}L_{g}(F^{\frac{n-2}{2}}v).$
Now we set
$u=F^{\frac{n-2}{2}}\widetilde{u}.$
We obtain
$\displaystyle L_{g}u$ $\displaystyle=$ $\displaystyle
F^{\frac{n+2}{2}}L_{F^{2}g}\widetilde{u}$ $\displaystyle=$ $\displaystyle
F^{\frac{n+2}{2}}{\lambda}_{\infty}|\widetilde{u}|^{N-2}\widetilde{u}$
$\displaystyle=$ $\displaystyle{\lambda}_{\infty}|u|^{N-2}u.$
This shows that $u$ is a solution on $(M\amalg S^{n}\setminus S^{\prime},g)$
of the following equation
$L_{g}u={\lambda}_{\infty}|u|^{N-2}u.$
Moreover, using Step 1 and the fact that
$\int_{N}u_{\theta}^{N}dv_{g_{\theta}}=1$, the function $u$ satisfies
$\displaystyle\int_{M\amalg S^{n}}u^{N}\,dv_{g}$ $\displaystyle=$
$\displaystyle\int_{M\amalg S^{n}\setminus
S^{\prime}}{\widetilde{u}}^{N}\,dv_{g}$ $\displaystyle=$
$\displaystyle\lim_{b\to 0}\lim_{{\theta}\to
0}\int_{U^{N}(b)}u_{{\theta}}^{N}\,dv_{g_{{\theta}}}$ $\displaystyle=$
$\displaystyle 1.$
###### Step 3.
Removal of the singularity
The next step is to show that $u$ is a solution on all $M\amalg S^{n}$ of
$\displaystyle L_{g}{u}={\lambda}_{\infty}|u|^{N-2}u.$ (12)
To prove this fact, we will show that for all ${\varphi}\in C^{\infty}(M\amalg
S^{n})$, we have
$\int_{M\amalg S^{n}}L_{g}u{\varphi}\,dv_{g}=\int_{M\amalg
S^{n}}{\lambda}_{\infty}|u|^{N-2}u{\varphi}\,dv_{g}.$
First, we have
$\displaystyle\int_{M\amalg S^{n}}uL_{g}{\varphi}\,dv_{g}$ $\displaystyle=$
$\displaystyle\int_{M\amalg
S^{n}}uL_{g}({\varphi}-\chi_{\epsilon}{\varphi}+\chi_{\epsilon}{\varphi})\,dv_{g}$
$\displaystyle=$ $\displaystyle\int_{M\amalg
S^{n}}uL_{g}(\chi_{\epsilon}{\varphi})\,dv_{g}+\int_{M\amalg
S^{n}}uL_{g}((1-\chi_{\epsilon}){\varphi})\,dv_{g},$
where
$\displaystyle\left|\;\begin{matrix}\chi_{\epsilon}=1\hfill&\hbox{if
}d_{g}(x,S^{\prime})<{\epsilon},\\\ \\\ \chi_{\epsilon}=0\hfill&\hbox{if
}d_{g}(x,S^{\prime})\geq 2{\epsilon},\\\ \\\
\left|d\chi_{\epsilon}\right|<\frac{2}{{\epsilon}}.\end{matrix}\right.$
Since $(1-\chi_{\epsilon})$ is compactly supported in $M\amalg S^{n}\setminus
S^{\prime}$, we have
$\displaystyle\int_{M\amalg
S^{n}}uL_{g}((1-\chi_{\epsilon}){\varphi})\,dv_{g}$ $\displaystyle=$
$\displaystyle\int_{M\amalg
S^{n}}(L_{g}u)(1-\chi_{\epsilon}){\varphi}\,dv_{g}$ $\displaystyle\to$
$\displaystyle\int_{M\amalg S^{n}}L_{g}u{\varphi}\,dv_{g}=\int_{M\amalg
S^{n}}{\lambda}_{\infty}|u|^{N-2}u{\varphi}\,dv_{g}.$
Then, it remains to prove that
$\int_{M\amalg S^{n}}uL_{g}(\chi_{\epsilon}{\varphi})\,dv_{g}\rightarrow 0.$
We have
$\displaystyle L_{g}(\chi_{\epsilon}{\varphi})$ $\displaystyle=$
$\displaystyle C_{n}\Delta(\chi_{\epsilon}{\varphi})+{\mathop{\rm
Scal}}_{g}(\chi_{\epsilon}{\varphi})$ $\displaystyle=$ $\displaystyle
C_{n}\Delta\chi_{\epsilon}{\varphi}+C_{n}\Delta{\varphi}\chi_{\epsilon}+{\mathop{\rm
Scal}}_{g}(\chi_{\epsilon}{\varphi})-2\left\langle\nabla\chi_{\epsilon},\nabla{\varphi}\right\rangle$
$\displaystyle=$
$\displaystyle\chi_{\epsilon}L_{g}{\varphi}+C_{n}(\Delta\chi_{\epsilon}){\varphi}-2\left\langle\nabla\chi_{\epsilon},\nabla{\varphi}\right\rangle.$
According to Lebesgue Theorem, it holds that
$\int_{M\amalg S^{n}}u\chi_{\epsilon}L_{g}{\varphi}\,dv_{g}\rightarrow 0\text{
a.e.}$
Further, we have
$\displaystyle\left|\int_{M\amalg
S^{n}}uL_{g}(\chi_{\epsilon}{\varphi})\,dv_{g}\right|$ $\displaystyle\leq$
$\displaystyle\frac{C}{{\epsilon}^{2}}\int_{C_{\epsilon}}u\,dv_{g}$ (13)
$\displaystyle\leq$
$\displaystyle\frac{C}{{\epsilon}^{2}}\left(\int_{C_{\epsilon}}u^{2}\,dv_{g}\right)^{\frac{1}{2}}\left({\mathop{\rm
Vol}}(Supp(C_{\epsilon}))\right)^{\frac{1}{2}},$ (14)
where $C_{\epsilon}=\left\\{x\in M\amalg
S^{n};{\epsilon}<d(x,S^{\prime})<2{\epsilon}\right\\}=U^{N}(2{\epsilon})\setminus
U^{N}({\epsilon})$.
In addition, we get from (8) that
$\int_{N}{\widetilde{u}}^{2}\,dv_{F^{2}g}<+\infty,$
which implies that
$\int_{C_{\epsilon}}{\widetilde{u}}^{2}\,dv_{F^{2}g}<+\infty.$
Let us compute
$\displaystyle\int_{C_{\epsilon}}{\widetilde{u}}^{2}\,dv_{g_{{\theta}}}$
$\displaystyle=$
$\displaystyle\int_{C_{\epsilon}}\left(F^{\frac{n-2}{2}}\right)^{\frac{2n}{n-2}}F^{-(n-2)}u^{2}\,dv_{g}$
$\displaystyle=$ $\displaystyle\int_{C_{\epsilon}}F^{2}u^{2}\,dv_{g}<+\infty.$
We recall that $F=\frac{1}{r}$ on $C_{\epsilon}$. Coming back to (13), we
deduce
$\displaystyle\left|\int_{M}uL_{g}(\chi_{\epsilon}{\varphi})\,dv_{g}\right|$
$\displaystyle\leq$
$\displaystyle\frac{C}{{\epsilon}^{2}}\left(\int_{C_{\epsilon}}\frac{u^{2}F^{2}}{F^{2}}\,dv_{g}\right)^{\frac{1}{2}}\left({\mathop{\rm
Vol}}(C_{\epsilon})\right)^{\frac{1}{2}}$ $\displaystyle\leq$
$\displaystyle\frac{C}{{\epsilon}^{2}}\times{\epsilon}\times{\epsilon}^{\frac{n-k}{2}}=C{\epsilon}^{\frac{n-k}{2}-1}.$
Since $k\leq n-3,$ we have
$\frac{n-k}{2}-1>0,$
which implies that
$\int_{M\amalg S^{n}}uL_{g}(\chi_{\epsilon}{\varphi})\,dv_{g}\rightarrow 0.$
Finally, we get that $u$ is a solution on $M\amalg S^{n}$ of the equation
$L_{g}u={\lambda}_{\infty}|u|^{N-2}u.$
###### Step 4.
We have either $u\equiv 0$ on ${\doba{S}}^{n}$ either
${\lambda}_{\infty}\geq\mu({\doba{S}}^{n}).$
Note that the function $u$ verifies
$\displaystyle\int_{M\amalg S^{n}}|u|^{N}\,dv_{g}\leq 1.$ (15)
Since
$\displaystyle\int_{M\amalg S^{n}}|u|^{N}\,dv_{g}$ $\displaystyle=$
$\displaystyle\int_{M\amalg S^{n}}|\widetilde{u}|^{N}\,dv_{g_{{\theta}}}$
$\displaystyle\leq$
$\displaystyle\int_{N}|\widetilde{u}|^{N}\,dv_{g_{{\theta}}}$
$\displaystyle\leq$ $\displaystyle\lim_{{\theta}\to
0}\int_{N}|{u_{\theta}}|^{N}\,dv_{g_{\theta}}=1.$
Assume that $u\not\equiv 0$ on ${\doba{S}}^{n}$.
Setting $w=u_{|{\doba{S}^{n}}}$ and using equations (12) and (15), we have
$\displaystyle\mu({\doba{S}}^{n})\leq Y(w)$ $\displaystyle=$
$\displaystyle\frac{{\lambda}_{\infty}\int_{{\doba{S}}^{n}}w^{N}\,dv_{g}}{\left(\int_{{\doba{S}}^{n}}w^{N}\,dv_{g}\right)^{\frac{n-2}{n}}}$
$\displaystyle=$
$\displaystyle{\lambda}_{\infty}\left(\int_{{\doba{S}}^{n}}w^{N}\,dv_{g}\right)^{\frac{2}{n}}\leq{\lambda}_{\infty}.$
Then we obtain that ${\lambda}_{\infty}\geq\mu({\doba{S}}^{n})$ and hence, the
conclusion $1)$ of Theorem 6.3 Part 1 is true.
#### 6.2.2. Proof of Theorem 6.3 Part 2
We consider a function $v_{\theta}$ satisfying
$\displaystyle
L_{g_{\theta}}v_{\theta}=\mu_{\theta}|u_{\theta}|^{N-2}v_{\theta},$ (16)
with
$\int_{N}|v_{\theta}|^{N}\,dv_{g_{\theta}}=1.$
A first remark is the following: as in Lemma 7.6 of [4], we observe that
$U^{N}(b)$ is a $WS$-bundle for any $b>0$. Since $u_{\theta}$ satisfies
$\lim_{b\to 0}\limsup_{{\theta}\to 0}\sup_{U^{N}(b)}u_{\theta}=0.$
Then, for $b$ small enough, we have
$\mu_{\theta}\|u_{\theta}\|_{U^{N}(b)}^{N-2}\leq\frac{(n-k-2)^{2}(n-1)}{8(n-2)}.$
We then can apply Theorem 5.2 on $U^{N}(b)$ and the proof of Lemma 7.6 of [4]
shows that there exists numbers $c_{1},c_{2}>0$ independent of ${\theta}$ such
that
$\displaystyle\int_{N}|v_{\theta}|^{2}\,dv_{g_{\theta}}\leq c_{1}{\Arrowvert
v_{\theta}\Arrowvert}^{2}_{L^{\infty}(N)}+c_{2}.$ (17)
As a consequence, we get that
$\liminf_{{\theta}\to 0}\|v_{\theta}\|_{L^{\infty}(N)}>0.$
Indeed, assume that
$\lim_{{\theta}\to 0}\left\|v_{\theta}\right\|_{L^{\infty}(N)}=0.$
By Equation (17), we have
$\displaystyle 1=\int_{N}|v_{\theta}|^{N}\,dv_{g_{\theta}}$
$\displaystyle\leq$ $\displaystyle{\Arrowvert
v_{\theta}\Arrowvert}_{L^{\infty}(N)}^{N-2}\int_{N}|v_{\theta}|^{2}\,dv_{g_{\theta}}$
$\displaystyle\leq$ $\displaystyle{\Arrowvert
v_{\theta}\Arrowvert}_{L^{\infty}(N)}^{N-2}(c_{1}{\Arrowvert
v_{\theta}\Arrowvert}_{L^{\infty}(N)}^{2}+c_{2})\to 0,$
as ${\theta}\to 0$. This gives the desired contradiction. In the rest of the
proof, we will study several cases. In what follows, only Subcase II.1.2 will
be a big deal: Subcases I.1, I.2 and II.1 will be excluded by arguments mostly
contained in [4]. So we will just give few explanations for these cases.
###### Case I.
$\limsup_{{\theta}\to 0}\|v_{\theta}\|_{L^{\infty}(N)}=\infty$.
Set $m_{\theta}\mathrel{\raise
0.40903pt\hbox{\rm:}\mkern-5.2mu=}\|v_{\theta}\|_{L^{\infty}(N)}$ and choose
$x_{\theta}\in N$ with $v_{\theta}(x_{\theta})=m_{\theta}$. After taking a
subsequence we can assume that $\lim_{{\theta}\to 0}m_{\theta}=\infty$.
###### Subcase I.1.
There exists $b>0$ such that $x_{\theta}\in N\setminus U^{N}(b)$ for an
infinite number of ${\theta}$.
By taking a subsequence we can assume that there exists $\bar{x}\in M\amalg
S^{n}\setminus U(b)$ such that $\lim_{{\theta}\to 0}x_{\theta}=\bar{x}$. We
define $\tilde{g}_{\theta}\mathrel{\raise
0.40903pt\hbox{\rm:}\mkern-5.2mu=}m_{\theta}^{\frac{4}{n-2}}g_{\theta}$. For
$r>0$, [4] tells that for ${\theta}$ small enough, there exists a
diffeomorphism
${\Theta}_{\theta}:B^{n}(0,r)\to
B^{g_{\theta}}(x_{\theta},m_{\theta}^{-\frac{2}{n-2}}r)$
such that the sequence of metrics
$({\Theta}_{\theta}^{*}(\tilde{g}_{\theta}))$ tends to the flat metric
$\xi^{n}$ in $C^{2}(B^{n}(0,r))$, where $B^{n}(0,r)$ is the standard ball in
$\doba{R}^{n}$ centered in $0$ with radius $r$. We let
$\tilde{u}_{\theta}\mathrel{\raise
0.40903pt\hbox{\rm:}\mkern-5.2mu=}m_{\theta}^{-1}u_{\theta},$
${\tilde{v}}_{\theta}\mathrel{\raise
0.40903pt\hbox{\rm:}\mkern-5.2mu=}m_{\theta}^{-1}v_{\theta}$ and we have
$\displaystyle L_{\tilde{g}_{\theta}}{\tilde{v}}_{\theta}$ $\displaystyle=$
$\displaystyle{\lambda}_{\theta}{\tilde{u}}_{{\theta}}^{N-2}{\tilde{v}}_{\theta}$
$\displaystyle=$
$\displaystyle\frac{{\lambda}_{\theta}}{m_{\theta}^{N-2}}{u_{\theta}}^{N-2}\tilde{v}_{\theta}.$
Since $\left\|u_{\theta}\right\|_{L^{\infty}(N)}\leq C,$ it follows that
$\left\|L_{\tilde{g}_{\theta}}\tilde{v}_{\theta}\right\|_{L^{\infty}(N)}$
tends to $0$. Applying Lemma 4.1, we obtain a solution $v\not\equiv 0$ of the
following equation on $\doba{R}^{n}$ $L_{\xi^{n}}v=0$. Since ${\mathop{\rm
Scal}}_{\xi^{n}}=0$, $v$ is harmonic and admits a maximum at $x=0$. As a
consequence, $v$ is constant equal to $v(0)=1$. This is a contradiction, since
$\|v\|_{L^{N}}\leq 1.$
###### Subcase I.2.
For all $b>0$ it holds that $x_{\theta}\in U^{N}(b)$ for ${\theta}$
sufficiently small.
We proceed as in Subcase I.2 in [4]. As in Subcase I.1 above, we get from
Lemma 4.1 a function $v$ which is harmonic on $\doba{R}^{n}$ and admits a
maximum at $x=0$. This is again a contradiction.
###### Case II.
There exists a constant $C_{0}$ such that $\|v_{\theta}\|_{L^{\infty}(N)}\leq
C_{0}$ for all ${\theta}$.
By (17), there exists a constant $A_{0}$ independent of ${\theta}$ such that
$\|v_{\theta}\|_{L^{2}(N,g_{\theta})}\leq A_{0}.$ (18)
We split the treatment of Case II into two subcases.
###### Subcase II.1.
$\limsup_{b\to 0}\limsup_{{\theta}\to 0}\sup_{U^{N}(b)}v_{\theta}>0$.
Again mimicking what is done in [4], we obtain from Lemma 4.1 a function $v$
which is a solution of $L_{G_{c}}v=0$ on $\doba{R}^{k+1}\times S^{n-k-1},G$
for some $c\in[-1,1]$ where $G_{c}=e^{2cs}\xi^{k}+ds^{2}+\sigma^{n-k-1}$. In
Subcases I.1 and I.2, we used the fact that
$\frac{{\lambda}_{\theta}}{m_{\theta}^{N-2}}$ tends to $0$ to show that at the
limit $L_{G_{c}}v=0$. Here, the argument is different: first we set
$\alpha_{0}:=\frac{1}{2}\limsup_{b\to 0}\limsup_{{\theta}\to
0}\sup_{U^{N}(b)}v_{\theta}>0$. Then, we can suppose that there exists a
sequence of positive numbers $(b_{i})$ and $({\theta}_{i})$ such that
$\sup_{U^{N}(b_{i})}v_{{\theta}_{i}}\geq\alpha_{0},$
for all $i$. To simplify, we write ${\theta}$ for ${\theta}_{i}$ and $b$ for
$b_{i}$. Take $x^{\prime}_{\theta}\in\overline{U^{N}(b_{\theta})}$ such that
$v_{\theta}(x^{\prime}_{\theta})\geq\alpha_{0}.$
For $r,r^{\prime}>0$, we define
$U_{\theta}(r,r^{\prime}):=B^{{\widetilde{h}}_{t_{\theta}}}(y_{\theta},e^{-f(t_{\theta})}r)\times[t_{\theta}-r^{\prime},t_{\theta}+r^{\prime}]\times
S^{n-k-1}.$
As in [4], the function $v$ is obtained as the limit of $v_{\theta}$ on each
$U_{\theta}(r,r^{\prime})$ (with $r,r^{\prime}>0$). The fact that
$L_{G_{c}}v=0$ follows from the observation that
$\sup_{U_{\theta}(r,r^{\prime})}|u_{\theta}|=0,$
hence
$|u_{\theta}|^{N-2}v_{\theta}\to 0\text{ uniformly on
}U_{\theta}(r,r^{\prime}).$
###### Subcase II.2.
$\lim_{b\to 0}\limsup_{{\theta}\to 0}\sup_{U^{N}(b)}v_{\theta}=0$.
By the same method than in Subsection 6.2.1, we obtain that there is a
function $v$ solution of the following equation
$L_{g}v=\mu_{\infty}|u|^{N-2}v,$
such that
$\int_{N}v^{N}\,dv_{g}\leq 1.$
Suppose that $v\not\equiv 0$ on $\doba{S}^{n}$, then we have
$\displaystyle\mu(\doba{S}^{n})\leq
Y(v)=\mu_{\infty}\frac{\int_{\doba{S}^{n}}u^{N-2}v^{2}\,dv_{g}}{(\int_{\doba{S}^{n}}v^{N}\,dv_{g})^{\frac{2}{N}}}=0$
since $u\equiv 0$ on $S^{n}$. This is a contradiction. This proves that
$v\not\equiv 0$ on $\doba{S}^{n}$. By the same argument than in Part 1, we
have $\int_{M}|v|^{N}dv_{g}=1$. We finally obtain that the function $v$
satisfies all the desired conclusions of Theorem 6.3 Part 2.
### 6.3. Proof of Theorem 6.1
Let $(g_{\theta})$ the sequence of metrics defined on $N$ as in Section 6.1.
Step 1: For ${\theta}$ small enough, we show that if
${\lambda}_{k}(M,g)>0\Rightarrow{\lambda}_{k}(N,g_{\theta})>0,$
where ${\lambda}_{k}$ is the $k^{th}$ eigenvalue associated to the Yamabe
equation.
###### Remark 6.5.
Note that this step implies that the existence of a metric with positive
${\lambda}_{k}$ is preserved by surgery of dimension $k\in\\{0,\cdots,n-3\\}$.
This is an alternative proof of a result already contained in [8].
We proceed by contradiction and we suppose that
${\lambda}_{k}(N,g_{\theta})\leq 0.$ Let $u_{\theta}$ be a minimizing solution
of the Yamabe problem. By referring to [10], there exists functions
$v_{{\theta},1}=u_{\theta},v_{{\theta},2},\cdots,v_{{\theta},k}$ solution of
the following equation on $N$
$L_{g_{{\theta}}}v_{{\theta},i}={\lambda}_{{\theta},i}u_{\theta}^{N-2}v_{{\theta},i},$
where
${\lambda}_{{\theta},i}={\lambda}_{i}(N,u_{\theta}^{N-2}g_{\theta}),$
such that
$\int_{N}{v_{{\theta},i}}^{N}\,dv_{g_{\theta}}=1\text{ and
}\int_{N}{u_{\theta}}^{N-2}v_{{\theta},i}v_{{\theta},j}\,dv_{g_{\theta}}=0\text{
for all }i\neq j.$
By conformal invariance of the sign of the eigenvalues of the Yamabe operator
(see [10]), we have
${\lambda}_{{\theta},i}={\lambda}_{i}(N,u_{\theta}^{N-2}g_{\theta})\leq 0.$
Moreover, by construction, it is easy to check that
${\lambda}_{{\theta},1}=\mu_{\theta}$ where $\mu_{\theta}=\mu(N,g_{\theta})$
is the Yamabe constant of the metric $g_{\theta}$. The main theorem in [4]
implies that $\lim_{{\theta}\to 0}{\lambda}_{{\theta},1}=\lim_{{\theta}\to
0}\mu_{\theta}>-\infty$. It follows that there exists a constant $C>0$ such
that $-C\leq{\lambda}_{{\theta},1}\leq\cdots\leq{\lambda}_{{\theta},k}\leq 0$.
Then, for all $i$, ${\lambda}_{{\theta},i}$ is bounded and by restricting to a
subsequence we can assume that ${\lambda}_{\infty,i}:=\lim_{{\theta}\to
0}{\lambda}_{{\theta},i}$ exists. Parts 1) and 2) of Theorem 6.1 give the
existence of functions $u=v_{1},\cdots,v_{k}$ defined on $M,$ with $v_{i}\neq
0$ for all $i$ such that $F^{\frac{n-2}{2}}v_{{\theta},i}$ tends to $v_{i}$ in
$C^{1}$ on each compact set $K\subset M\amalg S^{n}\setminus S^{\prime}$. The
functions $v_{i}$ are solutions of the following equation
$L_{g}v_{i}={\lambda}_{\infty,i}u^{N-2}v_{i}.$
Moreover, we have
$\int_{M}|v_{i}|^{N}\,dv_{g}\leq 1\text{ and }\lim_{b\to
0}\limsup_{{\theta}\to
0}\int_{U^{N}_{\epsilon}(b)}|v_{{\theta},i}|^{N}\,dv_{g}=0.$
Let us show that for all $i\neq j$, we get that
$\int_{M}u^{N-2}v_{i}v_{j}\,dv_{g}=0.$
Set
$\widetilde{u}_{\theta}=F^{\frac{n-2}{2}}u_{\theta},$
and
$\widetilde{v}_{{\theta},i}=F^{\frac{n-2}{2}}v_{{\theta},i}.$
For $b>0$ small, we have for $i\not=j$
$\displaystyle\int_{M\setminus{U(b)}}u^{N-2}v_{i}v_{j}\,dv_{g}$
$\displaystyle=\lim_{{\theta}\to 0}\int_{M\setminus U(b)=N\setminus
U^{N}_{\epsilon}(b)}\widetilde{u}_{\theta}^{N-2}\widetilde{v}_{{\theta},i}\widetilde{v}_{{\theta},j}dv_{g}$
$\displaystyle=\lim_{{\theta}\to 0}\int_{M\setminus U(b)=N\setminus
U^{N}_{\epsilon}(b)}u_{\theta}^{N-2}v_{{\theta},i}v_{{\theta},j}dv_{g_{\theta}}$
where we used $dv_{g_{\theta}}=F^{n}dv_{g}$. Using now the fact that
$\int_{N}u_{\theta}^{N-2}v_{{\theta},i}v_{{\theta},j}\,dv_{g_{\theta}}=0$, we
get
$\displaystyle\left|\int_{M\setminus U(b)}u^{N-2}v_{i}v_{j}\,dv_{g}\right|$
$\displaystyle=$ $\displaystyle\left|\lim_{{\theta}\to
0}\int_{N\setminus{U^{N}_{\epsilon}(b)}}u_{{\theta}}^{N-2}v_{{\theta},i}v_{{\theta},j}\,dv_{g_{\theta}}\right|$
$\displaystyle=$ $\displaystyle\lim_{{\theta}\to
0}\left|\int_{U^{N}_{\epsilon}(b)}u_{\theta}^{N-2}v_{{\theta},i}v_{{\theta},j}\,dv_{g_{\theta}}\right|.$
We write
$\displaystyle\left|\int_{U^{N}_{\epsilon}(b)}u_{\theta}^{N-2}v_{{\theta},i}v_{{\theta},j}\,dv_{g_{\theta}}\right|$
$\displaystyle\leq$
$\displaystyle\left(\int_{U_{\epsilon}^{N}(b)}u_{{\theta}}^{N}\,dv_{g_{\theta}}\right)^{\frac{N-2}{N}}\left(\int_{U_{\epsilon}^{N}(b)}|v_{{\theta},i}|^{N}\,dv_{g_{\theta}}\right)^{\frac{1}{N}}$
$\displaystyle\left(\int_{U_{\epsilon}^{N}(b)}|v_{{\theta},j}|^{N}\,dv_{g_{\theta}}\right)^{\frac{1}{N}}.$
Using the assertion
$\lim_{b\to 0}\limsup_{{\theta}\to
0}\int_{U^{N}_{\theta}(b)}v_{{\theta},i}^{N}\,dv_{g_{\theta}}=0.$
we obtain that
$\lim_{b\to 0}\limsup_{{\theta}\to
0}\left|\int_{U^{N}_{\epsilon}(b)}u_{\theta}^{N-2}v_{{\theta},i}v_{{\theta},j}\,dv_{g_{\theta}}\right|=0.$
We get finally that
$\left|\int_{M}u^{N-2}v_{i}v_{j}\,dv_{g}\right|=\lim_{b\to
0}\left|\int_{M\setminus U(b)}u^{N-2}v_{i}v_{j}\,dv_{g}\right|=0\text{ for all
}i\neq j.$
We now write
$\displaystyle 0<{\lambda}_{k}(M,g)$ $\displaystyle\leq$
$\displaystyle\sup_{(\alpha_{1},\cdots,\alpha_{k})\neq(0,\cdots,0)}F(u,\alpha_{1}v_{1}+\cdots+\alpha_{k}v_{k})$
$\displaystyle=$
$\displaystyle\sup_{(\alpha_{1},\cdots,\alpha_{k})\neq(0,\cdots,0)}\frac{\int_{M}(\alpha_{1}v_{1}+\cdots+\alpha_{k}v_{k})L_{g}(\alpha_{1}v_{1}+\cdots+\alpha_{k}v_{k})\,dv_{g}}{\int_{M}{u}^{N-2}(\alpha_{1}v_{1}+\cdots+\alpha_{k}v_{k})^{2}\,dv_{g}}$
$\displaystyle=$
$\displaystyle\sup_{(\alpha_{1},\cdots,\alpha_{k})\neq(0,\cdots,0)}\frac{\alpha_{1}^{2}\int_{M}v_{1}L_{g}v_{1}\,dv_{g}+\cdots+\alpha_{k}^{2}\int_{M}v_{k}L_{g}v_{k}\,dv_{g}}{\alpha_{1}^{2}\int_{M}u^{N-2}v_{1}^{2}\,dv_{g}+\cdots+\alpha_{k}^{2}\int_{M}u^{N-2}v_{k}^{2}\,dv_{g}}$
$\displaystyle=$
$\displaystyle\sup_{(\alpha_{1},\cdots,\alpha_{k})\neq(0,\cdots,0)}\frac{\alpha_{1}^{2}{\lambda}_{\infty,1}\int_{M}u^{N-2}v_{1}^{2}\,dv_{g}+\cdots+\alpha_{k}^{2}{\lambda}_{\infty,k}\int_{M}u^{N-2}v_{k}^{2}\,dv_{g}}{\alpha_{1}^{2}\int_{M}u^{N-2}v_{1}^{2}\,dv_{g}+\cdots+\alpha_{k}^{2}\int_{M}u^{N-2}v_{k}^{2}\,dv_{g}}$
$\displaystyle\leq$ $\displaystyle 0,$
since each ${\lambda}_{\infty,i}\leq 0$. This gives the desired contradiction.
###### Remark 6.6.
Note that, for $i\geq 2$ it could happen that
$\int_{M}u^{N-2}v_{i}^{2}dv_{g}=0$ if $M$ is not connected.
Step 2: Conclusion
Since $\mu_{2}(M,g)>0,$ from Step 1, we get that $\mu_{2}(N,g_{\theta})>0.$
Assume $\mu_{2}(N,g_{\theta})<\mu(\doba{S}^{n})$ (otherwise, we are done).
Using [10] we construct a sequence $(v_{\theta})$ solution of
$L_{g_{\theta}}v_{\theta}=\mu_{2}(N,g_{\theta})|v_{\theta}|^{N-2}v_{\theta},$
such that
$\int_{N}v_{\theta}^{N}\,dv_{g_{\theta}}=1.$
By Theorem 6.3 Part 1), this holds that $\lim_{{\theta}\to
0}\mu_{2}(N,g_{\theta})\geq\Lambda_{n}$ (and the conclusion of Theorem 6.1 is
true) or there exists a function $v$ solution on $M$ of the equation:
$L_{g}v=\mu_{\infty}|v|^{N-2}v,$
with $\mu_{\infty}=\lim_{\theta}\mu_{2}(N,g_{\theta})\geq 0$ and
$\int_{M}|v|^{N}\,dv_{g}=1.$
This is what we assume until now.
As explained in Paragraph 6.1.1, we can assume that $\mu(g)\not=0$.
Case 1: $\mu(g)<0$.
Assume that $M$ is connected (so is $N$) and let us prove that $v$ has a
changing sign. We suppose by contradiction that $v\geq 0.$ The maximum
principle gives that $v>0$. Let $u$ be a positive solution of the Yamabe
equation on $M,$ i.e.
$L_{g}u=\mu(g)u^{N-1}.$
Since $v>0$, we can write:
$L_{g}v=\underbrace{\mu_{\infty}}_{\geq 0}|v|^{N-2}v=\mu_{\infty}v^{N-1}.$
Multiplying the second equation by $u$ and integrating, we get
$\underbrace{\mu(g)}_{<0}\int_{M}u^{N-1}v\,dv_{g}=\int_{M}L_{g}uv\,dv_{g}=\int_{M}uL_{g}v\,dv_{g}=\underbrace{\mu_{\infty}}_{\geq
0}\int_{M}v^{N-1}u\,dv_{g}.$
This gives a contradiction. Then $v$ have a changing sign and this implies
that
$\mu_{2}(M,g)\leq\sup_{\alpha,\beta}F(v,\alpha v^{+}+\beta
v^{-})=\mu_{\infty}.$
If $M$ is now disconnected, then the Yamabe minimizer $u$ is positive on a
connected component of $M$. If $uv\not\equiv 0$, the same proof holds. If
$uv\equiv 0$ then
$\mu_{2}(M,g)\leq\sup_{\alpha,\beta}F(v,\alpha u+\beta v)=\mu_{\infty}$
In any case, the conclusion of Theorem 6.1 is true.
Case 2: $\mu(M,g)>0$.
Then, ${\lambda}_{1}(N,g_{\theta})>0$. In [10], it is established that the
sign of the eigenvalues of the Yamabe operator is conformally invariant.
Consequently, ${\lambda}_{1}(N,v_{\theta}^{N-2}g_{\theta})>0$. Set
$\mu_{1}={\lambda}_{1}(N,v_{\theta}^{N-2}g_{\theta})$ and let $u_{\theta}$ be
associated to $\mu_{1}$. Since associated to the first eigenvalue of the
Yamabe operator, $u_{\theta}$ is positive on at least one connected component
of $N$ (and $0$ on the other). In addition, $u_{\theta}$ is a solution of the
equation
$L_{g_{\theta}}u_{\theta}=\mu_{1}|v_{\theta}|^{N-2}u_{\theta},$
such that
$\int_{N}u_{\theta}^{N}\,dv_{g_{\theta}}=1\text{ and
}\int_{N}|v_{\theta}|^{N-2}u_{\theta}v_{\theta}\,dv_{g_{\theta}}=0.$
Using Theorem 6.3 Step 2), there exists a function $u$ solution on $M$ of the
following equation
$L_{g}u=\mu_{\infty,1}|v|^{N-2}u,$
where $\mu_{\infty,1}:=\lim_{\theta}\mu_{1}$. Note that this limit exists
after a possible extraction of a subsequence since
$0\leq\mu_{1}\leq\mu_{2}(N,g_{\theta})$. Proceeding as in Step 1, we show that
$\displaystyle\int_{M}|v|^{N-2}uv\,dv_{g}=0.$ (19)
By maximum principle and since $u_{\theta}>0$, $u>0$ on at least one connected
component of $M$. Then, $u$ and $v$ satisfy the equations
$L_{g}u=\mu_{\infty,1}|v|^{N-2}u,$
and
$L_{g}v=\mu_{\infty}|v|^{N-2}v.$
These equations implies that $\mu_{\infty,1}$ and $\mu_{\infty}$ are some
eigenvalues of the generalized metric $|v|^{N-2}g$ (see [10]). Since positive,
$u$ is associated to the first eigenvalue of $L_{|v|^{N-2}g}$ i.e.
$\mu_{\infty,1}={\lambda}_{1}(M,|v|^{N-2}g)$. Hence,
$\mu_{\infty,1}\leq\mu_{\infty}$.
Finally, we obtain that
$\displaystyle\mu_{2}(M,g)$ $\displaystyle\leq$
$\displaystyle{\lambda}_{2}(|v|^{N-2}g){\mathop{\rm
Vol}}_{|v|^{N-2}g}(M)^{\frac{2}{n}}=\mu_{\infty}$
since
${\mathop{\rm Vol}}_{|v|^{N-2}g}(M)=\int_{M}|v|^{N}dv_{g}=1$
and since $\mu_{\infty,1}\leq\mu_{\infty}$ are associated to two non
proportional eigenfunctions in the metric $|v|^{N-2}g$ (thanks to Relation
(19)) where we recall that $\mu_{\infty}=\lim_{{\theta}\to
0}\mu_{2}(N,g_{\theta}).$ This proves Theorem 6.1.
###### Remark 6.7.
The reason why we need $\mu(g)\not=0$ is the following. If $\mu(g)=0$, the
proof of Case 1 clearly does not lead to a contradiction. So, we would like to
apply the method used in Case 2 above. For this, we need that
${\lambda}_{1}(v_{\theta}^{N-2}g_{\theta})$ is bounded. When $\mu(g)>0$, this
holds true since
$0\leq{\lambda}_{1}(v_{\theta}^{N-2}g_{\theta})\leq{\lambda}_{2}(v_{\theta}^{N-2}g_{\theta})=\mu_{2}(N,g_{\theta})\to\mu_{\infty}.$
If $\mu(g)=0$, one cannot say nothing about the sign of
${\lambda}_{1}(v_{\theta}^{N-2}g_{\theta})$. In particular, if it is negative,
we were not able to prove that ${\lambda}_{1}(v_{\theta}^{N-2}g_{\theta})$ is
bounded from above and the proof breaks down.
## 7\. Some applications
In this section, we establish some topological applications of Theorem 1.6.
### 7.1. A preliminary result
We have
###### Proposition 7.1.
Let $V$, $M$ be two compact manifolds such that $V$ carries a metric $g$ with
${\mathop{\rm Scal}}_{g}=0$ and $\sigma(M)>0,$ then
$\sigma_{2}(V\amalg M)\geq\min(\mu_{2}(g),\sigma(M))>0.$
Proof: On $V\amalg M$, let $G={\lambda}g+\mu h$, where ${\lambda}$ and $\mu$
are two positive constants and for a small ${\epsilon}$, $h$ is a metric such
that $\sigma(M)\leq\mu(M,h)+{\epsilon}$. We have
$\displaystyle{\mathop{\rm Spec}}(L_{G})$ $\displaystyle=$
$\displaystyle{\mathop{\rm Spec}}(L_{{\lambda}g})\cup{\mathop{\rm
Spec}}(L_{\mu h})$ $\displaystyle=$ $\displaystyle{\lambda}^{-1}{\mathop{\rm
Spec}}(L_{g})\cup\mu^{-1}{\mathop{\rm Spec}}(L_{h})$ $\displaystyle=$
$\displaystyle\\{{\lambda}^{-1}{\lambda}_{1},{\lambda}^{-1}{\lambda}_{2},\cdots\\}\cup\\{\mu^{-1}{\lambda}_{1}^{\prime},\mu^{-1}{\lambda}_{2}^{\prime},\cdots\\}$
where ${\lambda}_{i}$ (resp. ${\lambda}_{i}^{\prime}$) denotes the $i$-th
eigenvalue of $L_{g}$ (resp. $L_{h}$). The assumption we made allows to claim
that ${\lambda}_{1}=0$, ${\lambda}_{2}>0$ and ${\lambda}_{1}^{\prime}>0$.
Hence, we deduce that
${\lambda}_{2}(L_{G})=\min\\{{\lambda}^{-1}{\lambda}_{2},\mu^{-1}{\lambda}_{1}^{\prime}\\}$.
We know that
${\mathop{\rm Vol}}_{G}(V\amalg M)={\lambda}^{\frac{n}{2}}{\mathop{\rm
Vol}}_{g}(V)+\mu^{\frac{n}{2}}{\mathop{\rm Vol}}_{h}(M).$
$\bullet$ For $\mu=1$ and ${\lambda}\to+\infty$, we have
${\lambda}_{2}(L_{G})={\lambda}^{-1}{\lambda}_{2}.$
$\displaystyle{\lambda}_{2}(L_{G}){{\mathop{\rm
Vol}}_{G}}^{\frac{2}{n}}(V\amalg M)$ $\displaystyle=$
$\displaystyle{\lambda}^{-1}{\lambda}_{2}\left(C+{\lambda}{{\mathop{\rm
Vol}}_{g}}^{\frac{2}{n}}(V)\right)$ $\displaystyle\to_{{\lambda}\to+\infty}$
$\displaystyle{\lambda}_{2}{{\mathop{\rm
Vol}}_{g}}^{\frac{2}{n}}(V)=\mu_{2}(g).$
$\bullet$ For ${\lambda}=1$ and $\mu\to+\infty$, in this case
${\lambda}_{2}(L_{G}))=\mu^{-1}{\lambda}_{1}^{\prime}.$
Hence
$\displaystyle{\lambda}_{2}(L_{G}){{\mathop{\rm
Vol}}_{G}}^{\frac{2}{n}}_{g}(V\amalg M)$ $\displaystyle=$
$\displaystyle\mu^{-1}{\lambda}_{1}^{\prime}\left(C+\mu{{\mathop{\rm
Vol}}_{h}}^{\frac{2}{n}}\right)$ $\displaystyle\to_{\mu\to+\infty}$
$\displaystyle{\lambda}_{1}^{\prime}{{\mathop{\rm
Vol}}_{h}}^{\frac{2}{n}}=\mu(M,h)\geq\sigma(M)-{\epsilon}.$
Finally we get that
$\sigma_{2}(V\amalg M)\geq\min(\mu_{2}(g),\sigma(M)).$
###### Remark 7.2.
1. (1)
It is known that if $\sigma(M)>0$ and $\sigma(N)>0$, then
$\sigma(M\amalg N)=\min(\sigma(M),\sigma(N)),$
where $M\amalg N$ is the disjoint union of $M$ and $N$. (see [4]).
2. (2)
Let $V$ with $\sigma(V)\leq 0$, then for $k\geq 2$
$\sigma_{2}(\underbrace{V\amalg\cdots\amalg V}_{k\hbox{ times }}\amalg M)\leq
0.$
Indeed, let any metric $g=g_{1}\amalg g_{2}\amalg\cdots\amalg g_{k}\amalg
g_{n}$ on $V\amalg\cdots\amalg V\amalg M$. Let $v_{i}$ be functions associated
to ${\lambda}_{1}(g_{i})$ which is non-negative by assumption. The functions
$\tilde{v_{i}}=0\amalg\cdots
0\amalg\underbrace{v_{i}}_{i^{th}\hbox{factor}}\amalg\hskip
2.84544pt0\cdots\amalg 0$ are linearly independent and satisfy
$L_{g}(\tilde{v_{i}})={\lambda}_{1}(g_{i})v_{i}$ and thus are eigenfunctions
of $L_{g}$. This implies that ${\lambda}_{k}(g)\leq 0$ and since $k\geq 2$,
${\lambda}_{2}(g)\leq 0$.
This remark explains the condition $|{\alpha}(M)|\leq 1$ in Corollary 1.7: it
is used to ensure that $M$ is obtained from a model manifold $V\amalg N$ with
a number of factors $V$ (where $V$ carries a scalar flat metric and
$\sigma(N)>0$) not larger than 1. We recall that the $\alpha$-genus is an
homomorphism from the spin cobordism ring $\Omega_{*}^{{\mathop{\rm Spin}}}$to
the real $K$-theory ring $KO_{*}(pt),$
$\alpha:\Omega_{*}^{{\mathop{\rm Spin}}}\rightarrow KO_{*}(pt).$
It is important that $\alpha$ is a ring homomorphism, i.e. for any connected
closed spin manifolds $M$ and $N$, $\alpha(M\amalg N)=\alpha(M)+\alpha(N)$ and
$\alpha(M\times N)=\alpha(M).\alpha(N).$
Noting that $KO_{n}(pt)$ vanishes if $n=3,5,6,7$ mod $8$, is isomorphic to
$\mathbb{Z}$ if $n=0,4$ mod $8$ and is isomorphic to
$\mathbb{Z}/{2\mathbb{Z}}$ if $n=1,2$ mod $8$. Recall also that $\alpha$ is
exactly the $\hat{A}$-genus in dimensions $0$-mod $8$ and equal to
$\frac{1}{2}\hat{A}$-genus in dimensions $4$ mod $8$.
In [9], Proposition 3.5 says that in dimensions $n=0,1,2,4$ mod $8$, there
exists a manifold $V$ such that $\alpha(V)=1$ and $V$ carries a metric $g$
such that ${\mathop{\rm Scal}}_{g}=0$.
$\bullet$ When $\alpha(M)=0$ then Thm A in [20] applies and
$\sigma(M)\geq\alpha_{n}$ where $\alpha_{n}$ depending only on $n$.
###### Theorem 7.3.
Let $M$ be a spin manifold, if $\alpha(M)=0$, this is equivalent to the
existence of a manifold $N$ cobordant to $M$ such that the scalar curvature of
$N$, ${\mathop{\rm Scal}}_{g}$ is positive.
Remember that a cobordism is a manifold $W$ with boundary whose boundary is
partitioned in two, $W=M\amalg(-N)$.
###### Theorem 7.4.
If $M$ is cobordant to $N$ and if $M$ is connected then $M$ is obtained from
$N$ by a finite number of surgeries of dimension $0\leq k\leq n-3.$
###### Proposition 7.5.
Let $M$ be a spin, simply connected, connected manifold of dimension $n\geq
5$, if $n=0,1,2,4$ mod $8$ and $|\alpha(M)|\leq 1$, then
$\sigma_{2}(M)\geq\alpha_{n},$
where $\alpha_{n}$ is a positive constant depending only on $n$.
Proof: Proposition 3.5 in [9] gives us that for each $n=0,1,2,4$ mod $8$,
$n\geq 1$, there is a manifold $V$ of dimension $n$ such that $V$ carries a
metric $g$ such that ${\mathop{\rm Scal}}_{g}=0$ and $\alpha(V)=1$.
$\bullet$ First case: If $\alpha(M)=0$, then $M$ is cobordant to a manifold
$N$ such that ${\mathop{\rm Scal}}_{g}$ on $N$ is positive. In this case we
can obtained $M$ from $N$ by a finite number of surgeries of dimension $k\leq
n-3$. Hence, by Corollary $\sigma(M)\geq c_{n}$ with $c_{n}$ is a positive
constant depending only on $n$.
$\bullet$ Second case: If $\alpha(M)=1$, then $\alpha(M\amalg(-V))=0$, so
there exists a manifold $N$ with ${\mathop{\rm Scal}}_{g}>0$ such that
$M\amalg(-V)$ is cobordant to $N$ which is equivalent to say that $M$ is
cobordant to $V\amalg N$. Consequently $M$ can be obtained from $V\amalg N$ by
a finite number of surgeries of dimension $k\leq n-3$. Applying the main
theorem of this paper, we get the desired result.
## References
* [1] R.A. Adams and J.J.F. Fournier, _Sobolev spaces_ , 2nd edition, Pure and Applied Mathematics, 140, Academic Press, Amsterdam, 2003.
* [2] B. Ammann, M. Dahl, and E. Humbert, _Low-Dimensional surgery and the Yamabe invariant_ , arXiv:1204.1197.
* [3] by same author, _Square-Integrability of solutions of the Yamabe equation_ , arXiv:1111.2780.
* [4] by same author, _Smooth Yamabe invariant and surgery_ , Prépublications de l’institut Elie Cartan, 58 pages, 2008.
* [5] by same author, _The conformal Yamabe constant of product manifolds_ , to appear in Proc. AMS.
* [6] B. Ammann and E. Humbert, _The second Yamabe invariant_ , J. Funct. Anal., 235 (2006), No 2, p. 377-412.
* [7] T. Aubin, _Some nonlinear problems in Riemannian geometry_ , Springer Monographs in Mathematics, Springer-Verlag, Berlin, 1998.
* [8] C. Bär and M. Dahl, _Small eigenvalues of the conformal Laplacian_ , Journal: Geometric and Functional Analysis GAFA - Volume 13, Issue 3, p. 483-508.
* [9] by same author, _Surgery and the spectrum of the Dirac operator_ , J. Reine Angew. Math., 552 (2002), 53–76.
* [10] S. El Sayed, _Second eigenvalue of the Yamabe operator and applications_ , arXiv:1204.1268.
* [11] D. Gilbarg and N. Trudinger, _Elliptic partial differential equations of second order_ , Grundlehren der mathematischen Wissenschaften, no. 224, Springer-Verlag, 1977.
* [12] M. Gromov and H. B. Lawson, _The classification of simply connected manifolds of positive scalar curvature_ , Ann. of Math. (2) 111 (1980), 423–434, MR0577131, Zbl 0463.53025.
* [13] E. Hebey, _Introduction à l’analyse non-linéaire sur les variétés_ , Diderot Editeur, Arts et Sciences, out of print, no longer sold, 1997.
* [14] T. Kato, _Perturbation theory of linear operators_ , Reprint of the 1980 edition, Springer Verlag, Berlin, Heidelberg (1995).
* [15] O. Kobayashi, _Scalar curvature of a metric with unit volume_ , Math. Ann. 279 (1987), 253–265, MR0919505, Zbl 0611.53037.
* [16] A. A. Kosinski, _Differential manifolds_ , Pure and Applied Mathematics, vol. 138, Academic Press Inc., Boston, MA, 1993.
* [17] J. M. Lee and T. H. Parker, _The Yamabe problem_ , Bull. Am. Math. Soc., New Ser. 17 (1987), 37–91.
* [18] J. Petean and G. Yun, _Surgery and the Yamabe invariant_ , Geom. Funct. Anal. 9 (1999), 1189–1199, MR1736933, Zbl 0976.53045.
* [19] R. Schoen and S.-T. Yau, _On the structure of manifolds with positive scalar curvature_ , Manuscripta Math. 28 (1979), 159–183, MR0535700. Zbl 0423.53032.
* [20] S. Stolz, _Simply connected manifolds of positive scalar curvature_ , Ann. Math., II. Ser. 36 (1992), 511–540.
|
arxiv-papers
| 2012-11-28T14:48:43 |
2024-09-04T02:49:38.603695
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Safaa El Sayed",
"submitter": "Safaa el Sayed",
"url": "https://arxiv.org/abs/1211.6617"
}
|
1211.6730
|
# Tau Decays in Pythia $8$
Philip Ilten School of Physics, University College Dublin
###### Abstract
As of version $8.150$ of Pythia, the isotropic decay model of $\tau$-leptons
has been replaced with sophisticated $\tau$-lepton decay machinery. The decays
and spin correlations for $\tau$-leptons in Pythia $8$ are described,
including the spin correlation algorithm, the available $\tau$-lepton
production processes, the $\tau$-lepton decay models, the user interface, and
the implementation.
###### keywords:
Monte Carlo , tau decays , polarization , hadronic currents
††journal: Nuclear Physics B Proceedings
## 1 Introduction
The role of $\tau$-leptons in Higgs boson measurements [1] and beyond the
Standard Model searches [2] is becoming increasingly important, due to the
enhanced coupling of the $\tau$-lepton in many of these physics models.
Consequently, it is necessary for current Monte Carlo generators to ensure
accurate modeling of $\tau$-lepton decays.
Prior to version $8.150$ of Pythia [3], $\tau$-lepton decays in Pythia were
performed using a leptonic or generic hadronic current matrix element without
including spin correlations, and more sophisticated $\tau$-lepton modeling was
only possible through external packages such as Tauola [4]. Now, in Pythia
version $8.150$ and above, fully modeled hadronic currents with spin
correlations are available, based on prior $\tau$-lepton modeling work in
Tauola and Herwig++ [5]. Currently, all known $\tau$-lepton decays with a
branching fraction greater than $0.04\%$ are modeled.
In this review, the spin correlation algorithm used in Pythia for
$\tau$-leptons is described, as well as the available $\tau$-lepton production
mechanisms, summarized in Table 1, and $\tau$-lepton decays, given in Table 2.
An introduction to the user interface, as well as the technical implementation
is given.
## 2 Correlations
The spin correlation algorithm used in Pythia for $\tau$-lepton decays is
based on the algorithm proposed by Collins [6] and Knowles [7], and expanded
by Richardson [8]. The algorithm separates spin correlations from the hard
process, parton shower, and hadronization phases of the Monte Carlo generator,
while maintaining full correlations, and can be broken into the following
steps.
1. 1.
The $2\to n$ hard process is generated according to its matrix element
$\mathcal{M}$.
2. 2.
One of the outgoing particles is selected and its helicity density matrix is
calculated,
$\displaystyle\rho_{\lambda_{j}\lambda_{j}^{\prime}}^{j}={}$
$\displaystyle\rho_{\kappa_{1}\kappa_{1}^{\prime}}^{1}\rho_{\kappa_{2}\kappa_{2}^{\prime}}^{2}\mathcal{M}_{\kappa_{1}\kappa_{2};\lambda_{1}\ldots\lambda_{n}}\mathcal{M}_{\kappa_{1}^{\prime}\kappa_{2}^{\prime};\lambda_{1}^{\prime}\ldots\lambda_{n}^{\prime}}^{*}$
(1) $\displaystyle\times\prod_{k\neq
j}\mathcal{D}_{\lambda_{k}\lambda_{k}^{\prime}}^{k}$
and the trace is normalized. Here, $\rho^{1,2}$ are the helicity density
matrices of the incoming particles with helicity $\kappa_{1,2}$, $\mathcal{M}$
is the matrix element with outgoing particle helicities $\lambda_{k}$, and
$\mathcal{D}^{k}$ are the decay matrices of the outgoing particles,
initialized to the identity. If the particle is from a decay, the helicity
density matrix is the same as above but without $\rho^{2}$ and $\kappa_{2}$.
3. 3.
The selected particle is decayed using the weight,
$\mathcal{W}=\rho_{\lambda_{0}\lambda_{0}^{\prime}}\mathcal{M}_{\lambda_{0};\lambda_{1}\ldots\lambda_{n}}\mathcal{M}_{\lambda_{0}^{\prime};\lambda_{1}^{\prime}\ldots\lambda_{n}^{\prime}}^{*}\prod_{k=1,n}\mathcal{D}_{\lambda_{k}\lambda_{k}^{\prime}}^{k}$
(2)
where $\rho$ is the helicity density matrix of the decaying particle with
helicity $\lambda_{0}$ and the decay matrix element $\mathcal{M}$.
4. 4.
Steps 1 through 2 are performed until a decay is reached with no unstable
particles.
5. 5.
The decay matrix of the last decayed particle is calculated,
$\mathcal{D}_{\lambda_{0}\lambda_{0}^{\prime}}=\mathcal{M}_{\lambda_{0};\lambda_{1}\ldots\lambda_{n}}\mathcal{M}_{\lambda_{0}^{\prime};\lambda_{1}^{\prime}\ldots\lambda_{n}^{\prime}}^{*}\prod_{k=1,n}\mathcal{D}_{\lambda_{k}\lambda_{k}^{\prime}}^{k}$
(3)
and the trace is normalized.
6. 6.
An undecayed particle from the decay above is randomly selected and steps 1
through 3 are repeated.
7. 7.
Step 6 is repeated until all unstable particles are decayed.
## 3 Production
Spin correlated $\tau$-lepton decays can be generated in Pythia from a variety
of hard processes, summarized in Table 1. The spin of the $\tau$-lepton can
also be provided from an externally generated hard process, or explicitly set
for all $\tau$-leptons or $\tau$-leptons from a specified parent.
For electroweak processes, spin correlations are fully handled if the incoming
fermions producing the $\gamma$, $Z$, or $W$ are known. If the $W$ is not
produced from fermions or the fermions are unavailable, the $\tau$-lepton is
given a polarization of $\mathcal{P}=-1$, e.g. $H\to W\to\tau{\nu_{\tau}}$.
For $\gamma$ or $Z$ hard processes not produced from fermions or with
unavailable fermion information, the $\gamma$ or $Z$ is assumed to be
unpolarized, e.g. $H\to ZZ\to 4\tau$.
Figure 1: Distribution of the fractional energy of the pion from a
$\tau^{-}\rightarrow\nu_{\tau}\pi^{-}$ decay in the rest frame of the
producing boson. The $\tau$-leptons are produced from $Z$ (dashed black), $H$
(dotted red), $W$ (dash-dotted green), and $H^{\pm}$ (dash-dot-dotted blue)
bosons.
Because the Higgs is spin zero, the production information is not necessary
for calculating $\tau$-lepton spin correlations. However, the type of Higgs
must be known. Currently, spin correlations of $\tau$-leptons produced from
the $\mathcal{CP}$-even ($H$, $h^{0}$, $H^{0}$), $\mathcal{CP}$-odd ($A^{0}$),
or charged ($H^{\pm}$) Higgses are calculated.
The $\tau$-lepton can also be produced from a variety of $B$ and $D$ meson
decays. For these decays, the constituent quarks of the meson are
approximated, and the spin correlations are calculated using an
$f\bar{f}^{\prime}\to W\to f\bar{f}^{\prime}$ matrix element. For any
$\tau$-lepton from an unknown process, the $\tau$-lepton is assumed to be
unpolarized.
The slope of the energy distribution for pions from
$\tau^{-}\rightarrow\nu_{\tau}\pi^{-}$ decays in the rest frame of the parent
bosons is proportional to the average $\tau$-lepton polarization. In Figure 1
this distribution, from Pythia simulation, is given for $\tau$-leptons
produced from electroweak and Higgs processes. Note the the opposite
polarization of $\tau$-leptons produced from a $W$ and $H^{\pm}$, the
unpolarized $\tau$-leptons produced from neutral Higgses, and the slight
polarization of $\tau$-leptons produced from $Z$ bosons at a $pp$ collider.
Type | Processes
---|---
electroweak | $f\bar{f}\to\gamma\to f\bar{f}$, $f\bar{f}\to Z\to f\bar{f}$,
$f\bar{f}\to\gamma^{*}/Z\to f\bar{f}$, $f\bar{f}^{\prime}\to W\to
f\bar{f}^{\prime}$,
$Z\to f\bar{f}$, $W\to f\bar{f}^{\prime}$
Higgs | $H\to f\bar{f}$, $h^{0}\to f\bar{f}$, $H^{0}\to f\bar{f}$
$A^{0}\to f\bar{f}$, $H^{\pm}\to f\bar{f}^{\prime}$
other | $B/D\to f\bar{f}^{\prime}+X$
Table 1: Production mechanisms in Pythia for which full spin correlations of
$\tau$-lepton decays are automatically calculated.
## 4 Decays
The matrix element for the decay of the $\tau$-lepton can be written as,
$\mathcal{M}=\frac{g_{w}^{2}}{8m_{W}^{2}}L_{\mu}J^{\mu}$ (4)
where $g_{W}$ is the $\mathit{SU(2)}$ coupling, $m_{W}$ the $W$ mass,
$L_{\mu}$ the leptonic current of the $\tau$-lepton, and $J_{\mu}$ a leptonic
or hadronic current dependent upon the decay. The $\tau$-lepton current
$L_{\mu}$ is $\bar{u}_{\nu_{\tau}}\gamma_{\mu}(1-\gamma^{5})u_{\tau}$ where
$u_{\tau}$ is the $\tau$-lepton spinor, dependent upon momentum and helicity,
and $\bar{u}_{\nu_{\tau}}$ is the $\tau$-lepton neutrino spinor. A full list
of the available $J_{\mu}$ currents modeled in Pythia is given in Table 2.
For the two-body decays of the $\tau$-lepton, $J_{\mu}$ is $fq$, where $f$ is
a constant and $q$ is the momentum of the hadron. For the three-body leptonic
decays, $\tau^{-}\rightarrow\nu_{\tau}e^{-}{\bar{\nu}_{e}}$ and
$\tau^{-}\rightarrow\nu_{\tau}\mu^{-}{\bar{\nu}_{\mu}}$, $J_{\mu}$ is of the
same form as $L_{\mu}$. Two hadronic three-body decay models are available, a
decay via a vector resonance and a decay via a vector and scalar resonance.
Four-body $\tau$-lepton decays in Pythia are implemented in four different
models. The primary four-body decays,
$\tau^{-}\rightarrow\nu_{\tau}\pi^{0}\pi^{0}\pi^{-}$ and
$\tau^{-}\rightarrow\nu_{\tau}\pi^{-}\pi^{-}\pi^{+}$, are modeled using the
CLEO fit. These decays can also be performed using a generic hadronic four-
body model from Decker, et al. The four-body decays with kaons in the final
state are calculated using a model from Finkemeier and Mirkes.
The five-body decays of the $\tau$-lepton to pions are produced with the
Novosibirsk model, a phenomenological fit of four pion production from
electron-positron annihilation. The six-body decays of the $\tau$-lepton are
handled through a model proposed by Kühn and Wa̧s.
A comparison of the hadronic invariant mass distributions, generated with
Pythia, between the dominant three, four, five, and six-body decays of the
$\tau$-lepton is given in Figure 2. The
$\tau^{-}\rightarrow\nu_{\tau}\pi^{0}\pi^{0}\pi^{-}$ distribution was
generated using the default CLEO model. The expected upwards shift for higher
multiplicity decays can be seen, as well the distinct difference in shapes due
to the differing propagators of the models.
Figure 2: Distribution of the invariant mass of the hadronic current for the dominant three-body $\pi^{0}\pi^{-}$ (dashed black), four-body $\pi^{0}\pi^{0}\pi^{-}$ (dotted red), five-body $\pi^{0}\pi^{-}\pi^{-}\pi^{+}$ (dash-dotted green), and six-body $\pi^{-}\pi^{-}\pi^{-}\pi^{+}\pi^{+}$ (dash-dot-dotted blue) decays of the $\tau$-lepton. Multiplicity | Model | $\mathcal{M}$ | Decay Products
---|---|---|---
$2$-body | single hadron | | 1521 | $\pi^{-}$, $K^{-}$
$3$-body | leptonic | | 1531 | $e^{-}{\bar{\nu}_{e}}$, $\mu^{-}{\bar{\nu}_{\mu}}$
Kühn and Santamaria | [9] | 1532 | $\pi^{0}\pi^{-}$, $K^{0}K^{-}$, $\eta K^{-}$
Finkemeier and Mirkes | [10] | 1533 | $\pi^{-}\bar{K}^{0}$, $\pi^{0}K^{-}$
$4$-body | CLEO | [11] | 1541 | $\pi^{0}\pi^{0}\pi^{-}$, $\pi^{-}\pi^{-}\pi^{+}$
Finkemeier and Mirkes | [12] | 1542 | $K^{-}\pi^{-}K^{+}$, $K^{0}\pi^{-}\bar{K}^{0}$, $K_{S}^{0}\pi^{-}K_{S}^{0}$, $K_{L}^{0}\pi^{-}K_{L}^{0}$, $K_{S}^{0}\pi^{-}K_{L}^{0}$,
$K^{-}\pi^{0}K^{0}$, $\pi^{0}\pi^{0}K^{-}$, $K^{-}\pi^{-}\pi^{+}$,
$\pi^{-}\bar{K}^{0}\pi^{0}$
Decker, et al. | [13] | 1543 | $\pi^{0}\pi^{0}\pi^{+}$, $\pi^{-}\pi^{-}\pi^{+}$, $K^{-}\pi^{-}K^{+}$, $K^{0}\pi^{-}\bar{K}^{0}$, $K^{-}\pi^{0}K^{0}$,
$\pi^{0}\pi^{0}K^{-}$, $K^{-}\pi^{-}\pi^{+}$, $\pi^{-}\bar{K}^{0}\pi^{0}$,
$\pi^{-}\pi^{0}\eta$
Jadach, et al. | [14] | 1544 | $\gamma\pi^{0}\pi^{-}$
$5$-body | Novosibirsk | [15] | 1551 | $\pi^{0}\pi^{-}\pi^{-}\pi^{+}$, $\pi^{0}\pi^{0}\pi^{0}\pi^{-}$
$6$-body | Kühn and Wa̧s | [16] | 1561 | $\pi^{0}\pi^{0}\pi^{-}\pi^{-}\pi^{+}$, $\pi^{0}\pi^{0}\pi^{0}\pi^{0}\pi^{-}$, $\pi^{-}\pi^{-}\pi^{-}\pi^{+}\pi^{+}$
Table 2: Summary of available $\tau$-lepton decay models in Pythia $8$ sorted
by multiplicity. For each model the internal Pythia matrix element mode
identifier ($\mathcal{M}$) is given, as well as the decays available through
the model and the reference of the model. The implicit ${\nu_{\tau}}$ is
omitted for brevity.
## 5 Interface
Documentation for $\tau$-lepton decays in Pythia can be found under the Tau
Decays subsection of Particle Decays in the Pythia HTML manual. The
$\tau$-lepton decay mechanism is configured by setting the mode
sophisticatedTau of ParticleDecays. This mode default is 1, where the decays
are performed using the sophisticated $\tau$-lepton decay machinery. When
sophisticated decays are not necessary, or a comparison with the isotropic
Pythia model is needed, the mode can be set to 0.
If the mode is set to 1, and a LHEF file is read, the spin of the
$\tau$-lepton will be set to the SPINUP information. However, if no SPINUP
information is provided, and the $\tau$-lepton is from a known production
process, Pythia will automatically calculate the polarization. If the
$\tau$-lepton is from an unknown process the sophisticated decay will be
performed assuming the $\tau$-lepton is unpolarized.
It is possible to force the polarization of the $\tau$-leptons and override
either the Pythia calculated polarization or the polarization from a LHEF
file. If the mode is set as 2, all $\tau$-leptons produced from a parent with
an identification code specified by tauMother will be given a polarization set
by the parameter tauPolarization. If mode 3 is selected, all $\tau$-leptons
will be produced with the polarization set by tauPolarization.
The models used to decay the $\tau$-leptons for each channel can be changed by
switching the matrix element mode of the channel. The syntax takes the form
15:channel:meMode = mode where 15 is the $\tau$-lepton particle identification
code, channel specifies the decay channel number as listed under Particle
Data, and mode is the new matrix element mode. For example, the default CLEO
model used for the $\tau^{-}\rightarrow\nu_{\tau}\pi^{0}\pi^{0}\pi^{-}$ decay
can switched to the Decker, et al. model using 15:9:meMode = 1543. However,
caution is advised in switching models, as it is possible to set a channel to
use an incompatible matrix element, causing unpredicted behavior.
## 6 Implementation
The implementation of the $\tau$-lepton decay machinery in Pythia is intended
to be complete and fast, yet easily extensible for new decay models or
production processes. A helicity framework is available in the Pythia source
files HelicityBasics where methods for calculating the helicity dependent
wavefunctions for particles, both fermions and bosons, is provided, as well as
the corresponding $\gamma$-matrices.
Using this framework, the helicity matrix element for the decay of a
$\tau$-lepton into a pion,
$\mathcal{M}=\bar{u}_{\nu_{\tau}}\gamma_{\mu}(1-\gamma^{5})u_{\tau}f_{\pi}q_{\pi}^{\mu}$
(5)
can be written in pseudo-code as,
$\displaystyle\mathcal{M}=\sum_{\mu}$
$\displaystyle{\nu_{\tau}}\mathtt{.waveBar(}\lambda_{\nu_{\tau}}\mathtt{)*GammaMatrix(}\mu\mathtt{)}$
(6)
$\displaystyle\mathtt{*(1-GammaMatrix(5))*}\tau\mathtt{.wave(}\lambda_{\tau}\mathtt{)*}f_{\pi}$
$\displaystyle\mathtt{*GammaMatrix(4)(}\mu\mathtt{,}\mu\mathtt{)*Wave4(}q_{\pi}\mathtt{)(}\mu\mathtt{)}$
where $\gamma^{4}$ provides the $+---$ metric.
Both the production matrix elements and $\tau$-lepton decay matrix elements
are written using this framework and are provided in the
HelicityMatrixElements files of the Pythia source. For the $\tau$-lepton decay
matrix elements, it is only necessary to implement the $J_{\mu}$ current. The
spin correlation and decays of the $\tau$-leptons are implemented in the
TauDecays files of the Pythia source.
## 7 Conclusion
Fully modeled $\tau$-lepton decays with spin correlations are now available in
Pythia, using fully extensible code. Further development of $\tau$-lepton
production mechanisms and decay models is underway, and feedback on requested
features is welcome.
## 8 Acknowledgments
The author would like to acknowledge Torbjörn Sjöstrand and the Pythia team
for all of their invaluable help. The funding for this project was provided by
Lund University and MCNet through a Marie Curie grant, contract MRTN-
CT-2006-035606.
## References
* [1] ATLAS Collab., Tech. Rep. ATLAS-CONF-2012-160.
* [2] CMS Collab., Tech. Rep. CMS-PAS-HIG-12-050.
* [3] T. Sjöstrand, et al., Comput.Phys.Commun. 178 (2008) 852–867. arXiv:0710.3820.
* [4] P. Golonka, et al., Comput.Phys.Commun. 174 (2006) 818–835. arXiv:hep-ph/0312240.
* [5] D. Grellscheid, P. Richardson. arXiv:0710.1951.
* [6] J. C. Collins, Nucl.Phys. B304 (1988) 794.
* [7] I. Knowles, Comput.Phys.Commun. 58 (1990) 271–284.
* [8] P. Richardson, JHEP 0111 (2001) 029. arXiv:hep-ph/0110108.
* [9] J. H. Kühn, A. Santamaria, Z.Phys. C48 (1990) 445–452.
* [10] M. Finkemeier, E. Mirkes, Z.Phys. C72 (1996) 619–626. arXiv:hep-ph/9601275.
* [11] CLEO Collab., Phys.Rev. D61 (2000) 012002. arXiv:hep-ex/9902022.
* [12] M. Finkemeier, E. Mirkes, Z.Phys. C69 (1996) 243–252. arXiv:hep-ph/9503474.
* [13] R. Decker, et al., Z.Phys. C58 (1993) 445–452.
* [14] S. Jadach, et al., Comput.Phys.Commun. 76 (1993) 361–380.
* [15] A. Bondar, et al, Comput.Phys.Commun. 146 (2002) 139–153. arXiv:hep-ph/0201149.
* [16] J. H. Kühn, Z. Wa̧s, Acta Phys.Polon. B39 (2008) 147–158. arXiv:hep-ph/0602162.
|
arxiv-papers
| 2012-11-28T20:27:49 |
2024-09-04T02:49:38.618469
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Philip Ilten",
"submitter": "Philip Ilten",
"url": "https://arxiv.org/abs/1211.6730"
}
|
1211.6759
|
EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN)
CERN-LHCb-DP-2012-003
Performance of the LHCb RICH detector at the LHC
The LHCb RICH group111Authors are listed on the following pages.
The LHCb experiment has been taking data at the Large Hadron Collider (LHC) at
CERN since the end of 2009. One of its key detector components is the Ring-
Imaging Cherenkov (RICH) system. This provides charged particle identification
over a wide momentum range, from 2–100 GeV$/c$. The operation and control,
software, and online monitoring of the RICH system are described. The particle
identification performance is presented, as measured using data from the LHC.
Excellent separation of hadronic particle types $(\pi,$ K, p) is achieved.
(To be submitted to EPJC)
LHCb RICH collaboration
M. Adinolfi4, G. Aglieri Rinella3, E. Albrecht3, T. Bellunato2, S. Benson7, T.
Blake3,9, C. Blanks9, S. Brisbane10, N.H. Brook4, M. Calvi2,b, B. Cameron9, R.
Cardinale1,a, L. Carson9, A. Contu10,e, M. Coombes4, C. D’Ambrosio3, S.
Easo6,3, U. Egede9, S. Eisenhardt7, E. Fanchini2,b, C. Fitzpatrick7, F.
Fontanelli1,a, R. Forty3, C. Frei3, P. Gandini10, R. Gao10, J. Garra Tico5, A.
Giachero2, V. Gibson5, C. Gotti2, S. Gregson5, T. Gys3, S.C. Haines5, T.
Hampson4, N. Harnew10, D. Hill10, P. Hunt10, M. John10, C.R. Jones5, D.
Johnson10, N. Kanaya3, S. Katvars5, U. Kerzel5, Y.M. Kim7, S. Koblitz3, M.
Kucharczyk2,b, D. Lambert7, R.W. Lambert7,f, A. Main7, M. Maino2, S. Malde10,
N. Mangiafave5, C. Matteuzzi2, G. Mini’1, A. Mollen3, J. Morant3, R.
Mountain11, J.V. Morris6, F. Muheim7, R. Muresan10,d, J. Nardulli6, P. Owen9,
A. Papanestis6, M. Patel9, G.N. Patrick6, D.L. Perego2,b, G. Pessina2, A.
Petrolini1,a, D. Piedigrossi3, R. Plackett9, S. Playfer7, A. Powell10, J.H.
Rademacker4, S. Ricciardi6, G.J. Rogers5, P. Sail8, M. Sannino1,a, T.
Savidge9, I. Sepp9, S. Sigurdsson5, F.J.P. Soler8, A. Solomin4, F. Soomro9, A.
Sparkes7, P. Spradlin8, B. Storaci3,c, C. Thomas10, S. Topp-Joergensen10, N.
Torr10, O. Ullaland3,b, K. Vervink3, D. Voong4, D. Websdale9, G. Wilkinson10,
S.A. Wotton5, K. Wyllie3, F. Xing10, R. Young7.
1Sezione INFN di Genova, Genova, Italy
2Sezione INFN di Milano Bicocca, Milano, Italy
3European Organization for Nuclear Research (CERN), Geneva, Switzerland
4H.H. Wills Physics Laboratory, University of Bristol, Bristol, United Kingdom
5Cavendish Laboratory, University of Cambridge, Cambridge, United Kingdom
6STFC Rutherford Appleton Laboratory, Didcot, United Kingdom
7School of Physics and Astronomy, University of Edinburgh, Edinburgh, United
Kingdom
8School of Physics and Astronomy, University of Glasgow, Glasgow, United
Kingdom
9Imperial College London, London, United Kingdom
10Department of Physics, University of Oxford, Oxford, United Kingdom
11Syracuse University, Syracuse, NY, United states of America
aUniversità di Genova, Genova, Italy
bUniversità di Milano Bicocca, Milano, Italy
cNow at Physik-Institut, Universität Zürich, Zürich, Switzerland
dNow at Horia Hulubei National Institute of Physics and Nuclear Engineering,
Bucharest-Magurele, Romania
eNow at Sezione INFN di Cagliari, Cagliari, Italy
fNow at Nikhef National Institute for Subatomic Physics and VU University
Amsterdam, Amsterdam, The Netherlands
## 1 Introduction
LHCb [1] is one of the four major experiments at the LHC, and is dedicated to
the study of CP violation and the rare decay of heavy flavours. It is a
forward spectrometer designed to accept forward-going $b$-and $c$-hadrons
produced in proton-proton collisions. The layout of the spectrometer is shown
in Fig. 1. The subdetectors of LHCb are described in detail in Ref.[1].
The RICH system of the LHCb experiment provides charged particle
identification over a wide momentum range, from 2 to 100 GeV$/c$. It consists
of two RICH detectors that cover between them the angular acceptance of the
experiment, 15–300 mrad with respect to the beam axis. The LHC accelerator
started at the end of 2009 and ran at a centre-of-mass energy of 7 TeV until
the end of 2011, followed by 8 TeV in 2012. The luminosity rapidly increased
and at the end of 2010 reached the nominal operating value for the LHCb
experiment, $2\times 10^{32}$ cm-2 s-1. This paper describes the performance
of the RICH system and also its alignment and calibration using data. Many
LHCb results have already fully exploited the RICH capabilities [2].
The paper is structured as follows: the requirements for particle
identification are discussed in Sect. 1, and a brief description of the RICH
detectors is given in Sect. 2. The alignment and calibration of the detectors
are described in Sect. 3. Section 4 gives an overview of the software used in
the RICH reconstruction, particle identification and detector performance,
followed in Sect. 5 by the conclusions.
### 1.1 Requirements for particle identification
The primary role of the RICH system is the identification of charged hadrons
($\pi$, K, p).
Figure 1: Side view of the LHCb spectrometer, with the two RICH detectors
indicated
One of the major requirements for charged hadron identification in a flavour-
physics experiment is for the reduction of combinatorial background. Many of
the interesting decay modes of $b$\- and $c$\- flavoured hadrons involve
hadronic multibody final states. At hadron colliders like the LHC, the most
abundant produced charged particle is the pion. The heavy flavour decays of
interest typically contain a number of kaons, pions and protons. It is
therefore important in reconstructing the invariant mass of the decaying
particle to be able to select the charged hadrons of interest in order to
reduce the combinatorial background.
The second major use of the particle identification information is to
distinguish final states of otherwise identical topology. An example is the
two-body hadronic decays, B$\rightarrow h^{+}h^{-}$, where $h$ indicates a
charged hadron [3]. In this case there are many contributions, as illustrated
in Fig. 2, including B${}^{0}\rightarrow\pi^{+}\pi^{-}$,
B${}^{0}_{s}\rightarrow$ K+K-, and other decay modes of the B0, B${}^{0}_{s}$
and $\Lambda_{b}$. A signal extracted using only kinematic and vertex-related
cuts is a sum over all of the decay modes of this type (Fig. 2 left), each of
which will generally have a different CP asymmetry. For a precise study of CP-
violating effects, it is crucial to separate the various components. This is
achieved by exploiting the high efficiency of the RICH particle identification
(Fig. 2 right).
Figure 2: Invariant mass distribution for B$\rightarrow h^{+}h^{-}$ decays
[3] in the LHCb data before the use of the RICH information (left), and after
applying RICH particle identification (right). The signal under study is the
decay B${}^{0}\rightarrow\pi^{+}\pi^{-}$, represented by the turquoise dotted
line. The contributions from different $b$-hadron decay modes
(B${}^{0}\rightarrow$ K$\pi$ red dashed-dotted line,
B${}^{0}\rightarrow$3-body orange dashed-dashed line, B${}_{s}\rightarrow$ KK
yellow line, B${}_{s}\rightarrow$ K$\pi$ brown line, $\Lambda_{b}\rightarrow$
pK purple line, $\Lambda_{b}\rightarrow$ p$\pi$ green line), are eliminated by
positive identification of pions, kaons and protons and only the signal and
two background contributions remain visible in the plot on the right. The grey
solid line is the combinatorial background
Another application of charged hadron identification is for an efficient
flavour tagging [4]. When studying CP asymmetries or particle-antiparticle
oscillations, knowledge of the production state of the heavy-flavoured
particles is required. This can be achieved by tagging the
particle/antiparticle state of the accompanying hadron. Heavy-flavoured
particles are predominantly produced in pairs. One of the most powerful means
of tagging the production state is by identifying charged kaons produced in
the $b\rightarrow c\rightarrow s$ cascade decay of the associated particle.
Such tagged kaons (as well as kaons from the $b$ fragmentation when a
B${}^{0}_{s}$ is created), have a soft momentum distribution, with a mean of
about 10 GeV/$c$. Particle identification down to a few GeV/$c$ can therefore
significantly increase the tagging power of the experiment.
The typical momentum of the decay products in two-body $b$ decays is about 50
GeV/$c$. The requirement of maintaining a high efficiency for the
reconstruction of these decays leads to the need for particle identification
up to at least 100 GeV/$c$. The lower momentum limit of about 2 GeV/$c$
follows from the need to identify decay products from high multiplicity B
decays and also from the fact that particles below this momentum will not pass
through the dipole magnetic field (4 Tm) of the LHCb spectrometer.
A further example of the requirements for particle identification in LHCb is
its use in the trigger. LHCb has a high performance trigger system [5], that
reduces the event rate from the 40 MHz bunch crossing frequency down to about
2 kHz that can be written to storage. This is achieved in two steps. The first
trigger level is implemented in hardware and is based on high transverse
energy deposits in the calorimeter and high transverse momentum detected by
the muon system, to reduce the rate to 1 MHz. All detectors are then read out
into a CPU farm where a high level trigger (HLT, see Fig. 3 ) decision is made
fully in software. The RICH reconstruction is fast enough to contribute to
this trigger. An example is the online selection of the $\phi$ particle, which
is present in many of the decay modes of interest.
## 2 The RICH detectors
### 2.1 Detector description
A description of the LHCb RICH detectors is given in Ref.[1]. Only the major
features are summarized here. In the forward region, covered by the LHCb
spectrometer, there is a strong correlation between momentum and polar angle,
with the high-momentum particles produced predominantly at low polar angles.
As a result, the RICH system has two detectors. RICH 1 covers the low and
intermediate momentum region 2 - 40 GeV$/c$ over the full spectrometer angular
acceptance of 25–300 mrad. The acceptance is limited at low angle by the size
of the beampipe upstream of the magnet. RICH 2 covers the high-momentum region
15–100 GeV$/c$, over the angular range 15–120 mrad.
To limit its overall volume, RICH 1 is placed as close as possible to the
interaction region. It is located immediately downstream of the silicon-
microstrip vertex locator (VELO), as shown in Fig. 1. To minimize the material
budget there is no separate entrance window, and the RICH 1 gas enclosure is
sealed directly to the exit window of the VELO vacuum tank. The downstream
exit window is constructed from a low-mass carbon-fibre/foam sandwich. RICH 2
is placed downstream of the magnet, since the high momentum tracks it measures
are less affected by the magnetic field. In this way it can be placed after
the downstream tracking system in order to reduce material for the measurement
of the charged tracks. The entrance and exit windows are again a foam sandwich
construction and skinned with carbon-fibre and aluminium, respectively.
Both RICH detectors have a similar optical system, with a tilted spherical
focusing primary mirror, and a secondary flat mirror to limit the length of
the detectors along the beam direction. Each optical system is divided into
two halves on either side of the beam pipe, with RICH 1 being divided
vertically and RICH 2 horizontally. The vertical division of RICH 1 was
necessitated by the requirements of magnetic shielding for the photon
detectors, due to their close proximity to the magnet. The spherical mirrors
of RICH 1 (4 segments) are constructed in four quadrants, with carbon-fibre
structure, while those of RICH 2 (56 segments), and all flat mirrors (16 and
40 segments in RICH 1 and RICH 2 respectively), are tiled from smaller mirror
elements, employing a thin glass substrate. A reflectivity of about 90% was
achieved for the mirrors, averaged over the wavelength region of interest,
200–600 nm. The total material budget for RICH 1 is only about 8% $X_{0}$
within the experimental acceptance, whilst that of RICH 2 is about 15%
$X_{0}$.
Fluorocarbon gases at room temperature and pressure are used as Cherenkov
radiators; C4F10 in RICH 1 and CF4 in RICH 2 were chosen for their low
dispersion. The refractive index is respectively 1.0014 and 1.0005 at 0o C,
101.325 kPa and 400 nm. About 5% CO2 has been added to the CF4 in order to
quench scintillation in this gas [6].
The momentum threshold for kaons to produce Cherenkov light in C4F10 is 9.3
GeV$/c$. Particles below this momentum would only be identified as kaons
rather than pions in veto mode, i.e. by the lack of Cherenkov light associated
to the particle. To maintain positive identification at low momentum and in
order to separate kaons from protons, a second radiator is included in RICH 1:
a 50 mm thick wall made of 16 tiles of silica aerogel [7] at the entrance to
RICH 1. The refractive index is $n=1.03$ and the light scattering length is
around 50 mm at 400 nm in pure N2. The aerogel is placed in the C4F10 gas
volume and a thin glass filter is used on the downstream face to limit the
chromatic dispersion.
The Cherenkov photons emitted by charged particles passing through the RICH
radiators are focused into ring images on the photon detector planes, situated
outside of the spectrometer acceptance. A novel hybrid photon detector (HPD)
was developed in collaboration with industry specifically for application in
the LHCb RICH system [8]. The HPDs employ vacuum tubes with a 75 mm active
diameter, with a quartz window and multialkali photocathode. The
photoelectrons are focused onto a silicon pixel array, using an accelerating
voltage of -16 kV. The pixel array is arranged in 32 columns and 32 rows,
giving a total of 1024 pixels per tube. The pixel size is 2.5$\times$2.5 mm2
at the level of the photocathode. A total of 484 HPDs are close-packed to
cover the four photodetector planes. Two planes are employed in each RICH,
with 196 tubes used in RICH 1 and 288 in RICH 2. The photodetector planes are
separated from the radiator gas volumes by quartz windows, and the
photodetector volumes are maintained in an atmosphere of CO2. The front-end
electronics chip is encapsulated within the HPD vacuum tube, and bump-bonded
to the silicon pixel sensor, which results in extremely low noise (typically
150 e- RMS per pixel for a signal of 5000 e- [9]). The tubes also feature high
detection efficiency, with an active area fraction of about 82%. The quantum
efficiency is about 30% at 270 nm.
### 2.2 Detector operation
The operation of the RICH detectors is fully automated and is controlled by
the Experiment Control System (ECS) [10]. The RICH ECS has been built using
components from the Joint Controls Project framework [11], developed by CERN
and the four main LHC experiments. The ECS uses predefined sequences for
normal detector operation, allowing non-experts to operate the detectors.
Automated actions protect the equipment when monitored parameters fall outside
the range of accepted values. Sensors are used as input to the LHCb Detector
Safety System which put the detectors in a safe state in case of a major
malfunction of the control system.
The RICH ECS also collects environmental information that is required by the
RICH reconstruction software. There are systems to monitor movements of the
RICH mirrors, monitor the quality of the gas radiators, and log the
temperature and pressure of the radiators in order to correct the refractive
index. Changes in temperature and pressure, which necessitate the re-
calculation of the refractive index of the gas radiators, are automatically
propagated to the Conditions Database [12].
The RICH detectors and the data recorded are monitored at several stages of
the data-acquisition and reconstruction chain to identify any potential
problems as early as possible. Figure 3 illustrates the online data-flow,
highlighting the dedicated monitoring and calibration farms for analyses using
fully reconstructed events.
Figure 3: RICH data-flow through the online system. Events selected by the L0
trigger are sent to the High Level Trigger (HLT) farm and, if they pass this
trigger requirements, are sent to storage. A fraction of these events
(typically 10%) is also sent to the monitoring farm. Online monitoring
algorithms examine the data for irregularities and send messages to the slow-
control (ECS) that can trigger automatic actions. Special triggers are sent
directly to the calibration farm bypassing the High Level Trigger
Low-level processes monitor the data integrity during data recording by cross-
checking the various data-banks and reporting any irregularities. Higher-level
monitoring algorithms use a neural network [22] to identify Cherenkov rings
using information from the RICH detectors only. On rare occasions, an
individual HPD may lose synchronisation with the rest of the detector and
transmit spurious data for each event. It is found that the performance of the
particle identification is affected only marginally by a few units of
malfunctioning HPDs222The number of malfuctioning HPDs is considered
acceptable if it is less than 6 peripheral tubes, or one central tube., and it
is usually more effective to continue recording data and reset those affected
front-end components during the next run initialization. In order to prevent
inefficiencies during data-taking due to anomalously busy events, the online
monitoring task automatically detects these cases and the read-out electronics
discards all data prior to transmission.
Special calibration triggers can be sent to the photodetectors during normal
data-taking to activate a pre-defined test pattern of hits. This provides a
continuous test of the response of all photodetectors, especially in low-
occupancy regions. As these calibration triggers are sent during gaps in the
bunch-train structure of the LHC beam, these events contain no activity
related to proton-proton interactions. These “empty” events can also be used
to evaluate noise that would be present in the detector during data-taking.
The online monitoring allows the full event reconstruction of a sizeable
fraction of the events being recorded to be processed online. This allows the
monitoring of the reconstructed Cherenkov angle, as well as the alignment of
the RICH detectors with respect to the tracking system.
## 3 Alignment and calibration
The tasks of spatial alignment of the RICH detectors and the calibration of
the refractive indices of the radiators are performed with data using high
momentum charged particles. In addition, the alignment of several mirror
segments and the purity of the gas radiators are also monitored using systems
that can provide information independently and during periods when there are
no collisions.
### 3.1 Time alignment
In order to maximise the photon collection efficiency of the RICH, the HPD
readout must be synchronised with the LHC bunch crossing to within a few
nanoseconds. This procedure is referred to as ”time alignment” in the
following. Individual HPDs vary in timing due to variations in drift time of
the electrons within the silicon sensor. HPD readout is triggered by a 25 ns
wide strobe pulse distributed by the LHCb network of optic fibres and detected
by the RICH Level-0 (L0) boards [13]. A RICH L0 board supervises the
triggering, timing and control of the HPDs, with two HPDs serviced by a single
board. HPDs that share a L0 board were chosen to have similar timing
characteristics in order to optimise the time alignment.
Three features on the timing profile are defined: the $turn$-$on$ point is the
delay between optimal readout efficiency and the strobe pulse at which 90% of
the peak photon collection efficiency is observed, the $turn$-off point is the
delay at which the profile drops below 90%, and the midpoint is the average
delay between these values.
The global time alignment of L0 boards has been performed in several steps
both prior to and during running at the LHC. The initial alignment was
performed in the absence of beam using a pulsed laser, resulting in a relative
alignment of the HPDs in each photodetector plane. The global time alignment
to the LHCb experiment is achieved with pp collisions using the LHCb first
level trigger. The distribution of the midpoints can be seen in Fig. 4,
showing that the HPDs have been time aligned to $\sim$ 1 ns.
Figure 4: Distribution of the midpoints in RICH 1 (left) and RICH 2 (right)
after time alignment with pp collisions. The RMS deviations of the HPDs are
approximately 1 ns
### 3.2 Magnetic distortions
Inside an HPD, photoelectrons travel up to 14 cm from the photocathode to the
silicon anode. This device is therefore sensitive to stray magnetic fields
from the LHCb spectrometer magnet. All HPDs in RICH 1 and RICH 2 experience
some residual fringe field: the magnetic shields surrounding the HPDs reduce
it from initial values of up to 60(15) mT in RICH 1(RICH 2), to a maximum
value of $\simeq$ 2.4 mT in RICH 1, and values ranging between 0.2 - 0.6 mT in
RICH 2. The resulting distortion of the images are mapped and corrected when
reconstructing Cherenkov angles.
A characterisation procedure has been developed to correct for magnetic
distortion effects and restore the optimal resolution. Different
implementations are used for RICH 1 and RICH 2 as the two detectors have
different geometries and experience different field configurations.
#### 3.2.1 RICH 1
The distortions of the HPD images are corrected using a dedicated calibration
system. The mapping system consists of two identical hardware arrangements,
one for each of the upper and lower HPD boxes [14]. Each system consists of an
array of green light-emitting diodes mounted on a carbon-fibre support that
spans the width of the HPD box. This “light bar” attaches at each end to a
pair of synchronised stepper-motors that facilitates the illumination of the
entire HPD array. The light bar comprises 19 LED units each with an array of
$5\times 28$ LEDs, 2.5 mm apart. A passive collimator unit is mounted flush to
the LED array such that light from each LED is channeled down an individual
collimator with 0.3 mm aperture.
Figure 5: Spatial residuals demonstrating the resolution with which the light
spots of the test pattern in RICH 1 are identified. The plot shows the
distance from the measured light spot centre to the nearest test point. The
dotted and solid lines are before and after the calibration respectively,
along the $x$ direction (left) and along $y$ (right) of the anode plane,
projected on the photocathode plane. The solid line is the Gaussian fit
The distortion is mapped by comparing the pattern of light spots in magnet-on
and magnet-off data. The direction of the magnetic field is predominantly
longitudinal with respect to the tube axis. The field effect causes a rotation
of the image about the central axis of the HPD, and a modest expansion of the
image. The residual transverse component of the field displaces the centre of
the photocathode image.
The result of the parametrisation is demonstrated in Fig. 5, showing the
residual positional uncertainty due to the magnetic distortion after the
correction procedure. The resolution after correction is $\sigma\simeq 0.2$
pixel (0.5 mm), significantly smaller than the irreducible uncertainty of
$\sigma\simeq 0.29\mathrm{\;pixel}$ (0.72 mm), originating from the finite HPD
pixel size.
#### 3.2.2 RICH 2
The magnetic distortion correction for RICH 2 uses a system based on the
projection of a light pattern onto the plane of HPDs using a commercial light
projector. The projected pattern is a suitable grid of light spots. The
algorithm to reconstruct the position of a light spot builds a cluster of hits
and the cluster centre is calculated. A resolution better than the pixel size
is achieved. Comparing the position of the light spots with and without the
magnetic field makes it possible to measure and parametrise the magnetic field
distortions [15, 16].
The distortion mainly consists of a small rotation (on average $\lesssim
0.1\mathrm{\;rad}$) of the test spots around the HPD axis. This rotation
varies from HPD to HPD, depending on the HPD position. No measurable variation
of the radial coordinate of the light spots was detected. The parameters
extracted using either orientation of the magnetic field also apply, with a
sign inversion, to the opposite magnetic field polarity. By applying the
correction procedure, the resolution on the position of the light spot
improves from $\sigma\simeq 0.33\mathrm{\;pixel}$ to $\sigma\simeq
0.19\mathrm{\;pixel}$ (0.47 mm) (see Fig. 6). For comparison the pixel size
resolution is $\sigma=0.29\mathrm{\;pixel}$ (0.72 mm).
Figure 6: Spatial residuals demonstrating the resolution with which the light
spots of the test pattern in RICH 2 are identified. The plot shows the
distance from the measured light spot centre to the nearest test point. The
dotted and solid lines are before and after the calibration respectively. Most
of the photodetectors of RICH 2 are in a region free from magnetic field
residual values (region around $x$=0 of the dotted histogram). Where these are
different from zero, the distorsions induced are visible in the two satellite
peaks of opposite sign (the magnetic field changes sign in the upper and lower
part of the photodetector matrix plane). The left plot is the measurement
along the $x$, the right plot along $y$ of the anode plane, projected on the
photocathode plane. The solid line is the Gaussian fit
### 3.3 Detector alignment
In order to reconstruct the Cherenkov angle associated with the individual
photons as accurately as possible, a number of components must be aligned with
an accuracy of 0.1 mrad with respect to the LHCb tracking system. The
alignment procedure calculates the misalignments of the various detector
components in a sequential process. First, the whole RICH detector is aligned
with the global LHCb coordinate system, followed by each detector half, each
mirror segment and finally each HPD. This includes aligning the silicon
sensors inside the HPDs to the whole RICH detector. One has to know the
position of the centre of each HPD photocathode on the anode. The silicon
sensors are aligned by mapping an image of the photocathode. This procedure
does not require the reconstruction of the Cherenkov angle. The relative HPD
alignment can also be corrected using data from the magnetic distortion
measurements. After these steps, the alignment procedure uses the
reconstructed Cherenkov angle of $\beta\approx 1$ particles to align the whole
RICH detector, the HPD panels, and eventually the 4 (56) spherical and 16 (40)
flat mirror segments in RICH 1 (RICH 2).
Any misalignment of the RICH detectors with respect to the tracking system is
observed as a shift of the track projection point on the photodetector plane
from the centre of the corresponding Cherenkov ring. This shift is observed by
analysing the Cherenkov angle, $\theta_{C}$, as a function of the azimuthal
Cherenkov angle $\phi$, defined as the angle of the pixel hit in the
coordinate system of the photodetector plane, with the projected track
coordinate at the origin. The angle $\theta_{C}$ is independent of the angle
$\phi$ for a well aligned detector, whilst a misaligned system would result in
a sinusoidal distribution as shown in Fig. 7.
In practice, distributions of $\Delta\theta_{C}$ against $\phi$ are plotted
where $\Delta\theta_{C}=\theta_{C}-\theta_{0}$ and $\theta_{0}$ is the
Cherenkov angle calculated from the momentum of the track and the refractive
index of the radiator. Any systematic shift away from the value $\theta_{0}$
is observable as a shift in $\Delta\theta_{C}$.
The $\Delta\theta_{C}$ distribution is then divided into slices in $\phi$. For
each slice, a one dimensional histogram of $\Delta\theta_{C}$ is fitted with a
Gaussian plus a second order polynomial background and the peak of the
distribution is extracted. The mean of each slice fit is then used to fit a
sinusoidal distribution given by
$\Delta\theta_{C}=\theta_{x}\cos(\phi)+\theta_{y}\sin(\phi).$
Figure 7: $\Delta\theta_{C}$ plotted as a function of the azimuthal angle
$\phi$ and fitted with $\theta_{x}\cos(\phi)+\theta_{y}\sin(\phi)$, for one
side of the RICH 2 detector. The left-hand plot is prior to alignment, and
shows a dependency of the angle $\theta_{C}$ on the angle $\phi$. The right-
hand plot is after the alignment correction, and $\Delta\theta_{C}$ is uniform
in $\phi$
The final fit is shown in Fig. 7; the extracted values of $\theta_{x}$ and
$\theta_{y}$ correspond to a misalignment on the HPD detector plane in the $x$
and $y$ direction respectively.
The alignment of the mirror segments has the extra complication that every
photon is reflected twice, and so the data must be separated into samples
which have unique spherical and flat mirror combinations. For this procedure,
only photons that can be uniquely associated to a given mirror pair are used.
Mirror segments are identified by considering photons to have been emitted at
both the start and end of the gas radiators. If the mirror segments reflecting
the photons are the same in both cases, the photon trajectory is considered
unambiguous and is used for the alignment of mirror segments.
The mirror arrangement in RICH 1 allows for alignment using a sequential
approach as described above, where the spherical mirrors are aligned first,
followed by the planar mirrors. This is possible because photons reaching a
particular planar mirror can only be reflected from a single spherical mirror
[17]. In RICH 2 the larger number of spherical/planar mirror combinations
makes the use of a sequential method impossible. The alignment of the RICH 2
mirror segments is performed by solving a set of simultaneous equations to
extract all the alignment parameters of all the mirrors. One iteration of this
method is required to obtain the final mirror alignment.
### 3.4 Refractive index calibration
The refractive index of the gas radiators depends on the ambient temperature
and pressure and the exact composition of the gas mixture. It can therefore
change in time, and this affects the performance of the particle
identification algorithms. The ultimate calibration of the refractive index is
performed using high momentum charged particle tracks in such a way that the
distribution of $\Delta\theta_{C}$ peaks at zero.
The calibration of the refractive index of the aerogel is performed using
tracks with momentum $p>10$ GeV/$c$ passing through each tile. It is found not
to change as a function of time.
### 3.5 Monitoring hardware
There are additional monitoring tasks, independent from the methods described
above.
The four spherical mirrors in RICH 1 and 20 of the mirror segments in RICH 2
are monitored for stability using laser beams and cameras. For each monitored
mirror there is an optic fibre with a lens to provide a focused beam, a beam
splitter, a mirror and a camera. The beam splitter creates two beams. The
reference beam is incident directly onto the camera. The second beam is
reflected to the camera via the monitored mirror. A comparison of the relative
position of these light spots tracks possible movement of the mirror.
The purity of the gas radiators is monitored by measuring the speed of sound
in the gas. A 50 kHz ultrasonic range finder is used. The gas to and from the
detector is monitored with a precision of about 1% for a binary gas mixture. A
gas chromatograph is periodically used for high precision measurements. Any
variation in time, after correction for temperature effects, is likely due to
changes in the composition of the gas.
After correcting for all the parameters monitored as a function of time as
described in this section, the detector behaviour is very stable, as shown in
Fig. 8.
Figure 8: The Cherenkov angular resolution (c.f. Sect. 4.2), after all
corrections have been applied, as a function of run number. a) for RICH 1 and
b) for RICH 2. The period of time covered on the x-axis corresponds to about 8
months of running
## 4 Performance
### 4.1 Data reconstruction
The LHCb software is based on the Gaudi Framework [18, 19] which provides a
flexible and configurable C++ Object Oriented framework. This flexibility
allows the same software to be used in a variety of different RICH
applications, ranging from the online monitoring, the utilization of the RICH
in the final stages of the higher level trigger, and providing the full
offline event reconstruction. This section describes the processing steps of
the RICH data.
#### 4.1.1 HPD data reconstruction
The first stage of the data processing chain is to decode the raw data, as
read out from the detector, to offline storage. This produces a list of the
HPD pixels that have been hit in each event. The next step is to apply various
data cleaning algorithms to the list of active pixels for each HPD. HPD data
are rejected if the HPD occupancy, which on average is $\sim$ 1%, exceeds a
tuneable maximum value of 20%, to exclude excessively large events.
Finally, the position of the photon hit is reconstructed on the HPD plane.
This procedure corrects for the alignment of the HPDs within the LHCb
detector, the electrostatic focusing parameters of the HPD tubes, and the
corrections for the magnetic field (Sect. 3.2).
#### 4.1.2 Cherenkov photon candidate reconstruction
The tracking system of LHCb provides detailed coordinate information on the
passage of charged particles through the LHCb spectrometer, and with this
information the trajectory of each particle through the three RICH radiator
volumes can be determined. This allows the computation of an assumed emission
point of the photon candidates for each track. As the exact emission point of
each photon is unknown (and can be anywhere along the particle trajectory
through the radiator), the mid-point of the trajectory in the radiator is
taken.
The candidate photons for each track are determined by combining the photon
emission point with the measured hit positions of the photons. Once the photon
candidates have been assigned, quantities such as the Cherenkov angle
$\theta_{C}$, can be computed. A full analytical solution of the RICH optics
is used, which reconstructs the trajectory of the photon through the RICH
optical system, taking into account the knowledge of the mirror and HPD
alignment [20].
### 4.2 Cherenkov angle resolution
The distribution of $\Delta\theta_{C}$, calculated for each photon with
respect to the measured track, is shown for the RICH 1 and RICH 2 gas
radiators in Fig. 9 after the alignment and calibration procedures have been
performed.
By fitting the distribution with a Gaussian plus a polynomial background, the
Cherenkov angle resolution is determined to be $1.618\pm 0.002$ mrad for C4F10
and $0.68\pm 0.02$ mrad for CF4. These values are in reasonable agreement with
the expectations from simulation [21] of $1.52\pm 0.02$ mrad and $0.68\pm
0.01$ mrad in RICH 1 and RICH 2, respectively.
Figure 9: Single photoelectron resolution for the RICH 1 (left) and RICH 2
(right) gases, as measured in data for high momentum charged particles. The
red line describes the background as determined from the fit using a
polynomial function together with the Gaussian for the signal Figure 10:
Single photoelectron resolution for the aerogel as measured in 2011 data with
the pp$\rightarrow$pp$\mu^{+}\mu^{-}$ events. The red line describes the
background as determined from the fit using a polynomial function together
with two Gaussians for the signal
The performance of the aerogel radiator has been studied with data collected
in 2010 and 2011. The data have been first used to calibrate the refractive
indices of individual tiles. Figure 10 shows the deviation $\Delta\theta_{C}$
in the four aerogel tiles located around the beampipe, which cover more than
90% of the acceptance. The $\Delta\theta_{C}$ distribution of the photons is
measured using good quality tracks with momentum above 10 GeV$/c$. The peak is
not symmetric, and the $\sigma$ from the FWHM gives an average value of about
5.6 mrad (the events used for this estimate are all pp collisions, not the
ones used in Fig. 10). This value is about a factor of 1.8 worse than the
simulation. This discrepancy is, at least partially, explained by the
absorption by the very porous aerogel structure of the C4F10 with which it is
in contact.
A new aerogel enclosure which isolates the aerogel from the C4F10 gas in RICH
1 is installed for the 2012 running.
### 4.3 Photoelectron yield
Figure 11: Distribution of $\Delta\theta_{C}$ for C4F10. This plot is produced
from kaons and pions from tagged D${}^{0}\rightarrow$ K${}^{-}\pi^{+}$ decays
in data selected with the criteria described in the text
The photoelectron yield Npe is measured for two categories of RICH event: one,
referred to as a _normal_ event, is representative of nominal RICH running
conditions during LHCb physics data collection; the other, referred to as an
_ideal_ event, is a special event type with very low photoelectron backgrounds
and clean tracks with full, unobstructed Cherenkov rings.
The _normal_ event category uses an unbiased (in that the RICH detectors are
not used in the selection) track sample composed of kaons and pions
originating from the decay D${}^{0}\rightarrow$ K${}^{-}\pi^{+}$, where the D0
is selected from D${}^{*+}\rightarrow$D${}^{0}\pi^{+}$ decays. The kaons and
pions are required to have track momenta $p_{\rm K}>9.8$ GeV/c and $p_{\pi}>5$
GeV/c in the aerogel; $p_{\rm K}>37$ GeV/c and $p_{\pi}>30$ GeV/c in C4F10,
and $p_{\rm K}>74.8$ GeV/c, $p_{\pi}>40.4$ GeV/c in CF4. These cuts ensure
that all tracks have an expected Cherenkov angle close to saturation
($\beta\approx 1$).
The track sample of the _ideal_ event category is composed of muons selected
from pp$\rightarrow$pp$\mu^{+}\mu^{-}$ events. The events are required not to
have a visible primary vertex. The track momentum selection criteria of the
muons is the same as for pions in the _normal_ event category. A cut was
applied on the track geometry, such that at least half of the Cherenkov cone
associated to the track projects onto the HPD pixels. This selection avoids
losses owing to the cone intersecting with the beampipe, or projecting onto
the region outside the HPD acceptance and the gaps between the HPDs.
Npe is measured by fitting the $\Delta\theta_{C}$ distributions of the
photoelectrons. For each selected charged particle track, photoelectron hits
that lie within a $\Delta\theta_{C}$ range of $\pm 5\sigma$, where $\sigma$ is
the Cherenkov angle resolution, are retained. Photoelectrons that are
correctly associated with a track peak around $\Delta\theta_{C}=0$ and are
distributed as a Gaussian, while those from other tracks and background
sources form a non-peaking background, as shown in Fig. 11 obtained from
C4F10.
An initial fit is performed on the $\Delta\theta_{C}$ distribution aggregated
from all the selected tracks, using a probability density function (PDF)
composed of a Gaussian signal over a quadratic background. The
$\Delta\theta_{C}$ distribution of each individual track is then fitted with a
Gaussian signal over a linear background PDF, with the mean of the Gaussian
fixed at 0 and the width fixed to that obtained from the fit to the aggregated
$\Delta\theta_{C}$ distribution. The individual track Npe is then taken as the
number of photoelectron candidates under the Gaussian shape. The overall value
for Npe is taken as the mean of the distribution of the track Npe, with the
error corresponding to the standard error on the mean. Figure 12 shows the
data distributions at the basis of the measurement.
The validity of the Npe calculation method was assessed using simulated
samples of D${}^{*+}\rightarrow$ D0(K${}^{-}\pi^{+})\pi^{+}$ decays. The same
selection criteria were applied as in data and in addition the track geometry
selection was applied with the same criteria as for _ideal_ RICH events, to
allow a like-for-like comparison between simulation and _ideal_ data events.
The calculated value for Npe was compared to the true photoelectron yield,
which was taken by counting the number of photons associated to each track by
the simulation and then taking the average over all tracks.
To allow a like-for-like comparison of the true and calculated Npe values in
the simulation study, events were required to have less than 50 hits in the
Scintillator Pad Detector (SPD) [1], which gives an approximate measurement of
the charged track multiplicity in the event. It has been observed that the
measured Npe is lower for high track multiplicity events, which have high HPD
occupancies (more than $20\%$ in the central HPD’s in RICH 1 for events with
$>500$ charged tracks). This results in a loss of detected photoelectrons,
because instances where multiple photons hit the same pixel result in only one
photoelectron hit due to the binary HPD readout. This suppression of Npe was
not observed when an analog HPD readout was emulated in the simulation.
Table 1 shows the results of the analysis performed on real data and on the
simulation. In the simulated data, the calculated and true values of Npe are
in good agreement for all the radiators. This validates the accuracy of the
yield calculation. The Npe values for the _ideal_ events are less than those
from the simulation sample. The _normal_ events have values of Npe that are
less than those for _ideal_ events. This is mainly due to the higher charged
track multiplicities of the _normal_ events, reducing the Npe, and the track
geometry cut that is applied to the _ideal_ events increasing their Npe yield.
The aerogel Npe data values have a large uncertainty due to the large
background in the $\Delta\theta_{C}$ distributions and the additional
uncertainty in the shape of the signal peak.
The photoelectron yields are lower than those predicted by the simulation:
however, there is evidence that the yield in data can be increased by a few
percent in RICH 1 by retuning the setting of the HPD readout chip. This
retuning was found necessary for all HPDs by the fact that the trigger rate
went up significantly during 2011 running, resulting in a readout
inefficiency. Furthermore, the detailed description of the detector in the
simulation needs continous retuning, especially for a RICH detector where the
Cherenkov photons must interact with many detector elements. It must be
stressed however, that the smaller yield measured in data does not have a
consequence on the final particle identification performance, as described in
Sect. 5.4.
| N${}_{\text{pe}}$ from data | N${}_{\text{pe}}$ from simulation
---|---|---
Radiator | tagged D${}^{0}\rightarrow$ K${}^{-}\pi^{+}$ | pp$\rightarrow$ pp $\mu^{+}\mu^{-}$ | Calculated $N_{\text{pe}}$ | true $N_{\text{pe}}$
Aerogel | $5.0\pm 3.0$ | $4.3\pm 0.9$ | $8.0\pm 0.6$ | $6.8\pm 0.3$
C4F10 | $20.4\pm 0.1$ | $24.5\pm 0.3$ | $28.3\pm 0.6$ | $29.5\pm 0.5$
CF4 | $15.8\pm 0.1$ | $17.6\pm 0.2$ | $22.7\pm 0.6$ | $23.3\pm 0.5$
Table 1: Comparison of photoelectron yields (Npe) determined from
D${}^{*}\rightarrow$D${}^{0}\pi^{+}$ decays in simulation and data, and pp
$\rightarrow$ pp $\mu^{+}\mu^{-}$ events in data, using the selections and
methods described in the text
Figure 12: Individual track photon yield distributions for the C4F10 (left)
and CF4 (right) radiators. The plot is produced from kaons and pions from
tagged D${}^{0}\rightarrow$ K${}^{-}\pi^{+}$ decays in data selected with the
criteria described in the text
## 5 Particle identification performance
Determining the performance of the RICH Particle IDentification (PID), both
during and after data taking, is particularly important for analyses that
exploit RICH PID, for which knowledge of efficiency and misidentification
rates are required. Moreover, it enables comparison with expectations and
provides a benchmark against which to compare the effectiveness of alignment
and calibration procedures.
This section provides a description of the PID algorithms and the performance
obtained following analysis of data from the first LHC runs.
### 5.1 Particle identification algorithms
Figure 13: Distribution of the number of pixel hits per event in (a) RICH 1
and (b) RICH 2. An example of a typical LHCb event as seen by the RICH
detectors, is shown below the distributions. The upper/lower HPD panels in
RICH 1 and the left/right panels in RICH 2 are shown separately
In order to determine the particle species for each track, the Cherenkov angle
information must be combined with the track momentum measured by the tracking
system, as described in Sect. 4.1.2. The RICH detectors operate in a high
occupancy environment, as shown in Fig.13. To reconstruct such events
efficiently, an overall event log-likelihood algorithm is employed, where all
tracks in the event and in both RICH detectors are considered simultaneously
[20]. This allows for an optimal treatment of tracks where Cherenkov cones
overlap.
Since the most abundant particles in pp collisions are pions, the likelihood
minimisation procedure starts by assuming all particles are pions. The overall
event likelihood, computed from the distribution of photon hits, the
associated tracks and their errors, is then calculated for this set of
hypotheses. Then, for each track in turn, the likelihood is recomputed
changing the mass hypothesis to e, $\mu$, $\pi$, K and proton, whilst leaving
all other hypotheses unchanged. The change in mass hypothesis amongst all
tracks that gives the largest increase in the event likelihood is identified,
and the mass hypothesis for that track is set to its preferred value. This
procedure is then repeated until all tracks have been set to their optimal
hypotheses, and no further improvement in the event likelihood is found.
The procedure described above is CPU intensive for a large number of tracks
and HPD pixels, since the number of likelihood calculations increases
exponentially with the number of tracks. In order to counter this, some
modifications are made to the minimisation procedure to limit the number of
combinations, whilst still converging on the same global solution. During the
search for the track with the largest improvement to the event likelihood, the
tracks are sorted according to the size of their likelihood change from the
previous step, and the search starts with the track most likely to change its
hypothesis. If the improvement in the likelihood for the first track is above
a tuneable threshold, the search is stopped and the hypothesis for that track
is changed. Secondly, if a track shows a clear preference for one mass
hypothesis, then once that track has been set to that hypothesis, it is
removed in the next iterations. These modifications to the likelihood
minimisation dramatically reduce the CPU resources required.
The background contribution to the event likelihood is determined prior to the
likelihood algorithm described above. This is done by comparing the expected
signal in each HPD, due to the reconstructed tracks and their assigned mass
hypothesis, to the observed signal. Any excess is used to determine the
background contribution for each HPD and is included in the likelihood
calculation.
The background estimation and likelihood minimisation algorithms can be run
multiple times for each event. In practice it is found that only two
iterations of the algorithms are needed to get convergence. The final results
of the particle identification are differences in the log-likelihood values
${\Delta\log\mathcal{L}}$, which give for each track the change in the overall
event log-likelihood when that track is changed from the pion hypothesis to
each of the electron, muon, kaon and proton hypotheses. These values are then
used to identify particle types.
### 5.2 Performance with isolated tracks
A reconstructed Cherenkov ring will generally overlap with several others.
Solitary rings from _isolated_ tracks provide a useful test of the RICH
performance, since the reconstructed Cherenkov angle can be uniquely
predicted. A track is defined as _isolated_ when its Cherenkov ring does not
overlap with any other ring from the same radiator.
Figure 14 shows the Cherenkov angle as a function of particle momentum using
information from the C4F10 radiator for isolated tracks selected in data
($\sim 2\%$ of all tracks). As expected, the events are distributed into
distinct bands according to their mass. Whilst the RICH detectors are
primarily used for hadron identification, it is worth noting that a distinct
muon band can also be observed.
Figure 14: Reconstructed Cherenkov angle as a function of track momentum in
the $\rm C_{4}F_{10}$ radiator
### 5.3 PID calibration samples
In order to determine the PID performance on data, high statistics samples of
genuine K${}^{\pm},\pi^{\pm}$, p and $\bar{\rm p}$ tracks are needed. The
selection of such control samples must be independent of PID information,
which would otherwise bias the result. The strategy employed is to
reconstruct, through purely kinematic selections independent of RICH
information, exclusive decays of particles copiously produced and
reconstructed at LHCb.
The following decays, and their charge conjugates, are identified:
K${}^{0}_{\rm S}\rightarrow\pi^{+}\pi^{-}$, $\Lambda\rightarrow$p$\pi^{-}$,
D${}^{*+}\rightarrow$ D0(K${}^{-}\pi^{+})\pi^{+}$. This ensemble of final
states provides a complete set of charged particle types needed to
comprehensively assess the RICH detectors hadron PID performance. As
demonstrated in Fig. 15, the K${}^{0}_{\rm S}$, $\Lambda$, and D∗ selections
have extremely high purity.
Figure 15: Invariant mass distributions of the (a) K${}^{0}_{S}$, (b)
$\Lambda$ and (c) D0 calibration samples. The best fit probability-density-
function (pdf), describing both background and signal, is superimposed in blue
While high purity samples of the control modes can be gathered through purely
kinematic requirements alone, the residual backgrounds present within each
must still be accounted for. To distinguish background from signal, a
likelihood technique, called $\\!\\!{}_{\phantom{1}s}\mathcal{P}lot$ [23], is
used, where the invariant mass of the composite particle K${}^{0}_{\rm
S},\Lambda$, D0 is used as the discriminating variable.
The power of the RICH PID can be appreciated by considering the $\rm\Delta
log\mathcal{L}$ distributions for each track type from the control samples.
Figures 16(a-c) show the corresponding distributions in the 2D plane of
$\rm\Delta log\mathcal{L}(K-\pi)$ versus $\rm\Delta log\mathcal{L}(p-\pi)$.
Each particle type is seen within a quadrant of the two dimensional $\rm\Delta
log\mathcal{L}$ space, and demonstrates the powerful discrimination of the
RICH.
Figure 16: Distribution of $\rm\Delta log\mathcal{L}(K-\pi)$ against
$\rm\Delta log\mathcal{L}(p-\pi)$ for (a) pions, (b) kaons and (c) protons
extracted from the control samples
### 5.4 PID performance
Utilizing the log-likelihood values obtained from the control channels, one is
able to study the discrimination achievable between any pair of track types by
imposing requirements on their differences, such as $\Delta{\rm log}(K-\pi)$.
Figure 17 demonstrates the kaon efficiency (kaons identified as kaons) and
pion misidentification (pions misidentified as kaons), as a function of
particle momentum, obtained from imposing two different requirements on this
distribution. Requiring that the likelihood for each track with the kaon mass
hypothesis be larger than that with the pion hypothesis, i.e. $\rm\Delta
log\mathcal{L}(K-\pi)>0$, and averaging over the momentum range 2 - 100
GeV/$c$, the kaon efficiency and pion misidentification fraction are found to
be $\sim 95\%$ and $\sim 10\%$, respectively. The alternative PID requirement
of $\rm\Delta log\mathcal{L}(K-\pi)>5$ illustrates that the misidentification
rate can be significantly reduced to $\sim 3\%$ for a kaon efficiency of $\sim
85\%$. Figure 18 shows the corresponding efficiencies and misidentification
fractions in simulation. In addition to K/$\pi$ separation, both p/$\pi$ and
p/K separation are equally vital for a large number of physics analyses at
LHCb. Figure 19 demonstrates the separation achievable between protons and
pions when imposing the PID requirements $\rm\Delta\mathcal{L}(p-\pi)>0$ and
$\rm\Delta\mathcal{L}(p-\pi)>5$. Finally, Fig. 20 shows the discrimination
achievable between protons and kaons when imposing the requirements
$\rm\Delta\mathcal{L}(p-K)>0$ and $\rm\Delta\mathcal{L}(p-K)>5$.
Together, Figures 17, 19 and 20 demonstrate the RICH detectors ability to
discriminate any pair of track types, from the set of kaons, pions and
protons, albeit for the PID requirements quoted.
Figure 17: Kaon identification efficiency and pion misidentification rate
measured on data as a function of track momentum. Two different $\rm\Delta
log\mathcal{L}(K-\pi)$ requirements have been imposed on the samples,
resulting in the open and filled marker distributions, respectively Figure
18: Kaon identification efficiency and pion misidentification rate measured
using simulated events as a function of track momentum. Two different
$\rm\Delta log\mathcal{L}(K-\pi)$ requirements have been imposed on the
samples, resulting in the open and filled marker distributions, respectively
Figure 19: Proton identification efficiency and pion misidentification rate
measured on data as a function of track momentum. Two different $\rm\Delta
log\mathcal{L}(p-\pi)$ requirements have been imposed on the samples,
resulting in the open and filled marker distributions, respectively Figure
20: Proton identification efficiency and kaon misidentification rate measured
on data as a function of track momentum. Two different $\rm\Delta
log\mathcal{L}(p-K)$ requirements have been imposed on the samples, resulting
in the open and filled marker distributions, respectively
### 5.5 Performance as a function of event multiplicity
The current running conditions333The LHCb RICH detector was designed to run
with 0.6 interaction per bunch crossing. However the current operating
conditions have 1.6 interactions per bunch crossing. , with increased particle
multiplicities, provide an insightful glimpse of the RICH performance at high
luminosity running.
Figure 21 shows the pion misidentification fraction versus the kaon
identification efficiency as a function of (a) track multiplicity and (b) the
number of reconstructed primary vertices, as the requirement on the likelihood
difference $\rm\Delta log\mathcal{L}(K-\pi)$ is varied. The results
demonstrate, as expected, some degradation in PID performance with increased
interaction multiplicity. The $K/\pi$ separation is, however, extremely robust
right up to the highest interaction multiplicities and thus gives confidence
that the current RICH system is suitable for operation at the higher
luminosities foreseen in the future.
Figure 21: Pion misidentification fraction versus kaon identification
efficiency as measured in 7 TeV LHCb collisions: (a) as a function of track
multiplicity, and (b) as a function of the number of reconstructed primary
vertices. The efficiencies are averaged over all particle momenta
## 6 Conclusions
The RICH detector was designed specifically for the physics program of LHCb.
It has been in operation since the end of 2009. The RICH detector has operated
with high efficiency during these first three years of LHC running. It has
demonstrated a PID performance that is well up to design specifications and
that allows the extraction of physics results in all sectors of $b$ and $c$
quark decays, in particular of the rare phenomena which may allow the
discovery of new physics at the LHC.
The performance of the RICH particle identification has been studied with the
LHC collisions taken since the startup of the LHC machine. Studies of the
decays of K0, $\Lambda^{0}$ and D∗ provide a source of $\pi$, K, p identified
kinematically for which the RICH identification performance can be
established. The precise alignment and calibration procedures are crucial to
reach the designed performance. The Cherenkov angle resolutions are in good
agreement with the expected design performance for the gas radiators, and are
still being improved for the aerogel radiator.
## Acknowledgements
This complex detector could only be constructed with the dedicated effort of
many technical collaborators in the institutes forming the LHCb RICH
collaboration. A special acknowledgement goes to all our LHCb collaborators
who over the years have contributed to obtain the results presented in this
paper. 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); ANCS/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.
## References
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M. Moritz et al., Performance Study of New Pixel Hybrid Photon Detector
Prototypes for the LHCb RICH Counters, IEEE TRANSACTIONS ON NUCLEAR SCIENCE,
VOL. 51, NO. 3, JUNE 2004
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F. Fontanelli et al.,Nucl. Inst. Meth. A 604 (2009) 675
C. Arnaboldi et al.,NSS’07 IEEE Volume 1 (2007) 677
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* [15] R. Cardinale et al., JINST 6 (2011) P06010
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* [18] G. Barrand et al., GAUDI - A software architecture and framework for building HEP data processing applications, Comput.Phys.Commun.140:45-55,2001
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|
arxiv-papers
| 2012-11-28T21:45:15 |
2024-09-04T02:49:38.626161
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "M. Adinolfi, G. Aglieri Rinella, E. Albrecht, T. Bellunato, S. Benson,\n T. Blake, C. Blanks, S. Brisbane, N.H. Brook, M. Calvi, B. Cameron, R.\n Cardinale, L. Carson, A. Contu, M. Coombes, C. D'Ambrosio, S. Easo, U. Egede,\n S. Eisenhardt, E. Fanchini, C. Fitzpatrick, F. Fontanelli, R. Forty, C. Frei,\n P. Gandini, R. Gao, J. Garra Tico, V. Gibson, C. Gotti, S. Gregson, T. Gys,\n S.C. Haines, T. Hampson, N. Harnew, D. Hill, P. Hunt, M. John, C.R. Jones, D.\n Johnson, N. Kanaya, S. Katvars, U. Kerzel, Y.M. Kim, S. Koblitz, M.\n Kucharczyk, D. Lambert, R.W. Lambert, A. Main, M. Maino, S. Malde, N.\n Mangiafave, C. Matteuzzi, G. Mini', A. Mollen, J. Morant, R. Mountain, J.V.\n Morris, F. Muheim, R. Muresan, J. Nardulli, P. Owen, A. Papanestis, M. Patel,\n G.N. Patrick, D.L. Perego, G. Pessina, A. Petrolini, D. Piedigrossi, R.\n Plackett, S. Playfer, A. Powell, J.H. Rademacker, S. Ricciardi, G.J. Rogers,\n P. Sail, M. Sannino, T. Savidge, I. Sepp, S. Sigurdsson, F.J.P. Soler, A.\n Solomin, F. Soomro, A. Sparkes, P. Spradlin, B. Storaci, C. Thomas, S.\n Topp-Joergensen, N. Torr, O. Ullaland, K. Vervink, D. Voong, D. Websdale, G.\n Wilkinson, S.A. Wotton, K. Wyllie, F. Xing, R. Young",
"submitter": "Christopher Jones",
"url": "https://arxiv.org/abs/1211.6759"
}
|
1211.6799
|
# GalViz: Visualize Social Bookmark Management
Lilian Weng and Filippo Menczer
School of Informatics and Computing, Indiana University Bloomington, USA
# Context Visualization for Social Bookmark Management
Lilian Weng and Filippo Menczer
School of Informatics and Computing, Indiana University Bloomington, USA
###### Abstract
We present the design of a new social bookmark manager, named GalViz, as part
of the interface of the GiveALink system. Unlike the interfaces of traditional
social tagging tools, which usually display information in a list view, GalViz
visualizes tags, resources, social links, and social context in an interactive
network, combined with the tag cloud. Evaluations through a scenario case
study and log analysis provide evidence of the effectiveness of our design.
Keywords: social tagging, social links, Web content, visualization, interface,
layout, network, GiveALink
## 1 Introduction
Social tagging is a flexible and powerful way to assist users who
collaboratively organize and share Web resources, as is widely done in many
popular Web 2.0 applications. Various kinds of online resources can be tagged,
reflecting user interests and making such resources more easily accessible to
the public. The basic relationship in social tagging is defined by the triple,
a tripartite combination of a user, a tag and a resource. To clearly display
and manage a large number of triples is a pretty hard problem, since they
involve complicated relationships among objects of three different types.
Current social bookmark managers display collected resources in simple list
views, distributed across multiple pages. Such a linear display of resources
neglects contextual information and weakens the relationships between
resources and tags. Socio-semantic connections among users through shared
annotations driven by common interests are hidden, since most existing systems
completely separate the personal and global views. Semantic relationships
among related tags and among related resources are deemphasized as well. Our
design aims to ameliorate these problems for facilitating the management of
complex relationships hidden in the social annotations. In particular, the
design adds value to the user experience by visualizing tripartite
relationships as networks.
## 2 Motivation
Social tagging collections, also known as social bookmarks or _folksonomies_ ,
are generated by spontaneous behavior of users, helping people manage and
organize their resources [15]. Existing social tagging sites usually
incorporate components such as resource lists assisted with tag lists, tag
clouds and hierarchical clusterings. However, each display has deficiency
limiting its functionality and usability. The imperfection of current systems
encourages us to design and develop a better alternative application.
Figure 1: Interfaces of existing social bookmark managers: (a) Delicious.com
(b) Flickr (c) Firefox Bookmark Manager.
List View: List views are widely adopted for displaying a group of items of
the same type. The interface of Delicious, as an example, consists of a list
of resources and a list of all tags (Fig. 1a). The resource list occupies most
space of the page, emphasized as the main content. Tags are crowd-sourced
keywords for indexing resources, so the tag list is treated as an auxiliary
sidebar. In this case, resources and tripartite relationships are displayed
linearly. List views sequentially present the resources according to a certain
order, hiding the links among resources, other similar resources and
semantically related tags. Key contextual information, such as similar sites,
people who have interests in it, and their collections, could be very valuable
for fostering a better understanding of the site and the exploration of other
relevant and interesting objects; but it is missing.
Tag Cloud: The tag cloud is a visual depiction of tags in the whole collection
to facilitate tag browsing (Fig. 1b). Tags are arranged in alphabetical order;
font sizes and colors are configured to highlight the most popular ones. Tag
clouds provide users with a global view to learn which tags are used by most
people, but they possess several disadvantages too. Flat tag clouds are not
sufficient to provide a semantic, rich and multidimensional browsing
experience over large tagging spaces, due to their inefficient navigations,
downgrading of alternative views, and lack of emphasis on the semantic
relationships [11, 17]. Tag clouds obscure useful information by having a
strong bias towards very popular tags [7, 10]. They usually don’t have enough
space to contain all the tags, so a portion of resources still remain
inaccessible through the tag clouds [10].
Browser Bookmarks: The hierarchical display of tags is naturally embedded in
bookmark managers of Web browsers (Fig. 1c). The browser tagging system is
designed to be very similar to a file explorer, where tags are organized in a
folder-like hierarchical structure. Resource categorization is restricted by
the structure of tags, and only the child-parent relations get apparent.
Therefore, when the collections of bookmarks become larger, it is hard for
users to efficiently find a specific resource or discover related items [16].
Without support for cross references, each resource instance can only belong
to one folder. Although multiple copies of the same resource can be assigned
to different tags, they are distinct instances, leading to duplication and
inconsistency.
## 3 Related Work
There is sparse literature on the design principles of social tagging tools
[7, 9, 20]. Some work has been done on building hierarchy structure from the
social tagging to uncover the hidden child-parent semantics [4, 19].
Visualization, as a powerful type of social bookmarking tool, has been studied
and utilized in several existing designs. Visualization of hyperlinks between
Web pages was adopted to enhance adaptive navigation [21]. Cluster Map, a
social bookmark visualization tool, highlighted the relationships among users
and bookmarks to identify tag and community structures [12]. Unlike
ClusterMap, GalViz is designed to emphasize the semantic relationships between
tags and resources to help users manage existing resources and discover new
ones. Besides, the graphical interface was shown to be useful for distributed
collaborations and interactions on social bookmarking [6]. Graphical
visualization of concept networks was integrated into several Web applications
as an innovative interactive user interface [2, 3]. Most of such existing
applications display context of items of the same type, but GalViz is able to
provide the visualization of heterogenous networks among two different
objects, tags and resources.
GiveALink.org, as a research-oriented social tagging platform, broadly
examines several aspects of social tagging to foster the construction and
applications of socially driven semantic annotation networks. Previous
research includes the design of effective similarity relationships [14],
social spam detection [13], and social tagging games as incentive for
collecting high-quality annotations [22]. Former work in GiveALink on
exploratory navigation interfaces [8] and bookmark management [18] have
significant influence on the new design presented in the paper.
## 4 Design Goals
_GalViz_ is an innovative bookmark management tool in the _GiveALink_ system
(Fig. 2). We aimed to achieve several design goals, to help overcome the
weakness of existing systems and improve user experience. Network display,
contextual maps, and social relationships are three core concepts.
Figure 2: Implementation of _GalViz_ : (a) Choose from two modes, either the
traditional list view or the network visualization. (b) Tag cloud of all
personal tags; the font size of a tag is proportional to its popularity. (c) A
switch between the personal view and the social view. The social view displays
additional relevant resources and tags based on other people’s collections.
The local items are green nodes, and the global items are purple ones. (d) The
filter allows one to expand available resources in the network with more
related tags. (e) Visualization of tags and resources. (f) Contextual menu of
a node. The sample menu is for a local resource, so the user has full control
over it, including editing the title, changing the tags, or removing it. If
the resource doesn’t belong to the user, she has to add it to her local
collection prior to making further changes.
### 4.1 Facilitate Annotation Management with Visualization
_GalViz_ visualizes personal bookmarks in an interactive network. Each node
represents a Web resource or a tag; related objects are connected. The default
network is centered at selected tags, and gets expanded by relevant triples.
It is easier to interpret than pure text [5]. Furthermore, users only need to
concentrate on a small area of the visualization, and therefore they can
quickly spot the target node by the text or favicon. The network provides an
interactive way to easily edit or remove any node through a contextual menu.
The menu shows different options according to the characteristics of the node:
whether it is a resource or a tag, and whether it belongs to the user’s
personal collection. Dragging and dropping nodes create operation shortcuts. A
GalViz bookmarklet allows users to easily add new resources and view their
semantic context.
### 4.2 Provide Contextual Maps
Currently, neither browsers nor popular social tagging sites visualizes a
page’s semantic context beyond its content. Therefore, when users need to tell
what a page is about, they learn based on the information inside the page
itself. To improve this, GalViz strives to provide valued and helpful
knowledge by exposing links with correlated tags and resources. Users are
therefore provided with contextual maps of semantic relationships of a tag or
Web page in terms of related tags and pages. Users can use such a map as an
alternative channel to understand the content and context of the target, as
well as to navigate the Web by visiting or exploring related objects.
### 4.3 Highlight Social Relationships
In social tagging systems, no matter whether users communicate directly, they
are always potentially connected by shared interests in certain resources or
similar agreements on the resources’ descriptions. One limitation of the
current interface [1] is that personal resources and the global repository are
completely separated, and consequently users cannot easily discover their
social connections with others. People have different views of same resources,
so learning others’ annotations can help improve one’s understanding and
optimize one’s collection. GalViz brings the idea of social view instead of
global view. The social view expands the local (personal) resources and tags
of a user with globally related triples, making it easier to find out what
other people view as similar objects.
### 4.4 Do Not Overwhelm Users with Information
Information overload is a common problem faced by many crowdsourcing sites.
Once a large group of users get into the system and contribute to the
knowledge repository, it is impossible for anyone to browse every piece of
information. Therefore, customization of the results becomes necessary. To
avoid overwhelming users, we propose a set of content filters in the design to
limit the network expansion, and to switch between personal and social view.
## 5 Implementation
_GalViz_ incorporates two styles of browsing, the _list_ mode and the
_visualization_ mode, as well as a _bookmarklet_ (Fig. 3(a)) gadget helping
add new resources.
Visualization: The visualization of bookmarks as a network with rich
contextual information is the core design of GalViz. As Fig. 2 shows, the
visualization mode is formed of a tag cloud on the left and a main bookmark
network canvas on the right. Several options on the top endow users with the
ability to switch between different views or filter out the results.
List: We provide a traditional list view to assist users gradually accommodate
to the new interface. The list view is also used for evaluation by comparison
between the two modes. Similar to the visualization mode, the list interface
has a tag cloud on the left, and the main content on the right where resources
of selected tags are listed. Personal and social view separate the local
resources and globally popular ones, with a switch similar to Fig. 2(a).
Bookmarklet: The bookmarklet is a small applet, working as a normal bookmark
item but with customized javascript code to provide additional functions
beyond simply opening a Web page. The GalViz bookmarklet is an extension of
the visualization service providing easy access to the contextual map of the
current Web page.
## 6 Case Study
Scenario:
A friend recommends the site Engadget.com.
Goal:
To learn the content and context of _Engadget_ through the social tagging
system, and then annotate it with a set of proper tags in the personal
bookmark collection.
Solution:
Let us consider how a user accomplishes the task through _Delicious_ and
GalViz, respectively:
(1) Through Delicious: First, the user visits Delicious and searches for
“engadget.com”. She should be able to see top popular tags about Engadget.
Second, the user clicks on “Save this site” on the top right corner and fills
in the pop-up dialog box to tag the bookmark.
(2) Through GalViz: While browsing the Engadget site, the user can access
GalViz by simply clicking the GalViz bookmarklet (Fig. 3a). After choosing
“View context visualization” (Fig 3b), he can see the contextual information
about Engadget. Through the right-click menu, the target site, as the center
of the visualization, is added into the personal view (Fig. 3c) with tags
customized on the system recommendations (Fig. 3d). The final visualization
will be the personal view of Engadget associated with user tags and other
relevant Web resources (Fig. 3e).
Users need to type in the required information and read the text in detail to
look for proper links in _Delicious_. GalViz provides a more transparent and
embedded service through the bookmarklet, and visualizes the contextual
information in a simple but insightful way.
## 7 Preliminary Log Study
During an evaluation period, we assigned _GiveALink_ users with equal chances
to see either the list or the visualization mode when they navigated to the
bookmark manager. Users are allowed to do any task they attend to do. Our log
recorded 38,181 actions by 6,310 users. Analysis of the log reveals several
advantages of the visualization interface. For each mode, a series of
continuous clicks is defined as a session. As shown in Table 1, the
visualization can attract more usage, as there are more sessions in the
visualization mode (1,192) than in the list view (960). Each session in the
visualization mode lasts 172.3 seconds on average, 25 seconds shorter than the
average session length of the list. With the help of the interactive network
visualization, users are shown to interact more with the system during a
shorter time period. With a reasonable assumption that users start using the
_GiveAlink_ bookmark manager with a certain goal and leave once the goal is
achieved, we can infer that the visualization mode can facilitate user tasks
more efficiently. Clicks involving tag selection, resource selection and
content editing are considered as content-related actions. 83.1% of clicks in
the visualization mode are content-related, much higher than 57.6% in the list
mode. The visualization encourages more interactions with the content, as the
graph visualization simplifies the management operations with an intuitive
representation. Finally, the visualization and list view have approximately
equal chances to lose users to the other mode during usage sessions.
Table 1: Statistics through log analysis. Mode | List | Visualization
---|---|---
Number of sessions | 960 | 1,192
Time per session (sec) | 197.4 | 172.3
Clicks per session | 12.4 | 17.3
Content-related clicks | 57.6% | 83.1%
Switch to other mode | 32.3% | 33.4%
Figure 3: Find out the context of the site Engadget.com and add it into the
personal collection through GalViz: (a) Use GalViz bookmarklet; (b) Choose to
see the context; (c) Add it into the personal collection; (d) Customize
recommended tags; (e) Check the final visualization.
## 8 Conclusion and Future Work
To complement the weaknesses of current social tagging tools, we designed and
implemented a new social bookmark manager. It visualizes the tripartite
relationships in a more accessible and friendly display, and incorporates
richer contextual information. Our design goals include facilitating
annotation management with visualization, providing contextual maps, and
highlighting social relationships while avoiding information overwhelming. A
case study on a common scenario and the system log analysis suggest that these
goals have been achieved by our design and implementation. _GalViz_ smoothens
the process of examining the contextual information and adding new resources;
at the same time, users are encouraged to interact with the system and explore
the information more efficiently. The GalViz social bookmark manager
(GiveALink.org/collection/show) is part of the GiveALink platform.
We presented a set of preliminary evaluations. To learn more about the
limitations of _GalViz_ and possible improvements in future versions, a more
thorough and complete evaluation should be considered. As part of the future
work, we plan to employ Amazon’s _Mechanical Turk_ , a crowd-sourcing
marketplace, to measure the usability of the system with several specifically
designed tasks. Close user studies, including face-to-face conversations and
user behavior observations, are necessary as well for gaining a deeper
understanding of user requirements.
## References
* [1] Delicious, http://delicious.com.
* [2] Infomous, http://infomous.com.
* [3] Visuwords, http://www.visuwords.com.
* [4] G. Begelman, P. Keller, and F. Smadja. Automated tag clustering: Improving search and exploration in the tag space. In Proc. Intl. Conf. on World Wide Web, 2006.
* [5] E. H. Chi. Improving web usability through visualization. Usability and the Web, 2002.
* [6] M. Collomb and M. Hasceet. Synchronous cooperation and visualization for social bookmarking systems. In Proc. Intl. Conf. on Advanced Visual Interfaces, 2010.
* [7] S. Deutsch, J. Schrammel, and M. Tscheligi. Comparing Different Layouts of Tag Clouds: Findings on Visual Perception. Springer, 2009.
* [8] J. Donaldson, M. Conover, B. Markines, H. Roinestad, and F. Menczer. Visualizing social links in exploratory search. In Proc. ACM Conf. on Hypertext and Hypermedia, 2008.
* [9] U. Farooq, T. G. Kannampallil, Y. Song, J. M. Carroll, and L. Giles. Evaluating tagging behavior in social bookmarking systems: metrics and design heuristics. In Proc. Conf. on Supporting Group Work, 2007.
* [10] M. Gupta, R. Li, Z. Yin, and J. Han. An overview of social tagging and applications, chapter 16. Springer, 2011.
* [11] M. A. Hearst and D. Rosner. Tag clouds: Data analysis tool or social signaller? In Proc. Hawaii Intl. Conf. on System Sciences, 2008.
* [12] J. Klerkx and E. Duval. Visualising social bookmarks. In Journal of Digital Info., volume 10, 2009.
* [13] B. Markines, C. Cattuto, and F. Menczer. Social spam detection. In Proc. Intl. Workshop on Adversarial Info. Retrieval on the Web, 2009.
* [14] B. Markines, C. Cattuto, F. Menczer, D. Benz, A. Hotho, and G. Stumme. Evaluating similarity measures for emergent semantics of social tagging. In Proc. Intl. Conf. on World Wide Web, 2009.
* [15] D. Millen, J. Feinberg, and B. Kerr. Social bookmarking in enterprise. ACM Queue - Social Computing, 2005.
* [16] T. B. Passin. Browser bookmark management with topic maps. In Proc. Extreme Markup Languages, 2003.
* [17] E. Quintarelli, A. Resmini, and L. Rosati. Information architecture: Facetag: Integrating bottom-up and top-down classification in a social tagging system. Bulletin of American Society for Info. Sci. and Tech., 33:10–15, 2007.
* [18] H. Roinestad, J. Burgoon, B. Markines, and F. Menczer. Incentives for social annotation. In Proc. ACM SIGIR Intl. Conf. on Research and Development in Information Retrieval, page 838, 2009.
* [19] A. Shepitsen, J. Gemmell, B. Mobasher, and R. Burke. Personalized recommendation in social tagging systems using hierarchical clustering. In Proc. ACM Conf. on Recommender Systems, 2008.
* [20] A. Shiri. An examination of social tagging interface features and functionalities: An analytical comparison. Online Info. Rev., 2009.
* [21] M. Tomša and M. Bieliková. Hyperlinks visualization using social bookmarking. In Proc. ACM Conf. on Hypertext and Hypermedia, 2008.
* [22] L. Weng, R. Schifanella, and F. Menczer. Design of social games for collecting reliable semantic annotations. In Proc. IEEE Intl. Conf. on Computer Games: AI, Animation, Mobile, Interactive Multimedia, Educational & Serious Games, 2011.
|
arxiv-papers
| 2012-11-29T02:34:28 |
2024-09-04T02:49:38.638089
|
{
"license": "Public Domain",
"authors": "Lilian Weng and Filippo Menczer",
"submitter": "Lilian Weng",
"url": "https://arxiv.org/abs/1211.6799"
}
|
1211.6940
|
Choice Disjunctive Queries in Logic Programming
Keehang Kwon
Dept. of Computer Engineering, DongA University
Busan 604-714, Korea
[email protected]
Abstract: One of the long-standing research problems on logic programming is
to treat the cut predicate in a logical, high-level way. We argue that this
problem can be solved by adopting linear logic and choice-disjunctive goal
formulas of the form $G_{0}\oplus G_{1}$ where $G_{0},G_{1}$ are goals. These
goals have the following intended semantics: $choose$ the true disjunct
$G_{i}$ and execute $G_{i}$ where $i(=0\ {\rm or}\ 1)$, while $discarding$ the
unchosen disjunct. Note that only one goal can remain alive during execution.
These goals thus allow us to specify mutually exclusive tasks in a high-level
way.
keywords: Prolog, mutual exclusion, cut, linear logic, computability logic
## 1 Introduction
One of the long-standing research problems on logic programming is to treat
the extra-logical primitive in a high-level way. The advances of logic
programming have enriched Horn clauses with additional programming primitives
in a high-level way (higher-order programming, modules, local constants, etc).
Nevertheless some key constructs could not be dealt with in a high-level way,
in particular when we are concerned with mutual exclusion (and the cut
predicate).
Consequently, much attention [10, 11, 6] has been given to finding a semantics
that captures the cut predicate. However, these proposals are not $logical$ in
that a well-defined yet simple declarative meaning as well as its proof theory
are still missing, exposing low-level operational details.
In this paper, inspired by the work in [4], we propose a purely logical
solution to this problem. It involves the direct employment of linear logic
[2] to allow for choice-disjunctive goals. A choice-disjunctive goal is of the
form $G_{0}\oplus G_{1}$ where $G_{0},G_{1}$ are goals. (A more intuitive name
would be $choose(G_{0},G_{1})$.) Executing this goal with respect to a program
$D$ – $ex(D,G_{0}\oplus G_{1})$ – has the following intended semantics:
$\mbox{\rm choose a true one between}\ ex(D,G_{0}),ex(D,G_{1}).$
An illustration of this aspect is provided by the following definition of the
relation $son(X,Y)$ which holds if $Y$ is a son of $X$.:
$son(X,Y)$ ${\rm:-}$ | $(male(X)\land father(Y,X))\ \oplus$
---|---
| $(female(X)\land mother(Y,X)).$
The body of the definition above contains a mutually exclusive goal, denoted
by $\oplus$. As a particular example, solving the query $son(tom,Y)$ would
result in selecting and executing the first goal $male(tom)\land
father(tom,Y)$, while discarding the second one. The given goal will succeed,
producing solutions for $Y$. Of course, we can specify mutually exclusive
goals using cut in Prolog, but it is well-known that cuts complicates the
declarative meaning of the program [1]. Our language makes it possible to
formulate mutually exclusive goals in a high-level way. The class of choice
disjunctive goals is, in a sense, a high-level abstraction for the cut
predicate.
As seen from the example above, choice-disjunctive goals can be used to
perform mutually exclusive tasks. There are several linear logic languages [3,
12] in which goals of the form $G_{0}\oplus G_{1}$ are present. A common yet
problematic aspect of these works is their treatment of the $\oplus$-goals:
these goals are treated as inclusive-OR (or classical disjunctive) goals
rather than exclusive-OR ones:
$ex(D,G_{0}\oplus G_{1})\ {\rm if}\ ex(D,G_{0})\ \lor ex(D,G_{1})$
where $\lor$ represents classical disjunction. Hence, the declarative reading
of $\oplus$ – known as the machine’s choice – is $violated$ in these
languages.
A satisfactory solution can be obtained by adding the choice action, as
discussed above, to their execution model of $\oplus$ so that the execution
respects the declarative reading of $\oplus$, while maintaining provability.
Hence, the main difference is that, once a goal is chosen, the unchosen goal
will be discarded in our language, while it will remain alive (typically
through a creation of a choicepoint) in those languages.
This paper proposes Prolog⊕, an extension of Prolog with choice-disjunctive
operators in goal formulas. The remainder of this paper is structured as
follows. We describe Prolog⊕ in the next section. In Section 3, we present
some examples of Prolog⊕. Section 4 concludes the paper.
## 2 The Language
The language is a version of Horn clauses with choice-disjunctive goals. Note
that we disallow linear clauses here, thus allowing only reusable clauses. It
is described by $G$\- and $D$-formulas given by the syntax rules below:
| $G::=$ | $A\;|\;t=s\;|\;G\land G\;|\;\exists x\ G\;|\;G\oplus G$
---|---|---
| $D::=$ | $A\;|\;G\supset A\ \;|\;\forall x\ D\;|\;D\land D$
In the rules above, $t,s$ represent terms, and $A$ represents an atomic
formula. A $D$-formula is called a Horn clause with choice-disjunctive goals.
The logic programming paradigm such as Prolog was originally founded on the
resolution method. But this approach was difficult to extend to richer logics.
The use of sequent calculus allows us to overcome this limit. Furthermore,
uniform proofs [9] allows us to execute logic programs in an efficient way by
integrating two separate phases – the proof phase and the execution phase –
into a single phase. We adopt this approach below.
We will present a machine’s strategy for this language as a set of rules.
These rules in fact depend on the top-level constructor in the expression, a
property known as uniform provability[8, 9, 7]. Note that execution alternates
between two phases: the goal-reduction phase and the backchaining phase. In
the goal-reduction phase (denoted by $ex(D,G)$), the machine tries to solve a
goal $G$ from a clause $D$ by simplifying $G$ . The rule (6) – (9) are related
to this phase. If $G$ becomes an atom, the machine switches to the
backchaining mode. This is encoded in the rule (5). In the backchaining mode
(denoted by $bc(D_{1},D,A)$), the machine tries to solve an atomic goal $A$ by
first reducing a Horn clause $D_{1}$ to simpler forms (via rule (3) and (4))
and then backchaining on the resulting clause (via rule (1) and (2)).
Definition 1. Let $G$ be a goal and let $D$ be a program. Then the notion of
executing $\langle D,G\rangle$ – $ex(D,G)$ – is defined as follows:
* (1)
$bc(A,D,A)$. % This is a success.
* (2)
$bc((G_{0}\supset A),D,A)$ if $ex(D,G_{0})$.
* (3)
$bc(D_{1}\land D_{2},D,A)$ if $bc(D_{1},D,A)$ or $bc(D_{2},D,A)$.
* (4)
$bc(\forall xD_{1},D,A)$ if $bc([t/x]D_{1},D,A)$.
* (5)
$ex(D,A)$ if $bc(D,D,A)$.
* (6)
$ex(D,t=s)$ if $unify(t,s)$ % $t,s$ are terms.
* (7)
$ex(D,G_{0}\land G_{1})$ if $ex(D,G_{0})$ $and$ $ex(D,G_{1})$.
* (8)
$ex(D,\exists xG_{0})$ if $ex(D,[t/x]G_{0})$.
* (9)
$ex(D,G_{0}\oplus G_{1})$ if select a successful disjunct between
$ex(D,G_{0})$ and $ex(D,G_{1})$. % This goal behaves as exclusive-OR.
In the above rules, only the rule (9) is a novel feature. To be specific, this
goal first attempts to execute $G_{0}$, discarding $G_{1}$. If it succeeds,
then do nothing (and do not leave any choice point for $G_{1}$ ). If it fails,
then $G_{1}$ is attempted.
Implementing $G_{0}\oplus G_{1}$ poses no difficulties. For example, it can be
done by translating it to a Prolog disjunctive goal of the form
$(G_{0},!);G_{1}$ where ; denotes a Prolog disjuction. The cut then destroys
the choice point created for $G_{1}$ if $G_{0}$ succeeds. On the contrary, the
same goal $G_{0}\oplus G_{1}$ will be translated to $G_{0};G_{1}$ in other
linear logic languages.
The following theorem connects our language to linear logic. Its proof is
obtained from [3] and from the simple observation that our modified execution
rule preserves provability.
###### Theorem 1
Let $D$ be a program and let $G$ be a goal. Then, $ex(D,G)$ terminates with a
success if and only if $G$ follows from $D$ in intuitionistic linear logic.
Furthermore, it respects the declarative reading of the operator $\oplus$.
## 3 Examples
Let us first consider the relation $f(X,Y)$ specified by two rules:
* (1)
if $X<2$, then $Y=0$.
* (2)
if $X\geq 2$, then $Y=3$.
The two conditions are mutually exclusive which is expressed by using the cut
in traditional logic programming as shown below:
$f(X,0):-\ X<2,!.$ $f(X,3):-\ X\geq 2.$
Using cut, we can specify mutually exclusive goals, but cuts affect the
declarative meaning of the program. Our language makes it possible to
formulate mutually exclusive goals through the choice-disjunctive goals as
shown below:
$f(X,Y)$ ${\rm:-}$ | $(X\geq 2\land Y=3)\ \oplus$
---|---
| $(X<2\land Y=0)$
The new program, equipped with $\oplus$-goals, is more readable than the
original version with cuts, while preserving the same efficiency. A similar
example is provided by the following “max” program that finds the larger of
two numbers.
$max(X,Y,Max)$ ${\rm:-}$ | $(X\geq Y\land Max=X)\ \oplus$
---|---
| $(X<Y\land Max=Y)$
These two goals in the body of the above clause are mutually exclusive. Hence,
only one of these two goals can succeed. For example, consider a goal
$max(3,9,Max)$. Solving this goal has the effect of choosing and executing the
second goal $(3<9)\land Max=9$, producing the result $Max=9$.
As another example, we consider the relation $member(X,L)$ for establishing
whether $X$ is in the list $L$. A typical Prolog definition of $member(X,L)$
is shown below:
$member(X,[Y|L])$ ${\rm:-}$ | $(Y=X)\ \lor member(X,L)$
---|---
This definition is nondeterministic in the sense that it can find any
occurrence of $X$. Our language in Section 2 makes it possible to change
$member$ to be deterministic and more efficient: only one occurrence can be
found. An example of this is provided by the following program.
$member(X,[Y|L])$ ${\rm:-}$ | $(Y=X)\ \oplus member(X,L)$
---|---
As a final example, we consider the relation $rprime$ for establishing whether
the keyboard input data $X$ is prime or not. An example of this is provided by
the following program.
$rprime$ ${\rm:-}$ | $read(X)\land$
---|---
| $(prime(X)\land\ write(`prime^{\prime}))\oplus$
| $(composite(X)\land\ write(`composite^{\prime}))$
## 4 Conclusion
In this paper, we have considered an extension to Prolog with choice-
disjunctive goals. This extension allows goals of the form $G_{0}\oplus G_{1}$
where $G_{0},G_{1}$ are goals. These goals are particularly useful for
replacing the cut in Prolog, making Prolog more concise and more readable.
In the near future, we plan to investigate the connection between Prolog⊕and
Japaridze’s Computability Logic(CL)[4, 5]. CL is a new semantic platform for
reinterpreting logic as a theory of tasks. Formulas in CL stand for
instructions that can carry out some tasks. We plan to investigate whether our
operational semantics is sound and complete with respect to the semantics of
CL.
## References
* [1] I. Bratko, “Prolog:programming for AI ”, Addison Wesley, 2001 (3rd edition).
* [2] J.Y. Girard, “Linear Logic”, Theoretical Computer Science, vol.50, pp.1–102, 1987\.
* [3] J. Hodas and D. Miller, “Logic Programming in a Fragment of Intuitionistic Linear Logic”, Information and Computation, vol.110, pp.327–365, 1994.
* [4] G. Japaridze, “Introduction to computability logic”, Annals of Pure and Applied Logic, vol.123, pp.1–99, 2003.
* [5] G. Japaridze, “Sequential operators in computability logic”, Information and Computation, vol.206, No.12, pp.1443-1475, 2008.
* [6] J. Kriener and A. King, “RedAlert: Determinacy Inference for Prolog”, Theory and Practice of Logic Programming, vol.11, no.4-5. pp.182–196.
* [7] E. Komendantskaya and V. Komendantsky, “On uniform proof-theoretical operational semantics for logic programming”, In J.-Y. Beziau and A.Costa-Leite, editors, Perspectives on Universal Logic, pages 379–394. Polimetrica Publisher, 2007.
* [8] D. Miller, “A logical analysis of modules in logic programming”, Journal of Logic Programming, vol.6, pp.79–108, 1989.
* [9] D. Miller, G. Nadathur, F. Pfenning, and A. Scedrov, “Uniform proofs as a foundation for logic programming”, Annals of Pure and Applied Logic, vol.51, pp.125–157, 1991.
* [10] A. Porto, “A structured alternative to Prolog with simple compositional semantics”, Theory and Practice of Logic Programming, vol.11, No.4-5, pp.611-627, 2011.
* [11] A. Saurin, “Towards Ludics Programming: Interactive Proof Search”, International Conference on Logic Programming, pages 253–268. 2008.
* [12] M. D. Winikoff, “Logic Programming with Linear Logic”, PhD. Thesis, Univ. Melbourne, 1997\.
|
arxiv-papers
| 2012-11-29T15:04:31 |
2024-09-04T02:49:38.650103
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/",
"authors": "Keehang Kwon and Daeseong Kang",
"submitter": "Keehang Kwon",
"url": "https://arxiv.org/abs/1211.6940"
}
|
1211.6978
|
EXTENDED FERMIONIC $p$-ADIC $q$-INTEGRALS ON $\mathbb{Z}_{p}$ IN CONNECTION
WITH APPLICATIONS OF UMBRAL CALCULUS
By
†Serkan ARACI, †Mehmet ACIKGOZ and ‡Erdoğan ŞEN
†University of Gaziantep, Faculty of Arts and Science, Department of
Mathematics, 27310 Gaziantep, Turkey
‡Department of Mathematics, Faculty of Science and Letters, Namik Kemal
University, 59030 Tekirdağ, Turkey
[email protected]; [email protected]; [email protected]
Abstract
The purpose of this paper is to derive some applications of umbral calculus by
using extended fermionic $p$-adic $q$-integral on $\mathbb{Z}_{p}$. From those
applications, we derive some new interesting properties on the new family of
Euler numbers and polynomials. That is, a systemic study of the class of
Sheffer sequences in connection with generating function of the weighted
$q$-Euler polynomials are given in the present paper.
2010 Mathematics Subject Classification. Primary 05A10, 11B65; Secondary
11B68, 11B73.
Key Words and Phrases. Appell sequence, Sheffer sequence, Euler numbers and
polynomials, formal power series, fermionic $p$-adic $q$-integral on
$\mathbb{Z}_{p}$.
## 1\. Preliminaries
Suppose that $p$ be a fixed odd prime number. Throughout this work we use the
following notations, where $\mathbb{Z}_{p}$ we denote the ring of $p$-adic
rational integers, $\mathbb{Q}$ denotes the field of rational numbers,
$\mathbb{Q}_{p}$ denotes the field of $p$-adic rational numbers, and
$\mathbb{C}_{p}$ denotes the completion of algebraic closure of
$\mathbb{Q}_{p}$. Let $\mathbb{N}$ be the set of natural numbers and
$\mathbb{N}^{\ast}=\mathbb{N}\cup\left\\{0\right\\}$. The $p$-adic absolute
value is defined by $\left|p\right|_{p}=p^{-1}$. Also, we assume that
$\left|q-1\right|_{p}<1$ is an indeterminate. Let
$UD\left(\mathbb{Z}_{p}\right)$ be the space of uniformly differentiable
functions on $\mathbb{Z}_{p}$. For $f\in UD\left(\mathbb{Z}_{p}\right)$, the
fermionic $p$-adic $q$-integral on $\mathbb{Z}_{p}$ is defined by T. Kim, as
follows:
$I_{-q}\left(f\right)=\int_{\mathbb{Z}_{p}}f\left(\xi\right)d\mu_{-q}\left(\xi\right)=\lim_{n\rightarrow\infty}\frac{1}{\left[p^{n}\right]_{-q}}\sum_{\xi=0}^{p^{n}-1}\left(-1\right)^{\xi}f\left(\xi\right)q^{\xi}\text{.}$
(1.1)
where $\left[x\right]_{q}$ is $q$-analogue of $x$ defined by
$\left[x\right]_{q}=\frac{q^{x}-1}{q-1}\text{ {and}
}\left[x\right]_{-q}=\frac{1-\left(-q\right)^{x}}{1+q}\text{.}$
We want to note that $\lim_{q\rightarrow 1}\left[x\right]_{q}=x$ (for details,
see [1-31]).
By (1.1), we have
$qI_{-q}\left(f_{1}\right)+I_{-q}\left(f\right)=\left[2\right]_{q}f\left(0\right)$
(1.2)
where $f_{1}\left(\xi\right):=f\left(\xi+1\right)$ (for details, see [3],
[8]).
Let us consider Kim’s fermionic $p$-adic $q$-integral on $\mathbb{Z}_{p}$ in
the following form: for $\left|1-\zeta\right|_{p}<1$
$I_{-q}^{\zeta}\left(f\right)=\int_{\mathbb{Z}_{p}}\zeta^{\xi}f\left(\xi\right)d\mu_{-q}\left(\xi\right)=\lim_{n\rightarrow\infty}\frac{1}{\left[p^{n}\right]_{-q}}\sum_{\xi=0}^{p^{n}-1}\zeta^{\xi}f\left(\xi\right)\left(-1\right)^{\xi}q^{\xi}\text{,}$
(1.3)
where $I_{-1}^{\zeta}\left(f\right)$ are called extended fermionic $p$-adic
$q$-integral on $\mathbb{Z}_{p}$.
Let us now consider $f_{1}\left(\xi\right):=f\left(\xi+1\right)$, then we
develop as follows:
$\displaystyle-q\zeta I_{-q}^{\zeta}\left(f_{1}\right)$ $\displaystyle=$
$\displaystyle\lim_{n\rightarrow\infty}\frac{1}{\left[p^{n}\right]_{-q}}\sum_{\xi=0}^{p^{n}-1}\zeta^{\xi+1}\left(-1\right)^{\xi+1}f\left(\xi+1\right)q^{\xi+1}$
$\displaystyle=$ $\displaystyle
I_{-q}^{\zeta}\left(f\right)+\frac{\left[2\right]_{q}}{2}\lim_{n\rightarrow\infty}\left(-f\left(0\right)-\xi^{p^{n}}f\left(p^{n}\right)q^{p^{n}}\right)$
$\displaystyle=$ $\displaystyle
I_{-q}^{\zeta}\left(f\right)-\left[2\right]_{q}f\left(0\right)\text{.}$
Therefore, we have the following lemma.
###### Lemma 1.
For $f\in UD\left(\mathbb{Z}_{p}\right)$, we get
$q\zeta
I_{-q}^{\zeta}\left(f_{1}\right)+I_{-q}^{\zeta}\left(f\right)=\left[2\right]_{q}f\left(0\right)\text{.}$
Taking $f\left(\xi\right)=e^{t\left(x+\xi\right)}\in UD(\mathbb{Z}_{p})$ in
Lemma 1, then we introduce the following expression:
$\int_{\mathbb{Z}_{p}}\zeta^{\xi}e^{t\left(x+\xi\right)}d\mu_{-q}\left(\xi\right)=\frac{\left[2\right]_{q}}{q\zeta
e^{t}+1}e^{tx}=\sum_{n=0}^{\infty}E_{n,\zeta}^{q}\left(x\right)\frac{t^{n}}{n!}\text{,}$
(1.4)
where we call $E_{n,\zeta}^{q}\left(x\right)$ as weighted $q$-Euler
polynomials. In the special case, $x=0$,
$E_{n,\zeta}^{q}\left(0\right):=E_{n,\zeta}^{q}$ are called weighted $q$-Euler
numbers and the relation between weighted $q$-Euler numbers and weighted
$q$-Euler polynomials are given by
$E_{n,\zeta}^{q}\left(x\right)=\sum_{l=0}^{n}\binom{n}{l}x^{l}E_{n-l,\zeta}^{q}=\left(x+E_{\zeta}^{q}\right)^{n}\text{,}$
(1.5)
with the usual of replacing $\left(E_{\zeta}^{q}\right)^{n}$ by
$E_{n,\zeta}^{q}$ is used. By (1.4), we note that
$E_{n,\zeta}^{q}=\int_{\mathbb{Z}_{p}}\zeta^{\xi}\xi^{n}d\mu_{-q}\left(\xi\right)\text{
{and}
}E_{n,\zeta}^{q}\left(x\right)=\int_{\mathbb{Z}_{p}}\zeta^{\xi}\left(x+\xi\right)^{n}d\mu_{-q}\left(\xi\right)\text{.}$
(1.6)
By (1.4), we have
$E_{n,\zeta}^{q}\left(x\right)=\left[2\right]_{q}\sum_{m=0}^{\infty}\left(-1\right)^{m}q^{m}\zeta^{m}\left(m+x\right)^{n}\text{,
{for} }n\in\mathbb{N}^{\ast}\text{.}$ (1.7)
From this, we can define weighted $q$-Zeta function as follows:
$\lambda\left(s,x:q:\zeta\right)=\left[2\right]_{q}\sum_{m=0}^{\infty}\frac{\left(-1\right)^{m}q^{m}\zeta^{m}}{\left(m+x\right)^{s}}$
(1.8)
By (1.7) and (1.8), we derive the following equation (1.9):
$\lambda\left(-n,x:q:\zeta\right)=E_{n,\zeta}^{q}\left(x\right),\text{ {for
any} }n\in\mathbb{N}^{\ast}\text{.}$ (1.9)
When we set as $q=\zeta=1$ in (1.9) which reduces to
$\zeta_{E}\left(-n,x\right)=E_{n}\left(x\right)$
which is well known in [11].
By (1.3) and (1.4), we compute
$\displaystyle\int_{\mathbb{Z}_{p}}\zeta^{\xi}\left(x+\xi\right)^{n}d\mu_{-q}\left(\xi\right)$
$\displaystyle=$
$\displaystyle\lim_{m\rightarrow\infty}\frac{1}{\left[dp^{m}\right]_{-q}}\sum_{\xi=0}^{dp^{m}-1}\left(-1\right)^{\xi}\zeta^{\xi}\left(x+\xi\right)^{n}q^{\xi}$
$\displaystyle=$
$\displaystyle\frac{d^{n}}{\left[d\right]_{-q}}\sum_{j=0}^{d-1}\left(-1\right)^{j}\zeta^{j}q^{j}\left(\lim_{m\rightarrow\infty}\frac{1}{\left[p^{m}\right]_{\left(-q\right)^{d}}}\sum_{\xi=0}^{p^{m}-1}\left(-1\right)^{\xi}\left(\zeta^{d}\right)^{\xi}\left(q^{d}\right)^{\xi}\left(\frac{x+j}{d}+\xi\right)^{n}\right)$
$\displaystyle=$
$\displaystyle\frac{d^{n}}{\left[d\right]_{-q}}\sum_{j=0}^{d-1}\left(-1\right)^{j}\zeta^{j}q^{j}\int_{\mathbb{Z}_{p}}\zeta^{d\xi}\left(\frac{x+j}{d}+\xi\right)^{n}d\mu_{-q^{d}}\left(\xi\right)\text{,}$
where $d$ is an odd natural number. So from the above
$\int_{\mathbb{Z}_{p}}\zeta^{\xi}\left(x+\xi\right)^{n}d\mu_{-q}\left(\xi\right)=\frac{d^{n}}{\left[d\right]_{-q}}\sum_{j=0}^{d-1}\left(-1\right)^{j}\zeta^{j}q^{j}\int_{\mathbb{Z}_{p}}\zeta^{d\xi}\left(\frac{x+j}{d}+\xi\right)^{n}d\mu_{-q^{d}}\left(\xi\right)\text{.}$
(1.10)
By (1.6) and (1.10), we get
$E_{n,\zeta}^{q}\left(dx\right)=\frac{d^{n}}{\left[d\right]_{-q}}\sum_{j=0}^{d-1}\left(-1\right)^{j}\zeta^{j}q^{j}E_{n,\zeta^{d}}^{q^{d}}\left(x+\frac{j}{d}\right)\text{,}$
(1.11)
which plays an important role for studying regarding Measure theory on
$p$-adic analysis.
Let we use the following notations, where $\mathbb{C}$ denotes the set of
complex numbers, $\mathcal{F}$ denotes the set of all formal power series in
the variable $t$ over $\mathbb{C}$ with
$\mathcal{F}=\left\\{f\left(t\right)=\sum_{k=0}^{\infty}a_{k}\frac{t^{k}}{k!}\mid
a_{k}\in\mathbb{C}\right\\},$
$\mathcal{P}=\mathbb{C}\left[x\right]$ and $\mathcal{P}^{\ast}$ denotes the
vector space of all linear functional on $\mathcal{P}$, $\left\langle L\mid
p\left(x\right)\right\rangle$ denotes the action of the linear functional $L$
on the polynomial $p\left(x\right)$, and it is well-known that the vector
space operation on $\mathcal{P}^{\ast}$ is defined by
$\displaystyle\left\langle L+M\mid p\left(x\right)\right\rangle$
$\displaystyle=$ $\displaystyle\left\langle L\mid
p\left(x\right)\right\rangle+\left\langle M\mid p\left(x\right)\right\rangle,$
$\displaystyle\left\langle cL\mid p\left(x\right)\right\rangle$
$\displaystyle=$ $\displaystyle c\left\langle L\mid
p\left(x\right)\right\rangle,$
where $c$ is any constant in $\mathbb{C}$ (for details, see [12], [19], [13],
[14], [31]).
The formal power series are known by
$f\left(t\right)=\sum_{k=0}^{\infty}a_{k}\frac{t^{k}}{k!}\in\mathcal{F}$
which describes a linear functional on $\mathcal{P}$ as $\left\langle
f\left(t\right)\mid x^{n}\right\rangle=a_{n}$ for all $n\geq 0$ (for details,
see [12], [19], [13], [14], [31]). In addition to
$\left\langle t^{k}\mid x^{n}\right\rangle=n!\delta_{n,k},$ (1.12)
where $\delta_{n,k}$ is the Kronecker delta. If we take as
$f_{L}\left(t\right)=\sum_{k=0}^{\infty}\left\langle L\mid
x^{k}\right\rangle\frac{t^{k}}{k!},$
then we obtain
$\left\langle f_{L}\left(t\right)\mid x^{n}\right\rangle=\left\langle L\mid
x^{n}\right\rangle$
and so as linear functionals $L=f_{L}\left(t\right)$ (see [12], [19], [13],
[14], [31]). Additionally, the map $L\rightarrow f_{L}\left(t\right)$ is a
vector space isomorphism from $\mathcal{P}^{\ast}$ onto $\mathcal{F}$.
Henceforth, $\mathcal{F}$ will denote both the algebra of the formal power
series in $t$ and the vector space of all linear functionals on $\mathcal{P}$,
and so an element $f\left(t\right)$ of $\mathcal{F}$ will be thought of as
both a formal power series and a linear functional. $\mathcal{F}$ will be
called as umbral algebra (see [12], [19], [13], [14], [31]).
It is well-known that $\left\langle e^{yt}\mid x^{n}\right\rangle=y^{n}$.
Then, leads to the following
$\left\langle e^{yt}\mid p\left(x\right)\right\rangle=p\left(y\right)$
(see [12], [19], [13], [14], [15], [31]). We want to note that for all
$f\left(t\right)$ in $\mathcal{F}$
$f\left(t\right)=\sum_{k=0}^{\infty}\left\langle f\left(t\right)\mid
x^{k}\right\rangle\frac{t^{k}}{k!}$ (1.13)
and for all polynomial $p\left(x\right)$,
$p\left(x\right)=\sum_{k=0}^{\infty}\left\langle t^{k}\mid
p\left(x\right)\right\rangle\frac{x^{k}}{k!},$ (1.14)
(for details, see [12], [19], [13], [14], [31]). The order
$o\left(f\left(t\right)\right)$ of the power series $f\left(t\right)\neq 0$ is
the smallest integer $k$ for which $a_{k}$ does not vanish. It is considered
$o\left(f\left(t\right)\right)=\infty$ if $f\left(t\right)=0$. We see that
$o\left(f\left(t\right)g\left(t\right)\right)=o\left(f\left(t\right)\right)+o\left(g\left(t\right)\right)$
and
$o\left(f\left(t\right)+g\left(t\right)\right)\geq\min\left\\{o\left(f\left(t\right)\right),o\left(g\left(t\right)\right)\right\\}$.
The series $f\left(t\right)$ has a multiplicative inverse, denoted by
$f\left(t\right)^{-1}$ or $\frac{1}{f\left(t\right)}$, if and only if
$o\left(f\left(t\right)\right)=0$. Such series is called an invertible series.
A series $f\left(t\right)$ for which $o\left(f\left(t\right)\right)=1$ is
called a delta series (see [12], [19], [13], [14], [15], [31]). For
$f\left(t\right),g\left(t\right)\in\mathcal{F}$, we have $\left\langle
f\left(t\right)g\left(t\right)\mid p\left(x\right)\right\rangle=\left\langle
f\left(t\right)\mid g\left(t\right)p\left(x\right)\right\rangle$.
A delta series $f\left(t\right)$ has a compositional inverse
$\overline{f}\left(t\right)$ such that
$f\left(\overline{f}\left(t\right)\right)=\overline{f}\left(f\left(t\right)\right)=t$.
For $f\left(t\right),g\left(t\right)\in\mathcal{F}$ , we have $\left\langle
f\left(t\right)g\left(t\right)\mid p\left(x\right)\right\rangle=\left\langle
f\left(t\right)\mid g\left(t\right)p\left(x\right)\right\rangle$. By (1.13),
we have
$p^{\left(k\right)}\left(x\right)=\frac{d^{k}p\left(x\right)}{dx^{k}}=\sum_{l=k}^{\infty}\frac{\left\langle
t^{l}\mid
p\left(x\right)\right\rangle}{l!}l\left(l-1\right)\cdots\left(l-k+1\right)x^{l-k}\text{.}$
(1.15)
Thus, notice that
$p^{\left(k\right)}\left(0\right)=\left\langle t^{k}\mid
p\left(x\right)\right\rangle=\left\langle 1\mid
p^{\left(k\right)}\left(x\right)\right\rangle\text{.}$ (1.16)
By (1.15), we have
$t^{k}p\left(x\right)=p^{\left(k\right)}\left(x\right)=\frac{d^{k}p\left(x\right)}{dx^{k}}\text{.}$
(1.17)
So from the above
$e^{yt}p\left(x\right)=p\left(x+y\right)\text{.}$ (1.18)
Let $S_{n}\left(x\right)$ be a polynomial with $\deg S_{n}\left(x\right)=n$.
Let $f\left(t\right)$ be a delta series and let $g\left(t\right)$ be an
invertible series. Then there exists a unique sequence $S_{n}\left(x\right)$
of polynomials such that $\left\langle g\left(t\right)f\left(t\right)^{k}\mid
S_{n}\left(x\right)\right\rangle=n!\delta_{n,k}$ for all $n,k\geq 0$. The
sequence $S_{n}\left(x\right)$ is called the Sheffer sequence for
$\left(g\left(t\right),f\left(t\right)\right)$ or that $S_{n}\left(t\right)$
is Sheffer for $\left(g\left(t\right),f\left(t\right)\right)$.
The Sheffer sequence for $\left(1,f\left(t\right)\right)$ is called the
associated sequence for $f\left(t\right)$ or $S_{n}\left(x\right)$ is
associated to $f\left(t\right)$. The sheffer sequence for
$\left(g\left(t\right),t\right)$ is called the Appell sequence for
$g\left(t\right)$ or $S_{n}\left(x\right)$ is Appell for $g\left(t\right)$.
Let $p\left(x\right)\in\mathcal{P}$. Then we have
$\displaystyle\left\langle f\left(t\right)\mid xp\left(x\right)\right\rangle$
$\displaystyle=$ $\displaystyle\left\langle\partial_{t}f\left(t\right)\mid
p\left(x\right)\right\rangle=\left\langle f{\acute{}}\left(t\right)\mid
p\left(x\right)\right\rangle,$ (1.19) $\displaystyle\left\langle e^{y}+1\mid
p\left(x\right)\right\rangle$ $\displaystyle=$ $\displaystyle
p\left(y\right)+p\left(0\right),\text{ (see
\cite[cite]{[\@@bibref{}{Roman}{}{}]}).}$
Let $S_{n}\left(x\right)$ be sheffer for
$\left(g\left(t\right),f\left(t\right)\right)$. Then
$\displaystyle h\left(t\right)$ $\displaystyle=$
$\displaystyle\sum_{k=0}^{\infty}\frac{\left\langle h\left(t\right)\mid
S_{k}\left(x\right)\right\rangle}{k!}g\left(t\right)f\left(t\right)^{k},\text{
}h\left(t\right)\in\mathcal{F}$ $\displaystyle p\left(x\right)$
$\displaystyle=$ $\displaystyle\sum_{k=0}^{\infty}\frac{\left\langle
g\left(t\right)f\left(t\right)^{k}\mid
p\left(x\right)\right\rangle}{k!}S_{k}\left(x\right),\text{
}p\left(x\right)\in\mathcal{P},$
$\displaystyle\frac{1}{g\left(\overline{f}\left(t\right)\right)}e^{y\overline{f}\left(t\right)}$
$\displaystyle=$
$\displaystyle\sum_{k=0}^{\infty}S_{k}\left(y\right)\frac{t^{k}}{k!},\text{
for all }y\in\mathbb{C},$ (1.20) $\displaystyle
f\left(t\right)S_{n}\left(x\right)$ $\displaystyle=$ $\displaystyle
nS_{n-1}\left(x\right)\text{.}$
Also, it is well known in [31] that
$\left\langle f_{1}\left(t\right)f_{2}\left(t\right)\cdots
f_{m}\left(t\right)\mid
x^{n}\right\rangle=\sum\binom{n}{i_{1},\cdots,i_{m}}\left\langle
f_{1}\left(t\right)\mid x^{i_{1}}\right\rangle\cdots\left\langle
f_{m}\left(t\right)\mid x^{i_{m}}\right\rangle$ (1.21)
where
$f_{1}\left(t\right),f_{2}\left(t\right),\cdots,f_{m}\left(t\right)\in\mathcal{F}$
and the sum is over all nonnegative integers $i_{1},\cdots,i_{m}$ such that
$i_{1}+\cdots+i_{m}=n$ (see [31]).
In [19] and [20], Dere and Simsek have studied applications of umbral algebra
to special functions. They gave some new interesting links for further works
of many mathematicians in Analytic numbers theory and in modern classical
umbral calculus. Kim et al. have given some properties of umbral calculus for
Frobenius-Euler polynomials [12], Euler polynomials [13] and other special
functions [14]. Also, they investigated some new applications of umbral
calculus associated with $p$-adic invariants integral on $\mathbb{Z}_{p}$ in
[1].
By the same motivation, we also give some applications of umbral calculus by
using extended fermionic $p$-adic $q$-integral on $\mathbb{Z}_{p}$. From those
applications, we derive some interesting equalities on weighted $q$-Euler
numbers, weighted $q$-Euler polynomials and weighted $q$-Euler polynomials of
order $k$.
## 2\. On the extended fermionic $p$-adic $q$-integrals on $\mathbb{Z}_{p}$
in connection with applications of umbral calculus
Suppose that $S_{n}\left(x\right)$ is an Appell sequence for
$g\left(t\right)$. Then, by (1.20), we have
$\frac{1}{g\left(t\right)}x^{n}=S_{n}\left(x\right)\text{ if and only if
}x^{n}=g\left(t\right)S_{n}\left(x\right)\text{, }\left(n\geq
0\right)\text{.}$ (2.1)
Let us contemplate as follows:
$g_{q}\left(t\mid\zeta\right)=\frac{q\zeta
e^{t}+1}{\left[2\right]_{q}}\in\mathcal{F}\text{.}$
Therefore, we easily notice that $g\left(t\right)$ is an invertible series. By
(2.1), we have
$\sum_{n=0}^{\infty}E_{n,\zeta}^{q}\left(x\right)\frac{t^{n}}{n!}=\frac{1}{g_{q}\left(t\mid\zeta\right)}e^{xt}\text{.}$
(2.2)
By (2.2), we have
$\frac{1}{g_{q}\left(t\mid\zeta\right)}x^{n}=E_{n,\zeta}^{q}\left(x\right)\text{.}$
(2.3)
Also, by (1.20), we have
$tE_{n,\zeta}^{q}\left(x\right)=\left(E_{n,\zeta}^{q}\left(x\right)\right){\acute{}}=nE_{n-1,\zeta}^{q}\left(x\right)\text{,}$
(2.4)
By (2.3) and (2.4), we have the following proposition.
###### Proposition 1.
For $n\geq 0$, $E_{n,\zeta}^{q}\left(x\right)$ is an Appell sequence for
$g_{q}\left(t\mid\zeta\right)=\frac{\zeta qe^{t}+1}{\left[2\right]_{q}}$.
By (1.6), we derive that
$\displaystyle\sum_{n=1}^{\infty}E_{n,\zeta}^{q}\left(x\right)\frac{t^{n}}{n!}$
$\displaystyle=\frac{xg_{q}\left(t\mid\zeta\right)e^{xt}-g{\acute{}}_{q}\left(t\mid\zeta\right)e^{xt}}{g\left(t\right)^{2}}$
(2.5)
$\displaystyle=\sum_{n=0}^{\infty}\left(x\frac{1}{g_{q}\left(t\mid\zeta\right)}x^{n}-\frac{g{\acute{}}_{q}\left(t\mid\zeta\right)}{g_{q}\left(t\mid\zeta\right)}\frac{1}{g_{q}\left(t\mid\zeta\right)}x^{n}\right)\frac{t^{n}}{n!}$
Because of (2.3) and (2.5), we discover the following:
$E_{n+1,\zeta}^{q}\left(x\right)=xE_{n,\zeta}^{q}\left(x\right)-\frac{g{\acute{}}_{q}\left(t\mid\zeta\right)}{g_{q}\left(t\mid\zeta\right)}E_{n,\zeta}^{q}\left(x\right)\text{.}$
Therefore, we get the following theorem.
###### Theorem 1.
Let $g_{q}\left(t\mid\zeta\right)=\frac{\zeta qe^{t}+1}{2}\in\mathcal{F}$.
Then we have for $n\geq 0:$
$E_{n+1,\zeta}^{q}\left(x\right)=\left(x-\frac{g{\acute{}}_{q}\left(t\mid\zeta\right)}{g_{q}\left(t\mid\zeta\right)}\right)E_{n,\zeta}^{q}\left(x\right)\text{.}$
(2.6)
Moreover,
$\lambda\left(-n-1,x:q:\zeta\right)=\left(x-\frac{g{\acute{}}_{q}\left(t\mid\zeta\right)}{g_{q}\left(t\mid\zeta\right)}\right)\lambda\left(-n,x:q:\zeta\right)\text{.}$
where
$g{\acute{}}_{q}\left(t\mid\zeta\right)=\frac{dg_{q}\left(t\mid\zeta\right)}{dt}$.
From (1.6), it is easy to show that
$\sum_{n=0}^{\infty}\left(\zeta
qE_{n,\zeta}^{q}\left(x+1\right)+E_{n,\zeta}^{q}\left(x\right)\right)\frac{t^{n}}{n!}=\sum_{n=0}^{\infty}\left(\left[2\right]_{q}x^{n}\right)\frac{t^{n}}{n!}\text{.}$
By comparing the coefficients in the both sides of $\frac{t^{n}}{n!}$ on the
above, we develop the following:
$\zeta
qE_{n,\zeta}^{q}\left(x+1\right)+E_{n,\zeta}^{q}\left(x\right)=\left[2\right]_{q}x^{n}\text{.}$
(2.7)
From Theorem 1, we get the following equation (2.8):
$g_{q}\left(t\mid\zeta\right)E_{n+1,\zeta}^{q}\left(x\right)=g_{q}\left(t\mid\zeta\right)xE_{n,\zeta}^{q}\left(x\right)-g{\acute{}}_{q}\left(t\mid\zeta\right)E_{n,\zeta}^{q}\left(x\right)\text{.}$
(2.8)
So from above
$\left(\zeta qe^{t}+1\right)E_{n+1,\zeta}^{q}\left(x\right)=\left(\zeta
qe^{t}+1\right)xE_{n,\zeta}^{q}\left(x\right)-\zeta
qe^{t}E_{n,\zeta}^{q}\left(x\right)\text{.}$
Thus, we can write the following equation:
$\zeta
qE_{n+1,\zeta}^{q}\left(x+1\right)+E_{n+1,\zeta}^{q}\left(x\right)=\zeta
q\left(x+1\right)E_{n,\zeta}^{q}\left(x+1\right)+xE_{n,\zeta}^{q}\left(x\right)-\zeta
qE_{n,\zeta}^{q}\left(x+1\right)\text{.}$ (2.9)
From (2.7) (2.8) and (2.9), we can state following theorem.
###### Theorem 2.
For $n\geq 0$, then we have
$\zeta
qE_{n,\zeta}^{q}\left(x+1\right)+E_{n,\zeta}^{q}\left(x\right)=\left[2\right]_{q}x^{n}\text{.}$
(2.10)
###### Remark 1.
Assume that $S_{n}\left(x\right)$ is Sheffer sequence for
$\left(g\left(t\right),f\left(t\right)\right)$. Then Sheffer identity is
introduced by
$S_{n}\left(x+y\right)=\sum_{k=0}^{n}\binom{n}{k}P_{k}\left(y\right)S_{n-k}\left(x\right)=\sum_{k=0}^{n}\binom{n}{k}P_{k}\left(x\right)S_{n-k}\left(y\right)\text{,}$
(2.11)
where $P_{k}\left(y\right)=S_{k}\left(y\right)g\left(t\right)$ is associated
to $f\left(t\right)$ (for details, see [13], [19], [20], [31]).
On account of (1.4) and (2.11), then we have
$\displaystyle E_{n,\zeta}^{q}\left(x+y\right)$ $\displaystyle=$
$\displaystyle\sum_{k=0}^{n}\binom{n}{k}P_{k}\left(y\right)S_{n-k}\left(x\right)$
$\displaystyle=$
$\displaystyle\sum_{k=0}^{n}\binom{n}{k}E_{n-k,\zeta}^{q}\left(y\right)x^{k}\text{.}$
So we have
$E_{n,\zeta}^{q}\left(x+y\right)=\sum_{k=0}^{n}\binom{n}{k}E_{n-k,\zeta}^{q}\left(y\right)x^{k}\text{.}$
By (1.4), we easily see for $\alpha\left(\neq 0\right)\in\mathbb{C}:$
$E_{n,\zeta}^{q}\left(\alpha
x\right)=\frac{g_{q}\left(t\mid\zeta\right)}{g_{q}\left(\frac{t}{\alpha}\mid\zeta\right)}E_{n,\zeta}^{q}\left(x\right)\text{.}$
(2.12)
From (1.11) and (2.12), we readily derive for $d\equiv
1\left(\mathop{\mathrm{m}od}2\right):$
$\frac{g_{q}\left(t\mid\zeta\right)}{g_{q}\left(\frac{t}{d}\mid\zeta\right)}E_{n,\zeta}^{q}\left(x\right)=\frac{d^{n}}{\left[d\right]_{-q}}\sum_{j=0}^{d-1}\left(-1\right)^{j}\zeta^{j}q^{j}E_{n,\zeta^{d}}^{q^{d}}\left(x+\frac{j}{d}\right)\text{.}$
Let us consider the linear functional $f\left(t\right)$ that satisfies:
$\left\langle f\left(t\right)\mid
p\left(x\right)\right\rangle=\int_{\mathbb{Z}_{p}}\zeta^{\xi}p\left(\xi\right)d\mu_{-q}\left(\xi\right),$
(2.13)
for all polynomials $p\left(x\right)$. From (2.13), we readily see that
$f\left(t\right)=\sum_{n=0}^{\infty}\frac{\left\langle f\left(t\right)\mid
x^{n}\right\rangle}{n!}t^{n}=\sum_{n=1}^{\infty}\left(\int_{\mathbb{Z}_{p}}\zeta^{\xi}\xi^{n}d\mu_{-q}\left(\xi\right)\right)\frac{t^{n}}{n!}=\int_{\mathbb{Z}_{p}}\zeta^{\xi}e^{\xi
t}d\mu_{-q}\left(\xi\right)\text{.}$ (2.14)
Thus, we have
$f\left(t\right)=\int_{\mathbb{Z}_{p}}\zeta^{\xi}e^{\xi
t}d\mu_{-q}\left(\xi\right)=\frac{\left[2\right]_{q}}{\zeta qe^{t}+1}\text{.}$
(2.15)
Therefore, by (2.13) and (2.15), we arrive at the following theorem.
###### Theorem 3.
For $n\geq 0$, then we have
$\left\langle f\left(t\right)\mid
p\left(x\right)\right\rangle=\int_{\mathbb{Z}_{p}}\zeta^{\xi}p\left(\xi\right)d\mu_{-q}\left(\xi\right)\text{.}$
(2.16)
Also,
$\left\langle\frac{\left[2\right]_{q}}{\zeta qe^{t}+1}\mid
p\left(x\right)\right\rangle=\int_{\mathbb{Z}_{p}}\zeta^{\xi}p\left(\xi\right)d\mu_{-q}\left(\xi\right).$
(2.17)
Obviously that
$E_{n,\zeta}^{q}=\left\langle\int_{\mathbb{Z}_{p}}\zeta^{\xi}e^{\xi
t}d\mu_{-q}\left(\xi\right)\mid x^{n}\right\rangle\text{.}$ (2.18)
From (1.6) and (2.18), we see that
$\sum_{n=0}^{\infty}\left(\int_{\mathbb{Z}_{p}}\zeta^{\xi}\left(x+\xi\right)^{n}d\mu_{-q}\left(\xi\right)\right)\frac{t^{n}}{n!}=\int_{\mathbb{Z}_{p}}\zeta^{\xi}e^{\left(x+\xi\right)t}d\mu_{-q}\left(\xi\right)=\sum_{n=0}^{\infty}\left(\int_{\mathbb{Z}_{p}}\zeta^{\xi}e^{\xi
t}d\mu_{-q}\left(\xi\right)x^{n}\right)\frac{t^{n}}{n!}\text{.}$ (2.19)
By (1.6) and (2.20), we see that for $n\in\mathbb{N}^{\ast}$:
$E_{n,\zeta}^{q}\left(x\right)=\int_{\mathbb{Z}_{p}}\zeta^{\xi}\left(x+\xi\right)^{n}d\mu_{-q}\left(\xi\right)=\int_{\mathbb{Z}_{p}}\zeta^{\xi}e^{\xi
t}d\mu_{-q}\left(\xi\right)x^{n}\text{.}$ (2.20)
Consequently, we obtain the following theorem.
###### Theorem 4.
For $p\left(x\right)\in\mathcal{P}$, then we have
$\displaystyle\int_{\mathbb{Z}_{p}}\zeta^{\xi}p\left(x+\xi\right)d\mu_{-q}\left(\xi\right)$
$\displaystyle=$ $\displaystyle\int_{\mathbb{Z}_{p}}\zeta^{\xi}e^{\xi
t}d\mu_{-q}\left(\xi\right)p\left(x\right)$ $\displaystyle=$
$\displaystyle\frac{\left[2\right]_{q}}{\zeta
qe^{t}+1}p\left(x\right)\text{.}$
That is:
$E_{n,\zeta}^{q}\left(x\right)=\int_{\mathbb{Z}_{p}}\zeta^{\xi}e^{\xi
t}d\mu_{-q}\left(\xi\right)x^{n}=\frac{\left[2\right]_{q}}{\zeta
qe^{t}+1}x^{n}\text{.}$ (2.22)
For $\left|1-\zeta\right|_{p}<1$, we introduce weighted $q$-Euler polynomials
of order $k$ as follows:
$\displaystyle\underset{\text{k-times}}{\underbrace{\int_{\mathbb{Z}_{p}}\cdots\int_{\mathbb{Z}_{p}}}}\zeta^{\xi_{1}+\cdots+\xi_{k}}e^{\left(\xi_{1}+\cdots+\xi_{k}+x\right)t}d\mu_{-q}\left(\xi_{1}\right)\cdots
d\mu_{-q}\left(\xi_{k}\right)$ $\displaystyle=$
$\displaystyle\left(\frac{\left[2\right]_{q}}{q\zeta
e^{t}+1}\right)^{k}e^{xt}$ $\displaystyle=$
$\displaystyle\sum_{n=0}^{\infty}E_{n,\zeta}^{\left(k\right)}\left(x\mid
q\right)\frac{t^{n}}{n!}\text{.}$
where, for $x=0$, $E_{n,\zeta}^{\left(k\right)}\left(0\mid
q\right):=E_{n,\zeta}^{\left(k\right)}\left(q\right)$ are called weighted
$q$-Euler numbers of order $k$.
By (2), we have
$\displaystyle\underset{\text{k-times}}{\underbrace{\int_{\mathbb{Z}_{p}}\cdots\int_{\mathbb{Z}_{p}}}}\zeta^{\xi_{1}+\cdots+\xi_{k}}\left(\xi_{1}+\cdots+\xi_{k}+x\right)^{n}d\mu_{-q}\left(\xi_{1}\right)\cdots
d\mu_{-q}\left(\xi_{k}\right)$ $\displaystyle=$
$\displaystyle\sum_{i_{1}+\cdots+i_{k}=n}\binom{n}{i_{1},\cdots,i_{m}}\int_{\mathbb{Z}_{p}}\zeta^{\xi_{1}}\xi_{1}^{i_{1}}d\mu_{-q}\left(\xi_{1}\right)\cdots\int_{\mathbb{Z}_{p}}\zeta^{\xi_{k}}\xi_{k}^{i_{k}}d\mu_{-q}\left(\xi_{k}\right)$
$\displaystyle=$
$\displaystyle\sum_{i_{1}+\cdots+i_{k}=n}\binom{n}{i_{1},\cdots,i_{m}}E_{i_{1},\zeta}^{q}\cdots
E_{i_{k},\zeta}^{q}=E_{n,\zeta}^{\left(k\right)}\left(x\mid q\right)\text{.}$
Thanks to (2) and (2), we have
$E_{n,\zeta}^{\left(k\right)}\left(x\mid
q\right)=\sum_{l=0}^{n}\binom{n}{l}x^{l}E_{n-l,\zeta}^{\left(k\right)}\left(q\right)\text{.}$
(2.25)
From (2) and (2.25), we notice that $E_{n,\zeta}^{\left(k\right)}\left(x\mid
q\right)$ is a monic polynomial of degree $n$ with coefficients in
$\mathbb{Q}$. For $k\in\mathbb{N}$, let us assume that
$\displaystyle g_{q}^{\left(k\right)}\left(t\mid\zeta\right)$ $\displaystyle=$
$\displaystyle\left(\underset{\text{k-times}}{\underbrace{\int_{\mathbb{Z}_{p}}\cdots\int_{\mathbb{Z}_{p}}}}\zeta^{\xi_{1}+\cdots+\xi_{k}}e^{\left(\xi_{1}+\cdots+\xi_{k}\right)t}d\mu_{-q}\left(\xi_{1}\right)\cdots
d\mu_{-q}\left(\xi_{k}\right)\right)^{-1}$ $\displaystyle=$
$\displaystyle\left(\frac{\zeta
qe^{t}+1}{\left[2\right]_{q}}\right)^{k}\text{.}$
From (2), we note that $g_{q}^{\left(k\right)}\left(t\mid\zeta\right)$ is an
invertible series. On account of (2) and (2), we readily derive that
$\displaystyle\frac{1}{g_{q}^{\left(k\right)}\left(t\mid\zeta\right)}e^{xt}$
$\displaystyle=\underset{\text{k-times}}{\underbrace{\int_{\mathbb{Z}_{p}}\cdots\int_{\mathbb{Z}_{p}}}}\zeta^{\xi_{1}+\cdots+\xi_{k}}e^{\left(\xi_{1}+\cdots+\xi_{k}+x\right)t}d\mu_{-q}\left(\xi_{1}\right)\cdots
d\mu_{-q}\left(\xi_{k}\right)$ (2.27)
$\displaystyle=\sum_{n=0}^{\infty}E_{n,\zeta}^{\left(k\right)}\left(x\mid
q\right)\frac{t^{n}}{n!}\text{.}$
Also, we note that
$tE_{n,\zeta}^{\left(k\right)}\left(x\mid
q\right)=nE_{n-1,\zeta}^{\left(k\right)}\left(x\mid q\right)\text{.}$ (2.28)
By (2.27) and (2.28), we easily see that
$E_{n,\zeta}^{\left(k\right)}\left(x\mid q\right)$ is an Appell sequence for
$g_{q}^{\left(k\right)}\left(t\mid\zeta\right)$. Then, by (2.27) and (2.28),
we get the following theorem.
###### Theorem 5.
For $p\left(x\right)\in\mathcal{P}$ and $k\in\mathbb{N}$, we have
$\underset{\text{k-times}}{\underbrace{\int_{\mathbb{Z}_{p}}\cdots\int_{\mathbb{Z}_{p}}}}\zeta^{\xi_{1}+\cdots+\xi_{k}}p\left(\xi_{1}+\cdots+\xi_{k}+x\right)d\mu_{-q}\left(\xi_{1}\right)\cdots
d\mu_{-q}\left(\xi_{k}\right)=\left(\frac{\left[2\right]_{q}}{\zeta
qe^{t}+1}\right)^{k}p\left(x\right)\text{.}$ (2.29)
In the special case, the weighted $q$-Euler polynomials of degree $k$ are
derived by
$E_{n,\zeta}^{\left(k\right)}\left(x\mid q\right)=\left(\frac{2}{\zeta
qe^{t}+1}\right)^{k}x^{n}=\int_{\mathbb{Z}_{p}}\cdots\int_{\mathbb{Z}_{p}}\zeta^{\xi_{1}+\cdots+\xi_{k}}e^{\left(\xi_{1}+\cdots+\xi_{k}\right)t}d\mu_{-q}\left(\xi_{1}\right)\cdots
d\mu_{-q}\left(\xi_{k}\right)x^{n}$
Thus, we get
$E_{n,\zeta}^{\left(k\right)}\left(x\mid q\right)\sim\left(\left(\frac{\zeta
qe^{t}+1}{\left[2\right]_{q}}\right)^{k},t\right)\text{.}$
Let us take the linear functional $f^{\left(k\right)}\left(t\right)$ that
satisfies
$\left\langle f^{\left(k\right)}\left(t\right)\mid
p\left(x\right)\right\rangle=\int_{\mathbb{Z}_{p}}\cdots\int_{\mathbb{Z}_{p}}\zeta^{\xi_{1}+\cdots+\xi_{k}}p\left(\xi_{1}+\cdots+\xi_{k}\right)d\mu_{-q}\left(\xi_{1}\right)\cdots
d\mu_{-q}\left(\xi_{k}\right)\text{,}$ (2.30)
for all polynomials $p\left(x\right)$. Therefore, we compute as follows:
$\displaystyle f^{\left(k\right)}\left(t\right)$ $\displaystyle=$
$\displaystyle\sum_{n=0}^{\infty}\frac{\left\langle
f^{\left(k\right)}\left(t\right)\mid x^{n}\right\rangle}{n!}t^{n}$
$\displaystyle=$
$\displaystyle\sum_{n=0}^{\infty}\left(\int_{\mathbb{Z}_{p}}\cdots\int_{\mathbb{Z}_{p}}\zeta^{\xi_{1}+\cdots+\xi_{k}}\left(\xi_{1}+\cdots+\xi_{k}\right)^{n}d\mu_{-q}\left(\xi_{1}\right)\cdots
d\mu_{-q}\left(\xi_{k}\right)\right)\frac{t^{n}}{n!}$ $\displaystyle=$
$\displaystyle\int_{\mathbb{Z}_{p}}\cdots\int_{\mathbb{Z}_{p}}\zeta^{\xi_{1}+\cdots+\xi_{k}}e^{\left(\xi_{1}+\cdots+\xi_{k}\right)t}d\mu_{-q}\left(\xi_{1}\right)\cdots
d\mu_{-q}\left(\xi_{k}\right)$ $\displaystyle=$
$\displaystyle\left(\frac{\left[2\right]_{q}}{\zeta
qe^{t}+1}\right)^{k}\text{.}$
Therefore, the following theorem can be expressed.
###### Theorem 6.
For $p\left(x\right)\in\mathcal{P}$, we have
$\displaystyle\left\langle\int_{\mathbb{Z}_{p}}\cdots\int_{\mathbb{Z}_{p}}\zeta^{\xi_{1}+\cdots+\xi_{k}}e^{\left(\xi_{1}+\cdots+\xi_{k}\right)t}d\mu_{-q}\left(\xi_{1}\right)\cdots
d\mu_{-q}\left(\xi_{k}\right)\mid p\left(x\right)\right\rangle$
$\displaystyle=$
$\displaystyle\int_{\mathbb{Z}_{p}}\cdots\int_{\mathbb{Z}_{p}}\zeta^{\xi_{1}+\cdots+\xi_{k}}p\left(\xi_{1}+\cdots+\xi_{k}\right)d\mu_{-q}\left(\xi_{1}\right)\cdots
d\mu_{-q}\left(\xi_{k}\right)$
Furthermore,
$\left\langle\left(\frac{\left[2\right]_{q}}{\zeta qe^{t}+1}\right)^{k}\mid
p\left(x\right)\right\rangle=\int_{\mathbb{Z}_{p}}\cdots\int_{\mathbb{Z}_{p}}\zeta^{\xi_{1}+\cdots+\xi_{k}}p\left(\xi_{1}+\cdots+\xi_{k}\right)d\mu_{-q}\left(\xi_{1}\right)\cdots
d\mu_{-q}\left(\xi_{k}\right)\text{.}$
That is:
$E_{n,\zeta}^{\left(k\right)}\left(q\right)=\left\langle\int_{\mathbb{Z}_{p}}\cdots\int_{\mathbb{Z}_{p}}\zeta^{\xi_{1}+\cdots+\xi_{k}}e^{\left(\xi_{1}+\cdots+\xi_{k}\right)t}d\mu_{-q}\left(\xi_{1}\right)\cdots
d\mu_{-q}\left(\xi_{k}\right)\mid x^{n}\right\rangle\text{.}$
From (1.21), we notice that
$\displaystyle\left\langle\int_{\mathbb{Z}_{p}}\cdots\int_{\mathbb{Z}_{p}}\zeta^{\xi_{1}+\cdots+\xi_{k}}e^{\left(\xi_{1}+\cdots+\xi_{k}\right)t}d\mu_{-q}\left(\xi_{1}\right)\cdots
d\mu_{-q}\left(\xi_{k}\right)\mid x^{n}\right\rangle$ $\displaystyle=$
$\displaystyle\sum_{i_{1}+\cdots+i_{k}=n}\binom{n}{i_{1},\cdots,i_{m}}\left\langle\int_{\mathbb{Z}_{p}}\zeta^{\xi_{1}}e^{\xi_{1}t}d\mu_{-q}\left(\xi_{1}\right)\mid
x^{i_{1}}\right\rangle\cdots\left\langle\int_{\mathbb{Z}_{p}}\zeta^{\xi_{k}}e^{\xi_{k}t}d\mu_{-q}\left(\xi_{k}\right)\mid
x^{i_{k}}\right\rangle$
Therefore, we have
$E_{n,\zeta}^{\left(k\right)}\left(q\right)=\sum_{i_{1}+\cdots+i_{k}=n}\binom{n}{i_{1},\cdots,i_{m}}E_{i_{1},\zeta}^{q}\cdots
E_{i_{k},\zeta}^{q}\text{.}$
###### Remark 2.
Our applications for weighted Euler polynomials, weighted $q$-Euler numbers
and weighted $q$-Euler polynomials of order $k$ seem to be interesting for
evaluating at $q=\zeta=1$ which lead to Euler polynomials and Euler
polynomials of order $k$, are defined respectively by
$\displaystyle\sum_{n=0}^{\infty}E_{n}\left(x\right)\frac{t^{n}}{n!}$
$\displaystyle=$ $\displaystyle\frac{2}{e^{t}+1}e^{xt}\text{,}$
$\displaystyle\sum_{n=0}^{\infty}E_{n}^{\left(k\right)}\left(x\right)\frac{t^{n}}{n!}$
$\displaystyle=$
$\displaystyle\left(\frac{2}{e^{t}+1}\right)^{k}e^{xt}\text{.}$
Also, it is well known that they have representations in terms of fermionic
$p$-adic integral on $\mathbb{Z}_{p}$ as follows:
$\displaystyle E_{n}\left(x\right)$ $\displaystyle=$
$\displaystyle\int_{\mathbb{Z}_{p}}\left(x+\xi\right)^{n}d\mu_{-1}\left(\xi\right),$
$\displaystyle E_{n}^{\left(k\right)}\left(x\right)$ $\displaystyle=$
$\displaystyle\int_{\mathbb{Z}_{p}}\cdots\int_{\mathbb{Z}_{p}}\left(\xi_{1}+\cdots+\xi_{k}+x\right)^{n}d\mu_{-1}\left(\xi_{1}\right)\cdots
d\mu_{-1}\left(\xi_{k}\right)\text{.}$
## References
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* [7] T. Kim, On a $q$-analogue of the $p$-adic log gamma functions and related integrals, J. Number Theory 76 (1999), 320-329.
* [8] T. Kim, Symmetry of power sum polynomials and multivariate fermionic $p$-adic invariant integral on $\mathbb{Z}_{p}$, Russ. J. Math. Phys. 16, no. 1, 93-96 (2009).
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* [18] I. N. Cangul, V. Kurt, H. Ozden and Y. Simsek, On the higher-order $w$-$q$-Genocchi numbers, Adv. Stud. Contemp. Math. 19 (2009), no. 1, 39–57.
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* [28] S. Araci, M. Acikgoz, K. H. Park and H. Jolany, On the unification of two families of multiple twisted type polynomials by using $p$-adic $q$-integral on $\mathbb{Z}_{p}$ at $q=-1$, Bulletin of the Malaysian Mathematical Sciences and Society (In press).
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|
arxiv-papers
| 2012-11-29T16:48:50 |
2024-09-04T02:49:38.657351
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Serkan Araci, Mehmet Acikgoz and Erdo\\u{g}an \\c{S}en",
"submitter": "Serkan Araci",
"url": "https://arxiv.org/abs/1211.6978"
}
|
1211.7016
|
# A variational characterization of $J$-holomorphic curves
Claudio Arezzo, Jun Sun The Abdus Salam International Centre for Theoretical
Physics
Trieste, Italy and
Universitá di Parma, Italy. [email protected] CIRM, Fondazione Bruno Kessler,
Via Sommarive, 14 - Povo, I-38123
Trento (TN), Italy. [email protected]
###### Abstract.
In this paper, we prove that if the area functional of a surface $\Sigma^{2}$
in a symplectic manifold $(M^{2n},\bar{\omega})$ has a critical point or has a
compatible stable point in the same cohomology class, then it must be
$J$-holomorphic. Inspired by a classical result of Lawson-Simons, we show how
various restrictions of the stability assumption to variations of metrics in
the space ”projectively induced” metrics are enough to give the desired
conclusion.
###### Key words and phrases:
area functional, variation, stable, $J$-holomorphic.
Mathematics Subject Classification (2010): 53A10 (primary), 53D05 (secondary).
## 1\. Introduction
By a well-known extension of Wirtinger’s inequality we know that complex
submanifolds of Kähler manifolds minimize volume in their homology class. A
classical problem is to determine to which extent the converse holds. For
example, Lawson-Simons ([12]) proved that any stable minimal surface in
$\textbf{CP}^{n}$ is holomorphic. Siu-Yau proved the same result when the
ambient manifold has a metric of positive holomorphic bisectional curvature
([20]). Micallef ([14]) studied complete stable minimal surfaces in
$\textbf{R}^{4}$, and proved that, under some further assumptions, stable
minimal surfaces must be holomorphic with respect to some complex structure on
$\textbf{R}^{4}$. He also proved the analogue result for compact surfaces in
flat $4$-tori ([15]).
Since then a series of examples of stable symplectic minimal surfaces
(representing $(1,1)$-classes in homology) which are not holomorphic w.r.t.
any complex structure have been found under different curvature assumptions of
the ambient metric (see e.g. [3] for higher dimensional flat tori, [4] for
higher dimensional euclidean spaces, [2] for Kähler-Einstein surfaces of
negative curvature, [16] for K3 surfaces and [5] for Kähler-Einstein manifolds
of dimensions greater than $4$ and positive curvature).
In this paper we consider immersions of surfaces into symplectic manifolds
endowed with a compatible almost complex structure which are critical and
stable w.r.t. variations of the ambient metric. This seems a very natural
extension of the classical situation since we are using the metric just as a
tool to detect $J$-holomorphicity of a submanifold but we are not really
interested in any of its riemannian properties.
Allowing arbitrary deformations of the metric on the ambient space give rise
to a badly defined concept of critical point, as explained in the Appendix A.
Instead, as we have a symplectic form $\bar{\omega}$ on $M$, we restrict
ourselves to variations of the metric induced via the (tamed almost) complex
structure $J$ by special variations of the symplectic form in the same
cohomology class as $\bar{\omega}$.
Let us first recall some notations. For a compact symplectic manifold
$(M^{2n},\bar{\omega})$, it is known that (Corollary 12.7 of [7]) there always
exists an almost complex structure $J$ that is tamed and compatible with
$\bar{\omega}$. Namely, $\bar{\omega}(X,JX)>0$ for $0\neq X\in TM$ and
$\bar{\omega}(JX,JY)=\bar{\omega}(X,Y)$. Thus we can define the associated
Riemannian metric by
(1) $\bar{g}(X,Y)=\bar{\omega}(X,JY).$
Given the almost complex structure $J$, we have the splitting of the tangent
space $TM^{\textbf{C}}=T^{1,0}M\oplus T^{0,1}M$ and the splitting of the
cotangent space
$\Lambda^{1}_{\textbf{C}}M=\Lambda^{1,0}M\oplus\Lambda^{0,1}M$. Then by
definition, $\partial\psi$ and $\bar{\partial}\psi$ are just the components of
$d\psi\in\Lambda^{1}M$ in $\Lambda^{1,0}M$ and $\Lambda^{0,1}M$, respectively.
Set
(2) $d=\partial+\bar{\partial},\ \ d^{c}=\sqrt{-1}(\bar{\partial}-\partial).$
Let
$\mathcal{H}=\\{\rho\in\mathcal{C}^{\infty}(M,\mathbb{R})\mid\bar{\omega}_{\rho}:=\bar{\omega}+dd^{c}\rho\,\,\mbox{tames}\,\,J\\}$,
which is clearly a nonempty open subset of
$\mathcal{C}^{\infty}(M,\mathbb{R})$.
To each $\rho\in\mathcal{H}$ we can associate a riemannian metric
$\bar{g}_{\rho}$ on $M$ defined by
(3)
$\bar{g}_{\rho}(X,Y)=\frac{1}{2}\left(\bar{\omega}_{\rho}(X,JY)+\bar{\omega}_{\rho}(Y,JX)\right)$
Let $\Sigma$ be a closed real surface and
$F:\Sigma\to M$
be an immersion. We can then define
(4)
$\mathcal{A}(\rho)=\mbox{Area}(F(\Sigma),F^{*}(\bar{g}_{\rho}))=\int_{\Sigma}d\mu_{\rho}\,\,,$
where $d\mu_{\rho}$ is the volume form of the induced metric
$g_{\rho}:=F^{*}(\bar{g}_{\rho})$.
Clearly the functional $\mathcal{A}$ depends only on the metric
$\bar{g}_{\rho}$ (which, by (3) depends in turn on $\bar{\omega}_{\rho}$) and
not on the choice of refence metric $\bar{\omega}$ and the potential $\rho$.
For this reason we will often think of $\mathcal{A}$ as a functional on the
“tamed” subset of $[\bar{\omega}]$.
###### Definition 1.1.
Given an immersion $F:\Sigma^{2}\to(M,\bar{\omega},J,\bar{g})$, we say that
the area functional $\mathcal{A}$ has a critical point $\rho\in\mathcal{H}$ if
for any $\phi(t)\in\mathcal{H}$ with $\phi(0)=\rho$
$\mathcal{A}^{\prime}(0)=0.$
It is a simple consequence of Stokes’ Theorem that if $\Sigma$ is
$J$-holomorphic (even in the more general tamed situation), the functional
$\mathcal{A}$ is constant on $\mathcal{H}$. The first result in this paper
shows that the existence of a critical point of $\mathcal{A}$ is enough to
guarantee the $J$-holomorphicity:
Theorem 2.5: Let $(M^{2n},\bar{\omega},J)$ be a compact symplectic manifold
with compatible almost complex structure $J$ and $F:\Sigma^{2}\to M$ be an
immersion. If the area functional ${\mathcal{A}}$ has a critical point in
$\mathcal{H}$, then the immersion is $J$-holomorphic.
In light of our knowledge about the relationship between stable minimal
surfaces and holomorphic curves, it is natural to look at special properties
of the second variation of the functional $\mathcal{A}$:
###### Definition 1.2.
Given an immersion $F:\Sigma^{2}\to(M,\bar{\omega},J,\bar{g})$, we say that
$\rho\in\mathcal{H}$ is a stable point for the area functional ${\mathcal{A}}$
if
${\mathcal{A}}^{\prime\prime}(0)\geq 0$
for any $\phi(t)\in\mathcal{H}$, $\phi(0)=\rho$. Furthermore, if $J$ is
compatible with $\bar{\omega}_{\rho}$, then we say $\rho$ is a compatible
stable point.
Note that the definition of ${\mathcal{A}}$-stability (as well as all the
other stability notions we are going to study) does not require $\rho$ to be a
critical point of the area functional. Our next result shows that the
existence of a compatible stable point is also enough to guarantee the
$J$-holomorphicity:
Theorem 3.2: Let $(M^{2n},\bar{\omega},J)$ be a compact symplectic manifold
with compatible almost complex structure $J$ and $F:\Sigma^{2}\to M$ be an
immersion. If the area functional ${\mathcal{A}}$ has a compatible stable
point $\rho\in\mathcal{H}$, then the immersion is $J$-holomorphic.
As above, the converse is also true even without assuming that $J$ is
compatible with $\bar{\omega}$.
Checking the proof of the above theorem carefully, we see that the result is
also true in the complete noncompact case. In particular, this applies to
minimal submanifold in $\textbf{R}^{2n}$. In this case, we need the test
function to have compact support.
Theorem 3.4: Let $(M^{2n},\bar{\omega},J)$ be a complete noncompact symplectic
manifold with compatible almost complex structure $J$ and $F:\Sigma^{2}\to M$
be an immersion. If the area functional ${\mathcal{A}}$ has a compatible
stable point $\rho\in\mathcal{H}$, then the immersion is $J$-holomorphic.
The above Theorems show an intriguing analogy with of a classical picture
discovered by Sacks-Uhlenbeck [18] and Schoen-Yau [19] to construct minimal
surfaces, which, as they proved, can be generated by first fixing a metric on
the domain surface and finding an energy minimizing map, and then letting the
metric on the base vary. In our case minimizing w.r.t. the metric on the
target plays the corresponding role which not surprisingly encodes a similar
strategy since of course for what the area can detect the immersion is not an
independent parameter compared to the ambient metric.
It is clear that, while very simple to state, the induced existence problem is
very difficult to attack, since we introduced a parameter (the target metric)
which varies freely in an infinite dimensional noncompact space (the “Kähler
potentials” $\mathcal{H}$). We then try to identify some geometrically
meaningful finite dimensional subspaces or submanifolds of metrics which are
enough to detect holomorphicity.
When $M$ is an algebraic manifold we can embed it into some complex projective
space ${\textbf{CP}^{N}}$ holomorphically. Lawson-Simons’ result ([12]) tells
us that a submanifold is holomorphic if the second variation of the area
functional (built with the metric induced by the projective space) is
nonnegative under the holomorphic deformation of $M$ in ${\textbf{CP}^{N}}$.
The latter means that the target metric varies in some finitely dimensional
submanifold of metrics representing the original Käher class. Their result is
true for submanifolds of any even dimension but only for projectively induced
Kähler metrics (in particular representing a rational class).
We first consider the same case as Lawson-Simons’ under a slightly different
stability assumption. We call the area functional has a _linearly projectively
stable point_ if the variation of the metric on the target is linear in time
along the directions induced by holomorphic deformations of the complex
projective space (Lawson-Simons’ assumption can be stated as to require the
variations of the metric to live in this subspace for all time).
Theorem 4.1: Let $(M,\bar{\omega},J,\bar{g})$ be an algebraic manifold with
all structures induced by the projective space as above and $F:\Sigma^{2}\to
M$ be an immersion. If the area functional has a linearly projectively stable
point, then the immersion must be holomorphic with respect to the complex
structure $J$.
This can be seen as a mild modification of Lawson-Simons’ result in the
surface case, yet our proof differs significantly from theirs in that we
explicitly identify in the nonholomorphic case a Killing field which induces
an area-decreasing variation, while they had to work on the whole space of
Killing fields and using heavily the homogeneous space structure of the
projective space to average the variations of area.
The advantage of our proof of Theorem $5.1$ is that it generalizes to any
symplectic manifold with rational symplectic class and to any complex
projective manifold with any Kähler form. Indeed, let $(M,\bar{\omega},J)$ be
a symplectic manifold with rational symplectic class and compatible almost
complex structure. It is known that ([6]), there exists an approximately
$J$-holomorphic embedding of $M$ in to some complex projective space
$\textbf{CP}^{N_{k}}$. In the symplectic case, using as above the holomorphic
deformations of $\textbf{CP}^{N_{k}}$, we can extend the notion of linearly
projectively stable point to that of compatible linearly
${\mathcal{A}}^{k}$-stable point, which again means that the target symplectic
form varies (still linearly) in the direction of projectively induced forms.
Theorem 5.1: Let $(M^{2n},\bar{\omega},J_{M},\bar{g})$ be a symplectic
manifold as above and $F:\Sigma^{2}\to M$ be an immersion. There exists an
integer $K_{1}$, such that if the area functional has a compatible linearly
${\mathcal{A}}^{k}$-stable point for some $k\geq K_{1}$, then the immersion
must be $J_{M}$-holomorphic.
Using our second variation formula, we can show that for surface case, we can
obtain similar result for algebraic manifolds but whose Kähler form represents
any real class. We can define the notion of $k$-linearly projectively stable
point, which means that the second variation of the area functional is
nonnegative when the potential of the target metric varies along the
directions in the finitely dimensional linear subspace of the space of
potentials determined by the Killing vector fields of ${\textbf{CP}^{N_{k}}}$,
where this projective space is the ambient of a diagonal approximating
sequence of rational forms converging to the original class and the above
approximation described in the symplectic case (which in this case relies on a
famous Theorem by Tian [21]).
Theorem 6.1: Let $(M,J)$ be an algebraic manifold, and $\bar{\omega}$ be any
Kähler metric with $[\bar{\omega}]\in H^{2}(M,\textbf{R})\cap
H^{1,1}(M,\textbf{C})$ and $F:\Sigma^{2}\to M$ be an immersion. There exists
an integer $K_{2}$, such that if if the area functional has a $k$-linearly
projectively stable point for some $k\geq K_{2}$, then the immersion must be
holomorphic with respect to the complex structure $J$.
We finally underline that most of our arguments and results are likely to
carry over to higher dimensional submanifolds, and in fact to more general,
less regular, objects. This will be clarified in a forthcoming paper.
The following sections are organized as follows: in Section 2 and Section 3 we
compute the first and second variation formulas for the area functional under
deformation of target metrics and prove the first two results; in Section 4,
we study the case of an ambient algebraic manifold with induced structures and
linearly projectively stable point; in Section 5 and 6, we prove the
symplectic case with rational classes and the Kähler case with any real Kähler
class.
## 2\. Critical points of $\mathcal{A}$ and $J$-holomorphicity
With the setup given in the introduction, we now compute the first variation
of the area functional ${\mathcal{A}}$ and prove the first theorem.
Let $\\{x_{1},x_{2}\\}$ be local coordinates on $\Sigma$ and
$\bar{g}_{\rho}(t)$ a variation of $\bar{g}_{\rho}$ coming from a
$1$-parameter deformation of $\rho$ in $\mathcal{H}$. . Then
(5) $g_{\rho,ij}(t)=\bar{g}_{\rho}(t)\left(\frac{\partial F}{\partial
x_{i}},\frac{\partial F}{\partial
x_{j}}\right)=\frac{1}{2}\left\\{\bar{\omega}_{\rho}(t)\left(\frac{\partial
F}{\partial x_{i}},J\frac{\partial F}{\partial
x_{j}}\right)+\bar{\omega}_{\rho}(t)\left(\frac{\partial F}{\partial
x_{j}},J\frac{\partial F}{\partial x_{i}}\right)\right\\}.$
Set
(6)
$\nu_{\rho}(t)=\frac{\sqrt{det(g_{\rho,ij}(t))}}{\sqrt{det(g_{\rho,ij}(0))}}.$
Then $\nu_{\rho}(t)$ is well-defined independent of the choice of coordinate
system. Furthermore,
(7)
${\mathcal{A}}(t)=\int_{\Sigma}\sqrt{det(g_{\rho,ij}(t))}=\int_{\Sigma}\nu_{\rho}(t)\sqrt{det(g_{\rho,ij}(0))},$
and therefore
(8)
$\frac{d}{dt}|_{t=0}{\mathcal{A}}(t)=\int_{\Sigma}\frac{d}{dt}|_{t=0}\nu_{\rho}(t)\sqrt{det(g_{\rho,ij}(0))}.$
Denote $(g_{\rho}^{ij})=(g_{\rho,ij})^{-1}$. By (6), we have in any local
coordinate $\\{x_{1},x_{2}\\}$
(9) $\displaystyle\frac{d}{dt}|_{t=0}\nu_{\rho}(t)$ $\displaystyle=$
$\displaystyle\frac{1}{2}\sum_{i,j=1}^{2}g_{\rho}^{ij}(0)g_{\rho,ij}^{\prime}(0)$
$\displaystyle=$
$\displaystyle\frac{1}{2}\sum_{i,j=1}^{2}g_{\rho}^{ij}(0)\frac{d}{dt}|_{t=0}\left\\{\bar{g}_{\rho}(t)\left(\frac{\partial
F}{\partial x_{i}},\frac{\partial F}{\partial x_{j}}\right)\right\\}$
$\displaystyle=$
$\displaystyle\frac{1}{2}\sum_{i,j=1}^{2}g_{\rho}^{ij}(0)\frac{d}{dt}|_{t=0}\left\\{\bar{\omega}_{\rho}(t)\left(\frac{\partial
F}{\partial x_{i}},J\frac{\partial F}{\partial x_{j}}\right)\right\\}$
$\displaystyle=$
$\displaystyle\frac{1}{2}\sum_{i,j=1}^{2}g_{\rho}^{ij}(0)\bar{\omega}_{\rho}^{\prime}(0)\left(F_{x_{i}},JF_{x_{j}}\right).$
Therefore the first variation formula is given by
(10)
${\mathcal{A}}(0)=\frac{1}{2}\sum_{i,j=1}^{2}\int_{\Sigma}g_{\rho}^{ij}(0)\bar{\omega}_{\rho}^{\prime}(0)\left(F_{x_{i}},JF_{x_{j}}\right)d\mu_{\rho}.$
For our later use, let’s recall the following simple facts:
###### Lemma 2.1.
1. (1)
For any smooth function $\psi$ on $M$, we have
(11) $d^{c}\psi=-d\psi\circ J.$
2. (2)
For any smooth function $\psi$ on $M$ and any tangent vector fields $X,Y$ on
$M$, we have
(12)
$(dd^{c}\psi)(X,Y)=-(\overline{\nabla}^{2}\psi)(X,JY)+(\overline{\nabla}^{2}\psi)(Y,JX)+\langle\overline{\nabla}\psi,(\overline{\nabla}_{Y}J)X-(\overline{\nabla}_{X}J)Y\rangle.$
Here, $\langle\cdot,\cdot\rangle$ is any Riemannian metric on $M$ and
$\overline{\nabla}$ is its Levi-Civita connection.
Now we turn to $J$-holomorphic curves.
###### Definition 2.1.
Let $(M^{2n},J)$ be an almost complex manifold and $\Sigma$ be a surface. We
call an immersion $F:\Sigma\to(M,J)$ $J$-holomorphic if $J_{F(x)}$ maps
$F_{*x}(T_{x}\Sigma)$ onto itself for any point $x\in\Sigma$.
Stokes’ theorem immediately gives the following
###### Proposition 2.2.
If $F:\Sigma\to(M,J)$ be a $J$-holomorphic immersion then $\mathcal{A}$ is
constant on $\mathcal{H}$, in particular any
$\bar{\omega}_{\rho}\in(\Lambda^{2}M)^{+}\cap[\bar{\omega}]$ is both a
critical point and a stable point for the area functional ${\mathcal{A}}$.
Proof: By the definition of $J$-holomorphic immersion, we can easily see that
the almost complex structure $J$ on $M$ can induce an almost complex structure
$j$ on $\Sigma$, such that the immersion $F:(\Sigma,j)\to(M,J)$ is
$(j,J)$-holomorphic. That if,
(13) $J\circ F_{*}=F_{*}\circ j.$
Given any curve $\bar{\omega}(t)=\bar{\omega}+d\beta(t)$ which is tamed by
$J$, where $\beta(t)$ is a family of smooth 1-forms on $M$, we define the
associated Riemannian metric $\bar{g}(t)$ by (3). It suffices to show that
${\mathcal{A}}^{\prime}(t)=0$
for each $t$. In order to show this, for fixed $t$, at a given point $x$, we
take local coordinates $\\{x_{1},x_{2}\\}$ on $\Sigma$ such that
$\\{\partial_{x_{1}},\partial_{x_{2}}\\}$ is $g(t)$-orthonormal. Then by the
above computation, we get that
${\mathcal{A}}^{\prime}(t)=\frac{1}{2}\sum_{i=1}^{2}\int_{\Sigma}\bar{\omega}^{\prime}(t)\left(F_{x_{i}},JF_{x_{i}}\right)d\mu.$
By the choice of the local frame, it is easy to see that at the given point,
$j\partial_{x_{1}}=\pm\partial_{x_{2}}$. Without loss of generality, we assume
that $j\partial_{x_{1}}=\partial_{x_{2}}$,
$j\partial_{x_{2}}=-\partial_{x_{1}}$. By (13), we have
$\displaystyle\frac{1}{2}\sum_{i=1}^{2}\bar{\omega}^{\prime}(t)\left(F_{x_{i}},JF_{x_{i}}\right)$
$\displaystyle=$
$\displaystyle\frac{1}{2}[d\beta^{\prime}(t)(F_{*}(\partial_{x_{1}}),JF_{*}(\partial_{x_{1}}))+d\beta^{\prime}(t)(F_{*}(\partial_{x_{2}}),JF_{*}(\partial_{x_{2}}))]$
$\displaystyle=$
$\displaystyle\frac{1}{2}[d\beta^{\prime}(t)(F_{*}(\partial_{x_{1}}),F_{*}j(\partial_{x_{1}}))+d\beta^{\prime}(t)(F_{*}(\partial_{x_{2}}),F_{*}j(\partial_{x_{2}}))]$
$\displaystyle=$
$\displaystyle\frac{1}{2}[(F^{*}d\beta^{\prime}(t))(\partial_{x_{1}},\partial_{x_{2}})+(F^{*}d\beta^{\prime}(t))(\partial_{x_{2}},-\partial_{x_{1}})]$
$\displaystyle=$
$\displaystyle(d(F^{*}\beta^{\prime}(t)))(\partial_{x_{1}},\partial_{x_{2}}).$
Therefore, at this point,
$\frac{1}{2}\sum_{i=1}^{2}\bar{\omega}^{\prime}(t)\left(F_{x_{i}},JF_{x_{i}}\right)d\mu=(d(F^{*}\beta^{\prime}(t)))(\partial_{x_{1}},\partial_{x_{2}})dx_{1}\wedge
dx_{2}=d(F^{*}\beta^{\prime}(t)),$
which is a globally defined exact 2-form on $\Sigma$. As $\Sigma$ is closed,
by Stokes’ theorem, we see that ${\mathcal{A}}^{\prime}(t)=0$. This proves the
theorem. Q.E.D.
Our interest is in whether (and in which sense) the converse holds.
Let $(M^{2n},\bar{\omega},J,\bar{g})$ be a symplectic manifold with symplectic
form $\bar{\omega}$, compatible almost complex structure $J$ and associated
Riemannian metric $\bar{g}$. Recall that the Kähler angle $\alpha$ of a
surface $\Sigma^{2}$ in $M$ is defined by ([9])
$\bar{\omega}|_{\Sigma}=\cos\alpha d\mu_{\Sigma},$
where $d\mu_{\Sigma}$ is the induced volume form on $\Sigma$. The following
fact is well known:
###### Proposition 2.3.
Let $(M^{2n},\bar{\omega},J,\bar{g})$ be a symplectic manifold with compatible
almost complex structure $J$. Then $F:\Sigma\to M$ is $J$-holomorphic if and
only if $\sin\alpha\equiv 0$.
Our main result in this section is as follows:
###### Theorem 2.4.
Let $(M^{2n},\bar{\omega},J)$ be a compact symplectic manifold with compatible
almost complex structure $J$ and $F:\Sigma^{2}\to M$ be an immersion. If the
area functional ${\mathcal{A}}$ has a critical point in $\mathcal{H}$, then
the immersion is $J$-holomorphic.
Proof: By definition, there exists a smooth function $\rho$ on $M$, such that
$\bar{\omega}_{\rho}(0)=\bar{\omega}_{\rho}=\bar{\omega}+dd^{c}\rho\in(\Lambda^{2}M)^{+}\cap[\bar{\omega}]$
and
${\mathcal{A}}^{\prime}(0)=0$
for any
(14)
$\bar{\omega}_{\rho}(t)=\bar{\omega}_{\rho}+dd^{c}\varphi(t)\in(\Lambda^{2}M)^{+}\cap[\bar{\omega}]\,\,\mbox{with}\,\,\varphi(0)=0\,\,.$
We will first express the first variation formula (10) in terms of Kähler
angle. Note that in general, $J$ does not need be compatible with
$\bar{\omega}_{\rho}$. We denote $\alpha$ the Kähler angle define by
$(\bar{\omega},J,\bar{g})$.
Fix a point $x\in\Sigma$, it is easy to see that we can choose a
$\bar{g}$-orthonormal frame $\\{e_{1},e_{2},\cdots,e_{2n}\\}$ of $T_{x}M$,
such that $\\{e_{1},e_{2}\\}$ spans the tangent space of $\Sigma$,
$\\{e_{3},\cdots,e_{2n}\\}$ spans the normal space of $\Sigma$, and the almost
complex structure takes the form
(15) $J=\left(\begin{array}[]{cccc}(J_{1})_{4\times 4}&0_{4\times(2n-4)}\\\
0_{(2n-4)\times 4}&(J_{2})_{(2n-4)\times(2n-4)}\\\ \end{array}\right),$
where
(16) $J_{1}=\left(\begin{array}[]{cccc}0&\cos\alpha&\sin\alpha&0\\\
-\cos\alpha&0&0&-\sin\alpha\\\ -\sin\alpha&0&0&\cos\alpha\\\
0&\sin\alpha&-\cos\alpha&0\\\ \end{array}\right),$
and $J_{2}$ satisfies $J_{2}^{2}=-Id_{2n-4}$. Suppose
$\bar{\omega}_{\rho}(t)=\bar{\omega}_{\rho}+dd^{c}\varphi(t)$, then by (10),
we have
(17)
${\mathcal{A}}^{\prime}(0)=\frac{1}{2}\sum_{i,j=1}^{2}\int_{\Sigma}g_{\rho}^{ij}(0)(dd^{c}\psi)\left(e_{i},Je_{j}\right)d\mu_{\rho},$
where $\psi=\varphi^{\prime}(0)$, $g_{\rho,ij}=g_{\rho}(e_{i},e_{j})$ and
$(g_{\rho}^{ij})=(g_{\rho,ij})^{-1}$. Plugging (15) into (17) and using (12),
we finally get that
(18) $\displaystyle{\mathcal{A}}^{\prime}(0)$ $\displaystyle=$
$\displaystyle\frac{1}{2}\int_{\Sigma}g_{\rho}^{11}\left\\{(\overline{\nabla}^{2}\psi)(e_{1},e_{1})+\cos\alpha^{2}(\overline{\nabla}^{2}\psi)(e_{2},e_{2})+\sin^{2}\alpha(\overline{\nabla}^{2}\psi)(e_{3},e_{3})\right.$
$\displaystyle\ \ \ \ \ \ \
\left.+2\sin\alpha\cos\alpha(\overline{\nabla}^{2}\psi)(e_{2},e_{3})\right\\}d\mu_{\rho}$
$\displaystyle+\frac{1}{2}\int_{\Sigma}g_{\rho}^{22}\left\\{(\overline{\nabla}^{2}\psi)(e_{2},e_{2})+\cos^{2}\alpha(\overline{\nabla}^{2}\psi)(e_{1},e_{1})+\sin^{2}\alpha(\overline{\nabla}^{2}\psi)(e_{4},e_{4})\right.$
$\displaystyle\ \ \ \ \ \ \
\left.+2\sin\alpha\cos\alpha(\overline{\nabla}^{2}\psi)(e_{1},e_{4})\right\\}d\mu_{\rho}$
$\displaystyle+\int_{\Sigma}g_{\rho}^{12}\left\\{\sin^{2}\alpha(\overline{\nabla}^{2}\psi)(e_{1},e_{2})-\sin^{2}\alpha(\overline{\nabla}^{2}\psi)(e_{3},e_{4})\right.$
$\displaystyle\ \ \ \ \ \ \
\left.-\sin\alpha\cos\alpha[(\overline{\nabla}^{2}\psi)(e_{1},e_{3})+(\overline{\nabla}^{2}\psi)(e_{2},e_{4})]\right\\}d\mu_{\rho}$
$\displaystyle+\frac{1}{2}\int_{\Sigma}\left\\{\langle\overline{\nabla}\psi,g_{\rho}^{11}[(\overline{\nabla}_{Je_{1}}J)e_{1}-(\overline{\nabla}_{e_{1}}J)(Je_{1})]+g_{\rho}^{22}[(\overline{\nabla}_{Je_{2}}J)e_{2}-(\overline{\nabla}_{e_{2}}J)(Je_{2})]\rangle\right\\}d\mu_{\rho}$
$\displaystyle+\frac{1}{2}\int_{\Sigma}\left\\{\langle\overline{\nabla}\psi,g_{\rho}^{12}[(\overline{\nabla}_{Je_{1}}J)e_{2}-(\overline{\nabla}_{e_{2}}J)(Je_{1})+(\overline{\nabla}_{Je_{2}}J)e_{1}-(\overline{\nabla}_{e_{1}}J)(Je_{2})]\rangle\right\\}d\mu_{\rho}.$
Here, $\langle\cdot,\cdot\rangle=\bar{g}$ and $\overline{\nabla}$ is its Levi-
Civita connection. By Proposition 2.3, it suffices to show that
$\sin\alpha\equiv 0$ on $\Sigma$. We prove this by taking special $\psi$ in
the first variation formula (18). We identify $\Sigma$ with its image in $M$.
Denote $d$ the distance function of $M$ from $\Sigma$ with respect to the
metric $\bar{g}$. Namely, for $Q\in M$, $d(Q)=dist_{\bar{g}}(Q,\Sigma)$. Then
it is known that ([13]) $\eta=\frac{1}{2}d^{2}$ is smooth in a neighborhood of
$\Sigma$ in $M$.
The following result is known for $M=\textbf{R}^{2n}$ (Theorem 3.1 of [1]),
and it is easy to prove using computations in [13].
###### Proposition 2.5.
Let $\Sigma$ be a $C^{\infty}$ regular submanifold of a $C^{\infty}$
Riemannian manifold $M$, then for any $x_{0}\in S$, the hessian
$Hess(\eta)(x_{0})=\frac{1}{2}Hess(d^{2})(x_{0})$ represents the orthogonal
projection on the normal space to $S$ at $x_{0}$. Namely, for each $X,Y\in
T_{x_{0}}M$, we have
(19) $Hess(\eta)(X,Y)(x_{0})=\langle X^{\perp},Y^{\perp}\rangle,$
where $T_{x_{0}}M=T_{x_{0}}S\oplus N_{x_{0}}S$ and $X^{\perp}$ is the
projection of $X$ onto $N_{x_{0}}S$.
Now we can finish the proof of Theorem 2.4.
Proof of Theorem 2.4 (continued): We take $\psi$ to be a smooth function on
$M$ such that $\psi=\eta$ in a neighborhood of $\Sigma$ in $M$. Then we have
that $\overline{\nabla}\psi$ vanishes restricting on $\Sigma$. By the choice
of the frame and Proposition 2.5, for
$\bar{\omega}_{\rho}(t)=\bar{\omega}_{\rho}+dd^{c}\varphi(t)$ with
$\varphi^{\prime}(0)=\psi$ as above, we have from (18)
${\mathcal{A}}^{\prime}(0)=\frac{1}{2}\int_{\Sigma}\sin^{2}\alpha\left(g_{\rho}^{11}+g_{\rho}^{22}\right)d\mu_{\rho}.$
By our assumption, we must have ${\mathcal{A}}^{\prime}(0)=0$ for any $\psi$.
In particular, for this special choice of $\psi$, it implies that
$\sin\alpha\equiv 0$. This proves the theorem. Q.E.D.
## 3\. Compatible stable point and $J$-holomorphicity
In this section, we will compute the second variation formula for the
functional ${\mathcal{A}}$ (not necessarily at a critical point). Using this
formula, we show that if ${\mathcal{A}}$ has a compatible stable point, then
the immersion is $J$-holomorphic.
Let $\bar{\omega}_{\rho}(t)$ be a variation of $\bar{\omega}_{\rho}$ as in
(14). By (7), we have
(20)
$\frac{d^{2}}{dt^{2}}|_{t=0}{\mathcal{A}}(t)=\int_{\Sigma}\frac{d^{2}}{dt^{2}}|_{t=0}\nu_{\rho}(t)\sqrt{det(g_{\rho,ij}(0))},$
where $\nu(t)$ is defined by (6). To evaluate
$\frac{d^{2}}{dt^{2}}|_{t=0}\nu_{\rho}(t)$ at a given point $x$, we choose the
coordinate system $g_{\rho}$-orthonormal at $x$. Thus, we have
(21)
$\frac{d}{dt}\nu_{\rho}(t)=\frac{1}{2}g^{ij}_{\rho}(t)\frac{d}{dt}g_{\rho,ij}(t)\nu_{\rho}(t),$
and
(22)
$\frac{d^{2}}{dt^{2}}|_{t=0}\nu_{\rho}(t)=\frac{1}{2}\sum_{i=1}^{2}g_{\rho,ii}^{\prime\prime}(0)-\frac{1}{2}\sum_{i,j=1}^{2}\left(g_{\rho,ij}^{\prime}(0)\right)^{2}+\frac{1}{4}\left(\sum_{i=1}^{2}g_{\rho,ii}^{\prime}(0)\right)^{2}.$
By (9), we have
(23)
$\sum_{i=1}^{2}g_{\rho,ii}^{\prime}(0)=\sum_{i=1}^{2}\bar{\omega}_{\rho}^{\prime}(0)(e_{i},Je_{i}).$
By (5), we have
(24)
$g_{\rho,ij}^{\prime}(t)=\frac{1}{2}\left[\bar{\omega}_{\rho}^{\prime}(t)(F_{x_{i}},JF_{x_{j}})+\bar{\omega}^{\prime}_{\rho}(t)(F_{x_{j}},JF_{x_{i}})\right]$
so that
(25)
$g_{\rho,ij}^{\prime}(0)=\frac{1}{2}\left[\bar{\omega}_{\rho}^{\prime}(t)(e_{i},Je_{j})+\bar{\omega}_{\rho}^{\prime}(t)(e_{j},Je_{i})\right]$
By (24), we have
(26)
$\sum_{i=1}^{2}g_{\rho,ii}^{\prime\prime}(0)=\sum_{i=1}^{2}\bar{\omega}_{\rho}^{\prime\prime}(0)(e_{i},Je_{i}).$
Combining (20), (22), (23), (25) and (26) together, we have
(27) $\displaystyle{\mathcal{A}}^{\prime\prime}(0)$ $\displaystyle=$
$\displaystyle\frac{1}{2}\sum_{i=1}^{2}\int_{\Sigma}\bar{\omega}_{\rho}^{\prime\prime}(0)(e_{i},Je_{i})d\mu$
$\displaystyle-\frac{1}{8}\sum_{i,j=1}^{2}\int_{\Sigma}\left[\bar{\omega}_{\rho}^{\prime}(0)(e_{i},Je_{j})+\bar{\omega}_{\rho}^{\prime}(0)(e_{j},Je_{i})\right]^{2}d\mu$
$\displaystyle+\frac{1}{4}\int_{\Sigma}\left\\{\sum_{i=1}^{2}\bar{\omega}_{\rho}^{\prime}(0)(e_{i},Je_{i})\right\\}^{2}d\mu.$
By direct computation, we have
$\displaystyle\sum_{i,j=1}^{2}\left[\bar{\omega}_{\rho}^{\prime}(0)(e_{i},Je_{j})+\bar{\omega}^{\prime}_{\rho}(0)(e_{j},Je_{i})\right]^{2}-2\left\\{\sum_{i=1}^{2}\left[\bar{\omega}_{\rho}^{\prime}(0)(e_{i},Je_{i})\right]\right\\}^{2}$
$\displaystyle=$ $\displaystyle
2\left[\bar{\omega}_{\rho}^{\prime}(0)(e_{1},Je_{2})+\bar{\omega}_{\rho}^{\prime}(0)(e_{2},Je_{1})\right]^{2}+2\left[\bar{\omega}^{\prime}_{\rho}(0)(e_{1},Je_{1})-\bar{\omega}^{\prime}_{\rho}(0)(e_{2},Je_{2})\right]^{2}.$
Plugging this into (27) we get
(28) $\displaystyle{\mathcal{A}}^{\prime\prime}(0)$ $\displaystyle=$
$\displaystyle\frac{1}{2}\sum_{i=1}^{2}\int_{\Sigma}\bar{\omega}_{\rho}^{\prime\prime}(0)(e_{i},Je_{i})d\mu$
$\displaystyle-\frac{1}{4}\int_{\Sigma}\left[\bar{\omega}_{\rho}^{\prime}(0)(e_{1},Je_{2})+\bar{\omega}_{\rho}^{\prime}(0)(e_{2},Je_{1})\right]^{2}d\mu$
$\displaystyle-\frac{1}{4}\int_{\Sigma}\left[\bar{\omega}_{\rho}^{\prime}(0)(e_{1},Je_{1})-\bar{\omega}_{\rho}^{\prime}(0)(e_{2},Je_{2})\right]^{2}d\mu.$
In particular, when $\bar{\omega}_{\rho}(t)$ is given by (14), defining
$\frac{\partial\varphi}{\partial t}|_{t=0}=\psi,\ \
\frac{\partial^{2}\varphi}{\partial t^{2}}|_{t=0}=\eta,$
we get
$\frac{\partial\bar{\omega}_{\rho}(t)}{\partial t}|_{t=0}=dd^{c}\psi,\ \
\frac{\partial^{2}\bar{\omega}_{\rho}(t)}{\partial t^{2}}|_{t=0}=dd^{c}\eta.$
Then the second variation formula reads
(29) $\displaystyle{\mathcal{A}}^{\prime\prime}(0)$ $\displaystyle=$
$\displaystyle\frac{1}{2}\sum_{i=1}^{2}\int_{\Sigma}\left[(dd^{c}\eta)(e_{i},Je_{i})\right]d\mu_{\rho}$
$\displaystyle-\frac{1}{4}\int_{\Sigma}\left[(dd^{c}\psi)(e_{1},Je_{2})+(dd^{c}\psi)(e_{2},Je_{1})\right]^{2}d\mu_{\rho}$
$\displaystyle-\frac{1}{4}\int_{\Sigma}\left[(dd^{c}\psi)(e_{1},Je_{1})-(dd^{c}\psi)(e_{2},Je_{2})\right]^{2}d\mu_{\rho}.$
###### Remark 3.1.
When $M$ is a complete noncompact symplectic manifold and $\Sigma$ is a
complete submanifold, we can follow the same way to compute the first
variation and second variation formulas and give similar definitions as in
Definition 1.1 and Definition 1.2. In this case, we need the test function for
the variations of the target metric $\psi$ and $\eta$ to have compact support
on $M$.
Our main result in this section is:
###### Theorem 3.2.
Let $(M^{2n},\bar{\omega},J)$ be a compact symplectic manifold with compatible
almost complex structure $J$ and $F:\Sigma^{2}\to M$ be an immersion. If the
area functional ${\mathcal{A}}$ has a compatible stable point in
$[\bar{\omega}]$, then the immersion is $J$-holomorphic.
Proof: By definition, there exists a smooth function $\rho$ on $M$, such that
$\bar{\omega}_{\rho}(0)=\bar{\omega}_{\rho}=\bar{\omega}+dd^{c}\rho\in(\Lambda^{2}M)^{+}\cap[\bar{\omega}]$
and
${\mathcal{A}}^{\prime\prime}(0)\geq 0$
for any
$\bar{\omega}_{\rho}(t)=\bar{\omega}_{\rho}+dd^{c}\varphi(t)\in(\Lambda^{2}M)^{+}\cap[\bar{\omega}]$
with $\varphi(0)=0$. As in the previous section, we need to express the second
variation formula in terms of the Kähler angle of $\Sigma$ in $M$. As $J$ is
compatible with $\bar{\omega}_{\rho}$ by our assumption, we can define Kähler
angle $\alpha_{\rho}$ using $(\bar{\omega}_{\rho},J,\bar{g}_{\rho})$.
At a fixed point $x$ on $\Sigma$, we can take a $\bar{g}_{\rho}$-orthonormal
frame $\\{e_{1},e_{2},\cdots,e_{2n}\\}$ of $TM$ such that the almost complex
structure takes the form (15) (with $\alpha$ replaced by $\alpha_{\rho}$). By
direct computation, we have
$\displaystyle D_{1}$ $\displaystyle:=$
$\displaystyle(dd^{c}\psi)(e_{1},Je_{2})+(dd^{c}\psi)(e_{2},Je_{1})$
$\displaystyle=$
$\displaystyle\cos\alpha_{\rho}[-(dd^{c}\psi)(e_{1},e_{1})+(dd^{c}\psi)(e_{2},e_{2})]+\sin\alpha_{\rho}[-(dd^{c}\psi)(e_{1},e_{4})+(dd^{c}\psi)(e_{2},e_{3})]$
and
$\displaystyle D_{2}$ $\displaystyle:=$
$\displaystyle(dd^{c}\psi)(e_{1},Je_{1})-(dd^{c}\psi)(e_{2},Je_{2})$
$\displaystyle=$
$\displaystyle\cos\alpha_{\rho}\left[(dd^{c}\psi)(e_{1},e_{2})+(dd^{c}\psi)(e_{2},e_{1})\right]+\sin\alpha_{\rho}\left[(dd^{c}\psi)(e_{1},e_{3})+(dd^{c}\psi)(e_{2},e_{4})\right].$
By (12), we know that $(dd^{c}\psi)(X,Y)+(dd^{c}\psi)(Y,X)=0$. Therefore,
(30)
$D_{1}=\sin\alpha_{\rho}[-(dd^{c}\psi)(e_{1},e_{4})+(dd^{c}\psi)(e_{2},e_{3})]$
and
(31) $\displaystyle D_{2}$ $\displaystyle=$
$\displaystyle\sin\alpha_{\rho}\left[(dd^{c}\psi)(e_{1},e_{3})+(dd^{c}\psi)(e_{2},e_{4})\right]$
$\displaystyle=$
$\displaystyle\sin\alpha_{\rho}\left\\{\sin\alpha_{\rho}\left[(\overline{\nabla}_{\rho}^{2}\psi)(e_{1},e_{1})-(\overline{\nabla}_{\rho}^{2}\psi)(e_{2},e_{2})+(\overline{\nabla}_{\rho}^{2}\psi)(e_{3},e_{3})-(\overline{\nabla}_{\rho}^{2}\psi)(e_{4},e_{4})\right]\right.$
$\displaystyle\left.+2\cos\alpha_{\rho}\left[(\overline{\nabla}_{\rho}^{2}\psi)(e_{2},e_{3})-(\overline{\nabla}_{\rho}^{2}\psi)(e_{1},e_{4})\right]\right.$
$\displaystyle\left.+\langle\overline{\nabla}_{\rho}\psi,((\overline{\nabla}_{\rho})_{e_{3}}J)e_{1}-((\overline{\nabla}_{\rho})_{e_{1}}J)e_{3}+((\overline{\nabla}_{\rho})_{e_{4}}J)e_{2}-((\overline{\nabla}_{\rho})_{e_{2}}J)e_{4}\rangle_{\rho}\right\\}.$
where $\langle\cdot,\cdot\rangle_{\rho}=\bar{g}_{\rho}$ and
$\overline{\nabla}_{\rho}$ is its Levi-Civita connection. Here, we used (12)
again. We will prove Theorem 3.2 by taking special choices of the test
function $\psi$. Indeed, we will take the normal extension of some function on
$\Sigma$, which we will recall in the following. (For more details, see, for
example, Chapter XIV of S. Lang’s book [11].)
Let $M$ be a $2n$-dimensional Riemannian manifold and $\Sigma$ be a
$p$-dimensional submanifold of $M$ with the induced metric. Locally, we can
find a function $r>0$ on $\Sigma$ such that if $N_{r}\Sigma$ denotes the
vectors $w$ with norm $||w||<r(x)$ for $w\in N_{x}\Sigma$, then the
exponential map
$\exp:N\Sigma\to M$
given by
$w\mapsto\exp_{x}(w)\ \ for\ w\in N_{x}\Sigma$
gives an isomorphism of $N_{r}\Sigma$ with an open neighborhood of $\Sigma$ in
$M$. Given a function $f$ on $\Sigma$, we may extend $f$ to this tubular
neighborhood by making $f$ constant in the normal directions, that is, we
define
$f_{M}(\exp_{x}(w))=f(x).$
This extension will be called the normal extension of $f$ to a tubular
neighborhood of $\Sigma$.
In the following, we list some properties of $f_{M}$ without proof. Some
proofs of them and more properties can be found in the book [11].
###### Lemma 3.3.
(a) For vector fields $\xi$, $\eta$ on $\Sigma$, we have on $\Sigma$
$(\overline{\nabla}^{2}f_{M})(\xi,\eta)=(\nabla^{2}f)(\xi,\eta).$
(b) Let $\nu$ be a normal vector field on $\Sigma$, then on $\Sigma$
$(\overline{\nabla}^{2}f_{M})(\nu,\nu)=0.$
(c) Let $\nu$ be a normal vector field on $\Sigma$ and $\xi$ be a tangent
vector field on $\Sigma$, then
$(\nu\cdot f_{M})(x)=0\ \ \ \ for\ x\in\Sigma$
and thus
$\xi(\nu\cdot f_{M})(x)=0\ \ \ \ for\ x\in\Sigma.$
(d) Let $\xi$ be a tangent vector field on $\Sigma$, then
$(\xi\cdot f_{M})(x)=(\xi\cdot f)(x)\ \ \ \ for\ x\in\Sigma.$
Now we can prove Theorem 3.2.
Proof of Theorem 3.2 (continued): By definition, for any
$\bar{\omega}_{\rho}(t)=\bar{\omega}_{\rho}+dd^{c}\varphi(t)$ with $t$ small,
we have
${\mathcal{A}}_{\rho}^{\prime\prime}(0)\geq 0.$
If we take $\bar{\omega}_{\rho}(t)=\bar{\omega}_{\rho}+tdd^{c}\psi$, then
$dd^{c}\eta=\bar{\omega}_{\rho}^{\prime\prime}(0)=0$. In this case,
${\mathcal{A}}_{\rho}^{\prime\prime}(0)=-\frac{1}{4}\int_{\Sigma}D_{1}^{2}d\mu_{\rho}-\frac{1}{4}\int_{\Sigma}D_{2}^{2}d\mu_{\rho}\geq
0.$
Therefore, we must have
(32) $D_{1}(\psi)=D_{2}(\psi)\equiv 0,\ \ \ on\ \Sigma,$
for any $\psi\in C^{\infty}(M,\textbf{R})$. Fix any $q\in\Sigma$, we will
prove that $\sin\alpha(q)=0$ by taking special $\psi$ on $M$. At $q$, we
choose an $\bar{g}_{\rho}$-orthonormal frame $\\{e_{1},e_{2},\cdots,e_{2n}\\}$
of $T_{q}M$ such that $J$ takes the form (15). Set
$\displaystyle A(\psi)$ $\displaystyle=$
$\displaystyle(\overline{\nabla}_{\rho}^{2}\psi)(e_{1},e_{1})-(\overline{\nabla}_{\rho}^{2}\psi)(e_{2},e_{2})+(\overline{\nabla}_{\rho}^{2}\psi)(e_{3},e_{3})-(\overline{\nabla}_{\rho}^{2}\psi)(e_{4},e_{4}),$
$\displaystyle B(\psi)$ $\displaystyle=$ $\displaystyle
2\left[(\overline{\nabla}_{\rho}^{2}\psi)(e_{2},e_{3})-(\overline{\nabla}_{\rho}^{2}\psi)(e_{1},e_{4})\right],$
$\displaystyle C(\psi)$ $\displaystyle=$
$\displaystyle\langle\overline{\nabla}_{\rho}\psi,((\overline{\nabla}_{\rho})_{e_{3}}J)e_{1}-((\overline{\nabla}_{\rho})_{e_{1}}J)e_{3}+((\overline{\nabla}_{\rho})_{e_{4}}J)e_{2}-((\overline{\nabla}_{\rho})_{e_{2}}J)e_{4}\rangle_{\rho}.$
Then by (31),
(33)
$D_{2}(\psi)=\sin\alpha_{\rho}\left[\sin\alpha_{\rho}A(\psi)+\cos\alpha_{\rho}B(\psi)+C(\psi)\right].$
Taking any $f\in C^{\infty}(\Sigma,\textbf{R})$, we have the normal extension
$f_{M}$ of $f$ over a neighborhood of $\Sigma$ in $M$. Let $\psi\in
C^{\infty}(M,\textbf{R})$ such that $\psi=f_{M}$ in a neighborhood of
$\Sigma$. We will compute the restrictions of $A(\psi)$, $B(\psi)$ and
$C(\psi)$ to $\Sigma$ using Lemma 3.3. By (a) and (b) of Lemma 3.3, we see
that on $\Sigma$
$A(\psi)=(\nabla_{\rho}^{2}f)(e_{1},e_{1})-(\nabla_{\rho}^{2}f)(e_{2},e_{2}).$
Here, $\nabla_{\rho}$ is the Levi-Civita connection of the induced metric
$g_{\rho}$. By (c), (d) of Lemma 3.3 and Gauss formula, we have on $\Sigma$
$B(\psi)=2\left[(h^{3}_{21}-h^{4}_{11})e_{1}(f)+(h^{3}_{22}-h^{4}_{12})e_{2}(f)\right].$
and
$\displaystyle C(\psi)$ $\displaystyle=$
$\displaystyle\langle((\overline{\nabla}_{\rho})_{e_{3}}J)e_{1}-((\overline{\nabla}_{\rho})_{e_{1}}J)e_{3}+((\overline{\nabla}_{\rho})_{e_{4}}J)e_{2}-((\overline{\nabla}_{\rho})_{e_{2}}J)e_{4},e_{1}\rangle
e_{1}(f)$
$\displaystyle\langle((\overline{\nabla}_{\rho})_{e_{3}}J)e_{1}-((\overline{\nabla}_{\rho})_{e_{1}}J)e_{3}+((\overline{\nabla}_{\rho})_{e_{4}}J)e_{2}-((\overline{\nabla}_{\rho})_{e_{2}}J)e_{4},e_{2}\rangle
e_{2}(f).$
We define a function $f$ around $q$ on $\Sigma$ by
(34) $f\left(\tilde{\exp}_{q}(te_{1}(q)+se_{2}(q))\right)=t^{2}-s^{2},$
where the exponential map $\tilde{\exp}$ is defined using the induced metric
$g_{\rho}$ on $\Sigma$, and then extend $f$ to be a smooth function on the
whole $\Sigma$. By definition, we can easily obtain that
$(e_{1}f)(q)=\frac{d}{dt}|_{t=0}f\left(\tilde{\exp}_{q}(te_{1}(q))\right)=\frac{d}{dt}|_{t=0}t^{2}=0,$
$(e_{2}f)(q)=\frac{d}{ds}|_{s=0}f\left(\tilde{\exp}_{q}(se_{2}(q))\right)=\frac{d}{ds}|_{s=0}(-s^{2})=0.$
For the second derivative, we have (Page 344 of [11])
$(\nabla^{2}_{\rho}f)(e_{1},e_{1})(q)=\frac{d^{2}}{dt^{2}}|_{t=0}f\left(\tilde{\exp}_{q}(te_{1}(q))\right)=\frac{d^{2}}{dt^{2}}|_{t=0}t^{2}=2,$
$(\nabla^{2}_{\rho}f)(e_{2},e_{2})(q)=\frac{d^{2}}{ds^{2}}|_{s=0}f\left(\tilde{\exp}_{q}(se_{1}(q))\right)=\frac{d^{2}}{ds^{2}}|_{s=0}(-s^{2})=-2.$
Taking values on both sides of (33) at $q$ and using (32), we have
$\sin\alpha_{\rho}(q)=0.$ As $q$ is arbitrary, we know that
$\sin\alpha_{\rho}\equiv 0$ on $\Sigma$. Therefore, the immersion is
$J$-holomorphic. Q.E.D.
The proof of the above theorem relies just on local arguments. In fact the
same proof works also in the noncompact case (see Remark 3.1):
###### Theorem 3.4.
Let $(M^{2n},\bar{\omega},J)$ be a complete noncompact symplectic manifold
with compatible almost complex structure $J$ and $F:\Sigma^{2}\to M$ be an
immersion. If the area functional ${\mathcal{A}}$ has a compatible stable
point in $[\bar{\omega}]$, then the immersion is $J$-holomorphic.
## 4\. The algebraic case: linear projective stability
Let us now assume that the target manifold is an algebraic manifold that
embeds into some complex projective space $\textbf{CP}^{N}$ holomorphically
and isometrically, namely that there is an embedding
$\iota:(M,\bar{\omega},J,\bar{g})\to(\textbf{CP}^{N},\omega_{FS}^{N},J_{FS},g_{FS}^{N}),$
which is holomorphic, such that
(35) $\iota^{*}\omega_{FS}^{N}=\bar{\omega},\ \ \iota^{*}g_{FS}^{N}=\bar{g}.$
Denote by ${\mathcal{H}}_{N}$ and ${\mathcal{K}}_{N}$ the space of holomorphic
vector fields and Killing vector fields on $\textbf{CP}^{N}$. Then it is well-
known that ${\mathcal{H}}_{N}={\mathcal{K}}_{N}\oplus J{\mathcal{K}}_{N}$.
Given any $W\in J{\mathcal{K}}_{N}$, it will generate a one parameter family
of diffeomorphisms $\Phi_{t}$ of $\textbf{CP}^{N}$. It is known that there
exists a family of smooth functions $\phi(t)$ on $\textbf{CP}^{N}$, such that
$\tilde{\omega}(t)=\Phi_{t}^{*}\omega_{FS}^{N}=\omega^{N}_{FS}+dd^{c}\phi(t)$.
Set $\varphi(t)=\phi(t)\circ\iota$, which is a family of smooth functions on
$M$. Set $\dot{\varphi}=\frac{d}{dt}|_{t=0}\varphi(t)$.
###### Definition 4.1.
Given immersion $F:\Sigma^{2}\to(M,\bar{\omega},J,\bar{g})$, we call the area
functional ${\mathcal{A}}$ has a linearly projectively stable point at
$\rho\in\mathcal{H}$ if $\bar{\omega}_{\rho}$ is projectively induced and
${\mathcal{A}}^{\prime\prime}(0)\geq 0$
for any $\bar{\omega}_{\rho}(t)=\bar{\omega}_{\rho}+tdd^{c}\dot{\varphi}$,
where $\varphi(t)$ is defined with $\bar{\omega}$ replaced by
$\bar{\omega}_{\rho}$ as above.
###### Theorem 4.1.
Let $(M,\bar{\omega},J,\bar{g})$ be an algebraic manifold with all structures
induced by the projective space as above and $F:\Sigma^{2}\to M$ be an
immersion. If the area functional has a linearly projectively stable point,
then the immersion must be holomorphic with respect to the complex structure
$J$.
Proof: As $J$ is compatible with any Kähler metric in $[\bar{\omega}]$,
without loss of generality, we assume that $\rho\equiv 0$ so that
$\bar{\omega}_{\rho}=\bar{\omega}.$ We denote by $\alpha$ and $\tilde{\alpha}$
the Kähler angle of $F:\Sigma^{2}\to(M,\bar{\omega},J,\bar{g})$ and
$\iota\circ
F:\Sigma^{2}\to(\textbf{CP}^{N},\omega_{FS}^{N},J_{FS},g_{FS}^{N})$,
respectively. As the embedding is holomorphic and satisfies (35), we see that
$\alpha=\tilde{\alpha}$. Set $\psi=\dot{\varphi}$, then by (29), the second
variation formula for $\bar{\omega}(t)=\bar{\omega}+tdd^{c}\psi$ is given by
(36)
${\mathcal{A}}^{\prime\prime}(0)=-\frac{1}{4}\int_{\Sigma}D_{1}^{2}d\mu-\frac{1}{4}\int_{\Sigma}D_{2}^{2}d\mu,$
where
$D_{1}=\sin\alpha[-(dd^{c}\psi)(e_{1},e_{4})+(dd^{c}\psi)(e_{2},e_{3})]$
and
(37)
$D_{2}=\sin\alpha\left[(dd^{c}\psi)(e_{1},e_{3})+(dd^{c}\psi)(e_{2},e_{4})\right].$
By our assumption, we must have
(38) $D_{1}(W)=D_{2}(W)\equiv 0.$
Set $\tilde{\psi}=\dot{\phi}$ and define
(39)
$\hat{D}_{2}(W)=\sin\alpha\left[(dd^{c}\tilde{\psi})(\tilde{e}_{1},\tilde{e}_{3})+(dd^{c}\tilde{\psi})(\tilde{e}_{2},\tilde{e}_{4})\right].$
Here, $\\{\tilde{e}_{1},\cdots,\tilde{e}_{2N}\\}$ is an orthonormal frame of
$\textbf{CP}^{N}$ so that $\tilde{e}_{\sigma}=\iota_{*}e_{\sigma}$ for
$1\leq\sigma\leq 4$. It is easy to see that for this choice of frame, the
complex structure $J_{FS}^{N}$ also takes the form (15). (Recall that
$\tilde{\alpha}=\alpha$.) We can have another expression for $\hat{D}_{2}(W)$.
From
$\tilde{\omega}(t)=\Phi_{t}^{*}\omega_{FS}^{N}=\omega^{N}_{FS}+dd^{c}\phi(t)$,
we have $dd^{c}\tilde{\psi}=\tilde{\omega}^{\prime}(0)=L_{W}\omega^{N}_{FS}$.
By direct computation, we can obtain that
(40) $\displaystyle\hat{D}_{2}(W)$ $\displaystyle=$
$\displaystyle\sin\alpha\left\\{\sin\alpha\left[\langle\overline{\nabla}^{N}_{\tilde{e}_{1}}W,\tilde{e}_{1}\rangle-\langle\overline{\nabla}^{N}_{\tilde{e}_{2}}W,\tilde{e}_{2}\rangle+\langle\overline{\nabla}^{N}_{\tilde{e}_{3}}W,\tilde{e}_{3}\rangle-\langle\overline{\nabla}^{N}_{\tilde{e}_{4}}W,\tilde{e}_{4}\rangle\right]\right.$
$\displaystyle\left.+\cos\alpha\left[\langle\overline{\nabla}^{N}_{\tilde{e}_{2}}W,\tilde{e}_{3}\rangle+\langle\overline{\nabla}^{N}_{\tilde{e}_{3}}W,\tilde{e}_{2}\rangle-\langle\overline{\nabla}^{N}_{\tilde{e}_{1}}W,\tilde{e}_{4}\rangle-\langle\overline{\nabla}^{N}_{\tilde{e}_{4}}W,\tilde{e}_{1}\rangle\right]\right\\}.$
Here, $\overline{\nabla}^{N}$ is the Levi-Civita connection on
$(\textbf{CP}^{N},g_{FS}^{N})$. Suppose $W=JV$ for $V\in{\mathcal{K}}_{N}$,
then using the fact that
$\langle\overline{\nabla}^{N}_{\tilde{e}_{i}}V,\tilde{e}_{j}\rangle+\langle\overline{\nabla}^{N}_{\tilde{e}_{j}}V,\tilde{e}_{i}\rangle=0,$
we can easily obtain that
(41)
$\hat{D}_{2}(W)=\tilde{D}_{2}(JV)=-2\sin\alpha\left[\langle\overline{\nabla}^{N}_{\tilde{e}_{1}}V,\tilde{e}_{3}\rangle+\langle\overline{\nabla}^{N}_{\tilde{e}_{2}}V,\tilde{e}_{4}\rangle\right].$
In order to proceed further, we need the following key lemma:
###### Lemma 4.2.
For each point $q\in\Sigma\subset\textbf{CP}^{N}$, there exists a Killing
vector field $V_{q}\in{\mathcal{K}}_{N}$, such that
(42)
$\left[\langle\overline{\nabla}^{N}_{\tilde{e}_{1}}V_{q},\tilde{e}_{3}\rangle+\langle\overline{\nabla}^{N}_{\tilde{e}_{2}}V_{q},\tilde{e}_{4}\rangle\right](q)=1,$
and
$|\overline{\nabla}^{N}V_{q}|(q)\in\left[\frac{\sqrt{2}}{2},\sqrt{2}\right]$.
In fact, we can take the value of (42) to be any real number by choosing
appropriate Killing vector $V_{q}$.
Proof of Lemma 4.2: Denote the homogeneous coordinate on $\textbf{CP}^{N}$ by
$[Z_{0}:\cdots:Z_{N}]$, and suppose w.l.o.g. that $q=[1:0:\cdots:0]\in
U_{0}:=\\{[Z_{0}:\cdots:Z_{N}|Z_{0}\neq 0]\\}$. Affine coordinates on $U_{0}$
are given by
$z_{j}=\frac{Z_{j}}{Z_{0}},\ \ 1\leq j\leq N.$
Set $z_{j}=x_{j}+\sqrt{-1}y_{j}$. We also assume that the homogeneous
coordinate is chosen so that the Fubini-Study metric at $q$ is identity.
Namely,
$\langle\frac{\partial}{\partial x_{i}},\frac{\partial}{\partial
x_{j}}\rangle=\langle\frac{\partial}{\partial y_{i}},\frac{\partial}{\partial
y_{j}}\rangle=\delta_{ij},\ \ \langle\frac{\partial}{\partial
x_{i}},\frac{\partial}{\partial y_{j}}\rangle=0.$
Set $\pi:\textbf{C}^{N+1}\backslash\\{0\\}\to\textbf{CP}^{N}$ the natural
projection, i.e., $\pi(Z_{0},\cdots,Z_{N})=[Z_{0}:\cdots:Z_{N}]$. In the
homogeneous coordinate, the vector field
$\tilde{X}=\sum_{A,B=0}^{N}a^{AB}Z_{A}\frac{\partial}{\partial Z_{B}}$
is a holomorphic vector field on $\textbf{C}^{N+1}$, where
$a^{AB}\in\textbf{C}$. It is known that
$X=\pi_{*}(\tilde{X})$
is a holomorphic vector field on $\textbf{CP}^{N}$, and its real part or
imaginary part is a Killing vector field on $\textbf{CP}^{N}$. By direct
computation, we obtain that
$\displaystyle X$ $\displaystyle=$
$\displaystyle\sum_{j=1}^{N}\left\\{-a^{00}z_{j}-\sum_{i=1}^{N}a^{i0}z_{i}z_{j}+a^{0j}+\sum_{i=1}^{N}a^{ij}z_{i}\right\\}\frac{\partial}{\partial
z_{j}}$ $\displaystyle:=$
$\displaystyle\sum_{j=1}^{N}(A_{j}+\sqrt{-1}B_{j})\frac{\partial}{\partial
z_{j}}.$
Set
$V_{1}=Re(X)=\frac{1}{2}\sum_{j=1}^{N}\left\\{A_{j}\frac{\partial}{\partial
x_{j}}+B_{j}\frac{\partial}{\partial y_{j}}\right\\},\
V_{2}=Im(X)=\frac{1}{2}\sum_{j=1}^{N}\left\\{B_{j}\frac{\partial}{\partial
x_{j}}-A_{j}\frac{\partial}{\partial y_{j}}\right\\}=-J_{FS}^{N}V_{1}.$
We suppose $V_{1}$ is a Killing vector field. (The case for $V_{2}$ is
similar.) Suppose $a^{AB}=u^{AB}+\sqrt{-1}v^{AB}$, then it is easy to obtain
that
$A_{j}=-u^{00}x_{j}+v^{00}y_{j}-\sum_{i=1}^{N}u^{i0}(x_{i}x_{j}-y_{i}y_{j})+\sum_{i=1}^{N}v^{i0}(x_{i}y_{j}+x_{j}y_{i})+u^{0j}+\sum_{i=1}^{N}(u^{ij}x_{i}-v^{ij}y_{i}),$
$B_{j}=-v^{00}x_{j}-u^{00}y_{j}-\sum_{i=1}^{N}v^{i0}(x_{i}x_{j}-y_{i}y_{j})-\sum_{i=1}^{N}u^{i0}(x_{i}y_{j}+x_{j}y_{i})+v^{0j}+\sum_{i=1}^{N}(v^{ij}x_{i}+u^{ij}y_{i}).$
As ${\textbf{CP}^{N}}$ is a symmetric space, we know that for any Killing
vector field $V$ and tangent vector field $U$, we have
(43) $\overline{\nabla}^{N}_{U}V=[U,V].$
Suppose
$\tilde{e}_{\sigma}(q)=\left(\sum_{j=1}^{N}P_{\sigma}^{j}\frac{\partial}{\partial
x_{j}}+\sum_{j=1}^{N}Q_{\sigma}^{j}\frac{\partial}{\partial y_{j}}\right)(q),\
\ \ \ 1\leq\sigma\leq 4.$
Define four vector fields
$\xi_{\sigma}=\sum_{j=1}^{N}P_{\sigma}^{j}\frac{\partial}{\partial
x_{j}}+\sum_{j=1}^{N}Q_{\sigma}^{j}\frac{\partial}{\partial y_{j}},\ \ \ \
1\leq\sigma\leq 4.$
Then
(44)
$(\overline{\nabla}^{N}_{\tilde{e}_{1}}V_{1})(q)=(\overline{\nabla}^{N}_{\xi_{1}}V_{1})(q),\
\
(\overline{\nabla}^{N}_{\tilde{e}_{2}}V_{1})(q)=(\overline{\nabla}^{N}_{\xi_{2}}V_{1})(q).$
By direct computation, using the definitions of $V_{1}$, $\xi_{1}$, $\xi_{2}$,
and recalling that $z_{j}(q)=0$, we can obtain that
(45) $\displaystyle[\xi_{\sigma},V_{1}](q)$ $\displaystyle=$
$\displaystyle\frac{1}{2}\sum_{j=1}^{N}\left\\{P_{\sigma}^{j}(u^{jj}-u^{00})+\sum_{i\neq
j}P_{\sigma}^{i}u^{ij}+Q_{\sigma}^{j}(v^{00}-v^{jj})-\sum_{i\neq
j}Q_{\sigma}^{i}v^{ij}\right\\}\frac{\partial}{\partial x_{j}}$
$\displaystyle+\frac{1}{2}\sum_{j=1}^{N}\left\\{-P_{\sigma}^{j}(v^{00}-v^{jj})+\sum_{i\neq
j}P_{\sigma}^{i}v^{ij}+Q_{\sigma}^{j}(u^{jj}-u^{00})+\sum_{i\neq
j}Q_{\sigma}^{i}u^{ij}\right\\}\frac{\partial}{\partial y_{j}}.$
Written in matrix language, the coordinate of $[\xi_{\sigma},V_{1}](q)$ in
$\\{x_{i},\cdots,x_{N},y_{1},\cdots,y_{N}\\}$ is given by
$\frac{1}{2}(P_{\sigma}^{1},\cdots,P_{\sigma}^{N},Q_{\sigma}^{1},\cdots,Q_{\sigma}^{N})O$.
Here $O$ is a $2N\times 2N$ matrix given by
$O=\left(\begin{array}[]{cc}O_{1}&O_{2}\\\ -O_{2}&O_{1}\\\
\end{array}\right),$
where
$O_{1}=\left(\begin{array}[]{ccccc}u^{11}-u^{00}&u^{12}&\cdot&\cdot&u^{1N}\\\
u^{21}&u^{22}-u^{00}&\cdot&\cdot&u^{2N}\\\ \cdot&\cdot&\cdot&\cdot&\cdot\\\
\cdot&\cdot&\cdot&\cdot&\cdot\\\ u^{N1}&u^{N2}&\cdot&\cdot&u^{NN}-u^{00}\\\
\end{array}\right)$
and
$O_{2}=\left(\begin{array}[]{ccccc}v^{11}-v^{00}&v^{12}&\cdot&\cdot&v^{1N}\\\
v^{21}&v^{22}-v^{00}&\cdot&\cdot&v^{2N}\\\ \cdot&\cdot&\cdot&\cdot&\cdot\\\
\cdot&\cdot&\cdot&\cdot&\cdot\\\ v^{N1}&v^{N2}&\cdot&\cdot&v^{NN}-v^{00}\\\
\end{array}\right).$
Our goal is to choose an appropriate matrix $O$ (which determines the Killing
vector field $V_{1}$ and in turn the holomorphic vector field $X$) such that
$\left[\langle\overline{\nabla}^{N}_{\tilde{e}_{1}}V_{1},\tilde{e}_{3}\rangle+\langle\overline{\nabla}^{N}_{\tilde{e}_{2}}V_{1},\tilde{e}_{4}\rangle\right](q)\neq
0$. By (43), (44) and (45), we see that
(46)
$\displaystyle\left[\langle\overline{\nabla}^{N}_{\tilde{e}_{1}}V_{1},\tilde{e}_{3}\rangle+\langle\overline{\nabla}^{N}_{\tilde{e}_{2}}V_{1},\tilde{e}_{4}\rangle\right](q)$
$\displaystyle=$
$\displaystyle\frac{1}{2}\left\\{\langle(P_{1}^{1},\cdots,P_{1}^{N},Q_{1}^{1},\cdots,Q_{1}^{N})O,(P_{3}^{1},\cdots,P_{3}^{N},Q_{3}^{1},\cdots,Q_{3}^{N})\rangle\right.$
$\displaystyle+\left.\langle(P_{2}^{1},\cdots,P_{2}^{N},Q_{2}^{1},\cdots,Q_{2}^{N})O,(P_{4}^{1},\cdots,P_{4}^{N},Q_{4}^{1},\cdots,Q_{4}^{N})\rangle\right\\}.$
It is more convenient to write (46) in complex notation. Set
$\tilde{u}^{jj}=u^{jj}-u^{00}$, $\tilde{u}^{ij}=u^{ij}$ for $i\neq j$.
Similarly, set $\tilde{v}^{jj}=v^{jj}-v^{00}$, $\tilde{v}^{ij}=v^{ij}$ for
$i\neq j$. Then $O_{1}=(\tilde{u}^{ij})$ and $O_{2}=(\tilde{v}^{ij})$. By
direct computation, using (45), we get that
(47)
$[\xi_{\sigma},V_{1}](q)=Re\left\\{\sum_{i,j=1}^{N}(P_{\sigma}^{i}+\sqrt{-1}Q_{\sigma}^{i})(\tilde{u}^{ij}+\sqrt{-1}\tilde{v}^{ij})\frac{\partial}{\partial
z_{j}}\right\\}.$
We also have that
$\tilde{e}_{\sigma}(q)=2Re\left\\{\sum_{j=1}^{N}(P_{\sigma}^{j}+\sqrt{-1}Q_{\sigma}^{j})\frac{\partial}{\partial
z_{j}}\right\\}.$
Therefore,
(48)
$\displaystyle\left[\langle\overline{\nabla}^{N}_{\tilde{e}_{1}}V_{1},\tilde{e}_{3}\rangle+\langle\overline{\nabla}^{N}_{\tilde{e}_{2}}V_{1},\tilde{e}_{4}\rangle\right](q)$
$\displaystyle=$ $\displaystyle 2\left\\{\left\langle
Re\left(\sum_{i,j=1}^{N}(P_{1}^{i}+\sqrt{-1}Q_{1}^{i})(\tilde{u}^{ij}+\sqrt{-1}\tilde{v}^{ij})\frac{\partial}{\partial
z_{j}}\right),Re\left(\sum_{j=1}^{N}(P_{3}^{j}+\sqrt{-1}Q_{3}^{j})\frac{\partial}{\partial
z_{j}}\right)\right\rangle\right.$ $\displaystyle+\left.\left\langle
Re\left(\sum_{i,j=1}^{N}(P_{2}^{i}+\sqrt{-1}Q_{2}^{i})(\tilde{u}^{ij}+\sqrt{-1}\tilde{v}^{ij})\frac{\partial}{\partial
z_{j}}\right),Re\left(\sum_{j=1}^{N}(P_{4}^{j}+\sqrt{-1}Q_{4}^{j})\frac{\partial}{\partial
z_{j}}\right)\right\rangle\right\\}.$
Set
$R_{\sigma}=(R_{\sigma}^{1},\cdots,R_{\sigma}^{n})=(P_{\sigma}^{1}+\sqrt{-1}Q_{\sigma}^{1},\cdots,P_{\sigma}^{n}+\sqrt{-1}Q_{\sigma}^{n})$
and $\tilde{O}=(\tilde{u}^{ij}+\sqrt{-1}\tilde{v}^{ij})$. We will take
$\tilde{O}$ as follows:
(49) $\tilde{O}=(R_{3}^{1}\bar{R}_{1},\cdots,R_{3}^{n}\bar{R}_{1}).$
Then
(50) $|\tilde{O}|^{2}=|R_{1}|^{2}(|R_{3}^{1}|^{2}+\cdots|R_{3}^{n}|^{2})=1,$
and
$R_{\sigma}\tilde{O}=(\sum_{i=1}^{N}R_{\sigma}^{i}\bar{R}_{1}^{i})R_{3}=\left(\sum_{i=1}^{N}(P_{1}^{i}P_{\sigma}^{i}+Q_{1}^{i}Q_{\sigma}^{i})+\sqrt{-1}\sum_{i=1}^{N}(P_{1}^{i}Q_{\sigma}^{i}-Q_{1}^{i}P_{\sigma}^{i})\right)R_{3}.$
In particular, we have
$R_{1}\tilde{O}=R_{3}$
and
$R_{2}\tilde{O}=\sqrt{-1}\sum_{i=1}^{N}(P_{1}^{i}Q_{2}^{i}-Q_{1}^{i}P_{2}^{i})R_{3}.$
By direct computation and using (48), we have
(51)
$\displaystyle\left[\langle\overline{\nabla}^{N}_{\tilde{e}_{1}}V_{1},\tilde{e}_{3}\rangle+\langle\overline{\nabla}^{N}_{\tilde{e}_{2}}V_{1},\tilde{e}_{4}\rangle\right](q)$
$\displaystyle=$ $\displaystyle 2\left\\{\left\langle
Re\left(\sum_{j=1}^{N}R_{3}^{j}\frac{\partial}{\partial
z_{j}}\right),Re\left(\sum_{j=1}^{N}R_{3}^{j}\frac{\partial}{\partial
z_{j}}\right)\right\rangle\right.$ $\displaystyle+\left.\left\langle
Re\left(\sqrt{-1}\sum_{i=1}^{N}(P_{1}^{i}Q_{2}^{i}-Q_{1}^{i}P_{2}^{i})R_{3}^{j}\frac{\partial}{\partial
z_{j}}\right),Re\left(\sum_{j=1}^{N}R_{4}^{j}\frac{\partial}{\partial
z_{j}}\right)\right\rangle\right\\}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left\\{1+\sum_{i=1}^{N}(P_{1}^{i}Q_{2}^{i}-Q_{1}^{i}P_{2}^{i})\sum_{i=1}^{N}(P_{3}^{i}Q_{4}^{i}-Q_{3}^{i}P_{4}^{i})\right\\}.$
Recall that $J_{FS}^{N}(\frac{\partial}{\partial
x_{j}})=\frac{\partial}{\partial y_{j}}$, $J_{FS}^{N}(\frac{\partial}{\partial
y_{j}})=-\frac{\partial}{\partial x_{j}}$. By the definition of
$\tilde{e}_{\sigma}$, it is easy to see that
$\left[\langle\overline{\nabla}_{\tilde{e}_{1}}V_{1},\tilde{e}_{3}\rangle+\langle\overline{\nabla}_{\tilde{e}_{2}}V_{1},\tilde{e}_{4}\rangle\right](q)=\frac{1}{2}(1+\langle
J_{FS}^{N}\tilde{e}_{1},\tilde{e}_{2}\rangle\langle
J_{FS}^{N}\tilde{e}_{3},\tilde{e}_{4}\rangle).$
By the choice of complex structure (15), we see that
$\langle J_{FS}^{N}\tilde{e}_{1},\tilde{e}_{2}\rangle=\langle
J_{FS}^{N}\tilde{e}_{3},\tilde{e}_{4}\rangle=\cos\alpha(q).$
Therefore
(52)
$\left[\langle\overline{\nabla}^{N}_{\tilde{e}_{1}}V_{1},\tilde{e}_{3}\rangle+\langle\overline{\nabla}^{N}_{\tilde{e}_{2}}V_{1},\tilde{e}_{4}\rangle\right](q)=\frac{1}{2}(1+\cos^{2}\alpha(q))\in\left[\frac{1}{2},1\right].$
For the covariant derivative of $V_{1}$, we note that, by the definition of
$V_{1}$ and the fact that $z_{j}(q)=0$, we have
(53)
$|\overline{\nabla}^{N}V_{1}|^{2}(q)=\frac{1}{4}\sum_{j=1}^{N}(|\overline{\nabla}^{N}A_{j}|^{2}+|\overline{\nabla}^{N}B_{j}|^{2})=\frac{1}{2}\left\\{\sum_{i,j=1}^{N}((u^{ij})^{2}+(v^{ij})^{2})+(u^{00})^{2}+(u^{00})^{2}\right\\}.$
We now choose $O$ so that $u^{00}=u^{0j}=u^{i0}=0$, $v^{00}=v^{0j}=v^{i0}=0$
for $1\leq i,j\leq N$. Then by the definition of $\tilde{u}^{ij}$ and
$\tilde{v}^{ij}$, we obtain that $\tilde{u}^{ij}=u^{ij}$,
$\tilde{v}^{ij}=v^{ij}$. On the other hand, as
$\tilde{O}=(\tilde{u}^{ij}+\sqrt{-1}\tilde{v}^{ij})$, we have by (50) that
$\sum_{i,j=1}^{N}((\tilde{u}^{ij})^{2}+(\tilde{v}^{ij})^{2})=|\tilde{O}|^{2}=1.$
Therefore, by (53), we see that
(54)
$|\overline{\nabla}^{N}V_{1}|^{2}(q)=\frac{1}{2}\sum_{i,j=1}^{n}((u^{ij})^{2}+(v^{ij})^{2})=\frac{1}{2}.$
The proof of the lemma is completed for $V_{q}=V_{1}$ by replacing $\tilde{O}$
by $\frac{2}{1+\cos^{2}\alpha(q)}\tilde{O}$. From the proof, we can also see
that, by multiplying a constant to $\tilde{O}$, we can take a Killing vector
field $V_{q}$ such that (42) takes any real number. Q.E.D.
The proof of the theorem is by combining Lemma 4.2 with the following
observation:
###### Lemma 4.3.
Under our assumption, we have $D_{2}(W)=\hat{D}_{2}(W)$.
Proof of Lemma 4.3: By (37) and (39), it suffices to prove that
(55)
$(dd^{c}\psi)(e_{1},e_{3})+(dd^{c}\psi)(e_{2},e_{4})=(dd^{c}\tilde{\psi})(\tilde{e}_{1},\tilde{e}_{3})+(dd^{c}\tilde{\psi})(\tilde{e}_{2},\tilde{e}_{4}).$
Here, $\\{e_{1},e_{2}\\}$ is an any orthonormal frame of
$F_{*}(T_{q}\Sigma)\subset T_{F(q)}M$, and $\\{e_{3},e_{4}\\}$ is chosen so
that the complex structures $J$ takes the form (15). Furthermore,
$\tilde{e}_{\sigma}=\iota_{*}e_{\sigma}$ for $1\leq\sigma\leq 4$. On the other
hand, from $\varphi(t)=\phi(t)\circ\iota$, we see that
$\psi=\tilde{\psi}\circ\iota$. Also note that, as $\iota$ is holomorphic and
$dd^{c}=2\sqrt{-1}\partial\bar{\partial}$, we have
$\iota^{*}dd^{c}\tilde{\psi}=dd^{c}\iota^{*}\tilde{\psi}=dd^{c}\psi$. Thus,
$\displaystyle(dd^{c}\tilde{\psi})(\tilde{e}_{1},\tilde{e}_{3})+(dd^{c}\tilde{\psi})(\tilde{e}_{2},\tilde{e}_{4})$
$\displaystyle=$
$\displaystyle(dd^{c}\tilde{\psi})(\iota_{*}e_{1},\iota_{*}e_{3})+(dd^{c}\tilde{\psi})(\iota_{*}e_{2},\iota_{*}e_{4})$
$\displaystyle=$
$\displaystyle(\iota^{*}dd^{c}\tilde{\psi})(e_{1},e_{3})+(\iota^{*}dd^{c}\tilde{\psi})(e_{2},e_{4})$
$\displaystyle=$
$\displaystyle(dd^{c}\psi)(e_{1},e_{3})+(dd^{c}\psi)(e_{2},e_{4}).$
Q.E.D.
Now we can complete the proof of Theorem 4.1. By Lemma 4.2, for each point
$q\in\Sigma\subset\textbf{CP}^{n}$, there is a Killing vector field
$V_{q}\in{\mathcal{K}}$, such that (42) holds. Combining this with (41) and
Lemma 4.3, we see that
$D_{2}(JV_{q})(q)=\hat{D}_{2}(JV_{q})(q)=-2\sin\alpha(q)$. By (38), we must
have $\sin\alpha(q)=0$. As $q$ arbitrary, we know that $\sin\alpha\equiv 0$ on
$\Sigma$. Therefore, the immersion is holomorphic. Q.E.D.
## 5\. Symplectic Manifolds with rational symplectic forms
In this section, we extend the results in the previous section to the case
that the target manifold $M$ is a symplectic manifold with rational symplectic
class. Using the approximately $J_{M}$-holomorphic embedding of $M$ into some
complex projective space, we can define the notion of linearly
${\mathcal{A}}^{k}$-stable point and prove that if $p=2$, then the existence
of linearly ${\mathcal{A}}^{k}$-stable pair point implies
$J_{M}$-holomorphicity.
Let $(M^{2n},\bar{\omega},\bar{g},J_{M})$ be a compact symplectic manifold
with symplectic form $\bar{\omega}$, compatible almost complex structure
$J_{M}$ and associated Riemannian metric $\bar{g}$, such that for any $X,Y\in
TM$,
(56) $\bar{g}(X,Y)=\bar{\omega}(X,J_{M}Y).$
Since $\bar{\omega}$ defines a rational cohomology class, then by a Theorem of
Borthwick and Uribe (Theorem 1.1 of [6]), we known that there exists a
sequence of embeddings
(57)
$\iota_{k}:M\to(\textbf{CP}^{N_{k}},\omega_{FS}^{k},g_{FS}^{k},J_{FS}^{k}),$
such that, if we put
(58) $\bar{\omega}_{k}=\iota_{k}^{*}\omega_{FS}^{k},\ \
\bar{g}_{k}=\iota_{k}^{*}g_{FS}^{k},$
then for $k\geq k_{0}$
(59)
$\left|\left|\frac{1}{k}\bar{\omega}_{k}-\bar{\omega}\right|\right|_{C^{0}}\leq\frac{C_{1}}{k},$
and
(60)
$\left|\left|\frac{1}{k}\bar{g}_{k}-\bar{g}\right|\right|_{C^{0}}\leq\frac{C_{2}}{k},$
for some constants $C_{1}$ and $C_{2}$ and large integer $k_{0}$.
Recall the definition of Kähler angle:
$\bar{\omega}|_{\Sigma}=\cos\alpha d\mu_{g},\ \
\omega^{k}_{FS}|_{\Sigma}=\cos\alpha_{k}d\mu_{g_{k}}.$
Here, $\alpha_{k}$ is the Kähler angle of the immersion $\iota_{k}\circ
F:\Sigma^{2}\to(\textbf{CP}^{N_{k}},\omega_{FS}^{k},g_{FS}^{k},J_{FS}^{k})$.
More precisely,
$\displaystyle\bar{\omega}|_{\Sigma}=F^{*}\bar{\omega},$
$\displaystyle\omega^{k}_{FS}|_{\Sigma}=(\iota_{k}\circ
F)^{*}\omega^{k}_{FS}=F^{*}\bar{\omega}_{k},$ $\displaystyle g=F^{*}\bar{g},$
$\displaystyle g_{k}=(\iota_{k}\circ F)^{*}g^{k}_{FS}=F^{*}\bar{g}_{k}.$
By (60) and the fact that $d\mu_{\frac{1}{k}g_{k}}=\frac{1}{k}d\mu_{g_{k}}$,
we see that $\frac{1}{k}d\mu_{g_{k}}\to d\mu_{g}$ and combining with (59), we
see that
(61) $\cos\alpha_{k}\to\cos\alpha,\ \ \sin\alpha_{k}\to\sin\alpha\ uniformly\
on\ \Sigma.$
Set ${\mathcal{K}}_{k}$ the space of Killing vector fields on
$\textbf{CP}^{N_{k}}$. Given any holomorphic vector field $W\in
J_{FS}^{k}{\mathcal{K}}_{k}$, let $\Phi_{t}$ be the one-parameter family of
diffeomorphisms generated by $W$. Set
$\omega^{k}(t)=\Phi_{t}^{*}\omega_{FS}^{k}=\omega_{FS}^{k}+dd_{FS}^{c}\varphi(t)$
for a family of smooth functions $\varphi(t)$ on $\textbf{CP}^{N_{k}}$.
Note that $\frac{1}{k}\bar{\omega}_{k}$ and $\bar{\omega}$ are in the same
cohomology class. Thus, there exists a smooth one form $\gamma_{k}$ on $M$,
such that $\bar{\omega}=\frac{1}{k}\bar{\omega}_{k}+d\gamma_{k}$. We consider
a family of projectively induced symplectic forms on $M$ given by
$\bar{\omega}(t)=\frac{1}{k}\iota_{k}^{*}\omega^{k}(t)=\frac{1}{k}\iota_{k}^{*}\Phi_{t}^{*}\omega_{FS}^{k}=\frac{1}{k}\bar{\omega}_{k}+d(\frac{1}{k}\iota_{k}^{*}d_{FS}^{c}\varphi(t))\equiv\bar{\omega}+d\beta_{k}(t),$
where $\beta_{k}(t)=\frac{1}{k}\iota_{k}^{*}d_{FS}^{c}\varphi(t)-\gamma_{k}$
is a family of smooth 1-forms on $M$. We can then extend the definitions of
the associated metrics by (3) and the area functional ${\mathcal{A}}(t)$ by
(4).
###### Definition 5.1.
Given immersion $F:\Sigma^{2}\to(M,\bar{\omega},J,\bar{g})$, we call the area
functional ${\mathcal{A}}$ has a compatible linearly
${\mathcal{A}}^{k}$-stable point at $\rho\in\mathcal{H}$ if
$\bar{\omega}_{\rho}$ is compatible with $J$ and
${\mathcal{A}}^{\prime\prime}(0)\geq 0$
for any $\bar{\omega}(t)=\bar{\omega}+td\dot{\beta}_{k}$, where $\beta_{k}(t)$
is defined with $\bar{\omega}$ replaced by $\bar{\omega}_{\rho}$ in the above
construction.
The main result in this section is
###### Theorem 5.1.
Let $(M^{2n},\bar{\omega},J_{M},\bar{g})$ be a symplectic manifold as above
and $F:\Sigma^{2}\to M$ be an immersion. There exists an integer $K_{1}$, such
that if the area functional has a compatible linearly
${\mathcal{A}}^{k}$-stable point for some $k\geq K_{1}$, then the immersion
must be $J_{M}$-holomorphic.
Proof: As $\bar{\omega}_{\rho}$ is compatible with $J$ by assumption we may
assume, without loss of generality, $\rho\equiv 0$ so that
$\bar{\omega}_{\rho}=\bar{\omega}$. Suppose
$\dot{\beta}_{k}=\frac{\partial\beta_{k}(t)}{\partial t}|_{t=0}=\theta_{k}$ so
that $\bar{\omega}(t)=\bar{\omega}+td\theta_{k}$. Then by (28), we see that
(62) $\displaystyle{\mathcal{A}}^{\prime\prime}(0)$ $\displaystyle=$
$\displaystyle-\frac{1}{4}\int_{\Sigma}\left[(d\theta_{k})(e_{1},J_{M}e_{2})+(d\theta_{k})(e_{2},J_{M}e_{1})\right]^{2}d\mu$
$\displaystyle-\frac{1}{4}\int_{\Sigma}\left[(d\theta_{k})(e_{1},J_{M}e_{1})-(d\theta_{k})(e_{2},J_{M}e_{2})\right]^{2}d\mu.$
Set
$D_{1}=(d\theta_{k})(e_{1},J_{M}e_{2})+(d\theta_{k})(e_{2},J_{M}e_{1}),\ \
D_{2}=(d\theta_{k})(e_{1},J_{M}e_{1})-(d\theta_{k})(e_{2},J_{M}e_{2}).$
Then we have
$D_{1}=\sin\alpha[-(d\theta_{k})(e_{1},e_{4})+(d\theta_{k})(e_{2},e_{3})]$
and
(63)
$D_{2}=\sin\alpha\left[(d\theta_{k})(e_{1},e_{3})+(d\theta_{k})(e_{2},e_{4})\right].$
Here, at a fixed point $q$ on $\Sigma$, we take the orthonormal frame
$\\{e_{1},\cdots,e_{2n}\\}$ such that the almost complex structure $J_{M}$
takes the form (15).
Set
$\tilde{D}_{2}=(\omega^{k})^{\prime}(0)(e_{1}^{k},J_{FS}^{k}e_{1}^{k})-(\omega^{k})^{\prime}(0)(e_{2}^{k},J_{FS}^{k}e_{2}^{k}),$
where $\\{e_{1}^{k},e_{2}^{k}\\}$ is an any orthonormal frame of
$(\iota_{k}\circ F)_{*}(T_{q}\Sigma)\subset T_{(\iota_{k}\circ
F)(q)}{\textbf{CP}^{N_{k}}}$. It is easy to see that we can choose an
orthonormal frame $\\{e_{1}^{k},e_{2}^{k},\cdots,e_{2N_{k}}^{k}\\}$ of
$T_{(\iota_{k}\circ F)(q)}{\textbf{CP}^{N_{k}}}$ such that $J_{FS}^{N_{k}}$
takes the form (15). Then we have two expressions for $\tilde{D}_{2}$. On the
one hand, for $W=JV$ with $V\in{\mathcal{K}}_{k}$, we have (see (41))
(64)
$\tilde{D}_{2}(W)=\tilde{D}_{2}(JV)=-2\sin\alpha_{k}\left[\langle\overline{\nabla}^{k}_{e_{1}^{k}}V,e_{3}^{k}\rangle+\langle\overline{\nabla}^{k}_{e_{2}^{k}}V,e_{4}^{k}\rangle\right].$
Here $\overline{\nabla}^{k}$ is the covariant differential of
$(\textbf{CP}^{N_{k}},g_{FS}^{k})$. On the other hand, similar to (63), we
have
(65)
$\tilde{D}_{2}(W)=\sin\alpha_{k}\left[(dd_{FS}^{c}\psi)(e_{1}^{k},e_{3}^{k})+(dd_{FS}^{c}\psi)(e_{2}^{k},e_{4}^{k})\right],$
where $\psi=\dot{\varphi}$.
We will prove the theorem by contradiction. Suppose
$F:\Sigma^{2}\to(M^{2n},\bar{\omega},\bar{g},J_{M})$ is not
$J_{M}$-holomorphic, then there exists a point $q\in\Sigma\subset M$, such
that $\sin\alpha(q)\neq 0$. Without loss of generality, we assume that
$\sin\alpha(q)=a>0$. Since $\sin\alpha_{k}\to\sin\alpha$, we know that
$\sin\alpha_{k}(q)>\frac{a}{2}>0$, for $k\geq N_{0}$ for some integer $N_{0}$.
Next, we will examine the relation between $D_{2}(W)$ and $\tilde{D}_{2}(W)$.
Note that by the definition of $\beta_{k}$,
$d\theta_{k}=d(\frac{1}{k}\iota_{k}^{*}d_{FS}^{c}\psi)$. Therefore, by (63),
(66) $\displaystyle D_{2}(W)(q)$ $\displaystyle=$
$\displaystyle\sin\alpha\left[\frac{1}{k}(\iota_{k}^{*}dd_{FS}^{c}\psi)(e_{1},e_{3})+\frac{1}{k}(\iota_{k}^{*}dd_{FS}^{c}\psi)(e_{2},e_{4})\right]$
$\displaystyle=$
$\displaystyle\sin\alpha\left[(dd_{FS}^{c}\psi)(\frac{1}{\sqrt{k}}(\iota_{k})_{*}e_{1},\frac{1}{\sqrt{k}}(\iota_{k})_{*}e_{3})+(dd_{FS}^{c}\psi)(\frac{1}{\sqrt{k}}(\iota_{k})_{*}e_{2},\frac{1}{\sqrt{k}}(\iota_{k})_{*}e_{4})\right].$
By the choice of the local frame, we know that $\\{e_{1},e_{2}\\}$ is an any
orthonormal frame of $F_{*}(T_{q}\Sigma)\subset T_{F(q)}M$, and
$\\{e_{1}^{k},e_{2}^{k}\\}$ is an any orthonormal frame of $(\iota_{k}\circ
F)_{*}(T_{q}\Sigma)\subset T_{(\iota_{k}\circ F)(q)}{\textbf{CP}^{N_{k}}}$. By
(15), we see that at $q$
(67) $e_{3}=\frac{J_{M}e_{1}-\cos\alpha(q)e_{2}}{\sin\alpha(q)},\ \
e_{4}=-\frac{J_{M}e_{2}+\cos\alpha(q)e_{1}}{\sin\alpha(q)},$
and
(68)
$e^{k}_{3}=\frac{J_{FS}^{k}e_{1}^{k}-\cos\alpha_{k}(q)e_{2}^{k}}{\sin\alpha_{k}(q)},\
\
e_{4}^{k}=-\frac{J_{FS}^{k}e_{2}^{k}+\cos\alpha_{k}(q)e_{1}^{k}}{\sin\alpha_{k}(q)}.$
We now fix $e_{1}$ and $e_{2}$ and take
$e_{1}^{k}=\frac{\frac{1}{\sqrt{k}}(\iota_{k})_{*}e_{1}}{|\frac{1}{\sqrt{k}}(\iota_{k})_{*}e_{1}|_{g_{FS}^{k}}},\
\
e_{2}^{k}=\frac{\frac{1}{\sqrt{k}}(\iota_{k})_{*}e_{2}-\langle\frac{1}{\sqrt{k}}(\iota_{k})_{*}e_{2},e_{1}^{k}\rangle
e_{1}^{k}}{|\frac{1}{\sqrt{k}}(\iota_{k})_{*}e_{2}-\langle\frac{1}{\sqrt{k}}(\iota_{k})_{*}e_{2},e_{1}^{k}\rangle
e_{1}^{k}|_{g_{FS}^{k}}}.$
Note that
$g_{FS}^{k}(\frac{1}{\sqrt{k}}(\iota_{k})_{*}e_{i},\frac{1}{\sqrt{k}}(\iota_{k})_{*}e_{j})=\frac{1}{k}\bar{g}_{k}(e_{i},e_{j})\to\bar{g}(e_{i},e_{j})=\delta_{ij},\
\ as\ k\to\infty.$
Therefore, it is easy to see that
(69) $|\frac{1}{\sqrt{k}}(\iota_{k})_{*}e_{1}-e_{1}^{k}|_{g_{FS}^{k}}\to 0,\ \
|\frac{1}{\sqrt{k}}(\iota_{k})_{*}e_{2}-e_{2}^{k}|_{g_{FS}^{k}}\to 0,\ \ as\
k\to\infty.$
On the other hand, by Proposition 4.4 of [6], we know that
$|\frac{1}{\sqrt{k}}(\iota_{k})_{*}J_{M}e_{i}-\frac{1}{\sqrt{k}}J_{FS}^{k}(\iota_{k})_{*}e_{i}|\leq\frac{C}{\sqrt{k}}\to
0,\ \ as\ k\to\infty.$
Combining all the above together, we see that
(70) $|\frac{1}{\sqrt{k}}(\iota_{k})_{*}e_{3}-e_{3}^{k}|_{g_{FS}^{k}}\to 0,\ \
|\frac{1}{\sqrt{k}}(\iota_{k})_{*}e_{4}-e_{4}^{k}|_{g_{FS}^{k}}\to 0,\ \ as\
k\to\infty.$
We can now finish the proof. Comparing (64) and (65) and using Lemma 4.2, we
see that we can take a Killing vector field $V_{k}\in{\mathcal{K}}_{k}$, such
that for $W_{k}=J_{FS}^{k}V_{k}$, we have
$\left[(dd_{FS}^{c}\psi_{k})(e_{1}^{k},e_{3}^{k})+(dd_{FS}^{c}\psi_{k})(e_{2}^{k},e_{4}^{k})\right](q)=-2,$
for some smooth function $\psi_{k}$ on $\textbf{CP}^{N_{k}}$. On the other
hand,
$\displaystyle|[(dd_{FS}^{c}\psi_{k})(\frac{1}{\sqrt{k}}(\iota_{k})_{*}e_{1},\frac{1}{\sqrt{k}}(\iota_{k})_{*}e_{3})+(dd_{FS}^{c}\psi_{k})(\frac{1}{\sqrt{k}}(\iota_{k})_{*}e_{2},\frac{1}{\sqrt{k}}(\iota_{k})_{*}e_{4})]$
$\displaystyle-[(dd_{FS}^{c}\psi_{k})(e_{1}^{k},e_{3}^{k})+(dd_{FS}^{c}\psi_{k})(e_{2}^{k},e_{4}^{k})]|$
$\displaystyle\leq$
$\displaystyle|(dd_{FS}^{c}\psi_{k})(\frac{1}{\sqrt{k}}(\iota_{k})_{*}e_{1}-e_{1}^{k},\frac{1}{\sqrt{k}}(\iota_{k})_{*}e_{3})|+|(dd_{FS}^{c}\psi_{k})(e_{1}^{k},\frac{1}{\sqrt{k}}(\iota_{k})_{*}e_{3}-e_{3}^{k})|$
$\displaystyle+|(dd_{FS}^{c}\psi_{k})(\frac{1}{\sqrt{k}}(\iota_{k})_{*}e_{2}-e_{2}^{k},\frac{1}{\sqrt{k}}(\iota_{k})_{*}e_{4})|+|(dd_{FS}^{c}\psi_{k})(e_{2}^{k},\frac{1}{\sqrt{k}}(\iota_{k})_{*}e_{4}-e_{4}^{k})|$
$\displaystyle\leq$ $\displaystyle
C|dd_{FS}^{c}\psi_{k}|_{g_{FS}^{k}}\epsilon_{k}$
for some sequence $\epsilon_{k}\to 0$ and constant $C$ independent of $k$.
From
$\omega^{k}(t)=\Phi_{t}^{*}\omega_{FS}^{k}=\omega^{k}_{FS}+dd_{FS}^{c}\varphi(t)$,
we see that $dd_{FS}^{c}\psi_{k}=L_{W_{k}}\omega_{FS}^{k}$. Therefore,
$|dd_{FS}^{c}\psi_{k}|_{g_{FS}^{k}}\leq
2|\overline{\nabla}^{k}W_{k}|_{g_{FS}^{k}}=2|\overline{\nabla}^{k}V_{k}|_{g_{FS}^{k}}\leq
2\sqrt{2}.$
In the last inequality, we used Lemma 4.2 again. Hence, we have
$\displaystyle\left|(dd_{FS}^{c}\psi_{k})(\frac{1}{\sqrt{k}}(\iota_{k})_{*}e_{1},\frac{1}{\sqrt{k}}(\iota_{k})_{*}e_{3})+(dd_{FS}^{c}\psi_{k})(\frac{1}{\sqrt{k}}(\iota_{k})_{*}e_{2},\frac{1}{\sqrt{k}}(\iota_{k})_{*}e_{4})\right|(q)$
$\displaystyle\geq$
$\displaystyle|(dd_{FS}^{c}\psi_{k})(e_{1}^{k},e_{3}^{k})+(dd_{FS}^{c}\psi_{k})(e_{2}^{k},e_{4}^{k})|(q)-C|dd_{FS}^{c}\psi_{k}|_{g_{FS}^{k}}(q)\epsilon_{k}$
$\displaystyle\geq$ $\displaystyle 2-2\sqrt{2}C\epsilon_{k}\geq 1$
for $k\geq K_{1}\geq N_{0}$. By (66), we see that for $k\geq K_{1}$, we have
$|D_{2}(W_{k})(q)|\geq a>0.$
By (62), we see that for $\beta_{k}$ associated to such $W_{k}$, we have
${\mathcal{A}}^{\prime\prime}_{\bar{\omega}}(d\theta_{k})<0.$
This contradicts our assumption and the proof of the Theorem is completed.
Q.E.D.
## 6\. Kähler Manifolds with possibly non rational Kähler class
We now assume that $(M,J)$ is an algebraic manifold, that is, a submanifold of
some complex projective space. When $[\bar{\omega}]$ is a rational class and
$\bar{g}$ is the metric induced by the Fubini-Study metric Lawson-Simons
(Corollary 9 of [12]) proved that a submanifold of $M$ is holomorphic if the
second variation of the area is nonnegative with respect to holomorphic
deformation of $M$ in $\textbf{CP}^{N}$. In Sections 5, we showed in this case
that, existence of linearly projectively stable point also implies
holomorphicity. In this section we allow $[\bar{\omega}]$ to be any real
Kähler class and $\bar{g}$ any $J$-induced metric. Take any Kähler metric
$\bar{\omega}$ on $M$ with $[\bar{\omega}]\in H^{2}(M,\textbf{R})\cap
H^{1,1}(M,\textbf{C})$. Let $\bar{g}$ be the Riemannian metric associated to
$\bar{\omega}$ and $J$.
As $(M,J)$ is an algebraic manifold it is easy to see that there exists a
sequence of Kähler forms $\tau_{k}$ with $[\tau_{k}]\in
H^{2}(M,\textbf{Q})\cap H^{1,1}(M,\textbf{C})$, such that
(71) $||\tau_{k}-\bar{\omega}||_{C^{2}}\leq\varepsilon_{k},$
with $\varepsilon_{k}\to 0$ as $k\to\infty$. Here, the $C^{2}$ norm is taken
with respect to the metric $\bar{\omega}$. Since $[\tau_{k}]$ is rational,
there exists, for every $k\in\textbf{N}$, a holomorphic line bundle
$(L_{k},h_{k})\to M$ carrying a hermitian connection $D_{k}$ of curvature
$\frac{\sqrt{-1}}{2\pi}D_{k}^{2}=\tau_{k}$. In particular,
$c_{1}(L_{k})=[\tau_{k}]$. For each positive integer $m>0$, the hermitian
metric $h_{k}$ induces a hermitian metric $h_{k}^{m}$ on $L_{k}^{m}$. Choose
an orthonormal basis $\\{S_{k,0}^{m},\cdots,S_{k,N_{k,m}}^{m}\\}$ of the space
$H^{0}(M,L_{k}^{m})$ of all holomorphic global sections of $L_{k}^{m}$. Here,
the inner product on $H^{0}(M,L_{k}^{m})$ is the natural one induced by the
Kähler metric $\tau_{k}$ and the hermitian metric $h_{k}^{m}$ on $L^{m}_{k}$.
By Kodaira embedding theorem, there exists an integer $m_{k,0}$ such that if
$m\geq m_{k,0}$, then such a basis induces a holomorphic embedding
$\Psi_{k,m}$ of $M$ into $\textbf{CP}^{N_{k,m}}$ given by
(72) $\Psi_{k,m}:M\to\textbf{CP}^{N_{k,m}},\ \
\Psi_{k,m}(z):=[S_{k,0}^{m}(z):\cdots:S^{m}_{k,N_{k,m}}(z)].$
Let $\omega^{k}_{FS}$ be the standard Fubini-Study metric on
$\textbf{CP}^{N_{k,m}}$. Then $\frac{1}{m}\Psi_{k,m}^{*}\omega^{k}_{FS}$ is a
Kähler form on $M$ which lies in the same Kähler class as $\tau_{k}$. We call
$\frac{1}{m}\Psi_{k,m}^{*}\omega^{k}_{FS}$ the Bergman metric. A famous
Theorem proved by Tian ([21]) tells us that
(73)
$\left|\left|\frac{1}{m}\Psi_{k,m}^{*}\omega^{k}_{FS}-\tau_{k}\right|\right|_{C^{2}}\leq\frac{C}{\sqrt{m}}.$
Here the $C^{2}$ norm is taken with respect to the metric $\tau_{k}$ and the
constant $C$ depends on $\tau_{k}$. Because of (71), we can assume that the
constant is uniformly bounded with respect to $k$. Although the Bergman metric
$\frac{1}{m}\Psi_{k,m}^{*}\omega^{k}_{FS}$ depends on the Kähler metric
$\tau_{k}$, the set of Bergman metrics
(74)
${\mathcal{P}}_{k,m}:=\left\\{\frac{1}{m}\Psi_{k,m}^{*}\sigma^{*}(\omega^{k}_{FS})|\sigma\in
Aut(\textbf{CP}^{N_{k,m}})\right\\},$
is independent of the choice of $\tau_{k}$ in $[\tau_{k}]$ and
${\mathcal{P}}_{k}:=\cup^{\infty}_{m=1}{\mathcal{P}}_{k,m}$ is dense in
$[\tau_{k}]\cap Ka(M)$ in the $C^{2}$-topology induced by the one on
$\Lambda^{2}M$. Here, $Ka(M)$ is the space of Kähler metrics on $M$. It is
known that ${\mathcal{P}}_{k,m}$ has finite dimension for each $k$ and $m$.
Set
(75)
${\mathcal{Q}}_{k}:=\left\\{\frac{1}{m(k)}\Psi_{k,m(k)}^{*}\sigma^{*}(\omega^{k}_{FS})|\sigma\in
Aut(\textbf{CP}^{N_{k,m(k)}})\right\\},$
where $m(k)\geq m_{k,0}$ is a sequence of integers such that $m(k)\to\infty$
as $k\to\infty$.
Define
$V_{k}:=\\{\bar{\omega}\\}-\\{\tau_{k}\\}+{\mathcal{Q}}_{k}=\left\\{\bar{\omega}-\tau_{k}+\frac{1}{m(k)}\Psi_{k,m(k)}^{*}\sigma^{*}(\omega^{k}_{FS})|\sigma\in
Aut(\textbf{CP}^{N_{k,m(k)}})\right\\}$
Then $V_{k}$ is a finitely dimensional submanifold of $[\bar{\omega}]$. In
particular, for any $\sigma(t)\subset Aut(\textbf{CP}^{N_{k,m(k)}})$, there
exists a smooth function $\varphi(t)$ on $M$, such that
(76)
$\bar{\omega}(t):=\bar{\omega}-\tau_{k}+\frac{1}{m(k)}\Psi_{k,m(k)}^{*}\sigma(t)^{*}(\omega^{k}_{FS})=\bar{\omega}+2\sqrt{-1}\partial\bar{\partial}\varphi(t)=\bar{\omega}+dd^{c}\varphi(t).$
Denote by ${\mathcal{H}}_{N_{k,m(k)}}$ the space of holomorphic vector fields
on $\textbf{CP}^{N_{k,m(k)}}$.
###### Definition 6.1.
Given an immersion $F:\Sigma^{2}\to(M,\bar{\omega},J,\bar{g})$, we say that
the area functional $\mathcal{A}$ has a $k$-linearly projectively stable point
at $\rho\in\mathcal{H}$ if there exists a smooth function $\rho$ on $M$, such
that $\bar{\omega}_{\rho}\in Ka(M)$ and
${\mathcal{A}}^{\prime\prime}(0)\geq 0$
for any $\bar{\omega}(t)=\bar{\omega}+tdd^{c}\dot{\varphi}$, where
$\varphi(t)$ is given with $\sigma(0)=id$ and $\bar{\omega}$ replaced by
$\bar{\omega}_{\rho}$ in the above construction.
The main result in this section is as follows:
###### Theorem 6.1.
Let $(M,J)$ be an algebraic manifold, $\bar{\omega}$ be any Kähler metric and
$F:\Sigma^{2}\to M$ be an immersion. Then there exists an integer $K_{2}$,
such that if if the area functional has a $k$-linearly projectively stable
point at $\rho\in\mathcal{H}$ for some $k\geq K_{2}$, then the immersion is
$J$-holomorphic.
Proof: As $J$ is compatible with any Kähler metric in $[\bar{\omega}]$ we
assume, without loss of generality, that $\rho\equiv 0$ so that
$\bar{\omega}_{\rho}=\bar{\omega}.$ By (28), for
$\bar{\omega}(t)=\bar{\omega}+tdd^{c}\dot{\varphi}$, the second variation
formula is given by
(77)
${\mathcal{A}}^{\prime\prime}(0)=-\frac{1}{4}\int_{\Sigma}D_{1}^{2}d\mu-\frac{1}{4}\int_{\Sigma}D_{2}^{2}d\mu,$
where
$D_{1}=\bar{\omega}^{\prime}(0)(e_{1},Je_{2})+\bar{\omega}^{\prime}(0)(e_{2},Je_{1}),$
$D_{2}=\bar{\omega}^{\prime}(0)(e_{1},Je_{1})-\bar{\omega}^{\prime}(0)(e_{2},Je_{2}).$
As before, $\\{e_{1},e_{2}\\}$ is any orthonormal frame of
$F_{*}(T_{q}\Sigma)\subset T_{F(q)}M$ with respect to the Kähler metric
$\bar{\omega}$. Taking any curve $\sigma(t)$ in
$Aut(\textbf{CP}^{N_{k,m(k)}})$, let $W$ be the vector field on
$\textbf{CP}^{N_{k,m(k)}}$ generating $\sigma(t)$, then it is known that
$W\in{\mathcal{H}}_{N_{k,m(k)}}$. From (76), we see that
$\bar{\omega}^{\prime}(0)=dd^{c}\dot{\varphi}=\frac{1}{m(k)}\Psi_{k,m(k)}^{*}L_{W}(\omega^{k}_{FS}).$
Therefore,
(78) $\displaystyle D_{2}$ $\displaystyle=$
$\displaystyle(L_{W}\omega^{k}_{FS})(\frac{1}{\sqrt{m(k)}}(\Psi_{k,m(k)})_{*}e_{1},\frac{1}{\sqrt{m(k)}}(\Psi_{k,m(k)})_{*}(Je_{1}))$
$\displaystyle-(L_{W}\omega^{k}_{FS})(\frac{1}{\sqrt{m(k)}}(\Psi_{k,m(k)})_{*}e_{2},\frac{1}{\sqrt{m(k)}}(\Psi_{k,m(k)})_{*}(Je_{2}))$
On the other hand, for the second variation formula of $\Psi_{k,m(k)}\circ
F:\Sigma\to\textbf{CP}^{N_{k}}$, we have (see (40))
(79)
$\hat{D}_{2}=(L_{W}\omega^{k}_{FS})(\tilde{e}_{1},J_{FS}^{k}\tilde{e}_{1})-(L_{W}\omega^{k}_{FS})(\tilde{e}_{2},J_{FS}^{k}\tilde{e}_{2})$
By the definition of $\Psi_{k,m(k)}$ , we know that $\Psi_{k,m(k)}$ is
holomorphic. From (71) and (73), we see that the holomorphic embedding
$\Psi_{k,m(k)}:(M,\bar{\omega})\to(\textbf{CP}^{N_{k,m(k)}},\omega_{FS}^{k})$
satisfies that
$\left|\left|\frac{1}{m(k)}\Psi_{k,m(k)}^{*}\omega^{k}_{FS}-\bar{\omega}\right|\right|_{C^{2}}\to
0$
and
$\left|\left|\frac{1}{m(k)}\Psi_{k,m(k)}^{*}g^{k}_{FS}-\bar{g}\right|\right|_{C^{2}}\to
0.$
Then following the same argument as in the proof of Theorem 5.1, the proof is
complete.
Q.E.D.
Recently, Popovici ([17]) proved that, for a Kähler manifold with
transcendental Kähler class (not necessarily algebraic), one can also have a
Kodaira-type approximately holomorphic projective embedding theorem and a
Tian-type almost-isometry theorem. Similar to what we did above, we can define
another notion of $k$-LP stable point and use the same idea to prove that in
fact the existence of $k$-LP stable point implies holomorphicity. It is
interesting to underline that even when $M$ is algebraic, this and Theorem 6.1
provide two different approximation arguments.
## Appendix A
In the main part of this paper, we only consider variations of the symplectic
form $\bar{\omega}(t)$ in the same cohomology class. In this appendix, we
consider the case the target manifold $M$ is any Riemannian manifold. We will
show that in this general case, the concept ”${\mathcal{A}}$-stationary” is
not well-posed without restrictions on the type of variations of the metric.
Suppose $M$ is an $(n+p)$-dimensional Riemannian manifold and $\Sigma$ is a
$p$-dimensional submanifold of $M$ and a family of immersions
$F(t):\Sigma\to(M,\bar{g}(t)).$
Then the induced metric on $\Sigma$ is given by
$g(t)=F(t)^{*}\bar{g}(t).$
Set
(80) ${\mathcal{A}}(t)\equiv Area(F(t)(\Sigma),g(t))=\int_{\Sigma}d\mu(t).$
We will compute the first variation of ${\mathcal{A}}$.
Let $F_{t}$ restricted to $\Sigma$ be the variational vector field and
$\\{x_{i}\\}_{i=1}^{n}$ be the local coordinates on $\Sigma$. Then in local
coordinate,
(81) $g_{ij}(t)=\bar{g}(t)\left(\frac{\partial F}{\partial
x_{i}}(t),\frac{\partial F}{\partial x_{j}}(t)\right).$
Denote by $\overline{\nabla}$ and $\nabla$ the Levi-Civita connections on $M$
and $\Sigma$ respectively. Set
(82) $\nu(t)=\frac{\sqrt{det(g_{ij}(t))}}{\sqrt{det(g_{ij}(0))}}.$
Then $\nu(t)$ is well-defined independent of the choice of coordinate system.
Furthermore,
(83)
${\mathcal{A}}(t)=\int_{\Sigma}\sqrt{det(g_{ij}(t))}=\int_{M}\nu(t)\sqrt{det(g_{ij}(0))},$
and therefore
(84)
$\frac{d}{dt}|_{t=0}{\mathcal{A}}(t)=\int_{\Sigma}\frac{d}{dt}|_{t=0}\nu(t)\sqrt{det(g_{ij}(0))}.$
Suppose
(85) $\frac{\partial\bar{g}(t)}{\partial t}|_{t=0}=h.$
To evaluate $\frac{d}{dt}|_{t=0}\nu(t)$ at a given point $x$, we choose an
orthonormal coordinate system. Using this and the fact that
$\overline{\nabla}_{F_{t}}F_{x_{i}}-\overline{\nabla}_{F_{x_{i}}}F_{t}=[F_{t},F_{x_{i}}]=0$,
we get at $x$:
(86) $\displaystyle\frac{d}{dt}|_{t=0}\nu(t)$ $\displaystyle=$
$\displaystyle\frac{1}{2}\sum_{i=1}^{p}g_{ii}^{\prime}(0)=\frac{1}{2}\sum_{i=1}^{p}\frac{d}{dt}|_{t=0}\left\\{\bar{g}(t)\left(\frac{\partial
F}{\partial x_{i}}(t),\frac{\partial F}{\partial x_{i}}(t)\right)\right\\}$
$\displaystyle=$
$\displaystyle\frac{1}{2}\sum_{i=1}^{p}h\left(F_{x_{i}},F_{x_{i}}\right)+\bar{g}(\overline{\nabla}_{F_{t}}F_{x_{i}},F_{x_{i}})$
$\displaystyle=$
$\displaystyle\frac{1}{2}tr_{g}(F^{*}h)+\sum_{i=1}^{p}\bar{g}(\overline{\nabla}_{F_{x_{i}}}F_{t}^{T},F_{x_{i}})+\sum_{i=1}^{p}\bar{g}(\overline{\nabla}_{F_{x_{i}}}F_{t}^{N},F_{x_{i}})$
$\displaystyle=$
$\displaystyle\frac{1}{2}tr_{g}(F^{*}h)+div_{\Sigma}F_{t}^{T}-\bar{g}(F_{t},\textbf{H}),$
Therefore,
(87)
$\delta{\mathcal{A}}(F_{t},h)=\frac{1}{2}\int_{\Sigma}tr_{g}(F^{*}h)d\mu-\int_{\Sigma}\bar{g}(F_{t},\textbf{H})d\mu.$
We cannot expect that ${\mathcal{A}}^{\prime}(0)=0$ for any $F_{t}$ and any
$h$. In fact, we have
###### Proposition A.1.
Given $(F,\bar{g})$, for any variational vector field $F_{t}$, there exist
$h_{1}$ and $h_{2}$, such that $\delta{\mathcal{A}}(F_{t},h_{1})>0$ and
$\delta{\mathcal{A}}(F_{t},h_{2})<0$.
Proof: As $F$ and $\bar{g}$ is fixed, given any $F_{t}$, suppose
$\left|\bar{g}(F_{t},\textbf{H})\right|\leq C_{0}.$
We take $h_{1}=2(C_{0}+1)\bar{g}$ and $h_{2}=-2(C_{0}+1)\bar{g}$, then
$F^{*}h_{1}=2(C_{0}+1)g$ and $F^{*}h_{2}=-2(C_{0}+1)g$. Therefore,
$\delta{\mathcal{A}}(F_{t},h_{1})\geq\left[(p-1)C_{0}+p\right]Area(\Sigma)\geq
pArea(\Sigma)>0$
and
$\delta{\mathcal{A}}(F_{t},h_{2})\leq\left[-(p-1)C_{0}-p\right]Area(\Sigma)\leq-
pArea(\Sigma)<0.$
Q.E.D.
## References
* [1] L, Ambrosio and H. Soner, Level set approach to mean curvature flow in arbitrary codimension, J. Differential Geom., 43 (1996), no. 4, 693-737.
* [2] C. Arezzo, Minimal surfaces and deformations of holomorphic curves in Kähler-Einstein manifolds, Ann. Scuola Norm. Sup. Pisa Cl. Sci., (4) 29 (2000), no. 2, 473-481.
* [3] C. Arezzo and M. J. Micallef, Minimal surfaces in flat tori. Geom. Funct. Anal. 10 (2000), no. 4, 679-701.
* [4] C. Arezzo, M.J. Micallef and G.P. Pirola: Stable minimal surfaces of finite total curvature, Comm. in Analysis and Geometry. 10 (2002), no. 1, 11-22.
* [5] C. Arezzo and G. La Nave, Minimal two spheres and Kahler-Einstein metrics on Fano manifolds. Advances in Math. 191 (2005) 209-223
* [6] D. Borthwick and A. Uribe, Nearly Kählerian embeddings of symplectic manifolds, Asian J. Math., 4 (2000), 599-620.
* [7] A. Cannas da Silva, Lectures on symplectic geometry, Lecture Notes in Mathematics, 1764. Springer-Verlag, Berlin, 2001.
* [8] B. Clarke and Y. Rubinstein, Ricci flow and the metric completion of the space of Käbler metrics, Amer. J. Math. to appear.
* [9] S. S. Chern and J. Wolfson, Minimal surfaces by moving frams, Amer. J. Math., 105 (1983), 59-83.
* [10] T. Colding and W. Minicozzi, Minimal surfaces, Courant Lecture Notes in Mathematics, no. 4, New York University, 1998.
* [11] S. Lang, Fundamentals of differential geometry, Graduate Texts in Mathematics, 191. Springer-Verlag, New York, 1999.
* [12] H. B. Lawson and J. Simons, On stable currents and their application to global problems in real and complex geometry, Ann. of Math. (2), 98 (1973), 427-450.
* [13] K. Matsumoto, A note on the differentiablity of the distance function to regular submanifolds of Riemannian manifolds, Nihonkai Math J., 3 (1992), 81-85.
* [14] M. Micallef, Stable minimal surfaces in Euclidean space, J. Differential Geometry, 19 (1984), no. 1, 57-84.
* [15] M.J. Micallef, Stable minimal surfaces in flat tori, Complex Differential Geometry and Nonlinear Differential Equations 49 (1984), 73-78.
* [16] M.J. Micallef and J. Wolfson, Area Minimizers in a K3 surface and holomorphicity, Geom. Funct. Anal. 16 (2006), 437-452.
* [17] D. Popovici, Transcendental Kähler Cohomology Classes, arXiv:1201.0740.
* [18] J. Sacks and K. Uhlenbeck, Minimal immersions of closed Riemann surfaces, Trans. Amer. Math. Soc 271 (1982), 639-652.
* [19] R. Schoen and S.T. Yau, Existence of incompressible minimal surfaces and topology of three-manifolds with non-negative scalar curvature, Ann. of Math. 110 (1979), 127-142.
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|
arxiv-papers
| 2012-11-29T18:39:56 |
2024-09-04T02:49:38.667272
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Claudio Arezzo, Jun Sun",
"submitter": "Jun Sun",
"url": "https://arxiv.org/abs/1211.7016"
}
|
1211.7075
|
# Secure and Reliable Transmission with Cooperative Relays in Two-Hop Wireless
Networks
Yulong Shen14, Xiaohong Jiang2, Jianfeng Ma1 and Weisong Shi3 1School of
Computer Science and Technology, Xidian University, China 2School of Systems
Information Science, Future University Hakodate, Japan 3Department of
Computer Science, Wayne State University, USA
4Email:[email protected]
###### Abstract
This work considers the secure and reliable information transmission in two-
hop relay wireless networks without the information of both eavesdropper
channels and locations. While the previous work on this problem mainly studied
infinite networks and their asymptotic behavior and scaling law results, this
papers focuses on a more practical network with finite number of system nodes
and explores the corresponding exact results on the number of eavesdroppers
the network can tolerant to ensure a desired secrecy and reliability. For
achieving secure and reliable information transmission in a finite network,
two transmission protocols are considered in this paper, one adopts an optimal
but complex relay selection process with less load balance capacity while the
other adopts a random but simple relay selection process with good load
balance capacity. Theoretical analysis is further provided to determine the
exact and maximum number of independent and also uniformly distributed
eavesdroppers one network can tolerate to satisfy a specified requirement in
terms of the maximum secrecy outage probability and maximum transmission
outage probability allowed.
## I Introduction
Two-hop ad hoc wireless networks, where each packet travels at most two hops
(source-relay-destination) to reach its destination, has been a class of basic
and important networking scenarios [1]. Actually, the analysis of basic two-
hop relay networks serves as the foundation for performance study of general
multi-hop networks. Due to the promising applications of ad hoc wireless
networks in many important scenarios (like battlefield networks, emergency
networks, disaster recovery networks), the consideration of secrecy (and also
reliability) in such networks is of great importance for ensuring the high
confidentiality requirements of these applications. This paper focuses on the
issue of secure and reliable information transmission in the basic two-hop ad
hoc wireless networks.
Traditionally, the information security is provided by adopting the
cryptography approach, where a plain message is encrypted through a
cryptographic algorithm that is hard to break (decrypt) in practice by any
adversary without the key. While the cryptography is acceptable for general
applications with standard security requirement, it may not be sufficient for
applications with a requirement of strong form of security (like military
networks and emergency networks). This is because that the cryptographic
approach can hardly achieve everlasting secrecy, since the adversary can
record the transmitted messages and try any way to break them [2]. That is why
there is an increasing interest in applying signaling scheme in physical layer
to provide a strong form of security, where a degraded signal at an
eavesdropper is always ensured such that the original data can be hardly
recovered regardless of how the signal is processed at the eavesdropper. We
consider applying physical layer method to guarantee secure and reliable
information transmission in the two-hop wireless networks.
By now, a lot of research efforts have been dedicated to providing security
through physical layer methods. A power control scheme is proposed in [3] to
ensure that an eavesdropper can never reach its desired signal-to-noise-plus-
interference ratio (SINR). However, such scheme is not effective when the
eavesdropper has a better channel than the receiver. The technique of
artificial noise generation has also been widely explored to jam the
eavesdroppers and provide secure transmission in the relay communications
[4][5][6][7]. Recently, the cooperative jamming through node cooperation has
been demonstrated to be efficient in ensuring physical layer security
[8][9][10]. It is notable that these schemes generally reply on the knowledge
of eavesdropper channels and locations to jam eavesdroppers. In practice,
however, it is difficult to gain such information, specifically in untrusted
network environment. To address this constraint, a cooperative protocol based
on artificial noise generation and multi-user diversity has been proposed
recently in [11] to achieve secure transmission in two-hop wireless networks
without the knowledge of eavesdropper channels and locations. In particular,
the asymptotic behavior of such cooperative protocol in a network has been
reported there to illustrate how the number of eavesdroppers the network can
tolerate scales as the number of system nodes there tends to infinite.
This paper focuses on applying the relay cooperation scheme to achieve secure
and reliable information transmission in a more practical finite two-hop
wireless network without the knowledge of both eavesdropper channels and
locations. The main contributions of this paper as follows:
1) For achieving secure and reliable information transmission in a more
practical two-hop wireless network with finite number of system nodes, we
consider the application of the cooperative protocol proposed in [11] with an
optimal and complex relay selection process but less load balance capacity,
and also propose to use a new cooperative protocol with a simple and random
relay selection process but good load balance capacity.
2) Rather than exploring the asymptotic behavior and scaling law results, this
paper provides theoretic analysis of above both cooperative protocols to
determine the corresponding exact results on the number of independent and
also uniformly distributed eavesdroppers one network can tolerate to satisfy a
specified requirement in terms of the maximum secrecy outage probability and
maximum transmission outage probability allowed.
The remainder of the paper is organized as follows. Section II introduces the
system models and two cooperative transmission protocols considered in this
paper. Section III provides theoretical analysis and also related discussions
of the two protocols, and Section IV concludes this paper.
## II System Models and Transmission Protocols
### II-A Network Model
As illustrated in Fig.1 that we consider a network scenario where a source
node $S$ wishes to communicate securely with its destination node $D$ with the
help of multiple relay nodes $R_{1}$, $R_{2}$, $\cdots$, $R_{n}$. In addition
to these normal system nodes, there are also $m$ eavesdroppers $E_{1}$,
$E_{2}$, $\cdots$, $E_{m}$ that are independent and also uniformly distributed
in the network. Our goal here is to ensure the secure and reliable information
transmission from source $S$ to destination $D$ under the condition that no
real time information is available about both eavesdropper channels and
locations.
Figure 1: System scenario: Source $S$ wishes to communicate securely with
destination $D$ with the assistance of finite relays $R_{1}$, $R_{2}$,
$\cdots$, $R_{n}$ ($n$=4 in the figure) in the presence of passive
eavesdroppers $E_{1}$, $E_{2}$, $\cdots$, $E_{m}$ ($m$=4 in the figure).
Cooperative relay scheme is used in the two-hop transmission. A assistant node
is selected randomly as relay ($R_{1}$ in the figure).
### II-B Transmission Model
Consider the transmission from a transmitter $A$ to a receiver $B$, and denote
by $x_{i}^{\left(A\right)}$ the $i^{th}$ symbol transmitted by $A$ and denote
by $y_{i}^{\left(B\right)}$ the $i^{th}$ signal received by $B$. We assume
that all nodes transmit with the same power $E_{s}$, path loss between all
pairs of nodes is equal and independent, and the frequency-nonselective multi-
path fading from $A$ to $B$ is a complex zero-mean Gaussian random variable.
Under the condition that all nodes in a group of nodes, $\mathcal{R}$, are
generating noises, the $i^{th}$ signal received at node $B$ from node $A$ is
determined as:
$y_{i}^{\left(B\right)}=h_{A,B}\sqrt{E_{s}}x_{i}^{\left(A\right)}+\sum_{A_{i}\in\mathcal{R}}h_{A_{i},B}\sqrt{E_{s}}x_{i}^{\left(A_{i}\right)}+n_{i}^{\left(B\right)},$
where the noise $\left\\{n_{i}^{\left(B\right)}\right\\}$ at receiver $B$ is
assumed to be i.i.d complex Gaussian random variables with
$E{\left[\left|n_{i}^{\left(B\right)}\right|^{2}\right]}=N_{0}$, and
$\left|h_{A,B}\right|^{2}$ is exponentially distributed with mean
$E{\left[\left|h_{A,B}\right|^{2}\right]}$. Without loss of generality, we
assume that $E{\left[\left|h_{A,B}\right|^{2}\right]}=1$. The SINR $C_{A,B}$
from $A$ to $B$ is then given by
$C_{A,B}=\frac{E_{s}\left|h_{A,B}\right|^{2}}{\sum_{A_{i}\in\mathcal{R}}E_{s}{\left|h_{A_{i},B}\right|^{2}}+N_{0}/2}$
For a legitimate node and an eavesdropper, we use two separate SINR thresholds
$\gamma_{R}$ and $\gamma_{E}$ to define the minimum SINR required to recover
the transmitted messages for legitimate node and eavesdropper, respectively.
Therefore, a system node (relay or destination) is able to decode a packet if
and only if its SINR is greater than $\gamma_{R}$, while the transmitted
message is secure if and only if the SINR at each eavesdropper is less than
$\gamma_{E}$.
### II-C Transmission Protocols
We consider here two transmission protocols for secure and reliable
information transmission in two-hop wireless networks. The first protocol
(hereafter called Protocol 1) is the one proposed in [11], in which the
optimal relay node with the best link condition to both source and destination
is always selected for information relaying. Although this protocol is
attractive in the sense that it provide very effective resistance against
eavesdroppers, it suffers from several problems. The protocol 1 involves a
complicated process of optimal relay selection, which is not very suitable for
the distributed wireless networks, in particular when the number of possible
relay nodes is huge. More importantly, since the channel state is relatively
constant during a fixed time period, some relay nodes with good link
conditions are always preferred for information relaying, resulting a severe
load balance problem and a quick node energy depletion in energy-limited
wireless environment.
Based on these observations, we propose to use a simple and random relay
selection rather than the optimal relay selection to achieve a better load and
energy consumption balance among possible relay nodes. By modifying the
Protocol 1 to include the random relay selection process, the new transmission
protocol (hereafter called Protocol 2) works as follows.
1) _Relay selection_ : A relay node, indexed by $j^{\ast}$, is selected
randomly from candidate relay nodes $R_{j},j=1,2,\cdots,n$.
2) _Channel measurement between the selected relay and the other relays_ : The
selected relay $j^{\ast}$ broadcasts a pilot signal to allow each of other
relays to measure the channel from $j^{\ast}$ to itself. Each of the other
relays $R_{j},j=1,2,\cdots,n,j\neq j^{\ast}$ then knows the corresponding
value of $h_{R_{j},R_{j^{\ast}}}$.
3) _Channel measurement between destination $D$ and the other relays_: The
destination $D$ broadcasts a pilot signal to allow each of other relays to
measure the channel from $D$ to itself. Each of the other relays
$R_{j},j=1,2,\cdots,n,j\neq j^{\ast}$ then knows the corresponding value of
$h_{R_{j},D}$.
4) _Message transmission from source $S$ to the selected relay
$R_{j^{\ast}}$_: The source $S$ transmits the messages to $R_{j^{\ast}}$.
Concurrently, the relay nodes with indexes in $\mathcal{R}_{1}={\left\\{j\neq
j^{\ast}:|h_{R_{j},R_{j^{\ast}}}|^{2}<\tau\right\\}}$, transmit noise to
generate sufficient interference at eavesdroppers.
5) _Message transmission from the selected relay $R_{j^{\ast}}$ to destination
$D$_: Similar to the Step 4, the relay $R_{j^{\ast}}$ transmits the messages
to destination $D$. Concurrently, the relay nodes with indexes in
$\mathcal{R}_{2}={\left\\{j\neq j^{\ast}:|h_{R_{j},D}|^{2}<\tau\right\\}}$,
transmit noise to generate sufficient interference at eavesdroppers.
## III Theoretical Analysis
This section first defines the transmission outage and secrecy outage adopted
in this paper to depict transmission reliability and transmission secrecy, and
then provides theoretical analysis to determine the numbers of eavesdroppers a
network can tolerate based on the Protocol 1 and Protocol 2, respectively.
### III-A Transmission Outage and Secrecy Outage
For a transmission from the source $S$ to destination $D$, we call
transmission outage happens if $D$ can not decode the transmitted packet,
i.e., $D$ received the packet with SINR less than the predefined threshold
$\gamma_{R}$. The transmission outage probability, denoted as
$P_{out}^{\left(T\right)}$, is then defined as the probability that
transmission outage from $S$ to $D$ happens. For a predefined upper bound
$\varepsilon_{t}$ on $P_{out}^{\left(T\right)}$, we call the communication
between $S$ and $D$ is reliable if
$P_{out}^{\left(T\right)}\leq\varepsilon_{t}$. Notice that for the
transmissions from $S$ to the selected relay $R_{j^{\ast}}$ and from
$R_{j^{\ast}}$ to $D$, the corresponding transmission outage can be defined in
the similar way as that of from $S$ to $D$. We use $O_{S\rightarrow
R_{j^{\ast}}}^{(T)}$ and $O_{R_{j^{\ast}}\rightarrow D}^{(T)}$ to denote the
events that transmission outage from source $S$ to $R_{j^{\ast}}$ happens and
transmission outage from relay $R_{j^{\ast}}$ to $D$ happens, respectively.
Due to the link independence assumption, we have
$\displaystyle P_{out}^{\left(T\right)}=P\left(O_{S\rightarrow
R_{j^{\ast}}}^{(T)}\cup O_{R_{j^{\ast}}\rightarrow D}^{(T)}\right)$
$\displaystyle\ \ \ \ \ \ =P\left(O_{S\rightarrow
R_{j^{\ast}}}^{(T)}\right)+P\left(O_{R_{j^{\ast}}\rightarrow D}^{(T)}\right)$
$\displaystyle\ \ \ \ \ \ \ \ \ -P\left(O_{S\rightarrow
R_{j^{\ast}}}^{(T)}\right)\cdot P\left(O_{R_{j^{\ast}}\rightarrow
D}^{(T)}\right)$
Regarding the secrecy outage, we call secrecy outage happens for a
transmission from $S$ to $D$ if at least one eavesdropper can recover the
transmitted packets during the process of this two-hop transmission, i.e., at
least one eavesdropper received the packet with SINR larger than the
predefined threshold $\gamma_{E}$. The secrecy outage probability, denoted as
$P_{out}^{\left(S\right)}$, is then defined as the probability that secrecy
outage happens during the transmission from $S$ to $D$. For a predefined upper
bound $\varepsilon_{s}$ on $P_{out}^{\left(S\right)}$, we call the
communication between $S$ and $D$ is secure if
$P_{out}^{\left(S\right)}\leq\varepsilon_{s}$. Notice that for the
transmissions from $S$ to the selected relay $R_{j^{\ast}}$ and from
$R_{j^{\ast}}$ to $D$, the corresponding secrecy outage can be defined in the
similar way as that of from $S$ to $D$. We use $O_{S\rightarrow
R_{j^{\ast}}}^{(S)}$ and $O_{R_{j^{\ast}}\rightarrow D}^{(S)}$ to denote the
events that secrecy outage from source $S$ to $R_{j^{\ast}}$ happens and
secrecy outage from relay $R_{j^{\ast}}$ to $D$ happens, respectively. Again,
due to the link independence assumption, we have
$\displaystyle P_{out}^{\left(S\right)}=P\left(O_{S\rightarrow
R_{j^{\ast}}}^{(S)}\right)+P\left(O_{R_{j^{\ast}}\rightarrow D}^{(S)}\right)$
$\displaystyle\ \ \ \ \ \ \ \ \ -P\left(O_{S\rightarrow
R_{j^{\ast}}}^{(S)}\right)\cdot P\left(O_{R_{j^{\ast}}\rightarrow
D}^{(S)}\right)$
### III-B Analysis of Protocol 1
In the Protocol 1 proposed in [11], the relay node with the largest value of
$min\left(\left|h_{S,R_{j}}\right|^{2},\left|h_{D,R_{j}}\right|^{2}\right),j=1,2,\cdots,n$,
is selected as relay. Notice that the Protocol 1 can always guarantee the
reliable transmission from source $S$ to destination $D$, this is because the
parameter $\tau$ is set as $\tau=\sqrt{\frac{\log{n}}{8n\gamma_{R}}}$, which
ensures that the target SINR at the selected relay and destination can be
achieved to decode the transmitted messages. Thus, we only need to focus the
secrecy requirement $P_{out}^{\left(S\right)}\leq\varepsilon_{s}$ to determine
the corresponding the number of eavesdroppers the network can tolerate here.
Theorem 1. For the network scenario illustrated in Fig 1 with equal path loss
between all pairs of nodes, to guarantee the secrecy requirement
$P_{out}^{(S)}\leq\varepsilon_{s}$ by applying the Protocol 1, the number of
eavesdroppers $m$ the network can tolerate should satisfy the following
condition.
$m\leq\left(1-\sqrt{1-\varepsilon_{s}}\right)\cdot\left(1+\gamma_{E}\right)^{\sqrt{\frac{n\log{n}}{32\gamma_{R}}}}$
###### Proof.
Notice that $P_{out}^{\left(S\right)}$ is determined as
$\displaystyle P_{out}^{\left(S\right)}=P\left(O_{S\rightarrow
R_{j^{\ast}}}^{(S)}\right)+P\left(O_{R_{j^{\ast}}\rightarrow D}^{(S)}\right)$
$\displaystyle\ \ \ \ \ \ \ \ \ -P\left(O_{S\rightarrow
R_{j^{\ast}}}^{(S)}\right)\cdot P\left(O_{R_{j^{\ast}}\rightarrow
D}^{(S)}\right)$
Since the transmission process from source $S$ to the selected relay
$R_{j^{\ast}}$ is identical to that of from the selected relay $R_{j^{\ast}}$
to destination $D$, we have
$P\left(O_{S\rightarrow
R_{j^{\ast}}}^{(S)}\right)=P\left(O_{R_{j^{\ast}}\rightarrow D}^{(S)}\right)$
and
$P_{out}^{\left(S\right)}=2P\left(O_{S\rightarrow
R_{j^{\ast}}}^{(S)}\right)-\left[P\left(O_{S\rightarrow
R_{j^{\ast}}}^{(S)}\right)\right]^{2}$
To ensure $P_{out}^{\left(S\right)}\leq\varepsilon_{s}$, then should have
$P\left(O_{S\rightarrow R_{j^{\ast}}}^{(S)}\right)\leq
1-\sqrt{1-\varepsilon_{s}}$
From the reference [11], we notice that
$\displaystyle P\left(O_{S\rightarrow
R_{j^{\ast}}}^{(S)}\right)=P\left(\bigcup_{i=1}^{m}\left\\{C_{S,E_{i}}\geq\gamma_{E}\right\\}\right)$
$\displaystyle\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\leq\sum_{i=1}^{m}P\left(C_{S,E_{i}}\geq\gamma_{E}\right)$ $\displaystyle\ \ \
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \leq
m\cdot\left(\frac{1}{1+\gamma_{E}}\right)^{\sqrt{\frac{n\log{n}}{32\gamma_{R}}}}$
To guarantee the secrecy requirement, we just need
$\displaystyle
m\cdot\left(\frac{1}{1+\gamma_{E}}\right)^{\sqrt{\frac{n\log{n}}{32\gamma_{R}}}}\leq
1-\sqrt{1-\varepsilon_{s}}$
and thus
$\displaystyle
m\leq\left(1-\sqrt{1-\varepsilon_{s}}\right)\cdot\left(1+\gamma_{E}\right)^{\sqrt{\frac{n\log{n}}{32\gamma_{R}}}}$
∎
### III-C Analysis of Protocol 2
The parameter $\tau$ involved in the Protocol 2 determines whether the relay
and destination can receive the messages successfully and whether sufficient
noise is generated to suppress eavesdroppers. For the analysis of the Protocol
2, we first determine the range for the parameter $\tau$ to ensure both
secrecy requirement and reliability requirement, based on which we then
analyze the number of eavesdroppers a network can be tolerate by applying the
protocol.
Theorem 2. Consider the network scenario of Fig 1 with equal path loss between
all pairs of nodes, to ensure $P_{out}^{\left(T\right)}\leq\varepsilon_{t}$
and $P_{out}^{\left(S\right)}\leq\varepsilon_{s}$ by applying the Protocol 2,
the parameter $\tau$ must satisfy the following condition.
$\tau\in\left[-\log{\left[1+\frac{\log{\left(\frac{1-\sqrt{1-\varepsilon_{s}}}{m}\right)}}{\left(n-1\right)\log{\left(1+\gamma_{E}\right)}}\right]},\sqrt{\frac{-\log\left(1-\varepsilon_{t}\right)}{2\gamma_{R}\left(n-1\right)}}\right]$
###### Proof.
Notice that in the Protocol 2, a larger value of $\tau$ indicates that more
system nodes will generate noise to suppress the eavesdroppers. However, too
high noise will also interrupt the legitimate transmission. Therefore, the
parameter $\tau$ should be set properly to satisfy both reliability and
secrecy requirements.
$\bullet$ Reliability Guarantee
Notice that $P_{out}^{\left(T\right)}$ is determined as
$\displaystyle P_{out}^{\left(T\right)}=P\left(O_{S\rightarrow
R_{j^{\ast}}}^{(T)}\right)+P\left(O_{R_{j^{\ast}}\rightarrow D}^{(T)}\right)$
$\displaystyle\ \ \ \ \ \ \ \ \ \ -P\left(O_{S\rightarrow
R_{j^{\ast}}}^{(T)}\right)\cdot P\left(O_{R_{j^{\ast}}\rightarrow
D}^{(T)}\right)$
Because the transmission process from source $S$ to the selected relay
$R_{j^{\ast}}$ is identical to that of from the selected relay $R_{j^{\ast}}$
to destination $D$, we have
$P\left(O_{S\rightarrow
R_{j^{\ast}}}^{(T)}\right)=P\left(O_{R_{j^{\ast}}\rightarrow D}^{(T)}\right)$
and
$P_{out}^{\left(T\right)}=2P\left(O_{S\rightarrow
R_{j^{\ast}}}^{(T)}\right)-\left[P\left(O_{S\rightarrow
R_{j^{\ast}}}^{(T)}\right)\right]^{2}$
To ensure $P_{out}^{\left(T\right)}\leq\varepsilon_{t}$, we need
$P\left(O_{S\rightarrow R_{j^{\ast}}}^{(T)}\right)\leq
1-\sqrt{1-\varepsilon_{t}}$
Based on the definition of transmission outage probability, we have
$\displaystyle P\left(O_{S\rightarrow R_{j^{\ast}}}^{(T)}\right)$
$\displaystyle\ \ \ \ \ =P\left(C_{S,R_{j^{\ast}}}\leq\gamma_{R}\right)$
$\displaystyle\ \ \ \ \
=P\left(\frac{E_{s}\cdot|h_{S,R_{j^{\ast}}}|^{2}}{\sum_{R_{j}\in\mathcal{R}_{1}}E_{s}\cdot|h_{R_{j},R_{j^{\ast}}}|^{2}+N_{0}/2}\leq\gamma_{R}\right)$
$\displaystyle\ \ \ \ \ \doteq
P\left(\frac{|h_{S,R_{j^{\ast}}}|^{2}}{\sum_{R_{j}\in\mathcal{R}_{1}}|h_{R_{j},R_{j^{\ast}}}|^{2}}\leq\gamma_{R}\right)$
Compared to the noise generated by multiple system nodes, the environment
noise is negligible and thus is omitted here to simply the analysis. Notice
that $\mathcal{R}_{1}={\left\\{j\neq
j^{\ast}:|h_{R_{j},R_{j^{\ast}}}|^{2}<\tau\right\\}}$, then
$\displaystyle P\left(O_{S\rightarrow R_{j^{\ast}}}^{(T)}\right)\leq
P\left(\frac{|h_{S,R_{j^{\ast}}}|^{2}}{{|\mathcal{R}_{1}|}\tau}\leq\gamma_{R}\right)$
$\displaystyle\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
=P\left(|h_{S,R_{j^{\ast}}}|^{2}\leq\gamma_{R}{|\mathcal{R}_{1}|}\tau\right)$
$\displaystyle\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
=1-e^{-\gamma_{R}{|\mathcal{R}_{1}|}\tau}$
Since there are $n-1$ other relays except $R_{j^{\ast}}$, the expected number
of noise-generation nodes is given by $|\mathcal{R}_{1}|=\left(n-1\right)\cdot
P\left(|h_{R_{j},R_{j^{\ast}}}|^{2}<\tau\right)=\left(n-1\right)\cdot\left(1-e^{-\tau}\right)$.
Then we have
$\displaystyle P\left(O_{S\rightarrow R_{j^{\ast}}}^{(T)}\right)\leq
1-e^{-\gamma_{R}\left(n-1\right)\left(1-e^{-\tau}\right)\tau}$
Thus, to ensure reliability requirement, we just need
$1-e^{-\gamma_{R}\left(n-1\right)\left(1-e^{-\tau}\right)\tau}\leq
1-\sqrt{1-\varepsilon_{t}}$
That is,
$-\gamma_{R}\left(n-1\right)\left(1-e^{-\tau}\right)\tau\geq\frac{1}{2}\log\left(1-\varepsilon_{t}\right)$
By using Taylor formula, we have
$\tau^{2}\leq\frac{-\log\left(1-\varepsilon_{t}\right)}{2\gamma_{R}\left(n-1\right)}$
and thus
$\displaystyle\tau\leq\sqrt{\frac{-\log\left(1-\varepsilon_{t}\right)}{2\gamma_{R}\left(n-1\right)}}$
The above result indicates that
$\sqrt{\frac{-\log\left(1-\varepsilon_{t}\right)}{2\gamma_{R}\left(n-1\right)}}$
is the maximum value the parameter $\tau$ can take to ensure the reliability
requirement.
$\bullet$ Secrecy Guarantee
Notice that $P_{out}^{\left(S\right)}$ is given by
$\displaystyle P_{out}^{\left(S\right)}=P\left(O_{S\rightarrow
R_{j^{\ast}}}^{(S)}\right)+P\left(O_{R_{j^{\ast}}\rightarrow D}^{(S)}\right)$
$\displaystyle\ \ \ \ \ \ \ \ \ -P\left(O_{S\rightarrow
R_{j^{\ast}}}^{(S)}\right)\cdot P\left(O_{R_{j^{\ast}}\rightarrow
D}^{(S)}\right)$
Since the transmission process from source $S$ to the selected relay
$R_{j^{\ast}}$ is identical to that of from the selected relay $R_{j^{\ast}}$
to destination $D$, then we have
$P\left(O_{S\rightarrow
R_{j^{\ast}}}^{(S)}\right)=P\left(O_{R_{j^{\ast}}\rightarrow D}^{(S)}\right)$
and
$P_{out}^{\left(S\right)}=2P\left(O_{S\rightarrow
R_{j^{\ast}}}^{(S)}\right)-\left[P\left(O_{S\rightarrow
R_{j^{\ast}}}^{(S)}\right)\right]^{2}$
To ensure $P_{out}^{\left(S\right)}\leq\varepsilon_{s}$, we need
$P\left(O_{S\rightarrow R_{j^{\ast}}}^{(S)}\right)\leq
1-\sqrt{1-\varepsilon_{s}}$
According to the definition of secrecy outage probability, we know that
$\displaystyle P\left(O_{S\rightarrow
R_{j^{\ast}}}^{(S)}\right)=P\left(\bigcup_{i=1}^{m}\left\\{C_{S,E_{i}}\geq\gamma_{E}\right\\}\right)$
Thus, we have
$\displaystyle P\left(O_{S\rightarrow
R_{j^{\ast}}}^{(S)}\right)\leq\sum_{i=1}^{m}P\left(C_{S,E_{i}}\geq\gamma_{E}\right)$
Based on the Markov inequality,
$\displaystyle P\left(C_{S,E_{i}}\geq\gamma_{E}\right)$ $\displaystyle\ \ \ \
\ \leq
P\left(\frac{E_{s}\cdot|h_{S,E_{i}}|^{2}}{\sum_{R_{j}\in\mathcal{R}_{1}}E_{s}\cdot|h_{R_{j},E_{i}}|^{2}}\geq\gamma_{E}\right)$
$\displaystyle\ \ \ \ \ =E_{\left\\{h_{R_{j},E_{i}},j=0,1,\cdots,n+mp,j\neq
j^{\ast}\right\\},\mathcal{R}_{1}}$ $\displaystyle\ \ \ \ \ \ \ \ \
\left[P\left(|h_{S,E_{i}}|^{2}>\gamma_{E}\cdot\sum_{R_{j}\in\mathcal{R}_{1}}|h_{R_{j},E_{i}}|^{2}\right)\right]$
$\displaystyle\ \ \ \ \ \leq
E_{\mathcal{R}_{1}}\left[\prod_{R_{j}\in\mathcal{R}_{1}}E_{h_{R_{j},E_{i}}}\left[e^{-\gamma_{E}|h_{R_{j},E_{i}}|^{2}}\right]\right]$
$\displaystyle\ \ \ \ \
=E_{\mathcal{R}_{1}}\left[\left(\frac{1}{1+\gamma_{E}}\right)^{|\mathcal{R}_{1}|}\right]$
Therefore,
$P\left(O_{S\rightarrow
R_{j^{\ast}}}^{(S)}\right)\leq\sum_{i=1}^{m}\left(\frac{1}{1+\gamma_{E}}\right)^{|\mathcal{R}_{1}|}=m\cdot\left(\frac{1}{1+\gamma_{E}}\right)^{|\mathcal{R}_{1}|}$
To ensure the secrecy requirement, we just need
$\displaystyle
m\cdot\left(\frac{1}{1+\gamma_{E}}\right)^{|\mathcal{R}_{1}|}\leq
1-\sqrt{1-\varepsilon_{s}}$
or equally
$\displaystyle\left(\frac{1}{1+\gamma_{E}}\right)^{\left(n-1\right)\left(1-e^{-\tau}\right)}\leq\frac{1-\sqrt{1-\varepsilon_{s}}}{m}$
$\displaystyle\left(n-1\right)\left(1-e^{-\tau}\right)\geq-\frac{\log{\left(\frac{1-\sqrt{1-\varepsilon_{s}}}{m}\right)}}{\log{\left(1+\gamma_{E}\right)}}$
$\displaystyle e^{-\tau}\leq
1+\frac{\log{\left(\frac{1-\sqrt{1-\varepsilon_{s}}}{m}\right)}}{\left(n-1\right)\log{\left(1+\gamma_{E}\right)}}$
Then we have
$\displaystyle\tau\geq-\log{\left[1+\frac{\log{\left(\frac{1-\sqrt{1-\varepsilon_{s}}}{m}\right)}}{\left(n-1\right)\log{\left(1+\gamma_{E}\right)}}\right]}$
The above result shows that
$-\log{\left[1+\frac{\log{\left(\frac{1-\sqrt{1-\varepsilon_{s}}}{m}\right)}}{\left(n-1\right)\log{\left(1+\gamma_{E}\right)}}\right]}$
is the minimum value parameter $\tau$ can take to guarantee the secrecy
requirement.
∎
Based on the results of Theorem 2, we now can establish the following theorem
about the performance of Protocol 2.
Theorem 3. Consider the network scenario of Fig 1 with equal path loss between
all pairs of nodes. To guarantee $P_{out}^{\left(T\right)}\leq\varepsilon_{t}$
and $P_{out}^{\left(S\right)}\leq\varepsilon_{s}$ based on the Protocol 2, the
number of eavesdroppers $m$ the network can tolerate must satisfy the
following condition.
$\displaystyle
m\leq\left(1-\sqrt{1-\varepsilon_{s}}\right)\cdot\left(1+\gamma_{E}\right)^{\sqrt{\frac{-\left(n-1\right)\log\left(1-\varepsilon_{t}\right)}{2\gamma_{R}}}}$
###### Proof.
From Theorem 2 we know that to ensure the reliability requirement, we have
$\displaystyle\tau\leq\sqrt{\frac{-\log\left(1-\varepsilon_{t}\right)}{2\gamma_{R}\left(n-1\right)}}$
and
$\displaystyle\left(n-1\right)\left(1-e^{-\tau}\right)\leq\frac{-\log\left(1-\varepsilon_{t}\right)}{2\gamma_{R}\tau}$
To ensure the secrecy requirement, we need
$\displaystyle\left(\frac{1}{1+\gamma_{E}}\right)^{\left(n-1\right)\left(1-e^{-\tau}\right)}\leq\frac{1-\sqrt{1-\varepsilon_{s}}}{m}$
Thus,
$\displaystyle
m\leq\frac{1-\sqrt{1-\varepsilon_{s}}}{\left(\frac{1}{1+\gamma_{E}}\right)^{\left(n-1\right)\left(1-e^{-\tau}\right)}}$
$\displaystyle\ \ \ \
\leq\frac{1-\sqrt{1-\varepsilon_{s}}}{\left(\frac{1}{1+\gamma_{E}}\right)^{\frac{-\log\left(1-\varepsilon_{t}\right)}{2\gamma_{R}\tau}}}$
By let $\tau$ taking its maximum value, we get the following bound
$\displaystyle
m\leq\frac{1-\sqrt{1-\varepsilon_{s}}}{\left(\frac{1}{1+\gamma_{E}}\right)^{\sqrt{\frac{-\left(n-1\right)\log\left(1-\varepsilon_{t}\right)}{2\gamma_{R}}}}}$
That is,
$\displaystyle
m\leq\left(1-\sqrt{1-\varepsilon_{s}}\right)\cdot\left(1+\gamma_{E}\right)^{\sqrt{\frac{-\left(n-1\right)\log\left(1-\varepsilon_{t}\right)}{2\gamma_{R}}}}$
∎
### III-D Discussion
The two protocols considered in this paper have their own advantages and
disadvantages and thus are suitable for different network scenarios. For the
protocol 1 proposed in [11], it can achieve a better performance in terms of
the number of eavesdroppers can be tolerated. However, such protocol always
tend to select the optimal node with the best links to both source and
destination as the relay, so it involves a complex relay selection process,
and more importantly, it results in an unbalanced load and energy consumption
distribution among systems nodes. Thus, such protocol is suitable for small
scale wireless network with sufficient energy supply rather than large and
energy-limited wireless networks (like wireless sensor networks). Regarding
the Protocol 2, although it can tolerate less number eavesdroppers in
comparison with the Protocol 1, it involves a very simple random relay
selection process to achieve a very good load and energy consumption
distribution among system nodes. Thus, this protocol is more suitable for
large scale wireless network environment with stringent energy consumption
constraint.
## IV Conclusion
This paper explores reliable and secure information transmission through
multiple cooperative systems nodes in two-hop relay wireless network with
passive eavesdroppers of unknown channels and locations, for which two
transmission protocols are considered. For each protocol, theoretical analysis
has been provided to show the number of eavesdroppers the network can tolerate
subject to constraints on transmission outage probability and secrecy outage
probability. These two protocols, each has different performance in terms of
eavesdropper tolerance, load and energy consumption distribution among nodes,
and also relay selection complexity, are suitable for different network
scenarios depending on network scale and also energy consumption constraint
there.
## References
* [1] Narayanan, Sathya, Two-hop forwarding in wireless networks, dissertation for the degree of Doctor of philosophy, Polytechnic University, 2006
* [2] J. Talbot and D. Welsh, _Complexity and Crytography : An Introduction_ , Cambridge, 2006.
* [3] K. Morrison, and D. Goeckel _Power allocation to noise-generating nodes for cooperative secrecy in the wireless environment_. In the Forty Fifth Asilomar Conference on Signals, Systems and Computers (ASILOMAR), 275-279, 2011.
* [4] S. Goel, and R. Negi, _Guaranteeing secrecy using artificial noise_. IEEE transactions on wireless communications, 7(6):2180-2189, 2008.
* [5] L. Lai and H. El Gamal, _The relay-eavesdropper channel: Cooperation for secrecy_ , IEEE Trans. Inf. Theory, vol. 54, no. 9, pp. 4005 - 4019, Sept. 2008.
* [6] M. Yuksel and E. Erkip, _Secure communication with a relay helping the wiretapper_ , in Proc. 2007 IEEE Information Theory Workshop, Lake Tahoe, CA, Sept. 2007.
* [7] R. Negi and S. Goelm, _Secret communication using artificial noise_ , in Proc. IEEE Vehicular Tech. Conf, vol. 3, Dallas TX, pp. 1906-1910, Sept. 2005.
* [8] S. Vasudevan, S. Adams, D. Geockel, Z. Ding, D. Towsley, and K. Leung, _Multi-user diversity for secrecy in wireless networks_. In Information Theorem and Applications Workshop, 2009.
* [9] X. He and A. Yener, _Two-hop secure communication using an untrusted relay: A case for cooperative jamming_ , in Proc. 2008 IEEE Global Telecommunications Conference, New Orleans, LA, Nov. - Dec. 2008\.
* [10] L. Dong, Z. Han, A. Petropulu, and H. V. Poor, _Improving wireless physical layer security via cooperating relays_ , IEEE Trans. Sig. Proc., vol. 58, no. 3, pp. 1875-1888, Mar. 2010.
* [11] D. Goeckel, S. Vasudevan, D. Towsley, S. Adams, Z. Ding, and K. Leung, _Artificial noise generation from cooperative relays for everlasting secrecy in two-hop wireless networks, IEEE Journal on Selected Areas in Communications_ , 29(10):2067-2076, 2011.
|
arxiv-papers
| 2012-11-29T21:01:22 |
2024-09-04T02:49:38.680054
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Yulong Shen, Xiaohong Jiang, Jianfeng Ma and Weisong Shi",
"submitter": "Yulong Shen",
"url": "https://arxiv.org/abs/1211.7075"
}
|
1211.7109
|
841
Martin C. Weisskopf
11institutetext: NASA/Marshall Space Flight Center, Astrophysics Office,
Huntsville AL 35812, USA11email: [email protected]
# Monitoring the Crab Nebula with Chandra
A Search for the Location of the $\gamma$-Ray Flares
Martin C. Weisskopf 11
###### Abstract
Subsequent to announcements by the AGILE and by the Fermi-LAT teams of the
discovery of $\gamma$-ray flares from the Crab Nebula in the fall of 2010, an
international collaboration has been monitoring X-Ray emission from the Crab
on a regular basis using the Chandra X-Ray Observatory. Observations occur
typically once per month when viewing constraints allow. The aim of the
program is to characterize in depth the X-Ray variations within the Nebula,
and, if possible, to much more precisely locate the origin of the $\gamma$-ray
flares. In 2011 April we triggered a set of Chandra Target-of-Opportunity
observations in conjunction with the brightest $\gamma$-ray flare yet
observed. We briefly summarize the April X-ray observations and the
information we have gleaned to date.
###### keywords:
SNR: individual: Crab Nebula
## 1 Introduction
Since 2007, the AGILE and Fermi satellites have detected several $\gamma$-ray
flares in the $0.1-1$ ${\rm Ge\\!V}$ range from the Crab Nebula (Tavani et
al., 2011; Abdo et al., 2011; Striani et al., 2011a; Buehler et al., 2012).
The largest flares exhibit variability on timescales as short as hours. Prior
to the 2011-April flare, the only Crab $\gamma$-ray flare covered by a multi-
wavelength observing program was the 2010-September flare, which triggered
only post-flare observations in radio, optical, and X-ray bands. Despite the
brightness of the $\gamma$-ray flares, there has been no clear evidence for
correlated variations in radio (Lobanov, Horns & Muxlow, 2011; Weisskopf et
al., 2012), near-infrared (Kanbach et al., 2010; Weisskopf et al., 2012),
optical (Caraveo et al., 2010), or X-ray bands, as discussed here and in
Evangelista et al. (2010); Shaposhnikov et al. (2010); Tennant et al. (2010);
Ferrigno et al. (2010); Horns et al. (2010); Cusumano et al. (2011); Tennant
et al. (2011); Tavani et al. (2011); Striani et al. (2011b); Weisskopf et al.
(2012).
Figure 1 shows the Fermi-LAT light-curve for the 2011-April flare (Buehler et
al., 2012). For this flare the source doubled its $\gamma$-ray flux within
eight hours and reached a peak 30-times the average. The (assumed) isotropic
luminosity increased to $2\times 10^{37}$ erg/s in about $10$ hr and the
spectrum peaked at $\approx 400$ ${\rm Me\\!V}$. (See Buehler et al. (2012)
for details.) Notification as to the level of flaring prompted us to trigger
pre-approved Target-of-Opportunity observations with Chandra and Figure 1 also
indicates the times of these observations.
Figure 1: Fermi-LAT photon flux ($10^{-7}$ ph/(cm2 s)) above 100 ${\rm
Me\\!V}$ during the 2011-April flare as a function of time. Displayed data
extend somewhat beyond the time span shown in Buehler et al. (2012) but follow
the same data processing as described there. The vertical lines mark times of
the $5$ Chandra observations discussed in the text.
## 2 The X-ray Observations
The five observations (ObsIDs $13150-13154$) used the (back-illuminated) ACIS
S3 CCD approximately centered on the Crab pulsar, during and somewhat after
the 2011-April $\gamma$-ray flare. For these observations, the spacecraft was
dithered with an amplitude set to $1\arcsec$. Although standard processing
typically produces an aspect solution better than $0.5\arcsec$, this small
uncertainty can still introduce noticeable shifts amongst different data sets.
Thus, we re-registered images using the read-out streak and the hole in the
images produced by the severely piled-up pulsar.
Owing to the high flux, we used a special mode with 0.2-s frame time, which
limits the CCD read-out to a $300\times 300$ ACIS-pixel ($\approx
150\arcsec\times 150\arcsec$) subarray. Although each observation lasted about
10 ks, telemetry saturation reduced the effective integration time to
approximately 1200 s per observation. Despite the short frame time, regions of
high surface brightness suffer somewhat from pile-up effects. In view of
interstellar absorption at low energies and declining flux at high energies,
we limited the analysis to data in the energy range 0.5–8.0 ${\rm ke\\!V}$. We
then searched for X-ray variations.
## 3 X-ray Image Analysis
For each observation, we re-binned a $120\times 120$ ACIS-pixel image centered
on the pulsar into a $60\times 60$ array of $2\times 2$ ACIS pixels. Each of
these $I=3600$ “analysis pixels” is sufficiently large (about 1 square arcsec)
to enclose most of the Chandra point spread function anywhere in the field of
view. Figure 2 shows the counts per analysis pixel, summed over the 5
observations.
For each analysis pixel $i$, we calculated the mean count rate $r_{i}$
averaged over the $J=5$ observations, weighted111
$r_{i}=\sum_{j=1}^{J}\\{r_{ij}/\sigma_{ij}^{2}\\}/\sum_{j=1}^{J}\\{1/\sigma_{ij}^{2}\\}$
by the respective (counting-rate) statistical error $\sigma_{ij}$. For
evaluating the statistical significance of temporal variations over the $J=5$
observations, we compute222
$\chi_{i}^{2}=\sum_{j=1}^{J}\\{(r_{ij}-r_{i})^{2}/\sigma_{ij}^{2}\\}$.
$\chi_{i}^{2}$. Figure 2 also indicates the three pixels which showed the
highest significance based on $\chi_{i}^{2}$.
The most significant variation has $\chi_{i}^{2}=23.5$ on $\nu=(J-1)=4$
degrees of freedom. Such a fluctuation is expected statistically in at least 1
of 3600 pixels in 31% of realizations and thus is not considered terribly
significant. Based upon the $\chi^{2}$ probability distribution and the number
of “tries”, a 99%-confidence detection would require $\chi_{i,99\%}^{2}>31.2$
on $(J-1)=4$ degrees of freedom. While we detect no statistically significant
variations at 99% confidence, it is perhaps curious that the 3 most
significant variations occur at locations on the inner ring.
Other effects, such as changes in the roll angle of the read-out streak, can
lead to possibly spurious variability. This may be the case for the analysis
pixel with the most significant variation, which lies to the east of the
pulsar but adjacent to the average read-out streak (Figure 2).
### 3.1 Limits to the X-ray Flux
Neglecting for the moment the effects of pile-up, the photon spectral flux is
proportional to the count rate for an assumed spectral shape. Consequently,
any change in count rate corresponds to a proportionate change in the photon
spectral flux. Using the Chandra
PIMMS333http://asc.harvard.edu/toolkit/pimms.jsp for ACIS-S and an absorption
column $N_{\rm H}=3.1\times 10^{21}\ {\rm cm}^{-2}$, we determine this
constant of proportionality for an X-ray power-law photon index
$\Gamma_{x}=\frac{2}{3}$, 1, and 2: At $E_{x}=1$ ${\rm ke\\!V}$,
$N_{E}(E_{x})/r=$ 0.99, 1.26, and 2.46 $\times 10^{-3}$ ph/(cm2 s ${\rm
ke\\!V}$) per ct/s, respectively. Table 1 shows our calculations of the upper
limits to the photon spectral flux $N_{E}(E_{x})$, the energy spectral flux
$F_{E}(E_{x})$, and the indicative (isotropic) luminosity $EL_{E}(E_{x})=4\pi
D^{2}EF_{E}(E_{x})$ at $D=2$ kpc, for the analysis pixel with the most
significant X-ray variation. Correcting for pile-up has little effect in low-
count-rate regions, but would raise these flux upper limits by $10$% or so for
the high-count-rate regions.
Table 1: 99%-confidence upper limits for various parameters at 1 ${\rm
ke\\!V}$ for the analysis pixel with the most significant variation.
$\Gamma_{x}$ | $\frac{2}{3}$ | 1 | 2
---|---|---|---
$N_{E}~{}^{a}$ | 0.55 | 0.70 | 1.36
$F_{E}~{}^{b}$ | 0.88 | 1.12 | 2.18
$EL_{E}~{}^{c}$ | 0.42 | 0.54 | 1.05
$\Gamma_{x\gamma}$ | 1.20 | 1.22 | 1.27
a $10^{-4}$ ph/(cm2 s ${\rm ke\\!V}$)
b $10^{-13}$ erg/(cm2 s ${\rm ke\\!V}$)
c $10^{32}$ erg/s
Figure 2: Summed count image for the 5 Chandra observations during the
$\gamma$-ray flare. North is up and the pulsar is at (0,0). The nearly
horizontal streak through the location of the pulsar is the trailed (out-of-
time) image, resulting from the very short exposure of each pixel as the image
is read out. As the 5 observations occurred at slightly different roll angles,
the read-out streak is slightly blurred. The $\mathsf{X}$ symbols mark
locations of the 3 statistically most significant variations, with the most
significant being the one to the east of the pulsar.
### 3.2 Constraints on the X-ray to $\gamma$-ray Spectral Index
We can also compare the $\gamma$-ray data to the X-ray data to quantify the
implications of our lack of detection of time variations in the latter. The
approach compares a variability measure for the X-ray (1-${\rm ke\\!V}$)
photon spectral flux $\Delta N_{E}(E_{x})$ in each analysis pixel with the
analogous variability measure for the $\gamma$-ray (100-${\rm Me\\!V}$) photon
spectral flux $\Delta N_{E}(E_{\gamma})$. Specifically, we calculate the
sample standard deviation of the $\gamma$-ray spectral flux at 100 ${\rm
Me\\!V}$ using power-law fits to the 5 Fermi-LAT measurements that were
simultaneous with the 5 Chandra observations. For these 5 observations, the
mean and sample standard deviation of the photon spectral flux at 100 ${\rm
Me\\!V}$ are $1.21\times 10^{-10}$ and $5.77\times 10^{-11}$ ph/(cm2 s ${\rm
ke\\!V}$), respectively.
Based upon the sample standard deviation of the photon spectral flux at
$E_{x}=1$ ${\rm ke\\!V}$ for each X-ray analysis pixel and the measured
standard deviation, $5.77\times 10^{-11}$ ph/(cm2 s ${\rm ke\\!V}$) at
$E_{\gamma}=100$ ${\rm Me\\!V}$, we constrain the effective X-ray to
$\gamma$-ray photon index of the flaring component:
$\Gamma_{x\gamma}\equiv-\log[\Delta N_{E}(E_{\gamma})/\Delta
N_{E}(E_{x})]/\log[E_{\gamma}/E_{x}]$.
In that the $\gamma$-ray variations are statistically significant and the
X-ray variations are not, we compute 99%-confidence upper limits to
$\Gamma_{x\gamma}$ (Table 1 last row). The $99$%-confidence limits to
$\Gamma_{x\gamma}$ are marginally consistent with the low-energy extrapolation
of the $\gamma$-ray spectrum ($\Gamma_{\gamma}=1.27\pm 0.12$) of the flaring
component (Buehler et al., 2012).
## 4 Conclusions
Using Chandra, we acquired X-ray images of the Crab Nebula contemporaneous
with the 2011-April $\gamma$-ray flare. We tested for time variations amongst
the 5 pointings, each with an effective exposure time $\approx 1200$ s and a
minimum separation of 0.6 days. We did not detect statistically significant
X-ray variations; thus we can set only upper limits to any X-ray variations
associated with the $\gamma$-ray flare. As the Chandra ACIS images suffer
severe pile-up near the Crab pulsar, our search for variability in the X-ray
images was not sensitive to variations within the central $\approx 1.5\arcsec$
or so.
Comparing the upper limits to X-ray variations with the Fermi-LAT-measured
$\gamma$-ray variations, we set upper limits at $99$%-confidence to the
effective X-ray–$\gamma$-ray photon power-law index $\Gamma_{x\gamma}\leq
1.20$ to $\leq 1.27$, dependent upon assumptions about the X-ray index
$\Gamma_{x}$. As Fermi-LAT measures a $\gamma$-ray index
$\Gamma_{\gamma}=1.27\pm 0.12$ for the flaring component, it is statistically
possible that the flaring component’s spectrum extends as a simple power-law
from $\gamma$-rays to X-rays. Further, we note that our upper limit to
$\Gamma_{x\gamma}$ is consistent with transparent synchrotron emission, whose
photon index must be $>\frac{2}{3}$.
Elsewhere (Weisskopf et al., 2012) we present a more detailed analysis of the
X-ray data including the results of searches for variability within each
observation. Weisskopf et al. (2012) also discusses a Keck near-IR observation
of the inner knot ($\approx 0.65\arcsec$ from the pulsar) made in conjunction
with the 2011-April flare and a number of VLA observations searching for a
point source appearing either at an unusual location and/or contemporaneous
with the flare.
Although no “smoking gun” has been identified, we are encouraged that we have
identified a number of regions in the X-ray images that are possible
candidates. In addition we (see acknowledgments) have also established further
Target of Opportunity observations with Chandra and HST that will be triggered
at the onset of the next $\gamma$-ray flare. The X-ray observations will also
probe the region very close to the pulsar using the Chandra High-Resolution
Camera (HRC).
###### Acknowledgements.
I am very grateful to all of my colleagues participating in this
collaboration: Allyn F. Tennant, Jonathan Arons, Roger Blandford, Rolf
Buehler, Patrizia Caraveo, C. C. Teddy Cheung, Enrico Costa, Andrea de Luca,
Carlo Ferrigno, Hai Fu, Stefan Funk, Moritz Habermehl, Dieter Horns, Justin D.
Linford, Andrei Lobanov, Claire Max, Roberto Mignani, Stephen L. O’Dell, Roger
W. Romani, Edoardo Striani, Marco Tavani, Gregory B. Taylor, Yasunobu
Uchiyama, & Yajie Yuan.
## References
* Abdo et al. (2011) Abdo, A. A., et al. 2011, Science, 331, 739
* Buehler et al. (2012) Buehler, R., et al. 2012, ApJ, 749, eid 26
* Caraveo et al. (2010) Caraveo, P., et al. 2010, ATel 2903
* Cusumano et al. (2011) Cusumano, G., et al. 2011, ATel 3279
* Evangelista et al. (2010) Evangelista, Y., et al. 2010, ATel 2866
* Ferrigno et al. (2010) Ferrigno, C., et al. 2010, ATel 2994
* Horns et al. (2010) Horns, D., et al. 2010, ATel 3058
* Kanbach et al. (2010) Kanbach, G., et al. 2010, ATel 2867
* Lobanov, Horns & Muxlow (2011) Lobanov, A. P., Horns, D., & Muxlow, T. W. B. 2011, A&A, 533, eid A10
* Shaposhnikov et al. (2010) Shaposhnikov, N., et al. 2010, ATel 2872
* Striani et al. (2011a) Striani, E., et al. 2011a, ATel 3286
* Striani et al. (2011b) Striani, E., et al. 2011b, ApJ 741,L5
* Tavani et al. (2011) Tavani, M., et al. 2011, Science, 331, 736
* Tennant et al. (2010) Tennant, A., et al. 2010, ATel 2882
* Tennant et al. (2011) Tennant, A., et al. 2011, ATel 3283
* Weisskopf et al. (2012) Weisskopf, M.C., et al. 2012, submitted to ApJ
|
arxiv-papers
| 2012-11-29T22:34:52 |
2024-09-04T02:49:38.687918
|
{
"license": "Public Domain",
"authors": "Martin C. Weisskopf",
"submitter": "Martin C. Weisskopf",
"url": "https://arxiv.org/abs/1211.7109"
}
|
1211.7229
|
# $Z\to{\tau\tau}$ and $W\to\tau\nu_{\tau}$ Cross-Sections at the LHC
Philip Ilten
on behalf of the LHCb Collaboration
including results from ATLAS and CMS School of Physics, University College
Dublin
###### Abstract
Measurements of the $Z\to{\tau\tau}$ and $W\to\tau\nu_{\tau}$ cross-sections
at the LHC with data taken at $\sqrt{s}=7$ TeV are reported for the ATLAS,
CMS, and LHCb experiments. All results are found to agree with the Standard
Model.
###### keywords:
LHC , ATLAS , CMS , LHCb , electroweak , tau production
††journal: Nuclear Physics B Proceedings
## 1 Introduction
The production of $Z$ and $W$ bosons from $pp$-collisions, and their
subsequent decays to $\tau$-leptons at the Large Hadron Collider (LHC) not
only provides important tests of the Standard Model (SM), but also lays the
groundwork for the study of beyond the SM physics using $\tau$-lepton
signatures.
Decays of $Z$ bosons to $\tau$-lepton pairs are both a mechanism for
experimentally measuring hadronic $\tau$-lepton identification efficiencies
[1, 2] and a calibration channel for neutral Higgs searches [3, 4]. The final
states of $\tau$-leptons produced from $W$ bosons can be used to measure the
polarization of the $W$ [5] boson or to search for charged Higgses [6, 7].
In this review, the complete set of $Z\to{\tau\tau}$ and $W\to\tau\nu_{\tau}$
cross-sections, as measured by the ATLAS, CMS, and LHCb experiments on the
LHC, are reported using $2010$ and $2011$ datasets taken at
$\sqrt{s}=7~{}\mathrm{TeV}$. A full summary of the results, including
corresponding references, is provided in Table 1, and a comparison of the
theoretical agreement of the results is given in Figure 1.
All three detectors are fully instrumented with charged particle trackers,
electromagnetic and hadronic calorimeters, and muon systems. Both ATLAS [8]
and CMS [9] are general purpose detectors designed to cover a central
pseudorapidity range of $|\eta|<2.4$, while the LHCb [10] detector is a
forward arm spectrometer, purpose built for $B$-hadron physics, covering the
forward pseudorapidity range $2.0<\eta<5.0$.
Exp. | Analysis | $N_{\mathrm{obs}}$ | $N_{\mathrm{bkg}}$ | $\sigma\pm\mathrm{stat.}\pm\mathrm{syst.}\pm\mathrm{lumi.}$ | $\mathcal{L}$ | Data | Ref.
---|---|---|---|---|---|---|---
ATLAS | $Z\to$ | $\tau_{\mu}\tau_{\mu}$ | $90$ | $47$ | $0.96$ | $\pm$ | $0.22$ | $\pm$ | $0.12$ | $\pm$ | $0.03$ | nb | $36$ | $\mathrm{pb}^{-1}$ | $2010$ | [11]
| $\tau_{\mu}\tau_{e}$ | $1035$ | $56$ | $0.96$ | $\pm$ | $0.03$ | $\pm$ | $0.09$ | $\pm$ | $0.04$ | nb | $1.55$ | $\mathrm{fb}^{-1}$ | $2011$ | [12]
| $\tau_{\mu}\tau_{h}$ | $5184$ | $793$ | $0.91$ | $\pm$ | $0.01$ | $\pm$ | $0.09$ | $\pm$ | $0.03$ | nb | $1.55$ | $\mathrm{fb}^{-1}$ |
| $\tau_{e}\tau_{h}$ | $2600$ | $449$ | $1.00$ | $\pm$ | $0.02$ | $\pm$ | $0.13$ | $\pm$ | $0.04$ | nb | $1.34$ | $\mathrm{fb}^{-1}$ |
$W\to$ | $\tau_{h}\nu_{\tau}$ | $2335$ | $411$ | $11.1$ | $\pm$ | $0.3$ | $\pm$ | $1.7$ | $\pm$ | $0.4$ | nb | $34$ | $\mathrm{pb}^{-1}$ | $2010$ | [13]
CMS | $Z\to$ | $\tau_{\mu}\tau_{\mu}$ | $58$ | $23$ | $1.14$ | $\pm$ | $0.27$ | $\pm$ | $0.04$ | $\pm$ | $0.05$ | nb | $36$ | $\mathrm{pb}^{-1}$ | $2010$ | [14]
| $\tau_{\mu}\tau_{e}$ | $101$ | $14$ | $0.99$ | $\pm$ | $0.12$ | $\pm$ | $0.06$ | $\pm$ | $0.04$ | nb | |
| $\tau_{\mu}\tau_{h}$ | $517$ | $228$ | $0.83$ | $\pm$ | $0.07$ | $\pm$ | $0.19$ | $\pm$ | $0.03$ | nb | |
| $\tau_{e}\tau_{h}$ | $540$ | $346$ | $0.94$ | $\pm$ | $0.11$ | $\pm$ | $0.22$ | $\pm$ | $0.04$ | nb | |
$W\to$ | $\tau_{h}\nu_{\tau}$ | $372$ | $155$ | not measured | $18$ | $\mathrm{pb}^{-1}$ | $2010$ | [15]
LHCb | $Z\to$ | $\tau_{\mu}\tau_{\mu}$ | $124$ | $42$ | $77.4$ | $\pm$ | $10.4$ | $\pm$ | $8.6$ | $\pm$ | $2.7$ | pb | $1.03$ | $\mathrm{fb}^{-1}$ | $2011$ | [16]
| $\tau_{\mu}\tau_{e}$ | $421$ | $130$ | $75.2$ | $\pm$ | $5.4$ | $\pm$ | $4.1$ | $\pm$ | $2.6$ | pb | $1.03$ | $\mathrm{fb}^{-1}$ |
| $\tau_{e}\tau_{\mu}$ | $155$ | $57$ | $64.2$ | $\pm$ | $8.2$ | $\pm$ | $4.9$ | $\pm$ | $2.2$ | pb | $0.96$ | $\mathrm{fb}^{-1}$ |
| $\tau_{\mu}\tau_{h}$ | $189$ | $53$ | $68.3$ | $\pm$ | $7.0$ | $\pm$ | $2.6$ | $\pm$ | $2.4$ | pb | $1.03$ | $\mathrm{fb}^{-1}$ |
| $\tau_{e}\tau_{h}$ | $101$ | $37$ | $77.9$ | $\pm$ | $12.2$ | $\pm$ | $6.1$ | $\pm$ | $2.7$ | pb | $0.96$ | $\mathrm{fb}^{-1}$ |
Table 1: A complete summary of the individual $Z\to{\tau\tau}$ and
$W\to\tau\nu_{\tau}$ cross-section analyses of ATLAS, CMS, and LHCb using
$2010$ and $2011$ datasets. The experiment, final state, number of observed
events ($N_{\mathrm{obs}}$), number of background events ($N_{\mathrm{bkg}}$),
cross-section measurement ($\sigma$), integrated luminosity ($\mathcal{L}$),
dataset, and reference are given for each analysis.
## 2 $Z\to{\tau\tau}$
Four final states produced from the $\tau$-lepton decays of $Z\to{\tau\tau}$
events are considered by all three experiments: two muons
($\tau_{\mu}\tau_{\mu}$), a muon and an electron ($\tau_{\mu}\tau_{e}$), a
muon and a hadronic jet ($\tau_{\mu}\tau_{h}$), and an electron and a hadronic
jet ($\tau_{e}\tau_{h}$). Six backgrounds to these final states are
considered: Drell-Yan production of di-muon or di-electron pairs, $WW$ decays,
$t\bar{t}$ decays, $Z$ production with an associated jet, $W$ production with
an associated jet, and QCD multijet events.
### 2.1 Particle Selection
All final states are triggered by a muon with a $p_{\mathrm{T}}$ greater than
$9-15~{}\mathrm{GeV}$, except the $\tau_{e}\tau_{h}$ final state, which is
triggered by an electron with $E_{\mathrm{T}}$ greater than
$12-15~{}\mathrm{GeV}$ for ATLAS and CMS and $p_{\mathrm{T}}$ greater than
$10~{}\mathrm{GeV}$ for LHCb. An additional hadronic trigger is included by
ATLAS for the $\tau_{e}\tau_{h}$ final state, while the LHCb
$\tau_{\mu}\tau_{e}$ final state is split into a muon triggered final state
($\tau_{\mu}\tau_{e}$), and an electron and muon triggered final state
($\tau_{e}\tau_{\mu}$).
Muons are identified by requiring an isolated track associated with muon
system hits. Electrons are identified by requiring an isolated track
associated with electromagnetic calorimeter energy. Both one and three-pronged
hadronic $\tau$-lepton decays are identified within ATLAS and CMS using the
anti-$k_{\mathrm{T}}$ jet algorithm and requiring one or three charged
particles, whereas only single-pronged hadronic $\tau$-lepton decays are
identified within LHCb by requiring a single isolated track with an associated
hadronic calorimeter energy and minimal electromagnetic calorimeter energy.
### 2.2 Event Selection
The $Z\to{\tau\tau}$ signal produces a high mass back-to-back final state in
the transverse plane with missing energy (${{\not\mathrel{E}}_{T}}$) and a
$p_{\mathrm{T}}$ imbalance between the two $\tau$-lepton decay products due to
unreconstructed neutrinos. Additionally, the lifetime of the $\tau$-lepton
produces decay products with an enhanced impact parameter. Selection
requirements based on these five signatures are used by the three experiments
to separate signal from background.
For ATLAS, a visible mass selection requirement is placed on all four final
states, and a transverse mass requirement on the $\tau_{\mu}\tau_{h}$ and
$\tau_{e}\tau_{h}$ final states. A requirement on the transverse separation of
the $\tau$-lepton decay products and the ${{\not\mathrel{E}}_{T}}$ of the
event is used for all final states except $\tau_{\mu}\tau_{\mu}$, where
further requirements on the angular separation, $p_{\mathrm{T}}$ asymmetry,
and impact parameters of the two $\tau$-lepton decay products are applied.
For CMS, only a transverse mass requirement is applied to the
$\tau_{\mu}\tau_{e}$, $\tau_{\mu}\tau_{h}$, and $\tau_{e}\tau_{h}$ final
states. Due to the large Drell-Yan background to the $\tau_{\mu}\tau_{\mu}$
final state, requirements are placed on the visible mass, transverse
separation between the muons and ${{\not\mathrel{E}}_{T}}$, the muon
$p_{\mathrm{T}}$ asymmetry, and the impact parameter of the muons.
Unlike for ATLAS and CMS, missing energy cannot be measured within LHCb.
However, a high resolution vertex locator allows for strict requirements to be
placed on the $\tau$-lepton decay product impact parameters for the
$\tau_{\mu}\tau_{\mu}$, $\tau_{\mu}\tau_{h}$, and $\tau_{e}\tau_{h}$ final
states. Both a visible mass and transverse separation requirement is placed on
all final states, while an additional $p_{\mathrm{T}}$ asymmetry requirement
is also placed on the $\tau_{\mu}\tau_{\mu}$ final state.
### 2.3 Background Estimation
Drell-Yan production of lepton pairs is the primary background to the
$\tau_{\mu}\tau_{\mu}$ final state for all three experiments, as well as the
$\tau_{e}\tau_{h}$ final state for ATLAS and CMS. The Drell-Yan visible mass
shape is determined for ATLAS from simulation, while for both CMS and LHCb the
template is obtained with a reversed impact parameter requirement. For ATLAS
and LHCb the template is normalized to the on-shell $Z$ mass peak, and for CMS
normalized to an impact parameter side-band.
The QCD multijet background is large in the $\tau_{\mu}\tau_{h}$ and
$\tau_{e}\tau_{h}$ final states, and a visible mass shape is determined from
data for all three experiments by requiring candidates with same-sign charge.
The normalization is also taken from data, scaling the number of same-sign
events by the estimated opposite-sign/same-sign event ratio for the
background.
The $W$ with jets background mass shape is determined from simulation for all
three experiments and normalized using transverse mass side-bands for ATLAS
and CMS, and a same-sign side-band for LHCb. The $Z$ with jets visible mass
shape is taken from simulation for ATLAS and LHCb, and from a reversed impact
parameter requirement for CMS. The background is normalized using a visible
mass side-band for ATLAS, an impact parameter side-band for CMS, and a same-
sign side-band for LHCb.
For all three experiments the visible mass distributions for the $WW$ and
$t\bar{t}$ backgrounds are estimated from simulation. The normalization for
these backgrounds is also taken from simulation for ATLAS and LHCb, and from a
transverse mass side-band for CMS. Both the $WW$ and $t\bar{t}$ background
contributions are minimal for all final states.
### 2.4 Systematics
For both ATLAS and CMS the hadronic $\tau$-lepton identification efficiency
and energy scale is the primary systematic uncertainty for the
$\tau_{\mu}\tau_{h}$ and $\tau_{e}\tau_{h}$ final states, ranging from
$8\%-23\%$. In the $\tau_{\mu}\tau_{\mu}$ final state the primary uncertainty
is between $2\%-9\%$ from muon efficiency and acceptance, and for the
$\tau_{\mu}\tau_{e}$ final state is between $2\%-6\%$ from electron
efficiency.
For LHCb, the Drell-Yan background provides the largest systematic uncertainty
of $8\%$ to the $\tau_{\mu}\tau_{\mu}$ final state. Electron reconstruction
efficiency contributes the primary uncertainty to the $\tau_{\mu}\tau_{e}$,
$\tau_{e}\tau_{\mu}$, and $\tau_{e}\tau_{h}$ final states of $4\%$, while the
impact parameter selection efficiency provides a $2\%$ uncertainty to the
$\tau_{\mu}\tau_{h}$ final state.
### 2.5 Results
The measured cross-section, number of observed events, and number of
background events for each final state of all three experiments is given in
Table 1. Note the reduced statistics of the LHCb results, due to the
acceptance of the detector, but the enhanced purity of the
$\tau_{\mu}\tau_{\mu}$ final state.
The combined $Z\to{\tau\tau}$ cross-section measurement for each experiment is
given in Table 2, including the fiducial definition and predicted theory
result. The ATLAS and CMS theory predictions were calculated using Fewz [17],
while the LHCb prediction was calculated with Dynnlo [18]. The ATLAS combined
result does not include the $\tau_{\mu}\tau_{\mu}$ final state.
Exp. | $\sigma\pm\mathrm{stat.}\pm\mathrm{syst.}\pm\mathrm{lumi.}$ | $\sigma$ theory
---|---|---
ATLAS | $0.92$ | $\pm$ | $0.02$ | $\pm$ | $0.08$ | $\pm$ | $0.03$ | $\mathrm{nb}$ | $0.96$ | $\pm$ | $0.05$ | $\mathrm{nb}$
$66<M_{\tau\tau}<116~{}\mathrm{GeV}$ | Fewz
CMS | $1.00$ | $\pm$ | $0.05$ | $\pm$ | $0.08$ | $\pm$ | $0.04$ | $\mathrm{nb}$ | $0.97$ | $\pm$ | $0.04$ | $\mathrm{nb}$
$60<M_{\tau\tau}<120~{}\mathrm{GeV}$ | Fewz
LHCb | $71.4$ | $\pm$ | $3.5$ | $\pm$ | $2.8$ | $\pm$ | $2.5$ | $\mathrm{pb}$ | $74.3$ | $\pm$ | $2.1$ | $\mathrm{pb}$
$p_{\mathrm{T}}^{\tau}>20~{}\mathrm{GeV},2.0<\eta^{\tau}<4.5$ | Dynnlo
$60<M_{\tau\tau}<120~{}\mathrm{GeV}$
Table 2: Combined $Z\to{\tau\tau}$ and theoretical cross-section results for
the three experiments. The ATLAS result does not include the
$\tau_{\mu}\tau_{\mu}$ final state.
## 3 $W\to\tau\nu_{\tau}$
Due to the large $W\to\ell\nu_{\ell}$ background to leptonically decaying
$\tau$-leptons produced from $W$ bosons, only hadronic decays of the
$\tau$-lepton are considered for the ATLAS and CMS $W\to\tau\nu_{\tau}$
analyses. QCD jets, $W\to\ell\nu_{\ell}$, $W\to\tau_{\ell}\nu_{\tau}$, and $Z$
with jets events provide the primary backgrounds to this signal. Of the three
experiments, only ATLAS has performed a $W\to\tau\nu_{\tau}$ measurement. An
observation has been made with CMS, but without a cross-section measurement.
### 3.1 Particle Selection
A $p_{\mathrm{T}}>12~{}\mathrm{GeV}$ trigger on the hadronic $\tau$-lepton
combined with a ${{\not\mathrel{E}}_{T}}>20~{}\mathrm{GeV}$ trigger is used to
select events for ATLAS, while a $p_{\mathrm{T}}>20~{}\mathrm{GeV}$ hadronic
$\tau$-lepton trigger and ${{\not\mathrel{E}}_{T}}>25~{}\mathrm{GeV}$ trigger
is used for CMS.
Hadronic $\tau$-leptons are selected for ATLAS using a boosted decision tree
based on the collimation, impact parameter, and lead track $p_{\mathrm{T}}$
over electromagnetic calorimeter energy of the hadronic $\tau$-lepton
candidate. For the CMS selection, requirements are placed on the hadronic
$\tau$-lepton lead track $p_{\mathrm{T}}$, isolation, and associated muon
hits.
### 3.2 Event Selection
To eliminate $W$ and $Z$ with jets backgrounds, events with high
$p_{\mathrm{T}}$ leptons outside the hadronic $\tau$-lepton jet are rejected
for both ATLAS and CMS. A large ${{\not\mathrel{E}}_{T}}$ is also required for
both to reduce the QCD jet background. Additionally, a high
${{\not\mathrel{E}}_{T}}$ significance and transverse separation between the
$\tau$-lepton and ${{\not\mathrel{E}}_{T}}$ is required for ATLAS, while the
ratio of the $\tau$-lepton jet $p_{\mathrm{T}}$ to the $p_{\mathrm{T}}$ of the
remaining jets is required to be large for CMS.
### 3.3 Background Estimation
The $W$ and $Z$ background transverse mass shapes for both ATLAS and CMS are
determined and normalized using simulation. The QCD jet background transverse
mass shape and normalization is determined using an $ABCD$ method for both.
For ATLAS, requirements are made on the ${{\not\mathrel{E}}_{T}}$ significance
and hadronic $\tau$-lepton identification, while requirements on the
${{\not\mathrel{E}}_{T}}$ and ratio of $\tau$-lepton jet $p_{\mathrm{T}}$ to
the $p_{\mathrm{T}}$ of the remaining jets are used for CMS.
### 3.4 Systematics
The primary systematic uncertainty for the ATLAS $W\to\tau_{h}\nu_{\tau}$
cross-section measurement is due to the identification efficiency for the
hadronic $\tau$-lepton jets and estimated to be $10\%$. For CMS, no cross-
section measurement was made, and so no uncertainty analysis is available.
### 3.5 Results
The number of observed $W\to\tau\nu_{\tau}$ events and background events is
given in Table 1 for both ATLAS and CMS, as well as the measured ATLAS cross-
section. To determine the total cross-section, the ATLAS
$W\to\tau_{h}\nu_{\tau}$ cross-section is extrapolated from the measured
fiducial region to full acceptance, and divided by the experimentally known
branching fraction for $\tau$-leptons to hadrons. The result given in Table 1
agrees well with the theoretically predicted value of $10.5\pm
0.5~{}\mathrm{nb}$ calculated using Fewz.
## 4 Conclusion
A comparison of the measured cross-sections of Table 1 and Table 2 divided by
their predicted theoretical values is given in Figure 1. The red points
represent combined results and the black points individual final states. The
dark error bars correspond to statistical uncertainty, while the light error
bars correspond the combined systematic uncertainty and uncertainty due to the
integrated luminosity. The light yellow line represents a ratio of unity,
while the dark yellow band indicates the theoretical uncertainty of the
prediction.
There is good agreement between all measured cross-sections and their
theoretical values. Currently, ATLAS provides the most statistics for the
combined $Z\to{\tau\tau}$ cross-section measurement, while LHCb yields the
most precise cross-section measurement. The ATLAS $W\to\tau\nu_{\tau}$
measurement has a large systematic uncertainty and does not allow for a more
precise test of the $W\to\tau\nu_{\tau}$ to $W\to\mu\nu_{\mu}$ ratio measured
at LEP.
Figure 1: Ratios of the experimental cross-section measurements of Table 1 and
Table 2 to their expected theoretical values. The combined results are given
in red and the individual final states in black. The dark error bars are the
statistical uncertainty, while the light error bars are the combined
systematic uncertainty and uncertainty due to the integrated luminosity. The
dark yellow band indicates the theoretical uncertainty centered about the
light yellow line.
## References
* [1] ATLAS Collab., Tech. Rep. ATLAS-CONF-2011-152 (2011).
* [2] CMS Collab., JINST 7 (2012) P01001. arXiv:1109.6034.
* [3] ATLAS Collab., Tech. Rep. ATLAS-CONF-2012-094 (2012).
* [4] CMS Collab., Phys.Lett. B713 (2012) 68–90. arXiv:1202.4083.
* [5] ATLAS Collab., Eur.Phys.J. C72 (2012) 2062. arXiv:1204.6720.
* [6] ATLAS Collab., JHEP 1206 (2012) 039. arXiv:1204.2760.
* [7] CMS Collab., JHEP 1207 (2012) 143. arXiv:1205.5736.
* [8] ATLAS Collab., JINST 3 (2008) S08003.
* [9] CMS Collab., JINST 3 (2008) S08004.
* [10] LHCb Collab., JINST 3 (2008) S08005.
* [11] ATLAS Collab., Phys.Rev. D84 (2011) 112006. arXiv:1108.2016.
* [12] ATLAS Collab., Tech. Rep. ATLAS-CONF-2012-006 (2012).
* [13] ATLAS Collab., Phys.Lett. B706 (2012) 276–294. arXiv:1108.4101.
* [14] CMS Collab., JHEP 1108 (2011) 117. arXiv:1104.1617.
* [15] CMS Collab., Tech. Rep. CMS-PAS-EWK-11-002 (2011).
* [16] LHCb Collab., arXiv:1210.6289.
* [17] R. Gavin, Y. Li, F. Petriello, S. Quackenbush, Comput.Phys.Commun. 182 (2011) 2388–2403. arXiv:1011.3540.
* [18] S. Catani, M. Grazzini, Phys.Rev.Lett. 98 (2007) 222002. arXiv:hep-ph/0703012.
|
arxiv-papers
| 2012-11-30T12:37:07 |
2024-09-04T02:49:38.698435
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Philip Ilten",
"submitter": "Philip Ilten",
"url": "https://arxiv.org/abs/1211.7229"
}
|
1212.0020
|
11institutetext: The “Simion Stoilow” Institute of Mathematics of the Romanian
Academy
11email: [email protected] 22institutetext: Faculty of Mathematics and
Informatics, Sofia University “St. Kliment Ohridski”
22email: [email protected]
# Modal functional interpretations
(variant with extended††thanks: In the present paper we give a more detailed
treatment of the induction for naturals and we correct the typo in the
definition of the weak compatibility rule $\mathtt{CMP}$: on page 1382 of
[10], it is $s$ instead of $x$ and $t$ instead of $y$; we also give the
treatment of $\mathtt{CMP}$ under Dialectica light. Appendix)
Mircea-Dan Hernest The first author gratefully acknowledges financial support
by the Romanian National Agency for Scientific Research grant PN-II-RU-
TE-2011-3-0122.11 Trifon Trifonov The second author gratefully acknowledges
the financial support of the Bulgarian National Science Fund within project
DTK 02/69-2009.22
###### Abstract
We extend our light Dialectica interpretation [10] to usual and light modal
formulas and prove it sound for pseudo-modal arithmetics based on Gödel’s
${\sf T}$ and classical $S_{4}$. The range of this light modal Dialectica
interpretation is the usual (non-modal) classical Arithmetic in all finite
types. We also illustrate the use of the new tools for optimized program
synthesis with new examples.
This recent work comes in addition to the program extraction technology
outlined in our previous paper [10] by adding a useful device for combining
the effect of previous optimizations by semi- and non-computational
quantifiers in a compact one-step content eraser, namely the modal operator
$\Box$ (and its weak co-modality
$\Diamond^{\\!\mathnormal{c}}\mathnormal{\hskip 0.5pt\equiv\hskip
0.5pt}\lnot\Box\lnot$). Beside the seemingly cosmetic improvement, we bring
the following new result: while the modal propositional axioms of system
$S_{4}$ are realizable, the defining axiom of $S_{5}$ is generally not
realizable under (light) modal Dialectica.
The use and interpretation of modal operators in this paper were inspired by
work of Oliva (partly joint with the first author, see [9]) at the linear
logic sublevel, see [14, 15]. It is no coincidence that, at formulas level,
our interpretation of $\Box A$ is syntactically the same as Oliva’s modified
realizability interpretation of $!A$ in intuitionistic linear logic. However,
a bureaucratic detour would be needed in order to simulate $\Box A$ in terms
of $!A$, which seems less suitable for an efficient computer implementation.
The second author independently noticed the possibility of using the same
supra-linear modal operators for light program extraction in [18], see also
[19]. However, the initiative of studying the full employment of $\Box$ for
more efficient program synthesis in the formal context of a classical first-
order modal logic (in the sense of Schütte, [16]) belongs to the first author.
As we will see, for our extractive purposes it is useful to depart from
Schütte’s original semantics for quantified modal logic. E.g., the
propositional fragments of our first-order modal systems are no longer modal,
but purely boolean, as $\Box p\equiv p\equiv\Diamond^{\\!\mathnormal{c}}p$ for
propositional atoms $p$. We thus design _pseudo_ (i.e., non-standard) modal
arithmetics for program extraction, with relative soundness syntactically
given via our (light) modal functional interpretation by the target system,
namely classical predicate Arithmetic with higher-type functionals, in a
Natural Deduction presentation.
We stress the fact that we are only concerned with fragments of Arithmetics
without undecidable predicates: all propositional atoms of our systems are a
priori decidable.
For an easier presentation we will give up the ‘pseudo’ prefix. Throughout the
paper, our modal Arithmetics are pseudo-modal. Note that soundness of
Schütte’s predicate modal logics (e.g., $S^{\star}_{4}$) is proved non-
constructively, using models, see [16].
## 1 Arithmetical systems for Modal Dialectica extraction
We build upon functional arithmetical systems ${\sf NA}$ and (the light
annotated) $\sf NA_{\mathnormal{l}}$ from [10]. While _verifying system_ ${\sf
NA}$ basically is the Arithmetic $Z$ of Berger, Buchholz and Schwichtenberg
[4] in a slightly different presentation which is more suitable for light
functional synthesis and features full classical logic (without strong
existence) and full extensionality111As inherited from system $Z$, our ${\sf
NA}$ is mostly a Natural Deduction presentation of the so-called ‘negative
arithmetic’ from [20], basically a double-negation, Gödel-Gentzen embedding of
classical into Heyting Arithmetic ${\sf HA}^{\omega}$., its light counterpart
$\sf NA_{\mathnormal{l}}$ is only partly classical. Moreover, the _input
system_ $\sf NA_{\mathnormal{l}}$ is weakly extensional and its contraction
(and hence also induction) rule is restricted for soundness of the (light)
functional interpretation of $\sf NA_{\mathnormal{l}}$ into ${\sf NA}$. In
computing terms, the program synthesis algorithm provided by the light
Dialectica (of [10], as inherited from the one222The restriction on
extensionality is at its turn inherited from the pure Gödel’s functional
interpretation [1, 6], whereas the restriction on contraction was first added
by Hernest, as it was imposed by the necessity of decidability of the
translation of light contraction formulas. of [7]) terminates without error
only modulo the above-mentioned restrictions on Extensionality and
Contraction333These restrictions are more relaxed than those from the first
author’s PhD thesis and weaker than Gödel’s restriction on extensionality,
Kreisel’s avoiding of contraction in his Modified Realizability [12] and
Girard’s total elimination of contraction in his original Linear Logic [5]..
For (light) modal functional synthesis we will use the same verifying system
${\sf NA}$. The simpler input system $\sf NA^{\\!\mathnormal{m}}$ is obtained
by adding $\Box$ to a restricted variant of ${\sf NA}$. This modal Arithmetic
will be proved sound via the modal Dialectica interpretation. The fully-
fledged input system $\sf NA^{\\!\mathnormal{m}}_{\mathnormal{l}}$ adds to
$\sf NA^{\\!\mathnormal{m}}$ all light universal quantifiers and is a modal
extension of $\sf NA_{\mathnormal{l}}$; its soundness will be given by the
light modal Dialectica interpretation. We will not detail here the arithmetics
${\sf NA}$ and $\sf NA_{\mathnormal{l}}$, but rather refer the reader to [10].
We mostly enumerate the new items that are added in order to get $\sf
NA^{\\!\mathnormal{m}}$ and respectively $\sf
NA^{\\!\mathnormal{m}}_{\mathnormal{l}}$. (Systems ${\sf NA}$ and $\sf
NA_{\mathnormal{l}}$ are retaken in the Appendix section 5 .)
The sets of finite types $\mathnormal{T}\hskip 0.5pt,$ terms $\mathcal{T}$ (of
Gödel’s ${\sf T}$), formulas $\mathcal{F}$ (of ${\sf NA}$) and, with the
addition of $\Box$, formulas $\mathcal{F}^{m}$ of $\sf NA^{\\!\mathnormal{m}}$
and $\mathcal{F}^{m}_{l}$ of $\sf NA^{\\!\mathnormal{m}}_{\mathnormal{l}}$ are
defined as follows:
$\displaystyle\mathnormal{T}\quad$ $\displaystyle\rho,\sigma$ $\displaystyle\
\mathtt{::=}\quad\iota\;|\;o\;|\;(\rho\sigma)$ $\displaystyle\mathcal{T}\quad$
$\displaystyle s,t$ $\displaystyle\ \mathtt{::=}\quad
x^{\rho}\;|\;\mathtt{T}^{o}\;|\;\mathtt{F}^{o}\;|\;\mathtt{0}^{\iota}\;|\;\mathtt{S}^{\iota\iota}\;|\;\mathtt{If}^{o\rho\rho\rho}\;|\;\mathtt{R}^{\iota\rho(\iota\rho\rho)\rho}\;|\;(\lambda
x^{\rho}.\,t^{\sigma})^{\rho\sigma}\;|\;(t^{\rho\sigma}s^{\rho})^{\sigma}$
$\displaystyle\mathcal{F}\quad$ $\displaystyle A,B$ $\displaystyle\
\mathtt{::=}\quad\mathtt{at}(t^{o})\;|\;A\rightarrow B\;|\;A\land
B\;|\;\forall x^{\rho}A\;|\;\,\widetilde{\exists}x^{\rho}A\;\mathnormal{\
:\equiv\ }\;\lnot\forall x^{\rho}\lnot A$ $\displaystyle\mathcal{F}^{m}\quad$
$\displaystyle A,B$ $\displaystyle\
\mathtt{::=}\quad\mathtt{at}(t^{o})\;|\;A\rightarrow B\;|\;A\land
B\;|\;\forall x^{\rho}A\;|\;\,\Box
A\;|\;\Diamond^{\\!\mathnormal{c}}A\;\mathnormal{\ :\equiv\ }\;\lnot\Box\lnot
A$ $\displaystyle\mathcal{F}^{m}_{l}\quad$ $\displaystyle A,B$ $\displaystyle\
\mathtt{::=}\quad\mathtt{at}(t^{o})\;|\;A\rightarrow B\;|\;A\land
B\;|\;\forall x^{\rho}A\;|\;\Box
A\;|\;\,\forall_{\left\\{\emptyset,+,-,\pm\right\\}}x^{\rho}A$
Recall that we employ just two basic types: integers $\iota$ and booleans
$o\hskip 0.5pt,$ and use $\rho\sigma\tau$ for $(\rho(\sigma\tau))\hskip
0.5pt.$ Building blocks for terms are the usual constructors for booleans
($\mathtt{T},\mathtt{F}$) and integers ($\mathtt{0},\mathtt{S}$), case
distinction $\mathtt{If}$ and Gödel recursion $\mathtt{R}\hskip 0.5pt.$
The operator ${\sf FV}(\cdot)$ returns the set of free variables of its
argument $t\in\mathcal{T}$ or $A\in\mathcal{F}$. Atomic formulas are decidable
by definition, as they are identified with boolean terms. In particular, we
have decidable falsity $\bot\hskip 1.0pt\mathnormal{\hskip 0.5pt:\equiv\hskip
0.5pt}\hskip 1.0pt\mathtt{at}(\mathtt{F})$ and truth $\top\hskip
1.0pt\mathnormal{\hskip 0.5pt:\equiv\hskip 0.5pt}\hskip
1.0pt\mathtt{at}(\mathtt{T})\hskip 0.5pt.$ As usual, we abbreviate
$A\rightarrow\bot$ by $\lnot A\hskip 0.5pt.$
For the necessity operator $\Box$ we have the following enhanced introduction
rule, which applies to many more premise sequents than usual (as the context
$\Gamma$ may be inhabited, see also Remark 4 in Section 2 for an extended
motivation):
$\begin{array}[]{rl}\Box^{i}:&{tensy\vbox{\hbox spread0.0pt{\hskip 0.0pt plus
0.0001fil\hbox{\kern 3.8889pt\hbox{$\displaystyle\penalty 1\Gamma\hskip
0.5pt\vdash\hskip 0.5ptA$}}\hskip 0.0pt plus 0.0001fil}\hbox{\hbox{\kern
0.0pt\vrule
height=0.25002pt,depth=0.25002pt,width=34.19437pt\hbox{}}}\hbox{\kern
0.0pt\hbox{$\displaystyle\Gamma\hskip 0.5pt\vdash\hskip 0.5pt\Box
A$}}}}\end{array}$ , where $\Gamma$ is restricted depending on the
translation of the (sub)proof of the premise sequent, in ways that will be
described below for each of the two proof translations: modal and light modal.
The following axioms of modal propositional logic $S_{4}$ are part of $\sf
NA^{\\!\mathnormal{m}}$ and $\sf NA^{\\!\mathnormal{m}}_{\mathnormal{l}}$:
$\begin{array}[]{l@{\qquad\qquad\qquad}l}\mathtt{AxT}:\ \Box A\rightarrow
A&\mathtt{AxT^{c}}:\ A\rightarrow\Diamond^{\\!\mathnormal{c}}A\\\\[3.0pt]
\mathtt{Ax4}:\ \Box A\rightarrow\Box\Box
A&\mathtt{Ax4^{c}}:\Diamond^{\\!\mathnormal{c}}\Diamond^{\\!\mathnormal{c}}A\rightarrow\Diamond^{\\!\mathnormal{c}}A\\\\[3.0pt]
\lx@intercol\mathtt{AxK}:\ [\Box(A\rightarrow B)\land\Box A]\rightarrow\Box
B\hfil\lx@intercol\end{array}$
In fact only $\mathtt{AxT}$ is needed as axiom of our non-standard modal
systems. Of course, $\mathtt{AxT^{c}}$ and $\mathtt{Ax4^{c}}$ were
syntactically deducible from $\mathtt{AxT}$ and respectively $\mathtt{Ax4}$
already in the propositional modal system $S_{4}$, only using minimal logic
(the proof of $\mathtt{Ax4^{c}}$ also uses $\mathtt{AxK}$ and the empty-
context $\Box^{i}$). It turns out that also $\mathtt{Ax4}$ and $\mathtt{AxK}$
are easily deducible in $\sf NA^{\\!\mathnormal{m}}$/$\hskip 1.0pt\sf
NA^{\\!\mathnormal{m}}_{\mathnormal{l}}$ just from $\mathtt{AxT}$ (and only
using minimal logic), given our very liberal necessity introduction rule, see
Definition 3 below. Note that Stability $\lnot\lnot B\rightarrow B$ needs to
be restricted already for $\sf NA^{\\!\mathnormal{m}}$, due to the necessary
restriction on Contraction, see Remark 3 further below, Remark 5 in Section 2
and Section 3.1 of [10].
We denote by $A\rightarrow_{k}B\;\mathnormal{\hskip 0.5pt:\equiv\hskip
0.5pt}\;\Box A\rightarrow B$ the so-called ‘Kreisel implication’, since its
translation by modal Dialectica coincides with its Modified Realizability
interpretation.
###### Definition 1 (modal Dialectica interpretation)
_The interpretation does not change atomic 444Any decidable formula can (and
should) be given via its associated boolean term, e.g., one should rather use
$\mathtt{at}(\mathtt{Odd}(x))$ instead of the more verbose $\forall y(2y\neq
x)\hskip 0.5pt,$ which is refutation relevant in a somewhat artificial and
probably unintended way. formulas, i.e., $|\mathtt{at}(t^{o})|\mathnormal{\
:\equiv\ }\mathtt{at}(t^{o})\hskip 0.5pt.$ Assuming
$|A|^{\boldsymbol{x}}_{\boldsymbol{y}}$ and
$|B|^{\boldsymbol{u}}_{\boldsymbol{v}}$ are already defined,_
$\begin{array}[]{r@{\quad}c@{\quad}l@{\qqquad} r@{\quad}c@{\quad}l}|A\land
B|^{\boldsymbol{x},\boldsymbol{u}}_{\boldsymbol{y},\boldsymbol{v}}&\mathnormal{\hskip
0.5pt:\equiv\hskip
0.5pt}&|A|^{\boldsymbol{x}}_{\boldsymbol{y}}\land|B|^{\boldsymbol{u}}_{\boldsymbol{v}}\hfil\qquad\quad&|\forall
z\hskip 1.0ptA(z)|^{\boldsymbol{f}}_{z,\boldsymbol{y}}&\mathnormal{\hskip
0.5pt:\equiv\hskip 0.5pt}&|A(z)|^{\boldsymbol{f}z}_{\boldsymbol{y}}\\\\[8.0pt]
|A\rightarrow
B|^{\boldsymbol{f},\boldsymbol{g}}_{\boldsymbol{x},\boldsymbol{v}}&\mathnormal{\hskip
0.5pt:\equiv\hskip
0.5pt}&|A|^{\boldsymbol{x}}_{\boldsymbol{f}\boldsymbol{x}\boldsymbol{v}}\rightarrow|B|^{\boldsymbol{g}\boldsymbol{x}}_{\boldsymbol{v}}\hfil\qquad\quad&|\Box
A|^{\boldsymbol{x}}&\mathnormal{\hskip 0.5pt:\equiv\hskip
0.5pt}&\forall\boldsymbol{y}|A|^{\boldsymbol{x}}_{\boldsymbol{y}}\end{array}$
_As an immediate consequence,_
$\begin{array}[]{r@{\quad}c@{\quad}l}|\Diamond^{\\!\mathnormal{c}}A\,\mathnormal{\;\equiv\;}(\lnot\Box\lnot
A)|_{\boldsymbol{f}}&\mathnormal{\;\equiv\;}&\widetilde{\exists}\boldsymbol{x}|A|^{\boldsymbol{x}}_{\boldsymbol{f}\boldsymbol{x}}\\\\[6.0pt]
|A\rightarrow_{k}B\,\mathnormal{\;\equiv\;}(\Box A\rightarrow
B)|^{\boldsymbol{g}}_{\boldsymbol{x},\boldsymbol{v}}&\mathnormal{\;\equiv\;}&\forall\boldsymbol{y}|A|^{\boldsymbol{x}}_{\boldsymbol{y}}\rightarrow|B|^{\boldsymbol{g}\boldsymbol{x}}_{\boldsymbol{v}}\\\\[8.0pt]
|\widetilde{\exists}z\hskip 1.0ptA(z)\,\mathnormal{\;\equiv\;}(\lnot\forall
z\lnot
A(z))|^{Z,\boldsymbol{f}}_{\boldsymbol{g}}&\mathnormal{\;\equiv\;}&\lnot\lnot|A(Z\boldsymbol{g})|^{\boldsymbol{f}\boldsymbol{g}}_{\boldsymbol{g}(Z\boldsymbol{g})(\boldsymbol{f}\boldsymbol{g})}\end{array}$
###### Definition 2 (light modal Dialectica interpretation)
_The following are added to the above (the deduced translation of
$\widetilde{\exists}_{\emptyset}z$ is outlined below for use at the end of
Section 2):_
$\begin{array}[]{r@{\ \ }c@{\ \ }l@{\qquad}r@{\ \ }c@{\ \
}l}|\forall_{\\!+}z\hskip
1.0ptA(z)|^{\boldsymbol{f}}_{\boldsymbol{y}}&\mathnormal{\,:\equiv\,}&\forall
z\hskip 1.0pt|A(z)|^{\boldsymbol{f}z}_{\boldsymbol{y}}&|\forall_{\\!-}z\hskip
1.0ptA(z)|^{\boldsymbol{x}}_{z,\boldsymbol{y}}&\mathnormal{\,:\equiv\,}&|A(z)|^{\boldsymbol{x}}_{\boldsymbol{y}}\\\\[6.0pt]
|\forall_{\\!\emptyset}z\hskip
1.0ptA(z)|^{\boldsymbol{x}}_{\boldsymbol{y}}&\mathnormal{\,:\equiv\,}&\forall
z\hskip
1.0pt|A(z)|^{\boldsymbol{x}}_{\boldsymbol{y}}&|\widetilde{\exists}_{\emptyset}z\hskip
1.0ptB(z)|^{\boldsymbol{f}}_{\boldsymbol{g}}&\mathnormal{\;\equiv\;}&\widetilde{\exists}z\hskip
1.0pt|B(z)|^{\boldsymbol{f}\boldsymbol{g}}_{\boldsymbol{g}(\boldsymbol{f}\boldsymbol{g})}\end{array}$
###### Remark 1
The light modal translation of formulas only adds $|\Box
A|^{\boldsymbol{x}}\mathnormal{\,:\equiv\,}\forall\boldsymbol{y}|A|^{\boldsymbol{x}}_{\boldsymbol{y}}$
to our light functional translation from [10].
The definition of computation relevance of (light) modal formulas $A$ is
basically the same as for non-modal formulas, relative to the enhanced
syntactic context. Namely, $A$ is realization relevant also under (light)
modal Dialectica if the tuple of witness variables $\boldsymbol{x}$ of its
translation $|A|^{\boldsymbol{x}}_{\boldsymbol{y}}$ is not empty and similarly
$A$ is refutation relevant if the tuple of challenge variables
$\boldsymbol{y}$ is not empty. See Remark 1 in Section 3 of [10].
Correspondingly, $A$ is realization irrelevant if it is not realization
relevant (i.e., $\boldsymbol{x}$ is an empty tuple), and $A$ is refutation
irrelevant if it is not refutation relevant (i.e., $\boldsymbol{y}$ is an
empty tuple), see also the more technical Definition 1 in Section 2 of [10].
###### Definition 3 (Necessity Introduction)
The restriction on $\Box^{i}$ depends on programs synthesized from the proof
of the premise $A$ of this rule, unless all formulas in the context $\Gamma$
are refutation irrelevant or $A$ is refutation irrelevant, see the paragraph
following Theorem 2.1 in Section 2 below. Thus input proofs are inductively
defined together with their extracted programs (and their corresponding output
proofs).
###### Remark 2 (restriction violation for $\Box^{i}$)
In an automated interactive search for modal input proofs of a given
specification, we can temporarily allow $\Box^{i}$ and postpone the validity
check for when the proof of its premise is fully constructed. This approach
would be similar to the ‘nc-violations’ check in the actual MinLog system, see
[17], and to the so-called ‘computationally correct proofs’ from [19].
For efficiency reasons, we recommend the use of modal operators whenever
possible instead of the above partly (or non) computational quantifiers
$\forall_{\\!+}$, $\forall_{\\!-}$, $\forall_{\\!\emptyset}$ and
$\widetilde{\exists}_{\emptyset}$. Thus it makes sense to study the (pure)
modal Dialectica in itself, as the use of such light quantifiers may not be
necessary in many cases of interest. It should be much easier to construct a
purely modal (i.e., without light quantifiers) input proof, also for a (semi)
automated proof-search algorithm. Nevertheless, it is the light variant of
modal Dialectica which provides the larger range of possibilities,
particularly for situations where the simpler, ‘heavier’ modal Dialectica does
not suffice.
###### Remark 3 (Contraction restriction)
We upgrade the $\bigstar$ restriction from [10] on the computationally
relevant contractions (those on refutation relevant open assumptions $A$),
such that the interpretation $|A|$ must be decidable (rather than strictly
quantifier-free). In the new modal context one needs to take into account also
the translation of the necessity operator, as this introduces new quantifiers.
These may alter the decidability of the translated formula (relative to the
corresponding non-modal formula obtained by wiping out all instances of
$\Box$). E.g., let $T(x,y,z)$ be a decidable predicate s.t.
$H(x,y)\mathnormal{\ :\equiv\ }\widetilde{\exists}z\hskip 0.5ptT(x,y,z)$ is
not decidable (take Kleene’s $T$ predicate which is expressible in Peano
Arithmetic, hence also in ${\sf NA}$, so that $H$ expresses the Halting
Problem “program with code $x$ halts on input $y$”). Then $P(x)\mathnormal{\
:\equiv\ }\forall y\hskip 0.5pt\forall z\hskip 0.5pt\lnot\hskip 0.5ptT(x,y,z)$
can be a contraction formula, whereas $P^{\Box}(x)\mathnormal{\ :\equiv\
}\forall y\hskip 0.5pt\Box\hskip 0.5pt\forall z\hskip 0.5pt\lnot\hskip
0.5ptT(x,y,z)$ cannot, as its translation is $\forall z\hskip 0.5pt\lnot\hskip
0.5ptT(x,y,z)$, an undecidable formula, since ${\sf NA}\hskip 0.5pt\hskip
0.5pt\vdash\hskip 0.5pt\hskip 0.5pt|P^{\Box}(x)|_{y}\hskip
0.5pt\leftrightarrow\hskip 0.5pt\lnot\hskip 0.5ptH(x,y)$.
On the other hand, both $\forall z(3z\neq x)\land\forall y(2y\neq x)$ and
$\forall z(3z\neq x)\land\Box\hskip 1.0pt\forall y(2y\neq x)$ can be
contraction formulas, since $\forall y(2y\neq x)$ is decidable.
## 2 Modal and light modal functional interpretations
The following metatheorem gives the general pattern in which soundness
theorems for Dialectica-based interpretations can be expressed, in a Natural
Deduction setting.
###### Theorem 2.1 (general soundness for Dialectica interpretations; $[\
\mathrm{ISys},\,\mathrm{VSys}\ ]$)
Let $A_{0},A_{1},\ldots,A_{n}$ be a sequence of formulas of $\mathrm{ISys}$
with $\boldsymbol{w}$ all their free variables. If the sequent $\,\hskip
0.5pta_{1}\\!:\\!A_{1}\hskip 0.5pt,\ldots,\hskip 0.5pta_{n}\\!:\\!A_{n}\hskip
0.5pt\hskip
0.5pt\vdash\\!\\!\\!\raisebox{-2.58334pt}{\scriptsize$l$}\,A_{0}\,$ is
provable in $\mathrm{ISys}\hskip 0.5pt,$ then terms
$\boldsymbol{t}_{0},\ldots,\boldsymbol{t}_{n}$ can be automatically
synthesized from its formal proof, such that the translated sequent $\hskip
0.5pta_{1}\\!:\\!|A_{1}|^{\boldsymbol{x}_{1}}_{\boldsymbol{t}_{1}}\hskip
0.5pt,\ldots,\hskip
0.5pta_{n}\\!:\\!|A_{n}|^{\boldsymbol{x}_{n}}_{\boldsymbol{t}_{n}}\hskip
0.5pt\hskip 0.5pt\vdash\hskip
0.5pt|A_{0}|^{\boldsymbol{t}_{0}}_{\boldsymbol{x}_{0}}\,$ is provable in
$\mathrm{VSys}\hskip 0.5pt,$ where the following _free variable condition_ (c)
holds: ${\sf
FV}(\boldsymbol{t}_{i})\subseteq\\{\boldsymbol{w},\boldsymbol{x}_{0},\ldots,\boldsymbol{x}_{n}\\}$
and $\boldsymbol{x}_{0}\not\in{\sf FV}(\boldsymbol{t}_{0})\hskip 0.5pt.$ Here
$\boldsymbol{x}_{0},\ldots,\boldsymbol{x}_{n}$ are tuples of fresh variables,
s.t. equal avars share a common such tuple.
In [10] the above was thoroughly proved for $\mathrm{ISys}\equiv\sf
NA_{\mathnormal{l}}$ and $\mathrm{VSys}\equiv{\sf NA}$. Below we prove that
(meta)Theorem 2.1 remains valid also for the pairs $[\sf
NA^{\\!\mathnormal{m}},{\sf NA}]$ (modal Dialectica) and $[\sf
NA^{\\!\mathnormal{m}}_{\mathnormal{l}},{\sf NA}]$ (light modal Dialectica),
which share the same $\mathrm{VSys}\equiv{\sf NA}$.
We can now complete the definition of $\Box^{i}$: the restriction is that
$\boldsymbol{x}_{0}\not\in\cup_{i=1}^{n}{\sf FV}(\boldsymbol{t}_{i})$ in the
translated premise sequent $\hskip
0.5pta_{1}\\!:\\!|A_{1}|^{\boldsymbol{x}_{1}}_{\boldsymbol{t}_{1}}\hskip
0.5pt,\ldots,\hskip
0.5pta_{n}\\!:\\!|A_{n}|^{\boldsymbol{x}_{n}}_{\boldsymbol{t}_{n}}\hskip
0.5pt\hskip 0.5pt\vdash\hskip
0.5pt|A_{0}|^{\boldsymbol{t}_{0}}_{\boldsymbol{x}_{0}}\,\hskip 0.5pt.$ This
ensures that the introduction rule $\forall^{i}$ can be applied for variables
$\boldsymbol{x}_{0}$ and thus the conclusion sequent $\,\hskip
0.5pta_{1}\\!:\\!A_{1}\hskip 0.5pt,\ldots,\hskip 0.5pta_{n}\\!:\\!A_{n}\hskip
0.5pt\hskip 0.5pt\vdash\\!\\!\\!\raisebox{-2.58334pt}{\scriptsize$l$}\,\Box
A_{0}\,$ is witnessed by the same realizers as the premise.
###### Lemma 1 (interpretation of $S_{4}$ modal axioms)
Axioms $\mathtt{AxT}$, $\mathtt{AxT^{c}}$, $\mathtt{Ax4}$, $\mathtt{Ax4^{c}}$
and $\mathtt{AxK}$ are realizable in ${\sf NA}$ under the (light) modal
Dialectica translation.
Proof: The translation of $\mathtt{AxT}$ is $|\Box A\rightarrow
A|^{\boldsymbol{g}}_{\boldsymbol{x},\boldsymbol{y}}\equiv\forall\boldsymbol{v}|A|^{\boldsymbol{x}}_{\boldsymbol{v}}\rightarrow|A|^{\boldsymbol{g}\boldsymbol{x}}_{\boldsymbol{y}}$
and we can take $\boldsymbol{g}$ to be the identity
$\lambda\boldsymbol{x}.\,\boldsymbol{x}$. Similarly, the translation of
$\mathtt{AxT^{c}}$ is
$|A\rightarrow\Diamond^{\\!\mathnormal{c}}A|^{\boldsymbol{f}}_{\boldsymbol{x},\boldsymbol{y}}\equiv|A|^{\boldsymbol{x}}_{\boldsymbol{f}\boldsymbol{x}\boldsymbol{y}}\rightarrow\widetilde{\exists}\boldsymbol{u}|A|^{\boldsymbol{u}}_{\boldsymbol{y}}$
and we can take $\boldsymbol{f}$ to be the projection
$\lambda\boldsymbol{x}\boldsymbol{y}.\,\boldsymbol{y}$. For $\mathtt{Ax4}$ and
$\mathtt{Ax4^{c}}$ it is immediate that $|\Box A|\equiv|\Box\Box A|$ and also
$|\Diamond^{\\!\mathnormal{c}}A|\equiv|\Diamond^{\\!\mathnormal{c}}\Diamond^{\\!\mathnormal{c}}A|$,
thus the realizer is again the identity in both cases. In the translation of
$\mathtt{AxK}$ below, we take $\boldsymbol{U}\mathnormal{\hskip
0.5pt:\equiv\hskip
0.5pt}\lambda\boldsymbol{f},\boldsymbol{g},\boldsymbol{x}.\,\boldsymbol{g}\boldsymbol{x}$,
which can easily be proved to be a realizer.
$\begin{array}[]{r@{\quad}c@{\quad}l}|\mathtt{AxK}|&\equiv&[\,\forall\boldsymbol{x},\boldsymbol{v}(|A|^{\boldsymbol{x}}_{\boldsymbol{f}\boldsymbol{x}\boldsymbol{v}}\rightarrow|B|^{\boldsymbol{g}\boldsymbol{x}}_{\boldsymbol{v}})\
\land\
\forall\boldsymbol{y}|A|^{\boldsymbol{{x}^{\prime}}}_{\boldsymbol{y}}\,]^{\boldsymbol{f},\boldsymbol{g},\boldsymbol{{x}^{\prime}}}\
\rightarrow\
\forall\boldsymbol{{v}^{\prime}}|B|^{\boldsymbol{u}}_{\boldsymbol{{v}^{\prime}}}\quad\equiv\\\\[4.0pt]
&\equiv&[\,\forall\boldsymbol{x},\boldsymbol{v}(|A|^{\boldsymbol{x}}_{\boldsymbol{f}\boldsymbol{x}\boldsymbol{v}}\rightarrow|B|^{\boldsymbol{g}\boldsymbol{x}}_{\boldsymbol{v}})\
\land\ \forall\boldsymbol{y}|A|^{\boldsymbol{{x}^{\prime}}}_{\boldsymbol{y}}\
\rightarrow\
\forall\boldsymbol{{v}^{\prime}}|B|^{\boldsymbol{U}(\boldsymbol{f},\boldsymbol{g},\boldsymbol{{x}^{\prime}})}_{\boldsymbol{{v}^{\prime}}}\,]^{\boldsymbol{U}}_{\boldsymbol{f},\boldsymbol{g},\boldsymbol{{x}^{\prime}}}\end{array}$
Given the above Lemma and comments, we have completely established the
following:
###### Theorem 2.2 (soundness of modal Dialectica)
Theorem 2.1 $[\ \sf NA^{\\!\mathnormal{m}},\,{\sf NA}\ ]$.
###### Theorem 2.3 (soundness of light modal Dialectica)
Theorem 2.1 $[\ \sf NA^{\\!\mathnormal{m}}_{\mathnormal{l}},\,{\sf NA}\ ]$.
The next result pictures the limits of our modal extension of Dialectica
interpretation.
###### Theorem 2.4 (unrealizability of $S_{5}$ defining axiom)
Axiom
$\mathtt{Ax5}:\Diamond^{\\!\mathnormal{c}}A\rightarrow\Box\Diamond^{\\!\mathnormal{c}}A$
is generally not realizable under the (light) modal Dialectica translation.
Proof: The translation of $\mathtt{Ax5}$ is a formula of shape
$B(\boldsymbol{x})\rightarrow\forall\boldsymbol{y}B(\boldsymbol{y})$ which
only holds true when $\boldsymbol{x}$ is the empty tuple, special case when
$\mathtt{Ax5}$ requires no realizer at all.
Notice that $\Diamond^{\\!\mathnormal{c}}\widetilde{\exists}\boldsymbol{x}A$
is akin to Berger’s uniform existence $\\{\exists\boldsymbol{x}\\}A$ from [2],
where one does not care about the witness for $\exists\boldsymbol{x}$ (which
is actually deleted from the extraction). We can thus see
$\Diamond^{\\!\mathnormal{c}}$ as an extension of Berger’s tool to more
general formulas than just existential ones. On the other hand there are
situations when $\Box$ and $\Diamond^{\\!\mathnormal{c}}$ are too general
tools and separate annotations for each quantifier are a better answer for the
problem at hand. In some of these cases it may still be possible to use the
modal operators if one changes the input specification and its proof.
###### Remark 4 (Necessity Introduction revisited)
The usual restriction on the introduction rule for the necessity operator
$\Box^{i}$ is that $\Gamma\equiv\emptyset$. In the natural deduction
presentation of modal logic, $\Box^{i}$ cannot be unrestricted or
$A\rightarrow\Box A$ becomes a theorem, thus all occurrences of $\Box$
becoming redundant. Our restriction on $\Box^{i}$ is strictly weaker, as,
e.g., allows any context $\Gamma$ whose formulas are all refutation irrelevant
and any context at all if the conclusion is refutation irrelevant. Thus,
$A\rightarrow\Box A$ not only is possible in our pseudo-modal systems, it even
defines a very interesting class of formulas, see below.
###### Definition 4 (necessary formulas)
Formulas $A$ s.t. $\hskip 1.0pt\hskip 0.5pt\vdash\hskip 0.5pt\hskip
0.5ptA\rightarrow\Box A\,$ in $\sf NA^{\\!\mathnormal{m}}$ or $\sf
NA^{\\!\mathnormal{m}}_{\mathnormal{l}}$.
Also due to $\mathtt{AxT}$, it follows that $\hskip 1.0pt\hskip
0.5pt\vdash\hskip 0.5pt\hskip 0.5ptA\leftrightarrow\Box A\,$ for any necessary
formula, thus placing $\Box$ in front of such $A$ would be logically
redundant. We say that an occurrence of $\Box$ is meaningful (i.e., non-
redundant) in front of any formula that is not necessary.
Note that all refutation irrelevant formulas are necessary formulas. It is
easy to see that some of the refutation relevant formulas are necessary, e.g.,
$\forall\boldsymbol{x}\bot$ and $\forall\boldsymbol{x}\top$ (in fact any $A$
s.t. $\hskip 1.0pt\hskip 0.5pt\vdash\hskip 0.5ptA\hskip 1.0pt$ or $\hskip
1.0pt\hskip 0.5pt\vdash\hskip 0.5pt\hskip 0.5pt\lnot A\hskip 1.0pt$ in $\hskip
1.0pt\sf NA^{\\!\mathnormal{m}}$ or $\hskip 1.0pt\sf
NA^{\\!\mathnormal{m}}_{\mathnormal{l}}$). However, even if such formulas
syntactically do require challengers, these functionals turn out to be
redundant and can be soundly discarded by a $\Box$, without the need to change
any other component of the input proof. In fact, a formula $A$ is necessary
iff it can be proved equivalent (in $\hskip 1.0pt\sf NA^{\\!\mathnormal{m}}$
or $\hskip 1.0pt\sf NA^{\\!\mathnormal{m}}_{\mathnormal{l}}$) to a refutation
irrelevant formula $B$. Indeed, for a necessary $A$ take
$B\mathnormal{\,:\equiv\,}\Box A$. For the converse we can use the long
implication $A\rightarrow B\rightarrow\Box B\rightarrow\Box A$, where for the
last implication a contextless $\Box^{i}$ together with $\mathtt{AxK}$ was
used.
Therefore, the ‘necessary’ class captures those formulas whose negative
computational content can always be erased regardless of the context in which
they are used. On the other hand, there are cases when $\Box$ can soundly be
applied to a non-necessary formula, leading to cleaner and more efficient
extracted programs (see Section 3 below).
###### Remark 5 (modal vs. pseudo-modal)
It would appear that our input Arithmetic $\sf NA^{\\!\mathnormal{m}}$ is able
to prove new modal theorems and even sentences that are invalid in Schütte’s
semantics. On the other hand, our $\bigstar$ restriction on contraction is not
present in the usual first-order modal logic systems, thus some of the
classical modal theorems will no longer be theorems of $\sf
NA^{\\!\mathnormal{m}}$. Therefore, we say that our input systems are ‘pseudo-
modal’ rather than modal. See [13] for extensive comments on the design of
formalisms for predicate modal logic, particularly on the yet-unsatisfactory
definition of necessity introduction in Natural Deduction systems. Contraction
restriction notwithstanding, we give the optimal restriction for $\Box^{i}$ in
view of automated program synthesis. However, this does not solve the issue
for general, fully-fledged first-order modal logics.
### 2.1 Modal induction rule
As first argued in [9], induction (for natural numbers, but more generally
also for lists, as naturals $\iota$ are a particular case of inductively
defined lists) should rather be treated in a Modified Realizability style
whenever possible under Dialectica extraction. In our non-standard modal
context we can introduce the following modal induction rule of systems $\sf
NA^{\\!\mathnormal{m}}$ and $\sf NA^{\\!\mathnormal{m}}_{\mathnormal{l}}$,
which is defined with a Kreisel implication at the step:
$\begin{array}[]{c}{tensy\vbox{\hbox spread0.0pt{\hskip 0.0pt plus
0.0001fil\hbox{$\displaystyle\penalty 1\Gamma\ \hskip 0.5pt\vdash\hskip 0.5pt\
\Box A(\mathtt{0})\qquad\Box\Delta\ \hskip 0.5pt\vdash\hskip 0.5pt\ \Box
A(n)\;\rightarrow\;A(\mathtt{S}n)$}\hskip 0.0pt plus
0.0001fil}\hbox{\hbox{\kern 0.0pt\vrule
height=0.25002pt,depth=0.25002pt,width=187.75447pt\hbox{\kern 3.00003pt$\ {\
\mathtt{Ind^{m}_{\iota}}}$}}}\hbox{\kern
52.94556pt\hbox{$\displaystyle\Gamma\,,\,\Box\Delta\ \hskip 0.5pt\vdash\hskip
0.5pt\ \Box A(n)$}}}}\end{array}$
This is an upgrade of the similar rule from [9] (given at the linear logic
sublevel, see also [15]), as it allows for non-empty contexts. While the base
context $\Gamma$ is unrestricted, the step context $\Box\Delta$ is made
entirely of refutation irrelevant assumptions of shape $\Box D$. Thus the step
context restriction as for $\mathtt{Ind^{\iota}_{\mathnormal{l}}}$ (see
Appendix) is bluntly satisfied, since this only concerned refutation relevant
assumptions (whose translations in ${\sf NA}$ had to be quantifier-free, as
their decidability was needed for case distinction in their corresponding
challenge realizers). Note that if $D$ already is refutation irrelevant,
placing $\Box$ in front of $D$ is somewhat redundant. We could refine
$\mathtt{Ind^{m}_{\iota}}$ by splitting the step context into
${\Delta}^{\prime}$ which consists of refutation irrelevant assumptions not of
shape $\Box D$ and ${\Delta}^{\prime\prime}\mathnormal{\;\equiv\;}\Box\Delta$.
Nonetheless such ${\Delta}^{\prime}$ were made of necessary formulas (cf.
Definition 4).
The treatment of $\mathtt{Ind^{m}_{\iota}}$ under (light) modal Dialectica is
much easier than the one of $\mathtt{Ind^{\iota}_{\mathnormal{l}}}$. In fact
$\mathtt{Ind^{m}_{\iota}}$ is a good simplification of
$\mathtt{Ind^{\iota}_{\mathnormal{l}}}$ for situations when the whole context
is made entirely of refutation irrelevant assumptions but $A(n)$ is a
refutation relevant formula. The challenger for $A(n)$ in the step conclusion
would be unneededly produced during the treatment of such
$\mathtt{Ind^{\iota}_{\mathnormal{l}}}$, as it becomes no part of any of the
witnesses for the conclusion sequent. Placing $\Box$ in front of the
negatively positioned $A(n)$ thus ensures a minimal optimization brought by
$\mathtt{Ind^{m}_{\iota}}$, in this particular case simply by elimination of
redundancy: the conclusion witnessing terms are the same as for
$\mathtt{Ind^{\iota}_{\mathnormal{l}}}$.
A more serious optimization concerns the challengers of $|C|$ for refutation
relevant assumptions $C$ from the $\Gamma$ context. These are simply preserved
by $\mathtt{Ind^{m}_{\iota}}$, while under
$\mathtt{Ind^{\iota}_{\mathnormal{l}}}$ they had to include the challengers
for the step $A(n)$. If $A(n)$ were refutation irrelevant, it would still make
sense to use $\mathtt{Ind^{m}_{\iota}}$ instead of
$\mathtt{Ind^{\iota}_{\mathnormal{l}}}$, if one is not interested in the
challengers for the refutation relevant assumptions from the step context.
While for such particular $\mathtt{Ind^{\iota}_{\mathnormal{l}}}$ we already
have the preservation of challengers for refutation relevant assumptions
strictly from $\Gamma$, still challengers for the refutation relevant step
assumptions are more complex in the conclusion sequent (they include a
meaningful Gödel recursion, even though here a challenger for the step
negative $A(n)$ is no longer comprised since it does not exist). Thus
$\mathtt{Ind^{m}_{\iota}}$ can bring an improvement over
$\mathtt{Ind^{\iota}_{\mathnormal{l}}}$ by wiping out the step challengers
altogether, should these not be needed in the global construction of the
topmost realizers for the goal specification.
It turns out that $\mathtt{Ind^{m}_{\iota}}$ strictly optimizes
$\mathtt{Ind^{\iota}_{\mathnormal{l}}}$ in many (if not most) situations. Yet
$\mathtt{Ind^{\iota}_{\mathnormal{l}}}$ will have to be used also in our non-
standard modal context, practically whenever $\mathtt{Ind^{m}_{\iota}}$ simply
cannot be applied for the goal at hand.
## 3 Examples
The weak extensionality of modal input systems $\sf NA^{\\!\mathnormal{m}}$
and $\sf NA^{\\!\mathnormal{m}}_{\mathnormal{l}}$ can better be expressed by
means of the following modal compatibility axiom (the usual compatibility
axiom, but with the outward implication changed to a Kreisel implication)
$\mathtt{CmpAx^{m}}:\
\Box(\boldsymbol{x}=_{\rho}\boldsymbol{y})\,\rightarrow\,A(\boldsymbol{x})\rightarrow
A(\boldsymbol{y})$
By straightforward calculations, it is easy to see that $\mathtt{CmpAx^{m}}$
is realizable under (light) modal Dialectica by simple projection functionals,
with the verification in the fully extensional ${\sf NA}$ given by the
corresponding compatibility axiom $\mathtt{CmpAx}$, see [10].
In [9] the following class of examples was considered: theorems of the form
$\forall\boldsymbol{x}A\to\forall\boldsymbol{y}B\to\forall\boldsymbol{z}C$ (1)
possibly with parameters, where the negative information on $\boldsymbol{x}$
is irrelevant, while the one on $\boldsymbol{y}$ is of our interest. Then it
must be possible to adapt the proof of (1) to a proof in $\sf
NA^{\\!\mathnormal{m}}$ or $\sf NA^{\\!\mathnormal{m}}_{\mathnormal{l}}$ of
$(\Box\forall\boldsymbol{x}A)\to\forall\boldsymbol{y}B\to\forall\boldsymbol{z}C$.
As noticed by Oliva in [15], the Fibonacci example first treated with
Dialectica in [8] falls into this category.
Oliva also suggested an interesting example, which motivated the definition of
our positively computational quantifier $\forall_{\\!+}$ (see [10]): “Any
infinite set $P$ of natural numbers contains numbers which are arbitrarily
apart”. The claim can be formalized as follows:
$\forall x\widetilde{\exists}y(y>x\land P(y))\rightarrow\forall
d\widetilde{\exists}{n_{1},n_{2}}\big{(}n_{2}>n_{1}+d\land P(n_{1})\land
P(n_{2})\big{)}$ (2)
This statement can be proved only via a contraction on the premise, and as a
result $x$ is refuted by a term involving case distinction on $|P|$. However,
if only the witnesses of $n_{1}$ and $n_{2}$ are needed, then the redundant
challenge for $x$ can be discarded by using a $\Box$ in front of the premise,
effectively applying a Kreisel implication. This example is of the form (1)
and can be treated both with the hybrid Dialectica from [9] and with the
extended light Dialectica interpretation from [10].
The example can be extended so that the premise becomes more involved [19]:
$\forall
m(\widetilde{\exists}n\,Q(n,m)\rightarrow\widetilde{\exists}n_{1}\,Q(n_{1},\mathtt{S}m))\rightarrow\widetilde{\exists}n_{0}\,Q(n_{0},\mathtt{0})\rightarrow\widetilde{\exists}n_{2}\,Q(n_{2},\mathtt{S}\mathtt{S}\mathtt{0})$
(3)
Again, a contraction must be used, and two semi-computational quantifiers need
to be applied to erase the negative computational content:
$\forall_{\\!+}m(\widetilde{\exists}_{+}n\,Q(n,m)\rightarrow\widetilde{\exists}n_{1}\,Q(n_{1},\mathtt{S}m))\rightarrow\widetilde{\exists}n_{0}\,Q(n_{0},\mathtt{0})\rightarrow\widetilde{\exists}n_{2}\,Q(n_{2},\mathtt{S}\mathtt{S}\mathtt{0})$
(4)
However, this solution is not desirable, as the light annotations would only
apply to a special class of binary relations $Q$ for which the witness $n_{1}$
for $Q(n_{1},\mathtt{S}m)$ does not depend computationally on the witness $n$
for $Q(n,m)$ for any $m$, hence reducing the generality of the claim. One of
the solutions would be to extend the light annotations to implications as in
[19], however a much simpler and more elegant approach would be to use a
Kreisel implication. The negative content of the premise will be fully erased
and the positive one will be fully preserved, achieving a Modified
Realizability effect.
We will consider another relevant case study, known as the “integer root
example”, which was suggested by Berger and Schwichtenberg in [3]: “every
unbounded integer function has an integer root function”. The example can be
formalized as follows:
$\forall x\widetilde{\exists}y(f(y)>x)\rightarrow\forall
m\Big{(}f(\mathtt{0})\leq m\rightarrow\widetilde{\exists}n\big{(}f(n)\leq
m<f(\mathtt{S}n)\big{)}\Big{)}$ (5)
The claim can be proved by contradiction using induction on the formula
$f(n)\leq m$. However, in addition to computing the integer root, the (heavy)
Dialectica also extracts a complicated recursive counterexample for $x$, with
a case distinction on each step [19]. This term challenges the outermost
premise, which forms the refutation relevant induction context shared by the
base and the step formulas. The undesired negative content can be erased by a
Kreisel implication, which converts the context to a necessary one, allowing
the application of the modal induction rule. As a result, only the integer
root is extracted, and additional artifacts are omitted. Note that, in
contrast to the previous two examples, this proof is classical, so Modified
Realizability is not applicable in this case. However, using $\forall_{\\!+}x$
would still achieve the same cleaning effect [19].
## 4 Conclusions and future work
Modal Dialectica provides the means of using both Modified Realizability and
Gödel’s Dialectica at the same time for more efficient program extraction.
This was already the case for the hybrid Dialectica of [9], but here we
eliminate the detour to the linear logic sublevel. Disregarding the light
quantifiers, (pure) modal Dialectica represents (directly at the supra-linear
logic level) a good combination of the original proof interpretations, with
the possibility of carrying out both in a sound way on certain input proofs.
All one needs is that some implications of the input proof can be seen as
Kreisel implications.
A natural continuation of the work reported in this paper concerns the
addition to our input systems of strong (intuitionistic) elements. Besides the
strong $\exists$ and its light associated $\exists_{\emptyset}$ (originally
from [7] where it was denoted $\overline{\exists}$, see also [19]), strong
possibility $\Diamond$ also needs to be considered as the intuitionistic dual
of necessity $\Box$.
The following clauses would then be added to Definition 1 for getting the
strong modal Dialectica interpretation $|\exists z\hskip
1.0ptA(z)|^{z,\boldsymbol{f}}_{\boldsymbol{y}}\ \mathnormal{\hskip
0.5pt:\equiv\hskip 0.5pt}\ |A(z)|^{\boldsymbol{f}}_{\boldsymbol{y}}$ and
$|\Diamond A|_{\boldsymbol{y}}\ \mathnormal{\hskip 0.5pt:\equiv\hskip 0.5pt}\
\exists\boldsymbol{x}|A|^{\boldsymbol{x}}_{\boldsymbol{y}}$, and further
$|\exists_{\emptyset}z\hskip 1.0ptA(z)|^{\boldsymbol{x}}_{\boldsymbol{y}}\
\mathnormal{\hskip 0.5pt:\equiv\hskip 0.5pt}\ \exists z\hskip
1.0pt|A(z)|^{\boldsymbol{x}}_{\boldsymbol{y}}$ to Definition 2 in order to
obtain the strong light modal Dialectica interpretation.
Intuitionistic (light) modal arithmetical systems will first be considered at
input for ‘strong’ program synthesis. Then their enhanced classical
counterparts will be interpreted, modulo some negative translation. Such
systems will soundly extend $\sf NA^{\\!\mathnormal{m}}$ with $\Diamond$ and
$\exists$, and $\sf NA^{\\!\mathnormal{m}}_{\mathnormal{l}}$ also with
$\exists_{\emptyset}$. Nevertheless certain restrictions may need to be
applied on $\sf NA^{\\!\mathnormal{m}}$ and/or $\sf
NA^{\\!\mathnormal{m}}_{\mathnormal{l}}$ before attempting such extensions
with intuitionistic elements.
Acknowledgements: Our first reading of predicate modal logic was [16], a rare
small and complete presentation of the topic, recommended by Prof.
Schwichtenberg, to whom we are grateful. Thanks to Diana Ratiu for providing
us a copy of this book. Many thanks to Paulo Oliva for valuable comments on an
early draft of this paper.
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$\mbox{$\mathtt{T}\mathtt{Ax}$}:\quad\hskip 0.5pt\vdash\hskip 0.5pt\mathtt{at}(\mathtt{T})$ | $\mathtt{CmpAx}:\quad\hskip 0.5pt\vdash\hskip 0.5ptx=_{\rho}y\,\rightarrow\,A(x)\rightarrow A(y)$
---|---
Table 1: Basic axioms, with $\mathtt{CmpAx}$ replaced by $\mathtt{CMP}$ rule in $\sf NA_{\mathnormal{l}}$ $\hskip 0.5pta\\!:\\!A\hskip 0.5pt\hskip 0.5pt\vdash\hskip 0.5ptA\quad\mathrm{(id)}$ | tensy $\displaystyle\penalty 1\Gamma,[\hskip 0.5pta\\!:\\!A\hskip 0.5pt]\hskip 0.5pt\vdash\hskip 0.5ptB$ $\ \mathnormal{\rightarrow^{\\!i}}$ $\displaystyle\Gamma\hskip 0.5pt\vdash\hskip 0.5ptA\rightarrow B$ | tensy $\displaystyle\penalty 1\Gamma\hskip 0.5pt\vdash\hskip 0.5ptA\quad\Delta\hskip 0.5pt\vdash\hskip 0.5ptA\rightarrow B$ $\ \mathnormal{\rightarrow^{\\!e}}$ $\displaystyle\Gamma,\Delta\hskip 0.5pt\vdash\hskip 0.5ptB$ | tensy $\displaystyle\penalty 1\Gamma\hskip 0.5pt\vdash\hskip 0.5ptA$ $\ \forall^{i}$ $\displaystyle\Gamma\hskip 0.5pt\vdash\hskip 0.5pt\forall zA$
---|---|---|---
tensy $\displaystyle\penalty 1\Gamma\hskip 0.5pt\vdash\hskip 0.5ptA\land B$ $\ \mathnormal{\land^{\\!e}_{l}}$ $\displaystyle\Gamma\hskip 0.5pt\vdash\hskip 0.5ptA$ | tensy $\displaystyle\penalty 1\Delta\hskip 0.5pt\vdash\hskip 0.5ptA\land B$ $\ \mathnormal{\land^{\\!e}_{r}}$ $\displaystyle\Delta\hskip 0.5pt\vdash\hskip 0.5ptB$ | tensy $\displaystyle\penalty 1\Gamma\hskip 0.5pt\vdash\hskip 0.5ptA\quad\Delta\hskip 0.5pt\vdash\hskip 0.5ptB$ $\ \mathnormal{\land^{\\!i}}$ $\displaystyle\Gamma,\Delta\hskip 0.5pt\vdash\hskip 0.5ptA\land B$ | tensy $\displaystyle\penalty 1\Gamma\hskip 0.5pt\vdash\hskip 0.5pt\forall zA$ $\ \forall^{e}$ $\displaystyle\Gamma\hskip 0.5pt\vdash\hskip 0.5ptA[t/z]$
Table 2: Logical rules, with $z\not\in{\sf FV}(\Gamma)$ at $\forall^{i}$ and certain explicit contractions at $\mathnormal{\rightarrow^{\\!e}}$ and $\mathnormal{\land^{\\!i}}$ tensy $\displaystyle\penalty 1\Gamma\hskip 0.5pt\vdash\\!\\!\\!\raisebox{-2.58334pt}{\scriptsize$l$}\,A$ $\ \forall_{\\!\pm}^{i}$ $\displaystyle\Gamma\hskip 0.5pt\vdash\\!\\!\\!\raisebox{-2.58334pt}{\scriptsize$l$}\,\forall_{\\!\pm}zA$ | tensy $\displaystyle\penalty 1\Gamma\hskip 0.5pt\vdash\\!\\!\\!\raisebox{-2.58334pt}{\scriptsize$l$}\,A$ $\ \forall_{\\!+}^{i}$ $\displaystyle\Gamma\hskip 0.5pt\vdash\\!\\!\\!\raisebox{-2.58334pt}{\scriptsize$l$}\,\forall_{\\!+}zA$ | tensy $\displaystyle\penalty 1\Gamma\hskip 0.5pt\vdash\\!\\!\\!\raisebox{-2.58334pt}{\scriptsize$l$}\,A$ $\ \forall_{\\!-}^{i}$ $\displaystyle\Gamma\hskip 0.5pt\vdash\\!\\!\\!\raisebox{-2.58334pt}{\scriptsize$l$}\,\forall_{\\!-}zA$ | tensy $\displaystyle\penalty 1\Gamma\hskip 0.5pt\vdash\\!\\!\\!\raisebox{-2.58334pt}{\scriptsize$l$}\,A$ $\ \forall_{\\!\emptyset}^{i}$ $\displaystyle\Gamma\hskip 0.5pt\vdash\\!\\!\\!\raisebox{-2.58334pt}{\scriptsize$l$}\,\forall_{\\!\emptyset}zA$
---|---|---|---
tensy $\displaystyle\penalty 1\Gamma\hskip 0.5pt\vdash\\!\\!\\!\raisebox{-2.58334pt}{\scriptsize$l$}\,\forall_{\\!\pm}zA$ $\ \forall_{\\!\pm}^{e}$ $\displaystyle\Gamma\hskip 0.5pt\vdash\\!\\!\\!\raisebox{-2.58334pt}{\scriptsize$l$}\,A[t/z]$ | tensy $\displaystyle\penalty 1\Gamma\hskip 0.5pt\vdash\\!\\!\\!\raisebox{-2.58334pt}{\scriptsize$l$}\,\forall_{\\!+}zA$ $\ \forall_{\\!+}^{e}$ $\displaystyle\Gamma\hskip 0.5pt\vdash\\!\\!\\!\raisebox{-2.58334pt}{\scriptsize$l$}\,A[t/z]$ | tensy $\displaystyle\penalty 1\Gamma\hskip 0.5pt\vdash\\!\\!\\!\raisebox{-2.58334pt}{\scriptsize$l$}\,\forall_{\\!-}zA$ $\ \forall_{\\!-}^{e}$ $\displaystyle\Gamma\hskip 0.5pt\vdash\\!\\!\\!\raisebox{-2.58334pt}{\scriptsize$l$}\,A[t/z]$ | tensy $\displaystyle\penalty 1\Gamma\hskip 0.5pt\vdash\\!\\!\\!\raisebox{-2.58334pt}{\scriptsize$l$}\,\forall_{\\!\emptyset}zA$ $\ \forall_{\\!\emptyset}^{e}$ $\displaystyle\Gamma\hskip 0.5pt\vdash\\!\\!\\!\raisebox{-2.58334pt}{\scriptsize$l$}\,A[t/z]$
Table 3: Additional rules for $\sf NA_{\mathnormal{l}}\hskip 0.5pt,$ with extra restrictions on $\forall_{\\!+}^{i}\hskip 0.5pt,$ $\forall_{\\!-}^{i}$ and $\forall_{\\!\emptyset}^{i}$ tensy $\displaystyle\penalty 1\Delta,\hskip 0.5pta\\!:\\!A\hskip 0.5pt,\hskip 0.5pta\\!:\\!A\hskip 0.5pt\hskip 0.5pt\vdash\hskip 0.5ptB$ C $\displaystyle\Delta,\hskip 0.5pta\\!:\\!A\hskip 0.5pt\hskip 0.5pt\vdash\hskip 0.5ptB$ | tensy $\displaystyle\penalty 1\Delta,\hskip 0.5pta\\!:\\!A\hskip 0.5pt,\hskip 0.5pta\\!:\\!A\hskip 0.5pt\hskip 0.5pt\vdash\\!\\!\\!\raisebox{-2.58334pt}{\scriptsize$l$}\,B$ $\mathcal{C}_{l}$ $\displaystyle\Delta,\hskip 0.5pta\\!:\\!A\hskip 0.5pt\hskip 0.5pt\vdash\\!\\!\\!\raisebox{-2.58334pt}{\scriptsize$l$}\,B$
---|---
Table 4: Contraction anti-rules C for ${\sf NA}$ and (restricted) $\mathcal{C}_{l}$ for $\sf NA_{\mathnormal{l}}$ tensy $\displaystyle\penalty 1\Gamma\hskip 0.5pt\vdash\hskip 0.5ptA(\mathtt{T})\quad\Delta\hskip 0.5pt\vdash\hskip 0.5ptA(\mathtt{F})$ $\ \mathtt{Ind}_{o}$ $\displaystyle\Gamma,\Delta\hskip 0.5pt\vdash\hskip 0.5ptA(b)$ | tensy $\displaystyle\penalty 1\Gamma\hskip 0.5pt\vdash\hskip 0.5ptA(\mathtt{0})\quad\Delta\hskip 0.5pt\vdash\hskip 0.5ptA(n)\rightarrow A(\mathtt{S}n)$ $\ \mathtt{Ind}_{\iota}$ $\displaystyle\Gamma,\Delta\hskip 0.5pt\vdash\hskip 0.5ptA(n)$
---|---
Table 5: Induction rules, with $\Gamma\uplus\Delta$ instead of
‘$\Gamma,\Delta$’ and $\Delta$ restricted at $\mathtt{Ind}_{\iota}$ of $\sf
NA_{\mathnormal{l}}$
## 5 Appendix
We use a special Natural Deduction (abbreviated “ND”) presentation of our
systems, where proofs are represented as sequents $\Gamma\hskip
0.5pt\vdash\hskip 0.5ptB\hskip 0.5pt,$ meaning that formula $B$ is the root of
the ND tree whose leaves $\Gamma$ are typed assumption variables (abbreviated
“avars”) $\hskip 0.5pta\\!:\\!A\hskip 0.5pt\hskip 0.5pt.$ Here formula $A$ is
the type of the avar $a\hskip 0.5pt,$ and $\Gamma$ is a multiset (since there
may be more leaves labeled with the same $\hskip 0.5pta\\!:\\!A\hskip 0.5pt$).
### 5.1 The verifying system ${\sf NA}$
The logical rules of system ${\sf NA}$ are presented in Table 2 , with the
usual restriction on universal quantifier introduction $\forall^{i}$ that
$\begin{array}[]{rcl}z&\not\in&{\sf FV}(\Gamma)\hskip 1.0pt\mathnormal{\
:\equiv\ }\bigcup_{\hskip 0.5pta:A\hskip 0.5pt\in\Gamma}{\sf
FV}(A)\end{array}$
At $\mathnormal{\rightarrow^{\\!i}}\hskip 0.5pt,$ $[\hskip
0.5pta\\!:\\!A\hskip 0.5pt]$ denotes the multisubset of all occurrences of
$\hskip 0.5pta\\!:\\!A\hskip 0.5pt$ in the multiset of assumptions of the
premise sequent of $\mathnormal{\rightarrow^{\\!i}}\hskip 0.5pt.$ Thus $\hskip
0.5pta\\!:\\!A\hskip 0.5pt\not\in\Gamma\hskip 0.5pt,$ hence $\hskip
0.5pta\\!:\\!A\hskip 0.5pt$ is no longer an assumption in the conclusion
sequent of $\mathnormal{\rightarrow^{\\!i}}\hskip 0.5pt.$ In the ND tree, this
means that all the leaves labeled $\hskip 0.5pta\\!:\\!A\hskip 0.5pt$ are
inactivated (or “discharged” as one usually says in Natural Deduction
terminology).
Whereas in ${\sf NA}$ alone we could have safely let all contractions be
handled implicitly at $\mathnormal{\rightarrow^{\\!i}}\hskip 0.5pt,$ in
relationship with the architecture of input system $\sf NA_{\mathnormal{l}}$
(see Section 5.2) we are compelled to introduce for ${\sf NA}$ the
_contraction anti-rule_ C in association with $\mathcal{C}_{l}$ of $\sf
NA_{\mathnormal{l}}$, see Table 4. We refer to contraction as “anti-rule”,
rather than “rule” because, despite the sequent-like representation of our
calculi, in fact our formalisms are ND and in the ND directed tree the
representation of explicit contractions is by convergent arrows that go in the
direction which is reverse to the direction of all the other rules.
We find it convenient to introduce induction for booleans and naturals as the
rules presented in Table 5 . Here we assume that the induction variables
$b^{o}$ and respectively $n^{\iota}$ do not occur freely in $\Gamma\hskip
0.5pt,$ nor $\Delta\hskip 0.5pt,$ and that they do occur in the formula
$A\hskip 0.5pt.$
Computation in ${\sf NA}$ is expressed via the usual $\beta$-reduction rule
$(\lambda x.t)s\hookrightarrow t[x\mapsto s]\hskip 0.5pt,$ plus rewrite rules
defining the computational meaning of $\mathtt{If}$ and $\mathtt{R}\hskip
0.5pt:$
$\begin{array}[]{rlrl}\mathtt{If}\,\mathtt{T}\,s\,t&\hookrightarrow\;s&\qquad\qquad\mathtt{R}\,\mathtt{0}\,s\,t&\hookrightarrow\;s\\\\[2.0pt]
\mathtt{If}\,\mathtt{F}\,s\,t&\hookrightarrow\;t&\qquad\qquad\mathtt{R}\,(\mathtt{S}n)\,s\,t&\hookrightarrow\;t\,n\,(\mathtt{R}\,n\,s\,t)\end{array}$
Since this typed term system is confluent and strongly normalizing (cf. [17]),
we are free not to fix a particular evaluation strategy. For simplicity, we
assume that all terms occurring in proofs are automatically in normal form. In
fact, normalization is necessary only when matching terms in formulas. We only
avoid introducing equality axioms $\mathtt{AxEQL}$ as in [7] and skip the
corresponding easy applications of $\mathtt{CmpAx}$. When building proofs,
some computation is thus carried out implicitly, behind the scene.
Using recursion at higher types we can define any provably total function of
ground arithmetic, including decidable predicates such as equality
$\mathtt{Eq}_{\hskip 0.5pto}$ for booleans and $\mathtt{Eq}_{\hskip
0.5pt\iota}$ for natural numbers:
$\begin{array}[]{r@{\ \;}c@{\ \;}l}\mathtt{Eq}_{\hskip
0.5pto}^{ooo}&\mathnormal{\hskip 0.5pt:\equiv\hskip 0.5pt}&\lambda
x.\mathtt{If}\,x\,(\lambda y.y)\,(\lambda
y.\mathtt{If}\,y\,\mathtt{F}\,\mathtt{T})\\\\[3.0pt] \mathtt{Eq}_{\hskip
0.5pt\iota}^{\iota\iota o}&\mathnormal{\hskip 0.5pt:\equiv\hskip
0.5pt}&\lambda x.\mathtt{R}\,x\,\big{(}\lambda
y.\mathtt{R}\,y\,\mathtt{T}\,(\lambda
n,q^{o}.\mathtt{F})\big{)}\,\big{(}\lambda m,p^{\iota
o},y.\mathtt{R}\,y\,\mathtt{F}\,(\lambda n,q^{o}.p\,n)\big{)}\end{array}$
The $\mathtt{at}(\cdot)$ construction allows us to view boolean programs as
decidable predicates. Given $\mathtt{Ind}_{o}\hskip 0.5pt,$ its logical
meaning is settled by the truth axiom $\mbox{$\mathtt{T}\mathtt{Ax}$}\hskip
0.5pt,$ see Table 1 . In this way we can define predicate equality at base
types as $s=_{\sigma}t\ \mathnormal{\hskip 0.5pt:\equiv\hskip 0.5pt}\
\mathtt{at}(\mathtt{Eq}\,s\,t)$ for $\sigma\in\left\\{o,\iota\right\\}$ and
further at higher types extensionally as usual $s=_{\rho\tau}t\
\mathnormal{\hskip 0.5pt:\equiv\hskip 0.5pt}\forall
x^{\rho}(sx=_{\tau}tx)\hskip 0.5pt.$ It is straightforward to prove by
induction on $\rho$ that $=_{\rho}$ is reflexive, symmetric and transitive at
any type $\rho\hskip 0.5pt.$
To complete our system, we must include in ${\sf NA}$ also the compatibility
(i.e., extensionality) axiom $\mathtt{CmpAx}\hskip 0.5pt,$ see Table 1 . Note
that ex falso quodlibet ($\mathtt{EFQ}$) $\hskip 0.5pt\bot\rightarrow A\hskip
1.0pt$ and stability ($\mathtt{Stab}$) $\hskip 0.5pt\lnot\lnot A\hskip
0.5pt\rightarrow\hskip 0.5ptA\hskip 1.0pt$ are fully provable in ${\sf NA}$
(cf. [17], by induction on $A\hskip 0.5pt,$ using $\mathtt{T}\mathtt{Ax}$ and
$\mathtt{Ind}_{o}$).
### 5.2 The input system $\sf NA_{\mathnormal{l}}$
Light formulas $\mathcal{F}_{l}$ are built over usual formulas $\mathcal{F}$
of ${\sf NA}$ by adding the three light universal quantifiers: the non-
computational $\forall_{\\!\emptyset}$ and the two semi-computational
$\forall_{\\!+}$ and $\forall_{\\!-}\hskip 0.5pt.$ In order to stress the
distinction of $\sf NA_{\mathnormal{l}}$ from ${\sf NA}$ it is convenient to
rename ${\sf NA}$’s $\forall$ to $\forall_{\\!\pm}$ in $\sf
NA_{\mathnormal{l}}$ (which marks the whole computational content, both
positive and negative)
$\displaystyle\mathcal{F}_{l}\quad$ $\displaystyle A,B$ $\displaystyle\
\mathtt{::=}\quad\mathtt{at}(t^{o})\;|\;A\rightarrow B\;|\;A\land
B\;|\;\forall_{\diamond}\,x^{\rho}A\quad\text{ for
}\diamond\in\left\\{\emptyset,+,-,\pm\right\\}$
Thus, system $\sf NA_{\mathnormal{l}}$ refines the clone of ${\sf NA}$ (also
with $\mathtt{CMP}$ for $\mathtt{CmpAx}$ and $\mathcal{C}_{l}$ for C) with
introduction and elimination rules for the light quantifiers (see Table 3).
These are copies of the clone rules $\forall_{\\!\pm}^{e}$ and
$\forall_{\\!\pm}^{i}\hskip 0.5pt,$ but with the usual restriction ($\pm$) on
$\forall_{\\!\pm}^{i}$ that $z\not\in{\sf FV}(\Gamma)$ enhanced with the
following conditions referring to the LD-interpretation
* ($+$)
at the $\forall_{\\!+}^{i}$ rule, $z$ may be used computationally only
positively, i.e.,
$z$ must not be free in the _challengers_ of the LD-translation of
$\Gamma\hskip 0.5pt\hskip 0.5pt.$
* ($-$)
at the $\forall_{\\!-}^{i}$ rule, $z$ may be used computationally only
negatively, i.e.,
$z$ must not be free in the _witnesses_ of the LD-translation of $A\hskip
0.5pt\hskip 0.5pt.$
* ($\emptyset$)
at $\forall_{\\!\emptyset}^{i}\hskip 0.5pt,$ $z$ may not be used
computationally at all, i.e., both ($+$) and ($-$).
Notice that the restrictions ($+$), ($-$) and ($\emptyset$) assume knowledge
of the LD-interpretation of whole proofs, in their full depth, thus forcing
the definition of $\sf NA_{\mathnormal{l}}$ proofs to go inductively in
parallel with the LD-extraction of part of their computational content (namely
free variables of the extracted terms). We simultaneously define the classes
of _realization irrelevant_ $A_{\oplus}$ and _refutation irrelevant_
$A_{\ominus}$ formulas as follows:
$\displaystyle A_{\oplus},B_{\oplus}$ $\displaystyle\ \mathtt{::=}\quad$
$\displaystyle\mathtt{at}(t)\;|\;A_{\oplus}\land
B_{\oplus}\;|\;A_{\ominus}\rightarrow B_{\oplus}\;|\;\forall_{\diamond}x\hskip
1.0ptA_{\oplus}\quad\text{ for }\diamond\in\left\\{\emptyset,+,-,\pm\right\\}$
$\displaystyle A_{\ominus},B_{\ominus}$ $\displaystyle\ \mathtt{::=}\quad$
$\displaystyle\mathtt{at}(t)\;|\;A_{\ominus}\land
B_{\ominus}\;|\;A_{\oplus}\rightarrow
B_{\ominus}\;|\;\forall_{\diamond}x\hskip 1.0ptA_{\ominus}\quad\text{ for
}\diamond\in\left\\{\emptyset,+\right\\}$
One necessary change when adopting principles from ${\sf NA}$ is to replace
$\mathtt{CmpAx}$ with a weak compatibility rule. This is because Dialectica is
unable to interpret full extensionality (cf. [11, 20]). We here employ an
upgraded variant of the $\mathtt{CMP}$ rule from [7]:
${tensy\vbox{\hbox spread0.0pt{\hskip 0.0pt plus 0.0001fil\hbox{\kern
14.41693pt\hbox{$\displaystyle\penalty 1\Gamma_{\ominus}\hskip
0.5pt\vdash\\!\\!\\!\raisebox{-2.58334pt}{\scriptsize$l$}\,s=_{\rho}t$}}\hskip
0.0pt plus 0.0001fil}\hbox{\hbox{\kern 0.0pt\vrule
height=0.25002pt,depth=0.25002pt,width=72.24852pt\hbox{\kern 3.00003pt$\
\mathtt{CMP}_{\rho}$}}}\hbox{\kern
0.0pt\hbox{$\displaystyle\Gamma_{\ominus}\hskip
0.5pt\vdash\\!\\!\\!\raisebox{-2.58334pt}{\scriptsize$l$}\,B(s)\rightarrow
B(t)$}}}}$
where all formulas in $\Gamma_{\ominus}$ are refutation irrelevant.
The computationally irrelevant contractions in $\sf NA_{\mathnormal{l}}$ can
safely be handled implicitly at $\mathnormal{\rightarrow^{\\!i}}\hskip 0.5pt.$
The situation is different for those contractions whose formula is refutation
relevant (i.e., the computationally relevant contractions), as we want to
automatically ensure that their translation is decidable (instead of leaving
the task of decidability check to the user). We achieve this by including in
$\sf NA_{\mathnormal{l}}$ the _contraction anti-rule_ $\mathcal{C}_{l}$ (see
Table 4) for all formulas $A$ that are refutation relevant and ($\bigstar$) do
not contain any $\forall_{\\!+}\hskip 0.5pt,$ nor
$\forall_{\\!\emptyset}\hskip 0.5pt.$ This triggers the addition to ${\sf NA}$
of an explicit (unrestricted) contraction anti-rule C which is needed in the
construction of the verifying proof (it only applies to quantifier-free
formulas $|A|$). The restriction $\bigstar$ ensures that all contraction
formulas that require at least one challenger term for their LD-interpretation
will have quantifier-free (hence decidable) LD-translations. Their
decidability is necessary for attaining soundness. Being a purely syntactical
criterion, $\bigstar$ does not admit formulas whose LD-translations contain
quantifiers, but could nevertheless be decidable, e.g.,
$\mathtt{Odd}(x)\mathnormal{\;\equiv\;}\forall y(2y\neq x)\hskip 0.5pt.$
Moreover, in order to avoid having any computationally relevant contractions
implicit in $\mathnormal{\rightarrow^{\\!i}}\hskip 0.5pt,$ we constrain the
deduction rules of $\sf NA_{\mathnormal{l}}$ to disallow multiple occurrences
of refutation relevant assumptions in any of the premise sequents. Thus,
whenever a double occurrence of a refutation relevant assumption is created in
a conclusion sequent by one of the binary rules of $\sf
NA_{\mathnormal{l}}\hskip 0.5pt,$ such sequent cannot be directly a premise
for the application of an(other) $\sf NA_{\mathnormal{l}}$ rule: the anti-rule
$\mathcal{C}_{l}$ must be applied first, in order to eliminate the critical
double. If $\bigstar$ is not satisfied and yet $\hskip 0.5pta\\!:\\!A\hskip
0.5pt$ is a refutation relevant assumption occurring at least twice in some
conclusion sequent, this is a dead end: such sequent can only be the root of
the $\sf NA_{\mathnormal{l}}$ proof-tree.
While $\hskip 1.0pt\mathtt{EFQ}:\,\bot\hskip 0.5pt\rightarrow\hskip
0.5ptA\hskip 1.0pt$ remains fully provable also in $\sf NA_{\mathnormal{l}}$
(for all formulas $A\in\mathcal{F}_{l}$) the situation changes for $\hskip
1.0pt\mathtt{Stab}:\,\lnot\lnot A\hskip 0.5pt\rightarrow\hskip 0.5ptA\hskip
1.0pt$ in the case of many formulas $A$ that feature light quantifiers in
certain places. As noted in [7], the usual proof in ${\sf NA}$ of
$\mathtt{Stab}$ (constructed by induction on $A$) makes unavoidably use of
contractions over $\neg\neg(B\land C)$ for subformulas $(B\land C)$ of
$A\hskip 0.5pt,$ and these are subject to the $\bigstar$ restriction for
refutation relevant $B\land C\hskip 0.5pt.$ Even when such $B\land C$ obey
$\bigstar$, they may lead to the failure of restrictions ($+$), ($-$) or
($\emptyset$). On the other hand $\mathtt{Stab}$ is provable in $\sf
NA_{\mathnormal{l}}$ for $A\in\mathcal{F}$ or $A$ conjunction-free.
### 5.3 The light Dialectica interpretation (LD-interpretation)
With each formula $A$ of $\sf NA_{\mathnormal{l}}$ we associate its LD-
translation: a not necessarily quantifier-free formula
$|A|^{\boldsymbol{x}}_{\boldsymbol{y}}$ of ${\sf NA}$ where
$\boldsymbol{x},\boldsymbol{y}$ are tuples of fresh variables, not appearing
in $A\hskip 0.5pt.$ The variables $\boldsymbol{x}$ in the superscript are
called the _witness variables_ , while the subscript variables
$\boldsymbol{y}$ are called the _challenge variables_. Terms $\boldsymbol{t}$
substituting witness variables (like $|A|^{\boldsymbol{t}}_{\boldsymbol{y}}$)
are called _realizing terms_ or “witnesses” and terms $\boldsymbol{s}$
substituting challenge variables (like
$|A|^{\boldsymbol{x}}_{\boldsymbol{s}}$) are called _refuting terms_ or
“challengers”. Intuitively, the LD-interpretation of $A$ can be viewed as a
game in which first Eloise ($\exists$) and then Abelard ($\forall$) make one
move each by playing type-corresponding objects $\boldsymbol{t}$ and
$\boldsymbol{s}$ for the tuples $\boldsymbol{x}$ and respectively
$\boldsymbol{y}\hskip 0.5pt.$ Formula $|A|^{\boldsymbol{x}}_{\boldsymbol{y}}$
specifies the “adjudication relation”, here not necessarily decidable: Eloise
wins iff $\,{\sf NA}\hskip 0.5pt\vdash\hskip
0.5pt|A|^{\boldsymbol{t}}_{\boldsymbol{s}}\hskip 0.5pt\hskip 0.5pt.$ In our
light context as well, Eloise has a winning move whenever $A$ is provable in
$\sf NA_{\mathnormal{l}}\hskip 0.5pt:$ the LD-interpretation will explicitly
provide it from the input $\sf NA_{\mathnormal{l}}$ proof of $A$ as a tuple of
witnesses $\boldsymbol{t}$ (s.t. ${\sf FV}(\boldsymbol{t})\subseteq{\sf
FV}(A)$) together with the _verifying proof_ in ${\sf NA}$ of $\hskip
1.0pt\forall\boldsymbol{y}\hskip
0.5pt|A|^{\boldsymbol{t}}_{\boldsymbol{y}}\hskip 0.5pt$ (Eloise wins by
$\boldsymbol{t}$ regardless of the instances $\boldsymbol{s}$ for Abelard’s
$\boldsymbol{y}$).
###### Definition 5 (LD-translation of formulas)
The interpretation does not change atomic formulas, i.e.,
$|\mathtt{at}(t^{o})|\mathnormal{\hskip 0.5pt:\equiv\hskip
0.5pt}\mathtt{at}(t^{o})\hskip 0.5pt.$ Assuming
$|A|^{\boldsymbol{x}}_{\boldsymbol{y}}$ and
$|B|^{\boldsymbol{u}}_{\boldsymbol{v}}$ are already defined,
$\begin{array}[]{r@{\quad}c@{\quad}l}|A\land
B|^{\boldsymbol{x},\boldsymbol{u}}_{\boldsymbol{y},\boldsymbol{v}}&\mathnormal{\hskip
0.5pt:\equiv\hskip
0.5pt}&|A|^{\boldsymbol{x}}_{\boldsymbol{y}}\land|B|^{\boldsymbol{u}}_{\boldsymbol{v}}\\\\[6.0pt]
|A\rightarrow
B|^{\boldsymbol{f},\boldsymbol{g}}_{\boldsymbol{x},\boldsymbol{v}}&\mathnormal{\hskip
0.5pt:\equiv\hskip
0.5pt}&|A|^{\boldsymbol{x}}_{\boldsymbol{f}\boldsymbol{x}\boldsymbol{v}}\rightarrow|B|^{\boldsymbol{g}\boldsymbol{x}}_{\boldsymbol{v}}\quad.\end{array}$
The interpretation of the four universal quantifiers is (upon renaming, we
assume that quantified variables occur uniquely in a formula):
$\begin{array}[]{l@{\quad}c@{\quad}l@{\qqquad}l@{\quad}c@{\quad}l}|\forall_{\\!\pm}z\hskip
1.0ptA(z)|^{\boldsymbol{f}}_{z,\boldsymbol{y}}&\mathnormal{\hskip
0.5pt:\equiv\hskip
0.5pt}&|A(z)|^{\boldsymbol{f}z}_{\boldsymbol{y}}\hfil\qquad\quad&|\forall_{\\!+}z\hskip
1.0ptA(z)|^{\boldsymbol{f}}_{\boldsymbol{y}}&\mathnormal{\hskip
0.5pt:\equiv\hskip 0.5pt}&\forall z\hskip
1.0pt|A(z)|^{\boldsymbol{f}z}_{\boldsymbol{y}}\\\\[6.0pt]
|\forall_{\\!-}z\hskip
1.0ptA(z)|^{\boldsymbol{x}}_{z,\boldsymbol{y}}&\mathnormal{\hskip
0.5pt:\equiv\hskip
0.5pt}&|A(z)|^{\boldsymbol{x}}_{\boldsymbol{y}}\hfil\qquad\quad&|\forall_{\\!\emptyset}z\hskip
1.0ptA(z)|^{\boldsymbol{x}}_{\boldsymbol{y}}&\mathnormal{\hskip
0.5pt:\equiv\hskip 0.5pt}&\forall z\hskip
1.0pt|A(z)|^{\boldsymbol{x}}_{\boldsymbol{y}}\end{array}$
Since $|\bot|\mathnormal{\hskip 0.5pt\equiv\hskip 0.5pt}\bot\hskip 0.5pt,$ we
get
$\begin{array}[]{l@{\quad}c@{\quad}l@{\qqquad}l@{\quad}c@{\quad}l}|\lnot
A|^{\boldsymbol{f}}_{\boldsymbol{x}}&\mathnormal{\hskip 0.5pt\equiv\hskip
0.5pt}&\lnot|A|^{\boldsymbol{x}}_{\boldsymbol{f}\boldsymbol{x}}\hfil\qquad\quad&|\lnot\lnot
A|^{\boldsymbol{f}}_{\boldsymbol{g}}&\mathnormal{\hskip 0.5pt\equiv\hskip
0.5pt}&\lnot\lnot|A|^{\boldsymbol{f}\boldsymbol{g}}_{\boldsymbol{g}(\boldsymbol{f}\boldsymbol{g})}\end{array}$
It is straightforward to compute that
$\begin{array}[]{l@{\ }c@{\ }l@{\qquad}l@{\ }c@{\
}l}|\widetilde{\exists}_{\pm}z\hskip
1.0ptA(z)|^{Z,\boldsymbol{f}}_{\boldsymbol{g}}&\mathnormal{\hskip
0.5pt:\equiv\hskip
0.5pt}&\lnot\lnot|A(Z\boldsymbol{g})|^{\boldsymbol{f}\boldsymbol{g}}_{\boldsymbol{g}(Z\boldsymbol{g})(\boldsymbol{f}\boldsymbol{g})}&|\widetilde{\exists}_{+}z\hskip
1.0ptA(z)|^{\boldsymbol{f}}_{\boldsymbol{g}}&\mathnormal{\hskip
0.5pt:\equiv\hskip 0.5pt}&\widetilde{\exists}z\hskip
1.0pt|A(z)|^{\boldsymbol{f}\boldsymbol{g}}_{\boldsymbol{g}z(\boldsymbol{f}\boldsymbol{g})}\\\\[6.0pt]
|\widetilde{\exists}_{-}z\hskip
1.0ptA(z)|^{Z,\boldsymbol{f}}_{\boldsymbol{g}}&\mathnormal{\hskip
0.5pt:\equiv\hskip
0.5pt}&\lnot\lnot|A(Z\boldsymbol{g})|^{\boldsymbol{f}\boldsymbol{g}}_{\boldsymbol{g}(\boldsymbol{f}\boldsymbol{g})}&|\widetilde{\exists}_{\emptyset}z\hskip
1.0ptA(z)|^{\boldsymbol{f}}_{\boldsymbol{g}}&\mathnormal{\hskip
0.5pt:\equiv\hskip 0.5pt}&\widetilde{\exists}z\hskip
1.0pt|A(z)|^{\boldsymbol{f}\boldsymbol{g}}_{\boldsymbol{g}(\boldsymbol{f}\boldsymbol{g})}\end{array}$
The length and types of the witnessing and challenging tuples are uniquely
determined.
### 5.4 Light Dialectica treatment of extensionality
We here give the LD-interpretation of the weak compatibility rule
${tensy\vbox{\hbox spread0.0pt{\hskip 0.0pt plus 0.0001fil\hbox{\kern
14.41693pt\hbox{$\displaystyle\penalty 1\Gamma_{\ominus}\hskip
0.5pt\vdash\\!\\!\\!\raisebox{-2.58334pt}{\scriptsize$l$}\,s=_{\rho}r$}}\hskip
0.0pt plus 0.0001fil}\hbox{\hbox{\kern 0.0pt\vrule
height=0.25002pt,depth=0.25002pt,width=73.42677pt\hbox{\kern 3.00003pt$\
\mathtt{CMP}_{\rho}$}}}\hbox{\kern
0.0pt\hbox{$\displaystyle\Gamma_{\ominus}\hskip
0.5pt\vdash\\!\\!\\!\raisebox{-2.58334pt}{\scriptsize$l$}\,B(s)\rightarrow
B(r)$}}}}$
where all formulas in $\Gamma_{\ominus}$ are refutation irrelevant, i.e., the
negative position in their LD-translation is empty. By definition of equality
at higher types, $s=_{\rho}r$ is
$\forall\boldsymbol{z}.\,s\boldsymbol{z}=r\boldsymbol{z}$, a purely universal
formula. We are given that
$\begin{array}[]{rcl}\hskip
0.5pta_{1}\\!:\\!|A_{1}|^{\boldsymbol{x}_{1}}_{\boldsymbol{t}_{1}}\hskip
0.5pt,\ldots,\hskip
0.5pta_{n}\\!:\\!|A_{n}|^{\boldsymbol{x}_{n}}_{\boldsymbol{t}_{n}}&\hskip
0.5pt\vdash&|A_{0}|^{\boldsymbol{t}_{0}}_{\boldsymbol{x}_{0}}\end{array}$
where $|\Gamma_{\ominus}|\mathnormal{\hskip 0.5pt\equiv\hskip
0.5pt}\left\\{a_{1},\ldots,a_{n}\right\\}$,
$\boldsymbol{t}_{0}\mathnormal{\hskip 0.5pt\equiv\hskip
0.5pt}\boldsymbol{t}_{1}\mathnormal{\hskip 0.5pt\equiv\hskip
0.5pt}\ldots\boldsymbol{t}_{n}\mathnormal{\hskip 0.5pt\equiv\hskip
0.5pt}\sqcup$ (empty tuple), $A_{0}$ is $s=_{\rho}r$ and $\boldsymbol{x}_{0}$
corresponds to $\boldsymbol{z}$, thus the above is more conveniently rewritten
as
$\begin{array}[]{rcl}\hskip
0.5pta_{1}\\!:\\!|A_{1}|^{\boldsymbol{x}_{1}}\hskip 0.5pt,\ldots,\hskip
0.5pta_{n}\\!:\\!|A_{n}|^{\boldsymbol{x}_{n}}&\hskip
0.5pt\vdash&s\boldsymbol{x}_{0}=r\boldsymbol{x}_{0}\end{array}$
To this we can apply the generalization rule, as $\boldsymbol{x}_{0}$ are not
free in the translated context $|\Gamma_{\ominus}|$; indeed,
$\boldsymbol{x}_{0}$ are fresh variables and they could have appeared free
only via terms $\boldsymbol{t}_{1}$,…,$\boldsymbol{t}_{n}$, were these not
empty tuples (hence the need for restricting the original context). We thus
obtain $|\Gamma_{\ominus}|\,\hskip 0.5pt\vdash\hskip 0.5pt\,s=r$ and further
apply the extensionality Axiom to get $|\Gamma_{\ominus}|\,\hskip
0.5pt\vdash\hskip 0.5pt\,|B|(s)\rightarrow|B|(r)$. Note that the Axiom is
required here, as $|\Gamma_{\ominus}|$ may contain general formulas. With
$g\mathnormal{\ :\equiv\ }\lambda\boldsymbol{u}.\,\boldsymbol{u}$ and
$f\mathnormal{\ :\equiv\
}\lambda\boldsymbol{u},\boldsymbol{v}.\,\boldsymbol{v}$ we have thus
constructed a verifying proof
$\begin{array}[]{rcl}\hskip
0.5pta_{1}\\!:\\!|A_{1}|^{\boldsymbol{x}_{1}}\hskip 0.5pt,\ldots,\hskip
0.5pta_{n}\\!:\\!|A_{n}|^{\boldsymbol{x}_{n}}&\hskip
0.5pt\vdash&|B(s)|^{\boldsymbol{u}}_{f\boldsymbol{u}\boldsymbol{v}}\rightarrow|B(r)|^{g\boldsymbol{u}}_{\boldsymbol{v}}\
\,[\,\mathnormal{\hskip 0.5pt\equiv\hskip 0.5pt}\ \,|B(s)\rightarrow
B(r)|^{f,g}_{\boldsymbol{u},\boldsymbol{v}}\,]\end{array}$
The new realizing terms $f,g$ are closed, hence the free variable condition
trivially holds.
### 5.5 Light Dialectica treatment of induction for naturals
Since the induction rule (for naturals) corresponds to a virtually unbounded
number of contractions of each assumption from the step context $\Delta$ (cf.
[7], see Table 5), its clone in the system $\sf NA_{\mathnormal{l}}$ is
subject to a restriction like the one of $\mbox{$\mathcal{C}_{l}$}\hskip
0.5pt.$ Namely, we need to require that all refutation relevant avars in
$\Delta$ satisfy $\bigstar$. Moreover, since the contractions on
$a\in\Gamma\cap\Delta$ will be handled differently than for simple binary
rules like $\mathnormal{\rightarrow^{\\!e}}$ or
$\mathnormal{\land^{\\!i}}\hskip 0.5pt,$ it is more convenient to require that
naturals induction in $\sf NA_{\mathnormal{l}}$ implicitly contracts all its
refutation relevant assumptions (instead of using the explicit
$\mathcal{C}_{l}$) . We will use the notation $\Gamma\uplus\Delta$ for a
special multiset union in which refutation relevant assumptions appear only
once, even if they appear in both $\Gamma$ and $\Delta\hskip 0.5pt.$ Thus the
$\mathtt{Ind^{\iota}_{\mathnormal{l}}}$ rule of $\sf NA_{\mathnormal{l}}$ is
finally obtained by replacing ‘$\Gamma,\Delta$’ with ‘$\Gamma\uplus\Delta$’ in
the conclusion sequent of $\mathtt{Ind}_{\iota}\hskip 0.5pt.$
tensy $\displaystyle\penalty 1\Gamma\ \hskip
0.5pt\vdash\\!\\!\\!\raisebox{-2.58334pt}{\scriptsize$l$}\,\
A(\mathtt{0})\qquad\Delta\ \hskip
0.5pt\vdash\\!\\!\\!\raisebox{-2.58334pt}{\scriptsize$l$}\,\
A(n)\;\rightarrow\;A(\mathtt{S}n)$ $\ {\
\mathtt{Ind^{\iota}_{\mathnormal{l}}}}$ $\displaystyle\Gamma\uplus\Delta\
\hskip 0.5pt\vdash\\!\\!\\!\raisebox{-2.58334pt}{\scriptsize$l$}\,\ A(n)$
---
We are given
$\displaystyle|\Gamma|^{\boldsymbol{u}}_{\boldsymbol{\gamma}[\boldsymbol{y}]}$
$\displaystyle\hskip 0.5pt\vdash\hskip 0.5pt$
$\displaystyle|A(\mathtt{0})|^{\boldsymbol{r}}_{\boldsymbol{y}}$ (6)
and
$\displaystyle|\Delta|^{\boldsymbol{z}}_{\boldsymbol{\delta}[\boldsymbol{x};\boldsymbol{v}]}$
$\displaystyle\hskip 0.5pt\vdash\hskip 0.5pt$
$\displaystyle|A(n)|^{\boldsymbol{x}}_{\boldsymbol{t}\boldsymbol{x}\boldsymbol{v}}\
\rightarrow\ |A(\mathtt{S}n)|^{\boldsymbol{s}\boldsymbol{x}}_{\boldsymbol{v}}$
(7)
We show that
$\displaystyle\forall\boldsymbol{v}\hskip
1.0pt\big{(}\,|\Gamma\uplus\Delta|^{\,\boldsymbol{u}\uplus\boldsymbol{z}}_{\,\boldsymbol{\zeta}[n]\boldsymbol{v}}\;\rightarrow\;|A(n)|^{\boldsymbol{{t}^{\prime}}[n]}_{\boldsymbol{v}}\,\big{)}$
(8)
is a theorem of $\ {\sf NA}$ , where
$\displaystyle{t}^{\prime}[n]$ $\displaystyle\mathnormal{\ :\equiv\ }$
$\displaystyle\mathtt{R}\,n\,r\,(\lambda n.s)$ (9)
for every corresponding pair $\,\langle
r\in\boldsymbol{r}/s\in\boldsymbol{s}\rangle\,$ and
$\,\boldsymbol{\zeta}[n]\,$ will be constructed as functional terms depending
on $\,\boldsymbol{v}$ . We here intentionally use the same variable $\,n\,$
that occurs freely in $\,s\,$ and $\,t$ . Implicitly, just $\,{t}^{\prime}\,$
denotes $\,{t}^{\prime}[n]$ . Also $\,\boldsymbol{\zeta}\,$ will be
constructed as the collection of all $\,\boldsymbol{{\zeta}^{\prime}}\,$
(corresponding to $\,\Gamma\setminus\Delta\,$) and
$\,\boldsymbol{{\zeta}^{\prime\prime}}\,$ (corresponding to $\,\Delta\,$).
Let $\,\hskip 0.5ptb\\!:\\!B\hskip 0.5pt\,$ be a refutation relevant avar in
$\,\Gamma\uplus\Delta$ . Let
$\,\boldsymbol{{\gamma}^{\prime}}\in\boldsymbol{\gamma}\,$ and/or
$\,\boldsymbol{{\delta}^{\prime}}\in\boldsymbol{\delta}\,$ be the challengers
for $\;b\;$ in $\;\Gamma\;$ and/or $\,\Delta$ . If $\;b\;$ appears only in
$\,\Gamma\,$ (hence not in $\,\Delta\,$) we define
$\displaystyle\boldsymbol{{\zeta}^{\prime}}[n]$
$\displaystyle\mathnormal{\hskip 0.5pt:\equiv\hskip 0.5pt}$
$\displaystyle\mathtt{R}\,n\,(\lambda\boldsymbol{v}.\boldsymbol{{\gamma}^{\prime}}[\boldsymbol{v}])\,\big{(}\lambda
n,p,\boldsymbol{v}.p(\boldsymbol{t}\,\boldsymbol{{t}^{\prime}}\boldsymbol{v})\big{)}$
(10)
If $\ b\ $ appears in $\ \Delta$ , then the decidability of $\ |B|\ $ is
needed at each recursive step to equalize the terms $\
p(\boldsymbol{t}\,\boldsymbol{{t}^{\prime}}\boldsymbol{v})\ $ obtained by the
recursive call with the corresponding terms
$\boldsymbol{{\delta}^{\prime}}\hskip 0.5pt.$ Thus the right stop point of the
backwards construction is provided. In fact an implicit contraction over
$\,b\,$ happens at each inductive step and $\bigstar$ guarantees that
$\,|B|\,$ is decidable. For $\ b\in\Gamma\cap\Delta\ $ let
$\displaystyle\boldsymbol{{\zeta}^{\prime\prime}}[n]$
$\displaystyle\mathnormal{\hskip 0.5pt:\equiv\hskip 0.5pt}$
$\displaystyle\mathtt{R}\,n\,(\lambda\boldsymbol{v}.\boldsymbol{{\gamma}^{\prime}}[\boldsymbol{v}])\,\Big{(}\lambda
n,p,\boldsymbol{v}.\mathtt{If}(|B|^{\boldsymbol{{z}^{\prime}}}_{\boldsymbol{{\delta}^{\prime}}[\boldsymbol{{t}^{\prime}};\boldsymbol{v}]})\,\big{(}p(\boldsymbol{t}\,\boldsymbol{{t}^{\prime}}\boldsymbol{v})\big{)}\,\boldsymbol{{\delta}^{\prime}}[\boldsymbol{{t}^{\prime}};\boldsymbol{v}]\Big{)}$
(11)
and for $\ b\in\Delta\setminus\Gamma\ $ we define its
$\,\boldsymbol{{\zeta}^{\prime\prime}}[n]\,$ by replacing in (11) the
$\,\boldsymbol{{\gamma}^{\prime}}\,$ with canonical zeros. Here
$\,\boldsymbol{{z}^{\prime}}\,$ are the challenge variables corresponding to
formula $\,B$ . Notice that
$\displaystyle\hskip 0.5pt\vdash\hskip 0.5pt$ $\displaystyle\hskip
1.0pt{t}^{\prime}[\mathtt{S}n]\ =\ s\hskip 0.5pt{t}^{\prime}[n]$ (12)
$\displaystyle\hskip 0.5pt\vdash\hskip 0.5pt$ $\displaystyle\hskip
1.0pt\boldsymbol{{\zeta}^{\prime}}[\mathtt{S}n]\boldsymbol{v}\ =\
\boldsymbol{{\zeta}^{\prime}}[n](\boldsymbol{t}\,\boldsymbol{{t}^{\prime}}\boldsymbol{v})$
(13) $\displaystyle\hskip 0.5pt\vdash\hskip 0.5pt$ $\displaystyle\hskip
1.0pt\boldsymbol{{\zeta}^{\prime\prime}}[\mathtt{S}n]\boldsymbol{v}\ =\
\,\mathtt{If}\,(|B|^{\boldsymbol{{z}^{\prime}}}_{\boldsymbol{{\delta}^{\prime}}[\boldsymbol{{t}^{\prime}};\boldsymbol{v}]})\;\big{(}\boldsymbol{{\zeta}^{\prime\prime}}[n](\boldsymbol{t}\,\boldsymbol{{t}^{\prime}}\boldsymbol{v})\big{)}\
\,\boldsymbol{{\delta}^{\prime}}[\boldsymbol{{t}^{\prime}};\boldsymbol{v}]$
(14)
We attempt to extend (13) to the whole $\,\boldsymbol{\zeta}\,$ by proving
from (14) the following
$\displaystyle|B|^{\boldsymbol{{z}^{\prime}}}_{\boldsymbol{{\zeta}^{\prime\prime}}[\mathtt{S}n]\boldsymbol{v}}$
$\displaystyle\hskip 0.5pt\vdash\hskip 0.5pt$
$\displaystyle\boldsymbol{{\zeta}^{\prime\prime}}[\mathtt{S}n]\boldsymbol{v}\
=\
\boldsymbol{{\zeta}^{\prime\prime}}[n](\boldsymbol{t}\,\boldsymbol{{t}^{\prime}}\boldsymbol{v})$
(15)
We obtain this as an immediate consequence of
$\displaystyle|B|^{\boldsymbol{{z}^{\prime}}}_{\boldsymbol{{\zeta}^{\prime\prime}}[\mathtt{S}n]\boldsymbol{v}}$
$\displaystyle\hskip 0.5pt\vdash\hskip 0.5pt$
$\displaystyle|B|^{\boldsymbol{{z}^{\prime}}}_{\boldsymbol{{\delta}^{\prime}}[\boldsymbol{{t}^{\prime}};\boldsymbol{v}]}$
(16)
Assuming
$\,\neg|B|^{\boldsymbol{{z}^{\prime}}}_{\boldsymbol{{\delta}^{\prime}}[\boldsymbol{{t}^{\prime}};\boldsymbol{v}]}\,$
by (14) we get
$\,\boldsymbol{{\zeta}^{\prime\prime}}[\mathtt{S}n]\boldsymbol{v}=\boldsymbol{{\delta}^{\prime}}[\boldsymbol{{t}^{\prime}};\boldsymbol{v}]\,$
hence
$\,\neg|B|^{\boldsymbol{{z}^{\prime}}}_{\boldsymbol{{\zeta}^{\prime\prime}}[\mathtt{S}n]\boldsymbol{v}}\,$
and
thus (16) follows via $\,\mathtt{Stab}\,$ (which is fully available in the
verifying system).
We now prove (8) by an assumptionless induction on $\,n$ . Let
$\,\boldsymbol{\zeta^{*}}\,$ be the collection of all
$\,\boldsymbol{{\zeta}^{\prime}}\,$ and those
$\,\boldsymbol{{\zeta}^{\prime\prime}}\,$ corresponding to
$\,\Gamma\cap\Delta$ . For $\;n\mathnormal{\;\equiv\;}\mathtt{0}\;$ it is
sufficient that
$\displaystyle|\Gamma|^{\boldsymbol{u}}_{\boldsymbol{\zeta^{*}}[\mathtt{0}]\boldsymbol{v}}$
$\displaystyle\hskip 0.5pt\vdash\hskip 0.5pt$
$\displaystyle|A(\mathtt{0})|^{\boldsymbol{{t}^{\prime}}[\mathtt{0}]}_{\boldsymbol{v}}$
which follows from (6) since by definition (9) we have $\;\hskip
0.5pt\vdash\hskip
0.5pt\,\boldsymbol{{t}^{\prime}}[\mathtt{0}]=\boldsymbol{r}\;$ and by
definitions (10) and (11) we have $\ \hskip 0.5pt\vdash\hskip
0.5pt\,\boldsymbol{\zeta^{*}}[\mathtt{0}]=\lambda\boldsymbol{v}.\,\boldsymbol{\gamma}[\boldsymbol{v}]$
. Now given (8) we want to prove
$\displaystyle|\Gamma\uplus\Delta|^{\,\boldsymbol{u}\uplus\boldsymbol{z}}_{\,\boldsymbol{\zeta}[\mathtt{S}n]\boldsymbol{v}}$
$\displaystyle\hskip 0.5pt\vdash\hskip 0.5pt$
$\displaystyle|A(\mathtt{S}n)|^{\boldsymbol{{t}^{\prime}}[\mathtt{S}n]}_{\boldsymbol{v}}$
(17)
To (8) we apply $\
\forall^{e}_{\\![{\boldsymbol{v}\,\mapsto\,\boldsymbol{t}\,\boldsymbol{{t}^{\prime}}\boldsymbol{v}}]}\
$ and via easy deductions in $\,{\sf NA}\,$ we get
$\displaystyle|\Gamma\uplus\Delta|^{\,\boldsymbol{u}\uplus\boldsymbol{z}}_{\,\boldsymbol{\zeta}[n](\boldsymbol{t}\,\boldsymbol{{t}^{\prime}}\boldsymbol{v})}$
$\displaystyle\hskip 0.5pt\vdash\hskip 0.5pt$
$\displaystyle|A(n)|^{\boldsymbol{t}^{\prime}[n]}_{\boldsymbol{t}\,\boldsymbol{{t}^{\prime}}\boldsymbol{v}}$
(18)
With (13) and (15) we can rewrite (18) to
$\displaystyle|\Gamma\uplus\Delta|^{\,\boldsymbol{u}\uplus\boldsymbol{z}}_{\,\boldsymbol{\zeta}[\mathtt{S}n]\boldsymbol{v}}$
$\displaystyle\hskip 0.5pt\vdash\hskip 0.5pt$
$\displaystyle|A(n)|^{\boldsymbol{t}^{\prime}[n]}_{\boldsymbol{t}\,\boldsymbol{{t}^{\prime}}\boldsymbol{v}}$
(19)
In (7) we substitute $\ \boldsymbol{x}\,\mapsto\,\boldsymbol{{t}^{\prime}}[n]\
$ and get
$\displaystyle|\Delta|^{\boldsymbol{z}}_{\boldsymbol{\delta}[\boldsymbol{{t}^{\prime}};\boldsymbol{v}]}$
$\displaystyle\hskip 0.5pt\vdash\hskip 0.5pt$
$\displaystyle|A(n)|^{\boldsymbol{{t}^{\prime}}[n]}_{\boldsymbol{t}\boldsymbol{{t}^{\prime}}\boldsymbol{v}}\,\rightarrow\,|A(\mathtt{S}n)|^{\boldsymbol{s}\boldsymbol{{t}^{\prime}}[n]}_{\boldsymbol{v}}$
which gives (17) by means of easy $\,{\sf NA}\,$ deductions using (12) , (16)
and (19) .
We have treated the most general situation, with all context sets
$\,\Gamma\setminus\Delta$ , $\,\Gamma\cap\Delta\,$ and
$\,\Delta\setminus\Gamma\,$ inhabited by refutation relevant assumptions, and
conclusion formula $\,A\,$ accepting both witnesses and challengers. Many
particular situations amount to easier treatments, with simpler extracted
terms. These can be obtained as simplifications of the general witnesses and
challengers presented above, by means of the reduction properties of the empty
tuple, which was denoted $\,\epsilon\,$ in [17]. We outline below only those
particular cases which are relevant in connection with the modal induction
rule $\mathtt{Ind^{m}_{\iota}}\hskip 0.5pt.$
* •
If $\Gamma\cup\Delta$ contains no refutation relevant assumption, but $A(n)$
is refutation relevant, then terms $\,\boldsymbol{t}\,$ are no part of the
realizers for the conclusion sequent, in this case only
$\,\boldsymbol{{t}^{\prime}}$ . Hence $\,\boldsymbol{t}\,$ would be
redundantly produced and a mechanism is needed to prevent their construction.
This is ensured by $\Box$ in front of the step $A(n)$ at
$\mathtt{Ind^{m}_{\iota}}\hskip 0.5pt.$
* •
If $\,A(n)\,$ is refutation relevant, $\,\Delta\,$ has no refutation relevant
element but $\,\Gamma\,$ is refutation relevant inhabited, then
$\,\boldsymbol{\delta}\,$ and $\,\boldsymbol{{\zeta}^{\prime\prime}}\,$ are
empty. Yet
$\,\boldsymbol{\zeta^{*}}\mathnormal{\;\equiv\;}\boldsymbol{{\zeta}^{\prime}}\,$
has to be produced as (10) and includes $\,\boldsymbol{t}[n]$ , which is no
longer the case for $\,\mathtt{Ind^{m}_{\iota}}$ .
* •
If $\,A(n)\,$ is refutation irrelevant then $\,\boldsymbol{v}$ ,
$\,\boldsymbol{t}\,$ and
$\,\boldsymbol{t}\,\boldsymbol{{t}^{\prime}}\boldsymbol{v}\,$ are empty
tuples. Thus
$\,\boldsymbol{{\zeta}^{\prime}}\mathnormal{\;\equiv\;}\boldsymbol{{\gamma}^{\prime}}\,$
and (11) simplifies to (recall $\,n\not\in{\sf
FV}(\boldsymbol{{\gamma}^{\prime}})$ , $\,n\in{\sf
FV}(\boldsymbol{{t}^{\prime}})$ , and possibly $\,n\in{\sf
FV}(\boldsymbol{{\delta}^{\prime}})$)
$\displaystyle\boldsymbol{{\zeta}^{\prime\prime}}[n]$
$\displaystyle\mathnormal{\hskip 0.5pt\equiv\hskip 0.5pt}$
$\displaystyle\mathtt{R}\;n\;\boldsymbol{{\gamma}^{\prime}}\,\Big{(}\,\lambda
n,p\,.\;\mathtt{If}\,(|B|^{\boldsymbol{{z}^{\prime}}}_{\boldsymbol{{\delta}^{\prime}}[\boldsymbol{{t}^{\prime}}]})\
p\ \boldsymbol{{\delta}^{\prime}}[\boldsymbol{{t}^{\prime}}]\,\Big{)}$
#### 5.5.1 Modal induction rule - technical details
tensy $\displaystyle\penalty 1\Gamma\ \hskip 0.5pt\vdash\hskip 0.5pt\ \Box
A(\mathtt{0})\qquad\Box\Delta\ \hskip 0.5pt\vdash\hskip 0.5pt\ \Box
A(n)\;\rightarrow\;A(\mathtt{S}n)$ $\ {\ \mathtt{Ind^{m}_{\iota}}}$
$\displaystyle\Gamma\,,\,\Box\Delta\ \hskip 0.5pt\vdash\hskip 0.5pt\ \Box
A(n)$
---
We are given
$\displaystyle|\Gamma|^{\boldsymbol{u}}_{\boldsymbol{\gamma}}$
$\displaystyle\hskip 0.5pt\vdash\hskip 0.5pt$
$\displaystyle\forall\boldsymbol{y}\,|A(\mathtt{0})|^{\boldsymbol{r}}_{\boldsymbol{y}}$
(20)
and
$\displaystyle|\Box\Delta|^{\boldsymbol{z}}$ $\displaystyle\hskip
0.5pt\vdash\hskip 0.5pt$
$\displaystyle\forall\boldsymbol{{y}^{\prime}}\,|A(n)|^{\boldsymbol{x}}_{\boldsymbol{{y}^{\prime}}}\,\rightarrow\,|A(\mathtt{S}n)|^{\boldsymbol{s}\boldsymbol{x}}_{\boldsymbol{v}}$
Since $\,\boldsymbol{v}\,\not\in\,{\sf FV}(|\Box\Delta|^{\boldsymbol{z}})\,$
and $\,\boldsymbol{v}\,\not\in\,{\sf
FV}(\forall\boldsymbol{{y}^{\prime}}\,|A(n)|^{\boldsymbol{x}}_{\boldsymbol{{y}^{\prime}}})\,$
from the latter we easily obtain
$\displaystyle|\Box\Delta|^{\boldsymbol{z}}$ $\displaystyle\hskip
0.5pt\vdash\hskip 0.5pt$
$\displaystyle\forall\boldsymbol{{y}^{\prime}}\,|A(n)|^{\boldsymbol{x}}_{\boldsymbol{{y}^{\prime}}}\,\rightarrow\,\forall\boldsymbol{v}\,|A(\mathtt{S}n)|^{\boldsymbol{s}\boldsymbol{x}}_{\boldsymbol{v}}$
(21)
With $\,t[n]\,\mathnormal{\ :\equiv\ }\,\mathtt{R}\,n\,r\,(\lambda n.s)\,$ for
every corresponding pair $\,\langle
r\in\boldsymbol{r}/s\in\boldsymbol{s}\rangle\,$ we show by induction on
$\,n\,$ in $\,{\sf NA}\,$ with base context
$\,|\Gamma|^{\boldsymbol{u}}_{\boldsymbol{\gamma}}\,$ and step context
$\,|\Box\Delta|^{\boldsymbol{z}}\,$ that
$\displaystyle|\Gamma|^{\boldsymbol{u}}_{\boldsymbol{\gamma}}\ ,\
|\Box\Delta|^{\boldsymbol{z}}$ $\displaystyle\hskip 0.5pt\vdash\hskip 0.5pt$
$\displaystyle\forall\boldsymbol{v}\,|A(n)|^{\boldsymbol{t}[n]}_{\boldsymbol{v}}$
As $\,\boldsymbol{t}[\mathtt{0}]\mathnormal{\;\equiv\;}\boldsymbol{r}\,$ the
base is given by (20) and the step follows from (21) with
$\,\boldsymbol{x}\mapsto\boldsymbol{t}[n]\,$ since
$\,\boldsymbol{t}[\mathtt{S}n]\,\mathnormal{\;\equiv\;}\,\boldsymbol{s}\,\boldsymbol{t}[n]$
. Thus challengers $\,\boldsymbol{\gamma}\,$ are simply preserved for
$\,|\Gamma|\,$ and witnesses $\,\boldsymbol{t}[n]\,$ are easily constructed
for $\,|\Box A(n)|\,$ in the conclusion sequent of
$\,\mathtt{Ind^{m}_{\iota}}$ .
###### Remark 6
Our modal induction rule is equivalent to a special case of
$\,\mathtt{Ind}_{\iota}$ , since a $\,\Box\,$ can be placed in front of
$\,A(\mathtt{S}n)\,$ from the step sequent of $\,\mathtt{Ind^{m}_{\iota}}$ .
The equivalence of the two formulations for the step sequent can easily be
proved using $\,\mathtt{AxT}$ , $\,\mathtt{Ax4}$ , $\,\mathtt{AxK}\,$ and
$\,\Box^{i}$ . Extracted terms are the same and the verifying proof only gets
more direct.
|
arxiv-papers
| 2012-11-30T21:32:52 |
2024-09-04T02:49:38.712299
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/",
"authors": "Dan Hernest and Trifon Trifonov",
"submitter": "Dan HERNEST gm",
"url": "https://arxiv.org/abs/1212.0020"
}
|
1212.0051
|
# Prescribing the behavior of Weil-Petersson geodesics in the moduli space of
Riemann surfaces
Babak Modami Department of Mathematics, Yale University, 10 Hillhouse Ave,
New Haven, CT, 06511 [email protected]
###### Abstract.
We study the Weil-Petersson (WP) geodesics with narrow end invariant and
develop techniques to control length-functions and twist parameters along them
and prescribe their itinerary in the moduli space of Riemann surfaces. This
class of geodesics is rich enough to provide for examples of closed WP
geodesics in the thin part of moduli space, as well as divergent WP geodesic
rays with minimal filling ending lamination.
As an intermediate step we prove that hierarchy resolution paths between
narrow pairs of partial markings or laminations are stable in the pants graph
of the surface.
###### 2010 Mathematics Subject Classification:
Primary 30F60, 32G15, 37D40
The author was partially supported by NSF grant DMS-1005973
###### Contents
1. 1 Introduction
1. 1.1 Outline of the paper
2. 2 Curve complexes and hierarchical structures
1. 2.1 The pants and marking graph
2. 2.2 Hierarchies and their resolutions in the pants and marking graph
3. 2.3 $\Sigma-$hulls and their projections
3. 3 The Weil-Petersson metric and its synthetic properties
1. 3.1 End invariant
4. 4 Length-function control along uniformly bounded length Weil-Petersson geodesics
1. 4.1 Limits of sequences of uniformly bounded length WP geodesic segments
2. 4.2 Length-function versus twist parameter control
5. 5 Stable hierarchy paths
1. 5.1 Narrow pairs
2. 5.2 Stability
3. 5.3 Fellow traveling
6. 6 Itinerary of a Weil-Petersson geodesic segment
1. 6.1 Isolated annular subsurfaces
2. 6.2 Length-function control
7. 7 Laminations with prescribed subsurface coefficients
1. 7.1 Scheme I
2. 7.2 Scheme II
8. 8 Weil-Petersson Geodesics
1. 8.1 Weil-Petersson geodesic rays with prescribed itinerary
2. 8.2 Divergent geodesic rays
3. 8.3 Closed geodesics in the thin part
4. 8.4 A recurrence condition
## 1\. Introduction
The Weil-Petersson (WP) metric on the moduli space of Riemann surfaces is an
incomplete Riemannian metric with negative sectional curvatures asymptotic to
both $0$ and $-\infty$ in the completion. The WP metric is a metric of higher
rank in the sense of Gromov (see [BF06]). The WP geodesic flow is not
uniformly hyperbolic. These features prevent applying most of the standard
techniques available in the study of the global geometry and dynamics of
complete metrics with negative curvatures bounded away from zero and uniformly
hyperbolic dynamical systems to the WP metric and its geodesic flow. For some
of these techniques see [PP10], [Ebe72], [KH95]. The main theme of this paper
and the pioneer work of Brock, Masur and Minsky in [BM08],[BMM10],[BMM11] is
to apply combinatorial techniques from surface theory to study the global
behavior of WP geodesics.
In [BMM10] the authors introduce a notion of ending lamination for WP geodesic
rays. They show that the ending lamination determines the strong asymptotic
class of a WP geodesic ray recurrent to a compact subset of moduli space. In
[BMM11] a more explicit connection between the combinatorics of the ending
laminations of a WP geodesic and its behavior was established. They provide a
necessary and sufficient combinatorial condition for a WP geodesic to stay in
the compact part of moduli space. In this paper we prove the following two
results about the behavior of WP geodesics in the moduli space of Riemann
surfaces:
###### Theorem 1.1.
(Closed geodesic in the thin part) Given any compact subset of moduli space
$\mathcal{K}$, there are infinitely many closed Weil-Petersson geodesics not
intersecting $\mathcal{K}$.
###### Theorem 1.2.
(Divergent geodesic) Starting from any point in the moduli space there are
uncountably many divergent WP geodesic rays with minimal filling ending
lamination.
The WP volume of moduli space is finite, so by the Poincaré recurrence theorem
almost every WP geodesic ray is recurrent to a compact subset of moduli space.
However the second theorem above exhibits the abundance of WP geodesic rays
divergent in the moduli space. These geodesics diverge in the moduli space by
getting closer and closer to a chain of completion strata.
Given a WP geodesic $g:(a,b)\to\operatorname{Teich}(S)$ denote the end
invariant associated to its forward trajectory by $\nu^{+}=\nu^{+}(g)$ and the
one associated to its backward trajectory by $\nu^{-}=\nu^{-}(g)$. Here
$\nu^{-}$ and $\nu^{+}$ are either lamination or (partial) marking.
To each subsurface $Z\subseteq S$ that is not a three holed sphere, there is
an associated subsurface coefficient denoted by
$d_{Z}(\nu^{-},\nu^{+})$
which is the distance in the curve complex of $Y$ between the projections of
$\nu^{-}$ and $\nu^{+}$, for more details see $\S$2.
Subsurface coefficients are an analogue of continued fraction expansions which
provide for a coding of geodesics on the modular surface which is the moduli
space of one hold tori $\mathcal{M}(S_{1,1})$, see for example [Ser85].
Conjecturally these coefficients provide for extensive information about the
behavior of Weil-Petersson geodesics in the moduli space. The following
conjecture was proposed in [BMM10]:
###### Conjecture 1.3.
(Short Curve) Let $g$ be a Weil-Petersson geodesic with end invariant
$(\nu^{-},\nu^{+})$. Then
1. (1)
For every $\epsilon>0$ there is an $A>0$, such that if
$d_{Z}(\nu^{-},\nu^{+})>A$ then $\inf_{t}\ell_{\alpha}(g(t))\leq\epsilon$ for
every $\alpha\in\partial{Z}$.
2. (2)
For every $A>0$ there is an $\epsilon>0$, such that if
$\inf_{t}\ell_{\alpha}(g(t))\leq\epsilon$ then there is a subsurface
$Z\subsetneq S$ such that $\alpha\in\partial{Z}$ and
$d_{Z}(\nu^{-},\nu^{+})>A$.
Furthermore, it would be very useful to have estimates on the length of the
time interval that a curve is short (has length less than a given $\epsilon$)
along a WP geodesic using subsurface coefficients of the end invariants.
The WP metric exhibits different features in the thick and thin parts of
Teichmüller space. For instance in the thick part the sectional curvatures are
all bounded away from both 0 and $-\infty$, while in the thin part the WP
metric is almost a product metric. Therefore, to answer questions about the
global geometry and dynamics of the WP metric often one needs to determine the
itinerary of geodesics. By this we mean the thin regions of the Teichmüller
space that a WP geodesic $g$ visits, the order $g$ visits the regions and the
time $g$ spends in each one of these regions.
After the work of Masur-Minsky [MM99],[MM00] and Rafi [Raf05],[Raf], the above
conjecture holds for Teichmüller geodesics and provides for a complete picture
of the itinerary of Teichmüller geodesics in the moduli space. An important
ingredient which is missing here is an explicit description of the Riemann
surface along WP geodesics.
The underlying machinery to obtain control on the behavior of WP geodesics is
the Masur-Minsky machinery of hierarchies and their resolutions in pants and
marking graphs introduced in [MM00]. Given a pair of partial markings or
laminations a hierarchy (resolution) paths $\rho:[m,n]\to P(S)$,
$[m,n]\subseteq\mathbb{Z}\cup\\{\pm\infty\\}$, is a quasi-geodesic between
them in the pants graph with certain properties encoded in the pair and their
subsurface coefficients. For example corresponding to any subsurface $Z$ with
big enough subsurface coefficient, there is a subinterval of $[m,n]$ denoted
by $J_{Z}$ such that $\partial{Z}\subset\rho(i)$, for every $i\in J_{Z}$. In
Theorem 2.13 the list of properties is listed.
By a result of Brock (Theorem 3.3) the Bers pants decompositions along a WP
geodesic trace a quasi-geodesic in the pants graph of the surface. When such a
quasi-geodesic and a hierarchy path $\rho$ fellow travel each other (see
Definition 5.23) there is a correspondence between the parameters of the
hierarchy path and the parameters of the WP geodesic, which is roughly the
nearest point correspondence of fellow traveling paths. In this situation we
use the hierarchy machinery to determine the itinerary of WP geodesics. An
example is visiting the region where the curves in $\partial{Z}$ are short
over the time interval corresponding to $J_{Z}$.
An $\it{A-narrow}$ condition on the end invariant $(\nu^{-},\nu^{+})$ is a
constraint on the set of subsurfaces with big subsurface coefficient. More
precisely, the pair is $A-$narrow if every non-annular subsurface $Z\subseteq
S$ with
$d_{Z}(\nu^{-},\nu^{+})>A$
is a large subsurface i.e. is a connected subsurface with complement
consisting of only annuli and three holed spheres.
The $A-$narrow condition on end invariant implies uniform fellow traveling,
depending only on $A$, of a WP geodesic segment and a hierarchy path with end
points equal to the pair. Heuristically hierarchy paths with narrow end
invariant avoid quasi-flats in the pants graph corresponding to separating
multi-curves on a surface, and WP geodesics with narrow end invariant avoid
asymptotic quasi-flats in the WP metric corresponding to separating multi-
curves on a surface. In this paper we develop a control for length-functions
and twist parameters along geodesics with narrow end invarinat and show that
their itinerary mimic combinatorial properties of hierarchy paths. In order to
prove our main technical results we introduce the following notions:
Let $Z\subseteq S$ be a subsurface with $d_{Z}(\nu^{-},\nu^{+})$ sufficiently
big. We say that $Z$ has $(R,R^{\prime})-$bounded combinatorics over a
subinterval $[i_{1},i_{2}]\subset J_{Z}\subset[m,n]$ if for every non-annular
subsurface $Y\subsetneq Z$ the subsurface coefficient be bounded as
$d_{Y}(\rho(i_{1}),\rho(i_{2}))\leq R$
and for every annular subsurface $A(\gamma)$ with core curve $\gamma$ inside
$Z$ the subsurface coefficient be bounded as
$d_{\gamma}(\rho(i_{1}),\rho(i_{2}))\leq R^{\prime}$
This condition is a local version of the bounded combinatorics of the end
invariant in [BMM11]. There is shown that a geodesic with bounded
combinatorics end invariant stays in the thick part of moduli space. In the
direction of Conjecture 1.3 we prove:
###### Theorem 1.5.
(Short Curve)
Given $A,R,R^{\prime}>0$ and a sufficiently small $\epsilon>0$, there is
$\bar{w}=\bar{w}(A,R,R^{\prime},\epsilon)$ with the following property. Let
$g:[a,b]\to\operatorname{Teich}(S)$ be a WP geodesic segment with $A-$narrow
end invariant $(\nu^{-},\nu^{+})$. Let $\rho:[m,n]\to P(S)$ be a hierarchy
between $\nu^{-}$ and $\nu^{+}$. Assume that a large component domain of
$\rho$, $Z$ has $(R,R^{\prime})-$bounded combinatorics over an interval
$[m^{\prime},n^{\prime}]\subset J_{Z}$.
If $n^{\prime}-m^{\prime}\geq 2\bar{w}$, then for every $\alpha\in\partial{Z}$
we have
$\ell_{\alpha}(g(t))\leq\epsilon$
for every $t\in[a^{\prime},b^{\prime}]$, where $a^{\prime}$ and $b^{\prime}$
are the corresponding times to $m^{\prime}+\bar{w}$ and $n^{\prime}-\bar{w}$,
respectively.
Here $N$ is the parameter map from $[m,n]$ to $[a,b]$ which comes from the
fellow traveling of a the Bers curves along a WP geodesic with narrow end
invariant $(\nu^{-},\nu^{+})$ and a hierarchy path between
$(\nu^{-},\nu^{+})$, see $\S$5.3.
In $\S$6 we use bounded combinatorics intervals to isolate annular subsurface
along a hierarchy path. The twist parameter developed about an isolated
annular subsurface along a WP geodesic is comparable to the one along a fellow
traveling hierarchy path. This together with the length-function versus twist
parameter controls over uniformly bounded length WP geodesic segments we
develop in $\S$4 are the main technical tools in $\S$6 where we prove the
above theorem.
Using the control we develop on length-functions along WP geodesic segments
and by extracting limits of WP geodesic segments with narrow end invariant we
provide WP geodesics with prescribed itinerary in the moduli space (Theorem
8.5). Itineraries of these rays mimic the combinatorial properties of
hierarchy paths encoded in the end points and the associated subsurface
cefficients. Using these rays and our constructions of pairs of
laminations/markings with prescribed list of subsurface coefficients in $\S$7
we prove theorems 1.1 and 1.2 in $\S$8. Jeffery Brock has communicated us that
he also constructs divergent WP geodesics, [Bro]. Our constructions of pairs
of laminations/markings with prescribed list of subsurface coefficients in
$\S$7 is a kind of symbolic coding for laminations using subsurface
coefficients analogue of continued fraction expansions.
Fellow traveling property of hierarchy paths is a crucial part of the
combinatorial frame work to control length-functions along WP geodesics. In
$\S$5 we prove the following stability result for hierarchy paths in the pants
graph of a surface
###### Theorem 5.11.
(Stable hierarchy path) Given $A>0$ there is a function $d_{A}$ such that any
hierarchy path with $A-$narrow end points is $d_{A}-$stable in the pants
graph.
Here $d_{A}:\mathbb{R}^{\geq 1}\times\mathbb{R}^{\geq 0}\to\mathbb{R}^{\geq
0}$ is the quantifier of the stability, see Definition 5.1.
This theorem implies that the Bers pants along a WP geodesic and a hierarchy
path connecting the end invariants of the geodesic uniformly fellow travel
each other in the pants graph.
### 1.1. Outline of the paper
Section 2 is devoted to the background about curve complexes and some
important notions and results in the setting of pants and marking graphs. Here
we recall hierarchical structures on pants and marking graphs and their
resolutions introduced by Masur and Minsky in [MM00]. In Theorem 2.13 we list
the properties of hierarchy resolution paths. Moreover, we recall the
$\Sigma-$hulls and their projections from [BKMM12]. In Section 3 we give some
background about the WP metric and synthetic properties of WP geodesics.
In Section 4 we provide the proofs of refined version of some of the key
results in [Wol03] and [BMM11] which we need in this work. These are mainly
based on compactness arguments in the WP completion of Teichmüller space and
are consequences of Wolpert’s geodesic limit theorem. These results give us a
kind of control on development of Dehn twists versus change off length-
functions along uniformly bounded length WP geodesic segments.
In Section 5 we prove that hierarchy paths between narrow pairs are stable.
The proof will be through $\Sigma-$hulls and their stability properties.
In Section 6, we develop some new techniques to control length-functions and
twist parameters along WP geodesics fellow traveling hierarchy paths. In Lemma
6.6 we provide certain rough bounds for length of curves. In Theorem 6.1 using
convexity of length-functions and an inductive argument on the complexity of
subsurfaces we sharpen these bounds.
In Section 7 we provide pairs of marking or laminations with a prescribed list
of subsurface coefficients. This is a kind of symbolic coding for laminations
in terms of subsurface coefficients similar to continued fraction expansions.
In Section 8 we provide examples of WP geodesics with prescribed behaviors.
These results could be considered as a kind of symbolic coding for WP
geodesics. Here we use the control on length-functions from $\S$6 and infinite
stable hierarchy paths with a prescribed list of subsurface coefficients from
$\S$7.
###### Notation 1.4.
Given $K\geq 1$ and $C\geq 0$. Let $f,g:X\to\mathbb{R}$ be two functions on a
set $X$. Then $f\asymp_{K,C}g$ means that for every $x\in X$ we have
$\frac{1}{K}g(x)-C\leq f(x)\leq Kg(x)+C$
Acknowledgment: I am so grateful to my thesis advisor Yair Minsky for so many
invaluable discussions through which this work has evolved. I would like to
thank Jeffery Brock who generously contributed some of the ideas in Section 4.
I am grateful to Scott Wolpert for willingly answering my questions about the
Weil-Petersson metric.
## 2\. Curve complexes and hierarchical structures
Curve complex of a surface: Let $S=S_{g,b}$ be a connected, compact, oriented
surface with genus $g$ and $b$ boundary components. We define $\xi(S)=3g-3+b$
as the complexity of the surface.
The curve complex of $S$, denoted by $\mathcal{C}(S)$, serves to organize the
isotopy classes of essential, simple closed curves on $S$. Let $S$ be a
surface with $\xi(S)\geq 1$. To each isotopy class of essential simple closed
curves on $S$ (neither isotope to boundary nor a point) is associated a vertex
($0-$simplex) in $\mathcal{C}(S)$. When $\xi(S)>1$, an edge is associated to
disjoint pair of isotopy classes of curves. Similarly, a $k-$simplex is
associated to any $k+1$ pairwise disjoint isotopy classes of simple closed
curves. Here two isotopy classes are disjoint if there are curves in each of
them which are disjoint on $S$. We denote the $k-$skeleton of this complex by
$\mathcal{C}_{k}(S)$. When $\xi(S)=1$, $S$ is either a one holed torus or a
four holed sphere. Then $1-$simplices (edges) correspond to curves
intersecting respectively once and twice, which are the minimum possible
intersection number of curves on $S$, respectively.
We equip the curve complex with a distance by making each simplex Euclidean
with side lengths 1, and denote the distance by $d_{S}=d_{\mathcal{C}(S)}$. By
the main result of Masur-Minsky in [MM99] $\mathcal{C}(S)$ equipped with this
distance is a $\delta-$hyperbolic space in the sense of Gromov, where $\delta$
only depends on topological type of $S$.
An annular subsurface is an annulus $Y$ with essential core curve in $S$. The
purpose of defining complexes for annuli is to keep track of Dehn twisting
about their core curves. These complexes are quasi-isometric to $\mathbb{Z}$.
Let $Y$ be an annulus with core curve. Let
$\widetilde{Y}=\widetilde{S}/\langle\alpha\rangle$ be the annular cover of $S$
to which $Y$ lifts homeomorphically. There is a natural compactification of
$\widetilde{Y}$ to a closed annulus $\widehat{Y}$ which is the quotient by
$\langle\alpha\rangle$ of the compactification of the universal cover of $S$,
$\widetilde{S}=\mathbb{D}^{2}$ (the Poincaré disk) by the closed disk. A
vertex of $\mathcal{C}(Y)$ is associated to a path connecting the two boundary
components of $\widehat{Y}$ modulo isotopies that fix the endpoints (isotopy
classes of arcs relative to the boundary). An edge is associated to two
vertices which have representatives with disjoint interiors. Curve comlexes
can be made metr with a metric by assigning length $1$ to each edge. Let
$\alpha\in\mathcal{C}_{0}(S)$ be the core curve of $Y$. We write $Y=A(\alpha)$
and $\mathcal{C}(\alpha)=\mathcal{C}(Y)$.
###### Notation 2.1.
A curve on $S$ refers to the isotopy class of an essential simple closed curve
on $S$, that is a vertex of $\mathcal{C}_{0}(S)$. If there are representatives
of the isotopy classes of curves $\alpha$ and $\beta$ which are disjoint, then
$\alpha$ and $\beta$ are disjoint. Otherwise, $\alpha$ and $\beta$ overlap
each other which is denoted by $\alpha\pitchfork\beta$. A multi-curve consists
of the vertices of a simplex in $\mathcal{C}(S)$, whihc is a collection of
pair-wise disjoint curves. A curve system is a finite diameter subset of
$\mathcal{C}_{0}(S)$. Two multi-curves $\sigma$ and $\sigma^{\prime}$ are
disjoint if any pair of curves $\alpha\in\sigma$ and
$\alpha^{\prime}\in\sigma^{\prime}$ are disjoint, otherwise $\sigma$ and
$\sigma^{\prime}$ overlap.
Filling curve systems: We say that a curve system $\mu$ on $S$ (a finite
diameter subset of $\mathcal{C}_{0}(S)$) fills $S$ if $S\backslash\mu$
consists of only topological disk and annuli. Given a filling system $\mu$ any
$\gamma\in\mathcal{C}_{0}(S)$ overlaps $\mu$.
Let $\alpha,\beta\in\mathcal{C}_{0}(S)$. If $d_{S}(\alpha,\beta)\geq 3$ then
$\alpha$ and $\beta$ fill $S$. To see this suppose that $\alpha$ and $\beta$
do not fill $S$. Then there is a curve $\gamma$ disjoint from both $\alpha$
and $\beta$ which implies that $d_{S}(\alpha,\beta)\leq 2$.
Laminations: Let $S$ be a surface equipped with a complete hyperbolic metric.
A geodesic lamination on $S$ is a closed subset $\lambda$ of $S$ consisting of
disjoint, complete, simple geodesics. We denote the space of geodesic
laminations of a surface $S$ by $\mathcal{GL}(S)$. A geodesic lamination
$\lambda$ can be equipped with transversal measures: a transversal measure is
a measure on arcs transversal to $\lambda$ which is invariant under isotopies
of the surface $S$ relative to $\lambda$. A measure (geodesic) lamination
$\mathcal{L}=(\lambda,m)$ is the pair of a geodesic lamination $\lambda$,
called the support of $\mathcal{L}$ and $m$ a transverse measure of $\lambda$.
The space of measured geodesic laminations of $S$ equipped with the weak∗
topology is denoted by $\mathcal{ML}(S)$. For more detail see [CEG87] and
[Bon01].
$\mathbb{R}^{+}$ acts on $\mathcal{ML}(S)$ by rescaling the measure, that is
$s(\lambda,m)=(\lambda,sm)$. Then the quotient,
$\mathcal{PML}(S):=\mathcal{ML}(S)/\mathbb{R}$ is the space of projective of a
measured laminations, the projective class of a measured lamination
$\mathcal{L}$ is denoted by $[\mathcal{L}]$. We equip $\mathcal{PML}(S)$ with
the quotient of the weak∗ topology on $\mathcal{ML}(S)$.
A geodesic lamination $\lambda$ is filling $S$ if the connected components of
$S\backslash\lambda$ are only topological disks or annuli. $\lambda$ is
minimal if any geodesic ray in $\lambda$ is dense in $\lambda$. Denote by
$\mathcal{EL}(S)$ the set of projective classes of measured laminations
supported on minimal filling laminations. Equip $\mathcal{EL}(S)$ with the
topology induced from the topology of $\mathcal{PML}(S)$.
The Gromov boundary of a $\delta-$hyperbolic space has a standard topology,
see for example $\S$III.H.3 of [BH99]. Klarriech in [Kla] proves that
###### Theorem 2.2.
There is a homeomorphism $\Phi$ from the Gromov boundary of $\mathcal{C}(S)$
to $\mathcal{EL}(S)$.
Furthermore Klariech describes the relation between a sequence of curves going
off to the Gromov boundary of $\mathcal{C}(S)$ and the accumulation points of
the sequence in $\mathcal{PML}(S)$.
###### Theorem 2.3.
(Theorem 1.4 of [Kla]) If a sequence of vertices in $\mathcal{C}_{0}(S)$
converges to a point $\xi$ in the Gromov boundary of $\mathcal{C}(S)$ then
regarding each element of the sequence as a projective measured lamination
every accumulation point of the sequence in $\mathcal{PML}(S)$ is
topologically equivalent to $\Phi(\xi)$. This means that their supporting
geodesic laminations are topologically equivalent.
The following notions of subsurface projection and subsurface coefficient are
basic in the Masur-Minsky machinery of curve complexes and hierarchical
structures on the pants and marking graphs.
Subsurface projection: Given a non-annular subsurface $Y\subseteq S$ define
the subsurface projection
$\pi_{Y}:\mathcal{GL}(S)\to\mathcal{P(C}_{0}(Y))$
where $\mathcal{P(C}_{0}(Y))$ is the power set of $\mathcal{C}_{0}(S)$ as
follows: Given $\lambda\in\mathcal{GL}(S)$, let $\lambda\cap Y$ be the
collection of all essential curves and arcs of the intersection of $\lambda$
and $Y$ after identifying all the arcs and curves which are isotopic to each
other. Here the end points of arcs are allowed to move within the boundary of
$Y$. $\pi_{Y}(\lambda)$ consists of the boundary of a regular neighborhood of
any arc $a$ in $\lambda\cap Y$ and $\partial{Y}$ and all of the closed curves
in $\lambda\cap Y$. If $\lambda$ does not intersect $Y$ essentially then
$\pi_{Y}(\lambda)=\emptyset$.
Note that since $\mathcal{C}_{0}(S)\subset\mathcal{GL}(S)$, the above
projection in particular gives a projection
$\pi_{Y}:\mathcal{C}_{0}(S)\to\mathcal{P(C}_{0}(Y)).$
The projection for an annular subsurface $Y$ is defined as follows: If
$\lambda\in\mathcal{GL}(S)$ crosses the core of $Y$ transversely, then the
lift of $\lambda$ to $\widehat{Y}$ (the compactification of $\tilde{Y}$ the
annular cover of $Y$) has at least one component that connects the two
boundaries of $\widehat{Y}$. Then together these components make up a set of
diameter $1$ in $\mathcal{C}(Y)$. Let $\pi_{Y}(\lambda)$ be this set. If
$\lambda$ does not intersect $Y$ essentially (including the case that
$\lambda$ is the core of $Y$) then $\pi_{Y}(\gamma)=\emptyset$.
The projection of a curve system $\mu$ to a subsurface $Y$ is the union of
$\pi_{Y}(\alpha)$ for all $\alpha\in\mu$.
Subsurface coefficient: Let $\mu$ and $\mu^{\prime}$ be two laminations or
curves systems. Let $Y\subseteq S$ be a subsurface. We consider the following
notion of distance
(2.1)
$d_{Y}(\mu,\mu^{\prime}):=\min\\{d_{Y}(\gamma,\gamma^{\prime}):\gamma\in\pi_{Y}(\mu),\gamma^{\prime}\in\pi_{Y}(\mu^{\prime})\\}$
which provides for a useful notion of complexity of $\mu$ and $\mu^{\prime}$
from the point of view of the subsurface $Y$. We call it the Y subsurface
coefficient of $\mu$ and $\mu^{\prime}$.
For an annular subsurface $Y$ with core curve $\alpha$ we denote
$\pi_{\alpha}:=\pi_{Y}$ and define the annular coefficients of geodesic
laminations/curve systems $\mu$ and $\mu^{\prime}$ by (2.1) and denote it by
$d_{\alpha}$.
Subsurface coefficients play the role of continued fraction expansions for
coding of laminations (see 7).
The triangle inequality: Given a curve system $\mu\subset\mathcal{C}(Y)$ (a
finite diameter subsets of $\mathcal{C}_{0}(Y)$) denote by
$\operatorname{diam}_{Y}(\mu)$ the diameter of $\pi_{Y}(\mu)$ as a subset of
$\mathcal{C}(Y)$.
Let $\mu_{1},\mu_{2},\mu_{3}\subset\mathcal{C}(Y)$ be curve systems. Suppose
that $d_{Y}(\mu_{1},\mu_{2})=d_{Y}(\alpha,\beta)$ and
$d_{Y}(\mu_{2},\mu_{3})=d_{Y}(\beta^{\prime},\gamma)$, where
$\alpha\in\mu_{1}$, $\beta,\beta^{\prime}\in\mu_{2}$ and $\gamma\in\mu_{3}$.
Then by the triangle inequality we have that
$d_{Y}(\alpha,\beta)+d_{Y}(\beta,\beta^{\prime})+d_{Y}(\beta^{\prime},\gamma)\geq
d_{Y}(\alpha,\gamma)$. Now since
$d_{Y}(\beta,\beta^{\prime})\leq\operatorname{diam}_{Y}(\mu_{2})$ and
$d_{Y}(\alpha,\gamma)\geq d_{Y}(\mu_{1},\mu_{2})$ we conclude that
$d_{Y}(\mu_{1},\mu_{2})+d_{Y}(\mu_{2},\mu_{3})+\operatorname{diam}_{Y}(\mu_{2})\geq
d_{Y}(\mu_{1},\mu_{3}).$
We refer to this inequality as the triangle inequality.
Hausdorff convergence: Let $\lambda_{k}$ be a sequence of geodesic laminations
and $\xi$ be any accumulation point of $\lambda_{k}$’s in the Hausdorff
topology of closed subsets of $S$. The Hausdorff convergence and the
definition of subsurface coefficient imply that given subsurface $Z\subseteq
S$ and $\lambda^{\prime}\in\mathcal{GL}(S)$
$d_{Z}(\lambda^{\prime},\xi)\asymp_{1,1}d_{Z}(\lambda^{\prime},\lambda_{k}),$
for all $k$ sufficiently large.
###### Proposition 2.4.
Let $\mathcal{L}_{i}=(\lambda_{i},m_{i})$ be a sequence of measured
laminations. Suppose that $\mathcal{L}_{i}$ converge to
$\mathcal{L}=(\lambda,m)$ in the weak∗ topology. Let $\xi$ be the limit of a
subsequece $\lambda_{i}$’s in the Hausdorff topology of closed subset of the
surface. Then $\lambda\subseteq\xi$.
###### Proof.
Let $x\in\lambda$. Let $U$ be an open neighborhood of $x$. Let $a\subset U$ be
an arc transversal to $\lambda$. By weak∗ convergence
$\int_{a}\mathcal{L}_{i}\to\int_{a}\mathcal{L}$ as $i\to\infty$. Further
$\int_{a}\mathcal{L}>0$, which implies that for all $i$ sufficiently large
$\int_{a}\mathcal{L}_{i}>0$. Then $\lambda_{i}\cap a$, for otherwise
$\int_{a}\mathcal{L}_{i}=0$. So there is $x_{i}\in\lambda_{i}$ so that
$x_{i}\in U$. This implies that $x$ is in the Hausdorff limit of any
convergent subsequence of $\lambda_{i}$’s. ∎
###### Notation 2.5.
A subsurface refers to the isotopy class of a subsurface. If a subsurface $Y$
has a representative which is a subset of a representative of a surface $Z$,
then $Y$ is a subsurface of $Z$, we denote it by $Y\subseteq Z$. Given a
multi-curve $\sigma$, $\sigma\pitchfork Y$ means that at least one of the
curves in $\sigma$ is not isotopic to curve in the complement of $Y$. If
$\partial{Y}\pitchfork Z$ and $\partial{Z}\pitchfork Y$, we say $Y$ and $Z$
overlap, and denote it by $Y\pitchfork Z$.
### 2.1. The pants and marking graph
A pants decomposition $P$ on a surface $S$ is a maximal collection of pairwise
disjoint simple closed curves. A (partial) marking $\mu$ consists of a pants
decomposition which is the base of the marking and is denoted by
$\operatorname{base}(\mu)$ together with an element of $\mathcal{C}(\alpha)$
for (some) every $\alpha\in\operatorname{base}{\mu}$. The element in
$\mathcal{C}(\alpha)$ can be represented by a transversal curve
$\beta(\alpha)$ to $\alpha$ on $S$. A clean marking is a marking such that
each transversal curve $\beta(\alpha)$ ($\alpha$ is in the base) does not
intersect any curve in $P-\alpha$ and has minimal intersection number 1 or 2
with $\alpha$ depending on that the subsurface $S\backslash\\{P-\alpha\\}$ is
a once punctured torus or a four holed sphere.
Pants and marking graph: Each vertex of the pants graph is a pants
decomposition, each edge corresponds to two pants decompositions which differ
by an elementary move. Pants decomposition $P$ and $P^{\prime}$ differ by an
elementary move if $P^{\prime}$ is obtained from $P$ by replacing one curve
$\alpha\in P$ with a curve $\alpha^{\prime}\in\mathcal{C}_{0}(S\backslash
P-\alpha)$ with minimal intersection number (1 or 2) with $\alpha$ and fixing
the rest of curves in $P$. Assigning length $1$ to each edge defines the
distance $d$ on $P(S)$ and makes it a metric graph.
Each vertex of the marking graph is a marking, each edge corresponds to a pair
of markings which differ by an elementary move. An elementary move on a
marking roughly speaking is either an elementary move on the base of the
marking or is an interchange of a curve in the base and its transversal curve.
For more detail see [MM00]. Assigning length one to each edge defines the
distance $\tilde{d}$ and makes the marking graph a metric graph.
The following theorem plays an important role in organizing the so called
tight geodesics in curve complexes of a surface and its subsurfaces and
inductive construction of hierarchies in the pants and marking graphs of a
surface.
###### Theorem 2.6.
(Bounded Geodesic Image)[MM00] There exists a constant $G>0$ depending only on
the topological type of $S$ with the following property. Let $Y\subsetneq S$
be an essential, connected subsurface and let $g:I\to\mathcal{C}(S)$ be a
geodesic such that for every $i\in I$, $\pi_{Y}(g(i))\neq\emptyset$ i.e.
$g(i)\pitchfork Y$. Then we have
$\operatorname{diam}_{Y}(\\{g(i)\\}_{i\in I})\leq G$
In the rest of this subsection we recall some results in the context of pants
and marking graphs which we use often in this paper.
Distance formula: Let $A\geq 0$, define the cut-off function
$\\{\\}_{A}:\mathbb{R}\to\mathbb{R}^{\geq 0}$ by
$\\{a\\}_{A}=\Big{\\{}\begin{array}[]{cl}a&a\geq A\\\ 0&a<A\\\ \end{array}.$
Masur-Minsky in [MM00] proved the following quasi distance formula for the
pants and marking graph distance.
There exists a constant $M_{1}>0$ such that for any $A\geq M_{1}$, there are
$K\geq 1$ and $C\geq 0$ such that the distance between any two pants
decompositions $P$ and $Q$ is given by
(2.2) $d(P,Q)\asymp_{K,C}\sum_{\begin{subarray}{c}Y\subseteq S\\\
\operatorname{non-annular}\end{subarray}}\\{d_{Y}(P,Q)\\}_{A}.$
Note that the sum is only over non-annular subsurfaces. We call $A$ the
threshold constant and say that $K,C$ are the constants corresponding to the
threshold constant $A$.
###### Remark 2.7.
Given partial markings $\mu$ and $\mu^{\prime}$ if in (2.2) we sum over all
subsurfaces coefficients, including the annular subsurface coefficients, we
get the marking distance of $\mu$ and $\mu^{\prime}$.
###### Theorem 2.8.
(Behrstock Inequality)[Beh06]
There exists a positive constant $B_{0}$ with the property that for
subsurfaces $Y,Z\subseteq S$ such that $Y\pitchfork Z$ and a partial marking
$\mu$ such that $\mu\pitchfork Y$ and $\mu\pitchfork Z$ we have
$\min\\{d_{Y}(\partial{Z},\mu),d_{Z}(\partial{Y},\mu)\\}\leq B_{0}.$
We recall the Consistency Theorem of Behrstock and Minsky from [BKMM12]. Note
that there this theorem is stated and proved for markings. It is
straightforward to verify that all of their arguments go through for pants
decompositions, considering only non-annular subsurfaces (excluding all
annular subsurfaces).
###### Theorem 2.9.
(Consistency)([BKMM12] Theorem 4.3) Given $F_{1},F_{2}\geq 1$, there is a
constant $F>0$ with the following property. Let $(x_{Y})_{Y\subseteq S}$ be a
tuple where $x_{Y}\in\mathcal{C}(Y)$ for each non-annular subsurface
$Y\subseteq S$. Suppose that $(x_{Y})_{Y\subseteq S}$ satisfies the following
two conditions:
1. 1)
If $Y\pitchfork Z$ then
$\min\\{d_{Y}(x_{Y},\partial{Z}),d_{Z}(x_{Z},\partial{Y})\\}\leq F_{1}$, and
2. 2)
If $Y\subseteq Z$ then $d_{Y}(x_{Y},\pi_{Y}(x_{Z}))\leq F_{2}.$
Then there is a $P\in P(S)$ such that for every subsurface $Y\subseteq S$ we
have:
$d_{Y}(P,x_{Y})\leq F.$
###### Remark 2.10.
Note that given a partial marking $\mu$, the tuple of $x_{Y}=\pi_{Y}(\mu)$
satisfies the above two conditions with $F_{1}=B_{0}$ ($B_{0}$ is the constant
from Theorem 2.8) and $F_{2}=1$.
We recall the following relation on subsurfaces of $S$ from [BKMM12].
###### Definition 2.11.
(Partial order in pants and marking graphs) Fix a tuple $(x_{Y})_{Y\subseteq
S}$ where $x_{Y}\subset\mathcal{C}(S)$ satisfying the consistency conditions
in Theorem 2.9. Assume that $F_{1}>\min\\{F_{2},B_{0},G\\}$, where $B_{0}$ is
the constant from Theorem 2.8. We define the following two relations on
proper, connected, essential subsurfaces of $S$:
1. (i)
Given a positive integer $k$, $Y\prec_{k}Z$, if $Y\pitchfork Z$ and
$d_{Y}(x_{Y},\partial{Z})\geq k(F_{1}+4)$.
2. (ii)
Given a positive integer $k$, $Y\ll_{k}Z$, if $Y\pitchfork\partial{Z}$ and
$d_{Y}(x_{Y},\partial{Z})\geq k(F_{1}+4)$
$Y\pitchfork Z$ if $\partial{Y}\pitchfork Z$ and $\partial{Z}\pitchfork Y$, so
it is immediate from the definition that $Y\prec_{k}Z\Longrightarrow
Y\ll_{k}Z$, but not the other way round. Moreover, if $k>l$ then
$Y\prec_{k}Z\Longrightarrow Y\prec_{l}Z$, and similarly
$Y\ll_{k}Z\Longrightarrow Y\ll_{l}Z$. These notions of partial order are
closely related to the partial order of subsurface in Proposition 2.15. The
following theorem provides for some useful transitivity properties of the
partial order defined above.
###### Theorem 2.12.
([BKMM12] Lemma 4.1)
Given an integer $k>1$, let $(x_{Y})_{Y\subseteq S}$ be a tuple satisfying the
consistency conditions and consider the partial order defined by it.
Let $U,V$ and $W$ be subsurfaces such that $x_{U},x_{V},x_{W}\neq\emptyset$
then we have:
1. (1)
If $U\prec_{k}V$ and $V\ll_{2}W$ then $U\prec_{k-1}W$, also if $\mu$ is a
partial marking, and if $U\prec_{k}V$ and $V\ll_{2}\mu$, then
$U\prec_{k-1}\mu$.
2. (2)
If $U\ll_{k}V$ and $V\ll_{2}W$ then $U\ll_{k-1}W$, also if $\mu$ is a partial
marking, and if $U\ll_{k}V$ and $V\ll_{2}\mu$ then $U\ll_{k-1}\mu$.
3. (3)
If $U\pitchfork V$ and both $U\ll_{k}\mu$ and $V\ll_{k}\mu$ for a partial
marking $\mu$, then $U$ and $V$ are $\prec_{k-1}-$ordered, that is, either
$U\prec_{k-1}V$ or $V\prec_{k-1}U$.
### 2.2. Hierarchies and their resolutions in the pants and marking graph
The hierarchies of tight geodesics of curve complexes of subsurfaces of a
surface were introduced by Masur and Minsky in [MM00]. See also [Min10],
[BKMM12]. Hierarchy paths are quasi-geodesics in the pants and marking graph
of a surface, with quantifiers depending only on the topological type of $S$,
obtained by resolving hierarchies. Although in most of this paper we use only
hierarchy resolution paths and their properties given in Theorem 2.13, a good
understanding of the structure of hierarchies themselves will be extremely
useful for the reader to follow our arguments. Given a geodesic $h$ in the
curve complex of a subsurface $Y$, we refer to $Y$ as the support of $h$ and
denote it by $D(h)$. Let $(\mu^{-},\mu^{+})$ be a pair of partial makings or
laminations in $S$, a complete Hierarchy of tight geodesics $H$ inductively
associates to the pair $(\mu^{-},\mu^{+})$ a collections of tight geodesics in
the curve complex of $S$ and the curve complexes of its subsurfaces. A tight
geodesic in $\mathcal{C}(Y)$, is a geodesic with the property that any three
of its consecutive vertices fill the surface $Y$. Here we list the properties
which describe the inductive and layered structure of the hierarchy $H$, for
more detail see the above references.
1. (1)
There is a unique main geodesic $g_{S}$ with $D(g_{S})=S$, whose endpoints lie
on $\operatorname{base}(\mu^{-})$ and $\operatorname{base}(\mu^{+})$.
2. (2)
For any geodesic $h\in H$ other than $g_{S}$, there exists another geodesic
$k\in H$ such that, for some simplex $v\in k$, $D(h)$ is either a component of
$D(k)\backslash v$, or an annulus whose core is a component of $v$. We say
that $D(h)$ is a component domain of $k$.
3. (3)
A subsurface in $S$ can occur as the domain of at most one geodesic in $H$.
Infinite hierarchies: An infinite hierarchy is a hierarchy $H$ with associated
infinite tight geodesic rays or lines in the curve complexes of subsurfaces.
Here each of the end points $\mu^{\pm}$ is a union of minimal, filling
laminations supported on disjoint subsurfaces. Each lamination is the point at
infinity of an infinite geodesic of the hierarchy in the curve complex of the
subsurface supporting the geodesic (see Theorem 2.2). The existence of
infinite hierarchies is proved in [Min10].
Hierarchy resolution paths: These are path which comprise for a set of
transitive quasi-geodesics in the pants (marking) graphs of the surfaces $S$
with constants depending only on the topological type of the surface. In $\S$5
we prove a condition for stability of hierarchy resolution paths in the pants
graph.
A pants graph resolution of a hierarchy $H$ is a path in the pants graph
denoted by $\rho:[n,m]\to P(S)$, where
$[m,n]\subseteq\mathbb{Z}\cup\\{\pm\infty\\}$. For any $i\in[m,n]$,
$\rho(i+1)$ is obtained from $\rho(i)$ by an elementary move ($\rho(i+1)$ and
$\rho(i)$have pants distance $1$). Similarly a marking graph resolution of $H$
is a path of clean markings such that any two consecutive markings differ by
an elementary move. Given a hierarchy path $\rho$ we denote
$|\rho|=\\{\rho(i):i\in[n,m]\\}$
which is a subset of the pants graph. A resolution path of a hierarchy
consists of slices of the hierarchy. Each slice is the union of vertices of
geodesics in curve complexes of subsurfaces of $S$. The supporting domains of
these geodesics consist a tower of nested domains of $H$. In the following
theorem we some of the list properties of these quasi-geodesics (see also
[BMM11], [BM08]) which we will use frequently in this paper. The main feature
of these properties is that they are encoded in the subsurface coefficients of
the pair.
###### Theorem 2.13.
(Properties of pants hierarchy resolution paths)
Given partial markings or laminations $(\mu^{-},\mu^{+})$, there are hierarchy
(resolution) paths $\rho:[m,n]\to P(S)$
$([m,n]\subseteq\mathbb{Z}\cup\\{\pm\infty\\})$ with $\rho(m)=\mu^{-}$ and
$\rho(n)=\mu^{+}$ satisfying the following list of properties. In what follows
$M_{1},M_{2}>0$ $(M_{1}\geq 2M_{2})$ are constants depending only on the
topological type of $S$.
1. (1)
Given a component domain $Y$, there is a connected interval
$J_{Y}\subseteq[m,n]$ and a geodesic $g_{Y}\subset\mathcal{C}(Y)$ such that
for each $j\in J_{Y}$, $\partial{Y}\subset\rho(i)$ and there is a simplex
$v\in g_{Y}$ such that $v\in\rho(j)$ $(v\subset g_{Y}\cap\rho(j))$.
2. (2)
There is a constant $M_{1}>0$, depending only on the topological type of $S$,
such that an essential, non-annular subsurface $Y\subseteq S$ with
$d_{Y}(\mu^{-},\mu^{+})>M_{1}$ is a component domain of $\rho$.
3. (3)
(Monotonicity) Let $i,j\in J_{Y}$, $v=\rho(i)\cap g_{Y}$ and $w=\rho(j)\cap
g_{Y}$. Then $i\leq j$ if and only if $v\leq w$ as vertices along $g_{Y}$.
4. (4)
(Bounded projection) Let $Y$ be a component domain of $\rho$ and
$J_{Y}=[j^{-},j^{+}]$. If $i>j^{+}$ then $d_{Y}(\rho(i),\rho(j^{+}))\leq
M_{2}$, and if $i<j^{-}$ then $d_{Y}(\rho(i),\rho(j^{-}))\leq M_{2}$.
5. (5)
(Hausdorff distance bound) Let $W\subseteq S$ be a subsurface. For any
$i\in[m,n]$ there is an $x\in\operatorname{hull}_{W}(\mu^{-},\mu^{+})$ such
that
$d_{W}(\rho(i),x)\leq M_{2}$
also for any $x\in\operatorname{hull}_{W}(\mu^{-},\mu^{+})$ there is an
$i\in[m,n]$ such that the above bound holds. Here
$\operatorname{hull}_{W}(\mu^{-},\mu^{+})$ is the convex hull of
$\pi_{W}(\mu^{-})$ and $\pi_{W}(\mu^{+})$ in $\mathcal{C}(W)$. In other words,
the Hausdorff distance of $\operatorname{hull}_{W}(\mu^{-},\mu^{+})$ and
$\pi_{W}(|\rho|)$ in $\mathcal{C}(W)$ is less than $M_{2}$. Note that here we
do not necessarily assume that $W$ is component domain of $\rho$. When $W$ is
a component domain the statement follows from (4).
6. (6)
(No backtracking) Let $i,j,k\in[m,n]$ with $i\leq j\leq k$. Then for any
subsurface $Y\subseteq S$, $d_{Y}(\rho(i),\rho(k))+M_{2}\geq
d_{Y}(\rho(i),\rho(j))+d_{Y}(\rho(j),\rho(k))$.
In this theorem $J_{Y}\subset[m,n]$ consists of all $j\in[m,n]$ such that
$\rho(j)$ is a slice of $H(\mu^{-},\mu^{+})$ containing $(v,g_{Y})$, where
$g_{Y}\in H$ is the tight geodesic supported on $Y$ and $v\in g_{Y}$. The fact
that $J_{Y}$ is an interval was proved in Lemma 4.9 of [BCM12]. This explains
property (1). Property (2) is Lemma 6.1 (Large Link Lemma) in [MM00]. Property
(4) is a consequence of the Bounded Geodesic Image Theorem and is established
in the proof of Large Link Lemma. Property (5) is established in the proof of
Lemma 5.14 in [Min10]. Property (6) is a straight forward consequence of
properties (4) and (5). Property (3) is a consequence of the definition of
slices of hierarchy and the partial order of them defined using the partial
order of point geodesics of $H$ in $\S 5$ of [MM00].
###### Remark 2.14.
Given $(\mu^{-},\mu^{+})$, a marking hierarchy resolution path
$\tilde{\rho}:[m,n]\to M(S)$ $([m,n]\subseteq\mathbb{Z}\cup\\{\pm\infty\\})$
between $\mu^{-}$ and $\mu^{+}$ satisfies the same list of properties as pants
hierarchy resolution paths, besides in properties (1)-(6) subsurfaces $Y$ and
$W$ can be annular subsurfaces as well.
Partial order on subsurfaces along hierarchies is introduced in [MM00]. It
roughly gives the order in which the intervals $J_{Y}$ appear along the
component domains of a resolution of the hierarchy. In this paper we mainly
need the following weaker version of it.
###### Proposition 2.15.
(Order of subsurfaces) Fix the constant $M=M_{1}+B_{0}+4$. Let $Y,W\subseteq
S$ be subsurfaces. Suppose that $Y\pitchfork W$, $d_{Y}(\mu,\mu^{\prime})>4M$
and $d_{W}(\mu,\mu^{\prime})>4M$. Then one and only one of the following
inequalities holds.
1. (a)
$d_{Y}(\mu,\partial{W})\geq 2M$ and $d_{W}(\mu^{\prime},\partial{Y})>2M$.
2. (b)
$d_{W}(\mu,\partial{Y})\geq 2M$ and $d_{Y}(\mu^{\prime},\partial{W})>2M$.
If (a) holds we denote $Y<W$. If (b) holds we denote $W<Y$.
Furthermore, the relation $<$ has the following properties.
* •
It is transitive.
* •
Suppose that $W<Y$, $j\in J_{Y}-J_{W}$ and $i\in J_{W}-J_{Y}$, then $i\leq j$.
###### Proof.
Since $d_{Y}(\mu,\mu^{\prime})>4M$, by the triangle inequality either
$d_{Y}(\mu,\partial{W})\geq 2M$ or $d_{Y}(\mu^{\prime},\partial{W})\geq 2M$.
First assume that $d_{Y}(\mu,\partial{W})\geq 2M$ i.e. (a) holds. Then by
Theorem 2.8 (Behrstock Inequality) we get $d_{W}(\mu,\partial{Y})\leq
B_{0}<M$. So (b) can not hold. Furthermore, by the assumption of the
proposition, $d_{W}(\mu,\mu^{\prime})>4M$. The last two inequalities combined
by the triangle inequality imply that
$d_{W}(\mu^{\prime},\partial{Y})>3M-\operatorname{diam}_{W}(\mu)\geq 3M-2>2M$.
When $(b)$ holds a similar argument shows that $d_{Y}(\mu,\partial{W})\leq
B_{0}$, thus (a) does not hold and using the assumption
$d_{Y}(\mu,\mu^{\prime})>4M$ we get $d_{Y}(\mu^{\prime},\partial{W})>2M$.
To prove the transitivity of $<$, let $W<Y$ and $Y<Z$. Then as we saw above
$d_{Y}(\mu^{\prime},\partial{W})>2M$ and $d_{Z}(\mu^{\prime},\partial{Y})>2M$.
The second inequality and the Behrstock inequality imply that
$d_{Y}(\mu,\partial{Z})\leq M$. This and first inequality combined by the
triangle inequality imply that
$d_{Y}(\partial{W},\partial{Z})>M-2.$
But $M-2\geq 2$, thus $d_{Y}(\partial{W},\partial{Z})\geq 3$. So $\partial{Y}$
and $\partial{Z}$ fill $W$. Thus $\partial{Y}\pitchfork\partial{Z}$ and
$Y\pitchfork Z$. $M-2>B_{0}$ so by the above inequality we also have that
$d_{Y}(\partial{W},\partial{Z})>B_{0}$. Then by the Behrstock inequality
$d_{W}(\partial{Y},\partial{Z})\leq B_{0}$. Now we have
$\displaystyle d_{W}(\mu,\partial{Z})$ $\displaystyle\geq$ $\displaystyle
d_{W}(\mu,\mu^{\prime})-d_{W}(\partial{Y},\mu^{\prime})-d_{W}(\partial{Y},\partial{Z})-\operatorname{diam}_{W}(\partial{Y})-\operatorname{diam}_{W}(\mu^{\prime})$
$\displaystyle\geq$ $\displaystyle 4M-2B_{0}-3>2M.$
This finishes proving that $W<Z$.
We proceed to prove the last assertion of the proposition. Since $W<Y$,
$d_{W}(\partial{Y},\mu)\geq 2M$ so by Theorem 2.8, $d_{Y}(\partial{W},\mu)\leq
M$. Moreover, $i\in J_{W}-J_{Y}$ so $\rho(i)\supset\partial{W}$. Thus
$d_{Y}(\rho(i),\mu)\leq M(*).$
Now suppose that $i>j$, then since $j\in J_{Y}-J_{W}$, $j$ is greater than the
right end pint of the interval $J_{Y}$. So Theorem 2.13 (4) implies that
$d_{Y}(\rho(i),\mu^{\prime})\leq M$. This inequality combined by
$d_{Y}(\mu,\mu^{\prime})>4M$ with the triangle inequality implies that
$d_{Y}(\rho(i),\mu)>3M-2>M$
which contradicts the upper bound (*). Thus $i\leq j$ as was desired. ∎
In this paper we deduce almost all of the properties of hierarchy paths we
need from the Theorem 2.13 and Proposition 2.15. In a couple of occasions we
need some finer properties of hierarchies and their resolutions where we
provide a reference.
### 2.3. $\Sigma-$hulls and their projections
In this section we recall the subsets of pants graph called $\Sigma-$hull and
their projection introduced in [BKMM12]. The projection is coarsely the
closest point projection on the $\Sigma-$hull as a subset of the pants graph
of the surface. Note that in [BKMM12] these notions are introduced in the
context of marking graphs. Other places were variations of this projection is
used are [Beh06],[BM08],[BMM11]. Given a pair of partial markings or
laminations $(\mu^{-},\mu^{+})$ and $\epsilon>0$ define
$\Sigma_{\epsilon}(\mu^{-},\mu^{+}):=\\{P\in
P(S):d_{Y}(P,\operatorname{hull}_{Y}(\mu^{-},\mu^{+}))\leq\epsilon$ for every
non-annular subsurface $W\subseteq S\\}$
Here $\operatorname{hull}_{Y}(\mu^{-},\mu^{+})$ is the set of all geodesics in
$\mathcal{C}(Y)$ connecting $\pi_{Y}(\mu^{-})$ to $\pi_{Y}(\mu^{+})$. Note
that since $\mathcal{C}(Y)$ is $\delta_{Y}-$hyperbolic all of the geodesics
connecting $\pi_{Y}(\mu^{-})$ and $\pi_{Y}(\mu^{+})$ uniformly fellow travel
each other.
###### Theorem 2.16.
([BKMM12] Proposition 5.2) There is an $F>0$ depending only on the topological
type of the surface such that for every $\epsilon>F$ there is a coarse map
(projection)
$\Pi:P(S)\to\Sigma_{\epsilon}(\mu^{-},\mu^{+})$
with the following properties:
1. (1)
For every non-annular subsurface $Y\subseteq S$ we have
$d_{Y}(\Pi P,\operatorname{hull}_{Y}(\mu^{-},\mu^{+}))\leq F$
2. (2)
$\Pi\big{|}_{\Sigma_{\epsilon}(\mu^{-},\mu^{+})}$ is uniformly close to the
identity.
3. (3)
$\Pi$ is coarse-Lipschitz.
###### Remark 2.17.
This theorem in [BKMM12] is stated and proved in the context of marking graph.
But it is straightforward to versify that all of their arguments go through in
the context of pants graph excluding all annular subsurfaces.
The coarse Lipschitz refers to the fact that the projection is defined on the
vertices of the pants graph and does not say any thing when two points are
within distance $1$.
Using the distance formula (2.2) $\Pi P$ coarsely minimizes the distance
between $P$ and the $\Sigma_{\epsilon}(\mu^{-},\mu^{+})$ in the pants graph.
The main ingredient of the proof of Theorem 2.16 is to prove that there are
positive constants $F_{1}$ and $F_{2}$, depending only on the topological type
of the surface, such that the tuple $(x_{Y})_{Y\subseteq S}$, where each
$x_{Y}\in\mathcal{C}(Y)$ is a nearest point to $\pi_{Y}(P)$ on
$\operatorname{hull}_{Y}(\mu^{-},\mu^{+})$, satisfies the consistency
conditions of Theorem 2.9 (Consistency Theorem). Then the consistency theorem
implies that there is a constant $F>0$ and a pants decomposition, denoted by
$\Pi P$, such that $d_{Y}(\Pi P,\operatorname{hull}_{Y}(\mu^{-},\mu^{+}))\leq
F$ for every $Y\subseteq S$.
## 3\. The Weil-Petersson metric and its synthetic properties
We start with some basic facts about Teichmüller theory and the Weil-Petrsoon
(WP) metric, and through out will set up our notation. Let $S$ be a surface
with genus $g$ and $b$ boundary components. A point in the Teichmüller space
of $S$, denoted by $\operatorname{Teich}(S)$, is a complete, finite area
hyperbolic surface $x$ equipped with a diffeomorphism $h:S\to x$. The map $h$
is a marking for $x$. Two marked surfaces $h_{1}:S\to x_{1}$ and $h_{2}:S\to
x_{2}$ define the same point in $\operatorname{Teich}(S)$ if and only if
$h_{2}\circ h_{1}^{-1}:x_{1}\to x_{2}$ is isotopic to an isometry. The mapping
class group of the surface $S$, denoted by $\operatorname{Mod}(S)$, is the
group of isotopy classes of orientation preserving self diffeomorphisms of
$S$. $\operatorname{Mod}(S)$ acts on $\operatorname{Teich}(S)$ by remarking as
follows: an element $f\in\operatorname{Mod}(S)$ maps a marked surface $h:S\to
x$ to the marked surface $h\circ f:S\to x$. The quotient
$\operatorname{Teich}(S)/\operatorname{Mod}(S)$ is the moduli space of $S$
denoted by $\mathcal{M}(S)$. Given a point $f:S\to x$ in the Teichmüller space
we usually drop the marking map and denote it by $x$. We denote the point in
moduli space corresponding to the $\operatorname{Mod}(S)$ orbit of $x$ by
$\hat{x}$.
Given $\epsilon>0$, the $\epsilon-$thick part of Teichmüller space is
$\\{x\in\operatorname{Teich}(S):\operatorname{inj}(x)\geq\epsilon\\}$. Here
$\operatorname{inj}(x)$ is the injectivity radius of the hyperbolic surface
$x$. The $\epsilon-$thin part is
$\\{x\in\operatorname{Teich}(S):\operatorname{inj}(x)\leq\epsilon\\}$. Let
$\epsilon>0$ be small enough such that by the Collar lemma ($\S 4.1$ of
[Bus10]) on any complete hyperbolic surface there is not any pair of
intersecting closed geodesic with length less than or equal of $\epsilon$. Let
$\epsilon^{\prime}>0$. Given a multi-curve $\sigma$ (a simplex in
$\mathcal{C}(S)$) we define the following regions in the Teichmüller space:
* •
$U_{\epsilon}(\sigma):=\\{x\in\operatorname{Teich}(S):\ell_{\alpha}(x)\leq\epsilon$
for any $\alpha\in\sigma\\}$
* •
$U_{\epsilon,\epsilon^{\prime}}(\sigma):=\\{x\in\operatorname{Teich}(S):\ell_{\alpha}(x)\leq\epsilon$
for any $\alpha\in\sigma$ and $\ell_{\alpha^{\prime}}(x)>\epsilon^{\prime}$
for every $\alpha^{\prime}\notin\sigma\\}$
Here we recall some properties of the Weil-Petersson metric and its geodesics
which will be used in this paper. References for these material are [Wol03],
[Wol08],[Wol10], see them also for further references.
Given holomorphic quadratic differentials $\varphi,\psi\in
T_{x}^{*}\operatorname{Teich}(S)$ the Weil-Petersson $L^{2}$ co-product is
defined by
$\mathcal{R}e(\int_{x}\varphi\overline{\psi}\rho^{-2})$
where $\rho(z)^{2}|dz|^{2}$ is the hyperbolic metric of the marked hyperbolic
surface $x$. This co-product induces a norm on Teichmüller space via the
standard pairing of quadratic differentials and measurable Beltrami
differentials on $x$ whcih is defined by $\int_{x}\varphi\mu$. Any measurable
Beltrami differential presents a vector in $T_{x}\operatorname{Teich}(S)$ and
the Weil-Petersson metric on Teichmüller space is defined by the polarization
of this norm. In this paper we study the global behavior of geodesics of this
metric.
The Weil-Petersson metric is a Riemannian metric with negative sectional
curvatures which is invariant under the action of the mapping class group of
the surface. It is an incomplete metric, however it is geodesically convex.
The negative curvature and convexity imply that the completion of Teichmüller
space with respect to the WP metric $\overline{\operatorname{Teich}(S)}$ is a
$\operatorname{CAT}(0)$ space. For background about $\operatorname{CAT}(0)$
space see for example [BH99].
Length-functions: Given any $\alpha\in\mathcal{C}_{0}(S)$ the $\alpha-$length-
function
$\ell_{\alpha}:\operatorname{Teich}(S)\to\mathbb{R}^{+}$
assigns to $x\in\operatorname{Teich}(S)$ the length of the geodesic
representative of $\alpha$ on the marked hyperbolic surface $x$.
The notion of length-function has a natural extension to the space of measured
geodesic laminations (see [Bon01]). Given a measured geodesic lamination
$\mathcal{L}$ we denote the $\mathcal{L}-$length-function by
$\ell_{\mathcal{L}}(x)$.
Fenchel-Nielsen coordinates and twist parameters: Given a pants decomposition
$P$, a Fenchel-Nielsen (FN) coordinate system
$(\ell_{\gamma},\theta_{\gamma})_{\gamma\in P}$ maps $\operatorname{Teich}(S)$
to $\prod_{\gamma\in P}\mathbb{R}_{\gamma}^{+}\times\mathbb{R}_{\gamma}$. The
first coordinate of the pair
$\mathbb{R}_{\gamma}^{+}\times\mathbb{R}_{\gamma}$ is the $\gamma-$length-
function and the second coordinate is a twist parameter about $\gamma$. For
more detail about Fenchel-Nielsen coordinates and twit parameters see $\S 3$
of [Bus10]. We denote the Dehn twist about a curve $\gamma$ by
$\mathcal{D}_{\gamma}$, it is defined as follows: Let
$x\in\operatorname{Teich}(S)$. Given a pants decomposition $P$ with $\gamma\in
P$ fix a FN coordinate system, $\mathcal{D}_{\gamma}(x)$ is the point with all
coordinates equal to that of $x$ except
$\theta_{\gamma}(\mathcal{D}_{\gamma}(x))=\theta_{\gamma}(x)+2\pi$.
The Weil-Petersson completion of Teichmüller space and the completion strata:
The incompleteness of the Weil-Petersson metric is due to existance of finite
length paths in Teichmüller space along which length of a curve converges to
zero, [Wol10]. In [Mas76], Masur gives a concrete description of the
completion as the augmented Teichmüller space. The augmented Teichmüller space
consists of strata: Let $\sigma$ be a simplex in the augmented curve complex
$\widehat{\mathcal{C}(S)}=\mathcal{C}(S)\cup\emptyset$, a point in the
$\sigma-$stratum is a collection of marked hyperbolic metrics of connected
components of $S\backslash\sigma$, where for each curve in $\sigma$ a pair of
cusps is introduced. The topology is described via extended Fenchel-Nielsen
coordinate systems as follows: Given a pants decomposition $P$, the FN
coordinate system maps $\operatorname{Teich}(S)$ to $\prod_{\gamma\in
P}\mathbb{R}\times\mathbb{R}^{+}$. We extend the FN coordinate system
$(\ell_{\gamma},\theta_{\gamma})_{\gamma\in P}$ to allow length-functions take
value $0$ as well. Now take the quotient of $\prod_{\gamma\in
P}\mathbb{R}\times\mathbb{R}^{+}$ by identifying
$(0,\theta)\sim(0,\theta^{\prime})$ in each $\mathbb{R}^{+}\times\mathbb{R}$
factor. Let $\sigma\subset P$ then the topology near any point of the
$\sigma-$stratum is such that the map defined by the FN coordinate system is a
homeomorphism near that point.
In this topology each stratum $\mathcal{S}(\sigma)$ is the product of the
lower dimensional Teichmüller spaces of the connected components of
$S\backslash\sigma$.
Continuity of length-functions: In this paper we refer to the following
theorem as the continuity of length-functions. It is consequence of the fact
that the topology induced by the Weil-Petersson metric and the Chaubaty
topology of the Teichmüller space are the same. The Chaubaty topology is
defined using the fact that each point in Teichmüller space is the conjugacy
class of a representation of $\pi_{1}(S)$ into $PSL_{2}(\mathbb{R})$. For more
detail see the beginning of $\S 4$ of [Wol08].
###### Theorem 3.1.
(Continuity of length-functions) Suppose that a sequence of points $x_{n}\to
x$ as $n\to\infty$ in the the completion of Teichmüller space with respect to
the WP metric. Then for every $\alpha\in\mathcal{C}_{0}(S)$,
$\ell_{\alpha}(x_{n})\to\ell_{\alpha}(x)$ as $n\to\infty$.
Non-refraction of strata:
###### Theorem 3.2.
(Non-refraction)[DW03],[Wol03], [Yam04]
Let $\zeta:[0,T]\to\overline{\operatorname{Teich}(S)}$ be a WP geodesic
segment, and let $\sigma^{-}$ and $\sigma^{+}$ be the maximal simplicies in
$\mathcal{C}(S)$ such that $\zeta(0)\in\mathcal{S}(\sigma^{-})$ and
$\zeta(T)\in\mathcal{S}(\sigma^{+})$ then
$int(\zeta)\subset\mathcal{S}(\sigma^{-}\cap\sigma^{+})$
As a consequence of the Non-refraction Theorem Daskalopoulos and Wentworth in
[DW03] and Wolpert in [Wol03] show that any pseudo-Anosov element of the
mapping class group $f$ has an axis in the Teichmüller space equipped with the
WP metric. The axis is a bi-infinite WP geodesic
$Ax_{f}\subset\operatorname{Teich}(S)$ such that
$d_{\operatorname{WP}}(x,fx)=\inf_{y\in\operatorname{Teich}(S)}d_{\operatorname{WP}}(y,fy)$
for every $x\in\operatorname{Ax}_{f}$. The axis projects to a closed geodesic
in the moduli space.
Bers pants decomposition and Bers marking: By a result of Bers (see $\S 3$ of
[Bus10]) given a surface $S$ with $\chi(S)<0$, there is a constant $L_{S}>0$
(Bers constant) depending only on the topological type of $S$ such that any
complete finite area hyperbolic metric on $S$ posses a pants decomposition
(Bers pants decomposition) with the property that the geodesic representative
of any curve in the pants decomposition has length at most $L_{S}$. We call
any curve in a Bers pants decomposition a Bers curve. By the Collar Lemma
there are only finitely many Bers curves and consequently Bers pants
decompositions on a complete hyperbolic surface. Given
$x\in\overline{\operatorname{Teich}(S)}$, suppose that
$x\in\mathcal{S}(\sigma)$ ($\sigma$ is a simplex in
$\widehat{\mathcal{C}(S)}$). Then a Bers pants decomposition of $x$ is the
union of Bers pants decompositions of each of the connect components of
$S\backslash\sigma$ and $\sigma$. A Bers marking is a (partial) marking
obtained from a Bers pants decomposition by adding transversal curves with
representatives at $x$ of minimal length. We denote a Bers marking of
$x\in\overline{\operatorname{Teich}(S)}$ by $\mu(x)$. Given
$x\in\mathcal{S}(\sigma)$ the partial marking does not have any transversal to
the curves in $\sigma$.
By the following theorem of Brock the hierarchies of curves complexes and
their resolution in pants and marking graphs would play an essential role in
our study of the global behavior of WP geodesics:
###### Theorem 3.3.
(Quasi-isometric model)[Bro03] There are constants $K_{\operatorname{WP}}\geq
1$ and $C_{\operatorname{WP}}\geq 0$ depending only on the topological type of
$S$, such that the coarsely defined map
$Q:\overline{\operatorname{Teich}(S)}\to P(S)$
which assigns to $x$ a Bers pants decomposition $Q(x)$ is a
$(K_{\operatorname{WP}},C_{\operatorname{WP}})-$quasi-isometry.
Gradient of length-functions: Wolpert gives the following estimate for the
pairing of the gradients of length-functions:
###### Lemma 3.4.
[Wol08] The WP pairing of length-function gradients of curves $\alpha,\beta$
with disjoint geodesic representatives satisfies
$0<\langle\operatorname{grad}\ell_{\alpha},\operatorname{grad}\ell_{\beta}\rangle-\frac{2}{\pi}\ell_{\alpha}\delta_{\alpha\beta}=O(\ell_{\alpha}^{2}\ell_{\beta}^{2})$
where the constant of the $O$ notation depends only on $c_{0}>0$ with
$\ell_{\alpha},\ell_{\beta}\leq c_{0}$.
###### Corollary 3.5.
Given $c_{0}>0$, there is a two variable function d with the following
property. Given $l,a\in[0,c_{0}]$ such that $l>a\geq 0$. Let
$x,x^{\prime}\in\operatorname{Teich}(S)$ be such that $\ell_{\alpha}(x)\leq
l-a$ and $\ell_{\alpha}(x^{\prime})\geq l$. Then
$d_{\operatorname{WP}}(x^{\prime},x)\geq\textbf{d}(l,a)$.
###### Proof.
By Lemma 3.4 at $y\in\operatorname{Teich}(S)$ with $\ell_{\alpha}(y)\leq
c_{0}$,
$||\operatorname{grad}\ell_{\alpha}(y)||\leq(\frac{2}{\pi}\ell_{\alpha}(y)+O(\ell_{\alpha}(y))^{4})^{1/2}$
(*) where the $O$ notation constant depends only on $c_{0}$. Let $u$ be the WP
geodesic segment from $x$ to $x^{\prime}$, parametrized by arc-length. Let
$t^{*}$ be the first time that $\ell_{\alpha}(u(t))=l$. Then
$\ell_{\alpha}(u(t))\leq l$ for every $t\in[0,t^{*}]$. Using this bound and
integrating (*) we get
$a\leq|\ell_{\alpha}(u(t^{*}))-\ell_{\alpha}(u(0))|\leq\int_{0}^{t^{*}}||\operatorname{grad}\ell_{\alpha}(u(t))||dt\leq(\frac{2}{\pi}l+O(l)^{4})^{1/2}t^{*}.$
Hence
$t^{*}\geq\textbf{d}(l,a):=\frac{a}{\big{(}\frac{2}{\pi}l+O(l^{4})\big{)}^{1/2}}$.
∎
Using the estimates on pairings of the gradient of length-functions Wolpert
gives an asymptotic expansion for the WP metric near completion strata. These
expansion shows that there are asymptotic quasi-flats transverse to strata
corresponding to any pair of pinching curves. He also gives the following
estimate for the distance of a point in Teichmüller space and stratum.
###### Proposition 3.6.
(Corollary 4.10 of [Wol08]) Let $x\in\operatorname{Teich}(S)$ and $\sigma$ be
a multi-curve, then
$d_{\operatorname{WP}}(x,\mathcal{S}(\sigma))\leq\big{(}2\pi\sum_{\alpha\in\sigma}\ell_{\alpha}(x)\big{)}^{1/2}.$
Tangent cones of the Weil-Petersson completion of Teichmüller space: The
completion of Teichmüller space with the Weil-Petersson metric is a
$\operatorname{CAT}(0)$ space. Assigned to any point $p$ of a
$\operatorname{CAT}(0)$ space there is $AC_{p}$ the Alexandrov tangent cone
consisting of equivalence classes of geodesics $\zeta$ starting at the point
$p$. Two geodesics $\zeta$ and $\zeta^{\prime}$ starting at $p$ are equivalent
if their angle at $p$ in the sense of Alexandrov is equal to $0$. For more
detail about tangent cones see [BH99].
Given a multi-curve $\sigma$ on $S$ let $\chi$ be a full marking on
$S\backslash\sigma$ and consider the map
$\mathcal{L}(\zeta(t))=(\ell_{\alpha}^{1/2}(\zeta(t)),\ell_{\beta}^{1/2}(\zeta(t)))_{\alpha\in\sigma,\beta\in\chi}$
where $\zeta:[0,T]\to\overline{\operatorname{Teich}(S)}$ is a geodesic segment
with $\zeta(0)=p\in\mathcal{S}(\sigma)$. Then define
$\Lambda:AC_{p}\to\mathbb{R}^{+}$ by
$\Lambda(\zeta)=(2\pi)^{1/2}\frac{d}{dt}\Big{|}_{t=0}\mathcal{L}(\zeta(t)).$
Wolpert gives the following description of the WP Alexandrov tangent cone of
Teichmüller space at given point $p$, Theorem 4.18 of [Wol08].
###### Proposition 3.7.
(WP tangent cone) The map $\Lambda$ from the tangent cone of the WP metric at
$p$ to $\mathbb{R}_{\geq 0}^{|\sigma|}\times
T_{p}\operatorname{Teich}(S\backslash\sigma)$ is an isometry of tangent cones
with restriction of inner products. A WP geodesic $\zeta$ with $\zeta(0)=p$
and root length-function initial derivative
$\frac{d}{dt}\big{|}_{t=0}\ell_{\alpha}^{1/2}(\zeta(t))$ vanishing is
contained in the stratum $\\{\ell_{\alpha}=0\\}$,
$\mathcal{S}(\sigma)\subset\\{\ell_{\alpha}=0\\}$.
### 3.1. End invariant
In this subsection we recall the notion of end invariant for WP geodesics
introduced by Brock, Masur and Minsky in [BMM11].
###### Theorem 3.8.
(Convexity of length-functions)[Wol08] Given $\epsilon>0$ there is
$c=c(\epsilon)$ with the following property. Let
$g:(a,b)\to\operatorname{Teich}(S)$ be a WP geodesic parametrized by arc-
length and $\alpha\in\mathcal{C}_{0}(S)$. If for some $t\in(a,b)$
$\operatorname{inj}g(t)\geq\epsilon$ ($g(t)$ is a point in the
$\epsilon-$thick part of Teichmüller space), then we have
(3.1) $\ddot{\ell}_{\alpha}(g(t))\geq c\ell_{\alpha}(g(t)).$
Similar inequality holds for the length of any measured lamination
$\mathcal{L}$, which is
$\ddot{\ell}_{\mathcal{L}}(g(t))\geq c\ell_{\mathcal{L}}(g(t)).$
###### Remark 3.9.
The above estimates are local and only depend on the injectivity radius of the
surface $g(t)$.
###### Definition 3.10.
(Ending measured lamination) The weak∗ limit in $\mathcal{ML}(S)$ of any
weighted sequence of distinct Bers curves along a WP geodesic ray $r$ is an
ending measured lamination of $r$.
In [BMM11] the following notion of ending lamination for WP geodesic rays is
introduced, its existence relies on the convexity of length-functions along WP
geodesics and properties of $\operatorname{CAT}(0)$ spaces.
###### Definition 3.11.
(Ending Lamination) The union of pinching curves along a WP geodesic ray and
the geodesic laminations arising as supports of all ending measured
laminations of $r$ is the ending lamination of $r$.
###### Definition 3.12.
(End invariant of Weil-Petersson geodesics) To each open end of a geodesic
$g:(a,b)\to\overline{\operatorname{Teich}(S)}$ we associate an end invariant
which is a partial marking or a lamination. If the forward trajectory
$g|_{[0,b)}$ can be extended to $b$ such that
$g(b)\in\overline{\operatorname{Teich}(S)}$ then the forward end invariant
$\nu^{+}(g)$ is any Bers marking $\mu(g(b))$ ( there are finitely many of
them). Otherwise, $\nu^{+}(g)$ is the ending lamination of the forward
trajectory ray $g|_{[0,b)}$ which was defined above. We define the backward
end invariant $\nu^{-}(g)$ similarly by considering the backward trajectory
$g|_{(a,0]}$. We call the pair $(\nu^{-},\nu^{+})$, the end invariant of $g$.
Here we recall two properties of the ending measured laminations proved in
[BMM10]:
###### Lemma 3.13.
(Decreasing of length-functions along WP geodesic rays) Let $\mathcal{L}$ be
any ending measured lamination of a WP geodesic ray $r$, then
$\ell_{\mathcal{L}}(r(t))$ is a decreasing function.
###### Lemma 3.14.
Let $r_{n}\to r$ be a convergent sequence of rays in the WP visual sphere at
$x$. Then if $\mathcal{L}_{n}$ is any sequence of ending measured laminations
or weighted pinching curves for $r_{n}$, any representative
$\mathcal{L}\in\mathcal{ML}(S)$ of the limit of the projective classes
$[\mathcal{L}_{n}]$ in $\mathcal{PML}(S)$ has bounded length along the ray
$r$.
## 4\. Length-function control along uniformly bounded length Weil-Petersson
geodesics
In this section we study length-functions and twist parameters along sequences
of bounded length WP geodesic segments in the WP completion of Teichmüller
space.
In $\S$4.2 we will prove a modified version of Lemma 4.5 in [BMM11] about the
development of Dehn twists along sequences of uniformly bounded length WP
geodesic segments (Theorem 4.6). Corollaries 4.13 and 4.12 are somewhat
quantified versions of this theorem which provide us with a kind of twist
parameter versus length-function control along WP geodesic segments. This
control plays an important role in $\S$6 where we study the itinerary of WP
geodesics fellow traveling hierarchy paths.
The proof of Theorem 4.6 uses Wolpert’s characterization of limits of
sequences of uniformly bounded length WP geodesic segments in the Weil-
Petersson completion of Teichmüller space. In $\S$4.1 we state Wolpert’s
geodesic limit theorem and using suggestions of Jeff Brock will give an
improved version of it (Theorem 4.5). This improved version is crucial to
prove our results in $\S$4.2.
### 4.1. Limits of sequences of uniformly bounded length WP geodesic segments
In this subsection we provide a modified version of Wolpert’s geodesic limit
theorem. Given a multi-curve $\sigma$, denote by $\operatorname{tw}(\sigma)$
the subgroup of $\operatorname{Mod}(S)$ generated by the Dehn twists about the
curves in $\sigma$. Using the non-refraction property of the Weil-Petersson
completion strata (Theorem 3.2) and the fact that the quotient of any
$U_{\epsilon}(\sigma)$ by the action of $\operatorname{tw}(\sigma)$ is
compact, Wolpert provides the following characterization of the limits of
uniformly bounded length WP geodesic segments in the Teichmüller space. See
also [BMM11].
###### Theorem 4.1.
[Wol03] Let $\zeta_{n}:[0,T]\to\overline{\operatorname{Teich}(S)}$ be a
sequence of WP geodesic segments parametrized by arc-length of length $T$ in
the WP completion of the Teichmüller space. Then after possibly passing to a
subsequence there exist a partition of the interval $[0,T]$ by
$0=t_{0}<t_{1}<t_{2}<...<t_{k}<t_{k+1}=T$, simplices
$\sigma_{0},...,\sigma_{k+1}$, and simplices
$\tau_{i}=\sigma_{i-1}\cap\sigma_{i}$ ($i=1,...,k+1$) in
$\widehat{\mathcal{C}(S)}$ where $\sigma_{i}\subsetneq\tau_{i}$ for each
$1\leq i\leq k$, and a piecewise geodesic
$\hat{\zeta}:[0,T]\to\overline{\operatorname{Teich}(S)}$
with the following properties
1. (1)
$\hat{\zeta}((t_{i-1},t_{i}))\subset\mathcal{S}(\tau_{i})$, for $i=1,...,k+1$,
2. (2)
$\hat{\zeta}(t_{i})\in\mathcal{S}(\sigma_{i})$, for $i=0,...,k+1$,
3. (3)
There are elements $\psi_{n}\in\operatorname{Mod}(S)$ and
$\mathcal{T}_{i,n}\in\operatorname{tw}(\sigma_{i}-\tau_{i}\cup\tau_{i+1})$,
for $i=1,..,k$, such that after possibly passing to a subsequence
$\psi_{n}\circ\zeta_{n}(t)$ converges to $\hat{\zeta}(t)$ for every
$t\in[0,t_{1}]$ in $\overline{\operatorname{Teich}(S)}$ and for each
$i=1,..,k$, and $t\in[t_{i},t_{i+1}]$,
$\mathcal{T}_{i,n}\circ...\circ\mathcal{T}_{1,n}\circ\psi_{n}\circ\zeta_{n}(t)\to\hat{\zeta}(t)$
as $n\to\infty$. For convenience for each $i=0,1,...,k+1$ we define
(4.1)
$\varphi_{i,n}=\mathcal{T}_{i,n}\circ...\circ\mathcal{T}_{1,n}\circ\psi_{n}$
4. (4)
The elements $\psi_{n}$ are either trivial or unbounded and the elements
$\mathcal{T}_{i,n}$ are unbounded.
5. (5)
The piecewise geodesic $\hat{\zeta}$ is the minimal length path in
$\overline{\operatorname{Teich}(S)}$ joining $\hat{\zeta}(0)$ to
$\hat{\zeta}(T)$ and intersecting the strata
$\mathcal{S}(\sigma_{1}),\mathcal{S}(\sigma_{2}),...,\mathcal{S}(\sigma_{k})$
in order.
The following two lemmas which were suggested to us by Jeff Brock help us to
considerably improve the above picture of limits of uniformly bounded length
WP geodesic segments (see Theorem 4.5).
###### Lemma 4.2.
Given a sequence of WP geodesic segments
$\zeta_{n}:[0,T]\to\operatorname{Teich}(S)$, let the simplices
$\tau_{i},\;i=1,...,k+1,$ be as in Theorem 4.1. Then
$\tau_{1}=...=\tau_{k+1}$. We denote
(4.2) $\hat{\tau}\equiv\tau_{i},\;\;i=1,...,k$
###### Proof.
Let the piecewise geodesic path $\hat{\zeta}:[0,T]\to\operatorname{Teich}(S)$,
the partition $0=t_{0}<t_{1}<...<t_{k+1}=T$ and simplices
$\sigma_{i},\;i=0,...,k+1$ be as in Theorem 4.1. Let
$\delta<\min_{i=1,...,k+1}\frac{t_{i}-t_{i-1}}{2}$. Let $0\leq i\leq k$. By
Theorem 4.1 (1),
$\hat{\zeta}|_{[t_{i}-\delta,t_{i})}\subset\mathcal{S}(\tau_{i})$ and
$\hat{\zeta}|_{(t_{i},t_{i}+\delta]}\subset\mathcal{S}(\tau_{i+1})$. Moreover
by Theorem 4.1 (5) the concatenation of $\hat{\zeta}|_{[t_{i}-\delta,t_{i})}$
and $\hat{\zeta}|_{[t_{i},t_{i}+\delta]}$ is the distance minimizing path in
$\overline{\operatorname{Teich}(S)}$ joining $\hat{\zeta}(t_{i}-\delta)$ to
$\hat{\zeta}(t_{i}+\delta)$ and intersecting $\mathcal{S}(\sigma_{i})$. Recall
that $\tau_{i+1}\subseteq\sigma_{i}$, so $\alpha\in\sigma_{i}$, then as
Wolpert shows on page 328 of [Wol08] the following equality of the one-sided
derivatives of the square root of the $\alpha-$length-function holds at
$t=t_{i}$,
(4.3)
$\frac{d}{dt}\Big{|}_{t=t_{i}^{+}}\ell_{\alpha}^{1/2}(\hat{\zeta}|_{[t_{i},t_{i}+\delta]})=-\frac{d}{dt}\Big{|}_{t=t_{i}^{-}}\ell_{\alpha}^{1/2}(\hat{\zeta}|_{[t_{i}-\delta,t_{i}]}).$
$\hat{\zeta}|_{[t_{i},t_{i}+\delta]}\subset\overline{\mathcal{S}(\tau_{i+1})}$,
so $\ell_{\alpha}^{1/2}(\hat{\zeta}(t))=0$ for all $t\in[t_{i},t_{i}+\delta]$
and thus
$\frac{d}{dt}|_{t=t_{i}^{+}}\ell_{\alpha}^{1/2}(\hat{\zeta}|_{[t_{i},t_{i}+\delta]})=0$.
Then by (4.3)
$\frac{d}{dt}|_{t=t_{i}^{+}}\ell_{\alpha}^{1/2}(\hat{\zeta}|_{[t_{i},t_{i}+\delta]})=0$.
So by Proposition 3.7,
$\hat{\zeta}([t_{i}-\delta,t_{i}])\subset\overline{\mathcal{S}(\alpha)}$.
Moreover, again by Theorem 4.1 (1)
$\hat{\zeta}([t_{i}-\delta,t_{i}))\subset\mathcal{S}(\tau_{i})$. The last two
inclusions imply that $\alpha\in\tau_{i}$. This holds for every
$\alpha\in\tau_{i}$, so we conclude that $\tau_{i+1}\subseteq\tau_{i}$.
Exchanging the role of $\tau_{i}$ and $\tau_{i+1}$ a similar argument implies
that $\tau_{i}\subseteq\tau_{i+1}$. Thus $\tau_{i}=\tau_{i+1}$ for $i=1,...,k$
and we get $\tau_{1}=...=\tau_{k+1}$ as was desired. ∎
###### Lemma 4.3.
Let $\zeta_{n}:[0,T]\to\operatorname{Teich}(S)$ be a sequence of WP geodesic
segments parametrized by arc-length. Let $\mathcal{T}_{i,n},\;i=1,...,k,$ and
the simplices $\sigma_{i},\;i=0,...,k+1,$ be as in Theorem 4.1 and the simplex
$\hat{\tau}$ be as in (4.2). Let $1\leq i\leq k$. Suppose that
$\sigma_{i}-\hat{\tau}\neq\emptyset$ and $\gamma\in\sigma_{i}-\hat{\tau}$,
then the power of $\mathcal{D}_{\gamma}$ in $\mathcal{T}_{i,n}$ goes to
$\infty$ as $n\to\infty$.
###### Proof.
Let the elements of mapping class group $\varphi_{i,n},$ $i=1,...,k+1,$ be as
in (4.1). For $i=1,...,k$ define
$\sigma_{i,n}=\varphi_{i-1,n}^{-1}(\sigma_{i})$
also define the geodesic segments
$\zeta_{i,n}(t)=\varphi_{i-1,n}\circ\zeta_{n}(t)$ for $t\in[t_{i-1},t_{i+1}]$
where $0=t_{0}<t_{1}<...<t_{i}<t_{i+1}<...<t_{k+1}=T$ is the partition from
Theorem 4.1. We claim that
###### Claim 4.4.
There are $\epsilon_{1},\epsilon_{2}>0$ and $\delta>0$ depending only on the
sequence $\zeta_{n}$ such that for each $i=1,...,k$ and every $n$ sufficiently
large
1. (i)
$\ell_{\gamma}(\zeta_{i,n}(t_{i}\pm\delta))\leq\epsilon_{2}$ for every
$\gamma\in\sigma_{i}$, and
2. (ii)
The injectivity radius of the points $\zeta_{i,n}(t_{i}\pm\delta))$ is bounded
below by $\epsilon_{1}$ away from the collars of the curves in $\hat{\tau}$.
The claim follows from the limit picture of geodesics $\zeta_{n}$ and
continuity of length-functions. To see this, let
$\delta<\min_{i=1,...,k+1}\frac{t_{i}-t_{i-1}}{2}$. Fix $1\leq i\leq k$. By
Lemma 4.2 the two points $\hat{\zeta}(t_{i}\pm\delta)$ are in the stratum
$\mathcal{S}(\hat{\tau})$ and by Theorem 4.1 (2),
$\hat{\zeta}(t_{i})\in\mathcal{S}(\sigma_{i})$. Thus there are
$\epsilon^{\prime}_{1},\epsilon^{\prime}_{2}>0$ such that:
1. (i’)
$\ell_{\gamma}(\hat{\zeta}(t_{i}\pm\delta))\leq\epsilon^{\prime}_{2}$ for
every $\gamma\in\sigma_{i}$, and
2. (ii’)
The injectivity radius of the points $\zeta_{i,n}(t_{i}\pm\delta))$ is bounded
below by $\epsilon^{\prime}_{1}$ away from the collars of the curves in
$\hat{\tau}$.
Moreover, by Theorem 4.1 (3) for every $t\in[t_{i-1},t_{i}]$,
$\zeta_{i,n}(t)\to\hat{\zeta}(t)$ as $n\to\infty$. Thus by the continuity of
length-functions for $n$ sufficiently large the bounds (i) and (ii) at
$\zeta_{i,n}(t_{i}-\delta)$ for $\epsilon_{1}=\frac{\epsilon^{\prime}_{1}}{2}$
and $\epsilon_{2}=2\epsilon^{\prime}_{2}$ follow from the bounds in (i’) and
(ii’), respectively.
Since $\varphi_{i,n}=\mathcal{T}_{i,n}\circ\varphi_{i-1,n}$, by Theorem 4.1
(3) for every $t\in[t_{i},t_{i+1}]$,
$\mathcal{T}_{i,n}\circ\zeta_{i,n}(t)\to\hat{\zeta}(t)$ as $n\to\infty$. Then
by the continuity of length-functions from (i’) we obtain that
$\ell_{\gamma}(\mathcal{T}_{i,n}\circ\zeta_{i,n}(t_{i}+\delta))\leq\epsilon_{2}$
for every $\gamma\in\sigma_{i}$, and from (ii’) we obtain that the injectivity
radius of the point $\zeta_{i,n}(t_{i}-\delta))$ is bounded below by
$\epsilon^{\prime}_{1}$ away from the collars of the curves in $\hat{\tau}$.
It follows form Theorem 4.1 (3) and Lemma 4.2 that for each $n$,
$\mathcal{T}_{i,n}\in\operatorname{tw}(\sigma_{i}-\hat{\tau})$. So applying
$\mathcal{T}_{i,n}$ does not change the length of every curve in $\sigma_{i}$
and injectvity radius. Thus the bounds (i) and (ii) at
$\zeta_{i,n}(t_{i}+\delta)$ follow from the ones we just established. This
finishes the proof of the claim.
We proceed to prove the lemma. Fix $\gamma\in\sigma_{i}-\hat{\tau}$. Let
$h_{1}:S\to\zeta_{i,n}(t_{i}-\delta)$ be the marking of the surface
$\zeta_{i,n}(t_{i}-\delta)$. Let
$\beta\in\mathcal{C}_{0}(S\backslash\hat{\tau})$ be a curve which has minimal
intersection number (1 or 2) with $\gamma$ and does not intersect any curve in
$\sigma_{i}-\gamma$ so that $h_{1}(\beta)$ has minimal length at
$\zeta_{i,n}(t_{i}-\delta)$, see Figure 1.
Figure 1. $\beta\in\mathcal{C}_{0}(S\backslash\hat{\tau})$ is a curve which
has minimal intersection number one or two with $\gamma$ and does not
intersect any curve in $\sigma_{i}-\gamma$ so that has minimal length at
$\zeta_{i,n}(t_{i}-\delta)$.
Realize the curves $h_{1}(\beta)$ and $h_{1}(\gamma)$ as geodesics on
$\zeta_{i,n}(t_{i}-\delta)$. Denote the collar of $h_{1}(\gamma)$ by
$C(h_{1}(\gamma))$. Denote the length of each component of the boundary of the
collar by $C$. Further let $w$ be the width of the collar. Lifting the picture
to the universal cover the length of $h_{1}(\beta)\cap C(h_{1}(\gamma))$ is
bounded above by $w+2C$, see Figure 2. Further the length of $h_{1}(\beta)$
outside $C(h_{1}(\gamma))$ is bounded above by the diameter of
$\zeta_{i,n}(t_{i}-\delta)$ out side the collars of curves in $\hat{\tau}$
plus $2C$. By a compactness argument the diameter is bounded above by a
constant depending only on the injectivity radius of the surface out side the
collars of the curves in $\hat{\tau}$ and $C$. Then the length of
$h_{1}(\gamma)$ is bounded above by $w+2C+2C$ plus the diameter. Claim 4.4
(ii) provides the lower bound $\epsilon_{1}$ for the length of $\gamma$ and
consequently an upper bound for $C$ and $w$. Part (i) of the claim provides
the upper bound $\epsilon_{2}$ for the injectivity radius and consequently an
upper bound for the diameter. Thus there is $L$ depending only on
$\epsilon_{1},\epsilon_{2}$ so that
(4.4) $\ell_{\beta}(\zeta_{i,n}(t_{i}-\delta))\leq L.$
Let $h_{2}:S\to\zeta_{i,n}(t_{i}+\delta)$ be the marking of
$\zeta_{i,n}(t_{i}+\delta)$. $h_{2}(\beta)=h_{1}\circ\mathcal{T}_{i,n}(\beta)$
and $h_{2}(\gamma)=h_{1}(\gamma)$. Denote by $\mathcal{D}_{\gamma}$ the Dehn
twist about $\gamma$. Let $m_{i}$ be the power of $\mathcal{D}_{\gamma}$ in
$\mathcal{T}_{i,n}$. Realize $h_{2}(\gamma)$ and
$h_{2}(\mathcal{D}_{\gamma}^{m_{i}}(\beta))$ as geodesics. Lifting the picture
to the universal cover the length of $\mathcal{D}_{\gamma}^{m_{i}}(\beta)\cap
C(h_{2}(\gamma))$ is bounded above by
$|m_{i}|\ell_{\gamma}(\zeta_{i,n}(t_{i}+\delta))+w+2C$, see Figure 2. Further
its length outside the collar is bounded by the diameter of
$\zeta_{i,n}(t_{i}+\delta)$ out side the collars of the curves in $\hat{\tau}$
plus $2C$. Then the length of $\mathcal{D}_{\gamma}^{m_{i}}(\beta)$ is bounded
above by $w+|m_{i}|\ell_{\gamma}(\zeta_{i,n}(t_{i}+\delta))+2C+2C$ plus the
diameter. Suppose that $|m_{i}|$ is bounded by some $N>0$. Then as we saw
above Claim 4.4 gives us $L^{\prime}$ depending only on
$\epsilon_{1},\epsilon_{2}$ and $N$ so that
(4.5) $\ell_{\beta}(\zeta_{i,n}(t_{i}+\delta))\leq L^{\prime}.$
On the other hand, since $\ell_{\gamma}(\zeta_{i,n}(t_{i}))\to 0$ as
$n\to\infty$ and $\beta\pitchfork\gamma$, by the Collar lemma ($\S 4.1$ of
[Bus10]) we have that
(4.6) $\ell_{\beta}(\zeta_{i,n}(t_{i}))\to\infty$
as $n\to\infty$.
For $n$ sufficiently large, by (4.6),
$\ell_{\beta}(\zeta_{i,n}(t_{i}))>\max\\{L,L^{\prime}\\}$. But
$t_{i}-\delta<t_{i}<t_{i}+\delta$, then this bound, (4.4) and (4.5) violate
the convexity of the $\beta-$length-function along the WP geodesic segment
$\zeta_{i,n}$. This contradiction shows that the power of
$\mathcal{D}_{\gamma}$ in $\mathcal{T}_{i,n}$ is unbounded. ∎
Figure 2. The lift of the collar of $h_{2}(\gamma)$ and the lift of
$h_{2}(\mathcal{D}_{\gamma}^{m_{i}}(\beta))$ to the universal cover. The
length of intersection of $h_{2}(\mathcal{D}_{\gamma}^{m_{i}}(\beta))$ and the
collar is bounded above by length of the red path. The segment $cd$ has length
$|m_{i}|\ell_{\gamma}(\zeta_{i,n}(t_{i}+\delta))$. The length of the segments
$ab$ and $ef$ are bounded by $C$ the length of each component of the boundary
of the collar. The length of the segments $bc$ and $de$ are $w$ the width of
the collar.
Here for the purpose of reference in this paper we state the following
strength version of the geodesic limit theorem which essentially contains the
properties listed in Theorem 4.1 modified to incorporate Lemmas 4.2 and 4.3.
###### Theorem 4.5.
(Geodesic Limit) Let $\zeta_{n}:[0,T]\to\overline{\operatorname{Teich}(S)}$ be
a sequence of WP geodesic segments of length $T$. Then after possibly passing
to a subsequence there exists a partition of the interval $[0,T]$ by
$0=t_{0}<t_{1}<...<t_{k}<t_{k+1}=T$, and simplices
$\sigma_{0},...,\sigma_{k+1}$ in $\widehat{\mathcal{C}(S)}$ such that
$\sigma_{i}\cap\sigma_{i+1}=\hat{\tau}$, for $i=0,...,k$, and a piecewise
geodesic
$\hat{\zeta}:[0,T]\to\overline{\operatorname{Teich}(S)}$
with the following properties
1. (1)
$\hat{\zeta}((t_{i-1},t_{i}))\subset\mathcal{S}(\hat{\tau})$, for
$i=1,...,k+1$,
2. (2)
$\hat{\zeta}(t_{i})\in\mathcal{S}(\sigma_{i})$, for $i=0,...,k+1$,
3. (3)
There are elements $\psi_{n}\in\operatorname{Mod}(S)$ and
$\mathcal{T}_{i,n}\in\operatorname{tw}(\sigma_{i}-\hat{\tau})$, for
$i=1,..,k$, such that for every $t\in[0,t_{1}]$, $\psi_{n}\circ\zeta_{n}(t)$
converges to $\hat{\zeta}(t)$, and for each $i=1,..,k$, and every
$t\in[t_{i},t_{i+1}]$,
$\mathcal{T}_{i,n}\circ...\circ\mathcal{T}_{1,n}\circ\psi_{n}\circ\zeta_{n}(t)\to\hat{\zeta}(t)$
as $n\to\infty$. For convenience for every $i=0,...,k+1$ we define
(4.7)
$\varphi_{i,n}=\mathcal{T}_{i,n}\circ...\circ\mathcal{T}_{1,n}\circ\psi_{n}$
4. (4)
The elements $\psi_{n}$ are either trivial or unbounded. Moreover, for any
$1\leq i\leq k$ and $\gamma\in\sigma_{i}$ the power of $\mathcal{D}_{\gamma}$
in the element $\mathcal{T}_{i,n}$ goes to $\infty$ as $n\to\infty$.
5. (5)
The piecewise geodesic $\hat{\zeta}$ is the minimal length path in
$\overline{\operatorname{Teich}(S)}$ joining $\hat{\zeta}(0)$ to
$\hat{\zeta}(T)$ and intersecting the strata
$\mathcal{S}(\sigma_{1}),\mathcal{S}(\sigma_{2}),...,\mathcal{S}(\sigma_{k})$
in order.
By Lemma 4.2 $\hat{\tau}\equiv\sigma_{i}\cap\sigma_{i+1}$ for $i=0,...,k$.
Then part (1) follows from this and part (1) of Theorem 4.1. Part (4) follows
from part (4) in Theorem 4.1 and Lemma 4.3 .
### 4.2. Length-function versus twist parameter control
In this subsection we show that, roughly speaking, provided a lower bound for
the length of a curve $\gamma$ at the end points of a uniformly bounded length
WP geodesic segment $\zeta$, the higher Dehn twist about $\gamma$ forces
$\gamma$ to get shorter along $\zeta$ (Corollary 4.12). Moreover, in Corollary
4.13 we show that the shorter $\gamma$ gets along $\zeta$ the higher Dehn
twist develops about $\gamma$.
The main technical part of this subsection is the following modification of
Lemma 4.5 in [BMM11].
###### Theorem 4.6.
Given $\epsilon_{0},T$ and $s$ positive, let
$\zeta_{n}:[0,T_{n}]\to\operatorname{Teich}(S)$ be a sequence of Weil-
Petersson geodesic segments parametrized by arc-length of length $2s\leq
T_{n}\leq T$. Then given a sequence of curves $\alpha_{n}$ we have the
following:
1. (1)
If there are subintervals $J_{n}\subseteq[s,T_{n}-s]$ such that
1. (a)
$\sup_{t\in J_{n}}\ell_{\alpha_{n}}(\zeta_{n}(t))\geq\epsilon_{0}$, and
2. (b)
$\inf_{t\in J_{n}}\ell_{\alpha_{n}}(\zeta_{n}(t))\to 0$ as $n\to\infty$
then
$d_{\alpha_{n}}(\mu(\zeta_{n}(0)),\mu(\zeta_{n}(T_{n})))\to\infty$
as $n\to\infty$.
2. (2)
If
1. (a)
$\sup_{t\in[0,T_{n}]}\ell_{\alpha_{n}}(\zeta_{n}(t))\geq\epsilon_{0}$, and
2. (b)
$d_{\alpha_{n}}(\mu(\zeta_{n}(0)),\mu(\zeta_{n}(T_{n})))\to\infty$ as
$n\to\infty$
then
$\inf_{t\in[0,T_{n}]}\ell_{\alpha_{n}}(\zeta_{n}(t))\to 0$
as $n\to\infty$.
###### Remark 4.7.
Here we prove that the conditions in parts (1) and (2) hold for the sequence
of curves $\alpha_{n}$, then the conclusions hold for the sequence
$\alpha_{n}$ itself. In Lemma 4.5 of [BMM11] the conclusions are proved for
some sequence of curves $\beta_{n}$ such that each $\beta_{n}$ does not
overlap $\alpha_{n}$.
###### Proof.
Trimming the intervals slightly and changing the parameters $s$ and $T$ we may
assume that $T_{n}\equiv T$ for some $T\geq 2s$. After possibly passing to a
subsequence by Theorem 4.5, there exist a partition of $[0,T]$ with
$0=t_{0}<t_{1}<...<t_{k+1}=T$, simplices $\sigma_{i}$ for $i=0,...,k+1$ in
$\widehat{\mathcal{C}(S)}$ and a simplex $\hat{\tau}$, and a piecewise
geodesic path
$\hat{\zeta}:[0,T]\to\overline{\operatorname{Teich}(S)}$
so that $\hat{\zeta}([t_{i},t_{i+1}])$ is a geodesic segment in
$\mathcal{S}(\hat{\tau})$ joining the stratum $\mathcal{S}(\sigma_{i})$ to
$\mathcal{S}(\sigma_{i+1})$. Let
$\varphi_{i,n}=\mathcal{T}_{i,n}\circ...\circ\mathcal{T}_{i,1}\circ\psi_{n}$
be as in (4.7), where
$\mathcal{T}_{i,n}\in\operatorname{tw}(\sigma_{i}-\hat{\tau})$ and
$\psi_{n}\in\operatorname{Mod}(S)$ is either trivial or unbounded.
We start by setting up some notation. For each $i=0,...,k+1$ and $n\geq 1$ let
$\sigma_{i,n}=\varphi_{i,n}^{-1}(\sigma_{i})=\varphi_{i-1,n}^{-1}(\sigma_{i})$
be the pull backs of $\sigma_{i}$ to the $\zeta_{n}$ picture. For each
$i=1,...,k$ and $n\geq 1$ let
$\tau_{i,n}=\varphi_{i-1,n}^{-1}(\hat{\tau})$
For each $i=0,...,k+1$, choose a partial marking $\mu_{i}$ such that
1. (1)
$\sigma_{i}\subset\operatorname{base}(\mu_{i})$, and
2. (2)
$\mu_{i}$ restricts to a full marking of each connected component $Y\subseteq
S\backslash\hat{\tau}$ with complexity at least one.
For each $i=1,...,k+1$ and $n\geq 1$ define the pullback marking
$\mu_{i,n}^{-}=\varphi_{i-1,n}^{-1}(\mu_{i})$
and for each $i=0,...,k$ and $n\geq 1$ the pullback marking
$\mu_{i,n}^{+}=\varphi_{i,n}^{-1}(\mu_{i})$
Let $1\leq i\leq k$ and let $\gamma\in\sigma_{i}-\hat{\tau}$. In the following
three claims we will measure the twisting of these markings relative to
$\gamma_{n}=\varphi_{i,n}^{-1}(\gamma)$ and prove that
(4.8) $d_{\gamma_{n}}(\mu(\zeta_{n}(0)),\mu(\zeta_{n}(T)))\to\infty$
as $n\to\infty$.
###### Claim 4.8.
$d_{\gamma_{n}}(\mu^{-}_{j,n},\mu^{+}_{j,n})$ is bounded for any $j=1,...,k$
with $j\neq i$.
First note that $\operatorname{base}(\mu^{+}_{j,n})$ and
$\operatorname{base}(\mu^{-}_{j,n})$ both contain
$\sigma_{j,n}=\varphi_{j,n}^{-1}(\sigma_{j})=\varphi_{j-1,n}^{-1}(\sigma_{j})$
Now we verify that
(4.9) $\gamma_{n}\notin\sigma_{j,n},\;\;\text{for any $j\neq i$}$
Otherwise, the length of $\gamma_{n}$ would converge to $0$ at
$\zeta_{n}(t_{i})$ and $\zeta_{n}(t_{j})$, and hence by the convexity of the
length-functions on all of $[t_{i-1},t_{i}]$ or $[t_{i},t_{i+1}]$ (the first
if $j<i$ and the second if $j>i$). So by Theorem 4.5 (1),
$\gamma\in\hat{\tau}$. But this contradicts the choice of
$\gamma\in\sigma_{i}-\hat{\tau}$. So (4.9) holds.
$\mu_{j,n}^{-}$ and $\mu_{j,n}^{+}$ restrict to a full marking on
$S\backslash\sigma_{j,n}$. So by (4.9) $\gamma_{n}$ intersects $\mu_{j,n}^{-}$
and $\mu_{j,n}^{+}$ nontrivially, and consequently
$\pi_{\gamma_{n}}(\mu_{j,n}^{\pm})$ is nonempty. Let
$\widetilde{\mathcal{T}}_{j,n}=\varphi_{j,n}^{-1}\circ\mathcal{T}_{j,n}\circ\varphi_{j,n}$,
then $\mu_{j,n}^{-}=\widetilde{\mathcal{T}}_{j,n}(\mu_{j,n}^{+})$.
Furthermore, since $\mathcal{T}_{j,n}\in\operatorname{tw}(\sigma_{j})$,
$\widetilde{\mathcal{T}}_{j,n}$ is an element of
$\operatorname{tw}(\sigma_{j,n})$. So $\mu_{j,n}^{+}$ differs from
$\mu_{j,n}^{-}$ by composition of Dehn twists about the curves in
$\sigma_{j,n}$. By the definition of annular subsurface coefficients the
annular subsurface coefficient of a sequence of curves intersecting both
$\mu^{-}_{j,n}$ and $\mu^{+}_{j,n}$ could go to $\infty$, only if they are in
$\sigma_{j,n}$. But by (4.9) it is not the case. The claimed bound follows
from this contradiction.
###### Claim 4.9.
$d_{\gamma_{n}}(\mu^{+}_{j,n},\mu^{-}_{j+1,n})$ is bounded for $j=1,...,k$.
$\mu_{j}$ and $\mu_{j+1}$ are full markings of $S\backslash\hat{\tau}$, where
their marking distance is some finite number. Hence we may connect them with a
finite sequence of full markings of $S\backslash\hat{\tau}$. Applying
$\varphi_{j,n}^{-1}$ to this sequence we obtain a sequence of the same length
connecting $\mu_{j,n}^{+}$ to $\mu_{j+1,n}^{-}$ through full markings of
$S\backslash\tau_{j+1,n}$. Moreover, $\gamma_{n}\notin\tau_{j+1,n}$ (because
$\gamma\notin\hat{\tau}$), so all of the markings in the connecting sequence
intersect $\gamma_{n}$ nontrivially. Any two consecutive markings in the
sequence differ by an elementary move and each elementary move increases the
$A(\gamma_{n})$ subsurface coefficient by at most one, then the claimed bound
follows.
###### Claim 4.10.
(4.10) $d_{\gamma_{n}}(\mu^{-}_{i,n},\mu^{+}_{i,n})\to\infty\;\;\text{as
$n\to\infty$}$
$\varphi_{i,n}(\mu^{-}_{i,n})=\mathcal{T}_{i,n}(\mu_{i})$, so after applying
$\varphi_{i,n}$ to all of the curves in the subsurface coefficient in (4.10)
we get
$d_{\gamma}(\mathcal{T}_{i,n}(\mu_{i}),\mu_{i})$
Now $\mu_{i}$ is a fixed marking which contains $\gamma$ as well as a
transversal curve for $\gamma$. By Theorem 4.5 (4) $\mathcal{T}_{i,n}$
contains an arbitrarily large power of $\mathcal{D}_{\gamma}$. So we obtain
the claimed bound.
Combining the bounds established in claims 4.8, 4.9 and 4.10 with the triangle
inequality the bound (4.8) follows. Having this bound we proceed by proving
our theorem.
Proof of part (1): We show that after possibly passing to a subsequence there
is an $1\leq i\leq k$ and a curve $\gamma\in\sigma_{i}-\hat{\tau}$, such that
$\alpha_{n}=\varphi_{i,n}^{-1}(\gamma)$. Part (1) then follows from (4.8).
Let $J_{n}\subset[s,T-s]$ be the subintervals in the statement of part (1).
Passing to a subsequence we may assume that $J_{n}$’s converge to a
subinterval $J$. Since each $J_{n}\subset[s,T-s]$, then $J\subset[s,T-s]$. For
each $i=0,...,k+1$,
$\varphi_{i,n}\circ\zeta_{n}|_{[t_{i},t_{i+1}]}\to\hat{\zeta}|_{[t_{i},t_{i+1}]}$
and $\varphi_{i,n}$ is isometry of the WP metric. So the length of
$\zeta_{n}(J_{n})$’s converge to the length of $\hat{\zeta}(J)$. Now since
$\zeta_{n}$’s and $\hat{\zeta}$ are (piece-wise) geodesics parametrized by
arc-length it follows that the length of the intervals $J_{n}$’s converge to
the length of the interval $J$. Now we show that the length of all $J_{n}$’s
are uniformly bounded below. For each $n\geq 1$, by 1(a),
$\ell_{\alpha_{n}}(\zeta_{n}(t))$ achieves the value $\epsilon_{0}$ in $J_{n}$
and by 1(b), for $n$ sufficiently large $\inf_{t\in
J_{n}}\ell_{\alpha_{n}}(\zeta_{n}(t))\leq\frac{\epsilon_{0}}{2}$. Thus by
Corollary 3.5 the length of $J_{n}$ is at least
$\textbf{d}(\epsilon_{0},\frac{\epsilon_{0}}{2})>0$. This uniform lower bound
for the length of $J_{n}$ for all $n$ sufficiently large and the convergence
of the length of $J_{n}$’s to that of $J$ implies that $J$ has length at least
$\textbf{d}(\epsilon_{0},\frac{\epsilon_{0}}{2})$.
For each $n\geq 1$, let $t^{*}_{n}\in J_{n}$ be the time where $\inf_{t\in
J_{n}}\ell_{\alpha_{n}}(\zeta_{n}(t))$ is realized. There is an $0\leq i\leq
k+1$ such that for $n$ sufficiently large $t^{*}_{n}\in[t_{i},t_{i+1}]$ and
$t_{n}^{*}$ converge to some $t^{*}\in J\cap[t_{i},t_{i+1}]$.
First suppose that $t^{*}\neq t_{i},t_{i+1}$. By 1(b)
$\ell_{\alpha_{n}}(\zeta_{n}(t^{*}_{n}))\to 0$ as $n\to\infty$, so applying
$\varphi_{i,n}$ to $\ell_{\alpha_{n}}(\zeta_{n}(t_{n}^{*}))$ we have
(4.11)
$\ell_{\varphi_{i,n}(\alpha_{n})}(\varphi_{i,n}\circ\zeta_{n}(t_{n}^{*}))\to
0\;\text{as}\;n\to\infty.$
By Theorem 4.5 (3),
$\varphi_{i,n}\circ\zeta_{n}(t^{*}_{n})\to\hat{\zeta}(t^{*})$ as $n\to\infty$.
Moreover, the only curves with length $0$ at $\hat{\zeta}(t^{*})$ are the ones
in $\hat{\tau}$. Thus Theorem 3.1 (Continuity of length-functions) implies
that the curves in $\hat{\tau}$ are the only ones whose length at
$\varphi_{i,n}\circ\zeta_{n}(t^{*}_{n})$ converge to $0$. Then by (4.11) after
possibly passing to a subsequence $\varphi_{i,n}(\alpha_{n})=\beta$ for some
$\beta\in\hat{\tau}$. For each $l=0,...,k+1$, $\hat{\tau}\subseteq\sigma_{l}$,
so $\beta\in\sigma_{n}$. Now given $j=0,...,k+1$,
$\varphi_{j,n}\circ\varphi_{i,n}^{-1}$ is a composition of Dehn twists about
curves in $\sigma_{l}$, $l=j+1,...,i$ (see 4.7), so preserves $\beta$. Thus
$\varphi_{j,n}(\alpha_{n})=\beta$.
Given $t\in[0,T]$, $t\in[t_{j},t_{j+1}]$ for some $0\leq j\leq k$. By Theorem
4.5 (3) $\varphi_{j,n}(\zeta_{n}(t))\to\hat{\zeta}(t)$ as $n\to\infty$.
Moreover, $\ell_{\beta}(\hat{\zeta}(t))\equiv 0$ for all $t\in[0,T]$. Thus the
continuity of length-functions implies that
$\ell_{\beta}(\varphi_{j,n}(\zeta_{n}(t)))\to 0$ as $n\to\infty$. As we saw
above $\varphi_{j,n}(\alpha_{n})=\beta$, so
$\ell_{\varphi_{j,n}(\alpha_{n})}(\varphi_{j,n}(\zeta_{n}(t)))\to 0$ as
$n\to\infty$, then applying $\varphi_{j,n}^{-1}$ we have that
$\ell_{\alpha_{n}}(\zeta_{n}(t))\to 0$ as $n\to\infty$. But this contradicts
1(a).
So $t^{*}_{n}$ converges to either $t_{i}$ or $t_{i+1}$. Let $t^{*}_{n}\to
t_{i}(t_{i+1})$ as $n\to\infty$. Then
$\varphi_{i,n}(\zeta_{n}(t))\to\hat{\zeta}(t_{i})$
($\varphi_{i,n}(\zeta_{n}(t))\to\hat{\zeta}(t_{i+1})$) as $n\to\infty$. Note
that the only curves with length $0$ at
$\hat{\zeta}(t_{i})(\hat{\zeta}(t_{i+1}))$ are the ones in
$\sigma_{i}(\sigma_{i+1})$. Thus (4.11) and the convergence of length-
functions imply that
$\alpha_{n}=\varphi_{i,n}^{-1}(\gamma)(\varphi_{i+1,n}^{-1}(\gamma))$ for some
$\gamma\in\sigma_{i}-\hat{\tau}(\sigma_{i+1}-\hat{\tau})$, as was desired.
Proof of part (2): Suppose that for an $i$ with $\sigma_{i}\neq\emptyset$,
$\alpha_{n}\in\sigma_{i,n}$ for all $n\geq 1$. Applying $\varphi_{i,n}$ to
$\ell_{\alpha_{n}}(\zeta_{n}(t_{i}))$ we get
$\ell_{\varphi_{i,n}(\alpha_{n})}(\varphi_{i,n}(\zeta_{n}(t_{i})))$. By
Theorem 4.5 (3) we have that
$\varphi_{i,n}(\zeta_{n}(t_{i}))\to\hat{\zeta}(t_{i})$ as $n\to\infty$. Since
$\alpha_{n}\in\sigma_{i,n}:=\varphi_{i,n}^{-1}(\sigma_{i})$ we have that
$\varphi_{i,n}(\alpha_{n})\in\sigma_{i}$. Further the length of every curve in
$\sigma_{i}$ is $0$ at $\hat{\zeta}(t_{i})$. So the continuity of length-
functions implies that
$\ell_{\varphi_{i,n}(\alpha_{n})}(\varphi_{i,n}(\zeta_{n}(t_{i}))\to 0$ as
$n\to\infty$. Consequently, $\ell_{\alpha_{n}}(\zeta_{n}(t_{i}))\to 0$ as
$n\to\infty$. So the proof of part (2) would be complete if we show that for
some $i$ with $\sigma_{i}\neq\emptyset$, $\alpha_{n}\in\sigma_{i,n}$ for all
$n\geq 1$.
In contrary suppose that after possibly passing to a subsequence
$\alpha_{n}\notin\sigma_{i,n}$ for all $i=0,...,k+1$ and $n\geq 1$. Then
$\alpha_{n}$ intersects $\mu_{i,n}^{-}$ for $i=1,...,k+1$, and $\mu_{i,n}^{+}$
for $i=0,...,k$. For each $i$, let
$\widetilde{\mathcal{T}}_{i,n}=\varphi_{i,n}^{-1}\circ\mathcal{T}_{i,n}\circ\varphi_{i,n}$,
as before, as we saw earlier
$\widetilde{\mathcal{T}}_{i,n}\in\operatorname{tw}(\sigma_{i,n})$ and
$\mu_{i,n}^{-}=\widetilde{\mathcal{T}}_{i,n}(\mu_{i,n}^{+})$. So the only
annular subsurfaces of $\mu_{i,n}^{-}$ and $\mu_{i,n}^{+}$ which grow as
$n\to\infty$ are the ones with core curve in $\sigma_{i,n}$. Thus
$d_{\alpha_{n}}(\mu_{i,n}^{-},\mu_{i,n}^{+})$ is uniformly bounded for
$i=1,...,k$ and all $n\geq 1$.
Moreover, as we saw in the proof of Claim 4.9 the fact that for each
$i=0,...,k+1$, $\alpha_{n}$ intersects $\mu_{i,n}^{+}$ and $\mu_{i+1,n}^{-}$
implies that $d_{\alpha_{n}}(\mu_{i,n}^{+},\mu_{i+1,n}^{-})$ is uniformly
bounded for $i=1,...,k$ and all $n\geq 1$.
Combining the bounds form the above two paragraphs by the triangle inequality
we conclude that $d_{\alpha_{n}}(\mu_{0,n}^{-},\mu_{k,n}^{+})$ is uniformly
bounded above for all $n$. But $\mu(\zeta_{n}(0))=\mu_{0,n}^{-}$ and
$\mu(\zeta_{n}(T))=\mu_{k,n}^{+}$, so this bound contradicts assumption 2(b).
∎
###### Remark 4.11.
Fix a pants decomposition $P$ and let
$\big{(}\ell_{\gamma},\theta_{\gamma}\big{)}_{\gamma\in P}$ be a corresponding
Fenchel-Nielsen coordinates. Let $x\in\operatorname{Teich}(S)$ be a point with
$\theta_{\alpha}=\theta_{0}$ for some $\alpha\in P$. Let $\epsilon_{n}\to 0$.
Consider the following two sequences of WP geodesic segments:
* •
Let $x_{n}$ ($n\geq 1$) be a point with all coordinates equal to that of $x$,
except for $\theta_{\alpha}(x_{n})=\theta_{0}+2\pi n$ and
$\ell_{\alpha}(x_{n})=\epsilon_{n}$. Then the length of the WP geodesic
segments $[x,x_{n}]$ is uniformly bounded and
$d_{\alpha}(\mu(x),\mu(x_{n}))\to\infty$ as $n\to\infty$.
* •
Let $x_{n}$ ($n\geq 1$) be a point with all coordinates equal to that of $x$,
except for $\theta_{\alpha}=\theta_{0}$ and
$\ell_{\alpha}(x_{n})=\epsilon_{n}$. Then the length of WP geodesic segment
$[x,x_{n}]$ is uniformly bounded and $d_{\alpha}(\mu(x),\mu(x_{n}))=0$.
These two examples show that having a sequence of curves $\alpha_{n}$ whose
lengths converge to zero at the end points of a sequence of WP geodesic
segments, a-priori does not give any information about the growth of the
$A(\alpha_{n})$ subsurface coefficients of the Bers markings at the end points
of the geodesic segments. Thus the above theorem is the sharpest control of
length versus annular coefficient one could expect.
We are ready to prove the following two corollaries which are somewhat
quantified versions of Theorem 4.6. These corollaries provide us with a kind
of length-function versus twist parameter bounds over uniformly bounded length
WP geodesic segments which often will be used in $\S$6.
###### Corollary 4.12.
(large twist $\Longrightarrow$ short curve) Given $T,\epsilon_{0}$ and $N$
positive there is $\epsilon<\epsilon_{0}$ with the following property. Let
$\zeta:[0,T^{\prime}]\to\operatorname{Teich}(S)$ be a WP geodesic segment of
length $T^{\prime}\leq T$ such that
$\sup_{t\in[0,T^{\prime}]}\ell_{\gamma}(\zeta(t))\geq\epsilon_{0}$
If $d_{\gamma}(\mu(\zeta(0)),\mu(\zeta(T^{\prime})))>N$ then we have
$\inf_{t\in[0,T^{\prime}]}\ell_{\gamma}(\zeta(t))\leq\epsilon$
###### Proof.
The proof is by contradiction. Assume that the corollary does not hold. Then
there is a sequence of WP geodesic segments
$\zeta_{n}:[0,T_{n}]\to\operatorname{Teich}(S)$ parametrized by arc-length
with lengths $T_{n}\leq T$ and curves $\gamma_{n}\in\mathcal{C}_{0}(S)$, such
that
1. (a)
$\sup_{t\in[0,T_{n}]}\ell_{\gamma_{n}}(\zeta_{n}(t))\geq\epsilon_{0}$ for
every $n$,
2. (b)
$d_{\gamma_{n}}(\mu(\zeta_{n}(0)),\mu(\zeta_{n}(T_{n})))\to\infty$ as
$n\to\infty$,
and $\inf_{t\in[0,T_{n}]}\ell_{\gamma_{n}}(\zeta_{n}(t))>\epsilon$ for every
$n$. But this contradicts Theorem 4.6 (2).
∎
###### Corollary 4.13.
(short curve $\Longrightarrow$ large twist)
Given $\epsilon_{0},T,s$ and $\epsilon<\epsilon_{0}$ positive with $T>2s$,
there is an integer $N>0$ with the following property. Let
$\zeta:[0,T^{\prime}]\to\operatorname{Teich}(S)$ be a WP geodesic segment
parametrized by arc-length of length $T^{\prime}\in[2s,T]$,
$\gamma\in\mathcal{C}_{0}(S)$ and $J\subseteq[s,T^{\prime}-s]$ be a
subinterval such that:
$\sup_{t\in[0,T^{\prime}]}\ell_{\gamma}(\zeta(t))\geq\epsilon_{0}$
If $\inf_{t\in J}\ell_{\gamma}(\zeta(t))\leq\epsilon$, then
$d_{\gamma}(\mu(\zeta(0)),\mu(\zeta(T^{\prime})))>N$
###### Proof.
The proof is again by contradiction. Assume that the corollary does not hold.
Then there is a sequence of WP geodesic segments
$\zeta_{n}:[0,T_{n}]\to\operatorname{Teich}(S)$ parametrized by arc-length of
length $2s\leq T_{n}\leq T$ , $\gamma_{n}\in\mathcal{C}(S)$, subintervals
$J_{n}\subset[s,T_{n}-s]$ such that
1. (a)
$\sup_{t\in J_{n}}\ell_{\gamma_{n}}(\zeta_{n}(t))\geq\epsilon_{0}$ for every
$n$,
2. (b)
$\inf_{t\in J_{n}}\ell_{\gamma_{n}}(\zeta_{n}(t))\to 0$ as $n\to\infty$,
and $d_{\gamma_{n}}(\mu(\zeta_{n}(0)),\mu(\zeta_{n}(T_{n})))\leq N$ for every
$n$. But this contradicts Theorem 4.6 (1). ∎
###### Remark 4.14.
Note that these results have been proved using only compactness arguments in
the WP completion of Teichmüller space.
## 5\. Stable hierarchy paths
In this section we show that a certain class of hierarchy paths are stable in
the pants graph of the surface.
###### Definition 5.1.
($d-$stable subset) Given a function $d:\mathbb{R}^{\geq
1}\times\mathbb{R}^{\geq 0}\to\mathbb{R}^{\geq 0}$ a subset $\mathcal{Y}$ of a
metric space $\mathcal{X}$ is $d-$stable if for any $K\geq 1$ and $C\geq 0$
every $(K,C)-$quasi-geodesic $h$ with end points in $\mathcal{Y}$ is contained
in the $d(K,C)$ neighborhood of $\mathcal{Y}$. We call the function $d$ the
quantifier of the stability.
Here we summarize some of the results about stability of subsets of pants
graph of surfaces: Brock and Masur in [BM08] prove that when $\xi(S)=3$ the
pants graph of $S$ is strongly relatively hyperbolic with respect to the
quasi-flats corresponding to separating curves. The main ingredient of their
proof is that given a hierarchy path $\rho:[m,n]\to P(S)$ the subset of the
pants graph $X(\rho)=|\rho|\cup_{W}\\{P(W)\times P(W^{c})\\}$, where $W$ or
$W^{c}=S\backslash W$ is a component domain of $\rho$, is a stable subset of
the pants graph. Behrstock, Drutu and Mosher in [BDM09] study thick metric
spaces. These are metric spaces with rank at least $2$ where any two quasi-
flats are connected through a chain of quasi-flats with the property that any
two consecutive quasi-flats in the chain has coarse intersection of infinite
diameter. They show that thick metric spaces fail to be relatively hyperbolic
with respect to any collection of quasi-flats. Moreover, they observe that
$P(S)$ for $\xi(S)>3$ is a thick metric space and consequently is not
relatively hyperbolic with respect to any collection of quasi-flats. In
[BMM11] it is proved that hierarchy paths with bounded combinatorics end
points are stable.
Here we show that restricting the subsurfaces of a pair of partial markings or
laminations with subsurface coefficient bigger than a given $A>0$ to large
subsurfaces, implies stability of any hierarchy path $\rho$ between the pair
in the pants graph. We call such a pair $A-$narrow. Heuristically, these
hierarchy paths avoid quasi-flats in the pants graph corresponding to
separating multi-curves on the surface.
To be able to save considerable amount of work using results in the context of
$\Sigma-$hulls (see $\S$2.3) and present our results in a more general setting
we prove that $\Sigma-$hulls with the mentioned constraint on the subsurface
coefficients of their end points are stable. In $\S$5.2 we prove that for any
$\epsilon>F$ the $\Sigma_{\epsilon}-$hull of an $A-$narrow pair is
$d_{A}-$stable. Then the stability of hierarchy paths between the $A-$narrow
pair follows from the fact that the Hausdorff distance of a hierarchy path
between an $A-$narrow pair and the $\Sigma_{\epsilon}-$hull ($\epsilon$
sufficiently large) of the pair is bounded depending only on $A$ and
$\epsilon$. This is proved in Theorem 5.5.
### 5.1. Narrow pairs
In this subsection first we introduce the notion of an A-narrow pair of
makings or laminations. Then we will show that any hierarchy path between a
narrow pair and the $\Sigma_{\epsilon}-$hull ($\epsilon>0$ is sufficiently
large) of the pair have finite Hausdorff distance depending only on $A$ and
$\epsilon$.
###### Definition 5.2.
(Large subsurface) A connected essential subsurface $Z\subseteq S$ is called
large if any connected component of $S\backslash Z$ is either an annulus or a
three holed sphere.
###### Remark 5.3.
If $Z$ is not connected or $S\backslash Z$ has connected components other than
annuli and three holed spheres then $Z$ is not a large subsurface.
###### Definition 5.4.
($A-$narrow) A pair of partial markings or laminations $(\mu^{-},\mu^{+})$ is
called $A-$narrow if every non-annular subsurface $Z\subseteq S$ with the
property that
$d_{Z}(\mu^{-},\mu^{+})>A$
is a large subsurface of $S$.
Recall the constants $M_{1},M_{2}$ from Theorem 2.13 and $B_{0}$ from Theorem
2.8. We fix the constant $M=M_{1}+B_{0}+4$ in this subsection. Note that since
$M_{1}\geq 2M_{2}$, $M\geq M_{2}$.
###### Theorem 5.5.
($\Sigma-$hull of narrow pair)
Given $\epsilon>M$ and $A>4M+2\epsilon+12$, there is a constant
$\Delta=\Delta(A,\epsilon)$ with the following property. Given an $A-$narrow
pair $(\mu^{-},\mu^{+})$, the Hasudorff distance of the
$\Sigma_{\epsilon}(\mu^{-},\mu^{+})$ and any hierarchy path between
$(\mu^{-},\mu^{+})$ is less than $\Delta$.
###### Proof.
Let $\rho:[m,n]\to P(S)$ be a hierarchy path with $\rho(m)=\mu^{-}$ and
$\rho(n)=\mu^{+}$. Let $X\subseteq S$ be a non-annular subsurface. Let
$i\in[m,n]$. By Theorem 2.13 (5), there is
$y\in\operatorname{hull}_{X}(\mu^{-},\mu^{+})$ such that $d_{X}(\rho(i),y)\leq
M_{2}<\epsilon$ (by the assumption of the theorem on the value of $\epsilon$,
$\epsilon>M_{2}$). Thus by the definition of $\Sigma_{\epsilon}-$hull we have
that $\rho(i)\in\Sigma_{\epsilon}(\mu^{-},\mu^{+})$. Consequently
$|\rho|\subset\Sigma_{\epsilon}(\mu^{-},\mu^{+})$
We proceed to prove that $\Sigma_{\epsilon}(\mu^{-},\mu^{+})$ is contained in
a neighborhood of $|\rho|$.
###### Lemma 5.6.
Given $\epsilon>M$ and $A>4M+2\epsilon+12$, there is $d=d(A,\epsilon)$ with
the property that given $P\in\Sigma_{\epsilon}(\mu^{-},\mu^{+})$ there is
$\hat{j}\in[m,n]$ such that for every non-annular subsurface $X\subseteq S$ we
have
(5.1) $d_{X}(\rho(\hat{j}),P)\leq d$
###### Proof.
Let $X\subseteq S$ be a non-annular subsurface.
$P\in\Sigma_{\epsilon}(\mu^{-},\mu^{+})$, so there is
$x_{X}\in\operatorname{hull}_{X}(\mu^{-},\mu^{+})$ such that
$d_{X}(P,x_{X})\leq\epsilon$. By Theorem 2.13 (5) there is a vertex
$x^{\prime}_{X}\in\pi_{X}(|\rho|)$ ($|\rho|=\bigcup_{i\in[m,n]}\rho(i)$) with
$d_{X}(x_{X},x^{\prime}_{X})\leq M$. So by the triangle inequality,
$d_{X}(x^{\prime}_{X},P)\leq M+\epsilon$. Let $j\in[m,n]$ be such that
$\pi_{X}(\rho(j))\supset x^{\prime}_{Y}$, then
(5.2) $d_{X}(\rho(j),P)\leq M+\epsilon.$
Let $e=2A$. Then the subset of parameters
$\mathcal{E}_{X}(P)=\\{i\in[m,n]:d_{X}(P,\rho(i))\leq e\\}$ is non-empty.
Because any $j$ as above is in $\mathcal{E}_{X}$. Denote the minimum and
maximum of the set $\mathcal{E}_{X}(P)$ by $e_{X}^{-}$ and $e_{X}^{+}$
respectively. Let $E_{X}:=E_{X}(P)=[e_{X}^{-},e_{X}^{+}]$.
Recall the interval $J_{X}$ from Theorem 2.13 (1).
###### Claim 5.7.
Let $X$ be a component domain of $\rho$. Then
(5.3) $E_{X}\cap J_{X}\neq\emptyset$
$P\in\Sigma_{\epsilon}(\mu^{-},\mu^{+})$, so there is
$x_{X}\in\operatorname{hull}_{X}(\mu^{-},\mu^{+})$ such that
$d_{X}(P,x_{X})\leq\epsilon$. $X$ is a component domain of $\rho$, so by
Theorem 2.13 (4) there is $x^{\prime}_{X}\in g_{X}$ such that
$d_{X}(x_{X},x^{\prime}_{X})\leq M_{2}$. Then by the triangle inequality we
have $d_{X}(P,x^{\prime}_{X})\leq M+\epsilon$. Let $j\in J_{X}$ be such that
$\rho(j)\supset x^{\prime}_{X}$. Then
(5.4) $d_{X}(\rho(j),P)\leq M+\epsilon.$
Now since $M+\epsilon>e$ we have that $j\in E_{X}$. The claim is proved.
Let $j\in E_{X}$. Suppose that $X$ is a component domain of $\rho$. Let
$v=\rho(j)\cap g_{X}$, $u=\rho(e_{X}^{-})\cap g_{X}$ and
$w=\rho(e_{X}^{+})\cap g_{X}$. Then by Theorem 2.13 (3) (Monotonicity), $u\leq
v\leq w$ as vertices along $g_{X}$. Furthermore, since
$d_{X}(P,\rho(e_{X}^{-}))\leq e$ we have $d_{X}(P,u)\leq e+1$ and since
$d_{X}(P,\rho(e_{X}^{+}))\leq e$ we have $d_{X}(P,w)\leq e+1$. Then by the
triangle inequality
$d_{X}(u,w)\leq 2e+2+\operatorname{diam}_{X}(P)\leq 2e+2+1.$
So either $d_{X}(u,v)\leq e+2$ or $d_{X}(v,w)\leq e+2$. Suppose that
$d_{X}(u,v)\leq e+2$. Then since $d_{X}(P,u)\leq e$ and $\rho(j)\supset v$, by
the triangle inequality we get
(5.5) $d_{X}(\rho(j),P)\leq 2e+2$
Similarly, we get the above bound assuming that $d_{W}(v,w)\leq e+2$.
Now suppose that $X$ is not a component domain of $\rho$. Then
$d_{X}(\mu^{-},\mu^{+})\leq M$. Furthermore, by Theorem 2.13 (5) the Hausdorff
distance of $\pi_{X}(|\rho|)$ and $\operatorname{hull}_{X}(\mu^{-},\mu^{+})$
is bounded above by $M$. Therefore, $\operatorname{diam}_{X}(|\rho|)\leq
3M+2$. Thus
(5.6) $d_{X}(\rho(j),P)\leq e+3M+2$
To prove the lemma it suffices to show that
(5.7) $\bigcap_{\begin{subarray}{c}X\subseteq S\\\ \text{non-
annular}\end{subarray}}E_{X}\neq\emptyset.$
To see this let $\hat{j}\in\bigcap_{\begin{subarray}{c}X\subseteq S\\\
\text{non-annular}\end{subarray}}E_{X}$. Then by the bounds (5.5) and (5.6)
for every non-annular subsurface $X\subseteq S$,
$d_{X}(\rho(\hat{j}),P)\leq\max\\{2e+2,e+3M+2\\},$
which is the desired bound in (5.1).
Our strategy to prove that (5.7) holds is to verify that for any two non-
annular subsurfaces $Y$ and $W$ we have that
(5.8) $E_{Y}\cap E_{W}\neq\emptyset.$
Then Helly’s Theorem in one dimension (see [Eck93]) implies that the
intersection of all of the intervals
$\\{E_{X}\\}_{\begin{subarray}{c}X\subseteq S\\\ \text{non-
annular}\end{subarray}}$ is nonempty.
If $d_{Y}(\mu^{-},\mu^{+})\leq A$, then
$\operatorname{diam}_{Y}(\operatorname{hull}_{Y}(\mu^{-},\mu^{+}))\leq A+2$.
$P\in\Sigma_{\epsilon}(\mu^{-},\mu^{+})$ so there is
$x_{Y}\in\operatorname{hull}_{Y}(\mu^{-},\mu^{+})$ with
$d_{Y}(P,x_{Y})\leq\epsilon$. By Theorem 2.13 (5) for every $i\in[m,n]$ there
is $y\in\operatorname{hull}_{Y}(\mu^{-},\mu^{+})$ such that
$d_{Y}(\rho(i),y)\leq M_{2}$. By the bound on the diameter of the hull
$d_{Y}(x_{Y},y)\leq A+2$. The last three bounds combined by the triangle
inequality give us
$d_{Y}(\rho(i),P)\leq\epsilon+M_{2}+A+2\leq e.$
Thus $E_{Y}=[m,n]$, which obviously intersects $E_{W}$. If
$d_{W}(\mu^{-},\mu^{+})\leq A$, similarly we can conclude that $E_{W}=[m,n]$,
which implies that $E_{W}\cap E_{Y}\neq\emptyset$. Therefore, in the rest of
the proof we may assume that
* •
$d_{Y}(\mu^{-},\mu^{+})>A$, and
* •
$d_{W}(\mu^{-},\mu^{+})>A$.
In particular, since $A>M$ both $Y$ and $W$ are component domains of $\rho$.
Now consider the following collection of subsurfaces:
* •
$\mathfrak{L}:=\mathfrak{L}_{A}(\mu^{-},\mu^{+})=\\{X\subseteq S\;\;\text{non-
annular subsurface}:d_{X}(\mu^{-},\mu^{+})>A\\}$
For each $X\in\mathfrak{L}$ define
* •
$i_{X}^{-}=\max\\{i\in[m,n]:d_{X}(\rho(i),\mu^{-})\leq M\\}$ and
* •
$i_{X}^{+}=\min\\{i\in[m,n]:d_{X}(\rho(i),\mu^{+})\leq M\\}$.
###### Claim 5.8.
Let $J_{X}=[j_{X}^{-},j_{X}^{+}]$. Then $j_{X}^{-}\leq i_{X}^{-}\leq
i_{X}^{+}\leq j_{X}^{+}$ and we may write
(5.9) $[i_{X}^{-},i_{X}^{+}]\subseteq J_{X}$
By Theorem 2.13 (4) $d_{X}(\mu^{-},\rho(j_{X}^{-}))\leq M$ and
$d_{X}(\mu^{+},\rho(j_{X}^{+}))\leq M$. So we have that $i_{X}^{-}\geq
j_{X}^{-}$ and $i_{X}^{+}\leq j_{X}^{+}$. Then since $J_{X}$ is an interval,
$i_{X}^{-},i_{X}^{+}\in J_{X}$. Now we show that $i_{X}^{-}\leq i_{X}^{+}$. As
we just said, $d_{X}(\rho(j_{X}^{-}),\mu^{-})\leq M$. By the definition of
$i_{X}^{-}$, $d_{X}(\rho(i_{X}^{-}),\mu^{-})\leq M$. The last two inequalities
combined by the triangle inequality imply that
(5.10) $d_{X}(\rho(i^{-}_{X}),\rho(j_{X}^{-}))\leq 2M+2.$
Similarly we have that
(5.11) $d_{X}(\rho(i^{+}_{X}),\rho(j_{X}^{+}))\leq 2M+2.$
Furthermore, $d_{X}(\mu^{-},\mu^{+})>A>6M+12$, so by the bounds
$d_{X}(\mu^{-},\rho(j^{-}))\leq M$ and $d_{X}(\mu^{+},\rho(j^{+}))\leq M$ we
have that
(5.12) $d_{X}(\rho(j_{X}^{-}),\rho(j_{X}^{+}))>4M+12.$
Let $u=\rho(j_{X}^{-})\cap g_{X}$, $w=\rho(j_{X}^{+})\cap g_{X}$,
$v=\rho(i_{X}^{-})\cap g_{X}$ and $v^{\prime}=\rho(i_{X}^{+})\cap g_{X}$. By
(5.10),
$d_{X}(u,v)\leq
d_{X}(\rho(i_{X}^{-}),\rho(j_{X}^{-}))+\operatorname{diam}_{X}(\rho(i_{X}^{-}))+\operatorname{diam}_{X}(\rho(j_{X}^{-}))\leq
2M+4,$
by (5.11) $d_{X}(w,v^{\prime})\leq 2M+4$ and by (5.12) $d_{X}(u,w)\geq 4M+10$.
Then since $u<w$ as vertices along $g_{X}$ the last there inequalities imply
that $v<v^{\prime}$. Then Theorem 2.13 (3) (Monotonicity) implies that
$i_{X}^{-}\leq i_{X}^{+}$.
###### Claim 5.9.
Let $X\in\mathfrak{L}$. If $i\in[m,i_{X}^{-}]$ then
$d_{X}(\rho(i),\mu^{-})\leq 2M+3$. If $i\in[i_{X}^{+},n]$ then
$d_{X}(\rho(i),\nu^{+})\leq 2M+3$.
We prove the first part of the claim. The proof of the second part is similar.
Let $J_{X}=[j_{X}^{-},j_{X}^{+}]$. By Claim 5.8, $j_{X}^{-}\leq i_{X}^{-}$. If
$i\leq j_{X}^{-}$ then by Theorem 2.13 (4) we have $d_{X}(\mu^{-},\rho(i))\leq
M<2M+3$, which is the desired bound. Otherwise, $j_{X}^{-}\leq i\leq
i_{X}^{-}$. Note that $j_{X}^{-},i,i_{X}^{-}\in J_{X}$. Then let
$u=\rho(j_{X}^{-})\cap g_{X},v=\rho(i)\cap g_{X}$ and $w=\rho(i_{X}^{-})\cap
g_{X}$. Then by (5.10) we have
$d_{X}(u,w)\leq
2M+2+\operatorname{diam}_{X}(\rho(j_{X}^{-}))+\operatorname{diam}_{X}(\rho(i_{X}^{-}))=2M+4.$
Moreover, since $j_{X}^{-}\leq i\leq i_{X}^{-}$, by Theorem 2.13 (3)
(Monotonicity) $u\leq v\leq w$ as vertices along the geodesic
$g_{X}\subset\mathcal{C}(X)$. So by the above inequality either
$d_{X}(u,v)\leq M+2$ or $d_{X}(v,w)\leq M+2$. The former inequality implies
that $d_{X}(\rho(j_{X}^{-}),\rho(i))\leq M+2$. This inequality and
$d_{X}(\rho(j_{X}^{-}),\mu^{-})\leq M$ (Theorem 2.13 (4)) combined by the
triangle inequality imply that $d_{X}(\rho(i),\mu^{-})\leq
2M+2+\operatorname{diam}_{X}(\rho(j_{X}^{-}))\leq 2M+3$. The later inequality
implies that $d_{X}(\rho(i),\rho(j_{X}^{-}))\leq M+2$. This inequality and
$d_{X}(\mu^{-},\rho(j_{X}^{-}))\leq M$ (Theorem 2.13 (4)) combined by the
triangle inequality imply that $d_{Y}(\rho(i),\mu^{-})\leq 2M+3$. The first
part of the claim is proved.
Given $Y,W\in\mathfrak{L}$ the $A-$narrow condition implies that either
1. (1)
$Y\pitchfork W$,
2. (2)
$Y\supsetneq W$, or
3. (3)
$Y\subsetneq W$.
In what follows we discuss these three cases and in each case verify that
(5.8) holds.
Case 1: $Y\pitchfork W$.
$Y,W\in\mathfrak{L}$ and $A>4M$. So $d_{Y}(\mu^{-},\mu^{+})>4M$ and
$d_{W}(\mu^{-},\mu^{+})>4M$. Then by Proposition 2.15 either $Y<W$ or $W<Y$
(not both). Assume that $Y<W$. $W<Y$ can be treated similarly. Then by the
proposition we have the following two inequalities
(5.13) $\displaystyle d_{Y}(\mu^{-},\partial{W})$ $\displaystyle>$
$\displaystyle M,\;\text{and}$ (5.14) $\displaystyle
d_{W}(\mu^{+},\partial{Y})$ $\displaystyle>$ $\displaystyle M.$
We proceed to discuss the following three subcases depending on the values of
$d_{Y}(P,\mu^{-})$ and $d_{Y}(P,\mu^{+})$. In each case we verify that (5.8)
holds.
Case 1.1:
(5.15) $d_{Y}(P,\mu^{-})\leq 3M+\epsilon+4.$
We show that the following inclusion of intervals holds
(5.16) $E_{Y}\supseteq[m,i_{Y}^{-}]$
Let $i\in[m,i^{-}_{Y}]$, then by the first part of Claim 5.9 we have
$d_{Y}(\rho(i),\mu^{-})\leq 2M+3$. This inequality and (5.15) combined by the
triangle inequality give us
$d_{Y}(P,\rho(i))\leq 5M+\epsilon+7+\operatorname{diam}_{Y}(\mu^{-})\leq
5M+\epsilon+9$
Furthermore, $e=2A>5M+\epsilon+9$, so by the definition of $E_{Y}$ we have
$i\in E_{Y}$.
Now we show that the inclusion of intervals
(5.17) $E_{W}\supseteq[m,i_{W}^{-}]$
holds. To see this, note that since $Y\in\mathfrak{L}$ we have
$d_{Y}(\mu^{-},\mu^{+})>A>4M+2\epsilon+12$. By (5.14) and Theorem 2.8
(Behrstock Inequality) we have, $d_{Y}(\partial{W},\mu^{+})\leq M$. The last
two bounds and (5.15) combined by the triangle inequality imply that
$d_{Y}(P,\partial{W})>\epsilon>M$. Thus by the Behrstock inequality we have
(5.18) $d_{W}(\partial{Y},P)\leq M.$
By (5.13) and the Behrstock inequality we have that
$d_{W}(\partial{Y},\mu^{-})\leq M$. This inequality and (5.18) combined by the
triangle inequality give us
(5.19) $d_{W}(P,\mu^{-})\leq 2M+1$
Let $i\in[m,i_{W}^{-}]$, by the first part of Claim 5.9,
$d_{W}(\mu^{-},\rho(i))\leq 2M+3$. Combining this inequality and (5.19) by the
triangle inequality we get
$d_{W}(P,\rho(i))\leq 4M+4.$
Now since $e>4M+4$, by the definition of $E_{W}$, $i\in E_{W}$. So we conclude
that $E_{W}\supseteq[m,i_{W}^{-}]$.
The inclusion of intervals (5.16) and (5.17) together imply that $E_{W}\cap
E_{Y}\neq\emptyset$.
Case 1.2:
(5.20) $\min\\{d_{Y}(P,\mu^{-}),d_{Y}(P,\mu^{+})\\}>3M+\epsilon+4$
We show that
(5.21) $E_{Y}\cap[i_{Y}^{-},i_{Y}^{+}]\neq\emptyset$
To see this, note that $P\in\Sigma_{\epsilon}(\mu^{-},\mu^{+})$, then by
(5.4), there is $i\in[m,n]$ such that
(5.22) $d_{Y}(P,\rho(i))\leq M+\epsilon$
Now since $e>M+\epsilon$, by the definition of $E_{Y}$, $i\in E_{Y}$.
By (5.20), $d_{Y}(P,\mu^{-})>3M+\epsilon+4$. Combining this inequality and
(5.22) by the triangle inequality we have that
$d_{Y}(\rho(i),\mu^{-})>2M+4-\operatorname{diam}_{Y}(P)=2M+3$. Then by the
contrapositive of the first part of Claim 5.9 we conclude that $i>i_{Y}^{-}$.
By (5.20), $d_{Y}(P,\mu^{+})>3M+\epsilon+4$. Combining this inequality and
(5.22) by the triangle inequality we have that $d_{Y}(\rho(i),\mu^{+})>2M+3$.
Then by the contrapositive of the second part of Claim 5.9 we conclude that
$i<i_{Y}^{+}$. Therefore $i\in[i_{Y}^{-},i_{Y}^{+}]$.
Now we show that the inclusion of intervals
(5.23) $E_{W}\supseteq[m,i_{W}^{-}]$
holds. By (5.14) and Theorem 2.8 (Behrstock Inequality),
$d_{Y}(\partial{W},\mu^{+})\leq M$. Moreover, by (5.20),
$d_{Y}(P,\mu^{+})>3M+\epsilon+4$. These two bounds combined by the triangle
inequality give us
$d_{Y}(P,\partial{W})>2M+\epsilon+4-\operatorname{diam}_{Y}(\mu^{+})>M$.
Therefore, by the Behrstock inequality
(5.24) $d_{W}(P,\partial{Y})\leq M$
Having (5.24), the rest of the proof of the inclusion of intervals (5.23)
follows from exact the same lines given after (5.18) to prove the inclusion of
intervals (5.17).
###### Claim 5.10.
$i_{Y}^{+}\leq i_{W}^{-}$.
By (5.14) and the Behrstock inequality, $d_{Y}(\partial{W},\mu^{+})\leq M$. By
(5.9) $i_{W}^{-}\in J_{W}$, so $\rho(i_{W}^{-})\supset\partial{W}$. Thus
$d_{Y}(\rho(i_{W}^{-}),\mu^{+})\leq M$. Then since $i_{Y}^{+}$ is the maximal
time such that $d_{Y}(\rho(i),\mu^{+})\leq M$ we have $i_{Y}^{+}\leq
i_{W}^{-}$.
By Claim 5.10, $[i_{Y}^{-},i_{Y}^{+}]\subseteq[m,i_{W}^{-}]$. Then by (5.23),
$[i_{Y}^{-},i_{Y}^{+}]\subseteq E_{W}$. So (5.21) implies that $E_{Y}\cap
E_{W}\neq\emptyset$.
Case 1.3:
(5.25) $d_{Y}(P,\mu^{+})\leq 3M+\epsilon+4.$
(5.25) and the second part of Claim 5.9 using an argument similar to the one
for the proof of (5.16) in Case 1.1 imply that
(5.26) $E_{Y}\supseteq[i_{Y}^{+},n]$
For every $j\in J_{W}$, $\rho(j)\supset\partial{W}$. By (5.14) and the
Behrstock inequality, $d_{Y}(\partial{W},\mu^{+})\leq M$. So
$d_{Y}(\rho(j),\mu^{+})\leq M$. Thus by the definition of $i_{Y}^{+}$, $j\geq
i_{Y}^{+}$. Therefore $J_{W}\subseteq[i_{Y}^{+},n]$. Then by (5.26)
$J_{W}\subseteq E_{Y}$. Moreover, by (5.3) $J_{W}\cap E_{W}\neq\emptyset$. So
we conclude that $E_{W}\cap E_{Y}\neq\emptyset$.
Case 2: $W\subsetneq Y$.
Recall the constant $e=2A$. We consider the following two subcases depending
on the value of $d_{Y}(\partial{W},P)$.
Case 2.1: $d_{Y}(\partial{W},P)\leq e$.
Let $i\in J_{W}$, by Theorem 2.13 (1) $\partial{W}\subset\rho(i)$, so
$d_{Y}(\rho(i),P)\leq d_{Y}(\partial{W},P)\leq e$. Recall that
$E_{Y}=\\{i:d_{Y}(\rho(i),P)\leq e\\}$, thus $J_{W}\subseteq E_{Y}$. Moreover,
by (5.3) $J_{W}\cap E_{W}\neq\emptyset$. Thus $E_{W}\cap E_{Y}\neq\emptyset$.
Case 2.2:
(5.27) $d_{Y}(\partial{W},P)>e.$
$P\in\Sigma_{\epsilon}(\mu^{-},\mu^{+})$, then as we saw in the paragraph
before (5.4), there is $x_{Y}\in\operatorname{hull}_{Y}(\mu^{-},\mu^{+})$ such
that $d_{Y}(x_{Y},P)\leq\epsilon$ and by Theorem 2.13 (5), there is
$x^{\prime}_{Y}$ on $g_{Y}$ such that $d_{Y}(x^{\prime}_{Y},x_{Y})\leq M$.
Then by the triangle inequality,
(5.28) $d_{Y}(x^{\prime}_{Y},P)\leq M+\epsilon.$
Let $h$ be a geodesic in $\mathcal{C}(Y)$ connecting $\pi_{Y}(P)$ to
$x^{\prime}_{Y}$. We claim that $h$ does not intersect the $1-$neighborhood of
$\partial{W}$. Otherwise, there is a vertex $z\in h$ with
$d_{Y}(z,\partial{W})\leq 1$, see Figure 3. Then we have
$\displaystyle d_{Y}(P,\partial{W})$ $\displaystyle\leq$ $\displaystyle
d_{Y}(P,z)+d_{Y}(z,\partial{W})$ $\displaystyle\leq$ $\displaystyle
d_{Y}(P,x^{\prime}_{Y})+1\leq(M+\epsilon)+1$
The first inequality is the triangle inequality and the second inequality
follows since $d_{Y}(P,z)\leq d_{Y}(P,x^{\prime}_{Y})$ (see Figure 3) and
$d_{Y}(z,\partial{W})\leq 1$. The third inequality holds by (5.28). But this
upper bound contradicts the lower bound (5.27) given as the assumption of Case
2.2.
Figure 3. Case 2.2: If $h$ intersects the $1-$neighborhood of $\partial{W}$
then the distance between $\pi_{Y}(\partial{W})$ and $\pi_{Y}(P)$ would be
less than the lower bound in the assumption of Case 2.2.
By the above claim $\partial{W}$ intersects every vertex of $h$, so Theorem
2.6 (Bounded Geodesic Image Theorem) implies that
(5.29) $d_{W}(P,x^{\prime}_{Y})\leq G.$
$W\subsetneq Y$ and both $W$ and $Y$ are component domains of $\rho$. Let
$\phi_{g_{Y}}(W)$ be the footprint of $W$ on $g_{Y}$ consisting of the
vertices of $g_{Y}$ which do not overlap $W$, see $\S 4$ of [MM00] and
Definition 4.9 there. Then by Lemma 4.10 of [MM00], $\phi_{g_{Y}}(W)$ is a
sequence of $1,2$ or $3$ consecutive vertices of $g_{Y}$.
If $x^{\prime}_{Y}\in\phi_{g_{Y}}(W)$ then by the definition of foot print it
does not intersect $\partial{W}$, so $d_{Y}(\partial{W},x^{\prime}_{Y})\leq
1$. Furthermore, by (5.28), $d_{Y}(x^{\prime}_{Y},P)\leq M+\epsilon$. These
two bounds combined by the triangle inequality imply that
$d_{Y}(\partial{W},P)\leq\epsilon+M+1$. But this contradicts (5.27) because
$M+\epsilon+1<e$. So we conclude that $x^{\prime}_{Y}\not\in\phi_{g_{Y}}(W)$.
Therefore, either $x_{Y}^{\prime}>\max\phi_{g_{Y}}(W)$ or
$x^{\prime}_{Y}<\min\phi_{g_{Y}}(W)$ as vertices on the geodesic
$g_{Y}\subset\mathcal{C}(S)$. We proceed to discuss these two cases. Let $u$
be the initial vertex of the geodesic $g_{Y}$ and $v$ be its final vertex.
Case 2.2.1: $x^{\prime}_{Y}<\min\phi_{g_{Y}}(W)$.
By the definition of foot print $\partial{W}$ intersects every vertex of
$g_{Y}$ between $u$ and $x^{\prime}_{Y}$. Hence by Theorem 2.6
$d_{W}(x^{\prime}_{Y},u)\leq G.$
$j_{W}^{-}$ be the initial parameter of $J_{W}$. By condition $S3$ of slices
at the beginning of $\S 5$ of [MM00] there is $w\in\phi_{g_{Y}}(W)$ such that
$w\subset\rho(j_{W}^{-})$. Let $i\in J_{Y}$ be such that $\rho(i)\supset u$.
Then since $u\leq w$ on $g_{Y}$, Theorem 2.13 (3) (Monotonicity) implies that
$i\leq j_{W}^{-}$. Then Theorem 2.13 (4) implies that
$d_{W}(u,\mu^{-})\leq M_{2}.$
Combining the above two subsurface coefficient bounds and (5.29) by the
triangle inequality we get
(5.30) $d_{W}(P,\mu^{-})\leq M_{2}+2G\leq 2M$
The second inequality above follows from the fact that $M\geq 2G$. Because
$M_{2}\geq G$ (see Lemma 6.1 (Sigma projection) in [MM00]), $M_{1}\geq 2M_{2}$
and $M\geq M_{1}$.
Let $i\in[m,i_{W}^{-}]$. Then by the first part of Claim 5.9,
$d_{W}(\rho(i),\mu^{-})\leq 2M+3$. This bound and the bound (5.30) combined by
the triangle inequality imply that $d_{W}(\rho(i),P)\leq 4M+3$. Now since
$e>4M+3$, by the definition of $E_{W}$ we have $i\in E_{W}$ so
(5.31) $[m,i^{-}_{W}]\subseteq E_{W}.$
By (5.9) $i_{W}^{-}\in J_{W}$, so by condition $S3$ of slices in $\S 5$ of
[MM00], as before, there is $w\in\pi_{g_{Y}}(W)$ such that
$w\subset\rho(i_{W}^{-})$. Let $j\in J_{Y}$ with $\rho(j)\supset
x_{Y}^{\prime}$. $x_{Y}^{\prime}\leq w$ as vertices along $g_{Y}$. Hence by
Theorem 2.13 (3) (Monotonicity) we have that $j\leq i_{W}^{-}$. So by (5.31),
$j\in E_{W}$.
Furthermore, $d_{Y}(\rho(j),P)\leq d_{Y}(x_{Y}^{\prime},P)\leq\epsilon+M$ and
$e>\epsilon+M$ so by the definition of $E_{Y}$ any $j$ as above is in the
interval $E_{Y}$. Therefore $E_{W}\cap E_{Y}\neq\emptyset$.
Case 2.2.2: $x^{\prime}_{Y}>\max\phi_{g_{Y}}(W)$.
In this case similar to Case 2.2.1 we can first show that
$d_{W}(P,\mu^{+})\leq 2M$, which again using a similar argument implies that
$E_{W}\supseteq[i_{W}^{+},n]$. Then following the exact same lines we can
conclude that $E_{W}\cap E_{Y}\neq\emptyset$.
In summary in all of the above cases we verified that (5.8) holds. Thus as we
explained earlier the lemma follows from Helly’s Theorem in dimension one. ∎
We can now complete the proof of Theorem 5.5. For any
$P\in\Sigma_{\epsilon}(\mu^{-},\mu^{+})$ Lemma 5.6 provides $d>0$ and
$\hat{j}\in[m,n]$ such that the inequality (5.1) $d_{X}(\rho(\hat{j}),P)\leq
d$ holds for every non-annular subsurface $X\subseteq S$. Let the threshold
constant in the distance formula (2.2) be $\max\\{M_{1},d\\}$. Let $\Delta$ be
the additive constant in the distance formula corresponding to this threshold
constant. Then
$d(P,\rho(\hat{j}))\leq\Delta.$
The above bound shows that $\Sigma_{\epsilon}(\mu^{-},\mu^{+})$ is contained
in the $\Delta$ neighborhood of $|\rho|$. We earlier proved that
$|\rho|\subset\Sigma_{\epsilon}(\mu^{-},\mu^{+})$. These facts together imply
that the Hausdorff distance of $|\rho|$ and
$\Sigma_{\epsilon}(\mu^{-},\mu^{+})$ is bounded by $\Delta$. Note that $d$
depends only on $A$ and $\epsilon$, so $\Delta$ depends only on $A$ and
$\epsilon$.
∎
### 5.2. Stability
The following theorem is the main result of this subsection
###### Theorem 5.11.
(Stable hierarchy resolution path) Given $A>0$ there is a quantifier function
$d_{A}:\mathbb{R}^{\geq 0}\times\mathbb{R}^{\geq 1}\to\mathbb{R}^{\geq 0}$
such that any hierarchy path with $A-$narrow end points $\mu^{-}$ and
$\mu^{+}$ is $d_{A}-$stable in the pants graph.
First note that by Theorem 5.5 given an $A-$narrow pair $(\mu^{-},\mu^{+})$
the Hausdorff distance of $\Sigma_{\epsilon}(\mu^{-},\mu^{+})$ ($\epsilon>M$)
and a hierarchy path $\rho$ between $\mu^{-}$ and $\mu^{+}$ is bounded by the
constant $\Delta$ depending only on $A$ and $\epsilon$. Then if
$\Sigma_{\epsilon}(\mu^{-},\mu^{+})$ is $d-$stable the hierarchy path $\rho$
is stable with quantifier function $d+\Delta$. So it suffices to prove that
$\Sigma_{\epsilon}(\mu^{-},\mu^{+})$ is stable. Our strategy to prove the
stability of the $\Sigma_{\epsilon}-$hull is to show that the projection map
$\Pi$ onto the $\Sigma_{\epsilon}-$hull ($\epsilon>F$) defined in Theorem 2.16
has the following contraction property:
###### Definition 5.12.
(Contraction property) Given $R,B\geq 0$ and $0<\eta\leq 1$ a subset
$\mathcal{Y}$ of a metric space $\mathcal{X}$ is $(R,B,\eta)-$contracting if
there is a map $\Pi:\mathcal{X}\to\mathcal{Y}$ with the following property.
For every $x,y\in\mathcal{X}$ if $d(x,\Pi x)>R$ then
$d(x,y)\leq\eta d(x,\Pi y)\Longrightarrow d(\Pi x,\Pi y)\leq B$
If a map $\Pi:\mathcal{X}\to\mathcal{Y}$ satisfies the contraction property,
the coarse Lipschitz property, and coarsely preserves $\mathcal{Y}$, then for
any $K\geq 1$ and $C\geq 0$ the standard Morse lemma argument as is in the
proof of Lemma 7.1 of [MM99], gives a $d>0$, such that a $(K,C)-$quasi-
geodesic with end points in $\mathcal{Y}$ stays in the $d-$neighborhood of
$\mathcal{Y}$. In this way we get a quantifier function $d:\mathbb{R}^{\geq
1}\times\mathbb{R}^{\geq 0}\to\mathbb{R}^{\geq 0}$, depending only on $R,B$
and $\eta$ such that $\mathcal{Y}$ is $d-$stable in $\mathcal{X}$.
We take from [Mah10] the following properties of $\delta-$hyperbolic spaces
which are not necessarily locally compact (for example the curve complex).
These properties will be used in the proof of Lemma 5.18. Let $\mathcal{X}$ be
a $\delta-$hyperbolic space which is not necessarily locally compact.
###### Proposition 5.13.
Let $\zeta$ be a geodesic in $\mathcal{X}$. Given points $x,y$ in
$\mathcal{X}$ or its Gromov boundary. Let $x^{\prime},y^{\prime}$ be closet
points to $x$ and $y$ on $\zeta$, respectively. When $x$ is at the boundary,
let $\xi_{i}\in\mathcal{X}$ be a sequence of points with $\xi_{i}\to x$ as
$i\to\infty$ and let $x^{\prime}$ be the limit of $\xi_{i}$’s. Similarly for
$y$ and $y^{\prime}$. Then
1. (i)
A geodesic $[x,w]$ connecting $x$ to $w$ any point on $\zeta$ intersects the
$3\delta$ neighborhood of $x^{\prime}$.
2. (ii)
$d(x^{\prime},y^{\prime})\leq d(x,y)+12\delta$.
3. (iii)
(Tree like) Let $K=14\delta$ and $\delta^{\prime}=24\delta$, suppose that
$d(x^{\prime},y^{\prime})>K$ then we have
(5.32) $d(x,x^{\prime})+d(x^{\prime},y^{\prime})+d(y^{\prime},y)\leq
d(x,y)+\delta^{\prime}$
There is a function $b(a,\delta)$ increasing in both $a$ and $\delta$ with the
following property. Let $[x,x^{\prime}]$ and $[y,y^{\prime}]$ be geodesics
connecting $x$ to $x^{\prime}$ and $y$ to $y^{\prime}$, respectively. Let
$\mathcal{N}_{a}$ denote the $a-$neighborhood of subset of $\mathcal{X}$.
4. (iv)
If
$\mathcal{N}_{a}([x,x^{\prime}])\cap\mathcal{N}_{a}([y,y^{\prime}])\neq\emptyset$,
then $d(x,x^{\prime})\leq b$.
(i) and (iii) are respectively propositions 3.2 and 3.4 of [Mah10]. (iv) can
be proved by a slight modification of the proof given for Proposition 3.4 in
[Mah10].
###### Proof of part (ii).
By part (i) there is a point $z$ on $[x,y^{\prime}]$ with $d(z,x^{\prime})\leq
3\delta\;(1)$ and there is a point $z^{\prime}$ with
$d(z^{\prime},y^{\prime})\leq 3\delta\;(2)$. Without loss of generality
suppose that $d(x,z)\geq d(y,z^{\prime})\;(3)$. We claim that
(5.33) $d(z,z^{\prime})\leq d(x,y)+6\delta.$
Otherwise, $d(z,z^{\prime})>d(x,y)+6\delta(4)$. Now we have
$\displaystyle d(x,y^{\prime})$ $\displaystyle\leq$ $\displaystyle
d(x,y)+d(y,z^{\prime})+d(z^{\prime},y^{\prime})<d(z,z^{\prime})-6\delta+d(x,z)+3\delta$
$\displaystyle\leq$ $\displaystyle
d(z,z^{\prime})+d(x,z)-d(z^{\prime},y^{\prime})\leq
d(y^{\prime},z)+d(x,z)=d(x,y^{\prime})$
The second inequality follows from $(2),(3)$ and (4). The third inequality
follows from (2). The fourth inequality is the triangle inequality.
But then we have $d(x,y^{\prime})<d(x,y^{\prime})$, which is contradictory and
we obtain the claimed bound.
Finally, we have that
$d(x^{\prime},y^{\prime})\leq
d(x^{\prime},z)+d(z,z^{\prime})+d(z^{\prime},y^{\prime})\leq d(x,y)+12\delta,$
where the first inequality is the triangle inequality and the second one
follows from (1), (5.33) and (2). ∎
The constants $K$ and $\delta^{\prime}$ depend only on $\delta$ the
hyperbolicity constant of the metric space $\mathcal{X}$. Given $Y\subseteq
S$, denote by $K_{Y},\delta^{\prime}_{Y}$ the corresponding constants of the
curve complex of $Y$ which depend only on $\delta_{Y}$ and consequently the
topological type of $Y$.
We will also need the following elementary lemmas.
###### Lemma 5.14.
Given a point $z$ and a geodesic $\zeta$ in $\mathcal{X}$. Let $z^{\prime}$ be
a nearest point to $z$ on $\zeta$. Let $x$ be a point on $\zeta$ and
$x^{\prime}$ be a nearest point to $x$ on $[z,z^{\prime}]$ the geodesic
connecting $z$ to $z^{\prime}$. Then $d(z^{\prime},x^{\prime})\leq 6\delta$
###### Proof.
By Proposition 5.13 (i) the geodesic segment $[x,z^{\prime}]$ intersects the
$3\delta$ neighborhood of $x^{\prime}$ at a point $f$. We claim that
$d_{Z}(f,x^{\prime})\leq 3\delta$. For otherwise the path
$[x,f]\cup[f,z^{\prime}]$ would have length less than the length of
$[x,x^{\prime}]$ which contradicts the fact that $[x,x^{\prime}]$ minimizes
the distance between $x$ and $x^{\prime}$. Then by the triangle inequality
$d_{Z}(x^{\prime},z^{\prime})\leq d(x^{\prime},f)+d(f,z^{\prime})\leq
3\delta+3\delta=6\delta$. ∎
###### Lemma 5.15.
Given a point $z$ and a geodesic $\zeta$ in $\mathcal{X}$. Let $z^{\prime}$ be
a closest point to $z$ on $\zeta$, then for any $p$ on $\zeta$,
$d(p,z)+6\delta\geq d(p,z^{\prime})$.
###### Proof.
Extend $\zeta$ to an infinite geodesic $\bar{\zeta}$. By Proposition 5.13 (ii)
the diameter of the set of nearest points to $z$ on $\bar{\zeta}$ is at most
$12\delta$. Let $p^{\prime}$ be the point on $\bar{\zeta}$ so that the
distance of $p^{\prime}$ and the set is the same as the distance of $p$ and
the set. By symmetry the $d(p,z)=d(p^{\prime},z)$ and by the triangle
inequality $d(p,z)+d(z,p^{\prime})\geq d(p,p^{\prime})$. Thus $2d(p,z)\geq
d(p,p^{\prime})$. But $d(p,p^{\prime})+12\delta\geq 2d(p,z^{\prime})$. So we
get $2d(p,z)+12\delta\geq 2d(p,z)$, dividing both sides of this inequality by
$2$ we get the desired inequality. ∎
###### Lemma 5.16.
Let geodesics $f$ and $f^{\prime}$ $D$ fellow travel each other in
$\mathcal{X}$. Let $u$ and $\hat{u}$ be nearest points to a point $p$ on $f$
and $f^{\prime}$ respectively. Then $d(\hat{u},u)\leq 3D+6\delta$.
###### Proof.
Since $f$ and $f^{\prime}$ $D$ fellow travel each other, the nearest point to
$u$ on $f^{\prime}$ has distance at most $d(u,p)+D$ to $p$. Moreover,
$\hat{u}$ is a nearest point to $p$ on $f^{\prime}$ so $d(p,\hat{u})\leq
d(u,p)+D$. Similarly $d(p,u)\leq d(\hat{u},p)+D$.
Let $u^{\prime}$ be a nearest point to $u$ on $f^{\prime}$, then
$d(u^{\prime},u)\leq d$. Now
$d(p,u^{\prime})\leq d(p,u)+d(u,u^{\prime})\leq d(p,u)+2D.$
Now we prove that $d(u,u^{\prime})\leq 3\delta+2D$. By Proposition 5.13 (i),
there is a point $z$ on $[p,u^{\prime}]$ such that $d(z,\hat{u})\leq 3\delta$.
Then by the triangle inequality $d(z,p)\geq d(p,\hat{u})-d(\hat{u},z)\geq
d(p,\hat{u})-3\delta$. Thus
$\displaystyle d(z,u^{\prime})$ $\displaystyle=$ $\displaystyle
d(p,u^{\prime})-d(p,z)\leq d(p,u^{\prime})-(d(p,\hat{u})-3\delta)$
$\displaystyle=$ $\displaystyle d(p,u^{\prime})-d(p,\hat{u})+3\delta\leq
D+3\delta.$
Then by the triangle inequality $d(u^{\prime},u)\leq
d(u^{\prime},z)+d(z,u)\leq 3\delta+2D$. Finally $d(u,\hat{u})\leq
d(\hat{u},u^{\prime})+d(u^{\prime},u)\leq D+3\delta+3D=6\delta+3D$. ∎
Let $F$ be the constant from Theorem 2.16.
###### Theorem 5.17.
(Narrow hulls are contracting) Given $A>0$ and $\epsilon>F$, there are $R,B>0$
and $0<\eta\leq 1$ with the following property. Let $(\mu^{-},\mu^{+})$ be an
$A-$narrow pair, then $\Sigma_{\epsilon}(\mu^{-},\mu^{+})$ has the contraction
property with constants $R,B$ and $\eta$.
###### Proof.
We prove that the projection map onto $\Sigma_{\epsilon}(\mu^{-},\mu^{+})$
defined in Theorem 2.16 has the contraction property. Note that to have a
projection onto the hull the theorem requires that $\epsilon>F$. Indeed we
prove the contrapositive of the contraction property which says that for
$B,R>0$, if $d(P,\Pi P)>R$ then $d(\Pi P,\Pi Q)>B\Longrightarrow d(P,\Pi
P)>\eta d(P,Q)$.
###### Lemma 5.18.
Given $A>0$ and $\epsilon>F$, there are $B>0$ and $q=q(A)>0$ with the
following properties. Let $(\mu^{-},\mu^{+})$ be an $A-$narrow pair and
$\Pi:P(S)\to\Sigma_{\epsilon}(\mu^{-},\mu^{+})$ be the projection onto the
$\Sigma-$hull. Let $P,Q\in P(S)$ be such that $d(\Pi P,\Pi Q)>B$, then
(5.34) $d_{Z}(P,Q)\geq d_{Z}(P,\Pi P)-q$
for every subsurface $Z\subseteq S$.
Let us first see how this lemma implies the contraction property. Assume that
(5.34) holds for every subsurface $Z\subseteq S$ then by the distance formula
(2.2) we have
$\displaystyle d(P,Q)$ $\displaystyle\asymp_{K,C}$
$\displaystyle\sum_{\begin{subarray}{c}Z\subseteq S\\\ \operatorname{non-
annular}\end{subarray}}\\{d_{Z}(P,Q)\\}_{A}$ $\displaystyle\geq$
$\displaystyle\sum_{\begin{subarray}{c}Z\subseteq S\\\ \operatorname{non-
annular}\end{subarray}}\\{d_{Z}(P,\Pi P)-q\\}_{A}$
Recalling the definition of $\\{.\\}_{A}$, for any term in the last sum above
we have
$\\{d_{Z}(P,\Pi P)-q\\}_{A}\geq\frac{A}{A+q}\\{d_{Z}(P,\Pi P)\\}_{A+q}$
Moreover, for the threshold constant $A_{1}=A+q$ there are constants
$K_{1},C_{1}$ such that the distance formula (2.2) is written as
$\sum_{\begin{subarray}{c}Z\subseteq S\\\ \operatorname{non-
annular}\end{subarray}}\\{d_{Z}(P,\Pi P)\\}_{A+q}\asymp_{K_{1},C_{1}}d(P,\Pi
P)$
Therefore, we obtain
$d(P,Q)>\eta^{\prime}d(P,\Pi P)-c$
where $\eta^{\prime}=\frac{A}{K(A+q)K_{1}}$ and
$c=\frac{AC}{K_{1}(A+q)K}+C_{1}$.
Now let $R$ be large enough such that $\eta=\eta^{\prime}-\frac{c}{R}>0$. Then
for any $P\in P(S)$ such that $d(P,\Pi P)>R$, and any $Q\in P(S)$, we have
$\displaystyle d(P,Q)$ $\displaystyle>$ $\displaystyle\eta^{\prime}d(P,\Pi
P)-c=[(\eta^{\prime}-\frac{c}{R})d(P,\Pi P)]-c+(\frac{c}{R})d(P,\Pi P)$
$\displaystyle>$ $\displaystyle\eta d(P,\Pi P)$
This shows that for $R$ and $\eta$ as above the projection map
$\Pi:P(S)\to\Sigma_{\epsilon}(\mu^{-},\mu^{+})$ provided that $\epsilon>F$ and
$(\mu^{-},\mu^{+})$ is $A-$narrow satisfies the contrapositive of the
contraction property. So it has the contraction property. ∎
###### Proof of Lemma 5.18.
We set
(5.35)
$A^{\prime}:=A+12\delta+\delta^{\prime}+\bar{K}+2F+2M+2(F_{1}+4)+G+2(3D+6\delta)+2(D_{1}+1),$
as the threshold costant of the distance formula (2.2) in this proof.
$\delta=\max_{Y\subseteq S}\delta_{Y}$, $\bar{K}=\max_{Y\subseteq S}K_{Y}$ and
$\delta^{\prime}=\max_{Y\subseteq S}\delta^{\prime}_{Y}$, where $\delta_{Y}$
is the hypebolicity constant of the curve complex of a subsurface $Y$
$\mathcal{C}(Y)$. $K_{Y}$ and $\delta^{\prime}_{Y}$ are the constants from
Proposition 5.13 (iv) for $\mathcal{C}(Y)$. Note that $\delta_{Y},K_{Y}$ and
$\delta^{\prime}_{Y}$ depend only on the topological type of $Y$, so the above
maxima exist. $G$ is the bound from Theorem 2.6 (Bounded Geodesic Image
Theorem). $M=M_{1}+B_{0}+4$, where the constants $M_{1}$ is from Theorem 2.13
(2) and $B_{0}$ is form Theorem 2.8 (Behrstock Inequality). $F$ and $F_{1}$
are the constants in Theorem 2.16 (Consistency Theorem).
Further, $D=\max_{Y\subseteq S}D_{Y}$, where $D_{Y}$ is the fellow traveling
distance of two geodesics in $\mathcal{C}(Y)$ with end points are within
distance $2$ of each other. In particular, since the diameter of the
projection of a marking or a lamination to a subsurface is at most $2$, $D$ is
the fellow traveling distance of two geodesics in the convex hull of a pair of
markings/laminations in any subsurface. Let $D_{1}$ be the maxima over all
subsurfaces $Y$ of the fellow traveling distance of two geodesics with end
points within distance $F+(3D+6\delta)$ in $\mathcal{C}(Y)$. For the fellow
traveling property of geodesics in $\delta-$hyperbolic spaces see part III.H
of [BH99].
First note that if
$d_{Z}(P,\Pi P)\leq A^{\prime}+5(F_{1}+4)$
then for $q=A^{\prime}+5(F_{1}+4)$, (5.34) holds. Thus in the rest of the
proof we will assume that $Z$ is a subsurface with
(5.36) $d_{Z}(P,\Pi P)>A^{\prime}+5(F_{1}+4)$
We proceed to discuss the following two cases depending on the value of
$d_{Z}(\Pi P,\Pi Q)$.
Caes 1:
(5.37) $d_{Z}(\Pi P,\Pi Q)>A^{\prime}.$
Let $\hat{u}$ be a nearest point to $\pi_{Z}(P)$ on
$\operatorname{hull}_{Z}(\mu^{-},\mu^{+})$ and $\hat{w}$ be a nearest point to
$\pi_{Z}(Q)$ on the hull. Then by Theorem 2.16, $d_{Z}(\hat{u},\Pi P)\leq F$
and $d_{Z}(\hat{w},\Pi Q)\leq F$. Let $f$ be a geodesic in
$\operatorname{hull}_{Z}(\mu^{-},\mu^{+})$. Let $u$ and $w$ be nearest points
to $\pi_{Z}(P)$ and $\pi_{Z}(Q)$ on $f$, respectively. $\hat{u}$ lies on a
geodesic $\hat{f}$ and $\hat{f}$ and $f$ $D$ fellow travel each other. Then
Lemma 5.16 applied to geodesics $f$, $\hat{f}$ and the point $\pi_{Z}(P)$
implies that $d_{Z}(\hat{u},u)\leq 3D+6\delta$. Similarly we can show that
$d_{Z}(\hat{w},w)\leq 3D+6\delta$.
The four bounds in the previous paragraph combined by the triangle inequality
imply that
$\displaystyle d_{Z}(u,w)$ $\displaystyle\geq$ $\displaystyle d_{Z}(\Pi P,\Pi
Q)-2F-2(3D+6\delta)-\operatorname{diam}_{W}(\Pi P)-\operatorname{diam}_{W}(\Pi
Q)$ $\displaystyle\geq$ $\displaystyle A^{\prime}-2F-2(3D+6\delta)-2$
where the second inequality follows from (5.37). Then by the choice of
$A^{\prime}$ (5.35), $A^{\prime}-2F-2(3D+6\delta)-2\geq K_{Z}+1$. Thus by the
tree like property (5.32) we have that
$d_{Z}(P,Q)\geq d_{Z}(P,u)-\delta_{Z}^{\prime}-\operatorname{diam}_{Z}(P)\geq
d_{Z}(P,u)-\delta^{\prime}-1.$
Furthermore, by the triangle inequality and the bounds $d_{Z}(\Pi
P,\hat{u})\leq F$ and $d_{Z}(\hat{u},u)\ 3D+6\delta$ above we have
$d_{Z}(P,u)\geq d_{Z}(P,\Pi P)-d_{Z}(\Pi
P,\hat{u})-d_{Z}(\hat{u},u)-\operatorname{diam}_{Z}(\Pi P)\geq d_{Z}(P,\Pi
P)-F-(3D+6\delta)-1.$
Plugging the above inequality into the one before it we get
$d_{Z}(P,Q)\geq d_{Z}(P,\Pi P)-\delta^{\prime}-F-(3D+6\delta)-1.$
Thus (5.34) holds for $q=\delta^{\prime}+F+(3D+6\delta)+1$.
Case 2:
(5.38) $d_{Z}(\Pi P,\Pi Q)\leq A^{\prime}.$
Since the threshold constant of the distance formula is $A^{\prime}$,
$d_{Z}(\Pi P,\Pi Q)$ has no contribution to $d(\Pi P,\Pi Q)$. Let $K^{\prime}$
and $C^{\prime}$ be the constants in the distance formula corresponding to the
threshold constant $A^{\prime}$. Set $B=K^{\prime}+C^{\prime}$ as the
projection bound in the statement of Theorem 5.17. Then by the assumption of
the lemma we have that
$d(\Pi P,\Pi Q)>K^{\prime}+C^{\prime}.$
Then by the distance formula (2.2) there must be a subsurface $W\subseteq S$
with
(5.39) $d_{W}(\Pi P,\Pi Q)>A^{\prime}.$
Let $\hat{v}$ be a nearest point to $\pi_{W}(P)$ on
$\operatorname{hull}_{W}(\mu^{-},\mu^{+})$ and $\hat{z}$ be a nearest point to
$\pi_{W}(Q)$ on the hull. Then by Theorem 2.16, $d_{W}(\Pi P,\hat{v})\leq F$
and $d_{W}(\Pi Q,\hat{z})\leq F$. Let $f$ be a geodesic in
$\operatorname{hull}_{W}(\mu^{-},\mu^{+})$. Let $v$ and $z$ be nearest points
to $\pi_{W}(P)$ and $\pi_{W}(Q)$ on $f$, respectively. $\hat{v}$ lies on a
geodesic $\hat{f}$ in the hull. Then applying Lemma 5.16 to the geodesics $f$
and $\hat{f}$ and the point $\pi_{W}(P)$ implies $d_{W}(v,\hat{v})\leq
3D+6\delta$. Similarly we cans how that $d_{W}(z,\hat{z})\leq 3D+6\delta$.
The four bounds in the above paragraph and (5.39) combined by the triangle
inequality imply that
(5.40) $d_{W}(v,z)>A^{\prime}-2F-2(3D+6\delta)-\operatorname{diam}_{W}(\Pi
P)-\operatorname{diam}_{W}(\Pi Q)\geq A^{\prime}-2F-2(3D+6\delta)-2.$
Now let $f$ be the geodesic in the hull realizing the distance between
$\pi_{W}(\mu^{-})$ and $\pi_{W}(\mu^{+})$. Let $v$ and $z$ be nearest points
to $\pi_{W}(P)$ and $\pi_{W}(Q)$ on $f$, respectively. Then by (5.40) and
since $d_{W}(\mu^{-},\mu^{+})\geq d_{W}(v,z)$, we get
(5.41) $d_{W}(\mu^{-},\mu^{+})\geq A^{\prime}-2F-2-2(3D+6\delta)$
By the choice of $A^{\prime}$ in (5.35), $A^{\prime}-2F-2(3D+6\delta)-2>A$, so
we get
$d_{W}(\mu^{-},\mu^{+})>A$
By the assumption the lemma of the pair $(\mu^{-},\mu^{+})$ is $A-$narrow, so
the above inequality implies that the subsurface $W$ is a large subsurface.
This excludes the possibility that $W$ and $Z$ are disjoint subsurfaces. Thus
we need to discuss the following three subcases:
1. (2.1)
$W\pitchfork Z$,
2. (2.2)
$W\subsetneq Z$ and
3. (2.3)
$Z\subsetneq W$.
Case 2.1: $W\pitchfork Z$.
Let
* •
$k:=k(P)=\lfloor\frac{d_{Z}(P,\Pi P)-2}{F_{1}+4}\rfloor$ and
* •
$k^{\prime}=\lfloor\frac{A^{\prime}}{F_{1}+4}\rfloor$.
Here $\lfloor x\rfloor$ is the floor function, which assigns to
$x\in\mathbb{R}$ the largest integer less than or equal to $x$.
Dividing both sides of the inequality (5.36) by $F_{1}+4$ and subtracting
$\frac{2}{F_{1}+2}$ from both sides we get
$\frac{d_{Z}(P,\Pi
P)-2}{F_{1}+4}\geq\frac{A^{\prime}}{F_{1}+4}+5-\frac{2}{F_{1}+4}.$
Now since $0<\frac{2}{F_{1}+4}\leq 1$, taking the floor of both sides of the
above inequality we have that
(5.42) $k\geq k^{\prime}+4$
We claim that
(5.43) $d_{Z}(\Pi P,\partial{W})\leq(k^{\prime}+2)(F_{1}+4).$
Otherwise,
$d_{Z}(\Pi P,\partial{W})>(k^{\prime}+2)(F_{1}+4),$
then using $\Pi P$ to define a partial order, which means that we let
$x_{Y}=\pi_{Y}(\Pi P)$ in Definition 2.11, the last inequality can be written
as
(5.44) $Z\ll_{k^{\prime}+2}W.$
Moreover, by (5.39) and since $A^{\prime}>2(F_{1}+4)$ we have
(5.45) $W\ll_{2}\Pi Q.$
Having (5.44) and (5.45), by the transitivity property of $\ll$ (Theorem 2.12
(2)) we deduce that
$Z\ll_{k^{\prime}+1}\Pi Q,$
which means that
$d_{Z}(\Pi Q,\Pi P)\geq(k^{\prime}+1)(F_{1}+4)>A^{\prime}.$
But this lower bound contradicts (5.38) and our claim follows. Therefore, in
the rest of Case 2.1 we may assume that (5.43) holds.
By the choice of $k$ we have
$d_{Z}(\Pi P,P)-2\geq k(F_{1}+4).$
This inequality and (5.43) combined by the triangle inequality imply that
(5.46)
$d_{Z}(P,\partial{W})\geq(k-k^{\prime}-2)(F_{1}+4)+2-\operatorname{diam}_{Z}(\Pi
P)\geq(k-k^{\prime}-2)(F_{1}+4)$
Now using $P$ to define a partial order, which means that we let
$x_{Y}=\pi_{Y}P$ in Definition 2.11, (5.46) can be written as
(5.47) $Z\ll_{k-k^{\prime}-2}W$
note that by (5.42) $k-k^{\prime}-2\geq 1$.
Let $f$ be a geodesic in the hull. Let $v$ and $z$, as before, be nearest
points to $\pi_{W}(P)$ and $\pi_{W}(Q)$ on $f$, respectively. Then by (5.40)
and the choice of $A^{\prime}$, $d_{W}(v,z)\geq
A^{\prime}-2F-2-2(3D+6\delta)>\bar{K}>K_{W}$. Therefore, the tree like
property (5.32) implies that
$\displaystyle
d_{W}(P,Q)>d_{W}(v,z)-\delta^{\prime}-\operatorname{diam}_{W}(P)-\operatorname{diam}_{W}(Q)>A^{\prime}-2F-\delta^{\prime}-2(3D+6\delta)-4.$
Thus we have
(5.48) $W\ll_{m}Q$
for
$m=\lfloor\frac{A^{\prime}-2F-\delta^{\prime}-2(3D+6\delta)-4}{F_{1}+4}\rfloor$.
Note that by the choice of $A^{\prime}$ (5.35), $m\geq 2$.
Having (5.47) and (5.48), by the transitivity property of $\ll$ (Theorem 2.12
(2)) we deduce that
$Z\ll_{k-k^{\prime}-3}Q$
where by (5.42), $k-k^{\prime}-3\geq 1$. Therefore,
$\displaystyle d_{Z}(P,Q)$ $\displaystyle\geq$
$\displaystyle(\lfloor\frac{d_{Z}(P,\Pi
P)-2}{(F_{1}+4)}\rfloor-k^{\prime}-3)(F_{1}+4)$ $\displaystyle\geq$
$\displaystyle d_{Z}(P,\Pi P)-(k^{\prime}+4)(F_{1}+4)-2.$
So the inequality (5.34) holds for $q=(k^{\prime}+4)(F_{1}+4)+2$.
Case 2.2: $W\subsetneq Z$.
Let $\rho^{\prime}$ be a hierarchy path between $P$ and $\Pi P$. By (5.36) we
have that $d_{Z}(P,\Pi P)>A^{\prime}>M_{1}$, so by Theorem 2.13 (2) $Z$ is a
component domain of $\rho^{\prime}$. Let $x$ be a nearest point to
$\pi_{Z}(Q)$ on $\operatorname{hull}_{Z}(P,\Pi P)$. By Theorem 2.13 (5) there
is a pants decomposition $T$ of $\rho^{\prime}$ such that $d_{Z}(T,x)\leq
M_{2}$. Let $h$ be a geodesic in $\mathcal{C}(Z)$ connecting $\pi_{Z}(Q)$ to
$\pi_{Z}(T)$, see the left diagram of Figure 4.
Figure 4. Case 2.2: Left diagram: $x$ is a nearest point to $\pi_{Z}(Q)$ on
$\operatorname{hull}_{Z}(P,\Pi P)$ and $T$ is a slice of $\rho^{\prime}$ (a
hierarchy path between $P$ and $\Pi P$) with $d_{Z}(T,x)\leq M_{2}$.
$\pi_{Z}(\partial{W})$ is in the $1-$neighborhood of the geodesic $h$
connecting $\pi_{Z}(Q)$ to $\pi_{Z}(T)$ and any geodesic $k$ in
$\operatorname{hull}_{Z}(\mu^{-},\mu^{+})$. Right diagram: $y$ on
$\operatorname{hull}_{W}(P,\Pi P)$ is such that $d_{W}(y,T)\leq M_{2}$ and
$y^{\prime}$ is a nearest point to $y$ on the geodesic $l$ connecting
$\pi_{W}(P)$ to $v$ a nearest point to $\pi_{W}(P)$ on the $f$.
Since $T\in|\rho^{\prime}|$, by Theorem 2.13 (5) there is
$y\in\operatorname{hull}_{W}(P,\Pi P)$ such that
(5.49) $d_{W}(T,y)\leq M_{2}\leq M,$
see the right diagram of Figure 4.
Let $\hat{v}$ and $\hat{z}$, as before, be nearest points to $\pi_{W}(P)$ and
$\pi_{W}(Q)$ on $\operatorname{hull}_{W}(\mu^{-},\mu^{+})$ respectively. Let
$f$ be a geodesic in $\operatorname{hull}_{W}(\mu^{-},\mu^{+})$ and $v$ and
$z$, as before, be nearest points to $\pi_{W}(P)$ and $\pi_{W}(Q)$ on $f$,
respectively.
$y$ is on a geodesic connecting $\pi_{W}(P)$ to $\pi_{W}(\Pi P)$. By the
triangle inequality $d_{W}(v,\Pi P)\leq d_{W}(v,\hat{v})+d_{W}(\hat{v},\Pi
P)\leq(3D+6\delta)+F$, so this geodesic and any geodesic $l$ connecting
$\pi_{Y}(P)$ to $v$, $D_{1}$ fellow travel each other ($D_{1}$ is the fellow
traveling distance of two geodesics with end pints within distance
$(3D+6\delta)+F$ of each other). Let $y^{\prime}$ be a nearest point to $y$ on
$l$, then by the fellow traveling we have that
$d_{W}(y,y^{\prime})\leq D_{1}.$
Furthermore, by Proposition 5.13 (ii),
$d_{W}(y^{\prime},Q)\geq
d_{W}(v,z)-12\delta_{W}-\operatorname{diam}_{W}(Q)\geq d_{W}(v,z)-12\delta-1.$
Combing the above two inequalities with the triangle inequality we get
$d_{W}(Q,y)\geq d_{W}(v,z)-12\delta-D_{1}-1,$
see the right diagram of Figure 4. By (5.40) $d_{W}(v,z)\geq
A^{\prime}-2F-2(3D+6\delta)-2$, so we get
(5.50) $d_{W}(Q,y)\geq A^{\prime}-2F-2(3D+6\delta)-12\delta-D_{1}-3.$
Now combining (5.49) and (5.50) by the triangle inequality we get
$d_{W}(Q,T)\geq A^{\prime}-2F-2(3D+6\delta)-D_{1}-12\delta-3-M.$
Then by the choice of $A^{\prime}$ (5.35) we have that
(5.51) $d_{W}(Q,T)>G.$
Note that since $W\subsetneq Z$ we have $\partial{W}\pitchfork Z$. We claim
that
###### Claim 5.19.
$\pi_{Z}(\partial{W})$ is in the $1-$neighborhood of $h$ in $\mathcal{C}(Z)$.
Otherwise, $\partial{W}$ would intersect every vertex of the geodesic $h$,
which connects $\pi_{Z}(T)$ to $\pi_{Z}(Q)$. Then by Theorem 2.6 (Bounded
Geodesic Image) $\operatorname{diam}_{W}(h)\leq G$. This contradicts the lower
bound (5.51) and the claim follows.
Let $k\subset\operatorname{hull}_{Z}(\mu^{-},\mu^{+})$ be any geodesic
connecting $\pi_{Z}(\mu^{-})$ to $\pi_{Z}(\mu^{+})$. We claim that
###### Claim 5.20.
$\pi_{Z}(\partial{W})$ is in the $1-$neighborhood of $k$.
Otherwise, $\partial{W}$ would intersect every vertex of $k$, so Theorem 2.6
implies that
(5.52) $d_{W}(\mu^{-},\mu^{+})\leq G$
On the other hand, by (5.41) and the choice of $A^{\prime}$ in (5.35) we have
$d_{W}(\mu^{-},\mu^{+})\geq A^{\prime}-2F-2(3D+6\delta)-2>G$. But this
contradicts the upper bound (5.52) and the claim follows.
Let $\hat{u}$ be a closest point to $\pi_{Z}(P)$ on
$\operatorname{hull}_{Z}(\mu^{-},\mu^{+})$. Suppose that $\hat{u}$ is on a
geodesic $k$ in the hull. Let $n$ be a geodesic connecting $\pi_{Z}(P)$ to
$\hat{u}$. Let $m$ be a nearest point to $\pi_{Z}(\mu^{+})$ on the geodesic
$n$. When $\pi_{Z}(\mu^{+})$ is at infinity of $\mathcal{C}(Z)$, let
$\xi_{i}\in\mathcal{C}(Z)$ be a sequence of points with
$\xi_{i}\to\pi_{Z}(\mu^{+})$ as $i\to\infty$ and let $m$ be the limit of
$\xi_{i}$’s. See the left diagram of Figure 4. Then by Lemma 5.14
$d_{Z}(\hat{u},m)\leq 6\delta_{Z}\leq 6\delta$. This implies that $k$ and $t$
a geodesic between $m$ and $\pi_{Z}(\mu^{+})$, $D_{1}$ fellow travel each
other. By Claim 5.20 there is a point on $k$ within distance $1$ of
$\pi_{Z}(\partial{W})$. Then the fellow traveling implies that
$\pi_{Z}(\partial{W})$ is within distance $D_{1}+1$ of $t$.
Let $x^{\prime}$ be a nearest point to $\pi_{Z}(Q)$ on $n$ and let
$h^{\prime}$ be a geodesic connecting $\pi_{Z}(Q)$ to $x^{\prime}$, see the
left diagram of Figure 4. By the choice of $T$, $d_{Z}(x,T)\leq M$. The fact
that the end points of $n$ and any geodesic between $\pi_{Z}(P)$ and
$\pi_{Z}(\Pi P)$ are within distance $F$ of each other implies that they
$D_{1}$ fellow travel each other. Then $r$ the geodesic in
$\operatorname{hull}_{Z}(P,\Pi P)$ on which $x$ lies and the geodesic $n$ on
which $x^{\prime}$ lies, $D_{1}$ fellow travel each other. Moreover $x$ is a
nearest point to $\pi_{Z}(Q)$ on $r$ and $x^{\prime}$ a nearest point to
$\pi_{Z}(Q)$ on the geodesic $n$. Then Lemma 5.16 implies that
$d_{Z}(x,x^{\prime})\leq 3D_{1}+6\delta$. Then by the triangle inequality we
get
$d_{Z}(T,x^{\prime})\leq d_{Z}(T,x)+d_{Z}(x,x^{\prime})\leq M+3D_{1}+6\delta.$
This bound on the distance of end points of $h$ and $h^{\prime}$ implies that
they $D_{2}$ fellow travel each other. Here $D_{2}$ is the maximum over all
subsurfaces of the fellow traveling distance of two geodesics with end point
within distance $M+3D_{1}+6\delta$ of each other. By Claim 5.19 there is a
point on $h$ within distance $1$ of $\pi_{Z}(\partial{W})$. Then the fellow
traveling implies that $\pi_{Z}(\partial{W})$ is within distance $D_{2}+1$ of
$h^{\prime}$.
Let $a:=\max\\{D_{1}+1,D_{2}+1\\}$. Then by the conclusions of the above two
paragraphs $\pi_{Z}(\partial{W})$ is in the $a$ neighborhood of $h^{\prime}$
and $t$, see the left diagram of Figure 4. So Proposition 5.13 (iv) implies
that $d_{Z}(m,x^{\prime})\leq b(\delta_{Z},a)\leq b(\delta,a)$. Further recall
the bounds $d_{Z}(\hat{u},\Pi P)\leq F$, $d_{Z}(x^{\prime},x)\leq
3D_{1}+6\delta$ and $d_{Z}(\hat{u},m)\leq 6\delta$ we established earlier.
These bounds combined by the triangle inequality imply that
(5.53) $d_{Z}(x,\Pi P)\leq 2+(3D_{1}+6\delta)+F+b+6\delta$
$x$ is a nearest point to $\pi_{Z}(Q)$ on the geodesic
$r\subset\operatorname{hull}_{Z}(P,\Pi P)$, then Lemma 5.15 applied to the
geodesic $r$ and the points $\pi_{Z}(Q)$ and $\pi_{Z}(P)$ implies that
(5.54) $d_{Z}(P,Q)+6\delta\geq d_{Z}(P,x)-\operatorname{diam}_{Z}(P)\geq
d_{Z}(P,x)-1$
Now we have
$\displaystyle d_{Z}(P,Q)$ $\displaystyle\geq$ $\displaystyle
d_{Z}(P,x)-6\delta-1\geq d_{Z}(P,\Pi P)-d_{Z}(\Pi
P,x)-\operatorname{diam}_{Z}(\Pi P)-6\delta-2$ $\displaystyle\geq$
$\displaystyle d_{Z}(P,\Pi P)-F-(3D_{1}+6\delta)-b-6\delta-6\delta-3$
The first inequality is (5.54). The second one is the triangle inequality. The
third one follows from the bound (5.53). Consequently, (5.34) holds for
$q=F+(3D_{1}+6\delta)+12\delta+b+3$.
Case 2.3: $Z\subsetneq W$.
Note that $\partial{Z}\pitchfork W$. We claim that
###### Claim 5.21.
$\pi_{W}(\partial{Z})$ is in the $1-$neighborhood of any geodesic
$l\subset\operatorname{hull}_{W}(P,\Pi P)$ connecting $\pi_{W}(P)$ to
$\pi_{W}(\Pi P)$.
Otherwise, $\partial{Z}$ would intersect every vertex of $l$. Then Theorem 2.6
implies that $d_{Z}(P,\Pi P)\leq G<A^{\prime}$, but this contradicts (5.36).
Let $f$ be a geodesic in $\operatorname{hull}_{W}(\mu^{-},\mu^{+})$. Let $v$
and $z$ be nearest points to $\pi_{W}(P)$ and $\pi_{W}(Q)$ on $f$,
respectively. Let $k$ be a geodesic connecting $\pi_{W}(Q)$ to $\pi_{W}(\Pi
Q)$, and $k^{\prime}$ be a geodesic connecting $\pi_{W}(Q)$ to $z$. Let $l$ be
a geodesics connecting $\pi_{W}(P)$ to $\pi_{W}(\Pi P)$, and $l^{\prime}$ be a
geodesic connecting $\pi_{W}(P)$ to $v$, see the left diagram of Figure 5. The
end points of $k$ and $k^{\prime}$ are within distance $F$ of each other, so
$D_{1}$ fellow travel each other. Similarly $l$ and $l^{\prime}$, $D_{1}$
fellow travel each other. By Claim 5.21 and the fellow traveling of $l$ and
$l^{\prime}$ there is a point $\hat{p}$ on $l^{\prime}$ with
(5.55) $d_{W}(\partial{Z},\hat{p})\leq 1+D_{1}.$
By (5.40) and the choice of $A^{\prime}$ in (5.35), $d_{W}(v,z)\geq
A^{\prime}-2F-2(3D+6\delta)>K_{W}$. Then the tree like property (5.32) implies
that for every $q$ on $k^{\prime}$ we have that
$d_{W}(\hat{p},q)\geq d_{W}(v,z)-\delta^{\prime}_{W}.$
Again by (5.40) and the choice of $A^{\prime}$,
$d_{W}(v,z)-\delta^{\prime}_{W}\geq
A^{\prime}-2F-2(3D+6\delta)-\delta^{\prime}>2D_{1}+2$, so by the above
inequality we get
$d_{W}(\hat{p},q)>2D_{1}+2.$
Since $q$ on $k^{\prime}$ was arbitrary, $d_{W}(\hat{p},k^{\prime})>2D_{1}+2$.
This and (5.55) imply that $d_{W}(\partial{Z},k^{\prime})>D_{1}+1$. Finally,
the $D_{1}$ fellow traveling of $k$ and $k^{\prime}$ implies that
$d_{W}(\partial{Z},k)>1$. Therefore, $\partial{Z}$ intersects every vertex of
$k$. So Theorem 2.6 implies that
(5.56) $d_{Z}(Q,\Pi Q)\leq G.$
By the triangle inequality $d_{Z}(P,Q)\geq d_{Z}(P,\Pi Q)-d_{Z}(\Pi Q,Q)-1$.
Replacing (5.56) in this inequality we get
(5.57) $d_{Z}(P,Q)\geq d_{Z}(P,\Pi Q)-G-1.$
Let $f$ be a geodesics in $\operatorname{hull}_{Z}(\mu^{-},\mu^{+})$. Let $u$
and $w$ be nearest points to respectively $\pi_{Z}(P)$ and $\pi_{Z}(Q)$ on
$f$. Since $u$ is a nearest point to $\pi_{Z}(P)$ on $f$ and $w$ is on $f$ we
have $d_{Z}(P,w)\geq d_{Z}(P,u)$. Furthermore, $d_{Z}(\Pi P,u)\leq d_{Z}(\Pi
P,\hat{u})-d_{Z}(\hat{u},u)\leq F+(3D+6\delta)$ and $d_{Z}(\Pi Q,w)\leq
F+(3D+6\delta)$. So
$d_{Z}(P,\Pi Q)\geq d_{Z}(P,\Pi P)-2F-2(3D+6\delta).$
Plugging the last inequality into (5.57) we get
$d_{Z}(P,Q)\geq d_{Z}(P,\Pi P)-G-2F-2(3D+6\delta)-1$
(see the right digram of Figure 5). Thus (5.34) holds for
$q=G+2F+2(3D+6\delta)+1$.
Figure 5. Case 2.3: Left diagram: $\partial{Z}$ is in the $1-$neighborhood of
$l$. The geodesics $l$ and $l^{\prime}$, and $k$ and $k^{\prime}$,
respectively, $D_{1}-$fellow travel each other. $d_{W}(v,z)>2D_{1}+2+G$. Thus
$\partial{Z}$ intersects every vertex of the geodesic $k$ connecting
$\pi_{W}(Q)$ to $\pi_{W}(\Pi Q)$. Then the Bounded Geodesic Image Theorem
implies that $d_{Z}(Q,\Pi Q)\leq G$.
Establishing (5.34) in cases (1) and (2), we may conclude that it holds for
$q$ the maximum of the $q$’s we obtained in these two cases. This finishes the
proof of the lemma. ∎
###### Remark 5.22.
$R\to\infty$, $\eta\to 0$ and $B\to\infty$, as $A\to\infty$, so applying the
Morse lemma argument we get $\Sigma-$hulls with worse and worse stability
property. More precisely, there are $K\geq 1$ and $C\geq 0$ such that
$d_{A}(K,C)\to\infty$ as $A\to\infty$.
### 5.3. Fellow traveling
We start by the definition of fellow traveling of parametrized quasi-geodesics
in a metric space.
###### Definition 5.23.
(Fellow traveling) Given $D\geq 0$. Let $h_{1}:I_{1}\to\mathcal{X}$ and
$h_{2}:I_{2}\to\mathcal{X}$ be two parametrized quasi-geodesics. We say that
$h_{1}$ and $h_{2}$, $D-$fellow travel if
* •
For every $i\in I_{1}$ there is an $i^{\prime}\in I_{2}$ such that
$d(h_{1}(i),h_{2}(i^{\prime}))\leq D$ and vice versa. In other words, the
Hausdorff distance of $h_{1}(I_{1})$ and $h_{2}(I_{2})$ is bounded by $D$.
Given a subinterval $I^{\prime}_{1}\subset I_{1}$, we say that $h_{1}$,
$D-$fellow travels $h_{2}$ over $I^{\prime}_{1}$ if there is a subinterval
$I^{\prime}_{2}\subset I_{2}$ such that $h_{1}|_{I^{\prime}_{1}}$ and
$h_{2}|_{I^{\prime}_{2}}$, $D-$fellow travel as above.
Let $h_{1}$ be a $(K_{1},C_{1})-$quasi-geodesic and $h_{2}$ be a
$(K_{2},C_{2})$. Let $N_{h_{1},h_{2}}:I_{1}\to I_{2}$, be the map which
assigns to each $i\in I_{1}$ any $h_{2}(i^{\prime})$ where
$d(h_{1}(i),h_{2}(i^{\prime}))\leq D$. Then $N_{h_{1},h_{2}}$ is a
reparametrization of $h_{2}$ and for any $i,j\in I_{1}$,
$d_{Huas}(N_{h_{1},h_{2}}(i),N_{h_{1},h_{2}}(j))\asymp_{K,C}|i-j|$
Here $d_{Haus}$ denotes the Hausdorff distance of subintervals of the real
line, $\mathbb{R}$, $K=K_{1}K_{2}$ and
$C=\max\\{K_{2}C_{1}+C_{2}+2K_{2}D,K_{1}C_{2}+C_{1}+2K_{1}D\\}$. In
particular, the diameter of $N_{h_{1},h_{2}}(i))$ is bounded above by $C$.
Also the same holds exchanging $h_{1}$ and $h_{2}$.
###### Theorem 5.24.
Given $A>0$ there is a constant $D=D(A)$ with the following property. Let
$g:[a,b]\to\operatorname{Teich}(S)$ be a WP geodesic segment with $A-$narrow
end invariant $(\nu^{-},\nu^{+})$, and let $\rho$ be a hierarchy path between
$(\nu^{-},\nu^{+})$. Then $\rho$ and $Q(g)$, $D-$fellow travel.
###### Proof.
$Q(g)$ is a $(K_{\operatorname{WP}},C_{\operatorname{WP}})-$quasi geodesic in
$P(S)$ where $K_{\operatorname{WP}}$ and $C_{\operatorname{WP}}$ depend only
on the topological type of $S$ (Theorem 3.3). The hierarchy path $\rho$ is a
$(k,c)-$quasi-geodesic in $P(S)$ where $k$ and $c$ depend only on the
topological type of $S$ (see $\S$2). By Theorem 5.11 $\rho$ is $d_{A}-$stable.
So the Hausdorff distance of $Q(g)$ and $|\rho|$ is bounded by
$D=d_{A}(K_{WP},C_{WP})$ in $P(S)$. These are the conditions required by
Definition 5.23 so that $\rho$ and $Q(g)$, $D$ fellow travel each other. ∎
Let $g:[a,b]\to\operatorname{Teich}(S)$ and $\rho:[m,n]\to P(S)$. Suppose that
$Q(g)$ and $\rho$, $D-$fellow travel. Given $i\in[m,n]$ take the smallest
interval $I_{i}$ containing every $t\in[a,b]$ such that
$d(Q(g(t)),\rho(i))\leq D$. Then define the coarse map
$N_{\rho,g}:[m,n]\to[a,b]$ so that $N_{\rho,g}(i)$ is any $t\in[a,b]$ in the
$K_{\operatorname{WP}}(2D+1)+C_{\operatorname{WP}}$ neighborhood $I_{i}$. The
following proposition is a straightforward consequence of the definition of
$N_{\rho,g}$.
###### Proposition 5.25.
The coarse map $N_{\rho,g}:[m,n]\to[a,b]$ has the following properties:
* •
$|N_{\rho,g}(i)|$ is bounded by a constant depending only on the fellow
traveling distance $D$ and topological type of the surface.
* •
$\bigcup_{i\in[m,n]}N_{\rho,g}(i)$ covers $[a,b]$,
* •
There are $K\geq 1$ and $C\geq 0$ depending only on $D$ and the topological
type of $S$, such that for any $i,j\in[m,n]$ we have
$d_{Haus}(N_{\rho,g}(i),N_{\rho,g}(j))\asymp_{K,C}|i-j|.$
By Theorem 5.24 this proposition in particular applies to a WP geodesic
segment with narrow end invariant and a hierarchy path with the same end
points.
## 6\. Itinerary of a Weil-Petersson geodesic segment
Itinerary of a WP geodesic $g$ in Teichmüller space refers to the list of
short curves (curves with length less than a sufficiently small $\epsilon>0$),
the time intervals along $g$ over which each curve is short and the order in
which these intervals appear along $g$.
In this section we present our results on the control of length-functions and
twist parameters along WP geodesics with narrow end invariant.
Our main result asserts the following: Suppose that over an interval all of
the subsurface coefficients are bounded, except possibly those of some annular
subsurfaces whose core curves consist the boundary of a large subsurface $Z$.
Then the length of these curves are arbitrary short over a suitably shrunk
subinterval of the interval.
###### Theorem 6.1.
(Short Curve)
Given $A,R,R^{\prime}>0$ and a sufficiently small $\epsilon>0$, there is a
constant $\bar{w}=\bar{w}(A,R,R^{\prime},\epsilon)$ with the following
property. Let $g:[a,b]\to\operatorname{Teich}(S)$ be a WP geodesic segment
with $A-$narrow end invariant $(\nu^{-},\nu^{+})$. Let $\rho:[m,n]\to P(S)$ be
a hierarchy path between $\nu^{-}$ and $\nu^{+}$. Assume that a large domain
$Z$ has $(R,R^{\prime})-$bounded combinatorics over
$[m^{\prime},n^{\prime}]\subset J_{Z}$.
If $m^{\prime}-n^{\prime}\geq 2\bar{w}$, then for every $\alpha\in\partial{Z}$
we have
$\ell_{\alpha}(g(t))\leq\epsilon$
for every $t\in[a^{\prime},b^{\prime}]$, where $a^{\prime}\in
N(m^{\prime}+\bar{w})$ and $b^{\prime}\in N(n^{\prime}-\bar{w})$. Here
$N:=N_{\rho,g}$ be the parameter map from Proposition 5.25.
We prove this theorem at the end of $\S$6.2. This theorem is a partial
itinerary for WP geodesic segments with narrow end invariant.
### 6.1. Isolated annular subsurfaces
In this subsection we prove two combinatorial lemmas which together with the
fellow traveling property of hierarchy paths between narrow pairs (Theorem
5.24) provide us with a combinatorial frame work in which we will be able to
control length-functions along WP geodesics.
Bounded combinatorics: Given $R,R^{\prime}>0$ and a subsurface $Z$, we say
that $Z$ has $(R,R^{\prime})-$bounded combinatorics between a pair of partial
markings/laminations $\mu_{1}$ and $\mu_{2}$ if for any essential, proper,
non-annular subsurface $Y\subsetneq Z$ the subsurface projection be bounded as
$d_{Y}(\mu_{1},\mu_{2})\leq R,$
and for any annular subsurface with core curve $\gamma\in\mathcal{C}_{0}(Z)$
which intersects both $\mu_{1}$ and $\mu_{2}$ the subsurface coefficient be
bounded as
$d_{\gamma}(\mu_{1},\mu_{2})\leq R^{\prime}.$
If only the first bound holds we say that the $Z$ has non-annular $R-$bounded
combinatorics and if only the second bound holds we say that $Z$ has annular
$R^{\prime}-$bounded combinatorics.
###### Lemma 6.2.
(No backtracking) Let $\rho:[m,n]\to P(S)$ be a hierarchy path. Let
$[i_{1},i_{2}]\subseteq[m,n]$ and $i,j\in[i_{1},i_{2}]$ with $i<j$. Then for
every subsurface $Y\subseteq S$ we have that
(6.1) $d_{Y}(\rho(i_{1}),\rho(i_{2}))\geq d_{Y}(\rho(i),\rho(j))-2M_{2}.$
###### Proof.
We have that
$\displaystyle
d_{Y}(\rho(i_{1}),\rho(i))+d_{Y}(\rho(i),\rho(j))+d_{Y}(\rho(j),\rho(i_{2}))$
$\displaystyle\leq$ $\displaystyle
d_{Y}(\rho(i_{1}),\rho(j))+d_{Y}(\rho(j),\rho(i_{2}))+M_{2}$
$\displaystyle\leq$ $\displaystyle d_{Y}(\rho(i_{1}),\rho(i_{2}))+2M_{2}$
The first inequality follows from Theorem 2.13 (6) (no backtracking) for
$i_{1}<i<j$. The second inequality follows from no backtracking by considering
$i<j<i_{2}$. All of the terms in (6.1) are non negative, thus
$d_{Y}(\rho(i_{1}),\rho(i_{2}))\geq d_{Y}(\rho(i),\rho(j))-2M_{2}$. ∎
Now let $Z$ be a component domain of $\rho$ and $[i_{1},i_{2}]\subset J_{Z}$.
If $Z$ has $(R,R^{\prime})-$bounded combinatorics between $\rho(i_{1})$ and
$\rho(i_{2})$ then by (6.1), $d_{Y}(\rho(i),\rho(j))\leq R+2M_{2}$ for any
$Y\subseteq S$ and $d_{\gamma}(\rho(i),\rho(j))\leq R^{\prime}+2M_{2}$ for any
$\gamma\in\mathcal{C}_{0}(Z)$. In this situation we say that $Z$ has
$(R,R^{\prime})-$bounded combinatorics over the interval $[i_{1},i_{2}]$.
In the following lemma we show that over a subinterval of a hierarchy path
where a large subsurface $Z$ has non-annular bounded combinatorics
$\pi_{Z}\circ\rho$ ($\pi_{Z}$ is the $Z$ subsurface projection) is a
parametrization of the geodesic $g_{Z}\subset\mathcal{C}(Z)$ (see Theorem 2.13
(1)) as a quasi-geodesic with constants depending only on $R$.
###### Lemma 6.3.
Given $R>0$, there are $K_{R}\geq 1$ and $C_{R}\geq 0$ with the following
properties. Let $\rho:[m,n]\to P(S)$ be a hierarchy path and let $Z$ be a
large, non-annular domain with non-annular $R-$bounded combinatorics over
$[i_{1},i_{2}]\subset J_{Z}\subset[m,n]$. Then for any $i,j\in[i_{1},i_{2}]$
we have
$d(\rho(i),\rho(j))\asymp_{K_{R},C_{R}}d_{Z}(\rho(i),\rho(j))$
###### Proof.
Given a threshold constant $A\geq M_{1}$ for distance formula (2.2), then we
have
(6.3) $d(\rho(i),\rho(j))\asymp_{K,C}\sum_{\begin{subarray}{c}Y\subseteq S\\\
\operatorname{non-annular}\end{subarray}}\\{d_{Y}(\rho(i),\rho(j))\\}_{A}$
Note that since $Z$ has complement consisting of only annuli and three holed
spheres, every subsurface $Y$ contributing to the above sum either is a
subsurface of $Z$ or overlaps $Z$. Suppose that $Y\pitchfork Z$. Since $i,j\in
J_{Z}$, by Theorem 2.13 (1) $\rho(i)\supset\partial{Z}$ and
$\rho(j)\supset\partial{Z}$. Then $\rho(i)\pitchfork Y$ and $\rho(i)\pitchfork
Y$, so by the triangle inequalty
(6.4) $d_{Y}(\rho(i),\rho(j))\leq
d_{Y}(\rho(i),\partial{Z})+d_{Y}(\partial{Z},\rho(j))+2\operatorname{diam}_{Y}(\partial{Z})\leq
4.$
Now suppose that $Y\subsetneq Z$. With out loss of generality we may assume
that $i_{1}<i<j<i_{2}$. Then the non-annular $R-$bounded combinatorics of $Z$
over $[i_{1},i_{2}]$ and no backtracking (6.1) imply that
(6.5) $d_{Y}(\rho(i),\rho(j))\leq d_{Y}(\rho(i_{1}),\rho(i_{2}))+2M_{2}\leq
R+2M_{2}.$
Having the bounds (6.4) and (6.5) on the subsurface coefficients contributing
to the sum (6.3) if we let the threshold constant be
$A_{R}=\max\\{R+2M_{2},4\\}$ (note that it is larger than $M_{1}$) we get
$d(\rho(i),\rho(j))\asymp_{K_{R},C_{R}}d_{Z}(\rho(i),\rho(j))$
where $K_{R}$ and $C_{R}$ are the constants corresponding to the threshold
constant $A_{R}$ in the distance formula. ∎
Let $\rho$ and $Q(g)$, $D-$fellow travel each other and $N=N_{g,\rho}$ be the
parameter map from Proposition 5.25. Given $i,j\in[m,n]$ let $r\in N(i)$ and
$s\in N(j)$. Then the $D-$fellow traveling implies that
$d(\rho(i),Q(g(r)))\leq D$ and $d(\rho(j),Q(g(s)))\leq D$. An elementary move
on a pants decomposition $P$, replaces a curve $\alpha\in P$ with a curve
$\alpha^{\prime}$ with distance $1$ in the complexity $1$ subsurface which
$\alpha$ and $\alpha^{\prime}$ fill and does not change curves in $P-\alpha$.
Thus an elementary move changes the value of a non-annular subsurface
coefficient by at most $1$. Thus for any $P,Q\in P(S)$ and any non-annular
subsurface $Y\subseteq S$ we have that
(6.6) $d_{Y}(P,Q)\leq d(P,Q)$
Therefore, $d_{Y}(\rho(i),Q(g(r))\leq D$ and $d_{Y}(\rho(j),Q(g(s)))\leq D$.
So by the triangle inequality we obtain the comparison of subsurface
coefficients
$d_{Y}(\rho(i),\rho(j))\asymp_{1,2D}d_{Y}(Q(g(r)),Q(g(s))).$
But such a comparison a-priori does not hold for annular subsurfaces, so we
consider isolated annular subsurfaces along hierarchy paths. In Lemma 6.5 we
prove a comparison for the subsurface coefficient of an isolated annular
subsurface along $\rho$ and a $D-$fellow traveling WP geodesic depending only
on $D$.
###### Definition 6.4.
(Isolated annular subsurface) Given $w,r>0$. Let $\rho:[m,n]\to P(S)$ be a
hierarchy path. We say that an annular subsurface $A(\gamma)$ with core curve
$\gamma$ is $(w,r)-$isolated at $i\in[m,n]$, if there is a pants decomposition
$\widehat{Q}$ such that $\gamma\in\widehat{Q}$ and $d(\widehat{Q},\rho(i))\leq
r$. Moreover, there are large, non-annular component domains of $\rho$,
$Z_{1}$ and $Z_{2}$ with $\gamma\notin\partial{Z_{1}}$ and
$\gamma\notin\partial{Z_{2}}$ and intervals $I_{1}\subseteq J_{Z_{1}}$ and
$I_{2}\subseteq J_{Z_{2}}$ with $|I_{1}|,|I_{2}|\geq w$ and $\max I_{1}<i<\min
I_{2}$ such that $Z_{1}$ and $Z_{2}$ have non-annular $R-$bounded
combinatorics over $I_{1}$ and $I_{2}$, respectively. For the illustration of
isolation see Figure 6.
###### Lemma 6.5.
(Annular coefficient comparison) Given $D,r$ and $R$ positive, there are
constants $\textbf{w}=\textbf{w}(D,r,R)$ and $B=B(D)$ with the following
properties. Let $\rho:[m,n]\to P(S)$ be a hierarchy path. Assume that
$A(\gamma)$ is $(\textbf{w},r)-$isolated at $i$. Let $I_{1},I_{2}$ be as in
the definition of isolated annular subsurface and $i_{1}<\min I_{1}$ and
$i_{2}>\max I_{2}$. Let $g:[a,b]\to\operatorname{Teich}(S)$ be a WP geodesic
parametrized by arc-length such that $Q(g)$ $D-$fellow travels $\rho$. Let
$t_{1}\in N(i_{1})$ and $t_{2}\in N(i_{2})$, where $N:=N_{\rho,g}$ be the
parameter map from Proposition 5.25. Then we have the annular coefficient
comparison
(6.7)
$d_{\gamma}(Q(g(t_{1})),Q(g(t_{2}))\asymp_{1,B}d_{\gamma}(\rho(i_{1}),\rho(i_{2})).$
Furthermore, we have the following lower bound on the length of $\gamma$
(6.8)
$\min\\{\ell_{\gamma}(g(t_{1})),\ell_{\gamma}(g(t_{2}))\\}\geq\omega(L_{S}).$
Here $\omega(l)$ denotes the width of the collar of a simple closed geodesic
with length $l$ on a hyperbolic surface.
###### Proof.
Since $\gamma$ is isolated at $i$, there is a pants decomposition
$\widehat{Q}$ with $\gamma\in\widehat{Q}$ and $d(\widehat{Q},\rho(i))\leq r$.
By the $D-$fellow traveling $d(\rho(i_{1}),P)\leq D$ for every $P$ on the
geodesic connecting $\rho(i_{1})$ to $Q(g(t_{1}))$. Thus by (6.6) these two
bounds imply that $d_{Y}(\widehat{Q},\rho(i))\leq r$ and
$d_{Y}(Q(g(t_{1})),\rho(i_{1}))\leq D$, respectively, for every non-annular
subsurface $Y\subseteq S$ (*).
Let $\textbf{w}(D,r,R)=kK_{R}\big{(}D+r+C_{R}+3+2M_{2}\big{)}+kc$, where
$K_{R}$ and $C_{R}$ are the constants form Lemma 6.3 and $k,c$ are the quasi-
geodesic constants of $\rho$ depending only on the topological type of $S$.
Let $P$ be a pants decomposition on a geodesic in $P(S)$ connecting
$\rho(i_{1})$ to $Q(g(t_{1}))$, see Figure 6. Let $I_{1}=[j,j^{\prime}]$.
$\rho$ is a $(k,c)-$quasi-geodesic so
$d(\rho(j),\rho(j^{\prime}))\geq\frac{1}{k}|I_{1}|-c$. Then by the assumption
of the lemma about $I_{1}$, Lemma 6.3 implies that
$d_{Z_{1}}(\rho(j),\rho(j^{\prime}))\geq\frac{1}{K_{R}}(\frac{1}{k}|I_{1}|-c)-C_{R}$.
Now no backtracking (6.1) applied to $i_{1},j,j^{\prime}$ and $i$ implies that
$d_{Z_{1}}(\rho(i),\rho(i_{1}))\geq\frac{1}{K_{R}}(\frac{1}{k}|I_{1}|-c)-C_{R}-2M_{2}$.
Then we have
$\displaystyle d_{Z_{1}}(P,\widehat{Q})$ $\displaystyle\geq$ $\displaystyle
d_{Z_{1}}(\rho(i_{1}),\rho(i))-d_{Z_{1}}(\widehat{Q},\rho(i))-d_{Z_{1}}(\rho(i_{1}),P)$
$\displaystyle\geq$
$\displaystyle\frac{1}{K_{R}}\big{(}\frac{1}{k}|I_{1}|-c\big{)}-2M_{2}-C_{R}-r-D\geq
3.$
The first inequality is the triangle inequality. The second inequality follows
from the inequality we established above and the bounds (*). The last
inequality follows from the choice of w.
Now since $\gamma\in\widehat{Q}$ and $\gamma\notin\partial{Z_{1}}$, the above
inequality implies that any curve in $P$ and $\gamma$ fill the subsurface
$Z_{1}$ and intersect each other. Hence the projection to the annular
subsurface $A(\gamma)$ of each two consecutive pants decompositions
$P,P^{\prime}$ on the geodesic connecting $\rho(i_{1})$ to $Q(g(t_{1}))$ is
non-empty. Then by (6.6) $d(P,P^{\prime})\leq 1$. Thus the projection of the
geodesic to $A(\gamma)$ has diameter bounded above by $D$, and consequently
$d_{\gamma}(Q(g(t_{1})),\rho(i_{1}))\leq D$. Similarly, replacing $i_{1}$ by
$i_{2}$, $t_{1}$ by $t_{2}$ and $Z_{1}$ by $Z_{2}$ we get
$d_{\gamma}(Q(g(t_{2})),\rho(i_{2}))\leq D$. These two bounds and the triangle
inequality imply that the annular coefficient comparison (6.7) holds for
$B=2D$.
As we saw above $\gamma\pitchfork Q(g(t_{1}))$ and $\gamma\pitchfork
Q(g(t_{2}))$. Now since every curve in the pants decompositions $Q(g(t_{1}))$
and $Q(g(t_{2}))$ has length at most the Bers constant $L_{S}$, the lower
bound for the length of $\gamma$ (6.8) follows from the Collar lemma.
Figure 6. Isolated annular subsurface: $\gamma\in\widehat{Q}$ and
$d(\widehat{Q},\rho(i))\leq r$. Non-annular, large domains $Z_{1}$ and $Z_{2}$
have non-annular bounded combinatorics on the left and the right blue
subintervals, respectively, moreover $\gamma\notin\partial{Z}_{1}$ and
$\gamma\notin\partial{Z}_{2}$. Each interval has length at least $w$. Then we
say that $A(\gamma)$ is $(w,r)-$isolated at $i$.
∎
### 6.2. Length-function control
Corollaries 4.12 and 4.13 in $\S$4 provide a local control on development of
Dehn twists about a curve versus the change of its length along WP geodesic
segments. The control is uniform in the length of the geodesic segment and the
supremum of the length-function along the geodesic segment. Lemma 6.5 provides
a comparison for the subsurface coefficient of an isolated annular subsurface
between two point of a hierarchy path and the corresponding points of a fellow
traveling WP geodesic. The comparison is uniform in the fellow traveling
distance. The comparison lets us to pull back the annular coefficient from the
geodesic to the hierarchy path and use the combinatorial properties of the
hierarchy to control the length-functions along the geodesic using its end
invariant. As we will see in $\S$8 the control of length-functions would lead
to the control of the global behavior of WP geodesics.
###### Lemma 6.6.
(Rough bounds)
Given $D,R,R^{\prime}>0$, there are constants $w=w(D,R)$,
$\bar{\epsilon}=\bar{\epsilon}(D,R^{\prime})$ and $l=l(D,R)$ with the
following properties. Let $\rho:[m,n]\to P(S)$ be a hierarchy path. Assume
that a large domain $Z$ has $(R,R^{\prime})-$bounded combinatorics over
$[m^{\prime},n^{\prime}]\subset J_{Z}$. Let
$g:[a,b]\to\overline{\operatorname{Teich}(S)}$ be a WP geodesic parametrized
by arc-length such that $\rho$ and $Q(g)$, $D-$fellow travel. If
$m^{\prime}-n^{\prime}>2w$ then
1. (1)
$\ell_{\alpha}(g(t))\leq l$ for every $\alpha\in\partial{Z}$ and
2. (2)
$\ell_{\gamma}(g(t))>\bar{\epsilon}$ for every $\gamma\notin\partial{Z}$
for every $t\in[a^{\prime},b^{\prime}]$, where $a^{\prime}\in N(m^{\prime}+w)$
and $b^{\prime}\in N(n^{\prime}-w)$. Here $N:=N_{\rho,g}$ is the parameter map
from Proposition 5.25.
###### Proof.
We start by establishing the lower bound (2). Let
$r=K_{\operatorname{WP}}D+C_{\operatorname{WP}}$. Let $w=\textbf{w}(D,r,R)$,
where w is the constant form Lemma 6.5. Let $s=Kw+C$, where $K$ and $C$ are
the constants for the parameter map $N$ in Proposition 5.25. Assume that
$n^{\prime}-m^{\prime}>2w$. Let $i_{0}\in[m^{\prime}+w,n^{\prime}-w]$ and
$t_{0}\in N(i_{0})$. Pick $t_{0}^{+}\in N(i_{0}+w)$ such that
$t_{0}^{+}-t_{0}=s$ and $t_{0}^{-}\in N(i_{0}-w)$ such that
$t_{0}-t_{0}^{-}=s$. See Figure 7. We denote by $\mu(g(t_{0}))$ a (partial)
Bers marking of the surface $g(t_{0})$. Similarly, we denote (partial) Bers
markings of $g(t_{0}^{-})$ and $g(t_{0}^{+})$ by $\mu(g(t_{0}^{-}))$ and
$\mu(g(t_{0}^{+}))$, respectively.
Let $\gamma\notin\partial{Z}$. If $\ell_{\gamma}(g(t_{0}))\geq L_{S}$ then we
already have the lower bound at $g(t_{0})$. Otherwise, there is $Q_{0}$ a Bers
pants decomposition of $g(t_{0})$ such that $\gamma\in Q_{0}$. Then according
to Definition 6.4, by the assumption of the lemma, $A(\gamma)$ is
$(w,r)$-isolated at $i_{0}$ where the non-annular bounded combinatorics domain
on both sides is $Z$. So by Lemma 6.5 (6.8) we have
(6.9)
$\min\\{\ell_{\gamma}(g(t_{0}^{-})),\ell_{\gamma}(g(t_{0}^{+}))\\}\geq\omega(L_{S}).$
Let $B=B(D)$ be the constant from Lemma 6.5 (6.7). By the choice of
$t_{0}^{-}$ and $t_{0}^{+}$, $|t_{0}^{+}-t_{0}^{-}|=2s$. Then by Corollary
4.13 there is $\bar{\epsilon}\leq L_{S}$ depending only on $s$ and
$\omega(L_{S})$ such that if
$\inf_{t\in[t_{0}^{-},t_{0}^{+}]}\ell_{\gamma}(g(t))\leq\bar{\epsilon}$
then
(6.10)
$d_{\gamma}(\mu(g(t_{0}^{-})),\mu(g(t_{0}^{+})))>R^{\prime}+2M_{2}+B+8.$
Let $\mu_{1}$ and $\mu_{2}$ be marking slices of $H(\mu^{-},\mu^{+})$ such
that $\operatorname{base}(\mu_{1})=\rho(i_{0}-w)$ and
$\operatorname{base}(\mu_{2})=\rho(i_{0}+w)$. Then
$d_{\gamma}(\mu_{1},\mu_{2})+\operatorname{diam}_{\gamma}(\mu_{1})+\operatorname{diam}_{\gamma}(\mu_{2})\geq
d_{\gamma}(\rho(i_{0}-w),\rho(i_{0}+w))$. Further $A(\gamma)$ is
$(w,r)$-isolated at $i_{0}$ so again by Lemma 6.5 (6.7),
$d_{\gamma}(\mu_{1},\mu_{2})\geq
d_{\gamma}(\mu(g(t_{0}^{-})),\mu(g(t_{0}^{+})))-B-4.$
Then by (6.10) we have
(6.11) $d_{\gamma}(\mu_{1},\mu_{2})>R^{\prime}+2M_{2}+4.$
Since $m^{\prime}\leq i_{0}-w\leq i_{0}+w\leq n^{\prime}$ by no backtracking
(6.1) we have that $d_{\gamma}(\rho(m^{\prime}),\rho(n^{\prime}))\geq
d_{\gamma}(\rho(i_{0}-w),\rho(i_{0}+w))-2M$. Further
$\rho(i_{0}-w)\subset\mu_{1},\rho(i_{0}+w)\subset\mu_{2}$, so
$d_{\gamma}(\rho(m^{\prime}),\rho(n^{\prime}))\geq
d_{\gamma}(\mu_{1},\mu_{2})-2M-\operatorname{diam}_{\gamma}(\mu_{1})-\operatorname{diam}_{\gamma}(\mu_{2}).$
Thus (6.11) implies that
$d_{\gamma}(\rho(m^{\prime}),\rho(n^{\prime}))>R^{\prime}$. But this
contradicts the assumption of the lemma that the $\gamma$ annular coefficient
is bounded above by $R^{\prime}$. The lower bound $\bar{\epsilon}$ for the
length of $\gamma$ at $g(t_{0})$ follows form this contradiction.
Moreover by Proposition 5.25, $\bigcup_{i\in[m^{\prime}+w,n^{\prime}-w]}N(i)$
covers $[a^{\prime},b^{\prime}]$. Thus $\ell_{\gamma}(g(t))>\bar{\epsilon}$
for any $t\in[a^{\prime},b^{\prime}]$. This is the desired lower bound (2).
Note that $\bar{\epsilon}$ depends only on $s$ and is uniform along
$[a^{\prime},b^{\prime}]$ so $\bar{\epsilon}$ is uniform over
$[a^{\prime},b^{\prime}]$ and does not depend on the value of the parameter
$t_{0}$.
We proceed to establish the rough upper bound (1). We choose a hyperbolic
surface $x\in V_{L_{S}}(\rho(i_{0}))$ as follows: Pick
$\alpha\in\rho(i_{0})-\partial{Z}$ and let $\ell_{\alpha^{\prime}}(x)=L_{S}$
for every $\alpha^{\prime}\in\rho(i_{0})-\alpha$. If
$\ell_{\alpha}(g(t_{0}))\geq\frac{3L_{S}}{4}$, let
$\ell_{\alpha}(x)=\frac{L_{S}}{2}$. Otherwise,
$\ell_{\alpha}(g(t_{0}))<\frac{3L_{S}}{4}$, let $\ell_{\alpha}(x)=L_{S}$.
Observe that
1. (a)
$\rho(i_{0})$ is a Bers marking of $x$.
Let d be the function in Corollary 3.5, let
$d_{0}=\min\\{\textbf{d}(L_{S},\frac{L_{S}}{4}),\textbf{d}(\frac{3L_{S}}{4},\frac{L_{S}}{4})\\}$.
Then we have
2. (b)
$d_{\operatorname{WP}}(g(t_{0}),x)\geq d_{0}$.
Finally, observe that by changing the twist parameters about the curves in
$\rho(i_{0})$ we can obtain $x$ such that
3. (c)
a Bers marking of $x$, which we denote by $\mu_{x}$, is equal to a marking
slice of any hierarchy $H(\mu^{-},\mu^{+})$ and
$\operatorname{base}(\mu_{x})=\rho(i_{0})$. Here $\mu^{-}$ and $\mu^{+}$ are
the end points of $\rho$. There may be several such surfaces and we choose
one.
4. (d)
It follows from (a) that $x$ has a definite positive injectivity radius
$\operatorname{inj}(x)$ only depending on $L_{S}$.
Let $[x,g(t_{0})]$ be the WP geodesic connecting $x$ to $g(t_{0})$. Let
$u:[0,d]\to\operatorname{Teich}(S)$ be the parametrization of $[x,g(t_{0})]$
by arc-length such that $u(0)=x$ and $u(d)=g(t_{0})$. Then by the $D-$fellow
traveling of $\rho$ and $Q(g)$ the length of $u$ is bounded by
$d_{1}=K_{\operatorname{WP}}D+C_{\operatorname{WP}}$. Thus $d\leq d_{1}$. On
the other hand, by (b) $d\geq d_{0}$.
To get a rough upper bound for lengths of boundary components of $Z$ at
$g(t_{0})$ first we establish a lower bound for the length of every
$\gamma\notin\partial{Z}$ along $u$. Then the rough upper bound will follow
from a compactness argument.
Let $D^{\prime}=K_{\operatorname{WP}}d_{1}+C_{\operatorname{WP}}$, then
$D^{\prime}$ bounds form above the distance between $Q(x)=Q(u(0))$ and
$Q(u(t))$ for any $t\in[0,d]$.
In the rest of the proof let $w=w(D,D^{\prime},R)$. Let
$\bar{\epsilon}=\bar{\epsilon}(D,R)$ be the lower bound we established earlier
for $\ell_{\gamma}(g(t))$ for each $\gamma\notin\partial{Z}$ and every
$t\in[a^{\prime}+s,b^{\prime}-s]$. Here we make the following two choices:
Figure 7. The domain $Z$ has $(R,R^{\prime})-$bounded combinatorics over the
red interval. $y$ is the point along $[x,g(t)]$ where the infimum of the
length of $\gamma$ is minimum. So there is $Q_{y}$ a Bers pants decomposition
of $y$ such that $\gamma\in Q_{y}$. Thus $A(\gamma)$ is
$(w,D^{\prime})-$isolated at $i_{0}$.
1. (i)
Let $I^{-}=[t_{0}^{-},t_{0}]$ and $I^{+}=[t_{0},t_{0}^{+}]$. By (6.9),
$\ell_{\gamma}(g(t_{0}^{-}))\geq\omega(L_{S})$. Thus $\sup_{t\in
I^{-}}\ell_{\gamma}(g(t))\geq\omega(L_{S})$. Moreover, $|I^{-}|=s$. Then
Corollary 4.12 applied to $g|_{I^{-}}$ implies that there is $N_{0}$ depending
only on $s$ and $\omega(L_{S})$ such that if
$d_{\gamma}(\mu(g(t_{0}^{-})),\mu(g(t_{0})))>N_{0}$
then
$\inf_{t\in I^{-}}\ell_{\gamma}(g(t))\leq\bar{\epsilon}.$
Similarly, $\sup_{t\in I^{+}}\ell_{\gamma}(g(t))\geq\omega(L_{S})$ and
$|I^{+}|=s$. So the corollary applied to $g|_{I^{+}}$ implies that if
$d_{\gamma}(\mu(g(t_{0}^{+})),\mu(g(t_{0})))>N_{0}$ then $\inf_{t\in
I^{+}}\ell_{\gamma}(g(t))\leq\bar{\epsilon}$
2. (ii)
Fix $\gamma\notin\partial{Z}$. By the lower bound (2),
$\ell_{\gamma}(g(t_{0}))\geq\bar{\epsilon}$ and by (d) $\ell_{\gamma}(x)\geq
2\operatorname{inj}(x)$. The length of $u$ is at least $d_{0}$. Let
$e=\min\\{\operatorname{inj}(x),\frac{\bar{\epsilon}}{2}\\}$. Then Corollary
3.5 implies that there is $s_{0}$ with $2s_{0}<d_{0}<d$ with the property that
if $r\in[d-s_{0},d]$ then $\ell_{\gamma}(u(r))\geq e$, and if $r\in[0,s_{0}]$
then $\ell_{\gamma}(u(r))\geq e$.
By what we just said $\sup_{r\in[0,d]}\ell_{\gamma}(u(r))\geq e$. Then
Corollary 4.13 applied to $u$ implies that there is $\epsilon_{2}<e$ such that
if
(6.12) $\inf_{r\in[s_{0},d-s_{0}]}\ell_{\gamma}(u(r))\leq\epsilon_{2}$
then
(6.13) $d_{\gamma}(\mu_{x},\mu(g(t_{0})))>N_{0}+R^{\prime}+2M_{2}+B$
In what follows we prove the lower bound
$\inf_{t\in[0,d]}\ell_{\gamma}(u(t))>\epsilon_{2}$
for every $\gamma\notin\partial{Z}$, where $\epsilon_{2}$ is the constant we
chose in (ii). Indeed, we rule out the possibility that
$\inf_{r\in[0,d]}\ell_{\gamma}(u(t))\leq\epsilon_{2}$ for any
$\gamma\notin\partial{Z}$.
The proof is by contradiction. First note that $\mu_{x}$ is a Bers marking of
$u(0)=x$ and $\mu(g(t_{0}))$ is a Bers marking of $u(d)=g(t_{0})$. Moreover,
by (6.12) there is a point $y\in u([s_{0},d-s_{0}])$ such that $\gamma\in
Q(y)$.
Since the length of $u$ is less than $d_{1}$ we have $d(Q(y),\rho(i_{0}))\leq
D^{\prime}$. Recall that
$D^{\prime}=K_{\operatorname{WP}}d_{1}+C_{\operatorname{WP}}$.
Let $\mu_{1}$ and $\mu_{2}$, as before, be marking slices of
$H(\mu^{-},\mu^{+})$ with $\operatorname{base}(\mu_{1})=\rho(i_{0}-w)$ and
$\operatorname{base}(\mu_{2})=\rho(i_{0}+w)$. Then as we saw in the proof of
Lemma 6.5 there is $B:=B(D)$ such that
(6.14) $d_{\gamma}(\mu_{1},\mu(g(t_{0}^{-}))\leq B.$
By (c) $\operatorname{base}(\mu_{x})=\rho(i_{0})$. Thus (6.1) and the
assumption of the lemma that the annular coefficients are bonded over
$[m^{\prime},n^{\prime}]$ imply that
(6.15) $d_{\gamma}(\mu_{x},\mu_{1})\leq R^{\prime}+2M_{2}$
(6.13), (6.14) and (6.15) combined by the triangle inequality give us
$d_{\gamma}(\mu(g(t_{0})),\mu(g(t_{0}^{-})))>N_{0}$
So the choice of $N_{0}$ in (i) implies that
$\inf_{t\in[t_{0}^{-},t_{0}]}\ell_{\gamma}(g(t))\leq\bar{\epsilon}$. Moreover,
$A(\gamma)$ is $(w,D^{\prime})-$isolated at $i_{0}$ where the isolating domain
on both sides is $Z$. Then by Lemma 6.5 (6.7),
$d_{\gamma}(\mu_{2},\mu_{1})\geq
d_{\gamma}(\mu(g(t_{0}^{-})),\mu(g(t_{0}^{+})))-B-4$. So by (6.10) we have
$d_{\gamma}(\mu_{2},\mu_{1})>R^{\prime}+2M_{2}+4$
Recall that $\rho(i_{0}-w)\subset\mu_{1}$ and $\rho(i_{0}+w)\subset\mu_{2}$.
Then since $m^{\prime}<i_{0}-w<i_{0}+w<n^{\prime}$ no backtracking (6.1) and
the above bound imply that
$d_{\gamma}(\rho(m^{\prime}),\rho(n^{\prime}))>R^{\prime}$. This contradicts
the assumption of the lemma that the $\gamma$ annular coefficient over
$[m^{\prime},n^{\prime}]$ is bounded above by $R^{\prime}$.
Figure 8. $U_{L_{S}}(\rho(i_{0}))$ is the Bers region of $\rho(i_{0})$ in
$\operatorname{Teich}(S)$. $g$ intersects this region. By (b) there is $x$ in
$U_{L_{S}}(\rho(i_{0}))$ so that $d_{\operatorname{WP}}(x,g(t_{0}))>d_{0}$.
Thus the length of the WP geodesic segment connecting $x$ to $g(t_{0})$ is at
least $d_{0}$. So we may choose $s_{0}$ with $2s_{0}<d_{0}$ in (ii)
independent of $t_{0}$ (uniformly along $g$). Then by the lower bounds for the
injectivity radius of $g(t_{0})$ (part (2) of the theorem) and $x$ (d)
Corollary 4.13 applies to the geodesic segment $u$ and gives a uniform upper
bound for the length of the curves in $\partial{Z}$ at $g(t_{0})$.
Note that $\epsilon_{2}$ does not depend on the value of the parameters
$i_{0}\in[m^{\prime}+w,n^{\prime}-w]$ and $t_{0}\in N(i_{0})$, therefore it is
uniform along $[m^{\prime}+w,n^{\prime}-w]$.
We will finish establishing of the upper bound (2) by a compactness argument.
Let $u_{n}:[0,d_{n}]\to\operatorname{Teich}(S)$ be a sequence of WP geodesic
segments parametrized by arc-length with $d_{0}\leq d\leq d_{1}$. Let
subsurfaces $Z_{n}$ be such that
$\inf_{t\in[0,d_{n}]}\ell_{\gamma}(u_{n}(t))\geq\epsilon_{2}$ for every
$\gamma\notin\partial{Z}_{n}$. Moreover, let $t^{*}_{n}\in[0,d_{n}]$ and
$\hat{\alpha}_{n}\in\partial{Z}_{n}$ be such that
$\ell_{\hat{\alpha}_{n}}(u_{n}(t^{*}_{n}))\to\infty$ as $n\to\infty$. After
possibly passing to a subsequence and remarking (applying elements of the
mapping class group) we may assume that there is a subsurface $Z$ such that
$\inf_{t\in[0,d_{n}]}\ell_{\gamma}(u_{n}(t))\geq\epsilon_{2}$ for every
$\gamma\notin\partial{Z}$, $\hat{\alpha}\in\partial{Z}$ and a sequence
$t^{*}_{n}$ such that $\ell_{\hat{\alpha}}(u_{n}(t^{*}_{n}))\to\infty$ as
$n\to\infty$. We proceed to get a contradiction. Let
$\hat{u}:[0,d]\to\overline{\operatorname{Teich}(S)}$ be the geodesic limit of
$u_{n}$’s as in Theorem 4.5 (Geodesic Limit Theorem). Let the partition
$0=t_{0}<t_{1}<...<t_{k+1}=T$, simplices $\sigma_{0},...,\sigma_{k+1}$,
simplex $\hat{\tau}$ and the elements of mapping class group
$\varphi_{i,n}=\mathcal{T}_{i,n}\circ...\circ\mathcal{T}_{i,n}\circ\psi_{n}$
be as in the theorem. We claim that
* •
$\ell_{\alpha}(\psi_{n}(x))=\ell_{\alpha}(x)$ for every
$\alpha\in\partial{Z}$,
* •
$\sigma_{i}\subseteq\partial{Z}$ for $i=1,...,k+1$.
In the proof of Geodesic Limit Theorem $\psi_{n}$ is applied to keep
$u_{n}(0)$ in a compact subset of Teichmüller space. Now since
$\ell_{\alpha}(u_{n}(0))=L_{S}$ for every $\alpha\in\partial{Z}$, we may
choose each $\psi_{n}$ so that is supported on $S\backslash\partial{Z}$ i.e.
is identity in a regular neighborhood of $\partial{Z}$ and $\psi_{n}\circ
u_{n}(0)$ be in a compact subset of Teichmüller space where injectivity radius
of surfaces is at least $L_{S}$. We have the first bullet. We proceed to prove
the second bullet inductively. In the Geodesic Limit Theorem for each
$i=1,...,k+1$, $\sigma_{i}$ is the multi-curve determining the stratum that
the limit of $\varphi_{i-1,n}\circ u_{n}|_{[t_{i-1},d]}$ after possibly
passing to a subsequence intersects. By the lower bound on the length of each
$\gamma\notin\partial{Z}$ along $u_{n}([0,d])$ and since $\psi_{n}$ is
supported on $S\backslash\partial{Z}$, the limit of $\psi_{n}\circ
u_{n}|_{0,d]}$ intersects the stratum of a multi-curve
$\sigma_{1}\subseteq\partial{Z}$. Now let $i>1$. Suppose that
$\sigma_{j}\subseteq\partial{Z}$ for every $0\leq j\leq i$. Then since
$\mathcal{T}_{j,n}\in\operatorname{tw}(\sigma_{j}-\hat{\tau})$ for each
$j=1,...,i$, their composition does not change the isotopy class of any curve
in $\partial{Z}$, so $\varphi_{i,n}$ does not change the isotopy class of
every curve in $\partial{Z}$. Furthermore, the length of each
$\gamma\notin\partial{Z}$ along $u_{n}([t_{i},d])$ is bounded below. So we may
conclude that the limit of $\varphi_{i,n}\circ u_{n}|_{[t_{i},d]}$ as
$n\to\infty$ intersects the stratum of a multi-curve
$\sigma_{i+1}\subseteq\partial{Z}$.
The boundary curves of $Z$ are the only curves with possibly length $0$ along
$\hat{u}$ and $\hat{\alpha}$ does not intersect any of them. So the length of
$\hat{\alpha}$ is bounded along $\hat{u}$ by some $l_{0}>0$. By the second
bullet above, each $\varphi_{i,n}$ is the composition of $\psi_{n}$ and powers
of Dehn twists about curves in $\partial{Z}$. So the isotopy class of each
curve in $\partial{Z}$ is preserved by $\varphi_{i,n}$. After possibly passing
to a subsequence we may assume that $t^{*}_{n}\in[t_{i},t_{i+1}]$ for all $n$.
Then by the third part of the Geodesic Limit Theorem, $\varphi_{i,n}\circ
u_{n}(t^{*}_{n})\to\hat{u}(t^{*})$ as $n\to\infty$, so
$\ell_{\hat{\alpha}}(u_{n}(t^{*}_{n}))=\ell_{\hat{\alpha}}(\varphi_{i,n}\circ
u_{n}(t^{*}_{n}))$. So for $n$ sufficiently large
$\ell_{\hat{\alpha}}(u_{n}(t^{*}_{n}))\leq 2l_{0}$. This contradicts the
assumption that $\ell_{\hat{\alpha}}(u_{n}(t^{*}_{n}))\to\infty$ as
$n\to\infty$.
∎
###### Proposition 6.7.
Given $A,R,R^{\prime}>0$, there are constants $w=w(A,R)$ and
$\bar{\epsilon}=\bar{\epsilon}(A,R,R^{\prime})$ with the following properties.
Let $g:[a,b]\to\overline{\operatorname{Teich}(S)}$ be a WP geodesic segment
parametrized by arc-length with $A-$narrow end invariant $(\nu^{-},\nu^{+})$.
Let $\rho:[m,n]\to P(S)$ be a hierarchy path between $\nu^{-}$ and $\nu^{+}$.
Suppose that $S$ has $(R,R^{\prime})-$bounded combinatorics over
$[m^{\prime},n^{\prime}]\subseteq[m,n]$ and $m^{\prime}-n^{\prime}>2w$, then
$\operatorname{inj}(g(t))\geq\frac{\bar{\epsilon}}{2}$
for every $t\in[a^{\prime},b^{\prime}]$, where $a^{\prime}\in N(m^{\prime}+w)$
and $b^{\prime}\in N(n^{\prime}-w)$.
###### Proof.
Let $D=D(A)$ be the fellow traveling distance of $g$ and the hierarchy path
between $(\nu^{-},\nu^{+})$. By the assumption that $S$ is the subsurface with
$(R,R^{\prime})-$bounded combinatorics Lemma 6.6 implies that for every
$\gamma\in\mathcal{C}(S)$ we have $\ell_{\gamma}(g(t))\geq\bar{\epsilon}$ for
every $t\in[a^{\prime},b^{\prime}]$. So
$\operatorname{inj}(g(t))\geq\frac{\bar{\epsilon}}{2}$ on this interval. ∎
###### Remark 6.8.
Compare the above corollary with the main result of [BMM11] which asserts that
given $R>0$ there is an $\epsilon>0$ such that if the end invariant of a WP
geodesic $g$ is $(R,R)-$bounded combinatorics then $g$ stays in the
$\epsilon-$think part of Teichmüller space.
###### Proof of Theorem 6.1.
To get arbitrary short boundary curves we sharpen the rough upper bound
obtained in Lemma 6.6. This would be done in the following lemma. Here we use
convexity of length-functions along WP geodesics to get an arbitrary short
curve in the boundary of $Z$ over an arbitrary long interval. Then using the
fact that over this interval the geodesic is close to an stratum an inductive
argument gives us the upper bound for the length of all of the curves in the
boundary of $Z$.
###### Lemma 6.9.
Given $l,\bar{\epsilon}$ and $\epsilon\leq\bar{\epsilon}$, there is
$\bar{s}>0$ with the following property. Let
$g:[a^{\prime},b^{\prime}]\to\overline{\operatorname{Teich}(S)}$ be a WP
geodesic such that the length-function bounds
1. (1)
$\ell_{\alpha}(g(t))\leq l$ for every $\alpha\in\partial{Z}$, and
2. (2)
$\ell_{\gamma}(g(t))\geq\bar{\epsilon}$ for every $\gamma\notin\partial{Z}$
hold for every $t\in[a^{\prime},b^{\prime}]$. Furthermore, assume that
$b^{\prime}-a^{\prime}>2\bar{s}$. Then for every $\alpha\in\partial{Z}$ we
have
(6.16) $\ell_{\alpha}(g(t))\leq\epsilon$
for every $t\in[a^{\prime}+\bar{s},b^{\prime}-\bar{s}]$.
The theorem follows from this lemma. Let $D=D(A)$ be the fellow traveling
distance from Theorem 5.24. Then $Q(g|_{[a^{\prime},b^{\prime}]})$ and
$\rho|_{[m^{\prime},n^{\prime}]}$, $D-$fellow travel. Furthermore, $Z$ has
$(R,R^{\prime})-$bounded combinatorics over $[m^{\prime},n^{\prime}]$. Then by
Lemma 6.6 there are $l=l(D,R)$, $\bar{\epsilon}=\bar{\epsilon}(D,R^{\prime})$
and $w=w(D,R)$ with the property that if $m^{\prime}-n^{\prime}\geq 2w$ we
have
1. $(1^{\prime})$
$\ell_{\alpha}(g(t))\leq l$ for every $\alpha\in\partial{Z}$,
2. $(2^{\prime})$
$\ell_{\gamma}(g(t))\geq\bar{\epsilon}$ for every $\gamma\notin\partial{Z}$.
for every $t\in[a^{\prime},b^{\prime}]$, where $a^{\prime}\in N(m^{\prime}+w)$
and $b^{\prime}\in N(n^{\prime}-w)$. Now let $\bar{w}=K\bar{s}+C$. Then
$m^{\prime}-n^{\prime}>2\bar{w}$ guarantees that
$b^{\prime}-a^{\prime}>2\bar{s}$. Then by the above lemma if
$b^{\prime}-a^{\prime}>2\bar{s}$ we have the asserted length-function bound of
the theorem. ∎
###### Proof of Lemma 6.9.
The proof is by induction on $|\partial{Z}|$ the number of boundary components
of $Z$. When $\partial{Z}=\emptyset$, the lemma holds vacuously and provides
us with the base of induction.
###### Claim 6.10.
Given $\epsilon<\bar{\epsilon}$, there is $T>0$ such that if
$[c,d]\subseteq[a^{\prime},b^{\prime}]$ is a subinterval with $d-c>2T$ then
there is a curve $\alpha\in\partial{Z}$ such that
$\ell_{\alpha}(g(t))\leq\epsilon$
for some $t\in[c+T,d-T]$.
We show that if $g|_{[c,d]}$ stays in the $\epsilon-$thick part of Teichmüller
space then there is an upper bound $T=T(l,\epsilon)$ for the length of the
interval $[c,d]$. Fix $\hat{\alpha}\in\partial{Z}$. $g|_{[c,d]}$ is in the
$\epsilon-$thick part, thus by Theorem 3.8 (3.1) there is $c(\epsilon)>0$ for
which the differential inequality
$\ddot{\ell}_{\hat{\alpha}}(g(t))\geq\epsilon c(\epsilon)$
holds on the interval $[c,d]$. By $(1^{\prime})$ above
$\ell_{\hat{\alpha}}(g(t))\leq l$ for every $t\in[c,d]$. Thus by the Mean-
value Theorem if $d-c>\frac{2l}{\epsilon\sqrt{2c(\epsilon)}}$ then there is
$t^{*}\in[c,d]$ such that
$|\dot{\ell}_{\hat{\alpha}}(g(t^{*}))|<\epsilon\sqrt{2c(\epsilon)}$.
Integrating the above differential inequality we get
(6.17)
$\ell_{\hat{\alpha}}(g(t))\geq\ell_{\hat{\alpha}}(g(t^{*}))+\dot{\ell}_{\hat{\alpha}}(g(t^{*}))(t-t^{*})+\frac{1}{2}\epsilon
c(\epsilon)(t-t^{*})^{2}$
$\ell_{\hat{\alpha}}(g(t^{*}))>\epsilon$ and
$|\dot{\ell}_{\hat{\alpha}}(g(t^{*}))|<\epsilon\sqrt{2c(\epsilon)}$. So for
$\Delta$ the discriminant of the quadratic function on the right hand side of
(6.17) we have
$\Delta=(\dot{\ell}_{\hat{\alpha}}(g(t^{*})))^{2}-4(\frac{\epsilon
c(\epsilon)}{2})\ell_{\hat{\alpha}}(g(t^{*}))\leq(\epsilon\sqrt{2c(\epsilon)})^{2}-4(\frac{1}{2}\epsilon^{2}c(\epsilon))=0.$
This guarantee that the quadratic function is positive on $\mathbb{R}$.
As before $\ell_{\hat{\alpha}}(g(t))<l$ for every $t\in[c,d]$. Then by the
completing square we get $l\geq(\sqrt{\frac{\epsilon
c(\epsilon)}{2}}(t-t^{*})-\frac{\dot{\ell}_{\hat{\alpha}}(g(t^{*})}{\sqrt{2\epsilon
c(\epsilon)}})^{2}-\frac{\Delta}{2\epsilon c(\epsilon)}$. So using
$|\dot{\ell}_{\hat{\alpha}}(g(t^{*}))|<\epsilon\sqrt{2c(\epsilon)}$ and
$\Delta\leq 0$ we get
$|t-t^{*}|\leq\frac{\sqrt{2}}{\sqrt{c(\epsilon)}}+\frac{\sqrt{2l}}{\sqrt{\epsilon
c(\epsilon)}}$. Consequently
$b^{\prime}-a^{\prime}\leq\max\\{\frac{2l}{\epsilon\sqrt{2c(\epsilon)}},\frac{\sqrt{2}}{\sqrt{c(\epsilon)}}+\frac{\sqrt{2l}}{\sqrt{\epsilon
c(\epsilon)}}\\}$. So the claim holds for
$T:=T(l,\epsilon)=\max\\{\frac{2l}{\epsilon\sqrt{2c(\epsilon)}},\frac{\sqrt{2}}{\sqrt{c(\epsilon)}}+\frac{\sqrt{2l}}{\sqrt{\epsilon
c(\epsilon)}}\\}$.
###### Remark 6.11.
For $\epsilon$ sufficiently small,
$T(l,\epsilon)=\frac{2l}{\epsilon\sqrt{2c(\epsilon)}}$.
The contrapositive of what we just proved is that if $d-c>T$, then there are
curves which get shorter than $\epsilon$ at some time along $g|_{[c,d]}$.
Moreover, $[c,d]\subseteq[a,b]$ so by the bound (2) component curves of
$\partial{Z}$ are the only curves which can get shorter than
$\epsilon<\bar{\epsilon}$ along $g|_{[c,d]}$. Thus we conclude that if
$d-c>T$, then there is a time $t\in[c,d]$ and a curve $\alpha\in\partial{Z}$
such that $\ell_{\alpha}(g(t))\leq\epsilon$ as was desired.
###### Claim 6.12.
Given $L>0$, if $b^{\prime}-a^{\prime}>(2|\partial{Z}|+1)L+2T$ then there is a
curve $\alpha\in\partial{Z}$ such that $\ell_{\alpha}(g(t))\leq\epsilon$ on a
subinterval of $[a^{\prime},b^{\prime}]$ of length at least $L$.
Figure 9. $\alpha\in\partial{Z}$ is shorter than $\epsilon^{\prime}$ along
$g|_{[t_{\alpha}^{-},t_{\alpha}^{+}]}$,
$g^{\prime}:[a^{\prime\prime},b^{\prime\prime}]\to\mathcal{S}(\alpha)$
connects $x$ and $y$, the projections of respectively $g(t_{\alpha}^{-})$ and
$g(t_{\alpha}^{-})$ on $\mathcal{S}(\alpha)$. The length-function bounds
$(1^{\prime\prime})$ and $(2^{\prime\prime})$ hold for the subsurface
$Z^{\prime}=Z\cup A(\alpha)$ along $g^{\prime}$. So by the assumption of
induction the length of every $\alpha^{\prime}\in\partial{Z^{\prime}}$ is
shorter than $\frac{\epsilon}{2}$ along $g^{\prime}$ over a suitably shrunk
subinterval of $[a^{\prime\prime},b^{\prime\prime}]$.
Let the intervals $I_{1},...,I_{2|\partial{Z}|+1}$ with $|I_{i}|\geq L$ for
$i=1,...,2|\partial{Z}|+1$ consist a partition of the interval
$[a^{\prime}+T,b^{\prime}-T]$ into $2|\partial{Z}|+1$ subintervals. Claim 6.10
applied to each interval $[\min I_{i}-T,\max I_{i}+T]$ implies that there is a
time $t_{i}\in I_{i}$ at which a component curve of $\partial{Z}$ is shorter
than $\epsilon$. Now the pigeon-hole principle implies that there is a curve
$\alpha\in\partial{Z}$ and indices $i_{1},i_{2}$ and $i_{3}$ with
$i_{1}<i_{2}<i_{3}$ such that
$\ell_{\alpha}(g(t_{i_{1}})),\ell_{\alpha}(g(t_{i_{2}}))$ and
$\ell_{\alpha}(g(t_{i_{3}}))$ are less than $\epsilon$. Then by the convexity
of the $\alpha-$length-function along $g$, $\ell_{\alpha}(g(t))\leq\epsilon$
on $[t_{i_{1}},t_{i_{3}}]$. Moreover $t_{i_{3}}-t_{i_{1}}\geq|I_{i_{2}}|\geq
L$. So $[t_{i_{1}},t_{i_{3}}]$ is the claimed subinterval.
Let $\epsilon^{\prime}<\min\\{\bar{\epsilon},\epsilon\\}$, which will be
determined. Let $L=\bar{s}^{\prime}+2\sqrt{2\pi\epsilon^{\prime}}$, where
$\bar{s}^{\prime}$ will be determined too.
Now by Claim 6.12 if $b^{\prime}-a^{\prime}>(2|\partial{Z}|+1)L+2T$, then
there is a curve $\alpha\in\partial{Z}$ such that
(6.18) $\ell_{\alpha}(g(t))\leq\epsilon^{\prime}$
on an interval of length at least $L$. Denote this interval by
$[t_{\alpha}^{-},t_{\alpha}^{+}]$. Let $x$ and $y$ be the nearest points to
$g(t_{\alpha}^{-})$ and $g(t_{\alpha}^{+})$ on the $\alpha-$stratum,
respectively. Since the $\alpha-$stratum is convex, there is a WP geodesic
segment $g^{\prime}:[a^{\prime\prime},b^{\prime\prime}]\to\mathcal{S}(\alpha)$
parametrized by arc-length connecting $x$ to $y$ (see Figure 9).
Since $\ell_{\alpha}(g(t_{\alpha}^{-}))\leq\epsilon^{\prime}$ and
$\ell_{\alpha}(g(t_{\alpha}^{+}))\leq\epsilon^{\prime}$, Proposition 3.6 gives
us the upper bounds
$d_{\operatorname{WP}}(g(t_{\alpha}^{-}),x)\leq\sqrt{2\pi\epsilon^{\prime}}$
and
$d_{\operatorname{WP}}(g(t_{\alpha}^{+}),y)\leq\sqrt{2\pi\epsilon^{\prime}}$.
Moreover, $\overline{\operatorname{Teich}(S)}$ equipped with the WP metric is
a $\operatorname{CAT}(0)$ space. Therefore by the $\operatorname{CAT}(0)$
comparison the distance between any point on
$g([t_{\alpha}^{-},t_{\alpha}^{+}])$ and its nearest point on $g^{\prime}$ is
less than $\sqrt{2\pi\epsilon^{\prime}}$. Here we choose $\epsilon^{\prime}$
such that:
* •
$\sqrt{2\pi\epsilon^{\prime}}\leq\min\\{\textbf{d}(\bar{\epsilon},\frac{\bar{\epsilon}}{2}),\textbf{d}(2l,l),\textbf{d}(\epsilon,\frac{\epsilon}{2})\\}.$
Here d is the function from Corollary 3.5. By the choice of
$\epsilon^{\prime}$ we have the following length-function bounds
1. $(1^{\prime\prime})$
$\ell_{\alpha}(g^{\prime}(t))\leq 2l$ for every
$\alpha^{\prime}\in\partial{Z^{\prime}}$
2. $(2^{\prime\prime})$
$\ell_{\gamma}(g^{\prime}(t))\geq\frac{\bar{\epsilon}}{2}$ for every
$\gamma\not\in\partial{Z^{\prime}}$
for every $t\in[a^{\prime\prime},b^{\prime\prime}]$. Here $Z^{\prime}=Z\cup
A(\alpha)$.
Now since $Z^{\prime}$ is a large subsurface, by the above two length-function
bounds the assumption of the induction for the geodesic
$g^{\prime}:[a^{\prime\prime},b^{\prime\prime}]\to\overline{\operatorname{Teich}(S)}$
implies that there is $\bar{s}^{\prime}$ such that if
$b^{\prime\prime}-a^{\prime\prime}>2\bar{s}^{\prime}$ then for every
$\alpha^{\prime}\in\partial{Z^{\prime}}=\partial{Z}-\\{\alpha\\}$
(6.19) $\ell_{\alpha^{\prime}}(g^{\prime}(t))\leq\frac{\epsilon}{2}$
for every
$t\in[a^{\prime\prime}+\bar{s}^{\prime},b^{\prime\prime}-\bar{s}^{\prime}]$.
Let $\bar{s}=\bar{s}^{\prime}+2\sqrt{2\pi\epsilon}$. Given
$t\in[t_{\alpha}^{-}+\bar{s},t_{\alpha}^{+}-\bar{s}]$, let
$t^{\prime}\in[a^{\prime\prime},b^{\prime\prime}]$ be the nearest point to
$g(t)$ on $g^{\prime}|_{[a^{\prime\prime},b^{\prime\prime}]}$. Since
$d_{\operatorname{WP}}(g(t_{\alpha}^{-}),g^{\prime}(a^{\prime\prime}))\leq\sqrt{2\pi\epsilon}$
and
$d_{\operatorname{WP}}(g(t_{\alpha}^{+}),g^{\prime}(b^{\prime\prime}))\leq\sqrt{2\pi\epsilon}$
the $\operatorname{CAT}(0)$ comparison implies that
$d_{\operatorname{WP}}(g(t),g^{\prime}(t^{\prime}))\leq\sqrt{2\pi\epsilon}$.
Furthermore by the triangle inequality
$d_{\operatorname{WP}}(g^{\prime}(t^{\prime}),g^{\prime}(a^{\prime\prime}))\geq
d_{\operatorname{WP}}(g(t),g(t_{\alpha}^{-}))-d_{\operatorname{WP}}(g(t),g^{\prime}(t^{\prime}))-d_{\operatorname{WP}}(g(t_{\alpha}^{-}),g^{\prime}(a^{\prime\prime}))$,
so
$t^{\prime}-a^{\prime\prime}\geq(t-t_{\alpha}^{-})-2\sqrt{2\pi\epsilon}\geq\bar{s}^{\prime}$
and similarly
$b^{\prime\prime}-t^{\prime}\geq(t_{\alpha}^{+}-t)-2\sqrt{2\pi\epsilon}\geq\bar{s}^{\prime}$.
Thus
$t^{\prime}\in[a^{\prime\prime}+\bar{s}^{\prime},b^{\prime\prime}-\bar{s}^{\prime}]$.
Then by the bound (6.19) and the choice of $\epsilon^{\prime}$, Corollary 3.5
implies that for every curve $\alpha^{\prime}\in\partial{Z^{\prime}}$,
$\ell_{\alpha^{\prime}}(g(t))\leq\epsilon$ for every
$t\in[a^{\prime}+\bar{s},b^{\prime}-\bar{s}]$. Moreover
$[a^{\prime}+\bar{s},b^{\prime}-\bar{s}]\subset[t_{\alpha}^{-},t_{\alpha}^{+}]$,
so by (6.18), $\ell_{\alpha}(g(t))\leq\epsilon$ on this interval. We
established the bound for the length of all of the component curves of
$\partial{Z}$ on the interval $[a^{\prime}+\bar{s},b^{\prime}-\bar{s}]$. This
finishes the step of induction.
∎
## 7\. Laminations with prescribed subsurface coefficients
Our purpose in this section is to construct pair of partial markings or
laminations with a given list of subsurface coefficients. More precisely,
given a sequence of integers $\\{e_{i}\\}_{i}$ we will construct a pair of
laminations/markings $(\mu_{I},\mu_{T})$ such that there is a list of large
subsurfaces $\\{Z_{i}\\}_{i}$ with
$d_{Z_{i}}(\mu_{I},\mu_{T})\asymp_{K_{1},C_{1}}|e_{i}|$ where $K_{1},C_{1}$
depend on certain initial choices. Furthermore, there are constants m and
$\textbf{m}^{\prime}$ depending on the initial choices such the subsurface
coefficient of any subsurface which is not in the list of $Z_{i}$’s is bounded
above by m, and all annular subsurface coefficients are bounded above by
$\textbf{m}^{\prime}$. This is a kind of symbolic coding for laminations using
subsurface coefficients. Here we restrict the set of subsurface with a big
subsurface coefficient. This can be thought of as continued fraction
expansions with a specific pattern of decimal numbers. We will use these
constructions in $\S$8 to provide examples of WP geodesics with certain
behavior in the moduli space.
The construction uses compositions of powers of (partial) pseudo-Anosov maps.
A partial pseudo-Anosov map $f$ is a reducible element of mapping class group
which preserves the isotopy class of a collection of curves
$\\{\delta_{j}\\}_{j}$ on the surface $S$ and does not rearrange connected
components of $S\backslash\\{\delta_{j}\\}_{j}$. Moreover, the restriction of
$f$ to each of the connected components of $S\backslash\\{\delta_{j}\\}_{j}$
is a pseudo-Anosov maps. We say that this partial pseudo-Anosov map is
supported on $S\backslash\sigma$.
We start by some background about the action of (partial) pseudo-Anosov maps
on the curve complex of a surface and its subsurfaces also the space of
projective measured laminations.
The following proposition is a straightforward consequence of Proposition 4.6
in [MM99].
###### Proposition 7.1.
Let $f$ be a (partial) pseudo-Anosov map supported on a subsurface $Y\subseteq
S$. There is $\tau_{f}>0$ such that for every $\alpha\pitchfork Y$ and every
integer $e$ we have
$d_{Y}(\alpha,f^{e}\alpha)\geq\tau_{f}|e|.$
###### Lemma 7.2.
Let $f$ be a (partial) pseudo-Anosov map supported on a subsurface $X$. There
is a constant $\bar{\tau}_{f}>0$ such that for every
$\alpha\in\mathcal{C}(X)$,
$\limsup_{n\to\infty}\frac{d_{X}(\alpha,f^{n}\alpha)}{n}=\bar{\tau}_{f}$.
###### Proof.
Using the fact that $f$ is an isometry of $\mathcal{C}(X)$ and the triangle
inequality we have that for any positive integer $n$,
$d_{X}(\alpha,f^{n}\alpha)\leq\sum_{i=0}^{n-1}d_{X}(f^{i}\alpha,f^{i+1}\alpha)\leq
nd_{X}(\alpha,f\alpha)$. So $\frac{d_{X}(\alpha,f^{n}\alpha)}{n}\leq
d_{X}(\alpha,f\alpha)$. Thus
$\limsup_{n\to\infty}\frac{d_{X}(\alpha,f^{n}\alpha)}{n}$ is a finite number.
Furthermore, for any $\beta\in\mathcal{C}(X)$ with $\alpha\neq\beta$, by the
triangle inequality we have $d_{X}(f^{n}\beta,\alpha)-d_{X}(\alpha,\beta)\leq
d_{X}(\beta,f^{n}\beta)\leq
d_{X}(f^{n}\beta,\alpha)+d_{X}(\alpha,f^{n}\beta)$. Now diving both sides of
each of the two inequalities by $n$ and taking $\limsup$ we see that
$\bar{\tau}_{f}$ is the same for $\alpha$ and $\beta$ and consequently
independent of the choice of $\alpha$. Finally, by Proposition 7.1,
$\bar{\tau}_{f}\geq\tau_{f}>0$. ∎
We continue by reviewing some facts about the action of (partial) pseudo-
Anosov maps on $\mathcal{PML}(S)$. We essentially follow exposé 11 of [FLP79]
and $\S 3$ and appendix A of [Iva92]. Here we replace measured geodesic
laminations with measured geodesic foliations used in these two references.
The correspondence of measured foliations and measured geodesic lamination is
explained in [CB88]. Given a reducible element of the mapping class group by
Theorem 11.7 of [FLP79] there is a finite collection of simple closed curves
$\\{\delta_{j}\\}_{j=1}^{m}$ and subsurfaces $\\{X_{i}\\}_{i=1}^{n}$ such that
$S\backslash\\{\delta_{j}\\}_{j}=\sqcup_{i}X_{i}$ and the restriction of $f$
to each $X_{i}$ is either pseudo-Anosov or periodic. For a partial pseudo-
Anosov map we further suppose that the map does not rearrange connected
components of $S\backslash\\{\delta_{j}\\}_{j}$ and its restriction to each
connected component is a pseudo-Anosov map. Then as in exposé 11 of [FLP79]
for each $i$, there are measured laminations
$\mathcal{L}_{i}^{\pm}=(\lambda_{i}^{\pm},m_{i}^{\pm})$ the attracting and
repelling measured laminations of $f|_{X_{i}}$ and real numbers $s_{i}>1$ such
that for each $i$
* •
$f(\lambda_{i}^{+})=\lambda_{i}^{+}$ and $fm_{i}^{+}\geq s_{i}m_{i}^{+}$,
* •
$f(\lambda_{i}^{-})=\lambda_{i}^{-}$ and $fm_{i}^{-}\leq s_{i}^{-1}m_{i}^{-}$.
Moreover, both $\lambda_{i}^{\pm}$ are uniquely ergodic laminations on
$X_{i}$. In particular the support of $\lambda_{i}^{\pm}$ is minimal filling
on $X_{i}$.
Let $i:\mathcal{ML}(S)\times\mathcal{ML}(S)\to\mathbb{R}$ be the intersection
number defined for any pair of measured laminations (see $\S 2.7$ of [Iva92]).
Given a complete hyperbolic metric on $S$, let
$\ell:\mathcal{ML}(S)\to\mathbb{R}^{\geq 0}$ be the length-function (for the
definition see [Bon01]). Note that both $i$ and $\ell$ are homogeneous of
degree one in each of their variables. For example,
$i(s\mathcal{L},\mathcal{L}^{\prime})=si(\mathcal{L},\mathcal{L}^{\prime})$.
As in Appendix $A$ of [Iva92] (see also $\S 3$ of the book) let
$\Delta_{f}^{+}$ be the set of projective classes of measured geodesic
laminations $\\{\sum_{i=1}^{n}t_{i}\mathcal{L}^{+}_{i}:t_{i}\geq
0,\sum_{i}t_{i}>0\\}$. Also let $\Psi_{f}^{+}$ be the set of projective
classes of measured geodesic laminations $\\{\mathcal{L}\neq
0:i(\mathcal{L},\mathcal{L}_{i}^{+})=0\;\text{for all $i$}\\}$. Note that
$\Delta_{f}^{+}\subseteq\Psi_{f}^{+}$. Similarly, define the sets
$\Delta_{f}^{-}$ and $\Psi_{f}^{+}$ and note that
$\Delta_{f}^{-}\subseteq\Psi_{f}^{-}$.
Define the functions $L^{\pm}:\mathcal{PML}(S)\to\mathbb{R}^{\geq 0}$ by
$L^{\pm}([\mathcal{L}])=\frac{1}{\ell(\mathcal{L})}(\sum_{i=1}^{n}i(\mathcal{L},\mathcal{L}_{i}^{\pm})+\sum_{j=1}^{m}i(\mathcal{L},\delta_{j}))$.
Note that $(L^{+})^{-1}(0)=\Delta^{-}$ and $(L^{-})^{-1}(0)=\Delta^{+}$.
In $\S 3$ of [Iva92] is showen that given a compact subset
$K\subset\mathcal{PML}(S)\backslash\Psi^{+}$, there are constants
$c_{1},d_{1},d_{1}^{\prime}$ and $c_{2},d_{2},d_{2}^{\prime}$, depending only
on $f$ and $K$ such that any integer $n\geq 1$:
$\frac{i(f^{n}(\mathcal{L}),\delta_{j})}{\ell(f^{n}(\mathcal{L}))}\leq\frac{d_{1}^{\prime}}{c_{1}n-d_{1}}$
and
$\frac{i(f^{n}(\mathcal{L}),\mathcal{L}_{i}^{+})}{\ell(f^{n}(\mathcal{L}))}\leq\frac{s_{i}^{n}d_{2}^{\prime}}{c_{2}n-d_{2}}$.
Similarly, given a compact subset $K\subset\mathcal{PML}(S)\backslash\Psi^{-}$
there are constants $c_{1},d_{1},d_{1}^{\prime}$ and
$c_{2},d_{2},d_{2}^{\prime}$, depending only on $f$ and $K$ such that for any
$n\geq 1$,
$\frac{i(f^{n}(\mathcal{L}),\delta_{j})}{\ell(f^{n}(\mathcal{L}))}\leq\frac{d_{1}^{\prime}}{c_{1}n-d_{1}}$
and
$\frac{i(f^{n}(\mathcal{L}),\mathcal{L}_{i}^{-})}{\ell(f^{n}(\mathcal{L}))}\leq\frac{s_{i}^{n}d_{2}^{\prime}}{c_{2}n-d_{2}}$.
Consequently, for any integer $n\geq 1$,
$L^{+}(f^{n}([\mathcal{L}]))\leq\frac{s_{i}^{-n}d^{\prime}}{cn-d}+\frac{d^{\prime}}{cn-d}$.
Similarly, for any integer $n\geq 1$,
$L^{-}(f^{-n}([\mathcal{L}]))\leq\frac{s_{i}^{n}d^{\prime}}{cn-d}+\frac{d^{\prime}}{cn-d}$.
Using the above bounds for the functions $L^{\pm}$ and the fact that they are
continuous one may easily verify that the action of $f$ on
$\mathcal{PML}(S)\backslash\Psi_{f}^{-}\cup\Psi_{f}^{+}$ has a compact
fundamental domain, denoted by $K_{f}$. Furthermore, Ivanov proves in Theorem
A.2 of Appendix A of [Iva92] (see also Theorem 3.5 in $\S 3$ of the book) that
###### Theorem 7.3.
Let $U$ be an open subset and $K$ be a compact subset of $\mathcal{PLM}(S)$.
If $\Delta_{f}^{+}\subset U$ and
$K\subset\mathcal{PML}(S)\backslash\Psi_{f}^{+}$, then there is $N>0$ such
that $f^{n}(K)\subset U$ for any $n\geq N$. If $\Delta_{f}^{-}\subset U$ and
$K\subset\mathcal{PML}(S)\backslash\Psi_{f}^{-}$, then there is $N>0$ such
that $f^{-n}(K)\subset U$ for any $n\geq N$.
We proceed to prove a lemma which will give us certain upper bounds for
subsurface coefficients in $\S$7.1 and 7.2 where we construct laminations with
prescribed subsurface coefficients.
###### Lemma 7.4.
Let $f$ be a partial pseudo-Anosov map supported on a large subsurface
$X\subsetneq S$. Given a compact subset
$K\subset\mathcal{PML}(S)\backslash\Psi_{f}^{-}\cup\Psi_{f}^{+}$ there is a
constant $m=m(f,K)$, depending only on $f$ and $K$, such that for every
$\gamma\in K$, every subsurface $W\pitchfork\gamma$ which is neither $X$ nor
an annulus with core curve a component of $\partial{X}$ and every integer $e$
we have
(7.1) $d_{W}(\gamma,f^{e}(\gamma))\leq m$
###### Proof.
First suppose that $W$ is a non-annular subsurface. Since $X$ is a large
subsurface either $W\pitchfork X$, $W\subsetneq X$ or $W\supsetneq X$.
We proceed to establish the bound (7.1) when $W\pitchfork X$ or $W\subsetneq
X$. In this situation there are component curves of $\partial{W}$ which
overlap $X$.
In this proof let $K:=K_{f}$ be the fundamental domain for the action of $f$
on $\mathcal{PML}(S)\backslash\Psi_{f}^{-}\cup\Psi_{f}^{+}$. Then applying an
appropriate power of $f$ to $W$ we may assume that the projective class of all
of the component curves of $\partial{W}$ which overlap $X$ are in $K$.
Applying this power of $f$ to the subsurface coefficient in (7.1) we get the
subsurface coefficient
(7.2) $d_{W}(f^{e_{1}}(\gamma),f^{e_{2}}(\gamma))$
for some $e_{1}$ and $e_{2}$. So we need to bound (7.2) for any pair of
integers $e_{1}$ and $e_{2}$.
Fix a complete hyperbolic metric on $S$. Realize all curves and laminations as
geodesics in this metric. We claim that there is a constant $l_{1}>0$ and a
positive integer $N_{1}$ depending only on $K$ and $f$ such that for any
$\gamma\in K$ with $\gamma\pitchfork W$ and any $n\geq N_{1}$,
$f^{n}(\gamma)\cap W$ has length less than $l_{1}$. We proceed to get the
bound by essentially following the compactness argument given by Minksy in
[Min00] (see also Theorem 3.9 of [KL08]). Suppose that the claim does not
hold. Then after possibly passing to a subsequence we may assume that for each
$n$ there is a subsurface $W_{n}$, a curve $\gamma_{n}\in K$ with
$\gamma_{n}\pitchfork W_{n}$ such that the length of the collection of arcs
$\alpha_{n}:=f^{n}(\gamma_{n})\cap W_{n}$ goes to $\infty$ as $n\to\infty$.
Theorem 7.3 applied to teh fundamental domain of the action of $f$, $K$ and
arbitrary open subsets $U\supset\Delta_{f}^{+}$ implies that $[\alpha_{n}]$
($\alpha_{n}$ is equipped with the measure $i(\alpha_{n},.)$) converges into
the subset $\Delta^{+}_{f}$. Since $X$ is a large subsurface,
$\Psi^{+}_{f}=[\mathcal{L}^{+}]$, where $\mathcal{L}^{+}$ is the attracting
lamination of $f|_{X}$. Note that $\lambda^{+}$ the support of
$\mathcal{L}^{+}$ is minimal and fills $X$.
Let $\beta_{n}$ be a component curve of $\partial{W}_{n}$ which overlaps $X$
and consider the projective measured laminations $[\beta_{n}]\in K$
($\beta_{n}$ is equipped with the measure $i(\beta_{n},.)$). $K$ is a compact
subset of $\mathcal{PML}(S)\backslash\Psi_{f}^{-}\cup\Psi_{f}^{+}$, so after
possibly passing to a subsequence $[\beta_{n}]$ converge to a projective
lamination $[\mathcal{L}^{\prime}]$. Denote the support of
$\mathcal{L}^{\prime}$ by $\xi$. For each $n$, $\alpha_{n}$ is disjoint from
$\beta_{n}$, so $\xi$, the support of $\mathcal{L}^{\prime}$, is disjoint from
$\lambda^{+}$, the support of $\mathcal{L}^{\prime}$. Furthermore,
$\lambda^{+}\cup\partial{X}$ fills $S$, so $\xi$ is a sublamination of
$\lambda^{+}\cup\partial{X}$. But then $[\mathcal{L}^{\prime}]$ is in
$\Psi_{f}^{+}$, which contradicts the fact that $K$ misses $\Psi^{+}$. So we
conclude that the integer $N_{1}$ and the bound $l_{1}$ exist.
By a similar argument there exist a constant $l_{2}>0$ and a positive integer
$N_{2}$, depending only on $K$ and $f$, such that for any $\gamma\in K$ with
$\gamma\pitchfork W$ and any $n\geq N_{2}$, $f^{-n}(\gamma)\cap W$ has length
bounded above by $l_{2}$. Here after possibly passing to a subsequence
$f^{-n}(\gamma)$ converges into $\Psi_{f}^{-}$ and the rest of argument would
be similar to the above situation.
Furthermore, the set $\bigcup_{e=-N_{2}}^{N_{1}}f^{e}(K)$ is a compact subset
of $\mathcal{PML}(S)$ and the length function is continuous on
$\mathcal{PML}(S)$. So there is an upper bound $l_{3}$ for the length of
laminations in this set.
By the bounds we established above we conclude that for any $\gamma\in K$ the
length of $f^{n}(\gamma)\cap W$ is bounded above by
$l=\max\\{l_{1},l_{2},l_{3}\\}$. Here $l$ only depends on $f$ and $K$.
Since $X$ is a large subsurface each component of $\partial{W}$ is either in
$K$ (a compact subset of $\mathcal{PML}(S)$) or $\partial{X}$. Then since the
length-functions are continuous on $\mathcal{PML}(S)$ we have that the length
of $\partial{W}$ in the hyperbolic metric we fixed on $S$ is bounded above by
an $L$ depending only on $K$ and the subsurface $X$. Furthermore, for any
integer $e$ as we saw above the length of $f^{e}(\gamma)$ is bounded above by
$l$. By the Collar lemma (§4.1 [Bus10]) upper bound on the length of two
curves gives an upper bound on their intersection number. Then
$i(f^{e}(\gamma),\partial{W})\leq i_{0}$ for some $i_{0}$. This implies that
the number of arcs in $f^{e}(\gamma)\cap W$ is bounded above by $i_{0}$ as
well.
By the definition of subsurface projection from $\S$2 the length of the curve
in $\pi_{W}(f^{e}(\gamma))\subset\mathcal{C}_{0}(W)$ corresponding to an arc
of $f^{e}(\gamma)\cap W$ with length at most $l$ is bounded above by $2l$ plus
the length of $\partial{W}$. As we saw above the number of arcs in
$f^{e}(\gamma)\cap W$ is bounded by $i_{0}$ and the length of $\partial{W}$ is
bounded above by $L$. Thus the length of $\pi_{W}(f^{e}(\gamma))$ is bounded
by $i_{0}(2l+L)$.
Given $e_{1},e_{2}$ let $\gamma_{1}=f^{e_{1}}(\gamma)$ and
$\gamma_{2}=f^{e_{2}}(\gamma)$. As we saw above the length of
$\pi_{W}(\gamma_{1})$ and $\pi_{W}(\gamma_{2})$ are bounded above by
$i_{0}(2l+L)$. Then by the Collar lemma $i(\gamma_{1},\gamma_{2})\leq i_{1}$
for some $i_{1}$. Further by Lemma 2.1 of [MM99] we have that
$d_{W}(\gamma_{1},\gamma_{2})\leq 2i(\gamma_{1},\gamma_{2})+1$. So (7.2) is
bounded above by $2i_{1}+1$.
Now we proceed to establish the bound (7.1) for non-annular subsurface
$W\supsetneq X$. Here $\partial{X}$ has at least one component curve
$\beta\in\mathcal{C}(W)$. $\gamma\pitchfork W$, so for any integer $e$,
$f^{e}(\gamma)\pitchfork W$. Then by the triangle inequality we have
$d_{W}(f^{e}(\gamma),\gamma)\leq
d_{W}(f^{e}(\gamma),\beta)+d_{W}(\beta,\gamma).$
Applying $f^{-e}$, $d_{W}(f^{e}\gamma,\gamma)=d_{f^{-e}W}(\gamma,\beta)$. Here
we use the fact that since $\beta$ is a component curve of $\partial{X}$ and
$f$ is supported on $X$, $f^{-e}(\beta)=\beta$. Lemma 2.1 of [MM99] we have
$d_{f^{-e}W}(\gamma,\beta)\leq 2i(\gamma,\beta)+1$. So the first subsurface
coefficient above is bounded by $2i(\gamma,\beta)+1$. Further by Lemma 2.1 of
[MM99] the second subsurface coefficient is bounded above by
$2i(\gamma,\beta)+1$. Now $\beta,\gamma\in K$ and $K$ is a compact subset of
$\mathcal{PML}(S)$. Then since the intersection number $i$ is a continuous
function, $i(\gamma,\beta)\leq i_{2}$ for some $i_{2}$ depending only on $K$.
Then (7.2) is bounded above by $2(2i_{2}+1)$.
Finally suppose that $W$ is an annular subsurface whose core curve is not a
boundary curve of $X$. Applying an appropriate power of $f$ to $W$ we may
assume that the projective class of the core curve of $W$ overlaps $X$ is in
$K$ the fundamental domain of the action of $f$. Applying this power of $f$ to
the subsurface coefficient in (7.1) we get the subsurface coefficient (7.2)
for some $e_{1}$ and $e_{2}$. So we need to bound (7.2) for any pair of
integers $e_{1}$ and $e_{2}$.
Denote the core curve of $W$ by $\beta$. Define the angle between two curves
or laminations realized as geodesics in the metric we fixed on $S$ to be the
minimum of the smaller angle between them at their intersection points. We
proceed by a compactness argument similar to the one we gave to bound the
length of $f^{e}(\gamma)\cap W$ to show that there exists a constant
$\theta_{1}>0$ which bounds from below the angle between the curve
$f^{n}(\gamma)$ and $\beta$ for any $n\geq 0$. If there is not such a
$\theta_{1}$, there is a sequence of annular subsurfaces $W_{n}$ with core
curve $\beta_{n}$ and a sequence of curves $\gamma_{n}\in K$ such that the
angle between $\beta_{n}$ and $f^{n}(\gamma_{n})$ goes to $0$ as $n\to\infty$.
Theorem 7.3 implies that $[f^{n}(\gamma_{n})]\to[\mathcal{L}^{+}]$ as
$n\to\infty$. Further since $K$ is compact after possibly passing to a
subsequence, $[\beta_{n}]\to[\mathcal{L}^{\prime}]$ in $K$. Denote the support
of $\mathcal{L}^{\prime}$ by $\xi$. But then since the angle between
$\beta_{n}$ and $f^{n}(\gamma_{n})$ goes to $0$, $\xi$ would be a sub-
lamination of $\lambda^{+}\cup\partial{X}$. Recall that $\mathcal{L}^{+}$ is
the attracting lamination of $f|_{X}$ and its support $\lambda^{+}$ fills $X$.
Further $X$ is a large subsurface, so $\lambda^{+}\cup\partial{X}$ fills $S$.
Consequently $[\mathcal{L}^{\prime}]$ is in $\Psi_{f}^{+}$. But this
contradicts the fact that $K$ misses $\Psi_{f}^{+}$. Similarly we can show
that the angle between $\beta$ and $f^{-n}(\gamma)$, $n\geq 0$ is bounded
below by some $\theta_{2}>0$. Thus the angle between $f^{e}(\gamma)$ and
$\beta$ is bounded below by some $\theta_{0}:=\min\\{\theta_{1},\theta_{2}\\}$
for any integer $e$.
Having the lower bound $\theta_{0}$ for the angle, Lemma 2.6 of [KL08] applied
to the curves $f^{e_{1}}(\gamma)$ and $f^{e_{2}}(\gamma)$ and the annular
subsurface $W$ with core curve $\beta$ gives us an upper bound for
$d_{\beta}(f^{e_{1}}(\gamma),f^{e_{2}}(\gamma))$, depending only on
$\theta_{0}$ and the lower bound for the length of $\beta$. The length of
$\beta$ is bounded below by twice of the injectivity radius of the hyperbolic
metric which was fixed on the surface. So we get the desired bound on (7.2)
for annular subsurfaces as well.
∎
### 7.1. Scheme I
The construction of this subsection will be used in $\S\ref{subsec : div}$ to
provide examples of divergent WP geodesic rays and in $\S$8.3 to provide
examples of closed WP geodesics in the thin part of moduli space.
Let $\alpha$ and $\beta$ be two disjoint curves on $S$ such that
$S\backslash\alpha$, $S\backslash\beta$ and $S\backslash\\{\alpha,\beta\\}$
are large subsurfaces. Consider indexed large subsurfaces
$X_{0}=S\backslash\\{\alpha,\beta\\}$, $X_{1}=S\backslash\alpha$,
$X_{2}=S\backslash\\{\alpha,\beta\\}$ and $X_{3}=S\backslash\beta$. Note that
$X_{0}$ and $X_{2}$ are both the same subsurface
$S\backslash\\{\alpha,\beta\\}$ with different indices. Let
$f_{0},f_{1},f_{2}$ and $f_{3}$ be partial pseudo-Anosov maps supported on
$X_{0},X_{1},X_{2}$ and $X_{3}$, respectively, where $f_{0}=f_{2}$ (the same
partial psuedo-Anosov maps with different indices). Then in particular,
$f_{a}$, $a=0,1,2,3$, preserves each component of $\partial{X}_{a}$.
Furthermore, suppose that for $a=0,1,2,3$, if $\delta\in\partial{X_{a}}$ then
we have
(7.3) $d_{\delta}(\gamma,f_{a}^{e}\gamma)\leq 2$
for every $\gamma\pitchfork\partial{X}_{a}$ and any integer $e$.
To see that partial pseudo-Anosov maps as above exist we have that: Given a
partial pseudo-Anosov map $g$ supported on $X$ by Theorem 11.7 in exposé 11 of
[FLP79], $g$ is isotopic to some $f$ such that the restriction of $f$ to all
of the component curves of $\partial{X}$ is identity. This $f$ satisfies the
above annular coefficient bound.
Let $q_{0}:\mathbb{N}\to\\{0,1,2,3\\}$ be the function $q_{0}(i)\equiv i$ (mod
4). Let $q_{1}(i)=q_{0}(i+1)$, $q_{2}(i)=q_{0}(i+2)$ and
$q_{3}(i)=q_{0}(i+3)$. Let $q$ denote any of the functions $q_{0},q_{1},q_{2}$
and $q_{3}$ or the restriction of any of them to the set $\\{1,...,k\\}$,
where $k$ is a positive integer.
Let $q$ be as above. When the domain of $q$ is $\mathbb{N}$ let
$\\{e_{i}\\}_{i}$ be an infinite sequence of integers and when the domain of
$q$ is $\\{1,...,k\\}$ let $\\{e_{i}\\}_{i}$ be a sequence of integers with
$k$ elements. For simplicity of notation we some times denote the sequence
$\\{e_{i}\\}_{i}$ by $e$.
For any $i$ in the domain of $q$ set the subsurface
$Z_{i}(q,e)=f_{q(1)}^{e_{1}}...f_{q(i-1)}^{e_{i-1}}X_{q(i)}$
Let $\mu_{I}(q,e)$ be a marking whose base contains
$\\{\partial{X}_{a}\\}_{a=0,1,2,3}=\\{\alpha,\beta\\}$. Throughout the
following lemmas and propositions we assume that the domain of $q$ is
$\\{1,..,k\\}$ for some $k\geq 1$. We let
$\mu_{T}(q,e)=f_{q(1)}^{e_{1}}...f_{q(k)}^{e_{k}}\mu_{I}(q,e)$ and establish
several bounds on the subsurface coefficients of $\mu_{I}(q,e)$ and
$\mu_{T}(q,e)$.
When there is no confusion we drop the reference to $(q,e)$. For example we
denote $Z_{i}(q,e)$ by $Z_{i}$.
###### Remark 7.5.
The construction of this subsection and the estimates on subsurface
coefficients can be carried out in a more general setting. Here we restrict
ourself to be able to provide detailed step by step estimates and complete
arguments.
###### Lemma 7.6.
There are constants $K^{\prime}_{1}>0,C^{\prime}_{1}\geq 0$ and $E_{1}>0$,
depending only on the partial pseudo-Anosov maps $f_{0},f_{1},f_{2}$ and
$f_{3}$, and $\mu_{I}$ with the following properties. Given $q$ and
$\\{e_{i}\\}_{i}$ such that $|e_{i}|>E_{1}$ for any $i\in\\{1,...,k\\}$, we
have
1. (i)
For any $i\in\\{1,...,k\\}$,
(7.4) $d_{Z_{i}(q,e)}(\mu_{I}(q,e),\mu_{T}(q,e))\geq
K^{\prime}_{1}|e_{i}|-C^{\prime}_{1}$
2. (ii)
Let $k\geq 3$. Let $i,j\in\\{1,...,k\\}$ and $j\geq i+2$. Then
$Z_{i}(q,e)<Z_{j}(q,e)$ between $\mu_{I}(q,e)$ and $\mu_{T}(q,e)$.
###### Proof.
Our proof modifies the proof of Theorem 5.2 of [CLM12]. There the authors
assume that any two of the subsurfaces which support partial pseudo-Anosov
maps either overlap or are disjoint. But here $X_{0}(=X_{2}$) is a subsurface
of $X_{1}$ and $X_{3}$. As a result their argument does not go through
completely to prove the lemma and needs some modification. Furthermore, our
set up is different.
Proof of part part (i). The proof is by induction on $k$. Denote $\mu_{I}$ by
$\mu$. Set the constant
$K^{\prime}_{1}=\min\\{\tau_{a}:a=0,1,2,3\\}$.
Here $\tau_{a}=\tau_{f_{a}}$, $a=0,1,2,3$, is the constant from Proposition
7.1 for the partial pseudo-Anosov map $f_{a}$ supported on $X_{a}$.
Let $\eta=\max\\{d_{X_{a}}(\mu,f_{b}^{e}\mu):a,b\in\\{0,1,2,3\\},\;X_{a}\neq
X_{b}\;\text{and}\;e\in\mathbb{Z}\\}$. Note that since $\mu$ is fixed and
$X_{a}\neq X_{b}$ by Lemma 7.4 the maxima exists. Set the constant
$C^{\prime}_{1}=2(B_{0}+\eta+1)$
Here $B_{0}$ is the constant in Theorem 2.8 (Behrstock Inequality).
Let $\omega=\max\\{d_{W}(\mu,\partial{X}_{a}):W\subseteq
S\;\text{and}\;a=0,1,2,3\\}$. Note that the marking $\mu$ and subsurfaces
$\\{X_{a}\\}_{a=0,1,2,3}$ are fixed so the maxima exists. Set the constant
$E_{1}=\frac{B_{0}+\omega+4M+4+C_{1}^{\prime}}{K^{\prime}_{1}}$
By Lemma 7.4, $d_{X_{a}}(f^{e}\mu,\mu)\geq\tau_{a}|e|\geq K^{\prime}_{1}|e|$,
$a=0,1,2,3$, so we have the base of induction for $k=1$.
Suppose that for any function $q:\\{1,...,k^{\prime}\\}\to\\{0,1,2,3\\}$ with
$k^{\prime}<k$ as the beginning of this section the proposition holds. Fix
$i\in\\{1,...,k\\}$ and let $g=f_{q(1)}^{e_{1}}...f_{q(i-1)}^{e_{i-1}}$.
Applying $g^{-1}$ to $d_{Z_{i}}(\mu_{I},\mu_{T})$ we get
$d_{Z_{i}}(\mu_{I},\mu_{T})=d_{X_{q(i)}}(g^{-1}\mu,f_{q(i)}^{e_{i}}h\mu)$
where $h=f_{q(i+1)}^{e_{i+1}}...f_{q(k)}^{e_{k}}$. By the triangle inequality
the left hand side is bounded below by
(7.5) $d_{X_{q(i)}}(h\mu,f_{q(i)}^{e_{i}}h\mu)-d_{X_{q(i)}}(g^{-1}\mu,h\mu)-2$
By Proposition 7.1 the first term of (7.5) is bounded below by
$K^{\prime}_{1}|e_{i}|$. This gives us the multiplicative constant in (7.4).
To get the additive constant in (7.4) we proceed to bound the second term of
(7.5). By the triangle inequality it is bounded above by
(7.6) $d_{X_{q(i)}}(g^{-1}\mu,\mu)+d_{X_{q(i)}}(\mu,h\mu)+2$
First we show that $d_{X_{q(i)}}(\mu,h\mu)\leq\frac{C^{\prime}_{1}}{2}+1$. Let
$q^{\prime}(j)=q(j+i)$ for $j=1,...,k-i$ and $e^{\prime}(j)=e(j+i)$ for
$j=1,...,k-i$. Then by the definition
$Z_{2}(q^{\prime},e^{\prime})=f_{q(i+1)}^{e_{i+1}}X_{q(i+2)}$ so
$d_{f_{q(i+1)}^{e_{i+1}}X_{q(i+2)}}(\mu,h\mu)=d_{Z_{2}(q^{\prime},e^{\prime})}(\mu,h\mu)$
The assumption of induction applied to $q^{\prime}$ implies that the right
hand side subsurface coefficient is greater than or equal to
$K^{\prime}_{1}|e_{i+1}|-C^{\prime}_{1}$. So we have
(7.7) $d_{f_{q(i+1)}^{e_{i+1}}X_{q(i+2)}}(\mu,h\mu)\geq
K^{\prime}_{1}|e_{i+1}|-C^{\prime}_{1}.$
We claim that
###### Claim 7.7.
$\partial{X}_{q(i)}\pitchfork f_{q(i+1)}^{e_{i+1}}X_{q(i+2)}$.
To see this, first suppose that $q(i)=1$ then $X_{q(i)}=S\backslash\alpha$.
Further by the definition of $q$, $X_{q(i+2)}=S\backslash\beta$ and
$X_{q(i+1)}=S\backslash\\{\alpha,\beta\\}$. $f_{q(i+1)}$ preserves each
component of $\partial{X}_{q(i+1)}=\\{\alpha,\beta\\}$, so
$f_{q(i+1)}^{e_{i+1}}X_{q(i+2)}=S\backslash\beta$. Further
$\partial{X}_{q(i)}=\alpha$. Then since $\alpha\neq\beta$,
$\partial{X}_{q(i)}\pitchfork X_{q(i+1)}$.
If $q(i)=3$ then $X_{q(i)}=S\backslash\beta$ and the claim follows from a
similar argument replacing $\alpha$ by $\beta$.
Now suppose that $q(i)=0(2)$, then $X_{q(i)}=S\backslash\\{\alpha,\beta\\}$.
Further by the definition of $q$, $X_{q(i+2)}=S\backslash\\{\alpha,\beta\\}$.
Moreover, $f_{q(i+1)}$ is supported on $X_{q(i+1)}$, which is
$S\backslash\alpha(S\backslash\beta)$. Note that $\partial{X}_{q(i)}\pitchfork
X_{q(i+1)}$, because $\\{\alpha,\beta\\}\pitchfork
S\backslash\alpha(S\backslash\beta)$.
* •
If $f_{q(i+1)}$ is supported on $S\backslash\beta$, then by Proposition 7.1,
$d_{X_{q(i+1)}}(f_{q(i+1)}^{e_{i+1}}\partial{X}_{q(i)},\partial{X}_{q(i)})>K^{\prime}_{1}|e_{i+1}|>K^{\prime}_{1}E_{1}>4.$
This implies that $f_{q(i+1)}^{e_{i+1}}\alpha$ and $\alpha$ overlap. Now
$f_{q(i+1)}^{e_{i+1}}\alpha\in f_{q(i+1)}^{e_{i+1}}\partial{X}_{q(i)}$ so
$\partial{X}_{q(i)}$ overlaps $f_{q(i+1)}^{e_{i+1}}X_{q(i)}$.
* •
If $f_{q(i+1)}$ is supported on $S\backslash\alpha$, then by a similar
argument $f_{q(i+1)}^{e_{i+1}}\beta$ and $\beta$ overlap. Now
$f_{q(i+1)}^{e_{i+1}}\beta\in f_{q(i+1)}\partial{X}_{q(i)}$, so
$\partial{X}_{q(i)}\pitchfork f_{q(i+1)}^{e_{i+1}}X_{q(i+2)}$.
The proof of the claim is complete.
By the above claim in hand we may write
(7.8) $d_{f_{q(i+1)}^{e_{i+1}}X_{q(i+2)}}(\partial{X}_{q(i)},h\mu)$
This subsurface coefficient by the triangle inequality is bounded below by
(7.9)
$d_{f_{q(i+1)}^{e_{i+1}}X_{q(i+2)}}(h\mu,\mu)-d_{f_{q(i+1)}^{e_{i+1}}X_{q(i+2)}}(\mu,\partial{X}_{q(i)})-2$
The second term of (7.9) is bounded above by $\omega$ and by (7.7) the first
term of (7.9) is greater than or equal to $K^{\prime}_{1}|e_{i+1}|$. So if
$|e_{i+1}|>E_{1}$ then (7.9) is at least $B_{0}$ and consequently (7.8) is at
least $B_{0}$. So by Theorem 2.8 (Behrstock Inequality) we get
(7.10) $d_{X_{q(i)}}(f_{q(i+1)}^{e_{i+1}}\partial{X}_{q(i+2)},h\mu)\leq
B_{0}.$
Now by the triangle inequality
(7.11) $\displaystyle d_{X_{q(i)}}(\mu,h\mu)$ $\displaystyle\leq$
$\displaystyle
d_{X_{q(i)}}(\mu,f_{q(i+1)}^{e_{i+1}}\partial{X}_{q(i+2)})+d_{X_{q(i)}}(f_{q(i+1)}^{e_{i+1}}\partial{X}_{q(i+2)},h\mu)+1$
$\displaystyle\leq$ $\displaystyle\eta+B_{0}\leq\frac{C^{\prime}_{1}}{2}.$
The second inequality follows from the choice of $\eta$ and the bound (7.10).
The third one follows from the choice of $C^{\prime}_{1}$.
Further, let $q^{\prime}(j)=q(i+1-j)$ for $j=1,...,i$ and
$e^{\prime}(j)=-e(i+1-j)$ for $j=1,...,i$. Then the exact same lines we gave
avobe using this $q^{\prime}$ and $e^{\prime}$ implies that
$d_{X_{q(i)}}(g^{-1}\mu,\mu)\leq\frac{C^{\prime}_{1}}{2}$. This bound and
(7.11) give us the bound $C^{\prime}_{1}+2$ for (7.6). Plugging this into
(7.5) we get the bound $C^{\prime}_{1}$ as the additive constant for (7.4).
Proof of part (ii). The proof is by induction on $k$. The base of induction
for $k=3$ is obtained as follows.
We would like to show that $Z_{1}<Z_{3}$. We need to verify that the
conditions of Proposition 2.15 hold for $\mu_{I},\mu_{T}$ and subsurfaces
$Z_{1}$ and $Z_{3}$. First note that by part (i),
$d_{Z_{1}}(\mu_{I},\mu_{T})>K^{\prime}_{1}|e_{1}|-C_{1}^{\prime}>K^{\prime}_{1}E_{1}-C_{1}^{\prime}>4M$
and similarly $d_{Z_{3}}(\mu_{I},\mu_{T})>4M$.
We proceed to show that $Z_{1}\pitchfork Z_{3}$. If $q(1)=1$, then by the
definition of $q$, $Z_{1}(q,e)=S\backslash\alpha$,
$Z_{2}(q,e)=f_{1}^{e_{1}}S\backslash\\{\alpha,\beta\\}$,
$Z_{3}(q,e)=f_{1}^{e_{1}}f_{2}^{e_{2}}S\backslash\beta$. Furthermore, $f_{1}$
preserves $\alpha$ and $f_{2}$ preserves both $\alpha$ and $\beta$. So
$Z_{2}(q,e)=S\backslash\\{\alpha,f_{1}^{e_{1}}\beta\\}$
and$Z_{3}(q,e)=S\backslash f_{1}^{e_{1}}\beta$. Then applying $f_{1}^{-e_{1}}$
to $\partial{Z_{1}}=\alpha$ and $\partial{Z_{3}}=f_{1}^{e_{1}}\beta$, we get
$\alpha$ and $\beta$. Moreover, $S\backslash\alpha\pitchfork
S\backslash\beta$, so $Z_{1}\pitchfork Z_{3}$.
If $q(1)=3$ a similar argument implies that $Z_{1}<Z_{3}$. If $q(1)=0(2)$,
then $X_{q(1)}=S\backslash\\{\alpha,\beta\\}$,
$X_{q(2)}=S\backslash\alpha(S\backslash\beta)$ and
$X_{q(3)}=S\backslash\\{\alpha,\beta\\}$. So
$Z_{1}(q,e)=S\backslash\\{\alpha,\beta\\}$,
$Z_{2}(q,e)=f_{0}^{e_{1}}S\backslash\alpha$ ($f_{2}^{e_{1}}S\backslash\beta$)
and
$Z_{3}(q,e)=f_{0}^{e_{1}}f_{1}^{e_{2}}S\backslash\\{\alpha,\beta\\}(f_{2}^{e_{1}}f_{3}^{e_{2}}S\backslash\\{\alpha,\beta\\})$.
Furthermore, $f_{q(1)}=f_{0}(f_{2})$ preserves $\alpha$ and $\beta$, and
$f_{q(2)}=f_{1}(f_{3})$ preserves $\alpha(\beta)$. So
$Z_{1}(q,e)=S\backslash\\{\alpha,\beta\\}$, $Z_{2}(q,e)=S\backslash\alpha$
($S\backslash\beta$) and
$Z_{3}(q,e)=S\backslash\\{\alpha,f_{0}^{e_{1}}f_{1}^{e_{2}}\beta\\}(S\backslash\\{f_{2}^{e_{1}}f_{3}^{e_{3}}\alpha,\beta\\})$.
Apply $f_{1}^{-e_{2}}f_{0}^{-e_{1}}(f_{3}^{-e_{2}}f_{2}^{-e_{1}})$ to
$\partial{Z}_{1}$ and $\partial{Z}_{3}$. The resulting multi curves contain
the curves $\beta$ and $f_{1}^{-e_{2}}\beta$ ($\alpha$ and
$f_{3}^{-e_{3}}\alpha$), respectively. Moreover, by Proposition 7.1,
$d_{S\backslash\alpha}(\beta,f_{1}^{e_{1}}\beta)>\tau_{1}|e_{1}|>\tau_{1}E_{1}>4$
($d_{S\backslash\beta}(\alpha,f_{3}^{-e_{3}}\alpha)>\tau_{3}E_{1}>4$), so
$\beta$ and $f_{1}^{e_{1}}\beta$ overlap ($\alpha$ and $f_{3}^{-e_{3}}\alpha$
overlap). This implies that $Z_{1}\pitchfork Z_{3}$.
Now we may write $d_{Z_{1}}(\mu_{I},\partial{Z}_{3})$. Then since
$\mu_{I}\supset\partial{Z}_{1}$ by (7.15) we get
$d_{Z_{1}}(\mu_{I},\partial{Z}_{3})\geq
d_{Z_{1}}(\mu_{I},\mu_{T})-\operatorname{diam}_{Z_{1}}(\mu_{I})\geq
K^{\prime}_{1}|e_{1}|-C^{\prime}_{1}-2>K^{\prime}_{1}E_{1}-C^{\prime}_{1}-2>2M.$
Consequently by Proposition 2.15, $Z_{1}<Z_{3}$. This finishes establishing of
the base of induction.
Suppose that (ii) holds for every $q:\\{1,...,k^{\prime}\\}\to\\{0,1,2,3\\}$,
with $k^{\prime}<k$. Let $q_{init}(l)=q(l)$ for $l=1,...,k-1$ and
$e_{init,l}=e_{l}$ for $l=1,...,k-1$. Then
$\mu_{I}(q_{init},e_{init})=\mu_{I}(q,e)$ and
$Z_{j}(q_{init},e_{init})=Z_{j}(q,e)$ for $j=1,...,k-1$.
Let $q_{term}(l)=q(l+1)$ for $l=1,...,k-1$ and $e_{term,l}=e_{l+1}$ for
$l=1,...,k-1$.
First suppose that $i,j\in\\{1,...,k\\}$ and $j\geq i+2$. If $j<k$, then the
assumption of induction applied to $q_{init}$ and $e_{init}$ implies that
$Z_{i}(q_{init},e_{init})<Z_{j}(q_{init},e_{init})$, which by Proposition 2.15
means that $Z_{i}(q_{init},e_{init})\pitchfork Z_{j}(q_{init},e_{init})$ and
$d_{Z_{j}(q_{init},e_{init})}(\mu_{I}(q_{init},e_{init}),\partial{Z}_{i}(q_{init},e_{init}))>2M$.
But this implies that $Z_{i}(q,e)<Z_{j}(q,e)$.
If $i>1$, then by the assumption of the induction
$Z_{i-1}(q_{term},e_{term})<Z_{j-1}(q_{trem},e_{term})$. This means that
$Z_{i-1}(q_{term},e_{term})\pitchfork Z_{j-1}(q_{trem},e_{term})$ and
$d_{Z_{j-1}(q_{term},e_{term})}(\mu_{T}(q_{term},e_{term}),\partial{Z_{i-1}}(q_{term},e_{term}))>2M$
Now applying $f_{q(1)}^{e_{1}}$ to the subsurfaces
$Z_{i-1}(q_{term},e_{term}),Z_{j-1}(q_{trem},e_{term})$ and the marking
$\mu_{T}(q_{term},e_{term})$ we get the subsurfaces $Z_{i}(q,e),Z_{j}(q,e)$
and the marking $\mu_{T}(q,e)$. Thus $Z_{i}(q,e)\pitchfork Z_{j}(q,e)$ and
$d_{Z_{j}(q,e)}(\mu_{T}(q,e),\partial{Z_{i}}(q,e))>2M$
So by Proposition 2.15, $Z_{i}(q,e)<Z_{j}(q,e)$.
Now suppose that $i=1$ and $j=k$. When $k=4$, we may proceed as in the base of
induction and prove directly that $Z_{1}\pitchfork Z_{4}$ and
$d_{Z_{1}}(\mu_{I}(q,e),\partial{Z}_{4})>2M$, which implies that
$Z_{1}<Z_{4}$. The details are similar so we skip them. When $k\geq 5$, let
$l\in\\{3,...,k-2\\}$. Then as we saw above $Z_{1}<Z_{l}$ and $Z_{l}<Z_{k}$.
So by the transitivity of the relation $<$ on subsurfaces (Proposition 2.15)
we conclude that $Z_{1}<Z_{k}$.
∎
The second part of the above lemma does not say anything about the order of
two consecutive subsurfaces $Z_{i}$ and $Z_{i+1}$. The following lemma gives
an order for times in the two consecutive intervals $J_{Z_{i}}$ and
$J_{Z_{i+1}}$ (see Theorem 2.13).
###### Lemma 7.8.
Given $q$ and $e$. Let $i$ be such that $q(i)=1$ or $3$ ($Z_{i}$ has one
boundary curve). If $i>1$ and $j\in J_{Z_{i}}$ then $j\geq\min J_{Z_{i-1}}$.
If $j\in J_{Z_{i-1}}$ then $j\leq\max J_{Z_{i+1}}$.
###### Proof.
By the definition of $q$ and the subsurfaces $Z_{i}(q,e)$ we have
$\partial{Z_{i}}=\partial{Z_{i-1}}\cap\partial{Z_{i+1}}$.
By Lemma 7.6 (ii), $Z_{i-1}<Z_{i+1}$, so by Proposition 2.15,
$d_{Z_{i+1}}(\mu_{T},\partial{Z}_{i-1})>2M$. Then Theorem 2.8 implies that
$d_{Z_{i-1}}(\mu_{T},\partial{Z}_{i+1})\leq M$. Then since
$\partial{Z}_{i}\subset\partial{Z}_{i+1}$ we obtain
$d_{Z_{i-1}}(\mu_{T},\partial{Z_{i}})\leq
M+\operatorname{diam}_{Z_{i-1}}(\partial{Z_{i+1}})\leq M+1$. Since $j\in
J_{Z_{i}}$, $\rho(j)\supset\partial{Z}_{i}$, so we get that
$d_{Z_{i-1}}(\mu_{T},\rho(j))\leq M+1$. This inequality and
$d_{Z_{i-1}}(\mu_{I},\mu_{T})>4M$, combined by the triangle inequality imply
that $d_{Z_{i-1}}(\mu_{I},\rho(j))>3M-1$.
Now assume that $j\leq\min J_{Z_{i-1}}$, then by Theorem 2.13 (4),
$d_{Z_{i-1}}(\mu_{I},\rho(j))\leq M$, which contradicts the lower bound we
just proved. Thus $j\geq\min J_{Z_{i-1}}$.
The proof of $j\leq\max J_{Z_{i+1}}$ is similar. ∎
###### Proposition 7.9.
There are constants $\textbf{m},\textbf{m}^{\prime}>4M$ and $E_{2}>E_{1}$,
only depending on $f_{0},f_{1},f_{2}$ and $f_{3}$ and $\mu_{I}$ with the
following properties. Given $q$ and $\\{e_{i}\\}_{i}$ such that
$|e_{i}|>E_{2}$ for any $i\in\\{1,...,k\\}$, we have
1. (i)
For any non-annular subsurface $W$ which is neither $Z_{i}(q,e)$ for some $i$
nor $S$ we have $d_{W}(\mu_{I}(q,e),\mu_{T}(q,e))\leq\textbf{m}$.
2. (ii)
Given $\gamma\in\mathcal{C}_{0}(S)$ we have
$d_{\gamma}(\mu_{I}(q,e),\mu_{T}(q,e))\leq\textbf{m}^{\prime}$.
###### Proof.
Proof of part (i). If $d_{W}(\mu_{I},\mu_{T})\leq 4M$, then we already have
the upper bound. If not, $d_{W}(\mu_{I},\mu_{T})>4M$.
Let $l\in\\{1,...,k\\}$ with $q(l)=1$ or $3$. Then $Z_{l}$ is a subsurface
with one boundary curve. We claim that $\partial{Z}_{l-1}$ and
$\partial{Z}_{l+1}$ fill $Z_{l}$. To see this, suppose that
$X_{q(l)}=S\backslash\alpha$. When $X_{q(l)}=S\backslash\beta$ the proof is
similar. Let $g=f_{q(1)}^{e_{1}}...f_{q(l-1)}^{e_{l-1}}$. We have
$g^{-1}Z_{l}=S\backslash\alpha$. By the definition of $q$,
$X_{q(l-1)}=S\backslash\\{\alpha,\beta\\}$. Then
$g^{-1}\partial{Z}_{l-1}=f_{q(l-1)}^{-e_{l-1}}\partial{X}_{q(l-1)}=\\{f_{q(l-1)}^{-e_{l-1}}\alpha,f_{q(l-1)}^{-e_{l-1}}\beta\\}$.
Furthermore, $f_{q(l-1)}$ is supported on $X_{q(l-1)}$, so preserves each
component of $\partial{X}_{q(l-1)}=\\{\alpha,\beta\\}$ i.e.
$f_{q(l-1)}\alpha=\alpha$ and $f_{q(l-1)}\beta=\beta$. Thus
$g^{-1}\partial{Z}_{l-1}=\\{\alpha,\beta\\}$. Again by the definition of $q$,
$X_{q(l+1)}=S\backslash\\{\alpha,\beta\\}$, so
$g^{-1}\partial{Z}_{l+1}=\\{\alpha,f_{q(l)}^{e_{l}}\beta\\}$. Now
$\beta\pitchfork S\backslash\alpha$, so by Proposition 7.1, for
$|e_{l}|>E_{1}$,
$d_{S\backslash\alpha}(\beta,f_{q(l)}^{e_{l}}\beta)\geq\tau_{1}|e_{l}|>3$. So
$\beta$ and $f_{q(l)}^{e_{l}}\beta$ fill $S\backslash\alpha$. Recall that if
two curves have distance at least 3 in the curve complex of a surface then
their union fills the subsurface (see $\S$ 2). Thus $g^{-1}\partial{Z}_{l-1}$
and $g^{-1}\partial{Z}_{l+1}$ fill $g^{-1}Z_{l}$, and consequently
$\partial{Z}_{l-1}$ and $\partial{Z}_{l+1}$ fill $Z_{l}$.
By the definition of $q$ and the subsurfaces $Z_{l}$ we have
$\partial{Z}_{l}=\partial{Z}_{l-1}\cap\partial{Z}_{l+1}$. If
$\partial{W}\subseteq\partial{Z}_{l}$ then since $Z_{l}$ has one boundary
curve, $W=Z_{l}$. This is excluded by the assumption that $W$ is not a
subsurface in the list of $Z_{l}$’s. As we saw in the previous paragraph
$\partial{Z}_{l-1}$ and $\partial{Z}_{l+1}$ fill $Z_{l}$, so either
$W\pitchfork Z_{l-1}$ or $W\pitchfork Z_{l+1}$. Since $|e_{l}|>E_{1}$ ($E_{1}$
is the constant from Proposition 7.11 (i)) by Lemma 7.6 for each $l$,
$d_{Z_{l}}(\mu_{I},\mu_{T})>4M$. Also $d_{W}(\mu_{I},\mu_{T})>4M$. So by
Proposition 2.15 $W$ is ordered with respect to either $Z_{l-1}$ or $Z_{l+1}$.
Similarly since $\partial{Z}_{l+1}$ and $\partial{Z}_{l+3}$ fill $Z_{l+2}$,
$W$ is ordered with respect either $Z_{l+1}$ or $Z_{l+3}$. If the subsurfaces
$W$ is not ordered with respect to two subsurfaces with different indices
among $l-1,l+1$ and $l+3$, then $W$ is ordered only with respect to $Z_{l+1}$.
Now since $\partial{Z}_{l+3}$ and $\partial{Z}_{l+5}$ fill the subsurface
$Z_{l+4}$ a similar argument shows that $W$ is ordered with respect to either
$Z_{l+3}$ or $Z_{l+5}$. If the subsurface is not ordered with respect to two
subsurfaces with different indices among $l+1,l+3$ and $l+5$ then it is
ordered only with respect to $Z_{l+3}$. But then it would be ordered with
respect to $Z_{l+1}$ and $Z_{l+3}$. Thus we conclude that $W$ is ordered with
respect to either both $Z_{l-1}$ and $Z_{l+1}$, $Z_{l+1}$ and $Z_{l+3}$,
$Z_{l-1}$ and $Z_{l+3}$, $Z_{l+1}$ and $Z_{l+3}$, $Z_{l+1}$ and $Z_{l+5}$ or
$Z_{l+3}$ and $Z_{l+5}$.
Repeating this argument for every $l$ with $q(l)=1$ or $3$ we conclude that
$W$ is ordered in the list of $Z_{i}$’s as one of the following cases.
1. (1)
There is an index $i$ such that either $Z_{i-1}<W<Z_{i+1}$ or
$Z_{i-2}<W<Z_{i+2}$,
2. (2)
$W<Z_{i}$, where $i=1$ or $2$,
3. (3)
$Z_{i}<W$, where $i=k-1$ or $k-2$.
We proceed to establish the upper bound on $d_{W}(\mu_{I},\mu_{T})$ in Case
(1). First suppose that $Z_{i-2}<W<Z_{i+2}$.
Since $Z_{i-2}<W$, by Proposition 2.15, $d_{W}(\partial{Z_{i-2}},\mu_{I})\leq
M$. Similarly, since $W<Z_{i+2}$, $d_{W}(\partial{Z_{i+2}},\mu_{T})\leq M$.
Then by the triangle inequality and these two bounds we have
$\displaystyle d_{W}(\mu_{I},\mu_{T})$ $\displaystyle\leq$ $\displaystyle
d_{W}(\mu_{I},\partial{Z}_{i-2})+d_{W}(\partial{Z_{i-2}},\partial{Z_{i+2}})+d_{W}(\partial{Z}_{i+2},\mu_{T})+2$
$\displaystyle\leq$ $\displaystyle
d_{W}(\partial{Z_{i-2}},\partial{Z_{i+2}})+2M+2.$
Let $g=f_{q(1)}^{e_{1}}....f_{q(i-1)}^{e_{i-1}}$. Applying $g^{-1}$ to the
last subsurface coefficient above we get
$d_{g^{-1}W}(f_{q(i-1)}^{-e_{i-1}}f_{q(i-2)}^{-e_{i-2}}\partial{X_{q(i-2)}},f_{q(i)}^{e_{i}}f_{q(i+1)}^{e_{i+1}}\partial{X_{q(i+2)}})$
Recall that $\mu_{I}\supset\\{\partial{X_{a}}\\}_{a=0,1,2,3}$. Denote
$\mu_{I}$ by $\mu$. Since $\operatorname{diam}_{g^{-1}W}(\mu)=2$ the above
subsurface coefficient is bounded above by
$d_{g^{-1}W}(f_{q(i-1)}^{-e_{i-1}}f_{q(i-2)}^{-e_{i-2}}\mu,f_{q(i)}^{e_{i}}f_{q(i+1)}^{e_{i+1}}\mu)+4.$
By the triangle inequality this subsurface coefficient is bounded above by
(7.12) $\displaystyle
d_{g^{-1}W}(f_{q(i-1)}^{-e_{i-1}}f_{q(i-2)}^{-e_{i-2}}\mu,f_{q(i-2)}^{-e_{i-2}}\mu)$
$\displaystyle+$ $\displaystyle
d_{g^{-1}W}(f_{q(i-2)}^{-e_{i-2}}\mu,\mu)+d_{g^{-1}W}(\mu,f_{q(i)}^{e_{i}}\mu)$
$\displaystyle+$ $\displaystyle
d_{g^{-1}W}(f_{q(i)}^{e_{i}}\mu,f_{q(i)}^{e_{i}}f_{q(i+1)}^{e_{i+1}}\mu)+6.$
Note that we replaced $\partial{X}_{a}$ by the marking $\mu$ because then the
markings in the above sum overlap $g^{-1}W$ and each subsurface coefficient
makes sense.
###### Claim 7.10.
For any $a\in\\{0,1,2,3\\}$, $g^{-1}W\neq X_{a}$ where
$g=f_{q(1)}^{e_{1}}...f_{q(i-1)}^{e_{i-1}}$.
First assume that $X_{q(i)}=S\backslash\alpha$. If $g^{-1}W=X_{a}$ for some
$a\in\\{0,1,2,3\\}$ we get contradiction as follows:
* •
$g^{-1}W=S\backslash\alpha$. Since $X_{q(i)}=S\backslash\alpha$,
$W=gX_{q(i)}=Z_{i}$. This contradicts the fact that $W$ is not in the list of
$Z_{i}$’s.
* •
$g^{-1}W=S\backslash\beta$. Since $X_{q(i)}=S\backslash\alpha$ by the
definition of $q$, $X_{q(i-1)}=S\backslash\\{\alpha,\beta\\}$ and
$X_{q(i-2)}=S\backslash\beta$. We have
$\partial{Z}_{i-2}=gf_{q(i-1)}^{-e_{i-1}}f_{q(i-2)}^{-e_{i-2}}\beta$. Now
$f_{q(i-1)}$ preserves each component of
$\partial{X}_{q(i-1)}=\\{\alpha,\beta\\}$ (because is supported on
$X_{q(i-1)}$) and $f_{q(i-2)}$ preserves $\partial{X}_{q(i-2)}=\beta$. So
$\partial{Z}_{i-2}=g\beta=\partial{W}$. Thus $W=Z_{i-2}$. This contradicts the
fact that $W$ is not in the list of $Z_{i}$’s.
* •
$g^{-1}W=S\backslash\\{\alpha,\beta\\}$. By the definition of $q$,
$X_{q(i+1)}=S\backslash\\{\alpha,\beta\\}$, so
$\partial{Z}_{i+1}=\\{gf_{q(i+1)}^{e_{i+1}}\alpha,gf_{q(i+1)}^{e_{i+1}}\beta\\}$.
Moreover, $f_{q(i+1)}$ preserves each component of
$\partial{X}_{q(i+1)}=\\{\alpha,\beta\\}$, so
$\partial{Z}_{i+1}=\\{g\alpha,g\beta\\}=\partial{W}$. Thus $W=Z_{i+1}$, which
contradicts the fact that $W$ is not in the list of $Z_{i}$’s.
If we assume that $X_{q(i)}=S\backslash\beta$, then $g^{-1}W=X_{a}$ for any
$a\in\\{0,1,2,3\\}$ gives a contradiction as above. Finally we assume that
$X_{q(i)}=S\backslash\\{\alpha,\beta\\}$. If $g^{-1}W=X_{a}$ for some
$a\in\\{0,1,2,3\\}$ we get contradiction as follows:
* •
$g^{-1}W=S\backslash\\{\alpha,\beta\\}$. Since
$X_{q(i)}=S\backslash\\{\alpha,\beta\\}$, $W=Z_{i}$. This contradicts the fact
that $W$ is not in the list of $Z_{i}$’s.
* •
$g^{-1}W=S\backslash\alpha$. Since $X_{q(i)}=S\backslash\\{\alpha,\beta\\}$ by
the definition of $q$, either $X_{q(i+1)}=S\backslash\alpha$ or
$X_{q(i-1)}=S\backslash\alpha$.
First assume that $X_{q(i+1)}=S\backslash\alpha$. By the definition of $q$,
$\partial{X}_{q(i+2)}=\\{\alpha,\beta\\}$. We have
$\partial{Z}_{i+2}=gf_{q(i)}^{e_{i}}f_{q(i+1)}^{e_{i+1}}\partial{X_{q(i+2)}}$.
$f_{q(i)}$ preserves each component of $\partial{X}_{q(i)}=\\{\alpha,\beta\\}$
and $f_{q(i+1)}$ preserves $\partial{X}_{q(i+1)}=\alpha$, so we get
$\partial{Z}_{i+2}=\\{g\alpha,gf_{q(i+1)}^{e_{i+1}}\beta\\}$. Now
$\partial{W}=g\alpha$, therefore $Z_{i+2}\subsetneq W$. But this contradict
the fact that $W\pitchfork Z_{i+2}$.
Now assume that $X_{q(i-1)}=S\backslash\alpha$.
$\partial{Z_{i-2}}=gf_{q(i-1)}^{-e_{i-1}}f_{q(i-2)}^{-e_{i-2}}\partial{X}_{q(i-2)}$.
$f_{q(i-1)}$ preserves $\partial{X}_{q(i-1)}=\alpha$ and $f_{q(i-2)}$
preserves $\partial{X}_{q(i-2)}=\\{\alpha,\beta\\}$, so we get
$\partial{Z_{i-2}}=\\{g\alpha,gf_{q(i-1)}^{e_{i-1}}\beta\\}$. Now
$\partial{W}=g\alpha$, so $Z_{i-2}\subsetneq W$. But this contradicts the fact
that $W\pitchfork Z_{i-2}$.
* •
$g^{-1}W=S\backslash\beta$. We may get a contradiction similar to the previous
bullet.
We proceed to bound the terms of (7.12). For this purpose for partial pseudo-
Anosov maps $f_{a}$, $a=0,1,2,3$, let $\Psi^{\pm}_{a}=\Psi^{\pm}_{f_{a}}$ and
$\Delta^{\pm}_{a}=\Delta^{\pm}_{f_{a}}$ be the subsets of $\mathcal{PML}(S)$
defined at the beginning of $\S$7. We fix $U_{a}^{\pm}$ neighborhoods of
$\Delta_{a}^{\pm}$ in $\mathcal{PML}(S)$, $a=0,1,2,3$, such that
$\overline{U_{a}^{-}}$ the closure of $U_{a}^{-}$ and $\overline{U_{a}^{+}}$
the closure of $U_{a}^{+}$ are disjoint from $\Psi_{b}^{-}\cup\Psi_{b}^{+}$
for any $a,b\in\\{0,1,2,3\\}$ with $X_{a}\neq X_{b}$. Since $\mathcal{PML}(S)$
is compact (see [CB88]) each set $\overline{U^{\pm}_{a}}$, $a=0,1,2,3$, is
compact. Applying Theorem 7.3 to the pseudo-Anosov map $f_{a}$ and compact
sets $\pi_{X_{b}}(\mu)$, $\overline{U_{b}^{-}}$ and $\overline{U_{b}^{+}}$,
for any $a,b\in\\{0,1,2,3\\}$ with $X_{a}\neq X_{b}$, there exists
$E_{2}>E_{1}$ ($E_{1}$ is the constant from Lemma 7.6) such that
* •
$f_{a}^{n}(\pi_{X_{b}}(\mu))\subset U_{b}^{+}$ for all $n\geq E_{2}$ and
$f_{a}^{-n}(\pi_{X_{b}}(\mu))\subset U_{b}^{-}$ for all $n\geq E_{2}$ and
* •
$f_{a}^{n}\overline{U_{a}^{\pm}}\subset U_{b}^{+}$ for all $n\geq E_{2}$ and
$f_{a}^{-n}\overline{U_{a}^{\pm}}\subset U_{a}^{-}$ for all $n\leq E_{2}$.
Bounding the second term of (7.12): Because $q(i-2)\in\\{0,1,2,3\\}$ by Claim
7.10 $f_{q(i-2)}$ is not supported on $g^{-1}W$. Lemma 7.4 applied to the
curves in $\mu$ which overlap $W$ implies that the term is bounded above by
$m_{1}=\max_{a=0,1,2,3}m(f_{a},\mu)$. Note that $m_{1}$ only depends on the
pseudo-Anosov maps $f_{0},f_{1},f_{2}$ and $f_{3}$ and the marking $\mu$.
Similarly, we can show that the third term of (7.12) is bounded by $m_{1}$.
The first term of (7.12): By the choice of $E_{2}$ for $|e_{i-2}|>E_{2}$,
$f_{q(i-2)}^{-e_{i-2}}\mu\subset U_{q(i-2)}^{\pm}$. But
$\overline{U_{q(i-2)}^{-}}$ and $\overline{U_{q(i-2)}^{+}}$ are disjoint from
$\Psi_{q(i-1)}^{-}\cup\Psi_{q(i-1)}^{+}$. Further by Claim 7.10 $f_{q(i-1)}$
is not supported on $g^{-1}W$. Then Lemma 7.4 applied to the compact subsets
$\overline{U_{q(i-2)}^{\pm}}\subset\mathcal{PML}(S)\backslash\Psi_{q(i-1)}^{-}\cup\Psi_{q(i-1)}^{+}$
implies that the term is bounded above by
$m_{2}=\max_{a=0,1,2,3}m(f_{a},\overline{U_{a}^{\pm}})$. Similarly, we may get
the bound $m_{2}$ on the fourth term of (7.12).
Then the sum in (7.12) is bounded above by $4\max\\{m_{1},m_{2}\\}+6$.
Therefore, $d_{W}(\mu_{I},\mu_{T})\leq 4\max\\{m_{1},m_{2}\\}+2M+12$.
We proceed to establish the bound for $d_{W}(\mu_{I},\mu_{T})$ in Case (1)
when $Z_{i-1}<W<Z_{i+1}$. By Proposition 2.15 since $Z_{i-1}<W$,
$d_{W}(\mu_{I},\partial{Z}_{i-1})\leq M$ and since $W<Z_{i+1}$,
$d_{W}(\partial{Z}_{i+1},\mu_{T})\leq M$. Then by the triangle inequality we
obtain
$d_{W}(\mu_{I},\mu_{T})\leq d_{W}(\partial{Z}_{i-1},\partial{Z}_{i+1})+2M+2.$
Let $i\in\\{1,...,k\\}$ and $g=f_{q(1)}^{e_{1}}...f_{q(i-1)}^{e_{i-1}}$.
Applying $g^{-1}$ to the subsurface coefficient on the right hand side we get
$d_{g^{-1}W}(f_{q(i-1)}^{e_{i-1}}\partial{X}_{q(i-1)},f_{q(i)}^{e_{i}}\partial{X}_{q(i+1)}).$
Note that $\mu\supset\\{\partial{X}_{a}\\}_{a=0,1,2,3}$ so the above
subsurface coefficient is bounded above by
$d_{g^{-1}W}(f_{q(i-1)}^{e_{i-1}}\mu,f_{q(i)}^{e_{i}}\mu)+4$. By the triangle
inequality this subsurface coefficient is bounded above by
$d_{g^{-1}W}(f_{q(i-1)}^{e_{i-1}}\mu,\mu)+d_{g^{-1}W}(\mu,f_{q(i)}^{e_{i}}\mu)+4$
By Claim 7.10, $f_{q(i-1)}$ and $f_{q(i)}$ are not supported on $g^{-1}W$.
Then we may bound the two terms above as the second term of (7.12) by $m_{1}$.
Then we get the bound $2m_{1}+4$ for the sum. Therefore,
$d_{W}(\mu_{I},\mu_{T})\leq 2m_{1}+2M+6$.
We are left with cases (2) and (3).
Case (2): First assume that $W<Z_{2}$. $W<Z_{1}$ would be treated similarly.
By Proposition 2.15, $d_{W}(\partial{Z}_{2},\mu_{T})\leq M$. Then by the
triangle inequality and this bound we get
$\displaystyle d_{W}(\mu_{I},\mu_{T})$ $\displaystyle\leq$ $\displaystyle
d_{W}(\mu_{I},\partial{Z_{2}})+M+1$ $\displaystyle=$ $\displaystyle
d_{W}(\mu_{I},f_{1}^{e_{1}}\partial{X_{2}})+M+1$
Since $\mu:=\mu_{I}\supset\\{\partial{X}_{a}\\}_{a=0,1,2,3}$ and
$\operatorname{diam}_{W}(\mu)\leq 2$ the last subsurface coefficient is
bounded by $d_{W}(\mu,f_{1}^{e_{1}}\mu)+2$. Then since $f_{1}$ is not
supported on $W$, Lemma 7.4 gives us
$d_{W}(\mu,f_{1}^{e_{1}}\mu)\leq m_{1}+M+1,$
where $m_{1}=\max_{a=0,1,2,3}m(f_{a},\mu)$.
Therefore we obtain the bound $d_{W}(\mu_{I},\mu_{T})\leq m_{1}+2M+5$.
Case (3) can be treated exactly similar to Case (2) by considering
$q^{\prime}$ defined by $q^{\prime}(i)=q(k-i+1)$ for $i=1,...,k$ and the
sequence $e^{\prime}_{i}=e_{k-i+1}$ for $i=1,...,k$. Then we obtain the upper
bound $d_{W}(\mu_{I},\mu_{T})\leq m_{1}+M+5$.
Establishing the bounds in cases (1), (2) and (3), we may conclude that
$\textbf{m}=4\max\\{m_{1},m_{2}\\}+4M+12$ is the desired bound in (i) for the
subsurface coefficients of non-annular subsurfaces which are not in the list
of $Z_{i}$’s.
We proceed to obtain the bound on annular subsurface coefficients.
Proof of part (ii). If $d_{\gamma}(\mu_{I},\mu_{T})\leq 4M$ we already have
the bound. If not, then $d_{\gamma}(\mu_{I},\mu_{T})>4M$.
We show that if $q(l)=1$ or $3$ (when $\partial{Z}_{l}$ consists of one
curve), then $\partial{Z}_{l-3}$ and $\partial{Z}_{l-5}$ fill $Z_{l}$.
$X_{q(l)}$ is either $S\backslash\alpha$ or $S\backslash\beta$. First suppose
that $X_{q(l)}=S\backslash\alpha$. By the definition of $q$,
$X_{q(l-3)}=S\backslash\\{\alpha,\beta\\}$. Let
$g=f_{q(1)}^{e_{1}}...f_{q(l-4)}^{e_{l-4}}$, then
$g^{-1}\partial{Z}_{l-3}=\\{\alpha,\beta\\}$. Furthermore, by the definition
of $q$, $X_{q(l-5)}=S\backslash\\{\alpha,\beta\\}$, then
$g^{-1}\partial{Z}_{l-5}=\\{f_{q(l-4)}^{-e_{l-4}}f_{q(l-5)}^{-e_{l-5}}\alpha,f_{q(l-4)}^{-e_{l-4}}f_{q(l-5)}^{-e_{l-5}}\beta\\}.$
$f_{q(l-5)}$ preserves $\partial{X}_{q(l-5)}=\\{\alpha,\beta\\}$. So
$g^{-1}\partial{Z}_{l-5}$ contains the curve $f_{q(l-4)}^{-e_{l-4}}\beta$.
We have $\beta\pitchfork S\backslash\alpha$. Furthermore by the definition of
$q$, $f_{q(l-4)}$ is supported on $S\backslash\alpha$. So by Lemma 7.6 and the
choice of $E_{2}>E_{1}$, if $|e_{l}|>E_{2}$, then
$d_{S\backslash\alpha}(\beta,f_{q(l-4)}^{-e_{l-4}}\beta)>4$. This implies that
$\beta$ and $f_{q(l-4)}^{-e_{l-4}}\beta$ fill $S\backslash\alpha$. So
$g^{-1}\partial{Z_{l-3}}$ and $g^{-1}\partial{Z_{l-5}}$ fill $g^{-1}Z_{l}$.
This implies that $\partial{Z_{l-3}}$ and $\partial{Z_{l-5}}$ fill $Z_{l}$. If
$g^{-1}Z_{l}=S\backslash\beta$, a similar argument implies that
$\partial{Z_{l-3}}$ and $\partial{Z_{l-5}}$ fill $Z_{l}$.Thus given $\gamma$
we have
* •
$\gamma\pitchfork\partial{Z_{l-3}}$ or $\gamma\pitchfork\partial{Z_{l-5}}$
Similarly we can prove that $\partial{Z}_{l+3}$ and $\partial{Z}_{l+5}$ fill
$Z_{l}$. Thus given $\gamma$ we have that
* •
$\gamma\pitchfork\partial{Z_{l+3}}$ or $\gamma\pitchfork\partial{Z_{l+5}}$.
Given $l$ with $q(l)=1$ or $3$ since $|e_{l}|>E$ by Lemma 7.6,
$d_{Z_{l}}(\mu_{I},\mu_{T})>4M$. Also $d_{\gamma}(\mu_{I},\mu_{T})>4M$. Then
by the above two bullets Proposition 2.15 implies that $\gamma$ is ordered
with respect to $Z_{j}$ and $Z_{j^{\prime}}$, where $j=l-3$ or $l-5$ and
$j^{\prime}=l+3$ or $l+5$. If $Z_{j}<\gamma<Z_{j^{\prime}}$ we are in Case (1)
below. If $Z_{j^{\prime}}<\gamma$ by transitivity of $<$, $Z_{j}<\gamma$, then
if $Z_{j^{\prime}}<\gamma<Z_{j^{\prime\prime}}$, where
$j^{\prime\prime}=j^{\prime}+3$ or $j^{\prime}+5$, we are again in Case (1).
Otherwise we repeat the comparison until we end up in either Case (1) for some
$j$ or Case (3) below. If $\gamma<Z_{j}$, similarly by repeating the
comparison we will end up either in Case (1) for some $j$ or Case (2).
1. (1)
$Z_{j}<\gamma<Z_{j^{\prime}}$, where $j=i-3$ or $i-5$ and $j^{\prime}=i+3$ or
$i+5$,
2. (2)
$\mu_{I}<\gamma<Z_{i}$, where $i=3$ or $5$,
3. (3)
$Z_{i}<\gamma<\mu_{T}$, where $i=k-3$ or $k-5$.
We proceed by establishing the bound in each of these cases.
Case (1). By Proposition 2.15, $d_{\gamma}(\mu_{I},\partial{Z_{i-3}})\leq M$
and $d_{\gamma}(\mu_{T},\partial{Z}_{i+3})\leq M$. Having these bounds, by the
triangle inequality we get
$d_{\gamma}(\mu_{I},\mu_{T})\leq
d_{\gamma}(\partial{Z}_{i-3},\partial{Z}_{i+3})+2M+2$
So we only need to bound $d_{\gamma}(\partial{Z}_{i-3},\partial{Z}_{i+3})$.
Let $g=f_{q(1)}^{e_{1}}...f_{q(i-1)}^{e_{i-1}}$. Applying $g^{-1}$ to the
subsurface coefficient on the right hand side, we get
(7.13)
$d_{g^{-1}\gamma}(f_{q(i-1)}^{-e_{i-1}}f_{q(i-2)}^{-e_{i-2}}f_{q(i-3)}^{-e_{i-3}}\partial{X_{q(i-3)}},f_{q(i)}^{e_{i}}f_{q(i+1)}^{e_{i+1}}f_{q(i+2)}^{e_{i+2}}\partial{X_{q(i+3)}})$
Recall that $\mu=\mu_{I}\supset\\{\partial{X_{a}}\\}_{a=0,1,2,3}$. Then since
$\operatorname{diam}_{g^{-1}\gamma}(\mu)\leq 2$ the above subsurface
coefficient is bounded above by
$d_{g^{-1}\gamma}(f_{q(i-1)}^{-e_{i-1}}f_{q(i-2)}^{-e_{i-2}}f_{q(i-3)}^{-e_{i-3}}\mu,f_{q(i)}^{e_{i}}f_{q(i+1)}^{e_{i+1}}f_{q(i+2)}^{e_{i+2}}\mu)+4$.
By the triangle inequality this subsurface coefficient is bounded above by
$\displaystyle
d_{g^{-1}\gamma}(f_{q(i-1)}^{-e_{i-1}}f_{q(i-2)}^{-e_{i-2}}f_{q(i-3)}^{-e_{i-3}}\mu,f_{q(i-2)}^{-e_{i-2}}f_{q(i-1)}^{-e_{i-1}}\mu)$
$\displaystyle+$ $\displaystyle
d_{g^{-1}\gamma}(f_{q(i-1)}^{-e_{i-1}}f_{q(i-2)}^{-e_{i-2}}\mu,f_{q(i-2)}^{-e_{i-2}}\mu)+$
$\displaystyle d_{g^{-1}\gamma}(f_{q(i-2)}^{-e_{i-2}}\mu,\mu)$
$\displaystyle+$ $\displaystyle d_{g^{-1}\gamma}(\mu,f_{q(i)}^{e_{i}}\mu)+$
(7.14) $\displaystyle
d_{g^{-1}\gamma}(f_{q(i)}^{e_{i}}\mu,f_{q(i)}^{e_{i}}f_{q(i+1)}^{e_{i+1}}\mu)$
$\displaystyle+$ $\displaystyle
d_{g^{-1}\gamma}(f_{q(i)}^{e_{i}}f_{q(i+1)}^{e_{i+1}}\mu,f_{q(i)}^{e_{i}}...f_{q(i+2)}^{e_{i+2}}\mu)+10$
We proceed to bound the terms of (7.1). Let the subsets
$\Psi_{a}^{\pm},\Delta_{a}^{\pm}$ and open subsets of $\mathcal{PML}(S)$
$U_{a}^{\pm}\supset\Delta_{a}^{\pm}$, $a=0,1,2,3$, be as in the proof of part
(i). The third term: $f_{q(i)}$ is supported on $X_{q(i)}$. If $g^{-1}\gamma$
is a boundary curve of $X_{q(i)}$ then by (7.3) this term is bounded above by
$2$. Otherwise, $A(g^{-1}\gamma)$ is not equal to $X_{q(i)}$, because $X_{a}$,
$a=0,1,2,3$, is a non-annular subsurface. Then applying Lemma 7.4 to the
curves in $\mu$ which overlap $A(g^{-1}\gamma)$, we get the upper bound
$m_{2}$ as was explained in the proof of part (i) where the subsurface was
non-annular. We may bound the second term by $m_{1}$ similarly. The term
fourth: If $g^{-1}\gamma$ is a boundary curve of $X_{q(i+1)}$ then by (7.3)
this term is bounded by $2$. Otherwise, $A(g^{-1}\gamma)$ is not equal to
$X_{q(i+1)}$. Then for $|e_{i}|>E_{2}$, $f_{q(i)}^{e_{i}}\mu\in
U_{q(i)}^{\pm}$, so we may get the bound $m_{2}$ as in part (i).
Thus we obtain the bound $6\max\\{2,m_{1},m_{2}\\}+2M+16$ for the sum (7.13).
In all of the other cases $d_{\gamma}(\mu_{I},\mu_{T})$ is bounded above by a
sum like (7.1) with at most $10$ terms. Then we may bound each term of the sum
as the above paragraph. Since the number of terms of the sum in each case is
at most $10$ (7.13) is bounded above by $10\max\\{2,m_{1},m_{2}\\}+2M+24$.
Case (2). Suppose that $i=3$. By Proposition 2.15,
$d_{\gamma}(\mu_{T},\partial{Z}_{3})\leq M$. Then by the triangle inequality
$d_{\gamma}(\mu_{I},\mu_{T})\leq
d_{\gamma}(\partial{Z}_{3},\mu_{T})+d_{\gamma}(\mu_{I},\partial{Z}_{3})\leq
d_{\gamma}(\mu_{I},\partial{Z}_{3})+M+1.$
Now $\mu\supset\\{\partial{X}_{a}\\}_{a=0,1,2,3}$ and
$\operatorname{diam}_{\gamma}(\mu)\leq 2$ so
$d_{\gamma}(\mu_{I},\partial{Z}_{3})\leq
d_{\gamma}(\mu,f_{q(1)}^{e_{1}}f_{q(2)}^{e_{2}}\mu)+2$. Then by the triangle
inequality we have
$d_{\gamma}(\mu_{I},\partial{Z}_{3})\leq
d_{\gamma}(\mu,f_{q(2)}^{e_{2}}\mu)+d_{\gamma}(f_{q(2)}^{e_{2}}\mu,f_{q(1)}^{e_{1}}f_{q(2)}^{e_{2}}\mu)+2+2$
The first term above can be bounded as the third term of (7.13) by $2$ or
$m_{1}$. The second term can be bounded as the fourth term of (7.13) by $2$ or
$m_{2}$. So $d_{\gamma}(\mu_{I},\mu_{T})\leq 2\max\\{2,m_{1},m_{2}\\}+M+5$.
When $i=5$, with a similar argument we can obtain the bound
$d_{\gamma}(\mu_{I},\mu_{T})\leq 4\max\\{2,m_{1},m_{2}\\}+M+9$.
Case (3). Considering $q^{\prime}(i)=q(k-i+1)$ for $i=1,...,k$ and
$e^{\prime}_{i}=e_{k-i+1}$ for $i=1,...,k$ the argument of Case (2) for
$q^{\prime}$ and $e^{\prime}_{i}$ gives us the bound
$d_{\gamma}(\mu_{I},\mu_{T})\leq 5\max\\{2,m_{1},m_{2}\\}+M+9$.
Establishing the bounds in cases (1), (2) and (3) we conclude that
$\textbf{m}^{\prime}=10\max\\{2,m_{1},m_{2}\\}+4M+24$ is the desired bound in
(ii). ∎
###### Proposition 7.11.
There are $K_{1}\geq 1$ and $C_{1}\geq 0$ and $E>E_{2}$, depending only on the
partial pseudo-Anosov maps $f_{0},f_{1},f_{2}$ and $f_{3}$ and $\mu_{I}$ with
the following properties. Given $q$ and $\\{e_{i}\\}_{i}$ such that
$|e_{i}|>E$ for any $i\in\\{1,...,k\\}$, we have
(7.15) $d_{Z_{i}(q,e)}(\mu_{I}(q,e),\mu_{T}(q,e))\asymp_{K_{1},C_{1}}|e_{i}|$
###### Proof.
By Lemma 7.6, there are $K^{\prime}_{1},C^{\prime}_{1}$ and $E_{1}$, such that
when $|e_{i}|>E_{1}$ for all $i\in\\{1,...,k\\}$, we have
$d_{Z_{i}}(\mu_{I},\mu_{T})\geq K^{\prime}_{1}|e_{i}|-C^{\prime}_{1}$.
Let $\mu=\mu_{I}(q,e)$. Let $i\in\\{1,...,k\\}$ and
$g=f_{q(1)}^{e_{1}}...f_{q(i-1)}^{e_{i-1}}$. Applying $g^{-1}$,
$d_{Z_{i}}(\mu,\mu_{T})=d_{X_{q(i)}}(g^{-1}\mu,f_{q(i)}^{e_{i}}h\mu)$, where
$h=f_{q(i+1)}^{e_{i+1}}...f_{q(k)}^{e_{k}}$. Now by the triangle inequality
$d_{X_{q(i)}}(g^{-1}\mu,f_{q(i)}^{e_{i}}h\mu)\leq
d_{X_{q(i)}}(f_{q(i)}^{e_{i}}h\mu,h\mu)+d_{X_{q(i)}}(h\mu,g^{-1}\mu).$
In Lemma 7.6, we proved that $d_{X_{q(i)}}(h\mu,g^{-1}\mu)\leq
C^{\prime}_{1}$. So here we only need to show that for some $K_{1}\geq 1$,
$d_{X_{q(i)}}(f_{q(i)}^{e_{i}}h\mu,h\mu)\geq K_{1}|e_{i}|$.
Define $q^{\prime}(j)=q(j+i-1)$ for $j=1,...,k-i+1$ and
$e^{\prime}_{j}=e_{j+i-1}$ for $j=1,...,k-i+1$. Then the subsurface $X_{q(i)}$
is not in the list of subsurfaces $Z_{j}(q^{\prime},e^{\prime})$. So by
Proposition 7.9 (i) there are $E_{2}$ and m such that: if $|e_{i}|>E_{2}$ then
$d_{X_{a}}(\mu,h\mu)\leq\textbf{m}$. Note that this bound does not depend on
$e_{i}$.
Let
$\bar{\tau}_{a}:=\bar{\tau}_{f_{a}}=\limsup_{n\to\infty}\frac{d_{X_{a}}(\delta,f^{n}\delta)}{n}$
for every $\delta\in\mathcal{C}(X_{a})$, $a=0,1,2,3$, as in Lemma 7.2. Let
$a\in\\{0,1,2,3\\}$. Given $\delta\in\mathcal{C}(X_{a})$, there is
$N=N(\delta)$ such that if $n\geq N$, then
$d_{X_{a}}(\delta,f^{n}\delta)\leq(\bar{\tau}_{a}+1)n$. Moreover, note that
for $\delta^{\prime}\neq\delta$, $|N(\delta)-N(\delta^{\prime})|$ is bounded
by a constant depending only on the distance of $\delta$ and $\delta^{\prime}$
in $\mathcal{C}(X_{a})$ (see the proof of Lemma 7.2). Thus $N$ is the same for
all $\delta$ in the $\textbf{m}-$neighborhood of
$\pi_{X_{a}}(\mu)\in\mathcal{C}(X_{a})$. Then there exists $E>E_{2}$ (recall
that $E_{2}>E_{1}$) such that for any $n\geq E$,
$d_{X_{a}}(\delta,f_{a}^{n}(\delta))\leq(\bar{\tau}_{a}+1)n$ for every curve
$\delta$ in the $\textbf{m}-$neighborhood of $\pi_{X_{a}}(\mu)$ in
$\mathcal{C}(X_{a})$ and $a=0,1,2,3$. Now since $\pi_{X_{q(i)}}(h\mu)$ is in
the $\textbf{m}-$neighborhood of $\pi_{X_{q(i)}}(\mu)$, if $|e_{i}|>E$, we
have $d_{X_{q(i)}}(h\mu,f_{q(i)}^{e_{i}}h\mu)\leq(\bar{\tau}_{a}+1)|e_{i}|$.
Let $K_{1}=\max\\{\bar{\tau}_{a}+1,\frac{1}{\tau_{a}}:a=0,1,2,3\\}$ then as we
saw above $d_{X_{q(i)}}(\mu,h\mu)\leq K_{1}|e_{i}|$. Further, by Lemma 7.6 (i)
$d_{X_{q(i)}}(\mu,h\mu)\geq
K_{1}^{\prime}|e_{i}|-C_{1}\geq\frac{1}{K_{1}}|e_{i}|-C^{\prime}_{1}$. Thus
for constants $K_{1},C_{1}=C^{\prime}_{1}$ and $E$ the proposition holds.
∎
Lemma 7.6 and propositions 7.9 and 7.11 together prescribe the list of all
subsurface coefficients of a hierarchy path between $\mu_{I}(q,e)$ and
$\mu_{T}(q,e)$ (corresponding to $q$ and $\\{e_{i}\\}_{i}$ ). To generalize
this construction to infinite hierarchy paths, let
$q:\mathbb{N}\to\\{0,1,2,3\\}$ be as the beginning of this subsection and let
$e_{i}$ be a sequence of integers. For each $i\geq 1$ set the subsurface
$Z_{i}(q,e)=f_{q(1)}^{e_{1}}...f_{q(i-1)}^{e_{i-1}}\partial{X}_{q(i)}$. Define
$q^{k}:\\{1,...,k\\}\to\\{0,1,2,3\\}$ by $q^{k}(i)=q(i)$ for $i=1,...,k$, and
define the sequence $e^{k}_{i}=e_{i}$ for $i=1,...,k$. Let
$\mu\equiv\mu_{I}(q^{k},e^{k})$ be a marking containing
$\\{\partial{X}_{a}\\}_{a=0,1,2,3}$ and let $\mu_{k}=\mu_{T}(q^{k},e^{k})$ be
defined as before. For each $k\geq 1$, let $\delta_{k}$ be a curve in the base
of the marking $\mu_{k}$, after possibly passing to a subsequence
$\delta_{k}$’s converge to a lamination $\lambda$ in the Hausdorff topology of
closed subsets of the surface $S$. Note that by Proposition 2.4 $\lambda$
contains the support of any accumulation point of the projective classes
$[\delta_{k}]$ in the $\mathcal{PLM}(S)$ topology. Here each $\delta_{k}$ is
equipped with the transversal measure $i(\delta_{k},.)$.
###### Proposition 7.12.
There are constants $\textbf{m},\textbf{m}^{\prime}>4M$, $C_{1}\geq 0$,
$K_{1}\geq 1$, depending only on $f_{0},f_{1},f_{2}$ and $f_{3}$, and
$\mu_{I}$ with the following properties. Given $q$ and $\\{e_{i}\\}_{i}$ such
that $|e_{i}|\geq E$ for any $i$ in the domain of $q$, we have
1. (i)
For any integer $i$ in the domain of $q$,
$d_{Z_{i}(q,e)}(\mu_{I}(q,e),\mu_{T}(q,e))\asymp_{K_{1},C_{1}}|e_{i}|$.
2. (ii)
For any non-annular subsurface $W$ which is neither $Z_{i}$ for some $i$ nor
$S$ we have $d_{W}(\mu_{I},\mu_{T})\leq\textbf{m}$.
3. (iii)
For any $\gamma\in\mathcal{C}_{0}(S)$ we have
$d_{\gamma}(\mu_{I},\mu_{T})\leq\textbf{m}^{\prime}$.
4. (iv)
Given $i,j\geq 1$, if $j\geq i+2$, then $Z_{i}<Z_{j}$.
5. (v)
Let $i$ be such that $q(i)=1$ or $3$. If $j\in J_{Z_{i}}$ then $j\geq\min
J_{Z_{i-1}}$ and $j\leq\max J_{Z_{i+1}}$.
When the domain of $q$ is $\mathbb{N}$,
6. (vi)
$\mu_{T}(q,e)$ is a minimal filling lamination.
###### Proof.
When the domain of $q$ is $\\{1,...,k\\}$ for some $k\geq 1$ the subsurface
coefficient bounds (ii) and (iii) are already established Proposition 7.9 and
(i) in Propositions 7.11. Further, the order of domains (iv) is established in
Lemma 7.6 (ii) and (v) is proved in Lemma 7.8.
We proceed to establish the bounds when the domain of $q$ is $\mathbb{N}$.
Given $i\geq 1$, let $Z_{i}=f_{q(1)}^{e_{1}}...f_{q(i-1)}^{e_{i-1}}X_{q(i)}$.
By Proposition 7.11 we have that for every $k\geq i$,
$d_{Z_{i}}(\mu,\mu_{k})\asymp_{K_{1},C_{1}}|e_{i}|$. Then the Hausdorff
convergence of $\delta_{k}$’s to $\mu_{T}(q,e)$ ($\delta_{k}$ is a curve in
the base of $\mu_{k}$) implies that for all $k$ sufficiently large
$d_{Z_{i}}(\mu,\mu_{T})\asymp_{1,1}d_{Z_{i}}(\mu,\mu_{k})$. (i) is
established.
Let $W$ be a subsurface which is neither $Z_{i}$ for some $i\geq 1$ nor $S$.
By Proposition 7.9 (i) for any $k\geq 1$ we have that
$d_{W}(\mu,\mu_{k})\leq\textbf{m}$. Then the Hausdorff convergence of
$\delta_{k}$’s to $\mu_{T}(q,e)$ implies that for all $k$ sufficiently large
$d_{W}(\mu,\mu_{T})\asymp_{1,1}d_{W}(\mu,\mu_{k})$. (ii) is established.
Similarly Proposition 7.9 (ii) and the Huasdorff convergence imply that
$d_{\gamma}(\mu,\mu_{k})\leq\textbf{m}^{\prime}$, (iii) is established.
Given $k\geq 1$ and $i,j\leq k$ by Lemma 7.6 (ii) the order of the subsurfaces
$Z_{i}$ and $Z_{j}$ with respect to $\mu$ and $\mu_{k}$ is: $Z_{i}<Z_{j}$ if
$j\geq i+2$. By Proposition 2.15 this means that $Z_{i}\pitchfork Z_{j}$ and
the subsurface coefficient bounds $d_{Z_{i}}(\mu,\mu_{k})>4M$,
$d_{Z_{j}}(\mu,\mu_{k})>4M$ and $d_{Z_{i}}(\mu,\partial{Z}_{j})>2M$ hold. Then
the Hausdorff convergence of $\mu_{k}$ to $\mu_{T}$ implies that the first two
subsurface coefficient lower bounds hold for $\mu$ and $\mu_{T}(q,e)$ and (iv)
follows for $\mu$ and $\mu_{T}(q,e)$.
The proof of (v) in Lemma 7.8 uses the fact that $Z_{i}$ subsurface
coefficients are greater than or equal to $2M$, and the way that the
subsurfaces $Z_{i-1}$ and $Z_{i}$, and $Z_{i}$ and $Z_{i+1}$ intersect. By
part (i) the $Z_{i}$ subsurface coefficients of $\mu$ and $\mu_{T}$ are
greater than $2M$. Moreover, their pattern of intersection is the same as the
lemma. Then the proof of Lemma 7.8 goes through and gives us (v) when the
domain of $q$ is $\mathbb{N}$ holds as well.
For each $k\geq 1$ let $\rho_{k}$ be a hierarchy path between $\mu$ and
$\mu_{k}$. Let $\rho$ be a hierarchy path between $\mu$ and $\mu_{T}(q,e)$.
For any $i$ with $q(i)=1$ or $3$, by part (i),
$d_{Z_{i}}(\mu,\mu_{T}(q,e))>M$, so $Z_{i}$ is a component domain of $\rho$
with one boundary curve. Thus $\partial{Z}_{i}$ is a curve on the main
geodesic of $\rho$. Furthermore, $Z_{i}<Z_{j}$ implies that $\partial{Z}_{i}$
is before $\partial{Z}_{j}$ along the main geodesic of $\rho$. To see this,
note that since $\partial{Z}_{i}$ and $\partial{Z}_{j}$ are on the main
geodesic and each consist of one curve, the tight geodesics $g_{Z_{i}}$ and
$g_{Z_{j}}$ are time ordered as is defined in $\S 4$ of [MM00]. Then if
$\partial{Z}_{i}$ is after $\partial{Z}_{j}$ there would be an $m\in
J_{Z_{i}}$ so that $m>\max J_{Z_{j}}$ (see $\S 5$ of [MM00]), but this
contradicts the second bullet of Proposition 2.15. Therefore, the curves
$\partial{Z}_{i}$ converge to a point $\xi$ in the Gromov boundary of
$\mathcal{C}(S)$ as $i\to\infty$. This point by Theorem 2.2 determines a
projective measured lamination $[\mathcal{E}]$ with minimal filling support.
Furthermore, given $k\geq 1$, for each $1\leq i\leq k$,
$d_{Z_{i}}(\mu,\mu^{k})>M$, so $\partial{Z}_{i}$ is on the main geodesic of
$\rho_{k}$ as well. This implies that as $k\to\infty$,
$\delta_{k}\in\operatorname{base}(\mu_{k})$ converge to $\xi$ in the Gromov
boundary of $\mathcal{C}(S)$. By Theorem 2.3 the projective classes
$[\delta_{k}]$ after possibly passing to a subsequence converge to a
lamination with support equivalent to the support of $\mathcal{E}$. Then by
Proposition 2.4 . is contained in $\mu_{T}(q,e)$ which implies that
$\mu_{T}(q,e)$ is minimal filling.
∎
### 7.2. Scheme II
The construction of this subsection will be used in $\S$8.4 to provide
examples of recurrent WP geodesic rays.
Let $\alpha$ be a curve such that $S\backslash\alpha$ is a large subsurface.
Consider the indexed subsurfaces $X_{0}=S$ and $X_{1}=S\backslash\alpha$. Let
$f_{0},f_{1}$ be partial pseudo-Anosov maps supported on $X_{0}$ and $X_{1}$,
respectively. Define functions $q_{0}(i)\equiv i$ (mod 2) and
$q_{1}(i)=q_{0}(i+1)$. Let $q$ be any of $q_{0}$ and $q_{1}$ or their
restriction to $\\{1,...,k\\}$, where $k$ is a positive integer.
Let $q$ be as above. When the domain of $q$ is $\mathbb{N}$ let
$\\{e_{i}\\}_{i}$ be an infinite sequence of integers and when the domain of
$q$ is $\\{1,...,k\\}$ let $\\{e_{i}\\}_{i}$ be a sequence of integers with
$k$ elements.
For any $i$ in the domain of $q$ set the subsurface
$Z_{i}(q,e)=f_{q(1)}^{e_{1}}...f_{q(i-1)}^{e_{i-1}}X_{q(i)}$
Let $\mu_{I}(q,e)$ be a marking whose base contains
$\\{\partial{X}_{a}\\}_{a=0,1}=\\{\alpha\\}$. When the domain of $q$ is
$\\{1,...,k\\}$ for some $k\geq 1$ let
$\mu_{T}(q,e)=f_{q(1)}^{e_{1}}...f_{q(k)}^{e_{k}}\mu_{I}(q,e)$. When the
domain of $q$ is $\mathbb{N}$ define $q^{k}(i)=q(i)$ for $i=1,..,k$ and the
sequence $e^{k}_{i}=e_{i}$ for $i=1,...,k$. Let $\delta_{k}$ be a curve in the
base of $\mu_{T}(q^{k},e^{k})$, after possibly passing to a subsequence
$\delta_{k}$ converge to a lamination $\mu_{T}(q,e)$ in the a Hausdorff
topology of closed subsets of the surface.
In the following proposition we establish several bounds on the subsurface
coefficients of $\mu_{I}(q,e)$ and $\mu_{T}(q,e)$.
###### Remark 7.13.
The construction of this subsection and the estimates on subsurface
coefficients can be carried out in a more general setting. Here we restrict
ourself to be able to provide detailed step by step estimates and complete
arguments.
###### Proposition 7.14.
There are constants $\textbf{m},\textbf{m}^{\prime}>4M$, $C_{1}\geq 0$ and
$K_{1}\geq 1$ depending only on $f_{0},f_{1}$ and $\mu_{I}$ with the following
properties. Given $q$ and $\\{e_{i}\\}_{i}$ such that $|e_{i}|>E$ for any $i$
in the domain of $q$, we have
1. (i)
For any integer $i$ in the domain of $q$ with $q(i)=1$,
$d_{Z_{i}(q,e)}(\mu_{I}(q,e),\mu_{T}(q,e))\asymp_{K_{1},C_{1}}|e_{i}|.$
2. (ii)
Given $i,j\geq 1$, if $i<j$ and $q(i)=q(j)=0$ then $Z_{i}<Z_{j}$. If $j<i$ and
$q(i)=q(j)=0$ then $Z_{j}<Z_{i}$.
3. (iii)
For any non-annular subsurface $W$ which is neither $Z_{i}$ for some $i$ nor
$S$ we have $d_{W}(\mu_{I},\mu_{T})\leq\textbf{m}$.
4. (iv)
For any $\gamma\in\mathcal{C}_{0}(S)$ we have
$d_{\gamma}(\mu_{I},\mu_{T})\leq\textbf{m}^{\prime}$.
When the domain of $q$ is $\mathbb{N}$ we have
5. (v)
$\mu_{T}(q,e)$ is a minimal filling lamination.
###### Proof.
First we prove the statements (i) to (iv) when for some $k\geq 1$,
$q:\\{1,...,k\\}\to\\{0,1\\}$ and $\\{e_{i}\\}_{i=1}^{k}$ is a sequence of
integers with $k$ elements. Most of the details are similar to the ones given
in $\S$ 7.1 (Scheme I) so here we mainly sketch them and explain the necessary
modifications.
We set the constants: $K^{\prime}_{1}=\min_{a=0,1}\\{\tau_{a}\\}$ where the
constant $\tau_{a}:=\tau_{f_{a}}$ is from the Proposition 7.1,
$C^{\prime}_{1}=2(B_{0}+\eta)$, where
$\eta=\max\\{d_{X_{a}}(f_{b}^{e}\mu,\mu):a,b\in\\{0,1\\}\;\text{and }\;a\neq
b\\}$, and $E_{1}=\frac{B_{0}+4+4M+\omega+C_{1}^{\prime}}{K^{\prime}_{1}}$,
where $\omega=\max\\{d_{W}(\mu,\partial{X_{a}}):W\subseteq
S\;\text{and}\;a=0,1\\}$.
Let $l\in\\{1,...,k\\}$ be such that $q(l)=1$. Let
$g=f_{q(1)}^{e_{1}}...f_{q(l-1)}^{e_{l-1}}f_{q(l)}^{e_{l}}$. $f_{q(l)}$ is
supported on $S\backslash\alpha$ and preserves $\alpha$ and by the definition
of $q$, $X_{q(l+2)}=S\backslash\alpha$. Then applying $g^{-1}$ to
$d_{S}(\partial{Z}_{l},\partial{Z}_{l+2})$ we get
$d_{S}(\alpha,f_{q(l+1)}^{e_{l+1}}\alpha)$. $f_{q(l+1)}=f_{0}$ is supported on
$S$, so by Proposition 7.1,
$d_{S}(\alpha,f_{q(l+1)}^{e_{l+1}}\alpha)\geq\tau_{0}|e_{l+1}|>\tau_{0}E_{1}\geq
3$. Therefore,
(7.16) $d_{S}(\partial{Z}_{l},\partial{Z}_{l+2})\geq\tau_{0}|e_{l+1}|.$
Let $i\in\\{1,...,k\\}$ be such that $q(i)=1$. Then the proof of Lemma 7.6 (i)
goes through line by line and gives us
(7.17) $d_{Z_{i}(q,e)}(\mu_{I}(q,e),\mu_{T}(q,e))\geq
K^{\prime}_{1}|e_{i}|-C^{\prime}_{1}$
The only difference is that Claim 7.7 which asserts that
$\partial{X}_{q(i)}\pitchfork f_{q(i+1)}^{e_{i+1}}X_{q(i+2)}$ here is proved
as follows: $\partial{X}_{q(i)}=\alpha$ and by the definition of $q$,
$\partial{X}_{q(i+2)}=\alpha$. Then as we saw above
$d_{S}(\alpha,f_{q(i+1)}^{e_{i+1}}(\alpha))\geq 3$ which implies that
$\alpha\pitchfork f_{q(i+1)}^{e_{i+1}}\alpha$.
Then as part (ii) of Lemma 7.6 we may show that the subsurfaces with $q(i)=1$
are ordered. Here we establish the base of induction as follows: By (7.16)
sicne $|e_{2}|>E_{1}$ we have that $d_{S}(\partial{Z}_{1},\partial{Z}_{3})\geq
3$ and thus $\partial{Z}_{1}\pitchfork\partial{Z}_{3}$. Further by (7.17)
$d_{Z_{1}}(\mu_{I},\partial{Z}_{3})>2M$. Then by Proposition 2.15
$Z_{1}<Z_{3}$.
Proof of part (iii). Suppose that $d_{W}(\mu_{I},\mu_{T})>4M$ (otherwise we
already have the bound). Let $l$ be such that $q(l)=1$. Since
$|e_{l+1}|>E_{1}$, it follows from (7.16) that $\partial{Z_{l}}$ and
$\partial{Z_{l+2}}$ fill $S$. Thus $W$ is ordered with respect to either
$Z_{l}$ or $Z_{l+2}$. If $Z_{l}<W<Z_{l+2}$ then we are in case (1) below.
Otherwise, either $Z_{l}<Z_{l+2}<W$ or $W<Z_{l}<Z_{l+2}$. In the former
situation compare $W$ with $Z_{l+2}$ and $Z_{l+4}$ and in the later situation
with $Z_{l}$ and $Z_{l-2}$. Repeating this comparison we may conclude that $W$
is ordered in the list of subsurfaces $Z_{i}$ as one of the following cases.
1. (1)
There is an index $i$ such that either $Z_{i-1}<W<Z_{i+1}$ or
$Z_{i-2}<W<Z_{i+2}$,
2. (2)
$W<Z_{i}$, where $i=1$ or $2$,
3. (3)
$Z_{i}<W$, where $i=k-1$ or $k-2$.
First suppose that $Z_{i-2}<W<Z_{i+2}$. Then
$d_{W}(\mu_{I},\mu_{T})\leq d_{W}(\partial{Z}_{i-2},\partial{Z}_{i+2})+2M.$
Let $g=f_{q(1)}^{e_{1}}...f_{q(i-1)}^{e_{i-1}}$. Applying $g^{-1}$ to the
subsurface on the left hand side we obtain
$d_{g^{-1}W}(f_{q(i-1)}^{-e_{i-1}}f_{q(i-2)}^{-e_{i-2}}\partial{X}_{q(i-2)},f_{q(i)}^{e_{i}}f_{q(i+1)}^{e_{i+1}}\partial{X}_{q(i+2)})$.
Recall that $\mu_{I}\supset\\{\partial{X}_{a}\\}_{a=0,1,2,3}$. Denote
$\mu_{I}$ by $\mu$. Since $\operatorname{diam}_{g^{-1}W}(\mu)\leq 2$,
$d_{g^{-1}W}(f_{q(i-1)}^{-e_{i-1}}f_{q(i-2)}^{-e_{i-2}}\mu,f_{q(i)}^{e_{i}}f_{q(i+1)}^{e_{i+1}}\mu)+4$
bounds the last subsurface coefficient. By the triangle inequality this is
bounded above by
(7.18) $\displaystyle
d_{g^{-1}W}(f_{q(i-1)}^{-e_{i-1}}f_{q(i-2)}^{-e_{i-2}}\mu,f_{q(i-2)}^{-e_{i-2}}\mu)$
$\displaystyle+$ $\displaystyle
d_{g^{-1}W}(f_{q(i-2)}^{-e_{i-2}}\mu,\mu)+d_{g^{-1}W}(\mu,f_{q(i)}^{e_{i}}\mu)$
$\displaystyle+$ $\displaystyle
d_{g^{-1}W}(f_{q(i)}^{e_{i}}\mu,f_{q(i)}^{e_{i}}f_{q(i+1)}^{e_{i+1}}\mu)+6$
###### Claim 7.15.
For any $a\in\\{0,1\\}$, $X_{a}\neq g^{-1}W$, where
$g=f_{q(1)}^{e_{1}}...f_{q(i-1)}^{e_{i-1}}$.
First suppose that $X_{q(i)}=S\backslash\alpha$.
* •
$g^{-1}W=S\backslash\alpha$. Then $W=gX_{q(i)}=Z_{i}$, but this contradicts
the fact that $W$ is not in the list of $Z_{i}$’s.
* •
$g^{-1}W=S$. Then $W=S$, but this contradict the fact that $W$ is a proper
subsurface.
Now suppose that $X_{q(i)}=S$.
* •
$g^{-1}W=S\backslash\alpha$. $f_{q(i+1)}$ is supported on $S\backslash\alpha$
and preserves $\alpha$. Then
$W=gS\backslash\alpha=gf_{q(i-1)}^{-e_{i-1}}S\backslash\alpha=Z_{i-1}$, but
this contradicts the fact that $W$ is not in the list of $Z_{i}$’s.
* •
$g^{-1}W=S$ contradicts the fact that $W$ is a proper subsurface.
These contradictions show that $X_{a}\neq g^{-1}W$. The claim is proved.
Let $\Psi_{a}^{\pm}=\Psi_{f_{a}}^{\pm}$ and
$\Delta_{a}^{\pm}=\Delta_{f_{a}}^{\pm}$, $a=0,1$, be the subsets of
$\mathcal{PML}(S)$ as in the beginning of $\S$7. Fix neighborhoods
$U_{a}^{\pm}\supset\Delta_{a}^{\pm}$ such that $\overline{U_{a}^{-}}$ and
$\overline{U_{a}^{+}}$ be disjoint from $\Psi^{-}_{b}\cup\Psi_{b}^{+}$ for any
$a,b\in\\{0,1\\}$ with $a\neq b$. Applying Lemma 7.4 to the pseudo-Anosov map
$f_{a}$ and compact sets $\pi_{X_{b}}(\mu)$, $\overline{U_{b}^{-}}$ and
$\overline{U_{b}^{+}}$, for any $a,b\in\\{0,1\\}$ with $a\neq b$, there exists
$E_{2}>E_{1}$ such that
* •
$f_{a}^{n}(\pi_{X_{b}}(\mu))\subset U_{b}^{+}$ for all $n\geq E_{2}$ and
$f_{a}^{-n}(\pi_{X_{b}}(\mu))\subset U_{b}^{-}$ for all $n\geq E_{2}$, and
* •
$f_{a}^{n}\overline{U_{a}^{\pm}}\subset U_{b}^{+}$ for all $n\geq E_{2}$ and
$f_{a}^{-n}\overline{U_{a}^{\pm}}\subset U_{a}^{-}$ for all $n\geq E_{2}$.
Now using Claim 7.15 we may proceed as in the proof of part (i) of Proposition
7.9 and bound all of the terms in (7.18) by either
$m_{1}=\max_{a=0,1}m(f_{a},\mu)$ or
$m_{2}=\max_{a=0,1}m(f_{a},\overline{U_{a}^{\pm}})$. Then
$4\max\\{m_{1},m_{2}\\}+2M+16$ is an upper bound for $d_{W}(\mu_{I},\mu_{T})$.
Moreover we may get the bound $m_{1}+M+5$ for $d_{W}(\mu_{I},\mu_{T})$ in
cases (2) and (3) following the argument in Proposition 7.9. Thus
$\textbf{m}=4\max\\{m_{1},m_{2}\\}+4M+16$.
Proof of part (iv). By (7.16) $d_{S}(\partial{Z}_{l-2},\partial{Z}_{l})\geq 3$
so $\partial{Z}_{l}$ and $\partial{Z}_{l-2}$ fill $S$. Similarly
$d_{S}(\partial{Z}_{l+2},\partial{Z}_{l})\geq 3$ and $\partial{Z}_{l}$ and
$\partial{Z}_{l+2}$ fill $S$. Thus
* •
$\gamma\pitchfork\partial{Z}_{l}$ or $\gamma\pitchfork\partial{Z}_{l-2}$, and
* •
$\gamma\pitchfork\partial{Z}_{l}$ or $\gamma\pitchfork\partial{Z}_{l+2}$
Then similar to the proof of part (ii) of Proposition 7.9 we can show that
$\gamma$ is ordered in the list of $Z_{i}$’s as one of the following cases:
1. (1)
There is an index $i$ such that either $Z_{i-1}<W<Z_{i+1}$ or
$Z_{i-2}<W<Z_{i+2}$,
2. (2)
$\mu_{I}<\gamma<Z_{i}$, where $i=1$ or $2$,
3. (3)
$Z_{i}<\gamma<\mu_{T}$, where $i=k-1$ or $k-2$.
Then similar to the proof of part (ii) of Proposition 7.9 we may establish the
upper bound $\textbf{m}^{\prime}=4\max\\{2,m_{1},m_{2}\\}+4M+16$ for
$d_{\gamma}(\mu_{I},\mu_{T})$.
The lower bound (7.17) and part (iii) together as in the proof of Proposition
7.11 give us (i). Here we set $E>E_{2}$ such that for any integer $e$ with
$|e|>E$, $d_{X_{a}}(\delta,f_{a}^{e}(\delta))\leq(\bar{\tau}_{a}+1)|e|$ for
every $\delta$ in the $\textbf{m}-$neighborhood of $\pi_{X_{a}}(\mu)$ in
$\mathcal{C}(X_{a})$ and $a=0,1$. Then for
$K_{1}=\max\\{\frac{1}{\tau_{a}},\bar{\tau}_{a}+1\\}$, $C_{1}=C^{\prime}_{1}$
and $E$ (i) holds.
Establishing all of the bounds when the domain of $q$ is $\\{1,...,k\\}$ for
some $k\geq 1$, the bounds when the domain of $q$ is $\mathbb{N}$ and the fact
that $\mu_{T}$ is minimal filling, part (v), follow from the limiting argument
we gave in the proof of Proposition 7.12. The order of subsurface, part (ii),
also follows from the one when the domain of $q$ is finite we established
above and the limiting argument we gave in the proof of Proposition 7.12.
∎
## 8\. Weil-Petersson Geodesics
In this section we use the control on length-functions along WP geodesic
segments from $\S$6 and the pair of laminations/markings with prescribed list
of subsurface coefficients from $\S$7 to provide examples of Weil-Petersson
geodesics with certain behavior in the moduli space. For example divergent
rays in the moduli space and closed geodesics in the thin part of moduli
space. We also provide a recurrence condition for WP geodesics in terms of
ending laminations. Our results in this section can be considered as a kind of
symbolic coding of WP geodesics.
In [PWW10], the authors construct WP geodesics which are dense in the moduli
space when $\xi(S)=1$. Jeff Brock produces examples of divergent WP geodesic
rays with minimal filling ending lamination in any complexity. Both
constructions start with a piecewise geodesic in the Weil-Petersson completion
of Teichmüller space. Then applying high Dehn twists about curves in the
multi-curves which determine the strata that the piecewise geodesic intersects
manage to replace the piecewise geodesic with a piecewise geodesic in the
completion of Teichmüller space with arbitrary small exterior angles. Then
using a kind of shadowing lemma perturb it to a single WP geodesic in the
Teichmüller space. In these examples the relation between the itinerary of the
ray the end invariants and their associated subsurface coefficients is not
explicit.
These constructions are analogue of the ones for Teichmüller geodesics. Cheung
and Masur in [CM06] give examples of divergent Teichmüller geodesic rays with
uniquely ergodic vertical lamination. Rafi using the control on length
functions in terms of subsurface coefficients he developed along Teichmüller
geodesic constructs closed Teichmüller geodesics staying in the thin part and
divergent geodesic rays.
### 8.1. Weil-Petersson geodesic rays with prescribed itinerary
In this subsection by extracting limits of WP geodesic segments with end
invariants on a single infinite hierarchy path with narrow end points we
construct WP geodesics whose behavior mimic the combinatorial properties of
hierarchy paths.
Let $\nu\in\mathcal{GL}(S)$. Let $Z_{a}$, $a=1,...,m$, be the connected
components of $S\backslash\\{\text{closed leaves of}\;\nu\\}$ with
$\xi(Z_{a})\geq 1$ such that $\nu_{a}$ the restriction of
$\nu\backslash\\{\text{closed leaves of}\;\nu\\}$ to $Z_{a}$ is minimal
filling on $Z_{a}$. For $a=1,...,m$ fix measures $\mathcal{L}_{a}$ supported
on $\nu_{a}$. By Theorem 2.2 the projective class of the measure
$\mathcal{L}_{a}$ supported on $\nu_{a}$ is a point in the Gromov boundary of
$\mathcal{C}(Z_{a})$. Let $\gamma_{n,a}\in\mathcal{C}(Z_{a})$ be a sequence of
curves which converges to $\mathcal{L}_{a}$ as $n\to\infty$ in the weak∗
topology of $\mathcal{ML}(Z_{a})$. For $n\geq 1$ let $Q_{n}$ be a pants
decomposition containing $\bigcup_{a=1}^{m}\partial{Z}_{a}\cup\gamma_{n,a}$.
Let $[x,c_{n}]$ be the WP geodesic segment connecting a base point $x$ in the
interior of Teichmüller space to $c_{n}$, a maximally nodal hyperbolic surface
at $Q_{n}$. Denote the parametrization of $[x,c_{n}]$ by arc-length by
$r_{n}$. The proof of the following lemma essentially follows the proof of
surjectivity of weighted ending laminations of WP geodesic rays in the
Teichmüller space of surface with complexity $5$ given in $\S 4$ of [BM08].
###### Lemma 8.1.
(Infinite ray) After possibly passing to a subsequence the geodesic segments
$r_{n}$ converge to an infinite ray $r$ in the Weil-Petersson visual sphere at
$x$. Furthermore the forward ending lamination of $r$ contains $\nu_{a}$, for
all $a=1,...,m$, and the length of every curve $\alpha\in\partial{Z}_{a}$,
$a=1,...,m$, is decreasing along $r$.
###### Proof.
By Theorem 3.2 (Non-refraction Theorem) the interior of each one of the WP
geodesic segments $r_{n}$ is inside the Teichmüler space. Moreover, by the
local compactness of the WP metric at $x$ after possibly passing to a
subsequence the initial parts of $r_{n}$’s converge to a geodesic segment
$r_{\infty}$ starting at $x$. Let $r$ be the maximal geodesic ray in
Teichmüller space with initial part $r_{\infty}$. We prove that $r$ is an
infinite ray.
Let $s_{n}=\frac{1}{\ell_{Q_{n}}(x)}$, where $\ell_{Q_{n}}(x)=\max_{\gamma\in
Q_{n}}\ell_{\gamma}(x)$. Then for each integer $n$, and $t$ in the domain of
$r_{n}$ we have that $\ell_{s_{n}Q_{n}}(r_{n}(t))\leq 1$. This follows from
the convexity of length-functions along WP geodesics and the observation that
$\ell_{s_{n}Q_{n}}(x)=1$ and $\ell_{s_{n}Q_{n}}(c_{n})=0$. Now assuming that
$r(t)$ has finite length $T$ we get a contradiction.
After possibly passing to a subsequence $s_{n}Q_{n}$ converge to some
$\mathcal{L}\in\mathcal{ML}(S)$ in the weak∗ topology of $\mathcal{ML}(S)$.
Let $\mathcal{L}_{a}$ be the restriction of $\mathcal{L}$ to $Z_{a}$,
$a=1,...,m$. By the way we chose $Q_{n}$’s, the projective class of
$\pi_{Z_{a}}(Q_{n})$ converge to the projective class of a measure supported
on $\nu_{a}$. Then by Theorem 2.3 the support of $\mathcal{L}_{a}$ is a
measured lamination equivalent to $\nu_{a}$ and in particular fills $Z_{a}$.
Now let $\gamma$ be a curve with $i(\gamma,\mathcal{L})=0$. Then it is
disjoint from $\mathcal{L}_{a}$’s and since for each $a$, $\mathcal{L}_{a}$
fills $Z_{a}$, $\gamma$ is disjoint from all $Z_{a}$’s.
The length-function
$\ell_{\mathcal{L}}(x):\operatorname{Teich}(S)\times\mathcal{ML}(S)\to\mathbb{R}^{\geq
0}$ is continuous in both $x$ and $\mathcal{L}$ variables, which implies that
$\ell_{\mathcal{L}}(r(t))\leq 1$ for any $t<T$. Thus
(8.1) $\lim_{t\to T}\ell_{\mathcal{L}}(r(t))\leq 1.$
Let $\sigma$ be the maximal multi-curve such that
$r(T)\in\mathcal{S}(\sigma)$. Then for each simple closed curve
$\gamma\in\sigma$ we have that $i(\gamma,\mathcal{L})=0$. Otherwise, since
$\ell_{\gamma}(r(t))\to 0$ as $t\to T$, we would have that
$\ell_{\mathcal{L}}(r(t))\to\infty$ as $t\to T$, which contradicts the bound
(8.1). This as we saw in the previous paragraph implies that $\sigma$ is
disjoint from all $Z_{a}$’s, and thus
$\sigma\subseteq\bigcup_{a=1}^{m}\partial{Z}_{a}$
###### Claim 8.2.
$d_{\operatorname{WP}}(r(T),c_{n})\to\infty$ as $n\to\infty$.
Let $1\leq a\leq m$. By the choice of $Q_{n}$’s $[Q_{n}]\to[\mathcal{L}_{a}]$
as $n\to\infty$ and $[\mathcal{L}_{a}]$ is in the Grormov boundary of
$\mathcal{C}(Z_{a})$ so $d_{Z_{a}}(Q_{n},Q(r(T)))\to\infty$ as $n\to\infty$.
So by the distance formula (2.2) $d(Q(c_{n}),Q(r(T)))\to\infty$ as
$n\to\infty$. Then by Theorem 3.3 (Quasi-Isometric Model)
$d_{\operatorname{WP}}(r(T),c_{n})\to\infty$ as $n\to\infty$.
###### Claim 8.3.
There is $d>0$ such that
$d_{\operatorname{WP}}(c_{n},\overline{\mathcal{S}(\sigma)})$ for all $n$
sufficiently large.
Let $P_{n}$ be the pants decomposition consisting of the curves in
$\pi_{S\backslash\sigma}(Q_{n})$ and $\sigma$. First we show that
$d(Q_{n},P_{n})$ is uniformly bounded. Let $W\subseteq S$ be a non-annular
subsurface. Either $W\pitchfork S\backslash\sigma$ or $W\pitchfork\sigma$.
Suppose that $W\pitchfork S\backslash\sigma$, then let
$\alpha\in\pi_{S\backslash\sigma}(Q_{n})$ be such that $\alpha\pitchfork W$.
Then since $\alpha$ is in both $P_{n}$ and $Q_{n}$ by the triangle inequality
we have $d_{W}(P_{n},Q_{n})\leq d_{W}(P_{n},\alpha)+d_{W}(\alpha,Q_{n})\leq
2$. If not, then $W\pitchfork\sigma$. As we saw above
$\sigma\subseteq\bigcup_{a=1}^{m}\partial{Z}_{a}$ and by the choice of
$Q_{n}$’s for $n$ sufficiently large $\bigcup_{a=1}^{m}\partial{Z}_{a}\subset
Q_{n}$, so $\sigma\subseteq Q_{n}$. Thus again by the triangle inequality
$d_{W}(P_{n},Q_{n})\leq
d_{W}(P_{n},\sigma)+d_{W}(\sigma,Q_{n})+\operatorname{diam}_{W}(\sigma)\leq
3$. Thus $d_{W}(P_{n},Q_{n})\leq 3$ for every $W\subseteq S$. Let $C$ be the
additive constant corresponding to the threshold constant $3$ in the distance
formula (2.2). Then $d(P_{n},Q_{n})\leq C$ for all $n$ sufficiently large.
Let $x_{P_{n}}\in\operatorname{Teich}(S)$ be a point with a Bers pants
decomposition $P_{n}$. By the bound from the previous paragraph Theorem 3.3
implies that $d_{\operatorname{WP}}(c_{n},x_{P_{n}})\leq
K_{\operatorname{WP}}C+C_{\operatorname{WP}}$. Furthermore, $\sigma\subset
P_{n}$ so by Proposition 3.6,
$d_{\operatorname{WP}}(x_{P_{n}},\mathcal{S}(\sigma))\leq\sqrt{2\pi\sum_{\gamma\in\sigma}\ell_{\gamma}}\leq\sqrt{2\pi\xi(S)L_{S}}$.
Then by the triangle inequality
$d_{\operatorname{WP}}(c_{n},\mathcal{S}(\sigma))\leq d$, where $d$ is the sum
of the two upper bounds.
###### Claim 8.4.
For any $t^{\prime}>T$ the points $r_{n}(t^{\prime})$ converge into
$\overline{\mathcal{S}(\sigma)}$.
Let $c^{\prime}_{n}$ be the nearest point to $c_{n}$ on the $\sigma-$stratum.
Let $\eta_{n}$ be the parametrization of $[r(T),c_{n}]$ by arc-length with
$\eta_{n}(0)=r(T)$ and $\eta^{\prime}_{n}$ be the parametrization of
$[r(T),c^{\prime}_{n}]$ by arc-length with $\eta^{\prime}_{n}(0)=r(T)$. Since
$\mathcal{S}(\sigma)$ is geodesically convex
$\eta^{\prime}_{n}\subset\mathcal{S}(\sigma)$. By Claim 8.2
$d_{\operatorname{WP}}(r(T),c_{n})\to\infty$ as $n\to\infty$. By Claim 8.3,
$d_{\operatorname{WP}}(c_{n},\mathcal{S}(\sigma))\leq d$. Then the
$\operatorname{CAT}(0)$ comparison for the triangles with vertices
$r(T),c_{n}$ and $c^{\prime}_{n}$ implies that given any $s$,
$\eta_{n}(s)\to\eta^{\prime}_{n}(s)$ as $n\to\infty$.
Further $\eta_{n}$ and $[r_{n}(T),c_{n}]$ have the same end point $c_{n}$, and
$r_{n}(T)$ converges to $r(T)$ as $n\to\infty$. Then the
$\operatorname{CAT}(0)$ comparison for the triangle with vertices
$r(T),r_{n}(T)$ and $c_{n}$ implies that the Hausdorff distance between
$\eta_{n}$ and $[r_{n}(T),c_{n}]$ tends zero as $n\to\infty$. This together
with the previous paragraph imply that $r_{n}(t^{\prime})$ converges into the
$\sigma-$stratum.
By Claim 8.4 the points $r_{n}(t^{\prime})$ converge to a point
$y\in\overline{\mathcal{S}(\sigma)}$. By the Non-refraction Theorem the
interior of $[x,y]$ is inside the maximal stratum containing its endpoints
which is the Teichmüller space. By the $\operatorname{CAT}(0)$ comparison the
Hausdorff distance of $r|_{[0,t^{\prime}]}$ and $[x,y]$ goes to $0$ as
$n\to\infty$. So $r_{n}(T)$ converges to a point inside the Teichmüller space.
But this contradicts the assumption that $r_{n}(T)$ converge to a point in the
$\sigma-$stratum.
This finishes the proof of that the geodesic $r$ is an infinite geodesic ray.
We proceed to show that the forward ending lamination of $r$ contains
$\nu_{a}$ for $a=1,...,m$. Fix an $a$. Let $t_{n}\in[0,\infty)$ be a sequence
of times such that $d_{\operatorname{WP}}(r(t_{n}),c_{n})\leq d$. Let
$D=K_{\operatorname{WP}}d+C_{\operatorname{WP}}$, then
$d(Q(r(t_{n})),Q(c_{n}))\leq D$. Let $\alpha_{n}\in Q(r(t_{n}))$ be such that
$\alpha_{n}\pitchfork Z_{a}$, then by (6.6)
$d_{Z_{a}}(\alpha_{n},\gamma_{n})\leq D$ (*). Furthermore, $[\gamma_{n}]$
converge to the projective class of measured lamination $\mathcal{E}$ with
support $\nu_{a}$ in the Gromov boundary of $\mathcal{C}(Z_{a})$. Then the
bound (*) implies that after possibly passing to a subsequence $[\alpha_{n}]$
converges to $[\mathcal{E}]$. This follows for example from the definition of
convergence of sequences in the union of a $\delta-$hyperbolic space and its
boundary using the Gromov inner product, see $\S$III.H.3 of [BH99]. Now
$\alpha_{n}$ is a sequence of distinct Bers curves along $r$ so by the
definition of forward ending lamination, $\nu_{a}$ is contained in the forward
ending lamination of $r$.
To prove the last statement of the lemma fix $1\leq a\leq m$. Let
$\alpha\in\partial{Z}_{a}$. For $n$ sufficiently large
$\partial{Z}_{a}\subseteq Q_{n}$, so $\alpha$ is a pinching curve of $r_{n}$.
Furthermore $r_{n}$’s converge to $r$. Then Lemma 3.14 implies that the length
of $\alpha$ is decreasing along $r$.
∎
Let $(\nu^{-},\nu^{+})$ be a narrow pair, where $Z=S\backslash\\{\text{closed
leaves of $\nu^{+}$}\\}$ is a large subsurface and
$\nu^{\prime}=\nu^{+}\backslash\\{\text{closed leaves of $\nu^{+}$}\\}$ is a
minimal filling lamination on $Z$. Let $\rho$ be any infinite hierarchy path
between $\nu^{-}$ and $\nu^{+}$ as above. Recall that by Theorem 5.5 any two
such resolution paths fellow travel each other in the pants graph. The above
lemma provides us with an infinite ray, denoted the ray by $r_{\nu^{\pm}}$,
with forward ending lamination containing $\nu^{\prime}$. The following
theorem shows that the behavior of $r_{\nu^{\pm}}$ mimics the combinatorial
properties of hierarchy paths encoded in the end invariant listed in Theorem
2.13.
###### Theorem 8.5.
(Infinite ray with prescribed itinerary) Given $A,R,R^{\prime}>0$. Let
$(\nu^{-},\nu^{+})$ be an $A-$narrow pair. Let $\rho$ be a hierarchy path
between $\nu^{-}$ and $\nu^{+}$. Let
$r_{\nu^{\pm}}:[0,\infty)\to\operatorname{Teich}(S)$ be the corresponding
infinite WP geodesic ray.
Let $\bar{\epsilon}=\bar{\epsilon}(A,R)$ be the constant from Lemma 6.6 and
for an $\epsilon\leq\bar{\epsilon}$ let
$\bar{w}=\bar{w}(A,R,R^{\prime},\epsilon)$ be the constant form Theorem 6.1.
Assume that $Z$ a large component domain of $\rho$ has
$(R,R^{\prime})-$bounded combinatorics over an interval
$[m^{\prime},n^{\prime}]\subset J_{Z}$ with $n^{\prime}-m^{\prime}>2\bar{w}$.
Then
1. (1)
$\ell_{\gamma}(r_{\nu^{\pm}}(t))>\bar{\epsilon}$ for every
$\gamma\notin\partial{Z}$, and
2. (2)
$\ell_{\alpha}(r_{\nu^{\pm}}(t))\leq\epsilon$ for every $\alpha\in\partial{Z}$
for every $t\in[a^{\prime},b^{\prime}]$, where $a^{\prime}\in
N(m^{\prime}+\bar{w})$ and $b^{\prime}\in N(n^{\prime}-\bar{w})$. Here
$N:=N_{\rho,g}$ is the parameter map from Proposition 5.25.
Moreover, if $Z_{1}$ and $Z_{2}$ are subsurfaces as above,
$n^{\prime}_{1}<m^{\prime}_{2}$ implies that $b^{\prime}_{1}<a^{\prime}_{2}$.
###### Proof.
By the narrow assumption given any subsurface $Y\subseteq S$ which is not
large, $d_{Y}(\nu^{-},\nu^{+})\leq A$. So for any $n$ Theorem 2.13 (6) (no
backtracking) implies that $d_{Y}(\rho(0),\rho(n))\leq A+2M_{2}$. Thus the end
invariant of the geodesic segment $[x,c_{n}]$ is $A+2M_{2}$ narrow. Since $Z$
has $(R,R^{\prime})-$bounded combinatorics over $[m,n]$, by Lemma 6.6, there
are $\bar{\epsilon}$ and $w$ such that for any $\gamma\notin\partial{Z}$,
$\ell_{\gamma}(r_{n}(t))$ for every $t\in N(j)$ where
$j\in[m^{\prime}+w,n^{\prime}-w]$. Moreover, since $w\leq\bar{w}$ the bounds
hold on the interval $[a^{\prime},b^{\prime}]$. Furthermore, by Theorem 6.1,
for every $\alpha\in\partial{Z}$, $\ell_{\alpha}(r_{n}(t))\leq\epsilon$ for
every $t\in[a^{\prime},b^{\prime}]$. Since the geodesic rays $r_{n}$ converge
to $r_{\nu^{\pm}}$ point-wise, by Theorem 3.1(Continuity of length-functions)
on Teichmüller space, the same bounds on length-functions hold along
$r_{\nu^{\pm}}$. Finally, the statement about order of intervals follows from
fellow traveling. ∎
We will refer to this ray as a ray with prescribed itinerary.
### 8.2. Divergent geodesic rays
In this subsection we construct divergent Weil-Petersson geodesics in the
moduli space. A ray is divergent if eventually leaves every compact subset of
moduli space. The existence of uncountably many divergent rays starting at a
given point with minimal filling ending lamination is a consequence of our
construction in this subsection.
###### Definition 8.6.
A geodesic ray $r:[0,\infty)\to\mathcal{M}(S)$ is recurrent to a compact
subset $\mathcal{K}\subset\mathcal{M}(S)$, if there is a sequence of times
$t_{i}\to\infty$ as $i\to\infty$ such that $r(t_{i})\in\mathcal{K}$. A ray
$r:[0,\infty)\to\mathcal{M}(S)$ is divergent if it is not recurrent to any
compact subset of moduli space. In other words, $r$ is divergent if for every
compact set $\mathcal{K}\subset\mathcal{M}(S)$, there is a $T\geq 0$ such that
$r([T,\infty))$ does not intersect $\mathcal{K}$.
###### Proof of Theorem 1.2.
Let the indexed subsurfaces $X_{0},X_{1},X_{2}$ and $X_{3}$, and partial
pseudo-Anosov maps $f_{0},f_{1},f_{2}$ and $f_{3}$ supported on them
respectively be as in $\S$ 7.1. Recall that $X_{0}$ and $X_{2}$ are the same
subsurface with different indices. Let the function $q=q_{0}$ where
$q_{0}(i)\equiv i$ (mod 4) and the sequence of integers $e_{i}>E$ be as in
Proposition 7.12. For each $i\geq 1$ let
$Z_{i}=f_{q(1)}^{e_{1}}...f_{q(i-1)}^{e_{i-1}}X_{q(i)}.$
Fix a marking $\mu_{I}$ containing $\\{\partial{X}\\}_{a=0,1,2,3}$ and let
$\mu_{T}$ be as in Proposition 7.12. Then by part (i) of the proposition there
are constants $K_{1},C_{1}$ and $E>0$ such that if $|e_{i}|>E$ then
(8.2) $d_{Z_{i}}(\mu_{I},\mu_{T})\asymp_{K_{1},C_{1}}|e_{i}|.$
In particular, by the choice of $E$,
$d_{Z_{i}}(\mu_{I},\mu_{T})>4M.$
By Proposition 7.12 (ii) every subsurface $Z$ with
$d_{Z}(\mu_{I}(q,e),\mu_{T}(q,e))>\textbf{m}$ is in the list of the
subsurfaces $Z_{i}$ and consequently is a large subsurface. So the pair
$(\mu_{I},\mu_{T})$ is $\textbf{m}-$narrow. Let $\rho:[0,\infty]\to P(S)$ be a
hierarchy path between $\mu_{I}$ and $\mu_{T}$. Let
$r:[0,\infty)\to\operatorname{Teich}(S)$ be the WP geodesic ray with end
invariant $(\mu_{I},\mu_{T})$ and prescribed itinerary as in Theorem 8.5. By
Proposition 5.25 we have the parameter map $N$ from the parameters of $\rho$
to the parameters of $r$.
For each $i$ odd, let $k_{i}=\max J_{Z_{i}}\cap J_{Z_{i-1}}$ and $l_{i}=\min
J_{Z_{i}}\cap J_{Z_{i+1}}$ and suppose that $l_{i}>k_{i}$. For each $i$ even,
let $k_{i}=\min J_{Z_{i}}$ and $l_{i}=\max J_{Z_{i}}$. Note that when $i$ is
even $J_{Z_{i}}=[k_{i},l_{i}]$.
First we collect some subsurface coefficient bounds.
$i$ is even. Let $J_{Z_{i}}=[j^{-},j^{+}]$ (when $i$ is even $k_{i}=j_{i}^{-}$
and $l_{i}=j_{i}^{+}$). By Proposition 7.12 (ii) and no backtracking (6.1) for
every subsurface which is not in the list of $Z_{i}$’s we have that
(8.3) $d_{W}(\rho(j^{-}),\rho(j^{+}))\leq\textbf{m}+2M.$
By Proposition 7.12 (iv), for all $j\geq 1$ with $j-i\geq 2$, $Z_{i}<Z_{j}$
and for all $j$ with $i-j\geq 2$, $Z_{j}<Z_{i}$. Suppose that $Z_{i}<Z_{j}$.
Then by Proposition 2.15 $d_{Z_{j}}(\mu_{I},\partial{Z}_{i})\leq M$.
Furthermore $\partial{Z}_{i}\subset\rho(j^{-})$ and
$\partial{Z}_{i}\subset\rho(j^{+})$. So $d_{Z_{j}}(\mu_{I},\rho(j^{-}))\leq
M+\operatorname{diam}_{Z_{j}}(\rho(j^{-}))=M+1$ and
$d_{Z_{j}}(\mu_{I},\rho(j^{+}))\leq M+1$. Then by the triangle inequality
$d_{Z_{j}}(\rho(j^{-}),\rho(j^{+}))\leq
d_{Z_{j}}(\mu_{I},\rho(j^{+}))+d_{Z_{j}}(\mu_{I},\rho(j^{-}))+4\leq 2M+4.$
If $Z_{j}<Z_{i}$, similarly we can get the same bound. We recored these bounds
(8.4) $d_{Z_{j}}(\rho(j^{-}),\rho(j^{+}))\leq 2M+4.$
$i$ is odd. Let $J_{Z_{i}}=[j^{-},j^{+}]$. By Proposition 7.12 (iii) and no
backtracking (6.1) for every subsurface $W$ which is not in the list of
$Z_{i}$’s,
(8.5) $d_{W}(\rho(j^{-}),\rho(j^{+}))\leq\textbf{m}+2M.$
and
(8.6) $d_{W}(\rho(k_{i}),\rho(l_{i}))\leq\textbf{m}+2M.$
By Proposition 7.12 (iv), for all $j\geq 1$ with $j-i\geq 2$, $Z_{i}<Z_{j}$
and for all $j$ with $i-j\geq 2$, $Z_{j}<Z_{i}$. Then similar to the proof of
(8.4) we can get
(8.7) $d_{Z_{j}}(\rho(j^{-}),\rho(j^{+}))\leq 2M+4,$
and
(8.8) $d_{Z_{j}}(\rho(k_{i}),\rho(l_{i}))\leq 2M+4.$
By Theorem 2.13 (6)
$d_{Z_{i-1}}(\rho(k_{i}),\rho(l_{i}))+d_{Z_{i-1}}(\rho(l_{i}),\mu_{T})\leq
d_{Z_{i-1}}(\rho(k_{i}),\mu_{T})+M.$
Now by Theorem 2.13 (4), $d_{Z_{i-1}}(\mu_{T},\rho(k_{i}))\leq M$. Then by the
above inequality
(8.9) $d_{Z_{i-1}}(\rho(k_{i}),\rho(l_{i}))\leq 2M.$
Similarly,
$d_{Z_{i+1}}(\rho(k_{i}),\mu_{I})+d_{Z_{i+1}}(\rho(k_{i}),\rho(l_{i}))\leq
d_{Z_{i+1}}(\rho(l_{i}),\mu_{I})+M,$
then since $d_{Z_{i+1}}(\mu_{I},\rho(l_{i}))\leq M$ we get
(8.10) $d_{Z_{i+1}}(\rho(k_{i}),\rho(l_{i}))\leq 2M.$
Finally, for any $i$ by Proposition 7.12 (iii) and no backtracking for any
$\gamma\in\mathcal{C}_{0}(S)$,
(8.11) $d_{\gamma}(\rho(k_{i}),\rho(l_{i}))\leq\textbf{m}^{\prime}+2M.$
Now we proceed to estimate the length of $J$ intervals (see Theorem 2.13 (1))
using the above subsurface coefficient bounds.
For each $i$ even by the bounds (8.3) and (8.4) all of the subsurface
coefficients of $\rho(j^{-})$ and $\rho(j^{+})$ except that of $Z_{i}$ are
bounded above by $\textbf{m}+2M$. Let the threshold constant in the distance
formula (2.2) be $\textbf{m}+2M$ (note that it is larger that $M_{1}$) and let
$K_{2},C_{2}$ be the constants corresponding to this threshold constant. Then
we have that
$d(\rho(j^{-}),\rho(j^{+}))\asymp_{K_{2},C_{2}}d_{Z_{i}}(\rho(j^{-}),\rho(j^{+})).$
Similarly for each $i$ odd by the bounds (8.5) and (8.7) and the distance
formula we have that
$d(\rho(j^{-}),\rho(j^{+})))\asymp_{K_{2},C_{2}}d_{Z_{i-1}}(\rho(j^{-}),\rho(j^{+}))+d_{Z_{i}}(\rho(j^{-}),\rho(j^{+}))+d_{Z_{i+1}}(\rho(j^{-}),\rho(j^{+})).$
By the no backtracking (6.1) and Theorem 2.13 (4) each subsurface coefficient
on the right hand side is $(1,2M)$ comparable with the corresponding
subsurface coefficient of $\mu_{I}$ and $\mu_{T}$. For example
$d_{Z_{i-1}}(\rho(j^{-}),\rho(j^{+}))\asymp_{1,2M}d_{Z_{i-1}}(\mu_{I},\mu_{T})$.
Moreover $\rho$ is a $(k,c)-$quasi-geodesic where $k,c$ depend only on the
topological type of $S$. Let $K_{3}=kK_{2}$ and $C_{3}=k(C_{2}+2M)+c$. Then we
have:
For each $i$ even
(8.12) $|J_{Z_{i}}|\asymp_{K_{3},C_{3}}d_{Z_{i}}(\mu_{I},\mu_{T})$
and for each $i$ odd,
(8.13)
$|J_{Z_{i}}|\asymp_{K_{3},C_{3}}d_{Z_{i-1}}(\mu_{I},\mu_{T})+d_{Z_{i}}(\mu_{I},\mu_{T})+d_{Z_{i+1}}(\mu_{I},\mu_{T})$
We proceed to set the sequence $\\{e_{i}\\}_{i}$ and using the above estimates
finish the construction of divergent WP geodesic rays.
Let $\epsilon_{j}\to 0$ be a decreasing sequence. Let $R=\textbf{m}+2M$ and
$R^{\prime}=\textbf{m}^{\prime}+2M$. For each $j\geq 1$ let
(8.14) $y_{j}\geq 2\bar{w}(\textbf{m},R,R^{\prime},\epsilon_{j})$
where $\bar{w}$ is the constant from Theorem 8.5.
Let $K=K_{3}K_{1}$ and $C=K_{1}C_{1}+K_{1}K_{2}C_{3}$. Define the sequence
$\\{e_{i}\\}_{i}$ as follows: For $i$ even let $e_{i}=Ky_{\frac{i}{2}}+KC$,
and for $i$ odd let
$e_{i}=(K^{3}y_{\frac{i-1}{2}}+K^{2}C)+(K^{3}y_{\frac{i+1}{2}}+K^{3}C)$.
We investigate the pattern that the $J$ intervals overlap and estimate the
length of each interval $[k_{i},l_{i}]$.
$i$ is even. Recall that when $i$ is an even integer
$[k_{i},l_{i}]=J_{Z_{i}}$. Then (8.12) implies that
$l_{i}-k_{i}\geq\frac{1}{K_{3}}d_{Z_{i}}(\mu_{I},\mu_{T})-C_{3}$. Then by
(8.2) $l_{i}-k_{i}\geq\frac{1}{K_{3}K_{1}}|e_{i}|-\frac{C_{1}}{K_{3}}-C_{3}$.
Therefore, $l_{i}-k_{i}\geq y_{\frac{i}{2}}$.
$i$ is odd. By (8.12) and (8.2) $|J_{Z_{i-1}}|\leq K|e_{i-1}|+C$, so
$|J_{Z_{i}}\cap J_{Z_{i-1}}|\leq K|e_{i-1}|+C$. Similarly, $|J_{Z_{i+1}}|\leq
K|e_{i+1}|+C$ and $|J_{Z_{i}}\cap J_{Z_{i+1}}|\leq K|e_{i+1}|+C$. Furthermore,
by (8.13) and (8.2),
$|J_{Z_{i}}|\geq\frac{1}{K}(|e_{i-1}|+|e_{i}|+|e_{i+1}|)-C$.
By Proposition 7.12 (v) for every $j\in J_{Z_{i}}$, $j\geq\min J_{Z_{i-1}}$,
so $J_{Z_{i-1}}$ and $J_{Z_{i}}$ intersect as in Figure 10. Furthermore, by
the choice of $e_{i}$’s and the bounds above $|J_{Z_{i}}|>|J_{Z_{i}}\cap
J_{Z_{i-1}}|$, thus $l_{i-1}<k_{i}$. Similarly by Proposition 7.12 (v) for
every $j\in J_{Z_{i}}$, $j\geq\max J_{Z_{i+1}}$ so $J_{Z_{i+1}}$ and
$J_{Z_{i}}$ intersect as in Figure 10. Furthermore,
$|J_{Z_{i}}|>|J_{Z_{i}}\cap J_{Z_{i-1}}|$ and $l_{i}<k_{i+1}$.
Finally putting together the bounds on the length of $J_{Z_{i}}$ and its
intersection with the intervals $J_{Z_{i-1}}$ and $J_{Z_{i+1}}$ above, we
obtain $|J_{Z_{i}}-(J_{Z_{i-1}}\cup J_{Z_{i+1}})|=|J_{Z_{i}}|-|J_{Z_{i}}\cap
J_{Z_{i-1}}|-|J_{Z_{i}}\cap J_{Z_{i+1}}|\geq y_{\lfloor\frac{i}{2}\rfloor}$.
Therefore, $l_{i}-k_{i}\geq y_{\lfloor\frac{i}{2}\rfloor}$ and $l_{i}>k_{i}$.
See Figure 10.
Let $R=\textbf{m}+2M$ and $R^{\prime}=\textbf{m}^{\prime}+2M$. When $i$ is
even by the subsurface coefficient bounds (8.3), (8.4) and (8.11) $Z_{i}$ has
$(R,R^{\prime})-$bounded combinatorics over the interval $[k_{i},l_{i}]$. When
$i$ is odd by the bounds (8.6), (8.8), (8.9), (8.10) and (8.11) the subsurface
$Z_{i}$ has $(R,R^{\prime})-$bounded combinatorics over the interval
$J_{Z_{i}}=[k_{i},l_{i}]$.
For each $i\geq 1$ let $t_{i}\in N(j)$ for some
$j\in[k_{i}+\bar{w},l_{i}-\bar{w}]$. For any $i\geq 1$, as we saw above,
$l_{i}<k_{i+1}$ so we have that $t_{i}<t_{i+1}$. Theorem 8.5 applied to the
interval $[k_{i},l_{i}]$ implies that
$\ell_{\partial{Z_{i}}}(r(t_{i}))\leq\epsilon_{\lfloor\frac{i}{2}\rfloor},$
($\ell_{\partial{Z}}(x)=\max\\{\ell_{\alpha}(x):\alpha\in\partial{Z}\\}$).
By the definition of $q$ any two consecutive domains $Z_{i}$ and $Z_{i+1}$
have a boundary curve in common
$\delta_{i}:=\partial{Z_{i}}\cap\partial{Z_{i+1}}$. Then by the convexity of
length-functions we have
$\ell_{\delta_{i}}(r(t))\leq\epsilon_{\lfloor\frac{i}{2}\rfloor}\;\text{for
every}\;t\in[t_{i},t_{i+1}]$
Now since the intervals $[t_{i},t_{i+1}]$, $i\geq 1$, cover $[0,\infty)$, the
domain of $r$, and $\epsilon_{i}\to 0$, the systole of the surfaces along the
WP geodesic ray $r$ decreases and goes to 0.
By the Mumford’s compactness criterion every compact subset of moduli space is
contained in some $\epsilon-$thick part. So $\hat{r}$ the projection of $r$ to
the moduli space is a divergent geodesic ray. Finally, by Proposition 7.12
(vi), $\mu_{T}(q,e)$ is a minimal filling lamination, so the forward ending
lamination of $\hat{r}$ is minimal filling.
Note that there are uncountably sequences $\\{y_{j}\\}_{j\geq 1}$ which
satisfy inequality (8.14). Consequently, we have uncountably many divergent
rays starting from a given point in the moduli space. These geodesics are
distinct because have different forward ending lamination. ∎
Figure 10. The intersection pattern of the intervals $J_{Z_{i-1}}$,
$J_{Z_{i}}$ and $J_{Z_{i+1}}$ when $i$ is even. Over the red subintervals of
$J$ intervals the corresponding domain has bounded combinatorics. $k_{i}=\max
J_{Z_{i-1}}\cap J_{Z_{i}}$ and $l_{i}=\min J_{Z_{i+1}}\cap J_{Z_{i}}$.
### 8.3. Closed geodesics in the thin part
In this subsection we provide examples of closed Weil-Petersson geodesics
which stay in the thin part of moduli space.
###### Proof of Theorem 1.1.
Let the indexed subsurfaces $X_{0},X_{1},X_{2}$ and $X_{3}$, and partial
pseudo-Anosov maps $f_{0},f_{1},f_{2}$ and $f_{3}$ supported on them
respectively be as in $\S$ 7.1. Recall that $X_{0}$ and $X_{2}$ are the same
subsurfaces with different indices. Let the function $q=q_{0}$ and the
sequence of integers $e_{i}>E$ ($i\geq 1$) be as in Proposition 7.12. For each
$i\geq 1$ let
$Z_{i}=f_{q(1)}^{e_{1}}...f_{q(i-1)}^{e_{i-1}}X_{q(i)}.$
Fix a marking $\mu_{I}$ containing $\\{\partial{X}\\}_{a=0,1,2,3}$ and let
$\mu_{T}$ be as in Proposition 7.12.
Then we are in the set up of the proof of Theorem 1.2 in $\S$8.2. As we saw
there the pair $(\mu_{I},\mu_{T})$ is $\textbf{m}-$narrow where m is the
constant form Proposition 7.12. Let $\rho:[0,\infty]\to P(S)$ be a hierarchy
path between $\mu_{I}$ and $\mu_{T}$. Let
$r:[0,\infty)\to\operatorname{Teich}(S)$ be the WP geodesic ray with end
invariant $(\mu_{I},\mu_{T})$ and prescribed itinerary as in Theorem 8.5. Let
$N$ be the parameter map from Proposition 5.25.
As in the proof of Theorem 1.2 for each $i$ odd let $k_{i}=\max
J_{Z_{i-1}}\cap J_{Z_{i}}$ and $l_{i}=\min J_{Z_{i+1}}\cap J_{Z_{i}}$. For
each $i$ even, let $k_{i}=\min J_{Z_{i}}$ and $l_{i}=\max J_{Z_{i}}$.
Then all of the subsurface coefficient bounds (8.3)-(8.11) in the proof of
Theorem 1.2 hold.
Let $R=\textbf{m}+2M$ and $R^{\prime}=\textbf{m}^{\prime}+2M$. Given
$\epsilon>0$ let
(8.15) $y\geq 2\bar{w}(\textbf{m},R,R^{\prime},\epsilon).$
where $\bar{w}$ is the constant from Theorem 8.5.
Let the constants $K\geq 1$ and $C\geq 0$ be as in the proof of Theorem 1.2.
Define the periodic sequence $e_{i}$ with $e_{i}=e_{i^{\prime}}$ whenever
$i\equiv i^{\prime}$ (mod 4) and the first four terms
$e_{1}=2(K^{3}y+K^{3}C),e_{2}=Ky+KC,e_{3}=2(K^{3}y+K^{3}C),e_{4}=Ky+KC$.
The estimates (8.12) and (8.13) on the length of $J_{Z_{i}}$ intervals hold.
Then similar to the proof of Theorem 1.2 we can show that the $J$ intervals
intersect as in Figure 10, $l_{i}<k_{i+1}$ and $l_{i}-k_{i}\geq y$ for all
$i\geq 1$.
When $i$ is even by the subsurface coefficient bounds (8.3), (8.4) and (8.11)
$Z_{i}$ has $(R,R^{\prime})-$bounded combinatorics over the interval
$[k_{i},l_{i}]$. When $i$ is odd by the bounds (8.6), (8.8), (8.9), (8.10) and
(8.11) the subsurface $Z_{i}$ has $(R,R^{\prime})-$bounded combinatorics over
the interval $[k_{i},l_{i}]$.
Let $t_{i}\in N(j)$ where $j\in[k_{i}+\bar{w},l_{i}-\bar{w}]$. Theorem 8.5
applied to the interval $[k_{i},l_{i}]$ implies that
$\ell_{\partial{Z_{i}}}(r(t_{i}))\leq\epsilon$
Furthermore since $l_{i}<k_{i+1}$, $t_{i+1}>t_{i}$.
By the definition of $q$ any two consecutive domains $Z_{i}$ and $Z_{i+1}$
have a boundary curve in common
$\delta_{i}=\partial{Z_{i}}\cap\partial{Z_{i+1}}$. Then by the convexity of
length-functions we have that
$\ell_{\delta_{i}}(r(t))\leq\epsilon\;\text{for every}\;t\in[t_{i},t_{i+1}]$
Now since the intervals $[t_{i},t_{i+1}]$, $i\geq 1$, cover $[0,\infty)$, at
any time the systole of the surface along $r$ is less than $\epsilon$.
Therefore, $r$ stays in the $\epsilon-$thin part of Teichmüller space.
Consequently, $\hat{r}$ the projection of $r$ to the moduli space stays in the
$\epsilon-$thin part of moduli space. Furthermore, since $q$ and
$\\{e_{i}\\}_{i}$ are periodic $\hat{r}$ is a closed geodesic.
By Mumford’s compactness criterion there is $\epsilon_{0}>0$ such that
$\mathcal{K}$, the compact subset of moduli space, is contained in the
$\epsilon_{0}-$thick part of the moduli space and consequently is disjoint
from the $\epsilon_{0}-$thin part of the moduli space. Let $y$ be such that
(8.15) holds for $\epsilon_{0}$. Then our construction produces closed WP
geodesics not intersecting $\mathcal{K}$. There are infinitely many integers
$y$ which satisfy this condition and consequently there are infinitely many
closed geodesics not intersecting $\mathcal{K}$. These geodesics are distinct
because have different forward ending lamination. ∎
### 8.4. A recurrence condition
Given $A,R$ and $R^{\prime}$ positive. Let the constants $w=w(A,R)$ and
$\bar{\epsilon}=\bar{\epsilon}(A,R,R^{\prime})$ be form Proposition 6.7. The
following theorem is a straightforward consequence of the proposition.
###### Theorem 8.7.
(Recurrence condition) Let $(\mu^{-},\mu^{+})$ be an $A-$narrow pair. Let
$\rho$ be a hierarchy path between $\mu^{-}$ and $\mu^{+}$. Let
$[k_{i},l_{i}]$, $i\geq 1$, be a sequence of intervals with $l_{i}-k_{i}\geq
2w$ and $l_{i}<k_{i+1}$. Furthermore suppose that over each interval
$[k_{i},l_{i}]$, $S$ has $(R,R^{\prime})-$bounded combinatorics. Let $r$ be
the ray with prescribed itinerary with end invariant $(\mu^{-},\mu^{+}$). Then
$\operatorname{inj}r(t_{i})\geq\frac{\bar{\epsilon}}{2}$, where $t_{i}\in
N(j)$ for some $j\in[k_{i}+w,l_{i}-w]$.
The WP volume of the moduli space is finite. It follows for example from the
fact that given $\bar{\epsilon}>0$ the WP metric extends to the Delinge-
Mumford compactification of the moduli space, [Mas76]. So the Poincaré
Recurrence Theorem implies that almost every WP geodesic is recurrent to the
$\frac{\bar{\epsilon}}{2}-$thick part of moduli space. But here our
construction of recurrent WP geodesics to the $\bar{\epsilon}-$thick part of
moduli space only uses the combinatorial control we developed in this paper.
###### Theorem 8.8.
There are WP geodesic rays recurrent to the $\bar{\epsilon}-$thick part of
moduli space.
###### Proof.
Let the subsurfaces $X_{0}=S$ and $X_{1}$ and the pseudo-Anosov maps $f_{0}$
and $f_{1}$ supported on $X_{0}$ and $X_{1}$, respectively be as in $\S$ 7.2.
Let the function $q=q_{0}$ and the sequence of $e_{i}>E$ be as in Proposition
7.14. For each $i\geq 1$ let
$Z_{i}=f_{q(1)}^{e_{1}}...f_{q(i-1)}^{e_{i-1}}X_{q(i)}$. Let $\mu_{I}$ be a
marking whose base contains $\\{\partial{X}_{a}\\}_{a=0,1}$ and $\mu_{T}$ be
as in Proposition 7.14.
By Proposition 7.14 (i) we have that
(8.16) $d_{Z_{i}}(\mu_{I},\mu_{T})\asymp_{K_{1},C_{1}}|e_{i}|.$
By Propositions 7.14 (iii) every subsurface $Z$ with
$d_{Z}(\mu_{I}(q,e),\mu_{T}(q,e))>\textbf{m}$ is a large subsurface. So the
pair $(\mu_{I},\mu_{T})$ is ${\bf m}-$narrow.
Let $r$ be the ray with the end invariant $\mu_{I}$ and $\mu_{T}$. Let $N$ be
the parameter map form Proposition 5.25.
For $i$ even let $k_{i}=\max J_{Z_{i-1}}$ and $l_{i}=\min J_{Z_{i+1}}$. For
$i$ odd, let $k_{i}=\min J_{Z_{i}}$ and $l_{i}=\max J_{Z_{i}}$. Note that when
$i$ is odd $J_{Z_{i}}=[k_{i},l_{i}]$.
We have the following subsurface coefficient bounds:
$i$ is odd: Proposition 7.14 (iii) and (6.1) (no backtracking) imply that for
every non-annular subsurface $W$ which is not in the list of $Z_{i}$’s,
(8.17) $d_{W}(\rho(k_{i}),\rho(l_{i}))\leq\textbf{m}+2M.$
By Proposition 7.14 (ii) for every odd integer $j>i$, $Z_{i}<Z_{j}$, also for
every odd integer $j<i$, $Z_{j}<Z_{i}$. Then an argument similar to the one we
gave to prove (8.4) in the proof of Proposition 7.14 we obtain the bound
(8.18) $d_{Z_{j}}(\rho(k_{i}),\rho(l_{i}))\leq 2M+4$
$i$ is even: Proposition 7.14 (iii) and (6.1) (no backtracking) imply that for
every non-annular subsurface $W$ which is not in the list of $Z_{i}$’s,
(8.19) $d_{W}(\rho(k_{i}),\rho(l_{i}))\leq\textbf{m}+2M.$
$k_{i}=\max J_{Z_{i-1}}$ and $l_{i}=\min J_{Z_{i+1}}$, so
$J_{Z_{i-1}}\cap[k_{i},l_{i}]=\emptyset$ and
$J_{Z_{i+1}}\cap[k_{i},l_{i}]=\emptyset$. Then by Theorem 2.13 (4),
(8.20) $\displaystyle d_{Z_{i-1}}(\rho(k_{i}),\rho(l_{i}))\leq
2M,\;\text{and}$ $\displaystyle d_{Z_{i+1}}(\rho(k_{i}),\rho(l_{i}))\leq 2M.$
Further, for any odd $j<i-1$, $Z_{j}<Z_{i}$ and for any odd $j>i+1$,
$Z_{i}<Z_{j}$. Then an argument similar to one for the proof of (8.4) gives us
(8.21) $d_{Z_{j}}(\rho(k_{i}),\rho(l_{i}))\leq 2M+4$
By Propositions 7.14 (iv) and the no backtracking for every annular subsurface
$A(\gamma)$,
(8.22) $d_{\gamma}(\rho(k_{i}),\rho(l_{i}))\leq\textbf{m}^{\prime}+2M.$
We proceed to estimate the length of the intervals $[k_{i},l_{i}]$.
$i$ is odd: By the bounds (8.17) and (8.18) all of the subsurface coefficients
of $\rho(k_{i})$ and $\rho(l_{i})$ except that of $Z_{i}$ are bounded above by
$\textbf{m}+2M$. Let the threshold constant in the distance formula (2.2) be
$\textbf{m}+2M$ and let $K_{2},C_{2}$ be the constants corresponding to this
threshold constant. Then we get
$d(\rho(k_{i}),\rho(l_{i}))\asymp_{K_{2},C_{2}}d_{Z_{i}}(\rho(k_{i}),\rho(l_{i}))$
By the no backtracking
$d_{Z_{i}}(\rho(k_{i}),\rho(l_{i}))\asymp_{1,2M}d_{Z_{i}}(\mu_{I},\mu_{T})$.
Furthermore, $\rho$ is a $(k,c)-$quasi-geodesic where $k$ and $c$ only depend
on the topological type of the surface. Let
$K_{3}=kK_{2},C_{3}=k(C_{2}+2M)+c$. Then
(8.23) $l_{i}-k_{i}\asymp_{K_{3},C_{3}}d_{Z_{i}}(\mu_{I},\mu_{T}).$
$i$ is even: By the bound (7.16) in the proof of Proposition 7.14 we have that
(8.24) $d_{S}(\rho(k_{i}),\rho(l_{i}))\geq\frac{1}{K^{\prime}_{1}}|e_{i}|.$
where as in the proposition $K^{\prime}_{1}=\min\\{\tau_{a}:a=0,1\\}$.
By the bounds (8.19), (8.20) and (8.21) all of the subsurface coefficients of
the pair $\rho(k_{i})$ and $\rho(l_{i})$ are bounded above by $\textbf{m}+2M$.
Let $K_{2},C_{2}$ be the constants corresponding to the threshold constant
$\textbf{m}+2M$ in the distance formula (2.2). Then we have
$d(\rho(k_{i}),\rho(l_{i}))\asymp_{K_{2},C_{2}}d_{S}(\rho(k_{i}),\rho(l_{i})).$
Note that $\rho$ is a $(k,c)-$quasi-geodesic. Let $K^{\prime}_{3}=kK_{2}$ and
$C^{\prime}_{3}=kC_{2}+c$. Then we have
(8.25)
$l_{i}-k_{i}\asymp_{K^{\prime}_{3},C^{\prime}_{3}}d_{S}(\rho(k_{i}),\rho(l_{i})).$
We proceed to set the sequence $\\{e_{i}\\}_{i}$ and using the above estimates
finish the construction of recurrent WP geodesic rays.
Let $R=\textbf{m}+2M$ and $R^{\prime}=\textbf{m}^{\prime}+2M$. For any $i$
even let
$y_{i}\geq 2w(\textbf{m},R)$
Let $K^{\prime}=K^{\prime}_{1}K^{\prime}_{3}$ and
$C^{\prime}=K^{\prime}_{1}K^{\prime}_{3}C^{\prime}_{1}$. Define the sequence
$\\{e_{i}\\}_{i}$ such that for any $i$ even
$e_{i}=K^{\prime}y_{i}+C^{\prime}$. For the next paragraph $e_{i}$ could be
any integer when $i$ is odd.
Then the bounds (8.24) and (8.25) imply that $l_{i}-k_{i}\geq 2w$.
Furthermore, by the bounds (8.19), (8.18) and (8.22) $S$ has
$(R,R^{\prime})-$bounded combinatorics over the interval $[k_{i},l_{i}]$. Let
$t_{i}\in N(j)$ where $j\in[k_{i},l_{i}]$. Applying Theorem 8.7 (Recurrence
condition) the geodesic ray $r$ with end invariant $\mu_{I}$ and $\mu_{T}$ has
injectivity radius at least
$\frac{1}{2}\bar{\epsilon}=\frac{1}{2}\bar{\epsilon}({\bf m},R,R^{\prime})$ at
each $t_{i}$. Consequently, $\hat{r}$ the projection of $r$ to the moduli
space is recurrent the $\frac{\bar{\epsilon}}{2}-$thick part of the moduli
space.
Let $R=\textbf{m}+2M$ and $R^{\prime}=\textbf{m}^{\prime}+2M$, as before. Let
$\epsilon_{j}\to 0$ as $j\to\infty$. For each $i$ odd let
$y_{i}\geq
2\bar{w}(\textbf{m},R,R^{\prime},\epsilon_{\lfloor\frac{i}{2}\rfloor})$
Let $K=K_{1}K_{3}$, $C=K_{1}C_{1}+K_{1}K_{3}C_{3}$. Now in the sequence
$\\{e_{i}\\}_{i}$ we set above let for each $i$ odd, $e_{i}\geq Ky_{i}+C$.
Then the bounds (8.23) and (8.16) imply that $l_{i}-k_{i}>2\bar{w}$.
Furthermore, by the bounds (8.17), (8.20), (8.21) and (8.22), $Z_{i}$ has
$(R,R^{\prime})-$bounded combinatorics over the interval $[k_{i},l_{i}]$. Let
$t_{i}\in N(j)$ where $j\in[k_{i}+\bar{w},l_{i}-\bar{w}]$. Then Theorem 8.5
applied to the interval $[k_{i},l_{i}]$ implies that
$\ell_{\partial{Z}_{i}}(r(t_{i}))\leq\epsilon_{\lfloor\frac{i}{2}\rfloor}$.
Thus since $\epsilon_{j}\to 0$ as $j\to\infty$ the ray $\hat{r}$ is not
contained in any thick part of the moduli space. As a result $\hat{r}$ is
recurrent to the $\frac{\bar{\epsilon}}{2}-$thick part of the moduli space and
is not contained in any compact subset of the moduli space.
∎
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* [BDM09] Jason Behrstock, Cornelia Druţu, and Lee Mosher, _Thick metric spaces, relative hyperbolicity, and quasi-isometric rigidity_ , Math. Ann. 344 (2009), no. 3, 543–595.
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|
arxiv-papers
| 2012-12-01T01:05:41 |
2024-09-04T02:49:38.733457
|
{
"license": "Creative Commons Zero - Public Domain - https://creativecommons.org/publicdomain/zero/1.0/",
"authors": "Babak Modami",
"submitter": "Babak Modami",
"url": "https://arxiv.org/abs/1212.0051"
}
|
1212.0082
|
# Separability criterion for bipartite states and its generalization to
multipartite systems
Jie-Hui Huang1,∗, Li-Yun Hu1, Lei Wang2, and Shi-Yao Zhu3 1College of Physics
and Communication Electronics, Jiangxi Normal University, Nanchang 330022,
People's Republic of China
2College of Physics, Jilin University, Changchun 130021, People's Republic of
China
3Beijing Computational Science Research Center, Beijing 100084, People's
Republic of China
###### Abstract
A group of symmetric operators are introduced to carry out the separability
criterion for bipartite and multipartite quantum states. Every symmetric
operator, represented by a symmetric matrix with only two nonzero elements,
and their arbitrary linear combinations are found to be entanglement
witnesses. By using these symmetric operators, Wootters' separability
criterion for two-qubit states can be generalized to bipartite and
multipartite systems in arbitrary dimensions.
###### pacs:
03.67.Mn, 03.65.Fd
Quantum entanglement, introduced in the early days of quantum theory
EPR&Schrodinger , not only plays a vital role in differentiating quantum
mechanics from classical mechanics, but also acts as key resource in quantum
information processing. Thus the verification of quantum entanglement is
fundamentally important in quantum information science. Since the mathematical
description on quantum entanglement was first introduced by Werner in 1989
werner , many entanglement measures have been proposed, such as distillable
entanglement, entanglement cost distillation&cost , relative entropy of
entanglement entropy , Schmidt number Schmidt , concurrence concurrence0 ;
concurrence , negativity negativity , and so on horodecki . Recently, the
verification of multipartite entanglement and relevant issues are attracting
more and more attention residual ; ME owing to the booming interest on
quantum many-body systems manybody .
The convexity requirement for quantum entanglement, which is in accord with
the fact local operations with classical communication (LOCC) can not create
or increase quantum entanglement distillation&cost , makes it very hard to
verify the entanglement of mixed quantum states. Historically, positive
partial transpose (PPT) criterion, proposed by Peres in 1996 PPTA , provides a
necessary and sufficient condition for verifying the separability of $2\otimes
2$ and $2\otimes 3$ bipartite systems PPTB . But for higher dimensional
systems, how to distinguish entangled states from separable ones is still a
challenging question. In this Letter, we introduce a group of symmetric
operators to answer this question. It is shown that all these symmetric
operators and their arbitrary linear combinations are entanglement witnesses.
We use these symmetric operators to generalize Wootters' concurrence
concurrence0 ; concurrence to high dimensional bipartite generalization ;
Iconcurrence and multipartite systems multiconcurrence . More importantly,
based on these symmetric operators the separability criterion for bipartite
and multipartite states is presented in arbitrary dimensions, which is a
generalization of Wootters' separability criterion from the two-qubit case to
a general case.
In the simplest two-qubit system, a pure state
$|\psi>=\sum_{i=1}^{2}\sum_{j=1}^{2}a_{ij}|i_{1}j_{2}>$, where the subscripts
``$1$'' and ``$2$'' in the state vectors denote the two subsystems, is
separable when it is a tensor product of a pure state in subsystem ``$1$'' and
a pure state in subsystem ``$2$'' , thus the four coefficients
$a_{ij}~{}(i,j\in\\{1,2\\})$ have to satisfy the condition
$a_{11}a_{22}=a_{12}a_{21}$. In Ref.concurrence , Wootters introduced a
``spin-flip'' operator $\sigma_{y}$ (Pauli matrix) in the standard basis to
verify the separability of a two-qubit pure state. That is, a two-qubit pure
state $|\psi>$ is separable iff
$<\psi^{*}|\sigma_{y}\otimes\sigma_{y}|\psi>=0$. In the following, we call the
state vector $|\psi>$ and its complex conjugate $|\psi^{*}>$ orthogonal with
respect to the operator $O$, or simply $O$-orthogonal, if they satisfy the
relation $<\psi^{*}|O|\psi>=0$. Wootters' separability criterion for a two-
qubit pure state can then be stated as, it should be $O$-orthogonal to its
complex conjugate with the operator $O=\sigma_{y}\otimes\sigma_{y}$. Here the
antisymmetry of the ``spin-flip'' operator $\sigma_{y}$, i.e.
$\sigma_{y}^{T}=-\sigma_{y}$ (the superscript ``T'' stands for matrix
transpose hereafter), plays the essential role in verifying the separability
of a two-qubit state. This $O$-orthogonal relation between a separable two-
qubit pure state and its complex conjugate is violated by all entangled two-
qubit states.
Now we consider a general bipartite case, with the two subsystems having
dimensions $D_{1}$ and $D_{2}$, respectively. If a bipartite pure state
$|\psi>=\sum_{i=1}^{D_{1}}\sum_{j=1}^{D_{2}}a_{ij}|i_{1}j_{2}>$ is separable,
the four coefficients $a_{ij}$, $a_{ij^{\prime}}$, $a_{i^{\prime}j}$, and
$a_{i^{\prime}j^{\prime}}$ with $i,i^{\prime}\in\\{1,2,\cdots,D_{1}\\}$ and
$j,j^{\prime}\in\\{1,2,\dots,D_{2}\\}$, have to satisfy the condition
$a_{ij}a_{i^{\prime}j^{\prime}}=a_{ij^{\prime}}a_{i^{\prime}j}$, so that this
bipartite pure state can be formulated as a tensor product. This condition,
similar to the above two-qubit case, is equivalent to the following
$O^{(i,i^{\prime}:j,j^{\prime})}$-orthogonal relation,
$\displaystyle<\psi^{*}|O^{(i,i^{\prime}:j,j^{\prime})}|\psi>=0,$ (1a) with
$\displaystyle
O^{(i,i^{\prime}:j,j^{\prime})}=[\sigma_{1}^{(i,i^{\prime})}-(\sigma_{1}^{(i,i^{\prime})})^{T}]\otimes[\sigma_{2}^{(j,j^{\prime})}-(\sigma_{2}^{(j,j^{\prime})})^{T}].$
(1b)
Here the operator $\sigma_{1}^{(i,i^{\prime})}$
($\sigma_{2}^{(j,j^{\prime})}$) in the subsystem ``$1$'' (``$2$'') has only
one nonzero element ``1'' located at row $i$ ($j$) and column $i^{\prime}$
($j^{\prime}$) in the standard basis, no matter what dimensions it has. The
two antisymmetric (or skew-symmetric) operators,
$[\sigma_{1}^{(i,i^{\prime})}-(\sigma_{1}^{(i,i^{\prime})})^{T}]$ and
$[\sigma_{2}^{(j,j^{\prime})}-(\sigma_{2}^{(j,j^{\prime})})^{T}]$, which play
the same role as the antisymmetric operator $\sigma_{y}$ in the two-qubit
case, establish a relation among the four coefficients $a_{ij}$,
$a_{ij^{\prime}}$, $a_{i^{\prime}j}$, and $a_{i^{\prime}j^{\prime}}$.
Since the superscript numbers $i,i^{\prime}\in\\{1,2,\cdots,D_{1}\\}$ and
$j,j^{\prime}\in\\{1,2,\dots,D_{2}\\}$ in the above Eq.(1) can be chosen
arbitrarily, the $O^{(i,i^{\prime}:j,j^{\prime})}$-orthogonal relation (1a)
for a separable bipartite pure state, holds valid for all possible operators
$O^{(i,i^{\prime}:j,j^{\prime})}$ in the form (1b). Owing to the linear
dependence of the relation (1a) on the operators
$O^{(i,i^{\prime}:j,j^{\prime})}$, we even can replace the operators
$\sigma_{1}^{(i,i^{\prime})}$ and $\sigma_{2}^{(j,j^{\prime})}$ by a random
matrix $\sigma_{1}^{\text{(rdm)}}$ in subsystem ``$1$'' and a random matrix
$\sigma_{2}^{\text{(rdm)}}$ in subsystem ``$2$'' to construct,
$\displaystyle O^{\text{(Semi-
rdm)}}=[\sigma_{1}^{\text{(rdm)}}-(\sigma_{1}^{\text{(rdm)}})^{T}]\otimes[\sigma_{2}^{\text{(rdm)}}-(\sigma_{2}^{\text{(rdm)}})^{T}],$
(2)
and the $O^{\text{(Semi-rdm)}}$-orthogonal relation,
$<\psi^{*}|O^{\text{(Semi-rdm)}}|\psi>=0$, still holds true if only the pure
state $|\psi>$ is separable. Thus we can conclude that a bipartite pure state
in arbitrary dimensions is separable iff it and its complex conjugate are
orthogonal with respect to all symmetric operators formulated as a tensor
product of two antisymmetric operators in the two subsystems. To verify the
separability of a given bipartite pure state, we only need to pick out a
finite number of linearly independent symmetric operators and check the above
$O^{\text{(Semi-rdm)}}$-orthogonal relation. A violation of the
$O^{\text{(Semi-rdm)}}$-orthogonal relation by any operator $O^{\text{(Semi-
rdm)}}$ is enough to declare the existence of quantum entanglement.
Now we use a complete set of linearly independent symmetric operators
$O^{(i,i^{\prime}:j,j^{\prime})}$, which are directly constructed by
$\frac{1}{2}D_{1}(D_{1}-1)$ linearly independent antisymmetric operators
$[\sigma_{1}^{(i,i^{\prime})}-(\sigma_{1}^{(i,i^{\prime})})^{T}]$ in subsystem
``$1$'' and $\frac{1}{2}D_{2}(D_{2}-1)$ linearly independent antisymmetric
operators $[\sigma_{2}^{(j,j^{\prime})}-(\sigma_{2}^{(j,j^{\prime})})^{T}]$ in
subsystem ``$2$'' , to define the entanglement measure of a bipartite pure
state $|\psi>$,
$\displaystyle
C^{(2)}(|\psi>)=\sqrt{\sum_{i=1}^{D_{1}-1}\sum_{i^{\prime}=i+1}^{D_{1}}\sum_{j=1}^{D_{2}-1}\sum_{j^{\prime}=j+1}^{D_{2}}|<\psi^{*}|[\sigma_{1}^{(i,i^{\prime})}-(\sigma_{1}^{(i,i^{\prime})})^{T}]\otimes[\sigma_{2}^{(j,j^{\prime})}-(\sigma_{2}^{(j,j^{\prime})})^{T}]|\psi>|^{2}},$
(3)
where the superscript ``$(2)$'' denotes the bipartite system. This operator-
based entanglement measure for bipartite pure states, which we call _O
concurrence_ in the following, turns back to Wootters' concurrence for two-
qubit systems, and satisfies the basic requirements for a good entanglement
measure. For example, (i) it presents zero result for all tensor product
states, and positive results for entangled states; (ii) it remains invariant
under all local unitary transformations, and so on. In fact, this _O
concurrence_ is the same as measured in the following way,
$\displaystyle
C^{(2)}(|\psi>)=\sqrt{2-\text{tr}(\rho_{1}^{2})-\text{tr}(\rho_{2}^{2})},$ (4)
with $\rho_{1}$ ($\rho_{2}$) being the reduced density matrix of subsystem
``$1$'' (``$2$''), and ``$\text{tr}()$'' denoting matrix trace. Since the
equality $\text{tr}(\rho_{1}^{2})=\text{tr}(\rho_{2}^{2})$ holds for bipartite
pure states, our entanglement measure is equivalent to the entanglement
measure proposed in Ref.Iconcurrence , which is called _I concurrence_ owing
to its connection to the universal inverter. It is very interesting that two
measures present the same result for the bipartite entanglement, though they
are defined in two different ways and based on different ideas.
This _O concurrence_ can also be extended by convex roof to measure the
entanglement of mixed bipartite states. At present, we only concern the
separability condition of mixed bipartite states, rather than the evaluation
of their entanglement. According to Werner's criterion werner , a mixed
bipartite state $\rho$ is separable only when it can be written as a mixture
of separable pure states,
$\displaystyle\rho=\sum_{i,j}P_{ij}|\psi_{1}^{i}><\psi_{1}^{i}|\otimes|\psi_{2}^{j}><\psi_{2}^{j}|,$
(5)
with the probabilities $P_{ij}\geq 0~{}\text{and}~{}\sum_{i,j}P_{ij}=1$.
However, every mixed quantum state has countless types of pure-state
decomposition. Given a density matrix, it is usually very difficult to prove
the existence or nonexistence of a decomposition composed of product states.
In other words, we can easily construct a separable mixed quantum state by
mixing a group of separable states in the above way (5). However, it is very
difficult to recover this group of separable states, provided only the density
matrix.
We begin our investigation on the separability of a mixed $D_{1}\otimes D_{2}$
state with a Hermitian matrix $\sqrt{\rho}$, which is the square root of the
density matrix $\rho$. Every pure-state decomposition of the density matrix
$\rho$ is connected to a $D_{12}\times M$ ($D_{12}=D_{1}\times D_{2}\leq M$)
right-unitary transformation $U$ through the relation, $W=\sqrt{\rho}U$. Each
column vector of the matrix $W$, the $i$th column vector denoted as $|W_{i}>$
hereafter, is a ``subnormalized'' pure state in the $U$-decomposition, and the
original density matrix can then be written as
$\rho=\sum_{i=1}^{M}|W_{i}><W_{i}|$. Now we suppose that the density matrix
$\rho$ represents a separable mixed state, which means there exists at least
one $U$-decomposition so that all pure states in this decomposition are tensor
product states. That is to say, every column vector $|W_{i}>$ is
$O^{\text{(Semi-rdm)}}$-orthogonal to its complex conjugate, with the operator
$O^{\text{(Semi-rdm)}}$ defined in Eq.(2).
Since all column vectors $|W_{i}>$ satisfy the $O^{\text{(Semi-
rdm)}}$-orthogonal relation, $<W_{i}^{*}|O^{\text{(Semi-rdm)}}|W_{i}>=0$, for
a particular $U$-decomposition of the separable mixed state $\rho$, the
symmetric matrix $W^{T}O^{\text{(Semi-rdm)}}W$ is a hollow matrix whose
diagonal elements are all equal to zero. Equivalently speaking, a mixed
bipartite state $\rho$ is separable, iff there exists a right-unitary
transformation $U$, so that the symmetric matrix $U^{T}SU$, with
$\displaystyle S=(\sqrt{\rho})^{T}O^{\text{(Semi-rdm)}}\sqrt{\rho},$ (6)
is a hollow matrix for all operators $O^{\text{(Semi-rdm)}}$ in the form (2).
Here we emphasize the above symmetric matrix $S$ depends only on the given
density matrix and an operator constructed by two random matrices. Now a new
question arises, given a symmetric matrix $S$, whether there exists such a
right-unitary transformation $U$ so that $U^{T}SU$ is a hollow matrix?
We can find out the answer by using Wootters' method in Ref.concurrence . A
symmetric matrix $S$ can be transformed to a hollow matrix in the way of
$U^{T}SU$, only when its maximal singular value is no larger than the sum of
the rest singular values. Supposing $\\{\lambda_{i}\\}$ are singular values of
the symmetric matrix $S$ in decreasing order, the above statement is
equivalent to the following condition,
$\displaystyle\lambda_{1}\leq\sum_{i=2}\lambda_{i}.$ (7)
On the contrary, if the above condition (7) is not satisfied for even one
particular operator in the form (2), e.g.
$O^{\text{(particular)}}=\sum_{i=1}^{D_{1}-1}\sum_{i^{\prime}=i+1}^{D_{1}}\sum_{j=1}^{D_{2}-1}\sum_{j^{\prime}=j+1}^{D_{2}}c_{i,i^{\prime}:j,j^{\prime}}O^{(i,i^{\prime}:j,j^{\prime})}$,
the mixed state $\rho$ must be entangled. Please see Eq.(1b) for the
definition of the operator $O^{(i,i^{\prime}:j,j^{\prime})}$. Without loss of
generality, we assume the first coefficients $c_{12:12}$ has the maximal
modulus among all the complex coefficients
$\\{c_{i,i^{\prime}:j,j^{\prime}}\\}$. In this case, the average _O
concurrence_ of the pure states, i.e. column vectors $\\{|W_{k}>\\}$, in an
arbitrary $U$-decomposition,
$C^{(2)}_{U}(\rho)=\sum_{k=1}^{M}C^{(2)}(|W_{k}>)$, is larger than zero,
because
$C^{(2)}_{U}(\rho)\geq\frac{1}{\sqrt{|c_{12:12}|}}(\lambda_{1}-\sum_{j=2}^{D_{12}}\lambda_{j})$.
So, given a mixed bipartite state $\rho$, if the condition (7) is violated by
even one operator in the form (2), this mixed state is an entangled state. One
of main conclusions in this Letters can now be concluded, a mixed bipartite
state in arbitrary dimensions is separable only when the condition (7) for the
singular values of the symmetric matrix (6) is generally true for _all_
operators $O^{\text{(Semi-rdm)}}$ in the form (2). This is a necessary and
sufficient condition. On the one hand, if the condition (7) is valid for _all_
operators $O^{\text{(Semi-rdm)}}$ in the form (2), there must exist a unitary
transformation $U$ independent on the operator $O^{\text{(Semi-rdm)}}$
(because it is constructed by a random matrix $\sigma_{1}^{\text{(rdm)}}$ in
subsystem ``$1$'' , and a random matrix $\sigma_{2}^{\text{(rdm)}}$ in
subsystem ``$2$'' ), to satisfy all $O^{\text{(Semi-rdm)}}$-orthogonal
relations in the corresponding decomposition, and such a bipartite state is
separable; On the other hand, any violation of this condition, just as we
already shown, is enough to declare the existence of quantum entanglement. In
other words, every operator $O^{\text{(Semi-rdm)}}$ in the form (2) is a
witness witness of bipartite entanglement under the violation of the
condition (7).
The singular values $\\{\lambda_{i}\\}$ of the symmetric matrix $S$ (6) are
equal to the square root of the eigenvalues of the matrix $S^{{\dagger}}S$ SVD
, and as well the square root of the eigenvalues of the following matrix,
$\displaystyle\rho\left[O^{\text{(Semi-
rdm)}}\right]^{{\dagger}}\rho^{*}O^{\text{(Semi-rdm)}},$ (8)
which is very similar to the form used in the concurrence paper concurrence ,
and can be considered as its generalization in high dimensional systems.
At present, we can not verify the separability of a bipartite state in high
dimensions through finite tests on the condition (7), unless it is a pure
state. But we here provided an efficient way for verifying the entanglement of
a bipartite state, because any operator $O^{\text{(Semi-rdm)}}$ in the form
(2) is an entanglement witness under the violation of the condition (7). Our
numerical results show that the larger entanglement a bipartite state
contains, the fewer tests it usually requires to find a violation by randomly
generating the matrices $\sigma_{1}^{\text{(rdm)}}$ and
$\sigma_{2}^{\text{(rdm)}}$ in the form (2).
Now we come to the multipartite case. A multipartite pure state is said to be
entangled only when it cannot be written as a tensor product of the states in
the subsystems. This definition does not distinguish between ``truly
multipartite'' entanglement and low-partite entanglement
truemultientanglecriterion . For example, both the Greenberger-Horne-Zeilinger
state and the $W$ state in the three-qubit system are tripartite entangled
states according to the above definition, but only the former one has nonzero
$3$-tangle for ``truly tripartite'' entanglement residual . Our following
discussion on the separability criterion of multipartite states is based on
this definition of entanglement, no matter what kind of entanglement, ``truly
multipartite'' entanglement or low-partite entanglement, is contained.
For a separable multipartite pure state,
$\displaystyle|\psi>=|\psi_{1}>\otimes|\psi_{2}>\otimes\cdots\otimes|\psi_{N}>,$
(9)
each subsystem is separable with the rest subsystems as a whole. The inverse
situation is also true. Among a multipartite system, if every subsystem is
separable with the rest subsystems as a whole, this multipartite pure state is
separable. That is to say, multipartite entanglement can be featured by
bipartite entanglement. Thus the entanglement degree of a multipartite pure
state can be quantified by
$C^{(N)}(|\psi>)=\sqrt{\frac{1}{2}\sum_{k=1}^{N}|C^{(2)}(|\psi_{k\overline{k}}>)|^{2}}$,
where the superscript ``$(N)$'' denotes $N$-partite system, $C^{(2)}$ is the
bipartite entanglement defined in Eq.(3), and $|\psi_{k\overline{k}}>$ is the
bipartite version of the quantum state $|\psi>$ in Eq.(9) by considering
$(N-1)$ subsystems, excluding the $k$th one, as a whole. Although some other
bipartite entanglement, e.g. $C^{(2)}(|\psi_{(12)(\overline{12})}>)$, where
the first part $(12)$ is composed of the two subsystems ``$1$'' and ``$2$'',
and the other part $(\overline{12})$ is composed of the rest $(N-2)$
subsystems, can also be included to define other types of multipartite
entanglement, for example in Ref. songheshan , the participation of these
terms only affects the evaluation of entanglement, but does not change the
separability criterion. It means we can measure the multipartite entanglement
in a simpler way, where every involved bipartite entanglement is associated
with one single subsystem and the rest as a whole. The above entanglement
measure for multipartite states can also be described in terms of operators,
$\displaystyle
C^{(N)}(|\psi>)=\sqrt{\frac{1}{8}\sum_{k=1}^{N}\sum_{i,i^{\prime}=1}^{D_{1}}\cdots\sum_{j,j^{\prime}=1}^{D_{N}}|<\psi^{*}|O_{k\overline{k}}^{(i,i^{\prime}:\cdots:j,j^{\prime})}|\psi>|^{2}},$
(10)
where the symmetric operator,
$\displaystyle
O_{k\overline{k}}^{(i,i^{\prime}:\cdots:j,j^{\prime})}=\sigma_{1}^{(i,i^{\prime})}\otimes\cdots\otimes[\sigma_{k}^{(m,m^{\prime})}-(\sigma_{k}^{(m,m^{\prime})})^{T}]\otimes\cdots\otimes\sigma_{N}^{(j,j^{\prime})}+T.c.,$
(11)
is associated the bipartite system composed of the $k$th subsystem and all the
other subsystems as a whole, $D_{k}$ is the dimension degree of the $k$th
subsystem and _T.c._ means matrix transpose. This is generalized _O
concurrence_ for multipartite pure states, which, inherited from bipartite
entanglement, remains invariant under local unitary transformations. Under
this measure, the maximally-entangled $D$-dimensional $N$-partite Greenberger-
Horne-Zeilinger state
$|GHZ^{(N)}>=\frac{1}{\sqrt{D}}\sum_{i=1}^{D}|i_{1}i_{2}\cdots i_{N}>$ has
entanglement $\sqrt{N(1-\frac{1}{D})}$, and the generalized $N$-qubit $W$
state $|W^{(N)}>=\frac{1}{\sqrt{N}}(|100\cdots>+|010\cdots>+\cdots+|00\cdots
1>)$ has entanglement $\sqrt{2(1-\frac{1}{N})}$. Similar to the bipartite
case, the entanglement measure (10) has an equivalent but much simpler
version,
$\displaystyle
C^{(N)}(|\psi>)=\sqrt{N-\sum_{k=1}^{N}\text{tr}(\rho_{k}^{2})},$ (12)
where $\rho_{k}$ stands for the reduced density matrix of the $k$th subsystem.
This measure is a generalization of the _I concurrence_ Iconcurrence for
multipartite systems.
Since the present entanglement measure for multipartite systems is based on
the bipartite entanglement between single subsystems and the rest as a whole,
we can directly generalize some conclusions from bipartite case to the present
multipartite case. For example, a multipartite pure state is separable only
when it is orthogonal to its complex conjugate with respect to _all_ operators
in the form (11), and also their extended version,
$\displaystyle O_{k\overline{k}}^{\text{(Semi-
rdm)}}=\sigma_{1}^{\text{(rdm)}}\otimes\cdots\otimes[\sigma_{k}^{\text{(rdm)}}-(\sigma_{k}^{\text{(rdm)}})^{T}]\otimes\cdots\otimes\sigma_{N}^{\text{(rdm)}}+T.c.,$
(13)
where $\sigma_{k}^{\text{(rdm)}}$ is a random $D_{k}\times D_{k}$ matrix in
the $k$th subsystem.
For a separable multipartite state $\rho$, there exists a decomposition,
$W=\sqrt{\rho}U$, where each ``subnormalized'' pure state $|W_{i}>$ ($i$th
column vector of $W$) is a tensor product state in the form (9), thus is
$O_{k\overline{k}}^{\text{(Semi-rdm)}}$-orthogonal to its complex conjugate.
Similar to the above case, the matrix $W^{T}O_{k\overline{k}}^{\text{(Semi-
rdm)}}W$ has to be a hollow matrix, which means the singular values
$\\{\lambda_{i}\\}$ in decreasing order of the symmetric matrix,
$\displaystyle S=(\sqrt{\rho})^{T}O_{k\overline{k}}^{\text{(Semi-
rdm)}}\sqrt{\rho},$ (14)
have to satisfy the condition (7). Now we conclude the separability criterion
for a mixed multipartite state, that is, the condition (7) for the singular
values of the above matrix (14) holds valid for _all_ symmetric operators in
the form (13). A violation by any operator in the form (13) is enough to
declare the existence of multipartite entanglement. So every operator
$O_{k\overline{k}}^{\text{(Semi-rdm)}}$ in the form (13) is a witness of
multipartite entanglement under the violation of the condition (7). This huge
class of entanglement witnesses can help us efficiently verify multipartite
entanglement, even for mixed multipartite states. Here we also note that the
singular values $\\{\lambda_{i}\\}$ of the symmetric matrix (14) are the
square root of the eigenvalues of $S^{{\dagger}}S$, and as well the square
root of the eigenvalues of the following matrix,
$\displaystyle\rho\left[O_{k\overline{k}}^{\text{(Semi-
rdm)}}\right]^{{\dagger}}\rho^{*}O_{k\overline{k}}^{\text{(Semi-rdm)}},$ (15)
which is directly determined by the density matrix $\rho$ and an operator
$O_{k\overline{k}}^{\text{(Semi-rdm)}}$ constructed by random matrices in the
way (13).
To summarize, a group of symmetric operators with only two nonzero matrix
elements are introduced to carry out the separability criterion of a general
quantum state. Every symmetric operator, formulated as a tensor product of an
antisymmetric operator in one subsystem and an antisymmetric operator in other
subsystems as a whole, and their arbitrary linear combinations are
entanglement witnesses for multipartite quantum states. Based on these
symmetric operators, Wootters' separability criterion for two-qubit states is
generalized to multipartite systems in arbitrary dimensions. How to
distinguish all mixed entangled states from separable ones within a finite
number of tests is the next challenging question worthy of investigation. Its
answer might directly bring us a computable entanglement measure for mixed
multipartite states.
This work was supported by the national Natural Science Foundation of China
under Grant Nos. 11174118, 11174026 and 11264018, and the Natural Science
Foundation of Jiangxi Province, China under Grant No. 20114BAB212003. We also
thank Prof. J.P. Dowling for helpful discussions.
_Note added._ -After completion of this work, we became aware of a related
work by Chen, Ma, Gühne, and SeveriniZHchen .
## References
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* (18) R. A. Horn and C. R. Johnson, _Matrix Analysis_ (Cambridge University Press, New York, 1985), p. 205.
* (19) B. M. Terhal, Phys. Lett. A 271, 319 (2000); M. Lewenstein, B. Kraus, J. I. Cirac, and P. Horodecki, Phys. Rev. A 62, 052310 (2000); B. M. Terhal, Linear Algebr. Appl. 323, 61 (2001); F. G. S. L. Brandão, Phys. Rev. A 72, 022310 (2005); G. Tóth, and O. Gühne, Phys. Rev. Lett. 94, 060501 (2005).
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|
arxiv-papers
| 2012-12-01T07:52:20 |
2024-09-04T02:49:38.768341
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Jie-Hui Huang, Li-Yun Hu, Lei Wang, and Shi-Yao Zhu",
"submitter": "Ran Zeng",
"url": "https://arxiv.org/abs/1212.0082"
}
|
1212.0084
|
20xx Vol. 9 No. XX, 000–000
11institutetext: National Astronomical Observatories, Chinese Academy of
Sciences, Beijing 100012, China; [email protected]
Received [year] [month] [day]; accepted [year] [month] [day]
# A survey of CO and its isotope lines for possible cloud-cloud collision
candidates
Nan Li 11 Jun-Jie Wang 11
###### Abstract
In the ${}^{12}\\!$CO ($J$=1-0) survey for the 1331 cold IRAS sources (Yang et
al., 2002), 214 sources show profiles with multiple-peak profiles and are
selected as cloud-cloud collision candidates. In January 2005, 201 sources are
detected with ${}^{12}\\!$CO(1-0), ${}^{13}\\!$CO(1-0), and
C${}^{18}\\!$O(1-0) emission by the 13.7m telescope at Purple Mount
Observatory. This is the first CO and its isotope lines directed toward
possible cloud-cloud collision regions. According to the statistics of the 201
sources in Galactic distribution, the 201 sources show a similar distribution
to the parent sample (1331 cold IRAS sources). These sources are located over
a wide range of the Galactocentric distances, and are partly associated with
the star formation region. Based on preliminary criteria which describe the
spectrum properties of the possible cloud-cloud collision region (Vallee,
1995a, b), the 201 sources are classified into four types by the fit of the
spectral profiles between the optically thick and thin lines toward each
source. The survey is focused on the possible cloud-cloud collision regions,
and gives some evidences to help us with selecting the target region. Then we
will carry on the mapping and multi-wavelength study for the selected region
in future.
###### keywords:
stars: formation - ISM: clouds - ISM: molecules.
## 1 Introduction
In the star forming region, the collapse could be triggered by self-
gravitation or some external interaction mechanism. Beside the other external
shocks (e.g. supernova shocks, and the stellar winds around a previous
generation of OB stars etc. ), the cloud-cloud collision was also proposed as
the effective external influence mechanism in star formation. These processes
have been calculated with numerical simulation (Hausman, 1981; Gilden, 1984;
Lattanzio et al., 1985; Habe & Ohta, 1992). When CO and its isotope molecules
are used to be a sensitive probe to detect the cool molecular regions by
astronomers, many CO survey have been made to describe the the molecular cloud
distribution (Goldreich & Kwan, 1974; Frerking & Langer, 1982). For cloud-
cloud collision region, CO and its isotope line are also the good probes to
describe the cloud colliding region (Wang et al., 2004; Xin & Wang, 2008). The
IRAS point-source database is an effective indicator of the star forming
regions which are hidden by dust in visible wavelength. In this paper, we
choose the cold IRAS source according to our criteria to make the CO and its
isotope line survey. This is a relatively complete IRAS-based CO survey for
the the possible cloud-cloud collision candidates. Although there are some
observations were done in the in earlier times (Kerton et al., 2003), it only
cover a little part (42 sources) of our sample in ${}^{12}\\!$CO line. There
are no such results with ${}^{12}\\!$CO(1-0), ${}^{13}\\!$CO(1-0), and
C${}^{18}\\!$O(1-0) emission for the possible cloud-cloud collision candidates
ever before. And we have selected two possible cloud-cloud collision
candidates ( IRAS 02459+6029 and 22528+5936 ) from our survey to observe with
${}^{12}\\!$CO(2-1), ${}^{13}\\!$CO(2-1), and
${}^{12}\\!$CO(3-2)$,{}^{13}\\!$CO(3-2) by KOSMA in previous work (Li & Wang,
accepted by RAA, 2012). The CO mapping and multi-wavelength study show the
IRAS 02459+6029 could be a cloud-cloud collision sample and the IRAS
22528+5936 could be two separate clouds without colliding.
## 2 Sample selection
We build the sample list on the basis of the ${}^{12}\\!$CO(1-0) survey for
the cold IRAS sources (Yang et al., 2002). These 1331 sources are selected
from the cold IRAS sources according to the criteria as follows:
1.(a) $\alpha$(B1950) $\leq 8^{h}$, $\alpha$(B1950) $\geq 16^{h}$,
$\delta$(B1950) $\geq-35^{\circ}$; (b) $|b|$ $\leq 25^{\circ}$; or (c) $|b|$
$\geq 1^{\circ}$ when $l\leq 60^{\circ}$ or $l\geq 300^{\circ}$. Sources
coordinates should follow the conditions.(a) and (b) cover the sources in the
northern sky near the galactic plane and avoid the source confusion in CO from
the the Galactic central region.
2\. Sources should have been detected at least in three wave bands to specify
their infrared properties.
3.Sources colors over the 12, 25, and 60 $\mu$m bands satisfy with $\log$(F12/
F25) $\leq-$0.4 and $\log$(F25/ F60) $\leq-$0.4 or $\log$(F12/ F60) $\leq-$0.4
when the FQUAL25 = 1.These cold IRAS colors range are derived from statistic
data for the embedded sources associated with recent star formation (Beichman,
1986).
4\. These sources are no association with late-type stars, planetary nebula,
extra galaxies, and other kinds of sources unrelated with Galactic star
formation identified by IRAS PSC.
Then we pick up our sample list based on the criteria for possible cloud-cloud
collision candidate from the 1331 cold IRAS sources. According to a portion of
criteria given by Vallee (1995) which describe the spectrum properties of the
possible cloud-cloud collision region, we choose the samples which the line
profile features show the double peak or multipeak and the velocity of peaks
is adjacent. It means this kind of sources may have several velocity
components and it might include several colliding cloud clumps. The Figure 1
shows the CO(1-0) spectrum that we selected from 1331 cold IRAS sources.
Figure 1: The ${}^{12}\\!$CO(1-0) line profile of initially selected sample
for possible cloud-cloud collision from 1331 IRAS sources (Yang et al. 2002).
After this step, there are 214 sources selected finally. But only 201 sources
are detected in the ${}^{12}\\!$CO(1-0), ${}^{13}\\!$CO(1-0), and
C${}^{18}\\!$O(1-0) lines. The galactic distribution of the 201 sources are
described in the Figure 2. Most of the sources are located around the Galactic
plane. A little part of sources are concentrated toward the well-known local
star-forming regions, particularly toward the Perseus, Taurus, and Orion
regions at $l=160^{\circ}-200^{\circ}$ and $b=-10^{\circ}$ to $-25^{\circ}$.
## 3 Observations
These sources were already observed in ${}^{12}\\!$CO(1-0),
${}^{13}\\!$CO(1-0) and C${}^{18}\\!$O(1-0) with the 13.7m (45 foot)
Millimeter Telescope at the Qinghai Station of Purple Mountain Observatory,
CAS (Chinese Academy of Sciences), in January 2005. The beam size is 50$\pm$7
arcsec in azimuth and 54$\pm$3 arcsec in elevation. The pointing accuracy is
around 5 arcsec. We used the cooled mixer SIS receiver which have three
acousto-optical spectrometers (AOSs) working at 110.20, 109.78 and 115.27GHz
to get the three CO(1-0) lines simultaneously. The AOS has 1024 channels, and
its band width for ${}^{12}\\!$CO(1-0),${}^{13}\\!$CO(1-0) and
C${}^{18}\\!$O(1-0) lines are 145.330, 42.672 and 43.097 MHz. The system
temperature is 200-300K during observations. The noise level of antenna
temperature was 0.4K for ${}^{12}\\!$CO(1-0), 0.3K for ${}^{13}\\!$CO(1-0) and
0.2K for C${}^{18}\\!$O(1-0) typically. In this paper, we don’t apply main-
beam efficiency corrections to any data, and just show the data simply in the
original antenna temperature scale $T^{*}_{A}$.
We observed these sources in position-switching mode in order to obtain a
better baseline. Each source was scanned two times at least except some
sources are measured many times for the bad SNR (signal to noise ratio) in few
cases. The integration time was 2 minutes on every source. We identified a
source to be detected if its peak antenna temperature is larger than $3\sigma$
rms noise level. We observed the standard sources, such as W51D, S140, W3(OH),
NGC2264, etc., every 2 hours during observation. These standard sources served
as the secondary temperature standard. We use it to calibrate target sources
within a specific period of time for reducing the influence of any possible
short-period variation of the system.
The data reduction based on the software packages, CLASS (Continuum and Line
Analysis Single-dish Software) and GREG (Grenoble Graphic).
Figure 2: Galactic distribution of the 201 sources which are selected as
cloud-cloud collision candidates.
## 4 RESULTS
### 4.1 The statistic study of the selected CO sources
The 201 sources are observed in ${}^{12}\\!$CO(1-0), ${}^{13}\\!$CO(1-0) and
C${}^{18}\\!$O(1-0). Table 1 shows the basic parameters of 201 sources as
possible cloud-cloud collision candidates. Column (1) shows IRAS name of the
sources, (2) and (3) denote equatorial coordinates in the J2000 epoch. Columns
(4) and (5) are their Galactic coordinates, and (6) shows the peak temperature
of the antenna to each sources. Column (7) is the corresponding rms noise
level of the CO spectrum, and (8) contains the radial velocity of the sources
relative to the local standard of rest. In column (9), the FWHM of each line
derived from Gaussian fitting is presented. Column (10) lists the line span
measured at the 0 K level, column (11) gives the kinematic distance of each
source which derived from other paper (Yang et al., 2002).
The kinematic distances of the CO sources were plotted in Figure 3. The lack
of the inner part of the Galaxy is obvious. Because the serious contamination
in the CO line, we do not observe these sources in this region ( $|b|$
$<1^{\circ}$ when $l\leq 60^{\circ}$ or $l\geq 300^{\circ}$ ). We could
investigate the molecular emission such as high density tracers around these
sources in further observation.
Figure 3: Projected Galactocentric distribution of the 201 candidates.
Figure 4 demonstrates the $L-V_{LSR}$ diagram for the all CO sources that we
selected. The range of the $V_{LSR}$ from 91.73 to -80.09 $kms^{-1}$. The
distribution of $V_{LSR}$ conforms to the distribution of the 1331 sources
sample given by Yang (Yang et al., 2002). It basically follows the Galactic
rotation. A significant part of sources which associated with the star-forming
regions distributes over a large fraction of the Galactic plane. A large
portion of the sources are located in the second Galactic quadrant. Some of
them with the minus sign of the radial velocities and large amplitudes are
located in the Perseus arm possibly.
Figure 4: L-$V$ distribution of our survey sample
The statistic distribution of the peak antenna temperature of the 201 sources
is illustrated in Figure 5. The range of antenna temperature covers from 0.73
to 18.4 K and the average temperature is 3.8 K. The distribution of the line
intensity is monotonic generally from 2 K to 10 K, and the peak lies on the
2-3 K. The source IRAS 05394-0151 is detected the highest antenna temperature
of 18.4 K.
Figure 5: Statistical distribution of the $T^{*}_{A}$ of the 201 sources Table 1: Basic parameters of the 201 selected IRAS sources. In this table, 42 sources with asterisk were also observed by the FCRAO Outer Galaxy Survey (Kerton et al., 2003). Name | $\alpha(2000)$ | $\beta(2000)$ | $l$ | $b$ | $T_{A}^{*}$ | $T_{\sigma}$ | $V_{LSR}$ | $\delta V$ | $V_{span}$ | $D_{k}$
---|---|---|---|---|---|---|---|---|---|---
| | | (deg) | (deg) | (K) | (K) | ($kms^{-1}$) | | ($kms^{-1}$) | (kpc)
(1) | (2) | (3) | (4) | (5) | (6) | (7) | (8) | (9) | (10) | (11)
∗00021+6604 | 00 04 43.60 | +66 20 58.0 | 118.26373 | 3.901982 | 2.53 | 0.15 | -7.53 | 2.94 | 5.65 | 0.45
∗00056+6605 | 00 08 18.80 | +66 21 53.0 | 118.621328 | 3.854562 | 2.89 | 0.12 | -14.34 | 1.47 | 3.47 | 0.89
∗00070+6516 | 00 09 44.20 | +65 33 21.9 | 118.631671 | 3.033102 | 5.11 | 0.19 | -19.81 | 3.05 | 6.52 | 1.26
∗00116+6716 | 00 14 20.10 | +67 33 22.0 | 119.390751 | 4.939818 | 1.95 | 0.22 | -7.24 | 1.84 | 8.68 | 0.42
∗00289+6327 | 00 31 51.50 | +63 43 58.0 | 120.767534 | 0.943341 | 1.14 | 0.12 | -63.37 | 1.78 | 4.8 | 7.19
∗00342+6347 | 00 37 11.02 | +64 04 01.0 | 121.374013 | 1.238323 | 4.32 | 0.2 | -29.18 | 11.68 | 30.32 | 1.87
∗00412+6638 | 00 44 15.34 | +66 54 41.1 | 122.226176 | 4.049512 | 2.95 | 0.2 | -5.46 | 2.43 | 7.39 | 0.29
∗00455+6137 | 00 48 33.27 | +61 53 36.8 | 122.592344 | -0.97629 | 1.73 | 0.17 | -46.63 | 6.34 | 9.56 | 3.77
∗00468+6527 | 00 49 55.13 | +65 43 41.2 | 122.775675 | 2.856825 | 3.57 | 0.21 | -63.12 | 4.59 | 12.61 | 7.2
∗00484+6531 | 00 51 33.00 | +65 47 58.0 | 122.943481 | 2.927701 | 2.77 | 0.21 | -65.47 | 4.34 | 16.95 | 7.52
00519+6535 | 00 54 52.90 | +65 53 55.0 | 123.284026 | 3.02922 | 2.5 | 0.21 | -66.64 | 4.74 | 15.65 | 7.7
∗01166+6635 | 01 20 03.73 | +66 51 31.7 | 125.745473 | 4.143561 | 2.19 | 0.21 | -2.01 | 2.04 | 9.7 | 0.07
01584+6706 | 02 02 17.98 | +67 21 23.9 | 129.689952 | 5.419357 | 1.78 | 0.19 | -10.57 | 15.22 | 8.7 | 0.58
∗02220+6107 | 02 25 47.50 | +61 20 47.0 | 133.995193 | 0.517945 | 2.6 | 0.15 | -49.37 | 7.58 | 13.9 | 5.77
∗02244+6035 | 02 28 10.40 | +60 49 17.0 | 134.452953 | 0.131981 | 1.37 | 0.15 | -14.73 | 2.12 | 8.26 | 0.83
02425+6851 | 02 47 00.20 | +69 04 11.0 | 132.981891 | 8.484117 | 3.59 | 0.12 | -10.33 | 3.36 | 16.19 | 0.57
∗02455+6034 | 02 49 23.20 | +60 47 01.0 | 136.83675 | 1.136532 | 8.32 | 0.23 | -42.81 | 6 | 13.91 | 4.75
∗02459+6029 | 02 49 47.30 | +60 42 10.1 | 136.916324 | 1.08546 | 9.68 | 0.12 | -40.63 | 3.2 | 15.2 | 4
02485+6902 | 02 53 07.20 | +69 14 36.0 | 133.396432 | 8.881912 | 4.92 | 0.27 | -11.8 | 13.18 | 33.03 | 0.66
02499+6911 | 02 54 32.30 | +69 23 23.0 | 133.441133 | 9.069436 | 6.66 | 0.19 | -15.54 | 8.6 | 21.73 | 0.9
03183+6321 | 03 22 36.38 | +63 32 06.3 | 138.853953 | 5.43829 | 3.78 | 0.15 | -17.76 | 5.09 | 36.94 | 1.07
03211+5446 | 03 24 59.10 | +54 57 25.0 | 143.828825 | -1.560649 | 7.88 | 0.2 | -32.73 | 3.15 | 31.29 | 2.94
03245+3002 | 03 27 40.97 | +30 12 57.3 | 158.774061 | -21.57275 | 5.36 | 0.16 | 3.17 | 3.53 | 10 | $\ldots$
03248+6551 | 03 29 20.46 | +66 01 38.8 | 138.048528 | 7.905155 | 2.02 | 0.16 | -3.62 | 2.36 | 8.69 | 0.18
03260+3111 | 03 29 09.53 | +31 22 02.4 | 158.3008734 | -20.463184 | 11.9 | 0.34 | 7.53 | 4.1 | 10.4 | $\ldots$
03275+5450 | 03 31 23.40 | +55 00 35.0 | 144.558996 | -1.000203 | 3.39 | 0.25 | -30.54 | 2.71 | 17.4 | 2.57
03463+5331 | 03 50 08.30 | +53 41 00.0 | 147.540273 | -0.424337 | 2.72 | 0.18 | -8.02 | 2.11 | 7.4 | 0.49
04034+5107 | 04 07 13.80 | +51 15 58.0 | 151.08813 | -0.574353 | 2.62 | 0.15 | -26.2 | 6.2 | 9.54 | 2.59
04044+5110 | 04 08 14.40 | +51 18 53.0 | 151.172315 | -0.432188 | 4.25 | 0.2 | -26.2 | 3.1 | 10.9 | 2.63
04088+3834 | 04 12 12.31 | +38 41 59.9 | 160.326764 | -9.182407 | 2.73 | 0.2 | -3.44 | 1.78 | 5.7 | 0.3
04173+4328 | 04 20 53.25 | +43 36 15.0 | 158.047386 | -4.535014 | 2.15 | 0.44 | -8.82 | 4.63 | 7.4 | 0.76
04173+4524 | 04 20 56.60 | +45 31 08.0 | 156.698679 | -3.172826 | 3.15 | 0.15 | -22.87 | 2.98 | 15.2 | 3.09
04271+3502 | 04 30 25.80 | +35 09 13.0 | 165.472555 | -9.058621 | 5.05 | 0.3 | 0.37 | 7.04 | 15.7 | $\ldots$
04299+2915 | 04 33 03.80 | +29 21 43.0 | 170.23338 5 | -12.521205 | 2.54 | 0.17 | 3.04 | 4.98 | 8.48 | $\ldots$
04307+5209 | 04 34 38.70 | +52 15 23.0 | 153.398895 | 3.116812 | 0.94 | 0.11 | -7.96 | 3.13 | 5.65 | 0.57
04319+5056 | 04 35 50.00 | +51 02 38.0 | 154.41958 | 2.435062 | 1.72 | 0.15 | -3.16 | 2.45 | 5.2 | 0.21
04324+5102 | 04 36 15.92 | +51 08 08.1 | 154.397355 | 2.546939 | 4.1 | 0.14 | -37.55 | 4.9 | 15.65 | 6.9
04324+5106 | 04 36 21.04 | +51 12 55.0 | 154.347282 | 2.610404 | 6.74 | 0.25 | -36.02 | 5.37 | 12.2 | 6.48
04335+5110 | 04 37 24.50 | +51 16 32.0 | 154.41362 | 2.77363 | 4.97 | 0.16 | -34.89 | 3.81 | 15.6 | 6.22
04368+2557 | 04 39 52.92 | +26 03 06.8 | 173.8215694 | -13.527364 | 1.87 | 0.14 | 4.51 | 4.71 | 8.26 | $\ldots$
04381+2540 | 04 41 12.69 | +25 46 35.5 | 174.2357869 | -13.472849 | 2.63 | 0.21 | 6.12 | 2.03 | 4.8 | $\ldots$
04587+4411 | 05 02 20.20 | +44 16 03.0 | 162.47734 | 1.50271 | 2.37 | 0.17 | -0.64 | 5.27 | 7.5 | 0.04
05075+3755 | 05 10 59.98 | +37 59 27.2 | 168.471492 | -0.972151 | 1.29 | 0.17 | -33.51 | 14.25 | 18.69 | 29.82
05236+0620 | 05 26 18.36 | +06 22 57.1 | 197.05278 | -15.68507 | 0.73 | 0.14 | -4.53 | 2.21 | 3.04 | $\ldots$
05335+3609 | 05 36 53.23 | +36 10 50.9 | 172.877253 | 2.270042 | 6.55 | 0.15 | -16.79 | 6.1 | 11.7 | 15.18
05356-0530 | 05 38 06.50 | -05 28 54.0 | 209.4348116 | -18.796256 | 4.53 | 0.13 | 5.11 | 4.28 | 40.85 | 0.36
05363+3127 | 05 39 35.85 | +31 29 12.2 | 177.146733 | 0.237072 | 3.03 | 0.13 | -1.09 | 5.42 | 7.39 | 0.65
05366+3601 | 05 40 02.50 | +36 03 31.0 | 173.320658 | 2.743877 | 4.35 | 0.18 | -20.34 | 4.83 | 13.91 | $\ldots$
05375+3536 | 05 40 52.50 | +35 38 24.0 | 173.765722 | 2.665709 | 10.9 | 0.22 | -17.71 | 3.32 | 14.34 | 26.76
05375+3540 | 05 40 52.90 | +35 42 16.7 | 173.711494 | 2.701006 | 12.6 | 0.26 | -16.64 | 4.86 | 30 | 20.98
05379+3515 | 05 41 19.20 | +35 16 46.0 | 174.120141 | 2.552314 | 3.3 | 0.18 | -17.67 | 6.32 | 15.21 | $\ldots$
05381-0921 | 05 40 34.30 | -09 20 08.0 | 213.3860117 | -19.950587 | 6.68 | 0.15 | 1.56 | 4.8 | 18.26 | 0.08
05387-0924 | 05 41 04.13 | -09 23 19.5 | 213.4933593 | -19.862943 | 5.37 | 0.17 | 2.86 | 2.68 | 9.6 | 0.17
05394-0151 | 05 41 58.50 | -01 50 24.0 | 206.5052375 | -16.268355 | 18.4 | 0.16 | 8.61 | 4.99 | 15.21 | 0.67
05399+2631 | 05 43 03.00 | +26 33 00.0 | 181.73645 | -1.732799 | 1.26 | 0.18 | -2.62 | 2.09 | 9.1 | $\ldots$
05399-0121 | 05 42 27.70 | -01 20 02.0 | 206.0984668 | -15.924558 | 7.86 | 0.17 | 7.62 | 5.45 | 20.86 | 0.59
05399-1001 | 05 42 19.10 | -10 00 08.0 | 214.2230027 | -19.848847 | 3.19 | 0.3 | 1.46 | 2.12 | 10 | 0.07
05404-0220 | 05 43 00.57 | -02 18 45.4 | 207.0662752 | -16.259142 | 3.82 | 0.2 | 8.77 | 2.98 | 12.17 | 0.68
05411+3302 | 05 44 24.82 | +33 03 31.0 | 176.347927 | 1.930962 | 1.84 | 0.23 | -9.39 | 3.96 | 17 | 18.94
05413-0104 | 05 43 51.50 | -01 02 52.0 | 206.0054854 | -15.482164 | 5.81 | 0.19 | 5.2 | 8.84 | 28.25 | 0.39
05437-0343 | 05 46 14.80 | -03 41 58.0 | 208.7359072 | -16.180012 | 3.19 | 0.22 | 9.1 | 3.93 | 8.69 | 0.67
05445+0020 | 05 47 05.01 | +00 21 47.2 | 205.1093819 | -14.107369 | 10.8 | 0.37 | 9.9 | 8.38 | 29.6 | 0.81
05462-0124 | 05 48 47.21 | -01 23 52.3 | 206.9210946 | -14.555896 | 2.43 | 0.22 | 7.33 | 9.16 | 12.48 | 0.55
05575+2141 | 06 00 35.26 | +21 41 11.8 | 187.961159 | -0.772756 | 1.51 | 0.13 | -0.78 | 7.38 | 28.25 | $\ldots$
06084-0611 | 06 10 51.31 | -06 11 54.0 | 213.8826992 | -11.834592 | 6.68 | 0.19 | 13.97 | 3.11 | 17.8 | 0.94
06102+1537 | 06 13 04.60 | +15 36 31.0 | 194.710338 | -1.152557 | 3.48 | 0.2 | 15.1 | 5.27 | 11.3 | 3.32
06281+1039 | 06 30 50.30 | +10 37 22.0 | 201.143823 | 0.317507 | 4.02 | 0.25 | 5.41 | 2.49 | 10.4 | 0.45
06282+0423 | 06 30 52.44 | +04 21 24.5 | 206.705413 | -2.572085 | 4.06 | 0.21 | 12.37 | 10.46 | 22.16 | 0.94
06294+0352 | 06 32 07.52 | +03 50 07.5 | 207.312536 | -2.535412 | 5.32 | 0.32 | 15.99 | 2.38 | 10.4 | 1.26
06308+0402 | 06 33 31.10 | +04 00 07.0 | 207.324606 | -2.150318 | 6.91 | 0.13 | 14.34 | 3.06 | 15.21 | 1.1
06318+0420 | 06 34 32.54 | +04 17 52.1 | 207.179248 | -1.78752 | 3.6 | 0.18 | 11.52 | 5.5 | 14.77 | 0.85
06343+0425 | 06 37 00.80 | +04 23 04.0 | 207.385365 | -1.200596 | 1.99 | 0.16 | 4.66 | 2.02 | 10 | 0.3
06373+1053 | 06 40 05.70 | +10 51 10.0 | 201.985033 | 2.442405 | 2.56 | 0.34 | 8.17 | 1.73 | 8.3 | 0.69
06423+0006 | 06 44 52.90 | +00 03 33.0 | 212.131769 | -1.431586 | 1.14 | 0.22 | 44.34 | 5.84 | 7.78 | 6.71
06425+0038 | 06 45 07.50 | +00 35 36.0 | 211.684076 | -1.133751 | 2.14 | 0.17 | 41.47 | 7.3 | 11.3 | 6.21
06511-0507 | 06 53 36.78 | -05 11 15.0 | 217.79789 | -1.880434 | 1.24 | 0.2 | 31.68 | 4.92 | 6.95 | 2.53
06545-0251 | 06 57 02.20 | -02 55 51.0 | 216.178333 | -0.091748 | 2.8 | 0.1 | 22.41 | 2.69 | 10.43 | 1.52
07100-1110 | 07 12 24.47 | -11 15 33.5 | 225.326443 | -0.531399 | 3.01 | 0.16 | 17.02 | 3.59 | 13.5 | 0.98
07157-1830 | 07 17 56.30 | -18 36 21.0 | 232.456019 | -2.760925 | 1.59 | 0.19 | 33.62 | 2.62 | 6.09 | 2.2
07221-2544 | 07 24 12.90 | -25 49 58.0 | 239.54154 | -4.848271 | 4.21 | 0.22 | 24.27 | 4.46 | 13 | 1.54
07297-1926 | 07 31 58.30 | -19 32 31.0 | 234.851753 | -0.284031 | 0.76 | 0.18 | 40.78 | 3.47 | 3.91 | 2.93
08004-2815 | 08 02 30.10 | -28 24 01.0 | 246.013009 | 1.260662 | 2.42 | 0.34 | 54.63 | 13.22 | 15.21 | 5.44
17207-3404 | 17 24 04.20 | -34 07 12.0 | 353.15702 | 1.05134 | 7.65 | 0.22 | 8.45 | 9.72 | 36.95 | 21.81
17364-1946 | 17 39 23.50 | -19 47 51.0 | 7.05197 | 5.99769 | 4.11 | 0.2 | 8.08 | 7.14 | 14.78 | 15.22
17369-1945 | 17 39 56.00 | -19 46 36.0 | 7.13702 | 5.90015 | 2.93 | 0.21 | 10.2 | 2.26 | 4.4 | 14.81
17392-3309 | 17 42 34.40 | -33 10 37.0 | 356.04489 | -1.66466 | 3.13 | 0.27 | -1.09 | 8.49 | 11.3 | 16.51
17419-3150 | 17 45 12.20 | -31 51 22.0 | 357.46099 | -1.4453 | 1.88 | 0.2 | -0.7 | 14.06 | 16.85 | 16.54
17520-2731 | 17 55 14.20 | -27 32 25.0 | 2.28423 | -1.10303 | 2.3 | 0.42 | 0.4 | 3.9 | 7 | 16.69
17555-2136 | 17 58 33.00 | -21 36 19.0 | 7.79598 | 1.22415 | 3.01 | 0.2 | 22.02 | 6.51 | 17.82 | 13.17
18111-2028 | 18 14 05.00 | -20 27 56.0 | 10.56027 | -1.38257 | 1.55 | 0.24 | 32.21 | 6.4 | 24.77 | -12.79
18134-1942 | 18 16 21.60 | -19 41 31.3 | 11.49535 | -1.48395 | 11.5 | 0.4 | 9.16 | 5.37 | 20.86 | 15.41
18136-1347 | 18 16 28.60 | -13 46 33.0 | 16.71365 | 1.30475 | 6.64 | 0.22 | 18.93 | 4.86 | 9.13 | 14.47
18148-0440 | 18 17 29.80 | -04 39 38.0 | 24.88448 | 5.38459 | 1.96 | 0.09 | 8.08 | 1.84 | 3.91 | 14.88
18188-1631 | 18 21 43.10 | -16 30 00.0 | 14.91099 | -1.09768 | 6.16 | 0.43 | 22.04 | 4.57 | 24.34 | 14.14
18273+0034 | 18 29 53.06 | +00 36 06.4 | 31.00628 | 5.06938 | 3.1 | 0.35 | 6.97 | 2.7 | 8.3 | 14.16
18282-1529 | 18 31 09.70 | -15 27 32.0 | 16.88971 | -2.62209 | 4.78 | 0.24 | 17.05 | 2.98 | 6.51 | 14.65
18316-0602 | 18 34 20.84 | -05 59 42.4 | 25.6494 | 1.04977 | 5.2 | 0.28 | 43.38 | 15.26 | 23.91 | 12.38
18385-0755 | 18 41 17.90 | -07 52 51.0 | 24.76566 | -1.34825 | 1.73 | 0.23 | 54.49 | 8.82 | 14.78 | 11.77
18473+0131 | 18 49 55.60 | +01 35 04.0 | 34.170041 | 1.058437 | 2.3 | 0.27 | 91.73 | 7.35 | 17.38 | 7.96
18532+0420 | 18 55 45.40 | +04 24 47.0 | 37.351628 | 1.052219 | 1.81 | 0.19 | 9.77 | 2.95 | 13 | 12.92
18567+0700 | 18 59 13.60 | +07 04 47.0 | 40.119071 | 1.500643 | 7.98 | 0.33 | 28.52 | 3.07 | 12.6 | 11.36
18572+0057 | 18 59 49.00 | +01 01 34.0 | 34.800828 | -1.395936 | 4.9 | 0.15 | 41.95 | 12.96 | 27.39 | 11.45
18583+0136 | 19 00 55.60 | +01 40 58.0 | 35.511907 | -1.342749 | 5.16 | 0.17 | 30.3 | 5.02 | 23.91 | 12.05
19207+1809 | 19 23 01.10 | +18 14 59.0 | 52.70594 | 1.52982 | 2.12 | 0.19 | 21.34 | 2.05 | 6.96 | 8.96
19223+1826 | 19 24 35.10 | +18 32 16.0 | 53.13552 | 1.33831 | 2.47 | 0.16 | 20.62 | 1.45 | 3.48 | 8.89
19258+1919 | 19 28 00.30 | +19 25 15.0 | 54.29695 | 1.0465 | 2.15 | 0.16 | -49.35 | 3.47 | 7.82 | 14.83
19300+2158 | 19 32 13.49 | +22 04 56.6 | 57.10624 | 1.45707 | 2.01 | 0.35 | -61.84 | 3.98 | 7.8 | 16.25
19387+2658 | 19 40 48.80 | +27 05 10.0 | 62.42938 | 2.19103 | 0.97 | 0.25 | 4.99 | 3.5 | 6.1 | 7.46
19407+2454 | 19 42 49.30 | +25 01 41.0 | 60.86387 | 0.78037 | 1.38 | 0.17 | 9.78 | 2.5 | 3.48 | 7.54
19408+2554 | 19 42 54.10 | +26 01 22.0 | 61.73622 | 1.25897 | 3.42 | 0.16 | 3.64 | 2.03 | 5.21 | 7.74
19413+2349 | 19 43 27.78 | +23 56 53.5 | 59.9997 | 0.11656 | 8.06 | 0.15 | 21.1 | 2.33 | 14.35 | 6.89
19454+2625 | 19 47 31.70 | +26 33 17.0 | 62.71595 | 0.62746 | 1.33 | 0.15 | 0.64 | 7.61 | 12.6 | 7.69
19457+2357 | 19 47 55.10 | +24 04 45.0 | 60.623325 | -0.698201 | 3.4 | 0.21 | 28.59 | 3.22 | 5.65 | 5.89
19458+2442 | 19 48 00.14 | +24 50 18.6 | 61.288383 | -0.331204 | 3.59 | 0.13 | -12.69 | 6.64 | 9.99 | 8.95
19508+2705 | 19 52 54.80 | +27 13 43.0 | 63.907111 | -0.063466 | 1.5 | 0.16 | -16.68 | 11.31 | 16.08 | 8.57
19560+3135 | 19 58 03.14 | +31 44 07.0 | 68.341878 | 1.313376 | 1.79 | 0.28 | -65.02 | 5.81 | 14.4 | 13.85
20016+3243 | 20 03 34.30 | +32 52 13.0 | 69.921521 | 0.920929 | 1.4 | 0.12 | 15.32 | 4.89 | 6.52 | 3.78
20033+2848 | 20 05 23.90 | +28 56 42.0 | 66.812815 | -1.507827 | 5.39 | 0.13 | 10.48 | 2.94 | 13.48 | 5.7
20050+2720 | 20 07 06.70 | +27 28 53.0 | 65.780227 | -2.612263 | 2.17 | 0.15 | 4.07 | 8.29 | 13.47 | 6.59
20051+3016 | 20 07 06.70 | +30 25 09.0 | 68.256233 | -1.029066 | 2.25 | 0.14 | 7.37 | 3.07 | 12.17 | 5.55
20051+3435 | 20 07 04.50 | +34 44 45.0 | 71.895364 | 1.311607 | 5.5 | 0.16 | 9.53 | 6.56 | 18.69 | 4.03
20068+3328 | 20 08 49.91 | +33 37 32.2 | 71.149824 | 0.399907 | 3.74 | 0.26 | -17.43 | 2.79 | 7.8 | 6.84
20072+2720 | 20 09 20.10 | +27 29 25.0 | 66.054422 | -3.022408 | 3.11 | 0.16 | 4.62 | 5.37 | 13.48 | 6.47
20078+3254 | 20 09 49.70 | +33 03 19.0 | 70.784484 | -0.084988 | 1.29 | 0.17 | 8.64 | 6.09 | 14.34 | 4.58
20184+3936 | 20 20 15.10 | +39 45 36.0 | 77.524535 | 1.896355 | 3.44 | 0.14 | 1.22 | 4.95 | 14.34 | 3.35
20190+4011 | 20 20 49.30 | +40 20 48.0 | 78.070124 | 2.138794 | 2.18 | 0.13 | 12.5 | 1.69 | 3.91 | $\ldots$
20190+4102 | 20 20 47.50 | +41 12 06.8 | 78.772125 | 2.628169 | 8.2 | 0.14 | 1.85 | 2.76 | 10.86 | 2.82
20197+3745 | 20 21 38.80 | +37 55 06.0 | 76.16238 | 0.627096 | 2.44 | 0.19 | 5.23 | 3.6 | 12.2 | 3.1
20227+4154 | 20 24 31.40 | +42 04 17.0 | 79.88505 | 2.55185 | 5.28 | 0.21 | 5.7 | 5.73 | 27 | $\ldots$
20228+4215 | 20 24 34.40 | +42 25 01.0 | 80.17348 | 2.74284 | 12.5 | 0.17 | 3.96 | 3.34 | 15.22 | $\ldots$
20243+3752 | 20 26 10.35 | +38 02 42.1 | 76.77831 | -0.02992 | 2.88 | 0.21 | -2.41 | 6.24 | 11.73 | 4.06
20243+3853 | 20 26 11.61 | +39 03 32.8 | 77.60721 | 0.55431 | 4.72 | 0.21 | 7.88 | 5 | 15.21 | $\ldots$
20274+4219 | 20 29 12.90 | +42 29 45.0 | 80.7348 | 2.09116 | 1.73 | 0.11 | -6.6 | 4.61 | 11.3 | 3.54
20275+4001 | 20 29 24.90 | +40 11 21.0 | 78.88707 | 0.70905 | 6.91 | 0.17 | -6.63 | 3.65 | 26.5 | 3.96
20285+3939 | 20 30 20.43 | +39 49 34.1 | 78.69743 | 0.35234 | 3.56 | 0.17 | 8.89 | 2.6 | 6.65 | $\ldots$
20290+4052 | 20 30 50.80 | +41 02 25.0 | 79.73534 | 0.98979 | 3.72 | 0.34 | -2.88 | 4.63 | 7.83 | 3.31
20293+3952 | 20 31 12.90 | +40 03 21.0 | 78.9818 | 0.35239 | 3.28 | 0.17 | 5.77 | 14.1 | 31.7 | 2
20300+3909 | 20 31 52.40 | +39 19 33.0 | 78.46799 | -0.18141 | 3.23 | 0.2 | 7.24 | 2.37 | 13.5 | $\ldots$
20300+4058 | 20 31 49.90 | +41 09 00.0 | 79.93352 | 0.90487 | 3.62 | 0.21 | 11.98 | 2.99 | 6.5 | $\ldots$
20306+4005 | 20 32 27.92 | +40 16 08.2 | 79.29496 | 0.28601 | 4.09 | 0.18 | 0.06 | 5.02 | 13.91 | 2.99
20309+4257 | 20 32 38.90 | +43 07 32.0 | 81.61517 | 1.95351 | 1.89 | 0.16 | -10.07 | 3.16 | 5.65 | 3.72
20321+4112 | 20 33 55.12 | +41 22 49.5 | 80.35193 | 0.72635 | 3.85 | 0.27 | -2.25 | 7.9 | 19.99 | 3.05
20322+4031 | 20 34 06.40 | +40 41 22.0 | 79.81866 | 0.28551 | 2.6 | 0.18 | 8.94 | 2.08 | 7.4 | $\ldots$
20329+3846 | 20 34 48.40 | +38 56 49.0 | 78.50218 | -0.864 | 4.73 | 0.35 | 4.83 | 17.09 | 24.34 | 1.98
20332+4124 | 20 35 00.50 | +41 34 48.0 | 80.63395 | 0.68206 | 6.75 | 0.45 | -3.44 | 5.16 | 11.3 | 3.15
20333+4102 | 20 35 09.50 | +41 13 18.0 | 80.36393 | 0.44481 | 5.14 | 0.26 | -33.78 | 17.9 | 34.33 | 6.12
20343+4129 | 20 36 07.10 | +41 40 01.0 | 80.82803 | 0.56836 | 4.52 | 0.18 | 11.67 | 3.21 | 22.6 | $\ldots$
20346+4706 | 20 36 15.90 | +47 16 48.0 | 85.33469 | 3.9201 | 4.13 | 0.16 | 0.31 | 2.77 | 9.1 | $\ldots$
20350+4126 | 20 36 52.60 | +41 36 33.0 | 80.86728 | 0.42046 | 6.67 | 0.41 | -2.85 | 5.29 | 15.21 | 3
20364+3816 | 20 38 20.00 | +38 27 06.0 | 78.52138 | -1.71157 | 3.14 | 0.16 | 0 | 9.98 | 16.95 | 3.23
20436+5849 | 20 44 49.25 | +59 00 18.1 | 95.53324 | 9.97828 | 2.4 | 0.2 | -2.77 | 3.31 | 12.6 | 0.43
20489+4410 | 20 50 43.20 | +44 21 59.0 | 84.5993 | 0.14062 | 6.1 | 0.23 | 3.03 | 3.87 | 13.5 | $\ldots$
20527+4957 | 20 54 22.20 | +50 08 53.0 | 89.44418 | 3.36011 | 1.25 | 0.16 | 5.52 | 3.8 | 7 | $\ldots$
20550+5158 | 20 56 32.10 | +52 10 09.0 | 91.21236 | 4.40319 | 2.08 | 0.14 | -4.33 | 9.04 | 13.48 | 1.11
20553+5208 | 20 56 53.10 | +52 20 06.0 | 91.37395 | 4.46961 | 4.38 | 0.23 | -8.24 | 3.96 | 21.3 | 1.71
20565+5003 | 20 58 10.95 | +50 15 07.2 | 89.91898 | 2.96227 | 1.49 | 0.14 | -0.59 | 4.38 | 10.4 | $\ldots$
20582+7724 | 20 57 11.70 | +77 35 47.9 | 111.66994 | 20.22046 | 1.73 | 0.25 | -8.62 | 2.31 | 4.3 | 0.69
20588+5215 | 21 00 21.20 | +52 27 09.0 | 91.809158 | 4.144649 | 4.43 | 0.38 | -0.66 | 4.32 | 7.4 | $\ldots$
21005+5217 | 21 02 05.50 | +52 28 54.0 | 92.005844 | 3.964499 | 2.62 | 0.31 | -4.43 | 2.71 | 13 | 1.03
21007+4847 | 21 02 26.22 | +48 59 37.5 | 89.423433 | 1.615789 | 2.94 | 0.19 | -72.81 | 4.07 | 12.6 | 10.68
21007+5036 | 21 02 21.79 | +50 48 34.7 | 90.77693 | 2.82765 | 4.63 | 0.26 | -10.08 | 4.05 | 17.4 | 2.03
21020+4939 | 21 03 42.30 | +49 51 53.0 | 90.2118 | 2.04031 | 1.68 | 0.21 | -3.37 | 3.17 | 11.74 | 1.04
21025+4912 | 21 04 15.40 | +49 24 25.0 | 89.92983 | 1.66867 | 2.61 | 0.17 | -74.41 | 2.81 | 5.65 | 10.78
21036+4927 | 21 05 14.73 | +49 39 53.1 | 90.22895 | 1.72159 | 2.52 | 0.21 | 3.68 | 14.95 | 19.99 | $\ldots$
21101+5150 | 21 11 42.40 | +52 03 15.0 | 92.67952 | 2.58843 | 2.7 | 0.17 | -13.39 | 3.45 | 27.81 | 2.16
21106+5206 | 21 12 12.20 | +52 18 31.0 | 92.91732 | 2.70721 | 2.47 | 0.27 | -10.93 | 3.72 | 7 | 1.85
21131+5221 | 21 14 45.10 | +52 33 56.0 | 93.37155 | 2.60245 | 5.59 | 0.15 | -7.33 | 10.16 | 34.77 | 1.35
21136+5317 | 21 15 11.00 | +53 30 30.0 | 94.09809 | 3.20788 | 1.59 | 0.19 | -3.27 | 7.34 | 9.13 | 0.61
21169+6804 | 21 17 38.52 | +68 17 34.2 | 105.16593 | 13.15667 | 2.86 | 0.17 | 2.01 | 1.77 | 10.87 | $\ldots$
21182+4641 | 21 20 04.70 | +46 54 43.0 | 89.92545 | -1.96175 | 5.55 | 0.18 | 2.02 | 2.27 | 9.1 | $\ldots$
21184+5507 | 21 19 58.60 | +55 19 52.0 | 95.89396 | 3.97984 | 2.03 | 0.24 | -80.09 | 2.83 | 4.34 | 10.73
21202+5157 | 21 21 59.80 | +52 10 56.9 | 93.87299 | 1.54495 | 5.11 | 0.26 | -60.11 | 3.68 | 14.4 | 7.8
21207+5338 | 21 22 23.40 | +53 51 49.0 | 95.10064 | 2.69591 | 2.01 | 0.13 | -3.83 | 2.7 | 5.65 | 0.64
21307+5049 | 21 32 31.50 | +51 02 22.0 | 94.2617 | -0.41114 | 2.97 | 0.13 | -47.98 | 3.53 | 7.39 | 5.28
21334+5039 | 21 35 09.20 | +50 53 09.0 | 94.46296 | -0.80404 | 4.65 | 0.15 | -45.32 | 3.44 | 10 | 4.95
21340+5339 | 21 35 43.90 | +53 52 59.0 | 96.541839 | 1.356916 | 5.22 | 0.37 | -69.49 | 3.41 | 11.3 | 9.31
21360+5607 | 21 37 38.84 | +56 21 21.3 | 98.40111 | 3.011709 | 2 | 0.21 | 5.51 | 1.58 | 5.2 | $\ldots$
21379+5106 | 21 39 41.12 | +51 20 35.6 | 95.298186 | -0.937454 | 4.68 | 0.25 | -42.26 | 9.25 | 33.47 | 4.53
21498+7053 | 21 50 40.14 | +71 07 46.7 | 109.298672 | 13.215891 | 3.01 | 0.18 | -11.36 | 3.15 | 11.74 | 0.92
21548+5747 | 21 56 29.70 | +58 01 35.0 | 101.433682 | 2.6534 | 3.83 | 0.17 | -2.11 | 3.07 | 9.99 | 0.16
∗22045+5800 | 22 06 13.80 | +58 15 28.0 | 102.59963 | 2.064894 | 3.06 | 0.19 | -27.73 | 2.47 | 16.51 | 2.52
22126+7443 | 22 13 19.76 | +74 58 24.0 | 113.085841 | 15.213202 | 1.67 | 0.2 | -6.3 | 4.12 | 5.65 | 0.44
22199+6322 | 22 21 30.20 | +63 36 23.0 | 107.164645 | 5.423008 | 6.82 | 0.17 | -9.49 | 5.64 | 10.4 | 0.78
∗22202+6317 | 22 21 55.80 | +63 32 39.0 | 107.170717 | 5.34498 | 5.84 | 0.14 | -10.81 | 6.02 | 11.73 | 0.89
∗22262+5938 | 22 28 06.40 | +59 53 53.0 | 105.848255 | 1.868516 | 1.35 | 0.18 | -12.98 | 2.95 | 3.91 | 1.12
∗22321+5829 | 22 34 02.00 | +58 44 39.0 | 105.90725 | 0.49037 | 3.12 | 0.19 | -52.63 | 3.11 | 14.8 | 4.91
∗22345+5917 | 22 36 24.00 | +59 32 42.0 | 106.569534 | 1.033382 | 5.05 | 0.23 | -12.65 | 4.33 | 9.12 | 1.06
∗22528+5936 | 22 54 49.61 | +59 52 48.1 | 108.782348 | 0.251085 | 5.24 | 0.19 | -52.94 | 4.55 | 12.6 | 4.87
∗22567+6113 | 22 58 47.90 | +61 30 04.0 | 109.913964 | 1.509566 | 2.54 | 0.15 | -7.44 | 1.81 | 7.4 | 0.55
∗23065+5927 | 23 08 39.20 | +59 44 04.0 | 110.308161 | -0.594135 | 4.64 | 0.21 | -36.27 | 5.87 | 17.8 | 2.76
∗23077+5925 | 23 09 54.80 | +59 41 55.0 | 110.441108 | -0.688177 | 2.41 | 0.23 | -51.38 | 2.38 | 14.3 | 4.52
∗23079+5932 | 23 10 03.60 | +59 48 24.0 | 110.499383 | -0.595324 | 4.37 | 0.17 | -52.74 | 3.46 | 16.51 | 4.8
∗23091+6211 | 23 11 15.96 | +62 27 34.5 | 111.6394 | 1.804636 | 5.36 | 0.24 | -10.91 | 1.59 | 5.21 | 0.78
∗23215+5826 | 23 23 48.70 | +58 43 15.0 | 111.753786 | -2.236154 | 5.87 | 0.14 | -47 | 3.02 | 25.2 | 3.81
∗23330+6437 | 23 35 23.41 | +64 54 29.1 | 114.978684 | 3.217181 | 4.28 | 0.2 | -64.54 | 2.95 | 8.25 | 7.42
∗23331+6523 | 23 35 27.60 | +65 39 42.0 | 115.204924 | 3.936178 | 2.75 | 0.2 | -24.94 | 3.96 | 7.8 | 1.7
∗23486+5958 | 23 51 05.70 | +60 15 11.0 | 115.50882 | -1.74603 | 1.76 | 0.23 | -3.45 | 3.87 | 6.52 | 0.19
23494+6155 | 23 51 54.10 | +62 11 58.0 | 116.051402 | 0.125665 | 0.86 | 0.15 | -49.29 | 9.28 | 8.26 | 4.1
23500+6729 | 23 52 27.80 | +67 46 24.0 | 117.3816 | 5.53945 | 2.16 | 0.19 | -6.84 | 2.05 | 3.5 | 0.41
∗23504+6012 | 23 52 58.20 | +60 28 45.0 | 115.7862 | -1.57878 | 2.03 | 0.21 | -39.08 | 7.04 | 10 | 2.81
∗23592+6716 | 00 01 49.50 | +67 32 59.0 | 118.20797 | 5.13425 | 5.5 | 0.17 | -16.51 | 4.61 | 16.51 | 1.05
Table 1: $Continued$
We also use IRAS PSC and MSX PSC to investigate if there are infrared sources
around these CO sources within 5 arcmin. Table 2 demonstrates the number of
IRAS point source and MSX point source around the 201 sources, and notes the
association of the IRAS point source and MSX point source by n or a tags.
Column (1) is the IRAS name. In the columns (2) and (3) give the number of the
IRAS point source and MSX point source in the region within 5 arcmin centered
with these sources sample. Column (4) show the note for the association
between columns (2) and (3). Label a represents MSX point source overlap our
selected CO source. Label n means There is no MSX source in the position of
central source. The Columns (4) and (5) denotes the type and identification of
these 201 observed sources.
Table 2: Detected IRAS and MSX sources around the selected sources Name | IRAS Point Sources | MSX Point Sources | Note | Type | Identification
---|---|---|---|---|---
| (Sky coverage within 5 arcmin) | (Sky coverage within 5 arcmin) | | |
(1) | (2) | (3) | (4) | (5) | (6)
00021+6604 | 2 | 0 | | IR |
00056+6605 | 1 | 1 | n | IR |
00070+6516 | 1 | 1 | a | IR |
00116+6716 | 1 | 0 | | IR |
00289+6327 | 1 | 1 | a | IR |
00342+6347 | 2 | 1 | a | IR |
00412+6638 | 1 | 2 | n | G | 2MASX J00441534+6654411
00455+6137 | 2 | 2 | n | IR |
00468+6527 | 1 | 2 | n | YSO | 45P 26
00484+6531 | 2 | 1 | a | IR |
00519+6535 | 3 | 2 | a | HII | SH 2-183
01166+6635 | 1 | 0 | | IR |
01584+6706 | 1 | 0 | | G | 2MFGC 01551
02220+6107 | 1 | 0 | | IR |
02244+6035 | 2 | 0 | | DNe | LDN 1365
02425+6851 | 1 | 0 | | IR |
02455+6034 | 2 | 16 | a | HII |
02459+6029 | 2 | 12 | a | HII | KSP2003] J024947.30+604210.1
02485+6902 | 2 | 0 | | IR |
02499+6911 | 2 | 0 | | IR |
03183+6321 | 2 | 0 | | G | 2MASX J03223621+6332054
03211+5446 | 4 | 3 | n | HII |
03245+3002 | 2 | 0 | | IR |
03248+6551 | 1 | 0 | | G | 2MASX J03292042+6601389
03260+3111 | 3 | 1 | a | IR | NGC 1333 NED01
03275+5450 | 1 | 1 | a | IR |
03463+5331 | 1 | 0 | | IR |
04034+5107 | 2 | 2 | n | IR |
04044+5110 | 3 | 0 | | IR |
04088+3834 | 3 | 0 | | IR |
04173+4328 | 1 | 0 | | G | 2MASX J04205315+4336158
04173+4524 | 1 | 0 | | IR |
04271+3502 | 1 | 0 | | IR |
04299+2915 | 1 | 0 | | IR |
04307+5209 | 1 | 0 | | IR |
04319+5056 | 1 | 0 | | IR |
04324+5102 | 2 | 2 | a | IR | 2MASX J04361587+5108075
04324+5106 | 2 | 3 | a | IR | 2MASX J04362105+5112546
04335+5110 | 1 | 0 | | IR |
04368+2557 | 2 | 0 | | IR |
04381+2540 | 1 | 0 | | IR | 2MASX J04411273+2546360
04587+4411 | 1 | 2 | n | IR |
05075+3755 | 1 | 4 | n | IR | B2 0507+37
05236+0620 | 1 | 0 | | G | 2MASX J05261836+0622571
05335+3609 | 1 | 1 | a | G | 2MASX J05365322+3610508
05356-0530 | 2 | 1 | a | IR |
05363+3127 | 1 | 0 | | IR |
05366+3601 | 3 | 3 | a | IR |
05375+3536 | 2 | 6 | a | IR |
05375+3540 | 3 | 9 | a | HII | WN B0537.5+3540
05379+3515 | 1 | 0 | | IR |
05381-0921 | 2 | 0 | | IR |
05387-0924 | 1 | 0 | | Or* | V* V1792 Ori
05394-0151 | 8 | 7 | n | IR |
05399+2631 | 1 | 0 | | IR |
05399-0121 | 1 | 0 | | YSO | JCMTSF J054227.9-012003
05399-1001 | 3 | 0 | | IR |
05404-0220 | 1 | 0 | | Em* | HD 38087
05411+3302 | 1 | 0 | | G | 2MASX J05442482+3303312
05413-0104 | 2 | 0 | | IR |
05437-0343 | 1 | 0 | | IR |
05445+0020 | 4 | 0 | | HII | NGC 2071 NED01
05462-0124 | 2 | 0 | | IR |
05575+2141 | 1 | 0 | | IR |
06084-0611 | 3 | 4 | n | IR |
06102+1537 | 1 | 0 | | IR |
06281+1039 | 1 | 2 | a | Rne | NAME Steine GN J0630.8+1037
06282+0423 | 4 | 6 | a | IR |
06294+0352 | 1 | 1 | n | YSO | 2MASS J06320752+0350075
06308+0402 | 1 | 11 | a | YSO |
06318+0420 | 4 | 11 | a | IR |
06343+0425 | 2 | 1 | a | IR |
06373+1053 | 2 | 0 | | IR |
06423+0006 | 1 | 2 | n | IR |
06425+0038 | 1 | 0 | | IR |
06511-0507 | 1 | 0 | | IR |
06545-0251 | 1 | 0 | | IR |
07100-1110 | 1 | 1 | a | G | 2MASX J07122445-1115336
07157-1830 | 2 | 1 | n | IR |
07221-2544 | 3 | 2 | n | ISM | BRAN 23
07297-1926 | 3 | 2 | n | IR |
08004-2815 | 2 | 5 | a | ISM | BRAN 98
17207-3404 | 3 | 38 | a | IR |
17364-1946 | 1 | 0 | | IR |
17369-1945 | 1 | 0 | | IR |
17392-3309 | 3 | 11 | n | IR |
17419-3150 | 3 | 13 | a | IR |
17520-2731 | 5 | 14 | n | IR |
17555-2136 | 1 | 9 | n | IR |
18111-2028 | 2 | 8 | n | IR |
18134-1942 | 1 | 9 | a | HII |
18136-1347 | 1 | 4 | a | HII |
18148-0440 | 1 | 1 | a | YSO | NAME LDN 483 FIR
18188-1631 | 3 | 8 | n | IR |
18273+0034 | 2 | 1 | a | COR |
18282-1529 | 2 | 8 | n | IR |
18316-0602 | 1 | 12 | a | HII | GAL 025.65+01.05
18385-0755 | 2 | 9 | n | IR |
18473+0131 | 3 | 9 | n | IR |
18532+0420 | 3 | 7 | a | IR |
18567+0700 | 2 | 11 | a | HII | SH 2-75
18572+0057 | 2 | 4 | n | IR |
18583+0136 | 2 | 6 | n | IR |
19207+1809 | 1 | 4 | a | IR |
19223+1826 | 2 | 5 | a | IR |
19258+1919 | 2 | 0 | | IR |
19300+2158 | 2 | 6 | a | YSO | 2MASS J19321348+2204566
19387+2658 | 2 | 2 | a | IR |
19407+2454 | 2 | 2 | a | IR |
19408+2554 | 2 | 4 | a | IR |
19413+2349 | 2 | 8 | a | IR |
19454+2625 | 3 | 5 | n | IR |
19457+2357 | 2 | 4 | a | IR |
19458+2442 | 3 | 13 | a | Radio |
19508+2705 | 3 | 6 | n | IR |
19560+3135 | 2 | 5 | a | IR |
20016+3243 | 1 | 0 | | IR |
20033+2848 | 3 | 1 | a | IR |
20050+2720 | 2 | 4 | a | COR |
20051+3016 | 3 | 4 | a | IR |
20051+3435 | 1 | 6 | a | Star |
20068+3328 | 1 | 12 | a | IR |
20072+2720 | 3 | 2 | n | IR |
20078+3254 | 3 | 8 | a | IR |
20184+3936 | 1 | 8 | a | IR |
20190+4011 | 2 | 9 | n | IR |
20190+4102 | 1 | 6 | a | G | 2MASX J20204749+4112067
20197+3745 | 3 | 13 | a | IR |
20227+4154 | 3 | 1 | a | HII |
20228+4215 | 1 | 11 | a | IR |
20243+3752 | 2 | 3 | n | RadioS | B2 2024+37
20243+3853 | 3 | 5 | a | RadioS |
20274+4219 | 1 | 0 | | IR |
20275+4001 | 3 | 15 | a | YSO |
20285+3939 | 3 | 10 | a | RadioS |
20290+4052 | 1 | 6 | a | IR |
20293+3952 | 1 | 2 | n | IR |
20300+3909 | 2 | 11 | a | IR |
20300+4058 | 4 | 22 | a | IR |
20306+4005 | 2 | 27 | a | G | 2MASX J20322791+4016081
20309+4257 | 2 | 6 | n | IR |
20321+4112 | 4 | 8 | a | RadioS | 18P 61
20322+4031 | 1 | 6 | a | IR |
20329+3846 | 1 | 3 | n | IR |
20332+4124 | 2 | 10 | a | IR |
20333+4102 | 4 | 16 | a | IR |
20343+4129 | 1 | 6 | a | IR |
20346+4706 | 2 | 2 | n | IR |
20350+4126 | 5 | 22 | a | HII |
20364+3816 | 1 | 0 | | IR |
20436+5849 | 3 | 0 | | IR |
20489+4410 | 5 | 11 | a | Cloud | JCMTSF J205042.8+442157
20527+4957 | 2 | 3 | n | IR |
20550+5158 | 1 | 0 | | IR |
20553+5208 | 1 | 1 | a | IR |
20565+5003 | 2 | 4 | a | IR |
20582+7724 | 1 | 0 | | IR |
20588+5215 | 3 | 1 | a | TT* | 2MASS J21002140+5227094
21005+5217 | 1 | 2 | a | IR |
21007+4847 | 5 | 0 | | RadioS |
21007+5036 | 1 | 1 | a | G | 2MASX J21022178+5048346
21020+4939 | 1 | 2 | n | IR |
21025+4912 | 2 | 0 | | IR |
21036+4927 | 3 | 5 | a | HII |
21101+5150 | 2 | 0 | | IR |
21106+5206 | 2 | 1 | n | IR |
21131+5221 | 1 | 1 | a | IR |
21136+5317 | 2 | 1 | n | IR |
21169+6804 | 1 | 0 | | IR | 2MASX J21173843+6817340
21182+4641 | 2 | 0 | | IR |
21184+5507 | 1 | 2 | n | IR |
21202+5157 | 6 | 5 | n | RadioS |
21207+5338 | 1 | 0 | | IR |
21307+5049 | 1 | 1 | a | IR |
21334+5039 | 2 | 3 | a | HII | 2MASS J21350991+5053048
21340+5339 | 4 | 6 | a | IR |
21360+5607 | 2 | 0 | | IR |
21379+5106 | 1 | 2 | a | G | 2MASX J21394111+5120356
21498+7053 | 1 | 0 | | G | 2MASX J21504014+7107468
21548+5747 | 1 | 5 | n | IR |
22045+5800 | 2 | 1 | a | IR |
22126+7443 | 1 | 0 | | G | 2MASX J22131981+7458237
22199+6322 | 4 | 0 | | Dne | LDN 1204B
22202+6317 | 2 | 0 | | IR |
22262+5938 | 3 | 0 | | IR |
22321+5829 | 1 | 4 | a | PN | PN PM 1-338
22345+5917 | 1 | 1 | a | IR |
22528+5936 | 1 | 1 | a | G | 2MASX J22544960+5952481
22567+6113 | 1 | 2 | n | IR |
23065+5927 | 1 | 4 | a | IR |
23077+5925 | 1 | 12 | a | IR |
23079+5932 | 1 | 4 | a | IR |
23091+6211 | 2 | 1 | a | Em* | 2MASS J23111596+6227344
23215+5826 | 2 | 1 | a | IR |
23330+6437 | 2 | 5 | a | G | 2MASX J23352340+6454290
23331+6523 | 2 | 0 | | IR |
23486+5958 | 1 | 0 | | IR |
23494+6155 | 2 | 1 | a | IR |
23500+6729 | 2 | 0 | | IR |
23504+6012 | 1 | 10 | a | IR |
23592+6716 | 4 | 3 | n | IR |
Table 2: $Continued$
### 4.2 The CO and its isotope molecules emission from sources
A complete set of the ${}^{12}\\!$CO(1-0),${}^{13}\\!$CO(1-0) and
C${}^{18}\\!$O(1-0) line emissions toward 201 sources is published on line.
The ${}^{12}\\!$CO(1-0),${}^{13}\\!$CO(1-0) and C${}^{18}\\!$O(1-0) spectrum
are plotted in blue, green and red colors respectively. These sources are
classified into four types of the possible cloud-cloud collision region
candidates. Type 1 illustrates the lines with different optical thickness of
each source have the similar profile. The ${}^{12}\\!$CO, ${}^{13}\\!$CO and
C${}^{18}\\!$O($J$=1-0) spectra have double-peaked or multi-peaked main line
and the peaks are almost located at the same velocity. It contains 38 sources.
Several reasons could induce the double-peaked line profile in optically thick
${}^{12}\\!$CO lines, which includes self-absorption, outflow, infall,
rotation and two clouds configuration, etc. People have already compared and
contrasted the spectral profile of ${}^{12}\\!$CO with the profile of
${}^{13}\\!$CO and C${}^{18}\\!$O, and they summarize the feature in different
situation (Wang et al., 2004; Dobashi, 1993; Walsh et al., 2002; Xin & Wang,
2008). Toward this type of source, the profile of the optically thick
${}^{12}\\!$CO lines fits for the profiles of the optically thin
${}^{13}\\!$CO and C${}^{18}\\!$O very well. So the possibility of the self-
absorption, outflow, infall and rotation is remote. It could be the two
separated cloud in double-peaked profile region. We need to investigate in
multi-wavelength toward these possible cloud-cloud collision regions in
future. In type 2, the C${}^{18}\\!$O($J$=1-0) spectra have the bad SNR
(signal to noise ratio) toward these 71 sources. But the profile of the
optically thick ${}^{12}\\!$CO lines fits for the profiles of the optically
thin ${}^{13}\\!$CO well, also show the double-peaked or multi-peaked profile.
This type of the sources are still treated as possible cloud-cloud collision
candidates. Type 3 contains the spectra with ${}^{12}\\!$CO double-peaked or
multi-peaked main lines, the profiles of the ${}^{13}\\!$CO and C${}^{18}\\!$O
line fit the profile of the ${}^{12}\\!$CO line partly. Considered the
uncertain factors, these sources can’t be made a decision as the non cloud-
cloud collision candidates, we need further observation toward these 58
sources to confirm it. The last 34 sources are plotted in type 4. One part of
these sources all show the ${}^{13}\\!$CO, C${}^{18}\\!$O and ${}^{12}\\!$CO
lines with single peak. The other sources present the ${}^{13}\\!$CO and
C${}^{18}\\!$O lines with single peak, and the peak is located between the
double peak of the optically thick ${}^{12}\\!$CO lines. It suggests that this
type of optical thick profile might be induced by self-absorption, infall, etc
(Wang et al., 2004; Zhou, 1993). So we treat these sources as non cloud-cloud
collision candidates.
We also check the galactic distribution for the 4 types of sources in Figure
6. The 4 types of sources are located around the Galactic plane mostly and the
distribution have similarity with the whole sample. It means the distribution
toward the 4 types of the sources are uniform in our sample. Figure 7 plots
the kinematic distances of the 4 types of sources. The sources of type 1
locate in the first and second Galactic quadrant. Some sources of type 1 are
associated with the Perseus arm possibly. The sources of type 2 and 3 have the
similar configuration of the distribution. The sources in Type 4 just locate
around 5 kpc. Figure 8 demonstrates the $L-V_{LSR}$ diagram for the 4 types.
The distributions for 4 types are similar with the distribution of 201 sources
roughly. And the last statistic data for the peak antenna temperature of the
sources in 4 types show some different distribution in Figure 9. The sources
in Type 1 show a bimodal distribution. From 2 K to 7 K, there are large
dispersion. The $T_{A}^{*}$ range in type 2 are concentrated from 2 K to 5 K.
The peak of distribution in Type 3 lies on the 1-2 K. Type 4 shows a wide
dispersion during 0-7 K.
Our next work will carry on mapping observations toward these possible cloud
collision region. We have selected some sources from our survey list to
observe recently. We want to develop new methods to identify the cloud-cloud
collision region and get some new results of the star formation triggered by
cloud-cloud collision.
### 4.3 Derived parameters
Because of bad SNR (signal to noise ratio) with the ${}^{13}\\!$CO(1-0) line
toward some sources in type 2, 3 and 4, we only calculate the parameters of
the 38 sources in type 1. Derived parameters of the sources in type 1 have
been listed in Table 3. $T_{R}^{*}(^{12}CO)$ and $T_{R}^{*}(^{13}CO)$ are the
${}^{12}\\!$CO and ${}^{13}\\!$CO(1-0) radiation temperature:
$T^{*}_{R}=T^{*}_{A}/\eta_{mb}$, where T${}_{A}^{*}$ is the antenna
temperature corrected with atmospheric attenuation and other losses (these are
done by the observatory ), and $\eta_{mb}$ is the main beam efficiency.
$\upsilon_{13}$ is the Gaussian fit of the $V_{LSR}$(${}^{13}\\!$CO) and
$\Delta\upsilon_{13}$ is its full width at half-maximum (FWHM). The cores’
distances from us are obtained from literatures (Wouterloot et al., 1989). The
${}^{12}\\!$CO(1-0) line is usually considered as optically thick in massive
CO cores. Assuming local thermal equilibrium (LTE), the excitation temperature
$T_{ex}$ of the ${}^{13}\\!$CO(1-0) transition is the same as that of the
${}^{12}\\!$CO (Garden et al., 1991).
$T_{ex}=\frac{h\nu}{k}\\{\ln[1+(\frac{kT_{R}^{*}(^{12}CO)}{h\nu}+\frac{1}{\exp(\frac{h\nu}{kT_{bg}}-1)})^{-1}]\\}^{-1}$
(1)
Where the $T_{bg}$=2.7K. The ${}^{13}\\!$CO(1-0) transition is usually
optically thin. On the LTE assumption, the opacity of ${}^{13}\\!$CO(1-0) is
approximately (Guan et al., 2008)
$\tau(^{13}CO)\approx-\ln[1-\frac{T_{R}^{*}(^{13}CO)}{T_{R}^{*}(^{12}CO)}]$
(2)
Sometimes the ${}^{12}\\!$CO spectra are self-reversed by absorption from the
cold part of the cloud core outer. In that case the ${}^{13}\\!$CO opacity
will be underestimated. The ${}^{13}\\!$CO column density is given by (Garden
et al., 1991)
$N(^{13}CO)=\frac{3k}{8\pi^{3}B\mu^{2}}\frac{\exp[\frac{hBJ(J+1)}{kT_{ex}}]}{J+1}\times\frac{T_{ex}+hB/3k}{[1-exp(\frac{-h\nu}{kT_{ex}})]}\int\tau_{\nu}d\nu$
(3)
Where B is the rotational constant of ${}^{13}\\!$CO, $\mu$ is the permanent
dipole moment and $J$ is the rotational quantum number of the lower state.
We adopt the ${}^{12}\\!$CO and ${}^{13}\\!$CO abundances to be
$\frac{N(H_{2})}{N(^{12}CO)}=10000,\frac{N(^{12}CO)}{N(^{13}CO)}=89$ (4)
### 4.4 Infrared results
We do the color cuts used for the IRAS sources in our survey for cloud-cloud
collision candidates. The color cuts used here were derived from values in the
literature and were taken from studies of several types of objects: T Tauri
stars, low-mass and intermediate-mass YSOs, and massive YSOs. Table 4 lists
the color cuts we used. We use the criteria to select the YSOs candidates, but
it’s not a unique classification method. The choice of color cuts is merely
meant to be as inclusive as possible. All the results plotted by color-color
diagram are represented in Figure 10. The sources of the 3 types which are
considered as possible cloud-cloud collision candidates are marked on the map.
Object Type | [25-12] | [60-25] | [100-60] | Reference
---|---|---|---|---
T Tauri stars | 0-0.6 | $<$0.5 | … | 1, 2
Intermediate-mass YSOs | 0-1.0 | 0.4-1.0 | … | 3
High-mass YSOs | 0.17-1.5 | 0.23-1.13 | 0.08-0.57 | 4
0.86Where $[\lambda 1-\lambda 2]=\log(F1/F2)$. 0.86 (1)Harris, Clegg & Hughes
1988.(2)Prusti,Adorf, & Meurs, 1992. (3)Beichman et.al 1986. (4) Chan,Henning,
& Schreyer 1996.
According to the range of the IRAS cuts given by Table 4, we summarize that 3
T Tauri stars, 25 Intermediate-mass YSOs and 24 High-mass YSOs are identified
from the sources of type 1. There are 2 T Tauri stars, 30 Intermediate-mass
YSOs and 22 High-mass YSOs identified in type 2. In type 3, 23 Intermediate-
mass YSOs and 27 High-mass YSOs are identified from the 58 sources and no T
Tauri stars.
## 5 Conclusions
In this paper, according to the criteria for cloud-cloud collision sample, we
observed the ${}^{12}\\!$CO, ${}^{13}\\!$CO and C${}^{18}\\!$O($J$=1-0) lines
in 201 IRAS sources which are selected as cloud-cloud collision candidates.
The criteria are as follows :
1.A portion of criteria given by Vallee (1995) which describe the spectrum
properties of the possible cloud-cloud collision region.
2.The line profile features show the double peak or multipeak and the velocity
of peaks are adjacent.
These sources are located over a wide range of the Galactocentric distances,
and associated with the star formation region partly. Then the 201 sources are
classified into 4 types by the close fit between the profile of optically
thick ${}^{12}\\!$CO lines and the profiles of the optically thin
${}^{13}\\!$CO and C${}^{18}\\!$O. The IRAS flux in four bands have been used
to identify sources with the colors of YSOs. Considering the association of
the IRAS and the MSX PSC, the sources in type 1, 2 and 3 could be selected to
do the multi-wavelength mapping. To see if there is star formation triggered
by cloud-cloud collision. Toward two possible cloud-cloud collision candidates
( IRAS 02459+6029 and 22528+5936 ) selected from our survey, combining the CO
mapping and multi-wavelength study we prove that the IRAS 02459+6029 could be
a cloud-cloud collision sample and the IRAS 22528+5936 could be two separate
clouds without colliding (Li & Wang, accepted by RAA 2012). And we also
develop our new criteria to identify the cloud-cloud collision region in that
paper. This survey is very foundational and efficient for selected a possible
cloud-cloud collision region.
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|
arxiv-papers
| 2012-12-01T08:41:07 |
2024-09-04T02:49:38.775648
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Nan Li, Jun-Jie Wang",
"submitter": "Li Nan",
"url": "https://arxiv.org/abs/1212.0084"
}
|
1212.0159
|
Illumina Sequencing Artifacts Revealed by Connectivity Analysis of Metagenomic
Datasets
Adina Chuang Howe1,2, Jason Pell3, Rosangela Canino-Koning3, Rachel
Mackelprang4, Susannah Tringe4, Janet Jansson4,5 , James M. Tiedje1,2, and C.
Titus Brown1,3∗
1 Microbiology and Molecular Genetics, Michigan State University, East
Lansing, MI, USA
2 Plant, Soil, and Microbial Sciences, Michigan State University, East
Lansing, MI, USA
3 Computer Science and Engineering, Michigan State University, East Lansing,
MI, USA
4 Department of Energy (DOE) Joint Genome Institute, Walnut Creek, CA, USA
5 Lawrence Berkeley National Laboratory, Genomics Division, Berkeley, CA, USA
$\ast$ E-mail: [email protected]
## Abstract
Sequencing errors and biases in metagenomic datasets affect coverage-based
assemblies and are often ignored during analysis. Here, we analyze read
connectivity in metagenomes and identify the presence of problematic and
likely a-biological connectivity within metagenome assembly graphs.
Specifically, we identify highly connected sequences which join a large
proportion of reads within each real metagenome. These sequences show
position-specific bias in shotgun reads, suggestive of sequencing artifacts,
and are only minimally incorporated into contigs by assembly. The removal of
these sequences prior to assembly results in similar assembly content for most
metagenomes and enables the use of graph partitioning to decrease assembly
memory and time requirements.
## Introduction
With the rapid decrease in the costs of sequencing, we can now achieve the
sequencing depth necessary to study microbes from even the most complex
environments [1, 2]. Deep metagenomic sequencing efforts in permafrost soil,
human gut, cow rumen, and surface water have provided insights into the
genetic and biochemical diversity of environmental microbial populations [1,
3, 2] and their involvement in responding to environmental changes [4]. These
metagenomic studies have all leveraged _de novo_ metagenomic assembly of short
reads for functional and phylogenetic analyses. _De novo_ assembly is an
advantageous approach to sequence analysis as it reduces the dataset size by
collapsing the more numerous short reads into fewer contigs and enables
improved annotation-based approaches by providing longer sequences [5, 6].
Furthermore, it does not rely on the a priori availability of reference
genomes to enable identification of gene content or operon structure [1, 3].
Although _de novo_ metagenomic assembly is a promising approach for
metagenomic sequence analysis, it is complicated by the variable coverage of
sequencing reads from mixed populations in the environment and their
associated sequencing errors and biases [7, 8]. Several metagenome-specific
assemblers have been developed to deal with variable coverage communities,
including Meta-IDBA [9], MetaVelvet [10], and SOAPdenovo [11]. These
assemblers rely on analysis of local sequencing coverage to help build
assemblies and thus are sensitive to the effects of sequencing errors and
biases on coverage estimations of the underlying dataset. The effects of
sequencing errors on _de novo_ assembly has been demonstrated in simulated
metagenomes [12, 7, 8] and isolate genomes [13, 14], but these datasets do not
necessarily represent real metagenomic data. Specifically, these models
exclude the presence of known non-biological sequencing biases which hinder
assembly approaches [15, 16, 17].
In this study, we examine metagenomic datasets for the presence of artificial
sequencing biases that affect assembly graph structure, extending previous
work to large and complex datasets produced from the Illumina platform. We
characterize sequence connectivity in an assembly graph, identifying potential
sequencing biases in regions where numerous reads are connected together.
Within metagenomic datasets, we find that there exist highly connected
sequences which partially originate from sequencing artifacts. Moreover, these
sequences limit approaches to divide or partition large datasets for further
analysis, and may introduce artifacts into assemblies. Here, we identify and
characterize these highly connected sequences and examine the effects of
removing these sequences on downstream assemblies.
## Results
### Connectivity analysis of metagenome datasets
#### Presence of a single, highly connected lump in all datasets
We selected datasets from three medium to high diversity metagenomes from the
human gut [2], cow rumen [1], and agricultural soil (SRX099904 and SRX099905)
(Table 1). To evaluate the effects of sequencing coverage, we included two
subsets of the 520 million read soil metagenome containing 50 and 100 million
reads. We also included a previously published error-free simulated metagenome
based on a mixture of 112 reference genomes [8].
We evaluated read connectivity by partitioning reads into disconnected
components with a de Bruijn graph representation [18]. This approach
guarantees that reads in different partitions do not connect to each other and
permits the separate assembly and analysis of each partition. For each
metagenome, regardless of origin, we found a single dominant, highly connected
set of sequencing reads which we henceforth refer to as the “lump” of the
dataset (Table 1). This lump contained the largest subset of connected
sequencing reads and varied in size among the datasets, ranging from 5% of
total reads in the simulated metagenome to 75% of total reads in the human gut
metagenome. For the soil datasets, as sequencing coverage (e.g., the fraction
of reads mapped to an assembly) increased from 1.4 to 4.7 to 5.6%, the lump
size increased more dramatically from 7 to 15 to 35% of all reads, indicating
increasingly larger connectivity between sequences with more sequencing.
#### Characterizing connectivity in the dominant partition
We characterized the connectivity of sequences within each lump by estimating
the average local graph density from each k-mer (k=32 unless otherwise stated)
in the assembly graph (see Methods). Here, local graph density is a
measurement of total connected reads within a fixed radius. Sequences in the
identified metagenomic lumps were characterized by very high local graph
densities: between 22 to 50% of the total nodes in metagenomic lump assembly
graphs had average graph densities greater than 20 (Table 1). This indicates
that these nodes were in very nonlinear portions of the assembly graph and had
high connectivity. In comparison, 17% of the total nodes in the simulated lump
had an average local graph density greater than 20, and fewer than 2% of the
nodes in the entire simulated data (all partitions) set had an average graph
density higher than 20.
We next assessed the extent to which graph density varied by position along
the sequencing reads. The degree of position-specific variation of graph
densities was estimated by calculating the average local graph density within
ten steps of every k-mer by position in each read. In all environmental
metagenomic reads, we observed variation in graph density at the 3’-end region
of reads (Fig. 1). In soil metagenomes, we observed the most dramatic
variation with local graph density increasing in sequences located at the
3’-end of the reads. Notably, this trend was not present in the simulated
dataset.
Next, we performed an exhaustive traversal of the assembly graph and
identified the specific sequences within dense regions of the assembly graph
which consistently contributed to high connectivity. We observed that this
subset of sequences was also found to exhibit position-specific variation
within sequencing reads, with the exception of these sequences in the
simulated dataset (Fig. 2, solid lines). As with local density trends,
position-specific trends in the location of these sequences also varied
between metagenomes. As sequencing coverage increased among metagenomes, the
amount of 3’-end variation appeared to decrease (e.g., the soils) or increase
(e.g., rumen and human gut).
### Effects of removing highly connected sequences on assembly
#### Removal of highly connected sequences enables graph partitioning of
metagenome
Since these highly connected sequences exhibited position-specific variation
indicative of sequences of non-biological origin, we removed them and assessed
the effect of their removal on assembly (see Methods). We found that by
removing these k-mers, we could effectively break apart metagenomic lumps, and
the resulting largest partition of connected reads in each metagenome was
reduced to less than 7% of the total reads in the lump. Partitioning also had
the effect of significantly decreasing assembly time and memory usage [18].
#### Removing highly connected sequences resulted in minimal losses of
reference genes
We explored the extent to which the identified highly connected sequences
impacted assembly by first evaluating the effects of the removal of these
sequences from the simulated lump. The assembly of the reads in the original,
unfiltered simulated lump and that of the reads remaining after removing
highly connected sequences (the filtered assembly) were compared for three
assemblers: Velvet [19], Meta-IDBA [9], and SOAPdenovo [11]. Based on the
total assembly length of contigs greater than 300 bp, filtered assemblies of
the simulated metagenome resulted in a loss of between 4 - 16% of total
assembly length (Table 2). In general, the filtered assemblies contained fewer
total contigs than unfiltered assemblies, while the maximum contig size
increased in the Velvet assembly but decreased in the Meta-IDBA and SOAPdenovo
assemblies. Direct comparisons of the unfiltered and filtered simulated
metagenome assemblies found that the filtered assemblies comprised on average
89% of the unfiltered assemblies, and the unfiltered assemblies contained
nearly all (97%) of the filtered assembled sequences. Despite the removal of
over 3% of the total unique 32-mers in the simulated metagenome, the resulting
filtered assemblies lost only 3-15% of annotated original reference genes
(Table 3).
We next evaluated the effects of removing highly connected sequences in real
datasets. Similar to the simulated assemblies, the removal of highly connected
sequences for all metagenomes and assemblers resulted in a decrease of total
number of contigs and assembly length (Table 2). In general, filtered
assemblies were largely contained within unfiltered assemblies and comprised
51-87% of the unfiltered assembly. The observed changes in metagenomic
assemblies were difficult to evaluate as no reference genomes exist, and a
decrease in assembly length may actually be beneficial if it eliminates
contigs that incorporate sequencing artifacts. To aid in this evaluation, we
used the previously published set of rumen draft genomes from _de novo_
assembly efforts of high abundance sequences in the rumen metagenome [1].
Overall, we found that removal of highly connected sequences from the rumen
dataset resulted in 9-13% loss of sequences present in draft reference genomes
(Table 3).
#### Unfiltered assemblies contained only a small fraction of highly connected
sequences
To further study the effects of highly connected sequences, we examined their
incorporation into unfiltered assemblies. Except in the human gut sample,
fewer than 2% of highly connected sequences were incorporated by any assembler
(Table 4). Each assembled contig was divided into percentile bins and examined
for the presence of the previously identified highly connected sequences. We
found that contigs, especially in assemblies from Velvet and Meta-IDBA,
incorporated a larger fraction of these sequences at their ends relative to
other positions (Fig 3). The SOAPdenovo assembler incorporated fewer of the
highly connected sequences into its assembled contigs; in the simulated data
set, none of these sequences were assembled, and in the small soil data set
only 41 were assembled. For the human gut metagenome assemblies, millions of
the highly connected sequences were incorporated into assembled contigs,
comprising nearly 4% of all assembled sequences on Velvet contig ends (Fig 3).
#### Identifying origins of highly connected sequences in known reference
databases
For the simulated metagenome, we could identify the source of highly connected
k-mers using available reference genomes. Reference genes with multiple
perfect alignments to highly connected k-mers present in the dataset a minimum
of 50 times were identified (Table 5). Many of these sequences were from well-
conserved housekeeping genes involved in protein synthesis, cell transport,
and signaling. To determine possible biological sources of highly connected
sequences within real metagenomes, we compared the sequences shared between
the soil, rumen, and human gut metagenomes (a total of 241 million 32-mers).
Among these 7,586 shared sequences, we identified the closest reference
protein from the NCBI-nr database requiring complete sequence identity. Only
1,018 sequences (13%) matched existing reference proteins, and many of the
annotated sequences matched to genes conserved across multiple genomes. The
most abundant proteins conserved in greater than 3 genomes are shown in (Table
6), and largely encode for genes involved in protein biosynthesis, DNA
metabolism, and biochemical cofactors.
One potential cause of artificial high connectivity within metagenomes is the
presence of high abundance subsequences. Thus, we identified the subset of
highly connected k-mers which were also present with an abundance of greater
than 50 within each metagenome and their location in sequencing reads (Fig 2,
dotted lines). These high abundance k-mers comprised a very small proportion
of the identified highly connected sequences, less than 1% in the soils, 1.5%
in the rumen, and 6.4% in the human gut metagenomes, but the position-specific
variation of these sequences was very similar to the variation in the larger
set of highly connected k-mers.
We attempted to identify patterns within the sequences causing position
specific variation by examining the abundance distribution of 5-mers within
the highly abundant/highly connected 32-mers. There were significantly fewer
5-mers in the simulated sequences compared to those in metagenomes: 336 total
5-mers in the simulated data and from 425,572 to 221,085,228 total 5-mers in
the small soil and human gut datasets, respectively. In the simulated dataset,
the top ten most abundant unique k-mers made up 75% of the total 5-mers; in
contrast, in the metagenomes, k-mers were more evenly distributed: the top ten
most abundant 5-mers comprised less than 10% of the total 5-mers. The
cumulative abundance distribution of the ranked 5-mers shown in Fig. 4 shows
this even distribution in all of the real metagenomes. This suggests that
there is no single, easily-identifiable set of sequences at the root of the
highly connected component observed in real metagenomes.
## Discussion
### Sequencing artifacts are present in real metagenomes
Through assessing the connectivity of reads in several metagenomes, we
identified a disproportionately large subset of reads connected together
within an assembly graph, which we refer to as the “lump.” The total number of
reads in metagenomic lumps (7-75% of reads) was significantly larger than that
of simulated dataset (5% of reads) (Table 1). In the simulated data, this
component consists of reads connected by sequences conserved between multiple
genomes (Table 5). The larger size of this component within the soil, rumen,
and human gut metagenomes suggests that anomalous, non-biological connectivity
may be present within these lumps. Moreover, in the soil metagenomes, we
observed that for a 5% increase in sequencing coverage of the assembled
contigs, the amount of connectivity nearly doubled. While sequencing coverage
of assembled contigs increased slightly from 4.7 to 5.6% in the medium and
large soil metagenomes, the number of reads contained in the lump grew
significantly from 15 million to 182 million. Given the very high diversity
and very low coverage of these soil samples, the magnitude of the observed
increases in connectivity cannot be due simply to increased coverage. This
suggests the presence of non-biological features that falsely connect reads.
The superlinear increase in connectivity exhibited in these data sets
indicates that a form of preferential attachment is occurring in the graphs
[20]. This graph-theoretic phenomenon describes the results of a process where
highly connected nodes in a graph preferentially acquire new edges –
colloquially known as “the rich get richer.” In assembly, any systematic bias
towards producing specific subsequences from shotgun sequencing would lead to
a tendency to connect otherwise unrelated graph components; such a bias could
be biological (e.g. repeat present in multiple genomes or other highly
conserved DNA sequences), or non-biological (e.g., inclusion of sequencing
primers in reads or even a low-frequency trend towards producing specific
subsequences [21, 22, 23]).
We believe a significant component of the high connectivity that we see is of
non-biological origin. Shotgun sequencing is a random process and consequently
any position-specific variation within sequencing reads is unexpected and
probably originates from bias in sample preparation or the sequencing process
[15, 24, 16]. For the metagenomes studied here, we used two approaches to
examine characteristics of connectivity correlated to specific positions
within sequencing reads. First, we measured the connectivity of sequences at
specific positions within reads by calculating local graph density. Next, we
identified the specific k-mers which were consistently present in highly dense
regions of the assembly graph and evaluated their location within sequencing
reads. When these approaches were applied to the simulated dataset, we
observed no position-specific trends when assessing either local graph density
(Fig 1) or highly connected k-mers (Fig 2) as is consistent with the lack of
sequencing errors and variation in this dataset. In all real metagenomes,
however, we identified position-specific trends in reads for measurements of
both local graph density and the location of highly connected sequences,
clearly indicating the presence of sequencing artifacts. Although present in
all metagenomes, the direction of the variation varied between soil, rumen,
and human gut datasets, especially for the position-specific presence of
identified highly connected sequences. It is likely that there is a larger
presence of indirectly preferentially attached reads which are connected to
high coverage sequences of biological origins in higher coverage datasets,
such as the rumen and human gut. This preferential attachment of such reads
would result in increasing the number of total reads and consequently the
decrease the total fraction of highly connected k-mers (Fig 2, y-axis). This
trend is observed in the decreasing fractions of highly connected sequences at
the 3’ end of reads as sequencing coverage increased in the small, medium, to
large soil metagenomes and in the soil, rumen, to human gut metagenomes (Fig
2).
### Highly connected sequences are of unknown non-biological origin
We attempted to identify biological characteristics of highly connected
sequences. Among the highly connected sequences in the simulated dataset and
those shared by all metagenomes, we identified only a small fraction (13% in
simulated and less than 7% in metagenomes) which matched reference genes
associated with core biological functions (Table 5 and 6). This suggests that
the remaining sequences are either not present in known reference genes (i.e.,
repetitive or conserved non-coding regions) or originate from non-biological
sources. This supports the removal of these sequences for typical assembly and
annotation pipelines, where assembly is often followed by the identification
of protein coding regions.
Speculating that many of the highly connected sequences originated from high
abundance reads, we examined the most abundant subsequences. We found that
these subsequences (k-mers present more than 50x in the data set) displayed
similar trends for position-specific variation compared to their respective
sets of highly connected subsequences (Fig 2), indicating that they contribute
significantly to position-specific variation. We attempted to identify
signatures in these abundant, highly connected sequences from the simulated
and metagenomic datasets by looking at shorter k-mer profiles. In the
simulated dataset, we found that the total number of unique 5-mers was
significantly lower than in metagenomes and that the most abundant of these
5-mers comprised the large majority of the total. This result is consistent
with the presence of conserved biological motifs in the simulated dataset
which would result in a small number of highly abundant sequences; it would
also be consistent with the inclusion of sequencing primers in the data, were
this a real data set.
In contrast, within real metagenomic data, we found that the 5-mers are evenly
distributed and exhibit no specific sequence properties (Fig 4), making them
difficult to identify and evaluate. Most importantly, we were unable to
identify any characteristics that would explain their origin. In addition, a
G-C content analysis of the highly connective k-mers did not reveal any
systematic differences between the highly connected k-mers and the background
k-mer distribution.
When we reviewed the literature on random and systematic sequencing errors in
Illumina sequencing, we found many different types of sequencing errors: PCR
amplification errors prior to and during cluster generation; random sequencing
errors e.g. from miscalls of bases; sequencing errors triggered by specific
sequence motifs [25]; adaptor contamination; and post-adaptor read through. Of
these errors, only random sequencing errors and adaptor contamination and
readthrough would be biased towards the 3’ end of the read. However, random
sequencing error does not contribute to aberrant de Bruijn graph connectivity
[18], while adaptor contamination and readthrough would yield a sharply biased
5-mer distribution. The observed artifactual sequences thus do not match any
known set of random or systematic errors in Illumina sequencing.
Our current working hypothesis is that a low rate of false connections are
created by a low-frequency tendency towards producing certain k-mers in the
Illumina base calling software, as signal intensities decline. We cannot
verify this without access to the Illumina software or source code.
### Highly connected sequences are difficult to assemble
Not all of the observed connectivity within real metagenomes is artificial,
and our approach cannot differentiate between sequencing artifacts and real
biological connectivity. Therefore, removing highly connected sequences could
remove real biological signal in addition to sequencing artifacts. However, we
suspected that assemblers would be unable to generate contigs from highly
connected graph regions, and so even the real sequences would be
underrepresented in the assembly.
Indeed, very few highly connected sequences with abundances greater than 50
were incorporated into contigs (Table 4). Moreover, those which were assembled
were often disproportionately placed at the ends of contigs (Fig 3),
demonstrating that they terminated contig assembly. Although this trend was
observed for all three assemblers, it was more prevalent in the Velvet and
Meta-IDBA assemblers, highlighting differences in assembler heuristics.
One initial concern upon discovering this false connectivity was that these
artifacts might nucleate false assemblies, e.g. as seen with the cow rumen
[1]. While we only examined their effects on contig assembly and ignored
scaffolding issues, misassembly is an especially significant concern for
environmental metagenomics, where experimental validation of assemblies is
virtually impossible. However, our observation that k-mers from highly
connected graph regions are generally not present in contigs suggests that
these particular artifacts do not create false assemblies at a high rate.
### Filtered assemblies retained most reference genes
The advantages of removing highly connected sequences must be balanced against
consequences to resulting assemblies. We compared several metagenome
assemblies before and after the removal of these sequences. In comparing the
simulated dataset’s assemblies, the removal of highly connected sequences
resulted in very little loss of annotated reference genes (less than 1% total)
and some loss of assembled contigs ($\sim$ 15% of the final assembly). For the
rumen metagenome, we performed a partial evaluation of the assemblies using
available draft reference genomes. Similar to the simulated assemblies, we
observed only a small loss (less than 3% total) of rumen reference genomes
assembled (Table 3). In general, for all metagenomes, we observed $\sim$ 25%
loss in assembly after removing highly connected sequences, much more than
observed in assemblies of reference genes and genomes in the simulated and
rumen datasets. Some of this loss could be beneficial, resulting from removal
of sequencing artifacts. In addition, trimming or removing reads from already
low-coverage data sets could result in the loss of contigs due to the length
cutoff we use; as observed above, most of these highly connected k-mers fall
at the ends of contigs, and removing them could shorten the contigs enough to
lose them from our assembly. It is also possible, of course, that our approach
removes sequences which can accurately be assembled, but we cannot evaluate
this in the absence of reference genomes.
### Filtered reads can be assembled more efficiently
Our original motivation for analyzing connectivity was to assess the
practicality of partitioning metagenomes for later assembly, which can
dramatically reduce the memory requirements for assembly [18]. Not only did
removal of highly connected sequences eliminate problematic sequences, but it
resulted in the dissolution of the largest component and allowed us to
partition the metagenomes.
We compared the combined assembly of the partitioned sets of filtered reads to
the original lump dataset, for several assemblers. For the partitioned reads,
we were able to assemble subsets of reads in parallel, resulting in
significantly reduced time and memory requirements for assembly (Table 2). In
the case of the largest soil metagenome (containing over 500 million reads),
we could not complete the Meta-IDBA assembly of the unfiltered reads in even
100 GB of memory, but after removing highly connected sequences and
partitioning, the assembly could be completed in less than 2 GB of memory.
Using partitioned sets of reads for all metagenomes, we were also able to
efficiently complete multiple k-mer length assemblies (demonstrated with
Velvet) and subsequently merge resulting assembled contigs. For unfiltered
datasets, this was either impossible (due to memory limitations) or
impractical (due to excessive processing time).
## Conclusion
In this study, we characterize the connectivity of sequences in several
metagenomes to detect and characterize a set of likely sequencing artifacts.
These artifacts are surprisingly abundant, comprising 5-13% of the total
unique sequence in real data sets. Moreover, they have a significant impact on
the overall graph connectivity of the data sets, leading to components
containing as many as 75% of the reads in the human gut data set.
Despite the prevalence and impact of these sequences, removal of the highly
connected k-mers at the heart of these components leads to assemblies that are
significantly but not catastrophically smaller than the original assemblies.
The original unfiltered assemblies contain the majority of the filtered
assemblies, while the filtered assemblies generally contain 70-94% of the
unfiltered assemblies. The variability in these statistics between the
different assemblers (Table 2) demonstrates that the assemblers have at least
as large an effect on the content of the assemblies as our filtering
procedure.
We cannot reach strong conclusions about the impact of these highly connected
sequences on the correctness of the assembled contigs. In particular, in the
absence of complex metagenomes that have been characterized by approaches
other than short-read shotgun approaches, we have no very high quality
positive control. However, we present evidence that these highly connected
sequences primarily affect the ends of contigs and thus are unlikely to cause
incorrect contig assembly with high frequency.
Our original motivation in exploring metagenome connectivity was to enable
partitioning, an approach that leads to substantially greater scalability of
the assembly procedure. In this respect, we were successful. By applying
partitioning to filtered metagenome data, we were able to reduce the maximum
memory requirements of assembly (including the filtering stage) to well below
48 GB of RAM in all cases. This enables the use of commodity “cloud” computing
for all of our samples [26]. The decreased computational requirements for
assembly also enabled ready evaluation of different assemblers and assembly
parameters; as metagenome datasets grow increasingly larger, this ability to
efficiently analyze datasets and evaluate multiple assemblies will be
increasingly important.
More generally, our results demonstrate that metagenome assembly is still at
an early stage of technology development, particularly for low-coverage data
sets. Different assemblers and different filtering techniques yield
substantially different assembly statistics on these data sets. Moreover,
there are likely to be additional sources of sequencing artifacts lurking
within large sequencing data sets, suggesting that more and better
computational filtering and validation approaches need to be developed as
environmental metagenomics moves forward. Evaluating the assembly graph
connectivity created by reads will be a useful approach in the future.
## Methods
### Metagenomic datasets
All datasets, with the exception of the agricultural soil metagenome,
originate from previously published datasets. Rumen-associated sequences
(Illumina) were randomly selected from the rumen metagenome (read length 36 -
125 bp) available at ftp://ftp.jgi-psf.org/pub/rnd2/Cow_Rumen [1]. Human-gut
associated sequences (Illumina) of samples MH0001 through MH0010 were obtained
from
ftp://public.genomics.org.cn/BGI/gutmeta/ Raw_Reads [2] (read length 44 bp).
The simulated high complexity, high coverage dataset was previously published
[8]. Soil metagenomes (read lengths 76-113 bp) are in the SRA (SRX099904 and
SRX099905). All reads used in this study, with the exception of those from the
simulated metagenome, were quality-trimmed for Illumina’s read segment quality
control indicator, where a quality score of 2 indicates that all subsequent
regions of the sequence should not be used. After quality-trimming, only reads
with lengths greater than 30 bp were retained. All quality trimmed datasets,
including the previously unpublished agricultural soil metagenome, are
available on a public Amazon EC2 snapshot (snap-ab88dfdb). The sequencing
coverage of each metagenome was estimated as the fraction of reads which could
be aligned to assembled contigs with lengths greater than 500 bp. For the
coverage estimates, an assembly of each metagenome was performed using Velvet
(v1.1.02) with the following parameters: K=33, exp cov=auto, cov cutoff=0, no
scaffolding. Reads were aligned to assembled contigs with Bowtie (v0.12.7),
allowing for a maximum of two mismatches.
### de Bruijn graph analysis and partitioning software
We used the probabilistic de Bruijn graph representation previously described
by [18] to store and partition the metagenome assembly graphs. The khmer and
screed software packages are required for the analysis, and the versions used
for this publication are available at https://github.com/ged-
lab/khmer/tree/2012-assembly-artifacts and https://github.com/ged-
lab/screed/tree/2012-assembly-artifacts.
For metagenomes in this study, we used 4 x 48e9 bit bloom filters (requiring
24 GB RAM) to store the assembly graphs. The data processing pipeline used for
this analysis is available for public use on the Amazon Web Services public
EBS snapshot snap-ab88dfdb: data-in-paper/lumps and method-
examples/0.partitioning-into-lump.
The local graph density was calculated as the number of k-mers within a
distance of N nodes divided by N. In this study, N was equal to 10. For the
largest metagenomes, the human gut and large soil datasets, local graph
density was calculated on a randomly chosen subset of reads because of
computational limitations.
To identify specific highly connected sequences within the lump assembly
graphs, graph traversal to a distance of 40 nodes was attempted from marked
waypoints. If more than 200 k-mers were found within this traversal were
identified (i.e. a graph density $>5$), all k-mers within this traversal were
marked. If the same k-mers were consistently identified in other graph
traversals, up to five times, the k-mer was flagged as a highly connected
sequence. Aligning these k-mers to original sequencing reads, we identified
the position-specific location of these k-mers. Data and examples of scripts
used for this analysis are available on the Amazon EC2 public snapshot: data-
in-paper/density-bias, data-in-paper/hc-kmer-bias, method-examples/1.density-
analysis, method-examples/2.identifying-hc-kmers, and method-examples/3.hc-
kmer-analysis.
We identified the sources of highly connected k-mers from the simulated
metagenome by aligning them against reference genes originating from the 112
source genomes using Bowtie (v0.12.7) requiring exact matches. Highly
connected k-mers shared between all the metagenomes were also aligned against
the NCBI non-redundant genome database (ftp://ftp.ncbi.nih.gov/blast/db,
March, 1, 2011) using blastn [27], requiring an exact match over the entire
k-mer.
We also identified the subset of highly connected 32-mers which were present
greater than 50 times within lumps. Data used for this analysis are available
on the Amazon EC2 public snapshot: data-in-paper/lumps/HC-kmers/HA-HC-kmers
and method-examples/4.abundant-hc-kmers. These high abundance, highly
connected sequences were aligned to sequencing reads to demonstrate position
specific variation as described above. We evaluated the existence of short
k-mer (k=5) motifs within high abundance, highly connected k-mers which did
not have an exact match to the NCBI non-redundant database. Each identified
32-mer was broken up into shorter 5-mers, and the frequency of each unique
5-mer was calculated. Next, each unique 5-mer was ranked based on its
abundance, from high to low, and the cumulative percentage of total 5-mers is
shown in the resulting rank-abundance plot (Fig 4).
### _De novo_ metagenomic assembly
The lump within each dataset was assembled and referred to as the “unfiltered
assembly”. Additionally, highly connected sequences identified as described
above were were trimmed from sequencing reads and the remaining reads
partitioned and assembled, resulting in the “filtered assembly”. _De novo_
metagenomic assembly of reads was completed with Velvet (v1.1.02) with the
following parameters: velveth -short -shortPaired (if applicable to the
dataset) and velvetg -exp_cov auto -cov_cutoff 0 -scaffolding no [19]. For the
small and medium soil, rumen, and simulated datasets, Velvet assemblies were
performed at K=25-49, resulting contigs were dereplicated to remove contigs
with 99% similarity using CD-HIT (v 4.5.6, [28]), and final contigs were
merged with Minimus (Amos v3.1.0, [29]). For the largest soil and human gut
metagenomes, assemblies were performed at only K=33 due to the size of the
datasets and memory limitations. Additional assemblies were performed with
Meta-IDBA (v0.18) [9] : –mink 25 –maxk 50 –minCount 0 and with SOAPdenovo: -K
31 -p 8 max_rd_len=200 asm_flags=1 reverse_seq=0. After removal of highly
connected k-mers in metagenomic lumps, each filtered lump was partitioned into
separate disconnected subgraphs. Multiple subgraphs were grouped together such
that assembly could be performed in parallel on groups of sequences. Identical
assembly parameters and methods as described above were used for these
assemblies. Unfiltered and filtered assemblies were compared using the total
number of contigs, total assembly length, and maximum contig size. Additional,
the coverage of each assembly was calculated through estimating the average
base pair coverage of the BLAST alignment of each assembly to one another
(E-value less than 10-5) or, in the case of the simulated and rumen
assemblies, to reference genomes. The simulated and rumen reference genomes
were previously published in [1] and [8], respectively. Resulting assemblies
are available on the Amazon EC2 public snapshot: /data-in-paper/assembly*.
We examined incorporation and the location of the identified high abundant,
highly connecting k-mers within assembled contigs. Incorporation of these
sequences was evaluated by dividing assembled contigs into words of 32 bp
length and identifying exact matches between sequences and contig fragments.
The location of these k-mers within assembled unfiltered contigs was examined
by dividing each contig into 100 equally-sized regions. The fraction of highly
connecting k-mers which aligned exactly to each region was calculated for each
metagenome. Data and examples of scripts used for this analysis are available
on the Amazon EC2 public snapshot: method-examples/5.hc-kmer-contigs/.
## Acknowledgements
This project was supported by Agriculture and Food Research Initiative
Competitive Grant no. 2010-65205-20361 from the United States Department of
Agriculture, National Institute of Food and Agriculture and National Science
Foundation IOS-0923812, both to C.T.B. A.H. was supported by NSF Postdoctoral
Fellowship Award #0905961 and the Great Lakes Bioenergy Research Center
(Department of Energy BER DE-FC02-07ER64494). The work conducted by the U.S.
Department of Energy Joint Genome Institute is supported by the Office of
Science of the U.S. Department of Energy under Contract No. DE-AC02-05CH11231.
We thank Nick Loman and Lex Nederbragt for their helpful comments on the
paper.
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Table 1: The original size and proportion of highly connective 32-mers in the largest subset of partitioned reads (“lump”) in several medium to high complexity metagenomes. Read coverage was estimated with the number of aligned sequencing reads to Velvet-assembled contigs (K=33). The dominant lump, or largest component of each metagenome assembly graph, was found to contain highly connecting (HC) k-mers responsible for high local graph density. High density nodes refer to nodes with graph density greater than 20. | Sm Soil | Med Soil | Large Soil | Rumen | Human Gut | Sim
---|---|---|---|---|---|---
Total Reads (millions) | 50.0 | 100.0 | 520.3 | 50.0 | 350.0 | 9.2
Mapped to assembly (percent) | 1.4 | 4.7 | 5.6 | 10.3 | 3.5 | 14.8
Reads in Lump (millions) | 3.0 | 15.0 | 182.2 | 10.3 | 263 | 0.5
Lump Fraction (%) | 7% | 15% | 35% | 21% | 75% | 5%
HC 32-mers (millions) | 6.4 | 33.3 | 230.4 | 25.4 | 136.6 | 0.4
Total 32-mers (million) | 84.9 | 326.5 | 2,198.1 | 201.5 | 860.6 | 11.6
Fraction of HC 32-mers (%) | 8% | 10% | 10% | 13% | 16% | 3%
High Density Nodes (%) | 50% | 37% | 40% | 22% | 28% | 17%
Table 2: Total number of contigs, assembly length, and maximum contig size was
estimated for metagenomic datasets with multiple assemblers, as well as memory
and time requirements of unfiltered read assembly (UF). Filtered reads (F)
were processed in 24 GB of memory, and after filtering required less than 2 GB
of memory to assemble. Velvet assemblies of the unfiltered human gut and large
soil datasets (marked as *) could only be completed with K=33 due to
computational limitations. The Meta-IDBA assembly of the large soil metagenome
could not be completed in less than 100 GB.
| UF Assembly | F Assembly | UF Requirements
---|---|---|---
| (contigs / length / max size) | (contigs / length / max size) | Memory (GB)/Time (h)
_Velvet_ | | |
Small Soil | 25,470 / 16,269,879 / 118,753 | 17,636 / 10,578,908 / 13,246 | 5 / 4
Medium Soil | 113,613 / 81,660,678 / 57,856 | 79,654 / 54,424,264 / 23,663 | 18 / 21
Large Soil | 554,825 / 306,899,884 / 41,217 | 290,018 / 159,960,062 / 41,423 | 33 / 12*
Rumen | 92,044 / 74,813,072 / 182,003 | 72,705 / 49,518,627 / 34,683 | 11 / 14
Human Gut | 543,331 / 234,686,983 / 85,596 | 203,299 / 181,934,800 / 145,740 | 76 / 8*
Simulated | 11,204 / 6,506,248 / 5,151 | 9,859 / 5,463,067 / 6,605 | <1 / <1
_MetaIDBA_ | | |
---|---|---|---
Small Soil | 15,739 / 9,133,564 / 37,738 | 12,513 / 7,012,036 / 17,048 | <1 / <1
Medium Soil | 76,269 / 45,844,975 / 37,738 | 52,978 / 30,040,031 / 18,882 | 2 / 2
Large Soil | 395,122 / 228,857,098 / 37,738 | N/A | >116 / incomplete
Rumen | 60,330 / 47,984,619 / 54,407 | 48,940 / 33,276,502 / 22,083 | 12 / 3
Human Gut | 173,432 / 211,067,996 / 106,503 | 132,614 / 142,139,101 / 85,539 | 58 / 15
Simulated | 8,707 / 4,698,575 / 5,113 | 7,726 / 4,078,947 / 3,845 | <1 / <1
_SOAPdenovo_ | | |
---|---|---|---
Small Soil | 14,275 / 7,100,052 / 37,720 | 12,801 / 6,343,110 / 13,246 | 3 / <1
Medium Soil | 66,640 / 33,321,411 / 28,695 | 56,023 / 27,880,293 / 15,721 | 10 / <1
Large Soil | 412,059 / 215,614,765 / 32,514 | 334,319 / 171,718,154 / 41,423 | 48 / 11
Rumen | 62,896 / 40,792,029 / 22,875 | 55,975 / 34,540,861 / 19,044 | 5 / <1
Human Gut | 190,963 / 171,502,574 / 57,803 | 161,795 / 139,686,630 / 56,034 | 35 / 5
Simulated | 6,322 / 2,940,509 / 3,786 | 6,029 / 2,821,631 / 3,764 | <1 / <1
Table 3: Comparison of unfiltered (UF) and filtered (F) assemblies of various
metagenome lumps using Velvet, SOAPdenovo, and Meta-IDBA assemblers.
Assemblies were aligned to each other, and coverage was estimated (columns
1-2). Simulated and rumen assemblies were aligned to available reference
genomes (RG) (columns 3-4).
Velvet Assembler
Cov. of UF by F Cov. of F by UF Cov. of RG by UF Cov. of RG by F Simulated
85.4% 99.4% 5.4% 4.6% Small Soil 74.7% 98.8% - - Medium Soil 75.6% 98.4% - -
Large Soil 50.9% 86.6% - - Rumen 75.9% 98.8% 17.5% 14.8% Human Gut 80.0% 89.1%
- - Meta-IDBA Assembler
Cov. of UF by F Cov. of F by UF Cov. of RG by UF Cov. of RF by F Simulated
87.4% 94.4% 4.7% 4.1% Small Soil 75.7% 94.2% - - Medium Soil 67.7% 94.8% - -
Large Soil N/A N/A - - Rumen 70.8% 95.0% 17.5% 14.8% Human Gut 74.4% 99.4% - -
SOAPdenovo Assembler
Cov. of UF by F Cov. of F by UF Cov. of RG by UF Cov. of RF by F Simulated
94.0% 97.0% 3.0% 2.9% Small Soil 86.8% 96.1% - - Medium Soil 82.4% 96.0% - -
Large Soil 78.9% 94.5% - - Rumen 85.2% 97.8% 14.9% 13.6% Human Gut 85.4% 99.3%
- -
Table 4: Total number of abundant (greater than 50x) highly connective sequences incorporated into unfiltered assemblies | Velvet | SOAPdenovo | MetaIDBA
---|---|---|---
Small Soil | 0 (0.0%) | 41 (0.0%) | 8,717 (0.1%)
Medium Soil | 32,328 (0.1%) | 852 (0.0%) | 23,881 (0.1%)
Large Soil | 643,071 (0.3%) | 279,519 (0.1%) | N/A
Rumen | 45,721 (0.2%) | 14,858 (0.1%) | 33,046 (0.1%)
Human Gut | 4,661,447 (3.4%) | 1,749,347 (1.3%) | 5,528,054 (4.0%)
Simulated | 5,118 (1.4%) | 0 (0.0%) | 5,480 (1.5%)
Table 5: Annotations (against 112 reference genomes) of highly-connecting (HC) sequences identified in the simulated metagenome. | Number of HC sequences with annotation
---|---
ABC transporter-like protein | 306
Methyl-accepting chemotaxis sensory transducer | 210
ABC transporter | 173
Elongation factor Tu | 94
Chemotaxis sensory transducer | 51
ABC transporter ATP-binding protein | 44
Diguanylate cyclase/phosphodiesterase | 36
ATPase | 36
S-adenosyl-L-homocysteine hydrolase | 36
Adenosylhomocysteine And downstream NAD binding | 36
Ketol-acid reductoisomerase | 34
S-adenosylmethionine synthetase | 34
Elongation factor G | 34
ABC transporter ATPase | 33
Table 6: Annotations (against NCBI-nr database) of highly-connecting (HC) sequences identified in thee three soil, rumen, and human gut metagenomes. | Number of HC sequences with annotation
---|---
Translation elongation factor/GTP-binding protein LepA | 11
S-adenosylmethionine synthetase | 8
Aspartyl-tRNA synthetase | 8
Malate dehydrogenase | 7
V-type H(+)-translocating pyrophosphatase | 6
Acyl-CoA synthetase | 6
NAD synthetase | 5
Ribonucleotide reductase of class II | 4
Ribityllumazine synthase | 4
Heavy metal translocating P-type ATPase, copA | 3
GyrB | 3
Glutamine amidotransferase chain of NAD synthetase | 3
ChaC family protein | 3
Figure 1: The extent to which average local graph density varies by read
position is shown for the lump of various datasets.
Figure 2: The extent to which highly connecting k-mers (solid lines) and the
subset of highly abundant (greater than 50) k-mers (dashed lines) are present
at specific positions within sequencing reads for various metagenomes. Figure
3: When incorporated into an assembly, abundant (greater than 50 times),
highly connecting sequences (k-mers) were disproportionately present at the
ends of contigs. The total fraction of highly connecting k-mers which are
incorporated into each contig binned region. Figure 4: Rank abundance plot of
5-mers present in abundant, highly connected sequences in various datasets.
|
arxiv-papers
| 2012-12-01T20:57:59 |
2024-09-04T02:49:38.794399
|
{
"license": "Public Domain",
"authors": "Adina Chuang Howe, Jason Pell, Rosangela Canino-Koning, Rachel\n Mackelprang, Susannah Tringe, Janet Jansson, James M. Tiedje, C. Titus Brown",
"submitter": "C. Titus Brown",
"url": "https://arxiv.org/abs/1212.0159"
}
|
1212.0181
|
# Stochastic Volatility Regression for
Functional Data Dynamics
Bin Zhu and David B. Dunson∗
††footnotetext: ∗Bin Zhu is Tenure-Track Principal Investigator, Biostatistics
Branch, Division of Cancer Epidemiology and Genetics, National Cancer
Institute, National Institutes of Health, Rockville, MD 20852 (Email:
[email protected]_). David B. Dunson is Professor, Department of Statistical
Science, Duke University, Durham, NC 27708, (Email: [email protected]_).
Abstract
Although there are many methods for functional data analysis (FDA), little
emphasis is put on characterizing variability among volatilities of individual
functions. In particular, certain individuals exhibit erratic swings in their
trajectory while other individuals have more stable trajectories. There is
evidence of such volatility heterogeneity in blood pressure trajectories
during pregnancy, for example, and reason to suspect that volatility is a
biologically important feature. Most FDA models implicitly assume similar or
identical smoothness of the individual functions, and hence can lead to
misleading inferences on volatility and an inadequate representation of the
functions. We propose a novel class of FDA models characterized using
hierarchical stochastic differential equations. We model the derivatives of a
mean function and deviation functions using Gaussian processes, while also
allowing covariate dependence including on the volatilities of the deviation
functions. Following a Bayesian approach to inference, a Markov chain Monte
Carlo algorithm is used for posterior computation. The methods are tested on
simulated data and applied to blood pressure trajectories during pregnancy.
Key words: Diffusion process; Gaussian process; State space model; Stochastic
differential equation; Stochastic dynamic model; Stochastic functional data
analysis.
## 1 Introduction
Multi-subject functional data arise frequently in many fields of research,
including epidemiology, clinical trials and environmental health. Sequential
observations are collected over time for multiple subjects, and can be treated
as being generated from a smooth trajectory contaminated with noise. There are
a rich variety of methods available for characterizing variability and
covariate dependence in functional data ranging from hierarchical basis
expansions to functional principal components analysis (FPCA). In defining
models for functional data and in interpreting variability in trajectories,
either unexplained or due to covariates, the emphasis has been on differences
in the level and local trends. Dynamic features, such as velocity,
acceleration and especially volatility, are also important but receive much
less attention.
Analysis of functional data dynamics studies temporal changes in trajectories
and effects of covariates on these changes. For example, Wang et al. (2008)
used linear differential equations to model price velocity and acceleration in
eBay auctions and explored the auction subpopulation effect. Müller and Yao
(2010) modeled the velocity of online auction bids using empirical stochastic
differential equations with time-varying coefficients and a smooth drift
process. Zhu et al. (2011) inferred the rate functions of prostate-specific
antigen profiles using the proposed semiparametric stochastic velocity model,
in which rate functions are regarded as realizations of Ornstein-Uhlenbeck
processes dependent on covariates of interest.
This article investigates a different dynamic feature, the volatility, which
measures the conditional variance of trajectory changes over an infinitesimal
time interval. We propose a stochastic volatility regression (SVR) model with
Gaussian process (GP) priors used for the group mean and subject specific
deviation functions through stochastic differential equations (SDEs). We
further accommodate inference on covariate effects on volatility through
allowing the diffusion term of SDEs for deviation functions to depend on
covariates. Although volatility has been extensively studied through
stochastic volatility (SV) models in finance (Heston, 1993; Jacquier et al.,
2002; Shephard, 2005; Barndorff-Nielsen and Shephard, 2012), the setting,
model specifications and data features are distinct from ours. SV models
typically focus on a single volatility process which is time-varying and
associated with a price process for high-frequency finance data. More relevant
is the literature on multivariate SV models; for recent references, refer to
Loddo et al. (2011), Van Es and Spreij (2011), Ishihara and Omori (2012) and
Durante et al. (2012).
This setting differs from ours in that the focus is on multivariate time
series modeling instead of functional data analysis, with interest in the
joint volatility dynamics over time for the different assets. In contrast, we
are interested in studying variation across individuals in a time-constant
subject-specific volatility; that is, certain subjects may have very smooth
trajectories while other subjects have erratic trajectories. It is our
conjecture that such volatility heterogeneity is common in biomedical
settings, but is overlooked in analyzing data with models that implicitly
prescribe a single level of smoothness for all subjects. As data are sparse
and irregularly spaced in most studies, it is not surprising such behavior is
overlooked. However, the volatility in a biomarker may be as important or more
important than the overall level and trend in the biomarker. We provide
motivation through the following longitudinal blood pressure data set.
The Healthy Pregnancy, Healthy Baby Study (HPHB, Miranda et al., 2009)
collected longitudinal blood pressure (BP) measurements for pregnant women.
Blood pressures are measured at irregularly spaced times during the second and
third trimesters with the number of measurements per subject varying from 9 to
19. We are interested in estimating subject-specific volatilities of BP
trajectories and in identifying covariates associated with the volatility.
Figure 1LABEL:sub@fig:1a plots mean arterial pressure (MAP) trajectories for
twenty randomly selected normal women and women with preeclampsia,
respectively. Clearly the MAP trajectories among the preeclampsia group are
more wiggly than the ones in the normal group, which is also implied by
boxplots of log-transformed empirical volatilities in Figure
1LABEL:sub@fig:1b. To explore volatility differences among various groups in
addition to preeclampsia, we apply normal linear regression for log-
transformed empirical volatilities with the covariates race, mother’s age,
obesity, preeclampsia, parity and smoking. The results suggest that
preeclampsia and smoking (p-values 0.0005 and 0.002) are associated with
empirical volatility. This is a two-stage approach, which is useful as an
exploratory tool but ignores measurement errors and uncertainty in volatility
estimation.
(a)
(b)
Figure 1: (a) Mean arterial pressure (MAP) trajectories for twenty randomly
selected normal women and women with preeclampsia; (b) Log-transformed
empirical volatilities for women in the normal group and preeclampsia group.
$Y_{ij}$ denotes blood pressure for the $i$th woman at time $t_{ij}$, and
$U_{ij}=Y_{ij}-\bar{Y}_{j}$ is the deviation from the group mean blood
pressure $\bar{Y}_{j}$. The empirical volatility measures the fluctuation of
trajectories empirically and is defined as
$\frac{1}{n_{i}}\sum_{j=1}^{n_{i}-1}\frac{(U_{i,j+1}-U_{i,j})^{2}}{t_{i,j+1}-t_{i,j}}$
with $n_{i}$ the number of observations for the $i$th woman.
Additionally, empirical volatilities in Figure 1LABEL:sub@fig:1b are
heterogeneous even within the normal or preeclampsia group. This heterogeneity
will be largely omitted when we apply FDA methods with identical or similar
smoothness for individual functions within a group. Consequently, the wiggly
trajectories will be over-smoothed while the smooth trajectories will be
under-smoothed. We can potentially estimate the individual trajectories
separately but it is well known that borrowing of information will
dramatically improve performance for sparse functional data. In addition,
separate estimation does not allow for inferences on covariate effects and
unexplained variability in volatility.
As for the clinical question addressed, the previous FDA methods mainly focus
on the shift of blood pressure level and ignore examining the volatility of
blood pressure, which measures the haemodynamic stability and is crucial for
cardiovascular health. For example, a recent study shows that blood pressure
stability rather than blood pressure level is associated with increased
survival among patients on hemodialysis (Raimann et al., 2012). For the HPHB
study, we observe that preeclampsia is commonly accompanied by blood pressure
over-swinging. The joint effect of high blood pressure level and large
volatility may lead to the adverse birth outcomes, such as low birth weight
and preterm birth.
The remainder of the article is organized as follows. Section 2 specifies the
SVR model and discusses its properties. Section 3 develops an efficient Markov
chain Monte Carlo algorithm for posterior inference. Section 4 presents
simulation studies and the proposed method is applied to a real dataset in
Section 5. Finally, section 6 contains concluding remarks and future possible
extensions.
## 2 Stochastic Volatility Regression Model
### 2.1 The Model Specification
Suppose that $Y_{i}(t)$, $i=1,2,\dots,m$, is the observation of the $i$th
subject at time
$t\in\mathcal{T}_{i}=\\{t_{i,1},t_{i,2},\cdots,t_{i,n_{i}}<t_{U}\\}$ with
$\mathcal{T}_{i}$ the set of observation times before time $t_{U}$ for the
$i$th subject. We specify an observation equation for $Y_{i}(t)$ as
$Y_{i}(t)=M_{k_{i}}(t)+U_{i}(t)+\varepsilon_{i}(t),$ (1)
where $Y_{i}(t)$ is contaminated by the measurement error $\varepsilon_{i}(t)$
following a one-dimensional normal distribution with mean $0$ and variance
$\sigma^{2}_{\varepsilon}$. Assuming the $i$th subject belongs to the
${k_{i}}$th group (e.g. by race or treatment) with
${k_{i}}\in\\{1,2,\cdots,g\\}$, we include a ${k_{i}}$th group mean function
$M_{k_{i}}(t)=\textsf{E}\\{Y_{i}(t)\mid M_{k_{i}}(t)\\}$ in the observation
equation. In addition, the trajectory of the $i$th subject will unlikely
coincide with $M_{k_{i}}(t)$ and therefore the departure from $M_{k_{i}}(t)$
is addressed and represented by the subject-specific deviation function
$U_{i}(t)$ with $\textsf{E}\\{U_{i}(t)\\}=0$.
The volatility of the $i$th subject is defined as the conditional variance of
the $(q-1)$th order derivative of $U_{i}(t)$ over an infinitesimal time
interval. Namely, we denote the volatility
$\sigma^{2}_{U_{i}}=\mathop{\lim}\limits_{h\to
0}h^{-1}\textsf{E}\left[\left\\{D^{q-1}U_{i}(t+h)-D^{q-1}U_{i}(t)\right\\}^{2}\mid
D^{q-1}U_{i}(t)\right]$ with differential operator
$D^{q}=\frac{d^{q}}{dt^{q}}$. As volatility approaches zero, $U_{i}(t)$ would
be a roughly flat line. In contrast, increasing the value of volatility would
lead to a more wiggly $U_{i}(t)$ with a larger magnitude of fluctuation around
$M_{k_{i}}(t)$.
We specify Gaussian process priors for $M_{k_{i}}(t)$ and $U_{i}(t)$ using
SDEs which incorporate the group and individual volatilities
$\sigma^{2}_{M_{k_{i}}}$ and $\sigma^{2}_{U_{i}}$ :
$\displaystyle D^{p}M_{k_{i}}(t)$
$\displaystyle=\sigma_{M_{k_{i}}}\dot{W}_{k_{i}}(t),$ (2) $\displaystyle
D^{q}U_{i}(t)$ $\displaystyle=\sigma_{U_{i}}\dot{W}^{\prime}_{i}(t),$ (3)
where $p,q\in\mathbb{N}\geq 1$ and
$\sigma_{M_{k_{i}}},\sigma_{U_{i}}\in\mathbb{R}^{+}$; $\dot{W}_{k_{i}}(t)$ and
$\dot{W}^{\prime}_{i}(t)$ are independent Gaussian white noise processes with
$\textsf{E}\\{\dot{W}_{k_{i}}(t)\\}=\textsf{E}\\{\dot{W}^{\prime}_{i}(t)\\}=0$
and covariance function
$\textsf{E}\\{\dot{W}_{k_{i}}(t)\dot{W}_{k_{i}}(t^{\prime})\\}=\textsf{E}\\{\dot{W}^{\prime}_{i}(t)\dot{W}^{\prime}_{i}(t^{\prime})\\}=\delta(t-t^{\prime})$,
a delta function. We denote
$\boldsymbol{M}_{{k_{i}}0}=\\{M_{k_{i}}(0),D^{1}M_{k_{i}}(0),\dots,D^{p-1}M_{k_{i}}(0)\\}$
and $\boldsymbol{U}_{i0}=\\{U_{i}(0),D^{1}U_{i}(0),\dots,D^{q-1}U_{i}(0)\\}$
as the initial values of $M_{k_{i}}(t)$ and $U_{i}(t)$ as well as their
derivatives till orders $q-1$ and $p-1$ respectively. The volatility
$\sigma^{2}_{U_{i}}$ in SDE (3) is allowed to vary between subjects and across
covariates. In this article, we focus on a simple transformed mean
relationship, namely
$\log(\sigma^{2}_{U_{i}})\sim\mathsf{N}_{1}(\boldsymbol{x}^{\prime}_{i}\boldsymbol{\beta},\sigma^{2})$,
which can be extended to the more complex specifications with less restrictive
assumptions and to high-dimensional covariates.
The mean and covariance functions of Gaussian process priors for
$M_{k_{i}}(t)$ and $U_{i}(t)$ can be obtained by applying stochastic
integration to SDEs (2) and (3), resulting in the following lemma.
###### Lemma 1
$M_{k}(t)$, $k=1,2,\dots,g$, and $U_{i}(t)$, $i=1,2,\dots,n$, are the
summations of mutually independent Gaussian processes written as
$M_{k}(t)=M_{k0}(t)+M_{k1}(t)$ and $U_{i}(t)=U_{i0}(t)+U_{i1}(t)$ with
corresponding mean functions
$\textsf{E}\left\\{M_{k0}(t)\right\\}=\textsf{E}\left\\{M_{k1}(t)\right\\}=\textsf{E}\left\\{U_{i0}(t)\right\\}=\textsf{E}\left\\{U_{i1}(t)\right\\}=0$
and covariance functions
$\displaystyle\mathcal{K}_{M_{k0}}(s,t)$
$\displaystyle=\sigma^{2}_{M_{0}}\mathcal{R}_{M_{0}}(s,t)=\sigma^{2}_{M_{0}}\sum_{l=0}^{q-1}\phi_{l}(s)\phi_{l}(t),$
$\displaystyle\mathcal{K}_{M_{k1}}(s,t)$
$\displaystyle=\sigma^{2}_{M_{k}}\mathcal{R}_{M_{1}}(s,t)=\sigma^{2}_{M_{k}}\int_{\mathcal{T}}G_{q}(s,u)G_{q}(t,u)du,$
$\displaystyle\mathcal{K}_{U_{i0}}(s,t)$
$\displaystyle=\sigma^{2}_{U_{0}}\mathcal{R}_{U_{0}}(s,t)=\sigma^{2}_{U_{0}}\sum_{l=0}^{p-1}\phi_{l}(s)\phi_{l}(t),$
$\displaystyle\mathcal{K}_{U_{i1}}(s,t)$
$\displaystyle=\sigma^{2}_{U_{i}}\mathcal{R}_{U_{1}}(s,t)=\sigma^{2}_{U_{i}}\int_{\mathcal{T}}G_{p}(s,u)G_{p}(t,u)du,$
respectively, where $\phi_{l}(t)=\frac{t^{l}}{l!}$,
$G_{q}(s,u)=\frac{(s-u)_{+}^{q-1}}{(q-1)!}$ and
$s,t,u\in\mathcal{T}=[0,t_{U}]$.
Hence, we can represent the prior of $M_{k_{i}}(t)+U_{i}(t)$ as a hierarchical
Gaussian process,
$\displaystyle M_{k_{i}}(t)+U_{i}(t)\mid M_{k_{i}}(t)$
$\displaystyle\sim\mathcal{GP}(M_{k_{i}}(t),\mathcal{K}_{U_{i0}}(s,t)+\mathcal{K}_{U_{i1}}(s,t)),$
$\displaystyle M_{k_{i}}(t)$
$\displaystyle\sim\mathcal{GP}(0,\mathcal{K}_{M_{k0}}(s,t)+\mathcal{K}_{M_{k1}}(s,t)),$
where $\mathcal{GP}(M(t),\mathcal{K}(s,t))$ denotes a Gaussian process with
mean function $M(t)$ and covariance function $\mathcal{K}(s,t)$. Different
from the previous hierarchical Gaussian process prior (Park and Choi, 2010),
in which the covariance function is modeled as a squared exponential kernel
and is identical across the subjects within a group, here
$\mathcal{K}_{U_{i0}}(s,t)+\mathcal{K}_{U_{i1}}(s,t)$ is subject specific and
depends on covariates through $\sigma^{2}_{U_{i}}$.
To carry out Bayesian inference, we further specify the following prior
distributions for the hyperparameters. In particular,
$\boldsymbol{M}_{{k_{i}}0}\sim\mathsf{N}_{p}(\boldsymbol{0},\sigma^{2}_{M_{0}}\boldsymbol{I})$
with $\sigma^{2}_{M_{0}}=10^{4}$,
$\boldsymbol{U}_{i0}\sim\mathsf{N}_{q}(\boldsymbol{0},\sigma^{2}_{U_{0}}\boldsymbol{I})$,
$\sigma^{2}_{\varepsilon}\sim\textsf{invGamma}(a,b)$,
$\sigma^{2}_{M_{k}}\sim\textsf{invGamma}(a,b)$ and
$\sigma^{2}_{U_{0}}\sim\textsf{invGamma}(a,b)$, where $\textsf{invGamma}(a,b)$
denotes the inverse gamma distribution with shape parameter $a$ and scale
parameter $b$. The $\boldsymbol{\beta}$ and $\sigma^{2}$ follow the
independent Jeffreys’ prior, $f(\boldsymbol{\beta},\sigma^{2})\propto
1/\sigma^{2}$.
### 2.2 Double-Penalized Smoothing Spline
It is well known that the Bayes estimate with the integrated Wiener process
prior is identical to the smoothing spline estimate (Wahba, 1990). By similar
arguments, we can show that when the volatilities are given and
$\sigma^{2}_{M_{0}}$ and $\sigma^{2}_{U_{0}}$ go to infinity, the posterior
means of $M_{k}(t)$ and $U_{i}(t)$ are equivalent to the double-penalized
smoothing spline $\hat{M}_{k}(t)+\hat{U}_{i}(t)$, which is the minimizer of
the double-penalized sum-of-squares,
$\displaystyle\textsf{DPSS}=$
$\displaystyle\sum_{i=1}^{m}\frac{1}{n_{i}}\sum_{j=1}^{n_{i}}\left\\{Y(t_{ij})-M_{k_{i}}(t_{ij})-U_{i}(t_{ij})\right\\}^{2}+$
(4)
$\displaystyle\sum_{k=1}^{g}\lambda_{M_{k}}\int_{\mathcal{T}}\left\\{D^{p}M_{k}(t)\right\\}^{2}dt+\sum_{i=1}^{m}\lambda_{U_{i}}\int_{\mathcal{T}}\left\\{D^{q}U_{i}(t)\right\\}^{2}dt,$
where penalty terms $\int_{\mathcal{T}}\left\\{D^{p}M_{k}(t)\right\\}^{2}dt$
and $\int_{\mathcal{T}}\left\\{D^{q}U_{i}(t)\right\\}^{2}dt$ penalize the
roughness of $M_{k}(t)$ and $U_{i}(t)$ respectively, where the smoothness and
the fidelity to data are balanced by the smoothing parameters
$\lambda_{M_{k}}=\displaystyle\sum_{i:k_{i}=k}\frac{\sigma^{2}_{\varepsilon}}{n_{i}\sigma^{2}_{M_{k}}}$
and
$\lambda_{U_{i}}=\frac{\sigma^{2}_{\varepsilon}}{n_{i}\sigma^{2}_{U_{i}}}$.
Expression for $\hat{M}_{k}(t)$ and $\hat{U}_{i}(t)$ can be obtained
explicitly, as shown in the following Theorem.
###### Theorem 1
The smoothing splines $\hat{M}_{k}(t)$ and $\hat{U}_{i}(t)$ with
$t\in\mathcal{T}$ minimize the double-penalized sum-of-squares (4) and have
the forms
$\displaystyle\hat{M}_{k}(t)$
$\displaystyle=\sum_{l=0}^{p-1}\mu_{kl}\phi_{l}(t)+\sum_{j=1}^{n}\nu_{kj}\mathcal{R}_{M_{1}}(t_{j},t)=\boldsymbol{\mu}^{\prime}_{k}\boldsymbol{\phi}_{\mu}(t)+\boldsymbol{\nu}^{\prime}_{k}\boldsymbol{R}_{M_{1}}(t)$
$\displaystyle\hat{U}_{i}(t)$
$\displaystyle=\sum_{l=0}^{q-1}\alpha_{il}\phi_{l}(t)+\sum_{j=1}^{n_{i}}\gamma_{ij}\mathcal{R}_{U_{1}}(t_{ij},t)=\boldsymbol{\alpha}^{\prime}_{i}\boldsymbol{\phi}_{\alpha}(t)+\boldsymbol{\gamma}^{\prime}_{i}\boldsymbol{R}_{U_{i1}}(t)$
where
$\boldsymbol{\mu}_{k}=\\{\mu_{k0},\mu_{k1},\cdots,\mu_{k(p-1)}\\}^{\prime}$,
$\boldsymbol{\nu}_{k}=(\nu_{k1},\nu_{k2},\cdots,\nu_{kn})^{\prime}$,
$\boldsymbol{\alpha}_{i}=\left\\{\alpha_{i0},\alpha_{i1},\cdots,\alpha_{i(q-1)}\right\\}^{\prime}$
and
$\boldsymbol{\gamma}_{i}=(\gamma_{i1},\gamma_{i2},\cdots,\gamma_{in_{i}})^{\prime}$
are the coefficients for the bases
$\displaystyle\boldsymbol{\phi}_{\mu}(t)$
$\displaystyle=\\{\phi_{0}(t),\phi_{1}(t),\cdots,\phi_{p-1}(t)\\}^{\prime},\quad\boldsymbol{R}_{M_{1}}(t)=\\{\mathcal{R}_{M_{1}}(t_{1},t),\mathcal{R}_{M_{1}}(t_{2},t),\cdots,\mathcal{R}_{M_{1}}(t_{n},t)\\}^{\prime},$
$\displaystyle\boldsymbol{\phi}_{\alpha}(t)$
$\displaystyle=\\{\phi_{0}(t),\phi_{1}(t),\cdots,\phi_{q-1}(t)\\}^{\prime},\quad\boldsymbol{R}_{U_{i1}}(t)=\\{\mathcal{R}_{U_{1}}(t_{i1},t),\mathcal{R}_{U_{1}}(t_{i2},t),\cdots,\mathcal{R}_{U_{1}}(t_{in_{i}},t)\\}^{\prime},$
with
$t_{j}\in\mathcal{T}_{m}=\cup_{i=1}^{m}\mathcal{T}_{i}=\\{t_{j},\;j=1,2,\dots,n\\}$,
the index set of unique observation times among all $m$ subjects.
Given $\hat{M}_{k}(t)$ and $\hat{U}_{i}(t)$, the double-penalized sum-of-
squares (4) can be written as
$\displaystyle\textsf{DPSS}=$
$\displaystyle\sum_{i=1}^{m}\frac{1}{n_{i}}(\boldsymbol{Y}_{i}-\boldsymbol{\Delta}_{i}\boldsymbol{\phi}_{\mu}\boldsymbol{\mu}_{k_{i}}-\boldsymbol{\Delta}_{i}\boldsymbol{R}_{M_{1}}\boldsymbol{\nu}_{k_{i}}-\boldsymbol{\phi}_{\alpha_{i}}\boldsymbol{\alpha}_{i}-\boldsymbol{R}_{U_{i1}}\boldsymbol{\gamma}_{i})^{\prime}\times$
(5)
$\displaystyle(\boldsymbol{Y}_{i}-\boldsymbol{\Delta}_{i}\boldsymbol{\phi}_{\mu}\boldsymbol{\mu}_{k_{i}}-\boldsymbol{\Delta}_{i}\boldsymbol{R}_{M_{1}}\boldsymbol{\nu}_{k_{i}}-\boldsymbol{\phi}_{\alpha_{i}}\boldsymbol{\alpha}_{i}-\boldsymbol{R}_{U_{i1}}\boldsymbol{\gamma}_{i})+$
$\displaystyle\sum_{k=1}^{g}\lambda_{M_{k}}\boldsymbol{\nu}^{\prime}_{k}\boldsymbol{R}_{M_{1}}\boldsymbol{\nu}_{k}+\sum_{i=1}^{m}\lambda_{U_{i}}\boldsymbol{\gamma}^{\prime}_{i}\boldsymbol{R}_{U_{i1}}\boldsymbol{\gamma}_{i},$
where
$\displaystyle\boldsymbol{Y}_{i}$
$\displaystyle=\\{Y(t_{i1}),Y(t_{i2}),\cdots,Y(t_{in_{i}})\\}^{\prime},\quad\boldsymbol{\Delta}_{i}=(\delta_{jj^{\prime}})_{n_{i}\times
n},$ $\displaystyle\boldsymbol{\phi}_{\mu}$
$\displaystyle=\\{\boldsymbol{\phi}_{\mu}(t_{1}),\boldsymbol{\phi}_{\mu}(t_{2}),\cdots,\boldsymbol{\phi}_{\mu}(t_{n})\\}^{\prime},\quad\boldsymbol{R}_{M_{1}}=\\{\boldsymbol{R}_{M_{1}}(t_{1}),\boldsymbol{R}_{M_{1}}(t_{2}),\cdots,\boldsymbol{R}_{M_{1}}(t_{n})\\},$
$\displaystyle\boldsymbol{\phi}_{\alpha_{i}}$
$\displaystyle=\\{\boldsymbol{\phi}_{\alpha}(t_{i1}),\boldsymbol{\phi}_{\alpha}(t_{i2}),\cdots,\boldsymbol{\phi}_{\alpha}(t_{in_{i}})\\}^{\prime},\quad\boldsymbol{R}_{U_{i1}}=\\{\boldsymbol{R}_{U_{i1}}(t_{i1}),\boldsymbol{R}_{U_{i1}}(t_{i2}),\cdots,\boldsymbol{R}_{U_{i1}}(t_{in_{i}})\\}$
with $\delta_{jj^{\prime}}=1$ if $i$th subject has an observation at time
$t_{ij}=t_{j^{\prime}}$, $t_{ij}\in\mathcal{T}_{i}$,
$t_{j^{\prime}}\in\mathcal{T}_{m}$ and $\delta_{jj^{\prime}}=0$, otherwise.
The proofs of Theorem 1 and the following Corollary are included in Appendix
A.
###### Corollary 1
The $\boldsymbol{\mu}_{k}$, $\boldsymbol{\nu}_{k}$, $\boldsymbol{\alpha}_{i}$
and $\boldsymbol{\gamma}_{i}$ can be obtained through a backfitting algorithm
or the Gauss-Seidel method, iterating the following two steps until
convergence:
1. (a) for each $i$, $\boldsymbol{\hat{\alpha}}_{i}=(\boldsymbol{\phi}_{\alpha_{i}}^{\prime}\boldsymbol{S}_{U_{i}}^{-1}\boldsymbol{\phi}_{\alpha_{i}})^{-1}\boldsymbol{\phi}_{\alpha_{i}}^{\prime}\boldsymbol{S}_{U_{i}}^{-1}\boldsymbol{\tilde{Y}}_{i}$ and $\boldsymbol{\hat{\gamma}}_{i}=\boldsymbol{S}_{U_{i}}^{-1}\left\\{\boldsymbol{I}-\boldsymbol{\phi}_{\alpha_{i}}(\boldsymbol{\phi}_{\alpha_{i}}^{\prime}\boldsymbol{S}_{U_{i}}^{-1}\boldsymbol{\phi}_{\alpha_{i}})^{-1}\boldsymbol{\phi}_{\alpha_{i}}^{\prime}\boldsymbol{S}_{U_{i}}^{-1}\right\\}\boldsymbol{\tilde{Y}}_{i}$, where $\boldsymbol{S}_{U_{i}}=\boldsymbol{R}_{U_{i}1}+n_{i}\lambda_{U_{i}}\boldsymbol{I}$ and $\boldsymbol{\tilde{Y}}_{i}=\boldsymbol{Y}_{i}-\boldsymbol{\Delta}_{i}\boldsymbol{\phi}_{\mu}\boldsymbol{\hat{\mu}}_{k_{i}}-\boldsymbol{\Delta}_{i}\boldsymbol{R}_{M_{1}}\boldsymbol{\hat{\nu}}_{k_{i}}$;
2. (b) for each $k$, $\boldsymbol{\hat{\mu}}_{k}=(\boldsymbol{\phi}_{\mu}^{\prime}\boldsymbol{\Delta}^{\prime}\boldsymbol{S}_{M_{k}}^{-1}\boldsymbol{\Delta}\boldsymbol{\phi}_{\mu})^{-1}\boldsymbol{\phi}_{\mu}^{\prime}\boldsymbol{\Delta}^{\prime}\boldsymbol{S}_{M_{k}}^{-1}\boldsymbol{\tilde{Y}}_{k}$ and
$\boldsymbol{\hat{\nu}}_{k}=\boldsymbol{S}_{M_{k}}^{-1}\left\\{\boldsymbol{I}-\boldsymbol{\Delta}\boldsymbol{\phi}_{\mu}(\boldsymbol{\phi}_{\mu}^{\prime}\boldsymbol{\Delta}^{\prime}\boldsymbol{S}_{M_{k}}^{-1}\boldsymbol{\Delta}\boldsymbol{\phi}_{\mu})^{-1}\boldsymbol{\phi}_{\mu}^{\prime}\boldsymbol{\Delta}^{\prime}\boldsymbol{S}_{M_{k}}^{-1}\right\\}\boldsymbol{\tilde{Y}}_{k}$,
where
$\boldsymbol{S}_{M_{k}}=\boldsymbol{\Delta}\boldsymbol{R}_{M_{1}}+\lambda_{M_{k}}\boldsymbol{I}$,
$\boldsymbol{\tilde{Y}}_{k}=\displaystyle\sum_{i:k_{i}=k}\frac{1}{n_{i}}\boldsymbol{\Delta}_{i}^{\prime}\left(\boldsymbol{Y}_{i}-\boldsymbol{\phi}_{\alpha_{i}}\boldsymbol{\hat{\alpha}}_{i}-\boldsymbol{R}_{U_{i1}}\boldsymbol{\hat{\gamma}}_{i}\right)$
and
$\boldsymbol{\Delta}=\displaystyle\sum_{i:k_{i}=k}\frac{1}{n_{i}}\boldsymbol{\Delta}_{i}^{\prime}\boldsymbol{\Delta}_{i}$.
## 3 Posterior Computation
Although we can obtain $\hat{M}_{k}(t)$ and $\hat{U}_{i}(t)$ by the
backfitting algorithm outlined in Corollary 1 and estimate $\lambda_{M_{k}}$
and $\lambda_{U_{i}}$ through generalized cross validation (Chap. 4, Wahba,
1990), it is unclear how to allow $\lambda_{U_{i}}$ to depend on covariates.
In addition, when $n$ is large, it is computational infeasible to invert the
$n\times n$ matrix $\boldsymbol{S}_{M_{k}}$ involved in the backfitting
algorithm. Instead, we propose a Markov chain Monte Carlo (MCMC) algorithm for
posterior computation that solves these problems. The algorithm achieves
computational efficiency by leveraging on the Markovian property of SDEs and
samples $M_{k}(t)$ and $U_{i}(t)$ through the simulation smoother (Durbin and
Koopman, 2002), which requires the following Proposition.
###### Proposition 1
Let $X(t)$ denote a (r-1)th-order integral Wiener process, defined by the
stochastic differential equation $D^{r}X(t)=\dot{W}(t)$. Consequently, the
$\boldsymbol{X}_{j}=\\{X(t_{j}),D^{1}X(t_{j}),\cdots,D^{r-1}X(t_{j})\\}^{\prime}$,
$j=1,2,\cdots,n$, follows a state equation
$\boldsymbol{X}_{j+1}=\boldsymbol{G}_{j}\boldsymbol{X}_{j}+\boldsymbol{\omega}_{j},$
where
$\boldsymbol{G}_{j}=\sum_{k=0}^{r}\frac{\delta_{j}^{k}}{k!}\boldsymbol{C}^{k}$
and
$\boldsymbol{\omega}_{j}\sim\mathsf{N}_{r}(\boldsymbol{0},\boldsymbol{W}_{j})$
with
$\boldsymbol{W}_{j}=\int_{0}^{\delta_{j}}\exp\\{\boldsymbol{C}(\delta_{j}-u)\\}\boldsymbol{D}\boldsymbol{D}^{\prime}\exp\\{\boldsymbol{C}^{\prime}(\delta_{j}-u)\\}du$,
$\boldsymbol{C}=(c_{ll^{\prime}})_{r\times r}$, $c_{ll^{\prime}}=1$ when
$l^{\prime}=l+1$ and $c_{ll^{\prime}}=0$ otherwise,
$\boldsymbol{D}=(0,0,\cdots,1)^{\prime}$ and $\delta_{j}=t_{j+1}-t_{j}$.
The proof is in Appendix A. Finally, we outline the proposed MCMC as follows.
1. (1) Given $M_{k_{i}}(t_{ij})$, $\sigma^{2}_{\varepsilon}$ and $\sigma^{2}_{U_{i}}$, sample $U_{i}(t_{ij})$, $i=1,2,\cdots,m$, $j=0,1,\cdots,n_{i}$. Let $Y_{U_{ij}}=Y_{i}(t_{ij})-M_{k_{i}}(t_{ij})$ and the SVR model for the $i$th subject can be expressed as the following state space model (Jones, 1993; Durbin and Koopman, 2001), from which we can draw samples of $U_{i}(t_{ij})$ and its derivatives using the simulation smoother.
$\displaystyle Y_{U_{ij}}$
$\displaystyle=\boldsymbol{F}_{U_{ij}}\boldsymbol{U}_{{ij}}+\varepsilon_{U_{ij}},$
$\displaystyle\boldsymbol{U}_{{i(j+1)}}$
$\displaystyle=\boldsymbol{G}_{U_{ij}}\boldsymbol{U}_{{ij}}+\sigma_{U_{i}}\boldsymbol{\omega}_{U_{ij}},$
where $\boldsymbol{F}_{U_{ij}}=(1,0,\cdots,0)$,
$\boldsymbol{U}_{{ij}}=\\{U_{i}(t_{ij}),D^{1}U_{i}(t_{ij}),\cdots,D^{q-1}U_{i}(t_{ij})\\}^{\prime}$
and
$\varepsilon_{U_{ij}}\stackrel{{\scriptstyle\mathrm{i.i.d.}}}{{\sim}}\mathsf{N}_{1}(0,\sigma^{2}_{\varepsilon})$.
Similar to the $\boldsymbol{G}_{j}$, $\boldsymbol{\omega}_{j}$ and
$\boldsymbol{W}_{j}$ in Proposition 1, the $\boldsymbol{G}_{U_{ij}}$,
$\boldsymbol{\omega}_{U_{ij}}$ and $\boldsymbol{W}_{U_{ij}}$ follow the same
specifications with $r=q$.
2. (2) Given $U_{i}(t_{j})$, $\sigma^{2}_{\varepsilon}$ and $\sigma^{2}_{M_{k}}$, sample $M_{k}(t_{j})$, $k=1,2,\cdots,g$, $j=0,1,\cdots,n$. Similarly, we rewrite the SVR model for the $k$th group as the following state space model and then sample $M_{k_{i}}(t_{ij})$ and its derivatives by the simulation smoother.
$\displaystyle\boldsymbol{Y}_{M_{kj}}$
$\displaystyle=\boldsymbol{F}_{M_{kj}}\boldsymbol{M}_{kj}+\boldsymbol{\varepsilon}_{M_{kj}},$
$\displaystyle\boldsymbol{M}_{{k(j+1)}}$
$\displaystyle=\boldsymbol{G}_{M_{kj}}\boldsymbol{M}_{{kj}}+{\sigma}_{M_{k}}\boldsymbol{\omega}_{M_{kj}},$
where $\boldsymbol{Y}_{M_{kj}}=(Y_{M_{kj}}^{i})_{m\times 1}$,
$\boldsymbol{M}_{{kj}}=\\{M_{k}(t_{j}),D^{1}M_{k}(t_{j}),\cdots,D^{p-1}M_{k}(t_{j})\\}^{\prime}$,
$\boldsymbol{F}_{M_{kj}}=(F_{M_{kj}}^{il})_{m\times p}$ and
$\boldsymbol{\varepsilon}_{M_{kj}}=\text{diag}(\varepsilon_{M_{kj}}^{1},\varepsilon_{M_{kj}}^{2},\cdots,\varepsilon_{M_{kj}}^{m})$.
When $i$th subject has an observation at time $t_{j}$ and $k_{i}=k$,
$Y_{M_{kj}}^{i}=Y_{i}(t_{j})-U_{i}(t_{j})$, $F_{M_{kj}}^{i1}=1$ and
$\varepsilon_{M_{kj}}^{i}\sim\mathsf{N}_{1}(0,\sigma^{2}_{\varepsilon})$.
Otherwise, $Y_{M_{kj}}^{i}=F_{M_{kj}}^{il}=\varepsilon_{M_{kj}}^{i}=0$. The
$\boldsymbol{G}_{M_{kj}}$, $\boldsymbol{\omega}_{M_{kj}}$ and
$\boldsymbol{W}_{M_{kj}}$ are given by Proposition 1 with $r=p$.
3. (3a) Given $M_{k_{i}}(t_{ij})$ and $U_{i}(t_{ij})$, $i=1,2,\cdots,m$, $j=1,2,\cdots,n_{i}$, sample $\sigma^{2}_{\varepsilon}$ from the posterior distribution $\textsf{invGamma}\left(a+\frac{1}{2}\sum_{i=1}^{m}n_{i},b+\frac{1}{2}\sum_{i=1}^{m}\sum_{j=1}^{n_{i}}\left\\{Y_{i}(t_{ij})-M_{k_{i}}(t_{ij})-U_{i}(t_{ij})\right\\}^{2}\right)$.
4. (3b) Given $\boldsymbol{U}_{{i0}}$, sample $\sigma^{2}_{U_{0}}$ from the posterior distribution $\textsf{invGamma}\left(a+\frac{mq}{2},b+\frac{1}{2}\sum_{i=0}^{m}\boldsymbol{U}_{{i0}}^{\prime}\boldsymbol{U}_{{i0}}\right)$.
5. (3c) Given $\boldsymbol{M}_{{kj}}$, sample $\sigma_{M_{k}}^{2}$ from the posterior distribution
$\textsf{invGamma}\left(a+\frac{np}{2},b+\frac{1}{2}\sum_{j=0}^{n-1}(\boldsymbol{M}_{{k(j+1)}}-\boldsymbol{G}_{M_{kj}}\boldsymbol{M}_{{kj}})^{\prime}\boldsymbol{W}_{M_{kj}}^{-1}(\boldsymbol{M}_{{k(j+1)}}-\boldsymbol{G}_{M_{kj}}\boldsymbol{M}_{{kj}})\right)$.
6. (3d) Given $\boldsymbol{U}_{{ij}}$, $\boldsymbol{\beta}$ and $\sigma^{2}$, sample $\sigma^{2}_{U_{i}}$ using a Metropolis-Hasting algorithm. We choose $\sigma^{2}_{U_{i}}\sim\textsf{invGamma}(a,b)$ as the proposal prior distribution and a proposal $\sigma^{2*}_{U_{i}}$ can be easily drawn from $\textsf{invGamma}\left(a+\frac{n_{i}q}{2},b+\frac{1}{2}\sum_{j=0}^{n_{i}-1}(\boldsymbol{U}_{{i(j+1)}}-\boldsymbol{G}_{U_{ij}}\boldsymbol{U}_{{ij}})^{\prime}\boldsymbol{W}_{U_{ij}}^{-1}(\boldsymbol{U}_{{i(j+1)}}-\boldsymbol{G}_{U_{ij}}\boldsymbol{U}_{{ij}})\right)$ the corresponding proposal posterior distribution. The $\sigma^{2*}_{U_{i}}$ will be accepted with the following probability and discarded otherwise with $\sigma^{2}_{U_{i}}$ unchanged,
$\min\left\\{\frac{f_{\mathsf{LN}}(\sigma^{2*}_{U_{i}}\mid\boldsymbol{x}^{\prime}_{i}\boldsymbol{\beta},\sigma^{2})\prod_{j=0}^{n_{i}-1}f_{\mathsf{N_{q}}}(\boldsymbol{U}_{{i(j+1)}}-\boldsymbol{G}_{U_{ij}}\boldsymbol{U}_{{ij}}\mid\boldsymbol{0},\sigma^{2*}_{U_{i}}\boldsymbol{W}_{U_{ij}})f_{\mathsf{iG}}(\sigma^{2}_{U_{i}}\mid
a_{U_{i}},b_{U_{i}})}{f_{\mathsf{LN}}(\sigma^{2}_{U_{i}}\mid\boldsymbol{x}^{\prime}_{i}\boldsymbol{\beta},\sigma^{2})\prod_{j=0}^{n_{i}-1}f_{\mathsf{N_{q}}}(\boldsymbol{U}_{{i(j+1)}}-\boldsymbol{G}_{U_{ij}}\boldsymbol{U}_{{ij}}\mid\boldsymbol{0},\sigma^{2}_{U_{i}}\boldsymbol{W}_{U_{ij}})f_{\mathsf{iG}}(\sigma^{2*}_{U_{i}}\mid
a_{U_{i}},b_{U_{i}})},1\right\\},$
where $f_{\mathsf{LN}}$, $f_{\mathsf{N_{q}}}$ and $f_{\mathsf{iG}}$ denote the
log-normal, $q$-dimensional normal and inverse gamma probability density
functions respectively with $a_{U_{i}}=a+\frac{n_{i}q}{2}$,
$b_{U_{i}}=b+\frac{1}{2}\sum_{j=0}^{n_{i}-1}(\boldsymbol{U}_{{i(j+1)}}-\boldsymbol{G}_{U_{ij}}\boldsymbol{U}_{{ij}})^{\prime}\boldsymbol{W}_{U_{ij}}^{-1}(\boldsymbol{U}_{{i(j+1)}}-\boldsymbol{G}_{U_{ij}}\boldsymbol{U}_{{ij}})$.
7. (4) Given $\sigma^{2}_{U_{i}}$, sample $\boldsymbol{\beta}$ and $\sigma^{2}$. Let $\boldsymbol{Z}=(\log\sigma^{2}_{U_{1}},\log\sigma^{2}_{U_{2}},\cdots,\log\sigma^{2}_{U_{m}})^{\prime}$, $\boldsymbol{\hat{\beta}}=(\boldsymbol{X}^{\prime}\boldsymbol{X})^{-1}\boldsymbol{X}^{\prime}\boldsymbol{Z}$ and $\hat{\sigma}^{2}=\frac{(\boldsymbol{Z}-\boldsymbol{X}\boldsymbol{\hat{\beta}})^{\prime}(\boldsymbol{Z}-\boldsymbol{X}\boldsymbol{\hat{\beta}})}{m-k}$. We draw $\tau$ from Chi-squared distribution with $m-k$ degrees of freedom and set $\sigma^{2}=\frac{(m-k)\hat{\sigma}^{2}}{\tau}$ and then sample $\boldsymbol{\beta}$ from $\textsf{N}_{m}\left(\boldsymbol{\hat{\beta}},\sigma^{2}(\boldsymbol{X}^{\prime}\boldsymbol{X})^{-1}\right)$.
## 4 Simulation
We carry out two simulation studies to evaluate the performance of the
proposed method and compare it to alternative methods including nature cubic
spline (NCS, Wahba, 1990) and functional principal components analysis (FPCA,
Yao et al., 2005). The comparison focuses on performance in estimating the
trajectory $M_{k_{i}}(t)+U_{i}(t)$, the volatility $\sigma^{2}_{U_{i}}$ and
the coefficients $\boldsymbol{\beta}$.
The first simulation study is designed to investigate the consequence of
ignoring either similarity or heterogeneity of volatilities when they are
present. One hundred replicated datasets, each consisting of 100 trajectories,
are sampled from the SVR model, in which the log-transformed volatilities are
varying and normally distributed. More precisely, we choose
$\boldsymbol{\beta}=(0,0.6,2)^{\prime}$ and
$\boldsymbol{x}_{i}=(1,x_{i1},x_{i2})^{\prime}$ with $x_{i1}$ and $x_{i2}$
sampled from
$x_{i1}\stackrel{{\scriptstyle\mathrm{i.i.d.}}}{{\sim}}\mathsf{Bernoulli}(0.4)$
and
$x_{i2}\stackrel{{\scriptstyle\mathrm{i.i.d.}}}{{\sim}}\mathsf{N}_{1}(0,0.25)$
respectively. Given $\boldsymbol{\beta}$ and $\boldsymbol{x}_{i}$,
volatilities $\sigma^{2}_{U_{i}}$’s can be drawn from
$\log(\sigma^{2}_{U_{i}})\sim\mathsf{N}_{1}(\boldsymbol{x}^{\prime}_{i}\boldsymbol{\beta},1)$.
Along with $\sigma^{2}_{M_{1}}=\sigma^{2}_{M_{2}}=10$,
$\sigma^{2}_{\varepsilon}=1$, $p=2$ and $q=1$, $M_{1}(t)$, $M_{2}(t)$,
$U_{i}(t)$ and $\varepsilon_{i}(t)$ are sampled at
$t\in\\{0.2,0.4,\cdots,4\\}$ from equations (2) and (3) and the distribution
of measurement error $\varepsilon_{i}(t)$. Twenty percent of samples are
removed completely at random, resulting in on average 16 unequally spaced
observations per subject. Finally, the $i$th subject is randomly assigned to
one of the two groups with equal probability and $Y_{i}(t)$ is obtained from
observation equation (1).
The simulated datasets are analyzed by three methods, SVR, NCS and FPCA. We
first apply the SVR approach, using the proposed MCMC algorithm with 15,000
iterations and keeping every 5th of the last 10,000 samples for posterior
analysis. It takes about 80 minutes on a PC with 2.33GHz Intel(R) Xeon(R) CPU.
Posterior means are chosen as the estimates of $M_{k_{i}}(t)+U_{i}(t)$,
$\sigma^{2}_{U_{i}}$ and $\boldsymbol{\beta}$. Additionally, the trajectories
$M_{k_{i}}(t)+U_{i}(t)$’s are estimated by NCS for one subject at a time, and
by FPCA for all subjects within a group and separately by the group, taking
about 1 minute and 2 minutes on the same PC respectively. For NCS and FPCA
methods, we may also estimate covariate effects on volatility through a two-
stage method: estimating empirical volatility by
$\frac{1}{n_{i}}\sum_{j=1}^{n_{i}-1}\frac{(\hat{U}_{i,j+1}-\hat{U}_{i,j})^{2}}{t_{i,j+1}-t_{i,j}}$
in the first stage with $\hat{U}_{i,j}$ the estimate of $U_{i}(t)$ at time
$t_{ij}$; and in the second stage, empirical volatilities are regressed on
covariates to obtain the estimate of $\boldsymbol{\beta}$.
For each simulated dataset, we calculate average squared error (ASE) for the
trajectory
$\text{ASE}(M+U)=\frac{1}{m}\sum_{i=1}^{m}\frac{1}{n_{i}}\sum_{j=1}^{n_{i}}\left\\{\hat{M}_{k_{i}}(t_{ij})+\hat{U}_{i}(t_{ij})-M_{k_{i}}(t_{ij})-U_{i}(t_{ij})\right\\}^{2}$,
ASE for log volatility
$\text{ASE}\\{log(\sigma^{2}_{U})\\}=\frac{1}{m}\sum_{i=1}^{m}\left\\{log(\hat{\sigma}^{2}_{U_{i}})-log(\sigma^{2}_{U_{i}})\right\\}^{2}$,
and squared errors (SE) for coefficient estimates
$\text{SE}(\beta_{l})=(\hat{\beta}_{l}-\beta_{l})^{2}$, $l=0,1,2$. Table 1
reports means of $\text{ASE}(M+U)$, $\text{ASE}\\{log(\sigma^{2}_{U})\\}$ and
$\text{SE}(\beta_{l})$ across 100 replicate datasets. MASEs and MSEs by NCS
and FPCA approaches are significantly inflated, for example, being doubled and
tripled in $\text{MASE}(M+U)$ respectively, compared to SVR. We randomly
select a data set and take a close examination. We calculate the individual
ASE of the trajectory
$\frac{1}{n_{i}}\sum_{j=1}^{n_{i}}\left\\{\hat{M}_{k_{i}}(t_{ij})+\hat{U}_{i}(t_{ij})-M_{k_{i}}(t_{ij})-U_{i}(t_{ij})\right\\}^{2}$
and select the top four subjects with the largest individual ASEs with respect
to NCS and FPCA approaches respectively.
Figure 2 shows estimates of the trajectory for six subjects. The plots reveal
that the increased MASEs or MSEs by NCS and FPCA are caused by different
reasons. NCS approach, treating one trajectory a time, ignores the similarity
between the subjects within a group, leading to over fitting true trajectories
(e.g. Figure 2(d) and 2(e)) with both over and under estimated volatilities.
On the other hand, FPCA approach omits the heterogeneity of the subjects
within a group; inflated $\text{MASE}(M+U)$ are mainly contributed by a few
subjects whose trajectory fluctuates with significantly higher volatility but
is overly smoothed (e.g. Figure 2(b) and 2(d)); and under the assumption of
similar smoothness, the estimates of volatility are largely under estimated.
Although this simulation study is in favor of SVR approach by design, the
scenario we consider is nevertheless realistic in practice and the simulation
results reveal the drawbacks of omitting similarity or heterogeneity of
volatilities by alternative approaches.
Our second simulation study is conducted to evaluate the performance of SVR,
NCS and FPCA when volatilities are homogeneous with no covariate effects. As
in the first simulation study, 100 replicate datasets are generated, each
consisting of 100 trajectories at $t\in\\{0.2,0.4,\cdots,4\\}$; twenty percent
of data points are deleted completely at random; and subjects are assigned to
one of two groups with equal probability. The observations are generated from
$Y_{i}(t)=10\\{t+sin(t)\\}+0.6\alpha_{1i}cos(\pi t/10)+0.2\alpha_{2i}sin(\pi
t/10)+\varepsilon_{i}(t)$ for subjects in the first group and from
$Y_{i}(t)=10\\{t+cos(t)\\}+0.5\alpha_{1i}cos(\pi t/10)+0.3\alpha_{2i}sin(\pi
t/10)+\varepsilon_{i}(t)$ for the ones in the second group, with
$\alpha_{1i}\stackrel{{\scriptstyle\mathrm{i.i.d.}}}{{\sim}}\mathsf{N}_{1}(0,4)$,
$\alpha_{2i}\stackrel{{\scriptstyle\mathrm{i.i.d.}}}{{\sim}}\mathsf{N}_{1}(0,1)$
and
$\varepsilon_{i}(t)\stackrel{{\scriptstyle\mathrm{i.i.d.}}}{{\sim}}\mathsf{N}_{1}(0,1)$.
The SVR, NCS and FPCA approaches are applied and $\text{MASE}(M+U)$ is
presented in Table 1, in which SVR and FPCA approaches show close
$\text{MASE}(M+U)$, both smaller than the one by NCS approach. This suggests
that SVR is no worse than FPCA for the cases with homogeneous volatilities.
In short, through the two simulation studies, we demonstrate that SVR achieves
substantial gains in terms of reducing the ASEs or SEs of the estimates of the
trajectory, volatility and covariate effect when volatilities are
heterogeneous, and works at least as well as FPCA approach when volatilities
are homogeneous.
Table 1: The mean of squared errors or average square errors of the estimates of trajectory, volatility and covariate effect across 100 replicate datasets . | | | Case I | | | Case II
---|---|---|---|---|---|---
method | $M+U$ | $\log(\sigma^{2}_{U})$ | $\beta_{0}$ | $\beta_{1}$ | $\beta_{2}$ | $M+U$
SVR | 0.345 | 0.614 | 0.043 | 0.081 | 0.075 | 1.122
NCS | 0.609 | 1.297 | 0.089 | 0.165 | 1.724 | 1.477
FPCA | 1.099 | 2.966 | 1.144 | 0.185 | 1.969 | 1.185
(a) ID=17
(b) ID=30
(c) ID=31
(d) ID=39
(e) ID=44
(f) ID=86
Figure 2: The plots of observation ($\circ$) and trajectory at time $t_{ij}$
($\times$), as well as estimates of trajectory $M_{k_{i}}(t)+U_{i}(t)$ by SVR
(—), NCS ($---$) and FPCA ($\cdotp\cdotp\cdotp$) approaches, for six subjects
in one simulated dataset with the largest individual average squared errors
$\frac{1}{n_{i}}\sum_{j=1}^{n_{i}}\left\\{\hat{M}_{k_{i}}(t_{ij})+\hat{U}_{i}(t_{ij})-M_{k_{i}}(t_{ij})-U_{i}(t_{ij})\right\\}^{2}$.
## 5 Applications
It is a common practice to monitor the blood pressure of pregnant woman during
pregnancy. Despite the trend of blood pressure being well studied, its
fluctuation has been little addressed and the factors associated with the
fluctuation are largely unknown. In this analysis, we apply the proposed SVR
approach to analyze longitudinal blood pressure measurements in HPHB study,
aiming to investigate the stability of blood pressure trajectories and
identify the associated factors. The data set consists of 106 non-Hispanic
white (NHW) and 176 non-Hispanic black (NHB) women whose first blood pressure
measurement is collected before the 16th week of gestation and the last one no
earlier than the 37th week of gestation. Most of subjects have 9 ($35.10\%$ of
them), 10 ($29.28\%$) or 11 ($14.98\%$) measurements spaced at irregularly
times. The covariates we focused on include race as NHW vs NHB, mother’s age
group as age $>$ 35 vs age $\leq$ 35, obesity as yes vs no, preeclampsia as
yes vs no, parity group as parity $>$ 0 vs parity $=$ 0, and smoking as yes vs
no. We run the proposed MCMC algorithm for 15,000 iterations, discard the
first 5,000 and retain every 5th of the remaining samples for posterior
analysis. The trace plots and autocorrelation plots suggest the algorithm
converges fast and mixes well. Posterior summary of selected parameters is
presented in Table 2.
The panels LABEL:sub@fig:MAPMNHW and LABEL:sub@fig:MAPMNHB of Figure 3 show
posterior means and $95\%$ credible intervals of the average blood pressure
for NHW and NHB groups respectively, which share a common pattern: decreasing
till the late stage of the second trimester (about 20 to 25 weeks) and then
increasing toward the pre-pregnancy level. Within the ethnic group,
significant heterogeneity exists in terms of the stability of the blood
pressure trajectory at the individual level. As Figure 3
LABEL:sub@fig:MAPVolPost indicated, posterior means of volatility vary from
-0.5 to 2 in the logarithmic scale, suggesting some subjects’ trajectories are
parallel to the group blood pressure trajectory with very small volatilities
while others may significantly depart from the group blood pressure
trajectory.
Most interesting, we find that obesity and preeclampsia are associated with
blood pressure volatility, with their $95\%$ credible intervals not covering
zero in Figure 3 LABEL:sub@fig:MAPCoeff. This implies that pregnant women with
obesity and/or preeclampsia are more likely to demonstrate irregular patterns
of blood pressure relative to their ethnic group. We further examine the
characteristics of these subjects with extreme volatilities (results not
shown). Among the top eight subjects presenting with the largest volatilities,
most of them are NHB with obesity and preeclampsia, do not smoke and give
birth to a baby for the first time; half of them are younger than 35. For the
eight subjects with the smallest volatilities, they are surprising
homogeneous, i.e. all of them being NHW (except one) without obesity and
preeclampsia, younger than 35, not smoking and giving birth to a baby before.
Table 2: Blood pressure data: Posterior summary of parameters in the SVR model. Parameter | Mean | Mode | SD | 95% HPD invteral
---|---|---|---|---
$\sigma^{2}_{\varepsilon}$ | 17.807 | 17.818 | 0.694 | [16.389, 19.106]
$\sigma^{2}_{M_{1}}$ | 0.236 | 0.187 | 0.181 | [0.040, 0.556]
$\sigma^{2}_{M_{2}}$ | 0.204 | 0.162 | 0.148 | [0.042, 0.472]
$\sigma^{2}_{U_{0}}$ | 46.729 | 46.412 | 4.776 | [38.263, 56.619]
$\sigma^{2}$ | 0.734 | 0.741 | 0.333 | [0.082, 1.295]
(a)
(b)
(c)
(d)
Figure 3: The posterior means and $95\%$ highest posterior density (HPD)
credible intervals for (a) the blood pressure during the 2nd and 3rd
trimesters for non-Hispanic white group; (b) the blood pressure during the 2nd
and 3rd trimesters for non-Hispanic black group; (c) the volatility in the
logarithmic scale; (d) covariate effects.
## 6 Discussion
We have proposed a stochastic volatility regression model to investigate the
volatility and its association with covariates for multi-subject functional
data. As an important dynamic feature, volatility measures the stability of
the biological process. The analysis of volatility not only reveals its
heterogeneity among subjects but also its dependence on the covariates of
interest. Complementing the current FDA methods which mainly focus on the
trend of trajectory and its derivatives, the SVR method initiates the
exploration of stability of functional data. As illustrated with the blood
pressure data, our view is that substantial new insights can be obtained in a
rich variety of biomedical applications by studying volatility.
## Acknowledgments
This work was supported by Award Number R01ES017436 and R01ES17240 from the
National Institute of Environmental Health Sciences, by funding from the
National Institutes of Health (5P2O-RR020782-O3) and the U.S. Environmental
Protection Agency (RD-83329301-0), and by the Intramural Research Program of
the National Cancer Institute, National Institutes of Health. The content is
solely the responsibility of the authors and does not necessarily represent
the official views of the National Institute of Environmental Health Sciences,
the National Institutes of Health or the U.S. Environmental Protection Agency.
## Appendix A Appendix: Proofs of Theoretical Results
### A.1 Proof of Theorem 1
By the RKHS theory of the polynomial smoothing spline (Wahba, 1990, Section
1.2), we have
$\displaystyle{M}_{k}(t)$
$\displaystyle={M}_{k0}(t)+{M}_{k1}(t)=\sum_{l=0}^{p-1}\mu_{kl}\phi_{l}(t)+\sum_{j=1}^{n}\nu_{kj}\mathcal{R}_{M_{1}}(t_{j},t)+\eta_{M_{1}}(t),$
and
$\int_{\mathcal{T}}\left\\{D^{p}M_{k}(t)\right\\}^{2}dt=\boldsymbol{\nu}^{\prime}_{k}\boldsymbol{R}_{M_{1}}\boldsymbol{\nu}_{k}+\langle\eta_{M_{1}}(\cdot),\eta_{M_{1}}(\cdot)\rangle_{\mathcal{H}_{\mathcal{R}_{M_{1}}}},$
where $M_{0}(t)\in\mathcal{H}_{\mathcal{R}_{M_{0}}}$ and
$M_{1}(t)\in\mathcal{H}_{\mathcal{R}_{M_{1}}}$, the RKHSs
$\mathcal{H}_{\mathcal{R}_{M_{0}}}=\left\\{f(t):D^{p}f(t)=0,t\in\mathcal{T}\right\\}$
and $\mathcal{H}_{\mathcal{R}_{M_{1}}}=\left\\{f(t):D^{i}f(t)\text{ absolutely
continuous for
}i=0,1,\cdots,p-1,D^{p}f(t)\in\mathcal{L}_{2}(\mathcal{T})\right\\}$;
$\mathcal{R}_{M_{0}}(s,t)$ and $\mathcal{R}_{M_{1}}(s,t)$ are reproducing
kernels defined in Lemma 1;
$\eta_{M_{1}}(\cdot)\in\mathcal{H}_{\mathcal{R}_{M_{1}}}$ is orthogonal to
$\mathcal{R}_{M_{1}}(t_{j},\cdot)$ with inner product
$\langle\mathcal{R}_{M_{1}}(t_{j},\cdot),\eta_{M_{1}}(\cdot)\rangle_{\mathcal{H}_{\mathcal{R}_{M_{1}}}}=\int_{\mathcal{T}}D^{p}\mathcal{R}_{M_{1}}(t_{j},u)D^{p}\eta_{M_{1}}(u)du=0$
for $j=1,2,\cdots,J$;
$\mathcal{L}_{2}(\mathcal{T})=\left\\{f(t):\int_{\mathcal{T}}f^{2}(t)dt<\infty\right\\}$
is the space of squared integrable functions defined on index set
$\mathcal{T}$.
Similarly,
$\displaystyle{U}_{i}(t)$
$\displaystyle={U}_{i0}(t)+{U}_{i1}(t)=\sum_{l=0}^{q-1}\alpha_{il}\phi_{l}(t)+\sum_{j=1}^{n_{i}}\gamma_{ij}\mathcal{R}_{U_{1}}(t_{ij},t)+\eta_{U_{i1}}(t),$
and
$\int_{\mathcal{T}}\left\\{D^{q}U_{i}(t)\right\\}^{2}dt=\boldsymbol{\gamma}_{i}^{\prime}\boldsymbol{R}_{U_{i1}}\boldsymbol{\gamma}_{i}+\langle\eta_{U_{i1}}(\cdot),\eta_{U_{i1}}(\cdot)\rangle_{\mathcal{H}_{\mathcal{R}_{U_{1}}}},$
where $U_{i0}(t)\in\mathcal{H}_{\mathcal{R}_{U_{0}}}$ and
$U_{i1}(t)\in\mathcal{H}_{\mathcal{R}_{U_{1}}}$ with
$\mathcal{H}_{\mathcal{R}_{U_{0}}}=\left\\{f(t):D^{q}f(t)=0,t\in\mathcal{T}\right\\}$
and $\mathcal{H}_{\mathcal{R}_{U_{1}}}=\left\\{f(t):D^{i}f(t)\text{ absolutely
continuous for
}i=0,1,\cdots,q-1,D^{q}f(t)\in\mathcal{L}_{2}(\mathcal{T})\right\\}$ the RKHSs
with reproducing kernel $\mathcal{R}_{U_{0}}(s,t)$ and
$\mathcal{R}_{U_{1}}(s,t)$ defined in Lemma 1;
$\eta_{U_{i1}}(\cdot)\in\mathcal{H}_{\mathcal{R}_{U_{1}}}$ is orthogonal to
$\mathcal{R}_{U_{1}}(t_{ij},\cdot)$ for $j=1,2,\cdots,n_{i}$.
Hence, the double-penalized sum-of-squares (4) can be written as
$\displaystyle\textsf{DPSS}=$
$\displaystyle\sum_{i=1}^{m}\frac{1}{n_{i}}(\boldsymbol{Y}_{i}-\boldsymbol{\Delta}_{i}\boldsymbol{\phi}_{\mu}\boldsymbol{\mu}_{k_{i}}-\boldsymbol{\Delta}_{i}\boldsymbol{R}_{M_{1}}\boldsymbol{\nu}_{k_{i}}-\boldsymbol{\phi}_{\alpha_{i}}\boldsymbol{\alpha}_{i}-\boldsymbol{R}_{U_{i1}}\boldsymbol{\gamma}_{i})^{\prime}\times$
$\displaystyle(\boldsymbol{Y}_{i}-\boldsymbol{\Delta}_{i}\boldsymbol{\phi}_{\mu}\boldsymbol{\mu}_{k_{i}}-\boldsymbol{\Delta}_{i}\boldsymbol{R}_{M_{1}}\boldsymbol{\nu}_{k_{i}}-\boldsymbol{\phi}_{\alpha_{i}}\boldsymbol{\alpha}_{i}-\boldsymbol{R}_{U_{i1}}\boldsymbol{\gamma}_{i})+$
$\displaystyle\sum_{k=1}^{g}\lambda_{M_{k}}\boldsymbol{\nu}^{\prime}_{k}\boldsymbol{R}_{M_{1}}\boldsymbol{\nu}_{k}+\sum_{i=1}^{m}\lambda_{U_{i}}\boldsymbol{\gamma}^{\prime}_{i}\boldsymbol{R}_{U_{i1}}\boldsymbol{\gamma}_{i}+$
$\displaystyle\sum_{k=1}^{g}\lambda_{M_{k}}\langle\eta_{M_{1}}(\cdot),\eta_{M_{1}}(\cdot)\rangle_{\mathcal{H}_{\mathcal{R}_{M_{1}}}}+\sum_{i=1}^{m}\lambda_{U_{i}}\langle\eta_{U_{i1}}(\cdot),\eta_{U_{i1}}(\cdot)\rangle_{\mathcal{H}_{\mathcal{R}_{U_{1}}}},$
which is minimized at
$\langle\eta_{M_{1}}(\cdot),\eta_{M_{1}}(\cdot)\rangle_{\mathcal{H}_{\mathcal{R}_{M_{1}}}}=\langle\eta_{U_{i1}}(\cdot),\eta_{U_{i1}}(\cdot)\rangle_{\mathcal{H}_{\mathcal{R}_{U_{1}}}}=0$,
leading to $\eta_{M_{1}}(\cdot)=\eta_{U_{i1}}=0$.
### A.2 Proof of Corollary 1
Taking partial derivatives of double penalized sum-of-squares in Corollary 1
with respective to $\boldsymbol{\mu}_{k}$, $\boldsymbol{\nu}_{k}$,
$\boldsymbol{\alpha}_{i}$ and $\boldsymbol{\gamma}_{i}$ respectively and
setting them to zeros, we have
$\displaystyle\frac{\partial\;\textsf{DPSS}}{\partial\;\boldsymbol{\mu}_{k}}$
$\displaystyle=\displaystyle\sum_{i:k_{i}=k}\frac{1}{n_{i}}\boldsymbol{\phi}_{\mu}^{\prime}\boldsymbol{\Delta}_{i}^{\prime}(\boldsymbol{\Delta}_{i}\boldsymbol{\phi}_{\mu}\boldsymbol{\mu}_{k_{i}}+\boldsymbol{\Delta}_{i}\boldsymbol{R}_{M_{1}}\boldsymbol{\nu}_{k_{i}}+\boldsymbol{\phi}_{\alpha_{i}}\boldsymbol{\alpha}_{i}+\boldsymbol{R}_{U_{i1}}\boldsymbol{\gamma}_{i}-\boldsymbol{Y}_{i})=\boldsymbol{0},$
$\displaystyle\frac{\partial\;\textsf{DPSS}}{\partial\;\boldsymbol{\nu}_{k}}$
$\displaystyle=\displaystyle\sum_{i:k_{i}=k}\frac{1}{n_{i}}\boldsymbol{R}_{M_{1}}\boldsymbol{\Delta}_{i}^{\prime}(\boldsymbol{\Delta}_{i}\boldsymbol{\phi}_{\mu}\boldsymbol{\mu}_{k_{i}}+\boldsymbol{\Delta}_{i}\boldsymbol{R}_{M_{1}}\boldsymbol{\nu}_{k_{i}}+\boldsymbol{\phi}_{\alpha_{i}}\boldsymbol{\alpha}_{i}+\boldsymbol{R}_{U_{i1}}\boldsymbol{\gamma}_{i}-\boldsymbol{Y}_{i})+\lambda_{M_{k}}\boldsymbol{R}_{M_{1}}\boldsymbol{\nu}_{k}=\boldsymbol{0},$
$\displaystyle\frac{\partial\;\textsf{DPSS}}{\partial\;\boldsymbol{\alpha}_{i}}$
$\displaystyle=\frac{1}{n_{i}}\boldsymbol{\phi}_{\alpha_{i}}^{\prime}(\boldsymbol{\Delta}_{i}\boldsymbol{\phi}_{\mu}\boldsymbol{\mu}_{k_{i}}+\boldsymbol{\Delta}_{i}\boldsymbol{R}_{M_{1}}\boldsymbol{\nu}_{k_{i}}+\boldsymbol{\phi}_{\alpha_{i}}\boldsymbol{\alpha}_{i}+\boldsymbol{R}_{U_{i1}}\boldsymbol{\gamma}_{i}-\boldsymbol{Y}_{i})=\boldsymbol{0},$
$\displaystyle\frac{\partial\;\textsf{DPSS}}{\partial\;\boldsymbol{\gamma}_{i}}$
$\displaystyle=\frac{1}{n_{i}}\boldsymbol{R}_{U_{i1}}(\boldsymbol{\Delta}_{i}\boldsymbol{\phi}_{\mu}\boldsymbol{\mu}_{k_{i}}+\boldsymbol{\Delta}_{i}\boldsymbol{R}_{M_{1}}\boldsymbol{\nu}_{k_{i}}+\boldsymbol{\phi}_{\alpha_{i}}\boldsymbol{\alpha}_{i}+\boldsymbol{R}_{U_{i1}}\boldsymbol{\gamma}_{i}-\boldsymbol{Y}_{i})+\lambda_{U_{i}}\boldsymbol{R}_{U_{i1}}\boldsymbol{\gamma}_{i}=\boldsymbol{0},$
which lead to
$\displaystyle\boldsymbol{\phi}_{\mu}^{\prime}\boldsymbol{\Delta}\boldsymbol{\phi}_{\mu}\boldsymbol{\mu}_{k}+\boldsymbol{\phi}_{\mu}^{\prime}\boldsymbol{\Delta}\boldsymbol{R}_{M_{1}}\boldsymbol{\nu}_{k}$
$\displaystyle=\boldsymbol{\phi}_{\mu}^{\prime}\boldsymbol{\tilde{Y}}_{k},$
(6)
$\displaystyle\boldsymbol{R}_{M_{1}}\boldsymbol{\Delta}\boldsymbol{\phi}_{\mu}\boldsymbol{\mu}_{k}+(\boldsymbol{R}_{M_{1}}\boldsymbol{\Delta}+\lambda_{M_{k}}\boldsymbol{I})\boldsymbol{R}_{M_{1}}\boldsymbol{\nu}_{k}$
$\displaystyle=\boldsymbol{R}_{M_{1}}\boldsymbol{\tilde{Y}}_{k},$ (7)
$\displaystyle\boldsymbol{\phi}_{\alpha_{i}}^{\prime}\boldsymbol{\phi}_{\alpha_{i}}\boldsymbol{\alpha}_{i}+\boldsymbol{\phi}_{\alpha_{i}}^{\prime}\boldsymbol{R}_{U_{i1}}\boldsymbol{\gamma}_{i}$
$\displaystyle=\boldsymbol{\phi}_{\alpha_{i}}^{\prime}\boldsymbol{\tilde{Y}}_{i},$
(8)
$\displaystyle\boldsymbol{R}_{U_{i1}}\boldsymbol{\phi}_{\alpha_{i}}\boldsymbol{\alpha}_{i}+(\boldsymbol{R}_{U_{i1}}+n_{i}\lambda_{U_{i}}\boldsymbol{I})\boldsymbol{R}_{U_{i1}}\boldsymbol{\gamma}_{i}$
$\displaystyle=\boldsymbol{R}_{U_{i1}}\boldsymbol{\tilde{Y}}_{i},$ (9)
with
$\boldsymbol{\tilde{Y}}_{k}=\displaystyle\sum_{i:k_{i}=k}\frac{1}{n_{i}}\boldsymbol{\Delta}_{i}^{\prime}\left(\boldsymbol{Y}_{i}-\boldsymbol{\phi}_{\alpha_{i}}\boldsymbol{{\alpha}}_{i}-\boldsymbol{R}_{U_{i1}}\boldsymbol{{\gamma}}_{i}\right)$,
$\boldsymbol{\tilde{Y}}_{i}=\boldsymbol{Y}_{i}-\boldsymbol{\Delta}_{i}\boldsymbol{\phi}_{\mu}\boldsymbol{{\mu}}_{k_{i}}-\boldsymbol{\Delta}_{i}\boldsymbol{R}_{M_{1}}\boldsymbol{{\nu}}_{k_{i}}$
and
$\boldsymbol{\Delta}=\displaystyle\sum_{i:k_{i}=k}\frac{1}{n_{i}}\boldsymbol{\Delta}_{i}^{\prime}\boldsymbol{\Delta}_{i}$.
The solutions of $\boldsymbol{\alpha}_{i}$ and $\boldsymbol{\gamma}_{i}$ in
the step (a) can be obtained from equations (8) and (9), while the solutions
of $\boldsymbol{\mu}_{k}$ and $\boldsymbol{\nu}_{k}$ in the step (b) from
equations (6) and (7).
### A.3 Proof of Proposition 1
Based on the SDE $D^{p}X(t)=\dot{W}(t)$, we have
$D^{1}\boldsymbol{X}(t)=\boldsymbol{C}\boldsymbol{X}(t)+\boldsymbol{D}\dot{W}(t),$
where $\boldsymbol{X}=\\{X(t),D^{1}X(t),\cdots,D^{p-1}X(t)\\}^{\prime}$,
$\boldsymbol{C}=(c_{ii^{\prime}})_{p\times p}$, $c_{ii^{\prime}}=1$ when
$i^{\prime}=i+1$ and $c_{ii^{\prime}}=0$ otherwise, and
$\boldsymbol{D}=(0,0,\cdots,1)^{\prime}$.
It follows that
$\displaystyle\boldsymbol{X}_{j+1}$
$\displaystyle=\exp(\boldsymbol{C}\delta_{j})\boldsymbol{X}_{j}+\int_{0}^{\delta_{j}}\exp\\{\boldsymbol{C}(\delta_{j}-u)\\}\boldsymbol{D}\boldsymbol{\dot{W}}(t_{j}+u)du$
$\displaystyle=\boldsymbol{G}_{j}\boldsymbol{X}_{j}+\boldsymbol{\omega}_{j},$
where
$\boldsymbol{G}_{j}=\exp(\boldsymbol{C}\delta_{j})=\sum_{k=0}^{p}\frac{\delta_{j}^{k}}{k!}\boldsymbol{C}^{k}$
and
$\boldsymbol{\omega}_{j}\sim\textsf{N}_{p}\left(\boldsymbol{0},\boldsymbol{W}_{j}\right)$
with
$\displaystyle\boldsymbol{W}_{j}$
$\displaystyle=\int_{0}^{\delta_{j}}\exp\\{\boldsymbol{C}(\delta_{j}-u)\\}\boldsymbol{D}\boldsymbol{D}^{\prime}\exp\\{\boldsymbol{C}^{\prime}(\delta_{j}-u)\\}du$
as required.
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|
arxiv-papers
| 2012-12-02T03:13:59 |
2024-09-04T02:49:38.804223
|
{
"license": "Public Domain",
"authors": "Bin Zhu and David B. Dunson",
"submitter": "Bin Zhu",
"url": "https://arxiv.org/abs/1212.0181"
}
|
1212.0245
|
# A note on some top local cohomology modules
Majid Eghbali Islamic Azad University, branch Soofian, Soofian, Iran.
[email protected]
###### Abstract.
Let $\mathfrak{a}$ be an ideal of a $d$-dimensional commutative Noetherian
ring $R$. In this paper we give some information on some last non-zero local
cohomology modules known as top local cohomology modules in particular,
$H^{d-1}_{\mathfrak{a}}(R)$.
###### Key words and phrases:
Top local cohomology, Attached primes, One dimensional ideal
###### 2000 Mathematics Subject Classification:
13D45, 13C14.
## 1\. Introduction
Throughout this note, we assume $R$ is a commutative Noetherian ring and
$\mathfrak{a}$ is an ideal of $R$. Let $M$ be an $R$-module. For an integer
$i\in\mathbb{Z}$ let $H^{i}_{\mathfrak{a}}(M)$ denote the $i$-th local
cohomology module of $M$ with respect to $\mathfrak{a}$ as introduced by
Grothendieck (cf. [Gr] and [Br-Sh]). The first non-vanishing cohomological
degree of the local cohomology modules $H^{i}_{\mathfrak{a}}(R)$ is well
understood. It is the maximal length of a regular sequence on $R$ consisting
of elements of $\mathfrak{a}$. The last non-vanishing amount of local
cohomology modules, instead, is more mysterious. It is known as the
cohomological dimension of $\mathfrak{a}$ in $R$, denoted by
$\operatorname{cd}(\mathfrak{a},R)$. It is a well-known fact that
$\begin{array}[]{ll}\
\operatorname{grade}(\mathfrak{a},R)\leq\operatorname{cd}(\mathfrak{a},R)\leq\dim
R.\end{array}$
Let $c:=\operatorname{cd}(\mathfrak{a},R)$ then, $\mathfrak{a}$ can not be
generated by the elements fewer than $c$. Hartshorne-Lichtenbaum vanishing
Theorem gives a characterization for vanishing of $H^{d}_{\mathfrak{a}}(R)$
i.e. for the case $\operatorname{cd}(\mathfrak{a},R)\leq\dim R-1$. In [E-Sch,
Theorem 3.3] the author and Schenzel have expressed the isomorphism
$H^{d}_{\mathfrak{a}}(R)\cong H^{d}_{\mathfrak{m}\widehat{R}}(\widehat{R}/J)$
for a certain ideal $J$ of $\widehat{R}$. It indicates the properties of
$H^{d}_{\mathfrak{a}}(R)$ via $H^{d}_{\mathfrak{m}\widehat{R}}(\widehat{R})$,
which is more well-known.
Ogus [O, Corollary 2.11] in equicharacteristic $0$ and Peskine and Szpiro
[P-S, 5.5] in equicharacteristic $p>0$ generalized the vanishing Theorem of
Hartshorne [Ha] for the cohomological dimension of the complement of a
subvariety of projective space. Huneke and Lyubeznik [Hu-L, Theorem 2.9] gave
a new characteristic-free proof of it. They proved the following Theorem:
###### Theorem 1.1.
Let $(R,\mathfrak{m})$ be a regular local ring of dimension $d$ containing a
field and let $\mathfrak{a}$ be an ideal of $R$. Then the followings are
equivalent:
1. (1)
$H^{i}_{\mathfrak{a}}(R)=0$ for $i=d-1,d$.
2. (2)
$\dim R/\mathfrak{a}\geq 2\ $ and $\
\operatorname{Spec}(R/\mathfrak{a})\setminus\\{\mathfrak{m}\\}$ is formally
geometrically connected.
In general the case $\operatorname{cd}(\mathfrak{a},R)\leq\dim R-2$ is more
complicated than the first one. On the other hand, $H^{d-1}_{\mathfrak{a}}(R)$
is more mysterious and its properties are not so much known. It provides
motivation to do research on this subject. The aim of the present paper is to
obtain more information about the structure of $H^{d-1}_{\mathfrak{a}}(R)$ to
shed light on the obscure aspects of this issue. The layout of it is as
follows. In Section $2$, we examine
$\operatorname{Att}_{R}(H^{d-1}_{\mathfrak{a}}(R))$ the set of attached prime
ideals of $H^{d-1}_{\mathfrak{a}}(R)$. As a technical tool we use the concept
of colocalization introduced by Richardson in [R]. As the main result in
Section $2$ we get the following result (cf. Theorem 2.2 ):
###### Theorem 1.2.
Let $(R,\mathfrak{m})$ be a complete local ring of dimension $d$. Let
$\mathfrak{a}$ be an ideal of $R$. Assume that $H^{d-1}_{\mathfrak{a}}(R)$ is
representable and $H^{d}_{\mathfrak{a}}(R)=0$. Then
1. (1)
$\operatorname{Att}_{R}(H^{d-1}_{\mathfrak{a}}(R))\subseteq\\{\mathfrak{p}\in\operatorname{Spec}R:\dim
R/\mathfrak{p}=d-1,\operatorname{Rad}(\mathfrak{a}+\mathfrak{p})=\mathfrak{m}\\}\cup\operatorname{Assh}(R)$.
2. (2)
$\\{\mathfrak{p}\in\operatorname{Spec}R:\dim
R/\mathfrak{p}=d-1,\operatorname{Rad}(\mathfrak{a}+\mathfrak{p})=\mathfrak{m}\\}\subseteq\operatorname{Att}_{R}(H^{d-1}_{\mathfrak{a}}(R))$.
In Section $3$, we focus on $H^{d-1}_{\mathfrak{a}}(R)$ with a one-dimensional
ideal $\mathfrak{a}$. Among the other results in Theorem 3.1 we show the
relation of $H^{d-1}_{\mathfrak{a}}(R)$ with some known modules of the form
$H^{i}_{\mathfrak{m}}(R)$ as a main result. In Section $4$ we consider
$H^{d-1}_{\mathfrak{a}}(R)$ with an arbitrary ideal $\mathfrak{a}$. We define
a certain ideal $J$ to get an epimorphism
$H^{d-1}_{\mathfrak{a}}(R)\rightarrow H^{d-1}_{\mathfrak{m}}(R/J)$ (cf. 4.1).
Part of this work is the author’s doctoral dissertation at the MLU Halle-
Wittenberg. My thanks are due to my adviser, Professor Peter Schenzel who draw
my attention to the importance of $H^{d-1}_{\mathfrak{a}}(R)$, helpful
conversations during the preparation of this note and careful reading of the
first draft of my paper.
## 2\. Attached primes of $H^{d-1}_{\mathfrak{a}}(R)$
Throughout this section, let $(R,\mathfrak{m})$ be a local ring and $S$ a
multiplicative closed subset of $R$. Recently, A. S. Richardson [R] has
proposed the definition of colocalization of an $R$-module $M$ relative to $S$
as the $S^{-1}R$-module $S_{-1}M=D_{S^{-1}R}(S^{-1}D_{R}(M))$, where $D(-)$
stands for the Matlis duality functor. See also [E] for more information.
In the light of [R, Theorem 2.2], representable modules are preserved under
colocalization and in case $M$ is a representable module, we have
$\begin{array}[]{ll}\
\operatorname{Att}S_{-1}M=\\{S^{-1}\mathfrak{p}:\mathfrak{p}\in\operatorname{Att}M\text{
and }S\cap\mathfrak{p}=\emptyset\\},\ \ \ \ \ \ \ (\ast)\end{array}$
where the notation $\operatorname{Att}M$ is used to denote the set of attached
prime ideals of $M$. This section is based on the use of this property. We
examine the set of attached prime ideals of last non-vanishing value of local
cohomology.
###### Proposition 2.1.
Let $(R,\mathfrak{m})$ be a local ring, $\mathfrak{a}$ be an ideal of $R$ and
$\mathfrak{p}\in\operatorname{Spec}R$. Let $c$ be an integer such that
$H^{i}_{\mathfrak{a}}(R)=0$ for every $i>c$. Assume that
$H^{c}_{\mathfrak{a}}(R)$ is representable. Then, we have the followings
1. (1)
$\operatorname{Att}_{R_{\mathfrak{p}}}(^{\mathfrak{p}}H^{c}_{\mathfrak{a}}(R))\subseteq\\{\mathfrak{q}R_{\mathfrak{p}}:\dim
R/\mathfrak{q}\geq c,\mathfrak{q}\subseteq\mathfrak{p}\text{ and
}\mathfrak{q}\in\operatorname{Spec}R\\}$.
2. (2)
In case $R$ is complete, we get the following equality
$\begin{array}[]{ll}\
\operatorname{Att}_{R_{\mathfrak{p}}}(^{\mathfrak{p}}H^{\dim
R}_{\mathfrak{a}}(R))=\\{\mathfrak{q}R_{\mathfrak{p}}:\dim R/\mathfrak{q}=\dim
R\text{ ,
}\mathfrak{q}\subseteq\mathfrak{p},\operatorname{Rad}(\mathfrak{a}+\mathfrak{q})=\mathfrak{m}\text{
and }\mathfrak{q}\in\operatorname{Spec}R\\}.\end{array}$
Proof.
1. (1)
By virtue of $(\ast)$ we have
$\begin{array}[]{ll}\
\operatorname{Att}_{R_{\mathfrak{p}}}(^{\mathfrak{p}}H^{c}_{\mathfrak{a}}(R))=\\{\mathfrak{q}R_{\mathfrak{p}}:\mathfrak{q}\in\operatorname{Att}_{R}(H^{c}_{\mathfrak{a}}(R))\text{
and }\mathfrak{q}\subseteq\mathfrak{p}\\}.\end{array}$
Since $H^{c}_{\mathfrak{a}}(R/\mathfrak{q})$ is representable then,
$\begin{array}[]{ll}\
\operatorname{Att}_{R}(H^{c}_{\mathfrak{a}}(R/\mathfrak{q}))=\operatorname{Att}_{R}(H^{c}_{\mathfrak{a}}(R))\cap\operatorname{Supp}(R/\mathfrak{q})\end{array}$
(cf. [A-M, Lemma 2.11]), which contains $\mathfrak{q}$. Hence,
$\operatorname{Att}_{R}(H^{c}_{\mathfrak{a}}(R/\mathfrak{q}))\neq\emptyset$ it
implies that $H^{c}_{\mathfrak{a}}(R/\mathfrak{q})\neq 0$ and consequently
$\dim R/\mathfrak{q}\geq c$.
2. (2)
Put $d:=\dim R$. If $H^{d}_{\mathfrak{a}}(R)=0$ we are done. Then we assume
that $H^{d}_{\mathfrak{a}}(R)\neq 0$.
$"\subseteq"$: Let
$\mathfrak{q}R_{\mathfrak{p}}\in\operatorname{Att}_{R_{\mathfrak{p}}}(^{\mathfrak{p}}H^{d}_{\mathfrak{a}}(R))$.
As we have seen in part one, $H^{d}_{\mathfrak{a}}(R/\mathfrak{q})\neq 0$ so
$\dim R/\mathfrak{q}=d$ and by Hartshorne-Lichtenbaum vanishing Theorem
$\operatorname{Rad}(\mathfrak{a}+\mathfrak{q})=\mathfrak{m}$.
$"\supseteq"$: again using $(\ast)$ we have
$\begin{array}[]{ll}\
\operatorname{Att}_{R_{\mathfrak{p}}}(^{\mathfrak{p}}H^{d}_{\mathfrak{a}}(R))=\\{\mathfrak{q}R_{\mathfrak{p}}:\mathfrak{q}\in\operatorname{Att}_{R}H^{d}_{\mathfrak{a}}(R),\mathfrak{q}\subseteq\mathfrak{p}\\},\end{array}$
so it is enough to show that
$\mathfrak{q}\in\operatorname{Att}_{R}H^{d}_{\mathfrak{a}}(R)$.
As $\dim R/\mathfrak{q}=d$ and
$\operatorname{Rad}(\mathfrak{a}+\mathfrak{q})=\mathfrak{m}$, so Independence
Theorem implies that $H^{d}_{\mathfrak{a}}(R/\mathfrak{q})\neq 0$. Hence
easily one can see that
$\begin{array}[]{ll}\
\emptyset\neq\operatorname{Att}_{R}(H^{d}_{\mathfrak{a}}(R/\mathfrak{q}))=\operatorname{Att}_{R}(H^{d}_{\mathfrak{a}}(R))\cap\operatorname{Supp}_{R}(R/\mathfrak{q}).\
\ \ \ \ \ (\ast\ast)\end{array}$
In the contrary assume that
$\mathfrak{q}\notin\operatorname{Att}_{R}H^{d}_{\mathfrak{a}}(R)$. Then by
virtue of $(\ast\ast)$ there exists a prime ideal
$\mathfrak{q}_{0}\in\operatorname{Att}_{R}H^{d}_{\mathfrak{a}}(R)$ such that
$\mathfrak{q}_{0}\supset\mathfrak{q}$ and so $\dim R/\mathfrak{q}_{0}<d$. On
the other $\mathfrak{q}_{0}\in\operatorname{Att}_{R}H^{d}_{\mathfrak{a}}(R)$
if and only if
$\mathfrak{q}_{0}R_{\mathfrak{q}_{0}}\in\operatorname{Att}_{R_{\mathfrak{q}_{0}}}(^{\mathfrak{q}_{0}}H^{d}_{\mathfrak{a}}(R))$.
By virtue of part one $\dim R/\mathfrak{q}_{0}=d$ which is a contradiction.
Now the proof is complete.
$\ \ \ \ \ \ \ \Box$
After this preparation we can state the main result of this section as
follows:
###### Theorem 2.2.
Let $(R,\mathfrak{m})$ be a complete local ring of dimension $d$. Let
$\mathfrak{a}$ be an ideal of $R$. Assume that $H^{d-1}_{\mathfrak{a}}(R)$ is
representable and $H^{d}_{\mathfrak{a}}(R)=0$. Then
1. (1)
$\operatorname{Att}_{R}(H^{d-1}_{\mathfrak{a}}(R))\subseteq\\{\mathfrak{p}\in\operatorname{Spec}R:\dim
R/\mathfrak{p}=d-1,\operatorname{Rad}(\mathfrak{a}+\mathfrak{p})=\mathfrak{m}\\}\cup\operatorname{Assh}(R)$.
2. (2)
$\\{\mathfrak{p}\in\operatorname{Spec}R:\dim
R/\mathfrak{p}=d-1,\operatorname{Rad}(\mathfrak{a}+\mathfrak{p})=\mathfrak{m}\\}\subseteq\operatorname{Att}_{R}(H^{d-1}_{\mathfrak{a}}(R))$.
Proof.
1. (1)
Let $\mathfrak{p}\in\operatorname{Att}_{R}H^{d-1}_{\mathfrak{a}}(R)$, then
$\mathfrak{p}R_{\mathfrak{p}}\in\operatorname{Att}_{R_{\mathfrak{p}}}(^{\mathfrak{p}}H^{d-1}_{\mathfrak{a}}(R))$.
Hence by Theorem 2.1 $\dim R/\mathfrak{p}\geq d-1$.
When $\dim R/\mathfrak{p}=d$ it follows that
$\mathfrak{p}\in\operatorname{Assh}(R)$. In the case $\dim
R/\mathfrak{p}=d-1$, as
$\mathfrak{p}\in\operatorname{Att}_{R}H^{d-1}_{\mathfrak{a}}(R)$ and
$H^{d-1}_{\mathfrak{a}}(-)$ is a right exact functor so one can deduce that
$H^{d-1}_{\mathfrak{a}}(R/\mathfrak{p})\neq 0$. By Hartshorne-Lichtenbaum
vanishing Theorem there exists a prime ideal
$\mathfrak{q}\supseteq\mathfrak{p}$ of $R$ of dimension $d-1$ with
$\operatorname{Rad}(\mathfrak{a}+\mathfrak{q})=\mathfrak{m}$. So, we must have
$\mathfrak{q}=\mathfrak{p}$.
2. (2)
Let $\dim R/\mathfrak{p}=d-1$ and
$\operatorname{Rad}(\mathfrak{a}+\mathfrak{p})=\mathfrak{m}$, then Theorem
2.1(2) implies that
$\mathfrak{p}R_{\mathfrak{p}}\in\operatorname{Att}_{R_{\mathfrak{p}}}(^{\mathfrak{p}}H^{d-1}_{\mathfrak{a}}(R/\mathfrak{p}))$.
By $(\ast)$ we deduce that
$\mathfrak{p}\in\operatorname{Att}_{R}(H^{d-1}_{\mathfrak{a}}(R/\mathfrak{p}))$.
On the other hand the epimorphism
$\begin{array}[]{ll}\ H^{d-1}_{\mathfrak{a}}(R)\rightarrow
H^{d-1}_{\mathfrak{a}}(R/\mathfrak{p})\rightarrow 0\end{array}$
implies that
$\mathfrak{p}\in\operatorname{Att}_{R}(H^{d-1}_{\mathfrak{a}}(R))$.
$\ \ \ \ \ \ \ \Box$
It is noteworthy to say that in the situation of Theorem 2.2 if $\mathfrak{a}$
is an ideal of dimension one, the inclusion at (1) will be an equality, see
[Hel, Theorem 8.2.3].
## 3\. One dimensional ideal
Of particular interest are the so-called top local cohomology modules, that
is, $H^{t}_{\mathfrak{a}}(R)\neq 0$ where $H^{i}_{\mathfrak{a}}(R)=0$ for all
$i>t$. In the light of Theorem 1.1, vanishing of the local cohomology modules
$H^{i}_{\mathfrak{a}}(R)$, for $i=\dim R,\dim R-1$ paving the ground for
connectedness results. $H^{\dim R}_{\mathfrak{a}}(R)$ has been studied in
[E-Sch] and some of its properties have been realized. As
$H^{d-1}_{\mathfrak{a}}(R)$ is mostly not Artinian so it rarely could be
possible to control $H^{d-1}_{\mathfrak{a}}(R)$ via
$H^{d-1}_{\mathfrak{m}}(R)$, however in the next Theorem we try to examine the
$H^{d-1}_{\mathfrak{a}}(R)$ via some known modules of the form
$H^{i}_{\mathfrak{m}}(R)$ for a one-dimensional ideal $\mathfrak{a}$ .
###### Theorem 3.1.
Let $\mathfrak{a}$ be a one dimensional ideal of a $d$-dimensional local ring
$(R,\mathfrak{m})$. Suppose $\operatorname{Rad}(\mathfrak{a})$ is not a prime
ideal. Then
* (1)
Let $\widehat{R}$ the $\mathfrak{m}$-adic completion of $R$ be an integral
domain, then there exists an epimorphism
$\begin{array}[]{ll}\ H^{d-1}_{\mathfrak{a}}(R)\rightarrow
H^{d}_{\mathfrak{m}}(R)\rightarrow 0.\end{array}$
Moreover, in case $R$ is complete and unmixed, we have
$\operatorname{Ann}_{R}H^{d-1}_{\mathfrak{a}}(R)=0$.
* (2)
If $R$ is a Gorenstein ring and $\widehat{R}$ is a domain then, there exist a
prime ideal $\mathfrak{p}$ and an element $y\in R\setminus\mathfrak{p}$ to
have the following exact sequence
$\begin{array}[]{ll}\ 0\rightarrow H^{d-1}_{\mathfrak{a}+yR}(R)\rightarrow
H^{d-1}_{\mathfrak{a}}(R)\rightarrow
E_{R_{\mathfrak{p}}}(k(\mathfrak{p}))\rightarrow 0.\end{array}$
* (3)
If $H^{d}_{\mathfrak{a}}(R)=0$ then, for some element $x\in R$ with
$\operatorname{Rad}(\mathfrak{a}+xR)=\mathfrak{m}$ we get the following
epimorphism
$\begin{array}[]{ll}\ H^{d-1}_{\mathfrak{a}}(R)\rightarrow
H^{d-1}_{\mathfrak{m}}(R/xR)\rightarrow 0.\end{array}$
If $\mathfrak{a}=\mathfrak{p}$ is a prime ideal and $R$ is Cohen-Macaulay ring
then, there exists a short exact sequence
$\begin{array}[]{ll}\ 0\rightarrow H^{d-1}_{\mathfrak{p}}(R)\rightarrow
H^{d-1}_{\mathfrak{p}}(R)_{y}\rightarrow H^{d}_{\mathfrak{m}}(R)\rightarrow
0,\end{array}$
for an element $y\in\mathfrak{m}\setminus\mathfrak{p}$. Moreover
$H^{0}_{yR}(H^{d-1}_{\mathfrak{p}}(R))=0$ and
$H^{1}_{yR}(H^{d-1}_{\mathfrak{p}}(R))=H^{d}_{\mathfrak{m}}(R)$. In
particular, $H^{d-1}_{\mathfrak{p}}(R)$ is not Artinian.
Proof.
* (1)
Put
$\begin{array}[]{ll}\
\operatorname{Rad}(\mathfrak{a})=\mathfrak{p}_{1}\cap...\cap\mathfrak{p}_{n},\
\ \ \ n\geq 2\end{array}$
where $\mathfrak{p}_{i}$’s are distinct minimal prime ideals of $\mathfrak{a}$
for $i=1,...,n$. Therefore, $\mathfrak{p}_{i}$’s are not
$\mathfrak{m}$-primary. Choose $\mathfrak{p}_{t}$ for some integer
$t\in\\{1,...,n\\}$ such that
$\operatorname{Rad}(\mathfrak{p}_{t}+\cap^{n}_{j=1,j\neq
t}\mathfrak{p}_{j})=\mathfrak{m}$.
Put $\mathfrak{a}_{1}:=\mathfrak{p}_{t},\mathfrak{a}_{2}:=\cap^{n}_{j=1,j\neq
t}\mathfrak{p}_{j}$, by Mayer-Vietoris sequence we have the following exact
sequence
$\begin{array}[]{ll}\
H^{d-1}_{\mathfrak{a}_{1}\cap\mathfrak{a}_{2}}(R)\rightarrow
H^{d}_{\mathfrak{a}_{1}+\mathfrak{a}_{2}}(R)\rightarrow
H^{d}_{\mathfrak{a}_{1}}(R)\oplus H^{d}_{\mathfrak{a}_{2}}(R).\end{array}$
Hartshorne-Lichtenbaum vanishing Theorem implies that
$H^{d}_{\mathfrak{a}_{1}}(R)=0=H^{d}_{\mathfrak{a}_{2}}(R)$ then, we get the
following epimorphism
$\begin{array}[]{ll}\ H^{d-1}_{\mathfrak{a}}(R)\rightarrow
H^{d}_{\mathfrak{m}}(R)\rightarrow 0.\end{array}$
In the light of the above epimorphism and by the assumptions, it follows that
$\operatorname{Ann}_{R}H^{d-1}_{\mathfrak{a}}(R)=0$. To this end note that
$\operatorname{Ann}_{R}H^{d}_{\mathfrak{m}}(R)=0$ (see for example [E-Sch,
Theorem 4.2(i)]).
* (2)
In this part we follow the lines of an argument given in [Hu-K, Lemma 4.7].
Let
$\begin{array}[]{ll}\
\operatorname{Rad}(\mathfrak{a})=\mathfrak{p}_{1}\cap...\cap\mathfrak{p}_{n},\end{array}$
where $\mathfrak{p}_{i}$’s are distinct minimal prime ideals of $\mathfrak{a}$
for $i=1,...,n$. Choose $y\in\cap^{n}_{i=1,i\neq
j}\mathfrak{p}_{i}\setminus\mathfrak{p}_{j}$ for some integer
$j\in\\{1,...,n\\}$.
Mayer-Vietoris sequence implies the following long exact
$\begin{array}[]{ll}\ (H^{d-2}_{\mathfrak{a}}(R))_{y}\rightarrow
H^{d-1}_{\mathfrak{a}+yR}(R)\rightarrow
H^{d-1}_{\mathfrak{a}}(R)\rightarrow(H^{d-1}_{\mathfrak{a}}(R))_{y}\rightarrow
H^{d}_{\mathfrak{a}+yR}(R).\end{array}$
Note that
$\begin{array}[]{ll}\
\operatorname{Rad}(\mathfrak{a}+yR)=\mathfrak{p}_{1}\cap...\cap\mathfrak{p}_{j}^{\wedge}\cap...\cap\mathfrak{p}_{n},\end{array}$
by $\wedge$ we mean the omitting the $j$th component and
$\operatorname{Rad}(\mathfrak{a}R_{y})={\mathfrak{p}}_{j}R_{y}$ which is
maximal ideal of $R_{y}$. Hence,
$\begin{array}[]{ll}\
(H^{d-2}_{\mathfrak{a}}(R))_{y}=H^{d-2}_{\mathfrak{a}R_{y}}(R_{y})\cong
H^{d-2}_{\mathfrak{p}_{j}R_{y}}(R_{y})=0.\end{array}$
To this end note that $\dim
R_{y}=\operatorname{ht}\mathfrak{p}_{j}R_{y}=\operatorname{ht}\mathfrak{p}_{j}=d-1$
and $R_{y}$ is a Gorenstein ring.
As $\widehat{R}$ is a domain and
$\operatorname{Rad}(\mathfrak{a}+yR)\neq\mathfrak{m}$ then,
$H^{d}_{\mathfrak{a}+yR}(R)=0$ using Hartshorne-Lichtenbaum vanishing Theorem.
Finally we have the following isomorphism
$\begin{array}[]{ll}\
(H^{d-1}_{\mathfrak{a}}(R))_{y}=(H^{d-1}_{\mathfrak{a}}(R))_{y}\otimes_{R_{y}}R_{\mathfrak{p}_{j}}\cong
E_{R_{\mathfrak{p}_{j}}}(k(\mathfrak{p}_{j})).\end{array}$
It is worthy to note that $R_{\mathfrak{p}_{j}}\cong R_{y}$ (cf. [At-Mac,
Exercise 8, pp.44]), $R_{\mathfrak{p}_{j}}$ is a Gorenstein local ring and
$\operatorname{Rad}(\mathfrak{a}R_{\mathfrak{p}_{j}})=\operatorname{Rad}(\mathfrak{a})R_{\mathfrak{p}_{j}}=\mathfrak{p}_{j}R_{\mathfrak{p}_{j}}$.
Hence, we get the desired short exact sequence
$\begin{array}[]{ll}\ 0\rightarrow H^{d-1}_{\mathfrak{a}+yR}(R)\rightarrow
H^{d-1}_{\mathfrak{a}}(R)\rightarrow
E_{R_{\mathfrak{p}_{j}}}(k(\mathfrak{p}_{j}))\rightarrow 0.\end{array}$
* (3)
Put
$\begin{array}[]{ll}\
\operatorname{Rad}(\mathfrak{a})=\mathfrak{p}_{1}\cap...\cap\mathfrak{p}_{n},\end{array}$
Choose $x\in\mathfrak{p}_{t}$ for some integer $t\in\\{1,...,n\\}$ with
$\operatorname{Rad}(\mathfrak{a}+xR)=\mathfrak{m}$ to get the following short
exact sequence
$\begin{array}[]{ll}\ 0\rightarrow R/(0:_{R}x)\stackrel{{\scriptstyle
x}}{{\rightarrow}}R\rightarrow R/xR\rightarrow 0.\end{array}$
It yields the long exact sequence
$\begin{array}[]{ll}\ H^{d-1}_{\mathfrak{a}}(R/(0:_{R}x))\rightarrow
H^{d-1}_{\mathfrak{a}}(R)\rightarrow H^{d-1}_{\mathfrak{a}}(R/xR)\rightarrow
H^{d}_{\mathfrak{a}}(R/(0:_{R}x)),\end{array}$
where $H^{d}_{\mathfrak{a}}(R/(0:_{R}x))$ is zero by assumption. So, by
Independence Theorem we get the epimorphism
$\begin{array}[]{ll}\ H^{d-1}_{\mathfrak{a}}(R)\rightarrow
H^{d-1}_{\mathfrak{m}}(R/xR)\rightarrow 0.\end{array}$
For the last part, Choose $y\in\mathfrak{m}\setminus\mathfrak{p}$. So,
$\operatorname{Rad}(\mathfrak{p}+yR)=\mathfrak{m}$. Then, by Mayer-Vietoris
sequence we have
$\begin{array}[]{ll}\ (H^{d-2}_{\mathfrak{p}}(R))_{y}\rightarrow
H^{d-1}_{\mathfrak{p}+yR}(R)\rightarrow
H^{d-1}_{\mathfrak{p}}(R)\rightarrow(H^{d-1}_{\mathfrak{p}}(R))_{y}.\end{array}$
As $R$ is Cohen-Macaulay then, it yields the next short exact sequence
$\begin{array}[]{ll}\ 0\rightarrow
H^{d-1}_{\mathfrak{p}}(R)\rightarrow(H^{d-1}_{\mathfrak{p}}(R))_{y}\rightarrow
H^{d}_{\mathfrak{m}}(R)\rightarrow 0.\end{array}$
By virtue of [Br-Sh, Corollary 2.2.18] $\
H^{0}_{yR}(H^{d-1}_{\mathfrak{p}}(R))=0$ and
$H^{1}_{yR}(H^{d-1}_{\mathfrak{p}}(R))=H^{d}_{\mathfrak{m}}(R)$. Note that in
case $H^{d-1}_{\mathfrak{p}}(R)$ is Artinian, it follows that
$H^{d}_{\mathfrak{m}}(R)=0$ which is a contradiction. $\ \ \ \ \ \ \ \Box$
###### Remark 3.2.
* (1)
Keep the notations and hypotheses in Theorem 3.1(3), with some additional
assumptions we can remove the assumption $H^{d}_{\mathfrak{a}}(R)=0$. Let $x$
be in some minimal prime ideals of $\mathfrak{p}_{t}$ such that
$\operatorname{Rad}(\mathfrak{a}+x)=\mathfrak{m}$ then, we show that
$H^{d}_{\mathfrak{a}}(R/(0:_{R}x))=0$. Note that if $x\in\mathfrak{p}$ for all
$\mathfrak{p}\in\min R$ then, $\operatorname{Rad}(\mathfrak{a})=\mathfrak{m}$
which is a contradiction.
Let $(0)=\mathfrak{q}_{1}\cap...\cap\mathfrak{q}_{n}$ be a minimal primary
decomposition and $(x)=\mathfrak{q}_{1}\cap...\cap\mathfrak{q}_{r}$, $r<n$
such that $\dim R/\mathfrak{q}_{i}=d$ and
$\operatorname{Rad}(\mathfrak{a}+\mathfrak{q}_{i})=\mathfrak{m}$ for all
$i\in\\{1,...,r\\}$ but this is not true for all $\mathfrak{q}_{i}$ where
$r<i\leq n$.
If $H^{d}_{\mathfrak{a}}(R/(0:_{R}x))\neq 0$ then, there exists a prime ideal
$\mathfrak{P}$ containing $(0:_{R}x)$ such that $\dim R/\mathfrak{P}=d$ and
$\operatorname{Rad}(\mathfrak{a}+\mathfrak{P})=\mathfrak{m}$ by Hartshorne-
Lichtenbaum vanishing Theorem. Since $\mathfrak{P}\supseteq(0:_{R}x)$ so
$\mathfrak{P}\supseteq\mathfrak{q}_{i}$ for some $i\geq r+1$. Therefore $\dim
R/\mathfrak{q}_{i}=d$ and $\mathfrak{P}=\operatorname{Rad}(\mathfrak{q}_{i})$.
This implies that
$\operatorname{Rad}(\mathfrak{a}+\mathfrak{q}_{i})=\operatorname{Rad}(\mathfrak{a}+\mathfrak{P})=\mathfrak{m}$,
contracting the fact that $i\geq r+1$.
* (2)
Let $\mathfrak{a}$ be a one dimensional ideal of a $d$-dimensional local ring
$(R,\mathfrak{m})$. Suppose $\operatorname{Rad}(\mathfrak{a})$ is not a prime
ideal. If for some element $x$ in a minimal prime ideal of $R$,
$\operatorname{Rad}(\mathfrak{a}+xR)=\mathfrak{m}$ then, by what we have seen
in (1) and Theorem 3.1(3) we get
$\begin{array}[]{ll}\
\operatorname{Assh}(R/xR)\subseteq\operatorname{Att}_{R}(H^{d-1}_{\mathfrak{a}}(R)).\end{array}$
$\ \ \ \ \ \ \ \Box$
It follows by Hartshorne’s result, (cf. [Ha, p. 417]) that
${\operatornamewithlimits{\varinjlim}}_{n}\operatorname{Ext}^{s}_{R}(\mathfrak{m}^{n}/\mathfrak{a}^{n},N)=0$
for a finitely generated $R$-module $N$ of dimension $s$. It is natural to
investigate the vanishing case for
${\operatornamewithlimits{\varinjlim}}_{n}\operatorname{Ext}^{s-1}_{R}(\mathfrak{m}^{n}/\mathfrak{a}^{n},N)$.
To be more precise we have the following result:
###### Proposition 3.3.
Let $(R,\mathfrak{m})$ be a local ring and $M$ a finite $R$-module of
dimension $s$ and $\mathfrak{a}$ an ideal of dimension one. Then,
* (1)
There exists an epimorphism
${\operatornamewithlimits{\varinjlim}}_{n}\operatorname{Ext}^{s-1}_{R}(\mathfrak{m}^{n}/\mathfrak{a}^{n},M)\rightarrow
H^{1}_{xR}(H^{s-1}_{\mathfrak{a}}(M))$.
* (2)
If $H^{s-1}_{\mathfrak{a}}(M)=0$ then, it implies that
${\operatornamewithlimits{\varinjlim}}_{n}\operatorname{Ext}^{s-1}_{R}(\mathfrak{m}^{n}/\mathfrak{a}^{n},M)=0$.
Proof. As $\dim R/\mathfrak{a}=1$ then, there exists an element
$x\in\mathfrak{m}\setminus\mathfrak{a}$. By virtue of [Sch, Corollary 1.4]
there is the following exact sequence
$\begin{array}[]{ll}\ 0\rightarrow
H^{1}_{xR}(H^{s-1}_{\mathfrak{a}}(M))\rightarrow
H^{s}_{(\mathfrak{a},xR)}(M)\rightarrow
H^{0}_{xR}(H^{s}_{\mathfrak{a}}(M))\rightarrow 0.\end{array}$
As $H^{s}_{\mathfrak{a}}(M)$ is an Artinian module so every element of
$H^{s}_{\mathfrak{a}}(M)$ is annihilated by some power of $\mathfrak{m}$ and
also by some power of $x$, i.e.
$H^{0}_{xR}(H^{s}_{\mathfrak{a}}(M))=H^{s}_{\mathfrak{a}}(M)$.
On the other hand there is the long exact sequence
$\begin{array}[]{ll}\
{\operatornamewithlimits{\varinjlim}}_{n}\operatorname{Ext}^{s-1}_{R}(\mathfrak{m}^{n}/\mathfrak{a}^{n},M)\rightarrow
H^{s}_{\mathfrak{m}}(M)\rightarrow
H^{s}_{\mathfrak{a}}(M)\rightarrow{\operatornamewithlimits{\varinjlim}}_{n}\operatorname{Ext}^{s}_{R}(\mathfrak{m}^{n}/\mathfrak{a}^{n},M),\end{array}$
where
${\operatornamewithlimits{\varinjlim}}_{n}\operatorname{Ext}^{s}_{R}(\mathfrak{m}^{n}/\mathfrak{a}^{n},M)=0$
as it has been mentioned above.
Now combine the above results to get the following commutative diagram with
exact rows
$\begin{array}[]{cccccccc}H^{s-1}_{\mathfrak{a}}(M)&\rightarrow&{\operatornamewithlimits{\varinjlim}}_{n}\operatorname{Ext}^{s-1}_{R}(\mathfrak{m}^{n}/\mathfrak{a}^{n},M)&\rightarrow
H^{s}_{\mathfrak{m}}(M)&\rightarrow&H^{s}_{\mathfrak{a}}(M)&\rightarrow&0\\\
&&&\ \ \ \ \ \ \ \downarrow=&&\ \ \ \ \downarrow=&\\\
0&\rightarrow&H^{1}_{xR}(H^{s-1}_{\mathfrak{a}}(M))&\rightarrow
H^{s}_{\mathfrak{m}}(M)&\rightarrow&H^{s}_{\mathfrak{a}}(M)&\rightarrow&0\end{array}$
By snake lemma we get the epimorphism
$\phi:{\operatornamewithlimits{\varinjlim}}_{n}\operatorname{Ext}^{s-1}_{R}(\mathfrak{m}^{n}/\mathfrak{a}^{n},M)\rightarrow
H^{1}_{xR}(H^{s-1}_{\mathfrak{a}}(M))$. It is clear that $\phi$ is isomorphism
when $H^{s-1}_{\mathfrak{a}}(M)=0$, this proves the claim.
$\ \ \ \ \ \ \ \Box$
## 4\. A general case
In this Section we try to examine the structure of $H^{d-1}_{\mathfrak{a}}(R)$
for an ideal $\mathfrak{a}$ which is not in general of dimension one. For this
reason we consider the following case:
Let $(R,\mathfrak{m})$ be a local ring. Assume that
$\begin{array}[]{ll}\
\Sigma:=\\{\mathfrak{p}\in\operatorname{Spec}\widehat{R}:\dim\widehat{R}/\mathfrak{p}=d-1\text{
and
}\operatorname{Rad}(\mathfrak{a}\widehat{R}+\mathfrak{p})=\mathfrak{m}\widehat{R}\\}\end{array}$
is non-empty. Here $\widehat{R}$ is the $\mathfrak{m}$-adic completion of $R$.
When we have an ideal $\mathfrak{b}$ of $R$ of dimension $d-1$ such that
$H^{d-1}_{\mathfrak{a}}(R/{\mathfrak{b}})\neq 0$ then, by Hartshorne-
Lichtenbaum vanishing Theorem the above assumption will be verified.
###### Proposition 4.1.
Let $(R,\mathfrak{m})$ be a complete local ring of dimension $d>0$. Let
$\mathfrak{a}$ be an ideal of $R$ such that $H^{d}_{\mathfrak{a}}(R)=0$.
Suppose that $\Sigma:=\\{\mathfrak{p}\in\operatorname{Spec}R:\dim
R/\mathfrak{p}=d-1\text{ and
}\operatorname{Rad}(\mathfrak{a}+\mathfrak{p})=\mathfrak{m}\\}\neq\emptyset$.
Then
$\begin{array}[]{ll}\ H^{d-1}_{\mathfrak{a}}(R)\rightarrow
H^{d-1}_{\mathfrak{m}}(R/J)\rightarrow 0\end{array}$
where $J=(\cap_{\mathfrak{p}_{i}\in\Sigma}\mathfrak{p}_{i})$ is an ideal of
$R$. ($J=R$, if for every $\mathfrak{p}\in\operatorname{Spec}R$ $\dim
R/\mathfrak{p}\neq d-1$ or
$\operatorname{Rad}(\mathfrak{a}+\mathfrak{p})\neq\mathfrak{m}$).
Proof. Consider the short exact sequence
$\begin{array}[]{ll}\ 0\rightarrow J\rightarrow R\rightarrow R/J\rightarrow
0\end{array}$
which implies the following long exact sequence
$\begin{array}[]{ll}\ H^{d-1}_{\mathfrak{a}}(J)\rightarrow
H^{d-1}_{\mathfrak{a}}(R)\rightarrow H^{d-1}_{\mathfrak{a}}(R/J)\rightarrow
H^{d}_{\mathfrak{a}}(J).\end{array}$
As the last module in the above exact sequence is zero so, we get the
following epimorphism
$\begin{array}[]{ll}\ H^{d-1}_{\mathfrak{a}}(R)\rightarrow
H^{d-1}_{\mathfrak{a}}(R/J)\rightarrow 0.\end{array}$
By virtue of Independence Theorem we get the claim:
$\begin{array}[]{ll}\ H^{d-1}_{\mathfrak{a}}(R)&\rightarrow
H^{d-1}_{\mathfrak{a}}(R/J)\\\ &\cong H^{d-1}_{\mathfrak{a}+J/J}(R/J)\\\
&\cong H^{d-1}_{\mathfrak{m}}(R/J).\end{array}$
To this end note that $\operatorname{Rad}(\mathfrak{a}+J)=\mathfrak{m}$ $\ \ \
\ \ \ \ \Box$
Let $M$ denote a $s$-dimensional, finitely generated $R$-module. Here
$(R,\mathfrak{m})$ is a local ring. The functor $\widehat{\cdot}$ denotes the
completion functor.
For an $R$-module $M$ let $0=\cap_{i=1}^{n}Q_{i}(M)$ denote a minimal primary
decomposition of the zero submodule of $M.$ That is $M/Q_{i}(M),i=1,\ldots,n,$
is a $\mathfrak{p}_{i}$-primary $R$-module. Clearly
$\operatorname{Ass}_{R}M=\\{\mathfrak{p}_{1},\ldots,\mathfrak{p}_{n}\\}.$ It
is defined in [E-Sch] that $Q_{\mathfrak{a}}(M)=\cap_{\mathfrak{p}_{i}\in
U}Q_{i}(M)$, where $U=\\{\mathfrak{p}\in\operatorname{Ass}_{R}M|\dim
R/\mathfrak{p}=s\text{ and }\dim R/\mathfrak{a}+\mathfrak{p}=0\\}$ and
$P_{\mathfrak{a}}(M)$ as the intersection of all the primary components of
$\operatorname{Ann}_{R}M$ such that $\dim R/\mathfrak{p}=\dim M$ and $\dim
R/\mathfrak{a}+\mathfrak{p}=0$. Clearly $P_{\mathfrak{a}}(M)$ is the pre-image
of $Q_{\mathfrak{a}R/\operatorname{Ann}_{R}M}(R/\operatorname{Ann}_{R}M)$ in
$R$. It has been shown in [E-Sch, Corollary 3.5] that when $(R,\mathfrak{m})$
is a local ring there is a natural isomorphism
$H^{s}_{\mathfrak{a}}(M)\simeq
H^{s}_{\mathfrak{m}\widehat{R}}(\widehat{M})/P_{\mathfrak{a}}(\widehat{M})H^{s}_{\mathfrak{m}\widehat{R}}(\widehat{M}),$
where $P_{\mathfrak{a}}(\widehat{M})\subset\widehat{R}$ is the ideal.
###### Proposition 4.2.
Let $(R,\mathfrak{m})$ be a local ring of dimension $d>0$. Let $\mathfrak{a}$
be an ideal of $R$ such that $H^{d}_{\mathfrak{a}}(R)=0$. Let
$J=\cap_{\mathfrak{p}_{i}\in\Sigma}\mathfrak{p}_{i}$ be as above such that
$\Sigma$ is finite. Assume that $H^{d-1}_{\mathfrak{a}}(R)$ is Artinian. Then,
$\begin{array}[]{ll}\
H^{d-1}_{\mathfrak{a}}(R)/JH^{d-1}_{\mathfrak{a}}(R)\cong
H^{d-1}_{\mathfrak{m}\widehat{R}}(S)/KH^{d-1}_{\mathfrak{m}\widehat{R}}(S),\end{array}$
where $K:=P_{\mathfrak{a}}(\widehat{R}/J\widehat{R})\subset
S:=\widehat{R}/J\widehat{R}$ is the ideal.
Proof. First note that as $H^{d-1}_{\mathfrak{a}}(R)$ is an Artinian
$R$-module so, it admits a unique $\widehat{R}$-module structure. Then,
without loss of generality we may assume that $R$ is complete. It is clear
that $\dim R/J=d-1$ so, by assumption and Grothendieck’s non-vanishing Theorem
one can see that $H^{d-1}_{\mathfrak{a}}(R/J)\neq 0$. Hence, by the above
explanations there exists a natural isomorphism
$\begin{array}[]{ll}\ H^{d-1}_{\mathfrak{a}}(R/J)\cong
H^{d-1}_{\mathfrak{m}}(R/(P_{\mathfrak{a}}(R/J),J)).\end{array}$
Now the claim is clear as $H^{d}_{\mathfrak{a}}(R)=0$. $\ \ \ \ \ \ \ \Box$
###### Remark 4.3.
Assume that $0=\cap^{n}_{i=1}q_{i}$ is a minimal primary decomposition of the
zero ideal of a ring $R$ with dimension $d\geq 3$, where at least one of
$q_{i}$ for $1\leq i\leq n$ is of dimension one. Let $\mathfrak{a}$ be an
ideal of $R$ such that $\operatorname{Spec}R\setminus V(\mathfrak{a})$ is
disconnected. Then, there are ideals $\mathfrak{b}$ and $\mathfrak{c}$ such
that $\operatorname{Rad}(\mathfrak{b})\neq
0\neq\operatorname{Rad}(\mathfrak{c})$,
$\operatorname{Rad}(\mathfrak{b}\cap\mathfrak{c})=\operatorname{Rad}(0)$ and
$\operatorname{Rad}(\mathfrak{b}+\mathfrak{c})=\operatorname{Rad}(\mathfrak{a})$.
If we choose one of them to be one-dimensional component of $0$, say
$\mathfrak{b}$ then, Mayer-Vietoris sequence implies the following long exact
sequence
$\begin{array}[]{ll}\ H^{d-2}_{\mathfrak{b}\cap\mathfrak{c}}(R)\rightarrow
H^{d-1}_{\mathfrak{a}}(R)\rightarrow H^{d-1}_{\mathfrak{b}}(R)\oplus
H^{d-1}_{\mathfrak{c}}(R)\rightarrow
H^{d-1}_{\mathfrak{b}\cap\mathfrak{c}}(R),\end{array}$
where
$H^{d-2}_{\mathfrak{b}\cap\mathfrak{c}}(R)=0=H^{d-1}_{\mathfrak{b}\cap\mathfrak{c}}(R)$.
Hence, we get the following isomorphism
$\begin{array}[]{ll}\ H^{d-1}_{\mathfrak{a}}(R)\cong
H^{d-1}_{\mathfrak{b}}(R)\oplus H^{d-1}_{\mathfrak{c}}(R).\end{array}$
Moreover, in case $R$ is a Cohen-Macaulay ring $H^{d-1}_{\mathfrak{a}}(R)$ is
not finitely generated because, $H^{d-1}_{\mathfrak{b}}(R)$ is not finitely
generated.
###### Problem 4.4.
Let $(R,\mathfrak{m})$ be a local ring and $\mathfrak{a}$ be an ideal of $R$.
By virtue of [E, Theorem 1.1] in the case $0ptR\geq 2$, $\dim R=d$, we have
the following equivalent statements of formal local cohomology modules:
$\begin{array}[]{ll}\
{\operatornamewithlimits{\varprojlim}}_{n}H^{0}_{\mathfrak{m}}(R/\mathfrak{a}^{n})=0={\operatornamewithlimits{\varprojlim}}_{n}H^{1}_{\mathfrak{m}}(R/\mathfrak{a}^{n})\end{array}$
if and only if
$\begin{array}[]{ll}\
{\operatornamewithlimits{\varprojlim}}_{n}H^{0}_{\mathfrak{m}}(R/\mathfrak{a}^{n})\text{\
and \
}{\operatornamewithlimits{\varprojlim}}_{n}H^{1}_{\mathfrak{m}}(R/\mathfrak{a}^{n})\end{array}$
are Artinian.
When $R$ is a Gorenstein ring the above statements are equivalent to
$H^{d}_{\mathfrak{a}}(R)=0=H^{d-1}_{\mathfrak{a}}(R)$ for $d\geq 2$. Now it
implies to find more precise conditions for Artinianness of
${\operatornamewithlimits{\varprojlim}}_{n}H^{0}_{\mathfrak{m}}(R/\mathfrak{a}^{n})$
and
${\operatornamewithlimits{\varprojlim}}_{n}H^{1}_{\mathfrak{m}}(R/\mathfrak{a}^{n})$.
It should be noted that if
$\operatorname{Coass}({\operatornamewithlimits{\varprojlim}}_{n}H^{i}_{\mathfrak{m}}(R/\mathfrak{a}^{n}))\not\subset
V(\mathfrak{a})$ then,
${\operatornamewithlimits{\varprojlim}}_{n}H^{i}_{\mathfrak{m}}(R/\mathfrak{a}^{n})$
is not Artinian (cf. [E, Theorem 1.4 and Corollary 3.5]).
For more information on formal local cohomology modules see [Sch2].
## References
* [A-M] M. Aghapournahr and L. Melkersson, Cofiniteness and coassociated primes of local cohomology modules, Math. Scand., 105(7) (2009), 161-170.
* [At-Mac] M. F. Atiyah and I. G. Macdonald, Introduction to comutative algebra, Addison-Wesley 1969\.
* [Br-Sh] M. Brodmann and R.Y. Sharp, Local cohomology: an algebraic introduction with geometric applications, Cambridge Univ. Press, 60, Cambridge, (1998).
* [E] M. Eghbali, On Artinianness of formal local cohomology, colocalization and coassociated primes , Math. Scand. , to appear. arXiv:1011.4141 [math.AC].
* [E-Sch] M. Eghbali and P. Schenzel, On an endomorphism ring of local cohomology, Comm. Algebra, 40: 4295-4305, 2012.
* [Gr] A. Grothendieck, Local cohomology, Notes by R. Hartshorne, Lect. Notes in Math., 20, Springer, 1966.
* [Ha] R. Hartshorne, Cohomological dimension of algebraic varieties, Annals of Math. 88 (1968), 403-450.
* [Hel] M. Hellus, Local Cohomology and Matlis Duality, Habilitationsschrift, Leipzig, (2006).
* [Hu-K] C. Huneke and J. Koh, Cofiniteness and vanishing of local cohomology modules, Math. Proc. Camb. Phil. Soc. (1991), 110, 421-429.
* [Hu-L] C. Huneke and G. Lyubeznik, On the vanishing of local cohomology modules, Invent. Math. 102, 73-93(1990).
* [O] A. Ogus, Local cohomological dimension of algebraic varieties, Annals Math. 98 (1973) 327-365.
* [P-S] C. Peskine and L. Szpiro, Dimension projective finie et cohomologie locale, Publ. Math. I.H.E.S., 42, (1973), 323-395.
* [R] A. S. Richardson, Co-localization, co-support and local cohomology, Rocky Mountain J. of Math., 36, 5, (2006), 1679-1703.
* [Sch] P. Schenzel, On the use of local cohomology in algebra and geometry, in: J. Elias, J.M. Giral, R.M. Mir -Roig, S. Zarzuela (Eds.), Six Lectures in Commutative Algebra, Proceed. Summer School on Commutative Algebra at Centre de Recerca Matem tica, in: Progr. Math., vol. 166, Birkh user, (1998), pp. 241-292.
* [Sch2] P. Schenzel, On formal local cohomology and connectedness, J. Algebra, 315(2), (2007), 894-923.
|
arxiv-papers
| 2012-12-02T21:04:26 |
2024-09-04T02:49:38.814577
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Majid Eghbali",
"submitter": "Majid Eghbali",
"url": "https://arxiv.org/abs/1212.0245"
}
|
1212.0287
|
# Exploring Relay Cooperation for Secure and Reliable Transmission in Two-Hop
Wireless Networks
Yulong Shen14, Xiaohong Jiang2, Jianfeng Ma1 and Weisong Shi3
1School of Computer Science and Technology, Xidian University, China 2School
of Systems Information Science, Future University Hakodate, Japan 3Department
of Computer Science, Wayne State University, USA
4Email:[email protected]
###### Abstract
This work considers the problem of secure and reliable information
transmission via relay cooperation in two-hop relay wireless networks without
the information of both eavesdropper channels and locations. While previous
work on this problem mainly studied infinite networks and their asymptotic
behavior and scaling law results, this papers focuses on a more practical
network with finite number of system nodes and explores the corresponding
exact result on the number of eavesdroppers one network can tolerant to ensure
desired secrecy and reliability. We first study the scenario where path-loss
is equal between all pairs of nodes and consider two transmission protocols
there, one adopts an optimal but complex relay selection process with less
load balance capacity while the other adopts a random but simple relay
selection process with good load balance capacity. Theoretical analysis is
then provided to determine the maximum number of eavesdroppers one network can
tolerate to ensure a desired performance in terms of the secrecy outage
probability and transmission outage probability. We further extend our study
to the more general scenario where path-loss between each pair of nodes also
depends the distance between them, for which a new transmission protocol with
both preferable relay selection and good load balance as well as the
corresponding theoretical analysis are presented.
###### Index Terms:
Two-Hop Wireless Networks, Cooperative Relay, Physical Layer Security,
Transmission Outage, Secrecy Outage.
## I Introduction
Two-hop ad hoc wireless networks, where each packet travels at most two hops
(source-relay-destination) to reach its destination, have been a class of
basic and important networking scenarios [1]. Actually, the analysis of basic
two-hop relay networks serves as the foundation for performance study of
general multi-hop networks. Due to the promising applications of ad hoc
wireless networks in many important scenarios (like battlefield networks,
vehicle networks, disaster recovery networks), the consideration of secrecy
(and also reliability) in such networks is of great importance for ensuring
the high confidentiality requirements of these applications.
Traditionally, the information security is provided by adopting the
cryptography approach, where a plain message is encrypted through a
cryptographic algorithm that is hard to break (decrypt) in practice by any
adversary without the key. While the cryptography is acceptable for general
applications with standard security requirement, it may not be sufficient for
applications with a requirement of strong form of security (like military
networks and emergency networks). This is because that the cryptographic
approach can hardly achieve everlasting secrecy, since the adversary can
record the transmitted messages and try any way to break them [2]. That is why
there is an increasing interest in applying signaling scheme in physical layer
to provide a strong form of security, where a degraded signal at an
eavesdropper is always ensured such that the original data can be hardly
recovered regardless of how the signal is processed at the eavesdropper. We
consider applying physical layer method to achieve secure and reliable
information transmission in the two-hop wireless networks.
By now, a lot of research works have been dedicated to the study of physical
layer security based on cooperative relays and artificial noise, and these
works can be roughly classified into two categories depending on whether the
information of eavesdroppers channels and locations is known or not (see
Section V for related works). For the case that the information of
eavesdroppers channels and locations is available, a lot of transmission
schemes have been proposed to achieve the maximum secrecy rates while
optimizing the artificial noise generation and power control to reduce the
total transmission power consumption [3-19]. In practice, however, it is
difficult to gain the information of eavesdropper channels and locations,
since the eavesdroppers always try to hide their identity information as much
as possible. To alleviate such a requirement on eavesdroppers information,
some recent works explored the implementation of secure and reliable
information transmission in wireless networks without the information of both
eavesdropper channels and locations [20-28]. It is notable, however, that
these works mainly focus on exploring the scaling law results in terms of the
number of eavesdroppers one network can tolerate as the number of system nodes
there tends to infinity. Although the scaling law results are helpful for us
to understand the general asymptotic network behavior, they tell us a little
about the actual and exact number of eavesdroppers one network can tolerate.
In practice, however, such exact results are of great interest for network
designers.
This paper focuses on applying the relay cooperation to achieve secure and
reliable information transmission in a more practical finite two-hop wireless
network without the knowledge of both eavesdropper channels and locations. The
main contributions of this paper as follows.
* •
For achieving secure and reliable information transmission in a more practical
two-hop wireless network with finite number of system nodes and equal path-
loss between all pairs of nodes, we consider the application of the
cooperative protocol proposed in [21] with an optimal and complex relay
selection process but less load balance capacity, and also propose to use a
new cooperative protocol with a simple and random relay selection process but
good load balance capacity.
* •
Rather than exploring the asymptotic behavior and scaling law results, we
provide theoretic analysis for above two cooperative protocols to determine
the corresponding exact results on the number of eavesdroppers one network can
tolerate to meet a specified requirement in terms of the maximum secrecy
outage probability and the maximum transmission outage probability allowed.
* •
We further extend our study to the more general and practical scenario where
the path-loss between each pair of nodes also depends on their relative
locations, for which we propose a new transmission protocol with both
preferable relay selection and good load balance and also present the
corresponding theoretical analysis under this new protocol.
The remainder of the paper is organized as follows. Section II presents system
models and also introduces transmission outage and secrecy outage for the
analysis of transmission protocols. Section III considers two transmission
protocols for the scenario of equal path-loss between all pairs of nodes and
provides the corresponding theoretical analysis. Section IV further presents a
new transmission protocol and its theoretical analysis to address distance-
dependent path-loss issue. Section V introduces the related works and Section
VI concludes this paper.
## II System Models
### II-A Network Model
As illustrated in Fig.1 that we consider a network scenario where a source
node $S$ wishes to communicate securely with its destination node $D$ with the
help of multiple relay nodes $R_{1}$, $R_{2}$, $\cdots$, $R_{n}$. In addition
to these normal system nodes, there are also $m$ eavesdroppers $E_{1}$,
$E_{2}$, $\cdots$, $E_{m}$ that are independent and also uniformly distributed
in the network. Our goal here is to ensure the secure and reliable information
transmission from source $S$ to destination $D$ under the condition that no
real time information is available about both eavesdropper channels and
locations.
Figure 1: System scenario: Source $S$ wishes to communicate securely with
destination $D$ with the assistance of finite relays $R_{1}$, $R_{2}$,
$\cdots$, $R_{n}$ ($n$=4 in the figure) in the presence of passive
eavesdroppers $E_{1}$, $E_{2}$, $\cdots$, $E_{m}$ ($m$=4 in the figure).
Cooperative relay scheme is used in the two-hop transmission. A assistant node
is selected randomly as relay ($R_{1}$ in the figure).
### II-B Transmission Model
Consider the transmission from a transmitter $A$ to a receiver $B$, and denote
by $x_{i}^{\left(A\right)}$ the $i^{th}$ symbol transmitted by $A$ and denote
by $y_{i}^{\left(B\right)}$ the $i^{th}$ signal received by $B$. We assume
that all nodes transmit with the same power $E_{s}$, path-loss between all
pairs of nodes is independent, and the frequency-nonselective multi-path
fading from $A$ to $B$ is a complex zero-mean Gaussian random variable. Under
the condition that all nodes in a group of nodes, $\mathcal{R}$, are
generating noises, the $i^{th}$ signal received at node $B$ from node $A$ is
determined as:
$y_{i}^{\left(B\right)}=\frac{h_{A,B}}{d_{A,B}^{\alpha/2}}\sqrt{E_{s}}x_{i}^{\left(A\right)}+\sum_{A_{i}\in\mathcal{R}}\frac{h_{A_{i},B}}{d_{A_{i},B}^{\alpha/2}}\sqrt{E_{s}}x_{i}^{\left(A_{i}\right)}+n_{i}^{\left(B\right)}$
where $\alpha\geq 2$ is the path-loss exponent. The noise
$\left\\{n_{i}^{\left(B\right)}\right\\}$ at receiver $B$ is assumed to be
i.i.d complex Gaussian random variables with
$E{\left[\left|n_{i}^{\left(B\right)}\right|^{2}\right]}=N_{0}$, and
$\left|h_{A,B}\right|^{2}$ is exponentially distributed with mean
$E{\left[\left|h_{A,B}\right|^{2}\right]}$. Without loss of generality, we
assume that $E{\left[\left|h_{A,B}\right|^{2}\right]}=1$. The SINR $C_{A,B}$
from $A$ to $B$ is then given by
$C_{A,B}=\frac{E_{s}\left|h_{A,B}\right|^{2}d_{A,B}^{-\alpha}}{\sum_{A_{i}\in\mathcal{R}}E_{s}{\left|h_{A_{i},B}\right|^{2}d_{A_{i},B}^{-\alpha}}+N_{0}/2}$
For a legitimate node and an eavesdropper, we use two separate SINR thresholds
$\gamma_{R}$ and $\gamma_{E}$ to define the minimum SINR required to recover
the transmitted messages for legitimate node and eavesdropper, respectively.
Therefore, a system node (relay or destination) is able to decode a packet if
and only if its SINR is greater than $\gamma_{R}$, while the transmitted
message is secure if and only if the SINR at each eavesdropper is less than
$\gamma_{E}$.
### II-C Transmission Outage and Secrecy Outage
For a transmission from the source $S$ to destination $D$, we call
transmission outage happens if $D$ can not decode the transmitted packet,
i.e., $D$ received the packet with SINR less than the predefined threshold
$\gamma_{R}$. The transmission outage probability, denoted as
$P_{out}^{\left(T\right)}$, is then defined as the probability that
transmission outage from $S$ to $D$ happens. For a predefined upper bound
$\varepsilon_{t}$ on $P_{out}^{\left(T\right)}$, we call the communication
between $S$ and $D$ is reliable if
$P_{out}^{\left(T\right)}\leq\varepsilon_{t}$. Notice that for the
transmissions from $S$ to the selected relay $R_{j^{\ast}}$ and from
$R_{j^{\ast}}$ to $D$, the corresponding transmission outage can be defined in
the similar way as that of from $S$ to $D$. We use $O_{S\rightarrow
R_{j^{\ast}}}^{(T)}$ and $O_{R_{j^{\ast}}\rightarrow D}^{(T)}$ to denote the
events that transmission outage from source $S$ to $R_{j^{\ast}}$ happens and
transmission outage from relay $R_{j^{\ast}}$ to $D$ happens, respectively.
Due to the link independence assumption, we have
$\displaystyle P_{out}^{\left(T\right)}=P\left(O_{S\rightarrow
R_{j^{\ast}}}^{(T)}\cup O_{R_{j^{\ast}}\rightarrow D}^{(T)}\right)$
$\displaystyle\ \ \ \ \ \ =P\left(O_{S\rightarrow
R_{j^{\ast}}}^{(T)}\right)+P\left(O_{R_{j^{\ast}}\rightarrow D}^{(T)}\right)$
$\displaystyle\ \ \ \ \ \ \ \ \ -P\left(O_{S\rightarrow
R_{j^{\ast}}}^{(T)}\right)\cdot P\left(O_{R_{j^{\ast}}\rightarrow
D}^{(T)}\right)$
Regarding the secrecy outage, we call secrecy outage happens for a
transmission from $S$ to $D$ if at least one eavesdropper can recover the
transmitted packets during the process of this two-hop transmission, i.e., at
least one eavesdropper received the packet with SINR larger than the
predefined threshold $\gamma_{E}$. The secrecy outage probability, denoted as
$P_{out}^{\left(S\right)}$, is then defined as the probability that secrecy
outage happens during the transmission from $S$ to $D$. For a predefined upper
bound $\varepsilon_{s}$ on $P_{out}^{\left(S\right)}$, we call the
communication between $S$ and $D$ is secure if
$P_{out}^{\left(S\right)}\leq\varepsilon_{s}$. Notice that for the
transmissions from $S$ to the selected relay $R_{j^{\ast}}$ and from
$R_{j^{\ast}}$ to $D$, the corresponding secrecy outage can be defined in the
similar way as that of from $S$ to $D$. We use $O_{S\rightarrow
R_{j^{\ast}}}^{(S)}$ and $O_{R_{j^{\ast}}\rightarrow D}^{(S)}$ to denote the
events that secrecy outage from source $S$ to $R_{j^{\ast}}$ happens and
secrecy outage from relay $R_{j^{\ast}}$ to $D$ happens, respectively. Again,
due to the link independence assumption, we have
$\displaystyle P_{out}^{\left(S\right)}=P\left(O_{S\rightarrow
R_{j^{\ast}}}^{(S)}\right)+P\left(O_{R_{j^{\ast}}\rightarrow D}^{(S)}\right)$
$\displaystyle\ \ \ \ \ \ \ \ \ -P\left(O_{S\rightarrow
R_{j^{\ast}}}^{(S)}\right)\cdot P\left(O_{R_{j^{\ast}}\rightarrow
D}^{(S)}\right)$
## III Secure and Reliable Transmission under Equal Path-Loss
In this section, we consider the case where the path-loss is equal between all
pairs of nodes in the system (i.e., we set $d_{A,B}=1$ for all $A\neq B$). We
first introduce two transmission protocols considered for such scenario, and
then provide theoretical analysis to determine the numbers of eavesdroppers
one network can tolerate under these protocols.
### III-A Transmission Protocols
The first protocol we consider (hereafter called Protocol 1) is the one
proposed in [21], in which the optimal relay node with the best link condition
to both source and destination is always selected for information relaying.
Notice that although the Protocol 1 can guarantee the optimal relay node
selection, it suffers from several problems. Protocol 1 involves a complicated
process of optimal relay selection, which is not very suitable for the
distributed wireless networks, in particular when the number of possible relay
nodes is huge. More importantly, since the channel state is relatively
constant during a fixed time period, some relay nodes with good link
conditions are always preferred for information relaying, resulting in a
severe load balance problem and a quick node energy depletion in energy-
limited wireless environment.
Based on above observations, we propose to use a simple and random relay
selection rather than the optimal relay selection in Protocol 1 to achieve a
better load and energy consumption balance among possible relay nodes. By
modifying the Protocol 1, the new transmission protocol (hereafter called
Protocol 2) works as follows.
1) _Relay selection_ : A relay node, indexed by $j^{\ast}$, is randomly
selected from all candidate relay nodes $R_{j},j=1,2,\cdots,n$.
2) _Channel measurement_ : The selected relay $R_{j^{\ast}}$ broadcasts a
pilot signal to allow each of other relays to measure the channel from
$R_{j^{\ast}}$ to itself. Each of the other relays $R_{j},j=1,2,\cdots,n,j\neq
j^{\ast}$ then knows the corresponding value of $h_{R_{j},R_{j^{\ast}}}$.
Similarly, the destination $D$ broadcasts a pilot signal to allow each of
other relays to measure the channel from $D$ to itself. Each of the other
relays $R_{j},j=1,2,\cdots,n,j\neq j^{\ast}$ then knows the corresponding
value of $h_{R_{j},D}$.
3) _Message transmission_ : The source $S$ transmits the messages to
$R_{j^{\ast}}$, and concurrently, the relay nodes with indexes in
$\mathcal{R}_{1}={\left\\{j\neq
j^{\ast}:|h_{R_{j},R_{j^{\ast}}}|^{2}<\tau\right\\}}$ transmit noise to
generate interference at eavesdroppers. The relay $R_{j^{\ast}}$ then
transmits the messages to destination $D$, and concurrently, the relay nodes
with indexes in $\mathcal{R}_{2}={\left\\{j\neq
j^{\ast}:|h_{R_{j},D}|^{2}<\tau\right\\}}$ transmit noise to generate
interference at eavesdroppers.
_Remark 1_ : The parameter $\tau$ involved in the Protocol 1 and Protocol 2
serves as the threshold on path-loss, based on which the set of noise
generating relay nodes can be identified. Notice that a too large $\tau$ may
disable legitimate transmission, while a too small $\tau$ may not be
sufficient for interrupting all eavesdroppers. Thus, the parameter $\tau$
should be set properly to ensure both secrecy requirement and reliability
requirement.
_Remark 2_ : The two protocols considered here have their own advantages and
disadvantages and thus are suitable for different network scenarios. For the
protocol 1, it can achieve a better performance in terms of the number of
eavesdroppers can be tolerated (see Theorem 1). However, it involves a complex
relay selection process, and more importantly, it results in an unbalanced
load and energy consumption distribution among systems nodes. Thus, such
protocol is suitable for small scale wireless network with sufficient energy
supply rather than large and energy-limited wireless networks (like wireless
sensor networks). Regarding the Protocol 2, although it can tolerate less
number eavesdroppers in comparison with the Protocol 1 (see Theorem 2), it
involves a very simple random relay selection process to achieve a good load
and energy consumption distribution among system nodes. Thus, this protocol is
more suitable for large scale wireless network environment with stringent
energy consumption constraint.
### III-B Analysis of Protocol 1
We now analyze that under the Protocol 1 the number of eavesdroppers one
network can tolerate subject to specified requirements on transmission outage
and secrecy outage. We first establish the following two lemmas regarding some
basic properties of $P_{out}^{\left(T\right)}$, $P_{out}^{\left(S\right)}$ and
$\tau$, which will help us to derive the main result in Theorem 1.
_Lemma 1_ : Consider the network scenario of Fig 1 with equal path-loss
between all pairs of nodes, under the Protocol 1 the transmission outage
probability $P_{out}^{\left(T\right)}$ and secrecy outage probability
$P_{out}^{\left(S\right)}$ there satisfy the following conditions.
$\displaystyle P_{out}^{\left(T\right)}\leq
2\left[1-e^{-2\gamma_{R}\left(n-1\right)\left(1-e^{-\tau}\right)\tau}\right]^{n}$
$\displaystyle\ \ \ \ \ \ \ \ \
-\left[1-e^{-2\gamma_{R}\left(n-1\right)\left(1-e^{-\tau}\right)\tau}\right]^{2n}$
$\displaystyle P_{out}^{\left(S\right)}\leq
2m\cdot\left(\frac{1}{1+\gamma_{E}}\right)^{\left(n-1\right)\left(1-e^{-\tau}\right)}$
$\displaystyle\ \ \ \ \ \ \ \ \ \
-\left[m\cdot\left(\frac{1}{1+\gamma_{E}}\right)^{\left(n-1\right)\left(1-e^{-\tau}\right)}\right]^{2}$
The proof of Lemma 1 can be found in the Appendix A.
_Lemma 2_ : Consider the network scenario of Fig 1 with equal path-loss
between all pairs of nodes, to ensure
$P_{out}^{\left(T\right)}\leq\varepsilon_{t}$ and
$P_{out}^{\left(S\right)}\leq\varepsilon_{s}$ under the Protocol 1, the
parameter $\tau$ must satisfy the following condition.
$\displaystyle\tau\leq\sqrt{\frac{-\log\left[1-\left(1-\sqrt{1-\varepsilon_{t}}\right)^{\frac{1}{n}}\right]}{2\gamma_{R}\left(n-1\right)}}$
and
$\displaystyle\tau\geq-\log{\left[1+\frac{\log{\left(\frac{1-\sqrt{1-\varepsilon_{s}}}{m}\right)}}{\left(n-1\right)\log{\left(1+\gamma_{E}\right)}}\right]}$
###### Proof.
$\bullet$ Reliability Guarantee
To ensure the reliability requirement
$P_{out}^{\left(T\right)}\leq\varepsilon_{t}$, we know from the Lemma 1 that
we just need
$\displaystyle
2\left[1-e^{-2\gamma_{R}\left(n-1\right)\left(1-e^{-\tau}\right)\tau}\right]^{n}$
$\displaystyle-\left[1-e^{-2\gamma_{R}\left(n-1\right)\left(1-e^{-\tau}\right)\tau}\right]^{2n}$
$\displaystyle\leq\varepsilon_{t}$
Thus,
$\left[1-e^{-2\gamma_{R}\left(n-1\right)\left(1-e^{-\tau}\right)\tau}\right]^{n}\leq
1-\sqrt{1-\varepsilon_{t}}$
That is,
$-2\gamma_{R}\left(n-1\right)\left(1-e^{-\tau}\right)\tau\geq\log\left[1-\left(1-\sqrt{1-\varepsilon_{t}}\right)^{\frac{1}{n}}\right]$
By using Taylor formula, we have
$\displaystyle\tau\leq\sqrt{\frac{-\log\left[1-\left(1-\sqrt{1-\varepsilon_{t}}\right)^{\frac{1}{n}}\right]}{2\gamma_{R}\left(n-1\right)}}$
The above result indicates the maximum value the parameter $\tau$ we can take
to ensure the reliability requirement.
$\bullet$ Secrecy Guarantee
To ensure the secrecy requirement
$P_{out}^{\left(S\right)}\leq\varepsilon_{s}$ , we know from the Lemma 1 that
we just need
$\displaystyle
2m\cdot\left(\frac{1}{1+\gamma_{E}}\right)^{\left(n-1\right)\left(1-e^{-\tau}\right)}$
$\displaystyle-\left[m\cdot\left(\frac{1}{1+\gamma_{E}}\right)^{\left(n-1\right)\left(1-e^{-\tau}\right)}\right]^{2}$
$\displaystyle\leq\varepsilon_{s}$
Thus,
$\displaystyle
m\cdot\left(\frac{1}{1+\gamma_{E}}\right)^{\left(n-1\right)\left(1-e^{-\tau}\right)}\leq
1-\sqrt{1-\varepsilon_{s}}$
That is,
$\displaystyle\tau\geq-\log{\left[1+\frac{\log{\left(\frac{1-\sqrt{1-\varepsilon_{s}}}{m}\right)}}{\left(n-1\right)\log{\left(1+\gamma_{E}\right)}}\right]}$
The above result implies the minimum value parameter $\tau$ we can take to
guarantee the secrecy requirement.
∎
Based on the results of Lemma 2, we now can establish the following theorem
regarding the performance of Protocol 1.
Theorem 1. Consider the network scenario of Fig 1 with equal path-loss between
all pairs of nodes. To guarantee $P_{out}^{\left(T\right)}\leq\varepsilon_{t}$
and $P_{out}^{\left(S\right)}\leq\varepsilon_{s}$ under the Protocol 1, the
number of eavesdroppers $m$ one network can tolerate must satisfy the
following condition.
$\displaystyle
m\leq\left(1-\sqrt{1-\varepsilon_{s}}\right)\cdot\left(1+\gamma_{E}\right)^{\sqrt{\frac{-\left(n-1\right)\log\left[1-\left(1-\sqrt{1-\varepsilon_{t}}\right)^{\frac{1}{n}}\right]}{2\gamma_{R}}}}$
###### Proof.
From Lemma 2, we know that to ensure the reliability requirement, we have
$\displaystyle\tau\leq\sqrt{\frac{-\log\left[1-\left(1-\sqrt{1-\varepsilon_{t}}\right)^{\frac{1}{n}}\right]}{2\gamma_{R}\left(n-1\right)}}$
and
$\displaystyle\left(n-1\right)\left(1-e^{-\tau}\right)\leq\frac{-\log\left[1-\left(1-\sqrt{1-\varepsilon_{t}}\right)^{\frac{1}{n}}\right]}{2\gamma_{R}\tau}$
To ensure the secrecy requirement, we need
$\displaystyle\left(\frac{1}{1+\gamma_{E}}\right)^{\left(n-1\right)\left(1-e^{-\tau}\right)}\leq\frac{1-\sqrt{1-\varepsilon_{s}}}{m}$
Thus,
$\displaystyle
m\leq\frac{1-\sqrt{1-\varepsilon_{s}}}{\left(\frac{1}{1+\gamma_{E}}\right)^{\left(n-1\right)\left(1-e^{-\tau}\right)}}$
$\displaystyle\ \ \ \
\leq\frac{1-\sqrt{1-\varepsilon_{s}}}{\left(\frac{1}{1+\gamma_{E}}\right)^{\frac{-\log\left[1-\left(1-\sqrt{1-\varepsilon_{t}}\right)^{\frac{1}{n}}\right]}{2\gamma_{R}\tau}}}$
By letting $\tau$ to take its maximum value for maximum interference at
eavesdroppers, we get the following bound
$\displaystyle
m\leq\left(1-\sqrt{1-\varepsilon_{s}}\right)\cdot\left(1+\gamma_{E}\right)^{\sqrt{\frac{-\left(n-1\right)\log\left[1-\left(1-\sqrt{1-\varepsilon_{t}}\right)^{\frac{1}{n}}\right]}{2\gamma_{R}}}}$
∎
### III-C Analysis of Protocol 2
Similar to the analysis of Protocol 1, we first establish the following two
lemmas regarding some basic properties of $P_{out}^{\left(T\right)}$,
$P_{out}^{\left(S\right)}$ and $\tau$ under the Protocol 2.
_Lemma 3_ : Consider the network scenario of Fig 1 with equal path-loss
between all pairs of nodes, the transmission outage probability
$P_{out}^{\left(T\right)}$ and secrecy outage probability
$P_{out}^{\left(S\right)}$ under the Protocol 2 satisfy the following
conditions.
$\displaystyle P_{out}^{\left(T\right)}\leq
2\left[1-e^{-\gamma_{R}\left(n-1\right)\left(1-e^{-\tau}\right)\tau}\right]$
$\displaystyle\ \ \ \ \ \ \ \ \
-\left[1-e^{-\gamma_{R}\left(n-1\right)\left(1-e^{-\tau}\right)\tau}\right]^{2}$
$\displaystyle P_{out}^{\left(S\right)}\leq
2m\cdot\left(\frac{1}{1+\gamma_{E}}\right)^{\left(n-1\right)\left(1-e^{-\tau}\right)}$
$\displaystyle\ \ \ \ \ \ \ \ \ \
-\left[m\cdot\left(\frac{1}{1+\gamma_{E}}\right)^{\left(n-1\right)\left(1-e^{-\tau}\right)}\right]^{2}$
The proof of Lemma 3 can be found in the Appendix B.
_Lemma 4_ : Consider the network scenario of Fig 1 with equal path-loss
between all pairs of nodes, to ensure
$P_{out}^{\left(T\right)}\leq\varepsilon_{t}$ and
$P_{out}^{\left(S\right)}\leq\varepsilon_{s}$ under the Protocol 2, the
parameter $\tau$ must satisfy the following condition.
$\tau\in\left[-\log{\left[1+\frac{\log{\left(\frac{1-\sqrt{1-\varepsilon_{s}}}{m}\right)}}{\left(n-1\right)\log{\left(1+\gamma_{E}\right)}}\right]},\sqrt{\frac{-\log\left(1-\varepsilon_{t}\right)}{2\gamma_{R}\left(n-1\right)}}\right]$
###### Proof.
$\bullet$ Reliability Guarantee
To ensure the reliability requirement
$P_{out}^{\left(T\right)}\leq\varepsilon_{t}$, we know from Lemma 4 that we
just need
$\displaystyle
2\left[1-e^{-\gamma_{R}\left(n-1\right)\left(1-e^{-\tau}\right)\tau}\right]$
$\displaystyle-\left[1-e^{-\gamma_{R}\left(n-1\right)\left(1-e^{-\tau}\right)\tau}\right]^{2}$
$\displaystyle\leq\varepsilon_{t}$
That is,
$\displaystyle
1-e^{-\gamma_{R}\left(n-1\right)\left(1-e^{-\tau}\right)\tau}\leq
1-\sqrt{1-\varepsilon_{t}}$
By using Taylor formula, we have
$\displaystyle\tau\leq\sqrt{\frac{-\log\left(1-\varepsilon_{t}\right)}{2\gamma_{R}\left(n-1\right)}}$
$\bullet$ Secrecy Guarantee
Notice that the secrecy outage probability of Protocol 1 and Protocol 2 is
same. Thus, to ensure the secrecy requirement, we need
$\displaystyle\left(\frac{1}{1+\gamma_{E}}\right)^{\left(n-1\right)\left(1-e^{-\tau}\right)}\leq\frac{1-\sqrt{1-\varepsilon_{s}}}{m}$
Thus,
$\displaystyle\tau\geq-\log{\left[1+\frac{\log{\left(\frac{1-\sqrt{1-\varepsilon_{s}}}{m}\right)}}{\left(n-1\right)\log{\left(1+\gamma_{E}\right)}}\right]}$
The above result implies the minimum value parameter $\tau$ can take to
guarantee the secrecy requirement.
∎
Theorem 2. Consider the network scenario of Fig 1 with equal path-loss between
all pairs of nodes. To guarantee $P_{out}^{\left(T\right)}\leq\varepsilon_{t}$
and $P_{out}^{\left(S\right)}\leq\varepsilon_{s}$ based on the Protocol 2, the
number of eavesdroppers $m$ the network can tolerate must satisfy the
following condition.
$\displaystyle
m\leq\left(1-\sqrt{1-\varepsilon_{s}}\right)\cdot\left(1+\gamma_{E}\right)^{\sqrt{\frac{-\left(n-1\right)\log\left(1-\varepsilon_{t}\right)}{2\gamma_{R}}}}$
###### Proof.
From Lemma 4, we know that to ensure the reliability requirement, we have
$\displaystyle\tau\leq\sqrt{\frac{-\log\left(1-\varepsilon_{t}\right)}{2\gamma_{R}\left(n-1\right)}}$
and
$\displaystyle\left(n-1\right)\left(1-e^{-\tau}\right)\leq\frac{-\log\left(1-\varepsilon_{t}\right)}{2\gamma_{R}\tau}$
To ensure the secrecy requirement, we need
$\displaystyle\left(\frac{1}{1+\gamma_{E}}\right)^{\left(n-1\right)\left(1-e^{-\tau}\right)}\leq\frac{1-\sqrt{1-\varepsilon_{s}}}{m}$
Thus,
$\displaystyle
m\leq\frac{1-\sqrt{1-\varepsilon_{s}}}{\left(\frac{1}{1+\gamma_{E}}\right)^{\left(n-1\right)\left(1-e^{-\tau}\right)}}$
$\displaystyle\ \ \
\leq\frac{1-\sqrt{1-\varepsilon_{s}}}{\left(\frac{1}{1+\gamma_{E}}\right)^{\frac{-\log\left(1-\varepsilon_{t}\right)}{2\gamma_{R}\tau}}}$
By letting $\tau$ to take its maximum value for maximum interference at
eavesdroppers, we get the following bound
$\displaystyle
m\leq\left(1-\sqrt{1-\varepsilon_{s}}\right)\cdot\left(1+\gamma_{E}\right)^{\sqrt{\frac{-\left(n-1\right)\log\left(1-\varepsilon_{t}\right)}{2\gamma_{R}}}}$
∎
## IV Secure and Reliable Transmission under Distance-Dependent Path-Loss
In this section, we consider the more general scenario where the path-loss
between each pair of nodes also depends the distance between them. We first
introduce the coordinate system adopted in our discussion, and then propose a
flexible transmission protocol to achieve both the preferable relay selection
and good load balance under such distance-dependent path-loss scenario. The
related theoretic analysis is further provided to determine the number of
eavesdroppers one network can tolerate by adopting this protocol.
### IV-A Coordinate System
To address the distance-dependent path-loss, we consider a two-hop relay
wireless network deployed in a square of unit area and defined by the the
coordinate system shown in Fig.2, where the source $S$ located at coordinate
$\left(0,0.5\right)$ wishes to establish two-hop transmission with destination
$D$ located at coordinate $\left(1,0.5\right)$. In addition to the source $S$
and destination $D$, we assume that there are $n$ cooperative relays and $m$
eavesdroppers of unknown channels and locations independently and uniformly
distributed in the network area.
Figure 2: Coordinate system for the scenario where path-loss between pairs of
nodes is based on their relative locations.
### IV-B Transmission Protocol
Notice that under the distance-dependent path-loss scenario, the further the
distance between a transmitter and a receiver, the weaker the signal received
at the receiver. Thus, the system nodes located in the middle region between
source $S$ and destination $D$ are preferable relays. Based such observation,
we propose here a general and practical protocol (hereafter called Protocol 3)
to ensure both the preferable relay selection and good load balance for
distance-dependent path-loss scenario, which works as follow.
1) _Relay selection_ : Based on two parameters $a$ and $b$, $0\leq a\leq
0.5,0\leq b\leq 0.5$, we first define a relay selection region
$\left[a,1-a\right]\times\left[b,1-b\right]$ between source $S$ and
destination $D$. A relay node, indexed by $j^{\ast}$, is then selected
randomly from relays falling within the relay selection region.
2) _Channel measurement_ : Each of the other relays measures the channel from
the selected relay $R_{j^{\ast}}$ and destination $D$ by accepting the pilot
signal from $R_{j^{\ast}}$ and $D$ for determining the noise generation nodes.
3) _Two-hop transmission_ : The source $S$ and the selected relay
$R_{j^{\ast}}$ transmit the messages in two-hop transmission. Concurrently,
the relay nodes with indexes in $\mathcal{R}_{1}={\left\\{j\neq
j^{\ast}:|h_{R_{j},R_{j^{\ast}}}|^{2}<\tau\right\\}}$ in the first hop and the
relay nodes with indexes in $\mathcal{R}_{2}={\left\\{j\neq
j^{\ast}:|h_{R_{j},D}|^{2}<\tau\right\\}}$ in the second hop transmit noise
respectively to help transmission.
_Remark 4_ : In the Protocol 3, a trade off between the preferable relay
selection and better load balance can be controlled through the parameters $a$
and $b$, which define the relay selection region. As to be shown in Theorem 3
that by adopting a small value for both $a$ and $b$ (i.e., a larger relay
selection region), a better load balance capacity can be achieved at the cost
of a smaller number of eavesdroppers one network can tolerant.
### IV-C Analysis of Protocol 3
To address the near eavesdropper problem and also to simply the analysis for
the Protocol 3, we assume that there exits a constant $r_{0}>0$ such that any
eavesdropper falling within a circle area with radius $r_{0}$ and center $S$
or $R_{j^{\ast}}$ can eavesdrop the transmitted messages successfully with
probability 1, while any eavesdropper beyond such area can only successfully
eavesdropper the transmitted messages with a probability less than 1. Based on
such a simplification, we can establish the following two lemmas regarding
some basic properties of $P_{out}^{\left(T\right)}$,
$P_{out}^{\left(S\right)}$ and $\tau$ under this protocol.
_Lemma 5_ : Consider the network scenario of Fig 2, under the Protocol 3 the
transmission outage probability $P_{out}^{\left(T\right)}$ and secrecy outage
probability $P_{out}^{\left(S\right)}$ there satisfy the following conditions.
$\displaystyle
P_{out}^{\left(T\right)}\leq\left[1-e^{-\frac{\gamma_{R}{\tau\left(n-1\right)\left(1-e^{-\tau}\right)}}{\phi^{-\alpha}}\left(\varphi_{1}+\varphi_{2}\right)}\right]\left(1-\vartheta\right)+1\cdot\vartheta$
$\displaystyle P_{out}^{\left(S\right)}\leq
2m\left[\pi{r_{0}}^{2}+\left(\frac{1}{1+\gamma_{E}\psi{r_{0}}^{\alpha}}\right)^{\left(n-1\right)\left(1-e^{-\tau}\right)}\left(1-\pi{r_{0}}^{2}\right)\right]$
$\displaystyle\ \
-\left[m\left(\pi{r_{0}}^{2}+\left(\frac{1}{1+\gamma_{E}\psi{r_{0}}^{\alpha}}\right)^{\left(n-1\right)\left(1-e^{-\tau}\right)}\left(1-\pi{r_{0}}^{2}\right)\right)\right]^{2}$
here,
$\displaystyle\vartheta=\bigg{[}1-\left(1-2a\right)\left(1-2b\right)\bigg{]}^{n}$
$\displaystyle\varphi_{1}=\int_{0}^{1}\int_{0}^{1}\frac{1}{\left[\left(x-0.5\right)^{2}+\left(y-0.5\right)^{2}\right]^{\frac{\alpha}{2}}}dxdy$
$\displaystyle\varphi_{2}=\int_{0}^{1}\int_{0}^{1}\frac{1}{\left[\left(x-1\right)^{2}+\left(y-0.5\right)^{2}\right]^{\frac{\alpha}{2}}}dxdy$
$\displaystyle\phi=\sqrt{(1-a)^{2}+(0.5-b)^{2}}$
$\displaystyle\psi=\int_{0}^{1}\int_{0}^{1}\frac{1}{\left(x^{2}+y^{2}\right)^{\frac{\alpha}{2}}}dxdy$
The proof of the Lemma 5 can be found in the Appendix C.
_Lemma 6_ : Consider the network scenario of Fig 2, to ensure
$P_{out}^{\left(T\right)}\leq\varepsilon_{t}$ and
$P_{out}^{\left(S\right)}\leq\varepsilon_{s}$ by applying the Protocol 3, the
parameter $\tau$ must satisfy the following condition.
$\displaystyle\tau\leq\sqrt{\frac{-\log\left(\frac{1-\varepsilon_{t}}{1-\vartheta}\right){\phi^{-\alpha}}}{\gamma_{R}{\left(n-1\right)\left(\varphi_{1}+\varphi_{2}\right)}}}$
and
$\tau\geq-\log\left[1+\frac{\log{\left(\frac{\frac{1-\sqrt{1-\varepsilon_{s}}}{m}-\pi{r_{0}}^{2}}{1-\pi{r_{0}}^{2}}\right)}}{\left(n-1\right)\log{\left(1+\gamma_{E}\psi{r_{0}}^{\alpha}\right)}}\right]$
here, $\vartheta$, $\varphi_{1}$, $\varphi_{2}$, $\phi$ and $\psi$ are defined
in the same way as that in Lemma 5.
###### Proof.
$\bullet$ Reliability Guarantee
To ensure the reliability requirement
$P_{out}^{\left(T\right)}\leq\varepsilon_{t}$, we know from Lemma 5 that we
just need
$\displaystyle\left[1-e^{-\frac{\gamma_{R}{\tau\left(n-1\right)\left(1-e^{-\tau}\right)}}{\phi^{-\alpha}}\left(\varphi_{1}+\varphi_{2}\right)}\right]\left(1-\vartheta\right)+1\cdot\vartheta\leq\varepsilon_{t}$
that is,
$-\frac{\gamma_{R}{\tau\left(n-1\right)\left(1-e^{-\tau}\right)}}{\phi^{-\alpha}}\left(\varphi_{1}+\varphi_{2}\right)\geq\log{\left(\frac{1-\varepsilon_{t}}{1-\vartheta}\right)}$
By using Taylor formula, we have
$\displaystyle\tau\leq\sqrt{\frac{-\log\left(\frac{1-\varepsilon_{t}}{1-\vartheta}\right){\phi^{-\alpha}}}{\gamma_{R}{\left(n-1\right)\left(\varphi_{1}+\varphi_{2}\right)}}}$
$\bullet$ Secrecy Guarantee
To ensure the secrecy requirement
$P_{out}^{\left(S\right)}\leq\varepsilon_{s}$, we know from Lemma 5 that we
just need
$\displaystyle
2m\left[\pi{r_{0}}^{2}+\left(\frac{1}{1+\gamma_{E}\psi{r_{0}}^{\alpha}}\right)^{\left(n-1\right)\left(1-e^{-\tau}\right)}\left(1-\pi{r_{0}}^{2}\right)\right]-$
$\displaystyle\left[m\left(\pi{r_{0}}^{2}+\left(\frac{1}{1+\gamma_{E}\psi{r_{0}}^{\alpha}}\right)^{\left(n-1\right)\left(1-e^{-\tau}\right)}\left(1-\pi{r_{0}}^{2}\right)\right)\right]^{2}$
$\displaystyle\leq\varepsilon_{s}$
Thus,
$\displaystyle
m\cdot\left[\pi{r_{0}}^{2}+\left(\frac{1}{1+\gamma_{E}\psi{r_{0}}^{\alpha}}\right)^{\left(n-1\right)\left(1-e^{-\tau}\right)}\left(1-\pi{r_{0}}^{2}\right)\right]$
$\displaystyle\leq 1-\sqrt{1-\varepsilon_{s}}$
That is,
$\tau\geq-\log\left[1+\frac{\log{\left(\frac{\frac{1-\sqrt{1-\varepsilon_{s}}}{m}-\pi{r_{0}}^{2}}{1-\pi{r_{0}}^{2}}\right)}}{\left(n-1\right)\log{\left(1+\gamma_{E}\psi{r_{0}}^{\alpha}\right)}}\right]$
∎
Based on the results of Lemma 6, we now can establish the following theorem
about the performance of Protocol 3.
Theorem 3. Consider the network scenario of Fig 2. To guarantee
$P_{out}^{\left(T\right)}\leq\varepsilon_{t}$ and
$P_{out}^{\left(S\right)}\leq\varepsilon_{s}$ based on the Protocol 3, the
number of eavesdroppers $m$ the network can tolerate must satisfy the
following condition.
$\displaystyle
m\leq\frac{1-\sqrt{1-\varepsilon_{s}}}{\pi{r_{0}}^{2}+\left(1-\pi{r_{0}}^{2}\right)\omega}$
here,
$\displaystyle\omega=\left(1+\gamma_{E}\psi{r_{0}}^{\alpha}\right)^{-\sqrt{\frac{-\left(n-1\right)\log\left(\frac{1-\varepsilon_{t}}{1-\vartheta}\right)}{\gamma_{R}\left(\varphi_{1}+\varphi_{2}\right)\phi^{\alpha}}}}$
$\vartheta$, $\varphi_{1}$, $\varphi_{2}$, $\phi$ and $\psi$ are defined in
the same way as that in Lemma 5.
###### Proof.
From Lemma 6, we know that to ensure the reliability requirement, we have
$\displaystyle\tau\leq\sqrt{\frac{-\log\left(\frac{1-\varepsilon_{t}}{1-\vartheta}\right){\phi^{-\alpha}}}{\gamma_{R}{\left(n-1\right)\left(\varphi_{1}+\varphi_{2}\right)}}}$
and
$\left(n-1\right)\left(1-e^{-\tau}\right)\leq\frac{-\log{\left(\frac{1-\varepsilon_{t}}{1-\vartheta}\right)}}{\gamma_{R}\tau{\phi^{\alpha}}\left(\varphi_{1}+\varphi_{2}\right)}$
To ensure the secrecy requirement, we need
$\displaystyle
m\cdot\left[\pi{r_{0}}^{2}+\left(\frac{1}{1+\gamma_{E}\psi{r_{0}}^{\alpha}}\right)^{\left(n-1\right)\left(1-e^{-\tau}\right)}\left(1-\pi{r_{0}}^{2}\right)\right]$
$\displaystyle\leq 1-\sqrt{1-\varepsilon_{s}}$
Thus,
$\displaystyle
m\leq\frac{1-\sqrt{1-\varepsilon_{s}}}{\pi{r_{0}}^{2}+\left(\frac{1}{1+\gamma_{E}\psi{r_{0}}^{\alpha}}\right)^{\left(n-1\right)\left(1-e^{-\tau}\right)}\left(1-\pi{r_{0}}^{2}\right)}$
$\displaystyle\ \
\leq\frac{1-\sqrt{1-\varepsilon_{s}}}{\pi{r_{0}}^{2}+\left(\frac{1}{1+\gamma_{E}\psi{r_{0}}^{\alpha}}\right)^{\frac{-\log{\left(\frac{1-\varepsilon_{t}}{1-\vartheta}\right)}}{\gamma_{R}\tau{\phi^{\alpha}}\left(\varphi_{1}+\varphi_{2}\right)}}\left(1-\pi{r_{0}}^{2}\right)}$
By letting $\tau$ to take its maximum value for maximum interference at
eavesdroppers, we get the following bound
$\displaystyle
m\leq\frac{1-\sqrt{1-\varepsilon_{s}}}{\pi{r_{0}}^{2}+\left(1-\pi{r_{0}}^{2}\right)\omega}$
Here,
$\displaystyle\omega=\left(1+\gamma_{E}\psi{r_{0}}^{\alpha}\right)^{-\sqrt{\frac{-\left(n-1\right)\log\left(\frac{1-\varepsilon_{t}}{1-\vartheta}\right)}{\gamma_{R}\left(\varphi_{1}+\varphi_{2}\right)\phi^{\alpha}}}}$
∎
## V Related Works
A lot of research works have been dedicated to the implementation of physical
layer security by adopting artificial noise generation for cooperative
jamming. These works can be roughly classified into two categories depending
on weather the information of eavesdroppers channels and locations is known or
not.
For the case that the information of eavesdroppers channels and locations is
available, many methods can be employed to improve physical layers security by
optimizing the artificial noise generation and power control. In case that the
global channel state information is available, to achieve the goal of
maximizing the secrecy rates while minimizing the total transmit power, a few
cooperative transmission schemes have been proposed in [3][4][5], and for two-
hop wireless networks the optimal transmission strategies were presented in
[6][7]. With respect to small networks, cooperative jamming with multiple
relays and multiple eavesdroppers and knowledge of channels and locations was
considered in [8][9]. Even if only local channel information rather than
global channel state information is known, it was proved that the near-optimal
secrecy rate can achieved by cooperative jamming schemes [10][11]. Except
channel information, the relative locations were also considered for
optimizing cooperative jamming and power allocation to disrupt an eavesdropper
with known location [12][13]. In addition, L. Lai et al. established the
utility of user cooperation in facilitating secure wireless communications and
proposed cooperation strategies in the additive White Gaussian Noise (AWGN)
channel [14], R. Negi et al. showed how artificially generated noise can be
added to the information bearing signal to achieve secrecy in the multiple and
single antenna scenario under the constraint on total power transmitted by all
nodes [15]. The physical layer security issues in a two-way untrusted relay
system was also investigated with friendly jammers in [16][17]. The
cooperative communications in mobile ad hoc networks was discussed in [18].
Effective criteria for relay and jamming node selection were developed to
ensure nonzero secrecy rate in case of given sufficient relays in [19].
For the case that the information of eavesdropper channels and locations is
unknown, the works in [20][21] considered the secrecy for two-hop wireless
networks, the works in [22][23][24] considered the secrecy for large wireless
networks, and the further work in [25] considered the energy efficiency
cooperative jamming strategies. These works considered how cooperative jamming
by friendly nodes can impact the security of the network and compared it with
a straightforward approach based on multi-user diversity. They also proposed
some protocols to embed cooperative jamming techniques for protecting single
links into a large multi-hop network and explored network scaling results on
the number of eavesdroppers one network can tolerate. A.Sheikholeslami et al.
explored the interference from multiple cooperative sessions to confuse the
eavesdroppers in a large wireless network [28]. The cooperative relay scheme
for the broadcast channel was further investigated in [26][27].
## VI Conclusion
To achieve reliable and secure information transmission in a two-hop relay
wireless network in presence of eavesdroppers with unknown channels and
locations, several transmission protocols based on relay cooperation have been
considered. In particular, theoretical analysis has been conducted to
understand that under each of these protocols how many eavesdroppers one
network can tolerant to meet a specified requirement on the maximum allowed
secrecy outage probability and transmission outage probability. Our results in
this paper indicate that these protocols actually have different performance
in terms of eavesdropper-tolerance capacity and load balance capacity among
relays, and in general it is possible for us to select a proper transmission
protocol according to network scenario such that a desired trade off between
the overall eavesdropper-tolerance capacity and load balance among relay nodes
can be achieved.
## Appendix A Proof Of Lemma 1
###### Proof.
Notice that $P_{out}^{\left(T\right)}$ is determined as
$\displaystyle P_{out}^{\left(T\right)}=P\left(O_{S\rightarrow
R_{j^{\ast}}}^{(T)}\right)+P\left(O_{R_{j^{\ast}}\rightarrow D}^{(T)}\right)$
$\displaystyle\ \ \ \ \ \ \ \ \ \ -P\left(O_{S\rightarrow
R_{j^{\ast}}}^{(T)}\right)\cdot P\left(O_{R_{j^{\ast}}\rightarrow
D}^{(T)}\right)$
Based on the definition of transmission outage probability, we have
$\displaystyle P\left(O_{S\rightarrow R_{j^{\ast}}}^{(T)}\right)$
$\displaystyle\ \ \ \ \ =P\left(C_{S,R_{j^{\ast}}}\leq\gamma_{R}\right)$
$\displaystyle\ \ \ \ \
=P\left(\frac{E_{s}\cdot|h_{S,R_{j^{\ast}}}|^{2}}{\sum_{R_{j}\in\mathcal{R}_{1}}E_{s}\cdot|h_{R_{j},R_{j^{\ast}}}|^{2}+N_{0}/2}\leq\gamma_{R}\right)$
$\displaystyle\ \ \ \ \ \doteq
P\left(\frac{|h_{S,R_{j^{\ast}}}|^{2}}{\sum_{R_{j}\in\mathcal{R}_{1}}|h_{R_{j},R_{j^{\ast}}}|^{2}}\leq\gamma_{R}\right)$
Compared to the noise generated by multiple system nodes, the environment
noise is negligible and thus is omitted here to simply the analysis. Notice
that $\mathcal{R}_{1}={\left\\{j\neq
j^{\ast}:|h_{R_{j},R_{j^{\ast}}}|^{2}<\tau\right\\}}$, then
$\displaystyle P\left(O_{S\rightarrow R_{j^{\ast}}}^{(T)}\right)\leq
P\left(\frac{|h_{S,R_{j^{\ast}}}|^{2}}{{|\mathcal{R}_{1}|}\tau}\leq\gamma_{R}\right)$
$\displaystyle\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
=P\left(|h_{S,R_{j^{\ast}}}|^{2}\leq\gamma_{R}{|\mathcal{R}_{1}|}\tau\right)$
$\displaystyle\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \leq
P\left(H^{l}\leq\gamma_{R}{|\mathcal{R}_{1}|}\tau\right)$
where
$H^{l}=min\left(\left|h_{S,R_{j^{\ast}}}\right|^{2},\left|h_{D,R_{j^{\ast}}}\right|^{2}\right)$
is the largest random variable among the $n$ exponentially distributed random
variables
$min\left(\left|h_{S,R_{j}}\right|^{2},\left|h_{D,R_{j}}\right|^{2}\right),j=1,2,\cdots,n$.
From reference [29], we can get the distribution function of the
$min\left(|h_{S,R_{j}}|^{2},|h_{D,R_{j}}|^{2}\right)$ for each relay
$R_{j},j=1,2,\cdots,n$ as following,
$F_{min\left(|h_{S,R_{j}}|^{2},|h_{D,R_{j}}|^{2}\right)}\left(x\right)=\begin{cases}1-e^{-2x}\
\ \ &\text{$x>0$}\\\ 0\ \ \ &\text{$x\leq 0$}\\\ \end{cases}$
From reference [29], we can also get the distribution function of random
variable $H^{l}$ as following,
$F_{H^{l}}\left(x\right)=\begin{cases}\left[1-e^{-2x}\right]^{n}\ \ \
&\text{$x>0$}\\\ 0\ \ \ &\text{$x\leq 0$}\\\ \end{cases}$
Therefore, we have
$\displaystyle P\left(O_{S\rightarrow
R_{j^{\ast}}}^{(T)}\right)\leq\left[1-e^{-2\gamma_{R}{|\mathcal{R}_{1}|}\tau}\right]^{n}$
Since there are $n-1$ other relays except $R_{j^{\ast}}$, the expected number
of noise-generation nodes is given by $|\mathcal{R}_{1}|=\left(n-1\right)\cdot
P\left(|h_{R_{j},R_{j^{\ast}}}|^{2}<\tau\right)=\left(n-1\right)\cdot\left(1-e^{-\tau}\right)$.
Then we have
$\displaystyle P\left(O_{S\rightarrow
R_{j^{\ast}}}^{(T)}\right)\leq\left[1-e^{-2\gamma_{R}\left(n-1\right)\left(1-e^{-\tau}\right)\tau}\right]^{n}$
Employing the same method, we can get
$\displaystyle P\left(O_{R_{j^{\ast}}\rightarrow
D}^{(T)}\right)\leq\left[1-e^{-2\gamma_{R}\left(n-1\right)\left(1-e^{-\tau}\right)\tau}\right]^{n}$
Thus, we have
$\displaystyle P_{out}^{\left(T\right)}\leq
2\left[1-e^{-2\gamma_{R}\left(n-1\right)\left(1-e^{-\tau}\right)\tau}\right]^{n}$
$\displaystyle\ \ \ \ \ \ \ \ \ \
-\left[1-e^{-2\gamma_{R}\left(n-1\right)\left(1-e^{-\tau}\right)\tau}\right]^{2n}$
Similarly, notice that $P_{out}^{\left(S\right)}$ is given by
$\displaystyle P_{out}^{\left(S\right)}=P\left(O_{S\rightarrow
R_{j^{\ast}}}^{(S)}\right)+P\left(O_{R_{j^{\ast}}\rightarrow D}^{(S)}\right)$
$\displaystyle\ \ \ \ \ \ \ \ \ -P\left(O_{S\rightarrow
R_{j^{\ast}}}^{(S)}\right)\cdot P\left(O_{R_{j^{\ast}}\rightarrow
D}^{(S)}\right)$
According to the definition of secrecy outage probability, we know that
$\displaystyle P\left(O_{S\rightarrow
R_{j^{\ast}}}^{(S)}\right)=P\left(\bigcup_{i=1}^{m}\left\\{C_{S,E_{i}}\geq\gamma_{E}\right\\}\right)$
Thus, we have
$\displaystyle P\left(O_{S\rightarrow
R_{j^{\ast}}}^{(S)}\right)\leq\sum_{i=1}^{m}P\left(C_{S,E_{i}}\geq\gamma_{E}\right)$
Based on the Markov inequality,
$\displaystyle P\left(C_{S,E_{i}}\geq\gamma_{E}\right)$ $\displaystyle\ \ \ \
\ \leq
P\left(\frac{E_{s}\cdot|h_{S,E_{i}}|^{2}}{\sum_{R_{j}\in\mathcal{R}_{1}}E_{s}\cdot|h_{R_{j},E_{i}}|^{2}}\geq\gamma_{E}\right)$
$\displaystyle\ \ \ \ \ =E_{\left\\{h_{R_{j},E_{i}},j=0,1,\cdots,n+mp,j\neq
j^{\ast}\right\\},\mathcal{R}_{1}}$ $\displaystyle\ \ \ \ \ \ \ \ \
\left[P\left(|h_{S,E_{i}}|^{2}>\gamma_{E}\cdot\sum_{R_{j}\in\mathcal{R}_{1}}|h_{R_{j},E_{i}}|^{2}\right)\right]$
$\displaystyle\ \ \ \ \ \leq
E_{\mathcal{R}_{1}}\left[\prod_{R_{j}\in\mathcal{R}_{1}}E_{h_{R_{j},E_{i}}}\left[e^{-\gamma_{E}|h_{R_{j},E_{i}}|^{2}}\right]\right]$
$\displaystyle\ \ \ \ \
=E_{\mathcal{R}_{1}}\left[\left(\frac{1}{1+\gamma_{E}}\right)^{|\mathcal{R}_{1}|}\right]$
Therefore,
$P\left(O_{S\rightarrow
R_{j^{\ast}}}^{(S)}\right)\leq\sum_{i=1}^{m}\left(\frac{1}{1+\gamma_{E}}\right)^{|\mathcal{R}_{1}|}=m\cdot\left(\frac{1}{1+\gamma_{E}}\right)^{|\mathcal{R}_{1}|}$
Employing the same method, we can get
$P\left(O_{R_{j^{\ast}}\rightarrow D}^{(S)}\right)\leq
m\cdot\left(\frac{1}{1+\gamma_{E}}\right)^{|\mathcal{R}_{2}|}$
Since the expected number of noise-generation nodes is given by
$|\mathcal{R}_{1}|=|\mathcal{R}_{2}|=\left(n-1\right)\cdot\left(1-e^{-\tau}\right)$,
thus, we can get
$\displaystyle P_{out}^{\left(S\right)}\leq
2m\cdot\left(\frac{1}{1+\gamma_{E}}\right)^{\left(n-1\right)\cdot\left(1-e^{-\tau}\right)}$
$\displaystyle\ \ \ \ \ \ \ \ \
-\left[m\cdot\left(\frac{1}{1+\gamma_{E}}\right)^{\left(n-1\right)\cdot\left(1-e^{-\tau}\right)}\right]^{2}$
∎
## Appendix B Proof Of Lemma 3
###### Proof.
Similar to the proof of Lemma 1, we notice that $P_{out}^{\left(T\right)}$ is
determined as
$\displaystyle P_{out}^{\left(T\right)}=P\left(O_{S\rightarrow
R_{j^{\ast}}}^{(T)}\right)+P\left(O_{R_{j^{\ast}}\rightarrow D}^{(T)}\right)$
$\displaystyle\ \ \ \ \ \ \ \ \ \ -P\left(O_{S\rightarrow
R_{j^{\ast}}}^{(T)}\right)\cdot P\left(O_{R_{j^{\ast}}\rightarrow
D}^{(T)}\right)$
Based on the definition of transmission outage probability, we have
$\displaystyle P\left(O_{S\rightarrow
R_{j^{\ast}}}^{(T)}\right)=P\left(C_{S,R_{j^{\ast}}}\leq\gamma_{R}\right)$
$\displaystyle\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \leq
P\left(\frac{|h_{S,R_{j^{\ast}}}|^{2}}{{|\mathcal{R}_{1}|}\tau}\leq\gamma_{R}\right)$
$\displaystyle\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
=P\left(|h_{S,R_{j^{\ast}}}|^{2}\leq\gamma_{R}{|\mathcal{R}_{1}|}\tau\right)$
$\displaystyle\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
=1-e^{-\gamma_{R}{|\mathcal{R}_{1}|}\tau}$
Here $\mathcal{R}_{1}={\left\\{j\neq
j^{\ast}:|h_{R_{j},R_{j^{\ast}}}|^{2}<\tau\right\\}}$. Since the expected
number of noise-generation nodes is given by
$|\mathcal{R}_{1}|=\left(n-1\right)\cdot\left(1-e^{-\tau}\right)$. Then we
have
$\displaystyle P\left(O_{S\rightarrow R_{j^{\ast}}}^{(T)}\right)\leq
1-e^{-\gamma_{R}\left(n-1\right)\left(1-e^{-\tau}\right)\tau}$
Employing the same method, we can get
$\displaystyle P\left(O_{R_{j^{\ast}}\rightarrow D}^{(T)}\right)\leq
1-e^{-\gamma_{R}\left(n-1\right)\left(1-e^{-\tau}\right)\tau}$
Thus, we have
$\displaystyle P_{out}^{\left(T\right)}\leq
2\left[1-e^{-\gamma_{R}\left(n-1\right)\left(1-e^{-\tau}\right)\tau}\right]$
$\displaystyle\ \ \ \ \ \ \ \ \ \
-\left[1-e^{-\gamma_{R}\left(n-1\right)\left(1-e^{-\tau}\right)\tau}\right]^{2}$
Notice that the eavesdropper model of Protocol 1 is the same as that of
Protocol 2, the method for ensuring secrecy is identical to that of in Lemma
1. Thus, we can see that the secrecy outage probability of Protocol 1 and
Protocol 2 is the same, that is,
$\displaystyle P_{out}^{\left(S\right)}\leq
2m\cdot\left(\frac{1}{1+\gamma_{E}}\right)^{\left(n-1\right)\cdot\left(1-e^{-\tau}\right)}$
$\displaystyle\ \ \ \ \ \ \ \ \
-\left[m\cdot\left(\frac{1}{1+\gamma_{E}}\right)^{\left(n-1\right)\cdot\left(1-e^{-\tau}\right)}\right]^{2}$
∎
## Appendix C Proof Of Lemma 5
###### Proof.
Notice that two ways leading to transmission outage are: 1) there are no
candidate relays in the relay selection region; 2) the SINR at the selected
relay or the destination is less than $\gamma_{R}$. Let $A_{1}$ be the event
that there is at least one relay in the relay selection region, and $A_{2}$ be
the event that there are no relays in the relay selection region. We have
$\displaystyle
P_{out}^{\left(T\right)}=P_{out|A_{1}}^{\left(T\right)}P(A_{1})+P_{out|A_{2}}^{\left(T\right)}P(A_{2})$
Since the relay is uniformly distributed, the number of candidate relays is a
binomial distribution $\bigg{(}n,\left(1-2a\right)\left(1-2b\right)\bigg{)}$.
We have
$P(A_{1})=1-\vartheta$
and
$P(A_{2})=\vartheta$
where $\vartheta=\bigg{[}1-\left(1-2a\right)\left(1-2b\right)\bigg{]}^{n}$.
When event $A_{2}$ happens, no relay is available. Then
$\displaystyle P_{out|A_{2}}^{\left(T\right)}=1$
Thus, we have
$\displaystyle
P_{out}^{\left(T\right)}=P_{out|A_{1}}^{\left(T\right)}\left(1-\vartheta\right)+1\cdot\vartheta$
Notice that $P_{out|A_{1}}^{\left(T\right)}$ is determined as
$\displaystyle P_{out|A_{1}}^{\left(T\right)}=P\left(O_{S\rightarrow
R_{j^{\ast}}}^{(T)}\bigg{|}A_{1}\right)+P\left(O_{R_{j^{\ast}}\rightarrow
D}^{(T)}\bigg{|}A_{1}\right)$ $\displaystyle\ \ \ \ \ \ \ \ \ \ \ \
-P\left(O_{S\rightarrow R_{j^{\ast}}}^{(T)}\bigg{|}A_{1}\right)\cdot
P\left(O_{R_{j^{\ast}}\rightarrow D}^{(T)}\bigg{|}A_{1}\right)$
Based on the definition of transmission outage probability, we have
$\displaystyle P\left(O_{S\rightarrow R_{j^{\ast}}}^{(T)}\bigg{|}A_{1}\right)$
$\displaystyle\ \ \ \ \
=P\left(C_{S,R_{j^{\ast}}}\leq\gamma_{R}\bigg{|}A_{1}\right)$ $\displaystyle\
\ \ \ \
=P\left(\frac{E_{s}\cdot\frac{|h_{S,R_{j^{\ast}}}|^{2}}{d_{S,R_{j^{\ast}}}^{\alpha}}}{\sum_{R_{j}\in\mathcal{R}_{1}}E_{s}\cdot\frac{|h_{R_{j},R_{j^{\ast}}}|^{2}}{d_{R_{j},R_{j^{\ast}}}^{\alpha}}+\frac{N_{0}}{2}}\leq\gamma_{R}\bigg{|}A_{1}\right)$
$\displaystyle\ \ \ \ \ \doteq
P\left(\frac{\frac{|h_{S,R_{j^{\ast}}}|^{2}}{d_{S,R_{j^{\ast}}}^{\alpha}}}{\sum_{R_{j}\in\mathcal{R}_{1}}\frac{|h_{R_{j},R_{j^{\ast}}}|^{2}}{d_{R_{j},R_{j^{\ast}}}^{\alpha}}}\leq\gamma_{R}\bigg{|}A_{1}\right)$
Compared to the noise generated by multiple system nodes, the environment
noise is negligible and thus is omitted here to simply the analysis. Notice
that $\mathcal{R}_{1}={\left\\{j\neq
j^{\ast}:|h_{R_{j},R_{j^{\ast}}}|^{2}<\tau\right\\}}$, then
$\displaystyle P\left(O_{S\rightarrow
R_{j^{\ast}}}^{(T)}\bigg{|}A_{1}\right)\leq
P\left(\frac{|h_{S,R_{j^{\ast}}}|^{2}d_{S,R_{j^{\ast}}}^{-\alpha}}{\sum_{R_{j}\in\mathcal{R}_{1}}\tau
d_{R_{j},R_{j^{\ast}}}^{-\alpha}}\leq\gamma_{R}\bigg{|}A_{1}\right)$
As shown in Fig 2 that by assuming the coordinate of $R_{j}$ as
$\left(x,y\right)$, we can see that the number of noise generating nodes in
square $\left[x,x+dx\right]\times\left[y,y+dy\right]$ will be
$\left(n-1\right)\left(1-e^{-\tau}\right)dxdy$. Then, we have
$\displaystyle\sum_{R_{j}\in\mathcal{R}_{1}}\frac{\tau}{d_{R_{j},R_{j^{\ast}}}^{\alpha}}$
$\displaystyle\ \ \ \ \ \ \ \
=\int_{0}^{1}\int_{0}^{1}\frac{\tau\left(n-1\right)\left(1-e^{-\tau}\right)}{\left[\left(x-x_{R_{j^{\ast}}}\right)^{2}+\left(y-y_{R_{j^{\ast}}}\right)^{2}\right]^{\frac{\alpha}{2}}}dxdy$
where $\left(x_{R_{j^{\ast}}},y_{R_{j^{\ast}}}\right)$ is the coordinate of
the selected relay $R_{j^{\ast}}$,
$x_{R_{j^{\ast}}}\in\left[a,1-a\right],y_{R_{j^{\ast}}}\in\left[b,1-b\right]$
and $a\in\left[0,0.5\right],b\in\left[0,0.5\right]$.
Notice that within the network area, where relays are uniformly distributed,
the worst case location for the selected relay $R_{j^{\ast}}$ is the point
$\left(0.5,0.5\right)$, at which the interference from the noise generating
nodes is the largest; whereas, the best case location for the selected relay
$R_{j^{\ast}}$ is the four corner points $(a,b),(a,1-b),(1-a,b)$ and
$(1-a,1-b)$ of the relay selection, where the interference from the noise
generating nodes is the smallest. By considering the worst case location for
the selected relay $R_{j^{\ast}}$, we have
$P\left(O_{S\rightarrow R_{j^{\ast}}}^{(T)}\bigg{|}A_{1}\right)\leq
P\left(\frac{|h_{S,R_{j^{\ast}}}|^{2}d_{S,R_{j^{\ast}}}^{-\alpha}}{\tau\left(n-1\right)\left(1-e^{-\tau}\right)\varphi_{1}}\leq\gamma_{R}\bigg{|}A_{1}\right)$
Here
$\varphi_{1}=\int_{0}^{1}\int_{0}^{1}\frac{1}{\left[\left(x-0.5\right)^{2}+\left(y-0.5\right)^{2}\right]^{\frac{\alpha}{2}}}dxdy$
Due to $a\leq d_{S,R_{j^{\ast}}}\leq\sqrt{(1-a)^{2}+(0.5-b)^{2}}$, and let
$\phi=\sqrt{(1-a)^{2}+(0.5-b)^{2}}$, then
$\displaystyle P\left(O_{S\rightarrow R_{j^{\ast}}}^{(T)}\bigg{|}A_{1}\right)$
$\displaystyle\ \ \ \leq
P\left(\frac{|h_{S,R_{j^{\ast}}}|^{2}\phi^{-\alpha}}{\tau\left(n-1\right)\left(1-e^{-\tau}\right)\varphi_{1}}\leq\gamma_{R}\bigg{|}A_{1}\right)$
$\displaystyle\ \ \
=P\left(|h_{S,R_{j^{\ast}}}|^{2}\leq\frac{\gamma_{R}{\tau\left(n-1\right)\left(1-e^{-\tau}\right)\varphi_{1}}}{\phi^{-\alpha}}\bigg{|}A_{1}\right)$
$\displaystyle\ \ \
=1-e^{-\frac{\gamma_{R}{\tau\left(n-1\right)\left(1-e^{-\tau}\right)\varphi_{1}}}{\phi^{-\alpha}}}$
Employing the same method, we can get
$\displaystyle P\left(O_{R_{j^{\ast}}\rightarrow
D}^{(T)}\bigg{|}A_{1}\right)\leq
1-e^{-\frac{\gamma_{R}{\tau\left(n-1\right)\left(1-e^{-\tau}\right)\varphi_{2}}}{\phi^{-\alpha}}}$
here,
$\varphi_{2}=\int_{0}^{1}\int_{0}^{1}\frac{1}{\left[\left(x-1\right)^{2}+\left(y-0.5\right)^{2}\right]^{\frac{\alpha}{2}}}dxdy$
Then, we have
$\displaystyle P_{out|A_{1}}^{\left(T\right)}\leq
1-e^{-\frac{\gamma_{R}{\tau\left(n-1\right)\left(1-e^{-\tau}\right)}}{\phi^{-\alpha}}\left(\varphi_{1}+\varphi_{2}\right)}$
Thus, we have
$\displaystyle
P_{out}^{\left(T\right)}\leq\left[1-e^{-\frac{\gamma_{R}{\tau\left(n-1\right)\left(1-e^{-\tau}\right)}}{\phi^{-\alpha}}\left(\varphi_{1}+\varphi_{2}\right)}\right]\left(1-\vartheta\right)+1\cdot\vartheta$
Notice that $P_{out}^{\left(S\right)}$ is given by
$\displaystyle P_{out}^{\left(S\right)}=P\left(O_{S\rightarrow
R_{j^{\ast}}}^{(S)}\right)+P\left(O_{R_{j^{\ast}}\rightarrow D}^{(S)}\right)$
$\displaystyle\ \ \ \ \ \ \ \ \ -P\left(O_{S\rightarrow
R_{j^{\ast}}}^{(S)}\right)\cdot P\left(O_{R_{j^{\ast}}\rightarrow
D}^{(S)}\right)$
According to the definition of secrecy outage probability, we know that
$\displaystyle P\left(O_{S\rightarrow
R_{j^{\ast}}}^{(S)}\right)=P\left(\bigcup_{i=1}^{m}\left\\{C_{S,E_{i}}\geq\gamma_{E}\right\\}\right)$
Thus, we have
$\displaystyle P\left(O_{S\rightarrow
R_{j^{\ast}}}^{(S)}\right)\leq\sum_{i=1}^{m}P\left(C_{S,E_{i}}\geq\gamma_{E}\right)$
Based on the definition of $r_{0}$, we denote by $G_{1}^{(i)}$ the event that
the distance between $E_{i}$ and the source is less than $r_{0}$, and denote
by $G_{2}^{(i)}$ the event that distance between $E_{i}$ and the source is
lager than or equal to $r_{0}$. We have
$\displaystyle P\left(C_{S,E_{i}}\geq\gamma_{E}\right)$ $\displaystyle\ \ \ \
=P\left(C_{S,E_{i}}\geq\gamma_{E}\bigg{|}G_{1}^{(i)}\right)P\left(G_{1}^{(i)}\right)$
$\displaystyle\ \ \ \ \ \ \ \
+P\left(C_{S,E_{i}}\geq\gamma_{E}\bigg{|}G_{2}^{(i)}\right)P\left(G_{2}^{(i)}\right)$
$\displaystyle\ \ \ \ \leq
1\cdot\frac{1}{2}\pi{r_{0}}^{2}+P\left(C_{S,E_{i}}\geq\gamma_{E}\bigg{|}G_{2}^{(i)}\right)\left(1-\frac{1}{2}\pi{r_{0}}^{2}\right)$
of which
$\displaystyle P\left(C_{S,E_{i}}\geq\gamma_{E}\bigg{|}G_{2}^{(i)}\right)$
$\displaystyle\ \ \leq
P\left(\frac{|h_{S,E_{i}}|^{2}{r_{0}}^{-\alpha}}{\Gamma\int_{0}^{1}\int_{0}^{1}\frac{1}{\left[\left(x-x_{E_{i}}\right)^{2}+\left(y-y_{E_{i}}\right)^{2}\right]^{\frac{\alpha}{2}}}dxdy}\geq\gamma_{E}\bigg{|}G_{2}^{(i)}\right)$
where $\left(x_{E_{i}},y_{E_{i}}\right)$ is the coordinate of the eavesdropper
$E_{i}$. $\Gamma$ is the sum of $\left(n-1\right)\left(1-e^{-\tau}\right)$
independent exponential random variables.
From Fig 2 we know that the largest interference at eavesdropper $E_{i}$
happens when $E_{i}$ is located at the point $(0.5,0.5)$, while the smallest
interference at $E_{i}$ happens it is located at the four corners of the
network region. By considering the smallest interference at eavesdroppers, we
then have
$\displaystyle P\left(C_{S,E_{i}}\geq\gamma_{E}\bigg{|}G_{2}^{(i)}\right)$
$\displaystyle\ \ \ \ \ \ \leq
P\left(\frac{|h_{S,E_{i}}|^{2}{r_{0}}^{-\alpha}}{\Gamma\psi}\geq\gamma_{E}\right)$
$\displaystyle\ \ \ \ \ \
=P\left(|h_{S,E_{i}}|^{2}\geq\Gamma\gamma_{E}\cdot\psi\cdot{r_{0}}^{\alpha}\right)$
here
$\psi=\int_{0}^{1}\int_{0}^{1}\frac{1}{\left(x^{2}+y^{2}\right)^{\frac{\alpha}{2}}}dxdy$
Based on the Markov inequality,
$\displaystyle P\left(C_{S,E_{i}}\geq\gamma_{E}\bigg{|}G_{2}^{(i)}\right)$
$\displaystyle\ \ \ \ \ \leq
E_{\Gamma}\left[e^{-\Gamma\gamma_{E}\psi{r_{0}}^{\alpha}}\right]$
$\displaystyle\ \ \ \ \
=\left(\frac{1}{1+\gamma_{E}\psi{r_{0}}^{\alpha}}\right)^{\left(n-1\right)\left(1-e^{-\tau}\right)}$
Then, we have
$\displaystyle P\left(C_{S,E_{i}}\geq\gamma_{E}\right)$
$\displaystyle\leq\frac{1}{2}\pi{r_{0}}^{2}+\left(\frac{1}{1+\gamma_{E}\psi{r_{0}}^{\alpha}}\right)^{\left(n-1\right)\left(1-e^{-\tau}\right)}\left(1-\frac{1}{2}\pi{r_{0}}^{2}\right)$
Employee the same method, we have
$\displaystyle P\left(C_{R_{j^{\ast}},E_{i}}\geq\gamma_{E}\right)$
$\displaystyle\leq\pi{r_{0}}^{2}+\left(\frac{1}{1+\gamma_{E}\psi{r_{0}}^{\alpha}}\right)^{\left(n-1\right)\left(1-e^{-\tau}\right)}\left(1-\pi{r_{0}}^{2}\right)$
Notice that
$\displaystyle\frac{1}{2}\pi{r_{0}}^{2}+\left(\frac{1}{1+\gamma_{E}\psi{r_{0}}^{\alpha}}\right)^{\left(n-1\right)\left(1-e^{-\tau}\right)}\left(1-\frac{1}{2}\pi{r_{0}}^{2}\right)$
$\displaystyle=\pi{r_{0}}^{2}+\left(\frac{1}{1+\gamma_{E}\psi{r_{0}}^{\alpha}}\right)^{\left(n-1\right)\left(1-e^{-\tau}\right)}\left(1-\pi{r_{0}}^{2}\right)$
$\displaystyle\ \ \ \
-\frac{1}{2}\pi{r_{0}}^{2}\left[1-\left(\frac{1}{1+\gamma_{E}\psi{r_{0}}^{\alpha}}\right)^{\left(n-1\right)\left(1-e^{-\tau}\right)}\right]$
$\displaystyle\leq\pi{r_{0}}^{2}+\left(\frac{1}{1+\gamma_{E}\psi{r_{0}}^{\alpha}}\right)^{\left(n-1\right)\left(1-e^{-\tau}\right)}\left(1-\pi{r_{0}}^{2}\right)$
Therefore
$\displaystyle P\left(O_{S\rightarrow R_{j^{\ast}}}^{(S)}\right)\leq
P\left(O_{R_{j^{\ast}}\rightarrow D}^{(S)}\right)$ $\displaystyle\leq
m\left[\pi{r_{0}}^{2}+\left(\frac{1}{1+\gamma_{E}\psi{r_{0}}^{\alpha}}\right)^{\left(n-1\right)\left(1-e^{-\tau}\right)}\left(1-\pi{r_{0}}^{2}\right)\right]$
Then, we have
$\displaystyle P_{out}^{\left(S\right)}\leq
2m\left[\pi{r_{0}}^{2}+\left(\frac{1}{1+\gamma_{E}\psi{r_{0}}^{\alpha}}\right)^{\left(n-1\right)\left(1-e^{-\tau}\right)}\left(1-\pi{r_{0}}^{2}\right)\right]$
$\displaystyle\ \
-\left[m\left(\pi{r_{0}}^{2}+\left(\frac{1}{1+\gamma_{E}\psi{r_{0}}^{\alpha}}\right)^{\left(n-1\right)\left(1-e^{-\tau}\right)}\left(1-\pi{r_{0}}^{2}\right)\right)\right]^{2}$
∎
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* [14] L. Lai and H.E. Gamal, _”The relay-eavesdropper channel: Cooperation for secrecy,”_ IEEE Trans. Inf. Theory, vol. 54, no. 9, pp. 4005 - 4019, 2008.
* [15] R. Negi and S. Goelm, _”Secret communication using artificial noise,”_ In 62nd IEEE conference on Vehicular Technology Conference (VTC 2005), pp.1906-1910, 2005.
* [16] R. Zhang, L. Song, Z. Han and B. Jiao. _”Physical Layer Security for Two-Way Untrusted Relaying With Friendly Jammers, ”_ IEEE Transactions on vehicular technology, vol. 61, no. 8, pp. 3693-3704, 2012.
* [17] X. He and A. Yener, _”Two-hop secure communication using an untrusted relay: A case for cooperative jamming.”_ in Proceeding of 2008 IEEE Global Telecommunications Conference, pp.1-5, 2008.
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* [19] Z. Ding, M. Xu, J. Lu, and F. Liu, _”Improving Wireless Security for Bidirectional Communication Scenarios,”_ IEEE Transactions on vehicular technology, vol.61, no.6, pp.2842-2848, 2012\.
* [20] D. Goeckel, S. Vasudevan, D. Towsley, S. Adams, Z. Ding and K. Leung, _”Everlasting Secrecy in Two-Hop Wireless Networks Using Artificial Noise Generation from Relays,”_ In proceeding of International Technology Alliance Collaboration System (ACITA 2011), 2011\.
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* [23] C. Capar, D. Goeckel, B. Liu and D. Towsley, _”Secret Communication in Large Wireless Networks without Eavesdropper Location Information,”_ In Proceeding of IEEE INFOCOM 2012, pp.1152-1160, 2012.
* [24] Z. Ding, K. Leung, D. Goeckel and D. Towsley, _”Opportunistic Relaying for Secrecy Communications: Cooperative Jamming vs Relay Chatting,”_ IEEE Transactions on Wireless Communications, vol.10, no.6, pp.1725-1729, 2011.
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* [27] C. Capar and D. Goeckel, _”Network Coding for Facilitating Secrecy in Large Wireless Networks,”_ In 46th Annual Conference on Information Sciences and Systems (CISS 2012), pp.1-6, 2012.
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|
arxiv-papers
| 2012-12-03T05:42:01 |
2024-09-04T02:49:38.823192
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Yulong Shen, Xiaohong Jiang, Jianfeng Ma, Weisong Shi",
"submitter": "Yulong Shen",
"url": "https://arxiv.org/abs/1212.0287"
}
|
1212.0500
|
# Vector fields on C*-algebras, semigroups of endomorphisms and gauge groups
Innocenti V. Maresin
###### Abstract
What is a vector field on a C*-algebra is defined. Its relation to semigroups
of *-endomorphisms was researched. Some results given about those vector
fields and semigroups. There are also various constructions of semigroups
including one parametrized by the cone of the future in (3+1)-dimensional
Minkowski space. Physical interpretations will be presented in a separate
paper.
This work is supported by Russian Foundation for Basic Research,
grant 10-01-00178.
### 0.1 Introduction
The formalism of C*-algebras is known to be used for quantization and quantum
observables for a long time. Several ideas of differential geometry, such as
bundles, differential forms and integrals, admit their translations to
algebraic language. This paper presents one concept which was apparently
missing: vector fields. In Section 1 we define real vector fields on a
C*-algebra, a continuous semigroup of *-endomorphisms of the same algebra, the
relation between them, and basic examples. In Section 2 we present some less
trivial constructs, such as the commutator of vector fields and a semigroup
parametrized by several real variables. Section 3 gives the construction of
vector fields via representations of the algebra. Section 4 gives some
implications to semigroups, if there exist ones generated by vector fields
from Section 3.
### 0.2 Basic facts about C*-algebras
C*-algebra ${\mathcal{A}}$ is a complex Banach algebra with an antilinear
involution, which reverses the order of multiplication: ${(a\hskip
1.25ptb)}^{*}={b}^{*}\hskip 1.25pt{a}^{*}$. The norm must satisfy:
$||a||^{2}=||{a}^{*}\hskip 1.25pta||$
${{\mathcal{A}}_{herm}}$ is a real linear subspace of self-conjugate
(Hermitian) elements.
A *-endomorphism does not have to keep the norm. Though, one can prove that a
*-endomorphism cannot increase the norm of an element:
$||E({a}^{*}\hskip 1.25pta)||=||E(a)||^{2},\ ||E({a}^{*}\hskip 1.25pta\hskip
1.25pt{a}^{*}\hskip 1.25pta)||=||E(a)||^{4},\ ||E(({a}^{*}\hskip
1.25pta)^{4})||=||E(a)||^{8},\ \cdots$
## 1 Vector fields and real-parametrized endomorphism semigroups
1.1. A real-parametrized semigroup (or group) of *-endomorphisms is a family
of *-endomorphisms parametrized by non-negative (or real) numbers, such that:
$E_{\zeta}\operatorname{\circ}E_{\eta}=E_{\zeta+\eta}$ and
$E_{0}=\mathrm{Id}$. These semigroups will be referred simply as “endomorphism
semigroups”. Possible example is
$E_{t}(q)=\exp(-itg)q\exp(itg),g\in{{\mathcal{A}}_{herm}}$, and it is actually
the only possible example in finite dimensions. If the algebra is infinite-
dimensional, then other semigroups may exist, and for a semigroup not defined
by a conjugation with an exponent we cannot compute the derivative
${\frac{d}{dt}}E_{t}(q)$ for any element $q$. For most constructions of a
semigroup (it’s not a theorem), though, we can find such *-subalgebra dense in
${\mathcal{A}}$ that *-endomorphisms of a semigroup preserve it and
aforementioned derivative is defined for any $q$ from the subalgebra. It is
obvious that such semigroup, restricted to this subalgebra, satisfies the
following ordinary differential problem:
${\scriptstyle\forall
q\in{{\mathcal{A}}_{V}}:}{\frac{d}{dt}}E_{t}(q)=V(E_{t}(q))\,,{\scriptstyle\
\forall t:}E_{t}:{{\mathcal{A}}_{V}}\to{{\mathcal{A}}_{V}}\mbox{\small{} a
bounded linear map,}\ E_{0}=\mathrm{Id},\hskip 16.00008pt\mbox{(!)}$
where ${{\mathcal{A}}_{V}}$ is a *-subalgebra dense in ${\mathcal{A}}$ and
$V:{{\mathcal{A}}_{V}}\to{\mathcal{A}}$ is a fixed (unbounded) linear
operator. Another way to express this is a differential equation
$\dot{E}=V\operatorname{\circ}E$ where the differentiation is understood in
the strong operator topology (not norm topology) on bounded linear operators
on ${{\mathcal{A}}_{V}}$. Are there necessary conditions on $V$? First, it
must commute with *, because any *-endomorphism has to preserve this
operation. Second, by differentiation of the condition (!) we obtain the
product rule:
$\forall u,v\in{{\mathcal{A}}_{V}}:V(u\hskip 1.25ptv)=u\hskip
1.25ptV(v)+V(u)\hskip 1.25ptv\hskip 30.00005pt\mbox{(\\.{ })}$
In other words, the operator $V$ must be a differentiation on
${{\mathcal{A}}_{V}}\,$.
1.2. Definition: Let ${{\mathcal{A}}_{V}}$ be a dense *-subalgebra, and
$V:{{\mathcal{A}}_{V}}\to{\mathcal{A}}$ is a linear operator satisfying the
equation (˙). Then $V$ is referred to as a vector field on ${\mathcal{A}}$. If
more precision is required, the one can say that $V$ is a vector field on
${\mathcal{A}}$ defined in the subalgebra ${{\mathcal{A}}_{V}}\,$. A vector
field which commutes with * will be referred to as a real vector field.
1.3. Theorem: Given a real vector field $V$ defined in
${{\mathcal{A}}_{V}}\,$, an $\varepsilon>0$ and a solution of the equation (!)
for $t\in[0,\varepsilon)$. Then:
* •
The solution is a family of *-endomorphisms of ${{\mathcal{A}}_{V}}$ (and,
hence, of ${\mathcal{A}}$ by continuity).
* •
The solution, in given conditions, is unique.
* •
The solution extends to any $t\in[0,+\infty)$.
* •
The solution is continuous by both $q$ (on all ${\mathcal{A}}$) and $t$. In
other words, ${E^{V}}_{t}$ is continuous in the (strong) operator topology.
Note: The restriction that solutions are looked for in an operator space, not
as individual trajectories, allows to bypass uniqueness problems of ordinary
differential equations.
Proof. From the property (˙) of $V$ follows a differential equality for a
solution $E$ of (!):
$\forall u,v\in{{\mathcal{A}}_{V}}:{\frac{d}{dt}}E_{t}(u\hskip
1.25ptv)={\frac{d}{dt}}(E_{t}(u)\hskip 1.25ptE_{t}(v))$
By integration of it by $t$ from $t=0$ we get that $E_{t}(u\hskip 1.25ptv)$
always equals to $E_{t}(u)\hskip 1.25ptE_{t}(v)$ in ${{\mathcal{A}}_{V}}$,
which in turn extends to all ${\mathcal{A}}$ by continuity. Conservation of *
is proved similarly from the condition that $V$ is real. Suppose two solutions
$E$ and $\tilde{E}$. Then, for any $\kappa\in{\mathbb{C}}$: $(1-\kappa)\hskip
1.25ptE+\kappa\hskip 1.25pt\tilde{E}$ is also a solution of (!), each of which
we already know to be a family of endomorphisms. For given $t$ the dependence
on $\kappa$ can either define a complex line or a fixed endomorphism (if there
is no dependence). But complex lines of *-endomorphisms are impossible: either
because they preserve *, or because each of them has the norm not greater than
1. Hence, for any $t$: $E_{t}=\tilde{E}_{t}$.
Extension to all $[0,+\infty)$ is made by composition, from the Archimedean
property of real numbers and uniqueness of the solution. Continuity by $t$ for
any $q\in{\mathcal{A}}$ can be proved from density of ${{\mathcal{A}}_{V}}$
and $|E_{t}(q)|\leq|q|$ inequality, for example, by direct verification of
$\varepsilon$–$\delta$ definition of continuity. []
Hence, for any real vector field $V$ there are only two possibilities: either
it defines a semigroup of endomorphisms of ${\mathcal{A}}$, or the equation
(!) has no solution anywhere for $t>0$. The former case will be referred to as
“$V$ generates a semigroup”, or “$V$ is semigroup-generating”, and this
semigroup will be denoted as ${E^{V}}$. Informally, ${E^{V}}_{t}$ is the
exponential of $t\hskip 1.25ptV\,$.
The only obvious property of the set of semigroup-generating real vector
fields defined in ${{\mathcal{A}}_{V}}$ is that this set is a cone.
1.4. Proposition: If a vector field $V$ generates a semigroup, then it
commutes with all its endomorphisms:
$\forall t\geq 0:\ V\operatorname{\circ}{{E^{\displaystyle
V}}_{{}_{\displaystyle\ t}}}={{E^{\displaystyle V}}_{{}_{\displaystyle\
t}}}\operatorname{\circ}V\,.$
Proof: First of all, sides of this equality belong to the same space of
operators from ${{\mathcal{A}}_{V}}$ to ${\mathcal{A}}$, because ${E^{V}}$
preserves ${{\mathcal{A}}_{V}}\,$. From (!):
$V\operatorname{\circ}{{E^{\displaystyle V}}_{{}_{\displaystyle\
t}}}=\frac{d}{d\tau}{{E^{\displaystyle V}}_{{}_{\displaystyle\
\tau}}}|_{\tau=t}={{E^{\displaystyle V}}_{{}_{\displaystyle\
t}}}\operatorname{\circ}\left(\frac{d}{d\tau}{{E^{\displaystyle
V}}_{{}_{\displaystyle\ \tau}}}|_{\tau=0}\right)={{E^{\displaystyle
V}}_{{}_{\displaystyle\ t}}}\operatorname{\circ}V$
where the middle equality follows from the fact that ${E^{V}}$ is a real-
parametrized semigroup of (linear) endomorphisms. The derivative is understood
as pointwise on elements of the *-subalgebra ${{\mathcal{A}}_{V}}$. Or, more
“scientifically”, in the strong operator topology from ${{\mathcal{A}}_{V}}$
to ${\mathcal{A}}\,$. []
1.5. Example: let $M$ be a real ${\mathcal{C}}^{1}$-smooth manifold. We’ll
denote as ${\mathcal{C}}_{0}(M)$ the algebra of continuous complex-valued
functions vanishing at infinity with pointwise operations multiplication and *
(as the complex conjugation). “Vanishing at infinity” means that
$|f(\cdot)|\geq\varepsilon$ must be compact for any $\varepsilon>0$ – this
property is important to get a C*-algebra on a manifold which is not compact.
Then any vector field $V_{M}$ on $M$ translates to the vector field on
${\mathcal{C}}_{0}(M)$ defined in the *-subalgebra ${\mathcal{C}}_{0}^{1}(M)$
of continuously differentiable functions (vanishing at infinity with the
derivative, if needed) by tautological pointwise scalar product
$V_{{\mathcal{C}}^{1}}(f)(\xi):=(df)(\xi)\hskip 1.25ptV_{M}(\xi)$
of the 1-form of derivative of $f$ and the given vector field on $M$. If the
differential equation $\dot{\xi}=V(\xi)$ has for $t>0$ the family of solutions
depending continuously on the initial condition $x(0)$, namely,
$F:M\times[0,+\infty)\to M$ such that
$\forall\xi_{0},t:\frac{\partial}{\partial t}F(\xi_{0},t)=V(F(\xi_{0},t));\
F(\xi_{0},0)=\xi_{0}\,,$
then the corresponding endomorphism semigroup of ${\mathcal{C}}_{0}(M)$ is
expressed as:
$({E^{V}}_{t}(f))(\xi)=f(F(\xi,t))$
This admits generalizations beyond ${\mathcal{C}}^{1}$-smoothness (say, to
functions with cusps in a point where $V$ is not defined) with appropriate
tuning of the *-subalgebra.
1.6. Theorem: for the commutative C*-algebra ${\mathcal{C}}_{0}({\mathbb{R}})$
of continuous functions vanishing at infinity and its *-subalgebra
${\mathcal{C}}_{0}^{1}({\mathbb{R}})$ of continuously differentiable
functions, the set of semigroup-generating real vector fields defined in
${\mathcal{C}}_{0}^{1}({\mathbb{R}})$ is not convex.
Proof:
$V_{1}=(\sqrt[3]{x}+1)\frac{d}{dx},\ V_{2}=(\sqrt[3]{x}-1)\frac{d}{dx}$
Both vector fields generate semigroups, and even groups.
For $x>-1$ it may be expressed with a ${\mathcal{C}}^{1}$-substitution:
$y:=\int\frac{1}{V_{1}}=\int\frac{dx}{\sqrt[3]{x}+1}=3(\ln(\sqrt[3]{x}+1)-\sqrt[3]{x}+\frac{1}{2}x^{\frac{2}{3}}),\
y\in{\mathbb{R}}$
which gives $V_{1}={d}/{dy}$ and $({E^{V_{1}}}_{t}(f))(y)=f(y+t)$. Note that
the ${\mathcal{C}}^{1}$-smoothness of $y$ relatively to $x$ in $x=y=0$ is not
obvious in the given expression, but can be verified by differentiation by
$x$. The similar expression (infinitely smooth this time) exists for the
action of ${E^{V_{1}}}$ on $x<-1$. $V_{2}$ gives same substitutions save for
replacement of $x$ with $-x$ and $x>-1$ with $x<1$.
Red: graphs $y=V(x)\hskip 1.125ptdx$ for $V_{1},V_{2}\,$.
Purple: integral curves for $V_{1}\,$.
Indeed, the field
$\frac{V_{1}+V_{2}}{2}=\sqrt[3]{x}\frac{d}{dx}$
is not a semigroup-generating, since the differential equation
$\dot{x}=\sqrt[3]{x}$ with the initial condition $x(0)=0$ has many solutions
at $t>0$, for example $x=\pm(\frac{2}{3}t)^{\frac{3}{2}}$, where the solution
with the “+” sign is continuous with solutions for $x(0)>0$ and one with the
“$-$” sign is continuous with solutions for $x(0)<0$. This obliges
$({E^{(V_{1}+V_{2})/2}}_{t}(f))(0)$ to take values of $f$ simultaneously from
positive and negative points, hence such endomorphism is not possible. []
Certainly, the same is true for the algebra ${\mathcal{C}}_{0}$ of any real
interval (of whatever type: closed or open).
$y=V(x)\hskip 1.125ptdx$ for the red curve. $y=t$ for purple integral curves.
Replacement of ${\mathcal{C}}^{1}$ with ${\mathcal{C}}^{k}$ or
${\mathcal{C}}^{\infty}$ does not change much.
The set of semigroup-generating vector fields is not convex even in the
simplest case of infinite-dimensional C*-algebra and the most natural choice
of a dense *-subalgebra: smooth functions. Henceforth, one should not expect
that this set would be convex in more complicated algebras. This theorem means
that if we are, for some reason, interested in semigroup, then it cannot be
constructed easily from generic real vector fields, even if they generate
semigroups.
## 2 Endomorphism semigroups and gauge groups
2.1. Note that any element $g$ of the algebra defines the vector field,
defined in all ${\mathcal{A}}$, by adjoint representation as a Lie algebra:
$\operatorname{ad}_{g}(\cdot):={[{g},{\cdot}]}$. Somewhat counter-intuitively,
skew-Hermitian ($g+{g}^{*}=0$, i.e. imaginary) elements define real vector
fields,111 Apparently, the source of well-knows disparateness in
real/imaginary factors between mathematicians (who follow algebraic intuition)
and physicists (who might follow differential-geometric intuition). and, vice
versa, Hermitian elements define “imaginary” (i.e. exchanging Hermitian and
skew-Hermitian elements) vector fields, because the commutator of two
Hermitian elements is skew-Hermitian.
Any real vector field on ${\mathcal{A}}$ defined by $\operatorname{ad}_{g}$
(i.e. $g$ must be skew-Hermitian: $g+{g}^{*}=0$) generates a group of
*-endomorphisms. If the algebra has “1”, then this group can be obtained via
exponentiation. A C*-algebra is a Banach algebra, and we can compute the
exponent of an element, which of skew-Hermitian element will be unitary:
${(\exp(g))}^{*}\hskip 1.25pt\exp(g)=1=\exp(g)\hskip 1.25pt{(\exp(g))}^{*}\,$.
Unitary elements define (internal) automorphisms of the algebra by
conjugation;222 If there is no “1” in ${\mathcal{A}}$, then such automorphisms
also exist, although this is not a conjugation in such case. see below.
2.2. From here onwards we’ll assume that ${\mathcal{A}}$ is a unital algebra.
Let ${\mathcal{G}}$ be a closed subset of unitary elements, i.e. such $u$ that
${u}^{*}\hskip 1.25ptu=1=u\hskip 1.25pt{u}^{*}$ , which forms a group under
multiplication. Then $\operatorname{Ad}_{u}(\cdot):=u\hskip 1.25pt\cdot\hskip
1.25pt{u}^{*}$ is its action on ${\mathcal{A}}$ by C*-automorphisms. Denote as
${\mathfrak{g}}$ the tangent space of ${\mathcal{G}}$ in 1. It is easy to
check that it is a closed linear subspace,333 Over real numbers
${\mathfrak{g}}$ is a closed subspace of skew-Hermitian elements
($g+{g}^{*}=0$). Of course, the complex tangent space is the complexification
of the real tangent space. that ${{\mathfrak{g}}^{\mathbb{C}}}$ is closed
under the commutator operation, but it is not, generally, a *-subalgebra.
Though, exponents of elements of ${{\mathfrak{g}}^{\mathbb{R}}}$ belong to
${\mathcal{G}}$, because it is topologically closed.
2.3. Definition: Let $E$ be an endomorphism semigroup, and ${\mathcal{G}}$ be
a closed connected group of unitary elements. If $E_{t}$ preserves the tangent
space ${\mathfrak{g}}\ $444 It is not important whether $E$ has to preserve
real or complex tangent space ${\mathfrak{g}}$, because any skew-Hermitian
element always maps to a skew-Hermitian one by a *-endomorphism. for any
$t>0$, then we’ll refer to ${\mathcal{G}}$ as a gauge group for $E$.
Note that a *-endomorphism (if it is not an isomorphism) usually does not map
${\mathcal{G}}$ to ${\mathcal{G}}$, even if it preserves its tangent space
${\mathfrak{g}}$.
Definition: Let $V$ be a real vector field generating an endomorphism
semigroup ${E^{V}}$, ${\mathcal{G}}$ be its gauge group, and let the
intersection ${\mathfrak{g}}\cap{{\mathcal{A}}_{V}}$ be dense in
${\mathfrak{g}}$. Then we’ll refer to ${\mathcal{G}}$ as a gauge group for the
vector field $V$.
In this case, for any $g\in{\mathfrak{g}}\cap{{\mathcal{A}}_{V}}$ the
derivative ${\frac{d}{dt}}{E^{V}}_{t}(g)|_{t=0}$ (which equals to $V(g)$ by
1.3.) belongs to ${\mathfrak{g}}$, which means that $\forall
g\in{\mathfrak{g}}\cap{{\mathcal{A}}_{V}}:V(g)\in{\mathfrak{g}}$. The author
currently does not know whether this condition on $V$ is sufficient to have a
gauge group ${\mathcal{G}}$.
2.4. Theorem: Let $V$ be a semigroup-generating real vector field with a gauge
group ${\mathcal{G}}$. Then:
* •
The family $\operatorname{Ad}_{u}\operatorname{\circ}{E^{V}}_{t}$ of
*-endomorphisms, where $u\in{\mathcal{G}},\ t\geq 0\,$, is a semigroup.
* •
For every $g\in{\mathfrak{g}}$ the sum of vector fields
$V+\operatorname{ad}_{g}$ generates an endomorphism semigroup which lies
within aforementioned two-parameters semigroup.
Proof: Due to associativity of the composition, to prove the first statement
it’s sufficient to demonstrate that
$\forall t\geq 0,\ \forall\tilde{u}\in{\mathcal{G}}:\exists
u\in{\mathcal{G}}:{\operatorname{Ad}_{\displaystyle\,u}}\operatorname{\circ}{{E^{\displaystyle
V}}_{{}_{\displaystyle\ t}}}={{E^{\displaystyle V}}_{{}_{\displaystyle\
t}}}\operatorname{\circ}{\operatorname{Ad}_{\displaystyle\,\tilde{u}}}$
Since ${\mathcal{G}}$ is connected, we can check it only in a neighborhood of
1, say, for $||{\tilde{u}}-1||<1\,$ where natural logarithm is defined by its
Taylor series:
$\ln(1+x)=x-\frac{x^{2}}{2}+\frac{x^{3}}{3}-\frac{x^{4}}{4}+\cdots$
After a trivial observation that logarithm of $\tilde{u}$ has to belong to
${\mathfrak{g}}$ it becomes obvious that $u=\exp({E^{V}}_{t}(\ln{\tilde{u}}))$
is the solution – it directly follows from the fact that ${E^{V}}_{t}$ is a
*-endomorphism. We also got that the dependence of $u$ on both ${\tilde{u}}$
and $t$ is continuous.
Now we can realize that the semigroup generated by $V+\operatorname{ad}_{g}$
is expressed as
${{E^{\displaystyle\ V+\operatorname{ad}_{g}}}_{{}_{\displaystyle\
t}}}={\operatorname{Ad}_{\displaystyle\,\exp(\int_{0}^{t}{{E^{\displaystyle
V}}_{{}_{\displaystyle\ t-\tau}}}(g)\hskip 1.25ptd\tau)_{+}}}\
\operatorname{\circ}{{E^{\displaystyle V}}_{{}_{\displaystyle\ t}}}\hskip
30.00005pt\mbox{(\char 126\relax)}$
where $\exp(\ldots)_{+}$ is the ordered exponential.[3] For $0\leq s\leq t\,$,
it satisfies the identity:
$\exp(\int_{0}^{t}{{E^{\displaystyle V}}_{{}_{\displaystyle\
t-\tau}}}(g)\hskip 1.25ptd\tau)_{+}=\exp(\int_{0}^{s}{{E^{\displaystyle
V}}_{{}_{\displaystyle\ s-\tau}}}(g)\hskip 1.25ptd\tau)_{+}\hskip
1.25pt\exp(\int_{0}^{t-s}{{E^{\displaystyle V}}_{{}_{\displaystyle\
t-\tau}}}(g)\hskip 1.25ptd\tau)_{+}$
On the other hand, since ${E^{V}}_{s}$ is an endomorphism of the algebra, we
have:
${\operatorname{Ad}_{\displaystyle\,\exp(\int_{0}^{t-s}{{E^{\displaystyle
V}}_{{}_{\displaystyle\ t-\tau}}}(g)\hskip
1.25ptd\tau)_{+}}}={\operatorname{Ad}_{\displaystyle\,\exp({{E^{\displaystyle
V}}_{{}_{\displaystyle\ s}}}\,\int_{0}^{t-s}{{E^{\displaystyle
V}}_{{}_{\displaystyle\ t-s-\tau}}}(g)\hskip 1.25ptd\tau)_{+}}}=$
$={{E^{\displaystyle V}}_{{}_{\displaystyle\
s}}}\operatorname{\circ}{\operatorname{Ad}_{\displaystyle\,\exp(\int_{0}^{t-s}{{E^{\displaystyle
V}}_{{}_{\displaystyle\ t-s-\tau}}}(g)\hskip 1.25ptd\tau)_{+}}}$
With it, we immediately see that compositions of *-endomorphisms from (~)
behave as they must do it in a semigroup. So, ordering in the “abnormal”
direction, higher semigroup-parameter factors to right and lower to left, is
necessary to put factors with semigroup parameter close to $t$ adjacent to the
${E^{V}}_{t}$ endomorphism.
Initial condition from (!) is evident. The only thing we need to complete the
proof of (~) is to compute its derivative. Since we already know that the
formula (~) defines a semigroup, we can do it only at $t=0\,$:
${\frac{d}{dt}}\left({\operatorname{Ad}_{\displaystyle\,\exp(\int_{0}^{t}{{E^{\displaystyle
V}}_{{}_{\displaystyle\ t-\tau}}}(g)\hskip 1.25ptd\tau)_{+}}}\
\operatorname{\circ}{{E^{\displaystyle V}}_{{}_{\displaystyle\
t}}}\right)_{t=0}=$
$={\operatorname{ad}_{\displaystyle\,{\frac{d}{dt}}\exp(\int_{0}^{t}{{E^{\displaystyle
V}}_{{}_{\displaystyle\ t-\tau}}}(g)\hskip
1.25ptd\tau)_{+}|_{t=0}}}+V={\operatorname{ad}_{\displaystyle\,g}}+V\ .$
[]
2.5. Commutator of vector fields cannot be defined straightly since we defined
the codomain of $V$ to be a broader space than its domain. Though, if there
are two fields $Y:{\mathcal{A}}_{Y}\to{\mathcal{A}}$ and
$Z:{\mathcal{A}}_{Z}\to{\mathcal{A}}$ and there exists such dense *-subalgebra
${\mathcal{A}}_{YZ}$ that is a subset of both ${\mathcal{A}}_{Y}$ and
${\mathcal{A}}_{Z}$, $Y$ maps it into ${\mathcal{A}}_{Z}$ and $Z$ maps it into
${\mathcal{A}}_{Y}$, then the commutator may be defined in this narrower
*-subalgebra:
${[{Y},{Z}]}:=Y\operatorname{\circ}Z-Z\operatorname{\circ}Y$
The (˙) condition (see 1.1.) for ${[{Y},{Z}]}$ is satisfied:
${[{Y},{Z}]}(u\hskip 1.25ptv)=Y(u\hskip 1.25ptZ(v)+Z(u)\hskip
1.25ptv)-Z(u\hskip 1.25ptY(v)+Y(u)\hskip 1.25ptv)=$ $={\scriptstyle u\hskip
0.875ptY(Z(v))+Y(u)\hskip 0.875ptZ(v)+Z(u)\hskip 0.875ptY(v)+Y(Z(u))\hskip
0.875ptv-u\hskip 0.875ptZ(Y(v))-Z(u)\hskip 0.875ptY(v)-Y(u)\hskip
0.875ptZ(v)-Z(Y(u))\hskip 0.875ptv\ }=$ $=u\hskip
1.25pt{[{Y},{Z}]}(v)+{[{Y},{Z}]}(u)\hskip 1.25ptv\,$
Since real vector fields commute with *, it is obvious that the commutator of
two real vector fields is real; see notes in 2.1.
2.6. Lemma: Let $V_{1}$ be a semigroup-generating real vector field defined in
a dense *-subalgebra ${{\mathcal{A}}_{V}}$. Let $V_{2}$ be a vector field such
that the commutator ${[{V_{1}},{V_{2}}]}$ is defined in some (narrower) dense
*-subalgebra ${\mathcal{A}}_{V_{1}V_{2}}$ and is a bounded operator. Then:
${[{{{E^{\displaystyle V_{1}}}_{{}_{\displaystyle\
t}}}},{V_{2}}]}=\int\limits_{0}^{t}{{E^{\displaystyle
V_{1}}}_{{}_{\displaystyle\
\tau}}}\operatorname{\circ}{[{V_{1}},{V_{2}}]}\operatorname{\circ}{{E^{\displaystyle
V_{1}}}_{{}_{\displaystyle\ t-\tau}}}\hskip 1.25ptd\tau$
Note that if ${[{V_{1}},{V_{2}}]}={\operatorname{ad}_{\displaystyle\,c}}$ for
certain $c\in{\mathcal{A}}$, then
$\int\limits_{0}^{t}{{E^{\displaystyle V_{1}}}_{{}_{\displaystyle\
\tau}}}\operatorname{\circ}{[{V_{1}},{V_{2}}]}\operatorname{\circ}{{E^{\displaystyle
V_{1}}}_{{}_{\displaystyle\ t-\tau}}}\hskip
1.25ptd\tau={\operatorname{ad}_{\displaystyle\,\int\limits_{0}^{t}{{E^{\displaystyle
V_{1}}}_{{}_{\displaystyle\ \tau}}}(c)\hskip
1.25ptd\tau}}\operatorname{\circ}{{E^{\displaystyle
V_{1}}}_{{}_{\displaystyle\ t}}}$
because ${E^{V_{1}}}$ consists of endomorphisms of the algebra.
Proof: First we have to ensure that the operator family
${[{{E^{V_{1}}}_{t}},{V_{2}}]}$ is continuous by $t$ in the strong operator
topology from ${\mathcal{A}}_{V_{1}V_{2}}$ to ${\mathcal{A}}\,$. It is obvious
for the ${E^{V_{1}}}_{t}\operatorname{\circ}{V_{2}}$ term, but for
${V_{2}}\operatorname{\circ}{E^{V_{1}}}_{t}$ we have to check that
${E^{V_{1}}}_{t}$ preserves ${\mathcal{A}}_{V_{1}V_{2}}$, which follows from
the boundness of ${[{V_{1}},{V_{2}}]}$.555 More strictly, given
${\mathcal{A}}_{V_{1}V_{2}}\,$ might have to be extended to some broader
*-algebra, such that both ${V_{1}}\operatorname{\circ}{V_{2}}$ and
${V_{2}}\operatorname{\circ}{V_{1}}$ are defined and the (new) algebra is
preserved by ${E^{V_{1}}}$. Then, the formula can be checked by interpolation
between ${V_{2}}\operatorname{\circ}{E^{V_{1}}}_{t}$ and
${E^{V_{1}}}_{t}\operatorname{\circ}{V_{2}}$ by
${E^{V_{1}}}_{\tau}\operatorname{\circ}{V_{2}}\operatorname{\circ}{E^{V_{1}}}_{t-\tau}$
(say, for $\tau=k\hskip 1.25ptt/n,\ k=1\ldots n-1$, like in Riemann integral),
using continuity of ${E^{V_{1}}}_{t}$ by $t$. []
2.7. Theorem: In all conditions of 2.6., let $V_{2}$ generate its semigroup,
let both semigroups have a gauge group ${\mathcal{G}}$, and $\exists
c\in{{\mathfrak{g}}^{\mathbb{R}}}:{[{V_{1}},{V_{2}}]}={\operatorname{ad}_{\displaystyle\,c}}\,$.666
Note that “$\operatorname{ad}$” of any element of ${\mathcal{A}}$ is bounded.
Then ${E^{V_{1}}}_{t}\operatorname{\circ}{E^{V_{2}}}_{s}$ and
${E^{V_{2}}}_{s}\operatorname{\circ}{E^{V_{1}}}_{t}$ differ only by left
composition with “$\operatorname{Ad}$” of an element of ${\mathcal{G}}$.
From this theorem, remembering 2.4., immediately follows that the family
$\operatorname{Ad}_{u}\operatorname{\circ}{E^{V_{2}}}_{s}\operatorname{\circ}{E^{V_{1}}}_{t}$
of *-endomorphisms, where $u\in{\mathcal{G}},\ t\geq 0,\ s\geq 0\,$, is a
semigroup. In a more abstract sense, this means a principal bundle[4] over
$[0,+\infty)\times[0,+\infty)$ with a semigroup structure corresponding to
addition of 2-vectors, with the representation by *-endomorphisms and with the
left action777 The standard definition of a principal bundle requires a right
action, but for this construct, a left action is better suited. of
${\mathcal{G}}$. It is also possible to prove the statement which generalizes
the second statement of the theorem of 2.4., but we’ll made such
generalisation in Section 4 for a more general case.
Proof: Aforementioned “$\operatorname{Ad}$” term can be obtained by solving a
differential equation (by $s$) in ${\mathcal{G}}$, using Lemma 2.6. and the
fact that $V_{2}$ preserves the gauge algebra. []
## 3 Constructions of vector fields
This section will present possible constructions of vector fields via
representations of the algebra. Although in simple cases these fields generate
semigroups, the section does not contain theorems about it.
3.1. Assume that ${\mathcal{A}}$ has a representation on $L^{2}({\mathbb{R}})$
(with variable $x\in{\mathbb{R}}$). The unbounded skew-Hermitian operator
${\mathrm{d}}=\frac{d}{dx}$ of differentiation by $x\,$888 In quantum-
mechanical parlance, ${\mathrm{d}}=i\hskip 1.25pt\hat{p}/\hbar$. is defined
on ${\mathcal{C}}^{1}$-smooth functions, which are dense in $L^{2}$. Suppose
that in some dense *-subalgebra the operator $V(q)={{\mathrm{d}}}^{*}\hskip
1.25ptq+q\hskip 1.25pt{\mathrm{d}}={[{q},{{\mathrm{d}}}]}\,$ is defined. It
means that aforementioned operator on $L^{2}({\mathbb{R}})$ is bounded and
belongs to the representation. For example, such *-subalgebra would be
${\mathcal{C}}^{1}_{0}({\mathbb{R}})$ for
${\mathcal{A}}={\mathcal{C}}_{0}({\mathbb{R}})\,$. From the product rule for
$V$ we see that it is a vector field. If ${\mathcal{A}}$ is translation-
invariant, then $V$ generates a group of *-endomorphisms, called “shifts”.
3.2. The “translation” vector field $V(q)={{\mathrm{d}}}^{*}q+q{\mathrm{d}}$
may be expressed in a matrix form:
${\mathrm{J}}:=\left(\begin{matrix}\mathbf{1}\\\
{\mathrm{d}}\end{matrix}\right);\ \ \ \
{}{{\mathrm{J}}}^{*}=\left(\begin{matrix}\mathbf{1}&{{\mathrm{d}}}^{*}\end{matrix}\right);\
\ \ \ {}V(q)={{\mathrm{J}}}^{*}\hskip 1.25pt\left(\begin{matrix}0&q\\\
q&0\end{matrix}\right)\hskip 1.25pt{\mathrm{J}}$
Or, more complicated:
$S:=\left(\begin{matrix}0&\mathbf{1}\\\ \mathbf{1}&0\end{matrix}\right);\ \ \
\ {}V(q)={{\mathrm{J}}}^{*}\hskip 1.25ptS\hskip
1.25pt\operatorname{{\mathrm{diag}}}(q,q)\hskip 1.25pt{\mathrm{J}}$
From skew-Hermitianness of “${\mathrm{d}}$” we have ${{\mathrm{J}}}^{*}\hskip
1.25ptS\hskip 1.25pt{\mathrm{J}}=0$.
Now let us think that instead of $\operatorname{{\mathrm{diag}}}(q,q)$ we have
another representation $\pi$ of ${\mathcal{A}}$, in another Hilbert space
${\mathcal{D}}$, which commutes with certain operator $S$. Suppose that
${\mathrm{J}}:{\mathcal{H}}\to{\mathcal{D}}$ (transition operator) is such
unbounded operator with dense domain that the operator product
$V(q)={{\mathrm{J}}}^{*}\hskip 1.25ptS\hskip 1.25pt\pi(q)\hskip
1.25pt{\mathrm{J}}$
for any $q\in{{\mathcal{A}}_{V}}$ belongs to our (default) representation of
${\mathcal{A}}$ in ${\mathcal{H}}$.
This bears some resemblance to Alain Connes’[2] definition of differentiation,
by commutator with the operator $F$, although we, contrary, try to avoid
commutators.
It is easy to check that conditions
${{\mathrm{J}}}^{*}\hskip 1.25ptS\hskip 1.25pt{\mathrm{J}}=0;\hskip
40.00006pt\forall u,v\in{{\mathcal{A}}_{V}}:{[{u},{{{\mathrm{J}}}^{*}}]}\hskip
1.25ptS\hskip 1.25pt{[{{\mathrm{J}}},{v}]}=0$
are sufficient to make $V$ a vector field. Note that
${[{u},{{{\mathrm{J}}}^{*}}]}$ means
$u\operatorname{\circ}{{\mathrm{J}}}^{*}-{{\mathrm{J}}}^{*}\operatorname{\circ}\pi(u)$,
an operator from ${\mathcal{D}}$ to ${\mathcal{H}}$, and similarly
${[{{\mathrm{J}}},{v}]}:{\mathcal{H}}\to{\mathcal{D}}\,$. Also, if $S$ is
self-conjugated, then $V$ is real.
3.3. What appears after replacement of one-dimensional “${\mathrm{d}}$” with
something like Dirac operator? Suppose that ${\mathrm{J}}$ depends linearly on
the parameter from ${\mathbb{C}}^{2}$. Say, there are two operators
${\mathrm{J}}_{0},{\mathrm{J}}_{1}$ and we define
${\mathrm{J}}_{\phi}:=\phi^{L}\hskip 1.25pt{\mathrm{J}}_{L}\,.$ Then,
${V_{\phi\otimes{\phi}^{*}}}(q)={{\mathrm{J}}_{\phi}}^{*}\hskip 1.25ptS\hskip
1.25pt\pi(q)\hskip 1.25pt{\mathrm{J}}_{\phi}\,;\ \phi\in{\mathbb{C}}^{2}$
The use of tensor product symbol is motivated by an observation that the
vector field depends on parameters bilinearly:
$V_{L{L^{\prime}}}(q)={{{{\mathrm{J}}}^{*}}_{L^{\prime}}}\hskip 1.25ptS\hskip
1.25pt\pi(q)\hskip 1.25pt{{\mathrm{J}}_{L}}\,,\mbox{ or,}$
$V_{X}(q)=X^{L{L^{\prime}}}\hskip
1.25pt{{{{\mathrm{J}}}^{*}}_{L^{\prime}}}\hskip 1.25ptS\hskip
1.25pt\pi(q)\hskip 1.25pt{{\mathrm{J}}_{L}}\mbox{ where }X\mbox{ is a }2\times
2\mbox{ matrix.}$
We can get a 4-dimensional family of vector fields at the price of only
2-dimensional, because ${\mathrm{J}}$ and ${{\mathrm{J}}}^{*}$ are essentially
the same thing. Though, (for self-conjugated $S$) only fields parametrized by
Hermitian matrices will be real. Note that $\phi\otimes{\phi}^{*}$ defines a
$2\times 2$ Hermitian matrix with rank not greater than 1 and non-negative
trace. We postulated the dependence of ${\mathrm{J}}$ on 2 complex numbers,
and got a vector field depending on a $2\times 2$ matrix, which gave the
structure of (1+3)-dimensional Minkowski space known in Special Relativity.
This construction of the family of vector fields is speculative, but it leads
to interesting situation if we suppose that, in some theory, we got that
namely $V_{\phi\otimes{\phi}^{*}}={{\mathrm{J}}_{\phi}}^{*}\hskip
1.25ptS\hskip 1.25pt\pi(q)\hskip 1.25pt{\mathrm{J}}_{\phi}$ are semigroup-
generating, but not other fields in this 4-dimensional complex-linear family.
## 4 Endomorphism semigroup parametrized by a cone
4.1. The construction from the previous section gives the ${\mathbb{C}}^{2}$
of vector fields of the type $\phi\otimes{\phi}^{*}$, which is actually the
cone over a Riemann sphere, due to independence of the phase of $\phi$.
Suppose that all of those fields generate a semigroup.999 This is the case,
for example, if we defined translations in Minkowski space and our
representation is defined on functions on the cone of the past, or another
domain containing the whole cone of the past of each of its points. A shift
semigroup acts on a domain by filling “overflowed” space with zeros (not
explained in this version of the paper). Let us also suppose that those
fields and semigroups has ${\mathcal{G}}$ as a gauge group and commutators of
those fields lie in the gauge algebra.
Obviously, this construction corresponds to the geometry of the cone of the
past in the Minkowski space, where 4-coordinates and $2\times 2$ matrices
correspond as:
$X=\left(\begin{matrix}x^{0}+x^{3}&x^{1}-i\hskip 1.25ptx^{2}\\\ x^{1}+i\hskip
1.25ptx^{2}&x^{0}-x^{3}\end{matrix}\right)=x^{0}\hskip
1.25pt{\mathrm{I}}+\sum\limits_{k=1,2,3}x^{k}\hskip 1.25pt\sigma_{k}\,,\mbox{
where }\sigma_{k}\mbox{ are Pauli matrices.}$
Here, the semigroup of translations cannot generated by two 1-parameter
semigroups, so the result from 2.7. is not applicable. BTW, it cannot be
generated by any finite number of 1-parameter semigroups.
How can we generalize 2.7. for this geometry?
4.2. The spinor representation possibly gave some insight about origins of
semigroup-generating vector fields parametrized by the light cone (i.e. by
null directions) in the Minkowski space, but it is useless when we have to
prove something about semigroups. From here onwards we’ll forget anything
about spinors and think we have $n$ fields $V_{k}\,$ and some cone $\Phi$ in
${\mathbb{R}}^{n}$ such that all fields $x^{k}\hskip 1.25ptV_{k}\,,\ x\in\Phi$
generate a semigroup. For the example from 4.1. it would be
$\Phi=\left\\{\,x\in{\mathbb{R}}^{4}\,|\,x^{0}\geq 0\,,\
(x^{0})^{2}-(x^{1})^{2}-(x^{2})^{2}-(x^{3})^{2}=0\,\right\\}$
Let all these semigroups have the same gauge group ${\mathcal{G}}$.101010 This
version of the paper lacks an explanation how gauge groups may be related to
the spinor construction from Section 3. To be fixed. The condition about
commutators takes the form
${[{V_{k}},{V_{l}}]}={\operatorname{ad}_{\displaystyle\,c_{kl}}},\
c_{kl}\in{\mathfrak{g}}$
due to bilinearity of the commutator. Then:
* •
There exists a principal bundle $F\to{\hat{\Phi}}$ (where $\hat{\Phi}$ is a
convex hull) with the structure group ${\mathcal{G}}$ and with a semigroup
structure on it which corresponds to vector addition in $\hat{\Phi}$.
* •
There exists its representation ${\mathcal{E}}$ by *-endomorphisms of
${\mathcal{A}}$ which include all semigroups ${E^{\,x^{k}\hskip
0.875ptV_{k}}},\ x\in\Phi$ such that ${E^{\,x^{k}\hskip 0.875ptV_{k}}}_{t}$ is
parametrized by the point of $F$ corresponding to the point $t\hskip
1.25ptx^{k}$ of $\Phi$.
* •
The fiber over the 0 point is the group ${\mathcal{G}}$ itself with its group
structure matching the semigroup structure of the bundle. So, endomorphisms
from the same fiber differ only by left composition with an
$\operatorname{Ad}_{u}$.
* •
All (other) vectors $x\in\hat{\Phi}$ have their semigroups ${E^{\,x^{k}\hskip
0.875ptV_{k}}}$ included to $F$ by the same way.
Proof: By applying the theorem of 2.7. to two-dimensional subspaces of
${\mathbb{R}}^{n}$ generated by pairs of vectors from $\Phi$ we can construct
the bundle over such subspaces. Where subspaces intersect, consistency is
provided by existence and uniqueness of the corresponding real-parametrized
endomorphism semigroup. Iterating this construction $n-1$ times (or directly
applying 2.7.-like reasonings to $n$-tuples of vectors from $\Phi$) we extend
it to all $\hat{\Phi}$.
The vector fields $V_{k}$ (or, in other words, the field linearly parametrized
by an $n$-dimentional real space) actually define the connection on our
bundle, where $c_{kl}$ is its 2-form of curvature. The last (fourth) statement
is a manifestation of the fact that such connection can be integrated along
the path. []
## References
* [1] Dixmier, Jaques. Les C*-algèbres et leurs représentations. Paris, 1969
* [2] Connes, Alain. Noncommutative Geometry. Academic Press, San Diego, CA, 1994, 661 p., ISBN 0-12-185860-X.
* [3] http://en.wikipedia.org/wiki/Ordered_exponential
* [4] http://en.wikipedia.org/wiki/Principal_bundle
|
arxiv-papers
| 2012-12-03T19:27:28 |
2024-09-04T02:49:38.838655
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Innocenti Maresin",
"submitter": "Innocenti Maresin",
"url": "https://arxiv.org/abs/1212.0500"
}
|
1212.0510
|
# Phase retrieval by power iterations
Stefano Marchesini Advanced Light Source, Lawrence Berkeley National
Laboratory, Berkeley, CA 94720
###### Abstract
I show that the power iteration method applied to the phase retrieval problem
converges under special conditions. One is given the relative phases between
small non-overlapping groups of pixels of a recorded intensity pattern, but no
information on the phase between the groups of pixels. Numerical tests show
that the inverse block iteration recovers the solution in 1 iteration.
††preprint: LBNL-5939E
## I Introduction
Given a set of intensity measurements, $a^{2}\in\mathbb{R}^{M}$, an unknown
object $\psi\in\mathbb{C}^{N}$ represented by a complex $n\times n$ image
($N=n^{2}$), a known “illumination matrix” or support matrix ${\bf Q}$
($\mathbb{C}^{M\times N}$ matrix), a known propagation operator (typically
one, or a stack of 2D FFT operators) $\bm{F}$ of dimension $M\times M$ and set
of frames ${\bf z}\in\mathbb{C}^{M}$, which are related by:
${\bf z}=\bm{F}{\bf Q}\psi,\qquad|{\bf z}|=a.$
Our goal is to find $\psi$ or the intermediate variable ${\bf z}$, given
$\bm{F}$, ${\bf Q}$ and $a$. To do so, we need to find a phase $\phi$ such
that ${\bf z}=a\phi$ is in the range of $\bm{F}{\bf Q}$.
We can eliminate $\psi$ by using the operator $P_{\bf Q}$ to project a vector
${\bf z}$ onto the range of $\bm{F}{\bf Q}$:
$\displaystyle P_{\bf Q}$ $\displaystyle=$ $\displaystyle\bm{F}{\bf Q}({\bf
Q}^{\ast}{\bf Q})^{-1}{\bf Q}^{\ast}\bm{F}^{\ast}.$ (1)
Which ensures that the unknown vector $\psi$ can be obtained from the frame
${\bf z}$ by $\psi=({\bf Q}^{\ast}{\bf Q})^{-1}{\bf Q}^{\ast}\bm{F}^{\ast}{\bf
z}$.
A popular approach is to find a vector $z$ such that:
$\displaystyle\left\|[I-P_{Q}]{\bf z}\right\|^{2}$ $\displaystyle=$
$\displaystyle 0,$ (2) $\displaystyle\left\|[I-P_{a}]{\bf z}\right\|^{2}$
$\displaystyle=$ $\displaystyle\||{\bf z}|-a\|^{2}=0,$ (3)
are satisfied simultaneously, and where the Fourier magnitude projection
$P_{a}$ when applied to a vector ${\bf z}$, yields:
$\displaystyle P_{{\bf a}}{\bf z}=\bm{a}\frac{{\bf z}}{|{\bf
z}|},\quad\bm{a}=\mbox{Diag}\left(a\right)\ \ $ (4)
where division are intended as element-wise operations Marchesini _et al._
(2012); Marchesini (2007).
## II Phase optimization
Here we want to minimize Eq. 2 w.r.t. a phase vector $\phi$
($\phi_{i}^{\ast}\phi_{i}=1,\forall i$). That is, we want to find:
$\displaystyle\arg\min_{\phi}$ $\displaystyle\left\|[I-P_{\bf
Q}]\mathrm{Diag}(a)\phi\right\|^{2},$ $\displaystyle\arg\min_{\phi}$
$\displaystyle\phi^{\ast}\bm{a}\left[I-P_{\bf Q}\right]\bm{a}\phi,$ (5)
I discuss three approaches that relax the phase modulus condition
($\phi_{i}^{\ast}\phi_{i}=1\quad\forall i$) to synchronize the relative
phases.
### Power iteration
By changing variable ${\bf z}=\bm{a}\phi$, we write:
$\displaystyle\arg\min_{\bf z}\quad{\bf z}^{\ast}\left(I-P_{\bf Q}\right){\bf
z}$ (6)
By relaxing $(a_{i}^{2}|\phi_{i}|^{2}=a_{i}^{2},\forall i)$ and using $\|{\bf
z}\|=\|\bm{a}\phi\|^{2}=\|\bm{a}\|^{2}$, we can re-write Eq. (6) as finding
the eigenvector with largest eigenvalueSinger (2011). Since $\|{\bf
z}\|=\|\bm{a}\phi\|=\|a\|$ is constant, we rewrite Eq, (5) as:
$\displaystyle\arg\max_{\bf z}\quad{\bf z}^{\ast}P_{\bf Q}{\bf z},\quad$ (7)
we apply one step of power iteration:
$\displaystyle\nu^{\ell+1}=P_{\bf Q}{\bf z}$ (8)
We then form a projection on the unit torus to ensure that
$|\nu^{\ell+1}_{i}|=1$ (or $|{\bf z}_{i}|=a_{i}$) by element-wise
normalization:
$\displaystyle z^{(\ell+1)}=\bm{a}\frac{\nu}{|\nu|}$ $\displaystyle=$
$\displaystyle P_{a}P_{\bf Q}z^{(\ell)}$
Here we have obtained the classical alternating projection method, which is
known to stagnate with classical CDI but to converge (slowly) in ptychographic
imaging.
### Greedy phase optimization
Since the diagonal term ${\bf z}^{\ast}\mbox{Diag}\left(P_{\bf Q}\right){\bf
z}$ is also independent on the choice of $\phi$ (for $|\phi_{i}|=1$), one can
remove it when computing the power iteration:
$\displaystyle\nu^{(\ell+1)}=[P_{Q}-\mbox{Diag}\left(P_{Q}\right)]{\bf
z}^{(\ell)}$ (9)
After we apply the projection of $\phi^{\ell+1}$ to the unit torus, we obtain
the following update Waldspurger _et al._ (2012):
$z^{(\ell+1)}=P_{a}\left(P_{\bf
Q}-\mbox{Diag}\left(P_{Q}\right)\right)z^{(\ell)}$
In classical CDI, $P_{Q_{ii}}={\|{\bf Q}\|^{2}\over\|\bm{1}\|^{2}}$ is simply
the sum of the support volume (or area) normalized by the oversampled volume,
in ptychographic imaging $\mbox{Diag}\left(P_{\bf Q}\right)$ is the ratio of
intensities $P_{Q_{ii}}={\|Q_{i}\|^{2}\over{\|{\bf Q}\|^{2}}}$ for every pixel
of a frame $i$ generate by a submatrix $Q_{i}$. At the first iteration, using
data generated from the object in Fig. 6 with a random phase as a starting
guess, Eq. (9) appears to out-perform Eq. (6), however the two methods
converge to similar local minimum within ten iterations. The relaxations in
Eqs. (6,9) are similar. By removing diagonal components we change the
relaxation. In Eq. (6) we have $\|\bm{a}^{2}\phi\|$ constant, in Eq. (9)
$\|[1-\mbox{Diag}\left(P_{\bf Q}\right)]\bm{a}^{2}\phi\|$ is constant. However
$\mbox{Diag}\left(P_{Q}\right)$ is often constant and the two relaxations are
equivalent, giving more weight to high intensity values.
### Inverse iterationMarchesini _et al._ (2012).
If we solve the minimization problem (Eq. (5) with a different relaxation,
setting $\|\phi\|=k$ to a constant, we re-write the problem as
$\displaystyle\arg\max\phi^{\ast}H^{-1}\phi,\quad H={\bf a}[I-P]{\bf a}$ (10)
and apply the power iteration:
$\displaystyle H\nu^{(\ell+1)}=\phi^{\ell}$ (11)
This method is commonly referred to as inverse iteration and it is used to
find the smallest eigenvector of a matrix. We note however that any $\nu$
written in the following way:
$\nu=\left({\bf a}^{-1}P_{\bf Q}{\bf a}\right){\bf x}$ (12)
is an eigenvector with 0 eigenvalue of ${\bf a}(I-P_{\bf Q}){\bf a}$,
therefore the inverse iteration method cannot be applied directly.
When $H$ is singular, then instead of power iteration we may want to find the
smallest modification of the phase that is in the null space of $H$, which we
can write it as a re-weighted LSQ problem of the form:
$\displaystyle\arg\min_{\bf z}\left\|\frac{1}{{\bf a}}({\bf z}-{\bf
z}^{\ell})\right\|,\quad\text{s.t. ${\bf z}=P_{\bf Q}{\bf z}$}$ (13)
which provides a search direction toward the solution that differs from
standard projection algorithms. Another approach is to include additional
restrictions on $\nu$ before applying the inverse iteration as described in
the following.
### Inverse block iteration
In Marchesini _et al._ (2012) it was observed that computing the exact
solution to Eq. (11) after “binning”, or fixing the relative phase between
groups of pixels, improved convergence rate in large scale ptychographic
imaging. The use eigensolvers for the interferometric case was also suggested
in Alexeev _et al._ (2012), for the connection Laplacian of a graph.
Let us introduce a binning matrix $\bm{T}^{\ast}$ composed of a series of
masks $T_{i}$ that integrate over a region of dimension $M/k$ of the data (in
Fourier domain). For example, we can partition our data in 3, creating a tall
matrix of dimension $M\times 3$:
$\bm{T}=\left(\begin{array}[]{c}{\bm{1}_{M/3},\bm{0}_{M/3},\bm{0}_{M/3}}\\\
{\bm{0}_{M/3},\bm{1}_{M/3},\bm{0}_{M/3}}\\\
{\bm{0}_{M/3},\bm{0}_{M/3},\bm{1}_{M/3}}\end{array}\right)$
where
$T_{1}=(\bm{1}^{\ast}_{M/3},\bm{0}^{\ast}_{M/3},\bm{0}^{\ast}_{M/3})^{\ast}$
is a vector of length $M$.
We restrict our search of the solution to Eq. (11) by restricting $\nu$ to be:
$\displaystyle\nu=\mbox{Diag}\left(\phi^{\ell}\right)T\omega,$ (14)
If we multiply from the left by
$T^{\ast}\mbox{Diag}\left(\phi^{\ell}\right)^{\ast}$ in Eq. (11) we obtain the
inverse iteration step with initial 0-phase vector as first guess:
$\displaystyle\hat{H}^{(\ell)}\omega^{(\ell+1)}=\lambda_{1}\bm{1},\quad$ (15)
Where
$\hat{H}^{(\ell)}=T^{\ast}\mbox{Diag}\left(z^{(\ell)}\right)^{\ast}[I-P_{\bf
Q}]\mbox{Diag}\left(z^{(\ell)}\right)T,$
$\lambda_{1}$ is a scalar multiplicative factor, and $\bm{1}$ is a vector of
appropriate length ($=3$ in this example). By computing $\omega^{(\ell+1)}$
from Eq. (15), and $\nu^{\ell+1}$ from Eq. (14), and projecting on the unit
torus we obtain the update $\phi^{(\ell+1)}$:
$\displaystyle\phi^{(\ell+1)}=\frac{\nu^{(\ell+1)}}{|\nu^{(\ell+1)}|}=\mbox{Diag}\left(\phi^{(\ell)}\right)\frac{T\omega^{(\ell+1)}}{|T\omega^{(\ell+1)}|}.$
(16)
In the following section we’ll show an example of the inverse iteration
method.
## III Numerical example
Here $\psi$ consists of the cameraman image of $32\times 32$ pixels, embedded
in a matrix of $64\times 64$ pixels (Fig. 6). The “illumination matrix” is the
support of the object, ${\bf Q}=\mbox{Diag}\left(S\right)$. The support is 1
inside the $32\times 32$ box containing the image, and 0 otherwise. $({\bf
Q}^{\ast}{\bf Q})^{-1}$ is replaced by the pseudoinverse ${\bf Q}={\bf
Q}^{\ast}$. The Fourier transform of $\psi$ was perturbed by $32\times 32$
randomly distributed phases (Fig. 6), each multiplying a bin of $2\times 2$
pixels (Fig. 6). Upon perturbation, the image in real space (Fig. 6) cannot be
distinguished. Many iterations of Eq. 6 or Eq. 9 cannot converge (Fig. 6
showing Eq. (6 ) updates), while 1 iteration of Eqs. (15,16) converges to the
solution (Fig. 6).
## IV Conclusions
I have shown that power iteration methods can recover phase perturbations
under special circumstances. If one is given the relative phases between a
small group of pixels (binned) and a random perturbation of the phase between
all the groups of pixels (the bins), then the inverse block iteration can
recover the solution in 1 iteration. In Marchesini _et al._ (2012) it was
observed that the inverse block iteration improved convergence rate in large
scale ptychographic imaging. The inverse block iteration was also shown to
recover perturbations in the experimental geometry such as position errors and
intensity fluctuations. More work is needed to determine the optimal
combination of Eqs. (6,9,13,15,16), and the properties of $T$, in large scale
phase retrieval problems.
I acknowledge usefull discussions with Jeff Donatelli of UC Berkeley. This
work was stimulated by the Phase Retrieval workshop at the Erwin Schroedinger
International Institute for Mathematical Physics (ESI) organized by Karlheinz
Gröchenig and Thomas Strohmer. This work was supported by the Laboratory
Directed Research and Development Program of Lawrence Berkeley National
Laboratory under the U.S. Department of Energy contract number DE-
AC02-05CH11231.
## Disclaimers
This document was prepared as an account of work sponsored by the United
States Government. While this document is believed to contain correct
information, neither the United States Government nor any agency thereof, nor
the Regents of the University of California, nor any of their employees, makes
any warranty, express or implied, or assumes any legal responsibility for the
accuracy, completeness, or usefulness of any information, apparatus, product,
or process disclosed, or represents that its use would not infringe privately
owned rights. Reference herein to any specific commercial product, process, or
service by its trade name, trademark, manufacturer, or otherwise, does not
necessarily constitute or imply its endorsement, recommendation, or favoring
by the United States Government or any agency thereof, or the Regents of the
University of California. The views and opinions of authors expressed herein
do not necessarily state or reflect those of the United States Government or
any agency thereof or the Regents of the University of California.
## References
* Marchesini _et al._ (2012) S. Marchesini, A. Schirotzek, F. Maia, and C. Yang, (2012), arXiv:1209.4924 [physics.optics] .
* Marchesini (2007) S. Marchesini, Rev Sci Instrum 78, 011301 (2007), arXiv:physics/0603201 .
* Singer (2011) A. Singer, Applied and Computational Harmonic Analysis 30, 20 (2011), arXiv:0905.3174 .
* Waldspurger _et al._ (2012) I. Waldspurger, A. d’Aspremont, and S. Mallat, ArXiv (2012), arXiv:1206.0102 [math.OC] .
* Alexeev _et al._ (2012) B. Alexeev, A. S. Bandeira, M. Fickus, and D. G. Mixon, (2012), arXiv:1210.7752 [cs.IT] .
Figure 1: Object $\psi$ ($64\times 64$) used to simulate diffraction data
Figure 2: Each column of the matrix $T$ extracts an area of $2\times 2$
pixels out of an image of ($64\times 64$) pixels.
Figure 3: Random perturbation: ($64\times 64$) phases generated by $32\times
32$ random phases each spread over a bin of ($2\times 2$) pixels.
Figure 4: Image in real space ($\left|\bm{F}^{\ast}{\bf z}\right|$) after
random phase perturbation (Fig. 6)
Figure 5: Image in real space after random phase perturbation (Fig. 6), using
Eq. 6 updates ( $\left|\bm{F}^{\ast}(P_{\bm{a}}P_{S})^{1000}{\bf z}\right|$).
Figure 6: Image in real space after one step of Eqs. (15,16) update.
|
arxiv-papers
| 2012-12-03T19:57:32 |
2024-09-04T02:49:38.846523
|
{
"license": "Public Domain",
"authors": "Stefano Marchesini",
"submitter": "Stefano Marchesini",
"url": "https://arxiv.org/abs/1212.0510"
}
|
1212.0784
|
# Magnetized hot neutron matter: lowest order constrained variational
calculations
G.H. Bordbar 1,2 111Corresponding author. E-mail: [email protected],
Z. Rezaei1 1Department of Physics, Shiraz University, Shiraz 71454,
Iran222Permanent address,
and
2Research Institute for Astronomy and Astrophysics of Maragha, P.O. Box
55134-441, Maragha 55177-36698, Iran
###### Abstract
We have studied the spin polarized hot neutron matter in the presence of
strong magnetic field. In this work, using the lowest order constrained
variational method at finite temperature and employing $AV_{18}$ nuclear
potential, some thermodynamic properties of spin polarized neutron matter such
as spin polarization parameter, free energy, equation of state and effective
mass have been calculated. It has been shown that the strong magnetic field
breaks the symmetry of the free energy, leading to a magnetized equilibrium
state. We have found that the equation of state becomes stiffer by increasing
both magnetic field and temperature. The magnetic field dependence of
effective mass for the spin-up and spin-down neutrons has been investigated.
Neutron matter - strong magnetic field - equation of state
## I INTRODUCTION
Based on the supernova models, after the gravitational collapse of a
degenerate stellar core and the explosive ejection of outer layers material, a
protoneutron star would be born Haensel . Through the formation of the
protoneutron star, the system arrives at a temperature of about $20-50\ MeV$
Camenzind . This protoneutron star is hot, opaque to neutrinos, and larger
than an ordinary neutron star Haensel . After the formation of the
protoneutron star, neutrino emission is the dominant process in cooling of the
neutron star (mainly by URCA process and neutrino Bremsstrahlung) Camenzind .
Woltjer has predicted a magnetic field strength of order $10^{15}\ G$ for
neutron stars as a result of the magnetic flux conservation from the
progenitor star Woltjer . This in agreement with the experimental indication
that the surface magnetic field strength of magnetars can be of the order
$B_{magnetar}\approx 10^{14}-10^{15}\ G$ Thompson ; Lazzati . The magnetic
field can be distorted or amplified by some mixture of convection,
differential rotation, and magnetic instabilities Tayler ; Spruit . The
relative importance of these ingredients depends on the initial field strength
and the rotation rate of the star. For both convection and differential
rotation, the field and its supporting currents are not likely to be confined
to the solid crust of the star but instead distributed in most of the stellar
interior, which is mostly a fluid mixture of neutrons, protons, electrons, and
other more exotic particles Reisen . Thompson et al. argued that newborn
neutron stars probably combine vigorous convection and differential rotation,
making it likely that a dynamo process might operate in them Thompson2 . They
expected fields up to $10^{15}-10^{16}\ G$ in neutron stars with few-
millisecond initial periods. In the core of high density inhomogeneous
gravitationally bound neutron stars, the magnetic field strength can be as
large as $10^{20}\ G$ Ferrer . In addition, considering the formation of a
quark core in the high density interior of a neutron star, the maximum field
reaches up to about $10^{20}\ G$ Ferrer ; Tatsumi . According to the scalar
virial theorem based on Newtonian gravity, the magnetic field strength is
allowed to be up to $10^{18}\ G$ in the interior of a magnetar Lai . Moreover,
general relativity predicts the allowed maximum value of the neutron star
magnetic field to be $10^{18}-10^{20}\ G$ shapiro . By comparing the cooling
curves of neutron stars with the observational data, Yuan et al. obtained the
magnetic field strength of order $10^{19}\ G$ for many not so old neutron
stars Yuan .
The finite temperature and strong magnetic field in the interior of a
protoneutron star can influence different astrophysical quantities. Therefore,
to have a better understanding of different astrophysical phenomena such as
supernova explosion, thermal evolution and cooling of protoneutron stars,
gravitational wave emission spectrum from neutron star mergers, and to get the
more precise astrophysical quantities such as properties of very young hot
neutron stars and composition of neutron stars, one should consider the
neutron star matter at finite temperatures (excited neutron star matter) and
strong magnetic fields.
Since $\beta$-equilibrium leads to an increase in the number of neutrons in
neutron star matter, it is possible to approximate the neutron star matter by
the pure neutron matter. Many works have dealt with the study of dense neutron
matter at finite temperature Alonso2 ; Mukherjee ; Panda ; Rios2 ; Bombaci ;
Rios3 . Alonso et al. have used a field theoretical model for the analysis of
relativistic neutron matter Alonso2 . By solving the model in the renormalized
Hartree approximation, they have investigated the effect of central
temperature on the maxima of mass for stable configurations, the radii of the
configurations, and the gravitational red shift at the surface of a neutron
star. Panda et al. used a mean-field description of nonoverlaping nucleon bags
bound by the self-consistent exchange of $\sigma$, $\omega$, and $\rho$ mesons
to investigate the properties of neutron matter at finite temperature Panda .
They showed that by increasing the temperature, the equation of state becomes
stiffer. Within the framework of the Brueckner-Hartree-Fock formalism and
using the $AV_{18}$ nucleon-nucleon interaction for the spin polarized neutron
matter, Bombaci et al. found that an increase in the temperature moderately
affects the single-particle potentials Bombaci . In the Hartree-Fock
approximation using Skyrme type interactions for spin polarized neutron
matter, Rios et al. showed that the critical density at which the
ferromagnetism takes place, decreases by temperature Rios2 . Within the self-
consistent Green’s-function approach applying the CD Bonn and the $AV_{18}$
potential for neutron matter, Rios et al. found that the effect of dynamical
correlations on the macroscopic properties is rather insensitive to the
thermal effects Rios3 . The variational theory for fermions at finite
temperature and high density has been applied by Mukherjee to neutron matter
Mukherjee . It has been found that the temperature dependence of the
correlation operator is weak, but it is not negligible. Besides, it has been
shown that the first order phase transition due to neutral pion condensation
has a critical temperature of about $22MeV$ for neutron matter. The effect of
strong magnetic field on the properties of dense neutron matter has also been
considered. Perez-Garcia showed that for the neutron matter in the presence of
strong magnetic fields, in the Skyrme model, there is a ferromagnetic phase
transition at $\rho\sim 4\rho_{0}$, whereas it is forbidden in the $D1P$ model
Perez-Garcia . The results indicate that the effects of temperature on the
neutron magnetization remain moderate at temperatures up to about $T=40\ MeV$.
In the context of the Landau theory of normal Fermi liquids, using Skyrme and
Gogny effective interactions, some thermodynamical quantities such as
isothermal compressibility and spin susceptibility of pure neutron matter have
also been studied Perez-Garcia2 .
In our previous works, we have investigated the spin polarized neutron matter
Bordbar75 , symmetric nuclear matter Bordbar76 , asymmetric nuclear matter as
well as neutron star matter Bordbar77 and magnetized neutron matter Bordbar83
at zero temperature using lowest order constrained variational (LOCV) method
with the realistic strong interactions. We have also investigated the
thermodynamic properties of spin polarized neutron matter Bordbar78 ,
symmetric nuclear matter Bordbar80 , and asymmetric nuclear matter Bordbar81
at finite temperature with no magnetic field. In the present work, we
calculate different thermodynamic properties of spin polarized neutron matter
at finite temperature in the presence of strong magnetic field using LOCV
technique employing $AV_{18}$ potential.
## II LOCV formalism for the spin polarized hot neutron matter in the
presence of magnetic field
We consider a homogeneous system of $N$ interacting particles with $N^{(+)}$
spin-up and $N^{(-)}$ spin-down neutrons under the influence of a uniform
magnetic field, i.e. $\mathbf{B}=B\widehat{k}$. The number densities of spin-
up and spin-down neutrons are presented by $\rho^{(+)}$ and $\rho^{(-)}$
respectively. We introduce the spin polarization parameter, $\delta$, by
$\displaystyle\delta=\frac{\rho^{(+)}-\rho^{(-)}}{\rho},$ (1)
where $-1\leq\delta\leq 1$, and $\rho$ is the total number density of system.
The magnetization density of neutron matter is defined as
$\displaystyle m=\mu_{n}\delta\rho,$ (2)
where $\mu_{n}$ is the neutron magnetic moment. The total magnetization of a
given volume is also as follows
$\displaystyle M=\int mdV.$ (3)
In order to calculate the energy of this system, we use LOCV method as
follows: we consider a trial many-body wave function of the form
$\displaystyle\psi=\mathcal{F}\phi,$ (4)
where $\phi$ is the uncorrelated ground-state wave function of $N$ independent
neutrons, and $\mathcal{F}$ is a proper $N$-body correlation function. Using
Jastrow approximation Jastrow , $\mathcal{F}$ can be replaced by
$\displaystyle\mathcal{F}=S\prod_{i>j}f(ij),$ (5)
where $S$ is a symmetrizing operator. We consider a cluster expansion of the
energy functional up to the two-body term,
$\displaystyle
E([f])=\frac{1}{N}\frac{\langle\psi|H|\psi\rangle}{\langle\psi|\psi\rangle}=E_{1}+E_{2}\cdot$
(6)
The one-body term, $E_{1}$, is given by
$\displaystyle E_{1}=-\frac{MH}{N}+\sum_{i=+,-}\overline{\varepsilon}_{i},$
(7)
where the first term of Eq. (7), $H$ shows the external magnetic field. It
should be noted that we have used $B$ to present the magnetic field strength;
while the total magnetic field is the sum of the external magnetic field and
the induced magnetization, $B=H+4\pi M$. Because of the tiny value of the
neutron magnetic moment, we assume that the induced magnetization has a small
contribution to the total magnetic field. Consequently,
$\displaystyle B\sim H.$ (8)
Using Eqs. (2), (3) and (8), the one-body term can be written as
$\displaystyle E_{1}=-\mu_{n}B\delta+\sum_{i=+,-}\overline{\varepsilon}_{i}.$
(9)
The second term in Eq. (7), $\overline{\varepsilon}_{i}$ is as follows
$\displaystyle\overline{\varepsilon}_{i}=\sum_{k}\frac{\hbar^{2}k^{2}}{2m}\overline{n}_{i}(k,T,B,\rho^{(i)}),$
(10)
where $\overline{n}_{i}(k,T,B,\rho^{(i)})$ is the Fermi-Dirac distribution
function in the presence of magnetic field,
$\displaystyle\overline{n}_{i}(k,T,B,\rho^{(i)})=\frac{1}{e^{\beta[\overline{\epsilon}_{i}(k,T,B,\rho^{(i)})-\overline{\mu}_{i}(T,B,\rho^{(i)})]}+1}.$
(11)
In Eq. (11), $\overline{\epsilon}_{i}$ and $\overline{\mu}_{i}$ are the
single-particle energy of a neutron and the neutron chemical potential
respectively. The single-particle energy, $\overline{\epsilon}_{i}$, of a
neutron with momentum $k$ and spin projection $i$ in the presence of magnetic
field is approximately written in terms of the effective mass as follows Rios2
$\displaystyle\overline{\epsilon}_{i}(k,T,B,\rho^{(i)})$ $\displaystyle=$
$\displaystyle\left\\{\begin{array}[]{ll}\frac{\hbar^{2}k^{2}}{2m_{+}^{*}(T,\rho)}-\mu_{n}B+U_{+}(T,\rho^{(+)})&;~{}~{}i=+,\\\
\\\
\frac{\hbar^{2}k^{2}}{2m_{-}^{*}(T,\rho)}+\mu_{n}B+U_{-}(T,\rho^{(-)})&;~{}~{}i=-.\end{array}\right.$
(15)
In fact, we use a quadratic approximation for the single particle potential
incorporated in the single particle energy as a momentum independent effective
mass. $U_{i}(T,\rho^{(i)})$ is the momentum independent single particle
potential. The effective mass, $m_{i}^{*}$, is determined variationally
Modarres3 ; Modarres5 ; Modarres7 ; Modarres8 ; Friedman . The chemical
potential, $\overline{\mu}_{i}$, is also obtained by applying the constraint
$\displaystyle\sum_{k}\overline{n}_{i}(k,T,B,\rho^{(i)})=N^{(i)}.$ (16)
The two-body energy, $E_{2}$, is
$\displaystyle E_{2}$ $\displaystyle=$
$\displaystyle\frac{1}{2N}\sum_{ij}\langle ij\left|\nu(12)\right|ij-
ji\rangle,$ (17)
where
$\nu(12)=-\frac{\hbar^{2}}{2m}[f(12),[\nabla_{12}^{2},f(12)]]+f(12)V(12)f(12).$
$f(12)$ and $V(12)$ are the two-body correlation function and nuclear
potential respectively. In the LOCV formalism, the two-body correlation
function, $f(12)$, is considered as follows Owen ,
$\displaystyle f(12)$ $\displaystyle=$
$\displaystyle\sum^{3}_{k=1}f^{(k)}(r_{12})P^{(k)}_{12},$ (18)
where
$\displaystyle P^{(k=1-3)}_{12}$ $\displaystyle=$
$\displaystyle(\frac{1}{4}-\frac{1}{4}\sigma_{1}.\sigma_{2}),\
(\frac{1}{2}+\frac{1}{6}\sigma_{1}.\sigma_{2}+\frac{1}{6}S_{12}),\
(\frac{1}{4}+\frac{1}{12}\sigma_{1}.\sigma_{2}-\frac{1}{6}S_{12}).$ (19)
In Eq. (19), $S_{12}$ and $\sigma_{1}.\sigma_{2}$ are the tensor and Pauli
operators respectively. Using the above two-body correlation function and the
$AV_{18}$ two-body potential Wiringa , after doing some algebra, we find the
following equation for the two-body energy:
$\displaystyle E_{2}$ $\displaystyle=$
$\displaystyle\frac{2}{\pi^{4}\rho}\left(\frac{\hbar^{2}}{2m}\right)\sum_{JLSS_{z}}\frac{(2J+1)}{2(2S+1)}[1-(-1)^{L+S+1}]\left|\left\langle\frac{1}{2}\sigma_{z1}\frac{1}{2}\sigma_{z2}\mid
SS_{z}\right\rangle\right|^{2}\times$ (20) $\displaystyle\times\int
dr\left\\{\left[{f_{\alpha}^{(1)^{{}^{\prime}}}}^{2}{a_{\alpha}^{(1)}}^{2}(r,\rho,T)\right.\right.$
$\displaystyle\left.\left.+\frac{2m}{\hbar^{2}}(\\{V_{c}-3V_{\sigma}+V_{\tau}-3V_{\sigma\tau}+2(V_{T}-3V_{\sigma
T})-2V_{\tau z}\\}{a_{\alpha}^{(1)}}^{2}(r,\rho,T)\right.\right.$
$\displaystyle\left.\left.+[V_{l2}-3V_{l2\sigma}+V_{l2\tau}-3V_{l2\sigma\tau}]{c_{\alpha}^{(1)}}^{2}(r,\rho,T))(f_{\alpha}^{(1)})^{2}\right]+\sum_{k=2,3}\left[{f_{\alpha}^{(k)^{{}^{\prime}}}}^{2}{a_{\alpha}^{(k)}}^{2}(r,\rho,T)\right.\right.$
$\displaystyle\left.\left.+\frac{2m}{\hbar^{2}}(\\{V_{c}+V_{\sigma}+V_{\tau}+V_{\sigma\tau}+(-6k+14)(V_{t\tau}+V_{t})-(k-1)(V_{ls\tau}+V_{ls})\right.\right.$
$\displaystyle\left.\left.+2[V_{T}+V_{\sigma T}+(-6k+14)V_{tT}-V_{\tau
z}]\\}{a_{\alpha}^{(k)}}^{2}(r,\rho,T)\right.\right.$
$\displaystyle\left.\left.+[V_{l2}+V_{l2\sigma}+V_{l2\tau}+V_{l2\sigma\tau}]{c_{\alpha}^{(k)}}^{2}(r,\rho,T)+[V_{ls2}+V_{ls2\tau}]{d_{\alpha}^{(k)}}^{2}(r,\rho,T)){f_{\alpha}^{(k)}}^{2}\right]\right.$
$\displaystyle\left.+\frac{2m}{\hbar^{2}}\\{V_{ls}+V_{ls\tau}-2(V_{l2}+V_{l2\sigma}+V_{l2\sigma\tau}+V_{l2\tau})-3(V_{ls2}+V_{ls2\tau})\\}b_{\alpha}^{2}(r,\rho,T)f_{\alpha}^{(2)}f_{\alpha}^{(3)}\right.$
$\displaystyle\left.+\frac{1}{r^{2}}(f_{\alpha}^{(2)}-f_{\alpha}^{(3)})^{2}b_{\alpha}^{2}(r,\rho,T)\right\\},$
where $\alpha=\\{J,L,S,S_{z}\\}$ and the coefficients
${a_{\alpha}^{(i)}}^{2}$, $b_{\alpha}^{2}$, ${c_{\alpha}^{(i)}}^{2}$, and
${d_{\alpha}^{(i)}}^{2}$ are defined as
$\displaystyle{a_{\alpha}^{(1)}}^{2}(r,\rho,T)=r^{2}I_{L,S_{z}}(r,\rho,T),$
(21) $\displaystyle{a_{\alpha}^{(2)}}^{2}(r,\rho,T)=r^{2}[\beta
I_{J-1,S_{z}}(r,\rho,T)+\gamma I_{J+1,S_{z}}(r,\rho,T)],$ (22)
$\displaystyle{a_{\alpha}^{(3)}}^{2}(r,\rho,T)=r^{2}[\gamma
I_{J-1,S_{z}}(r,\rho,T)+\beta I_{J+1,S_{z}}(r,\rho,T)],$ (23) $\displaystyle
b_{\alpha}^{2}(r,\rho,T)=r^{2}[\beta_{23}I_{J-1,S_{z}}(r,\rho,T)-\beta_{23}I_{J+1,S_{z}}(r,\rho,T)],$
(24)
$\displaystyle{c_{\alpha}^{(1)}}^{2}(r,\rho,T)=r^{2}\nu_{1}I_{L,S_{z}}(r,\rho,T),$
(25)
$\displaystyle{c_{\alpha}^{(2)}}^{2}(r,\rho,T)=r^{2}[\eta_{2}I_{J-1,S_{z}}(r,\rho,T)+\nu_{2}I_{J+1,S_{z}}(r,\rho,T)],$
(26)
$\displaystyle{c_{\alpha}^{(3)}}^{2}(r,\rho,T)=r^{2}[\eta_{3}I_{J-1,S_{z}}(r,\rho,T)+\nu_{3}I_{J+1,S_{z}}(r,\rho,T)],$
(27)
$\displaystyle{d_{\alpha}^{(2)}}^{2}(r,\rho,T)=r^{2}[\xi_{2}I_{J-1,S_{z}}(r,\rho,T)+\lambda_{2}I_{J+1,S_{z}}(r,\rho,T)],$
(28)
$\displaystyle{d_{\alpha}^{(3)}}^{2}(r,\rho,T)=r^{2}[\xi_{3}I_{J-1,S_{z}}(r,\rho,T)+\lambda_{3}I_{J+1,S_{z}}(r,\rho,T)],$
(29)
with
$\displaystyle\beta=\frac{J+1}{2J+1},\ \gamma=\frac{J}{2J+1},\
\beta_{23}=\frac{2J(J+1)}{2J+1},$ (30) $\displaystyle\nu_{1}=L(L+1),\
\nu_{2}=\frac{J^{2}(J+1)}{2J+1},\ \nu_{3}=\frac{J^{3}+2J^{2}+3J+2}{2J+1},$
(31) $\displaystyle\eta_{2}=\frac{J(J^{2}+2J+1)}{2J+1},\
\eta_{3}=\frac{J(J^{2}+J+2)}{2J+1},$ (32)
$\displaystyle\xi_{2}=\frac{J^{3}+2J^{2}+2J+1}{2J+1},\
\xi_{3}=\frac{J(J^{2}+J+4)}{2J+1},$ (33)
$\displaystyle\lambda_{2}=\frac{J(J^{2}+J+1)}{2J+1},\
\lambda_{3}=\frac{J^{3}+2J^{2}+5J+4}{2J+1},$ (34)
and
$\displaystyle I_{J,S_{z}}(r,\rho,T)=\frac{1}{2\pi^{6}\rho^{2}}\int dk_{1}\
dk_{2}\
\overline{n}_{i}(k_{m},T,B,\rho^{(i)})\overline{n}_{j}(k_{m},T,B,\rho^{(j)})J_{J}^{2}(|k_{2}-k_{1}|r)\cdot$
(35)
In the above equation, $J_{J}(x)$ is the Bessel function.
Now, we minimize the two-body energy with respect to the variations in the
function $f_{\alpha}^{(i)}$ subject to the normalization constraint,
$\displaystyle\frac{1}{N}\sum_{ij}\langle
ij\left|h_{S_{z}}^{2}-f^{2}(12)\right|ij\rangle_{a}=0.$ (36)
The minimization subject to the above normalization constraint leads to the
normalization of the two body wave function to unity Owen . For the spin
polarized hot neutron matter, the function $h_{S_{z}}(r)$ is defined as
follows,
$\displaystyle h_{S_{z}}(r)$ $\displaystyle=$
$\displaystyle\left\\{\begin{array}[]{ll}\left[1-\left(\frac{\gamma_{i}(r)}{\rho}\right)^{2}\right]^{-1/2}&;~{}~{}S_{z}=\pm
1,\\\ \\\ 1&;~{}~{}S_{z}=0,\end{array}\right.$ (40)
where
$\displaystyle\gamma_{i}(r)=\frac{1}{2\pi^{2}}\int dk\
\overline{n}_{i}(k_{m},T,B,\rho^{(i)})J_{0}(kr)k^{2}.$ (41)
From the minimization of the two-body cluster energy, we get a set of coupled
and uncoupled differential equations BordMod98 . By solving these equations,
we can obtain the correlation functions to compute the two-body energy term,
$E_{2}$. As the final step, we calculate the free energy per particle, $F$, to
get different thermodynamic properties of spin polarized hot neutron matter,
$\displaystyle F(\rho,T,B)=E(\rho,T,B)-TS(\rho,T,B),$ (42)
where S is the entropy per particle,
$\displaystyle S(\rho,T,B)$ $\displaystyle=$
$\displaystyle-\frac{1}{N}\sum_{i=+,-}\sum_{k}\\{{[1-\overline{n}_{i}(k,T,B,\rho^{(i)})]\textrm{ln}[1-\overline{n}_{i}(k,T,B,\rho^{(i)})]}$
(43) $\displaystyle+$
$\displaystyle{\overline{n}_{i}(k,T,B,\rho^{(i)})\textrm{ln}[\overline{n}_{i}(k,T,B,\rho^{(i)})]}\\}.$
It should be noted that in our calculations, we introduce the effective
masses, $m_{i}^{*}$, as variational parameters Modarres3 ; Modarres5 ;
Modarres7 ; Modarres8 ; Friedman . We minimize the free energy with respect to
the variations in the effective masses, and then we obtain the chemical
potentials and the effective masses of spin-up and spin-down neutrons at the
equilibrium state. In our approach, the effective mass depends on both density
and temperature but it is independent of the momentum. The effective mass of a
quasiparticle near the Fermi surface for the spin polarized neutron matter at
low temperatures is also the static physical quantity of interest in the
context of Landau Fermi liquid theory Landau .
## III RESULTS and DISCUSSION
In Fig. 1, we present the free energy per particle of spin polarized neutron
matter versus the spin polarization parameter $\delta$. Fig. 1a shows that in
the presence of magnetic field, the free energy is not a symmetric function of
spin polarization parameter and the equilibrium configuration would experience
a net magnetization. Clearly, the effects of magnetic fields below $B\sim
10^{18}\ G$ are almost insignificant, but by increasing the magnetic field,
the equilibrium value of the spin polarization parameter (i.e. the
polarization that minimizes the free energy) and the free energy decrease,
leading to a more stable system. Fig. 1b indicates that the effect of
temperature on the fully spin polarized neutron matter is less than that of
the unpolarized one.
Fig. 2 presents the equilibrium value of the spin polarization parameter
versus density $\rho$. Fig. 2a shows that at low densities ($\rho\leq 0.2\
fm^{-3}$), the magnitude of the spin polarization parameter decreases by
increasing the temperature. However, at higher densities, the related values
of the spin polarization parameter at different finite temperatures are almost
identical to the one for zero temperature. This is due to smaller values of
$T/\varepsilon_{f}^{*}$ at high densities (Fig. 2b), in which
$\varepsilon_{f}^{*}$ is defined as follows,
$\displaystyle\varepsilon_{f}^{*}=\sum_{i=+,-}\frac{\rho^{(i)}}{\rho}\varepsilon_{fi}^{*},$
(44)
with
$\displaystyle\varepsilon_{f+}^{*}=\frac{\hbar^{2}k_{F}^{(+)^{2}}}{2m}-\mu_{n}B,$
(45)
and
$\displaystyle\varepsilon_{f-}^{*}=\frac{\hbar^{2}k_{F}^{(-)^{2}}}{2m}+\mu_{n}B.$
(46)
In the above equations, $\varepsilon_{fi}^{*}$ and
$k_{F}^{(i)}=(6\pi^{2}\rho^{(i)})^{\frac{1}{3}}$ are the Fermi energy and
Fermi momentum of neutrons with spin projection $i$ in the presence of the
magnetic field. It is evident from Eq. (44) that $\varepsilon_{f}^{*}$ gives
an average of the Fermi energy of magnetized neutron matter. Therefore, the
ratio $T/\varepsilon_{f}^{*}$ is a criterion for the fraction of particles
which are thermally excited Pathria , and how much the system is disordered.
In Fig. 3, we show the spin polarization parameter at the equilibrium state as
a function of the magnetic field $B$. At each temperature, the magnitude of
spin polarization parameter grows by increasing the magnetic field (Fig. 3a).
We have found that at strong magnetic fields, the effect of finite temperature
is more significant because the ratio $T/\varepsilon_{f}^{*}$ rises with the
increase in the magnetic field (Fig. 3b).
The free energy per particle at the equilibrium value of the spin polarization
parameter is presented in Fig. 4. It can be seen that at finite temperature,
the free energy is an increasing function of density (Fig. 4a). At low
densities, the rate of increase in the free energy varies by increasing the
density, but at high densities, this rate of increase is nearly constant.
Moreover, the effect of temperature on the free energy is more pronounced at
low densities. Fig. 4b shows that the free energy decreases by growing the
temperature nearly at the same rate for different magnetic fields. The free
energy decreases by increasing the magnetic field (Fig. 4c). We can see that
by increasing the magnetic field up to a value of about $B\simeq 10^{18}\ G$,
the free energy per particle slowly decreases, and then it rapidly decreases
for the magnetic fields greater than this value. This indicates that, above
$B\simeq 10^{18}\ G$, the effect of magnetic field on the free energy of the
spin polarized neutron matter becomes more important.
From the free energy per particle of magnetized neutron matter, $F$, we can
obtain the corresponding pressure of neutron matter using the following
relation,
$\displaystyle P(\rho,T,B)=\rho^{2}{\left(\frac{\partial
F(\rho,T,B)}{\partial\rho}\right)_{T,B}}.$ (47)
This equation of state (EoS) is plotted in Fig. 5. For each value of the
density, pressure increases by growing the magnetic field (Fig. 5a). This
stiffening of the equation of state is due to the inclusion of neutron
anomalous magnetic moments. From the astrophysical point of view, it should be
noted that this stiffening of the EoS leads to the larger value for the
maximum mass of neutron star Bordbar2006 ; Bordbar2009 ; Bordbar2011 . At each
density, the pressure at finite temperature is larger than that of zero
temperature (Fig. 5b). It means that the equation of state of neutron matter
becomes stiffer by increasing the temperature. Fig. 5c also shows that by
increasing the temperature, the pressure increases nearly at the same rate for
different magnetic fields.
Fig. 6 shows the effective mass corresponding with the equilibrium of the
system for the spin-up and spin-down neutrons as a function of the magnetic
field $B$. At low magnetic fields, the effective masses of spin-up and spin-
down neutrons are nearly identical because the effective masses of spin-up and
spin-down neutrons have the same values at $\delta\simeq 0$. Fig. 6 indicates
that the effective mass of spin-up (spin-down) neutrons decreases (increases)
by increasing the magnetic field in agreement with the results obtained in
Ref. Perez-Garcia2 . From the comparison of Fig. 3a and 6, we can see that the
shift in mass is due to the polarization of neutron matter. For the maximum
value of the magnetic field considered in this work, i.e. $5\times 10^{18}\
G$, and at $T=10\ MeV$ and $\rho=0.3\ fm^{-3}$, the equilibrium value of the
spin polarization parameter is about $\delta_{B_{max}}=-0.23$ and that
corresponds to a mass shift of an amount of $\bigtriangleup(m^{*}/m)\approx
0.02$ with respect to the unpolarized case.
## IV Summary and Concluding Remarks
We have investigated the effect of strong magnetic fields on the thermodynamic
properties of spin polarized hot neutron matter applying LOCV method and using
$AV_{18}$ potential. We have found that in the presence of a strong magnetic
field, the free energy is not a symmetric function of the spin polarization
parameter and the system is macroscopically magnetized. By increasing both
density and temperature, the magnitude of the equilibrium value of the spin
polarization parameter decreases. At low magnetic fields, the free energy
decreases very slowly by increasing the magnetic field, but at stronger
magnetic fields, the free energy decreases rapidly with the increase in the
magnetic field. It has been found that the equation of state becomes stiffer
by increasing the magnetic field. This stiffening of the EoS leads to the
larger value for the maximum mass of neutron star.
###### Acknowledgements.
We wish to thank the Research Institute for Astronomy and Astrophysics of
Maragha and Shiraz University Research Council.
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Figure 1: Free energy per particle as a function of the spin polarization
parameter $\delta$: (a) for the cases $B=0\ G$ (solid curve), $B=10^{18}\ G$
(dashed curve) and $B=5\times 10^{18}\ G$ (dashdot curve) at the fixed values
of the temperature, $T=10\ MeV$, and the density, $\rho=0.2\ fm^{-3}$, (b) for
the cases $T=0\ MeV$ (solid curve), $T=10\ MeV$ (dashed curve) and $T=20\ MeV$
(dashdot curve) at the fixed values of the magnetic field, $B=5\times 10^{18}\
G$, and the density, $\rho=0.2\ fm^{-3}$.
Figure 2: (a) Spin polarization parameter at the equilibrium state as a
function of the density $\rho$ for the cases $T=0\ MeV$ (solid curve), $T=10\
MeV$ (dashed curve) and $T=20\ MeV$ (dashdot curve), and a fixed value of the
magnetic field, $B=5\times 10^{18}\ G$. (b) Same as in the top panel but for
the ratio $T/\varepsilon_{f}^{*}$.
Figure 3: (a) Magnetic field dependence of the spin polarization parameter
$\delta$ at the equilibrium state for the cases $T=0\ MeV$ (solid curve),
$T=10\ MeV$ (dashed curve) and $T=20\ MeV$ (dashdot curve), and a fixed value
of the density, $\rho=0.2\ fm^{-3}$. (b) Same as in the top panel but for the
ratio $T/\varepsilon_{f}^{*}$.
Figure 4: Free energy per particle at the equilibrium state as a function of:
(a) the density $\rho$ for the cases $T=0\ MeV$ (solid curve), $T=10\ MeV$
(dashed curve) and $T=20\ MeV$ (dashdot curve), and a fixed value of the
magnetic field, $B=5\times 10^{18}\ G$, (b) the temperature $T$ for the cases
$B=0\ G$ (solid curve), $B=10^{18}\ G$ (dashed curve) and $B=5\times 10^{18}\
G$ (dashdot curve), and a fixed value of the density, $\rho=0.2\ fm^{-3}$, (c)
the magnetic field $B$ for the cases $T=0\ MeV$ (solid curve), $T=10\ MeV$
(dashed curve) and $T=20\ MeV$ (dashdot curve), and a fixed value of the
density, $\rho=0.2\ fm^{-3}$.
Figure 5: Pressure of spin polarized neutron matter as a function of: (a) the
density $\rho$ for the cases $B=0\ G$ (solid curve), $B=10^{18}\ G$ (dashed
curve) and $B=5\times 10^{18}\ G$ (dashdot curve), and a fixed value of the
temperature, $T=20\ MeV$, (b) the density $\rho$ for the cases $T=0\ MeV$
(solid curve), $T=10\ MeV$ (dashed curve) and $T=20\ MeV$ (dashdot curve), and
a fixed value of the magnetic field, $B=5\times 10^{18}\ G$, (c) the
temperature $T$ for the cases $B=0\ G$ (solid curve), $B=10^{18}\ G$ (dashed
curve) and $B=5\times 10^{18}\ G$ (dashdot curve), and a fixed value of the
density, $\rho=0.2\ fm^{-3}$.
Figure 6: Magnetic field dependence of the effective masses of spin-up (solid
curve) and spin-down (dashed curve) neutrons corresponding with the
equilibrium state at the fixed values of the temperature, $T=10\ MeV$, and the
density, $\rho=0.3\ fm^{-3}$.
|
arxiv-papers
| 2012-12-04T16:34:17 |
2024-09-04T02:49:38.866914
|
{
"license": "Public Domain",
"authors": "G.H. Bordbar and Z. Rezaei",
"submitter": "Gholam Hossein Bordbar",
"url": "https://arxiv.org/abs/1212.0784"
}
|
1212.0803
|
# Electronic thermal conductivity as derived by density functional theory
M. X. Chen Department of Physics, University of Wisconsin-Milwaukee,
Milwaukee, Wisconsin 53211, USA R. Podloucky Department of Physical
Chemistry, University of Vienna, Sensengasse 8/7, 1090 Vienna, Austria
###### Abstract
Reliable evaluation of the lattice thermal conductivity is of importance for
optimizing the figure-of-merit of thermoelectric materials. Traditionally,
when deriving the phonon mediated thermal conductivity
$\kappa_{ph}=\kappa-\kappa_{el}$ from the measured total thermal conductivity
$\kappa$ the constant Lorenz number $L_{0}$ of the Wiedemann-Franz law
$\mathbf{\kappa_{el}}=TL_{0}\sigma$ is chosen. The present study demonstrates
that this procedure is not reliable when the Seebeck coefficient $|S|$ becomes
large which is exactly the case for a thermoelectric material of interest.
Another approximation using $L_{0}-S^{2}$, which seem to work better for
medium values of $S^{2}$ also fails when $S^{2}$ becomes large, as is the case
when the system becomes semiconducting/insulating. For a reliable estimation
of $\kappa_{el}$ it is proposed, that a full first-principles calculations by
combining density functional theory with Boltzmann’s transport theory has to
be made. For the present study such an approach was chosen for investigating
the clathrate type-I compound Ba8Au6-xGe40+x for a series of dopings or
compositions $x$. For a doping of $0.8$ electrons corresponding to $x=0.27$
the calculated temperature dependent Seebeck coefficient agrees well with
recent experiments corroborating the validity of the density functional theory
approach.
###### pacs:
1,2,3
###### pacs:
72.15.Jf, 72.15.Eb, 71.20.-b
Thermal conductivity plays an important role for the thermoelectric
performance of a material as expressed by the figure-of-merit
$ZT=T$S${}^{2}\sigma/(\kappa_{el}+\kappa_{ph})$ which includes the Seebeck
coefficient $S$, the electrical conductivity $\sigma$, and the thermal
conductivity $\kappa=\kappa_{el}+\kappa_{ph}$ summing up the contributions of
electronic states and phonon mediated processes. Consequently, a low thermal
conductivity in combination with large values of $S$ and $\sigma$ are
desirable in order to achieve large values of $ZT$. Considerable efforts for
lowering $\kappa$ by reducing $\kappa_{ph}$ were made by utilizing structural
properties, such as building up superlattices venkatasubramanian_2001 ;
beyer_pbte_2002 ; caylor_enhanced_2005 ; boettner_aspects_2006 and
incorporating suitable filler atoms into structural cages sales_filled_1996 ;
morelli_low_1995 ; nolas_effect_1996 ; sales_thermoelectric_2000 ;
nolas_high_2000 ; nolas_skutterudites:_1999 ; lamberton_high_2002 ;
nolas_semiconducting_1998 . These concepts rely on the strong scattering of
heat-transporting phonon modes. However, neither $\kappa_{el}$ nor
$\kappa_{ph}$ are directly measured. Rather, $\kappa_{ph}$ is derived by
subtracting $\kappa_{el}$ from the measured total thermal conductivity, i.e.,
$\kappa_{ph}\approx\kappa^{meas.}-\kappa_{el}$ in which the electronic thermal
conductivity is estimated via the Wiedemann-Franz (WF) relation for simple
metals, $\kappa_{el}\approx T$L${}_{0}\sigma$ morelli_low_1995 ;
nolas_effect_1996 ; sales_thermoelectric_2000 ; nolas_high_2000 ;
nolas_skutterudites:_1999 ; lamberton_high_2002 ; ohtaki_hightemperature_1996
; li_high_1999 ; takahata_low_2000 ; toprak_impact_2004 ;
androulakis_nanostructuring_2006 ; tang_preparation_2007 ;
zhou_nanostructured_2008 ; shi_low_2008 ; li_high_2009 . In this expression,
L0 is a universal constant and does not depend on temperature and materials
properties. In the present work it is shown by a density functional theory
(DFT) study for a typical thermoelectric material that the application of the
WF law leads to unreliable estimates of $\kappa_{el}$ in particular when the
Seebeck coefficient of the material is large, which is exactly the case of
interest.
The present theoretical study is based on the same DFT concept as applied for
first-principles calculations of Seebeck coefficients (for example, see Ref.
zeiringer_2011, ). In the present work the WF law is generalized by
introducing a material and temperature dependent Lorenz tensor $\mathbf{L}$,
for which Boltzmann’s transport theory in combination with electronic
properties derived by DFT calculation is used. This procedure is applied for
the clathrate type-I compound Ba8Au6-xGe40+x, which is a prototypical
thermoelectric material and for which also very recent measurements of Seebeck
coefficients are available enabling a test of the validity of the present
theoretical approach.
For the present purpose, the WF law is generalized to
$\mathbf{\kappa_{el}}=T\mathbf{L\sigma}~{},$ (1)
in which the tensor $\mathbf{\kappa_{el}}$ is linearly related to the
conductivity tensor $\mathbf{\sigma}$ (as defined in Eq. 6) via the Lorenz
tensor $\mathbf{L}$. These quantities as well as the Seebeck tensor
$\mathbf{S}$ (Eq. 7) are second rank tensors nye_tensor . In accordance with
Boltzmann’s transport theory one derives tritt_thermal_2004
$\mathbf{\kappa_{el}}=\frac{1}{T}\left(\mathbf{K_{2}}-\mathbf{K_{1}}^{2}\mathbf{K_{0}}^{-1}\right)$
(2)
for which Eq. A is utilized for the definition of the tensors
$\mathbf{K_{n}}$. The Lorenz tensor $\mathbf{L}$ can now be formulated as
$\mathbf{L}=\mathbf{L_{1}}-\mathbf{S}^{2}$ (3)
whereby $\mathbf{L_{1}}$ is expressed as tritt_thermal_2004
$\mathbf{L_{1}}=\frac{1}{e^{2}T^{2}}\mathbf{K_{2}K_{0}}^{-1}~{}.$ (4)
For a free-electron like metal the second term at the right hand side of Eq. 3
is negligible, because $|S|$ is small. It is then obvious that the deviation
from the free-electron like behavior is caused by the Seebeck coefficient in
terms of $-\mathbf{S}^{2}$. At low temperatures the original WF law is a
reasonable approximation, i.e. $\mathbf{L_{1}}\approx\mathbf{L_{0}}$, whereby
$\mathbf{L_{0}}$ would be a tensor with constant coefficients
L${}_{0}=\frac{\pi^{2}k_{B}^{2}}{3}=2.44\times 10^{-8}W\Omega/K^{2}$,
involving Boltzmann’s constant $k_{B}$.
When assuming a constant relaxation time $\tau=const$ –as it is the standard
approach for first-principles calculations of the Seebeck coefficients (for
example, see Ref. zeiringer_2011, )– then $\tau$ cancels out in the components
of $\mathbf{L}$ since it appears in the numerator as well as in the
denominator in Eqs. 3 and 4. The same holds for the Seebeck coefficients
because of Eq. 7. For the following discussion it should be noted that the
crystal structure of the material under study is of cubic symmetry. As a
consequence of this high symmetry all second rank tensors are diagonal and the
three diagonal coefficients are equal. Therefore, only one coefficient needs
to be considered for each tensor. However, the derivations and calculations
can be done for a general crystal symmetry and for tensors with less symmetry
and more components. If the tensors are symmetry averaged (as needed for a
polycrystalline material), again only one coefficient needs to be considered.
What remains to be done is the DFT calculation of the electronic structure of
the actual material, Ba8Au6Ge40. For that purpose, the Vienna Ab initio
Simulation Package (VASP) kresse_efficiency_1996 ; kresse_efficient_1996 was
used for which the pseudopotentials were constructed according to the
projector augmented wave method bloechl_projector_1994 ; kresse_ultrasoft_1999
. The exchange-–correlation functional was parametrized in terms of the local
density approximation according to Ceperley and Alder ceperley_ground_1980 .
The valence state configuration for the construction of the pseudopotentials
included the 5s, 5p and 6s atomic states for Ba, the 6s and 5d states for Au,
and the 3d, 4s and 4p states for Ge. For the Brillouin zone integration a 5
$\times$ 5 $\times$ 5 grid of $\mathbf{k}$-points was found to be sufficiently
accurate concerning the relaxed structural parameters as mentioned in the
caption of Fig. 1(a). Electronic transport properties were derived by
utilizing the Boltzmann transport equations within the constant relaxation
time approximation as implemented in the program package BoltzTrap
zeiringer_2011 ; madsen_boltztrap._2006 ; chen_thesis_2012 . For that purpose
the Kohn-Sham eigenvalues $\varepsilon_{i}(\mathbf{k})$ were determined on a
very dense 25 $\times$ 25 $\times$ 25 grid of $\mathbf{k}$-points. For the
investigation of the thermoelectric properties of off-stoichiometric compounds
Ba8Au6-xGe40+x with varying $x$ or number of valence electrons (i.e. doping
$\Delta N$) the rigid-band construction was used which consists in shifting
the Fermi energy according to the doping without changing the underlying
electronic structure.
Figure 1: DFT results for Ba8Au6Ge40: (a) density of states (DOS) vs. energy
$E$. The Fermi levels of the undoped ($E=0$ for $x=0$) and the electron doped
($x=0.27,\Delta$N = 0.8) compound are indicated by full and dashed lines,
respectively. Doping was modelled within the rigid band approximation. The
type-I clathrate structure is cubic with space group Pm-3n. The relaxed cubic
lattice parameter is $a=10.70$ Å, and the site specific coordinates are 2a
(0,0,0) and 6c (0.25,0.5,0) for Ba, 6d (0.25,0.5,0) for Au, 16i (0.183, 0.183,
0.183) and 24k (0, 0.117, 0.309) for Ge. (b) constant Lorenz number $L_{0}$
(dashed red line), coefficient $L$ of the cubic Lorenz tensor (solid blue
line) and its approximation L1 (solid black line) (Eqs. 3, 4) at 300 K vs.
rigid-band doping $\Delta N$ (lower abscissa) or carrier concentration (upper
abscissa) $n$. Negative/positive values of $\Delta N$ or $n$ correspond to
hole/electron doping or carriers. For $\Delta N=2$ the Fermi energy falls into
the gap of the DOS. The inset shows the carrier-concentration dependent
behavior of the resistivity $\rho$, the inverse of the conductivity $\sigma$
as calculated according to Eq. 6 for a constant relaxation time $\tau$ = 1
$\times$ 10-14 s.
Fig. 1(a) shows the density of states (DOS) around Fermi energy. A gap of
about 0.3 eV occurs about 0.15 eV above the Fermi energy $E_{F}$ for the
undoped case. Electron dopings of up to $\Delta N=1.2$ places $E_{F}$ closer
to the gap where the DOS diminishes rather strongly. For such a situation a
large Seebeck coefficient is expected, as is indeed the case (see inset in
Fig. 2). Assuming two valence electrons for each Ba atom, one valence electron
for each Au atom and for valence electrons for each Ge atom, the total number
of valence electrons of Ba8Au6Ge40 amounts to 182. Finally, by adding $\Delta
N=2$ electrons the corresponding $E_{F}$ falls into a gap because then the
total number of valence electrons is 184, which is precisely the total number
of valence electrons of Ge46 in its clathrate structure, which at Fermi energy
has a rather large gap of about 1.6 eV. This is a remarkable feature since the
electronic structure of Ge in its diamond ground state structure reveals no
gap at all when a local approximation of the DFT exchange-correlation
interactions is utilized, as it is the case here. Such an underestimation of
the gap size is a well-known shortcoming of standard DFT calculations.
Assuming that the gap is preserved when filler atoms such as Ba are placed in
the voids of the clathrate structure and elements such as Au are substituting
Ge according to the composition Ba8Au6-xGe40+x, then the relation $x=\Delta
N/3$ can be established. The critical composition, i.e. the Fermi energy falls
into the gap, would then be $x_{crit.}=2/3$.
Fig. 1(b) depicts the coefficients $L$ and $L_{1}$ of the cubic Lorenz tensor
(see Eqs. 3 and 4) at 300 K as functions of doping. The inset shows the
carrier-density-dependency of the resistivity $\rho$ which is the inverse of
the conductivity $\sigma$ in Eq. 6. Significant deviations between the Lorenz
coefficients occur in the range of $0\leq\Delta N\leq 4$, which increase
strongly when the doping is near to $\Delta N=2$. At $\Delta N=2$ the
coefficients $L$ and $L_{1}$ are undefined because in the gap no electronic
states are available for transport and therefore at lower temperatures the
electrical conductivity $\mathbf{\sigma}$ is zero (or the resistivity has a
pole, see inset in Fig. 1(b) ), and as a consequence the Lorenz coefficients
reveal singularities. The comparison between $L_{1}$ and $L$ demonstrates the
influence of the Seebeck coefficient in terms of -$S^{2}$, which increases
dramatically as the Fermi level approaches the gap upon doping.
For thermoelectric applications, both $S$ as well as $\mathbf{\sigma}$ should
be large for obtaining a large power factor $S^{2}\mathbf{\sigma}$. For such a
purpose, a suitable carrier concentration is required. In the case of
Ba8Au6-xGe40+x, a doping $\Delta N$ up to 1.2 yields a carrier concentration
of about $n=-2\times 10^{20}$ e/cm3, which is within the desired range for
thermoelectric properties. The negative sign of $n$ refers to hole carriers,
i.e., Fermi energy is below the gap, a positive sign just refers to the
opposite case, i.e., Fermi energy above the gap.
For dopings $\Delta N>4,\Delta N<0$ the coefficients $L$ and $L_{1}$ nearly
coincide and their values become comparable to the constant Lorenz number but
$L_{0}$ could still be off by 20%. The results suggest that the approximation
by $L_{1}$ is not useful for thermoelectric materials with large Seebeck
coefficients.
Figure 2: DFT results for Ba8Au6Ge40: Coefficients $L$ (full lines) and
$L_{1}$ (dotted lines) of the Lorenz tensor as a function of temperature for
no doping (black) and for a doping of $\Delta N=0.8$ electrons (blue). The
constant Lorenz number L0 of the Wiedemann-Franz law is indicated by a dashed
horizontal line in red. The inset shows the corresponding DFT-derived Seebeck
coefficients in comparison to the experimental values of Ref.
zeiringer_phase_2011 .
Fig. 2 reveals the temperature dependency of $L_{1}$ and $L$ for no doping and
$\Delta N=0.8$. The inset compares DFT–derived and experimental Seebeck
coefficients revealing very good agreement between measurement and DFT
calculation for a doping of $\Delta N=0.8$. Such a doping refers to an off-
stoichiometry of $x=0.27$, which is within the experimental error measuring
the composition zeiringer_phase_2011 . Both $L_{1}$ and $L$ approach the WF
limit $L_{0}$ at low temperatures but deviate significantly at elevated
temperatures. Overall, $L$ and $L_{1}$ exhibit a strong temperature dependency
and the deviation of the constant value $L_{0}$ becomes large in particular
around 700 K, which is in the temperature range of technological applications.
As demonstrated by the inset of Fig.2 the assumption of a constant relaxation
time $\tau$ works well for the evaluation of the Seebeck coefficient.
Nevertheless, we tested an energy dependent relaxation time
$\tau(\varepsilon)=const\,\,\varepsilon^{-1/2}$ according to Ref. blatt_1968,
recalculating $L$ as well as $S$. This ansatz for $\tau(\varepsilon)$ is
supposed to model the scattering of electrons by acoustic phonons. The energy
$\varepsilon=|E_{F}-\varepsilon_{\vec{k},\nu}|$ was defined as the absolute
value of the difference of Fermi energy and the respective band energy for
band $\nu$ and vector $\vec{k}$.
Figure 3: DFT results for Ba8Au6Ge40: Seebeck coefficient $S(T)$ and the
component $L(T)$ of the full Lorenz tensor for no doping and dopings of
$\Delta N=0.8,1.2$. Results with constant relaxation time $\tau_{0}$ (full
lines) and with the energy-dependent ansatz
$\tau(\varepsilon)=const\,\,\varepsilon^{-1/2}$ (dotted lines).
By considering this specific $\tau(\varepsilon)$, based on the energy
dispersions of the two-band Kane approximation Huang et al. demonstrated for
Bi2Te3huang_ab_2008 , that a constant relaxation time yields a slightly larger
Lorenz number than the energy dependent ansatz. We performed similar
calculations for Ba8Au6Ge40 for the dopings $\Delta N=0,0.8,1.2$ by utilizing
the DFT-derived electronic structure. In this context, it should be noted that
for deriving the relation $\tau(\epsilon)\propto\epsilon^{-1/2}$ bands with
parabolic dispersions near Fermi energy are assumed, which is not really the
case for the material under study. Nevertheless, the results for $S(T)$ and
$L(T)$ in Fig. 3 show that the influence of the assumed energy dependency of
$\tau(\varepsilon)$ is rather small, although it becomes more pronounced at
elevated temperatures and larger dopings with Fermi energy approaching the
gap.
Figure 4: DFT derived electronic thermal conductivities $\kappa_{el}$ for
Ba8Au6-xGe40+x. Results for a doping of $\Delta N=0.8$ using the
approximations $L_{0}$,$L_{1}$,$L_{0}-S^{2}$ and $L$ of the full calculation.
$S$ is the Seebeck coefficient.
For finally deriving the electronic thermal conductivity $\kappa_{el}$ from
Eq. 1 the full Lorenz coefficient $L$ together with its approximations $L_{1}$
and $L_{0}$ are used. The results in Fig. 4 reflect the behavior of the Lorenz
coefficients in Fig. 2 showing again very significant deviations up to 40%
between the full calculation and the result involving the approximations
$L_{1}$ and $L_{0}$. Clearly, just using the simple WF law (i.e. $L_{0}$) may
lead to rather unreliable values for the electronic thermal conductivity, and
consequently for the lattice thermal conductivity when it is derived from the
measured total thermal conductivity by
$\kappa_{ph}=\kappa^{meas.}-\kappa_{el}$. The rather similar behavior of
thermal conductivities $\kappa_{el}$ in Fig. 4 as calculated with $L$ and the
approximation $L_{app.}=L_{0}-S^{2}$ would suggest to use $\kappa_{el}\approx
TL_{app}\sigma$ for a reasonable estimation of $\kappa_{el}$. This would have
the big advantage, that $\kappa_{ph}$ could be estimated from measurable
quantities, namely the Seebeck coefficient $S(T)$ and the electrical
conductivity $\mathbf{\sigma}$. This observation, however, is only useful as
long as $S^{2}$ is not too large and temperatures are sufficiently.
Figure 5: DFT derived electronic thermal conductivities $\kappa_{el}$ for
Ba8Au6-xGe40+x. Results for a series of dopings $\Delta N$ using
$L_{0}$,$L_{0}-S^{2}$, and $L$ as Lorenz coefficients. The corresponding
carrier concentrations are also indicated.
As Fig. 5 however reveals, for dopings $\Delta N\geq 0.8$ the use of the
approximation $L_{0}-S^{2}$ grossly underestimates the results obtained with
$L$, the Lorenz coefficient calculated without any approximation (apart from
the assumption of a constant relaxation time). The errors become larger the
nearer $E_{F}$ approaches the gap. This is attribute to the fact that $L_{1}$
deviates $L_{0}$ significantly when the system becomes
semiconducting/insulating as shown in the inset of Fig. 1(b). Likewise, only
using the constant $L_{0}$ of the Wiedemann-Franz results in significant
overestimations. Therefore the conclusion has to be made, that it is mandatory
to derive the electronic thermal conductivity from DFT calculations in
combination with Boltzmann’s transport theory in terms of the full Lorenz-
tensor $\mathbf{L}$. Otherwise the estimated $\kappa_{el}$ and, subsequently,
$\kappa_{ph}$ might be rather wrong, which is undesirable for optimizing
thermoelectric properties of a material.
###### Acknowledgements.
The authors gratefully acknowledge the support by the Austrian Science
Foundation FWF under project nr. P24380-N16. The DFT calculations were done on
the Vienna Scientific Cluster (VSC).
*
## Appendix A
The tensor needed in Boltzmann’s transport theory is
$\displaystyle\mathbf{K_{n}}$ $\displaystyle=$
$\displaystyle\frac{1}{4\pi^{3}}\sum_{i,\mathbf{k}}\tau_{i}(\mathbf{k})\mathbf{v}_{i}(\mathbf{k})\otimes\mathbf{v}_{i}(\mathbf{k})(\varepsilon_{i}(\mathbf{k})-\mu)^{n}$
$\displaystyle\left(-\frac{\partial
f(\mu,T,\varepsilon_{i})}{\partial\varepsilon_{i}}\right)$
in which $\tau_{i}(\mathbf{k})$ is the relaxation time of the electronic
states with band index $i$, energy eigenvalue $\varepsilon_{i}$ and band
velocities $\mathbf{v}_{i}$ for wave vector $\mathbf{k}$, whereas $f(\mu,T$
denotes the Fermi-Dirac distribution function for the chemical potential $\mu$
at temperature $T$.
The electrical conductivity tensor $\mathbf{\sigma}$ is given by
$\mathbf{\sigma}=e^{2}\mathbf{K_{0}}$ (6)
and the Seebeck tensor $\mathbf{S}$ is defined by
$\mathbf{S}=\frac{1}{eT}\mathbf{K_{1}K_{0}}^{-1}.$ (7)
## References
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|
arxiv-papers
| 2012-12-04T17:33:11 |
2024-09-04T02:49:38.874372
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "M. X. Chen and R. Podloucky",
"submitter": "Mingxing Chen",
"url": "https://arxiv.org/abs/1212.0803"
}
|
1212.0945
|
# Multiclass Diffuse Interface Models for
Semi-Supervised Learning on Graphs
Cristina Garcia-Cardona1, Arjuna Flenner2 and Allon G. Percus1
1Institute of Mathematical Sciences, Claremont Graduate University, Claremont,
CA 91711, USA
2Naval Air Warfare Center, Physics and Computational Sciences, China Lake, CA
93555, USA
[email protected], [email protected], [email protected]
###### Abstract
We present a graph-based variational algorithm for multiclass classification
of high-dimensional data, motivated by total variation techniques. The energy
functional is based on a diffuse interface model with a periodic potential. We
augment the model by introducing an alternative measure of smoothness that
preserves symmetry among the class labels. Through this modification of the
standard Laplacian, we construct an efficient multiclass method that allows
for sharp transitions between classes. The experimental results demonstrate
that our approach is competitive with the state of the art among other graph-
based algorithms.
## 1 INTRODUCTION
Many tasks in pattern recognition and machine learning rely on the ability to
quantify local similarities in data, and to infer meaningful global structure
from such local characteristics [Coifman et al., 2005]. In the classification
framework, the desired global structure is a descriptive partition of the data
into categories or classes. Many studies have been devoted to the binary
classification problems. The multiple-class case, where the data is
partitioned into more than two clusters, is more challenging. One approach is
to treat the problem as a series of binary classification problems [Allwein et
al., 2000]. In this paper, we develop an alternative method, involving a
multiple-class extension of the diffuse interface model introduced in
[Bertozzi and Flenner, 2012].
The diffuse interface model by Bertozzi and Flenner combines methods for
diffusion on graphs with efficient partial differential equation techniques to
solve binary segmentation problems. As with other methods inspired by physical
phenomena [Bertozzi et al., 2007, Jung et al., 2007, Li and Kim, 2011], it
requires the minimization of an energy expression, specifically the Ginzburg-
Landau (GL) energy functional. The formulation generalizes the GL functional
to the case of functions defined on graphs, and its minimization is related to
the minimization of weighted graph cuts [Bertozzi and Flenner, 2012]. In this
sense, it parallels other techniques based on inference on graphs via
diffusion operators or function estimation [Coifman et al., 2005, Chung, 1997,
Zhou and Schölkopf, 2004, Szlam et al., 2008, Wang et al., 2008, Bühler and
Hein, 2009, Szlam and Bresson, 2010, Hein and Setzer, 2011].
Multiclass segmentation methods that cast the problem as a series of binary
classification problems use a number of different strategies: (i) deal
directly with some binary coding or indicator for the labels [Dietterich and
Bakiri, 1995, Wang et al., 2008], (ii) build a hierarchy or combination of
classifiers based on the one-vs-all approach or on class rankings [Hastie and
Tibshirani, 1998, Har-Peled et al., 2003] or (iii) apply a recursive
partitioning scheme consisting of successively subdividing clusters, until the
desired number of classes is reached [Szlam and Bresson, 2010, Hein and
Setzer, 2011]. While there are advantages to these approaches, such as
possible robustness to mislabeled data, there can be a considerable number of
classifiers to compute, and performance is affected by the number of classes
to partition.
In contrast, we propose an extension of the diffuse interface model that
obtains a simultaneous segmentation into multiple classes. The multiclass
extension is built by modifying the GL energy functional to remove the
prejudicial effect that the order of the labelings, given by integer values,
has in the smoothing term of the original binary diffuse interface model. A
new term that promotes homogenization in a multiclass setup is introduced. The
expression penalizes data points that are located close in the graph but are
not assigned to the same class. This penalty is applied independently of how
different the integer values are, representing the class labels. In this way,
the characteristics of the multiclass classification task are incorporated
directly into the energy functional, with a measure of smoothness independent
of label order, allowing us to obtain high-quality results. Alternative
multiclass methods minimize a Kullback-Leibler divergence function [Subramanya
and Bilmes, 2011] or expressions involving the discrete Laplace operator on
graphs [Zhou et al., 2004, Wang et al., 2008].
This paper is organized as follows. Section 2 reviews the diffuse interface
model for binary classification, and describes its application to semi-
supervised learning. Section 3 discusses our proposed multiclass extension and
the corresponding computational algorithm. Section 4 presents results obtained
with our method. Finally, section 5 draws conclusions and delineates future
work.
## 2 DATA SEGMENTATION WITH THE GINZBURG-LANDAU MODEL
The diffuse interface model [Bertozzi and Flenner, 2012] is based on a
continuous approach, using the Ginzburg-Landau (GL) energy functional to
measure the quality of data segmentation. A good segmentation is characterized
by a state with small energy. Let $u(\boldsymbol{x})$ be a scalar field
defined over a space of arbitrary dimensionality, and representing the state
of the system. The GL energy is written as the functional
$E_{GL}(u)=\frac{\epsilon}{2}\int\\!|\nabla
u|^{2}\;d\boldsymbol{x}+\frac{1}{\epsilon}\int\\!F(u)\;d\boldsymbol{x},$ (1)
with $\nabla$ denoting the spatial gradient operator, $\epsilon>0$ a real
constant value, and $F$ a double well potential with minima at $\pm 1$:
$F(u)=\frac{1}{4}\left(u^{2}-1\right)^{2}.$ (2)
Segmentation requires minimizing the GL functional. The norm of the gradient
is a smoothing term that penalizes variations in the field $u$. The potential
term, on the other hand, compels $u$ to adopt the discrete labels of $+1$ or
$-1$, clustering the state of the system around two classes. Jointly
minimizing these two terms pushes the system domain towards homogeneous
regions with values close to the minima of the double well potential, making
the model appropriate for binary segmentation.
The smoothing term and potential term are in conflict at the interface between
the two regions, with the first term favoring a gradual transition, and the
second term penalizing deviations from the discrete labels. A compromise
between these conflicting goals is established via the constant $\epsilon$. A
small value of $\epsilon$ denotes a small length transition and a sharper
interface, while a large $\epsilon$ weights the gradient norm more, leading to
a slower transition. The result is a diffuse interface between regions, with
sharpness regulated by $\epsilon$.
It can be shown that in the limit $\epsilon\to 0$ this function approximates
the total variation (TV) formulation in the sense of functional ($\Gamma$)
convergence [Kohn and Sternberg, 1989], producing piecewise constant solutions
but with greater computational efficiency than conventional TV minimization
methods. Thus, the diffuse interface model provides a framework to compute
piecewise constant functions with diffuse transitions, approaching the ideal
of the TV formulation, but with the advantage that the smooth energy
functional is more tractable numerically and can be minimized by simple
numerical methods such as gradient descent.
The GL energy has been used to approximate the TV norm for image segmentation
[Bertozzi and Flenner, 2012] and image inpainting [Bertozzi et al., 2007,
Dobrosotskaya and Bertozzi, 2008]. Furthermore, a calculus on graphs
equivalent to TV has been introduced in [Gilboa and Osher, 2008, Szlam and
Bresson, 2010].
### Application of Diffuse Interface Models to Graphs
An undirected, weighted neighborhood graph is used to represent the local
relationships in the data set. This is a common technique to segment classes
that are not linearly separable. In the $N$-neighborhood graph model, each
vertex $z_{i}\in Z$ of the graph corresponds to a data point with feature
vector $\boldsymbol{x}_{i}$, while the weight $w_{ij}$ is a measure of
similarity between $z_{i}$ and $z_{j}$. Moreover, it satisfies the symmetry
property $w_{ij}=w_{ji}$. The neighborhood is defined as the set of $N$
closest points in the feature space. Accordingly, edges exist between each
vertex and the vertices of its $N$-nearest neighbors. Following the approach
of [Bertozzi and Flenner, 2012], we calculate weights using the local scaling
of Zelnik-Manor and Perona [Zelnik-Manor and Perona, 2005],
$w_{ij}=\exp\left(-\frac{||\boldsymbol{x}_{i}-\boldsymbol{x}_{j}||^{2}}{\tau(\boldsymbol{x}_{i})\;\tau(\boldsymbol{x}_{j})}\right).$
(3)
Here, $\tau(\boldsymbol{x}_{i})=||\boldsymbol{x}_{i}-\boldsymbol{x}^{M}_{i}||$
defines a local value for each $\boldsymbol{x}_{i}$, where
$\boldsymbol{x}^{M}_{i}$ is the position of the $M$th closest data point to
$\boldsymbol{x}_{i}$, and $M$ is a global parameter.
It is convenient to express calculations on graphs via the graph Laplacian
matrix, denoted by $\boldsymbol{L}$. The procedure we use to build the graph
Laplacian is as follows.
1. 1.
Compute the similarity matrix $\boldsymbol{W}$ with components $w_{ij}$
defined in (3). As the neighborhood relationship is not symmetric, the
resulting matrix $\boldsymbol{W}$ is also not symmetric. Make it a symmetric
matrix by connecting vertices $z_{i}$ and $z_{j}$ if $z_{i}$ is among the
$N$-nearest neighbors of $z_{j}$ or if $z_{j}$ is among the $N$-nearest
neighbors of $z_{i}$ [von Luxburg, 2006].
2. 2.
Define $\boldsymbol{D}$ as a diagonal matrix whose $i$th diagonal element
represents the degree of the vertex $z_{i}$, evaluated as
$d_{i}=\sum_{j}w_{ij}.$ (4)
3. 3.
Calculate the graph Laplacian: $\boldsymbol{L}=\boldsymbol{D}-\boldsymbol{W}$.
Generally, the graph Laplacian is normalized to guarantee spectral convergence
in the limit of large sample size [von Luxburg, 2006]. The symmetric
normalized graph Laplacian $\boldsymbol{L_{s}}$ is defined as
$\boldsymbol{L_{s}}=\boldsymbol{D}^{-1/2}\;\boldsymbol{L}\;\boldsymbol{D}^{-1/2}=\boldsymbol{I}-\boldsymbol{D}^{-1/2}\;\boldsymbol{W}\;\boldsymbol{D}^{-1/2}.$
(5)
Data segmentation can now be carried out through a graph-based formulation of
the GL energy. To implement this task, a fidelity term is added to the
functional as initially suggested in [Dobrosotskaya and Bertozzi, 2010]. This
enables the specification of a priori information in the system, for example
the known labels of certain points in the data set. This kind of setup is
called semi-supervised learning (SSL). The discrete GL energy for SSL on
graphs can be written as [Bertozzi and Flenner, 2012]:
$\displaystyle E_{GL_{\mathrm{SSL}}}(\boldsymbol{u})$ $\displaystyle=$
$\displaystyle\frac{\epsilon}{2}\langle\boldsymbol{u},\boldsymbol{L_{s}}\boldsymbol{u}\rangle+\frac{1}{\epsilon}\sum_{z_{i}\in
Z}F(u(z_{i}))$ (6) $\displaystyle+\sum_{z_{i}\in
Z}\frac{\lambda(z_{i})}{2}\;\left(u(z_{i})-u_{0}(z_{i})\right)^{2}$
In the discrete formulation, $\boldsymbol{u}$ is a vector whose component
$u(z_{i})$ represents the state of the vertex $z_{i}$, $\epsilon>0$ is a real
constant characterizing the smoothness of the transition between classes, and
$\lambda(z_{i})$ is a fidelity weight taking value $\lambda>0$ if the label
$u_{0}(z_{i})$ (i.e. class) of the data point associated with vertex $z_{i}$
is known beforehand, or $\lambda(z_{i})=0$ if it is not known (semi-
supervised).
Equation (6) may be understood as an example of the more general form of an
energy functional for data classification,
$\displaystyle
E(\boldsymbol{u})=||\boldsymbol{u}||_{a}+\frac{\lambda}{2}||\boldsymbol{u}-\boldsymbol{f}||_{b}^{p},$
(7)
where the norm $||u||_{a}$ is a regularization term and $||u-f||_{b}$ is a
fidelity term. The choice of the regularization norm $||\cdot||_{a}$ has non-
trivial consequences in the final classification accuracy. Attractive
qualities of the norm $||\cdot||_{a}$ include allowing classes to be close in
a metric space, and obtain segmentations for nonlinearly separable data. Both
of these goals are addressed using the GL energy functional for SSL.
Minimizing the functional simulates a diffusion process on the graph. The
information of the few labels known is propagated through the discrete
structure by means of the smoothing term, while the potential term clusters
the vertices around the states $\pm 1$ and the fidelity term enforces the
known labels. The energy minimization process itself attempts to reduce the
interface regions. Note that in the absence of the fidelity term, the process
could lead to a trivial steady-state solution of the diffusion equation, with
all data points assigned the same label.
The final state $u(z_{i})$ of each vertex is obtained by thresholding, and the
resulting homogeneous regions with labels of $+1$ and $-1$ constitute the two-
class data segmentation.
## 3 MULTICLASS EXTENSION
The double-well potential in the diffuse interface model for SSL flows the
state of the system towards two definite labels. Multiple-class segmentation
requires a more general potential function $F(u)$ that allows clusters around
more than two labels. For this purpose, we use the periodic-well potential
suggested by Li and Kim [Li and Kim, 2011],
$F(u)=\frac{1}{2}\,\\{u\\}^{2}\,(\\{u\\}-1)^{2},$ (8)
where $\\{u\\}$ denotes the fractional part of $u$,
$\\{u\\}=u-\lfloor u\rfloor,$ (9)
and $\lfloor u\rfloor$ is the largest integer not greater than $u$.
This periodic potential well promotes a multiclass solution, but the graph
Laplacian term in Equation (6) also requires modification for effective
calculations due to the fixed ordering of class labels in the multiple class
setting. The graph Laplacian term penalizes large changes in the spatial
distribution of the system state more than smaller gradual changes. In a
multiclass framework, this implies that the penalty for two spatially
contiguous classes with different labels may vary according to the (arbitrary)
ordering of the labels.
This phenomenon is shown in Figure 1. Suppose that the goal is to segment the
image into three classes: class 0 composed by the black region, class 1
composed by the gray region and class 2 composed by the white region. It is
clear that the horizontal interfaces comprise a jump of size 1 (analogous to a
two class segmentation) while the vertical interface implies a jump of size 2.
Accordingly, the smoothing term will assign a higher cost to the vertical
interface, even though from the point of view of the classification, there is
no specific reason for this. In this example, the problem cannot be solved
with a different label assignment. There will always be an interface with
higher costs than others independent of the integer values used.
Thus, the multiclass approach breaks the symmetry among classes, influencing
the diffuse interface evolution in an undesirable manner. Eliminating this
inconvenience requires restoring the symmetry, so that the difference between
two classes is always the same, regardless of their labels. This objective is
achieved by introducing a new class difference measure.
Figure 1: Three class segmentation. Black: class 0. Gray: class 1. White:
class 2.
### 3.1 Generalized Difference Function
The final class labels are determined by thresholding each vertex $u(z_{i})$,
with the label $y_{i}$ set to the nearest integer:
$y_{i}=\left\lfloor u(z_{i})+\frac{1}{2}\right\rfloor.$ (10)
The boundaries between classes then occur at half-integer values corresponding
to the unstable equilibrium states of the potential well. Define the function
$\hat{r}(x)$ to represent the distance to the nearest half-integer:
$\hat{r}(x)=\left|\frac{1}{2}-\\{x\\}\right|.$ (11)
A schematic of $\hat{r}(x)$ is depicted in Figure 2. The $\hat{r}(x)$ function
is used to define a generalized difference function between classes that
restores symmetry in the energy functional. Define the generalized difference
function $\rho$ as:
$\rho(u(z_{i}),u(z_{j}))=\left\\{\begin{array}[]{lll}\hat{r}(u(z_{i}))+\hat{r}(u(z_{j}))&&y_{i}\neq
y_{j}\\\ &&\\\
\left|\hat{r}(u(z_{i}))-\hat{r}(u(z_{j}))\right|&&y_{i}=y_{j}\end{array}\right.$
(12)
Thus, if the vertices are in different classes, the difference $\hat{r}(x)$
between each state’s value and the nearest half-integer is added, whereas if
they are in the same class, these differences are subtracted. The function
$\rho(x,y)$ corresponds to the tree distance (see Fig. 2). Strictly speaking,
$\rho$ is not a metric since it does not satisfy $\rho(x,y)=0\Rightarrow x=y$.
Nevertheless, the cost of interfaces between classes becomes the same
regardless of class labeling when this generalized distance function is
implemented.
Half-integerInteger$\hat{r}(x)$ Figure 2: Schematic interpretation of
generalized difference: $\hat{r}(x)$ measures distance to nearest half-
integer, and $\rho$ then corresponds to distance on tree.
The GL energy functional for SSL, using the new generalized difference
function $\rho$, is expressed as
$\displaystyle E_{MGL_{\mathrm{SSL}}}(\boldsymbol{u})$ $\displaystyle=$
$\displaystyle\frac{\epsilon}{2}\sum_{z_{i}\in Z}\sum_{z_{j}\in
Z}\frac{w_{ij}}{\sqrt{d_{i}d_{j}}}\,\left[\rho(u(z_{i}),u(z_{j}))\,\right]^{2}$
(13) $\displaystyle+\frac{1}{2\epsilon}\sum_{z_{i}\in
Z}\\{u(z_{i})\\}^{2}\,(\\{u(z_{i})\\}-1)^{2}$ $\displaystyle+\sum_{z_{i}\in
Z}\frac{\lambda(z_{i})}{2}\;\left(u(z_{i})-u_{0}(z_{i})\right)^{2}.$
Note that $\rho$ could also be used in the fidelity term, but for simplicity
this modification is not included. In practice, this has little effect on the
results.
### 3.2 Computational Algorithm
The GL energy functional given by (13) may be minimized iteratively, using
gradient descent:
$u_{i}^{m+1}=u_{i}^{m}-dt\,\left[\frac{\delta E_{MGL_{\mathrm{SSL}}}}{\delta
u_{i}}\right],$ (14)
where $u_{i}$ is a shorthand for $u(z_{i})$, $dt$ represents the time step and
the gradient direction is given by:
$\frac{\delta E_{MGL_{\mathrm{SSL}}}}{\delta u_{i}}=\epsilon
G(u_{i}^{m})+\frac{1}{\epsilon}F^{\prime}(u_{i}^{m})+\lambda_{i}\left(u_{i}^{m}-{u_{i}}_{0}\right)$
(15)
$G(u_{i}^{m})=\sum_{j}\frac{w_{ij}}{\sqrt{d_{i}d_{j}}}\left[\hat{r}(u_{i}^{m})\pm\hat{r}(u_{j}^{m})\right]\hat{r}^{\prime}(u_{i}^{m})$
(16)
$F^{\prime}(u_{i}^{m})=2\;\\{u_{i}^{m}\\}^{3}-3\;\\{u_{i}^{m}\\}^{2}+\\{u_{i}^{m}\\}$
(17)
Algorithm 1 Calculate $\boldsymbol{u}$
0: $\epsilon>0,dt>0,m_{\mathrm{max}}>0,K\mathrm{~{}given}$
0: $\mathrm{out}=\boldsymbol{u}^{m_{\mathrm{max}}}$
$\boldsymbol{u}^{0}\leftarrow rand((0,K))-\frac{1}{2},\ m\leftarrow 0$
for $m<m_{\mathrm{max}}$ do
$i\leftarrow 0$
for $i<n$ do
$u_{i}^{m+1}\leftarrow
u_{i}^{m}-dt\left(\epsilon\>G(u_{i}^{m})+\frac{1}{\epsilon}\>F^{\prime}(u_{i}^{m})+\lambda_{i}\left(u_{i}^{m}-{u_{i}}_{0}\right)\right)$
if $\mathrm{Label}(u_{i}^{m+1})\neq\mathrm{Label}(u_{i}^{m})$ then
$(v_{i})_{k}\leftarrow k+\\{u_{i}^{m+1}\\}$
$u_{i}^{m+1}\leftarrow(v_{i})_{k}\mathrm{~{}where~{}}k=\arg\min_{\;0\leq
k<K}\;\sum_{j}\frac{w_{ij}}{\sqrt{d_{i}d_{j}}}\,\left[\rho((v_{i})_{k},u_{j})\,\right]^{2}$
end if
$i\leftarrow i+1$
end for
$m\leftarrow m+1$
end for
The gradient of the generalized difference function $\rho$ is not defined at
half integer values. Hence, we modify the method using a greedy strategy:
after detecting that a vertex changes class, the new class that minimizes the
smoothing term is selected, and the fractional part of the state computed by
the gradient descent update is preserved. Consequently, the new state of
vertex $i$ is the result of gradient descent, but if this causes a change in
class, then a new state is determined.
Specifically, let $k$ represent an integer in the range of the problem, i.e.
$k\in[0,K-1]$, where $K$ is the number of classes in the problem. Given the
fractional part $\\{u\\}$ resulting from the gradient descent update, define
$(v_{i})_{k}=k+\\{u_{i}\\}$. Find the integer $k$ that minimizes
$\sum_{j}\frac{w_{ij}}{\sqrt{d_{i}d_{j}}}\,\left[\rho((v_{i})_{k},u_{j})\,\right]^{2}$,
the smoothing term in the energy functional, and use $(v_{i})_{k}$ as the new
vertex state. A summary of the procedure is shown in Algorithm 1 with
$m_{\mathrm{max}}$ denoting the maximum number of iterations.
## 4 RESULTS
The performance of the multiclass diffuse interface model is evaluated using a
number of data sets from the literature, with differing characteristics. Data
and image segmentation problems are considered on synthetic and real data
sets.
### 4.1 Synthetic Data
A synthetic three-class segmentation problem is constructed following an
analogous procedure used in [Bühler and Hein, 2009] for “two moon” binary
classification, using three half circles (“three moons”). The half circles are
generated in $\mathbb{R}^{2}$. The two top circles have radius $1$ and are
centered at $(0,0)$ and $(3,0)$. The bottom half circle has radius $1.5$ and
is centered at $(1.5,0.4)$. We sample 1500 data points (500 from each of these
half circles) and embed them in $\mathbb{R}^{100}$. The embedding is completed
by adding Gaussian noise with $\sigma^{2}=0.02$ to each of the 100 components
for each data point. The dimensionality of the data set, together with the
noise, make this a nontrivial problem.
Figure 3: Three-class segmentation. Left: spectral clustering. Right:
multiclass GL (adaptive $\epsilon$).
The difficulty of the problem is illustrated in Figure 3, where we use both
spectral clustering decomposition and the multiclass GL method. The same graph
structure is used for both methods. The symmetric graph Laplacian is computed
based on edge weights given by (3), using $N=10$ nearest neighbors and local
scaling based on the $M=10$ closest point. The spectral clustering results are
obtained by applying a $k$-means algorithm to the first $3$ eigenvectors of
the symmetric graph Laplacian. The average error obtained, over 100 executions
of spectral clustering, is 20% ($\pm 0.6\%$). The figure displays the best
result obtained, corresponding to an error of $18.67\%$.
The multiclass GL method was implemented with the following parameters:
interface scale $\epsilon=1$, step size $dt=0.01$ and number of iterations
$m_{\mathrm{max}}=800$. The fidelity term is determined by labeling 25 points
randomly selected from each class (5% of all points), and setting the fidelity
weight to $\lambda=30$ for those points. Several runs of the procedure are
performed to isolate effects from the random initialization and the arbitrary
selection of fidelity points. The average error obtained, over 100 runs with
four different fidelity sets, is 5.2% ($\pm 1.01\%$). In general terms, the
system evolves from an initially inhomogeneous state, rapidly developing small
islands around fidelity points that become seeds for homogeneous regions and
progressing to a configuration of classes forming nearly uniform clusters.
The multiclass results were further improved by incrementally decreasing
$\epsilon$ to allow sharper transitions between states as in [Bertozzi and
Flenner, 2012]. With this approach, the average error obtained over 100 runs
is reduced to 2.6% ($\pm 0.3\%$). The best result obtained in these runs is
displayed in Figure 3 and corresponds to an average error of 2.13%. In these
runs, $\epsilon$ is reduced from $\epsilon_{0}=2$ to $\epsilon_{f}=0.1$ in
decrements of 10%, with $40$ iterations performed per step. The average
computing time per run in this adaptive technique is 1.53s in an Intel Quad-
Core @ 2.4 GHz, without any parallel processing.
For comparison, we note the results from the literature for the simpler two
moon problem ($\mathbb{R}^{100}$, $\sigma^{2}=0.02$ noise). The best errors
reported include: 6% for p-Laplacian [Bühler and Hein, 2009], 4.6% for ratio-
minimization relaxed Cheeger cut [Szlam and Bresson, 2010], and 2.3% for
binary GL [Bertozzi and Flenner, 2012]. While these are not SSL methods the
last of these does involve other prior information in the form of a mass
balance constraint. It can be seen that both of our procedures, fixed and
adaptive $\epsilon$, produce high-quality results even for the more complex
three-class segmentation problem. Calculation times are also competitive with
those reported for the binary case (0.5s - 50s).
### 4.2 Image Segmentation
As another test setup, we use a grayscale image of size $191\times 196$, taken
from [Jung et al., 2007, Li and Kim, 2011] and composed of 5 classes: black,
dark gray, medium gray, light gray and white. This image contains structure,
such as an internal hole and junctions where multiple classes meet. The image
information is represented through feature vectors defined as
$(x_{i},y_{i},\mathrm{pix}_{i})$, with $x_{i}$ and $y_{i}$ corresponding to
$(x,y)$ coordinates of the pixel and $\mathrm{pix}_{i}$ equal to the intensity
of the pixel. All of these are normalized so as to obtain values in the range
$[0,1]$.
The graph is constructed using $N=30$ nearest neighbors and local scaling
based on the $M=30$ closest point. We use parameters $\epsilon=1$, $dt=0.01$
and $m_{\mathrm{max}}=800$. We then choose 1500 random points (4% of the
total) for the fidelity term, with $\lambda=30$. Figure 4 displays the
original image with the randomly selected fidelity points (top left), and the
five-class segmentation. Each class image shows in white the pixels identified
as belonging to the class, and in black the pixels of the other classes. In
this case, all the classes are segmented perfectly with an average run time of
59.7s. The method of Li and Kim [Li and Kim, 2011] also segments this image
perfectly, with a reported run time of 0.625s. However, their approach uses
additional information, including a pre-assignment of specific grayscale
levels to classes, and the overall densities of each class. Our approach does
not require these.
Figure 4: Image Segmentation Results. Top left: Original five-class image,
with randomly chosen fidelity points displayed. Other panels: the five
segmented classes, shown in white.
### 4.3 MNIST Data
The MNIST data set available at http://yann.lecun.com/exdb/mnist/ is composed
of 70,000 images of size $28\times 28$, corresponding to a broad sample of
handwritten digits 0 through 9. We use the multiclass diffuse interface model
to segment the data set automatically into 10 classes, one per handwritten
digit. Before constructing the graph, we preprocess the data by normalizing
and projecting into 50 principal components, following the approach in [Szlam
and Bresson, 2010]. No further steps, such as smoothing convolutions, are
required. The graph is computed with $N=10$ nearest neighbors and local
scaling based on the $M=10$ closest points.
An adaptive $\epsilon$ variant of the algorithm is implemented, with
parameters $\epsilon_{0}=2$, $\epsilon_{f}=0.01$, $\epsilon$ decrement 10%,
$dt=0.01$, and 40 iterations per step. For the fidelity term, 7,000 images
(10% of total) are chosen, with weight $\lambda=30$. The average error
obtained, over 20 runs with four different fidelity sets, is 7% ($\pm
0.072\%$). The confusion matrix for the best result obtained, corresponding to
a 6.86% error, is given in Table 1: each row represents the segmentation
obtained, while the columns represent the true digit labels. For reference,
the average computing time per run in this adaptive technique is 132s. Note
that, in the segmentations, the largest mistakes made are in trying to
distinguish digits 4 from 9 and 7 from 9.
For comparison, errors reported using unsupervised clustering algorithms in
the literature are: 12.9% for p-Laplacian [Bühler and Hein, 2009], 11.8% for
ratio-minimization relaxed Cheeger cut [Szlam and Bresson, 2010], and 12.36%
for the multicut version of the normalized 1-cut [Hein and Setzer, 2011]. A
more sophisticated graph-based diffusion method applied in a semi-supervised
setup (transductive classification), with function-adapted eigenfunctions, a
graph constructed with 13 neighbors, and self-tuning with the 9th neighbor
reported in [Szlam et al., 2008] obtains an error of 7.4%. Results with
similar errors are reported in [Liu et al., 2010]. Thus, the performance of
the multiclass GL on this data set improves upon other published results,
while requiring less preprocessing and a simpler regularization of the
functions on the graph.
Table 1: Confusion Matrix for the MNIST Data Segmentation. Obtained / True | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9
---|---|---|---|---|---|---|---|---|---|---
0 | 6712 | 3 | 39 | 10 | 6 | 36 | 57 | 10 | 61 | 28
1 | 1 | 7738 | 7 | 15 | 9 | 1 | 9 | 23 | 36 | 12
2 | 24 | 50 | 6632 | 95 | 65 | 17 | 16 | 63 | 65 | 30
3 | 13 | 16 | 84 | 6585 | 8 | 218 | 5 | 42 | 153 | 84
4 | 5 | 6 | 27 | 8 | 6279 | 32 | 13 | 59 | 43 | 305
5 | 21 | 6 | 13 | 128 | 27 | 5736 | 57 | 3 | 262 | 34
6 | 91 | 26 | 50 | 11 | 35 | 91 | 6693 | 0 | 45 | 1
7 | 6 | 6 | 31 | 97 | 26 | 15 | 0 | 6689 | 24 | 331
8 | 27 | 15 | 86 | 156 | 21 | 110 | 25 | 16 | 6065 | 66
9 | 3 | 11 | 21 | 36 | 348 | 57 | 1 | 388 | 71 | 6067
## 5 CONCLUSIONS
We have proposed a new multiclass segmentation procedure, based on the diffuse
interface model. The method obtains segmentations of several classes
simultaneously without using one-vs-all or alternative sequences of binary
segmentations required by other multiclass methods. The local scaling method
of Zelnik-Manor and Perona, used to construct the graph, constitutes a useful
representation of the characteristics of the data set and is adequate to deal
with high-dimensional data.
Our modified diffusion method, represented by the non-linear smoothing term
introduced in the Ginzburg-Landau functional, exploits the structure of the
multiclass model and is not affected by the ordering of class labels. It
efficiently propagates class information that is known beforehand, as
evidenced by the small proportion of fidelity points (4% \- 10% of dataset)
needed to perform accurate segmentations. Moreover, the method is robust to
initial conditions. As long as the initialization represents all classes
uniformly, different initial random configurations produce very similar
results. The main limitation of the method appears to be that fidelity points
must be representative of class distribution. As long as this holds, such as
in the examples discussed, the long-time behavior of the solution relies less
on choosing the “right” initial conditions than do other learning techniques
on graphs.
State-of-the-art results with small classification errors were obtained for
all classification tasks. Furthermore, the results do not depend on the
particular class label assignments. Future work includes investigating the
diffuse interface parameter $\epsilon$. We conjecture that the proposed
functional converges (in the $\Gamma$-convergence sense) to a total
variational type functional on graphs as $\epsilon$ approaches zero, but the
exact nature of the limiting functional is unknown.
## ACKNOWLEDGEMENTS
This research has been supported by the Air Force Office of Scientific
Research MURI grant FA9550-10-1-0569 and by ONR grant N0001411AF00002.
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|
arxiv-papers
| 2012-12-05T07:13:54 |
2024-09-04T02:49:38.891572
|
{
"license": "Public Domain",
"authors": "Cristina Garcia-Cardona, Arjuna Flenner and Allon G. Percus",
"submitter": "Allon G. Percus",
"url": "https://arxiv.org/abs/1212.0945"
}
|
1212.1028
|
# The commensurate magnetic excitations induced by band-splitting and Fermi
surface topology in n-type Cuprates
H. Y. Zhang1, Y. Zhou1, C. D. Gong${}^{2},^{1}$, and H. Q. Lin3 1National
Laboratory of Solid State Microstructure, Department of Physics, Nanjing
University, Nanjing 210093, China
2Center for Statistical and Theoretical Condensed Matter Physics, Zhejiang
Normal University, Jinhua 321004, China
3Department of Physics and the Institute of Theoretical Physics, Chinese
University of Hong Kong, Hong Kong, China
###### Abstract
The antiferromagnetic correlation plays an important role in high-Tc
superconductors. Considering this effect, the magnetic excitations in n-type
cuprates near the optimal doping are studied within the spin density wave
description. The magnetic excitations are commensurate in the low energy
regime and further develop into spin wave-like dispersion at higher energy,
well consistent with the inelastic neutron scattering measurements. We clearly
demonstrate that the commensurability originates from the band splitting and
Fermi surface topology. The commensurability is a normal state property, and
has nothing to do with d-wave superconductivity. The distinct behaviors of
magnetic excitation between the n-type and p-type cuprates are further
discussed. Our results strongly suggest the essential role of
antiferromagnetic correlations in the cuprates.
###### pacs:
71.27.+a, 75.40.Gb 74.72.Ek, 75.30.Fv
## I Introduction
The parent compounds of the high-$T_{c}$-temperature superconductors are
antiferromagnetic (AFM) Mott insulator. Superconductivity (SC) emerges when
charge carries (holes or electrons) are doped into the $CuO_{2}$ planes. As
well known that the clear electron-hole asymmetry is found in the phase
diagram. For hole doped case, the AFM and superconducting phases are separated
by spin glass phase. In contrast, the AFM phase extends over a much wider
range of doping and even coexists with SC in the electron doped
cupratesArmitage-RMP10 . Due to proximity of antiferrmagnetism and SC, it is
generally believed that there exists intrinsic link between these two phases.
The studies of the spin dynamics in n-type and p-type cuprates will shed light
on the mechanism of superconductivity.
One of the most available techniques to study the spin dynamics is the
inelastic neutron scattering (INS), which directly measures the magnetic
excitations (MEs). Compared with the well studied p-type cuprateHourglass ;
p-type-THEO , the investigations on the MEs in n-type cupratesYamada-PRL03 ;
Wilson-Nature06 ; Motoyama-Nature07 ; Fujita-PRL08 ; Fujita-JPCS08 ; Zhao-
NPHY11 are much less due to the technical reason. A robust feature of MEs in
n-type cuprates, i.e., the commensurate spin response, had been revealed by
these INS measurements. The commensurability, characterized by the strongest
intensity peaked at $\textbf{Q}=(\pi,\pi)$, covers for a wide low-energy
region near the optimal doped NCCOYamada-PRL03 and PLCCOWilson-Nature06 .
Further detecting shows that such commensurability in n-type cuprates exists
for a wide doping range from the underdoping to heavy overdopingFujita-PRL08 ;
Fujita-JPCS08 . More importantly, the commensurate MEs persist well above
superconducting critical temperature $T_{c}$, indicating its non-
superconducting origin. It gradually develops into the spin-wave-like
dispersion centered around the $Q$ point at higher energy, analogous to its
undoped parent compoundWilson-Nature06 . In contrast, The well known
’hourglass’ type magnetic dispersion had been discovered in the p-type
cuprates, where the commensurate peak can only be found at the resonance
energy. Therefore, the two types of cuprates exhibit distinct spin response,
indicating the intrinsic particle-hole asymmetry.
Theoretically, several works had been carried out to interpret the MEs
features of n-type cuprates. Adopting the single band description with
experimentally fitted parameters, Krüger et al. claimed that the fermiology
random phase approximation (RPA) approach with a momentum independent (or
weakly dependent) Coulomb repulsion cannot account the low-energy
commensurability. Their numerical results indicated that the MEs in n-type
case should be more incommensurate than that in p-type caseKruger-PRB07 . Such
conclusion may reveal the fact that the single band description is invalid in
the n-type cupratesFujita-PRL08 . Ismer et al. showed that this may be
improved by a strongly momentum dependent Coulomb repulsion with form of
$U_{q}=U_{0}(\cos q_{x}+\cos q_{y})$Ismer-PRL07 . However, such improvement is
more likely originated from its sharply peaked form at $(\pi,\pi)$, which
cannot be understood physically. Using a slave-boson mean-field approach, Li
et al. showed that the commensurability can be established in the SC stateLJX-
PRB03 . These above mentioned theoretical works are all based on the belief
that the d-wave SC is response for the low energy commensurability. It is hard
to image that the small superconducting gap can produce the wide energy range
commensurability. The most important is that the commensurate phenomenon is
also found in the normal state of n-type cuprates, where the d-wave SC
disappears. The MEs had also been discussed within the frame work of
coexisting of SC and AFMYQS-PRB05 . Commensurate ME had been subsequently
obtainedLJX-JPC10 . However, this result clearly contradicts with the stoner
crititionROWE-PRB12 ; Schrieffer-PRB88 . Furthermore, recent INS data reveal a
magnetic quantum critical point where the SC first appears, implying that the
coexistence may not existMotoyama-Nature07 . Additionally, as we mentioned
above, the commensurability also exists in the state without long-range AFM
order.
In this paper, we focus on the commensurate MEs in the n-type cuprates near
the optimal doping. Its energy region is closely related the to strength of
effective $(\pi,\pi)$-scattering. The commensurate peak disappears at magnetic
resonance energy $\omega^{M}_{res}$ and develops into the spin-wave like
dispersion. These features are qualitatively consistent with INS measurements.
This commensurability is a normal state property, and has nothing to do with
the superconductivity. We explicitly demonstrate that the commensurability is
originated from the band splitting and Fermi surface topology. Therefore, the
AFM correlation plays key roles in the n-type cuprates. The differences of MEs
between the p-type and n-type cuprates are further discussed for comparison.
A spin-density wave (SDW) description is adopted to investigate the n-type
cuprates near the optimal doping. Such description is first suggested by
Armitage et al. based on the ARPES measurements on $NCCO$Armitage-PRL01 . The
underlying Fermi surface disappears around the hot-spot near the optimal
doping, where the long-range antiferromagnetism is absent, strongly suggesting
the existence of a $\mathbf{{Q}=(\pi,\pi)}$-scattering. Parker et al. further
proposed an effective energy band with
$\xi_{k}^{\eta}=\epsilon_{k}^{\prime}+\eta\sqrt{\epsilon_{k}^{2}+V_{\pi,\pi}^{2}}$
($\eta=1$, and $-1$ for upper, and lower band, respectively)Parker-PRB07 ,
where $V_{\pi,\pi}$ is the strength of the effective $\mathbf{Q}$-scattering,
representing the influence of the SDW. $\epsilon_{k}$ and
$\epsilon_{k}^{\prime}$ is the inter- and intra-lattice hopping term. This
description well reproduces the $\sqrt{2}\times\sqrt{2}$ band folding and
Fermi surface reconstructionIkeda-PRB09 ; Matsui-PRL05 , and its applications
on the temperature evolution of optical conductivityZimmer and the Hall
coefficientDagon-PR give qualitative agreement with experiments. Now, the
model Hamiltonian is expressed as
$\displaystyle H$ $\displaystyle=$
$\displaystyle\sum_{k\sigma}\epsilon_{k}^{\prime}(d_{k\sigma}^{+}d_{k\sigma}+e_{k\sigma}^{+}e_{k\sigma})+\sum_{k\sigma}\epsilon_{k}(d_{k\sigma}^{+}e_{k\sigma}+hc.)$
(1) $\displaystyle-\sum_{k\sigma}\sigma
V_{\pi,\pi}(d_{k\sigma}^{+}d_{k\sigma}-e_{k\sigma}^{+}e_{k\sigma})\text{,}$
where, the two sublattices $D$ and $E$ with respective fermionic operator $d$
and $e$ are introduced due to SDWEXPLAIN .
$\epsilon_{k}=-2t(cosk_{x}+cosk_{y})$, and
$\epsilon_{k}^{\prime}=-4t^{\prime}cosk_{x}cosk_{y}-2t^{\prime\prime}(cos2k_{x}+cos2k_{y})-\mu$
with $t$, $t^{\prime}$, and $t^{\prime\prime}$ are the fitting parameters for
nearest-neighbor (NN), second-NN, and third-NN hoping. The summation is
restricted in the AFM Brillouin zone. The quasiparticle dispersion
$\xi_{k}^{\eta}$ can be obtained by the rotation transformation, with
corresponding weight factor $W^{\eta}=\frac{1}{2}(1+\eta sin2\theta_{k})$.
Here,
$cos2\theta_{k}=\frac{V_{\pi,\pi}}{\sqrt{\epsilon_{k}^{2}+V_{\pi,\pi}^{2}}}$,
and
$sin2\theta_{k}=\frac{\epsilon_{k}}{\sqrt{\epsilon_{k}^{2}+V_{\pi,\pi}^{2}}}$.
Here, we would like to emphasize that the long-range AFM order disappears near
the optimal doping. As pointed by Motoyama et al. that the Neel temperature
detected above $x=0.134$ in $NCCO$ originates from the region of samples that
were not fully oxygen-annealedMotoyama-Nature07 . This means that the genuine
long-range antiferromagnetism does not coexist with superconductivity.
However, the $2$-dimensional AFM correlation remains. Unlike only several
lattice-distant length in p-type cupratesKastner-RMP98 , the AFM correlation
is about tens lattice-distance in the optimal electron-doped cupratesMotoyama-
Nature07 . In this sense, the AFM correlations in the n-type cuprates is
similar to the long-range AFM order at least in the small scaling. Therefore,
using a slowly fluctuating SDW order to describe the long-range AFM
correlation is a considerable treatment. Though present SDW description is
analogous to the form in the AFM phaseYQS-PRB05 , the physics behind is
essentially different.
The spin susceptibility under random phase approximation is
$\chi_{q}\left(\omega\right)=\frac{\chi_{q}^{0}-U\left(\chi_{q}^{0}\chi_{q+Q}^{0}-\chi_{q,q+Q}^{0}\chi_{q+Q,q}^{0}\right)}{\left(1-U\chi_{q}^{0}\right)\left(1-U\chi_{q+Q}^{0}\right)-U^{2}\chi_{q,q+Q}^{0}\chi_{q+Q,q}^{0}}\text{,}$
(2)
with $U$ is a reduced Coulomb interaction due to the screening effectZhou-
PLA10 . The bare spin susceptibilities are
$\displaystyle\chi_{q,q}^{0}$ $\displaystyle=$
$\displaystyle\sum_{k}\sin^{2}\left(\theta_{k+q}+\theta_{k}\right)\left(F_{--}+F_{++}\right)$
$\displaystyle+\sum_{k}\cos^{2}\left(\theta_{k+q}+\theta_{k}\right)\left(F_{-+}+F_{+-}\right)$
$\displaystyle\chi_{q,q+Q}^{0}$ $\displaystyle=$ $\displaystyle\sum_{k}(\cos
2\theta_{k}-\cos 2\theta_{kq})(F_{--}-F_{++})$ (3)
$\displaystyle-\sum_{k}(\cos 2\theta_{k}+\cos
2\theta_{kq})\left(F_{-+}-F_{+-}\right)$
with $F_{\eta\eta^{\prime}}$ is
$\displaystyle F_{\eta\eta^{\prime}}$ $\displaystyle=$
$\displaystyle\frac{1}{4}\left(f_{kq}^{\eta}-f_{k}^{\eta^{\prime}}\right)\left(\frac{1}{\omega-\xi_{kq}^{\eta}+\xi_{k}^{\eta^{\prime}}}\right)\text{,}$
(4)
where $f_{k}=1/(1+e^{\xi_{k}/kT)}$ is the Fermi distribution function. In
numerically, the doping level is fixed at $x=0.15$, near the AFM quantum
critical pointMotoyama-Nature07 . $t=250\emph{meV}$,
$t^{\prime}=-50\emph{meV}$, and $t^{\prime\prime}=20\emph{meV}$ are
adoptedParker-PRB07 . The best fitted effective $\mathbf{{Q}-}$ scattering
strength is $V_{\pi,\pi}=100\emph{meV}$, and it will be adjusted for
necessary. The temperature is fixed at $T=0.2\emph{meV}$. We adopt a broaden
factor $\Gamma$ to calculate the spin susceptibility. The reduced Coulomb
interaction is about $600\emph{meV}\sim 760\emph{meV}$, which is about $2\sim
3t$. Our calculations are carried out on a mesh with $2048\times 2048$ k-point
in the full Brillouin zone.
Figure 1: The typical momentum distribution of MEs $\Im\chi_{q}(\omega)$ for
different energy $\omega$. Only the near $\mathbf{Q}$ region are shown for
clarification with $\frac{15\pi}{16}\leq q_{x}(q_{y})\leq\frac{17\pi}{16}$.
$U=700\emph{meV}$, the effective $\mathbf{(Q)}$-scattering strength
$V_{\pi,\pi}=100\emph{meV}$, and the broaden factor $\Gamma=10\emph{meV}$.
The typical energy-evolution of the MEs $\Im\chi_{q}(\omega)$ is shown in Fig.
1. In the low-energy regime below $18\emph{meV}$ (Fig. 1(a), (b)), the MEs are
incommensurate with strong intensity at diagonal directions. Simultaneously,
the intensity near $\mathbf{Q}$ enhances gradually. In the intermediate-energy
regime, the strongest intensity locates at the $\mathbf{Q}$ point, leading to
the the so-called commensurability (Fig. 1(c), (d)). It maintains up to an
critical energy about $88\emph{meV}$ (Fig.1(e)), where the strongest intensity
$\Im\chi_{\mathbf{Q}}(\omega)$ in the normal state can be found, and is
referred as the magnetic resonance $\omega_{res}^{M}$. The total energy range
for the commensurability is approximately $70\emph{meV}$ for $\Gamma=10meV$.
This magnetic resonance is directly related to the fact that the real part in
denominator of the RPA formula (Eq. 2) reduces to zero. Subsequently, it
evolves into a ring-like incommensurability in the high-energy region with its
radius expanding upon the further increased energy (Fig. 1(f)). For high
enough energy, the MEs are incommensurate with strong intensity at the
vertical directions (not shown).
Figure 2: The dispersion of the MEs $\Im\chi_{q}(\omega)$ along the high
symmetry direction. Upper panels are for vertical direction with $q_{y}=\pi$,
and lower panels are for the diagonal direction with $q_{y}=q_{x}$. From left
to right is $U=660\emph{meV}$, $700\emph{meV}$, and $760\emph{meV}$,
respectively. The effective $\mathbf{Q}$-scattering potential is
$V_{\pi,\pi}=100\emph{meV}$ and the damping rate $\Gamma=5\emph{meV}$. All
data had been renormalized by setting the strongest intensity at given
$\omega$ as unit, denoted by the white lines.
Such features can be more clear in dispersion of the MEs at high symmetry
scanning lines as shown in Fig. 2. A wide energy regime with commensurability
exists for all selected $U$, manifesting its universal nature. The low-energy
incommensurability increases slightly upon $\omega$. Hence the
commensurability cannot not be viewed as the overlap of two incommensurate
peaks. It is an intrinsic feature of n-type cuprates. The low-energy
incommensurability may be suppressed and even absent with enhanced $U$. For
example, when $U=0.76V$ (Fig. 2(c)), the MEs are still commensurate at low
enough energy. Correspondingly, the magnetic resonance energy
$\omega_{res}^{M}$ decreases down to $30\emph{meV}$ for $\Gamma=5meV$. The
experimental discovered commensurability in $NCCO$Yamada-PRL03 and
$PLCCO$Fujita-PRL08 ; Fujita-JPCS08 ; Fujita-JPSJ06 near the optimal doping
is more like similar to this case. The value of $U=760\emph{meV}$ is near the
AFM stability, consisting with the fact that the optimal doping is near the
AFM quantum critical pointMotoyama-Nature07 .
The possible energy range of commensurability is mainly determined by the
effective $\mathbf{Q}$-scattering potential $V_{\pi,\pi}$. For
$V=100\emph{meV}$ and $\Gamma=5\emph{meV}$, it is about $48\emph{meV}$. This
energy range decreases down to $10\emph{meV}$ when the $V=50\emph{meV}$.
However, the realistic energy range of commensurability may be substantially
reduced for strong $U$ due to the proximity of the AFM stability. It is only
$30\emph{meV}$ for stronger $U=760\emph{meV}$. For strong enough $U\geq
770\emph{meV}$ at given $V=100\emph{meV}$, the commensurability is entirely
suppressed and only the ring-like magnetic feature remains. This situation is
indeed an AFM state. Therefore, the ring-like feature at high-energy regime in
the electron-doped cuprates shares the same origin as that in their parents
compounds. In fact, those theories based on the long-range AFM orderLJX-JPC10
; YQS-PRB05 cannot account the commensurability found in $NCCO$Yamada-PRL03
and $PLCCO$Fujita-PRL08 due to Stone instability at $\omega=0$Schrieffer-
PRB88 , unless some special control parameter is adopted. The energy range of
commensurability also depends on the broaden factor $\Gamma$ as comparing the
data in Fig. 1 and Fig. 2 ((e) and (f)). However, this phenomenon is still
present even a small $\Gamma=1\emph{meV}$ is adopted, which is less than the
instrument resolution. Hence, the commensurability is an intrinsic and
universal property of the electron-doped cuprates in the normal state.
The main difference in present work from the previous theoretical
investigation by Krüger et al.Kruger-PRB07 is the influence of the AFM
correlation is taken into account. The SDW description take the place of the
single band description, producing a splitting two-band. Therefore, the
commensurability is a directly result of the band-splitting. This can be also
seem from the fact that commensurate energy region diminishes with the reduced
$V_{\pi,\pi}$ as we shown before. As we known that the AFM correlation weakens
with dopingZhou-PRB08 . In the heavy overdoping range, the AFM correlation
disappears, i.e., $V_{\pi,\pi}=0$, leading to the absence of band-splitting.
Our results is then same as the work of Krüger _et al._ , the MEs become
incommensurate, consisting with INS measurementsFujita-PRL08 .
Figure 3: The electron sturecture (upper panels) and Fermi surface (lower
panels). (a), and (c) are for n-type cuprates with SDW description described
in the text. (b), and (d) are for p-type cuprates with YRZ ansatzCarbotte-
PRB10 . The line thickness in (a), and (b) denotes the weight factor. The
doping density is fixed at $x=0.15$.
In the p-type cuprate, the MEs exhibit the well known ’hourglass’ dispersion,
the commensurate peak emerges at the single energy point $\omega_{res}$. The
underdoped p-type cuprates can also be described by the two-band description
due to the opening of pseudogapHole-twoband , where the two bands near the
antinodes are separated. Why do the two types of cuprates show significant
particle-hole asymmetry? As well known that the Fermi surface is an ’arc’ or
hole pocket near the node point in the underdoped p-type cupratesarc , while
it is an electron-pocket near antinodal point in the n-type cuprates even near
the optimal doping as shown in Fig. 3. In fact, the two bands coincides at
node point in p-type cuprates, which leads to the single-point
commensurability at resonance energy. Furthermore, this can also interpret the
commensurability exists in slightly overdoped n-type cuprates even the large-
three-pieced Fermi surfaceArmitage-PRL01 forms because of the band splitting
at nodes. Therefore, both the band splitting and Fermi surface topology play
the important roles in the universal commensurability in the n-type cuprates.
As we stressed before, they both originate from the AFM correlation. Together
with the previous theoretical works on the band structureIkeda-PRB09 ; Matsui-
PRL05 and transport propertiesZimmer ; Dagon-PR , we conclude that the AFM
correlation plays essential roles in the cuprates.
The commensurate MEs remain in the presence of superconductivity. We introduce
a phenomenological BCS-like pairing term
$-\sum_{k}\Delta_{k}\left(d_{k\uparrow}e_{-k\downarrow}+e_{k\uparrow}d_{-k\downarrow}+h.c.\right)$
with standard $d$-wave symmetry $\Delta_{k}=\Delta(\cos k_{x}-\cos k_{y})$.
The resultant MEs change little, consistent with the INS observations.
Therefore, the commensurability in n-type cuprate is a normal state property,
and has nothing to do with SC. The commensurability had been also obtained in
a single band description with d-wave superconductivityIsmer-PRL07 ; LJX-PRB03
. Though the d-wave pairing produces two bands. However, the superconducting
gap in the optimal doped n-type cuprates is only $3\sim 4meV$Shan-PRB08 ;
Dagan-PRL05 , too small to account for the wide energy range commensurability.
It seems that the commensurability comes from the strong peaked factor
$U_{q}$Ismer-PRL07 or $J_{q}$LJX-PRB03 rather than the $d$-wave
superconductivity in these theoretical investigations. More importantly, the
commensurability is a normal state property, which can also be discovered well
above the superconducting transition temperature $T_{c}$.
In conclusion, the magnetic excitations near the optimal doped n-type cuprates
are studied within a spin-density wave description. The main features of
magnetic excitations in the normal state are well established. Our analyses
clearly demonstrate that the band splitting and the Fermi surface topology are
the key for commensurability in n-type cuprates. This strongly suggests that
the antiferromagnetic correlation plays important roles in cuprates. We
emphasize that the commensurability is a normal state property, and has
nothing to do with superconductivity. The qualitative agreement between the
theoretical calculations and experimental data also suggests the validity of
the spin-density wave description near the optimal doping where the long range
antiferromagnetic order is absent. We also discuss the distinct behavior of
magnetic excitations in the n- and p-type cuprates.
This work was supported by NSFC Projects No. 10804047, 11274276, and A Project
Funded by the Priority Academic Program Development of Jiangsu Higher
Education Institutions. CD Gong acknowledges 973 Projects No. 2011CB922101. HQ
Lin acknowledges RGC Grant of HKSAR, Project No. HKUST3/CRF/09.
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|
arxiv-papers
| 2012-12-05T14:03:25 |
2024-09-04T02:49:38.901497
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "H. Y. Zhang, Y. Zhou, C. D. Gong, and H. Q. Lin",
"submitter": "Yuan Zhou",
"url": "https://arxiv.org/abs/1212.1028"
}
|
1212.1045
|
EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN)
CERN-PH-EP-2012-349 LHCb-PAPER-2012-039 10 January 2013
Measurement of ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ production in
$pp$ collisions at $\sqrt{s}=2.76\mathrm{\,Te\kern-2.07413ptV}$
The LHCb collaboration†††Authors are listed on the following pages.
The production of ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ mesons is
studied with the LHCb detector using data from $pp$ collisions at
$\sqrt{s}=2.76\mathrm{\,Te\kern-1.00006ptV}$ corresponding to an integrated
luminosity of $71\mbox{\,nb}^{-1}$. The differential cross-section for
inclusive ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ production is
measured as a function of its transverse momentum $p_{\rm T}$. The cross-
section in the fiducial region $0<\mbox{$p_{\rm
T}$}<12{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ and rapidity $2.0<y<4.5$ is
measured to be $5.6\pm 0.1\,\mathrm{(stat)}\pm 0.4\,\mathrm{(syst)}\rm\,\upmu
b$, with the assumption of unpolarised ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}$ production. The fraction of ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}$ production from $b$-hadron decays is measured to be $(7.1\pm
0.6\,\mathrm{(stat)}\pm 0.7\,\mathrm{(syst)})\%$.
Submitted to JHEP
© CERN on behalf of the LHCb collaboration, license CC-BY-3.0.
LHCb collaboration
R. Aaij38, C. Abellan Beteta33,n, A. Adametz11, B. Adeva34, M. Adinolfi43, C.
Adrover6, A. Affolder49, Z. Ajaltouni5, J. Albrecht35, F. Alessio35, M.
Alexander48, S. Ali38, G. Alkhazov27, P. Alvarez Cartelle34, A.A. Alves
Jr22,35, S. Amato2, Y. Amhis7, L. Anderlini17,f, J. Anderson37, R.
Andreassen57, R.B. Appleby51, O. Aquines Gutierrez10, F. Archilli18, A.
Artamonov 32, M. Artuso53, E. Aslanides6, G. Auriemma22,m, S. Bachmann11, J.J.
Back45, C. Baesso54, V. Balagura28, W. Baldini16, R.J. Barlow51, C.
Barschel35, S. Barsuk7, W. Barter44, A. Bates48, Th. Bauer38, A. Bay36, J.
Beddow48, I. Bediaga1, S. Belogurov28, K. Belous32, I. Belyaev28, E. Ben-
Haim8, M. Benayoun8, G. Bencivenni18, S. Benson47, J. Benton43, A.
Berezhnoy29, R. Bernet37, M.-O. Bettler44, M. van Beuzekom38, A. Bien11, S.
Bifani12, T. Bird51, A. Bizzeti17,h, P.M. Bjørnstad51, T. Blake35, F. Blanc36,
C. Blanks50, J. Blouw11, S. Blusk53, A. Bobrov31, V. Bocci22, A. Bondar31, N.
Bondar27, W. Bonivento15, S. Borghi51, A. Borgia53, T.J.V. Bowcock49, E.
Bowen37, C. Bozzi16, T. Brambach9, J. van den Brand39, J. Bressieux36, D.
Brett51, M. Britsch10, T. Britton53, N.H. Brook43, H. Brown49, A. Büchler-
Germann37, I. Burducea26, A. Bursche37, J. Buytaert35, S. Cadeddu15, O.
Callot7, M. Calvi20,j, M. Calvo Gomez33,n, A. Camboni33, P. Campana18,35, A.
Carbone14,c, G. Carboni21,k, R. Cardinale19,i, A. Cardini15, H. Carranza-
Mejia47, L. Carson50, K. Carvalho Akiba2, G. Casse49, M. Cattaneo35, Ch.
Cauet9, M. Charles52, Ph. Charpentier35, P. Chen3,36, N. Chiapolini37, M.
Chrzaszcz 23, K. Ciba35, X. Cid Vidal34, G. Ciezarek50, P.E.L. Clarke47, M.
Clemencic35, H.V. Cliff44, J. Closier35, C. Coca26, V. Coco38, J. Cogan6, E.
Cogneras5, P. Collins35, A. Comerma-Montells33, A. Contu15, A. Cook43, M.
Coombes43, G. Corti35, B. Couturier35, G.A. Cowan36, D. Craik45, S.
Cunliffe50, R. Currie47, C. D’Ambrosio35, P. David8, P.N.Y. David38, I. De
Bonis4, K. De Bruyn38, S. De Capua51, M. De Cian37, J.M. De Miranda1, L. De
Paula2, W. De Silva57, P. De Simone18, D. Decamp4, M. Deckenhoff9, H.
Degaudenzi36,35, L. Del Buono8, C. Deplano15, D. Derkach14, O. Deschamps5, F.
Dettori39, A. Di Canto11, J. Dickens44, H. Dijkstra35, P. Diniz Batista1, M.
Dogaru26, F. Domingo Bonal33,n, S. Donleavy49, F. Dordei11, A. Dosil Suárez34,
D. Dossett45, A. Dovbnya40, F. Dupertuis36, R. Dzhelyadin32, A. Dziurda23, A.
Dzyuba27, S. Easo46,35, U. Egede50, V. Egorychev28, S. Eidelman31, D. van
Eijk38, S. Eisenhardt47, U. Eitschberger9, R. Ekelhof9, L. Eklund48, I. El
Rifai5, Ch. Elsasser37, D. Elsby42, A. Falabella14,e, C. Färber11, G.
Fardell47, C. Farinelli38, S. Farry12, V. Fave36, D. Ferguson47, V. Fernandez
Albor34, F. Ferreira Rodrigues1, M. Ferro-Luzzi35, S. Filippov30, C.
Fitzpatrick35, M. Fontana10, F. Fontanelli19,i, R. Forty35, O. Francisco2, M.
Frank35, C. Frei35, M. Frosini17,f, S. Furcas20, E. Furfaro21, A. Gallas
Torreira34, D. Galli14,c, M. Gandelman2, P. Gandini52, Y. Gao3, J. Garofoli53,
P. Garosi51, J. Garra Tico44, L. Garrido33, C. Gaspar35, R. Gauld52, E.
Gersabeck11, M. Gersabeck51, T. Gershon45,35, Ph. Ghez4, V. Gibson44, V.V.
Gligorov35, C. Göbel54, D. Golubkov28, A. Golutvin50,28,35, A. Gomes2, H.
Gordon52, M. Grabalosa Gándara5, R. Graciani Diaz33, L.A. Granado Cardoso35,
E. Graugés33, G. Graziani17, A. Grecu26, E. Greening52, S. Gregson44, O.
Grünberg55, B. Gui53, E. Gushchin30, Yu. Guz32, T. Gys35, C. Hadjivasiliou53,
G. Haefeli36, C. Haen35, S.C. Haines44, S. Hall50, T. Hampson43, S. Hansmann-
Menzemer11, N. Harnew52, S.T. Harnew43, J. Harrison51, P.F. Harrison45, T.
Hartmann55, J. He7, V. Heijne38, K. Hennessy49, P. Henrard5, J.A. Hernando
Morata34, E. van Herwijnen35, E. Hicks49, D. Hill52, M. Hoballah5, C.
Hombach51, P. Hopchev4, W. Hulsbergen38, P. Hunt52, T. Huse49, N. Hussain52,
D. Hutchcroft49, D. Hynds48, V. Iakovenko41, P. Ilten12, J. Imong43, R.
Jacobsson35, A. Jaeger11, E. Jans38, F. Jansen38, P. Jaton36, F. Jing3, M.
John52, D. Johnson52, C.R. Jones44, B. Jost35, M. Kaballo9, S. Kandybei40, M.
Karacson35, T.M. Karbach35, I.R. Kenyon42, U. Kerzel35, T. Ketel39, A.
Keune36, B. Khanji20, O. Kochebina7, I. Komarov36,29, R.F. Koopman39, P.
Koppenburg38, M. Korolev29, A. Kozlinskiy38, L. Kravchuk30, K. Kreplin11, M.
Kreps45, G. Krocker11, P. Krokovny31, F. Kruse9, M. Kucharczyk20,23,j, V.
Kudryavtsev31, T. Kvaratskheliya28,35, V.N. La Thi36, D. Lacarrere35, G.
Lafferty51, A. Lai15, D. Lambert47, R.W. Lambert39, E. Lanciotti35, G.
Lanfranchi18,35, C. Langenbruch35, T. Latham45, C. Lazzeroni42, R. Le Gac6, J.
van Leerdam38, J.-P. Lees4, R. Lefèvre5, A. Leflat29,35, J. Lefrançois7, O.
Leroy6, Y. Li3, L. Li Gioi5, M. Liles49, R. Lindner35, C. Linn11, B. Liu3, G.
Liu35, J. von Loeben20, J.H. Lopes2, E. Lopez Asamar33, N. Lopez-March36, H.
Lu3, J. Luisier36, H. Luo47, A. Mac Raighne48, F. Machefert7, I.V.
Machikhiliyan4,28, F. Maciuc26, O. Maev27,35, S. Malde52, G. Manca15,d, G.
Mancinelli6, N. Mangiafave44, U. Marconi14, R. Märki36, J. Marks11, G.
Martellotti22, A. Martens8, L. Martin52, A. Martín Sánchez7, M. Martinelli38,
D. Martinez Santos39, D. Martins Tostes2, A. Massafferri1, R. Matev35, Z.
Mathe35, C. Matteuzzi20, M. Matveev27, E. Maurice6, A. Mazurov16,30,35,e, J.
McCarthy42, R. McNulty12, B. Meadows57,52, F. Meier9, M. Meissner11, M.
Merk38, D.A. Milanes13, M.-N. Minard4, J. Molina Rodriguez54, S. Monteil5, D.
Moran51, P. Morawski23, R. Mountain53, I. Mous38, F. Muheim47, K. Müller37, R.
Muresan26, B. Muryn24, B. Muster36, P. Naik43, T. Nakada36, R. Nandakumar46,
I. Nasteva1, M. Needham47, N. Neufeld35, A.D. Nguyen36, T.D. Nguyen36, C.
Nguyen-Mau36,o, M. Nicol7, V. Niess5, R. Niet9, N. Nikitin29, T. Nikodem11, S.
Nisar56, A. Nomerotski52, A. Novoselov32, A. Oblakowska-Mucha24, V.
Obraztsov32, S. Oggero38, S. Ogilvy48, O. Okhrimenko41, R. Oldeman15,d,35, M.
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Palutan18, J. Panman35, A. Papanestis46, M. Pappagallo48, C. Parkes51, C.J.
Parkinson50, G. Passaleva17, G.D. Patel49, M. Patel50, G.N. Patrick46, C.
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Penso22,l, M. Pepe Altarelli35, S. Perazzini14,c, D.L. Perego20,j, E. Perez
Trigo34, A. Pérez-Calero Yzquierdo33, P. Perret5, M. Perrin-Terrin6, G.
Pessina20, K. Petridis50, A. Petrolini19,i, A. Phan53, E. Picatoste Olloqui33,
B. Pietrzyk4, T. Pilař45, D. Pinci22, S. Playfer47, M. Plo Casasus34, F.
Polci8, G. Polok23, A. Poluektov45,31, E. Polycarpo2, D. Popov10, B.
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Puig Navarro36, W. Qian4, J.H. Rademacker43, B. Rakotomiaramanana36, M.S.
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A.C. dos Reis1, S. Ricciardi46, A. Richards50, K. Rinnert49, V. Rives
Molina33, D.A. Roa Romero5, P. Robbe7, E. Rodrigues51, P. Rodriguez Perez34,
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Santacesaria22, C. Santamarina Rios34, E. Santovetti21,k, M. Sapunov6, A.
Sarti18,l, C. Satriano22,m, A. Satta21, M. Savrie16,e, D. Savrina28,29, P.
Schaack50, M. Schiller39, H. Schindler35, S. Schleich9, M. Schlupp9, M.
Schmelling10, B. Schmidt35, O. Schneider36, A. Schopper35, M.-H. Schune7, R.
Schwemmer35, B. Sciascia18, A. Sciubba18,l, M. Seco34, A. Semennikov28, K.
Senderowska24, I. Sepp50, N. Serra37, J. Serrano6, P. Seyfert11, M. Shapkin32,
I. Shapoval40,35, P. Shatalov28, Y. Shcheglov27, T. Shears49,35, L.
Shekhtman31, O. Shevchenko40, V. Shevchenko28, A. Shires50, R. Silva
Coutinho45, T. Skwarnicki53, N.A. Smith49, E. Smith52,46, M. Smith51, K.
Sobczak5, M.D. Sokoloff57, F.J.P. Soler48, F. Soomro18,35, D. Souza43, B.
Souza De Paula2, B. Spaan9, A. Sparkes47, P. Spradlin48, F. Stagni35, S.
Stahl11, O. Steinkamp37, S. Stoica26, S. Stone53, B. Storaci37, M.
Straticiuc26, U. Straumann37, V.K. Subbiah35, S. Swientek9, V. Syropoulos39,
M. Szczekowski25, P. Szczypka36,35, T. Szumlak24, S. T’Jampens4, M.
Teklishyn7, E. Teodorescu26, F. Teubert35, C. Thomas52, E. Thomas35, J. van
Tilburg11, V. Tisserand4, M. Tobin37, S. Tolk39, D. Tonelli35, S. Topp-
Joergensen52, N. Torr52, E. Tournefier4,50, S. Tourneur36, M.T. Tran36, M.
Tresch37, A. Tsaregorodtsev6, P. Tsopelas38, N. Tuning38, M. Ubeda Garcia35,
A. Ukleja25, D. Urner51, U. Uwer11, V. Vagnoni14, G. Valenti14, R. Vazquez
Gomez33, P. Vazquez Regueiro34, S. Vecchi16, J.J. Velthuis43, M. Veltri17,g,
G. Veneziano36, M. Vesterinen35, B. Viaud7, D. Vieira2, X. Vilasis-
Cardona33,n, A. Vollhardt37, D. Volyanskyy10, D. Voong43, A. Vorobyev27, V.
Vorobyev31, C. Voß55, H. Voss10, R. Waldi55, R. Wallace12, S. Wandernoth11, J.
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Williams45,46, M. Williams50,p, F.F. Wilson46, J. Wishahi9, M. Witek23, W.
Witzeling35, S.A. Wotton44, S. Wright44, S. Wu3, K. Wyllie35, Y. Xie47,35, F.
Xing52, Z. Xing53, Z. Yang3, R. Young47, X. Yuan3, O. Yushchenko32, M.
Zangoli14, M. Zavertyaev10,a, F. Zhang3, L. Zhang53, W.C. Zhang12, Y. Zhang3,
A. Zhelezov11, A. Zhokhov28, L. Zhong3, A. Zvyagin35.
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 Roma Tor Vergata, Roma, Italy
22Sezione INFN di Roma La Sapienza, Roma, Italy
23Henryk Niewodniczanski Institute of Nuclear Physics Polish Academy of
Sciences, Kraków, Poland
24AGH University of Science and Technology, Kraków, Poland
25National Center for Nuclear Research (NCBJ), Warsaw, Poland
26Horia Hulubei National Institute of Physics and Nuclear Engineering,
Bucharest-Magurele, Romania
27Petersburg Nuclear Physics Institute (PNPI), Gatchina, Russia
28Institute of Theoretical and Experimental Physics (ITEP), Moscow, Russia
29Institute of Nuclear Physics, Moscow State University (SINP MSU), Moscow,
Russia
30Institute for Nuclear Research of the Russian Academy of Sciences (INR RAN),
Moscow, Russia
31Budker Institute of Nuclear Physics (SB RAS) and Novosibirsk State
University, Novosibirsk, Russia
32Institute for High Energy Physics (IHEP), Protvino, Russia
33Universitat de Barcelona, Barcelona, Spain
34Universidad de Santiago de Compostela, Santiago de Compostela, Spain
35European Organization for Nuclear Research (CERN), Geneva, Switzerland
36Ecole Polytechnique Fédérale de Lausanne (EPFL), Lausanne, Switzerland
37Physik-Institut, Universität Zürich, Zürich, Switzerland
38Nikhef National Institute for Subatomic Physics, Amsterdam, The Netherlands
39Nikhef National Institute for Subatomic Physics and VU University Amsterdam,
Amsterdam, The Netherlands
40NSC Kharkiv Institute of Physics and Technology (NSC KIPT), Kharkiv, Ukraine
41Institute for Nuclear Research of the National Academy of Sciences (KINR),
Kyiv, Ukraine
42University of Birmingham, Birmingham, United Kingdom
43H.H. Wills Physics Laboratory, University of Bristol, Bristol, United
Kingdom
44Cavendish Laboratory, University of Cambridge, Cambridge, United Kingdom
45Department of Physics, University of Warwick, Coventry, United Kingdom
46STFC Rutherford Appleton Laboratory, Didcot, United Kingdom
47School of Physics and Astronomy, University of Edinburgh, Edinburgh, United
Kingdom
48School of Physics and Astronomy, University of Glasgow, Glasgow, United
Kingdom
49Oliver Lodge Laboratory, University of Liverpool, Liverpool, United Kingdom
50Imperial College London, London, United Kingdom
51School of Physics and Astronomy, University of Manchester, Manchester,
United Kingdom
52Department of Physics, University of Oxford, Oxford, United Kingdom
53Syracuse University, Syracuse, NY, United States
54Pontifícia Universidade Católica do Rio de Janeiro (PUC-Rio), Rio de
Janeiro, Brazil, associated to 2
55Institut für Physik, Universität Rostock, Rostock, Germany, associated to 11
56Institute of Information Technology, COMSATS, Lahore, Pakistan, associated
to 53
57University of Cincinnati, Cincinnati, OH, United States, associated to 53
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
pMassachusetts Institute of Technology, Cambridge, MA, United States
## 1 Introduction
This article presents the measurements of the differential inclusive
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ production cross-section as a
function of the ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ transverse
momentum, and of the fraction of ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}$ mesons coming from the decay of a $b$-hadron in $pp$ collisions at a
centre-of-mass energy of 2.76$\mathrm{\,Te\kern-1.00006ptV}$. The study is
based on a sample corresponding to an integrated luminosity of
$71\mbox{\,nb}^{-1}$ collected in March 2011 with an average of one visible
$pp$ interaction per recorded event. The main goal of this short run was to
provide a reference for the study of Pb-Pb interactions carried out at the
same centre-of-mass energy per nucleon-nucleon collision.
Studies of ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ production have been
performed by the LHC experiments using data taken at
$\sqrt{s}=7\mathrm{\,Te\kern-1.00006ptV}$[1, 2, 3, 4] as well as at lower
energies[5]. The data at $\sqrt{s}=2.76\mathrm{\,Te\kern-1.00006ptV}$ provide
an extra measurement to test theoretical models of
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ production in hadron collisions
and are also used to obtain a measurement of
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ production from $b$-hadron
decays.
The LHCb detector [6] 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.
Charged hadrons are identified using two ring-imaging Cherenkov detectors.
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 which
consists of five stations of alternating layers of iron and multiwire
proportional chambers, with the exception of the centre of the first station,
which uses triple-GEM detectors.
For the data used in this analysis, the VELO, which consists of two
retractable halves surrounding the interaction region, was positioned during
collisions with its sensitive area at a minimum distance of $13\,{\rm mm}$
from the beam instead of the nominal $8\,{\rm mm}$. This was necessary to
provide a larger aperture for the beam at the lower centre-of-mass energy of
2.76$\mathrm{\,Te\kern-1.00006ptV}$.
The trigger[7] 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. Only the triggers used in this analysis are
described here. At the hardware trigger level, a single muon candidate with
$p_{\rm T}$ larger than $0.8{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ is required.
In the first stage of the software trigger a simplified event reconstruction
is applied and one requires a $\mu^{+}\mu^{-}$ candidate with invariant mass
greater than 2.7${\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$. In the second
stage a full event reconstruction is performed and only events with a
$\mu^{+}\mu^{-}$ pair with invariant mass within
120${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ of the known
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ mass [8] are retained.
## 2 Event selection
The analysis strategy is based upon that described in Ref. [1]. Candidate
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ mesons are formed from pairs of
opposite-sign charged particles reconstructed in the fiducial region
$2<\eta<5$ by the full tracking system using algorithms adapted to the VELO at
its displaced position. Each particle must have $p_{\rm T}$ above
0.7${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ and be identified as a muon. The two
muons are required to originate from a common vertex, and only candidates with
a $\chi^{2}$ probability of the vertex fit larger than 0.5% are kept. Events
are selected in which at least one primary vertex is reconstructed from at
least three VELO tracks, excluding the two signal muon tracks from the
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ decay. A VELO track is required
to have at least three hits on a straight line in the radial strips of the
detector.
The Monte Carlo samples used for this analysis are based on the Pythia 6.4
generator [9] configured with the parameters detailed in Ref. [10]. The EvtGen
package [11] is used to generate hadron decays, in particular for
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ and $b$-hadrons. The
interaction of the generated particles with the detector and its response are
implemented using the Geant4 toolkit [12, *Agostinelli:2002hh] as described in
Ref. [14]. Radiative corrections to the decay
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\rightarrow\mu^{+}\mu^{-}$ are
generated using the Photos package [15]. The simulated position of the VELO
corresponds to that in the data.
## 3 Cross-section determination
The differential cross-section for ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}$ production in a $p_{\rm T}$ bin is given by
$\frac{{\rm d}\sigma}{{\rm d}\mbox{$p_{\rm
T}$}}=\frac{N\left({J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\rightarrow\mu^{+}\mu^{-}\right)}{\mathcal{L}\times\epsilon_{\rm
tot}\times{\cal B}\left({J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\rightarrow\mu^{+}\mu^{-}\right)\times\Delta\mbox{$p_{\rm T}$}}\,,$ (1)
where $N\left({J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\rightarrow\mu^{+}\mu^{-}\right)$ is the number of observed
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\rightarrow\mu^{+}\mu^{-}$ signal
decays in the given bin, $\epsilon_{\rm tot}$ the
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ detection efficiency per
$p_{\rm T}$ bin (including both acceptance and trigger), $\mathcal{L}$ the
integrated luminosity, ${\cal B}\left({J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\rightarrow\mu^{+}\mu^{-}\right)=(5.93\pm 0.06)\times 10^{-2}$ [8] the
branching fraction of the ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\rightarrow\mu^{+}\mu^{-}$ decay, and $\Delta\mbox{$p_{\rm T}$}$ the
$p_{\rm T}$ bin size.
The number of signal ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ mesons per
$p_{\rm T}$ bin is determined from an extended unbinned maximum likelihood fit
to the invariant mass distribution of the reconstructed
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ candidates in the interval
$3.0<M_{\mu\mu}<3.2{\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$, where the signal
is described by a Crystal Ball function [16] and the combinatorial background
by an exponential distribution. Figure 1 shows the
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ invariant mass distribution
together with the fit results for each $p_{\rm T}$ bin, where results for
$7<\mbox{$p_{\rm T}$}<12{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ are merged in
the last bin.
Figure 1: Dimuon mass distributions, with fit results superimposed, in bins of
$p_{\rm T}$. Results for $\mbox{$p_{\rm
T}$}>7{\mathrm{\,Ge\kern-0.90005ptV\\!/}c}$ are merged in the last bin.
There are two main sources that contribute to the inclusive
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ sample. Those produced at the
$pp$ collision point, either directly or from the decay of a directly produced
higher mass charmonium state, are called
$\mathrm{prompt}~{}{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$. The second
source, ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}~{}\mathrm{from}~{}b$,
are those produced in the decay of a $b$-hadron. Their production is displaced
from the $pp$ collision point because of the relatively large $b$ lifetime.
The two sources are statistically separated using the measured
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ pseudo-decaytime, defined as
$t_{z}=\frac{(z_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}-z_{\rm
PV})\times M_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}}{p_{z}}\,,$ (2)
where $z_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$ and $z_{\rm PV}$ are
the positions along the beam axis of the
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ decay vertex and of the primary
vertex refitted after removing the two muon tracks from the
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ candidate; $p_{z}$ is the
measured ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ momentum in the beam
direction and $M_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$ the known
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ mass [8]. Given that
$b$-hadrons are not fully reconstructed, the
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ momentum is used instead of the
exact $b$-hadron momentum and the $t_{z}$ variable provides a good estimate of
the $b$-hadron decaytime.
The fraction of ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}~{}\mathrm{from}~{}b$ is determined from a simultaneous fit to the total
pseudo-decaytime $t_{z}$ and $\mu^{+}\mu^{-}$ invariant mass. Due to the small
number of ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ candidates, the
fraction of ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}~{}\mathrm{from}~{}b$
is computed over the full $p_{\rm T}$ interval from 0 to 12
${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$. The signal decaytime distribution is
described by a delta function at $t_{z}=0$ for the prompt
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ component and an exponential
decay function for the ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}~{}\mathrm{from}~{}b$ component. The function describing the $t_{z}$
distribution of the signal is therefore
$f_{\rm signal}(t_{z};f_{\rm p},f_{\rm b},\tau_{\rm b})=f_{\rm
p}\,\delta(t_{z})+\theta(t_{z})f_{\rm b}\frac{e^{-\frac{t_{z}}{\tau_{\rm
b}}}}{\tau_{\rm b}}\,,$ (3)
where $\theta(t_{z})$ is the step function, $f_{\rm p}$ and $f_{\rm b}$ are
the fractions of prompt ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ and
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}~{}\mathrm{from}~{}b$ in the
sample, and $\tau_{\rm b}$ the $b$-hadron pseudo-lifetime. In the fit,
$\tau_{b}$ is fixed to the value of 1.52${\rm\,ps}$, as obtained from
simulation. The prompt and $b$ components of the signal function are convolved
with a triple-Gaussian resolution function
$f_{\rm
res}(t_{z};\mu,\sigma_{1},\sigma_{2},\sigma_{3},\beta,\beta^{\prime})=\frac{\beta}{\sqrt{2\pi}\sigma_{1}}\,e^{-\frac{(t_{z}-\mu)^{2}}{2\sigma_{1}^{2}}}+\frac{\beta^{\prime}}{\sqrt{2\pi}\sigma_{2}}\,e^{-\frac{(t_{z}-\mu)^{2}}{2\sigma_{2}^{2}}}+\frac{1-\beta-\beta^{\prime}}{\sqrt{2\pi}\sigma_{3}}\,e^{-\frac{(t_{z}-\mu)^{2}}{2\sigma_{3}^{2}}}.$
(4)
The parameter $\mu$ is the bias of the $t_{z}$ measurement, and $\beta$ and
$\beta^{\prime}$ the fractions of the first two Gaussian functions. The
background consists of random combinations of muons from semi-leptonic $b$ and
$c$ decays, which tend to produce positive $t_{z}$ values, as well as of mis-
reconstructed tracks from decays in flight of kaons and pions, which
contribute both to positive and negative $t_{z}$ values. The background
$t_{z}$ distribution is parameterised with an empirical function based on the
shape obtained from the ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ mass
sidebands. It is taken as the sum of a delta function and three exponential
components, two for positive $t_{z}$ and one for negative $t_{z}$. The
exponential parameter, $\tau_{\rm L}$, is common to the larger positive and
negative lifetime exponential components. The explicit form is
$f_{\rm bckg}(t_{z})=\left(1-f_{1}-f_{\rm
L}\right)\delta(t_{z})+\theta(t_{z})f_{1}\,\frac{e^{-\frac{t_{z}}{\tau_{1}}}}{\tau_{1}}+f_{\rm
L}\,\frac{e^{-\frac{|t_{z}|}{\tau_{\rm L}}}}{2\tau_{\rm L}},$ (5)
and is convolved with the same resolution function $f_{\rm res}$ as the
signal.
Figure 2: Distributions of the data with projections of the fit superimposed
for (left) the dimuon invariant mass and (right) $t_{z}$. The thick blue line
is the total fit function and the hatched area the background component.
The function used to describe the $t_{z}$ distribution is therefore
$\begin{split}f(t_{z};f_{\rm p},f_{\rm
b},\mu,\sigma_{1},\sigma_{2},\sigma_{3},\beta,\beta^{\prime},\tau_{\rm
b})=&\left(f_{\rm p}\,\delta(t_{z})+f_{\rm
b}\,\frac{e^{-\frac{t_{z}}{\tau_{\rm b}}}}{\tau_{\rm b}}+\left(1-f_{\rm
p}-f_{\rm b}\right)f_{\rm bckg}(t_{z})\right)\otimes\\\ &f_{\rm
res}(t_{z};\mu,\sigma_{1},\sigma_{2},\sigma_{3},\beta,\beta^{\prime})\,,\end{split}$
(6)
where all parameters except $\tau_{\rm b}$ are freely varied. The total fit
function is the sum of the products of the mass and $t_{z}$ fit functions for
the signal and background. Figure 2 shows the distributions of the dimuon
invariant mass and $t_{z}$ with the projections of the fit superimposed. The
invariant mass resolution is $13.0\pm
0.3{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$. The parameter $\mu$ describing
the bias of the $t_{z}$ resolution function is $2.3\pm 2.0$ fs and the RMS of
the $t_{z}$ resolution function is 84 fs. As a measure of the fit quality, a
$\chi^{2}$ is calculated using a binned event distribution. The resulting fit
probability for the $t_{z}$ distribution is 90%. The fit gives a total yield
of $3399\pm 65$ ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ signal decays.
The fraction of signal ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ coming
from $b$-hadron decays is measured to be $F_{\rm
b}=\frac{f_{b}}{f_{p}+f_{b}}=(6.7\pm 0.6)\%$. An absolute correction of
$0.4\%$ is applied based on simulation to take into account a bias produced by
events in which $b$-hadron decay products, other than the muons from the
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$, are wrongly used to
reconstruct the primary vertex. This leads to the result $F_{\rm b}=(7.1\pm
0.6)\%$ where the uncertainty is only statistical.
A simulated sample of inclusive, unpolarised
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ mesons is used to estimate the
geometrical acceptance in each $p_{\rm T}$ bin. The reconstruction efficiency,
which combines the ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ meson
detection, reconstruction and selection efficiencies, is also computed from
simulation as a function of $p_{\rm T}$ and is corrected to account for the
difference observed in the tracking efficiency between data and simulation at
$\sqrt{s}=7\mathrm{\,Te\kern-1.00006ptV}$. This correction is about 1%. The
efficiency of the hardware trigger is determined directly from data using a
large inclusive ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ sample at
$\sqrt{s}=7\mathrm{\,Te\kern-1.00006ptV}$ triggered and selected with the same
requirements as those used in this analysis: the efficiency is calculated in
small bins of the ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ transverse
momentum and rapidity and weighted according to the $p_{\rm T}$ and $y$
distributions as given by the simulation at
2.76$\mathrm{\,Te\kern-1.00006ptV}$. The efficiency of the software trigger,
which makes use of the VELO information, is determined from simulation since
the data at $\sqrt{s}=7\mathrm{\,Te\kern-1.00006ptV}$ were taken with the VELO
in the closed position. The total efficiency, calculated as the product of
acceptance, reconstruction and trigger efficiencies, and its components are
displayed in Fig. 3 as a function of $p_{\rm T}$. A non-zero polarisation of
the ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ at production can affect
the total efficiency [1]. The results quoted in this article assume that the
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ mesons are produced
unpolarised.
Figure 3: Acceptance (orange circles), reconstruction (upward-facing red
triangles), trigger (downward-facing green triangles) and total (blue squares)
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ efficiency, as a function of
$p_{\rm T}$. The efficiencies are evaluated from a Monte Carlo simulation in
which the ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ is produced
unpolarised.
## 4 Luminosity determination
To determine the integrated luminosity, an effective interaction rate is
continuously measured during data taking and an absolute calibration is
performed with a dedicated van der Meer (VDM) scan [17]. The strategy is
similar to that developed for the $\sqrt{s}=7\mathrm{\,Te\kern-1.00006ptV}$
running [18].
Table 1: Relative systematic uncertainties on the luminosity (%). Uncertainty
on relative normalisation
---
| Counter stability | 70.5
| $\mu$ variation among bunch crossings | 70.5
Uncertainty on absolute normalisation
| Statistical error of the VDM scan | 70.2
| Total beam current | 72.7
| Individual bunch population | 70.9
| Protons outside nominal bunches | 70.4
| Length scale calibration | 71.0
| Non-reproducibility in similar scans at $\sqrt{s}=7\mathrm{\,Te\kern-1.00006ptV}$ | 72.1
| Total uncertainty | 73.8
The VDM method exploits the ability to move the beams in both transverse
coordinates with high precision and thus to scan the colliding beams with
respect to each other. The limiting systematic uncertainty affecting the VDM
measurement arises from the knowledge of the number of protons in the
colliding bunch pairs. These are measured with two types of beam current
transformers installed in the LHC [19, *bct2, *bct3]. The DCCT (DC Current
Transformer) measures the total beam current, and is thus used to constrain
the total number of particles. The uncertainty associated with the DCCT
calibration is 2.7% [22, *bcnwg2, *Ohm:2009pf]. The other transformer, the
FBCT (Fast Beam Current Transformer) is used to measure the relative charges
of the individual bunches. The uncertainty in its offset and linearity
contributes a 0.9% uncertainty to the overall luminosity [22, *bcnwg2,
*Ohm:2009pf]. A small fraction of protons in the LHC may be captured outside
the nominally filled bunch slots. This contribution, which needs to be
subtracted from the DCCT measurement, is estimated to be 2.5% from the number
of beam-gas events in nominally empty bunch crossings. Due to the small number
of such events and uncertainties in the trigger efficiency, the subtraction
introduces a cross-section uncertainty of 0.4%. The uncertainty in the length-
scale calibration, which affects the beam separation values, contributes 1% to
the systematic uncertainty in the luminosity. Finally, a 2.1% uncertainty is
assigned to account for a non-reproducibility of the VDM results observed when
performing similar luminosity calibration measurements at
$\sqrt{s}=7\mathrm{\,Te\kern-1.00006ptV}$, as decribed in Ref. [18].
The integrated luminosity for the runs considered in this analysis is measured
to be $70.6\pm 2.7\mbox{\,nb}^{-1}$. A summary of the contributions to the
overall luminosity uncertainty is provided in Table 1. The uncertainties are
uncorrelated and therefore added in quadrature.
## 5 Systematic uncertainties
The different contributions to the systematic uncertainty affecting the cross-
section measurement are summarised in Table 2. Correction factors estimated
directly from data to take into account residual differences between
simulation and data are also detailed.
The influence of the choice of the fit function used to describe the shape of
the dimuon mass distribution is estimated by fitting the
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ invariant mass distribution
with the sum of two Crystal Ball functions. The relative difference of 2.2% in
the number of signal events is taken as systematic uncertainty.
A fraction of ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ events have a
lower mass because of the radiative tail. Based on Monte Carlo studies, 5% of
the ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ signal is estimated to be
outside the analysis mass window
($M_{\mu\mu}<3.0{\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$) and not counted as
signal. The fitted signal yields are therefore corrected, and an uncertainty
of 1% is assigned to the cross-section measurement based on a comparison
between the radiative tail observed in data and simulation.
To cross-check and assign a systematic uncertainty to the Monte Carlo
determination of the muon identification efficiency, the single track muon
identification efficiency is measured on data using a tag-and-probe method.
This method reconstructs ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$
candidates in which one muon is identified by the muon system (“tag”) and the
other one (“probe”) is identified by selecting a track with a minimum-ionising
energy deposition in the calorimeters. The absolute muon identification
efficiency is then evaluated on the probe muon, as a function of the muon
momentum and found to be larger than 95%. The ratio of the muon identification
efficiency measured in data to that obtained in the simulation is convolved
with the momentum distribution of muons from
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ to obtain an efficiency
correction. This factor is found to be $1.024\pm 0.011$ and is consistent with
being constant over the full ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$
transverse momentum and rapidity range; the error on the correction factor is
included as a systematic uncertainty.
Table 2: Relative systematic uncertainties on the cross-section results and on the fraction of ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ mesons from $b$-hadron decay (%). | Source | Systematic uncertainty
---|---|---
Correlated between bins
| Mass fits | $\phantom{1}2.2$
| Radiative tail | $\phantom{1}1.0$
| Muon identification | $\phantom{1}1.1$
| Tracking efficiency | $\phantom{1}0.8\ {\rm to}\ 1.1$
| Track $\chi^{2}$ | $\phantom{1}2.0$
| Vertexing | $\phantom{1}0.3$
| Model dependence | $\phantom{1}4.5$
| ${\cal B}({J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\rightarrow\mu^{+}\mu^{-})$ | $\phantom{1}1.0$
| Luminosity | $\phantom{1}3.8$
Uncorrelated between bins
| Trigger | $\phantom{1}1.6\ {\rm to}\ 7.7$
Applied only to ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ from $b$
fraction
| $t_{z}$ fit | $10.0$
Applied only to $\sigma(pp\rightarrow b\overline{}bX)$
| ${\cal B}(b\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}X)$ | $\phantom{1}8.6$
Studies at $\sqrt{s}=7\mathrm{\,Te\kern-1.00006ptV}$ have shown that the Monte
Carlo simulation reproduces the determination from data of the efficiency to
reconstruct the two muon tracks from the
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ decay within 0.8% to 1.1%,
depending on the ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ transverse
momentum. This difference is taken as a systematic uncertainty. An additional
uncertainty of 1% per track is assigned to cover differences in the efficiency
of the track $\chi^{2}$/ndf cut between data and simulation. Similarly, for
the selection based on the ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$
vertex $\chi^{2}$ probability, a difference below 0.3% is measured between the
cut efficiency computed in data and simulation, which is assigned as
systematic uncertainty. To take into account the model dependence of the
simulation in the efficiency calculation, the main parameters of the Pythia
6.4 generator related to prompt ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$
production were varied. These parameters define the minimum $p_{\rm T}$ cut-
offs for regularising the cross-section. A 4.5% effect on the total efficiency
was observed.
The hardware trigger efficiency is determined using a sample of events at
$\sqrt{s}=7\mathrm{\,Te\kern-1.00006ptV}$ that would still be triggered if the
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ candidate were removed. The
software trigger efficiency is obtained from the simulation. Its uncertainty
is evaluated by comparing true and measured trigger efficiency using a
trigger-unbiased sample of simulated ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}$ events.
Uncertainties related to the $t_{z}$ fit procedure are taken into account by
varying the slope of the exponential function of the
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}~{}\mathrm{from}~{}b$ component
by its uncertainty in the simulation (2%). The resulting 10% variation of the
number of ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}~{}\mathrm{from}~{}b$
is used as a systematic uncertainty that affects the measurement of $F_{\rm
b}$. The influence of the background parametrisation was studied by varying
the number of exponential functions in Eq. (5) and found to be negligible.
Furthermore, an uncertainty of 8.6% on the average branching fraction of $b$
decays to a final state containing a ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}$ meson contributes to the uncertainty on the extrapolation to the total
$b\overline{}b$ cross-section.
## 6 Results
Table 3: Differential cross-section ${\rm d}\sigma/{\rm d}\mbox{$p_{\rm T}$}$ at $\sqrt{s}=2.76\mathrm{\,Te\kern-0.90005ptV}$ for inclusive ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ production in bins of $p_{\rm T}$. The rapidity range covered is $2.0<y<4.5$. The first uncertainty is statistical and the second is systematic. $\mbox{$p_{\rm T}$}\,({\rm GeV}/c)$ | ${\rm d}\sigma/{\rm d}\mbox{$p_{\rm T}$}$ [nb/(${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$)]
---|---
$0$ | $-$ | $1$ | $1270$ | $\,\pm\,$ | $60$ | $\,\pm\,$ | $130\phantom{11111}$
$1$ | $-$ | $2$ | $1780$ | $\,\pm\,$ | $70$ | $\,\pm\,$ | $160\phantom{11111}$
$2$ | $-$ | $3$ | $1290$ | $\,\pm\,$ | $50$ | $\,\pm\,$ | $\phantom{1}90\phantom{11111}$
$3$ | $-$ | $4$ | $\phantom{1}\,700$ | $\,\pm\,$ | $40$ | $\,\pm\,$ | $\phantom{1}50\phantom{11111}$
$4$ | $-$ | $5$ | $\phantom{1}\,313$ | $\,\pm\,$ | $22$ | $\,\pm\,$ | $\phantom{1}24\phantom{11111}$
$5$ | $-$ | $6$ | $\phantom{1}\,142$ | $\,\pm\,$ | $13$ | $\,\pm\,$ | $\phantom{1}10\phantom{11111}$
$6$ | $-$ | $7$ | $\phantom{1\,1}61$ | $\,\pm\,$ | $\phantom{1}8$ | $\,\pm\,$ | $\phantom{11}4\phantom{11111}$
$7$ | $-$ | $12$ | $\phantom{1\,1}14$ | $\,\pm\,$ | $\phantom{1}2$ | $\,\pm\,$ | $\phantom{11}1\phantom{11111}$
The measured differential cross-section for inclusive
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ production as a function of
$p_{\rm T}$, after all corrections and assuming no polarisation, is given in
Table 3 and displayed in Fig. 4. The integrated inclusive cross-section for
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ production in the defined
fiducial region is
$\sigma\left({J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu},\,\mbox{$p_{\rm
T}$}<12\,{\mathrm{\,Ge\kern-1.00006ptV\\!/}c},\,2.0<y<4.5\right)\,=\,5.6\pm
0.1\pm 0.4\rm\,\upmu b.$
The first uncertainty is statistical and the second systematic. Studies
indicate that this result could change by up to 20% assuming fully
longitudinal or fully transverse ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}$ polarisation [1]. The fraction of
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}~{}\mathrm{from}~{}b$ is measured
to be
$F_{b}=(7.1\pm 0.6\pm 0.7)\%$
in the same acceptance range, $\mbox{$p_{\rm
T}$}<12{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ and $2.0<y<4.5$.
From the above results, one can deduce
$\sigma\left({J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}~{}\mathrm{from}~{}b,\,\mbox{$p_{\rm
T}$}<12\,{\mathrm{\,Ge\kern-1.00006ptV\\!/}c},\,2.0<y<4.5\right)\,=\,400\pm
35\pm 49\rm\,nb,$
in good agreement with the theoretical prediction of
$370^{+170}_{-110}\rm\,nb$, based on NLO calculations described in Ref. [25].
In addition, the total $b\overline{}b$ production cross-section is computed as
$\sigma(pp\rightarrow
b\overline{b}X)=\alpha_{4\pi}\,\frac{\sigma\left({J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu},\,\mbox{$p_{\rm
T}$}<12\,{\mathrm{\,Ge\kern-1.00006ptV\\!/}c},\,2.0<y<4.5\right)\times F_{\rm
b}}{2\,{\cal B}(b\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}X)},$
(7)
where the factor $\alpha_{4\pi}=6.3$ is an extrapolation factor of the cross-
section from the measured to the full kinematic region. This factor is
obtained using the simulation software described previously. The inclusive
$b\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}X$ branching fraction
is ${\cal B}(b\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}X)=(1.16\pm 0.10)\%$ [8]. The resulting total $b\overline{b}$ cross-
section is $\sigma(pp\rightarrow b\overline{b}X)=110\pm 9\pm 16\,\upmu{\rm
b}$. No systematic uncertainty has been included for the extrapolation factor
$\alpha_{4\pi}$ estimated from the simulation. The value of the extrapolation
factor given by NLO calculations is 6.1 [25].
Figure 4: Differential production cross-section for inclusive
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ production in the rapidity
range $2.0<y<4.5$ as a function of $p_{\rm T}$. The vertical error bars are
the quadratic sums of the statistical and systematic uncertainties.
## 7 Conclusions
The differential cross-section for inclusive
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ production is measured as a
function of the ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ transverse
momentum in the forward region, $2.0<y<4.5$. The analysis is based on a data
sample corresponding to an integrated luminosity of $71\mbox{\,nb}^{-1}$
collected by the LHCb experiment at the Large Hadron Collider at a centre-of-
mass energy of $\sqrt{s}=2.76\mathrm{\,Te\kern-1.00006ptV}$. The results
obtained are in good agreement with earlier measurements of the inclusive
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ production cross-section in
$pp$ collisions at the same centre-of-mass energy, performed by ALICE in the
region $2.5<y<4.0$ [26]. A first measurement of the production of
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ from $b$-hadron decays at
2.76$\mathrm{\,Te\kern-1.00006ptV}$ is also obtained.
## Acknowledgements
We thank M. Cacciari for providing theoretical predictions of the
$b\overline{b}$ production cross-section in the LHCb acceptance. 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);
ANCS/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
| 2012-12-05T15:16:30 |
2024-09-04T02:49:38.909608
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "LHCb collaboration: R. Aaij, C. Abellan Beteta, A. Adametz, B. Adeva,\n M. Adinolfi, C. Adrover, A. Affolder, Z. Ajaltouni, J. Albrecht, F. Alessio,\n M. Alexander, S. Ali, G. Alkhazov, P. Alvarez Cartelle, A. A. Alves Jr, S.\n Amato, Y. Amhis, L. Anderlini, J. Anderson, R. Andreassen, R. B. Appleby, O.\n Aquines Gutierrez, F. Archilli, A. Artamonov, M. Artuso, E. Aslanides, G.\n Auriemma, S. Bachmann, J. J. Back, C. Baesso, V. Balagura, W. Baldini, R. J.\n Barlow, C. Barschel, S. Barsuk, W. Barter, A. Bates, Th. Bauer, A. Bay, J.\n Beddow, I. Bediaga, S. Belogurov, K. Belous, I. Belyaev, E. Ben-Haim, M.\n Benayoun, G. Bencivenni, S. Benson, J. Benton, A. Berezhnoy, R. Bernet, M.-O.\n Bettler, M. van Beuzekom, A. Bien, S. Bifani, T. Bird, A. Bizzeti, P. M.\n Bj{\\o}rnstad, T. Blake, F. Blanc, C. Blanks, J. Blouw, S. Blusk, A. Bobrov,\n V. Bocci, A. Bondar, N. Bondar, W. Bonivento, S. Borghi, A. Borgia, T. J. V.\n Bowcock, E. Bowen, C. Bozzi, T. Brambach, J. van den Brand, J. Bressieux, D.\n Brett, M. Britsch, T. Britton, N. H. Brook, H. Brown, A. B\\\"uchler-Germann,\n I. Burducea, A. Bursche, J. Buytaert, S. Cadeddu, O. Callot, M. Calvi, M.\n Calvo Gomez, A. Camboni, P. Campana, A. Carbone, G. Carboni, R. Cardinale, A.\n Cardini, H. Carranza-Mejia, L. Carson, K. Carvalho Akiba, G. Casse, M.\n Cattaneo, Ch. Cauet, M. Charles, Ph. Charpentier, P. Chen, 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, G. Corti, B. Couturier,\n G. A. Cowan, D. Craik, S. Cunliffe, R. Currie, C. D'Ambrosio, P. David, P. N.\n Y. David, I. De Bonis, K. De Bruyn, S. De Capua, M. De Cian, J. M. De\n Miranda, L. De Paula, W. De Silva, P. De Simone, D. Decamp, M. Deckenhoff, H.\n Degaudenzi, L. Del Buono, C. Deplano, D. Derkach, O. Deschamps, F. Dettori,\n A. Di Canto, J. Dickens, H. Dijkstra, P. Diniz Batista, M. Dogaru, F. Domingo\n Bonal, S. Donleavy, F. Dordei, A. Dosil Su\\'arez, D. Dossett, A. Dovbnya, F.\n Dupertuis, 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, D. Elsby, A. Falabella, C.\n F\\\"arber, G. Fardell, C. Farinelli, S. Farry, V. Fave, D. Ferguson, V.\n Fernandez Albor, F. Ferreira Rodrigues, M. Ferro-Luzzi, S. Filippov, C.\n Fitzpatrick, M. Fontana, F. Fontanelli, R. Forty, O. Francisco, M. Frank, C.\n Frei, M. Frosini, S. Furcas, E. Furfaro, A. Gallas Torreira, D. Galli, M.\n Gandelman, P. Gandini, Y. Gao, J. Garofoli, P. Garosi, J. Garra Tico, L.\n Garrido, C. Gaspar, R. Gauld, E. Gersabeck, M. Gersabeck, T. Gershon, Ph.\n Ghez, V. Gibson, V. V. Gligorov, C. G\\\"obel, D. Golubkov, A. Golutvin, A.\n Gomes, 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, O.\n Gr\\\"unberg, B. Gui, E. Gushchin, Yu. Guz, T. Gys, C. Hadjivasiliou, G.\n Haefeli, C. Haen, S. C. Haines, S. Hall, T. Hampson, S. Hansmann-Menzemer, N.\n Harnew, S. T. Harnew, J. Harrison, P. F. Harrison, T. Hartmann, J. He, V.\n Heijne, K. Hennessy, P. Henrard, J. A. Hernando Morata, E. van Herwijnen, E.\n Hicks, D. Hill, M. Hoballah, C. Hombach, P. Hopchev, W. Hulsbergen, P. Hunt,\n T. Huse, N. Hussain, D. Hutchcroft, D. Hynds, V. Iakovenko, P. Ilten, J.\n Imong, R. Jacobsson, A. Jaeger, E. Jans, F. Jansen, P. Jaton, F. Jing, M.\n John, D. Johnson, C. R. Jones, B. Jost, M. Kaballo, S. Kandybei, M. Karacson,\n T. M. Karbach, I. R. Kenyon, U. Kerzel, T. Ketel, A. Keune, B. Khanji, O.\n Kochebina, I. Komarov, R. F. Koopman, P. Koppenburg, M. Korolev, A.\n Kozlinskiy, L. Kravchuk, K. Kreplin, M. Kreps, G. Krocker, P. Krokovny, F.\n Kruse, M. Kucharczyk, V. Kudryavtsev, T. Kvaratskheliya, V. N. La Thi, D.\n Lacarrere, G. Lafferty, A. Lai, D. Lambert, R. W. Lambert, E. Lanciotti, G.\n Lanfranchi, C. Langenbruch, T. Latham, C. Lazzeroni, R. Le Gac, J. van\n Leerdam, J.-P. Lees, R. Lef\\`evre, A. Leflat, J. Lefran\\c{c}ois, O. Leroy, Y.\n Li, L. Li Gioi, M. Liles, R. Lindner, C. Linn, B. Liu, G. Liu, J. von Loeben,\n J. H. Lopes, E. Lopez Asamar, N. Lopez-March, H. Lu, J. Luisier, H. Luo, A.\n Mac Raighne, F. Machefert, I. V. Machikhiliyan, F. Maciuc, O. Maev, S. Malde,\n G. Manca, G. Mancinelli, N. Mangiafave, U. Marconi, R. M\\\"arki, J. Marks, G.\n Martellotti, A. Martens, L. Martin, A. Mart\\'in S\\'anchez, M. Martinelli, D.\n Martinez Santos, D. Martins Tostes, A. Massafferri, R. Matev, Z. Mathe, C.\n Matteuzzi, M. Matveev, E. Maurice, A. Mazurov, J. McCarthy, R. McNulty, B.\n Meadows, F. Meier, M. Meissner, M. Merk, D. A. Milanes, M.-N. Minard, J.\n Molina Rodriguez, S. Monteil, D. Moran, P. Morawski, R. Mountain, I. Mous, F.\n Muheim, K. M\\\"uller, R. Muresan, B. Muryn, B. Muster, P. Naik, T. Nakada, R.\n Nandakumar, I. Nasteva, M. Needham, N. Neufeld, A. D. Nguyen, T. D. Nguyen,\n C. Nguyen-Mau, M. Nicol, V. Niess, R. Niet, N. Nikitin, T. Nikodem, S. Nisar,\n A. Nomerotski, A. Novoselov, A. Oblakowska-Mucha, V. Obraztsov, S. Oggero, S.\n Ogilvy, O. Okhrimenko, R. Oldeman, M. Orlandea, J. M. Otalora Goicochea, P.\n Owen, B. K. Pal, A. Palano, M. Palutan, J. Panman, A. Papanestis, M.\n Pappagallo, C. Parkes, C. J. Parkinson, G. Passaleva, G. D. Patel, M. Patel,\n G. N. Patrick, C. Patrignani, C. Pavel-Nicorescu, A. Pazos Alvarez, A.\n Pellegrino, G. Penso, M. Pepe Altarelli, S. Perazzini, D. L. Perego, E. Perez\n Trigo, A. P\\'erez-Calero Yzquierdo, P. Perret, M. Perrin-Terrin, G. Pessina,\n K. Petridis, A. Petrolini, A. Phan, E. Picatoste Olloqui, B. Pietrzyk, T.\n Pila\\v{r}, D. Pinci, S. Playfer, M. Plo Casasus, F. Polci, G. Polok, A.\n Poluektov, E. Polycarpo, D. Popov, B. Popovici, C. Potterat, A. Powell, J.\n Prisciandaro, V. Pugatch, A. Puig Navarro, W. Qian, J. H. Rademacker, B.\n Rakotomiaramanana, M. S. Rangel, I. Raniuk, N. Rauschmayr, G. Raven, S.\n Redford, M. M. Reid, A. C. dos Reis, S. Ricciardi, A. Richards, K. Rinnert,\n V. Rives Molina, D. A. Roa Romero, P. Robbe, E. Rodrigues, P. Rodriguez\n Perez, G. J. Rogers, S. Roiser, V. Romanovsky, A. Romero Vidal, J. Rouvinet,\n T. Ruf, H. Ruiz, G. Sabatino, J. J. Saborido Silva, N. Sagidova, P. Sail, B.\n Saitta, C. Salzmann, B. Sanmartin Sedes, M. Sannino, R. Santacesaria, C.\n Santamarina Rios, E. Santovetti, M. Sapunov, A. Sarti, C. Satriano, A. Satta,\n M. Savrie, D. Savrina, P. Schaack, M. Schiller, H. Schindler, S. Schleich, 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, P. Shatalov, Y. Shcheglov, T. Shears, L. Shekhtman, O. Shevchenko,\n V. Shevchenko, A. Shires, R. Silva Coutinho, T. Skwarnicki, N. A. Smith, E.\n Smith, M. Smith, K. Sobczak, M. D. Sokoloff, F. J. P. Soler, F. Soomro, D.\n Souza, B. Souza De Paula, B. Spaan, A. Sparkes, P. Spradlin, F. Stagni, S.\n Stahl, O. Steinkamp, S. Stoica, S. Stone, B. Storaci, M. Straticiuc, U.\n Straumann, V. K. Subbiah, S. Swientek, V. Syropoulos, M. Szczekowski, P.\n Szczypka, T. Szumlak, S. T'Jampens, M. Teklishyn, E. Teodorescu, F. Teubert,\n C. Thomas, E. Thomas, J. van Tilburg, V. Tisserand, M. Tobin, S. Tolk, D.\n Tonelli, S. Topp-Joergensen, N. Torr, E. Tournefier, S. Tourneur, M. T. Tran,\n M. Tresch, A. Tsaregorodtsev, P. Tsopelas, N. Tuning, M. Ubeda Garcia, A.\n Ukleja, D. Urner, U. Uwer, V. Vagnoni, G. Valenti, R. Vazquez Gomez, P.\n Vazquez Regueiro, 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.\n Waldi, R. Wallace, S. Wandernoth, J. Wang, D. R. Ward, N. K. Watson, A. D.\n Webber, D. Websdale, M. Whitehead, J. Wicht, D. Wiedner, L. Wiggers, G.\n Wilkinson, M. P. Williams, M. Williams, F. F. Wilson, J. Wishahi, M. Witek,\n W. Witzeling, S. A. Wotton, S. Wright, S. Wu, K. Wyllie, Y. Xie, F. Xing, Z.\n Xing, Z. Yang, R. Young, X. Yuan, O. Yushchenko, M. Zangoli, M. Zavertyaev,\n F. Zhang, L. Zhang, W. C. Zhang, Y. Zhang, A. Zhelezov, A. Zhokhov, L. Zhong,\n A. Zvyagin",
"submitter": "Patrick Robbe",
"url": "https://arxiv.org/abs/1212.1045"
}
|
1212.1192
|
# Using external sources of bilingual information for on-the-fly word
alignment
Miquel Esplà-Gomis, Felipe Sánchez-Martínez and Mikel L. Forcada
_Departament de Llenguatges i Sistemes Informàtics_
_Universitat d’Alacant, E-03071 Alacant, Spain_
[email protected], [email protected], [email protected]
(Technical report, December 7, 2012)
###### Abstract
In this paper we present a new and simple language-independent method for
word-alignment based on the use of external sources of bilingual information
such as machine translation systems. We show that the few parameters of the
aligner can be trained on a very small corpus, which leads to results
comparable to those obtained by the state-of-the-art tool GIZA++ in terms of
precision. Regarding other metrics, such as alignment error rate or
$F$-measure, the parametric aligner, when trained on a very small gold-
standard (450 pairs of sentences), provides results comparable to those
produced by GIZA++ when trained on an in-domain corpus of around 10,000 pairs
of sentences. Furthermore, the results obtained indicate that the training is
domain-independent, which enables the use of the trained aligner _on the fly_
on any new pair of sentences.
## 1 Introduction
### 1.1 The need for word [position] alignment
Corpus-based translation technologies use information obtained from existing
_segment pairs_ , that is, pairs of text segments which are a translation of
each other —such as (_Give the book to me_ , _Donne-moi le livre_)—, to
perform a translation task. These pairs of segments are usually, but not
always, _sentence_ pairs, and to be able to translate new, unseen text
segments, the information in them is usually generalized after performing
_word alignment_. The task of word alignment consists in determining the
correspondence between the words (actually word positions) in one segment and
those in the other segment. After word alignment, smaller sub-segment
_translation units_ , such as (_le livre_ , _the book_), can be extracted.
These translation units have a prominent role in state-of-the-art statistical
machine translation (SMT, (Koehn,, 2010)), and are usually referred to as
_phrase pairs_ in the SMT literature.
The most widely used alignment method is based on the so-called IBM models by
Brown et al., (1993) and the HMM-based alignment model by Vogel et al.,
(1996), both implemented in the free/open-source GIZA++ tool (Och and Ney,,
2003).111http://code.google.com/p/giza-pp/[last visit: 30th August 2012]
Roughly, these methods, which were devised for building word-based SMT
systems, establish correspondences between the word positions in one segment
and the word positions in the other segment of the pair by using iterative
expectation-maximization (EM) training on large sets of segment pairs called
_parallel corpora_ (also _translation memories_ in computer-aided translation,
CAT).
The two key components of the EM approach to word alignment are: (a) the
building of probabilistic dictionaries that model the correspondence between
the words (not word positions) in one language and those in the other
language, independently of the actual segment pairs in which they were found;
and (b) the building of rather sophisticated statistical _alignment models_
which explicitly model _fertility_ (the maximum number of words with which a
word can be aligned) and _reorderings_ , and that use the probabilistic
dictionaries to describe the alignment in each segment pair. EM iterations
improve these two probabilistic models alternatively by approximately
assigning an increasing likelihood to the training corpus in each iteration;
the quality of the estimation and the training time both increase with the
size of the parallel corpus (roughly linearly, (Toral et al.,, 2012)). The
resulting probability models are then used to extract the best word-position
alignment, usually called just _word alignment_ , in each sentence pair.
### 1.2 The need for _on-the-fly_ word [position] alignment
While the state-of-the-art approach to word alignment is appropriate as a
first step when building an SMT system, it may happen to be unfeasible because
the parallel corpus available is not large enough to get accurate word
alignments, or because it is too costly in terms of time. This is actually the
case when one needs to _word-align_ a few new segment pairs _on the fly_ ,
that is, instantaneously, for instance, when performing CAT using translation
memories, as in the case of the works by Kranias and Samiotou, (2004) and
Esplà-Gomis et al., (2011).222For the use of word-position alignment
information in CAT, see Esplà-Gomis et al., (2011) and Kuhn et al., (2011).
There is, of course, the possibility of using a probabilistic alignment model
previously trained on another, ideally related, parallel corpus to align the
word positions in the new segment pairs; however, these pre-trained alignment
models may not be generally available for every possible domain or task.
We describe alternative ways to perform _word-position_ alignment on a segment
pair, on the fly and on demand, by using readily available sources of
translation units, which we will refer to as _sources of bilingual
information_ (SBI); for instance, existing (on-line) machine translation
systems. Information from the SBI is initially used to discover
correspondences between variable-length sub-segments in the pair of segments
to align, and then processed to obtain word-position alignments. The word-
position alignments are obtained by applying a probabilistic word-position
model whose parameters have to be trained on a parallel corpus; no assumptions
are made about the pair of languages involved. The corpus, as it will be
shown, need not be related to the new segment pairs being word aligned;
parameters are therefore transferable across text domains. In addition, there
is a particular choice of parameters that completely avoids the need for
training and has an intuitive “physical” interpretation, yielding reasonably
good results.
### 1.3 Related work
In addition to the IBM models and the HMM alignment model previously
mentioned, one can find in the literature different approaches to the problem
of word-position alignment. In this section we focus on those approaches that
make use of SBI in some way; for a complete review of the state of the art in
word alignment the reader is referred to Tiedemann, (2011).
Fung and Mckeown, (1997) introduces the use of a bilingual dictionary as a SBI
to obtain an initial alignment between _seed words_ in a parallel corpus.
These seed words are chosen so that they cannot have multiple translations (in
both languages) and are frequent enough to become useful references in both
texts of the parallel corpus. These initial alignments are then used to align
the other words appearing around them in the parallel texts using an heuristic
method similar to the one introduced by Rapp, (1999).
Liu et al., (2005) propose the use of a _log-linear_ (maximum-entropy style)
model (Berger et al.,, 1996) to combine the IBM model 3 alignment model with
information coming from part-of-speech taggers and bilingual dictionaries; the
work was later extended to include new features and a new training procedure
(Liu et al.,, 2010). The main differences between their work and the one
presented here are: (i) we do not rely on any previously computed alignment
model; (ii) we use any possible SBI which may relate multi-word segments, and
(iii) they model the word-position alignment task as a _structured prediction
problem_ (Tiedemann,, 2011, p. 82) that generates the whole alignment
structure, whereas we model each association of positions independently. We
will further discuss this last difference in the next section.
## 2 The alignment model
The method we present here uses the available sources of bilingual information
(SBI) to detect parallel sub-segments in a given pair of parallel text
segments $S$ and $T$ written in different languages. Once sub-segment
alignments have been identified, the word-position alignments are obtained
after computing the probability $p(j,k)$ of every pair of word positions
$(j,k)$ being aligned. For the computation on these probabilities a set of
feature functions are used which are based on the sub-segment alignments
observed.
We define the probability $p(j,k)$ as follows:
$p(j,k)=\displaystyle\exp\left(\sum_{p=1}^{n_{F}}\lambda_{p}f_{p}(j,k)\right)\left(\displaystyle\sum_{k^{\prime}}\sum_{j^{\prime}}\exp\left(\sum_{p=1}^{n_{F}}\lambda_{p}f_{p}(j^{\prime},k^{\prime})\right)\right)^{-1}$
(1)
where (a) the source-side position indexes $j$ (also $j^{\prime}$) can take
values from 1 to $|S|$, but also be NULL, and target-side position indexes $k$
(also $k^{\prime}$) can take values from 1 to $|T|$, and also be NULL, but
never simultaneously to a source-side index (alignments from NULL to NULL are
not possible); and (b) $f_{p}(j,k)$ is the $p$-th feature (see below) relating
the $j$-th word of the source sentence $S$ and the $k$-th word of the target
sentence $T$. This is a maximum-entropy-style function that is always in
$[0,1]$ and that has the property that
$\sum_{k}\sum_{j}p(j,k)=1$
when summing for all valid index pairs. The probabilities $p(j,k)$ may be
interpreted as the probability that someone who does not know the languages
involved links position $j$ in $S$ and $k$ in $T$ after looking at the set of
translation pairs provided by the SBI which happen to match sub-segments in
$S$ and $T$.
This model is similar to the one proposed by Liu et al., (2005) and later by
Liu et al., (2010) as discussed in the previous section. One important
difference between both models is that these authors formulate the alignment
as a _structured prediction problem_ in which the probability for a pair of
segments is computed for the whole set of word-position alignments
$a=\\{(j,k)\\}$; that is, the probability of a word-position alignment $(j,k)$
gets influenced by the rest of word-positions alignments for that pair of
segments. In contrast, we model each word-position alignment independently.
This may be less expressive but has interesting advantages from the
computational point of view when searching for the best set of word-position
alignments for a pair of segments.
#### Sub-segment alignment.
To obtain the sub-segment alignments, both segments $S$ and $T$ are segmented
in all possible ways to obtain sub-segments of length $l\in[1,L]$, where $L$
is a given maximum sub-segment length measured in words. Let $\sigma$ be a
sub-segment from $S$ and $\tau$ a sub-segment from $T$. We consider that
$\sigma$ and $\tau$ are aligned if any of the available SBI confirm that
$\sigma$ is a translation of $\tau$, or vice versa.
Suppose the pair of parallel segments $S$=_Costarà temps solucionar el
problema_ , in Catalan, and $T$=_It will take time to solve the problem_ , in
English. We first obtain all the possible sub-segments $\sigma$ in $S$ and
$\tau$ in $T$ and then use machine translation as a SBI by translating the
sub-segments in both translation directions. We obtain the following set of
sub-segment alignments:
_temps_ | $\leftrightarrow$ | _time_
---|---|---
_problema_ | $\leftrightarrow$ | _problem_
_solucionar el_ | $\to$ | _solve the_
_solucionar el_ | $\leftarrow$ | _to solve the_
_el problema_ | $\leftrightarrow$ | _the problem_
It is worth noting that multiple alignments for a sub-segment are possible, as
in the case of the sub-segment _solucionar el_ which is both aligned with
_solve the_ and _to solve the_. In those cases, all the sub-segment alignments
available are used. Figure 1 shows a graphical representation of these
alignments.
Figure 1: Sub-segment alignments.
#### Features.
The information provided by the sub-segment alignments is used to build the
features that are combined to compute the probabilities $p(j,k)$ through eq.
(1). This feature functions are based on the function
$\mathrm{cover}(j,k,\sigma,\tau)$, which equals 1 if sub-segment $\sigma$
_covers_ the $j$-th word in $S$ and $\tau$ _covers_ the $k$-th word in $T$,
and 0 otherwise. In particular, by considering sub-segments $\sigma$ and
$\tau$ of lengths $m$ and $n$ varying from 1 to the maximum sub-segment length
$L$ we define the following set of $L^{2}$ features, one feature for each
possible combination of lengths $(m,n)\in[1,L]\times[1,L]$:
$f_{(m-1)L+n}=\sum_{(\sigma,\tau)\in
M(S,T),|\sigma|=m,|\tau|=n}\mathrm{cover}(j,k,\sigma,\tau),$
where $|x|$ stands for the length of sub-segment $x$ measured in words.333One
may also split this feature set to treat each different SBI separately or even
lift the restriction on the source and target lengths $m$ and $n$, and build
new features depending only on $n$ and $m$, respectively.
#### Alignment computation.
To get the word-position alignments of a pair of segments $S$ and $T$ we
follow a greedy method that makes two simplifying assumptions:
* •
each word position $j$ in $S$ is aligned to either a single word position $k$
in $T$ or to NULL (source-to-target alignment);
* •
then, independently, each word position $k$ in $T$ is aligned to either a
single word position $j$ in $S$ or to NULL (target-to-source alignment).
Therefore all possible alignments of sentences $S$ and $T$ have exactly
$|S|+|T|$ alignments. The total probability of each such alignment $a$ is
$p(a)=\prod_{(j,k)\in
a}p(j,k)=\prod_{j=1}^{|S|}p(j,k^{\star}(j))\times\prod_{k=1}^{|T|}p(j^{\star}(k),k),$
(2)
where each position $j$ in $[1,|S|]$ aligns to a single position
$k^{\star}(j)$ in $[1,|T|]\cup\\{\mathrm{NULL}\\}$, and each position $k$ in
$[1,|T|]$ aligns to a single position $j^{\star}(k)$ in
$[1,|S|]\cup\\{\mathrm{NULL}\\}$. It may be easily shown that if we choose
$j^{\star}(k)=\left\\{\begin{array}[]{cl}\mathrm{arg}\max_{1\leq
j\leq|S|}p(j,k)&\mbox{if }p(j,k)>1/Z\\\
\mathrm{NULL}&\mbox{otherwise}\end{array}\right.$ (3)
and
$k^{\star}(j)=\left\\{\begin{array}[]{cl}\mathrm{arg}\max_{1\leq
k\leq|T|}p(j,k)&\mbox{if }p(j,k)>1/Z\\\
\mathrm{NULL}&\mbox{otherwise}.\end{array}\right.$ (4)
the resulting alignment probability is the highest possible. The case
$p(j,k)=1/Z$ where $Z$ is the normalizing factor on the right side of eq. (1)
occurs when no evidence has been found for that particular position pair
$(j,k)$, i.e. $\textrm{cover}(j,k,\sigma,\tau)$ is zero; in that case, we
decide to align these words to NULL. In case of finding two equiprobable
alignment candidates for a given word, the one closest to the diagonal is
chosen.
Note that the above alignments may be considered as two separate sets of
asymmetrical alignments that may be symmetrized as is usually done with
statistical alignments. The union alignment is the whole set of $|S|+|T|$
alignments; the intersection and _grow-diagonal-final-and_ (Koehn et al.,,
2003) alignments can also be readily obtained from them.
#### Training.
To get the best values of $\lambda_{p}$ we try to fit our alignments to the
reference alignments $\hat{a}_{m}$ in a training corpus $C$ of $n_{S}$
sentences. We do this in basically two ways.
The first one consists in maximizing the probability (actually the logarithm
of the probability) of the whole training corpus $C$:
$\log p(C)=\sum_{m=1}^{n_{S}}\sum_{(j,k)\in\hat{a}_{m}}\log p(j,k;m)$ (5)
where indexes $j$ and $k$ can be NULL as explained above (unaligned words in
the reference alignment $\hat{a}_{m}$ are assumed to be aligned to NULL).
Sentence index $m$ has been added to the probability function for clarity.
Eq. (5) is differentiable with respect to the parameters $\lambda_{p}$, which
allows for gradient ascent training, with each component of the gradient
computed as follows:
$\frac{\partial
E}{\partial\lambda_{p}}=\sum_{m=1}^{n_{S}}\sum_{(j,k)\in\hat{a}_{m}}\left(f_{p}(j,k;m)-\sum_{(j^{\prime},k^{\prime})\in\hat{a}_{m}}p(j^{\prime},k^{\prime};m)f_{p}(j^{\prime},k^{\prime};m)\right),$
(6)
where sentence index $m$ has been also added to $f_{p}(j,k)$ for the sake of
clarity.
The second approach tries to minimize directly an alignment error measure that
indicates how much a discretized, symmetrized alignment obtained by our method
departs from the alignments observed in the training corpus: for instance, the
alignment error rate (AER) (Och and Ney,, 2003) or $1-F$ where $F$ is the
$F$-measure (Manning and Schütze,, 1999, Ch. 8.1), much as it is done by (Liu
et al.,, 2010). Discretization renders these error measures non-
differentiable; therefore, we resort to using general-purpose function
optimization methods such as the multidimensional simplex optimization of
(Nelder and Mead,, 1965).444Liu et al., (2010) use MERT instead.
With the two approaches the number of trainable parameters is small (of the
order of $L^{2}$, where $L$ is the maximum sub-segment length considered),
therefore reasonable results may be expected with a rather small training
corpus and a SBI covering well the sentence pairs. This is because no word
probabilities have to be learned but only parameters to produce word-position
alignments using information from the SBIs.
### 2.1 An intuitive aligner that does not need training
There is a set of parameters for the model described above that has an
intuitive “physical” interpretation, and that yields reasonable results, as
shown in Section 3. This set of parameters could be used as a starting point
for optimization or as a first approximation.
If one chooses $\lambda_{(m-1)L+n}=({mn})^{-1}$, eq. (1) may be rewritten as:
$p(j,k)=\exp(P_{jk}(S,T,M(S,T))\left(\sum_{j^{\prime}}\sum_{k^{\prime}}\exp(P_{j^{\prime}k^{\prime}}(S,T,M(S,T))\right)^{-1}$
where the _alignment presssure_ $P_{jk}(S,T,M(S,T))$ between the $j$-th word
in $S$ and the $k$-th word in $T$ is
$\displaystyle P_{jk}(S,T,M(S,T))=\sum_{(\sigma,\tau)\in
M(S,T)}\frac{\mathrm{cover}(j,k,\sigma,\tau)}{|\sigma|\cdot|\tau|}$
where $M(S,T)$ is the set of sub-segment alignments detected for the pair of
parallel segments $S$ and $T$. If either $j$ or $k$ are NULL,
$\mathrm{cover}(j,k,\sigma,\tau)$ is zero.
Intuitively, each $P_{jk}$ may be seen as the _pressure_ applied by the sub-
segment alignments on the word pair $(j,k)$; so the wider the surface
($|\sigma||\tau|$) covered by a sub-segment alignment, the lower the
contribution of that sub-segment pair to the total pressure on $(j,k)$.555If
just those $L^{2}$ features are used and the system is trained on a parallel
corpus, the value $mn\lambda_{(m-1)L+n}$ may be considered as the “effective
weight” of $m\times n$ sub-segment pairs. Clearly, the higher the pressure
$P_{jk}$, the higher the probability $p(j,k)$ is. In the absence of sub-
segment information for any of the $(j,k)$’s of a particular segment pair, all
probabilities are equal: $p(j,k)=\frac{1}{(|S||T|+|S|+|T|)}$. The _pressures_
are zero when either $j$ or $k$ is NULL.
Following our example, the alignment pressures for the words covered by the
sub-segment alignments are presented in Figure 2. The word pair (_temps_
,_time_) is only covered by a sub-segment alignment (_temps_ , _time_), so the
surface is 1 and the alignment pressure is $P_{2,4}=1$. On the other hand, the
word pair (_the_ ,_el_) is covered by three sub-segment alignments:
(_solucionar el_ , _solve the_), (_solucionar el_ , _to solve the_), and (_el
problema_ , _the problem_); therefore, the _alignment pressure_ is
$P_{4,7}=1/4+1/6+1/4=2/3\simeq 0.67$.
Figure 2: Alignment pressures.
In this simple model, the alignment pressures $P_{jk}$ themselves may then be
used instead of the probabilities $p(j,k)$ to obtain word-position alignments
as described at the end of Section 2.
As in the case of the general alignment model defined at the beginning of this
section, the alignment is performed both from source-to-target and from
target-to-source following the same procedure. Figure 3 shows the Catalan-to-
English and the English-to-Catalan word alignments for the running example. As
can be seen, words _to_ and _solve_ in English have the same alignment score
for words _solucionar_ and _el_ in Spanish, respectively. Therefore, the
alignments closest to the diagonal are chosen; in this case, _to_ is aligned
with _solucionar_ , and _solve_ is aligned with _el_ (not a very good
alignment). In the other direction of the alignment, the situation is similar
for word _solucionar_ in Spanish and words _solve_ and _the_ in English (the
resulting alignment is better here).
Figure 3: Resulting Catalan-to-English and English-to-Catalan word alignments.
Figure 4 shows two possible symmetrized word alignments obtained by computing,
in the first case, the intersection of the alignments shown in figure 3, and,
in the second case, the the widely-used _grow-diagonal-final-and_ heuristic of
Koehn et al., (2003), which, in this case, coincides with the union of the
alignments.
Figure 4: Two possible symmetrized word alignments, the first one using the
intersection heuristic and the second one using the _grow-diagonal-final-and_
heuristic.
## 3 Experiments
In this section we describe the experimental setting designed for measuring
the performance of the alignment models described in Section 2. Two different
experimental scenarios were defined in order to measure (a) the quality of the
alignments obtained when using training corpora with several levels of
reliability, and (b) the domain independence of the weights trained for the
parametric aligner (P-aligner).
#### Gold-standard experiment.
For this experiment, we used the EPPS _gold standard_ (Lambert et al.,, 2005),
a collection of 500 pairs of sentences extracted from the English–Spanish
Europarl parallel corpus (Koehn,, 2005) and hand-aligned at the word level
using two classes of alignments: _sure_ alignments and _possible_
alignments.666Once the sub-segment alignments were obtained, the gold standard
was lowercased to maximise the recall in the alignment process. This corpus
was used for performing several evaluations:
* •
_parametric alignment model (defined in Section 2)_: we evaluated this model
by using the gold standard corpus both for training and testing using a
10-fold cross-validation strategy. Therefore, for each fold we had 450 pairs
of sentences as a training set and 50 pairs of sentences as a test set. We
tried the two methods defined in Section 2 for training: optimization of eq.
(5) by using a gradient ascent algorithm (Duda et al.,, 2000), and minimizing
directly the alignment error rate (AER) by using the _simplex_ algorithm
(Nelder and Mead,, 1965). Increasingly large sets of bilingual sub-segments
were used by defining different values of the maximum sub-segment length $L$
in $[1,5]$.
* •
_pressure aligner (defined in Section 2.1)_: Since this alignment model does
not require training it was directly evaluated on the gold standard.
Increasingly large sets of bilingual sub-segments were used by defining
different values of the maximum sub-segment length $L$ in $[1,5]$.
* •
_GIZA++ trained on the EPPS gold standard_ : GIZA++ (Och and Ney,, 2003) was
used as a baseline by repeating the previously described 10-fold cross-
validation strategy.777The _test-corpus_ option in GIZA++ was used to train
the alignment models with one corpus and then align another one. Although it
is obvious that 450 pairs of parallel sentences is not enough for obtaining
high quality alignment models with this tool, this results are useful to
measure the performance of the models proposed when using a very small
training corpus.888To train GIZA++, the default configuration was used: 5
iterations of model 1 and hidden Markov model and 3 iterations of models 3 and
4.
* •
_GIZA++ trained on a large corpus_ : In this experiment a larger corpus was
used to train GIZA++ models: the English–Spanish parallel corpus provided for
the machine translation task at the Seventh Workshop on Statistical Machine
Translation (WMT12, Callison-Burch et al., (2012)), which includes the
Europarl parallel corpus, from which the gold standard is extracted. In this
way, it is possible to compare the models proposed in this work with the use
of the state-of-the-art tool GIZA++, which is commonly used in this scenario.
This corpus is provided already aligned at the sentence level and, before
training the alignment models, it was tokenised and lowercased, and sentences
longer than 50 words were removed.999This preprocessing was performed by using
the scripts provided by the Moses MT toolkit: https://github.com/moses-
smt/mosesdecoder/tree/master/scripts[last visit: 30th August 2012]
Since all the alignment models proposed in this experiment are asymmetric
(i.e. they must be trained from English to Spanish and from Spanish to English
separately) we experimented three different symmetrization methods:
intersection, union, and _grow-diagonal-final-and_ (Koehn et al.,, 2005).
#### GIZA++ alignments as a reference.
This second experiment focuses on measuring the re-usability of the weights
trained for the parametric alignment model. In this case, we used three
different corpora, all of them extracted from the translation memory published
by the _European Commission Directorate-General for Translation_ (European
Commission,, 2009).101010http://langtech.jrc.it/DGT-TM.html[last visit: 30th
August 2012] This translation memory is a collection of documents from the
_Official Journal of the European Union_ 111111http://eur-lex.europa.eu[last
visit: 30th August 2012] which are provided aligned at the sentence level.
These documents are indexed by using a set of domain codes121212http://eur-
lex.europa.eu/RECH_repertoire.do[last visit: 30th August 2012] which can be
used to identify the documents belonging to the same domain. Following this
method, we extracted three subsets from this translation memory belonging to
the domains: _elimination of barriers to trade_ (code 02.40.10.40), _safety at
work_ (code 05.20.20.10), and _general information of public contracts_ (code
06.30.10.00). These corpora were chosen because they have similar sizes
(between 15894 and 13414 pairs of sentences) and they belong to clearly
different domains, as evidenced by the cosine similarity measure131313The
cosine similarity was computed on the lowercased corpora, removing the
punctuation signs and the stopwords defined in the Snowball project:
http://snowball.tartarus.org/algorithms/english/stop.txt,
http://snowball.tartarus.org/algorithms/spanish/stop.txt[last visit: 30th
August 2012] presented in Table 1.141414As a reference, note that if we split
any of these three corpora into two parts and compute the cosine similarity
between them, the results obtained are around 0.98.
corpus en | 05.20.20.10 | 06.30.10.00
---|---|---
02.40.10.40 | 0.21 | 0.18
06.30.10.00 | 0.15 |
corpus es | 05.20.20.10 | 06.30.10.00
---|---|---
02.40.10.40 | 0.22 | 0.18
06.30.10.00 | 0.13 |
Table 1: Cosine similarity for both the English (en) and the Spanish (es)
documents in the corpora released by the _European Commission Directorate-
General for Translation_ that we used.
For this experiment, we followed these steps:
* •
GIZA++ was used to align the three corpora and these alignments were taken as
reference alignments;
* •
using the three reference alignments as training corpora, three different sets
of weights were obtained for the parametric aligner and each of these sets of
weights was used to align the other two corpora and also the same corpus on
which the weights were trained;
* •
the resulting alignments were compared with the reference alignments to
evaluate the re-usability of the weights in out-of-domain alignment tasks.
In addition, the GIZA++ alignment models obtained as a byproduct of the
computation of the reference alignments were also used to align the test
corpora. We used the resulting alignments as a point of comparison for the
alignments produced by the parametric aligner.
The experiments were performed by using: a range of values for the maximum
sub-segment length $L$, both the simplex and gradient ascent algorithms for
optimizing the weights of the parametric aligner, and the three symmetrization
methods previously commented. The best results were obtained with $L=5$ and
the _grow-diagonal-final-and_ symmetrization heuristic (Koehn et al.,, 2003).
#### Evaluation metrics.
For evaluating the different experiments defined in this section we used the
_Lingua-AlignmentSet_ toolkit151515http://gps-
tsc.upc.es/veu/personal/lambert/software/AlignmentSet.html [last visit: 30th
August 2012] which computes, for a pair of alignment set ($A$) and
corresponding gold standard ($G$), the precision ($P$), recall ($R$), and
$F$-measure ($F$) (Manning and Schütze,, 1999, Ch. 8.1), defined as usual:
$P=|A\cap G|/|A|\hskip 56.9055ptR=|A\cap G|/|G|\hskip 56.9055ptF=2PR/(P+R)$
These measures are computed (a) only for the _sure_ alignments and (b) both
for _sure_ and _possible_ alignments. In addition, the alignment error rate
(AER) is computed by combining sure and possible alignments in the following
way:
$AER=1-\frac{|A\cap G_{\mathrm{sure}}|+|A\cap G|}{|A|+|G_{\mathrm{sure}}|}.$
#### Sources of bilingual information.
We used three different machine translation (MT) systems to translate the sub-
segments from English into Spanish and vice versa, in order to get the sub-
segment alignments needed to obtain the features for the models defined in
Section 2:
* •
_Apertium_ :161616http://www.apertium.org [last visit: 30th August 2012] a
free/open-source platform for the development of rule-based MT systems
(Forcada et al.,, 2011). We used the English–Spanish MT system from the
project’s
repository171717https://apertium.svn.sourceforge.net/svnroot/apertium/trunk/apertium-
en-es/ [last visit: 30th August 2012] (revision 34706).
* •
_Google Translate_ :181818http://translate.google.com [last visit: 30th August
2012] an online MT system by Google Inc. (translations performed in July
2012).
* •
_Microsoft Translator_ :191919http://www.microsofttranslator.com [last visit:
30th August 2012] an online MT system by Microsoft (translations performed in
July 2012).
It is worth noting that the Apertium system is oriented to closely-related
pairs of languages; furthermore, the Spanish–English language pair is not as
mature as other pairs in Apertium; therefore, it is expected to produce
translations of lower quality compared with other state-of-the-art systems as
indicated by observed BLEU scores. For the gold-standard experiment, these
three MT systems were used. For the experiments using the translation memories
released by the _European Commission Directorate-General for Translation_ ,
only Apertium and Google could be used, given the huge amount of sub-segments
to be translated and the restrictions in the Microsoft Translator API.
## 4 Results and discussion
This section presents the results obtained in the experiments described in the
Section 3. Table 2 shows the results in terms of precision ($P$), recall
($R$), $F$-measure ($F$) and alignment error rate (AER) obtained by both the
parametric aligner (P-aligner) described in Section 2, the “pressure” aligner
described in Section 2.1, and GIZA++ both when using a 10-fold cross-
validation strategy on the gold standard corpus and when using the corpus from
the WMT12 workshop for training the alignment models. It is worth noting that
the results computed using the 10-fold cross-validation (_P-aligner
probability optimization_ , _P-aligner AER optimization_ , and _GIZA++ trained
on the gold standard_) are presented as the average of the results obtained in
each fold. The parametric aligner was both trained by using all the alignments
available in the training sets and only using the sure ones. The results of
the parametric aligner (best AER in the 27%–29% range) overcame, as expected,
the results obtained by the “pressure” aligner (AER around 32%), since the
weights were trained on a gold standard and not fixed beforehand.202020The
results of the “pressure” aligner come however surprisingly close. As can be
appreciated, both the P-aligner and the “pressure” aligner overcame the
results by GIZA++ trained on the gold standard for all the metrics used (AER
around 55%). This is easily explainable given the small size of the corpus
used to train the alignment models with GIZA++. In any case, this shows the
convenience of our model when using a very reduced training corpus. Finally,
the alignments from GIZA++ trained on the WMT12 corpus obtained the best
results in terms of F-measure and AER (16%). If precision and recall are
compared, one can see that the precision in both GIZA++ and the parametric
aligner are quite similar but GIZA++ obtains better results in recall. This is
an interesting result, since this means that, for tasks like CAT (Esplà-Gomis
et al.,, 2011), where precision is more relevant than the recall, the
parametric aligner may be as useful as GIZA++. Also, this means that using
more (or better) sources of bilingual information could help to obtain closer
results to those obtained by GIZA++ in recall and, consequently, in F-measure
and AER. To understand these results better, a complementary experiment was
performed by using several sub-sets from the WMT12 corpus with different
sizes. We found out that, to obtain the same results produced by the P-aligner
in terms of AER, GIZA++ requires an in-domain training corpus with a size
between 5,000 pairs of sentences (AER 29.5%) and 10,000 pairs of sentences
(AER 26.2%). This confirms that GIZA++ requires a considerably larger training
corpus than that needed by the proposed approach and, as a consequence, it
would be quite difficult to use it for aligning sentences on the fly or for
small amounts of corpora.
There are some differences in the results obtained for the P-aligner depending
on the training method used: the model trained through the maximization of the
total alignment probability obtained higher results in precision (91% versus
75%), whereas the model trained by minimizing the AER provided better results
for recall (65% versus 56%). Although the results for F-measure and AER are
very similar, they happen to be slightly better when using the minimization of
the AER, as expected, since in this case the evaluation function is directly
optimized during the training process.
| $L$ | $P_{s}$ | $R_{s}$ | $F_{s}$ | $P$ | $R$ | $F$ | $\mathrm{AER}$
---|---|---|---|---|---|---|---|---
GIZA++ trained on the gold standard | 48.0% | 40.0% | 43.6% | 52.2% | 30.4% | 38.4% | 54.5%
GIZA++ 5,000 sentences of WMT12 corpus | 66.6% | 66.2% | 66.4% | 74.7% | 52.0% | 61.3% | 29.5%
P-aligner probability optimization | 1 | 86.0% | 44.2% | 58.3% | 89.9% | 32.4% | 47.6% | 40.3%
2 | 88.3% | 52.9% | 66.1% | 92.2% | 38.7% | 54.5% | 32.5%
3 | 90.1% | 55.7% | 68.8% | 94.0% | 40.7% | 56.8% | 29.7%
4 | 91.0% | 56.4% | 69.6% | 94.9% | 41.2% | 57.4% | 28.9%
5 | 91.4% | 56.2% | 69.6% | 95.2% | 41.1% | 57.3% | 29.0%
P-aligner AER optimization | 1 | 81.6% | 52.5% | 63.9% | 85.6% | 38.6% | 53.2% | 34.6%
2 | 71.7% | 60.0% | 65.3% | 78.7% | 46.1% | 58.1% | 31.5%
3 | 73.7% | 63.8% | 68.4% | 81.4% | 49.5% | 61.4% | 28.1%
4 | 75.3% | 64.5% | 69.5% | 82.7% | 49.7% | 62.0% | 27.1%
5 | 74.8% | 65.4% | 69.8% | 82.4% | 50.6% | 62.6% | 26.7%
“pressure” aligner | 1 | 80.4% | 39.1% | 52.6% | 85.0% | 28.9% | 43.1% | 45.9%
2 | 70.9% | 54.0% | 61.3% | 76.9% | 40.9% | 53.4% | 36.1%
3 | 69.8% | 58.0% | 63.3% | 76.7% | 44.5% | 56.3% | 33.6%
4 | 69.2% | 59.0% | 63.7% | 76.3% | 45.5% | 57.0% | 33.0%
5 | 69.1% | 59.4% | 63.9% | 76.3% | 45.8% | 57.3% | 32.8%
GIZA++ 10,000 sentences of WMT12 corpus | 69.2% | 69.8% | 69.5% | 77.7% | 54.8% | 64.3% | 26.2%
GIZA++ trained on whole WMT12 | 77.2% | 80.6% | 78.9% | 87.3% | 63.7% | 73.7% | 16.0%
Table 2: Average values of precision ($P$), recall ($R$), $F$-measure ($F$),
and alignment error rate (AER) for the alignments obtained with GIZA++ (when
trained both on the gold standard and several portions of the WMT12 parallel
corpus), and the parametric aligner (P-aligner) trained by optimizing the
total alignment probabilities, and by optimizing the AER, for different values
of the maximum sub-segment length $L$. The results obtained by the “pressure”
aligner are also reported. The training of the parametric aligner was
performed by using only the sure alignments.
Finally, Table 3 shows the results obtained for the experiment with the
translation memories from the _Official Journal of the European Union_ , which
is aimed at measuring the domain-independence of the weights trained for the
parametric aligner. The table shows, for the parametric aligner (using both
training methods) and GIZA++, the results obtained when training on one of the
corpora and aligning the other two corpora. The results reported in this table
were obtained by using sub-segments of length $L=5$, as this setting provided
the best results. As in the previous experiments, the symmetrization technique
used was _grow-diagonal-final-and_ (Och and Ney,, 2003). As can be seen, the
results for all the parametric aligners compared are quite similar for all the
systems and all the training/test corpora (AER in the range 27%–34%). It is
worth mentioning that in this particular experiment the alignments produced by
GIZA++ are being used as a gold standard for evaluation, which could be unfair
for our system, since some correct alignments from the P-aligner could be
judged as incorrect. Nevertheless, when the corpora used for testing is
different from that used for evaluation, the parametric aligners obtain better
results than GIZA++ (AER in the range 30%–40%), but the most important finding
is the relative uniformity in the results when using different corpora for
training and aligning. This shows that the weights learned from a corpus in a
given domain can be re-used to align corpora in different domains. This is a
very desirable property, as it would imply that, in a real application, once
the aligner is trained, it can be used for aligning any new pair of sentences
_on the fly_.
| training | test | $P$ | $R$ | $F$ | $\mathrm{AER}$
---|---|---|---|---|---|---
P-aligner probability optimization | 02.04.10.40 | 02.04.10.40 | 73.12% | 66.61% | 69.71% | 30.29%
05.20.20.10 | 79.3% | 68.4% | 73.4% | 26.6%
06.30.10.00 | 77.0% | 65.5% | 70.8% | 29.3%
05.20.20.10 | 02.04.10.40 | 72.8% | 64.2% | 68.2% | 31.8%
05.20.20.10 | 79.61% | 66.29% | 72.34% | 27.66%
06.30.10.00 | 78.2% | 63.6% | 70.1% | 29.9%
06.30.10.00 | 02.04.10.40 | 71.9% | 63.9% | 67.7% | 32.4%
05.20.20.10 | 78.6% | 65.5% | 71.5% | 28.5%
06.30.10.00 | 77.5% | 63.2% | 69.6% | 30.4%
P-aligner AER optimization | 02.04.10.40 | 02.04.10.40 | 73,1% | 60,3% | 66,1% | 33,9%
05.20.20.10 | 80.5% | 65.5% | 72.3% | 27.8%
06.30.10.00 | 78.3% | 63.0% | 69.8% | 30.2%
05.20.20.10 | 02.04.10.40 | 71.2% | 64.6% | 67.8% | 32.3%
05.20.20.10 | 79,7% | 67,4% | 73,1% | 26,9%
06.30.10.00 | 76.1% | 63.0% | 68.9% | 31.1%
06.30.10.00 | 02.04.10.40 | 70.5% | 68.8% | 69.6% | 30.4%
05.20.20.10 | 75.6% | 70.2% | 72.8% | 27.2%
06.30.10.00 | 74,6% | 67,4% | 70,8% | 29,2%
GIZA++ | 02.04.10.40 | 02.04.10.40 | 83.2% | 81.7% | 82.5% | 17.5%
05.20.20.10 | 71.3% | 64.3% | 67.6% | 32.4%
06.30.10.00 | 67.5% | 62.4% | 64.9% | 35.1%
05.20.20.10 | 02.04.10.40 | 70.9% | 61.9% | 66.1% | 33.9%
05.20.20.10 | 90.0% | 89.6% | 89.8% | 10.2%
06.30.10.00 | 72.9% | 68.0% | 70.3% | 29.7%
06.30.10.00 | 02.04.10.40 | 64.1% | 55.8% | 59.7% | 40.4%
05.20.20.10 | 70.2% | 63.6% | 66.8% | 33.2%
06.30.10.00 | 87.4% | 87.4% | 87.4% | 12.6%
Table 3: Precision ($P$), recall ($R$), $F$-measure ($F$), and alignment error
rate (AER) for the alignments obtained with the parametric aligner (P-aligner)
trained by optimizing the total alignment probabilities, the P-aligner trained
by optimizing the AER, and GIZA++ when using corpora from different domains
for training and testing.
## Concluding remarks and future work
In this work we have described a new approach for word alignment based on the
use of sources of bilingual information that makes no assumptions about the
languages of texts being aligned. Two alignment methods have been proposed:
(a) an intuitive and training-free aligner based on the idea of the pressure
exerted on the word-pair squares of a sentence-pair rectangular grid by the
bilingual sub-segments (rectangles) covering words in both sentences to be
aligned, and (b) a more general maximum-entropy-style (“log-linear”)
parametric aligner which may be seen as a generalization of that aligner. A
set of experiments was performed to evaluate both approaches, comparing them
with the state-of-the-art tool GIZA++. The results obtained show that the
models proposed obtain results comparable to those obtained by the state-of-
the-art tools in terms of precision. Although GIZA++ obtains better results in
recall and in general measures, such as $F$-measure and AER (16%), the
parametric aligner overcomes GIZA++ (AER 54%) when using a small training
corpus. In addition, the results show that the weights trained for the
parametric aligner can be re-used to align sentences from different domains to
the one from which they were trained. In this case the new approach provides
better results than GIZA++ when aligning out-of-domain corpora. This means
that it is possible to use the proposed alignment models to align new
sentences _on the fly_ , which can be specially useful in some scenarios as
the case of computer-aided translation (CAT).
As a future work, we plan to perform wider experiments including other pairs
of languages and also other sources of bilingual information. Note that the
parameters of the parametric MT-based aligner proposed here could also be
intrinsically optimized according to the overall performance of a larger task
using alignment as a component, such as _phrase_ -based SMT.
#### Acknowledgements:
Work partially supported by the Spanish Ministry of Science and Innovation
through project TIN2009-14009-C02-01, and by the Universitat d’Alacant through
project GRE11-20.
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|
arxiv-papers
| 2012-12-05T22:10:04 |
2024-09-04T02:49:38.922450
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Miquel Espl\\`a-Gomis, Felipe S\\'anchez-Mart\\'inez, Mikel L. Forcada",
"submitter": "Mikel Forcada Dr.",
"url": "https://arxiv.org/abs/1212.1192"
}
|
1212.1230
|
# Finite Factor Representations of Higman-Thompson groups
Artem Dudko email: [email protected] Department of Mathematics, Stony
Brook University, NY Konstantin Medynets email: [email protected] Department
of Mathematics, U.S. Naval Academy, Annapolis, MD
###### Abstract
We prove that the only finite factor-representations of the Higman-Thompson
groups $\\{F_{n,r}\\}$, $\\{G_{n,r}\\}$ are the regular representations and
scalar representations arising from group abelianizations. As a corollary, we
obtain that any measure-preserving ergodic action of a simple Higman-Thompson
group must be essentially free. Finite factor representations of other classes
of groups are also discussed.
## 1 Introduction
The goal of this paper is to describe finite (in the sense of Murray-von
Neumann) factor-representations of the Higman-Thompson groups (see Section 3
for the definition). The discussion of historical importance of these groups
and their various algebraic properties can be found in [1], [2], and [11]. The
following is the main result of the present paper.
Theorem. Let $G$ be a group from the Higman-Thompson families $\\{F_{n,r}\\}$,
$\\{G_{n,r}\\}$ and $\pi$ be a finite factor representation of $G$. Then
either $\pi$ is the regular representation or $\pi$ has the form
$\pi(g)=\rho([g])\mathrm{Id},$
where $[g]$ is the image of $g$ in the abelianization $G/G^{\prime}$,
$G^{\prime}$ is the commutator of $G$,
$\rho:G/G^{\prime}\rightarrow\mathbb{T}$ is a group homomorphism and
$\mathrm{Id}$ is the identical operator in some Hilbert space.
Since finite factor representations are in one-to-one correspondence with
positive definite class functions (termed characters, see Definition 2.1),
this shows that the characters of any Higman-Thompson group $G$ are convex
combinations of the regular character and characters of its abelianization
$G/G^{\prime}$. The structure of group characters has implications on
dynamical properties of group actions. Suppose that a group $G$ admitting no
non-regular/non-identity characters acts on a probability measure space
$(X,\mu)$ by measure-preserving transformations. Setting
$\chi(g)=\mu(Fix(g))$, $Fix(g)=\\{x\in X:g(x)=x\\}$, one can check that the
function $\chi$ is a character. So our results imply that any non-trivial
ergodic action of $G$ on a probability measure space $(X,\mu)$ is essentially
free, i.e. $\mu(Fix(g))=0$ for every $g\in G\setminus\\{e\\}$, see Theorem
2.11.
In our proofs, we mostly utilize the fact that the commutators of Higman-
Thompson groups have no non-atomic invariant measures on the spaces where they
are defined. This means that the orbit equivalence relations generated by
their actions are compressible [8]. This observation allows us to state the
main result in terms of dynamical properties of group actions (Theorems 2.9
and 2.10) — transformation groups whose actions are “compressible” (Definition
2.5) do not admit non-regular $II_{1}$-factor representations, except for
possible finite-dimensional representations. This dynamical formulation allows
us to apply the main result to other classes of transformation groups (Section
3).
In [13] Vershik suggested that the characters of “rich” groups should often
arise as $\mu(Fix(g))$ for some invariant measure $\mu$. Thus, this paper
confirms Vershik’s conjecture in the sense that the absence of non-trivial
invariant measures implies the absence of non-regular characters. We also
mention the paper [5], where Vershik’s conjecture was established for full
groups of Bratteli diagrams.
The structure of the paper is the following. In Section 2 we build the general
theory of finite factor representations for groups admitting compressible
actions. In Section 3, we apply our general results to the Higman-Thompson
groups and to the full groups of irreducible shifts of finite type [9].
## 2 General Theory
In this section we show that if a group admits a compressible action on a
topological space, then this group, under some algebraic assumptions, has no
non-trivial factor representations. We will start with definitions from the
representation theory of infinite groups.
###### Definition 2.1.
A character of a group $G$ is a function $\chi:G\rightarrow\mathbb{C}$
satisfying the following conditions
* 1)
$\chi(g_{1}g_{2})=\chi(g_{2}g_{1})$ for any $g_{1},g_{2}\in G$;
* 2)
the matrix $\left\\{\chi\left(g_{i}g_{j}^{-1}\right)\right\\}_{i,j=1}^{n}$ is
nonnegatively defined for any $n$ and $g_{1},\ldots,g_{n}\in G$;
* 3)
$\chi(e)=1$. Here $e$ is the group identity.
A character $\chi$ is called indecomposable if it cannot be written in the
form $\chi=\alpha\chi_{1}+(1-\alpha)\chi_{2}$, where $0<\alpha<1$ and
$\chi_{1},\chi_{2}$ are distinct characters.
For a unitary representation $\pi$ of a group $G$ denote by
$\mathcal{M}_{\pi}$ the $W^{*}$-algebra generated by the operators of the
representation $\pi$. Recall that the commutant $S^{\prime}$ of a set $S$ of
operators in a Hilbert space $H$ is the algebra $S^{\prime}=\\{A\in
B(H):AB=BA\text{ for any }B\in S\\}$.
###### Definition 2.2.
A representation $\pi$ of a group $G$ is called a factor representation if the
algebra $\mathcal{M}_{\pi}$ is a factor, that is
$\mathcal{M}_{\pi}\cap\mathcal{M}_{\pi}^{\prime}=\mathbb{C}\mathrm{Id}$.
The indecomposable characters on a group $G$ are in one-to-one correspondence
with the finite type111The classification of factors can be found in [12,
Chapter 5]. factor representations of $G$. Namely, starting with an
indecomposable character $\chi$ on $G$ one can construct a triple
$\left(\pi,H,\xi\right)$, referred to as the Gelfand-Naimark-Siegal (abbr.
GNS) construction. Here $\pi$ is a finite type factor representation acting in
the space $H$, and $\xi$ is a unit vector in $H$ such that
$\chi(g)=(\pi(g)\xi,\xi)$ for every $g\in G$, see, for example, [5, Sect.
2.3]. Note that the vector $\xi$ is cyclic and separating for the von Neumann
algebra $\mathcal{M}_{\pi}$. The latter means that if $A\xi=0$ for some
$A\in\mathcal{M}_{\pi}$, then $A=0$.
###### Remark 2.3.
We note that each character defines a factor representation up to quasi-
equivalence. Two unitary representations $\pi_{1}$ and $\pi_{2}$ of the same
group $G$ are called quasi-equivalent if there is an isomorphism of von
Neumann algebras
$\omega:\mathcal{M}_{\pi_{1}}\rightarrow\mathcal{M}_{\pi_{2}}$ such that
$\omega(\pi_{1}(g))=\pi_{2}(g)$ for each $g\in G$. For example, all $II_{1}$
factor representations of an amenable group are hyperfinite [3, Corollary 6.9
and Theorem 6] and, hence, generate isomorphic algebras. At the same time,
they might be not quasi-equivalent.
Suppose that $G$ is an infinite conjugacy class (abbr. ICC) group. Then its
left regular representation $\pi$ generates a $II_{1}$-factor and the function
$\chi(g)=(\pi(g)\delta_{e},\delta_{e})$ is an indecomposable character (termed
the regular character).
###### Definition 2.4.
We will say that a group $H$ has no proper characters if $\chi$ being an
indecomposable character of $H$ implies that either $\chi$ is identity
character given by
$\chi(g)=1\;\;\text{for every}\;\;g\in G$
or the regular character defined as
$\chi(g)=0\;\;\text{if}\;\;g\neq e\;\;\text{and}\;\;\chi(e)=1.$
We notice that for non-ICC groups the regular characters are decomposable.
Fix a regular Hausdorff topological space $X$. Notice that any two distinct
points of $X$ have open neighborhoods with disjoint closures. To exclude
trivial counterexamples to our statements we assume that the set $X$ is
infinite. Suppose that a group $G$ acts on $X$. For a group element $g\in G$,
denote its support by $supp(g)=\overline{\\{x\in X:g(x)\neq x\\}}$.
###### Definition 2.5.
We will say that the action of $G$ on $X$ is _compressible_ if there exists a
base of the topology $\mathfrak{U}$ on $X$ such that
* (i)
for every $g\in G$ there exists $U\in\mathfrak{U}\,$ such that
$\mathrm{supp}(g)\subset U$;
* (ii)
for every $U_{1},U_{2}\in\mathfrak{U}\,$ there exists $g\in G$ such that
$g(U_{1})\subset U_{2}$;
* (iii)
for every $U_{1},U_{2},U_{3}\in\mathfrak{U}$ with
$\overline{U}_{1}\cap\overline{U}_{2}=\varnothing$ there exists $g\in G$ such
that $g(U_{1})\cap U_{3}=\varnothing$ and $\mathrm{supp}(g)\cap
U_{2}=\varnothing$.
* (iv)
for any $U_{1},U_{2}\in\mathfrak{U}$ there exists $U_{3}\in\mathfrak{U}$ such
that $U_{3}\supset U_{1}\cup U_{2}$.
###### Remark 2.6.
Suppose that $X$ is a Polish space. If an action of $G$ on $X$ is
compressible, then the $G$-action has no probability invariant measure. The
latter is equivalent to the $G$-orbit equivalence relation being compressible
(see [8] and references therein). This observation motivates our terminology.
The following result relates dynamical properties of group actions to the
functional properties of group characters.
###### Proposition 2.7.
Let $G$ be a countable group admitting a compressible action by homeomorphisms
on some regular Hausdorff topological space $X$. Then for every non-regular
indecomposable character $\chi$ of $G$ there exists $g\neq e$ such that
$|\chi(g)|=1$.
###### Proof.
Consider a proper indecomposable character $\chi$ of $G$. Assume that
$|\chi(g)|<1$ for all $g\neq e$. Let $(\pi,H,\xi)$ be the GNS-construction
associated to $\chi$.
(1) We notice that the definition of the compressible action implies that
$\chi$ has the multiplicativity property in the sense that if
$U_{1},U_{2}\in\mathfrak{U}\,$ and $g,h\in G$ are such that
$\mathrm{supp}(g)\subset U_{1},\mathrm{supp}(h)\subset
U_{2}\;\text{and}\;\overline{U}_{1}\cap\overline{U}_{2}=\varnothing$
then
$\chi(gh)=\chi(g)\chi(h).$ (1)
Indeed, find an increasing sequence of finite sets $F_{n}\subset G$ with
$\bigcup_{n}F_{n}=G$. Then by the conditions (i) and (iv) of Definition 2.5,
we can find open sets $V_{n}\in\mathfrak{U}$ such that
$V_{n}\supset\bigcup\limits_{f\in F_{n}}\mathrm{supp}(f).$
By the condition (iii) there exist elements $r_{n}\in G$ such that
$r_{n}(U_{1})\cap V_{n}=\varnothing\;\text{and}\;\mathrm{supp}(r_{n})\cap
U_{2}=\varnothing.$
Then $r_{n}hr_{n}^{-1}=h$ and $supp(r_{n}gr_{n}^{-1})\cap supp(f)=\varnothing$
for every $f\in F_{n}$. Passing to a subsequence if needed, we can assume that
$\pi(r_{n}gr_{n}^{-1})$ converges weakly to an operator
$Q\in\mathcal{M}_{\pi}$. Notice that $tr(Q)=\chi(g)$. Since the operator $Q$
commutes with $\pi(F_{n})$ for every $n$, we get that $Q$ belongs to the
center of $\mathcal{M}_{\pi}$. Therefor, $Q$ is scalar and
$Q=\chi(g)\mathrm{Id}$. Thus
$\chi(gh)=\lim_{n\to\infty}(\pi(r_{n}ghr_{n}^{-1})\xi,\xi)=(Q\pi(h)\xi,\xi)=\chi(g)\chi(h).$
(2) We claim that for any $\varepsilon>0$ and any open set $U$ there exists
$g\in G$ with $supp(g)\subset U$ and $|\chi(g)|<\varepsilon$. Indeed, fix an
element $h\neq e$ and $n\in\mathbb{N}$. Find $n$ subsets
$V_{1},\ldots,V_{n}\in\mathfrak{U}\,$ such that
$\overline{V}_{j}\cap\overline{V}_{k}=\varnothing$ for $j\neq k$. By
assumptions (i) and (ii) we can choose elements $g_{1},\ldots,g_{n}\in G$ such
that $g_{j}(\mathrm{supp}(h))\subset V_{j}$ for each $j$. Set
$f=(g_{1}hg_{1}^{-1})(g_{2}hg_{2}^{-1})\cdots(g_{n}hg_{n}^{-1}).$
By multiplicativity, we obtain that $\chi(f)=\chi(h)^{n}$. Choosing $n$
sufficiently large we get an element $f$ with $|\chi(f)|<\varepsilon$. By
assumptions (i) and (ii) we can find an element $g$ conjugate to $f$ with
$\mathrm{supp}(g)\subset U$, which proves the claim.
(3) Consider an element $g\in G,\;g\neq e$. Find an open set $U$ with
$\overline{g(U)}\cap\overline{U}=\varnothing$. Fix $\varepsilon>0$ and
$n\in\mathbb{N}$. Using the condition (ii) and (iv) of Definition 2.5, we can
find subsets $U_{1},\ldots,U_{n},V_{1},\ldots,V_{n}\in\mathfrak{U}$ with
pairwise disjoint closures such that $g(V_{i})\subset U_{i}\subset U$ for each
$i$. Find $h_{j}\in G,j=1,\ldots,n$ supported by $U_{j}$ with
$|\chi(h_{j})|<\varepsilon$. Set $\xi_{j}=\pi(h_{j}gh_{j}^{-1})\xi$. Then for
$i\neq j$, the multiplicativity of $\chi$ implies that
$\displaystyle(\xi_{i},\xi_{j})$ $\displaystyle=$
$\displaystyle\chi(h_{j}g^{-1}h_{j}^{-1}h_{i}gh_{i}^{-1})$ $\displaystyle=$
$\displaystyle\chi(h_{j}(g^{-1}h_{j}^{-1}g)(g^{-1}h_{i}g)h_{i}^{-1})$
$\displaystyle=$
$\displaystyle\chi(h_{j})\chi(g^{-1}h_{j}^{-1}g)\chi(g^{-1}h_{i}g)\chi(h_{i}^{-1}).$
As $|\chi(h_{j})|<\varepsilon$, we obtain that
$|(\xi_{j},\xi_{i})|<\varepsilon$. Thus,
$\|\xi_{1}+\ldots+\xi_{n}\|\leq\big{(}n+n(n-1)\varepsilon\big{)}^{\frac{1}{2}}.$
Since $(\xi_{l},\xi)=\chi(g)$ for each $l$, we have
$|\chi(g)|=\tfrac{1}{n}|(\xi_{1}+\xi_{2}+\ldots+\xi_{n},\xi)|\leq\tfrac{1}{n}\big{(}n+n(n-1)\varepsilon\big{)}^{\frac{1}{2}}.$
When $n$ goes to infinity, we obtain
$|\chi(g)|\leq\varepsilon^{\frac{1}{2}}.$
Since $\varepsilon>0$ is arbitrary, we conclude that $\chi(g)=0$. Thus, $\chi$
is the regular character. ∎
###### Lemma 2.8.
Let $G$ be a simple group and $\chi$ be a character. If $|\chi(g)|=1$ for some
$g\in G,g\neq e$, then
$\chi(s)=1\;\;\text{for all}\;\;s\in G.$
In particular, if $\chi$ is not the identity character, then $|\chi(s)|<1$ for
all $s\neq e$.
###### Proof.
Let $c=\chi(g)$, $|c|=1$. Consider the GNS construction $(\pi,H,\xi)$
corresponding to $\chi$. Using the Cauchy-Schwarz inequality and the fact that
the vector $\xi$ is separating, we obtain that
$(\pi(g)\xi,\xi)=c\;\;\Rightarrow\;\;\pi(g)\xi=c\xi\;\;\Rightarrow\pi(g)=c\mathrm{Id}.$
Take an arbitrary element $h\in G$ which does not commute with $g$ and set
$s=hgh^{-1}g^{-1}$. Then $\pi(s)=\mathrm{Id}$. It follows that
$\pi(s_{1})=\mathrm{Id}$ for all $s_{1}$ from the normal subgroup generated by
$s$. Since $G$ is simple, we get that $\pi(g)=\mathrm{Id}$ for every $g\in G$.
Thus, $\chi$ is the identity character. ∎
As a corollary of Lemma 2.8 and Proposition 2.7 we immediately obtain the
following result.
###### Theorem 2.9.
Let $G$ be a simple countable group admitting a compressible action on a
regular Hausdorff topological space $X$. Then $G$ has no proper characters.
Let $G$ be a group. For a subgroup $R$ of $G$ and an element $g\in G$ set
$C_{R}(g)=\\{hgh^{-1}:h\in R\\}$. Denote by $N(R)$ the normal closure of $R$
in $G$, i.e., the subgroup of $G$ generated by all elements of the form
$grg^{-1},g\in G,r\in R$.
###### Theorem 2.10.
Let $G$ be a group and $R$ be an ICC subgroup of $G$ such that
* (i)
$R$ has no proper characters;
* (ii)
for every $g\in G\setminus\\{e\\}$, there exists a sequence of distinct
elements $\\{g_{i}\\}_{i\geq 1}\subset C_{R}(g)$ such that $g_{i}^{-1}g_{j}\in
R$ for any $i,j$.
Then each finite type factor representation $\pi$ of $G$ is either regular or
has the form
$\pi(g)=\omega([g]),$
where $\omega$ is a finite factor representation of $G/N(R)$ and $[g]\in
G/N(R)$ is the coset of the element $g$.
###### Proof.
Consider an indecomposable character $\chi$ of $G$. Let $(\pi,H,\xi)$ be the
GNS-construction associated to $\chi$.
(1) Consider the restriction of $\pi$ onto the subgroup $R$. Set
$H_{{}_{R}}=\overline{Lin(\pi(R)\xi)}$. Since the restriction of $\chi$ on $R$
is a character and the only indecomposable characters of the group $R$ are the
regular and the identity characters, we can decompose the space $H_{{}_{R}}$
into $R$-invariant subspaces $H_{1}$ and $H_{2}$ (possibly trivial) such that
$H_{{}_{R}}=H_{1}\bigoplus H_{2}$ with $\pi(R)|H_{1}$ being the identity
representation and $\pi(R)|H_{2}$ being the regular representation.
The orthogonal projections $\\{P_{i}\\}$ onto $H_{i}$, $i=1,2$ belong to the
center of the algebra generated by $\pi(R)$. In particular, $P_{i}$ lies in
the algebra $\mathcal{M}_{\pi}$. Furthermore,
$\chi(g)=\alpha\chi_{id}(g)+(1-\alpha)\chi_{reg}(g)\;\;\text{for all}\;\;g\in
R,$
where $\chi_{id}$ is the identity character, $\chi_{reg}$ is the regular
character, and $\alpha\in[0,1]$. If $\alpha\neq 0,1$, we can write down the
vector $\xi$ as
$\xi=\alpha^{\frac{1}{2}}\xi_{1}+(1-\alpha)^{\frac{1}{2}}\xi_{2},$ (2)
where $\xi_{1}\in H_{1}$, $\xi_{2}\in H_{2}$ are unit vectors such that
$(\pi(h)\xi_{1},\xi_{1})=\chi_{id}(h)=1,\;\;(\pi(h)\xi_{2},\xi_{2})=\chi_{reg}(h)=\delta_{h,e}\;\;\text{for
all}\;\;h\in R.$
For convenience, if $\alpha=0$, we set $\xi_{1}=0,\xi_{2}=\xi$, if $\alpha=1$,
we set $\xi_{1}=\xi,\xi_{2}=0$. Observe that
$H_{i}=\overline{Lin(\pi(R)\xi_{i})}$, $i=1,2$.
(2) Assume that $H_{2}\neq\\{0\\}$. Consider an arbitrary element $g\in G$,
$g\neq e$. By our assumptions there exists a sequence of elements
$\\{h_{n}\\}\in R\setminus\\{e\\}$ such that
$h_{m}^{-1}g^{-1}h_{m}h_{n}^{-1}gh_{n}\in R$ for all $n$ and $m$ and elements
$h_{n}^{-1}gh_{n}$ are pairwise distinct. Set $g_{m}=h_{m}^{-1}gh_{m}$. Since
$g_{m}^{-1}g_{n}\in R\setminus\\{e\\}$, we get that
$(\pi(g_{n})\xi_{2},\pi(g_{m})\xi_{2})=\chi_{reg}(g_{m}^{-1}g_{n})=0.$
This shows that $\pi(g_{m})\xi_{2}\to 0$ weakly. Observe also that
$\displaystyle(\pi(g_{n})\xi_{2},\xi_{2})$ $\displaystyle=$
$\displaystyle(\pi(g_{n})P_{2}\xi,P_{2}\xi)=tr(P_{2}\pi(h_{n}^{-1}gh_{n})P_{2})$
$\displaystyle=$ $\displaystyle
tr(\pi(h_{n}^{-1})P_{2}\pi(g)P_{2}\pi(h_{n}))=tr(P_{2}\pi(g)P_{2}).$
Since the latter is independent of $n$ and $\pi(g_{n})\xi_{2}\to 0$, we
conclude that
$(\pi(g)\xi_{2},\xi_{2})=tr(P_{2}\pi(g)P_{2})=0.$
Set $H_{0}=\overline{Lin(\pi(G)\xi_{2})}$. Then $\pi(G)|H_{0}$ is quasi-
equivalent to the regular representation. Since $\pi$ is a factor
representation, we conclude that $\pi$ is the regular representation.
(3) Assume that $H_{2}=\\{0\\}$. Then $\xi=\xi_{1}$ and $\pi(h)=\mathrm{Id}$
for every $h\in R$. Therefor, $\pi(g)=\mathrm{Id}$ for all $g\in N(R)$. This
means that the representation $\pi$ factors through the quotient $G/N(R)$ and
defines a finite type factor representation $\omega$ of $G/N(R)$ such that
$\pi(g)=\omega([g])$ for all $g\in G$. ∎
Recall that a finite factor representation of a group $G$ is of type $I$ if
the von Neumann algebra of the representation is isomorphic to the algebra of
all linear operators in some finite-dimensional Hilbert space. We say that an
action of group $G$ on a measure space $(Y,\mu)$ is trivial if $g(x)=x$ for
every $g\in G$ and $\mu$-almost every $x\in X$. The following result shows
that any ergodic action of a group admitting no characters must be essentially
free, that is $\mu(Fix(g))=0$ for all $g\in G\setminus\\{e\\}$.
###### Theorem 2.11.
Assume that every finite factor representation of a countable ICC group $G$ is
either regular or of type $I$ and that there is at most a countable number (up
to quasi-equivalence) of finite factor representations of $G$. Then every
faithful ergodic measure-preserving action of $G$ is essentially free.
###### Proof.
Consider an ergodic action of $G$ on a measure space $(Y,\mu)$. Set
$\widetilde{Y}=\\{(x,y)\in Y\times Y|x=g(y)\;\text{for some}\;g\in G\\}.$
For a Borel set $A\subset\widetilde{Y}$ and a point $x\in Y$, set
$A_{x}=\\{(x,y)\in A\\}$. Define a $\sigma$-finite measure $\widetilde{\mu}$
on $\widetilde{Y}$ by $\widetilde{\mu}(A)=\int_{Y}card(A_{x})d\mu(x)$. Given a
function $f\in L^{2}(\widetilde{Y},\widetilde{\mu})$ and a group element $g\in
G$, set
$(\pi(g)f)(x,y)=f(g^{-1}x,y).$
Then $\pi(g)$ is a unitary operator on the Hilbert space
$L^{2}(\widetilde{Y},\widetilde{\mu})$. Denote by $\xi$ the indicator function
of the diagonal of $Y\times Y$. Set $H=\overline{Lin\\{\pi(G)\xi\\}}$. We note
the von Neumann algebra $\mathcal{M}_{\pi}$ generated by $\pi(G)$, restricted
to $H$, is of finite type. We refer the reader to [4] for the details. Since
the group $G$ has at most a countable number of finite factor representations,
the representation $\pi$ decomposes into a direct sum (at most countable) of
factor representations.
Our goal is to show that the representation $\pi$ is regular. Then the
uniqueness of the trace implies that $(\pi(g)\xi,\xi)=0$ for every $g\neq e$.
Using the identity $\mu(Fix(g))=(\pi(g)\xi,\xi)$, we get that the action is
essentially free.
Suppose to the contrary that the decomposition of $\pi$ into factors contains
a non-regular factor representation $\pi_{1}$, which, by our assumptions,
generates a finite-dimensional von Neumann factor. Let $P_{1}$ be a projection
from the center of $\mathcal{M}_{\pi}$ such that $\pi_{1}(g)=P_{1}\pi(g)$ for
every $g\in G$. Set $\xi_{1}=P_{1}\xi$.
Since for every $g\in G$ the unitary operator
$(\pi^{\prime}(g)f)(x,y)=f(x,g^{-1}y)$ belongs to $\mathcal{M}_{\pi}^{\prime}$
and $\pi^{\prime}(g)\xi=\pi(g^{-1})\xi$, we have that
$\pi^{\prime}(g)\pi(g)\xi_{1}=\pi^{\prime}(g)\pi(g)P_{1}\xi=P_{1}\pi^{\prime}(g)\pi(g)\xi=P_{1}\xi=\xi_{1}$
for all $g\in G$. This implies that the function $h(x):=|\xi_{1}(x,x)|$ is
$G$-invariant and $\mu$-integrable on $Y$. By the ergodicity, we get that
$h(x)\equiv C$ on $Y$ for some constant $C$. Note that if $C=0$, then
$0=\int_{\widetilde{Y}}\xi_{1}(x,y)\xi(x,y)d\widetilde{\mu}(x,y)=(\xi_{1},\xi),$
which is impossible as the projection of $\xi$ onto $\xi_{1}$ is non-trivial.
Fix an orthonormal basis $\eta_{1},\ldots,\eta_{n}$ for
$\overline{Lin\\{\pi_{1}(G)\xi_{1}\\}}$. For a given $g\in G$, write
$\pi_{1}(g)\xi_{1}=\sum\limits_{j=1}^{n}\alpha_{j}(g)\eta_{j}$
for some $\alpha_{1}(g),\ldots,\alpha_{n}(g)$ with
$\sum|\alpha_{j}(g)|^{2}=|\xi_{1}|^{2}\leq 1$. Observe that
$\sum\limits_{j=1}^{n}\alpha_{j}(g)\eta_{j}(x,y)=(\pi_{1}(g)\xi_{1})(x,y)=(\pi(g)\xi_{1})(x,y)=\xi_{1}(g^{-1}x,y)$
for every $(x,y)\in\widetilde{Y}$. Since $|\xi_{1}(g^{-1}x,y)|=C$ for
$(x,y)\in\widetilde{Y}$ with $x=g(y)$, we conclude that
$\sum\limits_{j=1}^{n}|\eta_{j}(x,y)|\geq C>0$ for $(x,y)$ with $x=gy$ and,
thus, for any $(x,y)\in\widetilde{Y}$. This implies that the function
$\sum\limits_{j=1}^{n}|\eta_{j}(x,y)|$ is not integrable with respect to
$\widetilde{\mu}$. This contradiction yields that $\pi_{1}=0$ and, thus, the
representation $\pi$ is regular. ∎
We finish this section by giving examples of groups admitting no compressible
actions. We observe that even though the following proposition yields a result
similar to that of Theorem 2.10, the underlying assumptions are different and
not mutually interchangeable.
###### Proposition 2.12.
Let $G$ be a countable group with trivial center and such that every proper
quotient is finite or abelian. Assume that the group $G$ admits a compressible
action on a regular Hausdorff space. Then all finite (Murray von Neumann) non-
regular representations of $G$ are of type $I$.
###### Proof.
Consider a non-regular indecomposable character $\chi$ of $G$. Let
$(\pi,H,\xi)$ be the GNS-construction associated to $\chi$. By Proposition 2.7
there exists $g\neq e$ such that $|\chi(g)|=1$. Choose $h\in G$ not commuting
with $g$. Denote by $N$ the normal subgroup of $G$ generated by the element
$ghg^{-1}h^{-1}$. Using the arguments from the proof of Lemma 2.8 we obtain
that $\pi|N=\mathrm{Id}$. Thus, the representation $\pi$ of the group $G$
gives rise to the representation of $G/N$ with the same von Neumann algebra.
Recall that factor representations of abelian groups are scalar. ∎
If a group $G$ as in the proposition above has a measure-preserving action on
a measures space $(X,\mu)$ with $0<\mu(Fix(g))<1$ for some $g\neq e$, then, in
view of Theorem 2.11, such a group cannot have compressible actions. Examples
of such groups are full groups of even Bratteli diagrams, commutators of
topological full groups of Cantor minimal systems [5], and just infinite
branch groups [6].
## 3 Applications
In this section we show that the results established in the previous section
are applicable to the Hignam-Thompson groups and to the full groups of
irreducible shifts of finite type.
### 3.1 The Higman-Thompson groups
###### Definition 3.1.
Fix two positive integers $n$ and $r$. Consider an interval $I_{r}=[0,r]$.
Define the group $F_{n,r}$ as the set of all orientation preserving piecewise
linear homeomorphisms $h$ of $I_{r}$ such that all singularities of $h$ are in
$\mathbb{Z}[1/n]=\\{\tfrac{p}{n^{k}}:p,k\in\mathbb{N}\\}$; the derivative of
$h$ at any non-singular point is $n^{k}$ for some $k\in\mathbb{Z}$.
Observe that the commutator subgroup of $F_{n,r}$ is a simple group and the
abelianization of $F_{n,r}$ is isomorphic to $\mathbb{Z}^{n}$ [1, Section 4].
Consider the subgroup $F_{n,r}^{0}$ of $F_{n,r}$ consisting of all elements
$f\in F_{n,r}$ with $supp(f)$ being a subset of $(0,r)$. Observe that (the
commutator subgroup) $F_{n,r}^{\prime}=(F_{n,r}^{0})^{\prime}$ [1, Section 4].
The following lemma shows that the commutator subgroup $F_{n,r}^{\prime}$
satisfies the assumptions of Theorem 2.9.
###### Lemma 3.2.
The base of topology
$\mathfrak{U}\,=\\{(a,b):[a,b]\subset(0,r),a,b\in\mathbb{Z}[\tfrac{1}{n}]\\}$
satisfies the conditions (i)-(iv) of Definition 2.4 for the action of the
group $R=(F_{n,r}^{0})^{\prime}$ on $(0,r)$. Thus, the action of $R$ is
compressible.
###### Proof.
The conditions (i) and (iv) of Definition 2.4 are clearly satisfied.
To check the condition (ii), consider intervals $U_{1}=(a,b)$ and
$U_{2}=(c,d)$ both in $\mathfrak{U}$. Replacing $U_{2}$ by a subinterval if
necessary we may assume that $\tfrac{b-a}{d-c}=n^{k}$ for some
$k\in\mathbb{Z}$. Since the function $\tfrac{a-x}{c-x}$ is continuous in $x$
for $x\neq c$, we can find $x\in\mathbb{Z}[\tfrac{1}{n}]$ such that
$0<x<\min\\{a,c\\}$ and $\tfrac{a-x}{c-x}=n^{k}$ for some $k\in\mathbb{Z}$.
Similarly, we can find $y\in\mathbb{Z}[\tfrac{1}{n}]$, $\max\\{b,d\\}<y<r$
such that $\tfrac{y-b}{y-d}=n^{k}$ for some $k\in\mathbb{Z}$. Let
$g:[0,r]\to[0,r]$ be the function such that
$g(0)=0,\;g(x)=x,\;g(a)=c,\;g(b)=d,\;g(y)=y,\;g(r)=r,$
and $g$ is linear on each of the line segments
$[0,x],[x,a],[a,b],[b,y],[y,r]$. Then $g\in R$ and $g(U_{1})=U_{2}$.
To check the condition (iii), we consider intervals $U_{i}=(a_{i},b_{i})$,
$i=1,2,3$ from $\mathfrak{U}$ such that
$\overline{U}_{1}\cap\overline{U}_{2}=\emptyset$. Without loss of generality,
assume that $a_{1}>b_{2}$. Set $a=a_{1},b=b_{1}$ and fix
$c,d\in\mathbb{Z}[\tfrac{1}{n}]$ with $\max\\{b_{2},b_{3}\\}<c<d<r$ and
$\tfrac{b-a}{d-c}=n^{k}$ for some $k\in\mathbb{Z}$. Construct $g$ as above
with $a_{1}>x>b_{2}$ and $r>y>d$. Then $\mathrm{supp}(g)\subset[x,y]$ and
$g(U_{1})=(c,d)$. Therefore, $g(U_{1})\cap U_{3}=\varnothing$ and
$\mathrm{supp}(g)\cap U_{2}=\varnothing$. ∎
Observe that all finite factor representations of abelian groups are scalar
representations, i.e. $\pi(g)=c_{g}\mathrm{Id}$, with $c_{g}\in\mathbb{T}$,
the unit circle. In particular, the indecomposable characters of abelian
groups are homomorphisms into $\mathbb{T}$.
###### Corollary 3.3.
(1) The group $F_{n,r}^{\prime}$ has no proper characters. (2) If $\chi$ is an
indecomposable character of $F_{n,r}$, then $\chi$ is either regular or
$\chi(g)=\rho([g])$, where $[g]$ is the image of $g$ in the abelianization of
$F_{n,r}$ and $\rho:\mathbb{Z}^{n}\rightarrow\mathbb{T}$ is a group
homomorphism.
###### Proof.
Statement (1) immediately follows from Lemma 3.2 and Theorem 2.9.
To establish the second result, we only need to check the condition (2) of
Theorem 2.10. Fix $g\in F_{n,r}\setminus\\{e\\}$. Find an interval $(a,b)$
with $g(a,b)\cap(a,b)=\emptyset$. Find a sequence of distinct elements
$\\{h_{n}\\}_{n\geq 1}\subset(F_{n,r})^{\prime}$ supported by $(a,b)$. Then
$(h_{n}^{-1}g^{-1}h_{n})(h_{m}^{-1}gh_{m})\in(F_{n,r})^{\prime}$ for any
$n\neq m$. This completes the proof. ∎
###### Definition 3.4.
Let $n$ and $r$ be positive integers. Define Higman’s group $G_{n,r}$ as the
group of all right continuous bijections of $[0,r)$ which are piecewise
linear, with finitely many discontinuities and singularities, all in
$\mathbb{Z}[1/n]$, slopes in $\\{n^{k}:k\in\mathbb{Z}\\}$, and mapping
$\mathbb{Z}[1/n]\cap[0,r)$ to itself.
Note that $F_{n,r}\subset G_{n,r}$. In fact, $F_{n,r}$ consists exactly of all
continuous elements $g\in G_{n,r}$. In [7] Higman showed that the commutator
subgroup $G_{n,r}^{\prime}{}$ is simple and that the abelianization of
$G_{n,r}$ is trivial for even $n$ and is $\mathbb{Z}/2\mathbb{Z}$ for odd $n$.
###### Lemma 3.5.
The groups $R=F_{n,r}^{\prime}$ and $G=G_{n,r}$ satisfy the conditions of
Theorem 2.10.
###### Proof.
Corollary 3.3 shows that the group $R$ has no proper characters. Consider an
arbitrary element $g\in G,g\neq e$. Choose an open interval $I$ such that
$I\cap g^{-1}(I)=0$ and $g$ is continuous on both $I$ and $g^{-1}(I)$. It
follows that for any two elements $r_{1},r_{2}\in R$ with
$\mathrm{supp}(r_{1})\subset I$, $\mathrm{supp}(r_{2})\subset I$ the element
$h=r_{2}g^{-1}r_{2}^{-1}r_{1}gr_{1}^{-1}\neq e$
is a continuous bijection of $[0,r)$. Observe that $h$ acts identically near
$0$ and $r$. It follows that $h\in R$ and the elements $r_{1}gr_{1}^{-1}$ and
$r_{2}gr_{2}^{-1}$ belong to the same coset of $G/R$. Since the group $R$ has
infinitely many elements supported by the set $I$, we immediately establish
the condition (ii). ∎
The following result is an immediate corollary of Theorem 2.10 applied twice
to the pairs $R=F_{n,r}^{\prime}$, $G=(G_{n,r})^{\prime}$ and
$R=F_{n,r}^{\prime}$, $G=G_{n,r}$.
###### Corollary 3.6.
(1) The group $G_{n,r}^{\prime}$ has no proper characters.
(2) If $\chi$ is an indecomposable character of $G_{n,r}$, then $\chi$ is
either regular or $\chi(g)=\rho([g])$, where $[g]$ is the image of $g$ in the
abeliazation of $G_{n,r}$ and
$\rho:G_{n,r}/G_{n,r}^{\prime}\rightarrow\mathbb{T}$ is a group homomorphism.
### 3.2 Full groups of irreducible shifts of finite type
We refer the reader to [9, Section 6] for the comprehensive study of full
groups of étale groupoids including the groups discussed below.
Let $(V,E)$ be a finite directed graph. Suppose that the adjacency matrix of
the graph is irreducible and is not a permutation matrix. For an edge $e\in
E$, denote by $i(e)$ the initial vertex and by $t(e)$ its terminal vertex. Set
$X=\\{\\{e_{n}\\}_{n\geq 1}\in E^{\mathbb{N}}:t(e_{k})=i(e_{k+1})\mbox{ for
every }k\in\mathbb{X}\\}.$
Equipped with the product topology, $X$ is a Cantor set. We note that the
space $X$ along with the left shift is called a one-sided subshift of finite
type, see [9] and references therein regarding relations with the symbolic
dynamics.
An $n$-tuple $(e_{1},\ldots,e_{n})\in E^{n}$ is called admissible if
$t(e_{k})=i(e_{k+1})$ for every $1\leq k\leq n-1$. Two admissible tuples
$\overline{e}=(e_{1},\ldots,e_{n})$ and $\overline{f}=(f_{1},\ldots,f_{m})$
are called compatible if $t(e_{n})=t(f_{m})$. Each admissible tuple
$\overline{e}=(e_{1},\ldots,e_{n})$ defines a clopen set
$U(\overline{e})=\\{x\in X:x_{i}=e_{i},\;i=1,\ldots,n\\}$. Such clopen sets
form the base of topology. Given two compatible admissible tuples
$\overline{e}_{1}$ and $\overline{e}_{2}$, define a continuous map
$\pi_{\overline{e}_{1},\overline{e}_{2}}:U(\overline{e}_{1})\rightarrow
U(\overline{e}_{2})$ as
$\pi_{\overline{e}_{1},\overline{e}_{2}}(\overline{e}_{1},x_{n+1},x_{n+2},\ldots)=(\overline{e}_{2},x_{n+1},x_{n+2},\ldots).$
###### Definition 3.7.
Following [9], we define the full group of $X$, in symbols $[[X]]$, as the set
of all homeomorphisms $g$ of $X$ for which there exists two clopen partitions
$X=\bigsqcup_{i=1}^{n}U(\overline{e}_{i})=\bigsqcup_{i=1}^{n}U(\overline{f}_{i})$
with $e_{i}$ and $f_{i}$ being compatible admissible tuples (possibly of
different lengths), $i=1,\ldots,n$, such that
$g|U_{\overline{e}_{i}}=\pi_{\overline{e}_{i},\overline{f}_{i}}$ for every
$i=1,\ldots,n$.
For a clopen subset $Y\subset X$, set $[[X|Y]]$ as the set of all $g\in[[X]]$
with $\mathrm{supp}(g)\subset Y$.
The following result was established in [9, Lemma 6.1 and Theorem 4.16]
###### Proposition 3.8.
For any clopen set $Y\subset X$, the commutator group $[[X|Y]]^{\prime}$ is
simple.
Fix an arbitrary point $x_{0}\in X$. Find an increasing sequence of clopen
sets $\\{Y_{n}\\}$ such that $X\setminus\\{x_{0}\\}=\bigcup_{n}Y_{n}$. Set
$R=\bigcup_{n}[[X|Y_{n}]]^{\prime}$. It follows from Proposition 3.8 that the
group $R$ is simple. Observe that the group $R$ consists of all elements
$g\in[[X]]^{\prime}$ equal to the identity on some neighbourhood of $x_{0}$.
Denote by $\mathcal{F}$ the set of all admissible tuples which are not
prefixes of $x_{0}$. Define $\mathfrak{U}$ as the family of all finite unions
of sets from $\\{U(\overline{e})\\}_{\overline{e}\in\mathcal{F}}$. Notice that
$\mathfrak{U}$ is a base of the topology on $X\setminus\\{x_{0}\\}$. One can
check that $\mathfrak{U}$ satisfies conditions (i)-(iv) of Definition 2.5 for
the action of $R$. Thus, using Theorem 2.9, we conclude that the group $R$ has
no characters. Considering $R$ as a subgroup of $G=[[X]]$, one can check that
the assumptions of Theorem 2.10 are satisfied. We leave the details to the
reader.
###### Corollary 3.9.
If $\chi$ is an indecomposable character of $[[X]]$, then $\chi$ is either
regular or $\chi(g)=\rho([g])$, where $[g]$ is the image of $g$ in the
abelianization of $[[X]]$ and $\rho:[[X]]/[[X]]^{\prime}\rightarrow\mathbb{T}$
is a group homomorphism.
To finish our discussion, we notice that the full group of the one-sided
Bernoulli shift over the alphabet with $n$ letters is isomorphic to $G_{n,1}$
[10].
Acknowledgements. We would like to thank R. Grigorchuk for the discussion of
Higman-Thompson groups and for his valuable comments.
## References
* [1] K. Brown, _Finiteness properties of groups_. J. Pure Appl. Algebra 44 (1987), no. 1-3, 45-75.
* [2] J.W. Cannon, J.W. Floyd, W.R. Parry, _Introductory notes on Richard Thompson’s groups._ Enseign. Math. (2) 42 (1996), no. 3–4, 215–256.
* [3] A. Connes, _Classification of injective factors. Cases $II_{1}$, $II_{\infty}$, $III_{\lambda}$, $\lambda\neq 1$_ Ann. of Math. (2) 104 (1976), no. 1, 73–115.
* [4] J. Feldman and C. Moore, _Ergodic Equivalence Relations, Cohomology, and Von Neumann Algebras. I._ Trans. of the AMS, 234, No. 2\. (1977), 289–324.
* [5] A. Dudko and K. Medynets, _On Characters of Inductive Limits of Symmetric Groups_ , arXiv:1105.6325, (2012).
* [6] R. Grigorchuk, _Some problems of the dynamics of group actions on rooted trees._ (Russian) Tr. Mat. Inst. Steklova 273 (2011), Sovremennye Problemy Matematiki, 72–191; translation in Proc. Steklov Inst. Math. 273 (2011), no. 1, 64–175
* [7] G. Higman, _Finitely presented infinite simple groups_. Notes on Pure Mathematics, No. 8 (1974). Department of Pure Mathematics, Department of Mathematics, I.A.S. Australian National University, Canberra, 1974. vii+82 pp.
* [8] R. Dougherty, S. Jackson and A.S. Kechris, _The structure of hyperfinite Borel equivalence relations_ , Trans. Amer. Math. Soc. 341 (1994), 193-225.
* [9] H. Matui, _Topological full groups of one-sided shifts of finite type_ , (2012) arXiv:1210.5800.
* [10] V. Nekrashevych, _Cuntz-Pimsner algebras of group actions_. J. Operator Theory 52 (2004), no. 2, 223-249.
* [11] M. Stein, _Groups of piecewise linear homeomorphisms_. Trans. Amer. Math. Soc. 332 (1992), no. 2, 477-514.
* [12] M. Takesaki, Theory of operator algebras I. Encyclopedia of Mathematical Sciences, vol. 124.
* [13] A. Vershik, _Nonfree Actions of Countable Groups and their Characters_ , arXiv:1012.4604 (2010).
|
arxiv-papers
| 2012-12-06T03:33:58 |
2024-09-04T02:49:38.932881
|
{
"license": "Public Domain",
"authors": "Artem Dudko and Konstantin Medynets",
"submitter": "Konstantin Medynets",
"url": "https://arxiv.org/abs/1212.1230"
}
|
1212.1255
|
# Uniqueness for Keller-Segel-type chemotaxis models
J. A. Carrillo, S. Lisini, E. Mainini José A. Carrillo, Department of
Mathematics, Imperial College London, South Kensington Campus, London SW7 2AZ,
UK. [email protected] Stefano Lisini, Dipartimento di Matematica “F.
Casorati”, Università degli Studi di Pavia, via Ferrata 1, 27100 Pavia, Italy.
[email protected] Edoardo Mainini, Università degli Studi di Genova,
Dipartimento di Ingegneria meccanica, energetica, gestionale e dei trasporti
(DIME) - sezione MAT. P.le Kennedy 1, 16129 Genova, Italy
[email protected]
###### Abstract.
We prove uniqueness in the class of integrable and bounded nonnegative
solutions in the energy sense to the Keller-Segel (KS) chemotaxis system. Our
proof works for the fully parabolic KS model, it includes the classical
parabolic-elliptic KS equation as a particular case, and it can be generalized
to nonlinear diffusions in the particle density equation as long as the
diffusion satisfies the classical McCann displacement convexity condition. The
strategy uses Quasi-Lipschitz estimates for the chemoattractant equation and
the above-the-tangent characterizations of displacement convexity. As a
consequence, the displacement convexity of the free energy functional
associated to the KS system is obtained from its evolution for bounded
integrable initial data.
††AMS 2010 subject classification: 35A15
## 1\. Introduction
The classical Keller-Segel (KS) model for chemotaxis is the system
$\left\\{\begin{array}[]{rl}\partial_{t}n&=\kappa\Delta
n-\chi\,\mathrm{div}\,(n\nabla c),\\\ \partial_{t}c&=\eta\Delta c+\theta
n-\gamma c.\end{array}\right.$
Here, $n$ is the number/mass density of a bacteria/cell population and $c$
represents the concentration of a chemical attractant that can suffer chemical
degradation and that it is produced by the cells themselves due to chemotactic
interaction. The parameters $\kappa,\chi,\eta,\theta,\gamma$ might be suitable
functions, assumed to be constant in this simplified model. We can perform a
time scaling and a suitable change of variables, that is $\tau=\kappa t$,
$\rho(x,\tau)=\frac{\theta\chi}{\eta\kappa}n(x,\tau/\kappa),v(x,\tau)=\frac{\chi}{\kappa}c(x,\tau/\kappa)$.
The system is therefore reduced to
$\left\\{\begin{array}[]{rl}\partial_{t}\rho&=\Delta\rho-\,\mathrm{div}\,(\rho\nabla
v),\\\\[5.69054pt] \varepsilon\partial_{t}v&=\Delta v+\rho-\alpha
v,\end{array}\right.$ (1.1)
where $\alpha\geq 0$ and $\varepsilon\geq 0$ are constants
($\alpha=\gamma/\eta,\>\varepsilon=\kappa/\eta$). In case $\varepsilon=0$, it
restricts to the classical parabolic-elliptic Patlak-KS model
$\left\\{\begin{array}[]{l}\partial_{t}\rho=\Delta\rho-\,\mathrm{div}\,(\rho\nabla
v),\\\\[5.69054pt] -\Delta v+\alpha v=\rho.\end{array}\right.$ (1.2)
For $\varepsilon>0$, the natural free energy functional associated to the
dynamics of the system (1.1) is
$\mathscr{F}_{\varepsilon,\alpha}(\rho,v):=\int_{\mathbb{R}^{d}}(\rho\log\rho-v\rho)\,dx+\frac{1}{2}\int_{\mathbb{R}^{d}}(|\nabla
v|^{2}+\alpha v^{2})\,dx\,.$ (1.3)
In the case $\varepsilon=0$, corresponding to (1.2), this Liapunov functional
is at least formally equivalent to
$\mathscr{F}_{0,\alpha}(\rho):=\int_{\mathbb{R}^{d}}(\rho\log\rho-\frac{1}{2}v\rho)\,dx\,$
(1.4)
with the convention that $v$ is obtained from the density $\rho$ by
$v=\mathcal{B}_{\alpha,d}\ast\rho$. Here, $\mathcal{B}_{\alpha,d}$ denotes the
Bessel kernel for $\alpha>0$ or the Newtonian kernel for $\alpha=0$, for any
dimension $d$. Therefore the role of the parameter $\varepsilon$ is to
discriminate between parabolic-parabolic and parabolic-elliptic system. Note
that the Liapunov functionals (1.3) and (1.4) are just formally equivalent
since the $L^{2}$-integrability of $\nabla\mathcal{B}_{\alpha,d}\ast\rho$
fails if $d=1,2$ and $\alpha=0$. Thus, even if the classical free energy
writing and valid for all cases when $\varepsilon=0$ is the one in (1.4), we
will prefer to work with the functional as in (1.3) even if $\varepsilon=0$,
with a suitable renormalization for the cases $d=1,2$ and $\alpha=0$ discussed
in Section 3.
Our main objective is the uniqueness of certain solutions, for both systems
(1.1) and (1.2). Let us introduce the notion of solution for the Cauchy
problems associated to (1.1) and (1.2) that we will consider in this work. We
denote by $\mathscr{M}_{2}(\mathbb{R}^{d};\mathfrak{m})$ the set of
nonnegative densities over $\mathbb{R}^{d}$ with mass $\mathfrak{m}$ and
finite second moment, i.e.,
$\mathscr{M}_{2}(\mathbb{R}^{d};\mathfrak{m}):=\left\\{\rho\in
L^{1}(\mathbb{R}^{d}):\rho\geq
0,\int_{\mathbb{R}^{d}}\rho(x)\,dx=\mathfrak{m},\int_{\mathbb{R}^{d}}|x|^{2}\rho(x)\,dx<+\infty\right\\}.$
###### Definition 1.1.
We say that a weakly continuous map $\rho\in
C_{w}([0,T];\mathscr{M}_{2}(\mathbb{R}^{d};\mathfrak{m}))$ is a bounded
solution to the Cauchy problem for (1.2), with initial datum
$\rho^{0}\in\mathscr{M}_{2}(\mathbb{R}^{d};\mathfrak{m})\cap
L^{\infty}(\mathbb{R}^{d})$, if
* i)
$\rho\in L^{\infty}((0,T)\times\mathbb{R}^{d})$ and $|x|^{2}\rho_{t}(x)\in
L^{\infty}((0,T),L^{1}(\mathbb{R}^{d}))$,
* ii)
$\rho_{0}=\rho^{0}$ and the first equation of (1.2) holds in the sense of
distributions on $(0,T)\times\mathbb{R}^{d}$, where
$v_{t}=\mathcal{B}_{\alpha,d}\ast\rho_{t}$ for all $t\in[0,T]$,
* iii)
$\rho_{t}\in W^{1,1}(\mathbb{R}^{d})$ for $\mathcal{L}^{1}$-a.e. $t\in(0,T)$
and
$\int_{0}^{T}\int_{\mathbb{R}^{d}}\frac{|\nabla\rho_{t}(x)|^{2}}{\rho_{t}(x)}\,dx\,dt<+\infty.$
(1.5)
###### Definition 1.2.
We say that the couple of functions $(\rho,v)$, with $\rho\in
C_{w}([0,T];\mathscr{M}_{2}(\mathbb{R}^{d};\mathfrak{m}))$ and $v\in
L^{2}((0,T);W^{1,\,2}(\mathbb{R}^{d}))$, is a bounded solution to (1.1) with
initial datum $(\rho^{0},v^{0})$,
$\rho^{0}\in\mathscr{M}_{2}(\mathbb{R}^{d};\mathfrak{m})\cap
L^{\infty}(\mathbb{R}^{d})$ and $v^{0}\in W^{1,2}(\mathbb{R}^{d})$, if
* I)
$\rho\in L^{\infty}((0,T)\times\mathbb{R}^{d})$ and $|x|^{2}\rho_{t}(x)\in
L^{\infty}((0,T),L^{1}(\mathbb{R}^{d}))$,
* II)
$\rho_{0}=\rho^{0}$, the first equation of (1.1) holds in the sense of
distributions on $(0,T)\times\mathbb{R}^{d}$, and $v$ is the unique solution
to the Cauchy problem for the forced parabolic equation
$\varepsilon\partial_{t}v-\Delta v+\alpha v=\rho$ over
$(0,T)\times\mathbb{R}^{d}$ in the standard sense, with initial datum $v^{0}$,
* III)
the property iii) of Definition 1.1 holds.
Let us emphasize that the main properties we need to get uniqueness of
solution are the boundedness of the densities and the Fisher information
(1.5). They together imply that the velocity field of the continuity equation
for the density $\rho$ is a well defined object belonging to the right
functional space, see section 2 for details. Moreover, the boundedness of the
density implies that we have a uniform in bounded time intervals estimate on
the quasi-Lipschitz constant of part of the velocity field. These are the
basic properties that imply the uniqueness for bounded solutions. Let us
finally mention that part of the strategy is related to the uniqueness of
solutions to fluid and aggregation equations developed in [34, 27, 5, 28, 19,
7, 29]. The main novelty here is the interplay between the diffusive and the
aggregation parts. The main results of this work are:
###### Theorem 1.3.
Let $T>0$ and let $\rho^{0}\in\mathscr{M}_{2}(\mathbb{R}^{d};\mathfrak{m})\cap
L^{\infty}(\mathbb{R}^{d})$. Let $\rho_{1},\rho_{2}$ be two bounded solutions
on $[0,T]\times\mathbb{R}^{d}$ to the Cauchy problem associated to (1.2), with
initial datum $\rho^{0}$. Then $\rho_{1}=\rho_{2}$.
###### Theorem 1.4.
Let $T>0$ and let $\rho^{0}\in\mathscr{M}_{2}(\mathbb{R}^{d};\mathfrak{m})\cap
L^{\infty}(\mathbb{R}^{d})$, $v^{0}\in W^{1,2}(\mathbb{R}^{d})\cap
W^{2,\infty}(\mathbb{R}^{d})$. Let $(\rho_{1},v_{1})$ and $(\rho_{2},v_{2})$
be two bounded solutions on $[0,T]\times\mathbb{R}^{d}$ of the Cauchy problem
associated to (1.1), with initial datum $(\rho^{0},v^{0})$. Then
$(\rho_{1},v_{1})=(\rho_{2},v_{2})$.
The proof of uniqueness as stated in Theorems 1.3 and 1.4 will be a
consequence of a more general property: we will show that bounded solutions
satisfies a strong gradient flow formulation by means of a family of evolution
variational inequalities. This formulation is similar to the one for semi-
convex functionals and implies a non-expansivity property of the distance
between two solutions. This non-expansivity property yields uniqueness. All
these results will be stated in Theorem 3.1. Moreover this formulation lead
also to a relaxed convexity property of the energy functional as stated in
Theorem 4.1.
There is a huge literature about the KS system and their variations, so we
just restrict here to discuss the main results concerning about bounded
solutions. In the classical parabolic-elliptic KS equation
$\varepsilon=\alpha=0$ and $d=2$, global in time bounded solutions in the
subcritical case $\mathfrak{m}<8\pi$ have been obtained joining the results in
[12, 24, 14]. Actually, the global existence of weak solutions satisfying all
properties in Definition 1.1 except the $L^{\infty}$ bound was obtained in
[12] while $L^{\infty}$-bounds in bounded time intervals can be obtained from
the results in [24, 14]. The same techniques could eventually be used to get
local in time bounded solutions for all masses, although such a result is not
present in the literature. Let us also mention the recent preprint [17] in
which the authors actually show that the $L^{\infty}$-norm of the solution
decays in time like for the heat equation in the subcritical case
$\mathfrak{m}<8\pi$ for more restricted initial data. $L^{\infty}$-apriori
estimates were obtained in the classical parabolic-elliptic KS equation
$\varepsilon=0$ with $d\geq 2$ and $\alpha\geq 0$ for small $L^{d/2}$ initial
data in [20, 21]. These results together with similar arguments as in [12] to
get the free energy dissipation property and thus the Fisher information
bounds, could lead to the existence of bounded solutions in these cases. We
emphasize that these $L^{\infty}$ estimates show that the solution in bounded
time intervals is bounded by a constant that depends only on the
$L^{\infty}$-norm of the initial data, the initial free energy, and the final
time. In particular, existence of bounded solutions is expected if
$\rho^{0}\in L^{\infty}(\mathbb{R}^{d})$, and this explains the presence of
such an assumption in the previous definitions.
Concerning the fully parabolic KS system, we find global in time solutions
satisfying all properties stated in Definition 1.2 except the $L^{\infty}$
bounds in [15] for $d=2$ and the subcritical mass case $\mathfrak{m}<8\pi$.
$L^{\infty}$-apriori estimates were obtained in [25] for the fully parabolic
case but in bounded domains. It is reasonable to expect that this strategy
should work for the whole space case, although it is not written as such in
the literature. Results in higher dimensions concerning solutions with
$L^{\infty}$ estimates for small initial data can be found in [8] but
estimates on the free energy dissipation are missing there. We finally refer
to [11, 13, 14, 22] for different results concerning the existence of
solutions satisfying the boundedness of the Fisher information and/or the
uniform bounds of the solutions for particular choices of $\varepsilon\geq 0$,
$\alpha\geq 0$, and nonlinear diffusions.
As mentioned before, Theorems 1.3 and 1.4 are based on the derivation of
quasi-Lipschitz estimates for the chemoattractant $v$. This is the reason
behind the additional assumption on the initial datum $v^{0}$ in Theorem 1.4.
We will clarify the use of quasi-Lipschitz estimates of the chemoattractant in
the next section together with a quick summary of the main properties of
optimal transport that we need in this work. Section 3 is devoted to show the
main uniqueness results, derived from a more general property of bounded
solutions for the Keller-Segel model. In fact, we will show that for bounded
solutions we can obtain evolution variational inequalities. In Section 4 we
show that these evolution variational inequalities lead to certain convexity
of the associated free energy functional. In order not to break the flow of
the argument, we postpone to Section 5 the rigorous derivation of the quasi-
Lipschitz estimates of the elliptic and parabolic equations for $v$. In
Section 5, we will also prove a strengthening of Theorem 1.4, with more
general initial data. Finally, Section 6 is devoted to show how to adapt these
arguments to Keller-Segel models with nonlinear diffusion.
## 2\. Preliminary notions
### 2.1. Some elliptic and parabolic regularity estimates
The proofs of our results are based on the technique used by Yudovich [34] for
treating uniqueness in the case of incompressible Euler equations for
fluidodynamics. In particular, we exploit a quasi-Lipschitz property for the
velocity field of the continuity equation for $\rho$ in (1.1) and (1.2). This
property comes from the regularity that $v$ gains being solution to the second
equation in (1.1) and (1.2).
Suppose first that $v=\mathcal{B}_{0,d}\ast\rho$. If $\rho\in L^{1}\cap
L^{\infty}(\mathbb{R}^{d})$, by exploiting some estimates of the Newtonian
potential, $\nabla v$ satisfies the following log-Lipschitz property (see [6]
and [30, Chapter 8], [32] and also [34]),
$|\nabla v(x)-\nabla v(y)|\leq C|x-y|(1-\log^{-}|x-y|),$
where $C$ is a suitable positive constant, depending only on
$\|\rho\|_{L^{1}}$ and $\|\rho\|_{L^{\infty}}$ and $\log^{-}$ denotes the
negative part of the natural logarithm function. As a consequence, we get the
estimate
$|\nabla v(x)-\nabla v(y)|^{2}\leq C^{2}\varphi(|x-y|^{2})$ (2.1)
for some new positive constant $C$, where $\varphi$ is the concave function on
$[0,\infty)$ defined as
$\varphi(x):=\left\\{\begin{array}[]{ll}x\log^{2}x&\mbox{if $x\leq
e^{-1-\sqrt{2}}$},\\\ x+2(1+\sqrt{2})e^{-1-\sqrt{2}}&\mbox{if
$x>e^{-1-\sqrt{2}}$}.\end{array}\right.$ (2.2)
Indeed, for large values of $|x-y|$ the estimate (2.1) is quite obvious, since
it is immediate to show that $\nabla\mathcal{B}_{0,d}\ast\rho$ is a bounded
function in the whole space with a direct estimate using the fact that
$\rho\in L^{1}\cap L^{\infty}(\mathbb{R}^{d})$. The log-Lipschitz property
itself can be justified through standard elliptic regularity, as we will do in
Section 5.
Analogous facts hold if we consider the equation $-\Delta v+\alpha v=\rho$,
appearing in (1.2), or more general uniformly elliptic operators, so that we
have
###### Proposition 2.1.
Suppose that $\rho\in L^{1}\cap L^{\infty}({\mathbb{R}^{d}})$ and $\alpha\geq
0$. Then $v=\mathcal{B}_{\alpha,d}\ast\rho$ satisfies the estimate (2.1),
where $C$ is a suitable positive constant, depending only on
$\alpha,d,\|\rho\|_{L^{1}(\mathbb{R}^{d})}$, and
$\|\rho\|_{L^{\infty}(\mathbb{R}^{d})}$.
About the parabolic equation for $v$ in (1.1), the quasi-Lipschitz property
also carries over, since formally inequality (2.1) translates in terms of the
parabolic metric to
$|\nabla v(t,x)-\nabla v(s,y)|^{2}\leq
C^{2}\varphi((|x-y|+|s-t|^{1/2})^{2})\quad\forall
x,y\in\mathbb{R}^{d},\;s,t\in[0,T].$ (2.3)
###### Proposition 2.2.
Suppose that $\rho\in L^{\infty}((0,T)\times{\mathbb{R}^{d}})$, $v^{0}\in
W^{2,\infty}(\mathbb{R}^{d})$ and $\alpha\geq 0$. If $v$ is the unique
solution to the Cauchy problem for the parabolic equation
$\partial_{t}v=\Delta v-\alpha v+\rho$ (in the standard sense of convolution
with fundamental solution), then $v$ satisfies (2.3), where $C$ is a suitable
positive constant, depending only on
$\alpha,d,\|v^{0}\|_{W^{2,\infty}(\mathbb{R}^{d})}$, and
$\|\rho\|_{L^{\infty}((0,T)\times{\mathbb{R}^{d}})}$.
For a more complete insight into these properties, it will be convenient to
recall some facts about the Zygmund class and its role in elliptic and
parabolic regularity. However, in order not to introduce some not really
necessary notation before the proof of our main results, we prefer to postpone
the proof of Proposition 2.1 and Proposition 2.2 to Section 5. Indeed, in
Section 5 we will develop a more rigorous discussion about the log-Lipschitz
estimates, and thanks to some refined parabolic regularity we will also prove
a slight strengthening of Theorem 1.4.
### 2.2. Elementary notions of optimal transport
Given $\rho_{0},\rho_{1}\in\mathscr{M}_{2}(\mathbb{R}^{d};\mathfrak{m})$, we
define the Wasserstein distance between $\rho_{0}$ and $\rho_{1}$ as
$W_{2}(\rho_{0},\rho_{1})=\left(\int_{\mathbb{R}^{d}}|x-\mathcal{T}(x)|^{2}\,\rho_{0}(x)\,dx\right)^{\frac{1}{2}},$
where $\mathcal{T}$ is the unique optimal transport map between $\rho_{0}$ and
$\rho_{1}$, that is, the map $\mathcal{T}:\mathbb{R}^{d}\to\mathbb{R}^{d}$
which minimizes $\int_{\mathbb{R}^{d}}|x-\mathcal{S}(x)|^{2}\,\rho_{0}(x)\,dx$
among all the Borel maps $\mathcal{S}:\mathbb{R}^{d}\to\mathbb{R}^{d}$
satisfying $\mathcal{S}_{\\#}\rho_{0}=\rho_{1}$. We recall that
$\mathcal{S}_{\\#}\rho_{0}=\rho_{1}$ means that
$\int_{\mathbb{R}^{d}}\varphi(x)\rho_{1}(x)\,dx=\int_{\mathbb{R}^{d}}\varphi(\mathcal{S}(x))\rho_{0}(x)\,dx$
for every continuous and bounded function
$\varphi:\mathbb{R}^{d}\to\mathbb{R}^{d}$.
The Wasserstein geodesic between $\rho_{0}$ and $\rho_{1}$ is the curve
$s\in[0,1]\mapsto\rho^{s}\in\mathscr{M}_{2}(\mathbb{R}^{d};\mathfrak{m})$
defined by the so-called displacement interpolation along the optimal
transport map $\mathcal{T}$ between $\rho_{0}$ and $\rho_{1}$, that is,
$\rho^{s}:=((1-s)\mbox{\boldmath$\mathfrak{i}$}+s\mathcal{T})_{\\#}\rho_{0}$.
In particular, for any $s$,
$\mathcal{T}_{s}:=(1-s)\mbox{\boldmath$\mathfrak{i}$}+s\mathcal{T}$ is the
optimal map between $\rho_{0}$ and $\rho^{s}$ and there holds
$W_{2}(\rho^{r},\rho^{s})=|s-r|W_{2}(\rho_{0},\rho_{1})$.
We recall a formula for the differentiation of the squared Wasserstein
distance along solutions of the continuity equation. Let
$t\in[0,T]\mapsto\rho_{t}\in\mathscr{M}_{2}(\mathbb{R}^{d};\mathfrak{m})$ be a
weakly continuous curve which is distributional solution of
$\partial_{t}\rho_{t}+\mathrm{div}\,(\xi_{t}\rho_{t})=0,$
for some Borel velocity field $\xi_{t}$ such that
$\int_{0}^{T}\|\xi_{t}\|^{2}_{L^{2}(\mathbb{R}^{d},\rho_{t};\mathbb{R}^{d})}\,dt<+\infty$.
Then the curve is absolutely continuous with respect to the Wasserstein
distance, [3, Theorem 8.3.1]. Then, for any
$\bar{\rho}\in\mathscr{M}_{2}(\mathbb{R}^{d};\mathfrak{m})$, it holds
$\frac{1}{2}\frac{d}{dt}W_{2}^{2}(\rho_{t},\bar{\rho})=\int_{\mathbb{R}^{d}}\langle\xi_{t}(x),x-\mathcal{T}_{t}(x)\rangle\,\rho_{t}(x)\,dx,\qquad\mbox{for
$\mathcal{L}^{1}$-a.e. $t\in(0,T)$},$ (2.4)
where $\mathcal{T}_{t}$ is the optimal map between $\rho_{t}$ and $\bar{\rho}$
(see [3, Theorem 8.4.7, Remark 8.4.8]).
Finally, let us recall an estimate relating the $2$-Wasserstein distance and
the $H^{-1}$ norm proved in [27, Proposition 2.8]. Given two nonnegative
densities with the same mass
$\rho_{1},\rho_{2}\in\mathscr{M}_{2}(\mathbb{R}^{d};\mathfrak{m})\cap
L^{\infty}(\mathbb{R}^{d})$, there holds
$\|\rho_{1}-\rho_{2}\|_{\dot{H}^{-1}(\mathbb{R}^{d})}\leq\max\\{\|\rho_{1}\|_{\infty},\|\rho_{2}\|_{\infty}\\}^{1/2}W_{2}(\rho_{1},\rho_{2}).$
(2.5)
Here we are letting $\dot{H}^{1}(\mathbb{R}^{d})$ be the space of Lebesgue
measurable functions $v:\mathbb{R}^{d}\to\mathbb{R}$ such that $\|\nabla
v\|_{L^{2}(\mathbb{R}^{d})}<+\infty$, so that $\dot{H}^{-1}(\mathbb{R}^{d})$
is defined by duality with functions having finite $L^{2}(\mathbb{R}^{d})$
norm of the gradient only. By the way, we can also consider the space
$H^{1}(\mathbb{R}^{d})=W^{1,2}(\mathbb{R}^{d})$. In fact, from the proof in
[27, Proposition 2.8] it is not difficult to see that the same estimate holds
considering the $H^{-1}(\mathbb{R}^{d})$ space given by duality with the full
norm $(\|\nabla
v\|^{2}_{L^{2}(\mathbb{R}^{d})}+\|v\|^{2}_{L^{2}(\mathbb{R}^{d})})^{1/2}$.
## 3\. Bounded solutions as gradient flows: EVI and uniqueness
The uniqueness Theorems 1.3 and 1.4 are consequences of a general result
interpreting bounded solutions to (1.1) (resp. (1.2)) as the trajectory of the
gradient flow of the functional (1.3) (resp. (1.4)) in the appropriate metric
setting. We prove that bounded solutions satisfy a family of evolution
variational inequalities (EVI). Among different notions of gradient flow in
metric sense, the EVI formulation is stronger than other formulations and
typically corresponding to a convex structure, as in [3, Theorem 11.2.1] for
the theory in the Wasserstein setting.
Notation for the energy functional. Before giving the proof, we introduce some
uniform notation for working with the full functional (1.3) even in the
parabolic-elliptic case. Let
$\rho\in\mathscr{M}_{2}(\mathbb{R}^{d};\mathfrak{m})\cap
L^{\infty}(\mathbb{R}^{d})$. We are considering the free energy functional
$\mathscr{F}_{\varepsilon,\alpha}(\rho,v):=\int_{\mathbb{R}^{d}}(\rho\log\rho-v\rho)\,dx+\frac{1}{2}\int_{\mathbb{R}^{d}}(|\nabla
v|^{2}+\alpha v^{2})\,dx\,,$
defined for $v$ being any $W^{1,2}(\mathbb{R}^{d})$ function if
$\varepsilon>0$. On the other hand, if $\varepsilon=0$ it is understood that
$v$ is given by $\mathcal{B}_{\alpha,d}\ast\rho$. Therefore the parameter
$\varepsilon$ only indicates if we are considering problem (1.1) or (1.2). In
particular, this writing of the functional as in (1.3) is valid in general,
even for $\varepsilon=0$, except for two particular cases:
$\varepsilon=\alpha=0$ and $d=1,2$, as discussed in the introduction. In these
two cases, we need to renormalize the free energy functional. Given
$\rho^{*}\in\mathscr{M}_{2}(\mathbb{R}^{d};\mathfrak{m})$ a smooth and
compactly supported density and $v^{*}=\mathcal{B}_{0,d}\ast\rho^{*}$, we
redefine (1.3) for $\varepsilon=\alpha=0$ and $d=1,2$ as
$\mathscr{F}_{0,0}(\rho,v):=\int_{\mathbb{R}^{d}}\left[\rho\log\rho-v(\rho-\rho^{*})\right]\,dx+\frac{1}{2}\int_{\mathbb{R}^{d}}|\nabla(v-v^{*})|^{2}\,dx-\frac{1}{2}\int_{\mathbb{R}^{d}}\rho^{*}v^{*}\,dx\,.$
(3.1)
Notice that $\nabla(v-v^{*})\in L^{2}(\mathbb{R}^{d})$, as $\rho-\rho^{*}$ has
zero mean, see [4, 32] for more details.
In the rest of this work, when referring to the free energy functional
$\mathscr{F}_{\varepsilon,\alpha}(\rho,v)$, we will be using (1.3) for any
$\varepsilon\geq 0,\alpha\geq 0$, except for $\varepsilon=\alpha=0$ and
$d=1,2$ where the free energy functional is given by (3.1).
Let us observe that now all the integrals involved in the definition of
$\mathscr{F}_{\varepsilon,\alpha}$ are well defined and finite for
$\varepsilon\geq 0,\alpha\geq 0$ and $\rho,v$ as above. The negative part of
the entropy term can be classically treated by the Carleman inequality, see
for instance [9, Lemma 2.2] where the second moment bound on the density is
used. The boundedness of the density controls the positive contribution of the
entropy term together with the integrability of $v\rho$ in case
$\varepsilon>0$ since $v\in W^{1,2}(\mathbb{R}^{d})$. For $\varepsilon=0$ the
integrability of $v\rho$ in case $\alpha>0$ is implied by the Newtonian
potential case $\alpha=0$ since the singularity of the Bessel potential at the
origin is the same. The integrability for $\alpha=\varepsilon=0$ and $d\geq 3$
results directly from the Hardy-Littlewood-Sobolev inequality for the
Newtonian potential. For $\alpha=\varepsilon=0$ and $d=1,2$ we use the
behavior at infinity of the density $\rho$. Actually, $\alpha=\varepsilon=0$
and $d=1$ is a trivial case since the Newtonian potential is given by
$\mathcal{B}_{0,1}(x)=|x|$. For $\alpha=\varepsilon=0$ and $d=2$ since
$\log(e+|x|^{2})\rho\in L^{1}(\mathbb{R}^{d})$ then $v\rho\in
L^{1}(\mathbb{R}^{d})$ using the logarithmic HLS inequality, see for instance
[10].
Notation for the ambient metric space. We let
$X_{\varepsilon}:=\mathscr{M}_{2}(\mathbb{R}^{d};\mathfrak{m})\times
L^{2}(\mathbb{R}^{d})$ endowed with the distance
$D^{2}(z_{1},z_{2})=D^{2}((\rho_{1},v_{1}),(\rho_{2},v_{2}))=W_{2}^{2}(\rho_{1},\rho_{2})+\varepsilon\|v_{1}-v_{2}\|^{2}_{L^{2}(\mathbb{R}^{d})}\,,$
with the convention that $X_{0}=\mathscr{M}_{2}(\mathbb{R}^{d};\mathfrak{m})$
and $D_{0}(z_{1},z_{2})=W_{2}(\rho_{1},\rho_{2})$. Moreover, for $z=\rho\in
X_{0}\times L^{\infty}({\mathbb{R}^{d}})$, $\mathscr{F}_{0,\alpha}(z)$ will be
understood to be $\mathscr{F}_{0,\alpha}(\rho,v)$ with
$v=\mathcal{B}_{\alpha,d}\ast\rho$, as usual when $\varepsilon=0$.
In the space $X_{\varepsilon}$ the metric derivative of an absolutely
continuous curve $t\mapsto z_{t}$ is denoted and defined by
$|z^{\prime}|_{D}(t)=\lim_{h\to 0}\frac{D(z_{t+h},z_{t})}{h},$
and it exists for $\mathcal{L}^{1}$-a.e. $t>0$. The local metric slope of the
functional $\mathscr{F}_{\varepsilon,\alpha}$ is defined by
$|\partial\mathscr{F}_{\varepsilon,\alpha}|_{D}(z):=\limsup_{D(\zeta,z)\to
0}\frac{(\mathscr{F}_{\varepsilon,\alpha}(z)-\mathscr{F}_{\varepsilon,\alpha}(\zeta))^{+}}{D(\zeta,z)}.$
These two abstractly defined objects are used to give the notion of curves of
maximal slope in general metric setting, see [2, §3], [3, Chapter 1]. The main
consequences of this gradient flow structure are summarized in the following
result.
Before stating the Theorem we define the function
$\omega:[0,+\infty)\to[0,+\infty)$ by
$\omega(x)=\sqrt{\mathfrak{m}x\varphi(\mathfrak{m}^{-1}x)},$ (3.2)
where $\varphi$ is defined in (2.2). Moreover, given a fixed $s_{0}>0$, we
define a strictly monotone continuous function
$G:[0,+\infty)\to[-\infty,+\infty)$ by
${G(s):=\int_{s_{0}}^{s}\frac{1}{\omega(r)}\,dr}$ for $s>0$ and $G(0)=-\infty$
(we observe that $G^{-1}:[-\infty,+\infty)\to[0,+\infty)$ is surjective).
###### Theorem 3.1.
Let $t\mapsto z_{t}=(\rho_{t},v_{t})$ be a bounded solution of problem (1.1)
for $\varepsilon>0$, starting from $z^{0}=(\rho^{0},v^{0})\in
X_{\varepsilon}\cap\big{(}L^{\infty}(\mathbb{R}^{d})\times(W^{1,2}(\mathbb{R}^{d})\cap
W^{2,\infty}(\mathbb{R}^{d}))\big{)}$, according to Definition 1.2. If
$\varepsilon=0$, let $z_{t}=\rho_{t}$ be a bounded solution to problem (1.2),
starting from $z^{0}=\rho^{0}\in X_{0}\cap L^{\infty}(\mathbb{R}^{d})$,
according to Definition 1.1. Then the three following properties hold:
* i)
The evolution variational inequality (EVI) formulation: for any
$\bar{z}=(\bar{\rho},\bar{v})\in
X_{\varepsilon}\cap\big{(}L^{\infty}(\mathbb{R}^{d})\times
W^{1,2}(\mathbb{R}^{d})\big{)}$ (reduced to $\bar{z}=\bar{\rho}\in X_{0}\cap
L^{\infty}(\mathbb{R}^{d})$ if $\varepsilon=0$), the map $t\mapsto
D^{2}(z_{t},\bar{z})$ is absolutely continuous and there exists a constant $C$
depending on $\|\rho\|_{L^{\infty}((0,T)\times\mathbb{R}^{d})}$,
$\|\bar{\rho}\|_{L^{\infty}(\mathbb{R}^{d})}$ and
$\|v^{0}\|_{W^{2,\infty}(\mathbb{R}^{d})}$, such that
$\frac{1}{2}\frac{d}{dt}D^{2}(z_{t},\bar{z})\leq\,\mathscr{F}_{\varepsilon,\alpha}(\bar{z})-\mathscr{F}_{\varepsilon,\alpha}(z_{t})+C\omega(D^{2}(z_{t},\bar{z}))\quad\text{for
$\mathcal{L}^{1}$-a.e. }t\in(0,T).$ (3.3)
* ii)
The energy dissipation equality (EDE) in metric sense: the map
$t\mapsto\mathscr{F}_{\varepsilon,\alpha}(z_{t})$ is locally Lipschitz
continuous and
$\frac{d}{dt}{\mathscr{F}_{\varepsilon,\alpha}}(z_{t})=-\frac{1}{2}|\partial{\mathscr{F}_{\varepsilon,\alpha}}|_{D}^{2}(z_{t})-\frac{1}{2}|z^{\prime}|_{D}^{2}(t)\quad\text{for
$\mathcal{L}^{1}$-a.e. }t\in(0,T).$ (3.4)
* iii)
The following expansion control property: given another bounded solution
$t\mapsto\zeta_{t}$, with initial datum $\zeta^{0}$ in the same space of
$z^{0}$ above, there exists a constant $C$, depending on
$\|\rho\|_{L^{\infty}((0,T)\times\mathbb{R}^{d})}$ and
$\|v^{0}\|_{W^{2,\infty}(\mathbb{R}^{d})}$ (and the same quantities associated
to $\zeta$), such that there holds
$D^{2}(z_{t},\zeta_{t})\leq
G^{-1}(G(D^{2}(z^{0},\zeta^{0}))+4Ct)\quad\text{for every }t\in[0,T).$ (3.5)
###### Proof.
We first introduce the auxiliary functional
$\Phi_{\varepsilon,\alpha}(\rho,v):=\int_{\mathbb{R}^{d}}(\rho\log\rho-v\rho)\,dx,$
for $\rho$ and $v$ being as in the definition of
$\mathscr{F}_{\varepsilon,\alpha}$ at the beginning of this section, so that
$\mathscr{F}_{\varepsilon,\alpha}(\rho,v)=\Phi_{\varepsilon,\alpha}(\rho,v)+\frac{1}{2}\int_{\mathbb{R}^{d}}(|\nabla
v|^{2}+\alpha v^{2})\,dx\,$
and
$\Phi_{0,0}(\rho,v)=\mathscr{F}_{0,0}(\rho,v)-\frac{1}{2}\int_{\mathbb{R}^{d}}|\nabla(v-v^{*})|^{2}\,dx+\frac{1}{2}\int_{\mathbb{R}^{d}}\rho^{*}v^{*}\,dx-\int_{\mathbb{R}^{d}}\\!\rho^{*}v\,dx\,\quad\mbox{for
$d=1,2$.}$ (3.6)
The proof is organized in four steps.
Step1. Quasi-Lipschitz Estimate implies control of the evolution of the
Wasserstein distance.- Thanks to the assumption (1.5), we learn that the
Fisher information
${\int_{\mathbb{R}^{d}}\frac{|\nabla\rho_{t}(x)|^{2}}{\rho_{t}(x)}\,dx}$ is
finite for $\mathcal{L}^{1}$-a.e. $t\in(0,T)$. Let
$\bar{\rho}\in\mathscr{M}_{2}(\mathbb{R}^{d};\mathfrak{m})\cap
L^{\infty}(\mathbb{R}^{d})$. Exploiting the differentiability properties of
the entropy functional, we can use the above-the-tangent formulation of
displacement convexity to get for $\mathcal{L}^{1}$-a.e. $t\in(0,T)$
$\int_{\mathbb{R}^{d}}\bar{\rho}(x)\log\bar{\rho}(x)\,dx-\int_{\mathbb{R}^{d}}\rho_{t}(x)\log\rho_{t}(x)\,dx\geq\int_{\mathbb{R}^{d}}\langle\nabla\rho_{t}(x),\mathcal{T}_{t}(x)-x\rangle\,dx,$
(3.7)
where $\mathcal{T}_{t}$ denotes the optimal transport map between $\rho_{t}$
and $\bar{\rho}$. We refer to [2, §3.3.1] for an intuitive proof of this fact,
and to [3, Chapter 10] for the theory in full generality. In particular, the
finiteness of the Fisher information of $\rho_{t}$ implies that the second
term is finite, so that this differentiation formula is meaningful. If
$\varepsilon>0$ (resp. $\varepsilon=0$), let $\bar{v}\in
W^{1,2}(\mathbb{R}^{d})$ (resp.
$\bar{v}=\mathcal{B}_{\alpha,d}\ast\bar{\rho}$). Take
$I_{t}:=\Phi_{\varepsilon,\alpha}(\bar{\rho},\bar{v})-\Phi_{\varepsilon,\alpha}(\rho_{t},v_{t})+\int_{\mathbb{R}^{d}}(\bar{v}(x)-v_{t}(x))\bar{\rho}(x)\,dx\,.$
Using the notation $x^{s}_{t}:=(1-s)x+s\mathcal{T}_{t}(x)$, $s\in[0,1]$, and
taking into account that
$\displaystyle\int_{\mathbb{R}^{d}}v_{t}(x)(\bar{\rho}(x)-\rho_{t}(x))\,dx$
$\displaystyle=\int_{\mathbb{R}^{d}}(v_{t}(\mathcal{T}_{t}(x))-v_{t}(x))\rho_{t}(x)\,dx$
$\displaystyle=\int_{\mathbb{R}^{d}}(v_{t}(x_{t}^{1})-v_{t}(x_{t}^{0}))\rho_{t}(x)\,dx=\int_{0}^{1}\frac{d}{ds}\int_{\mathbb{R}^{d}}v_{t}(x_{t}^{s})\rho_{t}(x)\,dx\,ds$
and (3.7), we obtain for $\mathcal{L}^{1}$-a.e. $t\in(0,T)$
$\displaystyle I_{t}$
$\displaystyle\geq\int_{\mathbb{R}^{d}}\langle\nabla\rho_{t}(x),\mathcal{T}_{t}(x)-x\rangle\,dx-\int_{\mathbb{R}^{d}}v_{t}(x)(\bar{\rho}(x)-\rho_{t}(x))\,dx$
$\displaystyle=\int_{\mathbb{R}^{d}}\langle\nabla\rho_{t}(x),\mathcal{T}_{t}(x)-x\rangle\,dx-\int_{0}^{1}\int_{\mathbb{R}^{d}}\langle\nabla
v_{t}(x^{s}_{t}),\mathcal{T}_{t}(x)-x\rangle\,\rho_{t}(x)\,dx\,ds$
$\displaystyle=\int_{\mathbb{R}^{d}}\langle\nabla\rho_{t}(x)-\rho_{t}(x)\nabla
v_{t}(x),\mathcal{T}_{t}(x)-x\rangle\,dx-\int_{0}^{1}\int_{\mathbb{R}^{d}}\langle\nabla
v_{t}(x^{s}_{t})-\nabla
v_{t}(x),\mathcal{T}_{t}(x)-x\rangle\,\rho_{t}(x)\,dx\,ds.$
Let us denote by $II_{t}$ the last term in the right hand side above. The
crucial point is to treat such term using the log-Lipschitz property of
$\nabla v$. Notice that, if $\varepsilon=0$, we are in the assumptions of
Proposition 2.1 and we apply (2.1), where the constant $C$ depends in
principle only on ($\mathfrak{m},\alpha,d$ and) the $L^{\infty}$ norm of
$\rho_{t}$, which we are assuming to be uniformly bounded on $(0,T)$. In the
case $\varepsilon>0$, still by the uniform space-time $L^{\infty}$ assumption
on $\rho_{t}$ and the $W^{2,\infty}$ assumption on $v^{0}$, we are in the
framework of Proposition 2.2, so that we can apply the estimate (2.3). In this
case the constant will depend also on ($\varepsilon$ and)
$\|v^{0}\|_{W^{2,\infty}(\mathbb{R}^{d})}$. Since $\varphi$ is concave, we can
also use the Jensen inequality, and letting
$\rho^{s}_{t}={x^{s}_{t}}_{\\#}\rho_{t}$ be the Wasserstein geodesic
connecting $\rho_{t}$ and $\bar{\rho}$ we have
$\displaystyle\left|II_{t}\right|$ $\displaystyle\leq
W_{2}(\rho_{t},\bar{\rho})\int_{0}^{1}\left(\int_{\mathbb{R}^{d}}|\nabla
v_{t}(x^{s}_{t})-\nabla v_{t}(x)|^{2}\rho_{t}(x)\,dx\right)^{1/2}\,ds$ (3.8)
$\displaystyle\leq
CW_{2}(\rho_{t},\bar{\rho})\int_{0}^{1}\left(\int_{\mathbb{R}^{d}}\varphi(|x^{s}_{t}-x|^{2})\rho_{t}(x)\,dx\right)^{1/2}\,ds$
$\displaystyle\leq\sqrt{\mathfrak{m}}CW_{2}(\rho_{t},\bar{\rho})\int_{0}^{1}\sqrt{\varphi(\mathfrak{m}^{-1}W_{2}^{2}(\rho_{t},\rho^{s}_{t}))}\,ds\leq\sqrt{\mathfrak{m}}CW_{2}(\rho_{t},\bar{\rho})\sqrt{\varphi(\mathfrak{m}^{-1}W_{2}^{2}(\rho_{t},\bar{\rho}))}\,.$
The last inequality holds since geodesic interpolation ensures
$\int_{\mathbb{R}^{d}}|x-x^{s}_{t}|^{2}\rho_{t}(x)\,dx=W_{2}^{2}(\rho_{t},\rho^{s}_{t})=s^{2}W_{2}^{2}(\rho_{t},\bar{\rho})$
for all $s\in[0,1]$ and since $\varphi$ is non decreasing. We recall that the
constant $C$ in (3.8) depends only on ($\varepsilon$, $\alpha,d$, the mass
$\mathfrak{m}$ and) the $L^{\infty}((0,T)\times\mathbb{R}^{d})$ norm of $\rho$
and, in the case $\varepsilon>0$, the $W^{2,\infty}(\mathbb{R}^{d})$ norm of
$v^{0}$. Inserting this in the estimate for $I_{t}$, we have for
$\mathcal{L}^{1}$-a.e. $t\in(0,T)$
$I_{t}\geq\int_{\mathbb{R}^{d}}\langle{\nabla\rho_{t}}(x)-\rho_{t}(x)\nabla
v_{t}(x),\mathcal{T}_{t}(x)-x\rangle\,dx-C\omega(W_{2}^{2}(\rho_{t},\bar{\rho}))\,,$
(3.9)
where $\omega$ is the function defined in (3.2). Since $\rho_{t}$ satisfies
the continuity equation
$\partial_{t}\rho_{t}+\mathrm{div}\,(\xi_{t}\rho_{t})=0\qquad\mbox{with
}\quad\rho_{t}\xi_{t}=-{\nabla\rho_{t}}+\rho_{t}\nabla v_{t}$
and (1.5), the uniform $L^{\infty}$ bound of $\rho_{t}$ implies that
$\int_{0}^{T}\|\xi_{t}\|^{2}_{L^{2}(\mathbb{R}^{d},\rho_{t};\mathbb{R}^{d})}\,dt<+\infty$.
Therefore $t\mapsto\rho_{t}$ is absolutely continuous with respect to $W_{2}$
and by (2.4)
$\frac{1}{2}\frac{d}{dt}W_{2}^{2}(\rho_{t},\bar{\rho})=\int_{\mathbb{R}^{d}}\langle\nabla\rho_{t}(x)-\rho_{t}(x)\nabla
v_{t}(x),\mathcal{T}_{t}(x)-x\rangle\,dx\qquad\mbox{for $\mathcal{L}^{1}$-a.e.
t}\in(0,T).$
Inserting this into (3.9), and recalling the definition of $I_{t}$, we finally
obtain
$\frac{1}{2}\frac{d}{dt}W_{2}^{2}(\rho_{t},\bar{\rho})\leq\Phi_{\varepsilon,\alpha}(\bar{\rho},\bar{v})-\Phi_{\varepsilon,\alpha}(\rho_{t},v_{t})+\int_{\mathbb{R}^{d}}(\bar{v}-v_{t})\bar{\rho}\,dx+C\omega(W_{2}^{2}(\rho_{t},\bar{\rho}))$
(3.10)
for $\mathcal{L}^{1}$-a.e. $t\in(0,T)$.
Step 2: EVI for the parabolic-parabolic case.- Recalling that $\bar{v}\in
W^{1,\,2}(\mathbb{R}^{d})$, observing that $\Delta v_{t}\in
L^{2}(\mathbb{R}^{d})$ for a.e.-$t\in(0,T)$ and using the elementary identity
$|a|^{2}-|b|^{2}=|a-b|^{2}+2\langle b,a-b\rangle$ for every
$a,b\in\mathbb{R}^{k}$, the variation of the second part of the functional
(1.3) (that is, $\mathscr{F}_{\varepsilon,\alpha}-\Phi_{\varepsilon,\alpha}$)
can be written as
$\displaystyle\frac{1}{2}\int_{\mathbb{R}^{d}}\left[|\nabla\bar{v}|^{2}-\right.$
$\displaystyle\left.|\nabla
v_{t}|^{2}+\alpha(\bar{v}^{2}-v_{t}^{2})\right]\,dx$ (3.11)
$\displaystyle=\int_{\mathbb{R}^{d}}(\alpha v_{t}-\Delta
v_{t})(\bar{v}-v_{t})\,dx+\frac{1}{2}\|\nabla(v_{t}-\bar{v})\|^{2}_{L^{2}(\mathbb{R}^{d})}+\frac{\alpha}{2}\|v_{t}-\bar{v}\|^{2}_{L^{2}({\mathbb{R}^{d}})}$
$\displaystyle=\int_{\mathbb{R}^{d}}(\rho_{t}-\varepsilon\partial_{t}v_{t})(\bar{v}-v_{t})\,dx+\frac{1}{2}\|\nabla(v_{t}-\bar{v})\|^{2}_{L^{2}(\mathbb{R}^{d})}+\frac{\alpha}{2}\|v_{t}-\bar{v}\|^{2}_{L^{2}({\mathbb{R}^{d}})}$
$\displaystyle=\int_{\mathbb{R}^{d}}\rho_{t}(\bar{v}-v_{t})\,dx+\frac{\varepsilon}{2}\frac{d}{dt}\|v_{t}-\bar{v}\|_{L^{2}(\mathbb{R}^{d})}^{2}+\frac{1}{2}\|\nabla(v_{t}-\bar{v})\|^{2}_{L^{2}(\mathbb{R}^{d})}+\frac{\alpha}{2}\|v_{t}-\bar{v}\|^{2}_{L^{2}({\mathbb{R}^{d}})}.$
Therefore, we deduce
$\displaystyle\mathscr{F}_{\varepsilon,\alpha}(\bar{\rho},\bar{v})-\mathscr{F}_{\varepsilon,\alpha}(\rho_{t},v_{t})$
$\displaystyle=\Phi_{\varepsilon,\alpha}(\bar{\rho},\bar{v})-\Phi_{\varepsilon,\alpha}(\rho_{t},v_{t})+\int_{\mathbb{R}^{d}}\rho_{t}(\bar{v}-v_{t})\,dx$
(3.12)
$\displaystyle\quad+\frac{\varepsilon}{2}\frac{d}{dt}\|v_{t}-\bar{v}\|_{L^{2}(\mathbb{R}^{d})}^{2}+\frac{1}{2}\|\nabla(v_{t}-\bar{v})\|^{2}_{L^{2}(\mathbb{R}^{d})}+\frac{\alpha}{2}\|v_{t}-\bar{v}\|^{2}_{L^{2}({\mathbb{R}^{d}})}.$
Now, we use again (3.10), leading to
$\displaystyle\frac{\varepsilon}{2}\frac{d}{dt}\|v_{t}-\bar{v}\|_{L^{2}(\mathbb{R}^{d})}^{2}$
$\displaystyle+\frac{1}{2}\frac{d}{dt}W_{2}^{2}(\rho_{t},\bar{\rho})\leq\,\mathscr{F}_{\varepsilon,\alpha}(\bar{\rho},\bar{v})-\mathscr{F}_{\varepsilon,\alpha}(\rho_{t},v_{t})+C\omega(W_{2}^{2}(\rho_{t},\bar{\rho}))$
(3.13)
$\displaystyle\quad+\int_{\mathbb{R}^{d}}(\bar{\rho}-\rho_{t})(\bar{v}-v_{t})\,dx-\frac{1}{2}\|\nabla(v_{t}-\bar{v})\|^{2}_{L^{2}(\mathbb{R}^{d})}-\frac{\alpha}{2}\|v_{t}-\bar{v}\|^{2}_{L^{2}({\mathbb{R}^{d}})}.$
By using the duality between $\dot{H}^{1}$ and $\dot{H}^{-1}$, the Young
inequality, and (2.5) we have
$\displaystyle\int_{\mathbb{R}^{d}}(\bar{\rho}-\rho_{t})(\bar{v}-v_{t})\,dx$
$\displaystyle\leq\|\bar{\rho}-\rho_{t}\|_{\dot{H}^{-1}(\mathbb{R}^{d})}\|\bar{v}-v_{t}\|_{\dot{H}^{1}(\mathbb{R}^{d})}\leq\frac{1}{2}\|\bar{\rho}-\rho_{t}\|^{2}_{\dot{H}^{-1}(\mathbb{R}^{d})}+\frac{1}{2}\|\nabla(\bar{v}-v_{t})\|^{2}_{L^{2}(\mathbb{R}^{d})}$
(3.14)
$\displaystyle\leq\frac{1}{2}QW_{2}^{2}(\bar{\rho},\rho_{t})+\frac{1}{2}\|\nabla(\bar{v}-v_{t})\|^{2}_{L^{2}(\mathbb{R}^{d})},$
where $Q$ is the largest of the $L^{\infty}$ norms of $\bar{\rho}$ and
$\rho_{t}$ over the time interval $(0,T)$. Taking into account that $\omega$
is given by (3.2) and that
$\sqrt{\mathfrak{m}\varphi(\mathfrak{m}^{-1}x^{2})}\geq x$ for every $x>0$,
combining (3.13) and (3.14) we get, up to introducing a new constant $C$,
$\displaystyle\frac{\varepsilon}{2}\frac{d}{dt}\|v_{t}-\bar{v}\|_{L^{2}(\mathbb{R}^{d})}^{2}+\frac{1}{2}\frac{d}{dt}W_{2}^{2}(\rho_{t},\bar{\rho})$
$\displaystyle\leq\,\mathscr{F}_{\varepsilon,\alpha}(\bar{\rho},\bar{v})-\mathscr{F}_{\varepsilon,\alpha}(\rho_{t},v_{t})+C\omega(W_{2}^{2}(\rho_{t},\bar{\rho}))-\frac{\alpha}{2}\|v_{t}-\bar{v}\|^{2}_{L^{2}({\mathbb{R}^{d}})}$
(3.15)
for a.e. $t\in(0,T)$. The new constant $C$ depends as usual on ($\varepsilon$
$\alpha$, $d$, $\mathfrak{m}$ and)
$\|\rho\|_{L^{\infty}((0,T)\times{\mathbb{R}^{d}})}$,
$\|v^{0}\|_{W^{2,\infty}(\mathbb{R}^{d})}$,
$\|\bar{\rho}\|_{L^{\infty}(\mathbb{R}^{d})}$.
Step 3: EVI for the parabolic-elliptic case.- When either $d\geq 3$ or
$\alpha>0$, we can repeat the proof of the parabolic-parabolic case, letting
$\varepsilon=0$ therein and recalling that $\bar{v}$ is no more an arbitrary
$W^{1,2}(\mathbb{R}^{d})$ function but is given by convolution with
$\bar{\rho}$. In particular we arrive to the corresponding of (3.13), and the
second line therein can now be estimated as follows. Using the inequality
$\|v\|_{H^{1}_{\alpha}(\mathbb{R}^{d})}\leq\|\rho\|_{H^{-1}_{\alpha}(\mathbb{R}^{d})}$
for $-\Delta v+\alpha v=\rho$, $\alpha>0$, where the notation is
$\|v\|^{2}_{H^{1}_{\alpha}(\mathbb{R}^{d})}:=\|\nabla
v\|^{2}_{L^{2}(\mathbb{R}^{d})}+\alpha\|v\|^{2}_{L^{2}(\mathbb{R}^{d})}$ (and
using $\dot{H}^{1}$ if $\alpha=0$), we get
$\int_{\mathbb{R}^{d}}(\bar{v}-v_{t})(\bar{\rho}-\rho_{t})\,dx\\!\leq\|\bar{v}-v_{t}\|_{H^{1}_{\alpha}(\mathbb{R}^{d})}\|\bar{\rho}-\rho_{t}\|_{H^{-1}_{\alpha}(\mathbb{R}^{d})}\leq\frac{1}{2}\|\bar{v}-v_{t}\|_{H^{1}_{\alpha}(\mathbb{R}^{d})}^{2}+\frac{1}{2}\|\bar{\rho}-\rho_{t}\|_{H^{-1}_{\alpha}(\mathbb{R}^{d})}^{2}.$
Moreover, recalling the estimate (2.5) (which works both in $\dot{H}^{-1}$ and
$H^{-1}_{\alpha}$) we have
$\|\bar{\rho}-\rho_{t}\|_{H^{-1}_{\alpha}(\mathbb{R}^{d})}\leq
QW_{2}^{2}(\bar{\rho},\rho_{t}),$
for all $t\in[0,T]$, where $Q$ is the largest of the $L^{\infty}$ norms of
$\bar{\rho}$ and $\rho_{t}$ over the time interval $[0,T]$. Inserting these
estimates in (3.13) we obtain
$\displaystyle\frac{1}{2}\frac{d}{dt}W_{2}^{2}(\rho_{t},\bar{\rho})\leq\,\mathscr{F}_{0,\alpha}(\bar{\rho})-\mathscr{F}_{0,\alpha}(\rho_{t})+C\omega(W_{2}^{2}(\rho_{t},\bar{\rho})),$
(3.16)
for $\mathcal{L}^{1}$-a.e. $t\in(0,T)$, where the constant $C$ depends only on
$\varepsilon$, $\alpha$, $d$, $\mathfrak{m}$,
$\|\rho\|_{L^{\infty}((0,T)\times{\mathbb{R}^{d}})}$,
$\|\bar{\rho}\|_{L^{\infty}(\mathbb{R}^{d})}$.
In the case $\alpha=0$, $d=1,2$, we have to consider the functional in (3.1).
By using the identity
$\frac{1}{2}\|\nabla(\bar{v}-v^{*})\|^{2}_{L^{2}(\mathbb{R}^{d})}-\frac{1}{2}\|\nabla(v_{t}-v^{*})\|^{2}_{L^{2}(\mathbb{R}^{d})}=\int_{\mathbb{R}^{d}}(\rho_{t}-\rho^{*})(\bar{v}-v_{t})\,dx+\frac{1}{2}\|\nabla(v_{t}-\bar{v})\|^{2}_{L^{2}(\mathbb{R}^{d})},$
with similar computations as in (3.11), this time considering
$\mathscr{F}_{0,0}(\rho,v)-\Phi_{0,0}(\rho,v)$ as obtained from (3.6), we can
still find (3.12) and conclude obtaining again (3.16).
Step 4: Conclusion.- We are ready to prove the three points in the statement
of the theorem. The proof of i) is a consequence of (3.15) for the case
$\varepsilon>0$, and (3.16) for the case $\varepsilon=0$, taking into account
that $\alpha\geq 0$ and that
$\omega(D^{2}(z_{t},\bar{z}))\geq\omega(W_{2}^{2}(\rho_{t},\bar{\rho}))$ being
$\omega$ increasing.
It is a standard fact that the gradient flow formulation in EVI sense implies
the one in EDE sense in (3.4). Indeed, the proof of ii) follows from (3.3)
and (3.5) and can be exactly carried out as in [2, Proposition 3.6].
The proof of (3.5) still follows from (3.3). Indeed we can apply [3, Lemma
4.3.4] (see also the argument of [3, Theorem 11.1.4]) and obtain that for
$\mathcal{L}^{1}$-a.e. $t\in(0,T)$
$\frac{1}{2}\frac{d}{ds}D^{2}(z_{s},\zeta_{s}){\Big{|}}_{s=t}\leq\,\frac{1}{2}\frac{d}{ds}D^{2}(z_{s},\zeta_{t}){\Big{|}}_{s=t}+\frac{1}{2}\frac{d}{ds}D^{2}(z_{t},\zeta_{s}){\Big{|}}_{s=t}\leq
2C\omega(D^{2}(z_{t},\zeta_{t})).$ (3.17)
Here, $C=\max\\{C_{1},C_{2}\\}$, where $C_{1}$ is the supremum on $s\in(0,T)$
of the constant in (3.3) for $z_{t}$ with $\bar{z}=\zeta_{s}$, which is finite
since $\zeta\in L^{\infty}((0,T)\times\mathbb{R}^{d})$, and $C_{2}$ is the
same inverting $z$ and $\zeta$. The estimate (3.17) implies
$\frac{d}{dt}D^{2}(z_{t},\zeta_{t})\leq\,4C\omega(D^{2}(z_{t},\zeta_{t})),\quad\text{for
$\mathcal{L}^{1}$-a.e. }t\in(0,T).$
Since the inequality
$y(t)\leq y(0)+4C\int_{0}^{t}\omega(y(s))\,ds$
entails that $y(t)\leq G^{-1}(G(y(0))+4Ct)$, we conclude. ∎
Proof of Theorems 1.3 and 1.4. The main theorems in the introduction are now a
straightforward consequence of the expansion control iii) in Theorem 3.1. Both
Theorems follow from the inequality (3.5) observing that
$G^{-1}(G(0)+4Ct)=G^{-1}(-\infty)=0$. $\square$
## 4\. $\omega$-convexity of the functional
In this section we show another consequence of the EVI formulation of bounded
solutions. For the functional $\mathscr{F}_{\varepsilon,\alpha}$ the following
relaxed $\omega$-convexity along geodesics holds, see [18] for
$\omega$-convexity of functionals on measures. We assume that bounded
solutions to (1.1) (resp. (1.2) for $\varepsilon=0$) verify that for some
$T>0$
$\|\rho_{t}\|_{L^{\infty}(\mathbb{R}^{d})}\leq
R_{T}(\rho^{0},v^{0}):=R\left(T,\|\rho^{0}\|_{L^{\infty}(\mathbb{R}^{d})},\mathscr{F}_{\varepsilon,\alpha}(\rho^{0},v^{0})\right),\quad\text{for
$\mathcal{L}^{1}$-a.e. }t\in(0,T)\,.$ (4.1)
This assumption has been proved in several cases, see the introduction for
more details.
###### Theorem 4.1.
Assume that bounded solutions for the evolutions (1.1) (resp. (1.2) for
$\varepsilon=0$) exist and verify (4.1). Then, for every $z^{0},z^{1}\in
X_{\varepsilon}\cap\big{(}L^{\infty}(\mathbb{R}^{d})\times(W^{1,2}(\mathbb{R}^{d})\cap
W^{2,\infty}(\mathbb{R}^{d}))\big{)}$ (reduced to $X_{0}\cap
L^{\infty}(\mathbb{R}^{d})$ if $\varepsilon=0$) and every geodesic
$s\in[0,1]\to z^{s}$ of the space $(X_{\varepsilon},D)$, connecting $z^{0}$ to
$z^{1}$ there holds for all $s\in[0,1]$
$\mathscr{F}_{\varepsilon,\alpha}(z^{s})\leq(1-s)\mathscr{F}_{\varepsilon,\alpha}(z^{0})+s\mathscr{F}_{\varepsilon,\alpha}(z^{1})+R_{T}\left[(1-s)\,\omega(s^{2}D^{2}(z^{0},z^{1}))+s\,\omega((1-s)^{2}D^{2}(z^{0},z^{1}))\right],$
where $R_{T}:=\max(R_{T}(z^{0}),R_{T}(z^{1}))$.
###### Proof.
We first remark that the set
$X_{\varepsilon}\cap\big{(}L^{\infty}(\mathbb{R}^{d})\times(W^{1,2}(\mathbb{R}^{d})\cap
W^{2,\infty}(\mathbb{R}^{d}))\big{)}$ (resp. $X_{0}\cap
L^{\infty}(\mathbb{R}^{d})$ if $\varepsilon=0$) is geodesically convex. This
is trivial for the part of the functional concerning $v$ while for the density
$\rho$ we use the classical displacement convexity of all the $L^{p}$ norms
[31]. Now, we take
$X_{\varepsilon}\cap\big{(}L^{\infty}(\mathbb{R}^{d})\times(W^{1,2}(\mathbb{R}^{d})\cap
W^{2,\infty}(\mathbb{R}^{d}))\big{)}$ (resp. $X_{0}\cap
L^{\infty}(\mathbb{R}^{d})$ if $\varepsilon=0$) as the set of initial data for
the evolutions (1.1) (resp. (1.2)).
Consider any $\bar{z}\in
X_{\varepsilon}\cap\left(L^{\infty}(\mathbb{R}^{d})\times
W^{1,2}(\mathbb{R}^{d})\right)$ (reduced to $\bar{z}\in X_{0}\times
L^{\infty}(\mathbb{R}^{d})$ if $\varepsilon=0$). Taking into account that for
a bounded solution $t\mapsto D^{2}(z_{t},\bar{z})$ is absolutely continuous
and $t\mapsto\mathscr{F}_{\varepsilon,\alpha}(z_{t})$ is decreasing by (3.4),
from (3.3) we obtain
$\frac{1}{2}D^{2}(z_{t},\bar{z})-\frac{1}{2}D^{2}(z_{0},\bar{z})\leq\,t(\mathscr{F}_{\varepsilon,\alpha}(\bar{z})-\mathscr{F}_{\varepsilon,\alpha}(z_{t}))+C\int_{0}^{t}\omega(D^{2}(z_{r},\bar{z}))\,dr$
(4.2)
for all $t\in[0,T]$. We denote by $z_{t}^{s}$ the bounded solution of (1.1) or
(1.2) starting from the initial datum $z^{s}$. We multiply by $(1-s)$ the
inequality in (4.2) for $z_{t}=z_{t}^{s}$ and $\bar{z}=z^{0}$ and we multiply
by $s$ the inequality in (4.2) for $z_{t}=z_{t}^{s}$ and $\bar{z}=z^{1}$.
Summing up the two inequalities we obtain
$\displaystyle\frac{1}{2}((1-s)D^{2}(z^{s}_{t},z^{0})+sD^{2}(z^{s}_{t},z^{1}))$
$\displaystyle-\frac{1}{2}((1-s)D^{2}(z^{s},z^{0})+sD^{2}(z^{s},z^{1}))$
$\displaystyle\leq\,$ $\displaystyle
t((1-s)\mathscr{F}_{\varepsilon,\alpha}(z^{0})+s\mathscr{F}_{\varepsilon,\alpha}(z^{1})-\mathscr{F}_{\varepsilon,\alpha}(z^{s}_{t}))$
$\displaystyle+C((1-s)\int_{0}^{t}\omega(D^{2}(z^{s}_{r},z^{0}))\,dr+s\int_{0}^{t}\omega(D^{2}(z^{s}_{r},z^{1}))\,dr).$
Using the fact that $s\mapsto z^{s}$ is a geodesic, the right hand side is
nonnegative, thus
$\displaystyle\mathscr{F}_{\varepsilon,\alpha}(z^{s}_{t})-(1-s)\mathscr{F}_{\varepsilon,\alpha}(z^{0})$
$\displaystyle-s\mathscr{F}_{\varepsilon,\alpha}(z^{1})$ $\displaystyle\leq
C((1-s)\frac{1}{t}\int_{0}^{t}\omega(D^{2}(z^{s}_{r},z^{0}))\,dr+s\frac{1}{t}\int_{0}^{t}\omega(D^{2}(z^{s}_{r},z^{1}))\,dr).$
The lower semi continuity of
$t\mapsto\mathscr{F}_{\varepsilon,\alpha}(z^{s}_{t})$ and the continuity of
$r\mapsto D^{2}(z^{s}_{r},z^{i})$, $i=0,1$ yield
$\mathscr{F}_{\varepsilon,\alpha}(z^{s})\leq(1-s)\mathscr{F}_{\varepsilon,\alpha}(z^{0})+s\mathscr{F}_{\varepsilon,\alpha}(z^{1})+C((1-s)\omega(D^{2}(z^{s},z^{0}))+s\omega(D^{2}(z^{s},z^{1}))).$
Since $s\mapsto z^{s}$ is a geodesic we have
$D^{2}(z^{s},z^{0})=s^{2}D^{2}(z^{1},z^{0})$ and
$D^{2}(z^{s},z^{1})=(1-s)^{2}D^{2}(z^{0},z^{1})$ and we conclude. ∎
## 5\. A refined result in Zygmund spaces
This section is devoted to give a rigorous justification of the estimates
stated in subsection 2.1. We will also give a slight improvement of Theorem
3.1 and Theorem 1.4 by guaranteing a suitable quasi-Lipschitz estimate under a
more general condition on the initial datum $v^{0}$. The right framework is
that of Zygmund spaces. These classes of functions were introduced in [35],
and they belong to the more general framework of Besov spaces.
#### Zygmund estimates and log-Lipschitz regularity in the elliptic case
We first introduce the basic Zygmund class $\Lambda_{1}({\mathbb{R}^{d}})$ as
the set of continuous bounded functions $f$ over $\mathbb{R}^{d}$ such that
$\sup_{x,y\in\mathbb{R}^{d}}\frac{|f(x)-2f((x+y)/2)+f(y)|}{|x-y|}<+\infty.$
It is well known that functions in the Zygmund class are in general not
Lipschitz, possibly nowhere differentiable. Indeed, functions in
$\Lambda_{1}(\mathbb{R}^{d})$ enjoy a log-Lipschitz modulus of continuity.
Therefore, for any $f\in\Lambda_{1}(\mathbb{R}^{d})$ there exists a positive
constant $C$ such that
$|f(x)-f(y)|\leq C|x-y||\log|x-y||\quad\forall x,y\in\mathbb{R}^{d},$
we refer for instance to [36, Chapter 2, §3]. More generally, following for
instance [33, Chapter 5] we may define the class $\Lambda_{n}(\mathbb{R}^{d})$
for any $n\in\mathbb{N}$ as follows. We let
$\Lambda_{0}=L^{\infty}(\mathbb{R}^{d})$ and we say that
$f\in\Lambda_{0}(\mathbb{R}^{d})$ belongs to $\Lambda_{n}(\mathbb{R}^{d})$,
$n\geq 2$, if $\nabla f\in\Lambda_{n-1}(\mathbb{R}^{d})$ or, equivalently, if
$f\in W^{n-1,\infty}(\mathbb{R}^{d})$ and all the derivatives of $f$ of order
$n-1$ belong to $\Lambda_{1}(\mathbb{R}^{d})$. In the usual notation of Besov
spaces, $\Lambda_{n}$ corresponds to $B^{n}_{\infty,\,\infty}$. In this
framework we have
Proof of Proposition 2.1. If $\alpha>0$, from the general theory on Bessel
potentials (see for instance [33, Chapter 5, §3-6]) we learn that by
convolution with the Bessel kernel $\mathcal{B}_{\alpha,d}$ we indeed get two
indices of regularity in $\Lambda_{n}$ spaces. Therefore, if $\rho\in
L^{\infty}(\mathbb{R}^{d})$, we indeed get that
$v=\mathcal{B}_{\alpha,d}\ast\rho$ belongs to $\Lambda_{2}(\mathbb{R}^{d})$,
and thus $\nabla v\in\Lambda_{1}(\mathbb{R}^{d})$ and (2.1) follows. For the
case $\alpha=0$ we address to the references mentioned in Section 2 (it is
also possible to directly check that $\nabla v\in L^{\infty}(\mathbb{R}^{d})$,
and then the Newtonian potential behaves like the Bessel potential near the
origin so that $\nabla v$ is also log-Lipschitz). $\square$
#### Zygmund estimates and log-Lipschitz regularity in the parabolic case
Let $T>0$. In this section, let us denote $Q_{T}:=(0,T)\times\mathbb{R}^{d}$
and then $\bar{Q}_{T}:=[0,T]\times\mathbb{R}^{d}$. In the half $d+1$
dimensional space, we consider the standard parabolic metric
$\delta((x,t),(y,s)):=\max\\{|x-y|,\sqrt{|t-s|}\\}.$
With respect to the parabolic metric, the definition of Zygmund spaces adapts
as follows. We have $\Lambda_{0}(\bar{Q}_{T}):=L^{\infty}(Q_{T})$, and
$\Lambda_{1}(\bar{Q}_{T})$ is the space of continuous bounded functions $f$
over $\bar{Q}_{T}$ such that there hold
$\sup_{{x,y\in\mathbb{R}^{d}}\atop{t\in[0,T]}}\\!\\!\frac{|f(x,t)-2f((x+y)/2,t)+f(y,t)|}{|x-y|}\;+\\!\\!\\!\sup_{{x\in\mathbb{R}^{d}}\atop{0\leq
s<t\leq
T}}\\!\\!\\!\frac{|f(x,t)-2f(x,(t+s)/2)+f(x,s)|}{|t-s|^{1/2}}<+\infty.$ (5.1)
Moreover, we say that $f\in L^{\infty}(Q_{T})$ belongs to
$\Lambda_{2}(\bar{Q}_{T})$ if
$\sup_{x\in\mathbb{R}^{d}\,,\,0\leq s<t\leq
T}\frac{|f(x,t)-2f(x,(t+s)/2)+f(x,s)|}{|t-s|}<+\infty$
and $\nabla f\in\Lambda_{1}(\mathbb{R}^{d})$. In particular, we see that
$f\in\Lambda_{2}(\bar{Q}_{t})$ implies $f\in
L^{\infty}((0,T);W^{1,\infty}(\mathbb{R}^{d}))$, with $\nabla f$ satisfying
(5.1), so that finally $f$ satisfies also (2.3).
When dealing with parabolic equations, it is suitable to consider spaces of
functions defined with respect to the parabolic metric, since it is natural to
deal with functions which have derivative up to order $k$ with respect to time
and $2k$ with respect to space. For classic results, we refer for instance to
[16] or to the monograph [26], where estimates are derived in Sobolev and
Hölder spaces of this kind, see Chapter 4 therein.
In [16] we find that if the forcing term of the heat equation has bounded mean
oscillation (BMO), still with respect to the parabolic metric, than the same
holds true for second order space derivatives and first order time derivatives
of the solution. This would be enough for deducing that first derivatives in
space are in the Zygmund class with respect to the parabolic metric and that
therefore they satisfy a log-Lipschitz estimate. The results in [16] deal only
with null initial datum, but they can be generalized to more general data with
suitable regularity requirements. Some extensions involving initial data in
Zygmund classes are found in [1, 23], based on direct estimates on fundamental
solutions. Summing up, we have
Proof of Proposition 2.2. Suppose that $v$ is the solution (convolution with
fundamental operator) of the forced heat equation $\partial_{t}v=\Delta
v+\rho$. Suppose $\rho\in\Lambda_{0}(\bar{Q}_{T})$ and
$v^{0}\in\Lambda_{2}(\mathbb{R}^{d})$. Then we have
$v\in\Lambda_{2}(\bar{Q}_{T})$. See [16] for the case $v^{0}=0$, see [23,
Theorem 4] for a general result. If we consider the second equation of (1.1)
with $\alpha>0$, the fundamental solution is just multiplied by a decaying
exponential at infinity and the same result carries over. Therefore, a
sufficient condition in order to have $v$ satisfying (2.3) is $v^{0}\in
W^{2,\infty}(\mathbb{R}^{d})$, because
$W^{2,\infty}(\mathbb{R}^{d})\subset\Lambda_{2}(\mathbb{R}^{d})$. $\square$
This gives a rigorous justification of the assumptions on the initial datum of
Theorem 1.4. However a refined analysis shows that this assumption can be
weakened, as we do next.
#### Initial datum in $\Lambda_{1}(\mathbb{R}^{d})$
We have to consider the weighted Zygmund space $\Lambda_{2}^{-1}(Q_{T})$,
defined as the corresponding space $\Lambda_{2}(\bar{Q}_{T})$, with the
addition of a time weight which is divergent as $t\to 0$. In particular,
locally in $Q_{T}$ functions in $\Lambda_{2}^{-1}(Q_{T})$ have the same
smoothness as the ones in $\Lambda_{2}(\bar{Q}_{T})$, but this regularity does
no more extend to the closure of $Q_{T}$. More precisely, by definition
$f\in\Lambda_{2}^{-1}(Q_{T})$ means that $f\in\Lambda_{1}(\bar{Q}_{T})$,
$\sup_{{x,y\in\mathbb{R}^{d}}\atop{t\in[0,T]}}\sqrt{t}\;\frac{|\nabla
f(x,t)-2\nabla f((x+y)/2,t)+\nabla f(y,t)|}{|x-y|}<+\infty$ (5.2)
and the second finite differences of $f$ and $\nabla f$ with respect to time
verify the corresponding estimates, as in the definition of
$\Lambda_{2}(\bar{Q}_{T})$, still with the addition of the weight $t^{1/2}$.
###### Theorem 5.1.
Let $T>0$. Let $\rho^{0}\in\mathscr{M}_{2}(\mathbb{R}^{d};\mathfrak{m})\cap
L^{\infty}(\mathbb{R}^{d})$ and $v^{0}\in\Lambda_{1}(\mathbb{R}^{d})\cap
W^{1,2}(\mathbb{R}^{d})$. Let $z_{t}=(\rho_{t},v_{t})$ be a bounded solution
on $[0,T]\times\mathbb{R}^{d}$ to the Cauchy problem for (1.1), according to
Definition 1.2, with initial datum $z^{0}=(\rho^{0},v^{0})$. For any reference
point
$\bar{z}=(\bar{\rho},\bar{v})\in(\mathscr{M}_{2}(\mathbb{R}^{d};\mathfrak{m})\cap
L^{\infty}(\mathbb{R}^{d}))\times W^{1,2}(\mathbb{R}^{d})$, the general EVI
holds
$\frac{1}{2}\frac{d}{dt}D^{2}(z_{t},\bar{z})\leq\,\mathscr{F}_{\varepsilon,\alpha}(\bar{z})-\mathscr{F}_{\varepsilon,\alpha}(z_{t})+Ct^{-1/2}\omega(D^{2}(z_{t},\bar{z}))\quad\text{for
$\mathcal{L}^{1}$-a.e. }t\in(0,T),$ (5.3)
for a constant $C$ depending on
$\|\rho\|_{L^{\infty}((0,T)\times{\mathbb{R}^{d}})}$,
$\|v^{0}\|_{\Lambda^{1}(\mathbb{R}^{d})}$,
$\|\bar{\rho}\|_{L^{\infty}(\mathbb{R}^{d})}$.
Moreover the EDE (3.4) holds, and the expansion control property holds in this
form: given another bounded solution $t\mapsto\zeta_{t}$ as above with initial
datum $\zeta^{0}\in(\mathscr{M}_{2}(\mathbb{R}^{d};\mathfrak{m})\cap
L^{\infty}(\mathbb{R}^{d}))\times(\Lambda_{1}(\mathbb{R}^{d})\cap
W^{1,2}(\mathbb{R}^{d}))$ there is
$D^{2}(z_{t},\zeta_{t})\leq
G^{-1}(G(D^{2}(z^{0},\zeta^{0}))+8C\sqrt{t})\quad\text{for every }t\in[0,T),$
(5.4)
where $C$ is a constant depending on
$\|\rho\|_{L^{\infty}((0,T)\times\mathbb{R}^{d})}$ and
$\|v^{0}\|_{\Lambda_{1}(\mathbb{R}^{d})}$ (and the same quantities associated
to $\zeta$). In particular, $z=\zeta$ if $z^{0}=\zeta^{0}$.
###### Proof.
Since we are in the hypotheses of [23, Theorem 4], $v$ belongs to
$\Lambda_{2}^{-1}(Q_{T})$, so that (5.2) above holds for $v$ and then, due to
the log-Lipschitz regularity in the Zygmund class, we deduce
$|\nabla v_{t}(x)-\nabla v_{t}(y)|\leq Kt^{-1/2}|x-y||\log|x-y||,$ (5.5)
for all $x\in\mathbb{R}^{d}$, $t\in(0,T)$, where $K$ is a suitable constant
depending only on $T$ and the data. Notice that from the definition of
$\Lambda_{2}^{-1}(Q_{T})$, it does not follow that $\nabla v\in
L^{\infty}(Q_{T})$. However, in [23, Theorem 4] it is also shown that, still
for $v^{0}\in\Lambda_{1}(\mathbb{R}^{d})$ and $\rho\in L^{\infty}(Q_{T})$, the
solution $v_{t}$ of the parabolic equation also satisfies
$\|\nabla v_{t}(\cdot)\|_{L^{\infty}(\mathbb{R}^{d})}\leq\bar{K}(1+|\log
t|),\quad t\in(0,T),$ (5.6)
again for some positive $\bar{K}$ depending on $\|\rho\|_{L^{\infty}(Q_{T})}$
and $\|v^{0}\|_{\Lambda_{1}(\mathbb{R}^{d})}$. Taking (5.6) into account, it
is clear that (5.5) can be improved (we do not relabel the constant) into
$|\nabla v_{t}(x)-\nabla v_{t}(y)|\leq Kt^{-1/2}|x-y|(1+\log^{-}|x-y|).$
Thus we deduce the weighted analogous of (2.3), that is
$|\nabla v_{t}(x)-\nabla
v_{t}(x)|^{2}\leq\frac{C^{2}}{t}\>\varphi(|x-y|^{2}),$ (5.7)
where $C$ is a new suitable positive constant depending on the data and
$\varphi$ is defined in (2.2). Following the line of the proof Theorem 3.1 we
reach the estimate (3.8) for $II_{t}$, which now has to be changed because we
have to use (5.7), obtaining
$|II_{t}|\leq
Ct^{-1/2}W_{2}(\rho_{t},\bar{\rho})\sqrt{\mathfrak{m}\varphi(\mathfrak{m}^{-1}W_{2}^{2}(\rho_{t},\bar{\rho}))}=Ct^{-1/2}\omega(W_{2}^{2}(\rho_{t},\bar{\rho})).$
We can repeat all the other steps which lead to (3.14), obtaining the
corresponding EVI with the additional weight $t^{-1/2}$, which directly lead
to (5.3). We conclude as in Step 4 of the proof of Theorem 3.1: from (5.3),
the EDE formulation (3.4) follows, still referring to [2, Proposition 3.6].
Moreover, (5.4) follows by (5.3) by
$\frac{d}{dt}D^{2}(z_{t},\zeta_{t})\leq\,4Ct^{-1/2}\omega(D^{2}(z_{t},\zeta_{t})),\quad\text{for
$\mathcal{L}^{1}$-a.e. }t\in(0,T).$
Indeed the inequality
$y(t)\leq y(0)+4C\int_{0}^{t}s^{-1/2}\omega(y(s))\,ds$
implies that $y(t)\leq G^{-1}(G(y(0))+8C\sqrt{t})$ as desired. Finally, the
uniqueness result follows since $G(0)=-\infty$ and $G^{-1}(-\infty)=0$. ∎
## 6\. The case of nonlinear diffusion
We show next how to adapt our techniques to more general aggregation diffusion
equations in a quite straightforward way. Let us consider the problem
$\left\\{\begin{array}[]{rl}\partial_{t}\rho&=\mathrm{div}\,(\rho\nabla
P(\rho))-\,\mathrm{div}\,(\rho\nabla v),\\\\[5.69054pt]
\varepsilon\partial_{t}v&=\Delta v+\rho-\alpha v,\end{array}\right.$ (6.1)
to which we associate the functional
$\mathscr{G}_{\varepsilon,\alpha}(\rho,v):=\int_{\mathbb{R}^{d}}(\Psi(\rho)-v\rho)\,dx+\frac{1}{2}\int_{\mathbb{R}^{d}}(|\nabla
v|^{2}+\alpha v^{2})\,dx,$ (6.2)
for all $\varepsilon>0$, $\alpha\geq 0$,
$\rho\in\mathscr{M}_{2}(\mathbb{R}^{d};\mathfrak{m})\cap
L^{\infty}({\mathbb{R}^{d}})$, $v\in W^{1,2}(\mathbb{R}^{d})$, where
${\Psi(\rho):=\int_{0}^{\rho}P(r)\,dr}$. We give the same restrictions as [3,
§9.3], the first one being
$\lim_{r\to 0}\frac{\Psi(r)}{r^{q}}>-\infty\quad\mbox{for some
}q>\frac{d}{d+2},$
a property ensuring that ${\int_{\mathbb{R}^{d}}\Psi(\rho)\neq-\infty}$.
Moreover, the crucial property to be satisfied by the new nonlinearity is the
displacement convexity, that is the map $r\mapsto r^{d}\Psi(r^{-d})$ is convex
and nondecreasing on $(0,+\infty)$. This notion, introduced in [31], is
stronger than convexity and corresponds for $C^{2}$ functions to the
inequality
$r^{-1}\Psi(r)-\Psi^{\prime}(r)+r\Psi^{\prime\prime}(r)\geq-\,\frac{1}{d-1}\,r\Psi^{\prime\prime}(r)\qquad\forall\;r\in(0,+\infty).$
The more relevant cases correspond to nonlinear diffusion of power kind.
Indeed, if
$\Psi(\rho)=\frac{1}{m-1}\,\rho^{m},\quad m\geq\frac{d-1}{d}$
the displacement convexity property holds. The case $m>1$ (resp. $m<1$)
correspond to a slow diffusion (resp. fast diffusion) in the equation. On the
other hand, the linear diffusion is recovered taking $P(\rho)=\log\rho$, it is
seen that in this case functional (6.2) is reduced, up to a constant, to
(1.3). Finally, let us mention that the free-energy functional in the
parabolic-elliptic case is similar to (1.4) and given by
$\mathscr{G}_{0,\alpha}(\rho,v):=\int_{\mathbb{R}^{d}}(\Psi(\rho)-\frac{1}{2}v\rho)\,dx,$
(6.3)
for $\rho\in\mathscr{M}_{2}(\mathbb{R}^{d};\mathfrak{m})\cap
L^{\infty}(\mathbb{R}^{d})$ and $v=\mathcal{B}_{\alpha,d}\ast\rho$. It can be
written as (6.2), taking into account the same renormalization as in (3.1), to
be done in the pathological cases $\varepsilon=\alpha=0$ and $d=1,2$.
The notion of bounded solution is completely analogous to Definitions 1.1 and
1.2, both for the parabolic-elliptic and the parabolic-parabolic case. Indeed,
the only point to adapt is the finiteness of the Fisher information, now
rewritten into the generalized version
$\int_{0}^{T}\int_{\mathbb{R}^{d}}|\nabla
P(\rho_{t}(x))|^{2}\,\rho_{t}(x)\,dx\,dt<+\infty.$ (6.4)
###### Corollary 6.1.
Theorem 1.3, Theorem 1.4, Theorem 3.1 and Theorem 5.1 hold for bounded
solutions to (6.1).
###### Proof.
The displacement convexity property makes the internal energy functional
$\rho\in\mathscr{M}_{2}(\mathbb{R}^{d};\mathfrak{m})\mapsto\int_{\mathbb{R}^{d}}\Psi(\rho(x))\,dx$
convex along Wasserstein geodesics, as shown in [3, §9.3]. This in turn gives
the possibility to write down a subdifferential inequality in Wasserstein
sense (for a definition see [3, §10.1.1]) as follows. Let
$\rho\in\mathscr{M}_{2}(\mathbb{R}^{d};\mathfrak{m})\cap
L^{\infty}(\mathbb{R}^{d})$ be such that $\int_{\mathbb{R}^{d}}|\nabla
P(\rho)|^{2}\,\rho\,dx$ is finite. Then
$\int_{\mathbb{R}^{d}}\Psi(\bar{\rho}(x))\,dx-\int_{\mathbb{R}^{d}}\Psi(\rho(x))\,dx\geq\int_{\mathbb{R^{d}}}\langle\nabla
P(\rho(x)),\mathcal{T}(x)-x\rangle\,\rho(x)\,dx,$ (6.5)
for any $\bar{\rho}\in\mathscr{M}_{2}(\mathbb{R}^{d};\mathfrak{m})$, where
$\mathcal{T}$ is the optimal transport map from $\rho$ to $\bar{\rho}$.
Convexity and differentiability of functionals defined on probability
densities, as the internal energy, are standard elements in the theory of
Wasserstein gradient flows. For the proof of inequality (6.5), which
characterizes the vector $\nabla P(\rho)$ as the Wasserstein subdifferential
of the internal energy functional, we refer to [2, §3.3.1] or to the general
theory in [3, §10.4.3].
On the other hand, (6.5) can be used to generalize the proof of Theorem 3.1.
Indeed, if $(\rho_{t},v_{t})$ solves (6.1) according to our notion of
solution, thanks to (6.4) $\rho_{t}$ satisfies the identity (6.5) for almost
any $t$. From this inequality, all the rest of the proof of Theorem 3.1 can be
carried out. Indeed, with the same notation therein, we obtain the
$\mathcal{L}^{1}$-a.e. $t\in(0,T)$ inequality
$\displaystyle I_{t}:=\mathscr{G}_{\varepsilon,\alpha}(\bar{\rho},\bar{v})$
$\displaystyle-\mathscr{G}_{\varepsilon,\alpha}(\rho_{t},v_{t})+\int_{\mathbb{R}^{d}}(\bar{v}(x)-v_{t}(x))\bar{\rho}(x)\,dx\geq$
$\displaystyle\int_{\mathbb{R}^{d}}\langle\rho_{t}(x)\nabla
P(\rho_{t}(x))-\rho_{t}(x)\nabla
v(x),\mathcal{T}_{t}(x)-x\rangle\,dx-C\omega(W_{2}^{2}(\rho_{t},\bar{\rho})),$
for any $\bar{\rho}\in\mathscr{M}_{2}(\mathbb{R}^{d};\mathfrak{m})\cap
L^{\infty}(\mathbb{R}^{d})$ and any $\bar{v}\in W^{1,2}(\mathbb{R}^{d})$ if
$\varepsilon>0$ or $\bar{v}=\mathcal{B}_{\alpha,d}\ast\bar{\rho}$ if
$\varepsilon=0$. This estimate substitutes (3.9) in the proof of Theorem 3.1.
The rest of the proofs is completely analogous. ∎
### Acknowledgements
The authors would like to thank Paolo Acquistapace and Lucilla Corrias for
several discussions about this work. JAC acknowledges support from the project
MTM2011-27739-C04-02 DGI (Spain) and 2009-SGR-345 from AGAUR-Generalitat de
Catalunya. JAC acknowledges support from the Royal Society through a Wolfson
Research Merit Award. This work was partially supported by Engineering and
Physical Sciences Research Council grant number EP/K008404/1. SL and EM has
been partially supported by the INDAM-GNAMPA project 2011 ”Measure solution of
differential equations of drift-diffusion, interactions and of Cahn-Hilliard
type”. EM has been partially supported by a postdoctoral scholarship of the
Fondation Mathématique Jacques Hadamard, he acknowledges hospitality from
Paris-Sud University.
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|
arxiv-papers
| 2012-12-06T08:28:21 |
2024-09-04T02:49:38.941954
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "J. A. Carrillo, S. Lisini, E. Mainini",
"submitter": "Jose A. Carrillo",
"url": "https://arxiv.org/abs/1212.1255"
}
|
1212.1290
|
# Indirect dark matter search with the ANTARES neutrino telescope
IFIC - Instituto de Fìsica Corpuscular - Edificio Institutos de Investigación,
Apartado de Correos 22085 E-46071 Valencia, Spain
E-mail on behalf of the ANTARES collaboration
###### Abstract:
Using the data recorded by the ANTARES neutrino telescope during 2007 and
2008, a search for high energy neutrinos coming from the direction of the Sun
has been performed. The neutrino selection criteria have been chosen so as to
maximize the rejection of the atmospheric background with respect to possible
signals produced by the self-annihilation of weakly interactive massive
particles accumulated in the centre of the Sun. After data unblinding, the
number of neutrinos observed was found to be compatible with background
expectations. The results obtained were compared to the fluxes predicted by
the Constrained Minimal Supersymmetric Standard Model, and 90% upper limits
for this model were obtained. Our limits are competitive with those obtained
by other neutrino telescopes such as IceCube and SuperKamiokande, which give
ANTARES limits for the spin-dependent WIMP-proton cross-section that are more
stringent than those obtained by direct search experiments.
## 1 Introduction
There is compelling evidence that $83$% of the matter in the Universe is in
the form of non-baryonic and non-relativistic matter that does interact weakly
with the ordinary matter, the so-called dark matter. Much of this evidence
comes from its gravitational effects on the motion of galaxies, clusters of
galaxies and from the large scale structure of the Universe. The existence of
dark matter is a key component of our present standard cosmological model, and
results from the study of the CMB anisotropies and gravitational lensing in
galaxy clusters further support its existence (for a review of the evidence,
candidates and constraints, see [1] and [2]).
One of the most favoured hypothesis is that dark matter is made of weakly
interacting massive particles (WIMPs) that are embedded in the visible,
baryonic part of the galaxies and surround them in the form of a halo. There
are a variety of candidates for WIMPs, among which those provided by theories
based on supersymmetry (SUSY) or universal extra dimensions (UED) attract a
great deal of interest. In some classes of SUSY, the lightest particle is
stable due to the conservation of R-parity that forbids its decay to standard
particles, making it a good candidate as a dark matter WIMP.
In addition to its gravitational effects, the search for evidence of the
existence of WIMPs is at present performed, on the one hand, by looking for
the recoiling products of the elastic scattering of dark matter particles off
normal baryonic matter in suitable detectors, the so-called direct searches,
and on the other hand, indirectly by the observation of the final products of
the possible annihilation of WIMPs that have accumulated in astrophysical
objects. WIMPs can scatter elastically in the Sun or the Earth and become
trapped in their gravitational potential wells, accumulating in sizeable
numbers over the age of the solar system, therefore a very wide region in the
Galaxy must have contributed, reducing the dependence on the detailed
structure of the dark matter halo. The WIMPs proposed by SUSY are Majorana
fermions that can self-annihilate, giving rise to standard particles.
Neutrinos will be at the end of the decay chain of the products of the WIMP
self-annihilation, they can escape these astrophysical objects and be detected
by neutrino telescopes on the Earth.
In this paper, the indirect search for dark matter looking for high energy
neutrinos coming from the Sun using the data recorded by the ANTARES neutrino
telescope in 2007 and 2008 is reported. The number of neutrinos observed is
compared to the neutrino fluxes predicted by the Constrained Minimal
Supersymmetric Standard Model (CMSSM) [3], a minimal supersymmetric extension
of the Standard Model with supersymmetry-breaking scalar and gaugino masses
constrained to be universal at the GUT scale.
The layout of the paper is as follows. In Section 2, the main features of the
ANTARES neutrino telescope and the reconstruction algorithm used in this work
are reviewed. In Section 3, the Monte Carlo simulations of the signal,
expected from the investigated WIMP models, and the background, expected from
atmospheric muons and neutrinos, are described. In Section 4, the method used
to optimise the criteria to select the sample of neutrino events and reduce
the background following a blind data strategy is presented. Finally, the
results obtained are discussed in Section 5, where limits on the models
investigated are imposed from the absence of a dark matter signal.
## 2 The ANTARES neutrino telescope
ANTARES is the first undersea neutrino telescope and the largest of its kind
in the Northern Hemisphere [4]. It is located at $2475$ m depth in the
Mediterranean Sea, $40$ km offshore from Toulon (France) at $42^{\circ}48$’ N
and $6^{\circ}10$’ E.
The telescope consists of $12$ mooring detection lines made up of 25 storeys
each. The standard storey is composed of a local control module that contains
the front-end and slow-control electronics and three optical modules (OMs)
that house a 10-inch photomultiplier. The OMs are looking 45∘ downwards in
order to optimise their acceptance to upgoing light and to avoid the effect of
sedimentation and biofouling. The length of the lines is 450 m and the
horizontal distance between neighbouring lines is 60-75 m. The absolute time
accuracy is ensured at the millisecond level by the UTC time provided by the
GPS system connected to the clock system of the detector. The relative time
between the elements of the detector is achieved by calibration of the lines
in the laboratory with short light pulses, by the use of a 25 MHz clock system
and by the operation of a series of optical beacons distributed along the
lines that emit short light pulses through the water. A relative timing of the
order of one nanosecond is reached. Additional information on the detector can
be found in reference [4].
Neutrinos are detected through the products of their interaction with the
matter inside or close to the detector. In the channel of neutrino observation
used in this work, a high energy neutrino interacts in the rock below the
detector producing a relativistic muon that can travel hundreds of meters and
cross the detector or pass nearby. This muon induces Cherenkov light when
travelling through the water, which is detected by the OMs. From the time and
position information of the hits in the OMs, the direction of the muon – which
at high energy is essentially that of the neutrino – can be reconstructed.
Data-taking started in 2007 when only 5 lines of the detector where installed.
The full detector was connected in May 2008 and has been operating ever since,
except for some periods in which repair and maintenance operations have taken
place. For this work the data taken during 2007 and 2008 has been used.
Results of other searches using this data-taking period can be found elsewhere
[5, 6, 7].
The reconstruction of the track from the position and time of the hits of the
Cherenkov photons in the OMs is a key ingredient of the physics analysis. In
this work, a dedicated fast algorithm to reconstruct the muon track is used
[8]. The algorithm is based on the minimization of a $\chi^{2}$-like quality
parameter, $Q$, that uses the differences of the expected and actual times of
the detected photons corrected by the effect of light absorption in water.
Monte Carlo simulations indicate that selecting tracks with a quality
parameter per degree of freedom of the fit smaller than $1.4$ results in a
purity of $90$% for upward reconstructed multi-line atmospheric neutrinos, the
contamination being misreconstructed downgoing atmospheric muon events, with
an angular resolution of about few degrees at energies of tens of GeV, driven
at $\sim 40$% by the kinematic of the neutrinos in low energy regime.
## 3 Signal and background simulation
The number of muon neutrinos as a function of their energy arriving at the
Earth’s surface from the Sun’s core, $dN_{\nu}/dE_{\nu}$, is computed using
the software package WimpSim [9]. The usual self-annihilation channels
($q\bar{q}$, $l\bar{l}$, $WW$, $ZZ$, Higgs doublets $\phi\phi^{*}$ and
$\nu\bar{\nu}$) were simulated for $17$ different WIMP masses from $10$ GeV to
$10$ TeV. Oscillations between the three neutrino flavours both in the Sun and
during their flight to Earth as well as $\nu$ absorption and $\tau$ lepton
regeneration in the Sun are taken into account.
For the CMSSM, three main self-annihilation channels were chosen for the
lightest neutralino, $\tilde{\chi}_{1}^{0}$, namely: a soft neutrino channel,
$\tilde{\chi}_{1}^{0}\tilde{\chi}_{1}^{0}\rightarrow b\bar{b}$, and two hard
neutrino channels, $\tilde{\chi}_{1}^{0}\tilde{\chi}_{1}^{0}\rightarrow
W^{+}W^{-}$ and
$\tilde{\chi}_{1}^{0}\tilde{\chi}_{1}^{0}\rightarrow\tau^{+}\tau^{-}$. Since
which of these three channels is dominant depends on the region of the CMSSM
parameter space being analysed [10], a 100% branching ratio was assumed for
all of them in order to explore them on an equal footing.
The main backgrounds for this search are muons and neutrinos produced in the
interaction of cosmic rays with the atmosphere. Downgoing atmospheric muons
dominate the trigger rate, which ranges from $3$ to$10$ Hz depending on the
exact trigger conditions. They are simulated using Corsika [11]. Upgoing
atmospheric neutrinos, which are recorded at a rate of $\sim$1 mHz (about four
per day) [4], are simulated according to the parametrisation of the
atmospheric $\nu_{\mu}$ flux from [12] in the energy range from $10$ GeV to
$10$ PeV.
The Cherenkov light produced in the vicinity of the detector is propagated
taking into account light absorption and scattering in sea water. The angular
acceptance, quantum efficiency and other characteristics of the PMTs are taken
from [13] and the overall geometry corresponded to the layout of the ANTARES
detector [4] according to the data taking periods (from $5$ to $12$ line
configurations).
## 4 Optimisation of the event selection criteria
The data set used in this analysis comprises a total of $2,693$ runs recorded
between the $27^{th}$ of January 2007 and the $31^{st}$ of December 2008,
corresponding to a total livetime of $\sim$294.6 days. The detector consisted
of $5$ lines for most of $2007$ and of $9$, $10$ and $12$ lines for $2008$.
Only upgoing events are kept in the analysis. The muon tracks are required to
have cos$\,\theta<$ 0.9998 in order to exclude those for which the fit stopped
at the boundary. The fit is required to use a number of hits greater than five
in at least two lines in order to ensure a non-degenerate 5-parameter fit with
a proper reconstruction of the azimuth angle.
The UTC time of the events is uniformly randomised on the period of the data
taking in order to estimate the background in the Sun’s direction from the
data itself. The local coordinates ($\theta$, $\phi$) are kept so as to
preserve the detector geometry in the optimisation of the selection criteria.
This procedure provides a means to follow a data blinding strategy while using
all the relevant information on the detector’s performance.
The values used in the event selection criteria for the quality parameter,
$Q$, and for the aperture of the search cone in the Sun’s direction, $\Psi$,
are optimised following the model rejection factor (MRF) technique [14]. For
each WIMP mass and each annihilation channel, the values of $Q$ and $\Psi$
used are those that optimised the average upper limit on the
$\nu_{\mu}+\bar{\nu}_{\mu}$ flux, $\bar{\phi}_{\nu_{\mu}+\bar{\nu}_{\mu}}$, as
defined by:
$\bar{\phi}_{\nu_{\mu}+\bar{\nu}_{\mu}}=\frac{\bar{\mu}^{90\%}}{\sum_{i}A_{eff}^{i}(M_{\rm
WIMP})\times T_{eff}^{i}}\,,$ (1)
where the index $i$ denotes the different periods in the detector
infrastructure (5, 9, 10 and 12 detection lines), $\bar{\mu}^{90\%}$ is the
average upper limit at $90$% confidence level (CL) computed from the time-
scrambled data set and using a Poisson distribution in the Feldman-Cousins
approach [15]; $T_{eff}^{i}$ is the livetime for each detector configuration
in $2007$-$2008$, namely: $\sim$134.6, $\sim$38, $\sim$39 and $\sim$83 days
for $5$, $9$, $10$ and $12$ lines respectively. The effective area averaged
over the neutrino energy, $A_{eff}(M_{\rm WIMP})^{i}$, is defined as (the
index $i$ is implied):
$A_{eff}(M_{\rm
WIMP})=\sum_{\nu,\bar{\nu}}\left(\frac{\int_{E_{\nu}^{th}}^{M_{\rm
WIMP}}A_{eff}(E_{\nu,\bar{\nu}})\,\frac{dN_{\nu,\bar{\nu}}}{dE_{\nu,\bar{\nu}}}dE_{\nu,\bar{\nu}}}{\int_{0}^{M_{\rm
WIMP}}\frac{dN_{\nu}}{dE_{\nu}}dE_{\nu}\,+\,\frac{dN_{\bar{\nu}}}{dE_{\bar{\nu}}}dE_{\bar{\nu}}}\right)\,,$
(2)
where $E_{\nu}^{th}=10$ GeV is the energy threshold for neutrino detection in
ANTARES, $M_{\rm WIMP}$ is the WIMP mass,
$dN_{\nu,\bar{\nu}}/dE_{\nu,\bar{\nu}}$ is the energy spectrum of the
neutrinos or the anti-neutrinos at the surface of the Earth for the three
channels of interest in this analysis, and $A_{eff}(E_{\nu,\bar{\nu}})$ is the
effective area of ANTARES as a function of the neutrino or anti-neutrino
energy. Due to their different cross-sections these two effective areas are
slightly different and therefore are studied separately.
Figure 1: Left: Distribution of the spatial angle $\Psi\in$[$0^{o}$,$20^{o}$]
of the event tracks with respect to the Sun’s diretion for the expected
background computed from the time-scrambled data (solid blue line) compared to
the data after the basic selection criteria (black triangles) . A $1\sigma$
Poisson uncertainty is shown for each data point (black cross). Right: $90$%
CL upper limit on the muon flux as a function of the WIMP mass in the range
$M_{\rm WIMP}\in$[$50$ GeV;$10$ TeV] for the three channels $b\bar{b}$
(green), $W^{+}W^{-}$ (blue) and $\tau^{+}\tau^{-}$ (red, ANTARES only). The
results from SuperKamiokande $1996-2008$ [17] (dotted lines) and IceCube-$40$
plus AMANDA $2001$-$2008$ [18] (dashed lines) are also shown.
## 5 Results and discussion
Once the optimised values of $Q$ and $\Psi$ were obtained using the time-
scrambled data, the data sample is unblinded. Figure 1 shows the distribution
of the spatial angle between the tracks of the events and the Sun’s direction
obtained after applying the basic selection criteria on the zenith angle and
the minimum number of hits and lines. A total of $27$ events were found within
a $20$ degrees spatial angle and no excess in the Sun’s direction above the
scrambled background are observed.
Using the values for the cuts obtained in the optimisation procedure, limits
on the $\nu_{\mu}+\bar{\nu}_{\mu}$ flux, $\phi_{\nu_{\mu}+\bar{\nu}_{\mu}}$,
can be extracted from the data according to:
$\phi_{\nu_{\mu}+\bar{\nu}_{\mu}}=\frac{\mu^{90\%}}{\sum_{i}A_{eff}^{i}(M_{\rm
WIMP})\times T_{eff}^{i}}\,,$ (3)
where $\mu^{90\%}$ is the upper limit at $90$% CL on the number of observed
events and the rest of variables have the same meaning as in Eq. 1.
The corresponding limits for muons are calculated using a conversion factor
between the neutrino and the muon fluxes
($\phi_{\mu}=\Gamma_{\nu\rightarrow\mu}\times\phi_{\nu_{\mu}+\bar{\nu}_{\mu}}$)
computed using DarkSusy [16]. Figure 1 (on the right) shows the 90% CL muon
flux limits $\phi_{\mu}$ for the channels $b\bar{b}$, $W^{+}W^{-}$ and
$\tau^{+}\tau^{-}$. The latest results from SuperKamiokande [17] and
IceCube-$40$ plus AMANDA [18] are also shown for comparison. Despite its
smaller detector volume, the ANTARES limits both in the soft and hard channels
are similar to those obtained by IceCube40 and AMANDA in the mass range
$M_{\rm WIMP}\in$[$50$;$100$] GeV. In this mass range most of the sensitivity
of the South Pole telescopes comes from AMANDA-II, whose effective area and
energy threshold were similar to those of ANTARES in its 12-line
configuration. Compared to the limits set by SuperKamiokande, those of ANTARES
are more stringent in the high mass region $M_{\rm WIMP}\,>\,150$ GeV.
Systematic uncertainties have been taken into account and included in the
evaluation of the limits using the approach of reference [19] by means of the
Pole software. The total systematic uncertainty on the detector efficiency is
around 20% and comes mainly from the efficiency and time resolution
uncertainties of the OMs, the total angular resolution and the absolute
pointing accuracy. This uncertainty translates into an increase of the upper
limit between 3% and 6% depending on the WIMP mass.
Figure 2: $90$% CL upper limits on the SD and SI WIMP-proton cross-sections
(left and right hand respectively) as a function of the WIMP mass in the range
$M_{\rm WIMP}\in$[$50$ GeV;$10$ TeV], for the three channels: $b\bar{b}$
(green), $W^{+}W^{-}$ (blue) and $\tau^{+}\tau^{-}$ (red), for ANTARES (solid
line) compared to the results of other indirect search experiments:
SuperKamiokande $1996-2008$ [17] (dotted lines) ,IceCube-$40$ plus AMANDA
$2001$-$2008$ [18] (dashed lines) and Baksan $1978-2009$ [22] (colored dot-
dashed line) and the result of the most stringent direct search experiments
(black): PICASSO $2007-2012$ [23] (three-dots-dashed line), KIMS $2011-2012$
[24] (two-dots-dashed line), SIMPLE $2004-2011$ [25] (one-dot-dashed line),
COUPP $2010-2011$ [26] (one-dot-large-dashed line) and XENON100 $2011-2012$
[27] (dashed line). For a comparison to the theoretical expectations, grid
scans corresponding to the CMSSM model have been added.
Assuming equilibrium between the WIMP capture and self-annihilation rates in
the Sun, the limits on the spin-dependent (SD) and the spin-independent (SI)
WIMP-proton scattering cross-sections can be obtained for the case in which
one or the other is dominant.
A conservative approach to the dark matter local halo was considered assuming
a gravitational effect of Jupiter on the Sun’s capture rate, which reduces
about $13$%($87$%) the SD capture and about $1$% ($12$%) the SI capture for
$1$($10$) TeV WIMPs. A local dark matter density of about $0.3$ GeV.cm-3 was
assumed. No additional dark matter disk that could enhance the local dark
matter density was considered (see [20] for a discussion).
The $90$% CL limits for the SD and SI WIMP-proton cross-sections extracted
from the channels $b\bar{b}$, $W^{+}W^{-}$ and $\tau^{+}\tau^{-}$ are
presented in Figure 2. The latest results from SuperKamiokande,
IceCube-$40$+AMANDA and Baksan together with the latest and the most stringent
limits from the direct search experiments PICASSO, KIMS, SIMPLE, COUPP and
XENON100 are also shown. The allowed parameter space from CMSSM model
according to the results from an adaptative grid scan performed with DarkSUSY
[16] are also shown. For a proper comparison all the limits presented in
Figure 2 are computed with a muon energy threshold at $E_{\mu}=1$ GeV. For
these figures, the shaded regions show a grid scan of the model parameter
space taking into account the last limits for the Higgs boson mass from ATLAS
and CMS merged together such as $M_{H}=125\pm 2GeV$ [21]. A relatively large
constraint on the neutralino relic density $0<\Omega_{CDM}h^{2}<0.1232$ is
used not to enclose the studied dark matter particle to only one possible
nature.
The neutrino flux due to WIMP annihilation in the Sun is highly dependent on
the capture rate of WIMPs in the core of the Sun, which in turn is dominated
by the SD WIMP-proton cross-section. This makes these indirect searches better
compared to direct search experiments such as KIMS and COUPP. This is not the
case for the SI WIMP-proton cross-section, where the limits coming from direct
search experiments such as CDMS and XENON100 are better thanks to their target
materials. Therefore, there is a sort of complementarity between both types of
searches.
## Acknowledgments
The authors acknowledge the financial support of the funding agencies: Centre
National de la Recherche Scientifique (CNRS), Commissariat à l’énegie atomique
et aux énergies alternatives (CEA), Agence National de la Recherche (ANR),
Commission Européenne (FEDER fund and Marie Curie Program), Région Alsace
(contrat CPER), Région Provence-Alpes-Côte d’Azur, Département du Var and
Ville de La Seyne-sur-Mer, France; Bundesministerium für Bildung und Forschung
(BMBF), Germany; Istituto Nazionale di Fisica Nucleare (INFN), Italy;
Stichting voor Fundamenteel Onderzoek der Materie (FOM), Nederlandse
organisatie voor Wetenschappelijk Onderzoek (NWO), the Netherlands; Council of
the President of the Russian Federation for young scientists and leading
scientific schools supporting grants, Russia; National Authority for
Scientific Research (ANCS), Romania; Ministerio de Ciencia e Innovación
(MICINN), Prometeo of Generalitat Valenciana and MultiDark, Spain; Agence de
l’Oriental and CNRST, Morocco. We also acknowledge the technical support of
Ifremer, AIM and Foselev Marine for the sea operation and the CC-IN2P3 for the
computing facilities.
## References
* [1] G. Bertone, D. Hooper, J. Silk, Phys.Rept., 2005, 405: pp. 279-390.
* [2] Particle Data Group, J. Phys. G 37, 070521 (2010). http://pdg.lbl.gov/2011/reviews/rpp2011-rev-dark-matter.pdf.
* [3] J. Ellis, K.A. Olive, C. Savage, and V.C. Spanos, Phys. Rev. D 81, 085004 (2010).
* [4] M. Ageron et al., ANTARES Collaboration, Nucl. Inst. and Meth. in Phys. Res. A 656 (2011) 11-38 [astro-ph/1104.1607].
* [5] J.A. Aguilar, ANTARES Collaboration, Phys. Lett, B696 (2011) 16.
* [6] S. Adrián-Martínez et al., ANTARES Collaboration, Ap. J. Letter 743 (2011) L14.
* [7] S. Adrián-Martínez et al., ANTARES Collaboration, Astropart. Phys. 35 (2012) 634.
* [8] J.A. Aguilar, ANTARES Collaboration, Astropart. Phys. 34 (2011) 652.
* [9] J. Edsjö, http://www.physto.se/ edsjo/wimpsim/.
* [10] S. Desai et al., Phys. Rev. D 70, 083523 (2004).
* [11] D. Heck et al., Report FZKA 6019 (1998), Forschungszentrum Karlsruhe; D. Heck and J. Knapp, Report FZKA 6097 (1998), Forschungszentrum Karlsruhe.
* [12] G. Barr et al., Phys. Rev. D 39 (1989) 3532; V. Agrawal et al., Phys. Rev. D 53 (1996) 1314.
* [13] P. Amram et al., [ANTARES Collaboration], Nucl. Instrum. Meth. A 484 (2002) 369.
* [14] G.C. Hill, K. Rawlins, Astropart. Phys., 2003, 19: pp. 393-402.
* [15] G.J. Feldman, R.D. Cousins, Phys. Rev., 1998, D 57: pp. 3873-3889.
* [16] P. Gondolo et al., J. Cosm. and Astropart. Phys., JCAP07, 008 (2004).
* [17] T. Tanaka et al., Astrophys. J. 742, 78 (2011).
* [18] R. Abbasi et al., Phys. Rev. D 85, 042002 (2012).
* [19] F. Tegenfeldt, J. Conrad A NIM A 539 (2005) 407-413; J. Conrad et al., Phys. Rev. D 67 (2003) 012002; J. Conrad (2006) [astro-ph/0612082v1].
* [20] G. Wikström and J. Edsjö, J. Cosm. and Astropart. Phys., JCAP04, 009 (2009).
* [21] O. Buchmeller et al. [hep-ph/1207.7315v1].
* [22] O. Suvorova et al., in proceedings of Dark Side of the Universe, PoS(DSU2012)042 [astro-ph/1211.2545].
* [23] S. Archambault et al., Phys. Lett. B 711 (2012) 153-161 [astro-ph/1202.1240].
* [24] H. S. Lee et al., Phys. Rev. Lett. 108, 181301 (2012) [astro-ph/1204.2646].
* [25] M. Felizardo et al., Phys.Rev.Lett. 108, 201302 (2012) [astro-ph/1106.3014].
* [26] E. Behnke et al., Phys.Rev. D 86, 052001 (2012) [astro-ph/1204.3094].
* [27] E. Aprile et al. [astro-ph/1207.5988].
|
arxiv-papers
| 2012-12-06T11:09:42 |
2024-09-04T02:49:38.953781
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Guillaume Lambard",
"submitter": "Guillaume Lambard",
"url": "https://arxiv.org/abs/1212.1290"
}
|
1212.1304
|
Hard two photon processes $\gamma\gamma\to{\overline{M}}_{2}M_{1}$ in QCD
Victor L. Chernyak
(e-mail: [email protected])
Budker Institut of Nuclear Physics SB RAS and Novosibirsk State University,
630090 Novosibirsk, Russia
Talk given at the Taipei International Workshop
"QCD in two photon processes"
2 - 4 October 2012, Taipei, Taiwan
Abstract
A short review of leading term QCD predictions vs those of the handbag model
for large angle cross sections $\gamma\gamma\to{\overline{P}}_{2}P_{1}$ ($P$
is the pseudoscalar meson $\pi^{\pm,\,o},\,K^{\pm,\,o},\,\eta$), and for
$\gamma\gamma\to{\overline{V}}_{2}V_{1}$ ($V$ is the neutral vector meson
$\rho^{o},\,\omega,\,\phi$), in comparison with Belle Collaboration measuments
Leading term QCD predictions
The general approach to calculations of hard exclusive processes in QCD was
developed in [1] and [2] (the operator expansions and resummation of Feynman
diagrams in the covariant perturbation theory), and in [3] (the resummation of
Feynman diagrams in the non-covariant light front perturbation theory in the
special axial gauge and in the basis of free on mass-shell quarks and gluons).
The review is [4].
In particular, the calculation of the large angle scattering amplitudes
$\gamma\gamma\to M_{2}M_{1}$ was considered in [5] (for symmetric meson wave
functions only, $\phi_{M}(x)=\phi_{M}(1-x)$) and [6] (for general wave
functions), see Fig.1.
Fig.1 Two typical lowest order Feynman diagrams for the leading term hard QCD
contributions to the large angle amplitude
$\gamma\gamma\rightarrow{\overline{M}}_{2}M_{1}$ (the dashed line is the hard
gluon exchange).
The leading term QCD expressions for cross sections look as (the example is
given for $\gamma\gamma\to K^{+}K^{-}$) :
$\frac{d\sigma(\gamma\gamma\rightarrow
M^{\dagger}M)}{d\cos\theta}=\frac{1}{32\pi
W^{2}}\,\frac{1}{4}\sum_{\lambda_{1},\lambda_{2}=\pm
1}\,\,\Bigl{|}M_{\lambda_{1}\lambda_{2}}\Bigr{|}^{2}\,,$
$M^{(lead)}_{\lambda_{1}\lambda_{2}}(W,\theta)=\frac{64\pi^{2}}{9W^{2}}\,\alpha\,{\overline{\alpha}}_{s}\,f_{P}^{2}\int_{0}^{1}dx_{s}\,\phi_{P}(x_{s})\int_{0}^{1}dy_{s}\,\phi_{P}(y_{s})\,T_{\lambda_{1}\lambda_{2}}(x_{s},\,y_{s},\,\theta)\,,$
$T_{++}=T_{--}=(e_{u}-e_{s})^{2}\,\frac{1}{\sin^{2}\theta}\,\frac{A}{D}\,,\quad
T_{+-}=T_{-+}\,,$
$T_{+-}=\frac{1}{D}\Biggl{[}\frac{(e_{u}-e_{s})^{2}}{\sin^{2}\theta}(1-A)+e_{u}e_{s}\frac{AC}{A^{2}-B^{2}\cos^{2}\theta}+\frac{(e_{u}^{2}-e_{s}^{2})}{2}(x_{u}-y_{s})\Biggr{]},\hskip
56.9055pt(1)$
$A=(x_{s}y_{u}+x_{u}y_{s}),\,\,B=(x_{s}y_{u}-x_{u}y_{s}),\,\,C=(x_{s}x_{u}+y_{s}y_{u}),\,\,D=x_{u}x_{s}y_{u}y_{s}\,,$
where: $x$ is the meson momentum fraction carried by quark inside the meson,
$x_{s}+x_{u}=1\,,\quad e_{u}=2/3,\quad e_{s}=e_{d}=-1/3\,,\,\,\phi_{P}(x)$ is
the leading twist pseudoscalar meson wave function (= distribution amplitude),
$f_{P}$ are the decay constants : $f_{\pi}\simeq 131\,{\rm
MeV}\,,\,\,f_{K}\simeq 161\,{\rm MeV}.$
The leading contribution to $d\sigma(\gamma\gamma\to\pi^{+}\pi^{-})$ can be
written as :
$\frac{s^{3}}{16\pi\alpha^{2}}\,\frac{d\sigma(\gamma\gamma\rightarrow\pi^{+}\pi^{-})}{d|\cos\theta|}\equiv\frac{|\Phi^{(eff)}_{\pi}(s,\theta)|^{2}}{\sin^{4}\theta}=\frac{|sF_{\pi}^{(lead)}(s)|^{2}}{\sin^{4}\theta}|1-\upsilon(\theta)|^{2},\hskip
85.35826pt(2)$
where $F_{\pi}^{(lead)}(s)$ is the leading term of the pion form factor :
$|sF^{(lead)}_{\pi}(s)|=\frac{8\pi\,{\overline{\alpha}}_{s}}{9}\,\Bigl{|}f_{\pi}\int_{0}^{1}\frac{dx}{x}\,\phi_{\pi}(x,\,{\overline{\mu}})\Bigr{|}^{2}\,,\hskip
213.39566pt(3)$
and $\upsilon(\theta)$ is due to the term $\sim AC$ in (1). We will compare
below the predictions of two frequently used models for
$\phi_{\pi}(x):\phi^{asy}(x)=6x(1-x)$ and
$\phi_{\pi}^{CZ}(x,\mu_{o})=30x(1-x)(2x-1)^{2},\,\mu_{o}\sim 1{\rm GeV}$ [7].
While the numerical value of $|sF^{(lead)}_{\pi}(s)|$ is highly sensitive to
the form of $\phi_{\pi}(x,\overline{\mu})$, the function $\upsilon(\theta)$ is
only weakly dependent of $\theta$ at $|\cos\theta|<0.6$ and, as emphasized in
[5], is weakly sensitive to the form of $\phi_{\pi}(x,\overline{\mu})$. For
the above two very different pion wave functions,
${\overline{\upsilon}}(\theta)\simeq 0.12$.
Red line : the asymptotic wave function $\phi_{\pi}^{\rm asy}(x)=6x(1-x)$
Blue line : the CZ wave function (at the low scale normalization point
$\mu_{o}\sim 1\,GeV$)
$\phi^{\rm CZ}_{\pi}(x,\mu_{o})=30x(1-x)(2x-1)^{2}$
Black line : the flat wave function $\phi_{\pi}(x,\mu_{o})=1$
$\int_{0}^{1}dx\,\phi_{\pi}(x,\mu/\Lambda_{QCD})=1$
Fig.2 Different models for the leading twist
pion wave function
Fig.3 The shape of the quarkonium model wave function $\phi(x,{\rm v}^{2})$.
Red line: ${\rm v}^{2}=1$ \- massless quarks (asymptotic). Blue line: ${\rm
v}^{2}=0.3$ \- charmonium. Black line: ${\rm v}^{2}=0.1$ \- bottomonium
$\int_{0}^{1}dx\phi_{\pi}(x,{\rm v}^{2})=1$.
The heavier is quark
the narrower is wave function
Red line : the model pion wave function
$\hskip 42.67912pt\phi_{\pi}(x)=30\,x_{d}x_{u}(x_{d}-x_{u})^{2}$
Blue line : the model kaon wave function
$\phi_{K}(x)=30\,x_{s}x_{u}\Bigl{[}0.6(x_{s}-x_{u})^{2}+0.08+0.1(x_{s}-x_{u})\Bigr{]}$
Fig.4 The kaon wave function is somewhat narrower than the pion one and
asymmetric :
the s-quark in the K-meson carries a larger part of the momentum fraction than
the u-quark
Therefore, the leading term QCD predictions for charged mesons
$\pi^{+}\pi^{-}$ and $K^{+}K^{-}$ look as
$d\sigma/d\cos\theta\sim{\overline{\alpha}}^{\,2}_{s}/(W^{6}\sin^{4}\theta)$.
The recent data from Belle [8] agree with the $\sim 1/\sin^{4}\theta$
dependence at $W\geq 3\,GeV$, while the angular distribution is somewhat
steeper at lower energies. The energy dependence at $2.4\,{\rm
GeV}<W<4.1\,{\rm GeV}$ was fitted in [8] as:
$\sigma_{o}(\pi^{+}\pi^{-},|\cos\theta|<0.6)=\int_{-0.6}^{+0.6}dz(d\sigma/dz)\sim
W^{-n}\,,\,\,\,n=(7.9\pm 0.4\pm 1.5)$ for $\pi^{+}\pi^{-}$, and $n=(7.3\pm
0.3\pm 1.5)$ for $K^{+}K^{-}$. However, the overall value $n\simeq 6$ is also
acceptable, see Fig.5 .
As for the absolute normalization, the $\pi^{+}\pi^{-}$ data are fitted [8]
with :
$\hskip 113.81102pt|\Phi_{\pi}^{(eff)}(s,\theta)|=(0.503\pm 0.007\pm
0.035)\,{\rm GeV}^{2}$.
Clearly, in addition to the leading terms $A^{(lead)}$, this experimental
value includes also all power corrections $\delta A$ to the
$\gamma\gamma\rightarrow\pi^{+}\pi^{-}$ amplitudes $A=A^{(lead)}+\delta A$.
These are different from corrections $\delta F_{\pi}$ to the genuine pion form
factor $F_{\pi}=F_{\pi}^{(lead)}+\delta F_{\pi}$. So, the direct connection
between the leading terms of $d\sigma(\pi^{+}\pi^{-})$ and $|F_{\pi}|^{2}$ in
(2),(3) does not hold on account of corrections.
Fig.5 a, b) Cross sections $\sigma_{o}(\gamma\gamma\to\pi^{+}\pi^{-})$ and
$\sigma_{o}(\gamma\gamma\to K^{+}K^{-})$ integrated over
the angular region $|\cos\theta|<0.6$, together with the $\sim(1/W)^{6}$
dependence line.
Fig.5 c) the cross section ratio $R_{\rm
exp}=\sigma_{o}(K^{+}K^{-})/\sigma_{o}(\pi^{+}\pi^{-})\simeq 0.9$.
Compare $R_{\rm exp}\simeq 0.9$ with the naive prediction for
$\phi_{\pi}(x)=\phi_{K}(x)\,:\,R=(f_{K}/f_{\pi})^{4}\simeq 2.3$ .
The experimental value
$|\Phi_{\pi}^{(eff)}(s,\theta)|=|1-\upsilon(\theta)||sF_{\pi}^{(lead)}(s)|\simeq
0.88|F_{\pi}^{(lead)}(s)|\simeq 0.5\,{\rm GeV}^{2}$ can be compared with
$0.88\cdot|s\,F_{\pi}^{(lead,CZ)}(s)|\simeq 0.4\,{\rm GeV}^{2}$ obtained with
$\phi_{\pi}(x,\overline{\mu}\sim 1\,GeV)=\phi_{\pi}^{CZ}(x)$ and taking here
the effective value of the coupling ${\overline{\alpha}}_{s}\simeq 0.4$. It is
seen that there is a reasonable agreement. At the same time, using
$\phi_{\pi}(x,\overline{\mu})\simeq\phi^{asy}(x)$ one obtains the much smaller
value $0.88\cdot|s\,F_{\pi}^{(lead,asy)}(s)|\simeq 0.13\,{\rm GeV}^{2}$.
Therefore, for the pion wave function $\phi_{\pi}(x,\mu)$ close to $\phi^{\rm
asy}(x)$ the leading term (i.e. without power corrections) calculation
predicts the cross section which is $\simeq 15$ times smaller than the data.
It seems that at $s=W^{2}=10-16\,{\rm GeV}^{2}$ the power corrections can not
cure so large difference.
Moreover, if power corrections were dominant numerically at these energies
then the cross sections $\sigma_{o}(\pi^{+}\pi^{-})$ and
$\sigma_{o}(K^{+}K^{-})$ will decay more like $\sim 1/W^{10}$, rather than
$\sim 1/W^{6}$.
The SU(3)-symmetry breaking, $d\sigma(K^{+}K^{-})\neq
d\sigma(\pi^{+}\pi^{-}),$ originates not only from different meson couplings,
$f_{K}\neq f_{\pi}$, but also from symmetry breaking effects in meson wave
functions, $\phi_{K}(x)\neq\phi_{\pi}(x)$, see Fig.4. (Let us recall: the
heavier is quark the narrower is wave function). These two effects tend to
cancel each other when using for the K-meson the wave function
$\phi^{CZZ}_{K}(x_{s},x_{u})$ obtained in [9] from the QCD sum rules. So,
instead of the naive original prediction $\simeq(f_{K}/f_{\pi})^{4}\simeq 2.3$
with $\phi_{K}(x)=\phi_{\pi}(x)$ in [5], the prediction in [6] for this ratio
is close to unity, and this agrees with the data from Belle [8] :
$\frac{\sigma_{o}(\gamma\gamma\rightarrow
K^{+}K^{-})}{\sigma_{o}(\gamma\gamma\rightarrow\pi^{+}\pi^{-})}=\left\\{\begin{array}[c]{ll}\displaystyle(f_{K}/f_{\pi})^{4}\simeq
2.3\scriptstyle&\begin{array}[c]{l}\rm{Brodsky,\,Lepage}\,\,{\rm\cite[cite]{[\@@bibref{}{BL}{}{}]}}\end{array}\par\\\
&\\\ \hline\cr&\\\ \simeq
1.06&\begin{array}[c]{l}\rm{Benayoun,\,Chernyak}\,\,{\rm\cite[cite]{[\@@bibref{}{Maurice}{}{}]}}\end{array}\\\
&\\\ \hline\cr&\\\ (0.89\pm 0.04\pm 0.15)\hskip 85.35826pt{\rm
Belle}\,\,\,{\rm\cite[cite]{[\@@bibref{}{Nakaz}{}{}]}}\end{array}\right.$
The leading terms in the cross sections for neutral particles are much smaller
than for charged ones, see (1). For instance, it was obtained in [6] that the
ratio $d\sigma^{(lead)}(\pi^{o}\pi^{o})/d\sigma^{(lead)}(\pi^{+}\pi^{-})$
varies from $\simeq 0.07$ at $\cos\theta=0$ to $\simeq 0.04$ at
$\cos\theta=0.6$, while the ratio
$\frac{d\sigma(\overline{K^{o}}K^{o})^{(lead)}}{d\sigma(\pi^{o}\pi^{o})^{(lead)}}\simeq
1.3\cdot(4/25)\simeq 0.2\,.$
Besides,
$\frac{\sigma_{o}^{(lead)}(K_{S}K_{S})}{\sigma_{o}^{(lead)}(K^{+}K^{-})}\simeq
0.005\,.$
It is seen that the leading contribution to $\sigma_{o}(K_{S}K_{S})$ is very
small. This implies that it is not yet dominant at present energies
$W^{2}<16\,{\rm GeV}^{2}$. I.e., the amplitude $M(\gamma\gamma\rightarrow
K_{S}K_{S})=a(s,\theta)+b(s,\theta)$ is dominated at these energies by the
non-leading term $b(s,\theta)\sim\varrho(\theta)(s_{o}/s)^{2}$, while the
formally leading term $a(s,\theta)\sim C_{o}f_{BC}(\theta)(s_{o}/s)$ has so
small coefficient $|C_{o}|\ll 1$, that $|b(s,\theta)|>|a(s,\theta)|$ at, say,
$s=W^{2}<12\,{\rm GeV}^{2}$.
Therefore, it has no much meaning to compare the leading term prediction of
[6], i.e.
$\frac{d\sigma(K_{S}K_{S})}{d\cos\theta}\sim|f_{BC}(\theta)|^{2}/W^{6}\quad{\rm
at}\,\,s\rightarrow\infty$
for the energy and angular dependence of $d\sigma(K_{S}K_{S})$ with the data
from Belle [10] at $W^{2}<15\,GeV^{2}$. Really, the only QCD prediction for,
say, $6\,{\rm GeV}^{2}<W^{2}<12\,{\rm GeV}^{2}$ is the expected energy
dependence
$\frac{d\sigma(K_{S}K_{S})}{d\cos\theta}\sim\frac{|b(s,\theta)|^{2}}{s}\sim\frac{|\varrho(\theta)|^{2}}{W^{10}}\,,$
while the angular dependence $|\varrho(\theta)|^{2}$ and the absolute
normalization are unknown. This energy dependence agrees with the results from
Belle [10], see fig.6 .
Fig. 6 (a) The total cross section $\sigma_{o}(\gamma\gamma\rightarrow
K_{S}K_{S})$ in the c.m. angular region
$\hskip 42.67912pt|\cos\theta|<0.6$ [10]. Here $n=(10.5\pm 0.6\pm 0.5)$ is the
$W$-dependence $\sigma_{o}(W)\sim 1/W^{n}$ .
Fig. 6 (b) The ratio $\sigma_{0}(K_{S}K_{S})/\sigma_{0}(K^{+}K^{-})$ versus
$W$ [10]
The dotted line DKV (Diehl-Kroll-Vogt) is the valence handbag model prediction
in the $~{}SU(3)$ symmetry limit [15]; the dashed BL and dash-dotted BC lines
are the Brodsky-Lepage ( with $\,\phi_{K}(x)=\phi^{\rm asy}(x)\,$) and
Benayoun-Chernyak ( with $\,\phi_{K}(x)=\phi^{CZZ}_{K}(x)\,$ from [9] )
leading term QCD predictions (for sufficiently large energy $W$).
The cross sections of other neutral particle productions were also measured by
Belle Collaboration [11, 12, 13], see Figs.7,8.
Fig. 7a Cross sections $\sigma_{o}(\gamma\gamma\to\pi^{0}\pi^{0})$
and $\sigma_{o}(\gamma\gamma\to\pi^{+}\pi^{-})$ for $|\cos\theta|<0.6$ ;
Fig. 7b Their ratio. The lines are fits to the
results in the energy region indicated.
The QCD predictions for this range of energies:
$\vspace*{-2mm}{}\sigma(\pi^{+}\pi^{-})\sim{\overline{\alpha}}_{s}^{\,2}/W^{6}\,,$
while the expected behavior of $\sigma(\pi^{o}\pi^{o})$ if the higher twist
terms are still dominant at $3<W<4\,GeV$ (and up to
the odderon contribution, see below) is $~{}\sigma(\pi^{o}\pi^{o})\sim
1/W^{10}$.
The handbag model prediction [15] :
${}R=\sigma(\pi^{o}\pi^{o})/\sigma(\pi^{+}\pi^{-})=0.5$
Fig.8a $W$ \- dependence of cross sections
$\sigma(\gamma\gamma\rightarrow\pi^{o}\pi^{o})$ and
$\sigma(\gamma\gamma\rightarrow\eta\pi^{o})\,,\,\,\,|\cos\theta|<0.8$.
The power low fit : $\sigma(\eta\pi^{o})\sim(1/W)^{n},\,\,~{}n=(10.5\pm 1.2\pm
0.5)$ [12]
Fig.8b The ratio of cross sections
$\sigma(\eta\pi^{o})/\sigma(\pi^{o}\pi^{o})$ [11, 12]
The energy dependencies of various cross sections measured and fitted by Bell
Collaboration are collected in the table.
The value of "n" in $\sigma_{\rm tot}\sim 1/W^{n}$ in various reactions
fitted in the $W$ and $|\cos\theta|$ ranges indicated
Process | n - experiment | $W$ range (GeV) | $|\cos\theta|$ | n - QCD | n - handbag | Ref
---|---|---|---|---|---|---
$\pi^{+}\pi^{-}$ | $7.9\pm 0.4\pm 1.5$ | $3.0-4.1$ | <0.6 | $\simeq 6$ | $\simeq 10$ | [8]
$K^{+}K^{-}$ | $7.3\pm 0.3\pm 1.5$ | $3.0-4.1$ | <0.6 | $\simeq 6$ | $\simeq 10$ | [8]
$K^{0}_{S}K^{0}_{S}$ | $10.5\pm 0.6\pm 0.5$ | 2.4 – 4.0 | <0.6 | $\simeq 10$ | $\simeq 10$ | [10]
$\eta\pi^{0}$ | $10.5\pm 1.2\pm 0.5$ | 3.1 – 4.1 | <0.8 | $\simeq 10$ | $\simeq 10$ | [12]
$\pi^{0}\pi^{0}$ | $\simeq 10$ | 2.5 – 3.0 | <0.8 | $\simeq 10$ | $\simeq 10$ | [11]
$\pi^{0}\pi^{0}$ | $8.0\pm 0.5\pm 0.4$ | 3.1 – 4.1 | <0.8 | $\simeq 10\,?$ | $\simeq 10$ | [11]
$\pi^{0}\pi^{0}$ | $\simeq 10$ | 2.5 – 3.0 | <0.6 | $\simeq 10$ | $\simeq 10$ | [11]
$\pi^{0}\pi^{0}$ | $6.9\pm 0.6\pm 0.7$ | 3.1 – 4.1 | <0.6 | $\simeq 10\,?$ | $\simeq 10$ | [11]
$\eta\eta$ | $7.8\pm 0.6\pm 0.4$ | 2.4 – 3.3 | <0.8 | $\simeq 10$ | $\simeq 10$ | [13]
The measured energy dependence of the $\pi^{0}\pi^{0}$ cross section is
similar to $K_{S}K_{S}$ and $\eta\pi^{o}$ cross sections at
$6<W^{2}<9\,GeV^{2}$, but behaves "abnormally" in the energy interval
$9<W^{2}<16\,GeV^{2}$.
Fig.9 $P_{1},\,P_{2}=\pi^{o},\,\eta,\,\eta^{\prime},\,\eta_{c}$
The additional hard contributions
for neutral pseudoscalar mesons
In attempt to understand this "abnormal" behavior of the $\pi^{0}\pi^{0}$
cross section we can recall that, unlike the $\sigma(\overline{K^{o}}K^{o})$
cross section, there are additional contributions to the
$\sigma(\pi^{o}\pi^{o})$ and $\sigma(\eta\pi^{o})$ cross sections shown in
Fig.9 (the odderon contribution becomes the leading one at sufficiently large
energies and small fixed angles).
The contribution of the diagram with the photon exchange to the amplitudes
$M_{\gamma}(\gamma\gamma\to\pi^{o}\pi^{o})$ is readily calculated and the
helicity amplitudes look as
$M_{\gamma}^{\pm\pm}=(4\pi\alpha)^{2}(e^{2}_{u}-e^{2}_{d})^{2}\,\frac{2f^{2}_{\pi}}{s}\,\Bigl{[}\,\int_{0}^{1}\frac{\phi_{\pi}(x)}{x}\,\Bigr{]}^{2}\,\Phi_{++}(\theta),\quad\Phi_{++}(\theta)=\frac{2(1+z^{2})}{(1-z^{2})^{2}}\,,$
$M_{\gamma}^{\pm\mp}=(4\pi\alpha)^{2}(e^{2}_{u}-e^{2}_{d})^{2}\,\frac{2f^{2}_{\pi}}{s}\,\Bigl{[}\,\int_{0}^{1}\frac{\phi_{\pi}(x)}{x}\,\Bigr{]}^{2}\,\Phi_{+-}(\theta),\quad\Phi_{+-}(\theta)=\frac{1+3z^{2}}{(1-z^{2})^{2}}\,,\quad
z=\cos\theta\,.$
As a result, using the pion wave function $\phi^{\rm
CZ}_{\pi}(x)=30x(1-x)(2x-1)^{2}$ the ratio is (really, this ratio is only
weakly dependent on the pion wave function form)
$\frac{\sigma_{\gamma}(\pi^{o}\pi^{o},|\cos\theta|<0.8)}{\sigma(\pi^{+}\pi^{-},|\cos\theta|<0.8)}\simeq
1\cdot\Bigl{(}\frac{\alpha}{{\overline{\alpha}}_{s}}\Bigr{)}^{2}\sim 0.5\cdot
10^{-3}\,,$
so that this contribution is very small and does not help.
The odderon contribution in Fig.9 has been calculated in [14] and looks at
$s\gg|t|\gg\mu^{2}_{o}$ as
$M_{++}=M_{--}\simeq-2.5M_{+-}=-2.5M_{-+}\simeq\frac{sf^{2}_{\pi}}{t^{2}}\,\,4\pi\alpha\Bigl{(}\,4\pi\overline{\alpha}_{s}\,\Bigr{)}^{3}\,\frac{5}{108\pi^{2}}\,I_{\pi\pi}$
$\hskip
85.35826ptI_{\pi\pi}=\int_{-1}^{1}d\xi_{1}\,\frac{\phi_{\pi}(\xi_{1})}{(1-\xi^{2}_{1})}\int_{-1}^{1}d\xi_{2}\,\frac{\phi_{\pi}(\xi_{2})}{(1-\xi^{2}_{2})}\,T_{\pi\pi}(\xi_{1},\xi_{2}),\hskip
85.35826pt(4)$
$T_{\pi\pi}(\xi_{1},\xi_{2})=\ln\Big{|}\frac{\xi_{1}+\xi_{2}}{1-\xi_{1}}\Big{|}\ln\Big{|}\frac{\xi_{1}+\xi_{2}}{1+\xi_{2}}\Big{|}+(\xi_{1}\rightarrow-\xi_{1}),\quad\xi_{1}=x_{1}-x_{2}\,,\quad\xi_{2}=y_{1}-y_{2}$
The numerical value of $I_{\pi\pi}$ in (4) is $I_{\pi\pi}\simeq 26.8$ [14] for
$\phi_{\pi}(\xi)=\phi^{\rm CZ}_{\pi}(\xi)$. Therefore, with
$\phi_{\pi}(\xi)=\phi^{\rm CZ}_{\pi}(\xi)$ :
$\frac{d\sigma^{(3\,\rm
gl)}}{dt}(\gamma\gamma\to\pi^{o}\pi^{o})\simeq\frac{0.7\,nb\,GeV^{6}}{t^{4}}\Bigl{(}\,\frac{{\overline{\alpha}}^{\,2}_{s}}{0.1}\,\Bigr{)}^{3},$
and using here ${\overline{\alpha}}_{s}\simeq 0.3$ :
$\sigma^{(3\,\rm
gl)}(\pi^{o}\pi^{o},\,|\cos\theta|<0.8)\simeq\left\\{\begin{array}[]{l l
l}\,\,23\cdot 10^{-2}\,nb&[\,\rm experiment:30\cdot 10^{-2}\,nb\,]\,\,{\rm
at}\,\,W=3\,GeV\\\ \,\,\,\,9\cdot 10^{-2}\,nb&[\,\rm experiment:\,\,\,8\cdot
10^{-2}\,nb\,]\,\,{\rm at}\,\,W=3.5\,GeV\\\ \,\,\,\,\,4\cdot
10^{-2}\,nb&[\,\rm experiment:\,\,\,3\cdot 10^{-2}\,nb\,]\,\,{\rm
at}\,\,W=4\,GeV\\\ \end{array}\right.$
Hence, according to these estimates with $\phi_{\pi}(\xi)=\phi^{\rm
CZ}_{\pi}(\xi)$, the odderon contribution is sufficiently large and may well
be responsible for a change of the behavior of $\sigma(\pi^{o}\pi^{o})$ at
$W>3$ GeV. At the same time, the numerical value of $I_{\pi\pi}$ in (4) is
$I_{\pi\pi}\simeq 7.4$ for $\phi_{\pi}(\xi)=\phi^{\rm asy}(\xi)$ and so the
value of $\sigma^{(3\,\rm gl)}(\pi^{o}\pi^{o})$ with
$\phi_{\pi}(\xi)=\phi^{\rm asy}(\xi)$ will be $\simeq 13$ times smaller.
In the $SU(3)$ symmetry limit $\sigma^{(3\,\rm
gl)}(\eta_{8}\pi^{o})/\sigma^{(3\,\rm gl)}(\pi^{o}\pi^{o})=2/3$. To estimate
the effects of $SU(3)$ symmetry breaking we use the same model wave function
of $\eta$ as those used in [20], i.e.
$|\eta\rangle=\cos\phi|n\rangle-\sin\phi|s\rangle,\,|n\rangle=|({\overline{u}}u+{\overline{d}}d)/\sqrt{2}\rangle,\,|s\rangle=|{\overline{s}}s\rangle$,
and with taking into account the $SU(3)$ symmetry breaking effects
distinguishing $|n\rangle$ and $|s\rangle$ wave functions. Then, instead of
$\phi^{\rm CZ}_{\pi}(\xi_{1})$ in (4) one has to substitute
$\phi^{\rm
CZ}_{\pi}(\xi_{1})\rightarrow\Bigl{[}\,\frac{\cos\phi}{3}\,\phi^{\rm
CZ}_{\pi}(\xi_{1})+\frac{f_{s}}{f_{\pi}}\frac{\sqrt{2}\sin\phi}{3}\,\phi^{\rm
asy}(\xi_{1})\,\Bigr{]}\,.$
Then, with $\phi\simeq 38^{o}$ and $f_{s}/f_{\pi}\simeq 1.3$, instead of
$I_{\pi\pi}=26.8$ with $\phi_{\pi}(\xi)=\phi^{\rm CZ}_{\pi}(\xi)$, the
corresponding integral will be $I_{\pi\eta}\simeq 11$ and $\sigma^{(3\,\rm
gl)}(\pi^{o}\eta)/\sigma^{(3\,\rm gl)}(\pi^{o}\pi^{o})\simeq 1/3$. It seems,
this additional suppression may be a reason why the odderon contribution is
still not seen clearly in $\sigma(\pi^{o}\eta)$ at $3<W<4\,GeV$ and
$|\cos\theta|<0.8$. The prediction is that it will be seen at somewhat higher
energies.
The handbag model
The hand-bag model [15] is a definite application of the general idea which
assumes that present day energies are insufficient for the leading terms QCD
to be the main ones. Instead, it is supposed that the soft nonperturbative
contributions dominate the amplitudes. The handbag model realizes applications
of this idea to a description of the large angle cross sections
$d\sigma(\gamma\gamma\rightarrow{\overline{M}}M)$.
As it is formulated in [15], the handbag model assumes that the above
described hard QCD contributions really dominate at very high energies only,
while the main contributions at present energies originate from the Fig.10a
diagram.
Fig.10a The overall picture of the handbag model contribution [15]
Fig.10b The standard lowest order Feynman diagram for the light cone QCD sum
rule
to calculate the soft valence handbag amplitude ${M_{\rm
handbag}(\gamma\gamma\to\pi^{+}\pi^{-})}$ [16, 17]
($A_{\mu}={\overline{u}}\gamma_{\mu}\gamma_{5}d$)
Here, the two photons interact with the same quark only, and these two active
${\overline{q}}$ and $q$ quarks carry nearly the whole meson momenta, while
the additional passive ${\overline{q}^{\prime}}$ and $q^{\prime}$ quarks are
wee partons which are picked out from the vacuum by soft non-perturbative
interactions [15]. Therefore, these soft form factors $R_{MM}(s)$ should be
power suppressed in QCD at sufficiently large $s\,:R_{MM}(s)\leq 1/s^{2}\,,\,$
in comparison with the leading meson form factors, $F_{M}(s)\sim 1/s\,$. But,
nevertheless, it is assumed that they are numerically dominant at present
energies for both charged and neutral mesons.
The energy dependence and the absolute normalization of the handbag amplitude
$M_{\rm handbag}(\gamma\gamma\\\ \rightarrow{\overline{M}}_{2}M_{1})$ is not
predicted in [15] but fitted to the data.
As for the angular dependence, it was also not really predicted in [15] in a
model independent way. The reason is that a number of special approximate
relations were used in [15] at intermediate steps to calculate the angular
dependence of the handbag amplitude. All these relations were valid, at best,
for the leading term only. But it turned out finally that their would be
leading term gives zero contribution to the amplitude, and the whole answer is
due to next power corrections, $\sim\Lambda_{QCD}^{2}/s$, which were not under
control in [15]. The "result" $M_{\rm handbag}(\gamma\gamma\to M_{2}M_{1})\sim
1/\sin^{2}\theta$ for the handbag amplitude in [15] is completely due to the
one especially (and arbitrary) chosen definite power suppressed term in the
amplitude while ignoring all other power corrections of the same order of
smallness.
The authors were fully aware of this arbitrariness [15]: "We must then at this
stage consider our result $M_{\rm handbag}\sim 1/\sin^{2}\theta$ as a model or
a partial calculation of the soft handbag contribution".
Hence, finally, the approach in [15] predicts neither energy and angular
dependencies nor the normalization of cross sections in a model independent
way.
Therefore, what only remains are the specific predictions of the handbag model
for the ratios of cross sections in the $SU(3)$ symmetry limit : there is only
one common valence handbag amplitude $M^{\rm val}_{\rm handbag}$ (the soft
non-valence handbag amplitudes are small, see below) :
$M(\pi^{+}\pi^{-})=M(\pi^{o}\pi^{o})=M(K^{+}K^{-})=\frac{5}{2}\,M(\overline{K^{o}}K^{o})=\frac{5}{2}\,M(K_{S}K_{S})=M^{\rm
val}_{\rm handbag}$
$\frac{5}{\sqrt{3}}\,M(\pi^{o}\eta_{8})=\frac{5}{\sqrt{6}}\,M(\pi^{o}\eta_{o})=\frac{5}{3}\,M(\eta_{8}\eta_{8})=\frac{5}{2}\,M(\overline{K^{o}}K^{o})=M^{\rm
val}_{\rm handbag}$
Due to these relations, the predictions of the handbag model for the ratios of
cross sections in comparison with the data look as (the red numbers are
specific predictions of the valence handbag model) :
$\frac{\sigma(K^{+}K^{-})}{\sigma(\pi^{+}\pi^{-})}=1\,(0.89\pm 0.04\pm
0.15)_{\rm
exp},\quad\frac{\sigma(\pi^{o}\pi^{o})}{\sigma(\pi^{+}\pi^{-})}={\color[rgb]{1,0,0}{\frac{1}{2}}}\,(0.32\pm
0.03\pm 0.05)_{\rm exp}$
$\frac{\sigma(K_{S}K_{S})}{\sigma(K^{+}K^{-})}={\color[rgb]{1,0,0}{0.08}}\,\,(0.13\,\rightarrow\,0.01,\,\rm{see\,Fig.6})_{\rm
exp},\quad\frac{\sigma(K_{S}K_{S})}{\sigma(\pi^{o}\eta_{8})}={\color[rgb]{1,0,0}{\frac{2}{3}}}\,\,(\,\sim
0.1\,)_{\rm exp},$
$\frac{\sigma(\eta_{8}\eta_{8})}{\sigma(\pi^{o}\pi^{o})}={\color[rgb]{1,0,0}{0.36}}\,(0.37\pm
0.04)_{\rm
exp},\quad\frac{\sigma(\pi^{o}\eta_{8})}{\sigma(\pi^{o}\pi^{o})}={\color[rgb]{1,0,0}{0.24}}\,(0.48\pm
0.06)_{\rm exp}$
Recalling that the angular dependence of the handbag amplitude $M_{\rm
handbag}\sim 1/\sin^{2}\theta$ used in [15] was a model form, it looks not so
surprising that the explicit calculation of the valence handbag amplitude
$M^{\rm val}_{\rm handbag}(W,\theta)$ in [16] (see also [17]) via the light
cone QCD sum rules [18],[19] gave a different angular dependence, $M^{\rm
val}_{\rm handbag}\sim{\rm const}$. These soft valence handbag contributions
to the cross sections calculated explicitly from the light cone QCD sum rules
in [16], see Fig. 7b, are definite functions of the energy and scattering
angle, and look as
$\hskip 85.35826ptd\sigma_{\rm
handbag}(\gamma\gamma\to{\overline{M}}_{2}M_{1})/d\cos\theta\sim{\rm
const}/W^{10}\hskip 113.81102pt(4)$
for all mesons, both charged and neutral.
Unfortunately, this angular behavior $d\sigma_{\rm
handbag}(\gamma\gamma\to{\overline{M}}_{2}M_{1})/d\cos\theta\sim\rm const$
disagrees with all data which behave similar to $\sim 1/\sin^{4}\theta$, and
the energy behavior $\sigma_{\rm
handbag}(\gamma\gamma\to{\overline{M}}_{2}M_{1})\sim 1/W^{10}$ disagrees with
the data for charged mesons $\pi^{+}\pi^{-}$ and $K^{+}K^{-}$, compatible with
$\sim 1/W^{6}$.
This energy behavior $\sim 1/W^{10}$ in $(4)$ is as expected (up to Sudakov
effects) in QCD for soft valence power corrections to the leading terms due to
the Feynman end-point mechanism (but one should remember that there is also a
number of hard valence power corrections in QCD with the same energy
dependence $\sim 1/W^{10}$, but with possibly different angular dependencies).
Fig.11 The diagrams to calculate the energy and angular dependencies of soft
handbag amplitudes (the valence Feynman mechanism)
The end-point region contributions of only two diagrams shown in Fig.11 (the
valence Feynman mechanism) are relevant for the standard valence handbag model
as it is formulated in [15] :
$\hskip 28.45274ptx_{1},\,y_{1}\rightarrow 1,\,\,\,\quad
k^{2}=x_{2}y_{2}S\sim\Lambda^{2}_{QCD}\,\,\rightarrow\,\,x_{2}\sim
y_{2}\sim\delta=(\Lambda^{2}_{QCD}/S)^{1/2}=\Lambda_{QCD}/W\ll 1$.
The direct calculation of these diagrams in Fig.11 gives for the hard kernel
$T^{(lead)}=const\,\frac{f^{2}_{\pi}}{S}\,\,\frac{(e_{1}e_{2})(x_{1}+y_{1})+2x_{1}y_{1}}{x_{1}x_{2}y_{1}y_{2}}\quad\rightarrow\quad
const\,\frac{f^{2}_{\pi}}{S}\,\,\frac{(e_{1}e_{2})+1}{x_{2}y_{2}}\,\hskip
85.35826pt(5)$
($e_{1}$ and $e_{2}$ are the photon polarization vectors), and for the soft
end-point region contributions to the whole valence handbag amplitude
$M_{\rm handbag}^{\rm
val,\,\pm\pm}\sim\frac{f^{2}_{\pi}}{S}\Biggl{[}\int_{0}^{\delta}dx_{2}\Bigl{(}\frac{\phi_{\pi}(x)}{x_{2}}\sim
const\Bigr{)}\Biggr{]}^{2}\sim\frac{f^{2}_{\pi}}{S}\,\delta^{2}\sim\frac{f^{2}_{\pi}\Lambda^{2}_{QCD}}{S^{2}=W^{4}}\gg
M_{\rm handbag}^{\rm val,\,\pm\mp}\,.\hskip 56.9055pt(6)$
Therefore, the angular and energy dependences of all cross sections resulting
from these soft valence contributions (the Feynman mechanism) look as
$\hskip 113.81102ptd\sigma_{handbag}/d\cos\theta\sim|M|^{2}/W^{2}\sim
1/W^{10}\,.\hskip 133.72786pt(7)$
The expressions (5)-(7) agree with the predictions of the light cone QCD sum
rules [16] not only in the energy and angular dependencies but also in the
photon helicities dependencies.
If we are interested only in the energy dependence of such soft end-point
region contributions, there is a simpler way to obtain it which does not
require the direct calculation of Feynman diagrams. This can be done as
follows.
1) There is the hard part of any such diagram and it is the amplitude of the
annihilation of two photons into a pair of active near mass-shell quarks with
each one carrying nearly the whole meson momenta, $A_{\rm
hard}(\gamma\gamma\to{\overline{q}}q)$. From the dimensional reasons it is
$A_{\rm hard}(\gamma\gamma\to{\overline{q}}q)\sim 1$.
2) All other parts of the Feynman diagrams are soft and, parametrically,
depend on the scale $\Lambda_{QCD}$ only. So, the energy dependence of the
soft valence end-point region contributions (i.e. the valence Feynman
mechanism) looks here as follows ($\,\phi_{2}(x_{1},x_{2})\sim x_{1}x_{2}\,$)
$R^{(\rm v,\,2)}_{MM}(W)\sim
M_{2}(\gamma\gamma\to{\overline{M}}M)\sim\int_{0}^{\delta}dx_{2}\,\phi_{2}(x)\int_{0}^{\delta}dy_{2}\,\phi_{2}(y)\Bigl{[}A_{\rm
hard}(\gamma\gamma\to{\overline{q}}q)\sim 1\Bigr{]}\sim$
$\sim\Bigl{[}\,\int_{0}^{\delta}dx_{2}\,x_{2}\int_{0}^{\delta}dy_{2}\,y_{2}\,\Bigr{]}^{\rm
n_{\rm wee}=1}\sim\Bigl{(}\,\delta^{4}\,\Bigr{)}^{\rm n_{\rm
wee}=1}\sim\Lambda_{QCD}^{4}/W^{4}\,.$
This method of obtaining the energy dependence of soft end-point contributions
will be used below to calculate the energy dependence of soft non-valence
handbag form factors originating from the 4-particle components of meson wave
functions.
The updated predictions of the handbag model for the
$\gamma\gamma\to{\overline{M}_{2}}M_{1}$ cross sections were given in the next
paper [23]. In comparison with the original paper [15], the main new element
in [23] is that (in the $SU(3)$ symmetry limit used in [23] and [15]) the
sizeable soft non-valence form factor $R^{\,\rm nv}_{\overline{M}M}(s)$ is
introduced now, in addition to the soft valence one $R^{\,\rm
v}_{\overline{M}M}(s)$ (the soft non-valence contributions were neglected in
[15]). Both functions, $R^{\,\rm v}_{{\overline{M}}M}(s)$ and $R^{\,\rm
nv}_{{\overline{M}}M}(s)$, are parameterized then in arbitrary forms with a
number of free parameters which are fitted in [23] to the data. 111 The form
factors $R^{\,\rm u}_{2\pi}(\rm s)$ and $R^{\,\rm s}_{2\pi}(s)$ used in [23]
are connected with those from [20] as: $R^{\,\rm u}_{2\pi}(s)=R^{\,\rm
v}_{2\pi}(s)+R^{\,\rm nv}_{2\pi}(s)$, $R^{\,\rm s}_{2\pi}(s)=R^{\,\rm
nv}_{2\pi}(s)$ .
As for the soft valence contributions to the cross sections and the soft
valence form factors $R^{\,\rm v}_{{\overline{M}}M}(s)$, these were estimated
numerically in [16] via the standard light cone QCD sum rules and were found
much smaller numerically (and with the expected suppressed power behaviour
$R^{\,\rm v}_{{\overline{M}}M}(s)\sim 1/s^{2}$) than the values fitted to data
in [15] and [23].
As for the soft non-valence contributions, the two types of such contributions
are presented in Fig.12 [20].
Fig.12a The leading power hard non-valence one-loop correction.
Fig.12b The leading contribution to the soft non-valence handbag form factor
$R^{\,\rm nv}_{{\overline{M}}M}(s)$.
The solid and dashed lines represent quarks and gluons.
It is worth noting that both non-valence contributions in Fig.12 are
$SU(3)$-flavor singlets in the $SU(3)$-symmetry limit. So, they contribute
equally to the amplitudes $\pi^{+}\pi^{-},\,\pi^{0}\pi^{0},\,K^{+}K^{-},\\\
\overline{K^{0}}K^{0}$ and $\eta_{8}\eta_{8}$, and don’t contribute to
$\eta_{8}\pi^{0}$.
The diagrams in Fig.12a constitute a small subset of all one-loop corrections
to the leading power contributions from the Born diagrams like those shown in
Fig.1 . If these leading power one-loop non-valence corrections to the Born
contributions were really significant, this will contradict then the data on
$K_{S}K_{S}$, see Fig.6 .
In particular, this hard non-valence one-loop correction was calculated, among
all others, in [21]. Its contribution into the cross section
$\sigma(\gamma\gamma\to K^{+}K^{-})$ (integrated over $|\cos\theta|<0.6$, and
with $\phi_{K}(x)=\phi^{\rm asy}(x)$ ) is [22] :
$\frac{\delta\sigma^{\rm
nv}}{\sigma}\simeq-\frac{{\overline{\alpha}}_{s}}{3\pi}\simeq-3\%\,,$
i.e., its contribution into the amplitude is: $\delta{\overline{A}}^{\,\rm
nv}/\,{\overline{A}}(K^{+}K^{-})\simeq-1.5\%$.
The leading term amplitude $|{\overline{A}}(K_{S}K_{S})|\simeq
0.15\,|{\overline{A}}(K^{+}K^{-})|$, see Fig.6. Hence, the rough estimate of
this non-valence one loop correction to the ${\overline{A}}(K_{S}K_{S})$
amplitude is :
$|\delta{\overline{A}}^{\,\rm nv}/\,{\overline{A}}(K_{S}K_{S})|\simeq 10\%$.
As for the soft non-valence handbag form factor $R^{\,\rm
nv}_{{\overline{M}}M}(s)$, it seems sufficient to say that the leading
contribution to it originates first from the Fig.12b two-loop correction
((without large logarithms) [20], so that the estimate looks as :
$\hskip 85.35826pt\frac{R^{(\rm nv,\,2)}_{MM}(s)}{R^{\,\rm
v}_{MM}(s)}\sim\Biggl{(}\frac{{\overline{\alpha}}_{s}}{\pi}\Biggr{)}^{2}\sim
0.01\,.\hskip 113.81102pt(8)$
Besides, there are also soft non-valence contributions from the 4-quark
components of the meson wave functions. For instance, the typical contribution
of the pion 4-quark components,
$|\pi^{+}\rangle_{4}\sim|({\overline{s}}s+{\overline{u}}u+{\overline{d}}d)\,{\overline{d}}u\rangle$,
is shown in Fig.13. The energy dependence of such contributions can be
obtained the same way as for the diagram in Fig.11
Fig.13 The additional contribution $R^{(\rm nv,\,4)}_{2\pi}(W)$ to the soft
non-valence handbag form factor. Here, one strange quark in $\pi^{+}$ and
$\pi^{-}$ carries nearly the whole pion momentum, while three other quarks are
wee partons ($\,\phi_{4}(x)\sim x_{1}x_{2}x_{3}x_{4},\,\,\,x_{1}\rightarrow
1,\,\,\,0<x_{2,3,4}<\delta\sim\Lambda_{QCD}/W$)
$R^{(\rm nv,\,4)}_{2\pi}(W)\sim
M_{4}(\gamma\gamma\to\pi^{+}\pi^{-})\sim\int_{0}^{\delta}dx_{2}dx_{3}dx_{4}\,\phi_{4}(x)\int_{0}^{\delta}dy_{2}dy_{3}dy_{4}\,\phi_{4}(y)\Bigl{[}A_{\rm
hard}(\gamma\gamma\to{\overline{s}}s)\sim 1\Bigr{]}$
$\sim\Bigl{[}\,\int_{0}^{\delta}dx\,x\int_{0}^{\delta}dy\,y\,\Bigr{]}^{\rm
n_{\rm wee}=3}\sim\Bigl{(}\,\delta^{4}\,\Bigr{)}^{n_{\rm
wee}=3}\sim\Lambda_{QCD}^{12}/W^{12}\,,$ $\hskip 99.58464pt\frac{R^{(\,\rm
nv,\,4)}_{2\pi}(W)}{R^{\,\rm
v}_{2\pi}(W)}\sim\Bigl{(}\frac{1\,GeV^{2}}{W^{2}}\Bigr{)}^{4}\sim 1.5\cdot
10^{-4}\,\,\,\,{\rm at}\,\,s=W^{2}=9\,GeV^{2}\,.\,\hskip 42.67912pt(9)$
Clearly, so small soft non-valence contributions, $R^{(\rm
nv,\,2)}_{MM}/R^{\,\rm v}_{MM}\sim 10^{-2}$ and $R^{(\rm
nv,\,4)}_{MM}/R^{\,\rm v}_{MM}\sim 10^{-4}$ at $W^{2}\simeq 9\,GeV^{2}$, can
be safely neglected and will not help.
In comparison,
$\frac{|R^{\,\rm nv}_{2\pi}(W^{2}=9\,GeV^{2})|}{|R^{\,\rm
v}_{2\pi}(W^{2}=9\,GeV^{2})|}\simeq 0.3$
was used in [23] to fit the data.
The cross sections $\gamma\gamma\rightarrow V_{1}^{o}V_{2}^{o}$
The contribution of diagrams like Fig.1 to the amplitude
$\gamma\gamma\to\rho^{o}\rho^{o}$ is also suppressed numerically in comparison
with $\gamma\gamma\to\rho^{+}\rho^{-}$. But there is the additional hard
leading twist one-loop contributions to
$\gamma\gamma\to\rho^{o}_{L}\rho^{o}_{L}$, see Fig.14, and it looks at
$s\gg|t|\gg\mu^{2}_{o}$ as [24][25] :
$M_{++}=M_{--}\simeq-2M_{+-}=-2M_{-+}\simeq
i\,\frac{sf^{2}_{\rho}}{t^{2}}\,(4\pi\alpha)\Bigl{(}4\pi{\overline{\alpha}}_{s}\Bigr{)}^{2}\frac{4}{9\pi}\,I_{\rho\rho}$
$I_{\rho\rho}=\int_{-1}^{1}d\xi_{1}\,\frac{\phi_{\rho}(\xi_{1})}{(1-\xi^{2}_{1})}\int_{-1}^{1}d\xi_{2}\,\frac{\phi_{\rho}(\xi_{2})}{(1-\xi^{2}_{2})}\,T_{\rho\rho}(\xi_{1},\xi_{2})$
$T_{\rho\rho}(\xi_{1},\xi_{2})=\xi_{1}\xi_{2}\ln\Big{|}\frac{\xi_{1}+\xi_{2}}{\xi_{1}-\xi_{2}}\Big{|}\,\,,\quad\xi_{1}=x_{1}-x_{2}=2x_{1}-1,\quad\xi_{2}=y_{1}-y_{2}=2y_{1}-1$
$\frac{d\sigma}{dt}(\gamma\gamma\to\rho^{o}_{L}\rho^{o}_{L})\simeq\frac{0.9\,nb\,GeV^{6}}{t^{4}},\quad\sigma(\rho\rho,|\cos\theta|\leq
0.8)\simeq\frac{300\,nb\,GeV^{6}}{W^{6}}$
(these numbers are for the $\rho$-meson wave function (11), see [9], for
$\phi_{\rho}\simeq\phi^{\rm asy}$ the cross section is $\simeq 6$ times
smaller),
$\hskip 28.45274pt\sigma(\rho^{o}\rho^{o},|\cos\theta|\leq
0.8)\simeq\left\\{\begin{array}[]{l l l}\,\,41\cdot 10^{-2}\,nb&\,\,{\rm
at}\quad W=3\,GeV\\\ \,\,16\cdot 10^{-2}\,nb&\,\,{\rm at}\quad W=3.5\,GeV\\\
\,\,\,\,\,7\cdot 10^{-2}\,nb&\,\,{\rm at}\quad W=4\,GeV\\\
\end{array}\right.\hskip 113.81102pt(10)$
Fig.14
The additional hard one-loop contribution to the amplitude
$\gamma\gamma\rightarrow V_{1}^{o}V_{2}^{o}$. Becomes dominant at high
energies and small fixed angles for longitudinally polarized vector mesons.
Fig.15 The model wave functions
Black line : the $\rho_{L}$-meson wave function
Blue line : the $\phi_{L}$-meson wave function
Red line : the asymptotic wave function
The contributions to other cross sections from the Fig.14 diagram
$\sigma_{o}(\rho^{o}\rho^{o}):\sigma_{o}(\rho^{o}\omega):\sigma_{o}(\omega\omega)\simeq
1:\frac{1}{5}:\frac{1}{80}$
The model leading twist $V_{L}=V_{\lambda=0}$ wave functions [9] are taken in
the form, see Fig.15
$\phi_{\rho}(\xi,\mu\sim 1\,GeV)\simeq\phi_{\omega}(\xi,\mu_{o}\sim
1\,GeV)\simeq\phi^{\rm
asy}(\xi)\Bigl{(}1+0.2\,C^{3/2}_{2}(\xi)\Bigr{)}=\phi^{\rm
asy}(\xi)\Bigl{(}0.70+1.5\,\xi^{2}\Bigr{)}$ $\phi_{\phi}(\xi,\mu_{o}\sim
1\,GeV)\sim\phi^{\rm
asy}(\xi)\Bigl{(}1+0.1\,C^{3/2}_{2}(\xi)\Bigr{)}=\phi^{\rm
asy}(\xi)\Bigl{(}0.85+0.75\,\xi^{2}\Bigr{)},\hskip 56.9055pt(11)$ $\phi^{\rm
asy}(\xi)=\frac{3}{4}(1-\xi^{2}),\quad f_{\rho}\simeq f_{\omega}\simeq
210\,MeV\,,\quad f_{\phi}\simeq 230\,MeV\,.$
At $W=4\,GeV\,,\,\,|\cos\theta|\leq 0.8\,\,:$
$\sigma(\omega\phi)\sim 3\,\sigma(\omega\omega)\sim
1.5\,\sigma(\phi\phi),\quad\sigma(\rho^{o}\rho^{o})\simeq 70\cdot 10^{-3}\,nb$
$\sigma(\omega\omega)\simeq 1\cdot 10^{-3}\,nb\quad\sigma(\omega\phi)\sim
3\cdot 10^{-3}\,nb,\quad\sigma(\phi\phi)\sim 2\,\sigma(\omega\omega)\sim
2\cdot 10^{-3}\,nb\,.$
The cross sections $\sigma(\omega\phi),\,\sigma(\phi\phi)$ and
$\sigma(\omega\omega)$ have been measured recently by Belle Collaboration
[26], see Figs.16-18.
Fig.16 The total cross section $\sigma(\omega\phi)$ in the c.m. angular region
$|\cos\theta|<0.8$ [26]
W-dependence : $\sigma_{\omega\phi}(W)\sim W^{-n},\,\,n=(7.2\pm
0.6_{stat}),\,\,\sigma(\omega\phi,\,W=4\,GeV)\simeq 2.5\cdot 10^{-3}\,nb$
Theory: $\sigma(\omega\phi,\,|\cos\theta|\leq 0.8)\simeq
12\,nb\,GeV^{6}/W^{6}\,,\,\,\sigma(\omega\phi,\,W=4\,GeV)\simeq 3\cdot
10^{-3}\,nb$
Fig.17 The total cross section $\sigma(\phi\phi)$ in the c.m. angular region
$|\cos\theta|<0.8$ [26].
W-dependence: $\sigma_{\phi\phi}(W)\sim W^{-n},\,\,n=(9.1\pm
0.6_{stat}),\,\,\sigma(\phi\phi,\,W=4\,GeV)\simeq 3.5\cdot 10^{-3}\,nb$
Theory: $\sigma(\phi\phi,\,|\cos\theta|\leq 0.8)\simeq
8\,nb\,GeV^{6}/W^{6},\,\,\sigma(\phi\phi,W=4\,GeV)\sim 2\cdot 10^{-3}\,nb$
Fig.18 The total cross section $\sigma(\omega\omega)$ in the c.m. angular
region $|\cos\theta|<1.0$
W-dependence: $\sigma_{\omega\omega}(W)\sim W^{-n},\,\,n=(8.4\pm
1.1_{stat}),\,\,\sigma(\omega\omega,\,W=4\,GeV)\simeq 15\cdot 10^{-3}\,nb$
Theory: $\sigma(\omega\omega,\,|\cos\theta|\leq 0.8)\simeq
4\,nb\,GeV^{6}/W^{6}\,,\,\,\sigma(\omega\omega,\,W=4\,GeV)\simeq 1\cdot
10^{-3}\,nb$
It is seen that there is an agreement between the predicted and measured cross
sections $\sigma(\omega\phi,\,|\cos\theta|\leq 0.8)$ and
$\sigma(\phi\phi,\,|\cos\theta|\leq 0.8)$ at $W=4\,GeV$.
The much larger measured cross section [26]
$\sigma(\omega\omega,\,|\cos\theta|<1)\gg\sigma(\phi\phi,\,|\cos\theta|\leq
0.8)\sim\sigma(\omega\phi,\,|\cos\theta|\leq 0.8)$
looks natural at the first sight as one expects it is dominated by the forward
region. But then it is strange that it decays so quickly,
$\sigma(\omega\omega,\,|\cos\theta|<1)\sim 1/W^{8}$.
But the authors said [27] :
"We do not observe many events in $0.9<|\cos\theta|<1.0$ angle range for
$\omega\omega$. Our measured cross section in the paper is for the whole angle
range in Fig.2c in the paper [26]. 222 Fig.18 here If we require
$|\cos\theta|<0.8$ for $\omega\omega$, the observed cross section dependence
on the energy in the high energy region is similar. This is only our
experimental observation. Due to very limited statistic, we can not measure
cross section for the range of $0.9<|\cos\theta|<1$ in high energy region".
Also, it is said in the article [26] that there is no detected events at
$|\cos\theta|>0.8$ for $\omega\phi$ and $\phi\phi$.
But if, due to experimental restrictions,
$\sigma(\omega\omega,\,|\cos\theta|<1)$ should be understood as
$\sigma(\omega\omega,\,|\cos\theta|<0.8)$, why then is it so large :
$\displaystyle\sigma(\omega\omega,\,|\cos\theta|<0.8)\sim
6\Biggl{(}\frac{\sigma(\omega\phi,\,|\cos\theta|<0.8)+\sigma(\phi\phi,\,|\cos\theta|<0.8)}{2}\Biggr{)}\quad?$
while the theory predicts
$\displaystyle\sigma(\omega\omega)<\sigma(\phi\phi)<\sigma(\omega\phi)\,.$
Short conclusions
1) The leading term QCD predictions
$d\sigma/d\cos\theta\sim{\overline{\alpha}}^{\,2}_{s}/(W^{6}\sin^{4}\theta)$
for charged mesons $\pi^{+}\pi^{-}$ and $K^{+}K^{-}$ agree reasonably well
with data both in the energy and angular dependences at energies
$W>2.5-3\,GeV$ [8]. The absolute values of cross sections agree reasonably
well with data [8] only for the wide pion (kaon) wave functions like
$\phi_{\pi,K}^{\rm CZ}(x)$ [7], while the asymptotic wave functions
$\phi_{\pi,K}(x)\simeq\phi^{\rm asy}(x)$ lead to cross sections one order
smaller than data.
2) In comparison with charged mesons, the QCD leading terms for neutral mesons
are much smaller so that non-leading terms may be dominant at present energies
$W<4\,GeV$ and in this case the energy dependence will be steeper,
$\sigma({\overline{M^{o}}}M^{o})\sim 1/W^{10}$. This agrees with the data on
$\sigma(K_{S}K_{S})$ [10] and $\sigma(\eta\pi^{o})$ [12], while
$\sigma(\pi^{o}\pi^{o})$ is consistent with $\sim 1/W^{10}$ at $6<W^{2}<$
$9-10\,GeV^{2}$, but behaves "abnormally" , $\sim(1/W)^{7-8}$, at
$10<W^{2}<16\,GeV^{2}$ [11].
This can have natural explanation as, unlike $\sigma(K_{S}K_{S})$, there is
the additional odderon contribution to $\sigma(\pi^{o}\pi^{o})$, see Fig.9.
With the pion wave function $\phi_{\pi}(x)=\phi^{\rm CZ}_{\pi}(x,\mu_{o})$ it
looks as
$\sigma^{(3\,\rm
gl)}(\pi^{o}\pi^{o},\,|\cos\theta|<0.8)\simeq\Bigl{(}\,\frac{{\overline{\alpha}}^{\,2}_{s}}{0.1}\,\frac{9\,{\rm
GeV}^{2}}{W^{2}}\Bigr{)}^{3}\cdot 0.3\,nb$
and may well be responsible for such an "abnormal" behavior of
$\sigma(\pi^{o}\pi^{o})$ at $W^{2}>9-10\,{\rm GeV}^{2}$. At the same time,
with $\phi_{\pi}(x)=\phi^{\rm asy}(x)$, the odderon contribution to the cross
section $\pi^{o}\pi^{o}$ will be $\simeq 13$ times smaller.
The role of the odderon contribution to $\eta\pi^{o}$ is somewhat smaller in
comparison with $\pi^{o}\pi^{o}$ and this may be a reason why it is still not
seen clearly in $\sigma(\eta\pi^{o})$ at $|\cos\theta|<0.8$ and
$10<W^{2}<16\,GeV^{2}$. The prediction is that it will be seen here at
somewhat higher energies.
3) In the Diehl-Kroll-Vogt approach [15], the handbag model gives definite
model independent predictions neither for the energy nor the angular
dependences of the cross sections $d\sigma(\gamma\gamma\to{\overline{M}}M)$.
The data are simply fitted with the model forms of amplitudes and a number of
free parameters.
4) The estimates of leading terms of the valence handbag amplitudes via the
standard light cone QCD sum rules [16] or from the valence Feynman mechanism
in QCD show that for all mesons, both charged and neutral, the soft handbag
cross sections behave as
$\frac{d\sigma_{\rm handbag}(\gamma\gamma\to
M_{2}M_{1})}{d\cos\theta}\sim\frac{const}{W^{10}}\,.$
This angular behavior disagrees with all data and the energy behavior
disagrees with the data for $(\pi^{+}\pi^{-})$ and $(K^{+}K^{-})$.
5) In the $SU(3)$ limit, the relatively large non-valence soft handbag form
factor, $R^{\,\rm nv}_{{\overline{M}}M}(s)\simeq\\\ 0.3R^{\,\rm
v}_{{\overline{M}}M}(s)$ at $s=9\,GeV^{2}$, parametrized in a model form with
a number of additional free parameters was used in the next paper of Diehl-
Kroll [23] to improve fits to the data. Unfortunately, so large soft non-
valence form factor contradicts the QCD estimates $R^{\,\rm
nv}_{{\overline{M}}M}(s)\sim({\overline{\alpha}}_{s}/\pi)^{2}R^{\,\rm
v}_{{\overline{M}}M}(s)\sim 0.01R^{\,\rm v}_{{\overline{M}}M}(s)$, see
(8),(9).
6) The very recent Belle data appeared on the cross sections of neutral vector
mesons $\sigma(\gamma\gamma\to\omega\phi,\,\phi\phi,\,\omega\omega)$ [26]. The
values of the cross sections $\sigma(\gamma\gamma\to\omega\phi)$ and
$\sigma(\gamma\gamma\to\phi\phi)$ at $W=4\,GeV$ and $|\cos\theta|<0.8$ are in
a reasonable agreement with the theory predictions, but the fitted energy
behavior at $2.7\,GeV<W<4\,GeV$ looks somewhat too steep (within large error
bars at $3.5<W<4\,GeV$), in comparison with the leading term QCD predictions
$\sigma(\gamma\gamma\to
V_{2}^{o}V_{1}^{o},|\cos\theta|<0.8)\sim{\overline{\alpha}}^{\,4}_{s}/W^{6}$.
But the value of the cross section $\sigma(\gamma\gamma\to\omega\omega)$ at
$W=4\,GeV$ and $|\cos\theta|<0.8$ looks too large in comparison with the
theory predictions.
I am grateful to organizers of this workshop Profs. Augustin E. Chen (NCU),
Hsiang-nan Li (AS) and Sadaharu Uehara (KEK), and to Prof. Hai-Yang Cheng (AS)
for a kind hospitality.
This work is supported in part by Ministry of Education and Science of the
Russian Federation and RFBR grant 12-02-00106-a.
## References
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Erratum: ibid B214 (1983) 547
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* [22] G. Duplancic, Private communication
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|
arxiv-papers
| 2012-12-06T12:05:42 |
2024-09-04T02:49:38.961231
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Victor L. Chernyak",
"submitter": "Victor Chernyak",
"url": "https://arxiv.org/abs/1212.1304"
}
|
1212.1406
|
3cm1.5cm3cm7cm
# The Maxflow problem and a generalization to simplicial complexes
Fabián Latorre
(2012)
Acknowledgements
Gracias a la cigueña y quienes, no siendo su decisión, han tenido que vivir en
mi tiempo y beber junto a mí y junto a Baco. Gracias por ser mis
contemporáneos, y que la casualidad nos haya llevado a conocernos y tal vez,
ver un poco más que un autómata el uno en el otro.
Gracias a mi asesor Mauricio Velasco a quien ha dedicado gran parte de su
tiempo a guíar este proyecto, a mis padres y hermanos.
###### Contents
1. 1 Introduction
2. 2 Preliminaries
1. 2.1 Flow in a network
2. 2.2 The problem of MAXFLOW
3. 3 MAXFLOW algorithms
1. 3.1 The Ford-Fulkerson algorithm
2. 3.2 The Goldberg-Tarjan algorithm
3. 3.3 Hochbaum’s pseudoflow
4. 4 Applications
1. 4.1 Hall’s Marriage Theorem
2. 4.2 Counting disjoint chains in finite posets
3. 4.3 Image segmentation
5. 5 A Generalization of Maxflow
1. 5.1 Preliminaries
2. 5.2 Higher Maxflow
3. 5.3 As an LP problem
4. 5.4 Further examples and conjectures
## 1 Introduction
The problem of Maxflow was formulated by T.E. Harris in 1954 while studying
the Soviet Union’s railway network, under a military research program financed
by RAND, Research and Development corporation. The research remain classified
until 1999. The Maxflow problem is defined on a _network_ which is a directed
graph together with a real positive capacity function defined on the set of
edges of the graph and two vertices $s,t$ called the _source_ and the _sink_.
A flow is another function of this type that respects capacity constraints and
a Kirchoff’s law type restriction on each vertex except the source and sink.
The net flow of a flow is defined as the amount of flow leaving the source.
The problem of maxflow is to find a flow with maximum net flow on a given
network. In the first section we will define clearly such concepts and present
basic results in the subject.
Throughout the second section, we will present three different algorithms for
the solution of Maxflow. In 1956 L. Ford and D. Fulkerson devised the first
known algorithm that solves the problem in polynomial time. The algorithm
works starting with the zero flow and finding paths from source to sink where
flow can be augmented preserving the flow and capacity restrictions. We then
analyze a more efficient algorithm developed by A. Goldberg and E. Tarjan in
1988. This algorithm works in a different fashion starting with a _preflow_ ,
a function saturating edges adjacent to the source, and then pushing excess of
flow to vertices estimated to be closer to the sink. At the end of the
algorithm the preflow becomes a flow and in fact, a maximum flow. Finally we
describe Dorit Hochbaum’s _pseudoflow_ algorithm, which is the most efficient
algorithm known to day that solves the Maxflow problem.
In the third section we show the usefulness of this subject and present three
applications of the theory of Network flow. First we show how well-known
theorems in combinatorics such as the Hall’s Marriage theorem can be proven
using Maxflow results. We then show how to find a set of maximal chains in a
poset with certain properties using the results in the previous sections.
Finally we describe an algorithm for _image segmentation_. an important
subject in computer vision, that relies on the relation between a maximum flow
and a _minimum cut_.
The problem of Maxflow is a widely developed subject in modern mathematics.
Efficient algorithms exist to solve this problem, that is why a good
generalization may permit these algorithms to be understood as a particular
instance of solutions in a wider class of problems. In the last section we
suggest a generalization in the context of simplicial complexes, that reduces
to the problem of Maxflow in graphs, when we consider a graph as a simplicial
complex of dimension 1.
## 2 Preliminaries
### 2.1 Flow in a network
There are many equivalent ways to define the objects needed to state our
problem. We will work with the following:
###### Definition 2.1.
A _network_ is a pair $(G,c)$ such that
1. i)
$G=(V,E)$ is a finite simple directed graph
2. ii)
$V=V^{\prime}\cup\\{s,t\\}$
3. iii)
for any $v\in V$, $(v,s)\not\in E$ and $(t,v)\not\in E$.
4. iv)
$c:E\longrightarrow{\mathbb{R}}_{+}$
We call $s$ and $t$ the _source_ and the _sink_ respectively. Condition iii)
means that there are no edges _into_ the source, and no edges _out_ of the
sink.
###### Definition 2.2.
For any simple directed graph we define the _incidence function_
$\phi:V\times E\longrightarrow\\{-1,0,1\\}$ as follows:
$\phi(v,e)=\left\\{\begin{array}[]{rl}1&:e=(v,x)\\\ -1&:e=(x,v)\\\
0&:\mbox{else}\end{array}\right.$
###### Definition 2.3.
A flow on a network $(G,c)$ is a function $f:E\longrightarrow{\mathbb{R}}_{+}$
such that:
1. i)
for any edge $e$, $f(e)\leq c(e)$
2. ii)
for any vertex $v\not=s,t$ _conservation of flow_ holds:
$\sum_{e\in E}f(e)\phi(v,e)=0$
###### Definition 2.4.
For a flow $f$ on a network $(G,c)$, we define the _net flow_ :
$|f|:=\sum_{e\in E}f(e)\phi(s,e)$
$|f|$ is the total amount flowing out of the source.
### 2.2 The problem of MAXFLOW
Given a network $(G,c)$ the MAXFLOW problem is to find a flow $f$ of maximum
net flow.
###### Theorem 2.5.
For any network $(G,c)$ there exists a flow $f$ of maximum net flow.
###### Proof.
Let $m:[|E|]\rightarrow E$ be an enumeration of the edges of $G$. Let ${\cal
F}$ be the set of feasible flows on $(G,c)$. The map $\psi:{\cal F}\rightarrow
A\subset{\mathbb{R}}^{|E|}$, $f\mapsto[f(m(i))]_{i}$ is a bijection between
${\cal F}$ and a subset of ${\mathbb{R}}^{|E|}$. Let ${\cal F}^{*}$ be the
image of $\cal F$ under this map. From definition 2.3, the edge capacity
constraints imply that $\cal F^{*}$ is bound, and the flow conservation
constraints imply that it is closed, hence $\cal F^{*}$ is compact. The map
$f^{*}\in{\cal F^{*}}\mapsto|\psi^{-1}(f^{*})|\in{\mathbb{R}}$ is a linear
map, hence continuous. As it is defined on a compact set, it achieves a
maximum value, say at $f_{\mbox{max}}$. $\psi^{-1}(f_{\mbox{max}})$ is then a
flow of maximum value.
∎
The previous theorem shows that, in fact, MAXFLOW is a linear programming
problem, the most important results of which can be proved with LP theory. We
discuss this formulation in detail in what follows.
###### Definition 2.6.
Let $G=(V,E)$ be a simple directed graph, $\phi$ its incidence function and
$(G,c)$ a network. Let $n:=|V|$, $m:=|E|$, $v:[n]\rightarrow V$ be an
enumeration of the vertices and $e:[m]\rightarrow E$ be an enumeration of the
edges. We define the _incidence matrix_ $\Phi_{v,e}$with respect to the
enumerations $v,e$ as
$\Phi_{v,e}(i,j)=\phi(v(i),e(j))$
Figure 1: a graph $G_{1}$ $\left[\begin{tabular}[]{ c c c c c }1&1&0&0&0\\\
-1&0&1&0&1\\\ 0&-1&0&1&-1\\\ 0&0&-1&-1&0\par\end{tabular}\right]$ Table 1:
Incidence matrix of $G_{1}$
From now on, we suppose a network $(G,c)$ has fixed enumerations $v,e$, of
vertices and edges. We take $n$ as the number of vertices, $m$ as the number
of edges and suppose that $v(1)=s$, $v(n)=t$, then we refer simply to the
incidence matrix as $\Phi$.
###### Lemma 2.7.
Given a network $(G,c)$, the problem of MAXFLOW is equivalent to the following
LP problem:
$\begin{array}[]{ccccccc}{\displaystyle\mathop{\max}_{x\in{\mathbb{R}}^{m}}\Phi_{1,j}\cdot
x}&;&\Phi^{*}x=0&;&I_{m}x\leq c^{*}&;&x\geq 0\end{array}$
where $\Phi_{1,j}$ is the first row vector of the matrix $\Phi$, $\Phi^{*}$ is
the matrix that results from $\Phi$ by deleting the first and last rows,
$I_{m}$ is the identity matrix of size $m$ and $c^{*}:=[c(e(i))]_{i}$ is the
vector of edge capacities.
###### Proof.
Follows from the definition 2.3 and the proof of theorem 2.5.
∎
###### Definition 2.8.
For a linear program (called the primal problem)
$\begin{array}[]{ccccc}{\displaystyle\mathop{\max}_{x}(c^{T}x)}&;&Ax\leq
b&;&x\geq 0\end{array}$
the dual program is defined as
$\begin{array}[]{ccccc}{\displaystyle\mathop{\min}_{y}(b^{T}y)}&;&A^{T}y\geq
c&;&y\geq 0\end{array}$
We will compute the dual program of MAXFLOW.
###### Definition 2.9.
Let $(G,c)$ be a network. A _cut_ is a partition of $V$ into two disjoint
subsets $(S,\bar{S})$ such that $s\in S$ and $t\in\bar{S}$. Let
$E^{\prime}:=\\{e\in E:e=(s,\bar{s})\,s\in S\,,\,\bar{s}\in\bar{S}\\}$. We
define the _capacity_ of the cut $(S,\bar{S})$
$C(S,\bar{S})=\sum_{e\in E^{\prime}}c(e)$
$C(S,\bar{S})$ is the sum of the capacities of the edges directed from $S$ to
$\bar{S}$. We say an edge $e$ _traverses_ the cut if $e=(s,\bar{s})\,s\in
S\,\bar{s}\in\bar{S}$
###### Lemma 2.10.
For any cut $(S,S^{\prime})$
$|f|=f(S,S^{\prime})-f(S^{\prime},S)$
where $f(X,Y)$ is the sum of the values of $f$ at the edges directed from $X$
to $Y$.
###### Proof.
$|f|=\sum_{e\in E}f(e)\phi(s,e)=\sum_{v\in S}\sum_{e\in
E}f(e)\phi(v,e)=\sum_{e\in E}\sum_{v\in S}f(e)\phi(v,e)$
$=\underbrace{\sum_{e:S\rightarrow S}\sum_{v\in
S}f(e)\phi(v,e)}_{=0}+\underbrace{\sum_{e:S\rightarrow S^{\prime}}\sum_{v\in
S}f(e)\phi(v,e)}_{=f(S,S^{\prime})}+\underbrace{\sum_{e:S^{\prime}\rightarrow
S}\sum_{v\in S}f(e)\phi(v,e)}_{-f(S^{\prime},S)}$
∎
###### Lemma 2.11.
Given a network $(G,c)$, for any flow $f$ and any cut $(S,\bar{S})$
$|f|\leq C(S,\bar{S})$
The problem of MINCUT is to find a cut of $(G,c)$ of minimum capacity. Two of
the most important results are the following
###### Theorem 2.12.
MAXFLOW=MINCUT. This means the net flow of a maximal flow is equal to the
capacity of a minimal cut.
In order to prove these results, we will see that the dual program of MAXFLOW
is a relaxation of the MINCUT problem and use the following:
###### Theorem 2.13.
_(weak duality)_. Let $x^{*}$ and $y^{*}$ be feasible solutions to a primal
problem and its dual, respectively, then
$c^{T}x^{*}\leq b^{T}y^{*}$
###### Proof.
For $y_{1},y_{2}\geq 0$ define the function
$g(y_{1},y_{2}):={\max_{x}}c^{T}x+y_{1}^{T}(b-Ax)+y_{2}^{T}x$
Clearly, for any feasible $x^{*}$ and $y_{1},y_{2}\geq 0$, $c^{T}x^{*}\leq
g(y_{1},y_{2})$.
Rearranging terms we have
$g(y_{1},y_{2})={\max_{x}}(c^{T}-y_{1}^{T}A+y_{2}^{T})x+y_{1}^{T}b$
where $y_{1}\in{\mathbb{R}}^{m}$ and $y_{2}\in{\mathbb{R}}^{n}$. Then we have
$g(y_{1},y_{2})=\left\\{\begin{array}[]{rl}y_{1}^{T}b&:c^{T}-y_{1}^{T}A+y_{2}^{T}=0\\\
\infty&:\mbox{else}\end{array}\right.$
Now minimizing over $y_{1},y_{2}\geq 0$ we have, for any feasible $x^{*}$
$c^{T}x^{*}\leq\min_{y_{1},y_{2}\geq 0}g(y_{1},y_{2})$
By the previous observation this is equivalent to $\min_{y_{1},y_{2}\geq
0}y_{1}^{T}b$ subject to $c^{T}-y_{1}^{T}A+y_{2}^{T}=0$. We see that
$c^{T}-y_{1}^{T}A+y_{2}^{T}=0\Rightarrow c^{T}-y_{1}^{T}A=-y_{2}^{T}\leq 0$
and this is equivalent to a single variable $y\geq 0$ such that
$c^{T}-y^{T}A\leq 0\Rightarrow c^{T}\leq y^{T}A$ and this is the dual problem,
as we wanted to show.
∎
###### Theorem 2.14.
_(strong duality)_. If the primal problem has an optimal solution $x^{*}$,
then the dual problem also has an optimal solution $y^{*}$ and
$c^{T}x^{*}=b^{T}y^{*}$
To find the dual of our problem, we state it in standard form
$\begin{array}[]{ccccc}{\displaystyle\mathop{\max}_{x\in{\mathbb{R}}^{m}}\Phi_{1,j}\cdot
x}&;&\left[\begin{array}[]{c}\Phi^{*}\\\ -\Phi^{*}\\\
I_{m}\end{array}\right]x\leq\left[\begin{array}[]{c}0_{n-2}\\\ 0_{n-2}\\\
c\end{array}\right]&;&x\geq 0\end{array}$ (1)
Where $0_{n}$ is the zero column vector of length $n$. Then we find the dual
to be
$\begin{array}[]{ccccc}{\displaystyle\mathop{\min}_{y\in{\mathbb{R}}^{2(n-2)+m}}\left[\begin{array}[]{c}0_{n-2}\\\
0_{n-2}\\\
c\end{array}\right]^{T}y}&;&\left[\begin{array}[]{ccc}\Phi^{*T}&-\Phi^{*T}&I_{m}\end{array}\right]y\geq\Phi_{1,j}^{T}&;&y\geq
0\end{array}$
After further inspection this is equivalent to $n-2$ unrestricted in sign
variables, one for each vertex $v(i)\neq s,t$ called $v_{i}$, and $m$
variables, one for each edge $e_{j}$ such that
$\begin{array}[]{ccccc}{\displaystyle\mathop{\min}_{[v,e]\in{\mathbb{R}}^{n-2+m}}c^{T}e}&;&\left[\begin{array}[]{cc}\Phi^{*T}&I_{m}\end{array}\right][v,e]^{T}\geq\Phi_{1,j}^{T}&;&e\geq
0\end{array}$ (2)
These restrictions translate to the following set of inequalities:
$v_{i}-v_{j}+e_{k}\geq 0:v(i)\neq s,t\,\,v(j)\neq s,t\,\,e(k)=(v(i),v(j)):i<j$
$-v_{i}+e_{k}\geq 1:v(i)\neq s,t\,\,e(k)=(s,v(i))$ $v_{i}+e_{k}\geq 0:v(i)\neq
s,t\,\,e(k)=(v(i),t)$ $e_{k}\geq 0\,\,\forall k\in\\{1,\ldots,m\\}$
We can define $v_{1}=-1$ and $v_{n}=0$ and we write all the equations in the
form
$v_{i}-v_{j}+e_{k}\geq 0:e(k)=(v(i),v(j)):i<j$
###### Lemma 2.15.
For any cut $(S,\bar{S})$ there exists a feasible solution of (2) such that
the value of the function at this feasible solution equals the capacity of the
cut.
###### Proof.
Let $(S,\bar{S})$ be a cut. Define $e_{k}=1$ if and only if $e(k)=(s,\bar{s})$
with $s\in S$, $\bar{s}\in\bar{S}$ and $e_{k}=0$ in any other case. $v_{i}=-1$
if and only if $v(i)\in S$ and $v_{i}=0$ in any other case. Then
$c^{T}e=C(S,\bar{S})$ and it is straightforward to check that the restrictions
hold.
∎
As a corollary we get
###### Corollary 2.16.
Lemma (2.11)
###### Lemma 2.17.
For an optimal solution $[v^{*},e^{*}]$ of (2) there exists a cut
$(S,\bar{S})$ such that $C(S,\bar{S})\leq c^{T}e^{*}$.
###### Proof.
Let $\chi\in[-1,0]$ be a random variable with uniform distribution. Define a
random variables for each edge by
$e(k)_{\chi}=\left\\{\begin{array}[]{rl}1&:v_{i}^{*}\leq\chi\leq v_{j}^{*}\\\
0&:\mbox{else}\par\end{array}\right.$
Note that this assignment defines a random cut. If $v_{i}^{*}<v_{j}^{*}$ then
${\mathbb{P}}(v_{i}^{*}\leq\chi<v_{j}^{*})=\min\\{1,v_{j}^{*}-v_{i}^{*}\\}\leq
v_{j}^{*}-v_{i}^{*}\leq e_{k}^{*}$ then by the restrictions of the problem, we
get
$\mathbb{E}(C(S,\bar{S}))=\sum c_{k}\mathbb{E}(e(k)_{\chi})\leq\sum
c_{k}e_{k}^{*}=c^{T}e^{*}$
As the expected value of the random cut capacity is less or equal to the
optimal value of the problem, there exists a cut of capacity less or equal to
the optimal value.
∎
This proves that the dual of MAXFLOW is in fact a relaxation of MINCUT and we
get, by strong duality
###### Corollary 2.18.
Theorem (2.12)
## 3 MAXFLOW algorithms
### 3.1 The Ford-Fulkerson algorithm
L.R, Ford Jr. and D.R. Fulkerson devised a polynomial time algorithm to
compute a maximal flow first published in 1962 [1]. We introduce some new
concepts needed to describe the algorithm, and prove some general facts about
it.
###### Definition 3.1.
Given a network $(G,c)$ and a flow $f$ we define $\bar{c},\bar{f}:V\times
V\rightarrow{\mathbb{R}}$
$\bar{c}(u,v)=\left\\{\begin{array}[]{rl}c(u,v)&:(u,v)\in E\\\
0&:\mbox{else}\end{array}\right.$
$\bar{f}(u,v)=\left\\{\begin{array}[]{rl}f(u,v)&:(u,v)\in E\\\
-f(u,v)&:(v,u)\in E\\\ 0&:\mbox{else}\end{array}\right.$
###### Lemma 3.2.
For a given flow $f$ on a network $(G,c)$, for any fixed vertex $u\neq s,t$
$\sum_{v\in V}\bar{f}(u,v)=0$
###### Remark 3.3.
In fact, it is equivalent to define a flow as a function $f$ mapping $V\times
V$ to the reals such that the equation in definition 3.2 holds, and such that
$f(u,v)=-f(v,u)$ for every pair of vertices. From now on we refer to a flow in
this sense, and we refer to $\bar{f},\bar{c}$ as $f$ and $c$ whenever it does
not cause confusion. Under this new definitions we have that the capacity of a
cut can be written as
$C(S,S^{\prime})=\sum_{u\in S,v\in S^{\prime}}c(u,v)$
and lemma (2.10) translates to
$|f|=\sum_{u\in S,v\in S^{\prime}}f(u,v)$
for any cut $(S,S^{\prime})$
###### Definition 3.4.
For a network $(G,c)$ and a flow $f$ we define the _residual capacity_ of a
pair of vertices $(u,v)$ as $c_{f}(u,v)=c(u,v)-f(u,v)$. Any such pair with
residual capacity greater than zero is called a residual edge. Note that the
residual capacity is always greater or equal to zero. We define the _residual
graph_ $G_{f}$ as the graph with vertex set that of $V(G)$ and edge set the
set of residual edges.
###### Definition 3.5.
Given a flow $f$ on a network $(G,c)$, an _augmenting path_ is a directed path
on $G_{f}$ from source to sink.
###### Lemma 3.6.
a flow $f$ is maximal if and only if there is no augmenting path on $G_{f}$.
###### Proof.
Suppose there is no augmenting path on $G_{f}$. Let $S$ be the set of vertices
$v$ such that there exists a directed path from $s$ to $v$ in $G_{f}$. Let
$S^{\prime}=V\backslash S$. $(S,S^{\prime})$ is then a cut. By definition of
S, we have that $c_{f}(u,v)=0\Rightarrow c(u,v)=f(u,v)$ for any $u\in S$,
$v\in S^{\prime}$ then we have, following remark (3.3)
$C(S,S^{\prime})=\sum_{u\in S,\,v\in S^{\prime}}c(u,v)=\sum_{u\in S,\,v\in
S^{\prime}}f(u,v)=|f|$
So $f$ is a maximal flow by Theorem (2.12) or Theorem(2.13).
Now suppose there is an augmenting path
$(v_{0}=s,v_{1},\ldots,v_{k-1},v_{k}=t)$. Let
$A=\min\\{c_{f}(v_{i},v_{i+1}):i=0,\ldots,k-1\\}>0$. Define $F:V\times
V\rightarrow{\mathbb{R}}$ as $F(v_{i},v_{i+1})=f(v_{i},v_{i+1})+A$,
$F(v_{i+1},v_{i})=f(v_{i+1},v_{i})-A$ and $F(u,v)=f(u,v)$ on any other pair of
vertices. One can easily check that $F$ is a flow, and that $|F|=|f|+A$ so $f$
is not a maximal flow.
∎
Now this theorem is the basic result needed to state the Ford-Fulkerson
algorithm. Starting with the zero flow, as long as there exists an augmenting
path with respect to such flow, we can increase the value of the flow by $A$
as defined in the above proof.
###### Lemma 3.7.
If $(G,c)$ is such that $c(u,v)\in\mathbb{Z}$ then the algorithm terminates.
###### Proof.
At each step of the algorithm, the value is increased by $A\geq 1$ so a
maximal flow is reached after a finite number of steps.
∎
As corollaries we get
###### Corollary 3.8.
If the capacities of a network are integers, then the value of the maximal
flow is an integer and there exists a maximal flow with $f(u,v)\in\mathbb{Z}$
for every edge $(u,v)$.
###### Corollary 3.9.
If the capacities of a network are rational numbers, then the algorithm
terminates.
In fact there are examples of networks with irrational capacities such that
the algorithm never terminates, moreover, the value of the flow in each step
does not converge to the actual value of the maximal flow, so our algorithm
must have as a condition that the capacity is at least a rational valued
function. Then, the running time of the algorithm depends on the way the
augmenting paths are chosen. There are many ways to find an augmenting path,
like the shortest augmenting path or the largest bottleneck (value of $A$)
augmenting path, that lead to a polynomial time algorithm.
### 3.2 The Goldberg-Tarjan algorithm
The Goldberg-Tarjan algorithm [2] is another polynomial time algorithm with a
different approach to the problem of finding a maximal flow. Instead of
increasing the flow along augmenting paths, it starts with a _preflow_ , which
is a function on $V\times V$ which satisfies _excess of flow_ at each vertex,
and then pushes _excess flow_ to edges closer to the sink. Next we formalize
these concepts following Goldberg-Tarjan’s article [2].
###### Definition 3.10.
Given a network $(G,c)$ a preflow is a function $f:V\times
V\rightarrow{\mathbb{R}}$ satisfying:
1. i)
$f(u,v)\leq c(u,v)$
2. ii)
$f(u,v)=-f(v,u)$
3. iii)
for any vertex $x\neq s$, $\sum_{v\in V}f(x,v)\geq 0$
###### Definition 3.11.
for a network $(G,c)$ and given a preflow $f$ on the network, we redefine the
residual capacity of $(u,v)\in V\times V$ as $c_{f}(u,v)=c(u,v)-f(u,v)$. If
$c_{f}(u,v)>0$ we call such pair a _residual edge_. We define the residual
graph as the directed graph having vertex set $V$ and edge set the set of
residual edges.
Note there are similarities with definition (3.4) but in this definition we
are working with a preflow rather than a flow.
###### Definition 3.12.
the _excess flow_ at a vertex $x\in V$ is defined as $\sum_{v\in V}f(x,v)\geq
0$.
###### Definition 3.13.
given a a _valid labeling_ on a network $(G,c)$ is a function
$d:V\rightarrow\mathbb{Z}_{\geq 0}\cup\\{\infty\\}$ such that $d(s)=n$,
$d(t)=0$ and $d(v)\leq d(w)+1$ for every residual edge $(u,v)$.
It can be shown that for any vertex $v$, if $d(v)<n$ then $d(v)$ is a lower
bound on the distance from $v$ to $t$ in the residual graph and if $d(v)\geq
n$ then $d(v)-n$ is a lower bound on the distance to $s$ in the residual graph
[2]. This labeling of the vertices permits the algorithm to push excess flow
to vertices that are estimated to be closer to the sink and, if needed, to
return flow to vertices estimated to be closer to the source.
###### Definition 3.14.
a vertex $v$ is called active if $d(v)<\infty$ and $e(v)\geq 0$.
Now we define the basic operations, push and relabel, that the main algorithm
uses.
Push. Let $(v,w)$ be such that $v$ is an active vertex, $c_{f}(u,v)>0$ and
$d(v)=d(w)+1$. Define $\delta=\min(e(v),c_{f}(v,w))>0$. Redefine
$f(v,w):=f(v,w)+\delta$, $f(w,v):=f(w,v)-\delta$, $e(v):=e(v)-\delta$ and
$e(w):=e(w)+\delta$.
Relabel. Let $v$ be an active vertex such that for any $w\in V$,
$c_{f}(v,w)>0\Rightarrow d(v)\leq d(w)$. Redefine
$d(v):=\min\\{d(w)+1\,:\,(v,w)\,\mbox{residual edge}\\}$.
As initial preflow we take the function $f$ such that for any $v\in V$,
$f(s,v)=c(s,v)$ and zero everywhere else. It is readily checked that this is a
preflow. As an initial labeling of the vertices we take $d(s)=n$ and zero
everywhere else. As long as there is an active vertex $v$, either an operation
of push or relabel is applicable to $v$. When there are no more active
vertices the algorithm terminates, and the preflow becomes a flow, and in
fact, it is maximal. Details of the proof of correctness and termination of
the algorithm can be found in [2]. We show only correctness assuming
termination.
###### Lemma 3.15.
If $f$ is a preflow and $d$ is any valid labeling for $f$ then the sink $t$ is
not reachable from $s$ in $G_{f}$.
###### Proof.
Suppose $(s=v_{0},\ldots,v_{k}=t)$ is a path from $s$ to $t$ in the residual
graph. Clearly $k\leq n$. Now $(v_{i},v_{i+1})$ is a residual edge for every
$i$. So by definition of valid labeling $d(v_{i})\leq d(v_{i+1}+1)$ so we have
$d(s)\leq d(v_{1})+1\leq\ldots\leq d(t)+k=k$ this contradicts the fact that
$d(s)=n$.
∎
Now recall lemma (3.6).
###### Theorem 3.16.
If the algorithm terminates and $d$ is a valid labeling for $f$ with finite
labels, then $f$ is a maximal flow.
###### Proof.
At the end of the algorithm there are no active vertices, as the labels are
finite, it means that all vertices have zero excess, so $f$ is a flow. By
lemma (3.15) and lemma (3.6) this flow is in fact maximal.
∎
One important remark about this algorithm is the fact that it always works (it
terminates and it is correct) no matter what type of capacity function we are
dealing with. The Ford-Fulkerson fails to terminate in some cases where the
capacity function is not rational. It is also important to note that the
algorithm relies only on _local_ operations, that means the operations depend
and modify only parameters related to a small part of the graph, this allows a
parallel implementation of the algorithm that takes advantage of multicore
processors. A special implementation of such algorithm terminates after
$O(n^{2}m)$ steps.
### 3.3 Hochbaum’s pseudoflow
Dorit Hochbaum’s _pseudoflow algorithm_ [3] is an algorithm with a different
approach to the maximum flow problem. Instead of directly finding a maximum
flow, it first solves the _maximum blocking cut_ problem, then a maximum flow
is recovered. Although the most complicated of the three, it is also the most
efficient. We follow [3]:
###### Definition 3.17.
A _pseudoflow_ $f$ on a given network $(G,c)$ is a function $f:V\times
V\rightarrow{\mathbb{R}}$ such that
1. i)
$f(v,w)=-f(w,v)$, $\forall(v,w)\in V\times V$
2. ii)
$0\leq f(v,w)\leq c(v,w)$, $\forall(v,w)\in E$
The concept of pseudoflow drops the conservation of flow constraint, preserves
the capacity constraint on the edges of the graph and the antisymmetry
constraint on $V\times V$. We define the residual capacity and residual graph
in the same manner we did with flows and preflows.
###### Definition 3.18.
For a directed, weighted, simple graph $G=(V,E)$ with weights $w_{v}$ for each
$v\in V$ and arc capacities $c(v,w)$ for each $(v,w)\in V\times V$, we will
define $G_{st}$ as a directed graph with vertex set
$V_{st}=V\cup\\{s,t\\}$,edge set $E_{st}=E\cup A(s)\cup A(t)$ where
$A(s)=\\{(s,v):w_{v}>0\\}$ and $A(t)=\\{(v,t):w_{v}<0\\}$ and arc capacities
$c(s,v)=w_{v}$, $c(v,t)=-w_{v}$ and the other arc capacities left unchanged.
Starting from $G_{st}=(V_{st},E_{st})$ we define the extended network
$G^{ext}$ as the graph obtained from $G_{st}$ by identifying $s,t$ as a single
vertex $r$ and adding the edges $(r,v),(v,r)$ for every $v\in
V_{st}\backslash\\{s,t\\}$.
We define the excess of flow at a vertex $e(v)$ as in definition ( 3.12).
Now we consider a pseudoflow $f$ on $G_{st}$ and a rooted spanning tree with
root $r$, $T$ of $G^{ext}$ such that
1. i)
$f$ saturates all arcs in $A(s)\cup A(t)$
2. ii)
For every arc in $E\backslash T$, $f$ is either zero or saturates the arc.
3. iii)
In every branch all downward residual capacities are strictly positive.
4. iv)
the direct children of $r$ are the only vertices that do not have zero excess.
###### Definition 3.19.
a spanning rooted tree with root $r$ of $G^{ext}$ that satisfies the previous
conditions is called a _normalized tree_. Note that this is an undirected
graph.
A child $r_{i}$ of $r$ is classified as:
1. i)
Strong if $e(v)>0$
2. ii)
Weak if $e(v)\leq 0$
A vertex $v$ is called weak or strong if it has a weak or strong ancestor,
respectively.
As we mentioned earlier, Hochbaum’s algorithm solves first the maximum
blocking cut problem, which we state next:
Problem: For a directed, weighted graph $G=(V,E)$ with vertex weights $w(v)$
for each vertex $v$, and arc capacity function $c(a,b)$ defined for every
$(a,b)\in E$, find $S\subset V$ such that
$surplus(S)=\sum_{v\in S}w(v)-\sum_{\begin{subarray}{c}a\in S\\\
b\in\bar{S}\end{subarray}}c(a,b)$
is maximum. Such a set is called a maximum surplus set and $(S,\bar{S})$ is
called a maximum blocking cut.
The key is to find the relation between a maximum blocking cut in $G$ and a
minimum cut in $G_{st}$. Given by the following lemma:
###### Lemma 3.20.
$\\{s\\}\cup S$ is the source set of a minimum cut in $G_{st}$ if and only if
$(S,\bar{S})$ is a maximum blocking cut in $G$.
This is proven in [3] following an article by Radzik [4]. The following lemma,
also found on the article [3], is fundamental for the correctness of the
algorithm.
###### Lemma 3.21.
For a normalized tree $T$ and pseudoflow $f$ on $G_{st}$ saturating $A(s)$ and
$A(t)$ and a set of strong vertices $S$, if the residual capacity of any edge
$(a,b)$ with $a\in S$ and $b\in\bar{S}$, $c_{f}(a,b)=0$ is zero then $S$ is a
maximum surplus set and $(S,\bar{S})$ is a maximum blocking cut.
For a normalized tree $T$ if the set of strong vertices $S$ satisfies the
condition in lemma (3.21) the tree is called _optimal_
The algorithm starts with a normalized tree related to a pseudoflow $f$ on
$G_{st}$. There are multiple choices of such a tree. We will start with a
_simple normalized tree_. It corresponds to a pseudoflow $f$ saturating $A(s)$
and $A(t)$ on $G_{st}$. In this normalized tree every vertex in $V$ forms an
independent branch. The set of strong vertices are those adjacent to the
source.
By lemma (3.21), it is desirable to reduce the residual capacity from strong
to weak vertices, therefore, with each iteration of the algorithm, a residual
edge from $S$ to $\bar{S}$ is chosen, this is called a _merger arc(edge)_. If
such an edge does not exist then the tree is optimal and the set of strong
vertices form a maximum blocking cut. If there is one, then such edge becomes
a new edge of the tree and the edge joining the root of the strong branch to
$r$ is removed from the tree. Then the excess of the root of the strong branch
is pushed upwards until it reaches the root of the weak branch. Note that this
path is unique.
It is not always possible to push the total of the excess along an edge.If
there is an edge, say $(a,b)$ that does not have enough residual capacity to
push the excess then such edge is removed (split) from the tree, $a$ (the tail
of the edge) becomes the root of a new strong branch with excess equal to the
excess pushed minus the residual capacity of the edge. This is done in such a
way so that the property that only roots of branches may have nonzero excess
is maintained through the running of the algorithm. The remaining excess at
$b$ continues to be pushed in the same fashion until it reaches the root of
the weak branch or until it reaches another edge that does not have enough
residual capacity and the process is repeated. This process assures that the
tree is normal at the end of each iteration.
Termination of the algorithm follows from the next lemma:
###### Lemma 3.22.
At each iteration of the algorithm either the total excess of the strong
vertices is strictly reduced or the number of weak vertices is reduced.
###### Proof.
Recall that from the properties of definition (3.19) we have that all downward
residual capacities of edges are positive. After appending a merger edge to
the tree and removing the edge joining the root of the strong branch $r_{s}$
to $r$, the path from $r_{s}$ to the weak branch becomes an _upward_ path with
positive residual capacity at each edge of the path, then some positive amount
of excess arrives at the weak branch that is being merged. Then either some
positive amount of excess arrives at the root of the weak branch and the total
excess is strictly reduced, or there is some edge in the weak branch without
enough residual capacity. In this case the edge is split and the tail of such
edge becomes a strong vertex. Note that if some weak vertex becomes strong in
this fashion, then all of its children, including the former strong branch,
becomes strong. Then if such operation takes place, the number of weak
vertices is strictly reduced.
∎
Now let $M^{+}=C(\\{s\\},V)$ be the sum of capacities in $A(s)$ and $M^{-}$ be
the sum of capacities in $A(t)$ then by the final comment in the previous
lemma we see that any iteration that reduces the total excess is separated
from another iteration of such type by at most $n$ iterations. Then it follows
immediately for integer capacities that
###### Corollary 3.23.
The complexity of the algorithm is $O(nM^{+})$
Now as the problem is symmetrical on $s$ and $t$ we find that by reversing all
directions of the edges of the graph and interchanging $s$ and $t$ we get an
equivalent problem so it follows again that for integer capacities
###### Corollary 3.24.
The complexity of the algorithm is $O(n*\min\\{M^{+},M^{-}\\})$
Correctness of the algorithm follows from lemma (3.21) as at the end of the
algorithm there are no merger arcs left.
Now in order to solve our initial problem we have to recover a maximum flow
from the pseudoflow and maximum blocking cut obtained after the algorithm
terminates, as it is not guaranteed that the pseudoflow becomes a flow after
termination. In what follows we describe how to recover such maximum flow.
###### Definition 3.25.
An $s-t$ path-flow on a network $(G,c)$ is a flow $f$ on $(G,c)$ such that the
edges carrying a strictly positive amount of flow form an $s-t$ path on
$(G,c)$. A cycle-flow on $(G,c)$ is a flow on $(G,c)$ such that the edges
carrying a strictly positive amount of flow form a directed cycle on $(G,c)$.
###### Theorem 3.26.
(Flow decomposition) Let $f$ be a flow on $(G,c)$, then $f$ can be decomposed
as the sum of at most $m$ $s-t$ path-flows and cycle-flows.
###### Proof.
Suppose $f$ is such that $|f|>0$ then there is some $(s,v_{1})\in E$ such that
$f(s,v_{1})>0$. If $v_{1}=t$ we have a directed $s-t$ path and we define a
flow $f_{0}$ carrying an amount of flow $f(s,v_{1})$ on such a path and zero
everywhere else. If $v_{1}\neq t$ then there exists some edge $(v_{1},v_{2})$
with some positive amount of flow as a result of conservation of flow. In this
way we construct an $s-t$ path (we may suppose it has no loops) and we define
the flow $f_{0}$ as carrying an amount of flow equal to the minimum of the
flow over the edges of this path and zero everywhere else, it is readily
checked that this is a feasible flow. $f^{\prime}=f-f_{0}$ is again a feasible
flow where $f(u,v)>0$ and $f^{\prime}(u,v)=0$. Using the same argument for
$f^{\prime}$ we arrive at a flow $f^{*}=f-f_{0}-\ldots f_{k}$ with zero net
flow. If $f^{*}$ is not the zero flow, then analogously to the previous
argument we may construct a cycle on the graph and define a cycle-flow as the
minimum over the flow of the edges on the cycle and zero everywhere else, this
is a feasible flow $f_{k+1}$ and $f^{*}-f_{k+1}$ has some new edge with zero
flow. We continue in such fashion and arrive at $f-f_{0}-\ldots f_{h}=0$ so
$f=f_{0}+\ldots f_{h}$ where $f_{i}$ is either a path-flow or a cycle-flow.
$h\leq m$ as at least the flow on one edge becomes zero in each step.
∎
###### Lemma 3.27.
(see [3]) For any strictly strong node there exists a residual path either to
the source or to some strictly weak node.
In order to use the flow decomposition theorem first we have to consider a
network $(G^{\prime},c^{\prime})$ related to $(G,c)$ such that the preflow $f$
becomes a feasible flow. This is done by considering a super source $\bar{s}$
and supersink $\bar{t}$, adding edges $(\bar{s},s)\cup\\{(\bar{s},v):\mbox{ v
is strictly weak}\\}$ with flow and capacity equal to the deficits on such
vertices, and edges $(t,\bar{t})\cup\\{(v,\bar{t}):\mbox{ v is strong}\\}$
with capacity and flow equal to the excesses on such vertices. The flow on any
other edge has the same value as the preflow. This function is now a feasible
flow on the network with source $\bar{s}$ and sink $\bar{t}$.
To get a feasible flow on the original network, we have to get rid of excesses
at strong nodes and deficits at strictly weak nodes. For any strong vertex
$v_{s}$, as long as $f(v_{s},\bar{t})>0$ we have that $(\bar{t},v_{s})\in
E_{f}$ is part of the residual network. Hence by lemma (3.27) we have a
residual path from $\bar{t}$ to $\bar{s}$ that contains the edge
$(\bar{t},v_{s})$. Increasing the flow on such path by an amount of $\delta$
equal to the minimum over the residual capacities of the path, actually
decreases the excess of $v_{s}$ by the same amount. After one such step,
either the vertex $v_{s}$ arrives at zero excess or this process can be
repeated by lemma (3.27). This is a process analogous to flow decomposition on
the reversed graph. After termination there are no vertices other than $t$
with positive excess.
In the same fashion, the remaining flow is decomposed until positive deficits
at strictly weak vertices are disposed. This must be done via $t$ as it is the
only vertex sending a positive amount of flow to $\bar{t}$. After termination
all vertices except $s$ and $t$ have nonzero excess. Deleting
$\bar{s},\bar{t}$ from the graph leaves us with a feasible flow on $G_{st}$.
###### Corollary 3.28.
A maximum flow can be recovered from an optimal normalized tree with
pseudoflow f.
###### Proof.
For an optimal tree we have $C_{f}(S,\bar{S})=0$ that is, there are no
residual edges directed from strong to weak vertices. Hence, following the
previous argument, excesses at strong vertices can be disposed using only
paths traversing strong nodes. Now there are no edges directed from a weak to
a strong vertex with positive flow, as otherwise the reverse edge would have
residual capacity greater than zero, a contradiction. So by the proof of
theorem (3.26) the remaining deficits at weak vertices are disposed using only
paths traversing weak vertices. It then follows that $C_{f}(S,\bar{S})=0$
after recovering a flow $f$ so that $c(v,w)=f(v,w)$ for $v\in S,w\in\bar{S}$
and as consequence $|f|=C(S,\bar{S})$. By lemma (3.20) and lemma (3.21)
$(S,\bar{S})$ is a minimum cut. This shows $|f|$ is maximum.
∎
## 4 Applications
There are many not so obvious applications of maximum flow algorithms and
results to different pure and applied topics, we show three interesting
problems that can be solved using the previous results.
### 4.1 Hall’s Marriage Theorem
Let $G=(V\cup W,E)$ be a bipartite graph, where $V\cap W=\emptyset$ and
$|V|=|W|=n$. Label the vertices in $V$ as $v_{1},\ldots,v_{n}$, and the
vertices in $W$ as $w_{1},\ldots,w_{n}$. A perfect matching on G is a
permutation $\sigma\in S_{n}$ such that $[v_{i},w_{\sigma(i)}]\in E$ for every
$i=1,\ldots,n$.
###### Definition 4.1.
Let $S\subset V$. $N(S):=\\{w\in W:[v,w]\in E\mbox{ for some }v\in S\\}$ is
the set of neighbors of S.
We prove the following using the Maxflow-Mincut theorem (2.12).
###### Theorem 4.2.
(Hall’s Marriage Theorem) A perfect matching exists if and only if
$\forall S\subset V,\,|S|\leq|N(S)|$
###### Proof.
Clearly such condition is necessary as $\sigma$ is injective. Suppose $\forall
S\subset V,\,|S|\leq|N(S)|$. Now we construct an $s-t$ network by directing
all edges $e\in E$ from $V$ to $W$, adding a source $s$ and sink $t$ and
appending the edges $\\{(s,v):v\in V\\}\cup\\{(w,t):w\in W\\}$. We set the
capacity of such new edges to 1, and the capacity of the original edges to
$n+1$. Let $S$ be a minimum cut on such network. We show that
$C(S,\bar{S})=n$. $C(S,\bar{S})\leq n$ as the cut $S^{\prime}=\\{s\\}$ has
capacity $n$ and $C(S,\bar{S})$ is minimum. Now we show $C(S,\bar{S})\geq n$.
Let $X=S\cap V$. $N(X)\subset W$ and if $N(X)\not\subset S$ then there would
be an edge crossing the cut, of capacity $n+1$ so $C(S,\bar{S})\geq n+1$ then
$N(X)\subset S\cap W$. On the other hand, all edges traversing the cut are of
the form $(s,v^{\prime})$ where $v^{\prime}\in V\backslash X$ or of the form
$(w^{\prime},t)$ where $w^{\prime}\in S\cap W$. Then
$C(S,\bar{S})=\sum_{v^{\prime}\in V\backslash
S}c(s,v^{\prime})+\sum_{w^{\prime}\in S\cap
W}c(w^{\prime},t)=\underbrace{|V\backslash S|}_{=n-|X|}+\underbrace{|S\cap
W|}_{\geq N(X)}\geq n-|X|+\underbrace{N(X)}_{\geq|X|}\geq n-|X|+|X|$
So the capacity of a minimum cut is $n$. By theorem (2.12) and (3.8) there
exists a maximum flow $f$ of integer values and net flow $|f|=n$. As there are
only $n$ edges out of the source $s$ and into the sink $t$, and they have
capacity $1$, they must be saturated. By conservation of flow and the fact
that the flow is integer, for any $v\in V$ there exists only one $w\in W$ such
that $f(v,w)=1$. Again by conservation of flow and integrality, for any $w\in
W$ there exists only one $v\in V$ such that $f(v,w)=1$. This shows that the
edges directed from $V$ to $W$ carrying a flow of $1$ define a perfect
matching on $G$.
∎
###### Corollary 4.3.
There exists a polynomial time algorithm that finds a perfect matching on a
bipartite graph.
### 4.2 Counting disjoint chains in finite posets
###### Definition 4.4.
A finite poset $P:=(P^{\prime},\leq)$ is a finite set $P^{\prime}$ together
with a partial order $\leq$ on $P^{\prime}$. We say that $P$ has $\hat{0}$ or
($\hat{1}$) if there exists an element $x\in P$ such that $x\leq y$ or $x\geq
y$ for any $y\in P$, respectively. A chain is a subset
$c:=\\{x_{0},\ldots,x_{n}\\}\subset P$ such that for any two elements
$x_{1},x_{2}$ either $x_{1}\leq x_{2}$ or $x_{2}\leq x_{1}$.
Given a finite poset $P$, we say that a chain $C$ is maximal if $C\cup\\{x\\}$
is not a chain for any $x\in P\backslash C$. Clearly any maximal chain
contains $\hat{0}$ and $\hat{1}$. We say that $y$ covers $x$ in the poset if
$x<y$ and there exists no $z$ such that $x<z<y$. We say that a set
$\\{C_{i}\\}$ of chains are cover-disjoint if whenever $y$ covers $x$ then
$\\{x,y\\}$ belongs to at most one chain $C_{i}$. We would like to find a
subset $S$ of the set of maximal chains, such that $S$ is cover-disjoint and
such that $|S|$ is maximum.
One of the possible ways of doing this is to work in a _greedy algorithm_
fashion, finding one of such chains and then repeating the process in the
remaining part of the poset. We note that this may not lead to a partition of
maximum size, as the example in Figure 2 suggests.
Figure 2: Two sets of maximal chains in a poset
Instead, we consider an associated network $P_{st}$ where $s=\hat{0}$,
$t=\hat{1}$, $V=P$ and $(x,y)\in E$ whenever $y$ covers $x$ . We define a
capacity function with value $1$ on every edge.
###### Lemma 4.5.
The maximum number of disjoint chains in $P$ is equal to the net flow of a
maximum flow in $P_{st}$.
###### Proof.
We first show that given a set of disjoint chains $C_{1},\ldots,C_{k}$ we can
find an associated feasible flow on $P_{st}$ with flow value equal to $k$.
$C^{\prime}_{i}\\{\hat{0},\hat{1}\\}\cup C_{i}$ forms a chain from $\hat{0}$
to $\hat{1}$ and a directed path from source to sink in $P_{st}$. Define a
flow $f_{i}$ in $P_{st}$ as having value 1 on the edges of $C^{\prime}_{i}$
and zero everywhere else. This is a feasible flow. $f=f_{i}+\ldots+f_{k}$ is a
function that satisfies conservation of flow and, as the chains were disjoint,
it also satisfies capacity constraints so it is a feasible flow. Each $f_{i}$
saturates one edge leaving the source, hence $|f|=k$. This shows that $k$ is
always less than the value of a maximum flow.
Now given a maximum flow $f$ on $P_{st}$ we construct a set of disjoint chains
of size $|f|$. We may assume $f$ has integer values by corollary (3.8). By
theorem (3.26) we may write $f=f_{1}\ldots f_{j}$ where each $f_{i}$ is an
$s-t$ path. As the flow has integer values, so do $f_{1},\ldots f_{j}$. As the
capacities are all equal to $1$ each $f_{i}$ must have net flow equal to one
and so the $f_{i}$’s do not intersect as they saturate all the path. Then
$\\{f_{1},\ldots,f_{j}\\}$ define a set of disjoint $\hat{0},\hat{1}$ chains.
Finally $|f|=|f_{1}|+\ldots+|f_{j}|=j$.
∎
### 4.3 Image segmentation
The problem of segmenting a given image is that of defining a partition of the
pixels as two sets, the foreground and the background, so that they form
coherent regions. There are multiple other problems defined under the label of
image segmentation. In the following we show how to define the problem and how
to solve it using algorithms of flow optimization in a network, following T.M.
Murali’s lecture notes [6]. Throughout this section we denote a directed edge
as $(v,w)$ and an undirected edge as $[v,w]$.
We define a finite undirected graph $G=(V,E)$ where $V$ is the set of pixels
of an image, $V\subset{\mathbb{Z}}_{+}\times{\mathbb{Z}}_{+}$ and the set of
edges $E$ comprises the set of neighbors for each pixel $(x,y)\in V$. The set
of neighbors of $(x,y)$ is
$N_{(x,y)}:=\\{(x+1,y),(x-1,y),(x,y+1),(x,y-1)\\}\cap V$. We define functions
$a:V\rightarrow[0,1]$, $v\mapsto a_{v}=\mbox{ probability that }v\mbox{ is in
the foreground}$, $b:V\rightarrow[0,1]$, $v\mapsto b_{v}=\mbox{ probability
that }v\mbox{ is in the background}$ and a penalty function
$p:E\rightarrow{\mathbb{R}}_{+}$, $[v,w]\mapsto p{[v,w]}=$ penalty for
defining $v,w$ for defining $v$ in the foreground and $w$ in the background.
Problem: Partition the set $V$ as two sets $A,B$ (foreground/background) such
that the function
$s(A,B):=\sum_{v\in A}a_{v}+\sum_{w\in
B}b_{w}-\sum_{\begin{subarray}{c}[v,w]\in E\\\
|A\cap[v,w]|=1\end{subarray}}p_{[v,w]}$
is maximized.The idea is that if $a_{v}>b_{v}$ it’s preferable to set $v$ as
in the foreground and if a pixel $v$ has most of it’s neighbors defined as in
the foreground, it is preferable to set $v$ as in the foreground also. Such
probabilities are given in the problem, however, different choices of such
values may lead to better or worse results in the segmentation of the image.
For instance if one is interested in isolating a small object in a big
background, the best choice is to take higher values for the probability
function $a_{v}$.
In order to construct such sets, one must define as foreground(background)
vertices those with higher probability of belonging to the
foreground(background), while reducing the total penalty of the boundary
between foreground and background. We want to formulate this problem as a
Mincut problem. In order to do this we have to overcome some difficulties,
namely, that of working with an undirected graph rather than a capacitated
network, and a function to be maximized rather than minimized.
###### Lemma 4.6.
Let $Q=\sum_{v\in V}a_{v}+b_{v}$ then
$s(A,B)=Q-\sum_{v\in A}b_{v}-\sum_{w\in
B}a_{w}-\sum_{\begin{subarray}{c}[v,w]\in E\\\
|A\cap[v,w]|=1\end{subarray}}p_{[v,w]}=Q-s^{\prime}(A,B)$
where
$s^{\prime}(A,B)=\sum_{v\in A}b_{v}+\sum_{w\in
B}a_{w}+\sum_{\begin{subarray}{c}[v,w]\in E\\\
|A\cap[v,w]|=1\end{subarray}}p_{[v,w]}$
Then maximizing $s(A,B)$ is the same as minimizing $s^{\prime}(A,B)$
Now we consider a directed graph $G^{*}=(V\cup\\{s,t\\},E^{*}\cup A(s)\cup
A(t)\\}$ where $E^{*}=\\{(v,w):[v,w]\in E\\}$, $A(s)=\\{(s,v):v\in V\\}$ and
$A(t)=\\{(v,t):v\in V\\}$ and we define a capacity function as $c(s,v)=a_{v}$,
$c(v,t)=b_{v}$ and $c(v,w)=p[v,w]$. We then have a network where the
source(sink) is connected to each pixel with such edge with capacity equal to
$a_{v}$ ($b_{v}$) and where each undirected edge $[v,w]$ of neighbor pixels is
replaced with two _antiparallel_ edges $(v,w),(w,v)$ both with capacity equal
to $p[v,w]$. Then it follows immediately that for a cut $(A,B)$ in such
network we have
$C(A,B)=\sum_{v\in A}b_{v}+\sum_{w\in
B}a_{w}+\sum_{\begin{subarray}{c}[v,w]\in E\\\
|A\cap[v,w]|=1\end{subarray}}p_{[v,w]}=s^{\prime}(A,B)$
By lemma (4.6) arrive at next result:
###### Corollary 4.7.
A minimum cut $(A,B)$ in $G^{*}$ solves the problem of image segmentation.
There’s only one difficulty left to overcome, as we must deal only with simple
graphs, we must replace the set of antiparallel edges. We do this by adding
for each pair of neighbor vertices $v,w$ two new vertices $c_{vw},c_{wv}$ and
replacing the antiparallel edges with the edges
$(v,c_{vw}),(c_{vw},w),(w,c_{wv}),(c_{wv},v)$ all with capacity equal to
$p[v,w]$. It use readily checked that it is equivalent to find a maximum flow
on this new graph.
Figure 3: How to get a simple graph
Then we solve the problem by finding a maximum flow on such network using
either the Ford-Fulkerson or Goldberg-Tarjan algorithm and then recovering a
minimum cut using theorem (3.6).
## 5 A Generalization of Maxflow
We would like to define a more general optimization problem that reduces to
the Maxflow problem on graphs and then try to generalize the optimization
algorithms studied on previous sections.
### 5.1 Preliminaries
###### Definition 5.1.
A simplex on a set $S$ is a finite subset $S^{\prime}\subset S$.
###### Definition 5.2.
A simplicial complex $\Delta$ on a set $S$ is a set of simplices on $S$ closed
under taking subsets. Elements of $\Delta$ are called faces. Maximal faces
(faces that are not subsets of any other face) of the complex are called
facets. Elements of a simplex $X$ are called its vertices. The dimension of a
face $X$ is defined as $\dim(X)=|X|-1$. The dimension of the complex
$\dim(\Delta)$ is defined as the maximum over the dimension of its faces.
###### Definition 5.3.
A simplicial complex $\Delta$ is called _pure_ if all facets have the same
dimension.
Given a network $(G,c)$ we can consider the graph $G^{*}$ that results from
appending the edge $(s,t)$ with infinite capacity and then, the problem of
finding a maximum flow on the original network is equivalent to finding a
maximum _circulation_ on $G^{*}$, that is a positive function on the edges of
$G^{*}$ satisfying capacity constraints and flow constraints on every vertex.
In this case the objective function $|f|$ is the amount flowing through the
edge with infinite capacity.
###### Definition 5.4.
Let $X=\\{v_{0},\ldots,v_{d}\\}$ be a simplex with $d\geq 1$. Consider the set
of _orderings_ of vertices of $X$,
$O(X):=\\{[v_{\sigma(0)},\dots,v_{\sigma(d)}]:\sigma\in S_{d+1}\\}$ modulo the
relation
$[v_{0},\ldots,v_{d}]=[v_{\sigma(0)},\dots,v_{\sigma_{(}d)}]\Leftrightarrow\sigma\mbox{
is even}$. This partitions the set in two equivalence classes that we call
_orientations_ of $X$. To choose an orientation for $X$ is to choose one of
such orientations, which we call the positive orientation and we say that $X$
is oriented. We denote an oriented simplex as $X=(v_{0},\ldots,v_{d})$.
Notation. For a simplicial complex $X$ we let $X^{(d)}$ be the set of its
$d$-dimensional simplices.
###### Definition 5.5.
For $d\geq 1$ Let $C_{d}$ be the free abelian group over the orderings of
elements in $X^{(d)}$ modulo the relations
$[v_{0},\ldots,v_{d}]=[v_{\sigma(0)},\dots,v_{\sigma(d)}]\Leftrightarrow\sigma\mbox{
is even}$ and
$[v_{0},\ldots,v_{d}]=-[v_{\sigma(0)},\dots,v_{\sigma(d)}]\Leftrightarrow\sigma\mbox{
is odd}$. $C_{0}$ is defined in the same fashion but notice that the relations
become trivial.
###### Definition 5.6.
The boundary operator $\partial_{d}:C_{d}\rightarrow C_{d-1}$ is a
homomorphism defined in the basis as
$\partial_{d}(X)=\sum_{i=0}^{d}(-1)^{i}[v_{0},\dots,v_{i-1},\hat{v_{i}},v_{i+1},\dots,v_{d}]$
where $\hat{v_{i}}$ means deleting such term.
Elements of $C_{d}$ are called $d$-chains. Elements of the subgroup
$\ker(\partial_{d})$ are called $d$-cycles.
### 5.2 Higher Maxflow
###### Definition 5.7.
A $d$-dimensional network is a triple $(X,T,c)$ where
1. 1.
$X$ is a simplicial complex of pure dimension $d$ all of whose facets have
chosen orientations.
2. 2.
$T\in X^{(d)}$ is a distinguished oriented simplex of dimension $d$ satisfying
the source condition:
* •
For every oriented $d$-simplex $\sigma$ which intersects $T$ in a
$(d-1)$-dimensional simplex $\tau$ the signs of $\tau$ in $\partial\sigma$ and
$\partial T$ are opposite.
3. 3.
$c:X^{(d)}\rightarrow\mathbb{R}_{+}$ is a function with $c(T)=\infty$.
###### Remark 5.8.
The source condition is a generalization to the assumption that on a network
$(G,c)$ every edge incident with the source is directed out of it and every
edge incident with the sink is directed into it. The source simplex $T$ is the
generalization of an appended edge directed from sink to source with infinite
capacity.
###### Definition 5.9.
A flow on a network $(X,T,c)$ is a function
$f:X^{(d)}\rightarrow\mathbb{R}_{+}$ satisfying the following properties:
1. 1.
$f$ is a weighted cycle, that is
$\sum_{\sigma\in X^{(d)}}f(\sigma)\partial{\sigma}=0$
2. 2.
For every $d$-dimensional simplex $\sigma$ we have $0\leq f(\sigma)\leq
c({\sigma})$
###### Remark 5.10.
The condition that $f$ is a weighted cycle is a generalization of the
conservation of flow condition. To see this we define the following:
###### Definition 5.11.
Let $v=\\{v_{0},\ldots,v_{d-1}\\}\in X^{(d-1)}$ be a $d-1$ dimensional simplex
of a simplicial complex $X$ of degree $d$. Fix an orientation
$[v]=[v_{0},\ldots,v_{d}]$ of v. Let $e\in X^{(d)}$
$\phi(v,e)=\left\\{\begin{array}[]{rl}1&:+[v]\mbox{ appears in }\partial e\\\
-1&:-[v]\mbox{ appears in }\partial e\\\ 0&:\mbox{else}\end{array}\right.$
This is, in fact, a generalization of the incidence function defined on
section (2.1). Then the condition that $f$ is a weighted cycle is equivalent
to:
_for any $d-1$ dimensional oriented simplex $v$ _
$\sum_{e\in X^{(d)}}\phi(v,e)f(e)=0$
###### Definition 5.12.
The amount carried by a flow $f$ in a $(d)$-dimensional network $(X,T,c)$ is
the number $f(T)$.
This comes from the fact that after appending the edge $(t,s)$ to a network
$(G,c)$ with flow $f$ one must define $f(t,s)=|f|$ so that conservation of
flow holds in all vertices including $s,t$.
(HMax-Flow.) The higher max flow problem asks to find the maximum possible
amount $f(T)$ which can be carried by a flow on a network $(X,T,c)$.
###### Remark 5.13.
A $(1)$-dimensional network is a capacitated graph ($T$ is the edge from $t$
to $s$) and HMax-Flow reduces to Max-Flow on graphs.
###### Remark 5.14.
If all the capacities are $1$ and $X$ is a triangulated orientable
$d$-manifold and $T$ is any top-dimensional simplex then every top-dimensional
cycle is a flow with $f(T)=1$. This is one HMax-flow. This follows directly
from the definition of oriented manifold.
### 5.3 As an LP problem
Even in higher dimension, the problem can be stated as a set of linear
equalities and inequalities in a finite dimensional vector space, so it can
also be stated as a linear program as we did in section (2.2). We continue
with the convention that $|n|=|X^{(d-1)}|$ is the number of _$(d-1)faces$_ ,
and $m=|X^{(d)}|-1$ is the number of _facets_ without considering the source
$T$. Then there are $2m$ flow restrictions, two for each edge, and $n$ flow
conservation restrictions, one for each vertex. As always we consider fixed
enumerations of the $(d-1)$ faces $v_{i}$ and of the facets $e_{i}$ (suppose
$v_{m+1}=T$). After fixing an orientation of the $(d-1)$ faces, we define the
matrix $[\partial]_{ij}=\phi(v_{i},e_{j})$, it has dimension $n\times m+1$.
Let $I_{m}$ be the identity matrix of dimension $m$ and $I_{m},0$ be such
matrix with a zero column vector appended as the rightmost column. Then we can
state the problem as :
$\begin{array}[]{ccccc}{\displaystyle\mathop{\max}_{x\in{\mathbb{R}}^{m+1}}x_{m+1}}&;&\left[\begin{array}[]{c}\partial\\\
-\partial\\\ I_{m},0\end{array}\right]x\leq\left[\begin{array}[]{c}0_{n}\\\
0_{n}\\\ c\end{array}\right]&;&x\geq 0\end{array}$ (3)
where $c\in{\mathbb{R}}^{m}$ is the vector $c_{i}=c(e_{i})$.
Computing the dual we find it can be stated as
$\min_{(v,e)\in{\mathbb{R}}^{n+m}}c^{T}e,\,\,\,\left[\partial^{T}\begin{array}[]{c}I_{m}\\\
0\end{array}\right](v,e)\geq e_{m+1},\,\,e\geq 0$
where $v\in{\mathbb{R}}^{n}$ and $e\in R^{m}$.
### 5.4 Further examples and conjectures
Example: Consider the three dimensional simplex. It is a simplicial complex
on the set $\\{1,2,3,4\\}$ where the elements of the complex are the faces of
the simplex.
Figure 4: Oriented simplicial complex, the tetrahedron $Sim_{2}$
Take the orientation as the induced by the outward normal. With this
orientation the oriented faces are $[234],[124],[143],[132]$. We choose the
orientations of the one dimensional faces to be
$[12],[14],[24],[13],[34],[23]$. The capacity on each facet is chosen to be 1.
In this example we drop the commas, dealing only with integers less than 10.
It is readily checked that $[132]$ satisfies the source condition so we may
take $T=[132]$. With the enumeration of facets and $(d-1)$ faces given by this
order we have that the incidence matrix is given by:
$\left[\begin{tabular}[]{ c c c c c }0&1&0&-1\\\ 0&-1&1&0\\\ -1&1&0&0\\\
0&0&-1&1\\\ 1&0&-1&0\\\ 1&0&0&-1\par\end{tabular}\right]$ Table 2: Incidence
matrix of $Sim_{2}$
Towards a good definition of a cut on a generalized $d$-dimensional network we
find feasible solutions of the dual of such problem. We are mostly interested
in integer solutions to the problem. In this example we find both $\partial$
and $\left[\partial^{T}\begin{array}[]{c}I_{m}\\\ 0\end{array}\right]$ to be
totally unimodular matrix so the existence of integer optimal solutions is
assured. We suggest the following definition.
###### Definition 5.15.
A cut on a generalized $d$-dimensional network $X$ is a partition
$(S,S^{\prime})$ of the $d-1$ faces of $X$.
From a cut we can construct a solution to the dual program of HMaxflow. Let
$(S,S^{\prime})$ be a cut. Assign to the first $n$ dual variables
$\lambda_{v}$ (where each one corresponds to a $d-1$ face $v$ of the complex)
the value $0$ if $v\in S$ and the value $1$ if $v\in S^{\prime}$. For each
facet $\sigma$ of the complex, define $+(\sigma)$ be the set of $d-1$ faces
such that $\tau$ induces their positive orientation and define $-(\sigma)$ in
the same fashion. In the dual program, the variables that correspond to the
$d-1$ faces are unrestricted in sign, while the variables that correspond to
the facets $\eta_{\sigma}$ must be nonnegative. There is one inequality in the
dual program for each facet that can be written in the form
$\sum_{v\in+(\sigma)}\lambda_{v}-\sum_{v\in-(\sigma)}\lambda_{v}+\eta_{\sigma}\geq
0$
for any $\sigma\neq T$, and
$\sum_{v\in+(T)}\lambda_{v}-\sum_{v\in-(T)}\lambda_{v}+\eta_{T}\geq 1$
So, given the cut and the assigned values to the $d-1$ faces variables we can
define the dual variables corresponding to the facets $\eta_{\sigma}$ as
$\eta_{\sigma}=\left\\{\begin{array}[]{rl}0&:\sum_{v\in+(\sigma)}\lambda_{v}-\sum_{v\in-(\sigma)}\lambda_{v}\geq
0\\\
-(\sum_{v\in+(\sigma)}\lambda_{v}-\sum_{v\in-(\sigma)}\lambda_{v})&:\mbox{else}\end{array}\right.$
for $\sigma\neq T$ and
$\eta_{T}=\left\\{\begin{array}[]{rl}0&:\sum_{v\in+(\sigma)}\lambda_{v}-\sum_{v\in-(\sigma)}\lambda_{v}-1\geq
0\\\
-(\sum_{v\in+(\sigma)}\lambda_{v}-\sum_{v\in-(\sigma)}\lambda_{v}-1)&:\mbox{else}\end{array}\right.$
In such way it is readily checked the solution is dual feasible. We define the
capacity of a cut $(S,S^{\prime})$ as the value of the dual objective function
at this solution, which is the weighted sum of the capacities of facets
$\sigma\neq T$, with weights $\eta_{\sigma}$. In such way the capacity of a
cut is always an upper bound to the value of a maximum flow, by weak duality.
It is natural to ask whether the minimum of the capacities over all cuts
equals the value of a maximum flow. We attempt to show this using an analogous
probability argument as in section (2).
As in section (3) we can extend the capacity function defined on $X^{(d)}$ to
the set of all orientations of elements of $X^{(d)}$ as $\bar{c}(x)=c(x)$ if
$x\in X^{(d)}$ and $\bar{c}(x)=0$ if $-x\in X^{(d)}$ and $\bar{f}(x)=f(x)$ if
$x\in X^{(d)}$ and $\bar{f}(x)=-f(-x)$ if $-x\in X^{(d)}$.
###### Definition 5.16.
The residual complex $X_{f}$ is the (multi) simplicial complex whose facets
are those $x\in X^{(d)}\cup-X^{(d)}$ such that the residual capacity of x
$c_{f}(x):=\bar{c}(x)-\bar{f}(x)>0$. By definition, if the capacity function
is not identically zero, the residual complex is a pure multisimplicial
complex of dimension $d$.
Recall from that from lemma (3.6), a flow is not maximal if there exists a
simple (no loops) path from $s$ to $t$ in the residual graph. After appending
the edge $(t,s)$ this reduces to: a flow is not maximal if there is no simple
cycle containing $(t,s)$ on the residual graph. This motivates the following
definition:
###### Definition 5.17.
Given a $(X,T,c)$ a $d$ dimensional network and a flow $f$ on $X$, an
augmenting cycle is $d$-cycle $\sigma$ such that $\sigma=\sum_{i}c_{i}X_{i}$
with $c_{i}\in{\mathbb{Z}}_{+}$ and $X_{i}\in X_{f}$ and such that $X_{i}=T$
for some $i$.
###### Lemma 5.18.
Let $(X,T,c)$ be a network and $f,f^{\prime}$ be two feasible flows in such
network. Then $f+f^{\prime}$ is positive and it is a weighted cycle.
###### Proof.
$\sum_{\sigma\in
X^{(d)}}(f+f^{\prime})(\sigma)\partial{\sigma}=\sum_{\sigma\in
X^{(d)}}f(\sigma)\partial{\sigma}+\sum_{\sigma\in
X^{(d)}}f^{\prime}(\sigma)\partial{\sigma}=0$
The fact that $f+f^{\prime}$ is positive is clear. ∎
###### Lemma 5.19.
A flow is not maximal if there exists an augmenting cycle.
###### Proof.
Let $A=\sum c_{i}X_{i}$ be an augmenting cycle with $X_{i}\in X_{f}$ and
$c_{i}$ positive. Let $m=\min\\{\frac{1}{c_{i}}c_{f}(X_{i})\\}$. Define a flow
as $\bar{f}(X)=f(X)+c_{i}m$ if $X=X_{i}$ for some $i$,
$\bar{f}(X)=f(X)-c_{i}m$ if $X=-X_{i}$ for some i and $\bar{f}(X)=f(X)$ else.
Now define $B=\\{X\in X^{(d)}:X=X_{i}\mbox{or}X=-X_{i}\\}$.
$\sum_{X\in X^{(d)}}\bar{f}(X)\partial(X)=\sum_{X\in X^{(d)}\backslash
B}f(X)\partial(X)+\sum_{X\in B}\bar{f}(X)\partial(X)$ $=\sum_{X\in
X^{(d)}}f(X)\partial(X)+\sum_{X\in B}\pm
c_{i}m\partial(X)=m\partial\left(\sum_{i}c_{i}X_{i}\right)=0$
Capacity constraints hold by definition of $m$. As $X_{i}=T$ for some $i$ the
value of the flow is strictly increased.
∎
We would like to prove the converse of this lemma to devise a first algorithm
for Higher Maxflow optimization.
###### Conjecture 5.20.
The converse of lemma (5.19) holds in general.
Figure 5: Oriented simplicial complex, the double tetrahedron $Sim_{2,2}$
Example Consider two tetrahedra oriented by the outward normal, with a common
facet $T$, over the vertex set $\\{1,2,3,4,5\\}$ We find the incidence matrix
to be
$\left[\begin{array}[]{ccccccc}0&0&1&0&0&1&-1\\\ -1&0&0&-1&0&0&1\\\
0&0&0&1&0&-1&0\\\ 1&0&-1&0&0&0&0\\\ 0&1&0&0&1&0&-1\\\ 0&0&0&0&-1&1&0\\\
0&-1&1&0&0&0&0\\\ 0&0&0&-1&1&0&0\\\ -1&1&0&0&0&0&0\\\ \end{array}\right]$
This matrix is also totally unimodular hence the problem has integer optima.
In various cases we find this optima to be the sum of the minimum capacities
over each tetrahedron. This optimum is achieved after finding two augmenting
cycles corresponding to each tetrahedron. When the flow is maximum there is no
augmenting cycle on the residual path.
###### Question 5.21.
When is the incidence matrix of a $d$ dimensional network totally unimodular?
We will cite some important theorems that will give us some insight about the
status of the question (5.21), and ultimately show that it is in fact false.
We find certain family of networks where it holds.
The map $\partial_{d}$ has a unique matrix representation with respect of
chosen basis. We denote such matrix as $[\partial_{d}]$. In fact it coincides
with the incidence matrix we have defined. The kernel of $\partial_{d}$ is
called the group of $p$-cycles and is denoted by $Z_{d}(X)$. The image of the
map $\partial_{d+1}$, denoted by $B_{d}(X)$ is a subgroup of $C_{d}$ called
the $d$-boundaries. We have that for any d $\partial_{d}\partial_{d+1}=0$
under composition so that $B_{d}\subset Z_{d}$. The $d$-dimensional homology
group is defined as the quotient $H_{d}:=Z_{d}/B_{d}$. For a simplicial
couplex $X$ we have that $Z_{d}(X),B_{d}(X),H_{d}(X)$ are finitely generated
abelian groups and $H_{d}(X)$ depends only on the homotopy type of $X$.
For a subcomplex $X_{0}\subset X$ we define the _group of relative chains_ of
$X$ modulo $X_{0}$ as the quotient $C_{d}(X)/C_{d}(X_{0}):=C_{d}(X,X_{0})$.
The map $\partial_{d}$ induces a map
$\partial_{d}^{(X,X_{0})}:C_{d}(X,X_{0})\rightarrow C_{d-1}(X,X_{0})$ which
also satisfies $\partial_{d}^{(X,X_{0})}\partial_{d+1}^{(X,X_{0})}=0$ So we
define the _relative homology groups_ as
$H_{d}(X,X_{0})=ker(\partial_{d}^{(X,X_{0})})/Im(\partial_{d+1}^{(X,X_{0})}):=Z_{d}(X,X_{0})/B_{d}(X,X_{0})$.
The following is a result by Dey-Hirani-Krishnamoorthy that gives a partial
result to our question:
###### Theorem 5.22.
[5] For a finite simplicial complex triangulating a $d+1$ dimensional compact
orientable manifold, $[\partial_{d+1}]$ is totally unimodular irrespective of
the orientations of the simplices.
The answer to question (5.21) is “not always”. Consider the following
counterexample:
Counterexample: [5] For certain simplicial complex triangulating the
projective plane, the matrix $[\partial_{2}]$ is not totally unimodular.
This might not be exactly a counterexample for our conjecture as we need to
prove existence of some facet that may work as source. However if we consider
the two dimensional sphere positively oriented, and a triangulation of such
manifold with consistent choice of orientations, then we may define any facet
of such complex as the source, and then joining this complex to the
triangulation of the projective plane by a vertex we find a submatrix of
$[\partial_{2}]$ that is not totally unimodular.
The following is a theorem also due to Dey-Hirani-Krishnamoorthy that
characterizes totally unimodular matrices arising from a boundary operator.
###### Theorem 5.23.
[5] $[\partial_{d}+1]$ is totally unimodular if and only if $H_{d}(Y,Y_{0})$
is torsion free, for all pure subcomplexes $Y,Y_{0}$ of $X$ of dimensions
$d+1$ and $d$ respectively, where $Y_{0}\subset Y$.
The following theorem yields another family of simplicial complexes where
total unimodularity holds, namely, the family of $d$ dimensional complexes
embeddable in ${\mathbb{R}}^{d}$.
###### Theorem 5.24.
[5] Let $K$ be a finite simplicial complex embedded in ${\mathbb{R}}^{d+1}$,
then $H_{d}(L,L_{0})$ is torsion free for all pure subcomplexes $L_{0}$ and
$L$ of dimensions $d$ and $d+1$ respectively.
It would be helpful to find more general families of simplicial complexes
where unimodularity holds. Using theorem (5.23) we may give an alternative
proof of the fact that for graphs, the incidence matrix is totally unimodular.
###### Theorem 5.25.
For a directed graph $G$, the incidence matrix $[\partial_{1}]$ is totally
unimodular.
###### Proof.
Suppose $G$ is connected. Let $L_{0}\subset L$ be two subcomplexes of $G$ of
dimension $0$ and $1$ respectively. We want to show that $H_{0}(L,L_{0})$ is
totally unimodular. Now $(L,L_{0})$ is a _good pair_ , this implies that
$H_{0}(L,L_{0})\simeq H_{0}(L/L_{0})$ which is a torsion free group isomorphic
to $\mathbb{Z}^{k}$ where k is the number of connected components of
$L/L_{0}$. The result follows by theorem (5.23).
∎
###### Definition 5.26.
Let $X$ be a simplicial complex. A pair of facets $(F,F^{\prime})$ of $X$ is a
leaf if for every face $H$ of $X$ we have $F\cap H\subset F^{\prime}$. A
simplicial tree is a simplicial connected complex $X$ such that every subset
of facets of $X$ contains at least one leaf.
###### Lemma 5.27.
Let $X$ be a pure simplicial complex of dimension 2, and suppose that for any
pure sub complex of dimension 2 $L\subset X$, $H_{1}(L)=0$, then
$[\partial_{2}]$ is totally unimodular.
###### Proof.
Let $L_{0}\subset L$ be pure sub complexes of dimension 1 and 2 respectively.
We have a long exact sequence in homology
$\ldots\rightarrow H_{1}(L)\rightarrow H_{1}(L,L_{0})\rightarrow H_{0}(L_{0})$
By exactness of the sequence we have that $H_{1}(L,L_{0})\rightarrow
H_{0}(L_{0})$ is an invective map. As $H_{0}(L_{0})$ is torsion free so is
$H_{1}(L,L_{0})$. The result follows by theorem (5.23).
∎
###### Lemma 5.28.
Let $X$ be a pure simplicial complex of dimension 2 such that there exist
disjoint facets $\\{T_{1},\ldots,T_{m}\\}$ such that
$X\backslash\\{T_{1},\ldots,T_{m}\\}$ is a simplicial tree. Then for any pure
subcomplex $L\subset X$ of dimension 2, $H_{1}(L)=0$.
###### Proof.
Let $L$ be a pure simplicial complex of dimension 2 with facets
$\\{F_{1},\ldots,F_{l}\\}$. Define $V$ as the pure simplicial complex with
facets $\\{F_{1},\ldots,F_{l}\\}\backslash\\{T_{1},\ldots,T_{,}\\}$. By
assumption, $V$ is a two dimensional subcomplex of a two dimensional
simplicial tree so it is also a simplicial tree. Any simplicial tree is
contractible and homology groups are homotopy invariant. Now for any $F$ facet
of $L$ that is not in $V$ we have two cases: Every one dimensional face of $F$
is in $V$ or not. In the first case, after contracting $V$, $F$ forms a
sphere. In the second case $F$ has at least a one dimensional facet not in $V$
and $F$ can be contracted to $F\cap V$. Hence after contracting $V$ we see
that $L$ is homotopy equivalent to a wedge of spheres so that $H_{1}(L)=0$.
∎
###### Corollary 5.29.
Let $X$ be a pure simplicial complex of dimension 2 such that there exist
disjoint facets $\\{T_{1},\ldots,T_{m}\\}$ such that
$X\backslash\\{T_{1},\ldots,T_{m}\\}$ is a simplicial tree. Then
$[\delta_{2}]$ is totally unimodular.
Conjecture 5.20 remains without answer.
## References
* [1] Ford, L.R., Fulkerson D.R. _Maximal flow through a network_ , Canadian Journal of Mathematics 8 (1962), 399-404.
* [2] Goldberg A. V., R. E. Tarjan. _A New Approach to the Maximum Flow Problem_. J. Assoc. Comput. Mach. 35 (1988), 921–940.
* [3] Hochbaum, Dorit. _The pseudoflow algorithm: a new algorithm for the maximum flow problem_ Operations Research July-August (2008), Vol. 58(4) 992-1009.
* [4] Radzik T. _Parametric flows, Weighted means of cuts, and fractional combinatorial optimization_. Complexity in Numerical Optimization, World Scientific, P. M. Pardalos Ed. (1993), 351–386
* [5] Dey, T. Hirani, A. Krishnamoorthy, B. _Optimal Homologous Cycles, Total Unimodularity and Linear Programming_. Proceedings of 42nd ACM Symposium on Theory of Computing. (2010).
* [6] Murali, T.M. _Data and algorithm analysis_. Lecture notes. available at http://courses.cs.vt.edu/$\sim$cs4104/murali/Fall09/
|
arxiv-papers
| 2012-12-05T19:16:54 |
2024-09-04T02:49:38.973440
|
{
"license": "Public Domain",
"authors": "Fabian Latorre",
"submitter": "Fabian Latorre",
"url": "https://arxiv.org/abs/1212.1406"
}
|
1212.1408
|
# Effects of spin density wave quantization on the electrical transport in
epitaxial Cr thin films
E. Osquiguil E. E. Kaul L. Tosi C. A. Balseiro Centro Atómico Bariloche
and Instituto Balseiro, Comisión Nacional de Energía Atómica, 8400 Bariloche,
Argentina
###### Abstract
We present measurements of the electrical resistivity, $\rho$, in epitaxial Cr
films of different thicknesses grown on MgO (100) substrates, as a function of
temperature, $T$. The $\rho(T)$ curves display hysteretic behavior in certain
temperature range, which is film thickness dependent. The hysteresis are
related to the confinement of quantized incommensurate spin density waves
(ISDW) in the film thickness. Our important finding is to experimentally show
that the temperature $T_{mid}$ where the ISDW changes from $N$ to $N$ \+ 1
nodes decreases as the film thickness increases. Identifying $T_{mid}$ with a
first order transition between ISDW states with $N$ and $N$ \+ 1 nodes, and
using a Landau approach to the free energy of the ISDW together with Monte
Carlo simulations, we show that the system at high temperatures explores all
available modes for the ISDW, freezing out in one particular mode at a
transition temperature that indeed decreases with film thickness, $L$. The
detailed dependence of $T_{mid}(L)$ seems to depend rather strongly on the
boundary conditions at the Cr film interfaces.
###### pacs:
75.30.Fv, 73.50.-h, 75.70.Ak
## I Introduction
Besides the superconducting state, itinerant antiferromagnetism is another
macroscopic manifestation of a collective quantum state of the charged
electrons in a metaloverhauser . Bulk Cr is the paradigmatic and unique simple
metal showing this electronic ground state. In single crystalline samples, an
incommensurate spin density wave (ISDW) forms below the Néel temperature
$T_{N}$ = 311 K, characterized by wave vectors ${\mathbf{Q}_{\pm}}$ =
$(2\pi/a_{Cr})(1\pm\delta)$ along the $\langle 100\rangle$ directionsfawcett .
Here $a_{Cr}$ = 2.88 Å is the lattice parameter of the Cr bcc unit cell and
$\delta$ = $a_{Cr}/\Lambda_{ISDW}$ is the deviation from commensurability,
with $\Lambda_{ISDW}$ the wavelength of the ISDW. This deviation is
temperature dependent,werner leading to a variation of $\Lambda_{ISDW}$ from
60 Å at 10 K to 78 Å at 310 K. Above the spin-flip temperature $T_{sf}$ = 123
K, the spin waves are transverse with the spin ${\mathbf{S}}$ perpendicular to
${\mathbf{Q}}$. Contrary to what happens in single crystals in which all three
crystallographic directions are equivalent, in thin Cr filmszabel the wave
vector ${\mathbf{Q}}$ orients perpendicular to the film surface as its
thickness decreasessonntag ; bodeker , and the ISDW’s are transverse for all
temperaturesbader below $T_{N}$. This confinement of the ISDW leads to the
quantization of the wave vector ${\mathbf{Q}}_{N}$, with $N$ an integer,
giving rise to very interesting hysteretic phenomena.
Fullerton et al.fullerton measured neutron scattering and electrical
transport in Cr/Cr97.5Mn2.5 (001) superlattices, where the CrMn layers, being
antiferromagnetic with a higher $T_{N}$, exert some stringent boundary
conditions on the Cr surfaces. They found that temperature cycling produced
irreversibility in the derivative of the resistivity and in the neutron
scattering scans. These features were attributed to a change in an odd number
of nodes in the ISDW inside the Cr layers which is compatible with the
magnetic coupling at the interfaces with the CrMn layers.
Recently, Kummamuru et al.kummamuru reported on transport results in
epitaxially grown Cr (100) thin films of different thicknesses. Here the
boundary conditions are different from those in the superlattices of Fullerton
et al. However, they found thermal hysteretic behavior of the Hall coefficient
and resistivity for thicknesses below 510 Å. They interpreted these results in
terms of ISDW domains with $N$ nodes (at high temperatures) or $N$+1 nodes (at
low temperatures) in the Cr film. For thicker films (3500 Å) this mechanism
disappears, and only an irreversibility in the $\rho(T)$ curves was seen in a
wider temperature range, and explained in terms of domain-wall scattering of
electrons.
If hysteretic phenomena are indeed related to a first order phase transition
taking place between domains of ISDW’s with $N$ and $N$+1 nodes inside the Cr
layers, it is natural to ask how the transition temperature depends on the Cr
thickness. Since for thicker films the number of available modes for the ISDW
increases in the whole temperature range where $\Lambda_{ISDW}(T)$ changes,
one would naively expect to find the hysteretic behavior at higher
temperatures in the thicker films. In this article we address these issues
both experimentally and theoretically, and demonstrate that exactly the
opposite behavior is found, i.e. the hysteretic region and the transition
temperature move towards lower temperatures as the film thickness is
increased. We therefore believe that this work throws new light on the
understanding of the behavior of ISDWs in confined geometries.
## II Experiment
Cr thin films were grown epitaxially on MgO (100) substrates using DC
Magnetron Sputtering in a similar way as that reported by Kummamuru et
al..kummamuru The films, with thicknesses between 270Å and 1100Å, were
characterized by x-ray diffraction showing rocking curves that have an angular
dispersion at FWHM around the [002] peak of $0.5^{\circ}$, indicating a very
good degree of epitaxy. Results from AFM scans show a mean surface roughness
of about 20Å, which is the third part of $\Lambda_{ISDW}$ at low temperatures.
All films were patterned using photolithography and chemical etching in a
convenient configuration to perform four terminal measurements of the
resistance. It was measured as a function of $T$ between 15 and 340K in a
commercial cryo-cooler. The error in the resistance measurement was 1/10000,
almost independent of $T$.
## III Results and Discussion
In Fig.1 we show the resistivity of the 270 Å film as a function of
temperature in the temperature region where the hysteretic behavior appears in
$\rho(T)$ while making a cooling-warming cycle. The inset shows the difference
between $\rho_{cooling}$ and $\rho_{warming}$ ($\rho_{c}-\rho_{w}$) normalized
by $\rho_{w}$ and expressed as a percentage. Clearly, in a rather small
temperature window, the resistivity while cooling down is higher than the
values of $\rho(T)$ when warming up the film by an amount that reaches 0.5% at
it’s maximum. This value is of the same order of magnitude than that reported
by Kummamuru et al.kummamuru
Figure 1: (Color online) Resistivity as a function of temperature for a
cooling-warming cycle for the 270 Å film. The inset shows the difference
$(\rho_{c}-\rho_{w})/\rho_{w}$, expressed as a percentage.
We observed that the derivative of the resistivity curves has an intrinsic
noise in the temperature region where the hysteresis is seen, which is not
related to the measurement method. We will address this issue in a forthcoming
article.ruido For the purpose of the physics we want to analyze here, it is
enough to work with the smoothed curves of $\partial\rho/\partial T$ as a
function of T as shown in Fig.2, corresponding to the data displayed in Fig.1.
Clearly, the hysteretic behavior seen in $\rho(T)$ is reflected also in the
derivatives of both, the cooling and warming cycles. The inset shows the
difference between them. We define the temperature at which the derivatives
are equal within the hysteretic zone (marked with an arrow) as $T_{mid}$(= 235
K for 270 Å). This coincides with the temperature at the maximum of
$(\rho_{c}-\rho_{w})/\rho_{w}$. The temperatures where the difference between
the derivatives has its maximum and minimum values, below and above $T_{mid}$,
are marked in the insets of Figs.1 and 2 using dotted vertical lines. We call
these temperatures $T_{L}$ and $T_{H}$ (212 K and 254 K respectively, for 270
Å) and define $\Delta T=T_{H}-T_{L}$. As we will show below, the $\Delta T$
region where the hysteresis occurs is film thickness dependent.
Figure 2: (Color online) Derivatives of the $\rho(T)$ curves shown in Fig. 1.
The inset shows the difference of $\partial\rho/\partial T$ between the
cooling and warming cycles.
To appreciate the mentioned behavior, we plot in Fig.3 $T_{mid}$ as a function
of film thickness $L$. Note that the bars are not error bars but represent the
irreversibility region width, $\Delta T$, as defined above. It can be observed
that there exists a systematic shift of the hysteretic region towards lower
temperatures as the film thickness is increased. In fact, $T_{mid}$ goes down
and tends to saturate as the film thickness becomes larger than 750 Å. For
comparison, we include the data from Kummamuru et al. kummamuru (squares). As
can be seen, their data are in very good agreement with our results. This
would be expected because both set of films have been grown following the same
procedure and over the same type of substrate. In the data of Fullerton et
al.fullerton a similar decreasing trend for $T_{mid}$ may be observed, but a
much steeper decrease of $T_{mid}$ with thickness seems to be present, at
least for the samples reported (63Å and 200Å). Since it is well known that the
magnetism at Cr surfaces strongly depends on conditions at its surface or
interface zabel , this different behavior might be related to the role played
by the CrMn layers in their Cr/CrMn superlattices.fishman
Other important features are worth to remark. In the inset of Fig. 3 we plot
the difference of the derivatives as a function of $T$ for films of increasing
thickness. For the sake of clarity, the vertical zero of each curve has been
shifted by an arbitrary offset. The important point here is not the absolute
value of $\partial\rho/\partial T$, but the temperature $T_{mid}$ at which
$\partial\rho_{c}/\partial T$ (cooling) equals $\partial\rho_{w}/\partial T$
(warming). Note that $\Delta T$ increases, while the amplitude of the
difference between $\partial\rho_{c}/\partial T$ and
$\partial\rho_{w}/\partial T$, measured at $T_{L}$ and $T_{H}$, decreases.
These two last observations give a clear clue on why the irreversible behavior
in $\rho(T)$ is not likely to be seen in thicker samples.
Figure 3: (Color online) Transition temperature as a function of film
thickness. Red squares correspond to data in ref..kummamuru The vertical bars
are not error bars but the width of the hysteretic region (see text). The
dashed lines are guides to the eye. Inset: Difference between the derivatives
of $\rho(T)$ curves for cooling and warming cycles of films with increasing
thicknesses. For clarity, the vertical zero of each curve (horizontal lines)
has been arbitrarily shifted. The arrows indicate the temperature $T_{mid}$
(see text).
If we identify $T_{mid}$ as the temperature at which a first order transition
takes place between a state with an ISDW with $N$ nodes to a new state with
$N$+1 nodes when the temperature is reduced, then our data clearly show that
the transition temperature decreases as the film thickness is increased,
tending to saturation for films thicker than 750 Å. In order to explain this
behavior of $T_{mid}(L)$, let us look more in detail the quantization of the
ISDW in thin films, which is important to understand the hysteresis of the
resistivity in cooling-warming cycles.
As a product of different internal interactions, $\delta$ depends on
temperature.werner It varies quite linearly from, $\delta$ = 0.0379 at $T$ =
300 K, to $\delta$ = 0.044 at $T$ = 200 K. As a direct consequence,
$\Lambda_{ISDW}$ changes from 76 Å to 65.5 Å in the same temperature range. At
much lower temperatures it saturates to a constant value $\delta_{c}$ = 0.048
giving $\Lambda_{ISDW}$ $\approx$ 60 Å.
Since the magnetic instability in the electron system depends on the
combination
$\cos(\mathbf{Q}_{+}\cdot\mathbf{r})+\cos(\mathbf{Q}_{-}\cdot\mathbf{r})$, it
is clear that the resulting beating has a fast variation modulated by a long
wavelength $(Q_{+}-Q_{-})/2=\frac{2\pi}{a/\delta}$. Thus, the ISDW in Cr has a
wavelength
$\Lambda_{ISDW}=\frac{2.88}{\delta}\text{\AA}.$ (1)
Figure 4: (Color online) Available ISDW modes for different film thicknesses
in the temperature region of interest. The temperature scale is the same for
all panels.
When the ISDW is confined in Cr films with non-magnetic interfaces, as is the
case in our Cr films, we expect a configuration with antinodes (or nodes)mibu
at the films’ surfaces. For this case, the available modes are separated by
half a wavelength. Given a film with thickness $L$, the allowed values for
$\Lambda_{N}$ are therefore $2L/N$, where $N$ is either the number of nodes
inside the film (antinodes at the surfaces), or antinodes in the film (nodes
at the surface).
In the window of wavelength values found in bulk chromium given by $\delta(T)$
(60 Å to 78 Å), there are only a few modes available for each film thickness.
Their number increases as $L$ increases. This is shown in Fig.4 where we plot
the allowed modes for each film thickness. The lines correspond to
$\Lambda(T)$ as calculated from equation 1, using a linear approximation for
$\delta(T)$ between 200 K and 300 K, obtained from the data in Ref.werner .
This approximation seems reasonable because in the thickness range in which
our samples lay (270-1100 Å) we do not see appreciable variations of the Néel
temperature, indicating that there is no strong dependence of $\delta$ with
thickness.fishman-rev ; rotenberg
As can be seen in Fig.4, in the the filmkummamuru of 175 Å, the transition
occurs between N = 4 and N = 5 at the temperature where the line corresponding
to $\Lambda_{5}\approx\Lambda_{ISDW}(T)$. Correspondingly, for the 270 Å film
the transition between N = 7 and N = 8 lies at a lower temperature. From this,
it seems that the transition temperature would decrease with $L$. This
reasoning, however, cannot be extended to the other film thicknesses based on
the information displayed in Fig.4 alone, because more modes than one become
available for the ISDW, and there is no way to choose a priori at which one
the transition would occur. In order to understand how the transition
temperature depends on $L$, in Section IV we propose a phenomenological model
which takes into account this variation of the number of permitted modes when
$L$ increases, and we also incorporate some dynamics.
## IV Model
### IV.1 The Free Energy of the ISDW
When the temperature is decreased below $T_{N}$, the wavelength of the ISDW in
the Cr film would try to follow it’s bulk behavior.werner However, as
discussed in the previous section: the ISDW is confined in a film of a given
thickness, the variation in temperature of $\Lambda_{ISDW}$ is quantized, and
the number of nodes varies discontinuously from $N$ to $N$ \+ 1, in a process
that involves a first order phase transition. Neutron diffractionfullerton as
well as X-ray experimentskummamuru confirm this physical picture in Cr/CrMn
multilayers and in Cr thin films respectively.
Let us consider a one-dimensional model of a transverse ISDW with
${\mathbf{Q}}$ = $q\hat{z}$, where $\hat{z}$ is the direction perpendicular to
the film’s surface. In this case the order parameter takes the form $\Psi$ =
$\Psi_{0}e^{iqz}$. We can then write an expression for the free energy (per
unit length) depending on $\Psi_{0}$ and $q$,
$\displaystyle F[\Psi_{0},q]$ $\displaystyle=$ $\displaystyle
F_{0}+a(T-T^{bulk}_{N})|\Psi|^{2}+b|\Psi|^{4}$ (2) $\displaystyle+$
$\displaystyle c|(\nabla-iq_{0})\Psi|^{2}+U_{surface},$
where $F_{0}$ is the part of the free energy which is independent of the ISDW,
$T^{bulk}_{N}$ = 311 K and $q_{0}$ is the wave vector that the ISDW would have
in the bulk material. Following McMillanmcmillan we include the term
$|(\nabla-iq_{0})\Psi|^{2}$ which gives a contribution proportional to
$(q-q_{0})^{2}\Psi_{0}^{2}$. This takes into account the fact that there
exists an energy cost in order to deform the ISDW with respect to it’s bulk
form. The key point is that $q_{0}$ varies in temperature and these variations
should be followed by the ISDW in order to keep the energy at a minimum.
There is, nevertheless, a competition between the energy paid to deform the
ISDW respect to the bulk, and the energy cost for not satisfying the boundary
conditions imposed by the surface. As already mentioned, the most natural way
to choose these boundary conditions for Cr films that are not in contact with
magnetic materials is to have nodes or antinodes at the surfaces. For both
cases $\Lambda_{N}=2L/N$, i.e. $q=\pi N/L=k_{N}$. Considering this, the
surface contribution to the free energy is given by a term proportional to
$(q-k_{N})^{2}\Psi_{0}^{2}$. The way to account for the different possible
available modes $\\{k_{N}\\}$ is simply by selecting the one that gives the
lowest contribution to the free energy at a given temperature (see Fig.4).
Then, we explicitly have for the free energy,
$\displaystyle F[\Psi_{0},q]$ $\displaystyle=$ $\displaystyle
F_{0}+a(T-T^{bulk}_{N})\Psi_{0}^{2}+b\Psi_{0}^{4}$ (3) $\displaystyle+$
$\displaystyle
c(q-q_{0})^{2}\Psi_{0}^{2}+U\min_{\\{k_{N}\\}}[(q-k_{N})^{2}\Psi_{0}^{2}].$
If the nodes (or antinodes) are pinned at the surface (enough large $U$), then
the free energy displays a series of minima corresponding to parabolic
potentials sitting close to each $k_{N}$. This is shown in Fig.5 for a fixed
amplitude of the order parameter $\Psi_{0}$ and adequate numeric parameters.
Note the change in the potential landscape with temperature. As $T$ is
decreased the barriers between modes increase and the free energy changes in
such a way to give preference to the modes with higher $q$’s, i.e. shorter
wavelengths and more nodes (antinodes) in the ISDW.
In this simple way we can understand how the ISDW’s wavelength is jumping from
$N$ to $N$ \+ 1 nodes and so on as the temperature decreases. Given the fact
that each of these jumps involves a first order transition, hysteresis is
expected in cooling-warming cycles.
Figure 5: Cuts of the two dimensional free energy, $F[\Psi_{0},q(T)]$, at
different temperatures and for a fixed value of $\Psi_{0}$. a) $T/T_{N}=0.96$,
b) $T/T_{N}=0.74$, c) $T/T_{N}=0.48$. Available modes correspond to a film of
$760$ Å.
### IV.2 Monte-Carlo Simulation Results
Figure 6: (Color online) Average $q$ as a function of temperature for
different film thicknesses. The fluctuations in $q$ at high temperatures
freeze to a fixed number of nodes at a temperature $T_{mid}$ which depends on
the film thickness (arrows). Figure 7: Freezing temperature from Fig.6 as a
function of film thickness. The dashed line is a guide to the eye.
One of the most important results of this work is the description of how the
temperature of the maximum hysteresis, $T_{mid}$, moves to lower values as the
thickness of the Cr films is increased. From the scheme shown in Fig.4 it can
be inferred that when many ISDW modes are available (larger $L$’s), a large
barrier should be needed in order to freeze the state of the system with the
largest number of nodes, otherwise the system would wander between all
available modes. Since the ISDW’s amplitude $\Psi_{0}$ increases with
decreasing temperature as well as the barriers do, lower temperatures are
needed to freeze the system in a high $k_{N}$ mode.
To better quantify this, we introduce some dynamics by means of a Monte-Carlo
simulation of the system. Here we have a two dimensional phase space
$(q,\Psi_{0})$ and an energy given by Eq.3. The resulting average value of $q$
after several runs is shown in Fig.6 for different film thickness as the
temperature is decreased. At high temperatures, the system explores all
possible $k_{N}$’s for each $L$, but as the temperature is lowered, the ISDW
chooses only one $k_{N}$ and stays there no matter how low the temperature
goes. This behavior defines a freezing temperature that we identify with
$T_{mid}$.
The values obtained from the results in Fig.6 for the freezing temperature
corresponding to each film thickness, are plotted in Fig.7. Clearly, the
freezing temperature follows the same trend as the experimental data for
$T_{mid}(L)$, i.e., it goes down as the film thickness is increased. The
saturation seen in the transition temperature in our data for $L$’s larger
than 750 Å naturally occurs in our Monte-Carlo simulations if we put,
mimicking the experimental results,werner a constant $\delta(T)$ at low
temperatures. With this simple phenomenological model we do not pretend to get
a quantitative agreement with our experimental data, but it is important to
understand the physics behind the observed hysteretic phenomena.
## V Conclusions
We have shown that the hysteretic behavior that appears in $\rho(T)$ in
epitaxial Cr thin films is a consequence of the quantization of a transverse
incommensurate spin density wave in the direction perpendicular to the film
surfaces, with a first order phase transition occurring between domains with
$N$ and $N$ \+ 1 nodes (or antinodes). The number $N$ at which the transition
manifests is film thickness dependent, being larger for thicker films. An
analysis of the one dimensional ISDW free energy, estimated using McMillan’s
approach for CDW’s, together with some dynamics introduced via Monte-Carlo
simulations, show that the potential barriers between allowed modes increases
as the system is cooled down. As a consequence of this, and for each $L$, at
high temperatures the system explores all available modes but remains frozen
in a high $k_{N}$ state at low enough temperatures. Therefore, this simple
phenomenological model explains in a natural way the experimental results
which show that the temperature at which the transition between $N$ and $N$ \+
1 modes takes place, decreases as the film thickness is increased.
## VI Acknowledgements
We would like to thank H. Pastoriza and K. Ogando for help with the patterning
of the samples. E.O., E.K. and C.A.B. are members of CONICET. L.T. has a
scholarship from CONICET. Work partially supported by PIP No.11220080101821
CONICET, PICT R1776 ANPCyT, and PIP No.1122008010111001 CONICET.
## References
* (1) A. W. Overhauser, Phys. Rev. 128, 1437 (1962).
* (2) E. Fawcett, Rev. Mod. Phys. 60, 209 (1988).
* (3) S. A. Werner et al., Phys. Rev. 155, 528 (1967).
* (4) H.Zabel, J. Phys.: Condens. Matter 11, 9303 (1999).
* (5) P. Sonntag et al., J. Magn. Magn. Mater. 183, 5 (1998).
* (6) P. Bödeker et al., Phys. Rev. Lett. 81, 914 (1998).
* (7) E. E. Fullerton et al., Phys. Rev. Lett. 77, 1382 (1996).
* (8) E. E. Fullerton et al., Phys. Rev. Lett. 91, 237201-1 (2003).
* (9) R. K. Kummamuru and Yeong-Ah Soh, Nature (London) 452, 859 (2008).
* (10) C. Balseiro et al., to be published.
* (11) R. S. Fishman and Z.-P. Shi, Phys. Rev. B 59, 13849 (1999).
* (12) K. Mibu et al., Phys. Rev. Lett. 89, 287202-1 (2002).
* (13) R. S. Fishman, J. Phys. Condens. Matter 13, R235 (2001).
* (14) Eli Rotenberg et al., New Jour. of Phys 7, 114 (2005).
* (15) W. L. McMillan, Phys. Rev. B 12, 1187 (1975).
|
arxiv-papers
| 2012-12-06T18:20:24 |
2024-09-04T02:49:38.985521
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "E. Osquiguil, E. E. Kaul, L. Tosi and C. A. Balseiro",
"submitter": "Leandro Tosi",
"url": "https://arxiv.org/abs/1212.1408"
}
|
1212.1411
|
11institutetext: Centro Atómico Bariloche, Comisión Nacional de Energía
Atómica - 8400 Bariloche, Argentina
Instituto Balseiro, Comisión Nacional de Energía Atómica and Universidad
Nacional de Cuyo - 8400 Bariloche, Argentina
Electronic transport phenomena in thin films Intrinsic properties of
magnetically ordered materials Spin waves
# Barkhausen-type noise in the resistance of antiferromagnetic Cr thin films
L. Tosi 1122 E. Osquiguil 1122 E. E. Kaul 1122 C. A. Balseiro 11221122
###### Abstract
We present an experimental study of the changes generated on the electrical
resistance $R(T)$ of epitaxial Cr thin films by the transformation of
quantized spin density wave domains as the temperature is changed. A
characteristic resistance noise appears only within the same temperature
region where a cooling-warming cycle in $R(T)$ displays hysteretic behavior.
We propose an analysis based on an analogy with the Barkhausen noise seen in
ferromagnets. There fluctuations in the magnetization $M(H)$ occur when the
magnetic field $H$ is swept. By mapping $M\rightarrow\Psi_{0}$ and
$H\rightarrow T$, where $\Psi_{0}$ corresponds to the order parameter of the
spin density wave, we generalize the Preisach model in terms of a random
distribution of resistive hysterons to explain our results. These hysterons
are related to distributions of quantized spin density wave domains with
different sizes, local energies and number of nodes.
###### pacs:
73.50.-h
###### pacs:
75.30.-m
###### pacs:
75.30.Ds
Among simple metals, Cr is the only one that shows an antiferromagnetic
electronic ground state corresponding to an incommensurate spin density wave
(SDW).[1] A single crystal of Cr cooled below the Néel temperature develops
domains whose SDW wave vector ${\mathbf{Q}}$ may be oriented along any of the
main crystallographic directions with the same probability. These domains are
separated by walls which can move back and forth with temperature, or that can
be suppressed by cooling in an applied magnetic field $H$.[2] Using coherent
x-ray diffraction Shpyrko et al. [3] measured the noise spectrum at different
temperatures in Cr single crystals produced by domain wall motion separating
domains with different ${\mathbf{Q}}$ orientations. Independently, Michel et
al. [4] observed the presence of spontaneous fluctuations on the resistance of
Cr films, resembling a telegraphic code. The resistance jumps were associated
to the thermal motion of two different domain walls: rotations of
${\mathbf{Q}}$ domains, and rotation of the polarization vector
${\mathbf{\eta}}$ within a single ${\mathbf{Q}}$ domain. However, it has been
shown that in thin films the SDW confinement leads to the quantization of the
wave vector ${\mathbf{Q}}_{N}$, $N$ being an integer, which orients
perpendicular to the film surfaces.[5, 6] This quantization is due to the
boundary conditions imposed by the film surfaces or interfaces. Rich
hysteretic phenomena in the electronic transport properties [7, 8] have been
ascribed to the development of domains with different ${\mathbf{Q}}_{N}$
corresponding to a different number of nodes of the SDW. Consequently, in this
case we would expect a different domains structure and dynamics compared to
that seen in single crystals or thick films.
In ferromagnetic materials, the existence of domain structures that can move
due to the action of an external field $H$ produce the well known Barkhausen
noise [9] in the magnetization $M(H)$. However, in antiferromagnets or SDW
systems the noise produced by domain wall motions as an external parameter is
varied, has been more elusive to measure due to the lack of a net magnetic
moment.
In this Letter we address an up to now unattended issue related to the
resistance changes generated by the movement or transformation of
antiferromagnetic domains that have different number of nodes (N or N+1) in
the quantized SDW state present in thin Cr films. We show that the whole
hysteretic behavior in $R(T)$ is highly reminiscent of that observed in
ferromagnetic materials where the control field is $H$, and the measured
magnitude is the magnetization $M(H)$. In these systems, Barkhausen noise has
been successfully explained in terms of magnetic hysterons [10, 11] using the
Preisach model [11]. In our transport measurements the control field is the
temperature $T$ and the measured magnitude is the film resistance $R(T)$.
Consequently we propose a mapping of $H\rightarrow T$ and of
$M\rightarrow\Psi_{0}$, where $\Psi_{0}$ is the order parameter associated
with the SDW. Using this we show that a simple phenomenological extension of
the Preisach model based on resistive hysterons [10] reproduce quite
satisfactorily the experimental data. With this model we extract relevant
conclusions about the origin of the hysteretic phenomena, the distribution of
energy barriers and the processes involved in antiferromagnetic domain
transformation with temperature. These findings may be relevant to
applications in spintronics, where the understanding of the dynamics of
antiferromagnetic domain walls is of importance to develop better pinning
layers, as well as in general problems of antiferromagnets in confined
geometries.
Figure 1: (color online) Experimental data: (a) Noise in the difference of
derivatives $dR_{c}/dT-dR_{w}/dT$ associated with the complete loop in a 550Å
thick film. The solid line corresponds to a smoothing of the data (see text).
The inset shows the difference of the data and the smoothing. (b) Difference
$\Delta R(T)$ of the resistance in cooling-warming cycles of different
amplitudes in $T$ (with final temperature $T_{f}$), for a 1100Å thick film.
(c) FORCs, L0 ($T_{f}=$ 50 K), L1 ($T_{f}=$ 150 K), L2 ($T_{f}=$ 185 K) and L3
($T_{f}=$ 225 K), calculated from the data in (b) with $\Delta R_{s}$ as
defined in the text.
Cr thin films were grown epitaxially on MgO (100) substrates using DC
magnetron sputtering. The films were characterized by x-ray diffraction
showing rocking curves with an angular dispersion at FWHM around the [002]
peak of $0.5^{\circ}$ indicating a very good degree of epitaxy. Results from
AFM scans show a mean surface roughness of about 20Å, which is the third part
of the SDW wavelength at low temperatures. The samples were patterned in a
four terminal configuration bar of 2 mm long and 140 $\mu$m wide using
photolithography and chemical etching. The resistance was measured using a dc
current of 10$\mu$A as a function of $T$ between 20 and 320 K in steps of 0.5
K in a commercial cryo-cooler. In each step the temperature was stabilized
within 30 mK giving an intrinsic error in the resistance measurement between
0.3 to 0.8 m$\Omega$, almost independent of $T$. Each $R(T)$ curve took about
24 hours of measurement.
In Fig.1 we present our main experimental findings. The changes in the
resistance produced by the cooling and warming of the sample are small jumps
($|\Delta R|\simeq$ 10-2 $\Omega$). Since the resistance is of the order of
tens of Ohms ($R$(300 K) $\simeq$ 63 $\Omega$) a plot of $R(T)$ shows no
appreciable features. Instead, in panel (a) we show an amplification of the
effect by plotting the difference of the resistance derivatives for a complete
(20 K to 320 K) cooling (c) and warming (w) cycle: $dR_{c}/dT-dR_{w}/dT$ for a
550 Å thick film. The solid line corresponds to a smoothing of the raw data
using the Savistky-Golay method. The inset shows the difference between the
raw and smoothed data. Although noise is seen in the whole temperature range,
it is evident that its amplitude strongly increases in the temperature window
where the irreversible behavior in $R(T)$ sets in (see inset). This clearly
indicates that the observed noise in $dR/dT$ is not exclusively due to thermal
fluctuations. We claim that it has to be related to the same mechanism that
governs the hysteretic behavior.
Neutron diffraction [12], x-ray diffraction [7], and electrical transport
measurements [7, 8], show that the hysteretic behavior in $R(T)$ and
$dR(T)/dT$ is due to the existence of antiferromagnetic domains with different
number of nodes $N$ in the confined SDW. These domains switch from $N$ to
$N+1$ as the temperature is decreased. As we will show and elaborate in more
detail below, we assume that the noise pattern seen in Fig.1 (a) is mainly due
to the switching and growth of these domains as the temperature of the film is
changed. From previous results [8] we know that the hysteretic region enlarges
in $T$ as the film thickness increases. Taking this into account, we performed
measurements of cooling-warming cycles of different amplitudes in $T$ in a Cr
film with a thickness of 1100Å for which the irreversible behavior embraces a
wider temperature range as compared to that found in thinner films. The cycles
were measured starting always at a temperature of 320 K, cooling down the film
to a given final temperature (which we call $T_{f}$), and then warming it up
again to 320 K. The results of such measurements plotted as $\Delta
R(T)=R_{c}-R_{w}$ are shown in panel (b) of Fig.1. Clearly, as $T_{f}$ is
increased the cycles reduce their amplitude, the maximum shifts to higher
temperatures, and the curves tend to coincide with the complete cycle curve at
high enough temperatures. From these data we may construct the equivalent to
the first order reversal curves (FORCs) [13] in ferromagnets. They are shown
in panel (c) and are defined as $\Delta R_{s}=R_{c}-R^{0}_{1/2}$ for the
cooling cycle and $\Delta R_{s}=R_{w}-R^{0}_{1/2}$ for the warming cycle,
where $R^{0}_{1/2}=(R^{0}_{c}+R^{0}_{w})/2$ is taken from the complete cycle
($T_{f}=50$ K).
The measured resistance depends on the domain distribution, the domain wall
structure and the energy barriers in a complex free energy landscape. Due to
the problem complexity we resort to a simple phenomenological model to
describe the physics behind the observed hysteretic behavior. We assume that
the building blocks of the model are resistive hysterons that correspond to
small hysteresis cycles in $R(T)$. These are the equivalent of magnetic
hysterons commonly used to explain $M(H)$ hysteresis in ferromagnets. There,
the Preisach model naturally incorporates the fact that the magnetization
increases with magnetic field and uses the magnetic hysterons to account for
the increase (decrease) of the total magnetic moment as the field increases
(decreases). In our case, as we show below, the resistive hysterons should
describe the transition from a high to a low resistance state as the
temperature is decreased in order to properly account for the experimental
data. From the detailed analysis of Kummamuru’s experimental data on
electrical transport [7] it was inferred that the state with N nodes has
larger resistance (or a smaller number of effective carriers) than the state
with N+1 nodes. Our recent measurements of the Hall coefficient as a function
of temperature also support this view showing hysteretic behavior with a
higher number of carriers while heating [14]. Although this seems counter-
intuitive (in the low temperature phase we would expect the amplitude of the
SDW to increase and consequently to reduce the number of carriers), numerical
simulations (in multi-layer systems with magnetic interfaces) show that the
amplitude of the order parameter is drastically reduced when the number of
nodes increases by one [15]. Based on this evidence, we consider that domains
with N+1 nodes have lower resistance than those with N nodes. With our
phenomenological description we attempt to account for the fact that the
resistance is given by the presence of domain walls and mostly by a
contribution from the domains. Hysterons represent a transition from states
with N (high $T$) to N+1 (low $T$) nodes. A second source of resistance noise
could be the redistribution of domain walls as growth of the domains implies
loss of domain-wall scattering. The shape of the resistance hysteresis loop
for the complete cooling-warming cycle shows that the dominant effect is the
change in the domains resistance: $R_{c}$ is always larger than $R_{w}$. If
the irreversibility were to be dominated by the domain-wall scattering of
pinned walls that jump as the domains grow, the shape of the resistance loop
would be different and we should expect a temperature interval where
$R_{w}>R_{c}$.
We characterize these hysterons with three independent parameters: the center
of the loop $T_{0}$, the half-width of the loop $D$ and the amplitude of the
change in resistance $Z$. See inset in Fig.2 (a).
Since we are interested in the derivative of $R(T)$ it is better to
mathematically describe the hysterons using an analytical function. We choose
$r(T)=Z\left[\tanh(\frac{T-T_{0}\pm D}{\alpha})+1\right]$, with $\alpha<D$,
that describes the switching between $2Z$ and $0$ at $T_{0}\mp D$. In order to
take into account the complex domain structure, we use a collection of
independent hysterons with random parameters (${T_{0},D,Z}$) each. Variations
stand for different domain sizes, local energies involved and different number
of nodes (N or N+1). Once defined a collection of $N_{h}$ hysterons, the total
resistance at a given temperature $R(T)$ is just the sum of the resistance of
each hysteron $r_{i}(T)$. We can then write
$R(T)=aT+b+\sum_{i=1}^{N_{h}}Z_{i}\left[\tanh\left(\frac{T-T_{0i}\pm
D_{i}}{\alpha}\right)+1\right]$ (1)
where the $+$ ($-$) sign stands for cooling (warming), and we have added a
linear trend $aT+b$ as the base where hysterons are mounted. This is a
reasonable assumption to account for the phonon and impurity scattering
contributions to the resistance in the temperature range of interest.
Figure 2: (color online) Theoretical model: (a) difference of the resistance
in cooling-warming cycles of different amplitudes in $T$. Inset: resistive
hysteron. (b) FORC curves L0M, L1M, L2M, and L3M obtained from the data in (a)
with $\Delta R_{s}$ as defined in the text.
The results obtained from this model are shown in Fig.2. Based on the shape of
the FORCs we adopted a random gaussian distribution for each of the parameters
$T_{0},D,Z$ with average values and standard deviations: $\langle
T_{0}\rangle=7$, $\sigma_{T_{0}}=0.6$; $\langle D\rangle=0.5$,
$\sigma_{D}=0.12$; and $\langle Z\rangle=10^{-4}$, $\sigma_{Z}=0.15\times
10^{-4}$. In panel (a) cooling-warming cycles have been computed using
different distributions of hysterons for each cycle. There are two main
effects of $N_{h}$ on $R(T)$, namely the enhancement of the difference between
$R_{c}$ and $R_{w}$ in the hysteresis region, and the smoothing of the curve
shape. When $N_{h}$ is very small, $N_{h}\sim 10^{1}$, discretization plays a
mayor role, giving a steps-shaped curve. To reproduce the features of our
experimental data we took $N_{h}\sim 700$. In Panel (b) we show the FORCs
obtained with the model.
Figure 3: (color online) Noise in $d\Delta R/dT$ for the loops shown in: (a)
Fig.1 (experiment), and (b) Fig.2 (model). Panel (c) shows the correlation
function $G_{ij}=G\langle Code_{i},Code_{j}\rangle/min\\{N_{i},N_{j}\\}$ as a
function of the window position. See text
.
A comparison of Figs.1 and 2 nicely shows that the results obtained with the
model are in very good agreement with the experimental data. This supports our
view of the hysteresis process as a complicated movement of domain-walls with
no memory of the path followed to reach the state with $N+1$ nodes by lowering
the temperature, or with $N$ nodes by increasing it.
In Fig.3 we display the noise in the loops L0 to L3, that is the peak
structure present in $d\Delta R/dT$. Panel (a) shows the experimental data
shifted vertically by an arbitrary offset for the sake of clarity. In panel
(b) the noise patterns computed using the model are plotted. A small random
gaussian noise has been included to account for the experimental thermal
fluctuations in $R(T)$. The similarity is remarkable and we may say, using the
analogy with ferromagnets, that this noise in $R(T)$ is the resistive version
in SDW systems of the Barkhausen noise characteristic of ferromagnetic
materials.[16] However, we expect some differences between the ferromagnetic
and SDW systems. In particular, the long-range dipolar interaction present in
ferromagnetic films plays an essential role in the domain dynamics.
Unfortunately, we do not have a direct access to observe the dynamics. Since
in our case the control parameter is the temperature, the dynamics of the
domain redistribution is affected by the way the sample thermalizes after each
temperature change. Therefore, we only have access to the final state at each
temperature.
Figs.3 (a) and (b), seem to indicate that each experimental noise pattern is
different.
In order to give a more qualitative analysis of this statement we digitalize
the data and introduce a function that allows us to obtain a code-bar type
identification of each pattern. We create our code-bar in the temperature axis
by dividing it in $M$ small intervals $T_{n}$. We subtract from each noise
pattern in Fig. 3(a) the smoothed curve to obtain a pattern as the one shown
in the inset of Fig. 1. Taking the highest (positive or negative) peak
amplitude in the resulting pattern we define the relevant scale that is
divided in intervals $W_{k}$ of fixed size $\Delta$. The interval $W_{k}$
extends from $k\Delta$ to $(k+1)\Delta$ with $k=0,1,...,N$. For the pattern of
each loop $L_{i}$ and each window $W_{k}$, the code function is defined as:
$\left\\{\begin{array}[]{cc}Code_{i,k}[T_{n}]=1,&if\ the\ peak\ amplitude\ \in
W_{k}\\\ Code_{i,k}[T_{n}]=0,&otherwise\end{array}\right.$ (2)
In this way, for each pattern and window we have an M-dimensional vector whose
components are 0 or 1. The square modulus
$G^{k}_{ii}=|Code_{i,k}[Tn]|^{2}=Code_{i,k}[Tn]\cdot Code_{i,k}[Tn]$ of each
code gives the total number of peaks with amplitude in the window $W_{k}$. In
order to compare the noise spectrum of the different loops we first note that
although all codes have the same dimension $M$, the first $N_{i}$ components
of loop $L_{i}$ (i=1,2,3) are zero by definition. In Fig. 3(c) we show the
normalized functions $G^{k}_{ii}/(M-N_{i})$ (upper curves) that are equivalent
to the histograms of the noise amplitude. The solid line is a fit with a
Gaussian function. In order to analyze the reproducibility of the noise in
different loops (Fig.3 (a)), we define the correlation function as the scalar
product of two code functions, i.e. $G^{k}_{ij}=Code_{i,k}[Tn]\cdot
Code_{j,k}[Tn]/(M-Nj)$ with $i<j$. The lower curves correspond to $G_{01}$
(full circles), $G_{02}$ (full diamonds) and $G_{03}$ (full squares). The
smaller correlation seen in all $G_{0i}$ correspond to stochastic correlations
among two different gaussian distributions, one for loop L0, and the others
for loops Li, with $i=1,2,3$. Correspondingly, the solid line is a fit with
the same upper Gaussian function but squared.
It is important to realize that if the noise patterns were exactly the same
for all loops, irrespective of its temperature span, all curves should
coalesce onto a single curve (as clearly is the case for the autocorrelation
functions $G_{ii}$). If instead, only part of the loops were correlated (for
example the common path when cooling down) the $G^{k}_{ij}$ should give larger
values than the correlations observed among two different gaussian
distributions. Therefore, this result illustrates in a more quantitative way
that each realization of the experiment follows a different path with respect
to the distribution of SDW domains, giving rise to different resistive noise
patterns in each different run. Finally, we mention that the noise amplitude
with quasi-Gaussian distribution is present for all studied films, and has a
standard deviation that increases with film thickness.
Another way to analyze in more detail the noise patterns, would be to
construct a FORC diagram [17] with the experimental data. This procedure would
allow to estimate the distribution of hysteron parameters directly from the
experiments. However, this task would demand more than a year for measuring a
reasonable number of loops, say 200, because each loop takes almost 50 hours
of measurement. The good agreement between experiments and simulations led us
to evaluate the FORC using our simple model. The resulting FORC diagram is
similar to the one obtained for systems of single domain ferromagnetic
particles (SDFP) with a narrow size distribution. In these systems each
particle is subject to the effect of and external magnetic field and to a
random field $H_{int}$ due to the dipolar interactions. In view of these
similarities, we can make an analogy between the Cr SDW resistivity hysteretic
behavior and the much simpler case of the SDFP. In the SDFP system each
hysteron corresponds to the orientation of one particle magnetization along
the external field direction. The random fields play an important role in
determining the shape of the FORC diagram. In the case under study here,
hysterons correspond to transitions between different configurations of
domains with N or N+1 SDW nodes. The role of the random fields is played by a
random distribution of $T_{0}$ which is crucial in order to reproduce the
experimental observations. Finally, the particle size distribution in the SDFP
case is here represented by the distribution of resistance jumps $Z$, and the
values of $D$ are related to the energy barriers strength between different
domain configurations.
In summary, we have shown experimentally that the resistance of thin Cr films
displays a characteristic noise in the same temperature region where
hysteretic behavior is seen. We presented an extension of the Preisach model
and the introduction of resistive hysterons in a simple phenomenological model
which reproduced quite satisfactorily the noise patterns in $dR/dT$ and
hysteresis in $R(T)$. These are mainly generated by the switching of a random
distribution of SDW domains with N or N+1 nodes in the spin density wave. The
main assumption supported by previous experiments [7] is that the domains with
N+1 nodes have a lower resistance than those with N nodes. Our results present
new evidence supporting this scenario. However, to our knowledge a microscopic
theory is still lacking and new theoretical efforts are needed to complete the
understanding of this interesting issue. We show that each realization of
cooling-warming cycles follows a different path with a different random
distribution of domains. We conclude that the system presents a narrow
distribution of $T_{0}$ that suggests a narrow distribution of meta stable
domain configurations. Our approach also opens the possibility to study domain
evolution as a function of temperature in other antiferromagnetic materials in
confined geometries.
###### Acknowledgements.
We thank fruitful discussions with E. Jagla. E.O., E.K. and C.A.B. are members
of CONICET. L.T. has a scholarship from CONICET. This work was partially
supported by CONICET, by PIP No 11220080101821 of CONICET, by PICT Nos
2006/483 and R1776 of the ANPCyT, Argentina.
## References
* [1] A. W. Overhauser Phys. Rev.12819621437.
* [2] Fawcett E. Rev. Mod. Phys.601988209.
* [3] Shpyrko O. G., Isaacs E. D., Logan J. M., Feng Yejun, Aeppli G., Jaramillo R., Kim H. C., Rosenbaum T. F., Zschack P., Sprung M., Narayanan S. Sandy A. R. Nature447200768.
* [4] Michel R. P., Israeloff N. E., Weissman M. B., Dura J. A. Flynn C. P. Phys. Rev. B4419917413.
* [5] Sonntag P., Bödeker P., Schreyer A., Zabel H., Hamacher K. Kaiser H. J. Magn. Magn. Mater.18319985.
* [6] Bödeker P., Hucht A., Schreyer A., Borchers J., Guthoff F. Zabel H. Phys. Rev. Lett.811998914.
* [7] Kummamuru R. K. Soh Yeong-Ah Nature (London)4522008859.
* [8] Osquiguil E., Kaul E. E., Tosi L. Balseiro C. A. Phys. Rev. B852012104410.
* [9] Barkhausen H. Z. Phys.201919401.
* [10] Krivoruchko V. N.,Melikhov Y. Jiles D. C. Phys. Rev. B692008180406(R).
* [11] Preisach F. Z. Phys.941935277.
* [12] Fullerton E. E., Robertson J. L., Prinsloo A. R. E., Alberts H. L. Bader S. D. Phys. Rev. Lett.912003237201.
* [13] Mayergoys I. D. IEEE Trans. Magn.MAG-221986603.
* [14] Kaul E. E. Osquiguil E. unpublished2012.
* [15] Shi Z. P. Fishman R. S. Phys. Rev. Lett.7819971351.
* [16] Kim Dong-Hung, Choe Sug-Bog Shin Sung-Chul Phys. Rev. Lett.902003087203.
* [17] Roberts A. P., Pike C. R. Verosub K. L. Jour. Geophys. Res.105200028461.
|
arxiv-papers
| 2012-12-06T18:28:42 |
2024-09-04T02:49:38.992367
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "L. Tosi, E. Osquiguil, E. E. Kaul and C. A. Balseiro",
"submitter": "Leandro Tosi",
"url": "https://arxiv.org/abs/1212.1411"
}
|
1212.1443
|
# Transparent ion trap with integrated photodetector
Amira M. Eltony [email protected] Shannon X. Wang Gleb M. Akselrod Peter F.
Herskind Isaac L. Chuang Center for Ultracold Atoms, Research Laboratory of
Electronics and Department of Physics, Massachusetts Institute of Technology,
Cambridge, Massachusetts, 02139, USA
###### Abstract
Fluorescence collection sets the efficiency of state detection and the rate of
entanglement generation between remote trapped ion qubits. Despite efforts to
improve light collection using various optical elements, solid angle capture
is limited to $\approx 10\%$ for implementations that are scalable to many
ions. We present an approach based on fluorescence detection through a
transparent trap using an integrated photodetector, combining collection
efficiency approaching 50% with scalability. We microfabricate transparent
surface traps with indium tin oxide and verify stable trapping of single ions.
The fluorescence from a cloud of ions is detected using a photodiode
sandwiched with a transparent trap.
Integrated trapped ion quantum computation systems have recently made advances
building on silicon chip technology by incorporating a variety of devices
including optical fibers VanDevender _et al._ (2010); Kim, Herskind, and
Chuang (2011); Brady _et al._ (2011), MEMs cantilevers Stick _et al._
(2005), and control electronics. This integration seeks to improve gate
fidelities, fault tolerance thresholds, and scalability, but state detection
efficiency relies on fluorescence detection, which is largely still
implemented with bulk optics and conventional photomultipliers or image-
intensified charge coupled detectors. Scaling-up to dense arrays of trapped
ions will require efficient light collection from many ions in parallel. For
distributed architectures, with remote atomic nodes connected by photons,
light collection is also a critical factor determining the efficiency of
remote entanglement generation Luo _et al._ (2009).
The conventional approach to light collection places a high numerical aperture
objective near the ion, and detects light with a photomultiplier or charge-
coupled detector located outside of the vacuum chamber, resulting in solid
angle capture of less than 5%. There are various proposals to enhance atom-
photon coupling within a scalable architecture. An array of Fresnel lenses in
place of a bulk lens can provide more efficient light collection from multiple
ions Streed _et al._ (2011); Jechow _et al._ (2011). Or, a micro mirror
embedded into a planar trap can improve solid angle collection from an ion
trapped above Herskind _et al._ (2011); Noek _et al._ (2010) to as much as
$\approx 10\%$ Merrill _et al._ (2011). Greater solid angle capture is
possible, at the cost of scalability, by placing the trapping site at the
focus of a spherical Shu _et al._ (2011) or parabolic Maiwald _et al._
(2012) mirror. Similarly, the ion can be trapped within a high-finesse optical
cavity for enhanced light collection into a single mode Mundt _et al._
(2002); Keller _et al._ (2004). Integration of a planar ion trap with an
optical fiber has been demonstrated Kim, Herskind, and Chuang (2011); Brady
_et al._ (2011), although the solid angle collection is low: $\approx 3.5\%$
VanDevender _et al._ (2010).
A different concept is to collect fluorescence _through_ a planar trap via the
underutilized $2\pi$ solid angle below the ion. A surface electrode trap made
of transparent materials would in principle allow for collection efficiency
approaching 50%. Combining such a transparent trap with an array of detectors
beneath opens up the possibility of massively parallel light collection.
Indium tin oxide (ITO), an electrical and optical conductor commonly used for
applications such as touch screens and LCD displays, is a natural choice for
transparent electrodes. But, trapping ions with ITO electrodes poses
significant difficulty because ITO has a resistivity about 1000 times higher
than metals typically used for trap electrodes, and ITO is an oxide, making
laser-induced charging of its surface a concern Wang _et al._ (2011).
Motivated by these challenges, we present results from transparent surface
electrode ion traps, fabricated using ITO. Two ITO traps are tested at
cryogenic temperatures (4 K and 77 K), where low pressure is achievable within
a short time frame, and heating of the ion’s motional state is suppressed
Labaziewicz _et al._ (2008). 88Sr+ ions are trapped 100 $\mu$m above the trap
surface, which is heat sunk to the 4 K stage of a bath cryostat. The
fluorescence emitted on the Doppler cooling transition (5S1/2
$\leftrightarrow$ 5P1/2) at 422 nm is collected for state detection. We
observe stable trapping in a first ITO trap. We then demonstrate a proof-of-
principle prototype for scalable fluorescence detection in a second ITO trap
by collecting light from a trapped ion cloud using a standard photodiode
mounted below. Finally, we propose a highly-efficient and compact
“entanglement unit” based on this design.
The trap geometry used is a well established five-electrode designLabaziewicz
_et al._ (2008) (see Figure 1(c)), with trap frequencies in the range of
0.8-1.3 MHz for an RF frequency of 35 MHz, and a trap depth of about 300 meV.
Trap fabrication begins with optical lithography using NR9-3000PY photoresist
on quartz substrate. Next, ITO is deposited by RF sputtering with argon gas at
a rate of 0.5 Å/s. Finally, the resist pattern is transferred to the ITO via
lift-off with RD6. The resulting optical transmission of the ITO samples
(including the polished quartz substrate), measured over a 4 mm2 area with a
422 nm light source at room temperature, averages to $\approx 60\%$, which is
about 10% lower than expected for commercial films Kim _et al._ (1999). The
measured resistivity of ITO varies from $1\times 10^{-5}$ $\Omega$m to
$2\times 10^{-5}$ $\Omega$m for different sputtering runs. To improve the
conductivity of the RF electrodes, an additional lithography step is performed
to deposit a thin gold layer on the RF electrodes only.
The first ITO trap, ITO-4K (see Figure 1(a)), has 400 nm of ITO for all trap
electrodes, and an additional 50 nm of gold on the RF electrodes only,
bringing the resistivity down to $\approx 2\times 10^{-8}$ $\Omega$m, which is
comparable to a fully-metal trap. The second ITO trap, ITO-PD (see Figure
1(b)), also has 400 nm of ITO for all trap electrodes, but only 5 nm of gold
on the RF electrodes for improved transparency ($\approx 60\%$ for 5 nm of Au
Stognij _et al._ (2003)) with a resistivity of $\approx 3\times 10^{-8}$
$\Omega$m. This second trap is sandwiched with a photodetector. For a proof-
of-concept demonstration, a commercially available PIN photodiode (Advanced
Photonix PDB-C613-2) is used. The photodiode efficiency drops significantly at
cryogenic temperatures due to carrier freeze-out. Measurements at 422 nm
reveal that the photodiode responsivity changes little from $\approx 0.1$ A/W
as it is cooled from room temperature to 77 K, but plummets to $\approx 0.01$
A/W at 4 K. For this reason, measurements with the photodiode are performed at
77 K.
Figure 1: (a) ITO-4K trap mounted in a CPGA; 50 nm of Au is visible on the RF
electrodes and contact pads for wire bonding. (b) ITO-PD trap mounted on
photodiode in a CGPA; with only 5 nm of Au on the RF electrodes, the
photodiode is visible through the trap. (c) Diagram of trap geometry showing
RF electrodes (red), ground electrodes (blue), and DC electrodes (green).
For ITO-4K, the estimated trap-surface temperature was 6 K. Single 88Sr+ ions
were stably trapped with a lifetime of several hours, comparable to metal
traps. No significant change in the micromotion amplitude was measured after
crashing either a 405 nm or a 461 nm laser beam with intensity $\approx 5$
mW/mm2 into the center of the trapWang _et al._ (2011) for 10 minutes,
indicating that charging is not a major problem for ITO traps operated at
cryogenic temperatures.
ITO-PD has an estimated trap-surface temperature of 77 K. Because the
photodiode used has no internal gain mechanism, the resulting picoamp-scale
photocurrent is difficult to distinguish from electrical noise, making lock-in
detection essential. The intensity of the laser addressing the 4D3/2
$\leftrightarrow$ 5P3/2 transition is chopped at 300 Hz, resulting in
modulation of the ion fluorescence as population is successively trapped then
pumped from the metastable 4D3/2 state during Doppler cooling. Inside the
cryostat, a custom preamplifier circuit mounted to the 77 K shield amplifies
the signal with low added noise before it is input to a lock-in amplifier
(Stanford Research Systems SR530) outside the chamber, as shown in Figure 2.
The expected signal for $\approx 50$ ions in this setup is estimated in Table
1 and compared with our conventional bulk optics and photomultiplier (PMT)
setup.
Figure 2: Apparatus for detection of ion fluorescence through a transparent trap using a photodiode mounted below. The acousto-optic modulator (AOM) is used to modulate the repumper for lock-in detection. | Light | Power | Detector | Photodiode | Lock-in
---|---|---|---|---|---
| collection | at the | quantum | current | amplifier
| efficiency | detector | efficiency | | output
ITO-PD | 30% | 60 pW | 30% | 6 pA | 120 mV
PMT | 5% | 10 pW | 20% | n.a. | n.a.
Table 1: Comparison of photodiode and photomultiplier (PMT) collection
efficiencies, with expected signal values for 50 ions, assuming a scattering
rate of $\approx 10^{7}$ photons/s per ion at 422 nm, resulting in $\approx
200$ pW of total fluorescence into $4\pi$ solid angle.
The pressure in the cryostat, when operated at 77 K, is insufficient ($\approx
1\times 10^{-7}$ Torr) for stable trapping, so a cloud of ions was continually
loaded. Figure 3(a) plots the photodiode voltage during initial loading of an
ion cloud against the photon counts for light collected at the same time using
bulk optics and a photomultiplier, indicating that the photodiode response is
proportional to the fluorescence rate. Variation in the photodiode signal is
likely due to scatter from the modulated repumper beam. Figure 3(b) compares
the photodiode voltage before and after an ion cloud was loaded, showing that
the ion cloud fluorescence is distinguishable from the background. Over a
measurement interval of a few minutes, the detected signal from the ion cloud
averages to $175\pm 49$ mV, which is consistent with the signal predicted
above for $\approx 50$ ions.
Figure 3: (a) Photodiode voltage and photomultiplier count rate, both
background subtracted, during loading of an ion cloud. Each point is averaged
over 30 seconds. (b) Histogram of photodiode voltages over a period of several
minutes without ions (red), and after loading an ion cloud (blue).
These experiments indicate that significant improvements in quantum state
detection are possible using transparent traps with integrated detectors.
Assuming a noiseless amplifier, our photodiode signal could be used to
distinguish between quantum states with greater than 99% fidelity with a 1 ms
integration time. More generally, nearly 50% solid-angle collection is
possible by using a photodiode with a large active area or focusing
fluorescence onto the photodiode using additional optics below the trap. Then,
the only losses before the detector occur in the ITO film and in the
substrate. For commercially available ITO films these losses could be as low
at 10% Kim _et al._ (1999). Replacing the photodiode by a device with an
internal gain mechanism such as the Visible Light Photon Counter (VLPC), which
has a quantum efficiency of 88% at 694 nm and 4 K McKay, Kim, and Hogue
(2009), would allow a total detection efficiency of nearly 40%, compared to
the typical 1-5% possible with a conventional photomultiplier and bulk optics.
For our system, this would mean reducing the time required for quantum state
detection with 99% fidelity from the current 200 $\mu$s to only 5 $\mu$s.
Figure 4: Diagram of proposed compact entanglement unit (see text).
Our measurements establish that ITO is a viable material to use for
microfabricated traps, and provide a first demonstration of light collection
from ions through a trap with an integrated photodetector. The ability to form
transparent traps opens up many possibilities to integrate ions with devices
to transfer and detect light efficiently in a scalable architecture. One
particularly interesting application is a compact entanglement unit, as shown
in Figure 4. Here, transparent traps mounted on adjacent faces of a beam
splitter house two (or more) ions to be entangled. Diffractive optics below
the traps overlap the images of the two ions on detectors at the opposite
faces of the cube, allowing for heralded entanglement generation between the
ions Luo _et al._ (2009). For current state-of-the-art experiments, relying
on bulk optics and light transmission in fibers, the total coupling efficiency
for photons is only $\approx 0.004$ Maunz _et al._ (2009), resulting in an
entanglement generation rate of only $\approx 2\times 10^{-3}$ s-1 (when the
experiment is repeated at 100 kHz, assuming a branching ratio of 0.005, and a
photodetector quantum efficiency of 15%). For the proposed compact
entanglement unit (neglecting losses due to reflection at interfaces, and
absorption in materials other than ITO), the coupling efficiency is $\approx
0.45$, resulting in a probability of entanglement generation of $\approx 30$
s-1, which is $\approx 10^{3}$ times higher than the best rates achieved to
date.
This work was supported by the MQCO Program with funding from IARPA, and by
the NSF Center for Ultracold Atoms.
## References
* VanDevender _et al._ (2010) A. VanDevender, Y. Colombe, J. Amini, D. Leibfried, and D. Wineland, Phys. Rev. Lett. 105, 023001 (2010).
* Kim, Herskind, and Chuang (2011) T. H. Kim, P. F. Herskind, and I. L. Chuang, Appl. Phys. Lett. 98, 214103 (2011).
* Brady _et al._ (2011) G. R. Brady, A. R. Ellis, D. L. Moehring, D. Stick, C. Highstrete, K. M. Fortier, M. G. Blain, R. A. Haltli, A. A. Cruz-Cabrera, R. D. Briggs, J. R. Wendt, T. R. Carter, S. Samora, and S. A. Kemme, Appl. Phys. B 103, 801 (2011).
* Stick _et al._ (2005) D. Stick, W. K. Hensinger, S. Olmschenk, M. J. Madsen, K. Schwab, and C. Monroe, Nat. Phys. 2, 36 (2005).
* Luo _et al._ (2009) L. Luo, D. Hayes, T. A. Manning, D. N. Matsukevich, P. Maunz, S. Olmschenk, J. D. Sterk, and C. Monroe, Fortschr. Phys. 57, 1133 (2009).
* Streed _et al._ (2011) E. Streed, B. Norton, A. Jechow, T. Weinhold, and D. Kielpinski, Phys. Rev. Lett. 106, 010502 (2011).
* Jechow _et al._ (2011) A. Jechow, E. W. Streed, B. G. Norton, M. J. Petrasiunas, and D. Kielpinski, Opt. Lett. 36, 1371 (2011).
* Herskind _et al._ (2011) P. F. Herskind, S. X. Wang, M. Shi, Y. Ge, M. Cetina, and I. L. Chuang, Opt. Lett. 36, 3045 (2011).
* Noek _et al._ (2010) R. Noek, C. Knoernschild, J. Migacz, T. Kim, P. Maunz, T. Merrill, H. Hayden, C. S. Pai, and J. Kim, Opt. Lett. 35, 2460 (2010).
* Merrill _et al._ (2011) T. J. Merrill, C. Volin, D. Landgren, J. M. Amini, K. Wright, S. Charles Doret, C.-S. Pai, H. Hayden, T. Killian, D. Faircloth, K. R. Brown, A. W. Harter, and R. E. Slusher, New J. Phys. 13, 103005 (2011).
* Shu _et al._ (2011) G. Shu, C.-K. Chou, N. Kurz, M. R. Dietrich, and B. B. Blinov, J. Opt. Soc. Am. B 28, 2865 (2011).
* Maiwald _et al._ (2012) R. Maiwald, A. Golla, M. Fischer, M. Bader, S. Heugel, B. Chalopin, M. Sondermann, and G. Leuchs, Phys. Rev. A 86, 043431 (2012).
* Mundt _et al._ (2002) A. Mundt, A. Kreuter, C. Becher, D. Leibfried, J. Eschner, F. Schmidt-Kaler, and R. Blatt, Phys. Rev. Lett. 89, 103001 (2002).
* Keller _et al._ (2004) M. Keller, B. Lange, K. Hayasaka, W. Lange, and H. Walther, Nature 431, 1075 (2004).
* Wang _et al._ (2011) S. X. Wang, G. Hao Low, N. S. Lachenmyer, Y. Ge, P. F. Herskind, and I. L. Chuang, J. Appl. Phys. 110, 104901 (2011).
* Labaziewicz _et al._ (2008) J. Labaziewicz, Y. Ge, P. Antohi, D. Leibrandt, K. Brown, and I. Chuang, Phys. Rev. Lett. 100, 013001 (2008).
* Kim _et al._ (1999) H. Kim, A. Piqué, J. S. Horwitz, H. Mattoussi, H. Murata, Z. H. Kafafi, and D. B. Chrisey, Appl. Phys. Lett. 74, 3444 (1999).
* Stognij _et al._ (2003) A. I. Stognij, N. N. Novitskii, S. D. Tushina, and S. V. Kalinnikov, Tech. Phys. 48, 745 (2003).
* McKay, Kim, and Hogue (2009) K. S. McKay, J. Kim, and H. H. Hogue, Opt. Express 17, 7458 (2009).
* Maunz _et al._ (2009) P. Maunz, S. Olmschenk, D. Hayes, D. Matsukevich, L.-M. Duan, and C. Monroe, Phys. Rev. Lett. 102, 250502 (2009).
|
arxiv-papers
| 2012-12-06T20:44:03 |
2024-09-04T02:49:38.999895
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Amira M. Eltony, Shannon X. Wang, Gleb M. Akselrod, Peter F. Herskind,\n and Isaac L. Chuang",
"submitter": "Amira Eltony",
"url": "https://arxiv.org/abs/1212.1443"
}
|
1212.1526
|
# Integral-type operators from Hardy space to Bloch space on the upper half-
plane
Ning Xu 111Corresponding author, e-mail: [email protected].
Department of Mathematics and Science, Huai Hai Institute of Technology,
Jiangsu, Lianyungang 222005, P. R. China
Abstract. The boundedness of compactness of integral-type operators from Hardy
space to Bloch space on the upper half-plane
$\Pi_{+}=\\{z\in\mathbb{C}:Imz>0\\}$ are characterized.
Keywords:Integral-type operator Upper half-plane Boundedness Compactness
1\. Introduction
Let $\Pi_{+}$ be the upper half-plane and $H(\Pi_{+})$ be the space of all
holomorphic functions on the $\Pi_{+}$. The Hardy spaces $H^{2}(\Pi_{+})$
consists of all $f\in H(\Pi_{+})$ such that
$\|f\|^{2}_{H^{2}(\Pi_{+})}=\sup_{y>0}\int^{\infty}_{-\infty}|f(x+iy)|^{2}dx<\infty$
where $y=Imz$.
The Bloch space $B_{\infty}(\Pi_{+})$ consists of all of $f\in H(\Pi_{+})$
such that
$B(f)=\sup_{z\in{\Pi_{+}}}Imz|f^{\prime}(z)|<\infty.$
A natural norm on the Bloch space can be introduced as follows
$\|f\|^{2}_{B_{\infty}(\Pi_{+})}=|f(i)|+B(f).$
With this norm the $B_{\infty}(\Pi_{+})$ becomes a Banach space.
For $f\in H(\Pi_{+})$,fix $x_{0}\in\Pi_{+}$ the integral-type operators
$J_{g}$ and $I_{g}$ are defined by
$J_{g}f(z)=\int_{z_{0}}^{z}f(\zeta)g^{\prime}(\zeta)d\zeta,z\in\Pi_{+}\ \ \ \
\ (1)$ $I_{g}f(z)=\int_{z_{0}}^{z}f^{\prime}(\zeta)g(\zeta)d\zeta,z\in\Pi_{+}\
\ \ \ \ (2)$
where $g\in H(\Pi_{+})$.
The boundedness and compactness of $J_{g}$ and $I_{g}$ have been introduced on
the unit dick in[1,8,10,11,20].The operators $J_{g}$ and $I_{g}$, as well as
their n-dimensional generalizations,acting on various spaces of analytic
functions, have been recently studied,for example, in[2-7,9,19].
The importance of the operators $J_{g}$ and $I_{g}$ comes from that
$J_{g}f+I_{g}f=M_{g}f-f(z_{0})g(z_{0})$
where the multiplication operator $M_{g}$ is defined by
$(M_{g}f)(z)=g(z)f(z)$.
Let $X$ and $Y$ be topological vector spaces whose topologies are given by
translation- invariant metrics $d_{X}$ and $d_{Y}$,respectively.It is said
that a linear operator $T:X\rightarrow Y$ is metrically bounded if there
exists a positive constant $K$ such that
$d_{Y}(Tf,0)\leq d_{X}(f,0)$
for all $f\in X$. When $X$ and $Y$ are Banach spaces, the metrically
boundedness coincides with the boundedness of operators between Banach spaces.
If we say that an operator is bounded we will regard that it is metrically
bounded. Operator $T:X\rightarrow Y$ is said to be metrically compact if it
takes every metric ball in $X$ into a relatively compact set in $Y$.
While there is a vast literature on composition and weighted composition
operators between spaces of holomorphic functions on the unit disk,there are
few papers on there operators on spaces of functions holomorphic on the upper
half-plane(see,12-18). In this paper,we consider the metrically boundedness
and compactness of (1),(2) acting from $H^{2}(\Pi_{+})$ to
$B_{\infty}(\Pi_{+})$ on the upper half-plane.
Throughout this paper, constants are denoted by $C$,they are positive and may
differ from one occurrence to the other. The notion $A\asymp B$ means that
there is a positive constant $C$ such that $C^{-1}B\leq A\leq CB$ .
2\. Auxiliary results
Lemma 1.Let $L=J_{g}(I_{g})$ and $g\in H^{\infty}(\Pi_{+})$.
Then$L:H^{2}(\Pi_{+})$ to $B_{\infty}(\Pi_{+})$ is metrically compact if and
only if for any bounded sequence $\\{f_{n}\\}_{n\in\mathbb{C}}$ in
$H^{2}(\Pi_{+})$ converging to zero on compact of $\Pi_{+}$, we have
$\lim_{n\rightarrow\infty}\|Lf_{n}\|_{B_{\infty}(\Pi+)}=0$.
The proof is standard which can be found in [14] and omitted here.
Lemma 2[14].Let $0<p<\infty$ and $0<a<b$. If $f\in H^{2}(\Pi_{+})$, then
$\lim_{z\rightarrow\infty z\in\Gamma_{a,b}}f(z)=0,$
where $\Gamma_{a,b}=\\{z\in\Pi_{+}:a\leq Imz\leq b\\}$.
Lemma 3[17]. Let $f\in H^{2}(\Pi_{+})$ then
$|f^{(n)}(z)|\leq
C\frac{\|f\|_{H^{2}(\Pi_{+})}}{(Imz)^{n+\frac{1}{2}}},n=0,1,2\ \ \ \ \ (3)$
3.Main results
Here we formulate and prove the main results of this paper.
Theorem 1. Let $g$ is holomorphic function of $\Pi_{+}$ and $g\in
H^{\infty}(\Pi_{+})$. Then
(a) $J_{g}:H^{2}(\Pi_{+})\rightarrow B_{\infty}(\Pi_{+})$ is bounded if and
only if
$M_{1}=\sup_{z\in\Pi_{+}}(Imz)^{\frac{1}{2}}|g^{\prime}(z)|<\infty.\ \ \ \ \ \
\ \ \ (4)$
(b) $I_{g}:H^{2}(\Pi_{+})\rightarrow B_{\infty}(\Pi_{+})$ is bounded if and
only if
$M_{2}=\sup_{z\in\Pi_{+}}\frac{|g(z)|}{(Imz)^{\frac{1}{2}}}<\infty.\ \ \ \ \ \
\ \ \ (5)$
Proof.(a)Assume that condition in(4) hold. Then
$\displaystyle\|J_{g}f\|_{B_{\infty}(\Pi_{+})}$ $\displaystyle=$
$\displaystyle|J_{g}f(i)|+\sup_{z\in\Pi_{+}}Imz|f(z)g^{\prime}(z)|$
$\displaystyle=$
$\displaystyle|\int^{i}_{z_{0}}f(\zeta)g^{\prime}(\zeta)d\zeta|+\sup_{z\in\Pi_{+}}Imz|f(z)g^{\prime}(z)|$
$\displaystyle\leq$ $\displaystyle\max_{|z_{0}|\leq|\zeta|\leq
1,\zeta\in\Pi_{+}}|f(\zeta)||g(i)-g(z_{0})|$ $\displaystyle+$
$\displaystyle\sup_{z\in\Pi_{+}}(Imz)^{\frac{1}{2}}|g^{\prime}(z)|\|f\|_{H^{2}(\Pi_{+})}.$
From $f\in H^{2}(\Pi_{+})$ and $g\in H^{\infty}(\Pi_{+})$, it follows that
$J_{g}:H^{2}(\Pi_{+})\rightarrow B_{\infty}(\Pi_{+})$ is bounded.
Now assume that $J_{g}:H^{2}(\Pi_{+})\rightarrow B_{\infty}(\Pi_{+})$ is
bounded. Choose
$f_{w}(z)=\frac{(Imw)^{\frac{3}{2}}}{\Pi^{\frac{1}{2}}(z-\bar{w})^{2}}\ \ \ \
\ (6)$
It is clear that $f_{w}\in H^{2}(\Pi_{+})$ and that
$\|f_{w}\|_{H^{2}(\Pi_{+})}=1$. Choose $w=z$, we obtain
$C\geq\|J_{g}f_{w}\|_{B_{\infty}(\Pi_{+})}\geq\sup_{w\in\Pi_{+}}Imw|f_{w}(w)g^{\prime}(w)|=\sup_{w\in\Pi_{+}}\frac{(Imz)^{\frac{1}{2}}|g^{\prime}(z)|}{4\Pi^{\frac{1}{2}}}$
Therefore we obtain (4).
(b) Assume that condition in (5) hold. Then
$\displaystyle\|I_{g}f\|_{B_{\infty}(\Pi_{+})}$ $\displaystyle=$
$\displaystyle|I_{g}f(i)|+\sup_{z\in\Pi_{+}}Imz|f^{\prime}(z)g(z)|$
$\displaystyle=$
$\displaystyle|\int^{i}_{z_{0}}f^{\prime}(\zeta)g(\zeta)d\zeta|+\sup_{z\in\Pi_{+}}Imz|f^{\prime}(z)g(z)|$
$\displaystyle\leq$ $\displaystyle|f(i)-f(z_{0})|\max_{|z_{0}|\leq|\zeta|\leq
1,\zeta\in\Pi_{+}}|g(\zeta)|$ $\displaystyle+$
$\displaystyle\sup_{z\in\Pi_{+}}(Imz)^{\frac{1}{2}}|g^{\prime}(z)|\|f\|_{H^{2}(\Pi_{+})}.$
From $f\in H^{2}(\Pi_{+})$ and $g\in H^{\infty}(\Pi_{+})$, it follows that
$I_{g}:H^{2}(\Pi_{+})\rightarrow B_{\infty}(\Pi_{+})$ is bounded.
Now assume that $I_{g}:H^{2}(\Pi_{+})\rightarrow B_{\infty}(\Pi_{+})$ is
bounded. Choose(6) and let $w=z$, we obtain
$C\geq\|I_{g}f_{w}\|_{B_{\infty}(\Pi_{+})}\geq\sup_{w\in\Pi_{+}}Imw|f^{\prime}_{w}(w)g(w)|=\sup_{w\in\Pi_{+}}\frac{|g(w)|}{4\pi^{\frac{1}{2}}(Imw)^{\frac{1}{2}}}.$
Therefore we obtain (5).
Theorem 2. Let $g$ is holomorphic function of $\Pi_{+}$ and $g\in
H^{\infty}(\Pi_{+})$.
(a) $J_{g}:H^{2}(\Pi_{+})\rightarrow B_{\infty}(\Pi_{+})$ is compact, then
$\lim_{r\rightarrow 0}\sup_{y<r}(Imz)^{\frac{1}{2}}|g^{\prime}(z)|=0.\ \ \ \ \
\ \ \ \ (7)$
(b) $I_{g}:H^{2}(\Pi_{+})\rightarrow B_{\infty}(\Pi_{+})$ is compact, then
$\lim_{r\rightarrow 0}\sup_{y<r}\frac{|g(z)|}{(Imz)^{\frac{1}{2}}}=0.\ \ \ \ \
\ \ \ \ (8)$
where it is understood that if $\\{z:Imz<r\\}$ is empty for some $r>0$, the
supremum is equal to zero.
Proof.(a) Suppose that $J_{g}:H^{2}(\Pi_{+})\rightarrow B_{\infty}(\Pi_{+})$
is compact and (7) does not hold. Then there exists a positive number $\delta$
and a sequence $\\{z_{n}\\}_{n\in\mathbb{N}}$ in $\Pi_{+}$ such that
$Imz_{n}\rightarrow 0$ and
$(Imz_{n})^{\frac{1}{2}}|g^{\prime}(z_{n})|>\delta,\ \ n\in\mathbb{N}.$
Choose
$f_{n}(z)=\frac{(Imw_{n})^{\frac{3}{2}}}{\Pi^{\frac{1}{2}}(z-\bar{w_{n}})^{2}}\
\ \ \ \ (9)$
and let $z_{n}=w_{n}$,then $f_{n}$ is norm bounded in $H^{2}(\Pi_{+})$ and
$f_{n}\rightarrow 0$ uniformly on compacts of $\Pi_{+}$ as $Imz_{n}\rightarrow
0$. By Lemma 2 it follows that a subsequence of $\\{J_{g}f_{n}\\}$ tends to 0
in $B_{\infty}(\Pi_{+})$. On the other hand,
$\|J_{g}f\|_{B_{\infty}(\Pi_{+})}\geq
Imw_{n}|f_{n}(w_{n})g^{\prime}(w_{n})|=\frac{1}{\sqrt{\pi}}(Imw_{n})^{\frac{1}{2}}|g^{\prime}(w_{n})|=\frac{\delta}{\sqrt{\pi}},$
which is a contradiction.
(b) Suppose that $I_{g}:H^{2}(\Pi_{+})\rightarrow B_{\infty}(\Pi_{+})$ is
compact and (8) does not hold. Then there exists a positive number $\delta$
and a sequence $\\{z_{n}\\}_{n\in\mathbb{N}}$ in $\Pi_{+}$ such that
$Imz_{n}\rightarrow 0$ and
$\frac{|g(z_{n})|}{(Imz_{n})^{\frac{1}{2}}}>\delta$
Choose (9) and let $z_{n}=w_{n}$, we obtain
$\|I_{g}f\|_{B_{\infty}(\Pi_{+})}\geq
Imw_{n}|f^{\prime}_{n}(w_{n})g(w_{n})|=\frac{1}{4\sqrt{\pi}}\frac{|g(w_{n})|}{(Imw_{n})^{\frac{1}{2}}}=\frac{\delta}{4\sqrt{\pi}},$
which is a contradiction.
Theorem 3. Let$g$ is holomorphic function of $\Pi_{+}$ ,$g\in
H^{\infty}(\Pi_{+})$ and $J_{g}:H^{2}(\Pi_{+})\rightarrow B_{\infty}(\Pi_{+})$
is bounded. Suppose that $g\in B_{\infty}(\Pi_{+})$. Then
(a) $J_{g}:H^{2}(\Pi_{+})\rightarrow B_{\infty}(\Pi_{+})$ is compact if
$\lim_{r\rightarrow 0}\sup_{y<r}(Imz)^{\frac{1}{2}}|g^{\prime}(z)|=0.\ \ \ \ \
\ \ \ \ (10)$
(b) $I_{g}:H^{2}(\Pi_{+})\rightarrow B_{\infty}(\Pi_{+})$ is compact if
$\lim_{r\rightarrow 0}\sup_{y<r}\frac{|g(z)|}{(Imz)^{\frac{1}{2}}}=0.\ \ \ \ \
\ \ \ \ (11)$
Proof.(a) Assume that (10) holds. Then for every $\epsilon>0$, there exists an
$M_{1}>0$ such that
$\sup_{y<r}(Imz)^{\frac{1}{2}}|g^{\prime}(z)|<\epsilon,\ \ \
wheneverImz<M_{1}.\ \ \ \ \ (12)$
Assume $\\{f_{n}\\}_{n\in\mathbb{N}}$ is a sequence in $H^{2}(\Pi_{+})$ such
that $\sup_{n\in\mathbb{N}}\|f_{n}\|_{H^{2}(\Pi_{+})}\leq M$ and
$f_{n}\rightarrow 0$ uniformly on compacts of $\Pi_{+}$ as
$n\rightarrow\infty$. Thus for $z\in\Pi_{+}$,such that $Imz<M_{1}$ and each
$n\in\mathbb{N},$ we have
$Imz|f_{n}(z)g^{\prime}(z)|\leq
C(Imz)^{\frac{1}{2}}|g^{\prime}(z)|\|f_{n}\|_{H^{2}(\Pi_{+})}\leq\epsilon CM.\
\ \ (13)$
By(3), we have
$|f_{n}(z)|\leq C\frac{\|f_{n}\|_{H^{2}(\Pi_{+})}}{(Imz)^{\frac{1}{2}}}\leq
C\frac{M}{(Imz)^{\frac{1}{2}}}.$
Thus there is an $M_{2}>M_{1}$ such that
$|f_{n}(z)|<\epsilon,\ \ \ \ WheneverImz>M_{2}.$
Hence for $z\in\Pi_{+}$ such that $Imz>M_{2}$,and each $n\in\mathbb{N}$, we
have
$(Imz)|f_{n}(z)g^{\prime}(z)|<\epsilon\|g\|_{B^{\infty}(\Pi_{+})}.\ \ \ \ \ \
(14)$
If $M_{1}\leq Imz\leq M_{2}$, then by Lemma 2,there exists an $M_{3}>0$ such
that
$|f_{n}(z)|<\epsilon,\ \ \ \ \ whenever|Rz|>M_{3}.$
Therefore, for each $n\in\mathbb{N}$, when $|Rz|>M_{3}$, we have
$Imz|f_{n}(z)g^{\prime}(z)|<\epsilon\|g\|_{B^{\infty}(\Pi_{+})}.\ \ \ \ \
(15)$
If $M_{1}\leq Imz\leq M_{2}$ and $|Rz|\leq M_{3}$, then there exists some
$n_{0}\in\mathbb{N}$ such that $|f_{n}(z)|<\epsilon$ for all $n\geq n_{0}$ and
so
$Imz|f_{n}(z)g^{\prime}(z)|<\epsilon\|g\|_{B^{\infty}(\Pi_{+})}.\ \ \ \ \
(16)$
Finally, we also have
$|J_{g}f_{n}(i)|=|\int_{z_{0}}^{i}f_{n}(\zeta)g^{\prime}(\zeta)d\zeta|\leq\max_{|z_{0}|\leq|\zeta|\leq
1,\zeta\in\Pi_{+}}|f_{n}(\zeta)||g(i)-g(z_{0})|\ \ \ (17)$
as $n\rightarrow\infty$.
Combining (12)-(17), we have$\|J_{g}f_{n}\|_{B_{\infty}(\Pi_{+})}<\epsilon C$,
for $n\geq n_{0}$ and some $C>0$. Thus by Lemma
1,$J_{g}:H^{2}(\Pi_{+})\rightarrow B_{\infty}(\Pi_{+})$ is compact.
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|
arxiv-papers
| 2012-12-07T03:28:33 |
2024-09-04T02:49:39.007995
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Xu Ning",
"submitter": "Xu Ning",
"url": "https://arxiv.org/abs/1212.1526"
}
|
1212.1738
|
# Filters for High Rate Pulse Processing
B.K. Alpert R.D. Horansky D.A. Bennett W.B. Doriese J.W. Fowler National
Institute of Standards and Technology, Boulder, Colorado 80305 A.S. Hoover
M.W. Rabin Los Alamos National Laboratory, Los Alamos, New Mexico 87545 J.N.
Ullom National Institute of Standards and Technology, Boulder, Colorado 80305
###### Abstract
We introduce a filter-construction method for pulse processing that differs in
two respects from that in standard optimal filtering, in which the average
pulse shape and noise-power spectral density are combined to create a
convolution filter for estimating pulse heights. First, the proposed filters
are computed in the time domain, to avoid periodicity artifacts of the
discrete Fourier transform, and second, orthogonality constraints are imposed
on the filters, to reduce the filtering procedure’s sensitivity to unknown
baseline height and pulse tails. We analyze the proposed filters, predicting
energy resolution under several scenarios, and apply the filters to high-rate
pulse data from gamma-rays measured by a transition-edge-sensor
microcalorimeter.
Keywords
baseline insensitivity, energy resolution, optimal filtering, pulse pile-up,
pulse tail insensitivity
PACS numbers
07.20.Mc, 07.05.Kf, 84.30.Sk.
U.S. government publication
Not subject to copyright.
The extraction of physical quantities from noisy data streams is ubiquitous in
the physical sciences. Examples include the determination of photon and
particle energies or incidence times in nuclear and particle physics. Raw data
records are invariably filtered to extract the quantity of interest with the
highest signal-to-noise, and extensive effort has gone into filter
development. One important example is so-called “optimal filtering,” for
isolated pulses with amplitude proportional to photon energy. Filters
constructed from the average pulse shape and the noise power spectral density
are convolved with pulse records to estimate pulse amplitudes Szymkowiak _et
al._ (1993). This filter is widely used in X-ray astrophysics Eckart _et al._
(2012) and direct searches for weakly interacting dark matter Ahmed _et al._
(2011).
Here, we propose and demonstrate a novel method for optimal-filter
construction for pulse processing. The previous optimal filter is shown to be
an example of a much larger class of filters that have new and useful
properties. For example, optimal filters can be constructed that are
orthogonal to exponential tails of prior pulses, prompted by the need to cope
with high-rate pulse data. Many applications of high-resolution photon
spectroscopy require very large photon counts for accurate characterization of
an absorption or emission spectrum across a broad energy band. For example,
isotopic analysis of nuclear materials for treaty verification requires
approximately $10^{9}$ photons in a spectrum between 60 keV and 260 keV to
achieve uncertainty of $10^{-3}$ Jethava _et al._ (2009). Low-temperature
detectors can reach this goal, within limited collection periods, only through
large arrays of elements operating at high photon count rates per element.
Consequently, operation at high count rates is an active topic of research Tan
_et al._ (2008, 2009, 2011); Alpert _et al._ (2012).
The new framework departs from prior algorithms in two respects: (1) noise
autocovariance is used in place of its mathematical dual, the noise power
spectral density, to avoid the discrete Fourier transform (DFT) and enable the
construction to be entirely in the time domain; and (2) the filter
optimization is subject to explicit constraints beyond maximization of signal-
to-noise ratio for isolated pulses, including for the filter length,
orthogonality to constants, and orthogonality to exponentials of one or more
decay rates. (The method is related to constrained optimization in some other
contexts. For example, a similar approach has recently been developed for
designing matched filters for wavefront sensing Gilles and Ellerbroek (2008).)
Orthogonality to exponentials can reduce or eliminate sensitivity to tails of
prior pulses. With these additional constraints imposed, the filters suffer
some loss of sensitivity for isolated pulses compared to filters optimized for
that case, but they compensate by retaining resolution with piled-up pulses
and by avoiding DFT artifacts, including artificial periodicity.
## Processing procedure.
Estimation of pulse amplitudes under standard filtering Szymkowiak _et al._
(1993); Moseley _et al._ (1988) is optimized for isolated pulses. Each pulse
is convolved with a filter and the maximum of the convolution (or a smoothed
maximum as provided by a quadratic polynomial fit to several values near the
maximum) provides an estimate of the pulse amplitude. The filter, in
principle, is constructed to minimize the variance of this estimate, given a
known pulse shape and known noise power spectrum.
## Continuous time model.
We assume a signal $f$ consists of a pair of pulses sitting on a baseline
$f(t)=a_{0}s(t-t_{0})+a_{1}s(t-t_{1})+b,$
where $s$ is the pulse shape, $a_{0}$ and $a_{1}$ are the pulse amplitudes,
$t_{0}$ and $t_{1}$ are the pulse arrival times, with $t_{0}<t_{1}$, and $b$
is the baseline. A noisy signal consists of signal plus noise,
$m(t)=f(t)+\eta(t),$ where the noise $\eta$ is assumed to be a realization of
a stationary stochastic process with a mean of zero and autocovariance
$R_{\eta}(\tau)=\int_{-\infty}^{\infty}\eta(t)\eta(t+\tau)dt.$
## Discrete time model.
The measurement apparatus obtains an approximation $m_{i}$ of $m(i\Delta)$ for
$i$ an integer, where $\Delta$ is the sample time spacing, as a convolution of
$m$ with a response function
$m_{i}=\int_{-\infty}^{\infty}m(i\Delta-t)w(t)dt,$
where $w$ is an approximate $\delta$-function centered at the origin with unit
integral. We define $f_{i},$ $s_{i},$ and $\eta_{i}$ analogously. Our
measurement model is then
$\displaystyle m_{i}$ $\displaystyle=f_{i}+\eta_{i}$
$\displaystyle=a_{0}s_{i-i_{0}}+a_{1}s_{i-i_{1}}+b+\eta_{i}.$ (1)
In this approximate model, arrival times
$t_{0}=i_{0}\Delta,\;t_{1}=i_{1}\Delta$ are assumed aligned with the samples,
and known, to avoid interpolation issues. The pulse shape
$s=(s_{0},\ldots,s_{n},\ldots)^{t}$ is approximated by averaging many pulses
to obtain the estimate $\hat{s}=(\hat{s}_{0},\ldots,\hat{s}_{n},\ldots)^{t}$,
normalized so $\max\hat{s}=1$, and the noise autocovariance
$r=(r_{0},\ldots,r_{n},\ldots)^{t}$, given by the expectation
$r_{k}=\mathbb{E}\left[\eta_{i}\eta_{i+k}\right]-\mathbb{E}\left[\eta_{i}\right]^{2}=\mathbb{E}\left[\eta_{i}\eta_{i+k}\right],$
(2)
is approximated by averaging products of pulse-free samples of the sensor
output to obtain the estimate
$\hat{r}=(\hat{r}_{0},\ldots,\hat{r}_{n},\ldots)^{t}$.
## Amplitude estimation.
The standard procedure assumes $a_{0}=0$, computes the discrete convolution
$(q\star m)_{i}=\sum_{j=0}^{n-1}q_{j}m_{i-j}$
of a given filter $q=(q_{0},\ldots,q_{n-1})^{t}$ with
$\ldots,m_{-1},m_{0},m_{1},\ldots,$ the discrete convolution of $q$ with
$\ldots,\hat{s}_{-1},\hat{s}_{0},\hat{s}_{1},\ldots,$ where $\hat{s}_{i}=0$
for $i<0,$ and estimates $a_{1}$ as the ratio of their maximums
$\hat{a}_{1}=\frac{\max_{i}(q\star m)_{i}}{\max_{i}(q\star\hat{s})_{i}}.$ (3)
## Estimate mean and variance.
We seek the mean and variance of the amplitude estimate $\hat{a}_{1}.$ We have
$\mathbb{E}\left[(q\star m)_{i}\right]=a_{0}\cdot(q\star
s)_{i-i_{0}}+a_{1}\cdot(q\star s)_{i-i_{1}}+b\sum_{j=0}^{n-1}q_{j}.$ (4)
We define $\bar{\imath}$ so that $(q\star s)_{\bar{\imath}}=\max_{i}(q\star
s)_{i}.$ Under assumptions of orthogonality to the prior tail and to
constants,
$(q\star s)_{\bar{\imath}+i_{1}-i_{0}}=0=\sum_{j=0}^{n-1}q_{j},$ (5)
we have
$\displaystyle\mathbb{E}\left[\hat{a}_{1}\right]$
$\displaystyle=\frac{\mathbb{E}\left[\max_{i}(q\star
m)_{i}\right]}{\max_{i}(q\star\hat{s})_{i}}$
$\displaystyle\approx\frac{\max_{i}\mathbb{E}\left[(q\star
m)_{i}\right]}{\max_{i}(q\star\hat{s})_{i}}\approx\frac{a_{1}\cdot(q\star
s)_{\bar{\imath}}}{(q\star\hat{s})_{\bar{\imath}}}\approx a_{1},$ (6)
where the approximations are equalities under somewhat restrictive conditions.
Toward a variance estimate,
$\mathbb{E}\left[m_{i-j}m_{i-k}\right]=\left(a_{0}s_{i-i_{0}-j}+a_{1}s_{i-i_{1}-j}+b\right)\\\
\times\left(a_{0}s_{i-i_{0}-k}+a_{1}s_{i-i_{1}-k}+b\right)+r_{j-k},$
where $r_{j-k}$ is the noise autocovariance of (2). Now
$\displaystyle{\rm Var}\left[\hat{a}_{1}\right]$
$\displaystyle=\mathbb{E}\big{[}{\hat{a}_{1}}^{\;2}\big{]}-\mathbb{E}\left[\hat{a}_{1}\right]^{2}$
$\displaystyle=\frac{\mathbb{E}\big{[}\max_{i}(q\star
m)_{i}^{\;2}\big{]}-\mathbb{E}\left[\max_{i}(q\star
m)_{i}\right]^{2}}{\max_{i}(q\star\hat{s})_{i}^{\;2}}$
$\displaystyle\approx\frac{\max_{i}\mathbb{E}\big{[}(q\star
m)_{i}^{\;2}\big{]}-\max_{i}\mathbb{E}\left[(q\star
m)_{i}\right]^{2}}{\max_{i}(q\star\hat{s})_{i}^{\;2}}$
$\displaystyle=\frac{q^{t}Rq}{[q^{t}\overline{s}]^{2}}\approx\frac{q^{t}\hat{R}q}{[q^{t}\overline{s}]^{2}}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\hat{\rm
V}{\rm ar}\left[\hat{a}_{1}\right],$ (7)
where the variance estimate $\hat{\rm V}{\rm ar}\big{[}\hat{a}_{1}\big{]}$ is
defined to be the last expression, $\hat{R}$ is the $n\times n$ estimated
covariance matrix with $\hat{R}_{jk}=\hat{r}_{j-k}=\hat{r}_{|j-k|},$ and
$\overline{s}=(\hat{s}_{i},\hat{s}_{i-1},\ldots,\hat{s}_{i-n+1})^{t}$ is the
length $n$ segment from $\hat{s}$ with
$q^{t}\overline{s}=\max_{i}(q\star\hat{s})_{i}$.
Figure 1: (color) Two scenarios, one with pile-up, are shown (top). From the pulse shape and noise autocovariance, a filter orthogonal to an exponential of tail decay, $\tau=3.2$ ms, is computed (inset, separate vertical scales). Convolution of the filter with the signal yields peaks of essentially constant height (bottom) and nearly eliminates pile-up dependence. Table 1: Dataset summary for four photon count rates. Sample spacing was 2.56 $\mu$s, full stream data were kept, and records were later formed, including for two lengths shown. Discards resulted from insufficient time between triggered pulses, signal drops below baseline, and rises in pre-trigger or post-peak attributed to (untriggered) nuisance pulses. Dataset | 1 | 2 | 3 | 4
---|---|---|---|---
Duration (s) | 4249.50 | 3096.93 | 3182.13 | 4583.17
Pulses triggered | 5496 | 6581 | 17872 | 60267
Rate (Hz) | 1.29 | 2.13 | 5.62 | 13.15
Records: 10.24 ms with 5.12 ms pre-trigger
Discards: | | | |
pulse starts ($>$1) | 143 | 201 | 1088 | 7610
SQUID unlock | 42 | 92 | 585 | 3648
early peak | 22 | 17 | 62 | 167
pre-trigger rise | 8 | 21 | 93 | 368
post-peak rise | 5 | 13 | 65 | 631
97 keV (raw height) | 1095 | 1286 | 3213 | 9607
Records: 25.60 ms with 6.40 ms pre-trigger
Discards: | | | |
pulse starts ($>$1) | 243 | 387 | 2463 | 17266
SQUID unlock | 40 | 90 | 537 | 2985
early peak | 19 | 17 | 58 | 138
pre-trigger rise | 8 | 23 | 105 | 606
post-peak rise | 19 | 36 | 223 | 1325
97 keV (raw height) | 1067 | 1228 | 2938 | 7565
## Filter optimization.
This expression for the variance of the amplitude estimate enables us to
design filters that minimize the estimated variance. $\hat{\rm V}{\rm
ar}\big{[}\hat{a}_{1}\big{]}$ is minimized at a stationary point of the
Lagrange function,
$\Lambda(q,\lambda)=q^{t}\hat{R}q-\lambda\left[q^{t}\overline{s}-1,\right]$
where $\lambda$ is a Lagrange multiplier to ensure that the scale of $q$
satisfies $q^{t}\overline{s}=1=\max\overline{s}.$ Setting the partial
derivatives of $\Lambda$ to zero and solving gives
$q=\frac{\hat{R}^{-1}\overline{s}}{\overline{s}^{t}\hat{R}^{-1}\overline{s}},\qquad\hat{\rm
V}{\rm
ar}\big{[}\hat{a}_{1}\big{]}=\left[\overline{s}^{t}\hat{R}^{-1}\overline{s}\right]^{-1}.$
(8)
This solution depends on the choice, made above tacitly, of the length $n$ of
the convolution filter $q$.
Extending beyond the prescription above, we optimize subject to stipulated
constraints. Orthogonality to constants or exponentials of particular decay
rates can be imposed by revising the Lagrange function. For orthogonality to
$k$ vectors $V=\left[v_{1}\cdots v_{k}\right],$
$\Lambda(q,\lambda,\gamma)=q^{t}\hat{R}q-\lambda\left[q^{t}\overline{s}-1\right]-q^{t}V\gamma,$
where $\gamma=(\gamma_{1},\ldots,\gamma_{k})^{t}$ are $k$ additional Lagrange
multipliers. The solution is
$q=\hat{R}^{-1}\overline{V}\left(\overline{V}^{t}\hat{R}^{-1}\overline{V}\right)^{-1}e_{1},\qquad\hat{\rm
V}{\rm ar}\big{[}\hat{a}_{1}\big{]}=q^{t}\hat{R}q,$ (9)
where $\overline{V}=[\overline{s}\;v_{1}\cdots v_{k}]$ and
$e_{1}=(1,0,\ldots,0)^{t}$ is of length $k+1.$
Figure 2: (color) Pulse spectrum is the absolute value of the discrete Fourier
transform (DFT) of the average of pulses, near 97.431 keV line, from the
highest-rate dataset. The noise spectrum is the average of the square of
absolute value of DFT of pulse-free records of TES output. Records are 25.6
ms. Figure 3: (color) Predicted resolution on an isolated pulse of four
filters is shown. The filters, determined from average pulse shape and noise
autocovariance (Fig. 2), include the standard DFT-computed filter with lowest
frequency bin set to zero Doriese _et al._ (2009) and proposed filters
orthogonal to constants and zero, one, or two exponentials ($\tau_{1}=6.0$ ms,
$\tau_{2}=1.5$ ms). Figure 4: (color) Energy histograms near the 97.431 keV
line, from the 2.13 Hz dataset with 10.24 ms record length, are shown for the
standard filter, the proposed filter orthogonal to constants, and the proposed
filter orthogonal to constants and exponentials ($\tau=6$ ms). Color denotes
the pulse arrival time lag since the previous pulse, averaged over the
histogram bin, and illustrates that filtering errors, concentrated in heavily
piled-up pulses, are nearly eliminated by the filter orthogonal to
exponentials.
Orthogonality to exponentials of a particular decay time constant enables
filters to be less sensitive to tails of prior pulses. Fig. 1 illustrates the
principle of these filters. Avoidance of the DFT, with an increase in filter
computation cost that is very mild for filter lengths up to $n\approx 10^{4},$
avoids false assumptions of signal and noise periodicity and yields
nonperiodic filters.
## Experiment.
Measurements were taken at NIST of photons from a 153Gd source with a single
transition-edge-sensor (TES) microcalorimeter Bennett _et al._ (2012), at
varied count rates (1.29, 2.13, 5.62, and 13.15 Hz), by placing the source at
four different distances from the detector. Essentially all pulses were
filtered; no attempt was made to selectively discard pulses to improve the
energy resolution. In extraction of pulse records from the data streams,
pulses were lost principally due to onset within the prior pulse record and to
occasional SQUID mode unlock. Statistics for these measurements are summarized
in Table 1. The following analysis focuses on pulses near the 97.431 keV
gamma-ray emission line of 153Gd.
Figure 5: (color) Energy histograms as in Fig. 4, except from 13.15 Hz
dataset. At this higher rate, the errors of the first two filters are much
more significant, as is the improvement offered by the third.
The noise spectrum and, for comparison, the spectrum of the average pulse are
plotted in Fig. 2. The noise spectrum and the DFT of the average pulse are
used to compute the standard filter. The autocovariance and the average pulse
(shown above in Fig. 1) are used to compute the proposed filters and the
predicted energy resolution of each. Fig. 3 shows predicted resolution versus
filter length for the proposed filters and the standard DFT-computed optimal
filter, with the lowest frequency bin set to zero to reduce sensitivity to
baseline drift. The standard filter and the proposed filter orthogonal to
constants would agree, absent discretization and periodicity artifacts due to
the DFT. This calculation is for isolated pulses; for piled-up pulses these
two filters suffer bias problems that are significantly reduced by the filters
orthogonal to exponentials. The filter orthogonal to two exponentials,
however, due to the additional constraint, suffers significant loss of
sensitivity at short to moderate filter lengths, and is not considered further
here.
The performance of the other three filters is compared on measured pulses, and
histograms near the 97.431 keV line are plotted for two different pulse rates
in Fig. 4 and Fig. 5. Each histogram was fit with a Gaussian plus a constant
to determine the energy resolution. The histogram bins are colored based on
the pulse arrival-time lag from the previous pulse, averaged over the bin,
demonstrating that errors in processing are due mainly to closely piled-up
pulses and are significantly ameliorated by the proposed filters orthogonal to
exponentials. This effect is pronounced at the higher pulse rate, yielding
much-enhanced peak height and reduced leakage for the filter orthogonal to
exponentials.
In Fig. 6 the output pulse rate, for the energy range $97.431\pm 0.100$ keV,
and energy resolution are compared for all four input count rates and the
three types of filter, for both short and long pulse records. At the highest
rate and for short pulse records, the filter orthogonal to both constants and
exponentials ($\tau=6$ ms) offers 45 % higher output rate than the standard
DFT-computed filter and 40 % higher than the filter orthogonal to constants
alone, at better energy resolution than either one.
Figure 6: (color) The output pulse rate and energy resolution are compared
across input count rates and three filter types ($\tau=6$ ms), for both short
(10.24 ms) and long (25.60 ms) pulse records. We note that the maximum output
pulse rate is considerably lower than the corresponding raw pulse rate,
because many raw pulses are due to spectral features other than the 97.431 keV
line.
One important issue regarding the filters orthogonal to exponentials concerns
their performance sensitivity to the choice of decay time constant $\tau$. The
average pulse, for the TES microcalorimeter tested, was well-approximated over
a 25.6 ms record by a linear combination of four exponentials, with decay time
constants $\tau=$ 0.018 ms, 0.144 ms, 0.963 ms, and 2.514 ms. If just the tail
is fit, however, the constants increase considerably. It is evident,
therefore, that no single decay rate is optimal for all arrival-time lags.
Nevertheless, for the full set of Poisson-distributed arrival time lags, the
performance of the filters is only mildly sensitive to the choice of time
constant. For the highest-rate data with short records, over the range
$\tau=3,\ldots,10$ ms, the $97.431\pm 0.100$ keV output pulse rate varied as
$1.568\pm 0.044$ Hz (mean and one standard deviation) and the energy
resolution varied as $128.3\pm 3.2$ eV, as compared with the $\tau=6$ ms
values of 1.640 Hz and 125.7 eV.
## Summary.
The proposed filter construction method, differing from the standard procedure
by being computed in the time domain and enabling filter optimization subject
to explicit length and orthogonality constraints, assumes linear superposition
of pulses and simple exponential decay of pulse tails. Although these
assumptions are satisfied rather imperfectly for the TES microcalorimeter
tested, the method yields notable improvement over standard filtering. Our
tests also point to additional, more specialized options, such as filters
optimized for a particular interval of arrival time lags since the previous
pulse. Such filters fit easily within this framework and underline that the
new approach has implications for pulse processing in a broad range of
applications.
We gratefully acknowledge support from the NIST Innovations in Measurement
Science program, the DOE Office of Nuclear Nonproliferation Research and
Development, and the DOE Office of Nuclear Energy.
## References
* Szymkowiak _et al._ (1993) A. E. Szymkowiak, R. L. Kelley, S. H. Moseley, and C. K. Stahle, J. Low Temp. Phys. 93, 281 (1993).
* Eckart _et al._ (2012) M. E. Eckart, J. S. Adams, C. N. Bailey, S. R. Bandler, S. E. Busch, J. A. Chervenak, F. M. Finkbeiner, R. L. Kelley, C. A. Kilbourne, F. S. Porter, J. P. Porst, J. E. Sadleir, and S. J. Smith, Journal of Low Temperature Physics 167, 732 (2012).
* Ahmed _et al._ (2011) Z. Ahmed, D. Akerib, S. Arrenberg, _et al._ , Physical Review Letters 106, 131302 (2011).
* Jethava _et al._ (2009) N. Jethava, J. N. Ullom, D. A. Bennett, W. B. Doriese, J. A. Beall, G. C. Hilton, R. D. Horansky, K. D. Irwin, E. Sassi, L. R. Vale, M. K. Bacrania, A. S. Hoover, P. J. Karpius, M. W. Rabin, C. R. Rudy, and D. T. Vo, IEEE Transactions on Applied Superconductivity 19, 536 (2009).
* Tan _et al._ (2008) H. Tan, D. Breus, W. Hennig, K. Sabourov, W. K. Warburton, W. B. Doriese, J. N. Ullom, M. K. Bacrania, A. S. Hoover, and M. W. Rabin, IEEE Nucl. Sci. Symp. Conf. Record, 1130 (2008).
* Tan _et al._ (2009) H. Tan, D. Breus, W. Hennig, K. Sabourov, J. W. Collins, W. K. Warburton, W. B. Doriese, J. N. Ullom, M. K. Bacrania, A. S. Hoover, and M. W. Rabin, AIP Conf. Proc. (American Institute of Physics) 1185, 294 (2009).
* Tan _et al._ (2011) H. Tan, W. Hennig, W. K. Warburton, W. B. Doriese, and C. A. Kilbourne, IEEE Trans. Applied Superconductivity 21, 276 (2011).
* Alpert _et al._ (2012) B. K. Alpert, W. B. Doriese, J. W. Fowler, and J. N. Ullom, Journal of Low Temperature Physics 167, 582 (2012).
* Gilles and Ellerbroek (2008) L. Gilles and B. L. Ellerbroek, Optics Letters 33, 1159 (2008).
* Moseley _et al._ (1988) S. H. Moseley, R. L. Kelley, R. J. Schoelkopf, A. E. Szymkowiak, D. McCammon, and J. Zhang, IEEE Trans. Nucl. Science 35, 59 (1988).
* Doriese _et al._ (2009) W. B. Doriese, J. S. Adams, G. C. Hilton, K. D. Irwin, C. A. Kilbourne, F. J. Schima, and J. N. Ullom, in _Low Temperature Detectors LTD 13_ , AIP Conference Proceedings, Vol. 1185, edited by B. Cabrera, A. Miller, and B. Young (2009) pp. 450–453.
* Bennett _et al._ (2012) D. A. Bennett, R. D. Horansky, D. R. Schmidt, A. S. Hoover, R. Winkler, B. K. Alpert, J. A. Beall, W. B. Doriese, J. W. Fowler, C. P. Fitzgerald, G. C. Hilton, K. D. Irwin, V. Kotsubo, J. A. B. Mates, G. C. O’Neil, M. W. Rabin, C. D. Reintsema, F. J. Schima, D. S. Swetz, L. R. Vale, and J. N. Ullom, Review of Scientific Instruments 83, 093113 (2012).
|
arxiv-papers
| 2012-12-07T23:09:24 |
2024-09-04T02:49:39.029093
|
{
"license": "Public Domain",
"authors": "B.K. Alpert, R.D. Horansky, D.A. Bennett, W.B. Doriese, J.W. Fowler,\n A.S. Hoover, M.W. Rabin, and J.N. Ullom",
"submitter": "Bradley Alpert",
"url": "https://arxiv.org/abs/1212.1738"
}
|
1212.1791
|
# Generative Models for Functional Data using Phase and Amplitude Separation
J. Derek Tucker [email protected] Wei Wu [email protected] Anuj Srivastava
[email protected] Naval Surface Warfare Center, Panama City Division - X13,
110 Vernon Avenue, Panama City, FL 32407-7001 Department of Statistics,
Florida State University, Tallahassee, FL 32306
###### Abstract
Constructing generative models for functional observations is an important
task in statistical functional analysis. In general, functional data contains
both phase (or $x$ or horizontal) and amplitude (or $y$ or vertical)
variability. Traditional methods often ignore the phase variability and focus
solely on the amplitude variation, using cross-sectional techniques such as
fPCA for dimensional reduction and data modeling. Ignoring phase variability
leads to a loss of structure in the data and inefficiency in data models. This
paper presents an approach that relies on separating the phase ($x$-axis) and
amplitude ($y$-axis), then modeling these components using joint
distributions. This separation, in turn, is performed using a technique called
elastic shape analysis of curves that involves a new mathematical
representation of functional data. Then, using individual fPCAs, one each for
phase and amplitude components, while respecting the nonlinear geometry of the
phase representation space; impose joint probability models on principal
coefficients of these components. These ideas are demonstrated using random
sampling, for models estimated from simulated and real datasets, and show
their superiority over models that ignore phase-amplitude separation.
Furthermore, the generative models are applied to classification of functional
data and achieve high performance in applications involving SONAR signals of
underwater objects, handwritten signatures, and periodic body movements
recorded by smart phones.
###### keywords:
amplitude variability , function alignment , function principal component
analysis , functional data analysis , generative model , phase variability
††journal: Computational Statistics & Data Analysis
## 1 Introduction
The problem of statistical analysis and modeling of data in function spaces is
important in applications arising in nearly every branch of science, including
signal processing, geology, biology, and chemistry. The observations here are
time samples of real-valued functions on an observation interval, and to
perform effective data analysis it is desirable to have a generative,
probabilistic model for these observations. The model is expected to properly
and parsimoniously characterize the nature and variability in the data. It
should also lead to efficient procedures for conducting hypothesis tests,
performing bootstraps, and making forecasts. An interesting aspect of
functional data is that underlying variability can be ascribed to two sources.
In a sample data the given functions may not be perfectly aligned and the
mechanism for alignment is an important topic of research. The variability
exhibited in functions after alignment is termed the amplitude (or $y$ or
vertical) variability and the warping functions that are used in the alignment
are said to capture the phase (or $x$ or horizontal) variability. A more
explicit mathematical definition of amplitude and phase variability will be
made in Section 2. It is imperative that any technique for analysis or
modeling of functional data should take both these variabilities into account.
### 1.1 Need for Phase-Amplitude Separation
Many of the current methods for analyzing functional data ignore the phase
variability. They implicitly assume that the observed functions are already
temporally aligned and all the variability is restricted only to the $y$-axis.
A prominent example of this situation is functional principal component
analysis (fPCA) (see e.g., Ramsay and Silverman (2005)) that is used to
discover dominant modes of variation in the data and has been extensively used
in modeling functional observations. If the phase variability is ignored, the
resulting model may fail to capture patterns present in the data and will lead
to inefficient data models.
Fig. 1 provides an illustration of this using simulated functional data. This
data was simulated using the equation $y_{i}(t)=z_{i}e^{-(t-a_{i})^{2}/2}$,
$t\in[-6,6],~{}i=1,2,\dots,21$, where $z_{i}$ is i.i.d.
$\mathcal{N}(1,(0.05)^{2})$ and $a_{i}$ is i.i.d. $\mathcal{N}(0,(1.25)^{2})$.
The top-left plot shows the original data; each sample function here is a
unimodal function with slight variability in height and a large variability in
the peak placement. One can attribute different locations of the peak to the
phase variability. If one takes the cross-sectional mean of this data,
ignoring the phase variability, the result is shown in the top-middle plot.
The unimodal structure is lost in this mean function with large amount of
stretching. Furthermore, if one performs fPCA on this data and imposes the
standard independent normal models on fPCA coefficients (details of this
construction are given later), the resulting model will lack this unimodal
structure. Shown in the top-right plot are random samples generated from such
a probability model on the function space where a Gaussian model is imposed on
the fPCA coefficients. These random samples are not representative of the
original data; the essential shape of the function is lost, with some of the
curves having two, three, or even more peaks.
The reason why the underlying unimodal pattern is not retained in the model is
that the phase variability was ignored. We argue that a proper technique is to
separate the phase and amplitude variability, using techniques for functional
alignment, and then develop a probability model for each component. While
postponing details for later, we show results obtained by a separation-based
approach in the bottom row. The mean of the aligned functions is shown in the
bottom left panel of Fig. 1. Clearly retained is the unimodal structure and so
do the random samples under a framework that model the phase and amplitude
variables individually. Some random samples are shown in the bottom right
panel, these displays are simply meant to motivate the framework and the
mathematical details are provided later in the paper. This example clearly
motivates the need for function alignment for modeling functional data that
contains phase variability.
(a) Original Data (b) Original Mean (c) Original Random Samples
(d) Aligned Mean (e) Samples from Proposed Models
Figure 1: Samples drawn from a Gaussian model fitted to the principal
components for the unaligned and aligned data.
### 1.2 Past Literature on Phase-Amplitude Separation
This brings up an important question: How to separate the phase and amplitude
components in a given dataset? While this is a topic of ongoing research, a
number of techniques have already been discussed in the literature. The main
difference between them lies in the choice of the cost function used in the
alignment process. The different cost functions suggested in the statistics
literature including area-under-the-curve matching (Liu and Muller (2004);
Tang and Muller (2008)), minimum second-eigenvalue (MSE) (Kneip and Ramsay
(2008)), moment based matching (MBM) (James (2007)), self-modeling
registration (Gervini and Gasser (2004)) and $k$-means alignment and
clustering (Sangalli et al. (2010b, a)).
In the meantime several other communities, often outside statistics, have
studied registration of functions in one or higher dimensions, e.g., in
matching MRI images (Christensen and Johnson (2001); Tagare et al. (2009); Beg
et al. (2005)), shape analysis of curves (Klassen et al. (2004); Michor and
Mumford (2006); Joshi et al. (2007); Srivastava et al. (2011a); Kurtek et al.
(2012); Younes et al. (2008); Younes (1998)), shape analysis of surfaces
(Kurtek et al. (2011a)), etc. The problem of curve registration is especially
relevant for phase-amplitude separation needed in functional data analysis
since the case for $\mathbb{R}^{1}$ is essentially that for real valued
functions! We will adapt a shape-analysis approach that has been termed
elastic shape analysis (Joshi et al. (2007); Srivastava et al. (2011a); Kurtek
et al. (2012)). Although these methods have been developed for alignment of
curves in $\mathbb{R}^{n}$, their application to functional data analysis has
been explained in Kurtek et al. (2011b); Kaziska (2011); Srivastava et al.
(2011b). The basic idea in this method is to introduce a mathematical
representation, called the square-root slope function or SRSF (details in the
next section) that improves functional alignment and provides fundamental
mathematical equalities that leads to a formal development of this topic.
The theoretical superiority of the elastic method comes from the following
fact: the alignment of functions is based on a cost term that is a proper
distance. Thus satisfying all desired properties in alignment, such as
symmetry (optimal alignment of $f$ to $g$ is same as that of $g$ to $f$),
positive definiteness (the cost term between any two functions is nonnegative
and it equals zero if and only if one can be perfectly aligned to the other),
and the triangle inequality. None of the current methods in the statistics
literature (e.g., Kneip and Ramsay (2008); James (2007); Liu and Muller
(2004); Gervini and Gasser (2004); Tang and Muller (2008)) have these
properties. In fact, most of them are not even symmetric in their alignment.
Additionally, many past methods perform component separation and fPCA in two
distinct steps, under different metrics, while in elastic shape analysis it is
performed jointly under the same metric. In addition to these theoretical
advantages, we have also emphasized the experimental superiority of elastic
curve analysis using a large number of datasets in this paper.
Another important issue, encountered in modeling phase variability, is to
characterize the geometry of the phase space. Generally speaking, phase
variability is represented by a warping function $\gamma$ that satisfies
certain properties such as boundary conditions, invertibility, smoothness, and
smoothness of its inverse. Ramsay and Silverman (2005) represent $\gamma$
using a basis set in the log-derivative space, i.e.,
$\log(\dot{\gamma}(t))=\sum_{i}\alpha_{i}b_{i}(t)$. Some others force $\gamma$
to be a piecewise linear function with positive derivatives (Liu and Muller
(2004)) and even linear functions (Sangalli et al. (2010b, a)). It becomes
clear that boundary conditions, combined with the smoothness and invertibility
requirements, restrict the set of allowable warping functions to a nonlinear
space. Although it seems natural, the use of nonlinear geometry of this set in
establishing a metric for comparing warping functions and for performing fPCA
has seldom been studied in the functional data analysis literature. Srivastava
et al. (2011a) have studied a square-root derivative representation, similar
to the one suggested by Bhattacharya (1943), for converting this set into a
sphere and analyzing warping functions as elements of a Hilbert sphere. The
paper Srivastava et al. (2007) demonstrates the advantages of using square-
root derivative over the log-derivative representation of warping functions.
### 1.3 Proposed Framework
After the separation of phase and amplitude components, we will define two
types of distances. One is a $y$-distance, defined to measure amplitude
differences between any two functions (and independent of their phase
variability) and computed as the $\mathbb{L}^{2}$ distance between the SRSFs
of the aligned functions. The other is an $x$-distance, or the distance on
their phase components, that measures the amount of warping needed to align
the functions. We will show that either of these distances provides useful
measures for computing summary statistics, for performing fPCA, and for
discriminating between function classes.
The main contribution of this paper is a modeling framework to characterize
functional data using phase and amplitude separation. The basic steps in this
procedure are: 1) Align the original functional data and obtain the aligned
functions (describing amplitude variability) and the warping functions
(describing phase variability). 2) Estimate the sample means and covariance on
the phase and amplitude, respectively. This step uses a nonlinear
transformation on the data to enable use of $\mathbb{L}^{2}$ norm (and cross-
sectional computations) for generating summary statistics (see Section 3); 3)
Based on the estimated summary statistics, perform fPCA on the phase and
amplitude, respectively; 4) Model the original data by using joint Gaussian or
non-parametric models on both fPCA representations; 5) As a direct
application, the model can be used to perform likelihood-based classification
of functional data. We will illustrate this application using several data
sets which include a simulated data set, a signature data set from Yeung et
al. (2004), an iPhone action data set from McCall et al. (2012), and a SONAR
data set.
The rest of this paper is organized as follows: Section 2 presents the
differential geometric approach for phase and amplitude separation adapted
from Srivastava et al. (2011a); Joshi et al. (2007) and explained in
Srivastava et al. (2011b); Kurtek et al. (2011b). Section 3 presents the
functional principal component analysis of these phase and amplitude
components, and statistical modeling of their principal coefficients. These
modeling results are presented in Section 4. Section 5 describes
classification of functional data using the developed models on real data
sets, and compares results with some conventional methods. Finally,
conclusions and observations are offered in Section 6. We have developed and R
package `fdasrvf` implementing the proposed functional alignment and fPCA
method Tucker (2012); this package is available on the CRAN archive.
## 2 Phase and Amplitude Separation Using Elastic Analysis
In this section, we adapt a method introduced for elastic shape analysis of
curves in Srivastava et al. (2011a); Joshi et al. (2007) to the problem of
functional data alignment. The details are presented in companion papers
Srivastava et al. (2011b); Kurtek et al. (2011b). This comprehensive framework
addresses three important goals: (1) completely automated alignment of
functions using nonlinear time warping, (2) separation of phase and amplitude
components of functional data, and (3) derivation of individual phase and
amplitude metrics for comparing and classifying functions. For a more
comprehensive introduction to this theory, including asymptotic results and
estimator convergences, we refer the reader to these two papers as we will
only present the algorithm here.
### 2.1 Mathematical Representation of Functions
Let $f$ be a real-valued function with the domain $[0,1]$; the domain can
easily be transformed to any other interval. For concreteness, only functions
that are absolutely continuous on $[0,1]$ will be considered; let
${\mathcal{F}}$ denote the set of all such functions. In practice, since the
observed data are discrete, this assumption is not a restriction. Also, let
$\Gamma$ be the set of boundary-preserving diffeomorphisms of the unit
interval $[0,1]$:
$\Gamma=\\{\gamma:[0,1]\to[0,1]|~{}\gamma(0)=0,~{}\gamma(1)=1,\gamma~{}\textnormal{is
a diffeomorphism}\\}$. Elements of $\Gamma$ play the role of warping
functions. For any $f\in{\mathcal{F}}$ and $\gamma\in\Gamma$, the composition
$f\circ\gamma$ denotes the time-warping of $f$ by $\gamma$. With the
composition operation, the set $\Gamma$ is a group with the identity element
$\gamma_{id}(t)=t$. This is an important observation since the group structure
of $\Gamma$ is seldom utilized in past papers on functional data analysis.
In a pairwise alignment problem, the goal is to align any two functions
$f_{1}$ and $f_{2}$. A majority of past methods use cost terms of the type
$(\inf_{\gamma\in\Gamma}\|f_{1}-f_{2}\circ\gamma\|)$ to perform this
alignment. Here $\|\cdot\|$ denotes the standard $\mathbb{L}^{2}$ norm.
However, this alignment is neither symmetric nor positive definite. To address
this and other related problems, Srivastava et al. (2011a) introduced a
mathematical expression for representing a function. This function,
$q:[0,1]\to\mathbb{R}$, is called the square-root slope function or SRSF of
$f$, and is defined in the following form:
$q(t)=\mbox{sign}(\dot{f}(t))\sqrt{|\dot{f}(t)|}\ .$
It can be shown that if the function $f$ is absolutely continuous, then the
resulting SRSF is square-integrable (see Robinson (2012) for a proof). Thus,
we will define $\mathbb{L}^{2}([0,1],\mathbb{R})$, or simply $\mathbb{L}^{2}$,
to be the set of all SRSFs. For every $q\in\mathbb{L}^{2}$ and a fixed
$t\in[0,1]$, the function $f$ can be obtained precisely using the equation:
$f(t)=f(0)+\int_{0}^{t}q(s)|q(s)|ds$, since $q(s)|q(s)|=\dot{f}(s)$.
Therefore, the mapping from ${\cal F}$ to $\mathbb{L}^{2}\times\mathbb{R}$,
given by $f\mapsto(q,f(0))$ is a bijection (see Robinson (2012)). The most
important property of this framework is the following. If we warp a function
$f$ by $\gamma$, the SRSF of $f\circ\gamma$ is given by:
$\tilde{q}(t)=(q,\gamma)(t)=q(\gamma(t))\sqrt{\dot{\gamma}(t)}$. With this
expression it can be shown that for any $f_{1},f_{2}\in{\mathcal{F}}$ and
$\gamma\in\Gamma$, we have
$\|q_{1}-q_{2}\|=\|(q_{1},\gamma)-(q_{2},\gamma)\|\ ,$ (2.1)
where $q_{1},q_{2}$ are SRSFs of $f_{1},f_{2}$, respectively. This is called
the isometry property and it is central in suggesting a new cost term for
pairwise registration of functions:
$\inf_{\gamma\in\Gamma}\|q_{1}-(q_{2},\gamma)\|$. This equation suggests we
can register (or align) the SRSFs of any two functions first and then map them
back to ${\mathcal{F}}$ to obtain registered functions. The advantage of this
choice is that it is symmetric, positive definite, and satisfies the triangle
inequality. Technically, it forms a proper distance111We note that restriction
of $\mathbb{L}^{2}$ metric to SRSFs of functions whose first derivative is
strictly positive, e.g., cumulative distribution functions, is exactly the
classical Fisher-Rao Riemannian metric used extensively in the statistics
community Rao (1945); Čencov (1982); Kass and Vos (1997); Efron (1975); Amari
(1985). on the quotient space $\mathbb{L}^{2}/\Gamma$. We mention that this
cost function has a built-in regularization term and does not require any
additional penalty term. Please refer to papers Srivastava et al. (2011b);
Kurtek et al. (2011b) for more details. In case one wants to control the
amount of warping or elasticity this can be done as described in Wu and
Srivastava (2011).
The isometric property in Eqn. 2.1 leads to a distance between functions that
is invariant to their random warpings:
###### Definition 1 (Amplitude or $y$-distance).
For any two functions $f_{1},\ f_{2}\in{\mathcal{F}}$ and the corresponding
SRSFs, $q_{1},q_{2}\in\mathbb{L}^{2}$, we define the amplitude or the
$y$-distance $D_{y}$ to be:
$D_{y}(f_{1},f_{2})=\inf_{\gamma\in\Gamma}\|q_{1}-(q_{2}\circ\gamma)\sqrt{\dot{\gamma}})\|.$
It can be shown that for any $\gamma_{1},\gamma_{2}\in\Gamma$, we have
$D_{y}(f_{1}\circ\gamma_{1},f_{2}\circ\gamma_{2})=D_{y}(f_{1},f_{2})$.
Optimization Over $\Gamma$: The minimization over $\Gamma$ can be performed in
many ways. In case $\Gamma$ is represented by a parametric family, then one
can use the parameter space to perform the estimation as Kneip and Ramsay
(2008). However, since $\Gamma$ is a nonlinear manifold, it is impossible to
express it completely in a parametric vector space. In this paper we use the
standard Dynamic Programming algorithm (Bertsekas (1995)) to solve for an
optimal $\gamma$. It should be noted that for any fixed partition of the
interval $[0,1]$, this algorithm provides the exact optimal $\gamma$ that is
restricted to the graph of this partition.
### 2.2 Karcher Mean and Function Alignment
In order to separate phase and amplitude variability in functional data, we
need a notion of the mean of functions. Basically, first we compute a mean
function and in the process warp the given functions to match the mean
function. Since we have a proper distance in $D_{y}$, in the sense that it is
invariant to random warping, we can use that to define this mean. For a given
collection of functions $f_{1},f_{2},\dots,f_{n}$, let
$q_{1},q_{2},\dots,q_{n}$ denote their SRSFs, respectively. Define the Karcher
mean of the given function as a local minimum of the following cost functions:
$\displaystyle\mu_{f}$ $\displaystyle=$ $\displaystyle\mathop{\rm
argmin}_{f\in{\cal F}}\sum_{i=1}^{n}D_{y}(f,f_{i})^{2}\ \ $ (2.2)
$\displaystyle\mu_{q}$ $\displaystyle=$ $\displaystyle\mathop{\rm
argmin}_{q\in\mathbb{L}^{2}}\sum_{i=1}^{n}\left(\inf_{\gamma_{i}\in\Gamma}\|q-(q_{i},\gamma_{i})\|^{2}\right)\
.$ (2.3)
(This Karcher mean has also been called by other names such as the Frechet
mean, intrinsic mean or the centroid.) These are two equivalent formulations,
one in the function space ${\mathcal{F}}$ and other in the SRSF space
$\mathbb{L}^{2}$, i.e.,
$\mu_{q}=\mbox{sign}(\dot{\mu}_{f})\sqrt{|\dot{\mu}_{f}|}$. Note that if
${\mu}_{f}$ is a minimizer of the cost function, then so is
${\mu}_{f}\circ\gamma$ for any $\gamma\in\Gamma$ since $D_{y}$ is invariant to
random warpings of its input variables. So, we have an extra degree of freedom
in choosing an arbitrary element of the set
$\\{\mu_{f}\circ\gamma|\gamma\in\Gamma\\}$. To make this choice unique, we can
define a special element of this class as follows. Let $\\{\gamma_{i}^{*}\\}$
denote the set of optimal warping functions, one for each $i$, in Eqn. 2.3.
Then, we can choose the $\mu_{f}$ to that element of its class such that the
mean of $\\{\gamma_{i}^{*}\\}$, denoted by $\gamma_{\mu}$, is $\gamma_{id}$,
the identity element. (The notion of the mean of warping functions and its
computation are described later in Section 3.1 and summarized in Algorithm 2).
The algorithm for computing the Karcher mean ${\mu}_{f}$ of SRSFs is given in
Algorithm 1, where the iterative update in Steps 2-4 is based on the gradient
of the cost function given in Eqn. 2.3.
Algorithm 1: Phase-Amplitude Separation
1. 1.
Compute SRSFs $q_{1},q_{2},\dots,q_{n}$ of the given $f_{1},f_{2},\dots,f_{n}$
and select $\mu=q_{i}$, where $i=\mathop{\rm argmin}_{1\leq i\leq
n}||q_{i}-\frac{1}{n}\sum_{j=1}^{n}q_{j}||$.
2. 2.
For each $q_{i}$ find the $\gamma_{i}^{*}$ such that
$\gamma_{i}^{*}=\mathop{\rm
argmin}_{\gamma\in\Gamma}\left(\|\mu-(q_{i}\circ\gamma)\sqrt{\dot{\gamma}}\|\right)$.
The solution to this optimization comes from the dynamic programming
algorithm.
3. 3.
Compute the aligned SRSFs using
$\tilde{q}_{i}\mapsto(q_{i}\circ\gamma_{i}^{*})\sqrt{\dot{\gamma_{i}^{*}}}$.
4. 4.
If the increment $\|\frac{1}{n}\sum_{i=1}^{n}\tilde{q}_{i}-\mu\|$ is small,
then continue. Else, update the mean using
$\mu\mapsto\frac{1}{n}\sum_{i=1}^{n}\tilde{q}_{i}$ and return to step 2.
5. 5.
The function $\mu$ represents a whole equivalence class of solutions and now
we select the preferred element ${\mu}_{q}$ of that orbit:
1. (a)
Compute the mean $\gamma_{\mu}$ of all $\\{\gamma_{i}^{*}\\}$ (using Algorithm
2). Then compute
${\mu}_{q}=(\mu\circ\gamma_{\mu}^{-1})\sqrt{\dot{\gamma_{\mu}^{-1}}}$.
2. (b)
Update $\gamma_{i}^{*}\mapsto\gamma_{i}^{*}\circ\gamma_{\mu}^{-1}$. Then
compute the aligned SRSFs using
$\tilde{q}_{i}\mapsto(q_{i}\circ\gamma_{i}^{*})\sqrt{\dot{\gamma_{i}^{*}}}$.
This procedure results in three items:
1. 1.
$\mu_{q}$, preferred element of the Karcher mean class
$\\{({\mu}_{q},\gamma)|\gamma\in\Gamma\\}$,
2. 2.
$\\{\tilde{q}_{i}\\}$, the set of aligned SRSFs, and
3. 3.
$\\{\gamma_{i}^{*}\\}$, the set of optimal warping functions.
From the aligned SRSFs, one can compute individual aligned functions using:
$\tilde{f}_{i}(t)=f_{i}(0)+\int_{0}^{t}\tilde{q}_{i}(s)|\tilde{q}_{i}(s)|\,ds$.
To illustrate this method we run the algorithm on the data previously used in
Kneip and Ramsay (2008). The individual functions are given by:
$y_{i}(t)=z_{i,1}e^{-(t-1.5)^{2}/2}+z_{i,2}e^{-(t+1.5)^{2}/2}$,
$t\in[-3,3],~{}i=1,2,\dots,21$, where $z_{i,1}$ and $z_{i,2}$ are i.i.d.
$\mathcal{N}(1,(0.25)^{2})$. (Note that although the elastic framework was
developed for functions on $[0,1]$, it can easily be adapted to an arbitrary
interval). Each of these functions is then warped according to:
$\gamma_{i}(t)=6\left(\frac{e^{a_{i}(t+3)/6}-1}{e^{a_{i}}-1}\right)-3$ if
$a_{i}\neq 0$, otherwise $\gamma_{i}=\gamma_{id}$ ($\gamma_{id}(t)=t$ is the
identity warping). Here $a_{i}$ are equally spaced between $-1$ and $1$, and
the observed functions are computed using $x_{i}(t)=y_{i}\circ\gamma_{i}(t)$.
A set of 21 such functions forms the original data and is shown in Panel
LABEL:sub@fig:simu_f of Fig. 2 with corresponding SRSFs in Panel
LABEL:sub@fig:simu_q. Panel LABEL:sub@fig:simu_qn presents the resulting
aligned SRSFs using our method $\\{\tilde{q}_{i}\\}$ and Panel
LABEL:sub@fig:simu_gam plots the corresponding warping functions
$\\{\gamma_{i}^{*}\\}$. The corresponding aligned functions
$\\{\tilde{f}_{i}\\}$ is shown in Panel LABEL:sub@fig:simu_fn. It is apparent
that the plot of $\\{\tilde{f}_{i}\\}$ shows a tighter alignment of functions
with sharper peaks and valleys, and thinner band around the mean. This
indicates that the effects of warping generated by the $\gamma_{i}$s have been
completely removed and only the randomness from the $y_{i}$s remain.
(a) Original SRSFs $\\{q_{i}\\}$
(b) Warped Data SRSFs $\\{\tilde{q}_{i}\\}$
(c) Warping Functions $\\{\gamma_{i}^{*}\\}$
(d) Original Data $\\{f_{i}\\}$
(e) Warped Data $\\{\tilde{f}_{i}\\}$
Figure 2: Alignment of the simulated data set using Algorithm 1.
We also compare the performance of Algorithm 1 with some published methods
including; the MBM method of James (2007) and the MSE method of Ramsay and
Silverman (2005) on a more difficult simulated data and a real SONAR data set.
The original simulated data are shown in Fig. 3(a) and the data consists of 39
unimodal functions which have been warped with equally-spaced centers along
the $x$-axis and have slight variation in peak-heights along the $y$-axis.
Fig. 3(b)-(d) present the alignment results for our elastic method, the MBM
method, and the MSE method, respectively. The original SONAR data are shown in
Fig. 3(e) and the data consists of 131 measured SONAR signals that contain
measurement ambiguity. Fig. 3(f)-(h) present the alignment results for our
elastic method, the MBM method, and the MSE method, respectively.
(a) $\\{f_{i}\\}$ (b) Elastic $\\{\tilde{f}_{i}\\}$
(c) MBM $\\{\tilde{f}_{i}\\}$ (d) MSE $\\{\tilde{f}_{i}\\}$
(e) $\\{f_{i}\\}$ (f) Elastic $\\{\tilde{f}_{i}\\}$
(g) MBM $\\{\tilde{f}_{i}\\}$ (h) MSE $\\{\tilde{f}_{i}\\}$
Figure 3: Comparison of alignment algorithms on a difficult unimodal data set
(top row) and a real SONAR data set (bottom row).
For the simulated data the elastic method performs the best while the MBM
method performs fairly well with a little higher standard deviation. The MBM
method and the MSE method both have a few numerical issues that lead to blips
in the functions. For the SONAR data only the elastic method performs well, as
MBM and MSE methods fail to align the data at all. We can also quantify the
alignment performance using the decrease in the cumulative cross-sectional
variance of the aligned functions. For any functional dataset
$\\{g_{i}(t),i=1,2,\dots,n,t\in[0,1]\\}$, let
$\mbox{Var}(\\{g_{i}\\})=\frac{1}{n-1}\int_{0}^{1}\sum_{i=1}^{n}\left(g_{i}(t)-\frac{1}{n}\sum_{i=1}^{n}g_{i}(t)\right)^{2}dt\
,$
denote the cumulative cross-sectional variance in the given data. With this
notation, we define:
$\mbox{Original Variance}=\mbox{Var}(\\{f_{i}\\}),\ \ \ \mbox{Amplitude
Variance}=\mbox{Var}(\\{\tilde{f}_{i}\\}),\ \mbox{Phase
Variance}=\mbox{Var}(\\{\mu_{f}\circ\gamma_{i}\\})\ .$
The phase- and amplitude-variances for the different alignment algorithms
shown in Fig. 3 is listed below in Table 1 with the simulated unimodal data on
the top two rows and the SONAR data on the bottom two rows:
Data | Component | Original Variance | Elastic Method | MBM | MSE
---|---|---|---|---|---
Unimodal | Amplitude-variance | 4.33 | 0.004 | 0.23 | .02
| Phase-variance | 0 | 4.65 | 4.31 | 4.54
SONAR Data | Amplitude-variance | 2.89e-5 | 1.53e-5 | 3.02e-5 | 2.42e-5
| Phase-variance | 0 | 1.48e-5 | 1.30e-5 | 1.36e-5
Table 1: The comparison of the amplitude variance and phase variance for
different alignment algorithms on the Unimodal and SONAR data set.
Based on its superior performance and theoretical advantages, we choose the
elastic method for separating the phase and amplitude components. For
additional experiments and asymptotic analysis of this method, please refer to
Srivastava et al. (2011b); Kurtek et al. (2011b).
## 3 Analysis and Modeling of Components
Having separated functional data into phase and amplitude components, we focus
on the task of developing their generative models.
### 3.1 Phase-Variability: Analysis of Warping Functions
First, we would like to study the phase-variability of the given functions,
available to us in the form of the warping functions $\\{\gamma_{i}^{*}\\}$
resulting from Algorithm 1. An explicit statistical modeling of the warping
functions can be of interest to an analyst since they represent the phase-
variability of the original data. As mentioned earlier, the space of warping
functions, $\Gamma$, is a nonlinear manifold and cannot be treated as a
Hilbert space directly. Therefore, we will use tools from differential
geometry to be able to perform statistical analysis and modeling of the
warping functions. This framework has been used previously but in different
application areas, e.g., modeling parameterizations of curves Srivastava and
Jermyn (2009) and studies of execution rates of human activities in videos
Veeraraghavan et al. (2009). It also relates to the square-root representation
of probability densities introduced by Bhattacharya (1943).
Figure 4: Depiction of the SRSF space of warping functions as a sphere and a
tangent space at identity $\psi_{id}$.
Let $\gamma_{1},\gamma_{2},\dots,\gamma_{n}\in\Gamma$ be a set of observed
warping functions. Our goal is to develop a probability model on $\Gamma$ that
can be estimated from the data directly. There are two problems in doing this
in a standard way: (1) $\Gamma$ is a nonlinear manifold, and (2) it is
infinite dimensional. The issue of nonlinearity is handled using a convenient
transformation which coincidentally is similar to the definition of SRSF, and
the issue of infinite dimensionality is handled using dimension reduction,
e.g., fPCA, which we will call horizontal fPCA. We are going to represent an
element $\gamma\in\Gamma$ by the square-root of its derivative
$\psi=\sqrt{\dot{\gamma}}$. Note that this is the same as the SRSF defined
earlier for $f_{i}$s and takes this form since $\dot{\gamma}>0$. The identity
map $\gamma_{id}$ maps to a constant function with value $\psi_{id}(t)=1$.
Since $\gamma(0)=0$, the mapping from $\gamma$ to $\psi$ is a bijection and
one can reconstruct $\gamma$ from $\psi$ using
$\gamma(t)=\int_{0}^{t}\psi(s)^{2}ds$. An important advantage of this
transformation is that since
$\|\psi\|^{2}=\int_{0}^{1}\psi(t)^{2}dt=\int_{0}^{1}\dot{\gamma}(t)dt=\gamma(1)-\gamma(0)=1$,
the set of all such $\psi$s is a Hilbert sphere $\mathbb{S}_{\infty}$, a unit
sphere in the Hilbert space $\mathbb{L}^{2}$. In other words, the square-root
representation simplifies the complicated geometry of $\Gamma$ to a unit
sphere. The distance between any two warping functions is exactly the arc-
length between their corresponding SRSFs on the unit sphere
$\mathbb{S}_{\infty}$:
$D_{x}(\gamma_{1},\gamma_{2})=d_{\psi}(\psi_{1},\psi_{2})\equiv\cos^{-1}\left(\int_{0}^{1}\psi_{1}(t)\psi_{2}(t)dt\right)\
.$
Fig. 4 shows an illustration of the SRSF space of warping functions as a unit
sphere.
The definition of a distance on $\mathbb{S}_{\infty}$ helps define a Karcher
mean of sample points on $\mathbb{S}_{\infty}$.
###### Definition 2.
For a given set of points
$\psi_{1},\psi_{2},\dots,\psi_{n}\in\mathbb{S}_{\infty}$, their Karcher mean
in $\mathbb{S}_{\infty}$ is defined to be a local minimum of the cost function
$\psi\mapsto\sum_{i=1}^{n}d_{\psi}(\psi,\psi_{i})^{2}$.
Now we can define the Karcher mean of a set of warping functions using the
Karcher mean in $\mathbb{S}_{\infty}$. For a given set of warping functions
$\gamma_{1},\gamma_{2},\dots,\gamma_{n}\in\Gamma$, their Karcher mean in
$\Gamma$ is $\bar{\gamma}(t)\equiv\int_{0}^{t}\mu_{\psi}(s)^{2}ds$ where
$\mu_{\psi}$ is the Karcher mean of $\sqrt{\dot{\gamma}_{1}}$,
$\sqrt{\dot{\gamma}_{2}}$, $\dots$, $\sqrt{\dot{\gamma}_{n}}$ in
$\mathbb{S}_{\infty}$. The search for this minimum is performed using
Algorithm 2:
Algorithm 2: Karcher Mean of Warping Functions
Let $\psi_{i}=\sqrt{\dot{\gamma}_{i}}$ be the SRSFs for the given warping
functions. Initialize $\mu_{\psi}$ to be one of the $\psi_{i}$s or simply
$w/\|w\|$, where $w=\frac{1}{n}\sum_{i=1}^{n}\psi_{i}$.
1. 1.
For $i=1,2,\dots,n$, compute the shooting vector
$v_{i}=\frac{\theta_{i}}{\sin(\theta_{i})}(\psi_{i}-\cos(\theta_{i})\mu_{\psi})$,
$\theta_{i}=\cos^{-1}\left(\left\langle\mu_{\psi},\psi\right\rangle\right)$.
By definition, each of these $v_{i}\in T_{\mu_{\psi}}(\mathbb{S}_{\infty})$.
2. 2.
Compute the average $\bar{v}=\frac{1}{n}\sum_{i=1}^{n}v_{i}\in
T_{\mu_{\psi}}(\mathbb{S}_{\infty})$.
3. 3.
If $\|\bar{v}\|$ is small, then continue. Else, update
$\mu_{\psi}\mapsto\cos(\epsilon\|\bar{v}\|)\mu_{\psi}+\sin(\epsilon\|\bar{v}\|)\frac{\bar{v}}{\|\bar{v}\|}$,
for a small step size $\epsilon>0$ and return to Step 1.
4. 4.
Compute the mean warping function using
$\bar{\gamma}(t)=\int_{0}^{t}\mu_{\psi}(s)^{2}ds$. Stop.
Since $\mathbb{S}_{\infty}$ is a nonlinear space (a sphere), one cannot
perform principal component analysis on it directly. Instead, we choose a
vector space tangent to the space, at a certain fixed point, for analysis. The
tangent space at any point $\psi\in\mathbb{S}_{\infty}$ is given by:
$T_{\psi}(\mathbb{S}_{\infty})=\\{v\in\mathbb{L}^{2}|\int_{0}^{1}v(t)\psi(t)dt=0\\}$.
In the following, we will use the tangent space at $\mu_{\psi}$ to perform
analysis. Note that the outcomes of Algorithm 2 include the Karcher mean
$\mu_{\psi}$ and the tangent vectors $\\{v_{i}\\}\in
T_{\mu_{\psi}}(\mathbb{S}_{\infty})$. These tangent vectors, also called the
shooting vectors, are the mappings of $\psi_{i}$s into the tangent space
$T_{\mu_{\psi}}(\mathbb{S}_{\infty})$, as depicted in Fig. 4. In this tangent
space we can define a sample covariance function:
$(t_{1},t_{2})\mapsto\frac{1}{n-1}\sum_{i=1}^{n}v_{i}(t_{1})v_{i}(t_{2})$. In
practice, this covariance is computed using a finite number of points, say
$T$, on these functions and one obtains a $T\times T$ sample covariance matrix
instead, denoted by $K_{\psi}$. The singular value decomposition (SVD) of
$K_{\psi}=U_{\psi}\Sigma_{\psi}V_{\psi}^{\mathsf{T}}$ provides the estimated
principal components of $\\{\psi_{i}\\}$: the principal directions
$U_{\psi,j}$ and the observed principal coefficients $\left\langle
v_{i},U_{\psi,j}\right\rangle$. This analysis on $\mathbb{S}_{\infty}$ is
similar to the ideas presented in Srivastava et al. (2005) although one can
also use the idea of principal nested sphere for this analysis Jung et al.
(2012).
As an example, we compute the Karcher mean of a set of random warping
functions. These warping functions are shown in the left panel of Fig. 5 and
their Karcher mean is shown in the second panel. Using the $\\{v_{i}\\}$’s
that result from Algorithm 2, we form their covariance matrix $K_{\psi}$ and
take its SVD. The first three columns of $U_{\psi}$ are used to visualize the
principal geodesic paths in the third, fourth, and fifth panels.
(a) (b)
(c) (d)
(e)
Figure 5: From left to right: a) the observed warping functions, b) their
Karcher mean, c) the first principal direction, d) second principal direction,
and e) third principal direction of the observed data.
### 3.2 Amplitude Variability: Analysis of Aligned Functions
Once the given observed SRSFs have been aligned using Algorithm 1, they can be
statistically analyzed in a standard way (in $\mathbb{L}^{2}$) using cross-
sectional computations in the SRSF space. This is based on the fact that
$D_{y}$ is the $\mathbb{L}^{2}$ distance between the aligned SRSFs. For
example, one can compute their principal components for the purpose of
dimension reduction and statistical modeling using fPCA. Since we are focused
on the amplitude-variability in this section, we will call this analysis
vertical fPCA.
Let $f_{1},\cdots,f_{n}$ be a given set of functions, and $q_{1},\cdots,q_{n}$
be the corresponding SRSFs, ${\mu}_{q}$ be their Karcher Mean, and let
$\tilde{q}_{i}$s be the corresponding aligned SRSFs using Algorithm 1. In
performing vertical fPCA, one should not forget about the variability
associated with the initial values, i.e., $\\{f_{i}(0)\\}$, of the given
functions. Since representing functions by their SRSFs loses this initial
value, this variable is treated separately. That is, a functional variable $f$
is analyzed using the pair $(q,f(0))$ rather than just $q$. This way, the
mapping from the function space $\mathcal{F}$ to
$\mathbb{L}^{2}\times\mathbb{R}$ is a bijection. In practice, where $q$ is
represented using a finite partition of $[0,1]$, say with cardinality $T$, the
combined vector $h_{i}=[q_{i}~{}~{}f_{i}(0)]$ simply has dimension $(T+1)$ for
fPCA. We can define a sample covariance operator for the aligned combined
vector $\tilde{h}=[\tilde{q}_{1}~{}~{}f_{i}(0)]$ as
$K_{h}(s,t)=\frac{1}{n-1}\sum_{i=1}^{n}E[(\tilde{h}_{i}(s)-\mu_{h}(s))(\tilde{h}_{i}(t)-\mu_{h}(t))]\
\ \ \ \in\mathbb{R}^{(T+1)\times(T+1)}\ ,$ (3.1)
where $\mu_{h}=[\mu_{q}~{}~{}\bar{f}(0)]$. Taking the SVD,
$K_{h}=U_{h}\Sigma_{h}V_{h}^{\mathsf{T}}$ we can calculate the directions of
principle variability in the given SRSFs using the first $p\leq n$ columns of
$U_{h}$ and can be converted back to the function space $\mathcal{F}$, via
integration, for finding the principal components of the original functional
data. Moreover, we can calculate the observed principal coefficients as
$\left\langle\tilde{h}_{i},U_{h,j}\right\rangle$.
One can then use this framework to visualize the vertical principal-geodesic
paths. The basic idea is to compute a few points along geodesic path
$\tau\mapsto\mu_{h}+\tau\sqrt{\Sigma_{h,jj}}U_{h,j}$ for $\tau\in\mathbb{R}$
in $\mathbb{L}^{2}$, where $\Sigma_{h,jj}$ and $U_{h,j}$ are the $j^{th}$
singular value and column, respectively. Then, obtain principle paths in the
function space $\mathcal{F}$ by integration as described earlier. Figure 6
shows the results of vertical fPCA on the simulated data set from Fig. 2. It
plots the vertical principal-geodesic paths of the SRSFs, $q_{\tau,j}$ for
$\tau=-2,-1,0,1,2$ and $j=1,2,3$ and the vertical principal-geodesic paths in
function space. The first 3 singular values for the data are: $0.0481$,
$0.0307$, and $0.0055$ with the rest being negligibly small. The first
principal direction corresponds to the height variation of the second peak
while the second principal component captures the height variation of the
first peak. The third principal direction has negligible variability.
---
Figure 6: Vertical fPCA of aligned functions in simulated data set of Fig. 2.
The first row shows the main three principal directions in SRSF space and the
second row shows the main three principal directions in function space.
### 3.3 Modeling of Phase and Amplitude Components
To develop statistical models for capturing the phase and amplitude
variability, there are several possibilities. Once we have obtained the fPCA
coefficients for these components we can impose probability on the
coefficients and induce a distribution on the function space ${\mathcal{F}}$.
Here we explore two possibilities: a joint Gaussian model and a non-parametric
model.
Let $c=(c_{1},\dots,c_{k_{1}})$ and $z=(z_{1},\dots,z_{k_{2}})$ be the
dominant principal coefficients of the amplitude- and phase-components,
respectively, as described in the previous two sections. Recall that
$c_{j}=\left\langle\tilde{h},U_{h,j}\right\rangle$ and $z_{j}=\left\langle
v,U_{\psi,j}\right\rangle$. We can reconstruct the amplitude component using
$q=\mu_{q}+\sum_{j=1}^{k_{1}}c_{j}U_{h,j}$ and
$f^{s}(t)=f^{s}(0)+\int_{0}^{t}q(s)|q(s)|ds$. Here, $f^{s}(0)$ is a random
initial value. Similarly, we can reconstruct the phase component (a warping
function) using $v=\sum_{j=1}^{k_{2}}z_{j}U_{\psi,j}$ and then using
$\psi=\cos(\|{v}\|)\mu_{\psi}+\sin(\|{v}\|)\frac{{v}}{\|{v}\|}$, and
$\gamma^{s}(t)=\int_{0}^{t}\psi(s)^{2}ds$. Combining the two random
quantities, we obtain a random function $f^{s}\circ\gamma^{s}$.
#### 3.3.1 Gaussian Models on fPCA Coefficients
In this setup the model specification reduces to the choice of models for
$f^{s}(0)$, $c$, and $z$. We are going to model them as multivariate normal
random variables. The mean of $f^{s}(0)$ is $\bar{f}(0)$ while the means of
$c$ and $z$ are zero vectors. Their joint covariance matrix is of the type:
$\left[\begin{array}[]{ccc}\sigma_{0}^{2}&L_{1}&L_{2}\\\
L_{1}^{\mathsf{T}}&\Sigma_{h}&S\\\
L_{2}^{\mathsf{T}}&S&\Sigma_{\psi}\end{array}\right]\in\mathbb{R}^{(k_{1}+k_{2}+1)\times(k_{1}+k_{2}+1)}$.
Here, $L_{1}\in\mathbb{R}^{1\times k_{1}}$ captures the covariance between
$f(0)$ and $c$, $L_{2}\in\mathbb{R}^{1\times k_{2}}$ between $f(0)$ and $z$,
and $S\in\mathbb{R}^{k_{1}\times k_{2}}$ between $c$ and $z$. As discussed in
the previous sections $\Sigma_{h}\in\mathbb{R}^{k_{1}\times k_{1}}$ and
$\Sigma_{\psi}\in\mathbb{R}^{k_{2}\times k_{2}}$ are diagonal matrices and are
estimated directly from the data. We will call this resulting probability
model on the fPCA coefficients as $p_{Gauss}$.
#### 3.3.2 Non-parametric Models on fPCA Coefficients
An alternative to the Gaussian assumption made above is the use of kernel
density estimation Silverman (1998), where the density of $f^{s}(0)$, each of
the $k_{1}$ components of $c$, and the $k_{2}$ components of $z$ can be
estimated using
$p_{ker}(x)=\frac{1}{nb}\sum_{i=1}^{n}\mathcal{K}\left(\frac{x-x_{i}}{b}\right)$
where $\mathcal{K}(\cdot)$ is the smoothing kernel, which is a symmetric
function that integrates to 1, and $b>0$ is the smoothing parameter or
bandwidth. A range of kernel functions can be used, but a common choice is the
Gaussian kernel.
## 4 Modeling Results
We will now evaluate the models introduced in the previous section using
random sampling. We will first estimate the means and the covariances from the
given data, estimate the model parameters, and then generate random samples
based on these estimated models. We demonstrate results on two simulated data
sets used in Figs. 2 and 3 and one real data set being the Berkeley growth
data222http://www.psych.mcgill.ca/faculty/ramsay/datasets.html. For the first
simulated data set, shown in Fig. 2, we randomly generate 35 functions from
the amplitude model and 35 domain-warping functions from the phase model and
then combine them to generate random functions. The corresponding results are
shown in Fig. 7, where the first panel is a set of random warping functions,
the second panel is a set of corresponding amplitude functions, and the third
panel shows their compositions. Comparing them with the original datasets
(Fig. 2) we conclude that the random samples are very similar to the original
data and, at least under a visual inspection, the proposed models are
successful in capturing the variability in the given data.
---
Figure 7: Random samples from jointly Gaussian models on fPCA coefficients of
$\gamma^{s}$ (left) and $f^{s}$ (middle), and their combinations
$f^{s}\circ\gamma^{s}$ (right) for Simulated Data 1. The last plot are random
samples if a Gaussian model is imposed on $f$ directly without any phase and
amplitude separation.
Furthermore, if we compare these sampling results to the fPCA-based Gaussian
model directly on $f$ (without separating the phase and amplitude components)
in the last panel of Fig. 7, we notice that our model is more consistent with
the original data. A good portion of the samples from the non-separated model
just contain three peaks or have a higher variation than the original data and
some barely represent the original data.
For the second simulated data set we use the data shown in Fig. 3 and perform
vertical and horizontal fPCA. As before, we randomly generate 35 functions
from the amplitude model and 35 domain-warping functions from the phase model
and then combine them to generate random functions. The corresponding results
are shown row of Fig. 8, where the first panel is a set of random warping
functions, the second panel is a set of corresponding amplitude functions, and
the last panel shows their compositions. Comparing them with the original data
in Fig. 3 we conclude that the random samples are very similar to the original
data and, under visual inspection, the proposed models are successful in
capturing the variability in the given data. In this example performing fPCA
directly on the function space does not correctly capture the data and fails
to generate any single unimodal function shown in the last panel.
---
Figure 8: Random samples from jointly Gaussian models on fPCA coefficients of
$\gamma^{s}$ (left) and $f^{s}$ (middle), and their combinations
$f^{s}\circ\gamma^{s}$ (right) for Simulated Data 2. The last panel shows the
random samples resulting from a Gaussian model imposed on $f$ directly.
For the Berkley growth data we again develop our phase and amplitude models
then randomly generate 35 functions from the amplitude model and 35 domain-
warping functions from the phase model. Then combine them to generate random
functions. The corresponding results are shown row of Fig. 9, where the first
panel is a set of random warping functions, the second panel is a set of
corresponding amplitude functions, and the last panel shows their
compositions. Comparing them with the original data set in the last panel we
conclude that the random samples are similar to the original data and, at
least under a visual inspection, the proposed models are successful in
capturing the variability in the given data.
---
Figure 9: From left to right: Random samples from jointly Gaussian models on
fPCA coefficients of $\gamma^{s}$ and $f^{s}$, respectively, and their
combinations $f^{s}\circ\gamma^{s}$ for the Berkley Growth Data. The last
panel shows the original data used in this experiment.
## 5 Classification Using Phase and Amplitude Models
An important use of statistical models of functional data is in classification
of future data into pre-determined categories. Since we have developed models
for both amplitude and phase, one or both can be used for classification and
analyzed for their performance. Here we use a classical setup: a part of the
data is used for training and estimation of model parameters while the
remaining part is used for testing. This partition is often random and
repeated many times to obtain an average classification performance.
Amplitude-Based Classification: As described earlier, we can impose a
probability model for the amplitude data using the principal subspace
associated with the aligned SRSFs. The actual model is imposed on the
principal coefficients $(c_{1},c_{2},\dots,c_{k_{1}})$, with respect to the
basis $U_{h,1},U_{h,2},\dots,U_{h,{k_{1}}}$. These basis elements, in turn,
are determined using the training data. We can select a $k_{1}$ such that the
cumulative energy
$\sum_{j=1}^{k_{1}}\Sigma_{h,jj}/\sum_{j=1}^{T+1}\Sigma_{h,jj}$ is above a
certain threshold, e.g., 90 percent. There are two choices of models: Gaussian
and kernel-density estimator. Classification is performed by constructing the
appropriate models for each class $C_{1},\cdots,C_{L}$ of the data. Then, for
a test sample $\tilde{h}_{j}\in\mathbb{R}^{T+1}$ project it to the principal
subspace using an orthonormal basis $U_{hl}\in\mathbb{R}^{(T+1)\times k_{1}}$,
one for each class, and calculate the likelihood under each class. The model
with the largest likelihood represents the class assigned to $\tilde{h}_{j}$.
Therefore, our classification rule is:
$\text{classify}(\tilde{h}_{j})=\mathop{\rm
argmax}_{C_{l}}~{}p_{a}(U^{\mathsf{T}}_{hl}\tilde{h}_{j}|K_{hl},\mu_{hl})\ \
,\ \ \mbox{where}\ \ p_{a}=p_{Gauss}\ \ \mbox{or}\ \ p_{ker}\ .$ (5.1)
Phase-Based Classification: Similarly, for the phase components, we can
represent the shooting vectors, $\\{v_{i}\\}$, in a lower order dimensional
space using the first $k_{2}$ columns of $U_{\psi}$. Where $k_{2}$ can be
chosen similar to $k_{1}$ as described above. Once again, we can either define
a Gaussian model or a kernel density estimator on these principal
coefficients. We can estimate the model parameters for each class
$C_{1},\cdots,C_{L}$ using the training data. Then, for a test sample’s
shooting vector $v_{j}$, we project it to each model’s subspace and calculate
the likelihood of $v_{j}$ under each pre-determined class. Therefore, our
classification rule is:
$\text{classify}(v_{j})=\mathop{\rm
argmax}_{C_{l}}~{}p_{\psi}(U^{\mathsf{T}}_{\psi l}v_{j}|K_{\psi l})\ \
\mbox{where}\ \ p_{\psi}=p_{Gauss}\ \ \mbox{or}\ \ p_{ker}\ .$ (5.2)
Joint Classification: Assuming independence we can combine the amplitude and
phase classification rules as,
$\text{classify}(\tilde{h}_{j},v_{j})=\mathop{\rm
argmax}_{C_{l}}~{}p_{a}(U^{\mathsf{T}}_{hl}\tilde{h}_{j}|K_{hl},\mu_{hl})p_{\psi}(U^{\mathsf{T}}_{\psi
l}v_{j}|K_{\psi l})$ (5.3)
and classification is as described previously.
In this section, we present the classification results on a signature data
Yeung et al. (2004), an iPhone-generated action data set from McCall et al.
(2012), and a SONAR data set using models developed using vertical and
horizontal fPCA.
### 5.1 Signature Data
In this section, we test our classification method on a signature recognition
data set from Yeung et al. (2004). The data was captured using a WACOM Intuos
tablet. The data set consists of signature samples from 40 different subjects
with 20 real signature samples of the subject and another 20 samples which are
forgeries of the subject’s signature. In our analysis we are going to
distinguish between the real and forgery signature for two of the subjects
using the tangential acceleration. The tangential acceleration is computed as
$A(t)=\sqrt{[X^{\prime\prime}(t)]^{2}+[Y^{\prime\prime}(t)]^{2}}$. To have a
robust estimate of the SRSF $\\{q_{i}\\}$, we first smooth the original
functions 100 times $\\{f_{i}\\}$ using a standard box filter $[1/4,1/2,1/4]$.
That is, numerically we update the signals at each discrete point by
$f_{i}(x_{k})\rightarrow\left(\frac{1}{4}f_{i}(x_{k-1})+\frac{1}{2}f_{i}(x_{k})+\frac{1}{4}f_{i}(x_{k+1})\right)$.
The smoothed acceleration functions are aligned in each class (real vs. fake)
using our alignment algorithm from Section 2. An example signature with 10
realizations is shown in Fig. 10 along with the corresponding acceleration
functions for both the real and fake signatures, the corresponding aligned
functions, and warping functions.
---
Figure 10: From left to right: the original signature samples for one of the
subjects, the corresponding tangential acceleration functions for both the
real and fake signatures, the corresponding aligned functions, and warping
functions.
Models were generated for the three classes as was outlined in Section 3.3 by
performing vertical and horizontal fPCA on the aligned data and the warping
functions, respectively. We then impose a multivariate Gaussian model,
$p_{Gauss}$, on the reduced data for each class, it is assumed here that the
cross-covariances $L_{1}$ and $L_{2}$ are zero. The threshold to select the
number of dimensions, $k_{1}$ and $k_{2}$, was set at 95%. Classification for
the amplitude component only was performed as described in Section 5 using the
classification rule in (Eqn. 5.1) and was evaluated using 5-fold cross-
validation. Similarly, the classification rule in (Eqn. 5.2) were used for the
phase component. Moreover, the joint classification was performed using (Eqn.
5.3). Table 2a presents the the mean and standard deviation (shown in
parentheses) of the classification rates from the cross-validation for the
three rules. As well as comparing to the standard $\mathbb{L}^{2}$ where
models were generated directly on the original data, dimension reduction with
fPCA, and imposing a multivariate normal distribution. Corresponding results
for another subject, U13 is presented in Table 2b.
| Gaussian | Kernel Density
---|---|---
amplitude only | 0.93 (0.07) | 0.78 (0.19)
phase only | 0.65 (0.16) | 0.75 (0.09)
phase and amplitude | 0.90 (0.05) | 0.80 (0.07)
standard $\mathbb{L}^{2}$ | 0.60 (0.14) | 0.55 (0.11)
(a) Subject U1
| Gaussian | Kernel Density
---|---|---
amplitude only | 0.75 (0.14) | 0.78 (0.21)
phase only | 0.50 (0.01) | 0.50 (0.01)
phase and amplitude | 0.58 (0.11) | 0.60 (0.10)
standard $\mathbb{L}^{2}$ | 0.50 (0.01) | 0.53 (0.06)
(b) Subject U13
Table 2: Mean classification rate and standard deviation (in parentheses) for
5-fold cross-validation on the signature data.
The classification rates have a low standard deviation indicating good
generalization, though we do have a little variation for the phase only model.
For both subjects the amplitude only rule greatly outperforms both the phase
only rule and the standard $\mathbb{L}^{2}$ with the best performance of 93%
and 75% for subjects U1 and U13, respectively. Since the phase only rule
performs poorly combining it with the amplitude only rule brings down the
overall performance. The alignment and modeling using a proper distance
improves the overall classification performance of the data. To compare the
results between $p_{Gauss}$ and $p_{kern}$, we classified the data again
forming models using $p_{kern}$ which was discussed in Section 5, where each
of the $k_{1}$ and $k_{2}$ components has an estimated density using a kernel
density estimator and independence is assumed. We used the Gaussian kernel
function and the bandwidth was selected automatically based upon the data
using the method presented by Botev et al. (2010). Classification using the
three classification rules was performed using 5-fold cross-validation. Table
2a and Table 2b present the the mean and standard deviation of the
classification rates from the cross-validation for the three rules as well as
comparing to the standard $\mathbb{L}^{2}$. Models were generated directly on
the original data using fPCA and the kernel density estimator for subjects U1
and U13, respectively. We see an improvement in the phase only method for
subject U1 and reduction in performance for the other methods, this suggest
the warping functions have some non-Gaussian behavior. However, for subject
U13 there is a minimal change between the Gaussian and kernel estimator.
### 5.2 iPhone Action Data
This data set consists of aerobic actions recorded from subjects using the
Inertial Measurement Unit (IMU) on an Apple iPhone 4 smartphone. The IMU
includes a 3D accelerometer, gyroscope, and magnetometer. Each sample was
taken at 60Hz, and manually trimmed to 500 samples (8.33s) to eliminate
starting and stopping movements and the iPhone is always clipped to the belt
on the right hand side of the subject. There is a total of 338 functions for
each measurement on the IMU and the actions recorded consisted of biking,
climbing, gym bike, jumping, running, standing, treadmill, and walking. With
the number of samples being 30, 45, 39, 45, 45, 45, 44, and 45, respectively
for each action. For more information on the data set the reader is referred
to McCall et al. (2012). For our experiments we used the accelerometer data in
the $x$-direction. Again, to have a robust estimate of the SRSF $\\{q_{i}\\}$,
we first smooth the original signals 100 times $\\{f_{i}\\}$ using the
standard box filter described in the previous section. As with the previous
data set, the smoothed iPhone data are aligned in each class (activity) using
our method. A selected subset of functions from three activities is shown in
Fig. 11 along with corresponding aligned functions and warping functions.
| |
---|---|---
| |
| |
Figure 11: Original iPhone functions for the walking, jumping, and climbing
activities in the first column (in corresponding descending order) with the
corresponding aligned functions and warping functions in the second and third
columns, respectively.
To perform the classification, models were generated for the 8 classes by
performing vertical and horizontal fPCA on the aligned data and the warping
functions then imposing a multivariate Gaussian on the reduced data for each
class. The threshold to select the number of dimensions, $k_{1}$ and $k_{2}$,
was set at 95%. Classification was performed as in the previous section. Table
3 presents the mean and standard deviation of the classification rates for the
cross-validation for all three rules as well as comparing to the standard
$\mathbb{L}^{2}$.
| Gaussian | Kernel Density
---|---|---
amplitude only | 0.60 (0.04) | 0.62 (0.05)
phase only | 0.34 (0.06) | 0.35 (0.06)
phase and amplitude | 0.62 (0.08) | 0.62 (0.07)
standard $\mathbb{L}^{2}$ | 0.12 (0.02) | 0.12 (0.02)
Table 3: Mean classification rate and standard deviation (in parentheses) for
5-fold cross-validation on the iPhone data.
The classification rates have a low standard deviation indicating good
generalization. The phase only rule and the amplitude only rule, drastically
out perform the standard $\mathbb{L}^{2}$ with the combination providing the
best performance at 62%. The alignment and modeling using a proper distance
improves the overall classification performance of the data. We again used the
kernel density estimator to compare the results with the Gaussian assumption
and the results are presented in Table 3. Using the kernel density estimator
we see only minor improvements in the phase only rule, suggesting the Gaussian
assumption is sufficient for this data.
### 5.3 SONAR Data
The data set used in these experiments was collected at the Naval Surface
Warfare Center Panama City Division (NSWC PCD) test pond. For a description of
the pond and measurement setup the reader is referred to Kargl et al. (2010).
The raw SONAR data was collected using a 1 - 30$kHz$ LFM chirp and data was
collected for nine proud targets that included a solid aluminum cylinder, an
aluminum pipe, an inert 81$mm$ mortar (filled with cement), a solid steel
artillery shell, two machined aluminum UXOs, a machined steel UXO, a de-
militarized 152$mm$ TP-T round, a de-militarized 155$mm$ empty projectile
(without fuse or lifting eye), and a small aluminum cylinder with a notch. The
aluminum cylinder is 2$ft$ long with a 1$ft$ diameter; while the pipe is 2$ft$
long with an inner diameter of 1$ft$ and 3/8 inch wall thickness.
The acoustic signals were generated from the raw SONAR data to construct
target strength as a function of frequency and aspect angle. Due to the
relatively small separation distances between the targets in the measurement
setup, the scattered fields from the targets overlap. To generate the acoustic
templates (i.e., target strength plot of frequency versus aspect), synthetic
aperture sonar (SAS) images were formed and then an inverse imaging technique
was used to isolate the response of an individual target and to suppress
reverberation noise. A brief summary of this process is as follows: The raw
SONAR data are matched filtered and the SAS image is formed using the
$\omega-k$ beamformer Soumekh (1999). The target is then located in the SAS
image and is windowed around selected location. This windowed image contains
the information to reconstruct the frequency signals associated with a given
target via inverting the $\omega-k$ beamformer Khwaja et al. (2005) and the
responses were aligned in rage using the known acquisition geometry. For the
nine targets, 2000 different data collections runs were done, and 1102
acoustic color templates were generated using the method described above from
the data set. From the acoustic color maps, one-dimensional functional data
was generated by taking slices at aspect value of $0^{\circ}$ and therefore
generating 1102 data samples. We will apply our method to this SONAR data,
where there are $n=1102$ SONAR signals with nine target classes and the
numbers of functions in the nine classes are
$\\{n_{i}\\}_{i=1}^{9}=\\{131,144,118,118,121,119,120,114,117\\}$ and are
sampled using 483 points. A selected subset of functions in each class from
the original data is shown in Fig. 12. We observe that the original data are
quite noisy, due to both the compositional and the additive noise, increasing
variability within class and reducing separation across classes. This
naturally complicates the task of target classification using SONAR signals.
Figure 12: Original SONAR functions in each of the 9 classes.
To again have a robust estimate of the SRSF $\\{q_{i}\\}$, we first smooth the
original signals 25 times $\\{f_{i}\\}$ using the standard box filter
described previously. As with the previous data sets, the smoothed SONAR data
are aligned in each class using our method. Models were generated for the
three classes by performing vertical and horizontal fPCA on the aligned data
and the warping functions then, imposing a multivariate Gaussian on the
reduced data for each class, with the aligned data shown in Fig. 13. The
threshold to select the number of dimensions, $k_{1}$ and $k_{2}$, was set at
90%. Table 4 presents the classification rates for the cross-validation for
all three rules as well as comparing to the standard $\mathbb{L}^{2}$.
| Gaussian | Kernel Density
---|---|---
amplitude only | 0.44 (0.03) | 0.47 (0.02)
phase only | 0.42 (0.02) | 0.43 (0.02)
phase and amplitude | 0.54 (0.03) | 0.53 (0.03)
standard $\mathbb{L}^{2}$ | 0.33 (0.01) | 0.34 (0.02)
Table 4: Mean classification rate and standard deviation (in parentheses) for
5-fold cross-validation on SONAR data.
Figure 13: Aligned and Smoothed SONAR functions in each of the 9 classes.
The classification rates have low standard deviation indicating good
generalization for the SONAR data. The phase only rule and the amplitude only
rule out perform the standard $\mathbb{L}^{2}$ with the combination providing
the best performance at 54%. The alignment and modeling using a proper
distance improves the overall classification performance of the data. We again
used the kernel density estimator to compare the results with the Gaussian
assumption and the results are presented in Table 4. Using the kernel density
estimator we see improvements in the classification results. However, nothing
is a dramatic improvement suggesting the Gaussian assumption is sufficient for
this data.
## 6 Conclusions
The statistical modeling and classification of functional data with phase
variability is a challenging and complicated task. We have proposed a
comprehensive approach that solves the problem of registering and modeling
functions in a joint, metric-based framework. The main idea is to use an
elastic distance to separate the given functional data into phase and
amplitude components, and to develop individual models for these components.
The specific models suggested in this paper use fPCA and imposition of either
multivariate Gaussian or nonparametric models on the coefficients. The
strengths of these models are illustrated in two ways: random sampling and
model-based classification of functional data. In the case of classification,
we consider applications involving handwritten signatures, motion data
collected using iPhones, and SONAR signals. We illustrate the improvements in
classification performance when the proposed models involving separate phase
and amplitude components are used.
## Acknowledgment
This research was supported by the Naval Surface Warfare Center Panama City
Division In-house Laboratory Independent Research program funded by the Office
of Naval Research (ONR) and was also supported in part by the grants NSF DMS
0915003 and NSF DMS 1208959. The authors would like to thank Dr. Frank Crosby
and Dr. Quyen Huynh at NSWC PCD for their technical support during this work
and the two anonymous referees and Associate Editor for their constructive
comments and suggestions, which led to a significant improvement of the paper.
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|
arxiv-papers
| 2012-12-08T13:41:50 |
2024-09-04T02:49:39.037637
|
{
"license": "Public Domain",
"authors": "J. Derek Tucker, Wei Wu, and Anuj Srivastava",
"submitter": "James Tucker",
"url": "https://arxiv.org/abs/1212.1791"
}
|
1212.1835
|
11institutetext: National Institute of Standards and Technology, Gaithersburg,
MD 20899-8422, USA
# EUV magnetic-dipole lines from highly-charged high-Z ions with an open $3d$
shell
D. Osin J.D. Gillaspy J. Reader Yu. Ralchenko [email protected]
(Received: date / Revised version: date)
###### Abstract
The electron beam ion trap (EBIT) at the National Institute of Standards and
Technology was used to produce highly-charged ions of hafnium, tantalum and
gold with an open $3d$ shell. The extreme-ultraviolet (EUV) spectra from these
ions were recorded with a flat-field grazing-incidence spectrometer in the
wavelength range of 4.5 nm to 25 nm. A total of 133 new spectral lines,
primarily due to magnetic-dipole transitions within the ground-state $3d^{n}$
configurations of the Co-like to K-like ions, were identified by comparing
energy-dependent experimental spectra with a detailed collisional-radiative
modeling of the EBIT plasma.
## 1 Introduction
Accurate knowledge of structure and spectra of highly-charged ions (HCI) is of
great importance for atomic physics, astrophysics, and other fields of
research Beyer_Shevelko_Intro_HCI_2003 ; Gillaspy_HCI_2001 . During the last
decade there was a surge in spectroscopic studies of HCI of heavy elements for
fusion applications 9823EL . This effort is primarily motivated by the
proposed use of tungsten as a plasma-facing material in the divertor region of
the international reactor ITER Hawryluk09 . Atoms of tungsten will be
sputtered from the divertor plates and are expected to deeply penetrate the
plasma core. The core temperatures on the order of 10-20 keV are not
sufficient to completely ionize tungsten, and therefore the partially ionized
atoms will strongly emit in the x-ray and extreme ultraviolet (EUV) ranges of
spectra. Although this will result in undesirable radiative power losses, on
the positive side, the measured radiation can be reliably used to diagnose
such plasma properties as, for example, temperature and density. This
potential application stimulated an extensive analysis of W spectra (see,
e.g., 16569EL and references therein), which resulted in identification of a
large number of new spectral lines and in the development of new techniques
for diagnostics of very hot plasmas 12366EL ; 16374EL ; 12263EL .
The spectroscopic properties of other high-Z elements ($Z\sim 70-80$) are also
a subject of active research. The interest in high-Z emission is not limited
to the isoelectronic analysis of spectra with a goal of a better understanding
of W emission. Since gold is one of the primary materials for hohlraums used
in the inertial confinement fusion experiments NIF2012 , its highly-charged
ions were studied in electron beam ion traps (EBITs) 12366EL ; 16347EL ;
11929EL ; 11933EL , laser-produced plasmas Reader_HC_Cu_Zn_1980 ;
Reader_HC_Ba_W_1981 ; 6846EL ; 10546EL ; 9170EL , and tokamaks 11118EL ;
9820EL ; Putterich_PhD_HCI_Tokamaks_2006 ; 12530EL . Also, the alloys
containing tantalum are considered to be another potential candidate for a
plasma-facing material in tokamaks, and therefore the physical properties of
Ta under the influence of hot plasmas are being examined too
Hirai_etal_TEXTOR_Ta_2003 . Our group has recently reported analysis of EUV
spectra of Hf, Ta and Au from the EBIT at the National Institute of Standards
and Technology (NIST) Draganic_etal_Hf_Ta_Au_2011 , where more than 100 new
spectral lines from 35- to 52-times ionized atoms were identified. Those
spectra are mainly due to the intrashell n=4–n=4 transitions in the ions with
an open $4s$ or $4p$ shells.
Here we present measurements and identifications of EUV spectra from even
higher (ion charge $z$=44-60) ions of Hf, Ta and Au with an open $3d$ shell in
the ground configuration. The analogous spectra from tungsten ions were
studied recently in Ref. 16374EL , where we found that practically all
spectral lines between 10 nm and 20 nm are due to the forbidden magnetic-
dipole (M1) transitions within the ground state configurations $3d^{n}$. It
was also found that many of the corresponding line ratios show high
sensitivity to electron density in the range typical for fusion plasmas and
thus can be used for diagnostics. In this work we continue application of the
detailed collisional-radiative modeling to the analysis of spectral line
intensities in EBIT plasmas and use it to identify the measured lines.
## 2 Experimental setup
The measurements of the EUV spectra of 3dn ions of Hf, Ta and Au were carried
out in the NIST EBIT, which produces an electron beam with a beam diameter of
about $60~{}\mu$m and a current density of about 3500 A cm-2. The electron
energy is controlled by applying a voltage to the central drift tube, and can
be precisely varied in a range between 100 eV and 30 keV. The uncertainty in
the electron beam energy is on the order of $\pm$ 50 eV, which is mainly
caused by the space charge of the beam 7624EL . The design and operation
principles of the EBIT are described in more detail elsewhere
Gillaspy_EBIT_1997 . In these measurements, the electron beam energy was
varied in a relatively narrow region between 4.0 keV and 6.5 keV for Hf and
Ta, and $5.0$ keV to $7.5$ keV for Au (Table 1). This set of energies allowed
us to produce spectra of all $3d^{n}$ ions of Hf, Ta and Au. The electron beam
current was kept constant at 150 mA for all energies, the central drift tube
potential was 220 V, and the ion loading duty cycle was 11 seconds throughout
the entire set of measurements.
The metallic ions were introduced into the trap using the metal vapor vacuum
arc ion source (MEVVA) Holland_MEVVA_2005 . Neutral gases, used mainly for
wavelength calibration, were loaded into the trap through the gas injection
system 12184EL .
The spectral window observed in the course of these measurements extended from
about 8 nm to 26 nm for Hf and Ta, and from 4.5 nm to 19.5 nm for Au. The
spectra were recorded with a flat-field grazing-incidence EUV spectrometer,
which is extensively described in Blagojevic_2005 . The emitted radiation was
collected from the 2-cm central region of the EBIT by a gold-coated spherical
mirror, which focused the light onto the spectrometer entrance slit. The EUV
spectrometer is equipped with a grazing incidence, aberration-corrected,
variable-line-spacing grating, which has 1200 lines mm-1 in the center, and
which produces a 2-D spectrally-dispersed image in a plane rather than on the
Rowland circle Harada_FFgrating_1980 , Kita_FFgrating_1983 . The vertical slit
was kept wide-open in order to preserve a constant resolving power of about
400, as observed in Blagojevic_2005 , and in order to collect as much light as
possible. The wide-open entrance slit, on the other hand, permits observation
of the EBIT ion cloud shifts, which contribute to systematic error in the
wavelength measurements. The spectra were recorded with a windowless nitrogen-
cooled back-illuminated charge-coupled-device (CCD) camera, which has a
spectroscopic type chip with $1320\times 400$ pixels of $20~{}\mu$m$\times
20~{}\mu$m each. The total spectrum at every beam energy is combined from 20
one-minute spectra. This allows for easy removal of cosmic ray traces. In
order to increase the signal-to-noise ratio, each individual spectral image is
integrated over pixel columns aligned parallel to the image of the trapped ion
cloud.
The measured spectra of Hf and Ta ions, observed in the region from 8 nm to 26
nm, were calibrated with known lines NIST_ASD of N3+-N6+, Kr17+-Kr33+,
O4+-O5+, Fe16+-Fe23+, and Xe39+-Xe43+. For the Au spectra, calibration lines
of Ar7+-Ar8+ and Ne5+-Ne7+ were used in addition to the O, Fe, and Xe lines.
It should be mentioned that some calibration lines of highly ionized Xe were
identified in our previous EBIT run Draganic_etal_Hf_Ta_Au_2011 (see also
XeTBP ). Both the MEVVA and the gas injector were utilized to introduce the
calibration elements into the EBIT trap. The calibration spectra were measured
in a relatively broad range of beam energies from 1250 eV to 9300 eV in order
to cover the entire spectral range.
The calibration lines were fitted with statistically-weighted Gaussian line
profiles. The calibration curve was obtained by the weighted fit of the line
center positions determined in CCD pixels to the known wavelengths using a
fourth-order polynomial. The weighting in the fit contained contributions from
the Gaussian fit uncertainties, the uncertainty in the assigned calibration
wavelengths, and the estimated systematic measurement uncertainty (a constant
value in CCD pixels, which was converted into wavelength using the known
dispersion of the spectrometer Blagojevic_2005 ). The quadrature sum of the
three uncertainty components yielded the final uncertainty of the wavelength.
The statistical uncertainties in line positions were typically less than 0.001
nm. Multiple wavelength values observed at different beam energies were weight
averaged. The total error in the final wavelength was taken to be the
quadrature sum of the total uncertainty from the calibration curve and the
statistical uncertainty from the weighted average. The final uncertainty of
the Hf, Ta and Au spectral lines was mainly on the order of 0.003 nm and
somewhat higher for blended lines.
Figures 1-3 show the evolution of the measured spectra of Hf, Ta and Au with
the beam energy. The signal counts in figures are the CCD analog-to-digital
units (ADU). The high signal-to-noise ratio obtained in the spectra was
important for identification of blended and weak spectral lines. The spectra
contain a number of lines from impurity elements. The NIST EBIT is routinely
used for various other studies including Xe deposition on surfaces, and
therefore xenon lines are almost always seen in our spectra. For instance,
several lines from the Xe ions with ion charge $z\approx$ 40 are observed
between 15 nm and 17 nm. Also, several lines from highly-charged O and Ar can
be identified as well. In some cases the impurity lines blend with the lines
from Hf, Ta and Au (e.g., Ar XV line at 22.115 nm with 22.091 nm line from Cr-
like Hf). However, their contribution can be reliably isolated using the
energy dependence of spectra and comparison with the simulated spectra which
do not include impurities. Some of the impurity lines are indicated in Fig. 1.
## 3 Collisional-radiative modeling and line identifications
As with our previous studies 12366EL ; 16374EL ; 9000EL ; 12286EL ; 14870EL ,
the spectral analysis and line identifications are based on detailed
collisional-radiative (CR) modeling. In 16374EL we comprehensively described
the basic principles of CR simulations for W ions with an open $3d$ shell, and
the present simulations follow that approach. Briefly, we start with
calculation of all required atomic data (energies, radiative probabilities and
electron-impact cross sections) using the Flexible Atomic Code (FAC) FAC . The
CR model includes singly-excited states up to at least $n$=5 (up to $n$=8 for
ions near closed shells) as well as doubly-excited states with $n$=3. In order
to reduce the number of states included, the $n\geq 4$ levels are combined
into “superterms,” which were introduced in 16374EL ; this results in more
manageable matrix sizes without loss of accuracy. The energies of the singly-
and doubly-excited states within $n$=3, which are the most important for the
present study, are improved by performing additional extensive calculations
which include all possible configurations of the $n=3$ complex as well as the
configurations described above. The calculated atomic data is then used as
input to the non-Maxwellian CR code NOMAD NOMAD , which calculates ionization
distributions (also taking into account charge exchange between ions and
neutrals in the EBIT), level populations, and spectral line intensities.
Finally, the theoretical spectrum is convolved with the spectrometer
efficiency curve and compared with the measured spectra. This procedure allows
us to identify the spectral lines using not only their wavelengths but also
their line intensities.
The calculated ionization potentials for the Ni-like to Cl-like ions of Hf, Ta
and Au are presented in Table 2. Also the Dirac-Fock results of Ref. 8672EL
are given for comparison. The two sets of ionization energies are seen to
agree within 0.4% or less. One can see then that the electron beam energies of
Table 1 are sufficient to produce the required ionization stages with $3d^{n}$
ground configurations.
The experimental spectra in Figs. 1–3 cover a quite large range of
wavelengths. This is achieved with a moderate spectral resolution
$\lambda/\delta\lambda\sim$ 400 that results in a number of blended lines. In
such cases, the corresponding wavelengths were determined from those spectra
where the abundance of one of the ions is small. This procedure is exemplified
by the line at $\sim$ 17.3 nm which was found to be very strong between 4350
eV and 5035 eV (Fig. 1). Closer inspection (see Fig. 4) shows that at lower
values of the beam energy $E_{b}$, the main contribution comes from the 17.263
nm line in the Mn-like ion, while at higher energies only the 17.300 nm line
in the Cr-like ion survives. Accordingly, the wavelengths of these two lines
were derived from the spectra measured at the lowest or highest energies of
the beam. Figure 1 also shows a strong line from Fe-like Hf47+ at 17.424 nm
that is blended with an impurity line at 17.438 nm. In order to analyze
possible impurities, we routinely measured EBIT spectra without the metal ion
injection. Such background spectra at approximately 5000 eV show a clear
presence of an impurity at 17.438 nm; the wavelength of this line was derived
from a Gaussian fit, and its contribution to the wide line profile with Hf
injection was accounted for in a two-Gaussian fit.
Tables 3-5 present the measured wavelengths, line identifications, theoretical
wavelengths and transition probabilities for 135 identified spectral lines,
133 of which are new. The identifications calculated by FAC are given in jj-
coupling, and the notations used are described in 16374EL . For such highly-
charged ions this type of coupling is the most appropriate. In the tables,
$l_{\pm}$ denotes an electron state with the total angular momentum $j$=$l\pm
1/2$, so that $d_{+}$ corresponds to $j$=5/2 and so on. Also, the wavelength
uncertainties in the tables are given in the units of the last significant
digit, so that 16.281(4) means 16.281$\pm$0.004.
### 3.1 Hf spectra
Table 3 presents 43 new spectral lines from Ni-like Hf44+ to Ar-like Hf54+.
Except for the four M1 lines within the excited configurations $3p^{5}3d^{2}$
of K-like and $3p^{5}3d$ of Ar-like ions, almost all other lines correspond to
the M1 transitions within the ground state configurations $3d^{n}$, $n$=1–9.
The only non-M1 transition identified in the Hf spectra is the electric-
quadrupole (E2) line at 16.149 nm in Ca-like Hf52+, for which the M1
transition is not allowed by the $|\Delta J|\leq 1$ selection rule. The
lowest-energy spectra in Fig. 1 also contain several lines from Cu-like and
Ni-like ions which have been analyzed in our previous work
Draganic_etal_Hf_Ta_Au_2011 . In the present work the M1 transition within the
excited configuration $3d^{9}4s$ of Ni-like Hf44+ was identified at
21.944$\pm$0.003 nm; the recent result of 21.9377 nm from the relativistic
many-body perturbation theory (RMBPT) Safronova_2007 thus reasonably agrees
with our value. The measured wavelength for the strong resonance line
$4s$–$4p$ in the Cu-like ion is 13.372$\pm$0.003 nm which excellently agrees
both with our previous result of 13.373$\pm$0.003 nm
Draganic_etal_Hf_Ta_Au_2011 and with the older tokamak measurement of
13.375$\pm$0.005 nm Putterich_PhD_HCI_Tokamaks_2006 .
The only two lines for which there exist theoretical wavelength calculations
are the $3d^{9}_{5/2}$–$3d^{9}_{3/2}$ transition in the Co-like ion and the
$3d_{3/2}$–$3d_{5/2}$ transition in the K-like ion. For the former, our
measured wavelength of 21.229$\pm$0.003 nm agrees within uncertainties with
the semiempirical prediction of 21.202$\pm$0.033 nm Ekberg_1987 , while the
calculated FAC value is slightly outside the experimental error bars. As also
was the case for the W measurements Draganic_etal_Hf_Ta_Au_2011 , the measured
wavelength of 18.158$\pm$0.003 nm for the M1 line in K-like ion agrees very
well with the multiconfiguration Dirac-Fock (MCDF) result of 18.1578 nm
obtained by Ali and Kim Ali_Kim_1992 .
The last column in Table 3 shows the calculated transition probabilities for
the identified lines of Hf. The M1 lines have A values in the range of
8$\times$104 s-1 to 3$\times$106 s-1. The transition probability for the
electric-quadrupole line at 16.149 nm is much smaller, about 2.8$\times$102
s-1, which exemplifies relative weakness of intraconfiguration E2 lines as
compared to M1 transitions.
### 3.2 Ta spectra
We indentified 50 new lines from thirteen ions from Ni-like Ta45+ to S-like
Ta57+ (Table 4). The identified lines correspond to intraconfiguration M1
transitions, although not all originate from the ground configurations: for
instance, all lines in Ar-, Cl-, and S-like ions are within the $3p^{n}3d$
configurations with $n$=3–5. Similar to the Hf case, the MCDF calculations of
Ali and Kim agree very well with our wavelength for the $3d$ line in the
K-like ion Ali_Kim_1992 , and Safronova et al.’s RMBPT data Safronova_2007
are very close to the measured wavelength in the Ni-like ion. The
semiempirical value of 19.816$\pm$0.032 nm for the M1 line in the Co-like ion
agrees with our measured wavelength of 19.843$\pm$0.003 nm within the stated
uncertainties.
### 3.3 Au spectra
Forty new spectral lines were identified for ions of Au, from Ni-like Au51+ to
K-like Au60+ (Table 5). The higher ion charges for the isoelectronic ions of
Au, as compared to Ta and Hf, result in smaller relative wavelength
separations between the analogous spectral lines. Therefore, the lines overlap
more often and are more difficult to analyze. This explains the relatively
smaller number of identified lines for Au. Similar to the Hf and Ta cases, the
RMBPT Safronova_2007 and MCDF Ali_Kim_1992 results for Ni-like and K-like
ions, respectively, agree well with the measured wavelengths, and again, the
semiempirical wavelength for the $3d^{9}$ intraconfiguration transition in the
Co-like ion agrees within uncertainties.
Two electric-quadrupole lines were identified in the measured spectra of Au.
As mentioned above, the transition probabilities for E2 lines are typically
much smaller than for M1 lines, and therefore they are more difficult to
observe due to collisional damping. Indeed, our calculations show that the
A-values for the E2 line at 11.993 nm in the V-like ion and for the E2 line in
the Sc-like ion are only 7.97$\times$102 s-1 and 1.55$\times$103 s-1,
respectively, while the A-values for M1 lines are on the order of $10^{6}$
s-1. The fact that such low-A lines were observed in the EBIT is due to their
high branching ratios: both E2 transitions are the strongest for their
corresponding upper levels.
We also identified two known lines in the Ni-like and Co-like ions of Au. The
newly measured wavelengths of 13.858$\pm$0.003 nm and 13.517$\pm$0.003 nm
agree within uncertainties with our previous results of 13.860$\pm$0.003 nm
and 13.522$\pm$0.003 nm Draganic_etal_Hf_Ta_Au_2011 . Thus, the recommended
averaged values for these lines are 13.8590$\pm$0.0020 nm for the $3d^{9}4s$
transition in the Ni-like Au51+ and 13.5195$\pm$0.0020 nm for the $3d^{9}$
transition in the Co-like Au52+.
## 4 Conclusions
In this paper we presented measurements and identifications of a large number
of EUV spectral lines originating from forbidden transitions, mostly within
the ground configurations, in the $3d^{n}$ ions of Hf, Ta, and Au. These
results provide a new accurate set of atomic data for highly-charged ions of
high-Z elements which are important for magnetic and intertial confinement
fusion research. Moreover, the measured wavelengths can also be used to test
the most advanced theories of atomic structure in the relativistic regime.
As was shown in our previous work 16374EL , the intensity ratios for similar
M1 lines in the $3d^{n}$ ions of W are very sensitive to electron densities in
the parameter range of magnetic fusion plasmas. Since the physical processes
affecting level populations for Hf, Ta and Au are the same as those for W, one
can expect similar sensitivity for the presently measured lines as well. The
only minor difference will be that the sensitivity range of densities for Hf
and Ta ions will be slightly shifted to lower densities while for Au it will
be shifted to higher densities. This results from a strong $z$-dependence of
the M1 radiative rates.
## 5 Acknowledgments
We are grateful to J. M. Pomeroy, J. N. Tan, and S. M. Brewer for assistance
during the early experimental phase of this work, to U. I. Safronova for
providing the energy levels of Ni-like ions from Ref. Safronova_2007 , and to
Y.A. Podpaly for valuable comments. This work is supported in part by the
Office of Fusion Energy Sciences of the U.S. Department of Energy.
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Figure 1: Experimental spectra for Hf. Nominal beam energies (in eV) are
shown in the upper corners. Some strong identified lines are indicated by
their isoelectronic sequences. Several impurity lines from Xe and O are
indicated as well.
Figure 2: Experimental spectra for Ta. Nominal beam energies (in eV) are
shown in the upper corners. Some strong identified lines are indicated by
their isoelectronic sequences.
Figure 3: Experimental spectra for Au. Nominal beam energies (in eV) are
shown in the upper corners. Some strong identified lines are indicated by
their isoelectronic sequences.
Figure 4: Hafnium spectra near 17.3 nm at the nominal beam energies between
4350 eV and 5035 eV. The unknown impurity line at 17.438 nm was prominent in
the background spectra.
Table 1: Nominal electron beam energies (in eV) used in the present work. Hf | 4005 | 4150 | 4350 | 4505 | 4650 | 4935 | 5035 | 5145 | 5350 | 5495 | 5645 | 5845 | 5995 | 6495 | |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
Ta | 4005 | 4150 | 4350 | 4550 | 4635 | 4835 | 5035 | 5150 | 5490 | 5640 | 5840 | 6005 | 6495 | | |
Au | 5035 | 5200 | 5350 | 5490 | 5650 | 5850 | 6000 | 6155 | 6345 | 6490 | 6650 | 6850 | 6995 | 7150 | 7350 | 7495
Table 2: Calculated ionization potentials (in eV) of Ni-like through Cl-like ions of Hf, Ta and Au. Sequence | Hf | Ta | Au
---|---|---|---
| | FAC | Ref. 8672EL | FAC | Ref. 8672EL | FAC | Ref. 8672EL
Ni | $3d^{10}$ | 3741.7 | 3741 | 3895.3 | 3898.7 | 4880.4 | 4888
Co | $3d^{9}$ | 3859.4 | 3858 | 4015.2 | 4014 | 5014.0 | 5013
Fe | $3d^{8}$ | 3984.9 | 3984 | 4143.1 | 4143 | 5156.0 | 5156
Mn | $3d^{7}$ | 4119.1 | 4118 | 4279.8 | 4278 | 5307.9 | 5307
Cr | $3d^{6}$ | 4248.2 | 4246 | 4411.3 | 4410 | 5453.7 | 5452
V | $3d^{5}$ | 4375.7 | 4372 | 4541.0 | 4537 | 5596.4 | 5594
Ti | $3d^{4}$ | 4576.9 | 4573 | 4749.1 | 4745 | 5850.8 | 5846
Sc | $3d^{3}$ | 4709.1 | 4703 | 4883.6 | 4877 | 5999.5 | 5994
Ca | $3d^{2}$ | 4853.3 | 4846 | 5030.4 | 5024 | 6162.3 | 6156
K | $3d$ | 4987.6 | 4980 | 5167.1 | 5159 | 6313.6 | 6305
Ar | $3p^{6}$ | 5359.8 | 5350 | 5545.9 | 5537 | 6732.4 | 6724
Cl | $3p^{5}$ | 5474.9 | 5468 | 5663.0 | 5655 | 6861.3 | 6854
Table 3: Identified spectral lines of highly-charged ions of hafnium. Level identifications include calculated level numbers. Blended lines are indicated by “b”. Notation a(b) for transition probabilities $A$ means $a\cdot 10^{b}$. The electric-quadrupole line is marked by asterisk. Other theoretical works: A – Safronova_2007 , B – Ekberg_1987 , C – Ali_Kim_1992 . Ion | Seq. | Conf. | Lower level | Upper level | $\lambda_{exp}$ | $\lambda_{th}$ | A
---|---|---|---|---|---|---|---
charge | | | No. | TermJ | No. | TermJ | (nm) | (nm) | (s-1)
44 | Ni | $3d^{9}4s$ | 3 | $((d_{+}^{5})_{5/2},s_{+})_{2}$ | 4 | $((d_{-}^{3})_{3/2},s_{+})_{1}$ | 21.944(3) | 22.0114, 21.9377A | 1.60(6)
45 | Co | $3d^{9}$ | 1 | $(d_{+}^{5})_{5/2}$ | 2 | $(d_{-}^{3})_{3/2}$ | 21.229(3) | 21.3176, 21.202(33)B | 1.65(6)
46 | Fe | $3d^{8}$ | 1 | $(d_{+}^{4})_{4}$ | 7 | $((d_{-}^{3})_{3/2},(d_{+}^{5})_{5/2})_{4}$ | 17.424(8) | 17.3934 | 6.90(5)
46 | Fe | $3d^{8}$ | 2 | $(d_{+}^{4})_{2}$ | 6 | $((d_{-}^{3})_{3/2},(d_{+}^{5})_{5/2})_{1}$ | 19.929(3) | 19.9516 | 1.15(6)
46 | Fe | $3d^{8}$ | 2 | $(d_{+}^{4})_{2}$ | 5 | $((d_{-}^{3})_{3/2},(d_{+}^{5})_{5/2})_{2}$ | 21.652(5) | 21.7390 | 1.29(6)
46 | Fe | $3d^{8}$ | 1 | $(d_{+}^{4})_{4}$ | 4 | $((d_{-}^{3})_{3}/2,(d_{+}^{5})_{5/2})_{3}$ | 21.775(3) | 21.8812 | 2.15(6)
47 | Mn | $3d^{7}$ | 1 | $(d_{+}^{3})_{9/2}$ | 9 | $((d_{-}^{3})_{3/2},(d_{+}^{4})_{4})_{11/2}$ | 17.263(3) | 17.2267 | 1.28(5)
47 | Mn | $3d^{7}$ | 1 | $(d_{+}^{3})_{9/2}$ | 5 | $((d_{-}^{3})_{3/2},(d_{+}^{4})_{4})_{9/2}$ | 19.403(3) | 19.4231 | 1.51(6)
47 | Mn | $3d^{7}$ | 3 | $(d_{+}^{3})_{5/2}$ | 10 | $((d_{-}^{3})_{3/2},(d_{+}^{4})_{2})_{7/2}$ | 20.851(3) | 20.8488 | 1.84(5)
47 | Mn | $3d^{7}$ | 2 | $(d_{+}^{3})_{3/2}$ | 8 | $((d_{-}^{3})_{3/2},(d_{+}^{4})_{2})_{1/2}$ | 21.390(3) | 21.4662 | 1.72(6)
47 | Mn | $3d^{7}$ | 1 | $(d_{+}^{3})_{9/2}$ | 4 | $((d_{-}^{3})_{3/2},(d_{+}^{4})_{4})_{7/2}$ | 21.687(3) | 21.7903 | 2.38(6)
47 | Mn | $3d^{7}$ | 2 | $(d_{+}^{3})_{3/2}$ | 7 | $((d_{-}^{3})_{3/2},(d_{+}^{4})_{4})_{5/2}$ | 21.916(3) | 21.9934 | 6.80(5)
48 | Cr | $3d^{6}$ | 1 | $(d_{+}^{2})_{4}$ | 10 | $((d_{-}^{3})_{3/2},(d_{+}^{3})_{3/2})_{3}$ | 17.300(3) | 17.2838 | 7.07(5)
48 | Cr | $3d^{6}$ | 1 | $(d_{+}^{2})_{4}$ | 8 | $((d_{-}^{3})_{3/2},(d_{+}^{3})_{9/2})_{5}$ | 19.437(3) | 19.4248 | 4.35(5)
48 | Cr | $3d^{6}$ | 2 | $(d_{+}^{2})_{2}$ | 9 | $((d_{-}^{3})_{3/2},(d_{+}^{3})_{3/2})_{2}$ | 19.721(3) | 19.7351 | 1.35(6)
48 | Cr | $3d^{6}$ | 1 | $(d_{+}^{2})_{4}$ | 5 | $((d_{-}^{3})_{3/2},(d_{+}^{3})_{9/2})_{4}$ | 22.091(3) | 22.1559 | 2.00(6)
48 | Cr | $3d^{6}$ | 1 | $(d_{+}^{2})_{4}$ | 4 | $((d_{-}^{3})_{3/2},(d_{+}^{3})_{9/2})_{3}$ | 22.754(3) | 22.8747 | 1.69(6)
49 | V | $3d^{5}$ | 1 | $(d_{+})_{5/2}$ | 10 | $((d_{-}^{3})_{3/2},(d_{+}^{2})_{0})_{3/2}$ | 13.487(3) | 13.4510 | 1.82(5)
49 | V | $3d^{5}$ | 1 | $(d_{+})_{5/2}$ | 9 | $((d_{-}^{3})_{3/2},(d_{+}^{2})_{2})_{7/2}$ | 16.275(3) | 16.2253 | 8.31(4)
49 | V | $3d^{5}$ | 1 | $(d_{+})_{5/2}$ | 5 | $((d_{-}^{3})_{3/2},(d_{+}^{2})_{2})_{3/2}$ | 19.583(3) | 19.6141 | 2.54(6)
49 | V | $3d^{5}$ | 1 | $(d_{+})_{5/2}$ | 3 | $((d_{-}^{3})_{3/2},(d_{+}^{2})_{4})_{7/2}$ | 20.114(3)b | 20.1278 | 1.05(6)
49 | V | $3d^{5}$ | 1 | $(d_{+})_{5/2}$ | 2 | $((d_{-}^{3})_{3/2},(d_{+}^{2})_{4})_{5/2}$ | 24.711(3) | 24.8671 | 2.19(6)
50 | Ti | $3d^{4}$ | 5 | $((d_{-}^{3})_{3/2},d_{+})_{3}$ | 17 | $((d_{-}^{2})_{0},(d_{+}^{2})_{4})_{4}$ | 15.105(3) | 15.0733 | 7.48(5)
50 | Ti | $3d^{4}$ | 3 | $((d_{-}^{3})_{3/2},d_{+})_{4}$ | 13 | $((d_{-}^{2})_{2},(d_{+}^{2})_{2})_{3}$ | 18.043(3) | 18.0451 | 1.19(6)
50 | Ti | $3d^{4}$ | 2 | $((d_{-}^{3})_{3/2},d_{+})_{1}$ | 7 | $((d_{-}^{2})_{2},(d_{+}^{2})_{4})_{2}$ | 19.148(3) | 19.1681 | 3.13(6)
50 | Ti | $3d^{4}$ | 3 | $((d_{-}^{3})_{3/2},d_{+})_{4}$ | 10 | $((d_{-}^{2})_{2},(d_{+}^{2})_{4})_{5}$ | 20.410(3) | 20.4503 | 1.10(6)
50 | Ti | $3d^{4}$ | 1 | $(d_{-}^{4})_{0}$ | 2 | $((d_{-}^{3})_{3/2},d_{+})_{1}$ | 22.325(3) | 22.4327 | 2.17(6)
50 | Ti | $3d^{4}$ | 3 | $((d_{-}^{3})_{3/2},d_{+})_{4}$ | 8 | $((d_{-}^{2})_{2},(d_{+}^{2})_{4})_{4}$ | 22.502(3) | 22.6346 | 1.98(6)
51 | Sc | $3d^{3}$ | 1 | $(d_{-}^{3})_{3/2}$ | 7 | $((d_{-}^{2})_{0},d_{+})_{5/2}$ | 13.706(3) | 13.6749 | 1.92(5)
51 | Sc | $3d^{3}$ | 5 | $((d_{-}^{2})_{2},d_{+})_{9/2}$ | 12 | $(d_{-},(d_{+}^{2})_{4})_{11/2}$ | 17.873(3) | 17.8904 | 9.60(5)
51 | Sc | $3d^{3}$ | 1 | $(d_{-}^{3})_{3/2}$ | 6 | $((d_{-}^{2})_{2},d_{+})_{1/2}$ | 18.150(6) | 18.1838 | 6.84(5)
51 | Sc | $3d^{3}$ | 2 | $((d_{-}^{2})_{2},d_{+})_{5/2}$ | 8 | $(d_{-},(d_{+}^{2})_{4})_{7/2}$ | 18.942(3) | 18.9797 | 3.29(6)
51 | Sc | $3d^{3}$ | 1 | $(d_{-}^{3})_{3/2}$ | 3 | $((d_{-}^{2})_{2},d_{+})_{3/2}$ | 19.639(3) | 19.6876 | 1.83(6)
51 | Sc | $3d^{3}$ | 1 | $(d_{-}^{3})_{3/2}$ | 2 | $((d_{-}^{2})_{2},d_{+})_{5/2}$ | 21.762(3) | 21.8713 | 2.26(6)
52 | Ca | $3d^{2}$ | 1 | $(d_{-}^{2})_{2}$ | 5 | $(d_{-},d_{+})_{4}$ | 16.149(4)* | 16.1575 | 2.82(2)
52 | Ca | $3d^{2}$ | 1 | $(d_{-}^{2})_{2}$ | 4 | $(d_{-},d_{+})_{2}$ | 16.884(3) | 16.9070 | 1.23(6)
52 | Ca | $3d^{2}$ | 1 | $(d_{-}^{2})_{2}$ | 3 | $(d_{-},d_{+})_{3}$ | 19.525(3) | 19.6012 | 2.48(6)
52 | Ca | $3d^{2}$ | 2 | $(d_{-}^{2})_{0}$ | 6 | $(d_{-},d_{+})_{1}$ | 22.158(3) | 22.2673 | 1.13(6)
53 | K | $3p^{5}3d^{2}$ | 6 | $((p_{+}^{3})_{3/2},(d_{-}^{2})_{2})_{7/2}$ | 19 | $((p_{+}^{3})_{3/2},d_{-}d_{+})_{9/2}$ | 14.576(3) | 14.5240 | 8.75(4)
53 | K | $3d$ | 1 | $(d_{-})_{3/2}$ | 2 | $(d_{+})_{5/2}$ | 18.158(3) | 18.2132, 18.1578C | 1.76(6)
53 | K | $3p^{5}3d^{2}$ | 6 | $((p_{+}^{3})_{3/2},(d_{-}^{2})_{2})_{7/2}$ | 9 | $((p_{+}^{3})_{3/2}d_{-}d_{+})_{9/2}$ | 21.316(3) | 21.4145 | 1.96(6)
54 | Ar | $3p^{5}3d$ | 5 | $((p_{+}^{3})_{3/2},d_{-})_{2}$ | 8 | $((p_{+}^{3})_{3/2},d_{+})_{3}$ | 16.693(3) | 16.7008 | 1.76(6)
54 | Ar | $3p^{5}3d$ | 4 | $((p_{+}^{3})_{3/2},d_{-})_{3}$ | 6 | $((p_{+}^{3})_{3/2},d_{+})_{4}$ | 19.958(3) | 20.0564 | 1.39(6)
Table 4: Identified spectral lines of highly-charged ions of tantalum. Level identifications include calculated level numbers. Blended lines are indicated by “b”. Notation a(b) for transition probabilities $A$ means $a\cdot 10^{b}$. Other theoretical works: A – Safronova_2007 , B – Ekberg_1987 , C – Ali_Kim_1992 . Ion | Seq. | Conf. | Lower level | Upper level | $\lambda_{exp}$ | $\lambda_{th}$ | A
---|---|---|---|---|---|---|---
charge | | | No. | TermJ | No. | TermJ | (nm) | (nm) | (s-1)
45 | Ni | $3d^{9}4s$ | 3 | $((d_{+}^{5})_{5/2},s_{+})_{2}$ | 4 | $((d_{-}^{3})_{3/2},s_{+})_{1}$ | 20.482(3) | 20.5428, 20.4758A | 1.96(6)
46 | Co | $3d^{9}$ | 1 | $(d_{+}^{5})_{5/2}$ | 2 | $(d_{-}^{3})_{3/2}$ | 19.843(3) | 19.9229, 19.816(32)B | 2.02(6)
47 | Fe | $3d^{8}$ | 1 | $(d_{+}^{4})_{4}$ | 7 | $((d_{-}^{3})_{3/2},(d_{+}^{5})_{5/2})_{4}$ | 16.436(3) | 16.4099 | 8.37(5)
47 | Fe | $3d^{8}$ | 2 | $(d_{+}^{4})_{2}$ | 6 | $((d_{-}^{3})_{3/2},(d_{+}^{5})_{5/2})_{1}$ | 18.667(3) | 18.6907 | 1.40(6)
47 | Fe | $3d^{8}$ | 2 | $(d_{+}^{4})_{2}$ | 5 | $((d_{-}^{3})_{3/2},(d_{+}^{5})_{5/2})_{2}$ | 20.204(3) | 20.2858 | 1.59(6)
47 | Fe | $3d^{8}$ | 1 | $(d_{+}^{4})_{4}$ | 4 | $((d_{-}^{3})_{3/2},(d_{+}^{5})_{5/2})_{3}$ | 20.322(3) | 20.4182 | 2.64(6)
48 | Mn | $3d^{7}$ | 1 | $(d_{+}^{3})_{9/2}$ | 9 | $((d_{-}^{3})_{3/2},(d_{+}^{4})_{4})_{11/2}$ | 16.281(4) | 16.2525 | 1.56(5)
48 | Mn | $3d^{7}$ | 5 | $((d_{-}^{3})_{3/2},(d_{+}^{4})_{4})_{9/2}$ | 15 | $((d_{-}^{2})_{2},(d_{+}^{5})_{5/2})_{9/2}$ | 17.640(3) | 17.6608 | 1.81(6)
48 | Mn | $3d^{7}$ | 3 | $(d_{+}^{3})_{5/2}$ | 11 | $((d_{-}^{3})_{3/2},(d_{+}^{4})_{2})_{5/2}$ | 18.169(3) | 18.1220 | 7.02(5)
48 | Mn | $3d^{7}$ | 1 | $(d_{+}^{3})_{9/2}$ | 5 | $((d_{-}^{3})_{3/2},(d_{+}^{4})_{4})_{9/2}$ | 18.210(3) | 18.2328 | 1.85(6)
48 | Mn | $3d^{7}$ | 3 | $(d_{+}^{3})_{5/2}$ | 10 | $((d_{-}^{3})_{3/2},(d_{+}^{4})_{2})_{7/2}$ | 19.508(3) | 19.5120 | 2.28(5)
48 | Mn | $3d^{7}$ | 2 | $(d_{+}^{3})_{3/2}$ | 8 | $((d_{-}^{3})_{3/2},(d_{+}^{4})_{2})_{1/2}$ | 19.968(3) | 20.0415 | 2.11(6)
48 | Mn | $3d^{7}$ | 1 | $(d_{+}^{3})_{9/2}$ | 4 | $((d_{-}^{3})_{3/2},(d_{+}^{4})_{4})_{7/2}$ | 20.215(3) | 20.3167 | 2.92(6)
48 | Mn | $3d^{7}$ | 2 | $(d_{+}^{3})_{3/2}$ | 7 | $((d_{-}^{3})_{3/2},(d_{+}^{4})_{4})_{5/2}$ | 20.417(3) | 20.4889 | 8.38(5)
48 | Mn | $3d^{7}$ | 3 | $(d_{+}^{3})_{5/2}$ | 7 | $((d_{-}^{3})_{3/2},(d_{+}^{4})_{4})_{5/2}$ | 23.031(3) | 23.1203 | 7.00(5)
48 | Mn | $3d^{7}$ | 3 | $(d_{+}^{3})_{5/2}$ | 6 | $((d_{-}^{3})_{3/2},(d_{+}^{4})_{2})_{3/2}$ | 23.938(3)b | 24.0887 | 1.38(6)
49 | Cr | $3d^{6}$ | 1 | $(d_{+}^{2})_{4}$ | 15 | $((d_{-}^{3})_{3/2},(d_{+}^{3})_{5/2})_{3}$ | 13.462(3) | 13.3992 | 3.39(5)
49 | Cr | $3d^{6}$ | 1 | $(d_{+}^{2})_{4}$ | 12 | $((d_{-}^{3})_{3/2},(d_{+}^{3})_{5/2})_{4}$ | 13.861(3) | 13.8207 | 3.13(4)
49 | Cr | $3d^{6}$ | 2 | $(d_{+}^{2})_{2}$ | 15 | $((d_{-}^{3})_{3/2},(d_{+}^{3})_{5/2})_{3}$ | 14.665(3) | 14.6071 | 4.07(5)
49 | Cr | $3d^{6}$ | 1 | $(d_{+}^{2})_{4}$ | 10 | $((d_{-}^{3})_{3/2},(d_{+}^{3})_{3/2})_{3}$ | 16.286(4) | 16.2870 | 8.47(5)
49 | Cr | $3d^{6}$ | 1 | $(d_{+}^{2})_{4}$ | 8 | $((d_{-}^{3})_{3/2},(d_{+}^{3})_{9/2})_{5}$ | 18.242(5) | 18.2370 | 5.35.(5)
49 | Cr | $3d^{6}$ | 2 | $(d_{+}^{2})_{2}$ | 9 | $((d_{-}^{3})_{3/2},(d_{+}^{3})_{3/2})_{2}$ | 18.486(3) | 18.5076 | 1.66(6)
49 | Cr | $3d^{6}$ | 1 | $(d_{+}^{2})_{4}$ | 5 | $((d_{-}^{3})_{3/2},(d_{+}^{3})_{9/2})_{4}$ | 20.602(3) | 20.6685 | 2.46(6)
49 | Cr | $3d^{6}$ | 1 | $(d_{+}^{2})_{4}$ | 4 | $((d_{-}^{3})_{3/2},(d_{+}^{3})_{9/2})_{3}$ | 21.144(3) | 21.2545 | 2.08(6)
49 | Cr | $3d^{6}$ | 2 | $(d_{+}^{2})_{2}$ | 6 | $((d_{-}^{3})_{3/2},(d_{+}^{3})_{3/2})_{1}$ | 22.214(3) | 22.3583 | 2.66(6)
50 | V | $3d^{5}$ | 1 | $(d_{+})_{5/2}$ | 10 | $((d_{-}^{3})_{3/2},(d_{+}^{2})_{0})_{3/2}$ | 12.787(3) | 12.7531 | 2.16(5)
50 | V | $3d^{5}$ | 1 | $(d_{+})_{5/2}$ | 9 | $((d_{-}^{3})_{3/2},(d_{+}^{2})_{2})_{7/2}$ | 15.374(3) | 15.3315 | 1.00(5)
50 | V | $3d^{5}$ | 1 | $(d_{+})_{5/2}$ | 5 | $((d_{-}^{3})_{3/2},(d_{+}^{2})_{2})_{3/2}$ | 18.352(3) | 18.3840 | 3.09(6)
50 | V | $3d^{5}$ | 1 | $(d_{+})_{5/2}$ | 3 | $((d_{-}^{3})_{3/2},(d_{+}^{2})_{4})_{7/2}$ | 18.841(3)b | 18.8579 | 1.30(6)
50 | V | $3d^{5}$ | 4 | $((d_{-}^{3})_{3/2},(d_{+}^{2})_{4})_{11/2}$ | 15 | $((d_{-}^{2})_{2},(d_{+}^{3})_{9/2})_{13/2}$ | 18.841(3)b | 18.8432 | 4.23(5)
50 | V | $3d^{5}$ | 4 | $((d_{-}^{3})_{3/2},(d_{+}^{2})_{4})_{11/2}$ | 13 | $((d_{-}^{2})_{2},(d_{+}^{3})_{9/2})_{11/2}$ | 20.368(3) | 20.4465 | 1.88(6)
50 | V | $3d^{5}$ | 1 | $(d_{+})_{5/2}$ | 2 | $((d_{-}^{3})_{3/2},(d_{+}^{2})_{4})_{5/2}$ | 22.869(3) | 23.0137 | 2.73(6)
51 | Ti | $3d^{4}$ | 5 | $((d_{-}^{3})_{3/2},d_{+})_{3}$ | 17 | $((d_{-}^{2})_{0},(d_{+}^{2})_{4})_{4}$ | 14.303(3) | 14.2704 | 9.03(5)
51 | Ti | $3d^{4}$ | 3 | $((d_{-}^{3})_{3/2},d_{+})_{4}$ | 10 | $((d_{-}^{2})_{2},(d_{+}^{2})_{4})_{5}$ | 19.073(3) | 19.1147 | 1.35(6)
51 | Ti | $3d^{4}$ | 1 | $(d_{-}^{4})_{0}$ | 2 | $((d_{-}^{3})_{3/2},d_{+})_{1}$ | 20.750(3) | 20.8492 | 2.69(6)
52 | Sc | $3d^{3}$ | 1 | $(d_{-}^{3})_{3/2}$ | 7 | $((d_{-}^{2})_{0},d_{+})_{5/2}$ | 12.987(7) | 12.9603 | 2.31(5)
52 | Sc | $3d^{3}$ | 1 | $(d_{-}^{3})_{3/2}$ | 6 | $((d_{-}^{2})_{2},d_{+})_{1/2}$ | 17.053(3) | 17.0818 | 8.37(5)
52 | Sc | $3d^{3}$ | 2 | $((d_{-}^{2})_{2},d_{+})_{5/2}$ | 8 | $(d_{-},(d_{+}^{2})_{4})_{7/2}$ | 17.758(3) | 17.8065 | 4.01(6)
52 | Sc | $3d^{3}$ | 1 | $(d_{-}^{3})_{3/2}$ | 3 | $((d_{-}^{2})_{2},d_{+})_{3/2}$ | 18.377(3) | 18.4346 | 2.24(6)
52 | Sc | $3d^{3}$ | 1 | $(d_{-}^{3})_{3/2}$ | 2 | $((d_{-}^{2})_{2},d_{+})_{5/2}$ | 20.247(3) | 20.3630 | 2.78(6)
53 | Ca | $3d^{2}$ | 1 | $(d_{-}^{2})_{2}$ | 4 | $(d_{-},d_{+})_{2}$ | 15.885(3) | 15.9086 | 1.49(6)
53 | Ca | $3d^{2}$ | 1 | $(d_{-}^{2})_{2}$ | 3 | $(d_{-},d_{+})_{3}$ | 18.250(3) | 18.3192 | 3.02(6)
53 | Ca | $3d^{2}$ | 2 | $(d_{-}^{2})_{0}$ | 6 | $(d_{-},d_{+})_{1}$ | 20.598(3) | 20.6941 | 1.40(6)
54 | K | $3d$ | 1 | $(d_{-})_{3/2}$ | 2 | $(d_{+})_{5/2}$ | 17.015(3) | 17.0662, 17.0145C | 2.14(6)
54 | K | $3p^{5}3d^{2}$ | 6 | $((p_{+}^{3})_{3/2},(d_{-}^{2})_{2})_{7/2}$ | 9 | $((p_{+}^{3})_{3/2},d_{-},d_{+})_{9/2}$ | 19.831(3) | 19.9134 | 2.43(6)
55 | Ar | $3p^{5}3d$ | 5 | $((p_{+}^{3})_{3/2},d_{-})_{2}$ | 8 | $((p_{+}^{3})_{3/2},d_{+})_{3}$ | 15.699(3) | 15.7090 | 2.12(6)
55 | Ar | $3p^{5}3d$ | 4 | $((p_{+}^{3})_{3/2},d_{-})_{3}$ | 6 | $((p_{+}^{3})_{3/2},d_{+})_{4}$ | 18.607(3) | 18.6975 | 1.71(6)
56 | Cl | $3p^{4}3d$ | 3 | $((p_{+}^{2})_{2},d_{-})_{5/2}$ | 7 | $((p_{+}^{2})_{2},d_{+})_{7/2}$ | 17.735(3) | 17.7940 | 1.51(6)
56 | Cl | $3p^{4}3d$ | 5 | $((p_{+}^{2})_{2},d_{-})_{7/2}$ | 8 | $((p_{+}^{2})_{2},d_{+})_{9/2}$ | 18.093(3) | 18.1197 | 1.88(6)
57 | S | $3p^{3}3d$ | 6 | $(p_{+},d_{-})_{3}$ | 7 | $(p_{+},d_{+})_{4}$ | 17.561(3) | 17.5900 | 2.14(6)
Table 5: Identified spectral lines of highly-charged ions of gold. Level identifications include calculated level numbers. Blended lines are indicated by “b”. Notation a(b) for transition probabilities $A$ means $a\cdot 10^{b}$. Electric-quadrupole lines are marked by asterisks. Other theoretical works: A – Safronova_2007 , B – Ekberg_1987 , C – Ali_Kim_1992 . Ion | Seq. | Conf. | Lower level | Upper level | $\lambda_{exp}$ | $\lambda_{th}$ | A
---|---|---|---|---|---|---|---
charge | | | No. | TermJ | No. | TermJ | (nm) | (nm) | (s-1)
51 | Ni | $3d^{9}4s$ | 3 | $((d_{+}^{5})_{5/2},s_{+})_{2}$ | 4 | $((d_{-}^{3})_{3/2},s_{+})_{1}$ | 13.858(3) | 13.8971, 13.8550A | 6.23(6)
52 | Co | $3d^{9}$ | 1 | $(d_{+}^{5})_{5/2}$ | 2 | $(d_{-}^{3})_{3/2}$ | 13.517(3) | 13.5678, 13.497(27)B | 6.38(6)
53 | Fe | $3d^{8}$ | 1 | $(d_{+}^{4})_{4}$ | 6 | $((d_{-}^{3})_{3/2},(d_{+}^{5})_{5/2})_{4}$ | 11.707(3) | 11.6996 | 2.52(6)
53 | Fe | $3d^{8}$ | 2 | $(d_{+}^{4})_{2}$ | 7 | $((d_{-}^{3})_{3/2},(d_{+}^{5})_{5/2})_{1}$ | 12.854(3) | 12.8748 | 4.38(6)
53 | Fe | $3d^{8}$ | 1 | $(d_{+}^{4})_{4}$ | 4 | $((d_{-}^{3})_{3/2},(d_{+}^{5})_{5/2})_{3}$ | 13.739(3) | 13.7943 | 8.38(6)
53 | Fe | $3d^{8}$ | 3 | $(d_{+}^{4})_{0}$ | 7 | $((d_{-}^{3})_{3/2},(d_{+}^{5})_{5/2})_{1}$ | 16.610(3) | 16.6990 | 1.86(6)
54 | Mn | $3d^{7}$ | 3 | $(d_{+}^{3})_{5/2}$ | 12 | $((d_{-}^{3})_{3/2},(d_{+}^{4})_{0})_{3/2}$ | 10.541(5) | 10.5155 | 2.49(6)
54 | Mn | $3d^{7}$ | 1 | $(d_{+}^{3})_{9/2}$ | 10 | $((d_{-}^{3})_{3/2},(d_{+}^{4})_{2})_{7/2}$ | 10.783(3) | 10.7761 | 3.89(5)
54 | Mn | $3d^{7}$ | 1 | $(d_{+}^{3})_{9/2}$ | 9 | $((d_{-}^{3})_{3/2},(d_{+}^{4})_{4})_{11/2}$ | 11.598(3) | 11.5917 | 4.80(5)
54 | Mn | $3d^{7}$ | 1 | $(d_{+}^{3})_{9/2}$ | 5 | $((d_{-}^{3})_{3/2},(d_{+}^{4})_{4})_{9/2}$ | 12.650(3)b | 12.6728 | 5.70(6)
54 | Mn | $3d^{7}$ | 1 | $(d_{+}^{3})_{9/2}$ | 4 | $((d_{-}^{3})_{3/2},(d_{+}^{4})_{4})_{7/2}$ | 13.627(3) | 13.6811 | 9.29(6)
55 | Cr | $3d^{6}$ | 1 | $(d_{+}^{2})_{4}$ | 15 | $((d_{-}^{3})_{3/2},(d_{+}^{3})_{5/2})_{3}$ | 9.910(3) | 9.8848 | 9.56(5)
55 | Cr | $3d^{6}$ | 1 | $(d_{+}^{2})_{4}$ | 12 | $((d_{-}^{3})_{3/2},(d_{+}^{3})_{5/2})_{4}$ | 10.117(3) | 10.0994 | 8.98(5)
55 | Cr | $3d^{6}$ | 2 | $(d_{+}^{2})_{2}$ | 15 | $((d_{-}^{3})_{3/2},(d_{+}^{3})_{5/2})_{3}$ | 10.652(3) | 10.6295 | 1.22(6)
55 | Cr | $3d^{6}$ | 2 | $(d_{+}^{2})_{2}$ | 14 | $((d_{-}^{3})_{3/2},(d_{+}^{3})_{5/2})_{2}$ | 10.764(3) | 10.755 | 7.40(5)
55 | Cr | $3d^{6}$ | 1 | $(d_{+}^{2})_{4}$ | 10 | $((d_{-}^{3})_{3/2},(d_{+}^{3})_{3/2})_{3}$ | 11.553(3) | 11.5501 | 2.39(6)
55 | Cr | $3d^{6}$ | 1 | $(d_{+}^{2})_{4}$ | 8 | $((d_{-}^{3})_{3/2},(d_{+}^{3})_{9/2})_{5}$ | 12.667(3) | 12.6774 | 1.73(6)
55 | Cr | $3d^{6}$ | 2 | $(d_{+}^{2})_{2}$ | 9 | $((d_{-}^{3})_{3/2},(d_{+}^{3})_{3/2})_{2}$ | 12.773(3) | 12.7997 | 5.20(6)
55 | Cr | $3d^{6}$ | 1 | $(d_{+}^{2})_{4}$ | 5 | $((d_{-}^{3})_{3/2},(d_{+}^{3})_{9/2})_{4}$ | 13.869(3) | 13.9155 | 7.98(6)
55 | Cr | $3d^{6}$ | 1 | $(d_{+}^{2})_{4}$ | 4 | $((d_{-}^{3})_{3/2},(d_{+}^{3})_{9/2})_{3}$ | 14.025(3) | 14.0868 | 6.85(6)
55 | Cr | $3d^{6}$ | 2 | $(d_{+}^{2})_{2}$ | 6 | $((d_{-}^{3})_{3/2},(d_{+}^{3})_{3/2})_{1}$ | 14.562(3) | 14.6410 | 8.98(6)
56 | V | $3d^{5}$ | 1 | $(d_{+})_{5/2}$ | 10 | $((d_{-}^{3})_{3/2},(d_{+}^{2})_{0})_{3/2}$ | 9.387(3) | 9.3698 | 5.75(5)
56 | V | $3d^{5}$ | 1 | $(d_{+})_{5/2}$ | 9 | $((d_{-}^{3})_{3/2},(d_{+}^{2})_{2})_{7/2}$ | 11.028(3) | 11.0158 | 2.96(5)
56 | V | $3d^{5}$ | 1 | $(d_{+})_{5/2}$ | 7 | $((d_{-}^{3})_{3/2},(d_{+}^{2})_{2})_{5/2}$ | 11.509(3) | 11.5065 | 9.17(4)
56 | V | $3d^{5}$ | 1 | $(d_{+})_{5/2}$ | 6 | $((d_{-}^{3})_{3/2},(d_{+}^{2})_{4})_{9/2}$ | 11.993(3)* | 11.9941 | 7.97(2)
56 | V | $3d^{5}$ | 1 | $(d_{+})_{5/2}$ | 5 | $((d_{-}^{3})_{3/2},(d_{+}^{2})_{2})_{3/2}$ | 12.647(3)b | 12.6749 | 9.52(6)
56 | V | $3d^{5}$ | 1 | $(d_{+})_{5/2}$ | 3 | $((d_{-}^{3})_{3/2},(d_{+}^{2})_{4})_{7/2}$ | 12.947(3) | 12.9689 | 4.22(6)
56 | V | $3d^{5}$ | 4 | $((d_{-}^{3})_{3/2},(d_{+}^{2})_{4})_{11/2}$ | 13 | $((d_{-}^{2})_{2},(d_{+}^{3})_{9/2})_{11/2}$ | 13.652(5) | 13.7051 | 6.11(6)
56 | V | $3d^{5}$ | 1 | $(d_{+})_{5/2}$ | 2 | $((d_{-}^{3})_{3/2},(d_{+}^{2})_{4})_{5/2}$ | 14.862(3) | 14.9435 | 9.34(6)
57 | Ti | $3d^{4}$ | 3 | $((d_{-}^{3})_{3/2},d_{+})_{4}$ | 13 | $((d_{-}^{2})_{2},(d_{+}^{2})_{2})_{3}$ | 11.870(4) | 11.8770 | 4.39(6)
57 | Ti | $3d^{4}$ | 3 | $((d_{-}^{3})_{3/2},d_{+})_{4}$ | 10 | $((d_{-}^{2})_{2},(d_{+}^{2})_{4})_{5}$ | 12.991(3) | 13.0196 | 4.30(6)
57 | Ti | $3d^{4}$ | 1 | $(d_{-}^{4})_{0}$ | 2 | $((d_{-}^{3})_{3/2},d_{+})_{1}$ | 13.781(3) | 13.8384 | 8.89(6)
58 | Sc | $3d^{3}$ | 1 | $(d_{-}^{3})_{3/2}$ | 7 | $((d_{-}^{2})_{0},d_{+})_{5/2}$ | 9.502(4) | 9.4918 | 6.44(5)
58 | Sc | $3d^{3}$ | 1 | $(d_{-}^{3})_{3/2}$ | 6 | $((d_{-}^{2})_{2},d_{+})_{1/2}$ | 11.887(5) | 11.9000 | 2.65(6)
58 | Sc | $3d^{3}$ | 1 | $(d_{-}^{3})_{3/2}$ | 5 | $((d_{-}^{2})_{2},d_{+})_{7/2}$ | 12.054(3)* | 12.0761 | 1.55(3)
58 | Sc | $3d^{3}$ | 1 | $(d_{-}^{3})_{3/2}$ | 3 | $((d_{-}^{2})_{2},d_{+})_{3/2}$ | 12.590(3) | 12.6212 | 7.14(6)
58 | Sc | $3d^{3}$ | 1 | $(d_{-}^{3})_{3/2}$ | 2 | $((d_{-}^{2})_{2},d_{+})_{5/2}$ | 13.501(3) | 13.5597 | 8.98(6)
59 | Ca | $3d^{2}$ | 1 | $(d_{-}^{2})_{2}$ | 4 | $(d_{-},d_{+})_{2}$ | 11.189(3) | 11.2089 | 4.49(6)
59 | Ca | $3d^{2}$ | 1 | $(d_{-}^{2})_{2}$ | 3 | $(d_{-},d_{+})_{3}$ | 12.444(3) | 12.4874 | 9.35(6)
59 | Ca | $3d^{2}$ | 2 | $(d_{-}^{2})_{0}$ | 6 | $(d_{-},d_{+})_{1}$ | 13.656(3) | 13.7184 | 4.65(6)
60 | K | $3d$ | 1 | $(d_{-})_{3/2}$ | 2 | $(d_{+})_{5/2}$ | 11.750(8)b | 11.7861, 11.7510C | 6.49(6)
60 | K | $3p^{5}3d^{2}$ | 6 | $((p_{+}^{3})_{3/2},(d_{-}^{2})_{2})_{7/2}$ | 9 | $((p_{+}^{3})_{3/2},d_{-})_{9/2}$ | 13.200(3) | 13.2556 | 7.95(6)
|
arxiv-papers
| 2012-12-08T21:06:41 |
2024-09-04T02:49:39.049728
|
{
"license": "Public Domain",
"authors": "D. Osin, J.D. Gillaspy, J. Reader and Yu. Ralchenko",
"submitter": "Yuri Ralchenko",
"url": "https://arxiv.org/abs/1212.1835"
}
|
1212.1899
|
# Extensions of Sperner and Tucker’s lemma for manifolds
Oleg R. Musin This research is partially supported by NSF grant DMS - 1101688.
###### Abstract
The Sperner and Tucker lemmas are combinatorial analogous of the Brouwer and
Borsuk - Ulam theorems with many useful applications. These classic lemmas are
concerning labellings of triangulated discs and spheres. In this paper we show
that discs and spheres can be substituted by large classes of manifolds with
or without boundary.
Keywords: Sperner’s lemma, Tucker’s lemma, the Borsuk-Ulam theorem.
## 1 Introduction
Throughout this paper the symbol ${\mathbb{R}}^{d}$ denotes the Euclidean
space of dimension $d$. We denote by ${\mathbb{B}}^{d}$ the $d$-dimensional
ball and by ${\mathbb{S}}^{d}$ the $d$-dimensional sphere. If we consider
${\mathbb{S}}^{d}$ as the set of unit vectors $x$ in ${\mathbb{R}}^{d+1}$,
then points $x$ and $-x$ are called antipodal and the symmetry given by the
mapping $x\to-x$ is called the antipodality on ${\mathbb{S}}^{d}$.
### 1.1 Sperner’s lemma
Sperner’s lemma is a statement about labellings of triangulated simplices
($d$-balls). It is a discrete analog of the Brouwer fixed point theorem.
Figure 1: A 2-dimensional illustration of Sperner’s lemma
Let $S$ be a $d$-dimensional simplex with vertices $v_{1},\ldots,v_{d+1}$. Let
$T$ be a triangulation of $S$. Suppose that each vertex of $T$ is assigned a
unique label from the set $\\{1,2,\ldots,d+1\\}$. A labelling $L$ is called
Sperner’s if the vertices are labelled in such a way that a vertex of $T$
belonging to the interior of a face $F$ of $S$ can only be labelled by $k$ if
$v_{k}$ is on $S$.
###### Theorem 1.1.
(Sperner’s lemma [15]) Every Sperner labelling of a triangulation of a
$d$-dimensional simplex contains a cell labelled with a complete set of
labels: $\\{1,2,\ldots,d+1\\}$.
There are several extensions of this lemma. One of the most interesting is the
De Loera - Petersen - Su theorem. In the paper [4] they proved the Atanassov
conjecture [1].
###### Theorem 1.2.
(Polytopal Sperner’s lemma [4]) Let $P$ be a polytope in ${\mathbb{R}}^{d}$
with vertices $v_{1},\ldots,v_{n}$. Let $T$ be a triangulation of $P$. Let
$L:V(T)\to\\{1,2,\ldots,n\\}$ be a Sperner labelling. Then there are at least
$(n-d)$ fully-colored (i.e. with distinct labels) $d$-simplices of $T$.
Meunier [8] extended this theorem:
###### Theorem 1.3.
Let $P^{d}$ be a $d$-dimensional PL manifold embedded in ${\mathbb{R}}^{d}$
that has bondary $B$. Suppose $B$ has $n$ vertices $v_{1},\ldots,v_{n}$. Let
$T$ be a triangulation of $P$. Let $L:V(T)\to\\{1,2,\ldots,n\\}$ be a Sperner
labelling. Let $d_{i}$ denote the number of edges of $B$ which are connected
to $v_{i}$. Then there are at least $n+\lceil\min_{i}\\{d_{i}\\}/d\rceil-d-1$
fully-labelled $d$-simplices such that any pair of these fully-labelled
simplices receives two different labellings.
### 1.2 Tucker’s lemma
Figure 2: A 2-dimensional illustration of Tucker’s lemma
Let $T$ be some triangulation of the $d$-dimensional ball ${\mathbb{B}}^{d}$.
We call $T$ antipodally symmetric on the boundary if the set of simplices of
$T$ contained in the boundary of ${\mathbb{B}}^{d}={\mathbb{S}}^{d-1}$ is an
antipodally symmetric triangulation of ${\mathbb{S}}^{d-1}$, that is if
$s\subset{\mathbb{S}}^{d-1}$ is a simplex of $T$, then $-s$ is also a simplex
of $T$.
###### Theorem 1.4.
(Tucker’s lemma [16]) Let $T$ be a triangulation of ${\mathbb{B}}^{d}$ that
antipodally symmetric on the boundary. Let
$L:V(T)\to\\{+1,-1,+2,-2,\ldots,+d,-d\\}$
be a labelling of the vertices of $T$ that satisfies $L(-v)=-L(v)$ for every
vertex $v$ on the boundary. Then there exists an edge in $T$ that is
complementary, i.e. its two vertices are labelled by opposite numbers.
Consider also the following version of Tucker’s lemma:
###### Theorem 1.5.
Let $T$ be a centrally symmetric triangulation of the sphere
${\mathbb{S}}^{d}$. Let
$L:V(T)\to\\{+1,-1,+2,-2,\ldots,+d,-d\\}$
be an equivariant (or Tucker’s) labelling, i.e. $L(-v)=-L(v)$). Then there
exists a complementary edge.
Tucker’s lemma was extended by Ky Fan [5]:
###### Theorem 1.6.
Let $T$ be a centrally symmetric triangulation of the sphere
${\mathbb{S}}^{d}$. Suppose that each vertex $v$ of $T$ is assigned a label
$L(v)$ from $\\{\pm 1,\pm 2,\ldots,\pm n\\}$ in such a way that $L(-v)=-L(v)$.
Suppose this labelling does not have complementary edges. Then there are an
odd number of $d$-simplices of $T$ whose labels are of the form
$\\{k_{0},-k_{1},k_{2},\ldots,(-1)^{d}k_{d}\\}$, where $1\leq
k_{0}<k_{1}<\ldots<k_{d}\leq n$. In particular, $n\geq d+1$.
In this paper we consider extensions of the Sperner, De Loera - Petersen - Su,
Tucker and Fan theorems for manifolds. We show that for all cases $d$-balls
and spheres can be substituted by $d$-manifolds with or without boundary.
## 2 Preliminaries
Throughout this paper we consider manifolds that admit triangulations. The
class of such manifolds is called piecewise linear (PL) manifolds. Note that a
smooth manifold can be triangulated, therefore it is also a PL manifold.
However, there are topological manifolds that do not admit a triangulation.
A topological manifold is a topological space that resembles Euclidean space
near each point. More precisely, each point of a $d$-dimensional manifold has
a neighbourhood that is homeomorphic to the Euclidean space of dimension $d$.
A compact manifold without boundary is called closed. If a manifold contains
its own boundary, it is called a manifold with boundary.
Smooth manifolds (also called differentiable manifolds) are manifolds for
which overlapping charts “relate smoothly” to each other, meaning that the
inverse of one followed by the other is an infinitely differentiable map from
Euclidean space to itself.
$M$ is called a piecewise linear (PL) manifold if it is a topological manifold
together with a piecewise linear structure on it. Every PL manifold $M$ admits
a triangulation: that is, we can find a collection of simplices $T$ of
dimensions $0,1,\ldots,d$, such that (1) any face of a simplex belonging to
$T$ also belongs to $T$, (2) any nonempty intersection of any two simplices of
$T$ is a face of each, and (3) the union of the simplices of $T$ is $M$. (See
details in [3].)
Note that the circle is the only one-dimensional closed manifold. Closed
manifolds in two dimensions are completely classified. (See details and proofs
in [14].) An orientable two-manifold (surface) is the sphere or the connected
sum of $g$ tori, for $g\geq 1$. For any positive integer $n$, a distinct
nonorientable surface can be produced by replacing $n$ disks with Möbius
bands. In particular, replacing one disk with a Möbius band produces the real
projective plane and replacing two disks produces the Klein bottle. The
sphere, the $g$-holed tori, and this sequence of nonorientable surfaces form a
complete list of compact, boundaryless two-dimensional manifolds.
Example 2.1. The real projective plane, ${\mathbb{R}P}^{2}$, can be viewed as
the union of a Möbius band and a disc. The correspondent model of the Möbius
band is shown in Fig. 3. Note that this model cannot be embedded to
${\mathbb{R}}^{3}$.
Figure 3: Möbius band. Diametrically opposite points of the inner boundary
circle are to be identified. The outer circle is the boundary of the Möbius
band.
Let $T$ be a triangulation of a PL manifold $M$. Then $T$ is a simplicial
complex. The vertex set of $T$, denoted by $V(T)$ is the union of the vertex
sets of all simplices of $T$.
Given two triangulations $T_{1}$ and $T_{2}$ of two PL manifolds $M_{1}$ and
$M_{2}$. A simplicial map is a function $f:V(T_{1})\to V(T_{2})$ that maps the
vertices of $T_{1}$ to the vertices of $T_{2}$ and that has the property that
for any simplex (face) $s$ of $T_{1}$, the image set $f(s)$ is a face of
$T_{2}$.
Note that the original Brouwer proof of his fixed point theorem that is based
on the concept of the degree of a continuous mapping. Let $f:M_{1}\to M_{2}$
be a continuous map between two closed manifolds $M_{1}$ and $M_{2}$ of the
same dimension. Intuitively, the degree is a number that represents the number
of times that the domain manifold wraps around the range manifold under the
mapping. Then $\operatorname{deg_{2}}2(f)$ (the degree modulo 2) is 1 if this
number is odd and 0 otherwise.
It is well known that the degree of a continuous map $f$ of a closed manifold
to a manifold is a topological invariant modulo 2 (see, for instance, [10] and
[7, pp. 44–46]). Therefore, the degree of $f$ is odd if any generic point in
the range of the map has an odd number of preimages.
Now we define $\operatorname{deg_{2}}2(f)$ more rigorously. Let $T_{1}$ be a
triangulation of a closed $d$-dimensional PL manifold $M_{1}$. Suppose that
$T_{2}$ is a triangulation of a $d$-dimensional PL manifold $M_{2}$. (We do
not assume that $M_{2}$ is closed.) Let $f:V(T_{1})\to V(T_{2})$ be a
simplicial map. Consider any $d$-simplex $s$ of $T_{2}$. Denote by $m$ the
number of preimages of $s$ in $T_{1}$. Then $\operatorname{deg_{2}}2(f)=1$ if
$m$ is odd and $\operatorname{deg_{2}}2(f)=0$ if $m$ is even. Since the parity
of $m$ does not depend on $s$, the degree of map modulo 2 is well defined.
Thus, the degree of a continuous map of a closed manifold to a manifold is a
topological invariant modulo 2.
Let $f:M_{1}\to M_{2}$ be a continuous map between two manifolds $M_{i}$ with
$d_{1}:=\dim(M_{1})\geq d_{2}:=\dim(M_{2})$. Then for a point $y\in M_{2}$ the
map $f$ is called transversal to $y$ (or generic with respect to $y$) if there
are open sets $U_{i}\subset M_{i}$ such that $U_{2}$ contains $y$,
$U_{2}=f(U_{1})$ and $U_{1}=f^{-1}(U_{2})$. In the case
$M_{2}={\mathbb{R}}^{d_{2}}$ and $y=0$, where $0$ is the origin of
${\mathbb{R}}^{d_{2}}$, $f$ is called transversal to zero.
Let $M$ be a closed PL-manifold. A simplicial map $A:M\to M$ is called a free
involution if $A(A(x))=x$ and $A(x)\neq x$ for all $x\in M$. A triangulation
$T$ of $M$ is called antipodal or equivariant if $A:T\to T$ is a simplicial
map. Let us call a pair $(M,A)$, where $A$ is a free simplicial involution as
${\mathbb{Z}}_{2}$-manifold.
Example 2.2. It is clear that $({\mathbb{S}}^{d},A)$ with $A(x)=-x$ is a
${\mathbb{Z}}_{2}$-manifold. Suppose that $M$ can be represented as a
connected sum $N\\#N$, where $N$ is a closed PL manifold. Then admits a free
involution. Indeed, $M$ can be “centrally symmetric” embedded to
${\mathbb{R}}^{k}$ with some $k$ and the antipodal symmetry $x\to-x$ in
${\mathbb{R}}^{k}$ implies a free involution $T:M\to M$ [12, Corollary 1]. For
instance, orientable two-dimensional manifolds $M^{2}_{g}$ with even genus $g$
and non-orientable manifolds $P^{2}_{m}$ with even $m$, where $m$ is the
number of Möbius bands, are ${\mathbb{Z}}_{2}$-manifolds.
Figure 4: The double torus that is centrally symmetric embedded to
${\mathbb{R}}^{3}$.
Suppose that $M$ admits a free simplicial involution $A$. We say that a map
$f:M\to{\mathbb{R}}^{d}$ is antipodal (or equivariant) if $f(A(x))=-f(x)$.
## 3 Extensions of Sperner’s lemma for manifolds
A $d$-simplex $S$ where each corner is labelled between 1 and $d+1$ such that
all labels are used exactly once is called fully labelled. Suppose that points
are added in $S$, then it may be triangulated, i.e. subdivided into smaller
$d$-simplices such that none of the smaller simplices contain any points: all
the points are corners of smaller simplices. This subdivision may be done in
many ways.
Now label all the interior points according to the following rule: an interior
point that is on a facet of the simplex must be given one of the labels of one
of the corners of that facet. The result is called a Sperner labelling.
Note that in this definition an interior point of $S$ can be labelled by any
label. So Sperner’s constraint is only for the boundary of $S$ that is
homeomorphic to ${\mathbb{S}}^{d-1}$. Let us extend this definition to any
closed manifold.
Definition. Let $K$ be a closed $m$-dimensional PL manifold with vertices
$V=\\{v_{1},\ldots,v_{n}\\}$ and faces $\\{F_{i}\\}$ of dimension from 1 to
$m$. Let $T$ be a triangulation of $K$ such that for any face (that is a
simplex) $F_{i}$ it is a triangulation of $F_{i}$. Suppose that the vertices
of $T$ have a labelling satisfying the following conditions: each vertex
$v_{k}$ of $V$ is assigned a unique label from $\\{1,2,\ldots,n\\}$, and each
other vertex $v$ of $T$ belonging to a face $F_{i}$ with vertices
$V(F_{i}):=\\{v_{i_{1}},\ldots,v_{i_{\ell}}\\}$ from $V$ is assigned a label
of one of the vertices of $V(F_{i})$. Such a labelling is called a Sperner
labelling of $T$.
Definition. We say that a $d$-simplex is a fully labelled cell or simply a
full cell if all its labels are are distinct.
Let $T$ be a triangulation of a $d$-dimensional PL manifold. Let
$L:V(T)\to\\{1,\ldots,n\\}$ be a labelling of $T$. Let $Q$ be a set of
$(d+1)$-subsets of $\\{1,\ldots,n\\}$. We denote by $\mathop{\rm
fc}\nolimits(L,T,Q)$ the number of fully labelled cells that are labelled as
labels in $Q$. In the case $n=d+1$ we denote $\mathop{\rm
fc}\nolimits(L,T):=\mathop{\rm fc}\nolimits(L,T,\\{1,\ldots,d+1\\})$.
Let $P$ be a set of $n$ points $p_{1},\ldots,p_{n}$ in ${\mathbb{R}}^{d}$.
Denote by $S(P)$ the collection of all simplices spanned by vertices
$\\{p_{i_{1}},\ldots,p_{i_{k}}\\}$ with $1\leq k\leq d+1$. Consider a point
$x\in{\mathbb{R}}^{d}$ and the set $S_{x}(P)$ of all simplices from $S(P)$
which cover $x$. If no such simplices exist, we write $S_{x}(P)=\emptyset$.
Denote this set of simplices by $\mathop{\rm cov}\nolimits_{P}(x)$ or just by
$\mathop{\rm cov}\nolimits(x)$.
Example 3.1. Let $P$ be a pentagon, see Fig. 5. Then
$\mathop{\rm cov}\nolimits(p_{1})=(123)\cup(124)\cup(125);\;\mathop{\rm
cov}\nolimits(p_{2})=(135)\cup(145)\cup(235)\cup(245);\;$ $\mathop{\rm
cov}\nolimits(p_{3})=(134)\cup(234)\cup(345);\;\mathop{\rm
cov}\nolimits(O)=(124)\cup(134)\cup(135)\cup(235)\cup(245);\;$
Figure 5: Pebbles and $\mathop{\rm cov}\nolimits(x)$ for a pentagon
Definition. Let $P:=\\{p_{1},\ldots,p_{n}\\}$ be points in ${\mathbb{R}}^{d}$.
Let $T$ be a triangulation of a closed PL manifold $M$ of dimension $m$. Let
$L$ be an $n$-labelling of $T$, i.e. a labelling (map)
$L:V(T)\to\\{1,2,\ldots,n\\}$. If for $v\in V(T)$ we have $L(v)=i$, then set
$f_{L,P}(v):=p_{i}$. Therefore, $f_{L,P}$ is defined for all vertices of $T$,
and it uniquely defines a simplicial (piecewise linear) map
$f_{L,P}:M\to{\mathbb{R}}^{d}$.
###### Theorem 3.1.
Let $P:=\\{p_{1},\ldots,p_{n}\\}\subset{\mathbb{R}}^{d}$. Suppose
$x\in{\mathbb{R}}^{d}$ is such that $\mathop{\rm cov}\nolimits_{P}(x)$
consists of $d$-simplices. Let $M$ be a a closed PL $d$-dimensional manifold.
Then any $n$-labelling $L$ of a triangulation $T$ of $M$ must contain an even
number of full cells which are labelled as simplices in $\mathop{\rm
cov}\nolimits_{P}(x)$.
###### Proof.
Consider $f_{L,P}:T\to{\mathbb{R}}^{d}$. It is easy to see that
$\operatorname{deg_{2}}2(f_{L,P})=0$. Indeed, if $y\in{\mathbb{R}}^{d}$ lies
outside of the convex hull of $P$ in ${\mathbb{R}}^{d}$, then
$f^{-1}_{L,P}(y)=\emptyset$. Therefore, for any point $x$ in
${\mathbb{R}}^{d}$ which is a regular value of $f_{L,P}$, we have
$|f^{-1}_{L,P}(x)|\equiv|f^{-1}_{L,P}(y)|\equiv 0\pmod{2}$. Thus, the number
of full cells which are labelled as simplices in $\mathop{\rm
cov}\nolimits_{P}(x)$ is even. ∎
For the classical case $n=d+1$ we have the following result (also see [6]):
###### Corollary 3.1.
Let $T$ be a triangulation of a closed PL manifold $M^{d}$. Any
$(d+1)$-labelling of $T$ must contain an even number of full cells.
###### Corollary 3.2.
Let $M$ be a $d$-dimensional compact PL manifold with boundary $B$. Let $B$ be
PL homeomorphic to the boundary of a $d$-simplex (i.e.
$B\cong{\mathbb{S}}^{d-1}$) with vertices $v_{1},\ldots,v_{d+1}$. Then any
$(d+1)$-labelling $L$ of a triangulation $T$ of $M$ such that $L(v_{i})=i$ and
$L$ is a Sperner labelling on the boundary $B$ must contain an odd number of
full cells.
Figure 6: Sperner’s lemma for the Möbius band
###### Proof.
We prove this corollary by induction on $d$. It is clear for $d=1$. Let $S$
denote a $d$-simplex. Two manifolds $M$ and $S$ can be glued together along
$B$. We denote the new manifold by $N$. Then $N$ is a closed manifold.
Corollary 3.1 implies that any $(d+1)$-labelling of any triangulation of $N$
has an even number of full cells.
Let us add to the vertices of $T$ one more vertex $q$ that is an internal
point of $S$. Let $C=$cone$(T|_{B})$ be the cone triangulation of $S$ with
vertex $q$. (Here $T|_{B}$ denote the triangulation $T$ on $B$.) Actually, $C$
consists of simplices formed by the union of all segments connecting the
points of $B$ with $q$ and the boundary triangulation $T|_{B}$. Then we obtain
the triangulation $\tilde{T}:=T\cup C$ of $N$.
Consider the following labelling $\tilde{L}$ on $\tilde{T}$. Let
$\tilde{L}(v):=L(v)$ for all $v\in V(T)$ and $\tilde{L}(q):=1$. Since
$\tilde{T}$ is a triangulation of $N$, we have that the number of full cells
$\mathop{\rm fc}\nolimits(\tilde{L},\tilde{T})$ is even.
By induction the face $F=v_{2}\ldots v_{d+1}$ of $B$ has an odd number of full
cells. Then $\mathop{\rm fc}\nolimits(\tilde{L},C)=\mathop{\rm
fc}\nolimits(L,T|_{F},\\{2,\ldots,d+1\\})$ is odd. Note that
$\mathop{\rm fc}\nolimits(\tilde{L},\tilde{T})=\mathop{\rm
fc}\nolimits(L,T)+\mathop{\rm fc}\nolimits(\tilde{L},C).$
Thus $T$ must contain an odd number of full cells. ∎
Note that for the case when $M$ is a $d$-simplex Corollary 3.2 is Sperner’s
lemma.
Now we show how the De Loera - Peterson - Su theorem follows from Theorem 3.1.
###### Corollary 3.3.
Let $P$ be a convex polytope in ${\mathbb{R}}^{d}$ with vertices
$p_{1},\ldots,p_{n}$. Let $M$ be a compact $d$-dimensional PL manifold with
boundary $B$. Let $B$ be piecewise linearly homeomorphic to the boundary of
$P$. Suppose $x\in{\mathbb{R}}^{d}$ is such that $\mathop{\rm
cov}\nolimits_{P}(x)$ consists of $d$-simplices. Then any $n$-labelling $L$ of
a triangulation $T$ of $M$ that is a Sperner labelling on the boundary must
contain an odd number of full cells which are labelled as simplices in
$\mathop{\rm cov}\nolimits_{P}(x)$. In other words, $\mathop{\rm
fc}\nolimits(L,T,\mathop{\rm cov}\nolimits(x))$ is odd.
###### Proof.
This corollary can be proved by similar arguments as Corollary 3.2. Indeed,
two manifolds $M$ and $P$ can be glued together along $B$. We denote the new
manifold by $N$. Then $N$ is a closed manifold.
Let $C:=$cone$(T|_{B})$ be the cone triangulation of $P$ with vertex $q$,
where $q$ is an internal point of $P$. Then we have the triangulation
$\tilde{T}:=T\cup C$ of $N$.
Consider the following labelling $\tilde{L}$ on $\tilde{T}$. Let
$\tilde{L}(v):=L(v)$ for all $v\in V(T)$ and $\tilde{L}(q):=1$. Now we show
that $\mathop{\rm fc}\nolimits(\tilde{L},C,\mathop{\rm cov}\nolimits(x))$ is
odd.
Consider the line in ${\mathbb{R}}^{d}$ passes through points $p_{1}$ and $x$.
By assumptions, this line intersects the boundary $B$ of the polytope $P$ in
two points $p_{1}$ and $y$, where $y$ is an internal point of some
$(d-1)$-simplex of $T|_{B}$ with distinct labels $\ell_{1},\ldots,\ell_{d}$.
Therefore, $y$ lies on the face $F=v_{\ell_{1}}\ldots v_{\ell_{d}}$. By
induction $\mathop{\rm fc}\nolimits(L,T|_{F},\\{\ell_{1},\ldots,\ell_{d})$ is
odd. Note that $\tilde{L}$ on $C$ contains only one labelling
$(1\ell_{1}\ldots\ell_{d})$ from $\mathop{\rm cov}\nolimits(x)$. Then
$\mathop{\rm fc}\nolimits(\tilde{L},C,\mathop{\rm
cov}\nolimits(x))=\mathop{\rm
fc}\nolimits(\tilde{L},C,\\{1,\ell_{1},\ldots,\ell_{d}\\})=\mathop{\rm
fc}\nolimits(L,T|_{F},\\{\ell_{1},\ldots,\ell_{d}\\})=1\mathop{\rm(mod}\nolimits
2).$
We have
$\mathop{\rm fc}\nolimits(\tilde{L},\tilde{T},\mathop{\rm
cov}\nolimits(x))=\mathop{\rm fc}\nolimits(L,T,\mathop{\rm
cov}\nolimits(x))+\mathop{\rm fc}\nolimits(\tilde{L},C,\mathop{\rm
cov}\nolimits(x)),$
where $\mathop{\rm fc}\nolimits(\tilde{L},\tilde{T},\mathop{\rm
cov}\nolimits(x))$ is even and $\mathop{\rm
fc}\nolimits(\tilde{L},C,\mathop{\rm cov}\nolimits(x))$ is odd. Thus
$\mathop{\rm fc}\nolimits(L,T,\mathop{\rm cov}\nolimits(x))$ is odd. ∎
###### Corollary 3.4.
Let $P$ be a convex polytope in ${\mathbb{R}}^{d}$ with vertices
$p_{1},\ldots,p_{n}$. Let $M$ be a compact $d$-dimensional PL manifold with
boundary $B$. Let $B$ be PL homeomorphic to the boundary of $P$. Then any
$n$-labelling of a triangulation of $M$ that is a Sperner labelling on the
boundary contains at least $n-d$ full cells.
For the case $M=P$ this statement is the polytopal Sperner lemma [4, Th. 1].
###### Proof.
A proof of this corollary follows from another theorem from [4, Th. 4]: Any
convex polytope $P$ in ${\mathbb{R}}^{d}$ with $n$ vertices contains a pebble
set of size $n-d$. (A finite set of points (pebbles) in $P$ is called a pebble
set if each $d$-simplex of $P$ contains at most one pebble interior to
chambers.)
Consider a pebble set $\\{p_{i}\\}$ of size $n-d$. Then for $i\neq j$ we have
$\mathop{\rm cov}\nolimits(p_{i})\cap\mathop{\rm
cov}\nolimits(p_{j})=\emptyset$. Thus Corollary 3.3 guarantees that there are
at least $n-d$ full cells. ∎
Remark. In fact, Meunier’s proof of his extension of the polytopal Sperner
lemma (De Loera - Peterson - Su’s theorem) is not based on the “pebbles set
theorem.” It is an interesting problem to find an extension of Meunier’s
theorem for manifolds.
Example 3.2. Consider the case when $P$ is a pentagon. In $P$ there are three
pebbles $p_{1},p_{2},p_{3}$, see Fig. 5. So the polytopal Sperner’s lemma
(Theorem 1.2) and Corollary 3.4 implies that there are at least three fully
labelled triangles. Actually, this statement can be improved.
Note that there are 10 5-labellings for triangles. Five of them are
consecutive: (123), (234), (345), (451), (512) and five are non-consecutive.
In fact, $\mathop{\rm cov}\nolimits(O)$ consists of non-consecutive
labellings, see Example 3.1. Then Corollary 3.3 implies the following
statement:
Any Sperner 5-labelling of a triangulation $T$ of a pentagon $P$ must contain
at least three full cells. Moreover, at least one of them is not consecutive
labelled.
## 4 Extensions of Tucker’s lemma for manifolds
Definition. Let $M$ be a closed PL $d$-dimensional manifold with a free
simplicial involution $A:M\to M$. We say that a pair $(M,A)$ is a BUT (Borsuk-
Ulam Type) manifold if for any continuous $g:M\to{\mathbb{R}}^{d}$ there is a
point $x\in M$ such that $g(A(x))=g(x)$. Equivalently, if a continuous map
$f:M\to{\mathbb{R}}^{d}$ is antipodal, then the zeros set $Z_{f}:=f^{-1}(0)$
is not empty.
In [12], we found several equivalent necessary and sufficient conditions for
manifolds to be BUT. For instance, $M$ is a BUT manifold if and only if $M$
admits an antipodal continuous transversal to zeros map
$h:M\to{\mathbb{R}}^{d}$ with $|Z_{h}|=2\pmod{4}$.
Let $T$ be any equivariant triangulation of $M$. We say that
$L:V(T)\to\\{+1,-1,+2,-2,\ldots,+d,-d\\}$
is an equivariant (or Tucker’s) labelling if $L(A(v))=-L(v)$).
An edge $e$ in $T$ is called complementary if its two ends are labelled by
opposite numbers, i.e. if $e=uv$, then $L(v)=-L(u)$.
###### Theorem 4.1.
A closed PL $d$-dimensional manifold $M$ with a free simplicial involution $A$
is BUT if and only if for any equivariant labelling of any equivariant
triangulation $T$ of $M$ there exists a complementary edge.
For the case $M={\mathbb{S}}^{d}$ this is Tucker’s lemma.
###### Proof.
Let $e_{1},\ldots,e_{d}$ be an orthonormal basis of ${\mathbb{R}}^{d}$. Any
equivariant labelling $L$ of a triangulation $T$ of $M$ defines a simplicial
map $f_{L}:T\to C^{d}$, where $C^{d}$ is the crosspolytope in
${\mathbb{R}}^{d}$ with vertex set
$\\{e_{1},-e_{1},e_{2},-e_{2},\ldots,e_{d},-e_{d}\\}$, where for $v\in V(T)$,
$f_{L}(v)=e_{i}$ if $L(v)=i$ and $f_{L}(v)=-e_{i}$ if $L(v)=-i$. (See details
in [7, Sec. 2.3].) In other words, $f_{L}=f_{L,C^{d}}$ (see Section 3).
Note that any fully labelled simplex contains a complementary edge. Therefore,
if $L$ has no complementary edges, then $f_{L}:T\to{\mathbb{R}}^{d}$ has no
zeros.
The reverse implication can be proved by the same arguments as equivalence of
the Borsuk-Ulam theorem and Tucker’s lemma in [7, 2.3.2], i.e. if there is
continuous antipodal map $f:M\to{\mathbb{S}}^{d-1}$ (i.e. $Z_{f}=\emptyset$)
then $T$ and $L$ can be constructed with no complementary edges. (See also
Theorem 4.2.) ∎
Theorem 4.1 and [12, Theorem 2] immediately imply:
###### Corollary 4.1.
Let $M$ be a closed PL manifold with a free involution $A$. Then $M$ is a BUT
manifold if and only if there exist an equivariant triangulation $\Lambda$ of
$M$ and an equivariant labelling of $V(\Lambda)$ such that
$f_{L}:\Lambda\to{\mathbb{R}}^{d}$ is transversal to zeros and the number of
complementary edges is $4k+2$, where $k$ is integer.
###### Corollary 4.2.
Let $T$ be a triangulation of a PL-compact $d$-dimensional manifold $M$ with
boundary $B$ that is homeomorphic to ${\mathbb{S}}^{d-1}$. Assume $T$ is
antipodally symmetric on the boundary. Let
$L:V(T)\to\\{+1,-1,+2,-2,\ldots,+d,-d\\}$ be a labelling of the vertices of
$T$ which satisfies $L(-v)=-L(v)$ for all vertices $v$ in $B$. Then there is a
complementary edge in $T$.
###### Proof.
Consider two copies of $M$: $M_{+}$ and $M_{-}$, where for $M_{+}$ we take a
given labelling $L$ and for $M_{-}$ we take a labelling $\bar{L}=(-L)$, i.e.
$\bar{L}(v)=-L(v)$. Since $L$ is antipodal on the boundary
$B={\mathbb{S}}^{d-1}$ the connected sum $N:=M\\#M$ with a free involution
$I:N\to N$, where $I(M_{+})=M_{-}$, is well defined. [12, Corollary 1] implies
that $N$ is BUT. Thus, from Theorem 4.1 follows that there is a complementary
edge. ∎
Figure 7: Tucker’s lemma for the Möbius band
Now we extend Theorem 4.1 for $n$-labellings.
###### Theorem 4.2.
Let $P=\\{p_{1},-p_{1},\ldots,p_{n},-p_{n}\\}$ be a centrally symmetric set of
$2n$ points in ${\mathbb{R}}^{d}$. Let points in $P$ be equivariantly labelled
by $\\{+1,-1,+2,-2,\ldots,+n,-n\\}$. Let $M$ be a closed PL $d$-dimensional
manifold with a free involution. Then $M$ is a BUT manifold if and only if for
any equivariant triangulation $T$ of $M$ and for any equivariant labelling
$L:V(T)\to\\{+1,-1,+2,-2,\ldots,+n,-n\\}$ there exists a simplex $s$ in $T$
such that $0\in f_{L,P}(s)$.
###### Proof.
If $M$ is BUT, then $f_{L,P}$ has zeros, so there is a simplex $s$ as
required.
Suppose $M$ is not BUT. Then there is a continuous antipodal
$h:M\to{\mathbb{S}}^{d-1}$. Let $T$ be an equivariant triangulation of $M$.
Let $Q$ denote the boundary of the convex hull of $P$ in ${\mathbb{R}}^{d}$.
Without loss of generality we may assume that $h:M\to Q$ and for any vertex
$v\in V(T)$ the image $f(v)$ has only one closest vertex $p$ in $Q$. Then set
$L(v):=L(p)$. This labelling implies that $f_{L,P}$ is an antipodal simplicial
map from $T$ to $Q$. Thus, $0$ in ${\mathbb{R}}^{d}$ is not covered by
$f_{L,P}$, a contradiction. ∎
## 5 Radon partitions and Ky Fan’s lemma for manifolds
In this section we show that Ky Fan’s lemma follows from Theorem 4.2.
Radon’s theorem on convex sets states that any set $S$ of $d+2$ points in
${\mathbb{R}}^{d}$ can be partitioned into two (disjoint) sets $A$ and $B$
whose convex hulls intersect. Moreover, if rank$(S)=d$, then this partition is
unique.
The partition $S=A\bigcup B$ is called the Radon partition of $S$.
Breen [2] proved that if $S$ is a $(d+2)$-subset of the moment curve $C_{d}$
in ${\mathbb{R}}^{d}$, then $S=A\bigcup B$ is the Radon partition if and only
if $A$ and $B$ alternate along $C_{d}$. Actually, Breen’s theorem can be
extended for convex curves in ${\mathbb{R}}^{d}$.
We say that a curve $K$ in $\mathbb{R}^{d}$ is convex if for every hyperplane
$K$ intersects it at no more than $d$ points. It is well known that the moment
curve $C_{d}$ is convex. In [11, Sec. 3] we considered several other examples
of convex curves.
Definition. Let $K=\\{x(t)=(x_{1}(t),\ldots,x_{d}(t)):t\in[a,b]\\}$ be a curve
in $\mathbb{R}^{d}$. Let $S=\\{x(t_{1}),\ldots,x(t_{d+2})\\}$, where
$a<t_{1}<t_{2}<\ldots<t_{d+2}<b$. We say that $A$ and $B$ alternate along $K$
if $S=A\bigcup B$, where $A=\\{x(t_{1}),x(t_{3}),\ldots\\}$ and
$B=\\{x(t_{2}),x(t_{4}),\ldots\\}$.
###### Theorem 5.1.
A curve $K$ in $\mathbb{R}^{d}$ is convex if and only if for any
$(d+2)$-subset $S$ of $K$ its Radon partition sets $A$ and $B$ alternate along
$K$.
###### Proof.
Let $K$ be convex and $S=\\{x(t_{1}),\ldots,x(t_{d+2})\\}$ be a $(d+2)$-subset
of $K$. Let $A\bigcap B$ be the Radon partition of $S$. If $A$ and $B$ do not
alternate along $K$, there are at most $d$ points $P=\\{x(\tau_{i})\\}$ which
separate $A$ and $B$ on $K$. If $r=|P|<d$, we add to $P$ $d-r$ points
$x(\tau)$ with $\tau\in(a,t_{1})$. Then $P$ defines a hyperplane $H$ which
passes through the points in $P$. Clearly, $H$ separates $A$ and $B$ in
$\mathbb{R}^{d}$. Thus, $A\bigcap B$ cannot be the Radon partition of $S$, a
contradiction.
Suppose that for any $(d+2)$-subset $S$ of $K$, its Radon partition sets $A$
and $B$ alternate along $K$. If $K$ is not convex, then there is a hyperplane
$H$ which intersects $K$ at $r\geq d+1$ points. Therefore, $H$ separates $K$
into $r+1$ connected components $C_{1},\ldots,C_{r+1}$. Let
$S=\\{x(t_{1}),\ldots,x(t_{d+2})\\}$, where $x(t_{i})\in C_{i}$. Since $A$ and
$B$ which alternate along $K$ are separated by $H$, the partition $S=A\bigcup
B$ is not Radon’s - a contradiction. ∎
Definition. Let $P$ be a convex polytope in $\mathbb{R}^{d}$ with $2n$
centrally symmetric vertices $\\{p_{1},-p_{1},\ldots,p_{n},-p_{n}\\}$. We say
that $P$ is ACS (Alternating Centrally Symmetric) $(n,d)$-polytope if the set
of all simplices in $\mathop{\rm cov}\nolimits_{P}(0)$, that contain the
origin $0$ of $\mathbb{R}^{d}$ inside, consists of edges $(p_{i},-p_{i})$ and
$d$-simplices with vertices {$p_{k_{0}},-p_{k_{1}},\ldots,(-1)^{d}p_{k_{d}}$}
and {$-p_{k_{0}},p_{k_{1}},\ldots,(-1)^{d+1}p_{k_{d}}$}, where $1\leq
k_{0}<k_{1}<\ldots<k_{d}\leq n$.
###### Theorem 5.2.
For any integer $d\geq 2$ and $n\geq d$ there exists ASC $(n,d)$-polytope.
###### Proof.
Let $q_{1},\ldots,q_{n}$ be points on a convex curve $K$ in
$\mathbb{R}^{d-1}$. Let $p_{i}=(q_{i},1)\in\mathbb{R}^{d}$. Denote by $P(n,d)$
a convex polytope with vertices $\\{p_{1},-p_{1},\ldots,p_{n},-p_{n}\\}$.
Clearly, $0\in(-p_{i},p_{i})$. Let $\Delta$ be a simplex spaned by vertices of
$P(n,d)$. Let $V(\Delta)=A\bigcup(-B)$, where $A$ and $B$ are vertices with
$x_{d}=1$. It is easy to see that $0\in\Delta$ if and only if $\mathop{\rm
conv}\nolimits(A)\bigcap\mathop{\rm conv}\nolimits(B)=\emptyset$, i.e.
$S=A\bigcup B$ is the Radon partition of $S$. Then Theorem 5.1 implies that
$A$ and $B$ alternate along $K$. Thus, $P(n,d)$ is an ASC $(n,d)$-polytope. ∎
Let $P$ be an ASC $(n,d)$-polytope. If we apply Theorem 4.2 for $P$, then we
obtain the following theorem.
###### Theorem 5.3.
Let $M$ be a BUT $d$-dimensional manifold with a free involution $A$. Let $T$
be any equivariant triangulation of $M$. Let
$L:V(T)\to\\{+1,-1,+2,-2,\ldots,+n,-n\\}$ be an equivariant labelling. Suppose
that there are no complementary edges in $T$. Then there are an odd number of
$d$-simplices with labels in the form
$\\{k_{0},-k_{1},k_{2},\ldots,(-1)^{d}k_{d}\\}$, where $1\leq
k_{0}<k_{1}<\ldots<k_{d}\leq n$.
For the case $M={\mathbb{S}}^{d}$ this theorem is Ky Fan’s combinatorial lemma
[5]. Actually, it is a new proof of this lemma.
## 6 Sperner and Tucker’s type lemmas for the case $m\geq d$
Now we consider extensions of the polytopal Sperner and Tucker lemmas for the
case when $d\leq\dim{M}=m$. In this case, the set of fully-colored
$d$-simplices defines certain $(m-d)$-submanifold $S$ of $M$. A natural
extension of Theorem 3.1 is that $S$ is cobordant to zero. We also consider an
extension of the Tucker lemma.
An $m$-dimensional manifold $M$ is called null-cobordant (or cobordant to
zero) if there is a cobordism between $M$ and the empty manifold; in other
words, if $M$ is the entire boundary of some $(m+1)$-manifold. Equivalently,
its cobordism class is trivial.
###### Theorem 6.1.
Let $P=\\{p_{1},\ldots,p_{n}\\}$ be a set of points in ${\mathbb{R}}^{d}$.
Suppose $y\in{\mathbb{R}}^{d}$ is such that $\mathop{\rm cov}\nolimits_{P}(y)$
consists of $d$-simplices. Let $M$ be a a closed PL $m$-dimensional manifold
with $m\geq d$. Then for any $n$-labelling $L$ of a triangulation $T$ of $M$
the set $S:=f_{L,P}^{-1}(y)$ is a null-cobordant manifold of dimension $d-m$.
Note that for $d=m$ this theorem yields Theorem 3.1. In this case $S$ consists
of even number of points.
###### Proof.
Let
$W:=M\times[0,1],\;M_{0}:=M\times\\{0\\}\mbox{ and }M_{1}:=M\times\\{1\\}.$
Let $f_{0}:=f_{L,P}:M_{0}\to{\mathbb{R}}^{d}$. Let us fix a point $q\neq y$ in
${\mathbb{R}}^{d}$ and set $f_{1}(x)=q$ for all $x\in M_{1}$.
Note that $f_{0}$ is transversal to $y$ and $f_{1}^{-1}(y)$ is empty. Let
$F(x,t):=(1-t)f_{0}(x)+tf_{1}(x)$
Then $F:W\to{\mathbb{R}}^{d}$ is transversal to $y$ with $F|_{M_{0}}=f_{0}$
and $F|_{M_{1}}=f_{1}$. Therefore, $Z_{F}:=F^{-1}(y)$ is a manifold of
dimension $(m+1-d)$.
Denote $Z_{i}:=Z_{F}\bigcap M_{i}=f_{i}^{-1}(y),\,i=0,1.$ It is clear that
$Z_{0}=S$ and $Z_{1}$ is empty. Thus, $Z_{0}$ is the boundary of $Z_{F}$ and
so it is a null-cobordant $(m-d)$-dimensional manifold. ∎
Now we extend the class of BUT manifolds.
Definition. We say that a closed PL-free $m$-dimensional
${\mathbb{Z}}_{2}$-manifold $(M,A)$ is a $\mathop{\rm BUT}\nolimits_{m,d}$ if
for any continuous $g:M\to{\mathbb{R}}^{d}$ there is a point $x\in M$ such
that $g(A(x))=g(x)$. Equivalently, if a continuous map
$f:M\to{\mathbb{R}}^{d}$ is antipodal, then the zeros set $Z_{f}:=f^{-1}(0)$
is not empty.
We obviously have
$\mathop{\rm BUT}\nolimits=\mathop{\rm BUT}\nolimits_{m,m}\subset\mathop{\rm
BUT}\nolimits_{m,m-1}\subset\ldots\subset\mathop{\rm BUT}\nolimits_{m,1}.$
Note that in our paper [12] we found a sufficient condition for $(M,\Lambda)$
to be a BUTm,d, see [12, Corollary 3].
Let $T$ be an antipodal triangulation of $M$ Any equivariant labelling
$L:V(T)\to\\{+1,-1,+2,-2,\ldots,+d,-d\\}$ defines a simplicial map $f_{L}:T\to
C^{d}$, where $C^{d}$ is the crosspolytope in ${\mathbb{R}}^{d}$ (see Section
4). It is easy to see that if $L$ has no complementary edges, then
$f_{L}:T\to{\mathbb{R}}^{d}$ has no zeros. It implies the following theorem.
###### Theorem 6.2.
Let $m\geq d$. Let $T$ be any equivariant triangulation of a $\mathop{\rm
BUT}\nolimits_{m,d}$ manifold $(M,A)$. Let
$L:V(A)\to\\{+1,-1,+2,-2,\ldots,+d,-d\\}$ be any equivariant labelling of $T$.
Then there exists a complementary edge in $T$.
This theorem is an extension of Theorem 4.1. When $m\geq d$, it is not hard to
extend other theorems and corollaries from Sections 4 and 5.
Acknowledgment. I wish to thank Arseniy Akopyan and Frédéric Meunier for
helpful discussions and comments.
## References
* [1] K. T. Atanassov, On Sperner’s lemma, Studia Sci. Math. Hungar., 32 (1996), 71-74.
* [2] M. Breen, Primitive Radon partitions for cyclic polytopes, Isr. J. Math., 15 (1973), 156-157.
* [3] J. L. Bryant, Piecewise linear topology, Handbook of geometric topology, 219-259, North-Holland, Amsterdam, 2002.
* [4] J. A. De Loera, E. Peterson, and F. E. Su, A Polytopal Generalization of Sperner’s Lemma, J. of Combin. Theory Ser. A, 100 (2002), 1-26.
* [5] K. Fan, A generalization of Tucker’s combinatorial lemma with topological applications. Ann. of Math., 56 (1952), 431-437.
* [6] K. Fan, Simplicial maps from an orientable n-pseudomanifold into ${\bf S}^{m}$ with the octahedral triangulation, J. of Combinatorial Theory, 2 (1967), 588-602.
* [7] J. Matoušek, Using the Borsuk-Ulam theorem, Springer-Verlag, Berlin, 2003.
* [8] F. Meunier, Sperner labellings: a combinatorial approach, J. of Combin. Theory Ser. A, 113 (2006), 1462-1475.
* [9] M. D. Meyerson and A. H. Wright, A new and constructive proof of the Borsuk-Ulam theorem, Proc. Amer. Math. Soc., 73 (1979), 134-136.
* [10] J. W. Milnor, Topology from the differentiable viewpoint, The University Press of Virginia, Charlottesville, Virginia, 1969.
* [11] O. R. Musin, Chebyshev systems and zeros of a function on a convex curve, Proc. Steklov Inst. of Math., 221 (1998), 236-246, arXiv:0903.1908.
* [12] O. R. Musin, Borsuk-Ulam type theorems for manifolds, Proc. Amer. Math. Soc. 140 (2012), 2551-2560.
* [13] T. Prescott and F. E. Su. A constructive proof of Ky Fan’s generalization of Tucker’s lemma, J. Combin. Theory Ser. A, 111 (2005), 257-265.
* [14] H. Seifert and W. Threlfall, Lehrbuch der Topologie, Teubner, 1934; Chelsea, 1947; translated as A Textbook of Topology, Academic Press, 1980.
* [15] E. Sperner, Neuer Beweis für die Invarianz der Dimensionszahl und des Gebietes, Abh. Math. Sem. Univ. Hamburg 6 (1928), 265-272.
* [16] A. W. Tucker, Some topological properties of the disk and sphere. In: Proc. of the First Canadian Math. Congress, Montreal, 285-309, 1945.
O. R. Musin
Department of Mathematics, University of Texas at Brownsville, One West
University Boulevard, Brownsville, TX, 78520
and
IITP RAS, Bolshoy Karetny per. 19, Moscow, 127994, Russia
E-mail address: [email protected]
|
arxiv-papers
| 2012-12-09T16:36:00 |
2024-09-04T02:49:39.062967
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Oleg R. Musin",
"submitter": "Oleg Musin",
"url": "https://arxiv.org/abs/1212.1899"
}
|
1212.1941
|
11institutetext: Moscow Institute of Physics and Technology
# Amortized communication complexity of an equality predicate.(Beta version)
Vladimir Nikishkin
(2012)
###### Abstract
We study the communication complexity of the direct sum of independent copies
of the equality predicate. We prove that the probabilistic communication
complexity of this problem is equal to $O(N)$; computational complexity of the
proposed protocol is polynomial in size of inputs. Our protocol improves the
result achieved in [11]. Our construction is based on two techniques: Nisan’s
pseudorandom generator [4] and Smith’s string synchronization algorithm [9].
## 1 Introduction
In this paper we study amortized communication complexity of the equality
predicate. We deal with the classic model of communication complexity with two
participants (Alice and Bob), who want to compute some function of the data
distributed between the participants. Alice and Bob can talk to each other via
a communication channel. We measure the number of bits that must be
transmitted between Alice and Bob to achieve the goal.
More specifically, let $f:\\{0,1\\}^{n}\times\\{0,1\\}^{n}\to\\{0,1\\}$ be a
function of two arguments. We assume that Alice is given the value of $x$, Bob
is given the value of $y$, and Alice and Bob communicate with each other to
compute the value $f(x,y)$. We denote by $C(f)$ deterministic communication
complexity of function $f$, i.e., the minimal number of bits that should send
by Alice and Bob to each other to get $f(x,y)$.
Further, let us denote by $f^{N}$ the direct sum of $N$ independent copies of
the initial function $f$. More formally, the two arguments of $f$ are an
$N$-tuples of values $(x_{1},\ldots,x_{N})$ and $N$-tuple of values
$(y_{1},\ldots,y_{N})$, and
$f^{N}(x_{1},\ldots,x_{N},y_{1},\ldots,y_{N})=(f(x_{1},y_{1}),\ldots,f(x_{N},y_{N})).$
We assume that Alice is given all values of $x_{i}$, and Bob is given all
values of $y_{i}$. Now ALice and Bob need to compute $f^{N}$, i.e., to get all
the values $f(x_{i},y_{i})$ for $i=1,\ldots,N$. Amortized communication
complexity of $f$ is defined as
$\mathrm{AC}(f)=\limsup\limits_{N\to\infty}\frac{f^{N}}{N}.$
It was proven in [11] that $AC(f)=\Omega(\sqrt{C(f)}-\log{n})$.
In a similar way, amortized communication complexity can be defined for
probabilistic version of communication complexity. In general, the properties
of amortized randomized communication complexity remains not well understood,
though several nontrivial particular examples are known. For instance, in [11]
it was proven that amortized randomized complexity of the _equality predicate_
is only $O(1)$, while it is known that for one individual predicate
$\mathrm{EQ}:\\{0,1\\}^{n}\times\\{0,1\\}^{n}\to\\{0,1\\}$
$\mathrm{EQ}(x,y)=\left\\{\begin{array}[]{rl}1,&\mbox{if }x=y,\\\ 0,&\mbox{if
}x\not=y\end{array}\right.$
randomized communication complexity is equal to $\Theta(\log n)$.
In our paper we construct a new randomized communication protocol for the
direct sum of the equality predicate. Our protocol is computationally
effective, i.e., it only requires polynomial computations for Alice and Bob.
Our construction improves the result from [11] in two respects. First of all,
we get a slightly better bound for the probability of error. And, second, our
protocol has a ‘‘modular’’ structure; it consist of several independent
gadgets, which make the construction more flexible. We hope that the same
technique can be applied to other problems of communication complexity.
Our principal result can be formulated as follows:
Main theorem. Probabilistic communication complexity (for the private coin
model) of a direct sum of $N$ equality predicates is equal to $O(N)$, with
probability of the error $P_{err}\leq O(2^{-c\frac{N}{\log^{2}N}})$. Moreover,
we explicitly construct a communication protocol that achieves this
communication complexity and requires only polynomial time computations on
Alice’s and Bob’s sides.
In our construction, we use several classic tools (N. Nisan’s pseudorandom
generator, BCH codes, deterministic synchronization protocol by A. Orlitsky)
and one relatively new construction (A.Smith’s probabilistic synchronization
protocol).
### 1.1 Model
We use three standard models of communication complexity: deterministic
communication protocols, randomized communication protocols with public random
bits, and randomized communication protocols with private random bits, see
Nisan’s and Kushlevitz’s textbook [1]. We done communication complexities for
these three models by $C_{det}$, $C_{pub}$, and $C_{priv}$ respectively.
Our principal construction is in a sense, an explicit and effective
implementation of the following theorem for some particular communication
protocol.
###### Theorem 1.1 (Comm. Compl. [1])
Let $f:\\{0,1\\}^{n}\times\\{0,1\\}^{n}\rightarrow\\{0,1\\}$ be a function of
two arguments. For every $\delta>0$ and every $\varepsilon>0$, it holds
$C_{priv}^{\varepsilon+\delta}(f)<C_{pub}^{\varepsilon}(f)+O(\log
n+\log\delta^{-1}).$
### 1.2 Our main problem
Equality predicate ($EQ_{n}$). First of all, we remind the following classic
problem of communication complexity. Alice and Bob each hold an n-bit string,
$x$ and $y$ respectively. They want to know whether $x=y$. Formally, they want
to compute the value of the predicate
$EQ_{n}(x,y)=\left\\{\begin{array}[]{rl}1,&\mbox{if }x=y,\\\ 0,&\mbox{if
}x\not=y.\end{array}\right.$
Direct sum of equality functions ($EQ_{n}^{N}$). Both Alice and Bob hold some
array of $N$ of $n$-bits blocks. Formally, inputs of Alice and Bob are bit
strings of length $n\cdot N$. But it is more instructive to represent the
input of Alice as $x=x^{1}\ldots x^{N}$ and the input of Bob as $y=y^{1}\ldots
y^{N}$, where $x_{i},y_{i}\in\\{0,1\\}^{n}$ for each $i=1,\ldots,N$. Then we
define $EQ_{n}^{N}$ as $EQ_{n}^{N}(x,y)=z\in\\{0,1\\}^{N}$, where each $i$-th
bit of $z$ is equal to $1$ iff $x^{i}=y^{i}$. Intuitively this means that
Alice and Bob wants to compute the value of the predicate $EQ_{n}$ for $N$
independent pairs of inputs $(x^{i},y^{i})$.
Communication complexity of the function $EQ_{n}^{N}$ is the main subject of
our paper. More precisely, we want to estimate the probabilistic communication
complexity of this function in the model with private sources of randomness.
## 2 The known results
### 2.1 Complexity of $EQ_{n}$ for different types of communication
protocols.
The predicate $EQ_{n}$ is pretty well studied, and its communication
complexity are well understood. Let us remind three different communication
protocols for this predicate.
##### 2.1.0.1 Deterministic model.
It is known that $C_{det}(EQ_{n})=n+1.$ The bound is achieved by a trivial
protocol: Alice transmits her string $x$ to Bob, Bob compares the two strings
$x$ and $y$ and sends back one-bit response, $1$ if the strings are equal and
$0$ otherwise. From the standard technique of fooling sets it follows that
this bound is tight, i.e., there is no protocols with communication complexity
less than $n+1$.
##### 2.1.0.2 Private coin model
For the randomized communication complexity with private sources of randomness
$C_{priv}^{\varepsilon}(EQ_{n})=O(\log\frac{n}{\varepsilon}).$ This bound is
achieved by several classic communication protocols. In what follows we
describe one of them. Alice and Bob view their inputs $x$ and $y$ as
$n$-digits binary representations of integers (between $0$ and $2^{n}-1$ ).
Alice chooses a prime number $p$ at random among the first $(n/\varepsilon)$
primes. She sends to Bob both $p$ and $x\mod p$. Bob verifies whether $x\mod
p=y\mod p$. If $x$ and $y$ are equal to each other modulo $p$, then Bob
returns outputs $1$, otherwise he returns $0$.
If $x=y$, then this protocol always return the correct result. If $x\not=y$,
then the difference $(x-y)$ has at most $n$ prime factors; hence, the protocol
returns the wrong answer with probability at most $\varepsilon$.
##### 2.1.0.3 Public coin model
For the randomized communication complexity with public sources of randomness
$C_{pub}^{\varepsilon}(EQ_{n})=O(\log\frac{1}{\varepsilon}).$
This bound for the communication complexity is achieved by the following
protocol. Alice and Bob jointly choose a random $n$-bit string $n$. Then Alice
computes the inner product $b=\langle x,r\rangle$ and transmits the result (a
single bit) to Bob. Bob checks whether $b=\langle y,r\rangle$ and outputs
"equal" if so and "not equal" otherwise. Obviously, if $x=y$, then the output
is always "equal." On the other hand, if $x\neq y$, then by the properties of
the inner product, $Pr[\langle x,r\rangle\neq\langle y,r\rangle]=\frac{1}{2}$.
Thus, Bob outputs "not equal" with probability $\frac{1}{2}$. To decrease the
probability to get the wrong answer, Alice and Bob should repeat these
procedure several times with several independently chosen random strings $r$.
If Alice and Bob repeat (in parallel or sequentially) $l$ times the discribed
procedure, then the probability that $\langle x,r\rangle\neq\langle
y,r\rangle$ for all $r_{1},\ldots,r_{l}$ is equal to $2^{-l}$. So, for
$l=\lceil\log 1/\varepsilon\rceil$ we reduce the probability to get an error
to $\varepsilon$, while communication complexity is $O(\log 1/\varepsilon)$.
### 2.2 Trivial generalizations for $EQ_{n}^{N}$
The protocols from the previous section can be easily adapted to get some
protocols for the direct sum of $N$ copies of $EQ_{n}$, i.e., for the function
$EQ_{n}^{N}$.
##### 2.2.0.1 Adaptation of the protocol from paragraph 2.1.0.2
We run the protocol independently for each pair of blocks$(x_{i},y_{i})$. The
probability to get a wrong answer _for at least one pair of blocks_ must be
bounded by $\varepsilon$. To this end we need to reduce the probability of an
error for each of $N$ pairs of blocks to be less than
$\epsilon^{\prime}=\varepsilon/N$. This results in communication complexity
$O(N(\log(n/\varepsilon^{\prime})))=O(N(\log n+\log N+\log 1/\varepsilon)))$.
Thus, from the trivial adaptation of the protocol from paragraph 2.1.0.2 we
get
$C_{priv}^{\varepsilon}(\mathrm{EQ}^{N})=O(N(\log n+\log N+\log
1/\varepsilon)))$
##### 2.2.0.2 Adaptation of the protocol from paragraph 2.1.0.3
We run the protocol from section 2.1.0.3 for each pair of blocks
$(x_{i},y_{i})$ independently. To guarantee than the total probability of the
error is bounded by $\varepsilon$, we need to reduce the probabilities of
errors for each pair of blocks to $\epsilon^{\prime}=\epsilon/N$. Then we get
$C_{pub}^{\epsilon}(\mathrm{EQ}^{N})=O(N(\log N+\log 1/\epsilon)))$
##### 2.2.0.3 From public to private randomness
The last protocol above can be transformed into a protocol with private
sources of randomness. Indeed, from theorem 1.1 we get immediately
$C_{priv}=O(N\cdot\log N+\log{\frac{1}{\epsilon}}+\log(n\cdot
N)+\log{\frac{1}{\delta}})=$ $=O(N\cdot\log N+\log n+\log
N+\log{\frac{1}{\epsilon}})=O(N\cdot\log N)$
Note that this communication protocol requires exponential computational
complexity (at least for the standard proof of theorem 1.1).
Can we reduce the obtained (rather trivial) bound $O(N\cdot\log N)$, hopefully
to $O(N)$? Can we achieve this bound with a communication protocol that
requires only poly-time computations? The answers both these questions are
positive. Construction of such a communication protocol is the main result of
this paper. Loosely speaking, we plan to do it in two steps. At the first
step, we construct more effective communication protocol for communication
model with public randomness (this part of our construction is based on ideas
of A. Smith). At the second step, we reduce this protocol with a public source
randomness to a protocol with private randomness. In some sense, this idea is
similar to the usual proof of theorem 1.1: we substitute the sequence of
random bits (shared by Alice and Bob) by a sequence of _pseudorandom_ bits,
which can be obtained as an output of a pseudo-random bits generator. A random
seed of this generator is rather short. So, one of participants can choose it
at random and then send to another participants. E.g., Alice choses a random
seeds and send it to Bob; then ALice and Bob apply the pseudo-random bits
generator to this same seed, and then both participants share the same long
string of pseudo-random bits. The sharp difference between our construction
and the standard general proof of theorem 1.1 is that we use an explicit and
effectively computable generator (the generator of N. Nisan).
Before we explain details of our construction, we remind the technical tools
used in our proof.
## 3 The tools used in our construction
### 3.1 Pseudorandom number generator
In our construction we need a pseudo-random generator that fools tests with a
bounded memory. Technically, we assume that a generator is a mapping
$G:\\{0,1\\}^{m}\rightarrow\\{0,1\\}^{n}$, and a test is a randomized Turing
machine with working space of some size $S$.
Definition. A function $G:\\{0,1\\}^{m}\rightarrow\\{0,1\\}^{n}$ is called a
pseudorandom generator is $\varepsilon$-robust for tests with space $S$, if
for every _statistical test_ $A$ with $S$ bits of working space
$|Pr_{y\in_{r}\\{0,1\\}^{n}}[A\mbox{ accepts
}y]-Pr_{y\in_{r}\\{0,1\\}^{m}}[A\mbox{ accepts }G(x)]|<\varepsilon.$
By _statistical test_ we mean the following construction.
Definition A statistical test with space $S(n)$ is a deterministic Turing
machine $M$ with three tapes: a working tape of size space $S(n)$, an
auxiliary read-only tape with some binary strings $a=(a_{1},...a_{n},...)$ (an
advice string), and a one-way input tape with an $n$-bits input $x$ (the
reading head on the input tape can move from the left to the right but cannot
move back to the left). We always assume that the length of $a$ should not be
greater than $exp(S(n))$. This machine returns input $1$ or input $0$. We
denote this result $M_{a}(x)$. Informally the output means that test
accepts/rejects $x$ given an advice string $a$.
Nisan suggested in [4] an explicit construction of a pseudo-random generator
that fools tests with small enough memory. This result of Nisan can be
formulated as follows.
###### Theorem 3.1
There exists a constant $c>0$ such that for any $R$ and $S$ there exists a
pseudorandom generator $G:cS\log R\to R$ (computable in time $poly(R)$) that
is $2^{-S}$-robust for statistical tests with $S$ bits of working space.
In section 4.3.3 we construct some statistical test, which verifies that a
(pseudo)random string $x$ is suitable for our communication protocol. Then, we
use the standard argument: our protocol with high probability returns the
correct answer when it runs with truly random public bits; futher, the
generator of Nisan fools our test; hence, given In section 4.3.3 we construct
some statistical test that tests that a random string is suitable for our
communication protocol. Then, we use the standard argument: the protocol with
high probability returns the correct answer if runs on truly random public
bits; futher, the generator of Nisan fools our test; hence, given pseudo-
random bits instead of truly random ones, the communication protocol must also
returns the correct answer with hight probability.
### 3.2 BCH codes
We also use in our construction the classic BCH-codes, see [12]. We do not
employ any specific properties of the construction of the BCH codes. We use
only the fact that $\forall m>3,t<2^{m-1}$ exists an explicit construction of
a linear code with parameters $[n,k,d]$ such that codeword length $n=2^{m}-1$,
the number of checksum bits is $n-k\leq mt$, and the minimal distance between
of the code is $d\geq 2t+1$. We also use the fact that the BCH codes can be
decoded efficiently (by Berlekamp-Messy algorithm), [13].
The BCH construction is not explicitly used in the article. Still, Orlitzky’s
construction 3.3 we do utilize, needs an error-correcting code, which is not
explicitly given in Orlitzky’s article. The BCH code is fits in his
construction and we are going to use it when refering to Orlitzky’s
construction.
### 3.3 Strings synchronization protocols
In our communication protocol we will need to solve the following auxiliary
problem. Let Alice and Bob each hold an $n$-bits string, $A$ and $B$
respectively. We assume that $A$ and $B$ differ from eahc other in at most $e$
positions. Alice and Bob want to exchange their inputs, i.e., Alice should get
string $B$, and Bob should get string $A$. We will call this problem by
_string synchronization problem_ (Alice and bob want to synchronize their
inputs).
Orlitsky suggested in [3] a deterministic communication protocol for the
problem of synchronization of a pair $n$-bits strings at the Hamming distance
at most $e$. Communication complexity of this protocol is $O(e\log{n})$. All
computations of Alice and Bob in this protocol run in polynomial time.
More formally, the theorem (formulation taken from [2]) looks the following:
###### Theorem 3.2
Given an error-corrcting code with parameters $(\alpha.R(\alpha))$ which
satisfies the following conditions:
1. 1.
It is linear.
2. 2.
An effective decoding algorithm exists.
A one-round communication protocol with communication complexity
$C=(1-R(\alpha))\cdot n$ can be constructed. Computational complexity of such
protocol is polynomial.
If using the BCH code(noted in section 3.2), the communication complexity of
this protocol is: $O(e\log{n})$
The protocol of Orlitsky makes sense if the distance $e$ between strings is
very small. In case $e=\Omega(n)$, communication complexity of the protocol of
Orlitsky is worse than the trivial bound $2n$.
Adam Smith suggested in [9] a randomized communication protocol for the
problem of strings synchronization, with asymptotically optimal bound for
communication complexity for the case $e=const\cdot n$. More precisely, Smith
proved that for every $\delta=\delta(n)=\Omega(\frac{\log\log n}{\sqrt{\log
n}})$ there exists an explicit family of communication protocols (with private
sources of randomness) that solve the problem for synchronization of $n$ bit
strings at that differ in at most $e$ positions, with communication complexity
$n(H(\frac{e}{n})+\delta)$ and error
$\varepsilon=2^{-\Omega(\frac{\delta^{3}n}{\log n})}$, where
$H(p)=p\log_{2}\frac{1}{p}+(1-p)\log\frac{1}{1-p}.$ Algorithms of Alice and
Bob in this protocol run in polynomial time.
## 4 The main result
### 4.1 Formulation
By $EQ_{n}$ we denote an equality predicate problem, by $EQ_{n}^{N}$ we denote
a direct sum of such problems.
###### Theorem 4.1
Probabilistic communication complexity (for the private coin model) of a
$EQ^{N}_{n}$ is equal to $O(N)$, with an error probability $P_{err}\leq
O(2^{-c\frac{N}{\log^{2}N}})$ if n < N. Moreover, there exists a protocol with
the required communication complexity and only polynomial time computations
for Alice and Bob.
### 4.2 Overview of the protocol
Our protocol runs as follows. First of all, Alice generates a string of truly
random bits of length $O(N)$ and send this string to Bob. They both generate
pseudo-random bits from this seed. In what follows, Alice and Bob use this
long string of pseudo-random bits.
Then, Alice and Bob iteratively calculate "checksums" (inner products with
pseudo-random string) for their $n$-bits blocks and synchronize strings of
resulting checksums using the probablilistic or the deterministic protocol
from section 3.3. As soon as some pair of non-equal blocks $X^{i}$, $Y^{i}$ is
revealed (if some checksums for these blocks are different), Alice and Bob
withdraw these blocks from the list of their bit strings and never test them
again. Thus, on each next iteration the fraction of non-equal pairs of blocks
(that are not discovered yet) becomes less and less.
On each next iteration, we make the length of checksums longer and longer, so
for each pair of non-equal blocks the probability to be discovered becomes
closer and closer to $1$. Hence, the fraction of (non-discovered) pairs of
non-equal blocks gradually reduces, and only pairs of equal blocks remain
untouched at their places. This means that on each next iteration the Hamming
distance between arrays of checksums (obtained by Alice and Bob respectively)
becomes less and less.
On each iteration Alice and Bob need to exchange the checksums computed for
their blocks of bits (inner products with the same pseudo-random bits). For
several first iterations (technically, for $\log\log N$ iterations) we use the
randomized synchronization protocol by Smith. Then we switch to the
deterministic protocol by Orlitsky.
In what follows we explain this protocol in more detail.
#### 4.2.1 Generation stage
Alice generates
$r=\log{(n\cdot N)^{4}}\cdot\log(2^{\frac{N}{\log{(n\cdot N)}}})=O(N)$
random bits and sends them to Bob. Then Alice and Bob apply Nisan’s
pseudorandom generator from section 3.1 and get $R=n^{2}N^{2}$ pseudo-random
bits. The length of the seed $r$ is chosen so that the generator is
$\varepsilon$-robust against tests with working space of size
$\frac{N}{\log{(n\cdot N)}}$.
#### 4.2.2 Probabilistic synchronization stage, steps $i=1,\ldots,\log\log
N$)
Synchronization protocols we use expect to know the distance between strings
in advance. As we may not know an initial distance between X and Y, we will
add N dummy, equal blocks to X and Y. This will guarantee the share of non
equal pairs to be less then 0.5. This is a coarse trick, but it will not
affect the asymptotic complexity of our protocol.
We repeat $\log\log N$ times the following procedure. We let
$\lambda_{i}=\frac{2^{i}}{\log N}$. Alice and Bob calculate checksums of
length $\lambda_{i}$ for each of their blocks (that are not yet proven to be
different). The checksum for each block consists of inner products modulo $2$
between this block and a new portion of pseudorandom bits generated on the
previous stage. Thus, the resulting checksums (for Alice and Bob) consists of
$\lambda_{i}N$ bits.
Then Alice and Bob exchange their checksums using the randomized protocol of
strings synchronization; when we apply this protocol, we assume that Alice’s
and Bob’s checksums differ from each other in a fraction at most $2^{-i}$.
When the checksums are exchanged, Alice and Bob remove from their lists the
blocks $X^{i}$, $Y^{i}$ whose checksums are not identical.
Note that for a pair of equal blocks $X^{i}$, $Y^{i}$, the checksums are
always equal. If blocks are not equal to each other, the chance to get all
equal checksums is about $2^{-\lambda_{i}}$ (this probability is not _exactly_
$2^{-\lambda_{i}}$ since Alice and Bob use not random but pseudo-random bits
to compute the inner products).
Typically, on each step the number of non-discovered pairs of non-equal blocks
$X^{i}$, $Y^{i}$ becomes more than twice less. We say that the $i$-th step of
the described procedure _fails_ , if at this stage Alice and Bob discover less
than $50\%$ of the pairs of non-equal blocks $X^{i}$, $Y^{i}$ (less than a
half of all pairs of non-equal blocks that was not discovered earlier). If at
least one step fails, we cannot guarantee correctness of the result of the
protocol. If no steps fail, then on each $i$-th step the arrays of checksums
of Alice and Bob differ from each other in a fraction at most $1/2^{i}$ of all
computed inner products.
Communication complexity of this stage is the sum of communication
complexities of copies of Smith’s protocol run for each step
$i=1,\ldots,\log\log N$:
$\sum_{i=1}^{\log{\log{N}}}H(1/2^{i})\lambda_{i}N=O(N).$
The last equation follows from the choice of $\lambda_{i}$ and the asymptotic
$H(\alpha)=\alpha\log(1-\alpha)+O(1)$ as $\alpha$ tends to $0$.
#### 4.2.3 Deterministic synchronization stage, steps $i=\log\log
N+1,\ldots,\log{N}$
At this stage we continue essentially the same procedure as at the prevoius
stage. On each step Alice and Bob get $\lambda_{i}=\frac{2^{i}}{\log^{2}N}$
bits of random checksums for each pair of blocks (that are not proven yet to
be not-equal) by computing the inner products with new portions of pseudo-
random bits; then Alice and Bob exchange the computed checksums. The
difference is only how the participants exchange their checksums. Now they use
the deterministic protocol by Orlitsky instead of the instead of probabilistic
protocol by Smith, see section 3.3.
Communication complexity of the deterministic protocol is about $\log N$ times
grater than complexity of the protocol by Smith. But nevertheless we can use
it since the value of $\lambda_{i}$ is reasonably small. The communication
complexity of this stage is
$\sum_{i=\log{\log{N}}}^{\log{N}}[\frac{N\lambda_{i}\log
N}{2^{i}}]=\sum_{i=\log{\log{N}}}^{\log{N}}{\frac{\log N\cdot
2^{i}}{\log^{2}N\cdot 2^{i}}\cdot N}=O(N).$
#### 4.2.4 Summary
When the described stages are completed, we believe that Alice and Bob has
found all pairs of non-equal blocks. In all remaining pairs $X^{i}$, $Y^{i}$
(in all pairs of blocks whose checksums at all steps of the protocol were
equal to each other) are considered equal.
### 4.3 Probability of an error
We need to estimate the probability of an error in our protocol. For
simplicity, let us assume at first, that instead of $R$ pseudo-random bits
Alice and Bob share $R$ independent and uniformly distributed random bits (so,
we temporarily switch to the model with a public source of randomness). Then
stages 4.2.3 and 4.2.3 make sense, and we can estimate the probability of an
error in the protocol.
The protocol may return a wrong answer because of the following reasons:
1. 1.
The probabilistic synchronization protocol of Smith’s fails at some stage.
2. 2.
Some of steps $i=1,\ldots,\log{N}$ fails since to many random checksums are
equal for non-equal pairs of blocks $X^{i}$, $Y^{i}$.
Let us bound probabilities of each of these bad events.
#### 4.3.1 Error in the probabilistic synchronization.
We sum up the probabilities of errors in Smith’s synchronization at each step
of our protocol:
$P(Err)=\sum_{i=1}^{\log\log N}O(2^{-(\frac{N}{\log N})})\leq
O(2^{-\frac{cN}{\log N}})$
for some constant $c>0$.
#### 4.3.2 The failure of because of checksums
Some step $i=1,\ldots,\log{N}$ fails if for more than a half of (not
discovered yet) pairs of non-equal blocks $X^{i}$, $Y^{i}$ all random
checksums turn out to be equal. We estimate the probability of this event with
Chernoff’s inequality. We may assume that after the first $(i-1)$ steps there
remain $N/2^{i}$ pairs of pairs of non-equal blocks.
We use our checksum scheme like some kind of a filter. That is - we "test"
each pair of blocks on equality using calculated checksum of length $\lambda$.
If the pair consists of non-equal blocks, then the test either successfully
discoveres it (with probability $1-2^{-\lambda}$) or not. The failure happens
if less then a half of tests succeed. We estimate the probability of failure
using Chernoff’s inequality.
Formulating differently: On each step i, the $\frac{N}{2^{i}}$(amount of
undiscovered pairs) tests are performed, each failing with probability
$2^{-\lambda_{i}}=$. The whole step fails if more than a half of the tests
fail.
For one step, this probability can be estimated:
$P({\mbox{Filtering out less then 1/2 pairs}})<2^{\frac{N}{2^{i}}D(q,p)}$
Where
$D(q,p)=q\ln{(\frac{q}{p})}+(1-q)\ln{(\frac{1-q}{1-p})},\mbox{ and
}q=\frac{1}{2},p=\frac{1}{2^{\lambda_{i}}}$
Substituting $\lambda\approx\frac{2^{i}}{\log^{2}N}$, this error probability
is less then
$P(Err_{i})\leq
O(2^{-\frac{N\lambda_{i}}{2^{i}}})=O(2^{-\frac{N}{\log^{2}N}})$
Summing error probabilities for all steps of stages 2 and 3:
$\sum_{i=1}^{\log{N}}O(2^{-\frac{N}{\log^{2}N}})\approx\log{N}\cdot
O(2^{-\frac{N}{\log^{2}N}})\approx O(2^{-\frac{N}{\log^{2}N}+\log{\log
N}})\leq O(2^{-c\frac{N}{\log^{2}N}})$
#### 4.3.3 Pseudorandom generator
In this section we construct a statistical test (see the definition in section
3.1) that simulates one step of our protocol. In a sense, this test verifies
that (pseudo)random bits are ‘‘suitable’’ for our communication protocol: they
do not cause the failure of the protocol at the $i$-th iteration.
The "advice strings" of this statistical test contains a sequence of pairs of
blocks $X^{i}$, $Y^{i}$ from the inputs Alice and Bob, that were not shown to
be not equal before iteration $i$. The test should work correctly for all
advice strings that correspond to the possible internal states of Alice and
Bob at the beginning of iteration $i$.
The input $x$ is a string of (pseudo)random bits that should be accepted or
rejected. The test must reject $x$ (for some advice string $a$), if our
communication protocol ‘‘fails’’ at the $i$-th iteration with this random bits
$x$ while Alice and Bob are given the blocks $X^{i}$, $Y^{i}$ corresponding to
the advice $a$.
The algorithm of the test is straightforward: it computes the checksums for
$X^{i}$ and $Y^{i}$ as it is done by our communication protocol at the $i$-th
iteration, with random bits $x$ shared by Alice and Bob, and compares the
corresponding checksums for Alice’s and Bob’s blocks. Note that the test does
not simulate the synchronization procedure (the sub-protocols following the
construction of Orlitsky and Smith).
The working space of our machine is $O(\frac{N}{\log^{2}N})$. This is enough
to simulate the computation of the checksums performed by our communication
protocol. The test accepts $x$, if in the simulation at least $50\%$ of non-
equal pairs of blocks are is successfully revealed, and rejects $x$ otherwise.
In other words, a teststring $x$ is rejected if it causes a failure at the
$i$-th iteration of the protocol.
Theorem 3.1 guarantees that Nisan’s pseudo-random generator fools this test.
Hence, for our protocol the probability of failure with pseudo-random bits is
not much greater than the probability of failure for truly random bits. More
precisely, difference between the probabilities of failure for random and
pseudo-random bits is at most
$2^{-S}=O(2^{-c\frac{N}{\log^{2}N}}).$
Technically, we should sum up this difference for all steps $i=1,\ldots,\log
N$, but this does not change the asymptotics: $P(Err)\leq\sum_{i=1}^{\log
N}O(2^{-\frac{N}{\log^{2}N}})=O(2^{-c_{3}\frac{N}{\log^{2}N}})$
#### 4.3.4 Summary
Probability of the error of our protocol consists of three parts: (1)
probability of an error in Smith’s protocol, (2) probability of failure with
truly random checksums on some step $i=1,\ldots,\log N$, and (3) the
additional probability of failure caused by the difference between truly
random and pseudorandom checksums:
$P(Err)=O(2^{-c_{1}(\frac{N}{\log
N})})+O(2^{-c_{2}\frac{N}{\log^{2}N}})+O(2^{-c_{3}\frac{N}{\log{N}}})=O(2^{-C\frac{N}{\log^{2}N}}).$
This concludes the proof of correctness of our communication protocol.
Compared to the previous results(presented in [11]):
$P(Err)=O(2^{-\sqrt{N}})$, our protocol provides better probability of error.
## References
* [1] Kushilevitz E., Nisan N. Communication Complexity Cambridge University Press, 1997.
* [2] Chuklin A. Effective protocols for low-distance file synchronization . arXiv:1102.4712, 2011
* [3] Orlitsky A. Interactive communication of balanced distributions and of correlated files, SIAM Journal on Discrete Mathematics. - 1993. - Vol. 6., Pp. 548 - 564
* [4] Nisan N. Pseudorandom Generators for Spacebounded Computation 1992, COMBINATORICA Akademiai Kiado - Springer-Verlag, COMBINATORICA 12 (4) (1992) 449-461
* [5] Nisan N., Widgerson N.Hardness vs. Randomness, Journal of Computer and System Sciences< Volume 49, Issue 2, October 1994, Pages 149–167
* [6] Canetti R., Goldreich O.Bounds on Tradeoffs between Randomness and Communication Complexity, 1990, Computational complexity, 1993, Volume 3, Issue 2, pp 141-167
* [7] Newman L.Private vs. Common Random Bits in Communication Complexity 1991, Information Processing Letters Volume 39, Issue 2, 31 July 1991, Pages 67–71
* [8] Impagliazzo R., Nisan N., Widgerson A. Pseudorandomness for Network Algorithms, Proceeding STOC ’94 Proceedings of the twenty-sixth annual ACM symposium on Theory of computing, Pages 356 - 364, ACM New York, NY, USA ©1994
* [9] Smith A., Scrambling Adversarial Errors Using Few Random Bits, Optimal Information Reconciliation , and Better Private Codes SODA ’07 Proceedings of the eighteenth annual ACM-SIAM symposium on Discrete algorithms. — Society for Industrial and Applied Mathematics Philadelphia, 2007. — Pp. 395 – 404.
* [10] Nisan N., Zukerman D.Randomness is Linear in Space, 1993 Journal of Computer and System Sciences Volume 52, Issue 1, February 1996, Pages 43–52
* [11] Feder T., Kushilevitz E., Naor M., Nisan N.Amortized Communication Complexity 1991, SIAM J. Comput., 24(4), 736–750. (15 pages)
* [12] Bose, R. C.; Ray-Chaudhuri, D. K. (March 1960), "On A Class of Error Correcting Binary Group Codes", Information and Control 3 (1): 68–79, ISSN 0890-5401
* [13] Berlekamp, Elwyn R. (1967), "Nonbinary BCH decoding", International Symposium on Information Theory, San Remo, Italy, Published in Information Theory, IEEE Transactions on, March 1968, Volume: 14 , Issue: 2, Page: 242
|
arxiv-papers
| 2012-12-10T00:13:19 |
2024-09-04T02:49:39.073277
|
{
"license": "Public Domain",
"authors": "Vladimir Nikishkin",
"submitter": "Vladimir Nikishkin",
"url": "https://arxiv.org/abs/1212.1941"
}
|
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